Geometry and physics of Wess-Zumino supergravity

PHYSICAL REVIEW D VOLUME 30, NUMBER 12 15 DECEMBER 1984

Geometry and physics of Wess-Zumino supergravity

Xizeng Wu Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 14 May 1984)

It is shown that the Kahler manifold of Wess-Zumino supergravity is the noncompact coset space SU(N,l)/[SU(N)XU( 1 )] and the Kahler metric is an Einstein metric. The mass sum rule turns out to be the same as that in global supersymmetry. This mass sum rule makes it possible to have supersymmetry (SUSYI-breaking and vanishing scalar potential. Effective theories with or without the grand-unified-theory (GUT) sector are derived from the Wess-Zumino supergravity. They do not have scalar mass terms and trilinear terms among the soft SUSY-breaking terms. Their relevance to the gravitino mass problem is suggested. In a simple SU(5) SUSY GUT model we also show how the scalar partners of quarks and leptons and gauginos get their masses radiatively. The stability problem of the gauge hierarchy is also discussed.

I. INTRODUCTION

As is well known, one of the great challenges facing particle physics is to find the principle governing the scalar interactions. In fact, many recent extensions of the Glashow-Weinberg-Salam standard model have centered on issues about scalar particles. While renormalization itself does not provide a guide, the stability of gauge hierar- chies invokes supersymmetry (SUSY) and supergravity, which relate bosons and fermions. One is thus led to consider supersymmetric extensions of the standard model with or without grand unification.

There exist theories where the scalar interactions are determined by a gauge principle. They are the extended supergravity theories. But, unfortunately, the picture these theories provide does not resemble the low-energy experimental world. On the other hand, there exists also the simple N = 1 s ~ ~ e r ~ r a v i t ~ , ~ , ~ which can somehow be thought of as an effective theory of the extended supergravity at lower energy.4 N = 1 supergravity seems to be able to accommodate low-energy physics and thus has drawn intensive attention recently. Ovrut and wess5 first proposed a mechanism to couple grand unified theories (GUT'S) with N = 1 supergravity. As a result of this coupling, soft SUSY-breaking terms are generated in the effective theories. Weinberg and others6 showed that the coupling with N = 1 supergravity can lift the degeneracy of the global SUSY vacua and trigger the SU(2)XU(1)- symmetry breaking. Since then many models based on N = 1 supergravity have been proposed7~8 by using the general formulations of N = 1 supergravity coupled with Yang-Mills fields developed by Cremmer et al. and by Witten and ~ a ~ ~ e r . ~

In N = 1 supergravity coupled with matter, the scalar interactions are described by a supersymmetric nonlinear u model. The configuration manifold of scalars is a Kahler m a n i f ~ l d . ~ The scalar potential is determined by the Kahler metric of this manifold and the s u ~ e r ~ o t e n t i a l

L .

one chooses. Although there is no principle to tell what to choose, the geometric symmetries may help us look for

judicious choices. Most of the models proposed so far are based on the so-called minimal coupling between scalars and supergravity, in which the Kahler manifold is flat.7 Some works9 discussed cases with a general Kahler metric and gave us some general features of the effective theories derived from N = 1 supergravity. These investigations seem to suggest that different Kahler metrics will give the same effective theory. But we believe that some nontrivial Kahler metrics could really make a difference. They may have some remarkable features that others lack.

Motivated by this idea, recently, we have proposed a theory of supersymmetric Yang-Mills matter non- minimally coupled to N = 1 s ~ ~ e r ~ r a v i t ~ . ' ~ In this theory, a nonflat Kahler metric is chosen. This Kahler metric is derived from Wess-Zumino s ~ ~ e r ~ r a v i t ~ ~ ~ ' ~ which is the simplest supergravity theory from the superspace point of view. We also found a new superpotential for the hidden sector, which gives a vanishing scalar potential and breaks supersymmetry with an undetermined gravitino mass. I t was the first known example with such a remarkable property. There must be some deep symmetries behind this theory. As pointed out later by-cremmer et al.,ll the hidden sector has a global noncompact SU(1,l) symmetry, which can be traced back to extended supergravities. What is more, as proposed quite recently by Ellis, Kounnas, and ~ a n o ~ o u l o s ' ~ in a series of papers, the flat potential-breaking supersymmetry can be used to construct "nonscale" SUSY models, in which all low- energy mass scales (m w , m 3/2) can be dynamically generated from the Planck mass mp. This may be progress in understanding the myth of the gauge hierarchy. Of course, it should be noted that in these nonscale SUSY models one also needs a nonminimal coupling between a gauge field and supergravity,12 so the supergravity theory on which the models are based is not Wess-Zumino supergravity.

