PHYSICAL REVIE% D VOL UM E 12, NUMBER 2 15 JULY 1975

Vacuum symmetry and the pseudo-Goldstone phenomenon*

Howard GeorgitLyman Laboratory, Harvard Universt'ty, Cambridge, Massachusetts 02138

A. PaisRockefeller University, New York, New York 10021

(Received 31 March 1975)

In the context of gauge theories with spontaneous symmetry breakdown induced by the Higgs phenomenon, asimple theorem is stated which gives conditions under which pseudo-Goldstone particles may occur. Theseconditions are broader than the occurrence of accidental symmetry and contain the latter case as a specialinstance. We analyze an example in which the gross features of the vector- and the scalar-meson mass spectraare determined by quantum effects.

I. INTRODUCTION

Field theories which contain spinless particleswhich are massless to zeroth order in a "natural"way' but which acquire mass due to radiative cor-rections may be of physical interest for severalreasons. First, they may give a clue as to theoccurrence of low-mass particles such as pions. 'Secondly, they may possibly play a role in theunderstanding of CI' violation as a natural quan-tum effect. ' The natural occurrence of such par-ticles has been realized in the context of gaugetheories with a spontaneous symmetry breakdowngenerated by the Higgs mechanism. Namely, aswas noted by steinberg, ' such particles may occurif the scalar field potential of the Lagrangian ex-hibits an accidental symmetry, i.e., a naturalsymmetry which is larger than the gauge symmetryof the theory. We shall denote this potential by

V(P, a), where P is a vector whose componentsft)~ are the set of scalar fields and where n denotesthe set of parameters which enters in the poten-tial. Note that the presence of accidental sym-metry is a property of V which is independent ofn (Of course .the ot are subject to the conditionof strict renormalizability of the theory. )

It is the purpose of this paper to state a theoremwhich broadens the options for obtaining naturallyparticles of this kind and to give a few examples.Once again, gauge theories with Higgs particlesprovide the frame; the accidental-symmetry caseas defined by Weinberg' is contained as a subclassin our case. In order not to be overcrowded withnew nomenclature we shall again denote such par-ticles a.s pseudo-Goldstone bosons (PGB's), theterm coined originally' for the more specific acci-dental-symmetry case. Our general theoremcovers in particular the instances noted by Hal-pern, ' by Danskin, ' and by Lieberman where suchPGB's were obtained seemingly without any deeper

reason, and also the local unlocking discussed byDuncan. ' As will soon be evident, our argumentssimply derive from an examination of the contentof the Goldstone theorem itself.

Let G be the local gauge group of the theory withdimension D(G). Let S be the surface in Q space(a finite-dimensional vector space) on which thepotential V(Q, a) assumes its minimum value in thetree approximation. That is to say, when thescalar fields Q; develop a zeroth-order vacuumexpectation value (P;)„„=P.;, then RES. Any trans-formation ~G carries S into itself. The vacuumexpectation value breaks G down to a subgroup Gq[with dimension D(G q}] which is the little group ofthe vector X. Let Sz be the D(S&)-dimensional sub-space of P space spanned by the tangents to thesurface S at the point X. Observe that the struc-ture of S depends in general on the n in V(@, o.)(see Sec. III). Moreover for fixed o., hence fixedS, D(Sz) may depend on the particular choice ofLES (see Sec. IV).

Theorem. The number of spinless particleswhich are massless in the tree approximation andwhich are not absorbed by the vector gauge fieldsis at least as large as the number n defined by

n=D(S),) —D(G)+D(G~) .

n is potentially the number of PGB's. It dependson the detailed particle content of the theorywhether all n are PGB's or whether some (or all)of them are in fact true surviving Goldstone bo-sons (GB's). (Examples are known' where a theorycontains both PGB's and a true GB.) With thisprovision well in mind we shall, for brevity only,refer to n as the number of PGB's.

In a large subclass of theories, the surface Scan be generated by acting on any A. in S with agroup G„.„.[with dimension D(G„,, )] of linear trans-formations mapping S into itself. Let G.,. q [withdimension D( G „, q}] be the subgroup which leave s

508

VACUUM SYMMETRY AND THE PSEUDO-GOLDSTONE. . . 509

A. unchanged. Then

D(S i) = D(G,„,.) —D(G,...~),so that

n = D(G„,„)—D(G„,„z)—D(G) +D(Gz) . (3)

V;(p(t), n)=0. Differentiating with respect to t

we obtain

V;t(g(t), n) y,'(t) =0,and therefore

Often [as for the familar SU(2}XU(1) modelsjG„, = G, G„.„. &

= Gz, but this is not always so.Indeed it is our main allegation (and examples tobe given will illustrate this) that G„„, which wewill call the vacuum symmetry, may be a largergroup than the symmetry G of the theory as a@hole.

