Models for Elementary Particles and the Nagoya School

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1 23 Progress of Theoretical Physics, Vol. 122, No. 1, July 2009 Models for Elementary Particles and the Nagoya School Yoshio Ohnuki Department of Physics, Nagoya University, Nagoya , Japan (Received April 8, 2009) A brief history of models of elementary particles that were mainly developed in Nagoya is reviewed on the basis of personal recollections, together with some related topics. Subject Index: 146, 136, The Sakata model At the 10th annual meeting of the Physical Society of Japan (Oct. 9 16, 1955), which was held at Tokyo University of Education, the composite model for elementary particles was presented by S. Sakata ( ) as an extra talk that was not part of the regular program. It was said that he obtained the idea for this model (which has been called the Sakata model) through discussions in the particle physics laboratory of Nagoya University (called E-ken ) in Japanese) at the end of September. The paper 1) was published in December of the next year, together with calculations that were done respectively by Tanaka, 2) Matumoto 3) and Maki. 4) In the Sakata model, all hadrons are assumed to be composite states which consist of a proton (p), a neutron (n), a Λ-particle (Λ) and their anti-particles, p, n, and Λ. Then, the isotopic spin of any hadron is ascribed to that of a nucleon N (= p, n), and the strangeness S is ascribed to the Λ particle. The Nakano-Nishijima 5) - Gell-Mann 6) relation, Q = I (S + N B), (1.1) was found to follow from the Sakata model, because this model implies the relations Q = N p, I 3 =(N p N n )/2, S = N Λ and N B = N p + N n + N Λ.Inrelationtothis, Sakata emphasized the importance of the role of the Λ-particle, in an analogy to the neutron, which, as shown by Heisenberg 7) and Iwanenko, 8) played an important role in elucidating the mysterious properties of atomic nuclei. For these reasons, the Sakata model was interpreted as providing a background understanding of hadron phenomena. He often used the word new atomism for this model, and based on this point of view, he anticipated a new theory to replace ordinary quantum field theory that could consistently describe the behavior of elementary particles. Before this time, during the period , the Nagoya group had tried to eliminate divergences in quantum field theory by applying the idea of the C-meson 9) and its generalization, 10) i.e., the method of mixed fields, which had been called by ) E and ken mean elementary particle and laboratory, respectively.

2 24 Y. Ohnuki Pauli a realistic approach, in contrast to the method of the regulator. 11) The C-meson is a scalar field coupled to electrons with coupling constant f. Then, it was shown through a second order perturbation calculation that the divergence of the electron self-energy could be cancelled out if the electro-magnetic coupling constant e satisfied the relation 2e 2 = f 2. Encouraged by this, Umezawa et al. 10) examined a general case in which various fields coexist with various interactions. However, it was concluded 12) that such an approach had limited applicability and could not completely eliminate divergences, in particular for vacuum polarizations. As a related approach the renormalization procedure, when applied to QED, was quite successful in accounting for existing experimental data. However, this alone did not necessarily imply the existence of a finite theory of quantum fields, since, as shown by Källen, 13) at least one of the renormalization constants in QED itself contains a divergence. Furthermore, Lee s nonrelativistic renormalizable model 14) and also Landau s QED calculation 15) cast some doubt on the consistency of the renormalization approach. In fact, the Lee model is exactly solvable, and it leads to a renormalization constant Z (> 0) given by Z =(g c /g) 2 = 1 1+g 2 (1.2) with g and g c being the bare and renormalized coupling constants, respectively. If g c (= Z 1/2 g) is assumed to be made finite through the renormalization, then we encounter the paradoxical relation Z =1 gc 2 < 0. Though it was found that the S-matrix is finite in this case, this computation clearly shows the inconsistency of the renormalization procedure; that is, the S-matrix becomes non-unitary. In QED, Landau explicitly constructed an approximate solution to the set of equations for propagators and vertices without use of a perturbation expansion. In this way he found that the renormalization constant Z 3 takes a form similar to (1.2) in the limit that the cutoff momentum diverges (Λ ). Because the renormalized charge e r is related to the bare charge e by e r = Z 1/2 3 e, we are led to a contradiction here as well. Fig. 1. Young members of E-ken, Feb 1968: T. Maskawa (leftmost in front row), M. Kobayashi (second from right in front row).

