Partkzb.s and Fe s

~r PhysikC

Z. Phys, C - Particles and Fields 33, 255-261 (1986)

9 Springer-Verlag 1986

A Model for a Light Graviphoton R. Barbieri and S. Cecotti Dipartimento di Fisica, Universita' di Pisa and I.N.F.N., Sezione di Pisa, 1-56100 Pisa, Italy Received 23 June 1986

Abstract. We describe an explicit N = 2 supergravity model where an arbitrarily light vector boson (" graviphoton") is coupled, with typical gravitational strength to matter hypermultiplets, possessing unbroken gauge interactions as well. We discuss: i) the mass and the couplings of the graviphoton; ii) the consistency of its coupling to a mass generated by the Higgs mechanism; iii) the actual composition of the graviphoton in terms of the original vector fields of the Lagrangian (e.g. its mixing with the photon).

1. Introduction

As especially emphasized by Scherk [-1], "the most obvious feature of extended supergravity models is that gravity in general will proceed not only through a tensorial exchange, but also through vectorial (and scalar) exchanges". The manifestation of such exchanges in a modification of Newton's law at short distances might actually lead to the discovery of supersymmetry in quite a different way than currently discussed via high energy collisions. Extended supergravity theories appear infact as an ideal framework where the observation of some "gravitational anomaly" can in principle be accomodated. Perhaps the geophysical determinations [2] of Newton's constant and their possible discrepancy with the Cavendishtype measurements of the same quantity - already constitute an example of such an observation. In this work we describe an explicit N = 2 supergravity model where an arbitrarily light vector boson (called "graviphoton") [l, 3] is coupled, with typical gravitational strength, to matter hypermultiplets, possessing unbroken gauge interactions as well. Supersymmetry is spontaneously broken and the cosmological constant is naturally vanishing (at the tree level).

As it is well known, the construction of a realistic model of all particle interactions based on a Lagrangian with an extended (N > 2) supersymmetry is a problematic task. This may not be however such a serious difficulty, since the relevant supersymmetry can be broken at a very large mass scale (~Mp). On the other hand a very light gravitationally coupled vector boson may be the (only) low-energy remnant of such an extended supersymmetry. This situation can even be compatible with a residual (independent) low energy N = 1 supersymmetry, as incorporated in popular models [4]. Leaving these general considerations aside, a number of more specific questions can be raised and addressed to, already in the model that we develop. The main ones are related to: i) The effects of supersymmetry breaking; ii) The consistency of the coupling of the "graviphoton" to a mass generated by the Higgs mechanism; iii) The actual composition of the "graviphoton" in terms of the original vector fields of the Lagrangian (e.g., its mixing with the photon); iv) The mass and the couplings of the "graviphoton". The answers that emerge from the model under consideration may have, at least for some of these questions, general validity. The plan of the paper is the following. We describe in Sect. 2 the Lagrangian of the model. In Sect. 3 we calculate the physical masses of the particles, in particular the gravitini and the vectors. In Sect. 4 we study the mixings and the couplings of the vectors to the electron. We consider in Sect. 5 the special situation for a light graviphoton. Conclusions are given in Sect. 6. The Appendix contains a detailed description of the two local supersymmetries of the model and its relationship with the superconformal tensor calculus.

