A Watershed: the Emergence Of

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experimental results on electron and neutrino deep-inelastic scat- received much attention recently as a means of constructing uni- tering gave strong evidence that charged partons are spin 1/2 par- fi ed and renormalizable theories of the weak and electromagnetic A watershed: the ticles [13] and that they have baryon number 1/3 [14], i.e. that charged interactions. In this note we report an investigation of the ultravio- partons are quarks. let (UV) asymptotic behaviour of such theories. We have found Quantum fi eld theory and the renormalization group: Martinus that they possess the remarkable feature, perhaps unique among Veltman and Gerardus ’t Hooft [15] brought powerful new tools to renormalizable theories, of asymptotically approaching free-fi eld the study of perturbative renormalization theory, leading to a more theory. Such asymptotically free theories will exhibit, for matrix emergence of QCD rigorous, quantitative formulation of gauge theories of electroweak elements between on-mass-shell states, Bjorken scaling. We there- interactions. Kenneth Wilson introduced a wealth of new ideas, fore suggest that one should look to a non-Abelian gauge theory conveniently though rather obscurely referred to as the renormali- of the strong interactions to provide the explanation for Bjorken zation group, into the study of quantum fi eld theory beyond the scaling, which has so far eluded fi eld-theoretic understanding.” limits of perturbation theory. He used these ideas with great suc- Thus the tension between scaling and quantum field theory David Gross and Frank Wilczek look cess to study critical phenomena. Neither of those developments might be resolved but only within a special, limited class of theo- related directly to the strong interaction problem but they formed ries. The paper surveys those possibilities and concludes: “One back at how QCD began to emerge in its an important intellectual background and inspiration. They particularly appealing model is based on three triplets of fermi- showed that the possibilities for quantum fi eld theory to describe ons, with Gell-Mann’s SU(3)xSU(3) as a global symmetry and an current form 40 years ago. physical behaviour were considerably richer than previously appre- SU(3) “colour” gauge group to provide the strong interactions. That ciated. Wilson [16] also sketched how his renormalization-group is, the generators of the strong-interaction gauge group commute ideas might be used to study short-distance behaviour, with spe- with ordinary SU(3)xSU(3) currents and mix quarks with the same In a recent article, Harald Fritzsch shared his perspective on the cifi c reference to problems in the strong interaction. isospin and hypercharge but different “colour”. In such a model history of the understanding of the strong interaction (CERN Cou- These various clues appeared to be mutually exclusive, or at least the vector mesons are neutral and the structure of the operator rier October 2012 p21). Here, we’d like to supplement that view. in considerable tension. The parton model is based on neglect of product expansion of electromagnetic or weak currents is (assum- Our focus is narrower but also sharper. We will discuss a brief interference terms whose existence, however, is required by basic ing the strong coupling constant but dramatic period during 1973–1974, when the modern theory principles of quantum mechanics. Attempts to identify partons is in the domain of attraction of the strong interaction – quantum chromodynamics, or QCD – with dynamical quarks [17] were partially successful but ascribed a of the origin!) essentially that emerged, essentially in its current form. While we were active par- David Gross and Frank Wilczek, when they received the Nobel much more intricate structure to protons than was postulated in the The conﬁ nement of the free quark model (up to ticipants in that drama, we have not relied solely on memory but prize in 2004. (Image credit: D Gross.) simplistic quark models and unambiguously required additional, of quarks calculable logarithmic correc- have carefully reviewed the contemporary literature. non-quark constituents. The confi nement of quarks contradicted contradicted all tions).*” This was the fi rst clear At the end of 1972 there was no fundamental theory of the strong Quarks and colour: A large body of strong-interaction phenom- all previous experience in phenomenology. Furthermore, such formulation of the theory that interaction – and no consensus on how to construct one. Proposals enology, including the particle spectrum and magnetic moments, behaviour could not be obtained within per turbative quantum fi eld previous we know today as QCD. The based on S-matrix philosophy, dual-resonance models, phenom- had been organized using the idea that mesons and baryons are theory. There were numerous technical challenges in combining experience in footnote indicated by * refers to enological quark models, current algebras, ideas about “partons” composite particles made from combinations of a small number of re-scaling transformations, as used in the renormalization group, phenomenology. additional work, which became and chiral dynamics – the logical descendant of Hideki Yukawa’s more fundamental constituents: quarks. This