<<

REVIEWS OF MODERN PHYSICS, VOLUME 77, JULY 2005

Nobel Lecture: The discovery of asymptotic freedom and the emergence of QCD*

David J. Gross Kavli Institute for Theoretical Physics, UCSB, Santa Barbara, California 93106, USA ͑Published 7 September 2005͒

INTRODUCTION largely mysterious. was divided into the study of the weak and the strong interactions, the two The progress of science is much more muddled than is mysterious forces that operate within the nucleus. In the depicted in most history books. This is especially true of case of the weak interactions, there was a rather success- theoretical physics, partly because history is written by ful phenomenological theory, but not much new data. the victorious. Consequently, historians of science often The strong interactions were where the experimental ignore the many alternate paths that people wandered and theoretical action was, particularly at Berkeley. down, the many false clues they followed, the many mis- They were regarded as especially unfathomable. In conceptions they had. These alternate points of view are hindsight this was not surprising since nature was hiding less clearly developed than the final theories, harder to her secrets. The basic constituents of ͑strongly understand, and easier to forget, especially as these are interacting particles͒ were invisible. We now know that viewed years later, when it all really does make sense. these are , but no one had ever seen a , no Thus reading history one rarely gets the feeling of the matter how hard protons were smashed into protons. true nature of scientific development, in which the ele- Furthermore, the “color” charges we now know are the ment of farce is as great as the element of triumph. source of the chromodynamic fields, the analogs of the The emergence of QCD is a wonderful example of the electric charge, were equally invisible. The prevalent evolution from farce to triumph. During a very short feeling was that it would take a very long time to under- period, a transition occurred from experimental discov- stand the nuclear force and that it would require revo- ery and theoretical confusion to theoretical triumph and lutionary concepts. Freeman Dyson had asserted that experimental confirmation. In this Nobel lecture I shall “the correct theory will not be found in the next hun- describe the turn of events that led to the discovery of dred years.” For a young graduate student, such as my- asymptotic freedom, which in turn led to the formula- self, this was clearly the biggest challenge. tion of QCD, the final element of the remarkably com- prehensive theory of physics—the . I shall then briefly describe the experi- mental tests of the theory and the implications of asymptotic freedom. Quantum field theory was originally developed for the treatment of electrodynamics, immediately after the completion of quantum mechanics and the discovery of PARTICLE PHYSICS IN THE 1960s the Dirac equation. It seemed to be the natural tool for The early 1960s, when I started my graduate studies at describing the dynamics of elementary particles. The ap- UC Berkeley, were a period of experimental supremacy plication of quantum field theory to the nuclear forces and theoretical impotence. The construction and utiliza- had important early success. Fermi formulated a power- ful and accurate phenomenological theory of beta decay, tion of major accelerators were proceeding at full steam. ͑ ͒ Experimental discoveries and surprises appeared every which though deficient at high energy was to serve as a few months. There was hardly any theory to speak of. framework for exploring the weak interactions for three The emphasis was on phenomenology, and there were decades. Yukawa proposed a field theory to describe the only small islands of theoretical advances here and nuclear force and predicted the existence of heavy me- there. Field theory was in disgrace; S-matrix theory was sons, which were soon discovered. On the other hand, in full bloom. Symmetries were all the rage. Of the four quantum field theory was confronted from the beginning forces observed in nature, only gravity and electromag- with severe difficulties. These included the infinities that netism were well understood. The other two forces, the appeared as soon as one went beyond lowest-order per- weak force responsible for radioactivity and the strong turbation theory, as well as the lack of any nonperturba- nuclear force that operated within the nucleus, were tive tools. By the 1950s the suspicion of field theory had deepened to the point that a powerful dogma emerged—that field theory was fundamentally wrong, *The 2004 was shared by David J. especially in its application to the strong interactions. Gross, H. David Politzer, and . This lecture is The procedure, developed by R. the text of Professor Gross’s address in conjunction with the Feynman, J. Schwinger, S. Tomanaga, and F. Dyson, award. which had eliminated the ubiquitous infinities that oc-

0034-6861/2005/77͑3͒/837͑13͒/$50.00 837 ©The Nobel Foundation, 2004 838 David J. Gross: The discovery of asymptotic freedom curred in calculations by expressing physical observables idea of permanently bound, confined constituents was in terms of physical parameters, was spectacularly suc- unimaginable at the time. Second, since the pion- cessful in . However, the physi- was so large, perturbative expansions cal meaning of renormalization was not truly under- were useless, and all attempts at nonperturbative analy- stood. The feeling of most was that renormalization was sis were unsuccessful. a trick. This was especially the case for the pioneering In the case of the weak interactions, the situation was inventors of quantum field theory. They were prepared somewhat better. Here one had an adequate effective at the first appearance of divergences to renounce their theory, the four-fermion Fermi interaction, which could belief in quantum field theory and to brace for the next be usefully employed, using perturbation theory to low- revolution. However, it was also the feeling of the est order, to organize and understand the emerging ex- younger leaders of the field, who had laid the founda- perimental picture of the weak interactions. The fact tions of perturbative quantum field theory and renor- that this theory was nonrenormalizable meant that be- malization in the late 1940s. The prevalent feeling was yond the Born approximation it lost all predictive value. that renormalization simply swept the infinities under This disease increased the suspicion of field theory. the rug, but that they were still there and rendered the Yang-Mills theory, which had appeared in the mid-1950s, notion of local fields meaningless. To quote Feynman, was not taken seriously. Attempts to apply Yang-Mills speaking at the 1961 Solvay conference ͑Feynman, theory to the strong interactions focused on elevating 1962͒, “I still hold to this belief and do not subscribe to global flavor symmetries to local gauge symmetries. This the philosophy of renormalization.” was problematic since these symmetries were not exact. Field theory was almost totally perturbative at that In addition non-Abelian gauge theories apparently re- time; all nonperturbative techniques that had been tried quired massless vector mesons—clearly not a feature of in the 1950s had failed. The path integral, developed by the strong interactions. Feynman in the late 1940s, which later proved so valu- In the Soviet Union field theory was under even able for a nonperturbative formulation of quantum field heavier attack, for somewhat different reasons. Landau theory as well as a tool for semiclassical expansions and and collaborators, in the late 1950s, studied the high- numerical approximations, was almost completely for- energy behavior of quantum electrodynamics. They ex- gotten. In a sense, the Feynman rules were too success- plored the relation between the physical electric charge ful. They were an immensely useful, picturesque, and and the bare electric charge as seen at infinitesimally intuitive way of performing perturbation theory. How- small distances. The fact that the electric charge in QED ever, these alluring qualities also convinced many that depends on the distance at which we measure it is due to all that was needed from field theory were these rules. “vacuum polarization.” The vacuum, the ground state of They diverted attention from the nonperturbative dy- a relativistic quantum-mechanical system, should be namical issues facing field theory. In my first course on thought of as a medium consisting of virtual particles. In quantum field theory at Berkeley in 1965, I was taught QED the vacuum contains virtual - that Field Theory=Feynman Rules. Today we know that pairs. If a charge is inserted into this dielectric medium, there are many phenomena, especially confinement in it distorts, or polarizes the virtual dipoles and this will QCD, that cannot be understood perturbatively. screen the charge. Consequently the charge seen at In the United States, the main reason for the aban- some distance will be reduced in magnitude, and the donment of field theory for the strong interactions was farther one goes the smaller the charge. We can intro- simply that one could not calculate. American physicists duce the notion of an effective charge, e͑r͒, which deter- are inveterate pragmatists. Quantum field theory had mines the force at a distance r.Asr increases, there is not proved to be a useful tool with which to make con- more screening medium, thus e͑r͒ decreases with in- tact with the explosion of experimental discoveries. The creasing r, and correspondingly increases with decreas- early attempts in the 1950s to construct field theories of ing r. The , which is minus the logarithmic the strong interactions were total failures. In hindsight derivative of the charge with respect to distance, is thus this was not surprising since a field theory of the strong positive. Landau and colleagues concluded, on the basis interactions faced two enormous problems. First, which of their approximations, that this effect is so strong that fields to use? Following Yukawa, the first attempts em- the physical charge, as measured at any finite distance, ployed pion and nucleon fields. Soon, with the rapid pro- would vanish for any value of the bare charge. They liferation of particles, it became evident that nothing stated: “We reach the conclusion that within the limits of was special about the nucleon or the pion. All the had- formal electrodynamics a point interaction is equivalent, rons, the strange baryons and mesons as well as the for any intensity whatever, to no interaction at all” ͑Lan- higher-spin recurrences of these, appeared to be equally dau and Pomeranchuk, 1955͒. fundamental. The obvious conclusion that all hadrons This is the famous problem of zero charge, a startling were composites of more fundamental constituents was result that implied for Landau that “weak-coupling elec- thwarted by the fact that, no matter how hard hadrons trodynamics is a theory, which is, fundamentally, logi- were smashed into one another, one had not been able cally incomplete” ͑Landau, 1955͒. This problem occurs to liberate these hypothetical constituents. This was not in any non-asymptotically-free theory. Even today, many analogous to the paradigm of atoms made of of us believe that many non-asymptotically-free theo- and or of nuclei composed of nucleons. The ries, such as QED, are inconsistent at very high energies.

