ELSEVIER Nuclear Physics A638 (1998) 35cM4c

N U C L E A R PHYSICS A

Symmetry Breaking, Chiral Dynamics, and Fermion Masses

Yoichiro Nambu Universi ty of Chicago, Chicago, Illinois, USA

A b s t r a c t

Spontaneous symmetry breaking (SSB) is one of the main paradig.ns of particle physics. I will briefly review 1) the general history of SSB, 2) the universal features and descriptions of gap formation and chirM dynamics in the BSC-type mechanisms in condensed matter, nuclear, and particle physics, and 3) some open problems including the mass hierarchy in the Standard Model and beyond.

1. Introduction

I am afraid I do not have anything new to offer to this conference o~ quark ma t t e r tha t deals with subjects largely outside of my expertise. .So I will here only t ry to give you a broad historical overview of spontaneous symmet ry breaking (SSB). It will be similar in nature to one I have given elsewhere recently, and I apologize in advance if some of you have heard it already.

The active and conscious use of the symmet ry principle in physics dates back to more than a century ago. It was Pierre Curie [1] who first showed (1894) how one may apply symmet ry group considerations to derive a kind of selection laws for physical effects in crystals. As an example he discussed the Wiedemann effect. Below I will reproduce his arguments.

A conducting cylinder suffers a twist t when a current J is passed and simultaneously a magnet ic field B is applied parallel to it. The behavior of & B and t under various reflection operations R are as follows:

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Rz: B - + B J - - + - J t - - ~ - t

R~ ,R~: B ~ - B J ~ J t - + - t

T: B --~ - B J --~ - J t -+ - t

f J

t "~

B J

J

Since t changes sign under both Rz and R~ (or Ry) whereas B or J alone does not, t is compatible with the environment only if J and B are present simultaneously. (Time reversal T is added here only for completeness.)

The thesis of Curie may be stated as follows. If one denotes the set of symmetries of the environment by Senv and that of an effect by Self, then S, fr = S, nv. In other words, the symmetries, or the lack thereof, of an effect must be already present in the environment for the effect to be possible. (Negative parity counts as violation, or absence, of reflection symmetry.)

The SSB, on the other hand, may be characterized by a self- diminu- tion of Senv to the level of Se~ : Sefr < SL where SL is the symmetry of the Lagrangian L of the system. It was not envisioned by Curie, but such phenomena were already known m classical systems. The spontaneous de- formation of a rotating baritropic body from a sphere to a Jacobi ellipsoid is an example, which clearly shows that symmetry breaking is a dynamical problem. More relevant examples for us, however, came after Curie. The ferromagnetism is the prototype of today's SSB, as was first explained by P. Weiss (1907)[2]. The Ising model [3] and the Heisenberg ferromagnetism [4] have since served us as the mathematical model of SSB. (It is no coincidence that Heisenberg made use of it later in his unified theory.[5])

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2. The characteristics of SSB

For my purposes I will characterize SSB by its impor tan t general features as follows.

1)Degeneracy of the ground s tate ft 2)The degrees of freedom N --+ oo.

This means tha t one is dealing with the the rmodynamic l imit . Fini te systems do also exhibit similar dynamical propert ies, but the SSB description becomes an exact and useful one only in infinite systems.

3)Superselection rule (contraction of the Hilbert space) This means that the Hilbert space of the system is buil t up from one of the ground states ~, and other Hilbert spaces built on other ground states become inaccessible because there are no local observables that can connect them.

Let G be the symmet ry group of a system, and the degenerate ground states ft be labeled by a set of indices {i}. The set {i} transfbrms as a representat ion of G. By choosing a par t icular element of {i}, the symmet ry of Hilbert space is reduced from G to a subgroup Ga which leaves {i} invariant. The elements S = exp(Q) of the coset G/Gfl are those that connect dif[erent vacua. If Q is represented by local fields as

Qv = ; iJ(x)d3x,

Sv = exp(Qv) = [ I s(x), V

S(x) = exp ( i J (x ) ) (])

Then the mat r ix element of each factor I< ft ' I S(x) I f~ >1< 1 in such a way tha t limv+oo < ft ' I Sv I Ft >--+ 0. In infinite systems the matr ix element may become nonzero only for a global change of the environment like the change of tempera ture . Different vacua may also exist as domains in different par ts of space,and environmental changes may cause phase transi t ions which change the s t ructure of the ground states.