approach, which had with gauge symmetry. the core of our two subsequent original pion-exchange idea – created a voluminous and rapidly its roots in the ideas of Murray Gell-Mann [6] and George Zweig [7], But the most concrete, quantitative tension, and the one whose papers [3, 4]. growing literature. None of those competing ideas, however, is reviewed in a nice book by J J J Kokkedee [8]. For the model to resolution ultimately broke the whole subject open, was the tension David Politzer’s paper [2] contains calculations of the renormali- offered a framework in which uniquely defi ned calculations lead- work, the quarks were required to have bizarre properties – quali- between the scaling behaviour observed experimentally at SLAC zation group coeffi cients for non-Abelian gauge theories with fer- ing to sharp, testable predictions could be carried out. It seemed tatively different from the proper ties of any known par ticles. Their and the basic principles of quantum fi eld theory. Several workers mions, broadly along the same lines as in our fi rst paper quoted possible that strong-interaction physics would evolve along the electric charges had to be fractional. They had to have an extra [18] expanded Wilson’s somewhat sketchy indications into a precise above [1]. It does not refer to the problem of understanding scaling in lines of nuclear physics: one would gradually accumulate insight internal “colour” degree of freedom [9,10]. Above all, they had to be mapping between calculable properties of quantum fi eld theories the hadronic strong interaction. (The reference to “strong interac- experimentally, and acquire command of an ever-larger range of confi ned. Extensive experimental searches for individual quarks and observable aspects of inclusive cross-sections. Specifi cally, tions” in the title is generic.) Politzer emphasized the importance phenomena through models and rules of thumb. An overarching gave negative results. Within the model quark–antiquark pairs this work made it clear that the scaling behaviour observed at of the converse of asymptotic freedom – that is, that the effective theory worthy to stand beside Maxwell’s electrodynamics or Ein- made mesons, while quark–quark–quark triplets made baryons; SLAC could be obtained only in quantum fi eld theories with very coupling grows at long distances. He remarks that this could lead stein’s general relativity was no more than a dream – and not a single quarks had to be much heavier than mesons and baryons – if, small anomalous dimensions. (Strict scaling, which is equivalent to surprises regarding the particle content of asymptotically free widely shared one. indeed, they existed at all. to vanishing anomalous dimensions, cannot occur in a non-trivial – theories and support dynamical symmetry breaking. Although Within less than two years the situation had transformed radi- Scaling and partons: The famous electroproduction experi- interacting – quantum fi eld theory[19] .) A few realized that approxi- we arrived at our results independently, we and Politzer learnt of cally. We had arrived at a very specifi c candidate theory of the ments at SLAC revealed, beginning in the late 1960s, that inclusive mate scaling could be achieved in an interacting quantum theory, each other’s work before publication, compared results, requested strong interaction, one based on precise, beautiful equations. And cross-sections did not exhibit the “soft” or “form factor” behaviour if the effective interaction approached zero at short distances. simultaneous publication and referred to one another. The paper we had specifi c, quantitative proposals for testing it. The theoreti- familiar in exclusive and purely hadronic processes (as explored up Anthony Zee called such fi eld theories “stagnant”(they are essen- by Howard Georgi and Politzer [5] adopts QCD without comment cal works [1–5] that were central to this transformation can be identi- to that time). Richard Feynman [11] interpreted these experiments tially what we now call asymptotically free theories) and he [20], and independently derives predictions for deviations from scaling fi ed, we think, with considerable precision. as indicating the existence of more fundamental point-like con- Kurt Symanzik [21] and Giorgio Parisi [22] searched for such theories. parallel to the corresponding parts of our papers [3, 4]. stituent particles within protons, which he called partons. His However, none found any physically acceptable examples. Indeed, First clues approach was intuitive, employing a form of impulse approxima- a powerful no-go result [23] demonstrated that no four-dimensional Further reﬂ ections Let us briefl y recall the key lines of evidence and thought that those tion. James Bjorken [12] arrived at related results earlier, using more quantum fi eld theory lacking non-Abelian gauge symmetry can be The preceding account omits several sidelights and near misses, works reconciled, synthesized and brought to fruition. They can formal operator methods (local current algebra). Current-algebra asymptotically free. and lots of prehistory. But, although it is incomplete, we do not be summarized under three headings: quarks and colour; scaling sum rules were derived using “quark–gluon” models with Abe- Our paper, submitted in April 1973 [1], alludes directly to these think it is distorted. ▲ and partons; quantum fi eld theory and the renormalization group. lian, fl avourless gluons. The agreement of these sum rules with motivating issues in its opening: “Non-Abelian theories have It may be appropriate to mention explicitly contributions by

