The superhiggs effect in superspacea

Volume 120B, number 1,2,3

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6 January 1983

THE SUPERHIGGS EFFECT IN SUPERSPACE M.T. GRISARU 1, M. ROCEK 2 and A. KARLHEDE 3 California Institute o f Technology, Pasadena, CA 91125, USA Received 14 September 1982 Revised manuscript received 27 September 1982

We discuss supersymmetry breaking for general systems of scalar and vector multiplets coupled to supergravity. Superspace-superfield methods are used to reduce the problem to one in global supersymmetry and permit an easy derivation of the supertrace formula relating boson and fermion masses after supersymmetry breaking.

1. Introduction. Although the very early work on supersymmetry was motivated by phenomenological considerations [1], with some notable exceptions [2] subsequent developments have been largely o f a formal nature. With the realization that supersymmetry may provide an explanation o f hierarchies in grand unified theories, there has been renewed interest in constructing realistic (broken) supersymmetry models. However, renormalizable globally supersymmetric models with spontaneous breaking are rigidly constrained by mass relations [3] that do not hold experimentally, whereas explicit (soft) breaking [4] has no fundamental justification (although it may describe an effective lagrangian theory). On the other hand, the pattern of spontaneous supersymmetry breaking changes significantly in the presence of supergravity. In an important recent work [5] a class of locally supersymmetric models was studied. Spontaneous supersymmetry breaking led to a superHiggs effect and a modified mass relation which need not be in conflict with experimental observations. Although the final results o f ref. [5] are simple, they were 1 Fairchild Scholar. On leave from Brandeis University, Waltham MA 02254. Supported in part by NSF grant no. PHY 79-20801. 2 Work supported in part by the National Science Foundation under grant SPI-8018080. 3 Work supported by the Swedish Natural Science Research Council. 110

found in a component formalism which required extensive computation. In this paper we generalize the results o f ref. [5] using a superfield formalism. This formalism provides an algebraically simpler setting for our investigations; in particular it allows us to recast the problem as a global supersymmetry problem which can be very easily handled. As in ref. [5] we consider a general system o f interacting scalar and vector multiplets coupled to N = 1 supergravity. The matter multiplets are described by chiral and (real) gauge scalar superfields cIg', V a, respectively; the supergravity multiplet is described by a real a x i a l - v e c t o r superfield H m and a chiral "compensating" superfield ~ [6] + 1. However, H m plays no direct role in the supersymmetry breaking mechanism or in the derivation of the mass formulae. Therefore all the relevant information can be extracted from a global nonrenormalizable system described b y ~, qsi, and V a. In section 2 we study spontaneous supersymmetry breaking for arbitrary models with only chiral superfields. We first explicitly evaluate the fermion and boson mass matrices and obtain an expression for the supertrace TrM2B -- TrM2F . We then rederive the result by using a general superfieM method which avoids the explicit evaluation o f mass matrices. In section 3 we extend our results by including gauge interactions. 4:1 For a review of superfield supergravity and other superspace techniques that we use, see ref. [7]. 0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

In section 4 we discuss the coupled matter-supergravity system. We conclude with a discussion of possible applications. In the appendix we calculate the supertrace from the mass matrices in the gauged case.

2. Global ehiral superfield systems. We consider a system o f N chiral superfields q¢" described by the superspace action S = f d4x d40 U(cb i, ~1) + f d4x d20 P ( ~ ' ) + h.c., (2.1) ,I~=q)(x,0,O),

/)k~=0,

~=(q))t,

D~=0.

This is the most general chiral superfield action that leads to a component action with no more than one (two) derivative(s) for femtion (boson) terms. The first term in S can be given a geometrical interpretation [8] : ~ , ~! can be thought of as coordinates of a complex manifold; its natural geometry is described by a metric uiJ -

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(a/acb i) (ala~j) U .

(2.2)

(Upper and lower indices are related by complex conjugation. We keep explicit all factors of the metric.) The manifold with this geometry is a K~hler manifold and U is the K,~hler potential. The metric is obviously invariant under gauge transformations U-> U + A(~I,) + A(q~),

(2.3)

as is the action S. Field redefinitions ~ ' = f(~I,) define holomorphic coordinate transformations on the manifold. We define

Ujl...]n _ a a a a U. tl'"tm oc~il "'" od~i m 0~/1 "'" O~/n

(2.4)

matter multiplets r/i/= 8iJ. If they include the chiral compensator ¢ of supergravity, rTi] -- (1, 1, ... - 1 ) . In normal gauge the curvature tensor of the manifold at D 0 is

RiJkt= U i / l .

