- Email: [email protected]
All-loop gauge couplings from anomaly cancellation in string effective theories
Physics Letters B 271 ( 1991 ) 307-313 North-Holland
PHYSICS LETTERS B
All-loop gauge couplings from anomaly cancellation in string effective theories Jean-Pierre Derendinger a, Sergio Ferrara b,~, Costas Kounnas c,2 and Fabio Zwirner b,3 a Institut de Physique, Universitk de Neuch&tel, CH-2000 Neuchdtel, Switzerland b TheoryDivision, CERN, CH-1211 Geneva 23, Switzerland ¢ Laboratoire de Physique Th~orique, Ecole Normale Sup&ieure, F- 75231 Paris, France
Received 3 September 1991
We derive, to all orders in perturbation theory, the Eg gauge coupling and the modified dilaton-axion Kiihler potential for the effective theories of a class of d= 4, N= 1 heterotic string models. The derivation relies on an extended version of the GreenSchwarz anomaly cancellation mechanism, and exploits target-space duality invariance. Although we deal with field-dependent effective gauge couplings and scales in a non-renorrnalizable supergravity theory, we derive for them a renormalization group equation as a relation among dynamical fields. When expectation values of these fields are considered, our results agree with those previously obtained in renormalizable theories with N= 1 global supersymmetry. We finally comment on possible generalizations of the present results.
A powerful tool to connect four-dimensional string solutions with physics at experimentally accessible energies is p r o v i d e d by the associated effective theories, which describe the low-energy d y n a m i c s o f the light string excitations. F o r potentially realistic models, and neglecting higher derivative terms, these are particular versions o f d = 4, N - - l supergravity. In the last years, intense theoretical effort has been devoted to the d e t e r m i n a t i o n o f these effective low-energy theories, using both explicit string calculations and general s y m m e t r y arguments. The first effective theories to be studied in detail were those reproducing the tree-level string predictions for scattering a m p l i t u d e s a m o n g light states. These classical effective theories, however, are not sufficient for a description o f low-energy physics. A striking example is p r o v i d e d by the effective gauge couplings ga, associated with the different factors o f the four-dimensional gauge group G = Ha Ga. At the Work supported in part by United States Department of Energy under Contract No. DE-AT03-88ER40384, Task E. 2 UPR du CNRS, asscoci6e/l l'Ecole Normale Sup6rieure et h l'Universit6 Paris-Sud. 3 On leave from Istituto Nazionale di Fisica Nucleate, Sezione di Padova, Padua, Italy.
classical level, the string coupling constant is universal, and is d e t e r m i n e d ( m o d u l o K a c - M o o d y levels) by the v a c u u m expectation value o f the four-dimensional dilaton field. To reproduce the differences among the low-energy effective gauge couplings, one needs to consider q u a n t u m corrections, in analogy with conventional grand unified theories. In the early days o f string m o d e l building, one-loop estimates o f the effective low-energy gauge couplings were obt a i n e d by using r e n o r m a l i z a t i o n group arguments at the level o f the effective field theory, i.e. considering only loop diagrams involving light states. F o r m o r e reliable estimates, however, one cannot neglect the effects o f loop diagrams involving the exchange o f heavy string modes. Recently, o n e - l o o p moduli-dep e n d e n t corrections to gauge coupling constants have been c o m p u t e d at the string level in a large class o f four-dimensional heterotic string models [ 1-8 ]. In this work, we generalize the previous results by introducing a framework to c o m p u t e the effective gauge couplings to all orders in the string perturbative expansion. We obtain an all-order result u n d e r the a s s u m p t i o n that no d i l a t o n - d e p e n d e n t correction to the low-energy effective lagrangian other than the a n o m a l y terms are generated in (string) perturba-
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
307
Volume 271, number 3,4
PHYSICS LETTERSB
tion theory. Our approach relies on some quantum string symmetries, namely the target-space dualities [ 9,10 ], which must be incorporated in the string-derived effective supergravities. The other important ingredient, which allows us to extract an all-loop result from explicit one-loop calculations, is the intimate connection [ 11], due to supersymmetry, between the anomaly of the dilatation current (related to the effective fl-functions) and the U ( 1 ) a-model anomalies associated to target-space duality transformations. It is precisely this supersymmetric relation between anomalies which causes a formal agreement between our determination of the field-dependent effective gauge couplings, derived for the non-renormalizable effective supergravity theory of strings as a relation among dynamical fields, and previous resuits obtained in renormalizable theories with N = 1 global supersymmetry [ 12-14 ], and using instanton calculus [ 15,16 ]. We consider here d = 4 , N = I heterotic string models. At the classical level, and in the standard formulation [ 17,18 ] with matter described by chiral multiplets only, their effective supergravity theories are characterized by a universal gauge kinetic function [ 19,20] Lb =~abS
( 1)
(we omit Kac-Moody levels, since they are usually equal to 1 in potentially realistic string vacua), by a K~ihler potential of the general form [ 20,21 ] K = - l o g ( S + S ) +/£(M, )Q; C, C ) ,
(2)
and by a superpotential w ( C ) . In eqs. ( 1 ) and (2), S is the dilaton-axion chiral superfield, whereas M and C represent the moduli and matter chiral superfields, respectively. In the following, we will consider only the moduli-dependent part of/£, freezing all the matter fields to zero. Limiting ourselves to (2,2) models, we can write the general formula [22] R = - l o g [ V~t,,)'V~2,, ~l,
(3)
where Vtl ,t) and VCz,~) are the volume factors of the internal manifolds for the ( 1,1 ) and (2,1 ) moduli, respectively (the former are associated with deformations of the K~ihler class, the latter with deformations of the complex structure). Under any classical symmetry transformation of the 308
21 November1991
tree-level effective action, the K~ihler potential K transforms as follows: (4)
K--.K+~o+~,
where ~0 is a holomorphic function of the chiral superfields. Since the classical supergravity lagrangian depends on K and w only via the combination fq=K+log [wl 2, the previous transformation can be compensated by a corresponding transformation of the superpotential w, (5)
w--,we -~ .
