Under the Spell of the Gauge Principle (Advanced Series in Mathematical Physics, Vol 19)

UNDER THE SPELL OF THE GAUGE PRINCIPLE

ADVANCED SERIES IN MATH~MATICALPHYSICS Editors-in4 harge H Araki (RIMS, Kyoto) V G Kac (Ml7) D H Phong (Columbia University) S-T Yau (Harvard University) Associate Editors L Alvarez-Game (CERA!) J P Bourgu~gnon(fcoie P o / ~ e c h n i ~ uPalaiseau) e, T Eguchi (University of Tokyo) B Julia (CNRS, Paris) F Wilczek (lnstitute for Advanced Study, Princeton)

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MathematicalAspects of String Theory edited by S-T Yau Bombay Lectures on Highest Weight Representations of Infinite DimensionalLie Algebras by V G Kac and A K Raina Kac-Moody and Virasoro Algebras: A Reprint Volume for Physicists edited by P Goddard and D Olive Harmonic Mappings, Twistors and 0-Models edited by P Gauduchon Geometric Phases in Physics edited by A Shapere and F Witczek Infinite DimensionalLie Algebras and Groups edited by V Kac Introductionto String Field Theory by W Siege/ Braid Group, Knot Theory and Statistical Mechanics edited by C N Yang and M L Ge Yang-Baxter Equations in Integrable Systems edited by M Jimbo New Developments in the Theory of Knots edited by T Kohno Soliton Equations and Hamiltonian Systems by L A Dickey Form Factors in Completely Integrable Models of Quantum Field Theory by F A Smirnov ~ o n - P e ~ u ~ a tQuantum ive Field Theory - Mathema~cal Aspects and Applications Selected Papers of Jurg Friihlich Infinite Analysis - Proceedingsof the RIMS Research Project 1991 edited by A Tsucbiya,T Eguchi and M Jimbo Braid Group, Knot Theory and Statistical Mechanics (11) edited by C N Yang and M L Ge Exactly Solvable Models and Strongly Correlated Electrons edited by V Korepin and F N L Ebler State of Matter edited by M Aizenman and H Araki

Advanced Series in Mathematical Physics voi. 19

UNDER THE SPELL OF THE GAUGE PRINCIPLE

G. 't Hooft i ~ s t ~ tfor ~ t Theoretical e Physics University of Utrecht The ~ e t h e r l a ~ ~ s

World Scientific New Jersey London ffongKong

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The author and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to reproduce the selected reprints found in this volume: Springer-Verlag(Trendsin EZemenThe American Physical Society (Phys.Rev. 0); fury Particle 7'Fyory); Uppsala University, Faculty of Science (Reporfs of the C E L S I U W N N E Lectures). While every effoa has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not

contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.

UNDER THE SPELL OF THE GAUGE PRINCIPLE Copyright 0 1994 by World Scientific Publishing Co. Re. ttd. All rights reserved. This book, or parts thereof;may not be reproduced in anyform or by any means,electronico r ~ c ~ i ci~~udingphotocopying, a ~ , recordingorany information storuge and retrieval system now known or to be invented, without writtenpermissionfrom the Pubiisher.

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To my mother

To the memory of my father

PREFACE Our view of the world that we live in has changed. And it continues to change, as we learn more. What became abundantly clear in the course of the twentieth century is that the terminology most suited to describe the Laws of Physics is no longer that of everyday life. Instead, we have learned to create a new language, the language of mathematics, to describe elementary particles and their constituents. To facilitate this work we often introduce familiar words from the ordinary world to describe some law, principle or quantity, such as “energy”, “force”, “matter”. But then these are given a meaning that is much more accurately defined than in our social lives. Sometimes the relation with the conventional meaning is remote, such as in “potential energy” and “action”. And sometimes they mean something altogether different, for instance “color”, or “charm”. The notion of “gauge invariance” also originated from a concept in daily life: instruments can be “gauged” by comparing them with other instruments, but the way they work is independent from the way they are gauged. The notion has been redefined with mathematical precision for theories in physics. Gauge symmetries, both “global” (i.e. space-time independent) and “local” (space-time dependent) became key issues in elementary particle physics. This book contains a selection of my work of over twenty years of research. The guiding principles in this work were symmetry and elegance of the magnificent edifice that we call our universe. And the most important symmetry was gauge symmetry. Tremendous successes have been achieved in the recent past by the application of advanced mathematics to physics. However, some of the papers published in recent physics journals seem to be doing nothing but preparing and extending fragile mathematical constructions without the slightest indication as to how these structures should be used to build a theory of physics. I think it is important always to keep in mind that the mathematics we use in trying to understand our world is just a tool, a language sharp and precise enough to help us where ordinary words fail. Mathematics can never replace a theory. And, although I love mathematics, none of my papers were intended to improve our understanding of pure mathematics.

vii

What I have always sought to do is quite the opposite. Using mathematics as a tool I have tried to identify the most urgent problems, the most baffling questions that axe standing in our way towards a better understanding of our physical world. And then the art is to select out of those the ones that are worth being further pursued by a theoretical physicist. More often than not I end up immersed in mathematical equations. The next question is then always: “HOWwill these equations help me answer the question I started off with? How do I interpret my mathematical expressions?’)In this light one has to read and understand the articles reproduced in this book. This book is a collection of what I consider to be the more salient chapters of my work. It is by no means meant to be complete. There axe several important subjects in quantum field theory on which I have also made investigations, in particular monopoles and statistics, instanton solutions, lattice theory) classical and quantum gravity, and fundamental issues in quantum mechanics. All these subjects were either too technical or too incomplete to be included in a review book such as this, but that does not mean that I would consider them to be less important. I also could have included more work that I did with my cO-authors, Martinus Veltman, Bernard de Wit, Peter Hasenfratz, Tevian Dray, Stanley Deser, Roman Jackiw, Karl Isler, Stilyan Kalitzin and others. In any case, if this book was supposed to be my “collected works” I sincerely hope it to be incomplete because I plan to continue my investigations. It is impossible to produce important contributions to science without the insight, advice, help and support of numerous colleagues and friends. Among them, of c o m e , the ccmuthors I just mentioned. Before Veltman became my teacher I learned a lot from N. G. van Kampen, who attempted to afflict me with his passion for precise arguments, discontent with half explanations, and total dedication to theoretical physics. Later I benefitted from so many inspiring discussions with equal-minded colleagues that I find it impossible to print out all their names. Often the importance of a discussion only became manifest much later at a time that I had forgotten who it was who told me. I thank them all.

...

Vlll

CONTENTS PREFACE

vii

Chapter 1. INTRODUCTION

1

Chapter 2. RENORMALIZATION OF GAUGE THEORIES Introductions ..................................................................... [2.1] “Gauge field theory”, in P m . of the Adriatic Summer Meeting on Particle Physics, eds. M. Martinis, S. Pallua, N. Zovko, Rovinj, Yugoslavia, Sep.-Oct. 1973,pp. 321-332 ................................ [2.2] with M. Veltman, “DIAGRAMMAR”, CERN Rep. 73-9(1973), reprinted in “Particle interactions at very high energies”, NATO Adv. Study Znst. Series B, vol. 4b, pp. 177-322 ............................... [2.3] “Gauge theories with unified weak, electromagnetic and strong interactions”, E.P.S. Znt. Conf. on High Energy Physics, Palermo, Sicily, June 1978 ........................................................

11 12

16

28

174

205 Chapter 3. THE RENORMALIZATION GROUP Introduction .................................................................... 206 [3.1] The Renormalization Group in Quantum Field Theory, Eight Graduate School Lectures, Doorwerth, Jan. 1988,unpublished ....... 208 Chapter 4. EXTENDED OBJECTS Introductions ................................................................... [4.1] “Extended objects in gauge field theories”, in Particles and Fields, eds. D. H. Bod and A. N. Kamal, Plenum, New York, 1978, pp. 165-198 .............................................................. [4.2] “Magnetic monopoles in unified gauge theories”, Nucl. Phys.

B79 (1974)276-284

....................................................

ix

251 252

254 288

Chapter 5. INSTANTONS

..

.

Introductions .. .. . . . .......... . ... ...... .. .,... . ... .. . .. . .. . . . . . . . . . .. .... ... [5.1] “Computation of the quantum effects due to a four-dimensional pseudoparticle”, Phys. Rev. D14 (1976) 3432-3450 .. .. ..., . . .. ..., .. [5.2] “HOWinstantons solve the U(1) problem”, Phys. Rep. 142 (1986) 357-387 . ... ... .......................... .... .. ...-..... ..... [5.3] “Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking”, In Recent Developments in Gauge Theories, Cargkse 1979, eds. G. ’t Hooft et aL, New York, 1990, Plenum, Lecture 111, reprinted in Dynamical Gauge Symmetry Breaking, A Collection of Reprints, eds. A. Farhi, and R. Jackiw, World Scientific, Singapore, 1982, pp. 345-367 . . . . ,.. .. . ... . . . .. .. ... . . .. . . .. .

.

.

.

.

.

297 298 302 321

352

Chapter 6. PLANAR DIAGRAMS Introductions . ,. ..... .. . . . , ... .. ... . .. . .. . .... ... . . . . .... ... ., . . . . . . .. .. . . . . . (6.11 “Planar diagram field theories”, in Progress in Gauge Field Theorp, NATO Adv. Study Inst. Series, eds. G . ’t Hoot%et al., Plenum, 1984, pp. 271-335 . ................................, ........... [6.2] “A two-dimensional model for mesons”, Nucl. Phys. B75 (1974) 461470 . ... .... ....... ..... .. .. . .... .. .., ....... . . . .. .. .. ,.. . . ..... Epilogue to the TwGDimensional Model .... . . . ...... . .... . .. . . . ...

375 376

Chapter 7. QUARK CONFINEMENT

455 456

.

.

.

. .

.. . . .

.

. .

.

378 443 453

Introductions .. ..... .... .... ....... .. ...... ... .... .... . . .. . ..... .. ... .. . ...... (7.11 “Confinement and topology in non-Abelian gauge theories”, Acta Phys. A w t r . Suppl. 22 (1980) 531-586 .. .... .. .. . ..... . . .... 458 (7.21 “The confinement phenomenon in quantum field theory”, 1981 Cargtse Summer School Lecture Notes on Fundamental Intemctions, eds. M. Uvy and J.-L. Basdevant, NATO Adv. Study Inst. Series B: Phys., vol. 85, pp. 639-671 ...................... ...... 514 [7.3) “Can we make sense out of “Quantum Chromodynamics”?” in The Whys of Subnuclear Physics, ed. A. Zichichi, Plenum, New York, pp. 943-971 ....................................................... 547

.

.

.

Chapter 8. QUANTUM GRAVITY AND BLACK HOLES htroduct ions . . ... .... .. ....... .. .... . .. .. .. . .......... ,..... . . . . . .... .. . .. i8.11 “Quantum gravity”, in %rids in Elementary Particle Theory, eds. H. Rollnik and K. Dietz, Springer-Verlag, 1975, pp. 92-113 . ... (8.21 “Classical N-particle cosmology in 2 1 dimensions”, Class. Quantum Gmv. 10 (1993) S79-S91 . . . ..... . . .. .. . .. ... . ... . . ..... ... . . [8.3] “On the quantum structure of a black hole”, Nucl. Phys. B256 (1985) 727-736 . . . .... ..... .. ... ..... . ........ . . .. . .. . .... ... . .. . .... . (8.41 with T. Dray, “The gravitational shock wave of a massless particle”, Nuc~.Phys. B253 (1985) 173-188 ..,..... . ............................

.

.

.

+

.

.

X

.

.

.

577 578

.

584 606 619 629

[8.5] “S-matrix theory for black holes”, Lectures given at the NATO

Adv.

Summer Inst. on New Symmetry Principles an Quantum Field Theory, eds. J. hohlich et al., Cargbse, 1992, Plenum, New York, pp. 275-294 . . .. .. .. .. .. . ....... .. ........ .. .. . .. .. ... . . ......... . .. ... . 645

Chapter 9. EPILOGUE 665 19.11 “Canthe ultimate laws of nature be found?”, Celsius-Limb Lecture, Feb. 1992, Uppsala University, Sweden, pp. 1-12 ... ....... .. 666

INDEX

678

xi

CHAPTER 1

INTRODUCTION When I began my graduate work, renormalization was considered by many physicists to be a dirty word. We had to learn about it because, inspite of its ugliness, renormalization seemed to work, at least in one limited area of particle physics, that of quantum electrodynamics. &normalization was thought to be ugly because the procedure was ill-formulated and apparently an ad hoc and imprecise cure for a fundamental shortcoming in the quantum field theories of the day. But for those who analyzed the situation sdliciently carefully the physical reasoning behind renormalization wasn’t that ugly at all. Indeed it seemed to be a quite natural fact that the fundamental interaction constants one puts into a theory do not need to be identical to the chargea and m m one actually measures. If the theory is formulated on a very fine but discretised lattice instead of a continuum of space and time, the rele tion between the input parameters and the measured quantities will be quite direct. But now it 80 happens that if we take the limit where the mesh size approaches to zero, the input parameters needed to reproduce the measured quantities walk off the scale. This seems to be no disaster as long as we can’t measure these input, or “bare”, parameters directly. Nowadays we know very well that such theories are not infinitely precise from a mathematical point of view, except possibly when the bare coupling strengths run to zero instead of infinity (such as in quantum chromodynamics). This latter, rare, property was not yet known about at the time, so in some sense the critics were right: our theories were mathematically empty. It is here that one needs more than just mathematical skill, namely insight into the requirements for our theories as needed to answer our questions. It would be too much to ask for infinite mathematical precision. If, in lieu of that, one could construct a theory that allows for an infinite crsgmptotic expansion in terms of some parameter that is measured by our experimental friends to be small, what may be achieved is a theory that is more than sufficiently precise to meet all our purposes.

My advisor and colleague, Tini Veltman, had an even more pragmatic view. He had reached the conclusion that Yang-Mills gauge field theories should be used in describing elementary particles by elimination: all the alternatives were uglier, and indeed quite a few aspects of the weak interactions pointed towards a renormalizable Yang-Mills theory for them. In any case, it was his perseverence that kept u8 on this track. Even though there was a moment that he was practically certain to have proven Yang-Mills theories to be nonrenormalizable (he had been studying the pure massive case and was not interested in the Higgs mechanism), he kept pointing towards the facts experimenters had told us about the weak interactions: Yang-Mills theories had to be right! How could gauge invariance have anything to do with renormalization? To analyze this question it is crucial to understand the physical requirements of a particle theory at short distance scales. Suppose we want a theory that describes all known particles at large distance scales, and essentially nothing else at small distances. If many new particle types would arise at very small distances, that is, if we would have a large series of increasingly massive particles, then we would be stuck with a large, in general infinite number of uncontrollable parameters. Such a theory would have zero predictive power. Thus, as measured at a short distance scale, what we want is a theory containing only very light particles. It is all a question of economy; we are dealing here with in some sense the most efficient theories of nature, carrying the least possible amount of structure at small distances. Ideally, we would like t o have a system in which besides the mass parameters also all coupling parameters have the dimension of a negative power of a length, so that at small distance scales the particles are nearly free. In a renormalizable theory we settle for the next best thing, namely dimensionless couplings. One can then prove a very powerful theorem: the onlv perturbative field theories with massless particles and dimensionless couplings am theories featuring fvndamental particles with spin 0, 1 or F’urthermore, the spin 1 fields have an invisible component: the longitudinal part decouples. All for the better, because the longitudinal parts mry essentially no kinetic terms, so they behave as ghosts, particles that would contribute in a disastrous way to the S matrix if they were real; unitarity and positivity of the energy would be destroyed. Now suppose we do want the particles at large distance scales to have a finite maas. Massive particles with spin 1 have a larger number of physical degrees of freedom than massless ones. For massless spin 1 particles only two of the four components of their vector field A,($) are independently observable (the two possible helicities of the radiation field); massive particles on the other hand carry three independent degrees of freedom: the values m, = - 1 , O and +l. If we would simply add a tiny mass term to the field equations we would give the bad ghost component of the vector fields at all distance scales the status of a physically observable field. The value of this field component is ill-controlled at short distances. Its wild oscilla tions would effectively produce strong interactions, more precisely, interactions with a negative mass dimension, or in other words: such a theory is nonrenormalizable.

3.

The only mechanism that can protect the ghost components from becoming observable under all circumstances is a complete local gauge invariance. This invariance must be =act at small distance scale, and therefore should also survive at large distance scale. The great discovery of Peter Higgs was that local gauge invariance is not at all incompatible with finite masses for the vector particles at large distance scales. All one needs to do is add physical degrees of freedom to the model that can play the role of the needed transverse field components. If these fields behave as scalar fields at short distances they don’t give rise to any ghost problem there. At large distances they conspire with the vector field to describe the three independent components of a massive spin 1 particle. The trick used is that although the local gauge invariance has to be exact, the gauge transformation rule can be more general than in a pure Yang-Mills system. It may also involve the scalar field, even if the energetically preferred value of this scalar field is substantially different from zero. Now if we had been dealing with merely a global gauge symmetry this would have been an example of spontaneous symmetry breaking: the vacuum, with its nonvanishing scalar field, is not gauge invariant. It was tempting to refer to the present situation as “spontaneous breakdown of local gauge symmetry”, but this would be somewhat inaccurate. In our theoriea the vacuum does not break local gauge invariance. Any state in Hilbert space that fails to be invariant under local gauge transformations is an unphysical state. Strictly speaking, the vacuum is entirely gauge invariant here. The reason why we can do our calculations in practice as if the symmetry were spontaneously broken is that one can choose the gauge condition such that the scalar field points in a (more or less) fixed direction. In view of the above, we prefer not to use the phrase “spontaneous local symmetry breakdown”, but rather the phrase “Higgs mechanism” to characterize this realization of the theory where the vector particles are massive. Space and time are continuous. This is how it has t o be in all our theories, because it is the only way known to implement the experimentally established fact that we have exact Lorentz invariance. It is also the reason why we must restrict ourselvea to renormaliaable quantum field theories for elementary particles. As a consequence we can consider unlimited scale transformations and study the behavior of our theories at all scales (Chapter 3). This behavior is important and turns out to be highly nontrivial. The fundamental physical parameters such as masses and coupling constants undergo an effective change if we study a theory at a different length and time scale, even the ones that had been introduced as being dimensionless. The reason for this is that the renormalization procedure that relates these constants to physically observable particle properties depends explicity on the mass and length scales used. It waa proposed by A. Peterman and E. C. G. Stueckelbergl, back in 1953,that the freedom to choose one’s renormalization subtraction points can be seen as an invariance group for a renormalizable theory. They called this group the “renormalization group”. In 1954 Murray Gell-Mann and Francis Low observed that ‘E. C. G. Stueckelberg and A. Peterman, Helu. Phva. Acta 20 (1953) 499.

the optimal choice of the subtraction depends on the energy and length scales at which one studies the system. Consequently it turned out that the most important subgroup of the renormalization group corresponds to the group of scale transformations. Later, Curtis G. Callan and independently Kurt Symanzik derived from this invariance partial differential equations for the amplitudes. The coefficients in these equations depend directly on the subtraction terms for the renormalized interaction parameters. The subtraction terms depend to some extent on the details of the subtraction scheme used. For gauge field theories Veltman and the author had introduced the so-called dimensional renormalization procedure. It turns out that if the subtraction terms obtained from this procedure are used for the renormalization group equations these equations simplify. Furthermore there is a purely algebraic relation between the dimensional subtraction terms and the original parameters of the theory. This enables us to express the scaling properties of the most general renormalizable theory directly in terms of all interaction constants via an algebraic master equation. In deriving this equation one can make maximal use of gauge invariance. One can extend this master equation in order to derive counter terms for nonrenormalizable theories such as perturbative quantum gravity, but it would be incorrect to relate these terms to the scaling behavior of this theory, because here the canonical dimension of the interaction parameter, Newton’s gravitational constant, does not vanish but is that of an inverse mass-squared. In the 1960’s and early 1970’s it was thought that all renormalizable field theories scale in such a way that the effective interaction strengths increase at smaller distance scales. Indeed, there existed some theorems that suggest a general law here, which were based on unitarity and positivity for the Feynman propagators. I had difficulties understanding these theorems, but only later realized why. I had made some preliminary investigations concerning scaling behavior, and had taken those theories I understood best: the gauge theories. Here the scaling behavior seemed to be quite different. Now that we have the complete algebra for the scaling behavior of all renormalizable theories we know that non-Abeiian gauge theories are the only renormalizable field theories that may scale in such a way that all interactions at small distances become weak. The reason why they violate the earlier mentioned theorems is that in the renormaliied formulation of the propagators the ghost particles play a fundamental role and the positivity arguments are invalid. This new development was of extreme importance because it enabled us to define theories with strong interactions at large distance scales in terms of a rapidly convergent perturbative formulation at small distances. The new theory for the strong interactions based on this principle was called “quantum chromodynamics”. According to this theory all hadronic subatomic particles are built from more elementary constituent particles called “quarks”. These quarks are bound together by a non-Abelian gauge force, whose quanta are called “gluons”. The crucial assumption was that quarks and gluons behave nearly as free particles as long as they stay close together, but attract each’other with strong binding forces if they are far apart. This fits with the renormalization group

properties of the system, but the fact that complete separation of the quarks from each other and the gluons is impossible under any circumstance could not be understood from arguments based on perturbative formulations of the theory. Only a nonperturbative approach could possibly explain this. Quantum chromodynamica is one of the very few renormalizable field theories in four spacetime dimensions that one can hope to understand nonperturbatively. But how could one possibly explain the confinement phenomenon? Permanent quark confinement must have everything to do with gauge invariance. In his early searches for an explanation the author hit upon a feature in gauge theories that at first sight is entirely unrelated to this problem: the existence of magnetic monopoles. Purely magnetically charged particles had been considered before in quantum field theories, notably by Paul A. M. Dirac in 1934. He had derived that, in natural units, the product of the magnetic charge unit g and the electric charge unit e had to be an integer multiple of 27r. This implies that one can never apply perturbative methods to the interactions of magnetic monopoles because they interact super strongly. What was discovered, also independently by Alexander M. Polyakov2 in Moscow, was that certain extended solutions of the non-Abelian field equations carry a magnetic charge that obeys Dirac’s quantization condition. These solutions behave like classical particles in the perturbative limit and the strong interactions among them are classical (i.e. unquantized) interactions. These monopoles are interesting in their own right, since they may be a feature of certain grand unified schemes for the fundamental particles and may play a role in cosmological theories. But also they may i n d d provide for a quark confinement mechanism. Quark confinement is now generally considered to be a consequence of superconductivity of the vacuum state with respect to monopoles. It is related to the Higgs mechanism if one exchanges electric with magnetic forces. Magnetic monopoles were a consequence of the topological properties of gauge theories. Are they the only consequences? Algebraic topology turns out to be a very rich subject, and indeed there is much more, as explained in Chapter 4. Magnetic monopoles are stable objects in three dimensions. If one search- for objects stable in one dimension one obtains domain walls. Stability in two dimensions is enjoyed by “vortices”. One may now also ask for topologically stable field structures in four dimensions rather than three. In a spacetime diagram such structures would show up as a special kind of “events”. An example of such a configuration that looked interesting had been considered by Alexander Belavin, Albert Schwartz, Yuri Tyupkin and A. Polyakov. But what was the phgsicai interpretation of such “events”? Would they change and enrich the theory as much as magnetic monopoles? They certainly would, as seen in Chapter 5. Events of this sort could be calculated to be extremely rare in most theories, except when the interactions are strong, or if the temperature is very high. In every respect they represented some kind of “tunneling transition”. But tunneling from what to where? At first sight the tunneling was merely from some gauge field configuration to a gaugerotated field configuration. What was special about that? One answer is: the side effects. The event goes associated with 2 ~ M. . poiyakov,

JETP ktt. ao (1974) 194.

an unusual transition between energy levels in the Dirac sea. The consequence of this transition is that in most of our standard theories some apparent conservation laws are violated in these events. For the weak interaction theories the violated conservation law is that of baryon number and lepton number. In the strong interaction the result is nonconservation of chiral U(1) charge. There are still some hopes that baryon number violation will be seen in future laboratory experiments (although I do not share this optimism), but chiral charge nonconservation in the strong interactions had already been observed for some time, causing confusion and embarassment among theorists who failed to understand what was going on. The new, “topoiogical” events, which we called “instantons”, explain this nonconservation in a satisfactory way. It was finally understood why the strong interactions render the 9 particle so much heavier than the pions. The mechanism that keeps quarks permanently bound together in quantum chromodynamics would be a lot easier to understand if we had a soluble version of QCD. Ordinary perturbation expansion, which had served us so well in the previous quantum field theories, was useless here because our question concerns the region of strong couplings. Does there not exist a simplified version of QCD that is exactly soluble, or better even, is the coupling constant expansion the only expansion we can perform, or does there exist some other asymptotic expansion? At first sight the coupling constant g seems to be the only variable available to expand in, but that is not true. There are at least two 0the1-s.~One variable is the dimension D of space-time. In 2 space-time dimensions the theory becomes substantially simpler; however it is not exactly soluble in 2 dimensions, and an expansion in D - 2 does not seem to be very enlightening. A very interesting expansion parameter is 1/N, where N is a parameter if we replace the color gauge group SU(3) by S U ( N ) . At first sight the theory does not seem to simplify at all as N + 00, since there still is an infinite class of highly complicated Feynman graphs. But it does, in a very special way. If we keep g2N = 0’ fixed we obtain a theory which, again, as expanded in powers of 3’ produces an infinite series of Feynman diagrams, and they are too complicated to sum up, even in the limit. But the simplification that does take place is a topological one: all surviving diagrams in the N + 00 limit are “planar”, which means that they can be drawn on a 2-dimensional plane without any crossings of the lines. The twedimensional structures one obtains this way remind one very much of string theory, such that the strings connect every quark with an antiquark. This is the topic of Chapter 6. But what is the use of this observation if one still cannot sum the required diagrams? First of all, the series of diagrams can be summed in 2 dimensions. In the exact theory for D = 2, N = 00, one obtains a beautiful spectrum of hadrons consisting of permanently confined quarks. It would be nice if now the expansion with respect to D - 2 could be performed, but that is not easy. There are two other reasons for further studying QCD in 4 dimensions in the N 4 00 limit. One is that one may consider the theory in a lightcone gauge. The stringy nature of the interactions is then fairly apparent and one may hope to obtain a sensible description 3A third involves the topological parameter 8, connected with instantons.

of hadrons this way.4 Another has to do with more rigorous mathematical aspects of the theory. One may expect namely that the diagrammatic expansion of the theory in terms of powers of B2 converges better than that of the full theory. If one counts the total number of diagrams one finds this to diverge only geometrically with the order, whereas this number diverges factorially in the full theory. Now if any single diagram would only give a contribution to the amplitudes that is strictly bounded by a universal limit, then one would have an expansion that converges within some circle of convergence in the complex 3 plane. But we are not that fortunate. The contribution of a single diagram is not bounded. This is because there were infinities that had to be subtracted. For each individual diagram the renormalization procedure does what it is supposed to do, namely produce a finite, hence useful expression. But when it comes to summing all those diagrams up, the result of these subtractions is a new divergence. Fortunately, the ultraviolet divergent diagrams only form a small subset of all diagrams, and maybe there is a different way to get these under control. Just because the total number of diagrams is much better behaved than in the full theory one can try to perform resummation tricks here. What we were able to prove is that, if some Higgs mechanism provides for masses so that also the infrared divergences are tamed, and if furthermore the coupling constant is small enough, then the resummation procedure named after Bore16 is applicable. Originally we did this in Euclidean space, but analytic continuation towards Minkmki space does not produce serious new problems. The only drawback is that the proof is only rigorous if the coupling constant 3 is so small that the theory is utterly trivial. From a physical point of view nothing new is added to what we already knew or suspected from ordinary perturbation theory. One does not obtain the hadron spectrum or even a confinement mechanism this way. It is rather from the more formal, mathematical point of view that this result is interesting, because it shows that this limit can be constructed in all mathematical rigor. A procedure to remove the constraint that there should be maases, or, equivalently, that Q should be kept small, is still lacking. Indeed, to solve that problem a better understanding of the vacuum state is needed. Atempts to Borel resum such theories bounce off against the fact that we do not know the vacuum expectation values of an infinite claw of composite operators. Then, in Chapter 7, we finally attack the confinement problem directly. Some details of this mechanism were first exposed when Kenneth G. Wilsona produced the l/g expansion of a gauge theory on a lattice. A linearly rising electric potential energy between two quarks then emerges naturally. As indicated earlier, confinement has to do with superconductivity of magnetic monopoles. But it is also directly related to gaugeinvariance. It had been argued by V. Gribov7 that the 4There was a recent report of substantial progreaa in lightcone QCD. See S. J. Brodsky and G. P. Lepage, in Perturbatiue Quantum Chmmod~mics,ed. A. H. Miiller (World Scientific, Singapore, 1989), p. 93. '&mile Borel, Lqom sur lea dries divergentes, Paris,Gauthier-Villers, 1901. 6K.G. Wilson, Phys. Rev. D10 (1974) 2445. 7V. N. Gribov, Nucl. Phys. B139 (1978) 1.

usual procedure for fixing the gauge freedom in non-Abelian gauge theories is not unambiguous. His claim that this ambiguity may have something to do with quark confinement was originally greeted with skepticism. But it turned out to be true. If one meticulously fixes the gauge in an unambiguous manner one finds a new p h e nomenon, namely a sea of color-magnetic monopoles and antimonopoles. It is these monopoles that may undergo Bose condensation. Whether Bose condensation actually occurs or not is not something one can establish from topological arguments alone, because the ordinary Higgs mechanism would be impossible to exclude, even if one has no fundamental scalar field at hand; composite scalar fields could also be used for the Higgs mechanism. So one has to look at dynamical properties of the theory, such as asymptotic freedom. What could be established was a precise description of the confinement mechanism in these terms, as well as alternative modes a theory such as QCD could condense into. Since these alternatives could not be ruled out we have not actually given a “proof” that QCD explains quark conhement. But the logical structure of the confinement mode is now so well understood that no basic mystery seems to be left. And thus elementary particle physics ran out of mysteries. The problem of uniting special relativity with quantum mechanics has been solved. The solution is called “renormalizable quantum field theory”, which in general includes gauge theories. The small distance divergence is in its most essential form only logarithmic, and the solution of that should be looked for by postulating hierarchies of field theories, each hierarchy d i d at its own characteristic distance scale. But this does not solve the biggest mystery of all: why do these theories work? &om a mathematical point of view they work because they are the most economical constructions in terms of physical degrees of freedom: we postulated the least amount of structure possible at small dietance scales. That postulate is what makes these theories so unique. But why should we postulate optimal economy? And what determines which structures exist at all at small distance scales? How does our chain of hierarchies get started? We all think we know where to look for the answer to such questions: the gravitational force. Gravity adds a new fundamental constant to our physical world, namely Newton’s constant. It is incredibly small, indicating that gravity dominates the natural forces at a tremendously tiny distance scale. At this distance scale, called the Planck scale, everything in physics will have to be reconsidered. It is here namely that all our presently known techniques fall short. Numerous attempts have been made to attack this problem. Basically all we want is unite geneml relativity with quantum mechanics. Now general relativity even without quantum mechanics is a highly complex theory, and the roles played by the concepts “energy” and “time” in general relativity are on the one hand equally crucial as, and on the other hand fundamentally different from the one8 they play in quantum mechanics. Well-known are the approaches where a new kind of symmetry is introduced, called “supersymmetry”. The resulting “supergravity theory” was potentially interesting because it seemed to be not as divergent as gravity alone. But I never gave this approach by itself much chance of success because it did

not address the most fundamental aspect of the difficulty, which is the fact that Newton’s constant has the dimensionality of a length squared, so that at distance scales shorter than the Planck length the gravitational force, supersymmetric or not, runs out of control. The same objection, though with a little more hesitation, can be brought against the “superstring” approach to quantizing gravity. The space-time in which the superstring moves is a continuous space-time, and yet we have a distance scale at which a smooth metric becomes meaningless. On the other hand a flat background metric is usually required at ultrashort distances, even in string theories. It is my conviction that a much more drastic approach is inevitable. Space-time cewes to make sense at distances shorter than the Planck length. Here again I reject a purely mathematical attack, particularly when the math is impressive for its stunning complexity, yet too straightforward to be credible. The point here is also that our problem is not only a mathematical one but more essentially physical as well: what is it precisely that we want to know, and what do we know already? There is a different, more “physical” way to see what goes wrong at small distances. Suppose we want to describe any physical phenomenon that is localized at a distance scale smaller than the Planck scale. According to an established paradigm in quantum mechanics this system will contain momenta that will be spread over more than one Planck unit of momentum, and its energy will be of the same order of magnitude. But then it is easy to see that this energy would be confined to within its Schwarzschild horizon, which will stretch beyond one Planck radius. Hence gravitational collapse must already have occurred: such a system is a black hole, and its size will be larger than a Planck unit. This proves that confining any system to within a Planck unit is impossible. Indeed, black holes form a natural barrier. Obviously they must be a central theme in any theory with gravity and quantum mechanics that boasts to be complete. This discussion is conspicuously missing in any superstring theory of quantum gravity, which is why I don’t believe these theories can be complete. In my attempts towards a better understanding of this problem I begin with black holes. How can their presence be reconciled with the laws of quantum mechanics? A brilliant discovery had been made before by Stephen Hawkings: quantum field theory gives black holes a behavior fundamentally different from unquantized general relativity: they radiate. Upon studying this system further one stumbles upon more surprises. It is a fascinating subject. One can imagine thought experiments that may ultimately lead to the resolution of our most fundamental questions. This is Chapter 8. One of the weaknesses a scientist may fall victim to later in his career is that he (or she) may begin to ponder about the deeper significance of his theories in a wider context, the direction they are heading to, the expectations they may hold for the more distant future. What would the most natural, the most satisfactory, the most complete results look like? Could there be an ulttimate theory? Somewhat surprising, perhaps, is that our present insights do indeed suggest that there may ‘S. W.Hawking, Commun. Math. Phys.

43 (1975) 199.

be such a thing as an ultimate description of all physical degrees of freedom and the laws according to which these should evolve. As argued above, space-time itself at the Planck length does not seem to allow for any further subdivision. Could an ultimate physical theory be formulated around that length scale? Our last chapter deals with such questions.

CHAPTER 2

RENORMALIZATION OF GAUGE THEORIES Introductions

.....................................................................

[2.1] “Gauge field theory”, in Pmc. of the Adriatic Summer Meeting on Particle Physics, eds. M. Martinis, S. Pallua, N. Zovko, Elovinj, Yugoslavia, Sep.-Oct. 1973, pp. 321-332 ................................ CERN Rep. 73-9 (1973), [2.2] with M. Veltman, “DIAGRAMMAR”, reprinted in “Particle interactions at very high energies”, NATO Adv. Study Inst. Series B, vol. 4b, pp. 177-322 ............................... [2.3] “Gauge theories with unified weak, electromagnetic and strong interactions”, E.P.S. Int. Conf. on High Energy Physics, Palermo, Sicily, June 1978 ........................................................

12

16

28

174

CHAPTER 2 RENORMALIZATION OF GAUGE THEORIES Introduction to Gauge Field Theory [2.1] The pioneering paper of Chen Ning Yang and Robert Mills9 in 1954 has inspired many physicists in the late 1960's to construct theories for the weak and the strong interactions. In the following papers it is assumed that the reader is somewhat familiar with this fundamental idea, namely to consider field equations that are invariant under symmetry transformations that vary from point to point in spacetime. It is basically just a generalization of the Maxwell equations which indeed have such a symmetry:

Q ( x ,t ) -t exp ieh(x,t) @(x,t ) , A J x , t ) A,(x, t ) - O p W , t ) 9 or a space-time dependent rotation in the complex plane. Yang and Mills replaced these by higher dimensional rotations, which in general are non-Abelian. It was clear however that the Yang-Mills equations would describe massless spin one particles, just as the Maxwell equations, and that these would interact as if they were electrically charged. And it was evident that such particles do not exist in the real world. This leads us to ask two questions: 1. Can one add a mass term to the Yang-Mills equations, such that they become physically more plausible? 2. Since power counting suggests renormalizability, can one renormalize the quantum version of this theory? As explained in the previous chapter, the answer to both these questions is yes, provided that one invokes what is now known as the Higgs mechanism. Without the Higgs mechanism the power counting argument would fail because the longitudinal components of the vector fields (which decouple in the massless case) have no kinetic part in their propagator and therefore interact nonrenormalizably. 'C.

N. Yang and R. L. Mills, Phys. Rev.

96 (1954)191.

But this had to be proved. The problem Veltman was studying in the late 1960's was that the contributions of the ghost particles, all in the longitudinal sector, did not seem to add up properly to give a unitary theory. The ghost problem itself had been studied by Richard Feynman for the massive case, and Bryce DeWitt, Ludwig Faddeev and Victor Popov in the massless case. It was not immediately realized that the massless theory is not simply the limit of a massive one where the mass is sent to zerolo because the transverse component would continue to contribute in loop diagrams: it can be pair-produced. The papers by Faddeev and P O P O Vmostly , ~ ~ in Russian, were not immediately available to us, but a short paper by them in Physics Letters12, in which they summarized their arguments, made their procedure quite clear to me. Basically the message was that the 5'-matrix amplitudes, just like all quantum mechanical amplitudes, are functional integral expressions. Like always in integrals, when one performs a symmetry transformation in the integrand, one has to keep track of the Jacobian factors. These are usually just big determinants. I decided to rewrite these determinants as Gaussian integrals, because then one can more easily read off what the Feynman rules for these are. These Feynman rules are just as if there are some extra complex scalar particles, but because of a sign switch we found that that these scalar particles had to be seen as fermions rather than bosons. And then the trick was that one had to combine the Faddeev-Popov prescription with the Higgs-Kibble mechanism. So we were dealing with two kinds of ghost fields rather than one. In papers (2.1) and (2.2) we consider this procedure as given. But we did not wish to trust the details of the functional integral approach, because the way it dealt with the renormalization infinities did not seem to be rational. Instead, we consider the perturbative formulation of the theory. We then prove that, order by order in the perturbation expansion and after renormalization, this theory indeed obeys all physically relevant requirements such as unitarity and causality. That it was wise t o be this careful soon became clear: there may sometimes be anomalies, and in that case the procedure does not work. A theory with anomalies in the local gauge sector is inconsistent. The fkst paper is an introduction to the second, illustrating the prescriptions in all get the same a simple example. In this example the three gauge bosons B1*293 mass M ,see eq. (2.11). That the masses are all the same is not a consequence of the local gauge symmetry but of a somewhat hidden global symmetry (2.7). This symmetry is explicitly broken by the A2 term, so that higher order corrections may cause a relative mass shift. We first see this mass shift in the fermions. The explicit is: expression for Peak cbreak = A 2F(Zl$l+

41x1) + cbreakJnt,

to be obtained by substituting (2.10) in the last part of (2.5). 'OH. van Dam and M. Veltman, Nucl. Phys. B2l (1970) 288. "L. D. FBddeev, Theor. Math. Phys. 1 (1969)3 (in Russian); Theor. Math. Phys. 1 (1969)1 (English translation). 12L.D. F'addeev and V. N. Popov, Phys. Lett. 26B (1967)29.

13

This symmetry structure is very special. It is one of the few systems with computable mass splittings for some of the elementary fields. This was the reason for my interest in this model.

Introduction to DIAGRAMMAR [2.2] Perturbation expansions with respect to small coupling constants are often looked upon as ugly but necessary tools, and repeatedly physicists attempt to avoid them altogether. It cannot be sufficiently emphasized however that perturbation expansions are an absolutely essential ingredient in quantized gauge field theories. Many of our cherished particle theories can only be defined perturbatively. This means that their treatment can only be considered as being mathematically rigorous if we consider all observable quantities as formal power series in terms of a small parameter. This is sometimes referred to as “nonstandard analysis”: we extend the field of numbers to the field of power expansions. The strength of this modification of our mathematics lies in the fact that the expansions need not have a finite radius of convergence (in general they don’t). If we substitute a finite number, say g2 for the coupling constants we can trust the series to be meaningful only as long as the next term is a smaller correction than the previous one. Later we will see that in most cases the series will behave as C N ( g 2 ) N N !This . then implies that observables cannot be computed more accurately than with margins of the order of e-1/C9a, in many cases more than good enough! It should be noted that shifts of vacuum expectation values are not difficult to deal with within this philosophy of nonstandard analysis. Most importantly, one can observe that the perturbation expansion for a quantum field theory is equivalent to dividing the amplitudes up in secalled Feynman diagrams. Feynman diagrams are nothing more than book-keeping devices for the various contributions to the amplitudes. They are central in the next paper, “DIAGRAMMAR”,written together with M. Veltman. With this word we intended to indicate that our prescriptions are nothing but the “grammatical rules” for working with diagrams. It also means “diagrams” in Danish. There are circumstances where one hopes to be able to do better than perturbation expansion. In asymptotically free theories one might be able to replace the perturbation parameter g2 by the much smaller number at an arbitrarily small distance scale, so that our margin should be tightened significantly, perhaps all the way to zero. Thus a theory such as QCD can perhaps be given a completely rigorous mathematical basis. This has never been proven, but it indeed seems to be plausible. Another point is that one expects phenomena that are themselves of the order of e-l/@, as explained in Chapter 5. These cannot be handled with Feynman diagrams. In “DIAGRAMMAR” we first show how to work with the diagrams, then how transformations in the field variables correspond to combinatorial manipulations of the diagrams, exactly as in expressions for functional integrals. If a series of diagrams is geometrically divergent (for instance if we sum the propagator inser-

tions, then unatady tells us exactly that the analytic sum is to be taken. We show how the Faddeev-Popov procedure is translated in diagrammatic language. Most importantly, it is now seen how and why these tricks continue to work when the theory is renormalized. The infinities cancel if the anomalies cancel. Then dimensional regularization and renormalization are explained. There are three steps to be taken. First we have to define what it means to have a non-integer number of dimensions. Fortunately this is unambiguous at the level of the diagrams - though not at all so “beyond” the perturbation expansion! Secondly we have to deal with integrations that diverge even at non-rational spece-time dimension near four; they are easily tamed by analytic continuation from the regions where the integrals do converge. In practice one does this by partial integrations. Thirdly we obeerve that precisely at integer dimensions some infinities are not tamed by partial integrations: the logarithmic ones. These show up as powers of l/e, where e = 4 - n, TI is the number of spacetime dimensions. They have to be cancelled out by inserting the proper counterterms into the Lagrangian. One then ends up with finite, physically meaningful expressions. In Section 11of “DIAGRAMMAR” a transformation is described that had been introduced before as “Bell-”reiman transformation”,by my c+author M. Veltman. The name is something of a joke. There is no reference to either Bell or Tkeiman. It would have been more appropriate to call them “Veltman transformations”. The Slavnov-Taylor identities play a crucial role in the renormalization proce dure of gauge theories. They get the attention they deserve in DIAGRAMMAR. The proofs here are as they were first derived. Nowadays a more elegant method exists: the Becchi-Rouet-Stora-Tyupkin quantization procedure. These authors observed that our identities follow from a global symmetry. Now I had tried such symmetries myself long before DIAGRAMMAR was written, but without success. The crucial, and brilliant, ingredient invented by BRST was that the symmetry generators had to be anticommuting numbers.

Introduction to Gauge Theories with Unifled Weak, Electromagnetic and Strong Interactions [!MI The last paper in this chapter needs little further introduction. It displays the enormous progress made in just a few years of renormalizable gauge theory. The J / @ particle, the last crucial ingredient needed to render credibility to what was to become the standard model, had just been found. I was a little too optimistic in applying asymptotic freedom to understand the J/$J energy levels, hence my underestimation of the splitting between the ortho and the para levels in Table 1. But these levels, at that time not yet seen, would soon be discovered and everything fell into place. Of course most of the more exotic models speculated about in this paper were ruled out during the years that followed. What remained was to be called the “Standard Model”. The solution to the eta problems, Section 10 was basically correct, but would be understood better in 1975,with the discovery of the instanton effects, see Chapter 5.

CHAPTER 2.1

GAUGE FIELD THEORY* G. ’t Hooft CERN - Geneva On leave from the University of Utrecht, Netherlands 1. INTRODUCTION

There are several possible approaches to quantum field theory. One may start with a classical system of fields, interacting throuqh non-linear equations of motion which are subsequently “quantized”. Alternatively, one could take the physically observed particles as a starting point; then define a Hilbert space, local operator fields, and an interaction Hamiltonian. More ambitious, perhaps, is the functional integral approach, which has the advantage of being obviously Lorentz covariant. All these approaches have one unpleasant and one pleasant feature in common. The unpleasant one is that in deriving the S-matrix for the theory, one encounters infinities of different types. In order to get rid of these, one has to invoke a rather ad hoc “renormalization procedure”, thus changing and undermining the theory halfway. The pleasant feature, on the other hand, is that one always ends up with a simple calculus for the S matrix: the Feynman rules. Few physicists object nowadays to the idea that these Feynman diagrams contain more truth than the underlying formalism, and it seems only rational to abandon the aforementioned principles and use the diagrammatic rules as a starting point. It is this diagrammatic approach to quantum field theory which we wish to advertise. The short-circuiting has several advantages. Besides the fact that it implies a considerable simplification, in particular in the case of gauge theories, one can simply superimpose the renormalization prescriptions on the Feynman rules. As f o r unitarity and causality, the situation has now been reversed: we shall have to investigate under which conditions these Feynman rules describe a unitary and causal theory. Within such a scheme many more or less doubtful or complicated theorems from the other approaches can be proved completely rigorously. Clearly, Feynman diagrams merely describe an asymptotic expansion of a theory for the coupling constants going to zero, and strict ly speaking, nothing is known about the theory with finite coupling constants. But the other approaches are not better in this respect, if it comes to calculations of physical effects. A really rigorous formulation of quantum field theory with finite interactions has not yet been given and w e are of the opinion that attempts at such a forProceedings of the Adriatic Summer Meeting on Particle Physics. Edited by M. Martinis et al. (North Holland, 1973).

16

G. ' t HOOFT mulation can only succeed if the perturbation expansion, formulated in the simplest possible way (Feynman diagrams) is well understood. In these notes, which must be considered as an introduction to the CERN report called "DIAGRAMMAR" 111, we shall outline the basic step8 that have to be taken in order to formulate and understand a gauge field theory. We shall illustrate our arguments with a simple example of a non-Abelian gauge model (which is not realistic physically). In Section 2 we give a review of the construction process of a model in general, showing the Brout-Englert-Higgs-Kibble phenomenon. In Section 3 the quantization problem is formulated and the essential steps in the proof of renormalizability are indicated. 2. CONSTRUCTION OF A MODEL In the construction of a model one must try to combine the experimental observations with the theoretical requirements. There is of course no logical prescription how to do that, and therefore we shall here only consider the theoretical principles 121 and leave it to the reader to alter our little example in such a way as to obtain finally physically more interesting results. a) First we choose the GAUGE GROUP. This is a group of internal symmetry transformations that depend on space and time 131. In our example this group will be SU(Z)XU(l) (at each space-time point). Let us denote an infinitesimal gauge transformation as eT, which is generated by an infinitesimal function of space-time, Aa(x), a 1,2,3,0. The condition that tha infinitesimal gauge transformations generate a qroup is

-

where [T(l), T(2)] 13 again an infinitesimal gauge transformation, generated by (2.2) Here the indices between the parentheses denote different choices for and fabc are structure constants of the group. In our example,

A,

fabc = cabc - 0

for a,b,c # 0 otherwise.

(Often we shall write gAa instead of Aa, where g is some coupling constant. ) By choosing the gauge group we also fix the set of vector particles: one for each generator T. In our example:

17

GAUGE FIELD THEORY a B,,(x),

a = 1,2,3

and

ACl(x)

.

b) Now we choose the other fields. They all must be representations of the gauge group. Example: A Bose field with "isospin" 1 (a two-component representation of S U ( 2 ) ) and "charge" 0 (a scalar for U ( 1 ) transformations). A Fermi field Qi with "isospin" 51 and "charge" 1; a Fermi field X with "isospin" 0 and "charge" 1. c) The GAUGE TRANSFORMATION LAW must satisfy the commutation rules (2.11, ( 2 . 2 ) . There is one way to satisfy the requirements. In our example the infinitesimal transformations are

z

B;'

=

B; - a

ha u3

$ ' = $ - 1 igl

1

+

glEabc A bB,c, ;

T ~ A ,~ $

as1 Q'=

X' =

Q

- z1

x +

3

1 T ~ A +~ Qig2Ao$ ig1 all

ig2AOX

.

, (2.31

d) Next we write down a Lagrange density, which we shall call the SYMMETRIC LAGRANGIAN, ginv(x). It is a functior. of the fields and their space-time derivatives at the point x. The corresponding Lagrange equations

6Y

6Ai(x)-

a

6g apAi(X) =

O

where Ai(x) is any of the fields Ba, u Av' $, Q i or X will describe to a first approximation the propagation of these fields, the quanta of which are the physical particles. The fact that we have a quantum theory will necessitate "higher order" corrections (the loops in the Feynman graphs). This Lagrangian must be invariant under the gauge transformations ( 2 . 3 ) . In our example we take

18

G. t ' HOOFT

D,,X =

a,,x +

ig A

2 u

x

.

(2.6)

~ l terms l in the Lagrangian must have dimension 4 o r less. Dimension is counted as follows: a derivative has dimension one, a Bose field 3 has dimension one and a Fermi field has dimension 5 Given that Bo1 1 se propagators behave as -2 and Fermi propagators as i; for Ikl + k this requirement will enable us to estimate the maximal "overall" divergence of an integral in a diagram, if only the external lines are known. For instance, the overall divergence of the diagram in fig. 1 is linear, independent of its internal topology (in fig. 1 there are 11 Fermion and 8 Boson propagators,

.

fermion ', ,*'

(-

1 i;)

----

boson

1 k2

( - -)

I'

5',,

vertices from

and ( $ * @ I 2 ,

respectively, (for instance).

Fig. 1

.

and 7 loops for integration, 7.4-11-1-8.2 = 1 ) e) Examine the GLOBAL SYMMETRIES that is, all symmetries that are independent of space time. In the example: Parity (PI, Charge conjugation (C), Time reversal (T), and, if X 2 + 0, an unexpected SU (2) symmetry:

0;

Qi + " E i j 0 3

+AziEij$;

+ A3i0i

t

(2.7)

A i space-time independent. The importance of the global symmetries is that all possible interaction terms that do not violate these symmetries nor the local

19

GAUGE FIELD THEORY gauge symmetry, must be present in the Lagrangian (2.5). This is why the term with X 1 must be present. 2 If p o p m and m >O, then the model contains massive Fermions X J, and @ particles. But note that our Lagrangian does not contain a mass term for the vector particles, simply because no gauge invariant mass term exists. So these spin-one particles are massless and they interact with each other. As a consequence there are huge infrared divergencies of a type that cannot be cured as in quantum electrodynamics. It is expected that the eventual solution of the quantized equations will be drastically different from the classical ones (2.4) and completely governed by these infrared effects (.conjecture: strongly interacting Regge particles), but nobody knows how to solve this problem.? Nevertheless, the model is interesting, and that is because we can take the variable uo2 to be negative. If we then assume for a moment that the fields in d. are classical, then the Hamilton density , obtained from , contains a term

The energy is minimal not if 4 4 , but if

Because of gauge invariance, we can always rotate until @i is parallel to the spinor (1,O). Small fluctuations of the various fields near this equilibrium state are now described by massive equations of motion, as we shall see in f. We can now proceed in two ways. Either we I) first quantize the "symmetric" theory and then construct the new vacuum, corresponding to this "equilibrium state", or we 11) first define new fields 4 , with the equilibrium value (2.9) subtracted, and then quantize this "asymmetric" theory. We choose possibility no. 11. Therefore: f) Shift those fields which have a vacuum expectation value 141. In the example (2.10)

Here the number F satisfies, up to possible quantum corrections, eq.

20

G.

't HOOFT

( 2 . 9 ) , and 2 and Ya a r e t h e new f i e l d v a r i a b l e s ( r e a l ) . The Lagrangfan I n terms of t h e new f i e l d s Is t h e NON-SYMMETRIC LAGRANGIAN. I n our example:

(2.11)

where M2 =

1 g:F2

6 = pi

+

; Mi = 2A1F2

AlF2

and (2.12)

.

Tha i n t e r a c t i o n tenn is r a t h e r complicated now and n o t so r e l e v a n t f o r t h e d i s c u s s i o n . I n o u r example I t Is

int =

- i glauYaedc B > ~ + l g Ba, , ( z ~ , , Y ~ - Ya au z) - 3 92B2u ( Z 2 + Y z ) - $ g M B 2u2 (2.13)

The tenn gbreak arises from t h e A 2 term and breaks t h e g l o b a l SU(2) symmetry ( t h e Fennion masses are a l s o s p l i t ) . that Note t h a t t h e r e1 is a subgroup of SU(2)local xsu ( 2 ) g l o b a l in (2.10) i n v a r i a n t . T h i s is t h e new symmetry leave8 t h e s p i n o r $), group, broken by t h e A 2 tenn. A 8 w e do n o t i n s i s t on eq. ( 2 . 9 ) f o r F, o u r s h i f t is f r e e , and w e g e t a new " f r e e parameter" B i n ( 2 . 1 1 ) . I n g e n e r a l , we s h a l l req u i r e t h a t a l l diagrams where t h e 2 p a r t i c l e vanishes i n t o t h e vacuum cancel ( f i g . 2 ) . I n lowest o r d e r t h i 8 corresponds to B-0.

21

GAUGE FIELD THEORY g) The obtained Lagrangian does not at all look invariant under the local gauge transformations, in particular the vector-mass term and the i 2 term. (Of course, the U(1) invariance here is obvious). But we can rewrite the original gauge transformation law (2.3) in terms of the new fields. We get the NEW GAUGE TRANSFORMATION LAW: Y,'

2'

a

Ya

+ 51

2 +

1

glcabc AbYc

5

glhaYa.

-

a

-

3 glAaZ - MAa

, (2.14)

No change in: BE'

= B~ 1

!

+ glEabcAb B,c,

.

P

In the general case, A;

= Ai

+ aA:%

+ g siajAaAj

,

(2.15)

where 1; are either coefficients with the dimension of a mass, or the derivative a for the vector fields. P Under these gauge transformations the Lagrangian is still invariant. Its nonsymmetric form is merely a consequence of the practitally arbitrary, nonsymmetric new coefficlents tt. Although we have much more bookkeeping to do now, the general principle, the invariance under a gauge transformation, is not changed by the shift. Only the transformation law is a little bit different. Note that the new t coefficients are of zeroth order in gl. h) The model contains PHYSICAL PARTICLES and GHOSTS. Ghosts correspond to those fields that can be turned away independently by gauge transformations. To determine which fields are ghosts it Ls sufficient to consider only the lowest-order parts of the gauge transformation, described by the t coefficients. In our example one can take a,,A,, = 0, and then either turn the Ya fields away or one of the spin-components of the vector fields WE. The first choice is obviously the most convenient one. Compare (2.14) ; we choose ha = 1 Ya+ + perturbation expansion in gl. Our model evidently contains three massive spin-one particles Ba a massless photon Ap (with only two lJ, possible helicities), one massive spinless particle 2, and three massive Fermions X and $i (of which the latter form a doublet). i) It is instructive to consider first the classical field equations. Just as in electrodynamics one must choose a gauge condition in order to remove the gauge freedom. In general, we can write

-

22

G. ' t HOOFT

-

this condition in the form ca(x)

0

,

(2.16)

where Ca(x) is some function of the fields that must not be gauge invariant. The number of components of Ca must be identical to the nwnber of generators in the gauge group, here four. In subsection h) we wou 1d have co co =

ca

-

avAIJ

E

or

ca

ya

a PAP aLIBIJ

One may impose such a gauge condition in an elegant way by replacing the original Lagrangian '&Inv by i .

p n v

-3

(cap

.

(2.17)

The equations of motion corresponding to this Lagrangian are fulfilled d4x. Let if any small variation of the fields does not change us choose as a small variation an infinitesimal gauge transformation, described by Ab(x) :

2-3-C a . bCa.

Ab

,

(2.18)

6Ab

-

where 6C" stands formally for the variation of Ca under a gauge trans Ad formatfon, and must be unequal to zero. Hence Ca must be zero. Applying other variations to the Lagrangian, we get back the original equations of motion, now with the gauge condition Ca= 0. In our example it is convenient to choose

(2.19)

where A is a free parameter. Consequently, after a gauge transformation,

more explicitly,

(2.20)

23

GAUGE FIELD THEORY The reason why this particular choice for Ca(x) is convenient: is that the bilinear interaction -M

Yaa,,Bt

in z i n v , eq. (2.11), is cancelled, so that the B and Y propagators IJ are decoupled. The importance of the gauge parameter A will be made clear in the next section. The model of our example becomes more interesting if we assign a U(1) charge not to X but to the field We then get something close151. Its gauge structure is a bit ly resembling the Weinberg model more complicated.

+.

3. THE MATHEMATICAL STRUCTURE OF GAUGE FIELD THEORY The models constructed along the lines given in the previous section are renormalizable (under one additional condition, see 11) in this section). For the proof of this statement several steps must be made. We indicate those here without going into any of the technical details (which can all be found in "DIAGRAMMAR"). We give the steps in an order which is logical in the mathematical sense. This is not the order in which the theorems have been derived, but the order in which a final proof should be given. Historically, the Feynman rules were first found by functional integral methods but the proof was heuristic and not very rigorous. A s our first step we shall 1) postulate the Feynman rules. These are always uniquely defined by a Lagrangian. As in the classical case, one must choose a gauge function c : ~ (XI ) , for which we shall again take (2.19) In our example. The Lagrangian however is not the one of the previous section, but

x= where describes a new "ghost particle", called Faddeev-Popov ghost, occurring only in closed loops, and it depends on the choice

of ca(x) :

where

Oa

is a new complex

24

G. ' t HOOFT The 0 particle may not occur among the external lines and because the vertices only connect two @ lines, it only occurs in single loopa. A further prescription is that there must be one more minus sign for eadr 4 loop ("wrong statistics"). Note that the @ corresponding to the Abelian group U ( 1 ) is a free particle and therefore drops out. The mass of the SU(21-0 depend. on the gauge parameter A . i i ) Reqularize the theory, that is, we must modify the theory slightly in terms of a small parameter, say c p such that all divergent integrals become finite and the physical situation is attained in the limit c+O. The modification must be as gentle as possible and should not destroy gauge invariance so that we can still use all our theorems. The most elegant method 171 is the dimensional regulariza4-d, d is the "number of space-time dition procedure, in which E mensions". What we mean by this can be formulated completely rigorously as long as we confine ourselves to diagrams (but that we have already decided). For some theories, however, containing y 5=y 1y 2 y 3y 4 or the tensor c o B y 6 * the extension to arbitrary dimensions cannot be made, and the method does not work. Indeed, such theories suffer, more often than not, from the so-called "Bell-Jackiw" anomalies 161. They are not renormalizable. iii) Now that all Green's functions are finite, we prove that the S matrix is independent of the choice of Ca(xL. For this we must consider non-local field transformations ("canonical transformations") and prove the Slavnov identities. These laentities are essential and from them the gauge independence of the S matrix follows. It is in the proof of the Slavnov identities where the "group property", eqe. (2.1) and (2.21, enters. iv) Find a set of gauge functions C:(x) such that a) the theory is "renormalizable by power counting" for (compare subsection 2d and fig. 1); all A < b) the S matrix is unitary in the space of states with energy*) E < X This can be established by means of the "cutting rules" ; c) the gauge functions must stay Lorentz-invariant.

-

-

.

..................................................................... * ) Note that the ghosts in our example all have mass AM. ..................................................................... (One could also require renormalizability and complete unitarity but give up Lorentz invariance for a while; this seems not very practi-

25

GAUGE FIELD THEORY -1.) Now we must consider the limit where the regulator parameter B approaches its physical value zero. v) Renormalize, according to some well-defined prescription. The most secure preacription is to add "local" counter terms to the Lagrangian that cancel one by one the divergencies in all possible diagrams. In the dimensional procedure they have the form of ordinary terms but with coefficients , , etc. It must be proved that 0 only local counter terms are sufficient to remove all divergencies, otherwise the cutting rules, e.g. unitarity are violated. Here we make use of the causality-dispersion relations (also derived from cutting rules). vi) Evidently, we altered the Lagrangian. Does this not spoil gauge invariance? We show that the new, "renormalized" Lagrmgian is invariant under new, "renormalized" gauge transformation laws. We must convince ourselves that the gauge group (in our example SU(2)xU(1) is not changed into another one (for example, U(l)xU(l)x xU(1) XU (1) ) In fact, the new laws are equivalent to the old ones for "renormalized" fields. It then follows that the theory is "multiplicitatively" renormalizable. We could only prove this if the gauge function Ca (x) has been chosen without bilinear field combinations, such as + a(B,)a 2 Ca(x) = a,B:

$

.

.

vii) Finally show that not only the regularized but also the Enormalized S matrix is independent of the choice of C"(x1, apart from the obvious fact that a variation of Ca(x) may have to be accompanied by a slight change of the original variables gl, M, etc. It is this latter remark which makes this point far from easy to deal with, but it follows from the "multiplicative" renormalizability in the case of linear C. REFERENCES 11

I

G. 't Hooft and M. Veltman, "DIAGRAMMAR", CERN 73-9 (1973), Chapter 2.2 of this book.

121 See also: B.W. Lee, Phys. Rev. D5 (1972) 823;

B.W. Lee and J. Zinn Justin, Phys. Rev. D5 (1972) 3121, 3137,

3155. 131 C.N. Yang and R.L. Mills, Phys. Rev. 96 (1954) 191.

26

G.

' t HOOFT

141 F. E n g l e r t and R. Brout, Phys. Rev. Letters 13 ( 1 9 6 4 ) 3 2 1 ;

P.W.

Higgs, Phys. Letters 12 ( 1 9 6 4 ) 132; Phys. Rev. Letters 13

( 1 9 6 4 ) 508; Phys. Rev. 145 ( 1 9 6 6 ) 1156; G.S.

Guralnik, C.R. Hagen and T.W.B.

Kibble, Phys. Rev. Letters

13 ( 1 9 6 4 ) 585;

T.W.B.

Kibble, Phys. Rev. 155 ( 1 9 6 7 ) 6 2 7 .

151 S. Weinberg, Phys. Rev. Letters 19 ( 1 9 6 7 ) 1264. 161 J . S . Bell and R. Jackiw, Nuovo Cimento 60A ( 1 9 6 9 ) 4 7 . 171 See however a l s o t h e approach of A. Slavnov, Theor. and Math.

Phys.

13 ( 1 9 7 2 ) 174.

*Reprinted from Proceedings of the Adriatic S m e r Meeting on Particle Physics, Rovinj, Yugoslavia, September 2 3 - O c t o b e r 5, 1973. *Note added: T h i s paper was w r i t t e n i n 1 9 7 3 , b e f o r e "asymptotic freedom" l e d t o t h e g e n e r a l acceptance of QCD. The c o n j e c t u r e , which r e f e r s t o gauge t h e o r i e s w i t h o u t Higgs mechanism, is now g e n e r a l l y b e l i e v e d t o be c o r r e c t .

27

CHAPTER 2.2

DIAGRAMMAR G. 't Hooft and M. Veltman CERN - European Organization for Nuclear Research - Geneva

Reprint of CERN Yellow Report 73-9 : Diagrammar by G . 't Hooft and M. Veltman with the kind permission of CERN

28

G. 't HOOFT and M. VELTMAN

1s

INTRODUCTION

W i t h t h e advent o f gauge t h e o r i e s i t became necessary t o r e c o n s i d e r many w e l l - e s t a b l i s h e d i d e a s I n quantum f i e l d t h e o r i e s . The c a n o n i c a l formalism, f o r m e r l y regarded as t h e most convent i o n a l and r i g o r o u s approach, has now been abandoned by many authors. The p a t h - i n t e g r a l concept cannot r e p l a c e t h e c a n o n i c a l f o r m a l i s m i n d e f i n i n g a theory, s i n c e p a t h i n t e g r a l s i n f o u r dimensions a r e meaningless w i t h o u t a d d i t i o n a l and r a t h e r ad hoc renormalization prescriptions. Whatever approach i s used, t h e r e s u l t i s always t h a t t h e S-matrix i s expressed i n terms of a c e r t a i n s e t of Feynman d i a grams. Few p h y s i c i s t s o b j e c t nowadays t o t h e i d e a t h a t diagrams c o n t a i n more t r u t h than t h e u n d e r l y i n g formalism, and i t seems o n l y r a t i o n a l t o t a k e t h e f i n a l s t e p and abandon o p e r a t o r formal i s m and p a t h i n t e g r a l s as i n s t r u m e n t s o f a n a l y s i s . Yet i t would be v e r y s h o r t s i g h t e d t o t u r n away completely from these methods. Many u s e f u l r e l a t i o n s have been derived. and many more may be i n t h e f u t u r e . What must be done I s t o p u t them on a s o l i d f o o t i n g . The s i t u a t i o n must be reversed: diagrams form t h e b a s i s from which e v e r y t h i n g must be d e r i v e d . They d e f i n e t h e o p e r a t i o n a l r u l e s , and t e l l us when t o worry about Schwinger terms, s u b t r a c t i o n s , and whatever o t h e r m y t h o l o g i c a l o b j e c t s need t o be introduced. The development of gauge t h e o r i e s owes much t o p a t h i n t e g r a l s and i t i s t e m p t i n g t o a t t a c h more t h a n a h e u r i s t i c v a l u e t o p a t h i n t e g r a l d e r i v a t i o n s . Although we do n o t r e l y on p a t h i n t e g r a l s i n t h i s paper, one may t h i n k of expanding t h e exponent of t h e i n t e r a c t i o n Lagrangian i n a T a y l o r s e r i e s , so t h a t t h e a l g e b r a o f t h e Gaussian i n t e g r a l s becomes e x a c t l y i d e n t i c a l t o t h e scheme of m a n i p u l a t i o n s w i t h Feynman diagrams. That would leave us w i t h p r e s c r i p t i o n i n t h e propat h e problems of g i v i n g t h e c o r r e c t IE g a t o r s , and t o f i n d a decent r e n o r m a l i z a t i o n scheme. There i s another aspect t h a t needs emphasis. From t h e o u t s e t t h e c a n o n i c a l o p e r a t o r f o r m a l i s m i s n o t a p e r t u b a t i o n theory. w h i l e diagrams c e r t a i n l y a r e p e r t u r b a t i v e o b j e c t s . Using d i a grams as a s t a r t i n g p o i n t seems t h e r e f o r e t o be a c a p i t u l a t i o n i n t h e s t r u g g l e t o go beyond p e r t u r b a t i o n theory. I t i s u n t h i n k a b l e t o accept as a f i n a l g o a l a p e r t u r b a t i o n theory, and i t i s n o t our purpose t o f o r w a r d such a n o t i o n . On t h e c o n t r a r y , i t becomes more and more c l e a r t h a t p e r t u r b a t i o n t h e o r y i s a v e r y u s e f u l d e v i c e t o d i s c o v e r equations and p r o p e r t i e s t h a t may h o l d t r u e even ift h e p e r t u r b a t i o n expansion f a i l s . There a r e a l r e a d y sever a l examples of t h i s mechanism: on t h e s i m p l e s t l e v e l t h e r e i s f o r i n s t a n c e t h e treatment o f u n s t a b l e p a r t i c l e s , w h i l e i f i t

29

DIAGRAMMAR

comes t o unfathomed depths t h e Callan-Symanzik equation may be quoted. A l l such treatments have i n common t h a t g l o b a l p r o p e r t i e s are e s t a b l i s h e d f o r diagrams and then e x t r a p o l a t e d beyond p e r t u r b a t i o n theory. G l o b a l p r o p e r t i e s are those t h a t h o l d i n a r b i t r a r y o r d e r of p e r t u r b a t i o n t h e o r y f o r t h e grand t o t a l o f a l l diagrams e n t e r i n g a t any g i v e n order. I t i s here t h a t v e r y n a t u r a l l y t h e concept of t h e g l o b a l diagram enter's: i t i s f o r a g i v e n o r d e r o f p e r t u r b a t i o n theory, f o r a g i v e n number o f e x t e r n a l l i n e s t h e sum of a l l c o n t r i b u t i n g diagrams. T h i s o b j e c t , v e r y o f t e n presented as a blob, an empty c i r c l e , i n t h e f o l l o w i n g pages, i s supposed t o have a s i g n i f i c a n c e beyond p e r t u r b a t i o n theor y . P r a c t i c a l l y all equations o f t h e c a n o n i c a l formalism can be r e w r i t t e n i n terms of such g l o b a l diagrams, thereby opening up t h e a r s e n a l o f t h e c a n o n i c a l f o r m a l i s m f o r t h i s approach.

A f u r t h e r deficiency i s r e l a t e d t o the divergencies o f t h e p e r t u r b a t i o n s e r i e s . T r a d i t i o n a l l y i t was p o s s i b l e t o make t h e t h e o r y f i n i t e w i t h i n t h e c o n t e x t o f t h e c a n o n i c a l formalism. F o r i n s t a n c e quantum electrodynamics can be made f i n i t e by means o f P a u l i - V i l l a r s r e g u l a t o r f i e l d s , r e p r e s e n t i n g heavy p a r t i c l e s w i t h wrong m e t r i c o r wrong s t a t i s t i c s . J u d i c i a l choice o f masses and c o u p l i n g constants makes e v e r y t h i n g f i n i t e and gauge i n v a r i a n t and t u r n s t h e c a n o n i c a l f o r m a l i s m i n t o a reasonably well-behaving machine, f r e e o f o b j e c t s such as 6[0), t o name one. U n f o r t u n a t e l y t h i s i s n o t t h e s i t u a t i o n i n t h e case o f gauge t h e o r i e s . There t h e most s u i t a b l e r e g u l a t o r method, t h e dimensionel r e g u l e r i z e t i o n scheme, i s d e f i n e d e x c l u s i v e l y f o r diagrams, and up t o now nobody has seen a way t o i n t r o d u c e a dimensional c a n o n i c a l f o r m a l i s m or p a t h i n t e g r a l . The v e r y concept of a f i e l d , and t h e n o t i o n o f a H i l b e r t space a r e t o o r i g i d t o a l l o w such g e n e r a l i z a t i o n s . The treatment o u t l i n e d i n t h e f o l l o w i n g pages i s n o t supposed t o .be complete, b u t r a t h e r meant as a f i r s t , more o r l e s s pedagogical attempt t o implement t h e above p o i n t o f view f e a t u r i n g g l o b a l diagrams as p r i m a r y o b j e c t s . The most i m p o r t a n t p r o p e r t i e s of t h e c a n o n i c a l as w e l l as p a t h i n t e g r a l f o r m a l i s m are r e d e r i v e d : u n i t a r i t y , c a u s a l i t y , Faddeev-Popov determinants, e t c . The s t a r t i n g p o i n t i s always a s e t of Feynemn r u l e s s u c c i n c t l y g i v e n by means o f a Legranglen. No d e r i v a t i o n o f these r u l e s i s g i v e n : corresponding t o any Lagrangian [ w i t h v e r y few l i m i t a t i o n s concern i n g i t s form1 t h e r u l e s a r e simply defined. Subsequently, Green's f u n c t i o n s , a Hilb81-t space and an S-matrix are d e f i n e d i n terms o f diagrams. Next we examine p r o p e r t i e s l i k e u n i t a r i t y and c a u s a l i t y o f t h e ' r e s u l t i n g theory. The b a s i c t o o l f o r t h a t are t h e c u t t i n g equations d e r i v e d i n t h e t e x t . The use of these equations r e l a t e s v e r y c l o s e l y t o t h e c l e s s i c a l work o f Bogoliubov, and Bogoliubov's d e f i n i t i o n o f c e u s a l i t y i s seen t o hold. The equations remain t r u e w i t h i n t h e framework o f t h e continuous dimension method: r e n o r m a l i z a t i o n can t h e r e f o r e be

30

G. 't HOOFT and M. VELTMAN

t r e a t e d B l a Bogoliubov. O f course, the c u t t i n g equations w i l l t e l l us i n general t h a t thb theory i s not u n i t a r y , unless t h e Lagrangian from which we s t a r t e d s a t i s f i e s c e r t a i n r e l a t i o n s . I n a gauge theory moreover, t h e S-matrix i e only u n i t a r y i n a "physical" H i l b e r t space, which I s a subspace o f the o r i g i n a l H i l b e r t space [ t h e one t h a t wae suggested by the form o f t h e Lagrangian).

To i l l u s t r a t e i n d e t a i l t h e complications o f gauge theory we have turned t o good o l d quantum electrodynamics. Even I f t h i s theor y lacks some of the complications t h a t may a r i s e i n the general case I t t u r n s out t o be s u f f i c i e n t l y s t r u c t u r e d t o show how everyt h i n g works. The m e t r i c used throughout the paper i s

The f a c t o r s 1 i n t h e f o u r t h components are only t h e r e f o r ease o f notation, and should not be revereed when t a k i n g the complex conjugate of a f o u r - v e c t o r

I n our Feynman r u l e s we have e x p l i c i t l y denoted t h e r e l e v a n t f a c t o r s (Zn14i, b u t o f t e n omitted t h e 6 f u n c t i o n s f o r energymomentum conservation.

31

DIAG RAMMA R

2. 2.1.

DEFINITIONS D e f i n i t i o n o f t h e Feynman Rules

The purpose o f t h i s s e c t i o n i s t o s p e l l o u t t h e p r e c i s e f o r m o f t h e Feynman r u l e s f o r a g i v e n Lagrangian. I n p r i n c i p l e , t h i s i s v e r y s t r a i g h t f o r w a r d : t h e propagators a r e d e f i n e d by t h e q u a d r a t i c p a r t o f t h e Lagrangian, and t h e r e s t i s represented by v e r t i c e s . As i s we13 known. t h e propagators a r e minus t h e i n v e r s e o f t h e o p e r a t o r found i n t h e q u a d r a t i c term, f o r example

1:

1:

=

-

1 ?-+(a2 -

-$(y'a,

2 m lg

+ m)$

-1 - i ~ ,l

*

(k2

+

(iyk + m1-I =

+

m2

-iyk + m k2 + m2

-

ie'

Customarily, one d e r i v e s t h i s u s i n g commutation r u l e s o f t h e f i e l d s , e t c . We w i l l s i m p l y s k i p t h e d e r i v a t i o n and d e f i n e t h e propagat o r , i n c l u d i n g t h e IE p r e s c r i p t i o n f o r t h e pole. Similarly, vertices arise. For instance, i f the i n t e r a c t i o n Lagrangian c o n t a i n s a term p r o v i d i n g f o r t h e i n t e r a c t i o n o f fermions and a s c a l a r f i e l d one has I 1

A

9

I n t h i s . and s i m i l a r cases t h e r e is no d i f f i c u l t y i n d e r i v i n g t h e r u l e s by t h e u s u a l c a n o n i c a l formalism, I f however d e r i v a t i v e s , o r worse n o n - l o c a l terms, occur i n L, t h e n c o m p l i c a t i o n s a r i s e . Again we w i l l s h o r t - c i r c u i t a l l d i f f i c u l t i e s and d e f i n e o u r v e r t i c e s , i n c l u d i n g n o n - l o c a l v e r t i c e s , d i r e c t l y f r o m t h e Lagrang i a n . Furthermore, we w i l l a l l o w sources t h a t can absorb o r e m i t p a r t i c l e s . They a r e an I m p o r t a n t t o o l i n t h e a n a l y s i s . I n t h e r e s t o f t h i s s e c t i o n we w i l l t r y t o d e f i n e p r e c i s e l y t h e Feynman rules f o r t h e g e n e r a l case, i n c l u d i n g f a c t o r s II, e t c . B a s i c a l l y t h e r e c i p e i s t h e s t r a i g h t f o r w a r d g e n e r a l i z a t i o n o f t h e simple cases shown above. The most g e n e r a l Lagrangian t o be discussed here i s

(2.11 The

and

d m o t e s e t s o f complex and r e a l f i e l d s t h a t may be

32

G. 't HOOFT and M. VELTMAN

scalar, spinor, vector, tensor, etc., f i e l d s . The index i stands f o r any spinor, Lorentz, i s o s p i n , etc., index. V and W are m a t r i x operators t h a t may contain d e r i v a t i v e s , and whose F o u r i e r transform must have an inverse. Furthermore, these inverses must s a t i s f y the Killen-Lehmann representation, t o be discussed l a t e r . The I n t e r a c t i o n Lagrangian XI ($*,$,41 i s any polynomial i n c e r t a i n coupling constants g as w e l l as t h e f i e l d s . This i n t e r a c t i o n L a grangian i s allowed t o be non-local, 1.e. not o n l y depend on f i e l d s I n the p o i n t x, b u t a l s o on f i e l d s a t other space t i m e p o i n t s XI,XI', The c o e f f i c i e n t s i n t h e polynomial expansion may be f u n c t i o n s o f x. The e x p l i c i t form o f a general t e r m inl: [ x ) I is

... .

[ d4Xld4x2

* $,

8

..., $, m

[x,), 1

..(x, ..., 4,

x1,x2,

%,I*. (XmL

[XnL

n

...1 ... .

(2.21

The u may contain any number o f d i f f e r e n t i a l operators working on the various f i e l d s . Roughly speaking propagators are defined t o be minus t h e inverse o f t h e F o u r i e r transforms o f V and W, and v e r t i c e s as t h e F o u r i e r transforms o f t h e c o e f f i c i e n t s a i n 1:

I'

The a c t i o n S i s defined by

I n 1: we make the replacement

-[

*

$I(xl 4i(x1

=

a ii

1

ikx

,

ci(kle i k x

,

d4k bi(kle

1 d4k

z . . . (x,

xl,

x2,

...1

=

33

DIAGRAMMAR

=

I

d4k d4kl

d4k2 , . . e , e

i k x + i k ( x - x l ) + i k (x-x21+ 1 2

... 0

(k,k1,k2 il...

The a c t i o n t i m e s I takes t h e form

w

i

each term i n t e g r a t e d over t h e momenta i n v o l v e d . The and c o n t a i n a f a c t o r i k ( o r -1k 1 f o r every d e r i v a t i v e a/ax a c t i n g t o t h e r i g h t [ l e f t l n i n V andnW, r e s p e c t i v e l y . The a c o n t k l n a factor i k f o r every d e r i v a t i v e a/ax a c t i n g on a f i e l d w i t h JlJ argument

A:.

The p r o p a g a t o r s are defined t o be:

-

-

w

-

.

Here i s r e f l e c t e d , 1.e. = W I n t h e r a r e case o f r e a l fermions t h e propagator mustw&$ mind& t h e i n v e r s e o f t h e a n t i symmetric p a r t o f W. Furthermore, t h e r e i s t h e u s u a l i e p r e s c r i p t i o n f o r t h e p o l e s o f these propagators. The momentum k i n Eqs. ( 2 . 5 ) i s t h e momentum flow i n t h e d i r e c t i o n of t h e arrow. The d e f i n i t i o n o f t h e v e r t i c e s i s :

x a

.[k,

il..

kl,

k2 *

...164(k

+ k

34

1

+

...1.

(2.6

1

,...I.

G. ‘t HOOFT and M. VELTMAN

The summation i s over a l l permutations o f the i n d i c e s and momenta indicated. The momenta a r e taken t o f l o w inwards. Any f i e l d $* corresponds t o a l i n e w i t h an arrow p o i n t i n g outwards; a f i e l d $ gives an opposite arrow. The 4 f i e l d s g i v e arrow-less l i n e s . The f a c t o r ( - 1 l p i s o n l y o f importance i f several fermion f i e l d occur. A l l fermion f i e l d s are taken t o anticommute w i t h a l l other fermion f i e l d s . There i s a f a c t o r -1 f o r every permutation exchang i n g two fermion f i e l d s . The c o e f f i c i e n t s u w i l l o f t e n be constants. Then the sum over permutations r e s u l t s simply i n a f a c t o r . I t i s convenient t o i n c l u d e such f a c t o r s already i n XI; f o r instance e,cx1

-3:6:

g$* ( x I 3$ ( X I

gives as vertex simply t h e constant g. As indicated, t h e c o e f f i c i e n t s a may be f u n c t i o n s of X. corresponding t o some a r b i t r a r y dependence on the momentum k i n (2.6). This momentum i s n o t associated w i t h any o f t h e l i n e s of the vertex. I f we have such a k dependence, i.e. t h e c o e f f i c i e n t a i s non-zero f o r some non-zero value o f k, then t h i s v e r t e x w i l l be c a l l e d a source. Sources w i l l be i n d i c a t e d by a cross o r other convenient n o t a t i o n as t h e need a r i s e s .

A diagram i s obtained by connecting v e r t i c e s and sources by means o f propagators i n accordance w i t h t h e arrow notations. Any diagram i s provided w i t h a combinatorial f a c t o r t h a t c o r r e c t s f o r double counting i n case i d e n t i c a l p a r t i c l e s occur. The comput a t i o n of these f a c t o r s i s somewhat cumberaorne; the r e c i p e i s given i n Appendix A. Further, i f f e n i o n s occur, diagrems are provided w i t h a sign. The r u l e i s as f o l l o w s :

i l there i s a minus s i g n f o r every closed ferrnion l o o p j i i l diagrams t h a t are r e l a t e d t o each other by the omission o r a d d i t i o n o f boson l i n e s have t h e same sign;

35

DIAGRAMMAR

i i i l diagrams r e l a t e d by t h e exchange of two fermion l i n e s , i n t e r nal or e x t e r n a l , have a r e l a t i v e m i n u s s i g n . E l e c t r o n - e l e c t r o n s c a t t e r i n g i n quantum-electrodynamics. Some diagrams and t h e i r r e l a t i v e s i g n a r e given i n t h e f i g u r e . The f o u r t h and f i f t h diagrams a r e r e a l l y t h e same diegram and should n o t be counted s e p a r a t e l y :

EXAMPLE 2.1.1

+l

+1

+1

m x -1

-1

T h i s r a i s e s t h e q u e s t i o n o f t h e r e c o g n i t i o n of t o p o l o g i c a l l y i d e n t i c a l diagrams. I n Appendix A some c o n s i d e r a t i o n s p e r t i n e n t t o t h e topology of quantum electrodynamics are p r e s e n t e d . 2.2 A Simple Theorem

THEOREM

3

Some Examples

Diagrams a r e i n v a r i a n t f o r t h e replacement

i n t h e Lagrangian (1.11, w h e r e X i s any m a t r i x t h a t may i n c l u d e d e r i v a t i v e s b u t m u s t have an i n v e r s e . T h e theorem i s l t r i v i a l t o prove. Any o r i e n t e d propagator obtains a f a c t o r X that cancels against t h e f a c t o r X occurring i n t h e v e r t i c e s . I t is l e f t t o t h e r e a d e r t o g e n e r a l i z e t h i s theorem t o i n c l u d e t r a n s f o r m a t i o n s of t h e $ J ' S and 4's.

Some examples i l l u s t r a t i n g our d e f i n i t i o n s a r e i n o r d e r . EXAMPLE 2 .2 .1 =

4

Charged s c a l a r p a r t i c l e s w i t h J,

$?a2 -

m21$

*

+

2:2:

[ a $1'

36

interaction

*

+ J

*

+

[2.71

$ J(ic).

G. 't HOOFT and M. VELTMAN

* Only 9 1x1 a n d ' 9 [ x l

i n the same p o i n t x occur, and i n accordance w i t h the d e s c r i p t i o n given a t the beginhing o f t h e previous s e c t i o n we have here a l o c a l Lagrangian. The f u n c t i o n s J [ x I end J* (XI ere source functions. Propagator:

k

[Zn14i k2 1

AF

1

1

+

m2

-

ic'

Vertex:

x

4 (2nl ig

( 8 f u n c t i o n f o r energy-momentum conservation is understood).

Sources :

*

Tkp f u n c t i o n s J [ k l , J [ k l are the F o u r i e r transforms o f J r x l and J [XI J[xl =

d4ke-ikxJ[kl.

Note t h a t k i s t h e momentum f l o w i n g from the l i n e I n t o t h e source. Some diagrams:

*

J (-p1 1 pi;

+

m2

-

ie

*

J [-pt) p:

+ m2

6 4 (PI

+

ic P2

-

J

[P,]

p:

+

m2

Pg

-

P4) x

37

J (P41

-

ic

p:

+

m2

-

ic

DIAGRAMMAR

* J(-pl)

J [ -p21

p2 + m2 - i E pz 1

x

1

7g

2

1 d4k

+

m

2

* [P3)

J

- i E p2 3

+

m2 - 1 s p i

1 k2 +

(P4)

J +

m2 - is

1

m2 - I E

[k

+

p1

+

p212

+

m2

-

iE'

The f a c t o r 1/2 I n t h e second case I s t h e c o m b i n a t o r l a l f a c t o r o c c u r r i n g because o f t h e i d e n t i c a l p a r t i c l e s I n t h e i n t e r mediate s t a t e . Free r e a l v e c t o r mesons

EXAMPLE 2.2.2

1:

E

- -41

[auwv

- a w l2 v u

- -21 JW2u*

I n terms o f f o u r r e a l f i e l d s

6 :Ma, a

(2.8) a = 1,

....

4.

The m a t r i x V i n momentum space I s

-6

(k2

+

m2 1

+

k,kB.

aB

-

The v e c t o r meson propagator becomes:

i

k

--

j

*FaB=

' 41

[ 2r

-1 (V

1

l a =~

[ZnI4i

6aB+k k /m k2

(2.91

+

m2

-

2 IE

G. 't HOOFT and M. VELTMAN

Indeed

68A (k2

r6a8

+

m21

-

-

k2 + m2

k2

-

m2

+

kakX

[k2

1~

m21

-

kakA

k2 2 3

=

6aX.

+

ic

k2 + m2 2

-

kakX

m

E l e c t r o n i n an e x t e r n a l electromagnetic f i e l d A

EXAMPLE 2.2.3

.t =

k8kX/m2

kaksl

1

+

+

-Il(vlJalJ

+

mlJ,

+

.

ieA,$y'$

(2.101

6

Note t h a t = $*y4. Because o f our l i t t l e theorem t h e m a t r i x y can be omitted i n g i v i n g r u l e s f o r t h e diagrams. Then:

k

As a f u r t h e r a p p l i c a t i o n o f our theorem we may s u b s t i t u e J,

-t

(-yva

v

+

ml$ t o g i v e

This gives the equivalent r u l e s :

39

U

4

DIAGRAMMAR

2.3

I n t e r n a l consistency

Two p o i n t s need t o be i n v e s t i g a t e d . F i r s t , t h e s e p a r a t i o n i n r e a l and complex f i e l d s is r e a l l y a r b i t r a r y , because f o r any complex f i e l d (a one can always write

4

= -

fi

(A

+

191,

where A and 6 a r e r e a l f i e l d s . The q u e s t i o n is whether t h e r e s u l t 8 w i l l be independent o f t h e r e p r e s e n t a t i o n chosen. T h i s indeed is t h e case, and may be b e s t e x p l a i n e d by c o n s i d e r i n g an example:

i

4 * (2nl i J i

4

( 2 ~ 1i J i

The diagram c o n t a i n i n g two sources i s :

On t h e o t h e r hand, l e t us w r i t e 41

1: +

-

+&v

2 1 ij6j

+iVijAj

1 * (JiAi

v5

+

*

iJi6i

+ AiJi

-

(l/fi)(A

+

16)

+ h V 6 - b V A + 2 1 ij 1 2 1 ij j

-

iE,J,I

G. 't HOOF1and M. VELTMAN

D e f i n i n g t h e r e a l f i e l d X,

we have

with

where t h e s u p e r s c r i p t s s and a denote t h e symmetrical and t h e antisymmetrical part, respectively. L e t Y now be t h e i n v e r s e o f V. Thus

VY

-

1

-

,

YV = 1.

Writing V Vs + V,' t e d VY w i t h YV

VSYS

+

Y = Y s + Ya one f i n d s ,

vBYa = 1

v9ya + \jays

I

comparing t h e r e f l e c -

,

yap

+ ysp

I

0

.

The i n v e r s e of t h e m a t r i x W is t h e r e f o r e

w-1 where Y

=

-

[ ;:] -ysa i Y

V-I.

,

*

The source-source diagram becomes

as before. C l e a r l y t h e s e p a r a t i o n i n t o r e a l and imaginary parts amounts t o s e p a r a t i o n i n t o symmetric and a n t i s y m m e t r i c p a r t s of t h e propagator.

41

DIAGRAMMAR

The second p o i n t t o be i n v e s t i g a t e d i s t h e q u e s t i o n of s e p a r a t i o n o f t h e q u a d r a t i c , 1.e. propagator d e f i n i n g , p a r t i n Thus l e t t h e r e be g i v e n t h e Lagrangian

1:

=

*

e.

*

4ivij4i

+

4iv;j4j -1

f o r the One can e i t h e r say t h a t one has a propagatorl-(V + V'] 4 - f i e l d , o r a l t e r n a t i v e l y a propagator - ( V ) and 8 v e r t e x : '

However, these t w o cases g i v e t h e same r e s u l t . Sumning over a l l p o s s i b l e i n s e r t i o n s o f t h e v e r t e x V ' one f i n d s :

1 - V

- 3v'

+ [- 1 V

1 I+ I- V

...

=

-

-v1 1

1 +

v'/v

= -

1

v

+

v"

T h i s demonstrates t h e i n t e r n a l consistency o f our scheme of d e f i n i t i o n s . We leave i t as an e x e r c i s e t o t h e reader t o v e r i f y t h a t combinational f a c t o r s check i f one makes t h e replacement 4 + [ A + i B ) / & , f o r i n s t a n c e f o r t h e diagrams:

2.4

D e f i n i t i o n o f t h e Green's F u n c t i o n s

L e t t h e r e be g i v e n a g e n e r a l Lagrangian o f t h e form (2.11. Corresponding t o any f i e l d we i n t r o d u c e source terms [we w i l l need many, b u t w r i t e o n l y one f o r each f i e l d )

* e+l+1 J* 1$ + $IJ,

+

4iKi

According t o t h e p r e v i o u s r u l e s such terms g i v e r i s e t o t h e following vertices:

42

G. 't HOOFT and M. VELTMAN

-

[2n14iK[kl

,

where t h e k dependence i m p l i e s F o u r i e r t r a n s f o r m a t i o n . Remember now t h a t t h e Lagrangian was a p o l y n o m i a l i n some c o u p l i n g constants. F o r a g i v e n o r d e r i n these c o u p l i n g constants we may consider t h e sum o f a l l diagrams connecting n sources. A l l n sources a r e t o be taken d i f f e r e n t , because we want t o be a b l e

t o v e r y t h e momenta independently. Each o f these diagrams, and a f o r t i o r i t h e i r sum, w i l l be of t h e f o r m

Ji [ k 13' (k21 1

I2

Ki n

[ k )Gi " 1

...

in(kl,

..., knI.

The f u n c t i o n Gi w i l l be c a l l e d t h e n - p o i n t Green's f u n c t i o n f o r t h e g i v e n e x t e r n a l l i n e c o n f i g u r a t i o n f o r t h e o r d e r i n t h e coupl i n g constants s p e c i f i e d , F a c t o r s (Zn14i from t h e source v e r t i c e s a r e i n c l u d e d . The k denote t h e momenta f l o w i n g f r o m t h e sources i n t o t h e diagrams. +he propagators t h a t connect t o t h e sources a r e included i n t h i s definition. The f i r s t example o f S e c t i o n 2.2 shows some diagrams c o n t r i b u t i n g t o t h e second-order f o u r - p o i n t Green's f u n c t i o n s . 2.5 D e f i n i t i o n o f t h e S-Matrix

s Some Examples

Roughly speaking t h e S-matrix o b t a i n s f r o m t h e Green's funct i o n s i n two steps: [ i l t h e momenta o f t h e e x t e r n a l l i n e a r e p u t on mess s h e l l , and [ii) t h e sources are normalized such t h a t they correspond t o emission o r a b s o r p t i o n o f one p a r t i c l e . Both these statements a r e somewhat vague. and we must p r e c i s e them, b u t they

43

DIAGRAMMAR

r e f l e c t t h e e s s e n t i a l p h y s i c a l c o n t e n t o f t h e reasoning below. Consider t h e diagrams connecting two sources:

The corresponding expression I s

The two-point Green's f u n c t i o n w i l l i n g e n e r a l have a p o l e (Or p o s s i b l y many s i n g l e p o l e s ) a t some v a l u e -PI2 o f t h e squared four-momentum k I f t h e r e i s no p o l e t h e r e w i l l be no c o r r e s ponding S-matrix element1 such w i l l be t h e case I f a p a r t i c l e becomes u n s t a b l e because of t h e I n t e r a c t i o n s . A t t h e p o l e t h e Green's f u n c t i o n w i l l be o f t h e form

.

G

[k,

4

k ' l = ( 2 ~ 1i S 4 [ k

+

ij

K k'l h k

(kl a

t

k2 +

-M

+ M

2

.

The m a t r i x r e s i d u e K can be a f u n c t i o n o f t h e components k , w i t h lJ t h e r e s t r i c t i o n thatijk2 = -M2. F i r s t we w i l l t r e a t t h e c u r r e n t s for emission of a p a r t i c l e , corresponding t o n ominy p a r t i c l e s o f t h e S-matrix. D e f i n e a new s e t o f c u r r e n t s ,+a', one f o r every non-zero eigenvalue o f K, 1 which a r e m u t u a l l y o r t h o g o n a l and e i g e n s t a t e s o f t h e m a t r i x K ( k )

(a)*J(bl Ji 1

=

0

If

a # b

*

( a l ( k ) I Kji(klJ:al(kl

IJd

=

,

1

f o r Integer spin

m

f o r half-integer spin

kO

T h i s i s p o s s i b l e o n l y ifa l l elgenvalues of K a r e p o s i t i v e .

44

. In

G. 't HOOF1and M. VELTMAN

t h e case o f negative eigenvalues n o n n a l i z a t i o n i s done w i t h minus t h e r i g h t - h a n d s i d e o f Eq. (2.141. The sources thus d e f i n e d a r e t h e p r o p e r l y normalized sources f o r emission o f a p a r t i c l e or a n t i p a r t i c l e [ t h e l a t t e r f o l l o w s from c o n s i d e r i n g )?[-k11. The p r o p e r l y normalized sources f o r absorption o f a p a r t i c l e o r a n t i p a r t i c l e f o l l o w by considering Eqs. (2.131 and t2.141 b u t w i t h k replaced by - k i n K. The above procedure defines t h e c u r r e n t s up t o a phase f e c t o r . We must t a k e care t h a t t h e phase f a c t o r f o r t h e emission of a c e r t a i n p a r t i c l e agrees w i t h t h a t f o r a b s o r p t i o n o f t h a t same p a r t i c l e . T h i s i s f i x e d b y r e q u i r i n g t h a t t h e two-point Green's f u n c t i o n p r o v i d d w i t h such sources has p r e c i s e l y t h e r e s i d u e 1 ( o r k o h l f o r k9 = -m2. The m a t r i x elements o f t h e m a t r i x S' (almost, b u t n o t e x a c t l y t h e S-matrix, see below1 f o r n i n g o i n g p a r t i c l e s and m outgoing p a r t i c l e s are d e f i n e d by

n

m

...,

...,

The energies k , k and p , 'mo are a l l p o s i t i v e . The minus signl?or t h e m88enta p% t h e Green's f u n c t i o n appears because i n t h e m a t r i x element t h e momenta o f t h e outgoing p a r t i c l e s a r e taken t o be f l o w i n g out, w h i l e i n our Green's f u n c t i o n s t h e convention was t h a t a l l momenta f l o w inwards.

EXAMPLE 2.5.1

Scalar p a r t i c l e s . Near t h e p h y s i c a l mass p o l e t h e two-point Green's f u n c t i o n w i l l be such t h a t G(k, k ' 1 4 [2nl icS4(k k'1

-

1 z2(k2

+

(2.161 N21'

.

For any k t h e p r o p e r l y normalized e x t e r n a l c u r r e n t i s J

-

Z, and t h e p r e a c r i p t i o n t o f i n d S ' - m a t r i x elements w i t h e x t e r n a l s c a l a r particles i s

45

EXAMPLE 2.5.2 Fermlons as i n OED. Near t h e p h y s i c a l mass p o l e t h e t w o - p o i n t Green’s f u n c t i o n w i l l be such t h a t

GI

(k, k’ 1

4

(271)1 s 4 ( k

-

k’l

1 =

2 2 [iyk

+

Ml

K

[k)

k2

+

6 (2.181 2’ M

with

Asset o f e i g e n s t a t e s of t h i s m a t r i x -3 p r o v i d e d by t h e s o l u t i o n s u ( k l of the Dirac equation

*

[lyk+M1u=O,

u

a

u

a

- 1 ,

a = l , 2 .

Because o f t h e n o r ~ ~ ) i z a t i ocno n d i t i o n Eq. (2.141, f o r the currents J (kl

(2.201 we must t a k e

(2.21 1 w i t h t h e D i r a c s p l n o r s normalized t o 1, see Eq. (2.201. Note t h a t t h e f a c t o r 1/2M cancels upon m u l t i p l i c a t l o n o f t h i s source w i t h t h e p r o p a g a t o r numerator [2.19). The a n t i p a r t i c l e emission c u r r e n t s a r e o b t a i n e d by c o n s i d e r i n g K [ - k l . The s o l u t i o n s a r e t h e a n t i p a r t i c l e s p l n o r s

ia(kl

,

a = 3, 4.

...

For o u t g o i n g a n t i p e r t i c l e s and p a r t i c l e s K I - k l end K [ k ) must be considered t o o b t a i g t h e proper a b s o r p t i o n c u r r e n t s . The s o l u t i o n s a r e t h e spinors u ( k l , a = 3, 4 and c a t k l , a = 1, 2.

46

G. ‘t HOOFT and M. VELTMAN

= 2M d i c t a t e s a m i n u s s i g n f o r t h e s o u r c e f o r emission of an a n t i p a r t i c l e (see Appendix A ) .

The phase c o n d i t i o n GaKua

Let u s now complete t h e S-matrix d e f i n i t i o n . T h e p r e s c r i p t i o n given above results i n z e r o when a p p l i e d t o t h e two-point f u n c t i o n , because there w i l l be two f a c t o r s K2 + M2 [one f o r every s o u r c e ) . T h u s we e v i d e n t l y do not o b t a i n e x a c t l y t h e S-matrix t h a t one has i n s u c h cases. T h i s d i s c r e p e n c y i s related t o t h e t r e a t m e n t of t h e o v e r - a l l d f u n c t i o n of energy-momentum conservat i o n , when p a s s i n g from S-matrix elements t o t r a n s i t i o n p r o b a b i l i t i e s . Anyway, t o g e t t h e S-matrix we m u s t a l s o e l l o w l i n e s where p a r t i c l e s go through without any i n t e r e c t i o n , and a s s o c i a t e a f a c t o r 1 w i t h s u c h lines. T h e s e p s r t i c l e s must. of c o u r s e , have a mass as given by t h e p o l e of t h e i r propagator.

Matrix elements of t h e S-matrix, i n c l u d i n g p o s s i b l e l i n e s going s t r a i g h t through, w i l l be denoted by graphs w i t h e x t e r n a l l i n e s t h a t have no t e r m i n a t i n g c r o s s . The convention i s : l e f t ere incoming p a r t i c l e s , r i g h t outgoing. Energy f l o w s from l e f t t o r i g h t . The d i r e c t i o n of t h e arrows i s of c o u r s e not r e l a t e d t o t h e d i r e c t i o n of t h e energy flow. We e m p h 8 S l Z e a g a i n t h a t t h e above S-matrix elements i n c l u d e diagrams c o n t a i n i n g i n t e r a c t i o n f r e e lines. For instance, t h e diagram shown i s included i n t h e 3-particle-in/3-particle-out Smatrix element. T h e d e f i n i t i o n of t h e S-matrix given above a p p l i e s if t h e r e i s no gauge symmetry. F o r gauge t h e o r i e s some of t h e propagators correspond t o ”ghost” p a r t i c l e s t h a t a r e essumed not t o have phys i c a l r e l e v a n c e . I n d e f i n i n g t h e S-matrix, t h e s o u r c e s m u s t be r e s t r i c t e d t o emit or absorb only p h y s i c a l p a r t i c l e s . Such sources w i l l be c a l l e d p h y s i c a l s o u r c e s and have t o be d e f i n e d i n t h e p r e c i s e c o n t e x t of t h e gauge symmetry. To show i n s u c h c a s e s t h a t t h e S-matrix i s u n i t a r y requires t h e n s p e c i a l e f f o r t . 2.6

D e f i n i t i o n of St T h e m a t r i x elements of t h e m a t r i x St a r e d e f i n e d as u s u a l by

< a l s t [ 0>

*

= 43ISla>

,

(2.22)

where t h e complex conjugation i m p l i e s a l s o t h e replacement i e i n the propagators.

47

+

-1e

D I AG RAMMAR

The m a t r i x elements of St can also be o b t a i n e d i n another way. In a d d i t i o n t o t h e Lagrangian 1: d e f i n i n g S, consider t h e conjugated Lagrangian f t . I t i s o b t a i n e d from 1: by complex conjugation and r e v e r s a l of t h e o r d e r o f t h e f i e l d s . The l a t t e r i s o n l y r e l e v a n t f o r fermions. This tt may be used t o d e f i n e another S-matrix; l e t 3 denote t h e matrix obtained i n t h e usual way from et, however with t h e o p p o s i t e s i g n f o r t h e i s i n t h e propagators and a l s o t h e replacement i + -i i n t h e n o t o r i o u s f a c t o r s ( h ) 4 i . We claim t h a t t h e m a t r i x elements o f 6 are equal t o t h o s e o f S+, In formula we g e t

< alsu,

iI+lB

>=
-il

le >.

T h e p r o o f rests m ai nl y on t h e o b s e r v a t i o n t h a t an incoming p a r t i c l e s o u r c e i s o b t a i n e d by c o n s i d e r i n g [ n o t a t i o n of S e c t i o n 2.5, Eq.

(2.1311:

and t h e complex c o n j u g a t e of an o u t g o i n g p a r t i c l e s o u r c e by s t u d y of:

or, equivalently,

where K gation

&’!

T h i s is i n d e e d w h a t c o r r e s onds t o complex c o n j u t=h K e pj ri o*p a g a t o r d e f i n i n g p a r t of 1:

?

Note t h e change of s i g n of t h e d e r i v a t i v e : w h a t worked t o t h e r i g h t works now t o t h e l e f t , w h i c h i m p l i e s a m i n u s s i g n . Vt i s o b t a i n e d by t r a n s p o s i t i o n and complex c o n j u g a t i o n of V. C l e a r l y complex c o n j u g a t i o n and exchange of incoming and o u t g o i n g s t a t e s c o r r e s p o n d s t o t h e use of Vt i n s t e a d of V. EXAMPLE 2 . 6 . 1

I:

=

-icy3

Fermion c o u p l e d t o complex s c a l a r f i e l d +

m)$

+

$*[a2

- mz)g

48

+

gi(1

+

y

5

*

,

G. 't HOOFT and M. VELTMAN

The lowest o r d e r S-matrix element I s :

According t o Eq.

(2.221

5 There are minus s i g n s because of y4, y exchange and 3 -P -3 changes except a4 + a4. I n c l u d i n g t h e s i g n change f o r [2,1 41, we o b t a i n f o r S:

which equals t h e r e s u l t f o r Si

found above.

t

To summarize, t h e m a t r i x elements o f S can be o b t a i n e d e i t h e r d i r e c t l y f r o m t h e i r d e f i n i t i o n (2.221, or by t h e use of d i f f e r e n t Feynmsn r u l e s . These new r u l e s can be o b t a i n e d f r o m t h e o l d ones by r e v e r s i n g a l l arrows i n v e r t i c e s and propagators and r e p l a c i n g a l l v e r t e x f u n c t i o n s and propagators by t h e i r complex conjugate ( f o r t h e propagators t h i s means u s i n g t h e H e r m i t i e n conjugate A l s o , t h e f a c t o r s ( Z n I 4 i and t h e l e terms i n t h e propagators). propagators e r e t o be complex conjugated. The i n - and o u t - s t a t e source f u n c t i o n s a r e d e f i n e d by t h e u s u a l procedure, i n v o l v i n g now t h e H e r m i t i e n conjugate propagators, L e t us f l n e l l y , f o r t h e seke of c l a r i t y , f o r m u l a t e t h e d e f l n i t i o n o f i n g o i n g and o u t g o i n g s t a t e s i n t h e diagram language f o r b o t h t h e o l d and t h e new r u l e s :

49

DIAGRAMMAR

out

in

antiparticle

antiparticle

)I--c-

Y_

particle

particle

I a> 2 . 7 Definition of Transition Probabilities

j

Cross-Sections and Lifetimes The S-matrix elements are the transition amplitudes of the theory. The probability amplitudes are defined by the absolute value squared of these amplitudes. Conservation of probability requires that the S-matrix be unitary

This property will be true only if the diagrams satisfy certain conditions, and we will investigate this in Section 6. In the usual way, lifetimes and cross-sections can be deduced from the transition probabilities. Consider the decay o f particle a into particles 1, 2, 3, n. The decay width r [ = inverse lifetime T I is d3P 1

1

- = r = / T

2Pl0t2lr1

t

x
d3Pn 3'

*",

64[P

zpn0t 2 n ~ 3

-

p1

t

P2 * - * -Pnl

2p o m r l

..., n>.a, ..., nlNla >, 2,

-

...'

4

(2.24)

t

where the matrix elements of M [and M 1 are those of S [and S 1 without the energy-momentum conservation 6 function

G. ‘t HOOF1and M. VELTMAN

...,

...,

The p p are t h e energies o f p a r t i c l e s a, I, 2. n. sec g i v e s r i n sec-1, p r o v i o l v l d i % ~r by nns 6.587 x 1022 MeV ded a l l was computed i n n a t u r a l u n i t s ( h = c 1) w i t h t h e MeV as the only u n i t l e f t .

-

Next, consider t h e s c a t t e r i n g o f a p a r t i c l e a on a p a r t i c l e

B a t r e s t g i v i n g r i s e t o a f i n a l s t a t e w i t h n p a r t i c l e s . The cross-section

u f o r t h i s process i s

+

Here pa i s t h e three-momentum of t h e incoming p a r t i c l e a i n t h e r e s t system of t h e B f f r t i c l e [ t a r g e t p a r t i c l e l . M u l t i p l y i n g by MeV cmI2, one o b t a i n s u i n cm2. ( h c I 2 = (1.9732 x 10-

51

DIAGRAMMAR 3.

DIAGRAMS AND FUNCTIONAL INTEGRALS

I n a l l p r o o f s we w i l l r e l y o n l y on combinatorics o f diagrams. However, t h e r u l e s and d e f i n i t i o n s may l o o k somewhat ad hoc, and t h e purpose o f t h i s s e c t i o n i s t o show a f o r m a l equivalence w i t h c e r t a i n i n t e g r a l expressions. Imagine a w o r l d w i t h o n l y a f i n i t e number of space-time N j p = 1, 4. F o r s i m p l i c i t y , we p o i n t s x , a = 1, consider’only t h e case o f r e a l boson f i e l d s . The a c t i o n S i s now

...,

....

where W i s taken t o be symmetric. D e r i v a t i v e s occur as d i f f e r e n c e s . The diagram r u l e s corresponding t o t h i s a c t i o n a r e as g i v e n I n S e c t i o n 2. Suppose now t h a t S i s r e a l i f I$ i s r e a l . The f o l l o w i n g theorem holds. The rules d e f i n e d i n S e c t i o n 2 f o r connected diagrams a r e p r e c i s e l y t h e same as t h e mathematical r u l e s t o o b t a i n t h e f u n c t i o n r d e f i n e d by

where t h e r i g h t - h a n d s i d e i s understood as a s e r i e s expansion i n terms o f t h e c o e f f i c i e n t s i n JI. Here C i s an a r b i t r a r y constant n o t depending on t h e sources i n 1: The s e t of a l l diagrams i n c l u I’ d i n g disconnected ones a r e o b t a i n e d by expanding elr. I n s t e a d o f Eq.

e

ir

=

c

/P4e

[3.2) we w i l l use t h e condensed n o t a t i o n

isr41

C3.31

The theorem i s easy t o prove. F i r s t c a l c u l a t e t h e i n t e g r a l f o r f r e e p a r t i c l e s , u s i n g f o r t h e a c t i o n t h e expression

(3.41 w i t h a r b i t r a r y source f u n c t i o n s

J. D e f i n e

52

G. ‘t HOOFT and M. VELTMAN

r0

c a n be f o u n d by meking a s h i f t i n t h e i n t e g r a t i o n v e r i a b l e s

so t h a t

T h e primed f i e l d s are n o t c o u p l e d t o t h e J: We f i n d

b e c a u s e t h e i n t e g r a l o v e r t h e (0’ i s now a s o u r c e i n d e p e n d e n t c o n s t a n t and c a n be a b s o r b e d I n t h e c o n s t a n t C I n E q . [3.31. N o t e t h e f a c t o r 112 b e c a u s e of i d e n t i c a l s o u r c e s . 0

We see t h a t I’ is n o t h i n g b u t t h e f r e e p a r t i c l e p r o p a g a t o r . Even t h e itz p r e s c r i p t i o n c0n be I n t r o d u c e d c o r r e c t l y ifw e i n t r o duce a smooth c u t - o f f f o r t h e i n t e g r a l f o r l a r g e v a l u e s of t h e f i e l d s (0

T h i s makes t h e i n t e g r a l well d e f i n e d e v e n i n d i r e c t i o n s w h e r e (0W+=O. result is t h e u s u a l I& a d d i t i o n t c W.

The

Let us now c o n s i d e r a l s o i n t e r a c t i o n terms. T h e p e r t u r b a t i o n expansion I s icL, r xal a i2 a b e = 1 + i LI(X ,$I + 2: LI(X ,(01LI(X ,OI... a a,b (3.101 or

1

r

1

.

w h e r e t h e s o u r c e s are i n c l u d e d i n S

we have

53

0

. From

t h e d e f i n i t i o n o f So

D IAGRAMMAR

a Consequently we can, replace everywhere the fields 4 1[x I by the derivative 6/6Ji[x 1

The remaining integral I s precisely the one computed before and is equal to exp Iiro1 with r0 given in Eq. (3.0). Expanding this exponential ir e o

=

1 -

-21 i 1 JW-IJ

+

8

i2[JW

-1 2 J)

... ,

(3.14)

we see that every term in Eq. 13.131 corresponds to a diagram with vertices and propagators as defined in Section 2. Diagrams not coupled to sources are called vacuum renormalizations and must be absorbed in the constant C. Note the combinatorial factors due to the occurrence of identical sources. The functional integral notation 13.31 for t h e amplitude in the presence of sources is elegant and compact, and it is very tempting to write the amplitudes of relativistic field theories in this way. Many theorists can indeed not resist this temptation. Let us see what is involved. In our definition r3.3) we restricted ourselves to a finite number o f space-time points. In any realistic theory this number is infinite end summations are to be replaced by integrations. So a suitable limiting procedure must be defined, but unfortunately these definitions cannot always be given in a manner free of ambiguities. Generally speaking, difficulties set in about at the same point that difficulties appear in the usual operator formalism, in particular we mention higher order derivatives o r worse non-localities in the Lagrangian. We shall, therefore, in this report not try to formulate such a definition but attach to the result of manipulations with functional integrals a heuristic value. Everything is to be verified explicitly by combinatorics of diagrams. But having verified certain basic algebraic properties we can happily manipulate these "path" integrals. One of the most interesting manipulations used with great advantage in connection with gauge theories is to change of field variables. Both local and non-local canonical transformations turn out to be correctly described by the path integral formulae, as we shall see later.

54

G. 't HOOFT and M. VELTMAN

We may f i n a l l y mention t h a t t h e v a r i o u s s i g n p r e s c r i p t i o n s , i n t h e case o f fermion f i e l d s . cannot be obtained i n a simple way, They can be c o r r e c t e d f o r by hand afterwards, o r i n a more sophist i c a t e d way an a l g e b r a o f anticommuting v a r i a b l e s can be i n t r o duced. Such work can be found i n t h e l i t e r a t u r e and i s q u i t e straightforward.

55

DIAGRAMMAR

4. KALLEN-LEHMANN REPRESENTATION

-1 and -W (see Eqs. [2,511 t h a t a r e d e f i ned a s propagators I n t h e t h e o r y a r e more p r e c i s e l y c a l l e d t h e b a r e propagators. T h i s i s c o n t r a s t t o t h e two-point Green’s f u n c t i o n t h a t when d i v i d e d by - ( 2 ~ ) * 6 4 ( k+ k’l i s c a l l e d t h e dressed propagator. Both t h e b a r e and d r e s s e d propagators a r e r e q u i r e d t o s a t i s f y t h e KallBn-Lehmenn r e p r e s e n t a t i o n , but i t w i l l t u r n out t h a t t h e dressed propagators s a t i s f y t h i s automat i c a l l y i f t h e b a r e propagators do. T h e q u a n t i t i e s -V

-1

-

Consider any propagator and decompose i t I n t o i n v a r i a n t f u n c t i o n s . For i n s t a n c e , f o r v e c t o r mesons

T h e KallBn-Lehmann r e p r e s e n t a t i o n f o r t h e I n v a r i a n t f u n c t i o n s i s

T h e f u n c t i o n s p ( s ’ 1 m u s t be r e a l . F o r b a r e propagators we w i l l i n s i s t t h a t t h e p ( s ’ 1 a r e a sum of 6 f u n c t i o n s

p(s’1

=

1i ai6(s’

-

mi)

f o r b a r e propagators (4.31

w i t h r e a l c o e f f i c i e n t s ai, and r e a l p o s i t i v e m

2 1‘

Now I n t r o d u c e t h e F o u r i e r transform of t h e propagators AFij(xl =

I d4kei k x A F i j ( k l

.

(4.41

Corresponding t o t h e decomposition of A ( k l i n i n v a r i a n t f u n c t i o n s F we w i l l have a decomposition i n v o l v i n g d e r i v a t i v e s . For i n s t a n c e , f o r v e c t o r mesons

where A and A a r e t h e F o u r i e r t r a n s f o r m s of f and f above. T h e staJement g h a t any f u n c t i o n f ( x l s e t i s f l e s * h e Kalf6n-Lehrnann

r e p r e s e n t a t i o n I s e q u i v a l e n t t o t h e statement

56

G. ‘t HOOFT and M. VELTMAN

f ( x i = e[xolf+(xi + e[-xolf-[xi

+

,

(4.61

-

where f [ f 1 i s a p o s i t i v e ( n e g a t i v e 1 energy f u n c t i o n

1 d4keikxekk

0 16(k2 + 9’1

.

(4.71 The p r o o f is v e r y simple. Using t h e F o u r i e r r e p r e s e n t a t i o n

(4.81

one d e r i v e s immediately t h e r e s u l t . Of course t h i s d e r i v a t i o n is o n l y c o r r e c t ift h e i n t e g r a l i n Eq. (4.21 e x i s t s , and t h i s i s i n g e n e r a l t h e case. I f n o t one must i n t r o d u c e r e g u l a t o r s . t o be discussed below. However, we need t h e r e p r e s e n t a t i o n (4.61 n o t o n l y f o r t h e i n v a r i a n t f u n c t i o n s , b u t also f o r t h e complete propagator such as i n Eq. (4.51. Whether t h i s i s t r u e depends on t,he convergence o f t h e d i s p e r s i o n i n t e g r a l s . If t h e f u n c t i o n s f and f go f o r x = 0 s u f f i c i e n t l y smoothly over i n t o one anot h e r , t h e n t h e exppession (4.61 has no p a r t i c u l a r s i n g u l a r i t y a t x = 0 and one may i g n o r e t h e a c t i o n of d e r i v a t i v e s on t h e e finctions.-Now f r o m t h e above e q u a l i o n (4.z) i t i s c l e a r t h a t f ( x o l = f I - x 1, so i n any case f [01 = f [01. T h i s is enough t o t r e a t t h e c8se of one d e r i v a t i v e such as o c c u r r i n g i n t h e case o f f e r m i o n propagators. Consider now

For x

0

= 0 we can do t h e k

-

C l e a r l y a f+ aofsuperconv8rgen t

0

integral

i n xo- 0 o n l y i f t h e d i s p e r s i o n i n t e g r a l is

F o r b a r e propagators, see Eq. (4.31, t h i s can o n l y be t r u e i f some o f t h e c o e f f i c i e n t s a a r e negative, some p o s o t i v e . T h i s i i m p l i e s t h e e x i s t e n c e of n e g a t i v e m e t r i c p a r t i c l e s , which i n t u r n

57

DIAGRAMMAR

may lead to unphysical results, such as negative lifetimes or cross-sections, or lack of unitarity depending on how one defines things. Let us now assume that the superconvergance equation (4.111 holds. Then one obtains Indeed

6(x0)aof+

-

6(xoiaof- = o ,

as well as

6"X01f+

- 6'(xolf-

=

0 ,

G. ‘t HOOF1and M. VELTMAN

5.

INTERIM CUT-OFF METHOD:

U N I T A R Y REGULATORS

Because of d i v e r g e n c i e s , t h e d e f i n i t i o n of diagrams given i n S e c t i o n 2 may be meaningless. Moreover, t h e p r o p a g a t o r s may n o t s a t i s f y t h e KallBn-Lehmann r e p r e s e n t a t i o n because o f l a c k of superconvergence (see S e c t i o n 41. To avoid such d i f f i c u l t i e s we i n t r o d u c e what we w i l l c a l l u n i t a r y r e g u l a t o r s . T h i s r e g u l a t o r method works f o r any t h e o r y i n t h e sense t h a t i t a l l o w s a proper d e f i n i t i o n of a l l diagrams and i s moreover very s u i t a b l e i n co n n e ct i o n w i t h t h e proof of u n i t a r y , c a u s a l i t y , etc. I t f a i l s i n the case of Lagrangians i n v a r i a n t under a gauge group, f o r w h i c h we w i l l l a t e r i n t r o d u c e a more s o p h i s t i c a t e d method. T h e p r e s c r i p t i o n i s e xc e e di ngly simple : c o n s t r u c t t h i n g s t h a t any p ropa ga tor i s r e p l a c e d by a s u m of propagator s. T h e e x t r a p r o p a g a t o r s correspond t o heavy p a r t i c l e s , and c o e f f i c i e n t s [ t h e ai i n Eq. (4.311 a r e chosen such t h a t t h e high momentum b e h av i o u r i s v e r y good. T h i s i s a c hi e ved a s f o l l o w s . I n t r o d u c e i n t h e o r i g i n a l Lagranglen l, Eq. (2.11, sets of r e g u l a t o r f i e l d s so

and 0;

i n t h e fol lowing way:

(5.11

x

The c o e f f i c i e n t s a and b and t h e mass m a t r i c e s Mx and m m u s t b e chosen so as t o x assure x t h e p r o p e r high momentum behaviour, The p r o p a g a t o r s f o r t h e r e g u l a t i n g f i e l d s $ RA ‘Fij

I

-a -1 rv

x

+

x -1 M lid

x

are [5.21

Because i n t h e i n t e r a c t i o n Lagranglen $ i s everywhere r e p l a c e d ( s i m i l a r l y $* and 91, only t h e f o l l o w i n g combination by + 2 of p r o p a g a t o r s w i l l oc c ur i n t h e diagrams,

- -1- I x, x1

1

(5.31

Choosing t h e a p p r o p r i a t e c o e f f i c i e n t s and mass m a t r i c e s a l l diagrams w i l l become f i n i t e . If e ve ry e i g e n v a l u e of t h e mass mat r i x i s made v e r y l a r g e , diagrams t h a t were f i n i t e b e f o r e t h e i n t r o d u c t i o n of r e g u l a t o r s w i l l converge t o t h e i r unregulated

59

DIAGRAMMAR

value.

EXAMPLE 5 . 1

Charged s c a l a r p a r t i c l e s w i t h

1 2

X I I J J + $ l

propagator:

interaction

(5.41

1 -

m 4 I k2

1

4

.

JI propagator:

$

4

-

1 [Za14i k2

1 +

2 ' m

1 +

m2

(5.61 +

M2

The combination o c c u r r i n g i n t h e diagrams is

1 -

MZ

(2n14i t k 2 + m21 (k2

(5.71 +

m2

+

M21

T h i s behaves as k-4 a t h i g h momenta. The diagrams shown i n Exemp l e 2.2.1 a r e now f i n i t e f o r l a r g e b u t f i n i t e M2.

G. 't HOOFT and M. VELTMAN

8.

6.1

CUTTING EQUATIONS

Preliminaries

I n order t o keep t h e work o f t h i s s e c t i o n transparent, we w i l l suppress indices, d e r i v a t i v e s , etc. I n p a r t i c u l a r , f o r v e r t i ces we r e t a i n only a f a c t o r ( 2 n i 4 i g i n momentum space, t h a t i s i g I n coordinate apace. There i s no d i f f i c u l t y whatsoever i n r e i n t r o d u c i n g t h e neceseary d e t a i l s .

I t i s assumed t h a t diagrams are s u f f i c i e n t l y regulated, so t h a t no divergencies occur. The s t a r t i n g p o i n t i s t h e decomposition of t h e propagator I n t o p o s i t i v e and negative energy p a r t s

with x

-

-

x,

x,.

Here we used t h e n o t a t i o n

I n view of t h e r e a l i t y o f t h e s p e c t r a l f u n c t i o n s p we have

A&

(6 31

Ail

Consequently

As ueual

1

e(x1 =

O(x1

+

2m

7

e 1Tx

1 =

dT

'0

i f xo < 0

-43

0r-x) = 1

i f xo

(6.51

.

The representation (6.51 can be v e r i f i e d by choosing t h e i n t e g r a t i o n contour w i t h a l a r g e s e m i - c i r c l e i n t h e upper (lower] complex

61

DIAGRAMMAR

p l a n e f o r xo p o s i t i v e ( n e g a t i v e ) . C o n s i d e r now a diagram w i t h n v e r t i c e s . Such a diagram rep r e s e n t s i n c o o r d i n a t e s p a c e an e x p r e s s i o n c o n t a i n i n g many prop a g a t o r s depending on arguments xl' x n We w i l l d e n o t e such an e x p r e s s i o n by

.... .

F(xl, x

2'

..., xn I .

For example, t h e t r i a n g l e diagram r e p r e s e n t s t h e f u n c t i o n :

F(xl. x

2'

x

3 3

= [igl

A31A23A,2

,

(6.6) Every diagram, when m u l t i p l i e d by t h e a p p r o p r i a t e s o u r c e f u n c t i o n s and i n t e g r a t e d o v e r a l l x c o n t r i b u t e s t o t h e S - m a t r i x . T h e cont r i b u t i o n t o t h e T-matrix, d e f i n e d by S = l + i T

(6.71

i s o b t a i n e d by m u l t i p l y i n g by a f a c t o r -1. U n i t a r i t y of t h e Sm a t r i x i m p l i e s an e q u a t i o n f o r t h e imaginary p a r t o f t h e so defined T matrix T

-

t

Tt = I T T .

(6.81

T h e T-matrix, or r a t h e r t h e diagrams. a r e a l s o c o n s t r a i n e d by t h e

r e q u i r e m e n t of c a u s a l i t y . As y e t nobody h a s found a d e f i n i t i o n of c a u s a l i t y t h a t c o r r e s p o n d s d i r e c t l y t o t h e i n t u i t i v e n o t i o n s i i n s t e a d f o r m u l a t i o n s have been proposed i n v o l v i n g t h e off-masss h e l l Green's f u n c t i o n s . We w i l l employ t h e c a u s a l i t y r e q u i r e m e n t i n t h e form proposed by Bogoliubov t h a t h a s a t l e a s t some i n t u i t i v e a p p e a l and i s most s u i t a b l e i n c o n n e c t i o n w i t h a d i a g r e m e t i c a n a l y s i s . Roughly s p e a k i n g B o g o l i u b o v ' s c o n d i t i o n can b e p u t a s f o l l o w s : i f a s p a c e - t i m e p o i n t x i s i n t h e f u t u r e w i t h res1 p a c t t o some o t h e r s p a c e - t i m e p o i n t x a t h e n t h e diagrams i n v o l and x can b e rewritten i n te?ms of f u n c t i o n s t h a t i n 2 v o l v e o s i t i v e energy f l o w f r o m x t o x o n l y . ving 2 1

xd

The trouble with t h i s d e f i n i t i o n i s t h a t space-time p o i n t s

62

G. 't HOOFT and M. VELTMAN

cannot be a c c u r a t e l y p i n p o i n t e d w i t h r e l a t i v i s t i c wave packets corresponding t o p a r t i c l e s on mass-shell. T h e r e f o r e t h i s d e f i n i t i o n cannot be f o r m u l a t e d as an S-matrix c o n s t r a i n t . I t can o n l y be used f o r t h e Green's f u n c t i o n s . Other d e f i n i t i o n s r e f e r t o t h e p r o p e r t i e s o f t h e f i e l d s . I n p a r t i c u l a r t h e r e i s t h e p r o p o s a l o f Lehmann, Symanzik and Zimmermann t h a t t h e f i e l d s commute o u t s i d e t h e l i g h t cone. D e f i n i n g f i e l d s i n terms o f diagrams, t h i s d e f i n i t i o n can be shown t o reduce t o Bogoliubov's d e f i n i t i o n . The f o r m u l a t i o n o f Bogoliubov c a u s a l i t y i n terms of c u t t i n g r u l e s f o r diagrams w i l l be g i v e n i n S e c t i o n 6.4.

6.2

The Largest T i m e Equation

...,

L e t us now consider a f u n c t i o n F [ x , x 1 corresn ponding t o some diagram. We define new & n k o n s F, where one o r more o f t h e v a r i a b l e s x 1' x a r e u n d e r l i n e d . Consider n

.*.,

F(xl,

xZ,

..*, xs i

. a * #

2,

...a

Xnl

.

T h i s f u n c t i o n 1 3 obtained from t h e o r i g i n a l f u n c t i o n f ollowing:

(6.91

F by t h e

1 1 A propagator Aki

i s unchanged i f n e i t h e r x n o r x are k 1 underlined. is r e p l a c e d by A i l ifx b u t n o t xi i s 111 A propagator Aki k underlined. i l l1 A propagator Aki i s r e p l a c e d by Aki i f x b u t n o t xk i s 1 underlined. i s replaced by A* i f x and x a r e i v 1 A propagator A ki k 1 ki underlined. V l F o r any u n d e r l i n e d x r e p l a c e one f a c t o r 1 by -1. Apart f r o m t h a t , t h e r u l e s f o r t h e v e r t i c e s remain unchanged.

-

[6.101 Equations (6.11 and (6.41 l e a d t r i v i a l l y t o an i m p o r t a n t is equation, t h e l a r g e s t t i m e equation. Suppose t h e t i m e x l a r g e r t h a n any o t h e r t i m e component. Then any functionif in which x i s n o t u n d e r l i n e d equals minus t h e same f u n c t i o n b u t 1 w i t h x now u n d e r l i n e d

1

(6.111

63

DIAGRAMMAR

The minus s i g n i s a consequence o f p o i n t ( v ) . I n view o f what f o l l o w s i t i s u s e f u l t o i n v e n t a diagramnatic r e p r e s e n t a t i o n o f t h e newly-defined f u n c t i o n s : Any f u n c t i o n F i s represented by a diagram where any v e r t e x corresponding t o an underl i n e d variable i s provided w i t h a c i r c l e . EXAMPLE 6.2.1

F(xl,

x

2’

L

If F(xl,

x

3

I

xzI

= (IgI

x31 i s g i v e n by Eq.

* A31A23A12

3 +

(6.6) t h e n (6.12)

A

I f t h e t i m e component of x3 i s l a r g e s t we have, f o r i n s t a n c e :

From such a diagram i t i s i m p o s s i b l e t o see i f a g i v e n l i n p donngcting a c i r c l e d t o an u n c i r c l e d v e r t e x corresponds t o a A or A f u n c t i o n . But due t o Eq. (6.21 t h e r e s u l t i s t h e same anyway. The i m p o r t a n t f a c t i s t h a t energy always f l o w s from t h e u n c i r c l e d t o t h e c i r c l e d v e r t e x , because of t h e 8 f u n c t i o n i n Eq. (6.21. O f course t h e r e i s no r e s t r i c t i o n on t h e s i g n o f energy f l o w f o r l i n e s connecting two c i r c l e d o r two u n c i r c l e d vertices. 6.3

Absorptive Part

To o b t a i n t h e c o n t r i b u t i o n of a diagram t o t h e S-matrix x 1 must be m u l t i p l i e d t h e corresponding f u n c t i o n F ( x I w i t h t h e a p p r o p r i a t e source f u r l c t i o n s f 8 r t h e i n g o i n g and o u t g o i n g l i n e s and i n t e g r a t e d over a l l x * F o r i n s t a n c e , t h e funct i o n F(xl, . . . I x 1 corresponding t o t h e diagram:

...,

6

G. 7 HOOF1 and M. VELTMAN

must be m u l t i p l i e d by i p x ip x -ik x -ik x 1 le 2 Se 13e 2 4

9

and subsequently i n t e g r a t e d over x

1' . * * x 6 ' The r e a t r i c t i o n t h a t Eq. (6.111 only holds i f x is larger than t h e o t h e r time components makes i t impossible t&obvrite down t h e analogue of Eq. (6.111 i n momentum space. We w i l l now write down an equation w h i c h f o l l o w s d i r e c t l y from t h e l a r g e s t time e q u a t i o n , but holds whatever t h e time o r d e r i n g of t h e v a r i o u s x Thus c o n s i d e r a f u n c t i o n F f x1' . * * a corresponding t o some diairam. We have

.

Xnl

T h e sumnation goes over a l l p o s s i b l e ways t h e t t h e v a r i a b l e s may be underlined. F o r i n e t a n c e , there i s one term without underlined v a r i a b l e s , n terms w i t h one underlined v a r i a b l e , etc. There i s a l s o one term, t h e l a s t , where a l l v a r i a b l e s a r e underlined. Under c e r t a i n c o n d i t i o n s , t o be d i s c u s s e d i n S e c t i o n 7 , t h a t term i s r e l a t e d t o t h e first term x 1 - F(xl, n - - ..., -

F(xl,

x2,

x2,

..., x n 1* .

(6.141

T h e proof of Eq. (6.131 i s t r i v i a l . Let one o f t h e X , s a y x have t h e l a r g e s t time component. T h e n on t h e l e f t - h a n d s i d eI'of Eq. (6.131 any diagram w i t h xi not underlined c a n c e l s a g a i n s t t h e term where i n a d d i t i o n xi i s underlined, by v i r t u e of t h e

l e r g e s t time equation. Now we can m u l t i p l y Eq. (6.131 by t h e a p p r o p r i a t e s o u r c e f a c t o r s and i n t e g r a t e over a l l X. We o b t a i n a set of f u n c t i o n s F depending on t h e varioua e x t e r n a l momenta and f u r t h e r i n t e r n a l momenta [loop momenta). Here, t h e bar on F denotes t h e F o u r i e r transform. T h e f u n c t i o n s I? a r e composed of A*, A and A* f u n c t i o n s

65

DIAGRAMMAR

(6.15)

m

(6.16)

We now o b s e r v e t h a t I n t h e r e s u l t i n g e q u a t i o n many terms w i l l be z e r o , due.,to c o n f l i c t i n g energy 0 f u n c t i o n s . l a k e , f o r example, a diagram”1 w i t h one u n d e r l i n e d p o i n t which I s n o t connected t o an o u t g o i n g l i n e :

+

Because of t h e 8 f u n c t i o n s i n t h e A- [ s e e Eq. [6.1511, energy I s f o r c e d t o flow towards t h e c i r c l e d v e r t e x . S i n c e energy c o n s e r v a t i o n h o l d s i n t h a t v e r t e x t h i s is i m p o s s i b l e and we conclude t h a t t h i s diagram is z e r o . The same is t r u e i f v e r t e x 5 I s c l r c l e d i n s t e a d . F u r t h e r , if t h e momenta p i and p2 r e p r e s e n t l n c o ming p a r t i c l e s , w h i c h I m p l i e s energy f l o w i n g from t h e o u t s i d e I n t o v e r t i c e s 1 and 6 , t h e n a l s o t h e diagrams w i t h v e r t e x 1 and/or v e r t e x 6 c i r c l e d a r e z e r o . Also t h e diagram w i t h 2 end 5 c i r c l e d i s z e r o , even i f now energy may f l o w i n e i t h e r d i r e c t i o n between 2 and 5. because a l l o t h e r l i n e s ending I n 2 or 5 f o r c e energy flow towards these v e r t i c e s . We t h u s come t o t h e f o l l o w i n g r e s u l t . A diagram c o n t a i n i n g c i r c l e d v e r t i c e s g i v e s r i s e t o a nonz e r o c o n t r i b u t i o n If t h e c i r c l e d v e r t i c e s form connected r e g i o n s t h a t c o n t a i n one or more o u t g o i n g l i n e s . And a l s o t h e u n c i r c l e d v e r t i c e s m u s t form connected r e g i o n s I n v o l v i n g incoming l i n e s .

T h u s , f o r example,

::I I n t h i s and I n t h e f o l l o w i n g diagrams,

incoming p a r t i c l e s a r e e t t h e l e f t , outgoing a t t h e r i g h t (energy f l o w s i n a t t h e l e f t , out a t t h e r i g h t ) .

66

G. 't HOOFT and M. VELTMAN

i s zero because i n v e r t e x 4 we have a c o n f l i c t i n g s i t u a t i o n . Examples o f non-zero diagrams are:

Note t h a t an i n g o i n g l i n e may be attached t o a c i r c l e d r e g i o n .

q

Since t h e c i r c l e d v e r t i c e s f o r m connected r e g i o n s we may drop t h e c i r c l e s and i n d i c a t e t h e r e g i o n w i t h t h e h e l p o f a boundary l i n e : = ~

Here i s an example of another diagram:

Note t h a t no s p e c i a l s i g n i f i c a n c e i s a t t a c h e d t o t h e c u t t i n g o f an e x t e r n a l l i n e . Taking t h e above i n t o account,

Eq.

t6.131 now reduces t o

16.17) Here f is,the F o u r i e r t r a n s f o r m o f t h e f u n c t i o n F w i t h o u t underl i n i n g s , F t h e F o u r i e r t r a n s f o r m of t h e f u n c t i o n F w i t h a l l v a r i a b l e s underlined. The f u n c t i o n s Fc correspond t o a l l non-zero diagrams c o n t a i n i n g b o t h c i r c l e d and u n c i r c l e d v e r t i c e s . They correspond t o a l l p o s s i b l e c u t t i n g s o f t h e o r i g i n a l diagram w i t h t h e p r e s c r i p t i o n t h a t f o r B c u t l i n e t h e propagator f u n c t i o n Afk) must be r e p l a c e d by A*[k) w i t h t h e s i g n such t h a t energy i s f o r c e d

67

DIAGRAMMAR

t o f l o w towards t h e shaded region. Equation (6.171 i s Cutkosky's cutting rule. Remembering t h a t t h e T - m a t r i x i s obtained by m u l t i p l y i n g by -1, we see t h a t Eq. (6.171 i s o f e x a c t l y t h e same s t r u c t u r e as t h e u n i t e r i t y equation (6.81. There i s one n o t a b l e d i f f e r e n c e : Eq. (6.171 h o l d s f o r a s i n g l e diagram, w h i l e u n i t a r i t y i s a p r o p e r t y t r u e f o r a t r a n s i t i o n amplitude, t h a t i s f o r t h e sum o f diagrams c o n t r i b u t i n g t o a g i v e n process. Equation (6.171 h o l d s f o r any t h e o r y described by a Legranglen, whether i t i s u n i t a r y o r not.+The Feynman r u l e s f o r are, however, d i f f e r e n t f r o m those f o r T ( S e c t i o n 2.61. Therefore, i f Eq. (6.171 i s t r u l y t o i m p l y u n i t a r i t y a number of p r o p e r t i e s must hold. T h i s w i l l be discussed l a t e r .

F!

6.4

Causality

Again c o n s i d e r any diagram, t h a t r e p r e s e n t s a f u n c t i o n x,l. Take any two v a r i a b l e s , say x and xj. L e t us F(x.,, suppose t h a t t h e t i m e component o f x i s l a r g i r than x oi. The f o l l o w i n g e q u a t i o n holds independently o f t h e t i m e o r d e r i n g o f t h e o t h e r t i m e components

Again terms cancel i n p a i r s . We do n o t need t h e diagrams where x i s u n d e r l i n e d , because we know f o r sure t h a t x i s never t h e 10 A g e s t time. Equation (6.181, when m u l t i p l i e d by t h e a p p r o p r i a t e source f u n c t i o n s and i n t e g r a t e d over a l l x except x and x , i s t h e s i n 1 g l e diagram v e r s i o n of Bogoliubov's c a u s a l i t y c o n d i i i o n . H i s notation i s

(6.191 Here t h e f i r s t t e r m d e s c r i b e s c u t diagrams [ i n c l u d i n g t h e case o f no c u t a t a l l - - t h e u n i t p a r t o f Sl w i t h x and x n o t c i r cled, and-the second t e r m denotes diagrams w i t h i x b u t jn o t xi c i r c l e d . S i s t h e S-matrix o b t a i n e d f r o m t h e conjdgate Feynman + r u l e s (1.0. ell v e r t i c e s u n d e r l i n e d ) , and w i l l o f t e n be equal t o S F u r t h e r g ( x I i s t h e c o u p l i n g constant, made i n t o a f u n c t i o n of space-time.

68

.

G. 't HOOFT and M. VELTMAN

> x Then S i m i l a r l y we can consider t h e case where x we have an equation where now x i s never t o beiflnder&ed. Separating o.Ff t h e t e r m w i t h nojv a r i a b l e u n d e r l i n e d we may combine equations, w i t h t h e r e s u l t F [ x ~ ,x

2'

...,

x

n

I = -e(x

-

x

-e[xio

-

x

I io

30

I

F(xl,

.... xk, ..., xn1-

F[x~,

..., - ...,

i

C0

j0 j

xk,

Xn).

The prime i n d i c a t e s absence o f t h e term w i t h o u t u n d e r l i n e d var i a b l e s . The index i i m p l i e s absence o f diagrams w i t h xi underlined. The summations i n Eq. (6.201 s t i l l c o n t a i n many I d e n t i c a l terms, namely those where n e i t h e r x n o r x a r e underlined. Also J these may be taken together t o g i v ei

in o t

X

c

iu n d e r l i n e d j not

F(xl,

em.,

x

n

I

t6.211

The f i r s t term on t h e r i g h t - h a n d s i d e of Eq. (6.21) i s a s e t o f c u t diagrams, w i t h x _and x always i n t h e unshaded region. They represent t h e product & w i t h Jhe r e s t r i c t i o n t h a t x and x are v e r t i c e s of S. We can now apply t h e same equation, w i t h t h ej same p o i n t s x and x , t o t h e diagrams f o r S i n t h i s product. i Doing t h i s as many dmes as necessary, t h e r i g h t - h a n d s i d e o f Eq. (6.211 can be reduced e n t i r e l y t o t h e sum o f two terms, one containing a function e(x - x 1 m u l t i p l y i n g a f u n c t i o n whose F o u r i e r transforms contaifig 0 fdF(ctions f o r c i n g energy f l o w from 1 t o j, t h e o t h e r c o n t a i n i n g t h e opposite combination. This i s p r e c i s e l y o f t h e form i n d i c a t e d i n S e c t i o n 6.1. L e t us now r e t u r n t o Eq. (6.211.

69

Introducing for 8 the Fourier

DIAGRAMMAR

r e p r e s e n t a t i o n Eq. ( 6 . 5 1 , we c a n see B as a n o t h e r k i n d of p r o p a g a t o r c o n n e c t i n g t h e p o i n t s x and x M u l t i p l y i n g by t h e a p p r o p r i a t e source f u n c t i o n and i n $ a g r e t d g over a l l xi we o b t a i n t h e following diagramnatic equation:

.

The b l o b s t a n d s f o r a n y d i a g r a m or c o l l e c t i o n of d i a g r a m s . The p o i n t s 1 and 2 i n d i c a t e two a r b i t r a r i l y s e l e c t e d v e r t i c e s . The "self i n d u c t a n c e " I s t h e c o n t r i b u t i o n d u e t o t h e 8 f u n c t i o n , and is obviously non-coveriant:

O f course, i n t h e diagrams on t h e r i g h t - h a n d s i d e summation

o v e r a l l c u t s w i t h t h e p o i n t s 1 and 2 i n t h e p o s i t i o n shown i s intended. T h i s i s p e r h a p s t h e r i g h t moment t o summarize t h e Feynman rules f o r t h e c u t diagrams. As an e x a m p l e we t a k e t h e s i m p l e scalar theory:

-I

P r o p a g a t o r i n unshadowed r e g i o n :

1 -

1

( Z n 1 4 i k2 + m2

-

IE

P r o p a g a t o r i n shadowed r e g i o n :

-- 1

1

( 2 S l 4 I k2 + m2 + I E Cut l i n e :

70

G. 't HOOFT and M. VELTMAN Vertex i n unshadowed r e g i o n :

4 (2s) i g .

4

Vertex i n shadowed r e g i o n :

-(2sr) i g .

F o r a spin-1/2 p s r t i c l e e v e r y t h i n g o b t a i n s by m u l t i p l y i n g w i t h t h e f a c t o r - i y k + m:

-I

1

-1yk

[2nl;?i k2

k

L

m

+

-

+ m2

i E

(6.251

The most simple a p p l i c a t i o n concerns t h e C 8 S e o f o n l y one propag a t o r connecting t w o sources. We w i l l l e t these sources emit and absorb energy, b u t we will n o t p u t a n y t h i n g on mass-shell. Indeed, nowhere have mass-shell c o n d i t i o n s been used i n t h e d e r l v e t l o n s . Thus c o n s i d e r :

Jk

k

i2

1

4

(271 i J l ( k l

-

k2 + m2

J2tk1

w i t h J 1 and J non-zero o n l y if k > 0. The u n i t a r i t y e q u a t i o n 2 0 (6.171 reads:

The complex c o n j u g a t i o n does apply t o e v e r y t h i n g except t h e sources J. The second t e r m on t h e r i g h t - h a n d s i d e i s zero, because o f t h e c o n d i t i o n ko > 0. The equation becomes

J

1 k2 + m2

- i~

-

1

4

(2aI i k2

71

+

rn2

IJ = +

I€

DIAGRAMMAR

4

Note t h a t t h e v e r t e x i n t h e shadowed r e g i o n g i v e s a f a c t o r -[2n1 I. With

1

a

-

ie

we see t h e e q u a t i o n h o l d s t r u e . A l s o EQ.

(6.221 can be v e r i f i e d :

i

We now o b t a i n ( n o t e t h e m i n u s s i g n f o r vertex i n shadowed r e g i o n )

- ie

+

8(-ko

+

po16[(k

-

p12

PO

+

mzl]+ p=o

.

The f o u r - v e c t o r p h a s z e r o s p a c e components [see e x p r e s s i o n (6.23)). The po i 8 t e g r a t i o n i s t r i v i a l and g i v e s t h e d e s i r e d re-

sult. 6.5

Dispersion Relations

E q u a t i o n s (6.221 a r e n o t h i n g b u t d i s p e r s i o n r e l a t i o n s , v a l i d f o r any diagram. Let r be t h e f o u r t h compoQent of t h e momentum f l o w i n g t h r o u g h t h e s e l f - i n d u c t a n c e . Let f [ T I be t h e f u n c t i o n c o r r e s p o n d i n g t o t h e cut diagrams e x c l u d i n g t h e r - l i n e i n t h e second term on t h e r i g h t - h a n d s i d e of Eq. (6.221, w i t h T d i r e c t e d from 1 t o 2. S i m i l a r l y f o r f [ T I . If f and f ’ r e p r e s e n t t h e l e f t hand s i d e and t h e f i r s t term on t h e r i g h t - h a n d s i d e , r e s p e c t i v e l y , w e have

72

G. 't HOOFT and M. VELTMAN

(6.261 O f course. a l l t h e funcpons f depend on t h e v a r i o u s e x t e r n a l momenta. The f u n c t i o n f [TI w i l l be zero f o r l a r g e p o s i t i v e T, namely as soon as r becomes l a r g e r than the t o t a l amount o f energy f l o w i n g i n t o t h e diagram i n t h e unshadowed region. S i m i l a r l y f [TI i s zero f o r l a r g e negative T.

The d i s p e r s i o n r e l e t i o n s Eq. c6.261 are very important i n connection w i t h renormalization. I f a l l subdivergencies o f a diagram have been removed by s u i t a b l e counter terms. then a l l cut diagrams w i l l be f i n i t e [ i n v o l v i n g products of subdiagrams w i t h c e r t a i n f i n i t e phase-space i n t e g r a l e l . According t o Eq. (6.261 t h e i n f i n i t i e s i n the diagram must then a r i s e because of non-converging dispersion i n t e g r a l s . S u i t a b l e subtractions. 1.e. counter terms, w i l l make t h e i n t e g r a l s f i n 1 t e . t

It may f i n a l l y be noted t h a t our d i s p e r s i o n r e l a t i o n s are very d i f f e r e n t from those u s u a l l y advertised. We do n o t disperse i n some e x t e r n a l Lorentz i n v a r i a n t . such as t h e centre-of-mass energy or momentum transfer.

A

'Note added: Indeed, a rigorous proof of multiplicative renormalizability of many theories, a t the expense of exclusively l o c a l counter terms a t a l l orders, can be constructed without too much trouble from these dispersion r e l a t i o n s .

73

DIAG RAMMAR

7.