As is well known, for the case of accidentalsymmetry there exists a formula' analogous toEq. (3) for the number n'of ,(potential) PGB's,namely

n' = D(G') - D(G'x) —D(G) +D(G ~), (4)

II. PROOF OF THE THEOREM

The tangent space S ), may be defined by con-sidering the set of differentiable curves p, (t) [forevery t, tj(t) is a vector in P space with compon-ents p, (t)j such that p(t)~S for t=[0, 1], g(0) =1,and the limit

p'(0) =lim p. '(t) (5}

exists and is nonvanishing. The vector p, '(0) is atangent to S at X, and the set of all such curvesgenerates all tangents. Now since S is the surfaceof minimum potential, V, (P, n) =0 for PHS [weuse the notation V, , . . . ; (P, n)=&"V(p, n)/&p; ~ ~ sQ; f. In particular

where G' is the accidental-symmetry group and

G), the corresponding little group. Since G' is asymmetry group of the entire potential V(P, n),it clearly contains G. In turn, G, .„. must containG'. The existence of a nontrivial G', that is a G'

which is larger than G, corresponds, for the casesanalyzed in the past, to G„„.= G'~G, in obviousnotation. In this sense, the accidental-symmetryequation (4) is a special case of Eq. (3).

We shall next do three things: First, we shallprove the theorem (Sec. II). Secondly, we shallexhibit situations where G„,, ~G even though thereis no accidental symmetry at all, thereby estab-lishing that accidental symmetry is a sufficient butnot a necessary condition for the presence ofPGB's (Sec. III). Thirdly, we will analyze an ex-ample where n&0 but where the surface S is notgenerated by a vacuum symmetry G„„.,-, so thatEq. (1) applies but not Eq. (3) (Sec. IV).

V„(A, n) ij, ,' (0) = 0 . (6)

III. VACUUM SYMMETRY

In this section we give an example of a modelwith no accidental symmetry but a nontrivial G„,.The models with PGB's mentioned earlier4 ' willreadily be seen as belonging to this subclassG,a,.~G. The example we have chosen to discussnext is not constructed to resemble physics butrather to be as simple as possible as an illustra-tion of vacuum symmetry.

But V;~(X, n) is the zeroth-order mass matrix ofthe spinless mesons, so Eq. (6) implies that eachtangent to S at ~ is an eigenvector of the mesonmass matrix with eigenvalue zero. Thereforethere are at least D(S~) massless spinless mesonsin the tree approximation. Since D(G) —D(G&)Goldstone bosons are absorbed by the Higgs mech-anism, there are at least D(Sq) —D(G) +D(Gq) spin-less mesons left massless in the tree approxima-tion, and the theorem is proved.

Consider a change of the parameters o, in thepotential, so that V(P, n) —V(P, n'). This changemaintains the symmetry G (a.nd any global sym-metries of the full theory). The surface of min-imum potential will change to S'. We will say thatS is a natural surface of minimum potential if forany sufficiently small change in a, the resultingsurface S' is similar to S in the following sense:There exists a one-to-one and onto map ofX~S-X'&S' such that G~=G~ and D(S&) =D(Sz ).In other words, S is natural if its shape is main-tained for a range of n, even though details (suchas the length of vectors ~&S) may change. If S isnatural, the masslessness of the spinless mesonsassociated with S & is also natural. In general, itmay be possible to find special choices of the pa-rameters n for which there are additional mass-less mesons, not associated with Sz. The secondderivative of V may vanish in some direction eventhough V is not constant over any finite interval(for instance, consider a massless scalar fieldwith quartic interactions). This is why we havebeen careful to state our theorem only as an in-equality. But we suspect that any such situationis unnatural in the sense that if the parameters inV are slightly changed, the spinless mesons notassociated with Sq will not remain massless.