It is important to keep in mind, however, that any classical symmetry transformation acts on the fermions ~ of the theory (gauginos J'a and matter fields ~ui), appropriately rescaled in order to have canonically normalized kinetic terms, in the form ofa chiral rotation %--. ~, exp [ - ~t(~o- ~) ] ,
(6)
where ~i is an appropriate real weight. As a result of this chiral rotation, any given classical symmetry of the theory can be violated at the quantum level, i.e. is potentially anomalous. For example, triangle graphs with external lines corresponding to two gauge bosons of Ga and one (composite) connection associated with the symmetry under consideration, will induce at one-loop anomaly terms of the form 1 ' a ~a/.tu - ~lCa(~O-~o)Fj,~F
ca-
1
,
4n2 ~ ~lTr Q2,
(7)
where Q~ is any of the generators of Ga, in the representation of the fermions ~1. As apparent from the previous formula, the coefficients ca can be easily calculated in the effective theory once one knows the weights ~i. Explicit string calculations are bound to give an identical result, except if some or all anomalies are cancelled by a generalized Green-Schwarz mechanism involving the massless states. The previous general argument finds a relevant application in the case of target-space duality symmetries, which are known to be good string symmetries to any order of the string (higher genus) perturbation theory [ 23 ]. It was recently shown [ 3,5,6,8 ] that in any (2,2) symmetric abelian orbifold model, all anomalies with respect to target-space modular in-
Volume 271, number 3,4
PHYSICS LETTERS B
variance are cancelled by the combination of two different mechanisms. The first one is purely field theoretical, and generalizes [ 3,4,6 ] the Green-Schwarz anomaly cancellation mechanism [24]. One introduces in the loop-corrected effective lagrangian a term of the form
Z ~p~,
- ~k:'~C.
(8)
a
where C u is a composite vector field and g25:~ is the Yang-Mills Chern-Simons form. The variation of (8) under a symmetry transformation gives
~ik(~-~)F~ff
au~ ,
(9)
with the constant k independent of the gauge group factor. Since the term (8) is also gauge variant, restoration of gauge invariance requires the presence of a linear supermultiplet [25 ], the gauge transformation of its antisymmetric tensor component compensating the gauge variation of the Cbern-Simons form. Since superstrings possess a single linear multiplet, the constant k in eq. (8) must be group independent. The second mechanism is a purely string phenomenon [2,5 ], which generates in the one-loop effective lagrangian, after integration of the string massive modes, additional terms of the form --
~lda(~-~)F/,~,F 1 '
--
a
--a/.t~'
,
(10)
where ~ is a holomorphic function of the scalar fields in the chiral multiplets, which under the symmetry transformation varies as ~-,c~-~,
(11)
and the d~ are constants which depend in general on the gauge group factor. In the (2,2) models under discussion, these local anomaly cancelling terms in the effective one-loop lagrangian involve automorphic functions of the target-space duality group [ 10,26,2]. From the field-theory point of view, the terms in eq. (10) can be seen as local "counterterms", necessary for the cancellation of target-space modular anomalies [3,5]. The condition for complete anomaly cancellation is then da+k=c~. A fact of main importance, which will be used in the following, is that once the anomaly coefficients are known, then, because of supersymmetry, the expressions of the field-dependent effective gauge coupling constants are completely determined up to field-inde-
21 November 1991
pendent terms. The latter are also fully controlled by the infrared divergences of the theory, due to the massless modes only [27]. To simplify the discussion, we consider here the case of the E8 gauge coupling constant and anomaly terms in (2,2) symmetric abelian orbifolds with no N = 2 fixed planes, e.g. Z3 and ZT. In this case, only gauginos undergo chiral rotations, and the composite connection associated to target-space modular invariance is actually proportional to the K~ihler connection. Moreover, the coefficients dEs are identically zero [6], and the full anomaly cancellation is achieved by means of the Green-Schwarz mechanism only [3 ]. To discuss anomalies and effective gauge couplings at once, the superfield formalism proves to be very useful. Using standard techniques [28,29 ] one can then extract all relevant component expressions. In the formalism in which the S chiral superfield is replaced by a linear multiplet L via a supersymmetric duality transformation [25], the tree-level effective lagrangian can be written as a superconformal density [ 30 ] ~=
-- ~
I
[ ( S ° S ° ) 3 / 2 e-g~ 2 £ - 1 / 2 1 D
+ ([So3WlF+h.c.),
(12)
where £ = L - Q - - L Y~a12a, g2a is the Chern-Simons superfield associated with the gauge group factor Ga and So is the compensating multiplet of supergravity ,t. To see the equivalence with the S-representation, it is sufficient [ 3 1 ] to replace £ with an unconstrained real superfield V and to introduce the lagrangian ~ = ~ ( £ ~ V) - [ (S+ 57) (V+g2) ]D-
( 13 )
The equation of motion for (S+~¢) indicates that V+ ~2 is a linear multiplet, and substituting the solution into eq. ( 13 ), one goes back to the previous description in terms of the linear multiplet, eq. ( 12). Solving the equations of motion with respect to V, one obtains an equivalent form of the lagrangian in Our conventions are such that the D-density - ~ [SoSo~(Z', ~ ) l D generates an Einstein term of the form - ½eRISol2~(Z, Z). We use the same notation for chiral superfields So or Z', and for their lowest complex scalar components.
309
Volume 27 l, number 3,4
PHYSICS LETTERS B
terms of the chiral supermultiplet S. Explicitly, the equation of motion for V reads
1( SoSo
U= ~ ~ )
=½(S-t-S).
(14)
As discussed later, eq. (14) is modified by one-loop and higher-loop effects. Since gauge kinetic terms are of the form - [ (S+S)I2]D = ~ [ S W a W a l F = - ~ ( S1 + S ) F u-,- , F
a
alzp
+h.c.
+...
(15)
satisfying eq. ( 1 ), we will make the identifications
1 ½( S + S ) - g2,
(16)
where go is the unrenormalized gauge coupling, and
1
where gE8 is the loop-corrected effective gauge coupling for the factor group E8. Eq. (14) is the obvious statement that go and gE~ coincide at the tree level. Also, we will see that the K~ihler potential for the dilaton, which at tree level takes the form =
ing dimension 2, and is invariant under target-space duality transformations. Both these properties also hold for £. Then the combination U defined in eq. (14), which is related both to the E8 gauge coupling constant and to the dilaton Khhler potential, is at the same time target-space duality invariant and dimensionless. To prepare the ground for the all-loop result, we first review the determination of the one-loop corrected gauge couplings and dilaton K~ihler potential, and the anomaly cancellation mechanism at the oneloop level [ 3 ]. In the case of Es, the triangular graph associated to the target-space duality anomaly can be represented, in superfield language, by a non-local lagrangian of the form 5eE~
r dE0 z I . T T a l . , l / a 1 ~2~2 = J 4 "' E8 v v E8 16 []
C(Es) 87~ 2
R+h.c. (21)
(17)
U = g~---~8,
Kdi,aton
21 November 1991
- l o g ( S + S ) =log( ½g2o),
(18)
Performing a Kfihler transformation gives rise to anomalies of the form C(Es) 64~z2 [ (~+ ~ ) ~ b - ~
+ (~0- ~ ) ~ f f a u ~ ] ,
which are compensated by the local Green-Schwarz term
or
= L~-T-2
Kdi~,~o~= -- log (2 U) = log ( ½g28),
(22)
i
log eK/3_lD .