UNITARITY

If t h e c u t t i n g e q u a t i o n (6.171,

diagrammatically represented

as :

c o r r e s p o n d i n g t o T - Tt f o l l o w i n g m u s t hold:

-

I Tt T , i s t o imply u n i t a r i t y , t h e

I 1 The diagrams i n t h e shadowed r e g i o n m u s t be t h o s e t h a t o c c u r + i n s+, i l l The A f u n c t i o n s m u s t be e q u a l t o what i s o b t a i n e d w h e n summing o v e r i n t e r m e d i a t e s t a t e s . R e f e r r i n g t o o u r d i s c u s s i o n of t h e m a t r i x St, i n S e c t i o n 2.6, we n o t e t h a t p o i n t (1) w i l l be t r u e i f t h e Lagrangian g e n e r a t i n g t h e S-matrix i s i t s own c o n j u g a t e . P o i n t ( i l l amounts t o t h e f o l l o w i n g . T h e two-point Green's f u n c t i o n , on which t h e d e f i n i t i o n of t h e S-matrix s o u r c e s was [see Eq. (2.121 and f o l l o w i n g ) . based, c o n t a i n e d a m a t r i x K I n c o n s i d e r i n g StS one w i l l ' i n c o u n t e r ( p a r t i c l e - o u t o f S, p a r t i c l e - i n of S ~ I :

-S

St

k

1 Kti j I-klJ*(') j

(klJ:(k)KQm(-k)

a

-

(7.11

i n t h e sum o v e r I n t e r m e d i a t e s t a t e s . The K are from t h e propagat o r s a t t a c h e d t o t h e s o u r c e s . Because of 1: = (CIconjugqte we* J(k1 t h e n K [-klJ * J , have K t m ( - k l = K Q [kl.. A l s o if J ( k l K ( - k ) showink t h a t J an! J * a r e t h e a p p r o p r i a t e e i g e n currents of S and St. If u n i t a r i t y i s t o be t r u e we r e q u i r e t h a t t h i s sum (7.11 o c c u r r i n g i n S'fS equals t h e m a t r i x K o c c u r r i n g when c u t t i n g im a propagator.

.

T h e proof of t h i s i s t r i v i a l . Suppose K i s diagonal w i t h T h e c u r r e n t d e f i n i n g e J d a t i o n s [2.13) and diagonal elements (2.141 imply t h a t tAe currents a r e o f t h e form

74

*

G. 't HOOFT and M. VELTMAN

There are no c u r r e n t s corresponding t o z8ro eigenvalues. Obviously

and t h i s remains t r u e i f one p r o v i d e s t h e c u r r e n t s w i t h phase factors, etc.

F o r s p i n - 1 / Z 4 p a r t i c l e s t h i n g s a r e s l i g h t l y more complicated, because o f y manipulations. For instance, one w i l l have 4

Kt[-kly4 = y K(k1

.

(7.31

Also t h e n o r m a l i z a t i o n o f t h e c u r r e n t s is d i f f e r e n t . One f i n d s t h e c o r r e c t expression when summing up particle-out/particle-in s t a t e s , b u t a minus s i g n e x t r a f o r antiparticle-out/antlpartlclei n s t a t e s . T h i s f a c t o r is found back i n t h e p r e s c r i p t i o n -1 f o r every f e r m i o n loop. A few examples a r e perhaps u s e f u l . EXAMPLE 7.1

Propagators and v e r t i c e s have been g i v e n b e f o r e . The a p p r o p r i a t e source f u n c t i o n s and r e l a t e d equations a r e g i v e n i n Appendix A.

-

There are f o u r two-point Green's functions:

k

ia[kl

-Iyk + k2

+

75

m2

ua(kl 2 2k0 ; a = 1, 2 4m

,

-

DIAGRAMMAR

k

-ia(kl

i k + m k

a

u (kl

+ m

2k0 ; 7

a - 3 , 4 .

4m

Note t h e minus s i g n f o r t h e incoming a n t i p a r t i c l e wave-function. S c a l a r p a r t i c l e s e l f - e n e r g y (we w r i t e a l s o 6 f u n c t i o n s ) :

2 - g ti4[k

-

k’l

I

d4P

-iyp p2 + m2

m

+

-

IE[ p

-iy[p

-

k12

-

kl + m +

m2

-

iE’

Note t h e minus s i g n f o r t h e closed f e r m i o n loop. Cut diagram (remember - [ 2 n 1 4 i f o r v e r t e x i n shadowed r e g i o n I :

Decay o f s c a l a r i n t o two fermions:

2 The s u p e r s c r i p t u now i n d i c a t e s a n t i p a r t i c l e s p i n o r . The complex conjugate, b u t w i t h k’ i n s t e a d of k, i s

The product o f t h e two summed over i n t e r m e d i a t e s t a t e s i s

76

G. ’t HOOFT and M. VELTMAN

Note t h e m i n u s s i g n f o r t h e q-spinor S i n c e po

=m

Bum.

we have

and similarly f o r q. The q i n t e g r a t i o n can be performed -[2n) 2 g2 641k

-

k’l

1 d4pBIpolb[p2

+

m21e[k0

-

pol x

w h i c h i n d e e d e q u a l s t h e result f o r t h e cut diagram. T h e minus

s i g n f o r t h e closed fermion loop appears here as a m i n u s s i g n i n f r o n t of t h e e n t i p a r t i c l e s p i n o r summation, One may wonder what happens i n t h e c a s e where an a n t i p a r t i c l e l i n e is c u t , but when there is no closed fermion loop. An example is provided by t h e a n t i p a r t i c l e self-energy as compared t o p a r t i c l e s e l f energy :

Somehow there m u s t a l s o b e an e x t r a m i n u s s i g n f o r t h e f i r s t diagram. Indeed i t is there, because t h e first diagram c o n t a i n s an incoming a n t i p a r t i c l e whose wave-function has a m i n u s s i g n . A l l t h i s demonstrates a t i g h t i n t e r p l a y between s t a t i s t i c s [minus s i g n f o r fermion loops) and t h e t r a n s f o r m a t i o n p r o p e r t i e s under Lorentz t r a n s o r m a t i o n s of t h e epinors. The l a t t e r r e q u i r e s the normalization t o energy d i v i d e d by mass a8 given before, and a l s o r e l a t e s t o t h e m i n u s s i g n f o r e n t i p a r t i c l e s o u r c e summation. If an i n t e g e r s p i n f i e l d is assigned Fermi s t a t i s t i c s i n t h e form of m i n u s s i g n s f o r l i n e i n t e r c h a n g e s t h e n u n i t a r i t y w i l l

be v i o l a t e d .

77

DIAGRAMMAR

8.

INDEFINITE METRIC

I f t h e numerator K o f a propagator has a n e g a t i v e eigenv a l u e a t t h e p o l e then u n i t a r i t y cannot h o l d because Eq. (7.21 cannot hold. U n i t a r i t y can be r e s t o r e d i f we I n t r o d u c e t h e convention t h a t a minus s i g n i s t o be attached whenever such a s t a t e appears. I t i s s a i d t h a t t h e s t a t e has n e g a t i v e norm, and t r a n s i t i o n p r o b a b i l i t i e s as w e l l as c r o s s - s e c t i o n s and l i f e t i m e s can now t a k e n e g a t i v e values. T h i s i s , of course, p h y s i c a l l y uncacceptable, and p a r t i c l e s corresponding t o these s t a t e s a r e c a l l e d ghosts. I n t h e o r i e s w i t h ghosts s p e c i a l mechanisms must be present t o assure absence o f u n p h y s i c a l e f f e c t s . I n gauge t h e o r i e s n e g a t i v e m e t r i c ghosts occur simultaneously w i t h c e r t a i n o t h e r p a r t i c l e s w i t h p o s i t i v e m e t r i c , i n such a way t h a t t h e t r a n s i t i o n p r o b a b i l i t i e s cancel. A l s o t h e second t y p e of p a r t i c l e , although completely decent, i s c a l l e d a ghost and has no p h y s i c a l s i g n i f i c a n c e .

78

I

G. 't HOOFT and M. VELTMAN

9.

DRESSED PROPAGATORS

The p e r t u r b a t i o n s e r i e s as f o r m u l a t e d up t o now w i l l i n g e n e r a l be d i v e r g e n t i n a c e r t a i n r e g i o n . Consider t h e case o f a s c a l a r p a r t i c l e i n t e r a c t i n ? w i t h i t s e l f and p o s s i b l y o t h e r p a r t i c l e s . L e t 6 4 ( k - k ' l r r k 1 denote t h e c o n t r i b u t i o n o f a l l self-energy diagrams t h a t cannot be separated i n t o two pieces by c u t t i n g one l i n e . These diagrams a r e c a l l e d i r r e d u c i b l e self-energy diagrams. The t w o - p o i n t Green's f u n c t i o n f o r t h i s scalar p a r t i c l e i s o f the form

iF

The f u n c t i o n i s c a l l e d t h e dressed propagator. T h i s i n c o n t r a s t t o t h e propagator of t h e s c a l a r p a r t i c l e , c a l l e d t h e bare p r o pagator. Ifwe denote t h i s b a r e propagator by A F we f i n d

-

A,=

AF

+

AFl'AF

+

AFrAFrAF

+

a * *

r9.a

Summing t h i s s e r i e s

1

r9.31

- rAF

corresponding t o t h e diagrams:

+

+

+

...

.

where t h e hatched b l o b s stand f o r t h e i r r e d u c i b l e s e l f - e n e r g y d i agr ams

r

The f u n c t i o n i s p r o p o r t i o n a l t o t h e c o u p l i n g constant o f t h e t h e o r y . I t i s c l e a r t h a t t h e p e r t u r b a t i o n s e r i e s converges o n l y i f r A F < 1. But ifAF has a p o l e f o r a c e r t a i n v a l u e o f t h e four-momentum then t h i s s e r i e s w i l l c e r t a i n l y n o t converge near t h i s pole, unless r happens t o be zero t h e r e . And i f we remember t h a t t h e d e f i n i t i o n o f t h e S-matrix i n v o l v e s p r e c i s e l y t h e behaviour of t h e propagators a t t h e p o l e s we see t h a t t h i s problem needs discussion. There are t w o p o s s i b l e s o l u t i o n s t o t h i s d i f f i c u l t y . The s i m p l e s t s o l u t i o n i s t o arrange t h i n g s i n such a way t h a t indeed I'

79

DIAGRAMMAR

i s z e r o a t t h e pole. T h i s can b e done by i n t r o d u c i n g a s u i t a b l e v e r t e x i n t h e Lagrangian. T h i s v e r t e x c o n t a i n s two s c a l a r f i e l d s and e q u a l s m i n u s t h e v a l u e of T a t t h e pole. F o r i n s t a n c e , suppose AF =

1

( Z n I 4 i k2

T h e f u n c t i o n l'lk

ro

r(kl =

+

2

1

(9.41

m2 - I E

+

1 can be expanded a t t h e p o i n t k2 (k2

+

2 m ITl

+

r2(k1.

2 and r a r e c o n s t a n t s , I' i s of o r d e r ( k + m2I2. 1 2 tRe Lagrangian a term t h a t l e a d s t o t h e v e r t e x :

r

I n s t e a d of

r

+

m2 = 0 (9.5)

Introduce i n

we w i l l now have a f u n c t i o n T':

+ r'

w i l l be o f o r d e r ( k 2 + m212 and t h e series (9.2) converges a t t h e pole. A c t u a l l y t h i s reasoning i s c o r r e c t only i n lowest o r d e r because t h e new v e r t e x can a l s o appear i n s i d e t h e hatched blob. Through t h e i n t r o d u c t i o n of f u r t h e r h i g h e r o r d e r v e r t i c e s of t h e t y p e ( 9 . 6 1 t h e r e q u i r e d result can be made accurate t o

a r b i t r a r y order. T h i s procedure, i n v o l v i n g mass and wave-function renormal i z a t i o n corresponding t o ro and r t y p e terms, r e s p e c t i v e l y , embodies c e r t a i n inconveniences. F h s t of a l l , i t may be t h a t I' and r c o n t a i n imaginary p e r t s . T h i s o c c u r s i f t h e p a r t i c l e 0 becomes h n s t a b l e because of t h e i n t e r a c t i o n . Such t r u l y p h y s i c a l e f f e c t s a r e p a r t of t h e c o n t e n t of t h e t h e o r y and t h e above n e u t r a l i z i n g procedure cannot be c a r r i e d through. Furthermore, i n t h e case of gauge t h e o r i e s , t h e freedom i n t h e c h o i c e of terms i n t h e Lagrangian i s l i m i t e d by gauge I n v a r i a n c e , and it is n o t sure t h a t t h e procedure can be c a r r i e d through without gauge invariance violation.

An a l t e r n a t i v e s o l u t i o n is t o u s e d i r e c t l y t h e s u m e d e x p r e s s i o n (9.3) f o r t h e p r o p a g a t o r s i n the diagrams. These diagrams must t h e n , of c o u r s e , c o n t a i n no f u r t h e r i n t e r n a l s e l f - e n e r g y p a r t s , t h a t is t h e y m u s t be s k e l e t o n diagrams. T h e f u n c t i o n r o c c u r r i n g i n t h e propagator m u s t t h e n be c a l c u l a t e d

80

G. 't HOOFT and M.VEkTMAN

w i t h a c e r t a i n accuracy i n t h e coupling constant. F o r instance, i n lowest order f o r any Green's f u n c t i o n t h e r e c i p e i s t o compute t r e e diagrams using bare propagators. I n t h e next order t h e r e are diagrams w i t h one closed loop [no self-energy loops) and bare propagators, and t r e e diagrams w i t h dressed propagators where r i s computed by considering one-closed-loop self-energy diagrams. I t i s c l e a r t h a t . t h 1 s r e c i p e leads t o a l l kinds of complications, which however do n o t appear t o be very profound. Mainly, kinematics and t h e p e r t u r b a t i o n expansion have t o be considered together. With respect t o renormalization t h e complications are t r i v i a l , because then only t h e beheviour o f the propagator f o r very l a r g e k2 i s of importance. But then t h e s e r i e s expansion Eq. (9.21 i s permitted.

For the r e s t of t h i s s e c t i o n we w i l l consider t h e i m p l i c e t i o n s o f t h e use o f dressed propagators f o r c u t t i n g r u l e s .

As a f i r s t step we note t h a t t h e dressed propagator s a t i s f i e s t h e KellBn-Lehmann representation. This f o l l o w s from t h e c a u s a l i t y r e l a t i o n Eq. 16.211, o r i n p i c t u r e Eq. [6.221 f o r t h e two-point Green's function, where x and x are taken t o be t h e in-source and out-source vertices!. repec&vely. The f i r s t term on t h e right-hand side o f these equations i s then zero. Indeed, t h i s gives p r e c i s e l y Eq. (6.11, the deCOmpOSltlDn of the (dressedlpropagator i n t o p o s i t i v e and negative frequency parts. The d e r i v a t i o n of the c u t t i n g r e l a t i o n s , which i s besed on t h i s decompoaition goes through unchanged. However, w i t h respect t o u n i t a r i t y t h e r e i s a f u r t h e r sublety. Using dressed propagators t h e p r e s c r i p t i o n i s t o use only diagrams wfthout self-energy i n s e r t i o n s , t h a t i s skeleton diagrams. Now the A f u n c t i o n corresponding t o a dressed propagator i s e v i d e n t l y obtained by c u t t i n g a dressed propagator. The dressed propagator l a a series, and c u t t i n g gives t h e r e s u l t (we use again the examp l e o f a scalar p a r t i c l e 1

-*

Re

bF

Re

iF = Re f2nI411k2

= iF(-Re

r1AF

pole part

+

[9.71

or 1 +

m2

+

4

ir/12n1 1

81

I

DIAGRAMMAR

where we used

k2 + m2 + i r / ( 2 n 1 4

%

Z[k2

+

f12

-

ial

+

0"k2

+

N2I21 (9.91

near t h e p o l e . Now Re r is o b t a i n e d when c u t t i n g t h e i r r e d u c i b l e self-energy diagrams. I n diagramnatic f o r m we have:

Note t h e occurrence of dressed propagators i n t h e second t e r m on the right. The s u b t l e t y h i n t e d a t ebove is t h e f o l l o w i n g . The m a t r i x S c o n t a i n s s k e l e t o n diagrams w i t h dressed propagators. C u t t i n g these diegrams a p p a r e n t l y r e s u l t s i n expressions o b t a i n e d when c u t t ng s e l f - e n e r g y diagrams. Indeed, these a r i s e a u t o m a t i c a l l y i n S S. I f S c o n t a i n s s k e l e t o n diagrams o f t h e t y p e :

+

t

then S S contains the type:

t

t

Even i f S and S c o n t a i n no r e d u c i b l e diagrams, t h e product S S n e v e r t h e l e s s has s e l f - e n e r g y s t r u c t u r e s . They correspond t o what i s o b t a i n e d by c u t t i n g B dressed propagator.

82

G. 't HOOFT and M. VELTMAN

10.

10.1

CANONICAL TRANSFORMATIONS

Introduction

I n t h i s s e c t i o n we s t u d y t h e b e h a v i o u r o f t h e t h e o r y u n d e r f i e l d t r a n s f o r m a t i o n s . F i e l d s by t h e m s e l v e s are n o t v e r y r e l e v a n t q u a n t i t i e s , from t h e p h y s i c a l p o i n t o f view. The S - m a t r i x i s supposed t o d e s c r i b e t h e p h y s i c a l c o n t e n t of t h e t h e o r y , and t h e r e i s no d i r e c t r e l a t i o n between S - m a t r i x and f i e l d s . Given t h e G r e e n ' s f u n c t i o n s f i e l d s may be d e f i n e d l t h e Green's f u n c t i o n s , however, can be c o n s i d e r e d as a r a t h e r a r b i t r a r y e x t e n s i o n o f t h e S - m a t r i x t o o f f - m a s s - s h e l l v a l u e s o f t h e e x t e r n a l momenta. W i t h i n t h e framework of p e r t u r b a t i o n t h e o r y i t i s p o s s i b l e . up t o a p o i n t , t o d e f i n e t h e f i e l d s by c o n s i d e r i n g t h o s e Green's f u n c t i o n s t h a t behave as smoothly as p o s s i b l e when g o i n g o f f m a s s - s h e l l . F o r gauge t h e o r i e s t h i s i s s t i l l i n s u f f i c i e n t t o f i x t h e t h e o r y , t h e r e b e i n g many c h o i c e s o f f i e l d s (and Green's f u n c t i o n s ) t h a t g i v e t h e same p h y s i c s CS-matrix) w i t h e q u a l l y smooth b e h a v i o u r . For g r a v i t a t i o n t h e s i t u a t i o n i s even more b e w i l d e r i n g , up t o t h e p o i n t of f r u s t r a t i o n . I n t h e s t u d y of f i e l d t r a n s f o r m a t i o n s p a t h i n t e g r a l s have been of g r e a t h e u r i s t i c v a l u e . T h e e s s e n t i a l c h a r a c t e r i s t i c s w i l l be shown i n t h e n e x t s e c t i o n . 10.2

Path Integrals

A few s i m p l e e q u a t i o n s form t h e b a s i s of a l l p a t h i n t e g r a l m a n i p u l a t i o n s , and w i l l be l i s t e d h e r e .

Let a be a complex number w i t h a p o s i t i v e n o n - z e r o i m a g i n a r y p a r t . F u r t h e r m o r e , z = x + i y i s a complex v a r i a b l e . We have *

/ldzeiaZ

m

=

-OD

2

m

I dx I

dye

2

i a [ x +y 1

[lo. 11

-0

Introducing polar coordinates t10.2)

I n c i d e n t a l l y , r e a l i z i n g t h a t t h e e x p r e s s i o n (10.11 i s a p u r e s q u a r e

Let now z be a complex n-component v e c t o r , and A a complex

83

DIAG RAMMAR

n x n m a t r i x . The g e n e r a l i z a t i o n o f Eqs.

*

\

dzl

...

i l z ,A21 dz e n

/

dzj

-\

dxj

nnin d e t [A)

[10.11, [10.21 i s C10.41

’

where

-0

, zJ

dyj

= xj

+

iyj

.

(10.51

-00

Equation [10.41 f o l l o w s t r i v i a l l y f r o m Eqs. (10.11, 110.21 i n t h e case where A is diagonal. Next n o t e t h a t t h e i n t e g r a t i o n measure i s i n v a r i a n t f o r u n i t a r y t r a n s f o r m a t i o n s U. To see t h i s write

U = A + iB

,

where A and B are real matrices.

-

-

AA + BB

1

-

BA

-

AB

The f a c t t h a t UUt = 1 i m p l i e s

(10.61

0

where t h e w i g g l e denotes r e f l e c t i o n . I f now zc = uz then x c = Ax - By and y’ = Ay + Bx. That i s , t h e 2n dimensional v e c t o r xl y transforms as

The determinant o f t h i s 2n x 2n t r a n s f o r m a t i o n m a t r i x is 1. I n f a c t , t h e m a t r i x i s orthogonal, because m u l t i p l i c a t i o n w i t h i t s transpose g i v e s 1, on account of t h e i d e n t i t i e s (10.61. Because o f t h e i n v a r i a n c e o f b o t h t h e i n t e g r a t i o n measure as w e l l as t h e complex s c a l a r p r o d u c t under u n i t a r y t r a n s f o r m a t i o n s , we conclude t h a t Eq. (10.51 h o l d s f o r any complex m a t r i x A t h a t cen be d i a g o n a l i z e d by means of a u n i t a r y t r a n s f o r m a t i o n and where a l l d i a g o n a l m a t r i x elements have a p o s i t i v e non-zero imaginary p a r t .

Consider now a p a t h i n t e g r a l i n v o l v i n g a Lagrangian depend i n g on a s e t of r e a l f i e l d s Ai [see Section31

(10.71

84

G. 't HOOFT and M. VELTMAN

where t h e a c t i o n S i s g i v e n by

S[A1

-1

d4d(A)

.

(10.81

Suppose we want t o use o t h e r f i e l d s 6 t h a t a r e r e l a t e d t o t h e i f i e l d s A as f o l l o w s i AI(x)

-

B,[xl

+

f tx,

i

B1

.

[10.91

The f, a r e a r b i t r a r y f u n c t i o n s ( a p a r t f r o m t h e f a c t t h a t Eq. (10.9f must be i n v e r t i b l e ) . They may depend on t h e f i e l d s B a t any sapce-time p o i n t i n c l u d i n g t h e space-time p o i n t x. According t o well-known r u l e s

&)Ai

= det

r 10.101

@Ai

= det

r 10.11)

or

The determinant i s simply t h e Jacobian o f t h i s t r a n s f o r m a t i o n . I t i s v e r y clumsy t o work w i t h t h i s determinant, e s p e c i a l l y if we r e a l i z e t h a t i t i n v o l v e s t h e f i e l d s a t every space-time p o i n t separately. F o r t u n a t e l y t h e r e e x i t s a n i c e method t h a t makes t h i n g s easy. L e t I$ be a complex f i e l d . According t o Eq. t10.41 we have

(10.121

where C i s an i r r e l e v a n t numerical f a c t o r .

and

(10.131 However, Eq. "J.111 i n v o l v e s a determinant, w h i l e Eq. [10.121 i s f o r t h e i n v e r s e of a determinant. We must I n v e r t Eq. (10.121. The expression on t h e r i g h t - h a n d s i d e o f Eq. [30.121 i s a p a t h i n t e g r a l of t h e t y p e considered i n S e c t i o n 3. I t i n v o l v e s a complex f i e l d 4 as w e l l as f i e l d s Bi t h a t appear s i m p l y as sour-

85

DIAGRAMMAR CBS.

T h e “ a c t i o n “ is [see Eq. [10.1111

T h e diagrams c o r r e s p o n d i n g t o Eq. [10.121 i n v o l v e a + - p r o p e g a t o r t h a t i s simply 1. There are only v e r t i c e s i n v o l v i n g two 4 - l i n e s :

i

j

c . c .

-ij

4

(2nl iY(k’,

’ k , 61

.

The b l o b c o n t a i n s 8 - f i e l d s , b u t w e have n o t i n d i c a t e d t h e m e x p l i c i t l y . The only diagrams t h a t can o c c u r are c l o s e d +-loops

i n v o l v i n g one, two or more Y - v e r t i c e s .

We w i l l write down t h e

f i r s t few: Zeroth order i n Y : F i r s t order

1

a

T h e e x p l i c i t l y w r i t t e n f a c t o r 1/2 i s a c o m b i n a t o r i e l f a c t o r . I t f o l l o w s a150 by t r e a t i n g t h e p a t h i n t e g r a l a l o n g t h e l i n e s i n d i t h c a t e d i n S e c t i o n 3. I n f a c t , i t i s e a s i l y e s t a b l i s h e d t h a t i n n o r d e r t h e r e w i l l be a c o n t r i b u t i o n of n t i m e s t h e f i r s t - o r d e r l o o p w i t h a f a c t o r l / n l S i m i l a r l y f o r t h e two-Y loop. I n t h i s way, i t i s s e e n t h a t t h e whole series adds up t o an e x p o n e n t i a l , w i t h i n t h e exponent only s i n g l e c l o s e d l o o p s

r 10.161

86

G. 't HOOFT and M. VELTMAN

T h i s can be e a s i l y i n v e r t e d simply by r e p l a c i n g r by -r. t h a t i s by t h e p r e s c r i p t i o n t h a t e v e r y c l o s e d 4 - l o o p m u s t be g i v e n a m i n u s s i g n . We so a r r i v e a t t h e e q u a t i o n

w i t h t h e a d d i t i o n a l r u l e t h a t e v e r y c l o s e d 4-loop m u s t be given

a m i n u s s i g n . S[+l i s g i v e n i n Eq. (10,141. F i n a l l y S"B1 f o l l o w s by s u b s t i t u t i n g t h e t r a n s f o r m e t i o n Eq. (10.9) i n t o t h e a c t i o n f o r 'the A - f i e l d , Eq. [ l 0 . 8 1 SOIB) = S[B

+

f(B11

.

[ l o . 181

Summarizing, t h e t h e o r y remains unchanged i f a f i e l d t r a n s f o r mation I s performed, provided c l o s e d loops of g h o s t p a r t i c l e s [ w i t h a m i n u s sign/loopl a r e a l s o introduced. The v e r t i c e s i n t h e g h o s t loop are determined by t h e t r a n s f o r m a t i o n law.

I n t h e f o l l o w i n g s e c t i o n s we w i l l d e r i v e t h i s same r e s u l t , however w i t h o u t t h e u s e of p a t h i n t e g r a l s . 10.3

Diagrams and F i e l d T r a n s f o r m a t i o n s

I n t h i s s e c t i o n we w i l l c o n s i d e r f i e l d t r a n s f o r m a t i o n s of t h e very s i m p l e s t type. T h i s we do i n o r d e r t o make t h e mechanism transparent. Consider t h e Lagranglen

.

1: [ 4 , 4 * A )

-2 4 i Vi j4 j

= 1

+

2

A

(a2 - m2 ) A i

1

+

eI[$l

+

Ji$i

+

HIAi-

(10.191 We assume V t o be symmetric. 1 i s any i n t e r a c t i o n Lagranglen n o t involving the A-fields; t h e l a g t e r a r e e v i d e n t l y free f i e l d s . T h e diagrams c o r r e s p o n d i n g t o t h i s Lagranglen a r e w a l l - d e f i n e d l i f n e c e s s a r y use can be made of t h e p r e v i o u s l y - g i v e n r e g u l a r i z a t i o n

87

DIAGRAMMAR procedure. The

+-

and A-propagators are:

The v e r t i c e s stemming f r o m

cI I n v o l v e + - l i n e s

We want t o study another Lagrengian by t h e replacement

i*

only.

obtained from

where a i s any m a t r i x . The A remain unchanged.

THEOREM The Green's f u n c t i o n s of t h e new Legrangian e0 a r e equal t o those o f t h e o r i g i n a l Legrangian. I n p a r t i c u l a r , A remains a f r e e f i e l d , t h a t I s a l l Green's f u n c t i o n s i n v o l v i n g an A as e x t e r n a l l e g a r e zeroI except f o r A-propagators connecting d i r e c t l y two A-sources. The proof of t h i s theorem is v e r y simple, and c o n s i s t s mainly of expanding t h e q u a d r a t i c p e r t of c O . We have ( d r o p p i n g e l l Indices1

Here t h e w i g g l e denotes r e f l e c t i o n . Rather t h e n t r y i n g t o i n v e r t t h e complete q u a d r a t i c p a r t we w i l l t r e a t as t h e propagator terms $,V4 and A l a 2 - m21A. The remaining terms i n t e r a c t i o n terms. We o b t a i n t h e f o l l o w i n g A-t$

matrix i n the p a r t only the a r e t r e a t e d as vertices:

G. 'r HOOFT and M. VELTMAN

They w i l l be c a l l e d " s p e c i a l v e r t i c e s " . I n a d d i t i o n t o these t h e r e w i l l be many v e r t i c e s i n v o l v i n g A - l i n e s , because o f t h e r e p l a c e ment (10.211 performed i n t h e i n t e r a c t i o n Lagrangian. Consider now any v e r t e x c o n t a i n i n g an A - f i e l d [ e x c l u d i n g t h e A-sources H I . Because t h e A - l i n e a r i s e s due t o t h e sub_stit u t i o n (10.211 t h e r e always e x i s t s a s i m i l a r v e r t e x w i t h t h e A - l i n e r e p l a c e d by a # - l i n e . B u t t h e A - l i n e can be connected t o t h a t v e r t e x a l s o a f t e r t r a n s f o r m a t i o n t o a # - l i n e v i a one o f t h e s p e c i e 1 v e r t i c e s . The sum of t h e two p o s s i b i l i t i e s i s zero. EXAMPLE 10.3.1 I n t h e transformed Lagrangian we have a new v e r t e x

c O , Eq.

(10.22)

4 (2n1 iJa

connecting an A - l i n e w i t h t h e source of t h e 4 - f i e l d . we have t h e o r i g i n a l v e r t e x :

I n addition,

and t h e n a l s o t h e diagram i n v o l v i n g one s p e c i a l v e r t e x . The f a c t o r V i n t h e v e r t e x cancels:

a g a i n s t t h e propagator, l e a v i n g a minus sign. The diagram cancels a g a i n s t t h e p r e v i o u s one. F o r t h e t w o - p o i n t Green's f u n c t i o n a t t h e zero EXAMPLE 10.3.2 loop l e v e l we have:

+

va

BV

19

w i t h a4 , e t c .

c---*-m---4+

Only t h e f i r s t t e r m survives:

a l l otheas cancel i n p a i r s .

Diagrams w i t h loops i n v o l v i n g A - l i n e s a l s o cancel s i n c e t h e y i n v o l v e v e r t i c e s a l r e a d y discussed. The above theorem can e a s i l y be g e n e r a l i z e d t o t h e case o f more complicated transformations, such as

89

DIAGRAMMAR

+++

+ f(Al

w i t h f any f u n c t i o n o f t h e A f i e l d s . Consider t h e t r a n s f o r m a t i o n

EXAMPLE 10.3.3

The f o l l o w i n g s p e c i a l v e r t i c e s a r e g e n e r a t e d [we l e a v e t h e f a c t o r s [ Z n 1 4 i as und e rs to o d 1 : 0

Va

,0

N:\ 0

\

0

\

\

aVa

Again, by t h e same t r i v i a l mechanism as b e f o r e we have, f o r example:

Ja

0

0

0

a:--

'

\

+

J

Va

0

=O -:---V\

\

\

The s i t u a t i o n becomes more c o m p l i c a t e d i f we a l l o w t h e f u n c t i o n f a l s o t o depend on t h e + - f i e l d s . even i n a n o n - l o c a l way. 10.4

L o c a l and N o n - l o c a l T r a n s f o r m a t i o n s

S t a r t i n g a g a i n f r o m t h e L a g r a n g i a n (10.191 we c o n s i d e r now the general substitution

I$ -f

I$

+

f[A,

+I

.

We w i l l now have s p e c i a l v e r t i c e s i n v o l v i n g t h e u n s p e c i f i e d f u n c t i o n f . The f u n c t i o n f w i l l be r e p r e s e n t e d as a b l o b w i t h a c e r t a i n number o f A- and + - l i n e s o f w h i c h we w i l l i n d i c a t e o n l y t h r e e e x p l i c i t l y . The p r o p a g a t o r m a t r i x V i s always t h e r e as a f a c t o r and i s i n d i c a t e d b y a d o t . The + - l i n e a t t a c h e d a t t h a t p o i n t w i l l be c a l l e d t h e o r i g i n a l @ - l i n e o f t h e s p e c i a l v e r t e x .

90

G. 't HOOFT and M. VELTMAN

Now t h i s o r i g i n a l $ - l i n e may be connected t o any o f t h e o l d v e r t i c e s o f t h e t h e o r y o r t o another o r i g i n a l $ - l i n e . Again t h e c a n c e l l a t i o n mechanism works. EXAMPLE 10.4.1

No new f e a t u r e a r i s e s i n these cases. However, if an o r i g i n a l $ - l i n e i s connected t o any o f t h e o t h e r Ip-lines o f 8 s p e c i a l vert e x (except t h e f V f v e r t e x t h a t i s c a n c e l l e d o u t a l r e a d y as shown above) no c a n c e l l a t i o n occurs. F o r example, t h e r e i s n o t h i n g t h a t cancels t h e f o l l o w i n g c o n s t r u c t i o n :

Thus we g e t a non-zero e x t r a c o n t r i b u t i o n o n l y ift h e o r i g i n a l # - l i n e i s connected t o another # - l i n e o f t h e s p e c i a l v e r t e x . A l l t h i s means t h a t t h e new Lagrangian c o n t a i n s all t h e c o n t r i b u t i o n s o f t h e o r i g i n a l Lagrangian p l u s a new k i n d o f diagram where t h e o r i g i n a l $ - l i n e s a r e connected t o one o f t h e o t h e r - l i n e s o f t h e s p e c i a l vertex. Such diagrams c o n t a i n a t l e a s t one closad l o o p o f s p e c i a l v e r t i c e s w i t h "wings" o f old v e r t i c e s as w e l l as s p e c i a l v e r t i c e s .

ell

EXAMPLE 10.4.2

Some new diagrams:

Considering these diagrams one immediately n o t i c e s t h a t t h e f a c t o r s V i n t h e s e c i a l v e r t i c e s a r e always m u l t i p l i e d w i t h t h e propagators -V-'. T h i s then g i v e s as n e t r e s u l t t h e simple propa-

91

DIAGRAMMAR

g a t o r f a c t o r -1 f o r tne " g h o s t " c o n n e c t l n g t h e f vertices. see (10.141.There is no momentum dependence i n t h a t p r o p a g a t o r . If now t h e s p e c i a l v e r t i c e s a r e l o c a l , i . e . o n l y p o l nomial dependence on t h e momenta and no f a c t o r s s u c h as l / k , t h e n these v e r t i c e s can be r e p r e s e n t e d by a p o i n t and t h e c l o s e d l o o p moment u m i n t e g r a t i o n becomes t h e i n t e g r a l o v e r a polynomial. W i t h i n t h e r e g u l a r i z a t i o n scheme t o be i n t r o d u c e d l a t e r such i n t e g r a l s a r e zero, and n o t h i n g s u r v i v e s .

Eq.

Y

I f , however, t h e f i e l d t r a n s f o r m a t i o n i s n o n - l o c a l t h e s e new diagrams s u r v i v e and g i v e an a d d i t i o n a l c o n t r i b u t i o n w i t h r e s p e c t t o what w e h a d from t h e o r i g i n a l Lagrangian. How can w e g e t back t o t h e o r i g i n a l Green's f u n c t i o n s ? A t f i r s t s l g h t i t seems t h a t we m u s t slmply s u b t r a c t t h e s e d i a g r a m s , 1.e. i n t r o d u c e new v e r t i c e s i n t h e t r a n s f o r m e d Lagrangian t h a t produce p r e c i s e l y s u c h diagrams, b u t w i t h t h e o p p o s i t e s i g n . T h i s however i g n o r e s t h e p o s s i b i l i t y t h a t tne o r i g i n a l t$-line of a s p e c i a l v e r t e x c o n n e c t s t o any o f tnese new v e r t i c e s . T h e c o r r e c t s o l u t l o n i s q u i t e simple: i n t r o d u c e v e r t i c e s t h a t r e p r o duce t h e c l o s e d loops o n l y , w i t h o u t t h e "wings", and g i v e each of t h o s e c l o s e d l o o p s a m l n u s s i g n . T h u s a t t h e Lagrangian l e v e l t h e e x t r a t e r m s are as d e p i c t e d below:

-e\

Now t h e r e is a l s o t h e p o s s i b i l i t y t h a t t h e o r i g i n a l $ - l i n e o f a s p e c i a l v e r t e x c o n n e c t s t o t h e s e c a u n t e r l o o p s . I n t h i s way c o u n t e r "winged" diagrams arise a u t o m a t i c a l l y and need n o t t o be i n t r o d u c e d by hand.

92

G. 't HOOFT and M. VELTMAN The f o l l o w i n g c a n c e l l a t i o n occurs i n case o f a EXAMPLE 10.4.3. Lagrangian i n c l u d i n g c o u n t e r c l o s e d loops:

=o

The crosses denote c o u n t e r c l o s e d loops. The s o l u t i o n may t h u s be summarized as f o l l o w s . S t a r t f r o m a Lagrangian .?,($I. Perform t h e t r a n s f o r m a t i o n g ( x l + @l(x) + f [ x , $1. Add a ghost Lagranglen fgBost. T h i s gkost Lagrangian I must be such t h a t i t g i v e s r i s e t o wingless'' c l o s e d loops of f u n c t i o n s f, w i t n one o f t h e e x t e r n a l l i n e s removed, a t which p o i n t another ( o r t h e samel f u n c t i o n f i s attached. The connect i o n i s by means o f a propagator - l / 1 2 n 1 4 i . O f course, every f u n c t i o n a l s o c a r r i e s a f a c t o r [2n14i. F u r t h e r t h e r e i s a minus s i g n f o r every such c l o s e d l o o p .

ns i n d i c a t e d , t h e v e r t i c e s i n such loops a r e t h e f w i t h one l i n e taken off. Symtmlically

Pictorially

T h i s agrees indeed p r e c i s e l y w i t h t h e r e s u l t found w i t h t h e h e l p o f path integrals.

I t may be t h a t t h e r e a d e r 1s somewhat w o r r i e d about combin a t o r i a l f a c t o r s i n these c a n c e l l a t i o n s . A well-known theorem states t h a t comblnatorial f a c t o r s are impossible t o explain; everyone must convince himself t h a t t h e above ghost ioop presc r i p t i o n leads e x a c t l y t o t h e r e q u i r e d c a n c e l l a t i o n s . There i s

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r e a l l y n o t h i n g d i f f i c u l t h e r e i we do n o t want t o suggest t h a t t h e r e i s . A good g u i d e l i n e 1s always g i v e n by t h e p a t h i n t e g r a l f o r m u l a t i o n . Another way 1s t o convince o n e s e l f t h a t , f o r every p o s s i b i l i t y o f s p e c i a l l i n e s and v e r t i c e s connecting up, t h e r e i s a s i m i l a r p o s s i b i l i t y a r i s i n g f r o m t h e gnost Lagrangian. That i s , t h e p r e c i s e t a c t o r i n f r o n t o f any p o s s i b i l i t y i s n o t r e l e vant, as l o n g as i t i s known t h a t i t i s t h e same as found i n t h e counterpart. 10.5

Concerning t h e Rigour of t h e D e r i v a t i o n s

There i s n o t h i n g mysterious about t h e p r e v l o u s deriVatiOnS -p r o v i d e d we remember t h a t we a r e working w i t h i n t h e c o n t e x t o f p e r t u r b a t i o n theory. Thus t h e t r a n s f o r m a t i o n s must be such t h a t t h e new Lagrangian i s o f t h e t y p e as d e s c r i b e d a t t h e beginning. In p a r t i c u l a r t h e q u a d r a t i c p a r t o f t h e new Lagrangien must be such t h a t propagators e x i s t . For instance, a s u b s t i t u t i o n of t h e form 4 + 4 $ is c l e a r l y i l l e g a l . I n s h o r t , t h e c a n o n i c a l t r a n s f o r m a t i o n must be i n v e r t i b l e .

-

There i s a f u r t h e r problem connected w i t h t h e IE prescript i o n . The propagators a r e n o t s t r i c t l y - V - I , i n t h e n o t a t i o n o f t h e p r e v i o u s s e c t i o n . b u t have been m o d i f i e d t h r o u g h t h e IE addit i o n . The f a c t o r s V appearing i n t h e s p e c i a l v e r t i c e s must t h e r e f o r e be m o d i f i e d a l s o such t h a t t h e k e y - r e l a t i o n L-V-lIV = -1 remains s t r i c t l y t r u e . I n connection w i t h gauge f i e l d t h e o r y t h e f u n c t i o n s f themselves c o n t a i n o f t e n f a c t o r s V - l , c a r e l e s s handl i n g o f such f a c t o r s may l e a d t o e r r o r s . As we w i l l see t h e c o r r e c t i.z p r e s c r i p t i o n f o r Faddeev-Popov ghosts can be e s t a b l i s hed by p r e c i s e c o n s i d e r a t i o n of tnese circumstances.

94

G. 't HOOFT and M. VELTMAN

11.

11.1

THE ELECTROMAGNETIC FIELD

L o r e n t z Gauge; Bell-Treiman T r a n s f o r m a t i o n s and Ward I d e n t i t i e s

The theorem about t r a n s f o r m a t i o n s of f i e l d s proved i n t h e f o r e g o i n g s e c t i o n a p p l i e s t o any Lagrangian I: and 1s q u i t e gener a l . We w i l l now e x p l o i t I t s consequences i n t h e c a s e of t h e e l e c t r o m a g n e t i c f i e l d Lagrengian, i n o r d e r t o d e r i v e i n p a r t i c u l a r t h e s o - c a l l e d " g e n e r a l i z e d Ward i d e n t i t i e s " f o r t h e Green's f u n c t i o n s of t h e t h e o r y .

We s t a r t c o n s i d e r i n g t h e f r e e e l e c t r o m a g n e t i c f i e l d c a s e , s i n c e i t c o n t a i n s a l l t h e main f e a t u r e s of t h e problems we want t o s t u d y . T h i s i n c l u d e s t h e case of i n t e r a c t i o n s w i t h o t h e r p a r t i c l e s 1e.g. e l e c t r o n s ) provided these i n t e r a c t i o n s are i n t r o duced i n a g a u g e - i n v a r i a n t way. r h e Lagrangian g i v i n g r l s e t o t h e Maxwell e q u a t i o n s i s

A s i s well known, i n t r y i n g t o a p p l y t h e c a n o n i c a l formalism t o

t h e q u a n t i z a t i o n of t h i s Lagrangien, many d i f f i c u l t i e s a r e e n c o u n t e r e d . Such d i f f i c u l t i e s a r e e s s e n t i a l l y d u e t o some redundancy o f t h e l e c t r o m a g n e t i c f i e l d v a r i a b l e s , meaning t h a t some combination of t h e m i s a c t u a l l y decoupled from t h e o t h e r ones and themselves

.

w i t h i n t h e scheme d e f i n e d i n S e c t i o n 2 , t o g e t ruaes f o r diagrams and t h e n a t h e o r y from a given Lagrangian, such d i f f i c u l t i e s m a n i f e s t themselves i n t h e f a c t t h a t t h e matrix V that d e f i n e s t h e p r o p a g a t o r has no i n v e r s e i n t n e c a s e of t h e i J Lagrangian r11.11. To avoid such problems c o n v e n t i o n a l l y one adds t o t h e Lagrang i a n 111.11, w i t h some m o t i v a t i o n s , a term - 1 / 2 1 3 A 1 2 . Then one lJlJ obtains

Now t h e p r o p a g a t o r i s well d e f i n e d :

~.--. k

v

1 k2

95

61JV

-

iE

DIAGRAMMAR

-

and we have t n e v e r t e x :

k

T h i s c e r t a i n l y d e f i n e s a theory, b u t i n f a c t we nave no i d e a i f t h e p h y s i c a l content I s as t h a t d e s c r i b e d by t h e Maxwell equations. T h i s is m a i n l y t h e s u b j e c t we want t o study now. I n d o i n g t h a t we w i l l encounter many o t h e r somewnat r e l a t e d problems which must be solved when c o n s t r u c t i n g a t n e o r y f o r t h e e l e c t r o m a g n e t i c f i e l d . The key t o a l l these problems i s p r o v i d e d by t h e general i z e d Ward i d e n t i t i e s . I n o r d e r t o d e r i v e these i d e n t i t i e s we add t o t h e Lagrang l e n (11.21 a f r e e r e a l s c a l a r f i e l d 8, coupled t o a source JB. We g e t

ei

x**(A,

- $ r a P A V 1’

=

+

J A

U P

+

1

B(a

2

-

.

2

J ~ B

p IB +

(11.31 We w i l l now p e r f o r m a B e l l - T r e i m a n t r a n s f o r m a t i o n . T h i s i s a c a n o n i c a l t r a n s f o r m a t i o n t h a t 1s i n form a l s o a gauge t r a n s formation. Here

A

U

+ A l

-

J

Ea

lJ

B

(11.41

depending on a parameter E. The replacement f11.41 111.31, g i v e s up t o f i r s t o r d e r i n t h i s parameter

L-*(A

lJ

-

Ea

lJ

8,

B I = .c*#rA,

BI

+ E[a

A

lJv

)a a B lJv

i n

l”,

Ea

BJ

lJlJ

Eq.

.

(11.51 As s t a t e d , t h e t r a n s f o r m a t i o n (11.41 i s i n form just a gauge transformation

A

U

+ A + a h , l J l J

(Il.EI

w i t h A any f u n c t i o n o f x. Note t h a t t h e f i r s t Lagrangian, Eq. 111.11, a p a r t from t h e source term is i n v a r i a n t under such a t r a n s f o r m e t i o n . On t h e c o n t r a r y , t h e second Lagrangian 111.21, and a f o r t i o r i Eq. l11.31, i s n o t gauge i n v a r i a n t because t h e added t e r m -1/21a,,AU12 is n o t . P r e c i s e l y t h e source t e r m and t h i s -1/2(alJAPJ2 term a r e r e s p o n s i b l e f o r t h e difference, i n Eq. (11.51, between t h e o r i g i n a l Lagrangian L**[AU, BJ and t h e

96

0.'t HOOFT and M. VELTMAN

transformed on@. L e t u s come back t o our l o c a l canonical transformation [11.4). The d i f f e r e n c e between t h e o r i g i n a l and t h e new Lagranglan gives rise t o t h e following e x t r a v e r t i c e s :

c(2nI 4 k,,J,,-

.

Here t h e dotted lines denote t h e 8 - f i e l d . and t h e s h o r t double l i n e a t t h e sources denotes t h e gauge f a c t o r a appearing i n t h e gauke transformation (11.61 i n f r o n t of A.'

.

(Note: A d e r i v a t i v e a a c t i n g on a f l e l d i n an I n t e r a c t i o n term of t h e Lagrangian. gibes i times t h e momentum of t n e f i e l d flowing towards t h e vertex.J 4 O f course t h e v e r t e x term ~[Znlk,,J would give no c o n t r l bution If t h e source of t h e elactromagnehc f l e l d J,, were a "gauge-invariant source". namely i f a,,Jcl = U. I n t h e discussion

which follows we want. h0WeVeI'. t o allow i n general a l s o "nongauge-invariant sources". whose four-divergence i s d i f f e r e n t from zero. As a consequence of t h e general theorem proved i n t h e f o r e going s e c t i o n . t h e 8 - f i e l d remains a f r e e f i e l d . j u s t as i n t h e theory described by t h e Lagranglan c". To f i r s t order i n E. considering t h e n-point Green's function w i t h one B source we have :

a a-

------

+

+

w@--

--rK

+

.....

97

-

DIAGRAMMAR

I n t h e f i r s t diagram t h e 6 - A v e r t e x is f o l l o w e d by a photon lJ propagator. One has:

Apart f r o m a s i g n t h i s Is p r e c i s e l y t h e same f a c t o r as o c c u r r l n g and we may use t h e same i n t h e a,,BJ,, v e r t e x 1eik,,[2nl4iJu1 notation f o r It:

The r e s u l t l n g Ward i d e n t i t i e s are then:

=

+... +

ro" %

(11.8) T h i s r e s u l t looks perhaps a l i t t l e strange, s i n c e u s u a l l y one tends t o f o r g e t t h e l i n e s t h a t go s t r a i g h t through w i t h o u t i n t e r a c t i o n . A s a m a t t e r o f f a c t , t h i s i s allowed o n l y i n t h e case o f gauge-Invariant sources, f o r which, s i n c e then k J = 0 , u u we have:

Equation [11.8) i s an e q u a t i o n f o r Green's f u n c t i o n s . O f course i n d e f i n i n g Green's f u n c t i o n s one should n o t employ p a r t i c u l a r p r o p e r t i e s o f sources, sucn as auJp = 0. As we w i l l see, t h e S-matrlx will be defined u s i n g g a u g e - i n v a r i a n t sources f o r t h e e l e c t r o m a g n e t i c f l e l d , and f o r such cases t h e r i g h t - h a n d s i d e o f Eq. (11.81 Is zero. We remark t h a t t h e War0 i d e n t i t y , Eq. [11.81, is t r i v i a l l y t r u e as I t stands i n t h e case o f no i n t e r a c t i o n t h a t we a r e considering. I t i s i n f a c t s u f f i c i e n t t o r e a l i z e t h a t here, f o r oxample f o r t h e f o u r - p o i n t Green's f u n c t i o n , we have:

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DIAGRAMMAR

It i s worth w h i l e t o n o t e t h a t t h e above d e r i v a t i o n goes through a l s o I f e l e c t r o n s , o r any o t h e r p a r t i c l e s , a r e p r e s e n t , provided t h e photon i s coupled t o t h e m i n a g a u g e - i n v a r i a n t way. Let u s c o n s i d e r i n some d e t a i l s t h e i n t r o d u c t i o n of t h e e l e c t r o n s i n t o t h e t h e o r y . The Lagrangian t o s t a r t rrom 1s

w h e r e L”(A,,, B) I s given i n tq. 111.31. The new p i e c e s o f - t h i s Lagrangian i n v o l v i n g t h e e l e c t r o n f i e l d , t h e s o u r c e terms .ley + $Je e x c e p t e d , are i n v a r i a n t under t h e t r a n s f o r m a t i o n 111.41 i f a l s o t h e e l e c t r o n f l e l d I s transformed

-

JI

-

+

$e

+lEeB

c1

-

$[1

+

iEeB1

Then, up t o f i r s t o r d e r i n

E,

+

2 O ~ E1

.

we have

The e x t r a v e r t i c e s of t h e transformed Lagrangian a r e t h e n t h e same a s t h o s e of t h e f r e e f i e l d case, g i v e n i n (11.71, t o g e t h e r w i t h t h e f o l l o w i n g ones i n v o l v i n g t n e e l e c t r o n s o u r c e s :

-(2nI4ljeLq

99

+

kllee

,

DIAGRAMMA R

Therefore, f o r Green's f u n c t i o n s i n v o l v i n g no e x t e r n a l e l e c t r o n l i n e s , t h e Ward i d e n t i t i e s a r e e x a c t l y t h e same as i n Eq. (11.81, even i f i n t h i s case t h e bubbles c o n t a i n any number of closed e l e c t r o n loops. When e x t e r n a l e l e c t r o n l i n e s a r e present, t h e Ward i d e n t i t i e s r e c e i v e a d d i t i o n a l c o n t r i b u t i o n s o f t h e form:

11.2

L o r e n t z Gauge: S-Matrix and U n i t a r i t y

L e t us now i n v e s t i g a t e t h e S-matrix, keeping I n mind f o r s i m p l i c i t y t h e f r e e e l e c t r o m a g n e t i c f i e l d Lagrangian 1:. Eq. 111.2'1. The m a t r i x K [see Sections 2 ana 71 i s here

From t h e c u t t i n g equation, S e c t i o n 7, we d e r i v e t h e f a c t t h a t u n i t a r i t y would h o l d i f t h e sources were normalized according t o Eq. ( 7 . 2 1 . nowever, t h e complex conjugate o f a f o u r - v e c t o r i s defined t o have an a d a i t i o n a l minus s i g n i n i t s f o u r t h component (see t h e I n t r o d u c t i o n ) , so we are forced t o a t t r l b u t e t o t h e f o u r t h component o f h v e c t o r p a r t i c l e e n e g a t i v e m e t r i c ( S e c t i o n 81. The sources Jtae, a = 1, 4 a r e chosen such t h a t U

...,

(12.91

= - 1 if

a - 4 .

On t h e o t h e r hand, i t I s w e l l known t h a t due t o gauge i n v a r i a n c e we do have a p o s i t i v e m e t r i c theory, b u t w i t h o n l y two photon p o l a r i z a t i o n s . i n t h e system where k = (0. 0 , K , l k o l , lJ we l a b e l t h e sources as f o l l o w s

100

5‘I)- 11, 0, 0, OJ d2) = ( 0 , 1, 0, 01 u Jl3] IJ

-

G. ’t HOOFT and M. VELTMAN

I

to, 0, 1, 01 ,

J[41 = (0, u, 0, 11 lJ

.

-

we now p o s t u l e t e t h a t o n l y t h e f i r s t two o f these sources e m i t p h y s i c a l photons. I n terms of a non-covariant o b j e c t z (0, 0, lJ - K , IK 1 obtained f r o m k DY s p a c e - r e f l e c t i o n , we have 0

v

whereas

Considering c e r t a i n s e t s of c u t diagrams we can e p p l y t h e Ward i d e n t i t i e s t o t h e l e f t - and t h e r i g h t - h a n d side, t o see t h a t t h e terms p r o p o r t i o n e l t o k end kv, r e s p e c t i v e l y cancel emong themselves. F o r i n s t a n c e , a#! t h e l e f t - h a n d s i d e o f a c u t diagram one can apply t h e Ward i d e n t i t y :

As b e f o r e t h e double l i n e i n d l c a t s s a f a c t o r ik,,. Note t h e t a l l p o l e r i z a t i o n s can occur a t t h e i n t e r m e a i a t e s t a t e s , 1.8. t h e o u t - s t a t e s f o r S end t h e i n - s t a t e s f o r St, b u t t h e o t h e r l i n e s ere physical.

I n our case t h e r i g n t - h a n d s i d e o f t h i s Ward i d e n t i t y i s zero because t h e sources on t h e r i g h t - h a n d s i d e absorb energy only, end t h e r e f o r e :

101

DIAGRAMMAR

-1 So, due t o t h e Ward i d e n t i t i e s , one may r e p l a c e t h e f a c t o r KPv i n t h e I n t e r m e d i a t e s t a t e s , found from t h e c u t t i n g e q u a t i o n , by

Jh13Jh11 + Jh21J$’3, w h i c h i m p l i e s u n i t a r i t y i n a H i l b e r t s p a c e w i t h o n l y t r a n s v e r s e photons. We m u s t make a s l i g h t d i s t i n c t i o n between p h y s i c a l s o u r c e s and g a u g e - i n v a r i a n t s o u r c e s . T h e combination

JIJ

= k

J(3’ 0

P

+

,Jt4’ P

= (0, 0, ko,

IKJ

i s gauge i n v a r i a n t because a,J,

= 0 , b u t on mass s h e l l t h i s s o u r ce i s p r o p o r t i o n a l t o kP and i t g i v e s no c o n t r i b u t i o n due t o t h e Ward I d e n t i t i e s , so d e s p i t e i t s gauge i n v a r i a n c e , I t i s unphysic a l , i n t h e s e n s e t h a t i t emits n o t h i n g a t a l l , not even g h o s t s .

11.3

Other Gauges: T h e Faddeev-Popov Ghost

tiefore going on, l e t u s come back f o r a moment and t r y t o i n t e r p r e t t h e Ward i d e n t i t y w e have proved. S t a r t i n g from a g a u g e - i n v a r i a n t Lagrangian, E q . 111.1), w e added t h e term

i n o r d e r t o d e f i n e a p r o p a g a t o r . T h e Ward i d e n t i t i e s f o l l o w by performing a Bell-Treiman t r a n s f o r m a t i o n , 1.e. a c a n o n i c a l t r a n s f o r m a t i o n t h a t i s a l s o a gauge t r a n s f o r m a t i o n . As we a l r e a d y n o t e d , o n l y t h e above term (and t h e s o u r c e term1 g i v e s rise t o a c o n t r i b u t i o n , namely ~ ( a , f i , l [ a 2 B l . T h i s i s t h e o n l y c o u p l i n g of t h e B - f i e l d f o r m a l l y a p p e a r i n g i n t h e t r a n s f o r m e d Lagrangian. Now t h e f a c t t h a t B remains a f r e e f i e l d , 1 . e . g i v e s z e r o when p a r t of a Green’s f u n c t i o n , i m p l i e s t h a t aPAP i s a l s o f r e e . T h i s i s a c t u a l l y t h e c o n t e n t o t t h e Ward i d e n t i t y . T h e r e f o r e we conclude t h a t t h e a d d i t i o n of t h e term -1/2ta A l 2 does P P n o t change t h e p h y s i c s of t h e t h e o r y . T h i s d e f i n e s o u r s t a r t i n g p o i n t . I f , due t o a gauge i n v a rlance, a Lagrangian Is s i n g u l a r , 1 . e . t h e p r o p a g a t o r s do n o t e x i s t , t h e n a “good” Lagrangian can be o b t a i n e d by adding a term

102

G. ’t HOOFT and M. VELTMAN

-4; 2 , wherc

C behaves under gauge transformations as C + C + + t A . Here t is any field-independent quantity that may contain derivatives. The argument given above, snowing that the addition of the term -1/2[a,,A,,12 does not modify the physical content of the theory, can equally well De applied here. C will appear t o - b e a free field, as can be seen by performing a Bell-Treiman transformation. However, for this simple recipe to be correct, as in the case explicitly considered C = apAU, one needs t, defined by the gauge transformetion C + C + tA, not to depend on any fields. The difficulty with Yang-Mills fields is that the gauge transformations are more complicated, so much so that no simple C with the required properties exists. [Actually there exists a choice of C for Yang-r.iills fields that is acceptable from this point of view., namely C = aA3, a + =. See R.L. Arnowitt and S.I. Fickler, Phys. Rev. 12/, 1821 (1962). I For this reason we will now study quantum electrodynamics using a gauge function C that has more complicated properties under gauge transformations. We take for understood the theory corresponding to the Lagrangian

Ihe sources will be taken to be gauge invariant and are included in Later we will consider also non-gauge-invariant sources.

xinv.

2

Let us now suppose that we would like to have a A,, + XA instead of a,,A,, for the function C. We can go from e!t above’ Lagrengian with C = +,Au to the case of C = + X A ~by means lJ of a non-local Bell-lreiman transtormation

A,,

+

A,,

+

xa -2ay[AvJ2 ,

or, more explicitly, x’JAtlx’)

103

,

[11.111

DIAGRAMMAR

Here we have a s i t u a t i o n as described i n Subsection 10.5. F o r t h e subsequent m a n i p u l a t i o n s t o be t r u e we must i n f a c t supply a i e t o t h e denominator. Note t h a t

-

1: formatf%,

1s unchanged under t h i s somewhat strange gauge t r a n s and

I n view o f t h e s t r u c t u r e o f our c a n o n i c a l t r a n s f o r m a t i o n we may expect t h a t a ghost Lagrangian must be added.

in

E

P e r f o r m i n g t h e t r a n s t o r m a t i o n (11.11) one g e t s t o f i r s t order t h e f o l l o w i n g s p e c i a l v e r t e x [see S e c t i o n I D ) :

Here, as usual, t h e d o t i n d i c a t e s t h e f a c t o r V whlch i s minus i t h e i n v e r s e photon propagator, and t h e Short dodble l i n e a f a c t o r I+,. The d o t t e d l i n e r e p r e s e n t s t h e f u n c t i o n A(x x’l, which i s j u s t l i k e a s c a l a r massless p a r t i c l e propagator. The shows t h a t t h e t h e o r y described ny theorem proved I n S e c t l o n IU t h e transformed LagrangIan remains unchanged i f we a l s o p r o v i d e f o r ghost loops, c o n s t r u c t e d by connecting t h e o r i g i n a l photon l i n e s t o one o f t h e o t h e r photon l i n e s o f t h e s p e c i a l v e r t e x . F o r example, we have:

-

Here we have c a n c e l l e d t h e photon propegetors a g a i n s t t h e dbts, so t h a t t h e new v e r t e x :

104

G. ‘t HOOFT and M. VELTMAN

-

appears which can be formed by i n t r o d u c i n g a massless complex f i e l d (I i n t e r a c t i n g w i t h t h e photon v i e t h e i n t e r a c t i o n Lagrangian e = 2A$ A,,a,$. A l l t h e c l o s e d loops c o n s t r u c t e d t h i s way must ha6e a minus s i g n I n f r o n t . We come t o t h e c o n c l u s i o n t h a t t h e Lagrangian

w i t h t h e p r e s c r i p t i o n t h a t every c l o s e d 9 l o o p g e t s a minus sign, reproduces t h e same Green’s f u n c t i o n s as oefore, when we had C = auAP and no ghost p a r t i c l e s . The (I p a r t i c l e i s c a l l e d t h e Faddeev-Popov ghost. rhe 10 p r e s c r i p t i o n f o r i t s propagator is t h e u s u a l one. I t 1s perhaps noteworthy t n a t t h e Faddeev-Popov ghost i s not t h e gnost w i t h propagator -1 o f S e c t i o n 10. I h e F-P ghost i s t h e I n t e r n a l s t r u c t u r e o f t h e t r a n s f o r m a t i o n f u n c t i o n , see t r a n s f o r m a t i o n [ I I . l I J . EXAMPLE 11.3.1

Wv-

The Feynman r u l e s from t h e Lagrangian (11.121

1 -

k

(Zn14i k2

1 4

*----a

[2n1 i k2

-

ic ’

-

(-1 f o r every IE c l o s e d loop1 ,

6x

Q

P

+ 6 6 a6

105

are:

BY

1 ,

DIAGRAMMAR

Let u s c o n s i d e r photon-photon s c a t t e r i n g i n t h e z e r o loop approximation. T h e f o l l o w i n g diagrams:

x+X.M+X a)

b)

C)

d)

m u s t g i v e z e r o . T h i s i s indeed t h e c a s e a1

-t

4

(2n1 2x6

aB

4 2 C J + 4(2nJ i h 6

[-kXl

a6

b

BY

1 7 21 ( 2 n l 4 2Xk 6 [2n1 1 K a ~6

4

= 412nl i A

2

,

Note t h a t I n t h e c o n t r i b u t i o n s from diagrams [ e l , I b J , [ c ) , many terms Vanish, because o u r e x t e r n a l s o u r c e s a r e t a k e n t o be gauge i n v a r i a n t . O f c o u r s e , t o g e t a c o n t r i b u t i o n from t h e Faddeev-Popov g h o s t , one h a s t o c o n s i d e r examples o f diagrams c o n t a i n i n g loops. Indeed, one n o t e s f o r i n s t a n c e :

A c t u a l l y t o prove t h i s c a n c e l l a t i o n we need a g a u g e - i n v a r i a n t r e g u l a r i z a t i o n method, w h i c h w l l l be provided i n t h e f o l l o w i n g . We want t o understand now whether a g e n e r a l p r e s c r i p t i o n can be g i v e n w h i c h , I n a g e n e r a l gauge d e f i n e d by t h e f u n c t i o n C ,

106

'

G. ’t HOOFT and M.VELTMAN

a l l o w s us t o w r i t e down immediately t h e Lagrangian f o r t h e ghost p a r t i c l e 4. 2 I n t h e case considered o f C = alJA,, + X A v , i t i s easy t o see t h a t t h e ghost Lagrangian i n 111.121 comes o u t by t h e f o l l o wing p r e s c r i p t i o n .

Take t h e f u n c t i o n C. under a gauge t r a n s f o r m a t i o n one has t o f i r s t order i n A

C + C + d A w i t h & some operator, which i n t n e a c t u a l case depends on t h e fields A then t h e ghost Lagrangian i s simply !J

.

Here

f r o m which 111.12)

follows.

The g e n e r a l i t y of t h i s p r e s c r i p t i o n can be understood by l o o k i n g a t o u r manipulations. The n o n - l o c a l t r a n s f o r m a t i o n 111.711 was taken such t h a t a A + a A + XA2. I f we consider a gauge transformation lJ lJlJ lJ

alJAIJ

+

a

ll lJ

2 +~ a h ,

t h e n we must s o l v e t h e equation

2 The f a c t t h a t t h e propagator l / k appeared I n our loops i s t h u s simply r e l a t e d t o t h e behaviour a A + a A + a2A. Secondly, t h e s p e c i a l v e r t i c e s are PlJ lJlJ

107

DIAGRAMMAR

where VUv I s I n t h e u s u a l way r e l a t e d t o t h e photon propagator. Then t h e v e r t l c e s appearing i n t h e ghost loops a r i s e f r o m t h e p r e s c r i p t i o n : remove one f a c t o r A, f r o m t h e expression AA2 and U r e p l a c e i t by a A. D i a g r a n m e t i c a l l y :

U

f a c t o r o f 2 p r o v i d e s f o r t h e two p o s s i b l e ways o f removing an A U - f i e l d f r o m AAE. T h i s is indeed p r e c i s e l y what one o b t a i n s t o f i r s t o r d e r I n A by s u b m i t t i n g XA2 t o a gauge t r a n s f o r m a t i o n , U see Eu. 111.131.

H

The beauty o f t h i s r e c i p e t o f i n d t h e ghost Lagrangian is t h a t t h e r e i s no r e f e r e n c e t o t h e f u n c t i o n C t h a t we s t a r t e d from. To c o n s t r u c t t h e ghost Lagranglan one needs o n l y t h e C s c t u a l l y r e q u i r e d . A l s o we need o n l y i n f i n i t e s i m a l gauge t r a n s formations. EXAMPLE 11.3.2 The above p r e s c r i p t i o n works a l s o C = i3 A t h a t we s t a r t e d from. I n f a c t we have VlJ

a~

i n t h e case of

+ a P A U + 2a ~ ,

U U meaning t h a t t h e ghost Lagrangian should be

w i t h no c o u p l i n g however o f t h e + - f i e l d t o t h e e l e c t r o m a g n e t i c f i e l d . Therefore, as we a l r e a d y know, no ghost l o o p I s t o be added I n t h i s case.

Even ift h e above m a n i p u l a t l o n s a r e q u i t e s o l i d one may have some doubts concerning t h e f i n a l p r e s c r i p t i o n f o r t h e ghost Lagrangian. To e s t a b l i s h c o r r e c t n e s s o f t h e theory, as we d i d i n Ward i d e n t i t i e s a r e needed. I n t h e g e n e r a l t h e case of C = aUH,,, case such i d e n t i t i e s a r e more complicated t h a n those g i v e n before and we w i l l c a l l them Slavnov-Taylor I d e n t i t i e s . 11.4

The Slavnov-Taylor I d e n t i t i e s

In t h i s s e c t i o n we w i l l a e r i v e t h e Slavnov-Taylor i d e n t i t i e s , u s i n g o n l y l o c a l c a n o n i c a l t r a n s f o r m a t i o n s . One can always keep i n mind t h e e x p l i c i t case o f C = a H + AA2, b u t we now s t a r t

,,u

108

U

G. 't HOOFT and M. VELTMAN adapting a general n o t a t i o n a n t i c l p a t m g t h e general case. L e t t h e r e be g i v e n a f u n c t i o n C, t h a t behaves under a gener a l gauge t r an s f o rma t i o n A + A + a h as f o l l o w s P l J U

c

+

c

rm

+

+

~ J +A

.

or**)

Here we s p l i t up t h e e a r l i e r f a c t o r It depending on t h e f i e l d A For C IJ and g = 2M a

.

v

-4a-

(11.14)

Ifi

H

lJlJ

+

+

iJ wizh XA

IJJ

WE

-

the p a r t have A a

ll'

Our s t a r t i n g p o i n t w i l l be t h e Lagrangian

w i t h t h e - l / l o o p p r e s c r i p t i o n f o r ghost loops. As before. we add t o t h i s Lagrangian B piece d e s c r i b i n g a f r e e f i e l d B

-21 ata 2 -

2 p IE + J ~ B

end perform t h e l o c a l Hell-Treiman t ransf or m et ion

A,,

+

Au

+

tuO

Working t o f i r s t order i n t h e f i e l a B

r11.1ti1 Here we used t h e transformation I

a

-

+

a

+ 10 +

;la

+

ore2)

.

Again we d i s t i n g u i s h t h e field-independent part 0 and t h e f i e l d dependent B . I n t h e a c t u a l case o f i = XA a we have lJlJ

i~

-

zxa

P

Ba

P J

d = D

I

109

DIAG RAMMAR T h e vertices i n v o l v i n g t h e 8 - f i e l a i n t h e t r a n s f o r m e d L a g r a n g i a n (11.16) a r e ( t h e o v e r - a l l f a c t o r ( Z V I 4 i is a l w a y s l e f t u n d e r s t o o d ) :

-

-i -cia, ----7 from

T h e v e r t e x c o r r e s p o n d i ? g t o 9 &I$ i s n o t d r a w n slnce I n - t h e actual case [C = +A,, + XAE) d is zero. I h e v e r t i c e s CmB a n d CRB a p p e a r w i t h a minus s i g n i n t h e Lagrangian: t h i s s i g n i s n o t included i n t h e v e r t e x d e f i n i t i o n and t h u s must b e p r o v i d e d s e p a r a t e l y . The s t a t e m e n t t h a t B r e m a i n s a f r e e f i e l d becomes t n e n t o f i r s t o r d e r i n the 8-vertices:

I n t h e s e c o n d term a t t h e r i g h t - h a n o s i d e we e x h i b i t e d e x p l i c i t l y t o which t h e 8 - l i n e is a t t a c h e d , t o g e t h e r w i t h t h e a s s o c i a t e d m i n u s s i g n . T h e b l o b s are b u i l t u p f r o m d i a g r a m s c o n t a i n i n g p r o p a g a t o r s a n d v e r t i c e s as g i v e n i n t h e p r e v i o u s s e c t i o n . T h e $ p a r t i c l e s go a r o u n d i n l o o p s o n l y , a n d s u c h l o o p s are i n c l u d e d i n t h e b l o b s .

t h e whole loop

I n t h e a b o v e e q u a t i o n t h e r e 1s o n e c r u c i a l p o i n t . C o n s i d e r t h e v e r y s l m p l e s t case, n o i n t e r a c t i o n a n d o n l y o n e p h o t o n s o u r c e . since C = alrA,, + XAE, one h a s t o z e r o t h o r d e r i n X :

110

G. 't HOOFT and M. VELTMAN

T h e first diagram [ s i m i l a r l y t o t h e second one) c o n t a i n s t h e B p r o p a g 3 t o r p o l e , a phOt?n p r o p a g a t o r p o l e , t h e v e r t e x f a c t o r

.

[Zn)4im and t c e f a c t o r t The l a s t diagram has o n l y thg B p o l e and a f a c t o r t,, = a,. ThN e q u a t i o n c a c be t r u e o n l y i f rn i s z e r o p r e c i s e l y on t h e photon p o l e . We haa m = a' + -k2, and we m u s t theref o r e always take [ 2 r I 4 i i = (ZnI4l[a2

+ iE) +

-[2n)4i1k2 - I € ) .

As w i l l be seen, t h i s m i s m i n u s t h e i n v e r s e of t h e g h o s t propaga-

t o r . I n t h e g e n e r a l c a s e , c o n s i d e r i n g lowest o r d e r i d e n t i t i e s , one m u s t check on t h e 1 i n t h e p r o p a g a t o r s i n t h e manner desc r i b e d here. We w i l l perform some m a n i p u l a t i o n s on Eq. [11.171. Consider t h e f i r s t diagram on t h e r i g h t - h a n d s i d e . Using:

-

= = C

6i

cc-

-av

c

t h e c o n t r i b u t i o n of t h i s aiagram becomes:

w i t h a new v e r t e x :

--

P

T r e a t i n g t h i s f i r s t v e r t e x a s a photon s o u r c e we see t h a t we can iterate, getting:

111

DIAGRAMMAR

ZJ means sum over a l l t h e photon sources. The l a s t diagram a r i s e s because t h e f i r s t v e r t e x i s t r e a t e d as a source. Some new v e r t i c e s e n t e r , wnose meaning i s unambigoualy defined. F o r example, we have:

4 which corre.spond p r e c i s e l y t o a v e r t e x 12n1 i e o r t o an l n t e r a c 4. I n p a r t i c u l a r , since t h i s term i s i n v a r i a n t t i o n t e r m $'*a,Ba under t h e I n t e r c k a n g e B -t $, we have:

---*,1

3

- - -4/ \

\

T h i s e q u a l i t y , t r i v i a l i n t h e case o f quantum electrodynamics, becomes much more t r i c k y i n t h e case o f non-Abelian gauge symmet r i e s . Then a s i m i l a r i d e n t i t y f o l l o w s f r o m t h e group s t r u c t u r e , and i s c l o s e l y r e l a t e d t o t h e Jacob1 i d e n t i t y f o r t h e s t r u c t u r e constants o f t h e group. The r i g h t - h a n d s i d e of Eq. [11.181 can now be i n s e r t e d i n p l a c e o f t h e f i r s t diagram on t h e r i g h t - h a n d s i d e o f t h e o r i g i n a l EQ. [11.171. We n o t e t h a t some c a n c e l l a t i o n s occur. The second diagram on t h e r i g h t - h a n d s i d e o f Eq. [11.171 c o n t a i n s a l s o t h e c o n f i g u r a t i o n where t h e ghost has no f u r t h e r i n t e r a c t i o n :

T h i s cancels p r e c i s e l y t h e l a s t diagram of t h e i t e r a t e d equation, a f t e r i n s e r t i o n i n Eq. 111.171.

112

G. 't HOOFT and M. VELTMAN This process can be repeated i n d e f i n i t e l y and one f i n a l l y a r r i v e s a t t h e equation:

The ghost l i n e going through t h e diagram may have zero, one, two, etc., v e r t i c e s of t h e type:

If one notes now t h a t t h e mass of t h e 8 - f i e l d i s an absolutely f r e e parameter i n our discus?ion. we can a l s o g i v e now t h e Bf i e l d t h e propagator - [ Z r 1 4 i m ' l . I n t h i s way we can simply reduce t h e f i r s t diagram on t h e right-hand s i d e o f Eq. [11.191 to:

and a l s o i n c l u a e i n I t t h e l a s t diagram o f Eq. that :

*-m-' --Q c

=

we get t h e i d e n t i t y :

113

-

[11.191. Observing

m

DIAGRAMMAR

-

Note: I n t h e case of C = a,A, t h i s equation coincides w i t h t h e Ward i d e n t i t y ~ 1 1 . d 1 . One has indeed:

4 -

&

= [ 2 ~ 1i J g

4 2 1271 i k

1-ik 1 =

lJ

-

k

K - t - 3 .

CL

so t h a t t h e f i r s t diagram of Eq.

(11.201 can a l s o be drawn as:

Furthermore, i n t h i s case o f C = and t h e n :

a

A r h e ghost does n o t i n t e r a c t lJlJ

E q u a t i o n [11.201 can be g e n e r a l i z e d as t'ollows. Suppose t h a t a source is coupled t o s e v e r a l e l e c t r o m a g n e t i c f i e l d s , f o r i n s t a n c e

JA2

P

or

JA4

P

o r even m o r e g e n e r a l t o a l o c a l b u t otherwise a r b i t r a r y f u n c t i o n R[A) o f the A Example: P

J[aPAlJ

+

.

AAZ lJ

+ KH

4

lJ

,

...I .

Ihe o n l y way t h l s comes i n t o t h e above d e r i v a t i o n i s through t h e behaviour o f t h i s whole term under gauge t r a n s f o r m a t i o n s . Suppose one has i n t h e Lagrangian [11.151 a l s o t h e c o u p l i n g

JR[A 1 1.1

.

Furthermore, under a gauge t r a n s f o r m a t i o n

G. 't HOOFT and M. VELTMAN

+ A

AIJ

lJ

+ a h ,

IJ

one has

c o n t a i n s t h e A depeni s t h e p a r t indgpenQe?t of t h e A and IJ dent p a r t s . Both r and p may contaffn d e r i v a t i v e s . Now, p e r f o r m i n g t h e B e l l - T r e i m a n t r a n s f o r m a t i o n w i t h t h e Bf i e l d one o b t a i n s t h e v e r t i c e s :

--

-x

from

~ r t,l

from

J&I

,

where t h e double l i n e denotes a c o l l e c t i o n o f p o s s i b i l i t i e s , inClUQing a t l e a s t one photon l i n e . The whole d e r i v a t i o n can be c a r r i e d through unchanged, and g i v e n many sources Ja coupled t o many f i e l d f u n c t i o n s Ra we g e t :

L

where r and p a r e d e f i n e d by t h e behaviour o f t h e f u n c t i o n s Ra under g8uge t r h s f o r m a t i o n s

The above i d e n t i t i e s a r e t h e Slavnov-Taylor i d e n t i t i e s . 11.5

Equivalence o f Gauges

From t h e precedlng s e c t i o n s we o b t a i n t h e f o l l o w i n g presc r i p t i o n for h a n d l l n g a t h e o r y w i t h gauge i n v a r i a n c e . F i r s t choose a non-gauge-invariant f u n c t i o n C and add -1/2C2 t o t h e Lagrangian. Next c o n s i d e r t h e p r o p e r t i e s o f C under i n f i n i t e s i m a l gauge transformations, f o r i n s t a n c e

115

DIAGRAMMAR

A ghost p a r t must a l s o be added t o t h e L a g r a n g i a n .

w i t h t h e p r e s c r i p t i o n of p r o v i d i n g a f a c t o r -1 f o r every c l o s e d 0 loop. C i e a r l y t h e choice of C i s m i l i t e d by t h e f a c t t h a t t h e o p e r a t o r m, d e f i n i n g t h e ghost propagator, must have an i n v e r s e . Moreover C must t o g e t h e r w l t h 1: d e f i n e n o n - s i n g u l a r A propagat o r s , b u t t h i s i s automatic if t""breaks t h e gauge i n v a r i a n c e . Suppose we had taken a s l l g h t l y d i f f e r e n t C

Now under a gauge t r a n s f o r m e t i o n C' -c C'

+

[k

+

i)A

+

~ [ r;)A +

,

where r and p are, r e s p e c t i v e l y . t h e f i e l d - i n d e p e n d e n t and f i e l d dependent p a r t s r e s u l t i n g From a gauge t r a n s f o r m a t l p n o f R. For example, ifR = A2 then R + R + 2A a A and we nave r = 0 and p = 2AIJau* IJ U P The ghost Lagrangian must be changed a c c o r d i n g l y . and we g e t

1' = J:

inv

-1 [c 2

+ ERI'

+

*

0 [ii +

i

+ EF.

+

(11.221

x'

We now prove t h a t t o f i r s t o r d e r i n E t h e s - m a t r i x generated by equals t h e s - m a t r i x generated by 1. l h i s is c l e a r l y s u f f i c i e n t t o have equivalence o f any two gauges t h a t can be connected by a s e r i e s of i n f i n l t e s i m a l steps.

L e t us compare t h e Green's f u n c t i o n s o f & * w l t h those o f e. The d i f f e r e n c e is g i v e n by Green's f u n c t i o n s c o n t a i n i n g one E-vert e x . From Eq. [11.221 these v e r t i c e s are:

116

G. 't HOOFT ond M. VELTMAN

The d i f f e r e n c e between an f same e x t e r n a l l e g s i s then:

d and

Green's f u n c t i o n w i t h t h e

We e x h i b i t e d e x p l i c i t l y t h e minus s i g n associated w i t h t h e ghost loops. upenlng up t h e t o p v e r t e x t h i s d i f f e r e n c e i s :

I f now a l l t h e o r i g i n a l sources are gauge i n v a r i e n t , namely a J,, = 0. we see t h a t t h i s d i f f e r e n c e i s zero as a consequence of !he slavnov-Taylor i d e n t i t i e s , Eq. 111.21). The diagrams i n Eq. (11.211, where t h e ghost l i n e i s attached t o such curre2ts. a,, g i v e ?o c o n t r i b u t i o ? , since f o r these currents we have r,, land r v J v = 01 and p 0. As t h e S-matrix i s defined on t h e b e s i s of gauge-invariant sources, we see t h a t t h e S-matrix i s i n v a r i a n t under a change o f gauge as given above.

-

-

11.6

I n c l u s i o n o f Electrons: Wave-Function Renormalizetion

The preceding discussion has been c a r r i e d out i n such 8 way t h a t t h e i n c l u s i o n of e l e c t r o n s changes p r a c t i c a l l y nothing. The main d i f f e r e n c e 1s t h a t we now must i n t r o d u c e sources t h a t e m i t or absorb electrons and such sources a r e n o t gauge i n v a r i a n t . This complicates somewhat t h e discussion o f t h e equivalence o f gauges. Thus conslder e l e c t r o n 8ource terms

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DIAGRAMMAR

u n d e r gauge t r a n s f o r m a t i o n s ( A

P

-P

A

P

+

a

U

A1

Let u s c o n s i d e r t h e d i f f e r e n c e of t h e Green's f u n c t i o n s of two d i f f e r e n t gauges i n t h e p r e s e n c e of an electron s o u r c e . For simp l i c i t y we w i l l o n l y draw one o f them e x p l i c i t l y . The SlavnovT a y l o r i d e n t i t y is:

J.

I h e f i r s t t h r e e terms t h e d i f f e r e n c e of t h e o b j e c t s o b t a i n e d when Green's f u n c t i o n (see

a r e p r e c i s e l y t h o s e found i n c o n s i d e r i n g Green's f u n c t i o n s , or more p r e c i s e l y af unfolding a vertex i n t h e d i f f e r e n c e o f t h e preceding s e c t i o n l .

F o l d i n g back t h e C and R s o u r c e t o o b t a i n a g a i n t h e t r u e d i f f e r e n c e of t h e Green's f u n c t i o n and u s i n g t h e Slavnov-Taylor i d e n t i t y we g e t :

I n g e n e r a l , s u c h a t y p e of dlagram w i l l have no p o l e a s one g o e s t o t h e e l e c t r o n m a s s - s h e l l . Thus p a s s i n g t o t h e S-matrix t h e c o n t r i b u t i o n of most diagrams will d i s a p p e a r . B u t i n c l u d e d i n t h e above s e t a r e also dlagrams o f t h e t y p e :

118

G. ‘t HOOFT and M. VELTMAN

w h i c h do have a pole. However, we m u s t not f o r g e t t h a t t h e Sm a t r i x d e f i n i t i o n I s based on t h e two-point f u n c t i o n . T h e s o u r c e s m u s t be s u c h t h e t t h e r e s i d u e of t h e two-point f u n c t i o n I n c l u d i n g t h e s o u r c e s i s one.

Consider t h e two-point f u n c t i o n f o r t h e e l e c t r o n :

A l s o t h i s f u n c t i o n w i l l change, and f o r any of t h e two s o u r c e s we w i l l have p r e c i s e l y t h e same change a s above:

I n accordance w i t h o u r d e f i n i t i o n of t h e S-matrix we m u s t r e d e f i n e o u r s o u r c e s s u c h t h a t t h e r e s i d u e of t h e two-point f u n c t i o n remains one8 t h a t I s

where 6

-

-&

value a t t h e poie o f

It i s seen t h e t i n c l u d i n g t h e r e d e f i n i t i o n of t h e s o u r c e s t h e S-matrix I s unchanged under 5 change of gauge.

I n conventional language. t h e above shows t h a t t h e e l e c t r o n wave-function r e n o r m a l i z a t i o n i s gauge dependent, b u t t h a t i s of no consequence f o r t h e S-matrix.

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DIAGRAMMAR

COPlBINATORIAL METHODS

12.

There ere essentially three levels of sophistication with which one can do combinatorics. On the first level one simply uses identities of Vertices that are a consequence o f gauge invariance. Example: electrons interacting with photons. The pert of the Lagrangian containing electrons is

JI

+ $ +

lee$ ,

-

JI

-

+

tp - leB$

,

+ A

P

+ a a P

Of course this LagrangIan remains invariant under th-se ransformations, but we want to understand this in terms o f diagrams. Une has, not cancelling anything, as extra terms

- i e [ Z n l 411-iyp

4

ie[2n1 i ( i y q

+

+

ml ,

ml ,

How does the cancellation manifest itself? In the Lagrangian one must write

120

G. 't HOOFT and M. VELTMAN

t o see i t . Here we take t h e f i r s t v e r t e x and w r i t e

Then one sees e x p l i c i t l y how t h e t h r e e v e r t i c e s together g i v e zero

4 -1el2nI i{iyq + l y k

+

m

-

iyq

- m - iyk)

-

0

-

and t h i s remains t r u e i f we replace m everywhere by m i c . Only then can one say t h a t t h e v e r t i c e s c o n t a i n f a c t o r s which a r e i n v e r s e propagators: now we know f o r sure t h a t t h e f o l l o w i n g Green's f u n c t i o n i d e n t i t y holds [ i n lowest order]:

where t h e short double l i n e i n d i c a t e s t h e i n v e r t e d propagator i n c l u d i n g i c . E x p l o i t i n g t h i s f a c t we obtain:

which i s a Ward i d e n t i t y given before. The above makes c l e a r how tne very f i r s t l e v e l combinatorics

is t h e b u i l d i n g block f o r t h e second-level combinatorics. This second-level combinatorics uses t h e f a c t t h a t a13 terms cancel when one has gauge Invariance. But t h i s f a c t must be v e r i f i e d e x p l i c i t l y by means o f f i r s t - l e v e l combinatorics t o a s c e r t a i n t h a t t h e i E p r e s c r i p t i o n i s conslstent w i t h t h e gauge invariance. I t is j u s t by such a type o f reasoning t n a t t h e ic i n t h e ghost propagator is f i x e d , as shown before. The t h i r d l e v e l o f comblnatorics is t h a t when one uses nonl o c a l canonlcal transformations. These can be used t o d e r i v e the Slavnov-Taylor i d e n t i t i e s d i r e c t l y , as was done by Slevnov. One p r e s c r i p t i o n : our procedure must be very c a r e f u l about t h e IE whereby t h i s i d e n t i t y was derived by means of f i r s t - and second€ rescription f o r l e v e l combinatorics shows t h a t t h e usual -Ip t h e ghost propagator i s t h e c o r r e c t one.

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DIAGRAMMAR

13.

REGULARIZATION AND RENORMALIZATION

13.1

General Remarks

As noted before, the regulation scheme beginning is not gauge invariant. We need a can also be used in case of gauge theories, Hbelian. An elegant method is the dimension which we Will discuss now.

introduced in the better scheme that Abelian or nonregularization scheme

A good regularization scheme must be such values of some parameters [the masses A in the methodl the theory is finite and well defined. ry obtains in a certain limit (masses oecoming one requires that quantities that were already regularization was introduced remain unchanged

that for certain unitary cut-off The physical theovery big), and finite before in this limit.

In order to obtain a finite physical theory it will be necessary to introduce counter-terms in the Lagrangian. H theory is said to be renormalitable if by addition of a finite number Of kinds of counterterms a finite physical theory results. This physical theory must of course not only be finite, but also unitary, causal, etc. With regard to a regularization and a renormallzation procedure for B gauge theory, some problems arise which are peculiar to this kind o f theory. As demonstrated before, the Ward identities (or more generally the Slavnov-Taylor identitiesl must hold, because they play a crucial role in proving unitarlty. Since regularization implies the introduction of new rules for diagrams. and because new vertices, corresponding to the renormalization countertens are added, it becomes problematic as to whether our final renormalized theory eatisfies Ward identities. A s far as the regularization procedure is concerned, a first step is to provide a scheme in which the Ward identities are satisfied for any value of the regularizetion parameters. This goal is obtained in the dlmensional regularization scheme. It must be stressed, however, that this does not guarantee that the counterterms satisfy Ward identitles and indeed, in general, one has considereble difficulty in proving ward Identities for the renormallzed theory. The point is this: in the unrenormalized theory there exists Ward identities, and an invariant cutoff procedure guarantees that the countertens satisfy certain relations. From these reletions one can derive new Ward identities satisfied by the renormalized theory. However, these Ward Identities turn out to be different: one may speak of renormalized Ward identities. That is, one can prove that the renormalized theory has a certain symmetry structure [giving rise to certain Ward identitiesl, but one has to show also that this symmetry is the same as that of the unrenormallzed theory. Let us remark once

122

G. 't HOOFT and M. VELTMAN

more t h a t Ward i d e n t i t i e s a r e needed f o r t h e renormalized t h e o r y because t h e y a r e needed i n p r o v i n g u n i t a r i t y . Indeed, Ward i d e n t i t i e s have n o t h i n g t o do w i t h r e n o r m a l i z a o i l i t y b u t e v e r y t h i n g t o do w i t h u n i t a r i t y . Another problem t o be considered is t h e f o l l o w i n g one. I f one admits t h e p o s s i b i l i t y t h a t t h e renormalized s y m e t r y i s d i f f e r e n t f r o m t h e unrenormalized one, t h e n f o r m a l l y t h e f o l l o wing can happen. I f one c a r r i e s tnrough a r e n o r m a l i z a t i o n program, one must f i r s t make a cholce o f gauge i n t h e unrenormalized theor y . Perhaps t h e n t h e symmetry o f t h e renormalized Lagrangian depends on t h e i n i t i a l choice o f gauge. Given a gauge t h e o r y one must show e x p l i c i t l y t h a t t h i s i s n o t t h e case, and t h a t t h e v a r i o u s renormalized Lagrangians b e l o n g i n g t o d i f f e r e n t gauge choices i n t h e unrenormalized t h e o r y a r e r e l a t e d by a change o f gauge w i t h r e s p e c t t o t h e renormalized symmetry. I n quantum electrodynamics t h l s problem i s s t a t e d as t h e gauge independence o f t h e renormalized theory. 13.2

Dimensional R e g u l a r i z a t i o n Method: One-Loop Diagrams

I n t h e dimensional r e g u l a r i z a t i o n scheme a parameter n i s i n t r o d u c e d t h a t i n some sense can be v i s u a l i z e d as t h e dimension 4 a f i n i t e theory r e s u l t s 1 the physical o f space time. F o r n t h e o r y o b t a i n s i n t h e l i m i t n = 4. As a f i r s t step, we d e f i n e t h e procedure f o r one-loop diagrams. The example we w i l l t r e a t makes c l e a r e x p l i c i t l y t h a t t h e dimensional method I n no way depends on t h e use o f Feynman parameters. A c t u a l l y , s i n c e f o r two or more c l o s e d loops u l t r a v i o l e t d i v e r g e n c i e s may a l s o be t r a n s f e r r e d f r o m t h e momentum i n t e g r a t i o n s t o t h e Feynman parameter i n t e g r a t i o n s t h e use of Feynman paremeters i n connection w i t h t h e dimensional r e g u l a r i z a t i o n scheme must be avoided. o r a t l e a s t be done v e r y j u d i c i o u s l y . The procedure for m u l t i l o o p diegrams w i l l be defined i n subsequent subsections. Consider a s e l f - e n e r g y diagram w i t h two s c a l a r i n t e r m e d i a t e p a r t i c l e s i n n dimensions:

I n t h e i n t e g r a n d t h e l o o p momentum p i s an n component v e c t o r .

123

DIAGRAMMAR

T h i s expression makes sense I n one-, two-, and three-dimensional space1 i n f o u r dimensions t h e i n t e g r a l i s l o g a r i t h m i c a l l y d i v e r gent. To e v a l u a t e t h i s i n t e g r a l we can go t o t h e k r e s t - f r a m e tk = 0, 0, 0, iu1. Next we can i n t r o d u c e p o l a r c o o r d i n a t e s i n t h e remaining space dimensions m

W

I

In

w

dPo

n-2

0

-OD

n

2n dw

I0 de2

dol 0

n s i n El2

2 dEly s i n

e3

0

.,.

1 x

2

1-Po

w2

+

+

m2

- i ~ J [ - ( p ~+

p12 + u2 + M2

- ic]

(13.2) Here w is t h e l e n g t h of t h e v e c t o r p i n t h e n - 1 dimensional subspace. The i n t e g r a n d has no dependence on t h e angles 8 1' and one can i n t e g r a t e u s i n g 'n-2'

'."

-1 1

(13.31

These and Other u s e f u l formulae w i l l be g i v e n i n Appendix 8 . The r e s u l t is

1 (-p20

+

u2 + m 2 j [ - ( p o

+

p12 + w2 + F I ~ I

1

.

113.41

T h i s I n t e g r a l makes sense a l s o f o r non-integer, I n f a c t also f o r complex n. We can use t h i s e q u a t i o n t o define I n i n t h e r e g i o n where t h e i n t e g r a l e x i s t s . And o u t s i a e t h a t r e g i o n we d e f i n e I n as t h e a n a l y t i c c o n t i n u a t i o n i n n o f t h i s expression.

-

A p p a r e n t l y t h i s expression becomes meaningless for n 5 1. f o r t h e n t h e w i n t e g r a l d i v e r g e s near w 0. or f o r ns 4 due t o t h e u l t r e v i o l e r behaviour. Tne lower l i m i t divergence i s n o t v e r y serious and m a i n l y a consequence o f our procedure. A c t u a l l y f o r n + 1 n o t o n l y does t h e i n t e g r a l diverge, b u t a l s o t h e r

124

G. 't HOOFT and M. VELTMAN

f u n c t i o n i n t h e denomlnator, so t h a t 11 f r o m Eq. t13.41 1 s an undetermined f o r m -/-. L e t US f i r s t f i x n t o be i n t h e r e g i o n where t h e above expression e x l s t s , for i n s t a n c e 1.5 < n < 1.75. Next we p e r f o r m a p a r t i a l i n t e g r a t i o n w i t h r e s p e c t t o w2

dw w

n-2

=

-21

2

dw

2 [n-31/2

[w 1

1 dw2

I -

2

2

2 (n-11/2 2 1w I " - I d @ d

.

For n i n t h e g i v e n domain t h e surface terms a r e zero. Using zrlz) = I'Iz + 11 and r e p e a t i n g t h i s o p e r a t i o n A t i m e s we o b t a i n

We have d e r i v e d t h i s e q u a t i o n f o r 1 < n < 4. However i t 1s meaningful a l s o f o r 1 - 2X < n < 4. The u l t r a v i o l e t behavlour is unchanged, b u t t h e divergence near w = 0 i s seen t o c a n c e l e g i a n s t t h e p o l e o f t h e 'l f u n c t i o n , Note t h a t

tnus f o r

z

-+

0 t h i s behaves as l / z .

The l a s t equation I s a n a l y t l c i n n f o r 1 - 2A < n < 4. Since i t c o i n c i d e s w i t h t h e o r i g i n a l I n for 1 < n < 4 i t must be equal t o t h e a n a l y t i c COntinUetiOn of I n o u t s i d e 1 < n < 4. C l e a r l y , we now have an e x p l i c i t sxpressron f o r I n f o r a r b i t r a r i l y s m a l l values of n ( t a k i n g X s u f f i c i e n t l y l a r g e ) . It I s e q u a l l y obvious t h a t i f t h e o r i g i n a l expression which we s t a r t e d from had been convergent, then i t s v a l u e would have been equal t o In f o r n 4 4. Moreover, c u t t i n g equations can be d e r i v e d u s i n g o n l y t i m e and energy components, and as long as t h e l a s t g i v e n expression for I n e x i s t s , and po and w i n t e g r a t i o n s can be exchanged C1.e. f o r n < 4 1 these c u t t i n g equations can be established. L e t us now see What happens f o r n >_ 4. Again we w i l l use t h e method o f p a r t i a l i n t e g r a t i o n s t o p e r f o r m t h e a n a l y t i c 0. F i r s t f i x n i n t h e c o n t i n u a t i o n . F o r s i m p l i c i t y we s e t X r e g i o n 1 < n < 4. Next we i n s e r t

-

9. dw

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DIAGRAMMAR

.

Next we p e r f o r m p a r t i a l i n t e g r a t i o n w i t h r e s p e c t t o po and Again i n t h e g i v e n domain t h e s u r f a c e terms a r e zero. F u r t h e r

x

E...)

=

2p0p

-2m2

x

{...I

+

2p2

-

2M

.

I n s e r t i n g t h i s I n t h e r i g h t - h a n d s i d e of Eq. In=-I -n + 6 -1; 2 n

(13.41

gives

113.6)

or

with

The i n t e g r a l

I; is convergent f o r

1 < n < 5.

126

Now t n e I g i v e n n

G. ‘t HOOFT and M. VELTMAN

b e f o r e and t h i s expression a r e equal f o r 1 < n < 41 s i n c e t h e l a s t expression I s a n a l y t i c f o r n e 5 w i t h a simple p o l e a t n = 4, i t must be equal t o t h e a n a l y t i c c o n t i n u a t i o n o f I n

.

The above procedure may be repeated i n d e f i n i t e l y . . t h a t In i s o f t h e form

One f i n d s

113.81

... .

where t h e f u n c t i o n f 1s well-behaved f o r a r b i t r a r i l y l a r g e n, The I’ f u n c t i o n shows simple p o l e s a t n = 4, 6, 8 , We see now why t h e l i m i t n + 4 cannot be taken: t h e r e i s a p o l e f o r n = 4. I t I s v e r y t e m p t i n g t o say t h a t one must i n t r o duce a counterterrn e q u a l t o minus t h e p o l e and i t s residue. But i f u n i t a r i t y i s t o be maintained t h i s counterterm may n o t have an imaginary p a r t , 1.e. i t must be a p o l y n o m i a l i n u, rn and m. Thus we must f i n d t h e form of t h e r e s i d u e o f t h e pole. I t w i l l t u r n o u t t o be of t h e r e q u i r e d form. l o show t h a t i t i s o f t h e p o l y n o m i a l form i n t h e g e n e r a l case o f many loops n e c e s s i t a t e s use o f t h e c u t t i n g equations. F o r t h e one-loop case a t hand we w i l l simply compute In s u i n g Feynman parameters. One has 1

In = dx 0

I

drip

1 [p2 + 2pkx + K2x

+

m2x

+

rn2 (1

-

XII

2 ’

S h i f t i n g i n t e g r a t i o n v a r i a b l e s (p’ = p + k x l , making t h e Wick r o t a t i o n , and i n t r o d u c i n g n-dimensional p o l a r coordinates, one computes

i n t h e f o r m o f Eq. (13.81. F o r I n t h i s way we have e x p l i c i t l y In n = 4, 6, 8, t h e i n t e g r a n d i s a simple polynomial, f o r n = 4 t h e i n t e g r a l g i v e s s i m p l y 1. Thus t h e p o l e t e r m i s

...,

PPLInl =

in2 2 -1 4 - n

(13.91

’

where PP stands f o r “ p o l e p a r t ” . Using t h e e q u a t i o n

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DIAGRAMMAR

one may compute

+

2 k xi1

-

XI] + C.

n e r e C is E constant r e l a t e d t 0 t h e n dependence Other t h a n I n t h e exponent o f t h e denominator, COntaining f o r i n s t a n c e I n n. f r o m vn/2. I n g e n e r a l C i s a p o l y n o m i a l j u s t as t h e p o l e p a r t o f I n . Since we could have taken as our s t a r t i n g p o i n t an I n m u l t i p l i e d by bn-4, where b I s any constant, we see t h a t C I s undetermined. T h i s I s t h e a r b i t r a r i n e s s t h a t always occurs i n connection w i t h r e n o r m a l i z a t i o n . We now must do some work t h a t w i l l f a c i l i t a t e t h e t r e a t m e n t o f t h e m u l t i l o o p case. L e t us consider t h e c u t t i n g equation. One has :

#

+ 2 2 = f 1k J = - I Z n I

/

dnp6[-po)a(p2 + m z l x

I n t h e r e s t frame

OD

x

d p o 6 L - p o l s [ - p ~ + w2 + m2 J 8 ( p o + p16(-2p0p + M2

-

m2

-W

113.10)

2

The f u n c t i o n f - ( k I can be o b t a i n e d by changing t h e s i g n s o f t h e arguments o f t h e 8 f u n c t i o n s .

I n c o o r d i n a t e space one has

The F o u r i e r t r a n s f o r m o f t h i s statement I s

128

-

p2 I .

G. 't HOOFT ond M. VELTMAN

+

-

1 2=i

OD

f-tk

-

TI

.

T can be considered as an n v e c t o r w i t h a l l components zero except t h e energy component. A 8 long as the w i n t e g r a l is s u f f i -

c i e n t l y convergent Li.e. n < 41 one may exchange f r e e l y the w end i n t e g r a t i o n . Doing t h e po and T I n t e g r a t i o n s one obtains o f courae t h e o l d r e s u l t f o r 1 . , which is n o t very i n t e r e s t i n g . L e t us t h e r e f o r e leave t h e o I n t e g r a t i o n i n f r o n t o f t h e w i n t e g r a l and compute the po i n t e g r a l . One f i n d s f o r the po i n t e g r a l i n f +

T

where

/(u2

-

-

m2

m212

-

4m2M 2

K'

The 0 f u n c t i o n expresses t h e f a c t t h a t p must be p o s i t i v e and ~ be p o s i t i v e I n order f o r t h e w i n t e g r a t i o n furthermore t h a t I C must t o g i v e a non-zero r e s u l t . For f - one obtalns t h e same. except f o r t h e change e I p M - m l + 0(-p - M - m l .

-

Also t h e w i n t e g r a t i o n can be done, and I s o f course independent of X ( t h i s f o l l o w s as ueual by considering t h e i n t e g r a l f o r 1 < n < 4 and doing t h e necessary p a r t i a l i n t e g r a t i o n s ) . For X 0 one f i n d s

-

obtains as given above. We must The complete f u n c t i o n f ( k l = In study

where

129

DIAGRAMMAR

4T'

Since

(13.121 where u i s p o s i t i v e and f i n i t e a t t h r e s h o l d T + m + m, t h i s i n t e g r a l i s n o t w e l l d e f i n e d f o r n 5 1. T h i s i s how t h e divergence a t w = U, p r e v i o u s l y found i n Eq. (13.41, m a n i f e s t s i t s e l f here. But t h i s i s again no problem, and r e a l l y due t o t h e f a c t t h a t o u r d e r i v a t i o n i s c o r r e c t o n l y f o r n > 1. We w i l l come back t o t h a t below. And t h e r e 1s no t r o u b l e i n c o n s t r u c t i n g +he a n a l y t i c c o n t i n u a t i o n t o s m a l l e r values o f n. l h i s can be done by p e r f o r m i n g p a r t i a l i n t e g r a t i o n s w i t h r e s p e c t t o t h e f a c t o r IT PI mJ i n Eq. (13.121.

-

-

For n

4, however. t h e I n t e g r a l d i v e r g e s f o r l a r g e values T h i s can be handled as f o l l o w s . The expression 113.121 f o r g L l ~ l i s n o t h i n g b u t a d i s p e r s i o n r e l a t i o n , and we may p e r form a s u b t r a c t i o n of

1.1.

before, by i n s e r t i n g d.r/dT i t may oe shown t h a t gIO1 has a p o l e a t n = 4. The remainder of g', however. i s p e r f e c t l y w e l l behaved f o r n < 5. Again we see t h a t t h e p o l e terms ( i n n = 41 have t h e p r o p e r polynomlal behaviour: they a r e l i k e s u b t r a c t i o n s I n a dispersion relation.

AS

We must now c l e a r up a f i n a l p o i n t , namely t h e q u e s t l o n o f i n t e g r a l near t h r e s h o l d . Consider as an t h e behaviour o f t h e example t h e f u n c t i o n

I n t h e complex T plane we have a p o l e a t T = 0 and f o r non-integer a c u t a l o n g t h e r e a l a x i s f r o m T = 1 t o T = m. M u l t i p l y t h i s

130

G. 't HOOFT and M. VELTMAN

f u n c t i o n w i t h [T - l~ around t h e p o i n t T = 2nifll~I

-

-1 and i n t e g r a t e o v e r a small c i r c l e One o b t a i n s

is)

u.

.

On t h e o t h e r hand t h e contour may be e n l a r g e d ; we.get

f(u1

=

1 2ni

LT

-

dT lJ

-

1E

(*

T

4

+

c o n t r i b u t i o n of t h e origin,

where C i s as i n t h e f o l l o w i n g diagram:

-

T h e c l r c l e a t i n f i n i t y may be ignored provided a < 4. S i n c e now t h e i n t e g r a n d has a q u i t e s i n g u l a r behaviour a t T 1. t h i s

p o i n t m u s t be t r e a t e d c a r e f u l l y . The c o n t o u r may be d i v i d e d i n t o a c o n t r l b u t l o n of a s m a l l c i r c l e w i t h r a d l u s E around t h i s p o i n t , end t h e rest.

I n c o n s i d e r i n g t h e i n t e g r a l over t h e c i r c l e , T may be s e t t o one except i n t h e f a c t o r (T - 11". Moreover, we may i n t r o d u c e t h e change o f v a r i a b l e T = T - 1. Writing T = cei6: r

On t h e o t h e r hand, t n e c o n t r i b u t i o n of t h e two c o n t o u r l i n e s from t h e c i r c l e t o some p o i n t b above and below t h e cut c o n t a i n s a part

131

DIAGRAMMAR

The i n t e g r a n d is t h e jump across t h e cut. T h i s c a n be i n t e g r a t e d t o give

T o g e t h e r w i t h t h e c o n t r i b u t i o n from t h e c i r c l e

1 b a + l Ie-ira - e l n u a + 1

]

T h i s i s i n d e p e n d e n t of E , and t h e l i m i t E + 0 c a n b e t a k e n . Note t h a t t h e result i s -2ni i n t h e l i m i t a + -1. as s h o u l d b e f o r a clockwise contour. I n the e x p r e s s i o n for f * i p ) (and f - t v ) ) we have n o t b o t h e r e d a b o u t t h e p r e c i s e b e h a v i o u r a t t h e s t a r t of t h e cut. T h i s is i n p r i n c i p l e a c c o u n t e d f o r by t h e 8 f u n c t i o n . T h i s 11 f u n c t i o n gives o n l y t h e c o n t r i b u t i o n a l o n g t h e c u t 1 t h e small c i r c l e h a s b e e n i g n o r e d . T h i s i s a l l o w e d o n l y i f n > 1 ( c o r r e s p o n d i n g t o a > -11. Otherwise one m u s t c a r e f u l l y s p e c i f y what happens a t t h r e s h o l d i n f + , f - , a n d i n t h e s u b s e q u e n t i n t e g r a l s over T. 2 To see i n d e t a i l how t h i s g o e s c o n s i d e r a f u n c t i o n o f T h a v i n g a c u t from T~ = t o T~ = a, b u t o t h e r w i s e a n a l y t i c land going s u f f i c i e n t l y fast t o z e r o a t i n f i n i t y l . I n t h e T p l a n e t h e f u n c t i o n h a s c u t s f r o m -a t o -00 a n d +a t o +a, T h e dispersion r e l a t i o n leads i n t h e T plane t o t h e following contour:

C o n s i d e r t h e r i g h t - h a n d s i d e c o n t o u r . The c u t starts a t t h e p o i n t = a. W e may w r i t e

T

w h e r e C& s t a n d s f o r t h e c i r c l e a t a w i t h r a d i u s 6. a n d t h e jump o v e r t h e c u t , 1.e.

132

P ~ 2TI

is

G. ‘t HOOF1and M. VELTMAN 1

I

+t?’J

= lim

6 ’+o

i

ftr’

+

16’1

-

f[r2

-

16’)

-

i

.

T h i s i s t h e p r e c i s e equivalent of our f’. It I s e s s e n t i a l t o first take t h e l i m i t 6’ 0 before t h e l i m i t 6 = 0. L e t u s now return t o t h e question of s u b t r a c t i o n s . It i s now p o s s i b l e t o t u r n t h e reasoning around. We know t h a t f o r In and i t s Fourier transforms t h e following p r o p e r t i e s hold:

...,

11 I n l k l has poles f o r n = 4, 6 , and tne <[XI pole f o r n = 4. 111 I n [ X ) = OfxoJf~Lxl + 81-XoJf,lXl f o r < 4. 1111 I ~ ~ , X fI f i t x l are Lorentz-invariant.

+

have no

fi

The f u n c t i o n s f n and are non-singular f o r n = 4: they a r e t h e cut diagrams and t h e s e are not aivergent f o r n = 4. Concerning point 1 1 1 1 , the d e r i v a t i o n of t h e c u t t i n g equation r e q u i r e s n < 4. O f couree It i s p o s s i b l e t o d e f i n e t h i n g s by a n a l y t i c continuation, but our d i s p e r s i o n - l i k e r e l a t i o n s as exhlbited above w i l l hold only f o r n < 4. From (11) and l i i i J i t follows immediately t h a t t h e pole i n n = 4 of I n t x l can a t most be a 6 function. T h e p r e c i s e reasoning i e as follows. Applying a Lorentz tranaformation t o [ i l l and i n s i s t i n g on Lorentz invariance we fl n d

for

x

0

10.

xto.

Consider nDW+the a n a l y t i c continuation of t i l l t o l a r g e r n values. If f and f - have no pole f o r n = 4 , t h e n t h e only way 0, 2 = 0 . t h a t I n t x l can have a pole f o r n = 4 is i n t h e point xo Remember now t h a t we nave shown t h a t t h e Fourier transform of I n 1s O f t h e form

x f i n i t e function f o r n n - 4

< 5

1.9. t h e Fourier transform of In o x l s t s f o r n < 5. Since t h e Fourier transform of t h e pole p a r t i s a function w h i c h i s nonzero only f o r xo = x = 0, t h e only p o s s i b i l i t y i s t h a t t h i s pole p a r t i s a polynomial i n t h e e x t e r n a l momenta [and t h e varlous masses i n t h e problem), 1.9. i n coordinate space a 6 function and d e r i v a t l v e s of 6 functions.

T h u s we know now t h a t t h e r e s i d u e of I n a t t h e pole 1s a polynomial. O i f f e r e n t i a t i n g I n w i t h r e s p e c t t o t h e e x t e r n a l momentum k. i n t h e region n < 4 w her e we can use a well-defined

133

DIAGRAMMAR

r e p r e s e n t a t l o n , see Eqs. (13.4) and (13.51, we see t h a t t h i s d e r i v a t i v e is f i n i t e f o r n = 4. I t f o l l o w s t h a t t h e p o l e p a r t I s independent o f k, which i s indeed what i s found before, Eq. (13.91. A l s o , t h i s i s a l l we need t o know f o r r e n o r m a l i z a t i o n purposes. 13.3

M u l t i l o o p Diagrams

We must now extend t h e method t o diagrams c o n t a i n i n g a r b i t r a r i l y many loops. T h i s i s q u i t e s t r a i g h t f o r w a r d . F i r s t n o t e t h a t e x t e r n a l momenta land sources) span a t most a f o u r - d i m e n s i o n a l space. We s p l i t n-dimensional space i n t o t h i s f o u r - d i m e n s i o n a l space and t h e r e s t

The & a r e t h e components o f t h e p i i n four-dimensional space. The i n t e g r a n d w i l l depend on t h e s c a l a r p r o d u c t s o f t h e with themselves a n t t h e e x t e r n a l vectors, and furthermore on t h e s c a l a r p r o d u c t s of t h e p i w i t h themselves. Again, t h e i n t e g r a t i o n over those angles t h a t do n o t appear i n t h e i n t e g r a n d may be performed. T h i s i s done as f o l l o w s . Consider t h e I n t e g r a l

..,

The argument o f t h i s I n t e g r a l i s a l r e a d y i n t e g r a t e d over p i + l p k i t h e r e f o r e t h e i n t e g r a l depends o n l y i n t h e s c a l a r p r o d u c t o f P i w i t h PI P i - 1 . These v e c t o r s .span an i-1 dimenslonal space, and we may w r i t e

...

(13.14) Now t h e i n t e g r a n d w i l l no l o n g e r depeEd on t h e d i r e c t i o n o f and I n t r o d u c i n g p o l a r c o o r d i n a t e s I n P space

F,

00

1 4-4-idwi(13.151

0

-

Tne oi r e p r e s e n t t h e l e n g t h s o f t h e v e c t o r s P.i dimensional space t h e v e c t o r s

134

I n t r o d u c e i n k-

G. ‘t HOOFT and M. VELTMAN

w

q1

=

* q2 =

2

0

0 0 b v iou 9 1y

113.161

but note

I13.17) w i t h w k running from

-OD to +w. we have t h e r e f o r e comparing E q s . (13.141 and (13.151 w i t h (13.16) and (13.171

dn-4Pk

’’In- k-41/2

where we used r(1/21 =

n-4-k

G.

Finally note

135

0(w,I

*

DIAGRAMMAR

Since wl,

wz,

B1wkl

-

ere p o s i t i v e we can w r i t e

etc.,

Btdet q l

.

We a r r i v e t h u s a t t h e e q u a t i o n

x

... dkq k [ d e t q) n-4-k B(det q l .

1 dkql

(13.181

Next we have

T h i s e q u a t i o n can be a r r i v e d e t by t e d i o u s work. J u s t w r i t e o u t t h e determinants w i t h t h e h e l p of t h e E symbol w i t h k i n d i c e s . Note t h a t f o r a = 0 t h e e q u a t i o n f o l l o w s t r i v i a l l y beceuse o f E

il...l

E

k

1

1

...

I k

= k:

Also f o r k = 1 the equation i s t r i v i e l . Now t h e i n t e g r a n d is a f u n c t l o n of t h e s c a l a r p r o d u c t s a i J = ( q l , qdJ. L e t us c o n s i d e r t h e o p e r a t i o n on such e f u n c t i o n o f the operetor

det

a . aq

One has

136

G. 't HOOF1and M. VELTMAN 1

k

2 tdet ql det w i t h these equations one can construct t h e a n a l y t i c c o n t i n u a t i o n of t h e above equation t o small values o f n

I dq l...dq

k

[det q )

1

D

ln-4J...ln-3-kl

n-4-k=

$

1

I dql" *dqk (n-4) (17-51..

dq l...dq

k

.

(n-3-k)

X

kdet q l

2 e Performing t h i s operation X times gives

x

I dkql

...

dkqKO[det q ) [ d e t q Jn-4-k+2A [det

-7 i a iJ (13.191

Rather than worrying whether t h i s d e r i v a t i o n i s c o r r e c t o r not we simply take t h i s equation as the d e f i n i t i o n of t h e k-loop diagrams f o r non-integer n. I t 1s n o t very d i f f i c u l t t o v e r i f y t h a t f o r f i n i t e diagrams the result coincides w i t h the above equation f o r n = 4 l s e e below). I n doing such work i t i s o f t e n advantageous t o go back t o t h e s p e c i a l coordinate system w i t h which we started1 the expression i n terms o f t h e ql i s mainly useful f o r invariance considerations. F o r instance s h i f t s o f i n t e g r a t i o n v a r i a b l e s such as

137

DIAGRAMMAR

l e a v e b o t h dkql and dkq2 a s well as d e t q unchanged as long as t h e i n t e g r a l s converge lwhich t h e y do f o r s u f f i c i e n t l y small n ) . Equation (13.1YI i s much t o o c o m p l l c a t e d f o r p r a c t i c a l work. and we w i l l i n s t e a d use t h e n o t a t i o n

A l l t h e above work I s t o show how t h i n g s c a n be d e f i n e d f o r n o n - i n t e g e r n.

Ihe f i n a l s t e p c o n s l s t s of showing t h a t t h e a l g e b r a i c p r o p e r t i e s n e c e s s a r y f o r dlagrammatlc a n a l y s i s hold f o r t h l s i n t h e same way a s t h e y do f o r i n t e g e r n. We have seen a l r e a d y t h a t t h i s i s so f o r s h i f t s of i n t e g r a t i o n v a r i a b l e s . Furthermore, clearly

(always assuming t h a t n i s chosen s u c h t h a t t h e I n t e g r a l s convergel.

N e x t t h e r e i s t h e f o l l o w i n g problem. I t seems t h a t t h e d e f i n i t i o n depends on t h e number o f l o o p s . Then i n c o n s i d e r i n g c u t t i n g e q u a t i o n s one h a s r e l a t l o n s i n v o l v i n g a l l k i n d s of p o s s i b i l i t i e s on both s i d e s of t h e c u t , and i t may seem hard t o u n d e r s t a n d what happens. I h e answer t o t h i s i s t h a t t h e d e f i n i t i o n I n v o l v e s f o r k l o o p s a k-dimensional q-space, b u t n o t h i n g p r e v e n t s u s from u s i n g a l a r g e r s p a c e . T h u s i n any e q u a t i o n one can use f o r k simply t h e l a r g e s t number of c l o s e d l o o p s t h a t o c c u r i n any one term. And t h e r e i s no d i f f i c u l t y i n d e c r e a s i n g t h e k used if t h e number of l o o p s is s m a l l e r . t n a n t h i s k : Simply perform t h e Integ r a t i o n s o v e r t h o s e d i r e c t i o n s t h a t od n o t a p p e a r I n t h e i n t e grand. Let u s now c o n s i d e r t h e l i m i t n -t 4 i n t h e c a s e where t h e o r i g i n a l i n t e g r a l e x i s t s . T h i s means t h a t t h e r e are no d i f f i c u l t i e s w i t h t h e u l t r a v i o l e t Dehaviour. Using Eq. (13.19) w i t h X s u f f i c i e n t l y l a r g e w e may first d o t h e q 1 i n t e g r a t i o n . Using c o o r d i n a t e s as d e s c r i b e d i n t h e b e g i n n i n g one a r r i v e s a t an i n t e g r a l of t h e t y p e as e n c o u n t e r e d i n t h e one-loop c a s e

138

G. 't HOOF1 and M. VELTMAN

Keeping n i n t h e neighbourhood o f f o u r i t is p o s s i b l e t o perform p a r t 1 8 1 i n t e g r a t i o n w i t h r e s p e c t t o w1 o n l y

I dwlo!-3

1

r[T1 t n

00

-

41

+

11

o E

[

-

$]f(u;)

-

2

Next we w r i t e wT-3 = w 1 . w 1 wl(l + E I n w 1 n 4. Since w 1 . d q = 1/2dw2, we f i n d

-

+

O ( E 11, w i t h E =

00

r[?1

tn

-1 41

I n o1

do: +

11

[-

2I +

$]f[w

-

We see t h a t t h e i n t e g r a l reduces t o what one would have w i t h q1 0, p l u s a f i n i t e amount p r o p o r t i o n a l t o n 4. Since q1 was t h e "unphysicaln p a r t o f t h e momentum p i we see t h a t we r e c o v e r I n t h i s way t h e expresslon t h a t we s t a r t e d from. Moreover, if t h e o r i g i n a l expression was f i n i t e , t h e n t h e r e s u l t f o r values of n c l o s e t o f o u r d i f f e r s by f i n i t e terms p r o p o r t i o n a l t o n 4.

-

-

-

4 , t h e n we a r e I n t e r e s t e d I f t h e i n t e g r a l does n o t e x i s t s f o r n i n c o n s t r u c t i n g an a n a l y t i c c o n t i n u a t i o n t o l a r g e r n values. This can be done as f o l l o w s . S e l e c t a number o f c l o s e d loops i n t h e diagram. C a l l t h e loop-momenta associated w i t h these loops p i ps [S loops s e l e c t e d l . Next i n s e r t

...

a(p I f a,

1=s o n which

j - 1 l=l d ( P l ) j

is symbolic f o r

and p e r f o r m p a r t i a l i n t e g r a t i o n s . The r e s u l t is t h e o r i g l n a l l n t e g r e l I p l u s en i n t e g r a l I' t h a t is b e t t e r convergent w i t h

139

DIAGRAMMAR

r e s p e c t t o th8 t o I we g e t

IOOPS

selected. S o l v i n g t h e Equation W i t h r e s p e c t

1 I=-I’ sn - X i n analogy w i t h Eq. (13.61, obtained i n t h e one-loop case. Here ( - A ] i s t h e number o b t a i n e d by c o u n t i n g t h e powers o f t h e momenta p s i n t h e lntegrand. That i s t h e v e r y n i c e t h i n g about pi t h i s method: t h e r e i s a d i r e c t r e l a t i o n between power c o u n t i n g and t h e l o c a t i o n of t h e p o l e s i n t h e complex n plane.

...

-

Here we f i n d a p o l e f o r n = X/s. If4s. X = 2 , 1, 0. we have q u a d r a t i c , l i n e a r , o r l o g a r i t h m i c d i v e r g e n c i e s w i t h r e s p e c t t o the p i ps i n t e g r a t i o n s .

...

I t i s c l e a r t h a t we now can have p o l e s f o r n = X/s, w i t h X and s I n t e g e r s . Diagrams w i t n many c l o s e d loops g i v e many p o l e s 4 2/s, i n t h e complex n plane. l h e f i r s t p o l e would be a t n 4 - l / s , 4 for q u a d r a t i c , etc.. divergent integrals, t h e next I s one [ g e n e r a l l y however two) u n i t s l/s f u r t h e r i n t h e d i r e c t i o n of i n c r e a s i n g n. e t c .

-

13.4

-

The Algebra o f n-Dimensional I n t e g r a l s

I t i s now v e r y i m p o r t a n t t o know how t h e p r e v i o u s l y d i s c u s sed combinatorics s u r v i v e s all t h e d e f i n i t i o n s . I n d o i n g v e c t o r a l g e b r a one manipulates v e c t o r s and Kronecker 6 symbols a c c o r d i n g t o the rules

P P VlJ ‘pv’va

2

= p

=ti

pa

, ,

a n . 61JV

The o n l y p l a c e where t h e dimension comes i n is i n t h e t r a c e o f t h e 6 symbol. T h i s 6 symbol a l s o appears n a t u r a l l y when performing i n t e g r a l s ~f o r i n s t a n c e [see Appendix B l

140

G. ‘t HOOFT and M. VELTMAN

{I’[

a - :)k,,kv

+

F b

-

1 - :)6yvIm

-

k21

1.

J

The i n d i c e s u, v are supposedly c o n t r a c t e d w i t h i n d i c e s p , v o f e x t e r n a l q u a n t i t i e s [such q u a n t i t i e s a r e zero i f t h e v a l u e o f t h e i n d e x i s l a r g e r t h a n f o u r ) or o t h e r i n t e r n a l q u a n t i t i e s . I n d o i n g combinatorics one w i l l c e r t a i n l y meet i d e n t i t i e s o f t h e form

-

6,,vPpPv

2 P

’

Now n o t e

A l l these i n t e g r a l s a r e computed a c c o r d l n g t o t h e p r e v i o u s l y g i v e n r e c i p e s . The i n d i c e s p, v simply s p e c i f y t w o e x t r a dimensions. t h e two-dimensional space spanned by t h e o b j e c t s w i t h which p and v s p e c i f y c o n t r a c t i o n s .

-

C l e a r l y t h e two equations a r e c o n s i s t e n t o n l y i f we use t h e n. rule 6 6 P V uv

-

A l l t h i s can a l s o be rephrased as follows. I f we t a k e t h e n, t h e n t h e a l g e b r a of t h e i n t e g r a l s I s t h e same as r u l e ,&, t h a t o f t h e integrands. The s i t u a t i o n i s somewhat more complicated i f t h e r e a r e fermions and y m a t r i c e s . Now y m a t r i c e s never occur i n f i n a l answers; o n l y t r a c e s occur. The o n l y r e l e v a n t r u l e s a r e

EYP. YV1 = 26,,v

4 (4=

u n i t matrix)

,

The numbers 2 and 4 a r e n o t d i r e c t l y r e l a t e d t o t h e d i m e n s i o n a l i t y o f space-time, and t h e y p l a y no r o l e i n c o m b l n a t o r i a l r e l a t i o n s . Tne 6 must of course be t r e a t e d as i n d i c a t e d above.

uv

141

DIAGRAMMAR

Some c o n s i d e r a t i o n s a r e i n o r d e r now t o e s t a b l i s h t h a t o u r r e g u l a r i z e d diegrams s a t i s f y Ward i d e n t i t i e s f o r any v a l u e of t h e parameter n. The c o m b i n a t o r i a l p r o o f o f Ward i d e n t i t i e s i n volves [IJ v e c t o r a l g e b r a and [ill s h i f t i n g o f i n t e g r a t i o n v a r i a b l e s . We have a l r e a d y shown i n s u b s e c t i o n 13.3 t h a t s h i f t i n g o f i n t e g r a t i o n v a r i a b l e s is a c t u a l l y allowed because o f t h e i n v a r i a n c e o f d e t q i n Eq. (13.191. I t is a l s o easy t o see now t h a t , i n t h e sense d e f l n e d above, t h e v e c t o r a l g e b r a goes through unchanged f o r any n. O i f f i c u l t i e s a r i s e as soon as i n t h e Ward i d e n t i t i e s t h s r e appear q u a n t i t i e s t h a t have t n e d e s i r e d p r o p e r t i e s o n l y i n f o u r dimensional space, l l k e y5 o r t h e completely antisymmetric t e n s o r tzUvpo. Then t h e scheme breaks down, and t h e r e a r e v i o l a t i o n s o f t h e Ward I d e n t i t i e s p r o p o r t i o n a l t o n - 4. I f t h e r e a r e i n f i n i t i e s ( 1 , s . p o l e s f o r n = 41 t h e r e may be f i n i t e v i o l a t i o n s o f t h e Ward i d e n t i t i e s i n t h e l i m i t n = 4. T h i s i s what happens i n t h e case o f t h e famous B e l l - J a c k i w - A a l e r anomalies.

13.5

Renormalization

Ever s i n c e t h e i n v e n t i o n o f r e l a t i v i s t i c quantum electmdynamics, work has been devoted t o t h e problem o f renormalizat i o n . Mainly, t h e r e i s t h e l i n e o f Bogoliubov-Parasiuk-HeppZimmerman and t h e l i n e of Stueckelberg-Petermann-BogoliubovE p s t e i n - G l a s e r . Of c o u r s e , both t r e a t m e n t s have a l o t i n comvon; t h e BPHZ method however seems a t a disadvantage i n t h e s e n s e t h a t t h e r e seem t o be unnecessary c o m p l i c a t i o n s . U n f o r t u n a t e l y t h e s e c o m p l i c a t i o n s are such a s t o i n h i b i t g r e a t l y t h e t r e a t m e n t o f gauge t h e o r i e s . Tne SPBEG method, on t h e o t h e r hand, can be taken over unchanged, and a l s o accomodates n i c e l y w i t h t h e dimensional r e g u l a r i z a t i o n scheme. The fundamental i n g r e d i e n t s a r e u n i t a r i t y and c a u s a l i t y p r e c i s e l y i n t h e form of t h e c u t t i n g equations. The o n l y c o m p l i c a t i o n t h a t remains i n t h e case o f gauge t h e o r i e s is t h e problem of dresses/bare propagators discussed i n S e c t i o n 9. It seems u n l i k e l y t h a t t h i s i s a fundamental d l f f i c u l t y . We w i l l sketch i n a rough way how r e n o r m a l i z a t i o n proceeds i n t h e sPBEG method. F o r more d e t a i l s we r e f e r t o t h e works of these authors.

I n t h e f o r e g o i n g a d e f l n i t i o n f o r diagrams a l s o f o r noni n t e g e r n [ = number o f dimensions) has been given. T n i s d e f i n i t i o n i s such t h a t :

I1 c u t t i n g equations h o l d f o r all n: 5 i i l Ward I d e n t i t i e s h o l d f o r a l l n, p r o v i d e d t h e E t e n s o r and y do n o t p l a y a r o l e i n t h e comDinatorics o f these Ward i d e n -

142

G. ‘t HOOFT and M. VELTMAN

tities;

iii) divergences manifest themselves as p o l e s for n = 4 in t h e complex n plane. I h e problem I s now t o add counterterms t o t h e Lagranglan such t h a t t h e p o l e s cancel out. The e s s e n t i a l p o i n t I s t h a t t h i s ha5 t o be done o r d e r by o r d e r I n p e r t u r b a t i o n theory, and s i n c e t h i s i s p r e c i s e l y t h e o r i g i n o f a l o t of t r o u b l e i n connection ’ w i t h gauge t h e o r i e s we w i l l focus a t t e n t i o n on t h a t p o i n t . c o n s i d e r one-loop self-energy diagrams I n quantum e l e c t r o magnetics. They are:

--

The Feynman r u l e s are: -1yp

1

1 ~ ~ p21 +~ m21

.-----.

-

v -

P

m

+

1

EW41 k2

E I

ti lJV

-

fe ’

They f o l l o w f r o m t h e Lagrangian

The f i r s t diagram g i v e s

Iy n

= -e

2

tr { y ” [ - ~ y [ p

1 dnP

(p2

+

m2

-

+

k)

IEI [ [ p

m1yv(-iyp

+ +

k12

+

m2

+

m))

- IE]

We must work o u t t h e t r a c e u s i n g o n l y { yuy v = 26,,,4 and t r ( y l yl v I = = 461Jv. The technique I s t o reduce a t r a c e o f X m a t r i c e s t o traces o f X 2 metrlces. For f o u r matrices

-

-

tr (y ~y y a yv J B -tr [ yay uyv yB I

143

+i

B61JQ6vB

- ...

a v B v = -tr[y y y y 1

+

DIAGRAMMAR

or u a v 6 tr ( y y y y J = 46tra6v6

-

46

6

+

trv a0

46

6

1.4 av

This gives

Using Feynman parameters and the equations o f Appendix B t h i s integral can be computed giving

-

In-

4ie

2 n/2 fl

r(21

r ( 9 ) 2 x t l

I

dx

[rn2

The q u a d r a t i c divergence ( p o l e a t n (1 n/21r(l n/2I = r [ 2 - n/2l.

-

-

+

-

-

(ktrkv

XJ

k2x11

-

-

k2 6 u v l

2-n/2

XI]

21 nas c a n c e l l e d o u t t h r o u g h

The second diagram g i v e s

From t h e a n t i c o m u t a t i o n r u l e s , and t h e r u l e 6

tr’

= n

Further

y”ypy”

= -y’y’yp

+

~ y =p [ Z

-

nlyp

( i f n = 4 t h i s reduces t o t h e well-known Chisholm r u l e . Such r u l e s a r e n o t v a l i d i n n-dimenslonal space]. l h e i n t e g r a l may now be worked o u t

: and :1 Both I

nave a p o l e a t n = 4. The r e s i d u e s a r e e a s i l y

144

G. 't HOOFT and M. VELTMAN

computed

PP~I:J

PP[I:l

-

8

in2e2

21n2e2

1 tkpkv n - 4

n-4

2 k &,,,,Ia

[4md- l y k l

.

If we i n t r o d u c e i n the Lagrangian (remember t h a t v e r t i c e s and terms i n the Lagrangian d i f f e r by a f a c t o r 12n14i t h e counterterms

z

-

then t h e two-point functions up t o order e w i l l be f r e e of poles f o r n = 4, and one can take the l i m i t n 4. To have a l l one-loop diagrams f i n i t e a c o u n t e r t e n t o cancel vertex divergenc i e s must be introduced.

Performing a similar calculation (simplified very much from the beginning i f only the pole part i s needed, s e e Appendix B) for the vertex diagram leads t o the counterterm 3 2 ie 8n (n

-

$YvJIAv 41

The remarkable f a c t i s t h a t t h e counterterms are gauge i n v a r i a n t by themselves. Tnis would n o t have been so ift h e c o e f f i c i e n t s of h a $ and -iAvWV$ had been d i f f e r e n t . I t w i l l become c l e a r t h a t t h i s is s p e c i a l t o t h i s gaugei i n the case o f a gauge where the ghost loops are non-zero the r e s u l t i s d i f f e r e n t , and t h e counterterms are i n g e n e r a n o t gauge i n v a r i a n t by themselves. There are now two separate questions t o be discussed. I n c l u d i n g t h e counterterms, a l l r e s u l t s up t o a c e r t a i n order are f i n i t e . T h i s order 1s such t h a t one has one closed loop b u t no "countervertex", o r a t r e e diagram i n c l u d i n g a t most one countervertex. An example i s e l e c t r o n - e l e c t r o n s c a t t e r i n g t o order e4 I n amplitude. Some examples of c o n t r i b u t i n g diagrams are:

145

DIAGRAMMAR

T h e c r o s s e s d e n o t e c o u n t e r t e r m s . T h e first and t h i r d diagrams ( a n d t h e second and f o u r t h 1 t o g e t k r are f i n i t e .

The q u e s t i o n i s now w h e t h e r i t i s p o s s i b l e t o make t h e t h e o r y f i n i t e up t o t h e next o r d e r i n e2 by i n t r o d u c i n g f u r t h e r c o u n t e r terms of t h e t y p e shown above. N e x t one may a s k how t o understand i n more d e t a i l t h e gauge s t r u c t u r e of t h e renormalized t h e o r y , 1.e. t h e Lagrangian i n c l u d i n g t h e c o u n t e r t e r m s . Moth q u e s t i o n s w i l l be i n v e s t i g a t e d now. 13.6

Overlapping Divergences

The problem is t h e f o l l o w i n g . Can t h e r e n o r m a l i z a t i o n procedure be carried through o r d e r by o r d e r . F i r s t we m u s t s t a t e more p r e c i s e l y what we mean, because i n t h e Lagranglan we have c o u n t e r t e r m s o f o r d e r e2 and e3. To t h i s purpose we i n t r o d u c e a p a r a m e t e r ns and a l l counterterrns found from t h e a n a l y s i s of one-loop diagrams g e t t h i s parameter a s c o e f f i c i e n t . T h u s we have now

If now i n a diagram t h e number of c l o s e d loops is i t n a n we may a s s o c i a t e w i t h each diagram a f a c t o r Li. The S m a t r i x 1s f i n i t e up t o f i r s t o r d e r i n [ L 4 n l , i n tne l i m i t n = 1, L = 1. T n u s a t most one c l o s e d loop no 0 v e r t e x , or no c l o s e d loop one n vertex. I t may p e r h a p s be noted t h a t if t h e number of i n g o i n g and outgoing lines i s given, t h e n specifying L + q i s equivalent t o s p e c l f y i n g a c e r t a i n o r d e r i n e. Let u s now c o n s i d e r diagrams of o r d e r [ L + q l Z , f o r i n s t a n c e photon s e l f - e n e r g y diagrams. There a r e diagrams of o r d e r L2, i . e . two c l o s e d l o o p s , of o r d e r Ln and of o r d e r 1’:

146

G. 't HOOFT and M. VELTMAN

Lz:-

+

a

t12 :

b

-

I t i s now necessary t o i n t r o d u c e a c l a s s i f i c a t i o n . W e divide a l l dlagrams i n t o two s e t s :

2

i l t h e s e t o f a l l diagrams of o r d e r L , Ln o r

lil

n2

t h a t can be disconnected by removing one p r o p a g a o t r i h e r e [a, e, f, and i l r t h e r e s t , c a l l e d t h e o v e r l a p p l n g diagrams.

s e t (11 a r e t h e non-overlapping dlagrams. No new divergences occur, as can be v e r i f i e d r e a d i l y . I n f a c t , on b o t h Bides o f t h e propagator i n q u e s t i o n one f i n d s p r e c i s e l y what has been made p r e v i o u s l y :

a + e + f + l =

-

(u+

l X l - 0 -

+

-

I

The o v e r l a p p i n g diagrams may c o n t a i n new divergencies, and we must t r y t o prove t h a t these new d i v e r g e n c i e s behave as l o c a l counterterms. I n p r i n c i p l e , t h e p r o o f i s v e r y simple and based on t h e use o f c u t t i n g equations. The f i r s t o b s e r v a t i o n i s t h a t a l l c u t diagrams (always t a k i n g t o g e t h e r diagrams o f a g i v e n o r d e r i n [L + n) on each s i d e o f t h e c u t 1 a r e f i n i t e , t h a t i s t h e r e i s no p o l e f o r n = 4. The p r o o f o f t h i s i s easy1 these c u t diagrams are o f t h e s t r u c t u r e o f a p r o d u c t o f diagrams o f lower o r d e r I n (L + nl, which a r e supposedly f i n i t e , and an i n t e g r a t i o n over i n t e r m e d i a t e s t a t e s . Since f o r g i v e n energy t h e a v a i l a b l e phase space i s f i n i t e t h e r e s u l t f o l l o w s . T h i s assumes t h a t t h e diagrams have n o - i n t e g r a b l e s i n g u l s r i t l e s i n phase space; t h e l a t t e r would correspond t o i n f i n i t e t r a n s i t i o n p r o b a b i l l t i e s i n lower order. which we t a k e n o t t o e x i s t . Now l e t us number t h e v e r t i c e s t o wnich t h e e x t e r n a l l i n e s are connected. Here t h e r e a r e o n l y two such v e r t i c e s , and we c a l l them 1 and 2. According t o our c u t t i n g r e l a t i o n s we have, i n t e g r a t i n g over a l l x except over x and x

1

147

2

DIAGRAMMAR

+

w i t h x = x2 - X.I Here d c o n t a i n s a l l c u t d i a g r a m w i t h x l i n t h e unshadowed r e g i o n , and A- a l l c u t diagrams w i t h x2 i n t h e unshadowed r e g i o n . T h i s i s of p r e c i s e l y t h e same s t r u c t u r e as discussed before1 we know t h a t i f A* a r e f r e e o f p o l e s f o r n = 4, t h e n f ( x 1 can o n l y have a p o l e p a r t t h a t i s a 6 f u n c t i o n o r d e r i v a t i v e o f a 6 f u n c t i o n . I n o t h e r words, t h e p o l e p a r t I n n = 4 must be a polynomial I n t h e e x t e r n a l momentum. F o r a r b i t r a r y diagrams t h i s must be t r u e f o r any combination o f " e x t e r n a l ' vertices v e r t i c e s t h a t have a t l e a s t one e x t e r n a l l i n e ] , and t h u s f o r any o f t h e e x t e r n a l momenta. Due t o t h e f a c t t h a t power c o u n t i n g an0 t h e e x i s t e n c e o f p o l e s f o r n = 4 a r e i n a one-one r e l a t i o n s h i p i t can e a s i l y be e s t a b l i s h e d t h a t t n e polynomials are a t most o f a c e r t a i n degree; here, photon s e l f energies, a t most of degree 2. To do a l l t h i s p r o p e r l y i t I s necessary t o d i s t i n g u i s h between o v e r - a l l divergences and subdivergences [ t h e former disappear ifany of t h e propagators i s opened), and show t h a t t h e r e a r e no subdivergences i f a l l t h e lower o r d e r counterterms a r e i n c l u d e d . Next i t must be shown t h a t a f t e r a c e r t a i n number o f d i f f e r e n t i a t i o n s w i t h r e s p e c t t o t h e e x t e r n a l momenta t h e o v e r - a l l divergence disappears. A l l t h i s i s t r i v i a l : opening up a propagator i n a self-energy diagram I s e q u i v a l e n t t o c o n s l d e r i n g diagrams w i t h f o u r e x t e r n a l l l n e s o f lower o r d e r i n [L + 9). thus a l r e a d y f i n i t e . O v e r - a l l divergences correspond t o o v e r - a l l power c o u n t i n g ( p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o a l l l o o p momentel. and d i f f e r e n t l a t i o n w i t h r e s p e c t t o any e x t e r n a l momentum lowers t h i s power by one. ( 0

13.7

The Order o f t h e P o l e s Renormalizetion may now be performed o r d e r by o r d e r i n

+

1L

+

rll. One s t a r t s w i t h dlagrems w i t h L1 (one c l o s e d l o o p l end

f i n d s t h e necessary counterterms. They g e t a f a c t o r r) i n t h e Lagrangian. Next one considers diagrams o f o r d e r L2 [ t w o c l o s e d loops) o r [Lr)J1 (one closed loop, one counter v e r t e x ] o r [ t r e e dlagrems, two c o u n t e r VertiCeSl. Tne t o t a l can be mede f i n i t e by adding f u r t h e r counterterms. Such terms g e t a f a c t o r n2 i n t h e Lagrangian. I n t h i s way one can go on and o b t a i n a counter Lagrangian i n t h e form of a power s e r i e s i n F o r r) = 1 t h e complete t h e o r y i s f i n i t e t t h e r e a r e no p o l e s f o r n = 41.

nz

.

We have a l r e a d y seen t h a t t h e terms of o r d e r rj c o n t a i n simple p o l e s only. We now want t o i n d i c a t e t h a t t h e terms of o r d e r n2 c o n t e i n p o l e s of t h e f o r m I n - 41-2, and furthermore, t h e c o e f f i c i e n t o f such a p o l e i s determined by t h e o r d e r r) terms. F i r s t of a l l , i t d r a t i c pole3 a t o r d e r t h e r e a r e no p o l e s a t i n two-loop diagrams.

i s t r i v i a l t o see t h a t t h e r e a r e no quaq2 i f t h e r e a r e no p o l e s a t o r d e r rj. If order n t h e n t h e r e are no subdivergences But o v e r - a l l divergences of diagrams o f

148

G. 't HOOFT and M. VELTMAN

2

order L are simple poles ( t h i s f o l l o w s from p a r t i e l i n t e g r a t i o n ) , 1.e. t h e r e are no quadratic poles. To see t n a t there are i n general quadratic poles consider t h e 5um o f phqton self-energy diagrams [ c l and ( d ) . The subi n t e g r a l i n [ c ) may be done and gives rise t o an expression of t h e form

where p i s t h e momentum going I n t o the sub-loop. This equation i s symbolic i n 80 fsr as t h 8 t we have i n d i c s t e d only t h e momentum dependence w i t h respect t o power counting. The above expression can then be obtained from dimensional arguments. The sum of d i a grams [ c l and I d ) i s o f t h e form

S i m p l i f i c a t i o n can go t o o f a r J we w r i t e p

-

(p2 + m2)1/2.

Next

and t h e i n t e g r a l becomes

Since

ri3

-

-r ( 4 - n l

nl = 3 - n

r[i -

,

$ ] = 2-n 2 r[2

we f i n d f o r t h e Qouble p o l e

1 2 ---m 1 3 - n 4 - n 4 - n

2 2 2 - n 4 - n 4 - n

2 ----m 2

I f we w r i t e 2

2

(m2~n-3 = m2 + m (n-41 I n m

,

149

2

_(4

-

$1,

1 n)'

2m2

DIAGRAMMAR

2 n/2-1 = m2 (m l

+

m2

n

-

4

2

2

,

2 we d i s c o v e r t h a t t h e I n m t e r m has no p o l e 8 2 - n

(4 - n l

2

-

4 i s zero. T h i s i s as should The r e s i d u e o f t h e p o l e f o r n be because a t e r m I n m2 i s non-local, and n o t a d m i s s i b l e as counterterm. Remember t h a t m2 stands f o r I n v a r i a n t s made up from e x t e r n a l momenta, masses, t e c .

The above argument i s very g e n e r a l and i n fact based o n l y on loop and power c o u n t i n g . I t can be extended t o a r b i t r a r y o r d e r , and a p r e c i s e r e l a t i o n between t h e c o e f f i c i e n t s o f t h e v a r i o u s h i g h e r o r d e r poles can be found. ( G . ' t Hooft, Nucl. Phys. 861 (1973) 455, see Chapter 3 o f t h i s book.) 13.8

Order-by-Order Renormalization Consider t h e f o l l o w i n g two-loop diagram:

---u-u

Suppose t h e s e l f - e n e r g y bubble is a f u n c t i o n o f n and t h e momentum k o f t h e form:

-0-

1

f [ k2 l = -

2

-

We have o m i t t e d a f a c t o r k 6

fl[k

2

1

+

f2[k

2

I

2 + (n-41f3[k l

kpkv. Th8 two-loop diagram g i v e s

VV

I f we simply throw away t h e p o l e p a r t s t h e r e s u l t is f o r n = 4

f;

+

2f f

1 3 '

150

.

G. 't HOOFT and M. VELTMAN

However, t h i s r e s u l t i s wrong because i t v i o l a t e s u n i t a r i t y . Suppose now we do order-by-order s u b t r a c t i o n . Then one has a f t e r t h e treatment o f one c l o s e d loop:

2

The p h y s i c a l r e s u l t i s t h e l i m i t n = 4 abd i s equal t o f 2 [ k I . C u t t i n g two-loop diagrams should t h e r e f o r e n o t i n v o l v e f3. Next i n c l u d e t h e p r o p e r counterterms and consider:

-o-O-+++e L = f2

+

O[n

-

41

+ -

.

Indeed one o b t a i n s t h e c o r r e c t r e s u l t c o n s i s t e n t w i t h u n i t a r i t y . T h i s demonstrates t h a t order-by-order r e n o r m a l i z a t i o n i s not e q u i v a l e n t t o t h r o w i n g away p o l e s and t h e i r r e s i d u e s o f t h e u n r e n o n e l l z e d S-matrix. I n a way t h a t i s a p i t y , because t h e l a s t method I s so much e a s i e r , But t h e r e I s no escape, one must do t h i n g s s t e p by step, t h a t i s o r d e r by order i n t h e parameter 11.

13.9

Renormalization and Slavnov-Taylor I d e n t i t i e s

I t has been shown t h a t t h e one-loop counterterms are gauge i n v a r i a n t by themwelves, I n t h e case o f L o r e n t z gauge quantum electrodynamics. The q u e s t i o n i s whether we can understand t h i s , and t o what e x t e n t t h i s is g e n e r a l phenomenon.

As a f i r s t step we n o t e t h a t c o m b i n e t o r i a l p r o o f s can be read backwards, a t l e a s t ift h e y a r e o f t h e l o c a l v a r i e t y . I n o t h e r words, Ward i d e n t i t i e s cen be t r a n s l a t e d backwards i n t o a symmetry Ift h e Lagrangian. Therefore, i f t h e Ward i d e n t i t i e s of t h e Legranglen I n c l u d i n g counterterms a r e I d e n t i c a l t o those w i t h o u t counterterms, then b o t h Lagranglans s a t i s f y t h e same gauge i n v a r i a n c e . L e t us c o n s i d e r t h e S-T i d e n t i t i e s o f t h e unrenormalized t h e o r y i n c l u d i n g e l e c t r o n sources:

We have drawn one photon and olie e l e c t r o n source e x p l i c i t l y , l e a v l n g o t h e r sources understood.

