PHYSICAL REVIE% D VOLUME 18, NUMBER 8 15 OCTOBER 1978

Qtslsstustt effeCtS On the Vaeuttsts gpfsttsstetrieS Of gauge SuperSymtstetry

Fran Nath and R. ArnowittDepartment of Physics, Northeastern University, Boston, Massachusetts 02115

(Received 3 April 1978)

Path-integral quantization of gauge supersymmetry is given which is manifestly consistent with globalsupersymmetry. The spontaneous-symmetry-brewing equations including quantum loop corrections todetermine the vacuum metric g o~a =(gg~iO) are deduced from the efFective potential. If solutionsexist for g „~which are globally supersymmetric, it is shown that a subgroup of Yang-Mills gaugesmust bepreserved. The full class of gauges t"(z) preserved under a globally supersymmetric gztsz& is determined.These are seen to be (i) the Einstein gauge, (ii) the preserved Yang-Mills gauges, and (iii) a supergravity-type gauge. All other gauges in t (z) are spontaneously broken. The analysis holds to arbitrary order inQtg

I. INTRODUCTION

Since the introduction of gauge supersymmetry, 'there has been a considerable amount of activityin the field of local supersymmetry. ' " Locallysupersymmetric gauge theories attempt to includeall the fundamental interactions of physics in asingle unified scheme. Thus the combination ofsupersymmetry and local gauge invariance leadsto exceptionally economical theories as the formerprinciple allows one to put all fields, Bose andFermi, into a single group multiplet, while thelatter transforms this into a gauge multiplet. Allfields become gauge fields. A framework thenexists where one may attempt to accommodategravitational, electromagnetic, weak, and stronginteractions. Gauge supersymmetry gauges super-symmetry by introducing a superspace

z"-=(x",e '}, tz=1, . . . , 4, a=1, . .. , N (1.1)of four ordinary space-time (Bose) coordinates x",and a set of N Majorana (i.e. , 4N fermionic} anti-commuting coordinates 8". (ct is the spinor indexand a the internal-symmetry index. ) The funda-mental gauge field is the single metric tensor su-perfield g»(z). Thus g„B contains all the fields(Bose and Fermi) of the theory. The gauge groupis the general coordinate group of superspace,z"=z" + $"(z) which leads to the gauge change

OgAB(z) Ag(c, B ( 1) $,AgcB g», c(

where the comma means right derivative, andtt = (0, 1) depending on whether A = (tt, a). Equation(1.3} represents the "group theoretical" unifica-tion of interactions, as these gauge transforma-tions automatically include Yang-Mills, Einstein,and supersymmetry transformations. If one fur-ther assumes that g» has an inverse, then thegauge invariance implies a unique set of (second

order) dynamical equations:

R» = )g», X= const, (1.3)where R» is the contracted curvature of super-space. Thus the gauge invariance then completelydetermines the "dynamical unification, " aside fromthe numerical value of the constant X [which hasdimensions of (mass}'].

One of the peculiarities of current ideas of thestructure of interactions is that it appears thatnature preserves some gauges perfectly [e.g. ,Maxwell and color SU(3)] while breaking othersto varying degrees (e.g. , weak-interaction gauges).This can be accounted for phenomenologically indifferent models by appropriately chosen Higgs po-tentials. However, the basic physical principlesthat determine the Higgs dynamics in conventionalYang-Mills models are at present unknown. It is,however, the unique feature of gauge supersym-metry that the gauge invariance uniquely deter-mines all the dynamics, and hence Eq. (1.3) mustalso include the Higgs interactions. Thus thetheory must determine without additional assump-tions which of the vector superfield of gauges $"(z)are preserved and which are broken. This ques-tion has been considered before, ' though only atthe tree level and only to the lowest orders in 8 'in the superfield expansion of ("(z). We presenthere some general theorems, valid with pll quan-tum corrections included, and analyzed for thefull superfield expansion of g»(z) and $"(z). Thefirst result concerns the vacuum expectation valueof the metric g„"B'=-(0~g»(z) ~0) which arises as aconsequence of spontaneous symmetry breaking ofgauge supersymmetry. We will show in Sec. IIIthe following:

1. Any solution of the spontaneous-symmetry-breaking equations for g„"~' which maintains globalsupersymmetry must also preserve a nonemptyset of Yang-Mills and/or Abelian internal-sym-

18 1978 The American Physical Society

2760 PRAN NATH AND R. ARNOWITT 18

metry gauges. Then when gauge supersymmetryspontaneously breaks to global supersymmetry, "there is automatically a set of preserved gauges,and the theory offers, for example, a possiblenatural explanation of the existence of quantumchromodynamics as a consequence of a more fun-damental structure.

Gauge supersymmetry contains a priori a hugenumber of gauges in the superfield expansion of)A(z). If the theory is to make physical sense,after spontaneous breaking not only must there bepreserved gauges to account for Maxwell andSU(3}' invariance, but also all other unwantedgauges must break. In this paper a f44ll analysisof this question is given when the spontaneousbreaking preserves global supersymmetry. Thegeneral form for a globally symmetric g„"~' iswritten down in Eq. (3.8) below. One may thenverify what gauges preserve this vacuum metric,i.e. , for what )A(z) does 5g„"B vanish under Eq.(1.2}. The following (group theoretical) result isestablished in Sec. IV below:

2. For a spontaneous-symmetry-breaking solu-tion which preserves global supersymmetry, onlythe following gauges of )A(z) are not broken: (i)the Einstein gauge of general relatively [arisingfrom the 8-independent part of $'(z)], (ii) a setof Yang-Mills gauges [arising from the linear 8 'part of $ '(z)], and (iii) A supergravity-type gauge.The above result shows that spontaneously brokengauge supersymmetry has precisely the type ofgauge structure can observes in nature. Thusthere are a large number of Einstein-type gaugesfrom higher 8" sectors in f'(z), but all these arebroken. The preserved Yang-Mills gauges arethose of the first result discussed above (whichwas a consequence of the dynamics of spontaneousbreaking). The existence of a supergravity-typegauge shows that at least group theoretically, su-pergravity is embedded in gauge supersymmetry. "It should be stressed that the above analysis isquite general, valid to all orders in 8 '.

