VI BRAZILIAN SYMPOSIUM on THEORETICAL PHYSICS Erasmo Ferreira, Belita Koiller, Editors

VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS Erasmo Ferreira, Belita Koiller, Editors

Elementary Particle Physics and Relativity CG. Bollini, J.J Giambiagi, A.A. Salam, S.W. Mac Dowell, CO. Escobar, R. Chanda, B. Schroer, Contributors VOLUME III

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO u

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Erasmo Ferreira, Belita Koiller, Editors

Elementary Particle Physics and Relativity CG. Bollini, J.J. Giambiagi, A.A. Saiam, S.W. Mac DoweSI. CO. Escobar, R. Chanda, B. Schroer, Contributors VOLUME III

CONSELHO NACIONAL DE DESENVOLVIMENTO OBITtFICOE TECNOLÓGICO Coordenação Editorial Brasília 1981 VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS

VOLUME III CNPq Comitê Editorial Lynaldo Cavalcanti de Albuquerque José Duarte de Araújo Itiro lida Simon Schwartzman Crodowaldo Pávan Cícero Gontijo Mário Guimarães Ferri Francisco Afonso Biato

Brazilian Symposium on Theoretical Physics, 6., Rio de Janeiro 1980. VI Brazilian Symposium on Theoretical Physics. Bra- sília, CNPq, 1981 3v. Bibliografia Conteúdo — v. 1. Nuclear physics, plasma physics and statistical mechanics. — v. 2. Solid state physics, biophysics and chemical physics. — v. 3. Relativity and elementary particle physics. 1. Física. 2. Simpósio. I. CNPq. II. Título. CDU 53:061.3(81) FOREWORD

The Sixth Brazilian Symposium on Theoretical Physics was held in Rio de Janeiro from 7 to 18 January 1980, consisting in lectures and seminars on topics of current interest to Theoretical Physics. The program covered several branches of modern research, as can be seen in this publication, which presents most of the material discussed in the two weeks of the meeting. The Symposium was held thanks to the generous sponsorship of Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Brazil), Academia Brasileira de Ciências, Fundação do Amparo a Pesquisas do Estado de . São Paulo (Brazil), Organization of American States, Sociedade Cultural Brasil- Israel (Brazil), Comissão Nacional de Energia Nuclear (Brazil), and the governments of France and Portugal. The meeting was attended by a large fraction of the theoretical physicists working in Brasil, and by about 30 scientists who come from other countries. The list of participants — about 250 — is given at the end of this book. / The Pontifical Catholic University of Rio de Janeiro was, for the sixth consecutive time, the host institution, providing excellent facilities and most of " the structural support for the efficient running of the activities. The community of Brazilian Theoretical Physicists is grateful to the sponsors, the host institution, the lecturers and the administrative and technical staff for their contributions to the cooperative effort which made the VI I Brazilian Symposium on Theoretical Physics a successful and pleasant event.

Organizing Comittee Guido Beck (CBPF) Carlos Bollini (CBPF) Hélio Coelho (UFPe) Darcy Dillenburg (UFRGS) Erasmo Ferreira (PUC/RJ) Henrique Fleming (USP) Í. Belita Koiller (PUC/RJ) I Roberto Lobo (CBPF) I Moysés Nussenzveig (USP) I Antônio F. Toledo Piza (USP) Ronald Shellard (IFT) Jorge A. Swieca (U.F. São Carlos e PUC/RJ) Zieli D. Thomé (UFRJ) VII

WELCOME ADDRESS

Ladies and Gentlemen: It is a great pleasure and honor for me to welcome to our University the participants of the Sixth Brazilian Symposium on Theoretical Physics. It is on such occasions that the true spirit of the university is manifested in the search for new knowledge and in the communication of new discoveries. Renowned physicists in several areas of Physics, and from leading institutions in Brazil and abroad, gather here to share their latest results of research, new hypotheses and unanswered questions. Through the exchange of new insights, suggestions and critical reactions, the academic community emerges, united in its commitment to truth and in its effort to contribute to the understanding of the world in which man lives. The understanding of Nature — which is the original meaning of Physics in Greek — is indeed the goal of this Symposium on Theoretical Physics. Events of recent years have taught us to be cautious in our dealing with Nature, since we do not understand all of its interrelations. The precipitate use of nature's forces, made possible by scientific progress, though intended to make the world more livable, threatens to destroy mankind itself. It is not possible to reduce nature to a single model and it is not possible to reduce history to a human project. The full understanding of nature demands humility and respect in dealing with it. Only under such conditions will the knowledge and use of nature be of serve to man. One of the basic goals of this Symposium is to show the fundamental unity of physical theory. The rediscovery of the deep unity of all the phenomena of nature is undoubtly a step forward in the understanding of nature, because understanding requires unification of many phenomena in one law or theory. But a unified theory, in order to be faithful to its object, must recognize the variety of levels through its use of analogy. It must respect the limits of its understanding of nature through the continual perfection and superseding of its own models. It must admit the existence of other dimensions and other approaches to reality, through openess to interdisciplinary dialogue. It is through the pluralism of viewpoints, consciously accepted as relative and interchangeable, that the global vision of reality progressively emerges, and converges to its final goal, which is man in all his complexity and his self-transcendence. May this attitude - so proper to the spirit of the university - be present in all the work of this Symposium. VIII

In closing I would like to thank all the Brazilian and foreign foundations and government agencies which are sponsoring this meeting. May the bold initiative of the Department of Physics at PUC in hosting this Symposium, and the generous efforts of its organizers be rewarded by the scientific and human growth of all the participants. Thank you.

Rev. João A. Mac Dowell, S.J. Rector, Pontifícia Universidade Católica do Rio de Janeiro IX

INDEX

Volume I NUCLEAR PHYSICS, PLASMA PHYSICS AND STATISTICAL MECHANICS 1. The Generator Coordinate Method in Nuclear Physics- B Giraud- 2. Collective Motion and the Generator Coordinate Method - E. J.V. Passos 3. Some New Radiation Processes in Plasmas - C. S. Wu 4. Magnetohydrodynamics and Stability Theory - P. H. Sakanaka 5. Free Energy Probability Distribution in the SK Spin Class Model - Gerard Toulouse and Bernard Derrida 6. Connectivity and Theoretical Physics! Some Applications to Chemistry -HE. Stanley, A. Coniglio, W. Klein and J. Ferreira 7. Random Magnetism - C. Tsallis

Volume II SOLID STATE PHYSICS, BIOPHYSICS AND CHEMICAL-PHYSICS 1. Scattering of Polarized Low-Energy Electrons by Ferromagnetic Metals -J. S. Helman 2. Phase Separations in Impure Lipid Bilayer Membranes - F. de Verteuil, D. A. Pink, E. B. Vadas and M. J. Zuckermann 3. Recent Progress in Biphysics - G. Bemski 4. Electronic Aspects of Enzymatic Catalysis - R. Ferreira and Marcelo A. F. Gomes

Volume III RELATIVITY AND ELEMENTARY PARTICLE PHYSICS 1. Gauge Fields and Unifying Ideas - C. G. Bollini 2. On the Personality of Prof. Abdus Salam - J. J. Giambiagi 3. Gauge Unification of Fundamental Forces - A. Salam 4. Geometric Formulation of Supergravity - S. W. MacDowell 5. Weak Interactions - R. Chanda 6. Applications of Quantum Chromodynamics to Hard Scattering Processes - C. O. Escobar 7. Topological Aspects in Field Theory and Confinement - B. Schroer

AUTHOR INDEX - AT THE END OF THE VOLUME XI

CONTENTS

Carlos G. Bollini GAUGE FIELDS AND UNIFYING IDEAS 295

Juan J. Giambiagi I ON THE PERSONALITY OF PROF. ABDUS SALAM 305

Abdus Salam GAUGE UNIFICATION OF FUNDAMENTAL FORCES 1. Fundamental Particles, Fundamental Forces and Gauge Unification 312 2. The Emergency of Spontaneously Broken SU (2) x U (1) Gauge Theory 313 3. The Present and Its Problems 321 4. Direct Extrapolation from the Electroweak to the Electronuclear 324 j 5. Elementarity: Unification with Gravity and Nature of Charge 328 Appendix 1 — Examples of Grand Unifying Groups 335 ,i Appendix 2 336 I References 337

Samuel W. MacDowell GEOMETRIC FORMULATION OF SUPERGRAVITY I 1. Introduction 345 ..•'•:•'- 2. Unified Gauge Theories in Special Relativity 349 3. Unified Theories in General Relativity 354 4. The Formalism of Fiber Bundles 359 5. Fiber Bundle Formulation of Einstein's Theory of Gravitation 3 70 6. Fiber Bundles Over Graded Manifolds 374 7. Construction of Simple Supergravity 380

8. Construction of Extended SO2 and SO3 Supergravity 383 9. Critical Evaluation of Supergravity 393 ii: 10. Appendix A —Glossary of Notations and Definitions 394 1 11. Appendix B — Gauge Transformation of Vector Potentials 398 ; 12. References 400

RajatChanda WEAK INTERACTIONS .••_- 1. Introduction 403 \ . 3. Gauge Theory: Unification of E—M and Weak Interactions 404 4. Gauge Invariant Introduction of Masses for W ±, Z . Higgs Mechanism 405 XII

4. Effective Lagrangian 406 5, Introduction of Hadrons: Quark Mixing 407 6. Generalization of GIM Mechanism to n-doubiets: Flavor Conservation in Neutral Currents 407 7. Motivation for Including the Sixth Quark t 408

8. Limits on the Mixing Angles 62 and 03 409 9. Generalized Rules for Weak Decay of Heavy Particles 411 10. Charged Currents Neutrino Reactions 412 11. Violation of Bjorken Scaling: a Test of QCD 413 12. Neutral Currents: i>N, QV and eN Reactions 414 13. Important Unsolved Problem 416 14. Outlook 417 References 417

Carlos 0. Escobar APPLICATIONS OF QUANTUM CHROMODYNAMICS TO HARD SCATTERING PROCESSES 1. Introduction 419 2. The Parton Model 420 3. Quantum Chromodynamics 422 4. QCD Corrections to the Simple Parton Model 425 5. Drell-Yan Phenomenology 435 6. Beyond the Leading Order 438 7. Future Prospects 445 8. Conclusions 450 References 451

Bert Schroer TOPOLOGICAL ASPECTS IN FIELD THEORY AND CONFINEMENT 1. Introductory Remarks 453 2. Mathematical Methods 455 3. Models of Gauge Theory. Confinement and U (1) Problem 464 References and Footnotes 467

LIST OF PARTICIPANTS 469

SYMPOSIUM LECTURERS 295

GAUGE FIELDS AND UNIFYING IDEAS

C. G. Bollini Centro brasileiro de Pesquisas Físicas, Rio de Janeiro

First of all I would like to define the level of this talk. Some excellent lectures have already being given about unifying theories. And I think they were sufficient to cater even for the needs of experts in fiber bundles.

My intention is now twofold: to give more emphasis to the physi. cal and geometrical meaning of gauge fields, and to address my talk to the non expert, that is to say using an intuitive language or reasoning.

We all know that physical interactions can be divided into four classes or categories: The (classical) long range interactions : Gravity and Electromagnetism. The short range Strong interaction. And the Weak interaction responsible for 3-decay.

In principle all these interaction can have their own laws and have been studied independently of each other. However there is a widespread belief that they are but different aspects of a grand unified interaction, from which all the others can be derived as special cases.

Let us first consider Gravitation and the important step towards unification given by Einstein's "General Relativity".

Special Relativity is born when we impose invariance of physi. cal laws under Lorentz transformations. Such a transformation is a linear relation, with constant coefficients, between two inertial frames. We shall call it-a "Global Lorentz Transformation" or simply G.L.T..

When we think about G.L.T. we soon arrive at some interesting questions.

The necessary global inertial frame: does it exist? Or can we only have infinitesimal versions of inertial frames? - much in the same way'as only infinitesimal patches of curved surfaces can be said to be "planes". Further, why must we accept a transformation which is the 296 C. Bollini same everywhere? Will it not be more logical to permit the transformation to vary from point to point? Will it not be necessary to consider local Lorentz transformations (L.L.T.) instead of global ones ? All these points lead us to the consideration of Lorentz transformations whose coefficients are arbitrary functions of the coordinates. As soon as this possibility is accepted we are faced with an interesting problem.

For a G.L.T., two vectors a , b , which are parallel in one frame

a,, = X b, (1)

remain parallel in the other

a{, = * K • (2)

But when we allow the transformation to vary from point to point, the equality only holds at one and the same point. At different points a and b transform differently and in general when eg. (1) is true, eg. (2) is false.

This means that parallelism at different points looses its meaning; it is frame dependent. For the very same reason, the usual derivative of a vector field v (x) ceases to have any physical significance. Thus, the situation becomes unacceptable. We feel that someone or something should be capable of saying which is the vector parallel to a given one. That someone who, so to say, keeps us informed, at every point on local parallelism is, precisely, the "gauge field".

Let us write down some relations to formalize that point.

Suppose va is a "constant" vector for a given ("barred") frame of reference (the components are independent of position). After a local a - coordinate dependent - Lorentz transformation with coefficients L a(x), P we will have a vector field:

va(x) = La.(x) v6 , vB = (LS r1 va(x) (3) p a In a neighboring point we have:

a a w a a a , v (xtdx) = v (x) + dx a v (x) = v (x) + dx^ 3 L (x) v GAUGE FIELDS 297 Now, shall we say that va(x+dx) is the vector at x+dx which is parallel to va(x) at x ? Hardly! To accept that is to accept that the frame in which va(x) H va (constant) is a privileged frame. In other words, if eg. (4) is chosen to be the definition of parallelism, then a the transformation L o(x) to the barred frame is essential, and with it p the barred frame itself becomes privileged. Different frames are no longer equivalent and invariance under L.L.T. is lost. The only vay out is to accept the existence of an independent entity, unattached to any ct B —1 particular frame, not necessarily having the form 3 L „ (x) . (L ) Let us call it a "gauge field", or a field of "parallel transport"

r" (x) (5) y»p

This field is "in charge" of local parallelism. The vector va' ', at x+dx, which is parallel to the vector va at x, is: (compare with (4)) •

a a u a B V 'I(x+dx) = v (x) + dx r B(x) v (x) (6)

The gauge field (5) contains all what is needed to identify neighboring parallel vectors. In particular the law of inertia, which states that the tangent vector Ê0— , for the movement xa(T) of a point ax particle, keeps always parallel to itself, means that the tangent vector at T + dt

(r+dx) - f£ (x, + ^ dx , (7) coincides with (cf.(6))

1 dx! ' = «^ + dxH r« dx£ (8) dx dx uB dx K '

Thus equating (7) to (8), and dividing by dx, we get:

2 a u B d x _ dx ro dx ,„. 7F= dT ^ — (9) Which is Einstein's "geodesic" version of the law of inertia. The left hand side of èq. (9) is the acceleration of the particle. The right hand side is the "force" (inertial or gravitacional) produced by the gauge field in orderto keep the particle on an inertial movement; always parallel to itself. The gauge field ra „ unifies inertial and gravitational WP force. The particle does not know whether a gravitational field exists or not, it just obeys the order of the gauge field to move always in 298 C. Bollini the direction of its velocity four vector.

The important poperty of the gauge field of giving sense to the motion of local parallelism (see eq. (6) ) allows one to reintroduce the concept of physical or geometrical derivative of a vector field. In fact, if v1" (x) is a vector field, what is the physical meaning of 9 va(x)? None! To calculate the change of a vector when we go to a neighboring point we must first know what is "no change" and this inplies the knowledge of a parallel vector in the vicinity of the chosen point. In other words, given a vector field v (x), to calculate the actual change at x+dx, we must compare the first line of (4), namely:

a a U a v (x+dx) = x (x) + ãx ay v (x) with (6) , the vector at (x+dx) which is parallel to the one at x. Taking the difference we find:

va(x+dx) - vf, "(x+dx) = dxu 3 va(x) - dxp ra . v(x) II M UP or, dropping some indices:

v(x+dx) - vi,(x+dx) = dxu(3 -r ) v(x) = dxMD v(x) (10) where

D = 3 - r (ID u V u is the actual or real derivative. It is the only derivative with phys^L cal or geometrical meaning. It measures the actual change of the vector field.

It is interesting that the law of parallel transport implied by the gauge field is not necessarily integrable. This means that when the vector, defined by eq. (6), is again transfered by parallelism to another point x+dx+âx, the result does not coincide in general with the vector one obtains by transporting it in the reversed order, that is to say first by 5x and then by dx. The difference is oorportional to

The commutador GAUGE FIELDS [DU ' Dv] =yV

(there are some hidden indices not shown in the formula) defines the "curvature" tensor. When a vector, parallel - transported by two differ ent routes does not coincide at the meeting point, it is because we have curvature in between. When the tensor (12) is zero everywhere, par allelism is path-independent and the space is said to be flat, r then contains purely inertial forces. In all other cases the gauge field contains gravitational forces.

So much for Gravitation. Let us now consider Electromagnetism.

In quantum mechanics a particle is represented by a complex wave function (j>(x). We know that a constant phase factor e (a=cte) is physically irrelevant. That means that the transformation í -»• eia $ leaves physics unchanged. This is a global phase transormation (gauge transformation of the first kind). Can we repeat here what we said before about G.L.T. ? Certainly! ("mutatis mutandis").

Why must we only accept phase transformation which are everywhere constant? Will it not be necessary to consider local phase transformation? After all, physical effects are propagated locally, from a point to a neighbouring one. Again, as soon as we consider this possibility we face a problem similar to that of local parallelism. If phases are changed arbitrarily at different points, then no comparison is possible between the values (the phases) of a wave function a two different points. Even the notion of derivative loses its physical meaning. For example, for a pure coordinate dependent phase we have

eia(x+dx) = eia(x)+ ^ i°t( _ icc(x) ia (X) ,-ia(x) ^ia (x) y < ue _ ia(x) ia(x) e + djc i 3 a (x) £ (13) U

(compare the middle line of (13) with eq. (4)).

So any wave function $(x), which is gauge transformed with the factor e1 , suffers an additional change (in going from x to x+dx) proportional to i 3 ot(x) . This vector can have any value we please at any given point. Who is going to tell us the particular value needed to have "phase parallelism"? The answer is: the gauge field! This is an objective physical entity which is the messenger for local phase comparison. It is a vector field A , not necessarily of the form 3 a , C. Bollini which gives a meaning to the idea of phase parallelism.

For a wave function (x) we define the "phase parallel" wave function at x+dx, to be (compare with eq. (6) and eq. (13))

|,(x+dx) dxy i (14)

The actual change of a wave function is obtained subtracting (14) from:

Thus obtaining:

u 4>(x+dx) - (x+dx) = dx (3jj- i Ay) <|>(x) (15)

Or, with

Dp S V

(x+dx) (17)

Eqs. (16), (17) define the actual derivative of a wave function. It is the only derivative with physical meaning.

If the wave function obeys some deferential equation, for example Dirac's one, then the operator (16) should be the derivative operator appearing in the equation:

(iy.3 - m)

(iy.3 - (19)

We then immediately recognize the gauge field A as the electro magnetic four potential. The electromagnetic vector potencial is the messenger for phase parallelism. Just as for gravitation this parallelism need not be integrable. When it is not, we say that space presents "phase curvature" and this curvature is measured by the tensor (compare with (12)) . GAUGE FIELDS 301 v = [v °J - \ \ - \ \ (2o) which is nothing but the tensor of electromagnetic field intensities. In this sense, the electric and magnetic fields are the physical manifestation of phase curvature.

Let us now consider the theory of strong interaction. Nowadays ;•• - we think that the basic particles are "quarks" which can be found in i three different, so called "color", states. However, there is complete ! symmetry between these states. In other words, there is no physical change when we transform the quark states by any (3x3) unitary unimodular matrix. Such matrices have eight independent elements and can be represented in the exponential form: exp (i a G. ) , where G. (k=l,2,...,8) are eight independent "generators" for the given representation,and the ak are eight parameters. Instead of having a single phase, we now have " eight phases. The symmetry (SO(3)) is global if these phases are con- } stants. But just as for the cases so far considered, we can argue about :, the possibility (or necessity) of taking the phases to be eight arbitrary functions of the coordinates. Comparison of quark fields at different *j points no longer has any physical meaning; unless an independent entity, the gauge field, tell us how to calculate actual local variations. Sup- '£ pose, for simplicity, that the quark field q is constant, and we subject ij it to a SU(3) local gauge transformation:

;. q(x) = U(x)q ; q = u"1 q

r f. Then:

V 1 q(x+dx)= q(x) + dx" 3p Ü q = q(x) + dx 3y U.lf q(x) (21)

Now, the actual change for the quark field (in going from x to x+dx) is not ãxdxvu 3 U Uif 1. ThThid s quantity depends on a particular gauge, a particular matrix (U) .

í • If we want local invariance, no particular matrix should be y previleged. An independent entity, B (x) must replace the quantity |. 3 U(x) U~1(x). This object is the SU(3) gauge field: the messanger

5ii: for "color parallelism"

fp q'I (x+dx) = q(x) + dxu B (x) q(x) (22)

[-?' (compare with (6) and (14)). 302 C. Bollini The gauge field B (x) is a matrix in the color indices of the \,r quark fields. It can be expressed as a combination of the eight gener- pf- ators G, :

B (x) = i B* (x) G, (23)

B (x) is then composed of eight vectors fields B (x). The actual change in the quark field is obtained by subtracting

r.) = q(x) + dx 8pq(x)

from (22)

W q(x~dx) - q,*, (x-dx) = dx Oy-B^) q(x)

Or:

q(x+dx) - qi,(x+dx) = dxU D q(x) (24)

with

D = 3 - B (25) V U U D is the actual physical derivative operator to act on the quark field. In the equation of motion for the quark field D is the derivative operator to be used.

The gauge field B coaches the wave function to move in such a way that color is "inertially" transported. As in previous cases, the parallel transport of color is not necessarily integrable. When it is, the space is "color flat". If not, we.have "color curvature" measured by the tensor

which is also a matrix in color indices.

Let us now tackle the last of the interactions mentioned at the begining: the Weak interaction.

For the sake of clarity we shall consider a so called "lepton generation". For example the electron e and the corresponding neutrino vi. GAUGE FIELDS 303 We shall assume these particles to be massless without worrying about the process by means of which the electron adquires mass.

The neutrino is known to be left-handed; i.e. it has negative

helicity. Electron states can be divided into positive helicity eR, and

negative helicity eL- It seems natural to group together the left handed

states eL with the neutrino to form a two component left handed object I = V* I . The other state of the electron remains a single (right handed)

object eR.

At this stage there exists complete symmetry between the two components of SU. This means that we can transform & by any two by two unitary unimodular matrix without any physical consequence. Let us call

SU(2)L the group of such transformations. Also, the phase of eR can be arbitrarily changed without physical effects. The corresponding group

is U(1)R. Both transformations acting simultaneously, are simbolized by SU(2) x U(l). When the lepton fields are changed by any transformation of SU(2) x U(l), physics remain invariant. t The matrices of SU(2) can be represented in the exponential V" form using the three Pauli matrices: exp (i a. x.). The three phases jí.. a. (i=l,2,3) and the phase fi of the right handed state, form a set of (', four phases whose values are physically irrelevant. If these phases are i-r constants we have a global SU(2) x U(l) transformation. Once again, if f- we accept phases which are arbitrary functions of the coordinates, we f' need a gauge field to "parallel transport" these "chiral" phases.

I We need not enter into details. We shall have a vector gauge \{\{\ field for each of the phases. Three of these four fields acquire mass ';-•( simultaneously with the lepton (electron) . They form the so called : \ ±0 IS_ intermediate vector bosons W and Z . They are responsible for the :•:£•! "weak" part of the interaction. The fourth one remains massless and can çVsj be identified with the electromagnetic field A -We then recognize that ••) Electromagnetism and Weak interactions can be considered to be included in the SU(2) x U(l) case just examined, receiving thus a unified treatment. Summarizing: All known interactions are such that there are some properties which, so to speak, are parallel transported along the path of the particles. Those "inertially carried" properties are associ ated with a symmetry under a local group. Unification is achieved when these groups are succesfully included into a bigger group of local 304 C. Bollini transformations. Automatically the new group is married (till death to they part) with the corresponding gauge fields. They give sense to local comparison of the bases for the group. They instruct the particles on how to move in such a way that the property in question is compared and preserved from point to point. 305

ON THE PERSONAUTY OF PROF. ABDUS SALAM

J.J.Giambiagi Centro Brasileiro de Pesquisas Físicas

The invitation of prof. Erasmo Ferreira to make some comments on the personality of prof. Abdus Salam came to me as a great honour and as a great pleasure as he is an old friend of mine.

We physicists of Latin America are paying this homage to him as he has been honoured with the Nobel Prize. In celebrating his triumph we are celebrating the triumph of one of us, as he is a representative of the physicists of the third world. This remark about prof. Salam being a representative of the third world marks an important characteristic of his personality. He has lived most of his life in the first world, but he has remained loyal to the third world devoting a good part of his time to the effort of developing science and promoting progress in it. As a matter of fact, he is not a new-comer to the arena field of the Nobel Prize. He has been "dribbling" in it for a long time. After making a brilliant entrance in the world of physics through a series of papers on renormalization in field theories he made a famous report on Dispersion Relations in the Rochester meeting at CERN in 1956. It was some time later that he produced a paper on Y5 invariance of weak interactions theories. This, he sent to Pauli asking for his comments. As an answer, he received a post- card saying: Why do you waste your time? Why don't you think on something better? So, he dropped the manuscript in the drawer of 306 J.J. Giambiagi

his desk. The content of this paper was essentially the same as that of Yang and Lee on violations of parity in weak interactions, which appeared in Physical Review a few months later and won the Nobel Prize for its authors. When Salam saw it he rescued his own paper and sent it for publication in Nuclear Physics. But it was already too late for him. This is probably one of the reasons why he always gives encouraging words to anybody - particularly young physicists - who come to him to discuss nevi ideas. He is always helpful to push you to carry them on.

Few years later, when Abdus Salam was chairman in the Imperial College, Yuval Ne-eman came to him as a graduate student, wanting to work for a PhD degree. Then Salam suggested him to look into the representations of SU(3) and its applications to particle-physics. After many discussions, finally Ne-eman appeared at Salam's office with a draft of a manuscript and with the name of Salam at the top of it. To which Salam said: erase my name, you did most of the work. This paper came to be one of the most important of the decade. Ne-eman was seriously considered to share the Nobel Prize together with Gell-Mann who also published a paper on a similar subject. One can possibly argue that the reason why Ne-eman was not nominated was essentially that it was his thesis work: just the beginning of his career. But this was not the case of Salam. So, for the second time Salam was very near around the Prize. This is why I said that he was not a foreigner to the arena field of the prize. Finally, his success came with the unified theory of weak and electromagnetic interactions, success which we celebrate today. Of course, in between he obtained the most distinguished prizes from U.S., Europe and the Soviet Union. But his scientific activity, ON THE PERSONALITY OF A. SALAM 307 although deep and intense only shows one of the aspects of his complex personality. It is far from exhausting it.

As I said before, his loyalty to the third world is a making characteristic of his personality. One of the main

driving forces behind him.

It was mainly due to his own effort - and that of prof. P. Budini that the Center in Trieste was created. Salam was appointed its Director, and it was under his leadership that the Institute has acquired its present spirit of international cooperation between ail nations, irrelevant of religions or political system and also of helping mostly the developing countries. The idea is that science is an essential ingredient of progress and that no technological "know how" will be developed unless in parallelism with basic research. Both must go together, parallel, in order to have stable scientific institutions. This is an important philosophical attitude towards science in the third world. He has also stressed the urgent need for the scientists tc do something of immediate use in promoting welfare for so many millions of people living in proverty, besides their own basic research.

I am sure that prof. Salam will use all the prestige (and in a sense, power) that the Nobel Prize brings with it to promote peaceful international cooperation and the progress of science in the developing countries. I am also sure he will do whatever he can to diminish hunger and misery in the third world. 309

GAUGE UNIFICATION OF FUNDAMENTAL FORCES

Abdus Saiam International Centre for Theoretical Physics Trieste, Italy

In June 1938, Sir George Thomson, then Professor of Physics at Imperial College, London, delivered his 1937 Nobel Lecture. Speaking of Alfredo Nobel, he said: "The idealism which permeated his character led him to... (being) as much concerned with helping science as a whole, as individual scientists. ... The Swedish people under the leadership of the Royal Family and through the medium of the Royal Academy of Sciences have made Nobel Prizes one of the chief causes of the growth of the prestige of science in the eyes of the world ... As a recipient of Nobel's generosity, I owe sincerest thanks to them as well as to him."

I am sure I am echoing my colleague's feelings as well as my own, in reinforcing what' Sir George Thomson said — in respect of Nobel's generosity and its influence on the growth of the prestige of science. Nowhere is this more true than in the developing world. And it is in this context that I have been encouraged by the Permanent Secre tary of the Academy — Professor Carl Gustaf Bernhard — to say a few words' before I turn to the scientific part of my lecture.

Scientific thought and its creation is the common and shared heritage of mankind. In this respect, the history of science, like the history of all civilization, has gone through cycles. Perhaps I can illustrate this with an actual example.

Seven hundred and sixty years ago, a young Scotsman left his native glens to travel south to Toledo in Spain. His hame was Michael, his goal to live and work at the Arab Universities of Toledo and Cordova, where the greatest of Jewish scholars, Moses bin Maimoun, had taught a generation before.

Michael reached Toledo in 1217 AD. Once in Toledo, Michael formed the ambitious project of introducing Aristotle to Latin Europe , translating not from the original Greek, which he knew not, but from the Arabic translation then taught in Spain. Prom Toledo, Michael travelled to Sicily, to the Court of Emperor Frederick II. 3\o Saiam Visiting the medical school at Salerno, chartered by Frederick in 1231, Michael met the Danish physician, Henrik Harpestraeng — later to become Court Physician of Eric IV Waldemarsson. Henrik had come to Salerno to compose his treatise on blood-letting and surgery. Henrik's sources ware the medical canons of the great clinicians of Islam,Al-Razi and Avicenna, which only Michael the Scot could translate for him.

Toledo's and.Salerno's schools, representing as they did the finest synthesis of Arabic, Greek, Latin and Hebrew scholarship, were some of the most memorable of international assays in scientific collaboration. To Toledo and Salerno came scholars not only from the rich countries of the East, like Syria, Egypt, Iran and Afghanistan, but also from developing lands of the West like Scotland and Scandinavia. Then, as now, tiiere were obstacles to this international scientific concourse, with an economic and intellectual disparity between different parts of the world. Men like Michael the Scot or Henrik Harpestraeng were singularities. They did not represent any flourishing schools of research in their own countries. With all the best will in the world their teachers at Toledo and Salerno doubted the wisdom and value of training them for advanced scientific research. At least one of his masters counselled young Michael the Scot to go back to clipping sheep and to the weaving of woollen cloths.

In respect of this cycle of scientific disparity, perhaps I can be more quantitative. George Sarton, in his monumental five - volume History of Science chose to divide his story of achievement in sciences into ages, each age lasting half a century. With each half century he associated one central figure. Thus 450 BC - 400 BC Sarton calls the Age of Plato; this is followed by half centuries of Aristotle,of Euclid, of Archimedes and so on. From 600 AD is the Chinese half century of Hsiiah Tsang, from 650 to 700 AD that of I-Ching, and then from 750 AD to 1100 AD — 350 years continuously — it is the unbroken succession of the Ages of Jabir, Khwarizmi, Razi, Masudi, Wafa, Biruni and Avicenna, and then Omar Khayam — Arabs, Turks, Afghans and Persians — men belonej ing to the culture of Islam. After 1100 appear the first Western names; Gerard of Cremona, Roger Bacon — but the honours are still shared with the names of Ibn-Rushd (Averroes), Moses Bin Maimoun, Tusi and Ibn-Nafis — the man who anticipated Harvey's theory of circulation of blood. No Sarton has yet chronicled the history of scientific creativity among the pre-Spanish Mayas and Aztecs, with• their invention of the zero, of the calendars of the moon and Venus and of their diverse pharmocological dis_ coveries, including quinine, but the outline of the story is the same — GAUGE UNIFICATION 311 one of undoubted superiority to the Western contemporary correlates.

