PERSPECTIVE

The discovery of asymptotic freedom and the emergence of QCD

David J. Gross* Kavli Institute For Theoretical Physics, University of California, Santa Barbara, CA 93106-0430

he progress of science is much was divided into the study of the weak that a powerful dogma emerged—that more muddled than is depicted and the strong interactions, the two field theory was fundamentally wrong, in most history books. This is mysterious forces that operate within especially in its application to the strong especially true of theoretical the nucleus. In the case of the weak in- interactions. Tphysics, partly because history is written teractions, there was a rather successful The renormalization procedure, devel- by the victorious. Consequently, histori- phenomenological theory, but not much oped by R. Feynman, J. Schwinger, ans of science often ignore the many new data. The strong interactions were S. Tomanaga, and F. Dyson, which had alternate paths that people wandered where the experimental and theoretical eliminated the ubiquitous infinities that down, the many false clues they fol- action was, particularly at Berkeley. occurred in calculations by expressing lowed, the many misconceptions they They were regarded as especially unfath- physical observables in terms of physical had. These alternate points of view are omable. In hindsight, this was not parameters, was spectacularly successful in less clearly developed than the final the- surprising since nature was hiding her Quantum Electrodynamics (QED). How- ories, harder to understand and easier secrets. The basic constituents of had- ever, the physical meaning of renormaliza- to forget, especially as these are viewed rons (strongly interacting particles) were tion was not truly understood. The feeling years later, when it all really does make invisible. We now know that these are of most was that renormalization was a sense. Thus, reading history one rarely quarks, but no one had ever seen a trick. This was especially the case for the gets the feeling of the true nature of quark, no matter how hard protons were pioneering inventors of quantum field the- scientific development, in which the ele- smashed into protons. Furthermore, the ory. They were prepared at the first ap- ment of farce is as great as the element ‘‘color’’ charges we now know are the pearance of divergences to renounce their of triumph. source of the Chromodynamic fields, belief in quantum field theory and to The emergence of QCD is a wonder- the analogs of the electric charge, were brace for the next revolution. However, it ful example of the evolution from farce equally invisible. The prevalent feeling was also the feeling of the younger leaders to triumph. During a very short period, was that it would take a very long time of the field, who had laid the foundations a transition occurred from experimental to understand the nuclear force and that of perturbative quantum field theory and discovery and theoretical confusion to it would require revolutionary concepts. renormalization in the late 1940s. The theoretical triumph and experimental Freeman Dyson had asserted that ‘‘the prevalent feeling was that renormalization confirmation. In this Nobel lecture, I correct theory will not be found in the simply swept the infinities under the rug, shall describe the turn of events that led next hundred years.’’ For a young gradu- but that they were still there and rendered to the discovery of asymptotic freedom, ate student such as myself, this was the notion of local fields meaningless. To which in turn led to the formulation of clearly the biggest challenge. quote Feynman, speaking at the 1961 QCD, the final element of the remark- Solvay conference (1), ‘‘I still hold to this ably comprehensive theory of elemen- Quantum Field Theory belief and do not subscribe to the philoso- tary particle physics—the Standard Quantum field theory was originally phy of renormalization.’’ Model. I shall then briefly describe the developed for the treatment of Electro- Field theory was almost totally pertur- experimental tests of the theory and the dynamics, immediately after the com- bative at that time; all nonperturbative implications of asymptotic freedom. pletion of quantum mechanics and the techniques that had been tried in the discovery of the Dirac equation. It 1950s had failed. The path integral, de- Particle Physics in the 1960s seemed to be the natural tool for de- veloped by Feynman in the late 1940s, The early 1960s, when I started my scribing the dynamics of elementary par- which later proved so valuable for a graduate studies at UC Berkeley, were a ticles. The application of quantum field nonperturbative formulation of quantum period of experimental supremacy and theory to the nuclear forces had impor- field theory as well as a tool for semi- theoretical impotence. The construction tant early success. Fermi formulated a classical expansions and numerical ap- and utilization of major accelerators powerful and accurate phenomenologi- proximations, was almost completely were proceeding at full steam. Experi- cal theory of beta decay, which (al- forgotten. In a sense, the Feynman rules mental discoveries and surprises ap- though deficient at high energy) was to were too successful. They were an im- peared every few months. There was serve as a framework for exploring the

hardly any theory to speak of. The em- weak interactions for three decades. *E-mail: [email protected]. phasis was on phenomenology, and Yukawa proposed a field theory to de- Adapted from Les Prix Nobel, 2004. © 2004 by the Nobel there were only small islands of theoret- scribe the nuclear force and predicted Foundation ical advances here and there. Field the- the existence of heavy mesons, which Editor’s Note: This article is a version of David Gross’ Nobel ory was in disgrace; S-Matrix theory was were soon discovered. On the other Lecture ‘‘The Discovery of Asymptotic Freedom and the in full bloom. Symmetries were all of hand, quantum field theory was con- Emergence of QCD.’’ The 2004 Nobel Price in Physics was the rage. Of the four forces observed in fronted from the beginning with severe awarded to Drs. Gross, Frank Wilczek, and H. David Politzer for their discovery of asymptotic freedom in the theory of nature, only gravity and electromagne- difficulties. These included the infinities the strong interaction. The Nobel Foundation graciously tism were well understood. The other that appeared as soon as one went be- has granted us permission to reprint this article. The Nobel two forces, the weak force responsible yond lowest order perturbation theory, Lectures provide examples of successful approaches to ma- for radioactivity and the strong nuclear as well as the lack of any nonperturba- jor scientific problems. However, in recent years, these lectures have rarely been read, perhaps because of the force that operated within the nucleus, tive tools. By the 1950s, the suspicion of difficulty in obtaining the collections. By reprinting this were largely mysterious. Particle physics field theory had deepened to the point lecture, we hope to broaden their exposure.