151

DIAGRAMMAR

Since these i d e n t i t i e s a r e t r u e f o r any v a l u e o f t h e dimens i o n a l parameter n, t h e y a r e i n p a r t i c u l a r t r u e f o r t h e p o l e p a r t s . Consider now one-loop diagrams. The p o l e p a r t s i n t h e above i d e n t i t y are:

11 t h e p o l e p a r t s found i n t h e Green's f u n c t i o n s themselvest ii) t h e p o l e p a r t a r i s i n g from a c l o s e d loop, i n v o l v i n g t h e e l e c t r o n - g h o s t v e r t e x shown i n t h e l a s t diagram. Now t h i s l a s t v e r t e x i s defined by t h e behaviour o f t h e Lagrangian under gauge transformations, b u t - i t i s n o t p r e s e n t I n t h e Lagrangian i t s e l f . I n 1: one o n l y f i n d s J$J if we i n t r o d u c e counterterms i n t h e Lagrangian t h a t make t h e Green's f u n c t i o n s f i n i t e [ i n c l u d i n g Green's f u n c t i o n s w i t h i n g o i n g and o u t g o i n g ghost l i n e s ) t h e n t h e above S-T i d e n t i t y can remain t r u e o n l y if we throw away t h e p o l e p a r t of t h e t y p e [ill.That i s , f o r t h e renormalized Lagrangian, t h e S-T i d e n t i t i e s t a k e t h e form:

Here t h e double cross stands f o r minus t h e p o l e p a r t o f any diagram such as:

C l e a r l y we can understand t h i s i d e n t i t y as an S-T i d e n t i t y i f we r e d e f i n e t h e behaviour of t h e term SJ, under gauge t r a n s f o r m a t i o n s . I f we say t h a t under a renormalized gauge t r a n s f o r m a t i o n J) t r a n s forms as

$'J,

+

ieh$

+

Z in - 4

Jw

where 2 is minus t h e r e s i d u e of t h e p o l e p a r t of t h e t y p e [ii), then t h e above i d e n t i t y i s a g a i n p r e c i s e l y an S-T i d e n t i t y . We conclude t h a t t h e Lagrangian i n c l u d i n g counterterms i s i n v a r i a n t f o r renormalized gauge transformations. To make t h e statement more p r e c i s e we must add on a f a c t o r t o t h e p o l e t e r m i n t h e renormalized gauge t r a n s f o r m a t i o n . Since we allow as a f i r s t s t e p o n l y one c l o s e d loop t h e statement is o n l y t r u e up t o f i r s t o r d e r i n n. I t i s u n f o r t u n a t e l y somewhat Q

152

G. 't HOOFT and M. VELTMAN complicated t o extend t h i s work t o a r b i t r a r y o r d e r i n

n,

b u t we

w i l l do i t i n t h e n e x t s e c t i o n . I n quantum electrodynamics t h i n g s are never r e a l l y complicated, because i n t h e u s u a l gauges [Lorentz, Landau, etc.1 t h e r e i s no ghost v e r t e x , ergo t h e r e are no p o l e p a r t s o f t h e t y p e ( i l l . Then t h e renormalized Lagranglan i s i n v a r i a n t w i t h r e s p e c t t o gauge t r a n s f o r m a t i o n s t h a t transform an e l e c t r o n source c o u p l i n g as before, 1.0.

JI

+

JI

ieAJI

+

w i t h t h e same e as i n t h e unrenormalized Lagranglan. The r e s u l t i s i s i n v a r i a n t under t h e t r a n s f o r m a t i o n [ A ) ,

'unren

1:unren

+ 'counter

I s i n v a r i a n t under t h e t r a n s f o r m a t i o n [ A ) ,

therefore alone i s I n v a r i a n t .

'counter

I n t h e g e n e r a l case t h i s i s n o t t r u e .

Higher-Order Counterterrns

13.10

Doing t h i n g s o r d e r by o r d e r we suspect [end have shown t h i s t o be t r u e up t o f i r s t o r d e r i n 01 t h a t Lren is of t h e form

'

ren

a

'unren

- nz1

1:

counter

+

+'

counter

2

n x2

+

'

... ,

, etc., c o n t a i n f a c t o r s l / [ n - 4 1 . F o r q = 1 t h e is invariant n i t e . The renormallzed Lagranglen '(,I under gauge t r a n s f o r m a t i o n s of t h e f o r m A

P

+

I

A

J

+

(to+ t Q

1

+

t2q2 +

...) A

where t h e t and t ' c o n t a i n f a c t o r s l / [ n

153

,

- 41.

DIAGRAMMAR

Let us s uppos e t h i s l a t t e r s t a t e m e n t t o be true up t o o r d e r k in

n. C o n s i d e r now:

1 1 d i a g r a m s c o n t a i n i n g o n l y vertices o f t h e u n r e n o r m a l i z e d Le g r a n g i an w i t h k + 1 c l o s e d loops, s u c h d i a g r a m s e x h i b i t p o l e s up t o d e g r e e k + 1 and s a t i s f y t h e u n r e n o r m a l i z e d S-T i d e n t i t i e s , ii) d i a g r a m s c o n t a i n i n g c o u n t e r t e r m s ( c o r r e s p o n d i n g t o v e r t i c e s L k , to tk, b u t w i t h a t l e a s t one of c l o s e d loop: i i i l d i a g r a m s o f o r d e r q k + l [ c o n t a i n i n g t h u s no c l o s e d l o o p l .

...

...

tb

... ti)

Example f o r two c l o s e d l o o p phot on s e l f - e n e r g y di agram s:

-

S e t (1) = d i a g r a m s [ a ) , ( b l , [ c ) . Set [ i l l diagrams [dl t o ( h l . S e t [ i i i l = di agr am ( 1 1 p l u s a new d i a g r a m o f t h e form t h a t is needed t o remove t h e d i v e r g e n c i e s of t h e o v e r l a p p i n g d i a g r a m s . Now (1) + [ i l l + I i i i ) g i v e a f i n i t e t h e o r y as n + 4 and q + 1. T h a t i s s i m p l y how t h e y were d e f i n e d . Next ( 1 1 + [ i i l s a t i s f y Ward i d e n t i t i e s [ t h e t h e o r y was assumed t o be i n v a r i a n t up t o o r d e r qk). T h e r e f o r e t h e r e s i d u e s o f t h e p o l e of set [ i l l 1 s a t i s f y t h e S-T i d e n t i t i e s . Because t h e d i a g r a m s [ill) are t r e e d i a g r a m s t h e di m ens i on n a p p e a r s nowhere e x c e p t i n t h e p o l e f a c t o r s l / l n - 41. T h e r e f o r e t h e d i a g r a m s [ i i i l s a t i s f y t h e S-T i d e n t i t i e s , which means i n v a r i a n c e o f t h e r e n o r m a l i z e d L agrangi an u n d e r t h e r e n o r m a l i z e d gauge t r a n s f o r m a t i o n c o n s i d e r i n g terms of o r d e r n k - I o n l y . T h i s c o m p l e t e s t h e proof by i n d u c t i o n .

Acknowledgements The a u t h o r s are d e e p l y i n d e b t e d t o O r . R . B a r b i e r i , who h e lp ed i n w r i t i n g a f i r s t a p p r o x i m a t i o n t o t h i s r e p o r t . One of t h e a u t h o r s (M.V.1 w i s h e s t o e x p r e s s h i s g r a t i t u d e f o r t h e h o s p i t a l i t y and p l e a s a n t at m os pher e e n c o u n t e r e d a t t h e S c u o l a Normale a t P i s a ; t h e l e c t u r e s g i v e n t h e r e have been a f i r s t t e s t of t h e p r e s e n t material. F i n a l l y , w e would l i k e t o t h a n k t h e CERN Document Reproduct i o n S e r v i c e f o r t h e i r e x c e l l e n t work on t h i s r e p o r t ( i n p a r t i c u l a r . Mme P a u l e t t e E s t i e r f o r t h e s peedy and accurate t y p i n g o f t h e t e x t ) , and h e E . Pons s en o f t h e CERN MSC Drawing O f f i c e f o r h e r e f f i c i e n t co-operation.

G. 't HOOFT and M. VELTMAN

APPENDIX A

FEYNMAN RULES The most f r e q u e n t l y encountered Feynman r u l e s w i l l be summar i z e d here. A l s o c o m b i n e t o r i a l f a c t o r s w i l l be discussed. The e x t e r n a l sources t o be employed f o r S-matrix d e f i n i t i o n must be normalized such t h a t t h e y e m i t o r absorb one p a r t i c l e . The normal i z a t i o n formulae f o l l o w f r o m t h e propagators1 t h e " i n g o i n g and o u t g o i n g l i n e wave-functions" i n d i c a t e d below a r e t h e p r o d u c t o f those sources, t h e associated propagator and t h e a s s o c i a t e d masss h e l l f a c t o r k2 + m2. Spin-0 p a r t i c l e s

-

Propagator:

1

IZa14i k2

1 +

m2

- is

[A. 11

I n shadowed r e g i o n :

--

1

1

IA.21

12a14i k2+ m2+ I s

'

Cut propagator:

IA.31

Wave-function:

1.

(A.41

Spin-1/2 p a r t i c l e s

-

Propagator:

1

-iyk+m

I Z d 4 i k2

+

m2

IA.51

k

- ic

c

I n shadowed r e g i o n : -iyk + m 2 ' 1 2 ~ 1 k2+ ~ 1 m + IE

1

IA.61

-I_

155

DIAGRAMMAR

Cut propagator:

Ingoing particle wave-function : Ingoing antiparticle wave-function:

ua(klq

I

a = 1, 2

-iia(klq

8

a = 3, 4

Outgoing particle wave-function:

i i a ( k l q : a = 1, 2

Outgoing antiparticle wave-function:

ua(klq :a

3, 4

=

1

1.

J

(A.91

Note the minus sign f o r the incoming antiparticle. The momentum k is directed inwards for incoming particles and outwards for outgoing particles. A S usual i = u*y4. 8re solutions of the Dirac equation [note that = + kO

(iy’k

’

+

tnlua(k1

0

,

a

-

-

1, 2

, (A.101

.

+ mlue(kl 0 , B = 3, 4 IJ In the 4 x 4 representation, with y 5 = y 1y 2y 3y 4

f-iy’k

-1

0 (A.111

1 80

’ these solutions are given in the fallowing table:

156

1

G. 't HOOFT and M. VELTMAN f u1 [ k l L

1

0

m -

X

2k0

m ko kg +

kl

+

ik2

m

+

ko

The arrows denote s p i n up/down assignments i n t h e k r e s t frame. Normalization:

[ A . 121

Spin eummations:

I n connection w i t h p a r i t y P, charge conjugation C , and timer e v e r s a l T, t h e f o l l o w i n g m a t r i c e s and transformation p r o p e r t i e s a r e o f relevance. The m a t r i c e y4 is t h e transformation m a t r i x connected w i t h space r e f l e c t i o n

157

DIAGRAMMAR

uat-Tt,

4 a +

kol = - y u [k, kol ,

031, 2

,

a n t i p a r t i c l e a = 3. 4

.

particle

a + 4 a + u ( - k , ko) = - y u (k, k 1 , p = 1 . 2 , 3

(A.141

Y Y Y

lJ’4.

y’

T h e m a t r i x C t r a n s f o r m s a n i n c o m i n g p a r t i c l e i n t o an i n c o r n i n g a n t i p a r t i c l e , etc. ua[kl

-

a - 4 .

B - 1

-C;’[kl a = 3 , 8 - 2 a - 1 , 8 - 4

;‘[kI

tA.151

= uB[k1C-’

a-2, B - 3

c-1yac c

-1 5

-a

5 - y

Y C ’ Y

[-

-

transpose)

-5

-

#

2 4 -1 where C = Y Y , C = C = -C.

I n c o n n e c t i o n w i t h t i m e - r e v e r s a l we h a v e t h e m a t r i x D

-4-8 +

= Dy u ( k , kol

a = l , B = 2

T h i s c h a n g e s s p i n , d i r e c t i o n of three-momentum a n d f u r t h e r m o r e exchanges i n - and o u t - s t a t e s ;

G. 't HOOFT and M. VELTMAN

'0 0

1'

0

0 0 - 1 0

C =

0 1

6 1 -

UV

[Zn14i

D =

0 0 '

0

.-1

0 0

+

1

0

0'

- 1 0

0

0

'0

,

0

0

,0

0

-1

0,

k kv/mz

-

k2+ m2

IE

I n shadowed r e g i o n :

--

1

6

+ k k / m llv ' + m 2+ i c

r-

2

lJV

(2nl I k

(A.171

0 1 '

Cut propagator:

P a r t i c l e wave-functions:

e (k1 ll

with

*

e (k1e (kl = 1 and u

l

l

k e

lJlJ

= 0

.

There a r e o n l y t h r e e wave-functions, because t h e propagator m a t r i x has one eigenvalue zero. I n t h e k r e s t system t h e v a r i o u s assignments are

Ingoing

Spin-z component zero: e

lJ

-

(0, 0, 1, 01

Out going

e

lJ

= [OD

0, 1. 01

F o r o u t g o i n g p a r t i c l e s we must t a k e these expressions t o have

159

.

DIAGRAMMAR

c o r r e c t phases. Indeed, t h e r e s i d u e o f t h e t w o - p o i n t spin-up/ spin-up amplitude i s equal t o one. Combinatorial f a c t o r s These a r e b e s t e x p l a i n e d by c o n s i d e r i n g a few examples. L e t t h e i n t e r a c t i o n Lagrangian f o r a s c a l a r f i e l d be

The v e r t i c e s a r e :

The lowest o r d e r s e l f - e n e r g y diagram is:

Draw two p o i n t s x1 and x and draw i n each of these p o i n t s t h e a 2 vertex:

Now count i n how many ways t h e l i n e s can be connected w i t h t h e same t o p o l o g i c a l r e s u l t . E x t e r n a l l i n e 1 can be a t t a c h e d i n s i x , a f t e r t h a t l i n e 2 i n t h r e e ways. A f t e r t h a t t h e r e are two ways t o connect t h e remaining l i n e s such t h a t t h e d e s i r e d diagram r e s u l t s . Thus t h e r e are a l t o g e t h e r 6 x 3 x 2 combinations. Now d i v i d e by t h e p e r m u t a t i o n a l f a c t o r s o f t h e v e r t i c e s (here 3! f o r each a v e r t e x ) . F i n a l l y d i v i d e by t h e number o f permutations o f t h e p o i n t s x t h a t have i d e n t i c a l v e r t i c e s . Here 2: The t o t a l result i s

--

6 x 3 ~ 2 - 1 3: 3: 2: 2 As another example c o n s i d e r t h e diagram:

160

G. 't HOOFT and M. VELTMAN

There a r e t h r e e x p o i n t s :

w w w XI

x2

x3

L i n e 1: s i x ways. L i n e 2: f o u r ways. Then we have f o r i n s t a n c e :

XI

x3

There a r e 6 x 3 x 2 ways t o connect t h e r e s t such as t o g e t t h e d e s i r e d topology. We must d i v i d e by v e r t e x f a c t o r s [3: 3: 4:) and by 2: [ p e r m u t a t i o n o f t h e i d e n t i c a l v e r t e x p o i n t s XI and x2). The r e s u l t i s 6

x

3:

4

~

31

4:

6 ~ 2:

3

I

1 ~ 2 2 .

-

F i n a l example: two i d e n t i c a l sources connected by a sce'lar l i n e :

J-J

Factor:

2:1

.

F o r two n o n - d e n t i c a l sources t h e f a c t o r i s 1:

Factor:

J1J-2

1

.

Topology o f quantum electrodynamics We now show t h a t t h e v e r t i c e s o f any diagram of quantum electrodynamics can be numbered i n a unique way. L e t t h e e x t e r knm n a l momenta be k

...

Step 1 S t e r t a t t h e e l e c t r o n l i n e w i t h t h e lowest momentum index. F o l l o w t h e arrow o f t h e l i n e . Number t h e v e r t i c e s c o n s e c u t i v e l y . Step 2 Go back t o t h e f i r s t v e r t e x [ o r lowest numbered v e r t e x of which t h e photon l i n e was n o t e x p l o i t e d ) . F o l l o w t h e photon l i n e . We a r r i v e t h e n e i t h e r a t a v e r t e x t h a t i s a l r e a d y numbered, a t an e x t e r n a l photon l i n e , or a t another e l e c t r o n l i n e . I n t h e f i r s t

161

DIAGRAMMAR

two casesr t a k e t h e n e x t v e r t e x a l o n g t h e e l e c t r o n l i n e o f s t e p 1 and r e s t a r t a t s t e p 1. When a r r i v i n g a t a new v e r t e x , number a g a i n c o n s e c u t i v e l y f o l l o w i n g t h e e l e c t r o n l i n e along t h e arrow. When h i t t i n g t h e end, o r an a l r e a d y numbered v e r t e x r go back a g a i n s t t h e arrow and number a l l t h e v e r t i c e s on t h e e l e c t r o n l i n e b e f o r e t h e v e r t e x t h a t was t h e entrance p o i n t o f t h a t l i n e . A f t e r t h a t r e s t a r t 2. Example :

1

- 11

2 lo

3

I 12

162

G. 't HOOFT and M. VELTMAN

APPENDIX

B SOME USEFUL FORMULAE

with 0

s

81

s

ns

e x c e p t 0 <_ 0 1 <_ 2n. If f ( x 1 depends o n l y on

one may perform t h e i n t e g r a t i o n over a n g l e s u s i n g

(8.21

leading t o

Keeping t h e p r e s c r i p t i o n s and d e f i n i t i o n s o f S e c t i o n 13 i n mind, t h e f o l l o w i n g e q u a t i o n s hold f o r a r b i t r a r y n

163

DIAGRAMMAR

I dnP

[p2

+

2kp

+

2 a m 1

(m2

I

P$VPA

dnP

I dnP

[p2 + 2kp

+

m2Ia

n/2 in 2 a-n/2 (m2 - k 1

2 P P

[p2 + 2kp

n/2 ill

+

m21a

2 2 a-n/2 Im - k 1

1 -

-

k21]

,

X

rI a1

-I-kpl

x

r(a1

The above equations c o n t a i n i n d i c e s p , v, A . These i n d i c e s a r e understood t o be c o n t r a c t e d w i t h a r b i t r a r y n - v e c t o r s ql, 92, e t c . I n computing t h e i n t e g r a l s one f i r s t i n t e g r a t e s over t h e p a r t o f n-space o r t h o g o n a l t o t h e Vectors k, 91, 92, etC., u s i n g Eqs. IB.11 t o (8.4). A f t e r t h a t t h e expressions a r e meaningful a l s o f o r n o n - i n t e g e r n. Note t h a t f o r m a l l y Eqs. IB.61 t o IB.101 may be obtained from Eq. (8.51 by d i f f e r e n t i a t i o n w i t h r e s p e c t t o k, o r by u s i n g p2 = [p2 + 2pk + m21 2pk m2,

-

-

To show t h a t i n t e g r a l s over polynomials g i v e zero w i t h i n t h e dimensional r e g u l a r i z a t i o n scheme i s v e r y simple. Consider, f o r example,

where a i s some i n t e g e r g r e a t e r t h a n o r equal t o zero. According t o Eq. [13.191 i n t h e case of o n l y one loop, we have

164

G. 't HOOFT and M. VELTMAN

By partial integrations (see Subsection 13.21 we get now

which gives zero for X > a. A nice example, suggested by 8. Lautrup is the following. Consider the following integral

I IJ

=

I d4k*

k

, (k + m 1

which gives zero, because of symmetric integration, if one regularizes, for example, as follows

I

-*

d4kky[ [k2

1!

1

1

m212

+

(k2

+

2 21 A 1

It ia also zero in the dimensional cut-off scheme according to Eq. (B.61.

Let us now shift the Integration variable, forgetting about regulators k

Iy

=

1 d4k

+ P

[[k + p12

+

m

22'

I

Expanding the denominators, we get

which by symmetric i n t e g r a t i o n (k V 2

IJ

d4k

+

0, kVkv

2 + O(p 1 f 0

[k2+ m2I3

+.+Vvk 2 ) gives

.

Using dimensional regularization, which means k u [6 /n1k2, we get from Eq. tB.121

-t

lJV

165

+

0 but k k + lJv

DIAGRAMMAR

2 otp 1

+

.

From Eqs. (B.51 and (B.71

I dnk

2 n/2-3

n/2

1

(m

= ir

4 3

I

k2 [k2

2 3 m

9 I

r(31

(k2 + m 2 I 3

dnk

-

= in

n/2

2 n/2-2 (m 1

2 z

r(31

'

Then

1

I n t h e l i m i t n + 4, remembering t h a t

r[zi

z-t-n

=

(-11 n

n:

1

z + n '

t h e c o e f f i c i e n t o f t h e p, t e r m t u r n s o u t t o be e x a c t l y zero. O f course, f r o m Eqs. lE.51 t o (8.7). I i n Eq. [8.111 g i v e s zero lJ t o any o r d e r i n p. I n computing p o l e p a r t s i t Is v e r y advantageous t o develop denorninetors. Take Eq. [ B . 5 1 . F o r a = 2 we f i n d t h e p o l e p e r t 2

PP[(E.51,

a = 21 =

-2in n-4

zo

(8.13) a

T h i s 20 is a b a s i c f a c t o r . Every l o g e r i t h m i c a l l y d i v e r g e n t i n t e g r a l has t h i s f a c t o r , and f u r t h e r vectors, 6 f u n c t i o n s . etc.

PP

I dnP

'aPB [p2 +

2 3 m I

-

1

zo

4

6ciB

166

tB.14)

G. 't HOOFT and M. VELTMAN

The f o u r on t h e r i g h t - h a n d s i d e f o l l o w s f r o m symmetry considerat i o n s i t h e c o e f f i c i e n t f o l l o w s because m u l t i p l i c a t i o n w i t h 6ap g i v e s t h e p r e v i o u s i n t e g r a l . We leave i t t o t h e reader t o f i n d t h e g e n e r a l equation.

F o r o t h e r t h a n l o g a r i t h m i c a l l y d i v e r g e n t i n t e g r a l s t h e denom i n a t o r must be developed. F o r i n s t a n c e

pp

dnP

(P

+

'a 2pk + m2 ) 2

(8.16)

-kaZo

11' $

where we used Eq. (B.141 t o g e t h e r w i t h 2 2

1 (P

+ 2pk

+

m I

= 4

-

4

+

.'.-.I}

.

P

L i n e a r l y , q u a d r a t i c a l l y , etc.. d i v e r g e n t i n t e g r a l s t h a t have no dependence on masses o r e x t e r n a l momenta can be p u t equal t o zero. The r e s u l t Eq. f r o m Eq.

(8.161 c o i n c i d e s w i t h w h a t can be deduced

(8.6).

167

DIAGRAMMAR

APPENDIX C DEFINITION OF THE FIELOS FOR DRESSED PARTICLES

A l s o t h e m a t r i x elements of f i e l d s and p r o d u c t s o f f i e l d s (such as encountered i n c u r r e n t s ) can be d e f i n e d i n terms of diagrams. I t i s then p o s s i b l e t o d e r i v e , o r r a t h e r v e r i f y , t h e equations o f motion f o r t h e f i e l d s . T h i s p r o v i d e s f o r t h e l i n k between diagrams and t h e c a n o n i c a l o p e r a t o r formalism. Since t h i n g s tend t o be t e c h n i c a l l y complicated we w i l l l i m i t o u r s e l v e s t o a simple case, namely t h r e e r e a l s c a l a r f i e l d s i n t e r a c t i n g i n t h e most simple way. The Lagrangian i s taken t o be

1

1

-

+ gABC + JAA + J B + JcC B

.

A(a

2

-

2 mA)A + 1 B ( a2 2

=

-

mBIB 2

+

-21 C r a 2

- m2C I C

+

(C.11

b b The bare propagators w i l l be denoted by t h e symbols A FA' 'FB t h e dressed prnpagators by AFA, +B and For ex amp 1e

aFc.

I -

1 [ 2 n 1 4 i k2

*FA

and

1 +

mi

-

i~ '

1

4 (2nl i Z i [ k 2

1 +

2

MA)

-

2

rlA(k

1

-

ic

The p o l e p a r t of t h e dressed propagator p l a y s an i m p o r t a n t r o l e and w i l l be denoted by AF

The r e s u l t (C.2) has been o b t a i n e d as f o l l o w s [see S e c t i o n 9, i n p a r t i c u l a r Eq. (9.311. The f u n c t i o n rA[k2) i s t h e sum o f a l l i r r e d u c i b l e self-energy diagrams f o r t h e A - f i e l d . The dressed propagator i s of t h e form (k2 + m i r A ) - ' # T h i s expression w i l l 2 have a p o l e for some v a l u e of k2, say f o r k2= -MA. Then we can expand rA around t h e p o i n t k2 = -MZ

-

168

2 TA[k I

-

G. 't HOOFT and M. VELTMAN

6mi

+

2 (k2 + NAIFA + rlA(k21

, (C.41

2

Z i = l - F A ,

2 6mA = m A

is of order (k2 where 'I leads t A A Eq. (C.21.

+

-

2 MA

,

M i l 2 . I n s e r t i o n o f t h i s expression

Next t o t h e propagators we d e f i n e e x t e r n a l l i n e f a c t o r s N (k21, etc. They are t h e r a t i o o f t h e dressed propagators and t a e i r pole parts 2 NA(k I

'FA

tc.51

'A'FA

2 I n t h e l i m i t k2 = -MA t h i s i s p r e c i s e l y the f a c t o r occurring i n e x t e r n a l l i n e s when passing from Green's f u n c t i o n t o S-matrix. F i n a l l y we have t h e important A+ and A- f u n c t i o n s

f AA =

1 1 nni3Z; 8 [ f k

0

2 16(k2 + MA]

.

Consider now any Green's f u n c t i o n i n v o l v i n g a t l e a s t one A - f i e l d source. For a l l except t h i s one source we f o l l o w the procedure as used i n o b t a i n i n g t h e S-matrix, t h a t i s a l l dressed propagators and associated sources a r e replaced by f a c t o r s N and the mass-shell l i m i t is taken. For t h e s i n g l e d out A - f i e l d source we replace t h e dressed propagator and source by N ~ ( k 2 1 , b u t do n o t take t h e l i m i t k2 = -M2. The F o u r i e r transform w i t h respect t o k o f t h e f u n c t i o n so obtained i s defined t o be t h e m a t r i x element ( f o r a given order i n t h e coupling constant w i t h t h e appropriate i n - and out-states1 o f an operator denoted by

(The n o t a t i o n used here should n o t be confused w i t h n o t a t i o n s of t h e type used i n Section 9.1 I t i s , roughly speaking, obtained f r o m t h e S-matrix by t a k i n g o f f one e x t e r n a l A - l i n e and r e p l a c i n g t h a t l i n e by t h e f a c t o r ~ A ( k 2 1 . Diagramnatically:

169

f

-

-

w i t h the notation:

,

,

*--+

2

NA(k ) ( C . 71

A-,

8-, C-line.

I t i s t o be noted t h a t t h e propagators used a r e completely dressed operators, and t h e r e f o r e self-energy I n s e r t i o n s a r e n o t t o be contained i n Eq. [C.71. I n p a r t i c u l a r t h e r e a r e no c o n t r i b u t i o n s of t h e type:

(C.81

2 However, t h e f a c t o r NA[k ) i m p l i e s r e a l l y t h e i n s e r t i o n of i r r e d u c i b l e s e l f - e n e r g y p a r t s . Working o u t Eq. ( C . 5 1 we see (compare Eq. (C.41)

1+

(I.

( k2 1

A

-

i(2rr1 4 (mA 2

-

MA) 2

-

*

Diagrammatically:

6 = (271 i [ m A

-

MA]

+

4

(2n1 i [ k 2

+

2 MAIFA

.

Remember t h a t T A s t a r t s w i t h a B- and a C - l i n e , and t h a t NA i s attached t o a 8-C v e r t e x (see Eq. (C.711. We see t h a t t h e r i g h t - h a n d s i d e o f Eq. (C.7) c o n s i s t s of s k e l e t o n diagrams s t a r t i n g w i t h an amputated 8-C v e r t e x , a p a r t f r o m t h e 6 - c o r r e c t i o n .

all

We now d e f i n e t h e product o f t h i s o b j e c t and t h e m a t r i x St. I t i s obtained by connecting diagrams of 6S/6A t o diagrams of St

170

G. 't HOOFT and M. VELTMAN

by meang o f A

+

functions:

t

T h i s i s a c o l l e c t i o n of c u t diagrams, w i t h S corresponding t o t h e p a r t i n t h e shadowed r e g i o n . T h i s d e f i n i t i o n of t h e p r o d u c t i s t h e same as t h a t encountered i n t h e expression S h .

I t t u r n s o u t t h a t t h e d i f f e r e n t i a t i o n symbol 6/6 i n W 6 A has more than f o r m a l meaning. W i t h t h e h e l p o f t h e c u t t i n g equations i t is easy t o show t h a t

The second t e r m has t h e p o i n t x t o t h e r i g h t o f t h e c u t t i n g l i n e . T h i s can be expressed f o r m a l l y by w r i t i n g 6'(StS)/GA[xl = 0, which i s what i s expected ifu n i t a r i t y holds, StS = 1. One may speak o f g e n e r a l i z e d u n i t a r i t y , because t h e A - l i n e i s o f f massshell. The A - f i e l d c u r r e n t j A ( x ) i s d e f i n e d by

[C. 121 By v i r t u e o f Eq.

(C.111 i t f o l l o w s t h a t j A [ x l i s Hermitian.

To d e f i n e t h e m a t r i x elements o f t h e f i e l d A c o n s i d e r t h e e q u a t i o n o f motion

C. 131 T h i s i s not d i r e c t l y t h e e q u a t i o n o f motion t h a t one would w r i t e down g i v e n t h e Lagrangian (C.11, because we have t h e mass fli [ d e f i n e d by t h e l o c a t i o n of t h e p o l e o f t h e dressed propagator) instead o f mi. The e q u a t i o n o f motion (C.131 can be r e w r i t t e n as an i n t e g r a l eouation

171

DIAGRAMMAR

The r e t a r d e d A f u n c t i o n i s

ARA[x)

- -1

1

4

1

ikx d4ke

2

(2nl i ZA

T h i s f u n c t i o n i s zero u n l e s s x

2 k2 + MA

0

-

(C.151 iEko

> 0. I n f a c t

I n passing, we n o t e t h e i d e n t i t i e s

AR

-

A+

AR

+

A-

-

*

-AF

, (C.171

'F

Equation (C.141 defines t h e A - f i e l d i n terms o f diagrams. I t s a t i e f i e s t h e weak o r a s y m p t o t i c d e f i n i t i o n

lim x

< a l A ( x 1 IB, =

lim x

+-w

0

< alAin(xl

1s >.

+-w

0

-

The f i e l d A i n ( x l i s a f r e e f i e l d s a t i s f y i n g t h e e q u a t i o n o f motion [C.131 w i t h j 0. We w i l l w r i t e Eq. (C.141 i n terms o f diagrams, and t o t h a t purpose we must i n t r o d u c e t h e "ordered product". We w r i t e

1 Aln(x1 zA

=

1 1 t ( X I S :S + A [xfStS - - :A ZA i n ZA in

+ ZA

I d4XcAi[X -

X41

6s' 6Atx'I

'

(C.18)

+

The double d o t s i m p l y t h a t A t n 1 s n o t t o be connected by a A - l i n e t+OS t , Below t h i s w i l l be shown d i a g r a m m a t i c a l l y . The f u n c t i o n AA has been d e f i n e d before. Due t o t h e presence o f t h i s A+ o n l y t h e mass-shell v a l u e o f t h e f a c t o r N i s r e q u i r e d i n 6St/6A, and

172

G. 't HOOFT and M. VELTMAN

t h i s ia 1 1 2 ~ . Using Eq. (C.181 we can r e w r i t e t h e i n t e g r a l equation o f motion (C.14) i n t h e form r

tc. 191 Keeping i n mind Eq. (C.171 as w e l l a8 Eq. (C.51 we see t h a t A(x)/ZA can be p i c t u r e d as follows:

+ T h i s then i s t h e diagrammatic expression f o r t h e m a t r i x elements o f t h e f i e l d A, S i m i l a r expressions can be d e r i v e d f o r t h e f i e l d s B and C. A l l k i n d s o f r e l a t i o n s from canonical f i e l d theory can be d e r i v e d u s i n g these expressions. F o r i n s t a n c e

2

jA[X)

2

6m

'A'B'C

A B ( X I C ( X ) - A~[ x l zA

FA(a

-

2

2 ")

A(xI

zA

fC.21) w i t h FA and 6m from Eq. (C.41.

173

CHAPTER 2.3 GAUGE THEORIES WITH UNIFIED WEAK, ELECTROMAGNETIC, A N D STRONG INTERACTIONS*

G . ‘T HOOFT University of Utrecht

1. Introduction

Only half a decade ago, quantum field theory w a s considered as just one of the many different approaches t o particle physics, and there were many reasons not t o take it too seriously. In the first place the only possible “elementary” particles were spin zero bosons, spin fermions, and photons. All other particles, in particular the p , the N * , and a possible intermediate vector boson, had to be composite. To make such particles we need strong couplings, and that would lead us immediately outside the region where renormalized perturbation series make sense. And if we wanted to mimic the observed weak interactions using scalar fields, then we would need an improbable type of conspiracy between the coupling constants to get the V-A structure’). Finally, it seemed to be impossible t o reproduce the observed simple behaviour of certain inclusive electron-scattering cross sections under scaling of the momenta involved, in terms of any of the existing renormalizable theories2). N o wonder that people looked for different tools, like current algebra’s, bootstrap theories and other nonperturbative approaches. Theories with a non-Abelian, local gauge invariance, were known3), and even considered interesting and suggestive as possible theories for weak

4

*Ftapporteur’s talk given at the E.P.S. Iiiternational Conference on High Energy Physics, Palermo, Sicily, 23-28 June 1975. 0 European Physical Society

174

interactions4i5), but they made a very slow start in particle physics, because it seemed that they did not solve very much since unitarity and/or renormalizability were not understood and it remained impossible to do better than lowest order calculations. When finally the Feynman rules for gauge theories were settled6) and the renormalization procedure in the presence of spontaneous symmetry breakdown underst~od’-’~), it was immediately realized that there might exist a simple Gauge Model for all particles and all interactions in the world. The first who would find the Model would obtain a theory for all particles, and immortality. Thus the Great Model Rush began29~35~36*44-53~56~5a). First, one looks a t the leptons. The observed ones can easily be arranged in a symmetry pattern consistent with experiment6): SU(2) x U(1). But if we assume that other leptons exist which are so heavy that they have not yet been observed then there are many other possibilities. To settle the matter we have to look at the hadrons. The observed hadron spectrum is so complicated with its octuplets, nonets and decuplets that it would have been a miracle if they would fit in a simple gauge theory like the leptons. They don’t. To reproduce the nice SU(3) x SU(3) structure one is forced to take the “quarks”, the building blocks of the hadrons, as elementary fields. The existing hadrons are then all assumed to be composite. To bind those quarks together we need strong forces and here we are, back at our starting point. What have we won? We have won quite a lot, because the tools we can use, renormalized gauge theories, are much more powerful than the old renormalizable theories. Not only do we have indications that they exhaust all possible renormalizable interactionsz0) but there is also a completely new property: the behaviour of some of these theories under scaling of all coordinates and momentaz1). If you look at such a system of particles through a microscope, then what you see is a similar system of particles, but their interactions have reduced. The theory is “asymptotically free”z1-z3). The old theories always show a messy, strongly interacting soup when you look through the m i c r o ~ c o p e ~If~ )you . assume that any theory should be defined by giving its behaviour at small distances, then the old theories would be very ill-defined, contrary to the gauge theories. But when it comes to model building, then it is still awkward that the forces between the quarks are strong, because that makes gauge theories not very predictive and there are countless possibilities. I have seen theories with 3, 4, 6, 9, 12, 18 and more quarks. How should we choose among all these

175

different group structures? Before answering the question let us first make up the balance. What we are certain of is: 1) Gauge theories are renormalizable, if firstly the local symmetry is broken

2)

3)

4)

5)

6)

spontaneously, and secondly the Adler-Bell-Jackiw anomalies are arranged t o cancel. Global symmetries may be broken explicitly, so we can always get rid of G 01dstone b osons. We can make asymptotically free theories for strong interactions. Then the following statements are not absolute but have been learnt from general experience: The Higgs mechanism is an expensive luxury: each time we introduce a Higgs field we have to accept many new free parameters in the system. There are always more free parameters than there are masses in the theory, so we can never obtain a reliable mass relation for the elementary constituent particles. Of course, masses of composite systems are not free but can be calculated. To break large groups like SU(4) or SU(3) x SU(3) by means of the Higgs mechanism is hopelessly complicated. Theories with a small gauge group like SU(2) x U ( l ) or large but unbroken groups are in a much better shape.

Of course, these are practical arguments, that distinguish useful from useless theories. But do they also distinguish good from false theories? Personally I tend to believe this. I find it very difficult to believe that nature would have created as many Higgs fields as are necessary to break the big symmetry groups. I t is more natural to suppose that just one or two Higgs fields are present and some remaining local symmetry groups are not broken a t all. 2. Strong Interaction Theory

a. Towards permanent color binding

There is a general consensus on the idea that gauge vector particles corresponding to the color group SU(3)c can provide for the necessary binding force between quarks, which transform as triplet representations of this group. The states with lowest energy are all singlets. This theory explains the observed selection rules and the SU(6) properties of the hadrons. But now there are essentially two possibilities.

176

The first possibility is that SU(3)' is broken by the Higgs mechanism, so that the masses of all colored objects are large, but finite. The $J particles can be incorporated in this scheme: they may be the first colored objects, as you heard in the sessions on color theories. Besides the disadvantages of such theories I mentioned before, it is also difficult to arrange suppression of higher order contributions to KO - K O mixing and K L -+ p+p- decay25) and the theoriee are not asymptotically free. The alternative possibility is that SU(3)' is not broken at all. All colored objects like the quarks and the color vector bosons have strictly infinite r n a e ~ e s ~ ~I~suspect ~ ~ ) . that this situation can be obtained from the former one through a phase transition. Let me explain this. In the Higgs broken color theories there exist "solitons", objects closely related to the magnetic monopole solutions2")in the Georgi-Glashow Now let me continuously vary the parameter p2 in the Higgs potential

from negative to positive values (Fig. 1).

p2< 0 Fig. 1. The Hi=

potential before and after the phase transition.

We keep A fixed. Now the vacuum expectation value F H ~of~the . Higgs field 4 is first roughly proportional to 1p1 and so are the vector boson mass M v and the soliton mass, 4~Mv MI"-, !12

as indicated on the left-hand side of Fig. 2.

177

Fig. 2. The phase transition. M F

H = mass ~ ~ of unbroken ~ ~ Higgs field H = vac. ~ exp. ~ value ~ ~of Higgs field M , = mass of soliton Fs = vac. exp. value of soliton field g = color (electric) coupling constant j = soliton coupling constant ( g j = 2an)

What I assume is that when p2 becomes positive, it is the soliton’s turn t o develop a non-zero vacuum expectation value. Since it carries color-magnetic charges, the vacuum will behave like a superconductor for color-magnetic charges. What does that mean? Remember that in ordinary electric superconductors magnetic charges are confined by magnetic vortex lines; as described by Nielsen and Olesen3’). We now have the opposite: it is the color charges that are confined by “electric” flux tubes. So we think that after the phase transition all color non-singlets will be tied together by “strings” into groups that are color singlets. In this phase the Higgs scalars play no physical role whatsoever and we may disregard them now. The great shortcoming of this theory is that it is intuitive and as yet no mathematical framework exists. But there are various reasons to take it seriously.

Theoretically: i) we see it happen if we replace continuous space-time by a sufficiently coarse latticez7): the action becomes that of the Nambu string. ii) We see it happen in the only soluble asymptotically free gauge theories: Schwinger's mode131) and, even better, in the SU(o0) gauge theory in two space-time dimension^^^). Experimentally: a) the flux lines would behave like the dual string and thus explain the straight Regge-trajectories%). b) This is the simplest and probably the only asymptotically free theory that explains Bjorken behaviourz2). c) The theory is closely related to the rather successful MIT bag

In principle there exists an intermediate possibility: we probably have several phase-transitions when we go from unbroken SU(3) to for instance SU(2), U ( l ) and finally complete breaking. We will not consider the possibility that we are in SU(2) or U(1), but we must remember that from the low lying states it is difficult to deduce in what phase we really are. If we are in the unbroken phase then SU(3) color must commute with weak and electromagnetic SU(2) x U(1), and with SU(3) flavor'). Consequently, we are obliged to introduce charm35), or even more new quarks perhaps36). Now let us consider the 11, particles. b. Charmonium The most celebrated theory for $Jis that it is a bound state of a charmed quark and its a n t i p a r t i ~ l e ~Charmed ~). quarks are assumed to be rather heavy. The size of the bound state wave function will therefore be small and we can look a t the thing through a microscope (i.e. apply a scale transformation). Then we see rather small couplings, so we may use perturbation expansion to describe the system. The mathematics is do-able here! At first approximation the gluon gauge field behaves exactly like Maxwell fields, even the SU(3) structure constants can be absorbed in the coupling constant. We can calculate the

*)i.e. the familiar broken symmetry group that transforms p , n and X into each other. The word "flavor" has been proposed by Gell-Mann to denote both isospin and strangeness.

179

Table.

ORTHO 11

3.7

3.6

3.5

3.4

3.3

3.2

3.1

hadrons

3.0

T

T 2:'

1':

0"

1--

180

T

T

0-+

1 +-

J pc

Tsble.SU(4) predictions pew Meeona

charm

JP=0-

1

mass (GeV)

2

+

+

+ o

+ o

0

0

0 0

-1

JP = 1'

-

0

2t

+ 2+

t

2

+ o

5

t

2+

+

3.7

+ o + o

2t

+ 1

-

2+

3

2

3.1

+ o + o

0

2.5

0

annihilation rate and level splitting8 exactly as in positronium. The annihilation rate of the positronium vector state is

For charmonium the formula would be')

where a, = g1/3+ (the subscript s standing for "strong"). It fits well in the renormalization group theory if a, is around 1/3 at 2 GeV. That it is raised to the sixth power explains the stability of 4. Note that this *)The quark flavora in this paper are called p , n, p' and A. More modem is the nomenclature u, d , c a d u

181

is the rate with which the two quarks annihilate. It is independent (to first approximation) of the details of the hadronic final state3g). Now this gauge theory for strong interactions gives precise predictions on the other charmonium states (Table 1) and the charmed hadrons (Table 2). They can be found in some nice papers by Appelquist, De Rcjula, Politzer, Glashow and others3?). It is very tempting t o assume that this new quark is really the charmed one as predicted by weak interaction theories (see Sect. 3), but it could of course be an “unpredicted” quark. If the predictions from this theory come out to be roughly correct then that would be a great success both for the asymptotically free gauge theory for strong interactions, and for the renormalization theories that predicted charm. 3. T h e Weak Interactions

a . SU(2) x U ( l ) theories A simple SU(2) x U ( l ) pattern seems to be compatible with all experimental data on pure leptonic and semileptonic processes (neutral currents and charm), but with the pure hadronic weak interactions we still have a problem: there should be AZ = 3/2 and AZ = 1/2 transitions with similar strength and we only see A1 = 1/2. Also, these interactions seem to be somewhat stronger than other weak interaction processes. The traditional way to try to solve the problem is the possibility that A1 = 1/2 is “dynamically enhanced”. Seen through our “microscope” a t momenta of the order of the weak boson mass, the AZ = 1/2 and A I = 3/2 parts of the weak interaction Hamiltonian may be equal in strength, but when we scale towards only 1 GeV, then AZ = 1/2 may be enhanced through its renormalization group equations. This mechanism really works, but can only give a factor between 6 and at most 14 in the amplitudes40). But there are many uncertainties, since we do not know exactly from where to where we should scale, and what the importance is of the higher order corrections. I t has been argued that a similar mechanism might depress leptonic decay modes of charmed particles, thus giving them a better camouflage that prevents their detection41) and the mechanism could influence parity and isospin violation4’). An interesting alternative explanation of the AZ = 1/2 rule has recently been given by De Ri?jula, Georgi and G l a s h ~ w ~ ~In) .usual SU(2) x U ( l ) theories only the left-handed parts of the spinors may be SU(2) doublets, all

182

right-handed parts are singlets. These authors however take the right-handed parts of p' and n to form a doublet. This adds to the hadronic current P'Yp(1 - 75)nI (5) which is only observable in purely hadronic events or charm decays. Note that there is no Cabibbo rotation. We get among others a term,

+

cos 0, F ' 7 p ( 1 7 5 ) A E Y p ( 1 - 75)d (6) in the effective Hamiltonian. Note the absence of Cabibbo suppression by factors sin 8, and the AZ = 1/2 nature of this term.

Fig. 3. a) The new strangeness changing processes, b) diagram for the decay A'c) conventional picture of K- 4 27r.

4

2?r,

The predictions on neutrino production of charmed particles, and the charm decay rates are radically changed by this theory. We can have v p +p-Cf+, (7) etc. with no sin2 8, suppression, and pseudo AS = 2 processes can take place like UP + p-pi(+ DO

p-v-Kt (8) because of Do-Do mixing (as in K&o mixing). Experts however are skeptical about the idea. The charmed quark in this current has to violate Zweig's rule and the charmed quark loop must couple to gauge gluons before the current can contribute.

183

The idea is very recent and I have not yet seen any detailed calculations. We will soon know what the contribution of such diagrams can be. Furthermore, the K L - K S mass difference, when naively calculated in this model seems to be too large compared with experiment. b. Other and larger groups

SU(2) x U ( l ) theories do not really unify weak interactions and electromagnetism. The U( 1) group may be considered as the fundamental electromagnetic group, and its photon is merely somewhat mixed with the neutral component of the weak SU(2) gauge field. True unification occurs only then when we have a single compact group. 1. Heavy lepfons

The first example was originally invented to avoid neutral currents: the Georgi-Glashow O ( 3 ) modelz9). Now we have a large class of based on SU(2) x U ( l ) , 0 ( 3 ) , 0 ( 4 ) , O(4) x U(1), SU(3), SU(3) x U(1), SU(3) x SU(3) etc., all predicting new leptons. An extensive discussion of these models is given by Albright, Jarlskog and Tjia53). Table 3 gives the predicted leptons in these schemes. Albright, Jarlskog and Wolfenstein also analysed the possibilities t o detect such objects by neutrino productions4). Of course, we can always extend the lepton spectrum in other ways, for instance by adding new representation^^^), so that we get more members in theseries (,”), (L),etc.

(:),

2. The Pali-Salam Model56) Theoreticians are eagerly awaiting the discovery of the first heavy lepton. But that may take quite a while and in the meantime we search for more guidelines to disclose the symmetry structure of our world. One such guideline is that eventually we do not expect that baryons and leptons are essentially different. That is, they might belong to just one big multiplet. This, and other ideas of symmetry and simplicity led Pati and Salam to formulate their most recent “completely unified model”. Leptons and quarks from just one representation of SU(4) x SU(4), of which color SU(3) and weak SU(2) x U ( l ) are subgroups. If, as argued before, color is unbroken then we are free to mix the photon (also unbroken) with colored bosons, so there is no physical difference between the charge assignments

184

w3. Heavy leptons Gauge group

sup) x

Muonic leptons

cc-

U(1)

Refs. Weinberg, Salam'

UP

~4'8''

)

~

Bjorken and Llewellyn Smith no. s5') BCg and Zee50)

UP

cc-

MO

M+

"

P-

Mo

M+

u,,

IA-

Prentki and Z u m i n ~ ~ ~ )

M+

Mo

fi-

Bjorken and Llewellyn Smith no. 551) Prentki and Z ~ m i n o ~ ~ )

Bjorken and Llewellyn Smith no. 451)

Mi Georgi and GlashowZ9)

O(3)N SU(2)

P-

up

M+

Bjorken and Llewellyn Smith no. 651)

M+

Pais45)

M+

Pais'6)

Mo O(4)

= SU(2) x

SU(2)

"9

cc-

Mo

UP

cc-

Mo ~

~

cc-

SU(3) x U(1)

~~

UP

Schechter and Singer")

M-

M-

Mo MfIo

185

Aft

Albright, Jarlskog, T j i 8 )

Table 3. (Continued) ~~

Gauge group

Leptons

U(3)

Refs.

left

right

Eo

M+

E-

Eo

Eol

right

M"

E+

Eo

pt

Salm and Pati 4')

PP

Ve

E" EO'

E+

M-

MoI

M-

Mol

p-

left

e-

e+

EO)

MQI

E-

Salam and Pati ')

uc

UP

M"

p-

M+

Mo

eS

Mo MO'

p+

Eo'

E-

Y e , Yp

SU(3)

x SU(3)

e-

Acluman, Weinberg4')

P+

LO

-113 213

-1/3 213

-1/3

-1

213

-1

0

0

0

1

1

-1

because the photon may freely mix with components of the unbroken color vectors fields*).

*)If the integer-charge assignment is adopted however, then leptons couple directly to color gluom, and this would make interpretation of the experimentally observed ratio R = a(hadrons)/a(p+p-) in e t e - annihilation more complicated. So R must be computed from the non-integer charges. This 16plet yields R = 10/3.

186

4. The Higgs Scalars

The ugly ducklings of all Unified Theories are the Higgs scalars. They usually bring along with them as many free parameters as there are masses in the theory or more. This makes the theories so flexible that tests become very difficult. These scalars are needed for pure mathematical reasons: otherwise we cannot do perturbation expansions, and we have no other procedure a t hand to do accurate calculations. But do we need them phy~ically?~') Various attempts have been made to answer this question negatively. First of all they are ugly, and physics must be clean. But that is purely emotional. Then: we do not observe scalars experimentally. But: there is so much that we do not observe: quarks, I.V.B.'s, etc. Linde and Veltman raised the point that the scalars do something funny with gravity: their vacuum-expectation value gives the vacuum a very large energy-density. That should renormalize the cosmological constant. (Otherwise our universe would be as curved as the surface of an orange5"). On the other hand the net cosmological cosntant is very many orders of magnitude smaller than this. How will we ever be able to explain this miraculous cancellation? In this respect it is interesting to note an observation made by Z ~ m i n o ~ ~in) )supersymmetric : models there is no cosmological constant renormalization, even a t the one-loop level. Ross and Veltman then suggest that perhaps one should choose the scalars in such a way that the vacuum-energy density vanishes")). In Weinberg's model that means that one should add an isospin 3/2 Higgs field. Such a Higgs field could reduce neutral current interactions, and that would be welcome to explain several experiments. Personally I think that one has to consider the renormalization group to see what kind of scalars are possible. If we scale to small distanceslo6) then the theory has nearly massless particles. Only a nearly exact symmetry principle can explain why their masses are so small at that scale. For fermions we have chiral symmetry for this. Scalar particles can be forced to be massless if they are the Goldstone bosons of some global symmetry. They then have the quantum numbers of the generators of that symmetry group. Since all global symmetry groups must commute with local gauge groups, it is difficult to get light scalar particles that are not gauge singlets. According to this argument it is impossible to have scalar Higgs particles, except when they are strongly interacting, since in that case we cannot scale very far because of non-asymptotic-freedom.

187

A notable exception may be constructions using supersymmetries (see Sect. 5). In connection with Higgs’ scalars I want to clear up a generally believed misconception. It is not true that theories with a Higgs phenomenon in general cannot be asymptotically free. For many simple Higgs theories one can obtain asymptotic freedom, provided that certain very special relations are satisfied, between coupling constants that would otherwise be arbitrary62). These theories do not have a stable fixed point at the origin but I cannot think of any physical reason t o require that. Examples of such theories are supersymmetric theories, with possible supersymmetry breaking masses. Quark theories of this nature do not exist.

Fig. 4. Example of an unstable ultraviolet fixed point at the origin of parameter space. The arrows indicate the change of the coupling constants as momenta increase. The solid line is

the collection of asymptotically free theories.

For weak interactions however I do not think that asymptotic freedom would be a good criterion, because any change of the theory beyond, say 10000 GeV would alter the relations between coupling constants completely. 5. Supersymmetric Models

Symmetry arguments can be deceptive. In the nineteenth century physicists argued that the Moon must be inhabited by animals, plants and people. This was based on theological and symmetry arguments (earth-moon-symmetry), similar to the ones we use today.

188

Keeping this warning in mind, let us now consider the supersymmetric modelseg). For supersymmetry we have a special session. But an interesting attempt by F ~ a y e t ~ must ~ ) be mentioned here, who constructs a Weinberglike model with supersymmetry. There, fermions and bosons sit in the same multiplet. Thus the photon joins the electreneutrino. But which massless boeon should join the muon-neutrino? This problem has not been solved, to my knowledge. Even if supersymmetry would be ruled out by the overwhelming experimental evidence that fermion and boson masses are not the same, I think we do not have to drop the idea. Suppose that quantum field theory begins at the Planck length; we cannot go further than the Planck length as long as we do not know how to quantize General Relativity. And suppose that Quantum Field Theory is supersymmetric but the supersymmetry is very slightly broken (it is a global symmetry, 80 we may do that) and the breaking is described by a coefficient of the order of Suppose further that the representations happen to be such that there are no supersymmetric mass terms. Then at the mass scale of 1 GeV we would have all dimensionless couplings completely supersymmetric, but mas8 terms arise that break supersymmetry, because l(GeV)a is in our natural units defined by the Planck length. I would like to call this relaxed supersymmetry and it is conceivable that many interesting models of this nature could be built106). 6. PC-Breaking

Theories without scalars are automatically PC invariant. To describe PCbreaking we can either introduce elementary scalar fields with PC = -1 and put PGodd terms in the Lagrangian, or believe that all scalars are in principle composite. Then PC-breaking must be spontaneous as described by T.D. Leeea). Usually however the scalars are not specified, but only the currents66). De RGula, Georgi and Glashow observe that their chiral current can easily be modified to incorporate PGbreaking, a scheme already proposed by Mohapatra and Pati6') in 1972: A J = ~ ' 7 ~ ( 75)(n l - cos g5 i A sin g5), (9) g 5 < 1.

+

7. Quantum Gravity

Quantum gravity is still not understood, but an interesting formal interpretation is given by Christodouloum). He gives a completely new definition of

189

time, in terms of the distance in "superspace" between two three-dimensional geometries. But his formalism is not yet in a shape that enables one to give interesting physical predictions, and he has not yet considered the problem of infinities. Renormalizable interactions between gravity and matter have not yet been found69*70)but Berends and Gastmans find suggestive cancellations in the gravitational corrections t o anomalous magnetic moments of leptons71). Numerous authors tried to put terms like &R2 or/and &RPv RP" in the Lagrangian, and thus (re)obtain renormalizability. It is about as clever as jumping to the moon through a telescope. Personally, I am convinced that if you want a finite theory of gravity, you have to put new physics in72). Very interesting in that respect are the attempts by Scherk and Schwarz to start with dual models. 8. Dual Models

Though originally designed as models for strong interactions, the dual models have become interesting for other types of interactions as well. They are the only field theoretical scheme that start from an infinite mass spectrum, forming Regge-trajectories with slope a'. In the limit a' + 0 they can mimic not only many renormalizable theories but also gravitation (always in combination with matter fields). If the tachyon-problem and the 26-dimension problem can be overcome then one might end up with a big renormalizable theory that unifies e~erything~~). Scherk and Schwarz point out that for a' # 0 those models are equally or better convergent than renorma1izal.de theories. 9. Two-Dimensional Field Theories

Field theories in one-space and onetime dimension are valuable playgrounds for testing certain mathematical theorems that are supposed to hold also in four dimensions. There is no time now to discuss all the interesting developments of the past years74) but I do want to mention just three things. In a recent beautiful paper S. Coleman75)explains the complete equivalency between two seemingly different structures: the massive Thirring model on the one hand, and the SineeGordon model on the other. The solitons (extended particle solutions) in one theory correspond to the fermions of the other. Thus we get one of the very few theories that can be expanded both at small and at large values of the coupling constant.'] *)Note added: this result would later be challenged, the equivalence of these two models is not complete.

190

Solitons, and their quantization procedure have been studied in two and four dimensions by many g r o ~ p s ~ " ~ Even ~ ~ ~ in ' ~theories ). with weak coupling constants, solitons interact strongly (they have very large cross sections!) so they could make interesting candidates for an alternative strong-interaction theory. N o convincing soliton theory for strong interactions does yet exist, but several magnetic-monopole quark structures have been c o n ~ i d e r e d ~ ~ * ~ ~ ~ ~ " ) . Thirdly we mention the gauge theories in two dimensions with gauge group U(N) or SU(N). They are exactly soluble for either N = 1, mfermion = 0, or for: N + 00, mfermionarbitrary, in which case the 1/N expansion is possible. These theories exhibit most clearly the quark confinement effect, when color is ~ n b r o k e n ~ l tQuark ~ ~ ) . confinement is almost trivial in two spacetime dimensions because the Coulomb potential looks like Fig. 5 . V

Fig. 5. The Coulomb potential iu one space-like dimension.

QED in two dimensions (Schwinger's model) is not a good confinement theory because the electrons manage to screen electric charges completely, so the dressed electron is free, except for one bound state. At N + 00, probably a t all N > 1, we get an infinity of bound states, "mesons" that interact with strength proportional to 1/N.And they are on a nearly straight trajectory (actually a series of daughters because there is no angular momentum). 10. The Two Eta Problems The latter one-space onetime dimensional model has lots of physically interesting properties. One is, that we can test ideas from current algebra. As in its four-dimensional analogue, we have in the limit where the quark masses mp, m, + 0 an exact SU(2) x SU(2) symmetry, and the pion mass goes to zero in the s e n ~ e ~ ~ ~ ' ~ ~ )

191

m:

-, C(mp + m,) .

(10) The proportionality constant contains the RRgge slope. In two dimensions we have (11) C=gJ;;=l/&. So if m, is small, then mp and m, must be very small, in the order of 10 MeV. Now we can consider two old problems associated with the etameson. The first is that our theory seems to have U(2) x U(2) symmetry, not only SU(2) x SU(2). Thus, there should also be an isospin-zero particle, 11, degenerate with 80,whose mass should vanish. Experimentally however, mi > m:. Also we do not understand why the ?ro - q splitting goes according to TO = p p - fin; 0 = p p + nn despite of possible mass differences of p and n. This problem reappears every now and then in the literature. Fritzsch, GellMann and Leutwyler”) gave in 1972 a beautiful and simple solution: there is an Adler-Bell-Jackiw anomaly associated with the chiral U( 1) subgroups1):

a, j,”

= 2mj5

+ C C , , , ,T~(G,,G,~). ~~

(12)

The symmetry is broken and the eta-mass is raised. But then came the big confusing counter argument: we can redefine the axial current so that it is conserved, by writing

3:

(

= j[ - 2Ccp,,p Tr A, &Ap - apA,

2 + :g[Aa, Ap] , )

(13)

so there should be a massless eta after all. Considerable effort has been made

the past years to show that this counter argument is wrong. There are four counter counter arguments, all based on the fact that 3: is not gauge-invariant:

3;

is not gauge invariant, therefore the massless eta carries color, so it will be confined and thus removed from the physical spectrum8’). ii) The gluon field that occurs explicitly in the “corrected” equation for the axial current is a long-range field. The Goldstone theorem does not apply when long-range interactions are present. iii) In the Lorentz-gauge there is no explicit long-range Coulomb force. But then there are negative metric states and this eta may be a ghost.. It will be cancelled by other ghosts with wrong metricm). iv) But the simplest argument is that we can calculate the eta mass exactly in two dimensions and see what happens. In two dimensions there is an anomaly only in the U(l) fieldlo7): i)

8, jt = 2mj6

192

+ CrPuF,,,.

(14)

It is exactly this anomaly that raises the mass of the eta (it can be identified with Schwinger’s photon), despite of the fact that we can find a new axial current: 3,” = j,”- 2C~,uAu 9 (15) which is conserved+). The second eta problem is the decay

It breaks G-parity and thus isospin. If we assume this decay to be electromagnetic then current algebra shows us that it should be suppressed by factors of rn:/rni, which it does not seem to bem). I think there is a very simple explanation in terms of the present theory: the proton- and neutronquark masses are free parameters in our theory, not “determined” by electromagnetism. Their difference follows from known hadron mass splitting8 and tends to unmix Fp and fin, that is, mix T O and q . We get

with the correct order of magnitudea6). The essential difficulty with the current algebra argument was that all SU(2) breaking effects were assumed to be electromagnetic. That would be beautiful, we could do current algebra by replacing 6, by 6, iqA, everywherea7). In a gauge theory this is wrong. As I emphasized in the beginning, mass differences cannot be explained by electromagnetism alone, they are arbitrary parameters and must be fixed by experiment. We’ll have to live with that. Only for composite systems mass differences can be calculated. This suggests of course that we must make a theory with composite quarks. We leave that for a next generation of physicists. Note that the breaking of SU(2) x SU(2) is governed by proton- and neutron quark masses alone. They are of order 10-20 MeV and differ by 5 MeV or so. So the breaking term of SU(2) x SU(2) also breaks SU(2) rather badly.

+

*)This argument is not quite correct, because in two apace-time dimensions the Goldstone realisation of a continuow aymmetry is impossiblea4).However, if we let first N = 00, then m = 0 then we do get nevertheless the Goldstone mode. This ir possible because the mesonmenon interactions decrease like 1/N. Note that in any CMC there arc no parity doublets for large N.

193

That is why you may not use PCAC and isospin together to get factors of mf/mi in the decay q -+ 3 ~ ' ) . 11. Misellaneous a. Perturbation expansion

The renormalizability of the perturbation theory for gauge fields being well settled, there are still new developments. The dimensional renormalization procedure is the solution to all existence and uniqueness problems for the necessary gauge-invariant counterterms. But if one wishes to circumvent the continuation to non-integer number of dimensions then the combinatorics is very hard. The Abelian Higgs-Kibble model can now be treated completely to all orders within the Zimmermann normal product formalism"") if there are no massless particles. Slavnov identities can be satisfied t o all orders in this procedure also in the non-Abelian Higgs-Kibble model if the group is semi-simple (no invariant U(l) group) and if no massless particles are present. Unitarity and gauge invariance have explicitly been proven this way in a particular SU(2) model. Massless particles are very complicated this way, but advances have been made by Lowenstein and Becchig9) in certain examples of massless Yang-Mills fields, and Clark and Rouetgo) for the Georgi-Glashow model. It is noted by B. de Witg1) that there is a technical restriction on the allowable form of the gauge-fixing term, relevant for supersymmetric models. The gauge-fixing term cannot have a non-vanishing vacuum expectation value in lowest order, otherwise contradictions arise. Of course, the Slavnov identities make the vacuum expectation value of this term vanish automatically in the usual formulation. b. The background field method The algebra in gauge theories is often quite involved. For certain calculations it would be of great help if gauge-invariance could be maintained throughout the calculation. On the other hand we must choose a gauge condition, which by definition spoils gauge-invariance right from the beginning. The trick is now to use the so-called background gaugeg2): the fields are split into a cnumber, called background field, and a q-number, called quantum field. Only *)As yet

we have no satisfactory explanation for a possible

194

A I = 3 component in 1 -+ 3n.

the quantum part must be fixed by a gauge condition, whereas gauge-invariance for the c-numbers can be maintained. The method was very successful in the case of g r a ~ i t y ~ ’ *and ~ ~ has ) also been applied to calculate anomalous dimensions of Wilson operator^^^^^^). The method can be generalized for higher order irreducible graphsIo5). c. Two-loop-beta The Callan-Symanzik beta function has now been calculated up to two loops for several gauge theoriesg5tg6). We have P(g) = Ag3

+ Bg5 + O(g7) .

Now A can have either sign, and can also be very close to zero. In general, B will not vanish (it can be either positive or negative). We can then get either an U V or an IR stable fixed point close to zero. In certain supersymmetric models, A vanishes. The question is whether perhaps p(g) vanishes identically for such a model. The answer appears to be: no, because for these models B has now been calculated also and it is non-zerog6).*) d. The infrared problem Massless Yang-Mills theories are very infrared divergent. As explained, we expect extremely complicated effects to occur, like flux tube formation and color confinement. A general argument is presented by Patrascioiug7) and Swiecag8) that shows that if we have a local gauge-invariance and if we can have isolated regions in space (-particles) with non-vanishing total charge, then there must exist massless photons coupled to that charge. This is only proven for Abelian gauge symmetries, but if it would also hold for non-Abelian invariance, then the absence of massless colored photons must imply the absence of any colored particles (= color confinement). e. Symmetry restoration at high temperatures

Just like a superconductor that becomes normal when the temperature is raised above a certain critical value, so can the vacuum of the Weinberg model become “normal” a t a certain temperature”). The critical temperature is typically of the order of

Mw kT- e

*)Note added later: this is now known to be a mistake; p vanishes in all orders in these models.

195

This assumes that Hagedorn’s limit on high temperatures”’) is invalid. Indeed it is invalid in the present quark theories, but the specific heat of the vacuum is very high because there are so many color components of fields. Observe that for SU(N) theories Hagedorn might be correct in the limit N-roo.

f. Symmetry restoration at high external fields If we consider extremely strong magnetic fields then also the symmetry properties of the vacuum might change”’). One can speculate on restoration of color symmetry, e - p symmetry, parity or CP restoration, and vanishing of Cabibbo’s angle. In still larger fields formation of magnetic monopoles2’) would make the vacuum unstable, as in strong electric fields. g. Symmetry restoration at high densities

A very high fermion density means that $$ and I&$ have a vacuum expectation valuelo2). This also can have a symmetry restoring effect. T.D. Lee, Margulies and Wicklo3) argue that chiral SU(2) x SU(2) might be restored a t very high nuclear densities. Thus the mass of one nucleon would go t o zero and perhaps very heavy stable nuclei could be formed. The most recent calculations show a very remarkable phase transition at no more than twice the normal nuclear density. Although the result is of course model dependent, this work seems to predict stable large nuclei with binding energy of 150 MeV/nucleon. At still higher densities we can speculate on more transition points. Again we think that the quark picture is more suitable than Hagedorn’s picture”*).

12. Conclusions a. Unifying everything What I hoped to have made clear at this conference is that gauge fields are likely to describe all fundamental interactions including, in a sense, gravity. This is a breakthrough in particle physics and deserves t o be called: “unification of all interactions”.

196

b. Unifying nothing But when we consider our present theory of strong interactions, the unbroken color version, then we see that it is unlikely to be really unified with weak and electromagnetic interactions, unless we go at ridiculously high energies, because the gauge coupling constants probably still differ considerably, and SU(3)m'0' commutes with SU(2) x U(1). Also if we look at weak and electromagnetic interactions, we see that true unification has not yet been reached. At small distances strong interactions become weak, weak interactions become strong and electromagnetic ones stay electromagnetic, but no unification yet. Perhaps our knowledge of the particle spectrum is still far too incomplete to enable us to unify their interactions. I have given air to my own feeling that we are going in the wrong direction by choosing larger and larger gauge groups. Perhaps we can use the confinement mechanism again to build quarks and leptons from still more elementary building blocks (chirps, growls, etc.). Instead of "unifying" all particles and forces, it is much more important to unify knowledge.

Acknowledgement I would like to thank D.A. Rms and M. Veltman for many interesting discussions.

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28. 29. 30. 31. 32. 33. 34. 35. 36.

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37. A. De Rfijula and S.L. Glashow, Phys. Rev. Lett. 34 46 (1975). Th. Appelquist,

38. 39. 40.

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75-069 (1975). 57. Many attempts have been made to formulate “dynamical symmetry breakdown”, i.e. without the simple Higgs scalar. See for instance: T.C. Cheng and E. Eichten, SLAGPUB-1340 preprint (Nov. 1973); E.J. Eichten and F.L. Feinberg, Phys. Rev. D10 3254 (1974). F.Englert, J.M. Frkre and P. Nicoletopoulos, Bruxelles preprint (1975); T. Goldman and P. Vinciarelli, Phys. Rev. D10 3431 (1974).

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58. A.D. Linde, Pis’ma Zh. Eksp. Teor. Fiz. 19 (1974) 320 ( J E T P Lett. 19 (1974) 183). J. Dreitlein, Phys. Rev. Lett. 33 1243 (1974). M. Veltman, Phys. Rev. Lett. 34 777 (1975). 59. B. Zumino, Nucl. Phys. B89 535 (1975). 60. D.A. Ross and M. Veltman, Utrecht preprint (1 April 1975). 61. See also: H. Georgi, H.R. Quinn and S. Weinberg, Harvard preprint (1974). 62. M. Suzuki, Nucl. Phys. B83 269 (1974). E. Ma, Phys. Rev. D11 322 (1975). N.P.Chang, fhys. Rev. D10 2706 (1974). 63. See Abdus Salam’s talk on Supersymmetries at this conference. 64. P. Fayet and J. Iliopoulos, Phys. Lett. S1B 475 (1974). P. Fayet, Nucl. Phvs. B78 14 (1974); preprint PTENS 74/7 (Nov. 1974, Ecole Normale SupCrieure, Paris), Nucl. Phys. B O O 104 (1975). 65. T.D. Lee, Phys. Rev. D8 1226 (1973); Phys. Rep. Qc(Z). 66. See A. Pais and J. Primack, Phys. Rev. D8 3063 (1973) and references therein. 67. R.N.Mohapatra and J.C. Pati, Phys. Rev. D11 566 (1975). 68. D. Christodoulou, Proceedings ofthe Academy of Athens, 47 (Jan. 1972); CERN preprint TH 1894 (June 1974). 69. G. ’t Hooft and M. Veltman, Ann. Inst. If. Poinmrd 20 69 (1974). 70. S. Deser and P. van Nieuwenhuizen, Phys. Rev. Lett. 32 245 (1974), Phys. Rev. D10 401 (1974), Lett. Nuovo Cimento 11 218 (1974), f h y s . Rev. D10 411 (1974). S. Deser, H.S. Tsao and P. van Nieuwenhuizen, Phys. Lett. SOB 491 (1974), Pbys. Rev. D10 3337 (1974). 71. F.A. Berends and R. Gastmans, Leuven preprint (Oct. 1974). M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 3203 (1975). 72. See also: G. ’t Hooft, in Trends in Elementary Particle Theory, Proceedings of the International Summer Institute on Theoretical Physics in Bonn 1974, ed. by H.Rollnik and K. Dietz, Springer-Verlag p. 92. 73. J. Scherk and J.H. Schwarz, Nucl. Phys. B81 118 (1974); Phys. Lett. 67B 463 (1975). 74. D.J. Gross and A. Neveu, Phys. Rev. D10 3235 (1974). R.G. Root, Phys. Rev. D11 831 (1975). L.F. Li and J.F. Willemsen, f h y s . Rev. D10 4087 (1974). J. Kogut and D.K. Sinclair, Phys. Rev. D10 4181 (1974). R.F. Dashen, S.K. Ma and R. Rajaraman, Phys. Rev. D11 1499 (1975). S.K. Ma and R. Rajaraman, Phys. Rev. D11 1701 (1975). 75. S. Coleman, Harvard preprint (1974). B. Schroer, Berlin preprint FUB HEP 5 (April 1975). J. Friihlich, Phys. Rev. Lett. 34 833 (1975). 76. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10 4114, 4130, 4138 (1974). P. Vinciarelli, 1993-CERN preprint TH (March 1975).

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3026 (1975). 78. A.P. Balachandran, H. Rupertsberger and, J. Schechter, Syracuse preprint rep. no. SU-4205-41(1974).G. Parisi, Phys. Rev. D 1 1 970 (1975).A. Jevicki and P. Senjanovic, Phys. Rev. D 1 1 860 (1975).See also interesting ideas by L.D. Faddeev, preprint MPI-PAE/Pth 16, Max-Planck Institute, Miinchen, June 1974. 79. This follows immediately if we assume that a term

c(pp

+ an)

in the quark Lagrangian corresponds to a term CIO

in the sy,mmetric Lagrangian for a sigma model of the pion, with c and c' of the same order of magnitude. See ref. 107. 80. H. Fritzsch, M. Cell-Mann and H. Leutwyler, Phys. Lett. 47B 365 (1973); W. Bardeen, Stanford report (1974,unpublished). 81. J.S. Bell and R. Jackiw, Nuovo Cimento 60A 47 (1969). 82. J. Kogut and L. Susskind, Phys. Rev. D9 3501 (1974);D 1 0 3468 (1974);D 1 1 3594 (1975). 83. S. Weinberg, Phys. Rev. D 1 1 3583 (1975). 84. S. Coleman, Commun. Math. Phys. 31 259 (1973). 85. D. Sutherland, Phys. Lett. 23 384 (1966). J.S. Bell and D. Sutherland, Nucl. Phys. B4 315 (1968). 86. See for instance ref. 107, and W. Hudnall and J. Schlechter, Phys. Rev. D9 2111 (1974).I. Bars and M.B. Halpern, Phys. Rev. D 1 1 956 (1975). 87. M. Veltman and J. Yellin, Phys. Rev. 154 1469 (1967). 88. C. Becchi, A. Rouet and R. Stora, Phys. Lett. 6 2 B 344 (1974);lectures given at the International School of Elementary Particle Physics, Basko Polje, Yugoslavia Sept. 1974, and a t the Marseille Colloquium on Recent Progress in Lagrangian Field Theory and Applications, June 1974;preprint 75,p. 723 (Marseille, April 1975). 89. J. H. Lowenstein and W. Zimmermann, preprints MPI-PAE/PTh 5 and 6/75 (Miinchen, New York, March 1975); Commun. Math. Phys. 44 73 (1975). C. Becchi, to be published. 90. T.Clark and A. Rouet, to be published. 91. B. de Wit, Phys. Rev. D 1 2 1843 (1975). 92. B.S. DeWitt, Phys. Rev. 162 1195;1239 (1967).J. Honerkamp, Nucl. Phys. B 1 8 269 (1972); J. Honerkamp, Proceedings Marseille Conf., 19-23 June 1972. G. 't Hooft, Nucl. Phys. B62 444 (1973). G. 't Hooft, lectures given at the XIIth Winter School of Theoretical Physics in Karpacz, Poland, (Feb. 1975). M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phvs. Rev. D 1 2 3203 (1975).

202

93. S. Sarkar, Nucl. Phys. B82 447 (1974); B83 108 (1974). S. Sarkar and H. Strubbe, Nucl. Phys. BOO 45 (1975). 94. H. Kluberg-Stern and J.B. Zuber, preprint DPh-T/75/28 (CEN-Saclay, March 1975); Phys. Rev. D12 467, 482 (1975). 95. D.R.T. Jones, Nucl. Phys. B75 531 (1974). A.A. Belavin and A.A. Migdal, Gorky State University, Gorky, USSR, preprint (Jan. 1974). W.E. Caswell, Phys. Rev. Lett. 33 244 (1974). 96. D.R.T. Jones, Nucl. Phys. B87 127 (1975). S. Ferrara and B. Zumino, Nucl. Phys. B70 413 (1974). A. Salam and J. Strathdee, Phys. Lett. 61B 353 (1974). 97. A. Patrascioiu, Inst. Adv. Study, Princeton preprint COO 2220-45 (April 1975). 98. J.A. Swieca, NYU/TR4/75 preprint (1975, New York). 99. C.W. Bernard, Phys. Rev. DD 3312 (1974). L. Dolan and R. Jackiw, Phys. Rev. DO 3320 (1974). S. Weinberg, Phys. Rev. DO 3357 (1974). B.J. Harrington and A. Yildiz, Phys. Rev. D11 779 (1975). 100. R. Hagedorn, Nuovo Cimento Suppl. 3 147 (1965). 101. A. Salam and J. Strathdee, preprint IC/74/140, Trieste (Nov. 1974); IC/74/133 (Oct. 1974). 102. R.J. Harrington and A. Yildiz, Phys. Rev. D11 1705 (1975). A.D. Linde, Lebedev Phys. Inst. preprint no. 25 (1975). 103. T.D. Lee and G.C. Wick, Phys. Rev. DO 2291 (1974). T.D. Lee and M. Margulies, Phys. Rev. D11 1591 (1975). 104. J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 1353 (1975). 105. G. ’t Hooft, “The background field method in gauge field theories”, lectures given a t the XIIth Winter School of Theoretical Physics in Karpacz, Feb. 1975, and at the Symposium on Interactions of Fundamental Particles, Copenhagen, August 1975. 106. The general idea to use the Renormalization Group to relate the theory a t our mass scales of order 1 GeV to theories at very small distances (Planck length), also occurs in: H. Georgi, H.R. Quinn and S. Weinberg, Harvard preprint (1974), and M.B. Halpern and W. Siegel, Berkeley preprint (Feb. 1975). 107. G. ’t Hooft, lectures given a t the International School of Subnuclear Physics “Ettore Majorana”, Erice 1975.

203

CHAPTER 3 THE RENORMALIZATION GROUP

.

Introduction .................................................................. . 206 [3.1] The Renormalization Group in Quantum Field Theony, Eight G d u a t e School Lectum, Doomrth, Jan. 1988, unpublished ....... 208

205

CHAPTER 3

THE RENORMALIZATION GROUP Introduction to the Renormalization Group in Quantum Field Theory [3.1] My first calculations of renormalization coefficients for gauge theories date from 1970. But at that time the delicate gauge dependence of the counter terms were not yet completely understood and it was hard to avoid making err01-s.'~By 1972 however I learned about a procedure that would simplify things considerably. While at CERN I met J. Honerkamp14 who explained to me the secalled background field method, that had been applied by B. DeWitt in his early gravity calculations. The basic idea was due to Feynman. It simply amounts to writing all fields A* (which may include the gauge fields as well as possible scalars) as

where the "classical" fields A*)'' are required to obey the classical equations of motion, in the possible presence of sources, whereas Aav'J" are now the "quantized variables", so that they serve as the integration variables in a functional integral. My first reaction was that this could only be a book-keeping device that would neither change the physics nor the kind of calculations one will have to do. But then Honerkamp told me about the trick: the gauge condition needs to be imposed only on the quantum fields, while it may depend explicitly on the classical fields. While fixing the gauge degrees of freedom for the quantum field, one may keep the expressions gauge invariant with respect to background gauge transformations. If we now renormalize the one-loop diagrams the counterterms needed will automatically be gauge invariant. 13The sign of the p-function for gauge theories was clear to me, already then, but at that time I was unable to convince my advisor of the significance of this observation. 14J. Honerkamp, Nucl. Phys. B48 (1972) 169.

206

The method enables one to produce a complete algebra for all one-loop p functions for all renormalizable theories in four space-time dimensions. We also use it in perturbative quantum gravity, and Chapter 8.1 gives another introductory text to this method. It does not work so nicely for diagrams with more loops because the overlapping divergences require counterterms also containing the quantum fields, so that the advantages disappear. The paper of this chapter is an unpublished set of graduate lecture notes, which had to be polished a bit.

207

CHAPTER 3.1

THE RENORMALIZATION GROUP

IN

QUANTUM FIELD THEORY

Eight Graduate School Lectures Doorwerth,

25-29 January 1988

C. ' t Hooft

I n s t i t u t e f o r Theoretical Physics Princetonplein 5, P.O. Box 80.006 3508 TA

UTRECHT, The Netherlands

1 Introduction

The values one should assign t o the f r e e parameters of a quantum field types

theory,

such as masses

( h i , g,), depend

on

( m i ) and

the

coupling constants of

renormalization

procedure

various adopted.

Although the physical properties of these theories should not be scheme dependent,

the

calculational

procedures

indeed

do,

in

a

rather

non-trivial way, depend on the subtraction scheme. I t w a s proposed by Stueckelberg and Peterman'"

i n 1953 t h a t one

should require f o r any decent renormalization procedure t h a t its r e s u l t s be invariant under the group of non-linear free

parameters,

and

they

consistency conditions upon this

group

renormalizable

the

noted the

that

theories

this

requirement

renormalization

"Renormalization

field

transformations among these

meet

One

group". these

constants. finds

requirements

would

imply

They

called

that

by

modern

construction,

and most of the requirements are r a t h e r t r i v i a l , with one exception. Renormalization

is

always

associated

procedure, which needs some s o r t of cut-off

with

a

parameter A

regularization

.

One subgroup

of the renormalization group coincides exactly with a redefinition of A: (1.1)

h+Kh,

and one f i n d s t h a t t h i s subgroup corresponds

with the group of

transformations: we a r e comparing a theory with itself

scale

with all masses

and all momenta scaled by a common f a c t o r K. Just

because

t h e renormalization procedure

208

is r a t h e r delicate one

finds that even those coupling parameters A i that one would expect to be scale-invariant

since

are

dimensionless,

actually

transform

And s o it turned out that the renormalization group has

non-trivially. an

they

important

application

in

quantum

field

theory.

Stueckelberg

and

Peterman had been thinking about the general diffeomorfism group in the parameter space ; but all that remains nowadays is the

multi-dimensional simple, one-

dimensional group of

scale transformations. Nevertheless,

the name "Renormalization group" stuck. The renormalization group becomes particularly non-trivial when the scale

transformations

information

on

the

that

we

behavior

consider of

are

amplitudes

large. at

We

get

extremely

extra

large

or

extremely s m a l l external momenta that would otherwise be difficult to obtain. But t h i s not only holds for quantum field theory. The same or similar procedures can be applied to statistical models. Here one also uses

the

words

"renormalization groupoorz1although

the

rationale

for

that is somewhat more obscure: a model is scale transformed and then compared

with

the

rigorous

treatment

original. the

it

But

scale

is sometimes observed that

transformation

can

only

be

in

a

performed

one-way, from smaller scales to larger scale theories, so that the group is actually a semi-groupt3'!

In these lectures we briefly discuss renormalization. For the basic imput

of

our

theory

we

do

not

need

many

details,

such

as

the

combinatoric proofs of its c o n s i ~ t e n c y ' ~the ~ , uniqueness of the results t5,Bl , as consequences of requirements such as causality and unitarity and the

independence

of

physically measurable

chosen gauge fixing parameters of

"counterterms" on cut-off

171

.

quantities

from freely

A l l we want to know is the dependence

parameters A,

and the relations between

"infinities" that follow. We then discuss a simple theory with one coupling constant and one

m a s s parameter such as there

are

three

A44

different

and quantum electrodynamics. We note that kinds

of

theories:

asymptotically

free,

infrared free, and scale-invariant. Some calculations a r e done in the next chapters and then we discuss the

more general

theories,

with

an arbitrary number

of

masses

and

compute

the

coupling constants. Also

for

the

more

general

case

we

want

to

renormalization group coefficients, but now we need a more powerful

209

method. We f i n d t h a t the various coefficients can be put i n a simple algebra, as a consequence of various internal symmetries of our models. One then needs t o do only a few calculations t o obtain the most general s e t s of renormalization group coefficients, up t o one loop. Two-loop calculations a r e much too lengthy to discuss here in any detail,

we

but

coefficients

if

show we

that

want

it

to

is

important

a

give

rigorous

(asymptotically f r e e ) quantum field theories. raised

in

proofs

that

the

beginning:

allow u s

maybe we

to

know

the

definition

We touch

two-loop

of

upon

can define models

certain

the point

rigorously,

to sum the perturbative expansion

but

without being

troubled by its possibly very divergent nature a r e s t i l l lacking. Quantum

chromodynamics

renormalization

group.

Many

cannot

be

attempts

have

understood been

made

without to

the

improve our

formulation of t h i s theory, f o r instance by "resumming" its perturbation expansion. Most of t h i s subject goes beyond the scope of these lectures but

we

briefly

indicate

some

sources

of

the

bad

behavior

of

the

perturbative axpansion. 2. Renormalization

Even though a mathematically completely rigorous description of quantum f i e l d theory

in 3+1 dimensions is not known,

it

generally thought

is

t h a t many d i f f e r e n t quantum field systems exist: models t h a t combine the requirements

of

quantum

r e l a t i v i t y on

the

other.

mechanics In

these

on

the

one

lectures we

hand,

will

and

explain

special

why,

and

which d i f f i c u l t i e s we encounter. A c e n t r a l point in quantum field theory is t h a t although we do not

know how t o construct rigorous models with f i n i t e , non-trivial

coupling

s t r e n g t h s , there a r e many perturbative models. These a r e a l l defined by postulating the infinitesimal,

(renormalized: see l a t e r ) coupling s t r e n g t h s , h i , to be and

everything

want

we

to

know

about

the

system

is

expressed as perturbative s e r i e s in h i :

fthj} = f,

+ f,iXi

in which a l l coefficients fNij

..

masses

mj)

convergence

are

uniquely

*+

f,jjhihj +

(2.1)

,

(depending in some definite way On the

computable,

or divergence of

...

the

but

series.

210

nothing Indeed,

is

said

quite

about

the

generally one

expects divergence of the form

.. I

IfNlj so

that

for

finite

[BI

3

(2.2) as

(2.11,

expressions

hi

,

CodvN!

they

are

stand,

not

meaningful.

We w i l l not t r y t o prove eq. (2.2) in these lectures. Here only we note t h a t , if t r u e , it would imply t h a t (2.1) may define f{A,) within a t i n y "error bar":

it makes a lot of sense in perturbative f i e l d theory

the s e r i e s (2.1) as soon as the N + l s t term becomes larger

t o c u t off

than the Nth. This happens when

at which point

and if t h i s is a good measure f o r the uncertainty in

then indeed

f

such models may be quite acceptable physically as soon as the coupling s t r e n g t h s h i s t a y reasonably s m a l l . A

standard

way

to

a

construct

perturbative

quantum

theoretical model is t o start off with a Lagrange density construct

functional

integrals

for

the

wanted

2

amplitudes.

field

and then One

then

as they occur in the

discovers t h a t the "bare" coupling constants

ASi Lagrangian are not the appropriate parameters t o use in the expansion

series (2.11. The coefficients We

will

now

explain

f

would then be infinite.

Nij..

what

introduced some s o r t of cut-off

this

really

means.

number of ways. For one, we could simply postulate Hilbert

space

momentum we

k

require

are with

considered

out(kl,k2 ,... Ip1,p2,...

to

)in

pl0+pzO+ ... does not exceed

that

there

are

no

in

which

any

hold

rigorously

2Amar

we

has

ha,. S

spacelike

In t h i s case

matrix

elements

kl0+kzo+... =

, because only then we a r e guaranteed states

IQ)

forbidden particles i n the u n i t a r i t y condition

- - - - - #We use S t i r l i n g ' s formula: N!

for

f o r which the t o t a l energy,

intermediate

that

t h a t no states In

particle

Ikl>ba,, , f o r some large number

u n i t a r i t y only

Imagine

in our theory. This could be done i n a

.

3

21 1

containing any

of

the

(2.5)

This would make the theory f i n i t e but no longer Lorentz-invariant. would be u n i t a r y up till energies 2 k a x A

better

particles.

way

t o make the

theory f i n i t e

is t o

add

"unphysical"

These a r e particles t h a t contribute t o the u n i t a r i t y relation states 10) IQ)(QI is wrong, as if

( 2 . 5 ) intermediate

sign of

It

.

that

may have

"negative metric":

the

state is determined by the value,

(The normalization of a one-particle

and s i g n , of the residue of the pole in the propagator). Loop integrals

can

then

made

be

to

converge

so

fast

that

ha, of the previous paragraph can s a f e l y go

finite-momentum cut-off

t o i n f i n i t y such t h a t the l i m i t exists, which now is Lorentz by

construction.

we

If

choose

the

particles t o exceed a c e r t a i n number up till energies forbid

hpv

intermediate

subscript

PV

for

regularization m e t hod'

of

all

these

invariant unphysical

then we a l s o have u n i t a r i t y

because energy conservation

would

I Q) containing these "sick" particles. The Pauli

*' .

masses A,,

2hPV

or

states

stands

the

and

Villars

who

first

proposed

this

A t h i r d way t o make amplitudes f i n i t e is much more subtle and w i l l

be used very often here. We imagine changing the theory a l i t t l e b i t by postulating

I el

that

the

number

of

space-time

dimensions,

n,

deviates

s l i g h t l y from 4:

n = 4-E , c infinitesimal.

(2.71

I t is not possible t o give a mathematically rigorous definition of what t h i s means unless we formulate everything in perturbation expansions (or if we abandon translation invariance, see ref

n

In Feynman diagrams over space-time momenta

k

n

only occurs in the closed loop integrals

:

Sdnk and f o r large enough

1.

'lo]

+

Sd4-"k ,

such integrals may be well-defined,

212

(2.8)

if

the

integrand contains only explicit external momenta dimensional subgroup of Minkowski space

f (

n - d )/ 2

m

r(( n - d ) / 2 )

that span a

pi

d
:

n-d-1

Jddkl I d r 2 r

2

f{k,,pi;r }

.

(2.91

0

The subtle point is that the integral (2.9) may diverge just about

as badly as the original integral with n integer, but t h i s time there is a unique way to define its "finite part" a s long a s n is non-integer. most

cases

this

finite

part

can

simply

continuation of the integral from sometimes

it

is

also

defined as

the

analytic

values where it converges, but

n

infra-red

be

In

divergent

integrand at finite or zero values of

kl

because

of

and/or

r

poles

.

in

the

In the latter

case one can still define the finite part by splitting these divergent pieces apart, or by repeated partial integrations. What one now observes is that expressions, though finite at generic

n

+

l/c3

4

,

.

They a r e of the form

...

similar to

time

unless

"dimensionally regularized"

, may still produce poles a t (for one-loop diagrams), or 1/c2 , n

at higher loops. The divergent term l/c plays a role very l o g b a x or logApV in the other regularization schemes'.

Again, at sufficiently small this

1/c

these

because

there

is

the

E

phase

some

unitarity is only very slightly violated, space integrals

lnfra-red

(which always converge,

divergence

which

must

be

handled

separately) w i l l approach the physical values. Coming back

to the coefficients

of

fN i j . .

eq.

observes that, quite generally, they diverge, either a s 1/&

+

m

.

So

at

first

sight

seems

it

that

(2.11, one ELpv

these

m

now

or as

theories

are

inconsistent. However, the coefficients

that we started off with are not

A,

directly physically observable. Throughout, we w i l l adopt the philosophy (renormalization Ansatz) that one is allowed to choose these arbitrary numbers to vary with

A,,dx

or

Apv

'In

or

E

: for

instance, in the

those schemes one may also encounter higher divergences going a power of A (linearly or quadratically). These have been eliminated in the dimensional scheme. One might say that our "finite part" procedure eliminated them. as

213

dimensional procedure we substitute into the s e r i e s (2.1): (2.10) where now E

(or

a r e defined t o be t r u l y f i n i t e constants, independent of

XRi

if another regulator w a s chosen). The coefficients

A

of second o r t h i r d order in

of successively higher order in

At

sight

first

a

IVl

.

AR

seems odd

it

are

a‘:’

and the higher order coefficients

hR

that

are

Asi

both

infinite

and

infinitesimal. We have to keep in mind t h a t eq. (2.10) only makes sense in the l i m i t

A

R

+

0

f i r s t , then

group w i l l enable u s t o replace

+

E

0

.

Indeed, the renormalization

(2.10) by a n expression

in which the

l i m i t s may be reversed. Since both

and

hR

perturbation expansion in

As hR

a r e t o be considered infinitesimal the is equivalent t o the one in

is important t o note t h a t since the

.

As

a r e higher order in

a‘y’

But it h

the

expansions a r e term by term different. There is lots of a r b i t r a r i n e s s in (2.10). A l l we want is t h a t any f i n i t e value f o r measured

produces some f i n i t e values f o r the physically

hRi

quantities

f{hBj)

f N i j has poles at

functions

in

the

limit

+

E

0.

The

s e r i e s of

and s o does the s e r i e s (2.10). We

E=O

want to define the s e r i e s (2.10) in such a way t h a t , when substituted into

(2.1) we

divergences

at

get

new expansions E=O

.

If

in

this

is

that

hRi

possible

are

the

all

theory

free is

from called

renormalizable. To find out whether a theory is renormalizable is in principle very simple:

( i l Write down all interaction terms in the Lagrangian t h a t agree with the chosen symmetries of your model. ( i i ) Expect a l l

possible

terms

t o be required in the s e r i e s

(2.11) f o r the bare coupling constants, as long as these

( a ) have the correct symmetry transformation r u l e s under whatever symmetry t h a t w a s imposed on the model, and ( b ) have the correct dimension.

(.i.fi) I f ,

following

the

above

rules,

a

certain

subset

of

couplings does not get renormalized once they are put equal t o zero then, and only then, we can leave such couplings out.

214

Requirement (lib) follows from t h e assumption t h a t the i n f i n i t e p a r t s of

r1

cannot be chosen f r e e l y (two d i f f e r e n t choices cannot possibly both

a[

yield a f i n i t e theory,

but

1, so they should follow from t h e s t r u c t u r e of the

f

then

they should not depend on our choice of

units

for

lengths and masses. In general, if s p a c e - t h e has 4 dimensions, only a f i n i t e number of coupling

constants

constructed.

positive

or

dimensions

then

step

(iif) allows

Only a f i n i t e number of

o t h e r s out.

vanishing

can

They should s u f f i c e t o

coupling constants a r e l e f t .

Eq.

render

to

positive the

of

us

be

all

(2.10) then only contains a f i n i t e number order.

dimension

i f we choose all coupling constants of

But then,

vanishing

or

with

possible

everything f i n i t e .

leave terms

at each

They do.

One

condition is t h a t t h e r e a r e no "anomalies", but t h i s is j u s t the obvious (and

very

important)

requirement

that

also

our

regulator

procedure,

whichever we chose, obeys the chosen symmetries. The student should be cautioned t h a t the above is not a proof of the renormalizability of f i e l d theories, but must be seen as a summary of

the

result

l i t e r a t u r e [ 4-71

of

such

a

proof,

which

can

be

found

in

the

.

theory

3.

To i l l u s t r a t e our methods we could e i t h e r take the simplest theory which is

just

could

a single self-interacting

take

more

complicated

scalar,

cases

called

containing

theory,

fermions

and/or

or

we

gauge

f i e l d s . The disadvantage of the pure s c a l a r theory is t h a t t h e r e is no f i e l d renormalization at one loop, which makes it a l t t l e too simple. We w i l l take t h i s simplest case anyway, but we'll do as if there is a f i e l d renormalization. The Lagrangian is

Here

J(x)

is a space-time dependent source insertion which we put in

t h e r e f o r technical convenience. Later it w i l l be put equal t o zero. The sub-

or

superscript

B

stands

for

"bare"

or

unrenormalized.

These

q u a n t i t i e s a r e not d i r e c t l y observed. We write

(3.2)

215

where

hR , mR

and

are t r u l y f i n i t e . The normalization of

$R

J

is

reveiled

by

not yet f ixed. Physically computing

observable

vacuum

expectation

The first of

etc.

features

of

model

( O I A x ) 10)

values

these is zero.

the

are

, (OIJ(xl)J(x2)10)

The second gives

us

the

,

"physical"

propagator. In diagrams Fig. 1.

x

X

x+x

+

+

x

-x

+ .

Fig. 1

The f i r s t diagram f o r the self energy insertion gives only a momentumindependent contribution. It c a n be removed by a m a s s renormalization. The second diagram h a s a non-trivial momentum dependence but is of too high order in

he

f o r o u r considerations.

In theories with fermions or

gauge particles we have a l s o diagrams of the kind given in Fig. 2

We'll now pretend t h a t there is a diagram of t h i s kind in o u r example. In

4-E

numerator

dimensions the integral would typically give (the has

been

put

there just

for

dimensional

k2

reasons;

elaborated theories the expressions are more complicated) :

r2(p)=

k2

(2n)e-4f(XB/2)~d'-Ek 2 2 2 2 ( ( k + p ) + m - k ) ( k + m -ic) 1

= (2n)E-4i(hB/2)~d~~d4-Ek 0

( k-xpI2

2

2

( k 2 + m 2 + x ( 1 - x ) p -ic)

216

-

-

i n the

in

more

(3.5)

Here,

is Euler's constant:

7

= W E :- 1 + O ( c )

I'(c/2)

x

insertion of the integral over the variable

.

The

is the famous "Feynman

trick". Again, in our scalar theory the diagram of Fig. 2 actually does not occur. We have only the "seagull" diagram in Fig 1, which gives the p independent expression

For

pedagogical

reasons

we

pretend,

will

temporarily,

as

if

we

are

dealing with the more complicated expression (3.5) in our scalar field theory. There are two things in eq. (3.5) that a r e of importance to us: the f a c t that there emerges a logarithmic function of

. First

p2

term and the f a c t that we get a

l/c

look at the

1/c

term.

Let us substitute eqs. (3.2-4) into our Lagrangian. We get

+ higher orders.

In the

counter

terms

the

distiction

(3.7)

between

B

and

R

was

neglected

because t h a t can be absorbed in the higher order corrections. The source term was not included because we are free to choose its normalization. We now add the correction terms in

(3.7) to the

diagrammatic rules

treating them as if they were new interactions. We

see

that

extra

terms

should

be

added

to

the

self

energy

expression of the form

-

(2c"'/c)(p2+m2)

- 2b111m/c .

217

(3.8)

So i f we choose

(3.9)

then the divergences of the form

p

all momenta

.

,

= A B / 2 i d x (2-2x+3x2) m B (4nI2 0

b[”

in eq.

1/c2

(3.10)

(3.5) a r e cancelled f o r

If we work with (3.6) we must choose

(3.11) Notice t h a t the source is now coupled t o the choice f o r

Now consider with the The

, not

@

.

We discuss

the

logarithm

in

(3.5). I t

seemed t o come together

term, and, as we w i l l see, it is indeed closely related.

1/c

important

#R

a[’’ l a t e r .

point

that,

is

in dimensionally regularized

expressions,

the quantities inside the logarithm a r e not dimensionless! Clearly t h i s is r a t h e r

we would change our units

absurd, because if

momentum these

expressions

would

change.

However,

such

of

m a s s and

changes

can

easily be absorbed in finite renormalization terms in the Lagrangian, s o indeed

they

not

will

regularization schemes

be

physically

(momentum cut-off

observable. or

In

Pauli-Villars)

the

other

one always

gets terms such as (3.12)

replacing the logarithm of (3.5). One may now understand why we compared

1/c

with

logh

.

The renormalization procedure in these older schemes

is j u s t as in the dimensional scheme: we put counter terms proportional to

Alogh

(and higher

powers of

this

to

remove

the

higher

order

divergences) in the Lagrangian. In a n “apparently scale invariant“ theory, t h a t is, if would at f i r s t sight expect (3.5) t o scale like

p2

.

m = O , one

But t h i s is not

so, because of the logarithm. I t is unbounded above and below, s o at a

c e r t a i n value of

p2

the s e l f energy

218

(3.5)

w i l l vanish, whereas f o r

very large or very small the

logh

p2

the logarithm w i l l dominate. But because

terms can be removed (or altered) by renormalizations,

is a l s o t h e freedom t o choose at w i l l the value of self-energy

vanishes.

where the

This is how one may first observe t h a t a s c a l e

a theory

transformation of

p2

there

maps

transformation is accompanied by

the

theory

into

itself

( f i n i t e ) renormalizations

only of

if

this

the

kind

(3.3,4).

Also

h

needs renormalization. This is seen when we consider the

x+x+x

one-loop corrections t o t h e s c a t t e r i n g diagrams (Fig. 3)

=

a

b

C

Fig. 3 Consider one of the terms in Fig. 3, €or instance

b . It is given by the

integral

(3.13)

p is t h e momentum through the horizontal channel in Fig. 3 . The divergence at c=O can be removed, as before, by plugging in eq. Here

(3.21, which would give i n addition t o (3.13): (3.14)

Therefore we choose

219

,

a [ * ’ = 3h2/16r2 where t h e f a c t o r 3 came divergences of

from t h e

t h e t o t a l of

(3.15)

we want t o cancel the

fact that

the three diagrams

in Fig. 3. Again,

the

logarithm determines t h e s c a l i n g behavior of the theory when the m a s s e f f e c t s become unimportant. 4.

analysis

of

the

scaling

behaviour

dimensionally

of

renormalized

theories When

E

is a n i r r a t i o n a l number dimensionally regularized amplitudes can

be uniquely defined.

With t h i s we mean t h a t t h e mathematical procedure

t o e x t r a c t t h e “ f i n i t e p a r t ” of a n integral in a n y Feynman diagram is unambiguous. The deeper reason f o r t h i s

is t h a t the integrands, b u i l t

out of propagators and vertices a r e allways r a t i o n a l expressions in the momenta t o be integrated over. Integrals over p u r e polynomials and p u r e

p o w e r s such as

can all be postulated t o be zero. Convergent i n t e g r a l s a r e postulated t o have t h e i r usual values. This way one succeeds in defining uinambiguous subtractions

for

power-like

all

divergences.

Only

the

logarithmically

divergent i n t e g r a l , rdx/x ,

(4.2)

0

cannot be made f i n i t e because i n f r a r e d divergences ( x with u l t r a v i o l e t divergences (x

+

logarithmic divergences ( f o r half

If

LO).

+

0) a r e entangled

is i r r a t i o n a l t h e r e a r e no

E

integer, t h i r d integer

logarithmic

E

divergences show up at two, three loops). Consequently, all relations between diagrams t h a t can be proved via partial

integration,

combinatorics

of

redefinitions

diagrams,

will

of

hold

integration

automatically,

variables and

we

need

and not

worry about omitted boundary terms in integrals. This includes all Ward and

Slavnov-Taylor

identities,

but

it

also

includes

scaling

relations

between diagrams. Thus, at

E

*

0

all “naive“ s c a l i n g r e l a t i o n s obtained by counting

dimensions hold exactly.

220

What a r e the dimensions of the various parameters at functional

integral we exponentiate the action

S

E f

0 7 In a

, s o its dimension

must be chosen to be zero. We have S so the Lagrange density

J?

!!(XI ,

= Id4-'*

has dimension J? = p4-"

(4.3) E-4

.

Let u s write

!e0 ,

(4.4)

where p is an a r b i t r a r y quantity with the dimension of a m a s s (unit of

mass) and

J?,

is a dimensionless object.

The kinetic term in unit of 4

4 , therefore . We w r i t e

J?

, being

, carries the dimensionless

-&(&$I2

the dimensions must match. So we find the dimension

9B =

Y290

(4.5)

*

Similarly, B

B

Thus, although

A

.

, hB = pE hoB

m = pmo

is scale-invariant

(4.6)

in four dimensions,

hB

in

dimensions is not. T h i s should be kept in mind when one uses eq.

4-E

(3.2).

From now on we should insist that

AR

is kept dimensionless, but

then eq. (3.2) can only be written down i f we have some quantity p to our disposal.

I.(

w i l l be chosen to be some a r b i t r a r y mass; in fact it

w i l l be related t o t h a t value of

p2

f o r which the self-energy

or the

vertex-correction vanishes. Thus we now write

I t is important t o know that the series (4.7) a r e constructed in such

a

way

that

finite expansion coefficients

choices of the (infinitesimal) quantity

AR

.

are

R'

+ el& + e2e2 +

221

for

all

Therefore it should do no

harm if we substitute in (4.7)

hR = A

obtained

...

,

R

R'

+ f l c + f2E2 +

= m

m

... ,

but t h i s does give an entirely d i f f e r e n t s e r i e s : 111

c

hB = p E [

a

I-Vl'

E

V

+

lll

AR"

+

V=l

B

m

lll

=

p

[

a[Vl',-V

) ,

b[Vl'E-V

)

C

b

I-Vl'

E

lll

v

R"

+ m

+

v =1

s i m i l a r expressions f o r the bare fields coefficients a IVl' and b [ v l ' are functions of d i f f e r e n t from the original

m

AR"

; also

R"

and

(4.10)

V =1

and

R

(4.9)

v=1

a

IVl

and

and

AR'

R'

m

A

R

and

R

, quite AR

as functions of

brV1

donot coincide with

m

, where the new

+B

and

.

m

Clearly eqs (4.9) and (4.10) a r e a more general choice f o r the bare parameters than

The positive powers of

(4.7).

a r e not necessary.

E

In

(4.7)

the coefficients f o r the positive powers were chosen t o vanish.

They

can

allways

be

judicious choices f o r

adjusted and

ev

to

fv

zero

by

substituting

(4.8)

with

.

Since p is essentially our unit of mass, a l l logarithms obtained in our integrations will have e n t r i e s made dimensionless with powers of

.

log((p2+m2)/p2)

p:

This is why p w i l l determine the order of magnitude of

the momenta at which our loop corrections w i l l (nearly) vanish. A t those momenta

perturbation

expansion

at

momenta

very

converge

relatively

we want to understand

least the f i r s t few terms). If behaves

will

f a r away from

these

we

rapidly

(at

how the system

have

to

make

the

t r a n s i t i o n t o other p values. Therefore, it is of importance t o consider

scale transformations : p = p'(l+u)

, u infinitesimal.

(4.11)

This t u r n s expressions (4.7) into

V=l R

m

m

= p'(l+u)[mR +

c

b

IVl

( AR ,mR)c-')

.

(4.13)

v=1 Notice

that

eq.

(4.12)

is of

the form

222

(4.91,

not

(4.7).

In

order t o

bring it into the form (4.7) we have to make a transformation ( 4 . 8 ) . Let us t r y X R = (1-crc) This removes the positive powers of

SiR .

E

(4.14)

from (4.12):

(4.15)

a ['I stands f o r A m a s s changes :

where

aa["/8h

, but also the expression f o r the bare

(4.16)

Note that we had to write the coefficients a s functions of the newly

iR,

chosen

which is now kept fixed a s

E

+= 0 .

The series (4.15) and (4.16) and a similar one f o r the fields

4

which we won't need for the moment, are now to substitute the old eqs 14.7). We see t h a t the zeroth coefficients are

R

-R

rnR' = rnR + urn - oh b

111

.

(4.17)

I t is essential that, with (4.141, we are free to keep either

KR

or

fixed as

the unit of mass keep

E

XR

+= 0. They are both finite. When is fixed, and when

fixed, which implies that

hR'

the two descriptions in the l i m i t AR

.

from

XR

p'

p

hR

is chosen to be

is chosen we'll decide to

is fixed. We wish to compare

c=O , so in (4.17) we may put

xR =

The conclusion is that when we make the (infinitesimal) transition p

to

p'

with (4.111, we have to replace

hR and

rnR by:

(4.18)

223

(4.19)

R

R

R

a

III R

p(AR,mR)

The function function these

.

(4.20)

describes the anomalous dimension of

can

be

computed

renormalization.

from the

At

counter term; at two and [21,[31, ... bt21,131,...

more

the

counter

one-loop

loops

we

also

able t o determine the coefficients of

theory we have in lowest order in

and from the coefficient

.I1’

b“’

have

not

yet

p

AR

in only

coefficients E

poles

-2

,

before

E

-3

,

being

S t i l l , u and p

AR.

A (see (3.13):

(4.21)

in (3.111,

written down the

(4.7) a f t e r t h e replacement

the

the

= 3A2/16n2 ;

u ( A , m ) = -m + Am/32n

We

needed

is

poles

the lowest ones.

a r e uniquely prescribed as perturbation expansions in

P(A) =

it

have

One w i l l f i r s t have t o s u b s t r a c t these higher

In

AR ; the

terms

level

, which govern higher

a

etc.

R

a l s o has a canonical p a r t . From the above it follows t h a t

u

functions

dimensional

R

(A,ml-rn

u(h,m)=A-b ahR

+ p’ .

at

2

,

complete s e r i e s

E f

0 R

= hR’ + pu - &A u

mR = mR’

(4.22) that

replaces

we s u b s t i t u t e ;

+ uu

(4.23)

We get from (4.7):

(4.24)

131

d o u r 01 d i f f e r s by a f a c t o r 2m from Symanzik’s a . But in h i s notation usually he puts u=l, which we a r e here not allowed t o do.

224

and m

m 3 = p'[mR' +

C

-V

E

IVJ R'

[b

(A

,m 1 + cbfVJ- uh b R'

R'

IV+ll +

V=1 +

I)

pu b[I1 + au b':'

.

(4.25)

Note that, the way they are defined, all coefficients R

and

m

a

and

a r e dimensionless. Obviously, b["'

of

and

hR'

m

R'

a , b , hR

the functional dependence of

should be independent of

.

p'

As

argued before, the series (4.7) is the only choice of counter terms that should

lead

to

a

finite

set

of

amplitudes

in

limit

the

c

0

.

Therefore, all "correction terms" proportional to u should cancel. Those terms proportional to

Q

that a r e of zeroth order in

c

-1

were made to

cancel by construction, but f o r the higher order ones t h i s observation leads to important identities between the coefficients a ["I and b E V J.

Since f o r

u>l

the coefficients

a ["I

R 2

a r e at least of order (A )

the equations (4.26) completely determine the higher order poles once we know the lowest order ones.

ref[12 'we

In

stated

that

usually

m , and the

independent of the masses

the

coefficients

are

b["

are proportional to

a ["I

m.

This follows from simple power counting arguments when the divergent parts of integrals are considered a s functions of the momenta and the masses m

.

However in some more complicated models not only the masses

and coupling constants are renormalized. and not only a r e the fields # multiplicatively renormalized, but one may also find renormalizations of the

vacuum

expectation

values

of

the

fields,

so

that

one

should

substitute

# + b + C ,

(4.27)

c . For instance, if there is a #3 term next to the #4 term in the Lagrangian one might be inclined to adjust C so as to remove any terms in the Lagrangian that a r e linear in # . where

C

contains poles in

But in t h i s case the renormalized

#3

225

and

tP4

couplings are defined in

a more delicate way and then the above power counting argument breaks down, because a condition on

such a s

C

= 0 ,

<#J>

(4.28)

introduces logarithmic dependence on masses in the theory. Consider now the formal sum m

ca

hB = hR +

IVl

( AR , mR )c+

;

(4.29)

1 b[V'(hR,mR)&-".

(4.30)

U=l m

mB = mR +

v=1

Putting

We f i n d t h a t (4.26) also holds f o r

, and (4.26) can be rewritten as

u=O

+ a ( hR , m

R

~

+

E

am R

R

~

+ 1

+ a(h ,m)-R am

Equations of

t h i s s o r t a r e solved by

the space spanned by the parameters

-dhR(t) dt

I" I" h

= 0 ;

(4.32)

m

= 0 .

(4.33)

considering t r a j e c t o r i e s R

hR and

.

m ;

fj(AR,mR)-EhR

in

(4.34)

- d- t

(4.35)

Then

Z h B ( t ) = -ch

B

;

d

B

m

(tl =

-m

8

.

(4.36)

Let's take the case (as in most two-parameter theories)

6 = B, (hRI2 Then the dependence between

A

B

;

CL

hB and

=

= -1 + Alh AR

R

.

(4.37)

is

CAR

(4.38)

h R - c/B2 '

but we also require t h a t hB = hR + 0 ( h R l 2 ,

226

(4.39)

so

C = -c/BZ

;

h

B

ARC

=

-

c

Bzh

(4.40)

R

This expression h a s been computed without special conditions on the order of the l i m i t s

c S

Bz
negative unless

and c -. 0 . So it replaces the s e r i e s AR . Note t h a t in t h a t case hR becomes

hR -. 0

(3.2) or (4.7) when

.

In the l i m i t

we have

&=O 2 R

hB = - c / B Z + c 2 / ( B 2 h ) + s o it starts out with a value independent of

Actually, are

theories with

in trouble:

,

hR

.

negative values

t h e Hamiltonian w i l l not

(4.41)

f o r the coupling constant

be

bounded

from below.

But

is so t i n y these diseases (havivg t o do with tunnelling out

because

AB

of

unstable

the

...

vacuum)

are

not

noticed

at

any

finite

order

in

perturbation expansion. Also we must s t r e s s t h a t (4.40) is only valid as long as

hR

and

hB

s t a y small, because only then the perturbative

expansions have some validity. So if

B2>0

the negative value of (4.41)

cannot be t r u s t e d because we had t o integrate t h e d i f f e r e n t i a l euations a c r o s s t h e pole, where it cannot be correct. Now we could have taken the contour

t space t o be complex, avoiding the pole, but t h i s would

in

r e q u i r e u s t o consider complex s c a l e transformations and complex values of

the

coupling

well-behaved. and

hR

constant.

where

the

Hamiltonian

is

also

not

Thus, eq. (4.41) has a formal validity only when both

E

a r e infinitesimal, in which case t h e i r relative s i z e s a r e not

restricted.

Bz

We do see t h a t the sign of

is very important. If

t h a t all bare coupling constants at c=O

Bz
are positive, we c a l l o u r theory

a s y m p t o t i c a l l y free. Let u s consider t h i s case f o r a moment, and assume

that

fi

depends only on

R

hR , not on

m

(which is the case in most

t h e o r i e s ) , such t h a t (4.42) with

g(0) = B, < 0

. One

then proves from (4.32) and (3.39): ,R

hB = hR exp(-[

dh

1 h

227

-

1 &/m

(4.43)

Let u s take

-B(A) = B2 + B3A + B4A2 +

... ,

(4.44)

so that

E/E

=

e/B2 - eB3A/B,2 +

...

,

(4.45)

then t h i s t u r n s (4.43) into

which f o r s m a l l

E

A B = -&/B2 +*

becomes

% log XR +0(1))

2

loge +e2(-BZ2AR B2

loge)

(4.47)

; these

are

completely determined by the mathematics of the system: only with a

AB

B~~

The f i r s t two terms of

t h i s a r e independent of

t h a t approaches zero t h i s way when amplitudes. E

3

AR

loge

But

the

E

2

+

term

~4

O(E3

AR

the theory w i l l give f i n i t e

is a r b i t r a r y .

The

order

higher

, e t c . a r e therefore not important. They could

be

terms,

absorbed by

. We a l s o note t h a t eqs. (4.34) and (4.35) a r e is the s a m e equations

for

m

.

R

AR

and

mR

as eqs (4.19) t h a t describe the response of

t o a n infinitesimal scale transformation

For t h i s reason the l i m i t

E

+

0

p

+

AR

and

p ’ = p ( l + a ) , when

c=O

in some respects corresponds t o a

l i m i t towards s h o r t distance scales, j u s t like the l i m i t more conventional regulator method.

228

A

-+

m

for a

The renormalization group equation W e have not yet discussed the renormalization of the fields # ( X I as it follows from eq. (4.7). I t should be treated in exactly the same way

as the mass renormalizations. Thus, in addition to eq. (4.33) we have

Here,

a/a#

The differentiation

is lacking of course because the field

+

was outside the brackets in (4.7). Related to these equations is the change in the definition of the renormalized field

when the unit of m a s s p is changed. A transition

+R

(4.11) is associated with transformations (4.19) and:

The combined transformations (4.111, (4.19) and (5.3) should leave the

entire

description

of

physical

features

invariant

(renormalization group invariance) :

where

TR = ($R(pl)6R(pz) ... $R(pH))

is an "amputated" Green function.

This can be rewritten as

R

However, remember that

rn

, and

#R

in (5.31, were defined t o be

dimensionless. But it is tempting t o redefine them such t h a t they have t h e i r canonical dimensions:

pmR = mren p#R =

We then only keep the higher) of

OL

and

6'""

(physical m a s s ) ;

(physical field)

"anomalous parts"

y :

229

.

(coming from one

(5.61 loop and

g jp(a

R

w

+ rn) = a

r+l

;

(5.7)

N

7 .

This is the renormalization group equation as i t follows from the rules

of

dimensional

equations

renormalization.

used by Callan

1111

look a l i t t l e b i t different. (hence h i s

The

renormalization

and Symanzik[13' a r e equivalent

For Symanzik,

but

they

is the mass parameter

p

is identical t o one) and he does not have our

01

group

-

a

term, in

stead of which h i s equations have a non-vanishing r i g h t hand side.

For our purposes we f i n d the or

Gell-mann

Low

"

renormalization group f low-equations"

equations"41,

(4.111,

(4.19) and

(5.3)

more

informative: if

then

(5.10) (as s t a t e d before, in most cases

f3

rnren

is independent of

and

linear in mren). Three important cases must be distinguished:

(i)

theories

in

which

constants) all

4

(for at

f3

s t a y s positive.

dimensional

theories

gauge f i e l d s (such as

A+4

least one of

its coupling

This happens in most models; that

donot

contain

in

non-Abelian

, QED, and Yukawa theories). In

t h i s case the solution of (5.10) must show a strong, possibly divergent,

effective

coupling

constant

at

high

energies.

Suppose

@(A) = bh2

(5.11)

Then we would get (5.12)

230

N

a

is

(5.13)

This

is

the

so-called

Landau

singularity.

advized to study t h i s singularity in the approximation" bubbles

for

scalar

represents

field

faithfully

The

theories.

what

student

"chain of The

happens

component scalar field theory in the l i m i t

chain

in

of

an

+

N

is

bubble

OD

N

,

AN

fixed . (if) Theories

/3

in which a l l

coefficients s t a r t out negative.

This may happen in non-Abellan

gauge theories only

guaranteed

only in a pure non-Abelian

according

to

(5.22)

and

(5.13)

This

means

that

logarithmically.

(it is

gauge theory). Then, goes

A(p)

at

high

to

zero

energy

the

perturbative expansion becomes better and better! We s a w that in t h i s case also the bare coupling constants formally tend to zero. There is a general consensus that theories of t h i s type

must

have

a

rigorous

mathematical

footing

(see

introduction) but t h i s could be proven only in very special cases[1s1. The phenomenon is called A s y m p t o t i c f r e e d o m

wa s

perhaps

non-Abelian

the

most

theory

interactions.

compelling

such

There

as

no

is

reason

for

Landau

assuming

describing

€or

QCD

in

ghost

and

a

strong

the

far

ultraviolet. There is one in the infrared (at very low values of

p

1, but one may well imagine that t h i s region should be

properly

described

by

integrating

out

the

field

equations

defined at high p values, s o that the Landau ghost may be j u s t some s o r t of physical bound state. ( i l l ) In some very special cases one might find that

all

orders

in

supersymmetric dimensions

perturbation Yang-Mills

theory. theory

An and

P(A) = 0

example

N=4

is

superstrings

at

in

2

Such a theory must be s t r i c t l y scale-invariant.

Quite generally one then also finds that such a theory must also be conformally invariant. One

must

careful

be

to

observe

that

the

form

taken

by

the

renormalization group equations and the details of the expressions f o r the

a

and

/3

functions depend a lot on details of the regularization

scheme. In Pauli-Villars

logh

plays the role of

231

E

and much of the

r e s t goes the same way, except that many higher order coefficients w i l l look different. A simple way to see t h i s is as follows. Suppose

that

two schemes choose a different

convention

for

the

finite p a r t of the subtraction constants. Consequently, h

R

= h

R’

+ f(hR’),

(5.14)

f(h) = O(h2) , a given function. Now

where

2 ac AR a R’ ac A =

= f3(hR) = b2(hRI2+ b3(hRI3+ R’

R’

R’ 3

2

p ’ ( h 1 = b z ’ ( h 1 + b3’lh 1

+

...

;

... .

(5.15)

Assume f o r simplicity t h a t

f(A) = ch2 I t follows that (write

2 aa hR =

.

(5.16)

R’

h = h ):

b2(h2+2ch3+c2h4)+ b3(h3+3ch4+. . . I

a

= (l+Zch)&

+ b4A4 +

2

= (1+2ch)(b2’A + b3’A3 + b4’h4 +

...

=

...) ,

(5.17)

... .

(5.18)

or b2’ = bz ;

2

b3’ = b3 ;

b4’ = bl+cb,+c b2 ;

Observe t h a t b2 and b3 are “universal“: they a r e scheme independent;

bz

because it is the lowest order term, depending on the leading divergence

at one loop, which because

c

is the

same f o r all regulators;

happens to cancel out. This two-loop

b3 is

universal

coefficient f o r the

beta function ceases t o be universal if we have more than one coupling constant“”.

The student is suggested to f i n d a plausible substitution

(5.14) in that case such that the two-loop coefficients change.

Having the freedom t o redefine our coupling constants a s suggested in (5.14) we could decide to choose hard

t o convince oneself

(5.14) with

t h a t if

f(h)

b2+0

such t h a t

b,’=O

a substitution

.

I t is not

of

the form

f ( h ) = O ( h 3 ) can be found in such a way t h a t b,’=O,

This

c

n = 4 , 5, . . . m.

(5.19)

is unique apart from scale transformations which obviously

232

leave the functional form of

p(h)

invariant.

The words “ m i n i m a l subtraction” have already been reserved f o r the dimensional

renormalization counter terms

(2.10)

or

(which a r e

(4.7)

unique in contrast with the more arbitrary choice ( 4 . 9 ) ) . We might call

6 function

the choice that gives (5.191, implying the exact

(no further terms) ,

p ( h ’ ) = b2(k’l2 + b3(h’I3

(5.201

an “ultra-minimal’’ subtraction. A word of caution is of order. The original beta function

f3(hR)

is sometimes believed to have a fixed point:

so that if

fixed

point

h

R

= ho

is

the theory becomes scale-invariant. Of course t h i s

totally

unrelated

to

possible

a

truncated series (5.20). A t such a point will

tend to show singulariries.

whether or not

f3

But we do

It

is argued

point

of

the

the realation (5.14)

hR = A.

learn

that the

has a fixed point may depend on how

it becomes large.

when

fixed

question is defined

k

by many researchers

that

the

question whether the theory is scale lnvariant should not depend on such definitions,

and

that

therefore

existence

the

function should be scheme independent. known

how

to

give

any

rigorous

zeros

of

But we s t r e s s that

definition

of

4

f3

the

of

it

is not

dimensional

field

theories beyond the perturbation expansion. Indeed there is reason to hope t h a t a definition can be given precisely in the case that there is not such a zero, when the theory is asymptotically free that case condition (5.20) determines

@(h)
.

In

uniquely apart from scale

h

transformations. 6 The higher order terms in the many parameter case

The procedure of chapter 4 and the equations (4.26) can be generalized to an arbitrary number of parameters

hk

which may be found to have

dimension

in

n=4-&

dimensions. Some of these may be masses (for which

vanishes), others a r e three-

or four-particle

233

Pk

coupling constants (which

have

.

dk=O )

parameters

.

a

+,

We could even include fields

or gauge f i x i n g

@,

As before, we write m

c a : ’ ~ { ~ R } c - u ,]

X,B = pDk[hkR+

(6.2)

v=l

where in the brackets { are

allowed

to

depend.

coupling s t r e n g t h s

o r field s t r e n g t h s

A

a r e those objects on which these quantities

}

The

renormalization

of

physically

observable

should not depend on gauge fixing parameters

4, 1,9,

a

s o these form a s e t by themselves.

Consider the transformation p = p’(l+(r)

.

(6.3)

The equivalent of ( 4 . 1 2 1 , ( 4 . 1 3 ) is A:

R

+ &p(k)hk(r + p ( ) ( ) a y ( r +

= ( p ’ I D k ( l + d ( k ) ohk ) [

R

m

c ( a : u l + p ( k ) a : v + l l ( r ) & - u.]

+

(6.4)

V = l

Let us w r i t e

where the functions (6.4) is j u s t E

hk

R’

must be chosen such t h a t the first term in

Pk

.

The

t e r m should remove the positive powers of

t3

in (6.4). Eq. (6.4) t u r n s into

[

hkB = ( p ’ I D k ( I + d ( k ) U ‘hk )

+ Pkc +

c

0‘

r11

-

R’

ak,t p ( t ) A t

+

t

m

+

I11

p(k)ak

R’

+ p ( k ) a [k U + lI 0‘ +

c &a:f:q

[U+l

-

ak,t

t

V = l

I

p(l,htR’(r)E-”]

*

t

(6.6)

a IkU,lt stands f o r

Here,

aaLul/ahtR .

Thus we f i n d how to express @ , { A R } = -d(k)hE

+

in terms of the counter terms

Pk

c a:::

p(t)ht

R

-

I11 p(k)ak

#

ak :

(6.7)

t

and since a:”{h)

IUl

R’

(l+d(k)(r)ak { h }

in (6.6) should be the same functions as

in (6.21, we have the identities f o r

234

IUl

ak

:

replacing

our

akfV1 are

previous

polynomials

eqs. of

In

(4.26).

very

most

low degrees

in

cases

the

the

masses

coefficients and

other

dlmensionful parameters, which simplifies the use of equs. ( 6 . 8 ) .

7. An algebra for the one-loop counter terms a, 6 and 7

The first terms of the important.

coefficients a r e by far the most

They determine the leading scaling behaviour of the theory,

and whether it is asymptotically free or not. The counter terms needed t o make t h e one-loop Feynman diagrams f i n i t e a r e particularly simple to compute, although,

you start out doing first the complete one-loop

if

calculation it can still be quite cumbersome. So it w i l l be of help t h a t a universal

"master formula" can be written down t h a t gives us

all

necessary one loop counter terms once and f o r all. A l l 4 dimensional renorrnalizable field theories can be written

in

the form

a, p , v , i, j ,

where summation over repeated indices a r e a bunch of s c a l a r fields and G

PV

is implied.

@i

a r e (possibly c h l r a l ) fermions.

#i

a r e the gauge fields,

= a A' - a A' + P V V P

' C pv

where

pbc

is any combination of

P bc A b

~c ,A

~

(7.2)

coupling constants and s t r u c t u r e

functions t h a t s a t i s f i e s the Jacobi identity f o r some compact gauge Lie group. The function such as

V{@)

+n1'@~/2+ Ad4/4!

is any quartic polynomial in the f i e l d s

@

in a simple s c a l a r field theory leaving open

the question whether or not spontaneous symmetry breakdown occurs. The derivatives

where

D

again

P

a r e covariant derivatives,

possible

coupling

constants

235

were

absorbed

into

the

T

coefficients

.

U

and

These must s a t i s f y the u s u a l commutation r u l e s

may be s p l i t i n t o a l e f t handed and a r i g h t handed p a r t ) :

U

(and the

(7.4) The f u n c t i o n s in

and

S

a Yukawa theory.

must be l i n e a r i n

P

And

of

course

the

+

, f o r instance,

entire

S=m++g$

Lagrangian

must

be

i n v a r i a n t under t h e local gauge group. We can now add t o t h i s Lagrangian a counter term chosen

such

construct

that

A2

We a r e instance

it

absorbes

all

(one

loop)

A2

t h a t can be

infinities.

The

way

to

is as follows.

interested in the complete s e t of

we

could

insert

(gauge-invariant1

one-loop source

diagrams. terms

Lagrangian and consider t h e dependence of the vacuum-vacuum

For

in

the

transition

amplitude on the s t r e n g t h s of these sources, expanded to one-loop order. A

practical

way

to

compute

this

amplitude

is

to

shift

the

field

variables :

(7.5) where

the

"classical"

fields

are

c

numbers

satisfying

the

classical

equations of motion (with the source terms i n ) , and t h e "quantum" f i e l d s a r e integrated over. Let u s denote the bosonic quantum f i e l d s together

as

.

{pi)

Now motion

The Lagrangian is now expanded in these:

the

fact

implies

vanishes.

The

that

that,

the when

quadratic

term

classical

fields satisfy

integrated generates

over,

the

propagators

the

equations

of

linear

term

g1

for

the

quantum

f i e l d s and v e r t i c e s with two quantum f i e l d s and other background f i e l d s . The higher order terms neglected in (7.5) generate vertices t h a t cannot contribute t o one-loop graphs, s o we ignore them. We rewrite the last term of (7.61 as (7.7) Here,

N

and

M

depend on the c l a s s i c a l background f i e l d s . We w i l l see

236

to it that In

W

does not depend on the background fields.

general,

Lagrangian

(7.7) is

a

gauge

theory

and

should

be

quantized and renormalized according t o the usual rules. A gauge-fixing term w i l l be needed, and ghost f i e l s . As a gauge fixing term one could f o r instance take

with (7.9) The reason why t h i s choice is s o clever is that we fixed the gauge, but still kept invariance under the transformation

(7.10)

Consequently the

required counter

will

terms

also exhibit

invariance. Since we w i l l only need t h e i r dependence on

this

gauge

, the only

A;'

allowed counter terms a r e gauge invariant ones. This would normally not have

been

the

case,

since

renormalizations

field

(which

are

unobservable) may well depend on the gauge fixing. We put (7.8) and the associated ghost (also Invariant under (7.10)) In our Lagrangian (7.7). In general one can choose things such t h a t

flu ij Now

M

= -gpvg

ij

.

(7.11)

w i l l in general contain a mass term, but we w i l l not include it

M

in the propagator. Rather, we keep

a s a vertex, realizing that from

a certain order on ( a t sufficiently high powers of only

convergent diagrams,

whereas

we

divergent renormalization counter terms.

are

really

M 1 we w i l l have only

In other words,

interested if

in

there is a

mass m we expand the propagator, l/(k2+m2-f&) = l/(k2-ic)

-

m2/(k2-ic)2 + m 4 / ( k 2 - i ~ ) 3-

treating the correction terms as vertex insertions.

237

...

(7.12)

Consider now the one loop diagram of Fig. 4.

Fig. 4 The amplitude is 1 (6jk6jl+6j16jk)p Sd"k 4(2nI4i (k2-ie)((k+p)2-ic) If

n = 4-c

(7.13) '

t h i s is (7.14)

which has a pole at

c=O: (7.15)

A counter t e r m in

!A

= -(1/8n

A!!

The complete

A!!

is needed: 2

C )

.

MijMji/4

(7.16)

containing a l l possible combinations of

M

and

#

can only have a relatively s m a l l number of terms. This is because simple power counting arguments tell us t h a t they have t o be polynomials of dimension 4: Now

W

is of dimension 0,

dimension 2.

W

were non-trivial

If

of dimension 1 and

N

we could have a n

M

of

i n f i n i t y of

possible terms but now we must have

+ a4N N a N + P V P V

+ a,(N N

N 1 P P

+

P V

.

Now it is easy t o see t h a t we can put some r e s t r i c t i o n s on

(7.17)

M

and

N : M i j = M .J. l ;

Nyj = - N y i

by absorbing the symmetric p a r t of

a

fl

P

, into

(7.18)

M

.

However, one can do

much more by observing a new "gauge" symmetry t h a t is apparent i n o u r system. Write

238

(7.19) with (7.20)

X=M-Nruc-'. P

We then have invariance under

(7.21)

.

X' = X + [A,XI Here,

A

is infinitesimal.

Note t h a t t h e dimension of t h i s gauge group is as large as there

are f i e l d s in the system, s o it could be larger than the original gauge symmetry, and t h a t the symmetry is exact also a f t e r a gauge fixing term is added (This w a s made possible by the fact t h a t the background fields

the gauge fixing term is allowed t o depend on

transform non-trivially;

the background f i e l d s not in the same way as from the quantum f i e l d s ) .

AP

A consequence is t h a t

must have the same symmetry. The only

combination of terms from (7.17) invariant under (7.21) is

AP = (1/8n2e)Tr(aX2+bY Y

PW PW

)

,

(7.22)

where

Y PV

=

a - ~ a

+~N N

U P

P V

- N N .

(7.23)

V P

P V

From (7.13) we f i n d t h a t

a = a. = -1/4

.

(7.24)

Similarly"71 one can f i n d , by computing a few other diagrams, b = ( a o - a s ) / 2 = a,/2 The

excercise

can

be

extended

Lagrangian we then start off with is

239

= -1/24 to

. include

(7.251 f e r m i ~ n s " ~ ' . The

plus higher orders in the quantum f i e l d s which a r e irrelevant for one loop calculations. By making much use of our previous r e s u l t , eq. (7.221, and taking the fermion minus sign into account, one f i n a l l y obtains

+ H2/8 + H

H

/48)

,

(7.27)

PV

PV

where

We

see

that

this

expression

contains

a

few

more

simple

numerical

coefficients t h a t had t o be fixed by explicitly computing some diagrams. The r e s u l t obtained the form we would still hidden

in the expressions

s u b s t i t u t e V, S, P fields is

'pi

a dull

here, essentially algebraically,

like t o see it,

is not

because the background

X,

Y , F,

H,

... .

from (7.1) and the s t r u c t u r e constants i n here. The

in (7.7) are i n fact a combination of

@i and

is

lengthy

*"

in

W e now have t o

and

f i n a l l y obtains'

yet

fields are

process.

But

the

result

worth

Ae P

This

q'.

while.

One

: 2

(871 & ) A 2 =

Ca G b [ - ( l l / 1 2 ) c b + (1/24)Gb + (l/6)Gb1 - AV PJ P

(7.29)

-

$(AS+fr,AP)JI

,

i n which (7.30) (7.31) (7.32) (7.33)

240

V i = aV/a#i

; S,{ =

aS/aai ,

(7.34)

etc.

And writing

w

= S+iP

;

* w

= s-iP

,

(7.35)

one gets

*

AW = (1/4)W,iW,iW +(1/4)WW:iW,i + W,iW'w,i + W,i#jTr(S,iS,j + P,,Plj)

+(3/2)G W +(3/2)W$

+

.

(7.36)

One may wonder why there a r e no field renormalization terms such as c(aF#I2

. The

answer is that these can be rewritten a s (7.37)

-c#(a2#)

which,

via the equation of

motion of

the background

fields.

can be

expressed in the other terms. I t also can be noted that an infinitesimal renormalization of the field

#+

# , (7.38)

# ' = # + S #

adds to the Lagrangian terms proportional to the equation of motion:

!t+ Z + S $ X ,

where

X

vanishes

because

of

the

(7.39)

equation

of

motion.

So

field

renormalizations, whlch by the way need not be gauge-invariant, are not seen if we use our background field method. eqs (7.291

-

(7.36) are our master formula f o r all one-loop counter

terms. I t is very instructive to play with a few examples to see what kind of renormalization group flow equations they produce.

241

8 . Asymptotic freedom

Let u s look at the one-loop r e s u l t s , eqs (7.29)-(7.36) more closely. How should we use them? The term AV is easiest t o understand. Take the case

There is only one f i e l d , s o

v,,

=

a2v/a+’

=

i=j=l ;

4~4‘ + m2 ,

(8.2)

and the first term in (7.33) is

and including the f a c t o r

1/8n

2

E

we get (note t h a t

AA = (3/16n2~)A2,

A ( m ) = Arn/32n2c

A(m2)=2mAm 1: ;

(8.4)

compare eqs. (3.15) and (3.11). The constant $m4 in (8.3) is immaterial. If the f i e l d has

N

components, and

V=A(52)2/8 , (a convenient

choice of normalization) then

AV = A2(N/4

+ 2)($f2/4 ;

Ah = (N+8)A2/16n2~ ,

which reproduces (8.4) f o r our definition of

A

N=l

(8.6)

(8.7)

since we had a f a c t o r 3 difference in

.

The next t e r m in (7.33) contains

T2 . This represents the gauge

4

is of our gauge group. Suppose

coupling, 1.e. which representation t h a t in (7.3)

which would imply t h a t

4

is in the adjoint representation of the gauge

group, and

242

Taij = gc iaj .

(8.9)

Then

(8.10) contributing t o

AV

:

(8.11)

s o t h a t we get a negative counter t e r m

A A = -12Ag2/8n2c

.

(8.12)

The t h i r d term in (7.33) gives again a positive contrubution:

I t is easy t o see now t h a t i f we i n s i s t on keeping the Hamilton density positive in a purely scalar theory most

(at least one of

positive

@

them), because of

functions w i l l allways be the definite sign of

the

f i r s t term in (7.29). But we a l s o see t h a t t h i s is no longer obviously the case if gauge interactions a r e included. In practice however one can prove from all equations (7.29)-(7.36) (with much pain) t h a t i f we have scalars and only vectors

alone

is not

able

to

(no spinors) then

render

the

system

the middle term of

asymptotically

free

(7.29)

(all

p

negative). There a r e too many terms with the wrong sign. But now consider the pure gauge theory. W e then only have:

!!A

= -(11/12)CIabGaG b

fJv P

and s u r e l y the C a s i m i r

operator

Clab

/8n2c ,

(8.14)

is positive

(or zero in the

Abelian case). Mostly we diagonalize it:

clab = C,(a)aab , with

ClrO

constant

C,

g

.

Let u s simply w r i t e

C,(a) = C,

(8.15)

.

Since the coupling

w a s absorbed into the s t r u c t u r e constants

is proportional t o

2

g

.

For an

f

we have t h a t

SU(2) gauge theory it is

c, = 2g2 .

243

(8.16)

(8.14) renormalizes t h e f i e l d s

.

A it

chosen to

(One could have

w r i t e (7.29) d i f f e r e n t l y s o as t o avoid f i e l d renormalizations. notation

came

temporarily t h e

to

out

the

easiest.)

Let

1/8n2c into t h e coefficient

Cl

be

rewrite

us

but t h i s (absorbing

(8.17) (8.18) but then

(8.191

Ag = -(11/61CIg/8n2c , which is

-(11/3)g3/8n2e in

Thus,

beta

the

(8.20)

SU(2).

function

for

g

fermions and bosons w i l l tend t o t u r n

p

is

negative.

Note

that

both

in the other direction:

Fields in t h e adjoint representation have c2,3

= cl

but i n the elementary representation of

(8.22)

*

SU(N)

we have

C2,3 = C1/2N , (note t h a t complex

in

(8.23)

(8.22) and (8.23) we consider c h i r a l fermions. Massive,

fermions

have

twice

these

values;

scalar

furthermore,

fields

were taken t o be r e a l . If they a r e complex we a l s o need a f a c t o r two). If we have 11 massive fermions i n t h e fundamental representation ( f o r

SU12)) or

164

then

(for SU(3)

p w i l l switch sign.

Cl=O , can never be asymptotically f r e e . A

Thus, QED, which h a s

purely s c a l a r theory a l s o c a n ’ t , but a l s o a scalar-fermion Yukawa

couplings

cannot

(7.32) onlu the terms with In

gauge

theories

be

asymptotically

free.

This

is

theory with because

in

situation

is

2

U W have a negative s i g n . with

fermions

and

scalars

the

complicated. If one imposes the conditions t h a t all beta functions come

244

out negative one usually will f i n d t h a t all coupling constants should keep fixed ratios with each other under scale transformations:

because only then they won't r u n away in different directions. I t is an interestlng game to t r y t o construct such models. There a r e plenty of them. The r a t i o s algebraical

numbers

except

often t u r n out t o be r a t h e r ugly

hi

when

one

invokes

supersymmetry. Supersymmetric Yang-Mills scale-invariant couple

.

(N=4)

sufficiently

symmetries

such

as

is asymptotically f r e e ( M 2 ) or

In practice one finds t h a t the fermions must

strongly

to

the

gauge

particles

which

happens

typically if they a r e i n the adjoint representation or higher. So in QCD

it w i l l be hard to add s c a l a r s while keeping asymptotic freedom because the quarks a r e all in the fundamental representation. Perhaps t h i s is the reason why fundamental s c a l a r s t h a t couple to the strong gauge group cannot exist. Constructing weakly interacting systems while insisting on (8.24) , though an then

interesting game,

is probably

physically

irrelevant

the coupling constants r u n s o slowly t h a t the Landau

because ghost

is

extremely far away in the ultraviolet, where, certainly if it is beyond the point where gravitational forces a r e expected, none of our theories can be believed anyway. Similarly to our equations (7.29)-(7.36)

one can devise expressions

t h a t generate the two-loop contributions t o the beta functions. But even in t h i s compact algebraic notation the r e s u l t is very lengthy' " I . 9. Perturbation expansion at high orders and t h e renormalization group

The renormalization

group gives us a good idea about the force l a w

between quarks in QCD. The strength of the coupling constant increases with distance. So we get f o r the inter-quark force:

g(rl2 F(r) = 2 ' r with

245

and since

f3cO

we have a force law in which the strength of the force

decreases not as quickly with the distance as scales quarks i n t e r a c t quite weakly,

strongly indicated

by experimental

.

l/r2

A t s m a l l distance

and indeed t h i s f a c t , which w a s

observations

in

the

late

~ O ' S , has

been a major reason t o believe t h a t a non-Abelian gauge theory must be responsible f o r the strong interactions. But theory

also

for

(with

theoretical

fermions

and

reasons

possibly

an

also

asymptotically

scalars)

is

the

free

gauge

only

likely

theory f o r any strongly interacting s e t of particles. One would believe

at

that

least

at

very

tiny

distance

scales

the

theory

should

be

precisely defined. According to our own philosophy t h i s is only the case if

the coupling constant tends to zero.

Now in any weakly interacting

theory the coupling constants a r e s m a l l enough t o make the perturbation expansion useful even if absolute formal convergence is lacking. So they need not become any smaller at small distance scales. But for strongly interacting theories t h i s is necessary. But

even

in

asymptotically

free

theories

the

coupling

constant

tends to zero very slowly. Is t h i s convergence f a s t enough t o make the theory well-defined? The answer t o t h i s question is not known. A s should be evident from J . S m i t ' s lectures at t h i s graduate school it is often taken f o r granted that

if,

in

a

lattice

gauge

theory,

the

bare

coupling

constant

is

postulated t o tend t o zero according t o

as

a

3

0 , where

A

and

B

a r e determined by the renormalization

group, then a sensible l i m i t is obtained,

independent of the details of

the lattice. There is no proof

of

this,

a p a r t from numerical evidence,

which

however at best can give only indications of how the system w i l l tend t o behave.

Our problem is t h a t , even at s m a l l distances, all f e a t u r e s of

our system a r e defined as perturbation s e r i e s in

C"g2"n!

, as

far

as we

know.

The e r r o r ,

decreases rapidly a s the scale becomes

g2 , which diverge as

behaving a s

smaller.

decrease fast enough t o become irrelevant in the

exp(-1/Cg2)

Does t h i s uncertainty limit?

Or could the

e r r o r amplify as one integrates the equations back t o large distances?

246

The question t u r n s out t o be not easy, particularly if

we a r e dealing

with models that are not exactly solvable. One way t o attack t h i s problem offending

factors

n!

is t o search the source of

i n perturbation expansions.

When one

the

sums over

large s e t s of Feynman diagrams one discovers t h a t the number of diagrams with

n

loops increases like

One case is the so-called

.

n!

Only in special cases they donot.

spherical model. This is

components, where we let

N

(4')'

theory with

N

AN =

is

tend t o infinity, such t h a t

x

kept fixed. This model is exactly solvable. The 6 function can be given in closed form, and indeed the asymptotically f r e e if

xC0

does no harm. After all,

limiting theory is well-defined.

It

, but in t h i s case the negative sign of

A

is N

A

itself tends t o zero. The amount of action

needed f o r a c e r t a i n field configuration t o tunnel into a mode t h a t has negative energy is found t o increase indefinitely with

N , s o t h a t such

a tunnelling is inhibited in the l i m i t . A less t r i v i a l model is planar f i e l d theory. with two indices example is

and

i

SU(N)

which both can take

j

gauge theory.

i n f i n i t y , again keeping

AN

If

N

one now allows

g'N

or

Here, we have fields values. A typical

N

t o tend

to

fixed, one recovers all planar

Feynman diagrams. There are infinitely many of these, and nobody knows how

to

sum

these

geometrically with

analytically.

N

But

their

( t h a t is, without the

number

group,

only

grows

N! 1.

But t h e r e is another source f o r f a c t o r s i n detail using the renormalization

now

n!

This can be understood

but here we

give only a n

heuristic argument. Consider a self-energy the following typical

k

insertion diagram with one loop.

I t shows

dependence:

r(k2) Y Ak210gk2 ,

(9.4)

as we have seen, and we know t h a t the log is a c r u c i a l f e a t u r e following from the renormalization group. Now consider

n

of these diagrams connected in a s e r i e s t o the

propagator. This diagram then behaves as P(k2) = (k2-ic)-l An(logk2)"

.

(9.5)

Suppose t h a t , i n t u r n , t h i s diagram occurs inside a loop diagram, which

247

we have to integrate:

and assume t h a t the integral converges by power counting. If it didn't we make it converge by using subtractions. Thus, at large

k , the function

F(k)

tends t o

.

k-4

Assuming no

infrared divergence the integral behaves as

1

Substituting

k2

+ es ,

we get m I n!

Sds

.

(9.8)

0

This is how It

n!

appears t h a t can r u i n perturbation expansion.

should be

(9.6) can

be

clear

understood

that by

the

logarithmic

studying

scaling

behavior

in eqs.

via

renormalization

the

group. One then f i n d s t h a t the coefficients i n f r o n t of the obtained directly from the

6 function coefficients.

248

n!

(9.4)can be

References 1. 2. 3. 4. 5 6

7 8 9 10 11 12 13 14 15 16 17 18

E.C.G. Stueckelberg and A. Petennan, Helv. Phys. Acta 26 (1953) 499; N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959) see for instance: M.N. Barber, Phys. Repts 29C (1977) 1 N.G. van Kampen, private communication G. 't Hooft, Nucl. Phys. B33 (1971) 173; ibid. B35 (1971) 167; M. Veltman, Physica 29 (1963) 186 G. 't Hooft and M. Veltman, DIAGRAMMAR, CERN report 73/9 (1973), repr. in "Particle interactions at Very High Energies", NATO Adv. Study fnst. Series, Sect B, Vol. 4b, p. 177. G. 't Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318 W. Pauli and F. Villars, Rev. Mod. Phys. 21 (1949) 434 G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; C.G. Bollini and J.J. Giambiagi, Phys. Letters 40B (1972) 566; J.F. Ashmore, Nouvo Cimento Letters 4 (1972) 289 G.L. Eyink, 1987 Ph.D. Thesis, Bruxelles C.G. Callan, Phys. Rev. D2 (1970) 1541 G. 't Hooft, Nucl. Phys. B61 (1973) 455 K. Symanzik, Comm. Math. Phys. 18 (1970) 227; ibid. 23 (1971) 49; 45 (1975) 79 M. Gell-Mann and F. Low, Phys. Rev. 95 (1954) 1300 For instance, G. 't Hooft, Comm. Math. Phys. 86 (1982) 449; ibid. 88 (1983) 1 R. van Damme, Nucl. Phys. B227 (1983) 317 G. 't Hooft, Nucl. Phys. B62 (1973) 444 See ref. 16 and: G.'t Hooft, Nucl. Phys. B254 (1985) 11

249

CHAPTER 4

EXTENDED OBJECTS Introductiom ......... .. ..................... .......... ................ ... ...... 252 [4.1] “Extended objects in gauge field theories”,in Particles and Fields, eds. D. H. Bod and A. N. Kamal,Plenum, New York, 1978, pp. 165-198 .............................................................. 254 “Magnetic monopoles in unified gauge theories”, Nucl. Phys. [4.2] .. . .. ..... 288 B79 (1974) 276-284 .... .... .......... . ..............

.

...

25 1

.....

CHAPTER 4

EXTENDED OBJECTS Introduction to Extended Objects in Gauge Field Theories [4.1] Up till now we maintained that quantum field theories (in four dimensions) can onlg be understood perturbatively. What we really meant however was that it is the infinite series of successive quantum corrections that must be addressed term by term. Even though the renormalization group allows us to partially resum this expansion, the manipulations are still permissible only when the coupling constant(s) is (are) sufficiently small. Extended objects form another topic that suggests the possibility to transcend beyond the perturbation expansion. We get particles with masses inversely proportional to the coupling constants, and we get “instantons” that will produce tunneling effects with amplitudes depending exponentially on the inverse coupling constant. Indeed, phenomena described in this chapter may perhaps be called “nonperturbative”, but we always have to remember that what we compute will require an infinite series of multiloop corrections, as usual. So, again, we must always assume the coupling constants to be sufficiently small. The first paper in this chapter is an introduction to kinks, vortices, monopoles and instantons. This is a logical order. In its Section 4 it is explained how in certain gauge theories 2-dimensional objects (vortices) become unstable and how this leads to the existence of %dimensional things (monopoles). This is also the way I discovered them. Alexander Polyakov had followed a slightly different route. He had considered “hedgehog” configurations of scalar fields and asked whether one could construct particles this way. Following the arguments that I explain in Section 2, he found that gauge fields had to be added. That these things actually carry magnetic charge had been suggested to him by Lev Okun. Note that this paper was written before the third generation of quarks and leptons had been firmly established. In a conservative “Standard Model” at that time therefore instantons gave rise to processes with A B = AL = 2 rather than 3.