3 Models for Elementary Particles 25 In addition, at that time, non-renormalizable interactions, such as the 4-Fermi interaction of β decay, ), ) were believed to exist in nature. For these reasons, in Japan, a number of people thought that a deeper understanding of nature might be gained only through analyses of non-renormalizable interactions without recourse to field theories. In connection to this, Sakata frequently said that a revolution in particle physics is inevitable. He seemed to expect the appearance of a new theory, a so-called super-quantum mechanics, which describes the world on lengthscales shorter than the nucleon Compton wavelength. This point of view is probably related to the fact that no quantum mechanical description is explicitly given in his paper on the composite model. Furthermore, even if use was made of quantum field theory, it was not an easy task to numerically derive reliable results on the basis of the composite model, because the binding energies of composite states are too large. In any case, rapid progress along this line could not be expected. In 1958, Ogawa proposed a new idea to treat the composite model. It was published the next year. 18) He assumed that, in addition to the ordinary charge independence in the Sakata model, the strong interaction should be invariant under the exchanges p Λ and n Λ. Immediately after this proposal, it was found that Ogawa s symmetry is equivalent to invariance under transformations of the form p p n U n, (1.3) Λ Λ where U is an arbitrary (3 3) unitary matrix. The set of such matrices U forms the group U(3). With this result, it was thought quite important to construct the representation theory of this group for analyses of the composite states. Applying irreducible representations of U(3), Ikeda, Ogawa and Ohnuki 19) found the possible existence of octet pseudo-scalar mesons and also of resonances appearing in hadron scattering. ) At almost the same time Yamaguchi 21) in CERN was working extensively on phenomenological analyses of hadron interactions by applying Ogawa s symmetry. He also considered the possibility of an octet baryon 22) at the end of But this idea was not published. 2. Nagoya model At the Kiev Conference on high energy nuclear physics in 1959, Gamba, Marshak and Okubo 23) pointed out a symmetry of weak currents under the following exchange between Sakata s triplet and three leptons: p ν, n e, Λ μ. (2.1) ) Non-renormalizability of the 4-Fermi interaction was pointed out by Kamefuchi (1951). 16) ) A general criterion for renormalizability of interactions was discussed by Sakata, Umezawa and Kamefuchi (1951). 17) ) Some details of the U(3)-symmetry developed in Japan were reported by the present author 20) in a talk at the International Symposium pnλ 50, The Jubilee of the Sakata Model (2006).

4 26 Y. Ohnuki Motivated by this idea, Maki, Nakagawa, Ohnuki and Sakata 24) proposed a unified description of elementary particles in which the basic particles p, n and Λ are assumed to be composite states consisting of positively charged matter B +,alongwithν, e and μ, respectively. They were symbolically written p = B + ν, n = B + e and Λ = B + μ,whereb + did not necessarily mean a bose particle, but rather was regarded as a new kind of matter responsible for the strong interaction among hadrons. It was called B-matter or B-charge. Because in this model, B + exists within all the basic particles, p, n and Λ, it seemed favorable to describe the U(3) symmetry for strong interactions. In addition, the weak interaction was thought to be caused by leptons through the current j α = f { ēγ α (1 + γ 5 )ν + μγ α (1 + γ 5 )ν }. (2.2) Then, by combining B + with each of the leptons in (2.2) the baryon current was obtained as J α = j α B = f { nγ α (1 + γ 5 )p + Λγ α (1 + γ 5 )p }. (2.3) Hence, the effective weak Hamiltonian was written H weak = J α J α,withj α = j α + J α. This model was presented by the Nagoya group at the end of 1959 at a workshop of the Research Institute for Fundamental Physics, Kyoto University. It was called the Nagoya model. In it, however, the binding mechanism between leptons and B-matter is not explicitly represented. Sakata seemed to believe that the formulation of such a mechanism is probably a problem to be solved with a theory of super-quantum mechanics, which might exist behind the Nagoya model. In 