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two extremely eminent physicists (with collaborators) that are lenge. The most successful of these, based on direct numerical often cited together with papers 1–5 in ways that can be misleading. 0.5 solution of the equations (so-called “lattice gauge theory”) has April 2012 ’t Hooft, together with Veltman, had developed effective meth- gone far beyond demonstrating confi nement to yield sharp quanti- ods for calculating quantum corrections in non-Abelian gauge τdecays (N3LO) tative results for the mass spectrum and for many detailed proper- theories. They had worked out many examples, specifi cally includ- lattice QCD (NNLO) ties of hadrons. ing one-loop wave function and vertex divergences [24]. It would DIS jets (NLO) More generally, the dramatic success of a fully realized quantum 0.4 heavy quarkonia (NLO) not have been very diffi cult, as a technical matter, to re-assemble e+e– jets and shapes (res. NNLO) fi eld theory in yielding a wealth of striking physical phenomena pieces of those calculations to construct calculations of renormali- Z pole fit (N3LO) that are not evident in a linear approximation – including emer- zation group coeffi cients. ’t Hooft attests – and Symanzik corrobo- pp–→jets (NLO) gence of a dynamical scale (“mass without mass”), dynamical rated – that he announced a negative value of the β function for symmetry breaking, a rich physical spectrum and, of course, con- non-Abelian gauge theories with fermions at a conference in Mar- 0.3 fi nement – helped catalyse a renewed interest in the deep possibili-

seilles in the summer of 1972. Unfortunately, there is no record of (Q) ties of quantum fi eld theory. It continues to surprise us today. s this in the workshop proceedings, nor in the contemporary litera- α Prior to papers 1–5, the behaviour of matter at ultrahigh tem- ture, so there is no documentation regarding the exact content of the peratures and densities seemed utterly inaccessible to theoretical announcement or its context. It had no infl uence on papers 1–5. In 0.2 understanding. After these papers, it was understood instead to his contemporary work on the strong interaction, ’t Hooft adopted be remarkably simple. That circumstance opened up the earliest a completely different perspective from that of Gross-Wilczek and moments of the Big Bang to scientifi c analysis. It is the foundation Georgi-Politzer, a perspective from which it would be very dif- of what has become a large and fruitful fi eld: astroparticle physics. fi cult to arrive at QCD and its property of asymptotic freedom The equations of QCD are rooted in the same mathematics of Fig. 2. Hadrons emerging from high-energy collisions at large as we understand them today. Specifi cally, ’t Hooft’s work con- 0.1 gauge symmetry [27] that underlies the modern theory of elec- transverse momentum occur in nearly collinear “jets”.

sidered a spontaneously broken gauge theory with hadrons as the QCD αs(Mz) = 0.1184±0.0007 troweak interactions. They are worthy to stand beside Maxwell’s According to QCD the jets are initiated by quarks, antiquarks, fundamental objects, e.g. ρ mesons as gauge par ticles. His relevant equations; one might even say they are an enriched version of those and gluons, and inherit their energy and momentum. Pictured 1 10 100 publications immediately following papers 1–5 supply alternative Q(GeV) equations. It becomes possible to contemplate still more extensive here is an event from the CMS collaboration at the LHC, which methods for calculating renormalization group coeffi cients but do symmetries, unifying the different forces. The methods used to features six jets. not propose specifi c physical applications. Fig. 1. Asymptotic freedom, a principal dynamical property of establish asymptotic freedom – specifi cally, running couplings Two contributions involving Gell-Mann and collaborators are QCD, predicts the logarithmic decrease of the strong interaction – provide quantitative tools for exploring that idea. Intriguing, [20] A Zee 1973 Phys. Rev. D7 3630. sometimes cited as sources of QCD. The fi rst is the “Rochester coupling as energy increases or distance decreases. This fi gure encouraging results have been obtained along these lines. They [21] G Parisi 1973 Lett. N. Cim. 7 84. Conference” at Fermilab in the summer of 1972 [25]. It contains two shows the current agreement of QCD predictions with many suggest, in particular, the possibility of low-energy supersymme- [22] K Symanzik 1973 Lett. N. Cim. 6 2. relevant presentations, Gell-Mann’s summary talk and a contrib- experiments. try, such as might be observed at the LHC. [23] S Coleman and D J Gross 1973 Phys. Rev. Lett. 31 851. uted paper with Fritzsch, entitled “Current Algebra: Quarks and [24] G ’t Hooft and M Veltman 1972 Colloquium on Renormalization of Yang- What Else?” In the summary talk, SLAC scaling is mentioned and cally improvable way opened up many applications (fi gure 1). The ● Further reading Mills Fields and Applications to Particle Physics, Marseilles, 19 June 1972 37; interpreted in terms of “quarks, treated formally”. The discussion subject now called perturbative QCD, which refi nes and improves [1] D J Gross and F Wilczek 1973 Phys. Rev. Lett. 30 1343 (received 27 April CERN-TH-1571. is not rooted in quantum fi eld theory; indeed, most of the discus- parton model ideas, is a direct outgrowth of papers 1–5 but extends 1973). [25.] XVI International Conference on High-Energy Physics, Batavia, September sion of the strong interaction, by far, is given over to S-matrix and their scope almost beyond recognition. Perturbative QCD is the [2]. H D Politzer 1973 Phys. Rev. Lett. 30 1346 (received 3 May 1973). 