Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 www.elsevierphysics.com

Asymptotic Freedom and QCD–a Historical Perspective

David J. Grossa aKavli Institute for Theoretical Physics, University of California, Santa Barbara CA 93106-4030, USA I describe the theoretical scene in the 1960’s and the developments that led to the discovery of asymptotic freedom and to QCD.

1. INTRODUCTION there was a rather successful phenomenological theory, but not much new data. The strong in- It was a pleasure to attend Loops and Legs in teractions were where the experimental and theo- Quantum Field Theory, 2004, and to deliver a his- retical action was, particularly at Berkeley. They torical account of the origins of QCD. The talks were regarded as especially unfathomable. The delivered at this exciting meeting are a dramatic prevalent feeling was that it would take a very illustration of how far QCD has developed since long time to understand the strong interactions its inception thirty years ago. Current and forth- and that it would require revolutionary concepts. coming experiments are performing tests of QCD For a young graduate student this was clearly the with amazing precision, and theoretical calcula- major challenge. The feeling at the time was well tions of perturbative QCD are truly heroic. In expressed by Lev Landau in his last paper, called particular it was especially satisfying for me to “Fundamental Problems,” which appeared in a meet at this conference some of the people, who memorial volume to Wolfgang Pauli in 1959 [1] . over the last 30 years have calculated the two, In this paper he argued that quantum field the- − three and four loop corrections to the β function ory had been nullified by the discovery of the zero that we calculated to one loop order 31 years ago. charge problem. He said:

“It is well known that theoretical 2. THE THEORETICAL SCENE physics is at present almost helpless in I would like first to describe the scene in the- dealing with the problem of strong in- oretical particle physics, as I saw it in the early teractions.... By now the nullification 1960’s at Berkeley, when I started as a graduate of the theory is tacitly accepted even student. The state of particle physics was then by theoretical physicists who profess almost the complete opposite of today. It was a to dispute it. This is evident from the period of experimental supremacy and theoretical almost complete disappearance of pa- impotence. The construction and utilization of pers on meson theory and particularly major accelerators were proceeding at full steam. from Dyson’s assertion that the cor- Experimental discoveries and surprises appeared rect theory will not be found in the every few months. There was hardly any theory next hundred years.” to speak of. The emphasis was on phenomenol- Let us explore the theoretical milieu at this time. ogy, and there were only small islands of theoret- ical advances here and there. Field theory was 2.1. Quantum field theory in disgrace; S-Matrix theory was in full bloom. Quantum field theory was originally developed Symmetries were all the rage. The field was di- for the treatment of electrodynamics almost im- vided into the study of the weak and the strong mediately after the completion of quantum me- interactions. In the case of the weak interactions, chanics and the discovery of the Dirac equation.

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It seemed to be the natural tool for describ- semiclassical expansions and numerical approx- ing the dynamics of elementary particles. The imations, was almost completely forgotten. In application of quantum field theory had impor- a sense the Feynman rules were too successful. tant early success. Fermi formulated a power- They were an immensely useful, picturesque and ful and accurate phenomenological theory of beta intuitive way of performing perturbation theory. decay, which was to serve as a framework for ex- However these alluring qualities also convinced ploring the weak interactions for three decades. many that all that was needed from field theory Yukawa proposed a field theory to describe the were these rules. They diverted attention from nuclear force and predicted the existence of heavy the non-perturbative dynamical issues facing field mesons, which were soon discovered. On the theory. In my first course on quantum field theory other hand, the theory was confronted from the at Berkeley in 1965, I was taught that Field The- beginning with severe difficulties. These included ory = Feynman Rules. In the United States, the the infinities that appeared as soon as one went main reason for the abandonment of field theory beyond lowest order perturbation theory, as well was simply that one could not calculate. Ameri- as the lack of any non-perturbative understanding can physicists are inveterate pragmatists. Quan- of dynamics. By the 1950’s the suspicion of field tum field theory had not proved to be a useful theory had deepened to the point that a pow- tool with which to make contact with the explo- erful dogma emerged–that field theory was fun- sion of experimental discoveries. The early at- damentally wrong, especially in its application tempts in the 1950’s to construct field theories to the strong interactions. The renormalization of the strong interactions were total failures. In procedure, developed by Richard Feynman, Ju- hindsight this was not surprising since a field the- lian Schwinger, Sin-itiro Tomanaga and Freeman ory of the strong interactions faced two enormous Dyson, was spectacularly successful in Quantum problems. First, which fields to use? Following Electrodynamics. However, the physical mean- Yukawa, the first attempts employed pion and ing of renormalization was not truly understood. nucleon fields. Soon, with the rapid prolifera- The feeling of most was that renormalization was tion of particles, it became evident that noth- a trick. This was especially the case for the pi- ing was special about the nucleon or the pion. oneering inventors of quantum field theory (for All the hadrons, the strange baryons and mesons example Dirac and Wigner). They were prepared as well as the higher spin recurrences of these, at the first drop of an infinity to renounce their appeared to be equally fundamental. The obvi- belief in quantum field theory and to brace for the ous conclusion that all hadrons were composites next revolution. However it was also the feeling of of more fundamental constituents was thwarted the younger leaders of the field, who had laid the by the fact thatno matter how hard one smashed foundations of perturbative quantum field the- hadrons at each one had not been able to lib- ory and renormalization in the late ’40’s. The erate these hypothetical constituents. This was prevalent feeling was that renormalization simply not analogous to the paradigm of atoms made of swept the infinities under the rug, but that they nucleons and electrons or of nuclei composed of were still there and rendered the notion of local nucleons. The idea of permanently bound, con- fields meaningless. To quote Feynman, speaking fined, constituents was unimaginable at the time. at the 1961 Solvay conference [2], “I still hold to Second, since the pion-nucleon coupling was so this belief and do not subscribe to the philoso- large, perturbative expansions were useless. All phy of renormalization.” Field theory was almost attempts at non-perturbative analysis were un- totally perturbative at that time. The nonper- successful. In the case of the weak interactions, turbative techniques that had been tried in the the situation was somewhat better. Here one 1950’s had all failed. The path integral, devel- had an adequate effective theory–the four fermion oped by Feynman in the late 1940’s, which later Fermi interaction, which could be usefully em- proved so valuable for a nonperturbative formula- ployed, using perturbation theory to lowest order, tion of quantum field theory as well as a tool for to organize and understand the emerging experi- D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 195 mental picture of the weak interactions. The fact and Pomeranchuk, a generation of physicists was that this theory was non-renormalizable meant forbidden to work on field theory. One might that beyond the Born approximation it lost all wonder why the discovery of the zero charge prob- predictive value. This disease increased the sus- lem did not inspire a search for asymptotically picion of field theory. Yang-Mills theory, which free theories that would be free of this disease. had appeared in the mid 1950’s was not taken The answer, I think, is twofold. First, many other seriously. Attempts to apply Yang-Mills theory theories were explored–in each case they behaved to the strong interactions