252

Introduction to Magnetic Monopoles in Unified Gauge Theories [4.2] The next paper is the one in which the discovery of magnetic monopoles as valid solutions in gauge theories was published. It gave rise to a renewed interest in monopoles by theoreticians, experimentalists and cosmologists alike. Experimental detection has never been confirmed, but, as I had pointed out in the Palermo Conference, Chapter 2.3,monopoles play a significant role in theory anyway. Most crucial is their part in the quark confinement mechanism, see in particular Chapter 7. One later observation should be added to this paper. The coefficient C(p)waa computed by numerical variation techniques, and found to be close to one when p is small. It waa an important result of Prasad and Sommerfield15 that the equations simplify essentially when p = 0. The second order field equations then become the square of a first order equation (duality). This equation can be solved exactly and one then finds that C(0)= 1.

“M. K. P r d and C. M. Sommerfield, Php. Rev. Lett. 96 (1975) 760.

253

CHAPTER 4.1

EXTENDED OBJECTS IN GAUGE FIELD THEORIES G. 't Hooft University of Utrecht T h e Netherlands

1.

INTRODUCTION AND PROTOTYPES I N 1+1 DIMENSIONS

With the exception of the e l e c t r o n , the neutrino, and some o t h e r elementary p a r t i c l e s , a l l o b j e c t s i n our physical world are known t o be extended over some region i n space. Thus, the s u b j e c t "extended objects" is too l a r g e t o be covered i n j u s t four l e c t u r e s . However, most of these extended o b j e c t s may be considered as bound s t a t e s of more elementary c o n s t i t u e n t s . I f we exclude a l l those, then a very i n t e r e s t i n g small s e t of p e c u l i a r o b j e c t s remains: o b j e c t s t h a t cannot be considered as j u s t a bunch of p a r t i c l e s , b u t aa some smeared, but a l s o more o r less l o c a l i z e d , configuration of f i e l d s . These f i e l d s m u s t obey very s p e c i a l types of f i e l d equations t o allow such smeared lumps of energy ("solitons") t o be stable. These may be two a l t e r n a t i v e reasons f o r alump of field-energy t o be s t a b l e against decay: i) A conservation l a w might t e l l us t h a t t h e t o t a l mass-energy of the decay products would be l a r g e r than our o b j e c t i t s e l f . It is easy t o prove t h a t t h i s cannot happen i n a l i n e a r ( i . e . non i n t e r a c t i n g ) theory: I f e l e c t r i c charge is given by

Q

-

ie

j[O*aoO - (ao4*)dl

d3x

,

and the energy by

Particles and Fields, Edited by D. H.Bod and A. N.Kamal (Plenum 1978).

254

(1.1)

G. 't HOOFT

(1.3)

whereas a bunch of charged p a r t i c l e s a t r e s t , w i t h a t o t a l charge e q u a l t o Q , could have a t o t a l mass-energy n o t exceeding

+ Consequently, a f i e l d c o n f i g u r a t i o n w i t h a$ # 0 would n o t be s t a b l e by charge c o n s e r v a t i o n a l o n e . I n i n t e r a c t i n g f i e l d theori-es however, one may c o n s t r u c t many i n t e r e s t i n g s t a b l e lumps of f i e l d s simply because t h e above arguments do n o t apply when t h e r e a r e i n t e r a c t i o n s . These have been c o n s i d e r e d by T.D. Lee and coworkers' and w i l l n o t be t r e a t e d i n t h i s series of t a l k s . i i ) T o p o l o g i c a l s t a b i l i t y . I n t h e s e l e c t u r e s w e w i l l o n l y consid e r t o p o l o g i c a l l y s t a b l e s t r u c t u r e s . The s t a t e m e n t t h a t our o b j e c t s are r e a l l y f i e l d c o n f i g u r a t i o n s i m p l i e s t h a t w e c o n s i d e r f i e l d e q u a t i o n s w i t h o u t b o t h e r i n g about t h e f a c t t h a t small oscill a t i o n s about t h e s o l u t i o n s ought t o b e q u a n t i z e d .

To e x p l a i n t o p o l o g i c a l s t a b i l i t y w e f i r s t t u r n o u r a t t e n t i o n one t i m e dimension2. Consider a simple t o models i n one s p a c e s c a l a r f i e l d (I s a t i s f y i n g

-

t o which corresponds a Lagrangian:

(1.6)

The two cases w e c o n s i d e r are a>

V(4)

=

(0'-

F2)'

(see Fig: 1)

255

.

(1.8)

EXTENDED OBJECTS IN GAUGE THEORIES

F

Fig. 1.

Potential of the form X ( $ 2 - F 2 ) 2 / 4 !

I t i s customary then t o write

1

2

x

,2

v=Trn$

+-$-413+

with

b)

m2 =

-13 X

V($)

= A(1-

4

F ~;

g

COB

-

XF

= F

+ 4’ ,

$14

41

(1.9)

.

(1.10)

a) (“Sine-Gordon” model- see Fig. 2 ) . F

(1.11)

Here w e write

(1.12) SO

that

v

=

-21m 2 4 2

x 4” + ... -41

(1.13)

In both cases m stands for the mass of the physical p a r t i c l e , ant A i s the usual coupling constant-expansion parameter.

Fig. 2 .

Potential i n “Sine-Gordon” model.

0 . 't HOOFT

The energy of a f i e l d configuration is

H = kd3x

, (1.14)

The examples I gave h e r e have the property t h a t t h e r e is more than one way t o make a zero-energy state' ("vacuum"):

-

a)

$

b)

4 =

nF, nF

n - 2 1 ,

,

n =

..., -2,

-1, 0, 1, 2 ,

(1.15)

...

The world i n these models is assumed t o be a one-dimensional line. W e envisage t h e s i t u a t i o n t h a t a t one s i d e of t h e l i n e Q, has one value f o r n, a t the o t h e r s i d e a d i f f e r e n t value (see Fig. 3). W e can have a t r a n s i t i o n region c l o s e t o t h e o r i g i n which c a r r i e s energy :

4 C

n~

,n

integer ;

ax+ f 0

.

(1.16)

Assme t h a t t h e s i t u a t i o n is s t a t i o n a r y :

at4

*

0

.

(1.17)

We wish a f i n i t e t o t a l energy; m u s t approach zero a t x + 200. So 4 m u a t approach one i f i t s p o s s i b l e vacuum values a t x+*. Thus we f i n d a d i s c r e t e set of allowable boundary conditions.

I n a s t a t i o n a r y s o l u t i o n of t h e equation (1.6) a l s o t h e energy is an extremum. The lowest-energy configuration under t h e given boundary conditions a l s o s a t i s f i e s t h i s equation, It is obviously s t a b l e . I n our examples t h e s o l u t i o n s are e a s i l y given: (1.18)

'-w2 )X

Fig. 3.

A s t a t i c form for $ w i t h d i f f e r e n t boundary c o n d i t i o n s a t x + fa.

257

EXTENDED OBJECTS IN GAUGE THEORIES

(1.19) (1.20)

therefore 41.21) Case a):

1

,

$ ( x ) = F tanh-mx 2

(1.22)

Case b ) : mx $ ( x ) = -i;2F a r c t g e Note t h a t , as x

+

03,

1 1""

+

F(l

F tanh

-

. 2e-mx)

(1.23)

,

mx + F ( I - - e-mx) 2F arctg e IT IT

(1.24)

,

(1.25)

t h u s t h e vacuum v a l u e is reached e x p o n e n t i a l l y , w i t h t h e mass of t h e p h y s i c a l p a r t i c l e i n t h e exponent ( s e e Fig. 3). The t o t a l energy of t h e s e o b j e c t s ( " s o l i t o n s " ) i s

(1.26) Case a ) :

E

-

2 m 3 ~,~

Case b ) :

E =

h31x

.

They behave as p a r t i c l e s i n e v e r y s e n s e ; f o r i n s t a n c e one can show t h a t t h e r e l a t i o n s between energy, momentum and v e l o c i t y are a s i n d i c a t e d by r e l a t i v i t y theory. W e d i s t i n g u i s h two t y p e s of s t a b i l i t y requirements:

1) T o p o l o g i c a l s t a b i l i t y . Since t h e boundary c o n d i t i o n s i n our c a s e form a d i s c r e t e s e t t h e r e .can b e n o continuous t r a n s i t i o n

258

G. 't HOOFT

towards t h e vacuum boundary c o n d i t i o n (which i s t h e one f o r a bunch of o r d i n a r y elementary p a r t i c l e s ) .

2) S t a b i l i t y a g a i n s t s c a l i n g . Let us r e t u r n f o r a moment t o n s p a c e l i k e dimensions ( i n s t e a d of one). I n t h e s t a t i o n a r y case w e a l w a y s have H = S + V ,

s

=

1:

-(a

,

x

V = IV($)dnx

.

(1.27)

The energy must be s t a t i o n a r y under any i n f i n i t e s i m a l v a r i a t i o n . L e t us try one s p e c i a l v a r i a t i o n : (1.28) then

S

+

An"

S ;

V

+

h"v

,

(1.29)

W e must have

na

(S

+

V) = ( n - 2 ) S

+ nV

-

0

.

(1.30)

But S and V are b o t h p o s i t i v e . That i s only compatible w i t h (1.30) i n t h e case n - 1. Since t h e decomposition (1.27) is p o s s i b l e i n a l l scalar f i e l d t h e o r i e s , s c a l a r t h e o r i e s have only s t a t i o n a r y s o l i t o n s i n one space-dimension. I n two o r more dimensions i t would always be e n e r g e t i c a l l y f a v o r a b l e f o r a system t o s h r i n k u n t i l i t becomes p o i n t l i k e . That i s n o t an extended o b j e c t by definition. F i n a l l y , we emphasize t h a t d e s c r i p t i o n of p a r t i c l e s i n terms of q u a n t i z e d f i e l d s i n most cases is only p o s s i b l e i f t h e coupling is n o t t o o s t r o n g :

( i n 1+1dimensions X h a s t h e dimension o f a mass-squared). T h e r e f o r e , t h e mass o f s o l i t o n s , b e i n g p r o p o r t i o n a l t o m3/X, is always much g r e a t e r t h a n t h a t of t h e o r i g i n a l p a r t i c l e s i n t h e theory.

259

EXTENDED OBJECTS IN GAUGE THEORIES

2. SOLITONS I N 2 SPACE DIMENSIONS AND STRINGS I N 3 SPACE DIMENSIONS There is a n o t h e r reason why s c a l a r t h e o r i e s have no t o p o l o g i c a l s o l i t o n s i n 2 o r more s p a c e dimensions. W e can o n l y have t o p o l o g i c a l s t a b i l i t y i f t h e boundary c o n d i t i o n d i f f e r s from: I $ + c n a t a n t Imagine t h a t w e have s e v e r a l f i e l d components and a t 1x1 assme t h a t V($) h a s a whole continuum of minima. For i n s t a n c e w e could t a k e n f i e l d components (n= number of s p a c e - l i k e dimens i o n s ) and have V($) minimal i f = F. W e could t h i n k of mapping

8

+-.

-

as a boundary c o n d i t i o n . But t h e n

This would imply t o p o l o g i c a l s t a b i l i t y .

and

d i v e r g e s a t l a r g e x u n l e s s n = 1. T h i s i n f r a r e d divergence is c l o s e l y r e l a t e d t o t h e e x i s t e n c e o f m a s s l e s s Goldstone bosons i n such a theory. The o n l y way known a t p r e s e n t t o r e - o b t a i n a t o p o l o g i c a l l y s t a b l e s o l i t o n i n more t h a n one s p a c e l i k e dimension i s t o add a gauge f i e l d :

The space-components of t h e v e c t o r f i e l d A can b e arranged such t h a t (D$)' * 0 more r a p i d l y t h a n 1 / x 2 , and t h i s way w e can approach a p h y s i c a l vacuum so r a p i d l y t h a t t h e t o t a l energy converges. (Note a l s o t h a t by adding a gauge f i e l d t h e massless Goldatone boson d i s a p p e a r s , so t h e t h e o r y h a s b e t t e r i n f r a r e d convergence. There may s t i l l be a massless v e c t o r boson but t h a t is u s u a l l y less harmful, a s we w i l l see.) The s i m p l e s t t h e o r y w i t h a s o l i t o n i n 2 s p a c e dimensions is scalar quantum-electrodynamics i n t h e Higgs mode4 9 :

260

G. 't HOOFT

Here q is t h e unit of charge. The 3 + 1 dimensional analogue of t h i s model is w e l l known i n physics: i t describes the superconductor. JI is the s i m p l i f i e d f i e l d ("order parameter") t h a t e s s e n t i a l l y d e s c r i b e s two-electron bound states with t o t a l s p i n zero (hence i t is a s c a l a r boson f i e l d ) . A,, is the ordinary vector p o t e n t i a l . q = 2e. I n s i d e t h e superconductor the photon behaves as a massive particle and electromagnetic f i e l d s are of s h o r t range. Longrange magnetic f i e l d s are forbidden because i n order t o c r e a t e them we need p o t e n t i a l d i f f e r e n c e s (according t o Maxwell's laws) and those d o n o t occur i n s i d e a superconductor. But when placed i n a s t r o n g magnetic f i e l d a superconductor w i l l f i n d an e x c i t e d s t a t e i n which i t can allow magnetic f l u x t o penetrate. This f l u x w i l l go through the superconductor i n narrow tubes. The f l u x i n such a tube t u r n s out t o be quantized. I f w e consider a plane orthogonal t o these tubes then we g e t equations f o r extended p a r t i c l e - l i k e o b j e c t s i n t h i s plane. They are t h e 2 dimensional s o l i t o n s t o be considered now. The f l u x tubes ("vortices", " s t r i n g s " ) , i n the t h r e e dimensional world are derived from them by adding a t h i r d coordinate ( p a r a l l e l t o t h e axie) on which the f i e l d s do not depend. The vacuun is described by

-

144

F

(2.6)

W e construct our s o l i t o n just as i n t h e previous s e c t i o n ; w e choose t o approach a vacuum r a p i d l y a t + -, but i n such a way t h a t continuity r e q u i r e s a t r a n s i t i o n region a t x -t 0. Write

Is]

x1

xp

-

8

,

r sin 8

,

r

COB

then we choose as r

-t

=, JI

+

Fei e

.

Continuity r e q u i r e s t h a t JI produces a zero somewhere near the o r i g i n of x-space. Since the vacuum value of is F, i t c o s t s energy t o produce such a zero: w e w i l l o b t a i n a topologically s t a b l e lump of energy near t h e o r i g i n of x-space. L e t us look a t t h e lowest-energy configuration. Assume i t is time-independent and .A = 0. As a gauge condition we choose:

lJll

and, by symmetry arguments :

261

EXTENDED OBJECTS IN GAUGE THEORIES

One can e a s i l y check t h a t t h e above i s a s e l f - c o n s i s t e n t a n s a t z f o r a s o l u t i o n of t h e f i e l d e q u a t i o n s .

x

To o b t a i n t h e e q u a t i o n f o r and A i t is e a s i e s t t o reexpress t h e Lagrangian i n terms of and A:

I

x

m

/&d2z =

2nrdr(- i r 2 ( $ ) 2

- L(a)2 - ( i + q A r ) 2 x2 2 dr

0

- X2 ( ~ 2 - F 2 ) 2 ]-

m

nr2A21 0

I+(-) I

Assuming -C 0 w e may i g n o r e t h e boundary term. boundary c o n d i t i o n s a t r + 0 w e m u s t r e q u i r e

As f o r

(2.10)

At r

+ w

t h e t h i r d and f o u r t h terms i n (2.9) converge only i f

(2.11) Since our system is assumed t o be s t a t i o n a r y ( a / a t = 01, and t h e time-components of A,, v a n i s h , t h e energy w i l l be j u s t minus t h e Lagrangian.

I

m

E

=

6(r)dr

,

(2.12)

0

(2.13) W e m u s t f i n d t h e minimum of t h i s energy under t h e above boundary c o n d i t i o n s . The coupled d i f f e r e n t i a l e q u a t i o n s ,

262

G. 't HOOFT

(2.14)

are n o t e x a c t l y s o l u b l e . The s o l u t i o n i s s k e t c h e d i n Fig. 4. As i n t h e 1+ 1 dimensional case, w e e x p e c t t h a t t h e f i e l d s a t l a r g e d i s t a n c e from t h e o r i g i n d e v i a t e only e x p o n e n t i a l l y from t h e v a c a m v a l u e s , w i t h t h e masses of t h e two massive p a r t i c l e s ( v e c t o r boson and Higgs p a r t i c l e ) i n t h e exponents. Thus w e o b t a i n a w e l l behaved, t o p o l o g i c a l l y s t a b l e s o l i t o n , p a r t i c l e - l i k e i n 2 + 1 dimens i o n s , s t r i n g - o r v o r t e x - l i k e i n 3 + 1 dimensions.

What w i l l t h e mass o f t h i s p a r t i c l e be? Without s o l v i n g t h e e q u a t i o n s e x p l i c i t l y , we can make some s c a l i n g arguments. Put 2

F

2

2

= mH/2A = %/2q

where mH and masses, and

%

2

,

(2.15)

are, r e s p e c t i v e l y t h e Higgs and t h e v e c t o r boson

then

(2.17)

F i g . 4.

Sketch of the s o l u t i o n s t o Eq.

263

(2.14).

EXTENDED OBJECTS IN GAUGE THEORIES

Note t h e coupling c o n s t a n t d o w n s t a i r s . Of c o u r s e , i n 2 + 1 dimens i o n s , q 2 h a s t h e dimension o f a mass, hence , n o t %. I n 3 + 1 dimensions, t h e energy o f a s t r i n g is p r o p o r t i o n a l t o its length, and (2.17) g i v e s t h e energy p e r u n i t of l e n g t h , which is i n u n i t s of mass-squared.

4

I n t h e beginning of t h i s s e c t i o n w e a n t i c i p a t e d t h a t t h i s v o r t e x would be a magnetic f l u x tube. L e t us now v e r i f y t h a t by computing t h e magnetic f i e Id. A t s p a t i a l i n f i n i t y t h e v e c t o r p o t e n t i a l is found t o b e , from (2.111, Ai

- Eil x

-F

/qr 2

.

(2.18)

Therefore

f

Aidx

i

-t

2nlq

.

(2.19)

So, indeed, w e have a magnetic f i e l d going along t h e v o r t e x w i t h t o t a l f l u x 2n/q, t h i s i n s p i t e of t h e f a c t t h a t t h e photon had become massive which would imply t h a t e l e c t r o m a g n e t i c f i e l d s are only short-range. There is no way of producing a broken f r a c t i o n of t h a t f l u x , t h e r e f o r e magnetic f l u x i n a superconductor is q u a n t i z e d , by u n i t s

nle

2nlq

.

(2.20)

There is c l e a r l y also a p h y s i c a l r e a s o n f o r o u r v o r t e x o r s t r i n g t o b e s t a b l e : t h e magnetic f l u x i t c o n t a i n s cannot s p r e a d o u t i n t h e superconductor and i t is e x a c t l y conserved.

3.

THE BOUNDARY CONDITION

- HOMOTOPY CLASSES

The above w a s a p e d e s t r i a n way t o o b t a i n t h e v o r t e x i n a superconductor. W e w i l l now r e f o r m u l a t e t h e boundary c o n d i t i o n more p r e c i s e l y i n o r d e r t o b e a b l e t o produce more complicated o b j e c t s

later. r

-F

Remember w e chose o u r boundary c o n d i t i o n t o b e J, could have chosen

Q).W e

+

eieF a t

every choice of n would y i e l d a p a r t i c u l a r c o n f i g u r a t i o n w i t h lowest energy, dependent on n. I f t h e r e would have been any way

264

G. 't HOOFT

-

t o make a continuous t r a n s i t i o n from one n t o another, without violating lJll F a t l a r g e d i s t a n c e from t h e o r i g i n , then t h a t would cause an i n s t a b i l i t y f o r whatever s t a t e had the h i g h e r energy. But t h e r e is no way t o make t h i s continuous t r a n s i t i o n . W e s a y t h a t the d i f f e r e n t choices of n are homotopically d i f f e r e n t 2 . The boundary has t h e topological shape o f a c i r c l e . (In N dimensions i t is t h e SN-1 sphere.) The vectors

a l s o form a circle. Two continuous mappings of a circle onto a c i r c l e are c a l l e d "homotopic" i f one can continuously be transformed i n t o t h e other. A l l c l a s s e s of mappings t h a t are homotopic t o each o t h e r form a "homotopy class". For t h e mappings of a c i r c l e onto a c i r c l e , o r any S sphere onto an SN-1 sphere, these homotopy classes are l a b e r i a by an i n t e g e r n running from -OD t o m. I n order t o formulate the boundary case w e introduce two types of vacuum: of space o r space-time where a l l vector scalar f i e l d s have t h e i r vacuum value: i n a preassigned d i r e c t i o n i n isospace, direction:

condition i n t h e g e n e r a l

a supervacuum is a region f i e l d s vanish and a l l F and are pointing f o r instance the z-

-

Now i n a gauge theory, l i k e electromagnetism, we are allowed t o make gauge r o t a t i o n s ; f o r i n s t a n c e i n t h e model of the previous se c t i o n :

After t h i s , i n general space-time-dependent, gauge r o t a t i o n w e o b t a i n non-vanishing vector-potentials and t h e s c a l a r f i e l d s w i l l point i n a r b i t r a r y d i r e c t i o n in isospace. Physically, w e s t i l l have a vacum. This w e w i l l c a l l a normal vacum, but no longer supervacum. Any vacuum configuration is defined by specifying the gauge r o t a t i o n Q(x, t ) t h a t produces i t from t h e supervacum. Sometimes, i f t h e r e is no complete spontaneous symmetry breaking, Q is only defined up t o a space-time independent constant r o t a t i o n . W e will ignore t h a t f o r a moment.

265

EXTENDED OBJECTS IN GAUGE THEORIES

Returning t o t h e model of t h e p r e v i o u s s e c t i o n , b a s e d on t h e gauge group U(1), t h e gauge r o t a t i o n s Q are determined by a p o i n t on t h e u n i t c i r c l e , and so t h e g e n e r a l vacuum i s determined by an a n g l e t h a t may vary as a f u n c t i o n of space-time. Now, i n f i n i t y i n two dimensions is c h a r a c t e r i z e d by a n a n g l e 8 ( t h e d i r e c t i o n i n which w e go t o i n f i n i t y ) , so t h e boundary c o n d i t i o n is d e f i n e d by a mapping

of t h e u n i t circle o n t o an a n g l e 8. As w e saw b e f o r e , t h i s mapping m u s t be i n one of a d i s c r e t e s e t of homotopy c l a s s e s , l a b e l e d by -m
Because n m u s t s t a y i n t e g e r t h e r e cannot b e continuous t r a n s i t i o n s from a s t a t e c o n t a i n i n g n s o l i t o n s o f t h e same type i n t o a s t a t e c o n t a i n i n g fewer s o l i t o n s . P h y s i c a l l y t h i s is a g a i n t h e s t a t e m e n t t h a t magnetic f l u x going through a two-dimensional s u r f a c e i s conserved.

Now we a r e i n a p o s i t i o n t h a t w e can e a s i l y g e n e r a l i z e o u r model i n t o a similar model b u t w i t h a non-Abelian gauge f i e l d i n s t e a d of electromagnetism: t h e non-Abelian superconductor. We assume t h a t t h e gauge r o t a t i o n s ( 3 . 1 ) and ( 3 . 2 ) a r e r e p l a c e d by real, o r t h o g o n a l , 3 x 3 m a t r i x r o t a t i o n Q ( x ) , w i t h determinant one. They form t h e non-Abelian group S O ( 3 ) . W e assume, t h a t , a s i n t h e U(1) superconductor, a Higgs f i e l d o r set of Higgs f i e l d s w i t h i n t e g e r i s o s p i n b r e a k t h e symmetry spontaneously and completely. Does t h i s model allow f o r s t a b l e f l u x tubes a s i t s Abelian counterp a r t does? Do w e have s o l i t o n s i n two-dimensional s p a c e ? To a n a l y s e t h i s q u e s t i o n w e merely have t o i n v e s t i g a t e t h e homotopy c l a s s e s of t h e mappings of S O ( 3 ) matrices i n t h e p e r i o d i c space of a n g l e s 8. I t i s w e l l known t h a t t h e group SO(3) i s l o c a l l y isomorphic w i t h SU(2), t h e group of 2 x 2 u n i t a r y m a t r i c e s w i t h determinant one. The correspondence i s made by i d e n t i f y i n g > v e c t o r s w i t h traceless, 2 x 2 t e n s o r s . An SU(2) m a t r i x a c t i n g on t h e t e n s o r corresponds t o an SO(3) r o t a t i o n of t h e 3-vector. The SU(2) matrices are easier t o handle. posed as

a+

* a* 0

They can b e decom-

(3.3)

- i f apt. 11- 1

266

G. 't HOOFT

The requirements Gilt = 1 and d e t $2 = 1 correspond to:

ao,ag are real and

13

a2 g = l .

(3.4)

R=O C l e a r l y , t h e SU(2) m a t r i c e s form an S3 sphere ( t h e s u r f a c e of a sphere embedded i n 4 dimensions). The SO(3) matrices do not form an S, sphere, however. reason is t h a t an S U ( 2 ) r o t a t i o n

The

on s p i n o r s leaves 2 x 2 t e n s o r s i n v a r i a n t . Consequently, two oppos i t e p o i n t s on t h e S3-sphere correspond t o t h e came SO(3) m a t r i x (see Fig. 5). I t is t h i s f e a t u r e t h a t will allow a n o n - t r i v i a l homotopy c l a s s t o emerge. The homotopy c l a s s e s here a r e e n t i r e l y d i f f e r e n t from those i n t h e U( 1 ) model: any closed contour drawn on an S 3 sphere can be deformed continuously t o a dot ( s e e Fig. 6a), so t h e mappings of SU(2) onto t h e u n i t c i r c l e are c h a r a c t e r i z e d by only one homotopy c l a s s , namely t h a t of t h e supervacuum. But i f we a r e only concerned with SO(3) m a t r i c e s one may c o n s t r u c t one d i f f e r e n t homotopy c l a s s : a l l contours t h a t s t a r t on one p o i n t of t h e S, sphere and c l o s e by ending up a t t h e p o i n t opposite t o A ( s e e Fig. 6b). These a r e t h e only two homotopy c l a s s e s i n t h i s c a s e , t h i s i n c o n t r a s t with the s i t u a t i o n i n t h e ordinary superconductor where we had an i n f i n i t y of homotopy c l a s s e s .

What is t h e consequence for our S0(3)-superconductor? The e x i s t e n c e of one n o n - t r i v i a l homotopy c l a s s implies t h a t w e can

Fig. 5 .

The o p p o s i t e p o i n t s on SJ-sphere are i d e n t i f i e d .

267

EXTENDED OBJECTS IN GAUGE THEORIES

...........

.........

Fig 6.

............. ....

.........

Two (and t h e only two) homotopy c l a s s e s of closed p a t h s f o r SO(3). P a t h s i n (b) cannot be deformed i n t o t h e p a t h s i n ( a ) .

c o n s t r u c t a n o n - t r i v i a l boundary c o n d i t i o n f o r a s t a b l e s o l i t o n i n 2 dimensions. So our model does a d m i t a s t a b l e f l u x tube. However, now t h e r e is no longer an a d d i t i v e f l u x conservation l a w : i f two o f these s o l i t o n s come t o g e t h e r then t h e boundary of t h e t o t a l s y s t e m is given by a contour i n SO(3) space t h a t goes through t h e S3 sphere twice. That contour is i n t h e t r i v i a l homotopy c l a s s , t o g e t h e r with t h e boundary of a supervacuum. So t h e p a i r of s o l i t o n s is not t o p o l o g i c a l l y s t a b l e which means they may a n n i h i l a t e each other.

Thus, only single f l u x tubes are

s t a b l e . P a i r s o f v o r t i c e s , r e g a r d l e s s t h e i r o r i e n t a t i o n , may a n n i h i l a t e each o t h e r , forming a shower of o r d i n a r y p a r t i c l e s .

268

G. 't HOOFT

4.

STRINGS AND MONOPOLES

Next, l e t UB d i s c u s s a model very similar t o t h e p r e v i o u s one. Again our gauge group i s S0(3), b u t now we assume only one Higgs t r i p l e t , b r e a k i n g i t down t o U(1). T h i s i s t h e s o - c a l l e d GeorgiGlashow model6 t n t r o d u c e d f i v e y e a r s ago as a p o s s i b l e c a n d i d a t e f o r t h e weak i n t e r a c t i o n s . It c o n t a i n s a massless U ( 1 ) photon, a massive charged "intermediate v e c t o r boson" , and a n e u t r a l rnassiwe Higgs p a r t i c l e . Designed t o avoid t h e n e c e s s i t y of n e u t r a l c u r r e n t s and charm, a t t h e p r i c e o f having t o i n t r o d u c e heavy l e p t o n s , t h i s model was doomed t o become o b s o l e t e a8 a r e a l i s t i c d e s c r i p t i o n o f observed weak i n t e r a c t i o n s . However, b e i n g one of t h e very few t r u l y u n i f y i n g models i t h a s aome p e c u l i a r f e a t u r e s which w e now wish t o focus on. To create a s i t u a t i o n e x a c t l y as i n t h e p r e v i o u s model, w e c o n s i d e r an o r d i n a r y superconductor' i n a world whose high-energy b e h a v i o r i s d e s c r i b e d by t h e Georgi-Glashow model. The o r d e r parameter then p r o v i d e s f o r b r e a k i n g of t h e r e s i d u a l U ( 1 ) symmetry. According t o t h e p r e v i o u s s e c t i o n t h i s superconductor d i f f e r s from t h e o r d i n a r y superconductor by t h e f a c t t h a t two f l u x t u b e s p o i n t i n g i n t h e same d i r e c t i o n may a n n i h i l a t e e a c h o t h e r , because t h e gauge group i s S0(3), n o t U ( 1 ) . But p h y s i c a l l y t h i s looks very odd, s i n c e one would e x p e c t magnetic f l u x t o b e conserved. The only p o s s i b l e e x p l a n a t i o n of t h e b r e a k i n g up of (double) flux tubes is t h a t a p a i r of magnetic monopoles i s formed. They c a r r y magnetic charge +4n/e. T h i s way one i s n a t u r a l l y l e d t o a c c e p t t h e i d e a t h a t magnetic monopoles occur i n t h e Georgi-Glashow model. They are n o t confined t o t h e superconductor, which we had o n l y i n t r o d u c e d f o r pedagogical reasons. L e t us now focus on t h e near vacuum s u r r o u n d i n g t h i s magnetic monopole. At one s i d e , s a y t h e p o s i t i v e z - d i r e c t i o n , we assume a double magnetic flux tube t o emerge. Around t h i s f l u x tube w e do n o t have a supervacuum b u t a gauge r o t a t i o n of t h a t , which i s i n ~ ~ ( n2o t1a t i o n :

where Q, i s t h e a n g l e around t h e z-axis. magnetic f l u x coming i n , so n(z

+

-,$I

= indep.

($1

.

A t t h e o t h e r s i d e i s no

(4.2)

269

EXTENDED OBJECTS IN GAUGE THEORIES

According t o homotopy t h e o r y no d i s c o n t i n u i t y i n 62 is needed elsewhere (except a t t h e l o c a t i o n of t h e monopole, where no 62 i s needed because i t i s n o t a vacuum c o n f i g u r a t i o n ) . Indeed w e may c o n s t r u c t 51 everywhere a s follows:

+

ieI, o

sin

4 i

.

(4.3)

The Higgs f i e l d i n t h i s model i s an i s o v e c t o r , which i n a superv a c u m t a k e s t h e form

(4.4)

I n the surroundings of a s i n g l e monopole i t i s

where i t is understood t h a t t h e 62 of Eq. w r i t t e n i n SO(3) n o t a t i o n .

(4.3) h a s f i r s t been

I t i s c l e a r t h a t t h e boundary c o n d i t i o n (4.5) f o r t h e Higgs f i e l d l e a v e s a s t a b l e s o l i t o n a t t h e c e n t e r . I t i s a l s o clear from t h e p r e v i o u s d i s c u s s i o n t h a t t h i s s o l i t o n w i l l c a r r y a magn e t i c charge e q u a l t o 4nle.

W e s k i p f u r t h e r d i s c u s s i o n of t h e monopole i n t h e s e n o t e s because w e have l i t t l e t o add t o t h e e x i s t i n g l i t e r a t u r e 6 ’ e * g . J u s t n o t e t h a t , as w a s t h e c a s e f o r o t h e r s o l i t o n s , i t s mass is i n v e r s e l y p r o p o r t i o n a l t o t h e coupling parameter:

5. 5.1.

INSTANTONS Construction

We have seen extended o b j e c t s i n one, two and t h r e e dimensions. None of t h e models of t h e previous s e c t i o n s , i n which t h e s e o b j e c t s occur are of d i r e c t i n t e r e s t i n high energy p h y s i c s . The models

270

G. 't HOOFT of i n t e r e s t are two d i f f e r e n t types of gauge t h e o r i e s t h a t are most e x t e n s i v e l y s t u d i e d t h e s e days: t h e SU(2) X U ( 1 ) gauge theory f o r t h e weak and e l e c t r o m a g n e t i c i n t e r a c t i o n s , "spontaneously broken" t o U ( 1 ) by an i s o d o u b l e t Higgs f i e l d ' O a l l , and a pure gauge theory based on SU(3) f o r t h e s t r o n g i n t e r a c t i o n s 1 2 . The weak i n t e r a c t i o n gauge group h a s e s s e n t i a l l y a four component Higgs f i e l d because t h e Higgs d o u b l e t is complex. Therefore we expect a topologically s t a b l e object not i n th r e e b u t i n f o u r dimensions. Why? The boundary o f a f o u r dimensional world is t o p o l o g i c a l l y an S, sphere. Also t h e complex Higgs doublet

form an S, sphere. The mappings of S, s p h e r e s on S , s p h e r e s a r e a g a i n c h a r a c t e r i z e d by n o n - t r i v l a l homotopy classes, l a b e l e d by an i n t e g e r n t h a t can run from -m t o Thus w e can c o n s t r u c t boundary c o n d i t i o n s f o r a f o u r dimensional s p a c e under which configur a t i o n s w i t h n s o l i t o n - l i k e o b j e c t s are s t a b l e .

*.

In c o n t r a s t t o t h e cases i n lower dimensions, a l s o pure gauge t h e o r i e s allow f o r t o p o l o g i c a l l y s t a b l e s o l i t o n - l i k e o b j e c t s in f o u r dimensions. T h i s i s most e a s i l y understood when we c o n s i d e r t h e s i m p l e s t non-Abelian group SU(2). Remember t h a t one way t o d e f i n e a vacuum a t + t h e boundary of a r e g i o n is t o s p e c i f y t h e gauge r o t a t i o n s n ( x ) t h a t produce t h i s vacuum o u t of t h e supervacuum. The gauge r o t a t i o n s 51 themselves form an S3 s p h e r e ( s e e s e c t i o n 31, so h e r e t o o we have t h e n o n - t r i v i a l homotopy classes a t t h e boundary. I n f a c t , t h e S, s p h e r e of t h e gauge r o t a t i o n s 51 themselves, is t h e same as t h e S, s p h e r e of t h e complex d o u b l e t s

i n t h e pure gauge t h e o r y are e s s e n t i a l l y t h e same a s t h o s e i n t h e t h e o r i e s broken by Higgs isodoub l e ts so one can say t h a t t h e 4 dimensional s o l i t o n s

.

I n o r d e r t o d i s t i n g u i s h o u r 4-dimensional o b j e c t s from t h e 3-dimensional s o l i t o n s and t h e s t r i n g s and s u r f a c e s , we g i v e them a new name: "Instantons". The name emphasizes t h a t i f one of t h e f o u r dimensions is t i m e , t h e n t h e s e o b j e c t s occupy only a s h o r t time-interval (instant). We w i l l now argue t h a t t h e r e is an important q u a n t i t y t h a t is s e n s i t i v e t o t h e p r e s e n c e of i n s t a n t o n s :

27 1

EXTENDED OBJECTS IN GAUGE THEORIES

where Ga

vv

is t h e usual covariant c u r l of the gauge vector f i e l d

pt" :

GE is gauge i n v a r i a n t , and i t is a pure divergence:

However, as one can v e r i f y e a s i l y , $ is not gauge i n v a r i a n t . us see w h a t t h e consequence is of t h a t .

Let

Consider a region of space-time enclosed by a l a r g e S 3 sphere. It is i r r e l e v a n t as y e t whether t h e space-time is Euclidean o r Minkowskian; w e are only concerned about topological s t a b i l i t y and not y e t i n t e r e s t e d i n any equation of motion f o r the f i e l d s , not even the space-time metric. Suppose w e have a supervacuum a t t h e surrounding sphere : A a - O ,

P

therefore

\ = o .

(5.4)

Because of (5.4) a t the boundary, and because of (5.3), w e f i n d by means of Gauss' l a w :

regardless of what values the f i e l d s take i n s i d e t h e sphere. Now w e take a n o n - t r i v i a l homotopy class of gauge r o t a t i o n s t o g e t a new vacuum boundary condition:

272

G. ‘t HOOFT

I f w e write the vector f i e l d in an SU(2) n o t a t i o n , (5 7) then the transformation l a w is

W e f i n d t h a t t h e vacuum a t t h e boundary has

with

U

1a2.3

s

(5.10)

Now although the f i e l d (5.9) is p h y s i c a l l y equivalent t o t h e vacuum ( f o r i n s t a n c e , G i V a t t h e boundary vanishes), w e nevertheless g e t a non-vanishing value f o r K,,. This w a s possible because K,, is not gauge-invariant : 4

(5.11)

where 2

2

1x1 = x4

2

+ xi

.

I f we now apply Gauss’ l a w w e f i n d t h a t i n s i d e t h e S, sphere G PV cannot vanish everywhere because (5.12) boundary Merely because the vacuum a t our boundary i e t o p o l o g i c a l l y d i s t i n c t from t h e supervacum we g e t a n o n - t r i v i a l phyeical f i e l d configuration inside.

273

EXTENDED OBJECTS I N GAUGE THEORIES

Note t h a t t h e r e s u l t (5.12) h o l d s t r u e a l s o i f t h e SU(2) group c o n s i d e r e d would be embedded i n a l a r g e r gauge group. T h i s i m p l i e s t h a t i n s t a n t o n s remain t o p o l o g i c a l l y s t a b l e even i f t h e gauge group is e n l a r g e d . Remember t h e Abelian f l u x t u b e s t h a t could become u n s t a b l e i f the U ( 1 ) group were p a r t of a l a r g e r group; such a t h i n g does n o t happen f o r i n s t a n t o n s . Now l e t ua i n s e r t t h e f i e l d e q u a t i o n s . It is easiest t o cons i d e r f i r s t Euclidean a p a c e , where t h e boundary choice. (5.6) h a s nice rotational properties. Is t h e r e a s o l u t i o n t o the E u c l i d e a n and s t a y s r e g u l a r f i e l d e q u a t i o n s t h a t approaches (5.9) a t x 2 + a t the origin? W e try

(5.13) The f i e l d e q u a t i o n , DVGpv= 0 , now r e a d s dL 2 ( r 2 f ) = 4r f (d log r ) 2

-

12(r2f12 + 8(r2f13

.

(5.14)

T h i s e q u a t i o n i s s c a l e - i n v a r i a n t , which makes i t a s e a s y t o s o l v e as t h e one-dimensional s o l i t o n , whose e q u a t i o n i s t r a n s l a t i o n i n v a r i a n t . Besides t h e t r i v i a l s o l u t i o n , f = l / r 2 , w e can have 1 3 f(r)

- 1, 1

r

+ A

X arbitrary

,

(5.15)

which h a s t h e r e q u i r e d p r o p e r t i e s . The t o t a l a c t i o n f o r t h e s o l u t i o n i s

(5.16) Without a c t u a l l y computing S i t i s e a s y t o d e r i v e an i n e q u a l i t y f o r i t . I n E u c l i d e a n space Ga and a l l have r e a l components VV PV only. T h e r e f o r e

ca

(5.17) (5.18) because of ( 5 . 1 2 ) , and w e can choose e i t h e r s i g n .

For o u r s o l u t i o n , (5.19)

274

G. 't HOOFT

and w e f i n d 2

S = - 8 T /g

2

.

(5.20)

a "a The i n e q u a l i t y i s s a t u r a t e d , because GPv= % V f o r o u r solu_tion. Indeed, any s o l u t i o n of t h e ( f i r s t o r d e r ) e q u a t i o n GPv = G-,,,, automafica&ly s a t i s f i e s t h e second o r d e r e q u a t i o n D G lJ lJv = 0 because D,,G = 0 f o r any gauge f i e l d d e r i v e d from a v e c t o r potential 2).

('d.

5.2.

Euclidean F i e l d Theory and Tunneling

A s o l u t i o n of t h e classical f i e l d e q u a t i o n i n Euclidean s p a c e may seem t o be of l i t t l e p h y s i c a l r e l e v a n c e . What we r e a l l y would l i k e t o c o n s i d e r are i n s t a n t o n e v e n t s i n Minkowski space. Coneid e r a g a i n t h e S, boundary o f a compact r e g i o n i n Minkowski space. The mapping Q ( x ) a t t h e boundary may b e deformed a b i t , as long as we s t a y i n t h e same homotopy class. Thus, t h e i n s t a n t o n can be s e e n t o c y r e s p o n d t o a t r a n s i t i o n from a supervacuum a t t i - (where Q ( x ) 1) t o a gauge r o t a t e d vacuum a t t +=. There w e can have

-

-

(5.21)

A t i n t e r m e d i a t e t i m e s t h e r e must b e a r e g i o n where G,," # 0 , because o f Eq. (5.12). Because w e have a vacuum a t t h e beginning and a t t h e end t h i s f i e l d c o n f i g u r a t i o n cannot b e a s o l u t i o n of t h e c l a s s i c a l e q u a t i o n s . R a t h e r , i t should be c o n s i d e r e d as a quantum-mechanical t u n n e l i n g t r a n s i t i o n . How do we. compute t h e amplitude o f such a t u n n e l i n g t r a n s i t i o n ? I t is i n s t r u c t i v e t o compare t h i s problem w i t h t h e motion of a s i n g l e p a r t i c l e through a one-dimensional p o t e n t i a l w e l l : 2

H =

+ V(x)

,

(5.22)

i n a r e g i o n where V(x) >> E , assuming t h a t P l a n c k ' s c o n s t a n t is small. To f i r s t approximation

(5.23)

The amplitude f o r a t u n n e l i n g p r o c e s s is p r o p o r t i o n a l t o e' w i t h

275

EXTENDED OBJECTS IN GAUGE THEORIES

..

S - - Lni " X

42m(V-

E) dx

.

(5.24)

a

On t h e o t h e r hand, i f a t r a j e c t o r y i s i n a n e n e r g e t i c a l l y allowed r e g i o n , E > V, t h e n t h e wave f u n c t i o n o s c i l l a t e s . The number of o s c i l l a t i o n s i s g i v e n by

1

X.

b

2niI

X

pdx = 2nn

I

42m(E- V) dx

-2m

R

.*

(5.25)

a

R I s related t o the action integral:

I f we normalize t h e energy t o z e r o t h e n

R -

I

Ldt = S .

(5.26)

This i s t h e t o t a l a c t i o n of a motion from xa t o Xb w i t h g i v e n energy. Note t h a t the t u n n e l i n g p r o c e s s i s g i v e n by (5.24) which is t h e same e x p r e s s i o n e x c e p t t h a t t h e s i g n of V - E is f l i p p e d . The s i g n of t h e p o t e n t i a l i n : d2x

3V

(5.27)

mdP=ax'

is a l s o f l i p p e d i f w e r e p l a c e t by i t . I f t h e r e i s no c l a s s i c a l l y t i m e s then allowed movement a t z e r o energy from xa t o xb a t t h e r e i s an allowed t r a n s i t i o n f o r imaginary t i m e s . W e find that t h e t o t a l a c t i o n 3 f o r t h i s imaginary t r a n s i t i o n determines t h e t u n n e l i n g amplitude". I n quantum f i e l d t h e o r y t h a t corresponds t o r e p l a c i n g Minkowski s p a c e by Euclidean space. The p h y s i c a l t u n n e l i n g amplitude from t h e supervacuum t o a gauge r o t a t e d vacuum i n a d i f f e r e n t homotopy c l a s s i s governed t o f i r s t approximation by t h e t o c a l a c t i o n 3 of a classical s o l u t i o n i n E u c l i d e a n space. The i n e t a n t o n h a s , a s w e s a w , 5 = - 8 n Z / g 2 . So, w e w i l l f i n d amplitudes p r o p o r t i o n t o -8n2/g2 e

.

The s u p e r p o s i t i o n of

(5.28) t h e supervacuum and t h e gauge r o t a t e d vacua

276

G. ‘t HOOFT

g i v e s rise t o d i f f e r e n t p o s s i b i l i t i e s f o r t h e t r u e ground s t a t e of H i l b e r t space, l a b e l e d by an a r b i t r a r y a n g l e 0. D i s c u s s i o n o f this phenomenon i e t o be found i n Refs. 15 and 16.

5.3. Symmetry Breaking through I n s t a n t o n s The above h o l d s f o r a gauge t h e o r y w i t h o u t fermions. Inst a n t o n e t h e r e r e a r r a n g e t h e ground s t a t e of t h e theory. They do n o t show up as s p e c i a l events”. A new phenomenon o c c u r s i f fermions a r e coupled t o t h e theory. Consider some gauge-invariant fermion c u r r e n t J,,. I f t h e r e is a s t r o n g background gauge f i e l d Ga t h e n t h e r e w i l l b e a vacuum ,v ’ polarization :

That e f f e c t is computed i n a t r i a n g u l a r Feynman diagram ( s e e Fig.7). One of t h e e x t e r n a l l e g s denotes t h e space-time p o i n t x where J,,(x) is measured. The o t h e r two l e g s are t h e two gauge photon e x t e r n a l linee

.

Fig. 7 .

T r i a n g u l a r anomaly diagram.

277

EXTENDED OBJECTS IN GAUGE THEORIES

Let us assume, w i t h o u t y e t s p e c i f y i n g J , t h a t i t i s conserved according t o t h e Noether theorem. Then o u r diagram, which w e could call (5.30) should s a t i s f y (5.31) Gauge i n v a r i a n c e f u r t h e r d i c t a t e s t h a t only t r a n s v e r s e photons a r e coupled :

karuaL3

-

(5.32)

O

(5.33) Now t h e t r i a n g l e diagram, when computed, shows a n i n f i n i t y t h a t must be s u b t r a c t e d . I t can happen t h a t t h e r e a r e only two subt r a c t i o n c o n s t a n t s t o b e chosen. Eqs. ( 5 . 3 1 ) - ( 5 . 3 3 ) form t h r e e c o n d i t i o n s f o r t h e s e two c o n s t a n t s . They can b e incompatible. I n t h a t c a s e w e have t o keep gauge i n v a r i a n c e i n o r d e r t o keep r e n o r m a l i z a b i l i t y . Consequently (5.31) is The r e s u l t can be w r i t t e n a s (5.34)

Here N is an i n t e g e r determined by t h e d e t a i l s which were l e f t o u t i n my d i s c u s s i o n . T h i s i s c a l l e d t h e Adler-Bell-Jackiw-anomaly. Now c o n s i d e r Eq. ( 5 . 1 2 ) 2 0 . We f i n d t h a t an i n t e g e r number of charge unitsQ belonging t o t h e c u r r e n t J are consumed by t h e u instanton:

AQ

-

/d4x

auJU

= 2N

.

(5.35)

I n t h e s t r o n g - i n t e r a c t i o n gauge t h e o r y , a c u r r e n t J w i t h t h i s p r o p e r t y is t h e c h i r a l charge ( t o t a l number of quarks w i t h h e l i c i t y + minus quarks w i t h h e l i c i t y -). Since t h e r i g h t hand s i d e of (5.34) i s a s i n g l e t under f l a v o r SU(3) i t is t h e n i n t h a x i a l v e c t o r c u r r e n t which is nonconserved through t h e i n s t a n t o n . T h i s probably e x p l a i n s why t h e r e is no SU(3) s i n g l e t ( o r SU(2) s i n g l e t ) pseudos c a l a r p a r t i c l e as l i g h t as t h e pion2’ s 2 2 .

278

G. 't HOOFT

S i n c e t h e v e c t o r c u r r e n t s remain conserved, t h e i n s t a n t o n a p p e a r s t o f l i p t h e h e l l c i t y of one of each t y p e of fermion i n v o l v e d ( s e e Fig. 8 ) . T h i s symmetry b r e a k i n g e f f e c t o f t h e i n s t a n t o n c a n b e w r i t t e n i n terms of an e f f e c t i v e i n t e r a c t i o n Lagrangian. The o r i g i n a l c h i r a l symmetry w a s

"

(N) l e f t

(5.36)

U(N)right

I t i s broken i n t o

's

(N) l e f t

SU(N)right

'(''vector

*

(5.37)

An e f f e c t i v e Lagrangian t h a t does t h i s b r e a k i n g i s

(5.38) A c t u a l l y t h e e f f e c t i v e i n t e r a c t i o n is more complicated when c o l o r i n d i c e s are i n c l u d e d 2 2 , and C s t i l l c o n t a i n s powers o f g. I n t h e exponent, 0 i s an a r b i t r a r y a n g l e a s s o c i a t e d w i t h t h e symmetry breaking.

Fig. 8.

n The i n s t a n t o n a p p e a r s t o f l i p t h e h e l i c i t y of one of each t y p e of f e r m i o n i n v o l v e d .

279

EXTENDED OBJECTS IN GAUGE THEORIES

Now l e t us t u r n t o the i n s t a n t o n s i n t h e weak and electromagn e t i c guage theory SU(2) x U(1), with leptons, and non-srrange, s t r a n g e and charmed quarks which have c o l o r indices. This theory is anomaly-free, according t o i t s commercials, a s f a r as such is needed f o r renormalizability. Indeed, i n t r i a n g l e diagrams where a l l t h r e e e x t e r n a l l i n e s are gauge photons of whatever type, the anomalies (must) cancel. But i f one of t h e e x t e r n a l l i n e s is one of the f a m i l i a r conserved c u r r e n t s l i k e baryon o r lepton c u r r e n t ( t o which no known photons are coupled) then t h e r e a r e anomalies. W e f i n d , by i n s e r t i n g the anomaly equation (5.34) with the c o r r e c t values f o r N, t h a t one i n s t a n t o n gives

-

AQE

1

(electron-number nonconservation)

,

,

AQM = 1

(muon-number nonconeervation)

AQN = 3

(non-strange quark nonconservation)

-

AQc

3

,

(strangelcharm quark nonconservation)

.

I n t o t a l , two baryon and two lepton u n i t s a r e consumed by the i n s t a n t o n (strange and non-strange may mix through the Cabibbo angle). Thus w e can g e t the decays

+-NN + v v PN

-t

e v p

or

+-

pve

,

e p ’

PP

-t

e+p+

, etc.

The order of magnitude of these decays is given by

(5.39)

With a Weinberg angle s i n 2 % = .35, and assuming t h a t the weak intermediate vector boson determines t h e s c a l e , then we g e t l i f e t i m e s of the order of (give o r take many orders of magnitude) T

=

1oZz5s e c ,

(5.40)

corresponding t o one deuteron degay i n 10’ 3 7 universes.

280

G. 't HOOFT

6.

SOME REFLECTIONS ON CLASSICAL SOLUTIONS AND THE QUARK CONFINEMENT PROBLEM

C l e a r l y , t h e s t u d y o f n o n - t r i v i a l classical f i e l d configurat i o n s g i v e s us a welcome e x t e n s i o n of our knowledge and understandi n g of f i e l d t h e o r y beyond t h e u s u a l p e r t u r b a t i o n expansion. One non-perturbative problem is p a r t i c u l a r l y i n t r i g u i n g : quark confinement i n QCD. There h a s been wide-spread s p e c u l a t i o n t h a t t h e new. classical s o l u t i o n s are somehow r e s p o n s i b l e f o r t h i s phenomenon. Some a u t h o r s b e l i e v e t h a t some plasma of i n s t a n t o n s o r i n s t a n t o n l i k e o b j e c t s does t h e t r i c k . W e w i l l n o t go t h a t f a r . W e will however e x h i b i t some s i m p l e models w i t h i n t e r e s t i n g p r o p e r t i e s . They w i l l c l e a r l y show t h a t a "phase t r a n s i t i o n ' ' towards permanent confinement is n o t a t a l l a n absurd i d e a ( i t is a c t u a l l y much h a r d e r t o m a k e models w i t h ''nearly confinement" of quarks). Our f i r s t model is based on an SU(3) gauge t h e o r y i n 3 + 1 dimensions. The gauge group h e r e is n e i t h e r "color" "flavor", so l e t us c a l l i t "horror" ( o r : " t e r r o r " ) , f o r r e a s o n s t h a t w i l l become clear l a t e r ( t h e model does n o t d e s c r i b e t h e o b s e r v a t i o n s on quarks very w e l l ) . L e t us assume t h a t t h e SU(3) symmetry is spontaneously comp l e t e l y broken by a c o n v e n t i o n a l Higgs f i e l d . The Higgs f i e l d m u s t b e a %on-exotic" r e p r e s e n t a t i o n of h o r r o r (mathematically: a r e p r e s e n t a t i o n of S U ( 3 ) / 2 ( 3 ) ) , e.g. a n o c t e t o r d e c u p l e t representation. I n t h e following we w i l l show t h a t t h i s model admits, l i k e t h e U ( 1 ) , and t h e SU(2) analogues, s t r i n g - v o r t i c e s , b u t t h e s e a r e a g a i n d i f f e r e n t . A f t e r t h a t w e w i l l i n t r o d u c e q u a r k s and show t h a t t h e y b i n d t o t h e s t r i n g s i n t h e combination t h a t we a c t u a l l y see i n hadrons2 3 . To see t h e t o p o l o g i c a l p r o p e r t i e s o f v o r t i c e s i n t h i s model w e c o n s i d e r , as we d i d i n s e c t i o n 3, a two dimensional s p a c e , e n c l o s e d by a boundary which h a s t h e topology of a circle. The vacuum a t t h i s boundary is s p e c i f i e d by a gauge r o t a t i o n fl(e), so t h e number of t d p o l o g i c a l l y s t a b l e s t r u c t u r e s is g i v e n by t h e number of homotopy classes of t h e mapping e+fl(e). The group SU(3 h s an i n v a r i subgroup Z(3) g i v e n by t h e t h r e e elements I, eini931 and e' Our p h y s i c a l f i e l d s have been r e q u i r e d t o b e n v a r i a n t under Z(3). So, t h e mappings w i t h fl(2T)e'2ni/ N O ) are a c c e p t a b l e . They form two homotopy classes b e s i d e s t h e t r i v i a l one: Q(2n) fl(0). See Fig. 9. W e conclude t h a t t h e r e are s t a b l e v o r t i c e s . O p p o s i t e l y o r i e n t e d v o r t i c e s are d i f f e r e n t : t h e i r b o u n d a r i e s are i n d i f f e r e n t homotopy classes. But two v o r t i c e s o r i e n t e d i n t h e same d i r e c t i o n are in t h e same class as one v o r t e x i n t h e o t h e r d i r e c t i o n (see Fig. lo), because

4

"V5I.

-

281

EXTENOED OBJECTS I N GAUGE THEORIES

........

.................. ........ ....................... .......................... ............................. ............ ........... ................ . i ; ; i i i :*:::..... : i ; i i q...... I!;, :,;&ixi; .................. .................. :iiiiii ::.. ::::; y : i ; ; : ;##::;i i;ii:i:;;; ::::..:::::::::::.. .................... ....................................... ...................................... ................ >....................... .................................... ....................................... ................................... ................................... ................................. .............................. ................... .................. .:i;;iii;iiiiy;3cc'jiii:. ................. ............... ........ (a) Fig. 9 .

Fig. 10.

fl:

......... ............... ................... ....................... ......................... ........................... ........................... ............. ....... ............ ..... ...................... ........................ ......................... ................................ .............................. ................................. ........... :: ....:-;*:* +;;;;;;;;;;;;; ................................... ...................................... ....................................... ........................................ ...................................... ..................................... .. .................................. .................................... ................................... ..................... .::iiiiiiiiiiiii~iifi" .............. ..................... .................. ................. ............

4 (b)

<-

.................. .......... ..... ......................... ............................ . .......................... ............................. ............................ .:* :ii# x;::y-..:..::. ............. .:#:::::: .::, .::: :: ;::::;.. .!".:::::::::::::::::: ..::::::::::::::: ............... ..................................... ::::::::::::::: ":: ................................. ............ ............. ::::::::::::::::: ii;i!iii{iiiii: :::: :iii~;ii:.iiiil::..:;;~, ................................. ................................ ........................... ................. ........................ ...... ............... .................... ...................... ..................... ................... ............... ........

Three homotopy c l a s s e s of the mapping 8

y:::::::::::::::::'

(C) +

Q(8).

Decay of two p a r a l l e l v o r t i c e s i n t o a s i n g l e vortex i n opposite d i r e c t i o n .

282

G. 't HOOFT

e4.rri/3 = e-2ni/3, so two p a r a l l e l v o r t i c e s will decay i n t o an e n e r g e t i c a l l y more f a v o r a b l e s i n g l e v o r t e x i n t h e o t h e r d i r e c t i o n . Note t h a t t h r e e - s t r i n g connections are p o s s i b l e . Quarks are now i n t r o d u c e d as t h e end-points of s t r i n g s . They could be compared w i t h magnetic monopoles i n s i d e a superconductor. Note t h a t quarks d i f f e r from a n t i q u a r k s and because of t h e topolog i c a l p r o p e r t i e s of t h e s e s t r i n g s they only occur i n t h e form of "hadronic" bound s t a t e s (see Fig. 11).

We could go i n t o l e n g t h y d e t a i l s about a c t u a l c o n s t r u c t i o n of quark monopoles i n t h e model, f o r i n s t a n c e by embedding S U ( 3 ) / Z ( 3 ) i n a l a r g e gauge group t h a t does n o t c o n t a i n Z(3) i t s e l f . Then t h e s e monopoles themselves are allowed as classical s o l u t i o n s . But t h e model is n o t very v i a b l e from a p h y s i c a l p o i n t of v i e w because: (i) The "horror" degrees of freedom are n o t known t o e x i s t ;

where a r e t h e "horror"-vector

bosons e t c . ?

(ii) The Dirac monopoles w i l l have monopole charges of t h e o r d e r of 2n/gh0rror. So, i f quarks should behave as approximately

f r e e p a r t o n s i n s i d e hadrons, ghor or should b e s o l a r g e t h a t no p e r t u r b a t i o n expansion is l i k e l y t o make s e n s e , and (ffl) u n l i k e monopoles ( s e e s e c t i o n 4), quarks are l i g h t , r e l a t i v i s -

tic, particles. ( i v ) It i s n ' t Q.C.D. One t h i n g on t h e o t h e r hand is very clear in t h i s model: because t h e s t r i n g s are t o p o l o g i c a l l y s t a b l e quarks are a b s o l u t e l y and permanently confined.

Fig. 11. Allowed quark and a n t i q u a r k bound states.

283

EXTENDED OBJECTS IN GAUGE THEORIES

The n e x t model w e c o n s i d e r is e n t i r e l y d i f f e r e n t from t h e previous one. It is pure SU(N) c o l o r gauge t h e o r y (quantume claim chromodynamics) b u t i n 2 s p a c e one t i m e dimension. W t h a t t h e r e is a n e l e g a n t way t o formulate t h e quark confinement mechanism h e r e , although t h e dynamics is very complicated. For pedagogical. r e a s o n s l e t us f i r s t add a s e t o f Higgs f i e l d s t o t h e s y s t e m , which must b e a r e p r e s e n t a t i o n of SU(N)/Z(N) as i n t h e previous model. There is a complete spontaneous symmetry br'eakdown and obviously no quark confinement. L e t us a l s o l e a v e o u t the quarks now. Since w e have only 2 s p a c e dimensions t h e model has no f l u x tubes b u t i n s t e a d s o l i t o n p a r t i c l e s . The boundary o f t h i s space can b e i n one of N homotopy classes, so i f w e have more than N / 2 s o l i t o n s they may decay: t h e number of s o l i t o n s minus a n t i - s o l i t o n s is only conserved modulo N. W e now make a very important s t e p z 4 : d e f i n e an o p e r a t o r f i e l d $ G , t ) t h a t d e s t r o y s one s o l i t o n ( o r creates one a n t i - s o l i t o n ) a t G,t). There is n o t enough t i m e now t o formulate a p r e c i s e d e f i n i t i o n of $ ( t , t ) . It s u f f i c e s t o s t a t e t h a t in p a r t i c u l a r t h e Green's f u n c t i o n s

-

and

can be d e f i n e d a c c u r a t e l y i n terms of o r d i n a r y p e r t u r b a t i o n theory.25 W e can r e q u i r e t h a t $ @ , t ) produces o r d e s t r o y s o n 4 "bare" s o l i t p s : -+ t h e vacuum ( n o t supervacuum) remains a t a l l x ' w i t h Ix'- X I 1 E b u t a l a r g e f i e l d c o n f i g u r a t i o n is produced where I P ' - t l < E ; t h e n E -+ 0. Then (6.1) and ( 6 . 2 ) s a t i s f y t h e Wightman axioms. W e could assume t h a t t h e y may b e more o r less reproduced by an e f f e c t i v e Lagrangian:

Here M is t h e c a l c u l a b l e mass o f t h e s o l i t o n . There w i l l be many i n t e r a c t i o n ' t e r m s ; t h e one w e w r o t e down is n e c e s s a r y t o make (6.2) d i f f e r e n t from zero. Observe a Z(N) symmetry t h a t l e a v e s (6.11, (6.2) and ( 6 . 3 ) i n v a r i a n t :

Now what would happen i f we vary t h e Higgs p o t e n t i a l such t h a t i t s second d e r i v a t i v e a t t h e - o r i g i n t u r n s from n e g a t i v e t o

284

G. 't HOOFT p o s i t i v e ? The f i e l d 4 remains w e l l defined. The s o l i t o n mass M2 goes t o z e r o i f t h e Higgs mass goes t o zero. I t i s t h e r e f o r e n a t u r a l t o assume t h a t a f t e r t h e t r a n s i t i o n M 2 i n (6.3) h a s changed s i g n . Our e f f e c t i v e Lagrangian now could b e

W e imagine t h a t t h e n o u r Z ( N ) symmetry (6.4) w i l l b e spontaneously broken! We stress t h a t we do n o t y e t understand t h e d e t a i l s of t h e dynamics, b u t if t h i s assumption is c o r r e c t t h e n t h e o r i g i n a l model c o n f i n e s its quarks! Why? There a r e now N d i f f e r e n t v a c u m states f o r t h e f i e l d 4:

-

e

2lriIN


...

I f t h e r e are two d i f f e r e n t vacuum c o n f i g u r a t i o n s i n one p l a n e t h e n w e o b t a i n "Bloch walls" s e p a r a t i n g them. These Bloch walls can be c o n s i d e r e d t o be c l o s e d s t r i n g s . It i s n o t h a r d t o see t h a t i f we now add t h e quarks t o t h e o r i g i n a l model, t h e n t h e s e w i l l form endp o i n t s o f t h e s t r i n g s ("Dirac monopoles"). Since t h e s t r i n g s c a r r y energy p e r u n i t of l e n g t h t h e s e q u a r k s w i l l b e permanently confined by a l i n e a r p o t e n t i a l . W e l i k e t o look a t t h e t r a n s i t i o n from t h e SU(N) gauge theory t o t h e Z ( N ) scalar t h e o r y as a d u a l t r a n s f o r m a t i o n . The s o l i t o n s i n one t h e o r y are t h e elementary p a r t i c l e s of t h e o t h e r . W e n o t e a p e c u l i a r antagonism: i f t h e S U ( N ) symmetry is spontaneously broken, t h e Z ( N ) symmetry is i n t a c t and v i c e v e r s a . A f t e r t h e d u a l t r a n s f o r m a t i o n from t h e unbroken S U ( N ) t h e o r y t o t h e broken Z ( N ) t h e o r y , quark confinement i n t h e l a t t e r is obvious (once w e a c c e p t t h e symmetry breaking antagonism). 26 What about 3+1 dimensional quantmchromodynamics? Our f i n a l s p e c u l a t i o n is t h a t t h e "horror" model d i s c u s s e d i n t h e beginning of t h i s s e c t i o n i s i n a similar way t h e d u a l transform of quantmchromodynamics. The d u a l t r a n s formation i n 3'+1 dimensions is much more complicated t h a n i n 2 + 1 dimensions, b u t a g a i n , one can show t h a t c e r t a i n d u a l Green's f u n c t i o n s corresponding t o t h e d u a l gauge f i e l d may be c o n s t r u c t e d . The s u b j e c t is obscured by t h e f a c t t h a t w e are n o t a b l e t o const r u c t Lagrangians from t h e Green's f u n c t i o n s , and t h a t t h e h o r r o r coupling c o n s t a n t is i n v e r s e l y p r o p o r t i o n a l t o t h e c o l o r coupling c o n s t a n t so t h a t i t i s i m p o s s i b l e t o compare t h e two p e r t u r b a t i o n series. However, i f t h e s e d u a l t r a n s f o r m a t i o n s can b e g i v e n a more sound f o o t i n g t h e n t h e quark confinement phenomenon w i l l no l o n g e r l o o k as m y s t e r i o u s as i t does now.

285

EXTENDED OBJECTS IN GAUGE THEORIES

REFERENCES 1) T.D. Lee and G.C. Wick, Phys. Rev. D9, 2291 (1974); T.D. Lee and M. M a r g u l i e s , Phys. R e v , = , 1 5 9 1 (1975).

2)

S . Coleman, i n New Phenomena i n Subnuclear P h y s i c s ,

Internat i o n a l School of Subnuclear P h y s i c s " E t t o r e M a jorana-'I, Erice 1975, P a r t A, e d . A. Z i c h i c h i (Plenum P r e s s , New York, 1977).

3)

"State" h e r e d o e s n o t mean " s t a t e i n H i l b e r t space" b u t r a t h e r a s t a t i o n a r y classical f i e l d configuration.

4)

H.B. N i e l s e n and P. Olesen, Nucl. Phys. E, 45 (1973); H.B. N i e l s e n i n P r o c e e d i n g s of t h e AdraticSummer Meeting on P a r t i c l e P h y s i c s , R o v i n j , Yugoslavia 1973, e d . M. M a r t i n i s e t a l . (North Holland, Amsterdam, 1974).

5)

B. Zumino, i n R e n o r m a l i z a t i o n and I n v a r i a n c e i n Quantum F i e l d Theory, NATO Adv. Summer I n s t . , C a p r i 1973, e d . E . R . C a i a n i e l l o (Plenum P r e s s , New York, 1 9 7 4 ) .

6)

H. Georgi and S.L.

7)

For s i m p l i c i t y o u r o r d e r p a r a m e t e r i s chosen t o c a r r y a s i n g l e c h a r g e q=e, so that s i n g l e f l u x t u b e s are q u a n t i z e d by u n i t s a l e .

8)

G.

9)

B. J u l i a and A. Zee, Phys. Rev.

Glashow, Phys. Rev. L e t t .

32,

438 (1974).

I t Hooft, Nucl. Phys. E, 276 (1974); Nucl. Phys. 538 (1976); A. Polyakov, JETP L e t t . 20, 194 (1974).

10) S. Weinberg, Phys. Rev. L e t t .

11) G.

' t Hooft, Nucl. Phys.

m,

z, 2227 (1975).

2, 1264

E, 167

(1967).

(1971).

=,

12) J. Kogut and L. S u s s k i n d , Phys. Rev. 3501 (1974); K.G. Wilson, Phys. Rev. D10, 2445 (1974). 13) A.A.

B e l a v i n e t a l . , Phys. L e t t .

E, 8 5

(1975).

14) S. Coleman, i n The Why's o f Subnuclear P h y s i c s , Erice l e c t u r e n o t e s , Erice 1977.

15) R. J a c k i w and C. Rebbi, Phys. Rev. L e t t . 16) C . Callan,

37,

172 (1976).

R. Dashen and D. Gross, Phys. L e t t .

=,

334 (1976).

1 7 ) It i s t o b e remarked however t h a t t h e i n s t a n t o n e f f e c t s v i o l a t e p a r i t y c o n s e r v a t i o n i f 0 i s n o t a m u l t i p l e o f 71. P a r i t y v i o l a t i n g e v e n t s could then b e searched f o r .

286

G. 't HOOFT

177, 2426

18) S.L.

Adler, Phya. Rev.

19) J.S.

Bell and R. Jackiw, I1 Nuovo Cimento

(1969).

g, 47

(1969).

20) The f a c t o r i should be added i n (5.12) s i n c e w e are now d e a l i n g w i t h a Minkowski metric where time components of v e c t o r s are t a k e n t o b e imaginary. 21) H. F r i t z s c h , M. Gell-Mann and H. Leutwyler, Phys. L e t t . 365 (1973).

-

22) G. ' t Hooft, Phys. Rev. L e t t . B37, 8 (1976), and G. Phys. Rev. 3432, (1976).

m,

*,

' t Hooft,

23) F. E n g l e r t , L e c t u r e s given a t t h e C a r g h e Summer School, J u l y 1977. 24) S. Mandelstam, Phys. Rev. 25) G.

z,3026

(1975).

't Hooft, t o b e published.

26) Compare t h e d u a l t r a n s f o r m a t i o n i n t h e 2 dimensional I s i n g model: t h e ordered phase i s transformed i n t o t h e d i s o r d e r e d phase and v i c e versa.

287

CHAPTER 4.2

MAGNETIC MONOPOLES IN UNIFIED GAUGE THEORIES G. 't HOOFT CERN. Geneva

Received 31 May 1974 Abstract: I t is shown that in all those gauge theories in which the electromagnetic group U(1) is taken to be a subgroup of a larger group with a compact covering group, like SU(2) or SU(3), genuine magnetic monopoles can be created as regular solutions of the field equations. Their mass is calculable and of order 137 Mw, whereMw is a typical vector boson mass.

1. Introduction The present investigation is inspired by the work of Nielsen et al. [ 11, who found that quantized magnetic flux lines, in a superconductor, behave very much like the Nambu string [2]. Their solution consists of a kernel in the form of a thin tube which contains most of the flux lines and the energy; all physical fields decrease exponentially outside this kernel. Outside the kernel we d o have a transverse vector potential A , but there it is rotation-free: if we put the kernel along the z axis, then

A(x) a (v,-x, 0)/(x2 + Y 2 )

-

(1.1)

A(x) can be obtained by means of a gauge transformation n(cp)from the vacuum. Here cp is the angle about the z axis: SZ(0) = SZ(2n) = 1 .

It is obvious that such a string cannot break since we cannot have an end point: it is impossible to replace a rotation over 2n continuously by S2(cp) + 1. Or: magnetic monopoles do not occur in the system. Also it is easy to see that these strings are oriented: two strings with opposite direction can annihilate; if they have the same direction they may only join to form an even tighter string. Now, let 11ssuppose that the electromagnetism in the superconductor is in fact described by a unified gauge theory, in which the electromagnetic group U(l) is a subgroup of, say, SO(3). In such a non-Abelian theory one can only imagine nonoriented strings, because a rotation over 4n can be continuously shifted towards a fixed SZ. What happened with our original strings? The answer is simple: in an SO(3) gauge theory magnetic monopoles with twice the flux quantum (i.e., the Nuclear Physics B79 (1974) 276-284. North-Holland Publishing Company

288

G. 't Hooft, Magnetic monopoles

Schwinger [3,4] value), occur. Two of the original strings, oriented in the same direction, can now annihilate by formation of a monopole pair [ 5 ] . From now on we shall dispose of the original superconductor with its quantized flux lines. We consider free monopoles in the physical vacuum. That these monopoles are possible, as regular solutions of the field equations, can be understood in the following way. Imagine a sphere, with a magnetic flux @ entering at one spot (see fig. 1). Immediately around that spot, on the contour C, in fig. 1, we must have a magnetic potential field A , with $ ( A dx) = @. It can be obtained from the vacuum by applying a gauge transformation A :

A=VA. (1.2) This A is multivalued. Now we require that all fields, which transform according to $ + $ enih ,

(1 -3) to remain single valued, so must be an integer times 2n: we then have a complete gauge rotation along the contour in fig. 1. In an Abelian gauge theory we must necessarily have some other spot on the sphere where the flux lines come out, because the rotation over 2 k n cannot continuously change into a constant while we lower the contour C over the sphere. In a non-Abelian theory with compact covering group, however, for instance the group 0(3), a rotation over 4n may be shifted towards a constant, without singularity: we may have a vacuum all around the sphere. In other theories, even rotations over 2n

Fig. 1. The contour C on the sphere around the monopole. We deplace it from CO to C1, etc.,

until It shrinks at the bottom of the sphere. We require that there be no singularity at that point.

289

G. 't Hooft, Magnetic monopoles

may be shifted towards a constant. This is why a magnetic monopole with twice or sometimes once the flux quantum is allowed in a non-Abelian theory, if the electromagnetic group U(l) is a subgroup of a gauge group with compact covering group. There is no singularity anywhere in the sphere, nor is there the need for a Dirac string. This is how we were led to consider solutions of the following type to the classical field equations in a non-Abelian Higgs-Kibble system: a small kernel occurs in the origin of three dimensional space. Outside that kernel a non-vanishing vector potential exists (and other non-physical fields) which can be obtained from the vacuum* by means of a gauge transformation 52(8,cp). At one side of the sphere (cos8 -+ 1) we have a rotation over 4n,which goes to unity at the other side of the sphere (cos 8 + -1). For such a rotation one can, for instance, take the following SU(2) matrix:

Now consider one rotation of the angle cp over 277. At 8 = 0, this 52 rotates over 4n (the spinor rotates over 2n). At 8 = R , this 52 is a constant. One easily checks that

ant

=1

.

(1.5)

In the usual gauge theories one normally chooses the gauge in which the Higgs field is a vector in a fixed direction, say, along the positive z axis, in isospin space. Now, however, we take as a gauge condition that the Higgs field is 52(8,cp) times this vector. As we shall see in sect. 2, this leads to a new boundary condition at infinity, to which corresponds a non-trivial solution of the field equations: a stable particle is sitting at the origin. It will be shown to be a magnetic monopole. If we want to be conservative and only permit the normal boundary condition at infinity, with Higgs fields pointing in the z direction, then still monopole-antimonopole pairs, arbitrarily far apart, are legitimate solutions of the field quations.

2. The model We must have a model with a compact covering group. That, unfortunately, excludes the popular SU(2) X U(l) model of Weinberg and Salam [ 6 ] .There are two classes of possibilities. (i) In models of the type described by Georgi and Glashow [7], based on S0(3), we can construct monopoles with a mass of the order of 137 M w ,where Mw is the

* As we shall see this vacuum will still contain a radial magnetic field. This is because the incoming field in fig. 1 will be spread over the whole sphere.

290

G. 't Hooft,Magnetic monopoles

mass of the familiar intermediate vector boson. In the Georgi-Glashow model, < 53 GeVlc2. (ii) The Weinberg-Salam model can still be a good phenomenological description of processes with energies around hundreds of GeV, but may need extension to a larger gauge group at still-higher energies. Weinberg [8] proposed SU(3) X SU(3) which would then be compact. Then the monopole mass would be 137 times the mass of one of the superheavy vector bosons. We choose the first possibility for our sample calculations, because it is the simplest one. We take as our Lagrangian:

Mw

P=-'Ga 4

p

pu p v

1 2 2 Qa

-iDpQaDpQa

-21.1

-$X(Q2I2

(2.1)

where

G& = a,

w,"- a&

t e egbCw,bW; ,

DpQa = a p Q a +eeabcWiQc

(2.2)

FV; and Q, are a triplet of vector fields and scalar fields, respectively. We choose the parameter p2 to be negative so that the field Q gets a non-zero vacuum expectation value [6,7,9] : (Qa)2 = F2 ,

p2 = -$AF2

.

(2.3)

Two components of the vector field will acquire a mass:

Mw,;2 = eF

(2.4)

Y

whereas the third component describes the surviving Abelian electromagnetic interactions. The Higgs particle has a mass:

M~=JxF.

(2.5)

We are interested in a solution where the Higgs field is not rotated everywhere towards the positive z direction. If we apply the transformation !2 of eq. (1 S ) to the isospin-one vector F(O,O, 1) we get

F(sin 8 cos cp, sin 8 sin cp, cos 8).

(2 -6)

We shall take this isovector as our boundary condition for the Higgs field at spacel i e infinity. As one can easily verify, it implies that the Higgs field must have at least one zero. This zero we take as the origin of our coordinate system. We now ask for a solution of the field equations that is time-independent and spherically symmetric, apart from the obvious angle dependence. Introducing the vector

r* = (XYYYZ)

Y

r 2a = r2 ,

(2.7)

291

G. 't Hoof&Magnetic monopoles

we can write

where epab is the usual E symbol if p = 1,2,3, and E4& = 0. In terms of these variables the Lagrangian becomes m

L = JL'd3x =4n

r2 d r [ -r2

(Fr

-4rW dW - 6W2 - 2er2W3

0

t

iXF2r2Q2 - iXr4Q4 - :W4] ,

(2.9)

where the constant has been added to give the vacuum a vanishing action integral. The field equations are obtained by requiring L to be stationary under small variations of the functions W(r) and Q(r). The energy of the system is then given by

(2.10)

E=-L,

since our system is stationary. Before calculating this energy, let us concentrate on the boundary condition at r OQ. From the preceding arguments we already know that we must insist on +

Q(r) F / r .

(2.1 1)

+

The field W must behave smoothly, as some negative power of r :

W ( r ) + ar-" .

(2.12)

From (2.9) we find the Lagrange equation

= r2 [4r !.!

dr

t 12W t 6 e r 2 W 2 t 2e2r4W3 +2er2Q2t2e2r4WQ2]

.

(2.13)

So, substituting (2.1 1) and (2.12),

t 2eF2r2 t 2e2aF2r4-n

.

(2.14)

The only solution is

n=2,

(2.1 5)

a = -l/e

So, far from the origin, the fields are

292

G. 't Hmft, Magnetic monopoles

W;(x,

r) - Epabr&r2 , +

Qa(X9

t ) Fra/r . +

(2.16)

Now most of these fields are not physical. To find the physically observable fields, in particular the electromagnetic ones, F P v ,we must first give a gauge invariant definition, which will yield the usual definition in the gauge where the Higgs field lies along the z direction everywhere. We propose: (2.17)

because, if after a gauge rotation, Q, = IQ I(O,O, 1) everywhere within some region, then we have there

F~~ = a,w,3 - a,$

,

as one can easily check. (Observe that the definition (2.17) satisfies the usual Maxwell equations, except where QQ= 0; this is one other way of understanding the possibility of monopoles in this theory.) From (2.16), we get (see the definitions (2.2)): (2.18) (2.19)

(2.20)

Again, the E symbol has been defined to be zero as soon as one of its indices has the value 4 . So, there is a radial magnetic field

B~ = ra/er3 ,

(2.2 1)

with a total flux

4n/e. Hence, our solution is a magnetic monopole, as we expected. It satisfies Schwinger's condition eg- 1

(2.22)

(in units where h = 1). In sect. 4 , however, we show that in certain cases only Dirac's condition

e g = 1z n ,

n integer,

is satisfied.

293

G. 't Hooft, Magnetic monopoles

3. The m a s of the monopole

Let us introduce dimensionless parameters: w = W/F2e,

q =Q/F2e,

x =eFr,

0 = Ale2 = M$M$ .

(3.1)

From (2.9) and (2.10) we find that the energy E of the system is the minima! value of

txq d q t f q 2 t 2x2wq2 t x4w2q2 - iDx2q2

dx

]

+ ipx444 t $ .

(3.2)

The quantity between the brackets is dimensionless and the extremum can be found by inserting trial functions and adjusting their parameters. We found that the mass of the monopole (which is equal to the energy E since the monopole is at rest) is

4n Mm = - M w C ( P ) , e2

(3.3)

where C(0) is nearly independent of the parameter 0.It varies from 1.1 for = 0.1 to 1.44 for 0 = 10 *. Only in the Georgi-Glashow model (for which we did this calculation) is the parameter Mw in eq. (3 . 3 ) really the mass of the conventional intermediate vector bostm. In other models it will in general be the mass of that boson which corresponds to the gauge transformations of the compact covering group: some of the superheavies in Weinberg's SU(3) X SU(3) for instance.

4. Conclusions

The relation between charge quantization and the possible existence of magnetic monopoles has been speculated on for a long time [ 101 and it has been observed that the gauge theories with compact gauge groups provide for the necessary charge quantization [ l l ] . On the other hand, solutions of thefield equations with abnormally rotated boundary conditions for the Higgs fields have also been considered before [ 1,121. Nevertheless, it had escaped to our notion until now that magnetic monopoles occur among the solutions in those theories, and that their properties are predictable and calculable.

* These values may be slightly too high, as a consequence of our approximation procedure.

294

G. ’t Hooft, Magnetic monopoles

Our way of formulating the theory of magnetic monopoles avoids the introduction of Dirac’s string [3], We expect no fundamental problems in calculating quantum corrections to the solution although they might be complicated to carry out. The prediction is the most striking for the Georgi-Glashow model, although even in that model the mass is so high that that might explain the negative experimental evidence so far. If Weinberg’s SU(2) X U(l) model wins the race for the presently observed weak interactions, then we shall have to wait for its extension to a compact gauge model, and the predicted monopole mass will be again much higher. Finally, one important observation. In the Georgi-Glashow model, one may introduce isospin $ representations of the group SU(2) describing particles with charges ie. In that case our monopoles do not obey Schwinger’s condition, but only Dirac’s condition

*

qg=i ’

where q is the charge quantum and g the magnetic pole quantum, in spite of the fact that we have a completely quantized theory. Evidently, Schwinger’s arguments do not hold for this theory [ 131. We do have, in our model Aqg= 1,

where Aq is the charge-difference between members of a multiplet, but this is certainly not a general phenomenon. In Weinberg’s SU(3) X SU(3) the monopole quantum is the Dirac one and in models where the leptons form an SU(3) X SU(3) octet [ 141 the monopole quantum is three times the Dirac value (note the possibility of fractionally charged quarks in that case). We thank H. Strubbe for help with a computer calculation of the coefficient C(@, and B. Zumino and D. Gross for interesting discussions.

References [ 11 H.B. Nielsen and P. Olesen, Niels Bohr Institute preprint, Copenhagen,(May 1973); B. Zumino, Lectures given at the 1973 Nato Summer Institute in Capri, CERN preprint TH. 1779 (1973). [2] Y. Nambu, Proc. Int. Conf. on symmetries and quark models, Detroit, 1969 (Gordon and Breach, New York, 1970) p. 269; L. Susskind, Nuovo Cimento 69A (1970) 457; J.L. Gervais and B. Sakita, Phys. Rev. Letters 30 (1973) 716. [3] P.A.M. Dirac, Proc. Roy. SOC.A133 (1934) 60; Phys. Rev. 74 (1948) 817. [4] J. Schwinger, Phys. Rev. 144 (1966) 1087. [ 5 ] G.Parisi, Columbia University preprint CO-2271-29. [6] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264. 171 M. Georgi and S.L. Glashow, Phys. Rev. Letters 28 (1972) 1494. [8] S.Weinberg, Phys. Rev. D5 (1972) 1962.

295

G. 't Hooft, Magnetic monopoles [9]F. Englert and R. Brout, Phys. Rev. Letters 13 (1964)321; P.W. Higgs, Phys. Letters 12 (1964)132;Phys. Rev. Letters 13 (1964)508;Phys. Rev. 145 (1966) 1156; G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Letters 13 (1964)585. [ 101 D.M. Stevens, Magnetic monopoles: an updated bibliography, Virginia Poly. Inst. and State University preprint VPI-EPP-73-5 (October 1973). [ 11 1 C.N. Yang, Phys. Rev. D1 (1970)2360. I121 A. Neveu and R. Dashen, Private communication. [ 13) B. Zumino, Strong and weak interactions, 1966 Int. School of Physics, Erice, ed. A. Zichichi (Acad. Press, New York and London) p. 71 1. [14]A. Salam and J.C. Pati, University of Maryland preprint (November 1972).

296

CHAPTER 6 INSTANTONS Introductions ................. .................................. ................ [5.1] "Computationof the quantum effects due to a four-dimensional pseudoparticle",Phys. Rev. D14 (1976)3432-3450 ...... ........... [5.2] "How instantons solve the U(1) problem", Phys. Rep. 142 (1986)357-387 ....... .......................................,........... [5.3] "Naturalness,chiral symmetry, and spontaneous chiral symmetry breaking", In Recent Developments in Gauge Theories, Cargbe 1979,eds. G. 't Hooft et aZ., New York, 1990,Plenum, Lectutv 111, reprinted in Dynamical Gauge S p m e t r y B d n g , A Collection of Reprints, eds. A. Farhi, and R. Jackiw, World Scientific, Singapore, 1982,pp. 345-367 ...............-................

.

297

298 302 321

352

CHAPTER 5

INSTANTONS Introduction to Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle [5.11 An introduction to instantons was already given in the first paper of the previous chapter. They are topologically stable twists in four dimensions. The classical solution in Euclidean space describes tunneling phenomena. In contrast with all other amplitudes in quantum field theory the instanton related amplitudes are exponentially damped when the coupling constant is small. Now this confronted me with a problem. As explained earlier, quantum field theory cannot be trusted with infinite accuracy, because the loop expansion has a built-in deficiency: the divergence at very high orders, irrespective of the value of the coupling parameters. The margin of error in any amplitude due to this divergence in general cannot be avoided (with the possible exception of asymptotically free theories), and it is of the same order of magnitude as the instanton effects. Therefore, are the instanton effects real? Fortunately, the instanton effects violate a symmetry, so that they can clearly be distinguished from ordinary perturbative contributions which in the corresponding channels vanish strictly. The dynamical origin of the instanton effects due to tunneling seems to be straightforward. And there seems to be no obstacle against calculating them. It was not hard to estimate what the results of such calculations would be. But are these exponentially damped phenomena really completely selfconsistent? The true coupling constant of the theory must be the renormalized one. Is that also the parameter in the exponent? Are instanton effects really finite? This was my motivation for performing some very detailed calculations. I wanted to know exactly how the nuts and bolts fit together here, and what the possible surprises would be. I decided to calculate the prefactor in front of the exponent.

298

The surprise was that everything works out fine. There were quite a few hurdles to be taken. The prefactor can really be seen as a oneloop quantum correction in the exponent itself. To compute it we needed the renormalized determinants of infinite dimensional matrices M, and several tricks were needed to obtain these. We first had to remove the zero modes, and for that we had to understand what their physical intepretation is. There are zero modes corresponding to translations and scale transformations, but also three due to global gauge rotations. The latter can only be understood in relation with the gauge fixing procedure. The zero modes, eight in total (if the gauge group is S U ( 2 ) ) ,give rise to an extra factor g-’ in front of the exponent. The original paper contained a few minor errors. Now all errors act multiplic& tively on the final result, so as far as the final amplitude is concerned none of the errors was minor. But of course the real importance of our calculation was to see if it could be done at all, and if the answer would come out finite. The answer we found to this is of course not affected by small errors. Anyway, the version reproduced here has been reedited to remove all errors known to us. We owe some of these refinements to meticulous calculations by Ore and Hasenfratz.l6 The word “instanton” does not occur in the original publication. I had proposed this name in a previous paper, but the editors of Phys. Rev. Lett. and Phys. Rev. opposed the use of self-invented phrases. Their alternative (“Euclidean Gauge Field Pseudoparticle, EGFP for short”) was so ugly that the name instanton caught on quickly.

Introduction to H o w Instantons Solve the U(1) Problem [6.2] Because of their ‘honperturbative’’nature, instantons give rise to a number of controve=. A very important recent issue is the question whether the symmetry breaking effects associated with instantons continue to be exponentially damped when collision processes at very high energies are considered. A number of authors have presented calculations suggesting sharply increasing amplitudes, and at energies above a few tens of TeV’s the exponential could be completely cancelled so that the amplitudes could even hit the unitarity barrier. It is true that at such energies tunneling is formally no longer necessary. The transition could take place via so-called sphdemns. These are metastable field configurations resembling magnetic monopoles, but such that they can decay with equal probabilities into two channels with different winding numbers, hence different B and L. But that the amplitudes should become strong is a highly dubious claim. Precisely because Feynman diagrams can no longer be trusted (they diverge) c d c u b tions based on diagrammatic techniques may well be incorrect. My own attempt at constructing a physically reasonable scenario for the efficiency of sphaleron production at high energies suggests that exponential suppression factors, comparable to 16F. R. Ore, Phys. Rev. Dl8 (1977) 2577; A. Hasenfrats and P. Hasenfratz, Nucl. Phys. B193 (1981) 210.

299

the one at low energies, exp (-87r2/g2), continue to be present at all energies. Only in high temperature plasmas such as the one that must have existed in the very early universe, the conditions for efficient sphaleron production seem to be realized. So much for instantons in weak interaction theories. In the strong interactions they also occur, and here the exponential suppression is not important. The symme try violated by instantons is chiral symmetry. This observation finally provided for a satisfactory answer to a famous problem in QCD: the U(1) problem. According to its Lagrangian namely, QCD possesses a nearly perfect global (chiral) symmetry of the form U ( 2 )@ U ( 2 ) ,only broken by the light quark masses into the diagonal U ( 2 ) subgroup of this group. This group has eight independent generators. PhenomenG logically however, there is only a global SU(2) @ SU(2) @ U(1) chiral symmetry, a group with seven generators, spontaneously broken into SU(2) @ U(1). The three broken symmetry generators give rise to three Goldstone bosons, 7r+ , 7ro and 7r-. This means that one U(1) subgroup of U ( 2 )@ U ( 2 ) is missing. The corresponding Goldstone boson would be one with the quantum numbers of the q particle. The mystery is that for this the q particle seems to be too heavy (550 MeV). Why is it so much heavier than the pions (140 MeV)? Remember that we have to compare the squares of the masses rather than the masses themselves. A spurious interaction, notably absent in the QCD Lagrangian, must be responsible for this explicit chiral symmetry breaking. In my 1975 Palermo paper (Chapter 2.3) I already indicated what a number of authors had observed: chiral U(1) is explicitly broken by an anomaly. The only remaining problem is: how does this anomaly raise the mass of the q particle? The answer is that the extra interaction responsible for this is precisely the one effectuated by the instanton. Indeed, the effective interaction vertex due to instantons has precisely the quantum numbers of an q mass term. But things are not quite this simple. In a number of papers, Rodney Crewther criticised this point of view. He claimed that the theory, including all its instantons, still obeys what he called “anomalous Ward identities”, and that these identities prohibit an q mass term. These arguments seemed to be formal mathematics to me, not reflecting the true physical nature of the instanton-induced interactions. Then, in 1984, a pupil of Crewther’s, G. A. Christos, published a review paper in Physics R e p o 7 - t ~clarifying ~~ Crewther’s arguments against the instanton solution of the U(1) problem. This paper was written in a style that compelled me to react. The “anomalous Ward identities” do not refer to amplitudes relating to physically observable quantities, and do not have a proper large distance limit. The easiest way to see what happens is to write down effective Lagrangians that exhibit all symmetries and selection rules (including the spurious Ward identities) of the expected instanton interactions. This was published in another issue of Physics Reports, of which we include a reprint.

17G. A. Christos, Phys. Rep. 116 (1984) 251.

300

Introduction to Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking [5.3] The next paper is only indirectly related to instantons. It shows how topological winding numbers can be used to deduce information about the spectrum of light bound states in theories such as QCD. The paper, one of a series of lectures presented in a summer school in Cargbe in 1979, speculates on how the Standard Model may be extended beyond the magic threshold of 1 TeV. At present most particle physicists adhere to the theory that beyond that energy range particles form supersymmetric multiplets. This could be a natural way to protect the Higgs scalar from becoming too heavy. At the moment this is written there is however still no single direct evidence for supersymmetry, and certainly in 1979 there wasn’t. If there is no supersymmetry then there must be strong interactions in the TeV range. The symmetry pattern of these interactions must then be such that the presently observed particle spectrum is the spectrum of low-lying bound states. How do we compute this spectrum in a given theory? Take QCD as an example. At low energies we have a chird Q model. Chird symmetry is spontaneously broken, but in other versions of this dynamicd system c h i d symmetry may be realized in a different way. We could have massless nucleons instead of massless pions. How many massleas bound states would be protected by the symmetries? What are the rules for more complicated variations of the theme “QCD”?This is the question the paper addresses. And an important theorem is discussed: all triangle anomalies for all fermionic currents (in particular the ones to which no gauge fields are coupled) must be the same for the bound state spectrum as they are for the original “bare” constituent particles. This leads to index theorems constraining the possibilities for the bound state spectrum. The theorems were not powerful enough to constrain the spectrum completely, so I tried to deduce more theorems. They are discussed in Section 3.12. The latter however are mere conjectures. It turns out that they cannot be completely true because the resulting spectrum would have fractional occupation numbers which is clearly absurd. My own conclusion was therefore that the various chiral symmetries must be spontaneously broken 80 that instead of massless fermions we get massless scalars. This would kill the bound state theories for quarks and leptons, and, disappointingly, such was my conclusion. Many authors since then argued that the conjectures of Section 3.12 are too stringent. But even the anomaly matching conditions alone (which surely must be true) are so restrictive that no really attractive bound state model for quarks and leptons could be produced. Does this imply that we have to accept the supersymmetry scenario? Or is Nature simply too complicated for us to guess how things go from what we know at present? Only the h t u r e will tell how stupid we are now.

301

PHYSICAL REVIEW D 0 Amcrican Physical Society

CHAPTER 5.1

15 DECEMBER 1976

Computation of the quantum effects due to a four-dimensional pseudoparticle* G. ’t Hooft’ Physics hbomrorier, Harvard Universily, Cambridge, Masrachuserrs 02138 (Received 28 June 1976) A detailed quanlitativc calculation is carried out of the tunneling process dcwribed by the Belavin-PolyakovSchwan-Tyupkin field confi~umtion.A certain chiral symmetry is violated as a consequence of the AdlerBell-Jackiw anomaiy. ?he collective motions or the pseudoparticle and all contributions from single loops of scalar, spinor. and vector fields are takcn into acwunt. The result is an elTcctivc interaction Lagrangian for the rpinors

I. INTRODUCTION

When one attempts to construct a realistic gauge theory f o r the observed weak, electromagnetic, and strong interactions, one is often confronted with the difficulty that most simple models have too much symmetry. In Nature, many symmetries are slightly broken, which leads to, for instance, the lepton massea, the quark masses, and C P violation, These symmetry violations, either explicit o r spontaneous, have to be introduced artlflcially in the existing models. There 1s one occasion where explicit symmetry violation is a necessary consequence of the laws of relativistic quantum theory: the Adler-Bell-Jackiw anomaly. The theory we consider i s an SU(2) gauge theory with an arbitrary set of scalar fields and a number, N’, of massless fermions. The apparent chiral symmetry of the form UW’) X U(”, is actually broken down to SUyU/) X SU(N/) X U(1). This paper is devoted to a detailed computation of this effect. The most essential ingredient in our theory Is the localized classical solution of the field equations in Euclidean space-time, of the type found by Belavin st nl.’ Although the main objective of this paper 1s the computation of the resulting effective symmetrybreaking Lagrangian in a weak-interaction theory, we present the calculations in such a way that they can also be used for possible color gauge theories of stong interactions based on the same classical field configurations. For such theories our intermediate expressions (12.5) and (12.8) will be applicable. Our final results a r e (15.1) together with the convergence factor (15.8). Our general philosophy has been sketched in Ref. 2. We a r e dealing with amplitudes that depend on the coupling constant g in the following way:

The coefficient a, involves one-loop quantum cor-

302

rections, and it determines the scale of the amplitude. Clearly, then, to understand the main features of such an amplitude, complete understanding of all one-loop quantum effects is desired. For instance, if one changes from one renormalization subtraction procedure to another, so that 2-2+ O(g‘), then this leads to a change in (1.1) by an overall multiplicative constant. Thus, the renormalization subtraction point p may enter a s a dimensional parameter in front of our expressions. This is just one of the reasons to suggest that our results will also have interesting applications in strong- interaction color gauge theories. The underlying classical solutions only exist in Euclidean space, but they give rise to a particular symmetry- breaking amplitude that can easily be continued analytically to Minkowski space. We interpret this amplitude as the result of a certaln tunneling effect from one vacuum to a gauge-rotated vacuum. We recall that, Indeed, tunneling through a ba rrie r can sometimes be described by means of a classical solution of the equation of motion in the imaginary time direction.’Ps We compute in Euclidean space the vacuum- tovacuum amplitude in the presence of external sources, thus obtaining full Green’s functions. Of course, we must limit ourselves to gaugeinvariant sources only, but that wlll be no problem. It turns out to be trivial to amputate the obtained Green’s function and get the effective vertex. The various calculational steps a r e the following. We first give in Sec. II the functional integral expression for the amplitude, first in a conventional Feynman gauge: C,= 8,A,. Later, we go over to the so-called background gauge: C,= D,Ar. This i s actually only correct up to an overall factor, as will be explained in Sec. XI. It is just for pedagogical reasons that we ignore this complication for a moment. It is in this gauge that the quantum excitations take a simple form: “Spinorbit” couplings commute with the operator La

C O M P U T A T I O N O F T H E Q U A N T U M EFFECTS D U E TO A , . =

- t L,,L,,,

must correspond to a local, space-time-dependent counterterm in the Lagrangian. The effect of this counterterm i s computed. Using the result of Sec. IV we now find also the contributions of all nonvanishing eigenvalues for the vectors and spinors. But there a r e also vanishing eigenvalues. They a r e listed in Sec. VIII. For the vector fields, we have eight zero eigenvectors in addition to the ones computed via the theorem of Sec. IV. They a r e to be interpreted a s translations (Sec. W , dilatation (Sec. X), and isospin rotations (Sec. XI). The last need special care and can only be interpreted correctly when different gauge choices a r e compared. This leads to the factor mentioned in the beginning. In Sec. XI1 we combine the results so far obtained and add the fermions. This intermediate result may be useful to strong-interaction theortes. In Sec. Xm we reexpress the result in terms of the dimensionally renormalized coupling constant gD, a s opposed to the previous coupling constant which was renormalized in a Pauli-Villars manner. In Sec. XIV the external sources for the fermions a r e considered and the amputation operation for the Green's function is performed. We obtain the desired effective Lagrangian, but there is still one divergence. So far, we only had massless partlcles, and as a consequence of that there i s still a scale parameter p over which we must integrate. Asymptotic freedom gives a natural CUtDff for this integral in the ultraviolet direction, but there is still an infrared divergence. In weak-interaction theories the Higgs field is expected to provide for the infrared cutoff. Section XV shows how to compute this cutoff. The Appendix lists the properties of the symbols q, Ti which a r e used many times throughout these calculations.

where

so that we can look at eigenstates of L'. (In other gauges only total angular momentum j=t+spin isospin is conserved, not L a . ) In Sec. III we consider the quantum fluctuations described by an eigenvalue equation, t

nJ,-.W,

(1.3)

in order to compute det9n. Now this looks like an ordinary scattering problem (in 4 + 1 dimensions), and, indeed, we show that the product of all nonzero eigenvalues E, in some large box, can be expressed in terms of the phase shift q(k) a s a function of the wave number k. In Sec. IV we show that Eq. (1.3) i s essentially the same for scalars, spinors, and vectors, from which we derive the important result that the product of all nonzero eigenvalues is the same for scalar fields a s for each component of the spinor and vector fields. So, we turn to the (much easier) scalar case first. But even for scalars Eq. (1.3) has no simple solutions In terms of well-known elementary functions. We decide not to compute d e m by 801ving (l.Z), but we compute instead det[(l +ra)X(l tw')] The factors 1 t x a drop out if we divlde by the same the determinant coming from the vacuum (h, case An'= 0). The equation (1.4)

is a simple hypergeometric equation that can be solved under the given boundary condition (Sec.

.

V)

11. FORMULATION OF THE PROBLEM

Now we must find the product of all eigenvalues A, but that diverges badly even if we divide by the values they take in the vacuum. We must find a gauge-invariant regulator, and the regulator determinant must be calculable. Dimensional regularization is not applicable here, but we can use background Pauli-Villars regulators. They give messy equations unless they have a spacetime-dependent mass:

Let a field theory in four space-time dimensions be given by the Lagrangian

e = - t G:YC:Y

- D, *'Dub

- %,D,$ t ?,&

9, , (2.1)

where the gauge group i s SU(2): Giv=O,,Rv- B,AO,tgr,,A~A,A:.

(2.2)

The SU2) indices will be called isospin indices. The scalars*, taken to be complex, may contain several multiplets of arbitrary isospin:

The rules a r e formulated in Sec. V. In Sec. VI we compute the product of the eigenvalues using this regulator. In Sec. VII we make the transition to regulators with fixed mass by observing that a change toward fixed regulator mass

(2.3) The spinors (I itre taken to be isospin-b doublets.

303

C . 't HOOFT

ordinates. The integrand in (2.4) now becomes

The total number of doublets i s N'. Mass t erms and interaction terms between scal ars and spinors a r e irrelevant for the time being. We inserted a source term in a gauge-invariant way, with respect to which we will expand, in order to obtain Green's functions. The indices s , t = 1,. . . ,N', called flavor indices, label the different isospin multiplets. Isospin and Dirac indices have been suppressed. d,, must be diagonal in the isospin indices but may contain Dirac y matrices. The system (2.1) seems to have a chiral UyU') X UYJ') global symmetry, butactuallyhasanAdlerBell-Jackiw anomaly4associated with the chiral U(1) current, breaking the symmetry down to SU(N') X SU(") X U(1). The aim of this paper is to find that part of the amplitude that violates the chiral U(l) conservation. The functional integral expression for the amplitude is

t (D,AY)' - gA: '"c,,C::'A','" - D , 8 * D u 8 - G U D u +t @+- f C,'+ efbu

e(A") - t ( D , A:")'t

+W4'",8,$)',

S"= IC(Aa')d% = - 8nZ/$

,

D , , q '"= B,,At q U + g c a k A ~ E ' A ~ q U ,

(2.9)

etc. We abbreviate the integral over (2.7) by

sCL-k q ' m A ~ q u + i j ~ ,o*ml.o$+*ma&, (2.10)

where the last term describes the Faddeev-Popov ghost, Thus, expression (2.4) is (ignoring temporarily the collective coordinates, and certain f r o m the Gaussian integration; see factors Sec. u()

=JaADdl,WD@

{I[e- t

(2.8)

and the "covariant derivative" D, only contains the background field A:', for instance:

w =out (0I@I. xexp

(2.7)

where

CI2(A)+ B y (@)Id x } ,

(2.4)

to be expanded with respect to d. Here C,(A) is a gauge-fixing term, and a r e the corresponding ghost terms. A s is argued in Ref. 2, the U(1)-breaking part of this amplitude comes from that region of superspace where the A field approaches the solutions described in Ref. 1:

W = exp(- 84/gz)(detJn,)~L'Zdet~,(det;m,)-'

x det mr,,.

(2.11)

The determinants will be computed by diagonalization: J b $ = E i $8

(2.12)

after which we multiply all eigenvalues E. Since there are infinitely many very large eigenvalues, this infinite product diverges very badly. There are two procedures that will make it converge: (I) The vacuum- to-vacuum amplitude in the absence of sources must be normalized to 1, so that the vacuum state has norm 1. This implies that W must be divided by the same expression with A" = 0. (ii) We must regularize and renormalize. The dimensional procedure is not available here because the four-dimensionality of the classical solution is crucial. We will use the so-called background Paull-Villars regulators (Secs. IV and V). Taking a closer look a t the eigenvalue equations (2.12) as they follow from (2.7), we notice that the background field in there gives rise to couplings between spin, isospin, and (the four-dimensional equivalent of) orbital angular momentum, through the coefficients qaWvin (2.5). Now these couplings

where 2, and p a r e five free parameters associated with translatlon invariance and scale invariance. The coefficients 9 a r e studied in the Appendix. Conjugate to (2.5) we have its mi rror image, described by the coefficients 5 (see Appendix). Now these solutions form a local extremum of our functional integrand, and therefore it makes sense to consider separately that contribution to Win (2.4) that is obtained through a new perturbation expansion around these new solutions, taking the integrand there to be approximately Gaussian. The fields 8 , @,, and $ all remain infinitesimal so that their mutual interactions may be neglected in the first approximation. Of course, we must also integrate over the values of zll and p. This will be done by means of the collective- coordinate formalism.5 One writes and those values of A'" that correspond to translations or dilatations a r e replaced by collective co-

304

COMPUTATION O F THE QUANTUM EFFECTS DUE TO A

simplify enormously if we go over to a new gauge that explicitly depends on the background fielde G (A q"=) 0, AtqU

8,A~uU+g<,0A~C1A,6u".

(2.17)

(2.13)

Thus the third and eighth terms in (2.7) cancel. This choice of gauge will lead to one complication, to be dlscussed In See. XI: The gauge for the vacuum-to-vacuum amplitude in the absence of sources, used for normalization, in the region A, 0, is usually invorlant under global lsospin rotations, but the classical solution (2.5) and the gauge (2.19) are not, Associated with this will be three spurious zero eigenvalues of Sn, that cannot be dlrectly associated with global isospin rotations. The question is resolved in Sec. XI by careful comparison of the gauge C, with C, and some intermediate choices of gauge. There will be five other zero eigenvalues of X A that of course must not be inserted in the product of eigenvalues directly, since they would render expression (2.11) infinite. They exactly correspond to the infinitesimal translations and dilatations of the clnesical solution, and, (LB discussed before, must be replaced by the corresponding collective coordinates (Seca. M and X). The matrices 9R are now (ignoring temporarily the fermion source)

-

3nAAt"--

where 1.1 = (x - 2)'. This clearly displays the isospin-orbit coupling. The vector and splnor fields also have a spinisospin coupling. F a r the spinors we define the spin operators (2.181

satisfying

and (2.20)

For the vector fields we define SYP- ~ LA F, S;Ar=-tfq,,,A:V,

(2.21)

sIp=s:= t .

2gt,,G:,01A,0eu,

s,

For the scalar fields = &= 0. Thus right- and left- handed spinore a r e (3,O) and (0,f) representations of 50(4), and vectors a r e (i,t)representations. Scalars of course a r e (0,O) representations. h terms of the operators S and T, the spinisospin couplings turn out to be universal for all particles. Substituting the classlcal value for G$ in (2.14) we find

rn&=-DS+. In order to substitute the ClaSslcal solution (2.5) with z = 0 md p = 1 (generalization to other z and p will be straightforward), we introduce the apacetime operators (2.15)

-

8

-G= t { q , " x ' g ,

(2.22)

with

-

with 317=311, or X t or 311, or 32,. Observe the abEenC8 of spin-orblt and Isoepinorbit couplings that contain xu or 8 / 8 x , explicitly. It all goes via the orbital angular momentum operator L, and thnt implies that Lacommutes with 9R. This would not be so In other gauges. Further, 9R commutes wlth 3, = 8, + and & and 8,. Eigenvectors of 311 can thus be characterized by the quantum numbers

[L;,L:I*fo,,&L;,

L'EL,'.

L,P=

- t(x,e, - ~ ~ 8 , ) ~ .

(2.16)

They represent rotations in the huo invariant SU(2) subgmupa of the rotation group SO(4). Isoepin rotatione will be generated by the oper' tor the scalars, T.=f7. for the spinors, ators T

el+

305

G . ' t HOOFT s,

and s1 (both either 0 o r

t) ,

t (total isospin, arbitrary for the s c a l a r s ,

... , - t , I - sl - t + 1,. .. , I +s, + t , j , 3 = - j ,,... , + j , , sl==-S1).. . , + s a t 1 ; = - 1 , ... , + 1 .

a for the spinors,

1 for the vector and the ghost),

l=O,i,l,

j,=I

- s,

a s long a s j ,

20 ,

(2.23)

For normalization we need the corresponding operator 9R for the case that the background field is zero: (2.24)

111. DETERMINANTS AND PHASE SHIFTS

The eigenvalue equation (2.12) with I a s in (2.22) differs in no essential way from an ordinary Schrlldinger scattering problem. In this section we show the relation between the corresponding scattering matrix and the desired determinant. Temporarily, we put the system in a large spherical box with radius R. At the edge we have some boundary condition: either 9 ( R )= 0, or W(R)= O (or a linear combination thereof). Here rIr stands for any of the scalar, spinor, or vector fields. In the case W ( R )= 0 the vacuum operator 3lt, has a zero eigenvalue corresponding to Q =constant, and also the lowest eigenvalue of Sn may go to zero more rapidly than 1/R2 when R Such eigenvalues have to be considered separately (negative eigenvalues can be proved not to exist). We here consider al l other eigenvalues of N. They approach the ones of "2, if R -, We wish to compute the product

The level distance Ak = k(n + 1) - k(n) is in both cases, asymptotically for large R , (3.8)

We find that

- -.

-

provided that the integral converges at both ends. At k-0 the integral (3.9) converges provided that the interaction potential decreases faster than l/P a s r - m ; a t kthe integral converges if the interaction potential i s less singular than l/P a s r - 0. The latter condition is satisfied if we compare 311 and Sn, at the same values for the quantum number I; the firs t condition is satisfied if ( L , + T)' for the interacting matrix is set equal to La for the vacuum matrix. If we consider the combined effect of all values for Lz and (4+ T ) aboth for the vacuum and for the interacting case then we can split the integral (3.9) somewhere in the middle, and combine the k-- parts, so that we get convergence everywhere.

-

The scattering matrix S(k) =ez'"(b'will be defined by comparing the solution of Xly = kZO

(3.2)

with

Xo#,= k2rIro

(3.3)

both with boundary condition 9- CP' at r =0. Let r ~ r , ( ~ ) ~ ~ - 3 / Z ( ~ - I b ( ? ++o&b(r+o) > )

-

= 2Cr-3'a cosk(r+a) for large

Y

(3.4)

306

COMPUTATION O F THE QUANTUM EFFECTS DUE TO A

Z,j,,# generates two solutions with s;=$, l ' = l i $ , j;=jl,t ' = f . In terms of the operators L , S, and T the new solutions to the coupled equations

An easier way to get convergence is to regularize:

can be expressed in terms of the S=0 solutions a s follows: L:ts:=L,,

(3.10)

l'=l*f,

T '= T I J' = J

,

(4.5)

Regulators will be introduced anyhow, SO we will not encounter difficulties due to non-convergence of the integral in (3.9). It is easy to check explicitly that if 0 satisfies (2.22) with S,= 0, then the two wave functions Q' both satisfy (2.22)when the operators L , S , T a r e replaced by the primed ones. Asymptotically, for large r , W =( 8 / 8 7 ) * , and hence the phase shift q(k) is the same for the primed case as for the original case. Consequently, the integral over the phase shifts a s it occurs in (3.9) is the same for spinor and vector fields (with sI=i) a s it is for scalar fields (with s,=O). The above procedure becomes more delicate if E = 0. Indeed, although scalar fields can easily be seen to have no zero-eigenvalue modes, spinor and vector fields do have them. In conclusion, the nonzero eigenvalues for the vector and spinor modes a r e the same a s for the scalar modes, but the zero eigenvectors a r e different. In the following sections we compute the universal value for the product. Note that also the regularized expressions (3.10) a r e equivalent because the q(k) match for all k. The regulator of Eq. (3.10) corresponds to new fields with Lagranglans

IV. ELIMINATION OF THE SPIN DEPENDENCE

In Eq. (2.22) the operators T * L , and T - S , do not commute. Only in the case that ljl-Z

I =s+t

(as defined in 2.23) do they simultaneously diagonalize. If Ij,-llcstt and

jl+o,

I+O, s=+, t # ~ ,

then we have a set of coupled differential equations for two dependent variables. In any other case there would be no hope of solving this s e t of equations analytically, but here we can make use of a unique property of the equation

;mQ=E+,

(4.1)

which enables us to diagonalize it completely. If sI= 0 the equation could describe a left-handed fermion wlth isospin t:

c = - & ( D , B , ) ~ - ~ M ~ B , ~ - ~ B ; E , , , G : , B C , (4.7) (4.2)

for vectors,

y5J,=+J,.

-

But then we can define, if E # 0 ,

V-r-Dll

Y5$' =

a=- (o,,[)*o,~-&'[*~ for s c a a r s .

(4.9)

Within the background field procedure it is obvious that such regulator fields make the one-loop amplitudes finite. Later (Sec. X U ) we wfll make the link with the more conventional dimensiona1 regulators.

,

- $'

Now (I' has s; = i, and hence we found a solution for the set of coupled equations with s:=& from a solution of the simpler equation with sI= 0. The operater y* Din Eq. (4.3) does not commute with La,so if J, has a given set of quantum numbers ZJ,, t then Q 1s a superposition of a statewithl' = 1 t i and one with Z ' = 1 i. Now I does commute with L', so if we project out the state withZ'=l++ or I' I 4 thenwe get a new solution in both cases. Thus one solutionwith s, = 0 and quantum numbers

--

(4.8)

and

(4.3)

with

3n4J'= EJ,'

C = i [ - ( y -D)z M z ] x for spinors ,

V. A NEW EIGENVALUE EQUATION AND NEW REGULATORS

As stated in the Introduction, the solutions to the equations 3R-9 = EP even in the scalar case cannot be expressed in terms of simple elementary functions. But eventually we only need det3lZ/ d e m o , and this can be obtained in another way.

-

307

G . 'I HOOF?

det@ZhRJ= det(V/VJ

We write V=!(1+*3)fSL(l+r'),

.

The equation

(5.1)

V* = x*

vo=~(l+~)aue(l+~), and, formally,

corresponds to the expression

This Is a hypergeometrlc equation. The physical region is 1/(1+ Ra)< X 6 1. In the Hilbert space of square-integrable wave functions the spectrum i s now discrete, which implies that we can safely take the limit R-m. The solutions for rb a r e just polynomials:

We choose M a here so large that anywhere near the origin the regulator i s heavy. Far from the origin the classical solution i s expected to be close enough to the real vacuum, so that there the details of the regulators a r e irrelevant. Of course the regulator procedure affects the definition of the subtracted coupling constant. In Sec. VII we link the regulator (5.11) with the more acceptable one of Sec. IV, and In Sec. XIII we make the link with the dimensional regulator. The eigenvalues of the regulator a r e

*(x)=

a&,%",

(5.7)

Y.

where n Is defined by

A=:

(n+ I +j , + 1) (n + I + j , + 2) = la+ A .

-

n=0,1,2

,...,

(5.12)

.

If n = integer 0 then the series (5.7) breaks off. Otherwise rb is not square-integrable. So we find the eigenvalues A,= ( n + l +jl + 1 t ) (n+l+j, + 2 + t ) ,

( n + l + j , + 1 - I ) ( f l + l +j , + 2 + f)+nn'.

The regulators M , with i = 1,. . ,R a r e aa usual of alternating metric e , = i 1. Consequently, detsR i s replaced by

(5.9)

(5.13)

(5.10) This converges rapidly if

Po=t(t+l).

$

The vacuum case, Vo*=XoU, is solved by the same equation, but with j, = I , t = 0. The product of these eigenvalues, even when divided by the vacuum values, still badly diverges so we must regularize. The regulators of Sec. N a r e not very attractive here because they spoil the hypergeometric nature of the equations. More convenient here is a s e t of regulator fields with masses that all depend on space-time in a certain way. They a r e given by the Lagrangians (4.7)-(4.9) but with M a replaced by

el=- 1 ,

(5.14)

and

$ e , l n M , = - lnM=finite.

(5.15)

Let i = 0 denote the physical field, then

(5.11)

eo=l,

308

Mo=O,

$

e,=O,

etc.

(5.16)

C O M P U T A T I O N O F THE Q U A N T U M E F F E C T S D U E TO A . . .

replacing t with zero and adding an additional multiplicity 21 + 1, thus

VL THE REGULARIZED PRODUCT OF THE NONVANISHING EIGENVALUES

We now consider the logarithm of the regularized product of the nonvanishing eigenvalues, for a scalar field with total isospin t :

t).

x ln(sa+M,l-

with XYI = (n+I + j l t 1

- t ) (n+Z + j , + 2 + t ) + M t

(6.2)

(we imply that e, = 1 and M o= 0). The summation goes over the values of all quantum numbers. Now for given n , l , j , the degeneracy is @ j , + l ) ( 2 j a +1) = ( Z j , + 1) (21 + 1). The values of 1 , j , , and n a r e restricted by

-

with

AM'($)=

us2 +jl t 2 0 ,

$a).

(6.0)

Let u s first consider the regulator contribution. Then M i s large. We may c6nsider the logarithm 88 a slowly varying function, and approximate the summation by means of the Euler-Maclaurin formula,

~ = j , - 1 + 1 ) 0 , 1 5 2 t , n=O.

, s rt + u+ 4.

$;, (s3-sga)ln(sa+M,*-

a

(6.3)

[Later we will divide n(f) by the vacuum value lTo(t),which Is obtained by the same formulas as above and the following, but with 1 replaced by zero, and the degeneracy will be (2t+ 1) (21 + l)'.] We go over to the variables u and T a s given by (6.3) and s with

s = n + l+jl+ f

(6.7)

Now we interchange the summation over s and i , letting first s go from t + 3 to A and taking Ain the end. We get

- By"(%) * * .] ;1 ,

(6.4)

(6.10)

We find that and we obtain

A n ( @ =) indep(9) + $'(-

f M a - A'ln

- 4A -

A - A WL

M -4

- i lnM)

(-$) + o ( i ) .

++ 94(2 w +1)+0

The summation over u and r gives

(6.11)

X

The f i r s t term stands f o r an a r r a y of expressions, all independent of $, and is not needed because It cancels out in Eq. (6.8). For Ao(@)the s e r i e s (6.10) will not converge a t x = p 80 it cannot be used. After some purely algebraic manipulations we find

ln[S'+M,'-(t+t)']. (6.6)

The vacuum value lI,(t) is obtained from (6.6) by

(6.12)

309

G

HOOFT

' t

Now we insert (6.11) and (6.12) into (6.8): s(2t + 1 - s) (s - t - i)Ins

+ t ( l + 1)[4

$

S Ins -

2A21nA - 2AInA- i l n A + A 2 + i I n M - % t ( t + 1)- i

We made use ofC:e,=O, C : e , M , 2 = 0 , C:e,lnMiZ=- InM. The limit A - m exists. Defining R=

!!?(

$

s Ins

- 4 A' InA - $ A InA -

InA+ f AZ

= 0.248 154 471 ,

we find that 2t+l

(6.14)

,+

I(*+ +

In[II(t)/IIo(t)] = 7

1

1)( ;In

M

4R

- k t ( t + 1) - 2)

R is related to the Riemann zeta function b ( z ) as follows:

s ( Z + 1 - .)(a

*-

-1-

1

be absorbed by a counterterm in the Lagrangian, and hence is local in space-time. So we expect that, if we make a space-time-dependent change in the regulator ma s s , then this change can be absorbed by a space- time-dependent counterterm. Moreover, since our regulators are both gaugeinvariant, this counterterm is gauge invariant. F o r space- time-independent regulators, this counterterm can be computed by totally conventional methods:

R=h-t'(-l)

y=

2tt1

0.511215664 9 is Euler's constant,

and

(7.2)

= 0.931 548 254 315 844.

From locality we deduce that the same formula must also be true for space-time-dependent regulator mass p ( x ) , simply because no other gaugeinvariant, local expressions of the same dimensionality exist. Inserting the classical value for

VII. THE FIXED MASS REGULATOR

Equation (6.15) gives the regularized product of all nonvanishing eigenvalues of Jn. But the regulator used was a very unsatisfactory one, from a physical point of view, because the regulator mass p depends on space- time:

G,"

I

(7.3)

and expression (7.1) for p , we get This p must be interpreted as the subtraction point of the coupling constant g. Now g does not occur in II(t)&(#), but it does occur in the expression for the total action f o r the classical solution, and as we emphasized in the Introduction, any change in the subtraction procedure is important. The problem here i s that we wish to make a spacetime-dependent change in the subtraction point, from p to a fixed po. We solve that in the following way. The effect of a change in the regulator mass can

as"'=

-

J A S 8% l6 12=* t(f + 1)(2t+ 1) 3 2 ~ ~ x 9

= $ t ( f + l ) ( Z t + l ) :In

31 0

"+

2M

&

COMPUTATION OF THE QUANTUM EFFECTS DUE TO A , . .

In the expression

fixed mass regulators, a s defined in (4.9). The regulator masses M , in there must be such that

$91,

e,hM,=-inN,.

with ll(t)/II,(t) a s computed in (6.15), we must correct 9’with the above W ’ , in order to get the corresponding expression with g subtracted with l(t + 1) (21 + 1) In(n(t)/n,(fll= 3 [*l+ + 4R +

g

Expression (7.4) must be added to (6.15). Thus we get

I .

s(2t

We note that the coefficient of the regulator term in (6.15) has the correct value. It matches the coefficlent of (7.2) that has been computed independently. The regulator in this expression, (7.6), is the same a s the one used in (%lo), and so we can use the result of Sec. IV to do the spinor and vector fields. In Sec. N we proved that the nonzero eigenvalues for vector and spinor fields a r e the same a s for scalar flelds, but we must take some multlplicity factors into account. Equation (7.6) holds for one complex scalar multiplet with lsospin f . Fields wlth integer isospin may be real and then we have to multiply by 1. The vector field has four components but is real, and hence its value for In(n/n,,) is twice expression (7.6), with t = 1. The complex Faddeev-Popov ghost has Fermi statistics and contributes with one unit, but opposite sign. Thus, altogether, the vector field contribute6 just like one complex scalar with t = 1. For fermions we must compute d e t q , but the theorem of Sec. IV applies to at. The fermlona have four Mrac components. So, altogether, fermions contribute just like two complex scalars, but the slgn in In(fI/nd is opposite because of fermi etatletlcs. The above summarizes in words the complete contribution of all nonzero eigenstates to the functional determinants. But the spinor and vector flelds have a few more modes, with E m 0, and also the regulators have corresponding new modes, with E = p:.

-

VIII. THE ZERO EIGENSTATES

First we conslder the vector fields. We have sI= i, I 1. Careful study of the operator a, Eq. (2.22), enables us to list the square-intergrable zero elgenstates a s follows: (i)

jl‘t,

l z 0 : Q=(l+f)’,

(1,=s,=f),

(7.5)

(8.1)

multiplicity = (2 j , + 1) (2 j l + 1)= 4 .

+ 1-

s) (s

- t - 4)Lns- 4 1 ( t + 1) -

(ii) j , = o , I = + : * = r ( ~ + r ~ ) ’ * . There a r e two possibilities for the other quantum numbers: (a) j , = 0, multiplicity = 1 ,

(82)

(b) j , = 1 , multiplicity = 3 .

(8.3)

This completes the set of zero eigenstates. We interpret these a s follows. States (1) have jl=j, = i tthat is, the quantum numbers of an infinitesimal translation. The translations a r e considered in Sec. M. State (iia) is the only singlet. It will correspond to the infinitesimal dilatation, Sec. X. State (ilb) is just an anomaly. It will be discussed in Sec. XI. It is indirectly connected with infinitesimal global Isospin rotations. Spinors have similar sets of elgenstates, but their interpretation will be totally different. If t = 1, then the eigenstates a r e essentially the same a s the vector ones, but their multiplicity is half of that because s1 = 0. In this paper we limit ourselves to t = $ . Then there Is just one zero eigenstate: jl =o, I = j , = 0, Q = (1 +r2)-9‘a. (8.4) Its multiplicity is of course N’ If there a r e N’ flavors. It leads to an ”-fold zero in the amplitude (note that in (2.11) the amplitude ia proportional to det‘JR, and thus is proportional to the product of the eigenvalues of “t,; if we have Nf zero elgenvalues then W has an N’-fold zero]. But this zero will be removed if we switch on the fermlon source $in the Lagrangian (2.1). In Sec. X N we will construct the resulting Nf-point Green’s function. In strong-interaction theories the fermion mass will also remove this zero. The zero eigenstates must also be included in the regulator contributions. From Sec. Vn on, our regulator mass is fixed and is essentially equal to pw Every zero eigenvector of the operator 3ll, Eq. (2.22), will be accompanied by a factor pCa for the regulator (a zero eigenvector of Jrr, i s accompanied by a factor IllJ-‘).

31 1

$3

C . 't HOOFT

IX. COLLECTIVE COORDINATES: I . TRANSLATIONS

so, together with the regulator, these four modes yield the factor

Clearly, zero eigenvalues make no sense if they would be included in the products carefully computed in the previous sections. In the case of the vector fields, which we will now discuss, they would render the functional integral infinite because they a r e in the denominator. It merely means that the integration in those directions is not Gaussian. Let us f ir st consider the four modes (8.1). The angular dependence and index dependence can be read off from the quantum numbers. Written in full, the mode corresponds to the quantum field fluctuation (with arbitrarily chosen norm):

A tq"(v) = 2qauv(l+ r z ) - z ,u= 1 , .

.. , 4 .

( ( :)4

$)'(2z2)'d42

= 2'n'po4g-'d'2

I

The integral over the collective coordinates Z" will yield the total volume of space-time, if no massless fermions a r e present. If there are massless fermions, then we must include the sources 3, which break the translation invariance. In that case the z integration is rather like the integration over the location of an interaction vertex in a Feynman diagram in coordinate conflguration, as we will see in Sec. XIV.

(9.1)

X. COLLECTIVE COORDINATES: 2. DILATATIONS

This can be seen to be the s p a c e - t h e derivative of the classical solution up to a gauge transformation:

From the quantum numbers of the zero eigenstate (8.2) we deduce its angular and index dependence: A~q"=q~Ly~Y(1+~*)~2.

(9.2)

-

.

(10.2) (9.3)

Thus, going from the integration variable Aau in this direction to the collective variable p, we need a Jacobian factor:

The gauge transformation is there because our gauge-fixing term depends on the background field. If we want to replace the varlable DAaUin this particular zero- mode direction by the collective variables &', then we must insert the corresponding Jacobian factor'

The norm of the solution (10.1) is

(A ~q")2d4x = tra (9.4)

tau( u)A z '"(X)d ' X = 2n2QA ,

.

(10.4)

Thus, from this mode we obtain the factor

where AEq"(u)is the solution (9.1). The factor 2/g comes from the factor g/2 in (9.2). This way the renult ia independent of the normalization of AEq"(u) of (9.1). The factors 2r-'/' arise from the fact that we compare this integral with Gaussian integrals of the form JdAexp(-hA'), and in these Gaussian integrals the factors fithat go with each eigenvalue had been suppressed previously. We could also have drag& along all factors 6at each of the eigenvalues of the matrices VI,and then we would have noticed that the factors figoing with the corresponding modes of the regulators, which are still Gaunnian, would have been left over. In (9.4) we just indude these factors from the beginning. The norm of the solution (9.1) is $A

(10.1)

This is a pure infinitesimal dilatation of the classical solution:

with A"(v)= qevA.8(l+ x *)-I

(9.6)

a t p = 1. O u r system is not scale invariant because of the nontrlvial renormalization-group behavior. The complete p dependence for p # 1 will be deduced from simple dimensional arguments (including renormalization group) in Sec. W. XI. GLOBAL GAUGE ROTATIONS AND THE GAUGE CONDITION

Discussion of the legitimacy of the background gauge-fixlng term h a s been deliberately post poned to this section, because we wanted to derive first the existence of the three anomalous zero eigenstates (8.3). They have the explicit form

(9.5)

312

COMPUTATION OF THE QUANTUM EFFECTS DUE TO A.

elgenmode yields the factor

(arbitrary normalization) (11.1)

$:(b)=2~~.rrSwrxA(l+X2)-'.

(11.10)

They a r e a pure gauge artifact

$t(b)=4,$"(~), $'(b)=tl..rv~&"XA(1

Now we will study different gauges, and for that we need to change the boundary condition (we know from See. Ill that that will not affect the finite eigenvalues, but the zero eigenmodes change) into a gauge-invariant one: The ghosts 0 and gauge generators A must satisfy

(11.2) +Yl)-'

,

(11.3)

but $'(b) is not square-Integrable. What is going on? Note that $"(a), slnce they a r e x dependent, do not generate a closed algebra of gauge rotations. They may net be replaced by a collectlve coordinate Zor global isospin rotations. To analyze this sltuation we first go back to a background- independent gauge-fixing term,

-

(11.11)

where R is the radius of the box, and the vector fields

C;(%)= a8 ,( AY'+A ",")

a

ae, A,'u:

A,(R)=O, =A,,(R)=O,

(11.4)

where A, is the vector component parallel to the boundary. Now observe the following. The gauge term C,, Eq. (11.4), does not fix the gauge completely which can be seen in hvo ways: (a) Global isospin rotations a r e still an invariance; (b) one component of the gauge term C, is identleally zero:

where a is a free parameter. In this gauge we know exactly how to handle all zero eigenmodea: There a r e five for translations and dilatations and also three for global isospin rotations because global isospin is still an Invariance in this gauge. To understand the latter we put the system in a large spherical box with volume V and assume that all (vector and ghost) fields vanish on the boundary. Let A:@,%) = C b

f q(x)d4x = 0 .

(11.5)

(11.13)

This leaves us the possibility of adding a constant to C,, which is orthogonal to it, with which we fix the remaining global gauge

generate an infinitesimal global isospin rotation. Then there i s a zero eigenmode:

4TU(b)= W ; ( b ) = 2q,,,b~eeu~Y(1+%*)-I

(11.12)

c(%) = aB,A :'" .

(11.6)

The subscript 1 is to remlnd us that this is a solution i n the gauge C,. Similarly a s in the foregoing two sectlons, we can replace the integral over aA by an integral over the collective coordinates dA(b) by inserting the corresponding Jacobian factor

wlth a , free ~ parameters and +~,,(b,y)=DUOsb, as in (11.6). The integral over group space is now replaced by a Gaussian integral. The Gaussian volume is corrected for by the Faddeev-Popov ghost,

(11.7)

c:=

cwpe,,D,,+w

with

-K

~ab(~:u(b,y)D,C(~)d4]. b

(11.8)

(11.15)

so that the combined contrlbution of vector fields and ghosts is now independent of a and K : The zero eigenvalues a r e replaced by

which diverges as the volume V of space-time goes to infinity. The integral over the gauge rotation is just the volume of the group and yields

&-' (11.9)

= Klw

(11.16)

and

f'

@Q*=Kal.

where the factor comes from our normalizatlon of A, in (11.6). Thus, in this gauge, the zero

(Remember that the ghost, with the new boundary

313

G . 'I HOOFT

condition, has now an eigenstate @ = constant.) Thus, instead of (ll.lO), this gauge gives (Xfa)3(A~aor)-3~*

=(x/V)3/2.

and (1 1.24)

(1 1.17)

where no summation over b is implied. In this gauge we find the contribution from the lowest eigenmodes:

Conclusion: If the redundant eigenmodes a r e fixed by an additional component in the gauge-fixing term, then a correction factor is needed: Equation (11.10) divided by (11.17):

!$(

( ~ ~ ) 3 ( x p I o) -r3 / 2 =

Here V i s the volume of the spherical box. The background gauge C,=D,Azq" has a problem simllar to the gauge C,. Both the ghost (under the new boundary condition) and the vector field have one eigenvalue that vanishes like 1 / V as V(not l/i?', a s the other eigenvalues). Let $:(b) be the three ghost eigenstates and $&(b)=D,$:(b) be the vector ones. Let A, be the ghost eigenvalue =

- h,JI:(b) .

(11.25)

(11.26)

-

@$",(a)

~,3/2.

Using

(11.18)

we find A, = 4na/V.

(11.27)

Thus, together with the correction factor (11.18), we find the correct contribution for the three eigenmodes (9.3), together with that of their regulator:

(11.19)

It is easy to see that (11.20)

From (11.1) and (11.3) we see that this is O(l/V). It i s safer to have a gauge condition that fixes this gauge degree of freedom a s V--,

XII. ASSEMBLING THE VECTOR, SCALAR, AND SPINOR TERMS

The eight zero-eigenvalue modes for the vector field give the factors (9.6), (10.5), and (11.28). Multiplying these gives $:(b,%)J$~"(b,~~~~'(~)d~, 3 2'' x8g-' PO' d' zd p (12.1) (11.2 1) for p = 1 (later we will find the p dependence). although the result, (11.25), will turn out to remain The contributions from the nonvanishing eigenthe same even if K = 0 , Q = 1. The ghost Lagranmodes both for the scalar and for the vector gian is fields a r e essentially contained in formula (7.6). A s we saw before, the vector fields, combined with the Faddeev-Popov ghost, together count as two real, or one complex, scalar with t = 1. Let there be N a ( t ) scalar multiplets for each isospin t , where each complex scalar multiplet counts a s one, and each real scalar multiplet counts a8 one-half. Then from (7.6) we obtah the total contribution from vector and scalar nonzero modes:

C,(%)= aD,AO,qU(x)

KC

(12.2)

(12.3)

314

COMPUTATION O F T H E Q U A N T U M E F F E C T S D U E TO A , . .

and P I

& ( f )=C(1)[2R-

t w+t

s(2t+ 1- s) (s- 1 - 4)Ins

.+ 1)-

4.

(12.4)

The numerical values for C ( t ) and a(t)a r e listed in Table I. Combining (12.1), (12.2), and the classical action (2.8) gives the total amplitude in the absence of fermions: 2''noP

&zdp 7 ew{-

-

&a

~ + l n ( p g ) [ e - ) ~ ~ a ( t ) C (at(jl ] )-EN'(Wt)).

Note that the coefficient multiplying lnp, coincides with the usual Callan-Symanzik p coefficient for $ ( p d In such a way that (12.5)becomesindependent of the subtraction point p, if we choose d ( p J to obey the Gell-Mann- Low equation. We now also insert the p dependence if p # 1 by straightforward dimensional analysis. The interpretation of (12.5) Is best given in the language of path integrals: If 10) is the vacuis the gauge-rotated vacuum, then um, and (12.5) i s the total contribution to (6 10) from all paths in Euclidean space that have apseudoparticle located at z within d'z, having a scale between p and p+dp. The fermions can be introduced in two ways:

where Euclidean pseudoparticles form a p l a s m a like statistical ensemble.' For that, one also needs to extend from SU(2) to SU(3). We will not do that in this paper. In Sec. XW we consider case (11). In that case m must be replaced by the eigenvalue of the lowest mode a s it is perturbed by the source insertion. XIII. DIMENSIONAL RENORMALIZATION

la)

The regulators used in Sees. VII-W a r e what we call fixed mass Pauli-Villars regulators and they only make sense in the background-field formalism. They a r e given by the Lagrangians (4.7)-(4.9). In this section we wish to switch to another regulator scheme which is much more widely used in gauge theories: the dimensional method.' Let u s emphasize again that if one switches to another regulator, then that affects the definition of g ( p ) and that influences our calculation by an overall constant. We know that in the dimensional procedure the limit of large cutoff is replaced by a limit n 4, where n is the number of space-time dimensions, roughly in the following way:

(1) If they have a mass m << l/p then only the lowest eigenvalue will depend critically on m. (11) If they a r e rigorously massless then the lowest eigenvalue depends critically on the external source 8.

-

In this paper we iimit ourselves only to fermions with isosph f -t. In case (I) the contribution of the lowest modes will simply be

(12.6)

The nonvanishing eigenmodes are again obtained from (7.6), which represents the eigenvalues of X,'. Now we wish to compute detSn, and we take into account that the Dirac field has four components. Thus the fermion nonvanishing eigenmodes w i l l give em[$

~ ( 4 bcco+ ) ~ f a ( i ).]

(12.5)

I ~ A -4-n

.

+ finite.

In (12.5) and (12.7) the regulator mass p, plays the role of the cutoff A. Clearly then, the finite part in (13.1) will be relevant. In this section we derive that finite part, in ordinary perturbation TABLE I. Numerical values of the coefficients C(t) and qt)a8 they occur in the text.

(12.7)

Together with (12.8) we find the total fermion factor that multiplies (12.5),

0

0

0

fNfln(p,p) + ~ f a ( + ) l , (12.8)

f

1

%-rln 2 -Lr 6 I2

where we again inserted the factors p as they follow from dimensional arguments. Note that the well-known Callan-Symanzik /3 coefficient for go( pJ again matches the term in front of lnpw Equation8 (12.5) and (12.8) could be used a s a starting point for a strong-interaction color theory

1

4

~ft+fln~-y

p"'m"exp[-

(13.1)

+

10

20R+4In3 -;In

=0.145 873 = 0.443 307

2 -j(9=0.853182

R=$(lnZnty)*-i;?~~=0.2487&4471033784 1 -

315

The coefficient in front of Y,,Y,,, is obtained in the same way by comparing the integral

theory. It correaponds to a finite counterterm in the Lagrangian. It is easy to compute this finite counterterm when, again, one makes use of the background fields. There are some diagrams to be computed and the rest is algebra. This algebra is identical t o the algebra devised in Ref. 9. Symmetry arguments restrict the possible form of the finite counterterm in just two independent terms, X z and Y,,Y,,,, in the language of Ref. 9. The first of these is obtained by comparing the integral

&

-

1

in the two limits, where the signs e i are defined an in Eqs. (5.15) and ( 5 . 1 6 ) , replacing M by PO. This time we get

(13.2)

/ d " k i F G T

in the limits po co and n -+ 4. For definitenes8, we specify the theory a t n # 4 dimensions: All trivial factors (2.)' must also be replaced by (2x)", which leads t o the factor (2r)-" in (13.2). The integral (13.2) is in this limit

So we see that for both the X z terms and the Y,yYpvterms the substitution (13.4) is to be made, However, we have to remember that in dimensional regularization the number of the fields Aq,U is n rather than 4. For these fields one therefore must replace

(13.3) where y is again Euler's constant. So here Inpo

- 4-n 1

1 - -y+

2

1

-1114s. 2

Inpo (13.4)

-

-, 4 -1n - 1~ y + -21I n 4 n - l .

(13.7)

In conclusion, (12.5) is to be replaced by

(13.8)

If we define the subtracted coupling constant as in Ref' lo,

with A = -a(l)

11 -(ln4x 3

+

- y) + - = 7.05399103 (13.9) 1

3

and

A(#) = -a(l)

1 + -(ln 12

4 r - r)C(#). with a, depending only on g, but not on n or p, then we can make the following replacements in (13.8) and u3.11):

Numerically, A(0) = 0 , A(!.) = 0.30869069, A(1) = 1.09457662, 2 3 ( 13.10) A ( - ) = 2.48135610 2 Similarly for the fermion factor

pNfmNf exp

[-f N'

(hp+

A)-

N'B]

g,(d - E m ,

-W W ) .

1 lnp+ 4-n

(13.11)

Here the superscript B stands for the dimensional procedure which defines g$. We s e e that the expression in t e r m s of g E ( p ) differs slightly from the one in terms of g(po).

with 1 1 B =-24-) -(ln4r 2 3 = 0.35952290.

+

(13.14)

- 7) (13.12)

316

C O M P U T A T I O N O F T H E Q U A N T U M E F F E C T S D U E TO A , . . XIV. THE PERMION SOURCE AND THE GREEN3

in coordinate configuration over the vertex variable. Equation (14.7)should really be considered as our final result for the space-time dependence of the fermion Green’s function. But it would be enlight8ning If we could represent it in terms of an effective Lagrangian. We found that the effective Lagrangian can best flrst be written in the form

FUNCfION

We now consider the fermion zero eigenmode (Q.4), and assume that the fermlon mas8 (12.6) vanishes. In that case the source Sat in (2.1) must be taken into account, since the lowest eigenvalue wlll now mainly be determined by this source. W e determine the lowest eigenvalues E ( i ) , i m 1,. ,h” of the operator

..

%=-

(14.I)

hD,b#,+Q.,

(14.8)

by perturbatlon theory, taking 8 as the small perturbation. The method is the standard one (the author thanks S. Coleman for an enlightening

where w is some fixed Dirac spinor wlth isospin 4. Owing to Fermi statistics, the varioue terms of the determlnant in (14.1) wlll arise with the appropriate minus signs 60 that we may limit ourselves to sources a,, that are dtagonal ln s and t:

discuseion on thls point). The unperturbed, degenerate elgenmodes a r e (taking for simplielty P‘1) 4f(t)nC(l+la)-st%a6,t, (14.2) a = 1,2, s , t = 1,.

&,=aWJ,,.

..,”.

(14.9)

Here d may still contain Dirac matrices. In the presence of thts source, the amplitude from the effective lnteractlon (14.8) would be

The coefficients u’ contatn besldes the isospin index a a Dirac index. They satisfy +‘**.+Pa&O,

(14.10)

or

where the minus sign comes from the Ferml statistics and the S,(%) are the Dirac propagators for massless fermions in coordinate configuration,

(14.3) y,u

--u.

(14.4)

(14.11)

The coefficient C 18 determined by normalizing #, Comparing this with (14.71, at large 2,we find that we must require (leaving aside temporarily the other contributions to the overall constant)

C‘.u*u~crx(l + x y

=#/a

uu*u=i.

(14.5)

Let

(14.12)

4,

(

=

m)1% I NO)

$-jcr41+~’)-3~:4,(x)u~,

where we may sum over the isospin index a but not over the Dirac components. Now from (14.3) one can derive

(14.8)

then E ( i ) are the N f elgenvalues of H. We wish to compute

v

(14.13)

E(i)=deW

so we must require the w , to be such that

c a

W,GQ = 4(l+ Y,)

.

(14.14)

Thue, the w, a r e some parity reflection Of un. There is clearly no gauge-invarlant solution to (14,14), 80 our effective Lagrangian (14.8) is apparently not gauge invariant. But note that we only wish to reproduce the amplitude (14.7) for gauge-invariant currents a,,. Thus, any gauge rotation of (14.8) does the same job. We get a gauge-invariant So‘f if we average over the whole

(14.7)

For large 9, this amplitude has exactly the space-tlme structure of an N’-potnt Green’s function, where each source point is connected to the origin by two fermion lines. The integral over the collective coordinate E [whlch is at the origin in Eq. (14.1)] will correspond to the integration

317

G . * t HOOFT

acting on the right-handed spinors only. So in total we get So" of (14.15) plus its Hermitian conjugate. Note that we obtain products of fermion fields, such as (14.17), that violate only chiral U(1) invariance. They have the chiral-symmetry properties of the determinant of an N f X N f matrix in flavor space and a r e therefore still invariant under chiral SU(N') X SU(N'). The symmetry violation i s associated with an arbitrary phase factor c * ' ~in front of the effective Lagrangians. If other mass terms o r interaction terms occur in the Lagrangian that also violate chiral U ( l ) , then they may have a phase factor different from these. We then find that o u r effective Lagrangian may violate P invariance, whereas C invariance is maintained. Thus we find that not only U(1) invariance but also P C invariance can be violated by our effect.

group of gauge rotations: (14.15)

where the brackets ( ) denote the average for all gauge rotations of w. We can then derive (14.16)

(w,q=t%.,(1+YJ,

and, for instance,

(a

@P) (ail,))=

& ( 2 q 6: - %; )t"' x ~;"L(1+YS)Jlfl~PZ(1+Y5)$rl. (14.17)

The Lagrangian (14.15) only acts on the lefthanded spinors. The parity- reflected Euclidean pseudoparticles will give a similar contribution

XV. CONVERGENCE OF THE p INTEGRATION

The entire expression that we now have for the effective Lagrangian i s &*ff(z)&z=~10+SNf,,6+2Nfg-8&

p-5*SN~dpew -

all2

{

+A-

5:

etc.,

and the numbers A , A ( t ) , B , C ( t )a s defined before. The p dependence has been changed because the effective Lagrangian (14.8) i s not dimensionless. We see that this integral converges as p- 0 (except when there a r e very many scalars). But there Is an infrared divergence a s pIn an unbroken color gauge theory for strong interactions this is just one of the various infrared disasters of the theory to which we have no answer. But in a weak-interaction theory it 1s expected that the Higgs field provides for the cutoff. Let there be a Higgs field wlth lsospin q and vacuum expectation value F. Let its contribution to the original Lagrangian be

-.

sH=- D,+*D,O - v(o).

b-;7

"(t)C(t)

- :N f ]

I

N a ( t ) A ( t ) - N f B ]8(.1f r (T,w)(Z#,))+H.c.,

(15.1)

duced by scaling to smaller distances, until the action reaches the usual value 8na/gl when the field configuration is singular. On the other hand, it is clear that the quantum corrections, a s can be seen in (15.1), act in the opposite way. There must be a region of values for p where the quantum effects compete with the effects due to the Higgs fields. To handle this situation rigorously we alter slightly the philosophy of Sec. rl. In Euclidean space it is not compulsory to conslder only those classical fields for which the action is stationary. We will now look a t approximate solutions of the classical equations, so that the total action is only a slowly varying function of one collective parameter, p. We simply postulate the gauge field A to have the same configuration a s before, with certain value for p , and now choose the Higgs field configuration in such a way that the total actlon is extreme. Only those infinitesimal variations that a r e pure scale transformations donot leave the action totally invariant, but nevertheless the parameter p gets the full

with

<waGB)=i6,(1+y,),

+ WPP)

(15.2)

Formally, no classical solution exlsts now, because the Higgs Lagrangian tends to add to the total action of the pseudoparticle a contribution proportional to F*,but this can always be re-

318

COMPUTATION O F THE QUANTUM EFFECTS DUE TO A

treatment a s a collective variable. A s will be verified explicitly, the dominant values for p will be those where the quantum effects and the Hlggs contribution a r e equally important. Since the quantum effects a r e small we expect that there P<< VM,,

APPENDIX: PROPERTIES OF THE q SYMBOLS

The group SO(4) i s locally equivalent to SO(3) x SO(3). The antisymmetric tensors A yy in SO(4) having six components form a 3 + 3 representation of SO(3) X SO(3). The self-dual tensors

(15.3)

which implies that the Higgs particle may be considered a s approximately massless. Let u s scale toward p=l,

Harvard for their hospitality, encouragement, and discussions during the completion of this work.

A UY = tfUu.6A.6

l F 1 - 1 , MH*'"XF2<
The equation for this field will be approximately

Dab=O,

(All

transform as 3-vectors of one SO(3) group. We now define the q symbols, in a way very similar to the Dirac y matrices:

(15.4)

a'(?-- Q)= F a . The solution to that i s a zero-eigenvalue mode of the famlllar operator (2.22): is a covariant mapping of SO(3) vectors on selfdual SO(4) tensors. A convenient representation is

j,=O, l=j,=q,

(15.5)

L,~=L,

The contribution to the classical action is

Sn =

j

-

va4u=-

[- DuWDu* V(b)]d'x.