4) The Nambu-Golds tone (NG) modes The lost symmet ry is nevertheless visible in Ft since Qv is a finite observable in the Hilbert space Ha. It excites ~t to a zero mass s ta te since Qv tends to the invisible symmet ry operator Q in the large V limit. It is also the reason for the absence of infrared divergence for the NG modes.

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Gauge (local) symmetries are difficult to break since it is difficult to de- fine its thermodynamic limit except for its global subgroup, which can be broken. But the mixing of the NG modes and the gauge field causes them to combine into a massive gauge field. (The massless mode deconples from the asymptotic states.)

5) Finite systems If the degree of freedom is finite, the degeneracy is also finite but large. In this case one can still talk about quasi-SSB, quasi-selection rules, quasi-NG modes, etc. The SSB description is only a convenient and approximate one, like describing a resonance as a particle. It is sometimes achieved when one ignores part of the Hamiltonian. The perturbative part makes the symmetry operators visible by tunnelling between the vacua and lift their approximate degeneracy. In the SSC picture these states may be interpreted as the low- lying NG modes.that tend to restore the broken symmetry. The 3ahn-Teller effect [6] in polyatomic molecules and the Skyrme-Witten model (SW)[8] may be regraded as examples of this. In the latter, the kinetic energy restores rotational symmetry but creates a rotational band. (There is a close analogy between SW and the old strong coupling theory /7] of pion and nucleon.)

3. The BCS and GL mechanisms

The BCS (Bardeen-Cooper-Schrieffer) mechanism of energy (mass) gap formation is common to many physical phenomena, as such it is most relevant to particle physics. Here I enumerate its general features.

1)It is induced by attractive interactions between ferlnions, typically by short range interactions characterized by a characteristic mass scale A (cut- off scale), leading to a Cooper pair condensate < ~/,.~b >, or a complex order parameter, to give the fermion an effective mass gap a to originally massless fermion and results in a violation of fermion number and chirality.

2) The < ~ b > correlation also induces the "pion" (re), as the massless NG mode, and the "sigma" (er), as the massive Higgs mode. The ma.sses of 7r, fermion, and a satisfy an approximate mass formula

m ~ : m s:m~=O:l :2 (2)

for an S-wave paring; it is good in the weak coupling limit. The Yukawa and the self-coupling constants of the induced interactions for rr and a modes are also related in a similar manner in terms of a single coupling constant.

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For other forms of pairings, one finds more complicated formulas involving several massive modes.) Thus these tr iplets of modes have a common mass scale ~ which is usually small compared to the mass scale A of the original dynamics.

3) It is then possible to construct an effective GL(Ginzburg- I,andau) theory for the tr iplets of modes by feeding a single mass scale m and a single coupling constant f without referring to A. So the BCS theory becomes a specially simple case of the GL theory. Thus the mass scale m is the w~cunm expectat ion value v = < ~r > of the GL-Higgs field ~, and the fermion, masses are expressed as I??~ i = gi v. The mass and coupling relations are a i'eflecl ion of what 1 have called quasi -supersymmetry of this GL :Lagrangian. (In fa( l the relation can be derived in the language of supersymmetl:y.[9].

The following is a somewhat more detailed description of the above sl aie- ments and known instances of tile occurrence of the B( :S /GL mechanisln.

The BCS gap equation with a 4-fermion coupling C/M2('d,'(,) ~, or i1~

l inearized version G ~ b t b ¢ - M 2 4 ~ / 2 in terms of an auxil iarly bare lliggs field d), determines the fermion mass m, and tile Yukawa coupling ./:

1 = G / M 2 F ( A ) 2,1n(A/,7~), l / f 2 = _ l / 2 O F / O m 2 (3)

In relat ivist ic theories, the la t ter may also be interpreted to reflect the rellOl realization group behavior (positive beta flmction) of f2 by regarding m as a running mass, since the fermion loop integral controls both. The [i'.rmio~ loop also generates the kinetic energy and self-couplings of the lliggs [ield and make it a dynamical object in such a way to yield the mass proper inass ratio (in one-loop approximat ion) . The GL transcription of 1he th(~se eqm~tiolls is then done by writing down the usual nonlinear Higgs Lagrangial~ with the coupling constants thus determined.

