Particle Physics-A Los Alamos Primer

i' -- 

I i 

LA-UR-88-9114

Particle Physics 

A Los Alamos Primer

C./ 

Particle Phvsics 

J 

A Los Alarnos Primer 

Edited by Necia Grant Cooper and Geoffrey B. West 

Los Alamos National Laboratory 

CAMBRIDGE UNIVERSITY PRESS 

Cain bridge 

New York New Rochelle Melboitrne Sydney

Published by the Press Syndicate of the University of Cambridge 

The Pit1 Building. Trurnpington Street. Cambridge CB2 IRP 

32 East 57th Street. New York. NY 10022. USA 

10 Svarnford Road, Oakleigh. Melbourne 3166. Australia 

0 Cambridge University Press 1988 

First publi:ihed 1988 

Printed in the United States of America 

Lihrury cf Congress Cat~tlo~iii~-iii-Puhlic~itioii Dota 

Panicle physics 

An updated version of Los Alamos science, no. I I 

(surnrner/fiill 1984). 

Includes index. 

I. Panicles (Nuclear physics) I. Cooper. Necia 

Grant. II. West, Geoffrey B. 

QC793.P358 1987 539.7'21 87-10858 

British Librrirv Cotologuiiig irr Puhliccition Dcttct 

Particle physics : a Los Alarnos primer. 

I. Particles (Nuclear physics) 

1. Cooper. Necia Grant 2. West. Geoffrey B. 

539.7'21 QC793.2 

ISBN 0-52 1-34542- I hard covers 

ISBN 0-521-34780-7 paperback

General Editors 

Editor 

Associate Editors 

Designer 

Illustration and Production 

Necia Grant Cooper 

Geoffrey B. West 


Roger Eckhardt 

Nancy Shera 

Gloria Sharp 

Jim Cmz 

Anita Flores 

John Flower 

Judy Gibes 

Jim E. Lovato 

Lenny Martinez 

LeRoy Sanchez 

Mary Stovall 

Chris West

Contents 

Prefae to Los Alamos Science, Number 11, Summer/Falll984 - 

Introduction 

vi ii 

ix 

Theoretical Framework 

Scale and Dimension-From Animals to Quarks 2 

by Geofrey B. West 

Fundamental Constants and the Rayleigh-Riabouchinsky Paradox 12 

Particle Physics and the Standard Model 22 

by Stuart Raby, Richard C. Slansky, and Geoflrey B. West 

QCD on a Cray: The Masses of Elementary Particles ~ 

by Gerald Guralnik, Tony Warnock, and Charles Zemach 

41 

Lecture Notes-From Simple Field Theories to the Standard Model 54 

by Richard C. Slansky 

Toward a Unified Theory: An Essay on the Role of Supergravity in the Search for Unification 72 


Fields and Spins in Higher Dimensions 86 

Supersymmetry at 100 GeV 98 

by Stuart Raby 

vi 

Supersymmetry in Quantum Mechanics 102

The Family Problem 114 

by T. Goldman and Michael Martin Nieto 

Addendum: CP Violation in Heavy-Quark Systems 124 

Experimental Developments 

Experiments to Test Unification Schemes 128 

by Gary H. Sanders 

An Experimentalist’s View of the Standard Model 130 

Addendum: An Experimental Update 149 

The March toward Higher Energies 150 

by S. Peter Rosen 

Addendum: The Next Step in Energy 

LAMPF I1 and the High-Intensity Frontier 

by Henry A. Thiessen 

156 

158 

The SSC-An Engineering Challenge 164 

by Mahlon T. Wilson 

Science Underground-The Search for Rare Events 166 

by L. M. Simmons, Jr. 

Personal Perspectives 

Quarks and Quirks among Friends 180 

A round table on the history and future of particle physics with Peter A. Carruthers, Stuart Raby, 

Richard C. Slansky, Geoflrey B. West, and George Zweig 

Index 

196 

vii

Preface 

0 

n the cover a mandala of the laws of physics floats in the 

cosmos of reality. It symbolizes the interplay between the 

inner world of abstract creation and the outer realms of 

measurable truth. The tension between these two is the magic and the 

challenge of fundamental physics. 

According to Jung, the “squaring ofthe circle” (the mandala) is the 

archetype of wholeness, the totality of the self. Such images are 

sometimes created spontaneously by individuals attempting to integrate 

what seem to be irreconcilable differences within themselves. 

Here the mandala displays the modern attempt by particle physicists 

to bring together the basic forces of nature in one theoretical 

framework. 

The content of this so-called standard model is summarized by the 

mysterious-looking symbols labeling each force: U( 1) for electromagnetism, 

SU(2) for weak interactions, SU(3)c for strong interactions, 

and SL(2C) for gravity; each symbol stands for an invariance, 

or symmetry, of nature. Symmetries tell us what remains 

constant through the changing universe. They are what give order to 

the world. There are many in nature, but those listed on the mandala 

are special. Each is a local symmetry, that is, it manifests independently 

at every space-time point and therefore implies the 

existence of a separate force. In other words, local symmetries 

determine all the forces of nature. This discovery is the culmination 

of physics over the last century. It is a simple idea, and it turns out to 

describe all phenomena so far observed. 

Where does particle physics go from here? The major direction of 

present research (and a major theme of this issue) is represented by 

the spiral that starts at electromagnetism and turns into the center at 

gravity. It suggests that the separate symmetries may be encompassed 

in one larger symmetry that governs the entire universe-one symmetry, 

one principle, one theory. The spiral also suggests that including 

gravity in such a theory involves understanding the structure of 

space-time at unimaginably small distance scales. 

Julian Schwinger, whose seminal idea led to the modern unification 

of electromagnetic and weak interactions, regards the present 

emphasis on unification with skepticism: “It’s nothing more than 

another symptom of the urge that amicts every generation of 

physicists-the itch to have all the fundamental questions answered 

in their own lifetime.”* To others the goal seems tantalizingly close, 

an achievement that may be reached, if not this year-then maybe 

the next.. . . 

The hope of unification depends on a second theme of this issue, 

symbolized by the ants and elephants walking round the mandala. 

These creatures are our symbol of scaling, the sizing up and sizing 

down of physical systems. Strength (or any other quality, for that 

matter) may look different on different scales. But if we look hard 

enough, we can find certain invariances to changes in scale that 

define the correct variables for describing a problem. Why do ants 

appear stronger than elephants? Why does the strong force look weak 

at high energies? How could all the forces of nature be manifestations 

of a single theory? These are the questions explored in “Scale and 

Dimension-from animals to quarks,” a seductively playful article 

that leads us to one of the most important contributions to modem 

physics, the renormalization group equations of quantum field theory. 

The insights about scaling gained from these equations are 

important not only to elementary particle physics but also to phase 

transition theory and the dynamics of complex systems. 

All the articles in this issue were written by scientists who care to 

tell not only about their own research but about the whole field of 

particle physics, its stunning achievements and its probing questions. 

Outsiders lo this field hear the names of the latest new particles, the 

buzz words such as grand unification or supersymmetry, and the 

plans for the United States to regain its leadership in this glamorous, 

high tech area of big science. But what is the real progress? Why does 

this field continue to attract the best minds in science? Why is it a 

major achievement of human thought? From a distance it may be 

hard to tell-except that it satisfies some deep urge to undcrstand 

how the world works. But if one could be given a closer look at the 

technical content of this field, its depth and richness would become 

apparent. That is the aim ofthe present issue. 

The hardest job was defining the technical level. How could the 

framework of the standard model be appreciated by someone unfamiliar 

with symmetry principles? How could modern particle 

physics research, all of which builds on the standard model, be 

understood by someone unfamiliar with what everyone in the field 

takes for granted? We hope we have solved this problem by presenting 

some of the major concepts on several levels and in several 

different places. We even include our own reference material, a 

remarkably clear and friendly set of lecture notes prepared especially 

for this issue. 

As one who was trained in this field, I returned to it with some 

trepidation-to deal with the subject matter, which had been so 

difficult, and with the personalitites competing in the field, who 

sometimes ride roughshod over each other as they battle these unruly 

abstractions. Much to my delight and the delight of the Los 241amos 

Science staff, the experience of preparing this issue was immensely 

enjoyable and rewarding. The authors were enthusiastic about explaining 

and re-explaining, about considering the essence of each 

point one more time to make sure that the readers too would be able 

to grasp it. Their generosity and interest made it fun for us to learn. 

May this presentation also be a treat for you. 

*This quote appeared in “How the Universe Works” by Robert P. Crease and 

Charles C. Mann (The Atlantic Monthly, August, 1984), a fast-paced article 

about the history of the electoweak theory. 


1984 

viii

Introduction 

B 

eginning with the dramatic discovery of the J/y particle in 

1974, particle physics has gone through a remarkably productive 

and exciting period. Quantum field theory, developed 

during the 1940s and ‘50s but abandoned in the O OS, was reestablished 

as the language for formulating theoretical concepts. The 

unification of the weak and electromagnetic interactions via a socalled 

non-Abelian gauge theory could only be understood in this 

framework. A similar theory of the strong interactions, quantum 

chromodynamics, was also constructed during this period, and 

nowadays one refers to the total package of the strong and electroweak 

theories as “the standard model.” Over the last decade the 

predictions of the standard model have been spectacularly confirmed, 

so much so that it is now almost taken for granted as 

embodying all physics below about 100 GeV. The culmination of this 

exuberant period was the inevitable discovery in 1983 of the W* and 

Zo particles, the massive bosons predicted by the standard model to 

mediate the weak interactions. Although the masses of these particles 

were precisely those predicted by the SU(2) X U(l) electroweak 

theory, their discovery was almost anticlimactic, so accepting had the 

particle-physics community become of the standard model. Indeed, 

future research in particle physics is often referred to as “physics 

beyond the standard model,” an implicit tribute to the progress of the 

past decade. 

The development of the standard model during the 1970s brought 

with it a lexicon of new words and concepts-quark, gluon, charm, 

color, spontaneous symmetry breaking, and asymptotic freedom, to 

name a few. Supersymmetry, preons, strings, and worlds of ten 

dimensions are among the buzz words added in the ’80s. While 

scientists, engineers, and even many lay people will recognize some 

subset of these words, only a few have more than a superficial 

understanding of the profound achievement they denote. Add to this 

the demand by particle physicists for several billion dollars to build a 

super-accelerator in order to explore “physics beyond the standard 

model,” and one can sense the gap between the particle physicist and 

his “public” reaching irreparable proportions. On the other hand 

there remains an endless wonder and fascination in the public’s eye 

for such speculative conceptual ideas, which are more usually associated 

with the literature of science fiction than with Physical 

Review. 

It was with some of these thoughts in mind that a group of us at 

Los Alamos National Laboratory decided to put together a series of 

pedagogical articles explaining in relatively elementary scientific 

language the accomplishments, successes, and projected future of 

high-energy physics. The articles, intended for a wide scientific 

audience, originally appeared in a 1984 issue of Los Alamos Science, 

a technical publication of the Laboratory. Since that time they have 

been used as a teaching tool in particle-physics courses and as a 

reference source by experimentalists in the field. 

Particle Physics-A Los Alamos Primer is basically an updated 

version of the original Los Alamos Science issue. We believe it will 

continue to help educate undergraduate and graduate students as 

well as bridge the gap between experimentalists and theorists. We are 

also confident that it will help non-experts to develop a good feel for 

the subject. 

The text consists of eight “chapters,” the first five devoted to the 

concepual fiamework of modem particle physics and the last three to 

experiments and accelerators. Each is written by a separate author, or 

group of authors, and is to a large extent self-contained. In addition, 

we have included a round table among several particle physicists that 

addresses some of the broader issues facing the field. This discussion 

is in some ways a unique evaluation of the present status of particle 

physics. It is quite personal and idiosyncratic, sometimes irreverent, 

and occasionally controversial. For the nonexpert it is probably the 

place to begin! 

The first article addresses the question of scaling. In its broadest 

sense this lies at the heart of any attempt to unify into one theory the 

fundamental forces of Nature-forces seemingly so very different in 

strength. “Scale and Dimension-From Animals to Quarks” begins 

by reviewing in an elementary and somewhat whimsical fashion the 

whole question of scale in classical physics and then introduces the 

more sophisticated concept of the renormalization group. The renormalization 

group is really no more than a generalization of 

classical dimensional analysis to the area of quantum field theory: it 

answers the seminal question of how a physical system responds to a 

change in scale. The concept plays a central role in the modem view 

of quantum field theory and has been particularly successful in 

elucidating the nature of phase transitions. Indeed, it is from this 

vantage point that the intimate relationship between particle and 

condensed-matter physics has developed. Clearly, the manner in 

which physics evolves from one energy or length scale to another is of 

fundamental importance. 

The second article, “Particle Physics and the Standard Model,” 

addresses the question of unification with an elementary yet comprehensive 

discussion of how the famous electroweak theory is 

constructed and works. The role of internal symmetries and their 

incorporation into a principle of local gauge invariance and subsequent 

manifestation as a non-Abelian gauge field theory are explained 

in a pedagogical fashion. The other component of the 

standard model, namely quantum chromodynamics (QCD), the 

theory of the strong interactions, is similarly treated in this article. 

Again, the discussion is rather elementary, beginning with an exposition 

of the “old” SU(3) of the “Eightfold Way” and finishing with 

the field theory of quarks and gluons. For the more ambitious reader 

we have included a set of “lectures” by Richard Slansky that give

some of the technical details necessary in going “from simple field 

theories to the standard model.” Crucial concepts such as local gauge 

invariance, spontaneous symmetry breaking, and emergence of the 

Higgs particles that give rise to the masses of elementary particles are 

expressed in the mathematical language of field theory and should be 

readily accessible to the serious student of the field. These lectures 

are very clear and provide the reader with the explicit equations 

embodying the physics discussed in the article on the standard 

model. 

Following this review of accepted lore, we begin our journey into 

“physics beyond the standard model” with an essay on supergravity 

by Slansky entitled “Toward a Unified Theory.” In it he discusses 

some of the speculative ideas that gained popularity in the late 1970s. 

Among them are supersymmetry (a proposed symmetry between 

fermions and bosons) and the embedding of our four-dimensional 

space-time world in a larger number of dimensions. Supergravity, a 

theory that encompasses both of these ideas, was the first serious 

attempt to include Einstein’s gravity in the unification scheme. This 

article also includes a description of superstring theory, which has 

gained tremendous popularity just in the last year or so. Slansky 

explains how the shortcomings of the supergravity scenario are 

circumvented by basing a unified theory on elementary fibers, or 

strings, rather than on point particles. This area of research is in a 

state of flux at the moment, and it is still far from clear whether 

strings really will form the basis of the “final” theory. The problems 

are both conceptual and technical. Conceptually there is still no hint 

as to what principles are to replace the equivalence principle and 

general coordinate invariance, which form the bases of Einstein’s 

gravity. Technically, the mathematics of string theory is beyond the 

usual expertise of the theoretical physicist; indeed it is on the 

forefront of mathematical research itself. This may be the first time 

for a hundred years or more that research in physics and 

mathematics has coincided. Some may view this as a bad omen, 

others as the dawning of a new exciting age leading to the equations 

of the universe! Only time will tell. 

A less ambitious use of supersymmetry has been in the attempts 

to unify, without gravity, the electroweak and strong theories of the 

standard model. Stuart Raby, in his article “Supersymmetry at 

100 GeV,” discusses some of these efforts by concentrating on the 

phenomenological implications of a world in which every boson 

has a fermion partner and vice versa. These include a possible 

explanation for why proton decay, certainly one of the more 

dramatic predictions of grand unified theories, has not yet been 

seen. Supersymmetric phenomenology has served as an important 

guide for speculating about what can be seen at new accelerators. A 

special feature of this article is the selfcontained section “Supersymmetry 

in Quantum Mechanics,” in which Raby explains this 

novel space-time symmetry in a setting stripped of all field-theoretic 

baggage. 

One of the more mysterious problems in particle physics is “the 

family problem” described in an article of that title by Terry 

Goldman and Michael Nieto. The apparent replication of the 

electron and its neutrino in at least two more families differing only 

in their mass scales has remained a mystery ever since the discovery 

of the muon. This replication, exhibited also by the quarks, can be 

accommodated in unified theories, though no satisfactory explanation 

of the family structure, nor even a prediction of the total 

number of families, has been advanced. The phenomenology of this 

problem as well as some attempts to understand it are carefully 

reviewed. An addendum to the original article presents a slightly 

more technical discussion of how experiments involving the third 

quark family might extend our knowledge of CP violation. This 

symmetry violation remains perhaps the most mysterious aspect of 

the known particle phenomenology. 

The next three articles concern the experimental side of particle 

physics. Although the choice of Los Alamos experiments to illustrate 

certain points does reflect some parochial interests of the authors, 

these articles succeed in providing a broad overview of experimental 

methodology. In this era of elaborate detection techniques requiring 

extensive collaboration, it is often difficult for the uninitiated to 

unravel the complicated machinations that are involved in the 

experimental process. In “Experiments to Test Unification Schemes” 

Gary Sanders presents a very clear exposition of the physics input to 

this process. Indeed, as if to emphasize the departure from the world 

of theory, he has included a brief page-and-a-half pr6cis subtitled “An 

Experimentalist’s View of the Standard Model.” For the beginner this 

might be read immediately following the round table! Sanders describes 

in some detail four experiments designed specifically to test 

the standard model, all being conducted at Los Alamos. Each is a 

“high-precision” experiment in which, say, a specific decay rate is 

measured and compared with the value predicted by the standard 

model. These experiments are prototypical of the kind that have been 

and will continue to be done at accelerators around the world to push 

the theory to its limits. Most exciting, or course, would be the 

observation of some deviation from the standard model that could be 

associated with grand unification. However, in an addendum Sanders 

reports that no such deviations were seen in the data from the Los 

Alamos experiments and others. So far the standard model has stood 

the test of time. 

The following article by Peter Rosen, “The March toward Higher 

Energies,” surveys the high-energy accelerator landscape beginning 

with a historical perspective and finishing with a glimpse into what 

we might expect in the not-toodistant future. The emphasis here is 

on tests of the standard model and searches for new and exotic 

particles not included in it. The traditional methodology is quite 

simple: go for the highest energy possible. This has certainly been 

successhl in the past, and we have no reason to believe that it won’t 

be successful in the future. Thus, there is a push to build a giant 

superconducting supercollider (SSC) that could probe mass scales in 

excess of 20 TeV, or 2 X loi3 eV. We have also included a brief 

X

eport by Mahlon Wilson, an accelerator physicist, on some of the 

problems peculiar to the gigantic scale of the SSC. 

An alternative technique for probing high mass scales is to perform 

very accurate experiments in search of deviations from expected 

results, such as those described in Sanders’ article. Obviously, highintensity 

beams are the desired tool in this approach. A high-intensity 

machine has been proposed for b s Alamos, and another brief report 

by Henry Thiessen, also an accelerator physicist, describes that 

machine and some of the questions it might answer. The reports on 

the SSC and LAMPF I1 provide an idea of what is involved in 

designing tomorrow’s accelerators. 

The final article is a review by Mike Simmons on “science underground.” 

In it he discusses what particle physics can be learned from 

experiments performed deep underground to isolate rare events of 

interest. The most famous of these is the search for proton decay. 

Other experiments measure the flux of neutrinos from the sun and 

search for exotic particles (such as magnetic monopoles) in cosmic 

rays. These essential fishing expeditions use “beams” from the 

biggest accelerator of them all, namely the universe! 

Particle Physics-A LQS Alamos Primer thus provides the reader 

with a comprehensive, up-to-date introduction to the field of particle 

physics. Our belief is that it will be a useful educational guide to both 

the student and professional worker in the field as well as provide the 

general scientist with an insight into some of the recent accomplishments 

in understanding the fundamental structure of the universe. 

In conclusion we would like to thank the staff of Los Alamos 

Science for their invaluable help in making this primer lively and 

accessible to a wide audience. 

Geoffrey B. West 

1986 

xi

Particle Phvsics 

A Los Alamos Primer

“1 have multiplied visions and used similitudes.” - Hosea 7:lO 

In his marvelous book Dialogues Concerning 

Two New Sciences there is a remarkably clear 

discussion on the effects of scaling up the 

dimensions of a physical object. Galileo realized 

that ifone simply scaled up its size, the 

weight of an animal would increase significantly 

faster than its strength, causing it ultimately 

to collapse. As Galileo says (in the 

words of Salviati during the discorso of the 

second day), “. . . you can plainly see the 

impossibility of increasing the size of structures 

to vast dimensions . . . if his height be 

increased inordinately, he will fall and be 

crushed under his own weight.” The simple 

scaling up of an insect to some monstrous 

size is thus a physical impossibility, and we 

can rest assured that these old sci-fi images 

are no more than fiction! Clearly, to create a 

giant one “must either find a harder and 

stronger material . . . or admit a diminution 

of strength,” a fact long known to architects. 

It is remarkable thal. so many years before 

its deep significance could be appreciated, 

Galileo had investigated one of the most 

fundamental questions of nature: namely, 

what happens to a physical system when one 

changes scale? Nowadays this is the seminal 

question for quantum field theory, phase 

transition theory, the dynamics of complex 

systems, and attempts to unify all forces in 

nature. Tremendous progress has been made 

in these areas during the past fifteen years 

based upon answers to this question, and I 

shall try in the latter part of this article to give 

some flavor of what has been accomplished. 

However, I want first to remind the reader of 

the power of dimensional analysis in 

classical physics. Although this is stock-intrade 

to all physicists, it is useful (and, more 

pertinently, fun) to go through several examples 

that explicate the basic ideas. Be warned, 

there are some surprises. 

Classical Scaling 

Let us first re-examine Galileo’s original 

analysis. For similar structures* (that is, 

structures having the same physical 

characteristics such as shape, density, or 

chemical composition) Galileo perceived 

that weight Wincreases linearly with volume 

V, whereas strength increases only like a 

cross-sectional area A. Since for similar 

structures V a l3 and A 12, where 1 is some 

characteristic length (such as the height ofthe 

structure), we conclude that 

-_ 

Strength A 1 1 

tc-a-a - 

Weight V I WV~’ 

Thus, as Galileo noted, smaller animals “appear” 

stronger than larger ones. (It is amusing 

that Jerome Siege1 and Joe Shuster, the 

creators of Superman, implicitly appealed to 

such an argument in one of the first issues of 

their comic.+ They rationalized his super 

strength by drawing a rather dubious analogy 

with “the lowly ant who can support weights 

hundreds of times its own” (sic!).) Incidentally, 

the above discussion can be used to 

understand why the bones and limbs of 

larger animals must be proportionately 

stouter than those of smaller ones, a nice 

example of which can be seen in Fig. 1. 

Arguments of this sort were used extensively 

during the late 19th century to un- 

4

Scale and Dimension 

Fig. 1. Two extinct mammals: (a) Neohipparion, a small American horse and (b) 

Mastodon, a large, elephant-like animal, illustrating that the bones of heavier 

animals are proportionately stouter and thus proportionately stronger. 

derstand the gross features of the biological 

world; indeed, the general size and shape of 

animals and plants can be viewed as nature’s 

way of responding to the constraints of gravity, 

surface phenomena, viscous flow, and 

the like. For example, one can understand 

why man cannot fly under his own muscular 

power, why small animals leap as high as 

larger ones, and so on. 

A classic example is the way metabolic 

rate varies from animal to animal. A 

measure B of metabolic rate is simply the 

heat lost by a body in a steady inactive state, 

which can be expected to be dominated by 

the surface effects of sweating and radiation. 

Symbolically, therefore, one expects 

B a W213. The data (plotted logarithmically 

in Fig. 2) show that metabolic rate does 

*The concept of similitude is usually attributed to 

Newton, who first spelled it out in the Principia 

when dealing with gravitational attraction. On 

reading the appropriate section it is clear that this 

was introduced only as a passing remark and does 

not have the same profound content as the remarks 

of Galileo. 

?This amusing observation was brought to my attention 

by Chris Llewellyn Smith. 

*This relationship with a slope of 3/4 is known as 

Kleiber’s law (M. Kleiber, Hilgardia 6(1932):315), 

whereas the area law is usually attributed to Rubner 

(M. Rubner, Zeitschrift fur Biologie (Munich) 

19(1883):535). 

indeed scale, that is, all animals lie on a 

single curve in spite of the fact that an 

elephant is neither a blown-up mouse nor a 

blown-up chimpanzee. However, the slope of 

the best-fit curve (the solid line) is closer to 

3/4 than to 2/3, indicating that effects other 

than the pure geometry of surface dependence 

are at work.# 

It is not my purpose here to discuss why 

this is so but rather to emphasize the importance 

of a scaling curve not only for establishing 

the scaling phenomenon itself but for 

revealing deviations from some naive 

prediction (such as the surface law shown as 

the dashed line in Fig. 2). Typically, deviations 

from a simple geometrical or 

kinematical analysis reflect the dynamics of 

the system and can only be understood by 

examining it in more detail. Put slightly differently, 

one can view deviations from naive 

scaling as a probe of the dynamics. 

The converse of this is also true: generally, 

one cannot draw conclusions concerning 

dynamics from naive scaling. As an illustration 

of this I now want to discuss some 

simple aspects of birds’ eggs. I will focus on 

the question of breathing during incubation 

and how certain physical variables scale 

from bird to bird. Figure 3, adapted from a 

Scientijic American article by Hermann 

Rahn, Amos Ar, and Charles V. Paganelli 

entitled “How Bird Eggs Breathe,” shows the 

dependence of oxygen conductance K and 

pore length I (that is, shell thickness) on egg 

mass W. The authors, noting the smaller 

slope for /, conclude that “pore length 

probably increases slower because the eggshell 

must be thin enough for the embryo to 

hatch.” This is clearly a dynamical conclusion! 

However, is it warranted? 

From naive geometric scaling one expects 

that for similar eggs I a W”3, which is in 

reasonable agreement with the data: a best fit 

(the straight line in the figure) actually gives / 

oc @.4. Since these data for pore length agree 

reasonably well with geometric scaling, no 

dynamical conclusion (such as the shell being 

thin enough for the egg to hatch) can be 

drawn. Ironically, rather than showing an 

anomalously slow growth with egg mass, the 

data for / actually manifest an anomalously 

fast growth (0.4 versus 0.33), not so dissimilar 

from the example of the metabolic 

rate! 

What about the behavior of the conductance, 

for which K 0: @.9? This relationship 

can also be understood on geometric 

grounds. Conductance is proportional to the 

totd available pore area and inversely 

proportional to pore length. However, total 

pore area is made up of two factors: the total 

number of pores times the area of individual 

pores. If one assumes that the number of 

pores per unit area remains constant from 

bird to bird (a reasonable assumption consistent 

with other data), then we have two 

factors that scale like area and one that 

5

IO0 1 o2 io4 

Body Weight (kg) 

scales inversely as length. One thus expects 

K a ( W2/3)2/ W1I3 = W, again in reasonable 

agreement with the data. 

Dimensional Analysis. The physical content 

of scaling is very often formulated in 

terms of the language of dimensional analysis. 

The seminal idea seems to be due to 

Fourier. He is, ofcourse, most famous for the 

invention of “Fourier analysis,” introduced 

in his great treatise Theorie Analpique de la 

Chaleur, first published in Paris in 1822. 

However, it is generally not appreciated that 

this same book contains another great contribution, 

namely, the use of dimensions for 

physical quantities. It is the ghost of Fourier 

that is the scourge of all freshman physics 

majors, for it was he who first realized that 

every physical quantity “has one dimension 

proper to itself, and that the terms of one and 

the same equation could not be compared, if 

they had not the same exponent of 

dimension.” He goes on: “We have introduced 

this consideration . . . to verify the 

analysis. .. it is the equivalent of the fundamental 

lemmas which the Greeks have left us 

without proof.” Indeed it is! Check the 

dimensions!-the rallying call of all 

physicists (and, hopefully, all engineers). 

However, it was only much later that 

physicists began to use the “method of 

dimensions” to solve physical problems. In a 

famous paper on the subject published in 

Fig. 2. Metabolic rate, measured as heat produced by the body in a steady state, 

plotted logarithmically against body weight. An analysis based on a surface 

dependence for the rate predicts a scaling curve with slope equal to 2/3 (dashed 

line) whereas the actual scaling curve has a slope equal to 3/4. Such deviation from 

simple geometrical scaling is indicative of other effects at work. (Figure based on 

one by Thomas McMahon, Science 179(1973):1201-1204 who, in turn, adapted it 

from M. Kleiber, Hilgardia 6(1932):315.) 

L- 5 I 

I I ‘ e 

Oxygen Conductance 

L 

B loo/ 

m 

* 

’13 

L 10- 

a2 

a. 

(Slope = 0.9) . 

(Slope * 0.4) 

I= 

QI 

P 

3 0.01 I I I I 

1 10 100 1000 

Egg Mass (9) 

00 

- 

E 

-5 

en 

C 

3 

E! 0 

p. 

.I 

Fig. 3. Logarithmic plot of two parameters relevant to the breathing of birds’ eggs 

during incubation: the conductance of oxygen through the shell and thepore length 

(or shell thickness) as a function of egg mass. Both plots have slopes close to those 

predicted by simple geometrical scaling analyses. (Figure adapted from H. Rahn, 

A. Ar, and C. V. Paganelli, Scientific American 24O(February 1979):46-55.) 

.01 

6


Nature in 19 15, Rayleigh indignantly begins: 

“I have often been impressed by the scanty 

attention paid even by original workers in 

the field to the great principle of similitude. 

It happens not infrequently that results in the 

form of ‘laws’ are put forward as novelties on 

the basis of elaborate experiments, which 

might have been predicted a priori after a few 

minutes consideration!” He then proceeds to 

set things right by giving several examples of 

the power of dimensional analysis. It seems 

to have been from about this time that the 

method became standard fare for the 

physicist. I shall illustrate it with an amusing 

example. 

Most of us are familiar with the traditional 

Christmas or Thanksgiving problem of how 

much time to allow for cooking the turkey or 

goose. Many (inferior) cookbooks simply say 

something like “20 minutes per pound,” implying 

a linear relationship with weight. 

However, there exist superior cookbooks, 

such as the Better Homes and Gardens 

Cookbook, that recognize the nonlinear 

nature of this relationship. 

Figure 4 is based on a chart from this 

cookbook showing how cooking time t varies 

with the weight ofthe bird W. Let us see how 

Fig. 4. The cooking time for a turkey or 

goose as a logarithmic function of its 

weight. (Based on a table in Better 

Homes and Gardens Cookbook, Des 

MoinexMeridith Corp., Better Homes 

and Gardens Books, 1962, p. 272.) 

one can understand this variation using “the 

great principle of similitude.” Let T be the 

temperature distribution inside the turkey 

and To the oven temperature (both measured 

relative to the outside air temperature). T 

satisfies Fourier’s heat diffusion equation: 

arjat = KV~T, where K is the diffusion coeficient. 

Now, in general, for the dimensional 

quantities in this problem, there will be a 

functional relationship of the form 

where p is the bird’s density. However, 

Fourier’s basic observation that the physics 

be independent of the choice of units, imposes 

a constraint on the form of the solution, 

which can be discerned by writing it in terms 

of dimensionless quantities. Only two independent 

dimensionless quantities can be 

constructed: T/To and p (~t)~/~/ W. If we use 

the first of these as the dependent variable, 

the solution, whatever its form, must be 

expressible in terms of the other. The relationship 

must therefore have the structure 

T 

To 

(3) 

The important point is that, since the lefthand 

side is dimensionless, the “arbitrary” 

function f must be a dimensionless function 

of a dimensionless variable. Equation 3, unlike 

the previous one, does not depend upon 

the choice of units since dimensionless quantities 

remain invariant to changes in scale. 

Let us now consider different but 

geometrically similar birds cooked to the 

same temperature distribution at the same 

oven temperature. Clearly, for all such birds 

there will be a scaling law 

p(Kt)3‘2 

=constant. 

(4) 

If the birds have the same physical 

characteristics (that is, the same p and K), Eq. 

4 reduces to 

t = constant x w2I3, 

(5) 

reflecting, not surprisingly, an area law, As 

can be seen from Fig. 4, this agrees rather 

well with the “data.” 

This formal type of analysis could also, of 

course, have been camed out for the 

metabolic rate and birds’ eggs problems. The 

advantage of such an analysis is that it delineates 

the assumptions made in reaching 

conclusions like B W2I3 since, in principle, 

it focuses upon all the relevant variables. 

Naturally this is crucial in the discussion of 

any physics problem. For complicated systems, 

such as birds’ eggs, with a very large 

number of variables, some prior insight or 

intuition must be used to decide what the 

important variables are. The dimensions of 

these variables are determined by the fundamental 

laws that they obey (such as the diffusion 

equation). Once the dimensions are 

known, the structure of the relationship between 

the variables is determined by 

Fourier’s principle. There is therefore no 

magic in dimensional analysis, only the art of 

choosing the “right” variables, ignoring the 

irrelevant, and knowing the physical laws 

they obey. 

As a simple example, consider the classic 

problem of the drag force F on a ship moving 

through a viscous fluid of density p. We shall 

choose F, p, the velocity v, the viscosity of the 

fluid p, some length paraTeter of the ship I, 

and the acceleration due to gravity g as our 

7

_- v 

, 

variables. Notice that we exclude other 

variables, such as the wind velocity and the 

amplitude of the sea waves because, under 

calm conditions, these are of secondary importance. 

Our conclusions may therefore not 

be valid for sailing ships! 

The physics of the problem is governed by 

the Navier-Stokes equation (which incorporates 

Newton’s law of viscous drag, 

telling us the dimensions of p) and the gravitational 

force law (telling us the dimensions 

of g). Using these dimensions automatically 

incorporates the appropriate physics. Since 

we have limited the variables to a set of six, 

which must be expressible in terms of three 

basic units (mass M, length L, and time T), 

there will only be three independent 

dimensionless combinations. These are 

chosen to be P = F/p$12 (the pressure coeficient), 

R = vlp/p (Reynold’s number), and 

N, = $/lg (Froude’s number). Although any 

three similar combinations could have been 

chosen, these three are special because they 

delineate the physics. For example, Reynold‘s 

number R relates to the viscous drag 

on a body moving through a fluid, whereas 

Froude’s number NF relates to the forces 

involved with waves and eddies generated on 

the surface of the fluid by the movement. 

Thus the rationale for the combinations R 

and NF is to separate the role of the viscous 

forces from that of the gravitational: R does 

not depend on g, and F does not depend on 

p. Furthermore, Pdoes not depend on either! 

Dimensional analysis now requires that 

the solution for the pressure coefficient, 

Libster 1924 

0 Allan 1900 

1, Cottingen 1921 

a Cottingen 1926 

--- Results of higher pressures 1922-23 

1 o-2 1 oo 1 o2 lo4 

Reynolds Number R 

lo6 

Fig. 5. The scaling curve for the motion of a sphere through a 

fluid that results when data from a variety of experiments 

are plotted in terms of two dimensionless variables: the 

pressure or drag coefficent P versus Reynolds number R. 

(Figure adapted from AIP Handbook of Physics, 2nd edition 

(1963):section II, p. 253.) 

8


whatever it is, must be expressible in the 

dimensionless form 

The actual drag force F can easily be obtained 

from this equation by re-expressing it 

in terms of the dimensional variables (see 

Eq. 8 below). 

First, however, consider a situation where 

surface waves generated by the moving object 

are unimportant (an extreme example is 

a submarine). In this case g will not enter the 

solution since it is manifested as tht restoring 

force for surface waves. N,c can then be 

dropped from the solution, reducing Eq. 6 to 

the simple form 

P=f(R). (7) 

In terms of the original dimensional 

variables, this is equivalent to 

Historically, these last equations have been 

well tested by measuring the speed of different 

sizes and types of balls moving 

through different liquids. If the data are 

plotted using the dimensionless variables, 

that is, Pversus R, then all the data should lie 

on just one curve regardless of the size of the 

ball or the nature of the liquid. Such a curve 

is called a scaling curve, a wonderful example 

of which is shown in Fig. 5 where one sees a 

scaling phenomenon that varies over seven 

orders of magnitude! It is important to recognize 

that if one had used dimensional 

variables and plotted F versus I, for example, 

then, instead of a single curve, there would 

have been many different and apparently 

unrelated curves for the different liquids. 

Using carefully chosen dimensionless 

variables (such as Reynold’s number) is not 

only physically more sound but usually 

greatly simplifies the task of representing the 

data. 

A remarkable consequence of this analysis 

is that, for similar bodies, the ratio of drag 

Fig. 6. The time needed for a rowing boat to complete a 2000-meter course in calm 

coxditions as a function of the number of oarsmen. Data were taken from several 

international rowing championship events and illustrate the surprisingly slow 

dropoff predicted by modeling theory. (Adapted from T. A. McMahon, Science 

173(1971):349-351.) 

force to weight decreases as the size of the 

structure increases. From Archimedes’ principle 

the volume ofwater displaced by a ship 

is proportional to its weight, that is, W a l3 

(this, incidentally, is why there is no need to 

include W as an independent variable in 

deriving these equations). Combined with 

Eq. 8 this leads to the conclusion that 

F * 

- cc-. (9) 

W I 

This scaling law was extremely important in 

the 19th century because it showed that it 

was cost efective to build bigger ships, 

thereby justifying the use of large iron steamboats! 

The great usefulness of scaling laws is also 

illustrated by the observation that the 

behavior of P for large ships (I- W) can be 

derived from the behavior of small ships 

moving very fast (v- w). This is so because 

both limits are controlled by the same 

asymptotic behavior off(R) =f(vlp/p). Such 

observations form the basis of modeling theory 

so crucial in the design of aircraft, ships, 

buildings, and so forth. 

Thomas McMahon, in an article in Science, 

has pointed out another, somewhat 

more amusing, consequence to the drag force 

equation. He was interested in how the speed 

of a rowing boat scales with the number of 

oarsmen n and argued that, at a steady velocity, 

the power expended by the oarsmen E to 

overcome the drag force is given by Fv. Thus 

9

Using Archimedes’ principle again and the 

fact that both E and W should be directly 

proportional to n leads to the remarkable 

scaling law 

which shows a very slow growth with n. 

Figure 6 exhibits data collected by McMahon 

from various rowing events for the time t (a 

I/v) taken to cover a fixed 2000-meter course 

under calm conditions. One can see quite 

plainly the verification of his predicted 

law-a most satisfying result! 

There are many other fascinating and 

exotic examples of the power of dimensional 

analysis. However, rather than belaboring 

the point, I would like to mention a slightly 

different application of scaling before I turn 

to the mathematical formulation. All the examples 

considered so far are ofa quantitative 

nature based on well-known laws of physics. 

There are, however, situations where the 

qualitative observation of scaling can be 

used to scientific advantage to reveal phenomenological 

“laws.” 

A nice example (Fig. 7), taken from an 

article by David Pilbeam and Stephen Jay 

Gould, shows how the endocranial volume V 

(loosely speaking, the brain size) scales with 

body weight W for various hominids and 

pongids. The behavior for modern pongids is 

typical of most species in that the exponent 

a, defined by the phenomenological relationship 

V 0: W, is approximately 1/3 (for 

mammals a varies from 0.2 to 0.4). It is very 

satisfying that a similar behavior is exhibited 

by australopithecines, extinct cousins of our 

lineage that died out over a million years ago. 

However, as Pilbean and Gould point out, 

I 

our homo sapiens lineage shows a strikingly 

different behavior, namely: a % 5/3. Notice 

that neither this relationship nor the “standard” 

behavior (a % 1/3) is close to the naive 

geometrical scaling prediction of a = 1. 

These data illustrate dramatically the 

qualitative evolutionary advance in the 

brain development of man. Even though the 

reasons for a = 1/3 may not be understood, 

this value can serve as the “standard” for 

revealing deviations and provoking speculation 

concerning evolutionary progress: for 

example, what is the deep significance of a 

brain size that grows linearly with height 

versus a brain size that grows like its fifth 

power? I shall not enter into such questions 

here, tempting though they be. 

Such phenomenological scaling laws 

(whether for brain volume, tooth area, or 

some other measurable parameter of the fos- 

- 6 

0 

- 

1250 

1000 

0, 

E 75c 

3 

0 

> 

c. 

m 

I- 

S 

!? 

0” 500 

0 c 

w 

350 

a= 1.73 / /r 

/ 

I 

?‘ 

/ 

A 

I 

/ 

sil) can also be used as corroborative 

evidence for assigning a newly found fossil of 

some large primate to a particular lineage. 

The fossil’s location on such curves can, in 

principle, be used to distinguish an australopithecine 

from a homo. Notice, however, 

that implicit in all this discussion is knowledge 

of body weight; presumably, 

anthropologists have developed verifiable 

techniques for estimating this quantity. Since 

they necessarily work with fragments only, 

some further scaling assumptions must be 

involved in their estimates! 

Relevant Variables. As already emphasized, 

the most important and artful aspect of the 

method of dimensions is the choice of 

variables relevant to the problem and their 

grouping into dimensionless combinations 

that delineate the physics. In spite of the 

/ 0 Australopithecines 

/ A Homo Lineage 

Pongids 

30 40 50 75 100 

Body Weight (kg) 

Fig. 7, Scaling curves for endocranial volume (or brain size) as a function of body 

weight. The slope of the curve for our homo sapiens lineage (dashed line) is 

markedly different from those for australopithecines, extinct cousins of the homo 

lineage, and for modern pongids, which include the chimpanzee, gorilla and 

orangutan. (Adapted from D. Pilbeam and S. J. Gould, Science 

I86(19 74):892-901.) 

10


relative simplicity of the method there are 

inevitably paradoxes and pitfalls, a famous 

case of which occurs in Rayleigh’s 1915 

paper mentioned earlier. His last example 

concerns the rate of heat lost H by a conductor 

immersed in a stream of inviscid fluid 

moving past it with velocity v (“Boussinesq’s 

problem”). Rayleigh showed that, if K is the 

heat conductivity, C the specific heat of the 

fluid, 0 the temperature difference, and 1 

some linear dimension of the conductor, 

then, in dimensionless form, 

Approximately four months after Rayleigh’s 

paper appeared, Nature published an 

eight line comment (half column, yet!) by a 

D. Riabouchinsky pointing out that Rayleigh’s 

result assumed that temperature was a 

dimension independent from mass, length, 

and time. However, from the kinetic theory 

of gases we know that this is not so: temperature 

can be defined as the mean kinetic 

energy of the molecules and so is not an 

independent unit! Thus, according to 

Riabouchinsky, Rayleigh’s expression must 

be replaced by an expression with an additional 

dimensionless variable: 

a much less restrictive result. 

Two weeks later, Rayleigh responded to 

Riabouchinsky saying that “it would indeed 

be a paradox if thefurther knowledge of the 

nature of heat afforded by molecular theory 

put us in a worse position than before in 

dealing with a particular problem. . . . It 

would be well worthy of discussion.” Indeed 

it would; its resolution, which no doubt the 

reader has already discerned, is left as an 

exercise (for the time being)! Like all 

paradoxes, this one cautions us that we occasionally 

make casual assumptions without 

quite realizing that we have done so (see 

“Fundamental Constants and the Rayleigh- 

Riabouchinsky Paradox”). 

Scale Invariance 

Let us now turn our attention to a slightly 

more abstract mathematical formulation 

that clarifies the relationship of dimensional 

analysis to scale invariance. By scale invariance 

we simply mean that the structure 

of physical laws cannot depend on the choice 

of units. As already intimated, this is automatically 

accomplished simply by employing 

dimensionless variables since these 

clearly do not change when the system of 

units changes. However, it may not be immediately 

obvious that this is equivalent to 

the form invariance of physical equations. 

Since physical laws are usually expressed in 

terms of dimensional variables, this is an 

important point to consider: namely, what 

are the general constraints that follow from 

the requirement that the laws of physics look 

the same regardless of the chosen units. The 

crucial observation here is that implicit in 

any equation written in terms of dimensional 

variables are the “hidden” fundamental 

scales of mass M, length L, time T, and so 

forth that are relevant to the problem. Of 

course, one never actually makes these scale 

parameters explicit precisely because of form 

invariance. 

Our motivation for investigating this 

question is to develop a language that can be 

generalized in a natural way to include the 

subtleties of quantum field theory. Hopefully 

classical dimensional analysis and scaling 

will be sufficiently familiar that its generalization 

to the more complicated case will 

be relatively smooth! This generalization has 

been named the renormalization group since 

its origins b’e in the renormalization program 

used to ma f, e sense out of the infinities inherent 

in quantum field theory. It turns out 

that renormalization requires the introduction 

of a new arbitrary “hidden”. scale that 

plays a role similar to the role of the scale 

parameters implicit in any dimensional 

equation. Thus any equation derived in 

quantum field theory that represents a physical 

quantity must not depend upon this 

choice of hidden scale. The resulting con- 

straint will simply represent a generalization 

of ordinary dimensional analysis; the only 

reason that it is different is that variables in 

quantum field theory, such as fields, change 

in a much more complicated fashion with 

scale than do their classical counterparts. 

Nevertheless, just as dimensional analysis 

allows one to learn much about the behavior 

of a system without actually solving the 

dynamical equations, so the analogous constraints 

of the renormalization group lead to 

powerful conclusions about the behavior ofa 

quantum field theory without actually being 

able to solve it. It is for this reason that the 

renormalization group has played such an 

important part in the renaissance of quantum 

field theory during the past decade or so. 

Before describing how this comes about, I 

shall discuss the simpler and more familiar 

case of scale change in ordinary classical 

systems. 

To begin, consider some physical quantity 

F that has dimensions; it will, of course, be a 

function of various dimensional variables 

x,: F(xl,x2,. . .,x,,). An explicit example is 

given by Eq. 2 describing the temperature 

distribution in a cooked turkey or goose. 

11

Fundamental Constants and the 

L 

et us examine Riabouchinsky’s paradox a little more carefully 

and show how its resolution is related to choosing a system of 

units where the “fundamental constants” (such as Planck’s 

constant h and the speed of light c) can be set equal to unity. 

The paradox had to do with whether temperature could be used as 

an independent dimensional unit even though it can be defined as the 

mean kinetic energy of the molecular motion. Rayleigh had chosen 

five physical variables (length I, temperature difference 8, velocity v, 

specific heat C‘, and heat conductivity K) to describe Boussinesq’s 

problem and had assumed that there were four independent 

dimensions (energy E, length L, time T, and temperature 6). Thus 

the solution for T/T, necessarily is an arbitrary function of one 

dimensionless combination. To see this explicitly, let us examine the 

dimensions of the five physical variables: 

[h = L, [e] = e, = LT’, [q = EL-~B-’, 

and [K] = EL-’ TI@-’ . 

Clearly the combination chosen by Rayleigh, IvCIK, is dimensionless. 

Although other dimensionless combinations can be formed, they 

are not independent of the two combinations (1vCIK and TIT,) 

selected by Rayleigh. 

Now suppose, along with Riabouchinsky, we use our knowledge of 

the kinetic theory to define temperature “as the mean kinetic energy 

of the molecules” so that 6 is no longer an independent dimension. 

This means there are now only three independent dimensions and the 

solution will depend on an arbitrary function of two dimensionless 

combinations. With 6 a E, the dimensions of the physical variables 

become: 

[O= L, [e] = E, [v] = LT’, [CJ = LP3, and [K] = L-’T’. 

It is clear that, in addition to Rayleigh’s dimensionless variable, there 

is now a new independent combination, C13 for example, that is 

dimensionless. To reiterate Rayleigh: “it would indeed be a paradox 

if the further knowledge of the nature of heat . . . put us in a worse 

position than before . . . it would be well worthy of discussion.” 

Like almost all paradoxes, there is a bogus aspect to the argument. 

It is certainly true that the kinetic theory allows one to express an 

energy as a temperature. However, this is only useful and appropriate 

for situations where the physics is dominated by molecular considerations. 

For macroscopic situations such as Boussinesq’s problem, the 

molecular nature of the system is irrelevant; the microscopic 

variables have been replaced by macroscopic averages embodied in 

phenomenological properties such as the specific heat and conductivity. 

To make Riabouchinsky’s identification of energy with temperature 

is to introduce irrekvant physics into the problem. 

Exploring this further, we recall that such an energy-temperature 

identification implicitly involves the introduction of Boltzmann’s 

factor k. By its very nature, k will only play an explicit role in a 

physical problem that directly involves the molecular nature of the 

system; otherwise it will not enter. Thus one could describe the 

system from the molecular viewpoint (so that k is involved) and then 

take a macroscopic limit. Taking the limit is equivalent to setting 

k = 0 the presence of a finite k indicates that explicit effects due to 

the kinetic theory are important. 

With this in mind, we can return to Boussinesq’s problem and 

derive Riabouchinsky’s result in a somewhat more illuminating 

fashion. Let us follow Rayleigh and keep E, L, T, and 6 as the 

Each of these variables, including F itself, is 

always expressible in terms of some standard 

set of independent units, which can be 

chosen to be mass M, length L, and time T. 

These are the hidden scale parameters. Obviously, 

other combinations could be used. 

There could even be other independent 

units, such as temperature (but remember 

Riabouchinsky!), or more than one independent 

length (say, transverse and longitudinal). 

In this discussion, we shall simply 

use the conventional M, L, and T. Any 

generalization is straightforward. 

In terms of this standard set of units, the 

magnitude of each x, is given by 

x, = Mal L ~I TYI (15) 

The numbers a,, PI, and y, will be recognized 

12 

as “the dimensions” of xi. Now suppose we 

change the system of units by some scale 

transformation of the form 

A4 -+ 

M‘ = hMM , 

L- L’=hLL, 

and 

T + 

T‘ = hTT. 

Each variable then responds as follows: 

x, - XI’ = Z,(h)X, , (17) 

where 

and h is shorthand for h ~ h , ~ and , h ~. Since 

Fis itself a dimensional physical quantity, it 

transforms in an identical fashion under this 

scale change: 

where 

Here a, p, and y are the dimensions of F. 

There is, however, an alternate but equivalent 

way to transform from F to F‘, namely, 

by transforming each of the variables xi 

separately. Explicitly we therefore also have


adox 

independent dimensions but add k (with ons E@-’) as a new 

physical variable. The solution will now rbitrary function of 

two independent dimensionless variables: IvC/K and kCl? When 

Riabouchinsky chose to make C13 his other dimensionless variable, 

he, in effect, chose a system of units where k= 1. But that was a 

temble thing to do here since the physics dictates that k = O! Indeed, 

if k = 0 we regain Rayleigh’s original result, that is, we have only one 

dimensionless variable. It is somewhat ironic that Rayleigh’s remarks 

miss the point: “further knowledge oft 

molecular theory” does not put one in a 

the problem-rather, it leads to a micros 

C. The important point pertinent to the 

that knowledge of the molecular theory is irrelevant and k must not 

enter. 

The lesson here is an important one because it illustrates the role 

played by the fundamental constants. Consider Planck’s constant 

h h/2~ it would be completely inappropriate to introduce it into a 

problem of classical dynamics. For e any solution of the 

scattering of two billiard balls will dep acroscopic variables 

such as thc masses, velocities, friction coefficients, and so on. Since 

billiard balls are made of protons, it might be tempting to the purist 

to include as a dependent variable the proton-proton total cross 

section, which. of course. involves fi. This would clearly be totally 

inappropriate but is analogous to what Riabouchinsky did in 

Boussinesq’s problem. 

Obviously, if the scattering is between two microscopic “atomic 

billiard balls” then h must he included. In thiscase it is not only quite 

legitimate but often convenient to choose a system of units where 

h = I . However, having done so one cannot directly recover the 

g to h - 0. With h = 1, one is stuck in 

with k = I , one is stuck in kinetic theory. 

A similar situation obviously occurs in relativity: the velocity of 

light c must not occur in the classical Newtonian limit. However, in a 

relativistic situation one is quite at liberty to choose units where 

c= 1. Making that choice, though, presumes the physics involves 

relativity. 

The core of particle physics, relativistic quantum field theory, is a 

hanics and relativity. For this reason, 

a system of units in which h = c = 1 is 

manifesto that quantum mechanics and 

relativity are the basic physical laws governing their area of physics. 

In quantum mechanics, momentum p and wavelength k are related 

by the de Broglie relation: p= 27thlh; similarly, energy E and frequency 

w are related by Planck‘s formula: E = ha. In relativity we 

have the famous Einstein relation: E = mc?. Obviously if we choose 

= c = 1, all energies, masses, and momenta have the same units 

example, electron 

ev)), and these are the same as inverse 

r energies and momenta inevitab1.v 

correspond to shorter times and lengths. 

Using this choice of units automatically incorporates the profound 

physics of the uncertainty principle: to probe short space-time intervals 

one needs large energies. A useful number to remember is that 

IO-13 centimeter, or I fermi (fm), equals the reciprocal of 200 MeV. 

We then find that the electron mass (e 1/2 MeV) corresponds to a 

length of e 400 fm-its Compton wavelength. Or the 20 TeV 

(2 X lo’ MeV) typically proposed for a possible future facility 

corresponds to a length of lo-’* centimeter. This is the scale distance 

that such a machine will probe! 

Equating these two different ways ofeffecting 

a scale change leads to the identity 

of taking a/& and then setting h = 1. For and x[ = xi, so that Eq. 23 reduces to 

example, if we were to consider changes in 

the mass scale, we would use a/ahM and the 

chain rule for partial differentiation to amve 

at ax, 8x2 ax3 

az, aF 

az 

Ex;- 1=- F. 

r=l a7\M axi dhM 

aF aF aF 

alxl - + azxz - + a3x3 - + . . . 

aF 

+ a,x, - = aF. 

ax” 

As a concrete example, consider the equation 

E = mc?. To change scale one can either 

transform E directly or transform m and c 

separately and multiply the results appropriately-obviously 

the final result must 

be the same. 

We now want to ensure that the resulting 

form of the equation does not depend on h. 

This is best accomplished using Euler’s trick 

When we set AM = I , differentiation of Eqs. 

18 and 20 yields 

Obviously this can be repeated with hL 

and AT to obtain a set of three coupled partial 

differential equations expressing the fundamental 

scale invariance ofphysical laws (that 

is, the invariance of the physics to the choice 

of units) implicit in Fourier’s original work. 

These equations can be solved without too 

much difficulty; their solution is, in fact, a 

speciai case of the solution to the re- 

13

normalization group equation (given explicitly 

as Eq. 35 below). Not too surprisingly, 

one finds that the solution is precisely 

equivalent to the constraints of dimensional 

analysis. Thus there is never any explicit 

need to use these rather cumbersome equations: 

ordinary dimensional analysis takes 

care of it for you! 

Quantum Field Theory 

We have gone through this little mathematical 

exercise to illustrate the well-known 

relationship of dimensional analysis to scale 

and form invariance. I now want to discuss 

how the formalism must be amended when 

applied to quantum field theory and give a 

sense of the profound consequences that follow. 

Using the above chain of reasoning as a 

guide, I shall examine the response of a 

quantum field theoretic system to a change 

in scale and derive a partial differential equation 

analogous to Eq. 25. This equation is 

known as the renormalization group equation 

since its origins lay in the somewhat 

arcane area of the renormalization procedure 

used to tame the infinities of quantum field 

theory. I shall therefore have to digress 

momentarily to give a brief rCsumC of this 

subject before returning to the question of 

scale change. 

Renormalization. Perhaps the most unnerving 

characteristic of quantum field theory for 

the beginning student (and possibly also for 

the wise old men) is that almost all calculations 

of its physical consequences naively 

lead to infinite answers. These infinities stem 

from divergences at high momenta associated 

with virtual processes that are 

always present in any transition amplitude. 

The renormalization scheme, developed by 

Richard 1’. Feynman, Julian S. Schwinger, 

Sin-Itiro Tomonaga, and Freeman Dyson, 

was invented to make sense out of this for 

quantum electrodynamics (QED). 

To get a feel for how this works I shall 

focus on the photon, which cames the force 

associated with the electromagnetic field. At 

the classical limit the propagator* for the 

14 

photon represents the usual static I/r 

Coulomb potential. The corresponding 

Fourier transform (that is, the propagator’s 

representation in momentum space) in this 

limit is I/$, where q is the momentum carried 

by the photon. Now consider the 

“classical” scattering of two charged particles 

(represented by the Feynman diagram in Fig. 

8 (a)). For this event the exchange of a single 

photon gives a transition amplitude proportional 

to &/$, where eo is the charge (or 

coupling constant) occurring in the Lagrangian. 

A standard calculation results in 

the classical Rutherford formula, which can 

be extended relativistically to the spin-1/2 

case embodied in the diagram. 

A typical quantum-mechanical correction 

to the scattering formula is illustrated in Fig. 

8 (b). The exchanged photon can, by virtue of 

the uncertainty principle, create for a very 

short time a virtual electron-positron pair, 

which is represented in the diagram by the 

loop. We shall use k to denote the momentum 

carried around the loop by the two 

particles. 

There are, of course, many such corrections 

that serve to modify the I/$ single- 

*Roughly speaking, the photon propagator can be 

thought of as the Green’s function for the electromagnetic 

field. In the relativistically covariant 

Lorentz gauge, the classical Maxwell’s equations 

read 

0’ A(X) = j(x), 

where A(x) is the vector potential and j(x) is the 

current source term derived in QED from the motion 

of the electrons. (To keep things simple I am 

suppressing all space-time indices, thereby ignoring 

spin.) This equation can be solved in the standard 

way using a Green’s function: 

A(x) = Id‘x’ 

with 

0 2 G(x) = 6(x) . 

G(x’ -x) j(x‘), 

Now a transition amplitude is proportional to the 

interaction energy, and this is given by 

HI = Id4 x j(x) A(x) = 

Id‘ x Id‘x’ j(x) G(x-x‘) j(x‘) , 

photon behavior, and this is represented 

schematically in part (c). It is convenient to 

include all these corrections in a single multiplicative 

factor DO that represents deviations 

from the single-photon term. The “full” 

photon propagator including all possible 

radiative corrections is therefore Do/$. The 

reason for doing this is that DO is a 

dimensionless function that gives a measure 

of the polarization of the vacuum caused by 

the production of virtual particles. (The origin 

of the Lamb shift is vacuum polarization.) 

We now come to the central problem: 

upon evaluation it is found that contributions 

from diagrams like (b) are infinite because 

there is no restriction on the magnitude 

of the momentum k flowing in the loop! 

Thus, typical calculations lead to integrals of 

the form 

(26) 

which diverge logarithmically. Several 

prescriptions have been invented for making 

such integrals finite; they all involve “reg- 

illustrating how G “mediates” the force between 

two currents separated by a space-time interval 

(x-x’). It is usually more convenient to work with 

Fourier transforms of these quantities (that is, in 

momentum space). For example, the momentum 

space solution for G is C(q) = I/q2, and this is 

usually called the free photon propagator since it 

is essentially classical. The corresponding 

“classical” transition amplitude in momentum 

space is just j(q)(I/q’Jj(q), which is represented 

by the Feynman graph in Fig. 8 (a). 

In quantum field theory, life gets much more 

complicated because of radiative corrections as 

discussed in the text and illustrated in (b) and (c) 

of Fig. 8. The definition of the propagator is 

generally in terms of a correlation function in 

which a photon is created at point x out of the 

vacuum for a period x-x‘ and then returns to the 

vacuum’at point x’. Symbolically, this is represented 

by 

G(x-x’) - (vaclA(x’) A(x)lvac). 

During propagation, anything allowed by the 

uncertainty principle can happen-these are the 

radiative corrections that moke an exact calculation 

of G almost impossible.


ularizing” the integrals by introducing some 

large mass parameter A. A standard technique 

is the so-called Pauli-Villars scheme in 

which a factor A2/(k2+A2) is introduced 

into the integrand with the understanding 

that A is to be taken to infinity at the end of 

the calculation (notice that in this limit the 

regulating factor approaches one). With this 

prescription, the above integral is therefore 

replaced by 

A2 

= In - 

aq2 

The integral can now be evaluated and its 

divergence expressed in terms of the (infinite) 

mass parameter A. All the infinities 

arising from quantum fluctuations can be 

dealt with in a similar fashion with the result 

that the following series is generated: 

In this way the structure of the infinite 

divergences in the theory are parameterized 

in terms of A, which can serve as a finite 

cutoffin the integrals over virtual momenta.* 

The remarkable triumph of the renormalization 

program is that, rather than 

imposing such an arbitrary cutoff, all these 

divergences can be swallowed up by an infinite 

rescalingof the fields and coupling con- 

Fig. 8. Feynman diagrams for (a) the ChZSSiCal scattering Of two particles Of 

charge e,, (b) a typical correction that must be made to that scattering-here 

because of the creation of a virtual electron-positron pair-and (c) a diagram 

representing all such possible corrections. The matrix element is proportional for 

(a) to e:/q2 and for (c) to D,/q2 where D,.includes all corrections. 

*In this discussion I assumed, for simplicity, 

that the original Lagrangian was massless; that 

~ ~ ~ ~ t ~ 

plicate the discussion unnecessarily without giving 

any new insights. 

15

stants. Thus, afinite propagator D, that does 

not depend on A, can be derived from DO by 

rescaling if, at the same time, one rescales the 

charge similarly. These rescalings take the 

form 

D = Zflo and e = Zgo . (29) 

The crucial property of these scaling factors 

is that they are independent of the physical 

momenta (such as q) but depend on A in 

such a way that when the cutoff is removed, 

D and e remain finite. In other words, when 

A - m, ZD and Z, must develop infinities of 

their own that precisely compensate for the 

infinities of Do and eo. The original so-called 

bare parameters in the theory calculated 

from the Lagrangian (DO and eo) therefore 

have no physical meaning-only the renormalized 

parameters (D and e) do. 

Now let us apply some ordinary dimensional 

analysis to these remarks. Because 

they are simply scale factors, the 2’s must be 

dimensionless. However, the 2’s are functions 

of ,4 but not of q. But that is very 

peculiar: a dimensionless function cannot 

depend on a single mass parameter! Thus, in 

order to express the Z’s in dimensionless 

form, a new jnite mass scale p must be 

introduced so that one can write 

Z = Z(A2/p2,eo). An immediate consequence 

of renormalization is therefore to induce a 

mass scale not maniyest in the Lagrangian. 

This is extremely interesting because it 

provides a possible mechanism for generating 

mass even though no mass parameter 

appears in the Lagrangian. We therefore 

have the exciting possibility of being able to 

calculate the masses of all the elementary 

particles in terms ofjust oneofthem. Similar 

considerations for the dimensionless Ds 

clearly require that they be expressible as 

DO = Do(q2/A2,eo), as in Eq. 28, and 

D = D(q2/p2,e). (The dream of particle 

theorists is to write down a Lagrangian with 

no mass parameter that describes all the 

interations in terms ofjust onecoupling constant. 

The mass spectrum and scattering 

amplitudes for all the elementary particles 

16 

would then be calculable in terms of the 

value of this single coupling at some given 

scale! A wonderful fantasy.) 

To recapitulate, the physical finite renormalized 

propagator 12 is related to its bare 

and divergent counterpart DO (calculated 

from the Lagrangian using a cutoff mass) by 

an infinite rescaling: 

Similarly, the physical finite charge e is given 

by an infinite rescaling of the bare charge eo 

that occurs in the Lagrangian 

e = lim zc( $ , eo) eo 

A-n. 

Notice that the physical coupling e now depends 

implicitly on the renormalization 

scale parameter p. Thus, in QED, for example, 

it is not strictly sufficient to state that the 

fine structure constant a =: 11137; rather, 

one must also specifj the corresponding 

scale. From this point of view there is 

nothing magic about the particular number 

137 since a change of scale would produce a 

different value. 

At this stage, some words of consolation to 

a possibly bewildered reader are in order. It is 

not intended to be obvious how such infinite 

rescalings of infinite complex objects lead to 

consistent finite results! An obvious question 

is what happens with more complicated 

processes such as scattering amplitudes and 

particle production? These are surely even 

more divergent than the relatively simple 

photon propagator. How does one know that 

a similar rescaling procedure can be carried 

through in the general case? 

The proof that such a procedure does indeed 

work consistently for any transition 

amplitude in the theory was a real tour de 

force. A crucial aspect of this proof was the 

remarkable discovery that in QED only a 

finite number (three) of such rescalings was 

necessary to render the theory finite. This is 

tembly important because it means that 

once we have renormalized a few basic entities, 

such as eo, all further rescalings of 

more complicated quantities are completely 

determined. Thus, the theory retains predictive 

power-in marked contrast to the highly 

unsuitable scenario in which each transition 

amplitude would require its own infinite 

rescaling to render it finite. Such theories, 

termed nonrenormalizable, would apparently 

have no predictive power. High 

energy physicists have, by and large, restricted 

their attention to renormalizable theories 

just because all their consequences can, in 

principle, be calculated and predicted in 

terms of just a few parameters (such as the 

physical charge and some masses). 

I should emphasize the phrase “in principle” 

since in practice there are very few 

techniques available for actually carrying out 

honest calculations. The most prominent of 

these is perturbation theory in the guise of 

Feynman graphs. Most recently a great deal 

of effort, spurred by the work of K. G. 

Wilson, has gone into trying to adapt quantum 

field theory to the computer using lattice 

gauge theories.* In spite of this it remains 

sadly true that perturbation theory is our 

only “global” calculational technique. Certainly 

its success in QED has been nothing 

less than phenomenal. 

Actually only a very small class of renormalizable 

theories exist and these are 

characterized by dimensionless coupling 

constants. Within this class are gauge theories 

like QED and its non-Abelian extension 

in which the photon interacts with 

itself. All modern particle physics is based 

upon such theories. One of the main reasons 

for their popularity, besides the fact they are 

renormalizable, is that they possess the property 

of being asymptotically free. In such 

theories one finds that the renormalization 

group constraint, to be discussed shortly, 

requires that the large momentum behavior 

*In recent years there has been some effort to 

come to grips analytically with the 

nonperturbative aspects of gauge theories.


be equivalent to the small coupling limit; 

thus for large momenta the renormalized 

coupling effectively vanishes thereby allowing 

the use of perturbation theory to calculate 

physical processes. 

This idea was of paramount importance in 

substantiating the existence of quarks from 

deep inelastic electron scattering experiments. 

In these experiments quarks behaved 

as if they were quasi-free even though they 

must be bound with very strong forces (since 

they are never observed as free particles). 

Asymptotic freedom gives a perfect explanation 

for this: the effective coupling, though 

strong at low energies, gets vanishingly small 

as 4’ becomes large (or equivalently, as distance 

becomes small). 

In seeing how this comes about we will be 

led back to the question of how the field 

theory responds to scale change. We shall 

follow the exact same procedure used in the 

classical case: first we scale the hidden parameter 

(p, in this case) and see how a typical 

transition amplitude, such as a propagator, 

responds. A partial differential equation, 

analogous to Eq. 25, is then derived using 

Euler’s trick. This is solved to yield the general 

constraints due to renormalization 

analogous to the constraints of dimensional 

analysis. I will then show how these constraints 

can be exploited, using asymptotic 

freedom as an example. 

The Renormalization Group Equation. As 

already mentioned, renormalization makes 

the bare parameters occumng in the Lagrangian 

effectively irrelevant; the theory has 

been transformed into one that is now specified 

by the value of its physical coupling 

constants at some mass scale p. In this sense 

p plays the role of the hidden scale parameter 

M in ordinary dimensional analysis by setting 

the scale of units by which all quantities 

are measured. 

This analogy can be made almost exact by 

considering a scale change for the arbitrary 

parameter p in which p - h’/*p. This change 

allows us to rewrite Eq. 30 in a form that 

expresses the response of D to a scale change: 

(From now on I will use g to denote the 

coupling rather than e because e is usually 

reserved for the electric charge in QED.) 

The scale factor Z(h), which is independent 

of q2 and g, must, unlike the Zs of Eqs. 

30 and 3 I , befinite since it relates two finite 

quantities. Notice that all explicit reference 

to the bare quantities has now been 

eliminated. The structure of this equation is 

identical to Eq. 22, the scaling equation derived 

for the classical case; the crucial dij 

ference is that Z(h) no longer has the simple 

power law behavior expressed in Eq. 18. In 

fact, the general structure of Z(1) and g(p) are 

not known in field theories of interest. 

Nevertheless we can still learn much by converting 

this equation to the differential form 

analogous to Eq. 25 that expresses scale invariance. 

As before we simply take a/ah and 

set h = I, thereby deriving the so-called renormalization 

group equation: 

where 

and 

(33) 

(34) 

(35) 

Comparing Eq. 33 with the scaling equation 

of classical dimensional analysis (Eq. 25), we 

see that the role ofthe dimension is played by 

y. For this reason, and to distinguish it from 

ordinary dimensions, y is usually called the 

anomalous dimension of D, a phrase originally 

coined by Wilson. (We say anomalous 

because, in terms of ordinary dimensions 

and again by analogy with Eq. 25, D is actually 

dimensionless!) It would similarly have 

been natural to call p(g)/g the anomalous 

dimension of g, however, conventionally, 

one simply refers to p(g) as the p-function. 

Notice that p(g) characterizes the theory as a 

whole (as does g itself since it represents the 

coupling) whereas y(g) is a property of the 

particular object or field one is examining. 

The general solution of the renormalization 

group equation (Eq. 33) is given by 

D (f,g) = eA(g)j( f , (36) 

where 

and 

(37) 

The arbitrary function f is, in principle, fixed 

by imposing suitable boundary conditions. 

(Equation 25 can be viewed as a special and 

rather simple case of Eq. 33. If this is done, 

17

the analogues of y(g) and p(g)/g are constants, 

resulting in trivial integrals for A and 

K. One can then straightforwardly use this 

general solution (Eq. 36) to verify the claim 

that the scaling equation (Eq. 22) is indeed 

exactly equivalent to using ordinary dimensional 

analysis.) The general solution reveals 

what is perhaps the most profound consequence 

of the renormalization group, 

namely, that in quantum field theory the 

momentum variables and the coupling constant 

are inextricably linked. The photon 

propagator (D/q2), for instance, appears at 

first sight to depend separately on the 

momentum 4’ and the coupling constant g. 

Actually, however, the renormalizability of 

the theory constrains it to depend effectively, 

as shown in Eq. 36, on only one variable 

($dC(g)/bi2). This, of course, is exactly what 

happens in ordinary dimensional analysis. 

For example, recall the turkey cooking problem. 

The temperature distribution at first 

sight depended on several different variables: 

however, scale invariance, in the guise of 

dimensional analysis, quickly showed that 

there was in fact only a single relevant 

variable. 

The observation that renormalization introduces 

an arbitrary mass scale upon which 

no physical consequences must depend was 

first made in 1953 by E. Stueckelberg and A. 

Peterman. Shortly thereafter Murray Gell- 

Mann and F. Low attempted to exploit this 

idea to understand the high-energy structure 

of QED and, in so doing, exposed the intimate 

connection between g and $. Not 

much use was made of these general ideas 

until the pioneering work of Wilson in the 

late 1960s. I shall not review here his seminal 

work on phase transitions but simply remark 

that the scaling constraint implicit in the 

renormalization group can be applied to correlation 

functions to learn about critical exponents.* 

Instead I shall concentrate on the 

*Since the photon propagator is defined as the 

correlation function of two electromagnetic 

fields in the vacuum it is not diffielt to imagine 

that the formalism discussed here can be directly 

applied to the correlation functions of statistical 

physics. 

18 

particle physics successes, including 

Wilson’s, that led to the discovery that non- 

Abelian gauge theories were asymptotically 

free. Although the foci of particle and condensed 

matter physics are quite different, 

they become unified in a spectacular way 

through the language of field theory and the 

renormalization group. The analogy with dimensional 

analysis is a good one, for, as we 

saw in the first part of this article, its constraints 

can be applied to completely diverse 

problems to give powerful and insightful results. 

In a similar fashion, the renormalization 

group can be applied to any problem 

that can be expressed as a field theory (such 

as particle physics or statistical physics). 

Often in physics, progress is made by examining 

the system in some asymptotic regime 

where the underlying dynamics 

simplifies sufficiently for the general structure 

to become transparent. With luck, 

having understood the system in some extreme 

region, one can work backwards into 

the murky regions of the problem to understand 

its more complex structures. This is 

essentially the philosophy behind bigger and 

bigger accelerators: k.eep pushing to higher 

energies in the hope that the problem will 

crack, revealing itself in all its beauty and 

simplicity. ’Tis indeed a faithful quest for the 

holy grail. As I shall now demonstrate, the 

paradigm of looking first for simplicity in 

asymptotic regimes is strongly supported by 

the methodology of the renormalization 

group. 

In essence, we use the same modelingtheory 

scaling technique used by ship designers. 

Going back to Eq. 36, one can see 

immediately that the high-energy or shortdistance 

limit ($- m with g fixed) is iden- 

tical to keeping 4’ fixed while taking K + 

m. 

However, from its definition (Eq. 38), K 

diverges whenever p(g) has a zero. Similarly, 

the low-energy or long-distance limit (42 - 0 

while g is fixed) is equivalent to K + 

-m, 

which also occurs when p - 0. Thus knowledge 

of the zeros of 0, the so-called fixed 

points of the equation, determines the highand 

low-energy behaviors of the theory. 

If one assumes that for small coupling 

quantum field theory is governed by ordinary 

perturbation theory, then the p-function 

has a zero at zero coupling (g- 0). In 

this limit one typically finds p(g) = -b$ 

where b is a calculable coefficient. Of course, 

p might have other zeroes, but, in general, 

this is unknown. In any case, for small g we 

find (using Eq. 38) that K(g) = (2b$)-’, 

which diverges to either +m or -4, depending 

on the sign of b. In QED, the case originally 

studied by Cell-Mann and Low, b < 0 so that 

K - -m, which is equivalent to the lowenergy 

limit. One can think of this as an 

explanation of why perturbation theory 

works so well in the low-energy regime of 

QED: the smaller the energy, the smaller the 

effective coupling constant. 

Quantum Chromodynamics. It appears that 

some non-Abelian gauge theories and, in 

particular, QCD (see “Particle Physics and 

the Standard Model”) possess the unique 

property of having a positive b. This 

marvelous observation was first made by H. 

D. Politzer and independently by D. J. Gross 

and F. A. Wilczek in 1973 and was crucial in 

understanding the behavior of quarks in the 

famous deep inelastic scattering experiments 

at the Stanford Linear Accelerator Center. As 

a result, it promoted QCD to the star position 

of being a member of “the standard 

model.” With b > 0 the high-energy limit is 

I


related to perturbation theory and is therefore 

calculable and understandable. I shall 

now give an explicit example of how this 

comes about. 

First we note that no boundary conditions 

have yet been imposed on the general solution 

(Eq. 36). The one boundary condition 

that must be imposed is the known free field 

theory limit (g= 0). For the photon in QED, 

or the gluon in QCD, the propagator G 

(=D/$) in this limit is just I/$. Thus 

D($/p2,0) = 1. Imposing this on Eq. 36 gives 

Now when g - 0, y(g) = -a$, where a is a 

calculable coeficient. Combining this with 

the fact that B(g) = -62 leads, by way of Eq. 

37, to A(g) = (a/@ In g. Since K(g) = 

(26$)-‘, the boundary condition (Eq. 39) 

gives 

Defining the dimensionless variable in the 

functionfas 

it can be shown that with b > 0 Eq. 40 is 

equivalent to 

lim fix) = (26 In x)alZb. (42) 

X-m 

An important point here is that the x - m 

limit can be reached either by lettingg- 0 or 

by taking 42 - m 

calculable, so is the 2 - m 

. Since the g - 0 limit is 

limit. The free 

field (g- 0) boundary condition therefore 

determines the large x behavior offlx), and, 

once again, the “modeling technique” can be 

used-here to determine the large q2 

behavior of the propagator G. 

In fact, combining Eq. 36 with Eq. 42 leads 

to the conclusion that 

(43) 

This is the generic structure that finally 

emerges: the high-energy or large-$ behavior 

of the propagator G = D/$ is given by free 

field theory (I/$) modulated by calculable 

powers of logarithms. The wonderful miracle 

that has happened is that all the’powers of 

ln(A2/$) originally generated from the 

divergences in the “bare” theory (as illustrated 

by the series in Eq. 28) have been 

summed by the renormalization group to 

give the simple expression of Eq. 43. The 

amazing thing about this “exact” result is 

that is is far easier to calculate than having to 

sum an infinite number of individual terms 

in a series. Not only does the methodology 

do the summing, but, more important, it 

justifies it! 

I have already mentioned that asymptotic 

freedom (that is, the equivalence of vanishingly 

small coupling with increasing 

momentum) provides a natural explanation 

of the apparent paradox that quarks could 

appear free in high-energy experiments even 

though they could not be isolated in the 

laboratory. Furthermore, with lepton probes, 

where the theoretical analysis is least ambiguous, 

the predicted logarithmic modulation 

of free-field theory expressed in Eq. 43 

has, in fact, been brilliantly verified. Indeed, 

this was the main reason that QCD was 

accepted as the standard model for the strong 

interactions. 

There is, however, an even more profound 

consequence of the application of the renormalization 

group to the standard model 

that leads to interesting speculations con- 

cerning unified field theories. As discussed in 

“Particle Physics and the Standard Model,” 

QED and the weak interactions are partially 

unified into the electroweak theory. Both of 

these have a negative b and so are not 

asymptotically free; their effective couplings 

grow with energy rather than decrease. By the 

same token, the QCD coupling should grow 

as the energy decreases, ultimately leading to 

the confinement of quarks. Thus as energy 

increases, the two small electroweak couplings 

grow and the relatively large QCD 

coupling decreases. In 1974, Georgi, Quinn, 

and Weinberg made the remarkable observation 

that all fhree couplings eventually became 

equal at an energy scale of about IOl4 

GeV! The reason that this energy turns out to 

be so large is simply due to the very slow 

logarithmic variation of the couplings. This 

is a very suggestive result because it is extremely 

tempting to conjecture that beyond 

IOl4 GeV (that is, at distances below 

cm) all three interactions become unified 

and are governed by the same single coupling. 

