Origin of internal symmetry

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  • P H Y S I C A L R E V I E W D V O L U M E 1 5 , N U M B E R 4 1 5 F E B R U A R Y 1 9 7 i

    Origin of internal symmetry*

    R. Arnowitttf Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138

    Pran Nath Department of Physics, Northeastern University, Boston, Massachusetts 02115

    (Received 1 1 October 1976)

    The possibility that the internal-symmetry group is a consequence of the gauge invariance of a theory (rather than being phenomenologically chosen) is suggested. For a fully unified theory where all interactions are determined by the gauge invariance, this could come about as a consequence of spontaneous (or dynamical) breakdown. Thus the vacuum state after symmetry breakdown may preserve only a subgroup of a larger arbitrary group of the orig- inal unbroken equations. The above suggestion appears to be at least partly realized within the framework of gauge supersymmetry where the local gauge invariance determines all in- teractions via the field equations RAB= hgAB, A=const. Starting with an arbitrary internal- symmetry group, we obtain general conditions to determine the remaining unbroken symme- try group when the gauge supersymmetry spontaneously breaks to a vacuum state that is in- variant under a generalized global supersymmetry. For the case Az 0, if one further as- sumes that the vacuum state preserves parity, these conditions uniquely determine the re- maining unbroken internal-symmetry group to be the U ( l ) gauge group of Maxwell theory (as well as the Einstein general coordinate group). For the case A=O, the internal-symme- try group is only partly determined. However, the condition that a spontaneous breakdown occurs automatically causes a violation of parity, and thus affords a natural origin of this phenomenon for weak interactions. The structure of the pseudo-Goldstone bosons of the theory (which are absorbed by the vector mesons of the broken gauge invariances) is de- termined.

    In recent years , gauge theories have played an increasingly important role in effor ts to under- s tand high-energy phenomena. Any theory that a t - tempts t o unify the different physical interactions mus t d i scuss two not unrelated questions: the s t ruc ture of the dynamical interactions among the fundamental fields, and the nature of the internal- symmetry group that governs these interactions. The conventional Yang-Mills gauge theories shed considerable light on the f i r s t of these questions, a s they uniquely determine the interactions among the gauge vector mesons and greatly r e s t r i c t the other interactions (though the l a t t e r a r e not uniquely determined, par t icular ly those inter- actions that determine whether o r not a spontane- ous breakdown of a symmetry occurs). In th i s con- nection, theories based on gauge s ~ p e r s y m m e t r y " ~ may represen t a useful possibility, a s here all f ie lds a r e gauge fields and hence a l l interactions a r e essentially uniquely determined. A number of v e r y interest ing and important suggestions exis t in the l i t e ra ture concerning the second question (the internal-symmetry group). However, fo r the mos t p a r t these suggestions have had phenomeno- logical motivation. It would certainly be more de- s i rab le to have the internal-symmetry group de-

    termined in a m o r e fundamental way, e.g., by proper t i es of the gauge theory itself. It i s the pur - pose of this paper t o show that the condition of spontaneous symmetry breaking in gauge-super- symnletry theory does indeed determine at least part of the structure of the internal -symmetry group, and the possibility a r i s e s that a theory based on gauge supersymmetry can determine in- ternally (without additional phenomenological as- sumptions) i t s internal-symmetry group. While the ideas presented h e r e s t i l l r epresen t work in p rogress , i t i s heartening that the symmetry groups that a r i s e in the models allowed by gauge supersymmetry a r e actually those that appear to be relevant to the r e a l interactions of nature.

    Since the possibility of self-determination of in- ternal symmetr ies i s a r a t h e r unique (and unex- pected) feature of gauge supersymmetry, l e t u s begin by discussing in general t e r m s how this can come about. The basic assumption of gauge super - symmetry i s that a l l physical f ie lds a r e m e m b e r s of the multiplet of the single tensor superfield gA,(z). Here zA ( x " , ea") a r e the coordinates of the supersymmetry space, where x p a r e the usual Bose (space-time) coordinates and Baa, cr= 1,. . . , 4 , a = 1, . . . , N, a r e a s e t of N anticommuting Major- a n a F e r m i coordinate^,^ {B " " , eBb} =O. The most s t raightforward approach i s to consider the super-

  • 1034 R . A R N O W I T T A N D P R A N N A T H - 15

    symmetry space to be Riemannian, with gAB play- ing the role of the met r ic tensor: ds2 =dzAgABdzB. Then the fundamental gauge group of the theory i s the general coordinate transformation group in supersymmetry space

    and gAB transforms a s a second-order tensor5

    The gauge transformation 6gAB i s conventionally defined a s

    gauge supersymmetry a t the t r e e level appears to ar ise . ' (Discussion of this is given in Secs. V and VI.) Under this c ircumstance a concomitant break- down of s o m e of the internal-space symmetr ies of Eq. (1.6) occurs . The remaining unbroken sym- m e t r i e s a r e then the t rue internal-symmetry group of the theory. It i s in this way that gauge super - symmetry can determine i ts own internal-sym- met ry group.

