ELSEVIER

UCLEAR PHYSIC~

Nuclear Physics B (Proc. Suppl.) 52A (1997) 289-293

PROCEEDINGS SUPPLEMENTS

The problem of the stabilization of the dilaton in string theories

J.A. Casas ~ *

a Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA

The crucial problem of how the dilaton field is stabilized at a phenomenologically acceptable value in string theories remains essentially unsolved. We show that the usual scenario of assuming that the dilaton is fixed by the (SUSY breaking) dynamics of just the dilaton itself (dilaton dominance scenario) is inconsistent unless the K£hler potential receives very important perturbative or non-perturbative contributions. Then, the usual predictions about soft breaking terms are lost, but still is possible to derive model-independent predictions for them.

1. I n t r o d u c t i o n

The crucial problem of how the dilaton field, S, whose expectation value gives the tree level gauge coupling constant at the string scale [1] ((ReS) ~ g~i,~g) is stabilized at the experimen-

tally "observed" value, g .2 "~ 2, remains as s t r i n g - - a very challenging one. The dilaton potential is fiat at all orders in perturbation theory in the absence of SUSY breaking, which on the other hand must be broken itself by non-perturbative effects. This strongly suggests that the same dy- namics that breaks SUSY is also responsible for the stabilization of the dilaton (see ref.[2]).

In this sense a very economical and model- independent scenario is the so-called "Dilaton Dominance" [3,4] one. This consists in assum- ing that SUSY is broken in the dilaton sector. In other words, only the Fs auxiliary field is to take a non-vanishing VEV. Furthermore, The dilaton dependence of the Kiihler potential, h', is assumed to be sufficiently well 'approximated by the tree-leveL expression, A" = - l o g ( S + S). These assumptions are completed with the phe- nomenologically mandatory one that the super- potential W is ill such an (unknown) way that the minimum of the potential, V, lies at an ac-

--O )o ceptable value for the dilaton ((ReS) = g a t ~ r i , g -

*On leave of absence from In s t i t u t o de E s t r u c t u r a de Ia Ma te r i a CSIC, Se r rano 123, 28006 Madr id , Spain. This work has been s u p p o r t e d in par t by: the C IC YT , unde r c o n t r a c t s AEN95-0195; the E u r o p e a n Union, unde r con- t r ac t CHRX-CT92-0004 . 2Here we are a s s u m i n g t ha t the K a c - M o o d y level of the

and at vanishing cosmological constant (V = 0). The previous assumptions lead to some interest- ing relationships among the different soft terms, with an automatical implementation of universal- ity, something which is phenomenologieally wel- come for FCNC reasons [5].

The interest of this scenario raises two ques- tions, which directly concern the problem of the stabilization of the dilaton:

/) Is there any form of the superpotential W(S) (preferably ~vith some theoretical jus- tification) able to fulfill the previous re- quirements (i.e. a minimum of the scalar potential at (ReS) -~ 2 and V = 0)?

i/) How good is expected to be the tree-level approximation for K(S, S)?

Regarding the first question, (i), I will survey in sect.2 a simple analytical argument (which has been presented in ref.[6]) that proves that , even if the potential has a minimum at (ReS} ~- 2 and V = 0, there must exist an additional mini- mum (or unbounded from below direction) in the perturbative region of S-values for which V < 0. This call be proven for an arbitrary form W(S). Similarly, we will show that the very existence of such a (local) mininmm is forbidden unless a re- ally huge conspiracy of different contributions to W(S) takes place.

gauge g roup is h = 1, which is the mos t c o m m o n possibil- ity; o therwise (ReS) --- g ~ 2 , . i ~ 9 / k .

0920-5632(97)/$17.00 ¢3 1997 Elsevier Science B.V All rights reserved.

PII: S0920-5632(96)00579-8

290 JA. Casas/Nuclear Physics B (Proc. Suppl.) 52A (1997) 289 293

Concerning the second question, (ii), there are, unfortunately, indications that the stringy non- perturbative corrections to the Kghler potential may be sizeable [7,8]. In principle, this may be good news since it could help to avoid the above- mentioned problems. The trouble here is that very little is known about the form of these non- perturbative corrections. The possibilities and predictions of such a scenario are explored in sec- tions 3 and 4.

