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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 interactions via the field equations RAB= hgAB, A=const. Starting with an arbitrary internal- symmetry group, we obtain general conditions to determine the remaining unbroken symmetry group when the gauge supersymmetry spontaneously breaks to a vacuum state that is invariant 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 remaining 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-symmetry 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 determined.

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 interactions that determine whether o r not a spontaneous 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 phenomenological 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 internally (without additional phenomenological assumptions) 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 internal 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 breakdown 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 between 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 covariant second-order field equations l inear in the second 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 possibilities, (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 corresponding 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 coordinate 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 vacuum state , just a s Poincare invariance i s a vacuum-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 condition used to actually calculate the unbroken internal-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 fictitious 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 conservation 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.

11. GAUGE TRANSFORMATIONS

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-

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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

and

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 fundamental 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

111. RELATION BETWEEN INTERNAL-SYMMETRY GROUP AND SPONTANEOUS BREAKDOWN

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 ransformations. The very la rge group formed from these, however, i s not actually a symmetry of the physical theory a s the field equations support a spontaneous symmetry breaking of gauge supersymmetry. In this section we will d i scuss how this spontaneous breaking reduces the s i z e of the internal-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 (general 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 invariant (i.e., dg$), =0):

We have assumed h e r e that Qa have the s a m e dimensions 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 involving 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 ransformation, the corresponding local internal symmetry i s spontaneously broken. A fictitious Gold- stone boson then a r i s e s which s e r v e s to grow a m a s s f o r the associated vector meson. Thus, to determine the internal-symmetry group that s u r - vives after spontaneous breakdown, it i s sufficient to find those global t ransformations obeying 6gFi =0, which we now proceed to do.

It i s s implest to consider f i r s t the gauge change of g$ generated by Eq. (1.6). F r o m Eqs. (1.4) and (3.2) we find

6g;i clear ly vanishes fo r a l l ant isymmetr ic MA which commute with I?, . More generally, the sub-

algebra which p r e s e r v e s gzi a r e a l l those independent elements of O(N),13

M,. c;. MA, -6, = -121, ,, (3.4a)

which obey

One may easily verify that Eqs. (3.4) a l so guaran- tee that the r e s t of gla,C i s p reserved by global internal-symmetry t ransformations, i.e., 6g:,C = O =dg;$. Thus we a r r i v e a t the following basic r e - sult:

After spontaneous breakdown of gauge supersymmetry, the u?zbrokelz-iizte?qzal-svi11111etrv a l g e b ~ a i s the subalgebra of O(N) co~zstntcted f r o ~ ~ i all those lineal* coi7tbzt~ntio)z.s o f g e ~ z e r n t o r s rvhiclz coiiiii~ute zoith r,. In order to determine the internal-symmetry group in detail one needs two items: (i) the dimension N of the F e r m i space: and (ii) the f o r m of the mat r ix r, which en te rs in g;;";. Information on both these i t ems comes from the equations which determine g:," = (O(gA,lO); i.e., a t the t ree level g2,' i s a solution of Eq. (1.5). We will s e e explicitly in the examples of Secs. V and VI how the t r e e equations give information on both i t ems and hence allow at l eas t a par t i a l determination of the internal-sym- met ry group without the use of phenomenological assumptions.

One may include the odd-parity t ransformations Eq. (2.10) into the internal-symmetry algebra to fo rm a l a r g e r algebra with both even- and odd- pari ty transformations. One finds now that the subalgebra that p r e s e r v e s the vacuum met r ic (and hence remains unbroken) a r e the l inear combinations

which commute with r, :

Note that in general one may wri te r, = y , r and s o the odd-parity p a r t s of MJ lead to ant icommutators with r.

IV. GOLDSTONE-BOSON STRUCTURE

In this section we examine the origin of the fic- ti t ious Goldstone bosons that a r i s e a s a consequence of the spontaneous breakdown of internal symmetr ies . As will be seen, a fictitious Gold- stone boson a r i s e s for each internal symmetry whose global transformation does not p r e s e r v e gpi. Mass growth then s e t s in for the corresponding gauge vector meson, which exhibits the breakdown of the associated local gauge invariance.


