Vacuum degeneracy in four-dimensional string theories

Nuclear Physics B (Proc. Suppl.) 5B (1988) 185-191 North-Holland, Amsterdam

185

VACUUM DEGENERACY IN FOUR-DIMENSIONAL STRING THEORIES

Hans Peter NILLES

Theoretical Physics Division,

The vacuum

structure

in ten space-time

CERN, 1211 Geneva 23, Switzerland

of the heterotic string

dimensions

(d = i0) is quite

simple and completely understood. supersymmetric

theories

The space-time

come

in

only

two

links

explicit

techniques make

use

string

constructions

of supergravlty of

the

fact

vacua of the field

with

that

the

supersymmetric

theory limit of d = 4 N = I

versions with gauge group Es×E8

and S0(32) res-

superstring theories are all degenerate.

pectively.

is

particular

This

considers Attempts string

"uniqueness"

theories to

if

one

in less than ten dimensions.

investigate

theory

lost

with

possible

elementary

relations

particle

of

physics

new of

string

"nearby" flat

the

string

Yukawa couplings

N = i space-tlme

supersymmetry.

In the language

any

theory

number

we

of

in

can

need

Given a test

for

scalar

the

potential

As

input of

the

the superpotential)

for

In

knowledge

of

from

the

classification of all (2,0) or (2,2) world sheet

for exactly marginal operators and the classifi-

supersymmetric

cation of multicritical points.

field

left-handed

tion

co-ordinates.

is not yet complete.

and c = 22 for This

classifica-

The historical

deve-

neracy for the examined models.

racies

that

construction

models we find. Usually such models are found by

others

point

the explicit

which

shows that the closer we look,

construction

the more

of a four-dimensional

Some of the flat

directions can be identified with vacuum degene-

exact

lopment

we

have

already

to

cannot

been

of s t r i n g

the

existence

yet

construct

observed

of

flcation of the d = i0 heterotic string.

covered models are quite exciting.

But we

possibilities; which

are

there

might

consistent

techniques

do

not

be

even

allow

string

if

us

our

to

models present

prove

this

consistency. In

this

to

break

Yukawa indicate models

talk

obtained

in

L.

and F.

Iba~ez

to obtain

I

want

to

collaboration Quevedo

present with

using

results

A.

a method

small gauge

with

these

groups,

symmetries

and

they

gauge

Preliminary might group

the

models,

newly

The dis-

They allow us

rank gauge

couplings. that

of

new

explicitly.

phenomenological

prospects

in

t h e o r i e s 2. But

string model in terms of a generalized compacti-

do not know whether this procedure exhausts all

to a search

Our results indicate an enormous vacuum dege-

6R+22 L dimensional target space, i.e., ~ =

(2/3)c = 6 for the right-handed the

theories

amounts

of con-

formal

superconformal

this

language

of conformal field theory this would amount to a

with

theory,

then

by an examination

theory.

(i.e.,

fields.

field

the

field

ture

we

theories

directions

would require the knowledge of the vacuum struc(moduli space) of d = 4 string models with

theory

string

underlying

the

field theories I. We

smooth ways

flexibility

for

investigations

contain

realistic

SU(3)xSU(2)xU(1)

and

three generations of quarks and leptons.

Font,

But before we enter these details let us look

that

at the "modull space" of the d = 4 N = i string

0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

H.-P. Nilles / Vacuum degeneracy

186

models.

Inspired by Ref. 3, consider Fig. 1 as a

basis

for

discussions.

We

can

distinguish

actually tic

somewhat

string

by

ill-defined itself

is

since the heteroalready

left-rlght

asymmetric.

several classes of models.

2. COVARIANT LATTICE CONSTRUCTIONS 8'9 DEGENERATE CALABI-YAU

LATTICE CONSTRUCTION

In this method the geometrical interpretation is not as transparent as in the case of orbifold compactification; nection

to

theory.

well

as

conformal then

lattices

bosonized

modular

using

so

invariant

however,

is

and

the

obtained

by the

field

of self-

the extra hosons

fernLions

The

con-

conformal

is the construction

ghosts.

involved,

has a closer

two-dimensional

Its basis

dual Lorentzian as

the method

super-

models

are

construction. construction

More

of

the

FIGURE 1 world ....Moduli space" of d = 4, N = i heterotic string theories

sheet

supercurrents.