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 David J. Gross: The discovery of asymptotic freedom 839

In the case of QED this is only an academic problem, second principle of the bootstrap was that there were no since the trouble shows up only at enormously high en- elementary particles. The way to deal with the increas- ergy. However, Landau believed that this phenomenon ing number of candidates for elementary status was to was more general and would occur in all field theories. proclaim that all were equally fundamental; all were dy- Why? First, every theory they looked at had this prop- namical bound states of each other. This was called erty. But more importantly, I think, dielectric screening Nuclear Democracy and was a response to the prolifera- is a natural physical explanation of charge renormaliza- tion of candidates for fundamental building blocks. tion, and they were unaware of any simple physical rea- S-matrix theory had some notable successes, such as son for the opposite effect. Thus they assumed that the dispersion relations and the development of Regge pole problem of zero charge would arise in any field theory of theory. However, there were drawbacks to a theory that the , but here it was an immediate ca- was based on the principle that there was no theory, at tastrophe. In the Soviet Union this was thought to be a least in the traditional sense. Nonetheless, until 1973 it compelling reason why field theory was wrong, and cer- was not thought proper to use field theory without tainly inappropriate for the strong force. Landau de- apologies. For example, as late as the National Accel- creed that “We are driven to the conclusion that the erator Laboratory conference of 1972, Murray Gell- Hamiltonian method for strong interaction is dead and Mann ended his talk on quarks with the summary: “Let must be buried, although of course with deserved us end by emphasizing our main point, that it may well honor” ͑Landau, 1960͒. be possible to construct an explicit theory of hadrons, Under the influence of Landau and Pomeranchuk, a based on quarks and some kind of glue, treated as ficti- generation of physicists was forbidden to work on field tious, but with enough physical properties abstracted theory. Why did the discovery of the zero-charge prob- and applied to real hadrons to constitute a complete lem not inspire a search for asymptotically free theories theory. Since the entities we start with are fictitious, that would be free of this disease? The answer, I think, is there is no need for any conflict with the bootstrap or twofold. First, many other theories were explored—in conventional dual parton point of view” ͑Gell-Mann, each case they behaved as QED. Second, Landau had 1973͒. concluded that this problem was inherent in any quan- tum field theory, that an asymptotically free theory could not exist. V. S. Vanyashin and M. V. Teren’tev carried SYMMETRIES out a calculation of the charge renormalization of If dynamics was forbidden, one could at least explore charged vector mesons in 1964 ͑Teren’tev and Van- ͒ the symmetries of the strong interactions. The biggest yashin, 1965 . They got the magnitude wrong but did get advance of the early 1960s was the discovery of an ap- the correct sign and concluded that the result was ab- proximate symmetry of hadrons, SU͑3͒, by Gell-Mann surd. They attributed this wrong sign to the nonrenor- and Neeman ͑1964͒ and then the beginning of the under- malizability of charged-vector-meson theory. standing of spontaneously broken chiral symmetry. Since the relevant degrees of freedom, especially color, were THE BOOTSTRAP totally hidden from view due to confinement, the em- phasis was on flavor, which was directly observable. This If field theory could not provide the theoretical frame- emphasis was enhanced because of the success of SU͑3͒. work for the strong interactions what could? In the early Nowadays we realize that SU͑3͒ is an accidental symme- sixties a radically different approach emerged—S-matrix try, which arises simply because a few quarks ͑the up, theory and the bootstrap. The bootstrap theory rested down, and strange quarks͒ are relatively light compared on two principles, both more philosophical than scien- to the scale of the strong interactions. At the time it was tific. First, local fields were not directly measurable. regarded as a deep symmetry of the strong interactions, Thus they were unphysical and meaningless. Instead, and many attempts were made to generalize it and use it one should formulate the theory using the observable as a springboard for a theory of hadrons. The most suc- S-matrix elements measured in scattering experiments. cessful attempt was Gell-Mann’s algebra of currents, a Microscopic dynamics was renounced. Field theory was program for abstracting relations from a field theory, to be replaced by S-matrix theory; a theory based on keeping the ones that might be generally true and then general principles, such as unitarity and analyticity, but throwing the field theory away, “In order to obtain such with no fundamental microscopic Hamiltonian. The ba- relations that we conjecture to be true, we use the sic dynamical idea was that there was a unique S matrix method of abstraction from a Lagrangian field-theory that obeyed these principles. It could be determined model. In other words, we construct a mathematical without requiring any fundamental constituents or equa- theory of the strongly interacting particles, which may or tions of motion ͑Chew, 1963͒. In hindsight, it is clear that may not have anything to do with reality, find suitable the bootstrap was born from the frustration of being algebraic relations that hold in the model, postulate unable to calculate anything using field theory. All mod- their validity, and then throw away the model. We may els and approximations produced conflict with some compare this process to a method sometimes employed dearly held principle. If it was so difficult to construct an in French cuisine: a piece of pheasant meat is cooked S matrix that was consistent with sacred principles then between two slices of veal, which are then discarded” maybe these principles had a unique manifestation. The ͑Gell-Mann, 1964a͒. This paper made quite an impres-

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 840 David J. Gross: The discovery of asymptotic freedom sion, especially on impoverished graduate students like MY ROAD TO ASYMPTOTIC FREEDOM me, who could only dream of eating such a meal. It was a marvelous approach. It gave one the freedom to play I was a graduate student at Berkeley at the height of with the forbidden fruit of field theory and abstract what the bootstrap and S-matrix theory. My Ph.D. thesis was one wanted from it, all without having to believe in the written under the supervision of Geoff Chew, the main theory. The only problem was that it was not clear what guru of the bootstrap, on multibody N/D equations. I principle determined what to abstract. can remember the precise moment at which I was disil- The other problem with this approach was that it di- lusioned with the bootstrap program. This was at the verted attention from dynamical issues. The most dra- 1966 Rochester meeting, held at Berkeley. Francis Low, matic example of this is Gell-Mann and George Zweig’s in the session following his talk, remarked that the boot- hypothesis of quarks, the most important consequence strap was less of a theory than a tautology, “I believe of the discovery of SU͑3͒͑Gell-Mann, 1964b; Zweig, that when you find that the particles that are there in 1964͒. The fact was that hadrons looked as if they were S-matrix theory, with crossing matrices and all the for- malism, satisfy all these conditions, all you are doing is composed of ͑colored͒ quarks whose masses ͑either the showing that the S matrix is consistent with the world current quark masses or the constituent quark masses͒ the way it is; that is, the particles have put themselves were quite small. Color had been introduced by O. W. ͑ ͒ ͑ ͒ there in such a way that it works out, but you have not Greenberg 1964 , Y. Nambu 1965, 1968 and M. Y. Han necessarily explained that they are there” ͑Low, 1967͒. ͑ ͒ and Nambu 1965 . Nambu’s motivation for color was For example, the then popular finite-energy sum rules ͑ twofold; first to offer an explanation of why only what ͑whereby one derived relations for measurable quanti- we would now call͒ color singlet hadrons exist by postu- ties by saturating dispersion relations with a finite num- lating a strong force ͑but with no specification as to what ber of resonance poles on the one hand and relating kind of force͒ coupled to color, which was responsible these to the assumed Regge asymptotic behavior on the for the fact that color-neutral states were lighter than other͒ were not so much predictive equations as merely colored states. The second motivation, explored with checks of axioms ͑analyticity, unitarity͒ using models Han, was the desire to construct models in which the and fits of experimental data. I was very impressed with quarks had integer-valued electric charges. Greenberg’s this remark and longed to find a more powerful dynami- motivation was to explain the strange statistics of non- cal scheme. This was the heyday of current algebra, and relativistic quark-model hadronic bound states ͑a con- the air was buzzing with marvelous results. I was very cern of Nambu’s as well͒. He introduced parastatistics impressed by the fact that one could assume a certain for this purpose, which solved the statistics problem, but structure of current commutators and derive measurable results. The most dramatic of these was the Adler- clouded the dynamical significance of this quantum Weisberger relation that had just appeared ͑Adler, 1965; number. Weisberger, 1966͒. Clearly the properties of these cur- Yet quarks had not been seen, even when energies rents placed strong restrictions on hadronic dynamics. were achieved that were ten times the threshold for their The most popular scheme then was current algebra. production. The nonrelativistic quark model simply did Gell-Mann and R. Dashen were trying to use the com- not make sense. The conclusion was that quarks were mutators of certain components of the currents as a ba- fictitious, mathematical devices. If one had believed in sis for strong-interaction dynamics ͑Dashen and Gell- an underlying field theory it would be hard to maintain Mann, 1966͒. After a while I concluded that this this attitude, but it was certainly consistent with the approach was also tautological. All it did was test the bootstrap. With this attitude one could ignore the appar- validity of the symmetries of the strong interactions. ently insoluble dynamical problems that arose if one This was apparent for vector SU͑3͒, but was also true of tried to imagine that quarks were real. This attitude to- chiral SU͑3͒, especially as Weinberg and others inter- wards quarks persisted until 1973 and beyond. Quarks preted the current-algebra sum rules as low-energy theo- clearly did not exist as real particles; therefore they were rems for Goldstone bosons. This scheme could not be a fictitious devices ͑see Gell-Mann above͒. One might basis for a complete dynamical theory. I therefore stud- “abstract” properties of quarks from some model, but ied the less understood properties of the algebra of local one was not allowed to believe in their reality or to take current densities. These were model dependent, but that was fine. They therefore might contain dynamical infor- the models too seriously. For many this smelled fishy. I mation that went beyond statements of global symmetry. remember very well Steve Weinberg’s reaction to the Furthermore, as was soon realized, one could check sum rules Curtis Callan and I had derived using the one’s assumptions about the structure of local current quark- model. I described my work on deep- algebra by deriving sum rules that could be tested in inelastic scattering sum rules to Weinberg at a Junior deep-inelastic lepton- scattering experiments. J. Fellows dinner at Harvard. I explained how the small Bjorken’s 1966 paper, on the application of U͑6͒ϫU͑6͒, longitudinal cross section observed at SLAC could be particularly influenced me ͑Bjorken, 1966, 1968͒.Inthe interpreted, on the basis of our sum rule, as evidence for spring of 1968 Curtis Callan and I proposed a sum rule quarks. Weinberg was emphatic that this was of no in- to test the then popular “Sugawara model,” a dynamical terest since he did not believe anything about quarks. model of local currents, in which the energy-momentum