(2.6)

The (complex) component fields of (I¢"are defined by projection

Ai = q¢'10=0, J2ai=Da~lo=o , Fi =D2q¢'i0=0 • (2.7) We denote vacuum expectation values of the component fields by

a i = (Ai),

fl = (Fi),

(~i) = 0

(2.8)

The vacuum expectation values are obtained by solving the classical field equations for x-independent fields. Spontaneous supersymmetry breaking occurs if fl = 0 is not a solution of these equations. The action S leads to the superfield equations z32 cri + ei = o ,

(2.9)

and its hermitian conjugate. Taking vacuum expectation values and evaluating this at 0 = 0 using the definitions (2.7,8), we obtain in particular

U/(a)~. + Pi(a): 0 .

(2.10}

Further component equations are obtained by differentiating (2.9) with D 2 and evaluating at 0 = 0. We find

[Ui/k (a)f k + Pi/(a)lj q = 0 .

(2.11)

After Finding the vacuum solution(s), we can choose to work in normal gauge at the vacuum point. In that case the vacuum equations (2.10, 11) reduce to

~" +Pi = O, PiffJ = 0 .

(2.12)

IfPi/(a ) is nonsingular all f i = 0 and supersymmetry Using gauge transformations (2.3) and holomorphic coordinate transformations it is possible to go to a normalgauge where at any given point {I}0, {I}o (at 0---0)

Uil...im = U]l...Jn = O, U/. . = U['"4n= O t 1 ...t m

for all n , m , for all n, m 4: 1,

(2.5)

U ij = 1?iJ , with ~i I = (1, 1, ... - 1 , - 1 , ...) depending on the signature of the manifold. If the @ describe physical

is not broken. Conversely, if f) = 0 is not a solution of (2.10, 11) then supersymmetry is broken and Pq(a) must be singular. To find the mass matrices of the component fields we expand the superfield equations (2.9) to linearized order in fluctuations about the vacuum expectation values (in normal gauge) ~i = ~0 t + 1¢ /~2 [uiJ(~o, {I)O)~/. + Ui/.(~O, ~}O)~j] +pij(r~o)~j= O. (2.13) Applying D ~ , D 2 and evaluating at 0 = 0 we find, 111

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using the definitions (2.7) and the anticommutation relations (Da,/Sa) = i0 ak with 0 ac~Oaa = 2[3

~i +Pi]~j = o, iOc~c~id + Pij~ lc~ = 0 ,

(2.14)

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superfield action if we imagine performing a superfield one-loop calculation. We expand S to second order in quantum fields q)z, with the coefficients evaluated at the background classical values ~0z:

[ ~ i + gikJlflfk~j + Pijk fk~4[ + Nil ~[ = O .

S ~2) =fd4xd40 ~'/Ui/~'+ fd4xd20 Xij~i~/+b.c.,

Eliminating the auxiliary fields we identify the fermion and boson mass matrices:

where

(2.19)

MF =Pi/(a) '

M2=_( vid'7,I -e,k B

k ~i]k~k

(2.15)

e,jk ~iik~kfl _ ~ikPk] )"

If supersymmetry is broken Pi] has at least one zero eigenvalue and one of the corresponding massless fermions is the goldstino. We evaluate the graded trace of the mass matrix squared. This supertrace gives the mass relation StrM 2 = ~ ( - 1)2J(2J + 1)M2 /= TrM 2 - 2TrMvM ~ =

-2uikitflf~.

Xij = Pi] + ~2 Uij .

(We work in a general gauge.) The quadratically divergent term in the one-loop effective action is d4p

-2Rktftf ~ ,

(2.16)

(2.21)

and therefore the supertrace is StrM 2 =

-2f d40 Tr [In (U/)]

= - 2 (D2/3 2 [Tr In (uiJ)]) l0 =0 (2.22)

Eq. (2.21) can be understood as follows: S(2) can be rewritten as S (2) : f d4x d40

~](eV)i/di~i

(2,17)

(2.18)

We observe that in contrast to renormalizable models, spontaneous supersymmetry breaking can occur in a model with a single chiral multiplet, for example with U = cos(q5 + ~),p(qb) = qs, where -Tr < A < zr. The procedure above uses essentially a component approach and does not take advantage of the superfield formalism. There is a much easier way to evaluate the supertrace expression (2.17) without ever computing the component mass matrices. We recall that if the action (2.1) is expanded in components and used to evaluate the one-loop effective potential, the quadratically divergent part of it is proportional to the supertrace StrM 2 [10]. However, we can easily read offthis quadratically divergent term from the classical 112