( 19 )
(23)
keeps this last form at one loop but in terms of the one-loop effective gauge coupling, and received corrections at higher orders. We will derive below an exact, all-order formula for Kd~to~. To begin, we observe that the classical lagrangian ( 12 ) is invariant under the transformations
This gives the one-loop result discussed in refs. [ 3,4,6 ]. Taking into account eq. (23), and performing as before the substitution £--, V, one obtains
R-,R+~0+~,
= ~ + ~ s ( £ + v), and the equation of motion for Vbecomes
U=½(S+S)
So---~So e ~/3 ,
w~we
£~£.
-~ ,
(20)
In particular, target-space duality transformations have an action of this type, with a specific form of ~0. Therefore, under target-space duality transformations, So transforms with weight ½, whereas its scaling dimension is 1. The combination SoSo e -e/3 has scal310
C(Es)R 167t2
"
(24)
We have thus obtained the one-loop correction to eq. (14). Solving the above equation of motion leads to the one-loop corrected effective action. In the resulting S-field formulation, as announced before, the effective E8 gauge coupling becomes
1 ! (S+S) g28 - 2
C(Es) K, 16~r2
(25)
Volume271, number 3,4
PHYSICSLETTERSB
21 November 1991
and the K~ihler potential for the dilaton reads ~GS = I ~
Kdilaton= -l°g(S+'~- C(Es) K)=l°g( (26) identical to eq. (19). This was the result of ref. [ 3 ]. The Green-Schwarz term of eq. (23) can be further completed if we remember that in a general superconformal gauge the compensating multiplet So is also not inert under target-space duality transformations. As a consequence, the Green-Schwarz counterterm should also cancel, simultaneously, the anomaly due to So transformations. This is because physics must be independent of the particular gauge choice for So. It is immediate to see that there is only one possible modification of the one-loop GreenSchwarz counterterm, eq. (23), which is simultaneously compatible with target-space duality invariance and superconformal invariance, namely /£--- 3 log eX/3~3 log(eX/3£) -= 3 log(½SoSo) - 2 log(2U).
a a 1 ~ 2 ~ 2 3C(Es) WE8 WE8 16 [] 87r2
f
× [log(~)
I/lra
1
~f
[2eg/3£\
_ )l
-L
L (29)
The second term in the last member precisely cancels the anomaly induced by the non-local terms. The connection between the cancellation of targetspace modular anomalies and the E8 fl-function is now transparent. The B-function is the dilatation anomaly obtained by varying the compensator field So (see for example refs. [ 12,29 ] ). However, this is related to the duality anomaly, since So transforms under duality. To move from the linear multiplet formulation to the S-field formulation, after the correction given in eq. (29), one has to write
~a. = ~ + £%s(£-~ V).
(30)
U=½(S+S) C(Es) + ~ [log
U+c- 3 log(So~qo)],
where c = - ~ +3 ~ 5 l o g 2 .
1 g2 8
~ 2 ~ 2 3C(Es)
(32)
-
* (s+g) 2
C(Es) + ~ [ - l o g g 2 8 + c - 3 log(So~qo) ] . + log (½So,-qo)] + h.c.
(31)
Identifying as before U with 1/g28, where g2E8is the corrected Es effective coupling constant, one finds
× log(eR/3£) +h.c. ..,12/21 I l / i T a
"
Solving the equation of motion for V, one finds (27)
Indeed, eq. (27) is the most general term compatible with the one-loop scaling symmetry [32,31 ] and which reproduces the desired anomaly. We can now write the non-local contribution in a general superconformal gauge as L~Er'= f d20 i
_[-C(Es)I"
3£ log(eR/3£)]D
(33)
(28)
In the first expression, £ corrects the quantum dimension of e R/3 without spoiling the transformation properties of K under target-space duality. In the second expression, we have rearranged the terms to evidentiate the scale-invariant and duality-invariant combination 2 U defined in (14) (after replacement o f £ by V). The log (~So~qo) term contains the violation of duality as well as scaling invariances. Using the same scaling argument, the local counterterm ofeq. (23) must be modified as follows:
From the previous formula one can immediately derive the effective fl-function for the Es gauge group factor. Remembering that So~qo~ M 2 sets the unit of measure for masses, we can write
fl(ga) = -
1 dg~ g a d log(SoSo) '
(34)
and therefore, in the case of Es, -3C(E8)/16n 2 fl(gEs) =g3Es 1 - - - [ ~ g ~ E s
'
(35)
311
Volume 271, number 3,4
PHYSICSLETTERSB
which coincides with the formula of refs. [ 13,15 ]. One can also write the all-loop corrected Kahler potential for the dilaton field, Kdi,atonml0g[lg2s(l_l- ~C(Es),,2 E s ]
~--3q.J,
(36)
where now gEs is the solution of eq. (33). The relation between the fields U and ½(S+~q), which we have associated with the all-loop corrected and uncorrected Es gauge couplings in the effective supergravity theory of (2,2) four-dimensional string models, turns out to be exactly the relation between 1/g2 and 1/g~) in renormalizable, globally supersymmetric N = 1 Yang-Mills theories. In principle, there is no reason why it should be so, since we are working in a non-renormalizable gauge theory coupled to gravity, so a plausible explanation is required. In string theory, and in the corresponding effective supergravity theories, couplings and scales are dynamical variables, so relations between them must be interpreted as field equations. This is the meaning of eq. (33). On the other hand, the all-loop renormalization group equation obtained in global supersymmetry is a consequence of the general structure of the anomaly supermultiplet. This is the reason why the two results formally agree. In eq. (33), the running of the gauge coupling is encoded in So, because So itself determines the energy scale of the theory. In order to define a modular invariant effective gauge coupling, one should fix the superconformal gauge with a condition of the form So-go= M 2 e ' ¢ / 3
,
(37)
where M 2 is an arbitrary modular invariant function, which in particular can be taken to be a constant. To connect the modular invariant effective gauge coupling defined by eqs. (33), (37) with the running gauge coupling defined in the renormalizable, lowenergy effective theory in a given renormalization scheme (for example, MS or DR), one should impose on the two couplings an appropriate boundary condition, as discussed for example in refs. [ 1,3,7 ]. This would also allow to define, as usual, a modular invariant, renormalization group invariant scale Ae8, analogous to AQCD, which plays a role in the discussion of possible non-perturbative phenomena like gaugino condensation. The previous result is valid only for the Es gauge 312
21 November 1991
group factor, where only gaugino fields contribute to the modular anomaly, and in the absence of anomaly cancellation mechanisms other than the GreenSchwarz one. In more general cases, one has also to consider the associated massless charged matter fields, which are subject to a rescaling under duality transformations, and the composite connections associated to duality transformations are not simply the K~ihler connection but also involve the metric of the matter fields. Similarly, in globally supersymmetric theories the fl-function formula contains also the anomalous dimensions of the matter fields, and mixing is possible in the presence of different gauge group factors. Here we will not examine these complications; work along this line is in progress and will be presented elsewhere. The results of this paper have been presented at the Erice and Les Houches Summer Schools (July 1991 ) [331. One of us (F.Z.) would like to thank the Laboratoire de Physique Throrique de l'Ecole Normale Suprrieure, Paris for the kind hospitality during part of this work and the Istituto Nazionale di Fisica Nucleare, Sezione di Padova, for partial financial support.