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 David J. Gross: The discovery of asymptotic freedom 841

tensor was expressed as a product of currents. The hope with his parton picture of deep-inelastic scattering was that the algebraic properties of the currents and the ͑Feynman, 1969͒, a very picturesque and intuitive way of expression for the Hamiltonian in terms of these would describing deep-inelastic scattering in terms of assumed be enough to have a complete theory. Our goal was pointlike constituents—partons. It complemented the slightly more modest—to test the hypothesis by exploit- approach to deep-inelastic scattering based on the op- ing the fact that in this theory the operator-product ex- erator product of currents, and had the advantage of pansion of the currents contained the energy- being extendible to other processes ͑Drell and Yan, momentum tensor with a known coefficient. Thus we 1971͒. The parton model allowed one to make predic- could derive a sum rule for the structure functions that tions with ease, ignoring the dynamical issues at hand. I could be measured in deep-inelastic electron-proton felt more comfortable with the approach based on as- scattering ͑Callan and Gross, 1968a͒. In the fall of 1968, suming properties of current products at short distances, Bjorken noted that this sum rule, as well as dimensional and felt somewhat uneasy about the extensions of the arguments, would suggest the scaling of deep-inelastic parton model to processes that were not truly domi- scattering cross sections. This prediction was shortly nated by short-distance singularities. At CERN I stud- confirmed by the new experiments at SLAC, which were ied, with Julius Wess, the consequences of exact scale to play such an important role in elucidating the struc- and conformal invariance ͑Gross and Wess, 1970͒. How- ture of hadrons ͑Bloom et al., 1969͒. Shortly thereafter ever, I soon realized that in a field-theoretic context only Callan and I discovered that by measuring the ratio R a free, noninteracting theory could produce exact scal- ␴ ␴ ͓ ␴ ͑␴ ͒ ing. This became very clear to me in 1969, when I came = L / T where L T is the cross section for the scat- tering of longitudinal ͑transverse͒ polarized virtual pho- to Princeton, where my colleague C. Callan ͑and K. Sy- tons͔, one could determine the spin of the charged con- mansik͒ had rediscovered the renormalization-group stituents of the nucleon. We evaluated the moments of equations, which they presented as a consequence of a the deep-inelastic structure functions in terms of the scale invariance anomaly ͑Callan, 1970; Symansik, 1970͒. equal-time commutators of the electromagnetic current Their work made it abundantly clear that once one in- using specific models for these—the algebra of fields in troduced interactions into the theory, scaling, as well as which the current was proportional to a spin-one field on my beloved sum rules, went down the tube. Yet the ex- the one hand, and the quark-gluon model on the other. periments indicated that scaling was in fine shape. One In this popular model quarks interacted through an could hardly turn off the interactions between the Abelian gauge field ͑which could, of course, be massive͒ quarks, or make them very weak, since then one would coupled to baryon number. The gauge dynamics of the expect hadrons to break up easily into their quark con- gluon had never been explored, and I do not think that stituents, and no one ever observed free quarks. This the model had been used to calculate anything until paradox and the search for an explanation of scaling then. We discovered that R depended crucially on the were to preoccupy me for the following four years. spin of the constituents. If the constituents had spin zero ␴ ␴ or one, then T =0, but if they had spin 1/2, then L HOW TO EXPLAIN SCALING =0 ͑Callan and Gross, 1968b͒. This was a rather dra- ␴ matic result. The experiments quickly showed that L About the same time that all this was happening, was very small. string theory was discovered, in one of the most bizarre These SLAC deep-inelastic scattering experiments turns of events in the history of physics. In 1968 G. Ven- had a profound impact on me. They clearly showed that eziano ͑1968͒ came up with a remarkably simple formula the proton behaved, when observed over short times, as that summarized many features of hadronic scattering, if it was made out of pointlike objects of spin one-half. with Regge asymptotic behavior in one channel and nar- In the spring of 1969, which I spent at CERN, C. row resonance saturation in the other. This formula was Llewelynn-Smith and I analyzed the sum rules that fol- soon generalized to multiparticle S-matrix amplitudes lowed for deep-inelastic neutrino-nucleon scattering us- and attracted much attention. The dual-resonance ing similar methods ͑Gross and Llewelynn-Smith, 1969͒. model was born, the last serious attempt to implement We were clearly motivated by the experiments that were the bootstrap. It was only truly understood as a theory then being performed at CERN. We derived a sum rule of quantized strings in 1972. I worked on this theory for that measured the baryon number of the charged con- two years, at CERN and then at Princeton with Schwarz stituents of the proton. The experiments soon indicated and Neveu. At first I felt that this model, which captured that the constituents of the proton had baryon number many of the features of hadronic scattering, might pro- 1/3, in other words again they looked like quarks. I was vide the long sought alternative to a field theory of the then totally convinced of the reality of quarks. They had strong interactions. However, by 1971 I realized that to be more than just mnemonic devices for summarizing there was no way that this model could explain scaling, hadronic symmetries, as they were then universally re- and I felt strongly that scaling was the paramount fea- garded. They had to be physical pointlike constituents of ture of the strong interactions. In fact the dual- the nucleon. But how could that be? Surely strong inter- resonance model led to incredibly soft behavior at large actions must exist between the quarks that would smear momentum transfer, quite the opposite of the hard scal- out their pointlike behavior. ing observed. Also, it required for consistency features After the experiments at SLAC, Feynman came up that were totally unrealistic for the strong interactions—