256

2. Description of the Model The Lagrangian is a suitable extension of the one previously written down by Cecotti, Girardello and Porrati [5] to discuss an N = 2 supergravity model with arbitrarily split gravitini in Minkowsky space9 We simply introduce in their Lagrangian a "visible" sector, made of matter hypermultiplets in interaction with vector multiplets gauging an unbroken group G. The model so obtained has a positive semidefinite potential and supersymmetry broken (with vanishing cosmological constant) for any value of the scalar fields inside the domain of definition of the various kinetic terms9 In this sense it is reminiscent of the models with "sliding" gravitino mass already discussed [6] in N = 1 supergravity. The Lagrangian is not described by the usual N = 2 tensor calculus [7], but it is rather obtained by a suitable "contraction" [5] of standard anti-deSitter supergravity models with matter (see the Appendix). This is the only known way toget split gravitini and vanishing cosmological constant in N = 2 supergravity [8]. In view of the relation between the graviphoton mass M and the gravitino masses m L 2, M=lm~-m21, (see Sect. 3) the non-degeneracy of the gravitini is required to describe a light but not strictly massless graviphoton. Moreover, the "contraction" procedure is needed if we want an hidden sector with a "flat" potential (i.e. a no scale model) in presence of hypermultiplets, in turn essential [5] to obtain in N = 2 supergravity a vanishing cosmological constant without spin-0 tachyons. The origin of the model is anyhow not particularly relevant, since it is simply described by its physical fields and the Lagrangian. The physical fields can be grouped into: i) The minimal N = 2 supergravity multiplet, made of the vierbein e~,, a pair of gravitini r i = 1, 2, and a vector field Gu . ii) An "hidden" sector composed of a vector multiplet (H,, A~, q)), i = 1, 2, and a hypermultiplet ({=, b~), i = 1, 2, e = 1, 2. (As in the standard N = 2 tensor calculus, the 2 complex scalars of the hypermulfiplet are organized into a 2 x 2 matrix satisfying the "reality" condition (b~)* - b~-- eij ~P b~). iii) A "visible" matter sector including, as mentioned, matter hypermultiplets and vector gauge multiplets. For concreteness we shall deal with one "electron" hypermultiplet (e,, E~) and a " p h o t o n " vector multiplet (A,, 2~, o)). To get split gravitini, the hidden vector multiplet gauges, with coupling g, the internal SO(2), acting on the index e, of the hidden hypermultiplet. We shall call e the gauge coupling of the photon to the electron

R. Barbieri and S. Cecotti: A Model for a Light G r a v i p h o t o n

supermultiplet, a situation easily generalizable to any number of "visible" matter multiplets with gauge group G. For simplicity we consider "minimal" kinetic terms for all the vector multiplets. The model depends on a dimensionless angle 0, which can take any value different from k rc with k an integer9 As we shall see from the Lagrangian, 0 ~ kg is a singular limit which requires a (singular) redefinition of the fields. Nevertheless, the physical quantities to be discussed below (masses, properly normalized couplings) are continuous in this limit, so no special care is required9 Finally the model includes also a supersymmetric and groupinvariant mass parameter m for the "electron" hypermultiplet. The physical electron mass will also include a contribution from a Higgs-like term, which may infact be the only source of mass (m = 0). For the present discussi6n, the relevant pieces of the Lagrangian are: i) The (generalized) kinetic terms for the various fields (in the notation of [7]) e- 1 ~eki"= - 89189

7~ + h.c.)

+ C - 2 [ - l~3u qgl2+ co c3u~ bu tp + got3u ooc3uCo - ( C + Icola) la. col2] + F - 2 [ _ ID. b[=+ E~ D. bTD"E ~ b a ~ j i p ct t~ J + E~,DuEjD bb-(Fc~/~ + E~Ei) DuEjD Et~]

C -1

_i~.,

.~

+E=e i~{b+(Fa ~ 1 [2~O-e-2i~

G + g+t~v

-4L- si- O 2 cos 0r176 _ _ 4 sin 2 0

2 G+/4+u~

~u~ ~-

ie-i~ + 2p--co = H+ H+UV sin 0 Gu~F+"V+ 4 sin 2 0 --u~-4.__H

~_ + _ sin 0 _ . ~+ _F + U ~ +2Fu~F+'~+h.c.

c=~o+@-I~ol z, i

1

D, b~ = Ou b ~ - ~ D. Ei~-- 0. E ~-i

(~

F=Trb-- 89

(1)

2,

(2)

i

(H u e l i - G u a~2~),

m

). = #

A u + ~ G u tazpEi,

~, / e m \ D. e" = 17 e - ~ A. + ~- C.} i o-~peft.

(3)

R. Barbieri and S. Cecotti: A Model for a Light Graviphoton

257

V, denotes the Lorentz covariant derivative (with Otorsion) and, for the field strengths G,,, Hu,, Fu~ associated respectively to G,, H , and A u one has V~+ -- 89

L~=89

+ ~,),

V ~.

The full Lagrangian invariant under the supersymmetry transformations given in Appendix, includes derivative, Pauli and 4-fermion terms as well.

(4)

3. Masses

(We have added to the gravitino covariant derivative the composite U(2)-connections (see Appendix), which give rise to some of the derivative couplings present in the Lagrangian). ii) The scalar potential term

As seen from the scalar potential, in the classical vacuum state, the scalar electron fields do not get any vacuum expectation value, E~=0, whereas all other fields, go, b~ and co are left undetermined. They "slide", as so do the gravitino masses

e l~scal__~ - - V

ml = 2 g C - 1/2 F - 1(1 + sin 0) 1/2 : --C-1F-1]m+I//2ech]2[EI2--2e2F-ZIE[ 4.