After 1350, however, the developing world loses out except for the occasional flash of scientific work, like that of Ulugh Beg — the grandson of Timurlane, in Samarkand in 1400 AD; or of Maharaja Jai Singh of Faipur in 1720 — who corrected the serious errors of the then Western tables of eclipses of the sun and the moon by as much as six minutes of arc. As it was, Jai Singh's techniques were surpassed soon after with the development of the telescope in Europe. As a contemporary Indian chronicler wrote: "With him on the funeral pyre, expired also all science in the East." And this brings us to this century when the cycle begun by Michael the Scot turns full circle, and it is we in . the developing world who turn to the Westwards for science. As Al-Kindi wrote 1100 years ago: "It is fitting then for us not to be ashamed to acknowledge truth and to assimilate it from whatever source it comes to us. For him who scales the truth there is nothing of higher value than truth itself; it never cheapens nor abases him."

Ladies and Gentlemen,

It is in the spirit of Al-Kindi that I start my lecture with ',f

a sincere expression of gratitude to the modern equivalents of the Uni- |fFv, versities of Toledo and Cordova, '•which I have been privileged to be associated with - Cambridge, Imperial College and the Centre at Trieste. 312 A. Saiam I. FUNDAMENTAL PARTICLES, FUNDAMENTAL FORCES AND GAUGE UNIFICATION

The Nobel lectures this year are concerned with a set of ideas relevant to the gauge unification of the electromagnetic force with the weak nuclear force. These lectures coincide nearly with the 100 death-anniversary of Maxwell, with whom the first unification of forces (electric with the magnetic) matured and with whom gauge theories originated. They also nearly coincide with the 100th anniversary of the birth of Einstein — the man who gave us the vision of an ultimate unification of all forces. The ideas of today started more than twenty years ago, as gleams in several theoretical eyes. They were brought to predictive maturity over a decade back. And they started to receive experimental confirmation some six years ago. In some senses then, our story has a fairly long background in the past. In this lecture I wish to examine some of the theoretical gleams of today and ask the question if these may be the ideas to watch for maturity twenty years from now. From time immemorial, man has desired to comprehend the complexity of nature in terms of as few elementary concepts as possible Among his guests — in Feynman's words — has been the one for " wheels within wheels" — the task of natural philosophy being to discover the innermost wheels if any such exist. A second quest has concerned itself with the fundamental forces which make the wheels go round and enmesch with one another. The greatness of gauge ideas — of gauge field theories — is that they reduce these two quests to just one;elementary particles (described by relativistic quantum fields) are representations of certain charge operators, corresponding to gravitational mass, spin, flavour, colour, electric charge and the like, while the fundamental forces are the forces of attraction or repulsion between these same charges. A third quest seeks for a unification between the charges (and thus of the forces) by searching for a single entity, of which the various charges are components in the sense that they can be transformed one into the other. But are all fundamental forces gauge forces? Can they be understood as such, in terras of charges — and their corresponding currents — only? And if they are, how many charges? What unified entity are the charges components of? What is the nature of charge ? Just as Einstein comprehended the nature of gravitational charge in terms of space-time curvature, can we comprehend the nature of the other charges — the nature of the entire unified set, as a set, in terms of GAUGE UNIFICATION 313 something equally profound? This hriefly is the dream, much reinforced by the verification of gauge theory predictions. But before I examine the new theoretical ideas on offer for the future in this particular context, I would like your indulgence to range over a one-man, purely subjective, perspective in respect of the developments of the last twenty years themselves. The point I wish to emphasise during this part of my talk was well made by G. P. Thomson in his 1937 Nobel Lee - ture. G. P. said "... The goddess of learning is fabled to have sprung full grown from the brain of Zeus, but is is seldom that a scientific conception is born in its final form, or owns a single parent. More often it is the product of a series of minds, each in turn modifying the ideas of those that came before, and providing material for those that come after."

II. THE EMERGENCE OF SPONTANEOUSLY BROKEN SU(2] X UQ) r> GAUGE THEORY

I started physics research thirty years ago as an experimen- *__ tal physicist in the Cavendish, experimenting with tritium — deuterium íp scattering. Soon I knew the craft of experimental physics was beyond Hi me — it was the sublime quality of patience — patience in accumulating £• data, patience with recalcitrant equipment — which I sadly lacked. Re_ %~ luctantly I turned my papers in, and started instead on quantum field theory with Nicholas Kemmer in the exciting department of P.A.M. Dirac. The year 1949 was the culminating year of the Tomonaga- Schwinger-Feynman-Dyson reformulation of renormali zed Maxwell — Dirac gauge theory, and its triumphant experimental vindication. A field theory must be renormalizable and be capable of being made free of infi nities — first discussed by Waller — if perturbative calculations with it are to make any sense. More — a renormalizable theory, with no dimensional parameter in its interaction term, connotes somehow that the fields represent "structureless" elementary entities. With Paul Matthews, we started on an exploration of renormalizability of meson theories. Finding that renormalizability held only for spin-zero mesons and that these were the only mesons that empirically existed then , Cpseudoscalar pions, invented by Kemmer, following Yukawa) one felt thrillingly euphoric that with the triplet of pions (considered as the carriers of the strong nuclear force between the proton-neutron doublet) one might resolve the dilemma of the origin of this particular force which is responsible for fusion and fission. By the same token, the so-called weak nuclear force — the force responsible for B-radioactivi1y A. Saiam

(and described then by Fermi's non-renormalizable theory! had to be '0, mediated by some unknown spin-zero mesons if it was to be renornalizable. fÇ- If massive charged spin-one mesons were to mediate this interaction,the '"•- theory would be non-renormalizable, according to the ideas then. Now this agreeably renormalizable spin-zero theory for the pion was a field theory, but not a gauge field theory. There was no conserved charge which determined the pionic interaction. As is well known, shortly after the theory was elaborated, it was found wanting The (4 , -a) resonance A effectively killed it off as a fundamental theory; we were dealing with a complex dynamical system, not "structure less" in the field-theoretic sense. For me, personally, the trek to gauge theories as candidates for fundamental physical theories started in earnest in September 1956- the year I heard at the Seattle Conference Professor Yang expound- his \_, and Professor Lee's ideas on the possibility of the hitherto sacred principle of left-right symmetry, being violated in the realm of the weak nuclear force. Lee and Yang had been led to consider abandoning X' left-right symmetry for weak nuclear interactions as a possible resol - 7 ution of the (T,8) puzzle. I remember travelling back to London on an & American Air Force (MATS) transport flight. Although I had been granted, j^i for that night, the status of a Brigadier or a Field Marshal — I don't y quite remember which — the plane was very uncomfortable, full of r crying service-men's children — that is, the children were crying, not |v the servicemen. I could not sleep. I kept reflecting on why Nature - should violate left-right symmetry in weak interactions. Now the hall- mark of most weak interactions was the involvement in radioactivity *- phenomena of Pauli's neutrino. While crossing over the Atlantic, came back to me a deeply perceptive question about the neutrino which' Pro- fessor Rudolf Peierls had asked when he Was examining me for a Ph.D. a few years before. Peierls' question was: "The photon mass is zero because of Maxwell's principle of a gauge symmetry for electromagnetism; tell me, why is the neutrino mass zero?" I had then felt somewhat uncomfortable at Peierls, asking for a Ph.D. viva, a question of which he himself said he did not" know the answer. But during that comfort - less night the answer came. The analogue for the neutrino, of the gauge symmetry for the photon existed; it had to do with the massless -

ness of the neutrino, with symmetry under the Y5 transformation (later christened "chiral symmetry"). The existence of this /symmetry

for the massless neutrino must imply a combination (1 + y5l or Cl - Y5) for the neutrino interactions. Nature had the choice of an aesthetic - ally satisfying but a left-right symmetry violating theory, with a neutrino which travels exactly with the velocity of light; or alternatively GAUGE UNIFICATION 315 a theory where left-right symmetry is preserved, but the neutrino has a tiny mass — some ten thousand times smaller than the mass of the electron. It appeared at that time clear to me what choice Nature must have made. Surely, left-right symmetry must be sacrificed in all neu - trino interactions. I got'off the plane the next morning, naturally very elated. I rushed to the Cavendish, worked out the,Michel parame -

ter and a few other consequences of ys symmetry, rushed out again, got onto a train to Birmingham where Peierls lived. To Peierls I presented my idea; he had asked the original question; could he approve of the answer? Peierls1 reply was kind but firm. He said "I do not believe left-right symmetry is violated in weak nuclear forces at all. I would not touch such ideas with a pair of tongs." Thus rebuffed in Birmingham, like Zuleika Dobson, I wondered where I could go next and the obvious place was CERN in Geneva, with Pauli — the father of the neutrino — nearby in Zurich. At that time CERN lived in a wooden hut just outside Geneva airport. Besides my friends, Prentki and d'Espagnat, the hut contained a gas ring on which was cooked the staple diet of CERN — Entrecôte ã Ia creme. The hut also contained Professor Villars of MIT, who was visiting Pauli the same day in Zurich. I gave him my paper. He returned the next day with a message from the Oracle; "Give my regards to my friend Salam and tell him to think of something better". This was discouraging, but I was compensated by Pauii's excessive kindness a few months later, when Mrs. Wu's , Lederman's and Telegdi's experiments were announced showing that left-right symmetry was indeed violated and ideas similar to mine about chiral symmetry were expressed independently by Landau and Lee and Yang . I received Pauii's first somewhat apologetic letter on 24 January 1957. Thinking that Pauii's spirit should by now be suitably crushed, I sent him two short notes I had written in the meantime. These contained suggestions to extend chiral symmetry to electrons and muons, assuming that their masses were a consequence of what has come to be known as dynamically spontaneous symmetry breaking. With chiral symmetry for electrons, muons and neu - trinos, the only mesons that could mediate weak decays of the muons would have to carry spin one. Reviving thus the notion of charged in - termediate spin-one bosons, one could then postulate for these a type of gauge invariance which I called the "neutrino gauge". Pauii's re- '• action was swift and terrible. He wrote on 30 January 1957, then on j 18 February and later on 11, 12 and 13 March: "I am reading (along the shores of Lake Zurich) in bright sunshine quietly your paper..." "I am very much startled on the title of your paper 'Universal Fermi inter - r action'... For quite a while I have for myself the rule if a theoreti-

'•'i 316 A. Saiam cian says universal it just means pure nonsense. This holds particularly in connection with the Fermi interaction, but otherwise too, and now you too, Brutus, my son, come with this word..." Earlier, on 30 January, he had written "There is a similarity between this type of gauge invariance and that which was published by Yang and Mills... In the latter, of course, no ys was used in the exponent." And he gave me the full reference of Yang and Mills' paper; (Phys. Rev. 96_, 191 (1954)). I quote from his letter: "However, there are dark points in your paper regarding the vector field B . If the rest mass is infinite (or very large), how can this be compatible with the gauge' transformation

B •* B - 3UA?" and he concludes his letter with the remark: "Every reader will realize that you deliberately conceal here something and will ask you the same questions". Although he signed himself "With friendly regards", Pauli had forgotten his earlier penitence. He was clearly and rightly on the warpath. Now the fact that I was using gauge ideas similar to the Yang-Mills (non-Abelian SU(2) - invariant) gauge theory was no news to 9) me. This was because the Yang-Mills theory (which married gauge ideas of Maxwell with the internal symmetry SU(2) of which, the proton -neutron system constituted a doublet) had been independently invented by a Ph.D. pupil of mine, Ronald Shaw ' , at Cambridge at the same time as Yang and Mills had written. Shaw's work is relatively unknown; it remains buried in his Cambridge thesis. I must admit I was taken aback by Pauli's fierce prejudice against universalism — against what we would today call unification of basic forces — but I did not take this too seriously. I felt this was a legacy of the exasperation which Pauli had always felt at Einstein's somewhat formalistic attempts at unifying gravity with electromagnetism —forces which in Pauli's phrase "cannot be joined — for God hath rent them asunder". But Pauli was absoluted right in accusing me of darkness about the problem of the masses of the Yang-Mills fields; one could not obtain a mass without wantonly destroying the gauge symmetry one had started with. And this was particularly serious in this context, because Yang and Mills had conjectured the desirable renormalizability of their theory with a proof which relied heavily and exceptionally on the masslessness of their / spin-one intermediate mesons. The problem was to be solved only years later with the understanding of what is now known as the mechanism, but I will come back to this later. Be that as it may, the point I wish to make from this change with Pauli is that already in early 1957, just after the set of parity experiments, many ideas coming to fruition now, started to become clear. These are: i GAUGE UNIFICATION 317 1. First was the idea of chiral symmetry leading to a V-A theory. In those early days ray humble suggestion of this was limited to neutrinos, electrons and muons only, while shortly after, that year , Sudarshan and Marshak , Gell-Mann and Feynman , and Sakurai had the courage to postulate y5 symmetry for baryons as well as leptons, *) making this into a universal principle of physics. Concomitant with the (V-A) theory was the result that :Lf weak interactions are mediated by intermediate mesons, these mesons must carry spin one.

2. Second, was the idea of spontaneous breaking of chiral symmetry to generate electron and muon masses: though the price which those latter- day Shylocks, Nambu and Jona-Lasinio and Goldstone exacted for this (i.e. the appearance of massless scalars), was not yet appreciated.

3. And finally, though the use of a Yang-Mills-Shaw (non-Abelian)gauge theory for describing spin-one intermediate charged mesons was suggested already in 1957, the giving of masses to the intermediate bosons through spontaneous symmetry breaking, in a manner to preserve the renormaliz - ability of the theory, was to be accomplished only during a long period of theoretical development between 1963 and 1971.

Once the iang-Mills-Shaw ideas were accepted as relevant to the charged weak, currents — to which the charged intermediate mesons were coupled in this theory — during 1957 and 1958 was raised the question of what was the third component of the SO(2) triplet, of which the charged weak currents were the two members. There were the two alternatives: the electroweak unification suggestion, where the electro magnetic current was assumed to be this third component; and the rival suggestion that the third component was a neutral current unconnected with electroweak unification. With hindsight, I shall call these the Klein 16) (1938) and the Kemmer 17) (1937) alternatives. The Klein suggestion, made in the context of a Kaluza-Klein five-dimensional space -time, is a real tour-de-force; it combined two hypothetical spin - one charged mesons with the photon in one multiplet, deducing from the compactification of the fifth dimension, a theory which looks like Yang-Mills-Shaw's. Klein intended his charged mesons for strong interactions, but if we read charged weak mesons for Klein's strong ones,one

*) Today we believe protons and neutrons are composites of quarks, so that y5 sym - metry is now postulated for the elementary entities of today — the quarks. 318 A. Saiam obtains the theory independently suggested by Schwinger (1957) , though Schwinger, unlike Klein, did not build in any non-Abelian gauge aspects. With just these non-Abelian Yang-Mills gauge aspects very much to the fore, the idea of uniting weak interactions with electromag^ netisra was developed by Glashow and Ward and myself in late 1958. The rival Kemmer suggestion of a global SU(2) - invariant triplet of weak charged and neutral currents was independently suggested by 211 Bludman (1958) in a gauge context and this is how matters stood till 1960. To give you the flavour of, for example, the year 1960,there 22) is a paper written that year of Ward and myself with the statement "Our basic postulate is that it should be possible to generate strong , weak and electromagnetic interaction terms with all their correct sym - metry properties (as well as with clues regarding their relative strengths) by making local gauge transformations on the kinetic energy terms in the free Lagrangiar. for all particles. This is the statement of an ideal which, in this paper at least, is only very partially realized". I am not laying a claim that we were the only ones who were saying this, but I just wish to convey to you the temper of the physics of twenty years ago — qualitatively no different today from then. But what a quantitative difference the next twenty years made, first with new and far-reaching developments in theory — and then, thanks to CERN, Fermilab, Brookhaven, Argonne, Serpukhov and SLAC in testing itI So far a theory itself is concerned, it was the next seven years between 1961-67 which were the crucial years of quantitative comprehension of the phenomenon of spontaneous symmetry breaking and the emergence of the SO(2) x U(l) theory in a form capable of being tested. The story is well known and Steve Weinherg has already spoken about it. So I will give the barest outline. First there was the realization that the two alternatives mentioned above a pure electromag_ netic current versus a pure neutral current — Klein-Schwinger versus Kemmer-Bludruan — were not alternatives; they were complementary. As was noted by Glashow and independently later by Ward and myself , both types of currents and the corresponding gauge particles (W~,Z° and y) were- needed in order to build a theory that could simultaneously accomodate parity violation for weak and parity conservation for the electromagnetic phenomena. Second, there was the influential paper of Goldstone in 1961 which, utilizing a non-gauge self-interaction between scalar particles, showed that the price of spontaneous breaking of a continuous internal symmetry vias the appearance of zero mass scalars — a result foreshadowed earlier by Nambu. In giving a proof of this theorem with Goldstona I collaborated with Steve Weinberg , GAUGE UNIFICATION 319 who spent a year at Imperial College in London. ,^ I would like to pay here a most sincerely felt tribute to ?'{ him and to Sheldon Glashow for their warm and personal friendship. I shaw not dwell on the now well-known contributions of Anderson , Higgs , Brout & Henglert , Guralnik, Hagen and Kibble starting from 1963, which showed the way how spontaneous sym metry breaking using spin-zero fields could generate vector-meson masses, defeating Goldstone at the same time. This is the so-called Higgs mech anism. The final steps towards the electroweak theory were taken by Weinberg and independently myself (with Kibble at Imperial College tutoring me about the Higgs phenomena). We were able to com - plete the present formulation of the spontaneously broken SU(2) x U(l) theory so far as leptonic weak interactions were concerned — with one parameter sin2 9 describing all weak and electromagnetic phenomena and with one isodoublet Higgs multiplet. An account of this development was given during the contribution to the Nobel Symposium ( organized by Nils Svartholm and chaired by Lamek Hulthin held at Gothenburg after , some postponements, in early 1968) . As is well known, we did not then, J,^ and still do not, have a prediction for the scalar Higgs mass. [t '" Both Weinberg and I suspected that this theory was likely to -'•', be renormalizable. Regarding spontaneously broken Yang-Mills - Shaw : theories in general this had earlier been suggested by Englert, Brout \ 29) and Thiry . But this subject was not pursued seriously except at Utrecht, where the actual proof of renormalizability was given by •t Hooft 33) in 1971. This was elaborated further by that remarkable physicist the late Benjamin Lee 34) , working with Zinn Justin, and by 't Hooft and Veltman . This followed on the earlier basic advances in Yang-Mills calculational technology by Feynman , DeWitt , Faddeev and Popov , Mandelstam , Fradkin and Tyutin , Boulware 415 , Taylor 42> , Slavnov 43) , Strathdee 44' and Salam. In Coleman's eloquent phrase " 't Hooft's work turned the Weinberg-Sa'.am frog into ~~ an enchanted prince". Just before had come the GIM (Gla3how,Iliopoulos and Maiani) mechanism , emphasising that the existence of the fourth *) When I was discussing the final version of the SU(2) x U(l) theory and its possible renormalizability in Autumn 1967 during a post-doctoral course of lectures at Imperial College, Nino Zichichi from CEEN happened to be present. I was delighted because Zichichi had been badgering me since 1958 with persistent questioning of what theore - tical avail his precise measurements on (g—2) for the rauon as well as those of the muon lifetime were, when not only the magnitude of the electromagnetic corrections to weak decays was uncertain, but also conversely the effect of non-renormalizable weak interactions on "renormalized" electromagnetism was so unclear. 320 A- Saiam charmed quark (postulated earlier hy several authors) was essential to the natural resolution of the dilemma posed by the absence of strangeness-violating currents. This tied in naturally with the understanding of the steinberger-Schwinger-Rosenberg-Bell-Jackiw-Adler anomaly and its removal for SU(2) x U(l) by the parallelism of four quarks and four leptons, pointed out by Bouchiat, Iliopoulos and Meyer and inde - 47) pendently by Gross and Jackiw . , If one has kept a count, I have so far mentioned around fifty theoreticians. As a failed experimenter, I have always felt envious of the ambience of large experimental teams and it.gives me the greatest pleasure to acknowledge the direct or the indirect contributions of the "series of minds" to the spontaneously broken SU(2) x U(l) gauge theory. My profoundest personal appreciation goes to my collaborators at Imperial College, Cambridge and the Trieste Centre, Paul Matthews, John Ward, Jogesh Peti, John Strathdee, Tom Kibble and to Nicholas Kemmer. In retrospect, what strikes me most about the early part of this story is how uninformed all of us were, not only of each other's work, but also of work done earlier. For example, only in 1972 did I learn of Kemmer1s paper written at Imperial College in 1937. Kemmer's argument essentially was that Fermi's weak theory was not globally SU(2) invariant and should be made so — though not for its own sake but as a prototype for strong interactions. Than this year I learnt that earlier, in 1936, Kemmer's Ph.D. supervisor, Gregor Wentzel , had introduced (the yet undiscovered) analogues of lepto - quarks, whose mediation could give rise to neutral currents after a Fierz reshuffle. And only this Summer, Cecilia Jarlskog at Bergen rescued Oscar Klein's paper from the anonymity of the Proceedings of the International Institute of Intellectual Cooperation of Paris, and we learnt of his anticipation of a theory similar to Yang-Mills-Shaw's long before these authors. As I indicated before, the interesting point is that Klein was using his triplet, of two charged mesons plus the photon, not to describe weak interaction but for strong nuclear force unification with the electromagnetic — something our generation started on only in 1972 — and not yet experimentally verified. Even in this recitation I am sure I have inadvertantly left off some names of those who have in some way contributed to SU(2) x U(l). Perhaps the moral is that not unless there is the prospect of quantitative verification, does a qualitative idea make its impress in physics. And this brings me to experiment, and the year of the 49) Gargamelle . I still remember Paul Matthews and I getting off the train at Aix-en-Provence for the 1973 European Conference and foolishly deciding to walk with our rather heavy luggage to the student hotel GAUGE UNIFICATION 321 where we were billeted. A car drove from behind us, stopped, and the driver leaned out. This was Musset whom I did not know well personally then. He peered out of the window and said: "Are you Salara?" I said "Yes". He said: "Get into the car. I have news for you. We have found neutral currents." I will not say whether I was more relieved for being given a lift because of our heavy luggage or for the discovery of neu - tral currents. At the Aix-en-Provence meeting that great and modestman, Lagarrigue, was also present and the atmosphere was that of a carnival — at least this is how it appeared to me. Steve Weinberg gave the rapporteur's talk with T.D. Lee as the chairman. T.D. was kind enough to ask me to comment after Weinberg finished. That Summer Jogesh; Pati and I had predicted proton decay within the context of what is now called grand unification and in the flush of this excitement I am afraid I ignored weak neutral currents as a subject which had already come to a successful conclusion, and concentrated on speaking of the possible decays of the proton. I understand now that proton decay experiments are being planned in the United States by the Brookhaven, Irvine and Michigan and the Wisconsin-Harvard groups and also by a European collaboration to be mounted in the Mont Blanc Tunnel Garage n9 17. The later quantitative work on neutral currents at CERN, Fermilab. , Brookhaven, Argonne and Serpukhov is, of course, history, but a special tribute is warranted to the beautiful SLAC-Yale-CERN experiment of 1978 which exhibited the effective Z -photon interference in accordance with the predictions of the theory. This was foreshadowed by Barkov et al's experiments at Novosibirsk in the USSR in their exploration of parity-violation in the atomic potential for bismuth. There is the apocryphal story about Einstein, who was asked what he sould have thought if experiment had not confirmed the light deflection predicted by him. Einstein is supposed to have said, "Madam, I would have thought the Lord has missed a most marvellous opportunity." I believe, however, that the following quote from Einstein's Herbert Spencer lecture of 1933 expresses his, my colleagues and my own views more accurately. "Pure logical thinking cannot yield us any knowledge of the empirical world ; all knowledge of reality starts from experience and ends in it." This is exactly how I feel about the Gargamelle-SLAC experience.

III. THE PRESENT AND ITS PROBLEMS

Thus far we have reviewed the last twenty years and the emergence of SU(2] x U(l), with the twin developments of a gauge theory of basic interactions, linked with internal symmetries, and of the 322 A. Saiam spontaneous breaking of these symmetries. I shall first summarize the situation as we believe it to exist now and the immediate problems.Then we turn to the future.

1. To the level of energies explored, we believe that the followingsets of particles are "structureless" (in a field-theoretic sense) and, at least to the level of energies explored hitherto, constitute the ele - mentary entities of which all other objects are made.

SUC(3] triplets

IV V l Family I quarks leptons SU(2) doublets

3R' CY' ' Family II quarks leptons

3R, sv, s,B]

IV V Si Family III quarks leptons

|bR,

Together with their antiparticles each family consists of 15 or 16 two- component fermions (15 or 16 depending on whether the neutrino is massless or not). The third family is still conjectural, since the top

quark (t,,, tv, tn] has not yet been discovered. Does this family really follow the pattern of the other two? Are there more families? Does the fact that the families are replicas of each other imply that Nature has discovered a dynamical stability about a system of 15 (or 16) objects , and that by this token there is a more basic layer of structure under- 52) neath?

2. Note that quarks come in three colours; Red (R), Yellow (Y) and Blue (B) . Parallel with the electroweak SU(2) x U(l), a gauge field *)

theory (SUC(3)) of strong (quark) interactions (quantum chromodynamics,

*) "To my mind the most striking feature of theoretical physics in the last thirty-six years is the fact that not a single new theoretical idea of a fundamental natuse has been successful. The notions of relativistic quantum theory....of a great number of talented physicists. We live in a dilapidated house and we seem to be unable to move out. The difference between this house and a prison is hardly noticeable" - Res.Jost C1963) in Praise of Quantum Field Theory (Siena European Conference). GAUGE UNIFICATION 323

QCD) has emerged which gauges the three colours. The indirect discovery of the (eight) gauge bosons associated with QCD (gluons), has 54) already been surmised by the groups at DESY.

3. All known baryons and mesons are singlets of colour SU (3). This has led to a hypothesis that colour is always confined. One of the major unsolved problems of field theory is to determine if QCD - treated non- perturbatively — is capable of confining quarks and gluons.

4. In respect of the electroweak SU(2) x U(l), all known experimentson weak and electromagnetic phenomena below 100 GeV carried out to date agree with the theory which contains one theoretically undetermined parameter sin29 = 0.230 ± 0.009. The predicted values of the asso ciated gauge boson (W~ and Z°) masses are: IIL. = 77-84 GeV, m_ = 89-95 GeV for 0.25 i sin26 5 0.21.

5. Perhaps the most remarkable measurement in electroweak physics is 2 that of the parameter p = Í w KI I . Currently this has been detar- l 2 J mined from the ratio of neutral to charged current cross-sections. The predicted value p = 1 for weak iso-doublet Higgs is to be compared with the experimental p = 1.00 ± 0,02.

6. Why does Nature favour the simplest suggestion in SU(2) x U(l)theory **) of the Higgs scalars being iso-doublet? Is there just one physical Higgs? Of what mass? At present the Higgs interactions with leptons , quarks as well as their self-interactions are non-gauge interactions. For a three-family (6-quark) model, 21 out of the 26 parameters needed, are attributable to the Higgs interactions. Is there a basic principle,

*) The one-loop radiative corrections to p suggest that the maximum mass of leptons contributing to p is less than 100 GeV.

**) To reduce the arbitrariness of the Higgs couplings and to motivate their iso- doublet character, one suggestion is to use supersymmetry. Supersymmetry is a Fermi-

Bose symmetry, so that iso-doublet leptons like (ve>e) or (v ,v) in a sypersymmetric theory must be accompanied in the same multiplet by iso-doublet Higgs. Alternatively, one may identify the Higgs as composite fields associated with bound states of a yet new level of elementary particles and new forces (Dimopoulos & Susskind , Weinberg and 't Hooft) of which, at present low energy, we have no

cognisance and which may manifest themselves in the l-100NTeV range. Unfortunately , both these ideas at first sight appear to introduce complexities, though in the con - 324 A. Saiam as compelling and as economical as the gauge principle, which embraces the Higgs sector? Alternatively, could the Higgs phenomenon itself be a manifestation of a dynamical breakdown of the gauge symmetry.

7. Finally there is the problem of the families; is there a distinct SU(2) for the first, another for the second as well as a third SU(2) , with spontaneous symmetry breaking such that the SU(2) apprehended by present experiment is a diagonal sum of these "family" SU(2)'s? To state this in another way, how far in energy does the e-y universality (for example} extend? Are there more Z°'s than just one, effecti - vely differentially coupled to the e and the y systems? (If there are, this will constitute mini-modifications of the theory, but not a drastic revolution of its basic ideas.) In the next section I turn to a direct extrapolation of the ideas which went into the electroweak unification, so as to include strong interactions as well. Later I shall consider the more drastic alternatives which may be needed for the unification of all forces (including gravity) — ideas which have the promise of providing a deeper understanding of the charge concept. Regreetfully, by the same token, I must also become more technical and obscure for the non-specia. list. I apologize for this. The non-specialist may sample the flavour of the arguments, with the next section (Sec. IV) ignoring the Appendi- ces and then go on to Sec. V which is perhaps less technical.

IV. DIRECT EXTRAPOLATION FROM THE ELECTROWEAK TO THE ELECTRONUCLEAR

4.1 - The Three Ideas

The three ideas which have gone into the electronuclear — also called grand — unification of the electroweak with the strong nuclear force (ar * which date back to the period 1972-1974), are the following:

1. First: the psychological break (for us) of grouping quarks and leptons in the same multiplet of a unifying group G, suggested by 60) Pati and myself in 1972 . The group G must contain SU(2)xU(l)xSUc(3); must be non-Abelian, if all quantum numbers (flavour, colour, lepton ,

**) cont... text of a wider theory, which spans energy scales up to much higher masses, a satisfactory theory of the Higgs phenomena, incorporating these, may well emerge. GAUGE UNIFICATION 325 ,„. quark and family numbers) are to be automatically quantized and the re- tâ suiting gauge theory asymptotically free. ri'

2. <• Second: an extension, proposed by Georgi and Glashow (1974) which places not only (left-handed) quarks and leptons but also their antiparticles in the same multiplet of the unifying group. Appendix I displays some example of the unifying groups presently considered. Now a gauge theory based on a "simple" (or with discrete sym metries, a "semi-simple") group G contains one basic gauge constant This constant would manifest itself physically above the "grand unific- -*' ation mass" M, exceeding all particle masses in the theory — these themselves being generated (if possible) hierarchially through a suit - able spontaneous symmetry-breaking mechanism.

3. The third crucial development was by Georgi, Quinn and Weinberg (.1974) who showed how, using renorraalization group ideas , j/ ;; one could relate the observed low-energy couplings a(y) , as(jj) (u ~ lOOGeV) | to the magnitude of the grand unifying mass M and the observed value of 17,, sin29(p); (tan6 is the ratio of the U(l) to the SU(2) couplings). |.!

*) !' 4. If one extrapolates with Jowett , that nothing essentially : new can possibly be discovered — i.e. one assumes that there are no '<.. new features, no new forces, or no new "types" of particles to be dis - covered, till we go beyond the grand unifying energy M — then the Gaorgi, Quinn, Weinberg method leads to a startling result: this featureless "plateau" with no "new physics" heights to be scaled stretches to fant- astically high energies. More precisely, if sin26(y) is as large as 0.23, then the grand unifying mass M cannot be smaller than 1.3 x 1013GeV 63> . (Compare with Planck mass nip = 1.2 * 1019GeV related

*) The universal urge to extrapolate from what we know to-day and to believe that nothing new can possibly be discovered, is well expressed in the following:

"I come first, My name is Jowett I am the Master of this College, Everything that is, I know it If I dont't, it isn't knowledge" -

The Balliol Masque. 326 A- Saiam *) to Newton's constant where gravity must come in.) The result follows from the formula '

2 z lia ln M = sin 8(M) - sin 6(y) (I) 3lr v cos2e (M)

if it assumed that sin29(M) - the magnitude of sin28 for energies of the order of the unifying mass M - equals 3/8 (see Appendix II). This startling result will be examined more closely in Appendix II. I show there that it is very much a consequence of the assumption that the SU(2) x U(l) symmetry survives intact from the low regime energies y right upto the grand unifying mass M. I will also show that there already is some experimental indication that this assumption is too strong, and that there may be likely peaks of new physics at energies of 10 TeV upwards.