focused on elevating as QED. Second, Landau and Pomeranchuk con- global flavor symmetries to local gauge symme- cluded, I think, that this problem was inherent in tries. This was problematic since these symme- any quantum field theory, that an asymptotically tries were not exact. In addition non-Abelian free theory could not exist. gauge theories apparently required massless vec- tor mesons–clearly not a feature of the strong in- 2.2. The bootstrap teractions. In the Soviet Union field theory was The bootstrap theory rested on two principles, under even heavier attack, for somewhat different both more philosophical than scientific. First, lo- reasons. Landau and collaborators, in the late cal fields were not directly measurable. Thus they 1950’s, studied the high energy behavior of quan- were unphysical and meaningless. Instead, one tum electrodynamics. They explored the relation should formulate the theory using only observ- between the physical electric charge and the bare ables. The basic observables are the S-Matrix el- electric charge (essentially the electric charge that ements measured in scattering experiments. Mi- controls the physics at energies of order the ul- croscopic dynamics was renounced. Field theory traviolet cutoff). They concluded, on the basis was to be replaced by S-matrix theory; a theory of their approximations, that the physical charge based on general principles, such as unitarity and vanishes, for any value of the bare charge as we analyticity, but with no fundamental microscopic let the ultraviolet cutoff become infinite (this is Hamiltonian. The basic dynamical idea was that of course necessary to achieve a Lorentz invariant there was a unique S-Matrix that obeyed these theory) [3]. “We reach the conclusion that within principles. It could be determined without the the limits of formal electrodynamics a point inter- unphysical demand of fundamental constituents action is equivalent, for any intensity whatever, or equations of motion that was inherent in field to no interaction at all.” This is the famous theory In hindsight, it is clear that the bootstrap problem of zero charge, a startling result that im- was born from the frustration of being unable to plied for Landau that “weak coupling electrody- calculate anything using field theory. All mod- namics is a theory, which is, fundamentally, log- els and approximations produced conflicts with ically incomplete.” [4] . This problem occurs in some dearly held principle. If it was so difficult any non-asymptotically-free theory. Even today, to construct an S-Matrix that was consistent with many of us believe that a non-asymptotically-free sacred principles then maybe these general prin- theory such as QED, if taken by itself, is incon- ciples had a unique manifestation. The second sistent at very high energies. In the case of QED principle of the bootstrap was that there were no this is only an academic problem, since the trou- elementary particles. The way to deal with the in- ble shows up only at enormously high energy. creasing number of candidates for elementary sta- However in the case of the strong interactions, tus was to proclaim that all were equally funda- it was an immediate catastrophe. In the Soviet mental, all were dynamical bound states of each Union this was thought to be a compelling rea- other. This was called Nuclear Democracy, and son why field theory was wrong. Landau decreed was a response to the proliferation of candidates that [1] “We are driven to the conclusion that for fundamental building blocks. The bootstrap the Hamiltonian method for strong interaction is idea was immensely popular in the early 1960’s, dead and must be buried, although of course with for a variety of reasons. Superseding quantum deserved honor.” Under the influence of Landau field theory, it rested on the solid principles of 196 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 causality and unitarity. It was real and physi- ray Gell-Mann ended his talk on quarks with the cal. It promised to be very predictive, indeed to summary [10], provide a unique value for all observables. The bootstrap promised that this hope would be re- “Let us end by emphasizing our main alized already in the theory of the strong inter- point, that it may well be possi- actions. This is of course false. We now know ble to construct an explicit theory of that there are an infinite number of consistent hadrons, based on quarks and some S-Matrices that satisfy all the sacred principles. kind of glue, treated as fictitious, but One can take any non-Abelian gauge theory, with with enough physical properties ab- any gauge group, and many sets of fermions (as stracted and applied to real hadrons long as there are not too many to destroy asymp- to constitute a complete theory. Since totic freedom.) The hope for uniqueness must the entities we start with are ficti- wait for a higher level of unification. In Berke- tious, there is no need for any con- ley, as in the Soviet Union, S-Matrix theory was flict with the bootstrap or conven- supreme, and a generation of young theorists was tional dual parton point of view.” raised ignorant of field theory. Even on the calmer 2.3. Symmetries East Coast S-Matrix theory swept the field. For If dynamics was impossible, one could at least example, I quote Marvin Goldberger who said [7], explore the symmetries of the strong interac- “My own feeling is that we have tions. The biggest advance of the early 1960’s learned a great deal from field the- was the discovery of an approximate symme- ory... that I am quite happy to discard try of hadrons, SU(3), by Gell-Mann and Yuval it as an old, but rather friendly, mis- Neeman, and then the beginning of the under- tress who I would be willing to recog- standing of spontaneously broken chiral symme- nize on the street if I should encounter try. Since the relevant degrees of freedom, espe- her again. From a philosophical point cially color, were totally hidden from view due to of view and certainly from a practi- confinement, the emphasis was on flavor, which cal one the S-matrix approach at the was directly observable. This emphasis was en- moment seems to me by far the most hanced because of the success of SU(3). Nowa- attractive.” days we realize that SU(3) is an accidental sym- metry, which arises simply because a few quarks S-Matrix theory had some notable successes, the (the up, down and strange quarks) are relatively early application of dispersion relations and the light compared to the scale of the strong inter- development of Regge pole theory. However, actions. At the time it was regarded as a deep there were drawbacks to a theory that was based symmetry of the strong interactions, and many on the principle that there was no theory, at least attempts were made to generalize it and use it in the traditional sense. As Francis Low said [9], as a springboard for a theory of hadrons. The “The distinction between S-Matrix most successful attempt was Gell-Mann’s algebra theory and field theory is, on the one of currents [12]. In an important and beautiful hand, between a set of equations that paper, he outlined a program for abstracting re- are not formulated, and on the other lations from a field theory, keeping the ones that hand between a set of equations that might be generally true and then throwing the are formulated if you knew what they field theory away [12], were and for which you do not know “In order to obtain such relations whether there is a solution or not.” that we conjecture to be true, we Nonetheless, until 1973 it was not thought proper use the method of abstraction from to use field theory without apologies. For exam- a Lagrangian field theory model. In ple as late as the NAL conference of 1972, Mur- other words, we construct a mathe- D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 197

