Approximate Symmetries and Pseudo-Goldstone Bosons

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  • VOLUME 2g, NUMBER 25 PHYSICAL RK VIEW LKTTKRS 18 DECEMBER 1972

    K)

    KN)

    name -(1940).We should like to express our gratitude to the

    CERN Track Chamber Division, and speciallyto Professor Ch. Peyrou, Dr. R. Armenteros,and Dr. D. E. Plane, for supplying the film andthe beam. This work would not have been pos-sible without the encouragement and advices ofDr. R. Barloutaud, and one of us (J.G. ) wouldlike to thank Professor A, Berthelot and Dr.A. Leveque for their generous support at theCentre d'Etudes Nucleaires de Saclay where partof this work has been done.

    I I I I I I I I ) I I I

    2

    a

    FIG. 2. Best fit to the coupling constants gz and g&of the assumed y octet. The new resonant wave withunknown isospin is tried as a triplet of SU(2) andmarked F*.

    according to Eq. (7), the predicted partial widthsof the predicted, getting

    r(=-'- =-~) 5 Me V,F(='- &K) -27 MeV,I'(:"'-:"ri)-0.4 MeV,F(:"'-AK)-0.2 MeV.

    As a speculation we point out the possibility ofidentifying =' with the poorly known group of par-ticles summarized by Soding et al. ' under the

    R. Armenteros et al. , Nucl. Phys. B8, 195 (1968).CERN-HERA Group, Report No. CERN/HERA 70-6

    (unpublished) .S. Dado, Ph. D. thesis, Technion, Haifa, Israel, 1972

    (unpublished); S. Dado et al ., to be published.R. J. Litchfield et al. , Nucl. Phys. 330, 125 (1971).R. D. Tripp, in Strong Interactions, Proceedings of

    the International School of Physics 'Enrico Fermi, "Course XXX1TI, i966, edited by L. W. Alvarez(Acade-mic, New York, 1966), pp. 70 ]40

    6P. Soding et a/. , Phys. Lett. 39B, 1 (1972).For elastic amplitudes, Eq. (5) is derived in, e.g.,

    L. D. Landau and E. M. Lifshitz, Quantum Mechanics(Pergamon, New York, 1958), formula (119.10).

    J. M. Blatt and V. F. Weisskopf, Theoretical clearPhysics (Wiley, New York, 1952), p. 358.

    R. D. Tripp, in Proceedings of the Fourteenth International Conference on High Energy Physics, vienna,Austria, 1968, edited by J. Prentki and J. Steinberger(CERN Scientific Information Service, Geneva, Switzer-land, 1968).

    Approximate Symmetries and Pseudo-Goldstone Bosons*

    Steven WeinbergLaboratory for clearScience and Department of Physics, Massachusetts Institute of Technology,

    Cambridge, Massachusetts 02189(Received 27 October 1972)

    In theories with spontaneously broken local symmetries, renormalizability sometimesforces the scalar field interactions to have a larger group of symmetries than the gaugefield interactions. Symmetries can then arise in zeroth order which are violated by fi-nite higher-order effects, thus providing a possible natural explanation of the approxi-mate symmetries observed in nature. Such theories contain spinless bosons which be-have like Goldstone bosons, but which pick up a small mass from higher-order effects.

    Renormalizable field theories with spontaneous-ly broken local symmetries were suggested' as aframework for unifying the electromagnetic andweak interactions and for solving the divergencedifficulties of the weak interactions. It is becom-

    ing increasingly apparent" that such theoriescan also provide a solution to one other of thelong-outstanding problems of particle physics,the existence of approximate symmetries such asisospin conservation.

    1698

  • Vox.UME 2g, NUMszR 2$ PHYSICAL RK V I K W I.KIT KR S 18 DzcxMszR 1972

    The new opportunity for understanding this oldproblem arises from the circumstance that aLagrangian which is renormalizable, is invariantunder a local symmetry group Q, and contains adeflnlte set of fields tx'ansfol ming according toprescribed representations of Q, is often so con-strained by these conditions that in zeroth order,after spontaneous symmetry breaking, the mass-es and other physical parameters are found toobey certain symmetry relations, which are notmerely consequences of some unbroken subgroupof Q, but which nevertheless remain valid for allvalues' of the parameters in the Lagrangian. Inthis case corrections generally appear in highox'dex', but QO Doxie 8ckoss to f88 88% Otk-of d8%symmetry relations must be finite, because thetheory is renormalizable, and there are no counterterms available to absorb a divergence, shouldone occur.

    In a previous Letter' I discussed one class ofzeroth-order fermion mass relations, those aris-ing from limitations on the representation con-tent of the scalar fields coupled to the fermions.Since then, I have tried without success to findmodels of this type which could explain such ap-proximate symmetries as electron chirality andhadronic SU(3) @SU(3). Meanwhile, Georgi andGlashow' described a few examples of theories inwhich zeroth-order mass relations arise fromconstraints on the zeroth-order vacuum expecta-tion values of the scalax fields, rather than sole-ly from their representation content. It was notclear, however, whether this was an exceptionalor a widespread phenomenon, or whether it couldbe put to use in constructing models of the realworld.