510 HOWARD GEORGI AND A. PAIS 12

The gauge group is SO(3) &&SO(3) &&SO(3) = G. Thescalar mesons are three real 3 xs matrix fields

and Q3 1 which transform as (I);f - U& Q; fUf,where U„U„and U, are independent real 3 XB

orthogonal matrices. In other words, the scalarstransform like (3, 3, 1) + (1, 3, 3) +(3, 1, 3) under thegauge group. The potential (with I the 3 &&3 unitmatrix) is

I'(e, ~) = o ([tr(e 0; —~'f}]'+[tr(e.,e; ~'f)]'+[tr(e„e,', —~'&)]'k

+ ~.I [tr(4 .4 '. —~'»] [tr(4.,C —~'»]+[tr(4..4.'. —~'f)] [tr(6 4.;—u'I)]

+[tr(4, 0' —~'f}][«(d-0'- l 'f}])

+ .( [(4.4'- 'f}']+ [(e..4'. — ' }']+ [(4, 6'- ' }']]

+& {tr[(4' 4&. —u'f)(0 0 —p'f)]+tr[(4, 4., —p' )f(4 0.—p'I)]

+tr[(4, 0' —u'f}(CA., —~'f)]) . (7)

This is the most general quartic potential con-sistent with gauge invariance and with the addi-tional discrete symmetries $12and Q,2- —p, 2

If o., &( a, [ (or n, &-,' o, &0) and a,&[ n, ), the po-tential is minimized [V(oto} =OJ when

o o

(ot») = g 0 cose sine

0 —sin& cos]9

(10)

T T T T T T 24124 12 ~12412 423423 4 23423 431431 431431 I I.

The condition (8) is unaffected by multiplicationof any Q;; on the left or right by an orthogonalmatrix. Thus G„„=SO(3) xSO(3) xSO(3) xSO(3)&SO(3) XSO(3): The symmetry of the surface ofminimum potential is larger than the symmetryof the potential as a whole. A particular choiceof vacuum expectation values X satisfying (8}breaks the vacuum symmetry down to the diagonalsubgroups G„„,q = SO(3) &SO(3) &SO(3) (if (Q) = po,where 0 is orthogonal, multiplication on the leftby a general orthogonal matrix U and on the rightby 0 UO leaves (p) unchanged). Thus D(Sq)=D(G„„)—D(G„.„., q) =18 —9 =9. So nine spinlessmesons are massless in the tree approximation.For example, if (Q») = (Q») = (Q») = I, the anti-symmetric parts of the P's are the nine masslessfields.

Just as in an accidental-symmetry situation, theminimization of the zeroth-order potential is notsufficient to determine the physical structure ofthe theory. In the present case, this can be seenas follows. We can use condition (8) and G andthe discrete symmetries to choose

(9)

Then (P») cannot in general be completely dia-gonalized, but we can put it in the form

where the angle 8 is not determined by extremalconditions in the tree approximation. If ]9= 0,G&=SO(3) and there are three massless vectormesons and three PGB's. For other values of 8,Gq = SO(2), and there is only a single massless"photon" and one PGB.

Thus the angle 8 has to be determined by thequantum corrections to the potential, as discussedby Coleman and Weinberg. ' We have describedearlier' another situation where an angle was tobe determined by radiative corrections; it was acase of accidental symmetry, and we had occasionto note at that time that the qualitative propertiesof the vector-meson mass spectrum depended onquantum corrections to the potential. Not only isthis also true here, but in addition something new

happens. Indeed, now not only the vector-mesonmasses but the scalar-meson masses as well de-pend on 8. Hence scalar-meson loops must beincluded in the Coleman-Weinberg sum to deter-mine 8.

It should be clear from this example that vac-uum symmetry is a straightforward extension ofaccidental symmetry which considerably expandsthe options for producing PGB's. We have exhib-ited the simplest of an infinite class of theoriesof this kind. Such models can be constructed withalmost any gauge group of the form g&g~g&The cyclic symmetry is not essential; some ofthe subgroups need not be gauged (as in the pseudo-Goldstone pion examples), etc. There are also

VACUUM SYMMETRY AND THE PSEUDO-GOLDSTONE. . .

many examples which do not have the simple cyclicstructure of the model discussed here.

IV. A PECULIAR EXAMPLE

In this section we give a particular example (itsmain virtue is again its simplicity) in which thesurface of minimum potential is not obtainablefrom a vacuum symmetry. The model'0 has a U(1)gauge group and four complex scalar fields P]Q„and Q, transforming in the same way under agauge transformation: Q, —e' (II)„. The potential is

+3 40 4k

This is the most general gauge-invariant quarticpotential with the following properties: It is sym-metric under permutations of the four fields; thequartic terms are invariant under independent U(1)transformations of each of the four fields,f,- e' "(Ir),. The last condition is renormalizablebecause the gauge couplings are also invariantunder the larger global symmetry U(1) &&U(1) XU(1)xU(l). Thus all terms of dimension 4 in the La-grangian have this symmetry and it is only brokenby one of the mass terms, " the n, term in Eq.(11).