Encouraged by these new developments, we feel that it is necessary to make a systematic analysis of Wess- Zumino supergravity. In this paper we present a detailed analysis of the geometrical and physical features of

2462 01984 The American Physical Society

30 - GEOMETRY AND PHYSICS OF WESS-ZUMINO SUPERGRAVITY

Wess-Zumino supergravity coupled with matter. In Sec. 11, we present a geometrical analysis of Wess-

Zumino supergravity. We show that the Kahler manifold in this case is the noncompact coset space SU(N, l)/[SU(N) x U(1)] and the Kahler metric is an Ein- stein metric. In Sec. 111, we derive the mass sum rule, which turns out to be the same as that in global SUSY. It explains why we can construct a vanishing scalar potential, which also breaks supersymmetry. The effective theories with or without GUT sector are also derived. The physical implications of the effective theories are discussed. In Sec. IV a simple SU(5) SUSY GUT model is constructed for illustrating the physical aspects of the effective theory. Finally, we draw the conclusions in Sec. v.

11. GEOMETRIC FEATURES

Let us begin with the action of the globally supersymmetric Yang-Mills theory:2

where Si are the chiral fields transforming as a representation of a gauge group G, V is the gauge vector superfield, the Ws are the gauge-covariant chiral superfields which contain the vector field strength and are defined as

g (Si) is the superpotential, g is the gauge coupling constant, and D,, is the SUSY-covariant derivative, T a is the generator of G, and

Then the most general action of the coupled Yang-Mills and supergravity system is given by3

where R is the chiral scalar-curvature superfield, @ is a real function of st and S, and faB are chiral superfields transforming as a symmetric product of the adjoint representation of G. E is the vector density superfield and E =detE;, where E; are vielbein superfields. Using the tensor calculus and (3), Cremmer et al. found a general Lagrangian in component formali~m.~ The overall scalar potential is

where

Here zi is the scalar component of a chiral field Si. J(zi,z*') is the so-called Kahler potential. It also deter- mines the kinetic energy term of scalars:

where Dm is the covariant derivative. This kinetic energy term is the same as that in a nonlinear cr model, in which the configuration manifold of zi and z*' is a Kahler manifold.', l 3

A short digression to a Kahler manifold may be helpful for the following discussions. An N-dimensional complex manifold is a manifold with complex coordinates zi ( i = 1, . . . , N) and z*'. A complex manifold is Hermitian if the line element ds takes the form

where the metric matrix J; is Hermitian, ds2 is real. A Kahler manifold is a Hermitian manifold on which the metric can be written as13

J is called the Kahler potential. It is important to note that (1 1). does not determine J uniquely. The Kahler metric Jj is invariant under shifts of the Kahler potential:

where F(zi) is an analytic function of zi. One can also define the "Levi-Civita connection" and curvature tensor on a Kahler manifold. It is found that the "Ricci tensor" is given by

From (4)-(9) it is evident that the overall scalar potential and scalar kinetic energy terms depend on choices of J and fa8. In the case of the popular minimal coupling one chooses

In this case, Jj= - $ k2aij and R{=O thus the scalar kinetic energy term is canonical and the Kahler manifold is flat. The scalar potential is given by

2464 XIZENG WU 30 -

portant is that this simplicity provides us with some re- e-'v=;exp [ 5 z i z * i ] markable geometric and physical features.

From ( 19) the Kahler potential of Wess-Zumino supergravity is

J=31n [I-$ l ~ i 1 2 ] . (20) (15)

Da= +gz*i~a,JZ 1 J * This is a nonflat potential. The Kahler metric is

It should be noted that in this case @ is [

Therefore the minimal coupling looks complicated in superspace. Since there is no principle to tell how to choose J , it definitely is worthwhile to try other choices.

Before the general component Lagrangian of N = 1 supergravity coupled to Yang-Mills matter was given,3 Wess and Zumino proposed a superspace formulation of N = 1 supergravity, in which superspace is curved and the tangent space group is the Lorentz group.2 Based on an affine geometry of the curved superspace, the action of N = 1 supergravity coupled with Yang-Mills matter is given by2

Therefore the scalar kinetic energy term is not canonical. This Kahler metric looks like that of the CP" nonlinear u model,I4 but because of the minus sign for I zi 1 2, it is not the CP" Kkhler manifold. In fact, this Kahler manifold is noncompact, whereas the C p N Kahler manifold is compact. In order to understand what the corresponding Kahler manifold of Wess-Zumino supergravity is, let us consider a complex manifold M,

Tr - IpAp and identify all v that differ by a phase h EU( l ) , 2g2c2(A)

v=uh . (23) 6

+g(S)-- ,R +H.c. , k I (17) Here c N + ' is a complex space with N + 1 complex di-

mensions and i runs over 1, . . . , N. We will show that a

where manifold M defined by (22) and (23) has a Kahler metric like that of (21). In fact, we can introduce a U(1) connec-

E= - $ ( ~ ~ - 8 ~ ) ~ + e p , tion

g (S) = fiSi + ;mijsisj + f hijksisjsk . and define the U(1) covariant derivative