Thus, the strong, weak, and electromagnetic 

forces, which at low energies 

appear quite disparate, may actually be 

manifestations of the same field theory. The 

search for such a unified field theory (and its 

possible extension to gravity) is certainly one 

of the central themes of present-day particle 

physics. It has proven to be a very exciting 

but frustrating quest that has sparked the 

imagination of many physicists. Such ideas 

are, of course, the legacy of Einstein, who 

devoted the last twenty years of his life to the 

search for a unified field theory. May his 

dreams become reality! On this note of fantasy 

and hope we end our brief discourse 

about the role of scale and dimension in 

understanding the world-or even the universe-around 

us. The seemingly innocuous 

investigations into the size and scale of 

animals, ships, and buildings that started 

with Galileo have led us, via some minor 

diversions, into baked turkey, incubating 

eggs, old bones, and the obscure infinities of 

Feynman diagrams to the ultimate question 

of unified field theories. Indeed, similitudes 

have been used and visions multiplied. H 

19

Geoffrey B. West was born in the county town of Taunton in Somerset, 

England. He received his B.A. from Cambridge University in 1961 and 

his Ph.D. from Stanford in 1966. His thesis, under the aegis of Leonard 

Schiff, dealt mostly with the electromagnetic interaction, an interest he 

has sustained throughout his career. He was a postdoctoral fellow at 

Cornell and Harvard before returning to Stanford in 1970 as a faculty 

member. He came to Los Alamos in 1974 as Leader of what was then 

called the High-Energy Physics Group in the Theoretical Division, a 

position he held until I98 I when he was made a Laboratory Fellow. His 

present interests revolve around the structure and consistency of quantum 

field theory and, in particular, its relevance to quantum 

chromodynamics and unified field theories. He has served on several 

advisory panels and as a member of the executive committee of the 

Division of Particles and Fields of the American Physical Society. 

Further Reading 

The following are books on the classical application of dimensional analysis: 

Percy Williams Bridgman. Dimensional Analysis. New Haven: Yale University Press, 1963. 

Leonid lvanovich Sedov. Similarity and Dimensional Methods in Mechanics. New York: Academic Press, 

1959. 

Garrett Birkhoff. Hydrodynamics: A Study in Logic, Fact and Similitude. Princeton: Princeton University 

Press, 1960. 

DArcy Wentworth Thompson. On Growth and Form. Cambridge: Cambridge University Press, 1.917. This 

book is, in some respects, comparable to Galileo’s and should be required reading for all budding young 

scientists. 

Benoit B. Mandelbrot. The Fractal GeometryofNafure. New York: W. H. Freeman, 1983. This recent, very 

interesting book represents a modern evolution ofthe subject into the area of fractals; in principle, the book 

deals with related problems, though I find it somewhat obscure in spite of its very appealing format. 

Examples of classical scaling were drawn from the following: 

Thomas McMahon. “Size and Shape in Biology.” Science 179( 1973):1201-1204. 

Hermann Rahn, Amos Ar, and Charles V. Paganelli. “How Bird Eggs Breathe.” Scientific American 

24O(February 1979):46-55. 

20


Thomas A. McMahon. “Rowing: a Similarity Analysis.” Science 173( 1971):349-351. 

David Pilbeam and Stephen Jay Could. “Size and Scaling in Human Evolution.” Science 186(1974): 

892-901. 

The Rayleigh-Riabouchinsky exchange is to be found in: 

Rayleigh. “The Principle of Similitude.” Nature 95( 19 I5):66-68. 

D. Riabouchinsky. “Letters to Editor.” Nature 95( 191 5):591. 

Rayleigh. “Letters to Editor.” Nature 95( 19 I5):644. 

Books on quantum electrodynamics (QED) include: 

Julian Schwinger, editor. Selected Papers on Quanfurn Elecfrodynarnics. New York Dover, 1958. This book 

gives a historical perspective and general review. 

James D. Bjorken and Sidney D. Drell. Relativistic Quantum Mechanics. New York: McGraw-Hill, 1964. 

N. N. Bogoliubov and D. V. Shirkov. Introduction to the Theory of Quantized Fields. New York: 

Interscience, 1959. 

H. David Politzer. “Asymptotic Freedom: An Approach to Strong Interactions.” Physics Reports 

14( 1974): 129- 180. This and the previous reference include a technical review of the renormalization group. 

Claudio Rebbi. “The Lattice Theory of Quark Confinement.” Scientific American 248(February 

1983):54-65. This reference is also a nontechnical review of lattice gauge theories. 

For a review of the deep inelastic electron scattering experiments see: 

Henry W. Kendall and Wolfgang K. H. Panofsky. “The Structure of the Proton and the Neutron.” Scientific 

American 224(June 1971):60-76. 

Geoffrey B. West. “Electron Scattering from Atoms, Nuclei and Nucleons.” Physics Reports 

18( 1975):263-323. 

References dealing with detailed aspects of renormalization and its consequences are: 

Kenneth G. Wilson. “Non-Lagrangian Models of Current Algebra.” Physical Review l79( 1969): 1499-1 512. 

Geoffrey B. West. “Asymptotic Freedom and the Infrared Problem: A Novel Solution to the Renormalization-Group 

Equations.” Physical Review D 27( 1983): 1402-1405. 

E. C. G. Stueckelberg and A. Petermann. “La Normalisation des Constantes dans la Theorie des Quanta.” 

Helveticu Physica Acta 26( I953):499-520. 

M. Cell-Mann and F. E. Low. “Quantum Electrodynamics at Small Distances.” Physical Review 

95( 1954): 1300- I3 12. 

H. David Politzer. “Reliable Perturbative Results for Strong Interactions?’ Physical Review Letters 

30( 1973): 1346- 1349. 

David J. Gross and Frank Wilczck. “Ultraviolet Behavior of Non-Abelian Gauge Theories.” Physical 

Review Letters 30( 1973): 1343-1 346. 

21

c 

Particle Physics 

and the 

Standard Model 

by Stuart A. Raby, Richard C. Slansky, and Geoffrey B. West 

U 

ntil the 1930s all natural phenomena were 

presumed to have their origin in just two 

basic forces-gravitation and electromagnetism. 

Both were described by classical 

fields that permeated all space. These fields extended 

out to infinity from well-defined sources, 

mass in the one case and electric charge in the 

other. Their benign rule over the physical universe 

seemed securely established. 

As atomic and subatomic phenomena were explored, 

it became apparent that two completely 

novel forces had to be added to the list; they were 

dubbed the weak and the strong. The strong force 

was necessary in order to understand how the 

nucleus is held together: protons bound together in 

a tight nuclear ball ( centimeter across) must 

be subject to a force much stronger than electromagentism 

to prevent their flying apart. The 

weak force was invoked to understand the transmutation 

of a neutron in the nucleus into a proton 

during the particularly slow form of radioactive 

decay known as beta decay. 

Since neither the weak force nor the strong force 

is directly observed in the macroscopic world, 

both must be very short-range relative to the more 

familiar gravitational and electromagnetic forces. 

Furthermore, the relative strengths of the forces 

associated with all four interactions are quite different, 

as can be seen in Table 1. It is therefore not 

too surprising that for a very long period these 

interactions were thought to be quite separate. In 

spite of this, there has always been a lingering 

suspicion (and hope) that in some miraculous 

fashion all four were simply manifestations of one 

source or principle and could therefore be described 

by a single unified field theory. 

The color force among quarks and gluons is described by a generalization of the Lagrangian 6p of quantum 

electrodynamics shown above. The large interaction vertex dominating these pages is a common feature of the 

strong, the weak, and the electromagnetic forces. A feature unique to the strong force, the self-interaction of 

colored gluons, is suggested by the spiral in the background.

Table 1 

basic interactions are ob

Particle Physics and the Standard Model 

Fig. 1. The main features of the standard model. The strong 

force and the electroweak force are each induced by a local 

symmetry group, SU(3) and SU(2) X U(l), respectively. 

These two symmetries are entirely independent of each other. 

SU(3) symmetry (called the color symmetry) is exact and 

therefore predicts conservation of color charge. The SU(2) X 

U(1) symmetry of the electroweak theory is an exact sym- 

metry of the Lagrangian of the theory but not of the solutions 

to the theory. The standard model ascribes this symmetry 

breaking to the Higgs particles, particles that create a 

nonzero weak charge in the vacuum (the lowest energy state 

of the system). The only conserved quantity that remains 

after the symmetry breaking is electric charge. 

The spectacular progress in particle physics 

over the past ten years or so has renewed 

this dream; many physicists today believe 

that we are on the verge of uncovering the 

structure of this unified theory. The theoretical 

description of the strong, weak, and electromagnetic 

interactions is now considered 

well established, and, amazingly enough, the 

theory shows these forces to be quite similar 

despite their experimental differences. The 

weak and strong forces have sources 

analogous to, but more complicated than, 

electric charge, and, like the electromagnetic 

force, both can be described by a special type 

of field theory called a local gauge theory. 

This formulation has been so successful at 

explaining all known phenomenology up to 

energies of 100 GeV (1 GeV = io9 electron 

volts) that it has been coined “the standard 

model” and serves as the point of departure 

for discussing a grand unification of all 

forces, including that of gravitation. 

The elements of the standard model are 

summarized in Fig. I. In this description the 

basic constituents of matter are quarks and 

leptons, and these constituents interact with 

each other through the exchange of gauge 

particles (vector bosons), the modern 

analogue of force fields. These so-called local 

gauge interactions are inscribed in the language 

of Lagrangian quantum field theory, 

whose rich formalism contains mysteries 

that escape even its most faithful practitioners. 

Here we will introduce the central 

themes and concepts that have led to the 

standard model, emphasizing how its formalism 

enables us to describe all 

phenomenology of the strong, weak, and 

electromagnetic interactions as different 

manifestations of a single symmetry principle, 

the principle of local symmetry. As we 

shall see, the standard model has many 

arbitrary parameters and leaves unanswered 

a number of important questions. It can 

hardly be regarded as a thing of great 

beauty-unless one keeps in mind that it 

embodies a single unifying principle and 

therefore seems to point the way toward a 

grander unification. 

For those readers who are more 

mathematically inclined, the arguments here 

are complemented by a series of lecture notes 

immediately following the main text and 

entitled “From Simple Field Theories to the 

Standard Model.” The lecture notes introduce 

Lagrangian formalism and stress the 

symmetry principles underlying construc- 

25

tion of the standard model. The main 

emphasis is on the classical limit of the 

model, but indications of its quantum generalizations 

are also included. 

Unification and Extension 

Two central themes of physics that have 

led to the present synthesis are “unification” 

and “extension.” By “unification” we mean 

the coherent description of phenomena that 

are at first sight totally unrelated. This takes 

the form of a mathematical description with 

specific rules of application. A theory must 

not only describe the known phenomena but 

also make predictions of new ones. Almost 

all theories are incomplete in that they 

provide a description of phenomena only 

within a specific range of parameters. Typically, 

a theory changes as it is extended to 

explain phenomena over a larger range of 

parameters, and sometimes it even 

simplifies. Hence, the second theme is called 

extension-and refers in particular to the 

extension of theories to new length or energy 

scales. It is usually extension and the resulting 

simplification that enable unification. 

Perhaps the best-known example of extension 

and unification is Newton’s theory of 

gravity (1666), which unifies the description 

of ordinary-sized objects falling to earth with 

that ofthe planets revolving around the sun. 

It describes phenomena over distance scales 

ranging from a few centimeters up to 

102’ centimeters (galactic scales). Newton’s 

theory is superceded by Einstein’s theory of 

relativity only when one tries to describe 

phenomena at extremely high densities 

and/or velocities or relate events over cosmological 

distance and time scales. 

The other outstanding example of unification 

in classical physics is Maxwell’s theory 

of electrodynamics, which unifies electricity 

with magnetism. Coulomb (1785) had established 

the famous inverse square law for the 

force between electrically charged bodies, 

and Biot and Savart (1820) and Ampere 

(1820-1825) had established the law relating 

the magnetic field B to the electric current as 

well as the law for the force between two 

26 

electric currents. Thus it was known that 

static charges give rise to an electric field 

E and that moving charges give rise to a 

magnetic field B. Then in 183 1 Faraday discovered 

that the field itself has a life of its 

own, independent of the sources. A timedependent 

magnetic field induces an electric 

field. This was the first clear hint that electric 

and magnetic phenomena were manifestations 

of the same force field. 

Until the time of Maxwell, the basic laws 

of electricity and magnetism were expressed 

in a variety of different mathematical forms, 

all of which left the central role of the fields 

obscure. One of Maxwell’s great achievements 

was to rewrite these laws in a single 

formalism using the fields E and B as the 

fundamental physical entities, whose sources 

are the charge density p and the current 

density J, respectively. In this formalism the 

laws of electricity arid magnetism are expressed 

as differential equations that manifest 

a clear interrelationship between the two 

fields. Nowadays they are usually written in 

standard vector notation as follows. 

Coulomb’s law: 

Ampere’s law: 

V . E = 41tp/%; 

V X B = 4npoJ; 

Faraday’s law: v x E + aB/at = 0; 

and the absence of 

magnetic monopoles: V B = 0. 

The parameters EO and po are determined by 

measuring Coulomb’s force between two 

static charges and Ampere’s force between 

two current-carrying wires, respectively. 

Although these equations clearly “unite” 

E with B, they are incomplete. In 1865 Maxwell 

realized that the above equations were 

not consistent with the conservation of electric 

charge, which requires that 

v . J + ap/at = 0 , 

This inconsistency can be seen from 

Ampere’s law, which in its primitive form 

requires that 

V*J=(4xpo)-’V*(VXB) 0. 

Maxwell obtained a consistent solution by 

amending Ampere’s law to read 

aE 

V X B=4xpoJ + kclo-. at 

With this new equation, Maxwell showec 

that both E and B satisfy the wave equation 

For example, 

at2 

This fact led him to propose the elec 

tromagnetic theory of light. Thus, from Max 

well’s unification of electric and magnetic 

phenomena emerged the concept of elec 

tromagnetic waves. Moreover, the speed c o 

the electromagnetic waves, or light, is give1 

by (~po)-’/~ and is thus determined unique]! 

in terms of purely static electric and magne 

tic measurements alone! 

It is worth emphasizing that apart fron 

the crucial change in Ampere’s law, Max 

well’s equations were well known to natura 

philosophers before the advent of Maxwell 

The unification, however, became manifes 

only through his masterstroke of expressin) 

them in terms of the “right” set of variables 

namely, the fields E and B. 

Extension to Small Distance 

Maxwell’s unification provides an ac 

curate description of large-scale elec 

tromagnetic phenomena such as radic 

waves, current flow, and electromagnets 

This theory can also account for the effects o 

a medium, provided macroscopic concepti 

such as conductivity and permeability arc 

introduced. However, ifwe try to extend it tc 

very short distance scales, we run intc 

trouble; the granularity, or quantum nature 

of matter and of the field itself become! 

important, and Maxwell’s theory must bt 

altered. 

Determining the physics appropriate tc 

each length scale is a crucial issue and ha! 

been known to cause confusion (see “Funda. 

mental Constants and the Rayleigh.


Fig. 2. The wavelength of the probe must be smaller than the scale of the structure 

one wants to resolve. Viruses, which are approximately 1r5 centimeter in extent, 

cannot be resolved with visible light, the average wavelength of which is 5 X l(T5 

centimeter. However, electrons with momentum p of about 20 eV/c have de Broglie 

wavelengths short enough to resolve them. 

Riabouchinsky Paradox”). For example, the 

structure of the nucleus is completely irrelevant 

when dealing with macroscopic distances 

of, say, 1 centimeter, so it would be 

absurd to try to describe the conductivity of 

iron over this distance in terms of its quark 

and lepton structure. On the other hand, it 

would be equally absurd to extrapolate 

Ohm’s law to distance intervals of 

centimeter to determine the flow of electric 

current. Relevant physics changes with scale! 

The thrust of particle physics has been to 

study the behavior of matter at shorter and 

shorter distance scales in hopes of understanding 

nature at its most fundamental 

level. As we probe shorter distance scales, we 

encounter two types of changes in the phys- 

ics. First there is the fundamental change 

resulting from having to use quantum mechanics 

and special relativity to describe 

phenomena at very short distances. According 

to quantum mechanics, particles have 

both wave and particle properties. Electrons 

can produce interference patterns as waves 

and can deposit all their energy at a point as a 

particle. The wavelength k associated with 

the particle of momentum p is given by the 

de Broglie relation 

h A=- 

P’ 

where h is Planck‘s constant (h/2rr = h = 

1.0546 X erg . second). This relation is 

the basis of the often-stated fact that resolving 

smaller distances requires particles of 

greater momentum or energy. Notice, incidentally, 

that for sufficiently short wavelengths, 

one is forced to incorporate special 

relativity since the corresponding particle 

momentum becomes so large that Newtonian 

mechanics fails. 

The mamage of quantum mechanics and 

special relativity gave birth to quantum field 

theory, the mathematical and physical language 

used to construct theories of the 

elementary particles. Below we will give a 

brief review of its salient features. Here we 

simply want to remind the reader that quantum 

field theory automatically incorporates 

quantum ideas such as Heisenberg’s uncertainty 

principle and the dual wave-particle 

properties of all of matter, as well as the 

equivalence of mass and energy. 

Since the wavelength of our probe determines 

the size of the object that can be 

studied (Fig. 2), we need extremely short 

wavelength (high energy) probes to investigate 

particle phenomena. To gain some 

perspective, consider the fact that with visible 

light we can see without aid objects as 

small as an amoeba (about lo-* centimeter) 

and with an optical microscope we can open 

up the world of bacteria at about IO4 centimeter. 

This is the limiting scale of light 

probes because wavelengths in the visible 

spectrum are on the order of 5 X IO-’ centimeter. 

To resolve even smaller objects we can 

exploit the wave-like aspects of energetic 

particles as is done in an electron microscope. 

For example, with “high-energy” electrons 

(E = 20 eV) we can view the world of 

viruses at a length scale of about lo-’ centimeter. 

With even higher energy electrons 

we can see individual molecules (about I 0-7 

centimeter) and atoms (lo-* centimeter). To 

probe down to nuclear ( centimeter) 

and subnuclear scales, we need the particles 

available from high-energy accelerators. Today’s 

highest energy accelerators produce 

100-GeV particles, which probe distance 

scales as small as centimeter. 

This brings us to the second type of change 

27

in appropriate physics with change in scale, 

namely, changes in the forces themselves. 

Down to distances of approximately 

centimeter, electromagnetism is the dominant 

force among the elementary particles. 

However, at this distance the strong force, 

heretohre absent, suddenly comes into play 

and completely dominates the interparticle 

dynamics. The weak force, on the other 

hand, is present at all scales but only as a 

small effect. At the shortest distances being 

probed by present-day accelerators, the weak 

and electromagnetic forces become comparable 

in strength but remain several orders 

of magnitude weaker than the strong force. It 

is at this scale however, that the fundamental 

similarity of all three forces begins to emerge. 

Thus, as the scale changes, not only does 

each force itself change, but its relationship 

to the other forces undergoes a remarkable 

evolution. In our modern way of thinking, 

which has come from an understanding of 

the renormalization, or scaling, properties of 

quantum field theory, these changes in physics 

are in some ways analogous to the 

paradigm of phase transitions. To a young 

and naive child, ice, water, and steam appear 

to be quite different entities, yet rudimentary 

observations quickly teach that they are different 

manifestations of the same stuff, each 

associated with a different temperature scale. 

The modern lesson from renormalization 

group analysis, as discussed in “Scale and 

Dimension-From Animals to Quarks,” is 

that the physics of the weak, electromagnetic, 

and strong forces may well represent different 

aspects of the same unified interaction. 

This is the philosophy behind grand 

unified theories of all the interactions. 

Quantum Electrodynamics and 

Field Theory 

Let us now return to the subject of electromagnetism 

at small distances and describe 

quantum electrodynamics (QED), the 

relativstic quantum field theory, developed 

in the 1930s and 194Os, that extends Maxwell’s 

theory to atomic scales. We emphasize 

that the standard model is a generalization of 

Antiproton 

this first and most successful quantum field 

theory. 

In quantum field theory every particle has 

associated with it a mathematical operator, 

called a quantum field, that carries the particle’s 

characteristic quantum numbers. 

Probably the most familiar quantum number 

is spin, which corresponds to an intrinsic 

angular momentum. In classical mechanics 

angular momentum is a continuous variable, 

whereas in quantum mechanics it is restricted 

to multiples of ‘12 when measured in units 

of h. Particles with 5’2-integral spin (1/2, 3/2, 

5/2, ...) are called fermions; particles with 

integral spin (0, I , 2, 3, ... ) are called bosons. 

Since no two identical fermions can occupy 

the same position at the same time (the 

famous Pauli exclusion principle), a collel 

tion of identical fermions must necessari 

take up some space. This special property I 

fermions makes it natural to associate the1 

with matter. Bosons, on the other hand, ca 

crowd together at a point in space-time 1 

form a classical field and are naturally ri 

garded as the mediators of forces. 

In the quantized version of Maxwell’s thc 

ory, the electromagnetic field (usually in tt 

guise of the vector potential A,,) is a boso 

field that carries the quantum numbers of tk 

photon, namely, mass m = 0, spin s = 1, an 

electric charge Q = 0. This quantized field, t 

the very nature of the mathematics, aut1 

matically manifests dual wave-partic 

properties. Electrically charged particle 

28


f 

ig. 3. (a) The force between two electrons is described classically by Coulomb’s 

aw. Each electron creates a forcefield (shown as lines emanating from the charge 

(e) that is felt by the other electron. The potential energy V is the energy needed to 

ring the two electrons to within a distance r of each other. (6) In quantum field 

heory two electrons feel each other’s presence by exchanging virtual photons, or 

virtual particles of light. Photons are the quanta of the electromagnetic field. The 

eynman diagram above represents the (lowest order, see Fig. 5) interaction 

etween two electrons (straight lines) through the exchange of a virtual photon 

I 

wavy line). 

such as electrons and positrons, are also represented 

by fields, and, as in the classical 

theory, they interact with each other through 

the electromagnetic field. In QED, however, 

the interaction takes place via an exchange of 

photons. Two electrons “feel” each other’s 

presence by passing photons back and forth 

between them. Figure 3 pictures the interaction 

with a “Feynman diagram”: the straight 

lines represent charged particles and the 

wavy line represents a photon. (In QED such 

diagrams correspond to terms in a 

perturbative expansion for the scattering between 

charged particles (see Fig. 5). 

Similarly, most Feynman diagrams in this 

issue represent lowest order contributions to 

the particle reactions shown.) 

These exchanged photons are rather 

special. A real photon, say in the light by 

which you see, must be massless since only a 

massless particle can move at the speed of 

light. On the other hand, consider the lefthand 

vertex of Fig. 3, where a photon is 

emitted by an electron; it is not difficult to 

convince oneself that if the photon is massless, 

energy and momentum are not conserved! 

This is no sin in quantum mechanics, 

however, as Heisenberg’s uncertainty principle 

permits such violations provided they 

occur over sufficiently small space-time intervals. 

Such is the case here: the violating 

photon is absorbed at the right-hand vertex 

by another electron in such a way that, overall, 

energy and momentum are conserved. 

The exchanged photon is “alive” only for a 

period concomitant with the constraints of 

the uncertainty principle. Such photons are 

referred to as virtual photons to distinguish 

them from real ones, which can, of course, 

live forever. 

The uncertainty principle permits all sorts 

of virtual processes that momentarily violate 

energy-momentum conservation. As illustrated 

in Fig. 4, a virtual photon being 

exchanged between two electrons can, for a 

very short time, turn into a virtual electronpositron 

pair. This conversion of energy into 

mass is allowed by the famous equation of 

special relativity, E = mc2. In a similar 

fashion almost anything that can happen will 

29

happen, given a sufficiently small space-time 

interval. It is the countless multitude of such 

virtual processes that makes quantum field 

theory :so rich and so difficult. 

Given the immense complexity of the theory, 

one wonders how any reliable calculation 

can ever be made. The saving grace of 

quantum electrodynamics, which has made 

its predictions the most accurate in all of 

physics, is the smallness of the coupling between 

the electrons and the photons. The 

coupling strength at each vertex where an 

electron spews out a virtual photon is just the 

electronic charge e, and, since the virtual 

photon must be absorbed by some other 

electron, which also has charge e, the 

probability for this virtual process is of magnitude 

e’. The corresponding dimensionless 

parameter that occurs naturally in this theory 

is denoted by a and defined as e2/4nh e. It is 

approximately equal to 11137. The 

probabilities of more complicated virtual 

processes involving many virtual particles 

are proportional to higher powers of a and 

are therefore very much smaller relative to 

the probabilities for simpler ones. Put 

slightly differently, the smallness of a implies 

that perturbation theory is applicable, and 

we can control the level of accuracy of our 

calcu1ai:ions by including higher and higher 

order virtual processes (Fig. 5). In fact, quantum 

electrodynamic calculations of certain 

atomic and electronic properties agree with 

experiment to within one part in a billion. 

As we will elaborate on below, the quantum 

field theories of the electroweak and the 

strong interactions that compose the standard 

model bear many resemblances to 

quantum electrodynamics. Not too surprisingly, 

the coupling strength of the weak interaction 

is also small (and in fact remains small 

at all energy or distance scales), so perturbation 

theory is always valid. However, the 

analogue of a for the strong interaction is not 

always small, and in many calculations 

perturbation theory is inadequate. Only at 

the high energies above 1 GeV, where the 

theory is said to be asymptotically free, is the 

analogue of a so small that perturbation theory 

is valid. At low and moderate energies 

Fig. 4. A virtualphoton being exchanged between two electrons can, for a very shor 

time, turn into a virtual electron-positron (e+-e-) pair. This virtualprocess is on 

of many that contribute to the electromagnetic interaction between electricall 

chargedparticles (see Fig. 5). 

(for example, those that determine the 

properties of protons and neutrons) the 

strong-interaction coupling strength is large, 

and analytic techniques beyond perturbation 

theory are necessary. So far such techniques 

have not been very :successful, and one has 

had to resort to the nasty business of numerical 

simulations! 

As discussed at the end of the previous 

section, these changes in coupling strengths 

with changes in scale are the origin of the 

changes in the forces that might lead to a 

unified theory. For an example see Fig. 3 in 

“Toward a Unified Theory.” 

Symmetries 

One cannot discuss the standard model 

without introducing the concept of symmetry. 

It has played a central role in classifying 

the known particle states (the ground 

states of 200 or so particles plus excited 

states) and in predicting new ones. Just as the 

chemical elements fall into groups in the 

periodic table, the particles fall into multiplets 

characterized by similar quantum 

numbers. However, the use of symmetry in 

particle physics goes well beyond mere 

classification. In the construction of the stai 

dard model, the special kind of symmeti 

known as local symmetry has become tl 

guiding dynamical principle; its aesthetic ii 

fluence in the search for unification is ren 

iniscent of the quest for beauty among tk 

ancient Greeks. Before we can discuss th 

dynamical principle, we must first review tl 

general concept of symmetry in partic 

physics. 

In addition to electric charge and mas 

particles are characterized by other quantui 

numbers such as spin, isospin, strangenes 

color, and so forth. These quantum numbe 

reflect the symmetries of physical laws an 

are used as a basis for classification an1 

ultimately, unification. 

Although quantum numbers such as spi 

and isospin are typically the distinguishir 

features of a particle, it is probably less we 

known that the mass of a particle is som 

times its only distinguishing feature. For e 

ample, a muon (p) is distinguished from a 

electron (e) only because its mass is 2C 

times greater that that of the electron. I1 

deed, when the muon was discovered 1 

1938, Rabi was reputed to have made tk 

remark. “Who ordered that?” And the ta 

30


Electron Scattering 

Electromagnetic 

Interaction = eJpA, 

(Interaction)* a a 

(~nteraction)~ a a2 

(Interaction)6 a a3 

. 

where Jo = p 

A _ _ 

Fig. 5. As shown above, the basic interaction 

vertex of quantum electrodynamics 

is an electron current J’ 

interacting with the electromagnetic 

field A,,. Because the coupling strength 

a is small, the amplitude for processes 

involving such interactions can be approximated 

by a perturbation expansion 

on a free field theory. The 

terms in such an expansion, shown at 

left for electron scattering, are proportional 

to various powers of a. The largest 

contribution to the electron-scattering 

amplitude is proportional to a and 

is represented by a Feynmann diagram 

in which the interaction vertex appears 

twice. Successively smaller contributions 

arise from terms proportional to 

a’ with four interaction vertices, from 

terms proportional to a3 with six interaction 

vertices, and so on. 

(T), discovered in 1973, is 3500 times heavier 

than an electron yet again identical to the 

electron in other respects. One of the great 

unsolved mysteries of particle physics is the 

origin of this apparent hierarchy of mass 

among these leptons. (A lepton is a fundamental 

fermion that has no strong interactions.) 

Are there even more such particles? Is 

there a reason why the mass hierarchy among 

the leptons is paralleled (as we will describe 

below) by a similar hierarchy among the 

quarks? It is believed that when we understand 

the origin of fermion masses, we will 

also understand the origin of CP violation in 

nature (see box). These questions are frequently 

called the family problem and are 

discussed in the article by Goldman and 

Nieto. 

Groups and Group Multiplets. Whether or 

not the similarity among e, p, and ‘5 reflects a 

fundamental symmetry of nature is not 

known. However, we will present several 

possibilities for this family symmetry to introduce 

the language of groups and the 

significance of internal symmetries. 

Consider a world in which the three leptons 

have the same mass. In this world atoms 

with muons or taus replacing electrons 

would be indistinguishable: they would have 

identical electromagnetic absorption or 

emission bands and would form identical 

elements. We would say that this world is 

invariant under the interchange of electrons, 

muons, and taus, and we would call this 

invariance a symmetry ofnature. In the,real 

world these particles don’t have the same 

mass; therefore our hypothetical symmetry, 

if it exists, is broken and we can distinguish a 

muonic atom from, say, its electronic 

counterpart. 

We can describe our hypothetical invariance 

or family symmetry among the 

three leptons by a set of symmetry operations 

that form a mathematical construct called a 

group. One property of a group is that any 

two symmetry operations performed in succession 

also corresponds to a symmetry 

operation in that group. For example, replacingan 

electron with a muon, and then replacing 

a muon with a tau can be defined as two 

discrete symmetry operations that when 

performed in succession are equivalent to 

the discrete symmetry operation of replacing 

an electron with a tau. Another group property 

is that every operation must have an 

inverse. The inverse of replacing an electron 

with a muon is replacing a muon with an 

electron. This set of discrete operations on 

e, p, and T forms the discrete six-element 

group rr3 (with K standing for permutation). 

In this language e, p, and T are called a 

multiplet or representation of ~3 and are said 

to transform as a triplet under ~ 3 . 

Another possibility is that the particles e, 

p, and T transform as a triplet under a group 

of conlinuous symmetry operations. Consider 

Fig. 6, where e, p, and T are represented 

as three orthogonal vectors in an abstract 

31

three-dimensional space. The set of continuous 

rotations of the three vectors about three 

independent axes composes the group 

known as the three-dimensional rotation 

group and denoted by SO(3). As shown in 

Fig. 6, SO(3) has three independent transformations, 

which are represented by orthogonal 

3 X 3 matrices. (Note that n3 is a 

subset of SO(3).) 

Suppose that SO(3) were an unbroken 

family symmetry of nature and e, p, and r 

transformed as a triplet under this symmetry. 

How would it be revealed experimentally? 

The SO(3) symmetry would add an 

extra degree of freedom to the states that 

could be formed by e, p, and r. For example, 

the spatially symmetric ground state of 

helium, which ordinarily must be antisymmetric 

under the interchange of the two electron 

spins, could now be antisymmetric 

under the interchange of either the spin or 

the family quantum number of the two leptons. 

In particular, the ground state would 

have three different antisymmetric configurations 

and the threefold degeneracy 

might be split by spin-spin interactions 

among the leptons and by any SO(3) symmetric 

interaction. Thus the ground state of 

known helium would probably be replaced 

by sets of degenerate levels with small hyperfine 

energy splittings. 

In particle physics we are always interested 

in the largest group of operations that leaves 

all properties of a system unchanged. Since e, 

p, and T are described by complex fields, the 

largest group of operations that could act on 

this triplet is U(3) (the group of all unitary 3 

X 3 matrices Usatisfying UtU= I). Another 

possibility is SU(3), a subgroup ofU(3) satisfying 

the additional constraint that det U = I. 

This list of symmetries that may be 

reflected in the similarity of e, p, and 7 is not 

exhaustive. We could invoke a group of symmetry 

operations that acts on any subset of 

the three particles, such as SU(2) (the group 

of 2 X 2 unitary matrices with det U = I) 

acting, say, on e and p as a doublet and on r 

as a singlet. Any one of these possibilities 

may be realized in nature, and each possibility 

has different experimentally observable 

, 

i 

I t 

1 

Fig. 6. (a) The three leptons e, F, and T are represented as three orthogonal vecton 

in an abstract three-dimensional space. (b) The set of rotations about the thret 

orthogonal axes defines S0(3), the three-dimensional rotation group. SO(3) haJ 

three charges (or generators) associated with the infinitesimal transformatiom 

about the three independent axes. These generators have the same Lie algebra as the 

generators of the g,roup SU(2), as discussed in Lecture Note 4 following this article. 

32


onsequences. However, the known dif- 

:rences in the masses of e, p, and T imply 

iat any symmetry used to describe the 

imilarity among them is a broken symietry. 

Still, a broken symmetry will retain 

aces of its consequences (if the symmetry is 

broken by a small amount) and thus also 

provides useful predictions. 

Our hypothetical broken symmetry 

among e, p, and T is but one example of an 

approximate internal global symmetry. Another 

is the symmetry between, say, the neu- 

tron and the proton in strong interactions, 

which is described by the group known as 

strong-isospin SU(2). The neutron and 

proton transform as a doublet under this 

symmetry and the three pions transform as a 

triplet. We will discuss below the classifica- 

CP Violation 

T 

he faith of physicists in symmetries of 

nature, so shaken by the observation 

of parity violation in 1956, was soon 

restored by invocation of a new symmetry 

principle-CP conservation-to interpret 

parity-violating processes. This principle 

states that a process is indistinguishable from 

its mirror image provided 

mirror image are replaced 

cles. Alas, in 1964 this 

shattered with the results 

on the decay of neutral kaons. 

According to the classic analysis of M. 

Gell-Mann and A. Pais, neutral kaons cxist 

in two forms: Kg, with an even CP eigenvalue 

and decaying with a relatively short 

lifetime of IO-” second into two pions, and 

K:, with an odd CP eigenvalue and decaying 

with a lifetime of about 5 X lo-* second into 

three pions. CP conservati 

decay of the longer lived K 

But in an experiment at Brookhaven, J. 

Christenson, J. Cronin, V. Fitch, and R. 

Turlay found that about I in 500 Kf mesons 

decays into two pions. This first observation 

of CP violation has been confirmed in many 

other experiments on the neutral kaon system, 

but to date no other CP-violating effects 

have been found. The underlying mechanism 

of CP violation remains to be understood, 

and an implication of the phenomenon, 

the breakdown of time-reversal invariance 

(which is necessary to maintain 

CPT conservation), remains to be observed. 

30 

20 

10 

w 

0 

0.9996 

J 

494 < m* < 504 

I 

0.9998 

cos 8 

I 

1 .oooo 

Evidence for the CP-violating decay of Ki into two pions. Here the number 

of events in which the invariant mass (m *) of the decayproducts was in close 

proximity to the mass of the neutral kaon is plotted versus the cosine of the 

angle 0 between the Kf beam and the vector sum of the momenta of the 

decay products. The peak in the number of events at cos 0 z 1 (indicative of 

two-body decays) could only be explained as the decay of K; into two pions 

with a branchingratio of about 2 X (Adapted from “Evidence for the 

2rc Decay of the Kg Meson” by J. H. Christenson, J. W. Cronin, V. L. Fitch, 

and R. Turlay, Physical Review Letters 13(1964):138.) 

33

tion of strongly interacting particles into 

multiplets of SU(3), a scheme that combines 

strong isospin with the quantum number 

called strangeness, or strong hypercharge. 

(For a more complete discussion of continuous 

symmetries and internal global symmetries 

such as SU(2), see Lecture Notes 2 

and 4.) 

Exact., or unbroken, symmetries also play 

a fundamental role in the construction of 

theories: exact rotational invariance leads to 

the exact conservation of angular momentum, 

and exact translational invariance in 

space-time leads to the exact conservation of 

energy and momentum. We will now discuss 

how the exact phase invariance of electrodynamics 

leads to the exact conservation 

of electric charge. 

Global U(l) Invariance and Conservation 

Laws. In quantum field theory the dynamics 

of a system are encoded in a function of the 

fields called a Lagrangian, which is related to 

the energy of the system. The Lagrangian is 

the mosl. convenient means for studying the 

symmetries ofthe theory because it is usually 

a simple task to check if the Lagrangian 

remains unchanged under particular symmetry 

operations. 

An electron is described in quantum field 

theory by a complex field, 

and a positron is described by the complex 

conjugate of that field, 

Although the real fields wI and I+I~ are 

separately each able to describe a spin-% 

particle, the two together are necessary to 

describe a particle with electric charge.* 

The Lagrangian of quantum electrodynamics 

is unchanged by the continuous 

operation of multiplying the electron field by 

*The real fields VI and y12 are four-component 

Majorana fields that together make up the standard 

four-component complex Dirac spinor field. 

an arbitrary phase, that is, by the transformation 

w - eiAQW, 

where A is an arbitary real number and Q is 

the electric charge operator associated with 

the field. The eigenvalue of Q is -I for an 

electron and +1 for a positron. This set of 

phase transformations forms the global symmetry 

group U(1) (the set of unitary 1 X 1 

matrices). In QED this symmetry is unbroken, 

and electric charge is a conserved 

quantum number of the system. 

There are other global U(I) symmetries 

relevant in particle physics, and each one 

implies a conserved quantum number. For 

example, baryon number conservation is associated 

with a U(1) phase rotation of all 

baryon fields by an amount e', where B = 1 

for protons and neutrons, B = '1 for quarks, 

and B = 0 for leptons. Analogously, electron 

number is conserved if the field of the electron 

neutrino is assigned the same electron 

number as the field of the electron and all 

other fields are assigned an electron number 

of zero. The same holds true for muon number 

and tau number. Thus a global U(1) 

phase symmetry seems to operate on each 

type of lepton. (Possible violation of muonnumber 

conservation is discussed in "Experiments 

To Test Unification Schemes.") 

The Principle of Local Symmetry 

We are now ready to distinguish a global 

phase symmetry from a local one and examine 

the dynamical consequences that emerge 

from the latter. Figure 7 illustrates what hap- 

pens to the electron field under the global 

phase transformation w - e"Qy. For convenience, 

space-time is represented by a set 

of discrete points labeled by the index j. The 

phase of the electron field at each point is 

represented by an arrow that rotates about 

the point, and the kinetic energy of the field 

is represented by springs connecting the arrows 

at different space-time points. A global 

U( 1) transformation rotates every two-dimensional 

vector by the same arbitrary angle 

A: 0,- 0, + QA, where Q is the electric 

charge. In order for the Lagrangian to be 

invariant under this global phase rotation, it 

is clearly sufficient for it to be a function only 

ofthe phase differences (0, - 0,). Both the free 

electron terms and the interaction terms in 

the QED Lagrangian are invariant under this 

continuous global symmetry. 

A local U( I ) transformation, in contrast, 

rotates every two-dimensional vector by a 

different angle A,. This local transformation, 

unlike its global counterpart, does not leave 

the Lagrangian of the free electron invariant. 

As represented in Fig. 7 by the stretching and 

compressing of the springs, the kinetic 

energy of the electron changes under local 

phase transformations. Nevertheless, the full 

Lagrangian of quantum electrodynamics is 

invariant under these local U( I ) transformations. 

The electromagnetic field (A,) 

precisely compensates for the local phase 

rotation and the Lagrangian is left invariant. 

This is represented in Fig. 7 by restoring the 

stretched and compressed springs to their 

initial tension. Thus, the kinetic energy of the 

electron (the energy stored in the springs) ir 

the same before and after the local phase 

transformation. 

In our discrete notation, the full La- 

Fig. 7. Global versus local phase transformations. The arrows represent the phases 

of an electron field at four discrete points labeled by j = I, 2, 3, and 4. The springs 

represent the kinetic energy of the electrons. A global phase transformation does 

not change the tension in the springs and therefore costs no energy. A local phase 

transformation witlhout gauge interactions stretches and compresses the springs 

and thus does cost energy. However introduction of the gaugefield (represented by 

the white haze) exactly compensates for the local phase transformation of the 

electron field and the springs return to their original tension so that local phase 

transformations with gauge interactions do not cost energy. 

34

~ 


__ 

Phase 

I 

j=2 j'3 j=4 

Global Phase 

Transformation 

+(j) -+ eiQA $'(I) 

ei -+ ei + QA 

Q$' =+$ 

Q= +I 

iQ A . 

Local Phase J/(j)-+e ' $'(i) 


Oi-+Oj+QAi Q=+I 

I I 

Gauge Field I I 

Compensates for 

Local Phase 


I I 

Ai, -+ Ai, - Ai + A, 

35

and is 

invariant under the simultaneous transformations 

grangian is a function of e, - ek + A,& 

r-- 

I 

I 

The matrix with elements A,k is the discrete 

space-time analogue of the electromagnetic 

potential defined on the links between the 

points k and j. Thus, if one starts with a 

theory of free electrons with no interactions 

and demands that the physics remain invariant 

under local phase transformation of 

the electron fields, then one induces the standard 

electromagnetic interactions between 

the electron current .Ip and photon field A,,, 

as shown in Figs. 5 and 8. From this point of 

view, Maxwell’s equations can be viewed as a 

consequence of the local U(1) phase invariance. 

Although this local invariance was 

originally viewed as a curiosity of QED, it is 

now viewed as the guiding principle for constructing 

field theories. The invariance is 

usually termed gauge invariance, and the 

photon is referred to as a gauge particle since 

it mediates the U(1) gauge interaction. It is 

worth emphasizing that local U(1) invariance 

implies that the photon is massless 

because the term that would describe a 

massive photon is not itself invariant under 

local U(1) transformations. 

The local gauge invariance of QED is the 

prototype for theories of both the weak and 

the strong interactions. Obviously, since 

neither of these is a long-range interaction, 

some additional features must be at work to 

accouni. for their different properties. Before 

turning to a discussion of these features, we 

stress that in theories based on local gauge 

invariance, currents always play an important 

role. In classical electromagnetism the 

fundamental interaction takes place between 

the vector potential and the electron current; 

this is reflected in quantum electrodynamics 

by Feynman diagrams: the virtual photon 

(the gauge field) ties into the current 

produced by the moving electron (see Fig. 8). 

As will become clear below, a similar situation 

exists in the strong interaction and, 

more important, in the weak interaction. 

36 

Fig. 8. The U(I) local symmetry of QED implies the existence of a gauge field to 

compensate for the local phase transformation of the electrically charged matter 

fields. The generator of the U(I) local phase transformation is Q, the electric 

charge operator defined in the figure in terms of the current density Jo. The gauge 

field A,, interacts with the electrically charged matter fields through the current J p. 

The coupling strength is e, the charge of the electron. 

The Strong Interaction 

In an atom electrons are bound to the 

nucleus by the Coulomb force and occupy a 

region about IO-* centimeter in extent. The 

nucleus itself is a tightly bound collection of 

protons and neutrons confined to a region 

about IO-” centimeter across. As already 

emphasized, the force that binds the protons 

and neutrons together to form the nucleus is 

much stronger and considerably shorter in 

range than the electromagnetic force. Leptons 

do not feel this strong force; particles 

that do participate in the strong interactions 

are called hadrons.


a- 

I 

n 1961 M. Gell-Mann and independently 

Y. Ne’eman proposed a sys- 

tem for classifying the roughly one hundred 

baryons and mesons known at the t 

This “Eightfold Way” was based on 

SU(3) group, which has eight independent 

symmetry operations. According to this system, 

hadrons with the same baryon number, 

spin angular momentum, and parity and 

with electric charge, strangeness (or hypercharge), 

isotopic spin, and mass related by 

certain rules were grouped into large multiplets 

encompassing the already establis 

isospin multiplets, such as the neutron and 

proton doublet or the negative, neutral, and 

positive pion triplet. Most of the known 

hadrons fit quite neatly into octets. However, 

the decuplet partly tilled by the quartet of A 

baryons and the triplet of C( 1385) baryons 

lacked three members. Discovery of the 

E( 1520) doublet was announced in 1962, and 

these baryons satisfied the criteria for membership 

in the decuplet. This partial confirmation 

of the Eightfold Way motivated a 

search at Brookhaven for the remaining 

member, already named R- and predicted to 

be stable against strong and electromagnetic 

interactions, decaying (relatively slowly) by 

the weak interaction. Other properties 

predicted for this particle were a bary 

number of I, a spin angular momentum of 

3/2, positive parity, negative electric charge, a 

strangeness of-3, an isotopic spin of 0, and a 

mass ofabout 1676 MeV. 

A beam of 5-GeV negative kaons 

produced at the AGS was directed into a 

liquid-hydrogen bubble chamber, where the 

R- was to be produced by reaction of the 

kaons with protons. The tracks of the decay 

products of the new particle were then sought 

in the bubble-chamber photographs. In early 

1964 a candidate event was found for decay Analysis of the tracks for these two events 

of an R- into a x- and a So, one of three confirmed the predicted mass and strangepossible 

decay modes. Within several weeks, ness, and further studies confirmed the 

by coincidence and good fortune, another 0- predicted spin and panty. Discovery of the 

was found, this time decaying into a Ao and a R- estabIished the Eightfold Way as a viable 

K-, the mode now known to be dominant. description of hadronic states. W 

- -_ 

The R- was first detected in the bubble-chamber photograph reproduced above. 

A K- entered the bubble chamber from the bottom (track 1) and collided with a 

proton. The collision produced an R- (track 3), a K+ (track 2), and a KO, which, 

being neutral, left no track and must have decayed outside the bubble chamber. 

The R- decayed into a IC- (track 4) and a 5’. The 5’ in turn decayed into a A’ 

and a no. The A’ decayed into a IC- (track 5) and a proton (track 6), and the no 

veuy quickly decayed into two gamma rays, one of which (track 7) created an e-- 

e’pair within the bubble chamber. (Photo courtesy of the Niels Bohr Library of 

the American Institute of Physics and Brookhaven National Laboratouy.) 

37

Table of “Elementary Particles” 

BARYONS 

Spin-112 Octet 

Strong 

Isospin 

Mass 

, 

The mystery of the strong force and the 

structure of nuclei seemed very intractable as 

little as fifteen years ago. Studying the relevant 

distance scales requires machines that 

can accelerate protons or electrons to 

energies of 1 CeV and beyond. Experiments 

with less energetic probes during the 1950s 

revealed two very interesting facts. First, the 

strong force does not distinguish between 

protons and neutrons. (In more technical 

language, the proton and the neutron transform 

into each other under isospin rotations, 

and the Lagrangian of the strong interaction 

is invariant under these rotations.) Second, 

the structure of protons and neutrons is as 

rich as that of nuclei. Furthermore, many 

new hadrons were discovered that were apparently 

just as “elementary” as protons and 

neutrons. 

The table of “elementary particles” in the 

mid-1960s displayed much of the same complexity 

and symmetry as the periodic table of 

the elements. In 1961 both Cell-Mann and 

Ne’eman proposed that all hadrons could be 

classified in multiplets of the symmetry 

group called SU(3). The great triumph of this 

proposal was the prediction and subsequent 

discovery ofa new hadron, the omega minus. 

This hadron was needed to fill a vacant space 

in one of the SU(3) multiplets (Fig. 9). 

In spite of the SU(3) classification scheme, 

the belief that all of these so-called elementary 

particles were truly elementary became 

more and more untenable. The most contradictory 

evidence was the finite size of 

hadrons (about centimeter), which 

drastically contrasted with the point-like 

nature of the leptons. Just as the periodic 

table was eventually explained in terms of a 

few basic building blocks, so the hadronic 

zoo was eventually tamed by postulating the 

existence of a small number of “truly 

elementary point-like particles” called 

quarks. In 1963 Cell-Mann and, independently, 

Zweig realized that all hadrons 

could be constructed from three spin4 fermions, 

designated u, d, and s (up, down, and 

strange). The SU(3) symmetry that manifested 

itself in the table of “elementary particles” 

arose from an invariance of the La- 

38 

Y 

Y 

1. 

0. 

- 1. 

-1 -112 0 112 1 

’3 

in-312 DecuDlet 

-312 -1 -112 0 112 1 312 

- 

‘3 

IUS) 

1 I2 

0 

1 

1 12 

312 

1 

939 

A(1 f 16) 

C(1193) 

Z(1348) 

A( 1232) 

C *( 1385) 

1 I2 Z* (1 530) 

0 (1672) 

Fig. 9. The Eightfold Way classified the hadrons into multiplets of the 

symmetry group SU(3). Particles of each SU(3) multiplet that lie on a 

horizontal line form strong-isospin (SU(2)) multiplets. Each particle is 

plotted according to the quantum numbers I, (the third component of strong 

isospin) and strong hypercharge Y (Y = S + B, where S is strangeness and B is 

baryon number). These quantum numbers correspond to the two diagonal 

generators of SU(3). The quantum numbers of each particle are easily 

understood in terms of its fundamental quark constituents. Baryons contain 

three quarks and mesons contain quark-antiquark pairs. Baryons in the spin- 

3/2 decuplet are obtained from baryons in the spin-% octet by changing the 

spin and SU(3) flavor quantum numbers of the three quark wave functions. 

For example, the three quarks that compose the neutron in the spin-% octet can 

reorient their spins to form the A’ in the spin-3/2 decuplet. Similar changes in 

the meson quark-antiquark wave functions change the spin-0 meson octet into 

the spin-I meson octet.


MESONS 

Mass 

K(495) 

K(495) 

K* (892) 

w (783) 

P (770) 

K*(892) 

Strong 

Isospin 

1 /2 

0 

1 

1 /2 

112 

0 

1 

112 

Spin-0 Octet 

K0(d9 

K'(u5) 

1 212 0 112 1 

13 

Spin-I Oclet 

*+ - 

K*'(d3 K (us) 

-1 -112 0 112 1 

Quarks 

'3 

Electric 

Name Symbol Charge Y 

UP U 213 1/3 

Down d -1J3 113 

Strange 5 -113 -2131 

11 

-1 

O Y 

grangian of the strong interaction to rotations 

among these three objects. This global 

symmetry is exact only if the u, d, and s 

quarks have identical masses, which implies 

that the particle states populating a given 

SU(3) multiplet also have the same mass. 

Since this is certainly not the case, SU(3) is a 

broken global symmetry. The dominant 

breaking is presumed to arise, as in the example 

of e, p, and 'I, from the differences in the 

masses of the u, d, and s quarks. The origin of 

these quark masses is one of the great unanswered 

questions. It is established, however, 

that SU(3) symmetry among the u, d, 

and s quarks is preserved by the strong interaction. 

Nowadays, one refers to this SU(3) as 

ayuvor symmetry, with u, d, and s representing 

different quark flavors. This nomen-\, 

clature is to distinguish it from another and 

quite different SU(3) symmetry dossessed by 

quarks, a local symmetry that is associated 

directly with the strong force and has become 

known as the SU(3) of color. The theory 

resulting from this symmetry is called quantum 

chromodynamics (QCD), and we now 

turn our attention to a discussion of its 

properties and structure. 

The fundamental structure of quantum 

chromodynamics mimics that of quantum 

electrodynamics in that it, too, is a gauge 

theory (Fig. IO). The role of electric charge is 

played by three "colors" with which each 

quark is endowed-red, green, and blue. The 

three color varieties of each quark form a 

triplet under the SU(3) local gauge symmetry. 

A local phase transformation of the 

quark field is now considerably extended 

since it can rotate the color and thereby 

change a red quark into a blue one. The local 

gauge transformations of quantum electrodynamics 

simply change the phase of an 

electron, whereas the color transformations 

of QCD actually change the particle. (Note 

that these two types of phase transformation 

are totally independent of each other.) 

We explained earlier that the freedom to 

change the local phase of the electron field 

forces the introduction of the photon field 

(sometimes called the gauge field) to keep the 

Lagrangian (and therefore the resulting phys- 

39

~ 

, 

its) invariant under these local phase 

changes. This is the principle of local symmetry. 

A similar procedure applied to the 

quark field induces the so-called chromodynamic 

force. There are eight independent 

symmetry transformations that change the 

color of a quark and these must be compensated 

for by the introduction of eight 

gauge fields, or spin-I bosons (analogous to 

the single photon of quantum electrodynamics). 

Extension of the local U( 1) 

gauge invariance of QED to more complicated 

symmetries such as SU(2) and SU(3) 

was first done by Yang and Mills in 1954. 

These larger symmetry groups involve socalled 

non-Abelian, or non-commuting algebras 

(in which AB + BA), so it has become 

customary to refer to this class of theories as 

“non-Abelian gauge theories.” An alternative 

term is simply “Yang-Mills theories.” 

The eight gauge bosons of QCD are referred 

to by the bastardized term “gluon,” 

since they represent the glue that holds the 

physical hadrons, such as the proton, 

together. The interactions of gluons with 

quarks are depicted in Fig. 10. Although 

gluons are the counterpart to photons in that 

they have unit spin and are massless, they 

possess one crucial property not shared by 

photons: they themselves carry color. Thus 

they not only mediate the color force but also 

carry it; it is as if photons were charged. This 

difference (it is the difference between an 

Abelian arid a non-Abelian gauge theory) has 

many profound physical consequences. For 

example, becauie gluons carry color they can 

(unlike photons) interact with themselves 

(see Fig. IO) and, in effect, weaken the force 

of the color charge at short distances. The 

opposite effect occurs in quantum electrodynamics: 

screening effects weaken the 

effective electric charge at long distances. (As 

mentioned above, a virtual photon emanating 

from an electron can create a virtual 

electron-positron pair. This polarization 

screens, or effectively decreases, the electron’s 

charge.) 

The weakening of color charge at short 

distances goes by the name of asymptotic 

freedom. Asymptotic freedom was first ob- 

I 

r 

Gluon Self-I nteractions 

9 

‘b I 

3 Quark Colors 8 Colored Gluons 

Fig. 10. The SU(3) local color symmetry implies the existence of eight massless 

gaugefields (the gluons) to compensate for the eight independent local transformations 

of the colored quark fields. The subscripts r, g, and b on the gluon and quark 

fields correspond respectively to red, green, and blue color charges. The eight 

gluons carry color and obey the non-Abelian algebra of the SU(3) generators (see 

Lecture Note 4). The interactions induced by the local SU(3) color symmetry 

include a quark-gluon coupling as well as two types of gluon self interactions (one 

proportional to the couping g, and the other proportional to g:). 

I 

I 

I 

I 

I 

40


H 

ow can we extract answers fr 

QCD at energies below 1 Ge 

As noted in the text, the confinement 

of quarks suggests that weak-couphng 

perturbative methods arc not going to be 

successful at these energies. Nevertheless, if 

QCD is a valid theory it must explain the 

multiplicities, masses, and couplings of the 

experimentally observed strongly interacting 

particles. These would emerge from the theory 

as bound states and resonances of quarks 

and gluons. A valid theory must also account 

for the apparent absence of isolated quark 

states and might predict the existence and 

properties of particles (such as glueballs) that 

have not yet been seen. 

The most promising nonperturbative formulation 

of QCD exploits the Feynman path 

integral. Physical quantities are expressed as 

integrals of the quark and gluon fields over 

the space-time continuum with the QCD 

Lagrangian appearing in an exponential as a 

kind of Gibbs weight factor. This is directly 

analogous to the partition function formulation 

of statistical rnachanics. The path integral 

prescription for strong interaction 

dynamics becomes well defined mathematically 

when the space-time continuum is 

approximated by a discrete four-dimensional 

lattice of finite size and the integrals are 

evaluated by Monte Carlo sampling. 

The original Monte Carlo ideas of Metropolis 

and Ulam have now been applied to 

QCD by many researchers. These efforts 

have given credibility, but not confirmation, 

to the hope that computer simulations might 

indeed provide critical tests of QCD and 

significant numerical results. With considerable 

patience (on the order of many months 

ofcomputer tirne)a VAX 11/780can be used 

to study universes of about 3000 space-time 

points, Such a universe is barely large enough 

to contain a proton and not really adequate 

for a quantitative calculation. Consequently, 

with these methods, any result from a computer 

of VAX power is, at best, only an 

indication of what a well-done numerical 

simulation might produce. 

by Gerald Guralnik, Tony Warnock, and Charles Zemach 

physical and mathematical ingenuity to 

search out the best formulations of problems 

ming; and (3) a computer with the speed, 

memory, and input/output rate of the Cray 

Cray XMP. Using new meth 

with coworkers R. Gupta, J. 

A. Patel, we are examining gl 

renormalization group behavior, and the 

behavior of the theory on much larger lattices. 

The results to date support the belief 

that QCD describes interactions of the 

elementary particles and that these numerical 

methods are currently the most powerful 

means for extracting the predictive content 

of QCD. 

p meson 767 

Excited p 

I426 

6 meson 1154 

A, meson 1413 

Proton 989 

A baryon 

1 I99 

Couplings 

The calculations, which have two input 

parameters (the pion mass and the longrange 

quark-quark force constant in units of 

the lattice spacing), provide estimates of 

ble quantities. The accompaows 

some of our results on 

elementary particle masses and certain 

meson coupling strengths. These results represent 

several hundred hours of Cray timc. 

The quoted relative errors derive from the 

statistical analysis of the Monte Carlo calculation 

itself rather than from a comparison 

with experimental data. Significantly more 

computer time would significantly reducc 

the errors in the calculated masses and couplings. 

Our work would not have been possible 

without the support of C Division and many 

of its staff. We have received generous support 

from Cray Research and are particularly 

indebted to Bill Dissly and George Spix for 

contribution of their skills and their time. 

Calculated and experimental values for the masses and coupling 

strengths of some mesons and baryons. 

Masses 

fn 

fP 

Calculated Relative Experimental 

Value Error Value 

(MeV/c2) (O/O) (MeV/c2) 

18 

27 

15 

17 

23 

17 

769 

1300? 

983 

1275 

940 

1210 

121 21 93 

21 1 15 144 

41

I 

observable hadrons are necessarily colorless, 

whereas quarks and gluons are permanently 

confined. This is just as well since gluons are 

massless, and by analogy with the photon, 

unconfined massless gluons should give rise 

to a long-range, Coulomb-like, color force in 

the strong interactions. Such a force is clearly 

at variance with experiment! Even though 

color is confined, residual strong color forces 

can still “leak out” in the form of colorneutral 

pions or other hadrons and be responsible 

for the binding of protons and 

neutrons in nuclei (much as residual electromagnetic 

forces bind atoms together to 

form molecules). 

The success of QCD in explaining shortdistance 

behavior and its aesthetic appeal as 

a generalization of QED have given it its 

place in the standard model. However, confidence 

in this theory still awaits convincing 

calculations of phenomena at distance scales 

of lo-” centimeter, where the “strong” 

nature of the force becomes dominant and 

perturbation theory is no longer valid. (Lattice 

gauge theory calculations of the hadronic 

spectrum are becoming more and more reliable. 

See “QCD on a Cray: The Masses of 

E!ementary Particles.”) 

The Weak Interaction 

I 

served in deep inelastic scattering experiments 

(see “Scaling in Deep Inelastic Scattering”). 

This phenomenon explains why 

hadrons at high energies behave as if they 

were made of almost free quarks even though 

one knows that quarks must be tightly bound 

together since they have never been experimentally 

observed in their free state. The 

weakening of the force at high energies 

means that we can use perturbation theory to 

calculate hadronic processes at these 

energies. 

42 

The self-interaction of the gluons also explains 

the apparently permanent confinement 

of quarks. At long distances it leads to 

such a proliferation of virtual gluons that the 

color charge effectively grows without limit, 

forbidding the propagation of all colored 

,particles. Only bleached, or color-neutral, 

states (such as baryons, which have equal 

proportions of red, blue, and green, or 

mesons which have equal proportions of redantired, 

green-antigreen, and blue-antiblue) 

are immune from this c:onfinement. Thus all 

alpha particles, beta rays, and gamma rays. 

These three are now associated with three 

quite different modes of decay. An alpha 

particle, itself a helium nucleus, is emitted 

during the strong-interaction decay mode 

known as fission. Large nuclei that are only 

loosely bound by the strong force (such as 

uranium-238) can split into two stable 

pieces, one of which is an alpha particle. A 

gamma ray is simply a photon with “high” 

energy (above a few MeV) and is emitted 

during the decay of an excited nucleus. A 

beta ray is an electron emitted when a neutron 

in a nucleus decays into a proton, an 

electron, and an electron antineutrino (n-p 

+e-+ Cc, see Fig. 1 1). The proton remains in

I- 


Positive Weak Currents JLeak 

the nucleus, and the electron and its antineutrino 

escape. This decay mode is 

characterized as weak because it proceeds 

much more slowly than most electromagnetic 

decays (see Table I). Other 

baryons may also undergo beta decay. 

Beta decay remained very mysterious for a 

long time because it seemed to violate 

energy-momentum conservation. The free 

neutron was observed to decay into two 

particles, a proton and an electron, each with 

a spectrum of energies, whereas energymomentum 

conservation dictates that each 

should have a unique energy. To solve this 

dilemma, Pauli invoked the neutrino, a 

massless, neutral fermion that participates 

only in weak interactions. 

I 

Fig. ll. (a) Components of the charge-raising weak current Jtk 

are represented in 

the figure by Feynman diagrams in which a neutron changes into a proton, an 

electron into an electron neutrino, and a muon into a muon neutrino. The chargelowering 

current Jieak is represented by reversing the arrows. (b) Beta decay 

(shown in thefigure) and other low-energy weak processes are well described by the 

Fermi interaction Jieak X Jieak. The figure shows the Feynman diagram of the 

Fermi interaction for beta decay. 

The Fermi Theory. Beta decay is just one of 

many manifestations ofthe weak interaction. 

By the 1950s it was known that all weak 

processes could be concisely described in 

terms of the current-current interaction first 

proposed in 1934 by Fermi. The charged 

weak currents &ak and Jieak change the 

electric charge of a fermion by one unit and 

can be represented by the sum of the Feynman 

diagrams of Fig. I la. In order to describe 

the maximal parity violation, (that is, 

the maximal right-left asymmetry) observed 

in weak interactions, the charged weak current 

includes only left-handed fermion fields. 

(These are defined in Fig. 12 and Lecture 

Note 8.) 

Fermi’s current-current interaction is then 

given by all the processes included in the 

product (G~/fi)(J+weak X JGeak) where 

Jieak means all arrows in Fig. 1 la are reversed. 

This interaction is in marked contrast 

to quantum electrodynamics in which 

two currents interact through the exchange of 

a virtual photon (see Fig. 3). In weak 

processes two charge-changing currents appear 

to interact locally (that is, at a single 

point) without the help of such an intermediary. 

The coupling constant for this local 

interaction, denoted by GF and called the 

Fermi constant, is not dimensionless like the 

coupling parameter a in QED, but has the 

43

Left-Handed 

Particle State 

I 

2 + 

S 

Fig. 12. A Dirac spinor field can be decomposed into leftand 

right-handed pieces. A left-handed field creates two 

types of particle states at ultrarelativistic energies-u, a 

particle with spin opposite to the direction of motion, and urn 

an antiparticle with spin along the direction of motion. Only 

lef-handed fields contribute to the weak charged currents 

shown in Fig. 11. The left- and right-handedness (or 

chirality) of a field describes a Lorentz covariant decomposition 

of Dirac spinorfields. 

dimension of mass-’ or energy-’ . In units of 

energy, the measured value of GF”’ equals 

293 CeV. Thus the strength of the weak 

processes seems to be determined by a specific 

energy scale. But why? 

Predictions of the W boson. An explanation 

emerges if we postulate the existence of an 

intermediary for the weak interactions. Recall 

from Fig. 3 that the exchanged, or virtual, 

photons in QED basically correspond to 

the Coulomb potential a/r, whose Fourier 

transform is a/$, where q is the momentum 

of the virtual photon. It is tempting to suggest 

that the nearly zero range of the weak 

interaction is only apparent in that the two 

charged currents interact through a potential 

of the form a’[exp(-M~.r)]/r (a form originally 

proposed by Yukawa for the short- 

44 

range force between nucleons), where a’ is 

the analogue of a and the mass MI+/ is so large 

that this potential has essentially no range. 

The Fourier transform of this potential, 

a’/($ + ML), suggests that, if this idea is 

correct, the interaction between the weak 

currents is mediated by a “heavy photon” of 

mass Mw. Nowadays this particle is called 

the W boson; its existence explains the short 

range of the weak interactions. 

Notice that at low energies, or, equivalently, 

when M& >> $, the Fourier transform, 

or so-called propagator of the W 

boson, reduces to a’/(M&r), and since this 

factor multiplies the two currents, it must be 

proportional to Fermi’s constant. Thus the 

existence of the PV boson gives a natural 

explanation ofwhy GF is not dimensionless. 

Now, since both the weak and electro- 

magnetic interactions involve electric 

charge, these two might be manifestations ol 

the same basic force. If they were, then a’ 

might be the same as a and GF would be 

proportional to a/M$. Thus the existence 01 

a very massive W boson can explain not onl) 

the short range but also the weakness ofweak 

interactions relative to electromagnetic interactions! 

This argument not only predict: 

the existence of a W boson but also yields a 

rough estimate of its mass: 

M~ = = 25 GeV/c2. 

I 

beyond reach of the existing accelerators. 

physicists that a theoretical unification o


Table 2 

Multiplets and quantum numbers in the SU(2) X U(1) electroweak theory. 

Quarks 

-- 

1 UL 

I 

SU(2) Doublet 

I 

SU(2) Singlets 

SU(2) Doublet 

SU(2)Singlet 

Leptons 

Gauge Bosons 

1-6 i 

UR __ 

1- dR_i 

I(Vi)2 

I 

PL! 

- - 

eR I 

W+l 

SU(2) Triplet W3‘ 

’ w-/ 

SU(2) Singlet B- 1 

SU(2) Doublet 

c--- 

Higgs Boson 

__- 

‘p+ ’ 

Weak Weak Electric 

Isotopic Hypercharge Charge Q 

Charge I3 Y (=Z3 + ‘/2 Y) 

112 

- Y2 

0 

0 

‘/l 

- Y2 

0 

1 

0 

-1 

electromagnetic and weak interactions must 

be possible. Several attempts were made in 

the 1950s and 1960s, notably by Schwinger 

and his student Glashow and by Ward and 

Salam, to construct an “electroweak theory” 

in terms of a local gauge (Yang-Mills) theory 

that generalizes QED. Ultimately, Weinberg 

set forth the modern solution to giving 

0 

Y2 

--% 

‘/3 

Y3 

413 

-2/3 

-I 

-1 

-2 

0 

0 

0 

0 

1 

1 

- Y3 

2/3 

-% 

0 

-1 

-1 

1 

0 

-1 

masses to the weak bosons in 1967, although 

it was not accepted as such until ’t Hooft and 

Veltman showed in 1971 that it constituted a 

consistent quantum field theory. The success 

of the electroweak theory culminated in 1982 

with the discovery at CERN of the W boson 

at almost exactly the prediced mass. Notice, 

incidentally, that at sufficiently high 

0 

1 

0 

energies, where q2 >> ML, the Leak interaction 

becomes comparable in strength to the 

electromagnetic. Thus we see explicitly how 

the apparent strength of the interaction depends 

on the wavelength of the probe. 

The SU(2) X U(1) Electroweak Theory. 

Since quantum electrodynamics is a gauge 

theory based on local U(I) invariance; it is 

not too surprising that the theory unifying 

the electromagnetic and weak forces is also a 

gauge theory. Construction of such a theory 

required overcoming both technical and phenomenological 

problems. 

The technical problem concerned the fact 

that an electroweak gauge theory is 

necessarily a Yang-Mills theory (that is, a 

theory in which the gauge fields interact with 

each other); the gauge fields, namely the W 

bosons, must be charged to mediate the 

charge-changing weak interactions and therefore 

by definition must interact with each 

other electromagnetically through the 

photon. Moreover, the local gauge symmetry 

of the theory must be broken because an 

unbroken symmetry would require all the 

gauge particles to be massless like the photon 

and the gluons, whereas the W boson must 

be massive. A major theoretical difficulty 

was understanding how to break a Yang- 

Mills gauge symmetry in a consistent way. 

(The solution is presented below.) 

In addition to the technical issue, there 

was the phenomenological problem of choosing 

the correct local symmetry group. The 

most natural choice was SU(2) because the 

low-lying states (that is, the observed quarks 

and leptons) seemed to form doublets under 

the weak interaction. For example, a W- 

changes vc into e, v,, into p, or u into d (where 

all are left-handed fields), and the W+ effects 

the reverse operation. Moreover, the three 

gauge bosons required to compensate for the 

three independent phase rotations of a local 

SU(2) symmetry could be identified with the 

W’, the W-, and the photon. Unfortunately, 

this simplistic scenario does not 

work: it gives the wrong electric charge assignments 

for the quarks and leptons in the 

SU(2) doublets. Specifically, electric charge 

45

Q would be equal to the SU(2) charge f3, and 

the values of f 3 for a doublet are fY2. This is 

clearly the wrong charge. In addition, SU(2) 

would not distinguish the charges of a quark 

doublet (*/I and 4 3 ) from those of a lepton 

doublet (0 and -1). 

To get the correct charge assignments, we 

can either put quarks and leptons into SU(2) 

triplets (or larger multiplets) instead of 

doublets, or we can enlarge the local symmetry 

group. The first possibility requires 

the introduction of new heavy fermions to 

fill the multiplets. The second possibility 

requires the introduction of at least one new 

U( 1) symmetry (let's call it weak hypercharge 

Y), which yields the correct electric charge 

assignments if we define 

This is exactly the possibility that has been 

confirmed experimentally. Indeed, the electroweak 

theory of Glashow, Salam, and 

Weinberg is a local gauge theory with the 

symmetry group SU(2) X U(1). Table 2 gives 

the quark and lepton multiplets and their 

associated quantum numbers under SU(2) X 

U(I), and Fig. 13 displays the interactions 

defined by this local symmetry. There is one 

coupling associated with each factor of SU(2) 

X U( I), a couplingg for SU(2) and a coupling 

g'/2 for U( I). 

The addition of the local U( 1) symmetry 

introduces a new uncharged gauge particle 

into the theory that gives rise to the so-called 

neutral-current interactions. This new type 

of weak interaction, which allows a neutrino 

to interact with matter without changing its 

identity, had not been observed when the 

neutral weak boson was first proposed in 

1961 by Glashow. Not until 1973, after all 

the technical problems with the SU(2) X 

U( 1) theory had been worked out, were these 

interactions observed in data taken at CERN 

in 1969 (see Fig. 14). 

The physical particle that mediates the 

weak interaction between neutral currents is 

the massive Zo. The electromagnetic interaction 

between neutral currents is mediated by 

the familiar massless photon. These two 

46


physical particles are different from the two 

neutral gauge particles (Band W3) associated 

with the unbroken SU(2) X U(l) symmetry 

shown in Fig. 13. In fact, the photon and the 

Zo are linear combinations of the neutral 

gauge particles W3 and B: 

A=BcosBw+ W3sinBw 

and 

Zo = B sin Bw - W3 COS OW 

The mixing ofSU(2) and U( 1) gauge particles 

to give the physical particles is one result 

of the fact that the SU(2) X U( 1) symmetry 

must be a broken symmetry. 

Spontaneous Symmetry Breaking. The astute 

reader may well be wondering how a local 

gauge theory, which in QED required the 

photon to be massless, can allow the 

mediator of the weak interactions to be 

massive, especially since the two forces are to 

be unified. The solution to this paradox lies 

in the curious way in which the SU(2) X U( 1) 

symmetry is broken. 

As Nambu described so well, this breaking 

is very much analogous to the symmetry 

breaking that occurs in a superconductor. A 

superconductor has a local U( 1) symmetry, 

namely, electromagnetism. The ground state, 

however, is not invariant under this symmetry 

since it is an ordered state of bound 

electron-electron pairs (the so-called Cooper 

pairs) and therefore has a nonzero electric 

charge distribution. As a result of this asymmetry, 

photons inside the superconductor 

acquire an effective mass, which is responsible 

for the Meissner effect. (A magnetic field 

cannot penetrate into a superconductor; at 

the surface it decreases exponentially at a 

rate proportional to the effective mass of the 

photon.) 

In the weak interactions the symmetry is 

also assumed to be broken by an asymmetry 

of the ground state, which in this case is the 

“vacuum.” The asymmetry is due to an ordered 

state of electrically neutral bosons that 

carry the weak charge, the so-called Higgs 

bosons. They break the SU(2) X U(l) sym- 

47

metry to give the U( 1) of electromagnetism 

in such a way that the W' and the Zo obtain 

masses and the photon remains massless. As 

a result the charges 13 and Y associated with 

SU(2) X U(1) are not conserved in weak 

processes because the vacuum can absorb 

these quantum numbers. The electric charge 

Q associated with U( 1) of electromagnetism 

remains conserved. 

The asymmetry of the ground state is frequently 

referred to as spontaneous symmetry 

breaking; it does not destroy the symmetry of 

the Lagrangian but destroys only the symmetry 

of the states. This symmetry breaking 

mechanism allows the electroweak Lagrangian 

to remain invariant under the local 

symmetry transformations while the gauge 

particles become massive (see Lecture Notes 

3, 6, and 8 for details). 

In the spontaneously broken theory the 

electromagnetic coupling e is given by the 

expression e = gsin Ow, where 

sin2ew = g2/(g2 + gt2) . 

Thus, e and Ow are an alternative way of 

expressing the couplings g and g', and just as 

e is not determined in QED, the equally 

important mixing angle Ow is not determined 

by the electroweak theory. It is, however, 

measured in the neutral-current interactions. 

The experimental value is sin2 Ow = 0.224 k 

0.015. The theory predicts that 

These relations (which are changed only 

slightly by small quantum corrections) and 

the experimental value for the weak angle Ow 

predict masses for the W' and Zo that are in 

very good agreement with the 1983 observations 

of the W' and Zo at CERN. 

In the electroweak theory quarks and leptons 

also o.btain mass by interacting with the 

ordered vacuum state. However, the values 

of their masses are not predicted by the 

Fig. 14. Neutral-current interactions were first identified in 1973 in photographs 

taken with the CEUN Gargamelle bubble chamber. The figure illustrates the 

difference between neutral-current and charged-current interactions and shows the 

bubble-chamber signature of each. The bubble tracks are created by charged 

particles moving through superheated liquid freon. The incoming antineutrinos 

interact with protons in the liquid. A neutral-current interaction leaves no track 

from a lepton, only u track from the positivley charged proton and perhaps some 

tracks from pions. A charged-current interaction leaves a track from a positively 

charged muon only. 

48


rons or protons. U 

of the 

fortv- 

Beam 

49

theory but are proportional to arbitrary 

parameters related to the strength of the 

coupling of the quarks and leptons to the 

Higgs boson. 

The Higgs Boson. In the simplest version of 

the spontaneously broken electroweak 

model, the Higgs boson is a complex SU(2) 

doublet consisting of four real fields (see 

Table 2). These four fields are needed to 

transform massless gauge fields into massive 

ones. A massless gauge boson such as the 

photon has only two orthogonal spin components 

(both transverse to the direction of 

motion), whereas a massive gauge boson has 

three (two transverse and one longitudinal, 

that is, in the direction of motion). In the 

electroweak theory the W’, the W-, and the 

Zo absorb three of the four real Higgs fields 

to form their longitudinal spin components 

and in so doing become massive. In more 

picturesque language, the gauge bosons “eat” 

the Higgs boson and become massive from 

the feast. The remaining neutral Higgs field 

is not used up in this magic transformation 

from massless to massive gauge bosons and 

therefore should be observable as a particle 

in its own right. Unfortunately, its mass is 

not fixed by the theory. However, it can 

decay into quarks and leptons with a definite 

signature. It is certainly a necessary component 

of the theory and is presently being 

looked for in high-energy experiments at 

CERN. Its absence is a crucial missing link in 

the confirmation of the standard model. 

Open Problems. Our review of the standard 

model would not be complete without mention 

of some questions that it leaves unanswered. 

We discussed above how the three 

charged leptons (e, p, and T) may form a 

triplet under some broken symmetry. This is 

only part of the story. There are, in fact, three 

quark-lepton families (Table 3), and these 

three families may form a triplet under such 

a broken symmetry. (There is a missing state 

in this picture: conclusive evidence for the 

top quark / has yet to be presented. The 

bottom quark b has been observed in 

e’e-annihilation experiments at SLAC and 

so 

w-, w+, zo 

I 

n January 1983 two groups announced the results ofseparate searches at the CERN 

llider for the W- and W+ vector bosons of the electroweak 

headed by C. Rubbia and A. Astbury, reported definite 

identification, from among about a billion proton-antiproton collisions, of four W- 

decays and one W+ decay. The mass reported by this group (81 k 5 GeV/c2) agrees well 

predicted by the electro ode1 (82 k 2.4 GeVlc’). The other group, 

P. Darriulat and using nt detector, reported identification of four 

possible W* decays, again from among a billion events. The charged vector bosons 

were produced by annihilation of a quark inside a proton (uud) with an antiquark 

inside an antiproton (&a): 

and 

d+G-+ W- 

u+a+ 

w+. 

Since these reactions have a threshold energy equal to the mass of the charged bosons, 

the colliding proton and antiproton beams were each accelerated to about 270 GeV to 

provide the quarks with an average center-of-mass energy slightly above the threshold 

energy. (Only one-half of the energy of a proton or antiproton is camed by its three 

quark constituents; the other half is carried by the gluons.) Rubbia’s group distinguished 

the two-body decay of the bosons (into a charged and neutral lepton pair 

such as P’v,) by two methods: selection of events in which the charged lepton 

possessed a large momentum transverse to the axis of the colliding beams, and 

selection of events in which a large amount of energy appeared to be missing, 

presumably camed off by the (undetected) neutrino. Both methods converged on the 

same events. 

By mid 1983 each of the two groups had succeeded also in finding Zo, the neutral 

vector boson of the electroweak model. They reported slightly different mass values 

(96.5 k 1.5 and 91.2 k 1.7 GeVfc’), both in agreement with the predicted value of 94.0 

2 2.5 GeVfc’. For Zo the production and decay processes are given by 

ui- ;(or d+ 2)- Zo -f e- 

+ e+ (or p-i- p+) . 