COlnlnents. a) For type 1 superconductors. b) The symmet ry is N (part icle number) x.s (spin) ×l (orbital angular

mon~entmn).[10] The pairing is p-wave, i. e. I = 1, .s = 1. leading to i>ossibh" j = 0, I and 2 pairings. Two phases are known. The j = 0 pairing (lhe It phase) violat(,s N and reduces .q × [ l.o the diagonal j , a n d / = "2 l)airi,~g (th,,

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v = ~ a :> m / ma f

a)Superconduct iv i ty ,-~keV ~ lO-3eV 2mr ~ 10 -7

b)aHe superfluidity ,-~MeV 10-6eV see comments ~ 10 -12

c)Pair ing in nuclei ,,~ 50MeV -,~ MeV ~ M e V ~ 10 -2

d)QCD-Chira l dynamics 9 3 M e V ~ 300MeV 2~)~f ~ 3

e )Standard Model 246GeV m t = 174GeV ~ < 2mr? l / v /2

AB phase) violates all. In the B phase there are two pairs of 7r and a modes havin j = 0 and 1 corresponding respectively to the violation of N and s x l, and two j = 2a states satisfying ml 2 + m~ = 4m}.

c) Quasi-SSB in a finite system. The as are the IBM bosons [11] formed from the nucleons in the open shell of 2j + 1 states with the (approximate) symmet ry G = U(2j + 1), a subgroup G' of which undergoes an SSB to G". Two cases of j = 3 /2(G = U(4)) can be interpreted well in this manner , (G', G") = (O(6), O(5)), and (SU(3), 0(3)) .

d) As an effective theory of QCD, the BCS description creates const i tuent quark masses.The pseudoscalar mesons are quasi-NG modes, but the scalar a modes are yet to be clearly identified experimental ly. For 2 flavors the BCS Hamila tonian incorporates the U(1) anomaly; for 3 flavors, an effective U(1) chiral anomaly te rm may be added. The BCS description may also include vector type interactions to generate the vector mesons.(For a review, see [13]

e) The Higgs part of the Standard Model could be regarded as a GL description of the t ( topquark) - t interact ion which generates masses of the order of 100 GeV for the t (current mass ) nd the W and Z bosons. The Higgs boson mass is generally < 2rnt when the running of the Yukawa coupling is taken into account. Since other quarks have much smaller masses, only the top quark could be natural ly viewed as due to the BCS-GL mechanism within the Standard Model itself. This may be one more rat ionale for the necessity of going beyond the Standard Model.

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4. Hierarchical SSB

A n o t h e r in te res t ing fea ture of SSB is tha t it can e i ther be r epea ted suc- cessively leading to a chain of SSBs, or it m a y boo t s t r ap itself. I have singed

this out here for its possible impor tance .

1. Hierarchica l chains la ) T h e phonon chain, of which the second one corresponds to case a) in the

list above. A t o m s m - - - a tomic in te rac t ion - - -SSB1--+ crystal fo rmat ion - -

- p h o n o n s - - - e-e a t t r ac t ion SSB2--+ superconduc t iv i ty , energy gap and col lect ive s ta tes

lb) T h e hadronic chain l inking case c) to case d) massless quarks - - - Q C D - - - qq and qqq a t t r ac t ion SSB1--+ massive q's, N ' s and (a, Tr) - - - N - N a t t r ac t ion SSB2-+ nuclear fo rmat ion and N - N pair ing - - - IBM bosons as a quasi-a-Tr T h e hierarchical SSB was or iginal ly in t roduced in the n a m e of t umb l ing by

Dimopoulos , Raby and Susskind (DRS)[12] in G U T hierarchies. The agent of

a t t r a c t i v e in te rac t ion was the gauge field, and the SSB reduced the s y m m e t r y group and inf luenced the running of the coupl ing constant . ILl the a.bove

examples , on the o ther hand, the SSB-induced NG (phonon) or Higgs (a) m o d e is ma in ly responsible for the next SSB in the chain. This would seem to give a more general connect ion be tween the first and the second SSB than

the DRS scenario, and the na tu re of the second SSB can also be different. It would be in te res t ing to look for more possibil i t ies of the SSB chain.

2. Boo t s t r ap The second SSB could also be the same as the first. SSB:SSB1 = SSB2, in

which case one can say it is a boo ts t rap in the sense tha t the a t t r a c t i ve force

be tween fermions in the t- channel is the bound fermion pair in the s-channel

resul t ing f rom the SSB. This was one of the mot iva t ions behind so- called

top-condensa t ion mode l [14] of the Higgs boson in the S tandard Model in

which the Higgs field was viewed as a t - t bound state , and it makes the large and comparab l e masses of the top quark and tile W and Z gauge boson

ra ther natura l . (A l though numerica l predic t ion of the top mass tends to be too high, the t ru th of this mode l is as yet an open quest ion, in my opinion.)