4.2 - Tests of electronuclear grand unification

The most characteristic prediction from the existence of the ELECTRONUCLEAR force is proton decay, first discussed in the context of grand unification at the Aix-en-Provence Conference (1973) . For "semi-simple" unifying groups with multiplets containing quarks and leptons only, (but no antiquarks nor antileptons) the lepto-quark com - posites have masses (determined by renormalization group arguments), of the order of = 105 - 10G GeV . For such theories the characteristic proton decays (proceeding through exchanges of three lepto-quarks) con 29 31 serve quark number + lepton number, i.e. P = qqq •+ III, Tp - IO - 10 * years. On the contrary, for the "simple" unifying family groups like SU(5) 61' or SO(10) 67^ (with multiplets containing antiquarks and anti^ leptons) proton decay proceeds through an exchange of one lepto-quark into an antilepton (plus pions etc.) (P •+ T). An intriguing possibility in this context is that investi - gated recently for the maximal unifying group SU(16) - the largest

*) On account of the relative proximity of M ~ 1013 GeV to m_ Cand the hope of eventual unification with gravity), Planck mass EL is now the accepted "natural" mass scale in Particle Physics. With this large mass as the input, the great unsolved problem ^of Grand Unification is the "natural" emergence of mass hierarchies (mp, amp, a2nip,...) or

nip expC-c^/a), where cn's are constants. GAUGE UNIFICATION 327

group to contain a 16-fold fermionic family (q» $» • Jl" + ir+ + ir+) and P + 3i (e.g. N + 3v + TT°, P -f 2v + e+ + ir°), the relative magnitudes of these alternative decays being mcdel-deDendent on how precisely SU(16) breaks down to SUC3) x SU(2) x UCD• Quite clearly, it is the central fact of the existence of the proton decay for which the present generation of experiments must be designed, rather than for any specific-type of decay modes. Finally, grand unifying theories predict mass relations ,.. 68) like:

m, m m. — - s - p - n a m m m e p T

for 6 (or at most 8} flavours below the unification mass. The Important remark for proton decay and for mass relations of .the above type as well as for an understanding of baryon excess in the Universe ' , Is that for the present these are essentially characteristic of the fact of grand unification - rather than of specific models. "Yet each man kills the thing he loves" sang Oscar Wilde an- guishedly in his famous Ballad of the Reading Goal. Line generations of physicists before us, some in our generation also (through a direct extrapolation of the electroweak gauge methodology to the electronuclear) - and with faith in the assumption of no "new physics", which lead to a grand unifying mass - 1013 GeV - are beginning to believe that the end of the problems of elementarity as well as of fundamental forces is nigh. They may be right, but before we are carried away by this pros - pect, it is perhaps worth stressing that even for the simplest grand unifying model (Georgi and Glashow's SU(5) with just two Higgs (a 5 and

*) the calculation of baryon excess in the Universe - arising from a combination of CP and baryon number violations - has recently been claimed to provide teleological arguments for grand unification. For example, Nanopoulos has suggested that the

"existence of human beings to measure the ratio n_/n, (where nfl is the numbers of baryons and n, the numbers of photons in the Universe) necessarily imposes severe bounds 2 on this quantity: i.e. 10 "~ (tne/m )^

del based on the humble SUC2) x u(l) x SUC(3). We cannot feel proud.

V. ELEMENTARITY.:. UNIFICATION WITH GRAVITY AND NATURE OF CHARGE

In some of the remaining parts of this lecture I shall be questioning two of the notions which have gone into the direct extrapolation of Sec. IV — first, do quarks and leptons represent the correct elementary ' fields, which should appear in the matter Lagran- gian, and which are structureless for renormalizaibility; second, could some of the presently considered gauge fields themselves be composite ? This part of the lecture relies heavily on an address I was privileged to give at the European Physical Society meeting in Geneva in July this year 64).

; 5.1 - The Quest for Elementary, Prequarks (Preons and Pre-Preons)

While the rather large number (15) of elementary fields for the family group SU(5) already makes one feel somewhat uneasy, the number 561, for example, proposed in the context of the three-family tri - bal group SU(ll) or 2048 for SOÍ22) (see Appx. I Cof which presumably 3 x 15 = 45 objects are of low and the rest of Planckian mass) is posi- .;: tively baroque. Is there any basic reason for one's instinctive revulsion when faced with these vast numbers of elementary fields. ;' The numbers by themselves would perhaps not matter so much . .;' After all, Einstein in his description of gravity , chose to work with 10 fields (g (x)) rather than with just one (scalar field) as 72) ** Nordstrom had done before him. Einstein was not perturbed by the multiplicity he chose to introduce, since he relied on the sheet-anchor of a fundamental principle — (the equivalence principle) — which per mitted him to relate the 10 fields for gravity g with the 10 compo - nents of the physically relevant quantity, the tensor T of energy

*) I would like to quote Feynman in a recent interview to the "Omni" magazine: "As long as it looks like the vay things are built with wheels within wheels, then you are looking for the innermost wheel - but it might not be that way, in which case you are looking for whatever the hell it is you find I" In the same interview he remarks "a few years ago I was very sceptical about the gauge theories... I was expecting mist, and now it looks like ridges arid valleys after all." GAUGE UNIFICATION 329 and momentum. Einstein knew that nature was not economical of structures: only of principles of fundamental applicability. The question we must ask ourselves is this: Have we yet discovered such prin ciples in our question for elementary, to justify having fields with such large numbers of components as elementary. Recall that quarks carry at least three charges ( colour , flavour and a family number). Should one not, by now, entertain the notions of quarks (and possibly of leptons) as being composites of some more basic entities *' (PRE-QUARKS or PREONS), which each carry but one basic charge . These ideas have been expressed before but .they have become more compulsive now, with the growing multiplicity of quarks and leptons. Recall that it was similar ideas which led from the eight-fold of baryons to a triplet of (Sakatons and) quarks in the first place. The preon notion is not new. In 1975, among others, Pati , Salam and Strathdee ' introduced 4 chromons (the fourth colour corresponding to the lepton number) and 4 flavours, the basic group being SU(8)

— of which the family group SUp(4) x SUC(4) was but a subgroup. As an extension of these ideas, we now believe these preons carry magnetic charges and are bound together by very strong short-range forces, with quarks and leptons as their magnetically neutral composites . The im portant remark in this context is that in a theory containing both elec- trie and magnetic generalized charges, the analogues of the well-known Dirac quantization condition gives relations like, f^ = § for ttle strength of the two types of charges. Clearly, magnetic monopoles v; (g = 4^"4 - , fr- = -jisi ) of opposite polarity, are likely to bind much more £ tightly than electric charges, yielding composites whose non-elementary % nature will reveal itself only for very high energies. This appears to .;!•£ be the situation at least for leptons if they are composites. ;f-v; In another form the preon idea has been revived this year by :-Á Curtright and Freund 52', who motivated by ideas of extended supergravity ": I (to be discussed in the next subsection), reintroduce an SU(8) of 3 chro- ;• mons (R, Y, B), 2 flavons and 3 familons (horrible names). The family i;| group SU(5) could be a subgroup of this SU(8). In the Curtright-Freund

Cf(| scheme, the 3 * 15 = 45 fermions of SU(5) can be found among the

*1 One muse emphasise however that zero mass neutrinos are the hardest objects to conceive of as composites.

**) According to 't Hooft's theorem, a monopole corresponding to the SUT(2) gauge symmetry is expected to possess a mass with the lower limit — . Even if such mo t-, * nopoles are confined, their indirect effects must manifest themselves, if they exist . ""if (Note that — is very much a lower limit for a grand unified theory like SU(5) for which the monopole mass is a a times the heavy lepto-quarks mass.) 330 A. Saiam 8 + 28 + 56 of SU(8) (or alternatively the 3 x 16 = 48 of SO(1Q) among the vectorial 56 fermions of SU(.8)I. (The next succession after the preon level may be the pre-preon level. It was suggested at the Geneva Conference that with certain developments in field theory of comp - osite fields it could be that just two pre-preons may suffice. But at this stage this is pure speculation). Before I conclude this section, I would like to make a prediction regarding the course of physics in the next decaUe, extrapolating from our past experience of the decades gone by:

DECADE 1950-1960 1960-1970 1970-1980 1980

Discovery'in The strange The 8-fold Confirmation w, z, early part of p.articles way, fi of neutral Proton decay the decade currents

Expectation for SU(3) Grand Unification, the rest of the resonances Tribal Groups decade

Actual Hit the next May hit the preon discovery level of level, and composite elementarity structure of quarks with quarks

5.2 - Post-Planck Physics, Supergravity and Einstein's dreams

I now turn to the problem of a deeper comprehension of the charge concept (the basis of gauging) — which, in my humble view, is the real quest of particle physics. Einstein, in the last thirty-five years of his life lived with two dreams: one was to unite gravity with matter (the photon) — he wished to see the "base wood" (as he put it ) which makes up the stress tensor T on the right-hand side of his equation R - -^g R = -T transmuted through this union, into the "marble" of gravity on the left-hand side. The second (and the complementary) dream was to use this unification to comprehend.the nature of electric charge in terms of space-time geometry in the same manner as he had successfully comprehended the nature of gravitational charge in terms of spacetime curvature. GAUGE UNIFICATION 331 In case some one imagines that such, deeper comprehension is irrelevant to quantitative physics, let me adduce the tests of Einstein's theory versus the proposed modifications to it (Brans-Dicke ' for example). Recently C1976] , the strong equivalence principle (i.e. the proposition that gravitational forces contribute equally to the inertial and the gravitational masses) was tested to one part in 1012 (i.e. to the same accuracy as achieved in particle physics for 781 (g-2)e) through lunar-laser ranging measurements . These measure - merits determined departures from Kepler equilibrium distances, of the moon, the earth and the sun to better than ± 30cms. and triumphantly vindicated Einstein. There have been four major developments in realizing Einstein's dreams: 1) The Kaluza-Klein 791 miracle: An Einstein Lagrangian (scalar curvature) in five-dimensional space-time (where the fifth dimension is compactified in the sense of all fields being explicitly independent of the fifth co-ordinate) precisely reproduces the Einstein- Maxwell theory in four dimensions, the g ^Lv = 0,1,2,3) components of the metric in five dimensions being identified with the Maxwell field . A . From this point of view, Maxwell's field is associated with the í extra components of curvature implied by the (conceptual) existence of

,; the fifth dimension.

\\ 2) The second developement is the recent realization by Cremraer, Scherk, Englert, Brout, Minkowski and others that the compactification of the extra dimensions — (their curling up to sizes perhaps smaller than Planck length < 10~ cms. and the very high curvature

; I associated with them) — might arise through a spontaneous symmetry •' —43 breaking (in the first 10 seconds) which reduced the higher diraen - sional space-time effectively to the four-dimensional that we apprehend directly.

*) The following quotation from Einstein is relevant here. "We new realize, with, special clarity, how much in error are those theorists «ho believe theory comes indue tively from experience. Even the great Newton could not free himself from this error (Hypotheses non fingo)." This quote is complementary to the quotation from Einstein at the end of Sec. II.

**) The weak equivalence principle Cthe proposition that all but the gravitational force contribute equally to the inertial and the gravitational masses) was verified by Eotvos to l:108 and by Dicfce and Braginsky and Fanov to l:10E . 332 A. Saiam ,

3) So far we have considered Einstein's second dream, i.e. the unification of electromagnetism (and presumably of other gauge forces) with gravity, giving a space-time significance to gauge charges as corresponding to extended curvature in extra bosonic dimensions. A full realization of the first dream (unification of spinor matter with gravity and with other qauge fields) had to await the development of supergravity ' 82' — and an extension to extra fermionic dimensions of superspace 83^ (with extended torsion being brought into play in addition to curvature). I discuss this development later.

4) And finally there was the alternative suggestion by Wheeler andd Schemberg that electric charge- may be associated with space-time topology — with worm holes, with space-time Gruyère-cheesiGri - *) ness. This idelea 1has recently been developed by Hawking and his 85) collaborators

5.3 - Extended Supergravity, SU(8) preons and Composite Gauge Fields ;

f Thus far I have reviewed the developments in respect of FC Einstein's dreams as reported at the Stockholm Conference held in 1978 ':•' in this hall and organized by the Swedish Academy of Sciences. A remarkable new development was reported during 1979 by Julia and Cremmer which started with an attempt to use the ideas of Kaluza and Klein to formulate extended supergravity theory in a higher (compactified) space-time — more precisely in eleven dimensions. This development links up, as we shall see, with preons and composite Fermi fields — and even more important — possibly with the notion of composite gauge fields. Recall that simple supergravity is the gauge theory of 88) supersymmetry the gauge particles being the (helicity ±2) gravitons

*) The Einstein Lagrangian allows large fluctuations of metric and topology on Planck-length scale. Hawking has surmised that the dominant contributions to the path integral of quantum gravity come from metrics which carry one unit of topology OfL\ per Planck volume. On account of the intimate connection (de Rham, Atiyah-Singer) of curvature with the measures of space-time topology (Euler number, Pontryagin num - ber) the extended Kaluza-Klein and Wheeler-Hawking points of view may find consonance after all. \ GAUGE UNIFICATION 333

and (helicity + J-) gravitinos ' . Extended supergravity gauges supersymmetry combined with SOCN} internal symmetry. For N = 8,the (tribal) : supergravity multiplet consists of the following SO(8) families: '

Helicity ±2 1 ±§ S ±1 28 I- * Í I' ''•J. 0 70

As is well known, S0(8J is too small to contain SU(2) x u(l) x SUC(3) . Thus this tribe has no place for WT (.though Z° and y are contained) and no places for p or T or the t quark. This was the situation last year. This year, Cremmer and Julia 87} attempted to write down the N = 8 supergravity Lagrangian ex- plicity, using an extension of the Kaluza-Klein ansatz which states that extended supergravity (with SO(8) internal symmetry) has the same Lagrangian in four space-time dimensions as simple supergravity in (compactified) eleven dimensions. This formal — and rather formidable I ansatz — when carried through yielded a most agreeable bonus. The jl supergravity Lagrangian possesses an unsuspected SO(8) "local" internal J~' symmetry although one started with an internal SO(8) only. Í The tantalizing questions which now arise are the following.

I. 1) Could this internal SU(8) be the symmetry group of the i?. 8 preons (3 chromons, 2 flavons, 3 familons) introduced earlier? •? " 2) When SU(8) is gauged, there should be 63 spin-one fields. " > The supergravity tribe contains only 28 spin-one fundamental objects which are not minimally coupled. Are the 63 fields of SU(8) to be 1 identified with composite gauge fields made up of the 70 spin-zero ob- . '•.-. jects of the form V~ 3 V; Do these composites propagate, in analogy ' with the well-known recent result in CPn~ theories , where a com - ^ posite gauge field of this form propagates as a consequence of quantum •; '.! effects (quantum completion) ?

••' Í ip- The entire development I have described — the unsuspected '"• *} Supersyrometry algebra extends Poincare group algebra by adjoining to it super- = T ie •Vj symmetric charges Qg which transform bosons to fermions. {Q , Qfi} ûûâB' * \; '•] currents which correspond to these charges CQ and P ) are J and T - these are '•';-! essentially the currents which in gauged supersymmetry Ci.e. supergravity) couple to -'; the gravitino and the graviton, respectively. 334 A. Saiam extension of SO (.81 to SUC81 when extra compactified space-time diigensions are used — and the possible existence and quantum propagation of composite gauge fields — is of such crucial importance for the future prospects of gauge theories that one begins to wonder how much of the SU 3 extrapolation which took SU(2) x U(l) * OC ) into the electronuclear grand unified theories is likely to remain unaffected by these new ideas now unfolding. But where in all this is the possibility to appeal directly to experiment? For grand unified theories, it was the proton decay What is the analogue for supergravity? Perhaps the spin •=• massive gravitino, picking its mass from a super-Higgs effect 90) provides the answer. Fayet91 1 has shown that for a spontaneously broken globally super symmetric weak theory the introduction of a local gravitational inter - action leads to a super-Higgs effect. Assuming that supersymmetry breakdown is at mass scale nv., the gravitino acquires a mass and an effective interaction, but of conventional weak rather than of the gravitational strength — an enhancement by a factor of 103" . One may thus search for the gravitino among the neutral decay modes of J/i|i -the predicted rate being 10 - 10~ times smaller than the observed rate for J/IJJ •* e e . This will surely tax all the ingenuity of the great 92) men (and women) at SLAC and DESY. Another effect suggested by Scherk is antigravity — a cancellation of the attractive gravitational force with the force produced by spin-one gravi-photons which exist in all extended supergravity theories, Scherk shows that the Compton wave length of the gravi-photon is either smaller than 5 cms. or comprised between 10 and 850 metres in order that no conflict wii-h what is pre - sently known about the strength of the gravitational force. Let me summarize: it is conceivable of course, that there is indeed a grand Plateau — extending even to Planck energies. If so,the only eventual laboratory for particle physics will be the Early Univer- se, where we shall have to seek for the answers to the questions on the nature of charge. There may, however, be indications of a next level of structure around 10 TeV; there are also beautiful ideas (like, for example, of electric and magnetic monopole duality) which may manifest at energies of the order of a~ IIL. (= 10 TeV) . Whether even this level of structure will give us the final clues to the nature of charge, one cannot predict. All I can say is that I am for ever and continually being amazed at the depth revealed at each successive level we explore. I would like to conclude, as I did at the 1978 Stockholm Conference with a prediction which J.R. Oppenheimer made more than twenty-five years ago and which has been fulfilled to-day in a manner he did not live to see. More than anything else, it expresses the faith for the GAUGE UNIFICATION 335 future with which, this greatest of decades in.particle physics ends "Physics will change even more ... If it is radical and unfamiliar ... we think7"that' the future" will be only more radical and^not'less, only more strange and not more familiar, and that it will have its own new insights for the inquiring human spirit.". J.R. Oppenheimer, Reith Lectures BBC 1953).

APPENDIX I

EXAMPLES OF GRAND UNIFYING GROUPS

Semi-simple groups Multiplet Jxotic gauge particles Proton decay (with left-right GR l Lepto-quarks -*• (qS.) jepto-quarks •* W symmetry) MV [k +(Higgs) or Jnifying mass = Example [SU(6)F x G G- x G_ = 106 MeV Proton=qqq * ^Í

Simple groups Diquarks * (qq; qq + qi i.e. q Dileptons •* (IV) Examples l Lepto-quarks (q£) , (qJ.)Proton P=qqq * £ Family groups*ÍSU(5) or (S0(10) G * Also possible, q Unifying mass Tribal groups*|SU(10 or (SO(22) = 1013- 1015 GeVP •+ t,P * 31, I P •* 3?. L

*) Grouping quarks (q) and leptons (Í.) together, implies treating lepton number ar 93) the fourth colour, i.e. SU (3) extends to SUc(4) (Pati and Salam) A Tribal group, by definition, contains all known families in its basic representation. Favoured representations of Tribal SU(ll) (Georgi) and Tribal S0(22) ( Gell - 9 5} Mann etal.) contain 561 and 2048 fermions! 336 A. Saiam APPENDIX II

The following assumptions went into the derivation of the formula CD in the text. a) SUT(2) x OT -Cl) survives intact as the electroweak symmetry group from energies = p right upto M. This intact survival implies that one eschews, for example all suggestions that i) low-energy SUL(2) may be the diagonal sum of SO? (2), SU*X(2}, SV*11(2) , where I, II, III Jj Li J_I refer to the (three?l known families; ii) or that the UL R(l) is a sum of pieces, where UR(1) may have differentially descended from a (V + A) -symmetric SU_(2) contained in G, or iii) that U(l) contains a piece from a four-colour symmetry SUC(4) (with lepton number as the fourth colour) and with SUC(4) breaking at an intermediate mass scale to

SUQC3) x uc(l). b) The second assumption which goes into the derivation of the formula above is that there are no unexpected heavy fundamental fermions, 2 which might make sin S(M) differ from -5 — its value for the low mass *) fermions presently known to exist. c) If these assumptions are relaxed, for example, for the three 2 family group G = [SUF(6) x SUGC6QL + R, where sin e(M) = ^ , we find the grand unifying mass M tumbles down to 106GeV. d) The introduction of intermediate mass scales Cfor example , those connoting the breakdown of family universality, or of left-right symmetry, or of a breakdown of 4-colour SUC(4) down to SUC(3) x UC(D)

*) If one does not know G, one way to infer the parameter sin28(M) is from the formula:

2 ZT f 9 N + 3 Nn sinz6(M) ZQ2 [ 20 N + 12 N

Here N and No are the numbers of the fundamental quark and lepton SU(2) doublets q X. (assuming these are the only multiplets that exist). If we make the further assumption that N = N. (from the requirement of anomaly cancellation between quarks and leptons) we obtain sinz9(M) = -^ . This assumption however is not compulsive; for QQ\ example anomalies cancel also if (heavy) mirror fermions exist . This is the case

for [|U(6)14 for which sin29CM) = — . 28 GAUGE UNIFICATION 337 will as a rule push the magnitude of the grand unifying jnass M upwards. 96) • In order to secure a proton decay life, consonant with present em 97) pirical lower limits (-1030 years} this is desirable anyway. (T . for M - 1013GeV is unacceptably low - 6 x 1023 years unless there are 15 Higgs.) There is from this point of view, an indication of there being in Particle Physics one or several intermediate mass scales which can be shown to start from around lO^GeV upwards. This is the end result which I wished this Appendix to lead upto.

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í \j 345

GEOMETRIC FORMULATION OF SUPERGRAVITY *

Samuel W.MacDowell Yale University, New Haven, Ct. 06520

I. INTRODUCTION

A basic problem in constructing a field theory of elementary particles is to find out what are the underlying symmetries of the interactions of the fundamental fields.

A symmetry is an invariance of a physical law, under a group of transformations of the fields. This group is called the symmetry group.

A classical theory of fields, in general, is a theory in which a group of symmetries is realized on a set of functions, called fields, defined on a four dimensional space-time manifold. The time evolution of these fields is governed by differential equations, invariant under transformations of the symmetry group. These equations are derived from an action principle, where the action is a functional of the fields, defined in terms of a local Lagrangian density which is invariant under the group of transformations.

These symmetry transformations are of two kinds: i) space-time symmetries in which the transformations correspond to a change of the space-time coordinates,

ii) internal symmetries involving degrees of freedom other than the space-time coordinates.

If the parameters of the transformations of a internal symmetry group are space-time dependent, the theory is said to have a local gauge invariance and the symmetry group is called a gauge group.

•Research supported in part by the U.S. Department of Energy under Contract No. EY-76-C-02-3075. 346 S. MacDowell

The lnvariance of Maxwell's theory of Eiectronar.netism under Lorentz transformations led Einstein to formulate the theory of Special Relativity. Electrodynanic3 is also invariant under a local Ui group of gauge transformations.

A gauge invariant theory with a non-abelian SU2 gauge group was first formulated by C. N. Yang and R. L. Mills .

The principle of Equivalence of local frames of reference was the basis for the development of General Relativity in which gravitation is described by a symmetric tensor g , the metric tensor; the field equations are invariant under general

coordinate transformations. The group of space-time symmetries

is the Einstein group of general coordinate transformations with

non-vanishing Jacobian. A formulation of General Relativity as

a gauge theory with the Lorentz group as gauge group was first

given by Utiyama .

In Special Relativity the group of space-time symmetries

is the Poincaré group, which is the semi-direct product of the

Lorentz group S£(2C) and a four dimensional Abelian group of

translations. It can be obtained from the de Sitter groups

30(4,1) and SO(3,2) - Sp(4) by a Wigner-Inonii contraction3.

The total symmetry group is the direct product of the

Poincaré" group and of the internal symmetry group. The fields

are then classified according to irreducible representations

of this group. Fields that belong to a tensor representation of

Sl{2C) (that Is,a representation of S0(3,l)) are called bosonic;

those belonging to a spinor representation are called fermionlc.

An irreducible representation of the Poincaré group contains

either boson or fermlon fields.

Supersymmetry is a generalization of the concept of symmetry

which allows for an invariance under transformations in which SUPERGRAVITY 347 I) boson and fermion fields transform into each other. The generators of infinitesimal supersyir-ietry transfcrmaticns and the corresponding infinitesimal parameters belong to a splnc-r representation of the Lorentz group. The algebra of the generators is a

L with the superalgebra, a Z2-graded alcebra L = i-zvift « odd' following Lie products:

The fermionic generators of supersynmetry transformations r belong to L ,,. In the supersyrametric model of Wess and Zunlno the Lie product of two' supersymmetry generators is a translation. Therefore, in this model the Poincare* group is embedded in a supergroup. If a theory is invariant under local supersymmetry transformations, and the algebra of generators is some generalization of the Wess-Zunino algebra, then it is also invariant under local infiniteSinal translations, that is, under general coordinate transformations. Because such a theory would incorporate general relativity, it is called Supergravity. If, in addition to the local supersymmetry, it has local f.auce invariance under an internal symmetry group, i.t is called Extended Supergravlty. A Quantum Field Theory is obtained from the classical theory by r.eans of canonical quantization or by Peynman's path integral nethod. In the canonical quantization, the classical fields become operator-valued distributions and Poisson brackets are replaced by commutators for two-boson or one-boson one-fermion fields, and anti-commutators for two fermions. Equivalently, in the path Integral method 348 s- MacDowell boson fields are treated as comnutinc functions whereas

fernion fields are treated as anti-commutinc ones. The commuting or anticommutinp; property of the fields determines the statistics, namely, Bose-Einstein for bosons, Permi-Uirac

for fernions. Physical states are interpreted as rays in a complex

KUbert space and scalar products as probability anplitudes.

The symmetries of the classical theory are carried over to the

quantum theory, but they may be spontaneously broken if the

state of lowest energy, the vacuum, is not invariant under

transformations of the symmetry croup. Spontaneous symmetry

breaking; is expected to occur as an essential feature of

unified gauge theories.

In order to preserve the correct statistics for forrcions

and bosons, the fermionic parameters of a supersyrr.r.etry

transformation must be anti-comr.utin£ numbers (Grassmann numbers)

or antl-comnutin£ functions.

The probabilistic interpretation of scalar products in the

quantum theory requires that the fields transform as a module

for a unitary representation of the internal sy rase try p;roup.

Hence, if the number of fields is finite, the internal symmetry

croup is compact, since only compact groups have finite dimension

unitary representations.

These lecture notes are organised as follows:

Section II is a survey of the main steps in the development

of a field theory of elementary particles in which the internal

symmetries are incorporated as local cauge invariances.

In Section III we discuss some of the attempts towards a

unification of the elementary interactions within the framework

of General Relativity, and give a brief review of Supercravity

and Extended Supergravity. SDPERGRAVITY 349

The main scope of these lectures is to present a formulation of supergravity and extended supergravity in which both the pauge invariance and the invariance under General coordinate transformations are Riven a unified geometrical interpretation.

The appropriate mathematical formalism for this purpose is the theory of Fibre Bundles which is described in Section IV and applied to Einstein's theory of gravitation in Section V.

In Section VI the geometry of fibre bundles is generalised in order to accommodate the new ir.variance under supersymmetry transformations. This is done by considering fibre bundles over graded manifolds.

In Section VII we show how one can actually construct simple supergravity as a realisation of the symmetries of a bundle of frames, on the space of functions consisting; of the vector

components of the connection and "vielbein".

In Section VIII this method is applied to the construction of extended supergravity, taking as its starting point a hypothesis of minimal coupling. Complete results are given for models with SO2 and SO3 internal symmetry.

Finally , in the last section, we make some remarks on the prospects for a realistic description of the known interactions of the elementary particles within the context of supergravity.

II. UNIFIED GAUGE THEORIES IM SPECIAL RELATIVITY

A difficulty that arises in the quantization of a classical

field theory is the appearance of divergent integrals giving

rise to infinities. In order for the theory to make sense,

these infinities must be removed in a way that preserves the

symmetry. The process by which the infinities are removed, by 350 S. MacDowell r.eans of a redefinition of physical parameters, is called

Renormalization. A theory for which this process can be carried

through is called renormalizable. Renormalizability has been

a powerful guiding principle in the selection of models for the

description of elementary interactions.

The foremost example of a renormalizable theory is Quantum

Electrodynamics. Here, renormalizability depends crucially on

the local gauge invariance, together with the hypothesis of

minimal coupling.

On the other hand, in quantizing general relativity, one

finds that the power counting criterion for the convergence of

integrals in the perturbation expansion of the S-matrix is not

satisfied. Therefore, the renormalizability of this theory

depends on the cancellation of infinities in every order of the

perturbation expansion. It has been shown that, for the self-

interaction of the gravitational field alone, such cancellation

occurs in terms corresponding to Feynman diagrams with one

closed loop. For two or more loop diagrams, one has not been

able tr> show that all divergences cancel.

In view of this obstacle with the quantization of general

relativity, the field theory of Elementary Particles has developed

within the framework Of Special Relativity. This is justified

since, at the level of processes involving elementary particles

at energies as produced in the laboratory, the gravitational

interaction does not play any significant role and can be

completely ignored.

The lntractions known to date can be classified into four

categories: 1. Electromagnetlsm 2. V/eak interactions SUPERGRAVITY 351

3, Strong interactions

4. Gravitation

Of these, the Electromagnetic and Gravitational interactions are the best understood.

They are lone-range interactions mediated by massless spin-1 and spin-2 fields respectively.

The source of the Electromagnetic field is a conserved vector current, that of the gravitational field is a conserved energy momentum tensor density. In contrast, the Weak and Strong interactions are of very short range and the fields that mediate these interactions are not well known.

The experimental evidence on the weak interactions shows

that they give rise to current - current couplings of vector and axial-vector currents. In the first model of the weak interactions proposed by 6

Fermi , the interaction Lagrangian was taken as a sum of products

of vector and axial-vector currents. This model was later extended

to a universal theory of the weak interactions when it was realized

that the strongly interacting part of the strangeness preserving

vector current coincides with the isotopic spin current, which is 7 conserved by the strong interactions. As subsequently modified o by Cabibbo , this theory accounts for many of the experimental facts about the weak interactions.

However, a serious problem with this theory is that it is

not renormalizable; thus, it should be considered as a phenomeno-

logical model. The local gauge invariance of Electrodynamics,

together with the minimal coupling hypothesis, are essential

features for its renormallzability. Since in the weak inter-

actions we also have couplings of vector (and axial-vector),

currents, a natural solution to the problem of renonnalizability

would be to assume, in analogy with Electrodynamics, that 352 S. MacDowell the weak interactions are also gauge invariant and mediated by vector bosons. However, since this interaction has very short range, the intermediate vector bosons have to be very massive.

As already mentioned, one can have a gauge theory with a non-abelian compact symmetry group and a set of spin-one gauge fields belonging to the adjoint representation of the symmetry group. If the symmetry were unbroken, as in Electromagnetism, all of these gauge fields would have to be massless. If, on the other hand, the symmetry were spontaneously broken, then according 9 to a theorem due to Goldstone .there should exist massless particles called the Goldstone bosons. The absence of such

Goldstone bosons was a serious obstacle in interpreting the weak interactions as a non-abelian gauge theory with spontaneous

symmetry breaking. This difficulty was resolved by Higgs who

showed that Goldstone bosons do not emerge in a gauge theory if

the symmetry is spontaneously broken because, in the vacuum state,

certain scalar fields have non-zero expectation value. This

process of symmetry breaking in gauge theories is known as the

Kiggs mechanism.