1962, it was made clear by experiment that there exist two kinds of neutrinos, ν e and ν μ, which were believed to couple with charged leptons through a current of the form j α =(ēν e ) α +( μν μ ) α, (2.4) wherewehave( ψφ) α ψγ α (1 + γ 5 )φ. To realize consistency with experiments, Maki, Nakagawa and, Sakata 25) assumed that the true neutrinos, ν 1 and ν 2,with definite masses consist of linear combinations of the weak neutrinos, ν e and ν μ,in the following way: ν 1 = ν e cos δ + ν μ sin δ, ν 2 = ν e sin δ + ν μ cos δ, (2.5) with real δ. Hence, the weak current (2.4) was rewritten as j α =(ēν 1 ) α cos δ + ( μν 1 ) α sin δ (ēν 2 ) α sin δ +( μν 2 ) α cos δ. Corresponding to this, the baryon-lepton symmetry (2.1) was modified as p ν 1, n e, Λ μ, (2.6) which implies p = B + ν 1, n = B + e and Λ = B + μ. It is noted that no baryon was assumed to correspond to ν 2. Thus, for the baryonic weak current, we immediately have J α = j α B =( np) α cos δ +(Λp) α sin δ. (2.7)

5 Models for Elementary Particles 27 This expression is of the type proposed by Gell-Mann, Lévy 26) and Cabibbo. 27) This modified version of the Nagoya model, based on (2.4) (2.7) is called the new Nagoya model. Accordingto(2.4), the weak neutrinos ν e ( ν e )andν μ ( ν μ ) are emitted with accompanying e + (e )andμ + (μ ), respectively. However, as seen from (2.5) they are not eigenstates of the mass matrix. Hence, it was thought that for massive ν 1 and ν 2 there would occur an oscillation like ν e ν μ. Later, neutrino oscillation phenomena of this type were supported in Kamiokande experiments. 28) In 1961, we received a preprint from Gell-Mann 29), ) in which he proposed the socalled eightfold way, applying the group SU(3), and examined mass relations among hadrons. In the same year, Ne eman proposed using this group for classification of hadrons in the same manner as in the eightfold way. For mesons, the eightfold way and the U(3)-symmetry in the Sakata model lead to the same results and facilitate group theoretical classifications for them. However, the situation is quite different for baryons. During 1962 and 1963, it became clear from experiments that the U(3)-symmetry was inadequate to describe baryons in a consistent way. To overcome this difficulty, we attempted to build a model of hadrons by modifying the original form of the Sakata model. In fact, we believed that the eightfold way lacks a substantial background supporting the symmetry, unlike the Sakata model. In our attempts at model reconstruction, however, the charge and the baryon number of the basic particles were always assumed to be integers from the outset. The quartet model 31),32) was one of these models. In any case, we were not able to construct the quark model. In February, 1964, the Ω-minus particle, with strangeness 3, was discovered at the Brookhaven National Laboratory. The observed mass of 1686±12 MeV/c 2 is consistent with that predicted by the Gell-Mann-Okubo mass formula. 33) This was hard evidence for the validity of the eightfold way. In that year, Gell-Mann 34) and Zweig 35) proposed the quark model, in which three quarks, u, d and s, formthe fundamental representation of SU(3) and play the role of the basic constituents of hadrons, in place of the Sakata triplet. Though these quarks have fractional charge and fractional baryon number, the quark model seemed to provide a substantial back ground for hadron phenomena. Furthermore, in 1964, Sakita 36) and Gürsey and Radicati 37) proposed an SU(6) model to describe the spin and the unitary spin of hadrons in a unified way. It seemed to imply the reality of the quark model. 38), ) With these results, the baryon-lepton symmetry (2.6) was rewritten as u ν 1, d e, s μ, (2.8) and correspondingly, p, n and Λ in the new Nagoya model were replaced by u = ) The full paper 30) was published the next year. ) However, there remained a problem concerning the statistical properties of quarks. In the SU(6) model, the ground states of baryons were assumed to belong to the 56-plet representation. Because this representation is totally symmetric with respect to spin and unitary spin, the three quarks in the baryon could not take S states. To avoid this difficulty, Nambu 39), 40) andthenhori 41) introduced new degrees of freedom, which were named color by Gell-Mann 42) within the framework of the gauge field theory.