at the moment

can be visualized points

that these models

are a

that could be obtained via

orbifold compactification although this question

i. ORBIFOLD COMPACTIFICATIONS 4-7

on which

with

(2,0) and (2,2) world sheet supersymmetry can be

It seems

Orbifolds

models

found. All models have a gauge group of rank 22.

subset of the models

manifolds

Again

geometrically

have

as

been identified

is

not

completely

speculate

that

c l a r i f i e d I0'II.

this

method

due to the action of a discrete symmetry group.

models

We are interested here in "flat orbifolds"

under consideration).

that

with maximal

rank

gives

(i.e.,

One might

all

orbifold

22 in the case

can be obtained by dividing a six-torus T6 by a point

group

everywhere

P.

These

and

provide

for string theory. with The

(2,0) (2,2)

or

are

flat

a consistent

almost

background

They can give rise to models

(2,2)

models

objects

world

sheet

correspond

supersymmetry°

to

the

so-called

3. FREE FERMIONIC CONSTRUCTIONS 12-14 The

method

uses

Then

various

twisted

on

these

gauge

traints

of

modular

larities

can

manifold; overlap

in

be the

corresponding blown

up

diagram

orbifold

singu-

to give a Calabi-Yau they

region of "orbifolds"

constitute

the

and "Calabi-Yau".

Asymmetric orbifolds 7 are those models where the

"bosoniza-

6R+22 L bosons are fermionized.

imposed

The

of

tion" in two-dimensional field theories. All the extradimensional

standard embedding of the spin connection in the group.

the existence

supersymmetry. to be contained fold

boundary

fermions

invariance

Again

these

conditions

satisfying and

world

sheet

are

believed

in the class obtained

by orbi-

compactification.

For

models

are

the cons-

details

consult

the

talk of Antoniadis at this Conference.

left- and right-handed string components live on different

orbifolds.

In

that

sense,

one

can

regard d = 4 strings on orbifolds as a generalized

eompactification

string.

The

of

the

definition

of

d =

i0 heterotic

"asymmetric"

is

4. TENSOR PRODUCTS OF c < 3 MODELS 15'16 This

method

uses

superconformal

models

from

the discrete series with central charge c < 3 as building

blocks

for

the construction

of

space-

H.-P. Nilles / Vacuum degeneracy

time

supersymmetrlc

are tensored maly

models.

The

models

to give the correct conformal ano-

and further

riance

string

have

constraints

to

constructed

be

models

supersymmetry.

from modular inva-

satisfied. all

have

Although

Up

to

(2,2)

this

now

world

method

the sheet

is

187

investigate theory

whether

on

one

could

asymmetric

for~mlate

Calabi-Yau

string

manifolds.

Details of the construction and a description of the

properties

of

Calabi-Yau

manifolds

can

be

found in the talk of L~tken at this Conference.

not 6. DEGENERATE CALABI-YAU MODELS 22-24

very

geometrical

in

its

been

conjectured

that

construction,

this method

it

gives

has

equivalent to conformal field theories on smooth Calabi-Yau

By

models

manifolds 17. It seems, however, more

this we mean

that

lead

to

supersymmetry

strings

models which

on smooth manifolds

with

are

(2,0)-world

continuously

sheet

connected

to the (2,2)-models given above. The geometrical likely

that

orbifolds, simple

also as

examples

those

has 18

been

could

be

correspond

demonstrated

, leading

that the conformal folds

models

in

to some

construction of such manifolds seems to be quite c o m p l l c a t e d 22'23.

In i t e r a t i v e

obtain

is

procedure

to

to the c o n j e c t u r e such

models

again

provided

via

the

field theories on some orbi-

equivalent

to

those

of

some

construction

of

orbifolds

and

a

distortion

of

some parameters as discussed by Cvetic 24. smooth Calabi-Yau manifolds. While this interesting far

conjecture from

might

being

be correct,

able

to prove

we are still

it.

The

blob

in

7. DEGENERATE ORBIFOLDS I Here

Fig.

I would

have

to

be

readjusted

once

one

starts

with

any

orbifold

and

this examines the flat directions of the potential of

situation is clarified. the massless 5. CALABI-YAU MANIFOLDS 19' 20 The considerations class

of Ricci

of strings moving on this

flat K~hler manifolds with SU(3)

holonomy has been at the origin of many phenomenologically

inspired

investigation.