matical theory of the strongly inter- quarks had not been seen, even when energies acting particles, which may or may wereachieved that were ten times the threshold not have anything to do with real- for their production. This was not analogous ity, find suitable algebraic relations to atoms made of nuclei and electrons or to nu- that hold in the model, postulate their clei made of nucleons. The non-relativistic quark validity, and then throw away the model simply did not make sense. The conclusion model. We may compare this process was that quarks were fictitious, mathematical de- to a method sometimes employed in vices. With this attitude one could ignore the ap- French cuisine: a piece of pheasant parently insoluble dynamical problems that arose meat is cooked between two slices of if one tried to imagine that quarks were real. This veal, which are then discarded.” attitude towards quarks persisted until 1973 and beyond. Quarks clearly did not exist as real par- This paper made quite an impression, especially ticles, therefore they were fictitious devices (see on impoverished graduate students like me, who Gell-Mann above). One might “abstract” prop- could only dream of eating such a meal. It was erties of quarks from some model, but one was a marvelous approach. It gave one the freedom not allowed to believe in their reality or to take to play with the forbidden fruit of field theory, the models too seriously. abstract what one wanted from it, all without having to believe in the theory. The only prob- 2.4. Experiment lem was that it was not clear what principle de- This was a period of great experimental excite- termined what to abstract? The other problem ment. However, I would like to discuss an in- with this approach was that it diverted attention teresting phenomenon, in which theorists and ex- from dynamical issues. The most dramatic ex- perimentalists reinforced each other’s conviction ample of this is Gell-Mann and George Zweig’s that the secret of the strong interactions lay in the hypothesis of quarks [13], the most important high-energy behavior of scattering amplitudes at consequence of the discovery of SU(3). The fact low momentum transfer. Early scattering exper- was that hadrons looked as if they were com- iments concentrated, for obvious reasons, on the posed of (colored) quarks whose masses (either events that had the largest rates. In the case of the current quark masses or the constituent quark the strong interactions, this meant searching for masses) were quite small. Color had been intro- resonant bumps or probing near forward scatter- duced by Yoichiro Nambu [14] , M.Y. Han and ing, where the cross section was largest. It was Nambu [15] and O.W. Greenberg [16]. Nambu’s not at all realized by theorists that the secret of motivation for color was two-fold, first to offer hadronic dynamics could be revealed by experi- an explanation of why only (what we would now ments at large momentum transfer that probed call) color singlet hadrons exist by postulating a the short distance structure of hadrons. Instead, strong force (but with no specification as to what prompted by the regularities that were discov- kind of force) coupled to color which was respon- ered at low momentum transfer, theorists devel- sible for the fact that color neutral states were oped an explanation based on the theory of Regge lighter than colored states. The second motiva- poles. This was the only strong interaction dy- tion, explored with Han was the desire to con- namics that was understood, for which there was struct models in which the quarks had integer val- a real theory. Therefore theorists concluded that ued electric charges. Greenberg’s motivation was Regge behavior must be very important and for- to explain the strange statistics of non-relativistic ward scattering experiments were deemed to be quark model hadronic bound states (a concern the major tool of discovery. Regge theory was of Nambu’s as well). He introduced parastatis- soon incorporated into the bootstrap program as tics for this purpose, which equally well solved a boundary condition. In response to this theoret- the statistics problem, but clouded the dynam- ical enthusiasm, the interest of experimentalists ical significance of this quantum number. Yet in forward scattering was enhanced. Opportuni- 198 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 ties to probe the less easily accessible domains able quantities by saturating dispersion relations of large momentum transfer were ignored. Only with a finite number of resonance poles on the one much later, after the impact of the deep-inelastic hand and relating these to the assumed Regge scattering experiments that had been ridiculed by asymptotic behavior on the other) were not so many as unpromising, was it understood that the much predictive equations, but merely checks of most informative experiments were those at large axioms (analyticity, unitarity) using models and momentum transfers that probe short or light- fits of experimental data. I was very impressed like distances. It used to be the case that when a with this remark and longed to find a more pow- new accelerator was initiated one of the first and erful dynamical scheme. This was the heyday of most important experiments to be performed was current algebra, and the air was buzzing with the measurement of the total p-p cross section. marvelous results. I was very impressed by the Nowadays, this experiment is regarded with little fact that one could assume a certain structure of interest, even though the explanation of Regge current commutators and derive measurable re- behavior remains an interesting, unsolved and sults. The most dramatic of these was the Adler- complicated problem for QCD. Ironically, one of Weisberger relation that had just appeared [18]. the principal justifications for this experiment to- Clearly the properties of these currents placed day is simply to calibrate the luminosity of the strong restrictions on hadronic dynamics. The machine. most popular scheme then was current algebra. Gell-Mann and Roger Dashen were trying to use 3. MY ROAD TO ASYMPTOTIC FREE- the commutators of certain components of the DOM currents as a basis for strong interaction dynam- ics. After a while I concluded that this approach 3.1. From N/D to QCD was also tautological–all it did was test the valid- I was a graduate student at Berkeley at the ity of the symmetries of the strong interactions. height of the bootstrap and S-Matrix theory. My This was apparent for vector SU(3). However it Ph.D. thesis was written under the supervision was also true of chiral SU(3), especially as the of Geoff Chew, the main guru of the bootstrap, current algebra sum rules were interpreted, by on multi-body N/D equations. I can remem- Weinberg and others, as low energy theorems for ber the precise moment at which I was disillu- Goldstone bosons. This scheme could not be a sioned with the bootstrap program. This was basis for a complete dynamical theory. I stud- at the 1966 Rochester meeting, held at Berke- ied the less understood properties of the algebra ley. Francis Low, in the session following his talk, of local current densities. These were model de- remarked that the bootstrap was less of a theory pendent; but that was fine, they therefore might than a tautology [8], contain dynamical information that went beyond “I believe that when you find that the statements of global symmetry. Furthermore, as particles that are there in S-Matrix was soon realized, one could check ones’ assump- theory, with crossing matrices and all tions about the structure of local current alge- the formalism, satisfy all these con- bra by deriving sum rules that could be tested ditions, all you are doing is showing in deep-inelastic lepton-hadron scattering exper- that the S matrix is consistent with iments. James Bjorken’s 1967 paper [20,21]), on × the world the way it is; that is the the application of U(6) U(6), particularly in- particles have put themselves there in fluenced me. In the spring of 1968 Curtis Callan such a way that it works out, but you and I proposed a sum rule to test the then pop- have not necessarily explained that ular “Sugawara model,” a dynamical model of they are there.” local currents, in which the energy momentum tensor was expressed as a product of currents For example, the then popular finite energy sum [22]. The hope was that the algebraic proper- rules (whereby one derived relations for measur- ties of the currents and the expression for the D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 199