Associated with each spontaneously broken gaugeis a fictitious Goldstone field in g„z(z) whose ab-sorption gives rise to mass growth in the brokengauge field. The full analysis of this question issomewhat extensive and we defer the discussion toa second paper. " It is shown there that the fieldsof gAB(z) can be divided as follows: preservedgauge fields, broken gauge fields, fictitious Gold-stone fields, and Higgs fields. A complete analy-sis is carried out to arbitrary order in 0 ', andthe different type of field in each 8 sector identi-fied. This analysis represents the first step inobtaining the full particle content of gauge super-symmetry. It also completes the picture of spon-taneous breaking within the framework of global

supersymmetry, and shows that gauge supersym-metry has a normal Goldstone structure.

In order to carry out the analysis of this paper,it is necessary to show that one may quantizegauge supersymmetry in a fashion consistent withglobal supersymmetry. Such a path integral quan-tization is carried out in Sec. II. Conclusions aregiven in Sec. V where some comparisons betweenthe different types of local supersymmetry the-ory"" are discussed.

II. FUNCTIONAL INTEGRAL FORMALISMFOR GAUGE SUPERSYMMETRY

While a canonical quantization of gauge super-symmetry has not yet been carried out, quantiza-tion using the functional integral method proceedsin a straightforward fashion. " The formalism issimilar to the one used in globally supersymmetricgauge theories except here the analog of theWess-Zumino gauge" where normal quantizationwork does not exist, and the fermi fields are truegauge fields.

The metric superfield g„B(z) has an expansion in6) of the form

gAB(Z) —gAB(X) + gABa~(X)8

+ g(4d)) (X)8a). . .8a4BABoy. a4N (2.1)We define the set of ordinary space-time fieldvariables y, (x) to be

4~.(X6=(g A.'., .„(X8"The vacuum-to-vacuum transition amplitude Z isgiven by the functional integral"

(2.2)

Z = dgggd'ggdg *e ' (2.3}

where dg» is short for mdy„g„and g"* are Fad-deev-Popov ghost superfields, and the total actionI consists of

I=I,,„+I,+I, .Here I„„(g„B}is the classical action'

l„„(d„)=f d d —d [R+ (4N —2)X],

(2.4)

(2.5)

g„",' = &o~ g, (z) ~ 0& (2 5)

exists when spontaneous symmetry breaking oc-curs. A convenient gauge choice is the analog

IC is the gauge-fixing term, and I~ the correspond-ing ghost action.

To proceed further, one must specify the gaugecondition. We wish to do this in a fashion suchthat Z is manifestly consistent with global super-symmetry, assuming that a globally supersym-metric solution for

QUANTUM EFFECTS ON THE VACUUM SYMMETRIES. . .

of the DeDonder gauge condition in Einstein the-ory, where g„(o~) is introduced as a backgroundfield. %e have then for the gauge function C„ theform

A gAB)cg &( ) (gzcg ) IA q (2 I)

where g'""~ is the inverse of g„"~, and the notation"1A" means covariant differentiation with respectto the background metric g~(~'. The contributionto the action is

symmetry-breaking conditions is to use the generaleffective potential formalism. The total action,I(g» , qA. , ))A*; g„"B') of Eq. (2.4) is a function ofmetric superfield g», the ghost fields q„, q~*and depends parametrically on the quantity g„"3'(which we wish to evaluate). The general analysisof Jaekiw extends starightforwardly, and one maydefine the effective potential I'(gAB., g„"B') (whichalso depends parametrically on g„"B') whose ex-tremum determines g» = g„(o~'.

d~C" ~ C„~; C"=-g'" 'C, .

The verification that this gauge function fixes thegauge is given in the Appendix.

To calculate the I addeev-Popov ghost action, onemust make a gauge transformation on C„(g):

5I'

Qg — (o)~ ~AB ~AB

By straightforward analysis one finds

(3.1)

(3.2)

c„(g')= c„(g)+hf„,~B. (2.9) where I is the total action.S' is a solution of the

functional equationThe F. -P determinant arising from M» can be ex-ponentiated using the ghost superfields qA(z),q"*(z), whose integral spin components are treatedas fermions, and half-integral spin componentsas bosons in the functional integral. One finds

Ig dz g )zd g (g g)I gg

-'(-1)"(&(B;c)g ' ))Al

)q(d„,)=- ) fdq„,dq„dq"' dq()), (q. q )

where

Aw. (OhI g», &», n~, n(o)

I(gAB AB) )A) ) ) gAB)

AB~ A & AB» ~gwhere

«A;B) =- &A;B+ (-»"""&B.A

(2.10)

(2.11)

ln Eq. (3.3), one has made the usual translation inthe integration variable: g»= g»+h». As shownby Jackiw, "this implies

and ";8"means covariant derivative with respectto the full metric g», as usual.