The discovery of the Higgs mechanism paved the way for a

unified formulation of the Electromagnetic and Weak interactions

as described by a non-abelian gauge theory. This was done by 11 12 S. Weinberg and Abdus Salam who proposed such a theory with

a gauge group SU2 8 Ui. A large array of experimental tests lends support to this theory. The success of this unification

has encouraged physicists to extend the concept of gauge invariance

to the strong interactions. It should be pointed out that if

the SU2 8 U, theory is restricted to the interactions of leptons, 13 one finds that it has non-vanishing Adler-Jackiw-Bell anomalies, connected with the coupling of the gauge bosons to axial currents, SÜPERGRAVITY 353

Which preclude renormalization. It is only when the Quarks (which are thought to be the building blocks of the strongly interacting particles) are incorporated into the theory that these anomalies cancel and the theory becomes renornalizable. >:

In the quark theory of strong interactions one postulates ? an exact SU3-symmetry (called SUa of color). If again we follow r the analogy and assume that this SU3-group is a group of local ;- gauge invariances, then we have the corresponding gauge fields is, which are called Gluons, as the vector mesons that mediate the ' , strong Interactions. This theory of strong interactions is $r called Quantum-Chromodynamics (Q.C.D.). It has been rather '* ". successful in the understanding of the physics of the newly *-• discovered particles. Moreover, an important feature of a |..- non-abellan gauge theory of the strong interactions is asymptotic m. freedom.which gives a natural explanation of the scaling laws % of form factors at large momentum transfers (Bjorken scaling). -if

Therefore, present experimental evidence favors a description f( of the Electromagnetic, Weak and Strong interactions in terms ^~ of a gauge theory with gauge group SU3 S SU2 8 Ui,with SUj • Uj j'- spontaneously broken down to the electromagnetic U, gauge symmetry. Since this group ls not simple, such a theory has (at least) three independent parameters, the strong coupling constant, the electric charge and the weak interaction angle (the Weinberg angle). A unification of all the interactions would require a sinple group as a gauge group. The group SUS is the smallest sir.ple group which has SU3 8 SU2 8 Uj as a subgroup. A grand unified theory with SUS gauge group has been proposed by Glashow and Georgi. The known leptons and the quarks required to account for the symmetries of the strong interactions fit, so far, within three copies of two irreducible representations of SUS 354 s- HacDowell of dimensions 5 and 10 resnectively. If one embeds the group

SUS into an S0,0 gauge group, these representations (together with a singlet) belong to an irreducible spinor representation

of SOl0 of dimension 16.

Other unification schemes have also been proposed; in 16 particular, a model suggested by F. Gürsey et ai. uses the

exceptional group E6 as gauge group. These unified theories are supposed to account for all the interactions of elementary

particles with the exception of gravitation. In the next section

we review unified theories which build upon General Relativity.

They are based on Einstein's idea of extending the symmetry

group so as to obtain all the interactions of the physical space

as manifestations of its symmetries.

III. UNIFIED THEORIES IN GENERAL RELATIVITY

After the discovery of General Relativity and the first

few experimental astronomical observations confirming its

predictions (bending of light rays, advance of perihilion of

Mercury, red shift), Einstein and nany co-workers in the field

devoted much of their scientific research effort to extending

the geometrical self-coupling theory so as to Incorporate other

interactions. Particularly sought after was a unified description

of the electromagnetic and gravitational fields in geometrical

terms. 17 In the conformally invariant .theory proposed by Weyl ,

an invariance under local dilations was assumed, which was

appropriately called gauge invariance. The connection associated

with this one dimensional group of transformations was identified

with the electromagnetic gauge potential. As it turns out, the SÜPERGRAVITY 355 quantum theory of Electrodynamics is invariant under a phase transformation or a local Ui group of transformations, not under the non-compact group of dilatations. However, in spite of its inadequacy, the name of gauge invariance continued to be used to describe the invariance under a compact group of local transformations and the symmetry group has since been called a Gauge group. 18 Another interesting proposal was made by Kaluza .

He considered a five-dimensional Riemannian space with the

de Sitter group SO(i|,l) as the group of local invariances.

He postulates that the metric n:ijc(i,h=l.. .5) has cylindrical symmetry, that is, there exists a coordinate system on which

the metric is independent of the fifth cOUl'Uitjate. Kaluza also

ir.poses the constraint ri =1. These conditions are preserved by

a transformation

with g = n - n t-nV(- left invariant. Also one finds that

det|nlk| = det|g I is invariant under this transformation. The metric g is identified with Einstein's metric. The

transformation (1), which has the form of a gauge transformation,

suggests the interpretation of n ^ as a gauge potential. Further

more, calculating the scalar curvature corresponding to the

metric ilk, one finds that it coincides with the Lagrangian density of the Einstein-Maxwell generally covariant theory.

Kaluza's theory, albeit ingenious, does not, as pointed 19 out by Pauli , solve the problem of unification. First one can criticize the ad hoc assumption of cylindrical symmetry.

Secondly, it does not address itself to the more general problem

of unification in the sense of incorporating the sources of 356 S. MacDowell the electromagnetic and gravitational fields.

The fact that matter is basically described by fermion

fields raises some general questions concerning unification.

First, in order to incorporate these fields into a theory of

general relativity, it is essential to use the "vierbein"

formalism. This implies that the metric tensor is no longer a

fundamental field. Instead we have g = h h^,, with the

1 "vierbein" h as the fundamental field, and g.1 the Minkowski metric.

Furthermore, in order to preserve the local Lorentz

invariance of the theory, the condition of zero torsion has

to be abandoned: the fermions appear as a source of torsion.

Thus one is led to consider non-Riemannian geometries. Although

it might be possible to include in the theory other bosons

beside the spin-2 field g , simply by enlarging the group

of symmetries, a true unification with the fermions cannot be

achieved in this way.

The new concept, that at least in principle allows for

the inclusion of fermions in a unified manner, is the concept

of supersymmetry, in particular local supersymmetry

A theory with local supersymmetry was first formulated 20 by D. Freedman, P. van Nieuwenhuizen and S. Ferrara . It is a locally supersymmetric theory of a spin-2 field, the

"vierbein" h and a Majorana spin-â field ha. A more concise

formulation using the first order formalism was given by Deser

and Zumino . The Lagrangian is simply the sum of the Einstein

Lagrangian and the generally invariant form of the Rarita-

Schwinger Lagrangian for a Majorana spin-|- field with the

ordinary derivative 3 replaced by the covariant derivative

+ o w erc s 3U h [iJ]* ^ u * the Lorentz connection and SUPERGRAVITY 357 °rin tne Dlrao~a matrix in the Najorana representation. A first attempt at a geometric interpretation was given 22 by P. Mansouri and myself using a four-dimensional space- tine manifold and the supergroup OSp(l|4) as gauge group.

The action integral is of the form: S where: k 1J] a b Cij:i -EU] _ Ml + h AhV + h *h f R " R Einstein + n h 1W + h " ^6

Hence, when h = 0, S reduces to the Einstein action with a cosnological term plus the Eula?-Poincaré invariant of general relativity.

The sane approach was used by P. van Nieuwenhuizen and 23 P. Townsend to construct SO2 supergravity. The action turns out to have the same form as before, plus a term corresponding to the generally covariant Maxwell action. Extended supergravity oil PS has been constructed with internal symmetry groups S02 , SO, , SO»26 and SO,27. 28 It has been shown that a linear representation of a supersynmetry algebra with SO., internal symmetry, which does not contain representations of the Lorentz group with spin higher than two, exists only for Ii £ 8. Therefore, in this

sense, the SO8 theory is the most general supergravity theory that can be constructed having an orthogonal group as the internal

synmetry group. This would not be large enough to construct a realistic grand unified theory of all interactions. One would need at least an SOl0 internal symmetry group. Remarkably, 27 the SOg-theory constructed by Julia and Crenmer has an additional local SU8-symmetry; however, the gauge potentials 358 S. MacDowell associated with this symnetry are not elementary but composite _-

; fields made out of the spin-0 fields in the model. The SUa " group is large enough to contain the known symmetries of the strong, weak and electromagnetic interactions, but In this model the gluons and intermediate vector bosons would not be elementary vector fields.

In the latter part of these lectures I shall discuss the. geometric structure and give an alcebraic.construction of these theories, based on what is generally referred to as the superspaee 29 formalism. This formalism has been used by several authors with a variety of objectives and techniques.

Although I shall be concerned only with the construction of the classical theory, I would like to mention that an early and major motivation for developing supergravity was the expectation that a consistent quantum theory could also be formulated.

In general, the existence of a symmetry is responsible for elimination of divergences. Thus, electrodynamics and the Yang-

Mills theories are renormalizable because of gauge invariance.

Also it has been noticed that certain cancellations of divergences do occur'in supersymmetric theories.

In quantizing supergravity one finds the same kind of difficulties already encountered in the quantization of general relativity. However, it has now been shown that, in simple and 30 • j. SOí supergravity , one and two loop diagrams are finite.

In order to investigate the convergence of diagrams with more loops, new and more powerful techniques using superfields are being developed. SUPERGRAVITY 359 31 IV. THE FORMALISM OF FIBRE BUNDLES A common mathematical framework for a possible grand unification, which would encompass gravitation, is the theory of fibre bundles . Within this framework one can give a geonetrical formulation of gauge theories and, also, one can cast general relativity into the form of a gauge theory, with the Lorentz -group as gauge group. A Principal Fibre Bundle is a set (P,M,G) with the following structure: 1. P is a topological apace called the bundle space. 2. M is a manifold called the base. 3. G is a Lie group called the structure group or gauge group, which acts (freely and differentlably) to the left on P: GxP+P. 4. M is the quotient space of P by equivalence under G, and the projection II: P+M is c". 5. P is locally trivial, that is, for meH there is a neighborhood 17^ of m and a diffeomorphism l H~ (U1)*U1fiG given by p+(n(p),*1(p)), for

pen" (U^), where ^ is a C™ map: H~ (U1)+G such that 4*(SP) • g^Cp). for any gcG.

6. Let us denote by $., m the map $, at 11** (m) where

If meU,nu, the map , o4$ ~ :G*G is an element g(m) 1 J J J™ I,m of GjC* in U.OU.. For meM, n" (m) is called the fibre over m. Let F be a manifold ana let there be a right action of F: fS~leF for all geG and feF. An associated bundle to the principal bundle (P,M,G) with 360 S. MacDowell standard fibre P Is the quotient space of P8F by equivalence under G. Let {U.} be a complete set of coordinate neighborhoods in M. Por each U. one defines a coordinate basis as a set of vector fields {3 } satisfying the conditions:

C3y,av] «o (2) and such that they form a basis for the tangent space M of M at every The dual of the basis {3 } is the set of one-forms {dxw} such that by contraction dxu(3 ) * 6**. Let us now consider the tangent space at a point p in the bundle. Displacements along the fibre passing through p are generated by tangent vectors called vertical. Their projections onto the base manifold vanish. One can choose a complete set of vertical vectors {P. }, such that, under confutation, they form an algebra lsomorphic to the Lie algebra of G:

where {f. ') is a set of structure constants for the Lie algebra of G. The set {V. } is also a module for the adjoint representation of G. Now in order to obtain a complete basis for the tangent space at p, vie have to specify a complete set of horizontal vectors which generate displacements in the tangent space orthogonal to the fibre. The horizontal lift of a vector 3 in a coordinate basis is a vector SOPERGRAVITY 361 whose projection onto the base manifold is 3 , and which is "equivariant", that is, it commutes with the vertical vectors

This Implies that the functions h '(p) transform under G as the components of a vector in the adjoint representation of the Lie algebra of G. A. A, C. °B h,, (P} " fB C \

<6> ao W o « ^ The set of vectors V form a basis for the horizontal tangent space at p. The functions h (p) are called connection coefficients. The set of one-forms which is dual to the basis {p. ,V. } A A is {« °,dxw} where w a is a vertical one-form called the one-form w of the connection.. Iff {iJ{UA °,dx°,dxu} iss the dual of the basis '. ,3 } then, from the condition: A, U

it follows that:

A, A ,+ n A rp A, A, u • (1 u dx = B + h (8)

The curvature R is a horizontal two-form taking values in Aa the algebra {X. } of G, that is, R-R X. where: n* " 0

A « o J Ao l Bo COj, A, X «dill - r 01 AID fn . (9)

Then, since u is vertical, it follows that:

Therefore, since [P ,f ] is vertical, we have: 362 5. MacDçrçell

Hence,

The bundle of bases B(M) of a manifold M of dimension d, with structure group G£(d,R),is a principal bundle defined by

specifying a point pelT'CU) as p = {ra^.. .ed},where meU and

{eA } is a basis for Mm. The projection is n(p) • m, and the 1 B B action of the group is given by gp = {m,g JeR . ..g^e,, }, where fleG-£(d,R) is represented by the d-dimenslonal non-singular ~ matrix (g *). Ai Let {e,,. } be a d-dimensional set of independent vector I- i I * fields in U. Then we have e. = g.1 el (m) where (g.1) is a %• non-singular matrix and therefore represents an element.of J Bi ;' Gi(d.R). The map * can be defined by

(e' (m).e' Cm)) - gA - (13) *i °i AiBi where g. n is a constant non-singular matrix. Aiai A subspace of B(M), which is obtained when the matrices Bi ga are restricted to belong to a subgroup H of &£(d,R), is _ "i also a principal bundle with structure group H. In particular, if H is a subgroup of G£(d,R) which preserves the metric

g. B , that is: AiBi

for all gell, then the bundle B(H,H) is called a bundle of frames. The relevance of bundle of frames for physics lies in the SUPERGRAVITY 363 fact that in a bundle of frames the parallel transport of vectors along a path in the base manifold preserves the scalar products. Let Y(S) be a continuous, piecewise smooth curve in M,

parametrized by s and passing through a point mQ=Y(0). The

horizontal lift 7(s) of Y(S) through some point p0 in the fibre over m,, is a curve such that n(7(s)) = y(s) and a vector tangent to 7(s) at arW °F its points is horizontal. Now, in a bundle of bases, a point in ^(s) is given by p = ^(s),e,(s)...e (s)}.

l If t(0) = ta e. (0) is a tangent vector in M,, the parallel 1 A transport of t(0) along Y(S) 1S the vector t(s) = t e. (s).

The scalar product of two vectors tt and t2 parallel transported along -y(s) is given by:

l l (t1(s),t2(s)) = t^ t^ (eAi(s),eBi(s))

A, Bl. C1, D1, t, t2 gA (s)gB (s)gc D (15)

Thus, on account of (I1*), it follows that in a bundle of frames the scalar product is preserved. Aq Ai Let (Í2 ,h ) be the set of one-forms dual to the basis f« »ea )• The horizontal one-form h = {h '}, called the solder form, is completely specified by the choice of vector fields

Indeed, let e^ be given by:

Í i A i where (h Mm)) considered as a matrix is non-singular. Then 364 s• MacOowe11

The functions h l(p) are called "vielbeins". Since every group element g.1 has an inverse and (h (m)) is non-singular, then h l(p) has an inverse which vie denote by hJJ (p).

The metric tensor in M is defined by:

Al Bl g_= gA B h (m)-h (m) (18)

or, in a coordinate basis:

u U The scalar product of two vectors ti = t?3 » t2 = t2d

is then given by:

<*i.t«> " 8^*2 Í20)

We notice that in a bundle of frames, because of (1M),

we also have

A B £ - EAiBih i(p)-h i(p) , (n(P) = m) (21)

Let {X. } be a basis for the matrix representation of the

Lie algebra of H, acting as a group of transformations on the

vector basis {e. }. We introduce the connection one-form A * u=u °X in B(M,H) and define the covariant basis [V. ,f>. } Ro Ao Ai in the tangent space to the bundle B as the dual of the set of A. A. one-forms tw ,h }. Then one can easily verify that the

horizontal lift V of the vector 3 of a coordinate basis is given by:

so that SUPERGRAVITY 365 J J J (23)

The torsion T Is defined as the set of horizontal two forms ÍT *} given by:

Al A Bl dh - «o °Ah fA £\ (24) Q 1 where f. £l - (X. )°1.. A,B, A, B, -V. Since u B is vertical, the components T * * T ^n'^v^ of Ai T are readily obtained:

- »„*;• -

We shall consider, in particular, bundles of frames

honeomorphic to a group space G, with structure group a sub- 32,33 group H of G and base manifold homeomorphic to G/H . Let ÍX.} be the set of generators of the Lie algebra L A t of G in the adjoint representation. We have:

[xA,xB] = -fAg xc (26)

where f^ = UA)£. Also ÍXA> = {XA } 9 ÍXA }, where {XA }

Is a set of generators for the Lie algebra Lo of H and ÍX,.}

Is a basis for the vector space L, = L/L9. The Killing metric of L Is defined by:

f f "AB " - A? BS (27)

^We follow here the notation of reference 33 with some changes in convention which are listed at the end of Appendix A. 366 S. MacDowe11 with the following property: r

rA£nDC - fCAnBD " ° (28)

We Introduce a one-form $, taking values in t, defined by:

4 - u + h (29) where u • u °X. Is the connection one-form and h » h *X. *0 "I Is the solder form. The set of one-forms ( } is the dual of the covariant basis (P.). A 3«l

It is convenient to introduce the two-form . :

A F - F XA - d* (30) taking values In I, whose components in the covariant basis are given by: v

c • c where PAB = F (1>A,V^). Thus:

F P f P R F A,B, 'A.B.^A.BL " BtAt A,Bl« AlBl AlBl» AlB1

The Bianchi identities, which result from the associativity of the differential operators V., are equivalent to :

dF - 0 (32) which, in component form in the covariant basis, gives:

D (ABC) - I [OAPBD • P^g] - C (33) (ABC) where the sum is over cyclic permutations of the indices. In the standard formulation of gauge theories, the physical SUPERGRAVITY 367 fields are defined on a cross section of the bundle. A cross section of a coordinate neighborhood U of M is a c", one-to-one map x* U • P, such that n(x(m)) • m for all meU.

A cross section through a point p(eP with It(p() * n>,eU, f: can be defined by the set of first order differential equations: js

p 1 ft •• flp *(x)dx (3 !) • r r where the functions n (x) are C in U (and satisfy the required t:v integrabllity conditions), and the Initial data is x(m9) " pt. |, Then,, on a cross section, we have: If

p w u td ' - Q ' +.hj*dx - (tlj° + hp°)dx - Vu°dx (35) y. A. which defines the vector potential V (x) on the cross section x of the bundle. Notice in particular that, in the trivial cross section x(m) • m, V ° and h ° coincide. A gauge transformation corresponds to a change of cross fr section. In an infinitesimal gauge transformation we have: ':

B0C,

The derivation of this result is given in Appendix B. We shall here follow a somewhat different approach. We shall consider a class of equivalent bundles of frames 8 • {B(M,H)} over a common base manifold M (covered by the same set of coordinate neighborhoods) and with the same structure group H. Two bundles B and D1 in this class have connections 368 s« MacDowell and "yielbelns" related by:

fA A h (p)U'A(p) = h (p)UA(p) (37) where

BB CC J(p) - exp{XX (p)l>(p)l>(B(P)} l?l?(pA(p) expi-AexpiA ((p)Pc(p)} (38) with XB(p) satisfying the condition:

P. XB(p) - f. I XC(p) (39)

These relations may be considered as transformations of the connections and the "vlelbelns", with the coordinate basis Í3 } and the vertical basis ÍI>A } left unchanged. ^ In an Infinitesimal transformation with paramaters e j. one obtains:

A B «hj - 3ue + hJe FB; (40) and

These transformations close under commutation, [6i,623 'i» with the composition law :

Transformations with X '(p) " 0 are equivalent to gauge transformations. Given a cross, section x of the bundle B(M,H)e8, there exists a bundle B'(M,H) such that h|A»(m)

The composition law (42) has been incorrectly given in references 33 and 34 where the first two terms were omitted. SOTljRGRAVITY 369 and h'Al(m) coincide respectively with ^'(xtm)) and hAl(x(m)). Moreover, h1 (p) and h (p) are related by a transformation (37-39) F! with XBl(p) - 0. L On the other hand, transformations with parameters

XA(p) - Xu(m)hA(p) , m - lt(p) are equivalent to coordinate transformations In a single bundle. In a general coordinate transformation {{xw} + lx'v • fM(x)} '*•'- we have h|A(x«) - hA(x), P'A(x') - PA(x), or:

dxuhj(x) (43a) L and

F^(x') - P*c(x) (43b)

For an infinitesimal transformation x|U • xv + cu(x) we have:

A v b» .(x»> - hj(x) - -OuE (x))h*(x) (44) Then: A v v A h'J(x) - h (x) - -[(3ue (x).hJ(x) + c (x)3vh (x)] (45)

Similarly:

PBC(x) -

Now an infinitesimal transformation in the set of equivalent A V bundles (B(M,H)} with parameters e (p) - -E (m)hv(p), m - n(p), gives: 370 S. Macoowell and i\. >h F (18) fiFBC E X BÍ; " "e 3vFBC i

Therefore: 6hJ- h'J(x) - hJU)

which establishes the correspondence between infinitesimal general coordinate transformations in a single bundle and infinitesimal transformations within the set 8.

A choice of gauge in our formulation corresponds to the' selection of a subclass of 8 whose elements differ by transformations (37-39) with parameters XA(p) - Xv(m)h*(p).

V. FIBRE BUNDLE FORMULATION OF EINSTEIN'S THEORY OF GRAVITATION

General Relativity can be formulated within the formalism developed in the preceding section. For pure gravity one should look for a ten parameter group G, which contains the six paramete subgroup H « S£(2C) to be taken as the gauge group, and a four dimensional coset space G/Sl(2C) for the base manifold. Among the simple groups only two satisfy these conditions; namely, the covering groups of the deSitter groups S0(4,l) and SO(3,2). The covering group of SO(3,2) is the symplectic group Sp(4). These two de Sitter spaces differ in their topology: the coset space G/S£(2C) has three compact and one non-compact dimensions for the former,, one compact and three non-compact dimensions for the latter.

A bundle of frames B(M,H) is defined by a set of vector SOPERGRAVITY. 371 fields {e°(m),l>1...4} which are local Lorentz frames. The metric gl1 defined by (13) Is the Mlnkowskl metric n^. The structure constants {fAg} are those of either de Sitter group with A, • [1J], a pair of Indices with i<3„ and Ai « i. The elementary fields are the connection coefficients hCiJ] for the eauge group Si(2C) and the "vierbeins" h* defined by (16-17). The metric tensor g is defined by:

(50)

The covariant derivative of a vector field t*t (x)3x alonp the direction of the vector 3 is defined by:

u Vutdx - t||(x+dx) - t(x)

where U(x+dx) is the parallel translation of t at x+dx to x. The components of V t are given by.

X y X X X v p :vwt) dx •= t (x+dx) - t (x) + r j]V(x)t (x)dx (51)

where r uv(x) is the familiar Christoffel symbol. Vie shall now express 1< F wv(x) in terms of hv'- '-' and h^. Accordinp: to the definition of parallel transport alonp: a curve y(s) we have:

v v 1 Al t (s)3v - t (s)h* eAi - t (O)eAi (52)

so that alonp Y(S) we have:

But, since Y(S) is horizontal, then: 372 s- MacDowell

On the other hand if t Is parallel translated alone Y(S) we nave VVfc fs~ * °an d (51)pive3 ! |f (55)

Taking (5»O and (55) into (53) one obtains:

U p A p A AB P A

where specifically A, • [1J] and A, - 1. From (56) and (25) one immediately obtains

relating the torsion to the antisymmetric part of ru, . If T • 0, then h^ ^ can be determined in terms of h^ and the space is a Riemannian symmetric space. Then one can show that the only second order invariant equation for h that can be derived from an action principle is:

which is equivalent to Einstein's equation in the absence of external sources,with an arbitrary cosmological constant \. A solution with constant Riemann tensor RjÇ11^ • *fijn^ corresponds to a homogenous de Sitter space. Performing a Wigner-In6nu contraction -of the de Sitter, group down to the Poincarl group, the cosmologlcal tern is eliminated. Equation (58) together with the torsion constraint

constitute the unique complete set of 16 + 2k Invariant

\ i equations for tit» 16 components of the "vierbein" and 21 y SUPERGRAVITY 373 conponents of the connection, which are linear in the curvature and torsion. So far we have only described general relativity, within this framework, for the gravitational field alone. One possibility of incorporating other fields using this formalism is to consider an associated bundle whose fibre is a carrier space for a representation X' of the Lorentz group. The connection in the principal bundle induces a connection In the associated bundle with the covariant derivative defined by:

where the matrices {x! } are the generators of the Lie algebra of H in the representation X'. However, a truly unified theory, incorporating fermions, can only be achieved by embedding the Lorentz group in a supergroup containing in addition the subgroup of internal symmetries. In order to accomplish this goal within a geometrical framework, it will be necessary to consider more general spaces with the structure of supergroups, and to develop a formalism for fibre bundles defined over graded manifolds. This will be done in the next section, in which we use the concept of 35 graded manifold as developed by Kostant . We would like to point out that this concept is not sufficiently broad to allow for the notion of Superfield as Introduced by Salam and Strathdee However, It seens to be adequate for the construction of any classical supergravlty theory. 374 S. MacDowell VI. FIBRE BUNDLES OVER GRADED MANIFOLDS

We shall give here an overall description of fibre bundles on graded manifolds, without being concerned with mathematical rigor or completeness. We start with the notion of graded 35 manifold according to Kostant Consider a manifold M of dimension d and a vector space V of dimension n, over the reals, and define a Grassmann algebra G on V. Given a complete set of neighborhoods {U,} in M, for each IL one introduces a "sheaf" in {U^G}. as an algebra A which is the direct product of G and the space of C - functions on U.. An element acA is then a sum of elements of G whose coefficients are C°* - functions in U,. A graded manifold is a set (M,A) in which a coordinate system has been Introduced in the following way: For every neighborhood U* of M take a set {uv} of d elements of A, called the even coordinates, which are even functions of the generators of G, and a set {ua} of n elements of A, called the odd coordinates, which are odd functions of the generators of G, with the following properties: i) The Jacobian of the coefficients of the Identity of the even coordinates íuu} is non-vanishing throughout U.. il) The product of the n odd coordinates {ua} is non- vanishing throughout UJ. We remark that the even coordinates are commutative, the odd coordinates anti-commutative and the product of more than n factors of odd coordinates always vanishes. Hence any function of the coordinates can be expanded as a polynomial in the odd coordinates with coefficients which are functions of the even coordinates. SÜPERGRAVITY 375 The concept of differentiation is introduced in the following ,£, in way: " i) For even coordinates let xv be the coefficient of ,

the identity of A in U^ and let fu(x) - 3yf(x). Then the partial derivative of a function f(u) with respect p f to the even coordinate u is defined by 3.,f(u} = u(u). ii) The partial derivative 3 with respect to an odd coordinate ua is defined in the following way: The derivative of a sum of terms is the sura of the derivatives of each term. The derivative of a term which does not contain u° as a factor is zero. The left (right) derivative of a term that has u° as a factor is obtained by displacing u>a to the left (right) of the product of , { odd coordinates and removing it from the term. f b- These derivatives have the following properties: f:(

(61)

We say that the set (3u»3a) form a coordinate basis for the tangent space to the graded manifold. a 1# Vie attribute to the indices fti,a)a signature a = °* a ~

A vector e. » (h¥ 31( + h? 3 ) has a definite signature a. • 0(1) according to whether the functions hV are even (odd) and h? are odd (even) functions of the odd coordinates ua. Ai Vie shall only consider bases {e. } for the tangent space to the graded manifold in which there are d elements of even signature and n elements of odd signature in the set {e. }. Let us next define differential forms in graded manifolds. Let (t,...t ) be a p-fold set of tangents t-j^M- of given

signature a*. Denote by vP a set whose elements are (ti...t ). 376 S. MacDowell A p-form Is a p-linear map w: vP •*• A such that:

1) w(t1...t1...t1+j...tp)

(62)

11) if t± = hJ?K then: ... w - H 1 °J

(63)

A contraction of a p-form w with a vector t, Is a (p-1)- form w«. such that: •V «(t,tt...tp) - wt (ta...tp) (61)

The wedge product of a p-form w and q-form w' Is a (p + q)- form w" = WAW1 defined by:

v"(tl...t p+._q ) = E(semr)w(t AA ...tA. )w'(tAft ...tA , ) TI i p p+l p+q (65)

where IT is a permutation (A,...A A +,...A + ) of (l...p+q)

such that (A,

(66) r Mai nn +1 j .i (sgnir)

where n. and nu are respectively the" number of indices SUPERGRAVITY 377

>A A.(iíp) with odd and even signatures and such that Ai p+j' The exterior derivative of a p-form w is a (p+D-form dw defined by:

1 1 i 3 + I (-1) w([t1,tJ},t1...ti_1,t1+1...tJ_1,tj+1...tp+1)

(67) where n* is the number of elements of odd signature with index less than 1.

The definition of fibre bundle may be taken- over with

appropriate changes. In general one can have fibre bundles

where the base is a graded manifold and the structure group 1 is a super-group. Here, we restrict our analysis to the k cases of bundles over graded manifolds with a Lie group

as structure group.

If Ü = (uX,ua) is a coordinate system in (M.A), the

set (3i»3 ) is a coordinate basis for the tangent space to

the base manifold.

We consider a bundle of frames G(M,H) where G Is a

supergroup, H a Lie subgroup of G and M a graded manifold

homeomorphic to G/H. The Lie algebra L of G is a graded

algebra L = L ® ^add with generators {X,,}, whose Lie products are given by:

AB [XA,XBJ = -(-l) [XB,XA} = -fA°xc (68)

where ifAg) is a set of structure constants for the superalgebra

L and oA is the signature of the generator XA:aA = 0,1 according

£l L We to whether ^pL -evint 0(i(i' remark that the signature of a 378 S. MacDowell

generator XA of the Lie algebra £.„ of K Is o^ =0 because H is

a Lie group; but the signature of X, belonging to L, = L/l0

depends on whether Xft eLevcn or Xft EL^.

A carrier space for a linear representation X1 of G is

also a direct sum V = Ve ® Vo of subspaces of even and odd signature satisfying the conditions:

Let {U«|} be a basis for V of vectors of definite signature

o.,, andd (XI)., be the matrix representinrepresent g the generator XA of L. Then, for a simple supergroup, we have: {• °D» = (-1) - X&J&\ = -f(X')nAB (69) . I)

where n.B is the metric of i (Independent of the representation) and f(X') is the index of the representation. For simple Lie

groups one can always set f(X.1) = 1 for the adjoint representation c c ^XA^B = fÀB» 3O that nnji coincides with the Cartan-KIUIng metric. However, for simple supergroups, in some special instances the

index of the adjoint representation vanishes. In general, one

can use the fundamental, representation to define the metric.

The metric n»r, satisfies the following important Identity:

fAB"DC- fCAnBD=Q

Since the functions defined on graded manifolds depend on

antlcommutlng variables, some care must be taken in preserving

the order of factors. The connection coefficients arid "vielbeins" SUPERGRAVITY 379

h? ({A} = {X,ct}) are assured to te even or odd functions of the

odd coordinates ua according to whether (o^ + o ) = 0 or 1.

Likewise, the parameters XB in the transformations (37-39) or

the infinitesimal Daraneters eB in (40-41) are also even or odd

functions of ua according to whether Og = 0 or 1.

The symmetry of the metric (13) is Riven by

_ i 1N°A1°B1 e . (71)

The algebra of the vectors {V.} in the covariant basis for the tangent space to G(K,H) is as in (3D, with the commutator replaced by:

where F ^ = fC(.V.,V~).

The Bianchi identities (33) are replaced by:

" (Jc) ^^ CVBS + FABFE^ = °

The relation between the components of F in a coordinate

basis and in the covariant basis is piven by:

a +a B c A °l(E A A B n ) i i FEA = (-D i H\

where P£J = Av^VJ. In general, the concepts introduced and results obtained

in the ordinary theory of fibre bundles can be carried over to

bundles on p-.raded manifolds. One noticeable exception is that

the concept of Integration of forms, so useful in the geometry 380 S. MacDowell of manifolds, cannot be generalized to graded manifolds.

Althouph it is possible to define an invariant integral in a graded manifold, the invariant measure is not a form.

In the next section we' use the formalism developed here to construct a Supergravity model which is a locally super-

symmetric generalization of Einstein's theory.

VII. CONSTRUCTION OF SIMPLE SÜPERGRAVITY

In analogy with the formulation of general relativity as

the geometry of a bundle of frames, in a simple supergravity

model without internal symmetry one should look for a simple

supergroup which contains either SO(l!,l) or SO(3,2) as its

r.aximal subgroup. The only such supergroup is the ortho-

symplectic group OSpCljA) with 14 generators.

Vie shall then consider a bundle of frames with structure

group II - S£(2C) and base supermanifold M homeomorphlc to

OSp(l|M)/S£(2C). This is an eight-dimensional graded manifold

with four even coordinates {xv} and four odd coordinates {ea}

The bundle of frames G(M,H) is defined in terms of a set of

vector fields {e°(m),e (m)} which are, respectively, local

vector and spinor Lorentz frames.