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 842 David J. Gross: The discovery of asymptotic freedom massless vector and tensor particles. These features later vide only information about physics for large spacelike became the motivation for the hope that string theory momenta, which are of no direct physical interest. Also, might provide a comprehensive and unified theory of all before the discovery of asymptotic freedom, the ultra- the forces of nature. This hope remains strong. How- violet behavior was not calculable using perturbative ever, the relevant energy scale is not 1 GeV but rather methods, and there were no others. Thus it appeared 1019 GeV! that the provided a framework in The data on deep-inelastic scattering were getting bet- which one could discuss, but not calculate, the ter. No violations of scaling were observed, and the free- asymptotic behavior of amplitudes in a physically unin- field-theory sum rules worked. I remember well the 1970 teresting region. Kiev conference on high-energy physics. There I met S. Polyakov and S. Migdal, uninvited but already impres- THE PLAN sive participants at the meeting. Polyakov, Migdal, and I had long discussions about deep-inelastic scattering. By the end of 1972, I had learned enough field theory, Polyakov knew all about the renormalization group and especially renormalization-group methods, to tackle the explained to me that naive scaling could be right. Be- problem of scaling head on. I decided, quite deliberately, cause of renormalization the dimensions of operators to prove that local field theory could not explain the change with the scale of the physics being probed. Not experimental fact of scaling and thus was not an appro- only that, dimensionless couplings also change with priate framework for the description of the strong inter- scale. They approach at small distances fixed-point val- actions. Thus deep-inelastic scattering would finally ues that are generically those of a strongly coupled settle the issue as to the validity of quantum field theory. theory, resulting in large anomalous scaling behavior The plan of the attack was twofold. First, I would prove quite different from free-field-theory behavior. I re- that “ultraviolet stability,” the vanishing of the effective torted that the experiments showed otherwise. He re- coupling at short distances, later called asymptotic free- sponded that this behavior contradicts field theory. We dom, was necessary to explain scaling. Second, I would departed, he convinced, as many were, that experiments show that there existed no asymptotically free field theo- at higher energies would change, I that the theory would ries. The latter was to be expected. After all the para- have to be changed. The view that the scaling observed digm of quantum field theory, quantum electrodynamics at SLAC was not a truly asymptotic phenomenon was ͑QED͒, was infrared stable; the effective charge grew rather widespread. The fact that scaling set in at rather larger at short distances and no one had ever con- low momentum transfers, “precocious scaling,” rein- structed a theory in which the opposite occurred. If the forced this view. Thus the cognoscenti of the renormal- effective coupling were, contrary to QED, to decrease at ization group ͑Wilson, Polyakov, and others͒ believed short distances, one might explain how the strong inter- that the noncanonical scaling indicative of a nontrivial actions turn off in this regime and produce scaling. In- fixed point of the renormalization group would appear deed, one might suspect that this is the only way to get at higher energies. pointlike behavior at short distances. It was well under- Much happened during the next two years. Gerard ’t stood, due to Wilson’s work and its application to deep- Hooft’s spectacular work on the renormalizability of inelastic scattering, that one might expect to get scaling Yang-Mills theory reintroduced non-Abelian gauge in a quantum field theory at a fixed point of the renor- theories to the community ͑’t Hooft, 1967͒. The elec- malization group. However, this scaling would not have troweak theory of S. Glashow, S. Weinberg, and A. canonical, free-field-theory behavior. Such behavior Salam was revived. Field theory became popular again, would mean that the scaling dimensions of the operators at least in application to the weak interactions. The path that appear in the product of electromagnetic currents at integral reemerged from obscurity. Kenneth Wilson’s de- lightlike distances had canonical, free-field dimensions. velopment of the operator-product expansion ͑Wilson, This seemed unlikely. I knew that if the fields themselves 1971͒ provided a new tool that could be applied to the had canonical dimensions, then for many theories this analysis of deep-inelastic scattering. The Callan- implied that the theory was trivial, i.e., free. Surely this Symansik equations simplified the renormalization- was also true if the composite operators that dominated group analysis, which was then applied to the Wilson the amplitudes for deep-inelastic scattering had canoni- expansion ͑Symansik, 1971; Callan, 1972; Christ et al., cal dimensions. 1972; Callan and Gross, 1973͒. The operator-product By the spring of 1973, Callan and I had completed a analysis was extended to the light cone, the relevant re- proof of this argument, extending an idea of G. Parisi gion for deep-inelastic scattering ͑Jackiw et al., 1970; ͑1973͒ to all renormalizable field theories, with the ex- Leutwyler and Stern, 1970; Frishman, 1971; Gross, 1971; ception of non-Abelian gauge theories. The essential Christ et al., 1972͒. Most important from my point of idea was to prove that the vanishing anomalous dimen- view was the revival of the renormalization group by sions of the composite operators, at an assumed fixed Wilson ͑Wilson and Kogut, 1974͒. The renormalization point of the renormalization group, implied the vanish- group stems from the fundamental work of Stueckelberg ing anomalous dimensions of the fields. This then im- and Petermann ͑1953͒, Gell-Mann and Low ͑1954͒, and plied that the theory was free at this fixed point. The Bogoliubov and Shirkov ͑1959͒. This work was ne- conclusion was that naive scaling could be explained glected for many years, partly because it seemed to pro- only if the assumed fixed point of the renormalization

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 David J. Gross: The discovery of asymptotic freedom 843 group was at the origin of coupling space, i.e., the theory dynamical symmetry breaking of an Abelian gauge must be asymptotically free ͑Callan and Gross, 1973͒. theory, to the non-Abelian case. An important ingredi- Non-Abelian gauge theories were not included in the ent was the knowledge of the renormalization flow, to argument since both arguments broke down for these decide whether lowest-order perturbation theory could theories. The discovery of asymptotic freedom made this be a reliable guide to the behavior of the energy func- omission irrelevant. tional. Indeed, Politzer went ahead with his own calcu- The second part of the argument was to show that lation of the ␤ function for Yang-Mills theory. there were no asymptotically free theories at all. I had Our calculation proceeded slowly. I was involved in set up the formalism to analyze the most general renor- the other parts of my program and there were some malizable field theory of fermions and scalars—again ex- tough issues to resolve. We first tried to prove on gen- cluding non-Abelian gauge theories. This was not diffi- eral grounds, using spectral representations and unitar- cult, since to investigate asymptotic freedom it suffices ity, that the theory could not be asymptotically free, gen- to study the behavior of the ␤ functions in the vicinity of eralizing the arguments of Coleman and myself to this the origin of coupling-constant space, i.e., in lowest- case. This did not work, so we proceeded to calculate order perturbation theory ͑one-loop approximation͒.I the ␤ function for a Yang-Mills theory. Today this calcu- almost had a complete proof but was stuck on my inabil- lation is regarded as quite simple and even assigned as a ity to prove a necessary inequality. I discussed the issue homework problem in quantum field theory courses. At with Sidney Coleman, who was spending the spring se- the time it was not so easy. This change in attitude is the mester in Princeton. He came up with the missing ingre- analog, in theoretical physics, of the familiar phenom- dient and added some other crucial points—and we had enon in experimental physics whereby yesterday’s great a proof that no renormalizable field theory that con- discovery becomes today’s background. It is always sisted of theories with arbitrary Yukawa, scalar, or Abe- easier to do a calculation when you know what the result lian gauge interactions could be asymptotically free is and you are sure that the methods make sense. One ͑Coleman and Gross, 1973͒. A. Zee had also been study- problem we had to face was that of gauge invariance. ing this. He too was well aware of the advantages of an Unlike QED, in which the charge renormalization was asymptotically free theory and was searching for one. trivially gauge invariant ͑since the photon is neutral͒, the He derived, at the same time, a partial result, indicating renormalization constants in QCD were all gauge de- the lack of asymptotic freedom in theories with pendent. However, the physics could not depend on the SU͑N͒-invariant Yukawa couplings ͑Zee, 1973a͒. gauge. Another issue was the choice of regularization. Dimensional regularization had not really been devel- THE DISCOVERY OF ASYMPTOTIC FREEDOM oped yet, and we had to convince ourselves that the one- loop ␤ function was insensitive to the regularization Frank Wilczek started work with me in the fall of used. We did the calculation in an arbitrary gauge. Since 1972. He had come to Princeton as a mathematics stu- we knew that the answer had to be gauge invariant, we dent, but soon discovered that he was really interested could use gauge invariance as a check on our arithmetic. in particle physics. He switched to the physics depart- This was good since we both kept on making mistakes. ment, after taking my field theory course in 1971, and In February the pace picked up, and we completed the started to work with me. My way of dealing with stu- calculation in a spurt of activity. At one point a sign dents, then and now, was to involve them closely with error in one term convinced us that the theory was, as my current work and very often to work with them di- expected, nonasymptotically free. As I sat down to put it rectly. This was certainly the case with Frank, who func- all together and to write up our results, I caught the tioned more as a collaborator than a student from the error. At almost the same time Politzer finished his cal- beginning. I told him about my program to determine culation and we compared our results. The agreement whether quantum field theory could account for scaling. was satisfying ͑Gross and Wilczek, 1973b; Politzer, 1973͒. We decided that we would calculate the ␤ function for Why are non-Abelian gauge theories asymptotically Yang-Mills theory. This was the one hole in the line of free? Today we can understand this in a very physical argument I was pursuing. It had not been filled largely fashion, although it was certainly not so clear in 1973. It because Yang-Mills theory still seemed strange and dif- is instructive to interrupt the historical narrative and ex- ficult. Few calculations beyond the Born approximation plain, in modern terms, why QCD is asymptotically free. had ever been done. Frank was interested in this calcu- The easiest way to understand this is by considering the lation for other reasons as well. Yang-Mills theory was magnetic screening properties of the vacuum ͑Nielsen, already in use for the electroweak interactions, and he 1981͒. In a relativistic theory one can calculate the di- was interested in understanding how these behaved at electric constant ␧ in terms of the magnetic permeability, high energy. ␮, since ␧␮=1 ͑in units where c=velocity of light=1͒.In Coleman, who was visiting in Princeton, asked me at classical physics all media are diamagnetic. This is be- one point whether anyone had ever calculated the ␤ cause, classically, all magnets arise from electric currents function for Yang-Mills theory. I told him that we were and the response of a system to an applied magnetic working on this. He expressed interest because he had field is to set up currents that act to decrease the field asked his student, H. David Politzer, to generalize the ͑Lenz’s law͒. Thus ␮Ͻ1, a situation that corresponds to mechanism he had explored with Eric Weinberg, that of electric screening or ␧Ͼ1. However, in quantum-