In addition, both groups reported an asymmetry in the angular distribution of 

charged leptons from the many more decays of W- and W+ that had been seen 

since their discovery. This parity violation confirmed that the particles observed 

are truly electroweak vector bosons. W


Table 3 

The thre masses. Th note right- and left-handed 

particles 

Quark Mass 

(MeV/c2) 

Quarks Leptons Lepton Mass 

( MeV/c2) 

First Family 

5 

8 

UP 

down 

UL 

dL 

UR (v,}~ electron neutrino 0 

dR eL 6 electron 0.51 1 

Second Family 

1270 

175 

charm cL CR 

strange SL SR 

(v& 

PI+ PR 

muon neutrino 

muon 

0 

105.7 

Third Family 

45000 (?) 

4250 

top tL 

bottom bL 

tR (VJL tau neutrino 0 

bR TL TR tau 1784 

Cornell.) The standard model says nothing 

about why three identical families of quarks 

and leptons should exist, nor does it give any 

clue about the hierarchical pattern of their 

masses (the 7 family is heavier than the p 

family, which is heavier than the e family). 

This hierarchy is both puzzling and intriguing. 

Perhaps there are even more undiscovered 

families connected to the broken 

family symmetry. The symmetry could be 

global or local, and either case would predict 

new, weaker interactions among quarks and 

leptons. 

Table 3 brings up two other open questions. 

First, we have listed the neutrinos as 

being massless. Experimentally, however, 

there exist only upper limits on their possible 

masses. The most restrictive limit comes 

from cosmology, which rcquires the sum of 

neutrino masses to be less then 100 eV. It is 

known from astrophysical observations that 

most of the energy in the universe is in a 

form that does not radiate electromagnetically. 

If neutrinos have mass, they 

could, in fact, be the dominant form of 

energy in the universe today. 

Second, we have listed u and d, c and s, 

and t and b as doublets under weak SU(2). 

This is, however, only approximately true. 

As a result of the broken family symmetry, 

states with the same electric charge (the d, s, 

and b quarks or the u, c, and t quarks) can 

mix, and the weak doublets that couple to the 

W' bosons are actually given by IJ and d', 

c and s', and t and b'. A 3 X 3 unitary matrix 

known as the Kobayashi-Maskawa (K-M) 

matrix rotates the mass eigenstates (states of 

definite mass) d, s, and b into the weak 

doublet states d', s', and b'. The K-M matrix 

is conventionally written in terms of three 

mixing angles and an arbitrary phase. The 

largest mixing is between the d and s quarks 

and is characterized by the Cabibbo angle 

€IC (see Lecture Note 9),which is named for 

the man who studied strangeness-changing 

weak decays such as Co - p + e- + V,. The 

observed value of sin 8C is about 0.22. The 

other mixing angles are all at least an order of 

magnitude smaller. The structure of the K-M 

matrix, like the masses of the quarks and 

leptons, is a complete mystery. 

Conclusions 

Although many mysteries remain, the 

standard model represents an intriguing and 

compelling theoretical framework for our 

present-day knowledge of the elementary 

particles. Its great virtue is that all of the 

known forces can be described as local gauge 

theories in which the interactions are generated 

from the single unifying principle of 

local gauge invariance. The fact that in quantum 

field theory interactions can drastically 

change their character with scale is crucial to 

51

I 

and the electr 

v) 

f 103 

I] 

0 

C 

(c1 

C 

3.10 3.12 3.14 

Center-of-Mass Energy (GeV) 

Graph of the evidence for formation of J/yr in electron- graph of their evidence. (Photo courtesy of the Niels 

positron annihilations at SPEAR. (A rn SLAC Library of the American Instiru Physics 

e, Volume 7, Number 11, Novem 6.) 

I 

52


T 

I 

n I977 a group led by L. Lederman provided evidence for a fifth, or bottom, quark 

with the discovery of”, a long-lived particle three times more massive than J/y. In 

an experiment similar to that of Ting and coworkers and performed at the 

Fermilab proton accelerator, the group determined the number of events giving rise to 

muon-antimuon pairs as a function of the mass ofthe parent particle and found a sharp 

increase at about 9.5 GeV/cZ. Like the J /~I system, the ‘I’ system has been elucidated in 

detail from experiments involving electron-positron collisions rather than proton 

collisions, in this case at Cornell’s electron sto 

The existence ofthe bottom quark, and of a sixth, or top, quark, was expected on the 

basis of the discovery of the tau lepton at SPEAR in 1975 and Glashow and Bjorken’s 

I964 argument of quark-lepton symmetry. Recent results from high-energy protonantiproton 

collision experiments at CERN have been interpreted as possible evidence 

for the top quark with a mass somewhere between 30 and 50 GeV/c2. 

this approach. The essence of the standard 

model is to put the physics of the apparently 

separate strong, weak, and electromagnetic 

interactions in the single language of local 

gauge field theories, much as Maxwell put 

the apparently separate physics of 

Coulomb’s, Ampere’s, and Faraday’s laws 

into the single language of classical field theory. 

It is very tempting to speculate that, because 

of the chameleon-like behavior of 

quantum field theory, all the interactions are 

simply manifestations of a single field theory. 

Just as the “undetermined parameters” 

EO and po were related to the velocity of light 

through Maxwell’s unification of electricity 

and magnetism, so the undetermined 

parameters of the standard model (such as 

quark and lepton masses and mixing angles) 

might be fixed by embedding the standard 

model in some grand unified theory. 

A great deal of effort has been focused on 

this question during the past few years, and 

some of the problems and successes are discussed 

in “Toward a Unified Theory” and 

“Supersymmetry at 100 GeV.” Although 

hints of a solution have emerged, it is fair to 

say that we arc still a long way from for- 

mulating an ultimate synthesis ofall physical 

laws. Perhaps one of the reasons for this is 

that the role of gravitation still remains mysterious. 

This weakest of all the forces, whose 

effects are so dramatic in the macroscopic 

world, may well hold the key to a truly deep 

understanding of the physical world. Many 

particle physicists are therefore turning their 

attention to the Einsteinian view in which 

geometry becomes the language of expression. 

This has led to many weird and 

wonderful speculations concerning higher 

dimensions, complex manifolds, and other 

arcane subjects. 

An alternative approach to these questions 

has been to peel yet another skin off the 

onion and suggest that the quarks and leptons 

are themselves composite objects made 

of still more elementary objects called 

preons. After all, the prolifcration of quarks, 

leptons, gauge bosons, and Higgs particles is 

beginning to resemble the situation in the 

early 1960s when the proliferation of the 

observed hadronic states made way for the 

introduction of quarks. Maybe introducing 

preons can account for the mystery of flavor: 

e, p, and T, for example, may simply be 

bound states of such objects. 

Regardless of whether the ultimate understanding 

of the structure of matter, should 

there be one, lies in the realm of preons, 

some single primitive group, higher 

dimensions, or whatever, the standard 

model represents the first great step in that 

direction. The situation appears ripe for 

some kind of grand unification. Where are 

you, Maxwell? 

I 


Gerard ’I Hooft. “Gauge Theories of the Forces Between Elementary Particles.” Scieniific American, June 1980, pp. 104-137. 

Howard Georgi. “A Unified Theory of Elementary Particles and Forces.” Scienfijic American, April 1981, pp. 48-63. 

53

Lecture Notes 

fiom simple field theories to the standard model 


The standard model of electroweak and strong interactions 

consists of two relativistic quantum field theories, one to 

describe the strong interactions and one to describe the 

- electromagnetic and weak interactions. This model, which 

incorporates all the known phenomenology of these fundamental 

interactions, describes spinless, ~pin-~h, and spin-l fields interacting 

with one another in a manner determined by its Lagrangian. The 

theory is relativistically invariant, so the mathematical form of the 

Lagrangian is unchanged by Lorentz transformations. 

Although rather complicated in detail, the standard model Lagrangian 

is based on just two basic ideas beyond those necessary for a 

quantum field theory. One is the concept of local symmetry, which is 

encountered in its simplest form in electrodynamics. Local symmetry 

r - 

-- 

determines the form of the interaction between particles, or fields, 

that carry the charge associated with the symmetry (not necessarily 

the electric charge). The interaction is mediated by a spin-I particle, 

the vector boson, or gauge particle. The second concept is spontaneous 

symmetry breaking, where the vacuum (the state with no 

particles) has a nonzero charge distnbution. In the standard model 

the nonzero weak-interaction charge distribution of the vacuum is 

the source of most masses of the particles in the theory. These two 

basic ideas, local symmetry and spontaneous symmetry breaking, are 

exhibited by simple field theones. We begin these lecture notes with a 

Lagrangian for scalar fields and then, through the extensions and 

generalizations indicated by the arrows in the diagram below, build 

up the formalism needed to construct the standard model. 

~- 

I 

i 

@) Quarks 

Future Theories ? 

See Article on 

Unified Theories 

54


Fields, Lagrangians, 6~(tI,tz)= J2 [ G6q + 

and Equations 

of Motion 

We begin this introduction to field theory with one of the simplest 

theories, a complex scalar field theory with independent fields cp(x) 

and cpt(x). (cpt(x) is the complex conjugate of cp(x) if cp(x) is a classical 

field, and, if cp(x) is generalized to a column vector or to a quantum 

field, qt(x) is the Hermitian conjugate of cp(x).) Since cp(x) is a 

complex function in classical field theory, it assigns a complex 

number to each four-dimensional point x = (ct, x) of time and space. 

The symbol x denotes all four components. In quantum field theory 

cp(x) is an operator that acts on a state vector in quantum-mechanical 

Hilbert space by adding or removing elementary particles localized 

around the space-time point x. 

In this note we present the case in which cp(x) and cpt(x) correspond 

respectively to a spinless charged particle and its antiparticle of equal 

mass but opposite charge. The charge in this field theory is like 

electric charge, except it is not yet coupled to the electromagnetic 

field. (The word “charge” has a broader definition than just electric 

charge.) In Note 3 we show how this complex scalar field theory can 

describe a quite different particle spectrum: instead of a particle and 

its antiparticle of equal mass, it can describe a particle of zero mass 

and one of nonzero mass, each of which is its own antiparticle. Then 

the scalar theory exhibits the phenomenon called spontaneous symmetry 

breaking, which is important for the standard model. 

A complex scalar theory can be defined by the Lagrangian density, 

where d’cp = dcp/d9. (Upper and lower indices are related by the 

metric tensor, a technical point not central to this discussion.) The 

Lagrangian itself is 

The first term in Eq. la is the kinetic energy of the fields q(x) and 

cpt(s), and the last two terms are the negative of the potential energy. 

Terms quadratic in the fields, such as the -mzqtcp term in Eq. la. 

are called mass terms. If in2 > 0, then cp(s) describes a spinless 

particle and cpt(x) its antiparticle of identical mass. If rn2 < 0, the 

theory has spontaneous symmetry breaking. 

The equations of motion are derived from Eq. 1 by a variational 

method. Thus, let us change the fields and their derivatives by a small 

amount Scp(x) and Sd,q(s) = d,Scp(x). Then, 

‘2 a 2 a 2 a 2 

6qt + a(apcp)a~6~ - 

where the variation is defined with the restrictions Gcp(x,fl) = 6cp(x,tz) 

= Scpt(x,fl) =6cpt(x,f2) = 0, and Fcp(x) and 6cpt(x) are independent. The 

last two terms are integrated by parts, and the surface term is dropped 

since the integrand vanishes on the boundary. This procedure yields 

the Euler-Lagrange equations for cpt(x), 

and for cp(x). (The Euler-Lagrange equation for cp(x) is like Eq. 3 

except that cpt replaces cp. There are two equations because 6cp(x) and 

6cpt(x) are independent.) Substituting the Lagrangian density, Eq. la, 

into the Euler-Lagrange equations, we obtain the equations of motion, 

plus another equation of exactly the same form with cp(x) and 

cpt(x) exchanged. 

This method for finding the equations of motion can be easily 

generalized to more fields and to fields with spin. For example, a field 

theory that is incorporated into the standard model is electrodynamics. 

Its list of fields includes particles that carry spin. The 

electromagnetic vector potential A,(x) describes a “vector” particle 

with a spin of 1 (in units of the quantum of action h = 1.0546 X 

erg second), and its four spin components are enumerated by the 

space-time vector index p ( = 0, I, 2, 3, where 0 is the index for the 

time component and I, 2, and 3 are the indices for the three space 

components). In electrodynamics only two ofthe four components of 

A@) are independent. The electron has a spin of Y2, as does its 

antiparticle, the positron. Electrons and positrons of both spin projections, 

W 2 , are described by a field w(~), which is a column vector 

with four entries. Many calculations in electrodynamics are complicated 

by the spins of the fields. 

There is a much more difficult generalization of the Lagrangian 

formalism: if there are constraints among the fields, the procedure 

yielding the Euler-Lagrange equations must be modified, since the 

field variations are not all independent. This technical problem 

complicates the formulation of electrodynamics and the standard 

model, especially when computing quantum corrections. Our examination 

of the theory is not so detailed as to require a solution of 

the constraint problem. 

(3) 

55

Continuous 

Symmetries 

It is often possible to find sets of fields in the Lagrangian that can 

be rearranged or transformed in ways described below without 

changing the Lagrangian. The transformations that leave the Lagrangian 

unchanged (or invariant) are called symmetries. First, we 

will look at the form of such transformations, and then we will 

discuss implications of a symmetrical Lagrangian. In some cases 

symmetries imply the existence of conserved currents (such as the 

electromagnetic current) and conserved charges (such as the electric 

charge), which remain constant during elementary-particle collisions. 

The conservation of energy, momentum, angular momentum, and 

electric charge are all derived from the existence of symmetries. 

Let us consider a continuous linear transformation on three real 

spinless fields cp,(x) (where i = I, 2, 3) with cp;(x) = cp/(x). These three 

fields might correspond to the three pion states. As a matter of 

notation, cp(x) is a column vector, where the top entry is cpl(x), the 

second entry is cp2(x), and the bottom entry is cp3(x). We write the 

linear transformation of the three fields in terms of a 3-by-3 matrix 

U(E), where 

cP’(f) = U(E)(P(x) , (54 

The repeated index is summed from 1 to 3, and generalizations to 

different numbers or kinds of fields are obvious. The parameter E is 

continuous, and as E approaches zero, U(E) becomes the unit matrix. 

The dependence of x’ on x and E is discussed below. The continuous 

transformation U(E) is called linear since cp,(x) occurs linearly on the 

right-hand side of Eq. 5. (Nonlinear transformations also have an 

important role in particle physics, but this discussion of the standard 

model will primarily involve linear transformations except for the 

vector-boson fields, which have a slightly different transformation 

law, described in Note 5.) For N independent transformations, there 

will be a set of parameters E,, where the index a takes on values from 

1 to N. 

For these continuous transformations we can expand cp’(x‘) in a 

Taylor series about E, = 0; by keeping only the leading term in the 

expansion, Eq. 5 can be rewritten in infinitesimal form as 

Gcp(x) = cp’(x) - cp(x) = is”T,cp(x), (64 

where T,, is the first term in the Taylor expansion, 

with 6x = x‘ - x. The Tu are the “generators” of the symmetry 

transformations of cp(x). (We note that Gcp(x) in Eq. 6a is a small 

symmetry transformation, not to be confused with the field variations 

6cp in Eq. 2.) 

The space-time point x’ is, in general, a function of x. In the case 

where x’ = x, Eq. 5 is called an internal transformation. Although our 

primary focus will be on internal transformations, space-time symmetries 

have many applications. For example, all theories we describe 

here have Poincare symmetry, which means that these theories 

are invariant under transformations in which x‘ = Ax + b, where A is 

a 4-by-4 matrix representing a Lorentz transformation that acts on a 

four-component column vector x consisting of time and the three 

space components, and b is the four-component column vector of the 

parameters of a space-time translation. A spinless field transforms 

under Poincart transformations as cp’(x’) = cp (x) or 6cp = -bpd,cp(x). 

Upon solving Eq. 6b, we find the infinitesimal translation is represented 

by id,. The components of fields with spin are rearranged by 

Poincart transformations according to a matrix that depends on both 

the E’S and the spin of the field. 

We now restrict attention to internal transformations where the 

space-time point is unchanged; that is, 6xv = 0. If E, is an in- 

finitesimal, arbitrary function ofx, E&), then Eqs. 5 and 6a are called 

local transformations. If the ,E, are restricted to being constants in 

space-time, then the transformation is called global. 

Before beginning a lengthy development of the symmetries of 

various Lagrangians, we give examples in which each of these kinds 

of linear transformations are, indeed, symmetries of physical theories. 

An example of a global, internal symmetry is strong isospin, as 

discussed briefly in “Particle Physics and the Standard Model.” 

(Actually, strong isospin is not an exact symmetry of Nature, but it is 

still a good example.) All theories we discuss here have global Lorentz 

invariance, which is a space-time symmetry. Electrodynamics has a 

local phase symmetry that is an internal symmetry. For a charged 

spinless field the infinitesimal form of a local phase transformation is 

6cp(x) = i~(x)cp(x) and 6cpt(x) = -~E(X)~+(X), where cp(x) is a complex 

field. Larger sets of local internal symmetry transformations are 

fundamental in the standard model of the weak and strong interactions. 

Finally, Einstein’s gravity makes essential use of local spacetime 

Poincare transformations. This complicated case is not discussed 

here. It is quite remarkable how many types of transformations 

like Eqs. 5 and 6 are basic in the formulation of physical 

theoiies. 

Let us return to the column vector of three real fields cp(x) and 

suppose we have a Lagrangian that is unchanged by Eqs. 5 and 6, 

where we now restrict our attention to internal transformations. (One 

such Lagrangian is Eq. la, where cp(x) is now a column vector and 

cpt(x) is its transpose.) Not only the Lagrangian, but the Lagrangian 

density, too, is unchanged by an internal symmetry transformation. 

56


Let us consider the infinitesimal transformation (Eq. 6a) and calculate 

62' in two different ways. First of all, 6 9 = 0 if 69 is a symmetry 

identified from the Lagrangian. Moreover, according to the rules of 

partial differentiation, 

Then, using the Euler-Lagrange equations (Eq. 3) for the first term 

and collecting terms, Eq. 7 can be written in an interesting way: 

The next step is to substitute Eq. 6a into Eq. 8. Thus, let us 

define the current JE(x) as 

Then Eq. 8 plus the requirement that 6cp is a symmetry imply the 

continuity equation, 

(9) 

Jt;(x) must fall off rapidly enough as 1x1 approaches infinity that the 

integral is finite. 

If Q"(t) is indeed a conserved quantity, then its value does not 

change in time, which means that its first time derivative is zero. We 

can compute the time derivative of go([) with the aid of Eq. IO: 

d 

3 Q"(t) = d3x - aJ8(x) /d3x V . J"(x) = /J' - dS = 0. (12) 

at 

The next to the last step is Gauss's theorem, which changes the 

volume integral of the divergence of a vector field into a surface 

integral. If J"(x) falls off more rapidly than I/lxl* as 1x1 becomes very 

large, then the surface integral must be zero. It is not a always true 

that J"(x) falls off so rapidly, but when it does, Q"(t) = Q' is a 

constant in time. One of the most important experimental tests of a 

Lagrangian is whether the conserved quantities it predicts are, indeed, 

conserved in elementary-particle interactions. 

The Lagrangian for the complex scalar field defined by Eq. I has an 

internal global symmetry, so let us practice the above steps and 

identify the conserved current and charge. It is easily verified that the 

global phase transformation 

apqx) = 0 . (10) 

We can gain intuition about Eq. 10 from electrodynamics, since the 

electromagnetic current satisfies a continuity equation. It says that 

charge is neither created nor destroyed locally: the change in the 

charge density, J&), in a small region of space is just equal to the 

current J(x) flowing out of the region. Equation IO generalizes this 

result of electrodynamics to other kinds of charges, and so J@) is 

called a current. In particle physics with its many continuous symmetries, 

we must be careful to identify which current we are talking 

about. 

Although the analysis just performed is classical, the results are 

usually correct in the quantum theory derived from a classical 

Lagrangian. In some cases, however, quantum corrections contribute 

a nonzero term to the right-hand side of Eq. IO; these terms are called 

anomalies. For global symmetries these anomalies can improve the 

predictions from Lagrangians that have too much symmetry when 

compared with data because the anomaly wrecks the symmetry (it 

was never there in the quantum theory, even though the classical 

Lagrangian had the symmetry). However, for local symmetries 

anomalies are disastrous. A quantum field theory is locally symmetric 

only if its currents satisfy the continuity equation, Eq. 10. 

Otherwise local symmetry transformations simply change the theory. 

(Some care is needed to avoid this kind of anomaly in the standard 

model.) We now show that Eq. IO can imply the existence of a 

conserved quantity called the global charge and defined by 

provided the integral over all space in Eq. 11 is well defined; that is, 

leaves the Lagrangian density invariant. For example, the first term 

of Eq. 1 by itself is unchanged: dpcptdrq becomes a,(e-"qt)ap(B"cp) 

= dpcptdhp, where the last equality follows only if E is constant in 

space-time. (The case of local phase transformations is treated in 

Note 5.) The next step is to write the infinitesimal form of Eq. 13 and 

substitute it into Eq. 9. The conserved current is 

Jp(x) = i[(apcpt)cp - @,cp)cp+] > (14) 

where the sum in Eq. 9 over the fields cp(x) and cpt(x) is written out 

explicitly. 

If m2 > 0 in Eq. 1, then all the charge can be localized in space and 

time and made to vanish as the distance from the charge goes to 

infinity. The steps in Eq. 12 are then rigorous, and a conserved charge 

exists. The calculation was done here for classical fields, but the same 

results hold for quantum fields: the conservation law implied by Eq. 

I2 yields a conserved global charge equal to the number of cp particles 

minus the number of cp antiparticles. This number must remain 

constant in any interaction. (We will see in Note 3 that if m2 < 0, the 

charge distribution is spread out over all space-time, so the global 

charge is no longer conserved even though the continuity equation 

remains valid.) 

Identifying the transformations of the fields that leave the Lagrangian 

invariant not only-salisfies our sense of symmetry but also 

leads to important predictions of the theory without solving the 

equations of motion. In Note 4 we will return to the example of three 

real scalar fields to introduce larger global symmetries, such as SU(2), 

that interrelate different fields. 

57

I _______ 

~ 

~ 

Spontaneous 

Breaking of a 

Global Symmetry 

It is possible for the vacuum or ground state of a physical system to 

have less symmetry than the Lagrangian. This possibility is called 

spontaneous symmetry breaking, and it plays an important role in 

the standard model. The simplest example is the complex scalar field 

theory of Eq. la with tn' < 0. 

In order to identify the classical fields with particles in the quantum 

theory, the classical field must approach zero as the number of 

particles in the corresponding quantum-mechanical state approaches 

zero. Thus the quantum-mechanical vacuum (the state with no 

particles) corresponds to the classical solution cp(x) = 0. This might 

seem automatic, but it is not. Symmetry arguments do not 

necessarily imply that cp(x) = 0 is the lowest energy state of the 

system. However, ifwe rewrite cp(x) as a function of new fields that do 

vanish for the lowest energy state, then the new fields may be directly 

identified with particles. Although this prescription is simple, its 

I 

i 

I 

I 

t 

justification and analysis of its limitations require extensive use of nonzero Of q- 

the details of quantum field theory. 

The energy of the complex scalar theory is the sum of kinetic and 

potential energies of the cp(x) and cpt(x) fields, so the energy density is 

x = apcp+apcp + 'dcptcp + h(cp+cp)2 , (15) 

with h > 0. Note that dPcpt~,cp is nonnegative and is zero if cp is a 

constant. For cp = 0, .vV = 0. However, if m2 < 0, then there are 

nonzero values of cp(x) for which H < 0. Thus, there is a nonzero 

field configuration with lowest energy. A graph of .W as a function of 

lcpl is shown in Fig. 1. In this example ,W is at its lowest value when 

both the kinetic and potential energies ( V= m2cptcp + k(cptcp)') are at 

their lowest values. Thus, the vacuum solution for cp(s) is found by 

solving the equation a V/dcp = 0, or 

1 

________.I___ 

i 

1 

I 

__._..__-, 

Fig- 1. The Hamiltonian ,#f defined by &. 15 has minima at 

! 

I 

I ! 

, 

I 

Next we find new fields that vanish when Eq. 16 is satisfied. For 

example, we can set 

/ 

where the real fields p(x) and n(x) are zero when the system is in the 

lowest energy state. Thus p(x) and n(x) may be associated with 

particles. Note, however, that cp0 is not completely specified; it may 

lie at any point on the circle in field space defined by Eq. 16, as shown 

in Fig. 2. 

Suppose 90 is real and given by 

cpo = (--tn2/h)'f2. (18) 

Then the Lagrangian is still invariant under the phase transformations 

in Eq. 13. but the choice of the vacuum field solution is changed 

Fig. 2. The closed curve is the location of the minimum of 

V in the field space 9. 

by the phase transformation. Thus, the vacuum solution is not 

invariant under the phase transformations, so the phase symmetry is 

spontaneously broken. The symmetry of the Lagrangian is not a 

symmetry of the vacuum. (For m2 > 0 in Eq. I, the vacuum and the 

Lagrangian both have the phase symmetry.)

This Lagrangian has the following features. 

0 The fields p(x) and ~(x) have standard kinetic energy terms. 

0 Since m2 < 0, the term m2p2 can be interpreted as the mass term for 

the p(x) field. The p(x) field thus describes a particle with masssquared 

equal to Im'I, not - Im'I. 

0 The n(x) field has no mass term. (This is obvious from Fig. 2, 

which shows that Y(p,n) has no curvature (that is, a2V/arc2 = 0) in 

the ~(x) direction.) Thus, n(x) corresponds to a massless particle. 

This result is unchanged when all the quantum effects are included. 

0 The phase symmetry is hidden in V when it is written in terms of 

p(x) and n(x). Nevertheless, 2' has phase symmetry, as is proved 

by working backward from Eq. 20 to Eq. 16 to recover Eq. la. 

0 In theories without gravity, the constant term V' 0: m4/h can be 

ignored, since a constant overall energy level is not measurable. 

The situation is much more complicated for gravitational theories, 

where terms of this type contribute to the vacuum energy-momentum 

tensor and, by Einstein's equations, modify the geometry of 

space-time. 

0 The p field interacts with the K field only through derivatives of n. 

The interaction terms in Eq. 20 may be pictured as in Fig. 3. 

Fig. 3. A graphic representation of the last four terms of Eq. 

20, the interaction terms. Solid lines denote the p field and 

dotted lines the nfield. The interaction of three p(x)fields at 

a singlepoint is shown as three solid lines emanating from a 

single point. In perturbation theory this so-called vertex 

represents the lowest order quantum-mechanical amplitude 

for one particle to turn into two. All possible configurations 

of these vertices represent the quantum-mechanical 

amplitudes defined by the theory. 

We now rewrite the Lagrangian in terms of the particle fields p(x) 

and ~(x) by substituting Eq. 17 into Eq. 1. The Lagrangian becomes 

To estimate the masses associated with the particle fields p(x) and 

~(x), we substitute Eq. 18 for the constant cpo and expand 9 in powers 

of the fields ~(x) and p(x), obtaining 

Although this model might appear to be an idle curiosity, it is an 

example of a very general result known as Goldstone's theorem. This 

theorem states that in any field theory there is a zero-mass spinless 

particle for each independent global continuous symmetry of the 

Lagrangian that is spontaneously broken. The zero-mass particle is 

called a Goldstone boson. (This general result does not apply to local 

symmetries, as we shall see.) 

There has been one very important physical application of spontaneously 

broken global symmetries in particle physics, namely, 

theories of pion dynamics. The pion has a surprisingly small mass 

compared to a nucleon, so it might be understood as a zero-mass 

particle resulting from spontaneous symmetry breaking of a global 

symmetry. Since the pion mass is not exactly zero, there must also be 

some small but explicit terms in the Lagrangian that violate the 

global symmetry. The feature of pion dynamics that justifies this 

procedure is that the interactions of pions with nucleons and other 

pions are similar to the interactions (see Fig. 3) of the n(x) field with 

the p(x) field and with itself in the Lagrangian of Eq. 20. Since the 

pion has three (electric) charge states, it must be associated with a 

larger global symmetry than the phase symmetry, one where three 

independent symmetries are spontaneously broken. The usual choice 

of symmetry is global SU(2) X SU(2) spontaneously broken to the 

SU(2) of the strong-interaction isospin symmetry (see Note 4 for a 

discussion of SU(2)). This description accounts reasonably well for 

low-energy pion physics. 

Perhaps we should note that only spinless fields can acquire a 

vacuum value. Fields carrying spin are not invariant under Lorentz 

transformations, so if they acquire a vacuum value, Lorentz invariance 

will be spontaneously broken, in disagreement with experiment. 

Spinless particles trigger the spontaneous symmetry breaking 

in the standard model.

T ncIwnnn1n 

1s with 

Larger Global 

Symmetries 

7 

proaches zero. 

TO identify the generators T, with matrix elements ( T~)~~, we use a 

specific Lagrangian, 

9 

1 1 

= -p 

‘pi~ll’p, - 7 m2’picpi - (Cplcpl)’ 

L 

L 

In a theory with a single complex scalar field the phase transformation 

in Eq. 13 defines the “largest” possible internal symmetry since 

the only possible symmetries must relate q(x) to itself. Here we will 

discuss global symmetries that interrelate different fields and group 

them together into “symmetry multiplets.” Strong isospin, an approximate 

symmetry of the observed strongly interacting particles, is 

an example. It groups the neutron and the proton into an isospin 

doublet, reflecting the fact that the neutron and proton have nearly 

the same mass and share many similarities in the way that they 

interact with other particles. Similar comments hold for the three 

pion states (K’, no, and f), which form an isospin triplet. 

We will derive the structure of strong isospin symmetry by examining 

the invariance of a specific Lagrangian for the three real scalar 

fields q,(x) already described in Note 2. (Although these fields could 

describe the pions, the Lagrangian will be chosen for simplicity, not 

for its capability to describe pion interactions.) 

We are about to discover a symmetry by deriving it from a 

Lagrangian; however, in particle physics the symmetries are often 

discovered from phenomenology. Moreover, since there can be many 

Lagrangians with the same symmetry, the predictions following from 

the symmetry are viewed as more general than the predictions of a 

specific Lagrangian with the symmetry. Consequently, it becomes 

important to abstract from specific Lagrangians the general features 

ofa symmetry; see the comments later in this note. 

A general linear transformation law for the three real fields can be 

written 

where the sum on j runs from 1 to 3. One reason for choosing this 

form of U(E) is that it explicitly approaches the identity as E ap- 

Let us place primes on the fields in Eq. 22 and substitute Eq. 21 into 

it. Then 9 written in terms of the new cp(x) is exactly the same as Eq. 

22 if 

where tij, are the matrix elements of the 3-by-3 identity matrix. (In 

the notation of Eq. 5a, Eq. 23 is U(E)U~(E) = I.) Equation 23 can be 

expanded in E, and the linear term then requires that To be an 

antisymmetric matrix. Moreover, exp ( ~E~T~) must be a real matrix so 

that q(s) remains real after the transformation. This implies that all 

elements of the T, are imaginary. These constraints are solved by the 

three imaginary antisymmetric 3-by-3 matrices with elements 

where ~ 123 = +I and is antisymmetric under the interchange of 

any two indices (for example, ~ 321 = -1). (It is a coincidence in this 

example that the number of fields is equal to the number of independent 

symmetry generators. Also, the parameter E, with one index 

should not be confused with the tensor &&with three indices.) 

The conditions on U(E) imply that it is an orthogonal matrix; 3- 

by-3 orthogonal matrices can also describe rotations in three spatial 

dimensions. Thus, the three components of cpI transform in the same 

way under isospin rotation as a spatial vector x transforms under a 

rotation. Since the rotational symmetry is SU(2), so is the isospin 

symmetry. (Thus “isospin” is like spin.) The T, matrices satisfy the 

SU(2) commutation relations


[Tu, Tb] T,Th - TbTu = iE,hcTc . (25) 

Although the explicit matrices of Eq. 24 satisfy this relation, the T, 

can be generalized to be quantum-mechanical operators. In the 

example of Eqs. 21 and 22, the isospin multiplet has three fields. 

Drawing on angular momentum theory, we can learn other 

possibilities for isospin multiplets. Spin-J multiplets (or representations) 

have 25 + I components, where J can be any nonnegative 

integer or half integer. Thus, multiplets with isospin of '12 have two 

fields (for example, neutron and proton) and isospin-3/2 multiplets 

have four fields (for example, the A++, A+, A', and A- baryons of mass 

- 1232 GeV/c2). 

The basic structure of all continuous symmetries of the standard 

model is completely analogous to the example just developed. In fact, 

part of the weak symmetry is called weak isospin, since it also has the 

same ma?hematical structure as strong isospin and angular momentum. 

Since there are many different applications to particle theory of 

given symmetries, it is often useful to know about symmetries and 

their multiplets. This mathematical endeavor is called group theory, 

and the results of group theory are often helpful in recognizing 

patterns in experimental data. 

Continuous symmetries are defined by the algebraic properties of 

their generators. Group transformations can always be written in the 

form of Eq. 21. Thus, if Q, (a = I, .. . , N) are the generators of a 

symmetry, then they satisfy commutation relations analogous to Eq. 

25: 

multiplet of the symmetry. 

The general problem of finding all the ways of constructing equations 

like Eq. 25 and Eq. 26 is the central problem of Lie-group 

theory. First, one must find all sets ofj& This is the problem of 

finding all the Lie algebras and was solved many years ago. The 

second problem is, given the Lie algebra, to find all the matrices that 

represent the generators. This is the problem of finding all the 

representations (or multiplets) of a Lie algebra and is also solved in 

general, at least when the range of values ofeach E, is finite. Lie group 

theory thus offers an orderly approach to the classification of a huge 

number of theories. 

Once a symmetry of the Lagrangian is identified, then sets of n 

fields are assigned to n-dimensional representations of the symmetry 

group, and the currents and charges are analyzed just as in Note 2. 

For instance, in our example with three real scalar fields and the 

Lagrangian of Eq. 22, the currents are 

and, if m2 > 0, the global symmetry charge is 

where the quantum-mechanical charges Q, satisfy the commutation 

relations 

where the constants,f,h, are called the structure constants of the Lie 

algebra. The structure constants are determined by the multiplication 

rules for the symmetry operations, U(E~)U(&~) = U(&j), where ~3 

depends on E~ and E ~ Equation . 26 is a basic relation in defining a Lie 

algebra, and Eq. 2 I is an example of a Lie group operation. The Q,. 

which generate the symmetry, are determined by the "group" structure. 

The focus on the generators often simplifies the study of Lie 

groups. The generators Q, are quantum-mechanical operators. The 

(Tu)', of Eqs. 24 and 25 are matrix elements of Q, for some symmetry 

(The derivation of Eq. 29 from Eq. 28 requires the canonical commutation 

relations of the quantum cpl(x) fields.) 

The three-parameter group SU(2) has just been presented in some 

detail. Another group of great importance to the standard model is 

SU(3), which is the group of 3-by-3 unitary matrices with unit 

determinant. The inverse of a unitary matrix U is Ut, so UtU = I. 

There are eight parameters and eight generators that satisfy Eq. 26 

with the structure constants ofSU(3). The low-dimensional representations 

of SU(3) have I, 3, 6, 8, IO, . .. fields, and the different 

representations are referred to as 1,3,3,6,6,8, 10, m, and so on. 

61

I 

Local Phase 

Invariance and 

Electrodynamics 

The theories that make up the standard model are all based on the 

principle of local symmetry. The simplest example of a local symmetry 

is the extension of the global phase invariance discussed at the 

end of Note 2 to local phase invariance. As we will derive below, the 

requirement that a theory be invariant under local phase transformations 

irnplies the existence ofa gauge field in the theory that mediates 

or carries the “force” between the matter fields. For electrodynamics 

the gauge field is the electromagnetic vector potential A,(x) and its 

quantum particle is the massless photon. In addition, in the standard 

model the gauge fields mediating the strong interactions between the 

quarks are the massless gluon fields and the gauge fields mediating 

the weak interactions are the fields for the massive Zo and w‘ weak 

bosons. 

To illustrate these principles we extend the global phase invariance 

of the Lagrangian of Eq. I to a theory that has local phase invariance. 

Thus, we require 9’ to have the same form for cp’(x) and cp(x), where 

the local phase transformation is defined by 

cp’(x) = efE(‘)cp(x) . (30) 

The potential energy, 

already has this symmetry, but the kinetic energy, ?cptd,,cp, clearly 

does not. since 

Y does not have local phase invariance if the Lagrangian of the 

transformed fields depends on E(X) or its derivatives. The way to 

eliminate the d , ~ dependence is to add a new field A,(x) called the 

gauge field and then require the local symmetry transformation law 

for this new field to cancel the d , ~ term in Eq. 32. The gauge field can 

be added by generalizing the derivative a, to D,, where 

D,, = a,, - ieA,,(x) . (33) 

This is just the minimal-coupling procedure of electrodynamics. We 

can then make a kinetic energy term of the form (Dpcp)t(Dpcp) if we 

require that 

When written out with Eq. 33, Eq. 34 becomes an equation for A;(x) 

in terms ofA,,(x), which is easily solved to give 

I 

AG(x) = A,(x) + - e d,E(X) . (35) 

Equation 35 prescribes how the gauge field transforms under the local 

phase symmetry. 

Thus the first step to modifying Eq. 1 to be a theory with local 

phase invariance is simply to replace dp by D,, in 9. (A slightly 

generalized form of this trick is used in the construction of all the 

theories in the standard model.) With this procedure the dominant 

interaction of the gauge field AW(x) with the matter field cp is in the 

form of a current times the gauge field, ePA,, where J, is the current 

defined in Eq. 14. 

of the calculation is replacing dpcpta,,cp by (Dpcp)t(D,,cp). However, 

Spontaneous 

instead of simply substituting Eq. 17 for cp and computing 

(Dwcp)t(D,cp) directly, it is convenient to make a local phase trans- 

Breaking of Local formation first: 

1 

Phase Invariance q’(x) = + 90~ 

exp[lrr(x)/cpo~, (41) 

We now show that spontaneous breaking of local symmetry implies 

that the associated vector boson has a mass, in spite of the fact 

that A”,, by itself is not locally phase invariant. Much of the 

calculation in Note 3 can be translated to the Lagrangian of Eq. 38. In 

fact, the calculation is identical from Eq. 16 to Eq. 18, so the first new 

step is to substitute Eq. 17 into Eq. 38. The only significantly new part 

where cp(x) = [p(x) + cpo]/\/Z. (The local phase invariance permits us 

to remove the phase of cp(x) at every space-time point.) We 

emphasize the difference between Eqs. 17 and 41: Eq. 17 defines the 

p(x) and R(X) fields; Eq. 41 is a local phase transformation of cp(x) by 

angle n(x). Don’t be fooled by the formal similarity of the two 

equations. Thus, we may write Eq. 38 in terms of cp(x) = [~(x) + 

cpO]/fi and obtain ,, ’ ., 

.’ ,

This leaves a problem. If we simply replace d,cp by D,cp in the 

Lagrangian and then derive the equations of motion for A,, we find 

that A, is proportional to the current J,. The A, field equation has no 

space-time derivatives and therefore A,(x) does not propagate. If we 

want A, to correspond to the electromagnetic field potential, we must 

add a kinetic energy term for it to 9. 

The problem then is to find a locally phase invariant kinetic energy 

term for A,(x). Note that the combination of covariant derivatives 

D,Dv - D,D,, when acting on any function, contains no derivatives 

of the function. We define the electromagnetic field tensor of electrodynamics 

as 

It contains derivatives of A,. Its transformation law under the local 

symmetry is 

F;, = F,, . (37) 

Thus, it is completely trivial to write down a term that is quadratic in 

the derivatives of A,, which would be an appropriate kinetic energy 

term. A fully phase invariant generalization of Eq. la is 

the key to understanding the electroweak theory. 

We now rediscover the Lagrangian of electrodynamics for the 

interaction of electrons and photons following the same procedure 

that we used for the complex scalar field. We begin with the kinetic 

energy term for a Dirac field of the electron y, replace 3, by D, 

defined in Eq. 33, and then add - 1/4FpvF,,, where Fpv is defined in 

Eq. 36. The Lagrangian for a free Dirac field is 

where y, are the four Dirac y matrices and $ = ytyo. Straightening out 

the definition of the y, matrices and the components of w is the 

problem of describing a spin-% particle in a theory with Lorentz 

invariance. We leave the details ofthe Dirac theory to textbooks, but 

note that we will use some of these details when we finally write down 

the interactions of the quarks and leptons. The interaction of the 

electron field w with the electromagnetic field follows by replacing d, 

by D,. The electrodynamic Lagrangian is 

where the interaction term in i$ypD,y has the form 

We should emphasize that 9 has no mass term for A,(x). Thus, when 

the fields correspond directly to the particles in Eq. 38, the vector 

particles described by A,(x) are massless. In fact, ANA, is not invariant 

under the gauge transformation in Eq. 35, so it is not obvious 

how the A, field can acquire a mass if the theory does have local 

phase invariance. In Note 6 we will show how the gauge field 

becomes massive through spontaneous symmetry breaking. This is 

where JP= cypy is the electromagnetic current of the electron. 

What is amazing about the standard model is that all the electroweak 

and strong interactions between fermions and vector bosons are 

similar in form to Eq. 40b, and much phenomenology can be 

understood in terms of such interaction terms as long as we can 

approximate the quantum fields with the classical solutions. 

(At the expense of a little algebra, the calculation can be done the 

other way. First substitute Eq. 17 for cp in Eq. 38. One then finds an 

AV,n term in 9 that can be removed using the local phase transformation 

Ab = A, - [ I/(ecpo)]d,n, pf = p, and d = 0. Equation 42 

then follows, although this method requires some effort. Thus, a 

reason for doing the calculation in the order of Eq. 41 is that the 

algebra gets messy rather quickly if the local symmetry is not used 

early in the calculation ofthe electroweak case. However, in principle 

it makes little difference.) 

The Lagrangian in Eq. 42 is an amazing result: the n field has 

vanished from 9 altogether (according to Eq. 41, it was simply a 

gauge artifact), and there is a term l/2&p; MA, in 9, which is a mass 

term for the vector particle. Thus, the massless particle of the global 

case has become the longitudinal mode of a massive vector particle, 

and there is only one scalar particle p left in the theory. In somewhat 

more picturesque language the vector boson has eaten the Goldstone 

boson and become heavy from the feast. However, the existence of 

the vector boson mass terms should not be understood in isolation: 

the phase invariance of Eq. 42 determines the form of the interaction 

of the massive A, field with the p field. 

This calculation makes it clear that it can be tricky to derive the 

spectrum of a theory with local symmetry and spontaneous symmetry 

breaking. Theoretical physicists have taken great care to 

confirm that this interpretation is correct and that it generalizes to the 

full quantum field theory. 

63

where is the inverse of the matrix . With these requirements, 

it is easily seen that (D!'cp)t(D,cp) is invariant under the group 

of local transformations. 

The calculation of the field tensor is formally identical to Eq. 36, 

except we must take into account that A,(x) is a matrix. Thus, we 

define a matrix F,, field tensor as 

F,, E 

i [D ,D,] = a,A, - &A, - ie [A,,A,] . 

(49) 

There is a field tensor for each group generator, and some further 

matrix manipulation plus Eq. 26 gives the components, 

Thus, we can write down a kinetic energy term in analogy to 

electrodynamics: 

The locally invariant Yang-Mills Lagrangian for spinless fields coupled 

to the vector bosons is 

Just as in electrodynamics, we can add fermions to the theory in 

the form 

where D, is defined in Eq. 46 and y~ is a column vector with nfentries 

(nf = number of fermions). The matrices To in D, for the fermion 

covariant derivative are usually different from the matrices for the 

spinless fields, since there is no requirement that cp and I+I need to 

belong to the same representation of the group. It is, of course, 

necessary for the sets of To matrices to satisfy the commutation 

relations of Eq. 26 with the same set of structure constants. 

We will not look at the general case of spontaneous symmetry 

breaking in a Yang-Mills theory, which is a messy problem 

mathematically. There is spontaneous symmetry breaking in the 

electroweak sector of the standard model, and we will work out the 

steps analogous to Eqs. 41 and 42 for this particular case in the next 

Note.


The SU(2) X U(1) 

@ Et;;weak 

The main emphasis in these Notes has been on developing just 

those aspects of Lagrangian field theory that are needed for the 

standard model. We have now come to the crucial step: finding a 

Lagrangian that describes the electroweak interactions. It is rather 

difficult to be systematic. The historical approach would be complicated 

by the rather late discovery of the weak neutral currents, and 

a purely phenomenological development is not yet totally logical 

because there are important aspects of the standard model that have 

not yet been tested experimentally. (The most important of these are 

the details ofthe spontaneous symmetry breaking.) Although we will 

write down the answer without excessive explanation, the reader 

should not forget the critical role that experimental data played in the 

development of the theory. 

The first problem is to identify the local symmetry group. Before 

the standard model was proposed over twenty years ago, the electromagnetic 

and charge-changing weak interactions were known. The 

smallest continuous group that can describe these is SU(2), which has 

a doublet representation. If the weak interactions can change electrons 

to electron neutrinos, which are electrically neutral, it is not 

possible to incorporate electrodynamics in SU(2) alone unless a 

heavy positively charged electron is added to the electron and its 

neutrino to make a triplet, because the sum of charges in an SU(2) 

multiplet is zero. Various schemes ofthis sort have been tried but do 

not agree with experiment. The only way to leave the electron and 

electron neutrino in a doublet and include electrodynamics is to add 

an extra U(1) interaction to the theory. The hypothesis of the extra 

U( I ) factor was challenged many times until the discovery of the 

weak neutral current. That discovery established that the local symmetry 

of the electroweak theory had to be at least as large as SU(2) X 

U( 1). 

Let us now interpret the physical meaning of the four generators of 

SU(2) X U(I). The three generators of the SU(2) group are I+, I3, 

and I-, and the generator of the U(1) group is called Y, the weak 

hypercharge. (The weak SU(2) and U( I) groups are distinguished 

from other SU(2) and U( I) groups by the label “W.”) I+ and I- are 

associated with the weak charge-changing currents (the general definition 

of a current is described in Note 2), and the charge-changing 

currents couplc to the W+ and W- charged weak vector bosons in 

analogy to Eq. 40b. Both I3 and Yare related to the electromagnetic 

current and the weak neutral current. In order to assign the electron 

and its neutrino to an SU(2) doublet, the electric charge Qe” is 

defined by 

Q‘” = I3 + Y/2 , (55) 

so the sum ofelectric charges in an n-dimensional multiplet is nY/2. 

The charge of the weak neutral current is a different combination of 

I3 and Y, as will be described below. 

The Lagrangian includes many pieces. The kinetic energies of the 

vector bosons are described by Y Y . ~ in , analogy to the first term in 

Eq. 38. The three weak bosons have masses acquired through spontaneous 

symmetry breaking, so we need to add a scalar piece YSca~,, to 

the Lagrangian in order to describe the observed symmetry breaking 

(also see Eq. 38). The fermion kinetic energy Yfermion includes the 

fermion-boson interactions, analogous to the electromagnetic interactions 

derived in Eqs. 39 and 40. Finally, we can add terms that 

couple the scalars with the fermions in a term Yyukawa. One physical 

significance of the Yukawa terms is that they provide for masses of 

the quarks and charged leptons. 

The standard model is then a theory with a very long Lagrangian 

with many fields. The electroweak Lagrangian has the terms 

(The reader may find this construction to be ad hoc and ugly. If so, 

the motivation will be clear for searching for a more unified theory 

from which this Lagrangian can be derived. However, it is important 

to remember that, at present, the standard model is the pinnacle of 

success in theoretical physics and describes a broader range of natural 

phenomena than any theory ever has.) 

The Yang-Mills kinetic energy term has the form given by Eq. 52 

for the SU(2) bosons, plus a term for the U( I) field tensor similar to 

electrodynamics (Eqs. 36 and 38). 

where the U( I) field tensor is 

Fpv = a,B, - d,B, 

and the SU(2) Yang-Mills field tensor is 

(57) 

F;:,, = a,, w: - a,, w;: + g&,hc w; WC, , (59) 

where the E,/,< are the structure constants for SU(2) defined in Eq. 24 

and the W; are the Yang-Mills fields. 

65

@ continued 

SU(2) X U( 1) has two factors, and there is an independent coupling 

constani for each factor. The coupling for the SU(2) factor is called g, 

and it has become conventional to call the U( I ) coupling g'/2. The 

two couplings can be written in several ways. The U(1) of electrodynamics 

is generated by a linear combination of I3 and Y, and the 

coupling is, as usual, denoted by e. The other coupling can then be 

parameterized by an angle Bw. The relations among g, g', e, and Bw 

are 

. .- ., : 

e = gg'/\/$+g",.and . .. tan Bw = g'/g. (60) 

. . 

These definitions will be motivated shortly. In the electroweak theory 

both couplings must be evaluated experimentally and cannot be 

calculated in the standard model. 

The scalar Lagrangian requires a choice of representation for the 

scalar fields. The choice requires that the field with a nonzero 

vacuum value is electrically neutral, so the photon remains massless, 

but it must carry nonzero values of 13 and Y so that the weak neutral 

boson (the q) acquires a mass from spontaneous symmetry breaking. 

The simplest assignment is 

assignment that the cp doublet has Y = 1. After the spontaneous 

symmetry breaking, three of the four scalar degrees of freedom are 

"eaten" by the weak bosons. Thus just one scalar escapes the feast 

and should be observable as an independent neutral particle, called 

the Higgs particle. It has not yet been observed experimentally, and it 

is perhaps the most important particle in the standard model that 

does not yet have a firm phenomenological basis. (The minimum 

number of scalar fields in the standard model is four. Experimental 

data could eventually require more.) 

We now carry out the calculation for the spontaneous symmetry 

breaking of SU(2) X U( 1) down to the U( I ) of electrodynamics. Just 

as in the example worked out in Note 6, spontaneous symmetry 

breaking occurs when m2 < 0 in Eq. 62. In contrast to the simpler 

case, it is rather important to set up the problem in a clever way to 

avoid an inordinate amount ofcomputation. As in Eq. 41, we write 

the four degrees of freedom in the complex scalar doublet so that it 

looks like a local symmetry transformation times a simple form ofthe 

field: 

We can then write the scalar fields in a new gauge where the phases of 

cp(x) are removed: 

cp'(x) = exp [-irca(x)ra/2cpo]cp(x) = 

where cp+ has I, = Y2 and Y = 1, and cpo has I3 = --% and Y = I . Since 

cp does not have Y = -I fields, it is necessary to make cp a complex 

doublet, so (q~+)~ = -9- has I3 = -I12 and Y = -1, and (TO)+ has 13 = 

and Y =: -1. Then we can write down the Lagrangian of the scalar 

fields as 

where we have used the freedom of making local symmetry transformations 

to write cp'(x) in a very simple form. This choice, called 

the unitary gauge, will make it easy to write out Eq. 63 in explicit 

matrix form. Let us drop all primes on the fields in the unitary gauge 

and redefine WE by the equation 

where 

is the covariant derivative. The 2-by-2 matrices ra are the Pauli 

matrice:;. The factor of Y2 is required because the doublet representation 

of the SU(2) generators is s,/2. The factor of Y2 in the B, term 

is due 1.0 the convention that the U(1) coupling is g'/2 and the 

where the definition of the Pauli matrices is used in the first step, and 

the W' fields are defined in the second step with a numerical factor 

that guarantees the correct normalization of the kinetic energy of the 

charged weak vector bosons. 

Next, we write out the D,cp in explicit matrix form, using Eqs. 63, 

65, and 66: 

66


Finally, we substitute Eqs. 65 and 61 into Eq. 62 and obtain 

where p is the, as yet, unobserved Higgs field. 

It is clear from Eq. 68 that the Wfields will acquire a mass equal to 

g(p0/2 from the term quadratic in the W fields, ($/4)(p6WKWi. 

The combination g‘B, - gWi will also have a mass. Thus, we 

“rotate” the B, and Wi fields to the fields Z: for the weak neutral 

boson and A, for the photon so that the photon is massless. 

sin Ow -cos OW) (2:) 

sin ow 

( ?,’) = ( cos ow 

Our purpose here will be to write out Eq. 72 explicitly for the 

assignments. 

Consider the electron and its neutrino. (The quark and remaining 

lepton contributions can be worked out in a similar fashion.) The lefthanded 

components are assigned to a doublet and the right-handed 

components are singlets. (Since a neutral singlet has no weak charge, 

the right-handed component of the neutrino is invisible to weak, 

electromagnetic, or strong interactions. Thus, we can neglect it here, 

whether or not it actually exists.) We adopt the notation 

(73) 

where L and R denote left- and right-handed. Then the explicit 

statement of Eq. 72 requires constructing D, for the left- and righthanded 

leptons. 

Upon substituting Eqs. 69 and 70 into Eq. 68, we find that the 8 

mass is ‘12 cpo m2, so the ratio of the Wand 2 masses is 

Values for Mb, and MZ have recently been measured at the CERN 

proton-antiproton collider: MH~ 

= (80.8 f 2.7) GeV/c2 and Mz = 

(92.9 f 1.6) GeV/c2. The ratio Mw/Mz calculated with these values 

agrees well with that given by Eq. 71. (The angle Ow is usually 

expressed as sin20w and is measured in neutrino-scattering experiments 

to be sin2& = 0.224 f 0.015.) The photon field A, does not 

appear in YScalar, so it does not become massive from spontaneous 

symmetry breaking. Note, also, that the na(x) fields appear nowhere 

in the Lagrangian; they have been eaten by three weak vector bosons, 

which have become massive from the feast. 

The next term in Eq. 56 is 9’fermion. Its form is analogous to Eqs. 39 

and 40 for electrodynamics: 

The weak hypercharge of the right-handed electron is -2 so the 

coefficient of E, in the first term of Eq. 74 is (-g‘/2) X (-2) = g‘. We 

leave it to the reader to check the rest of Eq. 74. The absence ofa mass 

term is not an error. Mass terms are of the form $w = $LI+JR + $ R~L. 

Since wL is a doublet and cR is a singlet, an electron mass term must 

violate the SU(2) X U( 1) symmetry. We will see later that the electron 

mass will reappear as a result of modification of ,9’yukawa due to 

spontaneous symmetry breaking. 

The next task is exciting, because it will reveal how the vector 

bosons interact with the leptons. The calculation begins with Eq. 74 

and requires the substitution of explicit matrices for T~ Wi, WR, and 

wL. We use the definitions in Eqs. 66, 69, and 73. The expressions 

become quite long, but the calculation is very straightforward. After 

simplifying some expressions, we find that 9’lepton for the electron 

lepton and its neutrino is 

Y~~~~~~ = i$d,e + iiLyV,vL - e $eA, 

The physical problem is to assign the left- and right-handed fermions 

to multiplets of SU(2); the assignments rely heavily on experimental 

data and are listed in “Particle Physics and the Standard Model.” 

(75)

@ continued 

The first two terms are the kinetic energies of the electron and the 

neutrino. (Note that e = eL + eR.) The third term is the electromagnetic 

interaction (cf. Eq. 40) with electrons of charge -e, 

where e is defined in Eq. 60. The coupling ofA, to the electron current 

does not distinguish left from right, so electrodynamics does not 

violate parity. The fourth term is the interaction of the W' bosons 

with the weak charged current of the neutrinos and electrons. Note 

that these bosons are blind to right-handed electrons. This is the 

reason for maximal parity violation in beta decay. The final terms 

predict how the weak neutral current of the electron and that of the 

neutrino couple to the neutral weak vector boson Zo. 

If the left- and right-handed electron spinors are written out 

explicitly, with eL = %(I - y&, the interaction of the weak neutral 

current of the electron with the Zo is proportional to $'[(I - 

4sin2ew) - ys]eZ,. This prediction provided a crucial test of the 

standard model. Recall from Eq. 71 that sinZew is very nearly Ih, so 

that the weak neutral current of the electron is very nearly a purely 

axial current, that is, a current of the form &''Y5e. This crucial 

prediction was tested in deep inelastic scattering of polarized electrons 

and in atomic parity-violation experiments. The results ofthese 

experiments went a long way toward establishing the standard model. 

The tests also ruled out models quite similar to the standard model. 

We could discuss many more tests and predictions of the model 

based on the form of the weak currents, but this would greatly 

lengthen our discussion. The electroweak currents of the quarks will 

be described in the next section. 

We now discuss the last term in Eq. 56, gyukawa. In a locally 

symmetric theory with scalars, spinors, and vectors, the interactions 

between vectors and scalars, vector and spinors, and vectors and 

vectors are determined from the local invariance by replacing 8, by 

D,. In contrast, yyukawa, which is the interaction between the scalars 

and spinors, has the same form for both local and global symmetries: 

This form forpy,kawa is rather schematic; to make it explicit we must 

specify the multiplets and then arrange the component fields so that 

the form of yyukawa does not change under a local symmetry transformation. 

Let us write Eq. 76 explicitly for the part of the standard model we 

have examined so far: cp is a complex doublet of scalar fields that has 

the form in the unitary gauge given by Eq. 65. The fermions include 

the electron and its neutrino. If the neutrino has no right-handed 

component, then it is not possible to insert it into Eq. 76. Since the 

neutrino has no mass term in Ylepton, the neutrino remains massless 

in this theory. (If VR is included, then the neutrino mass is a free 

parameter.) The Yukawa terms for the electron are 

(77) 

where we have used the fact that &eL = &a = 0, and e = eL + 4( is 

the electron Dirac spinor. Note that Eq. 77 includes an electron mass 

term, 

so the electron mass is proportional to the vacuum value of the scalar 

field. The Yukawa coupling is a free parameter, but we can use the 

measured electron mass to evaluate it. Recall that 

m o e o 

Mu,= - = ~ - 81 GeV, 

2 2sin BW 

where e'/47t = 1/137. This implies that cpo = 251 GeV. Since nip.= 

0.0005 1 1 GeV, CY = 2.8 X IOp6 for the electron. There are more than 

five Yukawa couplings, including those for the p and 'I leptons and 

the three quark doublets as well as terms that mix different quarks of 

the same electric charge. The standard model in no way determines 

the values of these Yukawa coupling constants. Thus, the study of 

fermion masses may turn out to have important hints on how to 

extend the standard model.


@) Quarks 

the assumption of local symmetry leads to a Lagrangian whose form 

is highly restricted. As far as we know, only the quark and gluon fields 

are necessary to describe the strong interactions, and so the most 

general Lagrangian is 

(79) 

Discovery of the fundamental fields of the strong interactions was 

not straightforward. It took some years to realize that the hadrons, 

such as the nucleons and mesons, are made up of subnuclear constituents, 

primarily quarks. Quarks originated from an effort to provide 

a simple physical picture of the “Eightfold Way,” which is the SU(3) 

symmetry proposed by M. Gell-Mann and Y. Ne’eman to generalize 

strong isotopic spin. The hadrons could not be classified by the 

fundamental three-dimensional representations of this SU(3) but 

instead are assigned to eight- and ten-dimensional representations. 

These larger representations can be interpreted as products of the 

three-dimensional representations, which suggested to Gell-Mann 

and G. Zweig that hadrons are composed of constituents that are 

assigned to the three-dimensional representations: the u (up), d 

(down), and s (strange) quarks. At the time of their conception, it was 

not clear whether quarks were a physical reality or a mathematical 

trick for simplifying the analysis of the Eightfold-Way SU(3). The 

major breakthrough in the development of the present theory of 

strong interactions came with the realization that, in addition to 

electroweak and Eightfold-Way quantum numbers, quarks carry a 

new quantum number, referred to as color. This quantum number 

has yet to be observed experimentally. 

We begin this lecture with a description of the Lagrangian of a 

strong-interaction theory of quarks formulated in terms of their color 

quantum numbers. Called quantum chromodynamics, or QCD, it is 

a Yang-Mills theory with local color-SU(3) symmetry in which each 

quark belongs to a three-dimensional color multiplet. The eight 

color-SU(3) generators commute with the electroweak SU(2) X U( 1) 

generators, and they also commute with the generators of the Eightfold 

Way, which is a different SU(3). (Like SU(2), SU(3) is a recurring 

symmetry in physics, so its various roles need to be distinguished. 

Hence we need the label “color.”) We conclude with a discussion of 

the weak interactions of the quarks. 

The QCD Lagrangian. The interactions among the quarks are 

mediated by eight massless vector bosons (called gluons) that are 

required to make the SU(3) symmetry local. As we have already seen, 

where 

The sum on a in the first term is over the eight gluon fields Ai. The 

second term represents the coupling of each gluon field to an SU(3) 

current of the quark fields, called a color current. This term is 

summed over the index i, which labels each quark type and is 

independent ofcolor. Since each quark field y~; is a three-dimensional 

column vector in color space, D, is defined by 

where ha is a generalization of the three 2-by-2 Pauli matrices of 

SU(2) to the eight 3-by-3 Gell-Mann matrices of SU(3), and g, is the 

QCD coupling. Thus, the color current of each quark has the form 

$.aywv. The left-handed quark fields couple to the gluons with 

exactly the same strength as the right-handed quark fields, so parity is 

conserved in the strong interactions. 

The gluons are massless because the QCD Lagrangian has no 

spinless fields and therefore no obvious possibility of spontaneous 

symmetry breaking. Of course, if motivated for experimental 

reasons, one can add scalars to the QCD Lagrangian and spontaneously 

break SU(3) to a smaller group. This modification has been 

used, for example, to explain the reported observation of fractionally 

charged particles. The experimental situation, however, still remains 

murky, so it is not (yet) necessary to spontaneously break SU(3) to a 

smaller group. For the remainder of the discussion, we assume that 

QCD is not spontaneously broken. 

The third term in Eq. 79 is a mass term. In contrast to the 

electroweak theory, this mass term is now allowed, even in the 

absence of spontaneous symmetry breaking, because the left- and 

right-handed quarks are assigned to the same multiplet of SU(3). The 

numerical coeficients MIJ are the elements of the quark mass matrix; 

they can connect quarks of equal electric charge. The PQCD of Eq. 79 

permits us to redefine the QCD quark fields so that MIJ = m,6,. The

@ continued 

mass matrix is then diagonal and each quark has a definite mass, 

which is an eigenvalue of the mass matrix. We will reappraise this 

situation below when we describe the weak currents of the quarks. 

After successfully extracting detailed predictions of the electroweak 

theory from its complicated-looking Lagrangian, we might be 

expected to perform a similar feat for the PQ,, of Eq. 79 without too 

much difficulty. This is not possible. Analysis of the electroweak 

theory was so simple because the couplingsgand g’ are always small, 

regardless of the energy scale at which they are measured, so that a 

classical analysis is a good first approximation to the theory. The 

quantum corrections to the results in Note 8 are, for most processes, 

only a few percent. 

In QCD processes that probe the short-distance structure of 

hadrons, the quarks inside the hadrons interact weakly, and here the 

classical analysis is again a good first approximation because the 

couplingg, is small. However, for Yang-Mills theories in general, the 

renormalization group equations of quantum field theory require 

that g, increases as the momentum transfer decreases until the 

momentum transfer equals the masses of the vector bosons. Lacking 

spontaneous symmetry breaking to give the gluons mass, QCD 

contains no mechanism to stop the growth of g,, and the quantum 

effects become more and more dominant at larger and larger distances. 

Thus, analysis of the long-distance behavior of QCD, which 

includes deriving the hadron spectrum, requires solving the full 

quantum theory implied by Eq. 79. This analysis is proving to be very 

difficult. 

Even without the solution of YQCD, we can, however, draw some 

conclusions. The quark fields w, in Eq. 79 must be determined by 

experiment. The Eightfold Way has already provided three of the 

quarks, and phenomenological analyses determine their masses (as 

they appear in the QCD Lagrangian). The mass of the u quark is 

nearly zero (a few MeV/c*), the dquark is a few MeV/cZ heavier than 

the u, and the mass of the s quark is around 300 MeV/cz. If these 

results are substituted into Eq. 79, we can derive a beautiful result 

from the QCD Lagrangian. In the limit that the quark mass differences 

can be ignored, Eq. 79 has a global SU(3) symmetry that is 

identical to the Eightfold-Way SU(3) symmetry. Moreover, in the 

limit that the u, d, and s masses can be ignored, the left-handed u, d, 

and s quarks can be transformed by one SU(3) and the right-handed 

u, d, and s quarks by an independent SU(3). Then QCD has the 

“chiral” SU(3) X SU(3) symmetry that is the basis of current algebra. 

The sums of the corresponding SU(3) generators of chiral SU(3) X 

SU(3) generate the Eightfold-Way SU(3). Thus, the QCD Lagrangian 

incorporates in a very simple manner the symmetry results of 

hadronic physics of the 1960s. The more recently discovered c 

(charmed) and b (bottom) quarks and the conjectured t (top) quark 

are easily added to the QCD Lagrangian. Their masses are so large 

and so different from one another that the SU(3) and SU(3) X SU(3) 

symmetries of the Eightfold-Way and current algebra cannot be 

extended to larger symmetries. (The predictions of, say, SU(4) and 

chiral SU(4) X SU(4) do not agree well with experiment.) 

It is important to note that the quark masses are undetermined 

parameters in the QCD Lagrangian and therefore must be derived 

from some more complete theory or indicated phenomenologically. 

The Yukawa couplings in the electroweak Lagrangian are also free 

parameters. Thus, we are forced to conclude that the standard model 

alone provides no constraints on the quark masses, so they must be 

obtained from experimental data. 

The mass term in the QCD Lagrangian (Eq. 79) has led to new 

insights about the neutron-proton mass difference. Recall that the 

quark content of a neutron is uddand that of a proton is uud. If the u 

and d quarks had the same mass, then we would expect the proton to 

be more massive than the neutron because of the electromagnetic 

energy stored in the uu system. (Many researchers have confirmed 

this result.) Since the masses of the u and d quarks are arbitrary in 

both the QCD and the electroweak Lagrangians, they can be adjusted 

phenomenologically to account for the fact that the neutron mass is 

1.293 MeV/c2 greater than the proton mass. This experimental 

constraint is satisfied if the mass of the d quark is about 3 MeV/c2 

greater than that of the u quark. In a way, this is unfortunate, because 

we must conclude that the famous puzzle of the n-p mass difference 

will not be solved until the standard model is extended enough to 

provide a theory of the quark masses. 

Weak Currents. We turn now to a discussion of the weak currents of 

the quarks, which are determined in the same way as the weak 

currents of the leptons in Note 8. Let us begin with just the u and d 

quarks. Their electroweak assignments are as follows: the left-handed 

components 111. and dl form an SU(2) doublet with Y = %, and the 

right-handed components UR and d~ are SU(2) singlets with Y = 4/3


and -%, respectively (recall Eq. 55). 

The steps followed in going from Eq. 73 to Eq. 75 will yield the 

electroweak Lagrangian of quarks. The contribution to the, Lagrangian 

due to interaction of the weak neutral current .IF’ of the u and d 

quarks with Zo is 

where 

The reader will enjoy deriving this result and also deriving the 

contribution of the weak charged current of the quarks to the 

electroweak Lagrangian. Equation 83 will be modified slightly when 

we include the other quarks. 

So far we have emphasized in Notes 8 and 9 the construction of the 

QCD and electroweak Lagrangians for just one lepton-quark 

“family” consisting of the electron and its neutrino together with the 

u and d quarks. Two other lepton-quark families are established 

experimentally: the muon and its neutrino along with the c and s 

quarks and the T lepton and its neutrino along with the f and b quarks. 

Just like (v,)~ and eL, (v& and p~ and (v,)~ and TL form weak-SU(2) 

doublets; c?R, p~ and TR are each SU(2) singlets with a weak hypercharge 

of-2. Similarly, the weak quantum numbers ofcand sand of 

t and b echo those of u and d cL and SL form a weak-SU(2) doublet as 

do [R and bL. Like UR and dR, the right-handed quarks CR, SR, tR, and 

b~ are all weak-SU(2) singlets. 

This triplication of families cannot be explained by the standard 

model, although it may eventually turn out to be a critical fact in the 

development of theories of the standard model. The quantum 

numbers of the quarks and leptons are summarized in Tables 2 and 3 

in “Particle Physics and the Standard Model.” 

All these quark and lepton fields must be included in a Lagrangian 

that incorporates both the electroweak and QCD Lagrangians. It is 

quite obvious how to do this: the standard model Lagrangian is 

simply the sum of the QCD and electroweak Lagrangians, except that 

the terms occurring in both Lagrangians (the quark kinetic energy 

terms ic,ypdPyi and the quark mass terms yiMuyj) are included just 

once. Only the mass term requires comment. 

The quark mass terms appear in the electroweak Lagrangian in the 

form 9’yukawa (Eq. 77). In the electroweak theory quarks acquire 

masses only because SU(2) X U( I) is spontaneously broken. However, 

when there are three quarks of the same electric charge (such as 

d, s, and b), the general form of the mass terms is the same as in Eq. 

79, ciMijy,, because there can be Yukawa couplings between dand s, 

d and b, and s and 6. The problem should already be clear: when we 

speak of quarks, we think of fields that have a definite mass, that is, 

fields for which Mi is diagonal. Nevertheless, there is no reason for 

the fields obtained directly from the electroweak symmetry breaking 

to be mass eigenstates. 

The final part of the analysis takes some care: the problem is to find 

the most general relation between the mass eigenstates and the fields 

occurring in the weakcurrents. Wegive theanswer forthecaseoftwo 

families ofquarks. Let us denote the quark fields in the weak currents 

with primes and the mass eigenstates without primes. There is 

freedom in the Lagrangian to set u = u’ and c = c‘. If we do so, then 

the most general relationship among d, s, d‘, and s’ is 

The parameter €Jc, the Cabibbo angle, is not determined by the 

electroweak theory (it is related to ratios of various Yukawa couplings) 

and is found experimentally to be about 13”. (When the band 

t (=t‘) quarks are included, the matrix in Eq. 84 becomes a 3-by-3 

matrix involving four parameters that are evaluated experimentally.) 

The correct weak currents are then given by Eq. 83 if all quark 

families are included and primes are placed on all the quark fields. 

The weak currents can be written in terms of the quark mass 

eigenstates by substituting Eq. 84 (or its three-family generalization) 

into the primed version of Eq. 83. The ratio of amplitudes for s - u 

and d - u is tan 8c; the small ratio of the strangeness-changing to 

non-strangeness-changing charged-current amplitudes is due to the 

smallness of the Cabibbo angle. It is worth emphasizing again that the 

standard model alone provides no understanding of the value of this 

angle. 0 

71

A 

II throughout his history man has 

wanted to know the dimensions 

of his world and his place in it. 

Before the advent of scientific instruments 

the universe did not seem very 

large or complicated. Anything too small to 

detect with the naked eye was not known, 

and the few visible stars might almost be 

touched if only there were a higher hill 

nearby. 

Today, with high-energy particle accelerators 

the frontier has been pushed down 

to distance intervals as small as centimeter 

and with super telescopes to cosmological 

distances. These explorations 

have revealed a multifaceted universe; at 

first glance its diversity appears too complicated 

to be described in any unified manner. 

Nevertheless, it has been possible to 

incorporate the immense variety of experimental 

data into a small number of 

quantum field theories that describe four 

basic interactions-weak, strong, electromagnetic, 

and gravitational. Their mathematical 

formulations are similar in that each 

one can be derived from a local symmetry. 

This similarity has inspired hope for even 

greater progress: perhaps an extension of the 

present theoretical framework will provide a 

single unified description of all natural 

phenomena. 

This dream of unification has recurred 

again and again, and there have been many 

successes: Maxwell’s unification of electricity 

and magnetism; Einstein’s unification 

of gravitational phenomena with the 

geometry of space-time; the quantum-mechanical 

unification of Newtonian mechanics 

.with the wave-like behavior of matter; the 

quantum-mechanical generalization of electrodynamics; 

and finally the recent unification 

of electromagnetism with the weak 

force. Each of these advances is a crucial 

component of the present efforts to seek a 

more complete physical theory. 

Before the successes of the past inspire too 

much optimism, it is important to note that a 

unified theory will require an unprecedented 

extrapolation. The present optimism is generated 

by the discovery of theories successful 

7 4 

at describing phenomena that take place over 

distance intervals of order 10-l6 centimeter 

or larger. These theories may be valid to 

much shorter distances, but that remains to 

be tested experimentally. A fully unified theory 

will have to include gravity and therefore 

will probably have to describe spatial structures 

as small as centimeter, the fundamental 

length (determined by Newton’s 

gravitational constant) in the theory of gravity. 

History suggests cause for further 

caution: the record shows many failures resulting 

from attempts to unify the wrong, too 

few, or too many physical phenomena. The 

end of the 19th century saw a huge but 

unsuccessful effort to unify the description of 

all Nature with thermodynamics. Since the 

second law of thermodynamics cannot be 

derived from Newtonian mechanics, some 

physicists felt it must have the most fundamental 

significance and sought to derive the 

rest of physics from it. Then came a period of 

belief in the combined use of Maxwell’s electrodynamics 

and Newton’s mechanics to explain 

all natural phenomena. This effort was 

also doomed to failure: not only did these 

theories lack consistency (Newton’s equations 

are consistent with particles traveling 

faster than the speed of light, whereas the 

Lorentz invariant equations of Maxwell are 

not), but also new experimental results were 

emerging that implied the quantum structure 

of matter. Further into this century came the 

celebrated effort by Einstein to formulate a 

unified field theory of gravity and electromagnetism. 

His failure notwithstanding, 

the mathematical form of his classical theory 

has many remarkable similarities to the 

modern efforts to unify all known fundamental 

interactions. We must be wary that our 

reliance on quantum field theory and local 

symmetry may be similarly misdirected, although 

we suppose here that it is not. 

Two questions will be the central themes 

ofthis essay. First, should we believe that the 

theories known today are the correct components 

of a truly unified theory? The component 

theories are now so broadly accepted 

that they have become known as the “standard 

model.” They include the electroweak 

theory, which gives a unified description of 

quantum electrodynamics (QED) and the 

weak interactions, and quantum chromodynamics 

(QCD), which is an attractive candidate 

theory for the strong interactions. We 

will argue that, although Einstein’s theory of 

gravity (also called general relativity) has a 

somewhat different status among physical 

theories, it should also be included in the 

standard model. If it is, then the standard 

model incorporates all observed physical 

phenomena-from the shortest distance intervals 

probed at the highest energy accelerators 

to the longest distances seen by 

modern telescopes. However, despite its experimental 

successes, the standard model remains 

unsatisfying; among its shortcomings 

is the presence of a large number of arbitrary 

constants that require explanations. It remains 

to be seen whether the next level of 

unification will provide just a few insights 

into the standard model or will unify all 

natural phenomena. 

The second question examined in this essay 

is twofold: What are the possible strategies 

for generalizing and extending the standard 

model, and how nearly do models based 

on these strategies describe Nature? A central 

problem of theoretical physics is to identify 

the features of a theory that should be abstracted, 

extended, modified, or generalized. 

From among the bewildering array of theories, 

speculations, and ideas that have 

grown from the standard model, we will 

describe several that are currently attracting 

much attention. 

We focus on two extensions of established 

concepts. The first is called supersymmetry; 

it enlarges the usual space-time symmetries 

of field theory, namely, PoincarC invariance, 

to include a symmetry among the bosons 

(particles of integer spin) and fermions 

(particles of half-odd integer spin). One of 

the intriguing features of supersymmetry is 

that it can be extended to include internal 

symmetries (see Note 2 in “Lecture Notes- 

From Simple Field Theories to the Standard 

Model). In the standard model internal local 

symmetries play a crucial role, both for 

classifying elementary particles and for de-

___I__- 

Toward a Unified Theory 

t 

Gravity 

I 

Classical 

Origins 

Electra 

J 

Extended Supergravity 

\ 

Development of 

Gravitational Theories 

including Other Forces 

'1 

'---___----_-- ----- 

Fig. 1. Evolution of fundamental theories of Nature from the 

cfassicaffield theories of Newton and Maxwell to thegrandest 

theoretical conjectures of today. The relationships among 

these theories are discussed in the text. Solid lines indicate a 

direct and well-established extension, or theoretical generalization. 

The wide arrow symbolizes the goal of present 

research, the unification of quantum field theories with 

gravity. 

75

termining the form of the interactions among 

them. The electroweak theory is based on the 

internal local symmetry group SU(2) X U( I) 

(see Note 8) and quantum chromodynamics 

on the internal local symmetry group SU(3). 

Gravity is based on space-time symmetries: 

general coordinate invariance and local 

PoincarC symmetry. It is tempting to try to 

unify all these symmetries with supersymmetry. 

Other important implications of supersymmetry 

are that it enlarges the scope of the 

classification schemes of the basic particles 

to include fields ofdifferent spins in the same 

multiplet, and it helps to solve some technical 

problems concerning large mass ratios 

that plague certain efforts to derive the standard 

model. Most significantly, if supersymmetry 

is made to be a local symmetry, then it 

automatically implies a theory of gravity, 

called supergravity, that is a generalization of 

Einsttin’s theory. Supergravity theories require 

the unification of gravity with other 

kinds of interactions, which may be, in some 

future version, the electroweak and strong 

interactions. The near successes of this approach 

are very encouraging. 

The other major idea described here is the 

extension of the space-time manifold to 

more than four dimensions, the extra 

dimensions having, so far, escaped observation. 

This revolutionary idea implies that 

particles are grouped into larger symmetry 

multiplets and the basic interactions have a’ 

geometrical origin. Although the idea of extending 

space-time beyond four dimensions 

is not new, it becomes natural in the context 

of supergravity theories because these complicated 

theories in four dimensions may be 

derived from relatively simple-looking theones 

in higher dimensions. 

We will follow these developments one 

step further to a generalization of the field 

concept: instead of depending.on space-time, 

the fields may depend on paths in spacetime. 

When this generalization is combined 

with supersymmetry, the resulting theory is 

called a superstring theory. (The whimsicality 

of the name is more than matched by 

the theory’s complexity.) Superstring the- 

’76 

ories are encouraging because some of them 

reduce, in a certain limit, to the only supergravity 

theories that are likely to generalize 

the standard model. Moreover, whereas 

supergravity fails to give the standard model 

exactly, a superstring theory might succeed. 

It seems that superstring theories can be 

formulated only in ten dimensions. 