We begin by briefly reviewing the prototype s i t - uation of s c a l a r electrodynamics s o that we can easi ly recognize the "signal" fo r the appearance of a fictitious Goldstone part ic le in the m o r e in- t r i ca te gauge-supersymmetry equations. The scalar-electrodynamic Maxwell equations a r e

where cp ( x ) i s the complex s c a l a r field. The polar decomposition cp =H(x) exp[ief(x)] then yields

If a potential V ( q ) ex is t s allowing a spontaneous breakdown, H(x) =HbaC + h ( x ) , then 11(x) i s the Higgs meson, f (x) i s the fictitious Goldstone boson, and the "photon" grows m a s s M' =2e2(HVaC)2. The different par t i c les can be character ized by their Maxwell gauge transformations; i.e.,

while C, Z A , - a,f appearing in Eq. (4.2) i s gauge invariant. (More generally, fo r the non-Abelian case , 6C, will be independent of a, h.) Thus the fictitious Goldstone boson can be character ized by the appearance of an inhomogeneous h(x) t e r m in i t s gauge transformation law, o r alternately by the fact one can find a gauge (the unitary gauge) where the Goldstone boson i s se t to z e r o (and hence i s absorbed into the longitudinal mode of the vector meson).

As discussed in Sec. 111, there a r e two c lasses of internal-symmetry transformations in gauge supersymmetry that do not p r e s e r v e &,': (i) those generated by the symmetr ic elements of GL(N, R) (k$)= + ~ g ) ) , and (ii) those elements of the O(N) subalgebra which d o not commute with I?,. We will s e e that it i s precisely each of these t ransformations that produces a fictitious Goldstone boson. (i) S v i ~ z ~ v e t r i c elenlerzts ($2) = +iVft l ) . The fictitious

Goldstone bosons f o r the symmetr ic elements of GL(lV,R) a r i s e in the field F ( x ) of ga8 in Eq. (2.2). F r o m Eq. (3.2) one s e e s that F has the vacuum p a r t ( 0 1 ~ 1 0 ) = k . Thus s ince F ( x ) must be a sym- met r ic matr ix, one inay expand around the broken vacuum

Thus for the symmetr ic elements of GL(N, R) Eq. (2.6a) impliesL4

where

[and the s u m s in Eq. (4.5) run only over the sym-

met r ic elements of GL(N, R)]. ~ h u s f A ( x ) has the charac te r i s t i c fictitious-Goldstone t ransformation law.

Equation (4.5) shows that 6fA p o s s e s s e s an inhomogeneous t e r m 2h$)(x). Fur ther , the fact that there a r e a s many gauge functions x$)(x) a s fields f A ( x ) suggests that it should be possible to find a gauge (the unitary gauge) where a l l the f A ( x ) have been s e t to zero. That this i s indeed possible can b e seen by examining the jiizite gauge t ransformation

Using the finite transformation law Eq. (1.2), one easi ly finds for the met r ic of Eq. (2.2)

Usingonly symmetr ic gauge t ransformations (i?

=A, U=U) one can s e e that a C ex is t s which r e - duces I.'* to ze ro ,

i.e., one may solve Eq. (4.9) fo r k$)(x) in t e r m s of f A ( x ) .

The finite gauge t ransformations f o r B;(x) of Eq. (2.2) a r e

where B , ( x ) = B $ ( ~ ) M ~ . The vector mesons B$") associated with the symnletr ic M$) in B, ( x ) a r e the gauge mesons of the broken gauge. More general ly , if one defines C, ( x ) = C; (x)MA a s

then under a finite transformation one finds, f r o m Eqs. (4.8) and (4.10),

In the unitary gauge C; reduces precisely to B;('). The fact that the C, t ransformation law Eq. (4.12) contains no t e r m s involving a,A, implies that m a s s t e r m s in the C, equations can a r i s e (i.e., C, plays the r o l e of the combination A, - a, f in the siinple Abelian scalar-electrodynamics case). To s e e the m a s s t e r m s requi res a m o r e detailed examination of the dynamics, i.e., the s t ruc ture of T, of the vacuum met r ic Eq. (3.2) and the field equations (1.5). We will s e e in Sec. V that m a s s t e r m s do a r i s e in the C, degrees of f reedom, completing the verification of the spontaneous breakdown of the symmetr ic gauges of GL(N, R).

(ii) Broketz eleixents of O ( N ) . The second c lass of gauges possessing a spontaneous breakdown a r e


those element of O(N) which do not commute with r, [Eq. (3.4)]. Here, the associated fictitious Goldstone bosons res ide in the field F ( x ) of Eq. (2.2). Thus one may expand E ( x ) around i t s vacuum value (OIE 10) =I? [where r, = y,I? in Eq. (3.2)] a s

Here we have chosen a b a s i s such that M, = -k, a r e the genera tors of the subgroup of O(N) that r e - mains unbroken (i.e., [I?, M,] =O), and M i a r e the remaining elements of O(N). The fictitious Gold- stone bosons a r e then the $ (x) fields. To verify this, we show that a unitary gauge exis ts which s e t s $(x) to ze ro . Consider f i r s t the infinitesimal transformation Eq. (2.6b) generated by the broken e lements A =hi (x)Mi :