    In this note I will describe a broad class ofquite natural theories in which the vacuum expec-tation values of the scalar fields are subject tocertain constraints, not merely corresponding tounbroken subgroups of the underlying symmetrygroup Q, for all values' of the parameters in theLagrangian. These theories necessarily containa new kind of particle, a spinless meson whichbehaves like a Goldstone boson, but has a smallmass, and is not eliminated by the Higgs mecha-nism. I suggest that the pion and its relativesmay be just such 'pseudo-Goldstone" bosons.

    The spontaneous breakdown of a symmetrygroup 6 is in general manifested in the appear-ance of nonzero vacuum expectation values of amultiplet (perhaps reducible) of Hermitian spin-0 fields p, . The zeroth-order vacuum expecta-tion value A,. of q,. is determined by the condition

    that the G-invariant polynomial -P(y) appearingin the Lagrangian be stationary at p= A.:

    BP(y)/By, .=0 at y=A, .

    In general, we would not expect the solutions ofEq. (I}to exhibit any particular symmetry, ex-cept that A. might be invariant undex some unbro-ken subgroup 8 of 6," in which case no higher-order corrections could arise. There are, how-ever, theories in which constraints on A. arisebecause G invariance requires P(y) (but not therest of the Lagrangian) to be invariant under alarger group of pseudosymmetries Q.

    To see how this is possible, note that every 5-invariant polynomial in q may be expressed' asa polynomial function of certain "typical basicpolynomials" in y. For some y representations,the table of typical basic invariants for |"is thesame as for a larger group G, in which case gaypolynomial in p, invariant under G, is also in-variant under G. This is the case' for the seven-dimensional representation of the exceptionalLie group G, ; the only invariant is the "length"y&y;, so any 62-invariant function of p is alsoinvariant under the larger group O(7). More of-ten, the existence of pseudosymmetries dependsupon the renormalizability requirement, thatP(y) must be at most quartic If enough .of thetypical basic invariants are of higher than fourthorder in y, and are thereby excluded from P(y),then we may find that the ones remaining are allinvariant under some larger group G. For in-stance, consider the adjoint representation ofU{3), furnished by Hermitian traceless 3&3 ma-trices C. If we adjoin to U(3) a reflection R:4- 4, then the typical basic invariants are Tr42and Det4 . However, Det4 is of sixth order in4, and cannot appear in a quartic polynomial.Hence, every quartic polynomial in 4 which isinvariant under G = U(3) SR must be a functionof Tr4 alone, and thexefore must be invariant'under the larger group G = SO(8).

    When P(y} is invariant under a pseudosymme-try group 6, it is natux'al to find solutions ofEq. (I) for which A. is invariant under any givensubgroup 5 of G, for all values4 of the parame-ters in the Lagrangian. In particular, P(y) doesnot know that the whole Lagrangian is only invari-ant under G, not G, so there is no reason why 8must be contained within G. Those transforma-tions that are in both C and $ are the only trueunbroken symmetries in the theory; presumably

  • VOLUME 29, NUMBER 25 PHYSICAL RK VIEW LKTTKRS 18 DECEMBER 1972

    these are restricted to electromagnetic gaugetransformations in the real world. On the otherhand, those transformations that are in 5 but notin Q are symmetries of the zeroth-order vacuumexpectation values but not of the whole theory,and therefore may account for the approximatesymmetries observed in nature.

    The breaking of the pseudosymmetry group 6down to F gives rise to a spinless boson withvanishing zeroth-order mass for every indepen-dent generator of G not in S. Those bosons corre-sponding to generators of the true symmetrygroup G are true Goldstone bosons, and are elim-inated by the Higgs mechanism. ' However, therewill also be a boson for each independent genera-tor of G which is not in G or $; these are mass-less in zeroth order, but since they do not corre-spond to true symmetries of the whole Lagran-gian, they pick up a finite mass from higher-or-der effects. These are the pseudo-Goldstonebo sons.

    For instance, in the U(3) 8 example discussedabove, the only possible symmetry-breaking so-lution of Eq. (1) must have A, invariant under anSO(7) subgroup 8 of G. In this case the true sym-metry group formed by the overlap between Gand S can be either U(1) ISU(1) SU(1) or U(2)NIU(1), leaving either 21 3=18 or 21 5=16"approximate" symmetry generators belongingto 8 but not G. There are 28 21=7 generatorsof V not in S, to which correspond seven bosonswith vanishing zeroth-order mass. Of these,either 9 3 = 6 or 9 5 = 4 are true Goldstone bo-sons, and are removed by the Higgs mechanism, 'leaving behind either 7 6 = I or 7 4 = 3 pseudo-Goldstone bosons.

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