Figure 1 provides a road map for this 

essay, which journeys from the origins of the 

standard model in classical theory to the 

extensions of the standard model in supergravity 

and superstrings. These extensions 

may provide extremely elegant ways to unify 

the standard model and are therefore attracting 

enormous theoretical interest. It must be 

cautioned, however, that at present no experimental 

evidence exists for supersymmetry 

or extra dimensions. 

Review of the Standard Model 

We now review the standard model with 

particular emphasis on its potential for being 

unified by a larger theory. Over the last 

several decades relativistic quantum field 

theories with local symmetry have succeeded 

in describing all the known interactions 

down to the smallest distances that have 

been explored experimentally, and they may 

be correct to much shorter distances. 

Electrodynamics and Local Symmetry. Electrodynamics 

was the first theory with local 

symmetry. Maxwell’s great unification of 

electricity and magnetism can be viewed as 

the discovery that electrodynamics is described 

by the simplest possible local symmetry, 

local phase invariance. Maxwell’s addition 

of the displacement current to the field 

equations, which was made in order to insure 

conservation of the electromagnetic current, 

turns out to be equivalent to imposing local 

phase invariance on the Lagrangian of electrodynamics, 

although this idea did not 

emerge until the late 1920s. 

A crucial feature of locally symmetric 

quantum field theories is this: typically, for 

each independent internal local symmetry 

there exists a gauge field and its corresponding 

particle, which is a vector boson (spin-I 

particle) that mediates the interaction between 

particles. Quantum electrodynamics 

has just one independent local symmetry 

transformation, and the photon is the vector 

boson (or gauge particle) mediating the interaction 

between electrons or other charged 

particles. Furthermore, the local symmetry 

dictates the exact form of the interaction. 

The interaction Lagrangian must be of the 

form eJ~(x)A,(x), where P (x) is the current 

density of the charged particles and A,(x) is 

the field of the vector bosons. The coupling 

constant e is defined as the strength with 

which the vector boson interacts with the 

current. The hypothesis that all interactions 

are mediated by vector bosons or, equivalently, 

that they originate from local symmetries 

has been extended to the weak and 

then to the strong interactions. 

Weak Interactions. Before the present understanding 

of weak interactions in terms of 

local symmetry, Fermi’s 1934 phenomenological 

theory of the weak interactions had 

been used to interpret many data on nuclear 

beta decay. After it was modified to include 

parity violation, it contained all the crucial 

elements necessary to describe the lowenergy 

weak interactions. His theory assumed 

that beta decay (e.g., n - p + e- + ic) 

takes place at a single space-time point. The 

form of the interaction amplitude is a prod- 

uct of two currents PJ,, where each current 

is a product of fermion fields, and J’J, describes 

four fermion fields acting at the point 

of the beta-decay interaction. This amplitude, 

although yielding accurate predictions 

at low energies, is expected to fail at centerof-mass 

energies above 300 GeV, where it 

predicts cross sections that are larger than 

allowed by the general principles ofquantum 

field theory. 

The problem of making a consistent (renormalizable) 

quantum field theory to describe 

the weak interactions was not solved 

until the 1960s, when the electromagnetic 

and weak interactions were combined into a 

locally symmetric theory. As outlined in Fig.

oson 


Fermi Theory 

Electroweak Theory 

Chai 

Amp1 itude 

Changing Cu 

Amplitude 

Fig. 2. Comparison of neutrino-quark charged-current scattering in the Fermi 

theory and the modern SU(2) X U(1) electroweak theory. (The bar indicates the 

Dirac conjugate.) Thepoint interaction of the Fermi theory leads to an inconsistent 

quantum theory. The W -+ exchange in the electroweak theory spreads out the 

weak interactions, which then leads to a consistent (renormalizable) quantum field 

theory. JF) and JL-) are the charge-raising and charge-lowering currents, respectively. 

The amplitudes given by the two theories are nearly equal as long as the 

square of the momentum transfer, q2 = (pu- pd)', is much less than the square of 

the mass of the weak boson, ML). 

2, the vector bosons associated with the electroweak 

local symmetry serve to spread out 

the interaction of the Fermi theory in spacetime 

in a way that makes the theory consistent. 

Technically, the major problem with 

the Fermi theory is that the Fermi coupling 

constant, GF, is not dimensionless (C, = 

(293 GeV)-*), and therefore the Fermi theory 

is not a renormalizable quantum field theory. 

This means that removing the infinities 

from the theory strips it of all its predictive 

power. 

In the gauge theory generalization of 

Fermi's theory, beta decay and other weak 

interactions are mediated by heavy weak 

vector bosons, so the basic interaction has 

the form gWpJ, and the current-current interaction 

looks pointlike only for energies 

much less than the rest energy of the weak 

bosons. (The coupling g is dimensionless, 

whereas GF is a composite number that includes 

the masses ofthe weak vector bosons.) 

The theory has four independent local symmetries, 

including the phase symmetry that 

yields electrodynamics. The local symmetry 

group of the electroweak theory is SU(2) X 

U( I), where U( 1 ) is the group of phase transformations, 

and SU(2) has the same structure 

as rotations in three dimensions. The 

one phase angle and the three independent 

angles of rotation in this theory imply the 

existence of four vector bosons, the photon 

plus three weak vector bosons, W', Zo, and 

W-. These four particles couple to the four 

SU(2) X U(1) currents and are responsible 

for the "electroweak" interactions. 

The idea that all interactions must be derived 

from local symmetry may seem simple, 

but it was not at all obvious how to apply this 

idea to the weak (or the strong) interactions. 

Nor was it obvious that electrodynamics and 

the weak interactions should be part of the 

same local symmetry since, experimentally, 

the weak bosons and the photon do not share 

much in common: the photon has been 

known as a physical entity for nearly eighty 

years, but the weak vector bosons were not 

observed until late 1982 and early 1983 at the 

CERN proton-antiproton collider in the 

highest energy accelerator experiments ever 

77

performed; the mass of the photon is consistent 

with zero, whereas the weak vector bosons 

have huge masses (a little less than 100 

GeV/c*); electromagnetic interactions can 

take place over very large distances, whereas 

the weak interactions take place on a distance 

scale of about centimeter; and 

finally, the photon has no electnc charge, 

whereas the weak vector bosons carry the 

electric and weak charges of the electroweak 

interactions. Moreover, in the early days of 

gauge theories, it was generally believed, al- 

though incorrectly, that local symmetry of a ue 

>-- 

Lagrangian implies masslessness for the vector 

bosons. 

How can particles as different as the , 

photon and the weak bosons possibly be ~ - 

>-- 

’ Any Charged Particle 

Photon Electromagnetic U(1) 

(QED) 

Any Charged 

w+ 

Electroweak SU(2) x U( 1) 

unified by local symmetry? The answer is 

explained in detail in the Lecture Notes; we 

mention here merely that if the vacuum of 

a locally symmetric theory has a nonzero 

symmetry charge density due to the 

presence of a spinless field, then the vector 

boson associated with that symmetry acquires 

a mass. Solutions to the equations of 

dwe+ 

motion in which the vacuum is not invariant - 

under symmetry transformations are called 

Conjectured 

spontaneously broken solutions, and the vec- 

Strongtor 

boson mass can be arbitrarily large 

Electroweak 

without upsetting the symmetry of the La- I U 

Unification 

grangian. 

(Proton Decay) 

i 

In the electroweak theory spontaneous 

symmetry breaking separates the weak and 

electromagnetic interactions and is the most 

important mechanism for generating masses 

of the elementary particles. In the theories 

dicussed below, spontaneous symmetry 

breaking is often used to distinguish interactions 

that have been unified by extending 

symmetries (see Note 8). 

lhe range of validity of the electroweak 

theory is an important issue, especially when 

considering extensions and generalizations 

to a theory of broader applicability. “Range 

of validity’’ refers to the energy (or distance) 

sale over which the predictions of a theory 

arc: valid. The old Fermi theory gives a good 

account of the weak interactions for energies 

less than 50 GeV, but at higher energies, 

where the effect of the weak bosons is to 

78


Number of 

Relative 

Vector Range of Strength at Mass 

Bosons Force Low Enerav Scale 

1 (photon) Infinite 1/137 

8 (gluons) 

4 (3 weak 

bosons, 1 

photon) 

(Graviton) 

24 

10-13 cm 

10-15 cm 

(weak) 

Infinite 

I oWz9 cm 

__ 

. ... 

1 

I 0-5 

GFIf2 = 290 GeVlc2 

10-38 G$/2 = 1.2 X 10'9GeVfc2 

10-32 

.__ -~ - ~ 

"_.__._.I." ...... . 

I OI5 GeV/cZ 

Mass Scale: There is no universal definition of mass scale in particle physics. It is, 

however, possible to select a mass scale of physical significance for each of these 

theories. For example, in the electroweak and SU(5) theories the mass scale is 

associated with the spontaneous symmetry breaking. In both cases the vacuum value 

of a scalar field (which has dimensions of mass) has a nonzero value. In the weak 

interactions GF is related directly to this vacuum value (see Fig. 2) and, at the same 

time, to the masses of the weak bosons. Similarly, the scale of the SU(5) model is 

related to the proton-decay rate and to the vacuum value of a different scalar field. In 

the Fermi theory GF is the strengt 

the strength of the gravitational 

massless graviton, the origin of th 

be related to a vacuum value but 

eak interaction in the Sam 

on. However, in gravity 

value of GN is not well under 

precisely the way that GF is.) 

scale is defined in a completely diEerent way. Aside from the quark masses, the 

classical QCD Lagrangian has no mass scales and no scalar fields. However, in 

quantum field theory the coupling of a gluon to a quark current depends on the 

momentum carried by the gluon, and this coupling is found to be large for momentum 

transfers below 200 MeVlc. It is thus customary to select p = 200 MeV/c2 (where p is 

the parameter governing the scale of asymptotic freedom) as the mass scale for QCD. 

spread out the weak interactions in spacetime, 

the Fermi theory fails. The electroweak 

theory remains a consistent quantum field 

theory at energies far above a few hundred 

GeV and reduces to the Fermi theory (with 

the modification for parity violation) at 

lower energies. Moreover, it correctly 

predicts the masses of the weak vector bosons. 

In fact, until experiment proves otherwise, 

there are no logical impediments to 

extending the electroweak theory to an 

energy scale as large as desired. Recall the 

example of electrodynamics and its quantum-mechanical 

generalization. As a theory 

of light in the mid-19th century, it could be 

tested to about IO-' centimeter. How could it 

have been known that QED would still be 

valid for distance scales ten orders of magnitude 

smaller? Even today it is not known 

where quantum electrodynamics breaks 

down. 

Strong Interactions. Quantum chromodynamics 

is the candidate theory of the 

strong interactions. It, too, is a quantum field 

theory based on a local symmetry; the symmetry, 

called color SU(3), has eight independent 

kinds of transformations, and so the 

strong interactions among the quark fields 

are mediated by eight vector bosons, called 

gluons. Apparently, the local symmetry of 

the strong interaction theory is not spontaneously 

broken. Although conceptually 

simpler, the absence of symmetry breaking 

makes it harder to extract experimental 

predictions. The exact SU(3) color symmetry 

may imply that the quarks and gluons, which 

carry the SU(3) color charge, can never be 

observed in isolation. There seem to be no 

simple relationships between the quark and 

gluon fields of the theory and the observed 

structure of hadrons (strongly interacting 

particles). The quark model of hadrons has 

not been rigorously derived from QCD. 

One of the main clues that quantum 

chromodynamics is correct comes from the 

results of "deep" inelastic scattering experiments 

in which leptons are used to probe the 

structure of protons and neutrons at very 

short distance intervals. The theory predicts 

79

that at very high momentum transfers or, 

equivalently, at very short distances (


I 

I 

I 

+.‘ 

rm 

CI 

E 

0 

m 

C 

.- 

n 

J 

0 

0.1 

0.0’ 

I \ 

1 o2 10’5 

Mass (GeV/c2 1 

ig. 3. Unification in the SU(5) model. The values of the SU(2), U(l), and SU(3) 

mplings in the SU(5) model are shown as functions of mass scale. These values 

re calculated using the renormalization group equations of quantum field theory. 

t the unification energy scale the proton-decay bosons begin to contribute to the 

!normalization group equations; at higher energies, the ratios track together along 

te solid curve. If the high-mass bosons were not included in the calculation, the 

iuplings would follow the dashed curves. 

iodest efforts to unify the fundamental inractions 

may be an important step toward 

icluding gravity. Moreover, these efforts reuire 

the belief that local gauge theories are 

irrect to distance intervals around 

mtimeter, and so they have made theorists 

lore “comfortable” when considering the 

ctrapolation to gravity, which is only four 

rders of magnitude further. Whether this 

utlook has been misleading remains to be 

:en. The components of the standard model 

-e summarized in Table I. 

Iectroweak-Strong Unification 

rithout Gravity 

The SU(2) X U( I ) X SU(3) local theory is a 

:tailed phenomenological framework in 

hich to analyze and correlate data on elecoweak 

and strong interactions, but the 

ioice of symmetry group, the charge assignients 

of the scalars and fermions, and the 

dues of many masses and couplings must 

: deduced from experimental data. The 

-oblem is to find the simplest extension of 

is part of the standard model that also 

iifies (at least partially) the interactions, 

assignments, and parameters that must be 

put into it “by hand.” Total success at unification 

is not required at this stage because 

the range of validity will be restricted by 

gravitational effects. 

One extension is to a local symmetry 

group that includes SU(2) X U(1) X SU(3) 

and interrelates the transformations of the 

standard model by further internal symmetry 

transformations. The simplest example 

is the group SU(5), although most of the 

comments below also apply to other 

proposals for electroweak-strong unification. 

The SU(5) local symmetry implies new constraints 

on the fields and parameters in the 

theory. However, the theory also includes 

new interactions that mix the electroweak 

and strong quantum numbers; in SU(5) there 

are vector bosons that transform quarks to 

leptons and quarks to antiquarks. These vector 

bosons provide a mechanism for proton 

decay. 

Ifthe SU(5) local symmetry were exact, all 

the couplings of the vector bosons to the 

symmetry currents would be equal (or related 

by known factors), and consequently 

the proton decay rate would be near the weak 

decay rates. Spontaneous symmetry breaking 

of SU(5) is introduced into the theory to 

separate the electroweak and strong interactions 

from the other SU(5) interactions as 

well as to provide a huge mass for the vector 

bosons mediating proton decay and thereby 

reduce the predicted decay rate. To satisfy 

the experimental constraint that the proton 

lifetime be at least IO3’ years, the masses of 

the heavy vector bosons isn the SU( 5 ) model 

must be at least IOl4 GeV/c2. Thus, experimental 

facts already determine that the 

electroweak-strong unification must introduce 

masses into the theory that are 

within a factor of IO5 ofthe Planck mass. 

It is possible to calculate the proton lifetime 

in the SU(5) model and similar unified 

models from the values of the couplings and 

masses of the particles in the theory. The 

couplings of the standard model (the two 

electroweak couplings and the strong coupling) 

have been measured in low-energy 

processes. Although the ratios of the couplings 

are predicted by SU(5), the symmetry 

values are accurate only at energies where 

SU(5) looks exact, which is at energies above 

the masses of the vector bosons mediating 

proton decay. In general, the strengths of the 

couplings depend on the mass scale at which 

they are measured. Consequently, the SU(5) 

ratios cannot be directly compared with the 

values measured at low energy. However, the 

renormalization group equations of field theory 

prescribe how they change with the mass 

scale. Specifically, the change of the coupling 

at a given mass scale depends only on all the 

elementary particles with masses less than 

that mass scale. Thus, as the mass scale is 

lowered below the mass of the proton-decay 

bosons, the latter must be omitted from the 

equations, so the ratios of the couplings 

change from the SU(5) values. If we assume 

that the only elementary fields contributing 

to the equations are the low-mass fields 

known experimentally and if the protondecay 

bosons have a mass of IOl4 GeVlc’ 

(see Fig. 3), then the low-energy experimental 

ratios of the standard model couplings are 

predicted correctly by the renormalization 

group equations but the proton lifetime 

81

prediction is a little less than the experimental 

lower bound. However, adding a few 

more “low-mass’’ (say, less than IO’* 

GeV/c:’) particles to the equations lengthens 

the lifetime predictions, which can thereby 

be pushed well beyond the limit attainable in 

present -day experiments. 

Thus, using the proton-lifetime bound 

directly and the standard model couplings at 

low mass scale, we have seen that electroweak-strong 

unification implies mass 

scales close to the scale where gravity must 

be included. Even if it turns out that the 

electroweak-strong unification is not exactly 

correct, it has encouraged the extrapolation 

of present theoretical ideas well beyond the 

energies available in present accelerators. 

Electroweak-strong unified models such as 

SU(5) achieve only a partial unification. The 

vector bosons are fully unified in the sense 

that they and their interactions are determined 

by the choice of SU(5) as the local 

symmetry. However, this is only a partial 

unificaition. The choice of fermion and scalar 

multiplets and the choice of symmetrybreaking 

patterns are left to the discretion of 

the physicist, who makes his selections based 

on low-energy phenomenology. Thus, the 

“unification” in SU(5) (and related local 

symmetries) is far from complete, except for 

the vector bosons. (This suggests that theories 

in which all particles are more closely 

related to the vector bosons might remove 

some of the arbitrariness; this will prove to 

be the case for supergravity.) 

In summary, strong experimental evidence 

for electroweak-strong unification, 

such as proton decay, would support the 

study of quantum field theories at energies 

just below the Planck mass. From the vantage 

of these theories, the electroweak and 

strong. interactions should be the low-energy 

limit of the unifying theory, where “low 

energy” corresponds to the highest energies 

available at accelerators today! Only future 

experiments will help decide whether the 

standard model is a complete low-energy 

theory, or whether we are repeating the ageold 

error of omitting some low-energy interactions 

that are not yet discovered. Never- 

82 

theless, the quest for total unification of the 

laws of Nature is exciting enough that these 

words of caution are not sufficient to delay 

the search for theories incorporating gravity. 

Toward Unification with Gravity 

Let us suppose that the standard model 

including gravity is the correct set of theories 

to be unified. On the basis of the previous 

discussion, we also accept the hypothesis that 

quantum field theory with local symmetry is 

the correct theoretical framework for extrapolating 

physical theory to distances perhaps 

as small as the Planck length. Quantum 

field theory assumes a mathematical model 

of space-time called a manifold. On large 

scales a manifold can have many different 

topologies, but at short enough distance 

scale, a manifold always looks like a flat 

(Minkowski) space, with space and time infinitely 

divisible. This might not be the structure 

of space-time at very small distances, 

and the manifold model of space-time might 

fail. Nevertheless, all progress at unifying 

gravity and the other interactions described 

here is based on theories in which space-time 

is assumed to be a manifold. 

Einstein’s theory of gravity has fascinated 

physicists by its beauty, elegance, and correct 

predictions. Before examining efforts to extend 

the theory to include other interactions, 

let us review its structure. Gravity is a 

“geometrical” theory in the following sense. 

The shape or geometry of the manifold is 

determined by two types of tensors, called 

curvature and torsion, which can be constructed 

from the gravitational field. The 

Lagrangian of the gravitational field depends 

on the curvature tensor. In particular, Einstein’s 

brilliant discovery was that the 

curvature scalar, which is obtained from the 

curvature tensor, is essentially a unique 

choice for the kinetic energy of the gravitational 

field. The gravitational field calculated 

from the equations of motion then determines 

the geometry of the space-time 

manifold. Particles travel along “straight 

lines” (or geodesics) in this space-time. For 

example, the orbits of the planets are 

geodesics of the space-time whose geometry 

is determined by the sun’s gravitational field. 

In Einstein’s gravity all the remaining 

fields are called matter fields. The Lagrangian 

is a sum of two terms: 

the matter fields together with the gravitational 

field in something like a curvature 

scalar and thereby eliminate 9mat,er In addition, 

generalizing the graviton field in this 

way might lead to a consistent (renormalizable) 

quantum theory of gravity. 

There are reasons to hope that the problem 

of finding a renormalizable theory of gravity 

is solved by superstrings, although the prool 

is far from complete. For now, we discuss the 

unification of the graviton with othcr field: 

without concern for renormalizability. 

We will discuss several ways to find mani. 

folds for which the curvature scalar depend! 

on many fields, not just the gravitationa


apply to symmetries such as supersymmetry, 

with its anticommuting generators. 

These two loopholes in the assumptions of 

the theorem have suggested two directions of 

research in the attempt to unify gravity with 

the other interactions. First, we might suppose 

that the dimensionality of space-time is 

greater than four, and that spontaneous symmetry 

breaking of the PoincarC invariance of 

this larger space separates 4-dimensional 

space-time from the other dimensions. The 

symmetries of the extra dimensions can then 

correspond to internal symmetries, and the 

symmetries of the states in four dimensions 

need not imply an unsatisfactory infinity of 

states. A second approach is to extend the 

PoincarC symmetry to supersymmetry, 

which then requires additional fermionic 

fields to accompany the graviton. A combination 

of these approaches leads to the 

most interesting theories. 

Higher Dimensional Space-Time 

second dimension, which is wound up in a circle, becomes visible. If space-time has 

more than four dimensions, then the extra dimensions could have escaped detection 

ifeach is wound into a circle whose radius is less than centimeter. 

field. This generally requires extending the 4- 

dimensional space-time manifold. The fields 

and manifold must satisfy many constraints 

before this can be done. All the efforts to 

Jnifygravity with the other interactions have 

Jeen formulated in this way, but progress 

was not made until the role of spontaneous 

symmetry breaking was appreciated. As we 

low describe, it is crucial for the solutions of 

he theory to have less symmetry than the 

,agrangian has. 

In the standard model the generators of 

he space-time PoincarC symmetry commute 

vith (are independent of) the generators of 

he internal symmetries of the electroweak 

nd strong interactions. We might look for a 

local symmetry that interrelates the spacetime 

and internal symmetries, just as SU(5) 

interrelates the electroweak and strong internal 

symmetries. Unfortunately, if this 

enlarged symmetry changes simultaneously 

the internal and space-time quantum 

numbers of several states of the same mass, 

then a theorem of quantum field theory requires 

the existence of an infinite number of 

particles of that mass. However, this seemingly 

catastrophic result does not prevent the 

unification of space-time and internal symmetries 

for two reasons: first, all symmetries 

of the Lagrangian need not be symmetries of 

the states because of spontaneous symmetry 

breaking; and second, the theorem does not 

If the dimensionality of space-time is 

greater than four, then the geometry of spacetime 

must satisfy some strong observational 

constraints. In a 5-dimensional world the 

fourth spatial direction must be invisible to 

present experiments. This is possible if at 

each 4-dimensional space-time point the additional 

direction is a little circle, so that a 

tiny person traveling in the new direction 

would soon return to the starting point. Theories 

with this kind of vacuum geometry are 

generically called Kaluza-Klein theories.' 

It is easy to visualize this geometry with a 

two-dimensional analogue, namely, a long 

pipe. The direction around the pipe is 

analogous to the extra dimension, and the 

location along the pipe is analogous to a 

location in 4-dimensional space-time. If the 

means for examining the structure of the 

pipe are too coarse to see distance intervals 

as small as its diameter, then the pipe appears 

I-dimensional (Fig. 4). If the probe of 

the structure is sensitive to shorter distances, 

the pipe is a 2-dimensional structure with 

one dimension wound up into a circle. 

83

I 

The physically interesting solutions of 

Einstein’s 4-dimensional gravity are those in 

which, if all the matter is removed, spacetime 

is flat. The 4-dimensional space-time 

we see around us is flat to a good approximation; 

it takes an incredibly massive hunk of 

high-density (much greater than any density 

observed on the earth) matter to curve space. 

However, it might also be possible to construct 

a higher dimensional theory in which 

our 4-dimensional space-time remains flat in 

the absense of identifiable matter, and the 

extra dimensions are wound up into a “little 

ball.” \Ye must study the generalizations of 

Einstein’s equations to see whether this can 

happen, and if it does, to find the geometry of 

the extra dimensions. 

The Cosmological Constant Problem. Before 

we examine the generalizations of gravity in 

more detail, we must raise a problem that 

pervades all gravitational theories. Einstein’s 

equations state that the Einstein tensor 

(which is derived from the curvature scalar 

in finding the equations of motion from the 

Lagrangian) is proportional to the energymomentum 

tensor. If, in the absence of all 

matter and radiation, the energy-momentum 

tensor is zero, then Einstein’s equations are 

solved by flat space-time and zero gravitational 

field. In 4-dimensional classical general 

relativity, the curvature of space-time 

and the gravitational field result from a 

nonzero energy-momentum tensor due to 

the presence of physical particles. 

However, there are many small effects, 

such as other interactions and quantum effects, 

not included in classical general relativity. 

that can radically alter this simple 

picture. For example, recall that the electroweak 

theory is spontaneously broken, 

which means that the scalar field has a 

nonzero vacuum value and may contribute 

to the vacuum value of the energy-momentum 

tensor. If it does, the solution to the 

Einstein equations in vacuum is no longer 

flat space but a curved space in which the 

curvature increases with increasing vacuum 

energy. Thus, the constant value of the potential 

energy, which had no effect on the 

weak interactions, has a profound effect on 

gravity. 

At first glance, we can solve this difficulty 

in a trivial manner: simply add a constant to 

the Lagrangian that cancels the vacuum 

energy, and the universe is saved. However, 

we may then wish to compute the quantummechanical 

corrections to the electroweak 

theory or add some additional fields to the 

theory; both may readjust the vacuum 

energy. For example, electroweak-strong unification 

and its quantum corrections will 

contribute to the vacuum energy. Almost all 

the details of the theory must be included in 

calculating the vacuum energy. So, we could 

repeatedly readjust the vacuum energy as we 

learn more about the theory, but it seems 

artificial to keep doing so unless we have a 

good theoretical reason. Moreover, the scale 

of the vacuum energy is set by the mass scale 

of the interactions. This is a dilemma. For 

example, the quantum corrections to the 

electroweak interactions contribute enough 

vacuum energy to wind up our 4-dimensional 

space-time into a tiny ball about 

centimeter across, whereas the scale of the 

universe is more like IOzs centimeters. Thus, 

the observed value of the cosmological constant 

is smaller by a factor of IOs2 than the 

value suggested by the standard model. 

Other contributions can make the theoretical 

value even larger. This problem has the innocuous-sounding 

name of “the cosmological 

constant problem.” At present 

there are no principles from which we can 

impose a zero or nearly zero vacuum energy 

on the 4-dimensional part of the theory, although 

this problem has inspired much research 

effort. Without such a principle, we 

can safely say that the vacuum-energy 

prediction of the standard model is wrong. 

At best, the theory is not adequate to confront 

this problem. 

If we switch now to the context of gravity 

theories in higher dimensions, the difficult 

question is not why the extra dimensions are 

wound up into a little ball, but why our 4- 

dimensional space-time is so nearly flat, 

since it would appear that a large cosmological 

constant is more natural than a 

small one. Also, it is remarkable that the 

vacuum energy winding the extra 

dimensions into a little ball is conceptually 

similar to the vacuum charge of a local symmetry 

providing a mass for the vector bosons. 

However, in the case of the vacuum 

geometry, we have no experimental data tha 

bear on these speculations other than thc 

remarkable flatness of our 4-dimensiona 

space-time. The remaining discussion of uni 

fication with gravity must be conducted ir 

ignorance of the solution to the cosmologica 

constant problem. 

Internal Symmetries 

from Extra Dimensions 

The basic scheme for deriving local sym 

metries from higher dimensional gravity wa 

pioneered by Kaluza and Klein’ in the 1920$ 

before the weak and strong interactions wer 

recognized as fundamental. Their attempt 

to unify gravity and electrodynamics in fou 

dimensions start with pure gravity in fivi 

dimensions. They assumed that the vacuun 

geometry is flat 4-dimensional space-timl 

with the fifth dimension a little loop of de 

finite radius at each space-time point, just a 

in the pipe analogy of Fig. 4. The Lagrangiai 

consists of the curvature scalar, constructec 

from the gravitational field in fivi 

dimensions with its five independent com 

ponents. The relationship of a higher dimen 

sional field to its 4-dimensional fields is sum 

marized in Fig. 5 and the sidebar, “Field 

and Spin in Higher Dimensions.” The in 

finite spectrum in four dimensions include 

the massless graviton (two helicity compo 

nents of values +2), a massless vector bosoi 

(two helicity components of *I), a massles 

scalar field (one helicity component of 0: 

and an infinite series of massive spin-. 

pyrgons of increasing masses. (The tern 

“pyrgon” derives from mjpyoo, the Creel 

word for tower.) The Fourier expansion fo 

each component of the gravitational field i 

identical to Eq. 1 of the sidebar. Since thl 

extra dimension is a circle, its symmetry is I 

phase symmetry just as in electrodynamic 

1

.-' 


D-Dimensional 

Of Spin 

Field Relabeled 

in Terms of 

4-Dimensional Spin Ji 

Infinite 

Towers of 

4-Dimensional 

Fields 

Zero Mode(s) @;"(x) 

(Massless) * 

4-Dimensional 

Space-Time Directions 

i 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

'ig. 5. A field in D dimensions unifies fields of different 

Dins and masses in four dimensions. In step 1 the spin 

omponents of a single higher dimensional spin are resolved 

rto several spins in four dimensions. (The total number of 

omponents remains constant.) Mathematically this is 

chieved by finding the spins J,, Jb ... in four dimensions 

kat are contained in "spin-$', of D dimensions. Step 2 is 

the harmonic expansion of the 4-dimensional spin components 

on the extra dimensions, which then resolves a single 

massless D-dimensional field into an infinite number of 4- 

dimensional fields of varying masses. When the 4-dimensional 

mass is zero, the corresponding 4-dimensionalfield is 

called a zero mode. The 4-dimensionalfields with 4-dimensional 

mass form an infinite sequence ofpyrgons. 

he symmetry ofthis vacuum state is not the more realistic theories. The zero modes no low-mass charged particles. (Adding fer- 

-dimensional PoincarC symmetry but the (massless particles in four dimensions) are mions to the 5-dimensional theory does not 

irect product of the 4-dimensional Poincare electrically neutral. Only the pyrgons carry help, because the resulting 4-dimensional 

'oup and a phase symmetry. electric charge. The interaction associated fermions are all pyrgons, which cannot be 

This skeletal theory should not be taken with the vector boson in four dimensions low mass either.) Nevertheless, the 

:riously, except as a basis for generalizing to cannot be electrodynamics because there are hypothesis that all interactions are conse- 

85

i 

le in our worl 

at the identical points 

n set of fields. For exa 

-dimensional Ei 

“dimensional reduction.” The dimensionally reduced theory should 

quences ofthe symmetries of space-time is so 

attractive that efforts to generalize the 

Kaluza-Klein idea have been vigorously 

pursued. These theories require a more complete 

discussion of the possible candidate 

manifolds of the extra dimensions. 

The geometry of the extra dimensions in 

the absence of matter is typically a space with 

a high degree of symmetry. Symmetry requires 

the existence of transformations in 

which the starting point looks like the point 

reached after the transformation. (For example, 

thc environments surrounding each 

point on a sphere are identical.) Two of the 

most important examples are “group manifolds” 

and “coset spaces,” which we briefly 

describe. 

The tranformations of a continuous group 

86 

are identified by N parameters, where N is 

the number of independent transformations 

in the group. For example, N = 3 for SU(2) 

and 8 for SU(3). These parameters are the 

coordinates of an N-dimensional manifold. 

Ifthe vacuum values of fields are constant on 

the group manifold, then the vacuum solution 

is said to be symmetric. 

Coset spaces have the symmetry of a group 

too, but the coordinates are labeled by a 

subset of the parameters of a group. For 

example, consider the space S0(3)/S0(2). In 

this example, SO(3) has three parameters, 

and SO(2) is the phase symmetry with one 

parameter, so the coset space S0(3)/S0(2) 

has three minus one, or two, dimensions. 

This space is called the 2-sphere, and it has 

the geometry of the surface of an ordinary 

sphere. Spheres can be generalized to an 

number of dimensions: the N-dimensions 

sphere is the coset space [SO(N+ I)]/SO(N 

Many other cosets, or “ratios” of group! 

make spaces with large symmetries. It i 

possible to find spaces with the symmetrie 

of the electroweak and strong interaction! 

One such space is the group manifold SU(2 

X U(I) X SU(3). which has twelv 

dimensions. More interesting is the lower 

dimensional space with those symmetrie: 

namely, the coset space [SU(3) X SU(2) : 

U(I)]/[SU(2) X U(l) X U(I)], which ha 

dimension 8 + 3 + I - 3 - I - 1 = 7. (TI- 

SU(2) and the U(l)’s in the denorninatc 

differ from those in the numerator, so the 

cannot be “canceled.”)Thus, one might hor 

that (4 + 7 = I I)-dimensional gravity woul


Higher Dimensions 

describe the low-energy limit of the theory. 

The gravitational field can be generalized to higher (>5) dimensional 

manifolds, where the extra dimensions at each 4-dimensional 

space-time point form a little ball of finite volume. The mathematics 

requires a generalization of Fourier series to “harmonic” expansions 

on these spaces. Each field (or field component if it ha5 spin) unifies 

an infinite set of pyrgons. and the series may also contain some zero 

modes. The terms in the series correspond to fields of increasing 4- 

dimensional mass, just as in thc 5-d ensional example. The kinetic 

energy in the extra dimensions of each term in the series then 

corresponds to a mass in our space-time. The higher dimensional 

field quite generally describes mathematically an infinite number of 

4-dimensional fields. 

Spin in Higher Dimensions. The definition of spin in D dimensions 

depends on the D-dimensional Lorentz symmetry; Cdimensional 

Lorentz symmetry is naturally embedded in the D-dimensional 

symmetry. Consequently a D-dimensional field of a specific spin 

unifies 4-dimensional fields with different spins. 

Conceptually the description of D-dimensional spin is similar to 

that of spin in four dimensions. A massless particle of spin J in four 

dimensions has helicities +J and -Jcorresponding to the projections 

of spin along the direction of motion. These two helicities are singlet 

multiplets of the I -dimensional rotations that leave unchanged the 

direction of a particle traveling at the speed of light. The group of 1- 

dimensional rotations is the phase symmetry S0(2), and this method 

for identifying the physical degrees of freedom is called the “lightcone 

classification.” However, the situation is a little more com- 

plicated in five dimensions, where there are three directions orthogonal 

to the direction of the particle. Then the helicity symmetry 

becomes SO(3) (instead of S0(2)), and the spin multiplets in five 

dimensions group together sets of 4-dimensional helicity. For example, 

the graviton in five dimensions has five components. The SO(2) 

of four dimensions is contained in this SO( 3) symmetry, and the 4- 

dimensional helicities of the 5-dimensional graviton are 2, I, 0, -1, 

and -2. 

nerally, the light-cone symmetry that leaves the direction 

of motion of a massless particle unchanged in D dimensions is 

SO(D - 2), and the D-dimensional helicity corresponds to the multiplets 

(or representations) of SO(D - 2). For example, the graviton 

has D(D- 3)/2 independent degrees of freedom in D dimensions; 

thus the graviton in eleven dimensions belongs to a 44-component 

representation of SO(9). The SO(2) of the 4-dimensional helicity is 

inside the S0(9), so the forty-four componcnts of the graviton in 

eleven dimensions carry labels of 4-dimensional helicity as follows: 

one component of helicity 2, seven of helicity I, twenty-eight of 

helicity 0, seven of helicity -I and one of helicity -2. (The components 

of the graviton in eleven dimensions then correspond to the 

graviton, seven massless vector bosons, and twenty-eight scalars in 

four dimensions.) 

The analysis for massive particles in D dimensions proceeds in 

exactly the same way, except the helicity symmetry is the one that 

leaves a resting particle at rest. Thus, the massive helicity symmetry 

is SO(D- I). (For example, SO(3) describes the spin of a massive 

particle in ordinary 4-dimensional space-time.) These results are 

summarized in Fig. 5 of the main text. 

nify all known interactions. 

It turns out that the 4-dimensional fields 

nplied by the I I-dimensional gravitational 

eld resemble the solution to the 5-dimenonal 

Kaluza-Klein case, except that the 

-avitational field now corresponds to many 

lore 4-dimensional fields. There are methds 

of dimensional reduction for group 

ianifolds and coset spaces, and the zero 

lodes include a vector boson for each symietry 

of the extra dimensions. Thus, in the 

I + 7)-dimensional example mentioned 

>ow. there is a complete set of vector bosns 

for the standard model. At first sight this 

lode1 appears to provide an attractive uni- 

Eation ofall the interactions of the standard 

iodel; it explains the origins of the local 

mmetries of the standard model as space- 

time symmetries of gravity in eleven 

dimensions. 

Unfortunately, this 1 I-dimensional 

Kaluza-Klein theory has some shortcomings. 

Even with the complete freedom consistent 

with quantum field theory to add fermions, it 

cannot account for the parity violation seen 

in the weak neutral-current interactions of 

the electron. Witten’ has presented very general 

arguments that no I I-dimensional 

Kaluza-Klein theory will ever give the correct 

electroweak theory. 

Supersymmetry and Gravity in 

Four Dimensions 

We return from our excursion into higher 

dimensions and discuss extending gravity 

not by enlarging the space but rather by 

enlarging the symmetry. The local PoincarC 

symmetry of Einstein’s gravity implies the 

massless spin-2 graviton; our present goal is 

to extend the Poincari: symmetry (without 

increasing the number ofdirnensions) so that 

additional fields are grouped together with 

the graviton. However, this cannot be 

achieved by an ordinary (Lie group) symmetry: 

the graviton is the only known 

elementary spin-2 field, and the local symmetries 

of the standard model are internal 

symmetries that group together particles of 

the same spin. Moreover, gravity has an 

exceptionally weak interaction, so if the 

graviton carries quantum numbers of symmetries 

similar to those of the standard 

model, it will interact too strongly. We can 

87

accommodate these facts if the graviton is a 

singlet under the internal symmetry, but then 

its multiplet in this new symmetry must 

include particles of other spins. Supersymmetry’ 

is capable of fulfilling this requirement. 

Four-Dimensional Supersymmetry. Supersymmetry 

is an extension of the Poincart 

symmetry, which includes the six Lorentz 

generators M,,” and four translations P,,. The 

Poincart generators are boson operators, so 

they can change the spin components of a 

massive field but not the total spin. The 

simplest version of supersymmetry adds fermionic 

generators Q, to the Poincart generators; 

Qa transforms like a spin-% field 

under Lorentz transformations. (The index a 

is a spinor index.) To satisfy the Pauli exclusion 

principle, fermionic operators in 

quantum field theory always satisfy anticommutation 

relations, and the supersymmetry 

generators are no exception. In the algebra 

the supersymmetry generators Qo anticommute 

to yield a translation 

where fp is the energy-momentum 4-vector 

and the yg, are matrix elements of the Dirac 

y matrices. 

The significance of the fermionic generators 

is that they change the spin ofa state 

or field by t%; that is, supersymmetry unifies 

bosons and fermions. A multiplet of 

“simple” supersymmetry (a supersymmetry 

with one fermionic generator) in four 

dimensions is a pair of particles with spins J 

and .I- %; the supersymmetry generators 

transform bosonic fields into fermionic 

fields and vice versa. The boson and fermion 

components are equal in number in all supersymmetry 

multiplets relevant to particle theories. 

We can construct larger supersymmetries 

by adding more fermionic generators to the 

PoincarC symmetry. “N-extended” supersymmetry 

has N fermionic generators. By 

applying each generator to the state of spin J, 

88 

we can lower the helicity up to N times. 

Thus, simple supersymmetry, which lowers 

the helicity just once, is called N = 1 supersymmetry. 

N = 2 supersymmetry can lower 

the helicity twice, arid the N = 2 multiplets 

have spins J, J- %, and J - 1. There are 

twice as many J - lh states as J or J - 1, so 

that there are equal numbers of fermionic 

and bosonic states. The N = 2 multiplet is 

made up of two N == 1 multiplets: one with 

spins J and J- I12 and the other with spins 

J- %and J- 1. 

In principle, this construction can be extended 

to any N, but in quantum field theory 

there appears to be a limit. There are serious 

difficulties in constructing simple field theories 

with spin 5/2 or higher. The largest 

extension with spin 2 or less has N = 8. In N 

= 8 extended supersymmetry, there is one 

state with helicity of 2, eight with 3/2, 

twenty-eight with 1, fifty-six with 1/2, seventy 

with 0, fifty-six with -l/2, twenty-eight 

with -1, eight with 3/2 and one with -2. 

This multiplet with 256 states will play an 

important role in the supersymmetric theories 

of gravity or supergravity discussed 

below. Table 2 shows the states of N-extended 

supersymmetry. 

Theories with Supersymmetry. Rather ordinary-looking 

Lagrangians can have supersymmetry. 

For example, there is a Lagrangian 

with simple global supersymmetry 

in four dimensions with a single Majorana 

fermion, which has one component with 

helicity +1/2, one with helicity -l/2, and 

two spinless particles. Thus, there are two 

bosonic and two fermionic degrees of freedom. 

The supersymmetry not only requires 

the presence of both fermions and bosons in 

the Lagrangian but also restricts the types of 

interactions, requires that the mass 

parameters in the multiplet be equal, and 

relates some other parameters in the Lagrangian 

that would otherwise be unconstrained. 

The model just described, the Wess- 

Zumino model,3 is so simple that it can be 

written down easily in conventional field 

notation. However, more realistic supersym- 

metnc Lagrangians take pages to write down, 

We will avoid this enormous complication 

and limit our discussion to the spectra 01 

particles in the various theories. 

Although supersymmetry may be an exaci 

symmetry of the Lagrangian, it does not appear 

to be a symmetry of the world because 

the known elementary particles do not have 

supersymmetric partners. (The photon and a 

neutrino cannot form a supermultiplet because 

their low-energy interactions are different.) 

However, like ordinary symmetries, 

the supersymmetries of the Lagrangian do 

not have to be supersymmetries of the 

vacuum: supersymmetry can be spontaneously 

broken. The low-energy predictions 

of spontaneously broken supersymmetric 

models are discussed in “Supersymmetry 

at 100 GeV.” 

mediates the gravitational interaction; lowmass 

spin-% fermions dominate low-energy 

phenomenology; and spinless fields provide 

the mechanism for spontaneous symmetq 

breaking. All these fields are crucial to thc 

standard model, although there seems to br 

no relation among the fields ofdifferent spin 

A spin of 3/2 is not required phenomenologi 

cally and is missing from the list. If thl 

supersymmetry is made local, the resultin, 

theory is supergravity, and the spin-2 gravi 

ton is accompanied by a “gravitino” wit1 

spin 312. 

Local supersymmetry can be imposed on 

theory in a fashion formally similar to th 

local symmetries of the standard model, ex 

cept for the additional complications due t 

the fact that supersymmetry is a space-tim 

symmetry. Extra gauge fields are required t 

compensate for derivatives of the spact 

time-dependent parameters, so, just as fc 

ordinary symmetries, there is a gauge partic

Toward a Unified Theoty 

number of states of each helicity for each possible supermultiplet containing a 

graviton but with spin I 2. Simple supergravity (N = 1) has a graviton and 

gravitino. N = 4 supergravity is the s 

The overlap of the multi 

gives rise to large addi 

supergravities have the same list o 

symmetry implies that the N = 7 theory must have two multiplets (as for N < 

7), whereas N = 8 is the first and last case for which particle-antiparticle 

symmetry can be satisfied by a single multiplet. 

Helicity 1 2 3 4 5 6 701-8 

2 1 1 1 

312 1 2 3 

1 1 3 

1 12 

0 

-112 

-I 1 3 

-312 1 2 3 

-2 1 1 1 

Total 4 8 16 

ymmetry transformation. However, the 

auge particles associated with the supersymietry 

generators must be fermions. Just as 

he graviton has spin 2 and is associated with 

ie local translational symmetry, the gravino 

has spin 312 and gauges the local superymmetry. 

The graviton and gravitino form 

simple (N = 1) supersymmetry multiplet. 

‘his theory is called simple supergravity and 

i interesting because it succeeds in unifying 

le graviton with another field. 

The Lagrangian of simple supergravity4 is 

n extension of Einstein’s Lagrangian, and 

ne recovers Einstein’s theory when the 

-avitational interactions ofthe gravitino are 

,nored. This model must be generalized to a 

lore realistic theory with vector bosons, 

1 

4 

6 

4 

2 

4 

6 

4 

I 

32 

1 1 

5 6 

10 16 

11 26 

10 30 

11 26 

10 16 

5 6 

1 1 

64 128 

1 

8 

28 

56 

70 

56 

28 

8 

1 

256 

spin-% fermions, and spinless fields to be of 

much use in particle theory. 

The generalization is to Lagrangians with 

extended local supersymmetry, where the 

largest spin is 2. The extension is extremely 

complicated. Nevertheless, without much 

work we can surmise some features of the 

extended theory. Table 2 shows the spectrum 

of particles in N-extended supergravity. 

We start here with the largx extended 

supersymmetry and investigate whether it 

includes the electroweak and strong interactions. 

In N = 8 extended supergravity the 

spectrum is just the N = 8 supersymmetric 

multiplet of 256 helicity states discussed 

before. The massless particles formed from 

these states include one graviton, eight gravi- 

tinos, twenty-eight vector bosons, fifty-six 

fermions, and seventy spinless fields. 

N= 8 supergravity’ is an intriguing theory. 

(Actually, several different N = 8 supergravity 

Lagrangians can be constructed.) It 

has a remarkable set of internal symmetries, 

and the choice of theory depends on which of 

these symmetries have gauge particles associated 

with them. Nevertheless, supergravity 

theories are highly constrained and 

we can look for the standard model in each. 

We single out one of the most promising 

versions of the theory, describe its spectrum, 

and then indicate how close it comes to 

unifying the electroweak, strong, and gravitational 

interactions. 

In the N = 8 supergravity of de Wit- 

Nicolai theory6 the twenty-eight vector bosons 

gauge an SO(8) symmetry found by 

Cremmer and Julia.’ Since the standard 

model needs just twelve vector bosons, 

twenty-eight would appear to be plenty. In 

the fermion sector, the eight gravitinos must 

have fairly large masses in order to have 

escaped detection. Thus, the local supersymmetry 

must be broken, and the gravitinos 

acquire masses by absorbing eight spin-% 

fermions. This leaves.56 -’8 = 48 spin-% 

fermion fields. For the quarks and leptons in 

the standard model, we need forty-five fields, 

so this number also is sufficient. 

The next question is whether the quantum 

numbers of SO(8) correspond to the electroweak 

and strong quantum numbers and 

the spin4 fermions to quarks and leptons. 

This is where the problems start: if we 

separate an SU(3) out of the SO(8) for QCD, 

then the only other independent ’interactions 

are two local phase symmetries of U( I ) X 

U( I), which IS not large enough to include 

the SU(2) X U( I ) of the electroweak theory. 

The rest of the SO@) currents mix the SU(3) 

and the two U( 1)’s. Moreover, many of the 

fifty-six spin-% fermion states (or forty-eight 

if the gravitinos are massive) have the wrong 

SU(3) quantum numbers to be quarks and 

leptons.’ Finally, even if the quantum 

numbers for QCD were right and the electroweak 

local symmetry were present, the 

weak interactions could still not be ac- 

89

counted for. No mechanism in this theory 

can guarantee the almost purely axial weak 

neutral current of the electron. Thus this 

interpretation of N = 8 supergravity cannot 

be the ultimate theory. Nevertheless, this is a 

model of unification, although it gave the 

wrong sets of interactions and particles. 

Perhaps the 256 fields do not correspond 

directly to the observable particles, but we 

need a more sophisticated analysis to find 

them. For example, there is a “hidden” local 

SU(8) symmetry, independent of the gauged 

SO(8) mentioned above, that could easily 

contain the electroweak and strong interactions. 

It is hidden in the sense that the Lagrangian 

does not contain the kinetic energy 

terms for the sixty-three vector bosons of 

SU(8). These sixty-three vector bosons are 

composites of the elementary supergravity 

fields, and it is possible that the quantum 

corrections will generate kinetic energy 

terms. Then the fields in the Lagrangian do 

not correspond to physical particles; instead 

the photon, electron, quarks, and so on, 

which look elementary on a distance scale of 

present experiments, are composite. Unfortunately, 

it has not been possible to work 

out a logical derivation of this kind of result 

for N =- 8 supergravity.* 

In summary, N = 8 supergravity may be 

correct, but we cannot see how the standard 

model follows from the Lagrangian. The 

basic fields seem rich enough in structure to 

account for the known interactions, but in 

detail they do not look exactly like the real 

world. Whether N = 8 supergravity is the 

wrong theory, or is the correct theory and we 

simply do not know how to interpret it, is not 

yet known. 

Supergravity in Eleven 

Dimensions 

The apparent phenomenological shortcomings 

of N = 8 supergravity have been 

known for some time, but its basic mathematical 

structure is so appealing that many 

theorists continue to work on it in hope that 

90 

some variant will give the electroweak and 

strong interactions. One particularly interesting 

development is the generalization of N = 

8 supergravity in four dimensions to simple 

(N = I ) supergravity in eleven dimensions.’ 

This generalization combines the ideas of 

Kaluza-Klein theories with supersymmetry. 

The formulation and dimensional reduction 

of simple supergravity in eleven 

dimensions requires most of the ideas already 

described. First we find the fields of l l- 

dimensional supergravity that correspond to 

the graviton and gravitino fields in four 

dimensions. Then we describe the components 

of each of the 1 I-dimensional fields. 

Finally, we use the harmonic expansion on 

the extra seven dimensions to identify the 

zero modes and pyrgons. For a certain 

geometry of the extra dimensions, the 

dimensionally reduced, 1 I-dimensional 

supergravity without pyrgons is N = 8 supergravity 

in four dimensions; for other 

geometries we find new theories. We now 

look at each of these steps in more detail. 

In constructing the 1 I-dimensional fields, 

we begin by recalling that the helicity symmetry 

of a massless particle is SO(9) and the 

spin components are classified by the multiplets 

of SO(9). The multiplets of SO(9) are 

either fermionic or bosonic, which means 

that all the four-dimensional helicities are 

either integers (bosonic) or half-odd integers 

(fermionic) for all the components in a single 

multiplet. The generators independent of the 

SO(2) form an S0(7), which is the Lorentz 

group for the extra seven dimensions. Thus, 

the SO(9) multiplets can be expressed in 

terms of a sum of multiplets of SO(7) X 

S0(2), which makes it possible to reduce I I- 

dimensional spin to +dimensional spin. 

The fields of 1 I-dimensional, N= I supergravity 

must contain the graviton and gravitino 

in four dimensions. We have already 

mentioned in the sidebar that the graviton in 

eleven dimensions has forty-four bosonic 

components. The smallest SO(9) multiplet of 

I I-dimensional spin that yields a helicity of 

312 in four dimensions for the gravitinos has 

128 components, eight components with 

helicity 3/2, fifty-six with 1/2, fifty-six with 

ofN= 8 supergravity in four dimen~ions.~ In 

this case each of the components is expanded 

in a sevenfold Fourier series, one series for 

each dimension just as in Eq. 1 in the sidebar, 

except that ny is replaced by Cn,v,. The 

dimensional reduction consists of keeping 

only those fields that do not depend on any 

y,, that is, just the 4-dimensional fields corresponding 

to n, = n2 =. . . = n, = 0. Thus, 

there is one zero mode (massless field in four 

dimensions) for each component. The 

pyrgons are the 4-dimensional fields witb 

any n, # 0, and these are omitted in the 

dimensional reduction. 

The I I-dimensional theory has a simple 

Lagrangian, whereas the 4-dimensiona1, N = 

8 Lagrangian takes pages IO write down. Ir 

fact the N = 8 Lagrangian was first derived ir 

this way.’ It is easy to be impressed by : 

formalism in which everything looks simple 

This is the first of several reasons to tak 

seriously the proposal that the extri 

dimensions might be physical, not just I 

mathematical trick. 

The seven extra dimensions of the 11 

dimensional theory must be wound up into


Table 3 

The relation of si (N= 1) supergr 

supergravity in fo ensions. The 256 ents of the mass 

11-dimensional, N = 1 supergravity fa 

SO(9). The members of these multiplets have definite helicities in four 

dimensions. The count of helicity states is given in terms of the size of SO(7) 

multiplets, where SO(7) is the Lorentz symmetry of the seven extra dimensions 

in the 11-dimensional theory. 

4-Dimensional Helicitv 

n 2 312 1 112 0 -112 -1 -312 -2 

44 1 7 14-27 7 1 

84 21 7+35 21 

I28 8 8+48 8+48 8 

Total 1 8 28 56 70 56 28 8 1 

ase described above assumes that the little 

all is a 7-torus, which is the group manifold 

lade of the product of seven phase symietries. 

As a Kaluza-Klein theory, the seven 

ector bosons in the graviton (Table 3) gauge 

lese seven symmetries. Since the twentyight 

vector bosons of N= 8 supergravity can 

e the gauge fields for a local S0(8), it is 

iteresting to see if we can redo the dimenional 

reduction so that I I-dimensional 

upergravity is a Kaluza-Klein theory for 

0(8), the de Wit-Nicolai theory. Indeed, 

lis is possible. If the extra dimensions are 

ssumed to be the 7-sphere, which is the 

oset space SO(S)/S0(7), the vector bosons 

o gauge SO(8).io This is, perhaps, the ulmate 

Kaluza-Klein theory, although it does 

ot contain the standard model. The main 

ifference between the 7-torus and coset 

paces is that for coset spaces there is not 

ecessarily a one-to-one correspondence beween 

components and zero modes. Some 

omponents may have several zero modes, 

chile others have none (recall Fig. 5). 

There are other manifolds that solve the 

I-dimensional supergravity equations, aliough 

we do not describe them here. The 

iternal local symmetries are just those of the 

extra dimensions, and the fermions and bosons 

are unified by supersymmetry. Thus, 1 1- 

dimensional supergravity can be dimensionally 

reduced to one of several different 4- 

dimensional supergravity theories, and we 

can search through these theories for one that 

contains the standard model. Unfortunately, 

they all suffer phenomenological shortcomings. 

Eleven-dimensional supergravity contains 

an additional error. In the solution where the 

seven extra dimensions are wound up in a 

little ball, our 4-dimensional world gets just 

as compacted: the cosmological constant is 

about 120 orders of magnitude larger than is 

observed experimentally.” This is the cosmological 

constant problem at its worst. Its 

solution may be a major breakthrough in the 

search for unification with gravity. Meanwhile, 

it would appear that supergravity has 

given us the worst prediction in the history of 

modern physics! 

Superstrings 

In view of its shortcomings, supergravity 

is apparently not the unified theory of all 

elementary particle interactions. In many 

ways it is close to solving the problem, but a 

theory that is correct in all respects has not 

been found. The weak interactions are not 

exactly right nor is the list of spin4 fermions. 

There seems to be no good reason 

that the cosmological constant should be 

nearly or exactly zero as observed experimentally. 

The issue of the renormalizability 

of the quantum theory of gravity 

also remains unsolved. Supergravity improves 

the quantum structure of the theory 

in that the unwanted infinities are not as bad 

as in Einstein’s theory with matter, but 

troubles still appear. Newton’s constant is a 

fundamental parameter in the theory, and 4- 

fermion terms similar to those in Fermi’s 

weak interaction theory are still present. In N 

= 8 supergravity, which is the best case, the 

perturbation solution to the quantum field 

theory is expected to break down eventually. 

In spite of these difficulties we have 

reasons to be optimistic that supergravity is 

on the right track. It does unify gravity with 

some interactions and is almost a consistent 

quantum field theory. The line of generalization 

followed so far has led to theories that 

are enormous improvements, in a mathematical 

sense, over Einstein’s gravity. It 

would seem reasonable to look for generalizations 

beyond supergravity. 

Superstring theories may answer some of 

these questions. Just as the progress of supergravity 

was based on the systematic addition 

of fields to Einstein’s gravity, superstring 

theory can also be viewed in terms of the 

systematic addition of fields to supergravity. 

Although the formulation of superstring theory 

looks quite different from the formulation 

of supergravity, this may be partially 

due to its historical origin. 

Superstring theories were born from an 

early effort to find a theory of the strong 

interactions. They began as a very efficient 

means of understanding the long list of 

hadronic resonances. In particular, hadrons 

of high spin have been identified expenmentally. 

It is interesting that sets of hadrons of 

different spins but the same internal quantum 

numbers can be grouped together into 

91

“Regge trajectories.” Figure 6 shows examples 

of :Regge trajectories (plots of spin versus 

mass-squared) for the first few states of the A 

and N resonances; these resonances for 

hadrons of different spins fall along nearly 

straighl. lines. Such sequences appear to be 

general phenomena, and so, in the ’60s and 

early OS, a great effort was made to incorporate 

these results directly into a theory. 

The basic idea was to build a set of hadron 

amplitudes with rising Regge trajectories 

that satisfied several important constraints 

of quantum field theory, such as Lorentz 

invariance, crossing symmetry, the correct 

analytic properties, and factorization of resonance-pole 

residues. ’’ Although the theory 

was a prescription for calculating the 

amplitudes, these constraints are true of 

quantum field theory and are necessary for 

the theory to make sense. 

The constraints of field theory proved to 

be too much for this theory of hadrons. 

Something always went wrong. Some theories 

predicted particles with imaginary mass 

(tachyons) or particles produced with 

negative probability (ghosts), which could 

not be interpreted. Several theories had no 

logical difficulties, but they did not look like 

hadron theories. First of all, the consistency 

requirements forced them to be in ten 

dimensions rather than four. Moreover, they 

predicted massless particles with a spin of 2; 

no hadrons of this sort exist. These original 

superstring theories did not succeed in describing 

hadrons in any detail, but the solution 

of QCD may still be similar to one of 

them. 

In 1074 Scherk and Schwarzi3 noted that 

the quantum amplitudes for the scattering of 

the massless spin-2 states in the superstring 

are the same as graviton-graviton scattering 

in the simplest approximation of Einstein’s 

theory. They then boldly proposed throwing 

out the hadronic interpretation of the superstring 

and reinterpreting it as a fundamental 

theory of elementary particle interactions. It 

was easily found that superstrings are closely 

related to supergravity, since the states fall 

into supersymmetry multiplets and massless 

spin-2 particles are required.14 

.- 

E 

n 

v) 

1112 

912 

712 

512 

312 

1 I2 

Fig. 6. Regge trajectories in hadron physics. The neutron and proton (N(938)) 

on a linearly rising Regge trajectory with other isospin-% states: the N(l680) 

spin 5/2, the N(2220) of spin 9/2, and so on. This fact can be interpreted as meani 

that the N(1680), for example, looks like a nucleon except that the quarks are in 

F wave rather than a P wave. Similarly the isospin-3/2 A resonance at 1232 M 

lies on a trajectory with other isospin-3/2 states of spins 7/2, I I ~ 1512, , and so t 

The slope of the hadronic Regge trajectories is approximately (1 GeV/cZ)-’. 1 

slope of the superstring trajectories must be much smaller 

The theoretical development of superstrings 

is not yet complete, and it is not 

possible to determine whether they will finally 

yield the truly unified theory of all 

interactions. They are the subject of intense 

research today. Our plan here is to present a 

qualitative description of superstrings and 

then to discuss the types and particle spectra 

of superstring theories. 

Recent formulations of superstring theories 

are generalizations of quantum field 

theory.I5 The fields of an ordinary field t 

ory, such as supergravity, depend on 

space-time point at which the field 

evaluated. The fields of superstring the( 

depend on paths in space-time. At each n 

ment in time, the string traces out a path 

space, and as time advances, the str 

propagates through space forming a surf 

called the “world sheet.” Strings can 

closed, like a rubber band, or open, lik 

broken rubber band. Theories of both ty, 

92

__ 

- 

- , , 

i 


- ___I- 

(a) 

- ____ - ~ 

*2 

‘3 

___ , 

Fig. 7. Dynamics of closed strings. Thefigures show the string configurations at a 

sequence of times (in two dimensions instead of ten). In Fig. 7(a) a string in motion 

from times t, to t2 traces out a world sheet. Figure 7(b) shows the three closed string 

interaction, where one string at t, undergoes a change of shape until itpinches off at 

apoint at time t2 (the interaction time). At time t3 two strings arepropagating away 

from the interaction region. 

are promising, but the graviton is always 

associated with closed strings. 

Before analyzing the motion of a superstring, 

we must return to a discussion of 

space-time. Previously, we described extensions 

of space-time to more than four 

dimensions. In all those cases coordinates 

were numbers that satisfied the rules of ordinary 

arithmetic. Yet another extension of 

space-time, which is useful in supergravity 

and crucial in superstring theory, is the addition 

to space-time of “supercoordinates” 

that do not satisfy the rules of ordinary arithmetic. 

Instead, two supercoordinates ea and 

i 

Bp satisfy anticommutation relations 0,0, + 

= 0, and consequently 0,0, (with no sum 

on a) = 0. Spaces with this kind of additional 

coordinatc are called superspaces.’6 

At first encounter superspaces may appear 

to be somewhat silly constructions. Nevertheless, 

much of the apparatus of differential 

geometry of manifolds can be extended to 

superspaces, so applications in physics may 

exist. It is possible to define fields that depend 

on the coordinates of a superspace. 

Rather naturally, such fields are called superfields. 

Let us apply this idea to supergravity, 

which is a field theory of both fermionic and 

bosonic fields. The supergravity fields can be 

further unified if they are written as a smaller 

number of superfields. Supergravity Lagrangians 

can then be written in terms of 

superfields; the earlier formulations are recovered 

by expanding the superfields in a 

powcr series in the supercoordinates. The 

anticommutation rule 0,0, = 0 leads to a 

finite number of ordinary fields in this expansion. 

The motion of a superstring is described 

by the motion of each space-time coordinate 

and supercoordinate along the string; thus 

the motion of the string traces out a “world 

sheet” in superspace. The full theory describes 

the motions and interactions of 

superstrings. In particular, Fig. 7 shows the 

basic form of the three closed superstring 

interactions. All other interactions of closed 

strings can be built up out of this one kind of 

intera~tion.’~ Needless to say, the existence 

of only one kind of fundamental interaction 

would severely restrict theories with only 

closed strings. 

There is a direct connection between the 

quantum-mechanical states of the string and 

the elementary particle fields of the theory. 

The string, whether it is closed or open, is 

under tension. Whatever its source, this tension, 

rather than Newton’s constant, defines 

the basic energy scale of the theory. To first 

approximation each point on the string has a 

force on it depending on this tension and the 

relative displacement between it and 

neighboring points on the string. The prob- 

93

lem of unravelling this infinite number of 

harmonic oscillators is one of the most 

famous problems of physics. The amplitudes 

of the Fourier expansion of the string displacement 

decouple the infinite set of harmonic 

oscillators into independent Fourier 

modes. These Fourier modes then correspond 

to the elementary-particle fields. 

The quantum-mechanical ground state of 

this infinite set of oscillators corresponds to 

the fields of IO-dimensional supergravity. 

Ten space-time dimensions are necessary to 

avoid tachyons and ghosts. The excited 

modes of the superstring then correspond to 

the new fields being added to supergravity. 

The harmonic oscillator in three 

dimensions can provide insight into the 

qualitative features of the superstring. The 

maximum value of the spin of a state of the 

harmonic oscillator increases with the level 

of the excitation. Moreover, the energy 

necessary to reach a given level increases as 

the spring constant is increased. The superstring 

is similar. The higher the excitation of 

the string, the higher are the possible spin 

values (now in ten dimensions). The larger 

94 

the string tension, the more massive are the 

states ofan excited level. 

The consistency requirements restrict 

superstring theories to two types. Type I 

theories have IO-dimensional N = I supersymmetry 

and include both closed and open 

strings and five kinds of string interactions. 

Nothing more will be said here about Type I 

theories, although they are extremely interesting 

(see Refs. 14 and 1 s). 

Type I1 theories have N = 2 supersymmetry 

in ten dimensions and accommodate 

closed strings only. There are two N = 2 

supersymmetry multiplets in ten dimensions, 

and each corresponds to a Type I1 

superstring theory. We will now describe 

these two superstring theories. 

The Type IIA ground-state spectrum is the 

one that can be derived by dimensional reduction 

of simple supergravity in eleven 

dimensions to N = 2 supergravity in ten 

dimensions. Thus, if we continue to reduce 

from ten to four dimensions with the 

hypothesis that the extra six dimensions 

form a 6-torus, we will obtain N = 8 supergravity 

in four dimensions. The superstring 

theory adds both pyrgons and Regge recurrences 

to the 256 N = 8 supergravity fields, 

but it has been possible (and often simpler) 

to investigate several aspects of supergravity 

directly from the superstring theory. 

The classification ofthe excited IO-dimensional 

string states (or elementary fields of 

the theory) is complicated by the description 

of spin in ten dimensions. However, the 

analysis does not differ conceptually from 

the analysis of spin for 11-dimensional 

supergravity. The massless states, which 

form the ground state of the superstring, are 

classified by multiplets of SO@), and the 

excitations of the string are massive fields in 

ten dimensions that belong to multiplets of 

SO(9). The ground-state fields of the Type 

IIA superstring are found in Table 4. 

Thz Type IIB ground-state fields cannot 

be derived from 1 I-dimensional supergravity. 

Instead the theory has a useful phase 

symmetry in ten dimensions. The fields 

listed as occumng twice in Table 4 carry 

nonzero values of the quantum number associated 

with U( I). So far, the main application 

of the U(1) symmetry has been the


.. 

Y 

1 SO(9) Multiplets 1 

I 

712 - 

( 10-Dimensional Massive Spin States),,,o 

I 

10-Dimensional Mass2 

Fig. 8. The ground state andfirst Regge recurrence of fermionic slates in the 10- 

dimensional Type IIB superstring theory. There are a total of 256 fermionic and 

bosonic stales in the ground state. (The 56, contains the gravitino.) The first 

excited states contain 65,536 component fields. Half of these are fermions. (Each 

representation of the fermions shown above appears twice.) 

derivation of the equations of motion for the 

ground-state fields.” It will certainly have a 

crucial role in the future understanding of 

Type IIB superstrings. 

The quantum-mechanical excitations of 

the superstring correspond to the Regge recurrences, 

which are massive in ten 

dimensions; they belong to multiplets of 

SO(9). Thus, it is possible to fill in a diagram 

similar to Fig. 6, although the huge number 

of states makes the results look complicated. 

We give a few results to illustrate the 

method. 

The sets of Regge recurrences in Type IIA 

and IIB are identical. In Figure 8 we show the 

first recurrence of the fermion trajectories. 

(Note that only one-half of the 32,768 fer- 

mionic states of this mode are shown. The 

boson states are even messier.) The first excited 

level has a total of65,536 states, and the 

next two excited levels have 5,308,416 and 

235,929,600 states, respectively, counting 

both fermions and bosons. (Particle 

physicists seem to show little embarrassment 

these days over adding a few fields to a 

theory!) 

The component fields in ten dimensions 

can now be expanded into 4-dimensional 

fields as was done in supergravity. Besides 

the zero modes and pyrgons associated with 

the ground states, there will be infinite ladders 

of pyrgon fields associated with each of 

the fields of the excited levels of the superstring. 

I 

The zero modes in four dimensions have 

been investigated only for the 6-torus; in this 

case all the zero modes come from the 

ground states. There is one zero mode for 

each component field, since the dimensional 

reduction is done as a 6-dimensional Fourier 

series on the 6-torus. The answers for other 

geometries are not yet known. It may be that 

many more fields become zero modes (or 

have nearly zero mass) in four dimensions 

when the dimensional reduction is studied 

for other spaces. An important problem is 

the analysis of superstrings on curved spaces, 

which has not yet been definitively studied. 

Although not much progress has been 

made toward understanding the phenomenology 

of these superstring theories, there 

has been some formal progress. The theory 

described here may be a quantum theory of 

gravity. (It may take all those new fields to 

obtain a renormalizable theory.) Although 

local symmetries can be ruined by anomalies, 

Type I1 (and several Type I) superstrings 

satisfy the constraints. Also, the one-loop 

calculation is finite; there are no candidates 

for counter terms, so the theory may be 

finite. Of course, this promising result needs 

support from higher order calculations. 

These results give some encouragement 

that superstrings may solve some long-standing 

problems in particle theory; whether they 

will lead to the ultimate unification of all 

interactions remains to be seen. 

Postscript 

The search for a unified theory may be 

likened to an old geography problem. Columbus 

sailed westward to reach India believing 

the world had no edge. By analogy, we 

are searching for a unified theory at shorter 

and shorter distance scales believing the 

microworld has no edge. Perhaps we are 

wrong and space-time is not continuous. Or 

perhaps we are only partly wrong, like Columbus, 

and will discover something new, 

but something consistent with what we already 

know. Then again, we may finally be 

right on course to a theory that unifies all 

Nature’s interactions. 

95

AUTHORS 

Richard C. Slansky has a broad background in physics with more than a 

taste ofmetaphysics. He received a B.A. in physics from Harvard in 1962 

and then spent the following year as a Rockefeller Fellow at Harvard 

Divinity School. Dick then attended the University of California, 

Berkeley, where he received his Ph.D. in physics in 1967. A two-year 

postdoctoral stint at the California Institute of Technology was followed 

by five years as Instructor and Assistant Professor at Yale University 

(1969-1974). Dick joined the Laboratory in 1974 as a Staff Member in the 

Elementary Particles and Field Theory group of the Theoretical Division, 

where his interests encompass phenomenology, high-energy physics, and 

the early universe. 

References 

I. For a modern description of Kaluza-Klein theories, see Edward Witten, Nuclear 

Physics B186(1981):412 and A. Salam and J. Strathdee, Annals of Physics 

141( 1982):3 16. 

2. Two-dimensional supersymmetry was discovered in dual-resonance models by 

P. Ramond, Physical Review D 3( 197 1):2415 and by A. Neveu and J. H. Schwarz, 

Nuclear Physics B3 I( 197 1):86. Its four-dimensional form was discovered by Yu. 

A. Gol’fand and E. P. Likhtman, Journal of Experimental and Theoretical 

Physics Letters I3( 1971):323. 

3. J. Wess and B. Zumino, Physics Letters 49B(1974):52 and Nuclear Physics 

B70( 1974):39. 

4. Daniel Z. Freedman, P. van Nieuwenhuizen, and S. Farrara, Physical Review D 

13( 1976):3214; S. Deser and B. Zumino, Physics Letters 62B( 1976):335; Daniel 

Z. Freedman and P. van Nieuwenhuizen, Physical Review D 14( 1976):912. 

5. E. Cremmer and B. Julia, Physics Letters 80B( 1982):48 and Nuclear Physics 

BI 59( 1979): 14 I. 

96


6. B. de Wit and H. Nicolai, Physics Letters 108B(1982):285 and Nuclear Physics 

B208( 1982):323. 

7. This shortage of appropriate low-mass particles was noted by M. Cell-Mann in a 

talk at the 1977 Spring Meeting of the American Physical Society. 

8. J. Ellis, M. Gaillard, L. Maiani, and B. Zumino in Unification ofthe Fundamental 

Particle Interactions, S. Farrara, J. Ellis, and P. van Nieuwenhuizen, editors 

(New YorkPlenum Press, 1980), p. 69. 

9. E. Cremmer, B. Julia, and J. Scherk, Physics Letters 76B( 1978):409. Actually, the 

N = 8 supergravity Lagrangian in four dimensions was first derived by 

dimensionally reducing the N = 1 supergravity Lagrangian in eleven dimensions 

to N = 8 supergravity in four dimensions. 

10. M. J. Duff in Supergravity 81, S. Farrara and J. G. Taylor, editors (London: 

Cambridge University Press, 1982), p. 257. 

11. Peter (3.0. Freund and Mark A. Rubin, Physics Letters 97B( 1983):233. 

12. “Dual Models,” Physics Reports Reprint, Vol. I, M. Jacob, editor (Amsterdam: 

North-Holland, 1974). 

13. J. Scherk and John H. Schwarz, Nuclear Physics B8 I( 1974): I 18. 

14. For a history of this development and a list of references, see John H. Schwarz, 

Physics Reports 89(1982):223 and Michael B. Green, Surveys in High Energy 

Physics 3( 1983): 127. 

15. M. B. Green and J. H. Schwarz, Caltech preprint CALT-68-1090, 1984. 

16. For detailed textbook explanations of superspace, superfields, supersymmetry, 

and supergravity see S. James Gates, Jr., Marcus T. Grisaru, Martin Rocek, and 

Warren Siegel, Superspace: One Thousand and One Lessons in Supersymmetry 

(Reading, Massachusetts: Benjamin/Cummings Publishing Co., Inc., 1983) and 

Julius Wess and Jonathan Bagger, Supersymmetry and Supergravity (Princeton, 

New Jersey:Princeton University Press, 1983). 

17. John H. Schwarz, Nuclear Physics B226( 1983):269; P. S. Howe and P. C. West, 

1 Nuclear Physics B238( 1984): 18 1. 

97

Supersymmetrj 

S upersymmetry is a symmetry that connects particles of integral and half-integral spix 

Invented about ten years ago by physicists in Europe and the Soviet Union, supersymmetr 

was immediately recognizled as having amazing dynamical properties. In particula 

this symmetry provides a rational framework for unifying all the known forces betwee 

elementary particles-tlhe strong, weak, electromagnetic, and gravitational. Indeed, 

may also unify the separate concepts of matter and force into one comprehensiv 

framework. 

In the supersymmetric world depicted here, each boson pairs with a fermion partne, 

98 

\

There are two types of symmetries in 

nature: external (or space-time) symmetries 

and internal symmetries. Examples of internal 

symmetries are the symmetry of isotopic 

spin that identifies related energy levels of 

the nucleons (protons and neutrons) and the 

more encompassing SU(3) X SU(2) X U( 1) 

symmetry of the standard model (see “Particle 

Physics and the Standard Model”). 

Operations with these symmetries do not 

change the space-time properties of a particle. 

External symmetries include translation 

invariance and invariance under the Lorentz 

transformations. Lorentz transformations, 

in turn, include rotations as well as the 

special Lorentz transformations, that is, a 

“boost” or a change in the velocity of the 

frame of reference. 

Each symmetry defines a particular operation 

that does not affect the result of any 

experiment. An example of a spatial translation 

is to, say, move our laboratory (accelerators 

and all) from Chicago to New 

Mexico. ’We are, of zourse, not surprised that 

the resull of any experiment is unaffected by 

the move, and we say that our system is 

translationally invariant. Rotational invariance 

is similarly defined with respect to 

rotating our apparatus about any axis. Invariance 

under a special Lorentz transformation 

corresponds to finding our results unchanged 

when our laboratory, at rest in our 

reference frame, is replaced by one moving at 

a constant velocity. 

Corresponding to each symmetry operation 

is a quantity that is conserved. Energy 

and momentum are conserved because of 

time and space-translational invariance, respectively. 

The energy of a particle at rest is 

its mass (E = mc2). Mass is thus an intrinsic 

property of a particle that is conserved because 

of invariance of our system under 

space-time translations. 

Spin. Angular momentum conservation is a 

result of ILorentz invariance (both rotational 

and special). Orbital angular momentum refers 

to the angular momentum ofa particle in 

motion, whereas the intrinsic angular 

100 

momentum of a particle (remaining even at 

rest) is called spin. (Particle spin is an external 

symmetry, whereas isotopic spin, 

which is not based on Lorentz invariance, is 

not.) 

In quantum mechanics spin comes in integral 

or half-integral multiples of a fundamental 

unit h (h = h/2n: where h is Planck‘s 

constant). (Orbital angular momentum only 

comes in integral multiples of h.) Particles 

with integral values of spin (0, h, 2h, . . .)are 

called bosons, and those with half-integral 

spins (h/2, 3h/2, 5h,’2,. . .) are called fer-, 

mions. Photons (spin I), gravitons (spin 2), 

and pions (spin 0) are examples of bosons. 

Electrons, neutrinos, quarks, protons, and 

neutrons-the particles that make up ordinary 

matter-are all spin-% fermions. 

The conservation laws, such as those of 

energy, momentum, or angular momentum, 

are very useful concepts in physics. The following 

example dealing with spin and the 

conservation of angular momentum 

provides one small bit of insight into their 

utility. 

In the process of beta decay, a neutron 

decays into a proton, an electron, and an 

antineutrino. The antineutrino is massless 

(or very close to being massless), has no 

charge, and interacts only very weakly with 

other particles. In short, it is practically invisible, 

and for many years beta decay was 

thought to be simply 

n-p+ee- 

However, angular momentum is not conserved 

in this process since it is not possible 

for the initial angular momentum (spin 1/2 

for the neutron) to equal the final total 

angular momentum (spin 1/2 for the proton 

k spin 1/2 for the electron k an integral value 

for the orbital angular momentum). As a 

result, W. Pauli predicted that the neutrino 

must exist because its half-integral spin 

restores conservation of angular momentum 

to beta decay. 

There is a dramatic difference between the 

behavior of the two groups of spin-classified 

particles, the bosons and the fermions. This 

difference is clarified in the so-called spinstatistics 

theorem that states that bosons 

must satisfy commutation relations (the 

quantum mechanical wave function is symmetric 

under the interchange of identical 

bosons) and that fermions must satisfy anticommutation 

relations (antisymmetric wave 

functions). The ramification of this simple 

statement is that an indefinite number of 

bosons can exist in thp same place at the 

same time, whereas only one fermion can be 

in any given place at a given time (Fig. 1). 

Hence “matter” (for example, atoms) is 

made of fermions. Clearly, if you can’t put 

more than one in any given place at a time, 

then they must take up space. If they are also 

observable in some way, then this is exactly 

our concept of matter. Bosons, on the other 

hand, are associated with “forces.” For example, 

a large number of photons in the 

same place form a macroscopically observable 

electromagnetic field that affects 

charged particles. 

Supersymmetry. The fundamental property 

of supersymmetry is that it is a spacetime 

symmetry. A supersymmetry operation 

alters particle spin in half-integral jumps, 

changing bosons into fermions and vice 

versa. Thus supersymmetry is the first symmetry 

that can unify matter and force, the 

basic attributes of nature. 

If supersymmetry is an exact symmetry in 

nature, then for every boson of a given mass 

there exists a fermion of the same mass and 

vice versa; for example, for the electron there 

should be a scalar electron (selectron), for the 

neutrino, a scalar neutrino (sneutrino), for 

quarks, scalar quarks (squarks), and so forth. 

