Approximate Symmetries and Pseudo-Goldstone Bosons

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VOLUME 2g, NUMBER 25 PHYSICAL RK VIEW LKTTKRS 18 DECEMBER 1972K)KN)name -(1940).We should like to express our gratitude to theCERN 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 I2aFIG. 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, gettingr(=-'- =-~) 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 theR. 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 B8, 195 (1968).CERN-HERA Group, Report No. CERN/HERA 70-6(unpublished) .S. Dado, Ph. D. thesis, Technion, Haifa, Israel, 1972(unpublished); S. Dado et al ., to be published.R. J. Litchfield et al.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 ofthe International School of Physics 'Enrico Fermi, "Course XXX1TI, i966, edited by L. W. Alvarez(Acade-mic, New York, 1966), pp. 70 ]406P. 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.1698Vox.UME 2g, NUMszR 2$ PHYSICAL RK V I K W I.KIT KR S 18 DzcxMszR 1972The 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 conditionthat 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; presumablyVOLUME 29, NUMBER 25 PHYSICAL RK VIEW LKTTKRS 18 DECEMBER 1972these 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.Even when X is constrained by some subgroupP of a group of pseudosymmetries G, this doesnot immediately lead to a constraint on the fermi-on masses unless the Yukawa interaction is alsoinvariant under G. Indeed, for every solution Xof Eq. (1) there are an infinity of other solutionsA. obtained by acting on ~ with any transforma-tion g in 6. The A. for g in Q are physicallyequivalent, but the others are not, and in orderto determine which A. is the physical solution itis necessary to take higher-order corrections in-to account. ' When the Yukawa interaction is Qinvariant these complications have no effect onthe zeroth-order fermion mass relations, thoughthey do affect the mass splittings.There is in fact a large class of pseudosymme-tries which naturally leave the Yukawa interac-tions invariant. These arise from a phenomenonI would like to call the unlocking of representa-tions. Suppose the boson field multiplet y,. maybe decomposed into two separate multiplets y,and g~, transforming under the representationsDand D (perhaps themselves reducible) of G.In general, the table of "typical basic invariant"polynomials includes some involving both y andg, which are invariant under simultaneous t"trans formations on both y and g, but not underG'transformations on y or q separately. It some-times happens that all of these invariants, whichlock the transformations of X and g together,are higher than fourth order in the boson fields,and so are excluded from P(y). In this case Dand D become unlocked, and the polynomialP(y) becomes necessarily invariant under apseudosymmetry group 6= GN|Q, consisting ofindependent transformations on y and g. If all ofthe scalar fields which are allowed by Ginvari-ance to have Yukawa couplings are contained inone of the scalar multiplets, say y, then the Yu-kawa interactions will be invariant under both Q and (trivially) G , and the zeroth-order fermionmass matrix will therefore exhibit invariance un-der any subgroup S of t which leaves A. invariant.But the gauge couplings are only t" invariant, not6 invariant, and so any symmetry in p which isnot in G will be broken by higher-order weak andelectromagnetic effects, which also give a massto the pseudo-Goldstone bosons.As an example of unlocking, consider an SU(4)SU(4} model, in which the left- and right-hand-ed parts of a quark quartet form the representa-tions (4, 1) and (1, 4). Let X be a complex (4, 4*)boson multiplet, which is strongly coupled to thequarks and itself, and consequently has a rathersmall vacuum expectation value, of the order ofthe quark masses. Let q be a single real (20, 1)@(1, 20) multiplet, which cannot couple to thequarks, and has a large vacuum expectation val-ue, of the order of 300 GeV. Unlocking occursbecause the products y *yb and X yb contain onlythe representations (15, 15), (15, 1), (1, 15),(1, 1), (6, 6), and (10,10*), while the symmetricproducts g ~g, contain only the representations(20, 20}, (20, 1), (84, 1), (175, 1), (1, 20), (1, 84),(1, 175}, and (1, 1); the only representation incommon is (1, 1}, so any invariant formed froma ~b ~k~l ~a~b ~I ~& must be the product of aninvariant function of X times an invariant func-tion of g. It will therefore be natural to find solu-tions of Eq. (1) such that Xx breaks SU(4) SU(4)down, say, to U(3), while X" breaks SU(4) SU(4)all the way down to electromagnetic gauge in-1700VOLUME 29, NUMBER 25 PHYSICAL REVIEW LETTERS 18 DECEMBER 1972variance. " With A. x A. ", the vector-mesonmass matrix is dominated by A" terms, and weakand electromagnetic effects can produce finitecorrections to U(3} invariance. The true Gold-stone bosons are linear combinations of y and gfields, but are nearly pure g, and so the Higgsmechanism has only a small effect on the y prop. -agators. The pseudo-Goldstone bosons here areessentially just those y fields corresponding tosymmetries broken by A", i.e., to a pseudoscalaroctet, triplet, and singlet and a scalar triplet.It remains to be seen whether such models haveanything to do with reality, but at least they openup the possibility of an integration of currentalgebras, including soft-pion theorems, with thenew picture of weak and electromagnetic interac-tions.Inspection of the one-loop contributions to thefermion mass matrix shows explicitly that can-celations eliminate the divergences in correc-tions to all types of zeroth-order mass relations,including those discussed here as well as thosediscussed in Refs. 2 and 3. This and other mat-ters will be discussed in detail in a more com-prehensive article on approximate symmetriesnow in preparation.I am grateful for discussions with S. Coleman,H. Georgi, S. L. Glashow, R. Herman, R. Jackiw,B.Kostant, B.W. Lee, F. E. Low, and V. F.%eisskopf.*Work supported in part through funds provided bythe U. S. Atomic Energy Commission under ContractNo. AT (11-1)-8069.S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). Forsubsequent references, see B. W. Lee, in Proceedingsof the Sixteenth International Conference on High Ener-gy Physics, National Accelerator Laboratory, Batavia,Illinois, 1972 (to be published).'S. Weinberg, Phys. Rev. Lett. 29, 888 (1972).H. Georgi and S. L. Glashow, to be published."All values" here means all values within at leastsome finite range, as requried for strict renormaliz-ability. The physical solutions of Eq. (1) are those forwhich P(X) is a minimumH. Weyl, The Classical GrouPs (Princeton Univ.Press, Princeton, N. J., 1946), p. 274. This holds forall compact Lie groups, and for a great variety of non-compact groups.6B. Kostant, private communication.This example was suggested to me by S. Coleman,who informs me that it was originally noticed by P. N.Burgoyne.P. W. Higgs, Phys, Lett. 12, 182 (1964), and Phys.Rev. Lett. 13, 508 (1964), and Phys. Rev. 145, 1156(1966); F. Englert and R. Brout, Phys. Rev. Lett. 13,Ml (1964); G. S. Guralnik, C. H. Hagen, and T. W. B.Kibble, Phys. Rev. Lett. 13, 585 {1964).I am grateful to H. Georgi and S, Coleman for thisremark. This problem has been studied in anothercontext by S. Coleman and E. Weinberg, to be pub-lished.' In general, the large zeroth-order vacuum expecta-tion values of the q multiplet might be invariant undersome subgroup of G&. This would lead to a hierarchyof symmetry breaking and vector-meson masses, ofthe sort described by S. Weinberg, Phys. Rev, D 5,1962 (1972).


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