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Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise

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Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Michael Dine Department of Physics University of California, Santa Cruz Nambu Memorial Symposium, University of Chicago, 2016 Work with P. Draper, L. Stevenson-Haskins, D. Xu, Francesco D’Eramo. Important input from N. Seiberg Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise From my graduate student days, when I first encountered his work on symmetry breaking in the strong interactions, Nambu has been one of my intellectual heroes. This was reinforced during my years at City College, where I regularly heard stories of Nambu from Bunji Sakita, who himself was an admirer. While somewhat younger, Sakita often regaled me with stories of the War years in Japan, and both his experiences and Nambu’s. I finally got to know Nambu in the 1980’s, and all of my interactions with him were intellectually stimulating and enhanced by his charm and wit. I remember many conversations at Chicago, but remarks he made at the 1984 Argonne meeting on String Theory, which were thoughtful but cautionary, particularly stand out. My final interactions came shortly after his Nobel Prize. Like many, I sent him a congratulatory note, only to receive a "mailbox is full" message. A year later, though, I received the most thoughtful note reply. Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise In my own work, Nambu’s influence is perhaps heaviest in the area of string theory and in the appearance of light scalars in the case of continuous global symmetry breaking. His work with Jona-Lasinio is instructive in that it takes a model which in detail cannot be taken too seriously, but extracts important, universal features. Some of what I say today I hope can be viewed as a modest effort in this style. While much of my discussion will center on Nambu-Goldstone bosons, I will also consider some questions in strong dynamics, where fermionic condensates will play important roles. Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise P H YSI CAL R EVI EW VOLUME &22, NUMBER AI RII, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. P Y. NAMBU AND G. JONA-LASINIoj' The Enrico terms Institute for Nuclear StuCkes and the Department of Physics, The University of Chicago, Chicago, Illinois (Received October 27, 1960) It is suggested that the nucleon mass arises largely as a self-energy of some primary fermion field through the same mechanism as the appearance of energy gap in the theory of superconductivity. The idea can be put into a mathematical formulation utilizing a generalized Hartree-Fock approximation which regards real nucleons as quasi-particle excitations. We consider a simplified model of nonlinear four-fermion interaction which allows a p5-gauge group. An interesting consequence of the symmetry is that there arise automatically pseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion. In addition, massive bound states of nucleon number zero and two are predicted in a simple approximation. The theory contains two parameters which can be explicitly related to observed nucleon mass and the pion-nucleon coupling constant. Some paradoxical aspects of the theory in connection with the p5 trans- formation are discussed in detail. I. INTRODUCTION equations' 4: " 'N this paper we are going to develop a dynamical E4~= e lto~+40 (1.1 - theory of elementary particles in which nucleons and E0 ~*= eA ~*—+44~, mesons are derived in a unified way from a fundamental near the Fermi surface. is the component of the spinor field. In basic physical ideas, it has thus the 11„+ excitation corresponding to an electron state of mo- characteristic features of a compound-particle model, mentum and corresponding to but unlike most of the existing theories, dynamical P spin +(up), andri ~* a hole state of momentum and spin which means treatment of the interaction makes up an essential part p +, an absence of an electron of momentum — and spin of the theory. Strange particles are not yet considered. p —(down). is the kinetic energy measured from the The scheme is motivated by the observation of an eo Fermi surface; is a constant. There will also be an interesting analogy between the properties of Dirac g equation complex conjugate to describing particles and the quasi-particle excitations that appear Eq. (1), another of excitation. in the theory of superconductivity, which was originated type Equation gives the eigenvalues with great success by Bardeen, Cooper, and Schrieffer, ' (1) and subsequently given an elegant mathematical