Since no such degeneracies have been observed, 

supersymmetry cannot be an exact 

symmetry of nature. However, it might be a 

symmetry that is inexact or broken. If so, it 

can be broken in either of two inequivalent 

ways: explicit supersymmetry breaking in 

which the Lagrangian contains explicit terms 

that are not supersymmetric, or spontaneous 

supersymmetry breaking in which the Lagrangian 

is supersymmetric but the vacuum 

is not (spontaneous symmetry breaking is 

l

Supersymmetry ai 100 Ge V 

t lclB 

Fig. 1. (a) An example of a symmetric wave function for apair of bosons and (b) an 

antisymmetric wave function for a pair of fermions, where the vector r represents 

the distance between each pair of identical particles. Because the boson wave 

function is symmetric with respect to exchange (yB (r) = yB(-r)), there can be a 

nonzero probablity (vi) for two bosons to occupy the same position in space (r = 

0), whereas for the asymmetric fermion wave function (yF (r) = -yF (-r)) the 

probability (vi) of two fermions occupying the same position in space must be 

zero. 

explained in Notes 3 and 6 of “Lecture 

Notes-From Simple Field Theories to the 

Standard Model”). Either way will lift the 

boson-fermion degeneracy, but the latter way 

will introduce (in a somewhat analogous way 

to the Higgs boson of weak-interaction symmetry 

breaking) a new particle, the Goldstone 

fermion. (We develop mathematically 

some of the ideas of this paragraph in 

“Supersymmetry and Quantum Mechanics”.) 

A question of extreme importance is the 

scale of supersymmetry breaking. This scale 

can be characterized in terms of the so-called 

supergup, the mass splitting between fermions 

and their bosonic partners (8’ = Mi - 

M;). Does one expect this scale to be of the 

order of the weak scale (- 100 GeV), or is it 

much larger? We will discuss the first 

possiblity at length because if supersymmetry 

is broken on a scale of order 100 GeV 

W 

+F 

there are many predictions that can be verified 

in the next generation of high-energy 

accelerators. The second possibility would 

not necessarily lead to any new low-energy 

consequences. 

We will also discuss the role gravity has 

played in the description of low-energy 

supersymmetry. This connection betweeen 

physics at the largest mass scale in nature 

(the Planck scale: Mpl = ( ~c/C~)’/~ = 1.2 X 

IOl9 GeV/c*, where CN is Newton’s gravitational 

constant) and physics at the low 

energies of the weak scale (Mw 83 GeV/c’ 

where Mw is the mass of the W boson responsible 

for weak interactions) is both 

novel and exciting. 

Motivations. Why would one consider 

supersymmetry to start with? 

First, supersymmetry is the largest 

possible symmetry of nature that can com- 

bine internal symmetries and space-time 

symmetries in a nontrivial way. This combination 

is not a necessary feature of supersymmetry 

(in fact, it is accomplished by extending 

the algebra of Eqs. 2 and 3 in “Supersymmetry 

and Quantum Mechanics” to include 

more supersymmetry generators and 

internal symmetry generators). However, an 

important consequence of such an extension 

might be that bosons and fermions in different 

representations of an internal symmetry 

group are related. For example, quarks 

(fermions) are in triplets in the strong-interaction 

group SU(3), whereas the gluons (bosons) 

are in octets. Perhaps they are all related 

in an extended supersymmetry, thus providing 

a unified description of quarks and their 

forces. 

Second, supersymmetry can provide a theory 

of gravity. If supersymmetry is global, 

then a given supersymmetry rotation must 

be the same over all space-time. However, if 

supersymmetry is local, the system is invariant 

under a supersymmetry rotation that 

may be arbitrarily different at every point. 

Because the various generators (supersymmetry 

charges, four-momentum translational 

generators, and Lorentz generators for 

both rotations and boosts) satisfy a common 

dgebra of commutation and anticommutation 

relations, consistency requires that all 

the symmetries are local. (In fact, the anticommutator 

of two supersymmetry generators 

is a translation generator.) Thus different 

points in space-time can transform in 

different ways; put simply, this can amount 

to acceleration between points, which, in 

turn, is equivalent to gravity. In fact, the 

theory of local translations and Lorentz 

transformations is just general relativity, that 

is, Einstein’s theory of gravity, and a supersymmetric 

theory of gravity is called supergravity. 

It is just the theory invariant under 

local supersymmetry. Thus, supersymmetry 

allows for a possible unification of all of 

nature’s particles and their interactions. 

These two motivations were realized quite 

soon after the advent of supersymmetry. 

They are possibilities that unfortunately 

have not yet led to any reasonable prediccontinued 

on page I06 

101

Supers * 

in 

Quantum 

nlcs 

I 

intend to develop here Some of the algebra pertinent to the 

basic concepts OfsupersYmmetry. f will do this by showing an 

analogy between the quantum-mechanical harmonic oscillator 

and a bosonic field and a further analogy between the 

quantum-mechanical spin-% particle and a fermionic field. One 

result of combining the two resulting fields will be to show that a 

“tower” ofdegeneracies between the states for bosons and fermions is 

a natural feature ofeven the simplest of supersymmetry theories. 

A supersymmetry operation changes bosons into fermions and 

vice versa, which can be represented schematically with the operators 

QL and Q, and the equations 

Qilboson) = Ifemion), 

and 

Qalfermion) = Iboson), . 

example, the operation of changing a fermion to a boson and back 

again results in changing the position of the fermion. 

If supersymmetry is an invariance of nature, then 

[H, e,] = 0 , 

that is, (2, commutes with the Hamiltonian H of the universe. Also, 

in this case, the vacuum is a supersymmetric singlet (Q,Ivac) = 0). 

Equations 1 through 3 are the basic defining equations of supersymmetry. 

In the form given, however, the supersymmetry is solely 

an external or space-time symmetry (a supersymmetry operation 

changes particle spin without altering any of the particle’s internal 

symmetries). An extended supersymmetry that connects external and 

internal symmetries can be constructed by . expanding . the number of 

operators of Eq. 2. However, for our purposes, we need not consider 

that complication. 

’L 

- 

1.1 

(3) 

In the simplest version of SuPersYmmetV, there are four such 

operators Or generators of supersymmetry (Qc~ and the krmitian 

conjugate QI with a = 1, 2). Mathematically, the generators are 

Lorentz spinors satisfying fermionic anticommutation relations 

The Harmonic Oscillator. In order to illustrate the consequences 

of Eqs. I through 3, we first need to review the quantum-mechanical 

treatment ofthe harmonic oscillator, 

The Hamiltonian for this system is 

where pJ’ is the energy-momentum four-vector bo = H, pi = threemomentum) 

and the o,, are two-by-two matrices that include the 

Pauli spin matrices o‘ (o,, = (1, 6’) where I = 1. 2, 3). Equation 2 

represents the unusual feature of this symmetry: the supersymmetry 

operators combine to generate translation in space and time. For 

where p and q are, respectively, the momentum and position 

coordinates of a nonrelativistic particle with unit mass and a 2n/m 

period of oscillation. The coordinates satisfy the quantum-mechanical 

commutation relation 

102

Supersymmetry at 100 Ge V 

he well-known solution to the harmonic oscillator (the set of 

nstates and eigenvalues of HOsJ is most conveniently expressed 

terms of the so-called raising and lowering operators, ut and a, 

respectively, which are defined as 

Finally, we find that 

that is, the states In) have ene 

perator for ut and lowering operator for a. 

counting operator since ut u In) = n I n). 

The Bosonic Field. There is a simple analogy between the quantum 

oscillator and th 

written as 

scillators ( ~ f up], , where p is 

d which satisfy the commutation relation 

terms of these operators, the Hamiltonian becomes 

with eigenstates 

t 7) 

(8) 

Ifscalar= 9 Iw,(ufup (13) 

with the summation taken over the individual oscillators p. 

round state of the free scalar quantum field is called the 

(it contains no scalar particles) and is described mathematically 

by the conditions 

up Ivac) = 0 

and (14) 

(vaclvac) = 1 . 

where N,, is a normalization factor and 10) is the ground state 

satisfying 

UlO) = 0 

and 

IO)= 1 . 

is easy to show that 

din) = &TI ~n + I) 

and 

)= fi In- I), 

The uf and up operators create or annihilate, respectively, a single 

scalar particle with energy ha, (ha,= s2, 

where p is the 

momentum carried by the created particle and m is the mass). A 

scalar particle is thus an excitation of one particular oscillator mode. 

The Fermionic Field. The simple quantum-mechanical analogue of 

a spin-% field needed to represent fermions is just a quantum particle 

with spin ‘12. This is necessary because, whereas bosons can be 

represented by scalar particles satisfying commutation relations, 

fermions must be represented by spin-% particles satisfying anticommutation 

relations. 

A spin-% particle has two spin states: 10) for spin down and 11) for 

spin up. Once again we define raising and lowering operators, here bt 

and 6, respectively. These operators satisfy the anticommutation 

relations 

{b, bt) = (bbt + b’b) = 1 

103

tates illustrates 

same energy as their fermionic part 

Moreover, it is easy to see that 

relations 

104

Supersymmetry at 100 GeV 

First we may add a small symmetry breaking term to the Hamiltonian, 

that is, H - H + EH’. where E is a small parameter and 

Energy 

States 

[ W, Q] # 0 . (25) 

Boson Fermion 

0 10,0> 

hw I 1,0> 10,1> 

2hw I2,0> 11,1> 

3hw I3,0> 12,1> 

This mechanism is called euplicit symmetry breaking. Using it we can 

give scalars a mass that is larger than that of their fermionic partners, 

as is observed in nature. Although this breaking mechanism may be 

perfectly self-consistent (even this is in doubt when one includes 

gravity), it is totally ad hoc and lacks predictive power. 

The second symmetry breaking mechanism is termed spontaneous 

symmetry breaking. This mechanism is characterized by the fact that 

the Hamiltonian remains supcrsymmetnc, 

but the ground state does not, 

The boson- fermion degeneracy for exact supersymmetry in 

which thefirst number in In,m) corresponds to the state for 

the oscillator degree of freedom (the scalar, or bosonic, 

field) and the second number to that for the spin-% degree of 

freedom (the fermionic field). 

which areanalogous to Eq. 1 because they represent the conversion of 

a fermionic state to a bosonic state and vice versa. 

The above example is a simple representation ofsupersymmetry in 

quantum mechanics. It is, however, trivial since it describes noninteracting 

bosons (oscillators) and fermions (spin-% particles). Nontrivial 

interacting representations of supersymmetry may also be 

obtained. In some of these representations it it possible to show that 

the ground state is not supersymmetric even though the Hamiltonian 

is. This is an example of spontaneous supersymmetry breaking. 

Symmetry Breaking. If supersymmetry were an exact symmetry of 

nature, then bosons and fermions would come in degenerate pairs. 

Since this is not the case, the symmetry must be broken. There are 

two inequivalent ways in which to do this and thus to have the 

degeneracy removed. 

Supersymmetry can either be a global symmetry, such as the 

rotational invariance ofa ferromagnet, or a local symmetry, such as a 

phase rotation in electrodynamics. Spontaneous breaking of a 

global symmetry leads to a massless Nambu-Goldstone particle. In 

supersymmetry we obtain a massless fermion c, the goldstino. 

Spontaneous breaking of a local symmetry, however, results in the 

gauge particle becoming massive. (In the standard model, the W 

bosons obtain a mass Me = gV by “eating” the massless Higgs 

bosons, where g is the SU(2) coupling constant and Vis the vacuum 

expectation value of the neutral Higgs boson.) The gauge particle of 

local supersymmetry is called a gravitino. It is the spin-3/2 partner of 

the graviton; that is, local supersymmetry incorporates Einstein’s 

theory ofgravity. When supersymmetry is spontaneously broken, the 

gravitino obtains a mass 

by “eating” the goldstino (here GN is Newton’s gravitational constant 

and A,, is the vacuum expectation of some field that spontaneously 

breaks supersymmetry). 

Thus, if the ideas of supersymmetry are correct, there is an 

underlying symmetry connecting bosons and fermions that is “hidden” 

in nature by spontaneous symmetry breaking. W 

105

coniinuedfiorn page 101 

tions. Many workers in the field are, however, 

still pursuing these elegant notions. 

Recently a third motivation for supersymmetry 

has been suggested. I shall describe the 

motivation and then discuss its expected 

consequences. 

For many years Dirac focused attention on 

the “problem of large numbers” or, more 

recently, the “hierarchy problem.” There are 

many extremely large numbers that appear 

in physics and for which we currently have 

no good understanding of their origin. One 

such large number is the ratio of the gravitational 

and weak-interaction mass scales 

mentioned earlier (Mpl/Mw- IO”). 

The gravitational force between two particles 

is proportional to the product of the 

energy (or mass if the particles are at rest) of 

the two particles times GN. Thus, since GN cc 

l/M& the force between two W bosons at 

rest is proportional to M&/M~,I - This 

is to be compared to the electric force between 

W bosons, which is proportional to a 

= e2/(4nhc) - lop2, where e is the electromagnetic 

coupling constant. Hence gravitational 

interactions between all known 

elementary particles are, at observable 

energies, at least IO” times weaker than their 

electromagnetic interactions. 

The key word is observable, for if we could 

imagine reaching an energy of order Mplc2, 

then the gravitational interactions would become 

quite strong. In other words, gravitationally 

bound states can be formed, in pnnciple, 

with mass of order Mpl - IOl9 GeV. 

The Planck scale might thus be associated 

with particles, as yet unobserved, that have 

strong gravitational interactions. 

At a somewhat lower energy, we also have 

the grand unification scale (MG - IOi5 GeV 

or greater), another very large scale with 

similar theoretical significance. New particles 

and interactions are expected to become 

important at MG. 

In either case, should these new 

phenomena exist, we are faced with the ques- 

tion ofwhy there are two such diverse scales, 

Mw and Mpl (or MG), in nature. 

The problem is exacerbated in the context 

of the standard model. In this mathematical 

H 

Y 

Perturbation Mass 

Fig. 2. If A. (lefi) represents a perturbative mass correction for an ordinary particle 

H due to the creation of a virtual photon y, then a supersymmetry rorarion of the 

central region of the diagram will generate a second mass correction A, (right) 

involving the supersymmetric partners H and thephotino 7. If supersymmetry is an 

exact symmetry, then the total mass correction is zero. 

framework, the W boson has a nonzero mass 

Mw because of spontaneous symmetry 

breaking and the existence of the scalar particle 

called the Higgs boson. Moreover, the 

mass of the W and the mass of the Higgs 

particle must be approximately equal. Unfortunately 

scalar masses are typically extremely 

sensitive to the details of the theory 

at very high energies. In particular, when one 

calculates quantum mechanical corrections 

to the Higgs mass p~ in perturbation theory, 

one finds 

where 

zero, and 6p2 is the perturbative correction. 

The parameter a is a generic coupling constant 

connecting the low mass states of order 

Mw and the heavy states of order Miarge, that 

is, the largest mass scale in the theory. For 

example, some of the theorized particles with 

mass Mpl or MG will have electric charge and 

interact with known particles. In this case, a 

= e2/4nAc, a measure of the electromagnetic 

coupling. Clearly p~ is naturally very large 

here and not approximately equal to the 

mass of the W. 

Supersymmetry can ameliorate the problem 

because, in such theories, scalar particles 

are no longer sensitive to the details at high 

energies. As a result of miraculous cancellations, 

one finds 

In these equations pb is the zeroth order 

value of the Higgs boson mass, which can be This happens in the following way (Fig. 2). 

106

Supersymmetry af 100 GeV 

Table 1 

The Supersymmetry Doubling of Particles 

spi n-lh 

qu’arks 

spin-% 

leptons 


- 

(i) ; 

(There are two other quark-lepton families similar to this one.) 

spin- 1 

gauge bosons 

y, W*, Zo, g 

spin-0 H+ A- 

Higgs bosons ( H o) ( H o) 

spin-0 

scalar partner 

spin-0 

scalar partner 

spin-2 

graviton 8 

__- -- ___ - ... -.. 

G 

G 

For each ordinary mass correction, there will 

be a second mass correction related to the 

first by a supersymmetry rotation (the symmetry 

operation changes the virtual particles 

of the ordinary correction into their corresponding 

supersymmetric partners). Although 

each correction separately is proportional 

to a Mirge, the sum of the two corrections 

is given by Eq. 3. In this case, if & = 0, 

then pH = 0 and will remain zero to all orders 

in perturbation theory as long as supersymmetry 

remains unbroken. Hence supersymmetry 

is a symmetry that prevents scalars 

from getting “large” masses, and one can 

even imagine a limit in which scalar masses 

vanish. Under these conditions we say 

scalars are “naturally” light. 

How then do we obtain the spontaneous 

Global Supersymmetry 

Local Supersymmetry 

(5) ; 

IC 

Supersymmetric Partners 

spin-0 

squarks 

spin-0 

sleptons 

breaking of the weak interactions and a W 

boson mass? We remarked that supersymmetry 

cannot be an exact symmetry of 

nature; it must be broken. Once supersymmetry 

is broken, the perturbative correction 

(Eq. 3) is replaced by 

where A,, is the scale of supersymmetry 

breaking. If supersymmetry is broken spontaneously, 

then A,, is not sensitive to Mlarge 

and could thus have a value that is much less 

than Mlarge. This correction to the Higgs 

boson mass can then result in a spontaneous 

breaking of the weak interactions, with the 

standard mechanism, at a scale of order Ass 

Mlarge ‘ 

The Particles. We’ve discussed a bit of the 

motivation for supersymmetry. Now let’s 

describe the consequences of the minimal 

supersymmetric extension of the standard 

model, that is, the particles, their masses, and 

their interactions. 

The particle spectrum is literally doubled 

(Table I). For every spin-% quark or lepton 

there is a spin-0 scalar partner (squark or 

slepton) with the same quantum numbers 

under the SU(3) X SU(2) X U( 1) gauge interactions. 

(We show only the first family of 

quarks and leptons in Table I; the other two 

families include the s, e, b, and I quarks, and, 

for leptons, the muon and tau and their 

associated neutrinos.) 

The spin-l gauge bosons (the photon y, the 

weak interaction bosons W’ and Zo, and 

the gluons g) have spin-% fermionic partners, 

called gauginos. 

Likewise, the spin-0 Higgs boson, responsible 

for, the spontaneous symmetry breaking 

ofthe weak interaction, should have a spin-% 

fermionic partner, called a Higgsino. However, 

we have included two sets of weak 

doublet Higgs bosons, denoted H and H, 

giving a total of four Higgs bosons and four 

Higgsinos. Although only one weak doublet 

of Higgs bosons is required for the weak 

breaking of the standard model, a consistent 

supersymmetry theory requires the two sets. 

As a result (unlike the standard model, which 

predicts one neutral Higgs boson), supersymmetry 

predicts that we should observe two 

charged and three neutral Higgs bosons. 

Finally, other particles, related to symmetry 

breaking and to gravity, should be 

introduced. For a global supersymmetry, 

these particles will be a massless spin-% 

Goldstino and its spin-0 partner. However, 

in the local supersymmetry theory needed 

for gravity, there will also be a graviton and 

its supersymmetric partner, the gravitino. 

We will discuss this point in greater detail 

later, but local symmetry breaking combines 

the Goldstino with the gravitino to form a 

massive, rather than a massless, gravitino. 

In many cases the doubling of particles 

just outlined creates a supersymmetric partner 

that is absolutely stable. Such a particle 

107

_----- 


1 

I 

i 

I 

I 

L .___- _ _ _ ~ _ _ _ ~ _ _ 

_^__--_- ~- 

- 

Fig. 3. Examples of interactions between ordinary particles 

(lefl) and the corresponding interactions between an ordinav 

particle and two supersymmetric particles (right) 

_ _ _ _-_____ ~ ____ 

______ 

obtained by performing a supersymmetry rotation on the 

first interaction. 

could, in fact, be the dominant form of matter 

in our universe. 

following manner. 

Although an unbroken supersymmetry 

can keep scalars massless, once supersym- 

The Masses. What is the expected mass for metry is broken, all scalars obtain quantum 

the supersymmetric partners of the ordinary corrections to their masses proportional to 

particles? The theory, to date, does not make the supersymmetry breaking scale Ass, that is 

any firm predictions; we can nevertheless 

obtain an order-of-magnitude estimate in the 6p2 - a A:s , (5) 

which is Eq. 4 with the first negligible term 

dropped. If we demand the Higgs mass &- 

Zip2 to be of order Mb, then Ais - MZw/a is at 

most oforder 1000 GeV. Moreover, the mass 

splitting between all ordinary particles and 

their supersymmetric partners is again of 

order Mw,. We thus conclude that if supersymmetry 

is responsible for the large ratio 

108

Supersymmetry ut 100 Ge V 

I 

and e- and the photino 7) that experimentally would be easily recognizable. 

\ 

\ Jet 

Fig. 5. A process involving supersymmetric particles (a gluino and squarks 3 that 

generates two hadronic jets. 

I 

M,,/M,,., then the new particles associated 

with supersymmetry will be seen in the next 

generation of high-energy accelerators. 

The Interactions. As a result of supersymmetry, 

the entire low-energy spectrum of 

particles has been doubled, the masses of the 

new particles are of order Mw, but these 

masses cannot be predicted with any better 

accuracy. A reasonable person might therefore 

ask what properties, if any, can we 

predict. The answer is that we know all the 

interactions of the new particles with the 

ordinary ones, of which several examples are 

shown in Fig. 3. To get an interaction between 

ordinary and new particles, we can 

start with an interaction between three or- 

Fig. 4. A possible interaction involving supersymmetric particles (the selectrons ;+ dinary particles and rotate two of these (with 

a supersymmetry operation) into their supersymmetric 

partners. The important point is 

that as a result of supersymmetry the coupling 

constants remain unchanged. 

Since we understand the interactions of 

the new particles with the ordinary ones, we 

know how to find these new objects. For 

example, an electron and a positron can annihilate 

and produce a pair of selectrons that 

subsequently decay into an electron-positron 

pair and two photinos (Fig. 4). This process 

is easily recognizable and would be a good 

signal of supersymmetry in high-energy electron-positron 

colliders. 

Supersymmetry is also evident in the process 

illustrated in Fig. 5. Here one of the three 

quarks in a proton interacts with one of the 

quarks in an antiproton; the interaction is 

mediated by a gluino. The result is the generation 

of two squarks that decay into quarks 

and photinos. Because quarks do not exist as 

free particles, the experimenter should observe 

two hadronic jets (each jet is a collection 

of hadrons moving in the same direction 

as, and as a consequence of, the initial motion 

of a single quark). The two photinos will 

generally not interact in the detector, and 

thus some of the total energy of the process 

will be “missing”. 

The theories we have been discussing until 

now have been a minimal supersymmetric 

extension of the standard model. There are, 

109

however, two further extrapolations that are 

interestihg both theoretically and phenomenologically. 

The first concerns gravity and 

the second, grand unified supersymmetry 

models. 

Gravity. We have already remarked that 

supersymmetry may be either a global or a 

local symmetry. If it is a global symmetry, 

the Goldstino is massless and the lightest 

supersymmetric partner. However, if supersymmetry 

is a local symmetry, it necessarily 

includes the gravity of general relativity and 

the Goldstino becomes part of a massive 

gravitino (the spin-3/2 partner of the graviton) 

with mass 

P 

With Ass of order Mw/& or 1000 GeV, mG 

is extremely small (- lo-’’ times the mass of 

the electron). 

Recently it was realized that under certain 

circumstances A,, can be much larger than 

Mw, but, at the same time, the perturbative 

corrections 6p2 can still satisfy the constraint 

that they be of order Mb. In these special 

cases, supersymmetry breaking effects vanish 

in the limit as some very large mass 

diverges, that is, we obtain 

Fig. 6. The decay mode of the proton predicted by the minimal unification 

symmetry SU(5). The expected decay products are a neutral pion no and a positron 

e+. 

(7) 

instead of Eq. 5. An example is already 

provided by the gravitino mass tnG (where 

Miarge = Mp,). In fact, models have now been 

constructed in which the gravitino mass is of 

order .44wand sets the scale ofthe low-energy 

supegap 62 between bosons and fermions. 

In either case (an extremely small or a very 

large gravitino mass), the observation of a 

massive gravitino is a clear signal of local 

supersymmetry in nature, that is, the nontrivial 

extension of Einstein’s gravity or 

supegravi ty. 

Grand Unification. Our second extrapolation 

of supersymmetry has to do with grand 

110 

unified theories, which provide a theoretically 

appealing unification of quarks and 

leptons and their strong, weak, and electromagnetic 

interactions. So far there has 

been one major experimental success for 

grand unification and two unconfirmed 

predictions. 

The success has to do with the relationship 

between various coupling constants. In the 

minimal unification symmetry SU(5), two 

independent parameters (the coupling constant 

gG and the value ofthe unification mass 

MG) determine the three independent coupling 

constants (g,, g, and g‘) of the standardmodel 

SU(3) X SU(2) X U( I) symmetry. As a 

result, we obtain one prediction, which is 

typically expressed in terms of the weakinteraction 

parameter: 

d2 

sin2Qw = - 

g2+gz’ 

The theory of minimal SU( 5) predicts sin29w I 

= 0.2 1, whereas the experimentally observed 

value is 0.22 k 0.01, in excellent agreement. 

The two predictions of SU(5) that have 

not been verified experimentally are the existence 

of magnetic monopoles and proton 

decay. The expected abundance of magnetic 

monopoles today is crucially dependent on 

poorly understood processes occumng in the 

first second of the history of the universe. 

As a result, if they are not seen, we may 

ascribe the problem to our poor understand- 1 

ing of the early universe. On the other hand, 

if proton decay is not observed at the ex- 

I

Supersymmetry ut 100 Ge V 

p [Supersymmetry Proton Decay ) K" + 

N 

n pupersymmetry Neutron D eea3 KO + 7 

i 

1 

I 

where rnp is the proton mass. 

Recent experiments, especially sensitive 

to the decay modes of Eq. 9, have found 7p 2 

years, in contradiction with the prediction. 

Hence minimal SU(5) appears to be in 

trouble. There are, of course, ways to complicate 

minimal SU(5) so as to be consistent 

with the experimental values for both sin20w 

and proton decay. Instead of considering 

such ad hoc changes, we will discuss the 

unexpected consequences of making minimal 

SU(5) globally supersymmetric. The parameter 

sin2& does not change considerably, 

whereas MG increases by an order of 

magnitude. Hence, the good prediction for 

sin20w remains intact while the proton lifetime, 

via the gauge boson exchange process 

of Fig. 6, naturally increases and becomes 

unobservable. 

It was quickly realized, however, that 

other processes in supersymmetric SU(5) 

give the dominant contribution towards 

proton decay (Fig. 7). The decay products 

resulting from these processes would consist 

ofKmesons and neutrinos or muons, that is, 

-- 

fig. 7. The dominant proton-decay and neutron-decay modes predicted by supersymmetry. 

The expected decay products are K mesons (K' and KO) and neutrinos 

6). 

and so would differ from the expected decay 

products of n mesons and positrons. This is 

very exciting because detection of the 

products of Eq. 11 not only may signal 

nucleon decay but also may provide the first 

signal of supersymmetry in nature. Experiments 

now running have all seen candidate 

events of this type. These events are, however, 

consistent with background. It may 

take several more years before a signal rises 

up above the background. 

pected rate, then minimal SU(5) is in serious 

trouble. 

The dominant decay modes predicted by 

minimal SU(5) for the nucleons are 

p - 7COef 

and 

n - n-e+ 

(9) 

These processes involve the exchange ofa socalled 

X or Y boson with mass of order Mc 

(Fig. 6), so that the predicted proton lifetime 

T, is 

Experiments. An encouraging feature of the 

theory is that low-energy supersymmetry can 

be verified in the next ten years, possibly as 

early as next year with experiments now in 

progress at the CERN proton-antiproton collider. 

Experimenters at CERN recently dis- 

111

covered the W' and Z'bosons, mediators of 

the weak interactions, and produced many of 

these bosons in high-energy collisions between 

protons and antiprotons (each with 

momentum - 270 GeV/c). For example, 

Fig. 8 shows the process for the generation of 

a W- boson, which then decays to a highenergy 

electron (detectable) and a highenergy 

neutrino (not detectable). A single 

electron with the characteristic energy of 

about 42 GeV was a clear signature for this 

process. 

Let us now consider some of the signatures 

of supersymmetry for pi or pp colliders. A 

clear signal for supersymmetry are multi-jet 

events with missing energy. For example, 

events containing one, two, three, or four 

hadronic jets and nothing more can be interpreted 

as a signal for either quark or gluino 

production (Figs. 5 and 9). A two- or four-jet 

signal is canonical, but these events can look 

like one- or three-jet events some fraction of 

the time. 

There may also be events with two jets, a 

highenergy electron, and some missing 

energy. This is the characteristic signature of 

top quark production via W decay (Fig. lo), 

and thus such events may be evidence for top 

quarks. But there is also an event predicted 

by supersymmetry with the same signature, 

namely, the production of a squark pair (Fig. 

11). It would require many such events to 

disentangle these two possibilites. 

The CERN proton-antiproton collider 

began taking more data in September 1984 

with momentum increased to 320 GeV/c per 

beam and with increased luminosity. No 

clear evidence for supersymmetric partners 

has been observed. As a result, the so-called 

UA-1 Collaboration at CERN has put lower 

limits on gluino and squark masses of ap 

proximately 60 and 80 GeV, respectively. As 

of this writing it is apparent that the discovery 

of supersymmetric partners, and perhaps 

idso the top quark, must wait for the 

next generation of high-energy accelerators. 

Hopefully, it will not be too long before we 

learn whether or not the underlying structure 

of the universe possesses this elegant, highly 

unifying type of symmetry. 1 

Fig. 8. The generation, in a high-energy proton-antiproton collision, of a W- 

particle, which then decays into an electron (e-) and an antineutrino (G). 

Fig. 9. A proton-antiproton collision involving supersymmetric particles (gluinos 

g, squarks antisquarks and photinos T) that generates four hadronic jets. 

Jet 

Fig. 10. Two-jet events observed by the UA-1 Collaboration at CERN can be 

interpreted, as shown here, as a process involving top quark t production. 

112

Supersymmetry ut 100 Ge Y 

- - Y w+/A 

w+ \ 

U 

P 

- 

c 1 

9 q 

Jet 

\ 

Fig. 11. The same event discussed in Fig. 10, only here interpreted as a supersymmetric 

process involving squarks and antisquarks. 


Daniel Z. Freedman and Peter van Nieuwenhuizen, “Supergravity and the Unification of the Laws of 

Physics.” Scienirfic American (February 1978): 126-143. 

Stuart A. Raby did his undergraduate work at the University of Rochester, receiving his B.Sc. in 

physics in 1969. Stuart spent six years in Israel as a student/teacher, receiving a M.Sc. in physics from 

Tel Aviv University in 1973 and a Ph.D. in physics from the same institution in 1976. Upon 

graduating, he took a Research Associate position at Cornell University. From 1978 to 1980, Stuart 

was Acting Assistant Professor of physics at Stanford University and then moved over to a three-year 

assignment as Research Associate at the Stanford Linear Accelerator Center. He came to the 

Laboratory as a Temporary Staff Member in 1981, cutting short his SLAC position, and became a 

Staff Member of the Elementary Particles and Field Theory Group of Theoretical Division in 1982. 

He has recently served as Visiting Associate Research Scientist for the University of Michigan. He 

and his wife Michele have two children, Eric and Liat. 

113

:. . . . 

. . . . . . 

* 

. . . . . . . . . , 

. . , 

. . . . - . . 

. 

. . . . 

. I 

.. . .. . ... . . .... 

* ' 

. . . . . . . . . . ... .. . .. 

. . . 

. .' ... . . . . . . . . . - .. 

. . . . . . 

. . . . . . ..... . 

.'. . . . 

# 

. ...: 

.... 

.. ,. 

. . . 

....... 

... 

e . . 

.... 

. . 

.* . 

. .. 

. . 

. . 

.. . . 

. - 

. .

... 

:..The Family Problem 

by T. Goldman and Michael Martin Nieto 

The roster of elementary particles includes replicas, exact in every detail but mass, 

of those that make up ordinary matter. More facts are needed to explain this 

seemingly unnecessary extravagance. 

T 

he currently “standard” model of particle physics phenomenologically 

describes virtually all of our observations of 

the world at the level of elementary particles (see “Particle 

Physics and the Standard Model”). However, it does not 

explain them with any depth. Why is SU(3)c the gauge group of the 

strong force? Why is the symmetry of the electroweak force broken? 

Where does gravity fit in? How can all ofthese forces be unified? That 

is, from what viewpoint will they appear as aspects of a common, 

underlying principle? These questions lead us in the directions of 

supersymmetry and of grand unification, topics discussed in 

“Toward a Unified Theory.” 

Yet another feature of the standard model leaves particle physicists 

dissatisfied: the multiple repetitions of the representations* of the 

particles involved in the gauge interactions. By definition the adjoint 

representationt of the gauge fields must occur precisely once in a 

gauge theory. However, quantum chromodynamics includes no less 

than six occurrences of the color triplet representation of quarks: one 

for each of the u, c, t, d, s, and b quarks. The u, c, and t quarks have a 

common electric charge of */3 and so are distinguished from the d, s, 

and b quarks, which have a common electric charge of -%. But the 

quarks with a common charge are distinguished only by their dif- 

ferent masses, as far as is now known. The electroweak theory 

presents an even worse situation, being burdened with nine leftchiralS 

quark doublets, three left-chiral lepton doublets, eighteen 

right-chiral quark singlets, and three right-chiral lepton singlets (Fig. 

1). 

Nonetheless, some organization can be discerned. The exact symmetry 

of the strong and electromagnetic gauge interactions, together 

with the nonzero masses of the quarks and charged leptons, implies 

that the right-chiral quarks and charged leptons and their left-chiral 

partners can be treated as single objects under these interactions. In 

addition, each neutral lepton is associated with a particular charged 

lepton, courtesy of the transformations induced by the weak interaction. 

Thus, it is natural to think in terms ofthree quark sets (u and d, c 

and s, and t and b) and three lepton sets (e- and v,, p- and v~, and 7- 

and v,) rather than thirty-three quite repetitive representations. 

Furthermore, the relative lightness of the u and dquark set and of the 

e- and v, lepton set long ago suggested to some that the quarks and 

leptons are also related (quark-lepton symmetry). Subtle mathematical 

properties of modern gauge field theories have provided new 

backing for this notion of three “quark-lepton families,” each consisting 

of successively heavier quark and lepton sets (Table 1). 

*We give a geometric definition of “representation,” using as an example the 

SU(3)c triplet representation of; say, the up quark. (This triplet, the smallest 

non-singlet representation of Scl(3)c, is called the fundamental representation.) 

The members of this representation (U,d, Ublao and usreed correspond to the set 

of three vectors directed from the origin of a two-dimensional coordinate system 

to the vertices of an equilateral triangle centered at the origin. (The triangle is 

usually depicted as standing on a vertex.) The “conjugate” of the triplet 

representation, which contains the three anticolor varieties of the up quark with 

charge -%, can be defined similarly: it corresponds to the set of three vectors 

obtained by reflecting the vectors of the triplet representation through the origin. 

(The vectors of the conjugate representation are directed toward the vertices of 

an equilateral triangle standing on its side, like a pyramid.) The “group 

transformations” correspond to the set of operations by which any one of the 

quark or antiquark vectors is transformed into any other. 

t The “adjoint” representation of SU(3)c, which contains the ei&ht vector bosons 

(the gluons), is found in the ‘(product” of the triplet representation and its 

conjugate. This product corresponds to the set of nine vectors obtained by 

forming the vector sums of each member of the triplet representation with each 

member of its conjugate. This set can be decomposed into a singlet containing a 

null vector (a point at the origin) and an octet, the adjoint representation, 

containing two null vectors and six vectors directed from the origin to the 

vertices of a regular hexagon centered at the origin. Note that the adjoint 

representation is symmetric under reflection through the origin. 

#A massless particle is said to be le/f-handed (right-handed) if the direction of 

its spin vector is opposite (the same as) that of its momentum. Chiraliw is the 

Lorentz-invariant generalization of this handedness to massive particles and is 

equivalent to handedness for massless particles.

If the underlying significance of this 

grouping by mass is not apparent to the 

reader, neither is it to particle physicists. NO 

one has put forth any compelling reason for 

deciding which charge Y3 quark and which 

charge --%quark to combine into a quark set 

or for deciding which quark set and which 

charged and neutral lepton set should be 

combined in a quark-lepton family. Like 

Mendeleev, we are in possession of what 

appears to be an orderly grouping but 

without a clue as to its dynamical basis. This 

is one theme of “the family problem.” 

Still, we do refer to each quark and lepton 

set together as a family and thus reduce the 

problem to that of understanding only three 

families-unless, of course, there are more 

families as yet unobserved. This last is another 

question that a successful “theory of 

families” must answer. Grand unified theories, 

supersymmetry theories, and theories 

wherein quarks and leptons have a common 

substructure can all accommodate quark-lepton 

symmetry but as yet have not provided 

convincing predictions as to the number of 

families. (These predictions range from any 

even number to an infinite spectrum.) 

Such concatenations of wild ideas (however 

intriguing) may not be the best approach 

to solving the family problem. A more conservative 

approach, emulating that leading 

to the standard model, is to attack the family 

problem as a separate question and to ask 

directly if the different families are 

dynamically related. 

Here we face a formidable obstacle-a 

paucity of information. A fermion from one 

family has never been observed to change 

into a fermion from another family. Table 2 

lists some family-changing decays that have 

been sought and the experimental limits on 

their occurrence. True, a p- may appear to 

decay into an e-, but, as has been experimentally 

confirmed, it actually is transformed 

into a v,,, and simultaneously the e- and a 

appear. Being an antiparticle, the ie carries 

the opposite of whatever family quantum 

numbers distinguish an e- from any other 

charged lepton. Thus, no net “first-familines” 

is created, and the “second-familiness” 

Fig. I. The electroweak representations of the fermions of the standard model, 

which comprise nine left-chiral quark doublets, eighteen right-chiral quark 

singlets, three left-chiral lepton doublets, and three right-chiral lepton singlets. 

The subscripts r, b, and g denote the three color charges of the quarks, and the 

subscripts R and L denote right- and left-chiralprojections. The symbols d’, s’, and 

b‘ indicate weak-interaction mass eigenstates, which, as discussed in the text, are 

mixtures of the strong-interaction mass eigenstates d, s, and b. Since quantum 

chromodynamics does not include the weak interaction, and hence is not concerned 

with chirality, the SU(3), representations of the fermions are fewer in number: six 

triplets, each containing the three color-charge varieties of one of the quarks, and 

three singlets, each containing a charged lepton and its associated neutral lepton. 

of the original p- is preserved in the v,,. 

In spite of the lack of positive experimental 

results, current fashions (which are based 

on the successes ofthe standard model) make 

irresistible the temptation to assign a family 

symmetry group to the three known families. 

Some that have been considered include 

SU(2), SU(2) X U( I), SU(3), and U( 1) X U( 1) 

X U(1). The impoverished level of our understanding 

is apparent from the SU(2) case, 

in which we cannot even determine whether 

the three families fall into a doublet and a 

singlet or simply form a triplet. 

The clearest possible prediction from a 

family symmetry group, analogous to 

Mendeleev’s prediction of new elements and 

their properties, would be the existence of 

one or more additional families necessary to 

complete a representation. Such a prediction 

can be obtained most naturally from either of 

two possibilities for the family symmetry: a 

spontaneously broken local gauge symmetry 

1161

p-, 

The Family Problem 

Table 1 

Members of the three known quark-lepton families and their masses, Each 

family contains one particle from each of the four types of fermions: leptons 

with an electric charge of -1 (the electron, the muon, and the tau); neutral 

leptons (the electron neutrino, the muon neutrino, and the tau neutrino); 

quarks with an electric charge of 2/3 (the up, charmed, and top quarks); and 

quarks with an electric charge of -V3 (the down, strange, and bottom 

quarks). Each family also contains the antiparticles of its members. (The 

antiparticles of the charged leptons are distinguished by opposite electric 

charge, those of the neutral leptons by opposite chirality, and those of the 

quarks by opposite electric and color charges. For historical reasons only 

the antielectron has a distinctive appellation, the positron.) Family membership 

is determined by mass, with the first family containing the least 

massive example of each type of fermion, the second containing the next 

most massive, and so on. What, if any, dynamical basis underlies this 

grouping by mass is not known, nor is it known whether other heavier 

families exist. The members of the first family dominate the ordinary world, 

whereas those of the second and third families are unstable and are found 

only among the debris of collisions between members of the first family. 

First Family Second Family Third Family 

electron, e- 

0.511 MeV/c2 

electron neutrino, vr 

0.00002 MeV/c2 (?) 

up quark, u 

=5 MeV/c2 

down quark, d 

10 MeV/c2 

Table 2 

muon, p- 

105.6 MeVlc’ 

muon neutrino, vp 

50.5 MeVJc2 

charmed quark, c 

=1500MeV/c2 

strange quark, s 

70 MeV/c2 

tau, ‘E- 

1782 MeVJc’ 

tau neutrino, vT 

5 147 MeV/c2 

top quark, t 

240,000 MeVJc2 (?) 

bottom quark, b 

-4500 MeVJc2 

Experimental limits on the branching ratios for some family-changing 

decays. The branching ratio for a particular decay mode is defined as the 

ratio of the number of decays by that mode to the total number of decays by 

all modes. An experiment capable of determining a branching ratio for p+ - 

e+y as low as lo”* is currently in progress at Los Alamos (see “Experiments 

To Test Unification Schemes”). 

Branching Ratio 

Dominant 

Decay Mode (upper bound) Decay Mode($ 

10-10 pi e+v& 

10-12 p+ e+v,v,, 

I 0-7 

no-+ w 

10-8 

K+ n+no or p+v 

10-8 

K~ - x+x-no or n 8 nono 

10-5 c+ - pno 

i 

I 

or a spontaneously broken global symmetry.* 

What follows is a brief ramble 

(whose course depends little on detailed assumptions) 

through the salient features and 

implications of these two possibilities. 

Family Gauge Symmetry 

All of the unseen decays listed in Table 2 

would be strictly forbidden if the family 

gauge symmetry were an exact gauge symmetry 

as those of quantum electrodynamics 

and quantum chromodynamics are widely 

believed to be. Here, however, we do not 

expect exactness because that would imply 

the existence, contrary to experience, of an 

additional fundamental force mediated by a 

massless vector boson (such as a long-range 

force like that of the photon or a strong force 

like that of the gluons but extending to leptons 

as well as quarks). But we can, as in the 

standard model, assume a broken gauge symmetry. 

We begin by placing one or more families 

in a representation of some family gauge 

symmetry group. (The correct group might 

be inferred from ideas such as grand unification 

or compositeness of fermions. However, 

it is much more likely that, as in the case of 

the standard model, this decision will best be 

guided by hints from experimental observations.) 

Together, the group and the representation 

determine currents that describe interactions 

between members of the representation. 

(These currents would be conserved if 

the family symmetry were exact.) For example, 

if the first and the second families are 

placed in the representation, an electrically 

neutral current describes the transformation 

- 

e ++ just as the charged weak current of 

the electroweak theory describes the transformation 

e- ++ v,. Since the other family 

*In principle, we should also consider the 

possibilities of a discrete symmetry or an explicit 

breaking of family symmetry (probably caused by 

some dynamics of a fermion substructure). However, 

these ideas would be radical departures from 

the gauge symmetries that have proved so successful 

to date. We will not pursue them here. 

117

members necessarily fall into the same representation, 

the e- - p- current includes 

contributions from interactions between 

these other members (d - s, for example), 

just as the charged weak current for 

e- - v, includes contributions from p- - vp 

and 5- e+ v,. 

If we now allow the family symmetry to be 

a local gauge symmetry, we find a “family 

vector boson,” F, that couples to these currents 

(Fig. 2) and mediates the family-changing 

interactions. As in the standard model, 

the coupled currents can be combined to 

yield dynamical predictions such as scattering 

amplitudes, decay rates, and relations 

between different processes. 

Scale of Family Gauge Symmetry 

Breaking. Weak interactions occur relatively 

infrequently compared to electromagnetic 

and strong interactions because 

of the large dynamical scale (approximately 

100 GeV) set by the masses of the W’ and 

Zo bosons that break the electroweak symmetry. 

We can interpret the extremely low 

rate of family-changing interactions as being 

due to an analogous but even larger 

dynamical scale associated with the breaking 

of a local family gauge symmetry, that is, to a 

large value for the mass MF of the family 

vector boson. The branching-ratio limit 

listed in Table 2 for the reaction KL - p’ + 

e’ allows us to estimate a lower bound for 

MF as follows. 

Like the weak decay of muons, the KL - 

pe decay proceeds through formation of a 

virtual family vector boson (Fig. 3). The rate 

for the decay, r, is given by 

Note that the fourth power of MF appears in 

Eq. I just as the fourth power of Mw does 

(hiding in the square of the Fermi constant) 

in the rate equation for muon decay. (Certain 

chirality properties of the family interaction 

could require that two of the five powers of 

the k.aon mass rnK in Eq. 1 be replaced by the 

muon mass. However, since the inferred 

value of MF varies as the fourth root of this 

term, the change would make little numerical 

difference.) It is usual to assume that gfamily, 

the family coupling constant, is comparable 

in magnitude to those for the weak and electromagnetic 

interactions. This assumption 

reflects our prejudice that family-changing 

interactions may eventually be unified with 

those interactions. Using Eq. 1 and the 

branching-ratio limit from Table 2, we obtain 

MF 2 lo5 GeV/c2 . 

Such a large lower bound on MF implies that 

the breaking of a local family gauge symmetry 

produces interactions much weaker 

than the weak interactions. 

Alternatively, processes like KL - pe may 

be the result of family-conserving grand unified 

interactions in which quarks are turned 

into leptons. However, the experimental 

limit on the rate of proton decay implies that 

such interactions occur far less frequently 

than the family-vio’lating interactions considered 

here. 

Experiments with neutrinos, also, indicate 

a similarly large dynamical scale for the 

breaking of a local family gauge symmetry. A 

search for the radiative decay vp - v,+ y has 

yielded a lower bound on the v,, lifetime of 

lo5 (m,/MeV) seconds. If the mass of the 

muon neutrino is near its experimentally 

observed upper bound of 0.5 MeVlc’, this 

lower bound on the lifetime is greater than 

the standard-model prediction of approximately 

lo’ (MeV/rn,)5 seconds. Thus, some 

family-conservation principle may be suppressing 

the decay. 

More definitive information is available 

from neutrino-scattering experiments. 

Positive pions decay overwhelmingly ( IO4 to 

1) into positive muons and muon neutrinos. 

In the absence of family-changing interactions, 

scattering of these neutrinos on nuclear 

targets should produce only negative 

muons. This has been accurately confirmed: 

neither positrons nor electrons appear more 

frequently than permitted by the present systematic 

experimental uncertainty of 0.1 per- 

Fig. 2. Examples of neutral familychanging 

currents coupled to a family 

vector boson (F). Such couplings follow 

from the assumption of a local gauge 

symmetry for the family symmetry. 

cent. An investigation of the neutrinos from 

muon decay has yielded similar results. The 

decay of a positive muon produces, in addition 

to a positron, an electron neutrino and a 

muon antineutrino. Again, in the absence of 

family-changing interactions, scattering of 

these neutrinos should produce only electrons 

and positive muons, respectively. A 

LAMPF experiment (E-31) has shown, with 

an uncertainty of about 5 percent, that no 

negative muons or positrons are produced. 

The energy scale of Eq. 2 will not be 

directly accessible with accelerators in the 

I 118


Fig. 3. Feynman diagram for the family-changing decay K, - p- + e', which is 

assumed to occur through formation of a virtual family vector boson (F). The K, 

meson is the longer lived of two possible mixtures of the neutral kaon (KO) and its 

antiparticle (EO). Neither this decay nor the equallyprobable decay K, - p' + e- 

has been observed experimentally; the current upper bound on the branching ratio 

is IO-*. 

_- 

foreseeable future. The Superconducting 

Super Collider, which is currently being considered 

for construction next decade, is conceived 

of as reaching 40,000 GeV but is 

estimated to cost several billion dollars. We 

cannot expect something yet an order of 

magnitude more ambitious for a very long 

time. Thus, further information about the 

breaking of a local family gauge symmetry 

will not arise from a brute force approach but 

+ 

e 

rather, as it has till now, from discriminating 

searches for the needle ofa rare event among 

a haystack of ordinary ones. Clearly, the 

larger the total number of events examined, 

the more definitive is the information obtained 

about the rate of the rare ones. For 

this reason the availability of high-intensity 

beams of the reacting particles is a very 

important factor in the experiments that 

need to be undertaken or refined, given that 

they are to be carried out by creatures with 

finite lifetimes! 

For example, consider again the decay KL 

- pe. Since the rate of this decay vanes 

inversely as the fourth power of the mass of 

the family vector boson, a value of MI- in the 

million-GeV range implies a branching ratio 

lower by four orders of magnitude than the 

present limit. A search for so rare a decay 

would be quite feasible at a high-intensity, 

mediumenergy accelerator (such as the 

proposed LAMPF 11) that is designed to 

produce kaon fluxes on the order of 10' per 

second. (Currently available kaon fluxes are 

on the order of IO6 per second.) A typical 

solid angle times efficiency factor for an inflight 

decay experiment is on the order of IO 

percent. Thus, IO' kaons per second could be 

examined for the decay mode of interest. A 

branching ratio larger than lo-'* could be 

found in a one-day search, and a year-long 

experiment would be sensitive down to the 

lopJ4 level. Ofcourse, we do not know with 

absolute certainty whether a positive signal 

will be found at any level. Nonetheless, the 

need for such an observation to elucidate 

family dynamics impels us to make the attempt. 

Positive Evidence for Family 

Symmetry Breaking 

Thus, despite expectations to the contrary, 

we have at present no positive evidence in 

any neutral process for nonconservation of a 

family quantum number, that is, for familychanging 

interactions mediated by exchange 

ofan electrically neutral vector boson such as 

the Fof Figs. 2 and 3. Is it possible that our 

expectations are wrong-that this quantum 

number is exactly conserved as are electric 

charge and angular momentum? The answer 

is an unequivocal NO! We have-for 

quarks-positive evidence that family is a 

broken symmetry. To see this, we must 

examine the effect of the electroweak interaction 

on the quark mass eigenstates defined by 

the strong interaction. 

We know, for instance, that a Kf (= u + s) 

decays by the weak interaction into a p' and 

119

a vp and also decays into a n+ and a no (Fig. 

4). In quark terms this means that the u 

quark and the s quark in the kaon are coupled 

through a W+ boson. The two families 

(up-down and charmed-strange) defined by 

the quark mass eigenstates under the strong 

interaction are mixed by the weak interaction. 

Since the kaon decays occur in both 

purely leptonic and purely hadronic channels, 

they are not likely to be due to peculiar 

quark-lepton couplings. Similar evidence for 

family violation is found in the decays of D 

mesons, which contain charmed quarks. 

Weak-interaction eigenstates d' and s' 

may be defined in terms of the strong-interaction 

mass eigenstates d and s by 

coset sinec 

(.") = (-sin ec cos ec) ( P) (3) 

where e,, the Cabibbo mixing angle, is experimentally 

found to be the angle whose 

sine is 0.23 & 0.01. (The usual convention, 

which entails no loss of generality, is to assign 

all the mixing effects of the weak interaction 

to the down and strange quarks, leaving 

unchanged the up and charmed quarks.) The 

fact that the mass and weak-interaction 

eigenstates are different implies that a conserved 

family quantum number cannot be 

defined in the presence of both the strong 

and the weak interactions. We can easily 

show, however, that this conclusion does not 

contradict the observed absence of neutral 

family-violating interactions. 

lhe weak charged-current interaction describing, 

say, the transformation of a d' 

quark into a u quark by absorption of a W+ 

boson has the form 

(id' + CS')W+ = (ii, C) ( w+ w+) 0 (.") , 

which, after substitution of Eq. 3, becomes 

(4) 

(Ud' + &w+ = ii(dc0~ ec + ssin Bc)W+ 

+ ;(-&in e, + scos e,) w+ . 

(5) 

(Here we suppress details of the Lorentz 

algebra.) 

Fig. 4. Feynman diagrams for the decays of a positive kaon into (a) a positive muoi 

and a muon neutrino and (b) a positive and a neutral pion. The ellkse witr 

diagonal lines represents any one of several possible pathways for production of 4 

positive and a neutral pion from an up quark and an antidown quark. These decays 

in which the up-down and charmed-strange quark families are mixed by the weai 

interaction (as: indicated by sin 0, and cos %), are evidence that the family sym 

metry of quarks is a broken symmmetry. 

Because of the mixing given by Eq. 3, the 

statement we made near the beginning of this 

article, that no family-changing decays have 

been observed, must be sharpened. True, no 

s' - u decay has been seen, but, of course, 

the s - u decay implied by Eq. 5 does occur. 

Thus, "No family-changing decays of weal 

interaction family eigenstates have been 01 

served" is the more precise statement. 

The weak neutral-current interaction dl 

scribing the scattering of a d' quark when 

absorbs a Zo has a form like that of Eq. 4: 

120


KO -+ 

- 

d W+ d 

- 

S W- S 

- 

-C KO 

Gg. 5. Feynman diagram for a CP-violating reaction that transforms the neutral 

:aon into its antiparticle. This second-order weak interaction occurs through 

ormation of virtual intermediate states including either a u, c, or t quark. 

dtdt + ?st)Zo = (511, it)( zo zo) 0 ($) 

hce the Cabibbo matrix in Eq. 3 is unitary, 

:q. 6 is unchanged (except for the disapiearance 

of primes on the quarks) by subtitution 

of Eq. 3: 

(ad’ + ?s’)Zo = (ad+ Ss)Zo. (7) 

-bus, the weak neutral-current interaction 

loes not change d quarks into s quarks anynore 

than it changes d’ quarks into s’ quarks. 

t is only the presumed family vector boson 

if mass greater than IO5 GeV that may effect 

uch a change. 

Tamily Symmetry Violation and 

:P Violation 

The combined operation of charge con- 

Jgation and parity reversal (CP) is, like 

sarity reversal alone, now known not to be 

n exact symmetry of the world. An under- 

:anding of CP violation and proton decay 

rould be of universal importance to explain 

big-bang’’ cosmology and the observed ex- 

:ss of matter over antimatter. 

The generalization by Kobayashi and 

Maskawa of Eq. 3 to the three-family case is 

introduced in “Particle Physics and the Standard 

Model”; it yields a relation between 

family symmetry violation and CP violation. 

Although other sources of CP violation may 

exist outside the standard model, this relation 

permits extraction of information about 

violation of family symmetry from studies of 

CP violation. 

The phenomenon of CP violation has, so 

far, been observed only in the KO- Eo system. 

The CP eigenstates of this system are 

the sum and the difference of the KO and Eo 

states. The violation is exhibited as a small 

tendency for the long-lived state, KL , which 

normally decays into three pions, to decay 

into two pions (the normal decay mode of 

the short-lived state, Ks) with a branching 

ratio of approximately This tendency 

can be described by saying that the Ks and 

KL states differ from the sum and difference 

states by a mixing of order E: 

[Ks) z [KO) + (1 - E) [EO) 

and (8) 

IKL) = [KO)- (1- E) [KO). 

The quark-model analysis based on the work 

of Kobayashi and Maskawa and the secondorder 

weak interaction shown in Fig. 5 

predict an additional CP-violating effect not 

describable in terms of the mixing in Eq. 8; 

that is, it would occur even if E were zero. 

The effect, which is predicted to be of order 

E’, where E‘IE is about IO-*, has not yet been 

observed, but experiments sufficiently sensitive 

are being mounted. 

Both E and &’are related to the Kobayashi- 

Maskawa parameters that describe family 

symmetry violation. This guarantees that if 

the value of E’ is found to be in the expected 

range, higher precision experiments will be 

needed to determine its exact value . If no 

positive result is obtained in the present 

round of experiments, it will be even more 

important to search for still smaller values. 

In either case intense kaon beams are highly 

desirable since the durations of such experiments 

are approaching the upper limit of 

reasonability. 

Of course, in principle, CP violation can 

be studied in other quark systems involving 

the heavier c, b, and t quarks. However, these 

are produced roughly 10’ times less 

copiously than are kaons, and the CP-violating 

effects are not expected to be as large as in 

the case of kaons. 

Global Family Symmetry 

In our discussion of family-violating 

processes like K - pe, we have, so far, 

assumed the existence of a massive gauge 

vector boson reflecting family dynamics. The 

general theorem, due to Goldstone, offers 

two mutually exclusive possibilities for the 

realization of a broken symmetry in field 

theory. One is the development ofjust such a 

massive vector boson from a massless one; 

the other is the absence of any vector boson 

and the appearance of a massless scalar 

boson, or Goldstone boson. The possible 

Goldstone boson associated with family 

symmetry has been called the familon and is 

denoted byj As is generally true for such 

scalar bosons, the strength of its coupling 

falls inversely with the mass scale of the 

symmetry breaking. Cosmological argu- 

121

ments suggest a lower bound on the coupling 

ofapproximately IO-’* GeV-’ , a value very 

near (within three orders of magnitude) the 

upper bound determined from particle-physics 

experiments. 

The familon would appear in the two- 

body decays p - e + f and s - d +f: 

The 

latter can be observed in the decay K f (= u + 

S) -. xt (= u + 71, + nothing else seen. The 

familon would not be seen because it is about 

as weakly interacting as a neutrino. The only 

signal lhat the decay had occurred would be 

the appearance of a positive pion at the 

kinematically determined momentum of 227 

MeV/c. 

Such a search for evidence of the familon 

would encounter an unavoidable back- 

ground of positive pions from the reaction 

Kf - xf + v, + if, where the index i covers 

all neutrino types light enough to appear in 

the reaction. This decay mode occurs 

through a one-loop quantum-field correction 

to the electroweak theory (Fig. 6) and is 

interesting in itself for two reasons. First, it 

depends on a different combination of the 

parameters involved in CP violation and on 

the number N, of light neutrino types. Since 

N, is expected to be determined in studies of 

Zo decay, an uncertainty in the value of a 

matrix element in the standard-model 

prediction of the K+ - x+v,;, branching 

ratio can be eliminated. Present estimates 

place the branching ratio in the range between 

and IO-” times N,. Second, a 

discrepancy with the N, value determined 

from decay of the Zo , which is heavier than 

the kaon, would be evidence for the existence 

of at least one neutrino with a mass greater 

than about 200 MeVlc’. 

Fermion Masses and Family Symmetry 

Breaking 

The mass spectrum of the fermions is itself 

unequivocal evidence that family symmetry 

is broken. These masses, which are listed in 

Table I, should be compared to the W* and 

Zo masses of 83 and 92 GeVlc’, respectively, 

which set the dynamical scale of electroweak 

Fig. 6. Feynman diagram for the decay K + - n+ + vi + ii, where the index i covel 

all neutrino types light enough to appear in the reaction. The symbol Qi standsfc 

the charged lepton associated with vi and ii. 

interactions. (The masses quoted are the theoretical 

values, which agree well with the 

recently measured experimental values.) The 

very existence of the fermion masses violates 

electroweak symmetry by connecting doublet 

and singlet representations, and the 

variations in the pattern of mass splittings 

within each family show that family symmetry 

is broken. But since we neither know 

the mass scale nor understand the pattern of 

the family symmetiy breaking, we do not 

really know the relation between the mass 

scale of electroweak symmetry breaking and 

the fermion mass spectrum. It is possible to 

devise models in which the first family is 

light because the family symmetry breaking 

suppresses the electroweak symmetry breaking. 

Thus, the “natural” scale of electroweak 

symmetry breaking among the fermions 

could remain approximately 100 GeV/cZ, 

despite the small masses (a few MeV/c2) of 

some fermions. 

Experiments to establish the masses of the 

neutrinos are of great interest to the family 

problem and to particle physics in general. 

Being electrically neutral, neutrinos are 

unique among the fermions in possibly being 

endowed with a so-called Majorana mass* in 

addition to the usual Dirac mass. One approach 

to determining these masses is by 

applying kinematics to suitable reactions. 

For example, one can measure the end-point 

energy of the electron in the beta decay ’H- 

or of the muon in the decay nf 

’He + e- + ic 

- p+ + vp. 

Another quite different approach is to 

search for “neutrino oscillations.” If the ne1 

trino masses are nonzero, weak interactioi 

can be expected to mix neutrinos from di 

ferent families just as they do the quark 

This mixing would cause a beam of, sa 

essentially muon neutrinos to be tran 

formed into a mixture (varying in space ar 

in time) of electron, muon, and tau nei 

trinos. Detection of these oscillations wou 

not only settle the question ofwhether or ni 

neutrinos have nonzero masses but wou’ 

also provide information about the di 

ferences between the masses of neutrinc 

from different families. Experiments are i 

progress, but, since neutrino interactions a 

infamously rare, high-intensity beams a 

required to detect any neutrinos at all, I 

alone possible small oscillations in the 

family identity. (For details about the tritiui 

beta decay and neutrino oscillation expel 

ments in progress at Los Alamos, see “E 

periments To Test Unification Schemes.”) 

Conclusion 

The family symmetry problem is a fund 

mcntal one in particle physics, apparent 

without sufficient information available 

present to resolve it. Yet it is as crucial ar 

important a problem as grand unificatio 

*Majorana mass terms are not allowed for ele 

trically charged particles. Such terms induce tran 

formations of particles into antiparticles and 

would be inconsistent with conservation of elect) 

charge. 

122


and it may well be a completely independent may, however, be accessible in studies of rare intensity, mediumenergy accelerator could 

me. The known bound of IO5 GeV on the decays of kaons and other mesons, of CP beahighlyeffectivemeansofachievingthese 

scale of family dynamics is an order of mag- violation, and of neutrino oscillations. To needs. Unlike many other experimental 

nitude beyond the direct reach of any present undertake these experiments at the necessary questions in particle physics, those on the 

3r proposed accelerator, including the Super- sensitivity requires intense fluxes of particles high-intensity frontier are clearly defined. 

:onducting Super Collider. These dynamics from the second or later families. A high- We await the answers expectantly. W 


Howard Georgi. “A Unified Theory of Elementary Particles and Forces.” Scientific American, April 

1981, p. 48. 

T. (Terry) Goldman received a B.Sc. in physics and mathematics in 1968 

from the University of Manitoba and an A.M. and a Ph.D. in physics 

from Harvard in 1969 and 1973, respectively. He was a Woodrow Wilson 

Fellow from 1968 to 1969 and a National Research Council of Canada 

postdoctoral fellow at the Stanford Linear Accelerator Center from 1973 

to 1975, when he joined the Laboratory’s Theoretical Division as a postdoctoral 

fellow. In 1978 he became a staff member in the same division. 

From 1978 to 1980 he was on leave from Los Alamos as a Senior 

Research Fellow at California Institute of Technology, and during the 

academic year 1982-83 he was a Visiting Associate Professor at the 

University of California, Santa Cruz. His professional work has centered 

around weak interactions and grand unified theories. He is a member of 

the American Physical Society. 

Michael Martin Nieto received a B.A. in physics from the University of 

California, Riverside, in 1961 and a Ph.D. in physics, with minors in 

mathematics and astrophysics, from Cornell University in 1966. He 

joined the Laboratory in 1972 after occupying research positions at the 

State University of New York at Stony Brook; the Niels Bohr Institute in 

Copenhagen; the University of California, Santa Barbara; Kyoto University; 

and Purdue University. His main interests are quantum mechanics, 

coherence phenomena, elementary particle physics, and astrophysics. He 

is a member of Phi Beta Kappa and the International Association of 

Mathematical Physicists and a Fellow of the American Physical Society. 

e. For example, in I972 Nieto authored a survey of important experiments in particle physics that could be done at the then 

I 

ciety. 

123 

1

Addendum 

CP Violation 

in Heavy-Quark 

Systems 

H 

ere we extend the discussion of CP 

violation in “The Family Problem” 

to 

heavier quark systems. This requires 

generalizing the Cabibbo mixing matrix 

(&. 3 in the main text) to more than two 

families. The Cabibbo matrix relates the 

weak-interaction eigenstates of the ud and cs 

quark families to their strong-interaction 

mass eigenstates. Now, in general, the unitary 

transformation relating the weak and 

strong eigenstates among n families will have 

%n(n - 1) rotations and Y2(n - l)(n - 2) 

physical phases. 

We are interested in the generalization to 

three families since the third family, containing 

the t and b qua.rks, is known to exist. This 

extension of the Cabibbo mixing matrix i 

called the Kobayashi-Maskawa (K-M) ma 

trix after the two physicists who elucidate( 

the problem. They realized that the mixin] 

matrix for three families would naturall: 

encompass a parameterization of CP viola 

tion. The K-M matrix can be written as i 

product of three rotations (which can bc 

thought of as the Euler rotation angles o 

classical physics even though the conventiox 

is not the standard one) and a singlt 

physically meaningful phase (which can bc 

identified as the CP-violating parameter). Ir 

particular, we define the K-M matrix V foi 

the three quark families (ud, cs, and tb) ai 

follows: 

(;)=V(t) 9 

where 

1 0 0 1 0 0 

0 0 1 0 -s3 c3 

12,4

Addendum 

Note the form of V in Eq. A2. The first, 

third, and fourth matrices are rotations 

about particular axes. Except for the unusual 

convention, this is just a general orthogonal 

rotation in a three-dimensional Cartesian 

system. The si and the ci are the sines and 

cosines of the three rotation angles 8,. Note 

that the i = 1 rotation is the Cabibbo rotation 

Oc described in the text. 

What is new is the second matrix factor in 

Eq. A2, which contains the complex 

amplitude with phase 6 that parameterizes 

CP violation. Indeed, this is the factor that 

makes V not an orthogonal transformation 

but a unitary transformation. Vis still normpreserving, 

but contains phase information, 

something that quantum mechanics allows. 

In principle, another matrix U relates the 

weak and strong eigenstates of the u, c, and t 

quarks, and the product UtV describes the 

mixing of weak charged currents. However, 

we follow the standard convention and take 

U = I, thereby putting all of the physics of 

UtVinto Vitself. (Note that the unitarity of 

V produces a result equivalent to that given 

by Eq. 7: there are still no family-changing 

neutral currents.) Because Vis “really” Ut V, 

the rows of V can be labeled by the u, c, and t 

quarks. Thus, we can write Vas 

Physically, this means that the matrix elements 

Vu can be considered coupling constants 

or decay amplitudes between the 

quarks and the weak charged bosons W*. 

For example, V, = sin 81 = sin 8c is the left 

vertex in Fig. 4a of the main text, which can 

be considered a u quark “decaying” into an s 

quark. 

We know from experiment that sin 8, = 

0.23 f 0.01. But further, from recent 

measurements of the lifetime of the b quark 

and the branching ratio l-b JrbC, we know 

that €Iz and O3 are both small. That is, we 

have the information 

and 

These results imply that we can take c2 and c3 

to be unity and obtain the approximation 

(‘47) 

In terms of quark mixing, CP violation in 

the Ko-Ro system is described by a secondorder 

imaginary amplitude proportional to 

s233 sin 6. In other words, the upper 2 by 2 

piece of the matrix in Eq. A7 has this new 

imaginary contribution when compared with 

the Cabibbo matrix of Eq. 3. By using the 

Feynman diagram of Fig. 5 in the main text, 

the Ko-Ro transition-matrix element (traditionally 

called Mlz) can be calculated in 

terms of the weak-interaction Hamiltonian 

and the entries of the mixing matrix V. 

The older parameterization of CP violation, 

which involves the parameter E, is 

model-independent. It focuses only on the 

properties of CP symmetry and the kaons 

themselves. It does not even need quarks. 

The value of E is determined by experiments 

(see below) and is directly related to Mk2. It 

remains for a particular formalism (such as 

that described here) to successfully predict 

M12 in a consistent manner. In particular, 

within the K-M formalism it is hard to obtain 

a large enough value for the CP-violating 

amplitude E even if one assumes 6 = x/2, 

because sz and s3 are so small. In fact, agree- 

ment with the measured value of E cannot be 

obtained unless the mass of the t quark is 

equal to or greater than 60 GeV/c2. Because 

the t quark has not yet been found, this 

possibility remains open. 

One way in which CP violation is observed 

in the Ko-KO system was described in 

the main text. Another way is to detect an 

anomalous number of decays to leptons of 

the “wrong” sign. In the absence of mixing 

one ordinarily expects positively charged 

leptons from the KO parent and negatively 

charged leptons from the Io parent; that is, 

KO = ds decays into d(uau) or d(uQ+v), and 

Io = as decays into a(udi) or a(uQ-C), as 

shown in Fig. Al. However, to describe the 

propagation of a KO (or a Ro), it must be 

decomposed into KL and Ks states each of 

which is an approximate CP eigenstate containing 

approximately equal amplitudes of 

KO and KO. Since the KS lifetime is negligibly 

short, it is easy to design experiments to 

measure decays of the KL only. If CP were an 

exact symmetry, then the KO and Io components 

of the KL would have equal amplitudes 

and would each provide exactly the same 

number of leptonic decays; that is, just as 

many “wrong”-sign leptons would come 

From decays of the Ko component (the antiparticles 

of Fig. Al) as “right”-sign leptons 

come from decays of the KO component (Fig. 

AI). The deviation from exact equality is 

another measure of CP violation. 

What about CP violation in other neutralboson 

systems? If one does the same type of 

anaylsis as is often done for the kaon system, 

one can phenomenologically describe CP 

violation by 

where cpo is a neutral boson, Go is its conjugate 

under C, E? is the CP-violating parameter 

specific to that boson, and 

L(t) = %(exp[-(irnl + rl/2)t] 

k exp[-(imZ + Tz/2)t]] . 

(A9) 

125

Addendum 

(Here the labels “I” and “2” refer to the 

approximate CP eigenstates.) The value of 

IcI

Addendum 

W- 

-- - -.__ 

- -__ - -__. 

- __ -. _- --____ 

- ___--- - I 

Fig. A2. Feynman diagram for the mixing between B: and B: mesons induced by 

second-order weak interactions. This diagram is analogous to that presented in the 

main text (Fig. 5) for mixing in the KO-Ko system. 

momentum are the decay products of a comnon 

b6 quark pair. Such parent quark pairs 

host always appear as BBmeson pairs. 

Suppose there was little or no mixing beween 

B:and E:. Then one would expect the 

Ibserved ratio of the decays of B,B: pairs 

nto back-to-back muon pairs with the same 

:harge [(+ +) or (- -)] to the decays of B:B: 

to be 

,airs with opposite charge [(+-)I 

tbout 25 percent. This ratio is deduced by 

he following argument. Without mixing (a) 

he main contribution to unlike pairs comes 

?om the direct decay of both quarks (b - 

y-i and 6 - $+v), and (b) the main conribution 

to like pairs comes from one prinary 

decay and one secondary decay (for 

:xample, b - cp-i and 6 - - 3.p-i). The 

,elathe rates can be calculated from the 

:nown weakdecay parameters, and one obaim 

the value 0.24 for the ratio of like- to 

mlike-sign pairs. 

However, with mixing (such as that shown 

in Fig. A2) one can sometimes have 

processes like sb - scp-i and s6 - 3.b - 

kp-i. This transforms some of the expected 

unlike-sign events into like-sign events. In 

fact, for a mixing of 10 percent, this changes 

the ratio of like- to unlike-sign events from 

about Y4 to about 112. 

Indeed, the UAl experiment at CERN 

sees a ratio of 50 percent. This result can be 

explained only by a large mixing between 

B: and Bf, which overwhelms the tendency 

for the band 6 quarks to decay into oppositesign 

pairs. Since one needs significant mixing 

to observe CP violation, there is hope of 

learning more about CP, depending on the 

(as yet undetermined) values of the mass- 

matrix parameters for B!and B:(that is, m1, 

m2, r~, and r2). 

For further details of this fascinating subject, 

we recommend the review “Quark Mixing 

in Weak Interactions” by Ling-Lie Chau 

(Physics Reports 95: 1( 1983)). W 

127

Experiments to Test 

Flash chambers discharging like neon lights, giant spectrometers, stacks of 

crystals, tons ofplastic scintillators, thousands ofprecisely strung 

wires-all employed to test the ideas of unifiedfield theories. 

I 

t has long been a dream of physicists to produce a unified field theory of the 

forces in nature. Much of the current experimental work designed to test such 

theories occurs at the highest energies capable of being produced by the latest 

accelerators. However, elegant experiments can be designed at lower energies 

that probe the details of the electroweak theory (in which the electromagnetic and 

weak interactions have been partially unified) and address key questions about the 

further unification of the electroweak and the strong interactions. (See “An Experimentalist’s 

View of the Standard Model” for a brief look at the current status of 

the quest for a unified field theory.) 

In this article we will describe four such experiments being conducted at Los 

Alamos, often with outside collaborators. The first, a careful study of the beta decay 

of tritium, is an attempt to determine whether or not the neutrino has a mass and 

thus whether or not there can be mixing between the three known lepton families 

(the electron, muon, and tau and their associated neutrinos). 

Two other experiments examine the decay of the muon. The first is a search for 

rare decays that do not involve neutrinos, that is, the direct conversion across 

lepton families of the muon to an electron. The muon is a duplicate, except for a 

greater mass, of the electron, making such a decay seem almost mandatory. 

Detection ofa rare decay, or even the lowering of the limits for its occurrence, would 

tell us once again more about the mixing between lepton families and about possible 

violation of lepton conservation laws. At the same time, precision studies of 

ordinary muon decay, in which neutrinos are generated (the muon is accompanied 

by its own neutrino and thereby preserves muon number), will help test the stucture 

of the present theory describing the weak interaction, for example, by setting limits 

on whether or not parity conservation is restored as a symmetry at high energies. 

The electron spectrometer for the tritium beta decay experiment under 

construction. The thin copper strips evident in the entrance cone region to 

the right and at thefirst narrow region toward the center are responsible 

for the greatly improved transmission of this spectrometer. 

128

Unification Schemes 

by Gary H. Sanders 

I 

129

The intent of the fourth experiment is to 

measure interference effects between the 

neutral and charged weak currents via scattering 

experiments with neutrinos and electrons. 

If destructive interference is detected, 

then the present electroweak theory should 

be applicable even at higher energies; if constructive 

interference is detected, then the 

theory will need to be expanded, say by 

including vector bosons beyond those (the 

Zo and the L@) already in the standard 

model. 

Tritium Beta Decay 

In 1930 Pauli argued that the continuous 

kinetic energy spectrum of electrons emitted 

in beta decay would be explained by a light, 

neutral particle. This particle, the neutrino, 

was used by Fermi in 1934 to account quantitatively 

for the kinematics of beta decay. In 

1953, the elusive neutrino was observed 

directly by a Los Alamos team, Fred Reines 

and Clyde L. Cowan, using a reactor at Hanford. 

Though the neutrino has generally been 

taken to be massless, no theory requires neutrinos 

to have zero mass. The current experimental 

upper limit on the electron neutrino 

mass is 55 electron volts (ev), and the 

Russian team responsible for this limit 

claims a lower limit of 20 eV. The mass of the 

neutrino is still generally taken to be zero, for 

historical reasons, because the experiments 

done by the Russian team are extremely 

complex, and because masslessness leads to a 

pleasing simplification of the theory. 

A more careful look, however, shows that 

no respectable theory requires a mass that is 

identically zero. Since we have many neutrino 

flavors (electron, muon and tau neutrinos, 

at least), a nonzero mass would immediately 

open possibilities for mixing between 

these three known lepton families. 

Without regard to the minimal standard 

model or any unification schemes, the 

possible existence of massive neutrinos 

points out our basic ignorance ofthe origin of 

the known particle masses and the family 

structure of particles. 

An Experimentalist’s 

View of the 


T 

he dream of physicists to produce a 

unified field theory has, at different 

times in the history of physics, appeared 

in a different light. For example, one 

of the most astounding intellectual achievements 

in nineteenth century physics was the 

realization that electric forces and magnetic 

forces (and their corresponding fields) are 

different manifestations of a single electromagnetic 

field. Maxwell’s construction of 

the differential equations relating these two 

fields paved the way for their later relation to 

special relativity. 

QED. The most wccessful field theory to 

date, quantum electrodynamics (QED), appears 

to have provided us with a complete 

description of the electromagnetic force. 

This theory has withstood an extraordinary 

array of precision tests in atomic, nuclear, 

and particle physics, and at low and high 

energies. A generation of physicists has 

yearned for comparable field theories describing 

the remaining forces: the weak interaction, 

the strong interaction, and gravity. 

An even more romantic goal has been the 

notion that a single field theory might describe 

all the known physical interactions. 

Electroweak Theory. In the last two decades 

we have comt: a long way towards realizing 

this goal. The electromagnetic and weak 

interactions appear to be well described by 

the Weinberg-Salam-Glashow model that 

unifies the two fields in a gauge theory. (See 

“Particle Physics and the Standard Model” 

for a discussion of gauge theories and other 

details just briefly mentioned here.) This 

electroweak theory appears to account for 

the apparent difference, at low energies. between 

the weak interaction and the electromagnetic 

interaction. As the energy of an 

interaction increases, a unification is 

achieved. 

So far, at energies accessible to modern 

high-energy accelerators, the theory is supported 

by experiment. In fact, the discovery 

at CERN in 1983 of the heavy vector bosons 

UT+, W-, and Zo, whose large mass (compared 

to the photon) accounts for the relatively 

“weak” nature of the weak force, 

beautifully confirms and reinforces the new 

theory. 

The electroweak theory has many experimental 

triumphs, but experimental 

physicists have been encouraged to press 

ever harder to test the theory, to explore its 

range of validity, and to search for new fundamental 

interactions and particles. The experience 

with QED, which has survived 

decades of precision tests, is the standard by 

which to judge tests of the newest field theories. 

QcD. A recent, successful field theory that 

describes the strong force is quantum 

chromodynamics (QCD). In this theory the 

strong force is mediated by the exchange of 

color gluons and a coupling constant is determined 

analogous to the fine structure constant 

of the electroweak theory. 

Standard Model. QCD and the electroweak 

theory are now embedded and 

united in the minimal standard model. This 

model organizes all three fields in a gauge 

130

~ 

Experiments To Test Unification Schemes 

I 

Table 

The first three generations of elementary particles. 