- b i ( x ) =khi +.Pf j~ ih ' ) (4.14)

wheref,,MA = [ r , M,]. TO eliminate yi(x) to infin- i tes imal o rder requ i res finding Ai(x) such that 6yi = y'' - yi = -yi (x) . One can indeed solve Eq. (4.14) f o r A'(%) s ince f j i p o s s e s s e s a n inverse in the broken subspace. (This follows f rom the fact that no l inear combination C'M, commutes with r . ) F o r finite t rans forn~a t ions , E ( x ) t rans forms a s

where 6 =u-' f o r A = hiM,. The condition for the unitary gauge then i s that yti vanish, i.e., E' = r +yfrM, +y[:~$). One can s e e that Eq. (4.15) can b e solved f o r hi in t e r m s of yi by a n iteration scheme based on the above infinitesimal analysis a s the f i r s t approximation. The vector mesons can be expanded in the s a m e b a s i s a s B, ( x ) =B;M, +BCM, +BC(')M$). The gauge mesons a s - sociated with the broken symmetr ies a r e , of course, the B; which will grow masses . The r e - maining s c a l a r f ie lds of E (x) , i.e., y r ( x ) , & ( x ) remain nonzero a f te r transforming to the unitary gauge.

Finally we mention that a s imi la r t reatment of the breakdown when one includes the odd-parity t ransformations Eq. (2.10) can be given.

V. INTERNAL-SYMMETRY GROUP FOR X f 0 CASE

The analysis of the previous sect ions has shown that two i t ems a r e needed to uniquely determine the unbroken internal-symmetry group of gauge supersymmetry: (i) the number N of Majorana coordinates in the F e r m i space, and (ii) the ma- t r ix r,, = y, r of g;; [Eq. (3.2)]. Up to now the analysis has been general since N and I' have not been specified. However, information on both

these i t ems a r i s e dynamically s ince the vacuum met r ic i s to be determined by the field equations (1.5). A s mentioned in Sec. I, the re a r e only two independent possibilities, A + 0 and A = O . We con- s ider in this section the h f 0 c a s e (where the determination of the internal-symmetry group i s a l- mos t complete) and examine the A = O c a s e in Sec. VI .

The necessary condition that a spontaneous sym- m e t r y breaking occur a t the t r e e level i s that the vacuum met r ic gTi= (O/g,,jO) be a solution of Eq. (1.5). Inser t ing the fo rm (3.2) fo r g?; into the explicit expressions given in Appendix A for RAB one finds"

R,, =-k-2TrI',,I'v ,

Here the t race i s over both the Dirac and internal- symmetry space. Equation (1.5) thus implies two constraints on rp (Ref. 2):

Taking the t race of the f i r s t condition shows that the two relat ions a r e consistent fo r A # O only if Nf =2N,, where Nf i s the F e r m i dimension and Nb i s the Bose dimension, i.e., Nb = 4 and Nf =4N for physical space-time. The internal-symmetry index then i s uniquely determined to be'"

Thus sponianeor~s synzwzetry breaking requires tlznt the Ferrni space coordinates be a doublet of Majorana spinors2 Baa, a = 1, 2. (Alternatively, one could combine these to a single Dirac spinor, 0gi,,= em' - iQa5)

To extract the full information implied by Eq. (5.2), i t i s convenient to expand r, = y p r in a complete s e t of matr ices . Thus one may wri te

Since q r , i s symmetr ic , r(ssa) a r e (symmetr ic , ant isymmetr ic) mat r ices in the internal-symmetry space. F o r N = 2, one may expand I?'") in a complete s e t of Pauli m a t r i c e s E,, r = 0 , . . . , 3 ( E , = I), acting on the symmetry indices:

where P,?=Pr. The E , a r e the MA m a t r i c e s f o r this case. Inserting into Eq. (5.2) now yields two c l a s s e s of solutions:

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

In o rder to understand the significance of these solutions we note that t e r m s in r involving c , , E,, and c 3 violate pari ty (P) for the vacuum metr ic , while the c , and E , t e r m s violate charge conjugation (C). The condition l/3,1 =/pol implies maximal P and C violation, but P C conservation.17

We s e e that while the t ree- level spontaneous- breaking equations (5.2) do determine many of the p roper t i es of the vacuum metr ic , i t does not de- t e rmine r, uniquely. P r e s u n ~ a b l y , additional theoret ical pr inciples a r e s t i l l needed. However, if we phenomenologically impose the condition of par i ty conservation (which i s t rue at l eas t for par t of the known physical interactions) then J?, is uniquely determined to be