1972. dual-resonance ideas. The presentation with Fritzsch briefl y men- subject of several large textbooks, dozens of conference proceed- [3] D J Gross and F Wilczek 1973 Phys. Rev. D8 3633 (received 23 July 1973). [26] H Fritzsch, M Gell-Mann and H Leutwyler 1973 Phys. Lett. 47B 365 tions the possibility of using colour octet gluons, as one among sev- ings, etc. It has become the essential foundation for analysing [4] D J Gross and F Wilczek 1974 Phys. Rev. D9 980 (received 27 August 1973). (received 1 October 1973). eral possibilities for extending light-cone current algebra (again, experimental results from high-energy accelerators including, [5] H Georgi and H D Politzer 1974 Phys. Rev. D9 416 (received 30 July 1973). [27] C N Yang and R Mills 1954 Phys. Rev. 96 191. not within a quantum fi eld theory). notably, the LHC. It justifi es, in particular, the identifi cation of [6] M Gell-Mann 1964 Phys. Lett. 8 214. The second contribution [26] appeared after 1–5 and refers to “jets” with quarks and gluons (fi gure 2), and allows calculation of [7] G Zweig 1964 CERN-TH-401. Résumé them. From a historical perspective, what is particularly revealing their production rates. [8] J J J Kokkedee 1969 The Quark Model” (W A Benjamin). Une étape historique : l’émergence de la QCD about it is the comment: “For us, the result that the colour octet fi eld The paradoxical heuristics of the quark model, with its juxta- [9] O W Greenberg 1964 Phys. Rev. Lett. 13 598. theory model comes closer to asymptotic scaling than the colour position of free-particle properties with confi nement, became [10] M Y Han and Y Nambu 1965 Phys. Rev. 139B 1006. En 1972 il n’existait pas de théorie fondamentale de l’interaction singlet model is interesting, but not necessarily conclusive, since physically plausible and matured into a well posed mathematical [11] R Feynman 1971 in The Past Decade in Particle Theory p773(Gordon and forte, ni de consensus sur la voie à suivre pour en élaborer une. we conjecture that there may be a modifi cation at high frequencies problem [4]. For the growth of the effective coupling with increas- Breach). Or en moins de deux ans la situation s’est transformée de façon that produces true asymptotic scaling.” ing distance, together with the [12] J Bjorken 1969 Phys. Rev. 179 1547. radicale. Dans cet article, David Gross et Frank Wilczek évoquent As events unfolded, the most profound and most fr uitful aspects existence of formally mass- [13] C G Callan and D J Gross 1968 Phys. Rev. Lett. 22 156. la naissance de la chromodynamique quantique, ou QCD, la of QCD and asymptotic freedom proved to be their embodiment in less (colour) charged parti- [14] D J Gross and C Llewelyn-Smith 1969 Nucl. Phys. B 14 337. théorie moderne de l’interaction forte. Ils racontent en détail a rigorously defi ned, quantitatively precise quantum fi eld theory, This was the cles, brought the theory into [15] G ’t Hooft and M Veltman 1972 Nucl. Phys. B50 318. cette période brève, mais riche en rebondissements, des années which could be tested through its prediction of deviations from ﬁ rst clear uncharted territory. Because [16] K Wilson 1969 Phys. Rev. 179 1499. 1973–1974, qui a vu l’émergence de la QCD sous sa forme actuelle. scaling. Yet just those aspects are what the authors hesitated to uncancelled fi eld energy threat- [17] J Bjorken and E Paschos 1969 Phys. Rev. 185 197; S D Drell and T M Yan Leur propre contribution à cette séquence mémorable leur a accept, even after they had been analysed. formulation of ens to build up catastrophically, 1971 Ann. Phys. (NY) 66 578. d’ailleurs valu un prix Nobel en 2004. The emergence of a specifi c, precise quantum fi eld theory for the theory that it was plausible that only sin- [18] R Jackiw, R Van Royen and G West 1970 Phys. Rev. D 2 2473; H Leutwyler the strong interaction – featuring beautiful equations – marked a we know today glet states might emerge with and J Stern 1970 Nucl. Phys. B 20 77; Y Frishman 1971 Ann. Phys. (NY) 66 373; David Gross, Kavli Institute for Theoretical Physics, Santa Barbara, and watershed. Remarkable progress ensued on several fronts. fi nite energy. Essentially new D J Gross 1971 Phys. Rev. D 4 1059; N Christ, B Hasslacher and A Mueller 1972 Frank Wilczek, Massachusetts Institute of Technology. See their Nobel The realization that basic strong interaction processes at high as QCD. mathematical techniques were Phys. Rev. D6 3543. lectures for much additional material on the history and impact of asymptotic energy could be calculated in a practical, controlled and systemati- invented to address this chal- [19] C Callan and D J Gross 1973 Phys. Rev. D8 4383. freedom and QCD.

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