References [ 1] v.s. Kaplunovsky,Nuel. Phys. B 307 (1988) 145. [2] L.J. Dixon, V.S. Kaplunovskyand J. Louis, Nucl. Phys. B 355 (1991) 649. [3] J.-P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, preprint CERN-TH.6004/91 (February 1991) (and improved version, June 1991), Nucl. Phys. B, to appear. [4] G. Lopes Cardoso and B.A. Ovrut, preprint UPR-0464T (1991). [5] V.S. Kaplunovskyand J. Louis, as quoted in: J. Louis, talk Pascos 91 Conf. (Boston, MA, March 1991), SLAC-PUB5527 (1991). [ 6 ] I. Antoniadis, K.S. Narain and T.R. Taylor,preprint NUB3025, IC/-91-124, CPTH-A050.0491. [7] I. Antoniadis, J. Ellis, R. Laeaze and D.V. Nanopoulos, preprint CERN-TH.6136/91. [8] G. Lopes Cardoso and B.A. Ovrut, preprint UPR-0481T (1991). [9] K. Kikkawa and M. Yamasaki, Phys. Lett. B 149 (1984) 357; N. Sakai and I. Senda, Prog. Theor. Phys. 75 (1986) 692;
Volume 271, number 3,4
PHYSICS LETTERS B
V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B 287 (1987) 402; B. Sathiapalan, Phys. Rev. Lett. 58 (1987) 1597; R. Dijkgraaf, E. Verlinde and H. Verlinde, Commun. Math. Phys. 115 (1988)649; A. Giveon, E. Rabinovici and G. Veneziano, Nucl. Phys. B 322 (1989) 167; A. Shapere and F. Wilczek, Nucl. Phys. B 320 (1989) 669; M. Dine, P. Huet and N. Seiberg, Nucl. Phys. B 322 (1989) 301; J. Lauer, J. Mas and H.P. Nilles, Phys. Lett. B 226 (1989) 251; A. Giveon and M. Porrati, Phys. Lett. B 246 (1990) 54; Nucl. Phys. B 355 ( 1991 ) 422. [ 10] S. Ferrara, D. Liist, A. Shapere and S. Theisen, Phys. Lett. B225 (1989) 363. [ 11 ] S. Ferrara and B. Zumino, Nucl. Phys. B 87 ( 1975 ) 207. [ 12 ] M.T. Grisaru and P.C. West, Nucl. Phys. B 254 ( 1985 ) 249. [ 13] D.R.T. Jones, Phys. Lett. B 123 (1983) 45. [ 14] M.A. Shifman and A.I. Vainshtein, Nucl. Phys. B 277 (1986) 456. [15]V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 229 ( 1983 ) 381; Phys. Lett. B 166 (1986) 329. [16] D. Amati, K. Konishi, Y. Meurice, G.C. Rossi and G. Veneziano, Phys. Rep. 162 (1988) 169. [ 17 ] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Phys. Lett. B 116 (1982) 231; Nucl. Phys. B 212 (1983) 413. [ 18 ] J. Bagger and E. Witten, Phys. Lett. B 118 ( 1982 ) 103; J. Bagger, Nucl. Phys. B 211 (1983) 302. [ 19 ] M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B 156 (1985) 55. [20] J.-P. Derendinger, L.E. Ib~i~ez and H.P. Nilles, Nucl. Phys. B267 (1986) 365. [ 21 ] E. Witten, Phys. Lett. B 155 ( 1985 ) 151; S. Ferrara, C. Kounnas and M. Porrati, Phys. Lett. B 181 (1986) 263; M. Cveti6, J. Louis and B. Ovrut, Phys. Lett. B 206 (1988) 227; S. Ferrara and M. Porrati, Phys. Lett. B 216 (1989) 289;
21 November 1991
L.J. Dixon, V.S. Kaplunowsky and J. Louis, Nucl. Phys. B 329 (1990) 27. [22] S. Ferrara and A. Strominger, in: Strings '89, eds. R. Arnowitt, R. Bryan, M.J. Duff, D.V. Nanopoulos and C.N. Pope (World Scientific, Singapore, 1990) p. 245; P. Candelas and C. de la Ossa, Nucl. Phys. B 355 (1991) 455. [23] E. Alvarez and M.A.R. Osorio, Phys. Rev. D 40 (1989) 1150; D. Gross and 1. Klebanov, Nucl. Phys. B 344 (1990) 475. [24] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [ 25 ] S. Ferrara, J. Wess and B. Zumino, Phys. Lett. B 51 (1974) 239; W. Siegel, Phys. Lett. B 85 (1979) 333; S. Ferrara and M. Villasante, Phys. Lett. B 186 (1986) 85; P. Bin6truy, G. Girardi, R. Grimm and M. Miiller, Phys. Lett. B 195 (1987) 389; S. Cecotti, S. Ferrara and L. Giradello, Phys. Lett. B 198 (1987) 336; B. Ovrut, Phys. Lett. B 205 (1988) 455; B. Ovrut and C. Schwiebert, Nucl. Phys. B 321 ( 1989 ) 163; P. Bin6truy, G. Girardi and R. Grimm, Phys. Lett. B 265 (1991) 111. [26 ] S. Ferrara, C. Kounnas, D. Liist and F. Zwirner, Nucl. Phys. B365 (1991) 431; P. Fr6 and P. Soriani, preprint SISSA 90/91/EP. [ 27 ] T. Taylor and G. Veneziano, Phys. Lett. B 212 ( 1988 ) 147. [28] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, N J, 1983 ). [29] S.J. Gates Jr., M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin/Cummings, Menlo Park, CA, 1983 ). [30] T. Kugo and S. Uehara, Nucl. Phys. B 222 (1983) 125. [ 31 ] S. Cecotti, S. Ferrara and M. Villasante, Intern. J. Mod. Phys. A 2 (1987) 1839. [32] H.P. Nilles, Phys. Lett. B 180 (1986) 240. [33] S. Ferrara, lectures, in: Proc. Erice Summer School (July 1991 ), to appear; C. Kounnas, lectures, in: Proc. Les Houches Summer School (July 1991 ), to appear.