Here denotes the chiral density ~ u ~ e r f i e l d , ~ R the cur- Dv=dv-uA . (25) vature chiral field, Dp the covariant derivative, g (S) the ~h~~ the line element of M can be defined as superpotential, and V the gauge superfield. It should be noted that action in (17) is formulated in the I: term, d s 2 = 3 [ ( ~ v o ) t ( ~ v o ) - - ( ~ u i ) i ~ u i ] . (26) whereas the general action in (3) is formulated in the D

This ds2 is U(l) gauge invariant. To solve for the con- Using the component expansions given in Ref

straints (22) and (23), we introduce N complex coordinates and integrating out 8, we have derived the component La- such that

grangian'O and found that Wess-Zumino supergravity cor- responds to the choice

f ap = 6ak9

and

with w E c - O] and u E cN. Through some calculations, we get ds expressed in terms of independent variables ,

-3 [If we include linear terms - $c (D~D, - 8~ )so of gauge ds = [( 1 - I u I 2 ) d ~ tdu - I du 'u 1 2] . (28) singlets in the action (17), then a more general @ is ob- (1- lu

tained. For detail see Ref. 10. For our discussion in this If we rescale ui, paper, we take c =O.] From (3) and (19) it is evident that from the superspace point of view Wess-Zumino super-

k u . - z . gravity is the simplest and most straightforward extension ' - 6 6 "

of the globally supersymmetric action. What is more im- then ds2 reads

30 - GEOMETRY AND PHYSICS OF WESS-ZUMINO SUPERGRAVITY

where J; is given by (21). Therefore, we have shown that the Kahler manifold of Wess-Zumino supergravity coupled to matter is the manifold defined by (22), (26), (27), and (29). Because the noncompact group SU(N, 1) leaves

invariant, this Kahler manifold can be realized as the coset space SU( n, 1 )/[SU( N) x U( 1 I]. As a comparison, we recall that the C P ~ manifold can be realized as the coset space SU(N + l)/[SU(N) X U( I)].

It should be stressed that the coset space SU( N, 1 )/[SU( N) XU( 1 )] is a noncompact manifold, whereas C P ~ is compact: This noncompactness is important for us, because it is impossible for a compact manifold to have a nonzero and everywhere nonsingular superpotential, which, as we will see later, is an important in- gredient of the theory.

Using the Kahler metric J j we also can get the Ricci tensor. Through some calculations we find the deter- minant

Thus the Ricci tensor isI3

R!--- a a [lndet 1 J; 1 ] J - az. az*i

I

It shows that this Kahler manifold is nonflat and an Ein- stein space. In order to see what is special for this Ein- stein space, we should examine the scalar interactions.

111. FLAT POTENTIAL A N D EFFECTIVE THEORIES

In N = 1 supergravity coupled to matter the curvature property of the Kahler manifold has its implication for the mass sum rule. It is well known that in the globally supersymmetric theory the masses of fields obeys the mass sum ruleI5

where J is the spin of a field and mJ is its mass. In N = 1 supergravity, if the gauge field kinetic energy term is canonical (fa,9=6ap) and if there is no D term SUSY breaking ( ( D, ) =O), then the mass sum rule readsI3

From (41, at the vacuum ( V) =0, one has

and thus

and

Equation (37) holds regardless of the value of the gravitino mass ~ ~ 3 1 2 . This a remarkable result. It holds as long as ( v ) =O and (D, ) =O. This result means that for Wess-Zumino supergravity the mass sum rule is the same as that of a globally supersymmetric theory, regardless if supersymmetry is broken and what superpotential is chosen.

To explore the implication of this remarkable mass sum rule, let us first consider a simple case, in which only a chiral field S is coupled to Wess-Zumino supergravity. The scalar potential can be obtained from (4) and (20) as1'

where g is the superpotential, z is the scalar component of S, and z (x ) = A (x ) +iB (x) . From (37) one has in the case with broken SUSY

This result suggests that the scalar potential is at least lo- cally flat at the vacuum ( u ) =0, regardless of g. So it is not a surprise that we found1' a special superpotential

which gives a vanishing potential

and breaks supersymmetry with an undetermined gravitino mass1'

Here mo is a mass scale and a = k ( 3 ) / K 6 ( 1 a I < 11, which is undetermined in this case. It was the first known example with such a remarkable feature. Through a redefinition one can transform J and g into other equivalent forms." As a comparison, we want to note that in the case of the popular minimal7 coupling it is impossible to have a vanishing potential breaking supersymmetry. This is because in that case

and

Using (311, one obtains

2466 XIZENG WU - 30

Once supersymmetry is broken, at least one of mA and m~ is nonzero and potential cannot be vanishing. There- fore, the vanishing potential-breaking supersymmetry is a result of the nontrivial geometric features of Wess- Zumino supergravity.