(5)

m2 = 2g C - 1/2F- 1(1 -- sin 0) 1/2

iii) The bilinear terms in the fermionic fields e ~~f f = -- 2 ]/2g C - 1 F -

3/2

(7)

always different from zero for finite C and F, (2), as required by a positive definite kinetic Lagrangian. The ratio of the two gravitino masses is on the other hand only determined by the angle 0. For arbitrary values of the sliding fields, the kinetic Lagrangian couples in general the three vectors

(.~j __ 05~j) Aj b(~b _.[_E~ e ~)

+ 2 m C - ~F - Zeu(HJ--0) Zi) - E / e~ [e ~ + F - ' (E~ ~" + E~# E~ e~)]

--2 ]/2e F - 1/2 eij[~ + C-~ c.5(0)2 j - AJ)]

=-(Gu, H u, Au)

9 E~ e~ [e # + F -~ (E~ ~a + E/~ E~ eV)]

with a non diagonal metric matrix

Vj]/v 2 (go+ qS)- (e- 2i00)2 _~ e210c52)

z= 89

t_

- 2 cos 0(go + ~) + (e- i~ 4 sin 2 0

+

2 cos 0 (go+ ~ + (e-io 0)2 + e~O052) 4 sin 2 0

+ ei~ 2)

9ei~176

\

2silo /I

+ 4 sin 2 0

i(eiOcb-e-iO0))

(8)

2 sin 0 and a diagonal mass matrix

+ 2 V 2 g c - 1 / 2 F - 2 ff~ +

oa) Aob

9(~b + E~ e ~) -- 2(m + ~/2e oh) C - 1/2

1

V"J

M 2 = 2 g ; F -2

1

9 [Y + F - 1 E ] ( ~ - + E~ ~)]

.

(9)

0

96a~ [e t~+ F - 1Eg(~b + E b e~)] The mass eigenvalues me, m o satisfy

--2 l / 2 g C - 1/2 F -1 ~i. y Aia(~, + E~ e 7)

det(ZmZ-MZ)=O,

-- 2(m + ~/2ecb) C - 1/2 eU ~--i . TEaj e~r

9[ e r

[Ak + o3C- l(co, k+ Ak)]

m+ = Ira1 + m21

'Pua T { [ g A u Aij= t~ijW ie-i~ tTaij.

-~ h.c.

(11)

The massless eigenvalue is a consequence of the residual gauge invariance. (The electron scalars are the only non-neutral scalar fields under "electromagnetism"). Notice also the relation

"ii ~ p "2~ E~ ~ J . y

9 --le0)ejk Eia f l E a ~ E ~k, ]

m+ = 2 [ / 2 g C - 1 / Z F - l ( 1 +__cos0) 1/2 mo -- 0.

ie

9

(10)

which allows to determine them. They are

~i. 7

" (]//~ Au + rne-i~ eik Ekp a~rEQ(AJ--0) 2j)

----

r e = m + , m_, mo

(6)

(12)

between the massive vectors and the gravitini. As to the remaining physical fields, in the hidden sector we have two massive Majorana fermions de-

258

R. Barbieri and S. Cecotti: A Model for a Light Graviphoton

generate with the gravitini (the other two fermions make the massive gravitini) and 4 massless (real) scalars (since here too the remaining 2 real scalars are required to give mass to the 2 vector bosons). In the "visible" sector, supersymmetry is effectively unbroken, at least at the tree level, with a full massless photon multiplet and with a mass for the electron me= C-1/2

im+ l/~ecol"

(13)

We call the term proportional to eco in (13) a Higgsgenerated mass component. One readily checks that these masses, in view of (12), satisfy the sum rule Tr(--)F

2=0

(14)

for the cut vector propagator, w here Ira> denotes any of the three states corresponding to the different mass eigenvalues, (15), up' to irrelevant normalization factors Im+>=

Ira->=

--

Imo>

=

9

(17)

From

1 C 2 1 + c o s 0 ' = 89

(18)

and the explicit expression of the covariant derivative acting on the electron field one gets Q2 =21__(1 -[-COS 0) C -11m +]/2ecol 2

4. Mixing and Couplings of the Vectors

=89 +cos 0) meg

For the 3 mass eigenvalues m = m_+, mo of the vectors, the matrix

ZmZ--M 2

annihilates the corresponding eigenvectors, which can therefore easily be computed (up to a normalization factor). Let us discuss first the (simpler) case in which the vev of co is real. We have

m+ --+ Gt, + Hu + COAu m_ --+ Gu-Hu+COA • mo ~ A,.