%'e have not as yet given equations to determineg~~'. This will be done in the next section wherespontaneous symmetry breaking is considered.One may thus think of Z as depending parametri-cally on g„"B'. However, it is clear from Eqs. (2.8)and (2.10), that if the spontaneous-breaking equa-tions yield a globally supersymmetric g(„o~), thenZ is globally supersymmetrie. Thus the quantiza-tion scheme is manifestly consistent with globalsupersymmetl yy as desired.

III. SPONTANEOUS-SYMMETRY-BREAKING CONDIONS

In this section we examine the conditions im-posed upon the vacuum metric g„"B' of Eq. (2.6) dueto spontaneous symmetry breaking. %e will thenassume that the solution for g(„o~) including quantumcorrection is globally supersymmetric and seethat this implies the existence of preserved Yang-Mills gauges, as discussed in the Introduction.

A convenient way of obtaining the spontaneous-

1dhABdriAdqA~e' h„B(z)=-0z (3.4)

(provided O'I'/5g' is nonsingular) and so Eq. (3.4)reduces to (01h»(z)10) =0 when the extremumcondition Eq. (3.1) is imposed. This verifies thatEq. (3.1) correctly determines g» to be ~0 Ig„B10)

One may differentiate Eq. (3.2) and set g» equalto g„'~' to deter mine the equation to be satisfied byg„"B'. Using Eq. (3.4) one finds

G" (d')+(Z)' Jdq dq dq* d'„x [G (g'+h) G"'(go)]=0, (3.5)

where G = 5I/5g» and Z is the same functionalintegral as 8 with I replaced by I. The firstterm of Eq. (3.5) is just the tree approximation,and the remainder is the quantum loop corrections(and may be calculated in the usual way).

We now assume that the solution of Eq. (3.5)maintains global supersymmetry. The invarianceof the vacuum state 10), then requires that 5g„"B'vanishes for the global supersymmetry transfor-mations generated by the constant spinor X:

FRAN WATH AND R. AR50% JTT

("(z)= i)&.r "e,From Eq. (1.2) this implies

(0) ~o 0 (o)~A + +(o) ~a p

(o) g (o) yg ).~&,ai+g a,.& =0.Equation (3.7) has the general solution

(3.6)

(3 'I)

[r„r,]= 8 K~,tr [I"(,)'+ 1"(,&'] = X —)) ' .

(3.14b)

(3.14c)

Q) Q)Tf p I ( ) p T (3.15)

The I' „,& can be expanded in terms of the (sym-metric, antisymmetric} matrices of the internal-symmetry space:

&0&~

)0&

~.",'=(n~). ,+ («.).(«)„ (3.8)where I"„and K are matrices in the Dirac and in-ternal-symmetry space with symmetry properties

(~r„);,=(~r.).„(m).,= -(n~)., (3.8)Proper Lorentz invariance implies I"„=y„I and

I -=[r&„+)&"r, &], z =I&.,+)y'I)..„ (3.10)where r&, , & are real (symmetric, antisymmetric}matrices Rnd Ko ) reRl symmetric n1Rtrlces ln theinternal-symmetry space. By a linear transfor-mation in the e space, e '=(Me')™,it is alwayspossible to reduce K to unity, and we will assumeE = I in the following. "

The type of global supersymmetry is thus deter-mined by the nature of the internal-symmetrymatri. ces I, which are in turn determined by solv-ing Eq. (3.5). The quantum loop corrections arenontrivial to calculate. However, the quantizationprocedure of Sec. II guarantees that they alsopossess global supersymmetry. Calling the func-tional integral in Eq. (3.5) q», global supersym-metry requires that it has the same general formas g„"s& in Eq. (3.8). Thus one may write

q„„=)'))„„,q„.=&'(-e'er. ). ,q.,=())z')+ ))'(er„).(er")„ (3.11)

where (riK') = -()}K') and I'„ is the same matrixas in Eq. (3.8). ))' and K' can be expressed interms of the functional integral of Eq. (3.5) andso

X'=))'()) I' ) X'=Sr'()) I ) (3.12)

r„r"= —,'(x z ),trl', I'" = -4{&)—X') .

This reduces to

r(, ) + r (, ) = &&(). -Ko),

(3.13)

(3.14a)

Thus when global supersymmetry is maintained,g/l quantum loop corrections can be summarizedin terms of one constant A. ' and two symmetric in-ternal symmetry matrices: K' =Ho+

i@�'E,

'. Insert-ing Eq. (3.8) (with K= 1) and Eq. (3.12) into Eq.(3.5}, and using the tree expressions for the firsttwo terms' gives the following equations for I'„:

where the N' constants (a„p„jare, of course,just the vacuum expectation values of the Higgsfields for the theory. However, Eqs. (3.14a) and(3.14b), being symmetric in the internal-sym-metry space, each contain ,'N(X+ 1—)equations andso Eqs. (3.14} represent N(N+ 1)+ 1 equations tobe satisfied for the X' vacuum expectation values.This is clearly impossible unless some of theequations are actually empty, or equivalently re-dundant. Since the redundancy is independent ofthe matrix x'epresentation of T" ~, it implies thatit must be a consequence of an internal groupsymmetry.

We thus arrive at the following basic result:The only solutions of the spontaneous-symmetry-breaking equations (3.5) which are globally super-symmetric, are those which rigorously preservean internal- symmetry subgroup.