313
PHYSICS LETTERS B
All-loop gauge couplings from anomaly cancellation in string effective theories Jean-Pierre Derendinger a, Sergio Ferrara b,~, Costas Kounnas c,2 and Fabio Zwirner b,3 a Institut de Physique, Universitk de Neuch&tel, CH-2000 Neuchdtel, Switzerland b TheoryDivision, CERN, CH-1211 Geneva 23, Switzerland ¢ Laboratoire de Physique Th~orique, Ecole Normale Sup&ieure, F- 75231 Paris, France
Received 3 September 1991
We derive, to all orders in perturbation theory, the Eg gauge coupling and the modified dilaton-axion Kiihler potential for the effective theories of a class of d= 4, N= 1 heterotic string models. The derivation relies on an extended version of the GreenSchwarz anomaly cancellation mechanism, and exploits target-space duality invariance. Although we deal with field-dependent effective gauge couplings and scales in a non-renorrnalizable supergravity theory, we derive for them a renormalization group equation as a relation among dynamical fields. When expectation values of these fields are considered, our results agree with those previously obtained in renormalizable theories with N= 1 global supersymmetry. We finally comment on possible generalizations of the present results.
A powerful tool to connect four-dimensional string solutions with physics at experimentally accessible energies is p r o v i d e d by the associated effective theories, which describe the low-energy d y n a m i c s o f the light string excitations. F o r potentially realistic models, and neglecting higher derivative terms, these are particular versions o f d = 4, N - - l supergravity. In the last years, intense theoretical effort has been devoted to the d e t e r m i n a t i o n o f these effective low-energy theories, using both explicit string calculations and general s y m m e t r y arguments. The first effective theories to be studied in detail were those reproducing the tree-level string predictions for scattering a m p l i t u d e s a m o n g light states. These classical effective theories, however, are not sufficient for a description o f low-energy physics. A striking example is p r o v i d e d by the effective gauge couplings ga, associated with the different factors o f the four-dimensional gauge group G = Ha Ga. At the Work supported in part by United States Department of Energy under Contract No. DE-AT03-88ER40384, Task E. 2 UPR du CNRS, asscoci6e/l l'Ecole Normale Sup6rieure et h l'Universit6 Paris-Sud. 3 On leave from Istituto Nazionale di Fisica Nucleate, Sezione di Padova, Padua, Italy.
classical level, the string coupling constant is universal, and is d e t e r m i n e d ( m o d u l o K a c - M o o d y levels) by the v a c u u m expectation value o f the four-dimensional dilaton field. To reproduce the differences among the low-energy effective gauge couplings, one needs to consider q u a n t u m corrections, in analogy with conventional grand unified theories. In the early days o f string m o d e l building, one-loop estimates o f the effective low-energy gauge couplings were obt a i n e d by using r e n o r m a l i z a t i o n group arguments at the level o f the effective field theory, i.e. considering only loop diagrams involving light states. F o r m o r e reliable estimates, however, one cannot neglect the effects o f loop diagrams involving the exchange o f heavy string modes. Recently, o n e - l o o p moduli-dep e n d e n t corrections to gauge coupling constants have been c o m p u t e d at the string level in a large class o f four-dimensional heterotic string models [ 1-8 ]. In this work, we generalize the previous results by introducing a framework to c o m p u t e the effective gauge couplings to all orders in the string perturbative expansion. We obtain an all-order result u n d e r the a s s u m p t i o n that no d i l a t o n - d e p e n d e n t correction to the low-energy effective lagrangian other than the a n o m a l y terms are generated in (string) perturba-
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
307
Volume 271, number 3,4
PHYSICS LETTERSB
tion theory. Our approach relies on some quantum string symmetries, namely the target-space dualities [ 9,10 ], which must be incorporated in the string-derived effective supergravities. The other important ingredient, which allows us to extract an all-loop result from explicit one-loop calculations, is the intimate connection [ 11], due to supersymmetry, between the anomaly of the dilatation current (related to the effective fl-functions) and the U ( 1 ) a-model anomalies associated to target-space duality transformations. It is precisely this supersymmetric relation between anomalies which causes a formal agreement between our determination of the field-dependent effective gauge couplings, derived for the non-renormalizable effective supergravity theory of strings as a relation among dynamical fields, and previous resuits obtained in renormalizable theories with N = 1 global supersymmetry [ 12-14 ], and using instanton calculus [ 15,16 ]. We consider here d = 4 , N = I heterotic string models. At the classical level, and in the standard formulation [ 17,18 ] with matter described by chiral multiplets only, their effective supergravity theories are characterized by a universal gauge kinetic function [ 19,20] Lb =~abS
( 1)
(we omit Kac-Moody levels, since they are usually equal to 1 in potentially realistic string vacua), by a K~ihler potential of the general form [ 20,21 ] K = - l o g ( S + S ) +/£(M, )Q; C, C ) ,
(2)
and by a superpotential w ( C ) . In eqs. ( 1 ) and (2), S is the dilaton-axion chiral superfield, whereas M and C represent the moduli and matter chiral superfields, respectively. In the following, we will consider only the moduli-dependent part of/£, freezing all the matter fields to zero. Limiting ourselves to (2,2) models, we can write the general formula [22] R = - l o g [ V~t,,)'V~2,, ~l,
(3)
where Vtl ,t) and VCz,~) are the volume factors of the internal manifolds for the ( 1,1 ) and (2,1 ) moduli, respectively (the former are associated with deformations of the K~ihler class, the latter with deformations of the complex structure). Under any classical symmetry transformation of the 308
21 November1991
tree-level effective action, the K~ihler potential K transforms as follows: (4)
K--.K+~o+~,
where ~0 is a holomorphic function of the chiral superfields. Since the classical supergravity lagrangian depends on K and w only via the combination fq=K+log [wl 2, the previous transformation can be compensated by a corresponding transformation of the superpotential w, (5)
w--,we -~ .
