Journal of the Physical Society of Japan Vol. 76, No. 11, November, 2007, 111002 SPECIAL TOPICS #2007 The Physical Society of Japan

Frontiers of Elementary Particle Physics, the Standard Model and Beyond

From Yukawa’s Pion to Spontaneous Symmetry Breaking

Yoichiro NAMBU

Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, U.S.A. (Received February 6, 2007; accepted April 9, 2007; published November 12, 2007) An overview is given of the development of particle physics since the 1930s to the 1960s when the concept of spontaneous symmetry breaking (SSB) was established as an essential component of particle theory and eventually led to the Standard Model. A brief account of SSB as a general phenomenon in physics is also added.

KEYWORDS: particle physics, standard model, spontaneous symmetry breaking, gauge theory, Nambu–Gold- stone mode, Higgs, BCS, Ginzburg–Landau DOI: 10.1143/JPSJ.76.111002

pion.3) The difficulty of infinities was circumvented later by 1. Evolution of Particle Physics the idea of renormalization by Tomonaga, Schwinger, and What we now call particle physics started first as nuclear Feynman.4) physics in the 1930s, and I have once proposed to call Ernest To have a general perspective of what has happened since Lawrence and Hideki Yukawa the two founders of particle the 1930s, it is useful to recall the so-called three-stages physics:1) The first invented the cyclotron, the second theory5) by Taketani, who was a disciple and collaborator ‘‘invented’’ the meson theory, which together opened the of Yukawa. He argued that the advance of physics goes door to the exploration of ever higher energy phenomena through cycles of three stages: Faced with a set of new and to the discovery of particles of ever increasing masses. phenomena, one has first to find some regular patterns in The problems of partcles physics have consisted of three them; then one tries to understand or explain the regularities questions: qualitatively in terms of concrete models; in the last stage a) What are the elementary constituents of matter? a more precise and quantitative theory is developed. But b) What are the interactions, i.e., the forces or fields sooner or later new phenomena or discrepancies with the operating among them, and their basic symmetry established theory will show up, and one starts the three properties? stages over again. In actuality, these stages do not c) What is the mathematical formalism to deal with these necessarily proceed in sequence. The three kinds of attempts questions? may go on simultaneously. The path may take a zigzag The present standard model has answered most of these pattern. questions successfully up to the energy scales currently available: it contains the fundamental fermions for a), the 2. Search for Symmetries gauge fields for b), and the quantum field theory with Yukawa’s theory originally envisioned only a unique renormalization and spontaneous symmetry breaking for c), vector charged boson, but soon they found it necessary to but with the question of the auxiliary field (Higgs field) examine various spin and neutral versions, simultaneously sector still awaiting verification. developing theoretical formulations as they went along. The Up until the 1930s, physicists had known only two apparent equality of forces among protons and neutrons elementary matter particles, the electron and the proton, plus led Kemmer6) to introduce the concept of isospin symmetry. the photon, the quantum of the only force beside gravity. But problems related to the divergent properties of The nature of atomic nuclei and that of the weak processes derived nuclear forces made clearcut conclusions difficult. belonged to an unknown territory. But they thought they There was a confusion when the cosmic ray muon was had already found the fundamental constituents of matter. discovered7) and incorrectly but understandably identified It was a surprise and embarrassment when the neutron was with Yukawa’s meson. As more and more new particles, discovered. On the other hand, quantum mechanics had now called hadrons, showed up in cosmic ray events as successfully solved the mysteries of atomic phenomena well as in reactions in the more powerful accelerators, which involved energies of the order of electron volts, but there followed attempts to make sense of the rich spectrum it could not resolve the difficulties of infinite self-energies of particles. In this period they started to search for inherited from classical theory. When dealing with symmetry properties of reactions even though they were nuclear phenomena involving energies a million times only approximate symmetries. The isospin symmetry for higher than the atomic counterparts, they inclined to believe proton and neutron was extended to include the strange that quantum mechanics had to be replaced with a new hadrons,8) then to the flavor SU(3) symmetry.9) In parallel mechanics but did not conceive the possibility of new there also were attempts to understand these regularities elementary particles. It was Yukawa2) who showed that one in terms of constituent particles: the nucleon pair model can keep quantum field theory but admit new particles of the pion,10) generalized to include Lambda baryons,11) in understanding the problem of nuclear forces. His and finally the logically simpler quark model,12) which prediction was confirmed later by the discovery of the introduced a new set of fundamental constituents below the