2. Incons i s t enc ie s of the d i la ton d o m i n a t e d s c e n a r i o

Taking the tree-level expression for the Kiihler potential

I£ = - log(S + S) +/~'(T, T, ¢1, ¢,) , (1)

where T, ¢I denote generically all the rnoduli and mat ter fields respectively, the scalar potential in the dilaton-dominance assumption reads

1 {1(2 aeS)W - wI =-31WI 2} (2) V - 2 ReS

with Ws = OW/OS. If the scenario is realistic, the previous potential should have a minimum at a realistic value of S, say S0, with

1 m S 0 = ~ - - 2 , (3)

where, for simplicity of notation, 9 denotes the gauge coupling constant at the string scale (~ 1017 GeV). In addition, the vanishing of the cos- mological constant, i.e. V(So) = 0, implies

1(2 a e & ) W s ( S o ) - W(S0)I = ,/31W(S0)I (4)

and, thus

2 ReSo < W < 2ReSo (5)

v ~ + l - Ws s:so - v / ~ - 1

Performing the following change of variables

z = e - e s (~ arbitrary) (6)

the physical region of S, i.e. ReS > 0, is mapped into the circle of radius 1 in the z-plane. The "re- alistic minimum" point, z0 = e -os° , lies son:e- where inside the circle. In the new variable, the

functions l/V, Ws are written as

W ( S ) = a ( z )

W s ( S ) = ~ s ( z ) -- - Z z f f ( z ) (7)

and condition (5) becomes

2 ReX0 < [ f*(z) . . . . 2 ReS0 (8)

with ReS0 = - l o g Izol/J3. Let us consider now the function

a(z) p(z ) - a s ( z ) " (9)

Let us suppose for the moment that p(z) is an an- alytical function with no poles inside the physical region Izl < 1. Then, the maxinmm of Ip(z)[ in the region ]z] < ]z0[ must necessarily occur (prin- ciple of maximum) at some point ZM belonging to the boundary, namely the circle C -- {]z I = [z0]}. If we consider the larger region enclosed by the broader circle C' = {]z[ = [z~]}, with [z~[ > Iz0[, the new maximum of Ip(z)l must occur now at some point, say z = zl, belonging to the bound- ary C'. From (8) it is clear that at zl

I a (z ) • 2 ReX0 (10) I ; ( z : ) l = as(z) . . . . >

Taking the radius of C' so that ReS1 =

- l o g ]zl[//3 = ~ R e S 0 , we can write (10) as q3+1

~(z) ] W 2ReS~ = > - - (11)

f~s(z) . . . . I -~ss s=s, ~ - 1

Therefore, at S = S1 the potential (2) has a neg- ative value. On the other hand S1 still belongs to the perturbative region

ReS: _~ 0.27ReS0 (12)

which means a ~_ 0.14. The only way-out to the previous argument is to allow the function p(z) -

~ to have some pole in the region enclosed

by C'. But then ~ -+ oo near the pole and

necessarily a 2 ReS > ~ C T at some point with non-

zero D. Hence we arrive to the same conclusion.

The previous argument shows that the realis- tic minimum assumed to take place in the usual

JA. Casas/Nuclear Physics B (Proc. Suppl.) 52A (1997) 289-293 291

dilaton-dominated scenario can never correspond to a global minimum. This is not certainly the most desirable situation.

We can go a bit further and show that under very general assumptions, the realistic point S = So cannot correspond to V ( So ) = O.

From symmetry and analyticity arguments we know [7] that the non-perturbative superpoten- tial must take the form W = ~ i di e - ~ s , so it is reasonable to assume that at the realistic point (So '-~ 2) W is dominated by one of the terms, say W ~ e -~s (as it happens for instance in usual gaugino condensation). Then, the vanishing of A~o~ at S = So, eq.(4), implies

- a 2R~So 4(ReSo)2 - 0 . (13)

From (xa) we obtain a _~ ( v / 3 - 1)/4, absolutely incompatible with the requirement of a hierarchi- cally small SUSY breaking (note that (W) ~ 1 TeV requires a ~ 18).