www.pnas.org͞cgi͞doi͞10.1073͞pnas.0503831102 PNAS ͉ June 28, 2005 ͉ vol. 102 ͉ no. 26 ͉ 9099–9108 Downloaded by guest on September 28, 2021 mensely useful, picturesque, and intui- to apply Yang–Mills theory to the this property. But more importantly, I tive way of performing perturbation strong interactions focused on elevating think, dielectric screening is a natural theory. However, these alluring qualities global flavor symmetries to local gauge physical explanation of charge renormal- also convinced many that all that was symmetries. This was problematic, since ization, and they were unaware of any needed from field theory were these these symmetries were not exact. In simple physical reason for the opposite rules. They diverted attention from the addition, non-Abelian gauge theories effect. Thus, they assumed that the nonperturbative dynamical issues facing apparently required massless vector problem of zero charge would arise in field theory. In my first course on quan- mesons—clearly not a feature of the any field theory of the strong interac- tum field theory at Berkeley in 1965, I strong interactions. tion, but here it was an immediate ca- was taught that Field Theory ϭ Feynman In the Soviet Union, field theory was tastrophe. In the Soviet Union, this was Rules. Today, we know that there are under even heavier attack, for somewhat thought to be a compelling reason why many phenomena, especially confine- different reasons. Landau and collabora- field theory was wrong, and certainly ment in QCD, that cannot be under- tors, in the late 1950s, studied the high- inappropriate for the strong force. stood perturbatively. energy behavior of QED. They explored Landau decreed that ‘‘We are driven to In the United States, the main reason the relation between the physical elec- the conclusion that the Hamiltonian for the abandonment of field theory for tric charge and the bare electric charge method for strong interaction is dead the strong interactions was simply that as seen at infinitesimally small distances. and must be buried, although of course one could not calculate. American phys- The fact that the electric charge in QED with deserved honor’’ (4). icists are inveterate pragmatists. Quan- depends on the distance at which we Under the influence of Landau and tum field theory had not proved to be a measure it is due to ‘‘vacuum polariza- Pomeranchuk, a generation of physicists useful tool with which to make contact tion.’’ The vacuum, the ground state of was forbidden to work on field theory. with the explosion of experimental dis- a relativistic quantum mechanical sys- Why did the discovery of the zero coveries. The early attempts in the 1950s tem, should be thought of as a medium charge problem not inspire a search for to construct field theories of the strong consisting of virtual particles. In QED, asymptotically free theories that would interactions were total failures. In hind- the vacuum contains virtual electron– be free of this disease? The answer, I sight, this was not surprising since a positron pairs. If a charge is inserted think, is twofold. First, many other theo- field theory of the strong interactions into this dielectric medium, it distorts, ries were explored—in each case, they faced two enormous problems. First, or polarizes the virtual dipoles, and this behaved as QED. Second, Landau had which fields to use? Following Yukawa, will screen the charge. Consequently, concluded that this problem was inher- the first attempts used pion and nucleon the charge seen at some distance will be ent in any quantum field theory, that an fields. Soon, with the rapid proliferation reduced in magnitude, and the farther asymptotically free theory could not ex- of particles, it became evident that noth- one goes the smaller the charge. We can ist. V. S. Vanyashin and M. V. Terentev ing was special about the nucleon or the introduce the notion of an effective carried out a calculation of the charge pion. All of the hadrons, the strange charge, e(r), which determines the force renormalization of charged vector me- baryons and mesons as well as the at a distance r.Asr increases, there is sons in 1964 (5). They got the magni- higher spin recurrences of these, ap- more screening medium, thus e(r) de- tude wrong, but did get the correct sign peared to be equally fundamental. The creases with increasing r, and corre- and concluded that the result was ab- obvious conclusion that all hadrons were spondingly increases with decreasing r. surd. They attributed this wrong sign to composites of more fundamental con- The ␤-function, which is minus the loga- the nonrenormalizability of charged vec- stituents was thwarted by the fact that, rithmic derivative of the charge with tor meson theory. no matter how hard hadrons were respect to distance, is thus positive. smashed into one another, one had not Landau and colleagues concluded, on The Bootstrap been able to liberate these hypothetical the basis of their approximations, that If field theory could not provide the constituents. This was not analogous to this effect is so strong that the physical theoretical framework for the strong the paradigm of atoms made of nucle- charge, as measured at any finite dis- interactions, what could? In the early ons and electrons or of nuclei composed tance, would vanish for any value of the sixties, a radically different approach of nucleons. The idea of permanently bare charge. They stated: ‘‘We reach the emerged—S-Matrix theory and the bound, confined, constituents was un- conclusion that within the limits of for- bootstrap. The bootstrap theory rested imaginable at the time. Second, since mal electrodynamics a point interaction on two principles, both more philosophi- the pion–nucleon coupling was so large, is equivalent, for any intensity whatever, cal than scientific. First, local fields were perturbative expansions were useless, to no interaction at all’’ (2). not directly measurable. Thus, they were and all attempts at nonperturbative This is the famous problem of zero unphysical and meaningless. Instead, analysis were unsuccessful. charge, a startling result that implied one should formulate the theory using In the case of the weak interactions, for Landau that ‘‘weak coupling elec- the observable S-Matrix elements mea- the situation was somewhat better. Here trodynamics is a theory, which is, fun- sured in scattering experiments. Micro- one had an adequate effective theory, damentally, logically incomplete’’ (3). scopic dynamics was renounced. Field the four-fermion Fermi interaction, This problem occurs in any non- theory was to be replaced by S-Matrix which could be usefully used, using per- asymptotically-free theory. Even today, theory, a theory based on general prin- turbation theory to lowest order, to or- many of us believe that many non- ciples, such as unitarity and analyticity, ganize and understand the emerging asymptotically-free theories, such as but with no fundamental microscopic experimental picture of the weak inter- QED, are inconsistent at very high ener- Hamiltonian. The basic dynamical idea actions. The fact that this theory was gies. In the case of QED, this is only an was that there was a unique S-Matrix nonrenormalizable meant that, beyond academic problem, since the trouble that obeyed these principles. It could be the Born approximation, it lost all pre- shows up only at enormously high en- determined without requiring any funda- dictive value. This disease increased the ergy. However, Landau believed that mental constituents or equations of mo- suspicion of field theory. Yang–Mills this phenomenon was more general, and tion (6). In hindsight, it is clear that the theory, which had appeared in the mid would occur in all field theories. Why? bootstrap was born from the frustration 1950s, was not taken seriously. Attempts First, every theory they looked at had of being unable to calculate anything

9100 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0503831102 Gross Downloaded by guest on September 28, 2021 using field theory. All models and ap- for abstracting relations from a field sion was that quarks were fictitious, proximations produced conflicts with theory, keeping the ones that might be mathematical devices. If one had be- some dearly held principle. If it was so generally true and then throwing the lieved in an underlying field theory, it difficult to construct an S-Matrix that field theory away, ‘‘To obtain such rela- would be hard to maintain this attitude, was consistent with sacred principles, tions that we conjecture to be true, we but it was certainly consistent with the then maybe these principles had a use the method of abstraction from a bootstrap. With this attitude, one could unique manifestation. The second prin- Lagrangian field theory model. In other ignore the apparently insoluble dynami- ciple of the bootstrap was that there words, we construct a mathematical the- cal problems that arose if one tried to were no elementary particles. The way ory of the strongly interacting particles, imagine that quarks were real. This atti- to deal with the increasing number of which may or may not have anything to tude toward quarks persisted until 1973 candidates for elementary status was to do with reality, find suitable algebraic and beyond. Quarks clearly did not exist proclaim that all were equally funda- relations that hold in the model, postu- as real particles; therefore, they were mental; all were dynamical bound states late their validity, and then throw away fictitious devices (see Gell-Mann above). of each other. This was called Nuclear the model. We may compare this pro- One might ‘‘abstract’’ properties of Democracy, and was a response to the cess to a method sometimes used in quarks from some model, but one was proliferation of candidates for funda- French cuisine: a piece of pheasant not allowed to believe in their reality or mental building blocks. meat is cooked between two pieces of to take the models too seriously. For S-Matrix theory had some notable veal, which are then discarded.’’ (9). many, this smelled fishy. I remember successes, such as dispersion relations This paper made quite an impression, very well Steve Weinberg’s reaction to and the development of Regge pole the- especially on impoverished graduate stu- the sum rules Curtis Callan and I had ory. However, there were drawbacks to dents like me, who could only dream of derived using the quark–gluon model. I a theory that was based on the principle eating such a meal. It was a marvelous described my work on deep inelastic that there was no theory, at least in the approach. It gave one the freedom to scattering sum rules to Weinberg at a traditional sense. Nonetheless, until play with the forbidden fruit of field Junior Fellows dinner at Harvard. I ex- 1973, it was not thought proper to use theory, abstract what one wanted from plained how the small longitudinal cross field theory without apologies. For ex- it, all without having to believe in the section observed at SLAC could be in- ample, as late as the NAL conference of theory. The only problem was that it terpreted, on the basis of our sum rule, 1972, Murray Gell-Mann ended his talk was not clear what principle determined as evidence for quarks. Weinberg was on quarks with the summary: ‘‘Let us what to abstract? emphatic that this was of no interest end by emphasizing our main point, that The other problem with this approach since he did not believe anything about it may well be possible to construct an was that it diverted attention from dy- quarks. explicit theory of hadrons, based on namical issues. The most dramatic ex- quarks and some kind of glue, treated ample of this is Gell-Mann and George My Road to Asymptotic Freedom as fictitious, but with enough physical Zweig’s hypothesis of quarks, the most I was a graduate student at Berkeley at properties abstracted and applied to real important consequence of the discovery the height of the bootstrap and S-Matrix hadrons to constitute a complete theory. of SU(3) (10, 11). The fact was that theory. My Ph.D. thesis was written un- Since the entities we start with are ficti- hadrons looked as if they were com- der the supervision of Geoff Chew, the tious, there is no need for any conflict posed of (colored) quarks whose masses main guru of the bootstrap, on multi- with the bootstrap or conventional dual (either the current quark masses or the body N͞D equations. I can remember parton point of view’’ (7). constituent quark masses) were quite the precise moment at which I was disil- small. Color had been introduced by lusioned with the bootstrap program. Symmetries O. W. Greenberg (12), Y. Nambu (13, This was at the 1966 Rochester meeting, If dynamics was forbidden, one could at 14), and M. Y. Han and Nambu (15). held at Berkeley. Francis Low, in the least explore the symmetries of the Nambu’s motivation for color was two- session following his talk, remarked that strong interactions. The biggest advance fold; first to offer an explanation of why the bootstrap was less of a theory than a of the early 1960s was the discovery of only (what we would now call) color tautology, ‘‘I believe that when you find an approximate symmetry of hadrons, singlet hadrons exist by postulating a that the particles that are there in S- SU(3), by Gell-Mann and Y. Neeman strong force (but with no specification Matrix theory, with crossing matrices (8), and then the beginning of the un- as to what kind of force) coupled to and all of the formalism, satisfy all these derstanding of spontaneously broken color which was responsible for the fact conditions, all you are doing is showing chiral symmetry. Since the relevant de- that color neutral states were lighter that the S matrix is consistent with the grees of freedom, especially color, were than colored states. The second motiva- world the way it is; that is the particles totally hidden from view due to confine- tion, explored with Han, was the desire have put themselves there in such a way ment, the emphasis was on flavor, which to construct models in which the quarks that it works out, but you have not nec- was directly observable. This emphasis had integer valued electric charges. essarily explained that they are there’’ was enhanced because of the success of Greenberg’s motivation was to explain (16). For example, the then-popular fi- SU(3). Nowadays, we realize that SU(3) the strange statistics of nonrelativistic nite energy sum rules (whereby one de- is an accidental symmetry, which arises quark model hadronic bound states (a rived relations for measurable quantities simply because a few quarks (the up, concern of Nambu’s as well). He intro- by saturating dispersion relations with a down, and strange quarks) are relatively duced parastatistics for this purpose, finite number of resonance poles on the light compared to the scale of the strong which solved the statistics problem, but one hand and relating these to the as- interactions. At the time, it was re- clouded the dynamical significance of sumed Regge asymptotic behavior on garded as a deep symmetry of the this quantum number. the other) were not so much predictive strong interactions, and many attempts Yet quarks had not been seen, even equations, but merely checks of axioms were made to generalize it and use it as when energies were achieved that was (analyticity, unitarity) using models and a springboard for a theory of hadrons. ten times the threshold for their produc- fits of experimental data. The most successful attempt was Gell- tion. The nonrelativistic quark model I was very impressed with this remark Mann’s algebra of currents, a program simply did not make sense. The conclu- and longed to find a more powerful dy-