The first term i s (observing that XuAE1=o)

-

- pea,* d'x = - 4l?qP.

.

q... = ( - i ) 6 ~ 4 * 0 9 0 u y (15.7)

(A4)

The symbols qa, will then do the same with vectors of the other SO(3) group and tensors Bw that

(15.8)

We s e e that the second term in the exponent may be neglected at first approximation. Thus (15.8) multiplies the integrand in (15.1) and the p integration is now completely convergent. The integration over p yields a factor

c - 2) ,

(A31

,

'I.44'0.

The second term in (15.6) is of order IF', where A is a small coupling constant. If we scale back to arbitrary p, then the Higgs field factor in the total expression is

t(4l?qFP)*-'"l)X'-Cr(~~'+

%u

Let us also define

aU(PDub)d% =

expS"=exp[- 4a'qF'd- O(xF'p')].

%"*

(15.6) %u'=

if ~ , ~ = 1 , 2 , 3

(15.9)

where (15.10)

ACKNOWLEDGMENT

The author wishes to thank S. Coleman, R. Jacklw, C. Rebbi, and all other theorists a t

319

G. *I HOOFT

'Work supported In p a r t by the Natlonal Sclence Foundation under Grant No. MPS 76-20427. ton leave from the Unlverslty of Utrecht. IA. A. BBlavln el d.. Phys. Lett. E. 85 (1975). 'G. 't Hooft. Phys. Rev. Lett. 3. 8 (1978); R. Jackiw and C . Rebbl, ibfd. 37, 172 (1976); Phys. Rev. D U , 517 (1978); C. Callan, R. Dashen. and D. Cross. Phys. 334 (1076). See also F. R. Ore. Jr.. Phys. Lett. Rev. D (to b e publlshed). % Jscklw I. and 9. Coleman (prlvate communlcatlon). 'J. 8. Bell and R. Jacklw. Nuovo Clmento E,47 (1969); 9. L. Adler. Phys. Rev. 111. 2426 (1969). 'J. L. Germla and B. Saklta, Phye. Ftev. 0 2 . 2943 (1976); E. Tomboulla, ibdd. 2, 1678 (1976).

e.

'J. Honerkamp. Nucl. Phye. E, 289 (197% J. Honerkamp, In Proceedings of the Colloquium on Renomallzation of Yag-Mills Fields and Applications to Particle Physics, 1972. edlted by C. P. Korthals-Altes (C.N.R.S., Mareellle, France. 19721. 'A. M. Polyakov, Phys. Lett. K B , 82 (1975). 'Go. 't Hooft and M. Veltman, Nucl. Phye. E, 189 (1972);C. G . Bolllnl and J. J. Glamblagl. phys. Lett. 40B, 586 (1972); J. F. Ashmore, Lett. Nuovo Clmento 4, 289 (1972); G. 't Hooft and M. Veltman. Report No. CERN 73-9, 1973 (unpublished). ' G . 't Hooft. Nucl. Phys. 444.(1973). %. ' t Hooft. Nucl. Phye. F&, 455 (1973).

-

s,

320

CHAPTER 5.2

HOW INSTANTONS SOLVE THE U(l) PROBLEM

C . 't HOOIT

Institute for Theorerical Physics, Princetonplein 5, P.0. Box 80.006, 3508 T A Utrecht, The Netherlands

NORTH-HOLLAND

- AMSTERDAM

321

Editorial Note A review article should not lead to any major controversy even if it often implies that the author takes sides when conflicting results or approaches exist. Yet, when it covers a delicate and topical question, it may happen that some important controversy escapes the editor’s attention. The very important and difficult “U(1) problem” was reviewed by G.A. Christos in a Physics Reports article published in 1984. The present article by G. ’t Hooft is meant to be a critical supplement to this former review. The editor is thankful to the present author to thus help to clarify this difficult question for the readers of Physics Reports. M. Jacob

322

PHYSICS REPORTS (Review Section of Physics Letters) 142, No. 6 (1986) 327-387. North-Holland, Amsterdam

HOW INSTANTONS SOLVE THE U(1) PROBLEM

G.'t HOOFT lnrfihrfefor Theoretical PhyJics, Princeronplein S. P.O. Box M.lN$ 3508 T A Ufrechf. The Nefherlands

Received I8 April 1986

Conunu: 1. Introduction 2. A simple model

3. QCD 4. Symmetries and currents 5. Solution of the U(1) dilemma 6. Fictitmur symmetry 7. The "exactly conserved chiral charge" in a canonically quanthcd theory

360 362 361 370 373 375

8. Diagrammatic interpretation of the anomalous Ward identities 9. Conclusion Appendix A. The sign of A& Appendix E. ?he integration over cha instanton size p References

381 382 382 383 306

377

0 Elsevier Science Publishers BY.(North-Holland Physics Publishing Division)

323

C. $1 HooJt. How

insfanfons solve fhe U ( I ) problem

1. Introduction

In addition to the usual hadronic symmetries the Lagrangian of the prevalent theory of the strong interactions, quantum chromodynamics (QCD) shows a chiral U( 1) symmetry which is not realized, or at least badly broken, in the real world 111. Now although it was soon established that the corresponding current conservation law is formally violated by quantum effects due to the AdlerBell-Jackiw anomaly [2] it was for some time a mystery how effective U(l) violating interactions could take place to realize this violation, in particular because a less trivial variant of chiral U(l) symmetry still seemed to exist. Indeed, all perturbative calculations showed a persistence of the U(l) invariance. With the discovery of instantons [3], and the form the Adler-Bell-Jackiw anomaly takes in these nonperturbative field configurations, this so-called U(l) problem was resolved [4,5]. It was now clear how entire units of axial U(l) charge could appear or disappear into the vacuum without the need of (nearly) massless Goldstone bosons. In a world without instantons the q and q’ particles would play the role of Goldstone bosons. Now the instantons provide them with an anomalous contribution to their masses. In the view of most theorists the above arguments neatly explain why the q particle is considerably heavier than the pions (one must compare m: with mf), and q’ much heavier than the kaons. Not everyone shares this opinion. In particular Crewther [6] argues that Ward identities can be written down whose solutions would still require either massless Goldstone bosons or gauge field configurations with fractional winding numbers, whereas experimental evidence denies the first and index theorems in QCD disfavor the second. If he were right then QCD would seem to be in serious trouble. In a recent review article this dissident point of view was defended [7]. The way in which it refers to the present author’s work calls for a reaction. At first sight the disagreement seems to be very deep. For instance the sign of the axial charge violation by the instanton is disputed; there are serious disagreements on the form of the effective interaction due to instantons, and the Crewther school insists that chiral U(l) is only spontaneously broken whereas we prefer to call this breaking an explicit one. Now as it turns out after closer study and discussions [8], much of the disagreement (but not all) can be traced back to linguistics and definitions. The aim of this paper is to demonstrate that using quite reasonable definitions of what a “symmetry” is supposed to mean for a theory, the “standard view” is absolutely correct; chiral U(l) is explicitly broken by instantons, and the sign of AQ, is as given by the anomaly equation; the effective Lagrangian due to. instantons can be chosen to be local and polynomial in the mesonic fields, and the q and q’ acquire masses due to instantons with integer winding numbers. In order to make it clear to the reader what we are talking about we first consider a simple model (section 2) which we claim to be the relevant effective theory for the mesons in QCD (even though it is dismissed by ref. [7]). The model shows what the symmetry structure of the vacuum is and how the q particles obtain their masses. It also exhibits a curious periodicity structure with respect to the instanton 0 angle, which was also noted in [7] but could not be cast in an easy language because of their refusal to consider models of this sort. Now does the model of section 2 reflect the symmetry properties of QCD properly? Since refs. [6,7,8] express doubt in this respect we show in section 3 how it reflects the exactly defined operators and Green’s functions of the exact theory. The calculation of the q-mass is redone, but now in terms of QCD parameters. We clearly do not pretend to “solve” QCD, so certain assumptions have to be

324

G. ‘ I Hoof!, How

inslclnfons solvr rhr

U(I)problrm

made. The most important of these, quite consistent with all we know about QCD and the real world, is ( q q ) = F # 0.

(1.1)

Now refs. [6,7] claim that in that case one needs gauge field configurations with fractional winding numbers. Our model calculations will clearly show that this is not the case. How can it be then if our calculations appear to be theoretically sound and in close agreement with experiment, that they seem to contradict so-called “anomalous Ward identities”? To answer this question we were faced with unraveling some problems of communication. The current-algebraic methods of refs. [6,7] were maiuly developed before the rise of gauge theories but are subsequently applied to gauge-noninvariant sectors of Hilbert space. Their language is quite different from the one used in many papers on gauge theories [4] and due to incorrect “translations” several results of the present author were misquoted in 171. After making the necessary corrections we try to analyze, in our own language, where the problem lies. There are two classes of identities that one can write down for Green functions. One is the class of identities that follow from exactly preserved global or local symmetries. Local symmetries must always be exact symmetries. From those symmetries we get Ward identities (91 (in the Abelian case), or more generally Ward-Slavnov-Taylor [lo] identities, which also follow from the (exact) BecchiRouet-Stora [ll] global invariance. On the other hand we have identities which follow from applying field transformations which may have the form of gauge transformations but which do not leave the Lagrangian (or, more precisely, the entire theory) invariant. These transformations are sometimes called Bell-Treiman transformations [12] in the literature, but “Veltman transformations” would be more appropriate [131. The identities one gets reflect to some extent the dynamics of the theory and form a subclass of its Dyson-Schwinger equations. Thus when refs. [6,7] perform chiral rotations of the fermionic fields, which do not leave 0 invariant, they are performing a Veltman transformation, and their so-called “anomalous Ward identities” fall in this second class of equations among Green functions. It is in the second derivative with respect to 0 that refs. [6,7] claim to get contradiction with our model calculations [8]: the second derivative of an insertion of the form iOFF,

(1.2)

in the Lagrangian vanishes, whereas the simple “effective field theory” requires an insertion of the form errdet( qLqR)t h.c. ,

(1.3)

of which the second derivative produces the q mass. So they claim that in the real theory the q mass cannot be. explained that simply. In section 5 we explain why, in the modem formalism this problem does not arise at all. (1.2) should not be confused with ordinary Lagrange insertions and after resummation correctly reproduces (1.3). The canonical methods are not allowed if one tries first to quantize the gauge fields A,, and only afterwards the fermion fields. The fermions have to be quantized and integrated out first. The phenomenon of “variable numbers of canonical fermionic variables” resolves the dilemma. We conclude, that the q mass is what it should be and there is no U(l) problem.

325

G.‘f HooJf,How

inslonrons solve the

U(I ) problem

There is a further linguistic disagreement on whether the U(1) breaking of QCD should be called an explicit or a spontaneous symmetry breaking. In the effective Lagrangian model the symmetry is clearly broken explicitly. Several authors however refer to the 0 angle as a property of the vacuum [5] in QCD. Of course as long as one agrees on the physical effects one is free to use whatever terminology seems appropriate. We merely point out that all physical consequences of the instanton effects in QCD (in particular the absence of a physical Goldstone boson) coincide with the ones of an explicit symmetry breaking. Our 0 angle is as much a constant of Nature as any other physical parameter, to be compared for instance with the electron mass term which breaks the electron’s chiral invariance. Only if one adds nonphysical sectors to Hilbert space one may obtain an alternative description of the 0 angle as a parameter induced by boundary effects producing a spontaneous symmetry breakdown. However, any explicit symmetry breaking can be turned into “spontaneous” symmetry breaking by artificially enlarging the Hilbert space. One gets no physically observable Goldstone boson however, so, the most convenient place to draw the dividing line between spontaneous and explicit global symmetry breaking is between the presence or absence of a Goldstone boson. We explain this situation in section 6, where we show that any nonphysical symmetry can be forced upon a theory this way. In section 7 we show how the spurious U(1) symmetry that is used as a starting point in [7],actually belongs to this class. Section 8 shows that if one takes into account the enlarged Hilbert space with the variable 0 angles then the effective theory of section 2 neatly obeys the so-called anomalous Ward identities. The effective model shows the vacuum structure so clearly that all problems with “fractional winding numbers” are removed. The decay amplitude q-3n posed problems similar to that of the q mass. As explained correctly in [7] this problem is resolved as soon as the U(1) breaking is understood so no further discussion of this decay is necessary. It fits well with theory. Appendix A is a comment concerning the sign of axial charge violation under various boundary conditions. Appendix B discusses the instanton-induced amplitude. The contribution from “small” instantons can be computed precisely but the infrared cutoff is uncertain. (Rough) Estimates of the amplitude in a simple color SU(2) theory give quite large values, which confirms that instantons may affect the symmetry structure of QCD sufficiently strongly such as to explain the known features of hadrons. 2. A simple model

Before really touching upon some of the more subtle aspects of the “U(1) problem” we first construct a simple “effective Lagrangian” model. Whether or not this model truly reflects the symmetry properties of QCD (which we do claim to be the case) is left to be discussed in the following sections. For simplicity the model of this section will be discussed only in the treeapproximation. To be explicit we take the number of quark flavors to be two. Generalization towards any numbers of flavors (two and three are the most relevant numbers to be compared with the situation in the real world) will be completely straightforward at all stages in this section;. So we start with the “unbroken model” having global invariance of the form ‘See however the remark following eq. (2.24).

326

G. ‘ I Hwff, How

U(t),@U(L), ,

insrancorn solve h e U( 1 ) problem

with L = 2 ,

(2.1)

where the subscripts L and R refer to left and right, respectively. We consider complex meson fields with the quantum numbers of the quark-antiquark composite operator iRiqLi.They transform under (2.1) as:

4,; = u;4&’

(2.2)

I

to be written simply as

4’ = U L 4 U R ’ .

(2.3)

Since we have no hermiticity condition on 4, there are eight physical particles a, q, n, and a, (a=1,2,3):

4 = i ( a + is) + 4 (a+ i n ) * ‘I,

(2.4)

where 7’s’*3are the Pauli matrices. We take as our Lagrangian:

2’= -Tr dp&’p4t

- V(4).

A potential V, invariant under (2.1) is

VO(4)=-p2Tr&$’t i ( A l -A,)(Tr&$’)’+ ~A,Tr(&$’)’ 2

= - k (a2+q2

+ a2t a2)+1 A (a2+ q2 t a’ + a2)2 t4 ((aa+qn)’+ (aA a)’), 8

2

2

(2.7)

Assuming, as usual (a)=f,

(2.8)

a=fts,

we get, by taking the extremum of (2.7),

f’= 2p2/A,;

v, =

A

A + d))’ t -2 + sa t qa)’+ (aA a)’), 2 ((fa

(fs t t(s2 + q2 t a2

(2.10)

from which we read off m:=Alf2=2p2,

m:=O,

mf=A,f’,

m$=O.

(2.11)

There are four Goldstone bosons, as expected from the U ( 2 ) @ U ( 2 )invariance, broken down to U(2) by (2.8).

327

G.' I Hwff, How iiuracuons d u e the 4 1 ) probltm

We now consider two less symmetric additional terms in V:

v, = urnt u:; U, = '' m eix&q,

"

=

f meixTr 4 = im eix(ut ir)) ,

(2.12)

and:

v, = u* t u: ; U,= " K eie det( &qL)

"

= K eiedet I$

(2.13)

= K ei'((u t iq)' - (a t in)2).

Here, m, x, K and 0 are all free parameters. The terms between quotation marks are there just to show the algebraic structure, up to renormalization constants. Notice that V, still has the U(l) invariance

(2.14)

Therefore we are free to rotate

x-+x+o,

e+e+20.

(2.15)

(Here the 2 would be repiaced by L in a t h e o j with L flavors.) Consider first the theory with K = 0;

V=V,+V,.

(2.16)

Then we can choose o = n - x, and

V, So

= -mu.

(2.17)

x is unphysical. Equation (2.9)is replaced by f2

(2.18)

= Z~'IA, t 2 m 1 ~ , f .

Consequently, in (2.11) we get

m,1 = m i = mlf.

(2.19)

Note that with the sign choice of (2.17), f must be the positive solution of (2.18).

328

G. ‘ I Hoofr. How insraniom ~olvcrhc U(1)problcm

Now take the other case, namely m = 0;

v = v, t v, .

(2.20)

It is now convenient to choose

- e) .

w = f(m

So here the angle

(2.21)

e is unphysical.

v, = -2K(cr* + Trz - Q 2 - a’). f = 2p21h, + ~ K M., The masses of the light particles become

So the

K term contributes directly to the q mass and not to the pion mass. In case of more than two flavors the determinant in (2.13) will contain higher powers of 4, and extra factors f will occur in (2.24). But the mechanism generating a mass for the q’ will not really be different from that of the q. It is interesting to study the case when both m and K are unequal to zero. We then cannot rotate both x and 0 independently and one of the two angles is physical. Since obviously the model Lagrangian is periodic with period 277 in 0, its physical consequences will be periodic in x with period T , because of the invariance (2.15). In the general case we now also expect a nonvanishing value for

(2.25)

(v)=g. So we write 9 = g i h, where g is a c-number and h a field. The conditions for f and g are:

It will be convenient however to choose o in (2.15) such that g = 0. According to (2.26) one then must have msin x

+ 4 ~ sin f 0=Of

(2.23

(2.W (2.29)

329

G

'I Hoof(. How

imranfonssolvc fhr U( I

) problem

apart from an 9-s mixing. The effect of this mixing however goes proportional to 16x2sin' 0 = m' sin' y,

and therefore is of higher order both in Consider now the function

(2.30) K

and m.

F(w)= rn cos(,y t 0) t 2Kf cos(0 t 20).

(2.31)

Then (2.27) requires o to be a solution of F'(O) = 0

(2.32)

(after which we insert the replacement (2.15)), and (2.29) implies: fmt = F"(o)> 0 .

So,

(2.33)

must be chosen such that (2.31) takes its minimum value. Furthermore, the replacement ?r switches the sign of the first but not of the second term in F(o),so, if o is chosen to be the absolute minimum of (2.31), then indeed also o

w--, o t

(2.34)

except when there are two minima. In that case there is a phase transition with long range order (due to the massless pion) at the transition point. This phase transition is at x=n12,

sin 8 = - m / 4 ~ f , cose
(2.35)

or, if a rotation (2.15) is performed:

x=O, sin B = m/4xf, o< 8 < n12.

(2.36)

A further critical point may occur at

x=o e = r12

(2.37)

m=4Kf

330

G. ‘I Hooft. How

i n ~ l M fJOIVC ~ t ~ hc

U(1) problcm

where also m, vanishes. Of course these values of the parameters are not believed to be close to the ones describing real mesons. Phase transitions of this sort were indeed also described in ref. [7]. But because no specific model such as ours was considered, the periodicity structure in x and 0 was discussed in a somewhat untransparent way. Although obviously in our model we have a periodicity . if o in (2.15) shifts by an amount T , then in 0 with period 2 r , ref. [7] suspected a period 4 ~ Indeed m eiY- - m eiy and one might end up in an unstable analytic extension of the theory. Clearly the properties of the minimum of the potential F(o),eq. (2.31), can only have period 2~ in 0.

3. QCD If Quantum Chromodynamics with L quark flavors, were to have an approximate U(L), X U(L); symmetry only broken by quark mass terms, the model of the previous section with V = V, + V, could then conveniently describe the qualitative features of mesons, but with q and n almost degenerate. (In the case L > 2 one merely has to substitute the 2 x 2 matrix 4 by an L X L matrix.) That would be the case if somehow the effects of instantons could be suppressed. Let us now consider instantons and write in a shorthand notation the functional integral I for a certain mesonic amplitude in QCD. For the ease of the discussion we assume all integrations to be in Euclidean space-time:

AS we will see shortly, it is crucial that the integration over $ is done first and the one over A afterwards. Since we have no way of solving the theory exactly certain simplifying assumptions must be made. We now claim that the assumptions to be formulated next will in no way interfere with the known symmetry properties of the low-energy theory. Discussion of this claim will be postponed to sections 4-7. An (anti-)instanton is a field configuration of the A fields with the property (3.6)

33 1

G. ’t Hoofr, How

iwrmrotu solve fhr

U(I)problem

where AV is the volume of a space-time region. Outside AV we have essentially [FFvI= O but we cannot have [ A [ =0 there because then (3.6) would vanish as the integrand is a total derivative. The assumption we make is that the A integral can be split into an integral over instanton-locations and an integral over perturbative fluctuations around those instantons. We do this as follows. Let US divide space-time into four-dimensional boxes with volumes AV of the order of l(fm)‘. Each box may or may not contain one instanton or one anti-instanton. (There could be more than one instanton or anti-instanton in a single box, but we choose our boxes so small that such multi-instantons in one box become statistically insignificant.) The essential point is that since an instanton in a box AV will do nothing but gauge-rptate any of the fields outside AV, the instanton-numbers in each box are independent variables. Notice that at this point we do not require these twisted field configurations in the boxes to be exact solutions to the classical field configurations. This is why we have no difficulties confining each instanton to be completely inside one box, with only gauge rotations of the vacuum outside. Let us then write A = Ai,,%, +- SA

(3.7)

where Ainrlis due to the instantons only, then the integral over 6A will essentially commute with the integral over the instantons. The 6A integral is assumed to be responsible for the strong binding between the quarks. The confinement problem is not solved this way but is not relevant here since we decided to concentrate on low-energy phenomena only. Note that the integration over Aim,, is more than a summation over total winding number v. Rather, if we write v = u+ - v _

(3.8)

then the integral over A,,,, closely corresponds to integration over the locations of the v+ instantons and the v- anti-instantons. It is important that we restrict ourselves to instantons with compact support (namely, limited to the confines of the box AV in which they belong). A larger instanton, if it occurs, should be represented as a small one in one of the boxes, with in addition a tail that is taken care of by integration over SA. “Very large” instantons are irrelevant because they would be superimposed by small ones. In short, in eq. (3.71,Aim,,is defined to be a smooth field configura!ion that accounts for all winding numbers inside the boxes, and SA is defined to contribute to FF by less than one unit in each box. Now consider an isolated instanton located within one of our boxes, located at x = x , . What is discussed at length in the literature is the fact that the integration is now affected by the presence of a zero mode solution of the Dirac equation. If there were no other anti-instantons and no source term J then the fermionic integral, being proportional to the determinant of the operator yp(8,, t igA,), would vanish because of this one zero eigenvalue. If we do add the source t e h J$q5 the integral need not vanish. In ref. [4] it was derived that the instanton exactly acts as if it would contain a source for every fermionic flavor. Thus with one instanton located at x = x , the fermionic integral

+

has the same e#ed as the integral

332

G. ‘I Hoof(, How instanlorn solve rhr U(I ) problem

where K may be computed from all one-loop corrections [4].Indeed it was shown that the zero eigenmodes for all flavors which extend beyond the volume AV conveniently reproduce the fermionic propagators connecting xl with the sources J. The fact that (3.9) does have the same quantum selection properties as (3.10)can also be argued by realizing that a gauge-invariant regulator for the fermions had to be introduced, and instead of the lowest eigenmodes one could have concentrated on the much more localized highest fermionic states. The correctly regularized fermionic integral contains a mismatch by one unit for each flavor between the total number of left handed and right ha_nded fermionic degrees of freedom. Since this happens both for the fermions and the antifennions $ the determinant in (3.10)consists of products of L fermionic and L antifermionic fields. Since we do require that the SU(L), BSU(L), is kept unharmed by the instantons, the determinant is at first sight the only allowed choice for (3.10)but, actually, if one does not suppress the color and spin indices, one can write down more expressions with the required symmetry properties. Next consider u+ instantons, located at x = x i . Following a declustering assumption which, at least to the present author’s taste, is quite natural and does not require much discussion, we may assume these to act on the fermionic integrations as K“+

1W D$

”+ (3.11)

es+ndet(+L(xi) &(xi)). i- 1

Let us add the 8 dependence and integrate over the instanton locations xi:

(3.12) The denominator v+! is due to exchange symmetry of the instantons. We now extend our declustering assumption to the anti-instantons as well. This assumption was vigorously attacked in [6-81. Indeed one might criticize it, for instance by suggesting that “merons” play a more Crucial role [14].We insist however that the assumption in no way interferes with the symmetry properties of our model. We will see in sections 7 and 8 that the anomalous Ward identities will be exactly satisfied by our model. To avoid confusion let us also stress that our declustering assumptions refer to the QCD part of the metric only, not to the contributions of the fermions which we denote explicitly. So there is no disagreement at all with the findings of ref. [15]. Indeed, our approach here is closely analogous to theirs. Thus, consider u- anti-instantons. The complete instanton contribution to the functional integral is

(3.13) The summations are now easy to carry out:

which is precisely the effective interaction V, of eq. (2.13).The remaining integrals over the fermionic

333

G. 7

Hoofl, How inrronfonr solve fhc U ( 1 ) problem

fields J, and the perturbative fields SA may well result (n the effective Lagrangian model of section 2. Notice that, before we interchanged the Ainr,and $, $ integrations, we have made the substitution (3.10). This will be crucial for our later discussions. Once the substitution (3.10) has been made, the (A-field-dependent)extra fermionic degrees of freedom have been taken care of, and only then one is allowed to interchange the A and the J, integrations. This is how (2.13) follows from (3.14). 4. Symmetries and currents

Let us split the generators A, and A , for the U(L), X U(L), transformations into scalar ones, A' and A', and pseudoscalar ones, A: and A:, The infinitesimal transformation rules for the various fields considered thus far are:

In a theory with L flavors the factor 4 in eq. (4.8) must be replaced by 2L. In a classical field theory the currents are most easily obtained by considering transformations (4.1)-(4.8) with space-time dependent A,(x). Their effect on the total action can be written as SS = d4x(- F,A,(x) - JLJ,A,(x))

(4.9)

(here i = 1,. . . ,8). Since according to the equatio'ns of motion 6s = 0 for all choices of A,(x), one has

a,Jl(x) = F,(x) .

(4.10)

A Lagrangian which gives invariance under the space-time independent A, must have F, = 0, so that the current JL is conserved. We are now mainly concerned about the current JF5 associated with A!. The QCD Lagrangian (3.1) produces the current

334

G. ’ I Hoo/r, How Jp5

= i&,*

inrlenlonr m1w the

U(I ) problem

(4.11)

9

and, prior to quantization:

a, J,,,

=I

2irn&&.

(4.12)

As is well known, however, eq. (4.12) does not survive renormalization. Renormalization cannot be performed in a chirally invariant way and therefore the symmetry cannot be maintained, unless we would be prepared to violate the local color gauge-invariance. But violation of color gauge invariance would cause violation of unitarity, so, in a correctly quantized theory, (4.12) breaks down. A diagrammatic analysis [2] shows that, at least to all orders of the perturbation expansion, one gets

(4.13) with

e“

.

= bpy.gF:g

(4.14)

We read off that, if we may ignore the mass term, then in a space-time volume V with and u- anti-instantons

J d‘r d,, JF5 = -2iL(v+ - v-) .

u+

instantons

(4.15)

V

Here the factor i is an artefact of Euclidean space. Defining the charge Q, in a 3-volume V, by

Q,

=[ J,, d3x=i “3

I

J4d3x=Q, - QL,

(4.16)

“3

each instanton causes a transition*

AQ, = 2 L .

(4.17)

This is called the “naive” equation in ref. [7].Since we were working in a finite space-time volume V the nature of the “vacuum” has not yet entered into the discussion. Remarks on the language used here and in ref. (71 are postponed to appendix A. Now let us write the corresponding equation in our effective Lagrangian model. Here, J,,,

=2iTrW,,4*)4 - 4*3,,41

(4.18)

and * Apct fmm the dispused sign thext aic also diffetences in sign mmentiona with ref. 171.

335

G. ’I Hooff. How

inrfanrons rolvc fhc

U(I)problem

d P ~ P , , = - 2 m ( ~ c o s ~ + u s i n ~ ) + 1 6 ~ ( ~ ~ a - ~ ) c o s 6 + 8 ~ ( ~ Z + ~ 2 (4.19) -uZ-a*)sin

Before comparing this with eq. (4.13) of the QCD theory let us chirally rotate over an angle i x . The mass term in the original Lagrangian then becomes -$m$ cos y, - i$rny,S sin ,y

,

(4.20)

and then (4.13) becomes dPJP, = 2im$-y5$cos x

iLgz - 2m& sin x - 7 F;”F:”. 1 6 ~

(4.21)

Therefore the first term in (4.19) can neatly be matched with the first terms of (4.21). An issue raised in refs. [7,8] is that there is an apparent discrepancy if we try to identify the last terms of (4.19) with the last term of (4.21). The last term of (4.21) contains the color fields only and there is absolutely no 8 dependence here. But the last terms of (4.19) do show a crucial 0 dependence. It is essentially 8K

h ( e “ det 4) .

(4.22)

Where did the 8 dependence come from? One way of arguing would be that the 8 dependence of (4.22) is obvious. Chiral transformations are described by eq. (2.15) and any symmetry breaking term in a Lagrangian can obviously not be invariant at the same time. So the 8 dependence of (4.22) is as it has to be. The symmetry breaking in QCD is not visible in its Lagrangian but is due to the 8 dependence of the regularization procedure. However, although this argument may explain why (4.19) shows a 8 dependence and (4.21) does not, it does not explain why nevertheless these two theories can describe the same system. This is (partly) what the dispute is about. We claim that one can identify in the effective theory (4.23)

so that, if 8 = 0, one may identify Ffi with the 17 field. At the same time we would also like to put

6 = 4RRqL ,

(4.24)

but this 8 phase seems to be in disagreement with the canonical quantization procedure if A:, qR, qL and Q were to be considered as independent canonical variables. Another way of formulating this problem is that the right hand side of (4.23) seems to commute with the chiral charge operator Q, while the left hand side does not. Notice that if we could somehow suppress instantons essentially F p would vanish. The left hand side of (4.23) would vanish also, because K+O. This suggests one simple answer to our problem: equation (4.23) violates axial charge conservation, but that is to be expected in a theory where axial charge is not conserved. Unfortunately some physicists insist in considering the U(l)

336

G. ‘I Hoof, How inrcpnronr solve the U(1)problrm

violation by instantons as being “spontaneous” rather than explicit and therefore they rejected this simple answer. A rather curious attempt to bypass the problem was described in [7].They first propose to replace our V, of eq. (2.13) by

V: = K Tr(log(Cpl4’))’ .

(4.25)

But this also does not commute with the axial charge operator and furthermore the logarithm is not single-valued so (4.25) makes no sense at all. So then they propose (4.26)

It is not obvious how this expression should be read such that it does make sense. If it is equivalent to (4.25) then clearly no improvement has been achieved. The problem of a multivalued logarithm has merely been substituted by the problem of an infrared divergent integral in x space. Equation (4.26) is then a clear example of linguistical gymnastics that should be avoided: formally it appears to be chirally invariant, yet it is equivalent to the local term (4.29, which is not. We conclude in this section that the aforementioned problem is not solved by the logarithmic potentials V : of eqs. (4.29, (4.26). Let us call this problem the “U(1) dilemma”. The correct resolution of the U(l) dilemma will be given in the next sections. 5. Solution of the U(1) dikmma

We must keep in mind how and why an effective Lagrangian is constructed. The word “effective” is meant to imply that such a model is not intended to describe the system in all circumstances. Rather, the model gives a simplified treatment of the system in a given range of energies and momenta. In this case we are interested in energies and momenta lower than, say, 1GeV. Now the cQmplete theory contains variables at much higher frequencies. In as far as they play a role at lower energies, we must assume that they have been taken care of in the effective model. Consequently, the simple identification (4.24) is not correct as it stands. It should be read as

6= (

(5.1)

4RRqL)lOW lreepu.dcr ’

But what does “low frequency” mean? In a gauge theory the concept “frequency” need not be gauge-invariant. Therefore the splitting between high frequency and low frequency components of the quark field#must depend in general on the gluonic fields A,. This is why the contribution of the high frequency components of the quark fields to the axial current JP5 may depend explicitly on the A fields, a fact that is correctly expressed by the so-called “anomalous commutators” of [6,7].After integrating out the high frequency modes of the quark fields, but before integrating out the A fields, we have an expression for the axial current which has the following form: J,,,=2iTrWPCp*) 4 - Cp*~,4)+ JLAA).

(5.2)

It is &(A) which is responsible for the nontri@al axial charge of the quantity FF in (4.23). Let Q;be the charge corresponding to JLs. How does FF commute with Q;?

337

G. ' 1 H w f t , How inntantom solve the U(1)problem

Rather than FF itself, it is the integral over some space-time volume AV,

\ F F = .f V i " , -

(5.3)

AV

that is relevant in (4.13). (We use the short hand notation of eq. (3.4).) Let us take AV so small that

J FF=O

or tl

(5.4)

AV

(i) If FF = 0 then we are not interested in its quantum numbers. (ii) Whenever the right hand side of (5.3) is tl we have an amplitude in which k2L units of axial charge are created. (iii) The higher values of the right hand side are negligible. The creation or-annihilation of axial charges occurs because of the extra high frequency modes of or IGL and JIR that make the functional integral non-invariant. Let us call their contribution 2. JL, If vf = 1 then

where the subscripts under the multiplication symbols denote the numbers of variables to be integrated over. Then if

we have for all integrals over the anticommuting fields:

(5.7)

so that Z+e-Zi"L

Z.

(5.8)

In this discussion we only include the high frequency components of the JI fields. We see two things: the effective interaction 2 due to an instanton trapsforms exactly as our insertion U, of eq. (2.13), and-secondly that, in a simplified picture where FF takes integer values only (eq. (5.3)), the quantity FF, after inregration over the high frequency fermionic modes, transforms with a factor

(5.9)

e+2iuL,

338

G. ’I Hwff. How instomons solve

the

U(1)problem

so that there is no longer any conflict. with (4.23). The transformation rules (4.1-4.8) hold for the

effective fields. The terms containing A: tell us how the various fields commute with Q,:

(5.10) (5.11) (5.12) (5.13)

6. Fictitious symmetry

The chiral U(1) symmetry breaking in QCD is an explicit one because the functional measure Il D$ fails to be chirally invariant when regularized in a gauge-invariant way [21]. This neatly explains why no massless Goldstone bosons are associated with this symmetry. Yet in several treatizes the words “spontaneous symmetry breaking” are used. How can this be? Any broken global symmetry can formally be considered as a “spontaneously” broken one by a procedure consisting of two steps. (i) Enlarge the physically accessible Hilbert space by adding all those Hilbert spaces of systems that would be obtained by applying the phoney symmetry transformation: Q’=@xS

(6.1)

where @ is the original Hilbert space and S the space of physical constants describing symmetry breaking. (ii) Define the symmetry operator(s) as acting both in S and in 18. We then obtain transformations in 4’ that obviously leave the Hamiltonian H invariant. This procedure allows one to write down Ward identities for theories with symmetries broken explicitly by one or more terms in the Lagrangian. Since such identities were excessively used and advocated by Veltman in his early work on gauge theories with mass-insertions, we propose to refer to the above transformations as Veltman transformations [12]. Consider for example quantum electrodynamics with electron mass term

-m*&.,lbk

-

m6Rd$,

(6.2)

’

Then S is the space of complex numbers m. In this theory then, m is promoted to be an operator rather than a c-number. The chiral transformation

For the factor el’ see scction 7

339

G. ‘t Hooft. How

instantons solve the U ( J ) problem

is obviously an invariance of this theory. If in the “physical world” (m)=m=reaI then one could argue that the symmetry (6.3) is “spontaneously broken”. The canonical charge operator Q associated with (6.3) is now

which commutes with (6.2). Thus, Q is exactly conserved. But, since m is not a dynamical field, the new term cannot be written as an integral over 3-space, unless we enlarge the Hilbert space once again. Let us now consider a Feynman diagram in which the mass term (6.2) occurs perturbatively as a two-prong vertex. Let there be a diagram with v + insertions of the last term in (6.2), going with m, and v- of the first term, going with m * . We have

Only by brute force one could produce a current of which the fourth component would give a charge satisfying (6.6): j,, = J,,,

+ K,,

L,‘

_____

,,’

,I R t

Lt\. ”

__-- --

tl

v.

Fig. 1. Propagating ekctron (solid line) with mass terms. The propagators are expanded in m. yielding artificial panicles (sehhru, dotted lines) that any away two units of axial charge. but no energy-momentum. Total chiral charge Q, is conserved. Here Q,(f,)= Q,(t2)= 1.

340

G.’1 Hoojr. How insranrons solve rhc U(I ) probleni

Clearly, K, is not locally observtble. There is a nonobservable “Goldstone ghost” (the pole of D’).It goes without saying that Q, although exactly conserved, and J,, are not very useful for and the charge Qk as used in ref. [7] are precisely of this canonical formalism. Yet the current Ji5,rym form. This will be explained in the next section. A neat way to implement the symmetry (6.3) is to treat the parameter m as a field: the “schizon”, or “spurion”. as those auxiliary objects are sometimes called to describe explicit symmetry breaking, such as isospin breaking by electromagnetism. The schizon field has a nonvanishing vacuum expectation value (6.4). Diagrammatically, a propagating electron could be represented by a diagram (fig. 1). Defining Q = 2 2 for the schizons we see that Q is absolutely conserved. Of course Q is also “spontaneously broken”. 7. The “exactly conserved chiral charge” in a canonically quantized theory

The fictitious symmetry described in the previous section can be mimicked in a gauge theory in a way that looks very real. Consider instead of (4.11), the current J,,.,,,

=4

5

+ K,

Then, in the limit m+O, one has ap J,S.aym

=0 .

(7.2)

The corresponding charge, (7.3)

generates “exact” chiral transformations. How does this operator act in Hilbert space? To answer this question we must formulate the canonical quantization of the gluon field carefully. Conceptually the most transparent way is to first choose the temporal gauge: (7.4)

A,=O,

which leaves us formally the set of all states (A@), Jl(x), G(x)) at a given time t, where Jl and $ should be seen as Grassmann numbers. Let us call the Hilbert space spanned by all these states the “huge” Hilbert space. Then (7.4) leaves us invariance under all time-independent gauge transformations R = O(x), so that the Hamiltonian in this space is invariant under a group G composed of gauge transformations R(x) that may vary from point to point. This generates an invariance at each x, according to Noether’s theorem. Writing

341

G.‘ I hoof^. How

insronronr solve rhe

U(I ) problem

RIA, 4, 6)= I A ” J I ” ~ ” )

(7.5)

where the subscript f j indicates how the fields are gauge-transformed, we have

[H,R]=O. We can impose the gauge conditions of the second type:

nip) =IT)

I

for ail infinitesimal R, acting nontrivially only in a finite region of 3-space: AC(x) = A(x) t D,A(x) ,

A infinitesimal, and with compact support. States [ Y )satisfying (7.7) are said to be in the “large” Hilbert space (which is not as large as the “huge” one). Finally, we consider all 0 with nontrivial winding number v

R,lT)

= eiB’lY)

.

(7.8)

These states IT) are said to constitute the small, or physical Hilbert space at given 8. Now notice that JpS,rym does not commute with 0: [Jp5.rym’

01 =

iLgz

Dv’

(7.9)

~puap‘mp

Therefore, J,,,sy, cannot be considered to be an operator for states in the ‘‘large’’ Hilbert space. Acting on a state satisfying (7.7) it produces a state not satisfying (7.7). Now the charge Q5,sym of eq. (7.3) does commute with all R with v 0, but not with the others: 7

[Qs.sym*

41 =2iLvR, .

(7.10)

Therefore, QJ,rymdoes act as an operator in the large Hilbert space, but not in the physical Hilbert space, because it mixes different 0 values. We can write

(7.11)

6

We see that in every respect Q5,sym behaves as of the previous section, and JlrS,sym as j . A Goldstone boson would emerge in the theory if, besides the states satisfying (7.8), it could be possible to construct physical states in which 8 would depend on space-time: 7

e = e(x, .

(7.12)

Here it is obvious that (7.12) would be in contradiction with (7.7) and (7.8): If we would compare

342

G. ‘I Hooff, How insfuntons solve fhe U(J) problem

different 0“but with the same u, such that the support of that of d2) near x ( * ) , then the combination

a(’)would be in a region near d’),and

0l“’~‘’’-’

would have winding number zero. So the second gauge constraint would exclude any states for which fI(x(’))#fJ(x(’)). This is an important contrast with systems such as a ferromagnet, where local fluctuations are allowed, which, because of their large correlation lengths, correspond to massless excitations. Because of the similarity between (7.11)and (6.5) we can consider eie as a “schizon” field just as the electron mass term. Since 0 cannot have any space-time dependence this schizon field cannot carry away any energy or momentum, just as m in the previous section. Although Q5,1ym does not act in the “physical” Hilbert space, it is possible to write Ward identities [6,7] due to its formal conservation,

a,K, = --g2i F i Y q Y . 321~’

(7.15)

One can take

op = KAO) >

(7.16)

and assume

K1 = O

(7.17)

while putting D,+O.

(7.18)

[Qs,sym*

Now (7.17)is not obvious. Substituting (7.15)gives

On the other hand we showed in section 5 that F& has nontrivial chiral transformation properties. This however corresponds to

343

G. ‘ I Hooff, How iwlanlow so/w chc U ( / ) problem

Indeed,

where the right hand side follows from the substitution (4.23) and the commutation rule (5.12). C is a constant and f the expectation value. Equation (7.19) is a fundamental starting point of the discussions in refs. [6-81. The difference Now KO between (7.19) and (7.20) must apparently be made up by the contribution of KO to Q,,,,,. is not a physically observable field. Assigning to it the conventional commutation rules to be deduced from its composition in terms of color gauge fields is only allowed if one works in the “huge” Hilbert space including the gauge noninvariant states. into our model of All we have to do to incorporate the fictitious symmetry generated by Q,,,,,,, effective fields described in section 2, is to add a schizon field, enlarging the Hilbert space. Let us call the schizon field

s = ei’ .

(7.22)

Our new identification is

- 128di FF = 7Im(S det 4) , Lg

(7.23)

--jKofO

--

IKo-O

S

WE.2. Effective instanton action and its Q, symmetry properties. (a) Due lo fermionic zero modes 2L units of Q, are absorbed at lhc aite of the instanton. In the same time the charge generated by KO is not mnscrvcd. (b) In the effective theory the fennions are replaced by he +&Id. and KO by a sehizon S. The 4 carry 2 units of Q, each, and S has -2L units. S has a nonvanishing vacuum expeaation vslue.

344

C.

'I

HOO~I, How inrm~onssolve (he U(1)problem

and if we postulate, in addition to (5.10)-(5.13) for [Qs.sym,

SI = -

2

9

QJ,sym

also (see 7.11): (7.24)

~

then with (5.12) we find that F@ commutes with QJ,sym. Substituting e" det 4 by S det I$ in (2.13) we see that indeed our effective field theory obeys the fictitious symmetry generated by QS,,,,,,. It must therefore also obey the so-called anomalous Ward identities. See fig. 2. 8. Diagrammatic Interpretation of the anomalous Ward identities

The conclusion of the previous sections is that the model of section 2, with the substitution

cis+ s

(8.1)

obeys all anomalous Ward identities. It also exhibits in a very transparent way how the symmetries are now spontaneously broken. There are two vacuum expectation values:

Both break Q,,,,, conservation. We can now draw Feynman diagrams in the Wigner representation by explicitly adding the vacuum bubbles due to (8.2) and (8.3). See fig. 3. By summing over the bubble insertions (geometric series which are trivial to sum), one reobtains the Goldstone representation of the particles. The a bubbles tend to make the pions and eta massless, but the terms with K (and the quark masses m) contribute linearly to m2 for the various mesons. These diagrams clearly visualize where the masses come from and how the QS,,ymcharges are absorbed into the vacuum, In ref. [7] an apparent problem was raised by their equations (4.27) and (4.28): they suggest the

9p @$ :,V

.

0

ua:

A

n

A

77;2iP.L t--

*.

#, ,

K

Fig. 3. Fcynmrn rules in the Wiper mode. Thc u blob, render the pion and the q massless. But the S blob g i v e n mass to q. HCI we drew explicitly che 1' propagator (L= 3). Its mass comes from the i n s d o n at the right. Note that mi. is linear in (S) and in (a): m:. rf.

345

G.

‘I

Hoofl. How insronrons solve the U ( 1 ) problem

need for field configurations with fractional winding number u, which would correspond with the breaking up of our S field into components with smaller Q,,sym charges. In our diagrams we clearly see that there is no such need. If we have a Green function with an operator that creates only two chiral charges, xop = 2

then the sigma field can absorb these two, or add 2(L - 1) more and have them absorbed by S. The vacuum simply isn’t an eigenstate of Q, (nor Q5.,ym)as it was assumed. 9. Conclusion

The disagreements between the approach of the Crewther’s school to the U(l) problem, using anomalous Ward identities, and the more standard beliefs are not as wide as they appear. Their anomalous Ward identities, if applied with appropriate care, are perfectly valid for a simple effective field theory that clearly exhibits the most likely vacuum structure of QCD. It is important however to realize that the relevant exactly conserved chiral charge Q,,,ymis not physically observable, something which explains the need for the introduction of a spurion field S in the effective theory. It appears that the consequences of working with an unphysical symmetry were underestimated in ref. (71.Some of the difficulties signalled in [7] were due to the too strong assumption that the vacuum is an eigenstate of Q,.That such assumptions are unnecessary and probably wrong would have been realized if they had taken the effective theory more seriously. The fact that the effective theory of section 2 displays the correct symmetry properties does not have to mean that it is accurate. Indeed it could be that instantons tend to split into “merons” [16], a dynamical property that might be a factor in the spontaneous chiral symmetry breaking mechanism [14]. But these aspects do not affect the symmetry transformation properties of the fields under consideration. More fields, describing higher resonances, could have to be added. The baryonic degrees of freedom are most likely to be considered as extended solutions of the effective field equations (skyrmions). That these pkyrmions I171 indeed possess the relevant baryonic quantum numbers was discovered by Witten [lS]. The author thanks R.J. Crewther for his patience in extensive discussions, even though no complete agreement was reached. Appendix A. The sign of AQ,

In ref. [7] the present author’s work was claimed to b e i n error a t various places. Although some minor technical corrections on the computed coefficients in t h e quantum corrections due t o instantons were found (in the original publication see ref. [20]) and even some insignificant inaccuracies in the notation of a sign might occur, we stress here that none of those claims of ref. [7] were justified. In particular there are no fundamental discrepancies i n the sign of AQs. Let us here ignore the masses of t h e quarks. As formulated i n section 4, an instanton in a finite space-time volume V causes a transition with

346

G. ' I Hooff. How irufcmfons solve fhe U( I probkm

where Q, is the gauge-invariant axial charge. Since QS,,,,,,,, as defined in section 7, is now strictly conserved one obviously has AQs.,ym = 0

(A21

which is of course only defined in the ''large'' Hilbert space comprising all 0 worlds. Instead of (Al-A2), we read in ref. [7]: AQ,=O;

AQs,lym=-2L.

('43)

These are not the properties of a closed space-time volume such as we described, but represent the features of a Green's function where the asymptotic states are 0-vacua. Since QS,,ym contains explicitly the operator d l d 0 in the ''large'' Hilbert space (cf. eq. (7.11)), the @-vacuumis not invariant under Q,,,ym. This is why (A3) is not in conflict with Ql,symconservation. But we also see that (Al-A2) and (A3) hold under different boundary conditions (the reason why AQs = O for Green's functions in a &vacuum is correctly explained in ref. (71).

Appendix B. The integration over the instanton size p To get even a rough estimate of the size of our instanton's contribution to an amplitude requires lengthy calculations. The most essential ingredients for such a calculation are provided in our previous paper (5.11 (see refs. I191 for the original publication, and the corrections given in [20]). Let us take for simplicity SU(2) as our color gauge group. An instanton with given size p then gives rise to an effective interaction of the form

where N f is the number of fermions in the doublet representation, p is the size of the instanton, p is an arbitrary mass unit enabling us to obtain a renormalization group invariant expression, and N " ( t )is the number of scalar field representations with color t. The parameter C was computed in the previous paper, and it also depends on the scalars and fermions present:

where A, A(t), and B are numbers computed in [5.1].

347

G. 'f Hooff, f f o w inr/antom solve thc U(f)problem

W = exp( -8?r2/g2)(det

det

det W8,,

and the determinants can be computed by diagonalization:

It is clear that (B3)is highly divergent unless we formulate very precisely an appropriate subtraction procedure. A convenient method is to apply first a variety of Pauli-Villars regularization to the fields Aq"and $ and then add correction terms to be obtained by comparing this regularization scheme to for instance dimensional regularization. It is then found that (B3)is not the complete answer: there are zero eigenvalues of W A ,n+ and a&,,which have to be considered separately. The corresponding eigenmodes must be replaced by collective coordinates including an appropriate Jacobian for this transformation. Since (B3) must be compared with the vacuum transition (in absence of instantons) the collective coordinate integration is to be divided by a norm factor determined by a Gaussian integral. The result is the effectiveLagrangian (for the case that the color gauge group is SU(2)):

where N' is the number of fermions in the doublet representation, p is a scalar parameter for the instanton, p is an arbitrary mass unit enabling us to obtain a renormalization group invariant expression, and N'(f) is the number of scalar field representations with color f . Defining coefficients a(f) as in table 1, we have now: A = - a ( l ) + Y ( l o g 4 n - y ) + f =7.053W103 A(!) = -a(t) + A(log4a - y)C(f) A( 112) = 0.308 690 69 A(1) = 1.09457662 A(3/2) = 2.481 356 10 B = -24112) + f(log4n- y ) = 0.35952290.

348

G. ’I Hoo11. How imlanlotu solve the U(I ) problem

Table 1 I

C(l)

4)

0

0 1 4 10

0

I /2 1

312

R=

1

2R - 1 log2 - 17/72 8R t I log2 - 1619 20Rt410g3- j l o g 2 - 2 6 5 1 3 6

(log2nt 7 ) t

1 = I

1

J

=0.2487544?’/.

1

Finally, the spinors o are normalized by

z @.k 1(1 =

-t-

Y,)

(B7)

and required to be smeared in color space, such that for instance

(@.q3)= f 4 J l

-t- YA

(B8)

and, in the case L = N‘ = 2:

(i

(&o)(Lj$,))=

1

24 (2SB,:6? - 8 p ~ ) & ’ ’ 4 ; v + Y5)$%?(1

+ n)dP

(B9)

8-1

where s and t are flavor indices and a,,pi color indices. The p integral may seem to diverge in most interesting cases (N‘>1). Note however that it would be natural to choose

and substitute g-’ by the running value g( p)-*. At large p one might take g a p and thus improve the convergence. Of course the infrared end of the integral is quite uncertain because in our perturbative procedure the effects of confinement etc. have not been taken into account. This inhibits a precise evaluation of the amplitude. A rough estimate (for a color SU(2) theory) is obtained if we take at large p

(B11)

g 2 ( l / p ) + 16.rrp2u

where (Iis the string constant. Then quarks with color charge 112 at a distance p from each other feel a force

Our integral becomes, in the case N‘ = 2,

S“‘= 216.rr10eA-”2,

I-

exp - 8 r 2 / g 2 ( l / p ).

349

G. 'I Hoof/, How

insfontons solve /he U ( I ) problem

From (B11) we get p dp 2 g d g / l 6 m

and using z = l/g2 the integral in (B13) is

so that

where Q l is the Lagrangian (B9). The result is uncomfortably large, but then the approximations used here (eq. B l l ) could at best only be expected to yield the order of magnitude of the expected interaction, which is clearly a strong one. Note that our coupling constant is subtraction scheme dependent. Since we did not yet include two loop order effects the subtraction scheme used strongly effects the final result. A factor 47r difference in the definition of p in (Bl) reduces the coefficient in front of the effective action (B16) by a factor ( 4 ~ =) 26n3. ~ This is just to illustrate how sensitively the amplitude obtained depends upon the assumptions. References [l] S.L. Glubow, in: H a h md their inleraaionc. Ericc 1967. 4. A. Z i c h i (Acad. Rar, New York. 1968) p. 83, S.L. Glubow, R. J d w and S.4. Shci. Phys. Rev. 187 (1969) 1916; M. O e l l - M i ~ in: , h. Tbid TOpial Cod. on Plrtidc Physia, Honolulu 1969. edr. W.A. S i md S.F.Tuan (Western Periodicals, La Angela. 1970) p. 1, and b: Ekmcnlary Partidc Physics. Wadming 1972. ed. P. Urbrn (Springer-Verlag, 1972); Actr Phyrifl AuslrircP Suppl. I X (1972) 733; H. Fflczacb d M. GeU-MIIUI, ROC.XVI Intern. Conf. on H.E.P.. chiago 1972, cdr. J.D. Jackson lad A. RO~CIIS,Vol. 2, p. 135, md Phys. Lea. 47B (1973) 36.5. 121 S.L. Adkr, Php. Rev. 177 (1969) 2426; I.S. &u and R.J&w, Nwvo Cim. 6oA (1969) 4?; S.L. Adler and W.A. Budecn, Phys. Rev. 182 (1969) 1517. 131 A.A. BcIaVio, A.M. Pdyllrov, AS. S c h w m md Yu.S. Tyupkio, Phys. Lett. S9B (1975) 85. [4]G. 'I Hooh, Php. Rev. Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432; R. Jackiw md C. Rebbi, Php. Rev. Lett. 37 (1976) 172; C.G. Cdhn Jr., R.F. Darhen and D.J. O m , Phys. Lett. 63B (1976) 334; Phys. Rev. D17 (1978) 2717. IS] S. Cokmm. in: The Whyx of Subnudur Physics. Ericc 1977, ed. A. zifhichi (Pknum Prar. New York. 1979) p. 805. 161 R. Crcwther, Php. Lett. 70B (1W) 349; Riv. Nuovo Cim. 2 (1979) 63; R. Crcwther, in: F a md Rape*s of Gauge Theories, schladming 1978, ed. P. Urban (Springer-Verlag, 1978); A m Phys. Aualriaca suppl. XIX (1978) 47. 171 G.A. chrirtoa, Phya. Repom 116 (1984) 251. 181 R. Crewther, private communiation. 191 J.C. Wud. Phya. Rev. 78 (1993) 1824; Y. Takahashi. Nuovo cim. 6 (1957) 370. [la] A. Shmov, '(heor. .ad k t h . Phyr. 10 (1972) 153 (in Russian). transl. Them.md Math. Phys. 10. p. 99; J.C. Taylor, Nud. Phys. B33 (1971) 436.

350

G. ' I Hoofl, How

i ~ ~ f a ~ solve ~ o n the i U ( f ) problem

[ll) C. Becfhi, A. Rouet and R. Ston, Comm. Math. Phys. 42 (1975) lD, AM. Phys. (N.Y.) 98 (1976) 281. [lZl M. Velunm, Nuel. Php. B7 (I=) 637; Nuel. Php. 921 (1970) 288. 1131 Then u no paper by &I1 or Treiman on thh plrrieulu subject. See ref. [12]. Rubalrov-Clllan effcn. Novaibint prepiat 1986. [14] A.R. ulitnitrky. The diwnte c h i spllncny b r d i o g in OCD IIa manifatationof IIIC 1151 C. Let ud W. Budecn, Nud. Flip. B153 (1979) 210. 1161 C.G. CAlm JI., R.F. Duhen d D.J. Gron. Phys. Len. 669 (1977) 375; Php. Rev. D17 (1978) 2717; Phys. Lett. 789 (1978) Wn. 1171 T.H.R. Skyme, Proc. Roy. Soe. A260 (1%1) 127. [l8] E. Wittea, Nucl. Phya. B223 (1983) 422,433. 1191 0. 't Hwft, Phys. Rev. D14 (1976) 3432. (201 F.R. On,Php. Rev. D16 (1977) ZSV, 0. 'I H d , Phys. Rev. D18 (1978) 2199; A. Hlrcnfratz a d P. Huenhtz. Nud. pbp. Bl!B (1981) 210. [21) K. Fujiklwr, Flip. Rev. Lett. 42 (1979) 1195; Php. Rev. DZ1 (19210) 2848.

351

CHAPTER 5.3

NATURALNESS, CHIRAL SYMMETRY, AND SPONTANEOUS CHIRAL SYMMETRY BREAKING G . 't Hooft Institute for Theoretical Physics Utrecht, T h e Netherlands

ABSTRACT

A properly called "naturalness" is imposed on gauge theories. It is an order-ofmagnitude restriction that must hold at all energy scales IJ. To construct models with complete naturalness for elementary particles one needs more types of confining gauge theories besides quantum chromodynamics. We propose a search program for models with improved naturalness and concentrate on the possibility that presently elementary fermions can be considered as composite. Chiral symmetry must then be responsible for the masslessness of these fermions. Thus we search for QCDlike models where chiral symmetry is not or only partly broken spontaneously. They are restricted by index relations that often cannot be satisfied by other than unphysical fractional indices. This difficulty made the author's own search unsuccessful so far. As a by-product we find yet another reason why in ordinary QCD chiral syrmnetry must be broken spontaneously. 1111. INTRODUCTION The concept of causality requires that macroscopic phenomena follow from microscopic equations. Thus the properties of liquids and solids follow from the microscopic properties of molecules and atoms. One may either consider these microscopic properties to have been chosen at random by Nature, or attempt to deduce these from.even more fundamental equations at still smaller length and time scales. In either case, i t is unlikely that the microscopic equations contain various free parameters that are carefully adjusted by Nature to give cancelling effects such that the macroscopic systems have some special properties. This is a Dynamical Symmetry Breaking, A Collection of Reprints, Edited by A. Farhi and R. Jackiw (World Scientific. 1982).

352

G. 't HOOFT

philosophy which we would l i k e t o apply t o the u n i f i e d gauge theories: t h e e f f e c t i v e i n t e r a c t i o n s a t a l a r g e length scale, corresponding t o a low energy scale u l , should follow from the properties a t a much smaller length s c a l e , o r higher energy s c a l e ~ 2 without , the requirement t h a t various d i f f e r e n t parameters a t the energy s c a l e 1.12 match with an accuracy of the o r d e r of u , / p 2 . That would be unnatural. On t h e o t h e r hand, i f a t the energy s c a l e ~2 some parameters would be very small, say

then t h i s may s t i l l be n a t u r a l , provided t h a t t h i s property would e now conjecture t h a t not be s p o i l t by any higher o r d e r e f f e c t s . W t h e following d o p a should be followed: a t any energy s c a l e p , a physical parameter o r s e t of physical p a r m e t e r s a i ( p ) i s allowed t o be very small only i f the replacement ui(1.1) o would increase t h e synaDetry of the system. I n what follows t h i s is what w e mean by naturalness. It i s c l e a r l y a weaker requirement than t h a t of P. Diracl) who i n s i s t s on having no small numbers a t a l l . It i s what one expects i f a t any m a s s s c a l e p > p o some ununderatood theory with s t r o n g i n t e r a c t i o n s determines a spectrum of p a r t i c l e s with various good o r bad symmetry properties. I f a t p yo c e r t a i n parameters come out t o be small, say 10'5, then t h a t cannot be an accident; i t must be t h e consequence of a near synmetry.

-

-

-

-

For instance, a t a mass s c a l e p

-

50 GeV,

t h e e l e c t r o n mass me is IO? This is a small parameter. It i s acceptable because me = o would imply an a d d i t i o n a l c h i r a l a y m e t r y corresponding t o s e p a r a t e conservation of l e f t handed and r i g h t handed electron-like leptons. This guarantees t h a t a l l a r e proportional t o me i t s e l f . I n s e c t s . renormalizations of I112 nd I113 we compare naturalness f o r quantum electrodynamics and theory.

+t

Gauge coupling constants and o t h e r (sets o f ) i n t e r a c t i o n constants may be small because p u t t i n g them equal t o zero would t u r n the gauge bosona o r o t h e r p a r t i c l e s i n t o f r e e p a r t i c l e s so t h a t they are s e p a r a t e l y conserved. I f within a set of small parameters one i s s e v e r a l orders of m a ~ n i t u d esmaller than another then t h e smallest must s a t i s f y our "dogma" separately. Ad w e w i l l see, naturalness w i l l put the s e v e r e s t r e s t r i c t i o n on the occurrence of scalar p a r t i c l e s i n renormalizable theories. I n f a c t we conjecture t h a t t h i s is the reason why l i g h t , weakly i n t e r a c t i n g scalar p a r t i c l e s a r e not seen.

353

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

It is our aim to use naturalness as a new guideline to construct models of elementary particles (sect. 1114). In practice naturalness will be lost beyond a certain mass scale po, to be referred to as "Naturalness Breakdown Mass Scale" (NBMS). This simply means that unknown particles with masses beyond that scale are ignored in our model. The NBMS is only defined as an order of magnitude and can be obtained for each renormalizable field theory. For present "unified theories", including the existing grand unified schemes, it is only about 1000 GeV. In sect. 5 we attempt to construct realistic models with an NBMS some orders of magnitude higher. One parameter in our world is unnatural, according to our definition, already at a very low mass scale (voh. eV). This is the cosmological constant. Putting it equal to zero does not seem to increase the symnetry. Apparently gravitational effects do not obey naturalness in our formulation. We have nothing to say about this fundamental problem, accept to suggest that onty gravitational effects violate naturalness. Quantum gravity is not understood anyhow so we exclude i t from our naturalness requirements. On the other hand it is quite remarkable that all other elementary particle interactions have a high degree of naturalness. No unnatural parameters occur in that energy range where our popular field theories could be checked experimentally. We consider this as important evidence in favor of the general hypothesis of naturalness. Pursuing naturalness beyond 1000 GeV will require theories that are immensely complex compared with some of the grand unified schemes.

A remarkable attempt towards a natural theory was made by Dimopoulos and Susskind z), These authors employ various kinds of confining gauge forces to obtain scalar bound states which may substitute the Higgs fields in the conventional schemes. In their model the observed fermions are still considered to be elementary. Most likely a complete model of this kind has to be constructed step by step. One starts with the experimentally accessibleaspects of the Glashow-Weinberg-Salam-Ward model. This model is natural if one restricts oneself to mass-energy scales below 1000 GeV. Beyond 1000 GeV one has to assume, as Dimopoulos and Susskind do, that the Higgs field is actually a fermion-antifermion composite field. Coupling this field to quarks and leptons in order to produce their mass, requires new scalar fields that cause naturalness to break down at 30 TeV or so. Dimopoulos and Susskind speculate further on how to remedy this. To supplement such ideas, we toyed with the idea that (some of) the presently "elementary" fermions may turn out to be bound states of an odd number of fermions when considered beyond 30 TeV. The binding mechanism would be similar

354

G. 't HOOFT

to the one that keeps quarks inside the proton. However, the proton is not particularly light compared with the characteristic mass scale of quantum chrmodynamics (00). Clearly our idea is only viable if something prevented our "baryons" from obtaining a mass (eventually a small mass may be due t o some secondary perturbation). The proton ows its mass to spontaneous breakdown of chiral symmetry, or so it seems according to a simple, fairly successful model of the mesonic and baryonic states in QCD: the Gell-Mann-Llvy sigma model3). Is it possible then that in some variant of QCD chiral symmetry is not spontaneously broken, or only partly, so that at least some chiral syutnetry remains in the spectrum of fermionic bound states? In this article we will see that in general in SU(N) binding theories this is not allowed to happen, i.e. chiral symmetry must be broken spontaneously. 1112. NATURALNESS IN QUANTUM ELECTRODYNAMICS Quantum Electrodynamics as a renormalizable model of electrons (and muons if desired) and photons is an example of a "natural" field theory. The parameters a, m, (and m,,) may be small (and m,,) are very small at large p . independently. In particular The relevant symmetry here is chiral symmetry, for the electron and the muon separately. We need not be concerned about the Adler-Bell-Jackiw anomaly here because the photon field being Abelian cannot acquire non-trivial topological winding numbers4). There is a value of p where Quantum Electrodynamics ceases to be useful, even as a model. The model is not asymptotically free, so there is an energy scale where all interactions become strong: (1112) where Nf is the number of light fermions. If some world would be described by such a theory at low energies, then a replacement of the theory would be necessary at or below energies of order uo. 4 1113. 4 -THEORY

A renormalizable scalar field theory is described by the Lagrangian

P

- -lo,,+)2

Am242

-

1

A+

4

.

(1113)

the interactions become strong at

u

5

m exp(l6a 2/3X)

,

(1114)

but is it still natural there?

355

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

There are two parameters, X and m. Of these, X may be small because X = o would correspond to a non-interacting theory with total number of $ particles conserved. But is small m allowed? If we put m = o in the Lagrangian (1113) then the symmetry is not enhanced*). However we can take both m and X to be small, becauseif X = m o we have invariance under

-

(1115) This would be an approximate symmetry of a new underlying theory at energies of order po. Let the symmetry be broken by effects described by a dimensionless parameter E . Both the mass term and the interaction term in the effective Lagrangian (1113) result from these symmetry breaking effects. Both are expected to be of order E . Substituting the correct powers of p0 to account for' the dimensions of these parameters we have (1116)

Therefore,

(1117) This value is much lower than eq. (1124). We now turn the argument around: if any "natural" underlying theory is to describe a scalar particle whose effectiue Lagrangian at low energies will be eq. (III3), then its energy scale cannot be given by (1114) but at best by (1117). We say that naturalness breaks down beyond m / A . It must be stressed that these are orders of magnitude. For instance one might prefer to consider X/n2 rather than A to be the relevant parameter. po 'then has to be multiplied by T . Furthermore, X could be much smaller than E because A = o separately also enhances the synnnetry. Therefore, apart from factors n, eq. (1117) indicates a maximum value for po. Another way of looking at the problem of naturalness is by comparing field theory with statistical physics. The parameter m h would correspond to (T-Tc)/T in a statistical ensemble. Why would the temperature T chosen by Nature to describe the elementary particles be so close to a critical temperature Tc? If Tc o then T may not be close t o Tc just by accident. 1114. NATURALNESS IN THE WEINBERG-SAW-GIM MODEL The difficulties with the unnatural mass parameters only occur in theories with scalar fields. The only fundamental scalar *) Conformal symmetry is violated at the quantum level.

G. 't HOOF1

field that occurs in the presently fashionable models is the Higgs f'eld in the extended Weinberg-Salam model. The Higgs mass-squared, is up to a coefficient a fundamental parameter in the agrangian. It is small at energy scales v >>mw Is there an approximate symmetry if % + 07 With some stretch of imagination we might consider a Goldstone-type symmetry:

1,

+(XI + +(XI + const.

(1118)

However we also had the local gauge transformations: $(XI

+

.

n(x) $(XI

(1119)

The transformations QII8) and QII9) only form a closed group if we also have invariance under

+

But then it becomes possible to transform away completely. The Higgs field would then become an unphysical field and that is not what we want. Alternatively, we could have that (1118) is an approximate synmetry only, and it is broken by all interactions that have to do with the symmetry (1119) which are the weak gauge field interactions. Their strength is g2/4n 0(1/137). So .at best we can have that the symmetry is broken by 6(1/137) effects. Therefore

-

4 Also the A $ term in the Higgs field interactions breaks this symmetry. Therefore (11111) Now (11112) where FH is the vacuum expectation value of the Higgs field, known to )'eb FH

-

(2Gfi)-'/*

-

174 GeV

.

(11113)

We now read off that

*) Some numerical values given during the lecture were incorrect.

I here give corrected values.

357

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

This means t h a t a t energy scales much beyond FH our model becomes more and more u n n a t u r a l . A c t u a l l y , f a c t o r s of n have been omitted. I n p r a c t i c e one f a c t o r of 5 o r 10 i s s t i l l not t o t a l l y unacceptable. Notice t h a t t h e a c t u a l value of dropped o u t , except t h a t

..

(1111s) Values f o r

%

of j u s t a few G e V a r e u n n a t u r a l .

1115. EXTENDING NATURALNESS Equation (11114) t e l l s us t h a t a t energy s c a l e s much beyond 174 G e V t h e s t a n d a r d model becomes u n n a t u r a l . As long a s t h e Higgs f i e l d H remains a fundamental scalar n o t h i n g much can be done about t h a t . We t h e r e f o r e conclude, with Dimopoulos and Susskind2) t h a t t h e "observed" Higgs f i e l d must be composite. A n o n - t r i v i a l s t r o n g l y i n t e r a c t i n g f i e l d theory must be o p e r a t i v e a t 1000 GeV o r so. An obvious and indeed l i k e l y p o s s i b i l i t y i s t h a t t h e Higgs f i e l d H can be w r i t t e n as

H = ZGJl

,

(11116)

where'Z i s a r e n o r m a l i z a t i o n f a c t o r and J, i s a new quark-like o b j e c t , a fermion with a new c o l o r - l i k e i n t e r a c t i o n 2 ) . We w i l l r e f e r t o t h e o b j e c t as mete-quark having meta-color. The theory w i l l have a l l f e a t u r e s of QCD so t h a t we can copy t h e nomenclature of QCD with t h e p r e f i x "meta-". The Higgs f i e l d i s a meta-meson. It i s now tempting t o assume t h a t t h e meta-quarks transform t h e same way under weak SU(2) x U ( 1 ) as o r d i n a r y quarks. Take a doublet with left-handed components forming one gauge d o u b l e t and r i g h t handed components forming two gauge s i n g l e t s . The nktaquarks a r e massless. Suppose t h a t t h e meta-chiral symmetry i s broken spontaneously j u s t a s i n o r d i n a r y QCD. What would happen?

What happens is i n o r d i n a r y QCD w e l l d e s c r i b e d by t h e GellMann-Levy sigma model. The l i g h t e s t mesons form a q u a r t e t of r e a l transforming as a fields, 0 ij' 21ef t 2right r e p r e s e n t a t i o n of

S U ( 2 p f t 0 SU(2)right. Since the weak i n t e r a c t i o n only d e a l s with SU(2)left t h i s q u a r t e t can a l s o be considered a s one complex doublet r e p r e s e n t a t i o n of weak Su(2). In o r d i n a r y QCD we have

358

a a b s i j + i.r..n

4ij

,

1J

and vacuum =

&

f n * 9 1 MeV

.

(11118)

The complex doublet i s then

4i

--

"i

'vacuum

(11119)

and

We conclude then w e g e t expectation numbers. If match

. I

[A]

x

64 MeV

.

(11120)

t h a t i f w e t r a n s p l a n t t h i s theory t o t h e TeV range a s c a l a r doublet f i e l d w i t h a non-vanishing vacuum value f o r f r e e . A l l w e have t o do now is t o match the w e scale a l l QCD masses by a s c a l i n g f a c t o r K then we

F~ = 174 ~ e =v K 64 MeV i K

= 2700

.

(11121)

Now the mesonic s e c t o r of QCD i s u s u a l l y assumed t o be reproduced i n t h e I/N expansion 5 , where N i s t h e number of c o l o r s 3). The 4-meson coupling c o n s t a n t goes l i k e ( i n QCD we have N 1/N. Then one would expect

-

f T a f i . Therefore K

= 2700

(11122)

4,

(11123)

i f t h e metacolor group i s SU(N). Thus w e o b t a i n a model t h a t reproduces t h e W-mass and p r e d i c t s t h e Higgs mass. The Higgs i s t h e meta-sigma p a r t i c l e . The ordinary sigma i s a wide resonance a t about 700 MeV3), so t h a t we p r e d i c t

(11124) and i t w i l l be extremely d i f f i c u l t t o d e t e c t among o t h e r s t r o n g l y interacting objects.

359

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

1116. WHAT NEXT?

The model of t h e previous s e c t i o n i s t o our mind n e a r l y inevit a b l e , but t h e r e are problems. These have t o do w i t h the observed fermion masses. A l l l e p t o n s and quarks owe t h e i r masses t o a n i n t e r a c t i o n term of t h e form

sJlHJ, ,

(11125)

where g i s a coupling c o n s t a n t , J, i s t h e l e p t o n or quark and H i s t h e Higgs f i e l d . With (11116) t h i s becomes a four-fermion i n t e r a c t i o n , a fundamental i n t e r a c t i o n i n t h e new theory. Because i t i s non-renormalizable f u r t h e r s t r u c t u r e i s needed. In r e f . 2 t h e obvious choice i s made: a new "meta-weak i n t e r a c t i o n ' ' gauge theory e n t e r s with new super-heavy i n t e r m e d i a t e v e c t o r bosons. But s i n c e H i s a s c a l a r t h i s boson must be i n t h e crossed channe1,a r a t h e r awkward s i t u a t i o n . (See o p t i o n a i n Figure 1.) A simpler theory i s t h a t a new s c a l a r p a r t i c l e i s exchanged i n the d i r e c t channel. (See o p t i o n b i n Figure I . )

me ta-quarks

mete-quarks

mete-quarks Figure 1.

360

G. 't HOOFT

Notice t h a t i n both cases new scalar f i e l d s are needed because i n case a ) something must cause the "spontaneous breakdown" o f the new gauge symmetries. Therefore choice b) i s simpler. W e removed a Higgs s c a l a r and we get a s c a l a r back. Does naturalness improve? The answer i s y e s . The coupling constant g i n the i n t e r a c t ion GI1251 s a t i s f ies (11126)

Here g and g2 a r e the couplings a t the new v e r t i c e s , M, i s the new s c a l a r 1s mass, and Z i s from (IIK16) and i s of order 2-

4

f

(K

.

m,,l2

m

(11127)

(1800 G e V ) 2

'

Suppose t h a t the heaviest lepton o r quark i s about 10 GeV. For that fermion the coupling constant g i s

"f g *p

-

1/20

.

We g e t

Naturalness breaks do& a t

an improvement of about a f a c t o r 50 compared w i t h the s i t u a t i o n in sect. 1114. Presumably we a r e again allowed t o multiply by f a c t o r s l i k e 5 or 10, before g e t t i n g i n t o r e a l trouble. Before speculating on how t o go on from here t o improve naturalness s t i l l f u r t h e r we must assure ourselves t h a t a l l other a l l e y s a r e b l i n d ones. An i n t r i g u i n g p o s s i b i l i t y i s t h a t the presently observed fermions a r e composite. W e would g e t option c> Figure 2.

361

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

or

y

Fig. 2 The d o t t e d l i n e could be an o r d i n a r y weak i n t e r a c t i o n W or photon, t h a t breaks an i n t e r n a l symmetry i n t h e binding f o r c e f o r t h e new components. The new b i n d i n g f o r c e could e i t h e r act a t t h e 1 TeV or a t t h e 10-100 TeV range. It could e i t h e r be an e x t e n s i o n o f metac o l o r or be a ( c o l o r ) " or p a r a c o l o r f o r c e . Is such an i d e a v i a b l e ? C l e a r l y , compared w i t h t h e energy scale on which t h e b i n d i n g f o r c e s take p l a c e , t h e composite fermions must be n e a r l y massless. Again, t h i s cannot be an a c c i d e n t . The c h i r a l symmetry r e s p o n s i b l e f o r t h i s must be p r e s e n t i n t h e u n d e r l y i n g - t h e o r y . Apparently then, t h e underlying theory w i l l possess a c h i r a l syaunetry which i s not (or n o t completely)spontaneou8ly broken, b u t r e f l e c t e d i n t h e bound state spectrum i n t h e Wigner mode: some massless c h i r a l o b j e c t s and p a r i t y doubled massive fermions. This p o s s i b i l i t y i s most c l e a r l y d e s c r i b e d by t h e o-model as a model f o r t h e lowest bound states o c c u r r i n g i n o r d i n a r y quantum chromodynamics.

1117. THE

U

MODEL

The fermion system i n quantum chromodynamics shows an axial synunetry. To i l l u m i n a t e our problem l e t us c o n s i d e r t h e case of two f l a v o r s . The l o c a l c o l o r group i s SU(3),. The s u b s c r i p t c h e r e s t a n d s for c o l o r . The f l a v o r symmetry group i s SU(Z), @ U(1) where t h e s u b s c r i p t s L and R s t a n d s f o r SU(Z)L l e f t and r i g h t and t h e group elements must be chosen t o be spacet i m e independent. W e s p l i t t h e fermion f i e l d s J, i n t o l e f t and r i g h t components: J,

I(l+Y5)J,L+

4(1T5)JIR

(11129)

transforms as a 3c 8 I L @ 2, 0 Is

(11130)

C

$,

fBzL 8

l R 0 2s

J,L transforms as a 3

and

(11128)

where t h e i n d i c e s r e f e r t o t h e v a r i o u s groups. 1 s t a n d s f o r t h e Lorentzgroup S0(3,1), l o c a l l y e q u i v a l e n t t o SL(2,c) which has two

362

G. 't HOOFT

d i f f e r e n t complex doublet r e p r e s e n t a t i o n s 21 and 24 (corresponding t o the transformation law f o r t h e n e u t r i n o and a n t i n e u t r i n o , r e s p e c t i v e l y ) . The f i e l d s $L and $8 have the same charge under U ( I ) , whereas a x i a l U(1) group (under which they would have opposite charges) is absent because of i n s t a n t o n e f f e c t s 4 ) . The e f f e c t of t h e c o l o r gauge f i e l d s i s t o bind these fermions i n t o mesons and baryons a l l of which must be c o l o r s i n g l e t s . It would be n i c e i f one could d e s c r i b e t h e s e hadronic f i e l d s a s r e p r e s e n t a t i o n s of SU(2)L 8 SU(2)R @ U ( I ) and the Lorentz group, and then c a s t t h e i r mutual i n t e r a c t i o n s i n the form of an e f f e c t i v e Lagrangian, i n v a r i a n t under t h e f l a v o r symmetry group. I n the case a t hand t h i s is p o s s i b l e and t h e r e s u l t i n g c o n s t r u c t i o n i s a successful d one-time popular model f o r pions and nucleons: e have a nucleon d o u b l e t t h e u model3Y. W (11131)

where N

and

L

transforms a s a I

C

@2L@ lR 9 2

NR transforms as a l c @ lL

,

(III32a)

@2R 8 2g .

(III32b)

+

Further w e have a q u a r t e t of real scalar f i e l d s (a , n) which t r a n e f o m a s a 1 O zL 8 21 @ it. The 7.agrangian i s C

(11133)

Here V must be a r o t a t i o n a l l y i n v a r i a n t function.

-

-

Usually V i s chosen such t h a t i t s absolute+minimum i s away v and n 0 . Here v is from the o r i g i n . Let V be minimal a t a j u s t a c-umber. To o b t a i n the p h y s i c a l p a r t i c l e spectrum we w r i t e 0 - v +

(11134)

8

+ i n t e r a c t i o n terms

-

.

(11135)

C l e a r l y , i n t h i s c a s e t h e nucleons a c q u i r e a mass term m = gOv and t h e s p a r t i c l e has a mass m: 4v2V"(v2), whereas t t e pion remains s t r i c t l y massless. The e n t i r e mass of the pion must be due t o e f f e c t s t h a t e x p l i c i t l y break SU(2)L x SU(2)R , such as a small

363

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

51)

mass term m for the quarks (11128). We say that in this case the 9 flavor group sU(2)~@ sU(2)~ is spontaneously broken into the isospin group SU(2). Another possibility however, apparently not realised in ordinary quantum chromodynamics, would be that SU(2)L @SU(2)R is not spontaneously broken. We would read off from the Lagrangian (11133) that the nusleons N would form a massless doublet and that the four fields (a,n) could be heavy. The dynamics of other confining gauge theories could differ sufficiently from ordinary QCD so that, rather than a spontaneous symmetry breakdown, massless "baryons" develop. The principle question we will concentrate on is why do these massless baryons form the representation (III32), and how does this generalize to other systems. We would let future generations worry about the question where exactly the absolute minimum of the effective potential V will appear. 1118. INDICES We now consider any color group Gc. The fundamental fermions in our system must be non-trivial representation of Gc and we assume "confinement" to occur: all physical particles are bound states that are singlets under Gc. Assume that the fermions are all massless (later mass terms can be considered as a perturbation). We will have automatically some global symmetry which we call the flavor group GF. (We only consider exact flavor symmetries, not spoilt by instanton effects.) Assume that GF is not spontaneously broken. Which and how many representations of GF will occur in the massless fermion spectrum of the baryonic bound statgs? We must formulate the problem more precisely. The massless nucleons in (11133) being bound states, may have many massive excitations. However, massive Fermion fields cannot transform-as a 21: under Lorentz transformations; they must go as a 21: @ 2 1 . That is because a mass term being a Lorentz invariant proiuct of two fields at one point only links 21: representations with 21: representations. Consider a given representation T of Let p be the number of field multiplets transforming as r 821: and q be the number of field multiplets r @21. Mass terms that link the 21: with 21: fields are completely invariant and in general to be expected in the effective Lagrangian. But the absolute value of

+.

11-p-q

(11136)

is the minimal number of surviving massless chiral field multiplets. We will call 11 the index corresponding to the By definition this index must be a (positive or integer. In the sigma model it is postulated that

.

364

G. 't HOOFT

index (IL @ 2R)

(11137)

-

-1

index (r)- o for all other representations r. This tells us that if chiral synrmetry is not broken spontaneously one massless nucleon doublet emerges. We wish to find out what massless fermionic bound states will come out in more general theories. Our problem is: how does (11137) generalize? 1119. ABSENCE OF MASSLESS BOUND STATES WITH SPIN 312 OR HIGHER In states. develop this is Massive

the foregoing we only considered spin o and spin 112 bound

Is it not possible that fundamentally massless bound states with higher spin? I believe to have strong arguments that indeed not possible. Let us consider the case of spin 312. spin 312 fermionr are described by a Lagrangian of the form (11138)

Just like spin-one particles, this has a gauge-invariance if m

v))

+

J,)) + a P . l ( X )

+ 0:

(11139)

s

where n(x) is arbitrary. Indeed, massless spin 312 particles only occur in locally supersymmetric field theories. The field q(x) is fundamental1y unobservable. Now in our model JI would be shorthand for some composite field: J, -+ $$$. However: then all components of this, including 0, would be'observables. If m = o we would be forced to add a gauge fixing term that would turn rl into an unacceptable ghost particle*). We believe, therefore, that unitarity and locality forbid the occurrence of massless bound states with spin 312. The case for higher spin will not be any better. And so we concentrate on a bound state spectrum of spin 112 particles only.

*) Note added: during the lectures it was suggested by one attendant to consider only gauge-invariant fields as ,,Y,, =aRJlv awJ,,.

-

However, such fields must satisfy constraints:a [a&pw]=o. Composite field will never automatically satisfy such constraints.

365

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

11110. SPECTATOR GAUGE FIELDS AND -FERMIONS So far, our model consisted of a strong interaction color gauge theory with gauge group Gc, coupled to chiral fermions in various representations r of Gc but of course in such a way that the anomalies cancel. The fermions are all massless and form multiplets of a global symmetry group, called GF. For QCD this would be the flavor group. In the metacolor theory GF would include all other fermion symmetries besides metacolor.

In order to study the mathematical problem raised above we will add another gauge connection field that turns GF into a local symmetry group. The associated coupling constants may all be arbitrarily small, so that the dynamics of the strong color gauge interactions is not much affected. In particular the massless bound state spectrum should not change. One may either think of this new gauge field as a completely quantized field or simply as an artificial background field with possibly non-trivial topology. We will study the behavior of our system in the presence of this "spectator gauge field". As stated, its gauge group is GF. Note however, that some flavor transformations could be associated with anomalies. There are two types of anomalies: i) those associated with Gc x GF, only occurring where the color field has a winding number. Only U ( 1 ) invariant subgroups of GF contribute here. They simply correspond to small explicit violations of the syrmnetry. From now on we will take as GF only the anomaly-free part. Thus, for QCD with N flavors, GF is not U(N) x U(N) but

+

GF

-

SU(N)

8 SU(N) @ U ( 1 )

.

ii) those associated with GF alone. They only occur if the spectator gauge field is quantized. To remedy these we simply add "spectator fermions" coupled to GF alone. Again, since these interactions are weak they should not influence the bound state spectrum. Here, the spectator gauge fields and fermions are introduced as mathematical tools only. It just happens to be that they really do occur in Nature, for instance the weak and electromagnetic SU(2) x U ( 1 ) gauge fields coupled to quarks in QCD. The leptons then play the role of spectator fermions. 11111. ANOMALY CANCELLATION FOR THE BOUND STATE SPECTRUM Let us now resume the particle content of our theory. At small distances we have a gauge group Gc 8 GF with chiral fernions in several representations of this group. Those fermions which are

366

G. t HOGFT

trivial under Gc are only coupled weakly and are called "spectator fermions". All anomalies cancel, by construction. At low energies, much lower than the mass scale where color binding occurs, we see only the GF gauge group with its gauge fields. Coupled to these gauge fields are the massless bound states, forming new rhpresentations r of Gp, with either left- or right handed chirality. The numbers of left minus right handed f e d o n fields in the representations r are given by the as yet unknown indices Il(r). And finally we have the spectator fermions which are unchanged. We now expect these very 1ight.objects to be described by a new local field theory, that is, a theory local with respect to the large distance scale that we now use. The central theme of our reasoning is now that this new theory must again be anomaly free. We simply cannot allow the contradictions that would arise if this were not so. Nature must arrange its new particle spectrum in such a way that unitarity is obeyed, and because of the large distance scale used the effective interactions are either vanishingly small or renormalizable. The requirement of anomaly cancellation in the new particle spectrum gives us equations for the indices a ( r ) , as we will see.

The reason why these equations are sometimes difficult or impogsible to solve is that the new representations r must be different from the old ones; if Gc = SU(N) then r must also be faithful representations of Gp/Z(N). For instance in QCD we only allow for octet or decuplet representations of (SU(3))flavor, whereas the original quarks were triplets, However, the anomaly cancellation requirement, restrictive as it may be, does not fix the values of E(r) completely. We must look for additional limitations. 11112

@PELQUIST-CARAZZONE DECOUPLING AND N-INDEPENDENCE

A further limitation is found by the following argument. Suppose we add a mass term for one of the colored fermions.

Clearly this links one of the left handed fermions with one of the right handed ones and thus reduces the flavor group into G' C GF. Now let us gradually vary m from o to infinity. A famous tffeorern 5 ) tells us that in the limit m + all effects due to this massive quark disappear. All bound states containing this quark should also disappear which they can only do by becoming very heavy. And they can only become heavy if they form representations r' of Gb with total index &'(r') = 0 . Each representation r of % forms

-

367

+

CHIRAL SYMMETRY AND CHIRAL SYMMETRY BREAKING

an a r r a y of represen'tations r ' of G k . Therefore (11140)

Apparently t h i s expression must vanish. Thus w e found another requirement f o r the i n d i c e s E(r). The i n d i c e s w i l l be nearly b u t not q u i t e uniquely determined now. C a l c u l a t i o n s show t h a t t h i s second requirement makes our i n d i c e s t ( r ) p r a c t i c a l l y independent of the dimensions n i of GF. For i n s t a n c e , i f Gc = SU(3) and i f we have l e f t - and righthanded quarks forming t r i p l e t s and s e x t e t s then

.

t o the s e x t e t s . Gc i s where n1,2 r e f e r t o the t r i p l e t s and n anomaly-free i f 3,4 n1

-

n2 + 7(n3- n4) = o

.

(11142)

Here we have t h r e e independent numbers n i . If we w r i t e the r e p r e s e n t a t i o n s r a s Young tableaus then E(r) could s t i l l depend e x p l i c i t l y on n i . However, suppose t h a t someone would s t a r t a s approximation of Bethe-Salpeter type t o discover t h e zero mass bound s t a t e spectrum. He vould s t u d y diagrams such a s Fig. 3

Fig. 3 The r e s u l t i n g i n d i c e s E(r) would follow from tdpological p r o p e r t i e s of the i n t e r a c t i o n s represented by the blobs. I t is u n l i k e l y t h a t t h i s topology would be seriousZy e f f e c t e d by d e t a i l s such a s t h e contributions o f diagrams c o n t a i n i n g a d d i t i o n a l closed fermion loops. However, t h a t is the only way i n which e x p l i c i t n-dependence enters. I t i s t h e r e f o r e n a t u r a l t o assume a ( r ) toben-independent. This l a t t e r assumption f i x e s E(r) completely. What i s t h e r e s u l t of these c a l c u l a t i o n s ?

368

6. 't HOOFT

11113. CALCULATIONS

Let C be any (reducible or irreducible) gauge group. Let chiral fermions in a representation r be coupled to the gauge fields by the covariant derivative

a where Aa are the gauge fields and X (r) a set of matrices depending on the lJ representation r. Let the left-handed fermions be in the representations rt and the right-handed ones in rR. Then the anomalies cancel If

(11144)

The object dabc(r) = Tr{Xa(r),

X

b

(r)) lc(r) can be computed for any

r. In table 1 we give some examples. The fundamental representation ro is represented by a Young tableau: 0 Let it have n components. We take the case that Tr X(ro) = 0 . Write

.

Tr I(ro) = n

, ,

Tr X(r) = o Tr Xa(r) Xb(r) dabc (r)

-

Tr I(r)

-

-

N(r)

,

C ( r ) Tr Aa(ro) A b (ro) s

K(r) dabc(ro)

.

(11145)

We read off C and K from table 1. Now 11144 must hold both in the high energy region and in the low energy region. The contribution of the spectator fermions in both regions is the same. Thus we get for the bound states dabc(r)

-

n babC(roL) C

- dabc (roR))

(11146)

where a,b,c are indices of Gp and ro is the fundamental representation of %, We have the factor nc written explicitly, being the number of color components. Let us now consider the case G, = SU(3); Gp = SUL(n) @ SUR(n) @U(l). We have n "quarks" in the fundamental representations. The representations r of the bound states must be They are assumed to be built from three quarks, but we in +/Z(3). are free to choose their chirality. The expected representations

369

CHIRAL WMMETRY AND CHIRAL WMMETRY BREAKING

K(r) 1

1

1

-1

n+2

n-?: 4

(n+3) ( e 6 ) 2

n2-3

370

G. 't HOOFT

are given in table 2 , where also their indices are defined. Because of left-right symmetry these numbers change sign under interchange of left ++ right. Table 2 representation

index

representation

index

For the time being we assume no other representations. In eq. 11146 we may either choose a, b and c all t o be SU(n)L indices, or choose a and b to be SU(n)L indices and c the U(1) index. We get two independent equations:

1 4(n*3)(d6)kIf-1

ln(n*7)t2,+(n2-9)!t3 f

2

and

1 l(n+2)(n+3)k

If

f

-1 ln(h+3)k2?+(n

2

f

-3)k 3

-

3, if n

1,

if n

>2 , >

I

. (11147)

The Appelquist-Carazzone decoupling requirement, eq. ( I I I 4 0 ) , gives

us in addition two other equations:

For n

a,,

-

1,-

- E2-

>2

a2+

+

i3

-

0

+ !L3 = o

9

, both

if n

the general solution is

37 1

>2

.

(11148)

CHIRAL SYMMETRY A N D CHIRAL SYMMETRY BREAKING

= 11

PI+ =

-

e2+ = e2113

-

,

3 e - 31 ,

I 21-5.

(11149)

Here II i s s t i l l a r b i t r a r y . Clearly t h i s r e s u l t i s unacceptable. tle cannot allow any of the i n d i c e s L t o be non-integer. Only f o r t h e 2 (QCD with j u s t two f l a v o r s ) t h e r e i s another s o l u t i o n . case n I n t h a t c a s e 112- and 113 d e s c r i b e the same r e p r e s e n t a t i o n , and a,an empty r e p r e s e n t a t i o n . We g e t

-

- -

(11150)

According t o t h e o-model, I l l + t2+ 0 ; k = 1 . The a-model i s t h e r e f o r e a c o r r e c t s o l u t i o n t o our equations.

I n t h e previous s e c t i o n w e promised t o determine t h e i n d i c e s completely. This i s done by imposing n-independence for the more general case i n c l u d i n g a l s o o t h e r c o l o r r e p r e s e n t a t i o n s such a s s e x t e t s b e s i d e s t r i p l e t s . The r e s u l t i n g equations are not very i l l u m i n a t i n g , w i t h r a t h e r ugly c o e f f i c i e n t s . One f i n d s t h a t i n general no s o l u t i o n exists except when one assumes t h a t a l l mixed r e p r e s e n t a t i o n s have vanishing i n d i c e s . With mixed r e p r e s e n t a t i o n s we mean a product of two o r mre n o n - t r i v i a l r e p r e s e n t a t i o n s of two o r more non-fielian i n v a r i a n t subgroups of GF. I f now w e assume n-independence t h i s must a l s o hold i f the number of s e x t e t s i s zero. So L2+ and k2- must vanish. t1e g e t

a,+

- a,-

e3 =

1/9

,

-119

.

(I115 1 )

If a l l quarks were sextets, not t r i p l e t s , w e would g e t

el+ a3

=

-

a,-

In the case Gc

-

2/9

,

-219

.

(11152)

SU(5) t h e i n d i c e s were a l s o found. See t a b l e 3.

372

G. 't HOOFT

Rl+ =

5+= a3

5-

Table 3 i n d i c e s f o r Gc I25

Su(5)

I25

= 1/25

w1

= 1/25

This c l e a r l y suggest3 a g e n e r a l tendency f o r SU(N) c o l o r groups t o produce i n d i c e s fl/N o r 0 . 11114. CONCLUSIONS Our r e s u l t t h a t the i n d i c e s w e searched f o r are f r a c t i o n a l i s c l e a r l y absurd. 1Je n e v e r t h e l e s s pursued t h i s c a l c u l a t i o n i n o r d e r t o e x h i b i t the g e n e r a l philosophy of t h i s approach and t o f i n d out what a p o s s i b l e cure might be. Our s t a r t i n g p o i n t was t h a t c h i r a l syrmuetry i s n o t broken spontangpuely. Most l i k e l y t h i s i s untenable, W e find that explicit chiral a s s e v e r a l a u t h o r s have argued synnnetry i n QCD l e a d s t o t r o u b l e i n p a r t i c u l a r i f the number of f l a v o r s i s more than two. A d a r i n g c o n j e c t u r e i s then t h a t i n QCD t h e s t r a n g e quark, being r a t h e r l i g h t , is r e s p o n s i b l e f o r the spontaneous breakdown of c h i r a l symmetry. An i n t e r e s t i n g p o s s i b i l i t y is t h a t i n some generalized v e r s i o n s of QCD c h i r a l synrmetry is broken o n l y p a r t l y , l e a v i n g a few massless c h i r a l bound states. Indeed t h e r e a r e examples of models where our philosophy would then g i v e i n t e g e r i n d i c e s , b u t s i n c e w e must drop t h e requirement of n-dependence our r e s u l t was n o t unique and i t was always ugly. No such model seems t o reproduce anything reeembling t h e observed quark-lepton spectrum.

.

F i n a l l y t h e r e i s t h e remote p o s s i b i l i t y t h a t t h e paradoxes a s s o c i a t e d with higher s p i n massless bound states can be resolved. Perhaps t h e A(1236) p l a y s a more s u b t l e r o l e i n t h e a-model than assumed so f a r (we took i t t o be a p a r i t y doublet).

We conclude t h a t w e are unable t o c o n s t r u c t a bound s t a t e theory f o r the p r e s e n t l y fundamental fermions along t h e l i n e s

373

CHIRAL SYMMETRY A N D CHIRAL SYMMETRY BREAKING

s u g g e s t e d above. We thank R . van Damme f o r a c a l c u l a t i o n y i e l d i n g t h e i n d i c e s i n t h e c a s e G = SU(5). C

REFERENCES

-

1 . P.A.M.

2. 3. 4.

Dirac, N a t u r e 139 (1937) 323, P r o c . Roy. SOC. A165 (1938) 199, and i n : C u r r e n t T r e n d s i n t h e Theory o f F i e l d s , ( T a l l a h a s s e e 1978) AIP Conf. P r o c . No 4 8 , P a r t i c l e s and F i e l d s S u b s e r i e s No 15, e d . by L a n n u t i and Williams, p . 169. S. Dimopoulos and L. S u s s k i n d , Nucl. Phys. (1979) 237. M. Gell-Nann and M. Lgvy, Nuovo C i m . (1960) 705. B.W. Lee, C h i r a l Dynamics, Gordon a n d B r e a c h , New York, London, P a r i s 1972. G. ' t H o o f t , Phys. Rev. L e t t . 37 (1976) 8 ; P h y s . Rev. D 1 4 (1976) 3432. S . Coleman, "The Uses o f I n s t a n t o n s " , Erice L e c t u r e s 1977. R. J a c k i w and C. R e b b i , Phys. Rev. L e t t . 37 (1976) 172. C. C a l l a n , R. Dashen and D. Gross, Phys. L e t t . 63B (1976) 334. G. ' t H o o f t , Nucl. Phys. B72 (1974) 461. T. A p p e l q u i s t and J. C a r a z z o n e , Phys. Rev. D11 (1975) 2856. A. C a s h e r , C h i r a l Symmetry B r e a k i n g i n Q u a r k o n f i n i n g T h e o r i e s , T e l Aviv p r e p r i n t TAUP 734179 ( 1 9 7 9 ) .

-

-

5. 6. 7.

374

CHAPTER 6

PLANAR DIAGRAMS Introductions ................................................................... 16.11 “Planar diagram field theories”. in Progress in Gauge Field Theory. NATO Adv. Study I w t . Series. eds. G . ’t Hooft et al., Plenum. 1984. pp. 271-335 ............................................. [6.2] “A two-dimensional model for mesons”. Nucl . Phys . B75 (1974) 461470 .................................................................. Epilogue to the Two-Dimensional Model ..............................

375

376

378 443 453

CHAPTER 6

PLANAR DIAGRAMS Introduction to Planar Diagram Field Theories [6.1]

Consider an S U ( N ) gauge theory, possibly enlarged with scalars and/or fermions in the elementary or the adjoint representation. Take the limit g -+ 0, N + 00, g2N = ij2 fixed. In this limit only those Feynman diagrams survive that can be drawn on a plane without any crossings of the lines. We call these “planar diagrams”. Planar diagrams resemble the world sheets of strings, and 80 one is led to hope that perhaps the stringlike behavior of the gluon field confining quarks can be understood in this limit. Unfortunately, planar diagrams form a set that is still too large to be summed analytically by any known method. However, there is another important aspect to the planar diagrams. The perturbative expansion with respect to ij may converge much better than in theories with finite N. The reason for this is that the total number of planar diagrams is much smaller than the total of all diagrams. The only complication is that each individual diagram may give huge amplitudes at very high orders, because of the need for renormalization subtractions. The renormalized diagrams are finite, but huge. The bulk of the first paper in this chapter is devoted to the question of whether perturbation expansion defines a theory with infinite accuracy in the infinite N limit. The answer is affirmative, but we could prove this only if the coupling constant is sufficiently small and all masses sufficiently large, a physically uninteresting caw. So here we have a rigorously defined theory. Our construction is very technical. A casual reader may well be interested only in the first part where the planarity of the diagrams is proved, or possibly the less technical Section 16, where Bore1 summability is deduced. In Appendix A a curious model is discussed which is on the one hand an asymptotically free gauge theory and on the other hand there are a sufiicient number of scalar fields to render all particles massive. To achieve this it was compulsory to also have fermions present.

376

Appendix B is the well-known “spherical model”, a scalar N-component field in the N 00 limit. This model is extremely instructive. In this limit one may allow the coupling constant to be negative (it has to become negative at high energies!); at sufficiently large N the vacuum will become sufficiently stable. The model shows that if the original renormalized maas parameter vanishes there will still be a “spontaneously generated‘’ physical m a a for the particle. There is a (negative) lower limit for the allowed values of the renormalized mass, at which the bound state mass tends to zero. The dotted region in Figure 21 is an unphysical solution for m and M for the one given (negative) value of m R . --.)

Introduction to A Tw-Dimensional Model for Mesons [6.2] Since our original motivation for studying the N + 00 limit was our hope to find solvable versions of QCD and to understand quark confinement we now present one successful attempt. In one-space, one-time dimension this limit can indeed be solved. This is because there a gauge choice can be made such that the gauge field self-interactions disappear. The surviving planar diagrams are then the socalled rainbow diagrams,see Figure 2. They obey simple closed Dyson-Schwinger equations which can be solved. The propagator can be found in closed form, the mesonic bound states can be expressed as BetheSalpeter like integral equations, which can easily be solved numerically. At the end of this paper we add an epilogue.

377

CHAPTER 6.1

PLANAR DIAGRAM FIELD THEORIES Gerard 't Hooft Institute for Theoretical Physics Princetonplein, 5 P.O. Box 80.006 3508 TA Utrecht, The Netherlands

ABSTRACT

In this compilation of lectures field theories are considered which consist of N component fields qi interacting with N x N component matrix fields Ai' with internal (local or global) symmetry = Ng2 can group SIJ(N) or SO(N). The double expansion in 1/N and be formulated in terms of Feynman diagrams with a )ty structure. If the mass is sufficiently large and -g!lanarsufficiently small then the(extreme1y non-trivial) expansion in g2 at lowest order in 1/N is Borel summable. Exact limits on the behavior of expansion are derived. the Borel integrand for the

2

z2

1. INTRODUCTION

In spite of conbiderable efforts it is still not known how to compute physical quantities reliably and accurately in any fourdimensional field theory with strong interactions, It seems quite likely that if any strong interaction field theory exists in which accurate calculations can be done, then that must be an asymptotically free non-Abelian gauge theory. In such theories the small-distance structure is completely described by solutions of the renormalization group equations1; and there are reasons to believe that the continuum theory can be uniquely defined as a limit of a lattice gauge theory2, when the size of the meshes of the lattice tends to zero, together with che coupling constant, in a way prescribed by this renormalization group3. Indeed, one can prove using this formalism4 that this limit exists up to any finite order in the perturbation expansion for small coupling. However, this result has not been extended beyond perturProgress in Gauge Field Theory, NATO Adv. Srudy Inst. Series Edited by G 't Hoot3 er al. (Plenum, 1984).

378

G. 't HOQFT

bation expansion. It is important to realize that this might imply that theories such as "quantum chromodynamics" (QCD) are not based on solid mathematics, and indeed, it could be that physical numbers such as the ratio between the proton mass and the string constant do not follow unambiguously from QCD alone. In view of the qualitative successes of the recent Monte-Carlo computation techniques5 the idea that hadronic properties could be shaped by forces other than QCD alone seems to be far-fetched, but it would be extremely important if this happened to be the case. More likely, we may simply have to improve our mathematics to show that QCD is indeed an unambiguous theory. Either way, it will be important to extend our understanding of the suaunability aspects of higher order perturbation theory as well as we can. The following constitutes just such an attempt. There are two categories of divergences when one attempts to sum or resum perturbation expansion for a field theory in four space-time dimensions. One is simpJy the divergence due to the increasingly large numbers of Feynman diagrams to consider at higher orders. They grow roughly as n! at order g2n. This is a kind of divergence that already occurs if the functional integral is replaced by some ordinary finite-dimensional integral of similar type:

Here the diagrams themselves are bounded by geometric expressions but the numbers K(n) of diagrams of order n are such that the expansion is only asymptotic: I(g2)

+

1 K(n)

Cn g2n ;

n K(n)

+

n

a n!

where a is determined by one of the stationary points of the action S, called "instantons". However in four-dimensional field theories the diagrams themselves are not geometrically bounded. In some theories it has been shown6 that diagrams of the nth order that required k ultraviolet subtractions (with essentially k 5 n) can be bounded at best by

379

PLANAR DIAGRAM FIELD THEORIES

bn k! g2n

.

(1.3)

But since the total number of such diagrams grow at most as n!/k! we still get bounds of the form ( 1 . 2 ) for the total amplitude, however with a replaced by a different coefficient. This is a different kind of divergence sometimes referred to as ultraviolet 11renormalons'14 7. If a field theory is asymptotically free the corresponding coefficient b is negative and one might hope that the ultraviolet renormalons are relatively harmless. But in massless theories a similar kind of divergence will then be difficult to cope with: the infrared renormalone, which, as the word suggests, are due to a build-up of infrared divergences at very high orders: individual diagrams may still be convergentbuttheir sum diverges againwithn! The theories we will study more closely are governed by planar diagrams only. Their numbers grow only geometrically8 so that divergences due to instantons are absent. These diagrams are akin to but more complicated than Bethe-Salpeter ladder diagrams and by trying to sum them we intend to learn much about the renormalon divergences. Infinite color quantum chromodynamics is of course the most interesting example of a planar field theory but unfortunately our analysis cannot yet be carried out completely there. We do get bounds on the behavior of its Bore1 functions however (sect. 1 7 ) . Examples of large N field theories that we can handle our way are given in sect. 3 and appendix A. 2. F E Y "

RULES FOR ARBITRARY N

In order to show that the set of planar Feynman diagrams becomes dominant at large N values we first formulate a generic theory at arbitrary N, with a coupling constant g as expansion parameter in the usual sense. Let us express the fields as a finite number K of N-component vector? I@(x) (ail ,K; i = l , . N), and a small number D of N x N matrices AbiJ (x) , where b=I ,D and i,j=l,. ,N. Usually we will take 9 to be complex and AiJ to be Hermitean, in which case thes-try group will be U(N) or SU(N). The case that Jli are real and AiJ real and symmetric can easily be included after a few changes, in which case the symmetry group would be O(N) or SO(N), but the complex case seems to be more interesting from a physical point of view. In quantum chromodynamics K would be proportional to the number of flavors and D is the number of space-time dimensions plus two for the (non-Hermitean) ghost field.

,...

.., ,...

In general then the Lagrangian has the form

380

..

G. 't HOOFT

i(A,$,$*)

=

- a,.1.. Jlra(Mtb

abab

!& A A

+ gM1 abcAc + ig2M;bCdACAd)jlb

1 abcabc 12abcdabcd A A A A ) + ~ g R 1 A A A + T; g R2

,

(2.1)

with Aa.j 1 * (Aaji)*

.

Here the usual matrix multiplication rule with respect to the is implied, and Tr stands for trace with respect indices i,j, to these indices. The objects M and R carry no indices i,j but Furthermore M o , ~ and R o , ~may only "flavor" indices a,b,... contain the derivatives of $(XI or A(x). So the case that J, are fermions is included: then a,b,... may include spinor indices. The coupling constant g has been put in (2.1) in such a way that it is a handy expansion parameter.

...

.

...

In order to keep track of fhe indices i,j , it is convenient to split the fields AaiJ into complex fields for i > j and real fields for i = j. One can then denote an upper index by an incoming arrow and a lower index by an outgoing arrow. The propagator is then denoted by a double line. In fig. 1 , the A propagator stands for an AiJ propagator to the right if i > j ; an Aji propagator to the left if i < j and a real propagator if i = j. It is crucial now that the coefficients M and P in the Lagrangian respect the U(N) (or O ( N ) ) symmetry: they carry no Hence the vertices in the Feynman graphs only indices i,j, depend on these indices v i a Kronecker deltas. We indicate such a Kronecker delta in a vertex by connecting the corresponding index lines. Since we have a unitary invariance group these Kronecker deltas only connect upper indices with lower indices, therefore the index lines carry an or-Lentation which is preserved at the vertices. This is where the unitary case differs from the real orthogonal groups: the restriction to real fields with O(N) symmetry corresponds to dropping the arrows in Fig. 1: the index lines then carry no orientation.

... .

As for the rest the Feynman rules for computing a diagram are as usual. For instance, fernionic and ghost loops are associated with extra minus signs. In some theories (such as SU(N) gauge theories) we have an extra constraint: Tr Aa =

o

.

(2.3)

381

PLANAR DIAGRAM FIELD THEORIES

i

.

3

i b

Pab(k) = fRoI,k

(N x N matrix propagator)

3

i

i ba -

-I

Pia(k) = <Mo )ba

(N-vector propagatof)

C

a

Fig. 1: Feynman Rules at arbitrary N. The accolades { 1 stand for cyclic symmetrization with respect to the indices a,b,...

382

G. 't HOOFT

In that case an extra projection operator is required in the propagator:

The second term in (2.4) corresponds to an extra piece in the propagator, as given in Fig. 2. We will see later that such terms' are relatively unimportant as N + Q). For defining amplitudes it is often useful to consider source terms that preserve the (global) symmetry: + lJtb(x) Tr Aa(x)Ab(x)

pource = J ab (x)$*'(x)$~(x)

Q

.

(2.5)

The corresponding notation in Feynman graphs is shown in Fig. 3. 3. THE N

+QD

LIMIT AND PLANARITY

As usual, amplitudes and Green's functions are obtained by adding all possible (planar and non-planar) diagrams with their appropriate combinatorial factors. Note that, apart from the optional correction term in ( 2 . 4 ) , the number N does not occur in Fig. 1. But, of course, the number N will enter into expressions for the amplitudes, and that is when an index-line closes. Such an index-loop gives rise to a factor

l a ii I N .

(3.1)

i We are now in a position that we can classify the diagrams (with only gauge invariant sources as given by eq. (2.5)) according to their order in g and in N . Let there be given a connected diagram. First we consider the two-dimensional structure obtained by considering all closed index loops as the edges of little (simply connected) surface elements. All N x N matrix-propagators connecr these surface elements into a bigger surface, whereas the N-vector-propagators form a natural boundary to the total surface. In the complex case the total surface is an oriented one; in the real case there is no orientation. In both cases the total surface may be multiply connected, containing "worm holes". For convenience we limit ourselves to the complex (oriented) case, and we close the surface by attaching extra surface elements to all N-vector-loops.

-

Let that surface have F faces (surface elements), P lines (propagators) and V vertices. We haye F L + I Pt, where L is the number of N-vector-loops*) and I the number of index-loops; and we write (footnote: see next page)

383

+

PLANAR DIAGRAM FIELD THEORIES

Fig. 2: Extra term i n the propagator i f T r A i s to be projected out.

b Jab

(N-vector source)

JI

a

a

b

(matrix source)

Fig. 3: Invariant source insertions.

384

G. 't HOOFT

where V i s t h e number of n-point v e r t i c e s . (Vq is t h e number of B source i n s e r t i o n s ) . The diagram is now associated with a f a c t o r v3+2v4 I-Pt r =8 N Here Pt i s t h e number of t i m e s t h e second t e r m of ( 2 . 4 ) has been i n s e r t e d t o obtain t r a c e l e s s propagators. By drawing a dot a t each end of each propagator w e f i n d t h a t t h e t o t a l number of d o t s is

2~ = l n V n n

,

(3.3)

and eq. (3.2) can be w r i t t e n a s 2P-2V

r - 8

F-L- 2Pt N

(3.4)

Now we apply a well-known theorem of Euler: F-P+V= 2-2H,

(3.5)

where H counts t h e number of "wormholes" i n t h e surface and i s t h e r e f o r e always p o s i t i v e (a sphere has H = 0 , a t o r u s H = 1 ,

etc.).

And so, 2-2H-L-2Pt

4v3+vq

r = (g2N)

N

(3.6)

Suppose we take t h e l i m i t N + a0

, g + o , g2N = g2

I f t h e r e are N-vector-sources vector-loop:

(fixed)

.

(3.7)

then t h e r e must be a t l e a s t one N-

L > I .

The leading diagrams i n t h i s l i m i t have H = 0 , Pt = o and L = 1. They have one o v e r a l l m u l t i p l i c a t i v e f a c t o r N , and they a r e a l l planar: an open plane with t h e N-vector l i n e a t i t s edge (Fig. 4a).

*) "N-vector"

h e r e stands f o r N-component vector i n U(N) space, so t h e N-vector-loops a r e t h e quark loops i n quantum-chromodynamics.

385

PLANAR DIAGRAM FIELD THEORIES

Fig. 4: Elements of the c l a s s of leading diagrams i n the N - P l i m i t . a) I f vector sources are present. b) In the absence of vector sources ( e . g . pure gauge theory).

386

G. 't HOOFT

If there are only matrix sources then L 3 0 . The leading diagrams all have the topology of a sphere and carry an overall factor N2 (see Fig. 4b). We read off from eq. (3.6) that next to leading graphs are down by a factor I/N for each additional N-vector-loop (- quark loop in quantum chromodynamics) and a factor 1/N2 for each "wormhole". Also the difference between U(N) and SU(N) theories disappears as l/N2. It will be clear that this result depends only on the field variables being N-vectors and N x N matrices gnd the Lagrangian containing only single inner products or traces (not the pmduats of inner products and/or traces). Diagrams with L 1 and H o are the easiest to visualize. In the sequel we discuss convergence aspects of the summation of those diagrams in all orders of I. Our main examples are I ) U(N) (or SU(N)) gauge theories with fermions in the N representation; 2) purely Lorentz scalar fields, both in N and in N x N represen~ ,- T r Q 4 , tations of U(N). That theory will be called T ~ X $ I or if A is given the unusual sign. Both SU(N) gauge theory and - T ~ X I #are ~ asymptotically free9. The latter has the advantage that one may add a mass term, so that it is also infrared convergent. However the fact that X has the wrong sign implies that that theory only exists in the N + - limit, not for finite N, A model that combines all "good" features of the previous model is: 3) an SU(N) Higgs theory with N Higgs fields in the elementary representation, N fermions in the elementary representation and a fermion in the adjoint representation. A global SU(N) symmetry then survives. All vector, spinor and scalar particles * 1; are massive, and it is asymptotically free if

-

-

x2@

(3.8)

where h is a Yukawa coupling constant and X the Higgs self coupling. The reason for mentioning this made1 is that it is asymptotically free in the ultraviolet, and it is convergent in the infrared, so that our methods will enable us to construct it rigorously in the N -)QI limit (provided that masses are chosen sufficiently large and the coupling constant sufficiently small), and positivity of the Hamiltonian is guaranteed also for finite N so that there is every reason for hope that the theory makes sense also at finite N, contrary to the - X T ~ ~ I ~ theory. This model is described in appendix A.

4. THE SKELETON EXPANSION

- -

From now on we consider diagrams of the type pictured in 1 ) . They all have the same N dependence, so figure 4a (H 0 , L once we restricted ourselves to these planar diagrams only we may drop the indices i,j, and replace the double-line pro agators by single lines. Often we will forget the tilde (-) on gq because

...

387

PLANAR DIAGRAM FIELD THEORIES

.

the factor N is always understood. Only the (few) indices a,b,.. of eq, (2.1) , as far as they do not refer to the SU(N) group(a), are kept. The details of this surviving index structure are not important for what follows, as long as the Feynman rules (Fig. 1) are of the general renormalizable type,

Our first concern will be the isolation of the ultraviolet divergent parts of the diagrams. For t is we use an ancient It can be applied to 'any devicelo called "skeleton-expansion" * graph, planar or not, but for the planar case it is particularly useful.

5.

Consider a graph with at least five external lines. A oneparticle irreducible subgraph is a subset of more than one vertices with the internal lines that connect these vertices, that is such that if one of the internal lines is cut through then the subgraph still remains connected. We now draw boxes around all one-particle irreducible subgraphs that have four of fewer external lines. In general one may get boxes that are partially overlapping. A box is m d m a Z if it is not entirely contained inside a larger box.

Theorem: All maximal boxes are not-overlapping. This means that two different maximal boxes have no vertex in common. Proof: If two maximal boxes A and B would overlap then at least one vertex XI would be both in A and B. There must be a vertex xp in A but not in B, otherwise A would not be maximal. Similarly there is an x3 in B but not in A. Now A was irreducible, so that at least two lines connect xi with xp. These are external lines of B but not of A U B. Now B may not have more than 4 external lines. So not more than two external lines of A U B are also external lines of B. The others may be external lines of A. But there can also be not more than two of those. So A U B has not more than four external lines and is also irreducible since A and B are, and they have a vertex in common. So we should draw a box around A U B. But then neither A nor B would be maximal, contrary to our assumption. No planarity was needed in this proof.

The skeleton graph of the diagram is now defined by replacing all maximal boxes by single "dressed" vertices. Any diagram can now be decomposed into its "skeleton" and the "meat", which is the collection of all vertex and self-energy insertions at every two-, three- and four-leg irreducible subgraph. In particular the self-

*)

The method described here differs from Bjorken and Drell"

in that w e do not distinguish fermiona from bosons, so that also subgraphs with four external l'ines are contracted.

388

G. 't HOOFT

energy insertions build up the so-called dressed propagator. We call the dressed three- and fourvertices and propagators the "basic Green functions" of the theory. They contain all ultraviolet divergences of the theory. The rest of the diagram, the "skeleton" built out of these basic Green functions is entirely void of ultraviolet divergencies because there are no further (sub)graphs with four of fewer external lines, which could be divergent. The skeleton expansion is an important tool that will enable us to construct in a rigorous way the planar field theory. For, under fairly mild assumptions concerning the behavior of the basic Green functions we are able to prove that, given these basic Green functions, the sum of all skeleton graphs contributing to a certain amplitude i n Euotidean apaoe is abeolutely convergent (not only Bore1 sumPable). This proof is produced in the next 6 sections. Clearly, this leaves us to construct the basic Green functions themselves. A recursive procedure for doing just that will be given in sects. 12-15. Indeed we will see that our original assumptions concerning these Green functions can be verified provided the masses are big and the coupling constants small, with one exception: in the scalar -XTr+4 theory the skeleton expansion always converges even if the bare (minimally subtracted) mass vanishes! (sect. 18)

5. TYTE IV PLANAR F E Y "

RULES

We wish to prove the theorem mentioned in the previous section: given certain bounds for the basic Green functions, then the sum of all skeleton graphs containing these basic Green functions inside their "boxes" converges in the absolute. In fact we want a little more than that. In sects. 13-15 we will also require bounds on the total sum. Those in turn will give us the basic Green functions. We have to anticipate what bounds those will satisfy. In general one will find that the basic Green functions will behave much like the bare propagators and vertices, with deviations that are not worse than small powers of ratios of the various momenta. Note that all our amplitudes are Euclidean. First we must know how the dressed propagators behave at high and low momenta. The following bounds are required:

-

Here P (k) is the propagator. From now on we use the absolute value ab symbol for momenta to mean: lpl h.-' Then the field renormalization factor Z(k) is approximately:

389

PLANAR DIAGRAM FIELD THEORIES

where u is a coefficient that can be computed from perturbation expansion. The mass term in ( 5 . 1 ) is not crucial for our procedure but m in (5.2) can of course not be removed easily.

To write down the bounds on the three- and four-point Green functions in Euclidean space we introduce a convenient notation to indicate which external momenta are large and which are small. For any planar Green function we label not the external momenta but the spaces in between two external lines by indices 1,2,3,. which have a cyclic ordering. An external line has momentum

..

(5.3)

pi,i+l dEf pi -pi+l We have automatically momentum conservation,

(5.4) and the pi are defined up pi

+

pi+q

,

all i

to

an overall translation,

.

(5.5)

A channel (any in which possibly a resonance can occur) is gives by a pair of indices, and the momentum through the channel is given by

So we can look at the pi as dots in Euclidean momentum space, and the distance between any pair of dots is the momentum through some channel. If we write

(5.7)

So the brackets are around wmenta.that

form close clusters.

Our bounds for the three- and four-point functions are now defined in table 1 .

390

G. ‘t HOOFT

TabZe 1

Bounds for the 3- and 4-point dressedGreenfunctions. Zij stands for Z(pi-p*). All other exceptional mmentum configurations can be obtained by cyclic rotations and reflections of these. Ki are coefficients close to one.

Here a and @ are small positive coefficients. g(x) is a slowly varying running coupling constant. For the time being all we need is some g with y x Ki g(x) 6 8

for all x

,

(5.9)

where we also assume that possible summation over indices a,b,,.. is included in the K coefficients. Clearly the bare vertices would satisfy the bounds with a = B = 0 . Having positive a and B allows us to have any of the typical logarithmic expressions coming from the radiative corrections in these dressed Green functions. Indeed we will see later (sect, 13) that those logarithms will never surpass our power-laws. Table 1 has been carefully designed such that it can be reobtained in constructing the basic Green functions as we will see in sects. 12-14. irst we notice that the field renormalization factors Z(pi-pj)cancel against corresponding factors in our bounds for the propagator (5.1). The power-laws of Table 1 can be conveniently expressed in terms of a revised set of Feynman rules. These are given in Pig. 5. We call them type IV Feynman rules after a fourth attempt to refonnulate our bounds (types I, I1 and

F

391

-

PLANAR DIAGRAM FIELD THEORIES

1

(dressed elementary propagator)

(k2+rn2)

/L

kl

g[mX(

1 +3a

k2

1

2

Ikl I Ikp I ,Ikg I) I

(dressed 3vertex)

(composite propagator)

lkI2*

e k

1

1kl-O

(external line)

Fig. 5: Type IV Feynman Rules. lkl stands for

392

.

s

G. 't HOOFT

I11 occur in refs. 1 1 , 12 and are not neede here). The trick is , that represents to introduce a new kind of propagator, an exchange of two or more of the original particles in the diagram we started off with. The procedure adapted in these lectures deviates from earlier workll in particular by the introduction of the last two vertices in Fig. 5. Notice that they decrease whenever two of the three external momenta become large. It is now a simple exercise to check that indeed any diagram built from basic Green functions that satisfy the bounds of Table 1 can also be bounded by corresponding diagram(s) built from type IV Feynman rules. The four-vertex is simply considered as a sum of two contributions both made by connecting two three-point vertices with a type 2 propagator, and the factors lkl-" from the propagators in Fig. 5 are considered parts of the vertex functions (the mass term of the propagator may be left out; it is needed at a later stage). Type 2 propagators will also be referred to as "composite propagators"

.

Elementary power counting now tells us that the superficial degree of convergence, 2, of any (sublgraph with El external single linea and E2 external composite lines is given by 2 = (1-a)El + (2-B)Ez

-4.

(5.10)

Since we consider only skeleton graphs, all our graphs and subgraphs have

El + 2E2

>5

.

(5.1 1)

Thus, 2 is guaranteed to be positive if we restrict our coefficients by o < a < 115;

(5.12)

(Infrared convergence would merely require a < 1; B < 2, and is therefore guaranteed also.) So we know that with (5.12) all graphs and subgraphs are ultraviolet and infrared convergent. The theorem we now wish to prove is: the sum of all convergent type IV diagrams contributing to any given amplitude with 5 (or more) external lines converges in Euclidean space. It is bounded by the sum of all type IV tree graphs (graphs without closed loops) multiplied with a fixed finite coefficient.

A further restriction on the coefficients a and B will be necessary (eq. (8.15)).

393

PLANAR DIAGRAM FIELD THEORIES 6 . NUMBER OF TYPE IV DIAGRAMS

The total number G(E,L) of connected or irreducible planar diagrams with E external lines and L closed loops in any finite set of Feynman rules, is bounded by a power law (in contrast with the non-planar diagrams that contribute for instance to the Lth order term in the expansion such as ( 1 . 1 ) for a simple functional integral):

for some C1 and Cg. In some cases C1 and C2 can be computed exactly and even closed expressions for G(E,L) exist'. These mathematical exercises are beautiful but rather complicated and give us much more than we really need. In order to make these lectures reasonably selfsustained we will here derive a crude but simple derivation of ineq. (6.1) yielding C coefficients that can be much improved on, with a little more effort. Let us ignore the distinction between the two types of propagators and just count the total number G(E,L) of connected planar o 3 diagrams with a given configuration of E external lines and L closed loops. We have (see Fig. 6) G(E+l ,L) = G(E+2,L-I)

1

+

G(n+l ,Li)G(E+I-n,L-LI)

.

(6.2)

n,L1 G(E,L)

= o

G(2,o)

= 1

if

E < 2

or if

L <

o ;

.

(6.3)

n

E-n Fig. 6: Eq. (6.2)

We wish to solve, or at least find bounds for, G(E,L)

with boundary condition (6.3). A good guess is to try

394

from ( 6 . 2 j

G. ‘t HOOFT

(6.4)

which is compatible with ( 6 . 3 ) if

.

coc: > 1

(6.5)

Using the inequality k

4

1

1

(6.6)

2

nil n (k-n) we find that the r.h.8. of ( 6 . 2 ) will be bounded by E+2 L-1 c2

COCl

2 E+2 L

+

( E + I ) ~L~

16CoC1 C,

(6.7)

( E - I ) ~ ( L + I’ ) ~

which is smaller than E+1 L cocl c2

E~(L+I)‘

’

if

This is not incompatible with (6.5) although the best “solution“ to these two inequalities is a set of uncomfortably large values for C , , C, and C , . But we proved that they are finite.

The exact solution to eq. ( 6 . 2 ) is 2L (2E-2) !(2E+3L-4) ! L! (E-I)! (E-2)! (2E+2L-2)!

G(E’L)

’

(6. 10)

which we wiXl not derive here. Using (A+B)! A!B!

< 2A+B-I

’

(6.1 I )

we find that in ( 6 . I ) , C1

< 16

; Cp

< 16

.

(6.12)

395

PLANAR DIAGRAM FIELD THEORIES

For fixed E, in the limit of large L, Cq + 27/2

.

(6. 13)

Similar expressions can be found for the set of irreducible diagram. Since they are a subset of the connected diagrams we expect C coefficients equal to or smaller than the ones of eqs. (6. 12) and ( 6 . 1 3 ) . Limiting oneself to only convergent skeleton graphs will reduce these coefficients even further. We have for the number of vertices V V - E + 2 L - 2 ,

( 6. 14)

and the number of propagators P: ( 6 . 15)

P - V + L - I .

So, if different kinds of vertices and propagators are counted separately then the number of diagrams is multiplied with (6.16)

where C and C are some fixed coefficients. This does not alter V our result quafitatively Also if there are elementary 4-vertices then these can be considered as pairs of 3-vertices connected by a new kind of propagators, as we in fact did. So also in that case the numbers of diagrams are bounded by expressions in the form of eq. ( 6 . 1 ) .

.

7. THE W E S T FACETS We now wish to show that every planar type IV diagram with L loops is bounded by a coefficient CL times a (set of) type IV tree graph(s), with the same momentum values at the E external lines. This will be done by complete induction. We will choose a closed loop somewhere in the diagram and bound it by a tree insertion. Now even in a planar diagram some closed loops can become quite large (i.e. have many vertices) and it will not be easy to write down general bounds for those. Can we always find a "small" loop somewhere? We call the elementary loops of a planar diagram facets. Now Euler's theorem for planar graphs is:

Take an irriducible diagram. Write

396

G. 't HOOFT

(7.2)

L=1Fn, n

where F are the number of facets with exactly n vertices (or. It "tr corners 1. Let P-Pi+Pe,

(7.3)

where Pi is the number of internal propagators and Pe is the number of propagators at the edges of the diagram. Then, by putting a dot at every edge of each facet and counting the number of dote we get

1n

F~

n

-

2pi

+

.

pe

(7.4)

For the numbers V of n-point vertices we have similarly n Cnvn=2p+E, n

(7.5)

but in ourcase we only consider +point vertices (compare eqs. (6.14) and (6.15)):

3 v = 2P + E

.

Combining eqs. (7.1)

1 (n-6)Fn

= 2E

(7.6)

-

(7.6) we find

Pe

n

-6.

(7.7)

This equation tells us that if a diagram has L32E-8

(7.8)

then either it is a "seagull graph" (Pe 4 1 ) which we usually are not interested in, or there must be at least one subloop with 6 or fewer external lines: Fn > o

for some

n 9 6

.

(7.9)

So diagrams with given E and large enough L must always contain facets that are either hexagons or even smaller.

In fact we can go further:

theorem: if a planar graph (with only 3-vertices) and all its irreducible subgraphs have 2E

- Pe 2 6 then the entire graph obeys

(7.10)

397

PLANAR DIAGRAM FIELD THEORIES

This simple theorem together w i t h eq. (7.7) t e l l s u s t h a t any diagram with a number of loops L exceeding t h e bound of (7.10) must have a t l e a s t one elementary f a c e t w i t h 5 of fewer l i n e s a t t a c h e d t o i t . Although we could do without i t , i t i s a convenient theorem and now w e devote t h e rest of t h i s s e c t i o n t o i t s proof ( i t could be skipped a t f i r s t reading).

F i r s t we remark t h a t i f w e have t h e theorem proven for a l l irreducible graphs up t o a c e r t a i n o r d e r , then i t m u s t a l s o hold f o r reducible graphs up t o t h e same order. This i s because i f w e connect two graphs with one l i n e w e g e t a graph 3 with

I f L 1 , E l and L 2 , E 2 s a t i s f y (7.10) then s o do L 3 and E3 (remember t h a t E and L are i n t e g e r s and t h e smallest graph with L > o h a s E 6; propagators t h a t form two edges of a diagram are counted twice i n P ) e

.

Forthe i r r e d u c i b l e graphs w e prove ( 7 . 1 0 ) by a r a t h e r unusual e consider t h e o u t e r r i m induction procedure f o r p l a n a r graphs. W of an i r r e d u c i b l e graph and a l l t h e ( i n general not i r r e d u c i b l e ) graphs i n s i d e i t ( s e e Fig. 7 ) . Let t h e e n t i r e graph have E ext e r n a l l i n e s and Pe propagators a t i t s s i d e s . The subgraphs i i n s i d e t h e r i m have e i e x t e r n a l l i n e s and Pei propagators a t t h e i r s i d e s . We count: p e = ~ + l e i ,

(7.12)

i and t h e number o f loops L of t h e e n t i r e diagram i s

Now each f a c e t between t h e subgraphs and t h e r i m must have a t least 6 propagators as supposed, t h e r e f o r e

(7.14) b u t i f some of t h e subgraphs a r e s i n g l e propagators we need t o be more p r e c i s e

Pe +

1 Pe i

+ 2

1 ei

26

1 (ei-l) i

398

+ 6 +

2N2

,

(7.15)

G. ‘t HOOFT

I

Fig. 7: Proving the theorem of sect. 7. The number E counts the external lines of the entire graph. Pe the number of sides and ei and P,i do the same for the subgraphs 1 and2. N is the number of single propagators.

where N p is the number of single propagators, each of which contributes with e = 2 in eq. (7.131, and have Pei = 0 . Now we use (7.12) , and I

Pei

< 2ei

-6,

(7.16)

as required, whereas C(2e-6) + 2N2 = o for the single propagators, to arrive at (7.17)

E>Jei+6. 1

From the assumption that all subraphs already satisfy (7.10) we get, writing L1 = F L and El = 9 . :

l

i

1

399

PLANAR DIAGRAM FIELD THEORIES

(7.18) and from (7.13) L
.

(7.19)

With (7.17) which reads E > El + 6 we now see t h a t (7.10) again holds f o r t h e e n t i r e diagram. The quadratic expression i n (7.10) i s the sharpest t h a t can be derived from (7.17)-(7.19), and indeed l a r g e diagrams t h a t s a t u r a t e t h e i n e q u a l i t y can be found, by j o i n i n g hexagons i n t o c i r c u l a r p a t t e r n s . Our conclusion i s t h a t i f we wish t o use an induction procedure t o express a bound f o r diagrams with type I V Feynman rules and L loops i n terms of one with a smaller number L' loops we can t r y t o do t h a t by r e p l a c i n g successively t r i a n g l e s , qudrangles and/or pentagons by type I V t r e e i n s e r t i o n s , u n t i l the bound (7.10) i s reached. I n p a r t i c u l a r i f E = 5 t h i s leads us t o a t r e e diagram. The next t h r e e s e c t i o n s show how t h i s procedure works i n detail.

8. TRIANGLES Consider a ( l a r g e ) diagram with type I V Feynman r u l e s . We had already decreed t h a t i t and a l l i t s subgraphs a r e u l t r a v i o l e t and i n f r a r e d convergent (divergent subgraphs had been absorbed into the v e r t i c e s and propagators b e f o r e ) . With eqs. (5.10)-(5.12) this mean8 t h a t each subgraph has

in p a r t i c u l a r , t h e r e a r e no self-energy blobs. F i r s t w e use the i n e q u a l i t y of Fig. 8 t o replace composite propagators by ordinary dressed propagators one by one u n t i l ineq. (8.1) f o r b i d s any f u r t h e r such replacements. The i n e q u a l i t y i s r e a d i l y proven: we w r i t e f o r the!propagators with i t s v e r t i c e s : . so,

Fig. 8. A composite propagator i s smaller than an elementary one.

400

G. 't HOOFT

-.

where k is t h e momentum through t h e propagator, Ipl I 2 I k! Ip21 > l k l , and a i = a f o r a dressed propagator, and a i B-1 f o r a composite propagator. A t t h e l e f t hand s i d e y = B-1, and a t t h e r i g h t hand s i d e y = a. Clearly we have

a and B being both c l o s e t o zero due t o ( 5 . 1 2 ) .

Now consider all elementary t r i a n g l e loops i n o u r diagram. Under what conditions can we r e p l a c e them by type IV 3-vertices (Fig. 9 ) ? Due t o (8.1) t h e r e can be a t most one elementary (dressed e x t e r n a l l i n e , the o t h e r s are composite: E2 2 o r 3;

-

Fig. 9: Removal of elementary t r i a n g l e f a c e t s .

El

-

1 or

0.

a1 = a

We write or

8 -1

(8.4)

t o cover both cases. Now l e t us r e p l a c e t h e v e r t e x functions by bounde t h a t depend only on t h e momenta of t h e i n t e r n a l l i n e s :

with

Y

-

1+3a ; R ( ~ I) 2 3 a * ,

401

PLANAR DIAGRAM FtELD THEORIES y = 8+2a ; R(y)

or

= 1

.

(8.6)

The integral over the loop momentum is then bounded by 8 terms all of the form

where for a moment we ignored the mass term. It can b e added easily later. We have convergence for all integrals:

z

= 2

1 6 - 4 = 1-2p-al >

0

(8.8)

.

and 61 > I +a -1(2a+~)- 1(1+2a+al) = l(l-2a-al-8)

;

(8.9)

The integral (8.7) can be done using Feynman multiplicators:

(8.10)

Now i f lpl 3 141 > lp+ql (the other cases can be obtained by permutation) then lql > 4 Ipl , so our integrals are bounded by (8.11)

where C is the sum of integrals of the type

which can be further bounded (replacing ~ 1 x 2by 1~1x2)by (8.13)

if all integrals converge, of course. All entries in the

402

r

G. 't HOOFT

functions must be positive. In particular, we must have 2

-

62

- 63 = 6 1 - lz > o .

(8.14)

Now with (8.8) and (8.9) this corresponds to the condition: B>2a

,

(8.15)

this is the extra restriction on the coefficients a and t3 to be combined with (5.12), and which we already alluded to in the end of sect. 5 . A good choice may be u

-

0.1

,6

-

0.3

.

(8.16)

We conclude that we proved the bound of Fig. 9 , if a and B have values such as (8.16), and the number C in Fig. 9 is bounded by the sum of eight finite numbers in the form of eq. (8.13). 9. QUADRANGLES

We continue removing triangular facets from our diagram, replacing them by single 3-vertices, following the prescriptions of the previous sections. We get fewer and fewer loops, at the cost of at most a factor C for each loop. Either we end up with a tree diagram, in which case our argument is completed, or we may end up with a diagram that can still be arbitrarily large but only contains larger facets. According to sect. 7 there must be quadrangles and/or pentagons among these. Before concentrating on the quadrangles we must realize that there still may be larger subgraphs with only three external lines. In that case we consider those first: a minimal triangular subgraph is a triangular subgraph that contains no further triangular subgraphs. If our .diagram contains triangular subgraphs then we first consider a minimal triangular subgraph and attack quadrangles (latdr pentagons) in these. Otherwise we consider the quadrangles inside the entire diagram. 0 ) Let us again replace as many composite propagators ( 0 by single dressed propagators (-) 1 as allowed by ineq. (8.1) for each subgraph. Then one can argue that as a result we must get at least one quadrangle yomewhere whose own propa ators are all of not composite ( 0 a). This is the elementary type ( -1, because facets with composite propagators now must be adjacent to 4-leg subgraphs (elementary facets or more complicated), and then these in turn must have facet(s) with elementary propagators. Also (although we will not really need this) one may argue that there will be quadrangles with not more than one external composite propagator, the others elementary (the one exception is the case when one of the adjacent quadranglular subgraphs has itself only

'

403

PLANAR DIAGRAM FIELD THEORIES

+ b

C

Fig. 10. Inequality for quadrangles. a, b and c may each be 1 or2.

pentagons, but that case will be treated in the next section). As a result of these arguments, of all inequalities of the type given in Fig. 10 we only need to check the case that only one external propagator is composite, a = b = c * 1 . But in fact they hold quite generally, also in the other cases. This is essentially because of the careful construction of the effective Feynman rules of type IV in Fig. 5. Rather than presenting the complete proof of the inequalities of Fig. 10 (5 different configurations) we will just present a simple algorithm that the reader can use t o prove and understand these inequalities himself. In general we have integrals of the form

We could write this as a diagram in Fig. 1 1 , where the 6i at the propagators now indicate their respective powers. The vertices are here ordinary point-vertices, not the type IV rules. Now write

with

(9.3) Inserted in a diagram, this is the inequality pictured in Fig. 1 1 .

404

G. ‘t HOOFT

a

6

We use it for instance when p1-p2 is the largest momentum of all channels, and if W l

< 261

; wp

< 262

; wi

<2 ,

(9.4)

-

where Z is the degree of convergence of the diagram: 2 = 2C6i 4. We continue making such insertione, everytime reducing the diagram% t o a convenient momentum dependent factor times a less convergent diagram. Finally we may have 2

< 2 6 1 , z < 262 ,

(9.5)

for two of its propagators. Then we use the inequality pictured in Fig. 12: d4k

n

(k-pi)

-26.

I

i Notice that in (9.5) we have a strictly unequal sign, contrary to ineq. (9.4). This C can be computed using Feynman multiplicators,

405

PLANAR DIAGRAM FIELD THEORIES

Fig. 12. Inequality holding if 6 1 , ~are strictly larger than 42 (ineqs. (9.41, (9.5)).

much like in the previous section. We get (9.7)

We see that ineqs. (9.2) and (9.6) have essentially the same effect: if two propagators have a power larger than a certain coefficient they allow us to obtain as a factor a corresponding "propagator" for the momentum in that particular channel. This is how proving the ineqs. of Fig. 10 can be reduced to purely algebraic manipulations. We discovered that ineq. (8.15) is again crucial. We notice that if the internal propagators of the quadrangle were elementary ones then the superficial degree of convergence of any of the other subgraphs of our diagram may change slightly, since in eq. (5.10) 6 > 2a, but the left hand side of eq. (5.11) remains unchanged, because AE1 = -2AE2

,

(9.8)

so our condition that all subgraphs be convergent remains fulfilled after the substitution of the inequalities of Fig. 10.

However, if one of the internal lines of the quadrangle had been a composite one (-), 2 then a subgraph would become more divergent, because we are unable to continue our scheme with something like a three-particle composite propagator ( *). A crucial point of our argument is that we will never really need such a thing, if we attack the quadrangle subgraphs in the right order. 10. PENTAGONS. CONVERGENCE OF THE SKELETON EXPANSION

As stated before, the order in which we reduce our diagram

406

G. 't HOOFT

into a tree diagram is: 1 ) remove triangular facets; 2) remove triangular subgraphs if any. By complete induction we

3)

4) 5)

6)

prove this to be possible. Take a minimal triangular subgraph and go to 3; remove quadrangular facets as far as possible. If any cannat be removed because of a crucial composite propagator in them, then remove qudrangular subgraphs, After that we only have to remove the pentagons. If we happened to be dealing with a subgraph by branching at point 2 or 3, then by now that will have become a tree graph, because of the theorem in sect. 7. Go back to 1.

We still must verify point 5 . If indeed our whole diagram contains pentagons then we can replace all propagators by elementary ones. But if we had branched at steps 2 or 3 then the subgraphs we are dealing with may still have composite external propagator(s). In that case it is easy to verify that there will be enough pentagons buried inside our subgraphs that do not need composite external lines. In that case we apply directly the inequality of Fig. 13. The procedure for proving Fig. 13 is

Fig. 13. Inequality for pentagons.

exactly as described for the quadrangles in the previous section. Again the degree of divergence of any of the adjacent subgraphs has not changed significantly. This now completes our proof by induction that any planar skeleton diagram with 5 external lines is equal to CL times a diagram with type IV Feynman rules, where C is limited to fixed bounds. Since also the number of diagrams is an exponential function of L we see that for this set of graphs perturbation expansion in g has a finite radius of convergence. The proof given here is slightly more elegant than in Ref. 1 1 , and also leads to tree graph expressions that are more useful for our manipulations.

407

PLANAR DIAGRAM FIELD THEORIES

If the diagram has 6 or more external lines then still a number of facets may be left, limited by ineq. (7.10), all having 6 or more propagators. If we wish we can still continue our procedure for thes.e but that would be rather pointless: having a limited number of loops the diagram is finite anyhow. The difficulty would not so much be that no inequalities for hexagons etc,. could be written down; they certainly exist, but our problem would be that the corresponding number C would not obviously be bounded by one universal constant. This is why our procedure would not work for nonplanar theories where ineq. (7.10) does not hold. In the non-planar case however similar theorems as ours have been derived6. 1 1 . BASIC GREEN mTNCTIONS

The conclusion of the previous section is that if we know the "basic Green functions", with which we mean the two- three- and four-point functions, and if these fall within the bounds given in Table 1, then all other Green functions are uniquely determined by a convergent sum. Clearly we take the value for the bound g2 for the coupling constant (ineq. (5.9)) as determined by the inverse product of the coefficient C2 found in sect. 6 and the maximum of the coefficients in the ineqs. pictured in Figs. 9, 10 and 13, times acombinatorial factor. Now we wish not only to verify whether these bounds are indeed satisfied, but also we would like to have a convergent calculational scheme to obtain these basic Green functions. One way of doing this would be to use the Dyson-Schwinger equations. After all, the reason why those equations are usually unsoluble is that they contain all higher Green functions for which some rather unsatisfactory cut-off would be needed. Now here we are able to re-express these higher Green functions in terms of the basic ones and thus obtain a closed set of equations. These Dyson-Schwinger equations however contain the bare coupling constants and therefore require subtractions. It is then hard to derive bounds for the results which depend on the difference between two (or more) divergent quantities. We decided to do these subtractions in a different way, such that only the finite, renormalized basic Green functions enter in our equations, not the bare coupling constants, in a way not unlike the old "bootstrap" models. Our equations, to be called "difference equations" will be solved iteratively and we will show that our iteration procedure converges. So we start with some Ansatz for the basic Green functions and derive from that an improved set of values using the difference equations. Actually this yill be done in various steps. We start with assuming some function g(x) for the f l o a t i n g coupting conetrmt, where x is the momentum in the maximal channel (see Table 1):

G. 't HOOFT (11.1)

and a aet of functions g(i)(x) g(x)

-

def

max Ig(;)(X)l

i

with

.

(11.2)

Here g i)(x) is the set of independent numbers that determined the basic f,reen function at their "sylmnetry point": Ipi-p.I 3

= x

for all i,j

.

(11.3)

The index i in ghi) then simply counts all configurations in (11.3). With "in ependent" we mean that in some gauge theories we assume that the various Ward-Slavnov-Taylor' identities among the basic Green functions are fulfilled. This is not a very crucial point of our argument so we will skip any further discussion of these Ward or Slavnov-Taylor identities. If the values of the basic n-point Green functions (n = 3 or 4 ) at their symmetry points are Ai(x), then the relation between A. and g is:

i

1

,

( I I .4)

where K' are coefficients of order one, and Z(x) is defined in (5.1) "and (5.2). (We ignore for a moment the case of superrenormalizable couplings.) Our first Ansatz for gi(x) is a set of functions that is bounded by (11.21, with g(x) decreasing asymptotically to zero for large x as dictated by the lowest order term(s) of the renormalization group equations. We will find better equations for gi(x) as we go along. In any case we will require (11.5)

for some finite coefficient

iT.

Our first Ansat2 for the basic Green functions away from the symmetry points will be even more crude. All we know now is that they must satisfy the bounds of Table 1 . In general one may start with choosing (11.4) to hold even away from the symmetry points, and x

-

max Ip.-p.1 i ,j 1 3

.

( I 1.6)

PLANAR DIAGRAM FIELD THEORIES

After a few iterations we will get values still obeying the bounds of Table 1 , and with Uncertainties also given by Table 1 but K; replaced by coefficients 6Ki. Thus we start with

.

6Kio) = K.1

(11.7)

We will spiral towards improved AnsLttze for the basic Green functions in two movements: i) the "small spiral" is the use of difference equations to obtain improved values at exceptional momenta, g i v e n the values g i ( x l a t the symmetry points. These difference equations will be given in the next section. ii) The "second spiral" is the use of a variant of the Gell-MannLow equation to obtain improved functions g;(x) from previous Ansstze for g;(x), making use of the convergent "small spiral" at every step. What is also needed at every step here is a set of integration constants determining the boundary condition of this Gell-Mann-Low equation. It must be ensured that these are always such that g(x) in ineq. (11.2) remains bounded: g(x) < go

3

where go is limited by the coefficients Ki and the various coefficients C from sects. 6, 8, 9 and 10, as in ineq. (5.9). 12. DIFFERENCE EQUATIONS FOR BASIC GREEN FUNCTIONS

The Feynman rules of our set of theories must follow from a Lagrangian, as usual. For brevity we ignore the Lorentz indices and such, because those details are not of much concern to us. Let the dressed propagator be

P(P> = -GZ1(p)

,

(12.1)

and let the corresponding zeroth order, bare expressions be indicated by adding a superscript 0. In massive theories:

so

that P(p+k)

-

P(k) = P(p+k)Gq,(plk)k,,P(p)

.

(12.4)

This gives us the "Feynman rule" for the difference of two dressed

410

G. 't HOOFT

Fig. 14. Feynman rule for the difference of two dressed propagators. The 3-vertex at the right is the function G,,,(p Ik).

F i g . 15. Difference equation (12.6) for G

41 1

2lJ

PLANAR DIAGRAM FIELD THEORIES

Fig. 16.Some arbitrarily chosen terms in the skeleton expansion for G2VUX'

propagatore, depicted in Fig. 14, (Note that, in this section only, p and k denote external line momenta, not external loop momenta.)

We have also this Feynman rule for bare propagators. There Go follows directly from the Lagrangian: 2P

-

-

.

2p,, k,, GiP Continuing this way we define

- G2P(plk)

G2,,(plk+q)

(12.5)

G2,,,,(plklq)q,,

,

(12.6)

with Gipu

-6

VV

-

(12.7)

In Feynman graphs this is sketched in Fig. 15. Differentiating once more we get G2,,,,(plklq+r)

- G2,,,,(plklq)

G2,,vX(plklq r)rX

.