We will u s e Eq. (5.7) f o r the r e s t of this section. With the fo rm of r,, now fixed, we apply the gen-

e r a l formalism of Sec. I11 to determine the internal-symmetry group. The basic condition i s Eq. ( 3 . 6 ) . There is , of course, only one antisymnletric (real) mat r ix of GL(Z,R), i.e., E - ic , [which gen- e r a t e s 0(2)]. Thus the genera l f o r m of All i s

Using Eqs. (5.7) and (3.6) one then finds that C, mus t vanish. Thus the unbroken symmetry i s uniquely determined to be O(2) [or, alternately, using Dirac coordinates 8gi,, the symmetry i s ~ ( l ) ] . The relevant p a r t of g,, of Eq. (2.2) i s fo r this c a s e

where E, a r e the symmetr ic Pauli mat r ices , c, = (E,, E ,, c3). The gauges associated with B; ,A: ,B;' a r e broken and pseudo-Goldstone bosons a r i s e fo r these f ie lds a s described in the general analysis of Sec. IV. The field equations fo r the vector mesons come f rom the t e r m s l inear in 8" in the R,, =Xg,, components of Eq. (1.5). One finds m a s s t e r m s there of o rder p2//k2 - A f o r a l l the vector mesons associated with the broken gauges.""

The only unbroken-internal-symmetry gauge i s the U(l) gauge associated withA,(x) . One can s e e that A , can b e interpreted a s the photon field of the theory. Thus it i s possible to show that A , i s odd under charge conjugation, remains mass less , and has the cur ren t of the Dirac spinor field # a a ( ~ ) of Eq. (5.9) a s i t s source. (It i s interesting to note that the theory correct ly fo rces the "axial- vector photon" A; to become massive.) One thus

a r r i v e s a t the following resul t : T h e o r e ~ r . F o r parity-conserving vacuum s ta tes

with h# 0, the spontaneous-symmetry -breaking condition uniquely de te rmines the dimension of the internal-symmetry space to be N = 2 and r, =By, @ 2 = h k v 8 ) . The unbroken-internal-symmetry group i s determined to be the U(l) gauge group of Maxwell theory.

The above resu l t s a r e remarkable in that it i s a n example of a model which almost completely de te rmines i t s internal symmetry. Fur ther , the unbroken gauge turns out to be the physically c o r - r e c t one. In addition to the electromagnetic gauge invariance, the theory p o s s e s s e s the Einstein in- var iance, 5 , = 5" ( x ) , = 0, which a l s o i s p r e - se rved by the spontaneous breakdown. The spontaneous breaking discussed above has a second remarkable feature of a l s o producing the unification of the gravitational and electromagnetic phe- n o m e n a . ' ~ ~ The parameter A may be related to the Einstein constant by G, =e"/nh, where e i s the unit of e lectr ic charge, A m o r e detailed discussion of this i s given in Ref. 18,

VI. INTERNAL-SYMMETRY CROUP FOR X = 0 CASE

We consider in this section the internal-symme- t r y group f o r the al ternate dynamical possibility of A = O . Here we will s e e that the field equations R,, = O do not determine the dimensionality of the Fer tn i space. However, the spontaneous-symme- t ry-breaking conditions fo r X = O requ i re that the vacuum s ta te violate pari ty invariance. Thus the h =O c a s e automatically accounts fo r the breakdown of P and C invariance of weak interactions in a natural way (while s t i l l including the electromagnetic and gravitational unification of the X # 0 model of Sec. V).

Since h i s now zero , the spontaneous-breaking conditions (5.2) no longer determine N. However, one may st i l l proceed a s in Sec. V and inser t the expression Eq. (5.4) into Eqs. (5.2). Here Eq. (5.4) i s the most general f o r m preserv ing proper Lorentz covariance, the J?(') piece being the pari ty- p reserv ing par t of r and the riS) piece the pari ty- violating part . The t race condition of Eq. (5.2), T r r , I', = 0 , now yields

~r + ~r [ria)] = 0 . (6.1)

If the vacuum p r e s e r v e s pari ty , then r(')=0. How- ever , Eq. (6.1) would then imply that r(') a l s o vanishes, and one has only the t r ivial null s o l ~ t i o n ' ~

= O . We have then the following basic resul t : Theorein. F o r h = 0 , the spontaneous-symme-

try-breaklng condition requ i res that the vacuum s ta te violate pari ty conservation f o r a l l nonnull solutions.