It is important to keep in mind, however, that any classical symmetry transformation acts on the fermions ~ of the theory (gauginos J'a and matter fields ~ui), appropriately rescaled in order to have canonically normalized kinetic terms, in the form ofa chiral rotation %--. ~, exp [ - ~t(~o- ~) ] ,
(6)
where ~i is an appropriate real weight. As a result of this chiral rotation, any given classical symmetry of the theory can be violated at the quantum level, i.e. is potentially anomalous. For example, triangle graphs with external lines corresponding to two gauge bosons of Ga and one (composite) connection associated with the symmetry under consideration, will induce at one-loop anomaly terms of the form 1 ' a ~a/.tu - ~lCa(~O-~o)Fj,~F
ca-
1
,
4n2 ~ ~lTr Q2,
(7)
where Q~ is any of the generators of Ga, in the representation of the fermions ~1. As apparent from the previous formula, the coefficients ca can be easily calculated in the effective theory once one knows the weights ~i. Explicit string calculations are bound to give an identical result, except if some or all anomalies are cancelled by a generalized Green-Schwarz mechanism involving the massless states. The previous general argument finds a relevant application in the case of target-space duality symmetries, which are known to be good string symmetries to any order of the string (higher genus) perturbation theory [ 23 ]. It was recently shown [ 3,5,6,8 ] that in any (2,2) symmetric abelian orbifold model, all anomalies with respect to target-space modular in-
Volume 271, number 3,4
PHYSICS LETTERS B
variance are cancelled by the combination of two different mechanisms. The first one is purely field theoretical, and generalizes [ 3,4,6 ] the Green-Schwarz anomaly cancellation mechanism [24]. One introduces in the loop-corrected effective lagrangian a term of the form
Z ~p~,
- ~k:'~C.
(8)
a
where C u is a composite vector field and g25:~ is the Yang-Mills Chern-Simons form. The variation of (8) under a symmetry transformation gives
~ik(~-~)F~ff
au~ ,
(9)
with the constant k independent of the gauge group factor. Since the term (8) is also gauge variant, restoration of gauge invariance requires the presence of a linear supermultiplet [25 ], the gauge transformation of its antisymmetric tensor component compensating the gauge variation of the Cbern-Simons form. Since superstrings possess a single linear multiplet, the constant k in eq. (8) must be group independent. The second mechanism is a purely string phenomenon [2,5 ], which generates in the one-loop effective lagrangian, after integration of the string massive modes, additional terms of the form --
~lda(~-~)F/,~,F 1 '
--
a
--a/.t~'
,
(10)
where ~ is a holomorphic function of the scalar fields in the chiral multiplets, which under the symmetry transformation varies as ~-,c~-~,
(11)
and the d~ are constants which depend in general on the gauge group factor. In the (2,2) models under discussion, these local anomaly cancelling terms in the effective one-loop lagrangian involve automorphic functions of the target-space duality group [ 10,26,2]. From the field-theory point of view, the terms in eq. (10) can be seen as local "counterterms", necessary for the cancellation of target-space modular anomalies [3,5]. The condition for complete anomaly cancellation is then da+k=c~. A fact of main importance, which will be used in the following, is that once the anomaly coefficients are known, then, because of supersymmetry, the expressions of the field-dependent effective gauge coupling constants are completely determined up to field-inde-
21 November 1991
pendent terms. The latter are also fully controlled by the infrared divergences of the theory, due to the massless modes only [27]. To simplify the discussion, we consider here the case of the E8 gauge coupling constant and anomaly terms in (2,2) symmetric abelian orbifolds with no N = 2 fixed planes, e.g. Z3 and ZT. In this case, only gauginos undergo chiral rotations, and the composite connection associated to target-space modular invariance is actually proportional to the K~ihler connection. Moreover, the coefficients dEs are identically zero [6], and the full anomaly cancellation is achieved by means of the Green-Schwarz mechanism only [3 ]. To discuss anomalies and effective gauge couplings at once, the superfield formalism proves to be very useful. Using standard techniques [28,29 ] one can then extract all relevant component expressions. In the formalism in which the S chiral superfield is replaced by a linear multiplet L via a supersymmetric duality transformation [25], the tree-level effective lagrangian can be written as a superconformal density [ 30 ] ~=
-- ~
I
[ ( S ° S ° ) 3 / 2 e-g~ 2 £ - 1 / 2 1 D
+ ([So3WlF+h.c.),
(12)
where £ = L - Q - - L Y~a12a, g2a is the Chern-Simons superfield associated with the gauge group factor Ga and So is the compensating multiplet of supergravity ,t. To see the equivalence with the S-representation, it is sufficient [ 3 1 ] to replace £ with an unconstrained real superfield V and to introduce the lagrangian ~ = ~ ( £ ~ V) - [ (S+ 57) (V+g2) ]D-
( 13 )
The equation of motion for (S+~¢) indicates that V+ ~2 is a linear multiplet, and substituting the solution into eq. ( 13 ), one goes back to the previous description in terms of the linear multiplet, eq. ( 12). Solving the equations of motion with respect to V, one obtains an equivalent form of the lagrangian in Our conventions are such that the D-density - ~ [SoSo~(Z', ~ ) l D generates an Einstein term of the form - ½eRISol2~(Z, Z). We use the same notation for chiral superfields So or Z', and for their lowest complex scalar components.
309
Volume 27 l, number 3,4
PHYSICS LETTERS B
terms of the chiral supermultiplet S. Explicitly, the equation of motion for V reads
1( SoSo
U= ~ ~ )
=½(S-t-S).
(14)
As discussed later, eq. (14) is modified by one-loop and higher-loop effects. Since gauge kinetic terms are of the form - [ (S+S)I2]D = ~ [ S W a W a l F = - ~ ( S1 + S ) F u-,- , F
a
alzp
+h.c.
+...
(15)
satisfying eq. ( 1 ), we will make the identifications
1 ½( S + S ) - g2,
(16)
where go is the unrenormalized gauge coupling, and
1
where gE8 is the loop-corrected effective gauge coupling for the factor group E8. Eq. (14) is the obvious statement that go and gE~ coincide at the tree level. Also, we will see that the K~ihler potential for the dilaton, which at tree level takes the form =
ing dimension 2, and is invariant under target-space duality transformations. Both these properties also hold for £. Then the combination U defined in eq. (14), which is related both to the E8 gauge coupling constant and to the dilaton Khhler potential, is at the same time target-space duality invariant and dimensionless. To prepare the ground for the all-loop result, we first review the determination of the one-loop corrected gauge couplings and dilaton K~ihler potential, and the anomaly cancellation mechanism at the oneloop level [ 3 ]. In the case of Es, the triangular graph associated to the target-space duality anomaly can be represented, in superfield language, by a non-local lagrangian of the form 5eE~
r dE0 z I . T T a l . , l / a 1 ~2~2 = J 4 "' E8 v v E8 16 []
C(Es) 87~ 2
R+h.c. (21)
(17)
U = g~---~8,
Kdi,aton
21 November 1991
- l o g ( S + S ) =log( ½g2o),
(18)
Performing a Kfihler transformation gives rise to anomalies of the form C(Es) 64~z2 [ (~+ ~ ) ~ b - ~
+ (~0- ~ ) ~ f f a u ~ ] ,
which are compensated by the local Green-Schwarz term
or
= L~-T-2
Kdi~,~o~= -- log (2 U) = log ( ½g28),
(22)
i
log eK/3_lD .