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5. Fermion mass hierarchies

The above observations on hierarchical SSB bring me to nay final remarks on fermion hierarchy. Firs t of all, the Standard Model already contains a hierarchical s t ructure of fermion masses which span almost 6 orders of magni tude from the top quark in the third generation to the electron in the first, and if the neutrinos are included, probably another 6 orders or more. The top condensation model dealt with only the top quark mass, but could not explain the smallness and the variety of mass values of the rest of fermions.

Turning to the GUT theories, obviously one of the impor tan t and un- solved questions is the origin of the smallness of the electroweak scale com- pared to the GUTS, Planck, and superstr ing scales. From the electroweak to the supersymmetr ic SGUTS there is a gap of 13 orders of magni tude, and 16 orders to the Planck scale, which is a small part of the 60 orders of magni- tude that makes up the whole hierarchy of hierarchies from the present size of the universe to the Planck scale, theories and models without a natural and convincing explanation. But I will close by pointing out, for whatever it is worth, to a possible regulari ty among the quark masses that is purely empirical and so far unnoticed, yet invites some theoret ical speculation.

Since the quarks are not directly observable, their masses can only be the- oret ical ly inferred and depend on the renormalizat ion point #. But generally the quark masses corresponding to tt ~ l GeV (except for the top quark) are consistent with a quantized power law:

m = 2%n0, (4)

where m0 = 0.00,5 GeV and n takes the values given below. Perhaps one way

n ,n ~n 'n nz ~n n m u 0 .005 6 + 2 c 8 1.28 6 + 1 t 15 165 d 1 .01 6 - 2 s 5 .[6 6 - 1 b 10 5

to make sense of the dis t r ibut ion of 7z values is to observe that 7, changes by 6-t- 2 between the first and the second generation up/down quarks, and 6 :t: 1 between tile second and the third. This suggests that perhaps two kinds of U(1) quantum numbers are involved in determining m, which reminds one of the model of Frogat t and Nielsen [15]. (The separat ion of 7z vahtes gets a

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little more regular if u and d in the first generation are flipped [16]. The ,~ values are then spaced by 6 + 1 in the upper and lower entries respectively.)

R e f e r e n c e s

[l] P. Curie, J. de Phys. 3 (1894) 393.

[21 P. Weiss, J. de Phys. 6 (1902) 667 (19077).

[3] E. Ising, Z. Phys. 31 (25) 253.

[4] W. Heisenberg, Z. Phys. 49 (28) 619.

[5] H-P. Duel'r, W. Heisenberg., H. Mitter, S. Schlieder. and K. Y~,,u~,zal<i. Z. f. Naturforschung 14A (59) 441.

[6] H. A. Jahn and E. Teller, Proc. Phys. Soc. London, A161 (1937) 220.

[7] G. Wentzel, Helv. Phys. Acta 13 (1940) 269. (See Twentieth Ceutu,y Physics, eds. Brown et al.(Amer. Inst. Phys, 1995) vol.i, i).407.

[8] T. H. Skyrme, Proc. Roy. Soc. A 260 (1961) 127. E. Witten, Nucl. Phys. B 223 (1983) 422, 433.

[9] Y. Nambu, in Rationale of Beings, eds. K Ishikawa el al. (World S(i(ql tific, Singapore, 1986), p. 3.

[10] P. Woelfle, Physica 90B (1977) 96. Y. Nambu, Physica 15D (1985) 147.

[11] A. Arirna and F. Iachello, Ann. Phys. 99 (1976) 253. M. Mukerjee and Y. Nambu, Ann. Phys. 191 (89) 143.

[12] S. Dimopoulos, S. Raby, and L. Susskind, Nuc. Phys. B169..t93 (1980).

[13] T. Hatsuda and T. Kunihiro, Phys. Rep. 2-17 (1994) 221.

[14] Y. Nambu, in New Theories it,, physics. Proc. XI Warsaw Symposium on Elementary Particle Physics, ed. Z. A. Ajduk et al. (World Scienti/ic. Singapore, 1989), p. 1.

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V. Miransky, M. Tanabashi, and K. Yamawaki, Mod. Phys. Lett. A, (1989) 1043. W. Bardeen, C. Hill, and M. Lindner, Phys. Rev. D41 (1990) 1647 .

[15] C. D. Frogatt and H. B. Nielsen, Phys. Rev. B147 (1979) 277; B164 (1979) 114.

[16] S. M. Barr, Phys. Lett. l12B (1982) 219.