We investigate next whether this par i ty breakdown i s related to the one found in nature in weak interactions. Since the value of N i s not de te r - mined by the symmetry-breaking analysis , we consider here the s implest nontrivial c a s e of a double: of Dirac spinor coordinates (N =4). It i s convenient then to label the Majorana F e r m i coordinates a s

We will r e f e r to q a s the charge degree of f reedom and a a s the (weak) isospin label. (Thus the cor - responding Dirac sp inors a r e Bgy,, = Baa' - ieaa'.) We a l so le t 7, and E, be a complete s e t of Pauli m a t r i c e s acting in the isospin and charge spaces, respectively. One may now expand r(') and r(a) in t e r m s of the complete s e t of mat r ices T , X E ~ (@) i s symmetr ic and I?(') i s antisymmetric). Inser t ing these expansions into Eq. (5.2) allows u s to again extract the full content of the symmetry-breaking conditions fo r this case. There a r e a number of possible solutions. However, if we consider only those c a s e s which maintain electr ic-charge con- servat ion, Eq. (5.2) allow only two independent solutions:

where the p's a r e a r b i t r a r y constants of dimen- s ions of mass , and

a r e the chiral r ight (left) projection opera tors in Majorana n ~ t a t i o n . ' ~ The fac tors P+ and P - violate P and C m a ~ i r n a l l y , ' ~ but conserve2' PC. Thus Eq. (6.3a) automatically leads to a theory with maxi- m a l P and C violation (as required for weak inter- actions2') and we will r e s t r i c t our considerations to the choice

f o r the r e s t of this section. We consider next the analysis of Sec. HI t o de-

t e rmine what internal-symmetry gauges remain unbroken with the r, of Eq. (6.5). The m a t r i c e s M A required for the test of Eq. (3.6) a r e the s i x ant isymmetr ic elements ET,, re,, a = 0 , 1 , 3 where E, (7,) a r e symmetr ic and E = i e z (7 =ir2) a r e anti- symmetr ic . After some calculation, one finds that there a r e only two M, of the fo rm of Eq. (3.5) satisfying Eq. (3.6) (for 0 , p,+ 0)23:

T o s e e the significance of Eq. (6.6), le t u s con- s ider the ch i ra l U(2)x U(2) subalgebra of the

GL(N,R) genera tors of Eqs. (1.6) and (2.10). The ch i ra l a lgebra i s generated by p,P,, where

(p+ = -p,, p,?= 4). In chiral notation, Eq. (3.6) r e a d s

where rc = y, I' and M, i s a l inear combination of the ch i ra l generators . The S U ( ~ ) , x U(1), subgroup of U(2)x U(2) i s generated by the setz4 (e.g., f o r the lepton sector)

F o r the r, of Eq. (6.5), we s e e that [r,t ,] vanishes showing that I?, does not cause a breakdown of any of the SU(2),XU(l), gauges. On the other hand, the M A fo r the right chiral W, mesons a r e p,P+, m = l , 2, and these do not commute with r. Thus the W R mesons will grow m a s s e s due to rl, (presumably scaled by p,) . The SU(2),x U(1), group i s broken, however, by the kq,, p a r t of the vacuum met r ic (3.2). Thus the t, matr ices a r e not ant isymmetr ic , which i s needed t o p r e s e r v e gzg [according to Eq. (3.3)]. This will produce m a s s growth for the left ch i ra l WL mesons and the Z0 meson (this m a s s growth scaled by k). The WR mesons gain a s imi la r additional m a s s growth f r o m this source, perhaps making it heavier than W L (and s o perhaps explaining why left ch i ra l cur - r e n t s appear to dominate r ight ch i ra l cur ren ts in naturez5). However, only th ree of the four genera- t o r s of SU(2), x U(l), do not p r e s e r v e &,, (cor res - ponding to the m a s s growth of the Wi and ZO) and the electromagnetic gauge i s not broken (guaran- teeing that the photon remains massless) . This i s generated (e.g., f o r the lepton sector) by t, =t , -to =$ET, - $ E , which by Eq. (6.6) i s an unbroken gauge. Thus the c o r r e c t m a s s spec t rum in the SU(2), x U(l) sec tor a r i s e s . Unfortunately, Eq. (6.6) shows that there i s a l so a second unbroken gauge in U(2) x U(2) generated by t tep=e (e.g., the lepton number gauge). This i s of course an incor- r e c t prediction of the model. One may hope that this gauge i s broken a t a l a te r s tage dynamically.

The fact that only th ree of the four mesons of SU(2),x U(1), grow m a s s e s i s fur ther borne out by the fact that only th ree fictitious Goldstone bosons a r i s e in g,, in the analysis of Sec. IV.' However, the m a s s growth i s complicated by the fact that the t r e e contributions vanish. This i s because t r e e m a s s e s a r e proportional to I',rp, which by the spontaneous-breaking conditions (5.2) i s z e r o for the X = 0 theory. However, m a s s growth does begin a t the first-loop level,' though a full ana lys i s of the m a s s mat r ix has not been c a r r i e d out.