( 19 )
(23)
keeps this last form at one loop but in terms of the one-loop effective gauge coupling, and received corrections at higher orders. We will derive below an exact, all-order formula for Kd~to~. To begin, we observe that the classical lagrangian ( 12 ) is invariant under the transformations
This gives the one-loop result discussed in refs. [ 3,4,6 ]. Taking into account eq. (23), and performing as before the substitution £--, V, one obtains
R-,R+~0+~,
= ~ + ~ s ( £ + v), and the equation of motion for Vbecomes
U=½(S+S)
So---~So e ~/3 ,
w~we
£~£.
-~ ,
(20)
In particular, target-space duality transformations have an action of this type, with a specific form of ~0. Therefore, under target-space duality transformations, So transforms with weight ½, whereas its scaling dimension is 1. The combination SoSo e -e/3 has scal310
C(Es)R 167t2
"
(24)
We have thus obtained the one-loop correction to eq. (14). Solving the above equation of motion leads to the one-loop corrected effective action. In the resulting S-field formulation, as announced before, the effective E8 gauge coupling becomes
1 ! (S+S) g28 - 2
C(Es) K, 16~r2
(25)
Volume271, number 3,4
PHYSICSLETTERSB
21 November 1991
and the K~ihler potential for the dilaton reads ~GS = I ~
Kdilaton= -l°g(S+'~- C(Es) K)=l°g( (26) identical to eq. (19). This was the result of ref. [ 3 ]. The Green-Schwarz term of eq. (23) can be further completed if we remember that in a general superconformal gauge the compensating multiplet So is also not inert under target-space duality transformations. As a consequence, the Green-Schwarz counterterm should also cancel, simultaneously, the anomaly due to So transformations. This is because physics must be independent of the particular gauge choice for So. It is immediate to see that there is only one possible modification of the one-loop GreenSchwarz counterterm, eq. (23), which is simultaneously compatible with target-space duality invariance and superconformal invariance, namely /£--- 3 log eX/3~3 log(eX/3£) -= 3 log(½SoSo) - 2 log(2U).
a a 1 ~ 2 ~ 2 3C(Es) WE8 WE8 16 [] 87r2
f
× [log(~)
I/lra
1
~f
[2eg/3£\
_ )l
-L
L (29)
The second term in the last member precisely cancels the anomaly induced by the non-local terms. The connection between the cancellation of targetspace modular anomalies and the E8 fl-function is now transparent. The B-function is the dilatation anomaly obtained by varying the compensator field So (see for example refs. [ 12,29 ] ). However, this is related to the duality anomaly, since So transforms under duality. To move from the linear multiplet formulation to the S-field formulation, after the correction given in eq. (29), one has to write
~a. = ~ + £%s(£-~ V).
(30)
U=½(S+S) C(Es) + ~ [log
U+c- 3 log(So~qo)],
where c = - ~ +3 ~ 5 l o g 2 .
1 g2 8
~ 2 ~ 2 3C(Es)
(32)
-
* (s+g) 2
C(Es) + ~ [ - l o g g 2 8 + c - 3 log(So~qo) ] . + log (½So,-qo)] + h.c.
(31)
Identifying as before U with 1/g28, where g2E8is the corrected Es effective coupling constant, one finds
× log(eR/3£) +h.c. ..,12/21 I l / i T a
"
Solving the equation of motion for V, one finds (27)
Indeed, eq. (27) is the most general term compatible with the one-loop scaling symmetry [32,31 ] and which reproduces the desired anomaly. We can now write the non-local contribution in a general superconformal gauge as L~Er'= f d20 i
_[-C(Es)I"
3£ log(eR/3£)]D
(33)
(28)
In the first expression, £ corrects the quantum dimension of e R/3 without spoiling the transformation properties of K under target-space duality. In the second expression, we have rearranged the terms to evidentiate the scale-invariant and duality-invariant combination 2 U defined in (14) (after replacement o f £ by V). The log (~So~qo) term contains the violation of duality as well as scaling invariances. Using the same scaling argument, the local counterterm ofeq. (23) must be modified as follows:
From the previous formula one can immediately derive the effective fl-function for the Es gauge group factor. Remembering that So~qo~ M 2 sets the unit of measure for masses, we can write
fl(ga) = -
1 dg~ g a d log(SoSo) '
(34)
and therefore, in the case of Es, -3C(E8)/16n 2 fl(gEs) =g3Es 1 - - - [ ~ g ~ E s
'
(35)
311
Volume 271, number 3,4
PHYSICSLETTERSB
which coincides with the formula of refs. [ 13,15 ]. One can also write the all-loop corrected Kahler potential for the dilaton field, Kdi,atonml0g[lg2s(l_l- ~C(Es),,2 E s ]
~--3q.J,
(36)
where now gEs is the solution of eq. (33). The relation between the fields U and ½(S+~q), which we have associated with the all-loop corrected and uncorrected Es gauge couplings in the effective supergravity theory of (2,2) four-dimensional string models, turns out to be exactly the relation between 1/g2 and 1/g~) in renormalizable, globally supersymmetric N = 1 Yang-Mills theories. In principle, there is no reason why it should be so, since we are working in a non-renormalizable gauge theory coupled to gravity, so a plausible explanation is required. In string theory, and in the corresponding effective supergravity theories, couplings and scales are dynamical variables, so relations between them must be interpreted as field equations. This is the meaning of eq. (33). On the other hand, the all-loop renormalization group equation obtained in global supersymmetry is a consequence of the general structure of the anomaly supermultiplet. This is the reason why the two results formally agree. In eq. (33), the running of the gauge coupling is encoded in So, because So itself determines the energy scale of the theory. In order to define a modular invariant effective gauge coupling, one should fix the superconformal gauge with a condition of the form So-go= M 2 e ' ¢ / 3
,
(37)
where M 2 is an arbitrary modular invariant function, which in particular can be taken to be a constant. To connect the modular invariant effective gauge coupling defined by eqs. (33), (37) with the running gauge coupling defined in the renormalizable, lowenergy effective theory in a given renormalization scheme (for example, MS or DR), one should impose on the two couplings an appropriate boundary condition, as discussed for example in refs. [ 1,3,7 ]. This would also allow to define, as usual, a modular invariant, renormalization group invariant scale Ae8, analogous to AQCD, which plays a role in the discussion of possible non-perturbative phenomena like gaugino condensation. The previous result is valid only for the Es gauge 312
21 November 1991
group factor, where only gaugino fields contribute to the modular anomaly, and in the absence of anomaly cancellation mechanisms other than the GreenSchwarz one. In more general cases, one has also to consider the associated massless charged matter fields, which are subject to a rescaling under duality transformations, and the composite connections associated to duality transformations are not simply the K~ihler connection but also involve the metric of the matter fields. Similarly, in globally supersymmetric theories the fl-function formula contains also the anomalous dimensions of the matter fields, and mixing is possible in the presence of different gauge group factors. Here we will not examine these complications; work along this line is in progress and will be presented elsewhere. The results of this paper have been presented at the Erice and Les Houches Summer Schools (July 1991 ) [331. One of us (F.Z.) would like to thank the Laboratoire de Physique Throrique de l'Ecole Normale Suprrieure, Paris for the kind hospitality during part of this work and the Istituto Nazionale di Fisica Nucleare, Sezione di Padova, for partial financial support.