Gross PNAS ͉ June 28, 2005 ͉ vol. 102 ͉ no. 26 ͉ 9101 Downloaded by guest on September 28, 2021 namical scheme. This was the heyday of elucidating the structure of hadrons vantage of being extendible to other current algebra, and the air was buzzing (23). Shortly thereafter, Callan and I processes (27). The parton model al- with marvelous results. I was very im- discovered that, by measuring the ratio lowed one to make predictions with pressed by the fact that one could as- R ϭ ␴L͞␴T [where ␴L (␴T) is the cross ease, ignoring the dynamical issues at sume a certain structure of current section for the scattering of longitudinal hand. I felt more comfortable with the commutators and derive measurable re- (transverse) polarized virtual photons], approach based on assuming properties sults. The most dramatic of these was one could determine the spin of the of current products at short distances, the Adler–Weisberger relation that had charged constituents of the nucleon. We and felt somewhat uneasy about the ex- just appeared (17, 18). Clearly, the evaluated the moments of the deep- tensions of the parton model to pro- properties of these currents placed inelastic structure functions in terms of cesses that were not truly dominated by strong restrictions on hadronic dynam- the equal time commutators of the elec- short distance singularities. At CERN I ics. The most popular scheme then was tromagnetic using specific models for studied, with Julius Wess, the conse- current algebra. Gell-Mann and R. these—the algebra of fields in which the quences of exact scale and conformal Dashen were trying to use the commu- current was proportional to a spin-one invariance (28). However, I soon real- tators of certain components of the cur- field on the one hand, and the quark– ized that, in a field theoretic context, rents as a basis for strong interaction gluon model on the other. In this popu- only a free, noninteracting theory could dynamics (19). After a while, I con- lar model, quarks interacted through an produce exact scaling. This became very cluded that this approach was also tau- Abelian gauge field (which could, of clear to me in 1969, when I came to tological, all it did was test the validity course, be massive) coupled to baryon Princeton, where my colleague C. Callan of the symmetries of the strong interac- number. The gauge dynamics of the (and K. Symansik) had rediscovered the tions. This was apparent for vector gluon had never been explored, and I renormalization group equations, which SU(3), but was also true of chiral do not think that the model had been they presented as a consequence of a SU(3), especially as Weinberg and oth- used to calculate anything until then. scale invariance anomaly (29, 30). Their ers interpreted the current algebra sum We discovered that R depended cru- work made it abundantly clear that once rules as low-energy theorems for Gold- cially on the spin of the constituents. If one introduced interactions into the the- stone bosons. This scheme could not be the constituents had spin zero or one, ory, scaling, as well as my beloved sum a basis for a complete dynamical theory. then ␴T ϭ 0, but if they had spin 1͞2, rules, went down the tube. Yet, the ex- I, therefore, studied the less under- then ␴L ϭ 0 (24). This was a rather dra- periments indicated that scaling was in stood properties of the algebra of local matic result. The experiments quickly fine shape. But one could hardly turn current densities. These were model showed that ␴L was very small. off the interactions between the quarks, dependent; but that was fine, they there- These SLAC deep-inelastic scattering or make them very weak, since then one fore might contain dynamical informa- experiments had a profound impact on would expect hadrons to break up easily tion that went beyond statements of me. They clearly showed that the proton into their quark constituents, and no global symmetry. Furthermore, as was behaved, when observed over short one ever observed free quarks. This par- soon realized, one could check ones’ times, as if it was made out of point-like adox and the search for an explanation assumptions about the structure of local objects of spin 1͞2. In the spring of 1969, of scaling were to preoccupy me for the current algebra by deriving sum rules which I spent at CERN, C. Llewelynn- following four years. that could be tested in deep inelastic Smith and I analyzed the sum rules that lepton–hadron scattering experiments. followed for deep-inelastic neutrino– How to Explain Scaling J. Bjorken’s 1967 paper on the applica- nucleon scattering using similar methods About the same time that all this was tion of U(6)XU(6) particularly influ- (25). We were clearly motivated by the happening, string theory was discovered, enced me (20, 21). In the spring of experiments that were then being per- in one of the most bizarre turn of events 1968, Curtis Callan and I proposed formed at CERN. We derived a sum in the history of physics. In 1968, G. a sum rule to test the then popular rule that measured the baryon number Veneziano came up with a remarkably ‘‘Sugawara model,’’ a dynamical model of the charged constituents of the pro- simple formula that summarized many of local currents in which the energy ton. The experiments soon indicated features of hadronic scattering (31), momentum tensor was expressed as a that the constituents of the proton had with Regge asymptotic behavior in one product of currents. The hope was that baryon number 1͞3, in other words, channel and narrow resonance satura- the algebraic properties of the currents again they looked like quarks. I was tion in the other. This formula was soon and the expression for the Hamiltonian then totally convinced of the reality of generalized to multiparticle S-Matrix in terms of these would be enough to quarks. They had to be more than just amplitudes and attracted much atten- have a complete theory. Our goal was mnemonic devices for summarizing had- tion. The dual resonance model was slightly more modest—to test the hy- ronic symmetries, as they were then born, the last serious attempt to imple- pothesis by exploiting the fact that in universally regarded. They had to be ment the bootstrap. It was only truly this theory the operator product expan- physical point-like constituents of the understood as a theory of quantized sion of the currents contained the en- nucleon. But how could that be? Surely strings in 1972. I worked on this theory ergy momentum tensor with a known strong interactions must exist between for two years, at CERN and then at coefficient. Thus, we could derive a sum the quarks that would smear out their Princeton with Schwarz and Neveu. At rule for the structure functions that point-like behavior. first I felt that this model, which cap- could be measured in deep-inelastic After the experiments at SLAC, Feyn- tured many of the features of hadronic electron–proton scattering (22). man came up with his parton picture of scattering, might provide the long In the fall of 1968, Bjorken noted that deep inelastic scattering (26), a very pic- sought alternative to a field theory of this sum rule, as well as dimensional turesque and intuitive way of describing the strong interactions. However, by arguments, would suggest the scaling of deep-inelastic scattering in terms of as- 1971 I realized that there was no way deep inelastic scattering cross sections. sumed point-like constituents–partons. It that this model could explain scaling, This prediction was shortly confirmed by complemented the approach to deep and I felt strongly that scaling was the the new experiments at SLAC, which inelastic scattering based on the opera- paramount feature of the strong interac- were to play such an important role in tor product of currents, and had the ad- tions. In fact, the dual resonance model