Using

the

standard embedding of the spin connection in the gauge group

this is believed

string background with symmetry.

We

explicitly

are

the

full

to be a consistent

(2,2) world sheet super-

not

yet

able

string

to

theory

construct on

such

a

smooth Calabi-Yau background and have to rely on topological and

properties

sense this

arguments

it

is

of

not

background

string

models tively

on

the massless

The most

Calabi-Yau

backgrounds.

any

an exact

In that

doubt

that

solution

convincing

argument

for

from It

manifolds through

could

be itera-

a

"blowing-up"

(2,2)-models

on o r b i f o l d

would

also

all

the parameters

be

interesting

theory.

to

This

leads

to

new

models

both

in

case of symmetric and asymmetric orbifolds.

of

the Some

of the models can be given a geometrical interpretation folds.

as blown up versions

The models

turbatively massless exact

the

fields,

Given

one

(2,0) mani-

can be constructed

only per-

but

they

are

believed

to

be

in the same way as Calabi-Yau

the central

construction

models

for

in the vacuum expectation values of

solutions~

models.

might

of

rSle

of orbifolds

lower-dimenslonal

conjecture

that

eventually

the

in

string class

of

all

of

degenerate

orbifolds

the moduli

space of d = 4 N = 1 string models,

covers

although Fig. i does not yet indicate this. As discussed

in

is given by the fact that string

constructed

p r o c e d u r e 21

the number

states.

beyond

provides

theory.

this direction

that determine

proved

states with

this potential computed in the underlying string

above,

to 4 can be explicitly in classes in

the

fields. models

It

in classes

constructed

while

i

those

5-7 can only be obtained iteratively

background

in

the models

is

not

classes

values so 6

of

some

surprising and

7

massless

that

exist,

these

what

is

H.-P. Nilles / Vacuum degeneracy

188

surprising

is the huge number of such theories.

Even if we cannot construct

these models expli-

fermions cannot be continuously connected to the above

model.

the other hand,

we are not allowed to ignore their exis-

tence.

Such continuous adjustable parameters are

also welcome 25 in the discussion of the phenome-

weak interactions should proceed in a smooth 27 way • As a S t r a t e g y to c o n s t r u c t a realistic

nological

model

issues

a

consult

rence). us

prospects (for

to

of

four-dimensional

detailed

the talk

discussion

string

of

these

of Ross at this Confe-

The method used in cases 6 and 7 allows construct

a

full

class

of

continuously

breakdown

one

of

would

discussions

of

SU(2)×U(1)

therefore

a

"topological"

the

we would argue

citly,

theories

the

On

that

group

start

with

of

the

string

model

with

desired

properties,

e.g.,

the

correct

number of generations our knowledge,

of quarks and leptons.

To

there is no smooth way to change

connected models. Certain topological properties

such a chiral structure like the number of gene-

like

rations.

a

chlral

sentation Whenever this

cannot

of

the

be altered

fermion

in this

repre-

procedure.

a gauge group is broken in such a way

will

Higgs

structure

happen

in a smooth

mechanism.

transitions the case

had

way exhibiting

Examples been

found

of

such

already

a

smooth

earlier

in

of torus compactification 26 as well as

in the case

of orbifolds and Wilson lines 2. It

seems

to

often

related

us

that

such

to

smooth

mechanisms

transitions

that

allow

us

are to

lower the rank of the gauge group. In

contrast

to

this

smooth

A word of caution has to be added here

since

string

smooth

ways

radius

theory

of

a

compact

discussion.

we

then have

value

is

tally

different,

breakdown

mechanism

of

They occur whenever quantized

just

having,

structure.

and

for

A

zero

or

the

potential

continuously

and

identify

connected

structure

the flat directions the

models.

full

class

After

in of

breakdown

tion

of

a

unique

models

example,

gauge

class.

vacuum

This

only

is the

constructed

within

recipe

such

a

to follow.

string model and

consider non-vanishing vacuum expectation values (vev's) for the massless states that satisfy

a W = F = D -- 0

symmetry

(i)

of this type will not exhibit a Higgs

mechanism; are

like

chiral

to

of supersymmetry we would then expect the selec-

for different discrete values can look fundamen-

different

go

"topological"

to examine

connected

background

letting the

Having found now such representative

Take an explicitly

some

"singular"

by

dimension

models with the desired

supersymmetry breakdown string theories can also transitions.

some

chirality

inflnity 28. We exclude this possibility from our

exhibit

abrupt

allows

to change

in general projected

some

from

of

the gauge bosons

the massless

spectrum.