(12.8)

Of course GzP,,~can be computed formally in perturbation expansion. The rules for computing the new Green functions G G,,,,, GPYl are easy to establish. Let p1 be one of the external Y;op momenta as defined in eq. (5.3). For a Green function G(p1) we have

(12.9)

where fi(qi) are bare vertex and/or propagator functions adjacent to the external facet labeled by 1. The remainder F is independent of p1. We write

412

G. 't HOOFT

which is just the rule for taking the difference of two products. We find the difference of two dressed Green functions G in terms of the difference of bare functions f. Therefore the "Feynman rules" for the diagrams at the right hand sides of Figs. 14, 15 and the 1.h.s. of Fig. 16 consist of the usual combinatorial rules with new bare vertices given by the eqs. (12.5) and (12.7). These bare vertices occur only at the edge of the diagram. We see that the power counting rules for divergences in GzU,,~ are just as in 5-point functions in gauge theories. Since the global degree of divergence is negative we can expand in skeleton graphs. See Fig. 16, in which the blobs represent ordinary dressed propagators and dressed vertices or dressed functions G,, and G,,,,. Notice that one might also need G3,,(pl,p2lk) G3(P1 ,p2+k)

- G3 (P1 rP2)

G3p (P1 rP2 Ik) o k p

defined by (12.11)

In short, the skeleton expansion expresses G 2 p V ~but also G 3 ~ v etc. in terms of the few basic functions Gp,,, GpVv, G3,, and the basic Green functions G2,3,1+.Also the function GkU, defined similarly, can thus be expressed. The corresponding Feynman rules should be clear and straightforward. We conclude that the basic Green functions can in turn be expressed in terms of skeleton expansions, and, up to overall constants, these equations, if convergent, determine the Green functions completely. Notice that we never refer to the bare Lagrangian of the theory, so, perhaps surprisingly, these sets of equations are the same for all field theories. The difference between different field theories only comes about by choosing the integration constants differently

.

Planarity however was crucial for this chapter, because only planar diagrams have well defined "edges": the new vertices only occur at the edge of a diagram.

13. FINDING THE BASIC GREEN FUNCTIONS AT EXCEPTIONAL MOMENTA (THE "SMALL SPIRAL") In this section we regard the basic Green functions at their synmetry points as given, and use the difference equations of

413

PLANAR DIAGRAM FIELD THEORIES

Sect. 1 2 to express the values at exceptional momenta in terms of these. If pi-pj is the momentum flowing through the planarchannel ij, then in our difference equations we decide to keep p = max Ip;-p. 3

I

(13.1)

i,j

fixed. So the left hand side of our difference equations will show two Green functions with the same value for u, one of which may be exceptional and the other at its symmetry point, and therefore known. (Weuse the concept of "exceptional momenta" as in ref, 14.) Now the right hand side of these difference equations show a skeleton expansion of diagrams which of course again contain basic Green functions, also at exceptional momenta. But these only come in combinationsofhigher order, and the effect of exceptional mumenta is relatively small, so at this point one might already suspect that when these equations are used recursively to determine the exceptional basic Green functions then this recursion might converge. This will indeed be the case under certain conditions as we will show in sect. 15. Our iterative procedure must be such that after every step the bounds of Table I again be satisfied. This will be our guide to define the procedure. First we take the 4-point functions, and consider all cases of Table 1 separately. The right hand side of our difference equations (Fig. 16) contains a skeleton expansion to which we apply the theorem mentioned in the end of sect. 5 and proven in sects. 5-10: the skeleton expansion for any 5-point Green function converges and is bound by free diagrams constructedwith type IV Feynman rules, Since thq 5-point functions in Fig. 16 are irreducible, the internal lines in the resulting tree graph will always be composite propagators, as in the r.h.s. of Fig. 13. So we simply apply the type IV Feynman rules for 5 tree graphs to obtain bounds on the 5-point function in various exceptional regions of momentum space. Table 2 lists the results. The power of g 2 ( A 3 ) in the table applies where we consider the function Gqu. The other functions G3pv and G2,,vh have one and zero powers of g- ( A 3- ) , respectively. In front of all this comes a power-series of the form m

1

n= 1

c"g$

-

cg;(l-cs~)-l

(13.2)

which converges provided that

414

G. ‘t HOOFT

Table 2 Bounds for the irreducible 5-point function at some exceptional momentum values.

go

- yx

Ig(ll)l <

c-4

(13.3)

We now write an equation such as (12.8) as follows: E4{(((12)1

312 413)

G~,{((523)2 413) + -t

(Pl-PS),, G4p{(((12)1

5312 413)

Y

(13.4)

where r = p1-p5. In this and following expressions the tilde (-) indicates which quantities are being replaced by new ones in the iteration procedure. If the Ansatz holds for G4{ ((523)2 413) then the new exceptional function will obey

n

cg;

Choosing

, =- Y

(13.6)

1-cg; and considering that to a good apiroximation (since Ip1-p5l << Ip4-p 5 I : 245

(13.7)

z41 s

415

PLANAR DIAGRAM FIELD THEORIES

we find

(13.8) What is now needed is a bound €or the last term in (13.8).

Let (13.9)

x12 = lp121/m> 1 (remember that Ipl stands for 6 -1, and

(13.10) where u is defined in (5.2). When x12

> exp

(13.1 I )

( - u / 2 a ) = xo

this f is an increasing function, so that if xgm 6 A1

< A2

(13.12)

then

The range 1 < x < x o is compact, so there exists a finite number L such that (13.14) as soon aa x12 < x 5 2

-

(13.15)

So we find that after one iteration given by (13.4), coefficient satisfies

.

the new K2 (13.16)

K: < y + K ~ L

Similarly we derive

416

G. 't HOOFT

Kq
,

(13.17)

when the difference equation is used to express terms of G{ (523412). Also we use

Z;c{((1211 +

(341213) 'G;c{(52(34)2)3}

(p1-P5),

G4,,{ ((12)1 5(34)2)3)

E{ ( ( 1 2 )

1 3412) in

+

9

(13.18)

to find that after one step K: < y + K;L < y + YL + ~2 ;

(13.19)

and for the three-point function

K1
(13.20)

The remaining coefficients K4-6 must be computed in a slightly different way. Consider Kq. We replace p1 by p5 now in such a way that

and work with induction. Write

Inspecting Tables 1 and 2 we find now ( 1 3.23)

Applyink the same technique we compute the fifth exceptional configuration of Table 1 . We separate p1 from p 3 until A1 -.) A2. Then we separate alternatively p2 from p4 and p1 from p 3 keeping A1 m A 2 . This makes the rate of convergence slightly slower: (13.24)

417

PLANAR DIAGRAM FIELD THEORIES

where (13.26) We find

The sum can certainly be bounded:

1 < L'

Q

2L(1-2a-fl)-1

.

(1

3.28)

Therefore 2

K6 g m a x ( 1 , y L ' )

.

(13.29)

Thus all coefficients Ki have bounds that will be obeyed everywhere in the "small spiral" induction procedure. Note that these coefficients would b ow up if a,B + 0 . In particular in (13.11) we need a > 0. Only if go + o we can let a,B + 0. It will be clear from the above arguments that our bounds are only very crude. Our present aim was to establish their existence and not to find optimalbounds.

!

In sect. 15 we show that the "small spiral" of iterations for the exceptional Green functions, given the non-exceptional ones in (13.27), actually converges geometrically.

14. NON-EXCEPTIONAL MOMENTA (THE

1

*

SPIRALII) ~

~

~

In order to formulate the complete recursion procedure for determinihg the basic Green functions we need relations that link these Green functions at different symmetry points. Again the difference equations are used:

Here pi and pji) are external Loop momenta. They are nonexceptional. JWe use a shorthand notation for (14.1). Writing

418

~

~

G. 't HOOFT

(14.2) Similarly we have (14.3) These are just discrete versions of the renormalization group equations. The right hand side of (14.21, [not (14.3)Il is to be expanded in a skeleton expansion which contains all basic Green functions at all p , also away from their symmetry points. There we insert the values obtained after a previous iteration. It is our aim to derive from eq. (14.2) a Gell-Mann-Low equation' of the form

+

lg(p)l

N

Pi(d ,

(14.4)

where B(') are the first k coefficients of the B function, and they must coincide with the perturbatively computed B coefficients. Often (depending on the dimension of the coupling constant) only odd powers occur so that k N-2 is odd. The rest function p must satisfy

-

Ipi(dl

,Q

(14.5)

for some constant QN
(14.6)

(so that p' may be smaller than p), in such a way that the absolute value of each diagram contributes to % and their total sum remains finite. Now clearly eq. (14.2) is a difference equation, not a differential equation such as (14.4). Up till now differential equations were avoided because of infrared divergences. Just for ease of notation we have put (14.4) in differential firm because the mathematical convergence questions that we are to consider now are insensitive to this simplification. Consider the skeleton expansion of Gti) in (14.2). At each of the four external particle lines a factor g(pi) occurs with p i 2 p,

419

PLANAR DIAGRAM FIELD THEORIES

so it may seem easy to prove (14.4) from (14.2) with N = 3 or 4. However, we find it more convenient* to have an equation of the form (14.4) with N 6 7, qyd our problem is that the internal $,G might have momenta which are le4S thanu. vertices of the We will return to this question later in this section.

In proving the difference equation variant of (14.4) from (14.2) we have to make the transition from GI, to g2 and Gg to g, and this involves the coefficients Z(p), associated to the functions G2, by equations of the form G2(u) =

- u2Z-’(u)

s

G3(u) = ~z’~’~(~)g3(~);

G~(P)

(14.7)

Z-2(u>gc(u)

where gg, g4 are just various components of the coupling constant gi. In the following expressions we suppress these indices i when we are primarily interested in the dependence on u (= lpl at the symmetry point). Now from (14.2) and (14.3) we find not first order but third order differential equations for Gp, basically of the form (14.8)

where G2 XxA is just a shorthand notation for the combination of expandabie functions G~,A,,,, obtained after taking differences three times. Write (14.9) then

(14.10)

* Closer analysis shows that actually N = 3 or 4 is sufficient to prove unique solubility. Only if we wish an exact, non-perturbative definition of the free parameters we need the higher N values. Note that not only QN but also go may deteriorate as N increases.

420

G. ‘t HOOFT

Here A and B are free integration constants; A is usually determined by Lorentz invariance and B by the mass, fixed to be equal to m. In lowest order: A = m~p(m) ;

B

- lm2Ui(m) .

(14.12)

This strange-looking form of the integration constants is an artifact coming from our substitution of difference equations by differential equations. Using difference equations we can impose Lorentz invariance by spmetrization in momentum space, so that only one (for each particle) integration constant is left: the mass term. We choose at all stages iUp(m) = Z(m) 1.

-

A convenient way to implement eq. (14.12) is to formally define U2(p) = 2 if o 5 p 5 m, and replace the lower bound of the B = 0. integral in (14.11) by zero. Then after symmetrization: A

-

Equation (14.11) has a linearly convergent integral, whereas (14.10) is logarithmic. Together they determine the next iterative approximation to G In fact we have

.

(14.13)

and in f({g)), Z occurs only indirectly. So the iteration converges fastest if we replace (14.10) by

where the tilde denotes the new function Up(p). One can however also use (14.9) with Up replaced by We find

& ! 2’ ap

=

-

1

dr(I-T)uGp,AXX(tp)

m/u As stated before, the

zation.

.

52. (14.15)

@( f) terms have been removed by symmetri-

This equation allows us to remove the Z factors from the functions 6 3 . 4 and arrive at first order renormalization group integrodifferential equations for gi(p). For the 3-point functions we must write

421

PLANAR DIAGRAM FIELD THEORIES (14.16) (14.17)

(14.18)

A potential difficulty in writing down the renormalization group equation even for N = 4 is the convolutions in (14.15) and (14.18) which contain Green functions at lower v values, and so they depend on g(p') with p ' < 1-1. So a further trick is needed to derive ( 1 4 . 4 ) . This is accomplished by realizing that the integrals in (14.15) and (14.18) converge linearly in u . Suppose we require at every iteration step (see eq. 1 1 . 5 ) : (14.19)

and for some B then

<m,

go <

=.

Then it is easy to show that if

u1

5

p,

(14.20)

if

with C large enough and go small enough we can make & as small as we like. Inequality ( 1 4 . 2 0 ) is proven by differentiating with u . -This enables us to replace g(T1.l) by g(U) in ( 1 4 . 1 5 ) and ( 1 4 . 1 8 ) while the factor T-& does no harm to our integrals. So we find So

and @ E3 in terms of a power series of g(p). We bounds for @ l? au au must terminate the series as soon as the factors T-' accumulate to give T - ~ . This implies that N must be kept finite, otherwise go

+

0.

The same inequality (14.16) is used to go from N = 4 to N = 7 in these equations. If in a skeleton diagram a vertex is not associated with any external line, then it may be proportional to a factor g(p') with < p . But using ( 1 4 . 1 6 ) we see that it may be replaced by g(p) at the cost of a factor (p/ul)'. At most three of these extra factors are needed. If the three corresponding vertices are chosen not to be too far away from one of the external vertices of the diagram (which we can always arrange), then this just corresponds to inserting an extra factor

422

G. 't HOOFT

(Cr

at an external vertex. We now note that such factors still

leave our integrals convergent. In the ultraviolet of course the diagrams converge even better than they already did, and in the infrared our degree of convergence was at least I-a (or 2-8) as can easily be read off from eq. (5.10): adding the external'propagators to any diagram one demands Z

+ (2+2u)E1 + 28E2 < 4(Ei+E2-1)

,

(14.22)

Thus infrared convergence requires 1 - a -T& > o

( 14.23)

where T 5 5 is the number of times our inequality (14.20) was applied. From the above considerations we conclude that an equation of the form (14.4) can be written down for any finite N, such that % in inequality (14.5) remains finite. We do expect of course that QN might increase rapidly with N, but then we only want the equation for N 2 7. We are now in a position to formulate completely our recursive definition of the Green functions G2, C3, G4 of the theory: 1) We start with a given set of trial functions G2(u),

for determine gi(v) and behaviour G4(u)

G3(u),

the basic Green functions at their symmetry points. They our initial choice for the floating coupling constants the functions Z . ( p ) . We require their asymptotic to satisfy ( 5 . 2 f , (5.9) and (11.5) (= eq. (14.19)).

2) We also start with an Ansatz for the exceptional Green functions that must obey the bounds of Table 1 . 3) Usa the difference equations of sect. 13 to improve the exceptional Green functions (the new values are indicated by a tilde (-))'. These will again obey Table 1 as was shown in sect. 13. Repeat theprocedure. It will converge towards fixed values for the exceptional Green functions (as we will argue in sect. 15). This we call the "small spiral".

4) With these values for the exceptional Green functions we are now able to compute the right hand side of the renormalization group equation for G2, or rather Z-', from (14.151, using (14.20):

l(p2 is again bounded.

where ficients

Here Yijk are the one-loopy coefThis gives us inproved propagators. See sect. (15.b).

423

PLANAR DIAGRAM FIELD THEORIES 5) Now we can compute the right hand side of eq. (14.4). Before integrating eq. (14.4) it is advisable to apply Ward identities (if we were dealing with a gauge theory) in order to reduce the number of independent degrees of freedom at each p. As is well known, in gauge theories one can determine all subtraction constants this way except those corresponding to the usual free coupling constants and gauge fixing parameter^'^. So the number of unknown funotions gi(u) need not exceed the number of "independent" coupling constants of the theory*.

6) Eq. (14.4) is now integrated, giving improved expressions for gi(P). Now go back to 2. This is the "second spiral", which will be seen to converge towards fixed values of g;(p). The question of convergence of these two spirals is now discussed in the following section. 15. CONVERGENCE OF THE PROCEDURE a ) Except7:onaZ

Mmenta

In sect. 13 a procedure is outlined to obtain the Green functions at exceptional momenta, if the Green functions at the symmetry point are given. That procedure is recursive because eqs. (13.4), (l3.18), (13.22) and (13.25) determine the Green functions G2,3,4 in terms of the symmetry ones, and GI,,,, G3,,,,, G2*,,1. But the latter still contain the previous ansatz for G2,3,4. Fortunately it is easy to show that any error 6G2,3,4 will reduce in size, so that here the recursive procedure converges: Let us indicate the bounds discussed in sects. 5 and 13 as (15.1)

and assume that a fikst trial Gil)

has an error (15.2)

with some

&(')

5 2.

Now GI,,,, G3,,,,, G~,,vA also satisfy inequalities of the form Furthermore they were one order higher in g2. So we have

(15. I).

* We put "independent" between quotation marks because our requirement of asymptotic freedom usually gives further relations m n g various running coupling constants, see appendix A

424

G. 't HOOFT

(5.3) when the function GqP itself converges like

1 cngn

¶

and BI,,, is the bound for Gq,, itself, as given by Table 2. The procedure of sect. 13 can be applied unaltered to the error 6Gn in the Green functions. But there is a factor in front, (15.4) This gives €or the newly obtained exceptional Green functions an error l):IGS


(15.5)

and e(2) < c(') if we reduce the maximally allowed value for g, as given by (5.9), somewhat m r e :

~max+ 0.6527 gmax

.

(15.6)

We stress that the above argument is only valid as long as the Green functions at their symmetry points were kept fixed and are determined by g ( v ) , bounded by (15.6).

b ) The Z Factors

(15.7) (15.8) where the 0(g2) terms are again bounded by a coefficient times g2(u). These equations must be solved iteratively, because the right hand side of (15.7) contains skeleton expansions that again contain Z(u), hidden in the function l ( u 1 ) . It is not hard to convince oneself that such iterations converge. A change

425

PLANAR DIAGRAM FIELD THEORlES

(15.9)

yields a change in the function 1621 so

,< &"'g;

1 (PI)

bounded by

1,

(15.10)

that (15.11)

with c ( * ) < c ( l ) if go is small enough. c ) The

coupZing constants

We now consider the integro-differential equation ( 1 4 . 4 ) . The solution is constructed iteratively by solving

where the tilde denotes the next "improved"function gi(p). Our first Ansatz will be a solution of (15.12) with piig(v),vl replaced by zero. This certainly exists because the B coefficients are determined by perturbation expansion and therefore finite. The integration constants must be chosen such that for all u > m we have

where gmX is the previously determined maximally allowed value of g(p) and K is again a constant smaller than 1 to be determined later. In practice this requirement implies asymptotic freedom3: (15.14)

(It is constructive to consider also complex solutions.) If we now substitute this g(p) (15.12) we may find a correction: g(v)

g(ll) = g(v)

+

6g(v)

in the right hand side of eq.

,

(15.15)

for which we may require ISg(v)l

E gmax

for all p

.

426

(15.16)

G. 't HOOFT

We must start with: E+K

< 1

(15.17)

Will a recursive application of eq. (15.12) converge to a solution? Let the first Ansatz produce a change (15.16). The next correction is then, up to higher orders in 6g, given by

(where Mij is determined by differentiation of (15.12) with respect to gi(p). To estimate Sf(p) we must find a limit for the change in p . Our argument that l p l < came from adding the absolute values of all diagrams contributing to p , possibly after application of (14.20) several times. Replacing (14.20) then by

which indeed is true if g satisfies (15.18), or (15.20)

as can be derived from (14.20) and (l4.21), we find that we can write

,

1 6 P l < EC'

(15.21)

with C' slightly larger than C, and

ISf(v)l

2 &(N+l)C'

g(vIN

.

(15.22)

Now asymptotically, M.. (11) 13

-+

M!. /log v 1J

,

(15.23)

0 where Mi. is determined by one-loop perturbation theory. If there is only dne coupling constant it is the number 312. In the more general case we now assume it to be diagonalized:

M?. 1J

= M(i)6.. 1J

,

(15.24)

with one eigenvalue equal to 312. (Our arguments can easily be ex0 cannot be diagonalized, tended to the special situation when Mij in which case the standard triangle form must be used.) The asymp-

427

PLANAR DIAGRAM FIELD THEORIES

totic form of the solution to P

where p(i)

are integration constants. If M(i) <

choose v(i) = cases we get

If M(i)

QD.

N

>T -

3- I

then we

I we set p(i) * m. Then in both (15.26)

where C" is related to C' and the first 8 coefficient. In a compact set of p values where the deviation from (15.25) is appreciable we of course also have an inequality of the form (15.16).

-

If M(i) N / 2 - 1 then we simply pick another N value (which needs not be integer here), raising or lowering it by one unit. We see that we only need to consider N ( 4 . Comparing (15.26) with ( l S . l 6 ) , noting that C" is independent of A, we see that if (15.27)

then our procedure converges. Since C" stays constant or decreases with decreasing bx, we find that a finite gmX will satisfy (15.27). Also we should check whether i ( p ) satisfies the Ansatz ,j with unshanged 8 . This however is obvious from the construction of g through eq. (15.18). Notice that the masses are adjusted in every step of the iteration for the Z functions, by choosing A and B in section 14. They are necessary now because we wish to confine the integrals (15.25) at II > m, limiting the solutions g(p) to satisfy Ig(p) I 6 g.

(14.19)

16. BOREL SUMMABILITY~~ The fact that we obtained eq. (14.4) holding for m 6 P <= is central result. For simplicity of the following discussions we ignore the next-to-leading terms of the coefficients B. Let us = 3. We find that take the case that the leading ones have

OUK

(16.1)

G. 't HOOF7

(the next-to-leading Bcoefficients give an unimportant correction in the denominator of the form log logu2). The coefficients bi are fixed by the leading renormalization group B coefficients. We must choose C such that

lg2(v)I


if

(16.2)

p > m .

This is guaranteed if either

Now the requirement of asymptotic freedom is usually so stringent that the constant C is the only free parameter besides the masses and possible dimension I coupling constants (a situation corresponding to the necessity of choosing all p(i) in eq. (15.25)). But still we can do ordinary perturbation expansion, writing

-

sib) =

QI

(16.4)

big + Q(g3) ;

(16.5) where g is now a regular expansion parameter. The 0(g3) terms are fixed by the asymptotic freedom requirement and can be computed perturbatively. We now claim that perturbation expansion in g is indeed Borei-summable: (16.6a) 0

where the path C must be choosen to lie entirely in the region limited by eqs. (16.3) and (16.5) (see Fig. 17). Since in that region the Green functions G(g2) are approximately given by their perturbative values the integral (16.6b) will converge rapidly along this path, if Rez>o,

(16.7)

429

PLANAR DIAGRAM FIELD THEORIES

Im( 1 /g2>

Fig. 17. The integration path C in eq. (16.7).

and F(z) will be bounded by

IF(z)~ < A exp(IzI/gkax) , where hxis the allowed limit for

(16.8) g

as derived in the previous

sections. However, eqs. (16.7) and (16.8) are not quite sufficient to prove Bore1 summability because we also want to have analyticity of F(z) for an open region around the origin. That this requirement is met c?n be seen as follows. Let us solve a variant of equation ( 1 4 . 4 ) of the following form:

(16.9)

where e ( x ) is the step function. The coefficients B(') and the functional p are the same as before (constructed the same way via difference equations of sects. 12 and 13). Clearly we have, if P > A: g(u,A) = g(A,A). And eq. (16.1) now reads (16.10)

430

G . 't HOOFT

Our point is that this solution exists not only in the region (16.31, but also if Re C

<-

max (b )

- log A2

E

-R-log A'

.

(16.11)

gLX

We can now close the contour C (see F i g . 18).

Fig. 18. The contour C of eq. (16.13); G(g,Ai) are shaded.

the forbidden regions of

Now compare two different A values: A 1 and A p , and compare the Green functions computed with these two A values, both as a function of g; taking (16.12)

We take these Green functions at (possibly exceptional) momentum values p , but always such that A >> Ipl. If they are computed directly following our algorithm then a slight A-dependence may still exist, coming from two sources: one is the fact that as a functionat of g, may gi(u,A) depends on A because p(u,{g)) depend on g(u') with p' > A. But clearly, since all integrals involved in the construction of p converge we expect this dependence to go like

or probably

431

PLANAR DIAGRAM FIELD THEORIES

(16.13b) (because linearly convergent equations can often be made quadratically convergent by symmetrization); here c(g) C o if g+o. The second source of A-dependence comes from the application of difference equations to compute Green functions away from the synmetry point (sects. 12,13). Again, this error must behave like (l6.13), because of convergence of the integrals involved. The change in the Borel function F(z) can be read off from eq. (16.6b) (16.14) C

i f A2 2 A1. Under what conditions does this vanish in the limit A 1 + integral2'g

ranges from -R-log

w

? In our

(3 -$

to R. So if Re z =G o then (16.15)

then c(g) C 0 . This we may use on the far left of the If lg'21 + curve c. Therefore ineq. (16.14) can be written as

(Id. 16) By comparing a series of A values of the form An = 2", one easily sees that (16.16) guarantees convergence as soon as Re z > - 1

,

(16.17)

which is a quite large region of analyticity of F ( z ) . Note that = - 1 is the location of the first renormalon singularity4s7. Eq. (16.17) together with (16.8) implies complete Borel sumability of this theory. 17. THE MASSLESS THEORY

In the sections 14-16 the mass m2 had to be non-zero. This was the only way we could obtain the necessary ineq. (16.3a), us to draw the contour C. What hap ens if we take the 0 , considering not g(m2) but g(p ) at some fixed l~ as

+

!

432

G. 't HOOFT

expansion parameter? Our method to construct the entire theory then fails, but for some z values F(z) will still exist. The argument is analogous to the one of the previous section where we dealt with an ultraviolet problem. Now our difficulty is in the infrared. Comparing two different small values for m, we have (17.1) (17.2)

C

if mp z m 1 . Now if Re z > o then (17.3)

Thus ( I 7.4)

Now we have convergence as m2

+

o as long as (17.5)

Rez
-

and,indeed, p 1 is a point where an infrared renormalon singularity is to be expected. Thus, the Bore1 transform F ( z ) of the massless theory is analytic in the region - 1 < Rez < 1

.

(17.6)

This result guarantees that perturbation expansion in g2 diverges not worse than g2" n! but is clearly not enough for Bore1 summability. 18. THE

- A Tr

EaODEL

A special case is the pure scalar planar field theory, with just one coupling constant X and a mass m. If m is sufficiently large and X sufficiently small then our analysis applies, and we find that all planar Green functions are uniquely determined. However, in this special case there is m r e : the Green functions can be uniquely determined as long as the masses in a l l channels

433

PLANAR DIAGRAM FIELD THEORIES

are non-negative, and X i s n e g a t i v e . T h i s w a s d i s c o v e r e d by comparing w i t h t h e much s i m p l e r " s p h e r i c a l model" (appendix B) which shows t h e same p r o p e r t y . The argument i s f a i r l y s i m p l e . L e t us f i r s t t a k e t h e t h e o r y a t m2 v a l u e s which are so l a r g e t h a t e v e r y t h i n g i s w e l l - d e f i n e d . Now d e c r e a s e t h e "bare mass" m$ c o n t i n u o u s l y . The change i n t h e d r e s s e d p r o p a g a t o r P(k,m2) = -Gp1(k,m2)

(18.1)

is determined by

a am zG 2 (k,m2)

: G'(k,m2)

(18.2)

which w e take t o s t a r t o u t a t s u f f i c i e n t l y l a r g e .'R Now t h e Feynman r u l e s f o r G"(k,m2)

f

am2 a

G'(k,m2)

,

(18.3)

can e a s i l y be w r i t t e n down, j u s t l i k e t h o s e f o r (18.4) Since G;' and G l are s u p e r f i c i a l l y convergent w e c a n a g a i n e x p r e s s them i n terms.of a s k e l e t o n expansion c o n t a i n i n g o n l y t h e f u n c t i o n s G4 and -G> and t h e p r o p a g a t o r s P. They are a l l p o s i t i v e (remember t h a t G4 s t a r f s o u t as - A , w i t h X < o ) , am' aZZ integrals and sununations converge. Only GL h a s one s u r v i v i n g minus s i g n from d i f f e r e n t i a t i n g one p m p a g a t o r w i t h m Thus:

.

G'; > o ;

(18.5)

G I < o .

(18.6)

I f w e l e t m2 d e c r e a s e t h e n c l e a r l y G 2 w i l l s t a y p o s i t i v e and G)2 n e g a t i v e . T h e i r a b s o l u t e v a l u e s grow however, u n t i l a p o i n t is reached where e i t h e r t h e sum o f a l l diagrams w i l l no l o n g e r converge, o r t h e two-point f u n c t i o n G2 becomes z e r o . As soon as t h i s happens t h e t h e o r y w i l l b e i l l - d e f i n e d . A t a c h y o n i c p o l e tends t o develop, followed by c a t a s t r o p h e s i n a l l c h a n n e l s . The p o i n t w e wish t o make i n t h i s s e c t i o n however i s t h a t as long as t h i s does n o t happen, indeed a l l summations and i n t e g r a l s converge, so t h a t o u r i t e r a t i v e p r o c e d u r e t o produce t h e Green f u n c t i o n s w i l l a l s o converge. For a l l t h o s e v a l u e s of A and m2 t h i s t h e o r y w i l l b e Bore1 summable.

This r e s u l t o n l y h o l d s f o r t h e s p e c i a l case c o n s i d e r e d h e r e , namely - X T r $4 t h e o r y , because a l l s k e l e t o n diagrams t h a t con-

434

G. 't HOOFT

tribute to some Green function, carry the same sign. They can never interfere destructively. Notice that what also was needed here was convergence of the diagram expansion. Now we know that at finite N the non-planar graphs give a divergent contribution. Thus the "tachyons" will develop already at infinite m2: the theory is fundamentally unstable. Of course we knew this already: X after all has the wrong sign. The instantons that bring about the-decay of our "false vacuum" carry on action S proportional to -N/X which is finite for finite N.

19. OUTLOOK Apart from the model of sect, 18, the models we are able to construct explicitly now lack any appreciable structure, so they are physically not very interesting. Two (extremely difficult) things should clearly be tried to be done: one is the massless planar theories such as SU(=) QCD. Clearly that theory should show an enormously intricate structure, including several possible phase-transitions. We still believe that more and better understanding of the infrared renormalons that limited analyticity of our bore1 functions in sect. 17 could help us to go beyond those singular points and may possibly "solve" that model (i.e. yield a demonstrably convergent calculational scheme). Secondly one would try to use the same or similar skeleton techniques at finite N (non-planar diagrams). Of course now the skeleton expansion does not converge, but, in Borel-summing the skeleton expansion there should be no renormalons, and all divergences may be due entirely to instantonlike structures. More understanding of resummation techniques for these diagrams by saddle point methods could help us out. If such a program could work then that would enable us to write down SU(3) QCD in a finite (but small) box. QCD in the real world could then perhaps be obtained by gluing boxes together, as in lattice gauge theories. Another thing yet to be done is to repeat our procedure now in Minkowski apace instead of Euclidean space. Singling out the obvioue singularities in Minkowski space may well be not so difficult, so perhaps this is a more reasonable challenge that we can leave for the interested student. APPENDIX A. ASYMPTOTICALLY FREE INFRARED CONVERGENT PLANAR HIGGS MODEL In discussing examples of planar field theories €or which our analysis is applicable we found that pure SU(o0) gauge theory (with possibly a limited number of fermions) is asymptotically free as required, but unbounded in the infrared so that even the ultraviolet limit cannot be treated exactly (see sects. 14 and 15).

-

435

PLANAR DIAGRAM FIELD THEORIES

-X T r

(p4

i s asymptotically free aad can be given a mass term

ao t h a t i n f r a r e d convergence is a l s o guaranteed. A t N + Q O t h i s i s a f i n e planar theory, but a t f i n i t e N t h e vacuum i s unstable. The

only theory t h a t s u f f e r s from none of t h e s e d e f e c t s i s an SU(-) gauge theory i n which a l l bosons g e t a mass due t o t h e Higgs mechanism. But then a new scalar self-coupling occurs t h a t tends to be not asymptotically f r e e . Asymptotic freedom i s only secured i f , curiously enough, s e v e r a l kinds of fermionic degrees of freedom are added. The following model i s an example ( s i m i l a r examples can a l s o be constructed a t f i n i t e N, such as SU(2)). I n general a renormalizable model can be w r i t t e n as

where G i a t h e covariant c u r l , .+i is a set of s c a l a r f i e l d s and 6 a set of spinors. V is a q u a r t i c and W a l i n e a r polynomial i n 0. W e write

a =

Cb:

ua v ~

~v II -

a

abc b c ~APAV ~ +

g

-

= - T r ( I J ~ ~ U ~ U -2Tr(UaUb+Uaub) ~) s s P P

.

(A. 2)

The most compact way t o w r i t e t h e complete s e t of one-loop 8 functions i s t o express them i n terms of t h e one-loop counterLagrangian, [ 8 ~ ~ ( 4 - n ) l - ~ A Cwhere , AL has been found t o be1', a f t e r performing t h e necessary f i e l d renormalizations,

+

+

Vfj

i

y3

Vi(T2.#)i + T(+T%b+)2 3

1 1

+ 2 $(62w+WU2)g + 4i(V. +$W.$)Tr(S.S +P.P.) 2

J

J

- Tr(S2+P2) + Tr[S,P]2 .

1 j

1 1

(A. 3)

436

G. ' t HOOFT

Here Vi stands for aV/Wi, etc. The scaling behavior of the coupling constants is then detetmined by

(A. 4) We now choose a model with U(N)l cal x SU(N)global symmetry. Besides the gauge field we have a scafar 4s in the Nlocal X Nglobal representation, and two kinds of fermions:

JI(1)i

j

x N

in Nlocal

local

and JI

(2)a

i in N local

Nglobal

. We choose (A. 5)

and

i + h.c.

a

I.

(A.6)

Writing c;b

2p

= c;b._

; Cfb = 322 gab

,

(A. 7)

we find: -8n2

"2 a ap

. I

Az2

I

22 z 4

*

(A. 8)

'

and with NX; K2 = Nh2, by substituting (A.5) and (A.6) in the expression (A.3) after some algebra, and in the limit N +a: AG2

9

-$4

AT .t -2y

+

+

92 " p2 ; h g 3;2y - -3 i 4 -

4ci;2y + 4Z4

4 These equations (A.8)-(A.10)

.

(A. 9) (A. 10)

are ordinary differential equations whose solutions we can study. The signs in all terms are typical for any such models with three coupling constants g, X and h. Only the relative magnitudes of the various terms differ from one model to another. For asymptotic fre-edom we need that the second and last terms of (A.10) and the last of (A.9) are sufficiently large. Usually this implies that the fermions must be in a sufficiently large representation of the gauge group, which explains our choice for the f e d o n i c representations. Our model has an asymptotically free solution if all coupling constants stay in a fixed ratio with respect to each other:

X=XT2

N

A

;

N

A

(A. 1 I )

h-hg,

and then, from (A.8)-(A.10)

we see:

437

PLANAR DIAGRAM FIELD THEORIES

x = -!a (rn-5)

,

(A. 12)

So indeed we have a s o l u t i o n with p o s i t i v e A .

It now must be shown t h a t i n t h i s model a l l p a r t i c l e s can be made massive v i a t h e Higgs mechanism. W e consider spontaneous breakdown of SU(N)local x SU(N)global i n t o t h e diagonal SU(N)global subgroup. Take as a mass term (A. 13)

We can w r i t e V as

v

=

+

2 + k - ~ 2 6 ~ + ~ const. l

(A. 14)

Clearly t h i s i s minimal i f (A. IS)

o r a gauge r o t a t i o n thereof.

All vector bosons g e t an equal mass: -D*@ D@

*-

Mi = 2g2F2

(A. 16)

g2P2A2 ; P

,

(A. 17)

and of course t h e s c a l a r s g e t a mass:

$=

hF2

.

(A. 18)

Thus t h e mass r a t i o i s given by

J--

,112 = 0.6303

ma...

.

(A. 19)

This is a f i x e d number of t h i s theory, but it w i l l be a f f e c t e d by higher order corrections. The fermions can each be given a maas term: (A. 20) and the Yukawa f o r c e w i l l give a mixing of a d e f i n i t e strength. The model described i n t h i s Appendixiis probably t h e simplest completely convergent planar f i e l d theory with absolutely s t a b l e vacuum. It i s unlikely however t h a t it would have a d i r e c t physical s i g n i f i c a n c e .

438

G. 't HOOFT APPENDIX B. THE N-VECTOR MODEL IN THE N + W LIMIT (SPHERICAL MODEL). SPONTANEOUS MASS GENERATION When only W-vector fields are present (rather than NxN tensors)then the N - ) w limit is easily obtained analytically. This is the quite illustrative spherical model. Let the bare Lagrangian be

The only diagrams that dominate in the N of bubbles (Fig. 19).

+a+limit

are the chains

a Fig. 19. a) Dominating diagrams for the &point b) Mass renormalization. Let us r e m v e some factors N

hg = NAB/16n2

H~

b function.

by defining

.

(B. 2)

The diagrams of Fig. 19 are easily summed. Mass and coupling constant need to be renormalized. Dimensional tenoylization is appropriate hete. In terms of the finite constants XR(p) and mR(p), chosenlat some subtraction point p , and the infinitesimal & = 4-n, where n is the number of space-time dimensions, one finds :

and

The sum of all diagrams of type 19a gives an effective propagator of the form

439

PLANAR DIAGRAM FIELD THEORIES

where q is the exchanged momentum, y is Euler's constant, and

0

and m is the physical mass in the propagators of Fig. 19a; that is because these should include the renormalizations of the form of Fig. 19b.

0

- = - + Fig. 20.

Fig. 20 shows how m follows from

%:

From ( B . 3 ) we see that the renormalization-group invariant combination is

N

that inevitably XR < o at large p. Indeed, in this model the vacuum would become unstable as soon as N is made finite. In the limit N + a however everything is still fine. so

2 Now in Fig. 2 1 we plot both 9 and the composite mass M2, determined by the pole of F ( q 4 , as a function of the physical mass m2. We see that at negative % there are two solutions for m2, but one should be rejected because $ would be negative, an indication for an unstable choice of vacuum. The observation we w'sh to make in this appendix is that in the 3 we get an entirely positive 4-point function allowed region for 9 in Euclidean space LF(q) > 0 ) . If we chose-% 2 to be fixed and vary 2 1~ (or rather vary XB! then at 9 > o ai?Z X valu s are allowed, at only sufficiently small values. At 9 = o we see a negative

s

4

440

G. 't HOOFT

I I' Fig. 21. Mass ratios at given value for xR(p).

.

"spontaneous"Ngeneration of a finite value for m Perturbation expansion in XR would show the "infrared renormalon" difficulty. Apparently here the difficulty solves itself via this spontaneous mass generation.

REFERENCES 1.

2.

3.

4. 5.

6.

E.C.G. Stueckelberg and A. Peterman, Helv. Phys. Acta 26 (1953), 499; M. Gell-Mann and F. Low, Phys. Rev. 95 (1954) 1300. K.G. Wilson, Phys. Rev. D10, 2445 (1974). K.G. Wilson, in "Recent Developments in Gauge Theories", ed. by G. 't Hooft et al., Plenum Press,New York and London, 1980, p. 363. G. 't Hooft, Marseille Conference on Renormalization of YangMills Fields and Applications to Particle Physics, June 1972, unpublished; H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); D.J. Cross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). G . 't.Hooft, in "The Whys of Subnuclear Physics", Erice 1977, A. Zichichi ed. Plenum Press, New York and London 1979, p. 943. M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. Lett. 42, 1390 (1979). C. de Calan and V. Rivasseau, Conrm. Math.*Phys. 82, 69 (1981); 91, 265 (1983).

441

PLANAR DIAGRAM FIELD THEORIES

7) G. Parisi, Phys. .Lett. 76B, 65 (1978) and Phys. Rep. 49, 215 (1979). 8) J. Koplik, A. Neveu and S. Nussinov, Nucl. Phys. B123, 109 (1977). W.T. Tuttle, Can, J. Math. 14, 21 (1962). 9) See ref. 3. 10) J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (IkGraw-Hill, New York, 1964). 1 1 ) G. 't Hooft, Commun. Math. Phys. 86, 449 (1982). 12) G. 't Hooft, Commun. Math. Phys. 88, 1 (1983). 13) J.C. Ward, Phys. Rev. 78 (1950) 1824. Y. Takahashi, Nuovo Cimento 6 (1957) 370. A. Slavnov, Theor. and Math. Phys. 10 (1972) 143 (in Russian). English translation: Theor. and Math. Phys. 10, p. 99. J.C. Taylor, Nucl. Phys. B33 (1971) 436. 14) K. Symanzik, Comm. Math. Physics 18 (1970) 227; 23 (1971) 49; Lett. Nuovo Cim. 6 (1973) 77; "Small-Distance Behaviour in Field Theory", Springer Tracts in Modern Physics vol. 57 (G. HEhler ed., 1971) p. 221. 15) G. 't Hooft, Nucl. Phys. B33 (1971) 173. 16) G. 't Hooft, Phys. Lett. 119B, 369 (1982). 17) G. 't Hooft, unpublished. R. van D a m e , Phys. Lett. IlOB (1982) 239.

442

Nuclear Physics B75 (1974)461-470. North-Holland Publishing Company

CHAPTER 6.2

A TWO-DIMENSIONALMODEL FOR MESONS C . 't HOOFT CERN,Geneva Received 21 February 1974

Abstract: A recently proposed gauge theory for strong interactions, in which the set of planar diagrams play a dominant role, is considered in one space and one time dimension. In this case, the planar diagrams can be reduced to selfenergy and ladder diagrams, and they can be summed. The gauge field interactions resemble those of the quantized dual string, and the physical mass spectrum consists of a nearly straight "Regge trajectory".

It has widely been speculated that a quantized non-Abelian gauge field without Higgs fields, provides for the force that keeps the quarks inseparably together [ 1-41 Due to the infra-red instability of the system, the gauge field flux lines should

.

squeeze together to form a structure resembling the quantized dual string. If all this is true, then the strong interactions will undoubtedly be by far the most complicated force in nature. It may therefore be of help that an amusingly simple model exists which exhibits the most remarkable feature of such a theory: the infinite potential well. In the model there is only one space, and one time dimension. "liere is a local gauge group U(&'), of which the parameter N is so large that the perturbation expansion with respect to 1/Nis reasonable. Our Lagrangian is, like in ref. [4] ,

B

( x ) = -A*

I r r

(x) ;

The Lorentz indices p , v , can take the two values 0 and 1. It will be convenient to use

443

G. 't Hooft, Wodimensbnut model for mesons

light cone coordinates. For upper indices:

and for lower indices

A , =-

1

4

(A1 f A o ) , etc.,

where

PI = P

1 9

Po=-P

0

.

Our summation convention will then be as follows,

,

x P p = x ccPp =

X,P,

-

x+p+ + x - p - = x + p - + x - p + = x + p - + x - p +

(4) *

"he model becomes particularly simple if we imp0 tion:

the light-cone gauge c ndi-

A- = A + = O .

In that gauge we have

c+- = + A +

I

and

1:= -3Tr (a-A+)2 - q"(ya + M(")+ m-A+)q"

.

(7)

"here is no ghost in this gauge. If we take x+ as our t h e direction, then we notice that the field A + is not an independent dynamical variable because it has no time derivative in the Lagrangian. But it does provide for a (non-local) Coulomb force between the Fermions. "he Feynman rules are given in fig. 1 (using the notation of ref. [4]). The algebra for the y matrices is

Y+Y- + Y-Y+ = 2

(8 .b)

*

Since the only vertex in the model is proportional to y- and 72 = 0 , only that part of the quark propagator that is proportional to y+ will contribute. As a consequence

444

G. 'tHooft, 7bodhenmbmi modei for mesons

Fig. 1. Planar Feynman ruler in the light cone gauge.

we can eliminate the y matrices from our Feynman rules (see the righthand side of rig. I). We now consider the limit N + 00; g2N fixed, which corresponds fo taking only the planar diagrams with no Fermion loops (4). They are of the type of fig. 2. AU gauge field lines must be between the Fermion lines and may not cross each other. They are so much simpler than the diagrams of ref. [4], because the gauge fields do not interact with themselves. We have nothing but ladder diagrams with selfenergy insertions for the Fermions. Let us first concentrate on these selfcnergy parts. Let il'(k) stand for the sum of the irreducible selfcnergy parts (after having eliminated the y matrices). The dressed propagator is -ik(9) mZ i2k+k- - k- r(k) - ic b

0

P Fig. 2. Large diagram. a and b must have opposite U(N) charge, but need not be each other's antiparticle.

445

G. 't Hooft, I\uodimensional model for mesuns

AL=LaL Fig. 3. Equations for the planar selfenergy blob.

Since the gauge field lines must all be at one side of the Fermion line, we have a simple bootstrap equation (see fig. 3).

Observe that we can shift k, + p+ --* k, ,so

re)must be independent of p+, and (1 1)

Let us consider the last integral in (1 1). It is ultra-violet divergent, but as it is well known, this is only a consequence of our rather singular gauge condition, eq. ( 5 ) . Fortunately, the divergence is only logarithmic (we work in two dimensions), and a symmetric ultra-violet cut-off removes the infinity. But then the integral over k, is independent of r. It is

ni 2Ik-tp-1 so,

'

This integral is infra-red divergent. How should we make the infra-red cutsff ? One can think of putting the system in a large but finite box,or turning off the interactions at large distances, or simply drill a hole in momentum space around k = 0. We shall take X < lk_l< 00 as our integration region and postpone the limit h + 0 until it makes sense. We shall not try to justify this procedure here, except for the remark that our final result will be completely independent of A, so even if a more thorough discussion would necessitate a more complicated momentum cut-off, this would in general make no difference for our final result. We find from (1 2), that

446

G. 't Hooft. Two-ciimensionaImodel for mesons

and the dressed propagator is -ik-

Now because of the infra-red divergence, the pole of this propagator is shifted towards k, + = and we conclude that there is no physical single quark state. This will be confirmed by our study of the ladder diagrams, of which the spectrum has no continuum corresponding to a state with two free quarks. "he ladder diagrams satisfy a Bethe-Salpeter equation, depicted in fig. '4. Let Jl(p, r) stand for an arbitrary blob out of which comes a quark with mass m1 and

Fig. 4. Eq. (15).

momentum p, and an antiquark with mass m2 and momentum r - p . Such a blob satisfies an inhomogeneous bootstrap equation. We are particularly interested in the homogeneous part of this equation, which governs the spectrum of twoparticle states:

+2p+p-

tg nX Ip-l

- ie]-yJ'@,,

where 4

writing

we have for cp

447

t k,r) *

-

dk+ dk-

,

(15)

G. 't Hooft, 7bodimensionaI model for mewns

One integral has been separated. This was possible because the Coulomb force is instantaneous. The p + integral is only non-zero if the integration path is between the poles, that is, sgn (P- - r - ) = -sgn

(P-1 *

and can easily be performed. Thus, if we take r-

> 0, then

The integral in eq. (20) is again infra-red divergent. Using the same cut-off as before, we find

where the principal value integral is defined as

p F k -k)tdk -

'

= 'f

cp(k- t k)dkt ( k - t ie)'

cp(k- - k)dk-

'J

'

(22) (k-- ie)'

'

and is always finite. Substitutiiig (2 I ) into (20) we find

(2.3) l l i c infra-red cut-off dependence has disappeared ! In fact. we have here the exact forni of the Haniiltonian discussed i n ref. 141. Let us introduce diiiiensionless units: "Mi,'

a,.2 =-

' g

ntu;., -- - I

,

g?

7

-2r+r- = - p' ; A

'R

448

i7.J.

= .v ;.

('4)

G. ‘tHmft, livo-dimendoonrrlmodel for mesons

p is the mass of the two-particle state in units of g/+.

Now we have the equation

We were unable to solve this equation analytically. But much can be said, in particular about the spectrum. First, one must settle the boundary condition. At the boundary x = 0 the solutions I&) may behave like x*@1,with qcotgnB1+a1= o ,

(26) but only in the Hilbert space of functions that vanish at the boundary the Hamiltonian (the right-hand side of (25)) is Hennitcan:

In particular, the “eigenstate”

in the case a1 = a2,is not orthogonal to the ground state that does satisfy do) = dl)= 0. Also, from (27) it can be shown that the eigenstates @ with ~ ~ (= @(1) 0 ) =0 form a complete set. We conclude that this is the correct boundary condition *. A rough approximation for the eigenstates Ipk is the following. I h e integral in (25) gives its main contribution ify is close to x. For a periodic function we have

The boundary condition is q(0) = Cp(1) =O. So ifal ,a2 5- 0 then the eigenfunctions can be approximated by @(x)=sin&nx, with eigenvalues fi@) 2

& = 1 , 2,...(

(28)

= 8% .

(29)

This is a straight “Regge trajectory”, and there is no continuum in the spectrum! The approximation is valid for large k,so (29) will determine the asymptotic form Footnote see next page.

449

G. 't Hmft, Tbodimenswnal model for mesvns

of the trajectories whereas deviations from the straight line are expected near the origin as a consequence of the finiteness of the region of integration, and the contribution of the mass terms. Further, one can easily deduce from (27), that the system has only positive eigenvalues if a l .a2 > -1. For a1 = a2 = -1 there is one eigenstate with eigenvalue zero (9= 1). Evidently, tachyonic bound states only emerge if one or more of the original quarks were tachyons (see eq. (24)). A zero mass bound state occurs if both quarkshave mass zero. The physical interpretation is clear. "he Coulomb force in a onedimensional world has the form va

Ix,-xzl,

which gives rise to an insurmountable potential well. Single quarks have do finite dressed propagators because they cannot be produced. Only colourless states can escape the Coulomb potential and are therefore free of infra-red ambiguities. Our result is completely different from the exact solution of twodimensional massless quantum electrodynamics [S , 2 ] ,which should correspond to N = 1 in our case. The perturbation expansion with respect to l/N is then evidently not a good approximation; in two dimensional massless Q.E.D. the spectrum consists of only one massive particle with the quantum numbers of the photon. In order to check our ideas on the solutions of eq. (25), we devised a computer program that generates accurately the first 40 or so eigenvalues p 2 . We used a set of t Bg!Ck sin knx. The actrial functions of the type Axo1(l-x)2-@1 t E(~-xY~x*-P~ curacy is typically of the order of 6 decimal places for the lowest eigenvalues, decreasing to 4 for the 40th eigenvalue, and less beyond the 40th. A certain W.K.B. approximation that yields the formula 2 Pf,)

n+-

n2n t (ff1+ a*)log n t CS'(al

,ff2),

n = 0,1,

...

was confumed qualitatively (the constant in front of the logarithm could not be checked accurately). In fig. S we show the mass spectra for mesons built from equal mass quarks. In the case mq = m; = 1 (or u1 = a2 = 0) the straight line is approached rapidly, and the constant in eq. (30) is likely to be exactly an2. In fa.6 we give some results for quarks with different masses. The mass difference for the nonets built from two triplets are shown in two cases: (a)

m,=O;

(b) m l 10.80;

m2 = 0.200 ;

m3 = 0.400,

m2 = 1.OO ;

m3 = 1.20,

in units of g / / n . The higher states seem to spread logarithmically, in accordance with This will certainly not be the last word on the boundary condition. For a more thorough study we would have to consider the unitarity condition for the interactions proportional to 1/N. That is beyond the aim of this paper.

450

G. ’tHooft, Wodimensmnal model for mesons

0

In’

U’

,w2

3w’

-p2

Fig. 5:“Regge trajectories” for mesons built from a quark-antiquark pair with equal mass, m , varying from 0 to 2.1 1 in units of g/Jw. ’IXe squared mass of the bound states is in unitsg2/n. n = 4

n = 3

n.2

n:

1

n’

0

- p’ - nrc’

Fig. 6. Meson nonets built from quark triplets. The picture is to be interpreted just as the previous figure, but in order to get a better display of the mass differences the members of one nonet have been separated vertically, and the nth excited state has been shifted to the left by an amount nn2. In case (a) the masses of the triplet are m l = 0.00;m2 0.20;m3= 0.40 and in case (b) m l = 0.80; m2 = 1.00;m3 = 1.20. Again the unit of mass isg/Jn.

-

45 1

G. 't Hmft, Twodimensional model for mesons

eq. (30). But, contrary to eq. (30),it is rather the average mass, than the average squared mass of the quarks that determines the mass of the lower bound states. Comparing our model with the real world we find two basic flaws. First there are no transverse motions, and hence there exists nothing like angular momentum, nor particles such as photons. Secondly, at IV = 3 there exist also other colourless states: the baryons, built from three quarks or three antiquarks. In the 1/Nexpansion, they do not turn up. To determine their spectra one must use different approximation methods and we expect those calculations to become very tedious and the results difficult to interpret. The unitarity problem for finite N will also be tricky. Details on our numbers and computer calculations can be obtained from the author or G. Komen, presently at CERN. "he author wishes to thank A. Neveu for discussions on the two dimensional gauge field model; N.G. van Kampen, M.Nauenberg and H. de Groot, for a contribution on the integral equation, and in particular G. Komen, who wrote the computer programs for the mass spectra. References [ l ] M.Y.Han and Y.Nambu, Phys. Rev. 139B (1965) 1006. (21 A. Casher, J. Kogut and L. Susskind, Tel Aviv preprint TAW-373-73 (June 1973). [ 3 ] P. Olesen, A dual quark prison, Copenhagen, preprint NBI-HE-74-1 (1974). [4] C. 't Hooft, CERN preprint TH 1786, Nucl. Phys. B, to be published. (51 J. Schwinger,Phys. Rev. 128 (1962) 2425.

452

Epilogue to the Two-Dimensional Model Apparently we find quite realistic meson spectra. The quark masses are freeparameters, and in the limit m, -t 0 the pions (and eta's!) are massless, which indicates that the chiral symmetry is spontaneously broken. This spontaneous symmetry breakdown may be surprising in view of the fact that there are theorems forbidding such a breakdown in two dimensions. The reuon why these theorems are violated is that the massless mesons do not interact at all if N = 00. One may expect that at finite N infrared divergences due to the maasless pion and eta propagators will eventually restore chiral symmetry nonperturbatively. The paper gave rise to another controversy. The only gauge choice in which the solutions can be given in elegant and practically closed form is the light cone gauge. However in this gauge the zero modes seem to be treated incorrectly, a problem related to the rather dangerous light cone boundary conditions. All we say about this in the paper is that the pole in the propagator is cut off by a principal value integration prescription. That this is actually correct is not so easy to see. As usual, one has to ask what the physical significance of the danger is. To understand the infrared structure of our model we have to take periodic boundary conditions, and these are only consistent with our procedure if the periodic variable is z-. Now our "gauge condition" is A- = 0, but actually this implies also a constraint: the total magnetic field generated by the closed loop formed by our periodic onespace is constrained to be zero. This is the total magnetic flux caught by our loop. Physically, we expect in the infrared limit that the loop should be so large that this single constraint should become unimportant. The principal value prescription can then be seen to be related to an extra gauge condition for the zero modes. The point here is that besides A- = 0 we could fix the residual global gauge invariance by demanding A + ( k = 0) = 0. And finally, to judge whether the periodic boundary condition in the lightlike direction is legal, one has to check whether masslees modes that survive can move with the speed of light so that a,causal behavior might result. Well, we find that all observable objects, the physical meaons, come out to be massive (as long as rn, # 0), so there is no problem here. The h a l check comes when one notices that the light cone gauge in the left direction gives the same bound states as the light cone gauge in the right direction. Apparently our procedure was gauge independent. An interesting problem was raised by D. Gross in a private discussion with the author.18 Since we start off with a model with conservation of parity, all bound states should either be parity even or parity odd. The even states should be coupled to the current && and the odd states to the current &76$~2. One expects the bound states to be alternatinglyeven and odd under parity. In terms of the solutions of the integral equation (25) one derives that the scalar current couples to a solution p(z) uio the integral

/' (s 0

2

- z1-2 )p(z)dz,

"C. G . Callan, N. Coote, and D.J. Gross, Phys. Rev. D1S (1976) 1649.

453

and the pseudoscalar current via the integral

J

0

1

(22 2 + =)v(x)dx, 1-2

where ml and m2 are the masses of the quark and the antiquark inside the meaon. If the masses are equal it is evident that for the even solutions the first integral vanishes and for the odd ones the second integral. But if the masses are unequal this is far from evident. The reason why it is 80 difficult to see is that our light cone gauge condition is not parity invariant. Yet it is true. To see this, consider the product of the two integrals, and prove that it is equal to

where the operator H is, up to a constant, the right-hand side of Q. (25), and P is defined by rl

1

This commutator expectation value must vanish for the eigenstates of H, hence one of the two integrals must vanish. All states are either even or odd under parity. This result collfirm8 once again that our procedure is gauge independent.

454

CHAPTER 7

QUARK CONFINEMENT Introductions

...................................................................

I7.11 “Confinement and topology in non-Abeliau gauge theories”, Acta Phys. Austr. Suppl. 22 (1980) 531-586 ......................... [7.2] “The confinement phenomenon in quantum field theory”, 1981 Garghe Summer School Lecture Notes on findamental Intemctions, eds. M. IRvy and J.-L. Baedevant, NATO Adv. Study Inst. Series B: Phya., vol. 85, pp. 634-671 ............................ [7.3] “Can we make sense out of “Quantum Chromodynamics”?” in The Whys of Subnuclear Physics, ed. A. Zichichi,Plenum, New York, pp. 943-971 .......................................................

455

456 458

514

547

CHAPTER 7

QUARK CONFINEMENT Introduction to Confinement and Topology in Non-Abelian Gauge Theorieg [7.1] Introduction to the Confinement Phenomenon in Quantum Field Theory [7.2] Introduction to Can We M a k e Sense out of Quantum Chromodynamics? [7.3] How can we understand the behavior of quarks in the theory of Quantum Chre modynamics? Is “ionization” of mesons and baryons into “free quarks” indeed absolutely forbidden? What is the mechanism for this absolute and permanent verdict of uconfinementl’?And &ally, how do we calculate the details of the properties of mesons and baryons once the QCD Lagrangian is given? The answers to these questions came very gradually. Our understanding became more and more complete. Precise calculations are still very diflicult these days, but there seems to be no longer any fundamental obstacle. Confinement is a state of aggregation, comparable to the solid state, the liquid state, the gaseous state or the plasma state of ordinary materials. Other modes a system such as quantum chromodynamica could have condensed into, are various versions of Higgs phasea or the Coulomb phase. Only in the Coulomb phase physical massless vector particles would persist in the physical spectrum, but, as will be demonstrated, these particles would have little left of what was once their non-Abelian character; they could only survive disguised as more or less ordinary photons (an exception to this would be QCD with such a large number of fermions, that the sign of the /3 function is reversed). I wrote a number of papers on the subject, and a couple of summer school lecture notes. Three summer school lecture notes are reproduced here; for some of the technical details I refer the reader to the original papers. There is 80 much material to be discussed that the three papers reproduced here hardly overlap; they

show different approaches to the confinement problem. In the first two papers the notion of “dual transformations”, or the transition from order parameters to disorder parameters and back, plays a very important role. It is instructive to consider these transformations in a box with periodic boundary conditions at finite temperature T = l/P. This is done in the first paper, from Section VI onwards. Combining methods from field theory and statistical physics it is discovered there that powerful identities are obtained by rotating this box over 90’ in Euclidean space. One finds distinct possibilities for the system to behave at large distance: the Higgs mode, the Coulomb mode and the confinement mode. The first and the lest are each other’s dual. The last paper addresses the question of calculability. Its results are modest, if not disappointing. Even if our theory is asymptotically free, replacement of perturbation expansion by some guaranteedly convergent resummation procedure seem to be impossible. Only very modest formal improvement on the most naive perturbative expansions could be obtained. The much more pragmatic approach of Monte Carlo simulations on lattices seems to give much more satisfactory results in practice, but formal proofs that these methods converge to unique spectral data and transition amplitudes when the lattice becomes infinitely dense have never been given. I have nothing to add to this except the conjecture that indeed the Monte Carlo calculations define the theory rigorously seem to be quite plausible. A small correction must be made in this last paper. As was correctly pointed out to me by Miinster, the instanton singularities and the infrared renormalon singularities in QCD will not show up independently as suggested in Fig. 6, but they merge, due to a divergence of the integration over the instanton size parameters p.

457

Acta Physica Austriaca, Suppl. XXII, 531-586 (1980) 0 by Springer-Verlag 1980

CHAPTER 7.1 CONFINEMENT AND TOPOLOGY IN NON-ABELIAN + GAUGE THEORIES

G. 't HOOFT Institute for Theoretical Physics Princetonplein 5, P.O.Box 8 0 0 0 6 3 5 0 8 TA Utrecht, The Netherlands

ABSTRACT Pure non-Abelian theories with gauge group SU(N) are considered in 3 and 4 space-time dimensions. In 3 dimensions non-perturbative features invalidate ordinary coupling constant expansions. Disorder operators can be defined in 3 and 4 dimensions. Confinement is first. explained in 3 dimensions as a spontaneous breakdown of a topologically defined 2 (N) global symmetry of the theory. In 4 dimensions confinement can be seen as one of the various possible phases of the system by considering a box with periodic boundary conditions in the "thermodynamic limit". An exact duality equation allows either electric or magnetic flux tubes to be stable, but not both. Special attention is given to the explicit oc-

+Lecture

given at XIX.Internationale Universitstswochen ftir Kernphysik,Schladming,February 20 2 9 , 1980.

-

458

currence of instanton configurations with an instanton angle 0 in deriving duality.

I. INTRODUCTION

The first non-Abelian gauge theory that was recognized to describe interactions between elementary particles, to some extent, was the Weinberg-Salam-WardGIM model[ll.That model contains not only gauge fields but also a scalar field doublet H. Perturbation expansion was considered not about the point H = 0, but about the "vacuum value" F H = ( o ) . Such a theory is usually called a theory with "spontaneous symmetry breakdown" [23. In contrast one mkght consider "unbroken gauge theories" where perturbation expansion is only performed about a symmetric "vacuum". These theories are characterized by the absence of a mass term for the gauge vector bosons in the Lagrangian. The physical consequences of that are quite serious. 2 The propagators now have their poles at k =o and it will often happen that in the diagrams new divergences arise because such poles tend to coincide. These are fundamental infrared divergencies that imply a blow-up of the interactions at large distance scales. Often they make it nearly impossible to understand what the stable particle states are. A particular example of such a system is "Quantum Chromodynamics", an unbroken gauge theory with gauge group S U ( 3 ) , and in addition some fermions in the 3 representation of the group, called "quarks". We will

investigate the possibility that these quarks are permanently confined inside bound structures that do not carry gauge quantum numbers. First of all this idea is not as absurd as it may seem. The converse would be equally difficult to understand. Gauge quantum numbers are a priori only defined up to local gauge transformations. The existence of g l o b a l quantum numbers that would correspond to these local ones but would be detectakle experimentally from a distance is not at all a prerequisite. We are nevertheless accustomed to attaching a global significance to local gauge transformation properties because we are familiar with the theories with spontaneous breakdown. The electron and its neutrino, for example, are usually said to form a gauge doublet, to be subjected to local gauge transformations. But actually these words are not properly used. Even the words "spontaneous breakdown" are formally not correct for local gauge theories (which i.s why I put them between quotation marks). The vacuum never breaks local gauge invariance because it itself is gauge invariant. All states in the physical Hilbert space are gauge-invariant. This may be confusing so let me illustrate what I mean by considering the familiar Weinberg et a1 model. The invariant Lagrangian is

HereHlsthe scalar Higgs doublet. The gauge group is 0 SU(2)xU(1), to which correspond ka(Ga ) and A ( F u v ) u uv lJ The subscripts L and R denote left and right handed components of a Dirac field, obtained by the projection operators ; ( I f y,)

.

460

eR is a singlet; JI,

is a doublet.

D

stands for covariant derivative.

P

The function one takes

r l ( l HI

takes its minimum at1 HI=F. Usually

(1.2) F

and perturbs around that value:

H = (

+ h, ).

h2

One identifies the components of$Lwith neutrino and electron:

9,

=

.

vL (e 1 L

However, this model is n o t fundamentally different from a model with "permanent confinement". One could interpret the same physical particles as being all gauge singlets, bound states of the fundamental fields with extremely a strong confining forces, due to the gauge fields A,, of the group SU(2). We have scalar quarks (the Higgs field H ) and fermionic quarks (theJILfield) both as fundamental doublets. Let us call them q. Then there are "mesons" (qq) and "baryons" (qq). The neutrino is a "meson". Its field is the composite, SU(2l-invariant H* qL = F

vL

+

negligible higher order terms.

The eL field is a "baryon", created by the SU ( 2 ) invariant

461

Hi

=

I),~

FeL

+

...

I

the eR field remains an SU(2) singlet. Also bound states with angular momentum occur: The neutral intermediate vector boson is the "meson"

*

H D H = i -,F2A(3' P

2-

P

+ total derivative + higher orders I

(1.5)

if h7e split off the total derivative term (which corresponds to a spin-zero Higgs particle). The Wf are obtained from the "baryon"EijHiDPHj and the Higgs particle can also be obtained from H H. Apparently some mesonic and baryonic bound states survive perturbation expansion, most do not (only those containing a Higgs "quark" may survive). Is there no fundamental difference then between a theory with spontaneous breakdown and a theory with confinement? Sometimes there is. In the above example the Higgs field was a faithful representation of SU(2). This is why the above procedure worked. But suppose that all scalar fields present were invariant under the

*

center Z(N) of the gauge group SU(N), but some fermion fields were not. Then there are clearly two possibilites. The gauge symmetry is "broken" if physical objects exist that transform non-trivially under ZN' such as the fundamental fermions. We call this the Higgs phase. If on the other hand all physical objects are invariant under Z N' such as the mesons and the baryons, then we have permanent confinement. Quantum Chromodynamics is such a theory where these distinct possibilities exist. It is unlikely that one will ever prove from first principles that permanent

462

c o n f i n e m e n t t a k e s place, s i m p l y b e c a u s e o n e c a n a l w a y s i m a g i n e t h e Higgs mode t o o c c u r . I f n o f u n d a m e n t a l

scalar f i e l d s e x i s t t h e n one c o u l d i n t r o d u c e composite f i e l d s s u c h as

or

a n d p o s t u l a t e n o n v a n i s h i n g vacuum e x p e c t a t i o n v a l u e s f o r them:

= F

1 dab8

+

F2 d a b 3

I n t h a t case t h e r e would b e n o c o n f i n e m e n t . W h e t h e r or not F

1,2

a r e e q u a l t o z e r o w i l l c?.epend o n d e t a i l s of

t h e d y n a m i c s . T h e r e f o r e , dynamic!; n u s t be a n i n q r e d i e n t OF t h e c o n f i n e m e n t m e c h a n i s m ] n o t o n l y t o p o l o g i c a l . a r g u -

m e n t s . What we w i l l a t t e m p t i n t h i s l e c t u r e i s t o show t h a t t o p o l o g i c a l a r g u m e n t s i m p l y for t h i s t h e o r y t h e e x i s t e n c e o f phase r e g i o n s , s e p a r a t e d by s h a r p phase t r a n s i t i o n b o u n d a r i e s ( u s u a l l y of f i r s t o r d e r ) . One r e g i o n corresponds t o what i s u s u a l l y c a l l e d "spontaneo u s b r e a k d o w n " ] a n d w i l l be r e f e r r e d t o a s H i g q s p h a s e . Another c o r r e s p o n d s t o a b s o l u t e q i a r k confinement. S t i l l a n o t h e r phase e x i s t s which allows f o r l o n n r a n 9 e

Coulomb-like f o r c e s t o o c c u r .

(Coulomb p h a s e . )

I t i s i l l u s t r a t i v e t o c o n s i d e r f j r s t i!ure q a u g e

t h e o r i e s i n 3 s p a c e - t i m e d i m e n s i o n s . These c i r f e r i n two i m p o r t a n t w a y s f r o m t h e i r 4 d i m e n s i o n a l c o u n t e r p a r t s . F i r s t , they are n o t s c a l e - i n v a r i a n t i n t h e c l a s s i c a l l i m i t . A c o n s e q u e n c e of t h a t i s t h a t t h e y 20

463

n o t have a computable small c o u p l i n g c o n s t a n t expansion. Already a t f i n i t e o r d e r s i n t h e

coupling

c o n s t a n t g phenomena o c c u r t h a t a r e a s s o c i a t e d t o a

complex sort of vacuum instability. This is explained i n S e c t s 2 and 3 . Secondly, t h e t o p o l o g i c a l p r o p e r t i e s

a r e d i f f e r e n t . I n t h r e e dimensio n s a " d i s o r d e r pa-

rameter", b e i n g i3n o p e r a t o r - v a l u e d f i e l d

-+

@ (x,t )

can

be defined. I f

t h e n t h e r e i s a b s o l u t e confinement,

as w e e x p l a i n i n

S e c t s , 4 a n d 5 . The r e r n s j n i n g s e c t i o n s a r c d e v o t e d t o t h e four-dimen:;iorial

ca'

0.

They o v e r l a p t o some e x t e n t

l e c t u r e s q i v e r . a t ~ a r g S s e [ 3 1e x c e p t f o r a more e x p l i c i t c o n s i d e r a t i o n of e f f e c t s d u e t o i n s t a n t o n s a n d t h e i r a n g l e 8 . Here i t i s c o n v e n i e n t t o i n t r o d u c e t h e " p e r i o d i c box"

( a c u b i c o r r e c t a n g u l a r box w i t h p e r i o d i c or

pseudo-periodic

b o u n d a r y c o n d i t i o n s ) . A g a i n w e h a v e order

a n d d i s o r d e r o p e r a t o r s b u t now t h e y a r e d e f i n e d n o t o n

space-time p o i n t s ( x , t ) b u t o n l o o p s ( C , t ) . I n S e c t . 7 i t i s e x p l a i n e d how t o i n t e r p r e t t h e s e o p e r a t o r s i n

terms o f m a g n e t i c a n d e l e c t r i c f l u x o p e r a t o r s , a n d m a g n e t i c f l u x i s d e f i n e d i n t e r m s of t h e b o u n d a r y c o n d i t i o n s of t h e b o x . T h e r e a r e a l s o o t h e r c o n s e r v e d q u a n t i t i e s ( S e c t . 8 ) , to be i n t e r p r e t e d as electric f l u x ard j-nstanton angle. I n S e c t . 9 we c o n s i d e r t h e " h o t b o x " , a b o x a t f i n i t e temperature T = l/kB

,

a n d e x p r e s s t h e :free

e n e r g y F of s u c h a box i n t e r m s of L u n c t i o n a l i n t e g r a l s w i t h t w i s t e d p e r i o d i c boundary c o n d i t i o n s i n 4-dimensional Euclidean space. I n S e c t .

10 w e n o t i c e an

e x a c t d u a l i t y r e l a t i o n f o t t h e f r e e enercfy of e l e c t r i c

464

and magnetic fluxes. Section 11 shows that if electric confinement is assumed, magnetic confinement is excluded, and vice versa. The Coulomh phase realized for instance in the Georgi-Glashow model[4] is dually symmetric, as explained'in sect. 12.

1I.DIFFICULTIES IN THE PERTURBATION EXPANSION FOR QCD IN 2+1 DIMENSIONS Consider pure gauge theory in 2+1 dimensions. The Lagrangian is

with =

G;"

all

A:

- av

a

A~

+

gf

abc b c A,,A,,

.

(2.2)

The functional integrals to be studied are IDA

exp(ijL d2x' dt

+

source terms).

(2.3)

2

has to be dimensionless, Clearly ( L d x dt a therefore A,, has dimension (mass)' and g has dimension (mass)f Gauge-invariant quantities are ultra-violet convergent if they are regularized in a gauge invariant way (for instance by dimensional regularization). This is because the only possible counter terms would be

.

(2.4)

465

Which would all be of too high dimension. Conventionally

this would imply that all Green's functions in Euclidean space would be well defined perturbatively. The physical theory (in Minkowsky space) would then be obtained by analytic continuation. However, in our case we do have an infra-red problem, even in Euclidean space. Consider namely diagrams of the following type:

p-LP

(2.5)

k fP

Simple power counting tells us that the small selfenergy insertion is proportional to g 2 Ikl , where k is the momentum circulating in the large blob. Because of the two propagators the k integration has an infrared divergent part:

/d3k

-*

(remainder).

(k

(2.6)

-iE)

Here the divergence is o n l y logarithmic, but it becomes worse if more self-energy insertions occur in the kpropagator. How should one cure such divergences? To understand the physics of this infra-red di3 vergence let us consider on easier case first: 90 theory without mass term:

466

t h i s t i m e i n f o u r s p a c e - t i m e d i m e n s i o n s : Power c o u n t i n g

t e l l s u s t h a t 4 h a s d i m e n s i o n (mass) 111 and g h a s d i r.ension (mass) [ I ] . T h i s t h e o r y i s a l i t t l e b i t more a r t i f i c a l because a n u l t r a v i o l e t d i v e r g e n c e h a s t o b e subt r a c t e d by a m a s s - c o u n t e r t e r m : AL

=

g2 $2

,

(2.8)

but still we could consider s t a r t i n g perturbation theory w i t h no r e s i d u a l mass. I f w e t a k e t h e same d i a g r a m ( 2 . 5 ) t h e n now t h e s e l f - e n e r g y i n s e r t i o n b e h a v e s as g2 log / k /

(2.9)

I

and t h e k i n t e g r a t i o n i s a g a i n i n f r a - r e d d i v e r g e n t :

la4k log I k l ( r e m a i n d e r ) (k2-ic) 2

.

(2.10)

I n this case however t h e c u r e s e e m s o b v i o u s : we were doing p e r t u r b a t i o n expansion a t a very s i n g u l a r p o i n t . N o problem a r i s e s i f w e f i r s t i n t r o d u c e a s m a l l m a s s

term L + L

'

- -2 v 2

@2

,

(2.11)

and t h e n l e t P L S O i n t h e end. How a r e t h e i n f i n i t i e s such a s ( 2 . 1 0 )

" r e g u l a r i z e d " i f w e do t h a t ?

i s r e p l a c e d by k2+U2 t h e n t h e -CO d o e s n o t e x i s t . However o n e may a r g u e t h a t

I f k2 i n (2.10) limit

2

t h i s i n f i n i t y i s n o t v e r y p h y s i c a l . Suppose we sum diagrams o f t h e t y p e :

467

i

(2.12)

t h e n t h e k i n t e g r a t i o n i s o f t h e form

1

+a,g 2 l o g l k l + p 2 - i s

Clearly the l i m i t U

2

$0

(remainder).

(2.13)

e x i s t here. I t has a s m a l l

imaginary p a r t due t o a t a c h y o n i c p o l e i n t h e d r e s s e d p r o p a g a t o r , which c a n n o t be a d m i t t e d i n a r e a l t h e o r y , 3

b u t t h i s is probably due t o t h e f a c t t h a t $

t h e o r y is

u n s t a b l e and w e i g n o r e i t . The p o i n t i s t h a t t h e "summed" t h e o r y i s supposed t o be f r e e of i n f r a - r e d d i v e r g e n c e s . I n w r i t i n g down ( 2 . 1 3 )

we r e p l a c e d a s o m e t i m e s d i -

v e r g e n t sum of b u b l e s by a n a n a l y t i c and c o n v e r g e n t e x p r e s s i o n . T h i s e x p r e s s i o n c o u l d have been o b t a i n e d d i r e c t l y i f w e wrote t h e Dyson-Schwinger e q u a t i o n s f o r t h e s e diagrams : (2.14)

I f eq.

(2.14)

ai s used t h e n t h e diagram

..... ......... .......

........... ........... ...... ....... .....

.:::::::::.

(2.15)

y i e l d s ( 2 . 1 3 ) . Now ( 2 , 1 4 ) i s o n l y a t r u n c a t e d Dyson-

Schwinger e q u a t i o n . I f w e would u s e t h e c o m p l e t e s e t

468

of equations we would get the complete amplitude and this may be assumed to be free of divergences. (Also, in a good theory, free of tachyonic poles or cuts). A good theory should be a solution to its DysonSchwinger equations with dressed propagators not more singular than l / k 2 Let us return to pure gauge theory in 2+1 dimensions. We can now roughly estimate how the integral ( 2 . 6 ) has to be cut-off, by replacing the bare propagators by dressed ones. This is not exactly according to Dyson-Schwinger equations but good enough for our purpose. The dressed propagator is

.

(2.16) 2

+...

Let us assume that f(z) has an expansion l+alz + a2z but is non singular at z + m (Thlsasymptotic expansion for f is not quite right, as as ws will see later). Our integral is

.

(2.17)

where R ( k ) expand R:

is the non-singular remainder. Let us

R(k)

=

Ro

+

R

P

k

P

+

R

!Jv

k

P

kL,

+

...,

(2.18)

and split the integral in two pieces:

(2.19)

469

I

w i t h g2 <<

E

2 I(g 1 =

( 2 . 1 9 ) as

<< 1. W e w r i t e I,(EI9

2

)

+ I2

(El9

2

I t i s e a s y now t o e s t a b l i s h how I

and g2:

1 It2

2 4 d3k ( l + 2 a l g ma 2+ 9 +.

PI2

Ik(>E k =

1

n

-

2

(2.20)

..

4 m ~ ~ ( & +a g, l o g E -

behave for s m a l l

E

) ( Ro+Rpkp+Rpvkpkv..

.

)

4

a2 ...) E

(2.21) And i f w e w r i t e k

=

then

I-r -

1 Bn(g21n + n

-

2

4mg 2 ~

E

9

~ +

a,( log 2 E + 2

.. . I

9

(2.22)

T a k i n g t h e t w o series t o g e t h e r w e f i n d t h a t , of c o u r s e , the

E

d e p e n d e n c e d i s a p p e a r s I b u t t h e a s y m p t o t i c g2-ex-

p a n s i o n does n o t o n l y c o n t a i n p o w e r s o f g 2 :

470

(2.23) SO now w e found where t h e i n f r a - r e d

infinity of t h e 2 i n t e g r a l ( 2 . 6 ) g o e s : I ( g ) h a s no p r o p e r T a y l o r ex2 2 pansion ir. g E q . ( 2 . 2 3 ) shows terms w i t h l o g g One

.

.

2

consequence of t h a t i s t h a t f ( g /

Ik

I

i n (2.16) w i l l

a l s o d e v e l o p l o y a r i t h m s f u r t h e r down i n i t s e x p a n s i o n series. I f w e c o r r e c t t h e p r e v i o u s a n a l y s i s f o r t h i s w e f i n d h i g h e r powers o f l o g a r i t m s f u r t h e r down t h e series. S i n c e a and Ro a r e known, t h e c o e f f i c i e n t i n 1 f r o n t of t h e loqx-ithm i s w e l l d e t o r r d r e o . On t h e c t h e r hand, t h e c o e f f i c i e n t B 1 c a n o n l y be d e t e r m i n e d i f f ( z ) i s c o m i e t e l y known, b e c a u s e it must b e i n t e g r a t e d o v e r . I n g e n e r a l , a t any f i x e d power o f CJ

2

,

only t h e

l e a d i n g power of i o y g 2 h a s a w e l l d e t e r m i n e d c o e f f i c i e n t . A l l o t h e r c o e f f i c i e n t s c a n o n l y be e v a l u a t e d i f t h e complete n o n -p e rt u rb a t i v e known. A t h i g h e r o r d e r s

31SG

s o l u t i o n of f(z) i s

t h r e e - a+

more-pcint

Green f u n c t i o n s a r e r e q u i r e a . Thus, t h e c o e f f i c i e n t s f o r t h e n o n - l e s d i n g powers of l o g g 2 c a n n e v e r k-e d e t e r m i n e d by p e r t u r b a t i v e m e a n s

47 1

R

lGne.

111. VACUUM STRUCTURE IN TEE PERTURBFTIVE 2+1

DIMENSIONAL THEORY. GAUGE INVARIANCE Does t h e disease o b s e r v e d i n t h e p r e v i o u s s e c t i o n have a n y t h i n g t o d o w i t h c o n f i n e m e n t ? One m i g h t a r g u e t h a t t h e d r e s s e d p r o p a g a t o r does n o t e x i s t i n a t h e o r y w i t h c o n f i n e m e n t . However, s i n c e i t i s n o t gaugei n v a r i a n t , it might e x i s t i n c e r t a i n c o n v e n i e n t gauges, and w e do n o t have t o w o r r y a b o u t s p u r i o u s p o l e s or c u t s i n t h i s p r o p a g a t o r b e c a u s e t h e y need n o t c o r r e s p o n d t o p h y s i c a l states. I n fact t h e r e are r e a s o n s t o believe t h a t i n c e r t a i n c o v a r i a n t gauges t h e propagator i s a c t u a l l y v e r y smooth and c o n v e r g e n t . Confinement i s a p r o p e r t y o f t h e vacuum i n t h e f i e l d t h e o r y a n d , a s we w i l l a r g u e now, i t i s t h e vacuum

t h a t d e t e r m i n e s o u r unknown c o e f f i c i e n t s . The f i r s t unknown c o e f f i c i e n t w a s B1 i n ( 2 . 2 2 ) .

It is

Although unknown, i t s dependence on t h e r e m a i n d e r o f t h e g r a p h , R , i s v e r y s i m p l e : i t o n l y d e p e n d s on R ( k = o ) . T h i s i m p l i e s t h a t once it h a s been determined f o r one g r a p h , i t w i l l be known f o r a l l o t h e r g r a p h s as well. L e t u s t a k e t h e diagram w i t h R = l :

472

It corresponds to the computation of

Of course it is ultra-violet divergent, but that divergence can be assumed to be subtracted in the usual way. We conclude that if we assume some value for subtr. CO = vacuum

(3.3)

then all amplitudes can be expanded up to 2

2

r(kl...lg ) = I'o(kt-.*) + g r l ( k , * * * )

+ 0(g6) In particular, will need

r 02

.

(3.4)

depends on Co. At higher orders one

4

c*

=

c3

= cap


subtr

.

(x)> vacuum

'

subtr. vacuum

J

am

(3.5)

etc. One may observe that A2 in (3.3) is not gaugeinvariant. Therefore, the actual value of CO should 0 not affect the r2 of gauge-invariant Green functions. Indeed, gauge-invariant Green functions have Ro= 0, as a consequence of certain Ward-Takahashi identities.

473

But of course, the relevant quantities are

a

1

bl

= < G

uv

=

G

>

p"

<$JI>

subtr. vacuum subtr. vacuum

By power counting it is possible to tell where in the power series these coefficients center. a has dimension

1

3; b 1 has di.mension 2 ; a2 has dimension 5. And g2 has dimension 1. Therefore, al will enter at order q8; bl at order g 6 a:id a2 at order cj 1 2

.

One may hope that iterative procedures exist to determine these coefficiznts semi-perturbatively: determine the known coefficients for f(i); extrapolate (Padti?), then to find a decent function f o r z +. substitute this f ! z ) tc! find the next coefficients and repeat. Whatever one does, the procedure will not be as straightforward as the determination of <€I> from a Higgs Lagrangian. The unknowns this time form an infinite series. They are the vacuum expectation values of all composite operators. It may seem that these vacuum experation values are only needed in the 2+1 dimensional theory, and that the problems discussed here are actually irrelevant for the real world which is in 3+1 dimensions. This, we emphasize, is not true. In 3+1 dimensions the same problem occurs, however not within the ordinary perturbation expansion. This perturbation expansion namely diverges at high orders. If one tries to rearrange the perturbation expansion to obtain

474

convergent expressions, one f i n d s t h a t amplitudes can b e c o n v e n i e n t l y expresaeci by a B o r e 1 f o r m u l a :

2

m

B(z)

I’(g2) =

,“Ig

dz/g2

.

0

According t o t h e r e n o r m a l i z a t i o n g r o u p B(z) h a s dimension p r o p o r t i o n a l t o z . I f t h i s dimension c o i n c i d e s


,

etc. s i n g u l a r i t i e s arise and t h e i r t r e a t m e n t g o e s a l o n u t h e same l i n e s a s i n t h e p r e v i o u s s e c t i o n . We r e f e r t o r e f s . [ 5 , 6 1 f o r f u r t h e r details.

w i t h t h a t of

IV.

GDv>

2+1 DIMENSIONS NON PERTURBATIVELY:

TOPOLOGICAL OPERATORS AND THEIR GREEN FUNCTIONS I n t h i s s e c t i o n a new f i e l d i s i n t r o d u c e d t h a t

w i l l e n a b l e u s t o g e t a b e t t e r u n d e r s t a n d i n g of t h e p o s s i b l e vacuum s t r u c t u r e s i n 2 i 1 d i m e n s i o n s . To s e t up o u r a r g u m e n t s s t e p ty s t e p w e b e g i n w i t h a d d i n g a Higgs f i e l d [ T ] . T h e gauge g r o u p i s SU(N) and t h e

g a u g e symmetry i s s n o n t a n c o u a l y and c o m p l e t e l y b r o k e n . The H i g g s f i e l d s , H , must be a s e t o f u n i q u e r e p r e s e n t a t i o n s of S U ( N ) / Z ( N )

s u c h a s t h e o c t e t and d e c u p l e t

r e p r e s e n t a t i o n s o f SU(3)/ Z ( 3 )

. Thus

a l l ( v e c t o r and

scalar) f i e l d s are i n v a r i a n t under t h e c e n t e r Z ( N ) o f t h e gauge g r o u p SU(N) ( T h i s i s t h e subgroup o f matrices e ” in” 1, where n i s i n t e q e r . ) Q u a r k s , which are n o t i n v a r i a n t u n d e r Z (14) , a r e n o t y e t introduced a t t h i s stage. B e s i d e s t h e m a s s i v e p h o t o n s a n d Higgs p a r t i c l e ( s ) t h i s model c o n t a i n s o n e o t h e r c l a s s of p a r t i c l e s :

.

475

e x t e n d e d s o l i t o n s o l u t i o n s t h a t a r e s t a b l e k e c a u s e of

+.

a t o p o l o g i c a l c o n s e r v a t i o n law Consider, therefore, a r e g i o n R i n t w o - d i m e n s i o n a l s p a c e s u r r o u n d e d by a n o t h e r r e q i o n B where t h e e n e r g y d - . n s i t y i s z e r o (vacuum). I n B , t h e Hicjgs f i e l d H ( x )

satisfies

where F i s a f i x e d number. l h e r e must be a q a u g e rotation

R ( 2 ) so

that

where Ho i s f i x e d .

n(2)

i s determined up t o elements

. We

a l s o r e q u i r e a b s e n c e of s i n g u l a r i t i e s so R (2) i s c o n t i n u o u s . C o n s i d e r a c l o s e d c o n t o u r C ( 8 ) 2Tr and i n B p a r a m e t r i z e d by a n a n g l e 8 wit.h 0 2 8

of Z ( N )

C ( 0 ) = C(2n).

C o n s i d e r t h e case t h a t C goes c l o c k w i s e

a r o u n d R . S i n c e 3 i s n o t s i m p l y c o n n e c t e d w e may h a v e

with 0

5

n ( N t n i n t e g e r . Because o f c o n t i n u i t y ,

+For

n is

a g e n e r a l i n t r o d u c t i o n t o s o l i t o n s see e . g . Coleman [81. F o r a n i n t r o d u c t i o n t o t h i s p r o c e d u r e see ' t H o o f t [ 9 1 .

476

c o n s e r v e d . I f n f 0 and i f ke r e q u i r e a b s e n c e of s i n g u larities i n R then t h e f i e l d configuration i n R cannot b e t h a t o f t h e vacuum o r a gauge r o t a t i o n t h e r e o f , so t h e r e must he s o m f i n i t e amount of energy i n R. The f i e l d c o n f i g u r a t i o n w i t h l o w e s t e n e r g y E , i n t h e case n = 1 , describes a s t a b l e s o l i t o n w i t h mass M=E. I f N > 2 t h e n s o l i t c n s d i f f e r from a n t i s o l i t o n s ( o h i c h c o r r e s p o n d t o n=N-1) and t h e nunber o f s o l i t o n s minus a n t i s o l i t o n s is c o n s e r v e d modulo N. An a l t e r n a t i v e way t o r e p r e s e n t t h e f i e l d s c o r r e s p o n d i n g t o a s o l i t o n c o n f i g u r a t i o n is t o e x t e n d n(x') t o be also w i t h i n R (which, however, may be p o s s i b l e o n l y i f o-e admit a s i n g u l a r i t y f o r n a t some p o i n t x' i n €?) . iJe t h e n a p p l y n-1 t o t h e above f i e l d 0 c o n f i g u r a t i o n . T h i s has t h e a d v a n t a g e t h a t e v e r y o h e r e i n B w e keep H=Hol regardless o f t h e number n of s o l i t o n s i n R I b u t t h e p r i c e of t h a t is t o a l l o w f o r a singularii-y i n R. W e w i l l r e f e r t o t h i s a s t h e "second r e p r e s e n t a t i o n " o f t h e s o l i t o n , f o r l a t e r u s e . A s e t of operators @(;) i s now d e f i n e d h s f o l l o w s . L e t lAi(Z),H(Z)> be a s t a t e i n H i l b e r t s p a c e which is a n e i g e n s t a t e of t h e s p a c e components of t h e v e c t o r f i e l d s and t h e Higgs f i e l d s , w i t h Ai(3) and H(2) as g i v e n e i g e n v o l u e s Then

zo

.

+

[xol where Q gauge r o t a t i o n w i t h t h e p r o p e r t y t h a t f o r e v e r y closed c u r v e C ( e ) t h a t e n c l o s e s %o once w e have (4.5)

where t h e minus s i g n h o l d s f o r c l o c k w i s e and t h e

477

+ sign

+ for anticlockwise C. When xo is outside C, then

+ + +

[X,l

The singularity of R at x=x must be smeared over +O an infinitesimal region around xo but we will not consider this "renormalization" problem in this paper. + In what sense is #(x) a l o c a l operator? The operator formalism in qauge theories is most conveniently + formulated in the gauge Ao(x,t)=O. Then, timc-independent continuous gauge rotations D(") still form an invariance group. For physical states[$> however, (4.7)

where R is any single-valued gauge rotatiqn. So #(;) would have been trivial were it not that R[xo3has a + singularity at xo. The operator a(;) l o a d s physical states into physical states, and the details of Q apart from ( 4 . 5 ) and ( 4 . 6 ) are irrelevant. It is now easy to verify that

@(a@($) I$>

= O($)#(Z)

I$>

I$>

I

is physical state, because koth the left- and the right-hand sides of ( 4 . 9 ) are completely defined by the singularities alone of the combined gauge + rotations. Also, when R(x) is a conventional gaugeinvariant field operator composed of fields at 2 then obvious1y if

478

[R(g),@($)l=

0

for

but n o t n e c e s s a r i l y

x' # 9 ,

for Because o f

( 4 . 8 ) and ( 4 . 9 )

t

=

3

(4.9)

i s considered to be a

@

l o c a l f i e l d o p e r a t o r when i t 8 c t s on p h y s i c a l s t a t e s . From i t s d e f i n i t i o n i t must be c l e a r t h a t a b s o r b s one t o p o l o q i c a l u n i t , so w e s a y t h a t

@(;I

@(;I

is

t h e a n n i h i l a t i o n ( c r e a t i o n ) o p e r a t o r f o r o n e "bare" -b

s o l i t o n ( a n t i s o l i t o n ) a t x and

+

@ (x) i s t h e c r e a t i o n

( a n n i h i l a t i o n ) o p e r a t o r f o r one " b a r e " s o l i t o n (antisoliton)

.

L e t u s now i l l u s t r a t e how one c a n compute Green

functions involving

#(x) by o r d i n a r y s a d d l e - p o i n t

t e c h n i q u e s i n a f u n c t i o n a l i n t e g r a l . Let u s c o n s i d e r -I-

< T ( @ ( O , t l ) @ (o,O))>=f ( t l )by computing t h e c o r r e sponding f u n c t i o n a l - i n t e g r a l e x p r e s s i o n : ~ D A D He x p S ( A , H I

atl) =

C

exp

IDA-DH

s (A,H)

'

(4.10)

where C i s t h e set o f f i e l d c o n f i g u r a t i o n s where t h e f i e l d s make a sudden gauge j w . p a t t = O d e s c r i b e d by

Q[ol

(see ( 4 . 5 ) and ( 4 . 6 ) ) and a t t = t l t h e y jump back by a t r a n s f o r m a t i o n Q[ol-C f i e l d s r.iust he c o n t i n u o u s everywhere else. This w a s how f ( t

1

)

f o l l o w s froin t h e d e f i n i t i o n s

b u t i t i s more e l e g a n t t o t r a n s f o r m a l i t t l e f u r t h e r . By gauge t r a n s f o r m i n g back i n t h e r e g i o n 0 < t < t w e 1 g e t t h a t t h e f i e l d s are c o n t i n u o u s e v e r y w h e r e e x c e p t a t -b

x=O,O < t < t l . So t h e r e i s a Dirac s t r i n g [ l o 1 g o i n g

t o ( 0 , t ) . A t 0 < t < t l we o b t a i n i n t h i s way t h e s o l i t o n i n i t s "second r e p r e s e n t a t i o n " : a nonfrom (0,o)

479

trivial field configuration with a singalarity at the + origin. In short:<@(x)@ ( O ) > is obtained ky integrating over field configurations with a Dirac string in spacetime frcn 0 t o X. Let us compute f (t,) for Euclidean time tl = iT, T real. Let the field theory be pure SU(N) without Higgs Scalars. We must find the 1eastnecJative action configuration with the given Dirac strinq. The conditions ( 4 . 5 1 , ( 4 . 6 ) can be realized for an Abelian subqroup of gauge rotations a , so let us take

the Last diagonal element being different from the others because we mist. have dctR = 1 for all 8 . Here 8 Ls t h e angle around the time axis (remernber space-time is here t h m e diaer.slona1) The singularity+ at the Dirac string must be t h e one obtained when $2CO 1 acts on the vacuum.

.

The transformations ( 4 . 1 1 ) form an Abelian subgroup, so, as an ansatz, the field configuration with this string singularity may be chosen to be an Abelian subset of fields corresponding to this subgroup:

7l Xij a

A,,(x) a

= a,,(x) XiJ(N) ;

(4.12)

F

IJV

- au

av

-

av

ap

(4.13)

are t h e c o n v e n t i o n a l Gell-Mann matrices extended t o SU(N). Within t h i s set of f i e l d s w e j u s t have t h e l i n e a r Maxwell equation..;, and t h e Dirac s t r i n g i s t h e one c o r r e s p o n d i n g t o t w o o p p o s i t e l y charged Dirac monopoles, one a t 0 and one s t t l . T h e i r maqnetic c h a r g e s a r e +2n/gN. The t o t a l a.ction of this c o n f i g u r a t i o n i s Here A;

S = 3 1 N(N-l)lFpv F

PV

d 3x = (4.14)

where 2 , and Z 2 are t h e s e l f - e n e r g i e s of t h e monopoles, which d i v e r g e b u t c a n be s u b t r a c t e d , l e a v i n g a spncet i m e independent r e n o r m a l i z a t i o n c o n s t a n t . W e now assume t h a t (4.14) is i n d e e d a n a b s o l u t e extremum f o r t h e t o t a l a c t i o n of a l l f i e l d c o n f i g u r a t i o n s e think t h a t t h i s with t h e given s t r i n g s i n g u l a r i t y . W assumption is p l a u s i b l e b u t p r e s e n t no p r o o f . We t h u s o b t a i n a f i r s t approsirtiation t o (4.10) : &

tl = i-c

,

T

real

. (4.15)

Here A is a f i x e d c o n s t a n t o b t a i n e d a f t e r s u b t r a c t i o n of t h e i n f i n i t e s e l f - a c t i o n a t t h e s o u r c e s . Note t h e

481

p l u s s i g n i n o u r exponent due t o t h e a t t r a c t i v e f o r c e between t h e monopole-antimonopole p a i r . Computation of 2 t h e terms o f h i g h e r o r d e r i n g T s u f f e r s from t h e o b s t a c l e s mentioned i n sect. 2 . I n any case, a t l a r g e T we e x p e c t n o convergence. O f c o u r s e , when t h e Higgs

mechanism i s t u r n e d on t h e n t h e s o l i t o n a c q u i r e s a f i n i t e mass, s a y M , and t h e n a t l a r g e T we e x p e c t

t,

=

T

= real

iT,

.

(4.16)

j u s t a s any o r d i n a r y ( d r u s s e d ) p r o p a g a t o r , However, t h e r e are il.lso ir,ternc: t i m s , i n p a r t i c u l a r t h e N - s o l i t o n p r o c e s s C s , d e s c r i b e d e s s e n t i a l l y by

(4.17)

where N i s t h e g r o u p p a r a m e t e r . Again w e c h o o s e {xk} t o T h e r e i s some freedom i n c h o o s i n g t h e Dirac s t r i n g s , for i n s t a n c e w e can l e t one s t r i n g l e a v e a t e a c h p o i n t X ~ , . . . , X ~ and - ~ l e t t h e s e a l l assemble a t xN. Because of t h e modulo N c o n s e r v a t i o n l.zv~, :he N-1 q u a n t a coming i n a t x a r c e q u i v a l e n t t o one quantum leavirlg a t x N' To f i n d t h e f i e l d configuratj-on w i t h least n e g a t i v e a c t i o n we c o u l d t r y a g a i n t.he a n s a t z ( 4 . 1 2 ) b u t t h e n t h e r e s u l t i s t h a t X ~ , . . . , X ~ r- e~p e l each o t h e r and a l l a t t r a c t xfl, a n unsymmctric and t h e r e f o r e u n l i k e l y r e s u l t . Ohviously,. s n y f i e l d conf i q u r a t i o n where t h e sicjns i n t h e e x p o n e n t s s u c h as (11.15) a r e p o s i t i v e , c o r r e s p o n d i n g t o a t t r a c t i o n , w i l l T i v e much be E u c l i d e a n .

larger, therefore dominating, contributions to the such a field conamplitude. So let us try to produce figuration now. Observe that pure permutations of the N spirm components are good elements of S U ( N ) , so by pure gauge rotations we are allowed to move the unequal diagonal element of (4.11) up or down the diagonal. So let us now again choose one Dirac strring leaving at each of xl,. ,xN- 1 and all entering at xN’ but this time all these strings have their unequal diagonal element (see (4.11)) in a different position along the diagonal At xN the combined rotation is again of type (4.11) as one can easily verify, so this is a more symrietric configuration. Since the qzuge transformations we performed so far are ali diagonal elementc of SU(N) they actually form the subgroup[U ( 1 ) IN-’ of SU (N) which is Abelian still. Let us define

..

k = l,...N

.

(4.18)

with N k= 1 ’(k)

= o .

so one of thece X matrjces is actu&lly redurdant. Our present ansatz is

(4.19)

without bothering about the invariance

483

(4.20)

(4.21)

with (4.22)

Notice t h e chanrje of s i g n

ir,

(4.21) when f i e l d s o f

d i f f e r e r l t i n d e s o v e r l a p . We now ch(:ose

i1

I?i:-ac monopole

c o r r e s p o n d i n g t o t h e k t h subgroup U ( 1 ) a t x

SO

A(k)

(x)

k, i s j u s t t h e f i e l d o f o n e monopole a t x k (remember t h a t , b e c a u s e we s t a y i n t h e A b e l i a n s u b c l a s s o f f i e l d s , l i n e a r s u p e r p o s i t i o n s a r e a l l o w e d ) . The d i a g o n a l terms i n (4.21)

t l l e n o n l y c o n t r i b u t e t o t h e monopole “ s e l f - e n e r g y ”

(more p r e c i s e l y : s e l f - a c t i o n ) a n d o n l y c o n t r i b u t e t o t h e a l r e a d y e s t a b l i s h e d Z f a c t o r t h a t must be s u b t r a c t e d . Only t h e cross t e r m s g i v e n o n - t r i v i a l e f f e c t s :

Again w e assume t h a t t h i s r e p r e s e n t s t h e l e a s t n e g a t i v e a c t i o n configuration without presenting thecompleteproof.

484

A good check o f t h i s e q u a t i o n i s t h a t if N-1

points come close and t h e Nth s t a y s f a r away t h e n w e r e c o v e r ( 4 . 1 4 ) a p a r t from o v e r a l l n o r m a l i z a t i o n , e x a c t l y as one s h o u l d e x p e c t . So w e conclude:

f (x,

...,x

,

N

) = ~ ’ e x p [ -“ 1 2g2N k>P.

’ + Ixk-xBr

O(10cJ(j2T))

I. (4.24)

f o r E u c l i d e a n {X,). The above was maLn1.y to i l l u s t r a t e how t o compute i n l o w e s t - o r d e r p e r t u r b a t i o n e x p a n s i o n t h e Green f u n c t i o n s t h a t w e w i l l d i s c u s s i u r t h e r i n s e c t . 5 . Wedid n o t prove t h a t t h e f i e l d conf i a u r a t i o n s w e choose t o expand a b o u t a r s r e a l l y t h e minima:. a n c s (which w e do e x p e c t ) b u t t h e r e i s no need t o e l a b o r a t e on t h a t p o i n t f u r t h e r here.

V.

SPONTPNEOUS SYMMETRY BREAKING AND CONFINEMENT

I n s e c t . 4 w e found t h a t some SU(N) gauge t h e o r i e s i n 2+1 dimensions p o s s e s s a t o p o l o g i c a l quantum number, conserved modulo N , and t h a t Green f u n c t i o n s c o r r e sponding t o exchange o f t h i s quantum may be computed. Our f i e l d behaves as a l o c a l complex scalar f i e l d (real f o r S u ( 2 ) i n a l l respects. M e expect t h a t these Green f u n c t i o n s s a t i s f y a l l t h e usu3al Wightman axioms [ 11 1 except that they small d i s t a n c e s . p u t e d i n sect.4 with free scalar

are more s i n g u l a r t h a n u s u a l I n f a c t , t h e Green f u n c t i o n s c a n be e x a c t l y reproduced i n p a r t i c l e s and non-polynomial

485

at as c o m a theory sources

using superpropagator techniques [1?1. W e leave t h a t

as a n e x e r c i s e €or t h e r e a d e r . I n p u r e S U ( N 1 t h e t r o u b l e i s t h a t o n l y t h e quantum c o r r e c t i o n s g i v e s o m e i n t e r e s t i n g s t r u c t u r e t o t h e s e Green f u n c t i o n s . The terms o f h i g h e r o r d e r s i n c j 2 i n ( 4 . 1 5 ) and ( 4 . 2 4 )

correspond t o

quantum l o o p c o r r e c ! t ? o n s . They h a v e n o t y e t been c o m puted, t o t h e e x t e n t t h a t they can be comp u t e d . However, l e t u s assume t h a t t h e r e s u l t i n g Green f u n c t i o n s c a n be computed and a r e r o u g h l y g e n e r a t e d by some e f f e c t i v e LacJrimyian w i t h n e c e s s a r i l y s t o n g coupling, f o r instance:

- -X I (aN + N!

(@*IN)

-

X2V(@*@)

.

Here w e have assumed t h a t t h e Higgs mechanism ( e i t h e r by some e x p l i c i t o r Sy some d y n a m i c a l Higgs f i e l d ) makes a l l f i e l d s m a s s i v e , g e n e r a t i n g a l s o a l a r g e s o l i t o n m a s s M.

O f c o u r s e we expect many p o s s i b l e

i n t e r a c t i o n terms b u t o n l y t h e m o s t i m p o r t a n t o n e s

are added i n ( 5 . 1 ) : t h e h t e r m p r o d u c e s t h e 111 s o l i t o n i n t e r a c t i o n and i s r e s p o n s i b l e f o r t h e nonv a n i s h i n g r e s u l t (4.24) f o r t h e e x p r e s s i o n ( 4 . 1 7 ) , t h e N-point f u n c t i o n . The X 2 t e r m i s o f c o u r s e a l s o e x p e c t e d and i s i n c l u d e d i n ( 5 . 1 ) f o r r e a s o n s t h a t w i l l become c l e a r l a t e r . O b s e r v e now a Z ( E J ) g l o b a l s y m e t r y t h a t l e a v e s t h e L a g r a n g i a n ( 5 . 1 ) , a n d t h e Green f u n c t i o n s cons i d e r e d i n sect.4 i n v a r i a n t :

486

T h i s is simply t h e symmetry a s s o c i a t e d w i t h t h e

t o p o l o g i c a l l y conserved s o l i t o n quantum number. Modulo N c o n s e r v a t i o n laws c o r r e s p o n d t o Z ( N ) g l o b a l symmetries. As w e s a w , t h e mass t e r m i s e s s e n t i a l l y d u e t o 2 t h e Higgs mechanism. Roughly, V2 a :U where l-IH is t h e second d e r i v a t i v e of t h e Higgs p o t e n t i a l a t t h s o r i g i n . W e can now c o n s i d e r e i t h e r s w i t c h i n g o f € t h e Higgs f i e l d , o r j u s t changing t h e s i g n of p i so t h a t < H > c. What happens t o M 2 ( t h e s o l i t o n m a s s ) ? I f it s t a y s p o s i t i v e t h e n t h e p h y s i c a l s o l i t o n h a s n o t gone away and t h e t o p o l o g i c a l c o n s e r v a t i o n l a w r e n a i n s v a l i d . The +

symmetry of t h e t h e o r y is as i n t h e Higgs mode and we d e f i n e this node t o be a "dynamical Higgs mode". However, a v e r y good p o s s i b i l i t y is t h a t M2 a l s o s w i t c h e s its s i g n , so t h a t <#>+F#O.The t o p o i o g i c a l g l o b a l Z ( N ) symmetry may g e t s p o n t a n e o u s l y broken. T h i s mode can b e r e c o g n i z e d d i r e c t l y from t h e Green f u n c t i o n s of s e c t . 4 . The c r i t e r i o n is

I F I *= , l i m

rtt,),

t, = iT

.

(5.3)

TI-...

J u s t f o r amusement we might n o t e t h a t ( 4 . 1 5 ) i n d e e d seems t o g i v e F f 0 b u t of c o u r s e t h e auantum c o r r e c t i o n s may n o t b e n e g l e c t e d and we do n o t knc,!. a t p r e s e n t how t o a c t u a l l y lcompute t h e l i m i t (5.3). Spontaneous breakdown of t h e t o p o l o g i c a l Z ( H ) symmetry i s a new phase t h e system may c h o o s e , depending on t h e dynamics. W e may compare a bunch of m o l e c u l e s t h a t c h o o s e s t o be i n a g a s e o u s , l i q u i d or s o l i d phase depending on t h e dynamics and on t h e v a l u e s of c e r t a i n intensive parameters. L e t u s s t u d y this new phase more c l o s e l y . The vacuum w i l l now have a Z (PI) d e q e n e r a c y , t h a t i s , a n

487

N-fold degeneracy. Labeling these vacua by an index from 1 to N we have

(5.4)

Since the symmetry is discrete there are no Goldstone particles; all physical particles have some finite mass. Again we are able to construct a set of topologically stable objects: the Bloch walls that separate two different vacua. These vortex-like structures are stable because the vacua that surround them are stable (Bloch walls are vortex-like because our model is in 2 + 1 dimensions). The width of tne Bloch wall or vortex is roughly proportional to the inverse of the lowest mass of all physical particles, and therefore finite. The Bloch wall carries a definite amount of energy per unit of length. We will now show the relevance of the operator

A(C) = Tr P exp

4

+ k ig Ak(x) dx

C

,

(5.5)

for these Bloch walls. Here C is an arbitrary oriented contour in 2-dim. space and P stands for the path ordering of the integral. $j are the space components of the gauge vector field in the matrix notation. A(C) is a non-local gauge-invariant operator that does not commute with @ Let us explain that. As is well known, if C' is an open contour, then the operator + x2 + + k A(C',x,xZ) = P exp 1 ig Ak(j;)dx ,

.

+

x;c'

transforms under a gauge rotation

fl

as

Now the operator @(go) was defined by a 3auge transformation Si that is multivalued when followed + over a contour that encloses x So when we close O+ C ' to obtain C, and C encloses xo once,

.

+

+

P(C) = Tr A(C,x, ,xl1

I

(5.7)

then the value of A makes a jump b y a factor ex? (f21~i/N) + wnen the operator @(xJ acts, so P ( C ) Q ( Z o ) = @(~o)P(C)exp(2~in/N) , (5.a) where n counts the number of times that C winds around -+ x0 in a clockwise fashion minus the number of tines + it winds around xo anticlockwise. E q . (5.8) is an extension of eq.(4.9) for non-local o?erators A(C). As we will see, (5.8) can be generalized to 3+1 dimensions. Now let us inter?ret (5.8) in a framework where a(;) is diagonalized. Then, as we see, A(C) is an operator that causes a jump by a factor exp(2nin/N) of 4(;)for all 2 inside C. So A(C) causes a switch from one vacuum to another vacuum within C in the case that Z ( N ) is spontaneously broken. In other words A ( C ) creates a "bare" Bloch wall or vortex exactly at the curve C. Our model does not yet include quarks. Quarks are not invariant under the center Z(N), so they do a(;). This difficulty not admit a direct definition of is to be expected when one considers the physics of the system. The vortices that were locally stable

489

w i t h o u t q u a r k s may now become l o c a l l y u n s t a b l e d u e t o v i r t u a l quark-antiquark

p a i r c r e a t i o n . Most a u t h o r s

t h e r e f o r e c o n s i d e r quark confinement t o b e a b a s i c p r o p e r t y of t h e g l u e s u r r o u n d i n g t h e q u a r k s , i n wich q u a r k s must be i n s e r t e d p e r t u r b a t i v e l y . Such a p r o c e d u r e i s j u s t i f i e d by t h e e x p e r i m e n t a l e v i d e n c e ; a l l h a d r o n s c a n b e l a b e l e d a c c o r d i n g t o t h e number a n d t y p e s of q u a r k s t h e y c o n t a i n ; none o f them is s a i d

t o b e composed of a n u n s p e c i f i a b l e o r i n f i n i t e number o f q u a r k s . The number o f g l u o n s on t h e o t h e r hand

c a n n o t e a s i l y be g i v e n . W e s h o u l d n ’ t s a y it is z e r o f o r most h a d r o n s , b e c a u s e w e need t h e v e r y s o f t g l u o n s to provide t h e binding force. How t o i n t r o d u c e q u a r k s a t t h e p e r t u r b a t i v e l e v e l i s f u r t h e r e x p l a i n e d i n r e f . l 1 3 1 . T h e outcome i s t h a t q u a r k s a r e t h e end p o i n t of a v o r t e x . The conventional operator -b

X

2

[P e x p ( j +

k i g Pk(G)dx

IJI(X,)

I

(5.9)

creates n o t o n l y a quark p a i r b u t a l s o a v o r t e x i n between them. T h i s v o r t e x i s t o p o l o g i c a l l y s t a b l e i f < a > = F # 0. I f w e h a v e a c o n f i g u r a t i o n w i t h N q u a r k s t h e n @ makes a f u l l r o t a t i o n over 271 when it f o l l o w s a c l o s e d c o n t o u r a r o u n d . T h i s is why a “baryon” c o n s i s t i n g of N quarks i s n o t co n fi n ed t o a n y t h i n g e l s e . E v i d e n t l y , f o r r e a l b a r y o n s N must be 3. Our c o n c l u s i o n is a s follows. I n SU(N) g a u g e t h e o r i e s where a l l s c a l a r f i e l d s

are i n r e p r e s e n t a t i o n s

t h a t are i n v a r i a n t u n d e r t h e c e n t e r Z ( N ) o f S U ( N ) ( s u c h as o c t e t o r d e c u p l e t r e p r e s e n t a t i o n s of S U ( 3 ) ) , there exists a non-trivial topological global

invariance. If the Higgs mechanism breaks S U ( N ) completely then the vacuum is Z ( N ) invariant. However, we can also have spontaneous breakdown of Z ( N ) symmetry. If that breakdown is comnlete thenwe can have no Higgs mechanism for S U ( N 1 , because in that mode "colored" objects are germanently and completely confined by the infinitely rising linear gotentials due to the Bloch-wall-vortices. 5Je can also envisage the intermediate modes where a Higgs mechanism breaks S U ( N ) partly, and Z ( N 1 is partly broken. Finally, if neither Higgs' effect, nor spontaneous breakdown of Z ( N ) take place, then there must be massless particles causing complicated long range interactions as we will show more explicitly for the 3+1 dimensional case. That may either corrPsDond to a point where a higher order phase transition occurs, or to a new phase, e.q. tne Coulomb or Georgi-Glashow phase, where an effective Abelian photon field survives at lorgdiatances, see sect. 12. Eqs. ( 4 . 8 ) and 5 . 8 ) are the basic commutation relations satisfied by our topological fields @. They suggest a dual relationship between A and @. Indeed, one could start with a scalar theory exhibiting global Z ( N ) invariance and then define the topological operator A(C) through eq. ( 5 . 8 1 , but it is impossible to see this way that A can be written as the ordered exponent of an integral of a vector potential, and also the gauge grou? S U ( N ) cannot be recovered. As we will exnlain later, the center Z ( N ) is more basic to this all then the complete group S U ( N ) . A good name for the field 8 is the "disorder parameter" [14]since it does not commute with the other, usual, fields wich have been called order Z(N)

491

parameters in solid-state physics. The fact that in the quark confinement Fhase the degenerate vacuum states are eigenstates of this disorder narameter shows a close analogy with the superconductor where the vacuum state is an eigenstate of the order parameter.

VI. SU(N) GAUGE THEORIES IN 3 + 1 DIMENSIONS In the previous sections the construction of a scalar field and the successive formulation of the spontaneous breakdown of the topological Z(N) symmetry were only possible because the model was in 2 space, 1 time dimensions. Also the boundary between different but equivalent vacua can only serve as an vortex in 2+1 dimensions. It would have the topology of a sheet in 3+1 dimensions and therefore not be useful as a vortex of conserved electric flux. So in 3+1 dimensions the formulation of quark confinement must be considerably different from the 2+1 dimensional case. Nevertheless extension of our ideas to 3+1 dimensions is possiole. We concentrate on longe-range topological phenomena. One topological feature is the instanton, corresponding to a gauge field configuration with non-trivial Pontryagin or Second Chern Class numer. This however has no direct implication for confinement. What is needed for confinement is something with the space-time structure of a string, i.e. a two dimensional manifold in 4 dim. space-time. Instantons are rather event-like, i.e. zero dimensional and can far instance give rise to new types of interactions that violate otherwise apparent symmetries.

492

As we will see, they do Glay a role, though be it a subtle one. A topological structure which is extended in two dimensional sheets exists in gauge theories, as has been first observed by Nielsen, Olesen [15land Zumino C16l.They are crucial. We will exhibit them by comgactifying space-time. For the instanton it had been convenient to compactify space-time to a sphere For our purposes a hypertorus s4

s1

x

s1

s,

s1

is more suitable f171.0ne can also consider this to be a four dimensional cubic box with periodic boundary conditions. Inside, space-time is flat. Tne box may be arbitrarily large. To be explicit we put a pure SU(N) gauge theory in the box (no quarks yet). Now in the continuum theory the gauge fields themselves are representations of SU (NI/Z (N)I where 2 (N) is the center of the group SU(N) : Z(N) = (e2’in”I;

n = 0,. . . l ~ - ~ ~

.

Tnis is because any gauge transformation of the type (6.1) leaves A (X) invariant. A consequence of this is !J the existence of another class of topological quantum numbers in this box besides the familiar Pontryagin number. Consider the most general possible periodic boundary condition for A,,(x) in the box. Take first a plane {XlIX2) in the 12 direction with fixed values of x3 and x4. One may have

493

Here, a 1 ' a 2 are t h e p e r i o d s . RA s t a n d s s h o r t f o r lJ

The p e r i o d i c i t y c o n d i t i o n s f o r 51

( X I follow by

1,2 a t t h e c o r n e r s o f t h e box:

considering (6.2)

where Z i s some e l e m e n t of Z ( N )

.

One may now p e r f o r m c o n t i n u o u s gauge t r a n s f o r m a t i o n s on A,,(x)

I

(XI

1x2)

+

R ( x l 1 x 2 ) A (~x l , x 2 )

r

(6.5)

where R ( x , ~ x ~ ) ( n o n - p e r i o d i c ) c a n b e a r r a n g e d e i t h e r such t h a t n2(x1) I or s u c h t h a t R1(x2) -+ I , b u t n o t b o t h , because Z i n ( 6 . 4 ) remains i n v a r i a n t under (6.5) -+

as one can e a s i l y v e r i f y . W e c a l l t h i s e l e m e n t 2 ( 1 , 2 ) b e c a u s e t h e 1 2 p l a n e w a s c h o s e n . By c o n t i n u i t y Z ( l , 2 ) c a n n o t depend on x or x 4 . For e a c h ( p v ) d i r e c t i o n 3 s u c h a Z e l e m e n t e x i s t , t o be l a b e l e d by i n t e g e r s n

UV

d e f i n e d modulo N .

= -n

VlJ

'

Clearly t h i s gives d (d-1)

N

=

N6

t o p o l o g i c a l c l a s s e s o f gauge f i e l d c o n f i g u r a t i o n s . Note t h a t t h e s e classes d i s a p p e a r i f a f i e l d i n t h e fundamental r e p r e s e n t a t i o n o f S U ( N ) i s added t o t h e s y s t e m ( t h e s e f i e l d s would make u n a c c e p t a b l e jumps a t t h e b o u n d a r y ) . I n d e e d , t o u n d e r s t a n d q u a r k c o n f i n e m e n t i t is n e c e s s a r y

t o u n d e r s t a n d p u r e gauge s y s t e m s w i t h o u t q u a r k s f i r s t .

494

As we shall see, the new topological classes will imply the existence of new vacuum parameters besides the well-known instanton[l8] angle 0 The latter still exists in our box, and will be associated with a topological quantum number v , an arbitrary integer.

VII. ORDER AND DISORDER LOOP INTEGRALS To elucidate the physical significance of the we first concentrate on topological numbers n PV gauge field theory in a three dimensional periodic box with time running from - w to m . To be specific we will choose the temporal gauge,

(this is the gauge in wich rotation towards Euclidean space is particularly elegant). Space has the topology (S,I3

.

There is a n i n f i n i t e set o f homotopy classes

of closed oriented curves C in this space: C may wind any number of times in each of the three principal directions. For each curve C at each time t there is a quantum mechanical operator A(C,t) defined by

called Wilson loop or order parameter. Here P stands when the for path ordering of the factors A(2,t) exponents are expanded. The ordering is done with respect to the matrix indices. The A(x',t) are also operators in Hilbert space, but for different x' , same t, all A(2,t) commute with each other. ay analogy

495

with ordinary electromagnetism we say that A ( C ) m e a s u r e s magnetic flux t h r o u g h C , and in the same time c r e a t e s an electric flux line a l o n g C. Since A ( C ) is gaugeinvariant under purely periodic gauge transformations, our versions of magnetic and electric flux are gauge-invariant. Therefore they are not directly linked to the gauge Tovariant curl Ga ( 2 ) . PV

There exists a dual analogon of A ( C ) wich will be called B(C) or disorder loop operator f 1 3 1 . C is again 3 a closed oriented curve in ( S , ) A simple definition of B ( C ) could be made by postulating its equal-time commutation rules with A(C) :

.

P-(C')]

=

0;

[B(C), B ( C ' ) I

=

0;

[A(C),

A(C)B(C')

= B ( C ' ) A ( C ) exp 21~in/N,

(7.3)

where n is the number of times C ' winds around C in a certain direction. Note that n is only well defined if either C or C' is in the trivial homotopy class (that is, can be shrunk to a point by continuous deformations). Therefore, if C' is in a nontrivial class we must choose C to be in a trivial class. Since these commutation rules (7.3) determine B(C) only up to factors that commute with A and B , we could make further requirements, for instance that B ( C ) be a unitary operator. An explicit definition of B ( C ) can be given as follows. A s in sect. 4 , we go to the temporal gauge, A, = 0. We then must distinguish a "large Hilbert space" H of all field configurations A ( x ) from a C H. This ?f is defined "physical Hilbert space

a

496

to be the subspace of H of all gauge invariant states:

wheren is any infinitesimal gauge transformation in 3 dim. space. Often we will also writen for the corresponding rotation in H:

Now consider a pseudo-gauge transformation Q [C' 3 defined to be a genuine gauge transformation at all + pointsx $ C',but singular on C'. For any closed path X ( 8 ) with 0 < 6 < 21r twisting n times around C' we require (7.6)

This discontinuity is not felt by the fieldsA(g,t) wich are invariant under Z ( N ) . They do feel the singularity at C ' however. We define B(C') as

p1 but with the singularity at C' smoothened; this corresponds to some form of regularization, and implies that the operator differs from an ordinary gauge transformation. Therefore, even for I$> E H we have

For any regular gauge transformation ilwe have an Q'

497

such that

'L

Therefore, if I$> E ?f then B(C') I+> E H, and B(C') is gauge-invariant. We say that B(C') m a e s u r e s electric flux t h r o u g h C' and c r e a t e s a magnetic flux line d o n g C' . We now want to find a conserved variety of Non-Abelian gauge-invariant magnetic flux in the 3-direction in the 3 dimensional periodic box. One might be temped to l o o k for some curve C enclosing the box in the 1 2 direction so that A(C) maesures the flux through the box. That turns out not to work because such a flux is not guaranteed to be conserved. It is better to consider a curve C' in the 3-direction winding over the torus exactly once:

creates one magnetic flux line. But B(C') also changes the number n12 into n12 + 1. This is because

B(C')

makes a Z(N) jump according to ( 7 . 6 ) . If Q , , 2 ( g ) in ( 6 . 2 ) are still defined to be continuous then Z in ( 6 . 4 ) changes by one unit. Clearly,n12 measures the number of times an operator of the type B(C') has acted, i.e. is the number of magnetic flux lines created.n, also conserved by continuity. We simply define n

ij

= E

ijk mk

(7.10)

'

498

with % the -+ Note thatm (apart from + Here, m is

total magnetic flux in the k-direction. corresponds to the usual magnetic flux a numerical constant) in the Abelian case. only defined as an integer modulo N.

VIII. NON-ABELIAN GAUGE-INVARIANT ELECTRIC FLUX IN THE BOX As in the magnetic case, there exists no simple curve C such that the total electric flux through C, measured by B(C), corresponds to a conserved total flux through the box. We consider a curve C winding once over the torus in the 3-direction and consider the electric flux creation operator A(C). But first we must study aomc new conserved quantum numbers. Let I$> be a state in the before mentioned little Hilbert space % * Then, according to eq. (7.5) , I $> is invariant under infinitesimal gauge transformations n. But we also have some non-trivial homotopy classes of gauge transformations n. These are the pseudoperiodic ones :

n(al,x21x3) = R(0,x2,x3)Z1 I s2CxlIa21x3)= R(xl ,GIx3)Z2 I n(xl ,x2,a3) = R(xl,x2,C)Z3 I E center Z ( N ) of S U ( N ) , ‘1,2,3 (8.1) and also those nwhich are periodic but do carry a non-trivial Pontryagin number V , A little problem arises when we try to combine these two topological features. The z,,2,3 can be labeled by three integers k1,2,3between 0 and N :

zt =

2.rrikt/M

(8.2)

e

499

But how is v defined? The best definition is obtained if we consider a field configuration in a fourdimensional space, obtained by multiplying the box(S,)3 with a line segment:

Now choose a boundary condition: F(t=l)=RA(t=O). if the fields in between are continuous,

Then,

is uniquely determined by R . On S4this would be the integer v , N o w however, it needs not to be integer anymore because of the twists in the periodic boundary conditions for ( S , ) .We find

where v is integer and ;fi is the magnetization defined in the previous section. Notice that v is only well defined if and % are given as genuine integers, not modulo N. Taking this warning to heart, we write -b

any n in the homotopy class [ k , ~ ] . Notice that not only do the A (x) transform IJ smoothly under n [ g , v ] , since they are invariant under the Z ( N ) transformations of eq.(8.1) , but also their boundary conditions do not change. These ncommute therefore with the magnetic flux &. If two 52 satisfy the same equation (8.1) and have the same v , they may act differently on states of the big Hilbert space H, but since they differ only by regular gauge Ir transformations they act identically on states inH, Q[g,v]

for

d e f i n e d i n (7.5). W e may s i m u l t a n e o u s l y d i a g o n a l i z e t h e Hamiltonian H , t h e magnetic f l u x and Q [ k , v l :

z,

“,vIl$>

=

e

(8.5)

where o ( $ , v ) are s t r i c t l y conserved numbers. Now t h e 52 o p e r a t o r s form a group. D e f i n i n g f o r e a c h 52 t h e number P as i n ( 8 . 4 ) w e have

so

if +

3

is an i n t e g e r . W e f i n d t h a t w must be l i n e a r i n k and v :

where ei are i n t e g e r numbers d e f i n e d modulo N , and 8 is t h e f a m i l i a r i n s t a n t o n a n g l e , d e f i n e d t o l i e between 0 and 27l. Now l e t u s t u r n back t o A ( C ) d e f i n e d i n e q . ( 7 . 2 ) . I f C i s t h e curve considered i n t h e beginning of t h i s s e c t i o n , A ( C ) is n o t i n v a r i a n t under Q [ z , v l because

(8.10)

501

Therefore,

If

(8.12) (8.13)

(8.1 4 )

T h e r e f o r e A ( C ) i n c r e a s e s e3 by one u n i t :

i s a good i n d i c a t o r f o r e l e c t r i c f l u x i n t h e 3 - d i r e c t i o n , up t o a c o n s t a n t . I t is s t r i c t l y conserved.

e3

However if w e l e t 0

r u n from 0 t o

into + 2. It i s t h e r e f o r e more approptiate to identify

e e + -2

+

2.rr t h e n

turns

p h y s i c a l l y perhaps

+ m

(8.16)

as b e i n g t h e t o t a l e l e c t r i c f l u x i n t h e t h r e e d i r e c t i o n s of t h e box.

IXaFREE ENERGY OF A GIVEN FLUX CONFIGURATION Again w e f o l l o w r e f . [ 3 ] b u t f o r c o m p l e t e n e s s w e add

t h e P o n t r y a g i n w i n d i n g number v. L e t u s w r i t e down t h e f r e e e n e r g y F of a g i v e n

state

+ + (e,m,8)

a t t e m p e r a t u r e T = l/kf3:

502

Here H i s t h e Hamiltonian and

*H

the l i t t l e Hilbert i s simply s p a c e , P a r e p r o j e c t i o n o p e r a t o r s . P,($) d e f i n e d t o select a g i v e n set of n i j = mk: t h e t h r e e s p a c e - l i k e i n d i c e s of e q . ( 6 . 6 ) . P e ( e ) P e ( e ) i s d e f i n e d by s e l e c t i n g s t a t e s I $ > w i t h

ntLv1

2 n i (z.z)+ N

+(G-Z)+iev

$>= e

I$>

(9.2)

'

T h e r e f o r e Pe

--,1 'N

c e k,v

2 ~ -fi + - (Nk . e ) -

ei

+

-,

- (Nm . k ) - i e v

","I

.

(9.3)

Now e'8H i s t h e e v o l u t i o n o p e r a t o r i n imaginary t i m e d i r e c t i o n a t i n t e r v a l 8 , e x p r e s s e d by a f u n c t i o n a l i n t e g r a l o v e r a E u c l i d e a n box w i t h sides ( a l I a 2 , a 3 , B ) :

W e may f i x t h e gauge f o r

3 ( 2 ) (2)

f o r i n s t a n c e by

choosing = 0 A(2)2(XIYIO)

1 (X,O,O)

W e a l r e a d y had A 4 ( Z , t )

I

= 0

I

= 0

.

(9.5)

= 0. S i n c e o n l y s t a t e s i n ;f a r e con-

I

sidered, w e i n s e r t a l s o a p r o j e c t i o n o p e r a t o r DSl were R E 1 I is t h e t r i v i a l homotopy c l a s s .

503

"Trace" means that we integrate over all P. (1 (2) therefore we get periodic boundary conditions in the 4-direction. Insertions of I DQ means that we have s 1 E I

periodicity up to gauge transformations, in the completely unique gauge

Eq. (9.3) tells us that we have to consider twisted boundary conditions in the 41, 4 2 , 43 directions and Fourier transform:

Here W(~,;fi,v,a }is the Euclidean functional integrsll U with boundary condition:> f ixsc? by chnosinq n ij - i' jk mk ; ni4 = ki; a4 = B , snd a Pontryagin number U . Because of the gauqe choice ( 9 . 6 1 t h i s functional integral Rust include integration over the belonging to the Given homoton;7 classes as they determine the boundary conditions such as ( 6 . 2 ) . The definition of W is completely Euclidean symmetric. In the next chapter I show how to make use of this 0 symmetry with respect to rotation over 90 in Euclidean space.

X. DUALITY

The Euclidean symmetry in eq. (9.7) suggests to consider the following SO(4) rotation:

L e t us i n t r o d u c e a n o t a t i o n f o r t h e f i r s t t w o components

o f a vector:

x IJ %

x

= (H,x4) = (x1,x2)

t

=

.

.L

x We have, from e q .

I

(X,,X,)

(10.2)

(9.7):

Notice t h a t i n t h i s f o r m u l a t h e t r a n s v e r s e e l e c t r i c and m a g n e t i c f l u x e s a r e F o u r i e r t r a z s f o r m e d and i n t e r c h a n g e p o s i t i o n s . Notice a l s o t h a t , a p a r t from a s i g n d i f f e r e n c e , t h e r e i s a complete electricm a g n e t i c sylnmetry i n t h i s e x p r e s s i o n , i n s p i t e of t h e f a c t t h a t t h e d e f i n i t i o n o f F i n t e r m s of W w a s n o t

s o symmetric. Eq. (10.3) i s a n e x a c t p r o p e r t y of o u r s y s t e m . N o a p p r o x i m a t i o n w a s made. W e r e f e r t o i t as " dua 1it y 'I

.

XI.

LONG-DISTANCE BEHAVIOR COMPATIBLE W I T H DUALITY

Eq. ( 1 0 . 3 ) shows t h a t t h e i n s t a n t o n a n q l e 8 p l a y s no r o l e i n d u a l i t y . I t d o e s however a f f e c t

505

+

the physical iiiterpretation of e as electric flux, see (8.10). From now on we put 6 = 0 for simplicity, and omit it. Let us now assume that the theory has a mass gap. No massless particles occur. Then asymptotic behavior at large distances will be approached exponentially. Then it is excluded that + + +

-+

F(e,m,a,B) for all

-+

+

0,

exponentially as

a,B

+

,

+

e and m, wich would clearly contradict (10.3). This means that at least some of the flux configurations must get a large energy content as + m. These fl-ux lines apparently cannot a, B spread out and because they were created along curves C it is practically inescapable that they get a total energy wich will be proportional to their length : -+

E = lim F = B +w

pa

.

(11.1)

However, duality will never enable us to determine whether it is the electric or the magnetic flux lines that behave this way. From the requirement that W in ( 9 . 7 ) is always positive one can deduce the impossibility of a third option, namely that only exotic combinations of electric and magnetic fluxes behave as strings (provided 8 = 0 ) . For further information we must make the physically quite plausible assumption of factorizab ility 'I : (11.2)

506

Suppose that we have confinement in the electric domain:

where p is the fundamental string constant. Then we can derive from duality the behavior of Fm(G). First we improve (11.3) by applying statistical mechanics to obtain Fe for large but finite 8 . One obtains: e-8Fe(e1 te2,0,gt8) + C(a,8)

= fc

n++n- n+ +ny1 y2

1

'

2 ntlniln;lnil

-

ti N (n+-n;-e1)6N(ni-ni-e2). I

"1 tn2

Here

(11.4) -8~a1 y 1 = ha2a3 e I -8~a2 y 2 = Aala3 e

tiNW=

c

I

1 N-l

fj

e

2nikx/N

k=O

1

if

x = 0 (mod N)

(1 1.5)

= {

0 if

x = other integer number.

f (the The sum i s over all nonnegative integer values of ni 3 ) . The y's are Boltzmann orientations f are needed i f N factors associated with each string-like flux tube. We now insert this, with (11.21, into ( 1 0 . 3 ) putting e3 = m3 = 0. One obtains

e

-8Fm (m, m2 ,0,-+ a ,P 1 =

C'e

507

21 y a' cos(2ma?/N) a

(11.6)

w h e r e C' is a g a i n a c o n s t a n t a n d -pa2a3 y;

= hal@

e

y;

= ha2B

e

I

-Pala3 (11.7)

with

+ 2 nml -pa2a3 El ( m l I a ) = ~ ~ ( I - c o sN Iale

I

( 1 1.8)

and s i m i l a r l y f o r E 2 and E3. One reads o f f f r o m eq.

(11.8) t h z t t h e r e w i l l

b e n o m a g n e t i c c o n f i n e m e n t , b e c a u s e i f w e l e t t h e box become w i d e r t h e e x p o n e n t i a l f a c t o r

c a u s e s a r a p i d decrease of t h e e n e r g y of t h e m a g n e t i c f l u x . Notice t h e o c c u r r e n c e o f t h e s t r i n g c o n s t a n t p i n there. Of colirse we c o u l d e q u a l l y w e l l h a v e s t a r t e d from t h e presumption t h a t t h e r e w e r e magnetic c o n f i n e m e n t . One t h e n w o u l d c o n c l u d e t h a t t h e r e w o u l d be n o e l e c t r i c c o n f i n e m e n t , b e c a u s e t h e n t h e e l e c t r i c

f l u x w o u l d h a v e ar, e n e r g y g i v e n b y (11.8)

.

XII. THE COULOMB PHASE T o see w h a t m i g h t h a p p e n i n t h e a b s e n c e o f a

mass qap o n e c o u l d s t u d y t h e ( € i r s t ) Gctorgi-Slashow model [ 4 ] Here SU(2) i s " D r o k e n s p o n t a n e o u s l y " i n t o

U(1) by an isospin one Higgs field. Ordinary perturbation expansion tells us what happens in the infrared limit. There are electrically charged (the charged vector particles). particles: Wf They carry two fundamental electric flux units ("quarks" with isospin f would have the fundamental flux unit qo = f 51 e). There are also magnetically charged particles (monopoles, [19].They also carry two fundamental magnetic flux units: 2.rr 4a g=-=--. 90

(12.1)

A given electric flux configuration of k flux units would have an energy 2 2 (12.2) E = gok al 2a2a3 At finite 6 however pair creation of Wf takes place, so that we should take a statistical average over various values of the flux. Flux is only rigorously defined mociulo 2qo. We have W

2 (12.3)

Similarly, because of pair creation of magnetic monopoles 2 W An a (12.4) k=-m

e-a,L a3

These expressions do satisfy duality, eq. (10.3). This is easily verified when one observes that

and

Notice now that this model realizes the dual formula in a symmetric way, contrary to the case that there is a mass gag. This dually symmetric mode will be referred to as the "Coulomb phase" or "Georgi-Glashow phase 'I . Suppose that Quantum Chromodynamics would be enriched with two free paramaters that would not destroy the basic topological features (for instance the mass of some heavy scalar fields ii7 the adjoint representation). Then we .dould have a phase diagran as in the Figure below.

I

mode

lllvu

\

Fig. 1

510

'

Numerical calculations[20] suggest that the phase transition between the two confinement modes is a first order one. Real QCD is represented by one point in this diagram. Where will that point be? If it were in the Coulomb phase there would be long range, strongly interacting Abelian gluons contrary to experiment. In the IIiggs mode quarks would have finite mass and escape easily. It could be still in the Higgs phase but very close to the border line with the confinement mode. If the phase transition were a second order one then that would imply long range correlation effects requiring light physical gluons. Again, they are not observed experimentally. If, wich is more likely, the phase transition is a first order one then even close to the border line not even approximate confinement would take place: quarkswould be produced copiously. There is only one possibility: we are in the confinement mode. Electric tlux lines cannot spread out. Quark confinement is absolute.

51 1

REFERENCES 1. S. Weinberg, Phys. Rev. Lett.

19

(1967) 1264.

A. Salam and J. Ward, Phys. Lett. 1 3 (1964) 168. S.L. Glashow, J. Iliopulos and L. Maiani, Phys. Rev. D2 (1970) 1285. 2. E.S. A k r s and B.W. Lee, Physics Reports no. 1. S. Coleman in: Laws of Hadronic matter, Erice July 1973, ed. by A . Zichichi, Acad. Press, NY and London. 3 . G. It Hooft, lectures given at the Cargese Summer Institute on "Recent Developments in Gauge Theories", 1979 (Plenum, New York, London) ed. G. 't Hooft et al. Lecture no 11. 4. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 28 (1972) 1494. 5. G. It Hooft, in "The Whys of Subnuclear Physics",

6. 7.

8.

9.

10.

11.

12.

Erice 1977, ed. by A. Zichichi (Plenum, New York and London) p. 943. G. Parisi, lectures given at the 1977 Cargese Summer Institute. P. Higgs, Phys. Rev. 145 (1966) 1156. T.W.B. Kibble, Phys. Rev. 155 (1967) 1554. S. Coleman, in "New Phenomena in Subnuclear Physics", Erice 1975, Part A, ed. by A . Zichichi (Plenum press, New York and London). G. 't Hooft, in "Particles and Fields", Banff 1977, ed. by D.H. Boa1 and A.N. Kamal, (Plenum Press, New York and London) p. 165. P.A.M. Dirac, Proc. Roy. Soc. A133 (1931) 60; Phys. Rev. 2 (1948) 817. R.F. Streater and A.S. Wightman, PCT, spin and statistics, and all that (Benjamin, New York and Amsterdam, 1964). H. Lehmann and K. Pohlmeyer, Comm. Math. Phys. 20

512

(1971) 101; A.

Salam and J. Strathdee, Phys. Rev. E (1970)

3296. 13. G. 't Hooft, Nucl. Phys. B138 (1978) 1. 14. L.P. Kadanoff and H. Ceva, Phys. Rev. B3 (1971) 3918. B61 (1973) 15. H.B. Nielsen and P. Olesen, Nucl. Phys. -

45;ibidem B160 (1979) 380. 16. B. Zumino, in "Renormalization and Invariance in Quantum Field Theory", ed. E.R. Caianiello, (Plenum Press, New York) p. 367. 17. G. 't Hooft, Nucl. Phys. B153 (1979) 141. 18. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. C. Callan, R. Dashen and D. Gross, Phys. Lett.

63B

(1976) 334 and Phys. Rev. E (1978) 2717. 19. G. 't Hooft, Nucl. Physics E (1974) 276 A.M. Polyakov, JETP Lett. 20 (1974) 194. 20. M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. p20 (1979) 1915.

513

CHAPTER 7.2

THE CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY Gerard 't Hooft Instituut voor Theoretische Fysica der Rijksuniversiteit

De Uithof Utrecht,The Netherlands 0. ABSTRACT

In these written notes of four lectures it is explained how the phenomenon of permanent confinement of certain types of particles inside bound structures can be understood as a consequence of local gauge invariance and the topological properties of gauge field theories.

I. INTRODUCTION The lagrangian of a quantum field theory describes the evolution of a certain number of degrees of freedom of the system, called f i e l d s as a function of space and time. In many cases this evolution is only straightforward in a small region of space-time and therefore these fields should be interpreted only as the microscopic variables of the theory. Only in the simplest cases these microscopic variables also correspond to actual physical particles' (the macroscopic objects) but very often the connection is less straightforward for two reasons. One reason is that there may be a local gauge invariance. This is a class of transformations that transform a set of fields into another set of fields with the postulate that the new set describes the same physical situation as the old one. Therefore the physical fields only constitute some orthogonal subset of the original set of fields and one of the problems that we will study in these lectures is how to associate this subset to observable objects. However even these observable objects ("transient particles") may in some cases not yet be the macroscopic physical particles. That is one second point : various kinds of Bose-condensation may 1981 CargLse Summer School Lecture Notes on Fundamental Interactions. NATO A&. Study Inst. Series E: Phys.. Vol. 85, Edited by M.U v y and J.-L. Basdevant (Plenum).

514

G. 't HOOFT

take place after which the spectrum of physical particles may look entirely different once again. We will study these condensation phenomena as we go along. The most challenging application of our theoretical considerations is "quantum chromodynamics" or generalizations thereof. The "microscopic" lagrangian there contains only vector particles ("gluons") and spinors ("quarks") but neither of these are really physical. The spectrum of physical particles always consists of bound states of certain numbers of quarks and/or antiquarks with unspecified numbers of gluons. We will obtain a qualitative understanding of this transition from the microscopic to the macroscopic dynamical variables. The first models that we will consider may seem to be a far way off from that desired goal but studying them will turn out to be crucial for obtaining a suitable frame and language in order to put the more advanced systems in a proper perspective. 11.

SCALAR FIELD THEORY Let us consider the lagrangian of a complex scalar field theory : (2.1)

Equivalently we can use real variables :

r

We write $* rather than $ because here the fields are c-numbers, not operators, and the lagrangian must be seen in the context of a functional integral as described by B. de Wit in his lectures at this School. The lagrangian (2.1) describes a system with two species of Boseparticles both with the same mass m but distinguishable in the quantum mechanical sense. One can either consider $1 and $2 as the two species, both equal to their own antiparticles, or consider $ a8 a particle and +* as its distinct antiparticle. At this level these descriptions are equivalent. Obviously there is a synnetrgunder exchange of $ and $ , or 1 2 more generally

$**

e-iA$*

(2.3)

515

CONFINEMENT PHENOMENONIN QUANTUM FIELD THEORY

This is a group of rotations in the complex plane called U(1) and is only global, that is, A must be independent of space-time. By Noether's theorem the model contains a conserved current,

jU with

=

+*aU+ - (aU+*)+

(2.4)

aPjU = 0

(2.5)

e,+*

Classically (2.5) is only true if are required to obey the Euler-Lagrange equations generated by (2.1). Quantum mechanically ( 2 . 5 ) follows if we substitute for 4 and +* the corresponding operators + and e t

.

Our model is trivial in the directly scopic fields + and scopic physical 1particles (apart and the symmetry (2.3) is also a

+

111.

sense that, if m2 > 0, the microcorrespond to the expected macrofrom possible stable bound states) symmetry between these particles.

BOSE CONDENSATION

Bose condensation is a well-known phenomenon in quantum statistical physics. Just in order to make the connection with our case of interest let us first consider an ideal non-relativistic Bose gas. The states are

where

and 11,12,13 are positive integers. L is the length of a side of the box in which the particles are contaimed. The energy of the states is (3.3) where M is the mass of the (non-relativistic) particles. for the thermodynamic free energy F,

We take

(3.4)

516

G. 't HOOFT

where N = zn(ci), 6 1/T i T is the temperature in natural units, and

v is the chemical potential.

We can easily solve (3.4) : OD

e-m=m

~e k n=O

0(V

- k2/2M)n

1

. I

k 1

- exp

-

8 ( ~ k2/2M>

(3.5)

and in the limit of infinite volume :

The particle number density is

(3.7)

-

One easily notes that the formulas (3.5) (3.7) explode if the chemical potential v becomes larger than or equal to zero, a typical property of Bose gases. Since this is a+non-relativistic system it is convenient to in the following way : introduce a field +(XI

-+where a(g) is an operator that annihilates one particle with momentum k in the usual way.

The hamiltonian is then

(3.9)

where for convenience we included the chemical potential term this hamiltonian describes the complete system : "'e

80

-

= Tr e BH

It ir obvious

here that

should not be allowed to be positive.

517

that

CONFINEMENTPHENOMENONIN QUANTUM FIELD THEORY

Now however we can take into account the repulsive forces between the particles. When many particles are close together we expect an extra, poeitive contribution to H. A simple model for that is (3.11)

As long as is negative the But if p is positive then the stabilize the system.

term is just a small perturbation. term is the only one that can

Of course it is difficul't t o find the free energy of this revised system exactly, but an easy approximation, valid for h not too large, is to substitute #I by a c-number (as defined in (3.8) it was an operator), and subsequently minimize H . bte then find approximately the energy of the lowest eigenstate :

t -

(3.12)

4 o = P

(3.13)

(3.14)

(3.15)

(3.16) F E E

(3.17)

0

(3.18) to be contrasted with (3.7), As p turns from negative to take place; we suddenly get in the lowest eigenstate of It takes place whenever the potential becomes negative. small.

still approximately valid for
The model described by our hamiltonian (3.11) resembles somewhat the model of the previous section. Note however that the mass

518

G. 't HOOFT

M comes in the derivative term and never vanishes. In the relativisstic field theory M is replaced by 112, and -p by the mass-squared, m2. In that theory Bose-condensation can take place also : we must extend the allowed values for m2 to negative values, clearly a more profound change in the particle properties. But what we get in return is that in the relativistic model, the vacuum itself, without any external pressure, can become a Bose-condensate. IV.

GOLDSTONE PARTICLES

In the previous section we discussed Bose-condensation in a statistical system only in a sketchy we,y because we will not need were not canonical variables in the details (for instance, Q, and the usual sense). We will be more precise for the case that is more relevant to us : the relativistic complex scalar case. We return to the lagrangian (2.1) and now assume m2 to be negative. It is convenient to rewrite it as

-

2 2 AF < 0 automatically, and an irrelevant constant, where m = -112 F4 , has been added to the lagrangian. The hamiltonian density of the system is

x = n tn

+

i

ai4 ai4

+

X i

-

#Q, 4 F

2 2

(4.2)

where n,mi are the canonical momenta associate with o,Q,1. which i!dependent part has are now operators rather than fields. The +,$ ! an extremum for 2 ?Q, = F

(4.3)

from which

(4.4) where w is arbitrary but fixed to the same value everywhere in space. Now if there were no n t n term inacthen (4.4) would be the exact solution to the Schriidinger equation for the lowest energy state. Every between 0 and 2n would describe a lowest-energy

519

CONFINEMENT PHENOMENONIN QUANTUM FIELD THEORY

3

eigenstate of H t = / X d x, each with eigenvalue E0 = 0. The question is whether the n TI term causes sufficient fluctuations to lift this degeneracy. The answer is not so simple : in 1 space 1 time dimension, yes; in 2 or more space dimensions, no. So let us limit ourselves to 3 space + 1 time dimensions. Then the vacuum (= lowest energy state) is degenerate and characteriz d by a phase angle W. But ( 4 . 4 ) is not exactly valid due to the n TI term. We replace it by

-

F

< O l g l O > = F'eLW

(4.5)

where 10 > is the vacuum state and U i s the vacuum angle. From now on we will consider only the world surrounded by a vacuum with = 0. Further, F' is close to F. In fact, because of ultraviolet divergences, subtractions must be mdUe in F and,+, and we could choose these such that F' F. Actually our theory only makes sense if either h is chosen to be rather small and F of order l/&, or if an adequate ultraviolet cutoff has been introduced. The reasons for this are deeper field theoretic arguments connected with the renormalization group that I will not go into. Let us assume that h is rather small. Then the fluctuations of 6 around F are also relatively small and it makes sense to split (4.6)

+ ' F + r l

and the lagrangian becomes

(4.7) Writing L

This becomes (4.8)

where "int" stands for higher order terms in that one particle, ql, obtained a mass

M

rl

(4.9)

- F a

But its companion

n 1' 02' Notice now

2

became massless.