9102 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0503831102 Gross Downloaded by guest on September 28, 2021 lead to incredibly soft behavior at large scattering. Wilson’s development of the ing, that one might expect to get scaling momentum transfer, quite the opposite operator product expansion provided a in a quantum field theory at a fixed of the hard scaling observed. Also, it new tool that could be applied to the point of the renormalization group. required for consistency many features analysis of deep inelastic scattering. The However this scaling would not have that were totally unrealistic for the Callan–Symansik equations simplified canonical, free field theory behavior. strong interactions—massless vector and the renormalization group analysis, Such behavior would mean that the scal- tensor particles. These features later be- which was then applied to the Wilson ing dimensions of the operators that ap- came the motivation for the hope that expansion (34–37). The operator prod- pear in the product of electromagnetic string theory may provide a comprehen- uct analysis was extended to the light currents at light-like distances had ca- sive and unified theory of all of the cone, the relevant region for deep- nonical, free field dimensions. This forces of nature. This hope remains inelastic scattering (38–42). Most impor- seemed unlikely. I knew that, if the strong. However, the relevant energy tant from my point of view was the fields themselves had canonical dimen- scale is not 1 Gev, but rather 1019 Gev! revival of the renormalization group by sions, then for many theories, this im- The data on deep inelastic scattering Wilson (43). The renormalization group plied that the theory was trivial, i.e., were getting better. No violations of stems from the fundamental work of free. Surely this was also true if the scaling were observed, and the free- Gell-Mann and Low (44), Stueckelberg composite operators that dominated the field-theory sum rules worked. I remem- and Petermann (45), and Bogoliubov amplitudes for deep-inelastic scattering ber well the 1970 Kiev conference on and Shirkov (46). This work was ne- had canonical dimensions. high-energy physics. There I met S. glected for many years, partly because it By the spring of 1973, Callan and I Polyakov and S. Migdal, uninvited but seemed to provide only information had completed a proof of this argument, already impressive participants at the about physics for large space-like mo- extending an idea of G. Parisi (47) to all meeting. Polyakov, Migdal, and I had menta, which are of no direct physical renormalizable field theories, with the long discussions about deep inelastic interest. Also, before the discovery of exception of non-Abelian gauge theo- scattering. Polyakov knew all about the asymptotic freedom, the UV behavior ries. The essential idea was to prove that renormalization group and explained to was not calculable using perturbative the vanishing anomalous dimensions of me that naive scaling cannot be right. methods, and there were no others. the composite operators, at an assumed Because of renormalization, the dimen- Thus, it appeared that the renormaliza- fixed point of the renormalization sions of operators change with the scale tion group provided a framework in group, implied the vanishing anomalous of the physics being probed. Not only which one could discuss, but not calcu- dimensions of the fields. This then im- that, dimensionless couplings also late, the asymptotic behavior of ampli- plied that the theory was free at this change with scale. They approach, at tudes in a physically uninteresting fixed point. The conclusion was that small distances, fixed point values that region. naı¨ve scaling could be explained only if are generically those of a strongly cou- the assumed fixed point of the renor- pled theory, resulting in large anoma- The Plan malization group was at the origin of lous scaling behavior quite different By the end of 1972, I had learned coupling space, i.e., the theory must be from free field theory behavior. I re- enough field theory, especially renor- asymptotically free (48). Non-Abelian torted that the experiments showed oth- malization group methods, to tackle the gauge theories were not included in the erwise. He responded that this behavior problem of scaling head on. I decided, argument since both arguments broke contradicts field theory. We departed; quite deliberately, to prove that local down for these theories. The discovery he convinced, as many were, that exper- field theory could not explain the exper- of asymptotic freedom made this omis- iments at higher energies would change; imental fact of scaling and thus was not sion irrelevant. I that the theory would have to be an appropriate framework for the de- The second part of the argument was changed. The view that the scaling scription of the strong interactions. to show that there were no asymptoti- observed at SLAC was not a truly as- Thus, deep inelastic scattering would cally free theories at all. I had set up ymptotic phenomenon was rather wide- finally settle the issue as to the validity the formalism to analyze the most gen- spread. The fact that scaling set in at of quantum field theory. The plan of the eral renormalizable field theory of fer- rather low momentum transfers, ‘‘preco- attack was twofold. First, I would prove mions and scalars—again excluding cious scaling,’’ reinforced this view. that ‘‘UV Stability,’’ the vanishing of the non-Abelian gauge theories. This was Thus, the cognoscenti of the renormal- effective coupling at short distances, not difficult, since to investigate asymp- ization group (Wilson, Polyakov, and later called asymptotic freedom, was totic freedom, it suffices to study the others) believed that the noncanonical necessary to explain scaling. Second, behavior of the ␤-functions in the vicin- scaling indicative of a nontrivial fixed I would show that there existed no ity of the origin of coupling constant point of the renormalization group asymptotically free field theories. The space, i.e., in lowest order perturbation would appear at higher energies. latter was to be expected. After all of theory (one-loop approximation). I al- Much happened during the next two the paradigm of quantum field theory, most had a complete proof but was years. Gerhard ‘t Hooft’s spectacular QED, was infrared stable; the effective stuck on my inability to prove a neces- work on the renormalizability of Yang– charge grew larger at short distances sary inequality. I discussed the issue Mills theory reintroduced non-Abelian and no one had ever constructed a the- with Sidney Coleman, who was spending gauge theories to the community (32). ory in which the opposite occurred. If the spring semester in Princeton. He The electroweak theory of S. Glashow, the effective coupling were, contrary to came up with the missing ingredient, S. Weinberg, and A. Salam was revived. QED, to decrease at short distances, and added some other crucial points— Field theory became popular again, at one might explain how the strong inter- and we had a proof that no renormaliz- least in application to the weak interac- actions turn off in this regime and pro- able field theory that consisted of theories tions. The path integral reemerged from duce scaling. Indeed, one might suspect with arbitrary Yukawa, scalar, or Abelian obscurity. Kenneth Wilson’s develop- that this is the only way to get point-like gauge interactions could be asymptoti- ment of the operator product expansion behavior at short distances. It was well cally free (49). A. Zee had also been (33) provided a tool that could be ap- understood, due to Wilson’s work and studying this. He too was well aware of plied to the analysis of deep inelastic its application to deep inelastic scatter- the advantages of an asymptotically free