In order to describe real world, we also have to couple other fields to supergravity. Suppose we have N chiral fields and denote their scalar components as zl , . . . , zN, respectively, z 1, . . . , ZN - 1 consist of the matter sector scalars and zN consists of the hidden sector,' which is re- sponsible for SUSY breaking. We also couple Yang-Mills fields to supergravity and zl, . . . ,zAypl form a representation of the gauge group, zN is gauge singlet. Then we suppose the superpotential is

where g2 is given by (40). Coupling these fields to Wess- Zumino supergravity and using (4) and (201, we have the overall scalar potential aslo

where

In order to examine the physical implications of this theory, in principle, one can directly use (46). But, in many cases, it is convenient to use effective theories derived from (46).

There are two types of effective theories. One is an effective theory for grand unified theories (GUT'S). In this case, because MCUT/Mp- we can investigate GUT'S emerging from a spontaneously broken N = 1 local supersymmetry in the mathematical limit Mp+ a (k-0) with M, and m3/2 fixed.I6 Only in this limit is the theory renormalizable and well defined. This is equivalent to having neglected all interactions containing M ~ - ' explicitly. A deeper understanding of supersymmetry corrections containing inverse powers of Mp should, however, be undertaken so that we could trust the present approach as a consistent mathematical truncation of a more fundamental theory encompassing quantum gravity. In this limit we have1' (a = k (z ) /z/6)

After a rescaling of zi and chiral rotations of fermion fields1' the whole Lagrangian can be put exactly in the form of a global SUSY Lagrangian plus explicit and soft SUSY-breaking terms. This breaking is soft in the sense of not generating quadratic divergences. The scalar potential in terms of the rescaled fields is

where 3 A=*---(1- a 1 2)( l+a*)1 '2(1+a)-3/2m3/2 . (49)

T 2

Suppose the matter superpotential g l has the form

gl =g\3 '+gj2)+gj1)+g\0) , (50)

where g\3), g\2), g\ l ) , and g\O) are cubic, quadratic, linear, and constant terms, respectively. Then we have

their masses and vacuum expectation values (VEV's) are negligible compared with Mp or MGUT. Evidently, (5 1) is still valid. But it should be noted that if g l contains only cubic terms, then from (51) there is no soft-SUSY- breaking term at all. But if we believe there must be a superheavy sector coupled to the light sector, then for g , , there must be some quadratic terms for light fields, such as pzizj ( / ~ - m 3 / ~ ) , are the relic of the superheavy field^.^ So, accordingly, the soft-SUSY-breaking terms are generated. Actually, to convince the reader, we also can get the same results as (5 1) for the low-energy effective theory by following the method of Hall, Lykken, and ~ e i n b e r ~ . ~ These authors have derived a general formula for the low-energy effective theory by "integrating" out the heavy fields. The scalar potential for light fields is9

(5 1) +mG2 /zi I2+vo+ f D a 2 . (52)

Now let us consider another type of the effective theory-low-energy effective theory. In this case the (For the notation, see Ref. 9.) It should be stressed that fields appearing in the effective theory are all light fields, m3/2, m ;12, and m G2 are all independent of the matter

30 - GEOMETRY AND PHYSICS OF WESS-ZUMINO SUPERGRAVITY 2467

superpotential g l ; they only depend on the hidden-sector superpotential g2.

Compare (441, (46), (501, and (51) with (52). It is obvi- ous that in our case there do not exist trilinear terms among the soft-SUSY-breaking terms. Therefore one has

For evaluating m G2 it suffices to consider a light field z~ which does not occur in g. In this case, evidently the only possible source for the mass term m G2 / zl I is

but this term vanishes as the cosmological constant is sup- posed to be zero. Thus one has

Because m ;j2 is independent of g , , Eq. (24) holds for any g l . In fact, this result can be easily understood from the mass sum rule (37). Therefore, for the nonminimal coupling we choose, the scalar potential of the low-energy effective theory becomes

This result agrees with (51), which is derived simply by taking k-+O with m3/2 fixed. In Ref. 17 an explicit calculation of integrating out the heavy fields has been done and that result is also consistent with (5 1).

Now some remarks about (51) are in order. First, it is a natural generalization of the Ovrut-Wess me~hanism,~ in which supersymmetry is broken explicitly. But in our theory supersymmetry is broken spontaneously. Second, it was widely believed that m ;/2 and mG2 in (52) are al- ways roughly equal to the graviton mass.9 Among these soft-SUSY-breaking terms in (52), the scalar mass term mG2 I zi 1 deserves special attention. Ellis and Nano- poulos proved that1' because of the existence of this term scalar masses will be bounded below by ~ ~ 3 1 2 , if m G2 -in 3n. Since the Weinberg-Salam doubles will re- ceive such tree-level mass, the gravitino mass should be lighter than - 100 G~v.'' On the other hand, Pagels and Primack, and weinbergI9 have derived the bounds on the gravitino mass from standard cosmology: either m3/2 5 1 keV or m3/2 > lo4 GeV. So there is a contradiction between particle physics and the standard cosmology. This is the problem called the "gravitino mass problem."20 The inflationary cosmology seems to predict similar bounds on the gravitino mass and thus cannot give any help.20 Gaillard et aL2I proposed using global invariance of the superpotential to get massless scalars. Remarkably, here in (51) one has m;j2 =O. Thus at the tree level the scalar masses are not bounded below by the gravitino mass. Definitely it provides a new possibility to solve this gravitino mass problem. Also due to mG2 =0 and the mass sum rule (37), at the tree level it is impossible to make all scalar quarks and scalar leptons more massive

than their fermionic partners. Scalar quarks and scalar leptons have to get their mass through radiative corrections.