It could happen, however, that two o more terms of the form W -a s = ~ die ' cooperate at the particular region S ~- 2 to produce a more realistic scenario 3. It is interesting to show that for this to happen in a dilaton dominated sce- nario a really huge conspiracy must take place. From eq.(5) we see that the condition V(So) = 0 implies

IWsl ~ IWl (14)

Since W s = - ~ aidie - ~ s and the condition of hierarchically SUSY breaking requires ai > O(10), it is clear that a cancellation between terms with different exponents must occur inside W s for (14) to be fulfilled. Moreover, one has to demand that So corresponds to a minimum, which implies OV/OS -- 0 and the determinant of the Hessian matr ix 7/ > 0. These two conditions imply in turn [6,2]

(2 ReS)2iWssl = 21w I (15)

3This is the mechanism of the so-called racetrack models to generate SUSY breaking (see e.g. [2] and references therein). These models, however, lead natural ly to moduli dominance SUSY breaking ra ther than dilaton dominance.

2 ~ ( R e S ) 2 l W s s s [ ~ IWsl ~ lWl . (16) V3

These requirements are completely unnatural since the typical sizes of W s s and W s s s are a2W, a 3 W respectively, i.e. much larger that W. Therefore, two unpleasant fine-tunings must oc- cur at So to become a (local) minimum of the potential (for related work see ref.[9]).

To summarize the results of this section, the stan- dard (tree-level) di laton-dominated scenario can never correspond to a global minimum of the po- tential at V = 0. Similarly, under very general assumptions it cannot correspond to a local min- inmm either, unless a really big conspiracy of dif- ferent contributions to W ( S ) takes place.

3. T h e ro l e o f t h e K~ihler p o t e n t i a l a n d a p r e d i c t i v e s c e n a r i o

The previous results, plus the fact that the IGihler potential is likely to receive sizeable string non-perturbative corrections, strongly suggest to consider a more general scenario, as commented in the introduction. Thus, in this section we will study how far we can go with the usual assump- tion of "dilaton-doinin~nce" (i.e. only IFsI • 0), but leaving the Kghler potential arbitrary. The potential reads

V = K s ~ [Fs[ 2 - 3e K [l/V] 2 , (17)

w h e r e Fs -~ eK/2 { ( ] ( - 1 ) ¢S ( O ¢ ~ V + W ] ( ¢ ) * }

with ¢ running over all the chiral fields and the subindices denoting partial derivatives. If we also assume vanishing cosmological constant, then

IFsl ( K s g ) 1/2 = x/3c K/2 ]WI = v ~ m 3 / 2 . (18)

The effective low-energy soft part of the La- grangian (ill terms of tile canonically normalized fields) is given by

1 a ~- ~ E 771'21~)112 -- /~soft = ~M1/2AaAa + I

+ (19)

The values of the gaugino masses, M~/2, scalar

masses, rn~, and coefficients of the trilinear scalar

292 JA. Casas/Nuclear Physics B (Proc. Suppl.) 52A (I997) 289 293

t e r m s , AIJL, can be computed using general for- mulae [3,10] and eqs.(17-18) (for more details see ref.[6])

=

= [ 1

Here we have assumed that the Yukawa couplings appearing in the original superpotential, W, do not depend on the dilaton S. This is true at tree-level (and thus at the perturbative level) and, since they are parameters of the superpotential, they are not likely to be appreciably changed at the non-per turbat ive level [7]. The expression for the coefficient of the bilinear term, B, depends on the mechanism of generation of the/~ term, so we prefer to leave it as an independent param- eter. Finally, in order to fulfill the phenomeno- logical requirement of (approximate) universality [5] we have demanded that Kii does not get S- dependent contributions 4. Notice also that in the tree level limit Ks - - I KEg _ _ 1 2R eS ' 4 (B.~S) 2 , a n d

we recover from eqs.(20) the usual tree level rela- tions [3,4].

In eqs.(20) there are three unknowns (m3/2, Ks, Kss) and three soft breaking parameters (rn~, M1/2, A). On the other hand, the non-perturbat ive superpotential must take the form W = ~ i die -a~S, so , as ex- plained above, it is reasonable to assume that at the realistic point, S --~ 2, W is dominated by' one of the terms, say W ~ e -~s, as it happens for in- stance in usual gaugino cond'ensation. Thus, the condition Aco~ = 0, i.e. eq.(18), gives us a further constraint, namely

I - a + A'sl ~ - 3~Cs~ = 0 (21)

which relates the values of Ks , K s s and a. Notice that the latter is essentially fixed by the condition

4This , of course , m a y not occur . However, it is a c o m m o n a s s u m p t i o n of all ex i s t ing s t r i ng -based mode l s ( inc luding the usua l d i l a t o n - d o m i n a n c e model ) . It should be noted however t h a t the un ive r sa l i ty of the sca lar masses (unless t h a t of g a u g i n o masses ) is s o m e t h i n g imposed , r a the r t han ob t a ined f rom the model .

of a hierarchical SUSY breaking (a _~ 18), thus eqs.(20) and eq.(21) give a non-trivial scenario whose phenomenology could be investigated.