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 844 David J. Gross: The discovery of asymptotic freedom mechanical systems paramagnetism is possible. This is portant of all the suggestion that the strong interactions the case in non-Abelian gauge theories where the must be based on a color ͑Gross and Wil- are charged particles of spin 1. They behave as perma- czek, 1973b͒. nent color magnetic dipoles that align themselves paral- Our abstract reads: “It is shown that a wide class of lel to an applied external field, increasing its magnitude non-Abelian gauge theories have, up to calculable loga- and producing ␮Ͼ1. We can therefore regard the anti- rithmic corrections, free-field asymptotic behavior. It is screening of the Yang-Mills vacuum as paramagnetism. suggested that Bjorken scaling may be obtained from QCD is asymptotically free because the antiscreening of strong-interaction dynamics based on non-Abelian the gluons overcomes the screening due to the quarks. gauge symmetry.” The first paragraph reads: “Non- The arithmetic works as follows. The contribution to ␧ Abelian gauge theories have received much attention ͑in some units͒ from a particle of charge q is −q2 /3, aris- recently as a means of constructing unified and renor- ing from ordinary dielectric ͑or diamagnetic͒ screening. malizable theories of the weak and electromagnetic in- If the particle has spin s ͑and thus a permanent dipole teractions. In this note we report on an investigation of moment ␥s͒, it contributes ͑␥s͒2 to ␮. Thus a spin-one the ultraviolet asymptotic behavior of such theories. We gluon ͑with ␥=2, as in Yang-Mills theory͒ gives a contri- have found that they possess the remarkable feature, bution to ␮ of ␦␮=͑−1/3+22͒q2 =11/3q2; whereas a spin perhaps unique among renormalizable theories, of one-half quark contributes ␦␮=−͓−1/3+͑21/2͒2͔q2 asymptotically approaching free-field theory. Such as- =−2/3q2 ͑the extra minus arises because quarks are fer- ymptotically free theories will exhibit, for matrix ele- mions͒. In any case, the upshot is that as long as there ments of currents between on-mass-shell states, Bjorken are not too many quarks the antiscreening of the gluons scaling. We therefore suggest that one should look to a wins out over the screening of the quarks. The formula non-Abelian gauge theory of the strong interactions to for the ␤ function of a non-Abelian gauge theory is provide the explanation for Bjorken scaling, which has given by so far eluded field-theoretic understanding.” We had a specific theory in mind. Since the deep- d␣͑␮͒ ␣2 ␣3 ␤͑␣͒ϵ␮ = b + b + , ͑1͒ inelastic experiments indicated that the charged con- d␮ ␲ 1 ␲2 2 ¯ stituents of the nucleon were quarks, the gluons had to be flavor neutral. Thus the gluons could not couple to where ␣=g2 /4␲. flavor. We were very aware of the growing arguments for Our result was that ͑Gross and Wilczek, 1973b; the color quantum number—not just the quark-model Politzer, 1973͒ spectroscopy that was the original motivation of Han, 11 2 Nambu, and Greenberg, but the counting factor ͑of 3͒ ͫ ⌺ ͬ ͑ ͒ b1 =− CA − RnRTR . 2 ␲→ ␥ 6 3 that went into the evaluation of the 2 decay rate from the axial anomaly ͑this had been recently empha- Here CR is the eigenvalue of the quadratic Casimir op- sized by Bardeen, Fritzsch, and Gell-Mann, 1973͒ and erator in the representation R of SU͑N͒͑for the adjoint the factor of 3 that color provided in the total annihila- ͒ representation, CA =N , TR is the trace of the square of tion cross section. Thus the gluons could couple to color ͑ ͒͑ the generators for the representation R of SU N TA and all would be well. Hence we proposed ͑Gross and ͒ =N and for the fundamental representation, TF =1/2 , Wilczek, 1973b͒: “One particularly appealing model is and nR is the number of fermions in the representation based on three triplets of fermions, with Gell-Mann’s ͑ ͒ ͑ ͒ϫ ͑ ͒ ͑ ͒ R. In the case of SU 3 , as in QCD, CA =3, TF =1/2, and SU 3 SU 3 as a global symmetry and an SU 3 ‘color’ thus gauge group to provide the strong interactions. That is, ͑ ͒ the generators of the strong-interaction gauge group b1 =−11/2+nF/3. 3 commute with ordinary SU͑3͒ϫSU͑3͒ currents and mix Thus one can tolerate as many as 16 triplets of quarks quarks with the same isospin and hypercharge but dif- before losing asymptotic freedom. ferent ‘color.’ In such a model the vector mesons are ͑flavor͒ neutral, and the structure of the operator- product expansion of electromagnetic or weak currents NON-ABELIAN GAUGE THEORIES OF THE STRONG is essentially that of the free-quark model ͑up to calcu- INTERACTIONS lable logarithmic corrections͒.” Thus we proposed that the strong interactions be described by the theory we For me the discovery of asymptotic freedom was to- now call QCD! tally unexpected. Like an atheist who has just received a Callan and I had already discussed the appearance of message from a burning bush, I became an immediate logarithmic corrections to scaling in asymptotically free true believer. Field theory was not wrong—instead scal- theories ͑Callan and Gross, 1973͒. We analyzed deep- ing must be explained by an asymptotically free gauge inelastic scattering in an asymptotically free theory and theory of the strong interactions. Our first paper con- discovered that “in such asymptotically free theories na- tained, in addition to the report of the asymptotic free- ive scaling is violated by calculable logarithmic terms.” dom of Yang-Mills theory, the hypothesis that this could Thus we were well aware what the form of the scaling offer an explanation for scaling, a remark that there deviations would be in such a theory. Wilczek and I im- would be logarithmic violations of scaling, and most im- mediately started to calculate the logarithmic deviations

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 David J. Gross: The discovery of asymptotic freedom 845 from scaling. We were tremendously excited by the pos- little connection between the ‘free’ Lagrangian and the sibility of deriving exact experimental predictions from spectrum of states…. The infrared behavior of Green’s first principles that could conclusively test our asymp- functions in this case is determined by the strong- totically free theories of the strong interactions. We had coupling limit of the theory. It may be very well that this already evaluated the asymptotic form of the flavor non- infrared behavior is such so as to suppress all but color singlet structure functions, which were the easiest to cal- singlet states, and that the colored gauge fields as well as culate, at the time our Physical Review Letter ͑Gross and the quarks could be ‘seen’ in the large-Euclidean- Wilczek, 1973b͒ was written, but did not have room to momentum region but never produced as real include the results. We immediately started to write a asymptotic states” ͑Gross and Wilczek, 1973a͒. longer paper in which the structure of the theory would Steve Weinberg reacted immediately to asymptotic be spelled out in more detail and the dynamical issues freedom. He wrote a paper in which he pointed out that would be addressed, especially the issue of confinement. in an asymptotically free gauge theory of the strong in- In our letter we were rather noncommittal on this issue. teractions the order-␣ interactions produced by elec- We had tentatively concluded that Higgs mesons would troweak interactions can be calculated ignoring the destroy asymptotic freedom, but had only begun to ex- strong force, and he found that these effects do not vio- plore the dynamical consequences of unbroken color late conservation of parity and strangeness, in agree- symmetry. The only thing we were sure of was that “per- ment with observation, as long as there were no colored turbation theory is not trustworthy with respect to the scalars ͑Weinberg, 1973͒. This led him to suggest that a stability of the symmetric theory nor to its particle con- theory with unbroken color symmetry could explain why tent” ͑Gross and Wilczek, 1973b͒. Politizer’s paper ap- we do not see quarks and gluons. There is a slight dif- peared just after ours. He pointed out the asymptotic ference between our respective conjectures. Weinberg freedom of Yang-Mills theory and speculated on its im- argued that perhaps the infrared divergences, caused by plications for the dynamical symmetry breaking of these the masslessness of the gluons in an unbroken color theories. His abstract reads, “An explicit calculation gauge theory, would make the rate of production of non- shows perturbation theory to be arbitrarily good for the singlet states vanish. Today we believe in the existence deep Euclidean Green’s functions of any Yang-Mills of nonconfining, Coulomb phases, with unbroken color theory and of many Yang-Mills theories with fermions. symmetry, for some supersymmetric non-Abelian gauge Under the hypothesis that spontaneous symmetry theories. We argued that perhaps the growth of the ef- breakdown is of dynamical origin, these symmetric fective coupling at large distances, the infrared behavior Green’s functions are the asymptotic forms of the physi- of the coupling caused by the flip side of asymptotic cally significant spontaneously broken solution, whose freedom ͑later dubbed infrared slavery by Georgi and coupling could be strong.” No mention is made of either Glashow͒, would confine the quarks and gluons in color Bjorken scaling or of the strong interactions ͑Politzer, singlet states. 1973͒. In October 1973 Fritzsch, Gell-Mann, and H. Leut- In our second paper, written a few months later, we wyler submitted a paper in which they discussed the “ad- outlined in much greater detail the structure of asymp- vantages of the color octet gluon picture.” Here they totically free gauge theories of the strong interactions discussed the advantages of “abstracting properties of and the predictions for the scaling violations in deep- hadrons and their currents from a Yang-Mills gauge inelastic scattering ͑Gross and Wilczek, 1973a͒. The pa- model based on colored quarks and color octet gluons” per was delayed for about two months because we had ͑Fritzsch et al., 1973͒. They discussed various models and problems with the singlet structure functions—due to pointed out the advantages of each. The first point was the operator mixing of physical operators with op- already discussed at the National Accelerator Labora- erators. This problem was similar to the issue of gauge tory high-energy physics conference in August 1972. invariance that had plagued us before. Here the problem There Gell-Mann and Fritzsch had discussed their pro- was more severe. Physical operators, whose matrix ele- gram of “abstracting results from the quark-gluon ments were measurable in deep-inelastic scattering ex- model” ͑Fritzch and Gell-Mann, 1973͒. They discussed periments, mixed under renormalization with ghost op- various models and asked “We shall treat the vector erators that could have no physical meaning. Finally we gluon, for convenience, as a color singlet”? In October deferred the analysis of the singlet structure functions to 1973 Fritzsch, Gell-Mann, and Leutwyler also noted that a third paper, in which we resolved this issue ͑Gross and in the nonrelativistic quark model with a Coulomb po- Wilczek, 1974͒. We showed that, even though this mixing tential mediated by vector gluons the potential is attrac- was real and unavoidable, the ghost operators decou- tive in color singlet channels, which might explain why pled from physical measurements. In the second paper these are light, a point that had been made previously by we discussed in detail the choice between symmetry H. Lipkin ͑1973͒. They also noted the asymptotic free- breaking and unbroken symmetry and noted that “An- dom of such theories, but did not regard this as an argu- other possibility is that the gauge symmetry is exact. At ment for scaling since “we conjecture that there might first sight this would appear to be ridiculous since it be a modification at high energies that produces true would imply the existence of massless, strongly coupled scaling.” Finally they noted that the axial U͑1͒ anomaly vector mesons. However, in asymptotically free theories in a non-Abelian gauge theory might explain the noto- these naive expectations might be wrong. There may be rious U͑1͒ problem, although they could not explain