Third, the parameter A which is proportional to the gravitino mass depends on the mass scale m and the VEV of Z N . In order to determine them, one possibility is to use "nonscale" SUSY models, proposed by Ellis, Koun- nas, and ~ a n o ~ o u l o s . ~ ~ In this case one has to introduce nontrivial gauge-field kinetic energy terms CfaS#Sap) and suppose mp ~ r n GUT. One also can hope to determine the VEV of zN from the full theory (44), or from (44) plus its radiative corrections. More moderately, one also can set the value of A by hand in (511, and concentrate on the implications of the effective theory itself. In Sec. IV we will adopt this attitude and discuss some SUSY GUT models based on the effective theory (5 1 ).

IV. A SIMPLE SU(5) SUSY GUT MODEL

As mentioned above, the main feature of the effective theory (51) is the absence of the tree-level mass term m $2 1 zi / and trilinear term m ;/&j3'. In this section we will discuss a simple SU(5) SUSY model to examine the implication of the absence of these terms.

We want to show that through radiative corrections the scalar partner of quarks and leptons can all get much heavier than quarks and leptons. We also show how gau- gino masses can be generated radiatively. The stability of gauge hierarchy will be discussed and it will be shown how the absence of the trilinear terms make things easier. We do not claim the model presented here is very fancy or perfect. This model is a popular model, we just want to use it as a testing ground of our theory. To simplify discussion, we set A by hand in this effective theory.

Suppose we choose the superpotential g l as

where 2, H, H i , and U are chiral superfields in 24,5,5X and singlet representations of the SU(5) gauge group, respectively. C is a constant and we shall denote the GUT mass scales by m and m' -m. Of course, we also have gauge vector superfields W and quark-lepton superfields, MXy and My which are in the I_O and 5* representations of the SU(5) gauge group. But for simplifying the analysis we suppose the Yukawa coupling constant is zero and thus MXY and My do not appear in (55). This means that at the tree level all quarks and leptons and their scalar partners are massless. From (5 1) the scalar potential is given by

where the terms in brackets are the soft-SUSY-breaking terms. These terms are the relic of supergravity. The

2468 XIZENG WU

strength of this breaking is determined by A. Without loss of generality we take it as real, negative, and

1 A /m << 1. We expand V in powers of A/m and then search for minima. The results are given as follows.

(1) SU(5)-invariant VEV's:

( U ): arbitrary .

The value of Vat these VEV's is

(2) SU(4) X U( lbinvariant VEV's:

( U ) : arbitrary . In this case

( 5 8 )

For

d =[3+(1-8A/3m3L1)1/2]/12,

V reaches a local minimum

V O = ~ C A + g h l m 4 [i ] +O(A2/m2) . (60)

(3) SU(3) X SU(2) X U( 1)-invariant VEV's:

( U ) : arbitrary , ( H ) = ( H ' ) = ( M , ) = ( M ~ Y ) = O .

In this case

V=30h12m4b2[(b - 1 ) 2 + ~ / 3 m h l ] + 2 ~ c . (62)

For

b = $ [ 3 + ( 1 - - 3 ~ / 8 h ~ m ) ' / ~ ] , (63)

V reaches a local minimum

In this case I

(4) SU(3) X U,,( 1)-invariant VEV's:

After a quite tedious calculation, it is found that for

- ? ~ ~ A ( m ' - m ) / h ~ ~ ,

( 8 > =

s0=3m ,

~ ~ = ( m ' - m ) h h ~ ~ / ( 1 0 h , h ~ ~ m ) , a = 1 + ~ h ~ ~ [ 3 ( m ' / m - 1)- 10h32/h22]/(30hlh32m) ,

V reaches the minimum 10 ~ ~ = 2 c ~ + m ~ ( 1 0 h ~ ~ / m + [ - ~ - ( m ' - m ) ~ h ~ ~ / ( h ~ m ) ~ ] ~ ~ / m ~ ) .

( H X ) = (H; ) = A , S ; / T ~ .

2ma 2ma

2ma -3ma +e2

-3ma -e2,

h2 , ( U)=-(so-3m1), A3


So if one chooses

and

m l > m , then it is evident that

(VO)SU(~)> ( V O ) S U ( ~ ) ~ U ( ~ ) > ( v o ) ~ u ( 3 ) X ~ ~ ( 2 ) ~ ~ ( 1 ) > (vo)~~(3)xu , , ( l )=o

This is a nice result. First, as expected, in this model the vacuum is unique. The degeneracy of vacua encountered in global SUSY has been lifted by supergravity. Second, the physical SU(3) x U,,( 1) vacuum is located at the abso- lute minimum of energy and with a vanishing cosmological constant.