[In(U~)]

= --2 [Trin(uiJ)]klflfk = - 2 R k l f l j d¢.

where Rkl = Rilk i is the Ricci tensor of the manifold evaluated at ~ = ~ 0 " In an arbitrary gauge (2.17) still holds, and the Ricci tensor has the general form in terms of derivatives of U [9]

Rk 1 = [In det (Uii)]kl.

d40 Tr

r= =f (2n)4p-----

Since we are in normal gauge we can rewrite this as StrUt 2 =

(2.20)

+ t'jd4x d20 Xij~i~ d + h.c.

(2.23)

and can be thought of as describing chiral multiplets in a background "gauge multiplet" with gauge superfields

Vii = In (UiJ) ,

(2.24)

It is well known [11,12] that the only quadratically divergent term in the effective action is the "D"-term

TrD=Trfd40

V=fd4OTrIn(Ui]),

(2.25)

which is precisely what appears in (2.21). A more detailed argument, based directly on a supergraph analysis, is the following: We are looking for quadratically divergent one-loop graphs, with chiral lines in the loop and (Uil - 6iJ ) or Xij and ~,27 vertices. At each (Ui] - 8iJ ) vertex we have factors D 2. /3 2 acting on the propagators [12], whereas at the Xii

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vertex which is chiral we have only one/)2 factor (a D 2 factor at the antichiral vertex .,g/i). Propagators are proportional to p - 2 . (We treat ~ as massless; masses from Xi] are treated as vertices.) To have a quadratic divergence in an n-propagator loop we need n U~ vertices. They produce a total of (D2/)2) n factors; one factor D2/) 2 is needed for the 0-loop [12], while (D2/)2) n -1 ~ ( _ p 2 ) n - 1 leave us with a single p - 2 factor in the loop. On the other hand Xi, vertices do not give sufficient D's and can be droppe~ (but P is involved in determining the vacuum solution). From the Uvertices we obtain

r = = Tr ~ f

(2~p2

directly in any gauge, this is not a problem. However, it is possible to covariantize (3 .3) with respect to coordinate transformations and calculate the mass matrices in a convenient coordinate system. For details we refer to the appendix. We begin by deriving the equations for the vacuum expectation values. It is convenient to define ( Y a n g Mills) covariant component fields by covariant projection (these components are related to the usual ones by field redefinitions which do not affect physical quantities):

Ai = cb/[0=0, ~ J = V~ d/~'10=0, Fi=v2cbiio=o , (3.4) where the covariant derivatives are

-l(6i/-u/)nn

V~=e-VD~e V, Vd=fia, = f d 4 0 Trin(Ui] ) f d4p . a d(2/r)4p2

(2.26)

3. Globalgauge systems. We repeat in this section the analysis of section 2, including gauge superfields V = vaTa, where T a are generators of an arbitrary compact Lie group. The action is

fd4x d20 [p(q~') + 1 Qab(Cbi)waawb I + h.c., (3.1)

where

4] = ~k(eV)] k , waa = iD2(e-VD~eV) a "

(3.2)

The chiral quantities Qab = 6ab + O ( * ) can generate masses for the gauge fermions contained in V. ~Tr V is the global Fayet-Iliopoulos term. Gauge invariance of S requires

The auxiliary component field D of the vector multiplet is defined by

Da = - ~1( •v w.) ~ Io=o=½t(V"Wa)alo=o "-'. We will need the identity (da = (Da)) { V 2 0 2 4i )10=0 = da (Ta)i]aj •

(3.6)

(3.7)

1 ~]o~a w o l b = 0 , V2Ui + Pi + ~Qab.i

Ui(Ta)i ] 4 / - ½i 7~(aab W~ b) + ½i V&(O-ab~vkb) + ~ Tr T a = 0 .

(3.8)

The equations for the vacuum expectation values are obtained by evaluating at 0 = 0 the above equations, as well as that obtained by differentiating the first one with V 2. We fired +P; = o ,

vi]k fkfi + Uif(Ta)fk da-~k +eifJq + 1 Qab,idad b = 0,

rbi(Ta)ill@ - Ui(Ta)i] ~1. = O, ¢bi(Ta)ile] = O, cbi(Tc)i/Qde,] + (Tc)daQae + (Tc)eaQad = O.

