The Chiral Anomaly Roman Jackiw Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, USA

How a discrepancy in the current algebra formulation of the decay of the neutral proton into two photons was resolved in terms of a quantum mechanical mechanism for symmetry breaking that has become an essential ingredient of models in particle physics.

John Bell and I met and became ac­ pancies between theoretical predictions borator on charge particle formalism —- quainted in 1967, when I went to CERN and experimental verification. was at the table and asked about our for a year-long research visit, soon after John was particularly impressed with current interests. When we described to finishing my doctoral studies at Cornell. an analysis by his good friend Martinus him the π° → 2γ puzzle, he expressed At that time, particle physics theory Veltman, and also D. Sutherland, to the amazement that theorists should still be was dominated, as it happens from effect that the neutral pion could not pursuing a process that he, an expe­ time-to-time, by a single idea : there was decay into two photons if the charge- rimentalist, calculated almost twenty broad agreement among theorists what neutral and gauge invariant axial vector years earlier. He had found excellent the important problems were and how (chiral) current is conserved, as it was agreement with experiment, while also they should be solved, although one can then taken to be in current algebra appli­ noting a discrepancy between results hardly remember the details of the pro­ cations. Because the decay does, in obtained when the pion coupled to nu­ gramme. But attaching my scientific ac­ fact, occur in Nature, while the Suther- cleons by pseudovector and pseudo- tivity to a consensus was not my ambi­ land / Veltman argument appeared in­ scalar interactions (pions, nucleons and tion; I had much admired the indepen­ controvertible, John stressed that the photons were the only particles in dent attitude of one of my research su­ subject of current algebra must not be Steinberger's model, and it was belie­ pervisors at Cornell, Ken Wilson. So I closed until this puzzle is resolved and ved that equivalent results emerge for looked among the staff at CERN for so­ urged a study of the chiral current. pseudovector and pseudoscalar pion- meone who pursued interesting issues This was the second time I received nucleon coupling). that were neither "central" nor "impor­ such advice: in my final student days There, at that table, came to us the tant", and I was delighted to find such Wilson suggested a critical examination realization that Steinberger's calcula­ a scientist in John Bell. Moreover, he of the apparent conservation of the tion would be identical to the one per­ was generous in giving his time : he tole­ axial vector current in the Baker-John- formed in the dynamical framework of rated my coming to his office and ap­ son-Willey theory of massless electro­ the σ-model, which was constructed peared willing to discuss without limit. dynamics, with which he had his own to realize current algebra explicitly. We I appreciated the magnitude of his gene­ disagreements. reasoned that within the σ-model we rosity only years later when I too be­ I was therefore willing to research could satisfy the current algebraic as­ came installed in an office and people this topic, but since the existing dis­ sumptions of Sutherland / Veltman and began coming in and taking my time to cussions were straightforward and the also obtain good experimental agree­ talk about things. conclusions immediate, it was hard ment in view of Steinberger's result, to see how a useful probe could be thereby resolving the π° → 2γ puzzle. The Pion Decay Puzzle launched. I asked fellow theorists for There began for us a period of wide- Guided by Steinberger's paper (at suggestions but the subject did not that time we were not familiar with the ranging conversations — and not only spark interest. I do recall two mathe­ about physics — which acquainted me work of his contemporaries, H. Fukuda matically oriented colleagues, Henri and Y. Miyamoto, and only dimly aware with the many issues that concerned Epstein and Raymond Stora, offering a John. But nothing was then said about of subsequent contributions by J. diagnosis that in retrospect proved Schwinger), we quickly established that his work on quantum mechanics -— he prescient: in their opinion one could not did not at that time describe it to me and the amplitude describing correlations rely on current algebra analyses be­ between the three currents appearing in I did not know of it. Current algebra inte­ cause physicists treat cavalierly singu­ rested John very much. Within its fra­ the problem — two vector currents to lar products of distributions. But their which the two photons couple and one mework one can understand the low prognosis that a cure will be found if energy behaviour of elementary parti­ axial vector current to which the pion one uses rigorous rather than heuristic couples — is given in lowest order (one cles, without making a commitment to mathematics did not appeal. In fact, the a dynamical model, unknown in the loop) perturbation theory by the now decisive suggestion did not come from famous triangle graph depicted in Fig. 1. 1960's, while today's "standard mo­ a theorist but from an experimentalist. del" resists solution in the low energy The amplitude is determined by Feyn­ domain. The approach seemed suc­ Reformulating the σ-model man rules only up to an overall ambi­ cessful, complete and exhausted by the One of CERN's civilized activities, to guity, owing to ultraviolet divergences, late 1960's, yet there remained discre- which John frequently invited me, con­ even though the amplitude is finite. sists of taking an afternoon drink in Moreover, while the ambiguity may be Professor R. Jackiw is with the Center for the cafeteria, where we would continue resolved by enforcing current conserva­ Theoretical Physics, MIT, Cambridge, MA, our conversations together with people tion, it is impossible to maintain conser­ USA. He received his Ph.D. from Cornell and who joined us. On one occasion, Jack vation of all three currents, as was assu­ was CERN Fellow in 1966/67. Steinberger — John's friend and colla- med in the current algebra calculation. 