The metric (13) is (j^, ,Cafa) where n.j* is the Minkowski

metric and Ca{, (the charge conjugation matrix) and its ab inverse -C are real and antisymmetric, with the property:

£b Ca6C « «a* (75)

The matrices Ca^ and C are used to lower and raise spinor indices. SOPERGFAVITY 381

The Lorentz connections h*- •'•' and the "vlelbeins" (h ,ha) have, in a coordinate basis, two sets of components: i) Vector components: [jp

II) Spinor components: ha i0^»*1*-

Since the physical space is really four dimensional, the question now arises of interpreting the new fermion-like coordinates 8°.

We remark first that one wants to formulate a theory, based on an action defined as an Integral over the space-tine manifold, invariant under general coordinate transformations.

A necessary and sufficient condition to enforce the assumption *i that (h, ) is a non-singular matrix is that the matrices (h t ) A • u a and (h ) be themselves non-singular. Therefore, one can use det(h )d x as the invariant measure.

The Lagrangian may depend on the vector components of h and their space-time derivatives. But since the infinitesimal transformations of the spinor eomnonents contain 3 E and the a odd variables are not Integrated over, it follows that the only dependence of the Lagrangian on h must be through the covarlant C C A components F. n of F . Also, the only dependence on h in the 11 A C "

transformation laws of h Is through P. B • Since derivatives of a field with respect to 6n have to be considered as new

independent fields, it follows that all the components F. H Aitíi must be so constrained that they can be expressed by means of

(7*0, in terms of F^* ^yv'^uv^ and a subset of covarlant

components to be considered as independent fields.

We shall thus obtain a realization of the superalgebra

(72) In the space of functions h plus some additional fields ÍF} taken from {F „ }. The remaining components of {F V, } BjCn , ° BjCD , have to be expressed in term3 of F and the new fields F. 382 S. MacDowell An inspection of (7*0 with AE • pv shows that, in order to carry out this program, one has to impose a certain set of constraints which relate the components F,.»,c, of Fc to the independent AaO| fields {F} and, linearly, to the components F.^. Assuming no new fields F, these constraints take the form given in Table 1, which has been reproduced from ref. 33 for the general case of extended supergravity to be discussed in the next section.

TABLE 1. Covariant constraints for simple (and extended) supergravity. The t's are covariant parameters constructed out of the structure constants of.G> The X's are constants restricted by: ^ii^ii*^*^ = aiifi2jfiik5»4 for alx Permutations

(ijk ) of (1 2'3 *») and XM = X2l » X,, = \+% = 1. Note the slight change in notation (a,b,c,...-»•(*,B,Y) , (a,b,c,...*a,b,c...).

Win »"/,<„ •,

which reproduce those obtained in ref. (40), and are equivalent to the equations of motion in ref3. (22,23). Prom the equations of motion one can reconstruct the Lagrangian. It should be mentioned that only the equations of motion are invariant under the supersymmetry transformations (40-41). The Lagrangian is invariant under a set of transformations which coincide with (40-41) when the equations of motion are taken Into account. However, this new set does not close under commutation. In order to obtain a Lagrangian invariant under the transformations (40-41) one has to include some new independent fields P as auxiliary fields. These fields vanish as a result of the equations of motion. Although the problem of auxiliary fields is essential for the quantization of this theory, we shall not go further into a discussion of this subject here.

VIII. CONSTRUCTION OF EXTENDED S02 AND SO, SUPERGRAVITY

The extension of the construction of the previous section to a model with internal symmetry is conceptually straight- forward. One has to consider an orthosymnlectic supergroup OSp(Il|4) whose Lie subgroup is Sp(4) 6 SO(H), where SO(N) will be the internal symmetry group of local invariances. The bundle of frames has structure pr°up H » S£(2C) 8 EO(N) 384 S. McDowell and base supermanifold M homeomorphic to OSn(N| 1J)/(S£(2C) « SO(N)), which has four even coordinates íxV) and *»! odd coordinates {6°}. It is defined in terns of local frames {e!(ra),eò (m)}.

The metric (13) has symmetric components r\i, and anti-

Tne w symetric components C .n b. * ° sets of components of the connection have, in addition to the Lorentz components h ^ and possibly new sets of fields F belonging to Irreducible repre- * sentations of Pn _ with respect to the gauge group H. Following the procedure described in the previous section, one finds that the constraints in Table 1 for a meael without additional fields F are consistent with the Bianchl identities If the internal symmetry group Is S0(2) but not consistent, for symmetry groups SO(N) with N>2.^ For the S0(2) r.odel one obtains the sane set of equations of motion (76) as for simple supergravity plus the equation for the S0(2)-gauge field:-

=0 , ([ab] = [12]) (76a)

For SOdO-supereravity with N>0 one does not expect the constraints on Table 1 to *? consistent with the Bianchi identities. In fact, the linear representations of the (Poincaré contraction of the) superalgebra OSp(H|4) in a Rilbert space with a state of highest helicity s, singlet with respect to SO(II), contains a set of state vectors with helicity (s-n/2) in the completely antisymmetric tensor representation of SO(N) of rank n, for all values SUPERGRAVITY 385 of nSH . Thus, for s-2 and N-3 one expects a set of spln-if fields belonging to the representation of SO(N) antisymmetric in 3 indices. One can show that this field comes from the torsion component T 5^. Therefore, the constraints in Table 1 should be modified to include the dependence on the new fields. In analogy with the models of simple and S0(2) supergravity 31» we shall set

1 • 1 m J aa'bb ab ab * aai

as a starting point for the construction of extended supergravlty with N>2. The physical meaning of these constraints is that, if the equations of motion are derived fron an action principle in space-time, the Lagranglan 'contains the Parita-Schwinger term with a covariant derivative of h*a, and no other term depending on h*J6 which involves derivatives of fields belonging to nontrivial representations of the gauge group S£(2C). The constraints (77) are then an assumption of "Minimal Coupling". Next in the construction of SO(N)-models, we take the constraints (77) into the Bianchi identities and determine the components of the curvature and torsion that have a lower spinor index,in terms of the subset R.,« ,T..* and a set of independent and auxiliary fields. The Bianchi identities should also give the super- syrxietry transformations of the new fields, that is, their

first order covariant derivatives Pfta. When the auxiliary fields are expressed in terms of the independent fields, the Bianchi identities also give equations of motion for the latter, as well as equations of motion for the gauge fields. Most of this program involves decomposing each Bianchi identity (A,B,C) - 0 in terms of irreducible representations of S*(2C) 9 SO(N). The only technical difficulty is in finding 386 S. MacDowell the relation between auxiliary and independent fields for j. N>3 supergravity models. ' We outline here the results obtained from each Bianchi < identity1"1"1":

1. (aa,bb,cc) It determines the torsion T £? to be of the form:

The spin-is field ^^r^cji which belongs to the irreducible representation of SO» anti-symmetric in three indices, is an T ls a independent multiplet In the extended theories; aa(be) spin-% auxiliary multiplet, and one must impose the constraint:

T«Cbc] " ° (79)

wt/'ch forms a set of covarlant equations for the spinor components hj; c-' of the internal symmetry connection.

2. (aa,bb,cc)dd

i) It gives the supersymmetry transformations of the irreducible components of.T £jj. For N>3 one finds that T transformation of aa/bc).contains terms quadratic in This analysis follows closely ref. 34; however, there are several misprints and some mistakes in that reference which have been corrected here. SUPERGRAVITY 387 Therefore, It becomes a non-vanishinp auxiliary field. As for

N • 1,2 one has to impose a trace condition on Ttta/b )'•

CT + T + ab aa(bc) b(ac) xTc(ab))n - 0 (80)

where x Is a parameter. The simplest choice is x = which, for N - 1, yields the conventional model26.

T and T contaln II) The transformations of jr-hon nfbc) new multlplets of scalar and pseudo-scalar fields, which are irreducible components of R '•?. ^ frc'd'lc 6lven by

R°»ss - i I p + R \ K[abcd°» ] " F tabc ) flRaa,fabCcd] + Rac,6d[ab] (81)

K[abc]d " I (abc)v aa,bb[cd] " Rac,bd[ab]nc )Ys > (82)

The first is a set of Independent fields, the latter a set of auxiliary fields.

For N=8, Rfahcdi ls not lrreduclble» In order to obtain an irreducible realization of the supersymmetry algebra one has to impose the self-duality conditions:

REabc«i] " <* Í)EabcdalbI°tdI REá'b'c.d'] (83) ut! which turn out to be consistent with the Bianchi identities. 1 \ ill) The remaining Irreducible components of Rfla I; are determined in terms of Taa £jj and T^ 388 S. MacDowell

3. Ua,()b,i)J~

1) It determines R a ^ ^ in terms of and ii) It restricts to be of the form:

(84)

+ T'(ac)

where

i(ab)f[pq]J ab and

(86)

It determines the curvature °f ., and the torsion T?..

c ln erms of I) It defines the curvature •• frbtci]b t torsion components and the supersymmetry transformation

Ti[bcy

II) It gives the supersymmetry transformations of the bosonlc auxiliary fields ,T«ab) .TJab) One has to impose further constraints on the auxiliary fields,

compatible with their transformation laws, in order to determine

them in terms of the independent fields. Once this is done, the

following become equations of motion:

ill) for the spin-| field SUPERGRAVITY 389

" Ti[ce] +

3" (afie) a Cbc] Y*tTa(bd)T [cé] " Ta(cd)T [be])n ír[kí]

(87)

lv) for the spln-| field h*a:

^'i Jo) T.tVaCbe) - VTl(bo) + ? Ti(bc)> + TiaTd(bc)

(I Tj fi • i T^f^fjn^ - 0 (88)

dd 6. («a.if

which, combined with. (85) and (86) can be solved for T^. This enables one to determine the Lorentz connection hj" ^.

ii) It gives the equation of motion for the spin-2 field h :

(90) 390 S. MacDowell

R Cd i 111) It relates the Yann-Mllls curvature R±J a = 1j "' '[cd]a to T.p . -j and the auxiliary fields:

(OiTk[ab]

T fV

= 0 (91)

This equation determines the auxiliary field in terms of the Yann-Hills field strength Rlk and the independent fleld Ti[ab]'

iv) It gives the equation of motion for the spin-1 gauge field hjab]:

0 fTCrs] m,Crs].f ,1 „, I frnCrs] m,Crs],_ J VT[ab] " T[ab] )f[rs]k + TJk (T[ab] " T[ab] )f[rs]£

+ TJk fJ T[abc] " TTjk f

c Tke " Tkc

(92)

The constrairts (77,79) together with the Bianchi identities A A (1-6) determine all the components of R ° and T ' with at least one A A lower spinor index in terms of (R..°, T^.1) and the following sets:

independent fields: Td[abc]f R'[aJcd], T1[ab].

Auxiliary fields: Ttta(bc), H'^ T^], SÜPERGRAVITY 391

rp!; in' ips r" 5. mti

For N>4 (with the exception of N=8), new independent fields

will emerge as covariant derivatives V of ~ "

For N=2,3 one can set

T«a(be) " 0 (93>

and the remaining auxiliary fields become determined in terms of

the Independent fields by:

Rtabc]d °

[ab] f [poji ^Jk - I Rika "be + l(1?iTkCab] " Vl[ab3>

+ è(Ti[ac]Tk[bd] " Tk[ac]Tl[bd])l)C

nab tor zero; , T(ab) = u |

m5 - T'* . 1 ~ m „ -p mCode] mil Ti(ab) ~ TKab)- 12 ^ab A[cde]Y5flT • T[ab]

where TÍ bx = n . or zero corresponds to the values for de Sitter

or Poincaré supergravity. In order to obtain an irreducible realization of the symmetry

one has to specify the independent field T.r .-,. Again we invoke

the hypothesis of "minimal coupling" to establish the constraint

that determines this field, which is

Then for N=2,3, the equations (87), (88), (90) and (92) reduce to: 392 S. MacDowell

(92')

(89')

- [rs] _ J . 1 _, f0 m[abc] ., ij fCrs]k + 7 T[abc] fkpiT + Xt n

These are the eauations of motion for SO(2) and SO(3) super- ;| gravity. Prom them one can derive the space-time action, by integration of the functional equation: • í

\\ h3 5S r,l p [rs] _ J to - [rs] ,, J, . 19 h n = {( R f n n R f } + 12 n Jk r £j [rs]n ik " ij [rs]k ik

5ija ÈWb^ Vt(hJ) (96)

where the covariant components of the curvature and torsion have to

be expressed in terms of Trabe-| and the vector components h of h by means of the relation (7*0, with A£ •= pv. For N>3 we have not yet obtained a complete determination of the auxiliary fields. Once this is done one expects that a space- tine action may also be derived from a functional equation analogous to (96). f- SÜPERGRAVITY 393 IX. CRITICAL EVALUATION OP SÜPERGRAVITY

The most attractive feature of supergravity and extended supergravity lies in its potential for achieving a complete geometric unification of all the Interactions, Including gravitation, with all the fermionic and bosonic fields appearing as members of one supersymmetric raultiplet. The SOe-model is singled out, not only because it is the most general model in which all the elementary fields have spin<2 (and the gravitational field is the only one with spin-2), but also because all the fields with spin>|- are gauge fields. This last condition is important for the removal of states of zero or negative norm.

However, the phenomenology of the elementary particles

gives evidence for an underlying SU, 8 SU2 8 U, gauge group,

which is not a subgroup of S0e. Thus SO, will not be a

sufficiently large gauge proup unless the SUa symmetry of the weak and electromagnetic interactions is a dynamical symmetry. Moreover, since all the fields in this theory are real, charge conjugation is trivial. Thus It appears that supergravity models with internal symmetry based on the orthogonal group do not qualify well as realistic theories. An alternative is to consider models based on the supergroup A(M|N). For I-M, in one of its forms A(1||N) has SU(2,2) 9 U(N) as maximal Lie subgroup; SU(2,2) is the conformai group which contains Sl(2C) as a subgroup. These theories have been investigated 36 by P. van Nieuwenhuizen et al . They correspond to generalizations of Weyl's theory and have an intrinsic difficulty with causality. Hence, although supergravlty contains the necessary features of a truly unified theory, some elements are still missing for the construction of a realistic theory. 394 S. MacDowell

APPENDIX A - GLOSSARY OF NOTATIONS AND DEFINITIONS

P - principal bundle; p - an element of P: (peP)

H - base manifold or graded manifold; m - an element of 11: (meM)

G - Lie group or supergroup; 3 - an element of G: (geG)

H - Lie subgroup of G

V - Vector space

B(H) - Bundle of bases over M

B(K,H) - Bundle of frames with structure group H

Ad.G- Adjoint representation of a group

I - Lie algebra of G; Lo - Lie algebra of H; Lt - Vector space L/Lt A,B...E - group indices for the generators of L.

AB,BB...EQ - group indices for the generators of LQ

AliB1...E1 - group indices for the generators of I, {f.g} - set of structure constants for L

{f, R'} - set of structure constants for Le f ' C c {XA> - set of generators of 1 in AdX3: (Xft)B = fAB" {X. } - set of generators of L. in Ad.G: (X. )?." = f. ?." Ao A» Bo AoBo

1 {XA } - set of generators of L, in Ad.G: (XA )B» = fft g

{Pft } - basis for vertical tangent space in a fibre bundle which

under commutation form an algebra isomorphic to Lo IV.) = {0. ,0. } - Covariant basis-in a bundle of bases

A Afl A, D • u °Xft - one form of the connection (vertical) A, h • h X. - solder form (horizontal) A • - in + h = XA A A A {$ } «= i(o ,h } - basis of one-forms dual to {V.} r,A,u,v,p,o - indices of even coordinates in a supermanifold M

iCfi - indices' of odd coordinates in a supermanifold M

- coordinate basis in tangent space M SUPERGRAVITY 395

3 3 {V ,V } - horizontal lifts of Í u» a^í basis for horizontal tangent space P to the bundle P at p. t - vector in a tangent space,

{e,} - set of vector fields forming a basis for the vector

representation V-i,h) of S£(2C)

{e } - set of vector fields forming a basis for the spinor

representation [(2s,0) + (0,^)] of S£(2C)

i,j...r,s - vector indices for a representation of S£(2C):(i = I...1*)

a,b...e,d - spinor indices for a representation of S£(2C).(a = 1...4)

a,b...e,d - vector indices for a representation of the group

S0(N)(a = 1...N)

[ab] - pair of indices with a

index for the adjoint representation of SOfi, or vector indices a,b = 1...N, when it osnotes components of a

tensor antisymmetric in a,b

(ab) - vector indices denoting components of a tensor symmetric

in a,b

h - connection coefficients for ,'3£(2C)

" connection coefficients for S<">(N) h •* - "vierbeins"

h* - Rarita-Schwinger field (Majorana representation)

^Ci]^ ~ 3tructure constants for S£(2C)

" structure constants for SO(N)

^fCiJ]Ck£]»f[iJ]k>fki i ~ structure constants for

The structure constants for OS (M|il) -may.be written in terms of

the structure constants for 0SD(l|i)) and S0(N + 1): 396 S. MacDowell

[pq] r I f [ij] - b _ b f [1J] f i, \íkf]' í±3lktlkl »1[ij]a»Iia'Ittb »xabJ constants for OS,

structure constants for 80(11+1)

Cpq3 fa b bb bfib C ]]'r[ij]kflk »r[ab][«wi] •1CiJ]aa

. dã _ . d.d [lj] [ij] f i _ f i f [cd] _ fCcd] f r s r 1 n I xf [ab]cc ~ [ab]c £' aa16b b " ab ab» aa,bb aabb

structure constants for OS (ill1!)

C , = -C, ,(abC - -CCfaafaa:CbcbcC = Sb) = Charge conjugation matrix; C , a a. CLO a a CLC 0. antisymmetric metric used to lower and raise spinor Indices.

Definition of ys'.~

v 6 e Ysa f[iJ]b

.a.b be 1 sac

The netric of OSp(N|il) may be written in terms of the metric of OSp(l|t) and of S0(>5 + 1). Definine the metric through the fundamental representation with index f = *1, we have:

l3'cab) = Metrlc of OS

(nCab][cd3'nab) = Metrle of

Metrlc of e [c'd'3 f[cd]aneb " fab n[c'd'][cd]

Notice the following differences in notation in this paper and in reference 33:

The Lie algebra of a group G is denoted by L, instead of g. The elements of the group G are denoted by 9, instead of {?. tit The elements of H in the adjoint representation are denoted by g, . «0 ft The elements of H in the co3et representation G/H are denoted by g. » Mi Dirac spinor indices are denoted by a,b,c..., instead of a,B,y.... The structure constants fBA and functions Ffij? have been replaced

*>y -fj^g and -FAB which is equivalent to multiplication by the o.on signature factor (-1) * .

The generators Xft of G have been replaced by -XA. 398 S. MacDowell APPENDIX B - GAUGE TRANSFORMATION OP VECTOR POTENTIALS

Let us consider a Lie group G, whose elements p(x) are

obtainened by eexponentiatio: n of the elements X XA of the Lie algebra of G: AA g(X) - expXX XA (B.I)

Let us denote the composition law for the parameters X, corresponding to the operation of group multiplication:

by:

X* - **(*,,X) (B.2)

The generators t>A of the Lie algebra of G, acting as differential operators on G considered as a topological space, are given by :

- ujcxj -Is (B.3) x -o ií x

The basis of forms {0 } dual to the set of basis vectors

{PA> is then:

tlA - dXB U~Q

Therefora, on a cross section xi G*Rn we have:

8*,^^-;'.^* (B.5)

In an infinitesimal change of cross-section, corresponding to a local gauge transformation with infinitesimal parameters V. SUPERGRAVITY 399

£ (x), we have;

.A . B \A •• AXA (B.6) X + C e-0 so that &XA - cBUg.

Therefore:

3U,B A nCE,,D (B.7)

But from (3) and (B.3) one gets:

3U B 3U L r ,-iA _ - A (B.B) 3X

Therefore:

(B.9) 400 S. MacDowell

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16. F. Gursey, P. Ramond and P. Sikivie, Phys. Lett. 60B, 177 (1976). SUPERGRAVIVY 401 17. H. Weyü., S. B, preuss. Akad. Wiss., 465 (1918); Math. Z. £, 384 (1918). 18. Th. Kaluza, S. B. preu33. Akad. Wiss., 966 (1911); 0. Klein, Nature, Lond. 118., 516 (1926); 2. Phys. 37., 895 (1926). 19. W. Pauli, Theory of Relativity, Pergamon Press (1958), 230. 20. D. F?eedman, S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. D13. 321'4 (1976). 21. S. Deser and B. Zumino, Phys. Lett. 62B, 335 (1976). 22. 3. W. MacDowell and F. Kansouri, Phys. Rev. Lett 38, 739 (1977). A. H. Chamsedine and P. C. West, Nucl. Phys. B129, 39 (1977). 33, P, van Nieuwenhuizen and P. K. Townsend, Phys. Letters 6_7B, 439 (1977). 21. S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Letters ^1» 1669 (1976). 25. D. Freedman, Phys. Rev. Letters 38., 105 (1977); S* Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B, 35 (1977). 26. E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 7.4B, 409 (1978); E. Cremmer and J. Scherk, Nucl. Phys. B127, 259 (1978). 27. E. Cremmer and B. Julia, Nucl. Phys. B159, 141 (1979). 28. M. Gell-Mann and Y. Ne'eman, unpublished. 29. J. Wess and B. Zumino, Phys.Lett. 66B, 361 (1977); Phys. Lett. 79B. 39M (1978). J. Wess, R. Grimm and B. Zumino, Phy3. Lett. 73B, 415 (1978); Nucl. Phys. B152, 255 (1979). L. Brink', M. Gell-Mann, P. Ramond and J. Schwarz, Phys. Lett. 74B, 336,(1978); Phys. Lett. 76_B, 417 (1978). S. J. Gates, Jr.,- and J. A. Schapiro, Phys. Rev. D18, 2768 (1978). S. J. Gates, Jr., and W. Siegel, Nuel. Phys. B147, 77 (1979). W. Siegel, Nucl. Phys. B142. 301 (1978). S. J, Gates, "On the Geometry of Superspace" (M.I.T. thesis, 402 S. MacDowell preprint CTF-621, (1977) unpublished); Harvard preprints HUTP-78/A001 (1978), HUTP-78/AO28, unpublished. M. Brown and S. J. Gates, Harvard preprint HUTP-79/A002 (1979) i to be published in Annals of Physics. F. Mansouri and C, Schaer, Phys. Lett. 8_3B, 329 (1979). 30. M. Grisaru, P. van Nieuwenhuizen and J. A. M. Vermaseren, Phys. Rev. Letters 37_, 1662 (1976). M. Grisaru, Phys. Lett. B66_, 75 (1977). M. Grisaru and P. van Nieuwenhuizen, New Pathways in Theoretical Physics (Coral Gables, 1977), ed. B. B. Kursunoglu and A. Perlmutter. 31. N. Steemrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, N.J. (1951). R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York, London (1961). 32. Y. Ne'eman and T. Regge, Phys. Lett. 71B., 31 (1978); Rivista Del Nuovo Cimento Vol. 1, 5,, 1 (1978). 33. S. W. MacDowell, Phys.. Lett. 80B, 212 (1979). 34. S. Vf. MacDowell, Supergravity 77, North Holland Publishing Company, Amsterdam (1979), Ed. P. van Nieuwenhuizen and D. Z, Freedman. 35. B. Kostant, "Graded Manifolds, Graded Lie Theories and Prequantization", Lecture Notes in Mathematics 570, Springer Verlag (1977). 36. M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 69B. 301 (1977); Phys. Rev. Lett. 39, 1109 (1977); Phys. Rev. D17, 3179 (1978). S. Ferrara, M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, Nucl. Phys. B129. 125 (1977). 403

WEAK INTERACTIONS

RajatChandow Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil

1. Introduction

The traditional charged-current weak interaction Lagrangian which accounts for low-energy weak processes, especially non-charmed par ticie. decays is written as

where the weak current J, has a leptonic part I, and a hadronic piece h,. A " " i. can be explicitly written down in terms of leptonic fields

Z, = I YiCl-Ys^ + V Y}Cl-Y5)v,,- (2)

The hadronic currents are known to have left-handed V-A structure and possess., strangeness-conserving and strangeness violating pieces according to the Cabibbo parametrization

=0 As =1 hx = hf cosec + 4 l sinec (3)

The Cabibbo angle 8 = 139. Only 'first class1 AS=0 currents with the following G-parity transform - ation properties are known to occur in nature G Vf=0 G-1 = Vf=0 (4) G Af=° G"1 = -Af=° .

Hadronic currents satisfy conservation laws called conserved vector current (CVC) and Partially Conserved Axial-Vector Current (PCAC) hypothesis. They also satisfy Equal-Time Commutation (ETC) rules pro - posed by Gell-Mann . Consequences of these general rules of conservation laws coupled with the algebra of currents were worked out about fifteen years ago. 404 R. Chanda Although the Lagrangian Cll leads to many successful predictions at low energies, we can not accept it to form a basis for Lagrangian Quantum Field Theory for Weak Interactions. To see the problems associated with it, let us consider the purely leptonic process vT +e~-*v+u" at high energies. Since the currents have dimension 3 in mass units, the coupling constant Gp has dimension -2 (Gp = 1.05xl0~ /mM. Therefore the total cross-section

- G/S (5) Jtot where S is the usual Mandelstam variable. On the other hand unitarity sets and upper limit on it:

'tot « I (6)

Hence above a certain energy given by &\ = —\- or E^ = 300 GeV. , the linear rise given by (5) can not continue and one meets the 'unitarity crisis'. Also, the renormalizability criterion requires the coupling

constant to have positive dimension in mass units. Since GF has negative dimension, the Lagrangian (1) is non-renormalizable and one can not canpute

higher order weak processes such as KL - Kg mass difference.

To dampen the growth of a^ tiintroduce massive boson exchange W~. However, the longitudinal modes raay grow with energy cancelling out the damping coming from the propagator of r

2. Gauge Theory; Unification of E-M and Weak Interaction

Local gauge invariance requires massless gauge fields. Renorm- alizable Yang-Hills theory handles internal degrees of freedom (4)

Choice of Gauge symmetry; The currents in the electron sector are given below. Inclusion of other leptons is trivial.

The chiral projections are

+ j (Leptonic) WEAK INTERACTIONS 405 Eq. (7) can be rewritten as

M J^ = L Vu T3 L + |(L Vp L + R Yu R)

(13-current) (Y - current)

where t's are the usual isospin matrices. These current structures show that the minimal gauge group containing weak and electromagnetic inter - actions is SUW 8 U*. Vector gauge fields associated with them are res - pectively W* and B (i1 = 1, 2, 3) :

^ * P y (8)

Note that W^'' can not be the photon. The photon field A is a suitable linear combination of w ' and B ]i V

1 A u will have pure vector coupling to e~ with strength "e , the electronic charge, and decouple from v if

;f e = " . ; sin 6 r~T~k '9 = ~±— {10) 12 5 w 12 ; (g^g ) (g^g )^ sin8w

3 The orthogonal combination of w t and B is

Zw = oosew Mi!31 + sin8w By-

This couples to both e~ and v . Electroweak model requires neutral weak currents coupled to Z°.

The parameter &w is to be determined experimentally.

3. Gauge Invariant Introduction of Masses for W+, Z°. Higgs Mechanism

i9(3Í Introduce a charged scalar field 4>(x) = *, (x)+i*2(x)=p(x)e in presence of gauge fields:

L = - 7 Fyv + \ 'V'2+ po21*'' % 406 R- Chanda /'< where F is the gauge invariant gauge-field strength and D is the co- ~, variant derivative. If y* < 0, V(i))I = U2|| + M "f1 nas 3 minimum at i£? | (ji | = p = /-u§" /x = v ft Q. The ground state is SU2 doublet

pi =

then mf = tíos8w

A more complicated Higgs structure would change this relation.

4. Effective Lagrangian j-;

In the region of momentum transfer q2 << mA which corresponds fi' to present experimental situation we have an effective current - current '•', interaction with

= — (14) /2 (14) , (13) and (10) tells us

m» = I "/?e2 I 2 = 79 GeV (15)

and m- = 90 GeV. We have used the experimental value sin28 =.23. The neutral coupling strength of any Fermion f to Z° is

Note that the couplings depend on l3 assignment. WEAK INTERACTIONS 407 5. Introduction of Hadrons ; Quark Mixing.

AS = 0 neutral currents were found experimentally in v-induced reactions but JAS..J = 1 neutral currents are highly supressed as seen from the absence of processes like K° •*• ]iVi K •* •n e e etc. The Cabibbo currents can not account for these suppressions. Glashow et al (GIM) provided an explanation introducing the charmed quark C(Q=2/3=OP . In GIH scheme the charged current couplings among quarks are

u d1"

1 1 where d d cose + s sin6c d eose (17) s1 =-d sine + s cose or s1 -sin6

In this scheme, the neutral currents do not have the |AS| = 1 terms: ds" + 3s pieces cancel out. Experimentally the Cabibbo angle 8c = 13?. Discovery of the charmonium, charmed mesons D, F, charmed baryon í and the decay properties of D obeying the predicted selection rule AC S has led to the acceptance of this picture (7) .Experimentally r a\ • we have an upper limit on AC j4 0 neutral current

•+ v C <.O2. (18)

6. Generalization of GIM Mechanism to n-doublets: Flavor Conservation in Neutral Currents

Consider n-doublets of quarks (i = l...n)

di where d! =

+ A A (19)

The n 'up1 quarks have electric charge Q = 2/3, the down quarks have 2 Qd = -1/3. The matrix of unitary transformation 'A' needs n parameters for its characterization. Among the 2n quark fields there are 2n - 1 unobservable relative phases. Hence the number of observable 408 R. Chanda parameters are n2 - (2n-l) = Cn-11 For n=2 (GIM case) we have just one parameter 9 , the rotation angle in d-s plane. For n=3 case (Kobayashi-Maskawa) we have 4 observable parameters 3 of which may be chosen to be the Euler angles Q^i = 1/ 2, 3) in the d-s-b space; we are left with an observable phase :6. Thus, in K-M six quark model one can naturally incorporate the CP violation. In the general n-doublet case (n > 3) the number of phases 6^ which can not be removed by redefinition of fermion field phases is fn n2 n(n-l) _ Cn-1) (n-2) (n-l) - 2 jr •

7. Motivation for Including the Sixth Quark t.

The reasons, for expecting an yet undiscovered sixth flavor of quark called 't' are as follows:

(i) The spin \ T-lepton (1782 MeV) and the associated v,p are

established. Hence there are six leptons. mv < 100 MeV. The limit on

my will improve soon via the ir end-point energy in x •+ it v transition. The Michel parameter p from the decay electron momentum spectrum in x •* e v gives e expt = "72 ± "15

in good agreement with V-A coupling for T-\> i.e. p ~A = .75. The T-decay branching ratios also check with V-A coupling. Renormalizability of Salam-Weinberg model demanding cancell - ation of Adler-Bell-Jackiw anomalies require a quark doublet (t,b) with V-A coupling. The fifth quark flavor b is already well established .

(ii) The natural incorporation of CP violation is an attractive feature of 6-quark models. Fermion contents of the 6-quark 6-lepton model are the follow - ing left-handed SU^ doublets

c (20) s1

The quark mixings are givsn by WEAK INTERACTIONS 409

1 d S1C3 S!S3

s' 8 c C C C +S S e10 C C C S S eÍ6 (21) - l i 1 2 3 2 3 1 2 3- 2 3

b" -S,Sa cisac«-C2SselS ClS2C3+C2C3e10

where c. = cos6. and s. = sin9.. 6j = 9 = 139. Note that for 82 = 6. =6 = 0 the mixing matrix reducex to the GIM case:

c, 8, 0

0 0 1

All right handed Fermions are Suw singlets.

8. Limits On the Mixing Angles 92 and 93.

Sirlin has calculated radiative correction to the allowed vector decay of O1"1 (u -»• d) . Together with the experimental results on K. decay (u + s) , the experimental check on Cabibbo universality for u-d and u-s couplings

z 2 li-s- v =^u g- leads to

c2 + s2c2 = 1.004 ± .003 (22) with s-j = .05, we have

(s3) = .28 ± .21 (23) which determines the mixing between u and b quarks.

Bound on 92 is obtained indirectly by considering t-quark contribution to Kj^ -*• yjl or to KL-Kg mass difference:

The mass difference is a complicated function of slf s2, m and

f s2, mc, 410 R. Chanda U,C

u,c,t

f 9) Limits on |s2| depends on m^ ':

0 < |sj < .25C.19) (24)

for mt = 15(30) GeV.