The precise nature of the preserved internal-symmetry group of course requires a calculationof 1' and E' so that explicit solutions of Eqs. (3.14)can be obtained. However, in the next section wewill show how to determine what the internal-sym-metry gx'oup is once R solution fox' I ls obtained.This allows a discussion of possible phenomeno-logical aspects of gauge supersymmetry which arenot dependent on the details of the loop calcula-tions.

In the tree approximation, Eqs. {3.14} imply thatN = 2 if X 4 0.3 Kith quantum corrections, this isno longer necessary. Thus taking the trace of Eq.(3.14a) and comparing with Eq. (3.14c) yields

X=2()) —1')/(1 SC,',), (3.16)~here K,', =-(trÃ,')/N. Only if X' and K,', are "a.c-cidentally" equal will N equal 2. This allows thepossibility of attempting to build realistic modelsfor the XW 0 case. %e are currently examining the%= 9 and K= IO cases, which are the lowest valuesof N such that O(N) contains the minimal physicallyacceptable group SU(2) i xUr(I) x SU(3)'.

As in any field theory, one cannot rigorouslyknow a priori if the full quantum effective potentialpossesses a minimum and whether all the [n„P„].are determined since Eqs. (3.14) are nonlinear al-gebraic equations. In the tree examples of Ref.3 Sec. V. some of the parameters are left unde-termined. It is quite possible that loop correctionswill allow one to eliminate this arbitrariness of the

QUANTUM EFFECTS ON THE VACUUM SYMMETRIES ~ « ~

vacuum solutions. However, if it should turn outthat not all the fn, , p,] get determined by Eqs.(3.14), it would imply that a spontaneous breakingthat preserved global supersymmetry does notpossess Rn absolute minimum of the effective po-tential and one may be achieved only aftex globalsupersymmetry (dynamically) breaks (which isphysically necessary anyway).

(o)g(.A&,,)= 0g»&, e+ &,.gAe+g. e, A& -0»(O) A A (o) (o)

(0) A (0)g feA~, B] g eB,A~

where g„"~' is given by Eq. (3.8} (with X=1}. Equation (4.1) has the general solution

(4 I)

(4.2)

(4 3)

5.( )=8«(e)+ «.„(e)»" «..= -«.

The spontaneous symmetry breaking consideredin the previous section causes the simultaneousspontaneous breakdown of most of the gauges ofgauge supersymmetry. As remarked in the In-troduction, this is physically desirable, since veryfew gauges are observed in nature to be pexfectlyconserved. In the previous section, the dynamicsof spontRneous breakdown wRS coQsldex'ed. In thissection we consider the corresponding group theo-retical question: Given a solution of Eq. (3.5) forI @ %'hlch px'csex'ves global supersymmetry» whichof the gauges $"(z) are broken, and which are pre-served'P

%'e discuss this question in two stages. Firstwe detel'nlllle which $ (g) leRve the g~e of Eq(3.8) invariant: 5g„"8'=0. This set of $"(8) in-cludes of course the global supersymmetry trans-formation Eq. (3.6) [by the analysis which leadsto Eq. (3.8}l plus ail the other global transforma-tions which are invariances of the vacuum state~0). [Geometrically, the elements of this invar-iant set of ("(») are the superspace Killing vec-tors of the space defined by g~ol. ] We then exhibitthe fields in g~(z) which transform in conventionalnon-Abelian fashion under the local gauge generali-zation of the invax'iant global set,. These are thepxeserved gauge fields. The txansformations whichdo not preserve the vacuum metric are, of course,the broken symmetries. Each of the correspondinggauge fields for these should have an associatedflctltlous Goldstolle field 111 gpss(g) whose Rbsol'p-tion allows mass gxowth. This is explicitly ex-hlblted 1Q R second paper» w'here» Rs required»it is also seen that the preserved gauge fieldsabove have no associated fictitious GoldstonefieMs.

From Eq. (1.2), the requirement 6g„"8'=0 im-poses the following conditions on $"(»):

=0, E „=0 (4.V)from the linear and higher powers of g" in Eq.(4.2'), while the remaining equation (independentof »") reduces to

« „. 1«„„—(er"). 2i(~-r.).= 0 (4 8)EquRtioll (4.V) SRys that « „„ls illdepelldent, of 8(and hence generates a homogeneous I.orentz trans-formation) and all the higher»" terms in Eq. (4.6)are zero. One may simplify Eq. (4.8) by making thetransformation

(4 8)

where o""=-—,i[@",y"]. Then «„'obeys

« ', ,—2i(('r), = 0.In order that a solution for « „'(8) exist, Eq. (4.10)must be consistent with the integrability condl-tlOQ & ~ eB= -g ~ Be RJld henCe

While Eq. (4.V) says that f, ' is independent of »",it is a pnori an arbitrary function of 8:

]t(g) g (i(n) eaq, ,', eu„ (4.12)where the coefficients P, ""'are antisymmetric ina„.. . , n„. To learn further about the propertiesof &'(8) we make use of the remaining conditionEq. (4.3). Inserting in Eqs. (4.4), (4.V), and (4.8)into Eq. (4.3), one finds after some calculation thesimple result

k', B —$B, (4.3')

The antisymmetry of n, n„ in Eq. (4.12) thenimplies that $' is at most jrjnegy in 8 and that

(4.13)

(4.5)

and Bose indices are raised or lowered with g'„'„'= Ii„„. The content of Eq. (4.2) is most easily ob-tained by expressing it in terms of P, „:

—ig„„(8r"),-2i()r„) + $, ,=0, (4.2where E, (z) =- (~(z)I)~ . One next expands ], in apower sex'les 1Q g» l.e. »

t(») = ~"'(8)+ t.„'"(8)»"+-'. t'„'„'(8)»"0+ ~ ~ ~ (4.6)

and compares coefficients of different powers of»" in Eq. (4.2'). The requirement that I' have aninverse to represent a valid global supex'sym-metry" then leads to

FRAN NATH AND R. ARNOT ITT

[I'„,M] =0. (4.14)

where X', is a constant spinor and M,",are anti-symmetric matrices in the internal-symmetryspace. Inserting Eq. (4.13) into the integrabilitycondition Eq. (4.11) leads to the additional condi-tion on M:

could be eliminated by choice of gauge, sincesuch fields transform inhomogeneously.

The general treatment of the Goldstone fieldsis given in Ref. 18. %e consider here the pre-served gauges, and consider first the internalsymmetry transformations of Eq. (4.16). Thelocal gauge extension of these are

With these constraints on M Eq. (4.10) can nowbe integrated to yield $

'=-X„(x)(T„8) ', +=0, (4.18)

&„=~ &'&+ 2&Yr„e+~eMr„e, (4.15)

where @&a' is a constant (the Poincare transla-tions).

%e now assemble all the results using Eqs.(4.5), (4.9), and (4.15) to obtain all the trans-formations $"(z) which leave the vacuum metricg gg lnvarlant;

~«(z) =[8«&"+ z „x"]+

iver'8,

$'(z) =[,'ie —(o""8)] X+ —(M8),(4.16}

where M obeys Eqs. (4.13) and (4.14). We seefrom Eq. (4.16) that the invariances of the vacuumstate consist of (i) Poincare transformations {si-multaneously in the x" and 8 space), {ii) globalsupersymmetry transformations generated by A. ,and (iii) linear orthogonal internal-symmetrytransfoxmations in 8 space generated by M =M,'"+iy'M&&" obeying" Eq. (4.14). The above discus-sion shows that this exhausts the set of invariancesof the vacuum. All other transformations, e.g. ,those of higher order in 8 are spontaneouslybroken for a globally symmetric vacuum. Further,givell a solution of Eq. (3.5) fol' I «~ oils call detel'-mine from Eq. (4.14) which internal-symmetrytransformations are preserved. As was discussedin Sec. III, a set of preserved symmetries mustexist if dynamically solutions for I'„are to occur.

Each of the preserved global symmetries cannow be generalized to a Local symmetry which arethe preserved gauges. To see this let us insert

gAB gAB + AB { (4.17)into the right-hand side of Eq. (1.2). Here &ABcontains the dynamical fields. If 5"(z) is a guagetransformation whose global transformation is pre-served by g'~~~, then the tex'ms that survive in6g„~ are of two types: those involving bose grad-ients of g", i.e., of typeg~c$~ „and homogene-ous structures of type A„gy ~$ n These two struc-tures are precisely the terms one would get in aYang-Mills transformation. On the other hand,for the broken $ "(z) whose global analog does notpreserve g~~), there will be additional inhomogen-eous structures linear in $"(z) of the formg„"~$ and g'~ ~( . These terms indicate thath„~ also con4, infictitious Goldstone fields which

5a "=8 X„(x)-f, X~„', (4.20)where the Yang-Mills vector meson fields 8„'(x)enter in h„,(z) as (8T„F)+," [F{x)is the {matrix)field entering ing, z(z) as (&)E(x)}~]. Thus 8„"(x)are the gauge mesons of the preserved Vang-Mills gauges. There are no fictitious Goldstonebosons associated with these fields.

Next we generalize the Poincare translationsto the local Einstein gauge transformations:

$«(z)=$«(x), $ '=0.One finds from 5g (z) that"

5gz(x)=(.„+~„.„,

(4.21)

(4.22)

wllel'8 'tile El&lstelll field g (x) ls the 8-indePendentterm of g~(z). The I orentz transformations inthe 8 space of Eq. (4.16) generalizes to local vier-bein rotations (we use m, n for local vierbein in-dices):

] '=-'iz (x)(o "8} ' $"=0 e = z -(423)Examining, for example, 6g„under Eq. (4.23) al-lows one to determine the position of the vierbeinfie d B (x) and the vierbein affinity &B„(x)in themetric. These arise ing„{z) in the term (8A„)where' s

A, = il™e„(-x)--,'ia "&B (x). (4.24)Under Eq. (4.23) one finds the B„„and &B cor-rectly transform as'9

Iz~ffm &

6&Bmn«= Zmn. «+ [&B n«Zp + &dm «Z&n].(4.25}

Finally, we examine the last of the invax'iantgauges, the local generalization of global super-symmetry. To linear order this is:

~"= iX(x)I'"8; a ) a(X) (4.26)Examining 6g„„(z)under Eq. (4.26) allows one to

where T„are the independent generators of O(N)x O'(N) obeying Eqs. (4.13) and (4.14) and X„(x)are the gauge functions. The gauge transformationof g, (z) is

(4.19)

which leads to

18 QUANTUM EFFECTS ON THE VACUUM SYMMETRIES. . . 2765

find the supergravity-type spin--, field in g,„(z) inthe term i8I', „g„&(x). Under Eq. (4.26) one finds,to linear order, the supergravity transformationlaw" |i(:„'(z)=28„X'(z). Thus a supergravity-type gauge is embedded in gauge supersymmetryeven zoithout going to the limit'4 K-O.