(15)

Note that the three linear combinations in (15), corresponding to the mass eigenstates, are actually orthogonal - as they should - with respect to the scalar product induced by the matrix Z. However, if co t 0, the original fields (G~, H~, A~) are not Z-orthogonal. Then the mass eigenstates are related to the gauge connections by a non-orthogonal linear transformation. By this mechanism, as we are going to see, the massive vectors, through their Au-component, couple to the Higgs-generated mass of the electron. On the other hand, the fact that the massless vector remains a pure Au-field keeps the "electromagnetic" coupling universal. The squared couplings Q2+, Q2_, Qg of the electron to the physical vectors of definite mass m = m +, m _, mo respectively, are obtained from the residues a t the pole at p2 = m 2 of the one-vector exchange amplitude in electron-electron scattering. A quick way to get them is by making use of the representation

2~zia(ZPZ-- M2) =

Z

m=m+,m-,mo

2ni6(p2--m 2) [m> (m[ Zlrn> (m] (16)

Q~=e 2.

(19)

The gravitational-like coupling to the electron mass is shared between the two massive vectors, with the relation

Q2++Q2__ _ me2

(20)

valid for arbitrary 0. Equation (20) is of course reminiscent, in an unbroken N = 2 supersymmetric theory, of the relation [9] M 2=Z 2+Z 2

(21)

between the electric and magnetic central charges of the N = 2 supersymmetry algebra

{ Q~, 0~ } = ; ~ (~ij j[_ (~r GiJz -~ i?~ ei~Zs

(22)

and the hypermultiplet mass M. The interesting fact is that, in the broken supergravity model we can identify 2 massive vectors whose square couplings satisfy (21). One can show, using ~r-model arguments [10], that this identification can be done for a generic supergravity model, so our conclusions are largely model independent.

5. A Light Graviphoton With quasi-degenerate gravitini, (0 ~ 0), we can get an arbitrarily light vector with mass m 1-4- m 2

m_ = Ira1 -rn21 ~- 0 ~

=- Omg.

(23)

In view of (t9), since we have in mind, as we mentioned, heavy gravitini, this situation is however not quite satisfactory. Restoring the dimensionful gravitational constant G N, one would have an invisible cou-

R. Barbieri and S. Cecotti:A Model for a Light Gray/photon

259

Appendix. Supersymmetry Transformations and Relation with the Snperconformal Tensor Calculus

pling of this vector to the electron 2 2 1 [m_\ 2 Q - ~ GN me ~ I ~ g ) .

(24)

and the corresponding squared couplings to the electron

In this Appendix we complete the construction of the model by specifying how the two local supersymmetries act on the various fields and by showing the invariance of the complete Lagrangian, including the Paul/, derivative and 4-fermion couplings that we have not written down. The proof of invariance is most easily obtained by exploiting the relation of our model with those obtained by the superconformal tensor calculus [7], which are locally supersymmetric by construction. Our model extends a Lagrangian obtained from the tensor calculus by a "contraction" procedure, that is a suitable field redefinition along the lines of [5]. The supersymmetry transformations are

Q+ = :

- a i f i ea, , - e-i 7a OltiAI-ei~ I/llt,

We have not yet discussed, however, the most general case, allowing for a complex value of the field co = R + iL In such a case the composition of the eigenstates with mass m_+ becomes m+ ~ G u + H ~ +

(

R-l-

( m_ --~ G. - H u + R

os0)

~lnO

l+c~ ~ln 0

I Au

0

A.

(25)

[(1 + c o s O ) ( m + ] / 2 e R ) 2

+(1T-cosO)2e212+_2sinO(m+]/2eR)]/2eI]

(26)

with the relation (20) kept valid. In the 0--+ 0 limit, one has now a nonvanishing coupling of the light vector to the electron, since 2 2(eI) 2 m(2~gg 2 2) Q2-=GZme ( m + l / ~ e R ) 2 + Z ( e i ) 2 t - 0 GNm~ . (27) The strength of this coupling becomes infact identical to the gravitational one, if the full electron mass is of Higgs origin (m =0) and the vev of co is purely imaginary. In turn this corresponds to the maximal breaking of parity, a symmetry of the Lagrangian for 0 = 0, under which c~ ~ &.

a~'~=2v.d+

'