Hamiltonian in terms of these would be enough scattering using similar methods [27]. We were to have a complete theory. (This idea actually clearly motivated by the experiments that were works in the now very popular two-dimensional then being performed at CERN. We derived a conformal field theories). Our goal was slightly sum rule that measured the baryon number of more modest–to test the hypothesis by exploiting the charged constituents of the proton. The ex- the fact that in this theory the operator prod- periments soon indicated that the constituents of 1 uct expansion of the currents contained the en- the proton had baryon number 3 –in other words ergy momentum tensor with a known coefficient. again they looked like quarks. I was then totally Thus we could derive a sum rule for the structure convinced of the reality of quarks. They had to functions [23] that could be measured in deep- be more than just mnemonic devices for summa- inelastic electron-proton scattering. In the fall of rizing hadronic symmetries, as they were then 1968, Bjorken noted that this sum rule, as well as universally regarded. They had to be physical dimensional arguments, would suggest the scal- point-like constituents of the nucleon. But how ing of deep-inelastic scattering cross sections [24]. could that be? Surely strong interactions must This prediction was shortly confirmed by the new exist between the quarks that would smear out experiments at SLAC, which were to play such their point-like behavior. After the experiments an important role in elucidating the structure of at SLAC, Feynman came up with his parton pic- hadrons [25]. Shortly thereafter Callan and I dis- ture of deep-inelastic scattering. This was a very σL covered that by measuring the ratio, R = σT , picturesque and intuitive way of describing deep- (where σL (σT ) is the cross section for the scat- inelastic scattering in terms of assumed point- tering of longitudinal or transverse polarized vir- like constituents–partons [28]. It complemented tual photons), one could determine the spin of the the approach to deep-inelastic scattering based charged constituents of the nucleon [26]. We eval- on the operator product of currents, and had the uated the moments of the deep-inelastic structure advantage of being extendible to other processes functions in terms of the equal time commuta- [29]. The parton model allowed one to make pre- tors of the electromagnetic using specific mod- dictions with ease, ignoring the dynamical issues els for these–the algebra of fields in which the at hand. I felt more comfortable with the ap- current was proportional to a spin-one field on proach based on assuming properties of current the one hand, and the quark-gluon model on the products at short distances. I felt somewhat un- other. In this popular model quarks interacted easy about the extensions of the parton model to through an Abelian gauge field (which could, of processes that were not truly dominated by short course, be massive) coupled to baryon number. distance singularities. At CERN I studied, with The gauge dynamics of the gluon had never been Julius Wess, the consequences of exact scale and explored, and I do not think that the model had conformal invariance [30]. However, I soon real- been used to calculate anything until then. We ized that in a field theoretic context only a free, discovered that R depended crucially on the spin non-interacting theory could produce exact scal- of the constituents. If the constituents had spin ing. This became very clear to me in 1970, when 1 zero or one, then σT = 0, but if they had spin- 2 , I came to Princeton, where my colleague Curtis then σL = 0. This was a rather dramatic re- Callan(andKurtSymanzik)hadrediscoveredthe sult. The experiments quickly showed that σL renormalization group equations [33], [34], which was very small. These SLAC deep-inelastic scat- they presented as a consequence of a scale invari- tering experiments had a profound impact on me. ance anomaly [36]. Their work made it abun- They clearly showed that the proton behaved, dantly clear that once one introduced interactions when observed over short times, as if it was made into the theory, scaling, as well as my beloved out of point-like objects of spin one-half. In the sum rules, went down the tube. Yet the exper- spring of 1969, which I spent at CERN, Chris iments indicated that scaling was in fine shape. Llewelynn-Smith and I analyzed the sum rules But one could hardly turn off the interactions be- that followed for deep-inelastic neutrino-nucleon tween the quarks, or make them very weak, since 200 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 then one would expect hadrons to break up easily lieved that the non-canonical scaling indicative of into their quark constituents. Why then had no a non-trivial fixed point of the renormalization one ever observed free quarks? This paradox and group would appear at higher energies. Much the search for an explanation of scaling were to happened during the next two years. Gerhard preoccupy me for the following four years. ’t Hooft’s spectacular work [32] on the renormal- izability of Yang-Mills theory, reintroduced non- 3.2. How to explain scaling Abelian gauge theories to the community. The About the same time that all this was hap- electroweak theory of Sheldon Glashow, Wein- pening, string theory was discovered, in one of berg and Abdus Salam was revived. Field the- the most bizarre turn of events in the history of ory became popular again, at least in applica- physics. In 1968 Gabrielle Veneziano came up tion to the weak interactions. The path integral with a remarkably simple formula that summa- reemerged from obscurity. Kenneth Wilson’s de- rized many features of hadronic scattering. It velopment of the operator product expansion pro- had Regge asymptotic behavior in one channel vided a tool that could be applied to the analysis and narrow resonance saturation in the other of deep-inelastic scattering. Most important from [31]. This formula was soon generalized to multi- my point of view was the revival of the renormal- particle S-Matrix amplitudes and attracted much ization group by Wilson [37]. The renormaliza- attention. The dual resonance model was born, tion group stems from the fundamental work of the last serious attempt to implement the boot- Gell-Mann and Low [33], E. Stueckelberg and A. strap. It was only truly understood as a theory of Petermann [34] and Bogoliubov and Shirkov [35] quantized strings in 1972. I worked on this theory . This work was neglected for many years, partly for two years, first at CERN and then at Prince- because it seemed to provide only information tonwithJohnSchwarzandAndreNeveu.Atfirst about physics for large space-like momenta, which I felt that this model, which captured many of the are of no direct physical interest. Also, before features of hadronic scattering, might provide the the discovery of asymptotic freedom, the ultravio- long sought alternative to a field theory of the let behavior was not calculable using perturbative strong interactions. However by 1971 I realized methods, and there were no others. Thus it ap- that there was no way that this model could ex- peared that the renormalization group provided plain scaling, and I felt strongly that scaling was a framework in which one could discuss, but not the paramount feature of the strong interactions. calculate, the asymptotic behavior of amplitudes In fact the dual resonance model lead to incred- in a physically uninteresting region. Wilson’s de- ibly soft behavior at large momentum transfer, velopment of the operator product expansion pro- quite the opposite of the hard scaling observed. vided a new tool that could be applied to the Furthermore, it was clear that it required for con- analysis of deep-inelastic scattering. The Callan- sistency many features that were totally unreal- Symanzik equations simplified the renormaliza- istic for the strong interactions–massless vector tion group analysis, which was then applied to the and tensor particles. These features later became Wilson expansion [39,42] . The operator prod- the motivation for the hope that string theory uct analysis was extended to the light cone, the may provide a comprehensive and unified theory relevant region for deep-inelastic scattering [38] . of all the forces of nature. This hope remains Most influential was Wilson’s deep understanding strong. However the relevant energy scale is not of renormalization, which he was then applying to 1 GeV but rather 1019GeV ! The view that the critical behavior. Wilson gave a series of lectures scaling observed at SLAC was not a truly asymp- at Princeton in the spring of 1972 [40]. These totic phenomenon was rather widespread. The had a great impact on many of the participants, fact that scaling set in at rather low momen- certainly on me. tum transfers, “precocious scaling,” reinforced this view. Thus the cognoscenti of the renormal- ization group (Wilson, Polyakov, and others) be- D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 201