Gross PNAS ͉ June 28, 2005 ͉ vol. 102 ͉ no. 26 ͉ 9103 Downloaded by guest on September 28, 2021 theory and was searching for one. He ple and even assigned as a homework color magnetic dipoles that align them- derived, at the same time, a partial re- problem in quantum field theory selves parallel to an applied external sult, indicating the lack of asymptotic courses. At the time it was not so easy. field increasing its magnitude and pro- freedom in theories with SU(N) invari- This change in attitude is the analogue, ducing ␮ Ͼ 1. We can therefore regard ant Yukawa couplings (50). in theoretical physics, of the familiar the anti-screening of the Yang–Mills phenomenon in experimental physics vacuum as paramagnetism. QCD is The Discovery of Asymptotic Freedom whereby yesterday’s great discovery be- asymptotically free because the anti- Frank Wilczek started work with me in comes today’s background. It is always screening of the gluons overcomes the the fall of 1972. He had come to Prince- easier to do a calculation when you screening due to the quarks. The arith- ton as a mathematics student, but soon know what the result is and you are sure metic works as follows. The contribution discovered that he was really interested that the methods make sense. One prob- to ␧ (in some units) from a particle of in particle physics. He switched to the lem we had to face was that of gauge charge q is Ϫq2͞3, arising from ordinary physics department, after taking my invariance. Unlike QED, where the dielectric (or diamagnetic) screening. If field theory course in 1971, and started charge renormalization was trivially the particle has spin s (and thus a per- to work with me. My way of dealing gauge invariant (since the photon is manent dipole moment ␥s), it contrib- with students, then and now, was to in- neutral), the renormalization constants utes (␥s)2 to ␮. Thus, a spin one gluon volve them closely with my current work in QCD were all gauge dependent. (with ␥ ϭ 2, as in Yang–Mills theory) and very often to work with them di- However, the physics could not depend gives a contribution to ␮ of ␦␮ ϭ (Ϫ1͞3 rectly. This was certainly the case with on the gauge. Another issue was the ϩ 22)q2 ϭ 11͞3 q2; whereas a spin one- Frank, who functioned more as a collab- choice of regularization. Dimensional half quark contributes ␦␮ ϭϪ(Ϫ1͞3 ϩ orator than a student from the begin- regularization had not really been devel- (21͞2)2)q2 ϭϪ2͞3 q2 (the extra minus ning. I told him about my program to oped yet, and we had to convince our- ␤ arises because quarks are fermions). In determine whether quantum field theory selves that the one-loop -function was any case, the upshot is that, as long as could account for scaling. We decided insensitive to the regularization used. ␤ there are not too many quarks, the anti- that we would calculate the -function We did the calculation in an arbitrary screening of the gluons wins out over for Yang–Mills theory. This was the one gauge. Since we knew that the answer the screening of the quarks. The for- hole in the line of argument I was pur- had to be gauge invariant, we could use mula for the ␤-function of a non- suing. It had not been filled largely be- gauge invariance as a check on our Abelian gauge theory is given by cause Yang–Mills theory still seemed arithmetic. This was good since we both strange and difficult. Few calculations kept on making mistakes. In February d␣͑␮͒ ␣2 ␣3 ␤͑␣͒ ϵ ␮ ϭ ϩ ϩ beyond the Born approximation had the pace picked up, and we completed ␮ ␲ b1 ␲2 b2 ..., ever been done. Frank was interested in the calculation in a spurt of activity. At d this calculation for other reasons as well. one point a sign error in one term con- g2 Yang–Mills theory was already in use vinced us that the theory was, as ex- ␣ ϭ where ␲ . [1] for the electroweak interactions, and he pected, non-asymptotically free. As I sat 4 was interested in understanding how down to put it all together and to write Our result was that (51, 52) these behaved at high energy. up our results, I caught the error. At Coleman, who was visiting in Prince- almost the same time Politzer finished 11 2 ϭ Ϫͫ Ϫ ͸ ͬ ton, asked me at one point whether any- his calculation and we compared our b1 CA RnRTR . one had ever calculated the ␤-function results. The agreement was satisfying 6 3 for Yang–Mills theory. I told him that (51, 52). [2] we were working on this. He expressed Why are non-Abelian gauge theories interest because he had asked his stu- asymptotically free? Today we can un- Here CR is the eigenvalue of the qua- dent, H. David Politzer, to generalize derstand this in a very physical fashion, dratic Casimir operator in the represen- the mechanism he had explored with although it was certainly not so clear in tation R of SU(N) (for the adjoint Eric Weinberg, that of dynamical sym- 1973. It is instructive to interrupt the representation, CA ϭ N), TR is the trace metry breaking of an Abelian gauge historical narrative and explain, in mod- of the square of the generators for the theory, to the non-Abelian case. An im- ern terms, why QCD is asymptotically representation R of SU(N) (TA ϭ N and portant ingredient was the knowledge of free. The easiest way to understand this for the fundamental representation, the renormalization flow, to decide is by considering the magnetic screening TF ϭ 1͞2), and nR is the number of fer- whether lowest order perturbation the- properties of the vacuum (53). In a rela- mions in the representation R.Inthe ory could be a reliable guide to the be- tivistic theory, one can calculate the di- case of SU(3), as in QCD, CA ϭ N, havior of the energy functional. Indeed, electric constant, ␧, in terms of the TF ϭ 1͞2, and thus Politzer went ahead with his own calcu- magnetic permeability, ␮, since ␧␮ ϭ 1 ␤ ϭ ϭ ϭ Ϫ ͞ ϩ ͞ lation of the -function for Yang–Mills (in units where c velocity of light b1 11 2 nF 3. [3] theory. 1). In classical physics all media are dia- Our calculation proceeded slowly. I magnetic. This is because, classically, all Thus, one can tolerate as many as 16 was involved in the other parts of my magnets arise from electric currents and triplets of quarks before losing asymp- program and there were some tough the response of a system to an applied totic freedom. issues to resolve. We first tried to prove magnetic field is to set up currents that on general grounds, using spectral rep- act to decrease the field (Lenz’s law). Nonabelian Gauge Theories of the resentations and unitarity, that the the- Thus ␮ Ͻ 1, a situation that corre- Strong Interactions ory could not be asymptotically free, sponds to electric screening or ␧ Ͼ 1. For me, the discovery of asymptotic generalizing the arguments of Coleman However, in quantum mechanical sys- freedom was totally unexpected. Like an and me to this case. This did not work, tems paramagnetism is possible. This is atheist who has just received a message so we proceeded to calculate the ␤- the case in non-Abelian gauge theories from a burning bush, I became an im- function for a Yang–Mills theory. Today where the gluons are charged particles mediate true believer. Field theory was this calculation is regarded as quite sim- of spin one. They behave as permanent not wrong—instead, scaling must be ex-

9104 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0503831102 Gross Downloaded by guest on September 28, 2021 plained by an asymptotically free gauge but different ‘color’. In such a model In our second paper, written a few theory of the strong interactions. Our the vector mesons are (flavor) neutral, months later, we outlined in much first paper contained, in addition to the and the structure of the operator prod- greater detail the structure of asymptot- report of the asymptotic freedom of uct expansion of electromagnetic or ically free gauge theories of the strong Yang–Mills theory, the hypothesis that weak currents is essentially that of the interactions and the predictions for the this could offer an explanation for scal- free quark model (up to calculable loga- scaling violations in deep-inelastic scat- ing, a remark that there would be loga- rithmic corrections).’’ Thus, we pro- tering (55). The paper was delayed for rithmic violations of scaling and, most posed that the strong interactions be about two months because we had prob- important of all, the suggestion that the described by the theory we now call lems with the singlet structure func- strong interactions must be based on a QCD! tions—due to the operator mixing of color gauge theory (51). Callan and I had already discussed physical operators with ghost operators. Our abstract reads: ‘‘It is shown that a the appearance of logarithmic correc- This problem was similar to the issue of wide class of non-Abelian gauge theo- tions to scaling in asymptotically free gauge invariance that had plagued us ries have, up to calculable logarithmic theories (48). We analyzed deep inelastic before. Here the problem was more se- corrections, free-field asymptotic behav- scattering in an asymptotically free theory vere. Physical operators, whose matrix ior. It is suggested that Bjorken scaling and discovered ‘‘That in such asymptoti- elements were measurable in deep- may be obtained from strong-interaction cally free theories naive scaling is vio- inelastic scattering experiments, mixed dynamics based on non-Abelian gauge lated by calculable logarithmic terms.’’ under renormalization with ghost opera- symmetry.’’ The first paragraph reads: Thus, we were well aware what the form tors that could have no physical mean- ‘‘Non-Abelian gauge theories have re- of the scaling deviations would be in ing. Finally, we deferred the analysis of ceived much attention recently as a such a theory. Wilczek and I immedi- the singlet structure functions to a third means of constructing unified and ately started to calculate the logarithmic paper, in which we resolved this issue renormalizable theories of the weak and deviations from scaling. We were tre- (56). We showed that, even though this electromagnetic interactions. In this mendously excited by the possibility of mixing was real and unavoidable, the note we report on an investigation of deriving exact experimental predictions ghost operators decoupled from physical the UV asymptotic behavior of such from first principles that could conclu- measurements. In the second paper we theories. We have found that they pos- sively test our asymptotically free theo- discussed in detail the choice between sess the remarkable feature, perhaps ries of the strong interactions. We had symmetry breaking and unbroken sym- unique among renormalizable theories, already evaluated the asymptotic form metry and noted that ‘‘Another possibil- of asymptotically approaching free-field of the flavor non-singlet structure func- ity is that the gauge symmetry is exact. theory. Such asymptotically free theories tions, which were the easiest to calcu- At first, sight this would appear to be will exhibit, for matrix elements of cur- late, at the time our Physical Review ridiculous since it would imply the exis- rents between on-mass-shell states, Letter was written, but did not have tence of massless, strongly coupled vec- Bjorken scaling. We therefore suggest room to include the results. We immedi- tor mesons. However, in asymptotically that one should look to a non-Abelian ately started to write a longer paper in free theories these naive expectations gauge theory of the strong interactions which the structure of the theory would might be wrong. There may be little to provide the explanation for Bjorken be spelled out in more detail and the connection between the ‘free’ Lagrang- scaling, which has so far eluded field dynamical issues would be addressed, ian and the spectrum of states. . . The theoretic understanding.’’ especially the issue of confinement. In infrared behavior of Green’s functions We had a specific theory in mind. our letter, we were rather noncommittal in this case is determined by the strong- Since the deep-inelastic experiments in- on this issue. We had tentatively con- coupling limit of the theory. It may be dicated that the charged constituents of cluded that Higgs mesons would destroy very well that this infrared behavior is the nucleon were quarks, the gluons had asymptotic freedom, but had only begun such so as to suppress all but color sin- to be flavor neutral. Thus, the gluons to explore the dynamical consequences glet states, and that the colored gauge could not couple to flavor. We were of unbroken color symmetry. The only fields as well as the quarks could be very aware of the growing arguments thing we were sure of was that ‘‘pertur- ‘seen’ in the large-Euclidean momentum for the color quantum number. Not just bation theory is not trustworthy with region but never produced as real as- the quark model spectroscopy that was respect to the stability of the symmetric ymptotic states’’ (55). the original motivation of Han, Nambu, theory nor to its particle content.’’ (51) Steve Weinberg reacted immediately and Greenberg, but the counting factor Politizer’s paper appeared just after to asymptotic freedom. He wrote a pa- (of three) that went into the evaluation ours. He pointed out the asymptotic per in which he pointed out that in an of the ␲ 3 2␥ decay rate from the axial freedom of Yang–Mills theory and asymptotically free gauge theory of the anomaly (this had been recently empha- speculated on its implications for the strong interactions the order ␣ interac- sized by W. Bardeen, H. Fritzsch, and dynamical symmetry breaking of these tions produced by electroweak interac- Gell-Mann; ref. 54), and the factor of theories. His abstract reads: ‘‘An explicit tions can be calculated ignoring the three that color provided in the total calculation shows perturbation theory to strong force; and found that these ef- annihilation cross section. Thus, the glu- be arbitrarily good for the deep Euclidean fects do not violate conservation of par- ons could couple to color and all would Green’s functions of any Yang–Mills ity and strangeness, in agreement with be well. Thus, we proposed (51): ‘‘One theory and of many Yang–Mills theories observation, as long as there were no particularly appealing model is based on with fermions. Under the hypothesis colored scalars (57). This led him to three triplets of fermions, with Gell- that spontaneous symmetry breakdown suggest that a theory with unbroken Mann’s SU(3)xSU(3)as a global symme- is of dynamical origin, these symmetric color symmetry could explain why we try and a SU(3) ‘color’ gauge group to Green’s functions are the asymptotic do not see quarks and gluons. There is a provide the strong interactions. That is, forms of the physically significant spon- slight difference between our respective the generators of the strong interaction taneously broken solution, whose cou- conjectures. Weinberg argued that per- gauge group commute with ordinary pling could be strong.’’ No mention is haps the infrared divergences, caused by SU(3)x SU(3) currents and mix quarks made of either Bjorken scaling or of the the masslessness of the gluons in an un- with the same isospin and hypercharge strong interactions (52). broken color gauge theory, would make