This vacuum also gives a correct mass hierarchy. The masses of gauge bosons are given by

( M ~ ) , ~ = ~ ~ ( z + ) T ~ T P ( z ) . (76)

From (65) and (66) we find

M 2 = ~ , L 2 = 2 5 g 2 m 2 / 2 , Xfi

Thus

So if one takes

then the correct gauge hierarchy can be obtained. Now we consider the masses of scalar quarks, scalar

leptons, and gauginos. At the tree level, in this model, quarks and scalar quarks, lepton and scalar leptons, and gauginos are all massless. But the phenomenology teaches us that scalar quarks and scalar leptons must be massive, even in the massless limit of quarks and leptons. In fact, the existing experimental facts give some lower bounds on scalar-electron Fand scalar-quark masses:

m,> 16 GeV, m,> 15 GeV . (81)

To satisfy the phenomenology, one has to generate masses radiative for scalar quarks and scalar leptons.

We first consider the radiative mass corrections to the right-handed scalar electron that comes from the super-

I

graphs shown in Figs. 1 and 2. Figure 1 gives a contribution

(82) where (AH5) = A5/<2, and A 5 is given by (68), mz is the mass of the weak 2: boson and is given by (78), (FH5) is the VEV of the F component of H S field, and

So (82) gives a contribution 6mzR2 to the mass of the

right-handed scalar electron

where m is the mass of the weak W: boson and given by (781, Ow is the Weinberg-Salam angle. In the calculation the relation

has been used. If one chooses

1 A 1 Am1-m)- 1

and

then

FIG. 1. Supergraph inducing the AzR term. FIG. 2 . Another supergraph inducing the A ; ~ term.

2470 XIZENG WU

FIG. 3. Supergraph inducing the term.

Another contribution to the FR mass comes from Fig. 2. It contributes

6m- e~ 2=e2g22tan20W(~H5)2 / ( 16r2mZ2)

Besides, Fig. 3 contributes a term

e2&2tan28w ( ~i~ ) ( FH5 ) /( 16.rr2mZ )2 $ d4x AERFiR

(89)

to the action. But since in this model

(89) gives no contribution to the TR mass. It should be noted that Fig. 4 is zero, because the integration over 0 is zero. On the other hand, because FR is massless at the tree level, the wave-function renormalization does not contribute to mass correction. Putting all the contributions together, one draws the conclusion that the right- handed scalar electron gets a mass of the order of Mw. What is more, from (85) and (78) one finds

l A l - m w . (90)

So this is the value we should put in for I A I . Of course, according to (49), the gravitino mass is still not determined yet, because in the effective theory we do not know what the VEV of z, is. The VEV of z, should be determined in the full theory, as commented on at the end of Sec. 111.

Evidently, similar supergraphs will give contributions to the left-handed scalar electron, scalar neutrino, and scalar quarks. And they will have radiative masses of the order of Mw.

E ;f E R

FIG. 4. Supergraph giving a vanishing contribution to the ZR mass.

( b )

FIG. 5. Supergraph inducing the hth; term, 71 = rn he2.

At the tree level there are other kinds of massless fermions. They are fermionic partners of gluons and pho- tons: gluino 6 and photino F . 6 and 7 also get their masses radiatively. The main contributions to gluino masses come from the supergraphs shown in Fig. 5. Then one obtains contributions from Fig. 5:

where a, =h2/4.rr is the strong-interaction coupling constant. Thus the gluino gets a mass

6 m c = h a , I A I -10- 'asmw. (91)

In addition to Fig. 5, Fig. 6 gives a similar contribution

6mB- 10-'a, 1 A 1 - 10-'a,mw . (92)

As for the photino, the main contribution comes from Figs. 7 and 8, and a similar calculation leads to a mass

where a is the fine-structure constant. Since gluinos and photinos are massless at tree level, their wave-function re-

FIG. 6. Another supergraph inducing the hEh; term.

GEOMETRY AND PHYSICS OF WESS-ZUMINO SUPERGRAVITY 2471

FIG. 9. Supergraph inducing the Fu term, q =mA02.

tant point is that

h5m IAl -m l A ( >>mw 2

FIG. 7. Supergraph inducing the hyh, term, q=mA02

normalizations make no contribution to the mass corrections.

Now we turn to the problem of stability of the hierarchy set at the tree level by (77) and (78). One has to make sure that loop corrections will not destroy this hierarchy. In particular, one has to watch out for generation of the nondiagonal mass terms

J d 4 x I L 2 A H ~ ( a =4 ,5 ) H,:

(94)

in the action, where AHa and A are the scalar partners H.:

of the Higgs doublets H a and H i , respectively. If these terms are generated with masses much larger than M w , then the mass hierarchy will be destroyed.