3.3. The plan work and its application to deep-inelastic scat- By the end of 1972, I had learned enough field tering, that one might expect to get scaling in a theory, especially renormalization group methods quantum field theory at a fixed point of the renor- from Ken Wilson, to tackle the problem of scaling malization group. However this scaling would head on. I decided, quite deliberately, to prove not have canonical, free-field-theory-like behav- that local field theory could not explain the ex- ior. Such behavior would mean that the scal- perimental fact of scaling and thus was not an ing dimensions of the operators that appear in appropriate framework for the description of the the product of electromagnetic currents at light- strong interactions. Thus, deep-inelastic scatter- like distances had canonical, free field dimensions. ing would finally settle the issue as to the validity This seemed unlikely. I knew that if the fields of quantum field theory. The plan of the attack themselves had canonical dimensions, then for was twofold. First, I would prove that “ultravi- many theories this implied that the theory was olet stability,” the vanishing of the effective cou- trivial, i.e., free. Surely this was also true if the pling at short distances, later called asymptotic composite operators that dominated the ampli- freedom, was necessary to explain scaling. Sec- tudes for deep-inelastic scattering had canonical ond, I would show that there existed no asymp- dimensions. By the spring of 1973, Callan and I totically free field theories. The latter was to be had completed a proof of this argument, extend- expected. After all the paradigm of quantum ing an idea of Giorgio Parisi [41] to all renormal- field theory-Quantum Electrodynamics (QED)- izable field theories, with the exception of non- was infrared stable; in other words, the effective Abelian gauge theories. The essential idea was charge grew larger at short distances and no one to prove that the vanishing anomalous dimen- had ever constructed a theory in which the oppo- sions of the composite operators, at an assumed site occurred. Charge renormalization is nothing fixed point of the renormalization group, implied more (certainly in the case of QED) than vac- the vanishing anomalous dimensions of the fields. uum polarization. The vacuum or the ground This then implied that the theory was free at state of a relativistic quantum mechanical sys- this fixed point. The conclusion was that naive tem can be thought of as a medium of virtual scaling could be explained only if the assumed particles. In QED the vacuum contains virtual fixed point of the renormalization group was at electron-positron pairs. If a charge, e0, is put in the origin of coupling space– i.e., the theory must this medium, it polarizes it. Such a medium with be asymptotically free [42]. Non-Abelian gauge virtual electric dipoles will screen the charge and theories were not included in the argument since the actual, observable, charge e, will differ from both arguments broke down for these theories. e0 e0 as  ,where is the dielectric constant. Now  The discovery of asymptotic freedom made this is frequency dependent (or energy or distance de- omission irrelevant. The second part of the ar- pendent). To deal with this one can introduce the gument was to show that there were no asymp- notion of an effective coupling e(r), which governs totically free theories at all. I had set up the the force at a distance r.Asr increases, there is formalism to analyze the most general renormal- more medium that screens, thus e(r) decreases izable field theory of fermions and scalars– again with increasing r, and correspondingly increases excluding non-Abelian gauge theories. This was with decreasing r.Theβ-function, which is sim- not difficult, since to investigate asymptotic free- ply minus the derivative of log[e(r)] with respect dom it suffices to study the behavior of the β- to log(r), is therefore positive. If the effective functions in the vicinity of the origin of coupling coupling were, contrary to QED, to decrease at constant space, i.e., in lowest order perturbation short distances, one might explain how the strong theory (one-loop approximation). I almost had a interactions turn off in this regime and produce complete proof but was stuck on my inability to scaling. Indeed, one might suspect that this is the prove a necessary inequality. I discussed the is- only way to get point-like behavior at short dis- sue with Sidney Coleman, who was spending the tances. It was well understood, due to Wilson’s spring semester in Princeton. He came up with 202 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 the missing ingredient, and added some other cru- bation theory could be a reliable guide to the be- cial points –and we had a proof that no renormal- havior of the energy functional. Indeed, Politzer izable field theory that consisted of theories with went ahead with his own calculation of the β- arbitrary Yukawa, scalar or Abelian gauge inter- function for Yang-Mills theory. Our calculation actions could be asymptotically free [43]. Tony proceeded slowly. I was involved in the other Zee had also been studying this. He too was well parts of my program and there were some tough aware of the advantages of an asymptotically free issues to resolve. We first tried to prove on gen- theory and was searching for one. He derived, at eral grounds, using spectral representations and the same time, a partial result, indicating the lack unitarity, that the theory could not be asymptoti- of asymptotic freedom in theories with SU(N)in- cally free, generalizing the arguments of Coleman variant Yukawa couplings [44]. and me to this case. This did not work, so we proceeded to calculate the β-function for a Yang- 3.4. The discovery of asymptotic freedom Mills theory. Today this calculation is regarded Frank Wilczek started work with me in the fall as quite simple and even assigned as a homework of 1972. He had come to Princeton as a mathe- problem in quantum field theory courses. At the matics student, but soon discovered that he was time it was not so easy. This change in attitude is really interested in particle physics. He switched the analogue, in theoretical physics, of the famil- to the physics department, after taking my field iar phenomenon in experimental physics whereby theory course in 1971, and started to work with yesterday’s great discovery becomes today’s back- me. My way of dealing with students, then and ground. It is always easier to do a calculation now, was to involve them closely with my cur- when you know what the result is and you are rent work and very often to work with them di- sure that the methods make sense. One problem rectly. This was certainly the case with Frank, we had to face was that of gauge invariance. Un- who functioned more as a collaborator than a like QED, where the charge renormalization was student from the beginning. I told him about my trivially gauge invariant (because the photon is program to determine whether quantum field the- neutral), the renormalization constants in QCD ory could account for scaling. We decided that we were all gauge dependent. However the physics would calculate the β-function for Yang-Mills the- could not depend on the gauge. Another issue ory. This was the one hole in the line of argument was the choice of regularization. Dimensional I was pursuing. It had not been filled largely be- regularization had not really been developed yet, cause Yang-Mills theory still seemed strange and and we had to convince ourselves that the one- difficult. Few calculations beyond the Born ap- loop β-function was insensitive to the regulariza- proximation had ever been done. Frank was in- tion used. We did the calculation in an arbitrary terested in this calculation for other reasons as gauge. Since we knew that the answer had to be well. Yang-Mills theory was already in use for gauge invariant, we could use gauge invariance as the electro-weak interactions, and he was inter- a check on our arithmetic. This was good since ested in understanding how these behaved at high we both kept on making mistakes. In February energy. Coleman, who was visiting in Princeton, the pace picked up, and we completed the calcu- asked me at one point whether anyone had ever lation in a spurt of activity. At one point a sign calculated the β-function for Yang-Mills theory. error in one term convinced us that the theory I told him that we were working on this. He ex- was, as expected, non-asymptotically free. As I pressed interest because he had asked his student, sat down to put it all together and to write up our H. David Politzer, to generalize the mechanism results, I caught the error. At almost the same he had explored with Eric Weinberg–that of dy- time Politzer finished his calculation and we com- namical symmetry breaking of an Abelian gauge pared, through Sidney, our results. The agree- theory– to the non-Abelian case. An important ment was satisfying. A month or two after this ingredient was the knowledge of the renormaliza- Symanzik passed through Princeton and told us tion flow, to decide whether lowest order pertur- that ’t Hooft had made a remark in a question ses- D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 203 sion during a meeting at Marseilles the previous a contribution to µ of fall to the effect that non-Abelian gauge theories 11 worked in the same way as an asymptotically free δµ =(−1/3+22)q2 = q2; scalar theory he had been playing with.1 ’t Hooft 3 2 did not publish and apparently did not realize whereas a spin one-half quark contributes, the significance for scaling and meeting for the 1 2 strong interactions. Why are non-Abelian gauge δµ = −(−1/3+(2× )2)q2 = − q2. theories asymptotically free? Today we can un- 2 3 derstand this in a very physical fashion, although (the extra minus arises because quarks are it was certainly not so clear in 1973. It is in- fermions). In any case, the upshot is that as structive to interrupt the historical narrative and long as there are not too many quarks the anti- explain, in modern terms, why QCD is asymptot- screening of the gluons wins out over the screen- ically free. The easiest way to understand this is ing of the quarks. The formula for the β-function by considering the magnetic screening properties of a non-Abelian gauge theory is given by of the vacuum [47]. In a relativistic theory one can calculate the dielectric constant, ,interms ≡ d | β(α) µ α(µ) αbarefixed of the magnetic permeability, µ,sinceµ =1(in dµ 2 2 units where c=velocity of light=1). In classical α