O R I G I N O F I N T E R N A L S Y M M E T R Y 1041

VII. CONCLUSIONS

We have considered in the above discussion the suggestion that the internal-symmetry group of a theory may be determined "internally" by the theory ( a s a consequence of i t s basic gauge invariance) r a t h e r than being a n additional "external" hypothesis imposed upon the theory. Such a possibility appears to be conceivable fo r any sufficiently unified gauge theory s o that (i) the interactions a r e completely determined by the gauge invariance, and (ii) a priwi, the theory p o s s e s s e s a reason- ably a rb i t ra ry internal gauge group, In such a theory, the spontaneous-symmetry -breaking conditions uniquely determine which symmetr ies a r e broken, and thus could select f rom the initial a rb i - t r a r y gauge group, a unique unbroken subgroup. F o r the c a s e of gauge supersymmetry, the gauge invariance does indeed determine a l l the inter- act ions except f o r the value of A (i.e., e i ther X # 0 o r X=O). Also, the initial unbroken-symmetry group i s quite l a rge [GL(N, R)] and s o the theory comes c lose to meeting both c r i t e r ia .

While the spontaneous-symmetry-breaking anal- ys i s given h e r e for gauge supersymmetry de te r - mines a good deal about the internal symmetry, i t does not completely specify it. Thus in the c a s e X + 0 (where the determination i s a lmost complete) one mus t add an additional constraint concerning pari ty conservation, while the X = O c a s e does not determine the F e r m i dimension N (and perhaps one mus t a l so impose charge conservation). It i s somewhat promising that what has been determined by the spontaneous-symmetry-breaking arguments s o f a r , a r e in fact in accord with experiment. Thus, fo r the X P 0 case, one i s left with only the Maxwell and Einstein gauges unbroken (which, as ide f romperhaps color gauge invariance, a r e the only perfect gauge invariances in nature). F o r the X = O case, par i ty- and charge-conjugation viola- tions a r e automatically achieved in a fashion con- s is tent with SU(2) x U(1) (Ref. 24) (and hence with maximal P and C violation). It is a l so feasible to include a conserved color group to charac te r ize the s t rong interactions2 (producing a model where a l l four interactions a r e unified). However, the lack of uniqueness of the p resen t analysis in the determination of the internal-symmetry group im- p l ies the need for additional physical pr inciples to completely specify it. In this regard, it should be noted that the spontaneous-symmetry-breaking conditions given here a r e necessary ( ra ther than sufficient) conditions in that the condition for min- imizing the effective potential has not yet been c a r r i e d out. The condition of picking out only the minimum solution may further r e s t r i c t the internal-symmetry s t ructure.

Closely associated with the question of minimiz- ing the effective potential i s problem of tachyon solutions in the m a s s matrix. Thus expanding gAB around the vacuum solution [Eq. (1.7)] in the La- grangian' t o quadratic o rder allows one to calcu- l a te the m a s s matrix.26 In principle, one may investigate whether the (mass) "igenvalues a r e al l positive for the A# 0 c a s e (though in p rac t ice this would be a very difficult calculation owing to the l a rge s ize of the matrix). F o r the A = O case, m a s s growth vanishes a t the t r e e level and s o one has the additional complication of needing the f i rs t - loop correct ions. These questions a r e cur - rently under investigation in the s impler one- and two-dimensional gauge-super symmetry models.

The vacuum met r ic Eq. (3.2) contains two par t s , that depending on I?, and that depending on k. The parameter k plays a c ruc ia l ro le in gauge super - symmetry in that k must be nonzero f o r gAB t o have an inverse, and hence for the supersymmetry space to be Riemannian. By fixing the space to be Riemannian, the interaction s t ruc ture becomes unique (i.e., the only allowed field equations a r e R A E = kgAB) SO that the above ideas of using the gauge invariance to determine the internal symmetry can be put into operation. Thus none of the successful resu l t s of this paper would be forced upon the theory if one s ta r ted with k = 0 , and one indeed s e e s the singular nature of this point in the spontaneous-symmetry-breaking equations (5.2). However, a f te r the spontaneous symmetry breaking i s achieved, it may be possible to consider the l imit k- 0, and this l imit may in fact be related to the m o r e conventional global-supersymmetry ana- lyses,1° and to the non-Riemannian supersymmetry space introduced by Z ~ m i n o . ' ~ In part icular , it has been shown3 that a f te r spontaneous symmetry breaking that RAE =kgA, equations for A# 0 include the spin-%-spin-$ supergravity equationsg in the k- 0 limit. It i s thus of importance to examine this l imi t in a more general way.

APPENDIX A

In this appendix we summar ize f o r e a s e of ref- e rence s o m e of the mathematical fo rmulas of gauge supersymmetry. A m o r e detailed discussion can be found in Refs. 1 and 18.