520

G. ‘t HOOFT

The occurrence of a.massless particle as soon as the vacuum expectation value of a field 4 is not invariant under t e continuous symmetry (2.3) has been first observed by J. Goldstonele, and it is an exact property of the system, not related to our perturbative approximation (no higher order mass corrections). We conclude that after the phase-transition caused by Bose-condensation, the symmetry (2.3) is 8pOntatteOl48Zy broken (the degeneracy of $1 and $ 2 is not reproduced in ~ 1 ,rl2) and at the same time 8 messless particle appears: the Goldstone particle. Here we see the first example where the microscopic fields $,$* in the lagrangian do not reflect accurately the physical spectrum, but the transition towards the 0 fields was still very simple. It is correct to characterize the vacuum by < OlI$(;)lO

> = F # 0

and the vacuum is infinitely degenerate.

Characterization of the

Hfgg8 mode, next sectionm) will be very different! V.

THE HIGGS MECHANISM

We now switch on electromagnetic interactions*)simply by adding the Maxwell term to the lagrangian and replacing derivatives by covariant derivatives :

where F,,v =

-

allAV

D,,$ = (a,,

v

a

’

~ (5.2)

+ iqA,,)$

q is the electric charge of the particle $. Indeed the previously introduced current j eq (2.41, is now the conserved electromagnetic current. But the’invariance (2.3) can now be replaced by Q + e +

i W $ A

U

,

$*

+

-iA(x) $*

- l/q a,,A(x)

(5.3)

*)It may seem to a su erficial reader that these notes are just repeating the story ) of the early 70’s. However we are now not primarily interested in perturbative quantization but rather nonperturbative characterization of what happens.

I

521

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

which is a local invariance As already mentioned in the Introduction, the consequence of this local invariance is that only a subspace of all (@,$I*, A 1, namely the gauge-nonequivalent values, correspond to physxal'observables. Traditionally, one now proceeds by choosing a gauge fixing procedure so that most, if not all, degeneracy is removed. One then performs perturbation expansions in h and q2. At zero A and q 2 one can again ask whether or not < $ >o = F f 0 ?

(5.4)

and since the higher order corrections to < $ >0 are of higher order in A and q2 the qualitative distinction whether or not < $ > = 0 remains valid at every order. And so we get the local varPant on the Goldstone mechanism : after Bose-condensation of charged particles we get the Higgs mechanism 3 ) . However, there is a difficulty with (5.41, because $ is not gauge invariant. Smearing (5.4) over all of space-time may yield zero or not, depending on the gauge chosen. In a trivial gauge

(5.5) we have

always. We therefore propose to use criterion (5.4) o n l y in perturbative considerations, where it is correct (as good as it can be) but not as an absolute non-perturbative criterion for the Higgs mode. Another criterion that cannot be used is whether or not the vacuum is degenerate. The problem there is that transformation (5.3) yields physically equivalent states, contrary to its global equivalent ( 2 . 3 ) 1 1 Therefore all those vacuum states corresponding to different angles are now one and the same state. The vacuum is never degenerate if the symmetry is local. Local symmetries are never "spontaneously broken". Then why is this phrase so often used in connectionwith gauge theories ? Because, as I will show now, there certainly is such a thing as a Higgs mode and it usually can be described in some or other reasonable perturbation expansion around a Goldstone (= global) field theory. Let us return to perturbation theory momentarily. We then write as usual

+(XI = F +

(5.6)

dx)

522

G. ‘t HOOFT

(5.7)

A convenient renormalizable gauge is obtained by adding the gauge fixing term

so that

c +cc=

- +a1

A l2 u v

1 2 2 1 2 2 - p1 2A2 -~~,,q* D , , ~ - pnnl - pAV2 + int (5.9)

with MA

-

fi qF

M

rl

-

m

F

(5.10)

It is easily read off from this lagrangian that the vector particle %. The longitudinal component of the ts, which both cancel against the Faddeev-Popov-keWi tt ghost ‘ssY all having the same mass MA.

A has a mass MA and ‘1, a mass vector field A and ‘12 are h

So perturbation theory suggests that the Higgs theory behaves in a way very different from the symmetric or Coulomb theory : one of the two scalar fields $I disappears and the vector field obtains a mass so that the photon field is short-range only. This i8 a distinction that should survive beyond perturbation theory. Thus the criterion that electromagnetic forces become short-range is much more fundamental than either the vacuum value of the scalar field (5.4) or the “degeneracy of the vacuum”. But, there is yet another new phenomenon in the Higgs mode contrary to the “unbroken” or Coulomb mode. This is important because the above does not yet distinguish a Higgs theory from just any non-gauge theory with massive vector particles.

VI.

VORTEX TUBES

The non-relativistic version of the theory of the previous section is the superconductor : if electrically charged bosons (Cooper’s bound state of an electron pair) Bose-condense then there the electric fields become short-range. Also magnetic fields are repelled completely (Meissner effect). Except when they become too strong. Then, because of magnetic flux conservation, they have to be allowed in. What happens is that a penetrating magnetic field forms narrow flux tubes. These flux tubes carry a multiple of a

523

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

precisely defined quantum of magnetic flux. A 1 ttle amendement to the perturbative Higgs theory can explain this. By gauge transformations the "vacuum value" of the Higgs field I$ can be changed into

iw(x)

< @(XI >o = Fe

(6.1)

if w(x) is a continuous, differentiable function of space and time, However we can also consider perturbation theory around field configurations

where p(x) = F nearly everywhere, but at some points p may be zero. At such points w needs not be well defined and therefore in all the rest of space w could be multivalued. For instance, if we take a closed contour C around a zero of p(x) then following around C could give values that run from 0 to 27 instead of back to zero. The energy of such a field configuration is only finite provided that Di#(x) goes to zero sufficiently rapidly at m. Since a i # does not go to zero fast enough there must be a supplementary vector potential A;(x). The easiest way to find that is by taking a gauge transformation that is regular at but singular at the origin. 00

In the new gauge 3 . 0 may vanish rapidly at also. So in the oid gauge

0

and therefore Ai(x)

(6.4) which is the magnetic flux. One can compute the energy of the flux tube by assuming cylindrical symmetry and substituting (6.2) into (5.1). One then varies AV and p with the boundary condition (6.4) fixed, and minimize8 the hamiltonian derived from (5.1) One typically finds that the energy of a flux with length ll is 6 ) E

-

(6.4)

all

(6.6) If finally a magnetic field is admitted inside a superconductor it can only come in some multiple of these vortices, never spread

524

G. 't HOOFT

out because of the Meissner effect. Classically one may considerthis as an aspect of the infinite conductivity of the material that is only broken down in sufficiently strong magnetic fields. The stable vortex configurations that we discussed here were first derived in the relativistic theory by Nielsen, Olesen and Zumino6). The existence of these macroscopic stable objects can be used as another characterization of the Higgs mechanism. They should also survive beyond perturbation expansion.

VII.

DIRAC'S MAGNETIC MONOPOLES

At this stage it is U ful to introduce the notion of a single magnetic charge B la Diracj5. It is not (yet) a dynamic particle but just a source or sink of magnetic flux, a spectator particle not dynamically involved in the lagrangian of *he theory. A Dirac monopole can be visualized as the end point of an infinitely thin coil carrying a large electric current, The vector potential it is very large close to the coil, because of this electric current :

where @is the magnetic flux of the coil and the integral is over any contour going closely around it. Close t o the coil dx is small, therefore 2 becomes large. Nevertheless the effect of the coil on its surroundings comes only through the end points, if a gauge transformation exists that removes this large vector potential :

Such g! ge transformations A would be multivalued, but we require that elx in (5.3) remains single-valued. So the jumps that A is allowed to make are multiples of 2n. Therefore'the gauge transformation (7.2) turns single-valued field configurations into singlevalued field configurations if Q = 2nnfq

(7.3)

This is how Dirac found that the total amount of magnetic flux carried by a magnetic monopole must be quantized in units 2n/q where q is the smallest possible electric charge in the universe. This condition must be satisfied whenever we want a rotationally invariant quantized theory with magnetic monopoles and single valued fields. It is illustrative now to see what would happen with such a spectator particle inside a Higgs theory (or superconductor).

525

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

It is not accidental that the monopole quantum 2s/q coincides with the Nielsen-Olesen-Zumino vortex quantum. This implies that the monopole will be sitting at the end of an integer number of such vortices. Antimonopoles may be sitting at the other ends. Now the energy of such configurations is approximated by eq. ( 6 . 4 ) . It is proportional to their separation distance. And so we notice that the monopoles inside a superconductor are kept together by an infinite potential well, the potential being simply linearly proportional to their separation. This is the first observation of a confinement feature in quantum field theory, although the confined objects were as yet spectators, not any of the participants of the field equations. That will come later (sect. XIII).

VIII.

THE UNITARY GAUGE

So far we limited ourselves strictly to Abelian gauge theories. We knew what the microscopic 'field variables are, and now we know what particles and vortices survive at macroscopic distance scales. The latter depend critically on what kind, if any, of Bose condensation took place. Now in our introductioq we also mentioned the microscopic physical variables. Formally the space H of these variables is given by

H

=

R/G

(8.1)

where R is the space of field variables and G is the (local) gauge group. How do we enumerate the variables in H ? Traditionally one imposes a gauge condition on the fields in R, thus obtaining a subspace in R which could be representative for H. In section V we used the gauge fixing term (5.8). This is good enough if one intends to do perturbation expansion2~5). The ghost particles one obtainscanceleach other and can be dealt with. However we claim that if a non-perturbative characterization of the physical variables is required then this is not good enough. Imagine t.hat one tries to solve the Dyson-Schwinger equations of the theory in some nonperturbative way. Whenever a computed S-matrix element shows a pole one can never be sure whether er not this is due to a ghost or whether it is physical. Furthermore as we will see the ghosts will produce their own topological features called "phantom solitons" which are entirely non-physical. Therefore if we want some understanding of the physical variables we must go to a "unitary gauge" (agaugewith no ghosts) Often the axial gauge .A

= 0

(8.2)

is used in order to understand the physical Hilbert space. However,

526

G . 't HOOFT

this leaves invariance with respect to time-independent gauge transformations : (8.3) and so there is still a redundancy in our set of variables. It is not suitable for our purposes.

A completely ghost-free gauge can be formulated if we have a charged scalar field (0 (if no such field is present one may consider building such a field by composing, say, two fermion fields). We do not require the Higgs phenomenon to take place. Regardless what condensation takes place at large distance scales one can look at the gauge

This fixes the gauge function A locally, point by point in spacetime, contrary to gauges such as eq. (5.8) where the condition on A requires solving a second order partial differential equation (the cause of the ghosts). Within the unitary gauge (8.4) all components of the vector field A, are entirely observable. The complex scalar field I$ is reduced to a real field p that can only take positive values. This would be a convenient description of the space of microscopic physical field variables were it not for one deficiency in the condition (8.4) : the original space of variables R certainly allows the scalar field to vanish at certain points in space-time. These points, defined by (for any I$ E R )

+

have the topological structure of a set of closed curves in 3-space, or closed surfaces in 3 + 1 dimensional spaceytime. At these points the condition (8.4) becomes singular : if (0 = pel8 then we must choose A = -8

(8.6)

but the gradient of 6 is easily seen to explode close to a zero of

+ and therefore the vector potential 2, transforming as the gradient of A , will grow as the inverse power of the distance to this zero. Thus we find that the string-like structures, defined by (8.5) are separate degrees of freedom, giving a boundary condition on p (p * 0) and a prescribed singular boundary behavior of A,,. This completes our discussion of the microscopic physical degrees of

527

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

freedom for any Abelian gauge theory : we have observable vector fields % a truncated scalar field p (p > 0) and all possible closed strings *), on which there is a boundary condition for both p and A,, Note that only in the Higgs theory these physical variables are in the same time the macroscopic physical variables, although of course the macroscopic variables will be "dressed with a cloud of virtual particles" (the string becomes a vortex with finite thickness). In the "unbroken" Abelian theory of electromagnetism the macroscopic variables are harder t o discuss. It appears that our vortices "Bose-condense" to form long range, non-energetic magnetic field lines : the ordinary magnetic field f .

IX. PHANTOM SOLITONS The gauge (8.5) is called "unitary gauge" because in that gauge all surviving fields will be physically observable. Their quanta will all contribute in the unitarity relation (9.1)

However as soon as practical calculationp are considered smoother gauge conditions are required. (8.5) is hard to implement if 4 oscillates wildly at small distances. We will now argue that after a transition towards "smoother" gauge conditions not only ghost particles arise but also what we call "phantom solitons" : extended structures which are stable for topological reasons but nevertheless unphysical gauge artifacts 8). Intermediate between the "renormalizable gauge" (5.8) and the unitary gauge we could choose arg(4) +

Ka

A

u u

(9.2)

= 0

where arg ( 4 ) = Im(log 4 ) and K is an arbitrary gauge parameter. The gauge condition (9.2)is smoother than the unitary gauge because at small distances, by power counting, the second term dominates and we come close to the renormalizable Lorentz gauge. One finds ghosts in this gauge which propagate with a mass (9.3) so

for small

K

they become unimportant.

*)That is, strings without ends; they couldrun from

528

m

to

-.

G. ‘t HOOR

Now imagine a Nielsen-Olesen-Zumino vortex tube in the form of a closed curve, What do the field configurations in the U! e (9.2) look like ? The gauge(9.2) $ 8 that particular gauge for

W

-

1d4x(q”(arg(I$))*

+

~4)

(9.4)

has an extremum. Let us assume this is a minimum. The system then likes to arrange arg(4) to be zero as much as possible but not with too large vector potentials A IJ

.

Fig 1 In fig 1 we pictured a cross section of the vortex. The plane is intersected twice, in opposite directions. A t the intersection points the scalar field I$ makes one complete rotation, again in opposite directions. The configuration in the figure is close to the optimal gauge ( 9 . 2 ) , keeping Q as much as possible oriented towards the positive real axis. We see that the topology of the complete “twist” from top to bottom had to be preserved. This twist will cover an entire sheet spanned by the vortex. Certainly (9.2) will be obtained (i.e. (9.4)will be minimallif that sheet has minimal surface. The equations for the field configurations inside the sheet are easy to solve if the sheet is considered to be locally sufficiently flat. Clearly the sheet is a gauge artifact. We think that structures of this sort will further obscure the physical interpretation of whatever solutions will be found to the Dyson-Schwinger equations in the gauge (9.2) or completely renormalizable gauges. For instance, bubbles made out of these sheets will form a whole Regge-like family of phantom particles.

529

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

X. NON-ABELIAN GAUGE THEORY We now intend to perform the same procedure in non-Abelian gauge theories. For simplicity we restrict ourselves to the case that the gauge group is SU(N), with arbitrary-N. The microscopic field variables are : a matrix vector field l J ( x ) where p is a Lorentz indel! and i, j run from 1 to N; and t ere may be Dirac spinor fields $ i ~ ,$ i ~ where F is a flavor index. Scalar fields are not assumed to play a significant role but may be present too.

2

A n element of the gauge group G (here SU(N)) is a space-time dependent unitary matrix ~(x). Any transformation of the form

(10.1) is postulated to describe the same physical situation as before, but of course gives a different set of values to the microscopic variables. If R is the space of microscopic variables, then RIG is the space of microscopic p h y s i c a l variables. Again we ask the question how to categorize or enumerate these physical variables. In perturbation theory it is customary to impose a gauge condition, which implies that we find a subspace of R (the set of fields in R that satisfy the gauge condition) representative for RIG. A renormalizable gauge condition is

a~v

v

(10.2)

- 0

but it is easy to see that this subspace of R does not accurately describe the physical degrees of freedom in R/G (even though it is accurate enough for perturbation theory). To see this, consider an infinitesimal perturbation in R A +A P

L

J

+&A

(10.3)

lJ

where &A,, is entirely localized in a particular region &V of spacetime. We may however have

In order to impose the gauge condition (10.2) we now find an infinitesimal gauge transformation n = elgA that restores (10.2). It must

530

G. 't HOOFT

satisfy (10.5) but the inverse of the operator a,,D,, is non-local. So in our subspace of R satisfying the gauge condition we find a perturbation which is spread out all over space-time, well outside 6V. Clearly this perturbation outside 6 V is unphysical and in perturbation theory we learnt how to deal with this : the theory has "ghosts". We know claim that beyond perturbation theory these ghosts obscure the physical contents of our theory. Therefore we shall look for a "unitary gauge". Our strategy is to determine this gauge in two steps. Let L v - p n e of the largest Abelian subgroups of G, in our case L = U(I)

.

(10.6)

L = u(1IN-l

We call G/L the "non-Abelian part" of the gauge group and our first step will be to fix the "non-Abelian part" of the gauge redundancy. We choose a gauge condition C thatreduces the space R into a subspace that could be called (10.7) H1 = R/(G/L) If the gauge group G has N2-1 geperators and L has N-1 generators, then we choose N2-N real components for the gauge condition C, all invariant under L C G b u t not G itself. The second step is the choice of an N-1 component gauge condition that fixes the remaining invariance under L :

H

- Hl/L

(10.8)

R/G

But this second step is precisely the same as fixing the gauge in an ordinary Abelian gauge theory such as electromagnetism and is therefore much more trivial. To understand the physical contents of the theory one could jdst as well stop after obtaining (10.7) which is expected to describe Abelian charged particles and photons. A variant on the Lorentz condition that reduces R to R/(G/L) is easy to find : DOA'~

PlJ

=

o

(10.9)

0

there D is the L-covariant derivative, containing the diagonal part lJ is2the set of off-diagonal elements of the vector field A only. of A only. Eq. (lo.!) has indeed N -N components.

qh

lJ

This gauge suffers from the ghost problem as much as the ordinary Lorentz gauge, and is therefore not suitable for understanding all physical degrees of freedom.

531

CONFINEMENT PHENOMENONIN QUANTUM FIELD THEORY

XI. UNITARY GAUGE A unitary gauge must be picked in a way similar to the Abelian case. We need a field that transforms without derivatives under gauge transformations. We will limit ourselves to the case that this field, call it X, transforms as the adjoint representation under G.

x + nxn-l

(11.1)

Such a field namely can always be found. The simplest choice would be (11.2)

which is one of the components of the covariant curl G This UV' choice has the disadvantage of not being Lorentz-invariant. One may choose a composite field : (11.3) This however does not work if G = SU(2) because then X would be proportional to the identity matrix. We need a non-vanishing isovector part. We could choose * *

X1'

ik 2 'k

= G ~ v DGdv

(11.4)

but this choice looks rather complicated. Perhaps the most practical choice would be to take ones refuge to an extra scalar field in the theory, giving it a sufficiently high mass value so that the theory is not changed perceptibly at low energies. Our gauge condition will be that X is diagonal :

(11.5) where the eigenvalues Ai may be ordered : (11.6)

532

6. 't HOOFT

What is the subgroup of the gauge transformations n under which (11.5) is invariant ? If we require

x

X' = n m - l =

then

fx,nl therefore,

= 0

(11.7)

n is also diagonal

:

(11.8) and rince detn

1, we have

N C w i m O ill

(11.9)

Indeed, this is the largest Abelian subgroup L of G. ~

=

iw

(11 .lo)

e

then the diagonal part Ao of A lJ

'A lJ

+ AO

lJ

If we write

transforms as

lJ

1 - -aglJ

(11.11)

and the off-diagonal part ACh as ACh

lJ

ij + e i(wi

-

U

w j ) Ach

ij (11.12)

lJ

Apart from the gauge transformations (11.11) and (11.12) all our fields are physically observable. So our physical degrees of freedom are

- N-1 "massless" photons - 1/2 N(N-1) "massive" charged vector fields -N

...

scalar fields Ai with the restriction : A1 > ,I2> > AN.

There is of course another constraint : depending on our choice for

533

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

X, we have that

Of course we still have the satisfies (11.21, (11.31, or (11.4). local U(1)N-l symmetry to be removed by either conventional gauge fixing, or the procedure described in the previous chapters. XII.

A TOPOLOGICAL OBJECT

...,

So far we assumed that the eigenvalues .xl, ,% coincide nowhere. What if they do, at some set of points in space-time ? At those points, the invariance group is larger and a problem emerges with our enumeration procedure. We now argue that these exceptionnal points : a) are pointlike in 3-space, describing particle-like trajectories in space-time, and b) correspond to singularities in the fields 9J and $ . The argument for the singularity is similar to the Abelian case, but the fact that these things are particle-like differs from the Abelian case, where the singular points were string-like.

The reason why the singularities in the generic case are pointlike is that the dimensionality of the space of field variables with two coinciding eigenvalues A is three less than the space with non-coinciding eigenvalues. Therefore the dimensionality of the points in space or space-time where two eigenvalues coincide is three less than that of space-time itself. This statement holds for any generic N x N hermitian matrix field X(x). What is the physical nature of such a particle-like singulariwere ordered, we only need to consity ? Since the eigenvalues der the case thattwo successive A's coincide : (12.1) for certain j. Let us consider a close neighbourhood of such a point. Prior to the gauge-fixing we may take X to be (12.21, where Di and D2 may safely be considered to be diagonalized because the other eigenvalues did not coincide. The three fields Ea(x) are small because we are close to the point where they vanish. With respect to that SU(2) subgroup of SU(N) that corresponds to rotations among the jth and j + lst components, the fields form an isovector.

534

G. 't HOOFT

(12.2) One may write the center block as

x

AI +

++

(12.3)

E.0

yhere U are the Payli spin matrices. Close to a zero point of this E field: the field E has a hedgehog conf&uration. But gauge fixing and E away i.e. diagonalization of X, corresponds to rotating 2 such that E3 is positive ( A > A +l). Thus is our unitary gauge,

j

j

(12.4) By now the reader may recognize this field configuration as the one for a magnetic monopole9). Indeed, fixing the L-gauge as well cannot be done without accepting a string-like singularity connecting zeros of opposite signature : the Dirac string. The magnetic charges of the monopole ban most easily be characterized with respect to the U(1IN subgroup of the extended gauge group U(N) :

+ m = (0,

21 ..., 0, 5,- g,

0,

..., 0) (12.5)

where the f 2n/g are at the jth and j+lst position. g is here the fundamental electric charge of the elementary representation. We then see that m' actually only acts in the subgroup U(l)N/U(l) of SU(N) because the sum of all its charges vanishes. It is constructive to notice a subtile difference between this magnetic charge spectrum and the spectrum of the electrically charged gauge

535

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

particles ''A charges

..

'

if

= (0,

: from (11.12) we read off that these have electric

..., 0, +g, 0, ..., 0, -g, 0, ..., 0)

(12.6)

where 2 g occur at arbitrary, not necessarily adjacent, positions i and j. Again however the sum of the charges vanishes, so that we are really working in the Cartan group IJ(1IN-l, not U(l)N. Magnetic monopoles with

are possible only if three or more eigenvalues coincide. The dimensionality of such points is at-least 8 less than space-time so in general they do not occur at single points. Rather, they should be considered as bound states of "elementary" monopoles (12.5). We conclude that we arrive at a picture where an Abelian gauge theory is enriched with magnetic monopoles, but because of the slightly different charge spectrum this picture is in general not "self ddal". Contrary to the case discussed in section VII, the monopoles we have here are "dynamical", that is, they will inevitably take part in the dynamics of the system. XIII.

THE MACROSCOPIC VARIABLES

We have now arrived at a point where we could sketch a possible strategy for precise calculations for the dynamics of the system : 1) Consider the physical degrees of freedom in the space H1 = R/(G/L). We find N-1 sets of Maxwell fields, electrically charged fields (among which vector fields), and magnetically charged particles. The particular case of interest is now the possibility that magnetically charged particles "Bose-condensepl. If we ever are to understand such a mechanism in detail; the following step is probably necessary : 2 ) Eliminate the electric charges. With as much precision as possible we must compute all light-by-light scattering amplitudes and express them in term of an effective interaction lagrangian for the photon fields :

+ higher orders

r(Au) =

(13.1)

536

.

G t HOOFT

3) Now perform the "dual transformation". Since we have only Maxwell fields and magnetic charges interacting with them, we could replace 3 by 3 and 3 by -3, then introduce operator fields in the usual way for the monopole particles, which now look like ordinary electrically charged objects.

4) Work out the self-interactions among these magnetic monopoles. Set up a perturbation theory now in terms of 2T/g. Then the question is : 5) Does, in terms of this perturbation theory, Bose condensation occur among these monopoles ? Is it reasonable to start with (13.2) If so, then the vacuum is a magnetic superconductor. The monopoles formally have a negative mass-squared. In this magnetic superconductor electric charges are confined. The descriptions of section VII apply qualitatively, after the interchange electric * magnetic.

XIV.

THE DIRAC CONDITION IN THE ELECTRIC-MAGNETIC CHARGE SPECTRUM

Fig 2 represents the spectrum of possible charges in the case that the gauge group G is SU(2).

m

Fig 2

537

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

Horizontally is plotted the electric charge Q, vertically the magnetic charge 2. Elementary and bound state charges are indicated. The crosses represent a fundamental SU(2) doublet which may or may not have been added to the theory. The pure Maxwell field equations are now invariant under rotations of the figure about the origin. This is why one is always free to postulate that the fundamental fields carry no magnetic monopole charge. But does the lattice obtained in fig 2 have to be rectangular ? We will now argue that Dirac's quantization condition allows more kinds of lattices and "oblique" lattices indeed may result in a non-Abelian gauge theory. The Dirac condition for a magnetic charge quantum m and the electric charge quantum q was qm = 2nn

(14.1)

where n is integer. To be precise this correspopds to a quantization condition for the Lorentz force that the magnetically charged particle exerts on the electrically charged object. But now consider two particles 1 and 2 both with various kinds of magnetic and electric charges. Then the Lorentz force quantization corresponds to (14.2) where the index i refers to the label of the species of photons and 1-112is an integer relevant for particles 1 and 2, to be referred to as the Dirac quantum of particle (1) with respect to particle (2). In our specific case of SU(N) broken down to IJ(1)"l require

we usually

N

N

C

q;=O

1

m.=O

i=l

i=l

(14.3)

Our charge lattice in this case will be spanned by 2(N-1) basic charges, to be labelled by an index A = 1, ..., 2N-2. Because of the invariance

(14.4) we may always take mp) = 0 1

for A = 1,

...,

N-1

538

(14.5)

G . 't HOOFT

The gluons will provide us with a basis of electric charges: (A) A A+ 1 q i = g6i . - g6i

for A

-

1,

...* N-1

(14.6)

(The fundamental representation, if it occurs, could have (14.7) The magnetic monopoles have the remaining basic charges: =

,(A)

i

& &A+l-N g

i

- & &A+2-N 8

for A

N,

...*2N-2

%

(14.8)

It was Witten") who observed that monopoles mAy also carry electric charges. He found

p= e g L (A) mi , i

for A = N,

471

..., 2N-2 (14.9)

is the instanton angle of the theory, 0 < 8 < 2n. Notice where that for any value of 8 the Dirac condition (14.2) is fulfilled. It can be seen that this phenomenon, eq. (14.91, follows from the lagrangian

(14.10) there (14.11) and Ga

2

P V uv

4E

corresponds to

3".sa (14.12)

a

The canonical argument can be found in refs 101, 111, 8 ) . In the case of SU(2) the charge lattice indeed becomes tilted now (Pig 3). It is remarkable that if 8 runs from 0 to 271 then the charge lattice indeed turns back into itself, but the "elementary" monopole labelled by (2) in fig 3 is replaced by the "monopole gluon bound state" labelled (3). It seems that no fundamental distinction

539

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

m

will be possible between monopoles and monopole-gluon bound states. XV.

OBLIQUE CONFINEMENT

We can now ask which of the objects in the lattice of fig 3 will form a Bose-condensate. If it is a purely electrically charged object, (1) in fig 3 , then we get the familiar Higgs theory or electric superconductor. All charges that are not on the horizontal axis will then be confined by linear potentials, because of the arguments presented in section VIZ. By duality we now expect that also monopoles may conceivably undergo Bose condensation, for instance charge 9(2) in fig 3. One cannot however have both electric and magnetic charges Bose-condense because if the electric ones condense then the magnetic ones have m2 -+ +-, not negative, and vice versa. In the case of larger N, Bose condensation can only take place among charges forming anylinear sublattice of the original charge lattice, as long as its members all have vanishing Dirac quanta with respect to each other. Now if 8 is switched on, then point (3) gradually takes the place (2) and (3) cannot both Bose condense because they have a non-vanishing relative Dirac quantum. So it is either (2) or (3). It is likely that Bose condensation in the (2)-direction is replaced by condensation in the (31-direction at n < 8 < 37r. This would then be a phase transition in 8, possibly of first order, just like the transition between Higgs and confinement. of (2).

Various attempts at dynamical calculations however indicate that

G. 't HOOFT

a t e a n the confinement mechanism is not strong. An explanation could simply be that the monopoles then carry large electric charges and therefore may have larger self-energies contributing positively to their mass-squared Suggestions have been made that a t e = n the Higgs mode reappears i2) or a Coulomb mode (no Bose condensation at all).

I suggest yet a different condensation mode that could possibly occur in theories withe close t o n . If neither (2) nor (3) condense because they carry large (but opposite) electric charges , then perhaps ( 4 ) which is a bound state of these two with much smaller electric charge condenses. This would only be possible if the lattice is oblique (8 # 0) so this mode is referred to as "oblique confinement". A theory with oblique confinement shows some peculiar features. We stress that these will not occur in ordinary QCD because there we know that 0 5 0. OUT observations may be relevant for certain models with "technicolor" as we will show shortly. Returning to the case that our gauge group was SU(2), we first argue that the "quarks" (or "preons") in this oblique confinement mode are not confined in the usual sense. That is, if we attach a flavor quantum number to every type Qf preon, then physical particles transforming as the fundamental representation of the flavor group do occur. The preons are the crosses in fig 4 .

Fig 4.

541

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

Since they are not on the line connecting the origin to the condensed object (arrow) they are confined. However, a bound state of a monopole (without flavor quantum numbers) and one preon does occur on that line (@), and hence is physical, and has the same flavor as the preon itself. But a price had to be paid : spin and statistics-properties of the physical object are opposite to that of the original preon! If the preon is a fermion, the liberated object is a boson and vice-versa. This is a consequence of a general rule : if two particles have an odd relative Dirac quantum, then the o r b i t a l angular momentum in any bound state of the two is half-odd-integer, a well-known property of the SchrSdinger equation of an electrically charged particle in the field of a magnetic point-s~urcel~). That also the s t a t i s t i c s of the bound state eta an extra fermionic contribution has been shown by Goldhaberl4f. Let me give an outline of the argument. If we wish to consider the statistics of two identical composite states A and B, both composed of a magnetic monopole M and an electric charge Q, (fig 5 ) then we would like to separate the center-ofmass motions of A and B from the orbital motions of M and Q inside A and B.

Fig 5

542

G . 't HOOFT

The particles Q see two Dirac strings, ending at both M ' s . The M's see two Dirac strings ending at both Q's. Now the center-of-mass motion of A can only be split from the orbital motion of M and Q if the magnetic strings run in a direction opposite to that of the electric strings (see fig 51, so that if M hits an electric string, in the same timeQ hits a magnetic string. Now let us ignore the motion of M and Q inside A and B, but only consider A and B as a whole. Then A feels both strings at B (and vice versa), in fact, these two are connected in such a way that one string results, running from infinity to infinity. This could be expected because A and B have the same electromagnetic charge combination (they were identical) so their relative Dirac quantum vanishes. Their relative motion (A against B) is as if they only were electrically charged. Therefore we obtain the familiar Coulomb SchrBdinger equation by removing this complete string by a single gauge transformation :

i 4 ~ ~ +AB where

+

+AB

(15.1)

4AB is the angle by which A rotates around the B-string. Now notice that if we interchange A and B this angle is 180°,

so that

This is how one can see an extra minus sign appearing in the commutat ion properties of the particles A and B. XVI

.

FERMIONS OUT OF BOSONS AND VICE-VERSA

Some exotic models can be constructed if we use obliaue confinement as a starting point. Consider for instance an SU(3) gauge theory without fernions but with a scalar field 4 in the fundamental representation. Let 8 be close to T , and let us assume that the familiar Higgs mechanism takes place, described by

(16.1) (though formally incorrect, as is useful and adequate for our neously" into SU(2).. One heavy original octet of gauge fields particle and one SU(2) complex

explained in section V, the notation purpose). SU(3) then "breaks spontaneutral Higgs particle arises. The splits into one neutral heavy vector doublet of heavy vector particles.

543

CONFINEMENT PHENOMENON IN QUANTUM FIELD THEORY

The SU(2) triplet remains massless, and now we will assume that it produces oblique confinement in the SU(2) sector. The SU(2) doublet of heavy vector bosons will appear as physical particles but disguised as fermions! Here we have an example of fermionic gauge "bosons" without invoking anything resembling supersynrmetry. Another model worth considering is a simplistic weak interaction There "tc" stands theory, based upon SU(2)tc x (SU(2) x U(l))ew. for "technicolor" and ew for "electro-weak". Quarks and fermions are all in the usual representation of (SU(2) x U(l))ew and singlets under SU(2)tC. Now add one fermion multiplet transforming just like all other fermions under (SU(2) x U(l))ew but also as a 2 under tc. Assume oblique confinement. Then this fermion will be liberated, but disguised as a boson. Probably it will have a spin-zero component. Since it is in line with the condensing monopole bound states it may Bose-condense itself. So we obtain a scalar field transforming just as the fermions under (SU(2) x U(l))ew and a non-vanishing vacuum expectation value : a model for the Higgs particle. Indeed, its Bose condensation could well be responsible for the oblique confinement mode in the first place, so here it was not even necessary to consider the monopole-dyon bound state. Unfortunately, elegant as it may be, this model seems to suffer from the same shortcomings as the more conventional technicolor ideas : it is hard to reproduce the required Yukawa couplings between this scalar field and the other fermions.

XVII.

OTHER CONDENSATION MODES

It will be clear from the previous sections, by looking at the electric-magnetic charge lattice, that even more exotic forms of oblique confinement can be imagined. Just assume condensation of bound states with three or more monopoles. Such a condensation mode would be required for instance if we would wish to liberate the fundamental triplets in an SU(3) gauge theor?. A fundamental exercise tells us the that these triplets do not switch their spin-statistics properties ay. In any case, all these different confinement modes will be separated from each other by sharp phase transition boundaries, which should show up in the solutions of the theory when the parameter 9 is varied. In principle each point on the electric-magnetic charge lattice may correspond to a possible phase of the system. There may however be features which cannot easily be understood in terms of our intermediate physical degrees of freedom with Abelian electric and magnetic charges. We have in mind a condensation mode studied in more detail by Bais15). The simplest example is an SU(2) gauge theory with an isospin-two Higgs field, $ab (a, b = 1, 2, 3 ) .

544

G. 't H O O n

Let us assume (17.1) If we were just dealing with a global syrmmetry, we would say that SU(2) is spontaneously broken into a subgroup D (the invariance group of eq. (17.1)). Now D is the discrete subgroup of SU(2) corresponding to rotations of'spinors over 90" :

D =

(k

I,

k

ial,

k

io2,

2

ia3

(17.2)

If the aymetry is local then D is not really a global invariance of the vacuum. What we do see is that magnetic vortex tubes can be constructed which are characterized by the following boundary condition at infinity :

(17.3) where Sl is multivalued. If we go around the vortex once, then n turn8 into itself multiplied with an element D1 of D. These vortices are non-commuting. Physically this means the following (fig 6)

Fig 6

If two strings, characterized by different, non cormnuting elements D1 and D2, approach each other at right angles (fig 6a), then they cannot pass each other without leaving a connecting string D3 (fig 6b). The element D3 is given by -1 -1 D g = D D D D 1 2 1

(17.4)

2

545

CONFINEMENT PHENOMENONIN QUANTUM FIELD THEORY

Clearly the non-commuting properties of the original gauge group were crucial for underetanding this phenomenon, so that our Abelian physical variables are not useful here. Indeed, we could ask the question whether the dual of this "Bais mode" exists, with electric strings having similar properties ? As yet, the answer to that question is unknown. REFERENCES

19 154

1.

J. Goldstone, Nuovo Cim.

2.

F.S. Abers and B.W. Lee, Phys Reports ces therein.

3.

P.W. Higgs, Phys. Lett. l2, 132 (19691, Phys. Lett. l3, 508 (1964), Phys. Rev. 145, 1156 (1966) F. Englert, R. Brout, Phys. Rev. Lett. 12, 321 (1964) G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Lett. 585 (1964) T.W.B. Kibble, Phys. Rev. 155, 627 (1967)

4. B . S . DeWitt, Phys. Rev.

(1961)

162, (1967),

E,

1 (1973) and referen-

3

1195, 1239

5.

G. 't Hooft and M. Veltman, "DIAGRAMMAR", CERN report 7319 (1973)

6.

H.B. Nielsen, P. Olesen, Nucl. Phys. E , 45 (1973) B. Zumino, in "Renormalization and Invariance in Quantum Field Theory ed. E.R. Caianiello Plenum Press New York (1974) p. 367

7.

P.A.M. Dirac, Proc. Roy. Soc. A133, 60 (1931) Phys. Rev. 14,8 m 1 9 4 8 )

8.

G. 't Hooft, Nucl. Phys. B, to be published

9.

A.M. Polyakov JETP Lett. 20, 194 (1974). G. 't Hooft, Nucl. Phys. E , 276 (1974)

-

10.

E. Witten, Phys. Lett. 86B, 283 (1979)

11.

A. Salam and J. Strathdee, Lett. in Mathematical Physics

2,

505

(1980)

12.

C. Callan, private communication (1979)

13.

R. Jackiw and C. Rebbi, Phys. Rev. Lett. 1116 (1976) G. 't Hooft and P. Hasenfratz, Phys. Rev. Lett 2, 1119 (1976)

14.

A.F. Goldhaber, Phys. Rev. Lett.

15.

F.A. Bais, private comunication.

a,

546

36,

1122 (1976)

CHAPTER 7.3

C A N WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"? G.'t Hooft Institute for Theoretical Physics University of Utrecht, Netherlands

1.

INTRODUCTION "Quantum Chromodynamics" is a pure gauge theory of fermions

and vector bosons that is assumed to describe the observed strong interactions.

To get an accurate theory it is mandatory to go

beyond the usual perturbation expansion. Not only must we explore the mathematics of solving the field equations non-perturbatively; it is more important and more urgent first to find a decent formulation of these equations themselves, in such a way that it can be shown that the solution is uniquely determined by these equations.

We understand how to renormalize the theory to

any finite order in the perturbation expansion, but it is expected

1-5)

that this expansion will diverge badly, for any value

of the coupling constant. define the theory.

Thus the expansion itself does not yet

But the renormalization procedure is not known

to work beyond the perturbation expansion.

A clear example of the

possible consequences of such an unsatisfactory situation was the recent suprising demonstration

6-1 0 )

that all non-Abelian gauge theo-

ries have parameters 0 , in the form of an angle, that describe certain symmetry breaking phenomena in the theory, but never show up within the usual perturbation expansion because they occur in the The Whys of SubnuclearPhysics, Edited by A. Zichichi (Plenum).

547

G. 't HOOFT

combination

where g is the gauge coupling constant.

This discovery marks an

increase in our understanding of non-perturbative field theory but this understanding is not yet complete and, in principle, more of such surprises could await us. The situation can be compared with the infinity problems in the older theories of weak interactions. Those problems were solved by the gauge theories for which an acceptable and unique regularization and renormalization scheme was found. For our present strong interaction theory, again a "regularization scheme" must be found, this time for "regularizing" the infinities encountered in summing the perturbation expansion. An interesting attempt t o give a non-perturbative formulation of the (renormalized) theory is the introduction of a space-time 11-1 3 ) lattice in various ways But, here also, a proof of unique-

.

ness could not be given (will the continuum-limit yield one and only one theory?) and of course the 8 phenomenon mentioned before was not observed in the lattice scheme. So, the lattice theories are still a long way off from answering our fundamental questions.

It is more important for us to make use as much

as

possible

of the important pieces of information contained in the coupling constant expansion. Because of asymptotic freedom1 4 - 1 6 ) this expansion tells us precisely what happens at asymptotically large external momenta and it would be a waste to throw that information away. Which tools could we use to extend our definitions? One is study of the theory at complex values of the coupling constant.

a

That this is possible in some particular cases is explained in Section 4. We may find that Green's functions must become singular at certain points in the complex g2 plane and stay regular at others.

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"?

Suppose we would discover that there are only singularities on the real axis for - 5

g2 < -a (unfortunately the true situation is

much less favourable).

Then we could make a mapping:

with the inverse

td=..4QIL(4-u)-2 The whole complex plane is now mapped onto the interior of the unit circle, with the singularities at the edge. the perturbation expansion in terms of

U

If *we rewrite then we

instead of g

have convergence everywhere inside the circle*), which implies convergence for all g2

not too close to the negative real axis.

definite improvement.

A

Unfortunately, the structure for complex g2

we find in Section 4 is too complicated for this method to work: the origin turns out to be an essential singularity. Still, I shall show how this knowledge of the complex structure may be used to give a very slight improvement in the perturbation expansion (Section 9). The other tool to be used is the Bore1 resummation procedure, to be explained in Section 5. A new set of Green's functions" is considered whose perturbation expansion terms are defined to be the previous ones divided by (n-l)! where n is the order of the expansion terms. The singularities of these new'functions can be found and the analytic continuation procedure as sketched above can be applied to obtain better convergence. types of singularities:

There seem to emerge two

one due to the instantons in the theory,

the other due to renormalization phenomena. The latter are slightly controversia1,they are the only ones that should occur in

*)

We make use of a well-known theorem for analytic functions that says that the rate of convergence of an expansion around the origin is dictated by the singularity closest to the origin.

549

G. 't HOOFT

quantum electrodynamics, giving that theory of n! type divergence. The first few terms for the electron 8-2 do not seem to diverge the way suggested by this singularity. Formally we can improve the perturbation expansion for QED but in practice the "improvement" seems to be bad. Our work is not finished. What remains to be done is to prove that all singularities in the Borel variable have been found and to find a prescription how to deal with those singularities that are on the positive real axis. Finally, it must be shown that the integrals that link the Borel Green's functions with the original Green's functions make sense and a good theory is found (see Section 9). The problem of quark confinement can probably be related to certain singularities on the positive real axis, because these singularities arise from infra-red divergences (Section 8 ) . 2.

DEFINITION OF THE COUPLING CONSTANT AM) NASS fA3AMETEP.S IN T E W S OF M G E PfOMENTUM LIMITS This section contains the mathematical definitions of the pa-

rameters in the theory,

so

that the statements in the other sec-

tions can be made rigorous and free of unnecessary assumptions. It could be skipped at first reading. The dimensional renormalization scheme is a convenient way of defining a perturbation series of off-mass shell Green's functions with some coupling constant g (U) as an expansion parameter. The D 17-2 0 ) subscript D stands for dimensionally renormalized Each term of the perturbation series is finite. There is an arbitrari-

.

ness in the choice of the subtraction point U (which has the dimension of a mass). The theory is invariant under a simultaneous change in gD and

provided that

550

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMICS?

Here we show explicitly the minus sign for the first coefficient. This minus sign 'is a unique property of non-Abelian gauge fields and is responsible for "asymptotic freedom" (at increasing

we

get decreasing g2, see Refs. 14-16). The coefficients

61

and

62

are known

21-23)

.

Since we shall

try to go beyond perturbation expansion we must be aware of two facts:

First, the perturbation expansion is expected to diverge

for all g2 and is, therefore, at this stage, meaningless as soon as we substitute some finite value for g2.

Second, the dimensional

procedure has only been defined in terms of the perturbation expansion (Feynman diagrams).

Consequently, gi may not have any

meaning at all as a finite number.

The correct interpretation of

these series is that they are asymptotic series valid for infinitesimal g only, or, equivalently, valid only for asymptotically large momentum: p2 O(p2) + W. Thus, gi may not be so good to use

-

as a variable for a study of analytic structures at finite complex values. We shall now introduce another parameter gi, that may just as well be used instead of gi.

It is defined by the following require-

ments : When

+ w,

then (2.2a)

(2.2b)

551

G. 't HOOF1

The series in (2.2b) must stop after the second term.

In pertur-

bation theory these requirements have a unique solution for g2

R'

For instance, we get that the rest term in (2.2a) is

In the dimensional renormalization scheme also the mass parameter was cut-off-dependent (of course, one cannot define such a thing as a physical quark mass parameter, which would have heen cut-off independent) :

Again ve define

by

and

(2.5b)

where the series in (2.5b)

stops efter the a1 term.

In OCD the parameters al,

21-23) 61,

62

are known

(2.6)

552

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMIW?

We emphasize that the new parameters gR and

5

are better than the

previous ones, because for any theory for which the perturbation expansion is indeed an asymptotic expansion, they are completely finite and non-trivial. Of coutse, they still depend on the subtraction point p . At infinite p they coincide with other definitions;

at finite p they are finite because we can solve Eqs.

(2.2b) and (2.5b):

(2.7b)

Here PO and m o are integration constants.

They are invariant

Thus, po is a true parameter that

under the renormalization group.

fixes the gluon couplings, and has the dimensions of a mass. For each quark in the system we have a mass parameter mo. As must be clear from the derivations,

~0

and mo actually tell us how the

theory behaves at asymptotically large energies and momenta. The value of po for QCD is presumably of the order of the p mass, and mo will be a few MeV for the up and down quarks, 100 MeV or so for the strange quark, etc. drop the terms containing

82,

For simplicity, we will often

and the quark masses will be put

equal to zero.

3.

THE !tENORMALIZATION GROUP EQUATION Now let us consider the Green's functions of the theory.

For

definiteness, take only the two-point functions (dimensionally renormalized)

553

G. ‘t HOOFT

They satisfy a renormalization group equation24).

.

GD ( K ,Ly c , $ ) = O Here f3

R

(3.2)

is the truncated, finite f3 function for the constant g i

as it occurs in Eq. (2.2b), but y ( g R2 ) is still an infinite series. We wish to do something about that also. Let us first make clear how to interpret Eq. (3.2).

Consider the p versus g2 plane.

Sup-

pose we choose a special curve in that plane, where g2 R depends on U such that (3.3) then it follows from (3.2) that

or

That implies that, if we stay on one of tbe curves ( 3 . 3 ) , then (3.2) reduces to (3.5) which can easily be integrated. integration constant will

But the

still depend on k 2 and on the curve

554

CAN WE MAKE SENSE OUT OF "QUANTUM CHAOMOOYNAMICS"?

chosen, that is, on the constant see Eq. (2.7a).

~

0

which , we get in solving (3.3),

Thus we get

with

and for dimensional reasons, G(k2,po) can only depend on the ratio

k2/d. For our purposes it is now important to observe the following. The coefficients z o and but the terms zpgi +

z1

... in

are clearly very important as g2

+

0,

(3.7) can simply be absorbed in a rein (3.1). That way we get definition of the coefficients al, new, improved functions GR that can be written as

...

In a typical example where we study the time ordered product of two operators

-~ ( o y) ( o )

corresponding to the

Q

a d ii;(x)v(~)

channel, we have

555

G. 't HOOF1

(3.9)

4.

ANALYTIC STRUCTUXE FOR COMPLEX g2 In Eq. (3.8) we can write (neglecting for simplicity the

82

terms)

This is a function of one single parameter

Complex x corresponds to either complex k2, real g2 or complex g2, real k2.

Now, on physical grounds, we know what we should expect

at real g2, complex k2 (Fig. 1).

-

The singularities are at kZ real

and negative (i.e. Minkowskian). That is when x real + ( 8 1 / 2 ) * (2n t l)ni, n integer. Choosing now k2 real and positive we find the same singularities at

They are sketched in Fig. 2.

556

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMICS"?

Fig. 1

R

Expected analytic structure of G for complex k2. The wavy line is a cut (in Baryonic channels this cut starts away from the origin, in mesonic channels, since we have put mf = 0, the cut starts at zero because the pion is massless). The dotted crosses are singularities to be expected in the second Riemann sheet (resonances).

557

G. 't HOOFT

Fig. 2 Resulting analytic structure for complex g i . The single cut of Fig. 1 now reproduces many times on semi circles. These semi-circles are only slightly distorted due to the 8 2 term in Eq. (2.7a). The arrow shows the region where perturbation expansion is done. The cut on the left is due to the Z-factor in ( 3 . 8 ) .

The conclusion of this section is obvious: we find such a bad accumulation of singularities at the origin that the analytic continuation procedure given in the introduction will never work. We must look for a more powerful technique.

558

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS?

5.

BOREL RESUMMATION We now assume that our Green's functions can be written as a

Laplace transform of a special' type

F(z) .can be found perturbatively:

If

then (5.3) where the 6 function is understood to be included in the integral (5.1).

One can check this trivially by inspection.

The importance

of this is that the series (5.3) converges much faster than (5.2). Contrary to (5.2) it may very well have a finite radius of convergence (this is at present believed to be the case for all renormalizable field theories).

If F ( z ) can now be analytically con-

tinued to all real positive z, and if, for some g2, the integral (5.1) converges, then the series (5.2) is called Borel summable. Be will call F ( z ) the Borel function corresponding to the Green's R function G (g2). First, we wish to find out where we can expect singularities in F ( z ) .

Let us illustrate an interesting feature in the case of

an over-simplistic "field theory", namely field theory at one spacetime point.

Remember that field theoretical amplitudes can be writ-

ten as functional integrals, with a certain number of integration variables at each space-time point").

If we have one space-time

point and one field, then there is just one integration to do

559

G. 't HOOFT

(5.4)

-

...

V3x3 + V4x4 + and the factor g in front is just for convenience. g2 comes out this way as the usual perturbation parameter. Note the minus signs in the integrand. This anticipates that we shall always consider Field theory in Euclidean space-time. Let us rescale the fields, and the action,

where V(x)

Our integral becomes

Comparing this with the Bore1 expression (5.1), we immediately find F(z) :

Thus at given z we must find all solutions of S'(A)

can call Ai(z).

The result of the integral is

= z, which we

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"?

This outcome reveals the singularities in the z plane:

we must

find the solutions of (5.9)

to be recognized as the classical field equation of this system. At a solution

z of

(5.9) we have

(5.10)

-

and so we find a square root branch point at z = z. Consider now multidimensional integrals of the same type.

The

integral (5.7) then corresponds to an integral over the contours in + A space defined by the equation S'(1) = z. You get singularities only at those values of z that are equal to the total action S' of a solution of the equation (5.11) because those are the contours that shrink to a point (in the case of a local extremum) or have crossing points (saddle points). Again, Eq. (5.11) is nothing but the classical Lagrange equation for the fields

1.

Conclusion:

to find singularities in F we have

to search for finite solutions of the classical field equations in Euclidean space-time. singularity points

Their rescaled action S' corresponds to in the z place for the function F.

these singularities are branch points.

561

In general,

In our actual four-dimensional

G. 't HOOFT

field theories that are supposed to describe strong (or weak and electromagnetic) interactions; such solutions indeed occur, and are called "instantons", because they are more or less instantaneous 25,26,6,10)

and local in the Euclidean sense case of QCD) is S '

=

.

Their action (in the

8.rr2n,where n counts the "winding number"e - 1 0 )

and so we may expect singularities in the complex z plane at z = * 8n2n.

6.

UNIVERSALITY OF THE BOREL SINGULARITIES

The student might wonder whether the conclusions of the previous sections were not jumped to a little too easily.

The con-

nected Green's functions in field theories are not just multi(infinite) dimensional integrals but rather the ratio of such integrals with some source insertion and an integral for the vacuum, and then often differentiated with respect to those source insertions. Do all these additional manipulations not alter or replace these singularities and/or create new ones? Do different Green's functions perhaps not have their own singular points? Let us for

a

moment forget the renormalization infinities, to

which we devote a special section. Then the answer to these equations is reassuring. Multiplications, divisions, exponentiations can be carried out, after which we shall always find the singularities back in the same place as they were before, possibly with a different power behaviour. To understand this general property of Bore1 transforms, let us formulate some simple properties. Let

Then, if

562

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS?

then Fx.

And let

then

Here the E symbols are there just to tell us to leave the 6 symbols out of the integration. It is easy to show that if (6.3) is solved iteratively, then the series converges for all z, as long as F1 stays finite between E and z . Now note that we may choose the contours ( 0 , z ) so that they avoid singularities. Only if z is a singularity of either F1 or

Fz, or both, then FJ(z) in Eq. (6.2) will be singular. Also F~(z) in Eq. ( 6 . 3 ) is only singular if F (z) is singular. Note, however, that if a singularity lies between 0 and z then the contour can be chosen in two (or more) ways, and, in general, we expect the outcome to depend on that. Thus if we start with pure pole singularities, they will propagate as branch points in the other Bore1 functions. In quantum field theories, the Green's functions are related through many Schwinger-Dyson equations and Ward-Slavnov-Taylor identities. Since singularities survive the multiplications and divisions in these equations without displacement, they must occur in all Borel-Green's functions at the same universal values of z

563

G. 't HOOFT

(unless miraculous cancellations occur; I think one can safely exclude that possibility).

These singularities will, in general, be

of the branch-point type. In particular, those singularities that we obtained through the solutions of the classical equations, will stay at the same position for all Green's functions. 7.

SINGULARITIES IN F(z) DUE TO INSTANTONS Let us consider these classical equations in Euclidean space-

time for the various field theories. First take theory, for simplicity, without mass term. Rescaling the fields and action the usual way

'Q'=JX'4

s r - i S'

(7.1)

A

we have

It turns outz5) that a purely imaginary solution exists for the equations 6 SIC4

=

0 (here, %indicates derivative in the Euler

sense), namely

(7.3) Here p is an arbitrary scale parameter (after all, our classical

ac-

tion is scale invariant). In spite of this solution being purely imaginary, it is important to us because it indicates a singularity in F ( z ) away from the positive real axis. The corresponding value for S ' is

564

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMOOYNAMICS"?

S't

46 'It

2

(7.4)

So the singularity occurs at z =.-16n2, indeed away from the posi-

tive real axis.

Such singularities are relatively harmless, since

F is only needed for positive z.

We may invoke the analytic con-

tinuation procedure sketched in the Introduction to improve convergence for the series in z. Now, let us turn our attention to Quantum Chromodynamics.

Here

we have a real solution in Euclidean space:

(7.5)

are certain real ~oefficients~'~') and p is again a free a w scale parameter. One finds for the action

where

Thus z = 8a2 is a singularity on the positive real axis.

In fact,

we can also have n instantons far apart from each other, so we also expect singularities*) at z * 81~*n.Now, Green's functions are obtained from F ( z ) by integrating from zero to infinity, over the positive real axis.

Do the singularities on the real axis give un-

I think not, although the correct prescription will be complicated. A clue is the following. The singleinstanton contribution to the amplitudes has been computed directly surmountable problems?

*)

A more precise analysis suggests that only those multi-instantons with zero total winding number (that is, as many instantons as anti-instantons) will give rise to ordinary singularities that limit the radius of convergence of F(z). The others give discontinuities rather than singularities.

565

G. 't HOOFT

in the small coupling constant limit. A typical result goes 1ike27-30 )

That is already

a

Green's function, the one we would like

to

ob-

tain after integrating

A function F ( z ) that yields

Indeed, a "singularity" at

(7.7) exists

z = BIT'.

We see that, since all Green's

functions will show the same exponential in their g dependence, the universality theorem of the previous section is obeyed. What is important is that by first computing (7.7) one can short-circuit the problem of defining an integration over such singular points. Thus the instanton singularities at the right-hand side on the

real axis will not destroy our hopes of obtaining a convergent theory.

The reason is that the physics of the instanton is under-

stood. The situation is less clear for the other type of singularities that we discuss in the next section.

8.

OTHER SINGULARITIES IN F, In principle, the instanton-singularities in F can also be un-

derstood within the context of ordinary perturbation expansion, by We do not show the derivation here, but the following argument has been given. In the

a statistical treatment of Feynman diagrams31).

previous section, we have never bothered about the renormalization procedure that is supposed to make all diagrams finite. Suppose we

566

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"?

had a strictly finite theory, with bounded propagators, bounded integrals and all that.

Individual diagrams in such a theory are then

bounded by a pure power law as a function of their order n.

The

only way that factors n! can arise is because there are n! diagrams

at nth order and they may not cancel each other very well.

This is

how in the statistical treatment the instanton singularity occurs. But in realistic four-dimensional renormalizable field theories, the power law for individual Feynman diagrams no longer holds. example is quantum-electrodynamics.

A simple

We consider the diagrams of the

type shown in Fig. 3.

Fig. 3 Fourth member of a subclass of diagrams discussed in this section. It is the class of diagrams with n electron bubbles in a row, which in itself closes again a loop.

It is well known that each electron

bubble separately behaves for large k2 as

and each propagator as .')'k(

Thus, for large k2 the integrand in

the k variable behaves as

567

G. 't HOOFT

where a is some fixed power.

After having made the necessary sub-

tractions to make the integral converge, and in order to obtain physically relevant quantities, such as a magnetic moment, the leading coefficient a becomes 3 or larger. Let us replace log k2 by a new variable x, then (8.2) becomes proportional to

Thus the integral over x will grow as n

+

03

like

A more precise analysis shows that C should be proportional to the

first non-trivial f3 coefficient

In the expansion for F ( z ) the factor n! is removed, as usual.

It

is clear that a new singularity develops at

It seems to be a universal phenomenon for all field theories, and not related to any instanton solution. Our definition for

fil was

positive for asymptotically free theories and negative otherwise. So,

the singularity is at negative real z and therefore harmless if

our theory is asymptotically free, but for non-asymptotically free theories such as QED and real axis.

we have singularities on the positive

Since a detailed understanding of the ultraviolet

568

CAN WE MAKE SENSE OUT OF "OUANTUM CHROMODYNAMICS"?

behaviour of non-asymptotically free theories is lacking, there may exist no cure for these singularities then. This is in contrast with the instanton singularities. An important observation has been made by G. Parisi3').

The

ultraviolet behaviour of A$4 and QED are well understood in the limit N

+

=, where N is the number of field components. A syste-

matic study of the singular point (8.6) is then possible.

Parisi

found in A$4 theory a conspiracy between diagrams such that the first singularity at a = 3 cancels.

In total the integrals do be-

have as (8.2) but with a > 3, after all necessary sudtractions.

At

present, it is not understood whether this conspiracy is accidental for A$'

theory with N components, or whether it is a more general

phenomenon. It does seem that only the first singularity may be subject to such cancellations. In Figs. 4 and 5 we show the complex planes for the Borel variables z in

theory and in QED, respectively.

The singulari-

ties discussed in this section are called "renormalons" for short.

? 1

-6

-3

6 8 renormalms

4

instantons

The situation for QCD is more complex.

Not only do we have the

renomalons at points on the negative real axis but also there are such singularities on the positive real axis. infra-red divergence of the theory.

They are due to the

The mechanism is otherwise the

G. ‘t HOOFT

Fig. 5

Singularities for QED. Here the units are 3n, if a i s the original expansion parameter.

I R divcrpcncies

Fig. 6

Bore1 z plane for QCD. The circles denote IR divergences that might vanish or become unimportant in colour-free channels.

the same as discussed for the ultraviolet singularities (Fig. 6 ) . An interesting speculation is that these infra-red singularities are only surmountable in colourless channels, but the integration over these singularities becomes impossible in single quarkor gluon- channels.

It is likely that these singularities are re-

lated to the quark confinement mechanism.

570

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS?

9.

SECOND BOREL PROCEDURE Many features of the singularities in the complex z plane of

the Bore1 functions F ( z ) are still uncertain and ill-understood. But from the foregoing we derive some hopes that it will be possible to obtain F(z) for 0 5 z < as QCD.

00

for asymptotically free theories, such

The only thing to be investigated then is how the integral

in

behaves at-.

Does the integral converge? The answer t o this is

almost certainly: no.

Consider massless QCD and its singularities

in complex g2 plane as derived in Section 3. R there are singularities when

According to Eq. ( 3 . 3 )

where the real number may be arbitrarily large.

Substituting that

in Eq. (9.1) we find that

S-kz) 0 e

must diverge.

-

g ( d +

&(2a+4)) 2

dz

(9.3)

We had assumed that the singularities at finite z did

not give rise to divergences. So F ( z ) must diverge at large z worse than any exponential of z.

Note that (9.3) contains an oscillating

term, It is likely then, that at z

-f

m,

F ( z ) does not only grow

very fast, but also oscillates with periods 4/81 or fractions thereof.

Can we cure this disease? We have no further clue at hand which could provide us with any limit on the large z behaviour of F. But there is a way to express the unknown Green's functions in terms of

571

G. t' HOOFT

a more convergent integral than (9.3).

Let us treat the divergent

integral (9.1) on the same footing as the divergent perturbation expansions which we had before.

We consider a new, better converging

integral

(9.4) We may hope that this has a finite region of convergence, from which we can analytically continue.

Note the analogy between (9.1) and

(9.4) on the one hand, and (5.2) and (5.3) on the other. gral relation between W and G, analogous to (5.1),

The inte-

is

Now, remembering that instead of varying g2 we could vary k2, replacing

So that, ignoring the Z factor that distinguishes G from GR (see Eq. ( 3 . 8 ) ) , one gets

(9.7)

Now we can easily prove that, if our theory makes any sense at all, there may be no singularities in W(s) on the positive real axis, and the integral (9.7) must converge rapidly.

Thus,

(9.4) makes sense, then our problems are solved. f01lows :

572

if the

integral

The proof goes as

CAN WE MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"?

C(k2) satisfies a dispersion relation: it is determined by its imaginary part. We have, at k2 = -a2, a > 0,

where p is usually a positive spectral function. into (9.7) we get

Thus, p(a)

Substituting (9.8)

and W(w2) are each other's Fourier transform. The in-

verse of (9.9) is

(9.10)

A

possible singularity at a2

=

0 is an artifact of our simplifica-

tions and can be removed. It is important to observe that (9.10) severely limits the growth of W(s) at large s so that ( 9 . 7 ) is always convergent. Conclusion: this section results in an improvement on perturbation theory. The physically relevant quantities can be expressed in terms of integrals of the type (9.4), which converge better than the original ones of type (9.1). It is not known whether this improvement is sufficient, i.e., whether (9.4) actually converges in some neighbourhood of the origin. Even if the important open questions mentioned in these lectures cannot be answered we think that refinement of these techniques will lead to an improved treatment of strong coupling theories.

573

G. 't HOOFT

REFERENCES F.J. Dyson, Phys. Rev. 85 (1952) 861. L.N. Lipatov, Leningrad Nucl. Phys. Inst. report (1976) (unpublished). 3)

E. Brbzin, J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. D15 (1977) 1544 and (1977) 1558.

4)

G. Parisi, Phys. Letters 66B (1977) 167.

5)

C. Itzykson, G. Parisi and J.B. Zuber, Asymptotic estimates in Quantum Electrodynamics, CEN Saclay preprint. The singularities I discuss in Section 8 of my lectures were assumed to be absent in this paper.

G. 't Hooft, Phys. Rev. Letters 37 (1976) 8 . A.M. Polyakov, Phys. Letters 59B (1975) 82 and unpublished work. C.

Callan, R. Dashen and D. Gross, Phys. Letters 63B (1976) 334.

R. Jackiw and C. Rebbi, Phys. Rev. Letters 37 (1976) 172. See S. Coleman's lectures at this School. K.G. Wilson, Phys. Rev. D10 (1974) 2445. J. Kogut and L. Susskind, Phys. Rev. D11, 395 (1975). L. Susskind, lectures at the Bonn Summer School (1974). S.D. Drell, M. Weinstein, S. Yankielowicz, Phys. Rev. D14 (1976) 487 and DL4 (1976) 1627. G. 't Hooft, Marseille Conf. on Renormalization of Yang-Mills fields and applications to particle physics, June (1972) (unpublished). H.D. Politzer, Phys. Rev. Letters 30 (1973) 1346. D.J. Gross and F. Wilczek, Phys. Rev. Letters 30 (1973) 1343. G. 't Hooft and M. Veltman, Nuclear Phys. B44 (1972) 189. C.G. Bollini and J.J. Giambiagi, Phys. Letters 40B (1972) 566. J.F. Ashmore, Lettere a1 Nuovo Cimento 4 (1972) 289.

574

CAN W E MAKE SENSE OUT OF "QUANTUM CHROMODYNAMICS"? 20) G. 't Hooft, Nuclear Phys. B61 (1973) 455.

D.R.T.

Jones, Nuclear Phys. B75 (1974) 531.

A.A. Belavin and A.A. Migdal, Gorky State University prepreprint (January 1974).

W.E. Caswell, Phys. Rev. Letters 33, (1974) 244. S. Coleman, lectures given at the "Ettore Majorana" Int. School of Subnuclear Physics, Erice, Sicily (1971). Note: we drop the inhomogeneous parts of the renormalization group equation, which can be avoided according to later formulations on the renormalization group (Ref. 20). S. Fubini, Nuovo Cimento 34A (1976) 521.

A.A. Belavin et al., Phys. Letters 59B (1975) 85. G. 't Hooft, Phys. Rev. D14 (1976) 3432.

F.R. Ore, "How to compute determinants compactly", MIT preprint (July 1977). A.A. Belavin and A.M. Polyakov, Nordita preprint 7711. A.M. Polyakov, Nordita preprint 76/33 (Nuclear Phys. in press). C.M. Bender and T.T. Wu, Phys. Rev. Letters 27 (1971) 461; Phys. Rev. D7 (1972) 1620. 32)

G. Parisi, private communication.

575

CHAPTER 8

QUANTUM GRAVITY AND BLACK HOLES

.

Introductions ...................................... ....................,....... ‘Quantum gravity”,in Z h d s in Elementary Particle Theory, eds. H. hllnik and K. Dietz, Springer-Verlag, 1975,pp. 92-113 ..... ‘Claesical N-particle cosmology in 2 + 1 dimensions”, C7ass. Quantum Gmv. 10 (1993)S7SS91 ... .......... .... .... ............... “On the quantum structure of a black hole”, Nucl. Phys. B266 (1985)727-736 .......................................................... with T. Dray, “The gravitational shock wave of a massless particle”, Nucl. Phye. B26S (1985) 173-188 ........ .......................,.... %matrix theory for black holes”, Lectures given at the NATO Adv. Summer Inst. on New Symmetty Principles in Quantum Field 1992,Plenum, New York, Theory, eds. J. F’rohlich et al., Car-, pp. 275-294 ..............................................................

.

577

578 584 606 619 629

645

CHAPTER 8

QUANTUM GRAVITY AND BLACK HOLES Introduction to Quantum Gravity [8.1] The years 1970-1975 gave us a remarkable insight in the general features common to all elementary particle theories. They must contain vector fields in the form of gauge fields, and spinors and scalars that must form representations of the vector field gauge group. The spectrum of all possible particle types that populate our universe will be represented in the most economical way possible, and if more types exist that are as yet unidentified then there are two corners to search for them: either they are very weakly interacting or they have very high masses. The strength of these results is their completeness: we know exactly how to enumerate all possibilities. There are two weaknesses however. One is that there will be a built-in imprecision as explained earlier, because of the divergence of the loop expansion. We might try to avoid this by postulating asymptotic freedom, but this gives too stringent a restriction that can easily be made undone by suspecting as yet unknown particles in the hidden corners as just mentioned. The other weakness is that the number of different viable theories is still too large for comfort. Why did Nature make the choice she did? On first sight the gravitational force makes things much worse. Including gravity maka our theory hopelessly nonrenormalizable; infinite series of uncontrollable counter terms undermine our ability to do meaningful calculations. But worse still, we do not at all understand how to formulate precisely the first principles for such theories. We can write down mathematical equations, but upon closer examination these turn out to be utterly meaningless. This problem persists up to this day, in spite of the commercials from the superstring adherents. It is not hard to speculate on radically new theoretical ideas. I have toyed with discrete theories for space-time myself. But then one opens a Pandora’s box full of schemes and formalisms that are completely void when it comes to making any firm predictions. In my opinion the art is not to obscure our view by clogging the

578

scientific journals with speculations. The problem is to deduce from hard evidence features that have to be true for the world at Planckian time and distance scales. What do we know about the perturbative expansion with respect to Newton’s gravitational constant IC ? In the first paper of this chapter I address this question. It is shown that pure gravity, without any matter coupled to it, is one-loop renormalizable. This means that uncertainties due to nonrenormalizability and arbitrariness of counter terms are down by terms of order IC’.An important lesson learnt here is that the “field variable” g,,(xlt) is not quite observable directly. When we do particle scattering experiments admixtures to this tensor of the form R,, and Rg,, are both necessary for renormalization and undetectable experimentally. Note that these admixtures are of higher order in IC and therefore infinitesimal. There is as yet no clash with causality. If matter fields are added such admixtures do become visible when compared to similar admixtures in the other fields. Consequently these theories are not even oneloop renormalizable.

Introduction to Classical N-Particle Cosmology in 2 Dimensions [8.2]

+1

The previous paper treated quantum gravity in a way I would now call “conventional”, which means that we assumed the problem to be to find a consistent prescription for calculating elements of the scattering matrix, under circumstances where we can still speak of a simply connected space-time, locally smooth and asymptotically flat. But gravity raises a couple of conceptual problems when one tries to do better. One simple demonstration of this is to consider an extremely simplified version of gravity: gravity in 2 space- and onetime dimension. In such a world there are no gravitons, but the gravitational force does exist, and even though the Newtonian attraction between stationary objects vanishes the problems raised by the curvature of space-time are considerable. My aim was originally to obtain a theory that can be formulated exactly and has two limits: one is a we& coupling limit in which ordinary (scalar) fields are weakly coupled to gravity and the perturbative approach makes sense; the other is a limit where Planck’s constant vanishes such that we have a finite number of massive point particles gravitating in a closed universe. But the system of classical point particles gravitating in a universe that is closed due to this gravitational force exhibits a number of features that predict evil for attempts to “quantize” it. For one, it may not be possible to define an S matrix. The reason is that under a wide class of initial conditions the universe does not end up in an asymptotic state where particles fly away from each other in classical orbits. Instead of this the universe may end up in a “big crunch”. I discovered this in attempting to disentangle a paradox presented in a paper by J. R. Gott. He had argued that under certain conditions a pair of particles in this 2+1 dimensional

579

universe may be surrounded by a region where a test particle may follow a closed timelike trajectory in space-time, and even move backwards in time. I could not believe that a sound physical system should admit such an event, and my intuition turned out to be correct: before the closed timelike curve gets a chance to come into being a “big crunch” ends it all. In a complete universe the region described by Gott is an illegal analytic extension; it is shut off from the physical world by a shower of particles crunching together. The paper reproduced here explains what happens. I should add that I later found the transitions described in Figure 8 to be strictly speaking not complete. Other transitions are possible where polygons, as described in the paper, disappear altogether, and also sometimes an edge shrinks to zero but the system continues without the topological transition. These new possible transitions do not affect any of the conclusions however.lg

Introduction to On the Quantum Structure of a Black Hole [8.3] The previous paper demonstrates some of the difficulties that will form considerable obstacles when one wishes to formulate “quantum cosmology”. The big crunch as an asymptotic state is extremely complex. Numerical experiments later showed that this crunch is in some sense “chaotic”, and so we will not have an easy time enumerating the possible asymptotic states in a quantum mechanical “measuring process”. But one could argue that “quantum cosmology” is not our most urgent project. I think it is legitimate to ask first for a theoretical scheme describing scattering proceases in regions of the universe that are surrounded by asymptotically flat space time. This may not be possible in 2+1 dimensional worlds, but should be an option in 3+1 dimensions. Any finite number of particles with limited total energy will be surrounded by asymptotically flat spacetime sufficiently far away. So should a scattering matrix exist there? This seem to be plausible, at first sight. But now a new problem emerges, and it is an enormous one. We have to have at least four space-time dimensions. We better avoid taking more than four because then none of our ordinary models is even renormalizable. But in four dimensions the gravitational force, already before quantization, exhibits a fundamental instability: gravitational collapse can occur, and black holes can form. Black holes will be a new state of matter, and it will be impossible to avoid them. They are regular solutions of the gravitational equations applied exclusively in those regions of spacetime where we may expect that we know exactly how to formulate what happens. An event horizon may open up long before matter fields or particles here had a chance to interact in a way not covered by the Standard Model. Therefore it doesn’t help to postulate that perhaps “black holes do not exist”. They exist as legitimate solutions of the classical theory, and the real problem is to ask what they have to be replaced with in a completely quantized theory. ”G. ’t Hooft, Class. Quantum Gravity 10 (1993) 1023.

At first sight a black hole seems to be just an “extended solution”, of the kind I could have covered in my B d lectures, Chapter 4.1. But if that were 80 it should have been possible to construct all their excited quantum modes just by performing standard field theory in a black hole background. This calculation has been done, or rather, that is what the authors thought. The outcome is a big surprise: not at all a reasonable spectrum of black hole states seems to emerge. What one finds is a continuous spectrum, as if black holes were infinitely degenerate. An immediate clash arises with the determination of the black hole entropy, which can be derived from its thermodynamical properties and it is finite. So the background field calculations for black holes are incorrect. It turna out not to be hard to guess why they were incorrect: the gravitational interactions between ingoing and outgoing objects were neglected, in the first approximation. Under normal circumstances, as in conventional theories, one may always presume that such interactions can be postponed to later refinements, but gravity is not a conventional theory. We argue that these mutual interactions become infinitely strong and therefore may not be neglected even in the zeroth approximation. The big question is how to do things correctly then. One can do better, but a completely satisfactory approach is still missing. I do not expect that a “completely satisfactory” formalism will be found very soon, because it can probably not be given without a completely satisfactory theory of quantum gravity itself. But what we can do is ask very detailed questions, insist on seneible answers, and hope that these answers, even if incomplete, also furnish a better insight in the problem of quantizing gravity itself. Another way of phrasing our point is this: if the gravitational force gives us so many problems when it becomes strong, why not concentrate entirely upon the question on how to formulate the theory where it is as strong as possible, which is near black holes? The quantum black hole paradox began to intrigue me around 1984 and I began to study it in great detail. I am quite convinced that I made substantial progress in understanding its nature and its relevance for the entire theory of quantum gravity. I first learned that the real paradox already occurs at our side of the horizon, and that the “back reaction”, i.e. the mutual (mostly gravitational) interactions between outcoming Hawking particles and objects falling into the hole somewhat later, is of crucial importance to resolve the problem. I found out that in contrast to a widespread misconception, the resolution of the problem will not require fundamental nonlocal interactions, but that it is strongly connected to very fundamental issues in the interpretation of quantum mechanics. I also found that one probably has to give up the idea that the quantum state of the black hole as it is seen by the outside observer can be described independently of the adventures of an observer who cro88e8 the horizon to enter into the black hole. The inside region does not make my sense at all to the outside observer. One should never try to describe “super observers”, a hypothetical simultaneous registration of what happens both inside and outside. I was annoyed by the lack of interest in this problem and my work on it by most of the particle physics community and the general relativists, two groups who for a

581

long time were unaware of the mutual incompatability of their views. But now that I am compiling my papers, early 1993, all this has changed. The literature is now being flooded with papers dealing with precisely the problem that kept me busy already for so long. I cannot say that these are already at a stage of affecting my present ideas, but an acceleration is certainly taking place, and the chances that new breakthroughs will occur (probably not by renown physicists but more likely by some young student), are now better than ever. The following section is part of one of my first papers on black holes. The remainder of the paper is less relevant for the arguments that follow, and was therefore omitted. To understand the WKB approximation in Eq. (3.5) it is necessary to replace the coordinate r temporarily by U according to

r - 2M = e‘, because in this new coordinate the wave packets are much more regular than before and the WKB procedure much more accurate.

Introduction to the Gravitational Shock Wave of a Massless Particle [8.4]

So what is the gravitational interaction between ingoing and outgoing particles near a black hole horizon? This turns out to be sizeable even if the particles have very low energies as seen by a distant observer. It is large if the time interval between the ingoing and the outgoing object, as seen by the distant observer is of order M logM in Planck units or larger. That is still small compared to the total expected lifetime of the black hole. To find this result we need to compute the gravitational field of one of them. First take coordinates in which this is weak and therefore easy to compute. Then transform to any coordinate frame one wishes. It often becomes strong. The calculation is done in the next paper, written together with Tevian Dray. The result is that there is a plane associated to the particle’s trajectory in spacetime (the “shock wave”) such that geodesics of particles crossing this plane are shifted. The fact that the shift depends on the transverse distance makes it visible. The effect is much like a sonic boom. A fragile detector might break in pieces if it encounters such a shock wave.

Introduction to the S-Matrix Theory for Black Holes [8.5] Because of the associated shock wave any object falling into a black hole has an effect upon the outgoing “Hawking” particles. The effect of these shifts may seem to be minute, until one realizes that the phases of the wave functions of particles coming out very late are extremely sensitive to precisely shifts of this nature. Thus we can now understand why information going into a black hole is always transmitted onto

582

particles coming out, be it extremely scrambled. We can use this to construct a scheme for a unitary black hole S matrix. As long as we concentrate on effects at moderately large distances across the horizon all interactions that contribute to this S matrix, gravitational and otherwise, are known. So many details of the S matrix we were searching for can be derived. Unfortunately, to see that this matrix will be unitary, and to determine the dimensionality of the Hilbert space in which it acts, we need more information concerning short distance interactions. Our hope was that there are still elements of the theory of general relativity that we have not yet used, and that they would be sufficient to remove most of our present uncertainties. Our feeling is that a lot of further inprovements should be possible along these lines, but they have not (yet?) been realized. The paper gives a good impression of the way I think about the problem at present. The algebra at the end should be seen as speculative and somewhat mysterious in its lack of unitarity. The idea is that it should inspire others to find improved methods along similar lines.

583

CHAPTER 8.1

QUANTUM GRAVITY G. 't Hooft University of Utrecht, T h e Netherlands

1.

Introduction

The g r a v i t a t i o n a l f o r c e i s by f a r t h e weakest e l e m e n t a r y i n t e r a c t i o n between p a r t i c l e s . It i s s o weak t h a t o n l y c o l l e c t i v e f o r c e s between l a r g e q u a n t i t i e s o f matter are o b s e r v a b l e a t p r e s e n t , and i t i a e l e m e n t a r y b e c a u s e i t a p p e a r s t o obey a new symmetry p r i n c i p l e i n n a t u r e : t h e i n v a r i a n c e under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s .

E v e r s i n c e t h e i n v e n t i o n of quantum mechanics and g e n e r a l r e l a t i v i t y , p h y s i c i s t s have t r i e d t o " q u a n t i z e g r a v i t y " ' ) , and t h e f i r s t t h i n g they r e a l i z e d i s t h a t t h e theory contains n a t u r a l u n i t s o f length (L), t i m e ( T ) and mass (MI. I f K

=

m3/kg Sec2

6.67

i s the gravitational constant, then

and

L

=

=

T

=

G=

M

=

1.616

m

5.39

sec

= 1.221 = 2.177

lo2'

eV/c 2 g

But t h e n t h e t h e o r y c o n t a i n s a number o f o b s t a c l e s . F i r s t t h e r e are t h e c o n c e p t u a l d i f f i c u l t i e s : t h e meaning o f s p a c e and t i m e i n Eins t e i n ' s g e n e r a l r e l a t i v i t y as a r b i t r a r y c o o r d i n a t e s , i s v e r y d i f f e r e n t from t h a t o f s p a c e and time i n quantum mechanics. The m e t r i c t e n s o r g y v , which used t o b e always f i x e d and f l a t i n quantum f i e l d t h e o r y , now becomes a l o c a l dynamical v a r i a b l e . Advances have been made, from d i f f e r e n t d i r e c t i o n s 2 B 3 s 4 ) , t o d e v i s e a language t o f o r m u l a t e quantum g r a v i t y , b u t t h e n t h e n e x t problem arises: t h e t h e o r y c o n t a i n s e s s e n t i a l i n f i n i t i e s s u c h t h a t a f i e l d t h e o r i s t would s a y : i t i s n o t r e n o r m a l i z a b l e . T h i s problem, d i s c u s s e d i n d e t a i l i n s e c t i o n 12, may b e v e r y s e r i o u s . It may v e r y w e l l imply t h a t t h e r e e x i s t s no well d e t e r m i n e d , l o g i c a l , way t o combine Trendr in Elementary Particle Theory, Edited by H.Rollnik and K.Dietz (Springet-Vtrlag. 1975).

g r a v i t y w i t h quantum mechanics from f i r s t p r i n c i p l e s . And t h e n one i s led t o t h e q u e s t i o n : should g r a v i t y be q u a n t i z e d a t a l l ? After a l l , such quantum e f f e c t s would be small, t o o small perhaps t o b e e v e r measurable. Perhaps t h e t r u t h i s very d i f f e r e n t , b o t h from quantum t h e o r y and from general r e l a t i v i t y . Whatever one s h o u l d , o r should n o t do, o u r p r e s e n t p i c t u r e of what happens a t a l e n g t h s c a l e L and a time s c a l e T i a incomplete, and we would l i k e t o improve i t . We c l a i m t h a t i t i s very worthwhile t o t r y and improve o u r p i c t u r e s t e p by s t e p , as a p e r t u r b a t i o n expansion i n K. In t h e f o l l o w i n g it i s shown how t o apply t h e t e c h n i q u e s o f gauge f i e l d theory and g a i n some remarkable r e s u l t s . The s e c t i o n s 2-1) d e a l w i t h t h e c o n v e n t i o n a l t h e o r y of g e n e r a l r e l a t i v i t y , seen from t h e viewpoint o f a gauge f i e l d t h e o r i s t . I n s e c t i o n 5 i t i s i n d i c a t e d how q u a n t i z a t i o n could be c a r r i e d out i n p r i n c i p l e , b u t i n p r a c t i c e we need a more s o p h i s t i c a t e d formalism t o ease c a l c u l a t i o n s . T h i s formalism, t h e background f i e l d method215), i s e x p l a i n e d i n I n t h e s e s e c t i o n s we mainly d i s c u s s gauge t h e o r i e s , and s e c t i o n 6-11 g r a v i t y i s h a r d l y mentioned; g r a v i t y i s j u s t a s p e c i a l c a s e h e r e .

.

Back t o g r a v i t y i n s e c t i o n 12, where w e d i s c u s s numerical r e s u l t s . It i s shown there why o n l y pure g r a v i t y i s f i n i t e up t o t h e one-loop

corrections. 2.

Gauge Transformations

The u n d e r l y i n g p r i n c i p l e o f t h e t h e o r y of g e n e r a l r e l a t i v i t y i s invariance under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s , t’”

=

f”(X)

.

(2.1)

It i s s u f f i c i e n t t o c o n s i d e r i n f i n i t e s i m a l t r a n s f o r m a t i o n s , XI’’

=

xp

+ rl”(x)

,n

infinitesimal.

(2.2)

Or, i n o t h e r words, a f u n c t i o n A ( x ) i s transformed i n t o A’(x) = A(x + n ( x ) ) = A(x) + n*(x)a,A(x)

.

(2.3)

If A does n o t undergo any o t h e r change, t h e n i t i s c a l l e d a s c a l a r . W e Call t h e t r a n s f o r m a t i o n (2:j) simply a gauge t r a n s f o r m a t i o n , g e n e r a t e d

585

A

by t h e ( i n f i n i t e s i m a l ) gauge f u n c t i o n n ( x ) , t o b e compared w i t h YangMills i s o s p i n t r a n s f o r m a t i o n s , g e n e r a t e d by gauge f u n c t i o n s Aa(x). F o r t h e d e r i v a t i v e of A(x) w e have a,,A’(X)

= aPA(x) + n x DPa ~ ( x +) t-,’a,apA(x)

,

(2.4)

where 11’ s t a n d s f o r a ox, t h e u s u a l c o n v e n t i o n . Any o b j e c t A t r a n s f o r JP P P ming t h e same way, i. e.

w i l l b e c a l l e d a c o v e c t o r . We s h a l l a l s o have c o n t r a v e c t o r s B p ( x ) ( n o t e t h a t t h e d i s t i n c t i o n i s made by p u t t i n g t h e i n d e x u p s t a i r s ) , which t r a n s f o r m l i k e

by c o n s t r u c t i o n s u c h t h a t

t r a n s f o r m s as a scalar. S i m i l a r l y , one may h a v e t e n s o r s w i t h a n arb i t r a r y number of u p p e r and lower i n d i c e s . F i n a l l y , t h e r e w i l l be d e n s i t y f u n c t i o n s w(x) t h a t t r a n s f o r m l i k e

They e n a b l e us t o write i n t e g r a l s of s c a l a r s Iw(X)

A(X)

db(X)

j

which are c o m p l e t e l y i n v a r i a n t u n d e r l o c a l gauge t r a n s f o r m a t i o n s (under c e r t a i n boundary c o n d i t i o n s ) . F o r t h e c o n s t r u c t i o n of a complete gauge t h e o r y it is of i m p o r t a n c e t h a t t h e gauge t r a n s f o r m a t i o n s form a group. O f c o u r s e t h e y do, and h e n c e we have a J a c o b i i d e n t i t y . L e t u ( i ) be t h e gauge t r a n s f o r m a t i o n s g e n e r a t e d by n P ( i J x ) . Then i f

586

3.

The M e t r i c Tensor

I n much t h e same way as i n a gauge f i e l d t h e o r y 6 ) , we ask f o r a dynamical f i e l d t h a t f i x e s t h e gauge of t h e vacuum by htwing a nonvanishing vacuum e x p e c t a t i o n v a l u e . ( C o n t r a r y t o t h e Yang-Mills c a s e it seeme t o be i m p o s s i b l e t o c o n s t r u c t a r e a s o n a b l e "symmetric" t h e o r y . To t h i e end we choose a two-index f i e l d , g,,(x), which i s symmetric i n its indices, %U = gul, (3.1) and i t s vacuum e x p e c t a t i o n v a l u e i s
=

'o

(3.2)

uu

(our m e t r i c corresponds t o a p u r e l y imaginary time c o o r d i n a t e ) . With g,,,, , o r i t s i n v e r s e , g" , we can now d e f i n e l e n g t h s and timei n t e r v a l s a t each p o i n t i n space-time:

fal* = and g,,

g*va%u

J

can be used t o r a i s e o r lower i n d i c e s : AM

g""~"

,

, etc.

%,,A"

=

A,

!3.3)

Just as i n t h e Yang-Mills c a s e , w e c a n now d e f i n e c o v a r i a n t d e r i v a t i v e s : D,,A

'

= a A

( t h e d e r i v a t i v e of a s c a l a r t r a n s f o r m s as a v e c t o r ) ,

DpAv

=

aUAu

-

D'B"

=

a,,~"

+ r" vaB"

l'aUvAa

,

(3.4)

.

The f i e l d P t u i s c a l l e d t h e C h r i s t o f f e l symbol and i s y e t t o be d e f i n e d . First w e write down how i t should t r a n s f o r m under a gauge t r a n s f o r m a t i o n , such t h a t t h e above-defined c o v a r i a n t d e r i v a t i v e s be r e a l t e n s o r s :

"

=

riu

+

( o r d i n a r y terms f o r 3-index t e n s o r ) +

a,,a,,n

A

.

(3.5)

rlJV

W e see t h a t no harm i s done by making t h e r e s t r i c t i o n t h a t

r;,,

=

rt,, ,

587

(3.6)

because t h e symmetric p a r t of l' o n l y i s enough t o make ( 3 . 4 ) c o v a r i a n t . W e now d e f i n e t h e f i e l d 'l by r e q u i r i n g

= 0

*

(3.7)

(from which f o l l o w s : DiguV = 0 ) . We see t h a t we have e x a c t l y t h e r i g h t number of e q u a t i o n s . By w r i t i n g Eq. ( 3 . 7 ) i n full w e f i n d t h a t i t i s easy t o solve

Note t h a t r A i s n o t a c o v a r i a n t t e n s o r . UV

The c o v a r i a n t d e r i v a t i v e may b e u s e d j u s t l i k e o r d i n a r y d e r i v a t i v e s when a c t i n g on a p r o d u c t : D(XY) =

+

(DX)Y

,

XDY

(3. 9)

b u t two c o v a r i a n t d i f f e r e n t i a t i o n s do n o t n e c e s s a r i l y commute: #

D,,DVAa

.

D,D,A,

(3.10)

I n s t e a d , w e have:

-

D,,D,Aa

=

DvD,,Aa

(3.11)

RBavU AB

with

The comma d e n o t e s ' o r d i n a r y d i f f e r e n t i a t i o n . S i n c e t h e 1 . h . s . of e q . ( 3 . 1 1 ) i s c l e a r l y c o v a r i a n t and A B is a n a r b i t r a r y v e c t o r , RBav,, t r a n s f o r m s a s an o r d i n a r y t e n s o r , i n c o n t r a s t w i t h rBap. It i s c a l l e d t h e Riemann o r c u r v a t u r e t e n s o r ( s e e t h e s t a n d a r d

t e x t books). I n d i c e s can be r a i s e d o r lowered f o l l o w i n g ( 3 . 3 ) . Without p u t t i n g i n any f u r t h e r dynamical e q u a t i o n , one f i n d s t h e f o l l o w i n g identities, Ra0~6 = RaB~6

RaBy6;p

Ry6a0

'

RaydB

E

+

-

RaBtjy

3

R a ~ ~ =y 0 ,

=

+

R a B 6 ~ +; ~ R a ~ w ; ~

O

(3. 13)

.

The semicolon denotes covariant differentiation. Further, we define*)

which satisfy, according to (3.13):

The metric tensor also enables us to define a density function Gee (2.711,

In the quantum theory we shall encounter a fundamental problem: instead of g, we could go over to a new metric g;, with, for instance,

in a curved space there is some arbitrariness in the choice of metric (Section iP).We bypass this problem here.

So

4.

Dynamics

The question now is whether we can make the fields gU V propagate. Indeed we can, because we can construct a gauge invariant action integral

where

U

is to be identified with the usual gravitational constant:

For simplicity we shall take the units in which C

2

1611K=1

(4.3)

At a later stage one could put U back in the expressions to find that the expansion in numbers of closed loops will correspond to an expansion

*)Note that there is a sign difference compared to some earlier papers.

589

with respect t o

K.

One c a n a l s o add o t h e r f i e l d s i n t h e L a g r a n g i a n , f o r i n s t a n c e (4.4)

where 4 i s a s c a l a r f i e l d . We s h a l l n o t r e p e a t h e r e t h e u s u a l arguments t o show t h a t v a r i a t i o n of t h e L a g r a n g i a n ( 4 . 4 ) r e a l l y l e a d s t o t h e familiar g r a v i t a t i o n a l i n t e r a c t i o n s between masses, and t o u n f a m i l i a r i n t e r a c t i o n s between o b j e c t s w i t h a g r e a t v e l o c i t y ( " g r a v i t a t i o n a l magnetism"). The e q u a t i o n f o r t h e g r a v i t a t i o n a l f i e l d w i l l b e (4.5)

where T i s t h e u s u a l energy-momentum t e n s o r ( E i n s t e i n ' s e q u a t i o n ) . CtV The a c t i o n ( 4 . 1 ) has much i n common w i t h t h e a c t i o n

i n Yang-Mills t h e o r i e s . As w e s h a l l i n d i c a t e i n t h e n e x t s e c t i o n , a massless g r a v i t o n w i t h h e l i c i t y 5 2 w i l l p r o p a g a t e . N o t i c e t h a t w e have been l e d t o E i n s t e i n ' s t h e o r y o f g r a v i t y a l m o s t a u t o m a t i c a l l y . It seems t o b e t h e s i m p l e s t c h o i c e i f w e a s k f o r a t h e o r y w i t h i n v a r i a n c e under general coordinate transformations.

5.

Quantization The f i r s t t h i n g w e must do i s make a s h i f t

and c o n s i d e r A

uv

a s t h e quantum f i e l d s . Here t h e problem mentioned i n

t h e i n t r o d u c t i o n p r e s e n t s i t s e l f : what i f w e s t a r t w i t h

The answer i s t h a t as l o n g as w e take a s o u r e l e m e n t a r y f i e l d any l o c a l f u n c t i o n of t h e g U v , t h e o b t a i n e d p h y s i c a l a m p l i t u d e s w i l l b e t h e same. The t r a n s f o r m a t i o n from one f u n c t i o n ' ( f o r example, g,,,,) t o t h e o t h e r ( f o r example, g p v ) w i l l b e accompanied by a J a c o b i a n , o r c l o s e d l o o p s of f i c t i t i o u s p a r t i c l e s ( s e e t h e Z i n n - J u s t i n l e c t u r e s on gauge t h e o r i e s ) ,

590

But the p r o p a g a t o r s of t h e s e p a r t i c l e s are c o n s t a n t s or pure polynomials i n k, because t h e t r a n s f o r m a t i o n i s l o c a l . I f we now t u r n on t h e d i mensional r e g u l a r i z a t i o n procedure, which h a s t o b e used i n o r d e r t o get gauge i n v a r i a n t r e s u l t s , t h e n t h e i n t e g r a l s o v e r polynomials,

vanish7). T h i s i s t h e r e a s o n why i t makes no d i f f e r e n c e whether we s t a r t a r e n o t allowed. from Eq. ( 5 . 1 ) o r Eq. ( 5 . 2 ) . Non-local f u n c t i o n s o f g IJV These non-local t r a n s f o r m a t i o n s would g i v e rise t o f i c t i t i o u s p a r t i c l e s that do c o n t r i b u t e i n t h e c u t t i n g rules'), and t h e y are outlawed once u n i t a r i t y has been e s t a b l i s h e d for t h e choice ( 5 . 1 ) o r ( 5 . 2 ) . By choosing a convenient gauge, comparable w i t h t h e Coulomb gauge i n QED, it is indeed n o t d i f f i c u l t t o e s t a b l i s h t h a t t h e t h e o r y i s u n i t a r y , i n a H i l b e r t s p a c e w i t h massless p a r t i c l e s w i t h h e l i c i t y 2 2 . We can work out t h e b i l i n e a r p a r t o f t h e Lagrangian i n E q . ( 4 . 1 ) i n terms of t h e f i e l d s A *

& =-

,,V*

$a

) 2 6 $1a I J A a a ) 2

A ~r aB

+ 1 L2 + h i g h e r o r d e r s , 2 I J

(5.3)

with

L,,

= -21

IJa aa ~

-

aaApa *

(5.4)

For p r a c t i c a l c a l c u l a t i o n s i t seems t o be convenient t o choose the gauge (5.5) The Lagrangian i n t h i s g a u g e , x + & C , has as a k i n e t i c term

where W i s a m a t r i x b u i l t from 6-functions. The propagator i e t h e n

(5.7) Just i n o r d e r t o show the d i v e r g e n t c h a r a c t e r of t h e c o m p l i c a t i o n s involved, we show h e r e t h e Faddeev-Popov ghost') f o r t h i s gauge, o b t a i n e d by s u b j e c t i n g L,, t o an i n f i n i t e s i m a l gauge t r a n s f o r m a t i o n :

591

The L a g r a n g i a n (4.1), expanded i n powers of A w i t h t h e gaugePV ' f i x i n g term ( 5 . 5 ) and t h e g h o s t term ( 5 . 8 ) , form a p e r f e c t quantum t h e o r y . It i s , however, more c o m p l i c a t e d t h a n n e c e s s a r y , b e c a u s e gauge i n v a r i a n c e i s g i v e n up r i g h t i n t h e b e g i n n i n g by a d d i n g t h e bad terms ( 5 . 5 ) and ( 5 . 8 ) . The background f i e l d method, d i s c u s s e d i n t h e n e x t s e c t i o n s , i s much more e l e g a n t b e c a u s e gauge i n v a r i a n c e i s e x p l o i t e d i n a l l s t a g e s of t h e c a l c u l a t i o n s , t h u s s i m p l i f y i n g t h i n g s a l o t .

6.

A P r e l u d e f o r t h e Background F i e l d Technique:

Qauge I n v a r i a n t Source I n s e r t i o n s The methods d e s c r i b e d i n t h i s and t h e f o l l o w i n g s e c t i o n s are not o n l y s u i t a b l e f o r quantum g r a v i t y , b u t have a v e r y wide a p p l i c a b i l i t y , i n p a r t i c u l a r i n gauge t h e o r i e s , f o r i n s t a n c e f o r c a l c u l a t i o n s of r e n o r m a l i z a t i o n group c o e f f i c i e n t s . F i r s t , i t i s c o n v e n i e n t t o i n t r o d u c e t h e concept of a gauge i n v a r i a n t s o u r c e i n s e r t i o n . T h i s i s a n a r t i f i c i a l term i n t h e L a g r a n g i a n of t h e form

where J(x) i s a c-number s o u r c e f u n c t i o n and R ( x ) i s some gauge i n v a r i a n t c o m b i q a t i o n of f i e l d s , c o n t a i n i n g a l i n e a r p a r t and q u a d r a t i c o r h i g h e r o r d e r c o r r e c t i o n s . L e t us g i v e some examples:

i ) I n a gauge t h e o r y w i t h Higgs mechanism: (6.1)

where

i s t h e Higgs f i e l d and D 4 i s i t s c o v a r i a n t d e r i v a t i v e . We take U

apJp = 0. If

< + > = F ;

O

I F + @ ,

592

then (6.1) becomes

The first t e r m emits o r absorbs s i n g l e n e u t r a l v e c t o r p a r t i c l e s . The

other terms are h i g h e r o r d e r c o r r e c t i o n s , e m i t t i n g two or t h r e e p a r t i c l e s a t once. I n g e n e r a l one can f i n d f o r a l l p h y s i c a l p a r t i c l e s a similar gauge-invariant s o u r c e t h a t p r o d u c e s them p r e d o m i n a n t l y and one by one.

ii) I n a p u r e gauge t h e o r y ( i n momentum r e p r e s e n t a t i o n ) : J:(k)

.

I n t e g r a t i o n o v e r k, p a n d f o r q i s u n d e r s t o o d . The where k u J E ( k = 0 h i g h e r terms are d e t e r m i n e d by r e q u i r i n g gauge i n v a r i a n c e u n d e r

This i m p l i e s

A source i n s e r t i o n t h a t s a t i s f i e s t h e s e c o n d i t i o n s c a n b e w r i t t e n i n

a closed form, i n terms of a n a n t i s y m m e t r i c t e n s o r s o u r c e J E v ( x ) :

J;,,(X)

T [exp

g A A ( x ' ) dx'A]ab

Guv(x) b

,

(6.5)

path where Ga = a Aa uv U

-

a

A'

V P

+

g f a b c A,b, A,,C

'A:

=

, and

'abcAbX

AA s t a n d s f o r t h e m a t r i x

'

The i n t e g r a l i s a l o n g a p a t h from i n f i n i t y t o x. The symbol T s t a n d s for time o r d e r i n g a l o n g t h e p a t h . Expanding ( 6 . 5 ) g i v e s X

Jauv(X)

@Ev(X)

+ g fabc

J

Ab,(xI) dx' GEv(x) +

path

iii) In g r a v i t y one c a n do a similar t h i n g : a s o u r c e J""(.x) s a t i s f y i n g

...)

(6.6)

c a n b e coupled t o A

uv

i n e q . (5.1),

and one c a n add h i g h e r o r d e r c o r -

r e c t i o n s t o r e s t o r e gauge i n v a r i a n c e . The d e t a i l s are n o t v e r y r e l e v a n t f o r what f o l l o w s , b u t n o t e t h a t w e r e s t r i c t o u r s e l v e s t o t h e perturba-

tive theory. The a m p l i t u d e w i t h which a gauge i n v a r i a n t s o u r c e emits a s i n g l e p a r t i c l e , o b t a i n s h i g h e r o r d e r c o r r e c t i o n s , see F i g .

+

+

etc.

I.

- t(-)

Fig. 1 A s o u r c e can emit a s i n g l e p a r t i c l e i n d i f f e r e n t ways

The S - m a t r i x can now r e a d i l y be o b t a i n e d i n a gauge-independent way by c o n s i d e r i n g vacuum-vacuum t r a n s i t i o n s i n t h e p r e s e n c e o f a g a u g e - i n v a r i a n t s o u r c e . Then t h e e x t e r n a l l e g s are amputated and put on m a s s - s h e l l . From F i g . 1 i t w i l l b e c l e a r t h a t i n p r a c t i c e one can j u s t as w e l l c a l c u l a t e t h e amputated G r e e n ' s f u n c t i o n s d i r e c t l y and m u l t i p l y them w i t h some r e n o r m a l i z a t i o n f a c t o r 2 . T h i s f a c t o r 2 however may be gauge-dependent. The c o r r e c t f a c t o r can b e o b t a i n e d by n o r m a l i z i n g t h e imaginary p a r t o f t h e two-point f u n c t i o n 8 )

.

7.

The Background F i e l d Method

The background f i e l d method, u s e f u l f o r gauge t h e o r i e s , i s p r a c t i c a l l y i n d i s p e n s i b l e f o r quantum g r a v i t y . L e t t h e f u l l L a g r a n g i a n be

&(A)

ainV(A)

I

t J R(A)

-

t C2(A)

+.tF*-' ,

(7.1)

where &"'(A) i s t h e complete gauge i n v a r i a n t L a g r a n g i a n f o r a l l f i e l d s Ai. A s d e s c r i b e d i n t h e p r e v i o u s s e ' c t i o n , R ( A ) i s a gauge i n v a r i a n t f i e l d c o m b i n a t i o n , and J i s a c-number s o u r c e f u n c t i o n . C ( A ) i s t h e d e s c r i b e s t h e a s s o c i a t e d Feynman-deWittg a u g e - f i x i n g f u n c t i o n and Faddeev-Popov ghost' > 2 A l l i r r e l e v a n t i n d i c e s have been s u p p r e s s e d . The gauge t r a n s f o r m a t i o n law w i l l be w r i t t e n as

z'-'

594

A:

=

Ai

+ j:s

Aa A j

+ t:

(7.2)

Aa

where Aa(x) i s t h e i n f i n i t e s i m a l g e n e r a t o r . t and s are c o e f f i c i e n t s b u i l t from numbers and t h e s p a c e - t i m e d e r i v a t i v e a lJ*

We now i n t r o d u c e t h e n o t i o n of a c l a s s i c a l f i e l d A'', which is a f u n c t i o n of t h e c-number s o u r c e s J . U s ~ a l l y ~ one * ~ )d e f i n e s i t s J dependence by r e q u i r i n g t h a t Acl s a t i s f i e s t h e c l a s s i c a l e q u a t i o n s of motion. This i s s u f f i c i e n t as l o n g as we are i n t e r e s t e d i n diagrams w i t h a t most one c l o s e d l o o p . I n these n o t e s we s h a l l make t h a t r e s t r i c t i o n a l s o , b u t i t must be k e p t i n mind t h a t f o r a p p l i c a t i o n s of t h e s e t e c h n i q u e s a t s t i l l h i g h e r o r d e r s i t w i l l be more c o n v e n i e n t t o add quantum c o r r e c t i o n s t o t h e e q u a t i o n of motion. A t t h i s stage we r e q u i r e Acl a l s o t o be i n t h e gauge C(AC1)

.

= 0

(7.3)

Next, we p e r f o r m a s h i f t : (7.4)

A = A C 1 + O ,

where now 4 i s t h e new quantum f i e l d , and we r e w r i t e t h e L a g r a n g i a n i n terms of 0 :

Here zl i s l i n e a r i n 4; order i n

+.

Now, s i n c e Acl

z2i s

quadratic i n

and U(g3) i s of h i g h e r

s a t i s f i e s t h e e q u a t i o n of motion

&a

= o

6AC1 and C(AC1)=

+

0, a l l terms l i n e a r i n

+

(7.6)

cancel: (7.7)

595

So t h e s o u r c e J i s n o t coupled t o terms l i n e a r i n @, and t h e r e are no v e r t i c e s w i t h o n l y one @ - l i n e . T h e r e f o r e , t h e @ - l i n e s c a n only go around i n l o o p s , and i f w e c o n f i n e o u r s e l v e s t o one-loop diagrams t h e n we can n e g l e c t t h e terms Cf(@'). One l o o p diagrams now o n l y consist of a +-loop, w i t h b i l i n e a r i n s e r t i o n s o f c l a s s i c a l s o u r c e s depending on

and J. But remember t h a t Acl i s a f u n c t i o n of J , which can b e obt a i n e d by s o l v i n g t h e c l a s s i c a l e q u a t i o n s o f motion by i t e r a t i o n . This i t e r a t i o n p r o c e s s c o r r e s p o n d s t o a d d i n g a l l p o s s i b l e trees t o t h e s i n g l e l o o p . Thus we r e p r o d u c e t h e o r i g i n a l Feynman r u l e s , w i t h t h e o n l y change t h a t l o o p l i n e s are c a l l e d @ - l i n e s , and t r e e l i n e s are c a l l e d Acl-lines. Acl

8.

The BackRround F i e l d Gauge The r e l e v a n t p a r t o f t h e Lagrangian

( 7 . 5 ) is

T h i s is j u s t a n o r d i n a r y gauge f i e l d L a g r a n g i a n w h e r e Z j n v =

5

t

i s i n v a r i a n t u n d e r what w e s h a l l c a l l gauge t r a n s f o r m a t i o n s of t y p e 8 :

(8.21

The gauge i s f i x e d by t h e f u n c t i o n C ( A

c l t $I).

xinv

Now t h e r e i s a n o t h e r i n v a r i a n c e of , a l s o b r o k e n by t h i s C term. We c a l l t h i s a gauge t r a n s f o r m a t i o n o f t y p e C:

The power o f t h e p r e s e n t f o r m u l a t i o n i s t h a t we can go o v e r t o a d i f f e r e n t gauge f u n c t i o n C ( A c l , @ ) which b r e a k s t h e Q-gauge invariance ( a s i s n e c e s s a r y ) b u t p r e s e r v e s gauge i n v a r i a n c e o f t y p e C . F o r example: i f @ai s t h e quantum p a r t o f t h e v e c t o r f i e l d A a , t h e n t h e c h o i c e lJ

?J

Ca

=

Due@,"

,

(8.4)

where D,, i s t h e c o v a r i a n t d e r i v a t i v e i n t h e c l a s s i c a l ( C ) s e n c e :

c l e a r l y p r e s e r v e s C-invariance. Case of g r a v i t y . I f

Such a gauge i s a l s o p o s s i b l e i n t h e

we can take (8.5)

where

is t h e covariant derivative i n the c l a s s i c a l sence. Ir The one-loop ( i r r e d u c i b l e ) v e r t e x f u n c t i o n s o b t a i n e d i n t h i s gauge w i l l be C - i n v a r i a n t a l s o . It f o l l o w s t h a t t h e y s a t i s f y n o t only the Slavnov i d e n t i t i e s t h a t d e s c r i b e t h e Q gauge symmetry, b u t i n a d di t i on t h e much s i m p l e r Ward i d e n t i t i e s which are t h e d i r e c t ge ne ra l i z a t i o n s of t h o s e i n quantum e l e c t r o d y n a m i c s . Again, D

O f c o u r s e , t h e new Feynman r u l e s i n t h i s background f i e l d gauge are independent of o u r o r i g i n a l c h o i c e of t h e gauge C ( A ) i n e q . ( 7 . 1 ) . T h i s i m p l i e s t h a t w e can now a l s o drop t h e gauge c o n d i t i o n ( 7 . 3 ) f o r the c l a s s i c a l f i e l d s s i n c e i t can be r e p l a c e d by a n o t h e r , a r b i t r a r y , gauge c o n d i t i o n .

9.

A Simple Example o f t h e BackRround F i e l d Gauge:

P u r e Yang-Mills F i e l d s Although we are mainly i n t e r e s t e d i n quantum g r a v i t y , i t is much more i n s t r u c t i v e t o i l l u s t r a t e o u r methods i n s i m p l e r f i e l d t h e o r i e s . Let us c o n s i d e r p u r e Yang-Mills f i e l d s . The i n v a r i a n t L a gra ngia n i s

--

niv.,. with

G,"~ =

a,Af

Ga

,,v Ga, , v '

-ri

- aY

P

+

597

g fabcA:A;

(9.1)

.

The gauge i n v a r i a n c e i s

=

A':

a

A lJ t g f a b c A

b c

-

A~

.

a,,Aa

(9.2)

W e l e a v e aside t h e gauge i n v a r i a n c e s o u r c e s ( s e c t . 6 ) . We s h i f t

(9.3)

=

(where D

a

g f Acl)

t

Using (9.4) we get

(9.51 t

t o t a l derivative.

A c o n v e n i e n t background f i e l d gauge i s

zc = -

3 C2

=

-

71

a 2 (DIJQ,)

.

(9.6)

The g h o s t L a g r a n g i a n i s t h e n

.

up t o i r r e l e v a n t i n t e r a c t i o n s w i t h Q a lJ O f c o u r s e t h e g h o s t i s also C - i n v a r i a n t .

Note t h a t C - i n v a r i a n c e permits u s t o write t h e i n t e r a c t i o n s o f @ a and qa i n a v e r y condensed way. It is lJ t h i s f e a t u r e t h a t p r e v e n t s o v e r p o p u l a t i o n o f i n d i c e s i n t h e c a s e of gravity. The one-loop i n f i n i t i e s can be s u b t r a c t e d by a C - i n v a r i a n t term i n the Lagrangian. The o n l y c a n d i d a t e i s

598

counter

The index a s i m u l t a n e o u s l y g o v e r n s t h e i n f i n i t i e s o f t h e two, t h r e e and f o u r p o i n t v e r t i c e s , and i s t h e r e f o r e d i r e c t l y p r o p o r t i o n a l t o t h e Callan Symanzik & - f u n c t i o n 9 ) . To f i n d t h i s & - f u n c t i o n one t h e r e f o r e only needs t o i n v e s t i g a t e t h e two p o i n t f u n c t i o n , c o n t r a r y t o t h e conventional f o r m u l a t i o n where a l s o t h r e e p o i n t f u n c t i o n s had t o b e c a l c u l a t e d l o ) i n o r d e r t o e l i m i n a t e t h e Callan-Symanzik y- f u n c t i o n .

-

10.

A

Master Formula f o r a l l One-Loop I n f i n i t i e s

From t h e p r e c e e d i n g i t w i l l b e c l e a r t h a t any one-loop amplitude* can be o b t a i n e d from a L a g r a n g i a n b i l i n e a r i n a s e t o f f i e l d s 4:.I One can t h e n r e a r r a n g e t h e c o e f f i c i e n t s i n s u c h a way t h a t

.Z=Q

r-

where N , g and

+ Ntj+j)gpV(aV$i +

-$(a,,$i

x are

N S ~ ~ pi J Xijbj1 +

1

a r b i t r a r y f u n c t i o n s o f s p a c e time x

,

(10.11

. Further,

In d e a l i n g w i t h g r a v i t y 3 ) we found i t v e r y c o n v e n i e n t f i r s t t o c a l c u l a t e the i n f i n i t i e s of t h i s g e n e r a l L a g r a n g i a n , i n terms o f N , g and X O f c o u r s e , t h e background m e t r i c i s a l l o w e d t o b e c u r v e d . A f t e r w a r d s one may s u b s t i t u t e t h e d e t a i l s s u c h as: t h e way N and X depend on t h e background f i e l d s ; and t h e f a c t t h a t t h e i n d i c e s i , j a c t u a l l y s t a n d f o r p a i r s o f L o r e n t z - i n d i c e s . I n gauge t h e o r i e s t h e o b j e c t s N are m o s t l y background gauge f i e l d s and i n g r a v i t y t h e N c o n t a i n t h e C h r i s t o f f e l symbols r.

.

I n r e f . 3 ) , we u s e d d i m e n s i o n a l r e g u l a r i z a t i o n and c o n s i d e r e d the Pole8 a t n .* 4 ( n i s t h e number o f s p a c e - t i m e d i m e n s i o n s ) . From power counting arguments one e a s i l y deduces t h a t t h e r e s i d u e s o f t h e s e p o l e s can c o n s i s t o n l y of a l i m i t e d number of terms. T h i s number i s even more r e s t r i c t e d i f w e use t h e observation t h a t , whatever N , g o r X are, there i a a C-gauge symmetry: dp i s i n v a r i a n t u n d e r f

provided a Feynman-like'background

f i e l d gauge i s chosen.

599

N;(x)

=

X'(X)

N,,

-

X

+

a UA + A N AX

-

,

(10.2)

NUA

XA

where A i s a n i n f i n i t e s i m a l , a n t i s y m m e t r i c m a t r i x . A s t r a i g h t f o r w a r d c a l c u l a t i o n y i e l d s , t h a t a l l one-loop as n

+

infinities

4 a r e a b s o r b e d by t h e c o u n t e r - L a g r a n g i a n 3 )

A Z =

+ 1 x * - L

1

Y'UY,,v i 8 n 2 (n-4) % T r { $ t- i

120

R ,,U R v v I

+

'4

1 2 Rx

& R 2 1 }

where Yvv

=

aPNw

-

aVNu

+

NPNv

-

NUN,,

(10.4)

and = number o f f i e l d s . and R a r e t h e Riemann c u r v a t u r e t e n s o r s f o r t h e backUW ground m e t r i c . R a i s i n g and l o w e r i n g i n d i c e s by u s e o f t h e background gpU i s understood.

Tr I R

The f o r m u l a (10.3) can a l s o b e u s e d t o c a l c u l a t e master f o r m u l a i n t h e c a s e o f Fermions 9)

.

11.

a similar

S u b s t i t u t i n g E q u a t i o n s o f Motion.

I n some c a s e s t h e r e s u l t i n g f o r m u l a (10.3) i s n o t t h e end o f t h e story. One may s t i l l make u s e o f t h e i n f o r m a t i o n t h a t t h e background f i e l d s are n o t a r b i t r a r y , b u t s a t i s f y e q u a t i o n s of motion. I f , f o r i n s t a n c e , one f i n d s a c o n t r i b u t i o n i n A& p r o p o r t i o n a l t o

and t h e e q u a t i o n o f motion i s

D2 Acl- =

t h e n one may r e p l a c e (10.1) by

V(AC1)

,

(11.21

- Acl V(AC1) .

(11.3)

Addition of infinitesimal terms in the Lagrangian that vanish if the equation of motion is fulfilled, corresponds exactly to making an infinitesimal field renormalization: if the equation of motion is

then the terms in question must be

and we have

A t = A + EB is a field renormalization. Note that all terms in A& are always infinitesimal, because we neglect everything that comes from two-loop diagrams.

12.

Some Numerical Results.

Conclusions.

The master formula (10.3) has been applied to calculate the infinity structure in different cases. For pure gravitation we used 3 ) the gauge (8.57 and found from the gravitons

where

1E. =

1/8n2(n-4)

,

and from the ghosts

Here g,, and R,,,, are the classical metric and curvature .(At this level it is never neceseary t o consider quantum fields in the counter terms. We are a l s o not interested in the renormalization of the source terms). From power counting one would expect a third possible term of the form

601

RaB~6

RaBy6

.

It i n d e e d o c c u r s , b u t we made u s e o f t h e i d e n t i t y

35- (RuByG

R"BY~

-

2 3a11)

4 R ~ ~ R + ~ R 2"

t o t a l derivative

,

(12.4)

which i m p l i e s t h a t terms o f t h e form ( 1 2 . 3 ) c a n b e e l i m i n a t e d . So a l l t o g e t h e r we h av e

However, we s t i l l h av e t h e e q u a t i o n s o f motion o f t h e background f i e l d (see p r e v i o u s s e c t i o n ) , which is Rllv

0

;

R = O

(12.6)

T h i s means t h a t t h e i n f i n i t y v a n i s h e s i n p u r e g r a v i t y . It is i n t e r e s t i n g t o observe that the f i e l d renormalization t h a t corresponds t o t h e e l i m i n a t i o n o f t h i s i n f i n i t y ( s e e previous s e c t i o n ) i s o f an unusual type :

T h i s was t h e r e a s o n f o r o u r remark i n t h e end o f s e c t . 3 , Note t h a t e q . ( 1 2 . 4 ) was e s s e n t i a l f o r t h i s r e s u l t . Next w e s t u d i e d p u r e g r a v i t y w i t h a ( m a s s l e s s ) Klein-Gordon field $. The c l a s s i c a l e q u a t i o n s of motion are

(12.8)

T h e r e i s one t y p e of c o u n t e r t e r m which i s a l l o w e d by power c o u n t i n g and c a n n o t b e e l i m i n a t e d by s u b s t i t u t i o n o f t h e e q u a t i o n s o f motion:

The n e x t t h i n g t h a t h a s been t r i e d i s p u r e g r a v i t y w i t h i n a d d i t i o n

602

Maxwell f i e l d s 12 1 The L a g r a n g i a n i s

FpV =

.

a,,Av

-

.

avA,,

( 12.10)

The e q u a t i o n s o f mo t i o n are

(12.11)

From which

R = - "Ta 2 a

= 0

.

In p r i n c i p l e one may e x p e c t

(12.12) E x p l i c i t c a l c u l a t i o n shows however:

So we see t h a t t h e r e are some c a n c e l l a t i o n s : a2=a3

(12.14)

= O .

Indeed, t h e y o c c u r i n a m i r a c u l o u s way duri.ig t h e c a l c u l a t i o n a nd ha ve not yet been ex p l ai n ed . I n t h e c o u p l e d E i n s t e i n - Y a n g - M i l l s s y s t e m 13) t h e s e c a n c e l l a t i o n s p e r s i s t and many new c a n c e l l a t i o n s o c c u r . S t a r t i n g from t h e o b v i o u s g e n e r a l i z a t i o n o f t h e A b e l i a n L a g r a n g i a n ( 1 2 . 1 0 ) , one f i n d s A&=

l ~ { [ #+

R,,~R""

+ f 2 c2

603

$

I

( 12.15)

where f i s t h e gauge c o u p l i n g c o n s t a n t ,

(12.16) F i v e o t h e r c o e f f i c i e n t s e a c h happen t o v a n i s h . The s e c o n d term i n ( 1 2 . 1 5 ) i s o f t h e r e n o r m a l i z a b l e t y p e . It r e n o r m a l i z e s t h e gauge c o u p l i n g c o n s t a n t and f i x e s t h e Callan-Symanzik 6 - f u n c t i o n . F i n a l l y , a l s o t h e combined E i n s t e i n - D i r a c s y s t e m h a s b e e n i n v e s t i g a t e d and n o n r e n o r m a l i z a b l e i n f i n i t i e s h a v e been f o u n d a l s o 14)

.

The f a c t t h a t i n a l l t h e s e s y s t e m s where matter i n some form i s added t o p u r e g r a v i t y i n f i n i t i e s o f t h e n o n r e n o r m a l i z a b l e dimension s u r v i v e r e a l l y means t h a t t h e s e t h e o r i e s c a n n o t be r e n o r m a l i z e d i n the p e r t u r b a t i o n e x p a n s i o n . I n t h e c a s e o f p u r e g r a v i t y t h e i n f i n i t i e s have b e e n shown t o b e n o n - p h y s i c a l up t o t h e o n e - l o o p l e v e l . No c a l c u l a t i o n s h a v e been p e r f o r m e d t o i n v e s t i g a t e r e n o r m a l i z a b i l i t y i n t h e o r d e r o f two l o o p s . The c a l c u l a t i o n s o f Nieuwenhuizen a n d Deser show t h a t Perhaps t h i s i s a n i n d i c a t i o n flmiraculous’l c a n c e l l a t i o n s o f t e n o c c u r o f a new s o r t of symmetry t h a t w e are n o t aware o f . I n v e s t i g a t i o n o f t h i s symmetry c o u l d r e v e a l new r e n o r m a l i z a b l e models w i t h g r a v i t y .

.

Even so, a r e n o r m a l i z e d p e r t u r b a t i o n e x p a n s i o n would o n l y b e a small s t e p f o r w a r d . A t v e r y small d i s t a n c e s t h e g r a v i t a t i o n a l e f f e c t s must b e l a r g e , b e c a u s e o f t h e d i m e n s i o n of t h e g r a v i t a t i o n a l c o n s t a n t , 8 0 t h e e x p a n s i o n would b r e a k down a t small d i s t a n c e s anyhow. W e have t h e i m p r e s s i o n t h a t n o t o n l y a b e t t e r m a t h e m a t i c a l a n a l y s i s i s needed, b u t a l s o new p h y s i c s . What w e l e a r n e d ( s e e e q . ( 1 2 . 7 ) and t h e remarks i n t h e e n d o f s e c t . 3 ) i s t h a t h s u c h a t h e o r y t h e m e t r i c t e n s o r might n o t a t a l l b e s u c h a f u n d a m e n t a l c o n c e p t . I n any c a s e , i t s d e f i n i t i o n i a n o t unambiguous. REFERENCES

1.

R. P. Feynman, A c t a Phys. Polon. 2,697 ( 1 9 6 3 ) S. Mandelstam, Phys. Rev. 1580, 1604 ( 1 9 6 8 ) E. S. F r a d k i n and I. V. T y u t i n , Phys. Rev. D2, 2841 (1970) L. D. Faddeev and V. N. Popov,, Phys. L e t t e r s 258, 29 ( 1 9 6 7 )

a,

2.

3. 4.

5.

6. 1. 0.

9. 10. 11.

12.

B. S. D e W i t t , & R e l a t i v i t y , Groups and Topology, Summerschool o f Theor. P h y s i c s , Lea Houches, France, 1963 (Gordon and Breach, New York, London) B. S. D e W i t t , Phys. Rev. 162, 1195, 1239 (1967) G . * t Hooft a n d M. V e l t m a n , CERN p r e p r i n t TH 1723 (1973), t o be publ. i n Annales de 1 ' I n s t i t u t H e n r i P o i n c a r 6 D. C h r i s t o d o u l o u , p r o c e e d i n g s o f t h e Academy o f A t h e n s . 2 (20 J a n u a r y 1972) D. C h r i s t o d o u l o u , CERN p r e p r i n t TH 1894 (1974) J. Honerkamp, Nucl. Phys. 848, 269 (1972); J. Honerkamp, P r o c . Marseille Conf.,l9-23 June 1972, CERN p r e p r i n t TH 1558 G. ' t Hooft, Nucl. Phys. U, 167 (1971) G. ' t Hooft and M. Veltman, Nucl. Phys. 189 (1972) G. I t Hooft and M. Veltman, DIAGRAMMAR, CERN r e p o r t 73-9 (1973) 0. ' t Hooft, Nucl. Phys. 455 (1973) 0. I t Hooft, Nucl. Phys. &, 444 (1973) H. D. P o l i t z e r , Phys. Rev. Letters 1346 (1973) D. J . Gross and F. Wilczek, Phys. Rev. L e t t e r s E, 1343 (1973) R. Bach, Math. Z. 2, 110 (1921) C. Lanczos, Ann. Math. 2,842 (1938) S. Dcaer and P. van Nieuwenhuizen, P h y s . R e v . L e t t e r s , z , 2 4 5 ( 1974);

E,

m,

z,

Phys. Rev.

G,401

(1974)

13.

S. Deser, Hung-Sheng Tsao and P. van Nieuwenhuizen,

14.

3337 (1974) Phys. Rev. S. Deser and P. Van Nieuwenhuizen, Phys. Rev.

z,

605

U , 411

(1974)

Class. Quantum Grav. 10 (1993) S79491.

CHAPTER 8.2

Classical N-particle cosmology in 2 + 1 dimensions G ’t Hooft Institute for Theoretical Physics, University of Utrecht, PO Box 80 006, NL 3508 TA Utrecht, The Netherlands

+ 1 dimensional cosmology particlea arc topological defects in a universe that is nearly everywhere flat. We use a time dependent triangulation procedure of space to formulate the laws of evolution and to construct phase space of this system. A theorem is derived from which it follaws that many configurations may have a big bang in their past or a big crunch in their future. The dimensionality of phase space for N particles is 4N - 11 129, where g is the genus of 2-space. This suggests to consider ZN 6 69 pairs of canonical variables and one time variable. The quantum version of this model is speculated about. AbshcL In 2

+

- +

1. Introduction

In 2

+ 1 dimensional General Relativity [1,2] Einstein’s equations for the vacuum,

imply also that the entire Riemann tensor R;,,, vanishes. Hence the vacuum is flat. A particle at rest, at the position z = a, produces a curvature proportional to 6*(2 - a). From that the global structure of space-time surrounding the particle can easily be seen to be a cone, and it is described by excising a wedge out of the plane (see figure l), after which one identifies points at each side of the wedge (for instance the two arrows in figure 1 are identified). The angle of the wedge, the conical deficiency angle 0, is proportional to the mass M of the particle:

where G is Newton’s constant, which we will choose to be one from now on.

Purticle P

a)

b)

Figure 1. (U) Excised wedge near a particle P (shaded). (b) Diagrammatic notation.

0

1993 IOP Publishing Lld

606

If we wish to use Cartesian coordinates to describe locations in the neighbourhood of this particle we have to attach strings to each of these locations, and then the coordinates depend on how a string attached as indicated in figure 1 can also be described by the coordinates of 2' with string as drawn, provided that 2'

= a+

Q(2 - a )

(1.3)

where a is the location of P and 52 is the rotation

?b describe a moving particle, one has to Lorentz transform this space-time. The relationship between the two coordinate frames then becomes

(;J = L{ ( R0 O ("7) (:)p (bbu) 1)

=A(;)

+

+

where L is the h r e n Q transformation that gives the particle the velocity w from its rest frame. A is a new Lorentz transformation and b and bo are some shift vectors. Note that now the particle, its mass and its velocity, are determined by just giving an element of the Poinard group. However not all elements of the Poincarf? group specify a particle because the particle's space-time trajectory is given by the points satisfying (Z',t')

= (2,t)

(1.6)

an equation that for generic elements of the PoincarC group has no solutions, as we will see. Consider now WO particles, moving with respect to each other. Following a path around both we find that points I and I' are identified by the product of two transformations of the type (1.5) (see figure 2).

F i p n 3. Tko moving panicles, P and Q.

The lines in figure 2 indicate the separation between the different coordinate frames used. The relation between the different coordinate frames for the point 2, indicated as z and x', is obtained by the product of the transformations (1.5) for the 607

Classical N-particle cosmology in 2

+ 1 dimensions

two particles P and Q. Let us now consider the center of m m pame. This is the coordinate frame that is chosen such that

In general one can indeed find a Lorenk frame such that A,, has only nontrivial elements in the first 2x2 block, as indicated, so that it is a pure rotation. It is natural to define the corresponding rotation angle to be the center of mass energy. Also one can find a vector a such that it acts as the origin of this rotation so that it can be interpreted as the location of the particle. But the new thing is the additional rime shifr 61 as we encircle the two particles. It is not hard to veriQ that 61 can be identified as being the total angular momentum [2], The angles over which one rotates are additive, and so are the time shifts 6t. Thus one recovers the consenation laws of energy and angular momentum. A complication is now that one can no longer define the trajectory of the Center of mass as being the set of points invariant under (1.7). If the time shift is non-zero there are no invariant points. One has to go to the center of mass frame to locate the center of mass. It is not difficult to derive the center of mass energy. Since the trace of a Lorentz matrix is Lorentz invariant one finds the cosine of the deficiency angle for the center of mass by taking the trace of the product of the Lorentz matrices corresponding to the two individual particles. The outcome of that (simple) calculation is [2]: COS A m c m

= COS

Aml

cos mn2- sin nml sin 7rm2 cosh y

(1.8)

where m1,2are the masses of the particles 1 and 2; mcmis the Center of mass energy; y is the Lorentz boost parameter connecting the rest frame of 1 with that of 2. Note that in this, expression we have the trigonometric functions of half the deficiency angles in stead of the angles themselves. This is because it is more convenient to work with the SL(2, R) representation of the Lorentz group than it is with the S O ( 2 , l ) representation. We see that there only is a center of mass if the condition cosh7

cos7rn2 + 1 < cosnm, sin7rm1 sin7rm2

is satisfied. If this condition is nof met P and Q are said to be a Gott pair [3]. The identification of the Cartesian coordinate frames connected by a loop around the particles is a Lorentz boost rather than a rotation. Such a pair can therefore be surrounded by a closed timelike curve (CTC). If no other particles occur it cannot be avoided that the universe is surrounded by an unphysical boundary condition [4, 51, even though CTC will nof occur if we go sufficiently far to the past or to the future 161. An open universe has an acceptable boundary if the total energy (corresponding to the sum of all deficiency angles in one coordinate frame divided by 27r) is less than 1. A universe closes if its energy is exactly 2. It then has the topology of a sphere (S2, g = 0). But a torus ( g = 1) or even 2-dimensional spaces with higher genus are also possible. The total energy vanishes for g = 1 and is negative in higher genus spaces. This is not in contradiction with positivity of the rest masses of the

608

particles because when they move there can be negative energy in the surrounding gravitational fields as we shall see. S M Carroll et 41 [5] proposed to consider a universe where the following happens. At t = 0 both Ml and M2 decay simultaneously each into a pair of light particles with masses:

Ml -+ ml m2

M2

-

(1.10)

m3 m,.

One easily finds that the two particles ml and m2 created by the decay of M,move away from each other at some angle A( 1- Ml) (see figure 3). With respect to each other they can never violate condition (1.9) because their Center of mass energy is Ml. But ml and m3 can meet each other head-on, and thus their relative boost parameter y can exceed the value (1.9). Thus they may form a Gott pair. It turns out that this may only happen if tg2j?rM1 > tg2nm

+1

(1.11)

so that we must have f < M, < 1. This implies that the universe must be closed. So we add an ‘antipode particle’ X with mass M , = 2 - 2Ml. The problem raised

by these authors was the question whether closed timelike curves can then also arise

in this universe. X

M

A

-

Fwurc 3. M I

ml m2.

F@rc 4. Ihc CFG u n h i t y at t = 0.

2. ”iangulatlon and Cauchy surfaces

We now propose an approach to 2+1 dimensional gravity/cosmology that stam with construction of Cauchy cross sections of space-time. These are purely spacelike surfaces that are timesrdered, and together span the entire universe. We make optimal use of the fact that in between the particles space-time is flat. The ideal method is ‘triangulation’, which implies that we consider Cauchy surfaces that are built from entirely flat triangular 2-simplexes, glued together [7]. In practice however it is easier to take polygons rather than triangles, because in the generic case the vertex

Classical N-particle cosmology in 2

-+ 1 dimensions

points will not connect more than three simplexes together. The general picture of such a Cauchy surface is sketched in figure 5, although in practice often the polygons will look somewhat more complex Indeed, ultimately the most economic description will be using just one polygon, with rather elaborate prescriptions as to how the edges are to be sewn together. All particles in our theory must be at vertex points, but vertex points where no particles sit are of course also allowed. In general we will put a particle at a 1-vertex; at such a point the adjacent sides of the corresponding polygon are glued together leaving the appropriate deficiency angle. If all particles are at rest with respect to each other then the situation is easy to visualize. At all vertices we can take the angles to add up to exactly 2n, unless there is a particle present, and then the angles add up to 2n - 27rm. If the total mass exceeds 1, then the Cauchy surface closes, so that there must be further particles at the ‘antipodes’. The total mass then automatically adds up to 2.

F @ n 5. Cauchy surface built from polygons. The heavy dots are particles.

But if the particles move the situation is more complex. The polygons become deformed as a function of time, and in general the boundary between one polygon and another will involve a Lorenk transformation.. It is now very instructive to formulate the rules that the moving polygons have to obey, if we insist that at the seams between all pairs of polygons space-time remains locally flat. We consider in each polygon (which is actually a 3-simplex in space-time) a Cartesian coordinate frame. In general the sides all move in this frame, and their lengths change. Now in principle we could allow that on each polygon time runs at different speeds (as long as time runs forward), but we will impose the simplifying extra requirement that time runs equally fast at each polygon. In that case the evolution is uniquely determined, and furthermore the rules at the seams [7] become very simple: 1) The lengths of matching edges of two adjacent polygons, as measured in the coordinate frame of each, are equal. This is not a completely trivial statement because the matching goes with a Lorentz transformation. 2) The velocities with which the matching edges of two adjacent polygons move (always in a direction orthogonal to the orientation of the edge) in the coordinate frame of each, are the same, but the signs may differ. In general the signs are such that if the edge of one polygon recedes, the matching edge in the other one recedes also, because in the other case the matching becomes trivial. 610

3) The identification of points at two matching edges is such that a point moving in an orthogonal direction on one edge, remains identifled with a point moving orthogonally on the matching edge, as seen in the corresponding frames. 4) The vertices between three polygons move in such way that rules 1, 2 and 3 remain valid. When new vertices are created (one vertex may split into several), special attention should be paid to whether they represent flat space or particles. Often this comes out all right automatically because of energy-momentum conservation at such space-time points. The proof that polygons describing localfy flat space can only move according to the above rules is not difficult. We will not here repeat the arguments already given in 171. It is important to note that when a panicle moves relatively to the coordinate frame chosen for the polygon that it is in, its velocity vector is only allowed to be in the direction of the bisector of the cusp, see figure 6.

b

a

1

C

F#gurc6. (0) Particle at r s t . 'Ibe shaded mgion is excised from Bat space and the two edges are glued together. The two dots are points to be identified. (b) Particle moving downwards ( a m ) . We get the Lorenu contraction of picture (q), therefore the angle widens. Since the two dots in ( U ) are at the same height, they still will be at the same height in (4, and after the Lorentz transformation there will be no relative time shift. (c) Diagrammatic notation for a particle.

<

If is the particle's rapidity, we see that a moving particle is Lorentz contracted by a factor cosh<. Therefore it gets a widened deficiency angle p: tg 40 = tg( lrn)cosh f .

(2.1)

The boost parameter q describing the velocity thq with which the adjacent edges of the polygon widen or retract, is given by thq = sin ifl thf.

(2.2)

Later it will turn out to be useful to eliminate f or p out of these expressions. One then obtains coshq cos40 = coslrm sinh sin lrrn = sinh q .

(2.3~)

<

(2.36)

When three polygons meet at a vertex, see figure 7(a),and if there is no particle present precisely at.this vertex (in the generic case there is none), then we can use the fact that space-time is locally flat here (note that although we talk about three 61 1

Classical N-particle cosmology in 2

+ 1 dimenswns

different polygons, they could equally well be different regions of just one polygon, meeting at this particular vertex). The three coordinate frames can be seen as three points in a hyperbolic space (figure 7(b)). Since the sides of two adjacent polygons recede or widen with the same boost factor 7 in each coordinate frame, the distance between these two frames is the boost given by 27. Hence h hyperbolic space the lengths of the sides of the triangle formed by the frames I, I1 and 111 are 27,. The angles A, in this triangle correspond to the angles a,of the three polygons at the vertex, as follows: A, = A - 0,.

(2.4)

These observations make it easy to understand the relations at each vertex as they were reported in ref. 7: writing

and all cyclic permutations These are nothing but the trigonometric properties of the triangle of figure 7 ( b ) in hyperbolic space.

II

It will be important to observe the ranges of values the angles can take: not more than one of the three angles a, is allowed to exceed A. It is not hard to convince oneself that if there would be two angles ai > A then there would be points in space-time that occur at two different spots on our Cauchy surface, which iS not allowed. Now suppose one only knew the three boost parameters 7, then from (2.9) the three angles can be determined, except for an ambiguity a # 2n - a. But because the relative signs of sina, are fixed by (2.6), and no more than one a is allowed to exceed A, this ambiguity can be resolved completely. So the angles are

.

completely determined by boasts 7 ,

612

GI ' Hoofi

3. Evolution The polygons of the previous section must be seen as fixed time cross sections of 3-simpkxes. At any given time i they together form a Cauchy surface. We now may ask haw such a Cauchy surface evokes. There are several kinds of mutations that can take place at a given moment. First, an edge connecting two polygons (or different regions of a single polygon) may shrink to zero and be replaced by another edge, see figure &a). It is also possible that a particle hits an edge of the polygon it is h. It then crosses over into the next polygon (figure 8(b)). When the edge a particle is on shrinks to zero it hops into the adjacent polygon (figure 8(c). Finally, if one of the angles of a polygon is greater than A then that vertex may hit another edge and cause a 'vacuum cross-over' (see figure 8(d)). In all these cases the angles and boost parameters of the newly opened edges are all fked by the trigonometric equations, in particular (2.9) and (2.10). These !ix all properties of the edge #1, if the boosts q2, q3 and the relative angle a1of the other edges are known. It is easily understood why all these properties are fixed: if the boosts and the angles of the transformations between frames I and 11, frames I1 and IV, frames I1 and 111, and frames 111 and IV are well known, then of course the transformation leading from I to IV (see figure 8(a))is determined by that. And so are all other new edges in figure 8(a)-(d).

a)

4

I

X

I

V

111

b)

4

/'

d)

F h r e 8.

(U)

Exchange; (b) cmpt-over, (c) hop; (d) vacuum cmu-avcr.

In practice one needs no more than one single polygon, with identillcation rules 613

Classical N-particle cosmology in 2

+ 1 dimensions

at its edges. An example of a five-particle universe is pictured in figure 9(a). The identifications among the various edges are indicated by arrows.

Figure 9. ( 0 ) Polygon representing a universe at given time 1. (b) Diagrammatic notation of the same universe.

It is convenient to indicate the identification rules for the edges by drawing a diagram, obtained by drawing the polygon on an S, sphere. One can then glue the edges together. The seams obtained are shown in the diagram (figure 9(b)).

4. Crunch and bang theorem

An important property of 2+1 dimensional universes was indicated in [7]. It can actually be formulated in terms of a theorem:

Theorem. Let at one moment i = 1 , all edges of all polygons describing a closed or open universe be contracting, which means that all Lorentz boosts 77 at all edges are negative. (We will refer to this condition as the Crunch codifion),then this condition will remain satisfied at all times 1 > i,.

Note that the crunch condition implies that at all vacuum vertices (the 3-vertices containing no particle, so that (2.6)-(2.10) apply), the angles ai are convex: ai < K , because due to (2.6) all sinai have the same sign and because of the remark at the end of section 2. Where the particles are, the polygons have angles a = 2~ - p, where p is given by (2.1) or ( 2 . 3 ~ ) .Thus a > K if rn < and a < A if rn > So where we have particles with mass rn < the polygons are not convex. See figure 9(a). Evidently, we will not need figure 8(d). The proof of the theorem can be given by checking all possible mutations, figure 8(a)-(c), one by one. Consider figure 8(a). The angles all start out being convex (< n). If 77 at the newly opened edge were positive all four adjacent angles would have to be > K, which is clearly not allowed. We see right away that that is geometrically impossible. So this transition keeps the Crunch situation as it was; the newly opened edge has 7) < 0 just like the others. Next consider the cross-over, figure 8(b). This transition is only possible for a particle with mass < (otherwise it could only be involved with the hop transition,

4,

4

614

5.

G

‘1

Hoop

figure 8(c)). Again the angles start out being all convex. First look at the lower vertex that appears. If the new q that arises there would not be negative the two adjacent angles would come out larger than A, which is not allowed. We a n then repeat the argument for the q of the edge attached to the particle after it entered the other polygon. One could argue alternatively that the particle as it enters gets an additional boost in the direction of the new polygon it is entering. This way also one finds unambiguously that the q of the particle (see equation (2.2)) must again be negative. Thus at this transition also the Crunch condition remains valid. The case of the hop transition, figure 8(c), is again easy to check if the mass is 3 f. Then again it is the same geometrical argument that tells us that it is impossible for the two new angles at the vertex both to exceed A. If the mass is < 4 this argument does not work. But careful analysis shows the theorem still to hold there. It is again geometry. The particle is entering a polygon at a convex point, where the two edges are contracting. The particle must outrun the edges and hence it must be receding also. It must have a negative q. We herewith checked all possible transitions. The Crunch condition remains valid forever. Of course the theorem can be time-inverted. Reversing the sign of all q coefficients we see that if one has at t = t1 a situation such that all polygons are expanding at all sides, then this expansion must have existed at all times t < t,. This is why we call this theorem t h e ‘Crunch and Bang theorem’. If the Crunch condition is satisfied the universe must end in a Big Crunch. If the converse or ‘Bang condition’ is satisfied there must be a Big Bang in the past. The theorem applies to the CFG universe. At the moment that the two particles M i decay simultaneously into particles m1 all polygons satisfy the Crunch condition. The theorem tells us that the universe will continue to shrink forever. The particles will all approach each other with smaller and smaller impact parameters until they crunch. Before the crunch occurs there cannot possibly be a closed timelike curve simply because we were dealing with Cauchy surfaces all along. The purported CTC only is obtained when analytic extensions of space-time are considered beyond the Big Crunch. But we showed that the crunch singularity is an essential one; all particles meet each other there. Analytic extensions beyond that point are illegal. The question could be asked whether perhaps the occurrence of a crunch is frame-dependent. But since the particles approach each other with ever decreasing impact parameters, and since in practice we found that at each mutation the crunch accelerates, we found that this question must be answered in the negative. There is no timelike path for which the moment of the crunch, in terms of its eigen time, can be postponed. 5. Degrees of freedom

An important question arises in these constructions. We would like to know exactly how to characterize the ‘physical’ degrees of freedom of 2+ 1dimensional cosmologies.

This is particularly of importance if one wishes to ‘quantize’ such models. Several quantization schemes have been proposed. It would be illustrative if a scheme could be devised that starts off with the degrees of freedom of a universe containing a finite number of particles, replacing the Poisson brackets of this finite dimensional phase space by wmmutators, and it would be of importance to consider the large distance, low mass limit of such a model where it should approach an ordinary Fock space 615

Classical N-particle cosmology in 2

+ 1 dimenswns

consisting of a small number of non-interacting scalar particles, which we certainly know how to quantize. This question calls for the most economic way to represent a 2+1 dimensional universe with N particles at a given time. Consider a universe of topology S,,and containing N particles. Take coordinates in the rest frame of one of the panicles, say M,. We now construct a Cauchy surface at time 1, corresponding to a moment T in the eigen time of M,. We spread its borders further and further out until points would be included that can be connected by timelike curves. A prescription can be worked out corresponding to moving the borders of our polygon about until a situation is reached that the edges touch each other without any time shifts A typical situation is then shown in figure 9. The diagrammatic description of figure 9(6) is most appropriate. The construction allowed us furthermore to postulate that one particle is at rest = 0. The direction in which the cusp of this particle is rotated is then also immaterial. The remaining N - 1 particles are connected by a tree graph having L = 2( N - 1)- 3 lines and V = ( N - 1) - 2 vertices. All diagrams of this sort satisfy 2L

= 3 V + ( N - 1).

(5.1)

There is one boost parameter at each line, and these fix all angles at the vertices via the identities (2.6)-(2.10). This gives us (54

L=2N-5

dimensionless degrees of freedom. Now these also fix the cusp angles p where the particles are, via equation ( 2 . 3 ~ and ~ ) ~all other angles including the cusp of panicle Mu were determined. Now since the polygon must close into itself, all external angles, ?r - a i l must add up to 27r. This gives us one more constraint, and so we are left with

P=2N-6

(5.3)

degrees of freedom that determine the angles between the velocifiesof the particles. They can be compared with the momentum variables of the particles involved. Then the lengths of all lines are arbitrary, giving us again 2 N - 5 degrees of freedom with the dimension of a length. Now not only the velocity of particle M, was fixed to be zero, also its position with respect to the other particles is completely determined now by the geometrical exercise of closing the polygon Therefore we find Q=2N-5

(5.4)

degrees of freedom with the dimension of a length. It may seem to be surprising that Q is one bigger than P and that it is odd. This surprise disappears if one realizes that time 1 was introduced completely arbitrarily. If we look at the total number of degrees of freedom for cosmologies, where time is an undetermined parameter running over a certain domain of real numbers, we find that cosmologies span a P + Q - 1=4N

- 12

(5.5)

dimensional phase space. 616

In an earlier stage of this research the author thought that 2-spaces with nonvanishing genus should not be allowed, because Euler’s theorem would limit the total energy to be zero or negative. Now the total energy is obtained by adding up all deficiency angles of all particles and the angular deficits at all vertices where pofygons meet Indeed, moving particles all contribute here by amounts pi, all having the same sign. But the deficits at the vertices count in the other direction. If for instance the Crunch condition of the previous section is satisfied then it is easily seen that equation (2.8), with y > 1, gives cosa1

< cos(a*+

(lj)

* c(li> 27r.

(5.6)

Thus, the vacuum vertices in general have negative curvature and hence they tend to cancel and reverse the contributions of the particles. We may interpret this as a negative contribution to the energy by the gravitational field. Due to this phenomenon total curvature can easily be negative, so that we can have a torus (g = 1, total deficit angles zero) or even higher genus surfaces. In these spaces the particles cannot be at rest; they have to move and the situation is far from stationary. Careful study of the torus gave us that here P + Q = 4N

+ 1.

(5.7)

In the general case the number of degrees of freedom is

P + Q = 4N + 129 - 11.

(5.8)

The one extra positional degree of freedom is again the time parameter. In these less trivial space parameters with dimensions of angles and boosts on the one hand and those with dimensions of lengths and times are not so easy to distinguish anymore. An open question seems to be how exactly to characterize the topological shape of the spaces (5.3)-(5.8). Our difficulty here is that the transitions of figures 8(0)(d) are mappings of these spaces into themsehres; we would like to have definitions of Poisson brackets that are continuous under these mappings. Only then Poisson brackets may be replaced by commutators if one wishes to quantize the theory. It is natural to view the parameters counted by equation (5.3) as being the ‘momenta’, they would turn into ordinary momenta of particles in Fbck space in the large N limit, an ordinary non-interacting scalar particle model. The parameters counted by (5.4) are the coordinates plus a time variable. Considering the complicated topology of phase space it is however unclear to the present author whether and how a self consistent Fbck space can be set up describing a cosmology with a fixed number of particles. An open question is also whether or not these particles, interacting only gravitationally can be pair created or destroyed. Classically (in the h 0 limit) as well as in the large IV limit (where we have ordinary Fock space of free scalar particles), this does not happen. Existing quantization schemes [S]suggest that topology change occurs but are inconclusive as to whether particle pair creation and annihilation take place.

Acknowledgments

The author thanks R Jackiw, S Deser and A Guth for extensive correspondence leading to this work. He thanks H B Nielsen and G Gibbons for discussions concerning the last section. 617

Classical N-parricle cosmology in 2

+ 1 dimensions

References (11 Stanrszkiewin A 1963 Acta Phy& pdon U 734 Got1 J R and Alpert M 1984 Gar RcL Grm. 16 243 Giddings S, Abbot J and Kuchar K 1984 Gor RcL Grm 16 751 [2] k r S, Jackiw R and 't Hooft G 1984 Ann Plrys. 152 220 [3] Golt J R 1991 Phys Rn! Lcn 66 1126 [4] Deser S, Jackiw R and '1 Hoof1 G 1992 Phys. Rn! Lcn 68 267 [5] Carroll S M,Farhi E and Guth A 1992 Phys Rev. Lm 68 263 (and A Guth penonal communication) (61 Culler C 1992 Phya Rcv. D 45 487 ( ~ e ealso On A 1991 Phys Rn! D 44 R2214) [7] '1 Hooft G 1992 Clarr Qumtnun Grm. 9 1335 [8) Witten E 1988 NucL Phys B 311 46 Carlip S 1989 NucL Phys. B 324 lOa; 1990 Physics, Geommy and ToporogY (NATO AS1 Series B, Physics) 238 ed H C Iw (Nm York Plenum) p 541

618

Nuclear Physics B256 (1985) 727-736 Q North-Holland Publishing Company

CHAPTER 8.3 ON THE QUANTUM STRUCTURE OF A BLACK HOLE Gerard ’t HOOFT

Institute for Theoretical Physics, Princetonplein 5, PO Box 80.006, 3508 TA Utrecht, 7he Netherlands Received 26 October 1984 (Revised 30 November 1984) The assumption is made that black holes should be subject to the same rules of quantum mechanics as ordinary elementary particles or composite systems. Although a complete theory for reconciling this requirement with that of general coordinate transformation invariance is not yet in sight, a number of observations can be made and a general framework is suggested.