From the analysis of CP violation in the kaon system

+ M(KT -*• TT IT~) n . = +~Z 0 1 M(KL + IT IT TT J

- 2xlO-13 The six-quark model yields I (25) In. | = s2s3 sinfi. ,-29 Also, the electric dipole moment of neutron - 10 e.cm. is well below the experimental limit. Therefore, for kaons the results are similar to the superweak theory of CP violation ^ '. However, in this model CP- violation should be much larger in B° - B° system. This prediction may be tested at Cornell and may throw new light on the CP-puzzle. It must be remembered that there is a problem of how to eliminate or control the QCD induced strong CP-violation via

(26) *QCD

where Fyv is the gluon field strength. The arbitrary angular parameter 9 < 10~9 experimentally; otherwise it leads to too large electric dipole moment for neutron. T2ie axion or m,, = 0 option for eliminating this does fill not work . An ad hoc proposal to controll 8 exists: WEAK INTERACTIONS 411

6 eff = 6 + 8Q = 0 where 9Q = arg det M

(10) MQ being the quark mass matrix

9. Generalised Rules for Weak Decay of Heavy Particles

(27) 192TT3 where 9 is the x-y weaeak mixing angle factor. The kinematic factor F fe]= 1 m < 1 for 0 < 1. m

Special cases of weak decays of hadrons containing c, b and t quarks can be worked out. m__ Example: Decays of b-flavored particles involve working out 8, , ebcandFn^j- The result is:

r(b + ux) = sz s2 x .8 x 1015 sec"1. 1 3 (28) ex) =(s2 + sz + 2s s cos6) x .25xlO15 sec"1. Z 3 2 3

Using the limits on the angles 6. and 5

T(b) > 10" sec"1 or r(b) é-IO"11 sec.

Note r (b -v CX] » r (b + ux) . (29) 412 R- Chanda

Also, if 62 = 63 = 0, x(b) -+• "• or b becomes stable! Determination of b life-time (at Cornell) would provide new constraints on the mixing angles. XM is to be noted that in the 6-quark model b-decay and CP-violation become related. jl< One can calculate the decay rates of t-hadrons in a similar way. Dominant decay chains are:

t -* b •+ c •+ s (30) u -*• d

Mixing may also occur in the lepton sector but present inform-

ation on leptonn flavorr conservationn (pp 7•/* ey, v -V+T ,T —A ey.) and T life-time does not require such mixing.

10. Charged Current Neutrino Reactions

Consider v N -»• y~X and UN * p+x, where X is any hadronic sys- tern consistent with conservation laws. Let E be the incident energy , M and p be the mass "and 4-momentum of the target and q the momentum 2 2 transfered to the hadron. Defining.-q = Q i 0 and v•= Ei2, we intro duce the scale variables

2 x = -^jO and y = M-;0íxíl,0íyíl.

Differential cross-section can then be written down in terms of the nu - cleon structure functions:

d2gv'v 2 dx dy ± xF3) + |(P2 ? XF3) (1-y) ] (31) 3

All the dynamics of the process is contained in the structure functions í\^(x, Q2) which can be extracted from the experiments as follows:

(32) 2 v z v VP n d g d a xt3 a dx dy dx dy

In the quark-parton model Bjorken Scaling in the deep inelastic collision becomes, natural: F^s are expressed in terms of quark-parton distributions where x becomes the longitudinal momentum fraction carried by the WEAK INTERACTIONS 413 struck quark; for example x F3(x) = x(q(x) - q(*i) • v — Experimental results show that the y-distribution of -—— does not require right-handed couplings of u and d to heavy quarks and consequently charge-symmetry is obeyed. Also, the linear rise of the total cross-section t^ot with incident neutrino lab. energy Ey — (s = 2ME,, -) up to E, - ~ 200 GeV. tells us that in + > 30 GeV.

11. Violation of Bjorken Scaling: A Test of QCD.

2 2 Experimentally F2(x, Q ) and xF3(x, Q ) show, a small but significant Q2 dependence even after removing target mass and threshold effects. QCD predicts scale breaking because of the dynamics of strong interactions.

Consider moments of the structure-function xF3 :

í1 7 t1 n 1 M (Q2) = dx xn~z xF.Cx, O2) = dx x (q(x,Q2)-q(x,Q2)) n JO >Q ,1 , NS = dx x q (x, Q2) Jn Within the framework of QCD one can compute the Q2 dependence of the non singlet quark distribution function qNS(x,Q2). The result is(13)

2 2 dn Mn(Q ) = Cn(lnQ )~ ; dn > 0 for n 5 2 (34) or In M (Q2) = -d lnQ2 + c (341) n n n 1 Comparing logs of a pair of moments (n, n ) we find dn/dn" experimentally to be compared with the QCD result

1 +, 1 22 y 1 d 2 nCn+U I j Vdn' = 3zL_ (35) n • z 2 n'{n'+3J j£2 j

for various values of n and n1 one obtains remarkable agreement of (35) with BEBC measurement of M (Q2I. This constitutes the best available quantitative test of QCD with spin 1 g.luons . 414 R. Chanda 12. Neutral Currents: VN, ev and eN Sectors

Hi (i) \)-N Reactions; Space-time and isospin structure of hadronic neutral currents have been investigated using large amount of experimental data on

(a) Inclusive: v N+VUX, v N + \7 X (quark distributions needed)

(bl Elastic: v N -*%N, v H •+• v N (Elastic form factors needed)

Analysing these data one can determine the couplings of the neutral currents to u and d quarks (main constituents of the nucleon):

(Hu) C 35 07) Neutral G | L " ' * - CÜu)Rx(-.19 ±.06)1 ^06) (dd] x (-.40± .07) (dd) x(0 ± .1) I TL Rn '

This is to be compared with the Salam-Weinberg predictions for the 2 couplings gT D « (I3L n - CQ)T „ sin 6 with standard I3 values and 2 sin ew = .23

(uu)LM.35) (uu)Rx (-.15)

(37)

(3d)T x (-.43) (Hd)x (.07 ) li K

The new reactor experiment at Irvine ' on v d + v np determines the

relative signs of the couplings uL and uR and confirms the opposite signs found above.

Comments: (a) Space-time structure is a mixture of V-A and V+A. fl4) (b) Isospin structure v of the neutral current is determined from v p •*• ~v pir" and %f n ->• \7 pit processes at CERN. The A (1232) peak in pir° and pir mass distribution shows that the neutral current has a 1=1 part to transform I = 3j nucleon to I = 3/2 A.

(ii) v - e Scattering: Pure leptonic processes are free from strong interaction effects and provides direct confrontation with theory. Expe- WEAK INTERACTIONS 415 rimental difficulty arises because of extremely low cross sections. New Fermilab experiments (-16) on v e~ + v e~ C46 events) yield sinz9 = ,25 + .05, The small error is quite an achievement,

>J e~ •+ v e~ and v e" + v e" CReactor) are also consistent with the standard model within experimental errors.

(iii) e~ - N Neutral Current Reactions: Parity violation in el d •+ e~X comes from the interference of neutral weak and eletromag - • (17) netic interactions. The asymmetry parameter A -,(x,y) is defined by

d^ da I dlT5 y + alTdyJ A j(x,y) f 0 when there is parity violation. The target deuteron being isoscalar, the x-dependence in A is cancelled out. An order of magnitude estimate for A can be carried out in the following way

0L-°R~ 0(GFe2)

2 aT + dp - 0(e") + 0(G e )

-10"4

In general we can write

A _ _ , _ 1 J-- \-J--Jf / [ ^Qj

For isoscalar targets al and a2 are constants related to the neutral current couplings of the electron and the light quarks u and d. Recently y dependence of A/Q2 has beccrae available in the range

0-1 S y S °-4 , enabling one to separately determine at and a2. Results agree well with S-W prediction with

sin2e = .224 ± .012 ± .008 w stat syst.

and rules out more complicated models proposed earlier. Parity violation experiments in atomic transitions are still not in agreement with each other. Note, the heavy atom parity violation experiments determine a different combination of e-q neutral couplings

than that appearing in eT _d experiments. 4ie R. Chanda (iv) - e+ - e~ neutral current effects should show up in y-Z" interference in Bhabha scattering. N-N and e~ - e~ neutral current effects are more difficult to see.

13. Important Unsolved Problem

(i) The age-old AI = \ rule in non-leptonic weak interactions is still with us. L,N L has AI = \ and 3/2 pieces with equal weight. W flRI It has been suggested that the dynamical enhancement of Al = \ piece is due to QCD correction to the weak amplitude. V w-

Effective |As| =1 quark scattering operator

Perturbative estimate in the lowest order in a gives an enhancement fac tor of 4-5 for the AI = Jj piece. Experimental enhancement factor is about 20. It has been pointed out that a new set of pure M = \ terms are introduced by QCD, generically called penguine diagram:

gluoni

Still the enhancement is not enough.. WEAK INTERACTIONS ' 417 (li) Attempts have heen made to calculate the weak, mixing angles 6^ in tetras of masses inspired by the. empirical relation

tane c =

However, the mass spectrum is not understood. It may be remembered that electron mass and charge are not calculated in QED.

(iii) Number of generations of families of leptons and quarks remain unknown.

Civ) Higgs structure of the theory and Higgs meson masses are not known.

14. Outlook

(a) CESR machine at Cornell will explore in detail the.b-particle fa mi lies. (b) Search for t quark will continue at PETRA, PEP and LEP. (c) PP Colliding beam at CERN may find W* or even Z° via Drell - Yan mechanism qq +r , Z°. (d) At LEP, CERN,2° resonances should appear at E^ - 90 GeV. (Z°- factory, - 1986). (e) Evidence of proton' decay may set the scale for the 'Grand Unific ation'. (£} Cosmic ray studies will play important role in elucidating particle physics at high energies.

References 1. S. Weinberg, Proc. 19tíl Int. Conf. High Energy Physics Tokyo 1978. 2. S.L. Adler and R. Dashen, Current Algebras and Applications to Part- icle Physics, W.A. Benjamin (1968). 3. R. Marshak, Riazuddin and C. Ryan, Theory of Weak Interactions.Wiley_ -Interscience, 1969. 4. C.N. Yang and R. Mills, Phys. Rev. 9£ (.1954) 631. 5. Por details see S. Coleman, These Proceedings. 418 R. Chanda

6. S. Glashow, J. Illiopoliis and L. Maiani, Phys. Rev. D2 (1970), 1285. 7. K. Berkelman, Proc. 19th Int. Conf. High Energy Phys., Tokyo, 1978. 8. K. Winter, Proc. Int. Symp. on Lepton-Photon Interactions, Fermilab, 1979. 9. R. Shrock et.al., Phys. Rev. Letters 42_ (1979), 1589. 10. R.N. Mohapatra, Proc, 19 Int. Conf. High Energy Physics, Tokyo , 1978. 11. R. Pecci, ibid. 12. K. Tittel, Proc. 19th Int. Conf. High Energy Phys., Tokyo, 1978. 13. See, For example, J. Ellis in Proc. SLAC Summer Inst., 1978. 14. C. Ealtay, Proc. 19th Int. Conf. High Energy Phys., Tokyo, 1978. 15. P. Hung and J. Sakurai Phys. Letters 88 B, 91 (1979). 16. K. Winter, Proc. Int. Symp. on Lepton-Photon Interactions, Fermilab, (1979). 17. C. Prescot et.al., Phys. Letters 77 B (1978) 347; ibid 84 B (1979), 524. 18. M. Gaillard, Proc. Int. Symp. on Lepton-Photon Interaction, Fermilab (1979).

I 41S

APPLICATIONS OF QUANTUM CHROMODYNAMICS TO HARD SCATTERING PROCESSES

C. O. Escobar Instituto de Física Teórica, São Paulo, Brasil

I. Introduction

It is our aim in these lectures to review the applications of Quantum Chromodynamics (QCD) to hard scattering processes .Throughout the lectures we shall keep in mind the basic results and general features of the simple parton model , so that we can examine which of these jfea- tures survive after taking into account the interactions of QCD.

We start the lectures by reviewing the philosophy and basic formula of the parton model, applied to processes like lepton-nucleon deep inelastic scattering (DIS), e e~ annihilation into hadrons, production of massive lepton pairs in hadron-hadron collisions (Drell-Yan process), etc. We then present a very brief summary of QCD, the main purpose of which is to set notation and to suggest QCD as the leading candidate for a theory of the strong interactions

The main topic of these lectures are the modifications introduced by QCD to the parton model. We begin by examining the leading order scaling violations in DIS, using the approach of Altarelli and Parisi ' . The next step is to consider processes which are not light-cone dominated (e.g., the Drell-Yan process). It is for these processes that great theoretical advances have been made during the last couple of years , regarding the applicability of perturbative Quantum Chromodynamics. A very simple recipe for parton model calculations in the framework of QCD emerges as a result of these theoretical advances.

As a first test-ground for perturbative QCD off the light-cone, we consider the phenomenology of the Drell-Yan process*9* . But it has to be remarked that there are also important corrections implied by QCD, beyond the leading order in the colour coupling constant ' . This is our next subject and we illustrate the importance of higher-order corrections by considering DIS and the Drell-Yan mechanism again. Finally we devote a section to further applications of perturbative QCD and consider some recent developments which have to do with the resummation of double logarithimic terms in the perturbative serie, namely the small-p dis- 420 c- Escobar tribution of lepton pairs (7) and the prospect of measuring the triple gluon vertex by looking for a marked rise with energy, in the multiplici- + — ( 8) ty of heavy flavours produced in e e annihilation . II. The Partorv Model

In the parton model picture of a hard collision (i.e. cne involving large energy and momentum transfers), we view a rapidly moving hadron as an assembly of collinearly moving partons (point-like constituents) one of which undergoes the hard scattering with the probe, may it be an electromagnetic or weak current or even another energetic parton belonging to some other hadron (as is the case in large-p_ processes) . For the va- lidity of this impulse picture, it is crucial that we can talk about two time scales, a short time scale determined by the large momentum transfer in the collision (T ~ 5) and a long time scale set by the soft forces responsible for the binding of the constituents into hadrons (T - hadroni^mass scale) • Ifc is the clean separation of time scales that enables us to write the cross sections for hard processes in a convolution form, involving cross sections for collisions between partons and structure functions which have to do with the binding and are therefore -') not accessible to perturbation theory. A typical parton model expression for an inclusive cross-section of a hadron with momentum P is of the form, • i parton I-- da(P) ="Í t ÍI dB f,(B)do. (BP) (2.1) The functions f^B) give the distribution of partons of type i inside the hadron, carrying a fraction B of its momentum (this is governed by the long time scale) and dc^arton is the hard scattering cross section involving partons.

In the following we will apply the general form (2.1) for parton model cross sections to several hard processes so as to abstract their common features for later comparison with the QCD modified parton model.

1. Deep Inelastic Electron-Nucleon Scattering (Fig.l)

Dropping obvious kinematical factors, we have,

f r _ aH(x) = I eflx qj(x) + x qj(x) (2.2)

n where x = 3— , e, is the electric charge and(qJH are the distri- 2P.q x - x bution functions of (anti-) quarks of flavour i inside the hadron H (f HARD SCATTERING PROCESSES 421 is total number of flavours). So in DIS we directly measure the momentum distribution of quarks.

2. Semi-inclusive e+e~ •+ H+x (Fig.2)

Here we have (again dropping kinematical factors) ,

dg (z) = I e£ zfõg U) + z D3 (Z)1 (2.3)

2PH H where z ^ —^- and D_ (z) gives the probability of a quark i fragmenting into a hadron H, carrying a fraction z of its momentum.

3. Massive lepton-pairv ' (Fig.3)

where, T = M2 /s and

What are the common features in (2.2-2.4)? We immediately notice the following characteristics. ti a) They involve quark distributions xq.(x) or/and quark fragmentation H functions zD (z) which exhibit Bjorken scale, i.e. they depend solely i o of x and/or z but not Q .

b) Factorization: In x, and x2 in the case of (2.4) , in x and z for eH. •*• e + Hj + x, semi-inclusive electroproduction (Fig.4) ,

F H H IS 2 a° (X,z) = I e? xq^ix) z Da (z) + q H1'H2 l 1L qi J c) The distribution and fragmentation functions are process independent i.e. universal, we may use a quark distribution extracted from DIS in the calculation of the Drell-Yan cross section.

d) No mention is made in the above formula to partons other than quarks. This cannot be the whole story since there must be other quanta in order to bind quarks into hadrons. How can their presence be felt? 422 C. Escobar

Fig. 1 Parton model diagram Fig. 2 Parton model diagram for DIS. for semi-inclusive e+e~ -> H + x.

Fig. 3 Parton model diagram Fig. 4 Parton model diagram for for massive lepton semi-inclusive lepto- pair production. production.

e) The expressions above have a very simple probabilistic interpretation but in quantum mechanics we talk about amplitudes, so where are the interference terms?

After this brief presentation of the parton model we now turn to a quick review of QCD.

III. Quantum Chromodynamics

We will not review here the reasons leading tc the introduction of a new degree of freedom for the quarks, called colour, such that quarks come in three colours*3' (solving the spin-statistics problem in the quark HARD SCATTERING PROCESSES 423 model for baryons, improving the ratio of TT°->-2Y by a factor 3, improving the ratio R = a (e+e~ -»- hadrons/a(e e~ -> p+u~) by a factor 3 again) . Once the evidence for colour is accepted it is natural to construct a field theory in which this new quantum number plays a dynamical role and therefore to introduce colour dependent forces. The simplest possibility for achieving this is to make colour an exact local gauge symmetry with a gauge group SU(3) , and introduce eight coloured massless vector mesons as the carriers of colour forces (the so called gluons). As the gluons themselves are coloured we expect self-coupling among them and thus the theory must be qualitatively different from QED, as actual computations have indeed shown it to be.

yí Quantum Chromodynamics has the following Lagrangian density, |J<

where f labels the quark flavour (u,d,s,c,b, ?) , a = 1,...8 labels S the colour of the gluons and a,g = 1,2,3 label the quark's colours. jfeL

Ga the gluon field tensor is given by, j|f

Ga =3 Aa - 3 Aa + gf3^ Ab AC (3.2)

and D11., the gauge covariant derivative, acting on the quark colour components, is given by

Da8 " *a»\ ~ ig Ta6A P (3"3)

T . are the eight hermitean traceless matrices which generate SU(3) , , abc colour with f the SU(3) structure constants, we have b T ] - TC (3.4)

The Feynman rules from this Lagrangian are indicated below (in the r Feynman gauge) gluon propagator k2 C. Escobar 424

ghost propagator k2 + ie

i.—^ + m 6 quark propagator k2-mz+ie af5

Vertices

-ig (Ta) quark-gluon vertex 6

-g fabC[g)JV(k-q)X + gvX (q-r) V

+ gXv(rr-k)- v three gluon vertex

g fabc gluon-ghost vertex

fcdg(gvXgvp-gupgvX)

facg

four gluon vertex

The new features in this theory, at the level of Feynman rules is the appearence of self-couplings between the gluons and of colour factors in the vertices (we shall never worry about the ghosts, a proper choice of gauge will eliminate then) .

It is quite simple to deal with the colour factors, since they factorize outside the loop integrals and are expressed in terms of the eigenvalues of the quadratic Casimir operator C_ of the fundamental (quarks) re- HARD SCATTERING PROCESSES . 425 presentation and CA of the adjoint (gluons) representation. Next we give examples of such colour factors,

fiaBCF= 2 —- 6 o= 7 s o

aCd bcd 6a b. CA = f f = N fiab .

C. = 3 (N=3 colours) . A

There are also factors proportional to the number f of quark flavours, which are independent of Cp and Cft,

f TaB

Sab For the purposes of these lectures, the most important property of QCD is its asymptotic freedom , i.e., the vanishing of the effective coupling constant as we go to higher and higher momentum. This property immediately raises the hope of understanding why the parton model is such a good description of hard processes. It is, however, not a trivial task to obtain the parton model from an interacting field theory like QCD, even allowing for the fact that it is an asymptotically free field theory (we could think that at really large momentum transfers the interactions between quarks and gluons could be switched off). It is our main objective in the following to show why this is not trivial and how this goal can be achieved.

IV. QCD Corrections to the Simple Parton Model

i. Scaling Violations in DIS

Instead of applying the formalism of the Operator Product Expansion (OPE) plus Renormalization Group Equations (RGE) in the study of scaling violations in DIS, we shall follow a more intuitive treatment, 426 C. Escobar closer to the parton model language, due to Altarelli and Paris! (AP) , ;; which is however equivalent in leading order to the formal approach. ;T>

Remember that in the parton model Bjorken scaling was obtained thanks ,, to the fact that we neglected the interactions between partons and assumed a clean separation between short and large time scales. So for DIS (Fig.l), it is clear that the cross-section will be proportional to a ~ 6((ZP4q) ) - 6(x-z), thus leading to Bjorken scaling. However, in an interacting field theory we have to consider the radiative corrections to the Born cross section (fig.5). 2 Naively we would think that these corrections are down by order a (Q ) '., and since QCD is asymptotically free, t»g(Q ) << 1 for high enough Q and thus the radiative corrections would be negligible. Unfortunately this is not true because when calculating the cross section we get a factor log ^ (p is the momentum the incoming quark or gluon, p^ ~ 0 (300 MeV )) from the integral over the loop momentum and so the true ex- 2 9 O^ ' pans ion parameter is not a (Q ) but is instead a (Q^) log — which is •: s s p2 _"< 0(1). P ,' We will now show how the log Q factor comes about. When calculating cr- the cross section we square the amplitude shown in Fig.5, so for ; instance the term 5a when squared gives (Fig.6) a loop integral of the form, ç 4 2 2 2 d k fi(k )6((p+q-k) - m ) f(k) •í Rp-k)2 - m2l

4 2

but d k = dk0 k dk d(cos )d$, where 8 is the angle between k and p.

The dkQ and dk integrals are easily done using the two 6 functions and using, 2 2 (p-k) = m - 2EÜS + 2 |p| u cos 8 where a = k = |k| , the e integration is

1 d Í

But Ew ~ Q and 2Eu> - 2|p|iii - p , so we obtain a term proportional to log Q /p . We see that the logarithm comes from the angular region where 8=0, that is, from the region where the gluon is almost paralell to the quark. This is a well known example of mass singularity (emis- HARD SCATTERING PROCESSES «27

Fig. 5 Sane radiative corrections of order g , to the Born cross-section for DIS.

l-z

Fig. 6 A diagram which gives rise Fig. 7 The vertex q-Kj+g, responsible 02 to a log ^=2- term. for the scaling violations in leading order for non-singlet I distributions (see ec. 4.4).

Fig. 8 Radiative corrections which only contribute at z = l. 428 c- Escobar r.

sion of hard collinear massless quanta) dealt with by Kinoshita, Lee \(:_ and Nauenberg^ ' . fit.

We come to the conclusion that the quark distribution must depend on 9 2 Q and therefore it does change when we change Q , the resolving power of the probe, Q2 2 q(x,Q2) = 0(

p z Following Altarelli and Parisi, we introduce a function a+_ ( ) , which will govern the evolution in Q of the quark distribution and which has a simple interpretation as the probability of a quark emitting another quark with longitudinal momentum fraction- z (Fig.7).

From now on we consider flavour non-singlet distributions such as F^" eP eN " and F2 - Fj so that we do not have to worry about the gluon contribution to the evolution of the quark distributions (Fig.5c) . For this non-singlet distribution we can write with A.P , ' j Q2X,qN-S-(x,Q2) = "f

It is, at this stage, convenient to introduce moments of the distri- Í butions, defined as follows,

1 ( dx x11"1 qNS(x,Q2) (4.5) 0 Since the right hand side of (4.4) is a convolution of qNS and P taking the moment of the left hand side, we obtain,

" =- = — MK,(Q2) * K, (4.6) d log Q2 2TT ^ « where N 1 AJJ = I dz z ~ P_MI(x) (4.7) 0

2 I But ag(Q ) = — j to leading order i'" (11 - I f) log S_ HARD SCATTERING PROCESSES 429 therefore,

2 ^N = 2_ d log Q

2 2 Integrating this equation from Q to Q , we obtain,

(4.8)

where

(4.9)

So, we arrived at the same expression for the moments as obtained via the more formal approach of the OPE plus RGE . Here the d^, are the anomalous dimensions and once we know P-^-tz) they can be calculated q-»q using (4.7). So our next task is to find the function P (z). n (12) In QED, the Wei zs acker-Williams ' equivalent photon approximation, permits the calculation of the photon density inside an electron,

(4.10)

where

(4.11)

for z^O. The only difference in QCD, is a colour factor «- = E Ta T D pa so that we obtain (4.12)

and by z + 1-z, Wz) • I (4.13)

How do we deal with the singularity at z=l . It is reminiscent of infra-red singularities due to soft gluon emission and as usual they should be regularized by radiative corrections (Fig.8) which do only contribute at z=l. With these corrections we replace 1/1-z by a distribution denoted by — defined in the following way. c 430 - Escobar

dz dz (4.14) for any f(z) regular at z=l.

Now we examine the constraints which must be satisfied by the function P (z). Since quark number is conserved in the bremsstrahlung of gluons, 111*1 P (z) has to obey the following constraint, q->q

dz (4.15) but using the regularized form of P .(z) ((4.3) with y^ replaced by q-Ka+Cj ) we see that (4.15) is not satisfied and therefore we must include a term fi(l-z) in P _(z). So

WZ) = 3 the coefficient a is easily found by imposing (4.15) and we find a = 3/2, so finally we obtain 4 ll+zj (z) «L. + 1 6(1-2)1 (4.17) L-z)+ * J

With this we can easily compute the anomalous dimensions for the non- -singlet case according to (4.7), finding, N "2 I X j=2

To summarize, leading order QCD predicts for the non-singlet moments,

-id. N (4.19) log 4 where fl = - -— and A., as given by (4.18) .

The above form suggests two tests of scaling violations. Since 2 2 log ^(Q ) - - djj log Q + const, we can compare the logarithms of two moments Mj, and My and should find, if QCD is correct, a straight line with slope given by HARD SCATTERING PROCESSES 431

N 1 j=2 5" (4.20) [~ 2+ M(M+1) " 2 ^2 3" J

We show in Fig.9 the data from the BEBC*13' and CDHS(ll>) collaborations which are in good agreement with the QCD prediction. This test is sensitive to the spin of the gluons, a scalar gluon would give results which are in disagreement with the data, as indicated in Fig. 9.

A second possible test of QCD, based on (4.19), is to compare logarithmic versus power lav scaling violation. Consider the quantity [Mj^0- )~}~ N' leading order QCD predicts that it should vary linearly with log Q2. The intercept of this straight line gives the fundamental parameter of QCD A2 to leading order (see later for higher order corrections). Fig. 10 shows the data^13'11** which is again consistent with the QCD prediction and not consistent with a power law violation of scaling.

We have only considered the non-singlet structure functions, there are also possible and useful tests of QCD involving the singlet structure functions 16, but unfortunately we do not have time to go into these. We have also skipped several technical problems like the question of higher twist effects, Nachtmann versus Cornwall-Norton moments etc. We refer the reader to the literature, for detailed analysis of these problems(17).

ii. QCD Off the Light-Cone: A Recipe for Parton Model Calculations

We have just seen how to describe the scaling violations in DIS using a simple parton model language (the Altarelli-Parisi approach). However, for DIS this is not new since by using the OPE and RGE formalism we already knew how to sum the leading logarithms. The question is: can we do the same for processes which do not have an OPE, in particular processes with two initial hadrons, like PP •+ v~V" + x, PP •+ ir+ x at large - P_? The answer is yes 5 and the essence of the proof is to 2 2 show that the mass singularities (log Q /p ) factorize and can be absorbed into the definition of scaling violating distribution and fragmentation functions of incoming and outgoing hadrons, respectively. These singularities are universal in the sense that they are associated to the incoming (or outgoing fragmenting) quarks and gluons and thus do not depend on the particular hard process under consideration. O.I 250 1 ' 1 11 / 1

/ 200- / / z 0.01 / / / UJ / / / 5 150- / \/ N =4 i is o 0.01 h dN=0.775 A 4/100- r , 2 X/ _ Scolõr A _V. Gluons 1 50 - Si 0.001 i i i i I .. i' i i i I 0.01 0.1 0.1 - N =3 1.0 dN=0.6l7 LOG OF MOMENT $)> T , 1 i l 10 100 2 (u l) 2 2 Pig. 9 Logarithmic plots of moments of F3 (x,Q ) '" l Q (GeV ) The solid lines are the QCE predictions {eq. 4.20), the dashed lines are the results from a scalar gluon Fig. 10 Plots of Clog theory. Data fran BEBC (Ref. 13). Dashed lines are power-law fits. o

W to O 8- tu HARD SCATTERING PROCESSES ' 433 Most of the calculations ' are done in a special gauge, the so called transverse gauge (n-A = 0, n2 < 0) in which the sum over the polarxza- tions of the gluons is given by

k.Ti (n.k)!

We see that this gauge is a physical gauge, since it does only involve , the two transverse polarizations, i.e. for k2 = 0, -g Pvv = 2 and | k Pvv =0. In this gauge the diagrams giving leading mass singularities ; are of the ladder type (Fig.11) where the dots indicate vertex and r self-energy insertions. These diagrams are amenable to a parton-like description, while the diagrams involving the crossed ladders (Fig.12), which would invalidate the parton model interpretation, do not contri- : 'v. bute to leading order.

It is not difficult to see why this happens, if we remember that in a physical gauge, the vertex for emission of a gluon by a massless quark •.'- vanishes as B , the angle between the quark and the gluon, goes to zero. The propagator l/(p-k)2 , in Fig.13 behaves as 1/82 and the phase space ,| is proportional to 6 de. Thus in a transverse gauge, a diagram like ' Fig. 13 behaves as

(r f e ae log^i (4.22) •:J • J. r. P2 e2 e2 p2

- _ while a diagram like Fig.14 behaves as il • f H yi' 6 de — = finite (4.23) ]•: J e2 This simple argument has been generalized to all leading logarithmic orders '.

Another important remark is that the effect of vertex and self-energy ; insertions is to enable us to put at each vertex the effective coupling '•• •• 2 2 constant g(k^) , where k^ is the largest four momentum squared at the t vertex, [it should be stressed that the dominant region of integration .-V ' for the ladder diagram comes from the ordering of four momentum squared, ) ' p2 « k2 « k2 ... << k2 << Q2 ] .

', l" ; ; The up-shot of these studies is the following recipe for calculating --:% hard scattering cross sections for which there is one large invariant 434 C. Escobar

Fig. 12 A crossed-ladder diagram which does not contribute to the leading logarithm approximation. Fig. 11 The dominant diagram, whithin the leading logarithm approximation, for a hard process (represented by the cross) involving a momentum transfer Q . The dots indicate self- energy and vertex insertions.

Fig. 13 A diagram contributing to the Fig. 14 A diagram, «iiich in a leading logarithm approximation. transverse gauge, does not contribute to the leading logarithm approximation. HARD SCATTERING PROCESSES 435 Q2 » p2. A2. ). a. Use the parton model formula R*i b. Replace the scaling distribution and fragmentation functions by non- -scaling ones (at mass scale Q ) c. Convolute these with the lowest order cross-section for the parton

subprocess, calculated using the effective coupling constant as(Q ) .

For example, the Drell-Yan formula is modified as follows, da 4ira2 „ I ãQ' 9Q4 i (4.24)

Notice that to leading order in QCD, a simple feature of the parton model is still valid, namely, factorization in x^ and x2 (the factorization in x and z for semiinclusive electroproduction. Fig.4; is also valid after allowing for leading order QCD corrections).

In the next section we will study the phenomenology of the Drell-Yan process.

V. Drell-Yan Phenomenology

Looking at (4.24) which is a very simple expression, we can imediately devise some tests of the Drell-Yan model.

The first test is the approximate scaling of the Drell-Yan cross-section, that is Q4 ^ = F(t) (5.1) dCT where we have neglected small scaling violations in the quark distribution. We show in Fig.15, data from the CFS collaboration at FNAL , por Pj^ = 200, 300 and 400 GeV/c. The data supports the approximate scaling behaviour shown in (5.1). The best test for scaling would undoubtedly be to compare the FNAL data at /s = 20-27 GeV with the ISR data at /s = 60 GeV, the trouble being that there is a small overlap in T between FNAL and ISR.