Gross PNAS ͉ June 28, 2005 ͉ vol. 102 ͉ no. 26 ͉ 9105 Downloaded by guest on September 28, 2021 the rate of production of non-singlet tion. Before asymptotic freedom it tion. Moreover, one could, as advocated states vanish. Today we believe in the seemed that we were still far from a dy- by Wilson, study this possibility numeri- existence of non-confining, Coulomb namical theory of hadrons; afterward, it cally using Monte Carlo methods to con- phases, with unbroken color symmetry, seemed clear that QCD was such a the- struct the lattice partition function. How- for some supersymmetric non-Abelian ory. [The name QCD first appeared in a ever, the first quantitative results of this gauge theories. We argued that perhaps review by W. Marciano and H. Pagels program did not emerge until 1981. By the growth of the effective coupling at (60), where it was attributed to Gell- now, the program of calculating the had- large distances, the infrared behavior of Mann. It was such an appropriate name ronic mass spectrum has come close to its the coupling caused by the flip side of that no one could complain.] Asymp- goal, achieving now reliable results that fit asymptotic freedom (later dubbed infra- totic freedom explained scaling at short the low-lying spectrum to a few percent! red slavery by Georgi and Glashow), distances and offered a mechanism for Personally I derived much solace in would confine the quarks and gluons in confinement at large distance. Suddenly, the coming year from two examples of color singlet states. it was clear that a non-Abelian gauge soluble two-dimensional field theories. In October 1973, Fritzsch, Gell-Mann, theory was consistent with everything One was the (⌿៮ ⌿)2 theory that Neveu and H. Leutwyler submitted a paper in we knew about the strong interactions. and I analyzed and solved for large N which they discussed the ‘‘advantages of It could encompass all of the successful (62). This provided a soluble example of color octet gluon picture.’’ Here they strong interaction phenomenology of the an asymptotically free theory that un- discussed the advantages of ‘‘abstracting past decade. Since the gluons were fla- derwent dimensional transmutation, properties of hadrons and their currents vor neutral, the global flavor symmetries solving its infrared problems by generat- from a Yang–Mills gauge model based of the strong interactions, SU(3) ϫ ing a dynamical fermion mass through on colored quarks and color octet glu- SU(3), were immediate consequences of spontaneous symmetry breaking. This ons’’ (58). They discussed various mod- the theory, as long as the masses of the provided a model of an asymptotically els and pointed out the advantages of quarks (the mass parameters of the free theory, with no built in mass pa- each. The first point was already dis- quarks in the Lagrangian, not the physi- rameters. We could solve this model cussed at the NAL high-energy physics cal masses that are effectively infinite and check that it was consistent and conference in August 1972. There, Gell- due to confinement) are small enough. physical. The other soluble model was Mann and Fritzsch had discussed their Even more alluring was the fact that two-dimensional QCD, analyzed by program of ‘‘abstracting results from the one could calculate. Since perturbation t’Hooft in the large N limit (63). Two quark-gluon model’’ (7). They discussed theory was trustworthy at short dis- dimensional gauge theories trivially con- various models and asked, ‘‘shall treat tances many problems could be tack- fine color. This was realized quite early the vector gluon, for convenience, as a led. Some theorists were immediately and discussed for Abelian gauge theory, color singlet.’’ In October 1973, Fritzsch, convinced, among them Altarelli, the Schwinger model, by A. Casher, Gell-Mann, and Leutwyler also noted Appelquist, Callan, Coleman, Gaillard, Kogut, and Susskind, as a model for that in the nonrelativistic quark model R. Gatto, Georgi, Glashow, Kogut, Ben confinement in the fall of 1973 (64, 65). with a Coulomb potential mediated by Lee, Maiani, Migdal, Polyakov, Politzer, However, QCD2 is a much better exam- vector gluons the potential is attractive Susskind, Weinberg, and Zee. At large ple. It has a spectrum of confined in color singlet channels, which might distances, however, perturbation theory quarks which in many ways resembles explain why these are light, a point that was useless. In fact, even today after 31 the four-dimensional world. These ex- had been made previously by H. Lipkin years of study, we still lack reliable, ana- amples gave many of us total confidence (59). They also noted the asymptotic lytic tools for treating this region of in the consistency of the concept of con- freedom of such theories, but did not QCD. This remains one of the most im- finement. It clearly was possible to have regard this as an argument for scaling portant areas of theoretical particle a theory whose basic fields do not corre- since ‘‘we conjecture that there might be physics. However, at the time, the most spond to asymptotic states, to particles a modification at high energies that pro- important thing was to convince oneself that one can observe directly in the lab- duces true scaling.’’ Finally, they noted that the idea of confinement was not oratory. Applications of the theory also that the axial U(1) anomaly in a non- inconsistent. One of the first steps in began to appear. Two calculations of the Abelian gauge theory might explain the that direction was provided by lattice ␤-function to two loop order were per- notorious U(1) problem, although they gauge theory. I first heard of Wilson’s formed, with the result that, in the ϭϪ ͞ 2 Ϫ could not explain how, since the anom- lattice gauge theory (61) when I gave a notation of Eq. 1, b2 [(17 12)CA aly itself could be written as a total di- lecture at Cornell in the late spring of (1͞2)CFTFn Ϫ (5͞6)CATFn] (66, 67). vergence. [It required the discovery of 1973. Wilson had started to think of this Appelquist and Georgi (68) and Zee instantons to find the explanation of the approach soon after asymptotic freedom (69) calculated the corrections to the U(1) problem.] was discovered. The lattice formulation scaling of the eϩ Ϫ eϪ annihilation of gauge theory (independently pro- cross-section; Gaillard and Lee (70), and The Emergence and Acceptance of QCD posed by Polyakov) had the enormous independently Altarelli and Maiani (71), Although it was clear to me that the advantage, as Wilson pointed out in the calculated the enhancement of the ⌬I ϭ strong interactions must be described by fall of 1973, that the strong coupling 1͞2 nonleptonic decay matrix elements. non-Abelian gauge theories, there were limit was particularly simple and exhibited The analysis of scaling violations for many problems. The experimental situa- confinement. Thus, one had at least a deep-inelastic scattering continued (72), tion was far from clear, and the issue of crude approximation in which confine- and the application of asymptotic free- confinement remained open. However, ment was exact. It is a very crude approxi- dom, what is now called perturbative within a small community of physicists, mation, since to arrive at the continuum QCD, was extended to many new the acceptance of the theory was very theory from the lattice theory one must processes. rapid. New ideas in physics sometimes take the weak-coupling limit. However, The experimental situation developed take years to percolate into the collec- one could imagine that the property of slowly, and initially looked rather bad. I tive consciousness. However, in rare confinement was not lost as one went remember in the spring of 1974 attend- cases such as this there is a change of continuously from strong to weak lattice ing a meeting in Trieste. There I met perception analogous to a phase transi- coupling, i.e., there was no phase transi- Burt Richter who was gloating over the