The bas ic geometr ical quantity of gauge super - symmetry i s the invariant metr ic: ds" =dzAgA,dzB, z A ( x p , o a 5 ) . The anticommutativity of the F e r m i coordinates implies the symmetry conditions

where a , b = (0; 1) depending upon whether the a s - sociated index A , B = ( p ; a, a ) is (Bose; Fermi) .


Thus decomposing gA into Bose and F e r m i s e c - to rs , gas = (gp , ,&?patgad) Eq. (-41) impliesg,,,(z) =g,,(z) (as usual) while g,,, = -gap and g a ~ = - g ' ~ , . Note that g,, and g,, a r e commuting quantities, while the g,,, anticommute. The Hermiticity condition (ds2) t=ds2 imposes the following constraints on gAB(z):

g,,(z) =g:"(z) , gpa(z) = -g;~(z) (UY') @a ) (A21

gae(z ) = - ( ~ ~ O ) a ~ g ; a (z)(W0)6a . This imposes various real i ty conditions on the coefficients of the superfield expansions; e.g., g, ,(x) of Eq. (2.2) must be real , a s must the mat r ices r(Ssa) of Eqs. (5.4) and (5.5), etc.

The right covariant derivative of a contravariant vector , v ~ ; ~ defines the affinity r& :

v ~ ; ~ v ~ , ~ +vCr& 7 (A3)

where VASE 2 aRvA/azB i s the right ordinary derivative. The vanishing of the torsion in supersymme- t r y space implies that r:, i s a "symmetric" affinity, i.e., that it obey

(-I)"+ b+ab+c(a+b) r 2 B = q a t c r L ) (Pabc = - (A4)

The supersymmetry space i s Riemannian if, in addition t o Eq. (A4), r2, i s determined in t e r m s of the met r ic by gAB;C = O . One finds then

O(c+d) r : ~ zf[(-l)b(c+d)gAD,B +(~abc(-l) ~ B D , A

- ( - I ) & ~ A B , D I ~ ~ , (A51

where gAB i s the mat r ix inverse of gA8:

*Research supported in part by the National Science Foundation.

?work supported in part by the John Simon Guggenheim Memorial Foundation.

$011 sabbatical from Department of Physics, North- eastern University, Boston, Massachusetts 02115.

'P. Nath and R. Arnowitt, Phys. Lett. m, 171 (1975); R. Arnowitt, P. Nath, and B. Zumino, Phys. Lett. 56B, 81 (1975); R. Arnowitt and P. Nath, Gen. Relativ. - Gravit. 1, 89 (1976); P. Nath and R. Arnowitt, J. Phys. (Par is ) 37, C2 (1976); P. Nath, in Gauge Theories and iModern Field Theory , proceedings of the Boston Conference, 1975, edited by H. Arnowitt and P. Nath (MIT Pres s , Cambridge, Mass., 1976).

'R. Arnowitt and P. Nath, Phys. Rev. Lett. 36, L526 (1976).

3 ~ . Nath and R . Arnowitt, Phys. Lett. e, 7 3 (1976). 4 0 ~ r notation i s a s in Ref. 1. Thus cu, 8 = 1, ..., 4 i s the

Dirac index and we will suppress the multiplet labels a , b when no confusion occurs. The indices A, B run over the combined Bose index @ ) and Fermi index (04.

"he left CL) and right ( R ) derivatives in Eq. (1.2) a r e

One may verify direct ly that the explicit f o r m (A5) obeys Eq. (A4).

The curvature tensor may be defined a s in Bose space by paral le l- t ransport ing a vector around a closed infinitesimal curve. One finds the usual resu l t with some additional sign factors:

A s in Bose space, a Riemannian supersymmetry geometry i s completely determined by the single t ensor superfield gA,(z). Thus it f o r m s a f rame- work for a unified field theory with the leas t amount of a rb i t ra r iness ; i.e., a l l f ie lds a r e in a single representat ion of the gauge group [ztA =z'A k)].

F o r Riemannian geometry there a r e only two independent tensor contractions of the curvature:

RAE p o s s e s s e s the s a m e symmetry proper t i es as gAB does [Eq. (Al)]. The most general field equations which a r e second order and l inear in the second derivat ives a r e then

The explicit f o r m of Eq. (A9) in t e r m s of the single superfield gA, can thus be obtained using Eqs. (A5)) (A7), and (A8).

defined by dV(z) = (aR~/azA)dzA=dzA(a,v/azA). The distinction i s necessary owing to the anticommuting property of the oGa.