    To s e e a lit t le more clear ly the interplay be- tween the spontaneous symmetry breakdown and the internal-symmetry group, one may proceed in the usual fashion by expanding g,, around the new

    (1.3) vacuum:

    gAB(z) =gT," +IzAB(~), gTi2 (O/gA,/O) - (1.7) and for the infinitesimal t ransformations Eq. (1.2)

    At the t r e e level, g2; a r e solutions of Eq. (1.5). implies The solutions that have been found2 a r e invariant

    aRkC aLSC ~ R ~ A B (1 S ~ A B = ~ A C + 3 ~ C B + -z . (1.4) under a generalized global supersymmetry; i.e.,

    a t the t r e e level global supersymmetry e n t e r s into The field equations obeyed by g,, must of course be invariant under the gauge t ransformations Eq. (1.1) [for equivalently, Eq. (1.4)]. As in the Bose- space c a s e of gravitational theory, the only covari- ant second-order field equations l inear in the sec- ond derivat ives a r e

    where RAB i s the contracted curvature tensor of supersymmetry space. (HA, i s a second-order differential function of gAB whose p r e c i s e fo rm i s given in Appendix A.) We s e e h e r e how the gauge invariance almost completely determines a l l the dynamical interactions. Indeed, the only freedom i s the choice of h [which has dimensions of (mass) '] and there a r e only two independent pos- sibilities, (i) A + O and (ii) h=O. (We will d i scuss s o m e of the distinctions between these c a s e s la- ter . )

    Equation (1.4) descr ibes the gauge change of g,, f o r a r b i t r a r y gauge function SA(z). A s will be seen in Sec. 11, the special transforlnations correspond- ing to the usual i,zte~~7al-syr?z1izet?,~~ l ~ ~ ~ n r z s f o n ~ ~ a - ti0~z.s a r e l inear t ransformations on the F e r m i co- ordinate multiplet label:

    Here MA, A = 1, . . . , N', a r e a complete se t of N XN r e a l mat r ices in the F e r m i multiplet space and hA(x) the corresponding r e a l gauge functions. Pvior to any spontaneous symmetry breakdowns a l l the t ransformations of Eq. (1.6) a r e symmetr ies of the theory. However, a s has been discussed previously, a spontaneous symmetry breaking of

    gauge supersymmetry a s an invariance of the vac- uum state , just a s Poincare invariance i s a vac- uum-state property in gravitational theory. How- ever , a s will be seen, the '' vacuum metric' ' g?: i s invariant only under a subgroup of Eq. (1.6), the other elements of Eq. (1.6) being broken. Thus the unbroken internal-symmetry group i s the subgroup of Eq. (1.6) which p r e s e r v e s g y ' . This i s the con- dition used to actually calculate the unbroken in- ternal-symmetry group in Sec. 111.

    This paper i s organized as follows. In Sec. I1 the p roper t i es of gauge t ransformations in gauge supersymmetry a r e reviewed. The conditions de- fining the unbroken internal-symmetry group a r e deduced in Sec. 111. Some proper t ies of the fictit- ious Goldstone bosons for the broken-internal- space t ransformations a r e given in Sec. IV. Sec- tions V and VI a r e concerned with two examples fo r the c a s e s h+O and h =0 , respectively. (In the f o r m e r c a s e the conditions of Sec. I11 completely determine the internal space if par i ty conserva- tion i s assumed, while the h = O c a s e automatically implies pari ty violation.) Some concluding r e - m a r k s a r e given in Sec. VII.


    The anticommuting nature of the F e r m i coordin- a t e s implies that gAB(z) i s a finite polynomial in Ban. Thus one may expand g,, a s

    where the coefficients of the powers of Baa a r e ant isymmetr ic in their F e r m i indices, e.g.,

    (2) g A S o l a ~ b = -gABmoln- One may easily r e e x p r e s s each coefficient in tensor f o r m by introducing a com-

  • 15 - O R I G I N O F I N T E R N A L S Y M M E T R Y 1035

    plete s e t of mat r ices in the F e r m i space: A, = { q r , ~ MA), where r, a r e the 16 Dirac mat r ices and vaB = - (C- l )aB and C i s the charge-conjugation m a t r i ~ . ~ Thus one may wri te g(&aa, =g~) , , (~~ ) ) , , , , where A t ) a r e ant isymmetr ic in the total F e r m i space (i.e., A$)= {(qr,)(a)x M,"), (qr,)(')x ~ 2 ) ) ) . Similar expressions hold f o r the higher coefficients of Eq. (2.1).