9106 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0503831102 Gross Downloaded by guest on September 28, 2021 fact that R ϭ ␴eϩeϪ3hadrons͞␴eϩeϪ3␮ϩ␮Ϫ can see quarks and gluons. The way in quark masses, an excellent approxima- was increasing with energy, instead of which we see quarks and gluons, indi- tion for ordinary hadrons since the approaching the expected constant rectly through the effects they have on light quarks are so light). But through value. This was the most firm of all of our measuring instruments, is not much the dependence of the charge on dis- the scaling predictions. R must approach different from the way we see electrons. tance or energy, the theory produces a a constant in any scaling theory. In most dynamical mass scale. One defines the theories one cannot predict the value of Implications of Asymptotic Freedom mass scale of QCD to be the energy at the constant. However, in an asymptoti- The most important implication of as- which the charge equals some value, cally free theory the constant is pre- ymptotic freedom is QCD itself with say 1. Then, via this phenomenon of dicted to equal the sum of the squares point like behavior of quarks at short dimensional transmutation, all masses, of the charges of the constituents. distance and the strong confining force indeed all observables, are calculable Therefore, if there were only the three at large distance. But, in addition, as- in terms of the dynamically generated observed quarks, one would expect that ymptotic freedom greatly increased our mass scale. It is sometimes claimed R 3 3[(1͞3)2 ϩ (1͞3)2 ϩ (2͞3)2] ϭ 2. confidence in the consistency of quan- that the origin of mass is the Higgs However, Richter reported that R was tum field theory, produced the first ex- mechanism that is responsible for the increasing, passing through 2, with no ample of a theory with no adjustable breaking of the electroweak symmetry sign of flattening out. Now many of us parameters, enabled us to probe the that unbroken would forbid quark knew that charmed particles had to ex- very early history of the universe, and masses. This is incorrect. Most, 99%, ist. Not only were they required, indeed allowed us to extrapolate the standard of the proton mass is due to the ki- invented, for the GIM mechanism to model to high energy. netic and potential energy of the mass- work, but as C. Bouchiat, J. Illiopoulos, less gluons and the essentially massless and L. Maini (73), and independently Consistency of Quantum Field Theory. Tra- quarks, confined within the proton. R. Jackiw and I (74), showed, if the ditionally, fundamental theories of na- Thus, QCD provides the first example charmed quark were absent, the elec- ture have had a tendency to break down of a complete theory, with no adjustable troweak theory would be anomalous and at short distances. This often signals the parameters and with no indication within nonrenormalizable. Gaillard, Lee, and appearance of new physics that is dis- the theory of a distance scale at which it Rosner (75) had written an important covered once one has experimental in- must break down. Indeed, were it not for and insightful paper on the phenome- struments of high enough resolution the electroweak interactions and gravity, nology of charm. Thus, many of us (energy) to explore the higher energy we might be satisfied with QCD as it thought that since R was increasing, regime. Before asymptotic freedom, it stands. It is the best example we possess probably charm was being produced. In was expected that any quantum field of a perfect, complete theory. 1974, the charmed mesons, much nar- theory would fail at sufficiently high en- rower than anyone imagined (except for ergy, where the flaws of the renormal- The Early History of the Universe. The uni- Appelquist and Politzer; ref. 76), were ization procedure would appear. To deal verse has been expanding since the Big discovered, looking very much like with this, one would have to invoke Bang, thus early on it was hot and positronium, and easily interpreted as some kind of fundamental length. In an dense. To trace the history of the uni- Coulomb bound states of quarks. This asymptotically free theory, this is not verse, we must understand the dynamics clinched the matter for many of the re- necessarily the case; the decrease of the that operates when the universe was hot maining skeptics. The rest were proba- effective coupling for large energy and particles were very energetic. Be- bly convinced once experiments at means that no new physics need arise at fore the standard model, we could not higher energy began to see quark and short distances. There are no infinities go back further than 200,000 years after gluon jets. at all, the bare coupling is finite, and in the Big Bang. Today, especially since The precision tests of the theory—the fact it vanishes. The only divergences QCD simplifies at high energy, we can logarithmic deviations from scaling— that arise are an illusion that appears extrapolate to very early times, where took quite a while to observe. I remem- when one tries to compare, in perturba- nucleons melt and quarks and gluons ber very well a remark made to me by a tion theory, the finite effective coupling are liberated to form a quark–gluon senior colleague, in April of 1973 when at finite distances with the vanishing plasma. I was very excited, right after the dis- effective coupling at infinitely short covery of asymptotic freedom. He re- distances. Unification. One of the most important marked that it was unfortunate that our Thus, the discovery of asymptotic implications of asymptotic freedom is new predictions regarding deep-inelastic freedom greatly reassured us of the con- the insight it gave into the unification scattering were logarithmic effects, since sistency of four-dimensional quantum of all of the forces of nature. Almost it was unlikely that we would see them field theory. We can trust renormaliza- immediately after the discovery of as- verified, even if true, in our lifetime. tion theory asymptotically free theories, ymptotic freedom and the proposal of This was an exaggeration, but the tests even though perturbation theory is only the non-Abelian gauge theories of the did take a long time to appear. Confir- an asymptotic expansion, since it gets strong interactions, the first attempts mation only started to trickle in 1975–78 simpler in the regime of short distances. were made to unify all of the interac- at a slow pace. By now the predictions We are very close to having a rigorous tions. This was natural, given that one are indeed verified, in many cases to mathematical proof of the existence of was using very similar theories to de- better than 1%. Nowadays, when you asymptotically free gauge theories in scribe all of the known interactions. listen to experimentalists talk about four dimensions—at least when placed Furthermore, the apparently insur- their results, they point to their Lego into a finite box to tame the infrared mountable barrier to unification— plots and say, ‘‘Here we see a quark, dynamics that produces confinement. namely, the large difference in the here a gluon.’’ Believing is seeing, see- strength of the strong interactions and ing believes. We now believe in the No Adjustable Parameters. At first glance, the electroweak interactions—was seen physical reality of quarks and gluons; we QCD has only one parameter, the di- to be a low energy phenomenon. Since now believe in asymptotic simplicity of mensionless number that specifies the the strong interactions decrease in their interactions at high energies so we strength of the force (if we neglect the strength with increasing energy, these

Gross PNAS ͉ June 28, 2005 ͉ vol. 102 ͉ no. 26 ͉ 9107 Downloaded by guest on September 28, 2021 forces could have a common origin at (77). This is our most direct clue as to As I end I would like to thank not only the very high energy. H. Georgi, H. Quinn, where the next threshold of fundamen- Nobel Foundation, but Nature itself, who has given us the opportunity to explore her se- and S. Weinberg showed that the cou- tal physics lies, and hints that at this crets and the fortune to have revealed one of plings run in such a way as to merge immense energy all of the forces of her most mysterious and beautiful aspects— somewhere around 1014 to 1016 Gev nature, including gravity, are unified. the strong force.