V. CONCLUSIONS AND COMMENTS

In this paper we have considered some of theproperties of spontaneously broken gauge super-symmetry for the situation where the spontaneousbreaking maintains global supersymmetry. (Thusin gauge supersymmetry, global supersymmetryreemerges as a property of the vacuum state,rather than of the tangent space. ) Under thesecircumstances it has been possible to analyze whatgauges spontaneously break and which are pre-served. It was seen that the only preserved gaugeswere the Einstein gauge, a set of Yang-Millsgauges, and a supergravity-type gauge. Thus thetheory has naturally produced a pattern of con-served and broken gauges not dissimilar to whatone finds in nature. " For each broken gauge onemay explicitly find a fictitious Goldstone boson,whose absorption by the gauge field will give riseto mass growth. ' Thus the general propertiesof gauge supersymmetry in these matters is thatof a "normal" field theory. Further, the theoryappears to be free of a cosmological term (in ac-cord with experiment) while simultaneously pos-sessing minimal couplings of the conserved Yang-Mills gauges [e.g. , Maxwell or SU(3)'].

The dynamics of gauge supersymmetry is uniquebecause of two elements in the theory: The re-quirement of invariance under the general super-space group, and the requirement that g» have aninverse. One is then led to Eq. (1.3). It is thesethat produce a determination of the Higgs dynamicsand hence a determination of the symmetry break-ing. The fact that g» is nonsingular is also cru-cial in the appearance of the minimal couplings.The minimal couplings are proportional to K ofEq. (3.8), and they would vanish in the singularlimit' K-0. Thus the existence of an inversefor g» is fundamental for obtaining the advan-tageous aspects of gauge supersymmetry.

As we have seen, spontaneous breaking playsa rather fundamental role in gauge supersym-metry, and the physical interpretation of thetheory can only be made after spontaneous break-ing. In this context, quantum phenomena appearto play a crucial role and gauge supersymmetryappears to be a truly quantum theory, just astheories of confinement" are intrinsically quantumtheories. The phenomena of spontaneous break-ing has a direct bearing on the question of particle

content, and in particular quantum effects are ex-pected to be very important, as they will fill inthe many zero entries of the tree-approximationkinetic-energy and mass matrices. One of thedifficulties of gauge supersymmetry is that, whileit is based on only a single gauge supermultipletof fields, the number of fields contained in g»(z)is very large. On the other hand, it appearsphenomenologically that in the low energy domain(i.e. , energies ~10' GeV) the dynamics is governedby only a few constituents such as leptons, quarks,and gluons. While it is somewhat comforting thatthe analysis of the preserved gauges does show thatobjects such as these do indeed exist in gauge su-persymmetry, g»(z) contains many other addi-tional fields. The physical significance of theseis at present unclear. However, it is conjectoralto regard the energy region between 10' GeV and10" GeV (which is the domain of validity of a unified scheme) to be structureless. It is not unreasonable that there is structure in going from thelow energy to the Planck mass energy, but thisstructure is suppressed in the domain of energiesof «10' GeV where most of the current phenomen-ology of particle physics is centered. In gaugesupersymmetry such a desired suppression ofcomponents of the superfield g» could come aboutthrough the instrument of spontaneous breakdownwhich involves a superheavy mass M~ -10"GeVas one of the parameters in the breakdown. Thegeneral theorems contained in this paper and Ref.18 allow one to proceed now to examine phenomeno-logically the possibility of a hierarchy of interac-tions in gauge supersymmetry. "

If such an interpretation of the additional fieldscontained ing»(z) is valid, there still are, how-ever, a number of important theoretical questionsthat remain unsettled. There is no a priori princi-ple that guarantees that the theory is free of ghostsor tachyons. As discussed above, quantum cor-rections are needed to settle this question, and itseems possible now to actually perform them, us-ing the globally symmetric quantum theory of Sec.II. The theory also contains higher (o —', ) spin fields.However, it is remarkable as we have seen in Sec.IV, that none of these higher spin fields involvepreserved gauges, and hence they should becomemassive, and possibly superheavy.

An alternate scheme of locally supersymmetrictheories is SO(N), ¹ 8 extended supergravity. 'Here, in contrast to gauge supersymmetry onestarts out with a well-defined particle content,which is just a global SO(N) multiplet. The theoryis automatically ghost free. Supergravity is trulyremarkable in that there is finiteness at least atthe one- and two-loop levels. However, atpresent,the theory fails to give a physically acceptable uni-

2 i'66 PRAN NATH AND R. ARNO%ITT 18

fication of interactions. On the dynamical level,supergravity leads to experimental contradictioninvolving the cosmological constant and the vec-tor-meson gauge-coupling constant. Thus, if onelimits the cosmological constant to be less than thepreaent experimental upper limit, then the gaugecoupling becomes negligibly small, i.e. , in effectthere are no minimal electromagnetic couplings orcolor-gluon couplings in the theory. Further, onthe group-theoretical level, SO(N) supergravitywith N & 8 is inadequate for building realisticmodels involving electromagnetic, weak, andstrong interactions since even the minimal modelfor such interactions, i.e. , SU(2) && U(1) && SU(3)'cannot be accomodated within this framework. "This problem might be overcome, of course, byconsidering 1V&8 but then higher spins s ~ -', mustbe accommodated consistently within the super-gravity framework.