111002-1 J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS Y. NAMBU level of hadrons. But the problem of statistics, when the the role of the NG boson. The fact that the chiral symmetry hadrons are regarded as composites of quarks, led further is only approximate, and not gauged, makes the pion to the concept of quarks endowed with an SU(3) of massive for reasons different from the case of the weak color.13) The origin of strong interaction were then to be bosons. transferred to that among the quarks, and the hadrons would An alternative and in fact older description of super- become dynamical composites of them. conductivity was one by Ginzburg and Landau,19) which turned out to be a phenomenological (effective) representa- 3. Search for Dynamical Principles tion of the BCS theory. In this transcription the complex The problems c) of the mathematical formalism, may be scalar ‘‘Ginzburg–Landau’’ field is equivalent to the Higgs divided into two different kinds: field, representing a bound pair of electrons or holes, and its 1) Search for general and rigorous properties of local phase and amplitude components correspond respectively to quantum field theory independent of the particular the massless NG and the massive Higgs type excitations. As Lagrangian or perturbation theory, which led to results it happens, the collective excitations of both the NG and like the spin-statistics and CPT theorems. There was Higgs types do exist in all phenomena of the superfluidity also a period of groping for alternatives to quantum type. The precise transcription and utilization (or sometimes field theory which persisted even after the successes of independent derivations) of these condensed matter exam- renormalization. These attempts include Heisenberg’s ples to relativistic theories were carried out in due course by S-matrix theory, its descendants in the form of a number of people.20) The essential point of the problem dispersion theory and the Regge trajectory theory. It may be explained as follows. In London’s phenomenological i is important to emphasize that all these efforts have description, the static induced current ji (@ ji ¼ 0)ina played an important, if not direct, role in arriving at the superconductive medium is of the form (in arbitrary static standard model. gauge) 2) Establishment of the gauge principle for the funda- k ji ¼ KikA ; mental interactions among particles. Yukawa’s meson  q q ð1Þ theory was a phenomenological one from today’s K ðqÞ¼ i k Kðq2Þ: ik ik 2 viewpoint. The non-Abelian gauge theory of Yang and q Mills14) was a purely theoretical construct inspired by The Meissner effect implies that Kð0Þ 6¼ 0: There are two the isospin symmetry. The crucial problem for prac- possibilities as to the origin of the second term: Either it is tical applications was its masslessness. Another was of dynamic origin so that in the nonstatic case q2 would be that the flavor symmetries are only approximate replaced by q2 v2!2, implying some accoustic-type whereas gauge symmetries are exact. As for the weak (massless) collective excitations, or else the pole remains interaction, its true origins were only a subject of static. But the latter could not happen unless the 1=q2 speculations, but when its V-A nature was establish- singularity was built into the Hamiltonian from the begin- ed,15) there arose the possibility that it might also be ning. In fact it is the Coulomb interaction (of the electron attributed to a gauge field. As it has turned out, the component, the compensating background charge remaining gauge principle was then not to be applied to the flavor inert) that actually produces the pole In a relativistically symmetries of the hadrons, but to the strong (color), invariant medium, the above eq. (1) will be replaced by its electromagnetic and weak interactions at the level of relativistic version: quarks and leptons. j ¼ K j ; Regarding the problem of mass, actually there was the  q q ð2Þ long known plasma mode, i.e., the massive Coulomb K ðqÞ¼ Kðq2Þ 2 field, as well as its generalization to the transverse counter- q parts16) But these happened in ionized media, not in the The two alternatives still exist. It is due to the lack of relativistic ‘‘vacuum’’. A crucial hint to the understanding manifest Lorentz invariance of gauge potentials that the and resolution of these problems came from the theories second alternative escapes the NG massless boson theorem, 17) 2 2 2 of superconductivity. The BCS theory assumed a con- and KðqÞ takes the form K ¼ q =ðq mV Þ, mV being the densate of charged pairs of electrons or holes, hence gauge boson mass. It is now history that this mechanism the medium was not gauge invariant. There were found of mass generation of gauge bosons was most successfully intrinsically massless collective excitations of pairs (NG applied to the weak interaction sector, rather than the strong modes) that restored broken symmetries, and they turned interaction sector, in a form of Ginzburg–Landau–Higgs into the plasmons by mixing with the Coulomb field. The (G–L–H) description of spontaneous symmetry breaking Fermi sea of electrons in the BCS theory developed (SSB) in the electroweak unification of Glashow, Salam, and fermionic excitations with an energy gap, reminding Weinberg (GSW) for the hypothetical W and Z vector one of the mass gap of the nucleons (and the quarks and bosons, which were to be confirmed experimentally later. leptons). Transporting these results to relativistic theories The gauge theory of strong interactions sector also joined the required one to abandon the concept of the relativistic GSW theory later to complete the full Standard Model. It ‘‘void’’ as Dirac once did in the interpretation of his contains two SSB phenomena: the gauged one in the weak equation. The spontaneous symmetry breaking thus emerged interaction sector explicitly displayed in the G–L–H form to as a universal phenomenon in physics, and was first make the weak bosons massive, whereas the other ungauged applied to the chiral symmetry breaking18) and (the con- chiral symmetry in the strong interaction sector for light stituent) mass generation for nucleons. Yukawa’s pion plays quarks is only implicit and approximate.