1. Feynman, R. (1961) in The Quantum Theory of 22. Callan, C. G. & Gross, D. J. (1968) Phys. Rev. Lett 50. Zee, A. (1973) Phys. Rev. D 7, 3630–3636. Fields: The 12th Solvay Conference (Interscience, 21, 311–313. 51. Gross, D. J. & Wilczek, F. (1973) Phys. Rev. Lett. New York), p. 177. 23. Bloom, E. D., Coward, D. H., DeStaebler, H., 30, 1343–1346. 2. Landau, L. D. & Pomeranchuk, I. (1955) Dokl. Drees, J., Miller, G., Mo, L. W. & Taylor, R. E. 52. Politzer, H. D. (1973) Phys. Rev. Lett. 30, 1346– Akad. Nauk SSSR 102, 489–492. (1969) Phys. Rev. Lett. 23, 930–934. 1349. 3. Landau, L. D. (1955) in Niels Bohr and the Devel- 24. Callan, C. G. & Gross, D. J. (1968) Phys. Rev. Lett. 53. Nielsen, N. K. (1981) Am. J. Phys. 49, 1171–1178. opment of Physics, ed. Pauli, W. (Pergamon Press, 22, 156–159. 54. Bardeen, W. A., Fritzsch, H. & Gell-Mann, M. London), pp. 52–69. 25. Gross, D. J. & Llewelyn-Smith, C. (1969) Nucl. (1973) in Scale and Conformal Symmetry in Had- 4. Landau, L. D. (1960) in Theoretical Physics in the Phys. B 14, 337–347. ron Physics, ed. Gatto, R. (Wiley, New York), p. Twentieth Century: A Memorial Volume to Wolf- 26. Feynman, R. P. (1969) Phys. Rev. Lett. 23, 1415– 139. 1417. gang Pauli, eds. Fierz, M. & Weisskopf, V. F. 55. Gross, D. J. & Wilczek, F. (1973) Phys. Rev. D 8, 27. Drell, S. D. & Yan, T. M. (1971) Ann. Phys. 66, (Interscience, New York), pp. 245–247. 3633–3652. 578–623. 5. VanTerent’ev, M. V. & Vanyashin, V. S. (1965) 56. Gross, D. J. & Wilczek, F. (1974) Phys. Rev. D 9, 28. Gross, D. J. & Wess, J. (1970) Phys. Rev. D 2, Soviet Phys. JETP 21, 375–380. 980–993. 753–764. 6. Chew, G. (1963) S-Matrix Theory (W. A. Ben- 57. Weinberg, S. (1973) Phys. Rev. Lett. 31, 494–497. jamin, New York). 29. Callan, C. G. (1970) Phys. Rev. D 2, 1541–1547. 30. Symansik, K. (1970) Comm. Math. Phys. 18, 227– 58. Fritzsch, H., Gell-Mann, M. & Leutwyler, H. 7. Fritzsch, H. & Gell-Mann, M. (1972) Proceedings (1973) Phys. Lett. B 47, 365–368. of the XVI International Conference on High Energy 246. 31. Veneziano, G. (1968) Nuovo Cimento A 57, 190– 59. Lipkin, H. (1973) Phys. Lett. B 45, 267–271. Physics, eds. Jackson, J. D. & Roberts, A. (Na- 60. Marciano, W. & Pagels, H. (1978) Phys. Rep. C 36, tional Accellerator Laboratory, Batavia, IL), vol. 197. 32. ’t Hooft, G. (1967) Nucl. Phys. 35, 167–188. 137–276. 2, pp. 135–161. 33. Wilson, K. (1971) Phys. Rev. D 3, 1818–1846. 61. Wilson, K. (1974) Phys. Rev. D 10, 2445–2459. 8. Gell-Mann, M. & Neeman, Y. (1964) The Eight- 34. Callan, C. G., Jr. (1972) Phys. Rev. D 5, 3202–3210. 62. Gross, D. J. & Neveu, A. (1974) Phys. Rev. D 10, fold Way (W. A. Benjamin, New York). 35. Symansik, K. (1971) Comm. Math. Phys. 23, 49– 3235–3253. 9. Gell-Mann, M. (1964) Physics 1, 63–75. 86. 63. t’ Hooft, G. (1974) Nucl. Phys. B 72, 461–473. 10. Gell-Mann, M. (1964) Phys. Lett. 8, 214–215. 36. Christ, N., Hasslacher, B. & Mueller, A. (1972) 64. Casher, A., Kogut J. & Susskind, L. (1973) Phys. 11. Zweig, G. (1964) CERN Report No. TH401, 4R12. Phys. Rev. D 6, 3543–3562. Rev. Lett. 31, 792–795. 12. Greenberg, O. W. (1964) Phys. Rev. Lett. 13, 37. Callan, C. G. & Gross, D. J. (1973) Phys. Rev. D 8, 65. Casher, A., Kogut J. & Susskind, L. (1974) Phys. 598–602. 4383–4394. Rev. D 10, 732–745. 13. Nambu, Y. (1965) in Proceedings of the 2nd Coral 38. Jackiw, R., Van Royen, R. & West, G. (1970) Phys. 66. Caswell, W. (1974) Phys. Rev. Lett. 33, 244–246. Gables Conference on Symmetry Principles at High Rev. D 2, 2473–2485. 67. Jones, D. (1974) Nucl. Phys. B 75, 531–538. Energy, eds. Kursumoglu, B., Perlmutter, A. & 39. Frishman, Y. (1971) Ann. Phys. 66, 373–389. 68. Appelquist, T. & Georgi, H. (1973) Phys. Rev . D Sanmar, I. (Freeman, New York), pp. 274–285. 40. Leutwyler, H. & Stern, J. (1970) Nucl. Phys. B 20, 14. Nambu, Y. (1965) in Preludes in Theoretical Phys- 8, 4000–4002. 77–101. 69. Zee, A. (1973) Phys. Rev. D 8, 4038–4041. ics in Honor of V. F. Weisskopf, ed. De-Shalit, A., 41. Gross, D. J. (1971) Phys. Rev. D 4, 1059–1072. Feshbach, H. & Van Hove, L. (North-Holland, 70. Gaillard, M. K. & Lee, B. W. (1974) Phys. Rev. 42. Christ, N., Hasacher, B. & Mueller, A. (1972) Lett. 33, 108–111. Amsterdam), pp. 133–142. Phys. Rev D 6, 3543–3562. 71. Altarelli, G. & Maiani, L. (1974) Phys. Lett. B. 52, 15. Han, M. Y. & Nambu, Y. (1965) Phys. Rev. B 139, 43. Wilson, K. & Kogut, J. (1974) Phys. Rep. 12, 351–354. 1006. 75–200. 72. Gross, D. J. (1974) Phys. Rev. Lett. 32, 1071– 16. Low, F. (1966) in Proceedings of the International 44. Gell-Mann, M. & Low, F. (1954) Phys. Rev. 95, 1073. Conference on High Energy Physics (Univ. of Cal- 1300–1312. ifornia Press, Berkeley), p. 249. 45. Stueckelberg, E. & Petermann, A. (1953) Helv. 73. Bouchiat, C., Iliopoulpos, J. & Meyer, P. (1972) 17. Adler, S. (1965) Phys. Rev. B 140, 736–747. Phys. Acta 26, 499–520. Phys. Lett. B 38, 519–523. 18. Weisberger, W. I. (1966) Phys. Rev. 143, 1302– 46. Bogoliubov, N. N. & Shirkov, D. V. (1959) Intro- 74. Gross, D. J. & Jackiw, R. (1972) Phys. Rev. D 6, 1309. duction to the Theory of Quantized Fields (Inter- 477–493. 19. Dashen, R. & Gell-Mann, M. (1966) Phys. Rev. science, New York). 75. Gaillard, M. K., Lee, B. W. & Rosner, J. L. (1975) Lett. 17, 340–343. 47. Parisi, G. (1973) Lett. Nuovo Cimento 7, 84–87. Rev. Mod. Phys. 47, 277–310. 20. Bjorken, J. D. (1966) Phys. Rev. 148, 1467–1478. 48. Callan, C. G. & Gross, D. J. (1973) Phys. Rev. D 8, 76. Appelquist, T. & Politzer, H. D. (1975) Phys. Rev. 21. Bjorken, J. D. (1967) Current Algebra at Small 4383–4394. Lett. 34, 43–45. Distances, Varenna School Lectures (Varenna 49. Coleman, S. & Gross, D. J. (1973) Phys. Rev. Lett. 77. Georgi, H., Quinn, H. R. & Weinberg, S. (1974) School, Varenna, Italy), course XLI. 31, 851–854. Phys. Rev. Lett. 33, 451–454.

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