The most recent variant of local supergravity isbased upon gauging the superconformal and ex-tended superconformal algebra. "' Here theinternal symmetry group is U(N) [rather than theO(N} of supergravity) and so can now accommodatethe desired group of weak, electromagnetic, andstrong interactions. Further, being conformallyinvariant, there is no cosmological constant.However, this theory, being conformally invariantinvolves the Weyl rather than the Einstein gravi-tational theory. Further, while minimal couplingscan occur, they are not obviously of the right phys-ical type, '~ and the quadratic parts of the Lagran-gian possess ghosts. " Very possibly, the physicalsiginificance of conformal supergravity will be-come clearer if there is a spontaneous breakdownof the conformal invariance.

This research was supported in part by the Na-tional Science Foundation. Part of this work wasdone while the authors attended the Workshop onSupersymmetry, Aspen Center for Physics, 1977.

APPENDIX: GAUGE-FIXING CONDITION

IN GAUGE SUPERSYMMETRY

Here we show that in gauge supersymmetry itis always possible to set up the globally super-symmetric deDonder gauge defined by the condi-tion

C„—= hAB ICg"' —2(-1)"(hBCg"' ) I„=0.(Al)

To prove that this can be done, one shows thatstarting in an arbitrary gauge with fields hA'B thereexists a gauge transformation defined by $" [see

Eqs. (1.1) and (1.2)] which transform h„'B to hABwhich obey Eq. (Al}. We shall establish our re-sult only to linearized order in (A though it mayeasily be extended at least in a perturbation fash-ion to the full form where one retains the nonlinearterms as well. To linearized order we have

~AB ~AB ~A I]B ~ah ~B IA y (A2)

where q„=(-1}""b.Inserting Eq. (A2) into Eq.(Al) we have that $A obeys the condition

A (~AIBC Ibb~B IAC4

[(~B I C Ibc~CIB)g ]IA(-1)" (0)CB

where C„' is just the function CA with h» replacedby hA'B and does not vanish in general. Equation(A3) may be simplified to read

&AiB + CA- o, (A4)

where $A]B )A[Beg'" involves the superspaceD'Alembertian. Intr oducing the globally covariantderivative

D, = a. —i(I7y'8) a, ,we may write $A]B in the form

=( DD+Cl'))A+affinity terms.

(A5)

(A6)

One may now expand $A in a power series in 9:i„(B)=g ~„.,. . ..(x)a " e". (A 7)

n (&)— j r. (0)DC2n AB ] CDg (A8)

Equation (A7} shows once again that one has aninvertible kinetic energy and mass operator as isdesired. One abvious drawback of the deDondergauge, however, is that all fields appearing in Eq.(A8) involve two Bose derivatives. Consequentlythe existence of the Dirac equations for the spin--,'particles is masked in this gauge. However, thefirst order Dirac equations are actually containedin the gauge condition C =0 itself. Thus the Diracfields that are thought to represent the quarks andleptons arise from the quadratic 8 parts of h „in terms of the form B,8iy„g From Eq.. (Al) onesees that C contains the structure

(0)IAV g g u 8 (A 9)

which is the Dirac equation differential operator.

Equation (A4) then can be seen to have solutions forthe coefficients $A, ... (x) since the operator 0'ADIT' ' CR~—DD has an inverse.

The linearized equations in this gauge arisefrom the linearized part of RAB'.

18 QUANTUM EFFECTS ON THE VACUUM SYMMETRIES. . . 2767

'P. Nath and R. Arnowitt, Phys. Lett. 568, 171 (1975);R. Arnowitt and P. Nath, Gen. Relativ. Gravit. 7, 89(1976); P. Nath and R. Arnowitt, J. Phys. (Paris) 37,C2 (1976); P. Nath, in Gauge Theories and ModernEield Theory, proceedings of the conference, Boston,1975, edited by R. Arnowitt and P. Nath (MIT Press,Cambridge, 1976).

R. Arnowitt, P. Nath, and B. Zumino, Phys. Lett. 568,81 (1975).

R. Arnowitt and P. Nath, Phys. Rev. Lett. 36, 1526(1976); Phys. Rev. D 15, 1033 (1977); S. S. Chang, ibid.14, 447 (1976); P. Nath and R. Arnowitt, Nucl. Phys.8122, 30 (1977).

4R. Arnowitt, inDeeper Pathways in High Energy Phy-sics, proceedings of Orbis Scientiae, Univ. of Miami,Coral Gables, 1977, edited by A. Perlmutter and L. F.Scott (Plenum, New York, 1977); and in Proceedingsof the Iifth International Conference on ExperimentalMeson Spectroscopy at Boston, 1977, edited by E. vonGoeller and R. Weinstein (Northeastern Univ. Press,Boston, 1978); P. Nath, Northeastern Univ. ReportNo. NUB 2329 (unpublished).B. Zumino, in Gauge Theories and Modem +ieldTheory, proceedings of the conference, 1975, editedby R. Arnowitt and P. Nath (see Ref. 1); J. Wess andB. Zumino, Phys. Lett. 668, 361 (1977).

~D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferr-ara, Phys. Rev. D 13, 3214 (1976); S. Deser andB. Zumino, Phys. Lett. 628, 325 (1976); also seeDeeper Pathways in High Energy Physics (Ref. 4).

~M. T. Grisaru, P. van Nieuwenhuizen, and J. A. M.Vermaseren, Phys. Rev. Lett. 37, 1662 (1977);S. Deser, J. H. Kay, and K. S. Stelle, Phys. Rev.Lett. 38, 527 (1977).