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 846 David J. Gross: The discovery of asymptotic freedom

how, since the anomaly itself could be written as a total from the lattice theory one must take the weak-coupling divergence. ͓It required the discovery of instantons to limit. However, one could imagine that the property of find the explanation of the U͑1͒ problem.͔ confinement was not lost as one went continuously from strong to weak lattice coupling, i.e., there was no phase transition. Moreover, one could, as advocated by Wilson, THE EMERGENCE AND ACCEPTANCE OF QCD study this possibility numerically using Monte Carlo Although it was clear to me that the strong interac- methods to construct the lattice partition function. How- tions must be described by non-Abelian gauge theories, ever, the first quantitative results of this program did not there were many problems. The experimental situation emerge until 1981. By now the program of calculating was far from clear, and the issue of confinement re- the hadronic mass spectrum has come close to its goal, mained open. However, within a small community of achieving reliable results that fit the low-lying spectrum physicists the acceptance of the theory was very rapid. to a few percent! New ideas in physics sometimes take years to percolate Personally I derived much solace in the coming year into the collective consciousness. However, in rare cases from two examples of soluble two-dimensional field such as this there is a change of perception analogous to theories. One was the ͑⌿¯ ⌿͒2 theory that Neveu and I a phase transition. Before asymptotic freedom it seemed analyzed and solved for large N ͑Gross and Neveu, that we were still far from a dynamical theory of had- 1974͒. This provided a soluble example of an asymptoti- rons; afterwards it seemed clear that QCD was such a cally free theory that underwent dimensional transmuta- theory. ͓The name , or QCD, tion, solving its infrared problems by generating a dy- first appeared in a review by W. Marciano and H. Pagels namical fermion mass through spontaneous symmetry ͑1978͒, where it was attributed to Gell-Mann. It was such breaking. This provided a model of an asymptotically an appropriate name that no one could complain.͔ free theory, with no built-in mass parameters. We could Asymptotic freedom explained scaling at short distances solve this model and check that it was consistent and and offered a mechanism for confinement at large dis- physical. The other soluble model was two-dimensional tance. Suddenly it was clear that a non-Abelian gauge QCD, analyzed by ’t Hooft in the large N limit ͑’t Hooft, theory was consistent with everything we knew about 1974͒. Two-dimensional gauge theories trivially confine the strong interactions. It could encompass all the suc- color. This was realized quite early and discussed for cessful strong-interaction phenomenology of the past Abelian gauge theory, the Schwinger model, by Casher, decade. Since the gluons were flavor neutral, the global Kogut, and Susskind ͑1973, 1974͒ as a model for confine- ͑ ͒ flavor symmetries of the strong interactions, SU 3 ment in the fall of 1973. However QCD2 is a much bet- ϫSU͑3͒, were immediate consequences of the theory, as ter example. It has a spectrum of confined quarks which long as the masses of the quarks ͑the mass parameters of in many ways resembles the four-dimensional world. the quarks in the Lagrangian, not the physical masses These examples gave many of us total confidence in the that are effectively infinite due to confinement͒ are small consistency of the concept of confinement. It clearly was enough. Even more alluring was the fact that one could possible to have a theory whose basic fields do not cor- calculate. Since perturbation theory was trustworthy at respond to asymptotic states, to particles that one can short distances many problems could be tackled. Some observe directly in the laboratory. Applications of the theorists were immediately convinced, among them Al- theory also began to appear. Two calculations of the ␤ tarelli, Appelquist, Callan, Coleman, Gaillard, Gatto, function to two-loop order were performed, with the re- Georgi, Glashow, Kogut, Lee, Maiani, Migdal, Polyakov, sult that in the notation of Eq. ͑2͒, Politzer, Susskind, Weinberg, and Zee. At large dis- tances, however, perturbation theory was useless. In 17 1 5 b =−ͫ C2 − C T n − C T nͬ fact, even today after 31 years of study we still lack re- 2 12 A 2 F F 6 A F liable, analytic tools for treating this region of QCD. This remains one of the most important areas of theo- ͑Caswell, 1974; Jones, 1974͒. Appelquist and Georgi retical particle physics. However, at the time the most ͑1973͒ and Zee ͑1973b͒ calculated the corrections to the important thing was to convince oneself that the idea of scaling of the e+-e− annihilation cross section; Gaillard confinement was not inconsistent. One of the first steps and Lee ͑1974͒, and independently Altarelli and Maiani in that direction was provided by lattice gauge theory. I ͑1974͒, calculated the enhancement of the ⌬I=1/2 non- first heard of Wilson’s lattice gauge theory ͑Wilson, leptonic decay matrix elements. The analysis of scaling 1974͒ when I gave a lecture at Cornell in the late spring violations for deep-inelastic scattering continued ͑Gross, of 1973. Wilson had started to think of this approach 1974͒, and the application of asymptotic freedom, what soon after asymptotic freedom was discovered. The lat- is now called perturbative QCD, was extended to many tice formulation of gauge theory ͑independently pro- new processes. posed by Polyakov͒ had the enormous advantage, as The experimental situation developed slowly and ini- Wilson pointed out in the fall of 1973, that the strong- tially looked rather bad. I remember in the spring of coupling limit was particularly simple and exhibited con- 1974 attending a meeting in Trieste. There I met Burt finement. Thus one had at least a crude approximation Richter, who was gloating over the fact that R ␴ ␴ in which confinement was exact. It is a very crude ap- = e+e−→hadrons/ e+e−→␮+␮− was increasing with energy, in- proximation, since to arrive at the continuum theory stead of approaching the expected constant value. This

