A watershed: the emergence of QCD

28 January 2013

David Gross and Frank Wilczek look back at how QCD began to emerge in its current form 40 years ago.

David Gross and Frank Wilczek

In a recent article, Harald Fritzsch shared his perspective on the history of the understanding of the strong interaction (CERN Courier October 2012 p21). Here, we’d like to supplement that view. Our focus is narrower but also sharper. We will discuss a brief but dramatic period during 1973–1974, when the modern theory of the strong interaction – quantum chromodynamics, or QCD – emerged, essentially in its current form. While we were active participants in that drama, we have not relied solely on memory but have carefully reviewed the contemporary literature.

At the end of 1972 there was no fundamental theory of the strong interaction – and no consensus on how to construct one. Proposals based on S-matrix philosophy, dual-resonance models, phenomenological quark models, current algebras, ideas about “partons” and chiral dynamics – the logical descendant of Hideki Yukawa’s original pion-exchange idea – created a voluminous and rapidly growing literature. None of those competing ideas, however, offered a framework in which uniquely defined calculations leading to sharp, testable predictions could be carried out. It seemed possible that strong-interaction physics would evolve along the lines of nuclear physics: one would gradually accumulate insight experimentally, and acquire command of an ever-larger range of phenomena through models and rules of thumb. An overarching theory worthy to stand beside Maxwell’s electrodynamics or Einstein’s general relativity was no more than a dream – and not a widely shared one.

Within less than two years the situation had transformed radically. We had arrived at a very specific candidate theory of the strong interaction, one based on precise, beautiful equations. And we had specific, quantitative proposals for testing it. The theoretical works [1–5] that were central to this transformation can be identified, we think, with considerable precision.

First clues

Let us briefly recall the key lines of evidence and thought that those works reconciled, synthesized and brought to fruition. They can be summarized under three headings: quarks and colour; scaling and partons; quantum field theory and the renormalization group.

Quarks and colour: A large body of strong-interaction phenomenology, including the particle spectrum and magnetic moments, had been organized using the idea that mesons and baryons are composite particles made from combinations of a small number of more fundamental constituents: quarks. This approach, which had its roots in the ideas of Murray Gell-Mann [6] and George Zweig [7], is reviewed in a nice book by J J J Kokkedee [8]. For the model to work, the quarks were required to have bizarre properties – qualitatively different from the properties of any known particles. Their electric charges had to be fractional. They had to have an extra internal “colour” degree of freedom [9,10]. Above all, they had to be confined. Extensive experimental searches for individual quarks gave negative results. Within the model quark–antiquark pairs made mesons, while quark–quark–quark triplets made baryons; single quarks had to be much heavier than mesons and baryons – if, indeed, they existed at all.

Scaling and partons: The famous electroproduction experiments at SLAC revealed, beginning in the late 1960s, that inclusive cross-sections did not exhibit the “soft” or “form factor” behaviour familiar in exclusive and purely hadronic processes (as explored up to that time). Richard Feynman [11] interpreted these experiments as indicating the existence of more fundamental point-like constituent particles within protons, which he called partons. His approach was intuitive, employing a form of impulse approximation. James Bjorken [12] arrived at related results earlier, using more formal operator methods (local current algebra). Current-algebra sum rules were derived using “quark–gluon” models with Abelian, flavourless gluons. The agreement of these sum rules with experimental results on electron and neutrino deep-inelastic scattering gave strong evidence that charged partons are spin 1/2 particles [13] and that they have baryon number 1/3 [14], i.e. that charged partons are quarks.

Quantum field theory and the renormalization group: Martinus Veltman and Gerardus ’t Hooft [15] brought powerful new tools to the study of perturbative renormalization theory, leading to a more rigorous, quantitative formulation of gauge theories of electroweak interactions. Kenneth Wilson introduced a wealth of new ideas, conveniently though rather obscurely referred to as the renormalization group, into the study of quantum field theory beyond the limits of perturbation theory. He used these ideas with great success to study critical phenomena. Neither of those developments related directly to the strong interaction problem but they formed an important intellectual background and inspiration. They showed that the possibilities for quantum field theory to describe physical behaviour were considerably richer than previously appreciated. Wilson [16] also sketched how his renormalization-group ideas might be used to study short-distance behaviour, with specific reference to problems in the strong interaction.