α 2 = b1 + b2 + ... (1) physics all media are diamagnetic. This is be- π π cause, classically, all magnets arise from electric where currents and the response of a system to an ap- g2 plied magnetic field is to set up currents that act α = (2) to decrease the field (Lenz’s law). Thus µ<1, 4π a situation that corresponds to electric screening Our result was that or >1. However, in quantum mechanical sys- 11 2   tems paramagnetism is possible. This is the case b1 = − CA − nRTR (3) 6 3 in non-Abelian gauge theories where the gluons R are charged particles of spin one. They behave Here CR is the eigenvalue of the quadratic as permanent color magnetic dipoles that align Casimir operator in the representation R of themselves parallel to an applied external field SU(N) (for the adjoint representation CA = N, increasing its magnitude and producing µ>1. N 2−1 for the fundamental CF = ), TR is trace of We can therefore regard the anti-screening of the N the square of the generators for the representa- Yang-Mills vacuum as paramagnetism. QCD is 1 tion R of SU(N)(TA=NandTF = ), and nR asymptotically free because the anti-screening of 2 is the number of fermions in the representation the gluons overcomes the screening due to the R. In the case of a SU(3) gauge group such as quarks. The arithmetic works as follows. The 11 n QCD, CA =3,TF =2, and thus b1 = −[ − ]. contribution to  (in some units) from a particle 2 3 2 Thus one can tolerate as many as 16 triplets of − q of charge q is 3 , arising from ordinary dielec- quarks before losing asymptotic freedom. tric (or diamagnetic) screening. If the particle has spin s (and thus a permanent dipole moment 4. NON-ABELIAN GAUGE THEORIES 2 γs), it contributes (γs) to µ.Thusaspinone OF THE STRONG INTERACTIONS gluon (with γ = 2, as in Yang-Mills theory) gives For me the discovery of asymptotic freedom 1This scalar theory was ruled out, as Coleman and I ar- was totally unexpected. Like an atheist who has gued [43], since one could prove it had no ground state just received a message from a burning bush, I and therefore was unstable. became an immediate true believer. Field theory 2Symanziks paper, in the proceedings of the Marseilles meeting (1973), presents the issue of the ultraviolet be- wasn’t wrong–instead scaling must be explained havior of Yang-Milss theory as an open question. by an asymptotically free gauge theory of the 204 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 strong interactions. Our first paper contained, ons could couple to color and all would be well. in addition to the report of the asymptotic free- Thus we proposed [45]: dom of Yang-Mills theory, the hypothesis that “One particularly appealing model is this could offer an explanation for scaling, a re- based on three triplets of fermions, mark that there would be logarithmic violations with Gell-Mann’s SU(3) × SU(3) as of scaling and most important of all the sugges- a global symmetry and a SU(3) tion that the strong interactions must be based ‘color’ gauge group to provide the on a color gauge theory. The first paragraph of strong interactions. That is, the our paper reads [45]: generators of the strong interaction Non-Abelian gauge theories have re- gauge group commute with ordinary × ceived much attention recently as a SU(3) SU(3) currents and mix means of constructing unified and quarks with the same isospin and hy- renormalizable theories of the weak percharge but different ‘color’. In and electromagnetic interactions. In such a model the vector mesons are this note we report on an investigation (flavor) neutral, and the structure of the ultraviolet asymptotic behavior of the operator product expansion of such theories. We have found that of electromagnetic or weak currents they possess the remarkable feature, is essentially that of the free quark perhaps unique among renormalizable model (up to calculable logarithmic theories, of asymptotically approach- corrections).” ing free-field theory. Such asymptoti- The appearance of logarithmic corrections to scal- cally free theories will exhibit, for ma- ing in asymptotically free theories had already trix elements of currents between on- been discussed by Callan and me, in our work mass-shell states, Bjorken scaling. We on the need for an asymptotically free theory to therefore suggest that one should look obtain Bjorken scaling. We also analyzed deep- to a non-Abelian gauge theory of the inelastic scattering in an asymptotically free the- strong interactions to provide the ex- ory and discovered [42] planation for Bjorken scaling, which has so far eluded field theoretic un- “That in such asymptotically free the- derstanding.” ories naive scaling is violated by cal- culable logarithmic terms.” We had a specific theory in mind, namely what Thus we were well aware what the form of the became known as QCD. Since the deep-inelastic scaling deviations would be in such a theory. experiments indicated that the charged con- Wilczek and I had immediately started to cal- stituents of the nucleon were quarks, the gluons culate the logarithmic deviations from scaling. had to be flavor neutral. Thus the gluons could We were tremendously excited by the possibility not couple to flavor. We were very aware of the of deriving exact experimental predictions from growing arguments for the color quantum num- first principles that could conclusively test our ber. Not just the quark model spectroscopy that asymptotically free theories of the strong interac- was the original motivation of Han and Nambu tions. We had already evaluated the asymptotic and Greenberg [15], [16], but the counting fac- form of the flavor non-singlet structure functions, tor (of three) that went into the evaluation of the which were the easiest to calculate, at the time π0 → 2γ decay rate from the axial anomaly3,and our Physical Review Letter was written, but did the factor of three that color provided in the total not have room to include the results. We imme- e+ − e− annihilation cross section. Thus the glu- diately started to write a longer paper in which 3This had been recently emphasized by William Bardeen, the structure of the theory would be spelled out Harald Fritzsch and Gell-Mann [48]. in more detail and the dynamical issues would be D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 205 addressed, especially the issue of confinement. In may be little connection between the our letter we were rather noncommittal on this ‘free’ Lagrangian and the spectrum issue. We had tentatively concluded that Higgs of states.... The infrared behavior of mesons would destroy asymptotic freedom, but Green’s functions in this case is de- had only begun to explore the dynamical conse- termined by the strong-coupling limit quences of unbroken color symmetry. The only of the theory. It may be very well thing we were sure of was that [45] that this infrared behavior is such so as to suppress all but color singlet “. . . perturbation theory is not trust- states, and that the colored gauge worthy with respect to the stability of fieldsaswellasthequarkscouldbe the symmetric theory nor to its parti- ‘seen’ in the large-Euclidean momen- cle content .” tum region but never produced as real Politizer’s paper appeared with ours [46]. He asymptotic states.” pointed out the asymptotic freedom of Yang-Mills Steve Weinberg reacted immediately to asymp- theory and speculated on its implications for the totic freedom. He wrote a paper in which dynamical symmetry breaking of these theories. he pointed out that in an asymptotically free In our second paper, written a few months later, gauge theory of the strong interactions the non- we outlined in much greater detail the structure conservation of parity and strangeness can be cal- of asymptotically free gauge theories of the strong culated ignoring the strong interactions, and thus interactions and the