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 David J. Gross: The discovery of asymptotic freedom 847 was the most firm of all the scaling predictions. R must no adjustable parameters, enabled us to probe the very approach a constant in any scaling theory. In most theo- early history of the universe, and allowed us to extrapo- ries one cannot predict the value of the constant. How- late the Standard Model to high energy. ever, in an asymptotically free theory the constant is pre- dicted to equal the sum of the squares of the charges of A. Consistency of quantum field theory the constituents. Therefore, if there were only the three observed quarks, one would expect that R→3͓͑1/3͒2 Traditionally, fundamental theories of nature have +͑1/3͒2 +͑2/3͒2͔=2. However, Richter reported that R had a tendency to break down at short distances. This was increasing, passing through 2, with no sign of flat- often signals the appearance of new physics that is dis- tening out. Now many of us knew that charmed particles covered once one has experimental instruments of high had to exist. Not only were they required, indeed in- enough resolution ͑energy͒ to explore the higher-energy vented, for the Glashow-Illiopoulos-Maiani mechanism regime. Before asymptotic freedom it was expected that to work, but as C. Bouchiat, J. Illiopoulos, and P. Meyer any quantum field theory would fail at sufficiently high ͑1972͒, and independently R. Jackiw and I ͑Gross and energy, where the flaws of the renormalization proce- Jackiw, 1972͒ showed, if the charmed quark were absent dure would appear. To deal with this, one would have to the electroweak theory would be anomalous and non- invoke some kind of fundamental length. In an asymp- renormalizable. Gaillard, Lee, and Rosner had written totically free theory this is not necessarily the case; the an important and insightful paper on the phenomenol- decrease of the effective coupling for large energy ogy of charm ͑Gaillard et al., 1975͒. Thus many of us means that no new physics need arise at short distances. thought that, since R was increasing, probably charm There are no infinities at all, the bare coupling is finite, was being produced. In 1974 the charmed mesons, much and in fact it vanishes. The only divergences that arise narrower than anyone imagined ͓except for Appelquist are an illusion that appears when one tries to compare, and Politzer ͑1975͔͒, were discovered, looking very much in perturbation theory, the finite effective coupling at like positronium and easily interpreted as Coulomb finite distances with the vanishing effective coupling at bound states of quarks. This clinched the matter for infinitely short distances. many of the remaining skeptics. The rest were probably Thus the discovery of asymptotic freedom greatly re- convinced once experiments at higher energy began to assured us of the consistency of four-dimensional quan- see quark and gluon jets. tum field theory. We can trust renormalization theory in The precision tests of the theory, the logarithmic de- asymptotically free theories, even though perturbation viations from scaling, took quite a while to observe. I theory is only an asymptotic expansion, since it gets sim- remember very well a remark made to me by a senior pler in the regime of short distances. We are very close colleague, in April of 1973 when I was very excited, right to having a rigorous mathematical proof of the existence after the discovery of asymptotic freedom. He remarked of asymptotically free gauge theories in four that it was unfortunate that our new predictions regard- dimensions—at least when placed into a finite box to ing deep-inelastic scattering were logarithmic effects, tame the infrared dynamics that produces confinement. since it was unlikely that we would see them verified, even if true, in our lifetime. This was an exaggeration, B. No adjustable parameters but the tests did take a long time to appear. Confirma- tion only started to trickle in between 1975 and 1978 at a At first glance QCD has only one parameter, the di- slow pace. By now the predictions are indeed verified, in mensionless number that specifies the strength of the many cases to better than 1%. Nowadays, when you lis- force ͑if we neglect the quark masses, an excellent ap- ten to experimentalists talk about their results they proximation for ordinary hadrons since the light quarks point to their Lego plots and say, “Here we see a quark, are so light͒. But through the dependence of the charge here a gluon.” Believing is seeing, seeing believes. We on distance or energy, the theory produces a dynamical now believe in the physical reality of quarks and gluons; mass scale. One defines the mass scale of QCD to be the we now believe in the asymptotic simplicity of their in- energy at which the charge equals some value, say 1. teractions at high energies, so we can see quarks and Then, via this phenomenon of dimensional transmuta- gluons. The way in which we see quarks and gluons, tion, all masses, indeed all observables, are calculable in indirectly through the effects they have on our measur- terms of the dynamically generated mass scale. It is ing instruments, is not much different from the way we sometimes claimed that the origin of mass is the Higgs see electrons. mechanism that is responsible for the breaking of the electroweak symmetry that unbroken would forbid IMPLICATIONS OF ASYMPTOTIC FREEDOM quark masses. This is incorrect. Most, 99%, of the pro- ton mass is due to the kinetic and potential energy of the The most important implication of asymptotic free- massless gluons and the essentially massless quarks, con- dom is QCD itself with pointlike behavior of quarks at fined within the proton. short distance and the strong confining force at large Thus QCD provides the first example of a complete distance. But in addition, asymptotic freedom greatly in- theory, with no adjustable parameters and with no indi- creased our confidence in the consistency of quantum cation within the theory of a distance scale at which it field theory, produced the first example of a theory with must break down. Indeed, were it not for the elec-

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 848 David J. Gross: The discovery of asymptotic freedom troweak interactions and gravity, we might be satisfied e+-e− annihilation,” Phys. Rev. Lett. 34, 43–45. with QCD as it stands. It is the best example we possess Bardeen, W. A., H. Fritzsch, and M. Gell-Mann, 1973, “Light- of a perfect, complete theory. cone current algebra, ␲° decay, and e+-e− annihilation,” in Scale and Conformal Symmetry in Hadron Physics, edited by R. Gatto ͑Wiley, New York͒. C. The early history of the universe Bjorken, J. D., 1966, “Applications of the chiral U͑6͒ϫU͑6͒ The universe has been expanding since the big bang; algebra of current densities,” Phys. Rev. 148, 1467–1478. thus early on it was hot and dense. To trace the history Bjorken, J. D., 1968, “Current algebra at small distances,” Pro- ceedings of the International School of Physics, “Enrico of the universe we must understand the dynamics that Fermi,” Course XLI, Varenna School Lectures, Varenna operated when the universe was hot and particles were Italy, 1967, edited by J. Steinberger ͑Academic, New York͒. very energetic. Before the Standard Model we could not Bloom, E. D., et al., 1969, “High-energy inelastic e-p scattering go back further than 200 000 years after the big bang. at 6° and 10°,” Phys. Rev. Lett. 23, 930–934. Today, especially since QCD simplifies at high energy, Bogoliubov, N. N., and D. V. Shirkov, 1959, Introduction to the we can extrapolate to very early times when nucleons Theory of Quantized Fields ͑Interscience, New York͒. melted and quarks and gluons were liberated to form a Bouchiat, C., J. Iliopoulpos, and Ph. Meyer, 1972, “An quark-gluon plasma. anomaly free version of Weinberg’s model,” Phys. Lett. 38B, 519–523. D. Unification Callan, C. G., 1970, “Broken scale invariance in scalar field theory,” Phys. Rev. D 2, 1541–1547. One of the most important implications of asymptotic Callan, C. G., 1972, “Broken scale invariance and asymptotic freedom is the insight it gave into the unification of all behavior,” Phys. Rev. D 5, 3202–3210. the forces of nature. Almost immediately after the dis- Callan, C. G., and D. J. Gross, 1968a, “Crucial test of a theory covery of asymptotic freedom and the proposal of the of currents,” Phys. Rev. Lett. 21, 311–313. non-Abelian gauge theories of the strong interactions, Callan, C. G., and D. J. Gross, 1968b, “High-energy electro- the first attempts were made to unify all the interactions. production and the constitution of the electric current,” Phys. This was natural, given that one was using very similar Rev. Lett. 22, 156–159. Callan, C. G., and D. J. Gross, 1973, “Bjorken scaling in quan- theories to describe all the known interactions. Further- tum field theory,” Phys. Rev. D 8, 4383–4394. more, the apparently insurmountable barrier to Casher, A., J. Kogut, and L. Susskind, 1973, “Vacuum polar- unification—namely, the large difference in the strengths ization and the quark-parton puzzle,” Phys. Rev. Lett. 31, of the strong interactions and the electroweak 792–795. interactions—was seen to be a low-energy phenomenon. Casher, A., J. Kogut, and L. Susskind, 1974, “Vacuum polar- Since the strong interactions decrease in strength with ization and the absence of free quarks,” Phys. Rev. D 10, increasing energy, these forces could have a common 732–745. origin at very high energy. Georgi, Quinn, and Weinberg Caswell, W., 1974, “Asymptotic behavior of non-Abelian ͑1974͒ showed that the couplings run in such a way as to gauge theories to two-loop order,” Phys. Rev. Lett. 33, 244– merge somewhere around 1014–1016 GeV. This is our 246. most direct clue as to where the next threshold of fun- Chew, G., 1963, S-Matrix Theory of Strong Interactions ͑Ben- damental physics lies, and it hints that at this immense jamin, New York͒. energy all the forces of nature, including gravity, are uni- Christ, N., B. Hasslacher, and A. Mueller, 1972, “Light-cone fied. behavior of perturbation theory,” Phys. Rev. D 6, 3543–3562. Coleman, S., and D. J. Gross, 1973, “Price of asymptotic free- dom,” Phys. Rev. Lett. 31, 851–854. ACKNOWLEDGMENTS Dashen, R., and M. Gell-Mann, 1966, “Representations of lo- cal current algebra at infinite momentum,” Phys. Rev. Lett. As I end I would like to thank not only the Nobel 17, 340–343. Foundation, but nature itself, who has given us the op- Drell, S. D., and T. M. Yan, 1971, “Partons and their applica- portunity to explore her secrets and the fortune to have tions at high energies,” Ann. Phys. ͑N.Y.͒ 66, 578–623. revealed one of her most mysterious and beautiful Feynman, R. P., 1962, in The Quantum Theory of Fields, Pro- aspects—the strong force. ceedings of the Twelfth Conference on Physics at the Univer- sity of Brussels, October, 1961 ͑Interscience, New York͒. REFERENCES Feynman, R. P., 1969, “High-energy collisions of hadrons,” Phys. Rev. Lett. 23, 1415–1417. Adler, S., 1965, “Sum rules for axial vector Frishman, Y., 1971, “Operator products at almost light like renormalization in beta decay,” Phys. Rev. 140, B736–B747. distances,” Ann. Phys. ͑N.Y.͒ 66, 373–389. Altarelli, G., and L. Maiani, 1974, “Octet enhancement of non- Fritzsch, H., and M. Gell-Mann, 1973, “Current algebra: leptonic weak interactions in asymptotically free gauge theo- Quarks and what else?” in Proceedings of the XVI Interna- ries,” Phys. Lett. 52B, 351–354. tional Conference on High Energy Physics, The University of Appelquist, T., and H. Georgi, 1973, “e+-e− annihilation in Chicago and National Accelerator Laboratory, 1972, edited gauge theories of strong interactions,” Phys. Rev. D 8, 4000– by J. D. Jackson and A. Roberts ͑National Accelerator Labo- 4002. ratory, Batavia, IL͒, Vol. 2, p. 164. Appelquist, T., and H. D. Politzer, 1975, “Heavy quarks and Fritzsch, H., M. Gell-Mann, and H. Leutwyler, 1973, “Advan-