1. Introduction

In view of the fundamental nature of both the theory of general relativity and that of quantum mechanics there seems to be little need to justify any attempt to reconcile the two theories. Yet the essential academic interest of the problem may be not our only motive. It seems to become more and more clear that “ordinary” elementary particle physics, in energy regions that will be accessible to machines in the near future, is plagued by mysteries that may require a more drastic approach than usually considered: the so-called hierarchy problems, and the freedom to choose coupling constants and masses. A variant of these problems also occurs in quantum gravity. Here it is the mystery of the vanishing cosmological constant. It would not be the first time in history if a solution to these mysteries could be found by first contemplating gravity theory; after all, the very notion of Yang-Mills fields was inspired by general relativity. In this paper we will present an approach in which the cosmological constant problem is acute and so, any progress made might help us in particle physics also. We are not yet that far. The aim of the present paper is to set the scene, to provide a new battery of formulas that may become useful one day. To many practitioners of quantum gravity the black hole plays the role of a soliton, a non-perturbative field configuration that is added to the spectrum of particle-like objects only after the basic equations of their theory have been put down, much like what is done in gauge theories of elementary particles, where Yang-Mills equations with small coupling constants determine the small-distance structure, and solitons and instantons govern the large-distance behavior. Such an attitude however is probably not correct in quantum gravity. The coupling constant increases with decreasing distance scale which implies that the smaller the distance scale, the stronger the influences of “solitons”. At the Planck scale it may

619

G.’t Hoofi f Black hole

well be impossible to disentangle black holes from elementary particles. There simply is no fundamental difference. Both carry a finite Schwarzschild radius and both show certain types of interactions. It is natural to assume that at the Planck length these objects merge and that the same set of physical laws should cover all of them. Now in spite of the fact that the properties of larger black holes appear to be determined by well-known laws of physics there are some tantalizing paradoxes as we will explain further. Understanding these problems may well be crucial before one can proceed to the Planck scale. At present a black hole is only (more or less) understood as long as it is in a quantum mechanically mixed state. The standard picture is that of the Hawking school [l, 31, but a competing description exists [2] which this author was unable to rule out entirely, and which predicts a different radiation temperature. The question which of these pictures is right will not be considered in this paper; we leave it open by admitting a free parameter A in the first sections. Whatever the value of A, the picture is incomplete. If black holes show any resemblance with ordinary particles it should be possible to describe them as pure states, even while they are being born from an implosion of ordinary matter, or while the opposite process, evaporation by Hawking radiation, takes place. As we wiH argue, any attempt to “unmix” the black-hole configuration (that is, produce a density matrix with eigenvalues closer to 1 and 0) produces matter in the view of a freely falling observer. It is this “matter” that we will be considering in this paper. We start with the postulate that there exists an extension of Hilbert space comprising black holes, and that a hamiltonian can be precisely defined in this Hilbert space, although a certain amount of ambiguity due to scheme (gauge) dependence is to be expected. Although at first sight this postulate may seem to be nearly empty, it is directly opposed to conclusions of the Hawking school [l]. These authors apply the rule of invariance under general coordinate transformations in the conventional way. We believe that, although our present dogma may well prove wrong ultimately, it stands a good chance of being correct if we apply general coordinate transformations more delicately, and in particular if the distinction between “vacuum” and “matter” may be assumed to be observer dependent when coordinate transformations with a horizon are considered. Another point where the conventional derivations could be greeted with some scepticism is the role of the infalling observer. If the infalling observer sees a pure state an outside observer sees a mixed state. In our view this could be due simply to the fact that the “infalling observer” makes part of the system seen by the outsider: he himself is included in the outside Hilbert space. Clearly the very foundations of quantum mechanics are touched here and we leave continuation of this intriguing subject to philosophers, rather than accepting the simple-minded conclusion that transitions will take place between pure states and mixed states [ 1). Nowhere a distinction may be made between “primordial” black holes and black holes that have been formed by collapse. Now we will make use of space-time

620

G’t

Hm/r f Black hole

metrics that contain a future and a past horizon, and at first sight this seems to be a misrepresentation of the history of a black hole with a collapse in its past. However, just because no distinction is made one is free to choose whichever metric is most suitable for a description of the present state of a black hole. It may be expected that in a pure-state description of a black hole a collapsing past will be very difficult to describe if this collapse took place much longer ago than at time t = - M log M in Planck units. From now on a black hole will be defined to be any particle which is considerably heavier than the Planck mass and which shows only the minimal amount of structure (nothing much besides its Hawking radiation [3]) outside its Schwamchild horizon. The first part of this paper gives a number of pedestrian arguments indicating what kind of quantum structure one might expect in a black hole. Although they should not be considered as airtight mathematical proofs this author finds it extremely hard to imagine how the conclusions of sect. 2 of this paper could be avoided: (i) The spectrum of black holes states is discrete. The density of states for heavy black holes can be computed up to an (unknown but finite) overall constant. (ii) Baryon number conservation, like all other additive quantum numbers to which no local gauge field is coupled, must be violated. The first of these conclusions is related to the expected thermodynamic properties first proposed by Bekenstein [4]. The second has been discussed at length by him and also by Zeldovich IS]. In sect. 3 a naive model (“brick wall model”) is constructed that roughly reproduces these featyres. Being a model rather than a theory it violates the fundamental requirement of coordinate invariance at the horizon so it cannot represent a satisfactory solution to our problem, but in its simplicity it does show some important facts: it is the horizon itself, rather than the black hole as a whole that determines its quantum properties. In the second part of the paper*)we formulate a much more precise and satisfying approach. A set of coordinate frames and transformation laws is proposed. One simplification is made that presumably is fairly harmless: we assume that, averaged over a certain amount of time, ingoing and outgoing particles near a black hole are smeared equally over all angles 0, (p. Certainly this is true for a Hawking-radiating Schwarzschild black hole. Furthermore, ingoing things may be chosen to be spherically smeared. As a result we may take the gravitational interactions between ingoing and outgoing matter (our most crucial problem) to be rotationally invariant. This assumption appears to be quite suitable for a first attempt to obtain an improved theory, and in fact it makes our whole approach quite powerful. As stated before, how to interpret the coordinate transformations physically is yet another matter. The unorthodox interpretation that we proposed in ref. [2] is not ruled out (in fact it fits quite well in our description) but we do not insist on it. Rather, we allow the reader to draw his own conclusions here. *)

not reproduced in this book.

621

G.’t Hooft f Black hole

It is important to note that never space-times are considered that contain a “conical singularity” at the origin or elsewhere. Even in the formalism with A = 2 there is no such singularity. This is because the “identification” of the points x with the points --x is Minkowski space of ref. [ 2 ] only is made in the classical limit, not in the quantum theory. Quantization of the space-time metric gpv itself is not considered explicitly, since we attempt to deduce the black-hole’s properties from known laws of physics much beyond the Planck length. This may be incorrect: it could be that oniy by considering the full Hilbert space of all metrics a workable picture emerges. Our attitude is to first keep everything as simple as possible and only accept such complications when they clearly become unavoidable. 2. The black hole spectrum

A black hole can absorb particles according to the well-known laws of general relativity: the geodesics of particles with an impact parameter below a certain threshold will disappear into the horizon. Conversely, black holes may emit particles as derived by Hawking. They radiate as black bodies with a certain temperature:

in units where the gravitational constant, the speed of light, Planck’s constant and Boltzmann’s constant are put equal to one. As is explained in sect. 1 there may be reasons to put into question the usual argument that A = 1. There is an alternative theory with A =2, but the precise value of A is of little relevance to the following argument. If a black hole is to be compared with any ordinary quantum mechanical system such as a heavy atomic nucleus or any “black box” containing a number of particles and possessing a certain set of energy levels, which furthermore can absorb and emit particles in a similar fashion, then a conclusion on the density of its energy levels, p ( E ) , can readily be drawn. Imagine an object with energy AE being dropped into a black hole with mass E, so that the final mass is E + AE, where in Planck units

Let R be the bound on the impact parameter; usually R-2E.

Then the absorption cross section

U

(2.3)

is u=rrR2.

622

(2.4)

G.’t Hoofr Black hole

From Hawking’s result we conclude that the emission probability W is

w = TR’P,,

e-pnAE

,

(2.5)

where P A E is the density of states for a particle with energy A E per volume element, and PH is the inverse of the temperature TH. Now if the same processes can be described by a hamiltonian acting in Hilbert space, then we should have in the first case (T

+AEJTIE,

=[(E

A E ) ~ ’ ~+ ( EA E ) ,

(2.6)

where T is the scattering matrix, and in the second case W = I(E, AE~TIE+AE)I~P(E)PAE,

(2.7)

by virtue of the “golden rule”. The matrix elements in both cases should be equal to each other if PCT invariance is to be respected (the expressions hold for particles and antiparticles equally). So dividing the two expressions we find

p ( ~=)

c e4m’-’E2.

(2.9)

Now this result could also have been concluded from the usual thermodynamical arguments [4]. The entropy S is

S = 47rA-I E’+ const,

(2.10)

from which indeed (2.9) follows. We note that E’ in (2.9) and (2.10) measures the total area of the horizon. The constant in (2.9) and (2.10) is not known. Could it be infinite? We claim that this can only be the case if there exists a “lightest” stable black hole. Just compare eqs. (2.6) and (2.7) if E + A E represents the lightest black hole, and E and AE are ordinary elementary particles. If p( E + AE) is infinite but p ( E ) finite then, since (T must be finite, W vanishes. Since this object fails to obey the classical laws of physics (it ought to emit Hawking radiation), it cannot be much heavier than the Planck mass. Now it is very difficult to conceive of any quantum theory that can admit the presence of such infinitely degenerate “particles”. They are coupled to the graviton in the usual way, so their contributions to graviton scattering amplitudes and propagators would diverge with the constant C, basically because the probability for pair creation of this object in any channel with total energy exceeding the mass threshold is proportional to C. Clearly the above arguments assume that several familiar concepts from particle field theory such as unitarity, causality, positivity, etc. also apply to these extreme energy end length scales, and one might object against making such assumptions. But we considered this to be a reasonable starting point and henceforth assume C

623

G.'; H w f t 1 Black hole

to be finite. An interesting but as yet not much explored possibility is that C, though finite, might be extremely large. After all, large numbers such as MPlanck/ q,,oton are unavoidable in this area of physics. One might find interesting links with the 1/N expansion suggested by Weinberg [6], who however finds physically unacceptable poles in the N -* 00 limit. It is rather unlikely that C is exactly constant. One expects subdominant terms in the exponent. Also, we have not taken into account the other degrees of freedom such as angular momentum and electric (possibly also magnetic) charge. These can be taken into account by rather straightforward extrapolations. Other additive quantum numbers cannot possibly be conserved. The reason is that the larger black hole may absorb baryons or any other such objects in unlimited quantities. If indeed it allows only a finite number of quantum states then any assignment of baryon number will fail sooner or later. Indeed one of the paradoxes we will have to face is that starting with a theory that is invariant under rotations in baryon number space one must end up with a theory where this invariance is broken [5].

3. The brick wall model When one considers the number of energy levels a particle can occupy in the vicinity of a black hole one finds a rather alarming divergence at th-e horizon. Indeed, the usual claim that a black hole is an infinite sink of information (and a source of an ideally random black body radiation of particles [3,7]) can be traced back to this infinity. It is inherent to the arguments in the previous sections that this infinity is physically unacceptable. Later in this section we will see that, as is often the case with such infinities, the classical treatment of the infinite parts of these expressions is physically incorrect. The particle wave functions extremely close to the horizon must be modified in a complicated way by gravitational interactions between ingoing and outgoing particles (sects. 5 and 6). Before attempting to consider these interactions properly we investigate the consequences of a simple-minded cut-off in this section. It turns out to be a good exercise to see what happens if we assume that the wave functions must all vanish within some fixed distance h from the horizon: cp(x)=O

if

x ~ 2 M + h ,

(3.1)

where M is the black hole mass. For simplicity we take cp(x) to be a scalar wave function for a light ( m 4 1 a M ) spinless particle. Later we will give them a multiplicity 2 as a first attempt to mimic more closely the real world. In the view of a freely falling observer, condition (3.1) corresponds to a uniformly accelerated mirror which in fact will create its own energy-momentum tensor due to excitation of the vacuum. As in sect. 1 we stress that this presence of matter and energy may be observer dependent, but above all this model should be seen as an

624

G.’t H m f t

1 Black hole

elementary exercise, rather than an attempt to describe physical black holes accurately. Let the metric of a Schwarzschild black hole be given by

( ”;”)dr2+(I-- 2 y ) - 1 d r 2 + r 2 d n ’

d s 2 = - I--

(3.2)

Furthermore, we need an “infrared cutoff” in the form of a large box with radius L: if x = L .

q(x)=O

(3.3)

The quantum numbers are I, l3 and n, standing for total angular momentum, its z-component and the radial excitations. The energy levels E ( / , I,, n) can then be found from the wave equation

As long as M > 1 (in Planck units) we can rely on a WKB approximation. Defining a r3dial wave number k(r, I, E) by

k2 =

r2 r(r -2M)

((

E2-r-’I(I+1)-

1 -?)-I

m2

(3.5)

as long as the r.h.s. is non-negative, and k 2 = 0 otherwise, the number of radial modes n is given by L

drk(r,I,E)

m=j

(3.6)

2M+h

The total number N of wave solutions with energy not exceeding E is then given by der

a N = I ( 2 f + l ) d f ~ n= g ( E ) = 2+/:h

d r ( 1 - 2M 7 )

-’

where the I-integration goes over those values of I for which the argument of the square root is positive. What we have counted in (3.7) is the number of classical eigenmodes of a scalar field in the vicinity of a black hole. We now wish to find the thermodynamic properties of this system such as specific heat etc. Every wave solution may be occupied by any integer number of quanta. Thus we get for the free energy F at some inverse temperature p, (3.8)

625

G.'r HooJ / Black hole

or (3.9) and, using (3.7),

TDF =

I

(

dg( E) log 1 - e-PE)

(3.10) Again the integral is taken only over those values for which the square root exists. In the approximation m2<2M/p2h,

L*2M,

(3.11 )

we find that the main contributions are (3.12) The second part is the usual contribution from the vacuum surrounding the system at large distances and is of little relevance here. The first part is an intrinsic contribution from the horizon and it is seen to diverge linearly as h + 0. The contribution of the horizon to the total energy U and the entropy S are (3.13) (3.14) We added a factor 2 denoting the total number of particle types. Let us now adjust the parameters of our model such that the total entropy is

S=4h-'M2,

(3.15)

as in eq. (2.10), and the inverse temperature is

p =877h-'M.

(3.16)

This is seen to correspond to (3.17)

626

G.’I Hoofr J Black hole

Note also that the total energy is U=iM,

(3.18)

independent of 2, and indeed a sizeable fraction of the total mass M of the black hole! We see that it does not make much sense to let h decrease much below the critical value (3.17) because then more than the black hole mass would be concentrated at our side of the horizon. Eq. (3.17) suggests that the distance of the “brick wall” from the horizon depends on M, but this is merely a coordinate artifact. The invariant distance is r=2M+h

J dn=J r=2M

dr

-11 - 2 M l r

= 2 h M h = p . 90a

(3.19)

Thus, the brick wall may be seen as a property of the horizon independent of the size of the black hole. The conclusion of this section is that not only the infinity of the modes near the horizon should be cut-off, but also the value for the cut-off parameter is determined by nature, and a property of the horizon only. The model described here should be a reasonable description of a black hole as long as the particles near the horizon are kept at a temperature as given by (3.16) and all chemical potentials are kept close to zero. The reader is invited to investigate further properties of the model such as the average time spent by one particle near the horizon, etc. The model automatically preserves quantum coherence completely, but it is also unsatisfactory: there might be several conserved quantum numbers, such as baryon number*. What is wrong, clearly, is that we abandoned the principle of invariance under coordinate transformations at the horizon. The question that we should really address is how to keep not only the quantum coherence but also general invariance, while dropping all global conservation laws. References [ l ] S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D14 (1976) 2460; The unpredictability of quantum gravity, Cambridge Univ. preprinl (May 1982) [2] G. ’t Hooft, Ambiguity of the equivalence principle and Hawking’s temperature, 1. Georn. Phys. 1 (1984) 45 [3] S.W. Hawking, Particle creation by black holes, Comrn. Math. Pbys. 43 (1975) 199; J.B. Hartle and S.W. Hawking, Path-integral derivation of black hole radiance, Phys. Rev. D13 (1976) 2188; W.G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14 (1976) 870 [4] J.D. Bekenstein, Black holes and the second law, Nuovo Cim. Lett. 4 (1972) 737; Black holes and entropy, Phys. Rev. D7 (1973) 2333; Generalized second law of thermodynamics in black-hole physics, D9 (1974) 3292 [ 5 ] J.D. Bekenstein, Non existence of baryon number for static black holes, Phys. Rev. a5 (1972) 1239, 2403 ; Ya.B. Zeldovich, A new type of radioactive decay: gravitational annihilation of baryons, Sov. Phys. JETP 45 (1977) 9; A.D. Dolgor and Ya.B. Zeldovich, Cosmology and elementary particles, Rev. Mod. Phys. 53 (1981) 1 *)

One may postpone this difficulty by inserting explicitly baryon number violating interactions near t h e horizon.

627

G.’r Haaft f Black hole

[6] S . Weinberg, in General relativity, an Einstein centenary survey, ed. S.W. Hawking and W. Israel (Cambridge Univ. Press, 1979) 795 [7] W.G.Unruh and R.M.Wald, What happens when an accelerating observer detects a Rindler particle, Berkeley preprint

628

Nuclear Physics B253 (1985) 173-188 Publishing Company

@ North-Holland

CHAPTER 8.4

THE GRAVITATIONAL SHOCK WAVE OF A MASSLESS PARTICLE Tevian DRAY' and Gerard 't HOOFT

fnstituut voor Theoretischc Fysicq Ph'ncetonplein 5, Postbus SO.OW, 3508 TA Utrecht. n e Netherlands Received 20 August 1984 The (spherical) gravitational shock wave due to a massless particle moving at the speed of light along the horizon of the Schwanschild black hole is obtained. Special cases of our procedure yield previous results by Aichelburg and Sex1 [I] for a photon in Minkowski space and by Penrose [2] for sourceless shock wavui in Minkowski space. A new derivation of the (plane) shock wave of a photon in Minkowski space [I] involving explicit calculation of geodesics crossing the shock wave is also given in order to clarify the underlying physics. Applications to quantum gravity, specifically the possible effect on the Hawking temperature, are briefly discussed.

1. Introduction

There are various reasons why one may be interested in exact expressions for the gravitationalfield surrounding a particle whose mass is dominated by kinetic energy rather.than rest mass. For instance the first non-trivial gravitational effects to be seen in particle-particle interactions at extreme energies may be due to such fields. Our understanding of quantum gravity may be helped by considering these field configurations. A specific case of interest is the gravitational back-reaction and self-interaction of matter entering or leaving a black hole (Hawking radiation). At the black hole horizon the relative velocity of these particles approach that of light. Aichelburg and Sexl [l] considered the gravitational field of a massless psrticle in Minkowski space, and showed that the resulting space-time is a special case of a gravitational impulsive wave* [ 2 ] which is also an asymmetric plane-fronted gravitational wave [3]. Penrose [2] also gives explicit examples of sourceless gravitational impulsive waves in' Minkowski space. In this paper we first summarize the properties of the shock wave due to a massless particle in Minkowski space. We do this by presenting a new derivation of the results of Aichelburg and Sexl [l] involving explicit calculation of (null) geodesics crossing the shock wave. This enables the physical properties of such shock waves to be easily exhibited. We then determine, for a particular class of vacuum solutions to the Einstein held equations, the (necessary and sufficient) conditions for being able to introduce

' Supported by the Stichting voor Fundamenteel Ondenbek der Materie.

' Note that Penrose [2] reserves the term gravitational shock wave for a metric which is C'

whereas the metrics we consider are only Co. We will nevertheless use the term "shock wave" for what are. in the terminology of [2], impulsive waves.

629

T. Dray, G. ’ 1 Hoofr

/

Grauiraiional shock wave

Fig. I. The horizon shift (eq. ( 1 5 ) ) due to the field of a massless particle moving in the v-direction along the horizon of the Schwarzschild black hole. The amount of the shift depends on 8.

a gravitational shock wave via a coordinate shift*. These conditions include both constraints on the metric coefficients and on the form of the shift. In Minkowski space they reduce to the plane-fronted wave of Aichelburg and Sex1 [ l ] and, of course, to Penrose’s results [2] for sourceless waves. However, for Schwarzschild black holes we obtain something new: there is a (spherical) shock wave at the horizon due to a massless particle at the horizon. (See fig. I.) Throughout this paper we think of the massless particle as the limit of a fast-moving particle with negligible rest mass**; this limit is given explicitly for the Minkowski case. Fig. 1 can thus be interpreted as describing an ordinary particle with small mass falling into the black hole from the left, as seen by an outside observer (on the left) at very late times; the particle is then seen close to the horizon and boosted to high energies. The paper is organized as follows: in sect. 2 we summarize the situation for the (plane) shock wave due to a massless particle in Minkowski space and discuss the general physical features of such a wave. These results are based on a calculation of the null geodesics in such a space-time, which is given explicitly in appendix A. In sect. 3 we give the conditions, derived in appendix B, for a shock wave to be possible starting from a given “background” space-time. After showing that these conditions reduce to the correct ones [ 1,2] in Minkowski space we then obtain the (spherical) shock wave at the horizon of the Schwarzschild black hole due to a massless particle there. In sect. 4 we discuss our results. 2. Shock waves: an example

Aichelburg and Sex1 [ 13 (cf. eq. (A.37)) have shown that the gravitational field of a massless particle in Minkowski space is described by the metric d s 2 = -du^(dv*+4p ln(p2)6(C)dzi) + dx2+ d 3 ,

(1)

This is just the scissors-and-paste approach of Penrose [2] applied to more general space-times. ++ We assume that the particle has no electric charge and no angular momentum. However, for an elementary particle for example we do not expect the results to differ significantly from those we derive here.

630

T. Dray, G. ‘ t Hoof? f Gravitational shock wave

where p” = 2 + y.The particle moves in the 6 direction with momentum p. By calculating geodesics which cross the shock wave, which is located at U = 0, we obtain the following two physical effects of such a shock wave (see appendix A): geodesics have a discontinuityd6 at U = 0 and are refracted in the transverse direction. The shift A 6 is given by (cf. eq. (A.26)) 4GP In Pi -’ c’

which, for a photon, is

where we have put the units back in and where we have used E = pc = hu, where U is the frequency of the photon and Ipl is the Planck length. po is the value of p when the geodesic reaches U = 0. This shift is illustrated (for nonzero m and x = 0) in fig. 2; for m = 0 the shift occurs as a discontinuity at U = 0. Note that the presence of a length scale in the argument of the logarithm is merely a reflection of our choice of units and has no physical meaning. It represents a constant shift in 6 which can be transformed away by a suitable redefinition of 6 (eq. (A.] 1)). Furthermore, by the same procedure, the value of po for which A6 = 0 (here po= fp,) can be chosen arbitrarily far from the photon (po large). In any case, only the diference in A 6 for nearby geodesics is physically relevant. There is also a refraction effect described by (cf. eq. (A.36)) cot a +cot p = 4GP 3, c Po which, for a photon, is cota+cotp=-

41gl U CPO

This is illustrated (for x = 0) in fig. 3, where the angles a and p are defined.

a) Yo

<1

b) yo> 1

Fig. 2. The path of a null geodesic in the 512. 8) plane as described by eq. (2) for m < 1, PO* m, and (a) p0< 1, (b) po> I . The near region N and the far region F, as well as the shift AI?, are indicated.

631

T. Dray, G.‘ I Hoofr

Gruvitationu/ shock wove

?“

Y=Y,

(a)

TY I

r (b)

y = -y+y,

Fig. 3. The “spatial refraction” of null geodesics as described by eq. (3) for the two speciat cases (a) a = p, and (b) a = jlr.

Eqs. (2) and (3) are the central results for these shock waves and describe physical effects which should also occur in more general situations. Note that if the shift (2) were constant it could be removed by a coordinate translation and would therefore not be physically observable (cf. the discussion after eq. (2)). Also, a shift linear in the transverse distance p would not be observable since it could be removed by a Lorentz rotation of one of the flat half-spaces with respect to the other. However the shift (2) is logarithmic in p and leads to physically observable effects. The relative shift for nearby observers goes as the first derivative ( l / p ) while the relative refraction goes as the second derivative ( l/pz) of the shift*. See fig. 4.

Fig. 4. Four synchronized clocks were originally situated at rest at the corners of a rectangle. A fast particle approaches from the left. The situation is shown when the shock wave has passed two of the clocks. The one closest to the trajectory of the particles has been shifted to the right with respect to the other; its clock now runs behind the other. They are also moving towards the trajectory of the fast particle at different speeds (arrows). Only their relarioe velocity, which is always away from each other, is locally observable.

* Note that a

local observer can only detect the second derivative ( l/p2) of the shift.

632

1 Dray, G. ' t Hooft

Gravitational shock waue

3. General result Consider a solution of the vacuum Einstein field equations of the form dg2=2A(u, u)du du+g(u, u)h,(x') dx'dx'.

(4)

Under what conditions can we introduce a shift in U at U = 0 so that the resulting space-time solves the field equation with a photon at the origin p = 0 of the (xi) 2-surface and U = O? As shown in appendix B the answer is that at U = 0 we must have A.v = 0 = g.0 A Af-&f= g

L?

9

32?rpA2S(p),

(5)

where f=f(x') represents the shift in U, Af is the laplacian off with respect to the 2-metric h,, and the resulting metric is described by (B.2) or (B.4). Eqs. ( 5 ) represent our main result. We now turn to specific examples. For a plane wave due to a photon in Minkowski space we have d t 2 = - d u dv+dX2+dy2,

(64

and thus

The conditions on the metric are trivially satisfied, and the condition on the shift f is Af= - 1 6 ~ p S ( p ) ,

(7)

where p2=x2+y2. The solution of this equation, unique up to solutions of the homogeneous equation, is

f = -4p In p2 ,

(8)

which agrees precisely with Aichelburg and Sex1 [l] (cf. eqs. (2) and (A.26)). For a sourceless plane wave in Minkowski space we set p = O to obtain Af=O,

(9)

which agrees with Penrose [2]. For a spherical wave in Minkowski space we write the metric in the form d t 2 = - d u d u + ~ ( u - u ) 2 ( d 0 2 + s i n Odq?), 2 so that

633

(104

7: Dray. G. ' f Hooft

1 Gravitational

shock wave

But the derivatives of g are not identically zero at U = 0. Thus, there are no spherical waves (of this form) in Minkowski space. Physically this might seem mysterious because one expects spherical shock waves to arise in, e.g., the debris of a violent explosion. On closer inspection one concludes that there must be non-zero curvature behind such shock waves. However, note that Penrose [2] does exhibit the existence of sourceless spherical shock waves in Minkowski space but having a different form than our ansatz (eq. (B.2)). We now turn to a more interesting example, namely the Schwarzschild metric which in (null) Kruskal-Szekeres coordinates takes the form

so that

g=r2. r is given implicitly as a function of

U

and

U

by

so that all u-derivatives of r are proportional to U. Thus, the conditions on the metric coefficients A and g are satisfied at U = 0. Furthermore, since g,uu= A the condition on f becomes

where K = 29m4pe - ' and where we have arranged the coordinates so that the photon is at 8 = 0 = U. We now solve eq. (12) by expanding f in terms of spherical harmonics &( 6 , ~ ) . We see immediately that only spherical harmonics with m = 0 contribute; expressing these in terms of'the Legendre polynomials S ( x ) leads to

r+f f=K?

I ( , + 1)+ 1

P,(COSe l .

We can obtain an integral expression for f by using the generating function for the Legendre polynomials, namely

634

T. Dray, G.’ I

Hooft

1 Gravitational shock wave

and the fact that

r’ e“” cos ( G s ) d s =

1+; , + 1)+

where r = es, t o finally obtain cosh s -cos O ) ’ / *

ds.

We have not attempted to perform the integration explicitly. We note that the homogeneous equation (eq. (12) with p = 0) has no solution. In the limit of small B eq. (15) in appropriate coordinates reduces to eq. (8), with a well-determined value of the integration constant. 4. Discussion

The surprisingly simple geometric shape of a gravitational shock wave of massless particles in flat space can help us obtain a better understanding of gravitational interactions among particles at extreme energies. It is easy to argue that at extremely high energies interactions due to this shock wave will dominate over all quantum field theoretic interactions, simply because the latter will be postponed by an infinite time shift (due to the logarithmic singularitity in eq. (2), see fig. 4). This implies that cross sections at such energies will be entirely predictable. A problem arises if two such particles are considered, both accompanied by their shock waves, that meet and collide. The result of such a collision will be curved shock waves which obey the vacuum Einstein field equations only if space-time after the collision in the region between both shock waves is curved, so that we then have to deal with the full complexity of general relativity. We have here a limiting case of the general problem of black hole encounters which has been studied in detail by D’Eath [4] and Curtis [5]. On physical grounds the Schwarzschild result, eq. (13), should not be surprising. The flat space result (e.g. eq. (8)) can be obtained [ I ] by infinitely boosting a (massive) source particle. Now take an r = const observer in the usual Schwarzschild coordinates. Put a (nearly) massless particle at the horizon and wait. The observer will see a particle with an increasingly large boost! It is only natural to expect a similar result in both cases. The spherical nature of the wave in the Schwarzschild case (as opposed to the plane wave in Minkowski space) is merely a reflection of the spherically symmetric nature of the “boost” relating an r = const observer to Kruskal-Szekeres coordinates. Physically, one expects any (weak) plane wave approaching the black hole to become gradually more spherical, as seen by an outside observer, as it comes closer to the horizon. Returning to the picture of a particle of small mass falling into the black hole (see discussion after fig. I ) one expects small increase of the Schwarzschild radius

635

T. Dray, G. ’ I Hoofr 1 Gravitational shock

waue

of the black hole, together with a slight expansion of its furture horizon. This expansion then grows exponentially with the Schwarzschild time coordinate. This is what our “shock wave” here actually describes. Eq. (15) is in closed form the extent of the horizon expansion. One might speculate what effect this expansion has on the quantum nature of the vacuum and, in particular, Hawking radiation. We believe that the gravitational interaction between infalling matter and Hawking radiation, crucial for a deeper understanding of the quantum properties of black holes themselves [6], should be described using our expression for the horizon expansion. Finally, we are aware of the analogy with the electric field of a charged particle moving at the speed of light, which is similar to the gravitational field described here. Our gravitational shock wave can be compared to a limiting case of Cherenkov radiation. We thank Paul Shellard for bringing the work of D’Eath [4] to our attention, which then led us to the previous work of Aichelburg and Sex1 [ 13 and Penrose [2]. The computer calculations of the Ricci tenser were performed while one of us (T.D.) was a visitor at Queen Mary College, London. He is deeply indebted to Malcolm MacCallum and Gordon Joly for hospitality and assistance.

A PHOTON IN MINKOWSKI SPACE

Consider the linearized field of a point mass in Lorentz gauge: ds2=-

(I - - 2 3 d T 2 +(I + - 3( d x 2 + d y 2 + d Z 2 ) ,

with rn 4 R. This is the field of the particle as seen in its rest frame. Boost this rest frame with respect to coordinates ( 1 , x, y, z) via T = t cosh

- z sinh p ,

Z = - f sinh p

+ z cosh p ,

and simultaneously set

rn = 2 p e-@ for some constant p > 0. Introduce null coordinates u=r--z,

v=t+z.

636

(A.4)

T Dray, G. 't Hooft 1 Gravitational shock waw

ne momentum of the particle is pa = m[(cosh p)S: +(sinh /3)S3,

(A.5)

and thus lim p a = p( S:

8-03

+ 8 ; ) = 2pSz ;

(A.6)

in the limit the particle is massless and moves (at the speed of light) in the u-direction; p is its momentum and is kept finite (possibly large).

Writing the metric in

x, y) coordinates we obtain

(U, U,

with

The key idea is to notice that fim

(A.9)

ds2=-du

m- 0

( U +O.u.x,y)

fixed

which is flat although the coordinate

U'

satisfying

4p du dv' = du --

(A.lO)

14

suffers a discontinuity at U = 0 due to the absolute value sign. To make this somewhat more precise, introduce coordinates ( 2, i3, x, y ) by* G=u+

m2Z In (2R) PR

9

4pZ In (2R) (A.11) R * Note that (2, 6,x, y ) is obtained from (U, U, x, y) by adding ( 4 2 In ( 2 R ) I R ) p " .Then C=U+

R'=x'+y2+(;G-%)

2

,

4P lim

m+O (u#O.qx.y)lixcd

ds2= -dG di3+dx2+dy2.

(A.12) (A.13)

The motivation for these coordinates is as follows: in the limit, Z I R acts like a @-functionand reproduces the effect of the absolute value sign, while dR/ R = d u / u Furthermore, In R is finite at -0. The factor 2 is chosen for convenience. We use geometric units in which G = c = h = I ; all quantities are dimensionless.

637

T. Dray, G. ’ t Hoop / Gravitational shock wave

The metric (A.13) is flat. It remains to investigate its behavior near U =0, which, in the limit, is just U^ = 0. To do this we consider the behavior of (null) geodesics crossing U = 0 for m # 0 and then take the limit as m goes to zero”. We do this both in a “near” region, which collapses to U^ = 0 in the limit - this is just the rest frame of the particle - and.a “far” region, where U^ remains non-zero in the limit. The (linearized) geodesics of the metric (1) are given by

( $), L( -$),

T = E 1+ y z - zi, =

1

(A.14)

where the dot denotes derivatives with respect to the affine parameter A along the geodesic. We have assumed x = 0 without loss of generality; the constants E, L, M denote the energy, angular momentum, and rest mass of the test particle, respectively. In what follows we consider only “null” geodescis; i.e. we set M = O(rn2). Expanding y , Z and T in powers of m and considering only the terms linear in m we have Y=Yo+mY,,

2 = Zo+ mZ, ,

T = To+mTl,

(A.15)

and eqs. (A.14) now become

To= E ,

i,;+z:=E 2 , Yo20 - Z O Y O = L ,

TI= 2 E / Ro , joj,+

zoz,= 0 , 2L RO

Y o ~ l - ~ l ~ o + Y l ~ o - --, ~ o ~ l =

(A.16)

where R : = y i + Z i .

Penrose and MacCallum [7] describe some properties of such geodesics without actually calculating them. Penrose and Curtis have performed similar calculations of null geodesics crossing a shock wave, but as far as we know these have not been published [a].

T. Dray, G. 'I Hoofr 1 Gravitational shock wow

However, since m u = -( T- Z), 2P o =2( T + Z ),

(A.17)

Zo=-To=-E

(A.18)

m

we must require

if U, and thus 6,is to remain finite in the limit as m goes to zero. The second and third of eqs. (A.16) now yield YO=(),

yo= - L I E ,

(A.19)

Z,=O.

(A.20)

and the fifth of eqs. (A.16) implies

Thus m2E P %'

m

U=--+--

P E v=4p-, RO

(A.21)

Using eq. (A. 18) these can be integrated directly to give U=

mE m2 A -- In (Zo+R,,), P

U = -4p

P

In (Zo+R,) ,

(A.22)

where we have ignored an irrelevant integration constant, and thus

(A.23)

We now separate into a near region N and a far region F as follows: N={~A~c~/G},

F = {&S

639

mlkl
(A.24)

T. Dray, G. ’ I hoof^ /

Gravitationtll shock waw

Note further that lim C=O,

A*-a0

iim v^ = -4p In y i , **+a0

mE

lim ;=-A.

(A.25)

P

*-tm

n u s , there is a total shift in v^ given by

do^=-4p In y : .

(A.26)

Note that, in the limit as m goes to zero, A is infinite everywhere in F. Furthermore, in this limit U^ is identically-zero in N, whereas 6 is a good affine parameter in F along the geodesic. The shift (A.26) thus occurs, for small m, “essentially” only in N! Thus, in the limit as m goes to zero, the shift (A.26) occurs at U^ = O and represents a finite discontinuity in 6 along null geodesics! This can also be seen by calculating (A.27)

thus showing that in the limit as m goes to zero U* is constant in F, i.e. for nonzero U^. This is just a reflection of the fact that, in the limit, F is flat. This is illustrated in fig. 2. We now turn to the behaviour of y. We must solve the last of eqs. (A.16), which, on inserting eqs. (A.19) and (A.20) becomes

-

~ 1 2 0 Zoj, = -2

L/Ro .

(A.28)

The homogeneous equation clearly has the solution Y:=Mo

(A.29)

for any constant A ; it remains to find a particular solution. Multiplying eq. (A.28) by Zo yields EZy,-R,R,jl = 2 L E / R o .

(A.30)

But noticing that (A.31)

suggests an ansatz for y , as a power series in Ro. We thus obtain the particular solution (A.32)

T. Dray, G.' t Hooji 1 Gravitational shock wave

The general solution to eq. (A.28) is thus

,

y, = -*+Azo

(A.33)

Yo

and therefore Y =-

E +m [L

2

+

(A.34)

We are interested in the behaviour of y in the far field F for m small. We obtain (A.35) This behaviour is illustrated in fig. 3. In general we have cot a +cot p =-4P

(A.36)

Yo

for the angles a and p as defined in fig. 3. At this point several comments are in order. We have nor considered all geodesics which cross the shock wave, but only a sufficient number to determine how to glue the two flat half-space together. That this is sufficient follows from the existence of coordinates in which the metric is in fact continuous (see appendix B). Using the results of appendix B we see that lim ds2=-d6(d6+4p l n y i S ( 6 ) d 6 ) + d x 2 + d y 2

rn-0

( 1 -2e(u))+4p In 4 S ( u ) d u

(A.37) which of course reduce to.(A.I3) and (A.9) respectively for U # 0. The first of (A.37) is just the result of Aichelburg and Sexl [I] (their eq. (3.9)), but the second of (A.37) disagrees,with their eq. (3.10). Although this is at first disconcerting, a more careful analysis reveals the source of the discrepancy: we have taken a limit different from theirs. This can be seen by noting that for m + 0, U # 0, our original coordinates (1, z ) are related to their coordinates (t; 2 ) by infinire scale factors. Equivalently, note that the original Minkowski space given in (U, U, x, y ) coordinates is"pushed to infinity" in the resulting space-time given in (6, 6,x, y ) coordinates. Specifically, {U # 0;
Iu~

(c

641

T

Dray, G. ' t Hooft

Gravitational shock wave

Schwarzschild metric in isotropic coordinates and expanded in powers of m only the linear terms we consider would have survived. Appendix B CALCULATION OF THE RICCl TENSOR

We start with the metric d?=2A(u,

U ) du

du+g(u, u)h,(x') dx'dx',

(B.1)

which is assumed to satisfy the Einstein vacuum equations. We introduce a shock wave by keeping (B. 1) for U 0 but replacing U by U + f ( x ' ) for U > 0*:

-=

ds2= 2A( U, U + Of) du(du + O f i dx') + g( U, U

+ 6f)hy dx' dx' ,

where 8 = O( U ) is the usual step function. Changing to coordinates (I?, by

(B.2)

G, ii)defined

A

u=u,

G=u+t)f, 2' = x i

03.3)

9

we obtain ds2=2A(I?,6) du*(d$-d(I?)f dI?)+g(I?,G)h, di'dk',

(B.4)

where 6 = 6( U ) is the Dirac delta "function". We note that the metric ds2 given in (B.2) and (B.4) is in fact conrinuous, i.e. there exisr coordinates (U, 0, 2') such that the metric coefficients are continuous. A possible choice is given implicitly by

I?=a,

G=

A

0+6$-fUB27 hmnfmfn, g

where

This is not quite the standard ansatz gab= ( I - @ ) g i b+ @:b. However, in the examples considend here the corresponding Ricci tensors differ at worst by a term in R,, proportional to O( 1 - e), which is not physically relevant.

642

T. Dray, G.' I Hooft 1 Graoitarional shock

warn

However, the coordinates (B.5) are extremely unwieldy both for mathematical Computations and for preserving physical intuition. We will thus proceed as follows. Noting that the coordinate transformation between (0, 0, 3') and (U, U, x i ) coordinates is continuous, we will calculate the Ricci tensor for the metric (8.2). However since the metric (8.4) is much easier to work with, we will formally c ri transform to ( U , U, x ) coordinates, calculate Rd6, and then transform back to obtain Rob. Direct calculation yields* A

where Rf' is the Ricci tensor derived from h,, A is the Laplace operator associated with hi,, and 6 = S(u^). Blithely ignoring the 6' term we transform to (U, U, xi) coordinates and insert the vacuum equations (obtained by setting f = 0) to get

R,, = R t # = O , R,, = R,j;=O, R , = R g = - h g + i iit fa, A

we note that Taub [9] has given a systematic presentation of space-times with distribution-valued CUrVIturc

tensors.

643

T. Dray, G.'I Hooft 1 Gravitational shock wave

R,, = R;; i-2R;&

The stress-energy tensor for a massless particle located at the origin p = O of the ( x i ) 2-surface and at U = 0 is Tab=4pS(p)S(u)StG&,

(B.9)

where p is the momentum of the particle. Thus, the only non-zero component is

Ti, = 4pA2S(p)S(u).

(B.lO)

Inserting (8.8) and (B. 10) into the Einstein field equations, partially integrating the S' term, noting that e.g. A,G(U^ = 0) = Oe A,"(U = 0) = 0 yields precisely eqs. ( 5 ) . The calculation above was first done by hand and then checked using the algebraic manipulation computer system SHEEP, As a further check on the validity of working in the singular coordinates (U^, 6,2')(eq. (B.4)) SHEEP was also used to calculate the Ricci tensor directly in (U, U, x ' ) coordinates (eq. (B.8)), thus checking the original calculation of 't Hooft for the Schwarzschild case. The same answer, namely eqs. (9,was of course obtained in all cases. References [I] P.C. Aichelburg and R.U. Sexl. 1. Gen. Rel. Grav. 2 (1971)303. [ 2 ] R. Penrose, in General relativity: papers in honour of J.L. Synge, ed. L. O'Raifeartaigh (Clarendon.

Oxford, 1972) 101; in Battelle rencontres, ed. C.M. De Witt and J.A. Wheeler (Benjamin, New York, 1968) 198; in Differential geometry and relativity, ed. M. Cahen and M. Flato (Reidel, Dordrecht, 1976) 271 [3] J. Ehlers and W. Kundt, in Gravitation: an introduction to current research, ed. L. Witten (Wiley, New York, 1962)SSf;, H.W. Brinkman, Proc. Natl. Acad. Sci. (US)9 (1923) I [4] P.D. DEath, Phys. Rev. D18 (1978)990 [S] G.E. Curtis, J. Gen. Rel. Grav. 9 (1978)987,999 [6]G. 't Hooft, J. Geom. Phys. 1 (1984)45 [7] R. Penrose and M.A.H. MacCallum. Phys. Reports 6 (1973)270 [8] P.D. D'Eath, private communication [9]A.H. Taub, J. Math. Phys. 21 (1980)1423

644

CHAPTER 8.5

S-MATRIX THEORY FOR BLACK HOLES G. 't Hooft Institute for Theoretical Physics Princetonplein 5, P.O. Box 80.006 3508 TA Utrecht, The Netherlands ABSTRACT We explain the principles of the l a w s of physics t h a t we believe t o be applicable f o r the quantum theory of black holes. In particular, black hole formation and evolution should be described in terms of a scattering matrix. This way black holes at the Planck scale become indistinguishable from other particles. This S-matrix can be derived from known l a w s of physics. Arguments a r e put forward in favor of a dlscrete algebra generating the Hilbert space of a black hole with its surrounding space-time including surrounding particles. 1. INTRODUCTION

There is quite a bit of controversy (and confusion) regarding the nature of physical l a w governing a black hole. Some of the difficulties have t h e i r origin i n the deceptively clean picture given by the "classical" (here t h i s means "non-quantum mechanical") solutions of Einstein's equations of gravity in the case of gravitational collapse. The metric tensor describing the fabric of space-time appears to be smooth and well-behaved i n the vicinity of a region we c a l l the "horizon", a surface beyond which there a r e space-time points from which no information can reach the outslde world. I t seems t h a t one should be able t o apply standard techniques from particle theory here to derive what a d i s t a n t observer can perceive and, naturally, t h l s exercise h a s been done'. There is one innocent-looking assumption t h a t most practitioners then make. One observes that clouds of particles may venture into the "forbidden region", from which they can no longer escape or even emit any signal towards the outside world, and so one assumes t h a t the corresponding s t a t e s in Hilbert space may be treated the way one always does in quantum mechanics: the unseen modes a r e averaged over. Operators describing observations in the outside world a r e assumed to be diagonal in the sector of Hilbert space that is not seen, and hence i n all computations one is obliged to sum over all unseen modes. The immediate consequence of t h i s practice is that the outside world alone is not anymore described by a single wave function but by a density matrix2. Even if one starts with a "pure" wave function, sooner or l a t e r one f i n d s the system to be in a mixed state. I t 1s as if p a r t

645

of the wave functions "disappeared into the wormhole" ; information escaped. as if the system were linked t o a heat bath. With large black holes one can perform Gedanken experiments, and consider observers who move semiclassically in the neighborhood of a black hole horizon. If we attach some sense of "reality" to these observers the correctness of the above assumptions seems to be a n inescapable conclusion. But what if the hole is small, so that classical observers a r e too bulky to enter? O r let us a s k a questlon that is probably equivalent t o this: suppose one keeps track of a l l possible s t a t e s a black hole can be in, is it then still impossible to describe the hole in terms of pure quantum s t a t e s alone? W i l l the very tiny black holes evolve according to conventional evolution equations in quantum physics or is the loss of information a fundamental new feature, even f o r them? There is a big problem with any theory in which the loss of "quantum information" is accepted as a fundamental item. This is the fact t h a t all effective l a w s become fuzzy. I t is not d i f f i c u l t t o construct a n example of a theory in which pure s t a t e s evolve into mixed s t a t e s . Consider a system with a Hamiltonian that depends on a f r e e physical parameter a ( f o r instance the fine s t r u c t u r e constant). A s t a t e I@)o at t=O evolves into the state (1.11

at time

t

. The

expectation value f o r an operator

0

evolves into (1.2)

and from the @ dependence one can recover the information t h a t the system remained in a pure quantum state. But now assume t h a t there Is a n uncertainty in a . We only know the f i r s t few decimal places. There is a distribution of values f o r a , each with a probability P(a) . Our theory now predicts f o r the "expectation value" of 0 : (O),

=

I d a (0)t,a H a )

Tr p ( t ) O

(1.3)

;

(1.4)

This is a n impure density matrix, of the kind one obtains in doing calculations with black holes. The outcome of a by now standard calculation is a thermal distribution of outgoing particles. A thermal distribution is always a mixed state. I n our example we clearly see what the remedy is. The e x t r a uncertainty had nothing to do with quantum mechanics; the Hamiltonian w a s not yet known because of our incomplete knowledge of the l a w s of physics (in t h i s case the value of a 1. By doing e x t r a experiments or by working harder on the theory we can establish a more precise value f o r a , and thus obtain a more precise prediction f o r ( O ) , Returning with t h i s wisdom to the black hole, what knowledge w a s incomplete? Here I think one has a situation that is common to all macroscopic systems: because of the large number of quantum mechanical s t a t e s it w a s hopelessly difficult to follow the evolution of Just one such s t a t e precisely. One w a s forced to apply thermodynamics. The outcome of our calculations with black holes got the form of thermodynamic expressions because of the impossibility, in practice, to follow in detail the evolution of any particular quantum s t a t e . But t h i s does mean that our basic understanding of black holes at present I s incomplete. I n a statistical system such as a vessel containing a n ideal gas, we have In prfnclple a quantum theory that is precise enough to study pure quantum states. In particular, if we dilute t h e gas so much that a single atom remains. the thermodynamic

.

646

description w i l l no longer be correct, and we must use the r e a l quantum theory. Similarly, if we want to understand how a black hole behaves when it reaches the PIanck m a s s , we expect the thermodynamic expressions to break down. The importance of a good quantum mechanical description is t h a t it would enable us to link black holes with ordinary particles. The Planck region may well be populated by a lot of different types of fundamental particles. t h e i r "high energy l i m i t " w i l l probably consist of particles s m a l l enough and heavy enough to possess a horizon and thus be indistinguishable from black holes. What we want is a consistent theory t h a t covers all of t h i s region. If we had a "conventional" Schrodinger equation in t h i s region, it would be relatively straightforward (at least conceptually) t o extrapolate to large distance scales using renormalization group techniques, and recover the "standard model" (or morel ) There have been many proposals concerning the nature of our physical world near the Planck length. we have seen "supergravity", "string theory", "heterotic strings". e t cetera. My problem with these ideas is t h a t they seem to be ad hoc. The models a r e "postulated" and then afterwards the authors t r y to argue why things have to be t h i s way (basically the argument is that the new model is "more beautiful" than anything else known). I t would be a lot s a f e r if we could derive the only possible correct setting of variables and forces, directly from the presently established l a w s of physics. In these lectures we w i l l argue t h a t it is possible t o do t h i s , or at least to make a good s t a r t , by doing Cedanken experiments with black holes. The reason why black holes should be used as a s t a r t i n g point i n a theory of elementary particles is t h a t anything t h a t 1s tiny enough and heavy enough to be considered a n e n t r y in the spectrum of u l t r a heavy elementary particles (beyond the Planck mass), must be essentially a black hole. Black holes are defined as solutions of the classical, i.e. unquantited. Einstein equations of General Relativity3. This implies t h a t we only know how t o describe them reliably when they are considerably bigger than the Planck length and heavier than the Planck m a s s . What w a s discovered by Hawking' in 1975 is t h a t these objects radiate and therefore must decrease in size. I t is obvious t h a t they w i l l sooner or l a t e r enter the domain t h a t we presently do not understand. Curiously, it is not easy to see why Hawking's derivation of the thermal black hole radiation would not be exactly correct. Even i n a functional integral expression f o r t h i s calculation one might still expect wormhole configurations through which quantum information leaks W e w i l l now decide t o be towards a mystical "other universe"'. merciless: topologically non-trivial space-times a r e forbidden ( u n t i l f u r t h e r notice) so that, a t least a t the microscopic level, pure quantum mechanics can be restored. More precisely, what we require is first of all some quantum mechanically pure evolution operator, and secondly t h a t t h i s operator be consistent with all we know of large scale physics, i n particular general relativity. A t f i r s t s i g h t these requirements a r e in conflict with each other. General relativity predicts unequivocally that gravitational collapse is possible, and t h i s produces a horizon with all its difficulties. However, we claim t h a t a pure quantum prediction t h a t naturally blends into thermodynamic behavior in the large scale l i m i t is not at all lmposslble, but it is t r u e t h a t the requirement f o r t h i s to happen is extremely restrictive. Combining it with all we already know about large scale physics may well yield an unambiguous theory. Anyway, we know f o r s u r e t h a t the amendments needed a t the horizon all r e f e r t o Planck scale physics. As long as t h i s physics is not completely understood i t will

647

also be impossible to refute our theories on the ground of inconsistencies with known physics. So t h i s is our program. We assume that, as f o r all spatially confined systems, there exists such a thing as a "scattering matrix". One then t r i e s to reconcile t h i s scattering matrix with the l a w s of physics already known. We w i l l find that t h i s scattering matrix, to some extent, can be derived. More precisely: the exact quantum behavior a t large d i s t a n c e s c a l e s (the distance scales reached in present particle experiments) can b e derived uniquely. The problem is an apparent acausality. If we apply linearized quantum field theory in the black hole background it seems ununderstandable how information that is thrown into the black hole can reemerge as information in the outgoing s t a t e s . This is because the outgoing radiation originates at t = -m and the ingoing matter proceeds u n t i l t = + m , so the information had to go backwards in time. We simply claim t h a t precisely f o r t h i s reason linearized quantum field theory is inappropriate here. One must take gravitational (if not other) interactions between in- and outgoing matter into account. One way to interpret what happens then is to assume that there is a fundamental symmetry p r i n c i p l e , because matter inside the horizon is unobservable. One can then perform. a transformation that transforms away the singularity at t=+m , and produces one at t = - m . I t is of crucial importance to note t h a t what we a r e deriving is not only the (quantum) behavior of the black hole itself. I t is the entire system, black hole p l u s a l l surrounding particles, t h a t we a r e talking about. Using our (assumed) knowledge of physics at large distance scales we derive the properties of the black hole and a l l o t h e r f o r m s of matter at energies larger than the Planck energy. In ordinary quantum field systems behavior at s m a l l distance, or equivalently, at high energies, determines the behavior at large distances and low energies. In the present case the interdependence goes both ways, or, in other words, the whole construction w i l l be overdetermined. We expect stringent constraints of consistency, which, one might hope, may lead to a single unique theory. The point is t h a t the symmetry principle Just mentioned a f f e c t s matter in a n essential way, and t h u s may perhaps continue to be of relevance at the low energy domain. This is the motivation of t h i s work. It may lead t o "the unique theory". Even though our work is f a r from finished, we w i l l be able to show that there w i l l be a remarkable role f o r the old s t r i n g theorys. The mathematical expressions we derive a r e so similar to those of s t r i n g theory t h a t perhaps some of its results w i l l apply without any change. But both the physical interpretation and the derivations will be very different. As a consequence, the mathematics is not identical. One important difference is the s t r i n g constant (determining the masses of the excitations), which in our case turns out t o be imaginary'. In the usual s t r i n g theory one uses the obvious requirements of u n i t a r i t y and causality to derive t h a t the s t r i n g is governed by a local Lagrangean on the s t r i n g world sheet. To derive similar requirements f o r the s t r i n g s born from black holes is f a r from easy. This is presently what is holding u s back from considerations such as tachyon elimination and anomaly cancellation t h a t so successfully seem t o have given u s the superstring scenario. What we advertise is a careful though slow process establishing the correct demands f o r a f u l l black hole/string theory. If successful, one w i l l know exactly the rules of the game and the ways how to select good from false scenarios and models. 2. QUANTUM HAIR

Classical

black

holes

are

characterized

by

exactly

three

parameters3:

.

the mass M , the angular momentum L , and the electrfc charge Q If magnetic monopoles exist in nature then there w i l l be a fourth parameter. namely magnetic charge Q, , and if besides electromagnetism U(1) gauge fields then also t h e i r charges there a r e other long range correspond to parameters f o r the black hole. However, the existence of long range U(ll gauge fields other than L , Q (and electromagnetism seems to be rather unlikely. Then, since Qm ) a r e all quantized, the number of different values they can take is limited, and indeed one can argue convlncingly (more about t h i s in Ref') that the black hole can be in much more different quantum s t a t e s than the ones labeled by L and Q (and Qm 1, or in other words, the mass M must be a function of much more variables than these quantum numbers alone. An interesting attempt to formulate new quantum numbers f o r black holes w a s initiated by Preskill. Krauss, Wilczek and others?. They took as a model field theory a U(1) gauge theory in which the local symmetry Ne . In undergoes a Higgs mechanlsm v f a a Higgs field with charge addition one postulates the presence of particles with charge e In such a theory there exist vortices, much like the Abrikosov vortex in a super conductor. These vortices can be constructed as classical solutions with cylindrical symmetry, at which the Higgs field makes one f u l l rotation if one follows It around the vortex. The behavior near the vortex of particles whose charge is only e is more complicated. One finds that because of the magnetic flux in the Abrikosov vortex the fields of these particles undergo a phase rotation when they flow around the vortex, in such a way t h a t an Aharonov-Bohm effect is seen. The Aharonov-Bohm phase is 2n/N , or, if we take a particle with charge ne , t h i s phase w i l l be 2nn/N The importance of t h i s Aharonov-Bohm phase is t h a t it w i l l be detectable f o r any charged particle, at any distance from the vortex, in such a way that we w i l l detect its charge modulo N . This is surprising because there is n o long range gauge field present! An observer who can only detect large scale phenomena may not be able t o uncover the chemical composition of the particle, but he can A l l he needs is a vortex, which t o him determine I t s charge modulo N w i l l look Just like a Nambu-Goto string. Even if a particle were absorbed by a black hole, its electric charge would still reveal itself. Thus, charge modulo N is a quantum number t h a t w i l l survive even f o r black holes. I t must be a s t r i c t l y conserved charge. One can then formallze the argument using only s t r i n g s and charges modulo N , without ever referring to the original gauge field. Then there may exist many kinds of strings/vortices, so t h a t the black hole may have a rich spectrum of these pseudo-invisible but absolutely conserved charges. W i l l t h i s argument allow u s to specify all quantum numbers f o r a black hole? There are several reasons t o doubt this. One is t h a t a n extremely large number of different kinds of s t r i n g s must be postulated, which seems t o be a substantial departure from the Standard Model at large distance scales. Secondly, it is not at all obvious that it w i l l be possible t o do Aharonov-Bohm experiments with black holes. One then has to assume f f r s t that black holes indeed occur in well-defined quantum s t a t e s , j u s t like atoms and molecules. So t h i s argument that black holes have quantum h a i r is r a t h e r circular. In my lectures there is no need f o r the mechanism advertised by Preskill e t al. I t is neither necessary nor likely t h a t all quantum s t a t e s can be distinguished by means of some conserved quantum number(s). In my other lectures I use j u s t the assumption t h a t quantum s t a t e s exist, and nothing else. No large-scale s t r i n g s a r e needed.

.

.

.

649

3. DECAY INTO SMALL BLACK HOLES Due to Hawking r a d i a t i o n t h e black The i n t e n s i t y of t h e r a d i a t i o n will is t h e temperature, and t h e t o t a l Schwarzschlld black hole is 4nR2 ;

T = 1/8nM

hole looses energy, hence also m a s s . be proportional t o 'T , where T a r e a of t h e horizon, which f o r t h e R = 2M . Since one expects'

,

(3.1)

t h e m a s s l o s s should obey

The c o n s t a n t s C , C' depend on the number of independent p a r t i c l e types at t h e corresponding mass s c a l e , and t h i s w i l l v a r y s l i g h t l y with temperature; t h e c o e f f i c i e n t s w i l l however s t a y of o r d e r one (as long as M stays c o n s i d e r a b l y l a r g e r t h a n t h e Planck mass). Ignoring t h i s s l i g h t m a s s dependence of C' , one f i n d s

= C" ( t*o - t )+

M(t)

,

(3.3)

where to is a moment where t h e t h i n g explodes violently. Conversely, t h e lifetime of a n y given Schwarzschild black hole with mass M can be estimated t o be

ti

=

M3/3C'

.

(3.4)

Now t h i s is t h e time needed f o r t h e complete disappearance of t h e black hole. One may a l s o a s k f o r t h e average lifetime of a black hole i n a given quantum mechanical s t a t e , i.e. t h e average time between two Hawking emissions. A rough estimate reveals t h a t t h e wavelength of t h e average Hawking p a r t i c l e 1s of t h e o r d e r of t h e black hole r a d i u s R , and t h a t t h i s is also t h e expected average s p a t i a l d i s t a n c e between two Hawking p a r t i c l e s . Therefore t h e lifetime of a given quantum state is of o r d e r R , i.e. of o r d e r 1/M i n Planck u n i t s . I n t h e language of p a r t i c l e physics t h i s implies t h a t t h e r a d i a t i n g black hole is a resonance state t h a t i n a n S matrix would produce a pole at t h e complex energy value E

where value

C,

=

M

-

C, i/M

,

(3.4)

is a g a i n a c o n s t a n t of o r d e r one. This corresponds t o t h e

EZ = MZ - 2C3 iMp12

(3.5)

for t h e Mandelstam variable s . We s e e t h a t all black hole poles are expected t o be below the real axis of s at a u n i v e r s a l average d i s t a n c e of o r d e r one i n u n i t s of t h e Planck mass s q u a r e d . It is not altogether unreasonable t o assume t h a t a black hole is J u s t a pole i n t h e S matrix like a n y o t h e r t i n y physical object.

*As was pointed o u t by t h i s author', t h e derivation of t h i s f o r m u l a r e q u i r e s a n assumption concerning t h e i n t e r p r e t a t i o n of quantum wave f u n c t i o n s f o r p a r t i c l e s disappearing i n t o t h e black hole. Though p l a u s i b l e , one c a n imagine t h i s assumption t o be wrong, i n which c a s e t h e black hole temperature w i l l be d i f f e r e n t from (3.1).

650

4. THE S-MATRIX ANSATZ AND THE SHIFTING HORIZON

The problem with linearised quantum field background 1s that the ingoing particles then the outgoing ones. Hilbert space is then I#),, x IS)out . If we were to describe a overall Schr6dinger equation then these inallowed to be independent of each other. In the existence of an S-matrix: I#)our

= s

1S)i.

theory in the black hole seem to be independent of a product space, IS) = black hole that obeys an and out-spaces cannot be contrast, one would expect (4.1)

P

and with t h i s mapping of in- to out-states the degrees of freedom pictured in Fig. la are replaced by the ones of Fig. lb or Fig. Ic.

matter Out

rvtter

matter

In

in

a)

b)

-

Fin. 1 a) Linearised quantum field theory produces

a Hilbert space that is the product of two factors: I#) = l # ) ~ n x l # ) o u ~ Ingoing partlcles form the factor space (bl. and (c). x and y are the Kruskal outgoing ones form {I#)out) Coordinates f o r the Schwarzschild metric.

A s stated earlier. the reason why superimposing in- and outparticles as in Fig. la is incorrect is the breakdown of linearised quantum field theory at distances closer than a Planck length from the horlzon. Gravitational interactlons there become super strong. W e can obtain the black hole representations of Fig. l b and Fig. lc by adopting the following elementary procedure: I) Postulate the existence of an S matrix, and il) take interactions between in- and out-states into account, in particular the gravitational ones. We can look upon this procedure a s a new and more precise formulation of the general coordinate transformation from Kruskal coordinates3 to Schwarzschlld coordinates, or from f l a t space-time to Rindler' spacetime. There is no direct contradiction with anything we know about general relativity or quantum mechanics, but because of the crucial role attributed to the interactions the picture is only somewhat more complicated.

651

future

'.... t n f a l 1 i n g ".... p a r t I c 1 e

t~o r I z o n,.."' o r 1zon

past event hor I zon

Fig. 2. The horizon displacement. The most important ingredient of t h e g r a v i t a t i o n a l i n t e r a c t i o n s is the horizon shift". Consider a n y p a r t i c l e f a l l i n g i n t o t h e black hole. I t s g r a v i t a t i o n a l f i e l d is assumed t o be so weak t h a t a l i n e a r i s e d d e s c r i p t i o n of i t , d u r i n g t h e i n f a l l , is reasonable. The c u r v a t u r e s induced i n t h e Kruskal frame a r e i n i t i a l l y much smaller t h a n t h e Planck length. Now perform a time t r a n s l a t i o n ( f o r t h e e x t e r n a l o b s e r v e r ) . A t the origin of Kruskal space this corresponds to a Lorentz transformation : x

+

T-lx;

y

+

.

T y

(4.2)

The 3 f a c t o r s h e r e c a n grow very quickly, exponentially with e x t e r n a l time. The v e r y t i n y i n i t i a l c u r v a t u r e s soon become s u b s t a n t i a l , b u t a r e o n l y s e e n as s h i f t s i n t h e y coordinate, because y h a s expanded s u c h a lot. The r e s u l t is a representation of t h e space-time metric where t h e ingoing p a r t i c l e e n t e r s along the y a x i s , with a velocity t h a t h a s been boosted t o become very close t o t h e speed of l i g h t . I t s e n e r g y i n t e r m s of t h e boosted coordinates h a s become so huge t h a t t h e c u r v a t u r e became s i z a b l e . I t is described completely by s a y i n g t h a t two halves of t h e conventional Kruskal space a r e glued together along t h e y a x i s with a shift 6 y , depending e x p l i c i t l y on t h e a n g u l a r coordinates 6 a n d (p . The ca_lculation of t h e f u n c t i o n 6 y ( 6 , ( p ) is elementary. I n Rindler s p a c e , 6 y ( x ) is simplyt1 6y(;;)

=

-

4c,p log("x2) +

c ,

(4.3)

is Newton's c o n s t a n t , p is t h e ingoing p a r t i c l e momentum, where G, and C is a n a r b i t r a r y c o n s t a n t . In Kruskal space _the a n g u l a r dependence of 6y is a b i t more complicated t h a n t h e x dependence i n Rindler space. I t is f o u n d by i n s e r t i n g E i n s t e i n ' s equation, which is RIlv = 0 , everywhere except where t h e p a r t i c l e comes in. S t a r t i n g with a n a r b i t r a r y 6y as a n Ansatz, one f i n d s Einstein's equation t o correspond t o (1

-

A,,,v)

6y(6,9)

=

0

,

(4.4)

Ad,, is t h e a n g u l a r Laplacian. A t t h e angles 6,,(p, where t h e where p a r t i c l e e n t e r s we simply compare wlth t h e Rindler r e s u l t (4.3) t o obtain (1

-

Ad,.)

6~(6,(p)=

K

Pin 62(6,(p;6,,(po)

.

where K is a numerical c o n s t a n t r e l a t e d to Newton's c o n s t a n t . This equation c a n be solved:

652

(4.5)

(4.6)

where 6, is t h e a n g u l a r s e p a r a t l o n between Other e x p r e s s i o n s f o r f a r e f

=

K~

rm-*'dz

(COS~~-COSZ)-~

(6,(p)

e-'fl

and

;

(bo,(p0)

. (4.7)

61

andi2

[-c0se1) f = 2'-4 c o+s h 'id3 (fnv'3)

(4.8)

where P is a Legendre f u n c t i o n with complex index (conical f u n c t i o n ) . From (4.7) one sees d i r e c t l y t h a t f o r a l l angles 61 f is positive. 5. SPACE-TIME SURROUNDING THE BLACK HOLE

The h o r l t o n s h l f t discussed in the previous section is a n e s s e n t i a l lngredient i n t h e S-matrix construction. Wlthout it we would not be able to perform t h i s t a s k . Now we a r e . We will d i s c u s s t h i s c o n s t r u c t i o n in t h e n e x t c h a p t e r . F i r s t one h a s t o understand what t h e r e l e v a n t degrees of freedom are and where in space-time they live. Here, p a r t l y a n t i c i p a t i n g on o u r r e s u l t s , we observe t h a t the outgoing c o n f i g u r a t i o n s w i l l depend o n what goes i n , and with a s e n s i t i v i t y t h a t depends dt is the time i n t e r v a l as s e e n by exponentlally w i t h dt/4M , where t h e d i s t a n t observer. However, the p a r t i c l e s g o i n g o u t l a t e r t h a n a

horizon singularity

ingofng matter wtth momentum 6 p

O n s e t o f horizon

Fig. 3

653

lapse of time of t h e o r d e r of At = 4 M log ( M / M p , ) , c a n no longer be described at all. We w i l l assume, f o r simplicity, t h a t t h e s e p a r t i c l e s w i l l be completely determined ( i n the quantum mechanical s e n s e ) by e a r l l e r e v e n t s , o r i n o t h e r words, one is not allowed t o choose t h e s e states a n y way one pleases. Simply counting s t a t e s (as one c a n d e r i v e from t h e f i n i t e black hole entropy"l) one notes t h a t , given a l l i n g o i n g p a r t i c l e s , t h e outgoing p a r t i c l e s can be chosen f r e e l y only d u r i n g a n amount of time At , o r v i s e versa. Any i n g o i n g p a r t i c l e w i l l not only a f f e c t t h e outgoing p a r t i c l e s a f t e r a t i m e lapse of o r d e r At , but a l s o all t h a t come o u t l a t e r t h a n t h a t . These l i m i t a t i o n s i n choosing in- and outgoing s t a t e s a r e J u s t what t h e y would be f o r a n y macroscopic system s u c h as a f i n i t e s i z e box containing a g a s o r liquid, connected t o t h e outside world f o r i n s t a n c e v i a a t i n y hole.

6. CONSTRUCTION OF THE

S-MATRIX

The argument now goes as followss. 1. Consider one p a r t i c u l a r i n - s t a t e and one p a r t i c u l a r o u t - s t a t e . Assume t h a t someone gave u s t h e amplitude defined by sandwiching t h e S-matrix between t h e s e two s t a t e s : (inlout)

.

(6.1)

Both t h e in- a n d t h e o u t - s t a t e a r e described by giving a l l p a r t i c l e s i n some conveniently chosen wave packets. The ingoing wave packets look like

i

where i r u n s over all p a r t i c l e s involved, and fin are smooth f u n c t i o n s . We assume them t o be s h a r p l y peaked i n t h e a n g u l a r coordinates s o t h a t we know exactly where t h e p a r t i c l e s e n t e r i n t o t h e h o r i z o n (so t h e angufar coordinates and t h e radial momenta o f all p a r t i c l e s a r e s h a r p l y defined). Similarly t h e outgoing wave packets a r e

let u s consider a s m a l l change i n t h e ingoing s t a t e : t i n ' ) . This brings about a s h a r p l y defined s m a l l change i n t h e d i s t r i b u t i o n o f t h e r a d i a l momenta pin(6.(p) on the horizon: Now

{in)

+

pin This

8pi,

+

Pin

+

6Pin(o,(P)

.

(6.4)

now produces a n ( e x t r a ) horizon s h i f t , 8y(R)

=

Jf(R-R')

6Pin(R')

,

(6.5)

where f i s t h e Green f u n c t i o n computed i n t h e previous s e c t i o n and R s t a n d s s h o r t f o r (19,cp) ; R-n' s t a n d s f o r t h e angle between n and

R'

.

The horizon s h i f t (6.5) does not a f f e c t t h e thermal n a t u r e of t h e Hawking r a d i a t i o n , but i t does change t h e quantum s t a t e s . A l l out-wave f u n c t i o n s are s h i f t e d . ( 6 . 3 ) is replaced by

( t h e e f f e c t of t h e s h i f t on f is of

lesser iinportance). The s h i f t

654

6y

is assumed t o be so small that the outgoing particle is not thrown over the horizon. This requires t h a t we consider only those outgoing particles t h a t are already sufficiently f a r separated from the horizon, or: they a r e not the ones t h a t emerge later than the time interval A t , a s defined in the previous section, see Fig. 3. This is why the time interval At w a s necessary. Now such a restriction w i l l also imply t h a t we w i l l have t o reconsider the definition of inner products in our Hilbert space, and t h i s w i l l imply that the operator exp(-ipAut6y) might not be unitary. We w i l l temporarily ignore t h i s important observation. We observe t h a t the S-matrix element (6.1)is replaced:

pout(n) is the total outgoing momentum at the angular Here, coordinates n What we have achieved is that we have been able to compute another matrix element of S . Now simply repeat t h i s procedure many times. We then f i n d all matrix elements of S t o be equal t o

.

where N is one common unknown factor. Apart from an overall phase, N should follow from unitarity. The derivation of (6.8) ignores all interactions other than the gravitational ones. W e w i l l be able to do better than t h a t , but l e t us f i r s t analyze t h i s expression. What is unconventional in the S-matrix (6.8) is the f a c t t h a t the inand out-states must have been characterized exclusively by specifying the total radial momentum distribution over the angular coordinates on the horizon. If there a r e more parameters necessary to characterize these s t a t e s , these extra parameters w i l l not figure i n the S-matrlx. But t h i s would mean that two different s t a t e s IA) and IB) could evolve into the same s t a t e , so these e x t r a parameters w i l l not be consistent with unitarity. We cannot allow f o r other parameters than the total momentum distributions (unless more kinds of interactions a r e taken into account). Thus, if f o r the time being we only consider gravitational Ipin(n)) and the interactions. the in-states can be given as out-states as lpout(n)) . The operators pin(S,cp) commute f o r different values of 6 and cp , and their representations span the entire Hilbert space; the same f o r pout(S,cp). The canonically conjugated operators uin(n) , uout(RI a r e defined by the commutation rules [p{n(n),u*"(n')l =

-iaZ(n-n')

(6.9)

(and similarly f o r the out operators), or, (uln(n)lptn(n)) =

where

C exp i.Jd2n ~ ~ ~ ( n ) u ,~ ~ ( n )

C is a normalization constant. In terms of the U operators the

(uouttn) lutntn) )

(6.lo')

S-matrix is

= ppoutnpin exp(-iptnu

i n +fPoutuout

-1PoutfPin)

s

(6.11) which is a Gaussian functional integral over the functions pout and f1 of f is ~-'(1 - A Q ) , the outcome of p i n . Since the inverse

655

t h i s functional integral is (O,,~(R) lufn(n)) =

C exp[-frc-*JdZR

uln(n)

( 1 - 4 ) uout(R))

=

The last term In the brackets 1s something like a m a s s term and becomes subdominant if we concentrate on s m a l l subsections of the horizon. Therefore it w i l l be ignored from now on. Eq. (6.12) seems to be more fundamental than ( 6 . 8 ) because it is local in R . Fourier transforming back we get

We reobtain (6.8) written as a functional integral. I t is Illuminating to redefine Pout = P-

;

PIn = p*

;

u0.t

= x-

;

Uin

= -x+

,

(6.14)

and t o replace the angular coordinate R by a transverse-coordinate x , so t h a t if we define the transverse momentum components p 0 one can write

Suppose now t h a t both the in-state and the out-state a r e wrltten as s e t s containing a f i n i t e number of particles, havirng not only fixed longitudinal momenta but also transverse momenta pf . We then hav_e,_to convolute the amplitude (6.15) with transverse wave functions exp(iplxl) f o r each particle i . It becomes

This functional integral is very s i m i l a r to the functional integral f o r a s t r i n g amplitude, lncluding the integration over Koba-Nielsen variab1esl3, except f o r the unusual imaginary value f o r the s t r i n g constant:

T

=

8nCHi

.

(6.17)

7. ELECTROMAGNETISM

What happens if more interactions a r e included? The simplest t o handle t u r n out to be the electromagnetic forces. Suppose t h a t the particles t h a t collapsed t o form the black hole carried electric charges. The angular charge distribution w a s Pin(R) As so

.

(7.1)

in the previous section, we consider a s m a l l change in t h i s setting, pi,m

+ PlnlR) + Gp&)

*

(7.2)

The Bpln(fl) produces an extra contribution to potential at the horizon which is not difficult to computeJ .

where

ro is the radius of the horizon, and

AQA(n) =

6pin(n)

A(R)

the

vector

must satisfy

.

(7.4)

The field (7.3) is only non-vanishing on the plane x=O , where it cause8 a sudden phase rotation for all wave packets that go through. An outgoing wave undergoes a phase rotation eiQ*(*) ; A(R) = ro-I A(R)

.

(7.5)

This rotation must be performed for a l l outgoing particles charge Q , A l l together the outgoing wave is rotated as follows:

with

IPouc(n)*Pout(~)) +

(7.61 exp -1 ld211sd%' f1tn-Q') pout(n)bpln(n') x l ~ o u ~ ~ Q ) , ~ o u ,~(~))

where

f1(Q-n') is a Green function that satisfies =

A p f1(n-ll')

.

-KeB2(n-n')

(7.7)

is a numerical constant. And using arguments identical to the ones of the previous section we repeat the infinitesimal changes to obtain the S-matrix dependence on p&) and pout(n) : K,

( ~ o u t ( n ) , ~ o u t ( Q )lPin(Q)*Pin(Q))

= (7.8)

e-fl~pout(RIf(n-n' 1 pin(n')d2CW2n'

e -r~~p,.~cn) r,(n-n' 1 ptn(nD)d2mZn' Now let us replace

p o u t ( f l ) p i n ( ~ ' )by

-4 (~out(n)-~in(~)) [pout(n'

)-pin(Q'

1)

.

(7.9)

This differs from the previous expression by two extra terms in the exponent. one depending on pout(Q) only and the other dependlng on pin(R) only. These would correspond to external "wave function renormalizatlon factors" that do not describe interaction between the in- and the out-state. So we ignore them. The electromagnetic contribution in ( 7 . 8 ) can then be written as a functional integral of the form

Now it may also be observed that the charge distribution a combination of Dirac delta distributions,

'The unit included in

e pin

.

of

electric

charge of

657

the

p

is actually

ingoing particle

1s

here

p ( ~ )=

~ ~ i ~ z ( ~ ;- noii )=

nie

.

(7.11)

Therefore, i f we add a n integer multiple of 2n/e t o t h e f i e l d rP t h e i n t e g r a n d does not change. I n o t h e r words: @ is a periodic v a r i a b l e . 4 a c t s exactly Adding (7.10) t o (6.15) we notice t h a t t h e f i e l d as a f i f t h , periodic dimension. Hence, electromagnetism emerges n a t u r a l l y as a Kaluza-Klein theory.

8 . HILBERT SPACE

In t h i s s e c t i o n we b r i e f l y r e c a p i t u l a t e t h e n a t u r e of t h e Hilbert space i n which t h e s e S matrix elements a r e defined. As explained i n Section 4, t h e states whose momentum and charge d i s t r i b u t i o n over t h e horizon were given by p(R1 and p(Q) include all p a r t l c l e s i n the black hole's v i c i n i t y . But ( i f f o r simplicity we ignore electromagnetism) we c a n a l s o form a complete b a s i s i n terms of s t a t e s f o r which t h e canonical o p e r a t o r s x(n) a r e given. These x(R) (eq. 6.12) may be i n t e r p r e t e d as t h e coordinates of t h e horizon. Apparently. the precise shape of the horizon determines the state of the surrounding particlest Furthermore, t h e . in-horizon and t h e out-horizon do n o t commute. Therefore, t h e positions of t h e f u t u r e event horizon a n d t h e p a s t event horizon do not commute with each o t h e r . If we d e f i n e a "black hole" as a n object f o r which t h e location i n space-time of t h e f u t u r e event horizon is precisely determined, we c a n define a "white hole" as a s t a t e f o r which the p a s t event horizon is precisely determined. The white hole is a linear superposition of black holes (and vice v e r s a ) ; o p e r a t o r s f o r white holes do not commute with t h e ones f o r black holes. I n o u r opinion t h i s r e s o l v e s t h e i s s u e of white holes in general r e l a t i v i t y . Obviously, it is important t h a t t h e horizon of t h e q u a n t i z e d black hole is not taken t o be simply s p h e r i c a l l y symmetric. In a black hole with a h i s t o r y t h a t is not s p h e r i c a l l y symmetric, t h e o n s e t of t h e h o r i z o n , i.e. t h e point(s1 i n space-time (at t h e bottom of Fig. 3) where f o r t h e f i r s t time a region of space-time emerges from which no timelike geodesic can escape to 9* , h a s a complicated geometrical s t r u c t u r e . I t s mathematical c o n s t r u c t i o n h a s t h e c h a r a c t e r i s t i c s of a c a u s t i c . One might c o n j e c t u r e t h a t t h e topological d e t a i l s of t h i s c a u s t i c s p e c i f y t h e quantum state a black hole may be In. The f a c t t h a t t h e geometry of t h e ( f u t u r e or p a s t ) horizon s h o u l d determine t h e quantum state of the s u r r o u n d i n g p a r t i c l e s g i v e s rise t o i n t e r e s t i n g questions and problems. In o r d i n a r y quantum f i e l d t h e o r y t h e Hilbert space describing p a r t i c l e s i n a region of space-time is Fock space; an a r b i t r a r y , f i n i t e , number of particles with specified positions o r momenta together define a s t a t e . But now, close t o t h e h o r i z o n , a state must be defined by specifying t h e total momentum e n t e r i n g (or leaving) t h e horizon at a given s o l i d a n g l e R Apparently we are not allowed t o s p e c i f y f u r t h e r how many p a r t i c l e s t h e r e were, a n d what their o t h e r quantum numbers were. Together all these p o s s i b f l l t i e s form j u s t one state. So, o u r Hilbert space is set u p d i f f e r e n t l y from Fock space. The d i f f e r e n c e comes about of c o u r s e because we have s t r o n g g r a v i t a t i o n a l i n t e r a c t i o n s t h a t we a r e not allowed t o ignore. The b e s t way t o formulate t h e s p e c i f i c a t i o n s of o u r b a s i s elements h e r e is t o assume a l a t t i c e cut-off i n the space of s o l i d a n g l e s (one "lattice point" f o r each u n i t of horizon s u r f a c e area somewhat bigger t h a n AX (the Planck d i s t a n c e s q u a r e d ) , and then t o s p e c i f y t h a t t h e r e BZ . should be exactly one ingoing and one outgoing particle at each The momenta are given by t h e o p e r a t o r s pi,,(Q) and p o U t ( n ) (and t h e p,,,(Q) I . The in- and out-operators o f c o u r s e c h a r g e s by p i , ( Q ) and do not commute. AX is extremely s m a l l , t h e t o t a l i t y One may s p e c u l a t e t h a t s i n c e

.

658

of all these particles may be indistinguishable from a n ordinary Dirac sea f o r the large-scale observers. Also one may notice t h a t the way conventional s t r i n g theory deals inand outgoing particles is remarkably similar. Before with integrating over the Koba-Nielsen variables the s t r i n g amplitudes a l s o depend exclusively on the distribution of total inand outgoing momenta (see concluding remarks in Sect. 6 ) . If Jilbert space is constructed entirely from the operators p(;) and x(x) then these operators a r e hermitean by c o n s t r y t i o n . But we have also- seen t h a t in terms of ordinary Fock space p ( x ) , and hence also x(x) a r e probably not hermitean. There a r e different ways to approach t h i s hermiticity problem, but we s h a l l not elaborate here on t h i s point. 9. RELATION BETWEEN TERMS IN THE HORIZON FUNCTIONAL INTEGRAL AND BASIC

INTERACTIONS IN 4 DIMENSIONS In principle one can pursue our doctrine t o obtain more precise expressions f o r our black hole S matrix by including more and more interactions t h a t we actually know to exist from ordinary particle theory. We should be certain to obtain a result t h a t is accurate a p a r t a limitation in the angular resolution, because particle from interactlons a r e known only up to a certain energy. In t h i s section we indicate some qualitative results. The details of our "presently favored Standard Model" may well change i n due time. We w i l l denote anything used as a n input regarding the fundamental interactions among in- and outgoing particles near the horizon. at whatever scale. by the words "standard model". Suppose the standard model contains a scalar field. The e f f e c t s of t h i s field w i l l be f e l t by slowly moving particles at some distance from the horizon. But at the horizon itself these e f f e c t s a r e negligible. Consider namely a particle such as a nucleon, surrounded by a s c a l a r field such as a pion field. Close to the horizon t h i s particle w i l l be Lorentz boosted to tremendous energies. The scalar field configuration w i l l become more and more flattened. But unlike vector or tensor f i e l d s , its intensity w i l l not be enhanced (it is Lorentz invariant). So the cumulated effect on particles traversing it will tend t o zero. However, one effect due to the scalar field w i l l not go away. Suppose our standard model contains a Higgs field, rendering a U(1) gauge boson massive. This means that the electromagnetic field surrounding a f a s t electrically charged particle w i l l be of s h o r t range only. One can derive that the field equation (7.4) w i l l change into

(AQ-M,')A(~)

= 6p,,(R)

.

One may s a y t h a t the ingoing charge density pi,(n) charges coming from the Higgs particles. This implies t h a t the equations f o r the @ field obtain a m a s s term:

(9.1)

is screened by in Sect.

7 will

4 + 4 + A Note t h a t t h i s m a s s term breaks explicitly the symmetry This explicit symmetry breaking may be seen as a r e s u l t of the f i n i t e and constant value of the Higgs field at the origin of Kruskal space-time . Next, we may ask what happens if our standard model exhibits confinement. This means t h a t a t long distance scales no effect of the gauge field is seen and all allowed particles a r e neutral.

659

Confinement is u s u a l l y considered t o be t h e dually opposite of t h e Higgs mechanism: Bose condensation of magnetic monopoles. A magnetic monopole is a n object to which t h e end point of a Dirac s t r i n g is a t t a c h e d . A Dirac s t r i n g is a s i n g u l a r i t y in a gauge t r a n s f o r m a t i o n s u c h t h a t t h e gauge transformation makes one f u l l r o t a t i o n if we follow a loop a r o u n d t h e s t r i n g . W e must know how t o describe t h e operator f i e l d of a monopole at t h e horizon. Suppose a monopole entered at t h e s o l i d a n g l e Rl . This means t h a t a Dirac s t r i n g connects t o t h e black hole at t h a t point. The outgoing charged p a r t i c l e s undergo a gauge r o t a t i o n t h a t r o t a t e s a f u l l cycle if we follow a closed c u r v e around nl (an anti-monopole may n e u t r a l i z e t h i s elsewhere on t h e horizon). The gauge jump f o r t h e vector potential f i e l d A c a n be i d e n t i f i e d with t h e periodic f i e l d 0 of Sect 7 . So adding a n e n t e r i n g monopole t o t h e i n - s t a t e implies t h a t t h i s f i e l d @ is s h i f t e d by a n amount h(n) where A makes a f u l l cycle when followed over a loop a r o u n d nl. This is an operation t h a t is called disorder operator i n statistical p h y s i c s a n d f i e l d theory. This operator, 0, , is d u a l to t h e o r i g i n a l f i e l d 0 We find that the dual transformation electricity f--) magnetism corresponds t o t h e dualiky between @ and . Thus, if we have confinement, a m a s s term w i l l r e s u l t i n t h e e q u a t i o n s f o r 0, . I t explicitly breaks t h e symmetry @, -t @, + C And t h i s b a r s t h e transformation back t o 0 . Therefore, i f confinement occurs, the f i e l d 0 i s no longer well-defined, we have o n l y 0, , I t s m a s s w i l l be t h e glueball m a s s . I n Table 1 we list p e c u l i a r i t i e s of t h e mapping from 4 to 2 dimensions. The generators of local symmetry t r a n s f o r m a t i o n s i n 4 dimensions correspond t o t h e dynamic v a r i a b l e s i n 2 dimensions. Thus one expects t h a t if t h e s t a n d a r d model includes a g r a v i t i n o ( r e q u i r i n g a supersymmetry generator of s p i n $1 then a fermionic f i e l d v a r i a b l e w i l l emerge In 2 dimensions. But t h e above are merely q u a l i t a t i v e f e a t u r e s . They s h o u l d be t u r n e d i n t o precise q u a n t i t a t i v e r u l e s a n d p r i n c i p l e s , for which f u r t h e r work is needed.

.

.

10. OPERATOR ALGEBRA ON THE HORIZON A fundamental shortcoming of t h e procedure described above is t h a t t h e dimensionality_ of H i l b e r i space is i n f i n i t e from t h e start. The f u n c t i o n s p ( x ) and x(x) generate a n i n f i n i t e s e t of b a s i s elements. Y e t t h e black hole entropy, as calculated from Hawking r a d i a t i o n , is f i n i t e . Indeed, we have not yet been a b l e to reproduce Hawking r a d i a t i o n from o u r S-matrix. This is because we have ignored t h e t r a n s v e r s e components of t h e g r a v i t a t i o n a l s h i f t s , and t h e s t r i n g f u n c t i o n a l s we produced t h u s f a r only allow f o r i n f i n i t e s i m a l s t r i n g e x c i t a t i o n s . We s h a l l now t r y to improve our d e s c r i p t i o n of t h e b a s i s elements of Hilbert space. This we do by s e t t i n g up a n o p e r a t o r algebra. F i r s t we consider t h e a l g e b r a generated by the amplitudes we have. Our s t a r t i n g point h e r e is t h a t s t a t e s i n H i l b e r t s p a c e are uniquely determined by specifying a n y one of th? following f o u r p , ( x ) ,_ t h e outgoing f u n c t i o n s : th_e d i s t r i b u t i o n of ingoing momenta momenta p - ( x ) , t h e conjugated o p e r a t o r s x ' ( x ) , or x - ( x ) They obey the a l g e b r a

.

(10.1) (10.2)

(10.3)

(10.4)

(10.5)

Table 1

STANDARD m 3 D n IN 3+1 DIMENSIONS

String variables

Spin 2: BU"(X, t ) local gauge generator: u " ( x , t ) S p i n 1: A,(x. t ) local gauge generator: A( x , t

Scalar v a r i a b l e

(spin 0):

@(n)mod 2n/e

mod 2n/e

4(x, t )

Spin 0:

(spfn I):

X,(n)

0

0

INDUCED 2 DIMENSIONAL FIELD THEORY ON BLACK HOLE HORIZON

No f i e l d at a l l

H i g g s mechanism: "spontaneous" mass M A for vector f i e l d

0

Conflnernent i n vector f i e l d A,

e x p l i c i t symmetry breaking; @(n)g e t s same mass MA . 0 must be replaced by disorder

op. OD ; i t s symmetry broken. -

~~

0

Non-Abelian gauge t h e o r y

0

spin

4:

~~

only s c a l a r s 0 , corresponding t o Cartan subalgebra

I no f i e l d at a l l

feraions

B: g r a v i t i n o local gauge generator s p i n f

Spin

Spin

f fermion

(7)

The algebraic relations among &I and #(;'I a r e slightly more subtle because of the quantization of electric charge and the ensuing periodic boundary conditions on 0 . We w i l l disregard these from here on, The relations (10.1-5) are not infinitely accurate. This is because we neglected any gravitational curvature in the sideways directions, This is fine as long as transverse distance scales a r e kept considerably larger than the Planck scale. One may convince +one_self that t h i s implies neglecting higher orders in the derivatives a x - / a x Is there any way to obtain a more precise algebra? I t is natural to search f o r an algebra that is invariant under Lorentz transformations. One might hope that such an algebra could generate the correct dearees of freedom at the Planck scale (In particular quantized degrees of freedom). I t w a s proposed in Ref6 that H i l b e r t space on the horizon may be generated by the operator algebra of fundamental surface elements,

.

661

=

I#"(;)

E

eb

ax' a x y - . arb

(10.6)

aua

-

where the transverse coordinates x were replaced by more a r b i t r a r y s u r f a c e coordinates U', . The relations (10.1-5) may be used in the case U = x , when the derivatives a r e s m a l l . This means

The commutation r u l e s can then be rewritten in the form

c

[ZS"(a),ZS"(a')1

=

tT

E'""ArsA(a)s2(a-a~

,

(10.8)

A

which is written in such a way that it remains true in all coordinate frames. T is a constant ('string constant') equal to 8n i n Planck units. In stead of (10.1-51 we can take t h i s to be the equation t h a t generalizes to a r b i t r a r y surfaces. I t has the advantage of being linear in W . Now (10.8) is not a closed algebra, because the l e f t hand side still contains a summation. A complete algebra is obtained a s follows. Let K be I times the self dual part of W :

I t has three independent components:

K, =

i(w23

+

~ ' 4 )

;

K, =

i(w3'

+

~ 2 4 )

;

K, =

i(W'2

+

~ 3 4 )

.

(10.10) Now from (10.8) we derive that these obey a complete commutator algebra, [K,(a),Kb(a')l

=

.

fT~&~K~(~)62(~-~')

(10.11)

Apart from a complication to be mentioned shortly, t h i s is a local and complete algebra of the kind we were looking f o r . A t first sight it seems t o generate an infinite dimensional Hilbert space because the operators K , like the W , a r e distributions But let u s introduce t e s t functions f(u) , g ( u ) and define operators

.

then these s a t i s f y commutation rules: (10.13) Let us values angular (10.12)

L,"'

now r e s t r i c t to t e s t functions f(;) t h a t can only take the 0 or 1 . Then s a t i s f y the commutation rules of ordinary momentum operators. Note that f o r such an f the integral is nothing but a boundary integral:

=

IT-' $(x2dx3 + x'dx')

, etc.,

(10.14)

6f

where 6 f stands f o r the boundary of t_he support of f . We conclude t h a t f o r every closed curve 6f on U space we have three 'angular momentum' operators L e t f ) t h a t s a t i s f y the usual commutation r u l e s and

662

addition r u l e s f o r angular momenta. Given such a bunch of closed curves f i we can characterize the contribution of that part of the horizon to I i and m i These a r e Hilbert space by the usual quantum numbers discrete and so, in some sense, we seem to come close to our a i m of realizing a discrete Hilbert space f o r black holes. We note a n important resemblance with the loop variable approach to quantum gravityi5. L, a r e not Unfortunately, there is a snag. The operators anti-hermitean then hermitean. If we take x i to be hermitean and x' In the deflnition (10.6) W i j a r e hermitean and Wi4 anti-hermitean. Lat correspond to the anti-self dual p a r t s of Wuv. The Therefore, commutation r u l e s between L, and Lat a r e non-local (they follow from (10.1-5)). The operators Lz are hermitean, but not necessarily positive (they a r e only nonnegative f o r time-like surface elements). If we may assume the smallest surface elements to be timelike we can still build our surface using quantum numbers 1 , and m i but the s t a t e s we get are not properly normalized (it is f o r finding the norms of the s t a t e s t h a t we need hermitean conJugation). If

.

$ { 1 I , mi)

Li ,

a r e the basis elements constructed using the self dual operators and

4(lt.mi) the basis elements generated by the anti-self dual

L i t , then we have (10.15)

$ themselves, or the 4 themselves, a r e not orthonormal. rememkr our realization earlier t h a t actually t h e operators x+(G) and x-lx) are not hermitean, when we pass from the "horizon Hilbert space" to ordinary Fock space, because the s h i f t operators may move particles behind the horizon. I t is conceivable t h a t t h i s w i l l lead to hermiticity conditions altogether different from (10.15). But it is f a r from clear whether or not we actually obtained a complete representation of our Hilbert space.

but the

Now

REFERENCES 1.

2.

3.

4. 5.

S.W. Hawking, Commun. Math. Phys. 43 (1975) 199; J . B . Hartle and S.W. Hawking, Phys.Rev. D13 (1976) 2188; W.G. Unruh, Phys. Rev. D14 (1976) 870; R.M. Wald, Commun. Math. Phys. 45 (1975) 9 S . W . Hawking, Phys. Rev. D14 (1976) 2460; Commun. Math. Phys. 87 (1982) 395; S.W. Hawking and R. Laflamme, Phys. Lett. B209 (1988) 39; D.N. Page, Phys. Rev. Lett. 44 (19801 301, Gen. Rel. Grav. 14 (1987) 299; D.J. Gross, Nucl. Phys. B236 (1984) 349 C.W. Misner, K.S. Thorne and J.A. Wheeler. "Gravitation", Freeman, San Francisco, 1973; S . W . Hawking and G.F.R. E l l i s , "The Large Scale Structure of Space-time", Cambridge: Cambridge Univ. Press, 1973; E.T. Newman e t al. J. Math. Phys. 6 (1965) 918; B. Carter, Phys. Rev. 174 (1968) 1559; K.S. Thorne, "Black Holes: the Membrane Paradigm", Yale Univ. press, New Haven, 1986; S. Chandrasekhar, "The Mathematical Theory of Black Holes", Clarendon Press, Oxford University Press S . Coleman, Nucl. Phys. B310 (1988) 643; S.B. Giddings and A. Strominger,Nucl. Phys. B321 (1989) 4&1; ibid. B306 (1988) 890 P. Goddard, J . Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. E56 (1973) 109; M.B. Green, J.H. Schwarz and E. Witten, "Superstring

663

6.

7.

8. 9. 10. 11.

12. 13 14.

15.

Theory", Cambridge Univ. Press; D.J. Gross, e t a l , Nucl. Phys. B 256 (1985) 253 G. 't Hooft, Phys. Scrlpta T1S (1987) 143; fbid. T36 (1991) 247; Nucl. Phys. B33S (1990) 138; G. 't Hooft, "Black Hole Quantization and a Connection to String Theory" 1989 Lectures, Banff NATO A S I , Part 1, "Physics, Geometry and Topology, Series B: Physlcs Vol. 238. Ed. H.C. Lee, Plenum Press, New York (1990) 105-128; C. 't Hooft, "Quantum gravity and black holes", in: Proceedings of a NATO Advanced Study Institute on Nonperturbative Quantum Field Theory, CargBse, July 1987, Eds. G. ' t Hooft e t a l , Plenum Press, New York. 201-226 L. Kraus and F. Wilczek, Phys. Rev. Lett. 62 (1989) 1221; J. Preskill, L.M. Krauss, Nucl. Phys. B341 (1990) 50; L.M. Krauss. Gen. R e l . Grav. 22 (1990); S. Coleman, J.Preskll1 and F. Wllczek, preprint IASSNS-91/17 CALT-68-1717/ HUTP-91-AO16 C. 't Hooft, J. Ceom. and Phys. 1 (1984) 45 W. Rindler, Am.J. Phys. 34 (1966) 1174 T. Dray and G. 't Hooft. Nucl Phys. 8253 (1985) 173 W.B. Bonner, Commun. Math. Phys. 13 (1969) 163; P.C. Alchelburg and R.U. Sexl, Gen.Re1. and Gravitation 2 (19711 303 C . Lousto. private cornmunlcation 2. Koba and H.B. Nielsen, Nucl. Phys. B10 (1969) 633 , Ibid. B12 (1969) 517; B17 (1970) 206; 2. Phys. 229 (1969) 243 J . D . Bekenstein, Phys. Rev. D7 (1973) 2333; R.M. Wald, Phys. Rev. D20 (1979) 1271; G. ' t Hooft, Nucl. Phys. B256 (1985) 727; V.F. Mukhanov. "The Entropy of Black Holes", In "Complexity, Entropy and the Physics of Information, SFI Studies in the Sciences of Complexity, vol IX, Ed. W. Zurek, Addlson-Wesley, 1990. See also M. Schiffer, "Black Hole Spectroscopy", Sao Paolo preprlnt IFT/P 38/89 (1989) A . Ashtekar, Phys. Rev. D36 (1987) 1587; A. Ashtekar et a l , Class. Quantum Grav. 8 (1989) L185; C. Rovelli. C l a s s . Quant. Grav. 8 (1991) 297, ibid. 8 (1991) 317

664

CHAPTER 9

EPILOGUE [9.1] “Can the ultimate laws of nature be found?”, Celsius-Linnd Lecture, Feb. 1992, Uppala University, Sweden, pp. 1-12 .......,...

.

665

666

CHAPTER 9.1

CAN

THE

ULTIMATE LAWSOF NATURE BE FOUND?

Celsius/LinnB Lecture, Uppsala University, Sweden

by

Gerard ' t Hooft Institute f o r Theoretical Physics Princetonplein 5 , P.O. Box 80.006 3508 TA UTRECHT, The Netherlands

Summary: Physicists a r e probing ever smaller structures In the world of fundamental particles. Each time we dig deeper, we f i n d different kinds of particles ruled by different types of forces. Will there be an end to t h i s , can there be a n "ultimate law?" One thing we know f o r sure: at the distance scale of c m (the "Planck length") particles w i l l start feeling each others gravitational force, and t h i s force is s t r a n g e r and probably more fundamental than any other. Tiny "black holes" will make it impossible t o consider distances smaller than the Planck length. If there is an ultimate l a w of physics, it should probably be formulated a t t h i s distance scale.

This lecture is commemorated to two great scientists from Uppsala. Both of them have l e f t their marks in history. The f i r s t time I encountered the name of C a r l von LinnB w a s when I lived with my parents in the Hague, close to the sea shore. I used t o s t r o l l along the beach and collect s h e l l s . Back home I tried to identify my finds, and found t h a t many of them had beautiful Latin names with a mysterious "L" behind them. The books did not even bother to explain what the "L" meant. I t was supposed to be obvious. More than half of all shell species ever found on our beach were named and classified by the Linnaeus school and still c a r r y the names given by him. Anders Celsius not only rationalized the temperature scale, but is also known f o r quite a few important researches in astronomy. He managed t o check by observations Newton's prediction t h a t the e a r t h is flattened, the length of one a r c unit of the meridian being larger near the pole than near the equator. Like Linnaeus, he went t o Lapland t o do h i s research.

666

Before talking about "The Ultimate Laws of Physics" I have to explain what possibly can be meant with such a phrase. I t is probably not only in my country that physicists who talk about such things a r e accused of being arrogant, if not a bit crazy, by colleagues in other fields. Those who introduced the pretentious words "Theory of Everything" probably deserve such criticism. We a r e now giving an impression of searching f o r a Stone of Wisdom, and we seem t o pretend t h a t Physics might provide f o r ultimate answers to all questions. But of course t h i s Is not at all what we a r e trying to do. What we are after is most easily explained by giving a simple example of the kind of theories one might stumble upon when doing particle physics. In the e a r l y seventies it became fashionable among some physicists and mathematicians to play simple games on computers. Shortly a f t e r t h a t the desk top computers made these games accessible to anyone. One such game w a s "Conway's Came of Life"'. I t went as follows (see Fig. 1):

0

Fig.

= o

=

I

1. Evolution of a particular pattern in Conway's Game of Life, at three consecutive times. It will propagate diagonally over the lattice.

On a rectangular infinite lattice we have "cells", each of which c a r r i e s one b i t of information: the cell is s a i d to be "alive" if t h i s bit of information is a one; i t is "dead" if it is a zero. Then there is a clock. A t every tick of the clock the contents of each cell is being updated. For each cell the new s t a t u s , at time t + l , depends on the contents of itself and its nearest eight neighbors a t time t (Fig. 2).

Fig. 2. The cell in the center is updated depending on what w a s there before, and on what w a s in the eight surrounding c e l l s drawn here The r u l e is as follows:

-

If as If In

exactly 2 neighbors are alive, the cell in the center will s t a y i t was. exactly 3 neighbors are alive, the cell in the center will l i v e . all other cases the cell in the center will d i e .

In Fig. 1, I show a pattern that, a f t e r a while, r e t u r n s into i t s e l f , but at a different place: it moves! One can start with a n i n i t i a l configuration where several of these moving patterns are s e n t towards each other so t h a t they w i l l collide, and then study what happens. The

667

r e s u l t s may become r a t h e r complex, and without a computer it is hopeless t o calculate. The point I want to make is t h a t t h i s system is a "model universe". Imagine t h a t , given a large enough lattice and a patient enough computer, one can have "intelligent" creatures b u i l t out of these building blocks. They w i l l investigate the world they a r e i n , and perhaps ultimately discover the three fundamental "laws of physics" on which t h e i r universe is based. The question I wish to address now is: could it possibly be t h a t t h e universe we a r e i n ourselves has such a simple s t r u c t u r e , based on s u c h a simple universal Law, a Law t h a t is invariable and absolute? I s it conceivable that we w i l l ever be able to discover t h a t Law if it exists? Theoretical physicists a r e studying the universal l a w s t h a t govern o u r world, but it seems t h a t we a r e still very f a r away from the "smallest possible s t r u c t u r e s " , i.e. anything like the u n i t c e l l s in Conway's model. The l a w s of physics t h a t we did uncover could possibly l a w s t h a t describe r e g u l a r i t i e s , be considered as "effective l a w s " , c o r r e l a t i o n phenomena, of p a t t e r n s at very much l a r g e r distance scales and time scales than the smallest possible. I t should not come as a s u r p r i s e t h a t these l a w s seem to be much more complicated t h a n the Conway l a w s . If there is a smallest distance and time s c a l e , what could t h e laws of physics there be like? One might suspect t h a t these w i l l be something much more clever and beautiful than Conway's game. What can be s a i d about them? In a certain way physicists have already come close to finding universal l a w s that govern the t i n i e s t known $articles of matter. These laws a r e known as "The Standard Model" . The model is f a i r l y complicated, and it cannot predict the reaction of p a r t i c l e s and f i e l d s under all conceivable circumstances, but it seems to be a very good description of what is happening nearly everywhere in our universe. I will now give a rough description of t h i s model and its laws. The basic e n t i t i e s a r e "Dirac fermions" , elementary p a r t i c l e s t h a t show a c e r t a i n amount of spinning motion. This amount of spinning motion is measured in multiples of a fundamental constant of nature, Planck's constant divided by 2n : These particles a r e s a i d to have "spin 4". We start with particles t h a t can only move with the speed of light. For t h e s e p a r t i c l e s the a x i s of rotation can be seen to be always parallel t o t h e i r velocity vector, and one can derive t h a t t h i s statement is unique and independent of the velocity of the observer. This is why one can distinguish unambiguously particles t h a t spin "to t h e l e f t " from p a r t i c l e s t h a t spin "to the r i g h t " , with respect to t h i s axis. The f o r c e s these fermions e x e r t onto each other a r e now described by introducing another s e t of particles, the gauge bosons. These a r e p a r t i c l e s with spin one. they can e i t h e r be considered as the "energy quanta" of various kinds of e l e c t r i c and magnetic f i e l d s , or one can view t h e s e particles themselves as the transmitters of these forces. When t h a t picture is used one describes the force as being the consequence of an exchange of a gauge boson between two fermions: one fermion emits a boson and the other fermion absorbs it. If the mass of t h e boson is negligible then t h e efficiency of t h i s exchange process is inversely proportional to the square of the distance: t h e Coulomb force law. If t h e boson has a certain amount of m a s s when at r e s t , then the f o r c e it transmits will range only up to a distance inversely proportional to t h a t m a s s , and decrease very rapidly beyond t h a t

668

distance. An important feature of the gauge boson force between two fermions is t h a t before and a f t e r the emission of a gauge boson the fermion keeps the same helicity. This means t h a t a l e f t rotating particle remains left rotating, and a right rotating particle remains right rotating. This is of special importance f o r the neutrinos: neutrinos only exist as left rotating objects. Right handed neutrinos have never been observed (at l e a s t not f o r s u r e ) . In contrast, anti-neutrinos only come rotating towards the right, not left. By emitting a gauge boson a neutrino may t u r n into a n electron, but then the electron must rotate towards the left

.

I

generation

I

Fig. 3.

1

generation

II

[

generation

111

THE STANDARD MODEL

u(l)em

based on

Su(2),,,k x X SU(3)stiong Right handed neutrinos, which m y exfst, are indicated in dotted boxes.

W e can now make a listing of all fermions and gauge particles. See Fig. 3. Here we see that the fermions a r e divided into leptons and quarks. The left rotating objects a r e indicated by an “L“ and the right rotating ones by an “ R ” . For the anti-leptons and anti-quarks, which a r e not shown, the L and R a r e interchanged. Of all fermions only the quarks are sensitive to the forces of the eight gauge bosons in the box called “ S U ( 3 ) “ . This force is very strong and so all objects containing quarks w i l l be strongly interacting with other objects. A l l left rotating fermions are sensitive t o the forces from the S U ( 2 ) gauge bosons. Finally, all left rotating, and all electrically charged r i g h t rotating fermions feel the U ( 1 ) force fields. Now the standard model a l s o contains a spin 0 particle, the “Higgs particle”. It also transmits a force, but an important difference with the gauge boson forces is t h a t if a fermion exchanges a Higgs boson it has t o f l i p from l e f t rotating to right rotating or vice versa. Another difference is that the spin 0 particle can disappear s t r a i g h t into the vacuum. This implies t h a t some fermions can make spontaneous transitions from left t o right and vice versa. Technically, t h i s is the way one may

669

introduce mass f o r these fermions. A particle with mass moves slower than the speed of light and it turns out that f o r such particles the rotation axis cannot be kept parallel to the velocity vector. So t h i s mass can only be there if the particle has the ability to change i t s s p i n direction relative to its velocity vector. One f i n d s t h a t m a s s of a fermion is proportional to the coupling strength of t h a t fermion to the field of the-Higgs particle. The gravitational force could be added to our descript-\ ion of the Standard Model by a postulating a spin 2 particle, the graviton. The r u l e s f o r t h i s force a r e very s t r i c t l y prescribed by Einstein's Theory of " General Relativity, but only in a s f a r a s it a c t s collectively i on many particles. The details of multiple graviton exchange between individual particles a r e not completely understood, mainly because any effects due to such exchanges would be so tremend- ously weak that no experimental verification will C be possible i n a n y foreseeable future. The above is a qualitative description of the rules accord3 ing to which the particles in the Standard Model move. To formulate these rules in more precise mathematical terms, a s we were able t o do f o r Conway's Game of Life, would lead us way t .. beyond the scope of this I . lecture. In some sense the rules 6 I-, ' . -- '. a r e nearly as precise. .7\. Much of t h i s information became known in the early 70's. Fig. 4 . "Neutral current event": A dramatic new prediction of e- U,, * e- ZO U,, + e- u p t h i s scheme at that time were In a beam of n e u t r i n o s moving upwards, the effects due to exchange of an e l e c t r o n ( a r r o w ) is thrown i n an the newly predicted Zo componupwards d i r e c t i o n , l e a v i n g a t r a c e made ent of the gauge fields. One v i s i b l e i n a bubble chamber. of the f i r s t pictures made of an ( p h o t o from H . F a i s s n e r , Aachen) electron h i t by a muon-neutrino is shown i n Fig. 4 . I would like to point out the similarities between the Standard model and an ideal "Theory of Everything": a l l material objects in the universe have to move according to its rules. But of course the model is f a r from perfect. One reason is that we expect the model not to provide the equations of motion f o r matter under a l l conceivable circumstances. One can imagine energies, temperatures and matter densities t h a t are s o large t h a t the situation cannot be mimicked in any accelerator. Although o u r model could in principle be used to calculate what will happen,

..

-

I

. a

-

670

there a r e several indications that one should not take such predictions seriously. A second reason is t h a t t h e mathematics of the model is less than perfect. Even i f we had infinitely powerful computers t o our disposal, some phenomena cannot be computed with any reasonable precision. One has t o rely on certain perturbative approximation schemes that sometimes f a i l catastrophically. Finally, in contrast with models such a s Conway’s, our interactions depend on a number of “constants of Nature” t h a t are to be given as r e a l numbers. The precision with which these r e a l numbers are. known w i l l always be limited. There a r e essentially 20 independent numbers: 3 gauge coupling constants, corresponding to the strength with which the various kinds of gauge bosons couple to the fermions. The S U ( 3 ) coupling constant is much larger than the one f o r S U ( 2 ) and U(1). Then there are several interaction terms between the Higgs field and the fermionic f i e l d s (Yukawa terms). Many of them correspond to the masses of the various fermions: 3 lepton masses: me , mu and m, 6 quark masses: m,, , m,, , m, , m, , m, and mb . 4 quark mixing angles, determining f u r t h e r details of the decay of exotic particles. 1 topological angle 0, , a peculiarity relevant only f o r the strong interactions; as far as is known it is very close t o zero. 2 self-interaction parameters f o r the Higgs field. One of these determines the Higgs m a s s MH , the other determines the Higgs-to-vacuum transitions. In combinations with the other constants it produces the gauge boson masses.

.

This adds up t o 19 “constants of Nature“, which a r e incalculable; they have to be determined by experiment. We could then add Newton’s gravitational constant G, , but t h i s could be used t o f i x the as yet a r b i t r a r y scale f o r m a s s , length and t i m e . S t r i c t l y speaking there is also the so-called cosmological coupling constant which is also incalculable, but it may perhaps be s e t t o be identically zero: it would be the 2 0 t h parameter. In principle, the Standard Model can be applied at any length and time scale. Indeed, we have nearly scale-invariance. This means that when we study particles and their scattering properties under a magnification glass, we see very nearly the same as without the magnification glass. Only the masses of particles seem to become much smaller if we magnify them3. but also the (non-gravitational) coupling strengths w i l l not be exactly the same. This is a subtle effect. When we consider a certain exchange process, f o r example the emission of a photon by a fermlon, one might discover in the magnification glass t h a t t h i s process actually took place in several steps: there could have been a very short-lived particle created temporarily, t o be reabsorbed immediately after. This implies that the effective coupling strength one experiences without t h e magnification glass, is actually the r e s u l t of a more complicated effect of various couplings at the microscopic level. One f i n d s t h a t any coupling constant g is changed by a s m a l l amount, typically of order g3 , when viewed at different scales. And so it may happen t h a t a coupling constant changes gradually when we go to widely different time- and length scales. Calculations show t h a t the U ( 1 ) force becomes gradually stronger at very small distance scales, the SUE!) force becomes somewhat weaker and the S U ( 3 )

671

force goes down more rapidly. Since these changes a r e f a i r l y slow, one has t o go t o tremendously s m a l l length scales before all three couplings become about equal in strength. In the simplest version of the Standard cm, way beyond anything one can see Model t h i s happens at about with whatever microscope or accelerator. A consequence of the variations of the coupling strengths at very s m a l l length scales is that Coulomb's Law of the electric attraction between opposite charges is no longer exactly valid. In stead of

F

=

C Q l Q z / r 2

,

one gets a very tiny modification of the power 2. For the gravitational coupling constant the story is very different. Gravity acts upon the m a s s of a particle, not its charge. If we go to shorter distance scales, one would have to locate the position of a particle with increasing precision. Due t o the quantum mechanical uncertainty relation

Ax Ap

h

(= Planck's constant/2n)

,

such particles w i l l tend to move around with very large momentum p = mv and since the velocity v is limited to that of light, the masses m of our particles w i l l increase. Therefore, the gravitational force w i l l no longer behave as a l/r' force but r a t h e r become a l/r4 phenomenon. There will be a scale where the gravitational force between individual particles will become stronger than any of the other forces. A t t h i s scale the gravitational force w i l l be the dominant one. I t is easy to calculate where t h i s happens. Using the three fundamental constants of nature, namely

-

the speed of light, Planck's constant, Newton's constant,

c = 2 997 924 km sec-', h = h/2n = 1.054 588 x lo-'' erg sec, G = 6.672 x lo-* cm3 g-' sec-',

we find a fundamental unit of length,

1%

=

1.6 x

cm ;

one u n i t f o r t i m e ,

'

J 7 -

=

5.4 x 10-44 sec ;

and a u n i t f o r m a s s : = J

22 pg

=

1.2 x

loz2 M e V =

1.2 x 10" TeV

.

G

These u n i t s are called the Planck length, Planck time and Planck m a s s or energy. I t is f o r s t r u c t u r e s at these distance, time and m a s s s c a l e s t h a t

672

our Standard Model needs to be thoroughly revised. Fluctuations of the gravitational force here become uncontrollable. In any perturbative approach the higher order corrections become larger than the lower order ones. Technically what t h i s implies is t h a t any conventional candidate theory of "Quantum Gravity" becomes unrenormalizable: ill-defined effects from the higher order corrections cannot be absorbed into redefinitions of the interaction terms. This unrenormalizability problem is something one can t r y to cure and attempts of a l l s o r t s have been made. But the real problem of quantum gravity is not the unrenormalizability; it is the f a c t t h a t the perturbative expansion has a dimensionful expansion parameter (Newton's constant) and t h a t henceforth perturbative corrections a t length scales much smaller than the Planck length w i l l be divergent. The only cure t o that would have to be a theory in which no time and length scales exist smaller than the Planck u n i t s . Certainly the space-time continuum w i l l have to be either d r a s t i c a l l y different from what we a r e used to, or be abandoned altogether. Although I do not believe t h a t curing the unrenormalizability problem w i l l help u s much in understanding quantum gravity, let u s look where some of those attempts led us anyway. The f i r s t observation w a s t h a t theories with higher symmetries a r e usually better convergent than others. A symmetry t h a t w a s considered extremely promising w a s called "super symmetry4": a relationship between fermions (particles with spin integer + 4) on the one hand and bosons (integer spin) on the other. If one t r i e s to construct a n interaction with t h i s symmetry a s a basis one finds as a force carrying particle the g r a v i t h o , a particle with spin 9 I t s supersymmetric partner must be the graviton. and so one obtains a theory in which the gravitational force h a s a natural place. Super symmetry greatly reduced the renormalization ambiguities, but did not eliminate the problems completely. The most curious idea is called "String Theory'". According to t h i s theory all point like particles must be replaced by s t r i n g like objects, e i t h e r open s t r i n g s having two end points each, or l i t t l e closed loops. Such objects can interact f o r instance by joining end points together followed by a rupture somewhere else, or by exchanging loops of s t r i n g . The big advantage of the scheme as it w a s much advertised w a s t h a t in a quantum theory of s t r i n g s one did not have to specify where and when exactly the joining and rupture take place. The transitions are gradual, and the resulting expressions f o r the amplitudes, in particular the higher o r d e r corrections, seemed to become much l e s s singular and divergent than corresponding expressions in conventional particle theory. The mathematics of these exchanges is very s t r i c t , and it turned out t h a t the exchange of the lightest closed s t r i n g loops reproduces all by itself the gravitational force. I t w a s a discovery by Michael Green in England and John Schwarz in Caltech that calculations i n s t r i n g theory could be made consistent to a high degree provided t h a t at those very s m a l l distance scales space-time exhibits 26 dimensions, To describe also particles with half-integer spin such as the electron and the quarks it w a s necessary t o introduce, again, super symmetry, but only on the s t r i n g itself. A very ingenuous idea came from the Princeton school, namely t h a t all fermionic excitations (all those t h a t cause half-odd-integer contributions to the spin) r u n along the s t r i n g only i n one preferred direction. In that direction space-time must be 10-dimensional; in the other directions excitations can take

.

673

place in 26 dimensions. This s t r i n g is called the "heterotic s t r i n g " and the nice thing about i t is t h a t it has a cork-screw nature, making a distinction between left and right, j u s t like in the r e a l world. To explain that the r e a l world only has 4 "visible" dimensions one would have to assume t h a t the 22 (or 6) residual dimensions are "compactified". This means t h a t in those directions space is rolled up very tightly like a tube. That there could be such compactified dimensions had already be proposed by Theodore Kaluza and Oscar Klein in the e a r l y 20's and is hence called the "Kaluza-Klein Theory". In t h i s theory the momentum of a particle in one of the compactified directions corresponds to electric charge, and gravitational fields in space-time in those directions a r e to be interpreted a s the electro-magnetic and other similar fields. The most intriguing aspect of t h i s theory w a s t h a t it seems to admit no free parameters as constants of nature a t a l l . This means that everything should be calculable and it w a s suggested that its laws a r e as precisely defined a s in Conway's model. But t h i s was not quite true. The theory, a s i t w a s formulated, requires the application of perturbative expansions j u s t l i k e the older particle field theories, and here, like always, one has the problem that the calculations will not converge ultimately. The r e a l problem, as I see it, w a s that the theory is formulated in terms of continuous s t r i n g coordinates as fundamental variables. They form a highly infinite manifold, and a r e vastly d i f f e r e n t from the nice, discrete zeros and ones in Conw-ay's cellular automaton. Maybe one should be able to calculate everything w i t h s t r i n g theory, but presently we can't. I t is conceivable that s t r i n g theory can be saved. Some suspect t h a t it is a topological theory, which means that not the details of the continuum a r e relevant, but only the ways boundaries a r e connected together. Maybe s o , but we do not understand how t o t u r n such ideas into practical prescriptions. A t the very best the theory is incomplete. Important pages of the manual a r e missing. Rather than trying to postulate and guess what our world should be like at the Planck length, one could t r y to deduce t h i s from more rigorous observations and arguments. One such observation is the following: One cannot measure any distance with more precision than the cm. Suppose namely t h a t one t r i e s t o detect Planck length, 1.6 x s t r u c t u r e s of such a s m a l l size. Now in particle theory we have the well-known "uncertainty principle":

Ax-Ap

2

$h

,

where Ax is the uncertainty in position, A p the uncertainty in the momentum m v of a particle, and h Planck's constant divided by 2n . If Ax must be a s small as the Planck length, then Ap must be a s big as the Planck mass times the velocity of light c . Both the measuring device and the thing measured must possess s t r u c t u r e s as s m a l l as Ax and therefore contain momenta as large a s Ap . Now if two objects with t h i s much momentum collide against each other they c a r r y a t o t a l amount of energy which in the center of mass frame exceeds the Planck energy, loi6 TeV. If they approach each other at a distance smaller than the Planck length the mutual gravitational interaction will become s o strong t h a t a tiny black hole will be formed. The s i z e of a black hole is proportional to its m a s s , and t h i s one willstherefore have to be again larger than the Planck length. A black hole can be viewed as being a

674

conglomeration of pure gravitational energy and its attraction t o matter is s o s t r o n g that whatever object enters its immediate neighborhood w i l l be swallowed without leaving a trace. In particular, the "thing we wanted t o measure" w i l l disappear into the hole. Thus, whenever we wish to describe potential theories concerning s t r u c t u r e s with sizes smaller than the Planck length we r u n into the fundamental problem of black hole formation. This problem is a very special and unique one in theoretical physics, and therefore I have advocated f o r some time now t h a t we should make theoretical studies of 7 tiny black holes as best as we can By doing Gedankenexperimenten with these black holes we should be able to find out at least how to formulate purely logical restrictions of self consistency of whatever theory one might imagine to apply a t the Planck scale. Some beautiful r e s u l t s have already been obtained in doing this. In the by now conventional theory of gravity (Einstein's theory of general relativity) any conglomeration of a certain minimal amount of matter within a certain s m a l l enough volume contracts due t o its own gravitational field, and t h i s contraction must lead t o a n implosion that cannot be averted. Whatever kind of information there w a s inside the imploding cloud of matter, the residual black hole w i l l c a r r y exactly three numbers that e n l i s t all properties t h a t can ever be measured or transmitted: its total mass (or energy), its total electric charge (positive or negative), and its total angular momentum. Nothing can ever 8 come out of the black hole, not even light, which is why it is black . But t h a t w a s the conventional or "classical" theory. I t w a s Stephen 9 Hawking's great discovery early 1975 t h a t one can say more than t h a t i f one applies what is known about quantum theory. He found t h a t black holes do indeed e m i t matter, and one can compute precisely the probability f o r the black hole t o emit j u s t anything. The emission goes exactly as if the surface of the black hole is hot: it is the radiation emitted by any object with a certain definite temperature. Hawking computed the temperature and found it to be inversely proportional to the black hole mass. The t i n i e r the black hole, the lighter it is and therefore the hotter it is. Hawking's discovery seems to t e l l us t h a t black holes a r e much more like ordinary physical obJects than the ideal black holes as descrlbed in Einstein's general relativity. They emit radiation and therefore they decay, Just like any heavy radio-active nucleus. This gave u s the idea t h a t the tiniest black holes may perhaps become indistinguishable from the heaviest "elementary particles". After all, particles carry gravitational flelds j u s t like black holes do, they decay radio actively j u s t like black holes do, and If one tries t o describe the gravitational f feld surrounding an elementary particle one finds the same expression as for a black hole. The idea that particles and black holes may ultimately be the same things s e e m s to be s o natural t h a t the consequences of such a n important assumption w i l l come as a surprise. The assumption namely does not fit with t h e technical details of Hawking's calculation. He found a very important difference between black holes and ordinary quantum mechanical obJects. The laws of quantum mechanics and particle theory can be applied perfectly to any small region in the neighborhood of a black hole, but not at all t o the black hole itself! We have become used to the f a c t t h a t there a r e uncertainty relations in quantum mechanics t h a t render it impossible f o r a physicist to predict exactly how a particle w i l l behave under given circumstances. in stead, the particle physicist

.

675

can give very precise s a t i s t i c a l predictions: given a large number of particles and an experiment t h a t is repeated many times, he can predict with ever increasing precision what the distribution of the outcome will be. For t h i s he uses parameters called “constants of n a t u r e “ . The worrisome thing with black holes is t h a t these parameters themselves w i l l be shrouded by uncertainty relations. Even o u r predictions of the s t a t i s t i c s w i l l be fuzzy. To me it is clear t h a t t h i s is a consequence of our understanding of the physics of black holes being incomplete. Not only incomplete but also wrong! The prediction of an experiment repeated many times should be sharp! Now here we have a perfect example of a paradox in our understanding of physical law, a paradox of the kind that in the past led t o the discovery of fundamental new theories such as quantum mechanics and relativity. Not every physicist agrees with t h i s view. Hawking, like several others, c l a i m t h a t the fuzziness comes about because there is an i n f i n i t e c l a s s of “universes“ and we simply do not know which universe we are i n . Each of these universes has slightly different constants of nature. But t h i s is a n agnostic argument I f i n d d i f f i c u l t t o accept. In t r y i n g t o devise an “improved” theory I h i t upon very strange theories indeed. Requiring the black hole to behave a s a whole in accordance with the laws of quantum mechanics I found s t r i k i n g similarities between black holes and s t r i n g s . Their difference in appearance is merely superficial. Mathematically these objects a r e s t a r t l i n g l y similar. Conceptually both theoretical constructs are prohibitively d i f f i c u l t . I t seems to be absolutely necessary t h a t the theory t o be constructed should be of a “topological” kind, as explained e a r l i e r . One can argue t h a t the Hawking black hole can only support a limited amount of information. I t s internal structure must, somehow, be discrete. As if it contained an a r r a y of zeros and ones, exactly like Conway’s “game of l i f e “ . But then we must realize that if you look at a black hole from close you j u s t see empty space. Therefore one might suspect t h a t indeed space and t i m e themselves can c a r r y only limited amounts of information, and as such may show much more similarity with Conway’s game of life than ever expected. Unfortunately it is impossible and it w i l l continue to be t o perform experiments impossible f o r any foreseeable amount of time t h a t directly reveal the s t r u c t u r e of space-time at the Planck length. We a r e condemned to do only thought experiments and a r e asked to s o r t out all possible theories by logical reasoning alone. This is a long and painful process without any guarantee of success. A poem by Chr. Morgenstern, a s noted e a r l i e r by M. Veltman, should serve a s a warning here (with English t r a n s l a t i o n ) :

-

-

Es gibt viele Theorien Die s i c h jedem Check entziehen, Dfese aber kann mann checken: Elend wird s i e dann verrecken. Many theories are presented For which no t e s t can be invented. But one t e s t you will appreciate: In boredom they’ 11 disintegrate.

676

References

' J . H . Conway, 1970, unpublished; M. Cardner, Sci. Amer. 224 (19711, Feb, 112; March, ibid. 226 (19721, J a n . , 104; S . Wolfram, Rev. Mod. Physics 55 (1983) 601.

106;

April,

114;

There a r e many t e x t books on the Standard Model. See f o r instance: I.J.R. Aitchison and A . J . G . Hey, "Gauge Theories in Particle Physics", Adam Hilger, 1989. More pedestrian: N. Calder, "The Key to the Universe", BBC, London 1977. J

So t h i s is opposite to what one should expect from d a i l y l i f e experiences: the m a s s of a sand grain will seem to be much l a r g e r when inspected through a magnification glass.

4See for instance P.C. West, Introduction Supergravity, World Scientific 1986.

to

5See for instance M . B . Green, J . H . Schwarz and E. Theory", Cambridge University Press 1987.

Supersymmetry Witten,

and

"Superstring

'See f o r instance "Black Holes: the Membrane Paradigm, ed. by Thorne, R . H . Price and d.A. Macdonald, Yale University Press, 1986.

K.S.

' t Hooft, Nucl. Phys. B335 (1990) 138 ' t Hooft, Physica Scripta T36 (1991) 247 ' t Hooft, in "EPS-8, Trends in Physics 1991, Proceedings of the 8th General Conference of the European Phys. Soc., Sept. 4-6, 1990, p a r t I ,

.'G.

G. G.

p. 187. Hawking, Proc. Roy. Soc. London A 300 (19671 187 S.W. Hawking and R. Penrose, Proc. Roy. Soc. London A 314 (1970) 529 R. Penrose, Phys. Rev. Lett 14 (1965) 57

8S.W.

Hawking, Commun. Math. Phys. 43 (1975) 199J.B. Hartle and S . W . Hawking, Phys.Rev. D13 (1976) 2188 W.G. Unruh, Phys. Rev. D14 (1976) 870 S.W. Hawking, Phys. Rev. D14 (1976) 2460 S . W . Hawking and G. Gibbons, Phys. Rev. D1S (1977) 2738 S.W. Hawking, Commun. Math. Phys. 87 (1982) 395

'S.W.

677

INDEX

Abbot, J. 618 Abers, E.S. 198, 512 Achiman, Y. 186,200 Adler-Bell-Jsckiw anomaly

Belavin, Alexander A. 5, 203, 286, 320, 351, 575

Bell, John S. 27, 142, 192, 202, 287, 320, 142, 176, 192,

351

302, 324

Bell-Tkeiman transformation

15, 95-100, 103, 109, 115, 325 Bender, I. 199 Bender, C.M. 575 Berends, h i t s A. 190,201 Bernard, C.W. 203 Big Crunch 614, 615 BigBang 615 Bjorken, James D. 185, 200, 442

Adler, Stephen L. 142, 287, 351 Aichelburg, P.C. 629, 644 Aitchiin, I.J.R. 677 Albright, C.H. 184, 185, 200 Alpert, M. 618 Altarelli, Guido 200 anomalous dimension 224 anomaly cancellation condition 366 anti-instantons 332 Appelquiet-Carrazone decoupling 367 Appelquist, Thomas 182, 200, 374 Arnowitt, R.L. 103 Ashmore, J.F. 249, 574 Ashtekar, Abhay 664 asymptotic freedom 175, 227, 231,

black hole decay 650 entropy 581, 623, 626 quantum 9, 580,583,619-664 spectrum 622 black hole Hilbert space 658 Bloch walls 285, 488 Bludman, Sidney A. 198 Bogoliubov, N.N. 30, 62, 142, 249 Bollini, C.G. 249, 574 Bonner, W.B. 664 Borel, &mile 7 Borel procedure, second 571-573 Borel resummation 378, 428-435,

242-246, 376, 435-438, 548

Bach, R. 605 background field method

194, 206, 585,

592-604

background gauge 194,596-599 Bais, F. Alexander 544, 546 Balachandran, A.P. 202 Balian, R. 199 Barber, M.N. 249 Barbieri, R. 154 Bardeen, William 202, 351 Barnett, R.M. 199 Bars, Itahak 202 Becchi, C. 15, 194, 202, 351 B6g, M.A. 198,200 Bekenstein, J.D. 621, 627, 664

55S562

Borel singularities 562-571 Bose condensation 8, 516, 540 boundary conditions in a box 493495 bounds on planar diagrams 379, 390-393 Brbzin, Edouard 574 brick wall model 621, 624-627 Brinkman, H.W. 644 Brodsky, Stanley J. 7 Brout, Robert 17, 26, 296

678

Cabibbo, Nicola 200 Cahill, Kevin 202 Calder, N. 677 Callan, Jr., Curtis G. 4, 197, 202, 249,

Dashen, Robert F. 201, 286, 296,351, 513, 574

D’Eath, Peter D. 635, 636, 644 decalan, C. 441 de Groot, Henri 452 A I = 1/2 rule 182, 183 De Rtijula, Alvaro 182, 189, 200 Deser, Stanley viii, 201, 605, 617, 618 de Wit, Bernard 202 DeWitt, Bryce 13, 198, 202, 206, 605 difference equations for diagrams 410-413 Dimopoulos, Saves 358,374 Dirac condition (for electric and magnetic charges) 5, 293, 295, 537-540 Dirac spinor algebra 156-159 Dirac, Paul A.M. 5, 295,353, 374, 512,

286, 351,453,513, 574

canonical transformations 83-94 Carazzone, J. 374 Carlip, Steve 618 Carroll, Sean M. 609, 618 Carter, B. 663 Casher, A. 374, 452 Caswell, W.E. 203, 575 Cauchy surface 609-614 carnality 26, 66-72 Ceva, H. 513 Chandraeekhar, S . 663 Chang, Ngee Pong 201 charmonlum 17!4-182 Cheng, T.P. 185, 200 chiral charge Q6 335-351 Chisholm rule 144 Chodos, A. 199 Christ, Norman 197 Christodoulou, D. 189, 201, 605 Christoffel symbol 587 Chriatos, G.A. 300, 351 Clark, T. 202 closed timelike curve 608 Coleman, Sidney 190, 197, 199, 201, 202,

546

disorder operator (or parameter)

458, 464,

491

disorder loop operator 495499 dispersion relations 26, 72-73 divergences 143 overlapping 146-148 Dolan, Louise 203 Dolgor, A.D. 627 Dray, Tevian viii, 582, 629-644 Dreitlein, J. 201 Drell, Sidney D. 442, 574 drsesed propagators 79-82, 166-173 Drouffe, Jean-Michel 199 dual transformation 285, 504-508

286, 319, 351, 374, 476, 512, 574, 575,663 collective coordinates 312 Collins, J.C. 203 combinatorial factors 160-162 combinatorics 120, 121, 151, 160-162 complex coupling constant 556-558 compoeite propagators 393 confinement (of quarks) 5, 191, 281-285, 456-576 oblique 5404543 constants of nature 671 Conway, J.H. 677 con way'^ Game of Life 667 Cmte, N. 453 Cornwall, J.M. 199 Coulomb phase 510 counterterms 26, 122, 145-154, 218, 225, 235-241 Creutz, M. 441, 513 Crewther, Rodney 300, 320, 351 crunch and bang theorem 614 Curtis, G.E. 635, 644 Cutkosky’s rule 68 Cutler, C. 618 cutting equations 61-73, 128

earth-moon-symmetry 188 Ehlera, Jurgen 644 Eichten, E. 200 Ellis, John 200 Ellis, G.F.R. 663 Englert, Francois 17, 27, 200, 287, 296 Epstein, Henri 142 9apu

319

eta problems (see &o U(1) problem) 191-194, 324, 326

Euler’s theorem 385 exceptional momenta 413-418 Eyink, G.L. 249 Faddeev, Ludwig D. 13, 198, 202, 604 Fhddeev-Popov ghost 24, 102-108 Faissner, H. 670 Farhi, Edward 618 Fayet, Pierre 189, 201 Feinberg, F.L. 200 Ferrara, Sergio 203 Feynman diagrams 14 Feynman, Richard P. 13, 604

679

Feynman rulea 32-40, 105, 106, 155-158, 380-383,392, 444,445 F i c k h , S.I. 103 fictitious symmetry 339 floating coupling constant 408 flux, electric 178, 496, 498-502 magnetic 496, 498 Radkin, Efim S. 198, 604 Fkampton, Paul H. 199 F'rbre, J.M. 200 Ftitzsch, Harald 192, 199, 202, 287, 351 Fkohlich, Jorg 201 Fkonedal, C. 197 Fubini, Sergio 575 Fqjikawa, K. 351 functional (or path) integrda 52-55, 83 Gaillard, Mary K. 200 Gardner, M. 677 Gastmenrr, h y m o n d 190, 201 gauge condition 23, 103 gauge invariant source insertions 592 gauge transformation law 18, 22 Gell-Mann, Murray 3, 192, 197-199, 202, 249, 287, 351, 374 Gell-Mann-Uvy sigma model 358, 362 Georgi-Gleshow model 177, 182, 185, 269, 290-294,465, 508 Georgi, Howard 177, 182, 19S201, 203, 286,512 Genraie, Jean-Loup 295,320 ghosts 22, 24, 93, 116 Fadd~v-Popov 24, 102-108 Giambiagi, J.J. 249, 574 Gibbons, Gary 617, 677 Giddings, S. 618, 663 Glaser,V. 142 Glaahow, Sheldon L. 177, 182, 199,200, 286, 351, 512 global symmetries 19, 21 Goddard, P. 663 Goldhaber, A.F. 542, 546 Goldman, T. 200 Goldetone, JeErey 521,546, 663 Goldstone particles 324, 519 Gott, J. Richard I11 579, 618 gravitational shock wave 62Great Model Rush 175 Green, Michael B. 663, 673, 677 Green function 42 basic 408, 409 Gribov, V. 7 Griseru, Marc T. 201-203 Gross, David d. 199, 201, 202, 286, 295, 351, 441, 453, 513, 574,605, 663 Gupta, V. 185, 200

Guralnik, G.S. 27, 296 Guth, Alan 617,618 Hagedorn, R. 196, 203 Hagen, C.R. 27, 296 Halpern, M.B. 202, 203 Hen, M.Y. 452 Harari, H. 199 Harrington, Barry J. 200, 203 Hartle, Jim B. 627,663, 677 Hasenfrats, Anna 299 Hasenfratz, Peter 299, 546 Haaslacher, B. 201 Hawking radiation 620, 660 Hawking, Stephen 9, 620, 627, 647, 663, 677 Hepp, K h u s 142 Hey, Antony J.G. 677 Higgs mechanism 2, 8, 176, 177, 460462, 521 Higgs, Peter W. 3, 12, 26, 176, 296, 512 Hilbert space huge 341 large 342, 496 physical 31, 343, 496 homotopy classes 264-268, 271, 272, 281, 284,495 Honerkamp, J. 202, 203, 206, 605 horizon shift 630,633, 634, 652 horror 281,283 Hudnall, W. 202 Iliopoulos, John 198, 199, 201,512 impulsive wavw 629 indefinite metric 78 instantons 6, 270, 321 Isler, Karl viii Itzykson, Claude 199, 574

J/$J particle 15, 17S182 Jackiw, Roman viii, 10, 27, 142, 192, 202, 203, 286, 287, 319, 320, 351, 374, 513, 546, 574,617, 618 Jacobian 13, 85 Jacobs, L. 441, 513 JafTe, R.L. 199 Jarlskog, Cecilia 184, 185, 200 Jevicki, A. 202 Johnson, K. 199 Jones, D.R.T. 96, 203, 575 Julia, Bernard 199, 286 Kadanoff, L.P. 513 Kalitsin, Stilyan viii KiillBn-Lehmann representation 81

33,5&58,

Kaluea-Klein theory 658,674 Kibble, Th. W.B. 13,27, 296, 512 Kluberg-Stern, H. 203 Koba, Z. 664 Kogut, John 199,201,202,286,462,674 Komen, Gerbrand 452 Koplik, J. 442 Komer, J. 199 Krueloel-Smkeres coordmatw 634 Kuchar,K. 618 Kummer, W. 197 Kundt, W. 644

Mohapatra, Rabindra N. 189,201 monopoles (magnetic) 5, 269,270, 283, 288-296, 625 Mukhanov,V.F. 664 multiloop diagrems 1.34-140

n-dimensional integrale 140-142, 163-167, 216 Nambu, Yoichiro 199,288,296,452 naturalnew 352-374 naturalnew breakdown maw scale (NBMS) 354 Nauenberg, Michael 452 Neveu, Andre 201,296,442 Newman, E.T. 663 Nicoletopodos, P. 200 Nieleen, Holger B. 178,199,286, 295,493, 613,617,664 non-local tramformatione W 9 4 Nuseinov,S. 442

Laflemms, R. 663 Lanc!4oS,C. 605 large&-time equation 63 Lautrup, Benny 165 Le Guillou, J.C. 674 Lee, Benjamin W. 26, 198,200,374,612 Lee, T.D. 189,196,201,203,255,286 Lee, C. 351 Lehmsnn,H. 612 Lepege,G.P. 7 Leutwyler, Harry 192,202,287 Levin, D.N. 199 LBvy,Maurice 374 Li, L.F. 197,201 Linde, Audrei 187,201,203 Lipatov, L.N. 574 LleweUyn Smith, Chrii H. 185, 198-200 LoUet0,Carlce 664 Low,Francis 3, 197,249 Lowenstein, J.H. 202

oblique confinement 540-543 Okun,Lev 262 Oleaen, P o d 178, 199, 286,295,452,493, 513 onsloop W t i w 127,236-241, 699804 Ore, A. 200 Ore, F.R. 299, 351,576 overlapping divergences 146-148

Page, DonN. 663 Paia, Abraham 185,200,201 Parmiuk, 0.S. 142 Parhi, Giorgio 202,296,442,612,674, 575 Park,S.Y. 200 Pati, Jogwh C. 184,200,201,296 Patraecioiu, A. 195,203 Pauli, Wolfgang 30,212,249 Penrose, Floger 629,630,633,636,638, 644 Perl, MartinL. 200 Peny,M.J. 203 Petennan, An& 3,142,208,249 phantomsolitone 528 phyeicalparticles 22 planar field theory 247 P h c k d e 8,584, 619,652 Plancklength 10,666 Pohlmeyer, K. 612 Politmr, H.David 182,200,441,574,606 Polyakov, Alexander M. 5, 252, 351, 513, 674,5’76 Popov, Vidor N. 13,198,604 Powell, J.L. 200 Prasad, M.K. 253 Prentki, Jacques 186,200

Ma, E. 201 Ma, S.K. 201 MacCaUum, M.A.H. 638,844 M a c d o d d , D.A. 677 macromxpic variables 514,536 magnetic mnopolee 5,269,270,283, 288496,526 Maiaui, Lucian0 199,200,512 Mandebtam, Stanley 198,199,202,287, 604 Mani, H.S. 186,200 M a r g u h , M. lM,203,286 Maxwell equStiOM 12, 95 Meieenercdht 523 metonspectrum 460-463 meta-alor ( M tdmialor) 368 meta-quarka (or quinks) 358 metric termor g,,,, 579, 587-589 microampic variables 514,530 Migdal, A.A. 203, 575 2, 12,26, 198 M ~ Robert , M M , P e t e r 199 Miener, CluuIea W. 663

681

Preskill, J. 649,664 Price, R.H. 677 Primack, Joel 201 propagator 34, 37,38 bare 79 composite 393 dressed 79-82, 168-173 pseudo-gauge transformation 497

rainbow digrams 377 Rajaraman, R. 201 Rapidis, P. 200 Rebbi, C. 286,319,351,374,441,513,546, 574,663 regularization 25, 122-142 dimensional 122-154, 212,213 regulators, Pauli-Villars 30, 231 space-time dependent 307,308 unitary 59 renormalhation 25, 122, 142-154, 21C215 dimensional 194,220-228,315,316 renormalization group 4, 205-249 equation 230,41-28, 553-556 order by order 150, 151 renormalom, infrared 570 ultraviolet 380, 569, 570 Riemann curvature 588,606 Rindler, W. 664 Rivasseau, V. 441 Root, R.G. 201 h n e r , J. 200 RQSS, Douglas A. 187,197,201 Fbuet, A. 15, 194,202, 351 Rovelli, Carlo 664 Rupertsberger, H. 202

Schwinger, Julian 198, 199, 295,452 Schwinger’s model 179, 191 Senjanovic, P. 202 Segrb, G. 197 Sexl, R.U. 629,633,636,641,644 Shaw, R. 198 Shei, S A . 351 Shellard, Paul 636 Shirkov, D.V. 249 Siegel, W. 203 Sinclair, D.K. 201 sineGordon model 256 Singer, M. 200 Sirlin, A. 198 skeleton expansion 387-389 Skyrme, T.H.R. 351 Slavnov-Taylor identities 108, 115, 118, 151, 152, 194,325 SlaVnOV, Andrei 27,351,442 solitons 191,260-264,476, 477 Sommerfield, Charles 253 Soni, A. 200 sources 35,37,44, 161 physical 47, 592 sphalerons 299 spherical model 377,439-441 stability, against SC8hg 259 Standard Model 16,647,669 Staruszkiewicz, A. 618 Stora, Raymond 15,202,351 Strathdee, J. 203,513 Streater, R.F. 512 string theory 648 strings 26Ck264,269 Strominger, A. 663 Stueckelberg, E.C.G. 3, 142,208,249 supersymmetry 8 relaxed 189 Susskind, Leonard 199,202, 286,295,358, 374,452,574 Sutherland, D. 202 Suzuki, M. 201 Swieca, J.A. 203 Symanzik, Kurt 4, 195, 197,230,249,442

Sekita, B. 295, 320 Salam, Abdua 184,186,198,200,201,203, 296,513 Sarkar, Sarben 202, 203 scattering matrix 25,43-51, 100, 594 for black holes 645-664 Ansatz 651-653 Schechter, J. 185,200,202 Scherk, Joel 190,201 Schroer, Bert 197,201 Schwartz, Albert S . 5, 350,351 Schwarz, John H. 190,201,663,673,677

W a s h i , Y. 351,442 n u b , A.H. 644 Taylor, John C. 351,442 technicolor 358 theory of everything 670 Thorn, C.B. 199,663 Thorne, Kip S. 663,677 Tiktopoulos, G. 199 Tjia, M. 0. 184, 185,200 topological operators 475 topological stability 255, 258 topological singularities 534

quantum chromodynamics (QCD, color theory) 4, 176-179, 231, 281 quantum hair 648,649 quark confinement 5, 191,281-285, 456-576 oblique 54Ck543 Quinn, Helen R. 201,203

682

Tsao,Hung-Sheng 201,605 tunneling 275-277 Tuttle, W.T. 442 Tyupkin, Yuri S. 5, 15,351 Tyutin, I.V. 198, 604

Weinstein, M. 574 Weisskopf, Victor F. 199 West, P.C. 677 Wheeler, John A. 663 Wick, G.C. 196,203, 286 Wightman, Arthur S. 512 Wilczek, Frank 199,441,574,605,649,

U(l) dilemma 321,337-339 U(l) problem 300,323-375 Ueda, Y. 200

664 Willemsen, J.F. 201 Wilson, Kenneth G. 7, 197, 199,200,286,

unitarity 25, 74-77, 100 unitary regulators 59 unitary gauge 526-528, 532 Unruh, W.G. 627,628,677

441,574 Wilson loop operator 488 Witten, Edward 351,618,663,677 Wolfenstein, Lincoln 184,200 Wolfram, Steve 677 w u , C.C. 201 Wu, Tai Tsun 575

vanDam,Henk 13 van Damme, Ruud 249,374,442 van Kampen, Nicolaaa G. vii, 249,452 van Nieuwenh-n, Peter 201-203, 605 Veltman, Martinus J.G. 2,13,28-173,187, 197, 198, 201,249,351,574 vertex 37 vertices 34 Villars, F. 30,212,249 Vinciarelli, P. 200, 201 vortex 5, 178,261,264, 282,283,523 vortices, 261

Yang-Mills theory 2, 12 Yang, Chen Ning 2, 12,26,198,296 Yankielowicz, S . 574 Yildis, Asim 200,203

Z ( N ) symmetry 486-492 Zee, Antony 185,199,200,286 Zeldovich, Ya.B. 621,627 zero eigenstates 311 Zimmermann, W. 202 Zinn Justin, Jean 26,198,574 Zuber, Jean-Bernard 203,574 Zumino, Bruno 185, 187,200,201,203, 295,296,493,513

Wdd, Robert M. 628,664 Ward, J.C. 198,351,442 Ward identitiea 95-100, 108,142,324 anomalous 345,346 wave function renormalization 117-1 19 Weinberg, Steven 27, 197-203, 286, 295,

512,628

683