If Qy A2 and n, are positive, the surf ace ofminimum potential for V(P, a) given by Eq. (11)is defined by

k=1 to 4

(12}

To understand what is going on, we can look at apart of the surface of minimum potential in theplane 8, =8, +84 —2n shown in Fig. 1. The twosolid lines are distinct branches of S, 8, = m,

8, = 8, —w and 84 = m, 8, = 8, —m. The dotted point(0, v, v) is the intersection of these branches. Theother dotted points are intersection points of thebranches in the 8, = 8, + 8, —2m plane with otherbranches not in the plane. All the dotted points arephysically equivalent because of the permutationsymmetry.

Since we have already removed the degree offreedom associated with the gauge symmetry bychoosing (P,) = p, , the dimension of the slice of thetangent space shown in Fig. 1 is D(S ~) —D(G)+D(Gq) =D(S q) —1. In other words, it is the numberof PGB's. Except at the dotted points, the sliceis one-dimensional and there is one PGB in accordwith our naive counting. But at (0, v, w) the sliceis two-dimensional and there are two PGB's. Thedotted points correspond to the degenerate situa-tion in which all four vectors are parallel, and atthese special points there are two different direc-tions along the surface of minimum potential.

The existence of separate branches of S inter-secting at isolated points is what keeps us frombeing able to define a vacuum symmetry in thissituation. Any single branch is generated by aU(1) symmetry [plus the U(1) of over-all rotation],but this U(1} is not a vacuum symmetry becauseit does not map the other branches into themselves.

(Tr, 2

We can think of the (II}'s as vectors in the complexplane. Then Eq. (12) implies that the four vectorshave the same length g and that their vector sumis zero. Thus the four vectors must form a par-allelogram. At first glance, one might expect twomassless mesons in the tree approximation cor-responding to the two degrees of freedom of theparallelogram, rotation and deformation. But aswe will see, more careful analysis is required.

There is, of course, always one massless me-son arising from the breakdown of the U(1) gaugesymmetry. We can "factor" the correspondingdegree of freedom out of the surface S of minimumpotential by using the gauge freedom to choose

&e, e

FIG. 1. & in the 92 =83+ 64-2~ plane. Coordinates arelabeled (02, 03, 84).

512 HOWARD GEORGI AND A. PAIS 12

V. CONCLUDING QUESTIONS

The present investigation raises two questions,one new, one old. The first one is: Now thatbroader conditions for PGB's have been obtained,how near are we to having not only sufficient butalso necessary conditions for the naturalness ofparticles whose mass vanishes, but only to lead-ing order? The second one is: Are such particles

needed in the description of such physical phenom-ena as mentioned in Sec. I? The answer to eitherquestion is presently beyond us.

ACKNOWLEDGMENT

Conversations with S. Coleman, A. Duncan, S.Glashow, and J. Lieberman are gratefully ac-knowledged.

~Work supported in part by the U. S. Atomic EnergyCommission under Contract No. AT(11-1)-2232B and bythe National Science Foundation under Grant No.MPS73-05038-A01.

$Junior Fellow, Society of Fellows, Harvard University.In this paper we use the term "natural" in the techni-cal sense. See H. Georgi and A. Pais, Phys. Rev.D 10, 539 (1974).

S. Weinberg, Phys. Rev. Lett. 29, 1698 (1972).3H. Georgi and A. Pais, Phys. Rev. D 10, 1246 (1974).4M. Halpern, as quoted by I. Bars and K. Lane, Phys.

Rev. D 8, 1169 (1973), Sec. V B.~H. Danskin, Phys. Rev. D 10, 3501 (1974).~J. Lieberman, private communication.

A. Duncan, MIT Report No. MIT-CTP-361 (unpublished).B. de Wit, S.-Y. Pi, and J. Smith, Phys. Rev. D 10,4303 (1974).

9S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888(1973).This model was constructed by S. L. Glashow.K. Symanzik, in Fundamental Interactions at HighEnergies, edited by A. Perlmutter et al. (Gordon andBreach, New York, 1970). See also S. Coleman, inProperties of the Fundamenta/ Interactions, proceed-ings of the 1971 International Summer School "EttoreMajorana, " Erice, Italy, 1971, edited by A. Zichichi(Editrici Compositori, Bologna, 1973), p. 605.