forlnu- E„=a (e,'+y')-*'. (1.2) ' lation by Bogoliubov. The characteristic feature of the The two states of this quasi-particle are separated in BCS theory is that it produces an energy gap between 2 In the ground state of the system all energy by ~ E„~. the ground state and the excited states of a supercon- the quasi-particles should be in the lower (negative) w'hich ductor, a fact has been confirmed experimentally. energy states of Eq. (2), and it would take a finite The is caused due to the fact that the attractive gap energy 2)E„~ )~2~&~ to excite a particle to the upper phonon-mediated interaction between electrons produces state. The situation bears a remarkable resemblance to correlated pairs of electrons with opposite momenta and the case of a Dirac particle. The four-component Dirac spin near the Fermi surface, and it takes a finite amount equation can be split into two sets to read of energy to break this correlation. can be EP,=o"Pter+ res, Elementary excitations in a superconductor — conveniently described by means of a coherent mixture Egs ——o"Pigs+ nell r, of electrons and holes, which obeys the following E„=W (p'+nt') l, * Supported by the U. S. Atomic Energy Commission. where tPt and Ps are the two eigenstates of the chirality f' Fulbright Fellow, on leave of absence from Instituto di Fisica operator ys —y jy2y3y4. dell Universita, Roma, Italy and Istituto Nazionale di Fisica According to Dirac's original interpretation, the Nucleare, Sezione di Roma, Italy. world has all the electrons 'A preliminary version of the work was presented at the ground state (vacuum) of the Midwestern Conference on Theoretical Physics, April, 1960 (un- in the negative energy states, and to create excited published). See also Y. Nambu, Phys. Rev. Letters 4, 380 (1960); states (with zero particle number) we have to supply an and Proceedings of the Tenth Annual Rochester Conference on High-Energy Nuclear Physics, 1960 (to be published). energy &~2m. ' J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, In the BCS-Bogoliubov theory, the gap parameter @, 162 (1957). which is absent for free electrons, is determined es- 3 N. N. Bogoliubov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 58, self-consistent (Hartree-Fock) representa- 73 (1958) Ltranslation: Soviet Phys. -JETP 34, 41, 51 (1958)g; sentially as a N. N. Sogoliubov, V. V. Tolmachev, and D. V. Shirkov, A %em tion of the electron-electron interaction eGect. Methodin the Theory of Supercondlctivity (Academy of Sciences of 4 U, S.S.R., Moscow, 1958). J. G. Valatin, Nuovo cimento 7, 843 (1958). 345 Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Y. NAM BU AN 9 G. JONA —LASI NIO tendency for partial cancellation between contributions which lead to q'= 0, and C:D= 1 —2m'I (0):mI (0). from diferent mesons or nucleon pairs. From Eq. (4.8), we have 0(2m'I(0) (-.', . We already remarked before that the model treated here is not realistic enough to be compared with the (b) Put r„5—(iv„vs+2mv, q„/q')Fi(q') was show that actual nucleon problem. Our purpose to + (iv.v~ —iv qvnq. /q')F2(q') (A 3) a new possibility exists for 6eld theory to be richer and more complex than has been hitherto envisaged, even This is seen. to satisfy the integral equation if though the mathematics is marred by the unresolved divergence problem. In the subsequent paper we will attempt to generalize — the model to allow for isospin and finite pion mass, and F2= J~(q')/[I »(q')), (A.4) draw various consequences regarding strong as well as »(q') = 2m'I(q') —J'v(q') weak interactions. where J(q') was defined in Eq. (4.13). APPENDIX On the mass shell, F„5 reduces to We treat here, for completeness, the problem created by the coupling of pseudoscalar and pseudovector terms (iv„vg+ 2mv Sq„/q') F(q'), encountered in the text. As we have seen, such an effect F(q') = 1+F2 (q') = 1/[1 —» (q')). is not essential for the discussion of y~ invariance, but rather adds to complication, which however naturally For q'=0, we have J(q') =0 so that 1&F(0)=1/ appears in the ladder approximation. [1—2m'I(0)) &2. First let us write down the integral equation for a (c) From the structure of the inhomogeneous term, vertex part j. : it is clear that the scattering matrix is given by I'(p+ :q, p :q)-—— 2go(P )f(v ) +go(P')f(iv v ) 2zgp where I"5 is the pseudoscalar vertex function. =v(p+ ,q, p lq)+--v. Trl v.&(p'+ :q)- (2~)4 Again, from Eq. (A.1), I'6 is determined as k—)-)d'p'— x(lp'+ 'qp 'q)&-i:(p I'& —VS[1—2m'I(q'))/q'I(q') miV —qV5/q', (A.6) Sgo v~v. )»[vip(p'+kq) which has an entirely different behavior from the bare (2a-)4 y5 for small q'.
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