References [ 1] v.s. Kaplunovsky,Nuel. Phys. B 307 (1988) 145. [2] L.J. Dixon, V.S. Kaplunovskyand J. Louis, Nucl. Phys. B 355 (1991) 649. [3] J.-P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, preprint CERN-TH.6004/91 (February 1991) (and improved version, June 1991), Nucl. Phys. B, to appear. [4] G. Lopes Cardoso and B.A. Ovrut, preprint UPR-0464T (1991). [5] V.S. Kaplunovskyand J. Louis, as quoted in: J. Louis, talk Pascos 91 Conf. (Boston, MA, March 1991), SLAC-PUB5527 (1991). [ 6 ] I. Antoniadis, K.S. Narain and T.R. Taylor,preprint NUB3025, IC/-91-124, CPTH-A050.0491. [7] I. Antoniadis, J. Ellis, R. Laeaze and D.V. Nanopoulos, preprint CERN-TH.6136/91. [8] G. Lopes Cardoso and B.A. Ovrut, preprint UPR-0481T (1991). [9] K. Kikkawa and M. Yamasaki, Phys. Lett. B 149 (1984) 357; N. Sakai and I. Senda, Prog. Theor. Phys. 75 (1986) 692;
Volume 271, number 3,4
PHYSICS LETTERS B
V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B 287 (1987) 402; B. Sathiapalan, Phys. Rev. Lett. 58 (1987) 1597; R. Dijkgraaf, E. Verlinde and H. Verlinde, Commun. Math. Phys. 115 (1988)649; A. Giveon, E. Rabinovici and G. Veneziano, Nucl. Phys. B 322 (1989) 167; A. Shapere and F. Wilczek, Nucl. Phys. B 320 (1989) 669; M. Dine, P. Huet and N. Seiberg, Nucl. Phys. B 322 (1989) 301; J. Lauer, J. Mas and H.P. Nilles, Phys. Lett. B 226 (1989) 251; A. Giveon and M. Porrati, Phys. Lett. B 246 (1990) 54; Nucl. Phys. B 355 ( 1991 ) 422. [ 10] S. Ferrara, D. Liist, A. Shapere and S. Theisen, Phys. Lett. B225 (1989) 363. [ 11 ] S. Ferrara and B. Zumino, Nucl. Phys. B 87 ( 1975 ) 207. [ 12 ] M.T. Grisaru and P.C. West, Nucl. Phys. B 254 ( 1985 ) 249. [ 13] D.R.T. Jones, Phys. Lett. B 123 (1983) 45. [ 14] M.A. Shifman and A.I. Vainshtein, Nucl. Phys. B 277 (1986) 456. [15]V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 229 ( 1983 ) 381; Phys. Lett. B 166 (1986) 329. [16] D. Amati, K. Konishi, Y. Meurice, G.C. Rossi and G. Veneziano, Phys. Rep. 162 (1988) 169. [ 17 ] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Phys. Lett. B 116 (1982) 231; Nucl. Phys. B 212 (1983) 413. [ 18 ] J. Bagger and E. Witten, Phys. Lett. B 118 ( 1982 ) 103; J. Bagger, Nucl. Phys. B 211 (1983) 302. [ 19 ] M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B 156 (1985) 55. [20] J.-P. Derendinger, L.E. Ib~i~ez and H.P. Nilles, Nucl. Phys. B267 (1986) 365. [ 21 ] E. Witten, Phys. Lett. B 155 ( 1985 ) 151; S. Ferrara, C. Kounnas and M. Porrati, Phys. Lett. B 181 (1986) 263; M. Cveti6, J. Louis and B. Ovrut, Phys. Lett. B 206 (1988) 227; S. Ferrara and M. Porrati, Phys. Lett. B 216 (1989) 289;
21 November 1991
L.J. Dixon, V.S. Kaplunowsky and J. Louis, Nucl. Phys. B 329 (1990) 27. [22] S. Ferrara and A. Strominger, in: Strings '89, eds. R. Arnowitt, R. Bryan, M.J. Duff, D.V. Nanopoulos and C.N. Pope (World Scientific, Singapore, 1990) p. 245; P. Candelas and C. de la Ossa, Nucl. Phys. B 355 (1991) 455. [23] E. Alvarez and M.A.R. Osorio, Phys. Rev. D 40 (1989) 1150; D. Gross and 1. Klebanov, Nucl. Phys. B 344 (1990) 475. [24] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [ 25 ] S. Ferrara, J. Wess and B. Zumino, Phys. Lett. B 51 (1974) 239; W. Siegel, Phys. Lett. B 85 (1979) 333; S. Ferrara and M. Villasante, Phys. Lett. B 186 (1986) 85; P. Bin6truy, G. Girardi, R. Grimm and M. Miiller, Phys. Lett. B 195 (1987) 389; S. Cecotti, S. Ferrara and L. Giradello, Phys. Lett. B 198 (1987) 336; B. Ovrut, Phys. Lett. B 205 (1988) 455; B. Ovrut and C. Schwiebert, Nucl. Phys. B 321 ( 1989 ) 163; P. Bin6truy, G. Girardi and R. Grimm, Phys. Lett. B 265 (1991) 111. [26 ] S. Ferrara, C. Kounnas, D. Liist and F. Zwirner, Nucl. Phys. B365 (1991) 431; P. Fr6 and P. Soriani, preprint SISSA 90/91/EP. [ 27 ] T. Taylor and G. Veneziano, Phys. Lett. B 212 ( 1988 ) 147. [28] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, N J, 1983 ). [29] S.J. Gates Jr., M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin/Cummings, Menlo Park, CA, 1983 ). [30] T. Kugo and S. Uehara, Nucl. Phys. B 222 (1983) 125. [ 31 ] S. Cecotti, S. Ferrara and M. Villasante, Intern. J. Mod. Phys. A 2 (1987) 1839. [32] H.P. Nilles, Phys. Lett. B 180 (1986) 240. [33] S. Ferrara, lectures, in: Proc. Erice Summer School (July 1991 ), to appear; C. Kounnas, lectures, in: Proc. Les Houches Summer School (July 1991 ), to appear.
313