Family: 

Doublets 

Singlets: 

Quarks: 

Leptons: 

I 

( ;IL 

( 3), 

theory of electroweak and strong interactions. 

There are two classes of particles: spin- 

I/z particles called fermions (quarks and leptons) 

that make up the particles of ordinary 

matter, and spin- I particles called bosons 

that account for the interactions between the 

fermions 

In this theory the fermions are grouped 

asymmetrically according to the "handedness'' 

of their spin to account for the experimentally 

observed violation of CP symmetry 

Particles with right-handed spin are 

grouped in pairs or doublets; particles with 

left-handed spin are placed in singlets. The 

exchange of a charged vector boson can convert 

one particle in a given doublet to the 

other, whereas the singlet particles have no 

weak charge and so do not undergo such 

transitions. 

The Table shows how the model, using 

this scheme, builds the first three generations 

of leptons and quarks. Since each quark (u, d, 

c, s, t, and b) comes in three colors and all 

fermions have antiparticles, the model includes 

90 fundamental fermions 

The spin-l boson mediating the electromagnetic 

force is a massless gauge boson, 

that is, the photon y. For the weak force, 

there are both neutral and charged currents 

that involve, respectively, the exchange of 

the neutral vector boson Zo and the charged 

vector bosons W+ and W-. The color force 

of QCD involves eight bosons called gluons 

that carry the color charge. 

The coupling constants for the weak and 

electromagnetic interactions, gwk and gem, 

arc related by the Weinberg angle Ow. a mixing 

angle used in the theory to parametrize 

the combination of the weak and electromagnetic 

gauge fields. Specifically, 

Only objects required by experimental results 

are in the standard model, hence the 

term minimal. For example, no right-handed 

neutrinos are included. Other minimal assumptions 

are massless neutrinos and no 

requirement for conservation of total lepton 

number or of individual lepton flavor (that 

is, electron, muon, or tau number). 

The theory, in fact, includes no mass for 

any of the elementary particles. Since the 

vector bosons for the weak force and all the 

fermions (except perhaps the neutrinos) are 

known to be massive, the symmetry of the 

theory has to be broken. Such symmetrybreaking 

is accomplished by the Higgs mechanism 

in which another gauge field with its 

yet unseen Higgs particle is built into the 

theory. However. no other Higgs-type particles 

are included. 

Many important features are built into the 

minimal standard model. For example, lowenergy, 

charged-current weak interactions 

are dominated by V- :I (vector minus axial 

vector) currents; thus, only left-handed W' 

bosons have been included. Also, since neutrinos 

are taken to be massless, there are 

supposed to be no oscillations between neutrino 

flavors. 

There are many possibilities for extensions 

to the standard model. New bosons, 

families of particles, or fundamental interactions 

may be discovered, or new substructures 

or symmetries may be required. The 

standard model, at this moment, has no 

demonstrated flaws, but there are many potential 

sources of trouble (or enlightenment). 

GUT. One of the most dramatic notions 

that goes beyond the standard model is the 

grand unified theory (GUT). In such a theory, 

the coupling constants in the electroweak 

and strong sectors run together at 

extremely high energies (IOi5 to IOi9 gigaelectron 

volts (GcV)). All the fields are unified 

under a single group structure, and a new 

object, the X, appears to generate this grand 

symmetry group. This very high-energy mass 

scale is not directly accessiblc at any conceivable 

accelerator. To explore the wilderness 

between present mass scales and the 

GUT scale. alas, all high-energy physicists 

will have to be content to work as low-energy 

physicists. Some seers believe the wilderness 

will be a desert, devoid of striking new physics. 

In the likely event that the desert is found 

blooming with unexplored phenomena, the 

journey through this terra incognita will be a 

long and fruitful one, even ifwe ure restricted 

to feasible tools. 8 

131

The reaction studied by all of the experiments 

mentioned is 

3H + 

3He+ + e- + ;e. 

This simple decay produces a spectrum of 

electrons with a definite end point energy 

(that is, conservation of energy in the reaction 

does not allow electrons to be emitted 

with energies higher than the end point 

energy). In the absence of neutrino mass, the 

spectrum, including this end point energy, 

can be calculated with considerable 

precision. Any experiment searching for a 

nonzero mass must measure the spectrum 

with sufficient resolution and control of systematic 

effects to determine if there is a 

deviation from the expected behavior. 

Specifically, an end point energy lower 

than expected would be indicative of energy 

carried away as mass by the neutrino. 

In 1972 Karl-Erik Bergkvist of the University 

of Stockholm reported that the mass of 

the electron antineutrino cr was less than 55 

eV. This experiment used tritium embedded 

in an aluminum oxide base and had a resolution 

of 50 eV. The Russian team set out to 

improve upon this result using a better spectrometer 

and tritium bound in valine 

molecules. 

Valine is an organic compound, an amino 

acid. A molecular biologist in the Russian 

collaboration provided the expertise 

necessary to tag several of the hydrogen sites 

on the molecule with tritium. This knowledge 

is important since one of the effects 

limiting the accuracy of the result is the 

knowledge of the final molecular states after 

the decay. 

Also important was the accurate determination 

of the spectrometer resolution 

funcfion, which involved a measurement of 

the energy loss of the beta electrons in the 

valine. This was accomplished by placing an 

ytter-bium-169 beta source in an identical 

source assembly and measuring the energy 

loss of these electrons as they passed through 

the valine. 

The beta particles emitted from the source 

were analyzed magnetically in a toroidal beta 

spectrometer. This. kind of. spectrometer 

provides the largest acceptance for ,a given 

resolution of any known design, and the 

Russians made very significant advances. , 

The Los Alamos research group, as we shall 

see, has improved the spectrometer design 

even further. 

In 1980 the Russian group published a 

positive result for the electron antineutrino 

mass. After including corrections for the uncertainties 

in resolution and the final state: 

spectrum, they quoted a 99 per cent confidence 

level value of 

. . 

I4 < mie< 46 eV . 

The result was received with.great excitement, 

but two specific criticisms emerged. 

John J. Simpson of the University of Guelph 

pointed out that the spectrometer resolution 

was estimated neglecting the intrinsic 

linewidth of the spectrum .of the ytterbium-I69 

calibration source. The ex-, 

perimenters then measured the source 

linewidth to be 6.3 eV; their revised analysis 

lowered the best value of the neutrino mass 

from 34.3 to 28. eV. The basic result of a 

finite mass survives this reanalysis, according 

to the authors, but it should,be noted that 

the result is very sensitive to the calibration 

linewidth. Felix Boehm of the California Institute 

of Technology has observed that with 

an intrinsic linewidth ofonly 9 eV, the 99 per 

cent confidence level result would become 

consistent with zero. 

The second criticism related to .th.e assumption 

made about the energy of the final 

atomic states of helium-3. The valine 

molecule provides a complex environment, 

and the branching ratios into the 2s and 

Is states of heliumi3 are difficult to estimate. 

Thus the published result may prove to be 

false. 

This discussion illustrates the.difficulty of 

experiments of this kind. Each effort 

produces, in addition to the published measurement, 

a roadniap to the next generation 

experiment. The Russian team built upon its 

1980 result and produced a substantially improved 

apparatus that yielded a new meas- 

urement in 1983. 

The spectrometer was improved by adding 

an electrostatic field between the source and 

the magnetic spectrometer that could be used 

to accelerate the incoming electrons. The 

beta spectrum could then be measured, 

under conditions of constant magnetic field, 

by sweeping the electrostatic field to select 

different portions of the spectrum. This technique 

(originally suggested by the Los Alamos 

group) provides a number of advantages. 

The magnetic spectrometer always 

sees electrons in the same energy range, 

providing constant detection efficiency 

throughout the measured spectrum. The 

magnetic field can also be set above the beta 

spectrum end point with the electrostatic 

field accelerating electrons from decays in 

the source into the spectrometer acceptance. 

This reduces the background by a large factor 

by making the spectrometer insensitive to 

electrons from decays of tritium contamination 

in the spectrometer volume. 

Also, finite source size, which produces a 

larger image at the spectrometer focal plane, 

was optically reduced by improved focusing 

at the source, yielding a higher count rate 

with better resolution. 

The improved spectrometer had a resolution 

of25 eV, compared to 45 eV in the 1980 

experiment. Background was reduced by a 

factor of 20, and the region of the spectrum 

scanned was increased from 700 eV to 1750 

eV. 

The controversial spectrometer resolution 

function was determined using a different 

line of the ytterbium-I69 source, and the 

Russians measured its intrinsic linewidth to 

be 14.7 eV. They also studied ionization 

losses by measuring the ytterbium- I69 spectrum 

through varying thicknesses of valine, 

yielding a considerably more accurate resolution 

function. 

The data were taken in 35 separate runs 

and the beta spectrum (Fig. 1) was fit by an 

expression that included the ideal spectral 

shape and the experimental corrections. The 

best fit gave 

I 

132.


0.04 1 

18.50 18.58 18.66 

Kinetic Energy (keV) 

Fig. I. Electron energy spectrum for 

tritium decay. This figure shows the 

1983 Russian data as the spectrum 

drops toward an end point energy of 

about 18.58 keV. The difference in the 

best fit to the data (solid line) and the 

‘t for a zero neutrino mass (dashed 

ne) is a shift to lower energies that 

wresponds to a mass of about 33.0 eV. 

Figure adapted from Michael H. 

aaevitz, “Experimental Results on 

leutrino Masses and Neutrino Osillations, 

’’ page 140, in Proceedings 

f the 1983 International Symposium 

n Lepton and Photon Interactions at 

[igh Energies, edited by David G. 

:assel and David L. Kreinick (Ithaca, 

lew York:F;R. Newman Laboratory of 

ruclear Studies, Cornell University, 

983).) 

ith a 99 per cent confidence limit range of 

20 < mCc< 55 eV . 

hese results were derived by making 

trticular choices for the final state spectra. 

ifferent assumptions for the valine molecu- 

lar final states and the helium-3 molecular, 

atomic, and nuclear final states can produce 

widely varying results. 

The physics community has been rantalized 

by the prospect that neutrinos have 

significant masses. Lepton flavor transitions, 

neutrino oscillations, and many other 

phenomena would be expected if the result is 

confirmed. The range of systematic effects, 

however, urges caution and enhanced efforts 

by experimenters to attack this problem in an 

independent manner. There are currently 

more than a dozen groups around the world 

engaged in improved experiments on tritium 

beta decay. A wide range of tritium sources, 

beta spectrometers, and analysis techniques 

are being employed. 

The Tritium Source. In an ambitious attempt 

to use the simplest possible tritium 

source, a team from a broad array of technical 

fields at Los Alamos is attempting to 

develop a source that consists of a gas of free 

(unbound) tritium atoms. Combining diverse 

capabilities in experimental particle 

physics, nuclear physics, spectrometer design, 

cryogenics, tritium handling, ultraviolet 

laser technology, and materials science, this 

team has developed a nearly ideal source and 

has made numerous improvements in electrostatic-magnetic 

beta spectrometers. 

The two most significant problems come 

from the scattering and energy loss of the 

electrons in the source and from the atomic 

and molecular final states of the helium-3 

daughter. These effects are associated with 

any solid source. Thus the ideal source would 

appear to be free tritium nuclei, but this is 

ruled impractical by the repulsive effects of 

their charge. 

The next best source is a gas of free tritium 

atoms. Detailed and accurate calculations of 

the atomic final states and electron energy 

losses can be performed. Molecular effects, 

including final state interactions, breakup, 

and energy loss in the substrate, are 

eliminated. Since the gas contains no inert 

atoms, the effect of energy loss and scattering 

in the source are reduced accordingly. Even 

the measurement of the beta spectrometer 

resolution function is simplified. 

The forbidding technical problem of such 

a design is building a source rich enough and 

compact enough to yield a useful count rate. 

Only one decay in IO’ produces an electron 

with energy in the interesting region near the 

end point where the spectrum is sensitive to 

neutrino mass. 

The Los Alamos group was motivated by a 

1979 talk given by Gerard Stephenson, of the 

Physics and Theoretical Divisions, on neutrino 

masses. They recognized quite early, in 

fact before the 1980 Russian result, that 

atomic tritium would be a nearly ideal 

source. In their first design, molecular 

tritium was to be passed through an extensive 

gas handling and purification system 

and atomic tritium prepared using a discharge 

in a radio-frequency dissociator. The 

pure jet of atomic tritium was then to be 

monitored for beta decays. It was clear, however, 

that the tritium atoms needed to be 

used more efficiently. 

Key suggestions were made at this point 

by John Browne of the Physics Division and 

Daniel Kleppner of the Massachusetts Institute 

of Technology. Advances had been 

made in the production of dense gases of 

spin-polarized hydrogen. The new techniques-in 

which the atomic beam was 

cooled and then contained in a bottle made 

of carefully chosen materials observed to 

have a low probability for promoting recombination 

of the atoms-promised a possible 

intense source of free atomic tritium. The 

collaboration set out to develop and demonstrate 

this idea. Crucial to the effort was the 

participation of Laboratory cryogenics 

specialists. 

The resulting tritium source (Fig. 2) 

circulates molecular tritium through a radiofrequency 

dissociator into a special tube of 

aluminum and aluminum oxide. Because the 

recombination rate for this material near 120 

kelvins is very low, the system achieves 80 to 

90 per cent purity of atomic tritium. The 

electrons from the beta decay of the atomic 

tritium are captured by a magnetic field, and 

then electrostatic acceleration, similar to that 

employed by the Russians, is used to trans- 

133

I 

! 

Atomic Tritium Source Region 

Transport and Focusing 

Region 

I 

I 

! 

Electron Gun 

Superconducting 

! 

, 

I 

! 

! 

I 

I 

I 

! 

I 

I 

Fig. 2. The tritium source. Molecular tritium passes through 

the radiofrequency dissociator and then into a I-meterlong 

tube as a gas of free atoms. The tube-aluminum with a 

surface layer of aluminum oxide-has a narrow range 

around a temperature of 120 kelvins at which the molecular 

recombination rate is very low, permitting an atom to 

experience approximately 50,000 collisions before a 

molecule is formed. The resulting diffuse atomic gas fills the 

tube, and mercury-diffusion pumps at the ends recirculate it 

through the dissociator. Typically, the system achieves 80 fo 

90 per cent purity of atomic tritium. By measuring the 

spectrum when the dissociator is off; the contribution from 

the 10 to 20per cent contamination of molecular tritium can 

be determined and subtracted, resulting in a pure atomic 

tritium electron spectrum. 

A superconducting coil surrounds the tube with afield oj 

1.5 kilogauss. At one end the winding has a reflectingfield 

provided by a magnetic pinch. These fields capture electrons 

from beta decays with 95per cent efficiency. 

The other end of the tube connects to a vacuum region and 

has coils that transport and, importantly, focus an image oj 

the electrons into the spectrometer (Fig 3). The tube is held 

at a selecfed voltage between -4 and -20 kilovolfs, and 

electrons exit the source to ground potential. Thus, electrons 

from decays in the source tube are accelerated by a known 

amount to an energy above that of electrons from decays in 

port the electrons toward the spectrometer. 

During this transport, focusing coils and a 

collimator are used to form a small image of 

the electron source in the spectrometer. 

Development of this tritium source required 

solving an array of problems associated 

with a system that was to recirculate 

atomic tritium. Everything had to be extremely 

clean, and no organic materials were 

allowed; all surfaces are glass or metal. Conducting 

materials had to be used wherever 

insulators could collect charge and introduce 

a bias. The aluminum oxide coating in the 

tube is so thin that electrons simply tunnel 

through it, thus providing a conducting surface 

that does not encourage recombination. 

Special mercury-diffusion pumps and custom 

cryopumps, free of oil or other organic 

materials, had to be fabricated. Every part of 

the tritium source was an exercise in 

materials science. 

134 

The idea of using electrostatic acceleration 

at the output of the source was first proposed 

by the group at Los Alamos in 1980 and 

subsequently used in the measurement described 

in the 1983 Russian publication. Accelerating 

the electrons to an energy above 

that of electrons from tritium that decays in 

the spectrometer both strongly reduces the 

background and also improves the acceptance 

of electrons into the spectrometer. 

However, this technique necessitates a larger 

spectrometer. 

There are two other important systematic 

effects that need to be dealt with: the source 

image seen by the spectrometer should be 

small, and electrons produced by decays in 

the tube that suffer scattering off the walls 

have an energy loss that distorts the 

measured spectrum. The focusing coil and 

the final collimator address both effects, 

providing a small image. The only energy 

loss mechanism remaining is in the tritium 

gas itself, where losses are less than 2 eV. 

development of the Russian design. Electrons 

from the source pass through the entrance 

cone and are focused onto the spectrometer 

axis. One very significant improvement 

in the spectrometer is the design of the 

conductors running parallel to the spec. 

trometer axis that do this focusing. In thc 

Russian apparatus, the conductors were 

thick water-cooled tubes. Most electron: 

strike the tubes and, as a result of this loss


Thin Curved 

Conductors 

Image 

Points 

or 

Decay 

Electrons 

Cone 

Flow of Thick Thin 

Electrons Conductors Conductors 

the spectrometer. Additional pumps also sharply reduce the 

amount of tritium escaping into the spectrometer. 

Several sophisticated diagnostic systems monitor source 

output and stability. Beta detectors mounted in the focus 

region in front of the collimator measure the total decay rate 

from molecular and atomic tritium, whereas the fraction of 

tritium in molecular form is monitored by an ultraviolet 

(1 027 angstroms wavelength) laser system developed by 

members of Chemistry Division that uses absorption lines of 

molecular tritium. A high-resolution electron gun is used to 

monitor energy loss in both the gas and the spectrometer. 

This gun is also used to measure the important spectrometer 

esolution function directly. 

Fig. 3. The spectrometer. Electrons from the source (Fig. 2) 

that pass through the collimator (with an approximate 

aperture of 1 centimeter) open into a cone shaped region in 

the spectrometer with a maximum half angle of 30 degrees. 

Electrons between 20 and 30 degrees pass between thin 

conducting strips into the spectrometer and are focused onto 

the spectrometer axis. This focus serves as a virtual image of 

the source. Transmission has been greatly improved over the 

Russian design through the use of thin conductors in all 

regions of electron flow (see opening photograph for a view 

of these conductors). The final focal plane detector is a 

position-sensitive, multi- wire proportional gas counter, also 

an improvement over previous detectors. 

heir spectrometer has low transmission. 

The Los Alamos spectrometer uses thin 

!O-mil strips for each ofthe conductors in the 

.egion within the transport aperture. This 

ichieves an order of magnitude higher transnission, 

essential in yielding a useful count 

‘ate in an experiment with a dilute gas 

ource. 

Another benefit of the thin strips is that 

hey can be formed easily. In fact, optical 

~alculations accurate to third order dictate 

he curvature of the entrance and exit strips. 

The improved focusing properties of this 

irrangement yield an acceptance three times 

iigher than the Russian device with no comwomise 

in resolution. 

The experimenters expect to be taking 

lata throughout the latter part of 1984. They 

xpect an order of magnitude less backround 

and an order of magnitude larger 

eometric acceptance than the Russian ex- 

periment. The design calls for a resolution 

between 20 and 30 eV, with a sensitivity to 

neutrino masses less than IO eV. Even with 

their dilute gas source, they estimate a data 

rate in the region within 100 eV of the spectrum 

end point of about 1 hertz, fully competitive 

with rates obtained using solid 

sources. 

Many groups around the world are 

vigorously pursuing this measurement. No 

other effort. however, will produce a result as 

free of systematic problems as the Los Alamos 

project. Other experiments are employing 

solid sources or, at best, molecular 

sources. Many have adopted an electrostatic 

grid system that introduces its own problems. 

To date, no design promises as clean a 

measurement. This year may well be the year 

in which the problem of neutrino mass is 

settled. The quantitative answer willbe an 

important tool in uncovering the very poorly 

understood relations between lepton 

families. No deep understanding of the models 

that unify the forces in nature can be 

expected without precise knowledge of the 

masses of neutrinos. 

Rare Decays of the Muon 

The muon has been the source of one 

puzzle after another. It was discovered in 

1937 in cosmic radiation by Anderson and 

Neddermeyer and by Street and Stevenson 

and was assumed to be the meson of 

Yukawa’s theory of the nuclear force. 

Yukawa postulated that the nuclear force, 

with its short range, should be mediated by 

the exchange of a massive particle, a meson. 

This differs from the massless photon of the 

infinite-range electromagnetic force. The 

muon mass, about 200 times the electron 

mass, fit Yukawa’s theory well. 

135

It was only after World War I1 ended that 

measurements of the muon’s range in 

materials were found to be inconsistent with 

a particle interacting via a strong nuclear 

force. Discovery of the pion, or pi meson, 

settled the controversy. To this day, however, 

casual usage sometimes includes the 

erroneous phrase “mu meson”. 

With the resolution of the meson problem, 

however, the muon had no reason to be. It 

was simply not necessary. The muon appeared 

to be, in all known ways, a massive 

electron with no other distinguishing attributes. 

A famous quotation of I. I. Rabi 

summarized the mystery: “The muon, who 

ordered that?’ 

This question is none other than the 

family problem described earlier. Today, the 

mystery remains, but its complexity has 

grown. Three generations of fermions exist, 

and the mysterious relation of the muon to 

the electron is replicated in the existence of 

the tau, discovered in 1976 by Martin Per1 

and collaborators. The three generation 

scheme is built into the minimal standard 

model, but there is little insight to guide us to 

the ultimate number of generations. 

Is there a conservation number associated 

with each family or generation? Are there 

selection rules or fundamental symmetries 

that account for the apparent absence of 

some transitions between these multiplets? 

Vertical and horizontal transitions between 

quark states do occur. Processes involving 

neutrinos connect the lepton generations. 

Can the pattern of these observed transitions 

give us a clue as to why we are blessed with 

this peculiar zoology? Should we look harder 

for the processes we have not observed? 

Rabi’s question, in its most modern form, is 

a rich and bewildering one, and many experimental 

groups have taken up its 

challenge by pursuing high sensitivity studies 

of the rare and unobserved reactions that 

may connect the generations. 

With the muon and electron virtual 

duplicates of each other, it was expected that 

the heavier muon would decay by simple, 

neutrinoless processes to the electron. Transitions 

such as p+ - e+ e+ e-, p+ - e+ y, or 

’ 

Conservation ~aws: CL~ = Constant, ZL, = Constant, XL 

Allowed Decay: p+ - e+ v, c, 

p- Z - e- Z (where Z signifies that the 

interaction is with a nucleus) were expected. 

Estimates of the rates for these processes 

using second-order, current-current weak interactions 

gave results too small to observe. 

In fact, the results were much smaller than 

the 1957 limit for the branching ratio for p+ 

- e+ y, which was < 2 X IO-’ (a branching 

ratio is the ratio of the probability a decay 

will occur to the probability of the most 

common decay). 

A better early model appeared in 1957 

when Schwinger proposed the intermediate 

vector boson (now called W and observed 

directly in 1983) as the mediator of the 

charged-current weak interaction. With this 

model and under most assumptions, rates 

larger than the experimental limits were 

predicted for the three reactions. The failure 

to observe these decays required a dynamical 

suppression or a new conservation law. Despite 

the discussion to follow, the situation 

today has changed very little. The measured 

limits are more swingent, though, by many 

orders of magnitude. 

The first proposal for lepton number con- 

Forbidden Decays: 

p+ - e+ e+ e- I 

p-Z-e-Z 

p- 2 - e- (2-2) 

p+ - e+Vevu 

servation came in 1953. In fact, there have 

been three different schemes for conserving 

lepton number. The 1953 Konopinski- 

Mahmoud scheme cannot accommodate 

three lepton generations and has not 

survived. A scheme in which lepton number 

is conserved by a multiplicative law was 

proposed in 196 I by Feinberg and Weinberg, 

but this method is not the favored conservation 

law. An early experiment with a neutrino 

detector at the Clinton P. Anderson 

Meson Physics Facility in Los Alamos 

(LAMPF) has removed the multiplicative 

law from favor, and the current experiment 

to study neutrino-electron scattering, described 

later in this article, has set even more 

stringent limits on such a law. 

The most favored scheme is additive lepton 

number conservation, proposed in 1957 

by Schwinger, Nishijima, and Bludman. In 

this scheme, any process must separately 

conserve the sum of muon number and the 

sum of electron number. Table 1 shows the 

assignment of lepton numbers used. The extension 

to the third lepton flavor, tau, is 

obvious and natural. 

136


10-1 

e o e p+ +e+? 

e 5 p+ -+ e+e+e- 

.- 0 

0 

+ 

o p-Z -+ e-Z 

a m10-4 - a p+ 4 e+yy 

OI 

.- C 

Qlp 

0 

-6 

- 

m 10-7 

10-10 i- 

..$I *e 0 0 

A 

4 

0 e 

< 

1950 1960 1970 l! 

Year 

Fig. 4. The progressive drop in the experimentally 

determined upper limit 

for the branching ratio of several 

muon-number violating processes 

shows a gap in the late 1960s. Essentially, 

this gap was the result of a belief 

by particle physicists in lepton number 

conservation. 

These schemes require, as the table hints, a 

distinct neutrino associated with each lepton. 

In a 1962 experiment the existence of 

separate muon and electron neutrinos was 

confirmed. 

With a conservation law firmly entrenched 

in the minds of physicists, searches 

for decays that did not conserve lepton number 

seemed pointless. In a 1963 paper 

Sherman Frankel observed “Since it now 

appears that this decay is not lurking just 

beyond present experimental resolution, any 

further search . . . seems futile.” 

In retrospect it can be said that the particle 

physics community erred. The conclusion 

stated in the previous paragraph resulted in a 

nearly complete halt to efforts to detect 

processes that did not conserve lepton number-and 

this on the basis ofa law postulated 

without any rigorous or fundamental basis! 

It is easy to justify these assertions. Figure 

4 shows that the experimental limits on rare 

decays were not aggressively addressed between 

1964 and the late 1970s. This era of 

inattention ended abruptly when an experimental 

rumor circulated in 1977-an er- 

roneous report terminated a decade of theoretical 

prejudice almost overnight! This 

could not have been the case if lepton conservation 

was required by fundamental ideas. 

In 1977 a group searching for the process 

p+ - e’ y at the Swiss Institute for Nuclear 

Research (SIN) became the inadvertent 

source of a report that the decay had been 

seen. The experiment, sometimes referred to 

as the “original SIN” experiment, was an 

order of magnitude more sensitive than any 

prior search for this decay and eventually set 

a limit on the branching ratio of 1.0 X low9 . 

A similar effort at the Canadian meson factory, 

TRIUMF, produced a limit of 3.6 X 

at about the same time. 

The Crystal Box. The extraordinary controversy 

generated by the “original SIN” report 

motivated a Los Alamos group to attempt 

a search for p’ - e+ y with a sensitivity 

to branching ratios of about IO-”. This 

experiment was carried out in 1978 and 

1979, using several new technologies and a 

new type of muon beam at LAMPF, and 

yielded an upper limit of 1.7 X 10-”(90 per 

cent confidence level). That result stands as 

the most sensitive limit on the decay to date 

but should be surpassed this year by an experiment 

at LAMPF called the Crystal Box 

experiment. 

This experiment was conceived as the 

earlier experiment came to an end. By 

searching for three rare muon decays simultaneously, 

the experiment would be a major 

advance in sensitivity and breadth. Several 

new technologies would be exploited as well 

as the capabilities of the LAMPF secondary 

beams. 

In any search for a very rare decay, sensitivity 

is limited by two factors: the total 

number of candidate decays observed, and 

any other process that mimics the decay 

being searched for. The design of an experi- 

ment must allow the reliable estimate of the 

contribution of other processes to a false 

signal. This is generally done by a Monte- 

Carlo simulation of these decays that includes 

taking into account the detector 

properties. 

In the absence of background or a positive 

signal for the process being studied, the number 

of seconds the experiment is run translates 

linearly into experimental sensitivity. 

However, when a background process is detected, 

sensitivity is gained only as the square 

root of the running time. This happens because 

one must subtract the number ofbackground 

events from the number of observed 

events, and the statistical uncertainties in 

these numbers determine the limit. Generally, 

when an experiment reaches a level 

limited by background, it is time to think of 

an improved detector. 

The Crystal Box detector is shown in Fig 5. 

A beam of muons from the LAMPF accelerator 

enters on the axis and is stopped in 

a thin polystyrene target. This beam consists 

of surface muons-a relatively new innovation 

developed during the 1970s and employed 

almost immediately at LAMPF and 

other meson factories. 

Normal beams of muons are prepared in a 

three-step process: a proton beam from the 

accelerator strikes a target, generating pions; 

the pions decay in flight, producing muons; 

finally, the optics in the beam line are adjusted 

to transport the daughter muons to the 

experiment while rejecting any remaining 

pions. A more efficient way to collect lowmomentum 

positive muons involves the use 

of a beam channel that collects muons from 

decays of positive pions generated in the 

target, but the muons collected are from 

pions that have only just enough momentum 

to travel from their production point in the 

target to its surface. Stopped in the surface, 

their decay produces positive muons of low 

momentum, near 29 MeV/c (where c is the 

speed of light). This technique enables experimenters 

to produce beams of surface 

muons that can be stopped in a thin experimental 

target with rates up to a hundred 

times more than conventional decay beams. 

137

The muons stopped in the target decay 

virtually 100 per cent of the time by the 

mode 

p+ - e+ v, Gp , 

with a characteristic muon lifetime of 2.2 

microseconds. The Crystal Box detector accepts 

about 50 per cent of these decays and, 

therefore, must reject the positrons from several 

hu.ndred thousand ordinary decays OCcurring 

each second. At the same time the 

detector must select those decays that appear 

to be generated by the processes of interest. 

The Crystal Box was designed to simultaneously 

search for the decay modes 

p+ e+ e+ e- 

- e+y 

-e+yy. 

(Since the Crystal Box does not measure the 

charge of the particles, we shall not generally 

distinquish between positrons and electrons 

in our discussion.) 

The detector properties necessary for 

selecting final states from these reactions and 

rejecting events from ordinary muon decay 

are: 

1. Energy resolution-The candidate 

decays produce two or three particles whose 

energies sum to the energy of a muon at rest. 

The ordinary muon decay and most background 

processes include particles from several 

decays or neutrinos that remain undetected 

but carry away some of the energy. 

These processes are extremely unlikely to 

yield the correct energy sum. 

2. Momentum resolution- Given energy 

resolution adequate to accomplish the first 

requirement, vector momentum resolution 

requires a measurement of the directions of 

the particle trajectories. Since muons are 

stopped in the target, the decays being sought 

for will have vector momentum sums 

clustered, within experimental resolution, 

about zero. Particles from the leading background 

processes (p+ - e+ e+ e- v, ;, p+ - 

e+ y v, Gw, or coincidences of different ordinary 

muon decays) will tend to have non- 

138 

Fig. 5. The Crystal Box detector. (a) A beam of muons enters the detector on axis. 

Because these are low-momenta surface muons, a thin polystyrene target is able to 

stop them at rates up to 100 times more than conventional muon beams. The beam 

intensity is generally chosen to be between 300,000 and 600,000 muons per second 

with pulses produced at a frequency of 120 hertz and a net duty factor between 6 

and 10 per cent. Three kinds of detectors (drvt chamber, plastic-scintillation 

counters, and NaI(T1) crystals) surround the target. The detector elements are 

divided into four quadrants, each containing nine rows of crystals with a plastic 

scintillator in .front of each row. This combination of detectors provides information 

on the energies, times of passage, and directions of the photons and electrons 

that result from muon decay in the target. The information is used to filter from 

several hundred thousand ordinary decays per second the perhaps several per 

second that may be of interest. 

A sophisticated calibration and stabilization system was developed to achieve


and maintain the desired energy and time resolution for 4 X 106 seconds of data 

taking. Before a run starts, a plutonium-beryllium radioactive source is used for 

electron energy calibration. Also, a liquid hydrogen target is substituted 

periodically for the experimental target, and the photons emitted in the subsequent 

pion charge exchange are used for photon energy calibration. During data taking, 

energy calibration is monitored by a fiber optic flasher system that exposes each 

photomultiplier channel to a known light pulse. A small number of positrons are 

accepted from ordinary p+ - e+ v, Vll decays, and the muon decay spectrum cutoff 

at 52.8 MeV is used as a reference. 

(b) The inner dectector, the drift chamber, consists of 728 cells in 8 annular 

rings with about 5000 wires strung to provide the drift cell electrostatic geometry. 

A 5-axis, computer-controlled milling machine was used to accurately drill the 

array of 5000 holes in each end plate. These holes, many drilled at angles up to 

about 10 degrees, had to be located within 0.5 mil so that the chamber wires could 

be placed accurately enough to achieve a final resolution of about 1 millimeter in 

measuring the position of a muon decay in the target. The area of the stopping 

muon spot is about 100 cm2. (Photo courtesy Richard Bolton.) 

(c) The outer layer of the detector (here shown under construction) contains 396 

thallium-doped sodium iodide crystals and achieves an electron and photon energy 

resolution of 5 to 6 per cent. This layer is highly segmented so that the electromagnetic 

shower produced by an event is spread among a cluster of crystals. A 

weighted average of the energy deposition can then be used to localize the 

interaction point of the photons with a position resolution of about 2 cm. 

zero vector sums. 

3. Time resolution-Particles from the 

decay ofa single muon are produced simultaneously. 

A leading source of background for, 

say p+ - e+ e+ e-, is three electrons from the 

decay of three different muons. Such threebody 

final states are unlikely to occur simultaneously. 

Precision resolution in the time 

measurement, significantly better than I 

nanosecond, provides a powerful rejection of 

those random backgrounds. 

4. Position resolution-Decays from a 

single muon will originate from a single point 

in the stopping target. Sometimes other 

processes will add extra particles to an event. 

The ability to accurately measure the trajectory 

of each particle in an event is crucial if 

experimental triggers that have extra tracks 

or that originate in separate vertices are to be 

rejected. 

These parameters are used to filter 

measured events. In a sample of lo’* 

muons-the number required to reach 

sensitivities below the IO-” level-most of 

this filtering must be done immediately, as 

the data is recorded. The Crystal Box experiment 

is exposed to approximately 500,000 

muons stopping per second. The experimental 

“trigger” rate, the rate of decays that 

satisfy crude requirements, is about 1000 

hertz. The detector has been designed with 

enough intelligence in its hardwired logic 

circuits to pass events to the data acquisition 

computer at a rate of less than 10 hertz. In 

turn, the program in the computer applies 

more refined filtering conditions so that 

events are written on magnetic tape at a rate 

of a few hertz. 

Each condition used to narrow down the 

event sample to those that are real candidates 

provides a suppression factor. The combined 

suppression factors must permit the desired 

sensitivity. The design of the apparatus 

begins with the required suppressions and 

applies the necessary technology to achieve 

them. 

A muon that stops in the target and decays 

by one of the subject decay modes produces 

only electrons, positrons or photons. The 

charged particles (hereafter referred to as 

139

electrons) are detected by an 8-layer wire 

drift chamber (Fig. 5 (b)) immediately surrounding 

the target. The drift chamber 

provides track information, pointing back at 

the origin of the event in the target and 

forward to the scintillators and crystals to 

follow. Its resolution and ability to operate in 

the high flux of electrons from ordinary 

muon decays in the target have pushed the 

performance limits of drift chambers; the 

chamber wires were placed accurately 

enough to achieve a final resolution of about 

1 millimeter (mm) in measuring the position 

ofa muon decay in the target. 

Electrons are detected again in the next 

shell out from the target-a set of 36 plastic 

scintillation counters surrounding the drift 

chamber. These counters provide a measurement 

of the time of passage of the electrons 

with an accuracy of approximately 350 

picoseconds. This accuracy is extraordinary 

for counters of the dimensions required (70 

cm Icing by 6 cm wide by 1 cm thick) but is 

crucial to suppressing the random trigger 

background for the p+ - e+ e+ e- reaction. 

This performance is achieved by using two 

photomultiplier tubes, one at each end of the 

scintillator, and two special electronic timing 

circuits developed by the collaborators. 

The electrons and photons that pass 

through the plastic scintillators deposit their 

energy in the next and outermost layer of the 

detector, a 396-crystal array of thalliumdoped 

sodium iodide crystals. These crystals, 

acting as scintillators, provide fast precision 

measurement of both electron and photon 

energy (providing the energy and momenturn 

filtering described earlier) and localize 

the interaction point of the photons with a 

position resolution of about 2 cm. The use of 

such large, highly segmented arrays of inorganic 

scintillator crystals was pioneered in 

high-energy physics in the late 1970's by the 

Crystal Ball detector at the Stanford Linear 

Accelerator Center. This technology is now 

widespread in particle physics research, with 

detectors planned that involve as many as 

12,000 crystals. 

The sodium iodide array also provides 

accurate time measurements on the photons. 

1410 

A fast photomultiplier tube and electronics 

with special pulse shaping, amplification, 

and a custom-tailored, constant-fraction timing 

discriminator were melded into a system 

that gives subnanosecond accuracy. 

The major detector elements-the drift 

chamber, plastic scintillators and sodium 

iodide crystals-are used in logical combinations 

to select events that may be of interest. 

A p'- e' e+ e- event is selected when three 

or more non-adjacent plastic scintillators are 

triggered and energy deposit occurs in the 

sodium iodide rows behind them. The 

special circuits developed for the scintillators 

are used for this selection: one high-speed 

circuit insures that the three or more triggers 

are coincident within a very tight time interval 

(approximately 5 nanoseconds), the second 

circuit requires the three or more hits to 

be in non-adjacent counters. The last requirement 

suppresses events in which low 

momentum radiative daughters trigger adjacent 

counters or when an electron crosses the 

crack between two counters. 

An even more sophisticated trigger processor 

was constructed to insure that the three 

particles triggering the apparatus conform to 

a topology consistent with a three-body 

decay of a particle at rest. Thus, a pattern of 

tracks that, say, necessarily has net momentum 

in one direction (Fig. 6 (a)) is rejected, 

but a pattern with the requisite symmetry 

(Fig. 6 (b)) is accepted. This "geometry box" 

is an array of programmable read-only-memory 

circuits loaded with all legal hit patterns 

as determined by a Monte-Carlo simulation 

of the p+ - e+ e+ e- experiment. 

Finally, the total energy deposited in the 

sodium iodide must be, within the real-time 

energy resolution, consistent with the rest 

energy of a muon. 

The p+ - e+ and p'- e+ y y reactions 

are selected by combining an identified electron 

(a plastic scintillator counter triggered 

coincident with sodium iodide signals) and 

one or more photons (a sodium iodide signal 

triggered with no count in the plastic scintillator 

in front ofit). Also, these events must 

be in the appropriate geometric pattern (for 

example, directly opposite each other for p+ 

Fig. 6. (a) Apattern of tracks with net 

momentum is not consistent with the 

neutrinoless decay of a muon at rest, 

and such an event will be rejected, 

whereas an event with apattern such as 

the one in (b) will be accepted. 

- e+ y) and have the correct energy balance. 

The Crystal Box should report limits in the 

IO-" range on the three reactions of interest 

this calendar year. It will also be used during 

the next year in a search for the KO - y y y 

decay, which violates charge conjugation invariance. 

A search for only the p+ - e+ e+ e-


process is being carried out at the Swiss 

Institute for Nuclear Research with an ultimate 

sensitivity of IO-'* available in the 

next year. 

A third LAMPF p'- e' y experiment is 

planned after the Crystal Box experiment. 

With present meson factory beams and foreseeable 

detector technology, this next generation 

experiment may well be the final round. 

Neutrino-Electron Scattering 

e- 

e- 

/ 

'e 

e- 

Weak (Neutral Current) 

\ 

Weak (Charged Current) 

ig. 7. Examples of the electromagnetic and weak interactions in quantum field 

ieory. 

- 

e 

'* 

The unification of the electromagnetic and 

weak interactions is a treatment of physical 

processes described by the exchange of three 

fundamental bosons. The exchange of a 

photon yields an electromagnetic current, 

and the W' and Zo bosons are exchanged in 

interactions classified as charged and neutral 

weak currents, respectively. Figure 7 illustrates 

how quantum field theory represents 

these processes. 

A traditional method of probing electroweak 

unification in the standard model 

has been to determine the precise onset of 

weak effects in an interaction that is otherwise 

electromagnetic. Especially important 

are experiments-with polarized electron 

scattering at fixed target accelerators and 

more recent studies at electron-positron colliders-that 

probe the interference between 

the amplitudes of the electromagnetic and 

neutral-current weak interactions. Interference 

effects may be easier to observe than 

direct measurement of the small amplitudes 

of the weak interaction. 

An Irvine-Los Alamos-Maryland team is 

conducting a unique and novel search for 

another interference. They have set out to 

probe the pure1.v weak interference between 

the amplitudes of the charged and neutral 

currents. In the same way that electron scattering 

experiments search for interference 

between photon and Zo boson interactions, 

the Los Alarnos based experiment is searching 

for the interference between charged-current 

W interactions and neutral-current Zo 

interactions. 

This experiment is attempting a unique 

141

Fig. 8. The interaction between an electron and its neutrino 

can take place via either the neutral current (with a Z ') or 

the charged current (with a W-), which results in an interference 

term (2ANcur,,,,ACbrgrd) in the expression for the 

square of the total amplitude A,, An experiment at 

LAMPF willprobe this purely weak interference by studying 

v,-electron scattering. 

measurement because Los AIamos is currently 

the only laboratory in the world with 

the requisite source of electron neutrinos. 

Moreover, the experiment gains importance 

from the fact that comparatively little is 

known about the physics ofthe Zo relative to 

that of the W. 

The measurement is a simple variation on 

the electron-electron scattering experiments. 

To substitute the W current for the electromagnetic 

current, the experimenters 

substitute the electron neutrino v, as the 

projectile and set out to measure the frequency 

of electron-neutrino elastic scattering 

from electrons. While this is conceptually 

simple, it is, in fact, technically quite difficult. 

The experiment must yield a sufficiently 

precise measure of the frequency of 

these scatters to separate out theoretical 

predictions made with different assumptions. 

To illustrate how the experiment tests 

the standard model, we must examine the 

nature of the model's predictions for vde 

scal tering. 

142 

Electroweak theory obeys the group structure 

SU(2) X U(1). The SU(2) group has 

three generators, W', W-, and W3, which 

are the charged arid neutral vector bosons 

identified with the gauge fields. The U(1) 

group has a single neutral boson generator B. 

The familiar phenomenological neutral 

photon field is constructed from the linear 

combination 

A 

= W3 sin Ow + B cos OW, 

(where 9~ is the Weinberg angle, a measure 

of the ratio of the contributions of the weak 

and the electromagnetic forces to the total 

interaction). The phenomenological neutral 

current carried by the Zo is similarily constructed 

from 

Zo = W3 

cos Ow - B sin Ow. 

In the standard model the process 

v,+e--v,+e- 

can take place by the exchange of either the 

neutral-current boson Zo or the chargedcurrent 

boson W- (Fig. 8), resulting in the 

usual interference term for the probablity of 

a process that can take place in either of two 

ways. The question then is what form will 

this interference take. 

All models of the weak interaction that are 

currently considered viable predict a 

negative, or destructive, interference term. A 

model that can produce constructive interference 

is one that includes additional neutral 

gauge bosons beyond the 2'. Thus, the 

observation of a v,-e scattering cross section 

consistent with constructive interference 

would indicate a phenomenal change in our 

picture of electroweak physics. Since the 

common Zo with about the predicted mass 

was directly observed only last year, and 

since higher mass regions will be accessible 

during this decade, such a result would set off 

a vigorous search by the particle physics 

community. 

How will the traditional low-energy theory


/‘ 

/ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

1 

result of a two-body decay, is monoenergetic 

with an energy at about 30 MeV. The i,, 

spectrum has a cutoff energy at about 53 

MeV, and the v, spectrum peaks around 35 

or 40 MeV then falls off, also at about 53 

MeV. These three particles are the source of 

many possible measurements. 

The primary goal is the study of the v,-e 

elastic scattering already discussed. The detector, 

which we shall describe in more detail 

shortly, must detect electrons characteristic 

of the elastic scattering, that is, they should 

have energies between 0 and 53 MeV and lie 

within about 15 degrees of the forward direction 

(the tracks must point back to the neutrino 

source). 

Also, by selecting events with electrons 

below 35 MeV, the group will search for the 

first observation of an exclusive neutrinoinduced 

nuclear transition. The process 

V, + ”C -t e- + ‘*N 

0 10 20 30 40 50 

Neutrino Energy (MeV) 

Fig. 9. The energy spectra for the three types of neutrinos that result from the decay 

ofapositivepion (n+ - p+ + vP p+ - e+ + V, + ;,). 

of weak interactions (apparently governed by 

V - A currents) mesh with future observations 

at higher energies? The standard model 

prediction, which contains negative interference, 

is that the cross section for v,-e 

elastic scattering should be about 60 per cent 

of the cross section in the traditional V - A 

theory. The LAMPF experiment must 

measure the cross section with an accuracy of 

about 15 per cent to be able to detect the 

lower rate that would occur in the presence of 

interference and thus be able to determine 

whether interference effects are present or 

not. 

In addition, the magnitude of the interference 

is a function of sin20w, and a precise 

measurement of the interference constitutes 

a measurement of this factor. In fact, it is 

60 

statistically more efficient to do this with a 

neutral current process because the charged 

current contains sin2ew (= 0.25) summed 

with unity, whereas for the neutral current 

the leading term is sin20w. 

The Experiments. The LAMPF proton 

beam ends in a thick beam stop where pions 

(n+) are produced These pions decay by the 

process 

n+ - p+ + vp 

Le+ + v,. + Gp , 

yielding three types of neutrinos exiting the 

beam stop. The v, and ;, are each produced 

with a continuous spectrum (Fig. 9) typical 

of muon decay, whereas the vp spectrum, the 

would produce electrons with less than 35 

MeV energy that lie predominantly outside 

the angular region for the elastic scattering 

events. 

Another important physics goal, neutrino 

oscillations, can be addressed simultaneously. 

A process, called an “appearance,” in 

which the ip species disappears from the 

beam and i, appears, can be probed by 

searching for the presence of v, in the beam. 

This type of neutrino does not exist in the 

original neutrino source, so its presence 

downstream could be evidence for the 

i,, -i, oscillation. The experimental 

signature for such a process is the presence of 

isotropic single positrons produced by the 

reaction 

combined with a selection in energy of more 

than 35 MeV, which can be used to isolate 

these candidate events from the nuclear transition 

process discussed above. 

In all three of the processes studied, the 

technical problem to be solved is the separation 

of the desired events from competing 

143

ackground processes. The properties of the 

detector (Fig. IO) needed to do this include: 

1. Passive shielding-Lead, iron, and concrete 

are used to absorb charged and neutral 

cosmic ray particles entering the detector 

volume. However, the shield is not thick 

enough to insure that events seen in the inner 

detector come only from neutrinos entering 

the detector and not from residual cosmic 

ray backgrounds. The outer shield merely 

reduces the flux, consisting mainly of muons 

and hadrons from cosmic rays and of neutrons 

from the LAMPF beam stop. 

The LAMPF beam is on between 6 and 10 

per cent of each second so that the periods 

between pulses will provide an important 

normalizing measurement indicating how 

well the passive shielding works. 

2. Active anti-coincidence shield-This 

multilayer device is an active detector that 

surrounds the inner detector and serves 

many purposes. For example, muons from 

cosmic rays that penetrate the passive shield 

are detected here by being coincident in time 

with an inner detector trigger. This allows the 

rejection of these “prompt” muons, with less 

than lone muon in IO4 surviving the rejection. 

Data acquisition electronics that store 

the history of the anti-coincidence shield for 

32 microseconds prior to an inner detector 

trigger serve an even more complex purpose. 

This information is used to reject any inner 

detector electrons coming from a muon that 

stopped in the outer shield and that took up 

to 32 microseconds to decay. The mean 

muon lifetime is only 2.2 microseconds, so 

this is a very satisfactory way to reject such 

events. 

3. Inner converter-Photons penetrating 

the anti-coincidence layer, produced perhaps 

by cosmic rays or particles associated with 

the beam, strike an additional layer of steel 

and are either absorbed or converted into 

electronic showers that are seen as tracks 

connected to the edge of the inner detector. 

Such events are discarded in the data analysis. 

4. Inner detector-This module’s primary 

role is to measure the trajectory and energy 

deposition of electrons and other charged 

Fig. 10. The detector for the neutrino-electron scattering experiments. The outer 

layer of passive shielding (mainly steel) cuts down the flux of neutral solar 

particles. 

The anti-coincidence shield rejects muons from cosmic rays and electrons 

coming from the decay of muons stopped in the outer shield. It consists of four 

layers of drvt tubes, totaling 603 counters, each 6 meters long. A total of 4824 

wires provides LI fine-grained, highly effective screen, with an inefficiency (and 

therefore a suppression) of 2 X 1r5. 

Another steel layer, the inner converter, is used to reject photons from cosmic 

rays or otherparticles associated with the beam. 

The inner detector consists of 10 tons ofplastic scintillators interleaved with 4.5 

tons of tracking chambers. The plastic scintillators sample the electron energy 

every 10 layers of track chamber. There are 160 counters, each 75 cm by 300 cm by 

2.5 cm thick, and they measure the energy to about 10 per cent accuracy. The track 

chambers are a classic technology: they are flash chambers that behave like neon 

lights when struck by an ionizing particle, discharging in a luminous and climactic 

way. There are a phenomenal 208,000Jlash tubes in the detector, and they measure 

the electron tracks and sort them into angular bins about 7 degrees wide.


particles. Electron tracks are the signature of 

the desired neutrino reactions, but recoil 

protons generated by neutrons from the 

beam stop and from cosmic rays must also be 

detected and filtered out in the data analysis. 

The inner detector contains layers of plastic 

scintillators that sample the particle energy 

deposited along its path for particle identification 

and also provide a calorimetric measurement 

of the total energy. Trajectory measurement 

is provided by a compact system of 

flash chambers interleaved with the plastic 

scintillators. 

When this detector is turned on, it counts 

about 10’ raw events per day, mostly from 

cosmic rays. To illustrate the selectivity required 

of this experiment, a recent data run 

ofa few months is expected to produce somewhat 

less than 50 v,-e elastic scattering 

events. 

This highly segmented detector is 

necessarily extremely compact. The neutrino 

flux produced in the beam stop is emitted in 

all directions and therefore has an intensity 

that falls off inversely with the square of the 

distance. Thus there was a strong design 

premium for developing a compact, dense 

detector and placing it as close to the source 

as feasible. 

The detector is now running around the 

clock, even when the LAMPF beam is off 

(to pin down background processes). The 

data already taken include many v,-e events 

that are being reported, as are preliminary 

results on lepton number conservation and 

neutrino oscillations. Data taken with additional 

neutron shielding during the next year 

or two are expected to provide the precision 

test of the standard model that the experimenters 

seek. 

Beyond this effort, the beginnings of a 

much larger and ambitious neutrino program 

at Los Alamos are evident. A group 

(Los Alamos; University of New Mexico; 

Temple University; University of California, 

Los Angeles and Riverside; Valparaiso University; 

University of Texas) working in a 

new LAMPF beam line are mounting the 

prototype for a much larger fine-grained neutrino 

detector. Currently, a focused beam 

source of neutrinos is being developed that 

will eventually employ a rapidly pulsed 

“horn” to focus pions that decay to neutrinos. 

This development will be used to 

provide neutrinos for a major new detector. 

The group is not content to work merely on 

developing the facility but is using a preliminary 

detector to measure some key cross 

sections and set new limits on neutrino oscillations 

as well. 

Another group (Ohio State, Louisiana 

State, Argonne, California Institute of Technology, 

Los Alamos) is assembling the first 

components ofan aggressive effort to search 

for the i, appearance mode. Other physicists 

at the laboratory are preparing a solar neutrino 

initiative. 

The exciting field of neutrino research, 

begun by Los Alamos scientists, is clearly 

entering a golden period. 

Precision Studies of Normal 

Muon Decay 

The measurement of the electron energy 

spectrum and angular distribution from ordinary 

muon decay, 

p + 

e+ i,+ V,, , 

is one of the most fundamental in particle 

physics in that it is the best way to determine 

the constants of the weak interaction. These 

studies have led to limits on the V - A 

character of the theory. 

The spectrum of ordinary muon decay 

may be precisely calculated from the standard 

model. Built into the minimal standard 

model-consistent with the idea that everything 

in the model must be required by 

measurements-are the assumptions that 

neutrinos are massless and the only interactions 

that enter are of vector and axial vector 

form (that is, V- A, or equal magnitude and 

opposite sign). Lepton flavor conservation is 

also taken to be exact. 

This V - A structure of the weak interaction 

can be tested by precise measurements 

of the electron spectrum from ordinary 

muon decay. The spectrum is characterized 

(to first order in m,/m,, and integrated over 

the electron polarization) by 

dN a (3-2x) 

2 dx d (cos 0) 

+ -p-1 (4x-3) 

(: ) 

+ 1 2 

m 

‘ 

(I-x) 

- 

m,, x2 

where me is the electron mass, 8 is the angle 

of emission of the electron with respect to the 

muon polarization vector P,, m,, is the muon 

mass, and x’ is the reduced electron energy (x 

= 2E/m,, where E is the electron energy). The 

Michel parameters p, q, e, 6 characterize the 

spectrum. 

The standard model predicts that 

p=6=3/4, 5=1, and q=O. 

One can also measure several parameters 

characterizing the longitudinal polarization 

of the electron and its two transverse components. 

Table 2 gives the current world average 

values for the Michel parameters. These 

data have been used to place limits on the 

weak interaction coupling constants, as 

shown in Table 3. As can be seen, the current 

limits allow up to a 30 per cent admixture of 

something other than a pure V- A structure. 

Other analyses, with other model-dependent 

assumptions, set the limit below IO per cent. 

One of the extensions of the minimal standard 

model is a theory with left-right symmetry. 

The gauge symmetry group that embodies 

the left-handed symmetry would be 

joined by one for right-handed symmetry, 

and the charged-current bosons W’ and W- 

would be expanded in terms of a symmetric 

combination of fields W, and W,. Such an 

145

extension is important from a theoretical 

standpoint for several reasons. First, it 

restores parity conservation as a high-energy 

symmetry of the weak interaction. The wellknown 

observation of parity violation in 

weak processes would then be relegated to 

the status of a low-energy phenomenon due 

to the fact that the mass of the right-handed 

W is much larger than that of the left-handed 

W. Each lepton generation would probably 

require two neutrinos, a light left-handed one 

and a very heavy right-handed member. 

The dominance of the left-handed charged 

current at presently accessible energies 

would be due to a very large mass for W,, 

but the W, - WR mass splitting would still 

be small on the scale of the grand unification 

mass Mx. Thus the precision study of a weak 

decay such as ordinary muon decay or 

nucleon beta decay can be used to set a limit 

on the left-right symmetry of the weak interaction. 

With such plums as the V- A nature of 

the weak interaction and the existence of 

right-handed W bosons accessible to such 

precision studies, it is not surprising that 

several experimental teams at meson factories 

are carrying out a variety of studies of 

ordinary muon decay. One team working at 

the Canadian facility TRIUMF has already 

collected data and set a lower limit of 380 

GeV on the mass of the right-handed W. 

This was done with a muon beam of only a 

few MeV! 

Table 2 

Theoretical and1 experimental values for the weak-interaction Michel 

parameters. 

Michel 

Parameter 

P 

11 

5 

6 

_. . 

Table 3 

V-A 

Prediction 

Current 

Value 

3/4 0.752 k 0.003 

0 -0.12 k 0.21 

1 0.972 k 0.014 

Y4 0.755 k 0.008 

Expected 

Los Alamos Accuracy 

k 0.00023 

rt 0.006 1 

k 0.001 

rt 0.00064 

. . . . . . . ...-, 

- .. . .- 

Experimental limits on the weak-interaction coupling constants, including 

the expected limit for the Los Alamos Experiment. 

Constant Present Limit Expected Limit 

Axial Vector 

0.76 < gA < 1.20 0.988 < gA < 1.052 

Tensor gT < 0.28 gT < 0.027 

Scalar gs < 0.33 gs < 0.048 

Pseudo Scalar gp < 0.33 g~. < 0.048 

Vecto-axial Vector Phase (PVA = 180" Ifr 15" (PVA= 18Vrt2.6" 

i 

i 

I 

- I 

, 

The Time Projection Chamber. A Los 

Alamos - University of Chicago-NRC Canada 

collaboration is carrying out a 

particularly comprehensive and sensitive 

study of the muon decay spectrum using a 

novel and elaborate device known as a time 

projection chamber (TPC). 

The TPC (Fig. 11) is a very large volume 

drift chamber. In a conventional drift 

chamber, an array of wires at carefully determined 

potentials collects the ionization 

left in a gas by a passing charged particle. The 

time of arrival of the packet of ionization in 

the cell near each wire is used to calculate the 

path of the particle through the cell. The gas 

and the field in the cell are chosen so that the 

ionization drifts at a constant terminal velocity. 

Thus the calculation of the position from 

the drift time can be done accurately. Many 

drift chambers provide coordinate measurements 

accurate to less than 100 micrometers. 

On the other hand, a TPC uses the same 

drift velocity phenomenon but employs it in 

a large volume with no wires in the sensitive 

region. The path iofionization drifts en masse 

under the influence of an electric field along 

the axis of the chamber. The ionization is 

collected on a series of electrodes, called 

pads, on the chamber endcaps, providing 

precision measurement of trajectory charge 

and energy. The pad signal also gives a time 

measurement, relative to the event trigger, 

that can be used to reconstruct the spatial 

coordinate of each point on the trajectory. 

The TPC in the Los Alamos experiment is 

placed in a magnetic field sufficiently strong 

that the decay electrons, whose energies 

range up to about 53 MeV, follow helical 

paths. The magnetic field is accurate enough 

to make absolute momentum measurements 

of the decay electrons.


I ' 

Scintillator 

/ 

Deflector 1 

Iron Yoke ' 

Magnet 

Coils 

Removable 

Pole 

High-Voltage Electrodes 

Readout Plane 

(Series of Pads) 

Fig. 11. The time projection chamber (TPC), a device to study the muon decay 

spectrum. A beam of muons from LAMPF enters the TPC via a 2-inch beam pipe 

that extends through the magnet pole parallel to the magnetic field direction. 

Before entering the chamber, the muons pass through a IO-mil thick scintillator 

that serves as a muon detector. The scintillator is viewed, via fiber optic light 

gzides, by two photomultiplier tubes located outside the magnet. The thresholds for 

the discriminators on these photomultiplier channels are adjusted to produce a 

coincidence for the more heavily ionizing muons while the minimum-ionizing 

beam electrons are ignored. A deflector located in the beam line 2 meters upstream 

of the magnet produces a region of crossed electric and magnetic fields through 

which the beam passes. This device acts first as a beam separator, purifying the 

muon flux-in particular, reducing the number of electrons in the beam from 

about 200 to about 1.5 for every muon. The device also acts as a deflector, keeping 

additional particles out of the chamber by switching off the electric field once a 

muon has been observed entering the detector. The magnetic field in this detector is 

provided by an iron-enclosed solenoid, with the maximum field in the current 

arrangement being 6.6 kilogauss. The field has been carefully measured and found 

to be uniform to better then 0.6per cent within the entire TPC-sensitive volume of 

52 cm in length by 122 cm in diameter. The TPC readout, on the chamber endcaps, 

consists of 21 identical modules, each of which has 15 sense wires and 255 pads 

arranged under the sense wires in rows of I7pads each. The sense wires provide the 

high field gradient necessary for gas amplification of the track ionization. The 21 

modules are arranged to cover most of the 122-centimeter diameter of the chamber. 

A beam of muons from LAMPF passes 

first through a device that acts as a beam 

separator, purifying the muon flux 

(especially of electrons, which are reduced by 

this device from an electron-to-muon ratio of 

200:l to about 3:2). The device also acts as a 

deflector, keeping additional particles from 

entering the chamber once a muon is inside. 

With a proper choice of beam intensity, only 

one muon is allowed in the TPC at a time. 

Next the beam passes through a IO-mil thick 

scintillator (serving both as a muon detector 

and a device used to reject events caused by 

the remaining beam electrons) and continues 

into the TPC along a line parallel to the 

magnetic field direction. 

The requirement for an event to be triggered 

is that one muon enters the TPC during 

the LAMPF beam pulse and stops in the 

central lOcm ofthe drift region. The entering 

muon is detected by a signal coincidence 

from photomultipliers attached to the IO-mil 

scintillator (this signal operates the deflector 

that keeps other muons out). The scintillator 

signal must also be coincident-including a 

delay that corresponds to the drift time from 

the central IO cm of the TPC-with a high 

level signal from any of the central wires of 

the TPC. If no delayed coincidence occurs, 

indicating that the muon did not penetrate 

far enough into the TPC, or a high level 

output is detected before the selected time 

window, indicating that the muon 

penetrated too far, the event is rejected and 

all electronics are reset. Then 250 microseconds 

later (to allow for complete clearing 

of all tracks in the TPC) the beam is allowed 

to re-enter for another attempt. The event is 

also rejected if a second muon enters the 

TPC during the 200-nanosecond period required 

to turn off the deflector electric field. 

If the event is accepted, the computer 

reads 20 microseconds of stored data. This 

corresponds to five muon decay lifetimes 

plus the 9 microseconds it takes for a track to 

drift the full length of the TPC. 

The experiment is expected to collect 

about IOs muon decay events, at a trigger 

rate of 120 events per second, during the next 

year. Preliminary data have already been 

147

taken, showing that the key resolution for 

electron momentum falls in the target range, 

namely 4p/p is 0.7 per cent averaged over the 

entire spectrum. Figure 12 shows one of the 

elegant helica: tracks obtained in these early 

runs. 

Ultimately, this experiment will be able to 

improve upon the four parameters shown in 

Table 2, although the initial emphasis will be 

on p. In the context of left-right symmetric 

models, an improved measurement of p will 

place a new limit on the allowed mixing 

angle between W, and W, that is almost 

independent of the mass of the W,. 

Summary 

The particle physics community is aggressively 

pursuing research that will lead to 

verification or elaboration of the minimal 

standard model. Most of the world-wide activity 

is centered at the high-energy colliding 

beam facilities, and the last few years have 

yielded a bountiful harvest of new results, 

including the direct observation of the W* 

and Zo bosons. Many of the key measure- 

Fig. 12. An example of the typical helical track observed for a muon-decay event in 

an early run with the TPC. (The detector here is shown on end compared to Fig. 

11.) 

ments of the 1980s are likely to be made at 

the medium-energy facilities, such as 

LAMPF, or in experiments far from accelerators, 

deep underground and at reactors, 

where studies of proton decay, solar neutrino 

physics, neutrino oscillations, tritium beta 

decay, and other bellwether research is being 

camed out. 

Gary H. Sanders learned his physics on the east coast, starting at 

Stuyvesant High School in New York City, then Columbia and an A.B. in 

physics in 1967, and finally a Ph.D. from the Massachusetts Institute of 

Technology in 1971. The work for his doctoral thesis, which dealt with the 

photoproduction of neutral rho mesons on complex nuclei, was 

performed at DESY's electron synchrotron in Hamburg, West Germany 

under the guidance of Sam Ting. After seven years at Princeton University, 

during which time he used the beam at Brookhaven National 

Laboratory and Fermi National Accelerator Laboratory, he came west to 

join the Laboratory's Medium Energy Physics Division and use the 

beams at LAMPF. A great deal of his research has dealt with the study of 

muons and with the design of the beams, detectors, and signal processing 

equipment needed for these experiments. 

I 1413

Addendum 

An Experimental 

Update 

Neutrino-Electron Scattering. This experiment 

has now collected 121 f 25 v,e- scattering 

events, of which 9,' 25 are identified 

as v,e- scatterings. The resulting cross section 

agrees with the standard electroweak 

theory, rules out constructive interference 

between weak chargedcurrent and neutralcurrent 

interactions, and favors the existence 

of destructive interference between these two 

interactions. Additional data are being collected, 

and this result will be made considerably 

more precise. 

I 

n the two years since "Experiments to 

Test Unification Schemes" was written, 

each of the four experiments described 

has completed a substantial program of 

measurements and published its first results. 

In one case, the entire program is complete 

with final results submitted for publication. 

So far, all results are fblly consistent with the 

minimal standard model. Opportunities for 

theories with new physics have been substantially 

constrained. 

Tritium Beta Decay. Although dissociation 

into atomic tritium has not yet been employed 

to make a physics measurement, the 

study of the tritium beta decay spectrum 

using molelcular tritium has been completed. 

An upper limit of 26.8 eV (95 per cent confidence 

level) has been placed on the mass of 

the electron antineutrino, with a best fit 

value ofzero mass. This result is inconsistent 

with the best fit value most recently reported 

by the Russians (30 & 2 eV) and excludes a 

large fraction of their latest mass range of 17 

to 40 eV. Several other experiments, including 

those done by teams from Lawrence Livermore 

National Laboratory, the Swiss Institute 

for Nuclear Research, and a Japanese 

group, have also begun to erode the Russian 

claim of a nonzero neutrino mass. Improvements 

in these limits, including the Los Alamos 

atomic tritium measurement, are expected 

soon. 

Rare Decays of the Muon. The Crystal Box 

detector has completed its search for rare 

decays of the muon and, for the three 

processes sought, has published (at the 90 per 

cent confidence level) the following new 

limits: 

B(p+- e'yy) < 7.2 X lo-'' . 

An experiment at the Swiss Institute for Nuclear 

Research has also obtained a limit on 

the first process of about 2.4 X : O-'*. These 

four results place severe lower limits on the 

masses of new objects that could produce 

nonconservation of lepton number. For example, 

in one analysis of deviations from the 

standard model, the Crystal Box limit on 

p+ - e'y sets the scale for new interactions 

to lo4 TeV or higher. 

A new and far more ambitious search for 

p'- e'y has been undertaken by some 

members of the Crystal Box group together 

with a large group of new collaborators. 

Their design sensitivity is set at less than 1 X 

and their new detector is under construction. 

In addition, four other groups, using 

rare decays of the kaon, are searching for 

processes that violate lepton number conservation, 

such as KL - pe and KL - npe. 

Normal Muon Decay. After preliminary 

studies, the team using the time projection 

chamber to study normal muon decay decided 

to concentrate on the measurement of 

the p parameter (Table 2). Since this parameter 

is measured by averaging over the muon 

spin, it was not necessary to preserve the spin 

direction of the muon stopping in the 

chamber. The researchers used an improved 

entrance separator to rotate the spin perpendicular 

to the beam direction, and precession 

in the chamber magnetic field then averaged 

the polarization. They also took advantage of 

a small entrance scintillator to trigger the 

apparatus on muon stops. This technique 

purified the experimental sample but 

perturbed the muon spin, which, however, is 

acceptable for the measurement of p. A 

higher event rate was possible because the 

entrance scintillator signal eliminated the 

need for the beam to be pulsed. The scintillator, 

by itself, effectively ensured a single 

stopping muon. In this mode, the group collected 

5 X lo7 events, which are now being 

analyzed, and this sample is expected to 

sharpen the knowledge of p by a factor of 5. 

Limits on charged right-handed currents 

from a related measurement have now been 

reported by the TRIUMF group. 

The minimal standard model has, to date, 

survived these demanding tests. Where is the 

edge of its validity? We shall have to wait a 

little longer to find the answer. Experimentalists 

are already mounting the next round 

of detectors in this inquiry. 

149

I 

h 

6.2 

J 

q 

P 0 

n 

f 

D 

@ 

8 

by S. Peteg Rosen

When these questions have been 

answered, we may expect the cycle to repeat 

itself until we run out of resources-or out of 

space. So far the field of particle physics has 

been fixtunate: every time it seems to have 

reached the end of the energy line, some new 

technical development has come along to 

extend it into new realms. Synchrotrons such 

as the Bevatron and the Cosmotron, its sister 

and rival at Brookhaven, both represented 

an order-of-magnitude improvement over 

synchrocyclotrons, which in their time overcame 

relativistic problems to extend the 

energy of cyclotrons from tens of MeV into 

the hundreds. What allowed these developments 

was the synchronous principle invented 

independently by E. McMillan at 

Berkeley and V. Veksler in the Soviet Union. 

In a cyclotron a proton travels in a circular 

orbit under the influence of a constant magnetic 

field. Every time it crosses a particular 

diameter, it receives an accelerating kick 

from an rf electric field oscillating at a constanl 

frequency equal to the orbital frequency 

of the proton at some (low) kinetic 

energy. Increasing the kinetic energy of the 

proton increases the radius of its orbit but 

does not change its orbital frequency until 

the effects of the relativistic mass increase 

become significant. For this reason a 

cyclotron cannot efficiently accelerate 

proions to energies above about 20 MeV. 

The solution introduced by McMillan and 

Veksler was to vary the frequency of the rf 

field so that the proton and the field remained 

in synchronization. With such 

synchrocyclotrons proton energies of hundreds 

of MeV became accessible. 

In a synchrotron the protons are confined 

to a narrow range of orbits during the entire 

acceleration cycle by varying also the magnetic 

field, and the magnetic field can then be 

supplied by a ring of magnets rather than by 

the solid circular magnet of a cyclotron. 

Nevertheless, the magnets in early synchrotrons 

were still very large, requiring 10,000 

tons of iron in the case of the Bevatron, and 

for all practical purposes the synchrotron 

appeared to have reached its economic limit 

with this 6-GeV machine. Just at the right 

152 

time a group of accelerator physicists at 

Brookhaven invented the principle of 

“strong focusing,” and Ernest Courant, in 

May 1953, looked forward to the day when 

protons could be accelerated to 100 

GeV-fifty times the energy available from 

the Cosmotron--with much smaller 

magnets! In the meantime Courant and his 

colleagues contented themselves with building 

a machine ten times more energetic, 

namely, the AGS (Alternating Gradient 

Synchrotron). 

Courant proved to be most farsighted, but 

even his optimistic goal was far surpassed in 

the twenty years following the invention of 

strong focusing. The accelerator at Fermilab 

(Fermi National Accelerator Laboratory) 

achieved proton energies of 400 GeV in 

1972, and at CERN (Organisation Europeene 

pour Recherche NuclCaire) the SPS 

(Super Proton Synchrotron) followed suit in 

1976. Size is the most striking feature of 

these machines. Whereas the Bevatron had a 

circumference of 0.1 kilometer and could 

easily fit into a single building, the CERN 

and Fermilab accelerators have circumferences 

between 6 and 7 kilometers and are 

themselves hosts YO large buildings. 

Both the Fermilab accelerator and the SPS 

are capable of accelerating protons to 500 

GeV, but prolonged operation at that energy 

is prohibited by excessive power costs. This 

economic hurdle has recently been overcome 

by the successful development of superconducting 

magnets. Fermilab has now installed 

a ring of superconducting magnets in the 

same tunnel that houses the original main 

ring and has achieved proton energies of 800 

GeV, or close to 1 TeV. The success of the 

Tevatron, as it is called, has convinced the 

high-energy physics community that a 20- 

TeV proton accelerator is now within our 

technological grasp, and studies are under 

way to develop a proposal for such an accelerator, 

which would be between 90 and 

160 kilometers in circumference. Whether 

this machine, known as the SSC (Superconducting 

Super Collider), will be the terminus 

of the energy line, only time will tell; but if 

the past is any guide, we can expect some- 

thing to turn up. (See “The SSC-An Engineering 

Challenge.”) 

Paralleling the higher and higher energy 

proton accelerators has been the development 

of electron accelerators. In the 1950s 

the emphasis was on linear accelerators, or 

linacs, in order to avoid the problem of 

energy loss by synchrotron radiation, which 

is much more serious for the electron than 

for the more massive proton. The development 

of linacs culminated in the two-milelong 

accelerator at SLAC (Stanford Linear 

Accelerator Center), which today accelerates 

electrons to 40 GeV. This machine has had 

an enormous impact upon particle physics, 

both direct and indirect. 

The direct impact includes the discovery 

of the “scaling” phenomenon in the late 

1960s and of parity-violating electromagnetic 

forces in the late 1970s. By the 

scaling phenomenon is meant the behavior 

of electrons scattered off nucleons through 

very large angles: they appear to have been 

deflected by very hard, pointlike objects inside 

the nucleons. In exactly the same way 

that the experiments of Rutherford revealed 

the existence of an almost pointlike nucleus 

inside the atom, so the scaling experiments 

provided a major new piece of evidence for 

the existence of quarks. This evidence was 

further explored and extended in the ’70s by 

neutrino experiments at Fermilab and 

CERN. 

Whereas the scaling phenomenon opened 

a new vista on the physics of nucleons, the 

1978 discovery of parity violation in the 

scattering of polarized electrons by deuterons 

and protons closed a chapter in the history ol 

weak interactions. In 1973 the phenomenon 

ofweak neutral currents had been discovered 

in neutrino reactions at the CERN PS 

(Proton Synchrotron), an accelerator very 

similar in energy to the AGS. This discovery 

constituted strong evidence in favor of the 

Glashow-Weinberg-Salam theory unifying 

electromagnetic and weak interactions. During 

the next five years more and more 

favorable evidence accumulated until only 

one vital piece was missing-the demonstration 

of parity violation in electron-nucleon

the march toward higher energies 

The “string and sealing wax” version of a cyclotron. With this 4-inch device E. 0. 

Lawrence and graduate student M. S. Livingston successfully demonstrated the 

Peasibility of the cyclotron principle on January 2, 1931. The device accelerated 

wotons to 80 keV. (Photo courtesy of Lawrence Berkeley Laboratory.) 

eactions at a very small, but precisely 

iredicted, level. In a brilliant experiment C. 

’rescott and R. Taylor and their colleagues 

ound the missing link and thereby set the 

ea1 on the unification of weak and electronagnetic 

interactions. 

A less direct but equally significant impact 

f the two-mile linac arose from the electronositron 

storage ring known as SPEAR 

(Stanford Positron Electron Accelerating 

Ring). Electrons and positrons from the linac 

are accumulated in two counterrotating 

beams in a circular ring of magnets and 

shielding, which, from the outside, looks like 

a reconstruction of Stonehenge. Inside, 

enough rf power is supplied to overcome 

synchrotron radiation losses and to allow 

some modest acceleration from about 1 to 4 

GeV per beam. In the fall of 1974, the w 

particle, which provided the first evidence 

for the fourth, or charmed, quark was found 

among the products of electron-positron collisions 

at SPEAR; at the same time the J 

particle, exactly the same object as y, was 

discovered in proton collisions at the AGS. 

With the advent of J/w, the point of view 

that all hadrons are made of quarks gained 

universal acceptance. (The up, down, and 

strange quarks had been “found” experimentally; 

the existence of the charmed quark had 

been postulated in 1964 by Glashow and J. 

Bjorken to equalize the number of quarks 

and leptons and again in 1970 by Glashow, J. 

Iliopoulos, and L. Maiani to explain the apparent 

nonoccurrence of strangeness-changing 

neutral currents. 

The discovery of J/w, together with the 

discovery of neutral currents the year before, 

revitalized the entire field of high-energy 

physics. In particular, it set the building of 

electron-positron storage rings going with a 

vengeance! Plans were immediately laid at 

SLAC for PEP (Positron Electron Project), a 

larger storage ring capable of producing 18- 

GeV beams of electrons and positrons, and 

in Hamburg, home of DORIS (Doppel-Ring- 

Speicher), the European counterpart of 

SPEAR, a 19-GeV storage ring named 

PETRA (Positron Electron Tandem Ring 

Accelerator) was designed. Subsequently a 

third storage ring producing 8-GeV beams of 

positrons and electrons was built at Cornell; 

it goes by the name of CESR (Cornell Electron 

Storage Ring). 

Although the gluon, the gauge boson of 

quantum chromodynamics, was discovered 

at PETRA, and the surprisingly long lifetime 

of the b quark was established at PEP, the 

most interesting energy range turned out to 

be occupied by CESR. Very shortly before 

this machine became operative, L. Lederman 

and his coworkers, in an experiment at 

Fermilab similar to the J experiment at 

Brookhaven, discovered the T particle at 9.4 

GeV; it is the bquark analogue of J/w at 3.1 

GeV. By good fortune CESR is in just the 

right energy range to explore the properties of 

the T system, just as SPEAR was able to 

153

elucidate the I+I system. Many interesting results 

about T, its excited states, and mesons 

containing the b quark are emerging from 

this unique facility at Cornell. 

The next round for positrons and electrons 

includes two new machines, one a CERN 

storage ring called LEP (Large Electron- 

Positron) and the other a novel facility at 

SLAC called SLC (Stanford Linear Collider). 

LEP will be located about 800 meters under 

the Jura Mountains and will have a circumference 

of 30 kilometers. Providing 86-GeV 

electron and positron beams initially and 

later 130-GeV beams, this machine will be an 

excellent tool for exploring the properties of 

the W’ bosons. SLC is an attempt to overcome 

the problem of synchrotron radiation 

losses by causing two linear beams to collide 

head on. If successful, this scheme could well 

establish the basic design for future machines 

of extremely high energy. At present SLC is 

expected to operate at 50 GeV per beam, an 

ideal energy with which to study the Zo 

boson. 

High energy is not the only frontier against 

which accelerators are pushing. Here at Los 

Alamos LAMPF (Los Alamos Meson Physics 

Facility) has been the scene of pioneering 

work on the frontier of high intensity for 

more than ten years. At present this 800- 

MeV proton linac carries an average current 

of I milliampere. To emphasize just how 

great an intensity that is, we note that most of 

the accelerators mentioned above hardly 

ever attain an average current of 10 microamperes. 

LAMPF is one of three so-called 

meson factories in the world; the other two 

are highly advanced synchrocyclotrons at 

TRIUMF (Tri-University Meson Facility) in 

Vancouver, Canada, and at SIN (Schweizerisches 

Institut fur Nuklearforschung) near 

Zurich, Switzerland. 

The high intensity available at LAMPF 

has given rise to fundamental contributions 

in nuclear physics, including confirmation of 

the recently developed Dirac formulation of 

nucleon-nucleus interactions and discovery 

of giant collective excitations in nuclei. In 

addition, its copious muon and neutrino 

1 54 

A state-of-the art version of aproton synchrotron. Here at Fermilabprotons will be 

accelerated to an energy close to 1 TeV in a 6562-foot-diameter ring of superconducting 

magnets. Wilson Hall, headquarters of the laboratory and a fitting 

monument to a master accelerator builder, appears at the lower left. (Photo 

courtesy of Fermi National Accelerator Laboratory.) 

beams have been applied to advantage in 

particle physics, especially in the areas ofrare 

modes of particle decay and neutrino physics. 