J

(A.1) "

/

--88a;" To; g.iJ~# g j - - ~# ~]i

+ fermionic terms,

(A.2)

6 ~p= C1/2giAi,

(A.3)

09 = C 1/2 gi 2i,

(A.4)

a b~ = - 2F 1/2 [~-"e i + pab gij ~b eJ],

(A.5)

6E~[ = 2F 1/2 [U e/+ p=~eii ea d],

(A.6)

1 6 (G, + e- i0H,) = - C - 1g/7, (A ~-- (5 2j) ei~ + 2 C - 1/2 gi ttl#j ,~ij

(A.7) (A.8)

6Au=~iyu2jeij + l co3(G,+e-i~ 6A i = 2 C - 1/2 i~q~d - 2 ] / 2 g F - 1Aii d

6. Conclusions

_ 2 i m F - l L,-ka G2fl ~, L~.t~ j ej

We have discussed a toy-model of N = 2 supergravity in which an arbitrarily light "gray/photon" is coupled with typical gravitational strenght to an "electron". We are satisfied with the internal consistency of the model. On the other hand, we obviously do not know how this model could be incorporated into a realistic theory of particle physics. In turn this makes any real prediction virtually impossible. Still the idea [1] of having a very low energy remnant of a badly broken extended supersymmetry, manifesting itself in a modification of Newton's law at short distances, is appealing. Furthermore, in our opinion, it could very well be compatible with the present theoretical wisdom on supersymmetry and particle physics.

-- i(2 sin 0)- ~[e-i~

Acknowledgements. One of us (R.B.) is grateful to Sergio Ferrara for explanationsand discussions.

+ qS)(GT~- ei~

-- (~2 dO(G;,,_ e-iOH;o)] o'Pa g,ij F.j

+ (5F/-, o-P%~jd + fermionic terms,

(A.9)

2 i = 2 C - 1/2 ~ ~i 21/-2i e F - 1 E~ a~a E] ej + (F;, -- co G;~) a ~176 eij ej i e I~(2 sin 0)- 1 (co -- (5) (G2~ -- e- ioH2~) a p~eij d

+ fermionic terms, ~ a = -- F - 1 / 2 Dbagi + V 2 g C -

+ fermionic terms,

(A.10) 1 / 2 F - 1/2 A * a i g i

(A.11)

6 e ~ = F - 1 / 2 ~EC[ei + i(m + e V2co) C - 1/2F-- 1/2 a~a E~ eijej

+ fermionic terms

(A. 12)

260

R. Barbieri and S. Cecotti: A Model for a Light Graviphoton

A i = _ eiO(1 - t2)1/2 Ai[n~w

where, i ~%=~

i

,-,

~=(1

C-* au(e-(0)+ 5 C-*(e' G ~)

+ fermionic terms, ~ ui J -__ _ F - 1 +F-1

(A.13)

i (6=i D~ bj= - ~j=D~, b=)

(E~i D"+~,E.i=) + fermionic terms,

(A. 14)

m = (1 - &),/z m[. . . .

rli = V'~g C - ,/2 F - *A *iJ ej + i ( m + e ] / / 2 c o ) C - W 2 F - * 8UE j=a 2 ~ f l E f gk k (A.15)

- i e i~ T.;- = sin 0 C*/2 [G;~ - e - ~ ~

+ fermionic terms.

in the limit t 2 ~ 1, where t = g'/g [51, with g' the coupling of the parent graviphoton to the "compensator" hypermultiplet and g the coupling of the parent hidden vector to the hidden hypermultiplet. Also the coupling parameters are correspondingly rescaled g = ( 1 -- t2) 3/2 glnew

A* ij = (Au). = 6ij_ i e i~a~

+ fermionic terms.

(A.17)

- - t2)1/2 ~=]new

(A.16)

The parent (tensor calculus) model has the following structure. It has an hidden sector which is just the model with partial supersymmetry breaking in AdS space studied in [5], and a visible sector which is essentially arbitrary except for a certain scaling property for the analytic function f generating the kinetic terms of the vectors [111. Here we consider only minimal kinetic terms, since all the relevant mechanisms that we have to discuss are present already in this case. The physical fields (i.e. after the solution of the superconformal constraints [71) in the parent model are: i) the N = 2 gravitational multiplet; ii) an hidden vector multiplet (Hu, A~, z); iii) an hidden hypermultiplet (B~, ~=); iv) visible vector multiplets (A~, 2[, Wx); v) visible hypermultiplets (A~, G). I is an index of the adjoint representation of the (visible) gauge group G. For simplicity, in the text we have taken G = U(1), but the general case is simply a matter of adding indices and a few couplings involving the structure constants of the non-Abelian group G. Our contraction corresponds to the following (singular) redefinition of the above fields