P. Nath and R. Arnowitt, Phys. Lett. 658, 73 (1976).M. Gell-Mann, talk at Orbis Scientiae, Coral Gables,1978 (unpublished); P. Breitenlohner, Phys. Lett.678, 49 (1977); Nucl. Phys. 8124, 500 (1977) .D. Z. Freedman and A. Das, Nucl. Phys. 8120, 221(1977); S. MacDowell and F. Mansouri, Phys. Rev.Lett. 38, 1433 (1977).S. Deser and B. Zumino, Phys. Rev. Lett. 38, 1433(1977).V. P. Akulov, D. V. Volkov, and V. A. Soroka, Zh.Eksp. Teor. Fiz.-Pis'ma Red. 22, 396 (1975) [JETPLett. 22, 187 (1975)]; M. H. Friedman and Y. N.Srivastava, Phys. Rev. D 15, 1o26 (1977).

' A. Ferber and P. G. O. Freund, Nucl. Phys. 8122,170 (1977);J.C. Romano, A. A. Ferber, and P. G. O.Freund, ibid. 8126, 429 (1977).

4M. Kaku, P. K. Townsend, and P. van Nieuwenhuizen,Phys. Lett. 698, 304 (1977); Phys. Rev. Lett. 39, 1109(1977); S. Ferrara, M. Kaku, P. K. Townsend, andP. van Nieuwenhuizen, Nucl. Phys. 8129, 125 (1977);M. Kaku, P. K. Townsend, and P. van Nieuwenhuizen,report January, 1978 (unpublished).

'~For a more complete list of references see P. Fayetand S. Ferrara, Phys. Rep. 32C, 5 (1977); in Pro-ceedings of Orbis Scientiae, Coral Gables, 1977 (seeRef. 4); B. Zumino, in Proceedings of the 1977 Euro-pean Conference on Particle Physics, Budapest, edited

by L. Jenik and I. Montvay (CRIP, Budapest, 1978).' The existence of spontaneous-symmetry-breakin0;

solutions that maintain global supersymmetry at thetree level was demonstrated in Ref. 3.

' Presumably, the supergravity gauge will not be pre-served when global supersymmetry is broken at thenext stage symmetry breaking.

' R. Arnowitt and P. Nath, Northeastern UniversityReport No. NUB No. 2344 (unpublished).Similar results have been independently obtained byJ. G. Taylor, Kings College report, 1977 (unpub-lished).A. A. Slavnov, Teor. Mat. Fiz. xx, xxx (pox) [Theor.Math. Phys. 23, 3 (1975)];Nucl. Phys. 897, 155 (1975);8 de Wit, Phys. Rev. D 12, 1628 (1975); S. Ferraraand O. Piquet, Nucl. Phys. 893, 261 (1975); J.Honer-kamp, F. Krause, M. Scheunert, and M. Schlindwein,Nucl. Phys. 895, 397 (1975).J. Wess and B. Zumino, Nucl. Phy's. 878, 1 (1974).

2~Equation (2.3) omits a possible functional measurefactor which requires a canonical quantization toevaluate. The determination of the functional measureis generally complicated and even for quantum gravityhas only been evaluated rather recently; see in thiscontext M. Kaku and P. Senjanovic, Phys. Rev. D 15,1019 (1977); E. S. Fradkin and G. A. Vilkovisky, ibid.8, 4241 (1973); H. Leutwyler, Phys. Rev. 134, 81155(1964).

23R. Jackiw, Phys. Rev. D 9, 1701 (1974}.4The singular limit K- 0 has been shown to be closelyrelated to supergravity theory. See Ref. 8.R. Haag, J. T. Lopuszanski, and M. Sohnius, Nucl.Phys. 888, 25 (1975).That subgroups of the orthogonal groups O(N) & 0 (N)obeying Eq. (4.14) were symmetries of the vacuummetric was previously derived in Ref. 3. See alsoP. G. O. Freund, J. Math. Phys. 17, 424 (1976).

2~See second paper of Ref. 3, Sec. II.~ One may write in general e~~ = g „+h „(x). In a local

geodesic frame, e „=g~„and QOA„)o is just the vacuummetric g„'o . Thus gravity and supersymmetry arewelded together in the vierbein field e~(x).See the last paper of Ref. 3.

3 See also Ref. 8 which discusses the dynamics in theK—0 limit.

~C. G. Callan, R. Dashen, and D. Gross, Phys. Lett.638, 334 (1976); 668, 375 (1977); Phys. Rev. D 17,2717 (1978); R. Jackiw and C. Rebbi, Phys. Rev. Lett.37, 172 (1976).

+That hierarchy of gauge interactions can indeed existin conventional Yang-Mills unified gauges theoriesis shown in A. J. Buras, J.. Ellis, M. K. Gaillard, andD. V. Nanopoulos, Nucl. Phys. 8135, 66 (1978).

3 M. Gell-Mann, talk at Orbis Scientiae, Coral Gables,1977; American Physical Society Meeting, Washing-ton, D.C., 1977.

+In the U(1) superconformal theory (which one mightexpect to represent the unification of gravity andelectromagnetism) the gauge meson is an axial-vectorfield (Ref. 14).

35S. Ferrara and B. Zumino, CERN Report No. TH.1248(unpublished) .

Northeastern UniversityOctober 15, 1978Quantum effects on the vacuum symmetries of gauge supersymmetryPran NathR. ArnowittRecommended Citation