In our model the troublesome contribution comes from Fig. 9. It gives a contribution

where mHi2=25m2k22 and A is a cutoff. So this means that in the bare Lagrangian one should have a term

and (95) means a renormalization for h5. Here the impor-

This generated term will cause a nondiagonal mass term

Since m / A / >>Mw, the hierarchy set at the tree level will be destroyed. Besides, Fig. 10 gives a contribution

This will cause a shift of Au and E;, and therefore generate a term

I d4x ( ~ A ' ) ' / ' A ~ , , A , , : , (99)

which destroys the hierarchy, because

At first sight, it seems that we can discard the term k 3 ~ H ' H and then there are no graphs like Figs. 9 and 10. But this case, as we have proven in Ref. 6, gives ( m I A 1 ) ' IZ-m w , so / A I is too small to give radiative masses to scalar quarks and scalar leptons.

As mentioned before, models with the minimal coupling also have an unstable It has been recognized that the dimension-three SUSY-breaking operator plays an important role in upsetting the hierarchy.16 It has also been demonstrated that in order to get a stable hierarchy, an SU(5) SUSY GUT model with the minimal coupling must contain at least 24, z, 50, and 50 representations in the heavy sector of the theory.16 Thus the Higgs structure in such a model is very complicated. Or, for a low-energy effective theory, one could break the SU(2) x U(1) symmetry radiatively.'' l 2

In our theory there is no dimension-three SUSY-

FIG. 8. Another supergraph inducing the h,hy term. FIG. 10. Supergraph inducing the A L ~ term, q = m h 0 2 .

2472 XIZENG WU

breaking operator. So simpler solutions are expected. First, if one can somehow eliminate Figs. 9 and 10, then the hierarchy will be stable. To do so, we revised our action to be

where I?' and fi are a new pair of 5 and 5* fields, m"-m, and other notations are the same as previously. Evidently, the vacuum is still SU(3)XUe,(1) invariant. But since there are no propagators for I? "Hi, g : 'Hi, I?;Hi, and Hil? ', the graphs like Figs. 9 and 10 cannot be drawn and a stable hierarchy is obtained.

The second way out is to make use of a heavy SU(3) x S U ( ~ ) X U ( ~ ) singlet. For example, we can use a new superpotential

where is a 2_4 representation and other previous notations are kept. Using a perturbation expansion of the sca-

FIG. 1 1 . Supergraph inducing the Fno term, 11 =mAe2.

lar potential in powers of A/m, as in the calculation done in the previous section, it is found that the vacuum is still SU(3) X U,,( 1) invariant and

where

a = 1-A/3hlm +(-2/3hl+3h22/8hlh42[hlh2(e/m + l /hl)2+eh1/3m- f ] )h2 /m2 ,

~ ~ = h ~ ~ / 8 h ~ ~ h ~ ~ [ h ~ h ~ ( e / m + 1/h1)2+ehl/3m - f ] m ( ~ / m ) ~ ,

A5'= -5h2/2h1h42[hlh2(e/m + 1/h)2+ehl/3m - f ]A2 ,

e r m (mf-m)/A .

This vacuum gives the correct hierarchy at the tree level, Since / A I -Mw, these small shifts of Ano and Fa, only if one chooses m =Mx and / A I -Mw. On the other cause changes in the VEV's of A,, and A at most of hand, all R$ get masses -m from the coupling Tr(2n2) . H6

In fact, if one parametrizes f i as the order of Mw, so the tree-level hierarchy is preserved.

where Q ji , a are 3 X 1 matrices, then

mwt=4h32m2, ma_ _2=h32m2 , x , y

2 ma+ '=ma3 =qh32m2, mno2=h32m2 .

In this model there are graphs shown in Figs. 11 and 12. They contribute terms like that given by (96) and (98):

mA J d4x d28ClO and m h 2 J d4x An, ,

respectively. But due to the huge mass of no it turns out that these terms only cause small shifts of An, and Fa,:

V. CONCLUSION

In this paper, starting from the superspace action of Wess-Zumino supergravity, we have shown that the Kahler manifold of scalars is the noncompact coset space SU(N, l)/[SU(N) x U ( l ) ] and the Kahler metric is a non-

FIG. 12. Supergraph inducing the A,, term, r ] = m A02.


flat Einstein metric. We find that this geometry leads to a remarkable physical result: the mass sum rule in Wess- Zumino supergravity is the same as that in global supersymmetry. This makes the point clear why a SUSY- breaking and vanishing scalar potential can be constructed. We also derive the effective theories with or without the G U T sector and show that the scalar mass terms and trilinear terms are absent among the soft-SUSY-breaking terms. It is pointed out that this structure of the soft- SUSY-breaking terms may shed new light on the gravitino mass problem. In a simple SU(5) model we show how scalar quarks, scalar leptons, and gauginos can get their masses radiatively. The stability of gauge hierarchy in this model is also discussed.