Here

W

represents

the auxiliary

the

fields

superpotential

and

F(D)

of ehiral and gauge super-

multiplets Such an abrupt breakdown can, of course, revealed tions such

through

in

abrupt

logically =

IO,

N

connected obtained extra chiral

the

an examination

potential. transitions

to by

I, the torus

dimensions. fermions

EsxE8 d

=

flat direc-

Nonetheless to

acceptable models. =

of

obtain

N

is =

4,

and

a model any

(2)

the d

and

continuously Es×E 8 of

does

model

~W Fi =

needs

phenomeno-

compactification Such

one

For example,

model 4,

not be

not

with

model

the

D

*Ti~ = ~i ~j

j

(3)

six

allow chiral

i

where

(T)j

gauge

group.

represent It

is

the g e n e r a t o r s

advisable

to

first

of the satisfy

H.-P. Nilles / Vacuum degeneracy the

equation

solution the

of

for

the

D-terms.

these equations

l i t e r a t u r e 29 and

here.

anomalous

U(1)-group

where terms

term

for

necessarily

vev's

for

some

non-trivial known

arises

cancellation

lliopoulos then

have

I shall

A complication

anomaly

a

Fayet-

U(1).

This

appearance

of

fields that have

this

term,

of an

Green-Schwarz

the

under

a

them

case

anomalous

scalar

a

the

induce

the

such

repeat

the

to

charges

cases

this

for

been given in

not

in

leads

of

Methods

U(1).

however,

189

in the untwisted sector consists of three copies of

(84,1~+(1,14)_2+(1,64)i

(7,1)#/3

at

each

of

and

the

we

twenty

have

seven

one

fixed

points in the twisted sector. The 84 of SU(9) is the

three-index

Consider a v e v

antisymmetric



=

tensor

=

X ijk.

= 0 for

one of the three 84's. The equality of the three vev's are

ensures

not

the

induced

absence

since

of

the

D-terms.

only

F-terms

allowed

Yukawa

In

all

couplings are E=~y 84= 84~ 84y excluding a self-

does

not

coupling

of

gauge group SU(9) is broken to SU(3) 3. This flat

general in

directions.

one can easily satisfy D

fact

m~ny

solutions.

investigation which

flat

= 0 and finds tricky

is

the

of the equations W = 0 and F i = 0

requires

couplings

More

In

the

and

calculation

this

of

the

Yukawa

is also the place where

the

direction was already tigations models

field a

that

situation

actually

where

we

fields

in the untwisted

dure

with

these

an

explicit

couplings

Observe

that

the

trilinear

but

one mast

it

to start

string

can

be

is not

model

where

computed

enough

(renormalizable) have

this proceall

exactly.

to just Yukawa

compute

couplings

a control over all couplings

consider

correspondence the

method

field

perturbative

it does

points

which

occurs

exclusively

actually

low energy

and

constructed

string

as a background 2 can be v a r i e d continuously .

It is

important

the

known from earlier inves-

explicitly

information about the full string theory enters. therefore

in which

and could be understood

gauge Such

in

direction

vev.

This

the

a flat

a non-vanishing

the

of

gives

84 with

induce supersymmetry breakdown nor does it alter discussion

then

that

in all

vev's

cases

for

the

sector.