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1) Y. Nambu: Particle Accelerators (Gordon & Breach, Boston, 1990) 4. General Remarks on SSB Vol. 26, p. 1. 2) H. Yukawa: Proc. Phys.-Math. Soc. Jpn. 17 (1935) 48. The general properties of SSB may be characterized as 3) C. M. G. Lattes, H. Muirhead, G. P. S. Occhialini, and C. F. Powell: follows. Nature 159 (1947) 694. 1) Degeneracy of the ground state 4) S. Tomonaga: Prog. Theor. Phys. 1 (1946) 27; Z. Koba, T. Tati, and S. 2) The degrees of freedom N !1(the thermodynamic Tomonaga: Prog. Theor. Phys. 2 (1947) 101; Z. Koba, T. Tati, and S. limit). Finite systems do also exhibit similar dynamical Tomonaga: Prog. Theor. Phys. 2 (1947) 198; J. Schwinger: Phys. Rev. 73 (1948) 416; J. Schwinger: Phys. Rev. 74 (1948) 1439;R.P. properties, but the SSB description becomes an exact Feynman: Rev. Mod. Phys. 20 (1948) 367; R. P. Feynman: Phys. Rev. one only in infinite systems. 74 (1948) 1430. 3) Superselection rule (contraction of the Hilbert space) 5) See the articles by S. Sakata and M. Taketani in Prog. Theor. Phys. This means that the Hilbert space of the system is built Suppl. 50 (1971). up from one of the ground states, and other Hilbert 6) N. Kemmer: Phys. Rev. 52 (1937) 906. 7) S. H. Neddermeyer and C. Anderson: Phys. Rev. 51 (1937) 884;J.C. spaces built on other ground states become inaccessible Street and E. C. Stevenson: Phys. Rev. 52 (1937) 1003; Y. Nishina, M. because there are no local observables that can connect Takeuchi, and T. Ichimiya: Phys. Rev. 52 (1937) 1198. them. 8) M. Gell-Mann: Phys. Rev. 92 (1953) 833; T. Nakano and K. 4) The Nambu–Goldstone (NG) modes Nishijima: Prog. Theor. Phys. 10 (1953) 581. The lost symmetry is nevertheless visible in the form 9) M. Ikeda, S. Ogawa, and Y. Ohnuki: Prog. Theor. Phys. 22 (1959) 715; M. Gell-Mann: Phys. Rev. 125 (1962) 1067; Y. Nee´man: Nucl. of the NG modes. which are massless since in the Phys. 26 (1961) 222. long wave limit it reduces to the global symmetry 10) E. Fermi and C. N. Yang: Phys. Rev. 76 (1949) 1739. operation. 11) S. Sakata: Prog. Theor. Phys. 16 (1956) 686. The conscious use of the symmetry principle in physics 12) M. Gell-Mann: Phys. Lett. 8 (1964) 214; G. Zweig: CERN Rep. 8182/ dates back to the 19th century. Curie21) used symmetry TH, p. 401; G. Zweig: CERN Rep. 8419/TH, p. 412. 13) Y. Nambu and M.-Y. Han: Phys. Rev. 139 (1965) B1006. considerations to derive a kind of selection rules for physical 14) C. N. Yang and R. L. Mills: Phys. Rev. 96 (1954) 191. effects, an example of which is the Wiedemann effect: A 15) R. P. Feynman and M. Gell-Mann: Phys. Rev. 109 (1958) 193; conducting cylinder suffers a twist t when a current J is E. C. G. Sudarshan and R. E. Marshak: Phys. Rev. 109 (1958) 1860; passed and simultaneously a magnetic field B is applied J. J. Sakurai: Nuovo Cimento 7 (1958) 649. parallel to it. The behavior of t under various space 16) D. Bohm and D. Pines: Phys. Rev. 82 (1951) 625. 