(A. 18)

To understand the physical meaning of our construction, let us consider the rescalings in (A.17) in connection with the various AdS vacua of the parent theory. (See the first paper of Ref. (5)). For each vacuum of the parent model we have a consistent contraction. Since the phase of the field z is not fixed by the classical potential, we obtain in this way a family of inequivalent Lagrangians, labelled by an angle 0, corresponding to the contraction around the vacuum with t e i~

X = (1 - t2) 1/2

(A. 19)

and a cosmological constant of order l~gz A = 0 []~-)=

OC(]-

t2)ZgZ[n~w).

(A.20)

In terms of the new fields all the supersymmetry transformations and the Lagrangian are finite. For the field transformations this is straightforward, unlike the case of the Lagrangian, which has some potentially divergent terms. As in Ref. (5), and in the same notation, these terms are ~1 *~T " l a "~ .Iv - -~ Y ~ v _ u. ~y - - ~ N I*j X

-88lT~T ,lg

I

X

J

+ " (Tl~vijgq)2

0~1~, I1J*o I~;#vJ ~u 71Z--v~ij *

- - 88N H X ' Tui T q g ij

ffl~vJ

_lA6Nise-t~uvp~tF.v o l o i j ~

('dJ km

_ ~ y , s e - l e u , p . e ~. , .i ~.j e., % ~ ( ? . O ,~ X a q- 89 tFpm X I X 2) gkmq- h.c.}

Z = e i~[1 -- (1 - t 2) ~ol B~ = 5 ~ - ( 1 - t 2) b~

i - 8(1 - t 2) sin 0 G{e-*e~~

Aa = (1 - ?)*/~ Ea

--cos O(A, G A ~ + B ~ VpB~)]}

W I = (1 - t2) 1/2 (DI

+ terms finite as t z -+ 1.

1 Gu = (1 -- t2) 1/2 G'ulnew

1 H . = (1 -- tz) 1/2 Hul.r

GAr

(A.21)

Therefore the divergence is a surface term which vanishes by integration. This completes the proof that one can find derivative, Pauli and 4-fermion terms, such that our Lagrangian is invariant under the transformation (A.1-12).

R. Barbieri and S. Cecotti: A Model for a Light Graviphoton

References 1. J. Scherk, in: Unification of the fundamental particle interactions, p. 381. Eds. S. Ferrara, J. Ellis, P. van Nieuwenhuizen. New York: Plenum Press 1980 2. F. Stacey et al.: Univ. of Queensland preprint 1986, and references therein 3. S. Ferrara, J. Scherk, B. Zumino: Nucl. Phys. B121, 393 (1977); C. Zachos: Phys. Lett. 76B, 329 (1978) 4. For a review, see H. Nilles: Phys. Rep. 110, 1 (1984) 5. S. Cecotti, L. Girardello, M. Porrati: Phys. Lett. B151, 363 (1985); Phys. Lett. B168, 83 (t986) 6. E. Cremmer, S. Ferrara, C. Kounnas, D. Nanopoulos: Phys.

261 Lett. 133B, 61 (1983); J. Ellis, C. Kounnas, D. Nanopoulos: Nucl. Phys. B241, 406 (1984). For the N = 2 case, see E. Cremmer et al. : Nucl. Phys. B250, 385 (1985) 7. B. deWit, P. Lauwers, A. van Proyen: Nucl. Phys. B225, 569 (1985) and references therein 8. S. Cecotti, L. Girardello, M. PorratJ: Phys. Lett. 145B, 61 (1984); Nucl. Phys. B268, 295 (1986) 9. P. Fayet: Nucl. Phys. B149, 137 (1979); S. Ferrara, C. Savoy in: Supergravity '81, p. 1. Eds. S. Ferrara, J. Taylor. London: Cambridge University Press 1982; G. Gibbons, C. Hull: Phys. Lett. 109B, 190 (1982); C. Hull: Nucl. Phys. B239, 541 (1984) 10. L. Castellani et al.: Phys. Lett. B161, 91 (1985) 11. J. Derendinger, S. Ferrara, A. Masiero, A. van Proyen: Phys. Lett. 140B, 308 (1984)