V~a=--i{V~,Va}.(3.5)

The superfield equations that follow from (3.1) are:

s =fd4x d40 [V(~', 4.i) + ~ Tr V] +

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(3.3)

We have assumed that the ~'s have the usual Y a n g Mills gauge transformation properties of chiral superfields. This restricts the kind of coordinate transformations (field redefinitions) and K~ihler gauge transformations we can make and in general prevents us from going to normal gauge. In a general gauge computation of the mass matrices is complicated, but since we have a method for obtaining the supertrace

Ui(Ta)ila] + (Qab + Q-.ab)db + ~ Tr T a = O,

(3.9)

where U, Qab, P are evaluated with ~i _+a i. We obtain the supertrace by examining again the quadratic divergence in the one-loop effective action. To second order in quantum fields we have

S(2) = f d4x d40 [Vif~k(eV)k]~i + ~' (Qab + O~ab)~"aDaD2Da ~rb ] ,

(3.10)

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where we have dropped terms that do not contribute to the quadratic divergence. The argument is as before. All propagators go as p - 2 and therefore at each vertex of a one-loop diagram we need a D 2 and a/~2 factor. Thus the P terms are irrelevant, quantum ~"s from the first term or ~ ' s from Qab do not contribute, etc. The first term in (3.10) gives a contribution as before from Ui], and an additional contribution from e V. From the second term in (3.10) we obtain loops with factors n -1 [½( a + Q) - 1] n , and (DD2D) n (p2)n-IDD2D, leaving us with a single p - 2 factor as before. The final result is StrM 2 = - 2

fd40 (Tr [(VaTa)i/+ In (U/l)]

- Trln [½(Qab + (~ab)] }

= -- 2 (da(Ta)i i + R k l f l f k + da(Ta)i]-d] P i + Tr [Qk(Q+ Q) -10_.t(Q+ Q ) - I ] f / f k -

Tr [Q l ( a + Q) -1 ]

(Ta)/~ida},

(3.11)

where we have replaced fd40 -+ V2 ~2 and used [V a, 5 2 ] = W~, and (3.7). We have introduced the contracted Christoffel connection F i = In det (u/.k)] i = (U 1)k]Ui] k .

(3.12)

Using the gauge invariance relations (3.3) we can show that the last term in (3.11) is real. We also have

da(ra)iJ~j r i : da( Ta)/ ia]ri .

(3.13)

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An analogous situation arises in a nonsupersymmetric context, when scalar fields are coupled to gravity. The action is S = - ½ f d4x V~- JR(g) + gmnGif(A)3mAiOnA]

+ R V I ( A ) + V2(A)] .

(4.1)

To find the vacuum expectation values of the scalar fields we cannot ignore the gravitational field, in general R will have a nonzero expectation value that affects the masses and scalar potential. However, we need not consider the full Einstein system, it is sufficient to look for solutions of the form gmn = 02r?mn and treat the system of scalar fields o, A i (subject to the condition o ¢ 0 at all points). There are two possible situations: If (V 2) 4= 0 we have a nonzero cosmological constant, o -1 - 1 (v2)l/2x2 , and the vacuum values and the masses o f A i are shifted from their flat space values. If (V 2) = 0, the cosmological constant vanishes and a consistent solution is o = constant. In this case the gravitational field does modify flat space results. (This is not the case in supergravity: Even if the cosmological constant vanishes, the supergravity auxiliary fields modify global results.) Returning to the matter-supergravity system, we consider the action S = f d4x d4OE-1 (~o,H ) { - exp [ - ( ~ Tr V+ G)]

+ [ R - l ( g + - ~1 ar 3b

14, ~ a

T, Jb covVV~cov)+h.c.]},

(4.2)

4. Supergravity. We now apply our results to matter systems coupled to N = 1 supergravity. The supergravity multiplet is most conveniently described in terms of two superfields [6] : An axial-vector real gauge superfield H m , and a chiral scalar superfield % with a large superspace invariance group. H m contains the graviton and gravitino physical degrees of freedom whereas the spin zero complex auxiliary field S, the gravitino 9'trace (3'' ~b)L = ~ ,a&, the vierbein determinant e = det eam and the divergence of the axial vector auxiliary field A m are contained in tp (these last two are the real and imaginary parts of the 0-independent component of ¢). Since only these quantities are relevant for studying spontaneous supersymmetry breaking and Str M 2 , we can ignore the H m dependent terms in the lagrangian and work entirely with ¢ and the matter superfields in a global setting. This simplifies enormously the discussion. - - ,