17) J. Bardeen, L. N. Cooper, and J. R. Schrieffer: Phys. Rev. 108 (1957) reflection operations matches that of either B or J.Sohe 1175. argued that an effect is possible only if its symmetries 18) Y. Nambu and G. Jona-Lasinio: Phys. Rev. 122 (1961) 345. Seff are compatible with those of the environment Senv. 19) V. I. Ginzburg and L. D. Landau: Zh. Eksp. Teor. Fiz. 20 (1950) 1064 The SSB, on the other hand, may be characterized by a [in Russian]. 20) P. W. Anderson: Phys. Rev. 110 (1958) 827; F. Englert and R. Brout: self-diminution of Senv from SL, the symmetries of the Phys. Rev. Lett. 13 (1964) 321; P. W. Higgs: Phys. Rev. Lett. 13 Lagrangian L of the system. It was not envisioned by Curie, (1964) 508; P. W. Higgs: Phys. Rev. 145 (1966) 1156. but such phenomena were already known in classical 21) P. Curie: J. Phys. (Paris) 3 (1894) 393 [in French]. systems. The spontaneous deformation of a rotating bary- 22) P. Weiss: J. Phys. (Paris) 5 (1907) 70 [in French]. tropic body from a sphere to a Jacobi ellipsoid is an 23) H.-P. Duerr, W. Heisenberg, H. Mitter, S. Schrieder, and K. example, which clearly shows that symmetry breaking Yamazaki: Z. Phys. 31 (1928) 619 [in German]. 24) See, for example, Y. Nambu: Physica D 15 (1985) 147. is a dynamical problem. More relevant examples for us, however, came after Curie. The ferromagnetism is the prototype of today’s SSB, as was explained by the works of Yoichiro Nambu was born in Tokyo in 1921. He Weiss,22) Heisenberg,23) and others. Ferromagnetism have obtained his B.S. (1942) and D. Sc. (1952) degrees since served us as a standard mathematical model of SSB. It from the old Imperial University of Tokyo. He was research associate (1946–1949) at University is no coincidence that Heisenberg made use of it later in his of Tokyo, Professor (1950–1956) at Osaka City attempt at a unified theory. Examples of BCS-type SSB are University, a member (1952–1954) of the Institute superconductivity and the superfluidity of He3. They have for Advanced Study in Princeton, U.S.A., research quasifermions, NG and Higgs bosons satisfying simple mass associate (1954–1956), Associate Professor (1956– relations among them.24) 57), and Professor (1958 to present) at the Univer- sity of Chicago, U.S.A. He has worked on statistical Acknowledgement mechanics, superconductivity, and the various aspects of nuclear and particle theory, including nuclear forces, flavor physics, dispersion theory, This work was supported by the University of Chicago. spontaneous symmetry breaking phenomena, colored quarks, and hadronic string theory.

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