This one-to-one

reveals

a limitation of

theoretic method. in the vev's

of

It is a

the fields

not allow us to find multieritical are

away

in

explicit

strings propagating on orbifolds can be computed 30 the tools of c o n f o r m a l field theory

ground

Flat

the perturbative mechanism.

directions of

superpotential. pondence with tion

in

certain

conformal

investigation

the

Let

us

become

just

consider

the procedure.

the back-

New models - actually such models that cannot

corres-

of certain correla-

can be found along flat directions for vev's of

the

quite

of

in

exactly

corresponding of such an

complicated

and

we

refer

to Ref.

i.

a

simple

example

to

We choose

this

the twisted this

would

several

of

sector fields. correspond the

9's.

to

In the present model vev's

for

one

to be

embedding and

The charged matter

or

The dangerous W-terms that

might prevent flat directions are of the form

[84 x ~3 x 142] n the Z3-orbifold with non-standard gauge group SU(9)xO(14)xU(1).

of

gauge field 2 which one cannot recover in

yet be constructed as a complete string theory -

The details

discussion

the

value

present

the

presence

in

theory.

can

a thorough

illustrate

in

operators

for

require

couplings

These are in one-to-one

field

here

potential

Yukawa

the vanishing

functions

marginal

the

large

the

The

model shows an enhancement of the gauge group to SU(3)3xU(1) 2 for some

of

space.

scattering amplitudes for the physical states of

absence

construction

field

with an arbitrary number of external fields. The

using

string

far

(4)

H,-P. Nilles / Vacuum degeneracy

190

Giving

a vev to one of the 84's and one of the

14's we

find many

the 9's

provided we only allow certain combina-

tions for

flat directions

of fixed points details

for vev's

of

Generically

Ref.

i.

SU(9)

is

broken

of

new

the existence

consistent

connected

to

string

already

of large classes

models,

known

those

up v e r s i o n s

models

that we

(2,0)

Others

models.

above

do

not

Some

previously

of like

of (2,2)-models 21. Among

allow

interpretation

continuously

models.

these models had been considered the blown

found

a new

The

parameters and

yet

allow

the

flat

models rank

blow

up

described

geometrical

prospects They

in

explicit massless

the

vev's

directions.

fields,

of

Some

the

gauge

prefer

gauge

group

the

need

additional

symmetries

particularly SU(2)xU(1) metrize

giving a

us

small

for the

the

fields

along of

the

importance.

The

The a

smooth

to

Yukawa

such

varied.

higher-order trilinear

orbifold and made

fields

receive

a

flat

couplings

mechaWith

directions, can be

we

conti-

this happens because of

renormalizable

effective

the

realistic

the new

Usually

para-

of flexibility

to have a satisfactory

that

of

Previously

quark and lepton masses.

find

way,

breakdown

nism to generate

of

It

breakdown

it very difficult

the observation

all

least

opportunity

parameter.

a lack

of at

that superstring

in

of potentially

revealed

nuously

the

couplings

couplings,

non-vanishing

leading to

once some of the vev.

Together

three

These

in view

rank

of

tude

of

hypercharge.

deserve

gauge

proton

an

group.

all

fact, 3 of

U(1)'s

corresponding

Additional

and

In

to

There exists a multiwhere

that

decay

and

further

mechanism

show that model

become heavy in this process. with

of quarks

new

our

directions for

group

families

models

the

except

gauge

of

investigations

flat

a large number

with

Ref. 31 is very promising.

the

colour-triplets

Potential problems

neutrino

examination

are

to

of

masses

the

can

fermion

be

mass

matrix is under way.

new

not be large.

U(1)'s.

appears

suited

investigation models

these us

properties

less and less convincing

models

of give

calculation

the

seem to be of particular of

seems

of

an

among

perturbative

lower

preliminary

correct

simple

exciting.

constructed

models

l e p t o n s 31.

solved

quite

we might now try to understand

ago we

consideration

to

ones

as in the case of Calabi-Yau models,

couplings

and

ways

a

phenomenological are

time

string

geometrical

interpretation.

models

Some of

nice

the

suppressed

the quark and lepton masses.

broken~

like

have

of exponentially

SU(3)xSU(2)xU(1) 5 and

we find many of those flat direc-

indicating

Yukawa couplings,

to appear in this choice;

consult

completely with this procedure.

tions

with the possibility

On

the

more

interesting the

moduli

should, tion

theoretical

to obtain space

it would

be

a better understanding

of

sketched

of course,

fields)

degenerate might

cover

of

the

the

might

models

which space.

in string

fled on Calabi-Yau

manifolds

the question rent

to what

geometrical

valent

world sheet.