114

where E -1 is the superdeterminant of the vielbein and R is the scalar curvature superfield [6] * l. The supergravity action is given by the first term in the expansion of the exponential G (q~', ~j) is an arbitrary gauge invariant function of chiral superfields, with + defined by (3.2),g(qW') is a chiral function, and Wco ~av is • the (supergravity covariant) gauge field strength. In the case without gauge interactions G has a natural interpretation as a Kghler potential with gauge transformations G G + A(qW) + A(qb/.) generated by scalings oftp. The exp ( - ~ Tr V) factor is the local form of the Fayet lliopoulos term [13]. It is gauge invariant by virtue of a combined gauge transformation of V and superscale transformations o f E -1 . Its presence severely restricts the form of the g terms; they must be Rinvariant [14] so that the whole action is invariant

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under the superscale transformations of E -1 (see below). We have set the gravitational constant ~ = 1. When it is suitably restored, in the J¢ ~ 0 limit the action (4.2) becomes (3.1), with the identification

G,g~U,P. As discussed above, we can split off the terms independent o f l t m in which case we need only consider [6] *' (Wcov ~ ~0-3/2W)

S = f d 4 x d40 ( - ~ t p exp [ - ( ~ Tr V+ G)])

+ fd4xd20(¢3g+¼QabWSaWsb)+h.c.

(4.3)

Under the gauge transformation TrV-+Tr[V+i(A-A)],

G~G,

(/SkA=0),

the action is i nvariant if we rescale ~0-+ ~0exp ( - i f Tr A). Thus the local F a y e t - Iliopoulos term acts as a conventional gauge term for ~p. [The action (4.3) cannot have a Fayet-Iliopoulos term in the global sense.] If 4: 0, as noted above the form o f g ( q ~/) is extremely restricted; tp3g must be gauge invariant. We now analyze the global system (4.3) subject to the condition that the cosmological constant vanishes. We can then choose (o) = (~o)10=0 =/x = constant. With the identification -~exp(-~

Tr V)~0exp [-G(q~', ~j)] = U, ~o3g(cbi)=P, (4.4)

In det (UiJ) = -(iV + 1)G + In det (GiJ) ¢exp(

~TrV)~] ,

(4.5)

where differentiation is with respect to ~ = ~ e x p ( - ~ Tr V) and not ~. The supertrace can be read from(3.11). We find StrM 2= -2{da(Ta)i i

(N+a)~Yrd+Rklflfk

(N+ 1) [GiJ fl.fi + Gi(Ta)ii?ld a ] + da(Ta)iJ~iFi

R k l= [lndet(Gil)]kl ,

I "l= [lndet(Gi])]l ,

(4.7)

are the Ricci tensor and contracted connection for the Kghler manifold with potential G. In the above formula the sum in the supertrace rnus over the spin 0, 1/2, 1 fields of the matter multiplets and ~0. To understand the significance of the supertrace we must analyze the contributions of the components of ~o. Since the cosmological term vanishes, the spin zero component field Re o is massless and gives no contribution. The pseudoscalar Im o -= 0 is not recognizable as one of the fields of a supergravity multiplet; it replaces the divergence of the axial vector auxiliary field A s k . In the absence of gauge interactions p has exactly the same effect on the mass matrix as Ac&. For example (ignoring an overall factor/x2e - G ) - l (A 2 s&)2 + ½iAsk((Gi)as&ft i

_

(Gi)askAi)

1 [Ask _ ½i((Gi)ask~i _ (Gi)askAi)]2 1

- g ( ( G i) a s k A^ i - (G i) OskAi)2

(4.8)

and thus Ask contributes a term --~ ((Gi)Oskft i (G i) askAi)2 to the spin 0 kinetic term. Similarly we have

- ~1 (as& p) 2 + ~1 i ask p ((G i) Os&~li - (G i) aakA i) _

21 {aSk [p __ "21i((Gi)A~" _ (Gi)Ai)]}2

_ 1 ((Gi) ask,~i _ (Gi) 3skAi) 2 ,

(4.9)

ask ((Gi M i ) = (Gi)Ds&Ai , only if gauge invariance is not spontaneously broken (Dsk is the Yang-Mills covariant derivative). If we redo the argument we Fred that Ask gives a contribution 1 i - -g ((G)Ds5 A i - (G i ) D s k A i)2

whereas p gives

+ Tr [Qk(Q+ ~) 1 Ol(Q+ ~))-1 ]f/fk - Tr [Q_I(Q + Q) -1 ] (Ta)li~ida) ,

where

and thus p contributes exactly the same term. However, in the presence of gauge interactions, if gauge invariance is broken, this equivalence does not hold. This is most easily understood by noting that

we have the action of (3.1) (without the global Fayet-Iliopoulos term). We note that