6 0 ~ r Majorana spinors obey 8, = 6 1 ' ~ , ~ , , where 8, - ( ~ ~ y ~ ) ~ The se t (qr, ) @ ) = {q, 5y5, ~ ? p } a r e antisymmetric while (qr,)(S)={~ry", qupu) a r e symmetric. We use ~ ~ = i y ~ y ~ y ? ~ and {y,,,~,} =-2%,, where rho =-I.

h he different components of gAB(Z) obey the symme- t r ies gb,iz) =g,,,(z), gpa(Zf =-gslp(Z), and gGa(Z) =-gBa(z). Thus F ( x ) must be symmetric, but there is no constraint on E(x). We use the notation (&'A)&

=(iVk)ba. We write QrGsl -Q&- Qaa. he use of gauge transformations to determine the

physical significance of the various fields is a convenient device in gauge supersymmetry. Thus Eq. (2.8) represents a quark transformation while it i s possible to show that x:(x) of Eq. (2.2) is purely a gauge degree of freedom and can be se t to zero by a gauge transformation t"(z) quadratic in 8". As shown in Ref. 3, the field $ ( x ) of Eq. (2.2) is related to the spin-2 ''7' part of the supergravity" multiplet (Ref. 9) (which i s


a part of the general supergauge theory). 'D. Z. Freedman, P. van Nieuwenhuizen, and S. Fer-

r a r a , Phys. Rev. D z , 3214 (1976); S. Deser and B. Zumino, Phys. Lett. E, 335 (1976).

'OD. V. Volkov and V. A. Soraka, Zh. Eksp. Teor. Fiz. Pis 'ma Red. 18, 529 (1973) [ J E T P Lett. 18, 312 (1973)l; J. Wess and B. Zumino, Nucl. Phys. E, 39 (1974); A. Salam and J. Strathdee, Nucl. Phys. B76, 477 (1974).

11<3ne can of course sca le k to unity. However, a s discussed in Ref. 3 and by Woo and Zumino (Ref. 12), it is interesting to be able to consider the singular limit k - 0 .

1 2 ~ . Woo, Lett. Nuovo Cimento 13, 546 (1975); B. Zu- mino, in Gauge Theories and Modern Field Theory (Ref. 1).

13The fact that the invariance of the k r h B piece of the metric reduces the internal symmetry group to O(N) has also been noted by P. G. 0. Freund, J. Math. Phys. 17, 424 (1976). -

'"or the gauge transformations generated by the antisymmetric MA [of O(191 one has from Eq. (2.6a) b f A =fHsCfB(x)h6) ( X I (where B runs over the symmetric elements and C over the antisymmetric elements).

I 5 ~ h e analysis in this paragraph was done in collabora- tion with Dr. S. S. Chang.

''AS pointed out to us by R. Delbourgo, A) = 2 N b / 2 ~ w h e n Nb is even. Then the condition Af =2Nb has integer solutions for fi only when Nb =2, 4, and 8 (corresponding to N = 2 , 2, and 1).

1 7 ~ h e parity transformation in Majorana notation may be defined a s x' = -x" , on" = (cZY '8' while charge conjugation i s defined a s eaa = ( ~ ~ 8 ' ) a a . The properties of these transformations a r e discussed in more detail in Ref. 18.

"P. Nath and R. Arnowitt, Northeastern Univ. Report NUB No. 2302, 1976 (unpublished).

1 9 r = 0 is of course a solution of the spontaneous-symmetry-breaking equations (5.2) for A = 0. However, then the vacuum metric Eq. (3.2) no longer possesses a global supersymmetry invariance.

"The field equations being a p r i w i chirally symmetric, of course allow as many chiral right a s chiral left solutions for r p . Each possible solution, however, leads to a chirally asymmetric vacuum state.

'lone can find solutions where P C i s also spontaneously broken by considering possibilities with N > 4. These a r e currently being examined.

''The alternate possibility (6.3b) appears to be in gross violat~on of experiment a s it requires that the two states of the SU(2) doublet (e.g., v, and e for leptons have opposite chiral structure).

2 3 ~ o verify Eq. (6.6) one must take arbitrary linear combinations of the antisymmetric generators, and it is nontrivial that only two conserved quantities survive.

"s. Weinberg, Phys. Rev. Lett. 2, 1264 (1967); A. Salam, in Elementary Par t i c le Theory: Relat ivis t ic Groups and Analyticity (No be1 Symposium No. 8), edited by N. Svartholm (Wiley, New York, 1969).

251f we had chosen the alternate choice in Eq. (6.3a), r, =y,(ij +ij,.r,)P+, then w L would presumably be the heavier meson. That is, the argument does not explain why nature chooses V - A theory rather than V + A theory, but only that one o r the other should be domin- ant.

2 6 ~ general expression for the quadratic te rms in ha is given in Ref. 18. See also S. S. Chang, Phys. Rev. D g , 447 (1976).

'?B. Zumino, in Gauge Theories and hfodern Field Theory (Ref. 1).