76 Europhys. News 22 (1991) Fig. 7 — Triangle Feynman diagram for the three-current idea that today has also gained wide correlation amplitude that leads to quantum mechanical currency among physicists. symmetry breaking. Two vector currents V and one axial vector current A form the three corners, at which virtual fermions are created/annihilated. Propagation of the virtual In all my contacts with John I was fermions, indicated by solid lines, prevents the currents always made aware of his overwhel­ from being conserved. A similar effect arises with three ming intellectual precision and honesty. axial vector currents. These are the qualities that made him such an incisive critic and therefore a wonderful colleague. Precisely this atti­ Thus, we found that the σ-model's Dirac sea (R. Feynman), anomalous tude lay behind his scientific achieve­ symmetries, which underlie current al­ transformation properties of the func­ ments, which are informed by clarity of gebra and which should guarantee the tional integral measure (K. Fujikawa), observation about previously murky conservation of the respective currents, the necessary effect of high-energy mo­ subjects. cannot be maintained when the model des on low-energy physics (V. Gribov), The same attitude characterized his is quantized. In the absence of these quantum field theoretic manifestation approaches outside science, for exam­ symmetries, pion decay is no longer of Berry's phase, local version of the ple to social and political questions. considered to be forbidden. Atiyah-Singer index, and cosmologi­ Many physicists profess humane and cal properties of gauge groups (L. Fad- liberal values, but often these become Anomalous Symmetry Breaking deev). The last two mathematical con­ obscured by personal emotion and pre­ Our work resolved the π° → 2γ pro­ nections seeded a remarkable collabo­ judice. In the last quarter century, issues blem, by exposing a purely quantum ration between mathematics and phy­ of Vietnam, Ireland and Palestine offer a mechanical mechanism for symmetry sics, which is still flourishing. On the dramatic opportunity for displaying breaking, which is the modern interpre­ other hand, the physical world itself social conscience in search of justice. tation of Steinberger's discrepancy, and became threatened by the anomaly John recognized and spoke on these these days is described as "anomalous because, as Gerard't Hooft showed [2], matters clearly. Already in 1967/68, I breaking of a symmetry", although it catalyzes baryon decay, but fortuna­ heard him analyze America's role in once the surprise has worn off, it is tely at a sufficiently slow rate to cause Vietnam in terms that did not gain ac­ better named "quantum mechanical no immediate concern. ceptance until years later; his opinions symmetry breaking" [page 72, 6]. In spite of these wide-ranging gene­ on the two other tragedies remain in the The "anomaly" was substantiated by ralizations, John preferred the simple minority even today, but one hopes that S. Adler, who working independently triangle graph calculation [page 72, 6]. here too his ideas are merely ahead of came to a similar conclusion about the He always stressed the element of their time. (absence of) symmetries in massless choice that exists in resolving the calcu­ I liked John very much and together electrodynamics, and building on our lational ambiguity, thus putting different with many colleagues I shall miss him. work established, with W. Bardeen, the faces on the nature of the anomaly — He was an outstanding scientist and important fact that higher perturbative a freedom that is obscured in the more helped us do good science, which is one orders do not modify the one-loop cal­ abstract and high-powered approach. reason why we become physicists. culation of pion decay. Further confir­ Indeed John was rather diffident Moreover, many enter our field not only mation came from Wilson, who used about the entire matter. In this he show­ for the opportunity of exploring Nature our theory as a case study for his non- ed one of his striking qualities : modesty in its most fundamental workings, but Lagrangian models of current algebra, about his own work, praise for the work also for what we perceive as the purity based on his operator product expan­ of others, but skepticism in the face of and honesty of the profession. These sion. The early period of research on inflated claims, even if they were extol­ qualities sometimes get submerged by this subject culminated in a phenome­ ling his own contributions. When he pressure of personal ambition, strugg­ nological description of quantum me­ eventually described to me his famous ling for achievement and recognition, chanical symmetry breaking in terms of analysis of the foundations of quantum but John Bell never lost them, and in an effective Lagrangian, constructed by mechanics [page 72, 1], he called that this way he reminded us of the ethical J. Wess and B. Zumimo [1] who appa­ research "a hobby". rently were not aware of our result. motivation for becoming a physicist. In time, this work, which arose from Fractional Charge Quantum Numbers clearing up a corner of current algebra, After I left CERN in 1968, we had ACKNOWLEDGEMENT grew to affect much of particle physics. many occasions to meet and talk about This work is supported in part by the U.S. It became an important ingredient of interesting topics, but our discussions Department of Energy under contract num­ model building, both for speculative never again resulted in a joint publi­ ber DE-AC02 - 76ER03069. string theories and for the conventional cation. The closest we came to this "standard model", where, among other happened when I described to John the ADDITIONAL REFERENCES things, it enforces colour triality and phenomenon of fractional charge quan­ [1] For detailed discussion and a guide to the explains the numerical equality of quark tum number [3], which had become literature, see Treiman S., Jackiw R., Zumino and lepton degrees of freedom, thus physically relevant [4]. He found this B. and Witten E., Current Algebra and Ano­ predicting the existence of the elusive interesting but characteristically was at malies (Princeton University Press / World top quark [1]. Scientific, Princeton, NJ / Singapore) 1985. first skeptical that a fractional value [2] 't Hooft G., Phys. Rev. Lett. 37 (1976) 8. John maintained an amused interest could be an eigenvalue. Upon elabora­ [3] Jackiw R. and Rebbi C., Phys. Rev. D 13 as our calculation became transformed ting the precise circumstances in which (1976) 3398. in various contexts, and was shown a sharp observable arises, he published [4] Su W.-P., Schrieffer J.R. and Heeger A., to be a consequence of diverse physi­ with R. Rajaraman [page 72, 21] an ana­ Phys. Rev. Lett. 42 (1979) 1968 ; Jackiw R. cal and mathematical considerations lysis that contributed to the understan­ and Schrieffer J.R., Nucl. Phys. B190 (1981) [1] : symmetry breaking aspects of the ding and acceptance of this fascinating 253. Europhys. News 22 (1991) 77