+Nln[

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(4.6)

1 [auk ((Gi),~i _ (Gi)Ai)] 2 If gauge invariance is broken, the former gives a contribution ½](Gi)Aask (To) j ial i2 to the spin 1, Aas5, mass 115

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whereas the latter gives no such term. Thus in our supertrace calculation, we must subtract a term _ 6122e-G (Q + ~))-lab (G i)(Tb)] iai(ak ) (Ta)k~l, by hand. Finally the spin 1/2 component field of ~o (gravitino 7-trace) gives no contribution to the supertrace. This can be seen in two ways: We have explicitly computed the total fermion kinetic and mass terms and, after identifiying the Goldstino, have found that the goldstino-gravitino 7-trace fermions decouple from the rest of the fermion system, and that both have zero mass. (For this to occur it is essential that the ~0 field is a ghost, i.e. it appears with the wrong sign of the kinetic term.) A simpler argument is the following: By definition the Goldstino is the fermion that couples to the 7-trace of the gravitino. The condition that the Goldstino can be gauged away (U-gauge) implies that they both have zero mass in the gauge in which we are working. The supertrace (4.6) does not include the contribution from the physical massive gravitino. Obviously, in a physical gauge, the gravitino mass is related to that of its 7-trace. To fred it, we go to U-gauge by setting the goldstino to zero, and, since they do not couple to the 7-trace we also set all other fields to their vacuum values. Then the mass of the "/-trace is the mass of the superfield ~0in the linearized form of (4.3). We fred m.~ = 6/2e a .

(4.10)

Substituting the 7-trace in the massive RaritaSchwinger equation (i.e. writing the gravitino action including the mass term m3/2 ~m Omn ~n and substituting in it ~m ~ ,t,m7" ~), we identify

m3/2 = 3#e G .

(4.11)

We can include now the gravitino contribution -4m3/2 2 in the supertrace. We give the gravitational action the correct normalization by identifying ~2 = 3/2-2 e G .

(4.12)

To study the vacuum conditions for the system (4.3) it is convenient to rescale ~: ~o3g-+ ~03, and redefine G: 1

G -+ G + ~ In [g~exp(3~ Tr V)]. Note that the Fayet-Iliopoulos term has been absorbed 116

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into G, and that ~0is now inert under gauge transformations. The ~ T r d term in eq. (4.6) is adsorbed into the Gi(Ta)iJ-di d a term as a consequence ofR-invafiance o f g 3 ~ Tr (T a ) - ~i(T a )if a]

=

0,

(4.13)

and hence ~Tr(Ta) ---~1 (in [g~exp(3~v)]i(Ta)iJ~j = 0.

(4.14)

The vacuum conditions (3.9) become (fi = (F i) = (V 2qbil0=0))

3/22eG--/2Gi~.=O,

Gi/fl.+ 3/2eG Gi=O '

6su2 = (Qab + Qab) dadb ,

"2e-Gai(ra)i ]?j + (Qab + O-.ab)db = O,

3. eC aiF/ + + 9GieC(s

I/ + GiJ(ra)j daa /22ea) + ~/2-2ea Qab,idad b = O. (4.15)

The assumption that the cosmological constant vanishes is equivalent to the condition that these equations have a solution for constant/2. We have also the gauge invariance conditions (these hold for general values of the fields, not just at the vacuum point):

Gi(Ta)i/~ = Gi(Ta)]id~J '

(4.16)

dpi(Tc)iJ Qab,j + (Te)bd Qda + (Te)adQdb = O . In the case when g = 0 we cannot perform the above rescaling. However this case is not interesting since then supersymmetry is not broken, even in the presence of a Fayet Iliopoulos term. Using the vacuum equations, we can substitute in the supertrace. Including the gravitino and the Im o contributions we find the final expression 3/2 ( - 1)2J(zJ + 1)M2 = - 2 (da(Ta)i i +R/i~'f ] J=0 - (N-

1) [m 3/2 2 -

lg2(Q+O-)abdadb]

+ da(Ta)if~j F i + Tr [O_J(Q+ ~))-1Qj(Q + O.)-lftfJ - Tr [OJ(Q + ~))-1] (ra)jiaida)

.