The search for rare decay modes (such as 

p’ - e+ + y) remains high on the agenda of 

particle physics because our present failure 

to see them indicates that certain conservation 

laws seem to be valid. Grand unified 

theories of strong and electroweak interactions 

tell us that, apart from energy and 

momentum, the only strictly conserved 

quantity is electric charge. According to these 

theories, the conservation of all other quantities, 

including lepton number and baryon 

number, is only approximate, and violations 

of these conservation laws must occur, although 

perhaps at levels the minutest of the 

minute. 

Meson factories are ideally suited to the 

search for rare processes, and here at Los 

Alamos, at TRIUMF, and at SIN plans are 

being drawn up to extend the range of present 

machines from pions to kaons. (See 

“LAMPF I1 and the High-Intensity Frontier.”) 

Several rare decays of kaons can 

provide important insights into grand uni- 

fied theories, as well as into theories that 

address the question of W’ and Zo masses, 

and so the search for them can be expected to 

warm up in the next few years. 

Another reason for studying kaon decays 

is CP violation, a phenomenon discovered 

twenty years ago at the AGS and still today 

not well understood. Because the effects of 

CP violation have been detected only in 

kaon decays and nowhere else, extremely 

precise measurements of the relevant 

parameters are needed to help determine the 

underlying cause. In this case too, kaon factories 

are very well suited to attack a fundamental 

problem of particle physics. 

In the area of neutrino physics, LAMPF 

has made important studies ofthe identity of 

neutrinos emitted in muon decay and is now 

engaged in a pioneering study of neutrinoelectron 

scattering. High-precision measurements 

of the cross section are needed as a test 

of the Glashow-Weinbcrg-Salam theory and 

are likely to be a major part of the experimental 

program at kaon factories. 

While the main thrust of particle physics 

has always been carried by accelerator-based


experiments, there are, and there have 

always been, important experiments performed 

without accelerators. The first 

evidence for strange particles was found in 

the late 1940s in photographic emulsions 

exposed to cosmic rays, and in 1956 the 

neutrino was first detected in an experiment 

at a nuclear reactor. In both cases accelerators 

took up these discoveries to explore 

and extend them as far as possible. 

Another example is the discovery of parity 

nonconservation in late 1956. The original 

impetus came from the famous 7-0 puzzle 

concerning the decay of K mesons into two 

and three pions, and it had its origins in 

accelerator-based experiments. But the definitive’ 

experiment that demonstrated the 

nonconservation of parity involved the beta 

decay ofcobalt-60. Further studies of nuclear 

beta decay led to a beautiful clarification of 

the Fermi theory of weak interactions and 

laid the foundations for modern gauge theories. 

The history of this era reveals a remarkable 

interplay between accelerator and 

non-accelerator experiments. 

In more recent times the solar neutrino 

experiment carried out by R. Davis and his 

colleagues deep in a gold mine provided the 

original motivation for the idea of neutrino 

oscillations. Other experiments deep underground 

have set lower limits of order lo3’ 

years on the lifetime of the proton and may 

yet reveal that “diamonds are not forever.” 

And the limits set at reactors on the electric 

dipole moment of the neutron have proved 

to be a most rigorous test for the many 

models of CP violation that have been 

proposed. 

In 1958, a time of much expansion and 

optimism for the future, Robert R. Wilson, 

the master accelerator builder, compared the 

building of particle accelerators in this century 

with the building of great cathedrals in 

12th and 13th century France. And just as 

the cathedral builders thrust upward toward 

Heaven with all the technical prowess at 

their command, so the accelerator builders 

strive to extract ever more energy from their 

mighty machines. Just as the cathedral 

builders sought to be among the Heavenly 

Hosts, bathed in the radiance of Eternal 

Light, so the accelerator builders seek to 

unlock the deepest secrets of Nature and live 

in a state of Perpetual Enlightenment: 

Ah, but a man’s reach should exceed 

his grasp, 

Or what s a heaven for? 

Robert Browning 

Wilson went on to build his great accelerator, 

and his cathedral too, at Fermilab 

near Batavia, Illinois. In its time, the early to 

mid 1970s, the main ring at Fermilab was the 

most powerful accelerator in the world, and 

it will soon regain that honor as the Tevatron 

begins to operate. The central laboratory 

building, Wilson Hall, rises up to sixteen 

stories like a pair of hands joined in prayer, 

and it stands upon the plain of northcentral 

Illinois much as York Minster stands upon 

the plain of York in England, visible for 

miles around. Some wag once dubbed the 

laboratory building “Minster Wilson, or the 

Cathedral of St. Robert,” and he observed 

that the quadrupole logo of Fermilab should 

be called “the Cross of Batavia.” But Wilson 

Hall serves to remind the citizens of northern 

Illinois that science is ever present in their 

.lives, just as York Minster reassured the 

peasants of medieval Yorkshire that God 

was always nearby. 

The times we live in are much less optimistic 

than those when Wilson first made 

his comparison, and our resources are no 

longer as plentiful for our needs. But we may 

draw comfort from the search for a few nuggets 

of truth in an uncertain world. 

To gaze upfiom the ruins of the 

oppressive present towards thestars is 

to recognise the indestructible world of 

laws, tostrengthen faith in reason, to 

realise the “harmonia mundi” that 

transfuses all phenomena, and that 

never has been, nor will be, disturbed. 

Hermann Weyl, 1919 

S. Peter Rosen, a native of London, was educated at Merton College, Oxford, receiving from that 

institution a B.S. in mathematics in 1954 and both an M.A. and D.Phil. in theoretical physics in 1957. 

Peter first came to this country as a Research Associate at Washington University and then worked as 

Scientist for the Midwestern Universities Research Association at Madison, Wisconsin. A NATO 

Fellowship took him to the Clarendon Laboratory at Oxford in 1961. He then returned to the United 

States to Purdue University, where he retains a professorship in physics. Peter has served as Senior 

Theoretical Physicist for the U.S. Energy Research and Development Administration’s High Energy 

Physics Program, as Program Associate for Theoretical Physics with the National Science Foundation, 

and as Chairman of the U.S. Department of Energy’s Technical Assessment Panel for Proton 

Decay and on the Governing Board for the Lewes Center for Physics in Delaware. His association 

with the Laboratory extends back to 1977 when he came as Visiting Staff Member. He has served as 

Consultant with the Theoretical Division and as a member of the Program Advisory Committee and 

Chairman of the Neutrino Subcommittee of LAMPF. He is currently Associate Division Leader for 

Nuclear and Particle Physics of the Theoretical Division. Peter’s research specialties are symmetries 

of elementary particles and the theory of weak interactions.

Addendum 

The Next Step in Energy 

T 

wo years have passed since this article 

was written, and the high-energy 

physics commuinity is now poised to 

take the next major step forward in energy. 

Between now and 1990 a progression of new 

accelerators (see table) will raise the centerof-mass 

energy of proton-antiproton collisions 

to 2 TeV and that of electron-positron 

collisions to 100 GeV-more than enough to 

produce Wf and Zo bosons in large quantities. 

In addition, the HERA accelerator at 

DESY will enable us to collide an electron 

beam with a proton beam, producing over 

300 (3eV in the center of mass. Thus over the 

next five years we can look forward to a 

wealth of new data and much new physics. 

If the past is any guide, we can anticipate 

many surprises and discoveries of new 

phenomena as the energy of accelerators 

marches upward. But even if we are surprised 

by a lack of surprises, there is still 

much important physics to be explored in 

this new domain. As explained in the article 

by Raby, Slansky, and West, we have a "standard 

model" of particle physics that does a 

beautiful job of describing all known 

phenomena but has the unsatisfactory feature 

of requiring far too many arbitrary 

parameters to be put in by hand. It is therefore 

important to look for physics beyond the 

standard model, the discovery of which 

could lead us to a more general, more highly 

unified model with far fewer arbitrary 

parameters. 

One avenue for searching beyond the standard 

model is the precise measurement of 

properties of the particles it includes. For 

example, the model provides a well-defined 

relationship between the masses of the W' 

arid Zo bosons on the one hand and the weal 

neutral-current mixing angle 8 on the other. 

The theoretical corrections to this relationship 

due to virtual quantum mechanical 

processes (the so-called radiative corrections) 

can be reliably calculated in perturbation 

theory. To achieve the experimental 

precision of 1 percent required to test these 

calculations, we must observe many 

thousands of events. 

Because of their higher energies, both the 

Tevatron and SLC are expected to be much 

more copious sources of electroweak bosons 

than the Spps facility at which they were 

discovered. Whereas the SppS has produced 

approximately 200 Zo - e'e- and 2000 W' 

-t e*v events in a period of three to four 

years, the design luminosity of the Tevatron 

is such that it should yield 1500 Zo -+ p'pand 

15,000 Wf - P'vp in a good year. Even 

more impressive is SLC. which will produce 

3,000,000 Zo bosons per year at its design 

luminosity! So we look forward to precise 

determinations of the properties of the Wf 

bosons from the Tevatron and of the Zo 

boson from SLC. 

Another hndamental test for the standard 

model is the existence of a neutral scalar 

boson, a component of the Higgs boson 

multiplet responsible for generating the 

masses of the W* and Zo gauge bosons. 

While theory imposes a lower limit of a few 

GeV on the mass of this Higgs particle, it

~ shall 

Addendum 

gives no firm prediction for its magnitude, 

nor even an upper bound, and so we have to 

conduct a systematic search over a wide 

band of energies. Should the mass of the 

Higgs particle be less than that of the Zo, 

then we have a good chance of finding it at 

SLC through such processes as Zo - Hoy, 

Hoe+e-, and Hop+p-. If the Higgs particle is 

more massive than the Zo, then it may show 

up at the Tevatron through such decays as 

Ho - Zoy and Zop+p-. Should it prove to 

be beyond the range of the Tevatron, then we 

have to wait until the SSC comes on line 

in the mid 1990s. 

Another avenue for exploration beyond 

the standard model is the search for particles 

it does not include. For example, are there 

more than three families of fermions? 

Precisions studies of the width for Zo - 

XpVpvp will enable us to count the number of 

neutrino species, while apparently 

anomalous decays of the W+ will enable us 

to detect new charged leptons, provided of 

course that the lepton mass is less than that 

of the W+. In a similar way, decays of W* 

into jets of hadrons may reveal the existence 

of new heavy quarks, including the top quark 

required to fill out the existing three families 

of elementary fermions. A hint of the top 

quark was found by the UAl detector at the 

SppS, but there were too few events for it to 

be convincing. The much higher event rates 

of the new accelerators will be extremely 

useful in these searches. 

Besides the Higgs scalar boson and further 

replications of the known fermion families, 

there are hosts of new particles predicted by 

theories that unify the strong and electroweak 

interactions with one another and 

with gravity. Some of these ideas are discussed 

in “Toward a Unified Theory” and 

“Supersymmetry at 100 GeV.” Perhaps the 

most prevalent of such ideas is that of supersymmetry, 

which predicts that for every particle 

there exists a sypersymmetric partner, 

or sparticle, differing in spin by ‘12 and obeying 

opposite statistics. Thus for each fermion 

there exists a scalar boson, an s fermion, and 

for each gauge boson there exists a fermion 

called a gaugino. Some of these sparticles 

could be suficiently light, between a few 

GeV and a few tens of GeV, to be in the mass 

range that can be explored with the new 

accelerators. 

There may also be new interactions that 

appear only at the higher mass scales open to 

the new accelerators. Right-handed currents 

are absent from known low-energy weak interactions, 

possibly because the corresponding 

gauge bosons are much heavier than the 

W* and Zo of the standard model. Depending 

upon their masses, these gauge bosons 

could be produced at LEP I1 (an energy 

upgrade of LEP) or at the Tevatron; the tails 

of their propagators might even show up at 

SLC and at LEP itself. 

Another way of searching for new interactions 

is provided by the electron-proton collider 

HEM, located at DESY in Hamburg. 

There collision of 820-GeV protons with 30- 

GeV electrons provides 314 GeV in the center 

of mass and momentum transfers as large 

as los GeV’. Furthermore the electrons are 

naturally polarized perpendicular to the 

plane of the ring around which they rotate, 

and this polarization can easily be converted 

to a longitudinal direction, left-handed or 

right-handed. We know how the left-handed 

state will interact through the known W* 

and 2’; with the right-handed state we can 

see if a new type of weak interaction comes 

into play at these high energies. 

As the energy of accelerators increases, so 

the resulting collisions become less like interactions 

of the complicated hadronic structures 

that make up protons and antiprotons 

and more like collisions between the elementary 

constituents of these structures, namely 

quarks and gluons. In electron-positron collisions 

we begin with what we believe are two 

elementary and point-like objects. (Think of 

them as the ideal mass points of mechanics 

rather than the heavy balls found upon the 

billiard table.) The hadrons produced in lowenergy 

collisions tend to emerge over large 

angles, but, as the incident energy increases, 

so does the tendency for the hadrons to 

become highly correlated in a small number 

of directions. These correlated groupings of 

hadrons are called jets, and we believe that 

they signify, as closely as is physically 

possible, the production of quarks and 

gluons. These elementary objects cannot 

emerge from the collision regions as free 

particles because of the confinement 

properties and the color neutrality of the 

strong force of quantum chromodynamics. 

Instead they “hadronize” into jets of highly 

correlated groupings of color-neutral 

hadrons. In practice gluon jets, which were 

first discovered at the Doris accelerator in 

Hamburg, are slightly fatter than quark jets, 

which were originally found at PEP. 

From such a point of view, the Tevatron 

becomes a quarkquark, quark-gluon, and 

gluon-gluon collider, while HERA is an electron-quark 

collider. The reduction to 

elementary fermions and bosons enables us 

to interpret events much more simply than 

might otherwise be possible, but it does reduce 

the effective energy available for collisions. 

At the SppS, for example, gluons take 

up half of the energy of the proton, and so 

each quark has, on the average, one-sixth of 

the energy of the parent proton. At the SSC 

the fraction will be somewhat lower. We can 

therefore anticipate that in the decades to 

come there will be a strong impetus to push 

available energies well beyond that of the 

ssc. 

Theoretical motivation for the continued 

thrust toward higher energies may come 

from the notion of compositeness. Fifty 

years ago the electron and proton were 

thought to be elementary objects, but we 

know today that the proton is far from 

elementary. It is possible that in the coming 

period of experimentation we will discover 

that electrons also are not elementary, but 

are made up of other, more fundamental 

entities. Indeed there are theories in which 

leptons and quarks are all composite objects, 

made from things called rishons, or preons. 

Should this be the case, we will need energies 

much higher than that of the SSC to explore 

their properties. The lesson is very simple: at 

whatever energy scale we may be located, 

there is always much more to learn. Today’s 

elementary particles may be tomorrow’s 

atoms. 

157

LAMPF HI and the 

High-Intensity Frontier 

by Henry A. Thiessen 

os group has spent the past two years plan- 

Why Do We Need LAMPF 

igun: 1 shows a layout of the proposed facility. 

future, the possibility of family-changing interactions can 

800-MeV H- Injection Line 

45-GeV Main Ring 

1 

Fig. 1. LAMPF 11, the proposed addition to LAMPe, is 

designed toproduceprotons beams with a maximum energy 

of 45 GeV and a maximum current of 200 microamperes. 

These proton beams will provide intense beams of aidprotons, 

kaons, muons, and neutrinos for use in experiments 

important to both particle and nuclearphysics. The addition 

consists of two synchrotrons, both located 20 meters below 

the existing LAMPF linac. The booster is a 9-GeV, 60- 

hertz, 200-microampere machine fed by UMPF, and the 

main ring is a 45-GeV, 6-hertz, 40-microampere m 

Proton beams will be delivered to the main experime 

area of LAMPF (Area A) and to an area for experiments 

with neutrino beams and short, pulsed beams of other 

secondary particles (Area C). A new area for experiments 

with high-energy secondary beams (Area H) will be constructed 

to make full use of the 45-GeVproton beam.


will include the search for quark effects with the Drell-Yan process, 

the production of quark-gluon plasma by annihilation of antiprotons 

in nuclei, the extraction of nuclear properties from hypernuclei, and 

low-energy tests of quantum chromodynamics. 

1.2 

th 

0 0.2 0.4 0.6 0.8 

ig. 2. The “EMC effect” was first observed in data on the 

:uttering of muons from deuterium and iron nuclei at high 

tomentum transfer. The ratio of the two nucleon structure 

rnctions (FF(Fe) and FF(D)) deduced from these data by 

?garding a nucleus as simply a collection of nucleons is 

kown above as a function of x, aparameter representing the 

*action of the momentum carried by the nucleon struck in 

be collision. The observed variation of the ratio from unity 

I quite contrary to expectations; it can be interpreted as a 

tanifetation of the quark substructure of the nucleons 

)?thin a nucleus. (Adapted from J. J. Aubert et al. (The 

‘uropean Muon Collaboration), Physics Letters 

23B(1983):175.) 

estigated only with high-intensity beams of kaons and muons. And 

udies of neutrino masses and neutrino-electron scattering, which 

re among the most important tests of possible extensions of the 

andard model, demand high-intensity beams of neutrinos to commate 

for the notorious infrequency of their interactions. 

Here I take the opportunity to discuss some of the experiments in 

&ear physics that can be addressed at LAMPF 11. The examples 

X 

Quark Effects, A major problem facing today’s generation of nuclear 

physicists is to develop a model of the nucleus in terms of its 

fundamental constituents-quarks and gluons. In terms of nucleons 

the venerable nuclear shell model has been as successful at interpreting 

nuclear phenomena as its analogue, the atomic shell model, has 

been at interpreting Ithe structure and chemistry of atoms. But 

nucleons are known to be made of quarks and gluons and thus must 

possess some additional internal degrees of freedom. Can we see 

some of the effects of these additional degrees of freedom? And then 

can we use these observations to construct a theory of nuclei based on 

quarks and gluons? 

Defining an experiment to answer the first question is difficult for 

two reasons. First, we know from the success of the shell model that 

nucleons dominate the observable properties of nuclei, and when this 

model fails, the facts can still be explained in terms of the exchange of 

pions or other mesons between the nucleons. Second, the current 

theory of quarks and gluons (quantum chromodynamics, or QCD) is 

simple only in the limit of extremely high energy and extremely high 

momentum transfer, the domain of “asymptotic QCD.” But the 

world of nuclear physics is very far from that domain. Thus, theoretical 

guidance from the more complicated domain of low-energy QCD 

is sparse. 

To date no phenomenon has been observed that can be interpreted 

unambiguously as an effect of the quark-gluon substructure of 

nucleons. However, the results of an experiment at CERN by the 

“European Muon Collaboration”’ are a good candidate for a quark 

effect, although other explanations are possible. This group determined 

the nuclear structure functions for iron and deuterium from 

data on the inelastic scattering of muons at high momentum transfers. 

(A nuclear structure function is a multiplicative correction to the 

Mott cross section; it is indicative of the momentum distribution of 

the quarks within the nucleus.) From these structure functions they 

then inferred values for the nucleon structure function by assuming 

that the nucleus is simply a collection of nucleons. (If this assumption 

were true, the inferred nucleon structure function would not vary 

from nucleus to nucleus.) Their results (Fig. 2) imply that an iron 

nucleus contains more high-momentum quarks and fewer lowmomentum 

quarks than does deuterium. This was quite unexpected 

but was quickly corroborated by a re-analysis2 of some ten-year-old 

electron-scattering data from SLAC and has now been confirmed in 

great detail by several new experiment^.^.^ The facts are clear, but 

how are they to be interpreted? 

The larger number of low-momentum quarks in iron than in 

deuterium may mean that the quarks in iron are shanng their 

momenta, perhaps with other quarks through formation of, say, six- 

159

ocess anses 

r (Fig. 3). Since valence and sea 

es make different contributions 

ry beams of pions, kaons, 

the possible explanations 

help solve these problems by providing large numbers of events 

e valencequark composition of a neutron is udd, an 

fact, the A plays a role in studies of the nuclear environment

~ 

i 8 

! 2 13- 

er, 

K 

.- 

t 

t 

" 

I 

I 

; 

1 

I 

i 

I 

i 

8- 

and minimum disruption to the ongoing experimental programs at 

LAMPF. The designs of both of the new synchrotons reflect these 

goals. 

The booster, or first stage, will be fed by the world's best H- 

injector, LAMPF. This booster will provide a 9-GeV, 200-microamperr: 

beam ofprotons at 60 hertz. The 200-microampere current is 

the maximum consistent with continued use of the 800-MeV 

LAMPF beam by the Weapons Neutron Research Facility and the 

Protort Storage Ring. The 9-GeV energy is ideal not only for i 

into the second stage but also for production of neutrinos to be used 

in scattering experiments (Fig. 5). Eighty percent of the booster 

I 

I 

30 40 

Proton Momentum (GeV/c) 

I 

I 

all six dimensions than the injection requirements of LAMPF IT, 

lossless injection at a correct phase space is straightforward. 

The 45-GeV main ring is shaped like a racetrack for two reasons: it 

fits nicely on the long, narrow mesa site and it provides the 

straight sections necessary for efficient slow extraction. The 

solid curve is S 

based on various 

rate. The scatterz 

compromise minimizes the initial cost yet preserves 

kaons and antiprotons with energies up to 25 GeV. Such high 

energies should prove especially useful for the experiments 

tioned above on the Drell-Yan process and exclusive hadron interactions. 

The booster has a second operating mode: 12 GeV at 30 he 

100 microamperes with a duty factor of 30 percent. This 12-GeV 

mode will be useful for producing 

ring is delayed for financial reasons. 

The most difficult technical problem posed by LAMPF II is the if 

system, which must provide up to 10 megavolts at a peak power of 10 

megawatts and be tunable from 50 to 60 megahertz. Furthermore, 

tunxng must be rapid; that is, the band 

be on the order of 30 kilohertz. The ferrite-tuned rf systems used ia 

the past are typically capable of providing only 5 to 10 kilovolts pier 

gap at up to 50 kilowatts and, in addition, are limited by power 

dissipation in the ferrite tuners and plagued by strong, uncontrollable 

nonlinear effects. We have chosen to concentrate the modest development 

funds available at present on the rf system. A teststand is 

being built, and various ferrites are being studied to gain a beti.er 

understanding of their behavior. 

Cavity Frequency (MHz) 

applied in a buncher cavity developed by the Laboratory's Ac- 

perpendicular bias is a reduction in the ferrite losses by as much as 

similar results. 

162


two orders of magnitude (Fig. 6). Since the loss in the ferrite is 

proportional to the square of the voltage on each gap, reducing these 

losses is essential to achieving the performance required of the 

LAMPF 11 system. 

A collaboration led by R. Carlini and including the Medium 

Energy and Accelerator Technology divisions and the University of 

Colorado has made a number of tests of the perpendicular bias idea. 

Their results indicate that in certain ferrites the low losses persist at 

power levels greater than that needed for the LAMPF I1 cavities. A 

full-scale cavity is now being constructed to demonstrate that 100 

kilovolts per gap at 300 kilowatts is possible. This prototype will also 

help us make a choice of femte based on both rf performance and 

cost of the bias system. A full-scale, full-power prototype of the rf 

system is less than a year away. 

Conclusion 

This presentation of interesting experiments that could be camed 

out at LAMPF I1 is of necessity incomplete. In fact,’ the range of 

possibilities offered by LAMPF I1 is greater than that offered by any 

other facility being considered by the nuclear science community. Its 

funding would yield an extraordinary return. 

References 

I. J. J. Aubertfit al. (The European Muon Collaboration). “The Ratio of the Nucleon Structure 

Functions F2 for Iron and Deuterium.” Physics Letters 123B( 1983):275. 

2. A. Bodek et al. “Electron Scattering from Nuclear Targets and Quark Distributions in Nuclei.” 

Physical Revrew Letters 50( 1983): I43 1. 

3. A. Bodek et al. “Comparison of the Deephelastic Structure Functions of Deuterium and 

Aluminum Nuclei.” Physicd Review Letters 51( 1983):534. 

4. R. G. Arnold et al. “Measurements of the A Dependence of Deephelastic Electron Scattering 

from Nuclei.” Physical Review Letters 52(1984):727. 

5. T. A. Carey, K. W. Jones, J. B. McClelland, J. M. Moss, L. B. Rees, N. Tanaka, and A. D. Bacher. 

“Inclusive Scattering of 500-MeV Protons and Pionic Enhancement of the Nuclear Sea-Quark 

Distribution.” Physical Revrew Letters 53( 1984):144. 

6. I. R. Kenyon. “The Drell-Yan Process.” Reports on Progress in Physics 45( 1982):126l. 

7. D. Strottman and W. R. Gibbs. “High Nuclear Temperatures by Antimatter-Matter Annihilation.” 

Accepted for publication in Physics Letters. 

8. W. Briickner et al. “Spin-Orbit Interaction of Lambda Particles in Nuclei.” Physics Letters 

79B( 1978): 157. 

9. I$, BeFini et al. “A Full Set of Nuclear Shell Orbitals for the A Particle Observed in i2S and 

A Ca. 

IO. A. Bouyssy. “Strangeness Exchange Reactions and Hypernuclear Spectroscopy.” Physics htters 

84B( 1979):41. 

11. M. May et ai. “Observation of Levels in fC, ;*N, and ,fO Hypernuclei.” Physical Review 

Letters 47( 1981): 1 106. 

12. A review of these experiments was presented by G. Bunce at the Conference on the Intersections 

between Particle and Nuclear Physics, Steamboat Springs, Colorado, May 23-30, 1984.and will 

be published in the conference proceedings. 

163

he march toward higher energies 

Features of the three SSC designs considered in the Reference Designs 

Study. The 6.5-tesla design involves a conductor-dominated field with 

both beam tubes in a common cold-iron yoke that contributes slightly to 

shaping the field. In this design the dipole magnet, beam tubes, and 

yoke are supported within a single cryostat. The 5-tesla design involves 

a conductor-dominated dipole field with a heavy-walled iron cryostat to 

attenuate the fringe field. This single-bore design requires two separate 

rings of dipole magnets. The 3-tesla design is similar to the 6.5 tesla 

design except that the field is shaped predominantly by the cold-iron 

yoke rather than by the conductor. 

Dipole 

Total 

Dipole Magnet Accelerator Estimated 

Field Length Diameter cost 

(TI (ft) (mi) 6) 

6.5 51 

5 46 

3 460 

. thermal contraction of the components 

ithin the cryostats must be accommodated. 

eat leaks from power and instrumentation 

ads must be minimized, as must those from 

e magnet supports. (What is needed are 

ipports with the strength of an ox yoke but 

e substance of a spider web.) Alignment 

ill require some means for knowing the 

,act location of the magnets within their 

yostats. And ifa leak should develop in any 

. the piping within a magnet’s cryostat, 

ere needs to be a method for locating the 

ick” magnet and determing where within it 

e problem exists. 

Questions of safety, also, must be adessed. 

For example, the refrigerator locains 

every 2 to 5 miles around the ring are 

gical sites for personnel acccss, but is this 

ten enough? What happens if a helium line 

ould rupture? After all, a person can run 

ly a few feet breathing helium. Will it be 

.- 

18 2.72 billion 



necessary to exclude personnel from the tunnel 

when the system is cold, or can this 

problem be solved with, say, supplied-air 

suits or vehicles? 

Achieving head-on collisions of the beams 

presents further challenges. Each beam must 

be focused down to 10 microns and, more 

taxing, be positioned to within an accuracy of 

about 1 micron. It takes a reasonably good 

microscope even to see something that small! 

Will a truck rumbling by shake the beams out 

of a collision course? What will be the effect 

of earth tides or earthquakes? Does the 

ground heave due to annual changes in temperature 

or water-table level? How stable is 

the ground in the first place? That is, does 

part of the accelerator move relative to the 

remainder? Will it be desirable, or necessary, 

to have a robot system constantly moving 

around the ringrweaking the positions ofthe 

magnets? What would the robot, or any 

surveyor, use as a reference for alignment? 

These are but a few of the many issues that 

have been raised about construction and 

operation of the SSC. Resolving them will 

require considerable technology and ingenuity. 

In April of this year, the Department of 

Energy assigned authority over the SSC effort 

to Universities Research Association 

(URA), the consortium of fifty-four universities 

that runs Fermilab. URA, in turn, assigned 

management responsibilities to a 

separate board of overseers under Boyce 

McDaniel of Cornell University. This board 

selected Maury Tigner as director and 

Stanley Wojcicki of Stanford University as 

deputy director for SSC research and development. 

A headquarters is being established 

at Lawrence Berkeley National Laboratory, 

and a team will be drawn together to define 

what the SSC must do and how best that can 

be done. Secretary of Energy Donald Hodel 

has approved the release of funds to support 

the first year of research and development. 

Since the $20 million provided was about 

half the amount felt necessary for progress at 

the desired rate, shortcuts must be taken in 

reaching a decision on magnet type so that 

site selection can begin soon. 

Los Mamos has been involved in the efforts 

on the SSC since the beginning. We 

have participated in numerous workshops, 

collated siting information and published a 

Site Atlas, and contributed to the portions of 

the Reference Designs Study on beam 

dynamics and the injector. We may be called 

upon to provide the injector linac, kicker 

magnets, accelcrating cavities, and numerous 

other accelerator components. Our research 

on magnetic refrigeration has the potential 

of halving the operating cost of the 

cryogenic system for the SSC. Although the 

results of this research may be too late to be 

incorporated in the initial design, magnetic 

refrigerator replacements for conventional 

units would quickly repay the investment. 

165

UNDERGROUND 

R 

emarkable though it may seem, some of our most direct 

information about processes involving energies far 

beyond those. available at any conceivable particle accelerator 

and far beyond those ever observed in cosmic' 

rays may come from patiently watching a large quantity of water, 

located deep underground, for indications of improbable behavior of 

3 

its constituents. Equally remarkable, our most direct informatio 

about the energy-producing processes deep in the cores of stars come 

not from telescopes or satellites but from carefblly sifting a larg 

volume of cleaning fluid, again located deep underground, for indic 

tions of rare interactions with messengers from the sun. In wi 

follows we will explore some of the science behind these statemen 

learn a bit about how such experiments are Carried out, and vent1 

into what the future may hold. 

The experiments that we will discuss, which can be characteriz 

as searches for exceedingly rare processes, have two features 

common: they are carried out deep below the surface ofthu earth, a 

they involve a large mass of material capable of undergoing 

participating in the rare process in question. The latter feature an! 

from the desire to increase the probability of observing the proci 

within a reasonable length oftime. The underground sire is necess; 

to shield the exDeriment from secondary cosmic rays. These produ 

of the interactions of primary cosmic rays within our atmosphi 

would create an overwhelming background of confusing, mislead! 

"noise." Since about 75 percent of the secondary cosmic rays I 

extremely penetrating muons (resulting from the decays of pions a 

kaons), effective shielding requires overburdens on the order o 

kilometer or so ofsolid rock (Fig. I). 

What arc the goals ofthe experiments that make worthwhile thi 

journeys into the hazardous depths of mines and tunnels w 

complex, sensitive equipment? The largest and in many ways 

ost spectacular experiments-the searches for decay of protons

the search for rare 

neutrons-are aimed at understanding the basic interactions of 

nature. The oldest seeks to verify the postulated mechanism of stellar 

energy production by detecting solar neutrinos-the lone truthful 

witnesses to the nuclear reactions in our star’s core. Smaller experiments 

investigate double beta decay, the rarest process yet observed 

in nature, to elucidate properties of the neutrino. Muon “telescopes” 

will observe the ndmbers, energies, and directions of cosmic-ray 

muons to obtain information about the composition and energy 

spectra ofprimary cosmic rays. Large neutrino detectors will measure 

the upward and downward flux of neutrinos through the earth and 

hence search for neutrino oscillations with the diameter of the earth 

sa baseline. These detectors can also serve as monitors for signals of 

are galactic events, such as the intense burst of neutrinos that is 

xpected to accompany the gravitational collapse of a stellar core. 

A site that can accommodate the increasingly sophisticated techiology 

required will encourage the mounting of underground experinents 

to probe these and other processes in ever greater detail. 

The Search for Nucleon Instability 

The universe is thought to be about ten billion (IO’O) years old, and 

if this unimaginable span of time, the life of mankind has occupied 

iut a tiny fraction. The lifetime of the universe, while immense on 

he scale of the lifetime of the human species, which is itself huge on 

he scale of our own lives, is totally insignificant when compared to 

he time scale on which matter is known to be stable. It is now certain 

hat protons and (bound) neutrons have lifetimes on the order of IO3’ 

ears or more. Thus for all practical purposes these particles are 

otally stable. Why examine the issue any further? 

The incentive is one of principle. The mass of a proton or neutron, 

bout 940 MeV/c2, is considerably greater than that of many other 

iarticles: the photon (zero mass), the neutrinos (very small, perhaps 

ero mass), the electron (0.511 MeV/c’), the muon (IO6 MeV/c2), and 

the charged and neutral pions (140 MeV/cL and 135 MeV/c2), to 

name only the most familiar. Therefore, energy conservation alone 

does not preclude the possibility of nucleon decay. Bearing in mind 

Murray Gell-Mann’s famous dictum that “Everything not compulsory 

is forbidden,” we are obligated to search for nucleon decay 

unless we know ofsomething that forbids it. 

Conservation laws forbidding nucleon decay had been independently 

postulated by Weyl in 1929, Stueckelberg in 1938, and 

Wiper in 1949 and 1952. But Lee and Yang argued in 1955 that such 

laws would imply the existence of a long-range force coupled to a 

conserved quantum number known as baryon number. (The baryon 

number of a particle is the sum of the baryon numbers of its quark 

constituents, +% for each quark and -% for each antiquark. The 

proton and the neutron thus have baryon numbers of +I.) Lee and 

Yang’s reasoning followed the lines that lead to the derivation of the 

Coulomb force from the law of conservation of electric charge. 

However, no such long-range force is observed, or, more accurately, 

the strength of such a force, if it exists, must be many orders of 

magnitude weaker than that ofthe weakest force known, the gravitational 

force. Thus, although no information was available as to just 

how unstable nucleons might be, no theoretical argument demanded 

exact conservation of baryon number. 

Los Alamos has the distinction of being the site of the first searches 

for evidence of nucleon decay. In 1954 F. Reines, C. Cowan, and M. 

Goldhaber placed a scintillation detector in an underground room at 

a depth ofabout 100 feet and set a lower limit on the nucleon lifetime 

of lo2’ years. In 1957 Reines, Cowan, and H. Kruse deduced a greater 

limit of 4 X years from an improved version of the experiment 

located at a depth of about 200 feet (in “the icehouse,” an area 

excavated in the north wall of Los Alamos Canyon). Since these early 

Los Alamos experiments, the limit on the lifetime of the proton has

Nonconservation of baryon number is 

also favored as an explanation for a difficulty 

with the big-bang theory of creation of the 

universe. The difficulty is that the big bang 

supposedly created baryons and antibaryons 

in equal numbers, whereas today we observe 

a dramatic excess of matter over antimatter 

(and an equally dramatic excess of photons 

over matter). In 1967 A. Sakharov pointed 

out that this asymmetry must be due to the 

occurrence of processes that do not conserve 

baryon number; his original argument has 

since been elaborated in terms of grand unified 

theories by several authors. The very 

existence of physicists engaged in searches 

for nucleon decay is mute testimony to the 

baryon asymmetry of the universe and, by 

inference, to the decay of nucleons at some 

level. 

The recent resurgence of interest in the 

stability of nucleons arises in part from the 

success of the unified theory of electromagnetic 

and weak interactions by Glashow, 

Salam, and Weinberg. This non-Abelian 

gauge theory, which is consistent with all 

available data and correctly predicts the existence 

and strength of the neutral-current 

weak interaction and the masses of the Zo 

and W’ gauge bosons, involves essentially 

only one parameter (apart from the masses of 

the elementary particles). The measured 

value of this parameter (the Weinberg angle) 

is given by sin2Bw = 0.22 k 0.01. The success 

of the electroweak model gave considerable 

legitimacy to the idea that gauge theories 

may be the key to unifying all the interactions 

of nature. 

The simplest gauge theory to be applied to 

unifying the electroweak and strong interactions 

(minimal SU(5)) gave rise to two exciting 

predictions. One, that sin2ew = 0.2 I 5, 

agreed dramatically with experiment, and 

the other, that the lifetime of the proton 

against decay into a positron and a neutral 

pion (the predicted dominant decay mode) 

lay between 1.6 X IO2* and 6.4 X IO3’ years, 

implied that experiments to detect nucleon 

decay were technically feasible. 

Experimentalists responded with a series 

of increasingly sensitive experiments to test 

168 

this prediction of grand unification. What 

approach is followed in these experiments? 

Out of the question is the direct production 

of the gauge bosons assumed to mediate the 

interactions that lead to nucleon decay. (This 

was the approach followed recently and successfully 

to test the electroweak theory.) The 

grand unified theory based on minimal 

SU(5) predicts that the masses of these bosons 

are on the order of IOl4 GeV/c2, in 

contrast to the approximately 102-GeV/c2 

masses of the electroweak bosons and many 

orders of magnitude greater than the masses 

of particles that can be produced by any 

existing or conceivable accelerator or by the 

highest energy cosmic ray. Thus, the only 

feasible approach is to observe a huge number 

of nucleons with the hope of catching a 

few of them in the quantum-mechanically 

possible but highly unlikely act of decay. 

The largest of these experiments (the IMB 

experiment) is that of a collaboration including 

the University of California, Irvine, the 

University of Michigan, and Brookhaven 

National Laboratory. In this experiment 

(Fig. 2) an array of 2048 photomultipliers 

views 7000 tons of water at a depth of 1570 

meters of water equivalent (mwe) in the 

Morton-Thiokol salt mine near Cleveland, 

Ohio. The water serves as both the source of 

(possibly) decadent nucleons and as the medium 

in which the signal of a decay is generated. 

The energy released by nucleon decay 

would produce a number ofcharged particles 

with so much energy that their speed in the 

water exceeds that of light in the water (about 

0.75c, where c is the speed of light in 

vacuum). These particles then emit cones of 

Cerenkov radiation at directions characteristic 

of their velocities. The photomultipliers 

arrayed on the periphery of the water detect 

this light as it nears the surfaces. From the 

arrival times of the light pulses and the patterns 

of their intersections with the planes of 

the photomultipliers, the directions of the 

parent charged particles can be inferred. 

Their energies can be estimated from the 

amount of light observed, in conjunction 

with calibration studies based on the vertical 

passage of muons through the detector. (The 

Fig. 1. For some experiments the only 

practical way to sufficiently reduce the 

background caused by cosmic-ray 

muons is to locate the experiments deep 

underground. Shown above is the number 

of cosmic-ray muons incident per 

year upon a cube 10 meters on an edg 

as a function of depth of burial. B 

convention depths of burial in rocks a 

various densities are normalized t 

meters of water equivalent (mwe). Th 

depths of some of the experiments dis 

cussed in the text are indicated. 

impressive sensitivity of such an experimer 

is well illustrated by the information that th 

light from a charged particle at a distance c 

10 meters in water is less than that on th 

earth from a photoflash on the moon.) 

This “water Cerenkov” detection schem 

was chosen in part for its simplicity, in pal 

for its relatively low cost, and in part for il

.--- 

Science Underground 

\ 

Fig. 2. Schematic view of the IMB nucleon-decay detector. A total of 2048 5-inch 

photomultipliers are arrayed about the periphery of 7000 tons of water contained 

within a plastic-lined excavation at a depth of 1570 mwe in a salt mine near 

Cleveland, Ohio. The photomultipliers monitor the water for pulses of Cerenkov 

radiation, some of which may signal the decay of aproton or a neutron. (From R. 

M. Bionta et al., “IMB Detector-The First 30 Days,” in Science Underground 

(Los Alamos, 1982) (American Institute of Physics, New York, 1982)). 

high efficiency at detecting the electrons that 

are the ultimate result of the p - e+ + no 

decay. (The neutral pion immediately decays 

to two photons, which produce showers of 

electrons in the water.) Note, however, that 

although this two-body decay is especially 

easy to detect because of the back-to-back 

orientation of the decay products, it must be 

distinguished, at the relatively shallow depth 

of the IMB experiment, among a background 

of about 2 X IOs muon-induced events per 

day. (The lower limit on the proton lifetime 

predicted by minimal SU(5) implies a maximum 

rate for p - e+no of several events per 

day.) 

Another experiment employing the water 

Cerenkov detection scheme is being carried 

out at a depth of 2700 mwe by a collaboration 

including the University ofTokyo, KEK 

(National Laboratory for High-Energy Physics), 

Niigata University, and the University 

of Tsukuba. The experiment is located under 

Mt. lkenayama in the deepest active mine in 

Japan, the Kamioka lead-zinc mine of the 

Mitsui Mining and Smelting Co. Although 

the mass of the water viewed in this experiment 

(3000 tons) is substantiallly less than 

that in the IMB experiment, its greater depth 

of burial results in lower background rates. 

More important, 1000 20-inch photomultipliers 

are deployed at Kamioka (Fig. 3), in 

contrast to the 2048 5-inch photomultipliers 

at IMB. As a result, a ten times greater fraction 

of the water surface at Kamioka is covered 

by photocathode material, and the lightcollection 

efficiency is greater by a factor of 

about 12. Thus the track detection and 

identification capabilities of the Kamioka 

experiment are considerably better. 

To date neither the IMB experiment nor 

the Kamioka experiment has seen any candidate 

for p - e+lro. These negative results 

yield a proton lifetime greater than 3 X lo3’ 

years for this decay mode, well outside the 

range predicted by the grand unified theory 

based on minimal SU(5). Since this theory 

has a number of other deficiencies (it fails to 

predict the correct ratio for the masses of the 

light quarks and predicts a drastically incorrect 

ratio for the number of baryons and 

169

. 

-. 

photons produced by the big bang), it is 

therefore now thought to be the wrong unification 

model. Other models, at the current 

stage of their development, have too little 

predictive power to yield decay rates that can 

be uriambiguously confronted by experiment. 

The question of nucleon decay is now 

a purely experimental one, and theory awaits 

the guidance of present and future experiment!;. 

The cosmic rays that produce the interfering 

muons also produce copious quantities of 

neutrinos (from the decays of pions, kaons, 

and muons). No amount of rock can block 

these neutrinos, and some of them interact in 

the water, mimicking the effects of proton 

decay. Estimates of this background as a 

function of energy are based on calculations 

of the flux of cosmic-ray-induced neutrinos 

from the known flux of primary cosmic rays. 

Although these calculations enjoy reasonable 

confidence, no accurate experimental data 

are available as a check. Full analyses of the 

neutrino backgrounds in the proton-decay 

experiments will provide the first such verification. 

Whether new effects in neutrino astronomy 

will be discovered from the spectrum 

of neutrinos incident on the earth remains 

to be seen. Thus nucleon-decay experiments 

may open a new field, that of 

neutrino astronomy. 

The water Cerenkov experiments have detected 

several events that could possibly be 

interpreted as nucleon decays by modes 

other than e+no (Table I). It is also possible 

that these events are induced by neutrinos. 

Although their configurations are not easily 

explained on that basis, their total number is 

consistent with the rate expected from the 

calculated neutrino flux. 

A perusal of Table 1 shows that the IMB 

and Kamioka experiments yield different 

lifetime limits and do not see the same number 

of candidate events for the various decay 

modes. This is not surprising since the two 

also differ in aspects other than those already 

mentioned. The Kamioka experiment can 

more easily distinguish events with multiple 

tracks, such as p - p'q, which is immediately 

followed by decay of the q meson 

Fig. 3. Photograph of the Kamioka nucleon-decay detector under construction a? a 

depth of 2700 mwe in a lead-zinc mine about 300 kilometers west of Tokyo. 

Already instal/ed are the bottom layer of photomultipliers and two ranks of 

photomultipliers on the sides of the cylindrical volume. The wire guards around the 

photomultipliers protect the workers from occasional implosions. The upper ranks 

and top layer of photomultipliers were installed from rafts as the water level was 

increased. The detector contains a total of 1000 20-inch photomultipliers. (Photo 

courtesy of the Kamioka collaboration.) 

+ 

p-+ VK 

p -+ vn+ 

n - r+n- 

n-vK 

0 

& .I . Y. 

[Ol 8 .,' roi 

I 

170


(99.75%) p + p ---+ d + e' + v, o - O.Q WV, 607 x la'/cm2 * s 

or 

(0.25%) p + p + e-- d + v, 1.44 MeV, 1.5 X la'/cd * s 

d+p+'He+y 

(86%) 3 ~ + e 3 ~ - e 2p + 4 ~ e 

(1 4%) 

or 

3He + 4He -+ 'Be 

+ y 

(99.89~0) 7Be + e- --* 7Li + v, 0.86 MeV, 43 X l$/cm2 * s 

7~i + p-+ z4tie + y 

(O.O.li%) 

or 

7Bi? + p -. 'B + y 

'6 -+ 'Be* + e' + v, 

'Be* - 24He 

0 - 14.0 MeV, 0.056 X 1a'/em2 ' S 

Fig. 4. The proton-proton chain postulated by the standard solar model as the 

principal mechanism of energyproduction in the sun. The net result of this series of 

nuclear reactions is the conversion of fourprotons into a helium-4 nucleus, and the 

energy released is carried off by photons, positrons, and neutrinos. Predicted 

branching ratios for competing reactions are fisted. Some of the reactions in this 

chain produce neutrinos; the energies of these particles and theirpredictedjluxes at 

the earth are listed at the right. 

by a number of modes. On the other hand, 

the IMB experiment has been in progress for 

a longer time and is thus more sensitive to 

decay modes with long lifetimes. 

The IMB collaboration has recently installed 

light-gathering devices around each 

photomultiplier and will soon double the 

number of tubes with the goal of increasing 

the lightcollection efficiency by a factor of 

about 6. At Kamioka accurate timing circuits 

are being installed on each photomultiplier 

to record the exact times of arrival of the 

light signals. As a result, more and better data 

can be expected from both experiments. 

What else does the future hold? The European 

F6jus collaboration (Aachen, Orsay, 

Palaiseau, Saclay, and Wuppertal) has completed 

construction of a 912-ton modular 

fine-grained tracking calorimeter. This detector 

is located at a depth of 4400 mwe in a 

3300-cubic-meter laboratory excavated near 

the middle of the Frkjus Tunnel connecting 

Modane, France and Bardonnecchia, Italy. 

Its 114 modules consist of 6-meter by 6- 

meter planes of Geiger and flash chambers 

interleaved with thin iron-plate absorbers. 

The detector can pinpoint particle tracks 

with a resolution on the order of 2 millimeters, 

a 250-fold greater resolution than 

that of the water Cerenkov detectors. Data 

about energy losses of the particles along 

their tracks distinguish electrons and muons. 

To date the Frbjus collaboration has observed 

no candidate proton-decay events. 

The Soudan I1 collaboration (Argonne National 

Laboratory, the University of Minnesota, 

Oxford University, Rutherford-Appleton 

Laboratory, and Tufts University) has 

excavated an 1 1,000cubic-meter laboratory 

at 2200 mwe in the Soudan iron mine in 

northern Minnesota and is now constructing 

an 1 100-ton dense fine-grained tracking calorimeter. 

The detector will contain 256 modules, 

each 1 meter by 1 meter by 2.5 meters, 

incorporating thin steel sheets and high-resolution 

drift tubes in hexagonal arrays. The 

spatial resolution of the detector will be 

about 3 millimeters. Information about the 

ionization deposited along the track lengths 

will provide excellent particle-identification 

capabilities. Completion of the detector is 

scheduled for 1988, but data collection will 

begin in 1987. 

Because the Frkjus and Soudan I1 detectors 

view relatively small numbers of 

nucleons (fewer than 6 X lo3'), they can 

record reasonable event rates only for those 

decay modes (if any) with lifetimes considerably 

less than 10'' years. On the other hand, 

they have good resolution for high-energy 

cosmic-ray muons, and this feature will be 

put to good use in experiments of astrophysical 

interest. 

Despite the hopes for these newer experiments, 

the IMB and Kamioka results to date 

imply that accurate investigation of most 

nucleon decay modes demands multikiloton 

detectors with very fine-grained resolution. 

These second-generation detectors will be 

multipurpose devices, sensitive to many 

other rare processes. Realistically, they can 

be operated to greatest advantage only in the 

environment of a dedicated facility capable 

of providing major technical support. 

The Solar Neutrino Mystery 

The light from the sun so dominates our 

existence that all human cultures have 

marveled at its life-giving powers and have 

concocted stories explaining its origins. 

Scientists are no different in this regard. How 

do we explain the almost certain fact that the 

sun has been radiating energy at essentially 

the present rate of about 4 X loz6 joules per 

second for some 4 to 5 billion years? Given a 

solar mass of 2 X lom kilograms, chemical 

means are wholly inadequate, by many orders 

of magnitude, to support this rate of 

energy production. And the gravitational 

171

energy released in contracting the sun to its 

present. radius of about 7 x 10’ kilometers 

could provide but a tiny fraction of the 

radiated energy. The only adequate source is 

the conversion of mass to energy by nuclear 

reactions. 

This answer has been known for a generation 

or two. Through the work of Hans Bethe 

and others in the 1930s and of many workers 

since, we have a satisfactory model for solar 

energy production based on the thermonuclear 

fusion of hydrogen, the most abundant 

element in the universe and in most stars. 

The product of this proton-proton chain 

(Fig. 4) is helium, but further nuclear reactions 

yield heavier and heavier elements. 

Detailed models of these processes are quite 

successful at explaining the observed abundances 

of the elements. Thus it is possible to 

say (with W. A. Fowler) that “you and your 

neighbor and I, each one of us and all of us, 

are truly and literally a little bit of stardust.” 

The successes of the standard solar model 

may:, however, give us misplaced confidence 

in its reality. It is all very well to study 

nuclear reactions and energy transport in the 

laboratory and to construct elaborate computational 

models that agree with what we 

observe of the exteriors of stars. But what is 

the direct evidence in support of our story of 

what goes on deep within the cores of stars? 

The difficulties presented by the demand 

for direct evidence are formidable, to say the 

least. Stars other than our sun are hopelessly 

distant, and even that star, although at least 

reasonably typical, cannot be said to lie conveniently 

at hand for the conduct of experiments. 

Moreover, the sun is optically so 

thick that photons require on the order of 10 

million years to struggle from the deep interior 

to the surface, and the innumerable 

interactions they undergo on the way erase 

any memory of conditions in the solar core. 

Thus, all conventional astronomical obseiivations 

of surface emissions provide no 

direct information about the stellar interior. 

The situation is not hopeless, however, for 

several of the nuclear reactions in the protonproton 

chain give rise to neutrinos. These 

pihcles interact so little with matter that 

Fig. 5. A view of the solar neutrino experiment located at a depth of 4850 feet in the 

Homestake gold mine. The steel tank contains 380,000 liters of perchloroethylene, 

which serves as a source of chlorine atoms that interact with neutrinos from the 

sun. Nearby is a small laboratory where the argon atoms produced are counted. 

(Photo courtesy of R. Davis and Brookhaven National Laboratory.) 

they provide true testimony to conditions in 

the solar core. 

The parameters incorporated in the standard 

solar model (such as nuclear cross sections, 

solar mass, radius, and luminosity, 

and elemental abundances, opacities (from 

the Los Alamos Astrophysical Opacity 

Library), and equations of state) are known 

with such confidence that a calculation of the 

solar neutrino spectrum is expected to be 

reasonably accurate. At the moment only 

one experiment in the world-that of Raymond 

Davis and his collaborators from 

Brookhaven National Laboratory-attempts 

to measure any portion of the solar neutrino 

flux for comparison with such a calculation. 

Located at a depth of 4400 mwe in the 

Homestake gold mine in Lead, South Dakota, 

this experiment (Fig. 5) detects solar 

neutrinos by counting the argon atoms from 

the reaction 

V,+~~C~+~’A~+B , 

which is sensitive primarily to neutrinos 

from the beta decay of boron-8 (see Fig. 4). 

Since chlorine-37 occurs naturally at an 

abundance of about 25 percent, any compound 

containing a relatively large number 

of chlorine atoms per molecule and satisfying 

cost and safety criteria can serve as the 

target. The Davis experiment uses 380,000 

liters of perchloroethylene (CzCl4). 

You might well ask why this reaction occurs 

at a detectable rate. All the solar neutrinos 

incident on the tank of percholoroethylene 

have made the journey from the 

solar core to the earth and then through 4850 

feet of solid rock with essentially no interactions, 

and the neutrinos from the boron-8 

decay constitute but a small fraction of the 

total neutrino flux. What is the special feature 

that makes this experiment possible? 

Apart from the large number of target 

chlorine atoms, it is the existence of an excited 

state in argon-37 that leads to an excep 

tionally high cross section for capture by 

chlorine-37 of neutrinos with energies 

greater than about 6 MeV. Figure 4 shows 

that the only branch of the proton-proton 

chain producing neutrinos with such energies 

is the beta decay of boron-8. The standard 

solar model predicts a rate for the reaction 

of about 7 x per target atom per 

1’72


Fig. 6. Yearly averages of the flux of boron-8 solar neutrinos, 

as measured by the Homestake experiment. The 

discrepancy between the experimental results and thepredictions 

of the standard solar model has not yet been explained. 

(From R. Davis, Jr., B. T. Cleveland, and J. K. Rowley, 

“Report on Solar Neutrino Experiments, ” in Intersections 

Between Particle and Nuclear Physics (Steamboat Springs, 

Colorado), New York American Institute of Physics, 1984.) 

second (7 solar neutrino units, or SNUs), nique has been verified by continual scrutiny 

which corresponds in the Davis experiment over more than meen years. 

to an expected argon-37 production rate of The Homestake experiment has provided 

about forty atoms per month. 

the scientific world with a long-standing 

It may seem utterly miraculous that such a mystery: its results are significantly and consmall 

number of argon-37 atoms can be de- sistently lower than the predictions of the 

tected in such a large volume of target mate- standard solar model (Fig. 6). So what’s 

rial, but the technique is simple. About every wrong? 

two months helium is bubbled through the The first possibility that immediately sugtank 

to sweep out any argon-37 that has been gests itself, that the Davis experiment con- 

Formed. The resulting sample is purified and tains some subtle mistake, cannot be 

:oncentrated by standard chemical tech- eliminated. But it must be dismissed as uniiques 

and is monitored for the 35-day decay likely because of the careful controls in- 

If argon-37 by electron capture. Great care is corporated in the experiment and because of 

&en to distinguish these events by pulse the years of independent scrutiny that the 

ieight, rise time, and half-life from various experiment has survived. The possibility 

)ackground-induced events. As part of the that the parameters employed in the calculatxovery 

technique argon-36 and -38 are in- tion might be in error has been repeatedly 

erted into the tank in gram quantities or less examined by careful investigators seeking to 

o monitor the recovery efficiency (about 95 explain the mystery (and thereby make reercent). 

An artifidly introduced sample of putations for themselves). However, no one 

00 argon-37 atoms has also been recovered has suggested corrections that are large 

uccessfully. Indeed, the validity of the tech- enough to explain the discrepancy. 

Another possibility is that the standard 

solar model is wrong. The reaction that gives 

rise to boron-8 is inhibited substantially by a 

Coulomb barrier and is thus extraordinarily 

sensitive to the calculated temperature at the 

center of the sun. A tiny change in this 

temperature or a small deviation from the 

standard-model value of the solarcore composition 

would be sufficient to change the 

rate of production of boron-8 and thus the 

neutrino flux to which the Davis experiment 

is primarily sensitive. Although many 

“nonstandard” solar models predict lower 

boron-8 neutrino fluxes, none of these are 

widely accepted. In general, the only experimentally 

testable distinction among the 

nonstandard models lies in their predictions 

of neutrino fluxes. A complete characterization 

of the solar neutrino spectrum is needed 

to provide quantitative constraints on the 

standard solar model of the future. 

The explanation of the solar neutrino 

puzzle quite possibly lies in the realm of 

173

particle physics rather than solar physics, 

nuclear physics, or chemistry. The results of 

the Hornestake experiment have generally 

been interpreted on the basis of conventional 

neutrino physics. It is, however, not known 

with certainty how many species of neutrinos 

exist, whether they are massless, or whether 

they are stable. New information about these 

issues could drastically influence the interpretation 

of solar neutrino experiments. 

For example, Bahcall and collaborators 

have pointed out that it is possible for a more 

massive neutrino species to decay into a less 

massive neutrino species and a scalar particle 

(such as a Goldstone boson arising from 

spontaneous breaking of the symmetry associated 

with lepton number conservation). 

If a neutrino species less massive than the 

electron neutrino exists and if the lifetime of 

the electron neutrino is such that those with 

an energy of 10 MeV have a mean life of 500 

seconds (the transit time to the earth), then 

lower-energy electron neutrinos would decay 

before reaching the earth. The resulting reduction 

in the solar neutrino flux could be 

sufficient to explain the Davis results. Note 

that this explanation for the solar neutrino 

puzzle, in direct contrast to explanations 

based on nonstandard solar models, involves 

a great reduction in the flux of essentially all 

but the boron-8 neutrinos. 

Several other explanations of the solar 

neutrino puzzle are also based on speculated 

features of neutrino physics. One of these, 

“oscillations” among the various neutrino 

species, is discussed in the next section. 

Future Solar Neutrino 

Experiments 

Among the nonstandard solar models alluded 

to above are some that allow long-term 

variations in the rate of energy production in 

the solar core. Such variations violate the 

constraint on steady-state solar models that 

hydrogen be burned in the core at a rate 

commensurate with the currently observed 

solar luminosity. To test the validity of these 

models, a Los Alamos group has devised an 

experiment for determining an average of the 

174 

solar neutrino flux over the past several 

million years. 

The experiment, likle Davis’s, is based on 

an inverse beta decay induced by boron-8 

solar neutrinos, namely, 

The molybdenum target atoms must be 

located at depths such that the cosmic-rayinduced 

background of technetium isotopes 

is low compared to the solar neutrino signal. 

This condidtion is satisfied by a molybdenite 

ore body 1 100 to 1500 meters below Red 

Mountain in Clear Creek County, Colorado. 

The ore is currently being mined by AMAX 

Inc. at a depth in excess of about 1150 

meters. The long half-lives of technetium-97 

and -98 (2.6 million and 4.2 million years, 

respectively) have permitted their accumulation 

to a level (calculated on the basis of the 

standard solar model) of about 10 million 

atoms each per 2000 metric tons of ore. 

Fortuitously, the initial large-scale concentration 

of the technetium (into a rheniumselenium-technetium 

sludge) occurs during 

operations involved in producing 

molybdenum trioxide from the raw ore. The 

Los Alamos group has developed chemical 

and mass-spectrographic techniques for 

isolating and counting the technetium atoms 

in the sludge. The first results from the experiment 

should be available in late 1987. 

Much more remains unknown about solar 

neutrinos. In particular, we completely lack 

information about the flux of neutrinos from 

other reactions in the proton-proton chain. 

According to the standard solar model, the 

preponderance of solar neutrinos arises from 

the first reaction in the chain, the thermonuclear 

fusion of two protons to form a deuteron. 

A thorough test of the solar model 

must include measurement of the neutrino 

flux from this reaction, the rate of which, 

although essentially independent of the details 

of the model (varying by at most a few 

percent), involves the basic assumption that 

hydrogen burning is the principal source of 

solar energy. 

The preferred reaction for investigating 

the initial fusion in the proton-proton chain 

is 

v, + 7’Ga --+ 7’Ge + e- , 

which has a threshold of 233 keV, well below 

the maximum energy of the pp neutrinos. 

Calculations based on the standard solar 

model and the relevant nuclear cross sections 

predict a capture rate of about 110 SNU, of 

which about two-thirds is due to the pp reaction, 

about one-third to the electron-capture 

reaction of beryllium-7, and a very small 

fraction to the other neutrino-producing reactions. 

Several years ago members of the Homestake 

team, in collaboration with scientists 

from abroad, carried out a pilot experiment 

to assess a technique suggested for a solar 

neutrino experiment based on this reaction. 

Germanium-71 was introduced into a solution 

of over one ton of gallium (as GaC13) in 

hydrochloric acid. In such a solution 

germanium forms the volatile compound 

GeQ, which was swept from the tank with a 

gas purge. By fairly standard chemical techniques, 

a purified sample of GeH4 was 

prepared for monitoring the 1 l-day decay of 

germanium-7 1 by electron capture. The pilot 

experiment clearly demonstrated the feasibility 

of the technique. 

Why has the full-scale version of this important 

experiment not been done? The 

trouble, as usual, is money. The original 

estimates indicated that achieving an amp 

table accuracy in the measured neutrino flux 

would require about one neutrino capture 

per day, which corresponded to 45 tons of 

gallium as a target. Gallium is neither common 

nor easy to extract, and the cost ofb5 

tons was about $25,000,000, a sum that 

proved unavailable. Nor did the suggestion 

to “borrow” the required amount of gallium 

succeed (despite the fact that only one gallium 

atom per day was to be expended), and 

the collaboration disbanded. 

The chances of mounting a gallium experiment 

seem brighter today, however, since 

recent Monte Carlo simulations have shown 

that an accuracy of 10 percent in the


measured neutrino flux is possible from a 

four-year experiment incorporating improved 

counting efficiencies and reduced 

background rates and involving only 30 tons 

of gallium. 

The European GALLEX collaboration 

(Heidelberg, Karlsruhe, Munich, Saclay, 

Paris, Nice, Milan, Rome, and Rehovot) has 

received approval to install a 3(rton gallium 

chloride experiment in the Gran Sasso Laboratory 

(this and other dedicated underground 

science facilities are described in the next 

section) and sufficient funding to acquire the 

gallium. The collaboration has achieved the 

low background levels required for monitoring 

the decay of germanium-71 and has the 

counting equipment in hand. Progress awaits 

acquisition of the gallium, which will take 

several years. 

The Institute for Nuclear Research of the 

Soviet Academy of Sciences has 60 tons of 

gallium available for an experiment, and a 

chamber has been prepared in the Baksan 

Laboratory. As planned, this experiment 

uses metallic gallium as the target rather than 

GaCl3. However, after a novel initial extraction 

of the germanium, the experiment is 

similar to the gallium chloride experiment. 

Pilot studies have demonstrated the chemical 

techniques necessary for separating the 

germanium from the gallium, and counters 

are being prepared. In November 1986 the 

Soviet group and scientists from Los Alamos 

and the University of Pennsylvania agreed to 

collaborate on the experiment, which will 

begin in late 1987. 

The INR also plans to repeat the Davis 

experiment, increasing the target volume of 

perchloroethylene by a factor of 5. This will 

increase the signal proportionally. 

As mentioned above, a gallium experiment 

detects neutrinos from both proton 

fusion and beryllium-7 decay. To determine 

the individual rates of the two reactions requires 

a separate measurement of the neutrinos 

from the latter. A reaction that satisfies 

the criterion of being sensitive primarily 

to the beryllium-7 neutrinos is 

v, + 'lBr - "Kr + e- 

Results from this bromine experiment are 

important to an unambiguous test of the 

standard solar model. 

The chemical techniques needed for the 

bromine experiment are substantially identical 

to those employed in the chlorine-37 

experiment, and therefore the feasibility of 

this aspect of the experiment is assured. 

However, since krypton-81 has a half-life of 

200,000 years, counting a small number of 

atoms by radioactive-decay techniques is out 

of the question. Fortunately, another technique 

has recently been developed by G. S. 

Hurst and his colleagues at Oak Ridge National 

Laboratory. In barest outline the technique 

involves selective ionization of atoms 

of the desired element by laser pulses of the 

appropriate frequency. The ionized atoms 

can then readily be removed from the sample 

and directed into a mass spectrometer, where 

the desired isotope is counted. Repetitive 

application of the technique to increase the 

selection efficiency has been demonstrated. 

The standard solar model predicts that a 

few atoms of krypton-8 1 would be produced 

per day in a volume of bromine solution 

similar to that of the chlorine solution in the 

Davis experiment. This is a sufficient number 

for successful application of resonance 

ionization spectroscopy. However, two other 

problems must be addressed. Protons 

produced by muons, neutrons, and alpha 

particles may introduce a troublesome background 

via the "Br(p,n)"Kr reaction, and 

naturally occurring isotopes of krypton may 

leak into the tank of bromine solution and 

complicate the mass spectrometry. Davis, 

Hurst, and their collaborators have undertaken 

a complete assessment of the feasibility 

of the bromine-8 1 experiment. 

Other inverse beta decays have been suggested 

as bases for detecting solar neutrinos 

by radiochemical techniques. An experiment 

based on one such reaction, 

v,+~L~ -7Be+e- , 

is being actively developed in the Soviet 

Union by the INR. According to the standard 

solar model, the observed rate of the 

reaction will be about 46 SNU. 

Particularly appealing is the inverse beta 

decay 

~,+~~~~n--.~~~~n'+e- , 

which has an enormous predicted rate (700 

SNU according to the standard solar model) 

and is dominated by pp, ppe, and beryllium-7 

neutrinos. Moreover, the 3-microsecond 

half-life of the product, an excited state of 

tin-I 15, implies that the reaction could be 

the basis for real-time measurements of the 

solar neutrino flux. Unfortunately, indium-1 

15 is not completely stable, decaying 

by beta emission with a half-life of about 5 X 

lOI4 years. Electrons from the beta decay of 

indium-1 15 give rise to signals that can 

mimic the signature of its interaction with a 

solar neutrino (a prompt electron followed 3 

microseconds later by two coincident 

gamma rays). This background is difficult to 

overcome, and such an experiment has not 

yet been fully developed. 

As mentioned above, the source of the 

solar neutrino puzzle may lie not in imperfections 

of solar models but in our limited 

knowledge of neutrino physics. Neutrino oscillations, 

for example, could provide an explanation 

for the Davis results. This phenomenon 

is a predicted consequence of 

nonzero neutrino rest masses, and no theory 

compels an assignment of zero mass to these 

particles. 

If neutrinos are massive, the flavor 

eigenstates that participate in weak interactions 

need not be the same as the mass 

eigenstates that propagate in free space. The 

two types of eigenstates are related by a 

unitary matrix that mixes the various neutrino 

species. For the case of only two neutrino 

species, say electron and muon neutrinos, 

this relation is 

where v, and v,, and VI and v2 are flavor and 

mass eigenstates, respectively. According to 

175

the SchriMinger equation, the wave functions 

of vl and v2 acquire phase factors e8’1‘ 

and e-iE2‘ as they propagate. Therefore a pure 

Ve state (created by, say, the beta decay of 

boron-8) evolves with time (“oscillates”) 

into a state with a nonzero vp component. 

The probability Pve that v, remains at time t 

is given by 

Pve= 1 - sin2 2q3 sin2[(E2 - El) 2/21 , 

where El and E2 are the energies of v1 and v2. 

Thus P., differs from unity if and only if ml 

+ m2, since, in units such that the speed of 

light and Planck’s constant are unity, Ef=p2 

+ mf ForpB m2 > ml, 

rnt-rnt Am2 Am2 

E2-El 

--- x- 

2P 2p 2Ev ’ 

and the characteristic oscillation length (the 

distance over which P., undergoes one cycle 

of its variation) is proportional to Ev/Am2. 

The failure of numerous experiments to 

detect neutrino oscillations in terrestrial neutrino 

sources places an upper limit on Am2 of 

about 0.02 (eV)2. (The precise limits are joint 

limits on Am2 and the mixing angle e.) However, 

if Am2 -c 0.02 (eV2 (as some theoretical 

considerations suggest), oscillations 

would be undetectable in most terrestrial 

experiments and would most profitably be 

sought in lowenergy neutrinos at large distances 

from the source (distances comparable 

to the oscillation length). Unlike the 

terrestrial oscillation experiments to date, 

experiments designed to characterize the 

solar neutrino spectrum could effectively 

search for oscillations in solar neutrinos and 

be capable of lowering the upper limit on 

Am2 to perhaps lo-’’ (eV)2. 

Vacuum oscillations consistent with the 

standard solar model and the Davis experiment 

would require a rather large value of 

the mixing angle 8. However, Wolfenstein, 

Mikhaev, and Smirnov have recently 

pointed out a feature of neutrino oscillations, 

namely, their amplification by matter, that 

could accommodate the Davis results even if 

e is small, since it would greatly increase the 

probability that an electron neutrino 

176 

produced in the highdensity core of the sun 

emerge as a muon neutrino. The amplification 

is due to scattering by electrons and is 

therefore dependent upon electron density. 

(Scattering changes the phase of the 

propagating neutrino; its effect can be viewed 

as a change in either the index of refraction of 

the matter for neutrinos or in the potential 

energy (that is, effective mass) of the neutrino.) 

Observation of matterenhanced oscillations 

should be possible for values of 

Am2 between lo4 and lo-* (eV)2, a range 

inaccessible to experiments on terrestrial 

neutrino sources. 

The importance of the solar neutrino 

puzzle and the exciting possibility that its 

solution may involve fundamental prop 

erties of neutrinos have led to a number of 

recent proposals for real-time flux measurements. 

The Japanese protondecay group, 

together with researchers From Caltech and 

the University of Pennsylvania, is improving 

the Kamioka detector to observe the most 

energetic of the boron-8 solar neutrinos. The 

signal detected will be the Cerenkov radiation 

emitted by electrons in the water that 

recoil from neutrino scattering, receiving on 

average about half the neutrino energy. If the 

goal of a 7-MeV threshold for the detector is 

achieved, about 1 ‘scattering event should be 

observed every 2 days (as predicted on the 

basis of the Davis flux measurements). The 

directionality of the signal relative to the sun 

wil help distinguish scattering events from 

the isotropic background. Similar real-time 

flux measurements will also be possible with 

several of the second-generation detectors 

being built or planned for the Gran Sasso 

Laboratory. 

The Sudbury Neutrino Observatory collaboration 

(Queen’s, Irvine, Oxford, NRCC, 

Chalk River, Guelph, Laurentian, Princeton, 

Carleton) has proposed installing a 1000-ton 

heavy-water Cerenkov detector in the Sudbury 

Facility fix real-time flux measurements 

of a different type. Here the source of 

the Cerenkov radiation will be the electrons 

produced in the inverse beta decay 

ve + d - p -I- p + e-. Since the energy 

imparted to the electron is Ev - 1.44 MeV 

and the hoped-for threshold of the detector is 

about 7 MeV, the experiment will provide 

data on the higher energy portion of the 

boron-8 spectrum. About 8 events per day 

are expected to be recorded. The detector will 

be sensitive also to proton decay and to 

events induced by neutrinos from astrophysical 

sources and by muon neutrinos. 

Dedicated Underground Science 

Facilities 

For at least two decades scientists with 

experiments demanding the enormous 

shielding from cosmic rays afforded by deep 

underground sites have been setting up their 

apparatus in working mines. We owe a great 

debt to the enlightened mine owners who 

have allowed this pursuit of knowledge to 

take place alongside their search for valuable 

minerals. However, as the experiments increase 

in complexity, the need for more sup 

portive, dedicated facilities becomes more 

obvious. 

One argument in favor of a dedicated facility 

is simple but compelling: the need to 

have access to the experimental area controlled 

not by the operation of a mine or a 

tunnel but by the schedules of the experiments 

themselves. Another is the need for 

technical support facilities adequate to experiments 

that will rival in complexity those 

mounted at major accelerators. And not to 

be ignored is the need for accommodations 

for the scientists and graduate students from 

many institutions who will participate in the 

experiments. 

What should such a facility be like? The 

entryway should be large, and the experimental 

area should include at least several 

rooms in which different experiments 

can be in progress simultaneously. 

Provisions for easy expansion, ideally not 

only at the principal depth but also at greater 

and lesser depths, should be available. Another 

aspect that must be carefully planned 

for is safety. The underground environment 

is intrinsically hostile, and in addition some 

experiments may, like the Homestake experiment, 

involve large quantities of


4- 

To L'Aquila 

0 5 

Kilometers 

Fig. 7. The three large (-35,000- 

cubic-meter) experimental halls 

planned for the Gran Sass0 Laboratory 

are shown in afloorplan of the facility 

(top). Two of the three halls are now 

excavated. Also shown are the locations 

of the laboratory ofl the highway tunnel 

under the Gran Sass0 d'Italia and 

of the tunnel in central Italy. 

CORN0 GRAND€ 

(2912 rnl- 

10 

materials that pose hazards in enclosed 

spaces. Materials being considered for the 

bromine experiment, for example, include 

dibromoethane, and other experiments being 

planned involve cryogenic materials under 

high pressure and toxic or inflammable 

materials. Excellent ventilation and gas-tight 

entries to some areas are obvious requirements. 

Such dreams of dedicated facilities for underground 

science are now being realized. 

Italy, for example, recognized the op 

portunity offered by the construction several 

years ago of a new highway tunnel in the 

Apennines and incorporated a major underground 

laboratory (Fig. 7) under the Gran 

Sasso d'Italia near L'Aquila, which is about 

80 kilometers east of Rome. This location 

offers an overburden of about 5000 mwe in 

rock of high strength and low background 

radioactivity. Two of three large rooms (each 

about 120 meters by 20 meters by 15 meters) 

have been completed. Support laboratories 

and offices are located above ground at the 

west end of the tunnel. 

Because of its size, depth, support facilities, 

and ready access by superhighway, the 

Gran Sasso Laboratory is unrivaled as a site 

for underground science. In the spring of 

1985, about a dozen new experiments were 

approved for installation. Among these are 

experiments on geophysics, gravity waves, 

and double beta decay; the GALLEX solar 

neutrino experiment; the large-area (1400- 

square-meter) MACRO detector, which can 

be used in studies of rare cosmic-ray 

phenomena, high-energy neutrino and 

gamma-ray astronomy, and searches for 

magnetic monopoles; and the 6500-ton 

liquid-argon ICARUS detector, which will 

have unprecedented sensitivity to neutrinos 

of solar and galactic origin, proton decay, 

high-energy muons, and many other rare 

phenomena. As an example of the 

capabilities of ICARUS, in one year of operation, 

it will detect, with an accuracy of 10 

percent, a flux of boron-8 neutrinos more 

than twenty times smaller than the Davis 

limit, far below that allowed by any nonstandard 

solar model. 

177

For more than ten years the Soviet Union 

has maintained an underground laboratory 

for cosmic-ray experiments in the Baksan 

River valley near Mt. Elbrus, the highest 

peak of the Caucasus Mountains. A 460-ton 

cosmic-ray telescope and a double beta decay 

experiment are in place at about 800 mwe. 

This laboratory is being greatly expanded 

(Fig. 8). The horizontal entry has been extended 

3.6 kilometers under Mt. Andyrchi. 

There a 60-meter by 5-meter laboratory has 

been constructed to accommodate the Soviet 

gallium solar neutrino experiment and other 

smaller experiments. Further excavations 

are in progress to extend the adit an additional 

700 meters and provide a large room 

for the 3000-ton chlorine solar neutrino experiment. 

On a more modest scale Canada has 

proposed creation of an underground laboratory 

within the extensive and very deep excavations 

of the INCO Creighton No. 9 

nickel mine near Sudbury, Ontario. The 

company has suggested available sites at 

about 2100 meters where rooms as large as 

20 meters in diameter can be constructed. 

Within the United States all underground 

experiments are in working or abandoned 

mines. None of these sites offers any prospect 

for expansion into a full-scale underground 

laboratory to rival Gran Sasso, Baksan, 

or even Sudbury. In 198 1 and 1982 Los 

Alamos conducted a site survey and developed 

a detailed proposal to create a dedicated 

National Underground Science Facility 

at the Department of Energy’s Nevada 

Test Site. The proposal called for vertical 

entry by a l4foot shaft extending initially to 

3600 feet (approximately 2900 mwe) and 

optionally to 6000 feet, excavation of two 

large experimental chambers, and provision 

of surface laboratories and offices. The 

proposal was not funded, and there is no 

other plan to provide a dedicated site in the 

United States for the next generation of underground 

searches for rare events. 

Conclusion 

We have touched in detail upon only two 

of the fascinating experiments that drive 

Kilometers 

Fig. 8. The main experimental areas of the Baksan Laboratory are shown in a 

profile of Mt. .Andyrchi through the adit (top). Area A houses a large cosmic-ray 

telescope, area BI has been excavated for the gallium solar neutrino experiment, 

and area B2, when excavated, will house the 3000-ton chlorine solar neutrino 

experiment. Also shown is the location of the facility near Mt. Elbrus in the 

Kabardino- Balkarian Autonomous Soviet Socialist Republic. 

scientists deep underground. Such experiments 

are not new on the scene, but the large 

and sophisticated second-generation detectors 

being built open up a new era. These 

devices should not be regarded as apparatus 

for a single experiment but as facilities useful 

for a variety of observations. They may be 

able to monitor continuously the galaxy for 

rare neutrino-producing events or the sun for 

variations in neutrino flux and hence in 

energy production. The day may be approaching, 

as Alfred Mann is fond of saying, 

where we will be able, from underground 

laboratories, to take the sun’s temperature 

each morning to see how our nearest star is 

feeling. 

1178


AUTHORS 

L. M. Simmons, Jr., is Associate Division Leader for Research in the Laboratory’s Theoretical 

Division and was until 1985 Program Manager for the proposed National Underground Science 

Facility. He received a B.A. in physics from Rice University in 1959, an M.S. from Louisiana State 

University in 1961, and, in 1965, a Ph.D. in theoretical physics from Cornel1 University, where he 

studied under Peter Carmthers. He did postdoctoral work in elemetary particle theory at the 

University of Minnesota and the University of Wisconsin before joining the University of Texas as 

Assistant Professor. In 1973 he left the University of New Hampshire, where he was Visiting 

Assistant Professor, to join the staff of the Laboratory’s Theoretical Division Office.There he worked 

closely with Carmthers, as Assistant and as Associate Division Leader, to develop the division as an 

outstanding basic research organization while continuing his own research in particle theory and the 

quantum theory of coherent states. He has been, since its inception, coeditor of the University of 

California’s “Los Alamos Series in Basic and Applied Sciences.” In 1979 he originated the idea for 

the Center for Nonlinear Studies and was instrumental in its establishment. In 1980 he took leave, as 

Visiting Professor of Physics at Washington University, to work on strongcoupling field theories 

and their large-order behavior, returning in 1981 as Deputy Associate Director for Physics and 

Mathematics. While in that position, he developed an interest in underground science and began 

work as leader of the NUSF project. He is President of the Aspen Center for Physics and has also 

served that organization as Trustee and Treasurer. 