    A convenient way of extracting the content of the gauge t ransformation law (1.4) i s to expand both s i d e s in powers of O a . To exhibit how one pro- ceeds, we give a few t e r m s (but by no means all) in the superfield expansion of g,,(z):

    where Fa,(x) = F A ( x ) ( ~ ? ) ) , , i s a symmetr ic mat r ix in the internal-symmetry space and E,,(x) = E ~ ( X ) ( M ~ ) . ~ i s a n a r b i t r a r y matrix.7

    This s implest t ransformation i s the Einstein gauge transformation, character ized by a rb i t ra ry Bose coordinate changes:

    Inser t ing Eq. (2.3) into Eq. (1.4) one finds

    where D, i s the usual covariant derivative with respect to the met r ic g,,(x). [The other f ie lds of Eq. (2.2) t rans form a s their tensor indices indi- cate.] Thus g,,(x) can be identified a s the Ein- s tein gravitational field.

    One may a l s o calculate the gauge change (1.4) fo r the t ransformations (1.6). Thus for 6gaB one has

    Hence f rom the t e r m s of lowest o rder in Ba one has

    Similarly f rom 6g,, one finds


    wherefABc a r e the s t ruc ture constants of GL(N,R):

    [The transformation laws of other fields in Eq. (2.2) can be s imilar ly established.] The somewhat pecul iar gauge transformation law Eq. (2.6) fo r F ( x ) and E (x) will be seen in Sec. IV to be related to the fact that a f te r spontaneous breakdown these quantities contain the fictitious Goldstone boson fields associated with the broken symmetr ies . Equation (2.7) shows that B; t r ans forms a s the GL(N, R) Yang-Mills f ie lds under Eq. (1.6), while Eq. (2.8) shows that $ (x) t rans forms a s the funda- mental representation. Thus Eq. (1.6) generates the internal-symmetry gauge t ransformations fo r GL(N,R). This i s how internal symmetr ies enter into a gauge-supersymmetry theory.'

    We conclude our discussion of gauge proper t i es by noting that there a l s o exis ts a second s e t of internal-space gauge t ransformations which dif- f e r s f rom Eq. (1.6) by having opposite pari ty:

    Associated with these a r e a s e t of axial-vector- gauge mesons, which can be explicitly exhibited in the superfield expansion (2.2) by replacing F ( x ) by F ( x ) - F ( x ) + i y5G(x) and including the additional t e r m

    to g,,. The field B;, t r a n s f o r m s a s the regular representat ion under Eq. (1.6) and obeys

    6 B C 5 = a , ~ f ( x ) - f A B c h f ( x ) ~ E ( X I , (2.12)

    = f A B c h f ( x ) ~ z ,(x)

    under Eq. (2.10). We a l so have

    f o r the sca la r and pseudoscalar mesons of F, and


    In the preceding section it was established that Eq. (1.6) generated the internal-symmetry gauge group GL(N,R) f o r a general F e r m i space of N Majorana spinors. Similarly, Eq. (2.10) genera tes analogous odd-parity internal gauge t ransforma- tions. The very la rge group formed from these, however, i s not actually a symmetry of the physi- cal theory a s the field equations support a spon- taneous symmetry breaking of gauge supersym- metry. In this section we will d i scuss how this spontaneous breaking reduces the s i z e of the in- ternal-symmetry group.

    The type of spontaneous symmetry breaking that has been discussed2 i s one that p r e s e r v e s a (gen- eral ized) global supersymmetry. Thus the conven-

  • 1036 R . A R N O W I T T A N D P R A N N A T H 15 -

    tional global-supersymmetry transformationlo tU =i,3(byu8), ta =da leaves the following met r ic in- variant (i.e., dg$), =0):

    We have assumed h e r e that Qa have the s a m e di- mensions a s x p s o that p has dimensions of mass. k i s an a rb i t ra ry constant." The f o r m s of g2: = (OlgA,lO) which a r e t r e e solutions of Eq. (1.5) that have been obtained a r e 2

    Here r, i s an a rb i t ra ry matr ix in the Dirac and internal-symmetry space obeying7 ( v r u ) " = q r u . Thus the vacuum s ta te p o s s e s s e s a generalized global supersymmetry, with fly, replaced by r,. How Eq. (1.5) determines r,, will be discussed fur ther in Secs. V and VI. F o r now, we leave r, arb i t ra ry .

    The expansion of g?;(z) around the new vacuum, Eq. (1.7), shows that g-7: plays the role of a back- ground met r ic fo r the dynamical f ie lds hA,(z). Thus the internal-symmetry gauge transformations (1.6) which do not leave g2; invariant must be r e - lated to the broken internal symmetr ies . More precisely, when one calculates 6gFi using Eq. (1.4). one will find t e r m s proportional to XA and to a,XA. The a,k A t e r m s get absorbed into the gauge t ransformation of B: [of Eq. (2.7)]. The t e r m s in- volving hA correspo 'ndprecisely to the gauge change of gp; fo r a global internal symmetry. We will s e e explicitly in Sec. IV that unless g:: i s in- var iant under a global internal-symmetry t r...