The investigation made in this paper demonstrates how intimately the superspace geometry, Kahler manifold

geometry, and the physics of N = 1 supergravity are relat- ed t o each other. Of course, so far we d o not know if there are deeper reasons behind the remarkable features of Wess-Zumino supergravity. One guess is that the noncompact symmetry exhibited here may come from the extended s ~ ~ e r ~ r a v i t ~ . " , ~ ~ On the other hand, we also have to work on more elaborate models for a full exploitation of what this theory provides.

ACKNOWLEDGMENTS

The author would like to thank Professor Ngee-Pong Chang for a helpful discussion. This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under Contract No. DE-AC02- 76ER03069.

'For a review, see P. van Nieuwenhuizen, Phys. Rep. 68, 189 (1982).

2J. Wess and B. Zumino, Phys. Lett. 66B, 361 (1977); 74B, 51 (1978); 3. Wess and J. Bagger, Supersymmetry and Supergrav- ity (Princeton University Press, Princeton, New Jersey, to be published).

3E. Cremmer, B. Julia, J. Scherk, P. van Nieuwenhuizen, S. Fer- rara, and L. Girardello, Phys. Lett. 79B, 23 (1978); Nucl. Phys. B147 105 (1979); E. Cremmer, S. Ferrara, L. Girardel- lo, and A. van Proeyen, Phys. Lett. 116B, 231 (1982); Nucl. Phys. B212, 413 (1983); J. Bagger and E. Witten, Phys. Lett. 115B, 202 (1982); J. Bagger, Nucl. Phys. B211, 302 (1983).

4J. Ellis, M. K. Gaillard, L. Maiani, and B. Zumino, in Unijica- tion of the Fundamental Particle Interactions, edited by S. Ferrara, J. Ellis, and P. van Nieuwenhuizen (Plenum, New York, 1980), p. 69.

SB. Ovrut and J. Wess, Phys. Lett. 112B, 347 (1982); J. Wess, in Unification of the Fundamental Particle Interactions 11, edited by J. Ellis and S. Ferrara (Plenum, New York, 1983), p. 343.

6S. Weinberg, Phys. Rev. Lett. 48, 1776 (1982); R. Arnowitt and P. Nath, ibid. 49, 970 (1982); X. Wu, CCNY Report No. HEP-82/17 (unpublished).

'For a review, see P. Nath, R. Arnowitt, and C. H. Chamsed- dine, in Lectures at the Summer Workshop on Particle Phys- ics, Trieste, Italy, 1983 (unpublished) and references therein.

8R. Arnowitt, in Talk at 5th Workshop on Grand Unification, Brown University, 1984 (unpublished).

9L. Hall, J. Lykken, and S. Weinberg, Phys. Rev. D 27, 2359 (1983).

'ON. P. Chang, S. Ouvry, and X. Wu, Phys. Rev. Lett. 51, 327

(1983); X. Wu, Ph.D. thesis, City University of New York (1983).

"E. Cremmer, S. Ferrara, C. Kounnas, and D. V. Nanopoulos, Phys. Lett. 133B, 61 (1983).

I2J. Ellis, A. B. Lahanas, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 134B, 429 (1984); J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B241, 406 (1984); C. Kounnas, CERN Report No. TH3785, 1983 (unpublished); J. Ellis, in talk at 5th Workshop on Grand Unification, Brown Universi- ty, 1984 (unpublished). In Ellis's talk the Kahler metrics similar to ours were also proposed in a different context, and the supergravity models proposed in the talk are not based on Wess-Zumino supergravity.

I3B. Zumino, Phys. Lett. 87B, 203 (1979); L. Alvarez-Gaume and D. Z. Freedman, in Unification of the Fundamental Parti- cle Interactions (Ref. 4), p. 41; see also Bagger and Witten's papers in Ref. 2.

I4D. Maison, in Deuelopments in the Theory of Fundamental In- teractions, edited by L. Turko and A. Pekalski (Harwood Academic, Chur, Switzerland, 198 1).

I5S. J. Gates, M. T. Grisaru, M. Rocek, and W. Siegel, Super- space (Benjamin/Cummings, London, 1983).

I6S. Ferrara, D. Nanopoulos, and C. Savoy, Phys. Lett. 123B, 214 (1983).

I7S. Ouvry, Phys. Lett. 136B, 165 (1983). I8J. Ellis and D. V. Nanopoulos, Phys. Lett. 116B, 133 (1983). 19H. Pagels and J. Primack, Phys. Rev. Lett. 48, 223 (1982); S.

Weinberg, ibid. 48, 1303 (1982). 20L. M, Krauss, Nucl. Phys. B227, 556 (1983). 21M. Gaillard et al., Phys. Lett. 122B, 355 (1983).

Documents

Geometry and physics of Wess-Zumino supergravity