predictions for the scaling vi- is of order α, as observed. He also suggested that olations in deep-inelastic scattering [55]. Actually a theory with unbroken color symmetry could ex- the paper was delayed for about two months be- plain why we do not see quarks. There is a im- cause we had problems with the singlet structure portant difference between our respective conjec- functions–due to the operator mixing of physi- tures. Weinberg argued that perhaps the infrared cal operators with ghost operators. This problem divergences, caused by the masslessness of the was similar to the issue of gauge invariance that gluons in an unbroken color gauge theory, would had plagued us before. Here the problem was make the rate of production of non-singlet states more severe. Physical operators, whose matrix vanish.4 We argued that perhaps the growth of elements were measurable in deep-inelastic scat- the effective coupling at large distances, the in- tering experiments, mixed under renormalization frared behavior of the coupling caused by the flip with ghost operators that could have no physical side of asymptotic freedom5, would confine the meaning. Finally we deferred the analysis of the quarks and gluons in color singlet states. In Oc- singlet structure functions to a third paper [56], tober 1973 Fritzsch, Gell-Mann and H. Leutwyler in which we resolved this issue. We showed that, submitted a paper in which they discussed the even though this mixing was real and unavoid- “advantages of color octet gluon picture” [50]. able, the ghost operators decoupled from physical Here they discussed the advantages of measurements. In the second paper we discussed in detail the choice between symmetry breaking “abstracting properties of hadrons and unbroken symmetry and noted that [55] and their currents from a Yang-Mills gauge model based on colored quarks “Another possibility is that the gauge and color octet gluons.” symmetry is exact. At first, sight They discussed various models and pointed out this would appear to be ridiculous the advantages of each. The first point was al- since it would imply the existence of massless, strongly coupled vec- 4Today we believe in the existence in non-confining, tor mesons. However, in asymptot- Coulomb phases, with unbroken color symmetry, for some supersymmetric non-Abelian gauge theories. ically free theories these naive ex- 5Later dubbed infrared slavery by Georgi and Glashow pectations might be wrong. There [76], a name invented by Sidney Coleman. 206 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 ready discussed at the NAL high-energy physics clear that QCD7was such a theory. Asymptotic conference in August 1972. There Gell-Mann and freedom explained scaling at short distances and Fritzsch had discussed their program of “abstract- offered a mechanism for confinement at large dis- ing results from the quark-gluon model.” They tance. Suddenly it was clear that a non-Abelian discussed various models and asked, “Should we gauge theory was consistent with everything we regard the gluons as well as being color non- knew about the strong interactions. It could en- singlets.” They noted that if one assumed that the compass all the successful strong interaction phe- gluons were color octets then “an annoying asym- nomenology of the past decade. Since the gluons metry between quarks and gluons is removed.” In were flavor neutral, the global flavor symmetries that talk no dynamical theory was proposed and of the strong interactions, SU(3)× SU(3), were in most of the paper they “shall treat the vec- immediate consequences of the theory, as long tor gluon, for convenience, as a color singlet.” as the masses of the quarks were small enough.8 [10] In October 1973 Fritzsch, Gell-Mann and Even more alluring was the fact that one could Leutwyler also noted that in the non-relativistic calculate. Since perturbation theory was trust- quark model with a Coulomb potential mediated worthy at short distances many problems could by vector gluons the potential is attractive in be tackled. Some theorists were immediately con- color singlet channels, which might explain why vinced, among them Guido Altarelli, Tom Ap- these are light. This point had been made pre- pelquist, Callan, Coleman, Mary K. Gaillard, viously by Harry Lipkin [57]. They also noted R. Gatto, Georgi, Glashow, John Kogut, Ben the asymptotic freedom of such theories, but did Lee, Luciano Maiani, Migdal, Polyakov, Politzer, not regard this as an argument for scaling since Lennie Susskind, S. Weinberg, Zee. At large dis- “we conjecture that there might be a modifica- tances however perturbation theory was useless. tion at high energies that produces true scaling.” In fact, even today after thirty-one years of study Finally they noted that the axial U(1) anomaly we still lack reliable, analytic tools for treating in a non-Abelian gauge theory might explain the this region of QCD. This remains one of the most notorious U(1) problem, although they could not important, and woefully neglected, areas of the- explain how, since the anomaly itself could be oretical particle physics. However, at the time written as a total divergence.6 the most important thing was to convince oneself that the idea of confinement was not inconsistent. One of the first steps in that direction was pro- 5. THE EMERGENCE AND ACCEP- vided by lattice gauge theory. I first heard of TANCE OF QCD Wilson’s lattice gauge theory when I gave a lec- Although it was clear to me that the strong ture at Cornell in the late spring of 1973. Wilson interactions must be described by non-Abelian had started to think of this approach soon after gauge theories, there were many problems. The asymptotic freedom was discovered. The lattice experimental situation was far from clear, and the formulation of gauge theory (independently pro- issue of confinement remained open. However, posed by Polyakov) had the enormous advantage, within a small community of physicists the accep- as Wilson pointed out in the fall of 1973, that tance of the theory was very rapid. New ideas in the strong coupling limit was particularly simple physics sometimes take years to percolate into the and exhibited confinement [59]. Thus one had collective consciousness. However in rare cases at least a crude approximation in which confine- such as this there is a change of perception anal- ogous to a phase transition. Before asymptotic 7The name QCD first appeared in a review by Bill Mar- freedom it seemed that we were still far from a dy- ciano and Heinz Pagels [58], where it was attributed to namical theory of hadrons; afterwards it seemed Gell-Mann. It was such an appropriate name that no one could complain. 8I refer of course to the mass parameters of the quarks 6It required the discovery of instantons [52] to find the in the Lagrangian, not the physical masses that are effec- explanation of the U(1) problem [53,54]. tively infinite due to confinement. D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 207 ment was exact. It is a very crude approximation, corrections to the scaling of the e+ − e− anni- since to arrive at the continuum theory from the hilation cross section [66,67]. Gaillard and Lee lattice theory one must take the weak coupling [68], and independently Altarelli and Maiani [69], 1 limit. However one could imagine that the prop- calculated the enhancement of the ∆I = 2 non- erty of confinement was not lost as one went con- leptonic decay matrix elements. The analysis tinuously from strong to weak lattice coupling, of scaling violations for deep-inelastic scattering i.e., there was no phase transition. Moreover one continued [70], and the application of asymptotic could, as advocated by Wilson, study this pos- freedom, what is now called perturbative QCD, sibility numerically using Monte Carlo methods was extended to many new processes. The exper- to construct the lattice partition function. How- imental situation developed slowly, and initially ever, the first quantitative results of this program looked rather bad. I remember in the spring of did not emerge till the work of Creutz [60] in 1974 attending a meeting in Trieste. There I met 1981. The ambitious program of calculating the Burt Richter who was gloating over the fact that hadronic mass spectrum has still not attained its R = σe+e−→hadrons/σe+e−→µ+µ− was increasing goal, and still awaits the next generation of com- with energy, instead of approaching the expected puters. Personally I derived much solace in the constant value. This was the most firm of all coming year from two examples of soluble two- the scaling predictions. R must approach a con- dimensional field theories. One was the (ΨΨ)¯ 2 stant in any scaling theory. In most theories theory that Neveu and I analyzed and solved for however one cannot