(4.17)

The above formula extends that of ref. [5] and allows us to treat many cases of interest. In particular, in extended supergravity theories we encounter "non-

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minimal" G and Q terms. For example, in N = 4 supergravity, which contains one physical chiral multiplet, three vector multiplets, three (3/2, 1) multiplets and the supergravity multiplet, G~-ln(1

(b~,),

a~(1-(I))/(l+cb).

(4.18)

We cannot treat the a c t u a l N = 4 theory since a complete description of the (3/2, 1) multiplet is not available, but (4.18) suggests looking at a system with one scalar multiplet and n vector multiplets V a, coupled to N = 1 supergravity, with G as above and Qab = [(1 - ~b)/(1 + oh)] lab

,

(4.19)

We f'md, with G" = 02G/O • ~

= (1 - aa) -2 ,

(4.20)

the supertrace 3/2 ( - 1)2J(2J + 1)M2 = - 2 ( n + 2 ) G " f f J=0 = - 2 ( n + 2)m2/2 .

(4.21)

Note that the Q and R terms in (4.17) combine because ( 2 + 0 ) 2 = 4(G") -1 [(1 + 4))(1 + ~ ) ] - 2 . Unless a scalar potential g ( ~ ) is introduced, no supersymmetry breaking will occur. However, it is possible to add such a term i n N = 1 supergravity, and it may also be possible to do so in N = 4 supergravity [ 15]. Independent of the potential term, d = 0 because the scalar is singlet. For N > 4 the analogs of G and Q are expressed in terms of an overcomplete set of fields. We may expect however that Q and G are related such that det (GiJ) ~ det (Q + O.)h(did)h(cbi) , where h(q~) is a holomorphic function. In that case we may also expect a simple result for the supertrace. We can also construct models with a F a y e t Iliopoulus term and vanishing cosmological constant (such models have not been constructed before). For example, consider G = ~eV(b + l a 2 in (~eVq)) +YO/ + ½ in [(3 + X) (3 + X)],

(4.22)

where q~ and X are chiral fields, ~b transforming under

6 January 1983

the Fayet-Iliopoulos gauge transformation while X is inert and/3 is chosen so as to make the cosmological constant vanish (the potential and the Fayet-Iliopoulos term are included in G as the a-term). We find a solution to (4.15) w i t h d v~ 0 for some ffmite range of a (as can be verified by a perturbation expansion about a = 0). Our discussion has dealt exclusively with n = - 1 / 3 supergravity [6] * 1 with the minimal set of auxiliary fields. However, we can equally well handle n ~ - 1 / 3 , by using instead of ~0the appropriate compensators. We can also discuss the case with nonvanishing cosmological constant, in deSitter space, although the algebra becomes then more complicated. Finally, we can take a suitable global limit (K -+ 0) and study the lowenergy implications of spontaneous supersymmetry breaking in the presence o f supergravity, without ever having to introduce the full graviton-gravitino system. Part of this work was done while the authors were atttending the Summer Workshop at Ecole Normale Superieure. We wish to thank many of our colleagues, in particular E. Cremmer, D.Z.. Freedman, L. Girardello and W. Siegel for hlepful discussions. A p p e n d i x . The form of the Yang-Mllls gauge transformation 8(I)i = iAacbl(Ta)i i ,

8~ i = -iAa(Ta)ii~l,

(A.1)

fixes the coordinate system on the K~aler manifold. We can generalize to arbitrary coordinates by introducing holomorphic vectors ka i 8cI)i = Aaka i ,

8 ~ i = Aakai .

(A.2)

The requirements (3.3) for the action being gauge invariant then become Uika i + Uikai = 0,

Pik.a i= 0 , (A.3)

Qde,ikc i - i(Tc)daQa¢ - i(Tc)eaQad = O . It follows from (A.3) kai'J=kai,]=O,

that

ka i fulfill Killing's equations

kai;j + ( U - 1 ) i k k a l ~ ; k u 5 • = 0 .

(A.4)

The Yang-Mills gauge group is thus a subgroup o f the isometry group of the KLh.ler manifold. The gauge invariance (A.3) of the action fixes the K~aler gauge and in general makes it impossible to reach normal gauge (2.5). However, by performing a 117

PHYSICS LETTERS

Volume 120B, number 1,2,3

is obtained b y replacing kai ,i with kai ;i and Uikli with

coordinate transformation, we can go to normal coordinates where at a point (at 0 = 0)

U{.