6 28 Y. Ohnuki B 1/3 ν 1, d = B 1/3 e and s = B 1/3 μ, respectively, where B 1/3 denotes B matter with charge 1/3 and baryon number 1/3. Then the baryonic weak current (2.7)wasrewrittenintheform J α =( du) α cos δ +( su) α sin δ. (2.9) Thus, in this model, the curious properties of quarks are all reduced to those of B matter which, according to Sakata, might obey a super-quantum theory. 3. New developments In 1961, a breakthrough in quantum field theory was made by Nambu and Jona-Lasinio. 43) This was the discovery of the spontaneous breakdown of symmetry. According to their argument, there must appear a massless particle corresponding to spontaneous symmetry breaking. This massless particle was called the Nambu- Goldstone boson, and it can produce masses for gauge bosons via the Higgs mechanism 44) when absorbed into the gauge bosons. In , Weinberg 45) and Salam 46) proposed a model of the electroweak interaction for a system of leptons. The weak interaction derived therein is mediated by a gauge boson whose mass is produced by the Higgs mechanism in spontaneous breaking of U(2) U(1). Accordingly, this theory is less singular than the theory with the 4-Fermi interaction. In fact, in 1971 t Hooft 47) proved the renormalizability of gauge interactions for massive gauge fields. This implies that in the Weinberg-Salam model, the weak interactions can be treated within the same framework as QED. In 1971, Niu 48) discovered a new type of event in an emulsion chamber experiment, which strongly suggested the existence of a fourth quark. In fact, this was the discovery of the charm quark, c. Using this as the missing partner of ν 2 in the quark version of the new Nagoya model, the current (2.9) is found to be J α =( du) α cos δ ( dc) α sin δ +( su) α sin δ +( sc) α cos δ. (3.1) We are now able to incorporate these four quarks into the Weinberg-Salam model. In that case, however, they are no longer represented in terms of SU(3), and, instead, split into the two doublets (u, d) and (c, s). Each of these undergoes transformations of SU(2) ( SU(2) U(1)) and is called a generation. M. Kobayashi and T. Maskawa, who entered the graduate school of Nagoya University in 1967 and 1962, respectively, investigated the puzzle of CP violation, 49) which had been discovered in a K-meson decay experiment in In the Weinberg-Salam model with two generations, however, it is impossible to incorporate CP-violating interactions in a consistent manner with gauge invariance under SU(2) U(1). For this reason, Kobayashi and Maskawa introduced the third generation for quarks. With this, they finally discovered the general expression 50) for CP-violating weak current, which is represented by the so-called Kobayashi- Maskawa matrix. Since that time, the formulation of weak currents proposed by them has been completely supported by experiments. In 1973, when they wrote the paper on CP violation the third generation of quarks was not known and thus most people had not thought of such a possibility.

7 Models for Elementary Particles 29 Nevertheless, in solving the CP puzzle, Kobayashi and Maskawa employed a new generation of quarks in their theory without hesitation. Perhaps, the fact that they took this bold step is related to Sakata s philosophy, which asserts the importance of the role of unknown substances which may exist behind phenomena. Thus their efforts yielded greatly successful results that have had a profound influence on particle physics. References 1) S. Sakata, Prog. Theor. Phys. 16 (1956), ) S. Tanaka, Prog. Theor. Phys. 16 (1956), ) K. Matumoto, Prog. Theor. Phys. 16 (1956), ) Z. Maki, Prog. Theor. Phys. 16 (1956), ) T. Nakano and K. Nishijima, Prog. Theor. Phys. 10 (1953), 581. K. Nishijima, Prog. Theor. Phys. 12 (1954), 107; Prog. Theor. Phys. 13 (1955), ) M. Gell-Mann, Phys. Rev. 92 (1953), ) W. Heisenberg, Z. Phys. 77 (1932), 1. 