4. A n s a t z s fo r n o n - p e r t u r b a t i v e K~ihler p o t e n t i a l s

It is tempting to go a bit further in our analysis and explore the phenomenological capabilities of explicit (stringy) non-perturbat ive effects on K which have been suggested in the literature s .

From the arguments explained in ref.[7], it is enough to focuss our at tention on possible forms of K(S + S), exploring the chances of getting a global minimum at V = 0. From eq.(21) we can check that a successful generalized dilaton domi- nated scenario (i.e. with V = 0 at ReS ~_ 2) re- quires very sizeable non-perturbat ive corrections to K. This can be seen by considering the form of eq.(21) when K is substi tuted by the tree level ex- pression K = - log(S' + S). The fact that a _~ 18 implies that the non-per turbat ive "corrections" to Ks, Kss must indeed be bigger than the tree level values. If we further demand that the po- tential has a min imum at V = 0 this implies

(-a + K ' ' - 3I*'''-4 = 0

''' = 0

(at the realistic point ReS = 2), where tile primes denote derivatives with respect to ReS (the sec- ond equation is simply eq.(21)). It would be nice if the previous conditions could be fulfilled by some simple form of K.

The fact that the SUGRA lagrangian has an ex- ponential dependence on K (e.g. it can be written as £ = [e-K/a]O ) together with the fact that in the known examples (mainly orbifold compacti- tications) the perturbat ive corrections to K are small, suggest that a sensible decomposition for K is

e z" = e t ' ' ' - + e K - p , ( 2 3 )

where the first term corresponds to the tree level expression (I<-t~e~ = - l o g ( 2 ReS)) and A'~ v de-

5For o the r a t t e m p t s in th i s sense , see ref.[11].

JA. Casas/Nuclear Physics B (Proc. Suppl.) 52A (1997) 289-293 293

notes the non-perturbative contributions. The next step is to choose some plausible form for I(,~p(ReS) and to study the form of V, see eq.(17). The field theory contributions t o [(np have been evaluated in ref.[12], but for a realistic ease they turn out to be too tiny to appreciably mod- ify the tree-level results 6. On the other hand, according to the work of ref.[8], stringy non- perturbative effects may be sizeable (even for weak four-dimensional gauge coupling) and plau- sibly go as g-Pc -61g with p, b ,-~ O(1). Then a simple possibility is to take

e K 'v = d g-Pc -b/g (24)

where d, p, b are constants with p, b > 0 and 9 -2 = ReS.

The numerical results indicate that, plugging this ansatz, it is not difficult to get a mininmm of the potential using just one condensate with a ~_ 18 (thus guaranteeing the correct size of SUSY breaking) and sensible values for the d, p, b constants. More precisely, for d = -3 , p = 0, b = 1, there is a minimum of the potential at ReS = 2.1. Unfortunately, the negative value of d makes this example unacceptable. Another ex- ample with positive d is d = 7.8, p = 1, b = 1, which has a minimum at ReS = 1.8. However, as a general result, playing just with these simple forms for K, it seems impossible to get the min- inmm at V = 0. Anyway, it is impressive that just with one condensate and choosing very rea- sonable values for d, p and b (note also that there is no fine-tuning in the previous choices), a min- imum appears at the right value of the dilaton. Another typical characteristic of these examples is the appearance of singularities in the potential caused by zeroes of the second derivative of the KS~hler potential, K " , at some particular values of S. Of course, this can be cured by additional terms in K.

Finally, it would be worth to analyze the possi- ble implications of this kind of non-perturbative

6More precisely, the a u t h o r s of ref.[12] ob- ta in I( = - -3 log [(2 ReS) 1/3 4- e - K p / 3 4- de -(2 ReS)/2c], where I'[p d e n o t e s t he p e r t u r b a t i v e cor rec t ions to the tree- level express ion , d is an u n k n o w n c o n s t a n t and c is essen- t ial ly the coefficient of t he one- loop b e t a func t ion ( the n o n - p e r t u r b a t i v e s u p e r p o t e n t i M goes as e -3S/2c) .

K~ihler potentials for other different matters, such as the cosmological moduli problem [13,14].

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