A second test has to do with the beam-type dependence of (4.24). For beam particles containing antiquarks as valence constituents (IT'S, k's, M/Vs 1 1 02 0.3 0.4 05 0,6 o a 200 Ge V/c I 1 1— ü? 1 u - A 300 » . .1 * GeV/c - 3 400 GeV/c C> 11o- ' • 11 < > io2r O g LJJ J CM

E 10-32 _ TI ! 1 -o 43 1 -

- - D < 33 % IO" W- O -

CM •o 1 1 I • it 9 L4 0 1 b

à 1 11 ** 8 * CM t "O 0.6 1 1 1 . 1 > 1 .1 I 0.5 • 0.2 0.3 0.4 6 8.10 12 2 Fig. 15 Approximate scaling of the Drell-Yan cross- M (GeV/c ) section (eq. 5.1). Data from Ref. 19 at three n energies, lhe figure at the botton displays the ratio Fig. 16 The ratio of it induced to proton induced y y. pair of the cross-sections. cross-sections, as a function of the mass of the pair. Data fran Ref. 19. o & sv H

• -r- HARD SCATTERING PROCESSES 437 p's), the cross-sections for lepton pair production should be much larger than those involving sea-valence annihilation (nucleon-nucleon collisions). This intensity rule should be even more marked at large values of T , where we probe the quark distributions at large x. We show in Fig. 16 the ratio ir~P -»-p+i/" + x/PP ->-y%~ + x , which exhibits the qualitative behaviour expected in the Drell-Yan model. We should mention that the study of massive lepton-pair production enables us to determine the (20 21} quark distribution in hadrons other than the nucleons ' A third test of the Drell-Yan picture is to check the spin -1/2 nature of the quarks, by looking at the angular distribution of the leptons in a suitable frame(22) .

In the centre of mass of the lepton pair, assuming that they are produced via one-photon exchange, the angular distribution is given by,

2 2 — = WT(Q) (l+cos e) + WL(Q) sen e (5.2) -j dQ d cose

where Wm and WT are transverse and longitudinal structure functions, .1 T Jj | respectively. ' In order' to define the angle e, we must choose an axis in the lepton ']• pair center of mass frame. One familiar definition is called the Gottfried-Jacks on (G-J) fratie , the z axis being chosen along the ': beam direction. This frame would be a very convenient one, had the quarks no transverse momentum in which case the beam direction would -:.{ coincide with the quark direction of flight. We know however that ;•"• quarks do have a certain transverse momentum (more on this later) so another frame was introduced by Collins and Soper {-O-S) , in order to minimize the effect of the quark's transverse momentum. For our purposes there is little difference between the G-J and the C-S frames. If quarks have spin- 1/2, the longitudinal structure function W_ is zero, so we are left with A- O 45.3)

•'; The data is usually fitted with the form 1 + X cos 8 and for ir-nucleon

j ; collisions at Plab - 200 GeV/c and 3.5 GeV < Q < 9 GeV, the result of ';• the CIP collaboration(20) is X = 1.30 ± 0.23, using the C-S frame, while ' the NA3 collaboration*21) at the SPS finds X =0.80 ± 0.16 using the G-J frame and X = 0.85 ± 0.17 using the C-S frame. We see that these results support spin-1/2 quarks.

\l 1 438 C. Escobar The deviations from X=l, as well as the dependence of X on the rapidity of the lepton pair are important, since they could signal the presence of higher order effects, which we will discuss later on.

We conclude this very brief discussion of the Drell-Yan phenomenology, to which we shall return when discussing higher-order corrections in QCD, by saying that the data supports the naive parton model picture. The main task is to look for the QCD corrections in the experimental data.

VI. Beyond the Leading Order

i. Deep Inelastic Scattering

We have seen that in lowest order QCD predicts for the non-singlet moments, dxxN 1 dN " J " (6.1)

with -±_-2 I i N(N+1) j=2 3

But B and A^ are just the first terms in the expansion of the B function and of the anomalous dimension in powers of g,

(6.2) 2 3 5 •B(g ) -Bog - ,g +

Going to the higher orders in QCD imply calculating the next terms in (6.1). A typical diagram contributing to B, is shown in Fig. 17. The calculation of the anomalous dimensions to order g also involve two-loop diagrams (2"),

Caswell and Jones.(25K25>) first calculated ^ and found, Fig. 17 Typical two-loop diagram contributing to the. Bx = 102. - — f . B-function to order g5 (eq. 6.2). Due to the lack of space, we will illustrate the importance of higher order corrections by means of one simple example, the ambiguity in ;the HARD SCATTERING PROCESSES • 439 determination of A in lowest order QCD. (26)

When higher orders are considered, (6.1) gets modified.

(6.3)

where the C.T involve A»T and BT and were calculated by Flor atos et ai. and Bardeen et ai. A leading order calculation consists in neglecting the O(—=—j- ) term relative to the 0(logQ ) terra. But this is dangerous log ÇT , since the definition of A depends on the neglected terms 0( j). This is most easily seen by rescaling A, Og

= a A (6.4)

Then, using the leading order expression (6.1)

2 2 A"

(1 + .. ) (6.5)

But since we are neglecting the terms of order Q2//,2 we find that MJJ(Q ) = Mi(Q ) ! So we come to the conclusion that to leading order, different A's lead to the same Q dependence for the moments, so it is impossible to determine A unambigously to leading order.

To eliminate this ambiguity, we have to include higher order terms. r - ~\ Following Bardeen et al , we define a leading order parameter AT _,(N) , where -C» AL.0.(N) " A (6.6)

with this definitions the moments ') in (6.5) can be written as,

o2 "dN C.. N (1 + + .... ) - dog a.) b

(log (1 + 0(- ) L.O Q2 -dv = (log (6.7) L.O. 440 C. Escobar which is the same as the lowest order expression but now AL Q depends on N in a calculable way (6.6). This conclusion is very important for the phenomenology of QCD, since fits using the same A for different N are in principle wrong. The correct procedure is to either consider the higher-order corrections directly in the expression of Mg(Q ) (6.3),

or introduce these corrections into the definition of AL Q (N).

Fig. 18 shows the dependence of A on N, as given by equation (6.6), compared with an analysis of combined muon and electron deep inelastic data (see reference (27) for more details). Strictly speaking only

ratios A. n (N)/AT n (M) are significant, due to the fact that the coeffi cients Cj, depend on the subtraction scheme employed when regularizing

the theory. CN is of the form,

where y is an arbitrary number, independent of N but dependent on the (ZB) * * subtraction scheme . Another correction which is of some interest is introduced in the /29\ log MJJ vs log M^ plots. When higher-order terms are considered we obtain

2 2 dN F as 1 log MjjiQ ) = const + log W^(Q ) * •£• 1 + C^ •£ + (6.8)

where

and dM N are two-loop anomalous dimensions. The following table shows values of C^ for low values of M and N

H N CMN 2 4 0.42 4 6 0.21 6 8 0.15

These numbers show that we do indeed have a good perturbative expansion for the quantity log K^ vs log M^, with the corrections to the slope being of the order of 10% at moderate values of Q2.

The conclusion on higher-order corrections in DIS is that they are small, generally within the errors of present data (for more details see a HARD SCATTERING PROCESSES 441

p-n AN • F

1.0

0.5

Fig. 18 Experimental results for A.(N: )' compared.with the definition (6.6). From an analysis of Duke and Roberts (Ref. 27). 442 c- Escobar recent paper by Para and Sachrajda '). As we will see next, this conclusion is, unfortunately, not true for other processes, like massive lepton pair production. ii. Higher Order Corrections to the Drell-Yan Process.

In Section V we presented a brief review of the phenomenology of the Drell-Yan process, but neglected the' QCD corrections. In this section we consider these corrections and compare them with the available data.

The first obvious QCD effect is a small scaling violation to the simple 2 2—2 scaling behaviour (5.1), through the Q dependence ofqix^Q )q(x2rQ ). The available data is not precise enough to see this small violation. A second manifestation of QCD effects, is through the transverse momentum distribution of the lepton pair . In the simple parton model the leptons have a small and limited transverse momentum, = 300 MeV/c, due to the Fermi motion of the quarks inside the hadrons (this is sometimes called the intrinsic transverse momentum of the quarks). But in QCD there is another source of large transverse momentum, coming from the diagrams of Fig.19, for the processes, q'+g •+ Y*+ 5 and _ 2 q+q -»• g+Y*, which are down by O(CL (Q ) relative to the qq annihilation. They should however be important when we look at lepton pairs with large 2 2 transverse momentum, P* ~ O(Q ) • QCD makes an unambigous prediction for the behaviour of the average p+y- transverse momentum of the pair. At fixed T = M + , , < P,

a) b)

Fig. 19 Diagrams giving rise to large-PT muons pairs. a) Compton diagram: g+q •+• y*+q b) Annihilation diagram: qq -+ Y*g. HARD SCATTERING PROCESSES 443 increases with the energy, being given by the following expres- (32) 1 sion,v '

v 2 2 = as(Q ) /I" f (T,as(Q )) + non-perturbative term (6.10)

It is clean from phase space considerations, that at T-0 and 1, f(t) must vanish." In order to compare with the data, we should keep x fixed and vary the energy, unfortunately as we commented before, there is a very small overlap in T when going from FNAL (/i~ = 19-27 GeV) to ISR (/s~ = 60 GeV) energies, so we make use of the flatness of with T, which is confirmed experimentally and average the data over x, obtaining the plot of Fig. 20 showing versus /s~ . It is difficult to make a quantitative comparison with the QCD predictions for the slope due,

r CFS CHFMNP 200I 30I°t 400

=0.6 + 0.022 VS GeV/c 0 I 10 20 30 40 50 60 Vi (GeV) Fig. 20 Average transverse momentum versus /s" .

among other things, to the unknown gluon distribution. Also there is 2 2

an ambiguity in the value of a (Q ), in particular, do we use as(Q )

or ag(s)? The last question can only be answered when higher order corrections are included (see section VI i) .- Anyway, there is defi- < > nitely an. increase of PT with energy which is evidence for QCD effects in the Drell^Yan process. We will leave the discussion of the

small-PT region for the next section. Of course the perturbative cal-•

culation based on Fig; 19, cannot be extended to Pm=0 since the cross 2 section diverges there as 1/P_. We now consider a third and more disturbing effect of QCD in the Drell- -Yan process. The result given in (4.24) was obtained in the leading 444 c- Escobar log approximation. When the first order corrections to the leading loq calculation are considered, it is found that they are very large (3 3) and modify it as follows' ,

~ , Air dx,dx_ P - H. , H9 - 1 2 2 2 2 d£_= 4*«i _±_l j e2 q (x,,Q )qi (x2,Q ) ± dQ2 9s J xL x2 [U

2 + l+-*2 6(l-z) + as(Q )8(l-z)fq(z)J [ H H "I 1 2 2 2 j^ F2 (x1,Q )G (X2,Q ) + 1 ++ 2 *

2 x as(Q )e(l-z)fG(z)> (6.11)

where

— , F*(x,Q2) = x L e2[q?(x,Q2) + 2 x,xXl*2n i L and

4 S /i . 4TT2» . /i • 2 osfq(«) - % £ Id + ^)«(l-«) + 2(l+z )

+ —5 6 - 4z j (6.12) (1-zK

2 2 2 a f (z) = I Ü1 (z + (l-z) )log(l-z) + - z - 5z + 2. | (6.13) sG 2 2, 2 2

We show these corrections in Fig.21. The first term in f (z), 4 a 4ir2 " j2* (1+ -j—)6(l-z) comes from the vertex and soft gluon corrections shown in Fig.20.a. ' The other terms in f-iz) are due to hard brentnstrahlung

of gluons (Fig.21.b), while the term fG(z) corresponds to Fig.21.c. The gluori corrections are not very large, however the quark corrections f (z) are really large, the most important term being that proportional to 5(l-z) which renormalizes the usual Drell-Yan cross section (the term proportional to u2 comes from the continuation from space-like to HARD SCATTERING PROCESSES 445

* «1

Fig. 21 Corrections of order ag to the Drell-Yan process: a) and b): virtual and soft gluon corrections c) hard gluon bremsstrahlung d) gluon contribution.

time-like Q of (log S_) terms, a continuation that is necessary when p2 comparing DIS and the Drell-Yan process) . We show in Fig.22 the effect of f_(z) on the Drell-Yan cross-section. As we see, their effect, in the present range of x is to simulate a large change in the normalization of the "zero order" cross section. It is possible that this effect is being seen in experiments with pion beams at FNAL(20) and at the CERN-SPS(2l). It is however premature to reach any firm conclusions on this. If the first order corrections are already so large, can we believe that the next corrections are indeed negligible? Of course more study has to be devoted to this subject and a proper phenomenology of higher-order corrections has to be developed .

VII. Future Prospects

i. Summing Double Logs.

a. Small-Pt distribution of lepton- pairs

Up to now we have considered the production of massive lepton pairs with either P2 * Q2 (section VI) or P2 - 0(p2) << Q2 (section V). In both cases there is only one large invariant and therefore dangerous 446 C. Escobar

0.01 0.05

Fig. 22 Effect of O(a ) corrections to the ir~N cross section. From Altarelli et al (Ref. 33).

2 2 logarithms of the form log PÍ/Q could be neglected in the first case 2 o2

(PT - Q ), while in the second case the log *£_. terms were absorbed into the quark distributions giving rise to the QCD modification of the simple Drell-Yan expression (sec.eq. (4.24)). In this section we consider a new development in the perturbative approach to QCD, which consists in examining situations where there are more than one large invariant'. Consider, for instance, the production 2 2 of lepton pairs with transverse momentum P* , much smaller than Q but larger than P (or A2),that is, HARD SCATTERING PROCESSES 447 Q2 » P2 » P2 (7.1)

In this case we have to keep track, to all orders, of terms like log Pr/Q , with the additional difficulty that now it is not obvious which mass scale will govern the violation of scaling of the quark 2 2 2 2 distribution, do they come from log Q /P or from log PT/P ? This problem was investigated by Dokshitser, D'Yakonov and Troyan (DDT) , who arrived at the following expression for the cross section for pro- 2 2 ducing a massive lepton pair of mass Q , with transverse momentum PT at rapidity y, such that Q2 >> P2 >> P2 ~ A2,

2 H H O2 do _ 4™ a (_2 l( 2. - 2, 2. Q - y *• e. q.= \x. ,FTJ q. \x~,f-) 2 2 2 dQ dP dy 9sP 3 log PT i L + 1 «-+ 2| t T2(Ê- ) (7.2) a2

where,

2 T(X) = f dzz expp - -\ 3I-- asalog X (7.3)

We notice the following novel features in (7.2). First of all, it is

not Q but PT which sets the mass scale for the scaling violations in the q, q distributions, this results from summing the terms , eT ,n ( o nlog -j- ) s Pz Secondly, there is a Sudakov form factor T2 (Q2 /P_) 2 , which measures the probability of producing a virtual photon of mass Q2 , without emission of gluons having transverse momentum greater than P . The Sudakov form factor comes from the exponentiation of double logarithms of the form

Parisi and Petronzio ' have made a careful study of the DDT calculation

and concluded that as Q increases the PT distribution of the lepton pair is less and less sensitive to the intrinsic transverse momentum of the quarks. They claim it is possible to calculate the P distri- 44g C. Escobar bution.perturbatively, down to PT=0 • The physical reason behind this is shown in Fig.23, As Q increases it also increases the probability of multiple bremmstrahlung, which washs out any memory of the initial transverse momentum quarks may have. Then, asking for a pair with 2 1 PT >> A , is something that can be calculated in perturbation theory .

We show in Fig. 24, the result of a calculation by Parisi and Petrcnzio17 , 2 of the Pm distribution down to P™=0 (the parameter M is chosen such o p 2 as to avoid the singularity in s( T)) •

( 8) ii. Heavy flavour multiplicities at very high energies

The authors of Ref.8 have shown that in an transverse gauge (n.A=0), the main contribution to the heavy flavour multiplicity in e e~ annihilation comes from the ladder diagrams shown in Fig.23, where the final 2 2 gluon is off shell by QQ > 4m_. The summation of double logs log2 Q2)n yields for the multiplicity, 2 /2~ 2 1/4 Nn(Q ) - (log Ry)~ exp /const log %• (7.4) Q Q2 / Q2 which rises faster vJian any power of log Q but slower than any power 2 of Q . The above mentioned authors stressed that this result is very sensitive to the three gluon vertex and in Fig.25, taken from their paper, it is shown the behaviour of the multiplicity N (Q ) (eq.7.4) compared with the exact lowest order calculation, which does not involve the three gluon vertex. So, it is possible that at LEP energies this marked rise in the heavy flavour multiplicity becomes apparent, thus signaling the presence of the three gluon vertex, an unmistakable characteristic of ''

Among the more recent developments of perturbative QCD, we should also mention the interesting attempts that have beem made to extend it to exclusive processes, which for long have been thought to be outside the region of applicability of perturbative methods. We do not have the time to go into this important new branch and instead refer the reader to the literature(3S'. HARD SCATTERING PROCESSES 449

Big. 23 Multiple gluon bremsstrahlung in Drell-Yan process.

Fig. 24 Transverse manentum distribution of the lepton pair, theoretical predictions fran Parisi and Petronzio (Ref. 7) cairoared with the data of Yoh et al. (Ref. 18). (S = 750 Gev\ Q2 = 56 GeV2).

Fig. 25 The Qz dependence of the charm multiplicity in e e~ annihilation (from Ref. .8). The dashed line is the exact lowest order calculation. 450 C. Escobar

VIII. Conclusions

In closing these lectures we would like to call your attention to several points awaiting further clarification and for which we can expect that a great deal of work will be done in the coining years.

1. It would be nice to perform (or improve) higher order calculations for several processes (large-P_, Drell-Yan, etc) and see if they are indeed so large (see Section VI). Also a more careful phenomenology of these higher-order corrections has to be developed.

2. Can we finally forget about the intrinsic transverse momentum of partons, which has unfortunately vitiated much of the phenomenology

of QCD (in Drell-Yan, high-PT etc)? See Section VII i.

3. Is it possible to isolate the three gluon vertex, maybe at future machines (LEP, hadron colliders-multiplicity of heavy jclusters?) ? See Section VII ii.

4. We have not touched several other topics of great interest like Y-Y collisions 36 (a very clean place where to study QCD), three jet production in e e~ and lepton scattering 37 , etc.

Acknowledgements

I would like to thank Ronald Shellard for his careful reading of the manuscript. This work was supported by PINEP, Rio de Janeiro, under contract 522/CT. HARD SCATTERING PROCESSES 451 References 1. In preparing these lectures I have made extensive use of the following excellent lectures and reviews: C.T.Sachrajda: Parton Model Ideas and QCD, lecture given at the XIII Rencontre de Moriond (1978). Applications of QCD to Hard-Scattering Processes, lectures given at the CERN ACADEMIC TRAINING COURSE (1979). A.J.Buras: Asymptotic Freedom in DIS in th« Leading Order and Beyond, Fermilab preprint PUB-79/17-THY (1979), to appear in Rev.Mod.Phys. J.Ellis: Status of Perturbative QCD, invited talk at the 1979. Inter- national Symposium on Lepton and Photon Interactions, FNAL, Batavia, Illinois (1979) - CERN-TH-2744. Yu.L.Dokshitser, D.I. D'yakonov and S.I.Troyan: Inelastic Processes in QCD, SLAC Translation 183 (1978) . 2. R.P.Feynman, Photon-hadron interactions (Benjamin, New York, 1972). P.V.Landshoff and J.C.Polkinghorne, Phys. Rep. 5c, 1(1972). ' 3. For reviews, see, H.D.Politzer, Phys. Rep. 14c, 129(1974). W.Marciano and H.Pagels, PKys.Rep. 36c, 137(1978). 4. G.Altarelli and G.Parisi, Nucl.Phys. B126, 298(1977) . b. D.Araati, R.Petronzio and G.Veneziano, Nucl.Phys. B140, 54(1978) and B146, 29(1978). R.K.Ellis, H.Georgi, M.Machacek, H.D.Politzer and G.C.Ross, Nucl. Phys. B152, 285(1979). . A Yu.L.Dokshitzer, D.I.D'Yakonov and S.I.Troyan, Proc.of the 13 / Winter School of the Leningrad Institute of Nuclear Physics (available as SLAC translation N9 183 (1978)). : C.H.Llwelyn Smith, Acta Phys. Austriaca Suppl. XIX, 331 (1978) . H- 6. For a review of QCD beyond the leading order, see A.J.Buras (Ref.1). : 7. Yu.L.Dokshitzer et al (Ref.5) and Phys.Lett B79, 269 (1978) G.Parisi and R.Petronzio, Nucl.Phys. B154, 4T7~(1979) . 8. W.Furmanski, R.Petronzio and S.Pokorski, Nucl.Phys. B155, 253 (1979). 9. S.D.Drell and T.M,Yan, Phys.Rev.Lett. 25, 316(1970). •• t 10. For recent reviews of the OPE plus RGE approach to DIS, see A.Peterman, Phys. Rep. 53C, 157(1979). A.J.Buras (Ref.l). 11. T.Kinoshita, J.Math.Phys. 3, 650(1962). T.D.Lee and M.Nauenberg, PKys.Rev. 133, 1549(1964). 12. C.F.von Weizâcker, Z.Phys. 88, 612(1934). E.J.Williams, Phys. Rev. 45_~729 (1934) . 13. P.C.Bosetti et al., Nucl.Phys. B142, 1(1978). 14. J.G.H. de Groot et al., Z.Phys. Cl, 143(1979); Phys .Lett 82B, 292, 456 (1979). — 15. J.Ellis, Current trends in the theory of fields (eds.J.E.Lannatti and | P.K.Williams) (A.I.P., N.Y., 1978), p.81. I 16. E.Reya, Phys. Rev. Lett., £3, 8(1979). .;] 17. See, for instance, the following review I J.Ellis, Status of perturbative QCD, invited talk at the 1979 Interna- " tional Symposium on Lepton and Photon Interactions, FNAL, Batavia, Illinois (1979) - CERN- TH 2744. 452 Escobar IS. D.C.Hom et al.,Phys.Rev.Lett. 37, 1374(1976); J.K.Yoh et al., Phys. Rev.LettTTTT 684(1978) and 4J7 1083(1978) . 19. K.J.Anderson et al., Phys.Rev.Lett. ^2, 944(1979). 20. CIP Collaboration, Ref.19. 21. NA3 Collaboration, J.Badier et al., CERN preprint, EP 79-67,EP.79-68. 22. For a review see R.Stroynowski, SLAC-PUB-2423 (1979). 23. J.C.Collins and ü.E.Soper, Phys.Rev. D16, 2219 (1977). 24. E.G.Floratos, D.A.Ross and C.T.Sachrajda, Nucl.Phys. B129, 66(1977), 139, 545 (1978) and B152, 493(1979). W.A.Bardeen, A.J.Buras, D.W.Duke and T.Muta, Phys.Rev.D18, 3998 (1978). For reviews see Ref.10. 25. W.E.Caswell, Phys .Rev .Lett. 33_, 244(1974). D.R.T. Jones, Nucl.Phys. B75, 531(1974). : 26. M.Bacê, Phys.Lett. 78B, 132(1978). 27. W.A.Bardeen et al., (Ref.24). D.W.Duke and R.G.Roberts, Phys.Lett. 85B, 289(1979). 28. See A.J.Buras, ref.1 for more details on this point. 29. M.R.Pennington and G.G.Ross, Oxford Univ,preprint 23/79 (1979). '-• 30. A.Para and C.T.Sachrajda, preprint TH.2702-CERN (1979) . 31. G.Altarelli, G.Paris! and R.Petronzio} Phys.Lett. 76B, 351 and 356 (1978). si K.Kajantie and R.Raitio, Nucl.Phys. B139, 72 (1978). i F.Halzen and D.Scott, Phys .Rev.Lett.~W7 1117(1978). H.Fritzsch and P.Minkowski, Phys.Lett." 7_3B, 80(1978). (~ 32. See G.Altarelli, G.Parisi and R.Petronzio (Ref.31). I 33. J.Kubar-André and F.E.Paige, Phys.Rev. D19, 221(1978). G.Altarelli, R.K. Ellis and G.MartinellTT~Nucl.Phys. B143, 521(1978). Erratum B146, 544(1978) and Nucl.Phys. B157, 46K1979T7" J.Abad and B.Humbert, Phys.Lett. 78B, 627(1978); 80B, 433(1979). K.Harada, T.Kaneko and S.Sakai, NücT.Phys. B155, T5¥(1979) and ; Erratum to be published. B.Humpert and W.L.van Neerven, Phys.Lett. 84B, 327(1979) and 85B, 293U979); CERN preprint TH 2738(1979) and"1rratum. 34. For other proposals for measuring the three gluon vertex see: E.Reya, Phys.Rev.Lett. 43, 8(1979). I-. De Rüjula, B.Lautrup and R.Petronzio, Nucl.Phys. B146, 50(1979). 35. S.j.Brodsky and G.P.Lepage, SLAC preprint SLAC-PUB-2294 (1979); Phys.Lett. 87B, 359(1979); Phys.Rev.Lett. 43, 545(1979). G.Farrar an

TOPOLOCICAL ASPECTS W FIELD THEORY AND CONFINEMENT

Bert Sduoct Instituto de Física e Químki de Slo Carios, Unnenidade de Sio Paulo

1. Introductory Remarks

The hypothesis that the model of QCD is the correct description for strong interaction has been steadily gaining strength ever since nonabelian gauge theories became theoretically respectable as viable renormalizable field theories. The popularity which QCD enjoys is primarily related to its conceptual clarity, its simplicity in terms of a Lagrangian description and the conspicuous absence of alternative models which qualitatively explain as many experimental facts. By some divine justice the Lagrangian of QCD is easily written down, but the exploration of its physical consequence turned out to be one of the most difficult problems in theoretical physics up to now. The main reason is the complicated relation between particles and fields. The fact that this relation is more involved than in the "old" QFT is of course related to the confinement problem.

Let us remind ourselves briefly of the historical development of thoughts on the particle-field relation. In "ancient" days particles used to be identified with field quanta, i.e. Fourier-components of fields at a particular time. This led to confusion in the computation of radiative correction to lowest order perturbative processes. Clarity was restored by incorporating the S-matrix concept of Wheeler and Heisenberg into QFT, an achievement which goes under the name of LSZ (Lehmann Symanzik Zlmmermann) theory. The essential idea was that the remormalized Lagrangiean field describing dressed field quanta have a simple, asymptotic behaviour for large times.

LSZ A(x) • A±n (x) , ,i out

:(. The assumptions under which this asymptotic property was derived , were: 454 B. Schroer (a) spec V E forward light cone, with a finite gap between vacuum and one particle states as well as a gap to the continuum

<0| A(x) )p> * 0

(b) locality, i.e. spacelike (anti)commutation relation for the basic field.

Formally also bound states |b,q> with say <0| A(x) |b,q>=0 but <0|polyn.(A) |b,q> + 0 were incorporated into the LSZ frame, although no systematic approximation method for processes involving bound states had been proposed. It was believed that such a framework leads to complete description of particles (asymptotic completeness) and serves to describe strong interaction in which no physical zero mass particles play a role.

Bya slight modification one could also incorporate particles with permanent infrared-clouds so called infraparticles, i.e. the electron of QED. However, according to our best knowledge the LSZ framework in the above form is not valid for QCD. The reason is two-fold:

(a) there may exist particles (or infraparticles) whose interpolating fields are not constructable in the polynomial algebra of Lagrangian fields. (b) the quark resp. gluon fields may not have any accompanying particles: quarks may be confined and color may be screened.

The first phenomenon is not typical for gauges theories.lt has been first observed in the Sine-Gordon model. There are particles, called topological solitlons and kinks whose interpolating fields (though mostly local with respect to themselves) are non local with respect to the Lagrangian fields. The usually carry a (hidden) symmetry group of topological origin which may be continuous (U(l) in the case of the topo- 11 4 logical Sine-Gordon soliton ) or discrete Z- for the A_ kinds or the 2) t& Ising,-kinks ). Historically these particles have been treated first quasiclassically, but nowadays it is an accepted fact that the duality3) of statistical mechanics (Kramer-Wannier duality) is. the most adequate framework for soli tons and kinks. According to Mandelstam 4) and 't Hoof*5t ) there are dual fields (order-disorder fields) in QCD. It is not known whether they lead to new particles, i.e. particles which cannot be in- terpolated by Lagrangian fields and their gauge invariant composites. TOPOLOGY AND CONFINEMENT 455 The problems of confinement and screening has presently not ,/. been understood in the QCD4 model. Attempts have been made to obtain p insight into static quark confinement of corresponding lattice gauge theories . The use of the topologically significant order-disorder op- "' erators (similar to the lattice Ising model discussion) and topological boundary conditions ' was important for this recent progress. Another line of research is concerned with the physical properties of string models 9)'. Here differential geometric methods have been quite powerful tools . In those investigations the relation to the QCD. Lagrangian remains somewhat unclear. The most explicit picture has come from the n study of two-dimensional models (QED2, QCD2 and the CP models). In two dimensions the color neutrality of physical states is (kinematically) guaranteed. One can use such models for the study of the "quark screen ing versus the quark confinement problem". Furthermore the chiral U(l) symmetry breaking which is topological in origin and leads to the massive n - plasmon has been completely understood in such models•

In a condensed form the present frontiers of and problems of '• /, theoretical (non-lattice) QCD research may be described in the following ] way: r 1) Attempts to understand functional integrals. What is the (topological) structure of the field manifold over which the functional integral ; is extended? How does one obtain an insight into the differential geometry including the integration theory on such manifolds? 2) Systematic expansion methods: 1/N expansion. 3) Dual structures: functional integrals for order-disorder fields. 4) Study of particles and fields in two dimensional "solvable" models as a source of dynamical intuition.

In most of these problems topological ideas and methods play an important role. Here we will only review attempts concerning the ~~ understanding of continuous QCD or related continuous field theory models.

2. Mathematical Methods

In a situation in which the basic Lagrangian is simple and physical information is difficult to extract, the progress depends very much on the improvement of the mathematical machinery. The starting point of most mathematical investigations is the functional integral 456 B- Schroer representation of Euclidean correlation function, i.e. in QCD

e"SX

X = product of ijifij) and A 's

= 1 F2 + i F D 4 J pv J YVYV. UU* V V V

4 For perturbation theory one has to follow Faddejev and Popov's '{•• procedure of adding in addition to a gauge breaking term a Faddejev-Popov ghost contribution: S+S + S-,-, + S_ _ =S__ , u.r. r.Jr. r.Jr.

From S_ p one may obtain Feynman rules to which the process of renormalization may be applied: a redefinition of the coupling constant can be compensated by changes in the wave-function renormalization Z•:• . (and possibly the gauge fixing parameters) . The corresponding parametric differential equations contain a function 3(g) " 14) whose negative sign leads to asymptotic freedom : perturbative theory .:. is converted into an asymptotic expansion for small distances.The tacit v assumption here is that 6(g) in a small region around g = 0 really has •, a global (non-parturbative) meaning. However, the coupling constant g j- plays no role in the physical parametrization in terms of say the lowest ( . partial mass, all physical quantities in particular the partial spectrum do not depend on g. In the so called mass transmutated physical theory

:;. the meaning of B(g) is less clear. Nevertheless, asymptotic freedom f has had great importance for the structural understanding as well as |. for phenomenological applications. All attempts to obtain a hint about : the structure of the mass transmutated correlation functions from the perturbative infrared aspects have failed up to now. We will in the following briefly discuss several non-perturbative attempts to understand functional integrals. (1) Structural investigation of the field manifold

. QCO4 is as any gauge theory a geometric theory. There are ;• other models as for example the class of two-dimensional Sigma-models | which share with gauge theories a rich topological structure of their I ; field manifold. Consider for example the CPn model. Classically it •1 15) -) can be described in terms of a complex n-component vector field .

(x) = j • j with [ifi]2 =i()+ $ = 1 TOPOLOGY AND CONFINEMENT 457

The field manifold consists of a equivalence classes Ú ] : J."