Rev. Mod. Phys., Vol. 77, No. 3, July 2005 David J. Gross: The discovery of asymptotic freedom 849

tages of the color octet gluon picture,” Phys. Lett. 47B, 365– Landau, L. D., and I. Pomeranchuk, 1955, “Radiation of ␥-rays 368. upon collision of fast ␲-mesons with nucleons,” Dokl. Akad. ⌬ 1 Gaillard, M. K., and B. W. Lee, 1974, “ I= 2 rule for non- Nauk SSSR 102, 489. leptonic decays in asymptotically free field theories,” Phys. Leutwyler, H., and J. Stern, 1970, “Singularities of current Rev. Lett. 33, 108–111. commutators on the light cone,” Nucl. Phys. B 20, 77–101. Gaillard, M. K., B. W. Lee, and J. L. Rosner, 1975, “Search for Lipkin, H., 1973, “Triality, exotics and the dynamical basis of charm,” Rev. Mod. Phys. 47, 277–310. the quark model,” Phys. Lett. 45B, 267–271. Gell-Mann, M., 1964a, “The symmetry group of vector and Low, F., 1967, Proceedings of the International Conference on axial vector currents,” Physics ͑Long Island City, N.Y.͒ 1, 63. High Energy Physics 1966 ͑University of California, Berke- Gell-Mann, M., 1964b, “A schematic model of baryons and ley͒, p. 249. mesons,” Phys. Lett. 8, 214. Marciano, W., and H. Pagels, 1978, “Quantum chromodynam- Gell-Mann, M., 1973, “Current algebra: Quarks and what ics,” Phys. Rep. 36, 137–276. else?” in Proceedings of the XVI International Conference on Nambu, Y., 1965, “Dynamical symmetries & fundamental High Energy Physics, The University of Chicago and Na- fields,” in Proceedings of the Second Coral Gables Conference tional Accelerator Laboratory, 1972, edited by J. D. Jackson on Symmetry Principles at High Energy ͑Freeman, San Fran- ͑ and A. Roberts National Accelerator Laboratory, Batavia, cisco͒, p. 133. ͒ IL , Vol. 4, p. 135. Nambu, Y., 1966, “Systematics of hadrons in subnuclear phys- Gell-Mann, M., and F. Low, 1954, “Quantum electrodynamics ics,” in Preludes in Theoretical Physics: In Honor of V. F. at small distances,” Phys. Rev. 95, 1300–1312. Weisskopf, edited by A. de Shalit, H. Feshback, and L. van Gell-Mann, M., and Y. Neeman, 1964, The Eightfold Way Hove ͑North-Holland, Amsterdam͒. ͑ ͒ Benjamin, New York . Nielsen, N. K., 1981, “Asymptotic freedom as a spin effect,” Georgi, H., H. R. Quinn, and S. Weinberg, 1974, “Hierarchy of Am. J. Phys. 49, 1171–1178. interactions in unified gauge theories,” Phys. Rev. Lett. 33, Parisi, G., 1973, “Deep inelastic scattering in a field theory 451–454. with computable large-momenta behavior,” Lett. Nuovo Ci- Greenberg, O. W., 1964, “Spin and unitary-spin independence mento Soc. Ital. Fis. 7, 84. in a paraquark model of baryons and mesons,” Phys. Rev. Politzer, H. D., 1973, “Reliable perturbative results for strong Lett. 13, 598–602. interactions?” Phys. Rev. Lett. 30, 1346–1349. Gross, D. J., 1971, “Light cone structure of current commuta- Stueckelberg, E., and A. Petermann, 1953, “La normalisation tors in the gluon-quark model,” Phys. Rev. D 4, 1059–1072. des constantes dans la theorie des quanta,” Helv. Phys. Acta Gross, D. J., 1974, “How to test scaling in asymptotically free 26, 499. theories,” Phys. Rev. Lett. 32, 1071–1073. Symansik, K., 1970, “Small distance behavior in field theory Gross, D. J., and R. Jackiw, 1972, “The effect of anomalies on and power counting,” Commun. Math. Phys. 18, 227. quasi-renormalizable theories,” Phys. Rev. D 6, 477–493. Symansik, K., 1971, “Small-distance-behavior analysis and Wil- Gross, D. J., and C. Llewelyn-Smith, 1969, “High energy son expansions,” Commun. Math. Phys. 23, 49. neutrino-nucleon scattering, current algebra and partons,” ’t Hooft, G., 1967, “Renormalizable Lagrangians for massive Nucl. Phys. B 14, 337–347. Gross, D. J., and A. Neveu, 1974, “Dynamical symmetry Yang-Mills fields,” Nucl. Phys. B 35, 167–188. breaking in an asymptotically free theory,” Phys. Rev. D 10, ’t Hooft, G., 1974, “A planar diagram theory for strong inter- 3235–3253. actions,” Nucl. Phys. B 72, 461–473. Gross, D. J., and J. Wess, 1970, “Scale invariance, conformal Terent’ev, M. V., and V. S. Vanyashin, 1965, “The vacuum po- invariance and the high energy behavior of scattering ampli- larization of a charged vector field,” Sov. Phys. JETP 21, 380. tudes,” Phys. Rev. D 2, 753–764. Veneziano, G., 1968, “Construction of a crossing-symmetric Gross, D. J., and F. Wilczek, 1973a, “Asymptotically free gauge Regge-behaved amplitude for linearly rising trajectories,” theories. I,” Phys. Rev. D 8, 3633–3652. Nuovo Cimento A 57, 190. Gross, D. J., and F. Wilczek, 1973b, “Ultraviolet behavior of Weinberg, S., 1973, “Non-Abelian gauge theories of the strong non-Abelian gauge theories,” Phys. Rev. Lett. 30, 1343–1346. interactions,” Phys. Rev. Lett. 31, 494–497. Gross, D. J., and F. Wilczek, 1974, “Asymptotically free gauge Weisberger, W. I., 1966, “Unsubtracted dispersion relations theories. II,” Phys. Rev. D 9, 980–993. and the renormalization of the weak axial-vector coupling Han, M. Y., and Y. Nambu, 1965, “Three-triplet model with constants,” Phys. Rev. 143, 1302–1309. double SU͑3͒ symmetry,” Phys. Rev. 139, B1006–B1010. Wilson, K., 1971, “Renormalization group and strong interac- Jackiw, R., R. Van Royen, and G. West, 1970, “Measuring light tions,” Phys. Rev. D 3, 1818–1846. cone singularities,” Phys. Rev. D 2, 2473–2485. Wilson, K., 1974, “Confinement of quarks,” Phys. Rev. D 10, Jones, D., 1974, “Two-loop diagrams in Yang-Mills theory,” 2445–2459. Nucl. Phys. B 75, 531–538. Wilson, K., and J. Kogut, 1974, “The renormalization group Landau, L. D., 1955, “On the quantum theory of fields,” in and the ␧ expansion,” Phys. Rep., Phys. Lett. 12, 75–199. Niels Bohr and the Development of Physics, edited by W. Zee, A., 1973a, “Study of the renormalization group for small Pauli ͑Pergamon, London͒,p.52. coupling constants,” Phys. Rev. D 7, 3630–3636. Landau, L. D., 1960, “Fundamental problems,” in Theoretical Zee, A., 1973b, “Electron-positron annihilation in stagnant Physics in the Twentieth Century: A Memorial Volume to field theories,” Phys. Rev. D 8, 4038–4041. Wolfgang Pauli, edited by M. Fierz and V. F. Weisskopf ͑In- Zweig, G., 1964, CERN Report No. TH401, 4R12 ͑unpub- terscience, New York͒, p. 245. lished͒.

Rev. Mod. Phys., Vol. 77, No. 3, July 2005