These various clues appeared to be mutually exclusive, or at least in considerable tension. The parton model is based on neglect of interference terms whose existence, however, is required by basic principles of quantum mechanics. Attempts to identify partons with dynamical quarks [17] were partially successful but ascribed a much more intricate structure to protons than was postulated in the simplistic quark models and unambiguously required additional, non-quark constituents. The confinement of quarks contradicted all previous experience in phenomenology. Furthermore, such behaviour could not be obtained within perturbative quantum field theory. There were numerous technical challenges in combining re-scaling transformations, as used in the renormalization group, with gauge symmetry.

But the most concrete, quantitative tension, and the one whose resolution ultimately broke the whole subject open, was the tension between the scaling behaviour observed experimentally at SLAC and the basic principles of quantum field theory. Several workers [18] expanded Wilson’s somewhat sketchy indications into a precise mapping between calculable properties of quantum field theories and observable aspects of inclusive cross-sections. Specifically, this work made it clear that the scaling behaviour observed at SLAC could be obtained only in quantum field theories with very small anomalous dimensions. (Strict scaling, which is equivalent to vanishing anomalous dimensions, cannot occur in a non-trivial – interacting – quantum field theory [19].) A few realized that approximate scaling could be achieved in an interacting quantum theory, if the effective interaction approached zero at short distances. Anthony Zee called such field theories “stagnant”(they are essentially what we now call asymptotically free theories) and he [20], Kurt Symanzik [21] and Giorgio Parisi [22] searched for such theories. However, none found any physically acceptable examples. Indeed, a powerful no-go result [23] demonstrated that no four-dimensional quantum field theory lacking non-Abelian gauge symmetry can be asymptotically free.

The tension between scaling and quantum field theory might be resolved but only within a special, limited class of theories

Our paper, submitted in April 1973 [1], alludes directly to these motivating issues in its opening: “Non-Abelian theories have received much attention recently as a means of constructing unified and renormalizable theories of the weak and electromagnetic interactions. In this note we report an investigation of the ultraviolet (UV) asymptotic behaviour of such theories. We have found that they possess the remarkable feature, perhaps unique among renormalizable theories, of asymptotically approaching free-field theory. Such asymptotically free theories will exhibit, for matrix elements between on-mass-shell states, Bjorken scaling. We therefore suggest that one should look to a non-Abelian gauge theory of the strong interactions to provide the explanation for Bjorken scaling, which has so far eluded field-theoretic understanding.”

Thus the tension between scaling and quantum field theory might be resolved but only within a special, limited class of theories. The paper surveys those possibilities and concludes: “One particularly appealing model is based on three triplets of fermions, with Gell-Mann’s SU(3)xSU(3) as a global symmetry and an SU(3) “colour” gauge group to provide the strong interactions. That is, the generators of the strong-interaction gauge group commute with ordinary SU(3)xSU(3) currents and mix quarks with the same isospin and hypercharge but different “colour”. In such a model the vector mesons are neutral and the structure of the operator product expansion of electromagnetic or weak currents is (assuming the strong coupling constant is in the domain of attraction of the origin!) essentially that of the free quark model (up to calculable logarithmic corrections).*” This was the first clear formulation of the theory that we know today as QCD. The footnote indicated by * refers to additional work, which became the core of our two subsequent papers [3, 4].

David Politzer’s paper [2] contains calculations of the renormalization group coefficients for non-Abelian gauge theories with fermions, broadly along the same lines as in our first paper quoted above [1]. It does not refer to the problem of understanding scaling in the hadronic strong interaction. (The reference to “strong interactions” in the title is generic.) Politzer emphasized the importance of the converse of asymptotic freedom – that is, that the effective coupling grows at long distances. He remarks that this could lead to surprises regarding the particle content of asymptotically free theories and support dynamical symmetry breaking. Although we arrived at our results independently, we and Politzer learnt of each other’s work before publication, compared results, requested simultaneous publication and referred to one another. The paper by Howard Georgi and Politzer [5] adopts QCD without comment and independently derives predictions for deviations from scaling parallel to the corresponding parts of our papers [3, 4].