predict the value of the con- large N [61]. This provided a soluble example stant. However, in an asymptotically free theory of an asymptotically free theory that underwent the constant is predicted to equal the sum of the dimensional transmutation, solving its infrared squares of the charges of the constituents. There- problems by generating a dynamical fermion mass fore if there were only the three observed quarks, 1 2 1 2 2 2 through spontaneous symmetry breaking. This one would expect that R → 3[( 3 ) +(3 ) +(3 ) ]= provided a model of an asymptotically free the- 2. However Richter reported that R was increas- ory, with no built in mass parameters. We could ing, passing through 2, with no sign of flatten- solve this model and check that it was consis- ing out. Now many of us knew that charmed tent and physical. The other soluble model was particles had to exist. Not only were they re- two dimensional QCD, analyzed by t’Hooft in the quired, indeed invented, for the GIM mechanism large N limit [62]. Two dimensional gauge theo- to work, but as Claude Bouchiat, John Illiopou- ries trivially confine color. This was realized quite los and Maini [71] and Roman Jackiw and I early and discussed for Abelian gauge theory–the [72] showed, if the charmed quark were absent Schwinger model– by Aharon Casher, Kogut and the electro-weak theory would be anomalous and Susskind, as a model for confinement in the fall non-renormalizable. Gaillard, Lee and Jonathan of 1973 [63]. However QCD2 is a much better Rosner had written an important and insight- example. It has a spectrum of confined quarks ful paper on the phenomenology of charm [73]. which in many ways resembles the four dimen- Thus, many of us thought that since R was in- sional world. These examples gave many of us creasing probably charm was being produced. In total confidence in the consistency of the concept 1974 the charmed mesons, much narrower than of confinement. It clearly was possible to have anyone imagined9 were discovered, looking very a theory whose basic fields do not correspond to much like positronium–Coulomb bound states of asymptotic states, to particles that one can ob- quarks. This clinched the matter for many of the serve directly in the laboratory. Applications of remaining skeptics. The rest were probably con- the theory also began to appear. Two calcula- vinced once experiments at higher energy began tions of the β-function to two loop order were to see quark and gluon jets. The precision tests of performed [64,65], with the result that, in the no- the theory, the logarithmic deviations from scal- 17 2 1 5 tation of (3), b2 = −[ 12 CA− 2 CF TF n− 6 CATF n]. Appelquist and Georgi and Zee calculated the 9Except for Appelquist and Politzer [75]. 208 D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 ing, took quite a while to observe. I remember of the effective coupling for large energy means very well a remark made to me by a senior col- that no new physics need arise at short distances. league, in April of 1973 when I was very high, There are no infinities at all, the bare coupling right after the discovery of asymptotic freedom. is finite–indeed it vanishes. The only divergences He remarked that it was unfortunate that our that arise are an illusion that appears when one new predictions regarding deep-inelastic scatter- tries to compare, in perturbation theory, the fi- ing were logarithmic effects, since it was unlikely nite effective coupling at finite distances with the that we would see them verified, even if true, in vanishing effective coupling at infinitely short dis- our lifetime. This was an exaggeration, but the tances. Thus the discovery of asymptotic freedom tests did take a long time to appear. Confirma- greatly reassured one of the consistency of four- tion only started to trickle in in 1975 -78; and dimensional quantum field theory. One can trust then at a slow pace. By now the predictions are renormalization theory for an asymptotically free indeed verified, as we have heard at this meeting, theory, independent of the fact that perturbation in some cases to better than a percent. Nowadays, theory is only an asymptotic expansion, since it when you listen to experimentalists talk about gets better and better in the regime of short dis- their results they point to their lego plots and tances. We are very close to having a rigorous say, “Here we see a quark, here a gluon.” Believ- mathematical proof of the existence of asymptot- ing is seeing, seeing is believing. We now believe ically free gauge theories in four dimensions–at in the physical reality of quarks and gluons; we least when placed into a finite box to tame the now believe in asymptotic simplicity of their in- infrared dynamics that produces confinement. As teractions at high energies so we can see quarks far as we know, QCD by itself is a totally consis- and gluons. The way in which we see quarks and tent theory at all energies. Moreover, aside from gluons, indirectly through the effects they have on the quark masses it has no arbitrary, adjustable our measuring instruments, is not much different parameters.10 Indeed, were it not for the electro- from the way we see electrons. Even the objection weak interactions and gravity, we might be satis- that quarks and gluons can not be real particles, fied with QCD as it stands. since they can never be isolated, has largely been dissipated. If we were to heat the world to a tem- 6.2. Unification perature of a few hundred MeV, hadrons would Almost immediately after the discovery of melt into a plasma of liberated quarks and gluons. asymptotic freedom and the proposal of the non- Abelian gauge theories of the strong interactions, the first attempts were made to unify all the inter- 6. OTHER IMPLICATIONS OF actions. This was natural, given that one was us- ASYMPTOTIC FREEDOM. ing very similar theories to describe all the known 6.1. Consistency of quantum field theory interactions. Furthermore, the apparently insur- Traditionally, fundamental theories of nature mountable barrier to unification–namely the large have had a tendency to break down at short dis- difference in the strength of the strong interac- tances. This often signals the appearance of new tions and the electro-weak interactions –was seen physics that is discovered once one has experi- to be a low energy phenomenon. Since the strong mental instruments of high enough resolution (en- interactions decrease in strength with increasing ergy) to explore the higher energy regime. Be- energy these forces could have a common origin fore asymptotic freedom it was expected that any at very high energy. Indeed in the fall of 1974 quantum field theory would fail at sufficiently Georgi and Glashow proposed a unified theory, high energy, where the flaws of the renormal- based on the gauge group SU(5), which remark- ization procedure would appear. To deal with ably contained the gauge groups of the standard this, one would have to invoke some kind of fun- model as well as the quark and lepton multiplets damental length. In an asymptotically free the- orythisisnotnecessarily the case–the decrease 10This is one of the reasons it is so hard to solve. D.J. Gross / Nuclear Physics B (Proc. Suppl.) 135 (2004) 193–211 209 in an alluringly simple fashion [76].11 Georgi, He- What Else?,inProceedings of the XVI Int’l len Quinn and Weinberg [78] showed that the cou- Conference on High Energy Physics, Vol. 4, plingsruninsuchawayastomergesomewhere 135 (1972). around 1014 to 1016 GeV. This theory had the 11. Gell-Mann, M. and Neeman, Y. The Eight- great advantage of being tight enough to make fold Way (W.A. Benjamin Inc.) (1964). sufficiently precise predictions (proton decay and 12. Gell-Mann, M., The Symmetry Group of Vec- the Weinberg angle). It was a great stimulus tor and Axial Vector Currents, Physics, 1,63 for modern cosmology, since it implied that one (1964). could extrapolate the standard model to enor- 13. Gell-Mann, M., ASchematicModelof mously high energies that corresponded to very Baryons and Mesons, Phys. Lett. 8, 214 early times in the history of the universe. Al- (1964); Zweig, G., CERN Report No. TH401, though the SU(5) theory has been invalidated by 4R12 (1964) (unpublished). experiment, at least in its simplest form, the basic 14. Nambu,Y., Dynamical Symmetries and Fun- idea that the next fundamental threshold of uni- damental Fields, Proceedings of the Second fication is set by the scale where the strong and Coral Gables Conference on Symmetry Prin- electro-weak couplings become equal in strength ciples at High Energy , 133 (1965); ASystem- remains at the heart of most attempts at unifica- atics of Hadrons in Subnuclear OPhysics,in tion. Preludes in Theoretical Physics, (1965) 15. 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