.(.'(x) = eiA(x)

The classical CPn~ Lagrangian

*- D

depends only.on the equivalence classes M . It is renormalizable and conformaily invariant. The R can therefore be one-point compactified 2 2 and we may think of R as a S (use stereographic projection) .The field c manifold «-• , Lll n_x 5 : Sz CPnX

is an infinite dimensional manifold (a natural topology in terms of Sobolov norms may be introduced), which decomposes because of

iM .o-*, - m.

into disjoint submanifolds.

r- p rk

The existence of this decomposition, as it is well know, leads :; to an ambiguitam y in the quantization which can be parametrized by a phase angle 6 . In each sector 5^ there exist local minima: the- instantons

which form a finite dimensional manifold fk •

According to a theorem of G.B. Segal these instanton-mani-

folds for large k asymptotically approach í» k in the sense of homotopy theory

On the other hand the right hand side is

2 1 TTq (S + CP"'?") = irg+2 (CP"" ) q > 0 458 B« Schroer independent of k.

Via a standard fibre bundle argument applied to the fibring: S211"1 + CP11"1 we obtain Va (cpn"1] " V2(s2n "1)

We therefore have

IT (£) = (0 , q+2 2n-l q K \ < , q+2 = 2n-l [unknown but in general nontrivial for q+2 > 2n-l

From these mathematical remarks we collect the important facts

(a) The field manifold has a complicated homotopy (and homology) structure, it is far removed from the structure of a linear, i.e. free field manifold which is homotopically flat.

(b) The instanton manifolds f^ homotopically approximate -Fk for large k. However (at it becomes clear from a more detailed examination): Morse theory of cristical points is not applicable; one meeds a generalization including "quasi-critical points".

i.e. 3r becomes asymptotically flat.

This is the topological prerequisite for finding in the leading order 1/N a free field theory. There have been similar investigations of the QCD4 manifold space for the QCD4 manifold for n •* °° .This limiting manifold is known, but complicated.

The simplest model of topologically rich field manifold ' is the rigid body. The action to be used in a Feynman path integral description of the rigid body is an action on a path space in SU(2). In this case (and its SU(N) generalization) the Morse theory of extrema give a complete description of the homology groups and (via "spectral sequences") on the homotopy structure of the field manifold. However, the Morse theory does much more: It gives the exact value ( via a quasiclassical approach including all saddle points) of the functional

integral even though the action is not gaussian. One can use the SU(N) TOPOLOGY AND CONFINEMENT 459 rigid body, i-iorse theory to understand the Bott-Atiyah "theorem conjecture": the Yang Mills action on two dimensional surfaces is a perfect Morse function. Using formal intuitive arguments of Migdal one may expect that two-dimensional (renormalizable) sigma models bear a similar relation to nonabeli^n 4-dimensional gauge theories as the SU(N) rigid body to the two dimensional (superrenormalizabie) SU(N) gauge theory. At this exciting speculative remark one should, however, not surpress a sobering note: a "good" Morse theorv does not yet exist for 19) renormalizable two dimensional field theories as sigma models

We now comment on the 1/N expansion technique.

(2) The 1/N technique

As a typical example we take again the CPn model

Here the only change to the classical Lagrangian is the normal ization of (j); | |2 =~ ; one has to allow for wave function renornal - ization. Now _ — r _ -A [d$] [d<(>] [dx]e = [d$] [d"J [dx]

with <£A = £+(j3-iA)2

A z = iDy

D A =3 -i A V V V

This is the trick of introducing a "dummy" field in the functional integral, a standard method to convert a quadrilinear interaction (between the "s) into trilinear Yukawa form. It is now easy to see that

-N Seff

Seff =to ln 460 B. Schroer The minimum of S .. is easily determined: it occurs at ±L C 2 2 N . Aj = 0 , X = m = A e~ * A = cutoff. In lowest order we see the famous mass transmutation happening in this model. We now write

X = XC + X

and obtain Feynman rules in terms of < A A > and propagators. To obtain the parameter in the correlation functions one has to compute with

8 Q e 3 A S = Seff -i 98 , ^ | uv v u

The A-propagator is in lowest order:

. C

f This is the so-called "secret long range force" related to the confinement of the 's. This method works for all models in which the interaction can be cast (by the trick of dummy functional variables) into the form

3 í or 3

(D = (pseudo) scalar, 3 = (pseudo) vector)

where the 's are isoscalar.

The simplest model with pion isoscalar interaction is

I i where $ J are fields transforming, i.e. under the adjoint representation |. ] of U(N) . In that case, by integrating over the and using Fierz i-'oj identities the interaction takes the form20'

i'-'l

7 , S/.y TOPOLOGY AND CONFINEMENT 461 g2 [[ 2 _ 2 - - "I lnf 4 JL ç y u J g* ff _ — 2 2

It consists of a Gross-Neveu part and a nonlocal part.Now both parts are isoscalar, though the second part is nonlocal. One plays the trick with the dummy fields. In addition to the local dummy fields it, a and A for the Gross Neveu and Thirring coupling one has to introduce a nonlocal fields V (x,y) . Again the Fermion integration brings into the form V Sinf = N Sinf V J

but now the variational equation for the minimum has in addition to the constant value for a a nontrivial value for

V (q) = q V(q)

This is given bj a nonlinear integral equation which precisely corresponds to the summation of all planar two point function graphs for . Lack of time prevents me form saying more about the results obtained by this 1/N method for Yukawa type nonisoscalar interactions.

The method cannot be applied directly to QCD2 because of the occurance of trilinear and quadrilinear selfcouplings of the gluon field. It can formally be applied to the Euclidean functional integral in the Z = x-iy = 0 gauge (the Euclidean "light cone" gauge) however a gauge transformation into such a gauge is a GL(N,C) father than a SU(N) transformation. This creates some complication21' .

(3) The dual variable method

We first illustrate this method in a simple example (Kadanoff- 221 -Ceval) ': the Ising model A = - J Z a. a. a^ {±1}

•a(xn)>=f ií, "(«,)••.• 462 B. Schroer

The dual "disorder-variable" y is introduced with the help of an antiferromagnetic coupling on strings connecting the u's

A 1 u(y2)> -i t e" r

y. = dual lattice points,

p = A/p : i.e. sum over those links which are crossed by r.

Despite the apparent r dependence in their definition they are actually r-independent. The y's are very util in the understanding of the phase transition. They are of crucial importance in the construction of the scaling limit of the theory. The results of the scaling limit , i.e. the limit T •+• T , a •* o, such that correlation length Ç = const. 2) c are: (1) The product of order x disorder (at adjacent points) are described by the components of a massive free Majorana field ty = I|JC .. The dynamical variables a and y are given in terms of kink operators: u (x) My) = eiir9(x-Y) My) o(x)

V (x) = short distance limit U(x) i|i(y) y •+ x

With,, a minimaluty assumption on short distance behavior U and v are unique. U is of the form 0 = exp. bilinear (a, a ) and they are both local Bose fields of scale dimension 1/8.

(2) The order (disorder) variables a and y are given by:

T " Tc "* °+ u(x) = U(x) a(x) = V(x)

T - Tc -+• 0_ y(x) = i(c+c ) V(x) a(x) = (c+c+) U(x) where {c, c } = 1 , {c* , i|j*} = 0 , c = c* or c+

c is a (x independent) spurion operator which describes the two fold vacuum degeneracy of the 0_phase. The mixed correlation functions are double valued in the Euclidean region: one needs to introduce a ramified covering in order to talk about the Euclidean field theory of dual fields, TOPOLOGY AND CONFINEMENT 463 i 24) i.e. one is outside the Wightroan framework

It has up to now not been possible to take that scaling limit directly in the functional integrals: this is a common difficulty of all Zy theories for which the field values of the lattice model are discrete. However, in the case of U(l) duality the functional integrals of dual field theory can be directly discussed in the continuum limit Followinswing Marino and SwiecSwie a let us briefly consider such an example: free massless field

-ib f EWV 3 * (z)dz a(x) = ei%altl , u(x) = e * v u

They lead (application of canonical commutation relations) to | the dual commutation relations Iy y(x) a(y) =a(y) u(x) e2iTi 9(v"x) jf' s= fr |

The correlation function of a and p are represented by A Euclidean functional integrals with a bilinear action. Such integrals Eft are saturated by the classical confirgurations. An interesting corre- K lation function is the mixed correlation: m !r + + 2 = j I [d<(i] exp-J [ 3*<|> + (a+B) if»] d x r

a(Z) = -ia [ç(Z-x,)- ç(Z-x2)] y2 < P(Z) = -b j ey'a^ ç(Z-Z") dZ"

yir This is a system of (imaginary) charges and monopoles at the -*_ ends of I. As long as the o's do not cross F ,the functions are univalued; under a crossing of the string the interaction energy of the system 1 changes by 2ir is • If the number of a and a+ as well \i and y+ are not the same the correlation functions vanish because of boundary contribution lin e~c nR . The string is the position of the Euclidean branch cut. This string position is fictitious as long as a charge does not cross. Apart from the above mentioned crossing rules it always remains | fictitious. How define the dyon-operators 464 B. Schroer

ípj = lim CJ(X) y(y) y+x

i|>2 = lim a (x) p(y) y+x

with a suitable convention for the limit.

If we take the limit of the electric charge approaching the monopole in opposite angles,

then there remains only the ambiguity depending on whether the limit is taken on the same sheet or whether a a(relatively to the other a) has crossed the line of ramification. The two point function ijJ> has as many possible values as there are sheets, i.e. finitely many (if S is rational) or •» otherwise. The order of i() and Tp has e priori no meaning !• in the Euclidean regime. However for S = 1/2 one may link the order fi-, uniquely with the two classes of path-limits and on obtains the relation ;i

r = - -!

Hence two-dimensional fermi statistics has a topological origin. This U(l) duality structure exists in all two-dimensional models involving complex fermions. Therefore the bosonization of ferminons and the Coleman equivalence between the Sine Gordon model and the massive Thirring find their ultimate understanding in the framework of Euclidean duality: they are just special cases of the grand duality scheme.

The ZN duality is more diffici-lt to handle than the U(l) duality: one has not yet understood to write down continuous functional integrals although in special cases, as the Ising model, there are other methods to construct the scaling limit. In D = 4 dimensions there are formal difficulties (renormalization problem) in the U(l) case.

r ! 3. Models of Gauge Theory, Confinement and U(l) Problem

Interesting regorous results have been obtained by studying TOPOLOGY AND CONFINEMENT 465

two dimensional gauge theories. There already abelian gauge theories r-x ave 8-vacua and lead to confinement. It has been known for some time that massless QED, screenes and massive QED_ confines quarks. The confinement versus screening properties are most conveniently exposed by 26) introducing flavour into the model

I|J * \|). , i corresponds to the up(N) flavour group

screening; colorless states with fundamental flavour appear in i ("bleached" quarks, if they are one particle state)

confinement: there are no states with fundamental flavour in Tit . .

A solvable model with very realistic confinement properties is the SU (NJJJ- - "torns" model: it is obtained from QCD_ by only integrating over diagonal configuration, i.e. A .. ,. = 0 • for __ y #OII—oxag. mq = 0 3fpnys . contains only trivial representation of the discrete Z (N) subgroup (apart from certain 6 values for which confinement changes into screening). Most of the results can be obtained on the basis of the functional integration over fermions. In the presence of 9-vacua one has:

Fermion integration: -/-+• Mathews-Salam rules —*• modification due to Atiyah-Singer zero modes '

This method does not work if a fractional winding number : -i— I F ?£ integer becomes important. In that case uses the method of duality (dyonisation of f ), This method also _gives a nicer physical insight: the magnetic monopoles through the * A. become coupled to a background of magnetic sources, the fictitious Dirac string is converted via condensation of monopoles into a physical string. The effective potential computed with Wilson's formula:

•:. = c e - ^ 28) -; behaves different in the screening and confining case 1 466 B. Schroer However it does not reveal the tramatica difference between the two situations in the particle structure of ^ * If

Tn two dimensional models the fermion mass term is essential for confinement. However, even with this mass term, there exists a discrete set of 6 angels at which confinement goes over into screening .

It is currently believed that in 4 - dimensional teories the Wilson area-law expressing quark confinement will insure the (light) quark confinement of the full QCD theory.

There is a curious aspect of screening versus confinement if one considers quark propagation in relativistic gauges:

n2 í.n(x-y) 2 u x - y •+• °° e q

i.e. has the secret long range from behavior only in a gauge where - g to all orders in part one loons this pathological behavior.

If in QCD. by some non-perturbation method one can establish

then (by using sceleton expansions or Dyson equations) it seems very plausible that quark propagation increases exponentially for large distances. Again the logical connection of this observation with the structure of

In QCD^ there have been many controversies concerning the so called U(l)-problem: why and how the anomalous U(l) . . , current in the limit n = 0 leads to an y-plasmon (instead of a Goldstorian bosan)

A very neat discussion has been given recently by Roth and Swiça . They comple the U(l) chiral SU(N) Gross - Neven model

! l L -g |OFÍ) - (TTr*) l inf e TOPOLOGY AND CONFINEMENT 467 with QED,. Their findings are: (1) The infrared cloud of the chiral G.N. Model (in D = 2: Goldston bosons infrared cloud) is eaten up by massless gluous via the Schwinger-Higgs mechanism. The resulting massive cloud destroys chiral selection rule, the corresponding massive particle is the y (which appears already in lowest order jj) .

(2) The relevant gauge field configurations which lead to ^ 0 have fractional winding number = - . So the controvery represented 25) 26) in its extreme version by the names of Crewther and Witten is superficial at least in this model. Both of them are partially right: Witten on point (1) and Crewther on point (2).

I thank the organizers of the 6 Brazilian Symposium in Theoretical Physics, in particular Prof. Erasmo Ferreira for the invi, tation and the hospitaly. This manuscript was written during a breef stay at DESY. I thank the DESY theory group in particular Prof. H. Joos for inviting me.

References and footnotes

Here we give only those references wuich were directly used in preparing this talk. Via these references the reader may back to the often more important earlier papers.

1) S. Coleman, Phys. Rev. Dll, 2088 (1975) 2) M. Sato, T. Miwa and M. Jimbo RIMS Kyoto preprint 207 (1976), unpublished. B. Schroer and T.T. Truong, Nucl. Phys. B 154 (1979), 125 3) H.A. Kramers and G.H. Hannier, Phys. Rev. 60 (1941), 252 4) S. Mandelstam, Phys. Lett. 53B (1975), 476 5) G. 't Hooft, Nucl. Phys. B 138 (1978) 1 6) K. Wilson, Phys, Rev. D10 (1974), 2445 J. Kogut and L. Susskind, Phys. Rev. Dll (1975), 395 7) G. 't Hooft, Nucl. Phys. B153, 141 (.1979) 8) G. Mach. DESY-report 80/03 (Jan. 1980) 9) Y. Nambu, Phys. Lett. 80B (1979) 372 468 B- Schroer

10) M. Lüscher, K. Symanzih and P.H. Weisz DESY preprint DESY - 80/31 fe 11) J.A. Swieca, Kaiscoslauton lectures 1979, UPSC preprint 1979 K.D. Rothe and B. Schroer, Kaiserslautern lectures 1979, FU Berlin preprint 1979. ' A.D'Adda, P. Di Vecchia and M. Lüscher, Nucl. Phys. B146 (1978), 63 12) K.D. Rothe and J.A. Swieca UFSC preprint 1980 To be published in Phys. Revi. D. 13) C. Callen, Phys. Rev. D2, 1541 (1970) K. Symanzik, Commun. Math. Phys. 49_, 424 (1970) 14) 't Hooft unpublished remarks at the Marseilles Conference on Gauge Theories 1972 G.D. Gross and F. Wilszck, Phys. Rev. Lett. J30, 1343 (1973) H.D. Politzer, Phys. Rev. Lett. 29_» 1346 (1973) 15) H. Eichenher, Nucl. Phys. B146 (1978) 215 M. Lüscher, Phys. Lett. 78B (1978) 465 16) G.B. Segal, Inventiones Math. 2\, 213 (1973) M.F. Atiyah and J.D.S. Jones, Commun. Math. Phys. til, 97 (1978) I 17) L. Schulman, Phys. Rev. 1/76, 1558 (1968) > J.S. Dowker, J. Phys. A5, 936 (1975) 18) B. Schroer, in preparation '] 19) M. F. Atiyah and R. Bott ' Lectures presented at the 1979 Cargese Summer School ;• 20) M. Hartagsu, K.D. Rothe and B. Schroer, Nucl. Phys. B 1980 j 21) In the Minkowski-space version of 't Hooft ref 11) these problems ! are hidden in the boundary conditions for the gluon propagator 22) H. Ceva and L.P. Kadanoft, Phys. Rev. B3 (1971) 3918 23) B. Me Coy, C.A. Tracy and T.T. Wu, Phys. Rev. Lett. 38 (1977) D.B. Abraham, Phys. Lett. 61A (1977) 271 i R.Z. Bariev, Phys. Lett. £4A (1977) 169 24) The correlation functions become multivalued in the permited extended tube region 25) R. Crewther, Kaiserslautern lectures 1979 26) E. Witten, Harvard University preprint 1979. Lectures given the at 1979 Cargese Summer School. LIST OP PARTICIPANTS 469

CAMARGO, Sérgio Souza PUC, Rio de Janeiro CAMPOS, Mareio D'Olne UNICAMP, Campinas CANDOTTI, EniO UFRJ, Rio de Janeiro CAPARICA, Álvaro de Almeida IFQ, São Carlos CARNEIRO, Gilson PUC, Rio de Janeiro CARRIÇO, Artur S. UFRN, Natal CARVALHO, Hélio Freitas de UFRJ, Rio de Janeiro CARVALHO, Joel Câmara de UFRN, Natal CASTILHO, Caio Mario Castro de UFBa, Salvador CASTRO, Helena Maria Ávila de USP, São Paulo CENEVIVA, Carlos Augusto P. UNICAMP, Campinas CHAN, Roberto UFRJ, Rio de Janeiro CHANDA, Rajat UFRJ, Rio de Janeiro CHAVES, Carlos Maurício F. PUC, Rio de Janeiro CHAVES, F. Artur B. UFRJ, Rio de Janeiro CHIAN, Abraham INPE, São José dos Campos l COELHO, Hélio T. UFPe, Recife y COLEMAN, Sidney Harvard University, USA CONTINENTINO, Mucio UFF, Niterói COSTA, Rogério C. T. da IFQ, São Carlos COUTINHO, Maurício UFPe, Recife CRESSONI, José Carlos IFQ, São Carlos CRUZ, Pedro Paulo de Oliveira PUC, Rio de Janeiro CUDEN, Cyril UFBa, Salvador DAVIDOVICH, Luiz PUC, Rio de Janeiro DAVIDOVICH, Maria Augusta PUC, Rio de Janeiro DEZA, Roberto . Centro Atômico Bariloche, Argentina DIETZSCH, Olacio USP, São Paulo ' OILLENBURG, Darcy UFRS, Porto Alegre DONOVAN, Boris Francois UNICAMP, Campinas DREIFUS, Henrique Von USP, São Paulo DUSSEL, Guilherroo Comisión de Energia Atômica, Argentina EPELE, Luiz Universidad Nacional de La Plata, Argentin, ESCOBAR, Carlos O. IFT, São Paulo FALOMIR, Horacio Universidad Nacional de La Plata, Argentin FARIA, Armando Nazareno UFRJ, Rio de Janeiro FARIAS, Gil Aquino UFSCar, São Carlos FERREIRA, Erasmo PUC, Rio de Janeiro FERREIRA, Heloísa Helena IFQ, São Carlos FERREIRA, Maria Julita USP, São Paulo List of Participants 471

FERREIRA, Paulo Leal IFT, São Paulo FERREIRA, Ricardo UFSCar, São Carlos FERRER, Rodrigo Universidad de Chile, Santiago, Chile FONSECA JR., Cesar Augusto L. da CBPF, Rio de Janeiro FRANCISCO, Gerson IFT, São Paulo FREDERICO, Tobias USP, São Paulo FREIRE JR., Fernando PUC, Rio de Janeiro FREJLICH, Jaime UNICAMP, Campinas FULCO, Paulo UFRN, Natal FURTADO, Mario PUC, Rio de Janeiro GALEÃO, Alfredo Pio USP, São Paulo GALLAS, Jason A.c. UFSCat, Santa Catarina GARCIA, Rogério Lopes USP, São Paulo GARG, Vijayendra Kumar UPES, Vitória GEICKE, J.W.H. CTA, São José dos Campos GEIGER, Davi PUC, Rio de Janeiro GHILARDI, Antonio José Pio (JFGo, Goiânia GHIOTTO, Renato Carlos USP, São Paulo GIAMBIAGI, Juan Jose CBPF, Rio de Janeiro GIRAUD, Bertrand Centre d'Etudes Nucléaires, Saclay, França GIROTTI, Horacio 0. UFRS, Porto Alegre GITAHV, Vera Henriques USP, São Paulo GOMES, Affoo^ CBPF, Rio de Janeiro GOMES, Marcelo C. USP, São Paulo GOMES, Paulo Silveira UFF, Niterói GONÇALVES, Lindberg Lima UFCe, Fortaleza GREGORIO, Miguel A. UFRJ, Rio de Janeiro GUIMARÃES, Ademar Barbosa UFJF, Juiz de Fora GUIMARÃES, Renato Bastos UFF, Niterói GUSMÃO, Mercedes UFRS, Porto Alegre HARARI, Haim Weizman Institute, Israel HARARI, Roberto Saleh UNICAMP, Campinas HAUSER, Paulo Roberto UFSCat, Santa Catarina HELMAN, Jorge Instituto Politécnico Nacional, México HERMANN, Rosana USP, São Paulo HIPOLITO, Oscar UFSCar, São Carlos HORNOS, José Eduardo IFQ, São Carlos IGLESIAS, José Roberto UFRS, Porto Alegre JAYARAMAN, Jambunatha UFPb, João Pessoa KD4EL, Isidoro USP, São Paulo 472

KOBERLE, Roland IFQ, São Carlos KOILLER, Belita PUC, Rio de Janeiro KRMPOTIC, Francisco USP, São Paulo KURAK, Valerio USP, São Paulo LAMBERT, Franklin Vrije Universiteit Brusael,Bélgica LERNER, Eugênio UFRJ, Rio de Janeiro LOBO, Roberto CBPF, Rio de Janeiro LOZANO LEYVA, Manuel Universidad de Seville, Espanha LUCINDA, Jair USP, São Paulo MAC DOWELL, Samuel Wallace Yale University, USA MACEDO, Annita Mischan de M. UFRJ, Rio de Janeiro MACHADO, Luiz Eugênio UFSCar, São Carlos MACHADO, Waltair Vieira UNICAMP, Campinas MAGALHÃES, Maria Elisa da C. UFRJ, Rio de Janeiro MAIA, Marize P. UFRJ, Rio de Janeiro MAJLIS, Noberto UFF, Niterói MALBOUISSON, Luiz Augusto C. UFBa, Salvador MALTA, Coraci Pereira USP, São Paulo MARIZ, Ananias M. UFRN, Natal MARTINS, Maria Helena Pedra UFRJ, Rio de Janeiro MASPERI, Luis Centro Atômico de Bariloche, Argentina MEDERO, Ivone Maluf UFRS, Porto Alegre MELO, Severino Toscano do Rego UFPe, Recife MELLO, Eugênio Bezerra de - UFPb, João Pessoa MENDES, Rui Vilela CFMC, Lisboa, Portugal MENDONÇA, EvaIdo CBPF, Rio de Janeiro MENEZES, Leoner CBPF, Rio de Janeiro MIGHACO, Juan CBPF, Rio de Janeiro MI2RAHI, Salomon IFT, São Paulo MONDAINI, Rubem UFRJ, Rio de Janeiro MOSSIN, Cecilia Universidad de Salta, Argentina NADER, Alexandre A.G. PUC, Rio de Janeiro NASCIMENTO, Luis Carlos S. do UFRN, Natal NATALE, Adriano, A. IFT, São Paulo NEVES, Eduardo Jordão USP, São Paulo NOBRE, Fernando Dantas IFQ, São Carlos O'CARROL, Michael PUC, Rio de Janeiro OLINTO, Angela Villela PUC, Rio de Janeiro OLINTO, Antonio Cesar CBFF, Rio de Janeiro OLIVEIRA, Mario José de USP, São Paulo OLIVEIRA, Paulo Murilo de PUC, Rio de Janeiro List of Participants 473

OLIVEIRA, Sergio A. Carias de Universidade Estadual de Londrina ONODY, Roberto Nicolas IFQ, São Carlos OYAR2ON, Guilhermo Cabrera UNICAMP, Campinas PASSOS, Emerson José USP, São Paulo PENNA, Antonio F. UNXCAMP, Campinas PEREIRA, Flavio Irlneu M. PUC, Rio de Janeiro PEREIRA, Nelma Tarares UFRJ, Rio de Janeiro PÉREZ MUNGUIA, Gustavo Adolfo PUC, Rio de Janeiro PÊREZ, José Fernando USP, São Paulo PINTO, Marcus Venicius Cougo UFRJ, Rio de Janeiro PINTO, Maria José T.C. PUC, Rio de Janeiro PIRES, Antonio Sergio T. UFMG, Belo Horizonte PIZA, Antonio F. Toledo USP, São Paulo PLASCAK, João Antonio UFMG, Belo Horizonte PLEITEZ, Vicente IFT, São Paulo POL, Maria Elena PUC, Rio de Janeiro REIS, Fábio Gonçalves dos UNICAMP, Campinas ROBILLOTA, Manoel USP, São Paulo RODITI, Itzhak CBPF, Rio de Janeiro ROLF, Tobler ETH, Zurique, Suíça ROSA, Luis Pinguelli UFRJ, Rio de Janeiro ROTHE, Klaus Dieter OFSCar, São Carlos RÜGGIERO, José Roberto IFT, São Paulo UNICAMP, Campinas SAKANAKA, Paulo H. ICTP, Trieste, Itália SAIAM, Abdus UFRJ, Rio de Janeiro SANDOVAL, Elizabeth-Buteri UNICAMP, Campinas SANTIAGO, Marcos Antonio Natos CBPF, Rio de Janeiro SANTORO, Alberto Franco de Sá SANTOS, Filipe Duarte .Centro de Física Nuclear, Lisboa, Portugal SANTOS, Marcus Lacerda UFMG, Belo Horizonte te; SANTOS, Roberto Jorge V. dos UFAL, MaceiS SANTOS, Samuel UFRJ, Rio de Janeiro SARACENO, Marcos Comisión de Energia Atômica, Argentina SAITOVICH, Henrique CBPF, Rio de Janeiro SCHAPOSNIK, Fidel Universidad Nacional de La Plata, Argentina SCHOR, Ricardo UFMG, Belo Horizonte SCHROER, Bert IFQ, São Carlos SCHWEITZER, Paul A. PUC, Rio de Janeiro SELZER, Silvia UFF, Niterói SHELIARD, Ronald IFT, são Paulo SIGADO FILHO, Casalo UFRJ, Rio de Janeiro 474

SILVA JR., Eronides F. da UFPe, Recife SILVA, Marco Antonio A. da IFQ, São Carlos SILVA, Mariano Sabino CBPF, Rio de Janeiro SILVA, Paulo Roberto UFMG, Belo Horizonte SILVA, Rosane Riera PUC, Rio de Janeiro SILVEIRA, Adel da CBPF, Rio de Janeiro SILVEIRA, Enio Frota da PUC, Rio de Janeiro SIMÃO, Fernando R. Aranha CBPF, Rio de Janeiro SIMÕES, Tiago UFRS, Porto Alegre SOFIA, Hugo Maria Comisión de Energia Atômica, Argentina SOUZA, Moacyr Henrique G. e CBPF, Rio de Janeiro SRIVASTAVA, Prem CBPF, Rio de Janeiro STANLEY, H. Eugene Boston University, USA STEFFANI, Maria H. UFRS, Porto Alegre STILCK, Jurgen F. USP, São Paulo STUDART, Nelson UFSCar, São Carlos SWIECA, Jorge André UFSCar, São Carlos THOMÊ FILHO, Zieli D. UFRJ, Rio de Janeiro TIOMNO, Jayme PUC, Rio de Janeiro TORNAGHI,Alberto José da C. PUC, Rio de Janeiro TOULOUSE, Gerard Êcole Normais Superieuré, Paris, França TROPER, Amos CBPF, Rio de Janeiro TSALLIS, Constantino CBPF, Rio de Janeiro TURCHETTI, Giorgio .PUC, Rio de Janeiro UEDA, Mario INPE, São José dos Campos URBANO, José N. Universidade de Coimbra, Portugal URE, José Ernesto UFF, Niterói VAIDYA, Arvind Narayan ÜFRJ, Rio de Janeiro VALADARES, Eduardo de Campos UNICAMP, Campinas VALLE, José Wagner S. Syracuse University, USA VANHECKE, Francisco UFRN, Natal VASCONCELLOS, Cesar UFRS, Porto Alegre VENTURA, Ivan USP, são Paulo VIDEIRA, Antonio Luciano Leite PUC, Rio de Janeiro . VIOLINI, Galileo Universidad de Los Andes, Bogota, Colombia WEID, Jean Pierre von der PUC, Rio de Janeiro WEY JR., Clovis Ferraz USP, São Paulo WU, Ching S. . University of Maryland, USA ZAGURY, Nicim PUC, Rio de Janeiro ZIMERMAN, Abraham H. IFT, são Paulo List of Participants 475

, Maztin Mac Qill university, CanadS USP são Paulo I3. YOKOI, CarlOB S. r

V«S,v.

Si) SYMPOSIUM LECTURERS

;í i GEORGES BEMSKI , : Departamento de Física i Pontifícia Universidade Católica ; 22451 - Rio de Janeiro - RJ, Brasil | CARLOS G. BOLLINI Í ; Centro Brasileiro de Pesquisas Físicas %\ Av. Venceslau Bras, 71 ^\i 22290 — Rio de Janeiro - RJ, Brasil l-^í RAJATCHANDA IV Instituto de Física fe Universidade Federal do Rio de Janeiro •V' 22910 - Rio de Janeiro - RJ, Brasil I SIDNEY COLEMAN* I Harvard University and f Stanford Linear Accelerator Center í* P.O. Box 4349 g Stanford, California 94305, USA t CARLOS O. ESCOBAR • Instituto de Física Teórica i Rua Pamplona, 145 1? 01405 - SSo Paulo - SP, Brasil f- RICARDO FERREIRA Departamento de Física Universidade Federal de Pernambuco Recife, Pernambuco, Brasil J.J. GIAMBIABI Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil BERTRAND GIRAUD Centre d'Etudes Nucleaires de Saday Service de Physique Theorique Boite Postale n9 2 91190 - Gif-sur-Yvette, France | HAIM HARARI* Department of Nuclear Physics Weizmann Institute of Science Rehovot, Israel JORGE S. HELMAN Centro áe Investigacion y de Estúdios Avanzados Instituto Politécnico Nacional Apartado Postal 14740, México 14, D.F., México OSCAR HIPOLITO* Departamento de Física Universidade Federal de São Carlos 13560 - Sío Carlos - SP, Brasil SAMUEL W. MACDOWELL Department of Physics, Yale University New Haven - Connecticut 06520, USA EMERSON J. V. PASSOS Instituto de Física Universidade de São Paulo 05508 - Sío Paulo - SP, Brasil PAULO H. SAKANAKA Instituto de Física Universidade Estadual de Campinas 13100 - Campinas - SP, Brasil ABDUS SALAM International Center for Theoretical Physics P.O. Box 586 34100-Trieste, Italy BERT SCHROER Instituto de Física e Química de São Carlos Universidade de São Paulo 13560 - São Carlos - SP, Brasil H. EUGENE STANLEY Department of Physics, Boston University Boston, Massachusetts 02215, USA JAYMETIOMNO* Departamento de Física Pontifícia Universidade Católica 22451 - Rio de Janeiro - RJ, Brasil GÉRARD TOULOUSE Groupe de Physique des Solides de PEcole Nonnale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05, France CONSTANTINO TSALLIS Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil JOSÉ URBANO* Centro de Física Teórica Faculdade de Ciências e Tecnologia Universidade de Coimbra Coimbra, Portugal RUI VILELA MENDES* Centro de Física da Matéria Condensada Av. Gama Pinto, 2 1699 Lisboa Cedex, Portugal CHING S. WU Institute for Physical Science and Technology University of Maryland College Park, Maryland 20742, USA MARTIN J. ZUCKERMANN Department of Physics, McGill University w Montreal, P.Q. H3A 2T8, Quebec, Canada

Not included in this publication.

1 I

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