This

of

the

class

of

eventually We

need

a

field theories

theories

compacti-

to finally examine

extent

structure

superconfor~l

vev's

moduli

appear

I.

construc-

large

better control of those conformal that

Fig.

a stringy

in

of

orbifold all

in

include

(non-perturbative

massless

side,

models might

field

with diffe-

lead

theories

to equion

the

H.-P. Nilles / Vacuum degeneracy

191

17. D. Gepner, Exactly Solvable String Compactifications on Manifolds of SU(N)-Holonomy, Princeton Preprint PUPT-I066 (1987).

REFERENCES I. A. F o n t , L.E. I b a ~ e z , H.P. N i l l e s and F • Quevedo, Degenerate Or bifolds, CERN Preprint TH. 4969 (1988), to appear in Nucl. Phys. B. 2. L.E. Iba~ez, H.P. Nilles and F. Quevedo, Phys.Lett. 192B (1987) 332. 3. L. Dixon, Some world sheet properties of superstring com@actifications, on orbifolds and otherwise, Princeton University Preprint PUPT-1074 (1987).

18. L. Dixon, Private communication. 19. P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl.Phys. B258 (1985) 46. 20. For a review, see: Class of Calabi-Yau TH. 4900 (1987).

C.A. LUtken, Vacua, CERN

A Large Preprint

21. See, for example: M. Cvetic and L. Dixon, Blown-up Orbifolds, SLAC Preprint PUB-4113 (1987).

4. L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl.Phys. B261 (1985) 678 and Nucl. Phys. B274 (1978) 285.

22. E. Witten, Nucl.Phys. B268 (1986) 79.

5. M. Mueller (1986) 28.

23. J. Distler and B. Greene, String Compactifications, HUTP-87/A065 (1987).

and

E.

Witten,

6. L.E. Iba~ez, H.P. Nilles Phys.Lett. 187B (1987) 25.

Phys.Lett.

182B

and F. Quevedo,

7. K.S. Narain, M. Sarmadi and C. Vafa, Nucl. Phys. B288 (1987) 551. 8. W. Lerche, D. LUst and A.N. Nucl.Phys. B287 (1987) 477.

Schellekens,

9. For a review, see: W. Lerche and A.N. Schellekens, The Covariant Lattice Construction of Four-Dimensional Strings, CERN Preprint TH. 4925 (1987). i0. D. LUst and S. Theisen, Four-Dimensional Strings - Orbifolds and Covariant Lattices, Preprint MPI-PAE-PTh83/87 (1987). ii. A.N. Schellekens and N.P. Warner, Weyl Groups, Supercurrents and Covariant Lattices, CERN Preprint TH. 4916 (1987). 12. H. Kawai, D.C. Lewellen and S.-H.H. Tye, Phys.Rev.Lett. 57 (1986) 1832 and Nucl.Phys. B288 (1987) i. 13. I. Antoniadis, C. Bachas Nucl.Phys. B289 (1987) 87.

and

C.

Kounnas,

14. A method similar in spirit can be found in: A. Chamseddine and J.P. Derendinger, Nucl. Phys. B301 (1988) 381. 15. D. Gepner, Nuel.Phys. B287 (1987) iii. 16. K. Bardakci, E. Rahinovici and CERN Preprint TH. 4760 (1987).

B.

S~ring,

Aspects Harvard

of (2,0) Preprint

24. M. Cvetic, Phys. Rev.Lett. 59 (1987) 2829. 25. B. Greene, K.H. Kirklin, P.J. Miron G.G. Ross, Nucl.Phys. B278 (1986) 667.

and

26. K.S. Narain, Phys.Lett. 169B (1986) 41. 27. L . E . Iba~ez, J. Mas, H.P. N i l l e s F. Quevedo, Nucl.Phys. B301 (1988) 157.

and

28. H. Itoyama and T.R. Taylor, Phys.Lett. B186 (1987) 129. 29. F. Buccella, J.P. Derendinger, S. Ferrara and C.A. Savoy, Phys.Lett. II5B (1982) 375; R. Gatto and G. Sartori, Phys.Lett. 157B (1985) 389. 30. S. Hamidi and C. Vafa, Nucl. Phys. (1987) 465; L. Dixon, D. Friedan, E. M a r t i n e c S. Shenker, Nucl. Phys. B282 (1987) 13.

B279 and

31. L.E. Iba~ez, Jihn E. Kim, H.P. Nilles and F. Quevedo, Phys.Lett. 191B (1987) 282.