Further reflections

The preceding account omits several sidelights and near misses, and lots of prehistory. But, although it is incomplete, we do not think it is distorted.

It may be appropriate to mention explicitly contributions by two extremely eminent physicists (with collaborators) that are often cited together with papers 1–5 in ways that can be misleading.

’t Hooft, together with Veltman, had developed effective methods for calculating quantum corrections in non-Abelian gauge theories. They had worked out many examples, specifically including one-loop wave function and vertex divergences [24]. It would not have been very difficult, as a technical matter, to re-assemble pieces of those calculations to construct calculations of renormalization group coefficients. ’t Hooft attests – and Symanzik corroborated – that he announced a negative value of the β function for non-Abelian gauge theories with fermions at a conference in Marseilles in the summer of 1972. Unfortunately, there is no record of this in the workshop proceedings, nor in the contemporary literature, so there is no documentation regarding the exact content of the announcement or its context. It had no influence on papers 1–5. In his contemporary work on the strong interaction, ’t Hooft adopted a completely different perspective from that of Gross-Wilczek and Georgi-Politzer, a perspective from which it would be very difficult to arrive at QCD and its property of asymptotic freedom as we understand them today. Specifically, ’t Hooft’s work considered a spontaneously broken gauge theory with hadrons as the fundamental objects, e.g. ρ mesons as gauge particles. His relevant publications immediately following papers 1–5 supply alternative methods for calculating renormalization group coefficients but do not propose specific physical applications.

Two contributions involving Gell-Mann and collaborators are sometimes cited as sources of QCD. The first is the “Rochester Conference” at Fermilab in the summer of 1972 [25]. It contains two relevant presentations, Gell-Mann’s summary talk and a contributed paper with Fritzsch, entitled “Current Algebra: Quarks and What Else?” In the summary talk, SLAC scaling is mentioned and interpreted in terms of “quarks, treated formally”. The discussion is not rooted in quantum field theory; indeed, most of the discussion of the strong interaction, by far, is given over to S-matrix and dual-resonance ideas. The presentation with Fritzsch briefly mentions the possibility of using colour octet gluons, as one among several possibilities for extending light-cone current algebra (again, not within a quantum field theory).

The second contribution [26] appeared after 1–5 and refers to them. From a historical perspective, what is particularly revealing about it is the comment: “For us, the result that the colour octet field theory model comes closer to asymptotic scaling than the colour singlet model is interesting, but not necessarily conclusive, since we conjecture that there may be a modification at high frequencies that produces true asymptotic scaling.”

As events unfolded, the most profound and most fruitful aspects of QCD and asymptotic freedom proved to be their embodiment in a rigorously defined, quantitatively precise quantum field theory, which could be tested through its prediction of deviations from scaling. Yet just those aspects are what the authors hesitated to accept, even after they had been analysed.

The emergence of a specific, precise quantum field theory for the strong interaction – featuring beautiful equations – marked a watershed. Remarkable progress ensued on several fronts.

The realization that basic strong interaction processes at high energy could be calculated in a practical, controlled and systematically improvable way opened up many applications (figure 1). The subject now called perturbative QCD, which refines and improves parton model ideas, is a direct outgrowth of papers 1–5 but extends their scope almost beyond recognition. Perturbative QCD is the subject of several large textbooks, dozens of conference proceedings, etc. It has become the essential foundation for analysing experimental results from high-energy accelerators including, notably, the LHC. It justifies, in particular, the identification of “jets” with quarks and gluons (figure 2), and allows calculation of their production rates.

The paradoxical heuristics of the quark model, with its juxtaposition of free-particle properties with confinement, became physically plausible and matured into a well posed mathematical problem [4]. For the growth of the effective coupling with increasing distance, together with the existence of formally massless (colour) charged particles, brought the theory into uncharted territory. Because uncancelled field energy threatens to build up catastrophically, it was plausible that only singlet states might emerge with finite energy. Essentially new mathematical techniques were invented to address this challenge. The most successful of these, based on direct numerical solution of the equations (so-called “lattice gauge theory”) has gone far beyond demonstrating confinement to yield sharp quantitative results for the mass spectrum and for many detailed properties of hadrons.