1 ..din

= U(l"'ln = O ,

Rkl

foralln,rn:~l

i

ui j = ~;i.

S t r M 2 = - 2 (idakai ~ + Rklfk f l

(A.5)

Note that Uil•Am and U/1"''in are unrestricted as compared to n o r m a l gauge. The i n t r o d u c t i o n o f Killing vectors and, as a consequence o f this, the possibility to use n o r m a l coordinates facilitate computations. As an example of this we rederive the supertrace (3.11) b y calculating the linearized field equations for the c o m p o n e n t fields. These are obtained b y expanding the covariantized form o f the superfield equations (3.8) a r o u n d the v a c u u m and applying the operators 1, V ,V 2 to the first and 1 ,V,[V,V] to the second of the equations and evaluating at 0 = 0. We find (in n o r m a l coordinates) F / * Pi/..~/" = 0 ,

1. a ~ba iOu&~&i + Xa~kai+Pi]~al --~l~ab,i d =0, ~

"

[3A i + (idakai d+ Uikt/ fk f l - PikDkJ)~4i + (Pi/kf k + ~ Qab,ifdadb) ~ / + (Qab,i d b + ikai)D a = 0 , (Qab + Q,ab) 5 b + (aab, i db+ ikai ) j i + (Q.ab'id b - ikai)7t i = O, 1 i(Qa b + Qab) aa&~b& + (kai _l._~lQab,idb) f a i =0 + !2 f~ ~ab,i ¢i'~ba J (Qab + Qab)V a&F b a ~ - 2(kaikbi +kbikat)Absa " ~ " =O. (A.6) We use a n o r m a l i z a t i o n such that the vector wave e q u a t i o n is v~AFa~ - 2m2A~ & = 0. Eliminating the auxiliary fields we fired the mass matrices from which we obtain the supertrace (in normal coordinates) S t r M 2 = - 2 (idakaf i + Uikli fk fl + Tr [Qi(Q+ ~ ) ) - 1 0 j ( Q + O , ) - l ] f i ~ + i ( a + O,)-lab(kaiO, bc'id c -

kjab~,id~)).

(A.7)

A covariant formula, valid in any coordinate system,

118

6 January 1983

+ Tr [ai(a + Q) -1 QJ(Q + ~9)-1 ] f i ~ - i T r [Q,i(o + O~)-l]kai da) ,

(A.8)

where we have rewritten the last term using (A.3). This reduces to (3.11) in the coordinate system used in section 3 where kai ;i = -i(Ta)i i - i(Ta)ildJF i.

References [1] D.V. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109. [2] P. Fayet, in: Unification of the fundamental particle interactions, eds. S. Ferrara et al. (Plenum, New York, 1980). [3] S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403. [4] L. Girardello and M.T. Grisaru, Nucl. Phys. B194 (1982) 65. [5] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, CERN preprints TH.3312, 3348; R. Arnowitt, A.H. Chamseddine and Pran Nath, Northeastern U. preprints NuB-2559, 2565, 2569, 2570; see also E. Cremmer et al., Nucl. Phys. B147,(1979) 105. [6] W. Siegel and S.J. Gates, Nucl. Phys. B147 (1979) 77. [7] M. Ro~ek, in: Superspace and supergravity, eds. S.W. Hawking and M. Ro~ek (Cambridge U.P., Cambridge, 1981); S.J. Gates, M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace, to be published. [8] B. Zumino, Phys. Lett. 87B (1979) 203. [9] L. Alvarez-Gaum~ and D.Z. Freedman, in: Unification of the fundamental particle interactions, eds. S. Ferrara et al. (Plenum, New York, 1980). [10] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887; R. Jackiw, Phys. Rev. D9 (1974) 1686. [ 11 ] W. Fischler, H.P. Nilles, J. Polchinski, S. Raby and L. Susskind, Phys. Rev. Lett. 47 (1981) 757. [12] M.T. Grisary, M. Ro~ek and W. Siegel, Nucl. Phys. B159 (1979) 429. [13] K.S. SteUe and P.C. West, Nucl. Phys. B145 (1978) 175. [ 14 ] R. Barbieri, S. Ferrara, D.V. Nanopoulos and K.S. SteUe, Phys. Lett. l13B (1982) 219; M. Sohnius and P. West, Nucl. Phys. B203 (1982) 179. [ 15 ] S.J. Gates and B. Zwiebach, Gauged and ungauged N=4 supergravities in superspace, Caltech preprint CALT-68943 (1982)•