8) D. Iwanenko, Nature 129 (1932), ) S. Sakata and O. Hara, Prog. Theor. Phys. 2 (1947), ) H. Umezawa, J. Yukawa and E. Yamada, Prog. Theor. Phys. 4 (1949), 25; Prog. Theor. Phys. 4 (1949), 113. H. Umezawa and E. Yamada, Prog. Theor. Phys. 4 (1949), 231. H. Umezawa and R. Kawabe, Prog. Theor. Phys. 4 (1949), 423; Prog. Theor. Phys. 4 (1949), ) W. Pauli and F. Villars, Rev. Mod. Phys. 21 (1949), ) S. Sakata and H. Umezawa, Prog. Theor. Phys. 5 (1950), ) G. Källen, Dan. Mat. Fys. Medd. 27 Nr. 12 (1953). 14) T. D. Lee, Phys. Rev. 95 (1954), ) L. Landau et al, DAN. 95 (1954), ) S. Kamefuchi, Prog. Theor. Phys. 6 (1951), ) S. Sakata, H. Umezawa and S. Kamefuchi, Phys. Rev. 84 (1951), 154; Prog. Theor. Phys. 7 (1952), ) S. Ogawa, Prog. Theor. Phys. 21 (1959), ) M. Ikeda, S. Ogawa and Y. Ohnuki, Prog. Theor. Phys. 22 (1959), 715; Prog. Theor. Phys. 23 (1960), ) Y. Ohnuki, Prog. Theor. Phys. Suppl. No. 167 (2007), ) Y. Yamaguchi, Prog. Theor. Phys. Suppl. No. 11 (1959), 1; Prog. Theor. Phys. Suppl. No. 11 (1959), ) Y. Yamaguchi, letter to S. Sakata, Dec. 25, 1959, Sakata Memorial Library. 23) A. Gamba, R. E. Marshak and S. Okubo, Proc. Natl. Acad. Sci. USA 45 (1959), ) Z. Maki, M. Nakagawa, Y. Ohnuki and S. Sakata, Prog. Theor. Phys. 23 (1960), ) Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962), ) M. Gell-Mann and M. Lévy, Nuovo Cim. 16 (1960), ) N. Cabibbo, Phys. Rev. Lett. 10 (1963), ) Super Kamiokande Collaboration, Phys. Rev. Lett. 81 (1998), 1562; Phys. Lett. B 539 (2002), ) M. Gell-Mann, A Theory of Strong Interaction Symmetry, California Institute of Technology (March 15, 1961). 30) M. Gell-Mann, Phys. Rev. 125 (1961), ) Z. Maki, Prog. Theor. Phys. 31 (1964), 331; Prog. Theor. Phys. 31 (1964), ) Y. Hara, Phys. Rev. 134 (1964), B ) S. Okubo, Prog. Theor. Phys. 27 (1962), ) M. Gell-Mann, Phys. Lett. 8 (1964), ) G. Zweig, CERN Report 8182/TH. 401 (Jan. 11, 1964); CERN Report 8419/TH. 412 (Feb 21, 1964). 36) B. Sakita, Phys. Rev. 136 (1964), B ) F. Gürsey and L. A. Radicati, Phys. Rev. Lett. 13 (1964), 173.

8 30 Y. Ohnuki 38) Y. Ohnuki and A. Toyoda, Nuovo Cim. 36 (1965), ) Y. Nambu, Dynamical Symmetries and Fundamental Fields, Proceedings of 2nd Coral Gables Conf. on Symmetry Principles at High Energy, 133 (1965); Systematics of Hadrons in Subnuclear Physics, Preludes in Theoretical Physics (1965). 40) M. Y. Han and Y. Nambu, Phys. Rev. 139 (1965), B ) S. Hori, Prog. Theor. Phys. 36 (1966), ) M. Gell-Mann, Acta Phys. Austriaca, Suppl. 9 (1972), 733. H.Fritzsch,M.Gell-MannandH.Leutwyler,Phys.Lett.B47 (1973), ) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961), 345; Phys. Rev. 124 (1961), ) P. W. Higgs, Phys. Rev. 145 (1966), ) S. Weinberg, Phys. Rev. Lett. 19 (1967), ) A. Salam, Weak and Electromagnetic Interactions, Elementary Particle Theory, ed. N. Svartholm (John Wiley and Sons, New York, London, Sydney, 1968), p ) G. t Hooft, Nucl. Phys. B 35 (1971), ) K. Niu, E. Mikumo and Y. Maeda, Prog. Theor. Phys. 46 (1971), 1644; Conf. Paper. 12 th Int. Cos. Ray Conf. (Hobart, 1971), ) As for details on CP violation, see I. I. Bigi and A. I. Sanda, CP Violation (Cambridge University Press, Cambridge, U.K., 1999), ISBN ) M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973), 652.