The equations of QCD are rooted in the same mathematics of gauge symmetry

More generally, the dramatic success of a fully realized quantum field theory in yielding a wealth of striking physical phenomena that are not evident in a linear approximation – including emergence of a dynamical scale (“mass without mass”), dynamical symmetry breaking, a rich physical spectrum and, of course, confinement – helped catalyse a renewed interest in the deep possibilities of quantum field theory. It continues to surprise us today.

Prior to papers 1–5, the behaviour of matter at ultrahigh temperatures and densities seemed utterly inaccessible to theoretical understanding. After these papers, it was understood instead to be remarkably simple. That circumstance opened up the earliest moments of the Big Bang to scientific analysis. It is the foundation of what has become a large and fruitful field: astroparticle physics.

The equations of QCD are rooted in the same mathematics of gauge symmetry [27] that underlies the modern theory of electroweak interactions. They are worthy to stand beside Maxwell’s equations; one might even say they are an enriched version of those equations. It becomes possible to contemplate still more extensive symmetries, unifying the different forces. The methods used to establish asymptotic freedom – specifically, running couplings – provide quantitative tools for exploring that idea. Intriguing, encouraging results have been obtained along these lines. They suggest, in particular, the possibility of low-energy supersymmetry, such as might be observed at the LHC.

Further reading

[1] D J Gross and F Wilczek 1973 Phys. Rev. Lett. 30 1343 (received 27 April 1973).
[2]. H D Politzer 1973 Phys. Rev. Lett. 30 1346 (received 3 May 1973).
[3] D J Gross and F Wilczek 1973 Phys. Rev. D8 3633 (received 23 July 1973). [4] D J Gross and F Wilczek 1974 Phys. Rev. D9 980 (received 27 August 1973). [5] H Georgi and H D Politzer 1974 Phys. Rev. D9 416 (received 30 July 1973). [6] M Gell-Mann 1964 Phys. Lett. 8 214.
[7] G Zweig 1964 CERN-TH-401.
[8] J J J Kokkedee 1969 The Quark Model” (W A Benjamin).
[9] O W Greenberg 1964 Phys. Rev. Lett. 13 598.
[10] M Y Han and Y Nambu 1965 Phys. Rev. 139B 1006.
[11] R Feynman 1971 in The Past Decade in Particle Theory p773(Gordon and Breach).
[12] J Bjorken 1969 Phys. Rev. 179 1547.
[13] C G Callan and D J Gross 1968 Phys. Rev. Lett. 22 156.
[14] D J Gross and C Llewelyn-Smith 1969 Nucl. Phys. B 14 337.
[15] G ’t Hooft and M Veltman 1972 Nucl. Phys. B50 318.
[16] K Wilson 1969 Phys. Rev. 179 1499.
[17] J Bjorken and E Paschos 1969 Phys. Rev. 185 197; S D Drell and T M Yan 1971 Ann. Phys. (NY) 66 578.
[18] R Jackiw, R Van Royen and G West 1970 Phys. Rev. D 2 2473; H Leutwyler and J Stern 1970 Nucl. Phys. B 20 77; Y Frishman 1971 Ann. Phys. (NY) 66 373; D J Gross 1971 Phys. Rev. D 4 1059; N Christ, B Hasslacher and A Mueller 1972 Phys. Rev. D6 3543.
[19] C Callan and D J Gross 1973 Phys. Rev. D8 4383.
[20] A Zee 1973 Phys. Rev. D7 3630.
[21] G Parisi 1973 Lett. N. Cim. 7 84.
[22] K Symanzik 1973 Lett. N. Cim. 6 2.
[23] S Coleman and D J Gross 1973 Phys. Rev. Lett. 31 851.
[24] G ’t Hooft and M Veltman 1972 Colloquium on Renormalization of Yang- Mills Fields and Applications to Particle Physics, Marseilles, 19 June 1972 37; CERN-TH-1571.
[25.] XVI International Conference on High-Energy Physics, Batavia, September 1972.
[26] H Fritzsch, M Gell-Mann and H Leutwyler 1973 Phys. Lett. 47B 365 (received 1 October 1973).
[27] C N Yang and R Mills 1954 Phys. Rev. 96 191.

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