Michael Martin Nieto, W. C. Haxton, C. M. Hoffman, E. W. Kolb, V. D., Sandberg, and J. W. Toevs, 

editors. Science Underground (Los Alarnos, 1982). New York Amencan Institute of Physics, 1983. 

F. Reines. “Baryon Conservation: Early Interest to Current Concern.” In Proceedings of the 8th International 

Workshop on Weak Interactions and Neutrinos. A Morales, editor. Singapore: World Scientific, 1983. 

D. H. Perkins. “Proton Decay Experiments.” Annual Review of Nuclear and Particle Science 34( 1984): 1. 

R. Bionta et al. “The Search for Proton Decay.” In Intersections Between Particle and Nuclear Physics 

(Steamboat Springs, 1984). New York American Institute of Physics, 1984. 

R. Davis, Jr., B. T. Cleveland, and J. K. Rowley, “Report on Solar Neutrino Experiments.” In Intersections 

Beiween Particle and Nuclear Physics (Steamboat Springs, 1984). New York American Institute of Physics, 

1984. 

J. N. Bahcall, W. F. Huebner, S. H. Lubow, P. D. Parker, and R. K. Ulrich. “Standard Solar Models and the 

Uncertainties in Predicted Capture Rates of Solar Neutrinos.” Reviews of Modern Physics 54( 1982):767. 

R. R. Shafp, Jr., R. G. Warren, P. L. Aamodt, and A. K. Mann. “Prelifninary Site Selection and Evaluation 

for a National Underground Physics Laboratory.” Los Alamos National Laboratory unclassified release 

LAUR-82-556. 

S. P. Rosen, L. M. Simmons, Jr., R. R. Sharp, Jr., and M. M. Nieto. “Los Alamos Proposal for a National 

Underground Science Facility. In ICOBAN 84: Proceedings of the International Conference on Baryon 

Nonconservation (Park City, January I984), D. Cline, editor. Madison, Wisconsin: University of Wisconsin, 

1984. 

R. E. Mischke, editor. Intersections Between Particle and Nuclear Physics (Steamboat Springs. 1984). New 

York American Institute of Physics, 1984. 

C. Castagnoli, editor. “First Symposium on Underground Physics (St.-Vincent, 1985):’ I1 Nuovo Cimento 

9C( 1986): 1 I 1-674. 

J. C. Vander Velde. “Experimental Status of Proton Decay.” In First Aspen Winter Physics Conference, M. 

Block, editor. Annals of the New York Academy of Sciences 461( 1986). 

M. L. Cherry, K. Lande, and W. A. Fowler, editors. Solar Neutrinos and Neutrino Astronomy (Homestake, 

1984) AIP Conference Proceedings No. 126. New York American Institute of Physics, 1984. 

G. A. Cowan and W. C. Haxton. “Solar Neutrino Production of Technetium-97 and Technetium-98.’’ 

Science216(1982):SI. 

179

6 

hat could be worse 

than a bunch of 

physicists gathering 

in a corner at a 

cocktail party to discuss physics?’ asks Pete 

Carruthers. We at Los Alamos Science 

frankly didn’t know what could be 

worse. . .or better, for that matter. However 

we did find the idea of “a bunch of physicists 

gathering in a corner to discuss physics” 

quite intriguing. We felt we might gain some 

insight and, at the same time, provide them 

with an opportunity to say things that are 

never printed in technical journals. So we 

gathered together a small bunch of four, Pete 

Carruthers, Stuart Raby, Richard Slansky, 

and Geoffrey West, found them a corner in 

the home of physicist and neurobiologist 

George Zweig and turned them loose. We 

knew it would be informative; we didn’t 

know it would be this entertaining. 

WEST: Z have here a sort of “jiractalized” 

table of discussion, ihe,first topic being, 

“What isparticlephysr’cs, and what are its 

origins?” Perhaps the older gentlemen among 

us might want to answer that. 

CARRUTHERS: Everyone knows that older 

gentlemen don’t know what particle physics 

is. 

ZWEIG: Particle physics deals with the 

structure of matter. From the time people 

began wondering what everything was made 

of, whether it was particulate or continuous, 

from that time on we had particle physics. 

WEST: In that sense ofwondering about the 

nature of matter, particle physics started 

with the Greeks, if not observationally, at 

least philosophically. 

ZWEIG: I think one ofthe first experimental 

contributions to particle physics came 

around 1830 with Faraday’s electroplating 

experiments, where he showed that it would 

take certain quantities ofelectricity that were 

integral multiples ofeach other to plate a 

mole ofone element or another onto his 

electrodes. 

An even earlier contribution was Brown’s 

observation ofthe motion ofminute particles 

suspended in liquid. We now know the 

chaotic motion he observed was caused by 

the random collision ofthese particles with 

liquid molecules. 

RABY So Einstein’s study of Brownian motion 

is an instance of somebody doing particle 

physics? 

ZWEIG: Absolutely. There’s a remarkable 

description of Brown’s work by Darwin, who 

was a friend of his. It’s interesting that 

Darwin, incredible observer of nature 

though he was, didn’t recognize the chaotic 

nature of the movement under Brown’s 

microscope; instead, he assumed he was see- 

180

181

ing “the marvelous currents of protoplasm in 

some vegetable cell.” When he asked Brown 

what he was looking at, Brown said, “That is 

my little secret.” 

SLANSKY: Quite a bit before Brown, Newton 

explained the sharp shadows created by 

light as being due to its particulate nature. 

That’s really not the explanation from our 

present viewpoint, but it was based on what 

he saw. 

CARRUTHERS: Newton was only half 

wrong. Light, like everything else, does have 

its particulate aspect. Newton just didn’t 

have a way ofexplaining its wave-like 

behavior. That brings us to the critical concept 

offield, which Faraday put forward so 

clearly. You can speak ofparticulate structure, 

but when you bring in the field concept, 

you have a much richer, more subtle structure: 

fields are things that propagate like 

waves but materialize themselves in terms of 

quanta. And that is the current wisdom of 

what particle physics is, namely, quantized 

fields. 

Quantum field theory is the only conceptual 

framework that pieces together the concepts 

of special relativity and quantum theory, 

as well as the observed group structure 

of the elementary particle spectrum. All these 

things live in this framework, and there’s 

nothing to disprove its structure. Nature 

looks like a transformation process in the 

framework ofquantum field theory. Matter 

is not just pointy little particles; it involves 

the more ethereal substance that people 

sometimes call waves, which in this theory 

are subsumed into one unruly construct, the 

quantized field. 

ZWEIG: Particle physics wasn’t always 

quantized field theory. When I was a graduate 

student, a different philosophy governed: 

S-matrix theory and the bootstrap 

hypothesis. 

CARRUTHERS: That was a temporary 

aberration. 

ZWEIG: But a big aberration in our lives! S- 

matrix theory was not wrong, just largely 

irrelevant. 

RABY: If particle physics is the attempt to 

understand the basic building blocks of 

182 

nature, then it’s not a static thing. Atomic 

physics at one point was particle physics, but 

once you understood the atom, then you 

moved down a level to the nucleus, and so 

forth. 

WEST: Let’s bring it up to date, then. When 

would you say particle physics turned into 

high-energy physics? 

ZWEIG: With accelerators. 

SLANSKY: Well, it really began around 

19lOwith theuseofthecloudchambers to 

detect cosmic rays; that is how Anderson 

detected the positron in 1932. His discovery 

straightened out a basic concept in quantized 

field theory, namely, what the antiparticle is. 

CARRUTHERS: Yes, in 1926 Dirac had 

quantized the electromagnetic field and had 

given wave/particle duality a respectable 

mathematical framework. That framework 

predicted the positron because the electron 

had to have a positively charged partner. 

Actually, it was Oppenheimer who predicted 

the positron. Dirac wanted to interpret the 

positive solution of his equation as a proton, 

since there were spare protons sitting around 

in the world. To make this interpretation 

plausible, he had to invoke all that hankypanky 

about the negative energy sea being 

filled-you could imagine that something 

was screwy. 

SLANSKY: Say what you will, Dirac’s idea 

was a wonderful unification of all nature, 

much more wonderful than we can envisage 

today. 

ZWEIG: Ignorance is bliss. 

SLANSKY: There were two particles, the 

proton and the electron, and they were the 

basic structure ofall matter, and they were, 

in fact, manifestations ofthe same thing in 

field theory. We have nothing on the horizon 

that promises such a magnificent unification 

as that. 

RABY Weren’t the proton and the electron 

supposed to have: the same mass according to 

the equation? 

WEST: No, the negative energy sea was supposed 

to take care ofthat. 

CARRUTHERS: It was not unlike the present 

trick ofexplaining particle masses 

through spontaneous symmetry breaking. 

Dirac’s idea ofviewing the proton and the 

electron as two different charge states ofthe 

same object was a nice idea that satisfied all 

the desires for symmetries that lurk in the 

hearts oftheorists, but it was wrong. And the 

reason it was wrong, ofcourse, is that the 

proton is the wrong object to compare with 

the electron. It’s the quark and the electron 

that may turn out to be different states ofa 

single field, a hypothesis we call grand unification. 

WEST: Well, it is certainly true that highenergy 

particle physics now is cloaked in the 

language ofquantized field theory, so much 

so that we call these theories the standard 

model. 

CARRUTHERS: But I think we’re overlooking 

the critical role of Rutherford in inventing 

particle physics. 

WEST: The experiments ofalpha scattering 

on gold foils to discern the structure ofthe 

atom. 

ZWEIG: Rutherford established the 

paradigm we still use for probing the structure 

of matter: you just bounce one particle 

off another and see what happens. 

CARRUTHERS: In fact, particle physics is a 

continuing dialogue (not always friendly) between 

experimentalists and theorists. Sometimes 

theorists come up with something that 

is interesting but that experimentalists 

suspect is wrong, even though they will win a 

Nobel prize if they can find the thing. And 

what the experimentalists do discover is frequently 

rather different from what the 

theorists thought, which makes the theorists 

go back and work some more. This is the way 

the field grows. We make lots of mistakes,we 

build the wrong machines, committees decide 

to do the wrong experiments, and 

journals refuse to publish the right theories. 

The process only works because there are so 

many objective entrepreneurs in the world 

who are trying to find out how matter 

behaves under these rather extreme conditions. 

It is marvelous to have great synthetic 

minds like those of Newton and Oalileo, but 

they build not only on the work ofunnamed 

thousands of theorists but also on these 

countless experiments.

“To understand the universe that we feel and touch, 

even down to its minutiae, you don’t have to know a 

damn thing about quarks.” 

ROUND TABLE 

WEST: Perhaps we should tell how wepersonally 

got involved inphysics, what drives us, 

why we stay with it. Because it is an awfully 

difficult field and a very frustrating field. 

How do wefind the reality of it compared to 

our early romantic images? Let’s start with 

Pete, who’s been interviewed many times and 

should be in practice. 

CARRUTHERS: I was enormously interested 

in biology as a child, but I decided that 

it was too hard, too formless. So I thought I’d 

do something easy like physics. Our town 

library didn’t even have modern quantum 

mechanics books. But I read the old quantum 

mechanics, and I read Jeans and Eddington 

and other inspirational books filled 

with flowery prose. I was very excited about 

the mysteries of the atom. It was ten years 

before I realized that I had been tricked. I had 

imagined I would go out and learn about the 

absolute truth, but aftera little bit ofexperience 

I saw that the “absolute truth” of 

this year is replaced next year by something 

that may not even resemble it, leaving you 

with only some small residue of value. 

Eventually I came to feel that science, despite 

its experimental foundation and reference 

frame, shares much with other intellectual 

disciplines like music, art, and literature. 

WEST Dick, what about you? 

SLANSKY In college I listed myself as a 

physics major, but I gave my heart to 

philosophy and writing fiction. I had quite a 

hard time with them, too, but physics and 

mathematics remained easy. However, since 

I didn’t see physics as very deep, I decided 

after I graduated to look at other fields. I 

spent a year in the Harvard Divinity School, 

where I found myself inadvertently a 

spokesman for science. I took Ed Purcell’s 

quantum mechanics course in order to be 

able to answer people’s questions, and it was 

there that I found myself, for the first time, 

absolutely fascinated by physics. 

During that year I had been accepted at 

Berkeley as a graduate student in philosophy, 

but in May I asked them whether I could 

switch to physics. They wrote back saying it 

GEOFFREY B. WEST: “One of the great things that has happened in particle 

physics is that some of. . . the wonderful, deep questions. . . are being asked 

again . . . Somehow we have to understand why there is a weak scale, why there is 

an electromagnetic scale, why a strong scale, and ultimately why a grand scale.” 

would be fine. I don’t know that one can be 

such a dilettante these days. 

SCIENCE: Why were people in Divinity 

School asking about quantum mechanics? 

SLANSKY People hoped to gain some insight 

into the roles of theology and 

philosophy from the intellectual framework 

of science. In the past certain philosophical 

systems have been based on physical theories. 

People were wondering what had really 

happened with quantum mechanics, since no 

philosophical system had been built upon it. 

Efforts have been made, but none so successful 

as Kant’s with Newtonian physics, for 

example. 

CARRUTHERS: Particle physics doesn’t 

stand still for philosophy. The subject is such 

that as soon as you understand something, 

you move on. I think restlessness 

characterizes this particular branch of science, 

in fact. 

SLANSKY I never looked at science as 

something I wanted to learn that would be 

absolutely permanent for all the rest ofthe 

history of mankind. I simply enjoy the doing 

of the physics, and I enjoy cheering on other 

people who are doing it. It is the intellectual 

excitement of particle physics that draws me 

to it. 

ZWEIG: Dick, was there some connection, 

in your own mind, between religion and 

physics? 

183

“The realproblem was that you had a zoo ofparticles, 

with none seemingly more fundamental than any 

other.” 

SLANSKY: Some. One of the issues that 

concerned me was the referential 

mechanisms oftheological language. How 

we refer to things. In science we also have 

that concern, very much so. 

ZWEIC;: What do you mean by “How we 

refer to things”? 

SLANSKY When we use a word to refer to 

God or to refer to great generalizations in our 

experience, how does the word work to refer 

beyond the language? Language isjust a 

sound. How does the word refer beyond just 

the mere word to the total experience? I’ve 

never really solved that problem in my own 

mind. 

CARRUTHERS. When you mention the 

word God, isn’t there a pattern of signals in 

your mind that corresponds to the pattern of 

sound? Doesn’t God have a peculiar pattern? 

SLANSKY The referential mechanisms of 

theological language became a major concern 

around 1966, after I’d left Harvard Divinity. 

Before that the school was under the influerice 

of the two great theologians Paul 

Tillich and Reinhold Niebuhr. Their concern 

was with the eighteenth and nineteenth century 

efforts to put into some sort of theoretical 

or logical framework all of man and his 

nature. I found myselfswept up much more 

into theological and philosophical issues 

than into the study ofethics. 

ZWEIG: Do you think these issues lie in the 

domain of science now? Questions about 

what man is, what his role in nature is, and 

what nature itself is, are being framed and 

answered by biologists and physicists. 

SLANSKY: I don’t view what I am trying to 

do in particle physics as finding man’s place 

in nature. I think of it as a puzzle made ofa 

lot ofexperimental data, and we are trying to 

assemble the pieces. 

CARRUTHERS: But the attitudes are very 

theological, and often they tend to be 

dogmatic. 

SLANSKY: I would like to make a personal 

statement here. That is, when I go out for a 

walk in the mountains, enjoying the beauties 

of nature with a capital N, I don’t feel that 

that has any very direct relationship to formulating 

a theory of nature. While my per- 

184 

sonal experience may set my mind in motion, 

may provide some inspiration, I don’t 

feel that seeing the Truchas peaks or seeing 

wild flowers in the springtime is very closely 

related to my efforts tCJ build a theory. 

WEST: Along that line I have an apocryphal 

story about Hans and Rose Bethe. One summer’s 

evening when the stars were shining 

and the sky was spectacular, Rose was exclaiming 

over their beauty. Allegedly Hans 

replied, “Yes, but yoii know, I think I am the 

only man alive that knows why they shine.” 

There you have the difference between the 

romantic and the scientific views. 

RABY: Particle physics to me is a unique 

marriage ofphilosophy and reality. In high 

school I read the philosopher George 

Berkeley, who discusses space and time and 

tries to imagine what space would be like 

were there nothing in it. Could there be a 

force on a particle were there nothing else in 

space? Obviously a particle couldn’t move 

because it would have nothing to move with 

respect to. Particle physics has the beauty of 

philosophy constrained by the fact you are 

working with observable reality. For a science 

fair in high school I built a cloud 

chamber and tried to observe some alpha 

particles and beta particles. That’s the reality 

part: you can actually build an experiment 

and actually see some of these fundamental 

objects. And there are people who are 

brilliant enough, like Einstein, to relate ideas 

and thought to reality and then make predictions 

about how the world mu$t be. Special 

relativity and all the Gedanken experiments, 

which are basically philosophical, say how 

the world is. To me what particle physics 

means is that you can have an idea, based on 

some physical fact, that leads to some experimental 

prediction. That is beautiful, and 

I don’t know how you define beauty except 

to say that it’s in the eye ofthe beholder. 

ZWEIG How was science viewed in your 

family? 

RABY: No one understood science in my 

family. 

ZWEIG: Well, did they respect it even if they 

STUART A. RABY: “I think what particle physics means to me is this unique 

intermarriage of philosophy and reality. . . . Particle physics has the beauty of 

philosophy constrained by the fact that you are working with observable reality. 

. . . If you have a beautiful idea and it leads to a prediction that, in fact, 

comes true, that would be the most amazing thing. That you can understand 

something on such a fundamental level!”

“. b b b one thing that distinguishesphysics from 

philosophy is predictive powerb The quark model had a 

lot ofpredictive powerb” 

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didn’t understand it? 

RABY: I guess they accepted the fact that I 

would pursue what interested me. I’m the 

first one in my family to finish college, and 

that in itself is something big to them. My 

grandfather, who does understand a little, 

has read about Einstein. My grandfather’s 

interest in science doesn’t come from any 

particular training, but from the fact that he 

is very inventive and intuitive and puts 

radios together and learns everything by 

himself. 

ZWEIG: Was he respected for it? 

RABY By whom? My grandfather owned a 

chicken market, so he did these things in his 

spare time. 

WEST: That’s interesting. I have to admit I 

am another person who got into physics in 

spite of himself. I was facile in mathematics 

but more keen on literature. I turned to 

natural sciences when I went to Cambridge 

only because I had begun reading Jeans and 

Eddington and all those early twentieth century 

visionaries. They were describing that 

wonderful time of the birth of quantum mechanics, 

the birth of relativity, the beginning 

of thinking about cosmology and the origin 

ofthe universe. Wonderful questions! Really 

important questions that dovetailed into the 

big questions raised by literature. What is it 

all really about, this mysterious universe? 

The other crucial reason that I went into 

science was that I could not stand the world 

ofbusiness, the world of the wheelerdealer, 

that whole materialistic world. Somehow I 

had an image of the scientist as removed 

from that, judged only by his work, his only 

criteria being proof, knowledge, and wisdom. 

I still hold that romantic image. And that has 

been my biggest disappointment, because, of 

course, science, like everything else that involves 

millions ofdollars, has its own 

wheeler-dealers and salesmen and all the rest 

of it. 

My undergraduate experience at Cambridge 

was something of a disaster in terms 

ofphysics education, and I was determined 

to leave the field. I had become very interested 

in West Coastjazzand managed to 

obtain a fellowship to Stanford where, for a 

- . *. 

RICHARD A. SLANSKY: “It is the intellectual excitement of particle physics 

that draws me to it, really . . . . Ifindparticlephysics an intriguing effort to try to 

explain and understand, in a very special way, what goes on in nature . . . . I enjoy 

the effort. . . . I enjoy cheering on otherpeople who are trying. . . . I think of it as 

a puzzle made of a lot of experimental data, and we are trying to assemble all the 

pieces. ” 

year, I could be near San Francisco, North 

Beach, and that whole scene. Although at 

first I hated Palo Alto, my physics courses 

were on so much more a professional level, 

so much more an exciting level, that my 

attitude eventually changed. Somehow the 

whole world opened up. But even in graduate 

school I would go back to reading Eddington, 

whether he were right or not, because his 

language and way ofthinking were inspirational, 

as ofcourse, were Einstein’s. 

CARRUTHERS: Do you think our visions 

have become muddied in these modern 

times? 

WEST: I don’t think so at all. One ofthe 

great things that has happened in particle 

physics is that some of the deep questions are 

being asked again. Not that I like the 

proposed answers, particularly, but the questions 

are being asked. George, what do you 

say to all this? You often have a different 

slant. 

ZWEIG: My parents came from eastern 

Europe-they fled just before the second 

World War. I was born in Moscow and came 

to this country when I was less than two years 

old. Most of my family perished in the war, 

probably in concentration camps. I learned 

185

“It is an old Jewish belief that ideas are what really 

matter. If you want to create things that will endure, 

you create them in the mind of man.” 

at a very early age from the example of my 

father, who was wise enough to see the situation 

in Germany for what it really was, that 

it is very important to understand teality. 

Reality is the bottom line. Science deals with 

reality, and psychology with our ability to 

accept it. 

I grew up in a rough, integrated 

neighborhood in Detroit. Much of it subsequently 

burned down in the 1967 riot. I 

hated school and at first did very poorly. I 

was placed in a “slower” non-college 

preparatory class and took a lot of shop 

courses. Although I did not like being viewed 

as a second class citizen, I thought that operating 

machines was a hell of a lot more 

interesting than discussing social relations 

with my classmates and teachers. 

Eventually I was able to do everything that 

was asked of me very quickly, but the teachers 

were not knowledgeable, and classes were 

boring. In order to get along I kept my mouth 

shut. Occasionally I acted as an expediter, 

asking questions to help my classmates. 

At that time science and magic were really 

one and the same in my mind, and what 

child isn’t fascinated by magic? At home I 

did all sorts of tinkering. I built rockets that 

flew and developed my own rocket fuels. The 

ultimate in magic was my tesla coil with a six 

foot corona emanating from a door knob. 

College was a revelation to me. I went to 

the University of Michigan and majored in 

mathematics. For the first time I met teachers 

who were smart. And then I went to 

Caltech, a place I had never even heard of six 

months before I arrived. At Caltech I was 

very fortunate to work with Alvin 

Tollestrup, an experimentalist who later designed 

the superconducting magnets that are 

used at Fermilab. And I was exposed to 

Feynman and Cell-Mann, who were unbelievable 

individuals in their own distinctive 

ways. That was an exciting time. 

Shelly Glashow was a postdoc. Ken Wilson, 

Hung Cheng, Roger Dashen, and Sidney 

Coleman were graduate students. Rudy 

Mossbauer was down the hall. He was still a 

research fellow one month before he got the 

Nobel prize. The board of trustees called a 

GEORGE ZWEIG: “I learned at a very early age. . .that reality is the bottom line. 

Science deals with reality, andpsychology with our ability to accept it.” 

crash meeting and promoted him to full 

professorjust before the announcement. I 

remember pleading with Dan Kevles in the 

history department to come over to the physics 

department and record the progress, because 

science history was in the making, but 

he wouldn’t budge. “You can never tell what 

is important until many years later,” he said. 

CARRUTHERS: I’ve forgotten whether I 

first met all ofyou at Cal Tech or at Aspen. 

ZWEIG: Wherever Pete met us, I know 

we’re all here because of him. He was always 

very gently asking me, “How about coming 

to Los Alamos?’ Eventually I took him up 

on his offer. 

WEST: Before we leave this more personal 

side of the interview, I want to ask a question 

or two about families. Is it true that 

physicists generally come from middle class 

and lower backgrounds? Dick, what about 

your family? 

SLANSKY My father came from a farming 

family. Since he weighed only ninety-seven 

pounds when he graduated from high school, 

farm work was a little heavy for him. He 

entered a local college and eventually earned 

a graduate degree from Berkeley as a physical 

chemist. My mother wanted to attend 

medical school and was admitted, but back 

in those days it was more important to have 

186

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children. So I am the result rather than her 

becoming a doctor. 

CARRUTHERS: My father grew up on a 

farm in Indiana, was identified as a bright 

kid, and was sent off to Purdue, where he 

became an engineer. So I at least had somebody 

who believed in a technical world. 

However, when I finally became a professor 

at Cornell, my parents were a bit disappointed 

because in their experience only 

those who couldn’t make it in the business 

world became faculty members. 

WEST What about your parents, George? 

ZWEIG: Both my parents are intellectuals, 

people very much concerned with ideas. To 

me one ofthe virtues ofdoing science is that 

you contribute to the construction ofideas, 

which last in ways that material monuments 

don’t. It is an old Jewish belief that ideas are 

what really matter. If you want to create 

things that will endure, you create them in 

the mind of man. 

WEST What did your parents do? 

ZWEIG: My mother was a nursery school 

teacher. She studied in Vienna in the O OS, an 

exciting time. Montessori was there; Freud 

was there. My father was a structural engineer. 

He chose his profession for political 

reasons, because engineering was a useful 

thing to do. 

WEST Then all three ofyou have scientific 

or engineering backgrounds. My mother is a 

dressmaker, and my father was a professional 

gambler. But he was an intellectual 

in many ways, even though he left school at 

fifteen. He read profusely, knew everything 

superficially very well, and was brilliant in 

languages. He wasted his life gambling, but it 

was an interesting life. I think I became facile 

in mathematics at a young age just because 

he was so quick at working out odds, odds on 

dogs and horses, how to do triples and 

doubles, and so on. 

CARRUTHERS: Are we all firstborn sons? I 

think we are, and that’s an often quoted 

statistic about scientists. 

WEST Have we all retreated into science for 

solace? 

RABY: It’s more than that. At one time I felt 

divided between going into social work in 

PETER A. CARRUTHERS: “There’s no point in a full-blown essay on quantum 

field theory because it’s probably wrong anyway. That’s what fundamental science 

is all about-whateveryou’re doing is probably wrong. That’s how you know when 

you’re doing it. Once in a while you’re right, and then you’re a great man, or 

woman nowadays. I’ve tried to explain this before to people, but they’re very slow to 

understand. What you have to do is look back and find what has been filtered out as 

correct by experiments and a lot of subsequent restructuring. Right? But when 

you’re actually doing it, almost every time you’re wrong. Everybody thinks you sit 

on a mountaintop communing with Jung’s collective unconscious, right? Well you 

try, but the collective unconscious isn’t any smarter than you are.” 

187

“Why do the forces in nature have different 

strengths . That’s one of those wonderful deep questions 

that has come back to haunt us.” 

order to be involved with people or going 

into science and being involved with ideas. It 

was continually on my mind, and when I 

graduated from college, I took a year off to do 

social work. I worked in a youth house in the 

South Bronx as a counselor for kids between 

the ages of seven and seventeen. They were 

all there waiting to be sentenced, and they 

were very self-destructive kids. The best 

thing you could do was to show them that 

they should have goals and that they 

shouldn’t destroy themselves when the goals 

seemed out ofreach. For example, a typical 

goal was to get out of the place, and a typical 

reaction was to end up a suicide. I kept trying 

to tell these kids, “Do what you enjoy doing 

and set a goal for yourselfand try to fulfill 

that goal in positive ways.” In the end I was 

convinced by my own logic that I should 

return to physics. 

WEST Let’s discuss the way physics affects 

our personal lives now that we are grown 

men. Suppose you are at a cocktail party, and 

someone asks, “What do you do?’ “I am a 

physicist,” you say, “High-energy physics,” 

or “Particle physics.” Then there is a silence 

and it is very awkward. That is one response, 

and here is the other. “Oh, you do particle 

physics? My God, that’s exciting stuff! I read 

about quarks and couldn’t understand a 

word of it. But then I read this great book, 

The Tu0 oSphysics. Can you tell me what you 

do?’ I groan inwardly and sadly reflect on 

how great the communication gap is between 

scientists such as ourselves and the general 

public that supports us. We seem to have 

shirked our responsibility in communicating 

the fantastic ideas and concepts involved in 

our enterprise to the masses. It is a sobering 

thought that Capra’s book, which most of us 

don’t particularly like because it represents 

neither particle physics nor Zen accurately, is 

probably unique in turning on the layman to 

some aspects ofparticle physics. Whatever 

your views ofthat book may be, you’ve 

certainly got to appreciate what he’s done for 

the publicity ofthe field. As for me, I find it 

difficult to talk about this life that I love in 

two-line sentences. 

Now, the cocktail party is just a superficial 

188 

aspect of my social life, but the problem 

enters in a more crucial way in my relationship 

with my family, the people dear to 

me. Here is this work which I love, which I 

spend a majority of my time in doing, and 

from which a large number ofthe frustrations 

and disappointments and joys in my 

life come, and I cannot communicate it to 

my family except in an incredibly superficial 

way. 

SLANSKY: The cocktail party experiences 

that Geoffrey describes are absolutely 

perfect, and I know what he means about the 

family. Now that my children are older, they 

are into science, and sometimes they ask me 

questions at the dinner table. I try to give 

clear explanations, but I’m never sure I’ve 

succeeded even superficially. And my wife, 

who is very bright but has no science background, 

doesn’t hesitate to say that science in 

more than twenty-five words is boring. 

Sometimes, in fact, I feel that my doing 

physics is viewed by them as a hobby. 

CARRUTHERS: Socially, what could be 

worse than a bunch ofphysicists gathering in 

a corner at a cocktail party to discuss physics? 

RABY: I find there are two types ofpeople. 

There are people who ask you a question just 

to be polite and who don’t really want an 

answer. Those people you ignore. Then there 

are people who are genuinely interested, and 

you talk to them. Ifthey don’t understand 

what a quark is, you ask them if they understand 

what a proton or an electron is. If they 

don’t understand those, then you ask them if 

they know what an atom is. You describe an 

atom as electrons and a nucleus of protons 

and neutrons. You go down from there, and 

you eventually get to what you are studying-particle 

physics. 

WEST: Does particle physics affect your relationship 

with your wife? 

RABY: My wife is occasionally interested in 

all this. My son, however, is genuinely interested 

in all forms of physical phenomena and 

is constantly asking questions. He likes to 

hear about gravity, that the gravity that pulls 

objects to the earth also pulls the moon 

around the earth. I have to admit that I find 

his interest very rewarding. 

WEST: Maybe, since we’ve been given the 

opportunity today, we should start talking 

aboutphysics. Particlephysics has gone 

through a minirevolution since the discovery 

of thepsi/Jparticle at SLACand at 

Brookhaven ten years ago. Although not important 

in itself; that discovery confirmed a 

whole way of thinking in terms of quarks, 

symmetry principles, gauge theories, and 

unification. It was a bolt out of the blue at a 

time when the direction ofparticlephysics was 

uncertain. From then on, it became clear that 

non-Abelian gauge theories and unification 

were going to form the fundamental 

principles for research. Sociologically, there 

developed a unanimity in thefield, a unanimity 

that has remained. This has led us to 

the standard model, which incorporates the 

strong, weak, and electromagnetic interactions. 

SLANSKY: Yes, the standard model is a 

marvelous synthesis of ideas that have been 

around for a long time. It derives all interactions 

from one elegant principle, the principle 

oflocal symmetry, which has its origin 

in the structure ofelectromagnetism. In the 

1950s Yang and Mills generalized this structure 

to the so-called non-Abelian gauge the-

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ories and then through the '60s and '70s we 

learned enough about these field theories to 

feel confident describing all the forces of 

nature in terms ofthem. 

RABY: 1 think we feel confident with Yang- 

Mills theories because they are just a sophisticated 

version of our old concept of force. 

The idea is that all of matter is made up of 

quarks and leptons (electrons, muons, etc.) 

and that the forces or interactions between 

them arise from the exchange of special kinds 

ofparticles called gauge particles: the photon 

in electromagnetic interactions, the W' and 

Zo in the weak interactions responsible for 

radioactive decay, and the gluons in strong 

interactions that bind the nucleus. (It is believed 

that the graviton plays a similar role in 

gravity.) [The local gauge theory ofthe strong 

forces is called quantum chromodynamics 

(QCD). The local gauge theory that unifies 

the weak force and the electromagnetic force 

is the electroweak theory that predicted the 

existence ofthe W'and Zo.] 

ZWEIG: You are talking about a very limited 

aspect ofwhat high-energy physics has 

been. Our present understanding did not develop 

in an orderly manner. In fact, what 

took place in the early '60s was the first 

revolution we have had in physics since 

quantum mechanics. At that time, ifyou 

were at Berkeley studying physics, you 

studied S-matrix, not field theory, and when 

I went to Caltech, I was also taught that field 

theory was not important. 

SLANSKY Yes, the few people that were 

focusing on Yang-Mills theories in the '50s 

and early '60s were more or less ignored. 

Perhaps the most impressive ofthose early 

papers was one by Julian Schwinger in which 

he tried to use the isotopic spin group as a 

local symmetry group for the weak, not the 

strong, interactions. (Schwinger's approach 

turned out to be correct. The Nobel prizewinning 

SU(2) X U( 1) electroweak theory 

that predicted the W'and Zo vector mesons 

to mediate the weak interactions is an expanded 

version of Schwinger's SU(2) 

model.) 

WEST In retrospect Schwinger is a real hero 

in the sense that he kept the faith and made 

some remarkable discoveries in field theory 

during a period when everybody was 

basically giving him the finger. He was completely 

ignored and, in fact, felt left out ofthe 

field because no one would pay any attention. 

SCIENCE: Why was field theory dropped? 

CARRUTHERS: Now we reach a curious 

sociological phenomenon. 

RABY Sociological? I thought the theory 

wasjust too hard to understand. There were 

all those infinities that cropped up in the 

calculations and had to be renormalized 

away. 

CARRUTHERS I am afraid there is a phase 

transition that occurs in groups of people of 

whatever IQ who feverishly follow each new 

promising trend in science. They go to a 

conference, where a guru raises his hands up 

and waves his baton; everyone sits there, 

their heads going in unison, and the few 

heretics sitting out there are mostly intimidated 

into keeping their heresies to 

themselves. After a while some new religion 

comes along, and a new faith replaces the 

old. This is a curious thing, which you often 

see at football games and the like. 

WEST The myth perpetrated about field 

theory was, as Stuart said, that the problems 

were too hard. But if you look at Yang-Mills 

and Julian Schwinger's paper, for example, 

there was still serious work that could have 

been done. Instead, when I was at Stanford, 

Sidney Drell taught advanced quantum mechanics 

and gave a whole lecture on why you 

didn't need field theory. All you needed were 

Feynman graphs. That was the theory. 

RABY: The real problem was that you had a 

zoo of particles, with none seemingly more 

fundamental than any other. Before people 

knew about quarks, you didn't feel that you 

were writing down the fundamental fields. 

WEST In 1954 we had all the machinery 

necessary to write down the standard model. 

We had the renormalization group. We had 

local gauge theories. 

SLANSKY But nobody knew what to apply 

them to. 

SCIENCE: George, in 1963, when you came 

up with the idea that quarks were the constit- 

uents of the strongly interacting particles, did 

you think at all about field theory? 

ZWEIG: No. The history I remember is 

quite different. The physics community 

responded to this proliferation ofparticles by 

embracing the bootstrap hypothesis. No particle 

was viewed as fundamental; instead, 

there was a nuclear democracy in which all 

particles were made out ofone another. The 

idea had its origins in Heisenberg's S-matrix 

theory. Heisenberg published a paper in 1943 

reiterating the philosophy that underlies 

quantum mechanics, namely, that you 

should only deal with observables. In the 

case ofquantum mechanics, you deal with 

spectral lines, the frequencies of light emitted 

from atoms. In the case ofparticle physics, 

you go back to the ideas of Rutherford. 

Operationally, you study the structure of 

matter by scattering one particle off another 

and observing what happens. The experimental 

results can be organized in a kind 

of a matrix that gives the amplitudes for the 

incoming particles to scatter into the outgoing 

ones. Measuring the elements of this 

scattering, or S-matrix, was the goal ofexperimentalists. 

The work oftheorists was to 

write down relationships that these S-matrix 

elements had to obey. The idea that there 

was another hidden layer of reality, that there 

were objects inside protons and neutrons 

that hadn't been observed but were responsible 

for the properties ofthese particles, was 

an idea that wasjust totally foreign to the S- 

matrix philosophy; so the proposal that the 

hadrons were composed of more fundamental 

constituents was vigorously resisted. Not 

until ten years later, with the discovery ofthe 

psi/Jparticle, did the quark hypothesis become 

generally accepted. By then the 

evidence was so dramatic that you didn't 

have to be an expert to see the underlying 

structure. 

RABY: The philosophy ofthe bootstrap, 

from what I have read ofit, is a very beautiful 

philosophy. There is no fundamental particle, 

but there are fundamental rules ofhow 

particles interact to produce the whole spectrum. 

But one thing that distinguishes physics 

from philosophy is predictive power. The 

189

“It’s important to pick one fundamental question, push 

99 

on it, and get the right answer . 

quark model had a lot of predictive power. It 

predicted the whole spectrum of hadrons 

observed in high energy experiments. It is 

not because of sociology that the bootstrap 

went out; it was the experimental evidence of 

J/psi that made people believe there really 

are objects called quarks that are the building 

blocks ofall the hadrons that we see. It is this 

reality that turned people in the direction 

they follow today. 

CARRUTHERS: And because of the very 

intense proliferation ofunknowns, it is unlikely 

that the search for fundamental constituents 

will stop here. In the standard 

model you have dozens of parameters that 

are beyond any experimental reach. 

SCIENCE: But you have fewer coordinates 

now than you had originally, right? 

CARRUTHERS: If you are saying the 

coordinates have all been coordinated by 

group symmetry, then ofcourse there are 

many fewer. 

WEST: I think the deep inelastic scattering 

experiments at SLAC played an absolutely 

crucial role in convincing people that quarks 

are real. It was quite clear from the scaling 

behavior of the scattering amplitudes that 

you were doing a classic Rutherford type 

scattering experiment and that you were literally 

seeing the constituents ofthe nucleon. 

I think that was something that was extremely 

convincing. Not only was it 

qualitatively correct, but quantitatively 

numbers were coming out that could only 

come about if you believed the scattering was 

taking place from quarks, even though they 

weren’t actually being isolated. But let me 

say one other thing about the S-matrix approach. 

That approach is really quantum 

mechanics in action. Everything is connected 

with everything else by this principle of unitarity 

or conservation of probability. It is a 

very curious state ofaffairs that the quark 

model, which requires less quantum mechanics 

to predict, say, the spectrum ofparticles, 

has proven to be much more useful. 

SLANSKY Remember, though, there were 

some important things missing in the 

bootstrap approach. There was no natural 

way to incorporate the weak and the elec- 

190 

tromagnetic forces. 

WEST That picks up another important 

point; the S-matrix theory could not cope 

with the problem ofscale. And that brings us 

back to the standard model and then into 

grand unification. The deep inelastic scattering 

experiments focused attention on the 

idea that physical theories exhibit a scale 

invariance similar to ordinary dimensional 

analysis. 

One of the wonderful things that happened 

as a result was that all of us began to accept 

renormalization (the infinite rescaling of 

field theories to make the answers come out 

finite) as more than just hocus-pocus. Any 

graduate student first learning the renormalization 

procedure must have thought 

that a trick was being pulled and that the 

procedure for getting finite answers by subtracting 

one infinity from another really 

couldn’t be right. An element ofhocus-pocus 

may still remain, but the understanding that 

renormalization was just an exploitation of 

scale invariance in the very complicated context 

offield theory has raised the procedure 

to the level ofa principle. 

The focus on scale also led to the feeling 

that somehow we have to understand why 

the forces in nature have different strengths 

and become strong at different energies, why 

there are different energy scales for the weak, 

for the electromagnetic, and for the strong 

interactions, and ultimately whether there 

may be a grand scale, that is, an energy at 

which all the forces look alike. That’s one of 

those wonderful deep questions that has 

come back to haunt us. 

RABY: I guess we think ofquantum electrodynamics 

(QED) as being such a successful 

theory because calculations have been 

done to an incredibly high degree of accuracy. 

But it is hard to imagine that we will 

ever do that well for the quark interactions. 

The whole method of doing computations in 

QED is perturbative. You can treat the electromagnetic 

interaction as a small perturbation 

on the free theory. But, in order to 

understand what is going on in the strong 

interactions of quantum chromodynamics 

(QCD), you have to use nonperturbative 

methods, and then you get a whole new 

feeling about the content of field theory. 

Field theory is much richer than a 

perturbative analysis might lead one to believe. 

The study of scaling by M. Fisher, L. 

Kadanoff, and K. Wilson emphasized the

interrelation of statistical mechanics and 

field theory. For example, it is now understood 

that a given field theoretic model may, 

as in statistical mechanics systems, exist in 

several qualitatively different phases. 

Statistical mechanical methods have also 

been applied to field theoretic systems. For 

example, gauge theories are now being 

studied on discrete space-time lattices, using 

Monte Carlo computer simulations or analog 

high temperature expansions to investigate 

the complicated phase structure. There has 

now emerged a fruitful interdisciplinary 

focus on the non-linear dynamics inherent in 

the subjects offield theory, statistical mechanics, 

and classical turbulence. 

ZWEIG: Isn’t it true to say that the number 

ofthings you can actually compute with 

QCD is far less than you could compute with 

S-matrix theory many years ago? 

WEST: I wouldn’t say that. 

ZWEIG: What numbers can be experimentally 

measured that have been computed 

cleanly from QCD? 

RABY: What is yourdefinition ofclean? 

ZWEIG: A clean calculation is one whose 

assumptions are only those ofthe theory. Let 

me give you an example. I certainly will 

accept the numerical results obtained from 

lattice gauge calculations ofQCD as definitive 

ifyou can demonstrate that they follow 

directly from QCD. When you approximate 

space-timeasadiscreteset ofpoints lying in 

a box instead ofan infinite continuum, as 

you do in lattice calculations, you have to 

show that these approximations are legitimate. 

For example, you have to show that 

the effects of the finite lattice size have been 

properly taken into account. 

RABY: To return to the question, this is the 

first time you can imaginecalculating the 

spectrum ofstrongly interacting particles 

from first principles. 

ZWEIG: The spectrum of strongly interacting 

particles has not yet been calculated in 

QCD. In principle it should be possible, and 

much progress has been made, but operationally 

the situation is not much better than 

it was in theearly ’60s when the bootstrap 

was gospel. 

SLANSKY Yes, but that was a very dirty 

calculation. The agreement got worse as the 

calculations became more cleverly done. 

WEST The numbers from latticegauge theory 

calculations of QCD are not necessarily 

meaningful at present. There is a serious 

question whether the lattice gauge theory, as 

formulated, is a real theory. When you take 

the lattice spacing to zero and go to the 

continuum limit, does that give you the theory 

you thought you had? 

RABY: That’s the devil’sadvocate point of 

view, the view coming from the mathematical 

physicists. On the other hand, people 

have made approximations, and what you 

can say is that any approximation scheme 

that you use has given the same results. First, 

there are hadrons that are bound states of 

quarks, and these bound states have finite 

size. Second, there is no scale in the theory, 

but everything, all the masses, for example, 

can be defined in terms ofone fundamental 

scale. You can get rough estimates of the 

whole particle spectrum. 

WEST: You can predict that from the old 

quark model, without knowing anything 

about the local color symmetry and the eight 

colored gluons that are the gauge particles of 

the theory. There is only one clean calculation 

that can be done in QCD. That is the 

calculation of scattering amplitudes at very 

high energies. Renormalization group analysis 

tells us the theory is asymptotically free at 

high energies, that is, at very high energies 

quarks behave as free point-like particles so 

the scattering amplitudes should scale with 

energy. The calculations predict logarithmic 

corrections to perfect scaling. These have 

been observed and they seem to be unique to 

QCD. Another feature unique to quantum 

chromodynamics is the coupling ofthegluon 

to itselfwhich should predict the existence of 

glueballs. These exotic objects would provide 

another clean test of QCD. 

ZWEIG: I agree. The most dramatic and 

interesting tests ofquantum chromodynamics 

follow from those aspects of the 

theory that have nothing to do with quarks 

directly. The theory presumably does predict 

the existence ofbound states ofgluons, and 

ROUND TABLE 

furthermore, some of those bound states 

should have quantum numbers that are not 

the same as those ofparticles made out of 

quark-antiquark pairs. The bound states that 

I would like to see studied are these “oddballs,” 

particles that don’t appear in the 

simple quark model. The theory should 

predict quantum numbers and masses for 

these objects. These would be among the 

most exciting predictions of QCD. 

RABY: People who are calculating the 

hadronic spectrum are doing those sorts of 

calculations too. 

ZWEIG: It’s important to pick one fundamental 

question, push on it, and get the right 

answer. You may differ as to whether you 

want to use the existence ofoddballs as a 

crucial test or something else, but you should 

accept responsibility for performing calculations 

that are clean enough to provide meaningful 

comparison between theory and experiment. 

The spirit ofempiricism does not 

seem to be as prevalent now as it was when 

people were trying different approaches in 

particle physics, that is, S-matrix theory, 

field theory, and the quark model. The development 

of the field was much more Danvinian 

then. Peopleexplored many different 

ideas, and natural selection picked the winner. 

Now evolution has changed; it is 

Lamarckian. People think they know what 

the right answer is, and they focus and build 

on one another’s views. The value of actually 

testing what they believe has been substantially 

diminished. 

SLANSKY I don’t think that is true. The 

technical problems of solving QCD have 

proved to be harder than any other technical 

problems faced in physics before. People 

have had to back offand try to sharpen their 

technical tools. I think, in fact, that most do 

have open minds as to whether it is going to 

be right or wrong. 

WEST: What do you think about the rest of 

the standard model? Do we think the electroweak 

unification is a closed book, 

especially now that W’and Zo vector bosons 

have been discovered? 

SLANSKY: It is to a certain level ofaccuracy, 

but the theory itself is just a 

191

“It may be that all this matter is looped together in 

some complex topological web and that ifyou tear 

apart the Gordian knot with your sword of Damocles, 

something really strange will happen. ” 

phenome:nology with some twenty or so free 

parameters floating around. So it is clearly 

not the final answer. 

SCIENCE: What are these numbers? 

RABY: All the masses of the quarks and 

leptons are put into the theory by hand. Also, 

the mixing angle, the so-called Cabibbo 

angle, which describes how the charmed 

quark decays into a strange quark and a little 

bit of the down quark, is not understood at all. 

ZWEIC;: Operationally, the electroweak theory 

is solid. It predicted that the W’and Zo 

vector bosons would exist at certain masses, 

and they actually do exist at those masses. 

SLANSKY: The theory also predicted the 

coupling ofthe Zo to the weak neutral current. 

People didn’t want to have to live with 

neutral currents because, to a very high 

degree ofexperimental accuracy, there was 

no evidence for strangeness-changing weak 

neutral currents. The analysis through local 

symmetry seemed to force on you the existence 

ofweak neutral currents, and when 

they were observed in ’73 or whenever, it was 

a tremendous victory for the model. The 

electron has a weak neutral current, too, and 

this current has a very special form in the 

standard model. (It is an almost purely axial 

current.) This form ofthe current was established 

in polarized electron experiments at 

SLAC. Very shortly after those experimental 

results, Glashow, Weinberg, and Salam received 

the Nobel prize for their work on the 

standard model of electroweak interactions. I 

think that was the appropriate time to give 

the Nobel prize, although a lot of my colleagues 

felt it was a little bit premature. 

RABY: However, the Higgs boson required 

for the consistency of the theory hasn’t been 

seen yet. 

SLANSKY: A little over a year ago there 

were four particles that needed to be 

seen-now there is only one. The standard 

model theory has had some rather impressive 

successes. 

WEST Can we use this as a point of departure 

to talk about grand unification? Unification 

ofthe weak and electromagnetic 

interactions, which had appeared to be quite 

separate forces, has become the prototype for 

attempts to unify those two with the strong 

interactions. 

RABY In the standard model of the weak 

interactions, the quarks and the leptons are 

totally separate even though phenomenologically 

they seem to come in families. For 

example, the up and the down quarks seem 

to form a family with the electron and its 

neutrino. Grand unification is an attempt to 

unify quarks and leptons, that is, to describe 

them as different aspects of the same object. 

In other words, there is a large symmetry 

group within which quarks and leptons can 

transform into each other. The larger group 

includes the local symmetry groups of the 

strong and electroweak interaction and 

thereby unifies all the forces. These grand 

unified theories also predict new interactions 

that take quarks into leptons and vice versa. 

One prediction ofthese grand unified theories 

is proton decay. 

WEST: The two most crucialpredictions of 

grand unified theories are, first, rhat prorons 

are not perfectly stable and can decay and, 

second, that magnetic monopoles exist. 

Neither of these has been seen so far. Suppose 

rhey are never seen. Does fhaf mean the question 

of grand unification becomes merely 

philosophical? Also, how does that bear on the 

idea of building a very high-energy accelerator 

like fhe SSC (superconducting super 

collider) rhar will cost the taxpayer $3 billion? 

CARRUTHERS: Why should we build this 

giant accelerator? Because in our theoretical 

work we don’t have a secure world view; we 

need answers to many critical questions 

raised by the evidence from the lower 

energies. Even though I know that as soon as 

you do these new experiments, the number 

ofquestions is likely to multiply. This is part 

ofmy negative curvature view ofthe progress 

of science. But there are some rather 

primitive questions which can be answered 

and which don’t require any kind ofsophistication. 

For instance, are there any new particles 

ofwell-defined mass ofthe oldfashioned 

type or new particles with different 

properties, perhaps? Will we see the Higgs 

particle that people stick into theories just to 

make the clock work? If you talk to people 

who make models, they will give you a panorama 

of predictions, and those predictions 

will become quite vulnerable to proof if we 

increase the amount ofaccelerator energy by 

a factor of 10 to 20. Those people are either 

going to be right, or they’re going to have to 

retract their predictions and admit, “Gee, it 

didn’t work out, did it?” 

There is a second issue to be addressed, 

and that is the question ofwhat the fundamental 

constituents of matter are. We 

messed up thirty years ago when we thought 

protons and neutrons were fundamental. We 

know now that they’re structured objects, 

like atoms: they’re messy and squishy and all 

kinds of things are buzzing around inside. 

Then we discovered that there are quarks 

and that the quarks must be held together by 

glue. But some wise guy comes along and 

says, “How do you know those quarks and 

gluons and leptons are not just as messy as 

those old protons were’?’’ We need to test 

whether or not the quark itselfhas some 

composite structure by delivering to the 

quarks within the nucleons enough energy 

and momentum transfer. The accelerator 

acts like a microscope to resolve some fuzziness 

in the localization of that quark, and a 

whole new level ofsubstructure may be discovered. 

It may be that all this matter is 

looped together in some complex topological 

web and that ifyou tear apart the Gordian 

knot with your sword ofDamocles. something 

really strange will happen. A genie may 

pop out ofthe bottle and say, “Master, you 

have three wishes.” 

A third issue to explore at the SSC is the 

dynamics of how fundamental constituents 

interact with one another. This takes you 

into the much more technical area ofanalyzing 

numbers to learn whether the world view 

you’ve constructed from evidence and theory 

makes any sense. At the moment we have 

no idea why the masses of anything are what 

they are. You have a theory which is attractive, 

suggestive, and can explain many, many 

things. In the end, it has twenty or thirty

~ 

“If we can get people to agree on why we should be 

doing high-energy physics, then 1 think we can solve the 

problem ofprice. ’’ 

ROUND TABLE 

I 

parameters. You can’t be very content that 

you’ve understood the structure of matter. 

SLANSKY: To make any real progress both 

in unification of the known forces and in 

understanding anything about how to go 

beyond the interactions known today, a machine 

ofsomething like 20 to 40 TeV center 

of mass energy from proton-proton collisions 

absolutely must be built. 

SCIENCE: Will these new machines test 

QCD at the same time they test questions of 

unification? 

SLANSKY The pertinent energy scale in 

QCD is on the order of GeV, not TeV, so it is 

not clear exactly what you could test at very 

high energies in terms of the very nonlinear 

structure of QCD. Pete feels differently. 

CARRUTHERS: All I say is that you may be 

looking at things you don’t think you are 

looking at. 

WEST: Obviously all this is highly speculative. 

A question you are obligated to ask is at 

what stage do you stop the financing. I think 

we have to put the answer in terms ofa 

realistic scientific budget for the United 

States, or for the world for that matter. 

CARRUTHERS: Is there a good reason why 

the world can’t unify its efforts to go to higher 

energies? 

WEST: Countries are mostly at war with one 

another. They couldn’t stop to have the 

Olympic games together, so certainly not for 

a bloody machine. 

SLANSKY The Europeans themselves have 

gotten together in probably one ofthe most 

remarkable examples of international collaboration 

that has ever happened. 

WEST: Yes, I think the existence ofCERN is 

one ofthe greatest contributions ofparticle 

physics to the world. 

RABY: But the next step isgoing to have to 

be some collaborative effort of CERN, the 

U.S., and Japan. 

WEST: Our SSC is going to be the next step. 

But you are still not answering the question. 

Should we expect the government to support 

this sort of project at the $3 billion level? 

SLANSKY That’s $3 billion over ten years. 

RABY You can ask that same question of 

any fundamental research that has no direct 

application to technology or national security, 

and you will get two different answers. 

The “practical” person will say that you do 

only what you conceive to have some 

benefits five or ten years down the line, 

whereas the person who has learned from 

history will say that all fundamental research 

leads eventually either to new intellectual 

understanding or to new technology. 

Whether technology has always benefited 

mankind is debatable, but it has certainly 

revolutionized the way people live. I think 

we should be funded purely on those 

grounds. 

WEST Where do you stop? If you decide 

that $3 billion is okay or $ IO billion, then do 

you ask for $100 billion? 

ZWEIG: This is a difficult question, but if 

we can get people to agree on why we should 

be doing high-energy physics, then I think we 

can solve the problem of price. Although 

what we have been talking about may sound 

very obscure and possibly very ugly to an 

outside observer (quantum chromodynamics, 

grand unification, and twenty or 

thirty arbitrary parameters), the bottom line 

is that all ofthis really deals with a fundamental 

question, “What is everything made 

Of?” 

It has been our historical experience that 

answers to fundamental questions always 

lead to applications. But the time scale for 

those applications to come forward is very, 

very long. For example, we talked about 

Faraday’s experiments which pointed to the 

quantal nature of electricity in the early 

1800s; well, it was another half century 

before the quantum of electricity, the electron, 

was named and it was another ten years 

before electrons were observed directly as 

cathode rays; and another quarter century 

passed before the quantum of electric charge 

was accurately measured. Only recently has 

the quantum mechanics of the electron 

found application in transistors and other 

solid state devices. 

Fundamental laws have always had application, 

and there’s no reason to believe 

193

“I groan inwardly and sadly reflect on how great the 

communication gap is between scientists such as 

ourselves and the general public that supports us.” 

this will aot hold in the future. We need to 

insist that our field be supported on that 

basis. We need ongoing commitment to this 

potential for new technology, even though 

technology’s future returns to society are difficult 

to assess. 

CARRIJTHERS: Whenever support has to 

be ongoing, that’sjust when there seems to be 

a tendency to put it off. 

WEST: What’s another few years, right? 

Now I would like to play devil’s advocate. 

One of the unique things about being at Los 

Alamos is that you are constantly being 

asked to justify yourself. In the past, science 

has dealt with macroscopic phenomena and 

natural phenomena. (I am a little bit on 

dangerous ground here.) Even when it dealt 

with the quantum effects, the effects were 

macroscopic: spectroscopic lines, for example, 

and the electroplating phenomena. The 

crucial difference in high-energy physics is 

that what we do is artificial. We create rare 

states of matter: they don’t exist except 

possibly in some rare cosmic event, and they 

have little impact on our lives. To understand 

the universe that we feel and touch, 

even down to its minutiae, you don’t have to 

know a damn thing about quarks. 

ZWEIG: Maybe our experience is limited. 

Let me give you an example. Suppose we had 

stable heavy negatively-charged leptons, that 

is, heavy electrons. Then this new form of 

matter would revolutionize our technology 

because it would provide a sure means of 

catalyzing fusion at room temperature. So it 

is not true that the consequences ofour work 

are necessarily abstract, beyond our experience, 

something we can’t touch. 

WEST: This discussion reminds me of 

something I believe Robert Wilson said during 

his first years as director of Fermilab. He 

was before a committee in Congress and was 

asked by some aggressive Congressman, 

“What good does the work do that goes on at 

your lab? What good is it for the military 

defense ofthis country?” Wilson replied 

something to the effect that he wasn’t sure it 

helped directly in the defense ofthe country, 

but it made the country worth defending. 

Certainly, finding applications isn’t 

1194 

predominantly what drives people in this 

field. People don’t sit there trying to do grand 

unification, saying to themselves that in a 

hundred years’ time there are going to be 

transmission lines of Higgs particles. When I 

was a kid, electricity was going to be so cheap 

it wouldn’t be metered. And that was the 

kind ofattitude the AEC took toward science. 

I, at least, can’t work that way. 

SCIENCE: George, do you work that way? 

ZWEIG: I was brought up, like Pete, at a 

time when the funding for high-energy physics 

was growing exponentially. Every few 

years the budget doubled. It was absolutely 

fabulous. As a graduate student Ijust 

watched this in amazement. Then I saw it 

turn off, overnight. In 1965, two years after I 

got my degree from Caltech, I was in Washington 

and met Peter Franken. Peter said, 

“It’s all over. High-energy physics is dead.” I 

looked at him like he was crazy. A year later I 

knew that, in a very real sense, he was 

absolutely right. 

It became apparent to me that if I were 

going to get support for the kind of research I 

was interested in doing, I would have to 

convince the people that would pay for it that 

it really was worthwhile. The only common 

ground we had was the conviction that basic 

research eventually will have profound applications. 

The same argument I make in high-energy 

physics, I also make in neurobiology. lfyou 

understand how people think, then you will 

be able to make machines that think. That, in 

turn, will transform society. It is very important 

to insist on funding basic research on 

this basis. It is an argument you can win. 

There are complications, as Pete says; if 

applications are fifty years off, why don’t we 

think about funding twenty-five years from 

now? In fact, that is what we havejust heard: 

they have told us that we can have another 

accelerator, maybe, but it is ten or fifteen 

years down the road. 

SLANSKY: We really can’t build the SSC 

any faster than rhat. 

ZWEIG: They could have built the machine 

at Brookhaven. 

WEST: Let’s talk about that. How can you 

explain why a community who agreed that 

building the Isabelle machine was such a . 

great and wonderful thing decided, five years 

later, that it was not worth doing. 

SLANSKY: It is easy to answer that in very 

few words. The Europeans scooped the US. 

when they got spectacular experimental data 

confirming the electroweak unification. That 

had been one of our main purposes for building 

Isabelle. 

CARRUTHERS: If you want to stay on the 

frontier, you have to go to the energies where 

the frontier is going to be. 

ZWEIG: Some interesting experiments were 

made at energies that were not quite what 

you would call frontier at the time. CP violation 

was discovered at an embarrassingly low 

energy. 

SLANSKY: The Europeans already have the 

possibility ofbuilding a hadron collider in a 

tunnel already being dug, the large electronpositron 

collider at CERN. It is clear that the 

U.S., to get back into the effort, has to make a 

bigjurnp. Last spring the High Energy Physics 

Advisory Panel recommended cutting off 

Isabelle so the U.S. could go ahead in a 

timely fashion with the building of the SSC. 

WEST: If you were a bright young scientist, 

would you go into high-energy physics now? 

I think you could still say there is a glamour 

in doing theory and that great cosmic questions 

are being addressed. But what is the 

attraction for an experimentalist, whose 

talents are possibly more highly rewarded in 

Silicon Valley? 

RABY: It will become more and more difficult 

to get people to go into high-energy 

physics as the time scale for doing experiments 

grows an order of magnitude equal to 

a person’s lifetime. 

ZWEIG: Going to the moon was a successful 

enterprise even though it took a long time 

and required a different state of mind for the 

participating scientists. 

WEST Many of the great creative efforts of 

medieval life went into projects that lasted 

more than one generation. Building a great 

cathedral lasted a hundred, sometimes two 

hundred years. Some of the great craftsmen, 

the great architects, didn’t live to see their

“I consider doing physics something that causes me an 

enormous amount of emotional energy. Iget upset. I 

get depressed. I get joyful. ” 

ROUND TABLE 

work completed. 

As for going to high energies, I see us 

following Fermi’s fantasy: we will find the 

hydrogen atom of hadronic physics and 

things will become simpler. It is a sort of 

Neanderthal approach. You hit as hard as 

you can and hope that things break down 

into something incredibly small. Somewhere 

in those fragments will be the “hydrogen” 

atom. That’s the standard model. Some people 

may decide to back off from that 

paradigm. Lower energies are actually 

amenable. 

CARRUTHERS: I think that people have 

already backed OK Wasn’t Glashow going 

around the country saying we should do lowenergy 

experiments? 

WEST: Just to bring it home, the raison 

d’etre for LAMPF I1 is to have a low-energy, 

high-intensity machine to look for interesting 

phenomena. It is again this curious thing. 

We are looking at quantum effects by using a 

classical mode-hitting harder. The idea of 

high accuracy still uses quantum mechanics. 

I suppose it is conceivable that one would 

reorient the paradigm toward using the 

quantum mechanical nature of things to 

learn about the structure ofmatter. 

SLANSKY: Both directions are very important. 

SCIENCE: Is high-energy physics still attracting 

the brightest and the best? 

SLANSKY: Some of the young guys coming 

out are certainly smart. 

CARRUTHERS I think there is an increasing 

array of very exciting intellectual 

challenges and new scientific areas that can 

be equally interesting. Given a limited pool 

of intellectual talent, it is inevitable that 

many will be attracted to the newer disciplines 

as they emerge. 

ZWEIG: Computation, for example. Stephen 

Wolfram is a great example of someone 

who was trained in high-energy physics but 

then turned his interest elsewhere, and 

profitably so. 

CARRUTHERS: Everything to do with conceptualization-computers 

or theory of the 

mind, nonlinear dynamics advances. All of 

these things are defining new fields that are 

very exciting-and that may in turn help us 

solve some of the problems in particle physics. 

ZWEIG: That’s optimistic. What would 

physics have been like without your two or 

three favorite physicists? I think we would all 

agree that the field would have been much 

the poorer. The losses ofthe kind we are 

talking about can have a profound effect on a 

field. Theoretical physics isn’tjust the 

cumulative efforts ofmany trolls pushing 

blocks to build the pyramids. 

WEST: But my impression is that the work 

is much less individualized than it ever was. 

The fact that the electroweak unification was 

shared by three people, and there were others 

who could have been added to that list, is an 

indication. If you look at QCD and the standard 

model, it is impossible to write a name, 

and it is probably impossible to write ten 

names, without ignoring large numbers of 

people who have contributed. The grand unified 

theory, ifthere ever is one, will be more 

the result ofmany people interacting than of 

one Einstein, the traditional one brilliant 

man sitting in an armchair. 

SCIENCE: Was that idea ever really correct? 

WEST: It was correct for Einstein. It was 

correct for Dirac. 

SCIENCE: Was their thinking really a total 

departure? 

ZWEIG: The theory ofgeneral relativity is a 

great example, and almost a singular example, 

of someone developing a correct theoretical 

idea in the absence ofexperimental information, 

merely on the basis ofintuition. I 

think that is what people are trying to do 

now. This is very dangerous. 

RABY: Another point is that Einstein in his 

later years was trying to develop the grand 

unified theory ofall known interactions, and 

he was way off base. All the interactions 

weren’t even known then. 

WEST: Theorizing in the absence ofsupportive 

data is still dangerous. 

CARRUTHERS Particle physics, despite all 

ofits problems, remains one ofthe principal 

frontiers of modern science. As such it combines 

a ferment of ideas and speculative 

thoughts that constantly works to reassess 

the principles with which we try to understand 

some of the most basic problems in 

nature. If you take away this frothy area in 

which there’s an enormous interface between 

the academic community and all kinds of 

visitors interacting with the laboratory, giving 

lectures on what is the latest excitement 

in physics, then you won’t have much left in 

the way of an exciting place to work, and 

people here won’t be so good after awhile. W 

195

Index 

accelenitors 150-1 57 

AGS 52,150,152,153,158,188,194 

Bevatron 150,152 

CEIW 130,150,152,156,157,159,193, 

194 

CESR 153 

Cosmotron 152 

cyclotron 152, 153 

DORIS 153 

Fermilab 152,155 

HERA 156,157 

LAMPF 136, 137, 141-145, 147, 148, 154, 

158-163,195 

LEP 154,156,157 

linac 152, 154 

PEP 153 

PETRA 153 

SIN 137,141,154 

SLAC 18,140,152,159,188,190, 192 

SLC 154,156,157 

SPEAR 153 

SSC 152,157,164-165 

synchrocyclotron 152 

synchrotron 152 

Tevatron 152,155,156,157 

Tristan 156 

TRIUMF 137,146,154 

antiparticles 108, 131, 182 

antilepton 160 

antimuon 160 

antineutrino 100,112,132 

antiproton 28, 109,150, 158-160, 162 

antiquark 108,160,191 

antisquark 108,112,113 

asymmetry, matter-antimatter 168 

asymptotic freedom 16-18, 19,40,41 

axial vector 145.146 

bare parameters 16,17 

bare theory 19 

baryon 161 

Lambda 160,161 

number 37,167 

conservation of 34, 154, 167, 168 

Sigma 161 

beta decay 43,128,130,146,167 

intrinsic linewidth 132 

betaelectron spectrometer 133, 134 

toroidal 132 

resolution finctiori 132- 134 

beta-function 17 

big bang theory 168 

birds' eggs 5-7 

bone structure 4-5 

bosonicfield 102,103,105 

boson 25,28,40-43,69,76,79,98-112,13 1, 

141,142,146,153,158 

charged-current 141,142,145 

charged-vector 141,142 

heavy vector 130 

Goldstone 59,63, 121 

Higgs47,50,66, 101, 105-107, 131, 

156,157,192,194 

intermediate vector 136 

massless gauge 131 

neutralvector 131,141,142 

vector 25,40,43,69,76,130, 131, 145, 

158,191,192 

W*44,45,50,67,77-78, 101, 105-107, 

112, 130, 141, 142, 146, 148, 150, 

154,156,157,189,191,192 

2'46-48,50,67,77-78, 112, 130, 141, 

142, 148, 150, 154, 156, 157, 189, 

191,192 

brain size 10 

Cabibbo 

angle 51,71 

matrix 120 

chargedcurrents 130,131,141 

interference effects 130,141,142 

chirality 44 

color30,39,40, 131, 191 

charge 69,131 

force 131 

gluons 130 

conservation 34,100,136,137 

of energy loo., 132 

ofleptons 128,136,137,145 

continuity equation 57 

correlation fimctions 14, 18 

cosmic ray 144,145 

cosmic-ray astronomy 167 

cosmological constant problem 84,9 1 

CPviolation, 33, 121, 123, 131, 154-157, 194 

in B '-B system 123 

inKo-Rosystem 121, 156, 157 

relation to family symmetry breaking 

121,123 

Crystal Ball 140 

Crystal BOX 137-141 

deBroglie relation 27 

decuplet 37 

detector properties 138,139 

dimension 3 

anomalous 17 

dimensional analysis 4,6-14,17, 190 

dimensionless variables 7- 19 

doublets 131 

drag, V~SCOUS 7- 10 

Eightfold way 37,38 

electrodynamics 26 

electromagnetic 

coupling constant 13 1 

current 142 

field 14,34 

force 19,23,24, 135 

interaction 25, 128 

shower 139 

electromagnetism 23,28, 188 

electron 27,29, 100, 128, 131 

number 34,136 

scattering 

deep inelastic 17, 18,42, 190 

electronelectron I42 

electron-neutrino elastic 142 

polarized 14 1 

electron-positron colliders 141 

electron-positron pair I5 

electroweak theory 19,45-50,65-68,76-79, 

128, 130, 131, 141, 142, 168, 189, 192, 

194,195. See also forces and 

interactions, basic 

elementary particles, representations of 

in quantum chromodynamics 1 15 

in electroweak theory 1 15,116 

EMC effect 159,160 

end-point energy 132,133 

endocranial volume 10 

1196

families, quark-lepton 5 1,71, 115, 117, 136, gauge 

157,158 

fields 142 

familychanging interactions 116-122 

invariance 36 

family problem 31, 128, 130, 136 

prticles 189,191 

family-symmetry breaking 116- 122 

theory25,39,45,130,155,188,191 

Fermi 130 

non-Abelian (Yang-Mills) 40,45, 188, 

constant 43 

189 

theory 43 

gauginos 107 

fermionicfield 102, 103, 105 

geometry box 140 

fermion 28,98- 112, 131 

general relativity. See gravity, Einstein’s 

generations 136 

theory of 

Goldstone 101 

global invariance 34 

Feynman diagram 14- 16,29,189 

glueballs41,161,191 

fields in higher dimensions 86-87 

gluino 108, 109,112,131 

fine-structure constant 130 

gluons 19,25,40,42,79, 101, 108, 153, 

fixed points 18 

159-161, 189, 191, 192 

flavor 130 

goldstino 105, 107, 110 

symmetry 39 

Goldstone fermion 101 

forces and interactions 

grandunifiedtheory81-82,106,110,131, 

basic 24,74,78-79 

146, 154, 168- 170, 182, 190, 192- 195 

electromagnetic 19,23-26,28-30,31,36, graviton 80,82,84,93,189 

42,98, 110, 128, 130,135, 141,142, gravitino 105,107,110 

182, 188-190, 192 

massive 110 

electroweak 65-68,70-71, 128,130, 131, 

gravity 

189,192 

Einstein’s theory of 74,75,80-8 1,82,84 

gravitational 23,24,26,59,98, 110, 130, unification with other forces 82-95 

188,189 

group multiplets 31 

neutral current 46 

strengths of 24,28,30,40,43-45,66,70 hadron 36,80,91-92, 109, 144,160-162, 

strong 19,23-25,28,30,36,38-40,69-70, 189-191,195 

98, 110,128,130, 136, 188-192, 192 Heisenberg uncertainty principle 13, 14,29 

unification of23,25,28,30,44-46,53, helium-3 132 

72-95, 152, 153,168 

Hierarchy problem 106 

weak 19,23,24,28,42-45,98, 107,110, Higgs 

112, 128, 130, 136, 141-143, 145, bosons47,50,101,105-107,131,192,194 

146, 188-190, 192 

mechanism 131 

beta decay 42-43 

Higgsino 107,108 

Fermi theory of 43-44,76,77,79,155 hypercharge 37 

chargedcurrent 43,46-49,7 1 

hypernucleus 159- 161 

neutralcurrent 46-47,48,49,68,7 1, 

152,153 

interference effects 

right-handed 157 

between neutral and charged weak 

form invariance 11,14 

currents 130,141-143 

fundamental 

isotopic spin 30,37,189 

constants, 12,13 

scales 11 

J/Psi 52, 153, 154, 188-189 

symmetries 136 

jets, hadronic 109, 112, 157 

Kaluza-Klein theories 83,84,87 

kaons 158- 162 

Kobayashi-Maskawa matrix 5 1,123 

Lagrangian 14,17,34,100 

complex scalar field 55 

electroweak theory 65-68 

quantum chromodynamics 69-70 

quantum electrodynamics 63 

quantum field theory 25 

real vector fields 56-57 

standard model 71 

weak interactions of quarks 70-71 

Yang-Mills theories 64 

Lambda 160,161 

lepton 19,25,31,36,51, 107,131, 160, 189, 

192,194 

conservation 128,136,137,145,154 

families 128, 135, 136, 146 

flavor 133,136 

multiplets 136 

local gauge 

invariance 36 

theory 25,189 

transformations 39 

local symmetry 25,30,34,40 

magnetic 

monopoles 110, 192 

pinch 134 

Majorana Fields 34 

mass 

neutrino 128,130,133,135,146 

scale 13,15,16,18,101,106,107,131 

massive gravitino 110 

meikton 161 

meson 111, 135, 136,161,189 

metabolic rate 5-7 

Michelparameters 145, 146, 148 

modeling theory 9,18,19 

Monte Carlo 

calculation 162 

sampling 41 

simulation 137, 140,191 

modeling 134 

multiplets 136 

muon 111, 128,131, 135-141, 143-147, 

158-160,189 

branching ratio 132,136,137 

daughter 137 

decay 136, 138-141, 145-148, 155 

discovery 135 

lifetime 138,144 

Michelparameters 145,146,148 

number 34,136,137 

prompt 144 

range 136 

197

neutralweakcurrent46,48,130,131,141, 

X 92 

neutrino 100, 111, 112, 128, 130,136-146, 

155,158,159,162,192 

appearancemode 143,144 

astronomy 167, 170 

decay 173- 174 

electronneutrino 137,142,143,145 

flavors 130 

mass 122,128,130-133,135,159 

oscillations 122, 133, 143, 148, 155, 167, 

176 

physics 154 


solar 155,167 

flux measurements of 172- 177 

neutrinoelectron scattering 130,136, 

141-145, 159,162 

detector 144 

background 144 

neutron decay. See proton decay 

non-Abelian gauge theories 16,18,40,188 

nucleon decay 111,146 

octets 31 

!2-37,150 

Parity 

conservation 128,146 

violation 49,69,146 

Pauli exclusion principle 28, 104 

photinos 106,109,112 

photon 14, 16, 19,29,36,76-78, 100, 135 

pion 100,110, 136, 137, 143, 145,159,160, 

162 

decay 140,143,145 

pion dynamics 59, 152, 155 

Planck mass, length 80,82,83 

positrons 29,109- 111,138,139,143,182 

preons 53,157 

propagator 14- 19 

proton 

beam 137,158,160-162 

decay81-82, 110, 111, 148, 155,168, 192 

searches for 166-17 1 

pyrgons 84-87,95 

quantum chromodynamics 18,19,39-42, 

69-70,79-80, 130, 131, 159, 161, 190, 

191,193. See abo forces and 


quantum electrodynamics 14, 16-19,28-30, 

31,34,36,40,55,62-63,76,79, 130, 

189,190. See also forces and 


quantum field theory, 4,11,13,14,16,18, 

25,27,28,30,24,64, 141, 182, 187 

quark3, 17, 19,25,31,38,39,42,51, 100, 

107-110, 112, 131, 136, 150, 152-155, 

159-161, 180, 182-184, 188-192, 194 

confinement of 42 

families 5 1 

flavors 39 

masses of 69-71 

mixing of 5 1,71 

transitions between states 136 

rare decays 128 

limits 138 

ofthemuon 135,137 

Rayleigh-Ruabouchinsky Paradox 12- 13 

Regge 

recurrences 95 

trajectories 92 

renormalization'll, 14-18,28, 189-191 

group 11,16,18,19,189,191 

group equation 13,14,17 

rotation group 32 

rowing 9,lO 

scalar 146 

pseudo 146 

particles 100, 103, 105, 107, 108 

scale2-21,101, 107, 131,183,190,191 

energy 19 

invariance 11,13,190 

scaling 4-2 1,28,42,190 

classical 4 

curve 9 

scattering experiments 130,189,191 

inelastic 17,18,42,79-80, 190 

selectron 100, 109 

similitude $7 

singlets 131 

slepton 107 

!+matrix theory 189- 191 

sneutrino 100 

solar energy-production models 166, 167, 

171-172, 173, 174 

solar neutrino 145 

physics 148 

space-time manifold 82 

extension to higher dimensions, 76,83, 

84,86 

special relativity 27 

spin 30,87, 100, 131, 189 

spin-polarized hydrogen 133 

spin-statistics theorem 100 

quark 100,107-109,11,113 

standardmodel 18, 19,23,25,30,42,50-51, 

53,54,71,74,76-80, 100, 106,107, 

109, 110, 115, 130, 141, 142, 145, 159, 

182, 188- 190, 192 

minimal 130,131,136,145,148 

strangeness 30,37 

supergap 101 

supergravity 76,88-91,93,94, 101, 110 

superstring theories 76,82,91-95 

superspaces 93 

supersymmetry 74,76,88-90,98, 100-113, 

157 

in quantum mechanics 102- 105 

interaction 1 19 

spontaneous breaking 100,105,107,110 

supersymmetry rotation 106, 108 

symmetries; symmetry groups, multiplets, 

and operations 30-36,38-39,45-47, 

61,64,75,88,90,100,101,110,111, 

136,142,189 

Eightford Way 69,70 

of electroweak interactions 65,77 

ofhgrangians 56-57 

of quark-lepton interactions 8 1 

of strong interactions 69-70 

strong isospin 60-6 1,69 

weak isospin 6 1 

198

symmetry30,39,128,145,146,182,188-192 

boson-fermion 74 

broken 31,32,33,39,45,48, 100, 

105-107, 131, 182 

continuous 31,56 

CP33,131 

discrete 31 

exact 34 4 

external 100-102 

left-handed 145 

local 25,30,3436,39-40,46-47,54-56 

7475,188,192 

Lorentz in variance 56,59 

phase invariance 56,59,62-63,76 

Poincak 56,80 


spontaneous breaking of47-48,54,58-59 

62-63,66-68,78,81,83,88 

tau 30,128,136 

particle number 34,131 

time projection chamber (TPC) 146, 147 

tritium 128,132-135 

beta decay 128,130,148 

betadecay spectrometer 128, 132-135 

resolution function 132-134 

end-point energy 132, 133 

final state spectrum 132 

molecular 133-135 

recombination 133 

source 133-135 

W*. Seeboson. 

weak charge 131 

weak force 19,23-25,28 

currents49,131,141,192 

weakinteraction 42,107,110,112,128,136, 

141-143, 145, 146, 192 

constructive and destructive interference 

142,143 

coupling constant 131,145,146 

weak mixingangle 48,66,67,110,111,131, 

142,143 

weak scale 101 

Weinbergangle48,110,111,131,142,143 

Weinberg-Salam-Glashow model 130 

Yang-Mills theories 40,45, 188, 189 

Yukawa's theory 135 

Zo. Seeboson. 

zero modes 85,86,95 

ultraviolet laser technology 133 

uncertainty principle 13, 14,29 

underground science facilities 178-1 79 

unification26, 101, 141, 188, 190, 192-195 

T 53. 153-154 

vacuum state 58 

vector-axial 13 1, 145 

currents 143 

vector potential 28 

199

Particle Physics-A Los Alamos Primer

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