Modular invariant formulation of multi-gaugino and matter condensation

N

ELSEVIER

Nuclear Physics B 493 (1997) 27-55

Modular invariant formulation of multi-gaugino and matter condensation* Pierre Bin&my 1, Mary K. Gaillard 2, Yi-Yen Wu Department of Physics, University of California, and Theoretical Physics Group, 50A-5101, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Received 26 November 1996; accepted 25 February 1997

Abstract Using the linear multiplet formulation for the dilaton superfield, we construct an effective Lagrangian for hidden sector gaugino condensation in string effective field theories with arbitrary gauge groups and matter. Non-perturbafive string corrections to the K~ihler potential are invoked to stabilize the dilaton at a supersymmetry breaking minimum of the potential. When the cosmological constant is tuned to zero the moduli are stabilized at their self-dual points, and the vevs of their F-component superpartners vanish. Numerical analyses of one- and two-condensate examples with massless chiral matter show considerable enhancement of the gauge hierarchy with respect to the Es case. The non-perturbative string effects required for dilaton stabilization may have implications for gauge coupling unification. As a comparison, we also consider a parallel approach based on the commonly used chiral formulation. @ 1997 Elsevier Science B.V.

PACS: 11.30.Pb; 11.25.Sq; 11.10.Ef; 04.65.+e Keywords: Modular invariance; Linear multiplet; Multi-gaugino condensation

1. I n t r o d u c t i o n E f f e c t i v e L a g r a n g i a n s for g a u g i n o condensation in effective field theories f r o m superstrings w e r e first c o n s t r u c t e d by generalizing the w o r k o f Veneziano and Y a n k i e l o w This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-95-14797. Visiting Miller Professor. Permanent address: Laboratoire de Physique Th6oriqne et Hautes Energies (Laboratoire associ6 au CNRS-URA-D0063), Universit6 Paris-Sud, F-91405 Orsay, France. 2 Miller Professor, Fall 1996. 0550-3213/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PI1 S 0 5 5 0 - 3 2 1 3 ( 9 7 ) 00162-4

28

P. Bindtruy et aL/Nuclear Physics B 493 (1997) 27-55

icz [ 1] to include the dilaton [2] and gravity [3]. These constructions used the chiral formulation for the dilaton superfield. While the resulting Lagrangian has a simple interpretation [4] in terms of the two-loop running of the gauge coupling constant, it does not respect the modular invariance [5] of the underlying superstring theory. Modular invariance was recovered [6] by adding a moduli-dependent term to the superpotential that is reminiscent of threshold corrections [7] found in some orbifold compactifications. However, there is a large class of orbifolds that do not have moduli-dependent threshold corrections [ 8 ] ; moreover in all orbifold models, at least part of the modular anomaly is canceled by a Green-Schwarz counterterm [9], which must therefore be included. This has the unfortunate effect of destabilizing the dilaton. It was recently shown how to formulate gaugino condensation using the linear multiplet [ 10] formulation for the dilaton superfield, both in global supersymmetry [ 11,12] and in the superconformal formulation of supergravity [ 12]. In this case the superfield U which is the interpolating field for the Yang-Mills composite superfield (U -~ T r ( W ~ W ~ ) ) emerges as the chiral projection of a real vector supermultiplet V whose lowest component is the dilaton field g. Using the K~ihler superspace formalism of supergravity [ 13,14], which we use throughout this paper, it was subsequently shown [ 15] how to include the Green-Schwarz term for a pure Yang-Mills E8 hidden sector. In this case there are no moduli-dependent threshold corrections and there is a single constant the /?8 Casimir C - that governs both the Green-Schwarz term and the coupling renormalization. That model was studied in detail in Ref. [16], where it was found that the dilaton can be stabilized at a phenomenologically acceptable value with broken supersymmetry if non-perturbative terms [ 17,18] are included in the KS_hler potential, 3 but a sufficiently large gauge hierarchy is not generated. The advantage of the linear multiplet formulation of gaugino condensation is twofold. First, it is the correct string formulation since among the massless string modes are found the dilaton and the antisymmetric tensor field. Second, the traditional chiral formulation of gaugino condensation is incorrect in that it treats the interpolating field U ~- T r ( W ~ W , ) as an ordinary chiral superfield of K~ihler chiral weight w = 2. However, this is inconsistent [ 11,12,15] with the constraint -

( 79~ 79~ - 2 4 R t ) Tr( W ~ W . ) - (79a79 a - 2 4 R ) Tr( W a W a)

=

total derivative, (1.1)

where W'* is the Yang-Mills field strength chiral supermultiplet and the chiral superfield R is an element of the super-Riemann tensor. On the other hand, the superfield U considered as the chiral projection of the real vector superfield V automatically satisfies the constraint (1.1) with Tr(W'~142,,) --~ U. Moreover, the implementation of the Green-Schwarz anomaly cancelation mechanism is simpler in the linear multiplet formulation [ 15] and much closer in spirit to what happens at the string level. 3 A similar observationhas been made by Casas [ 19] in the context of the chiral formulation and without modular invariance.

P Bin(truy et al./Nuclear Physics B 493 (1997) 27-55

29

As mentioned above, our analysis in Ref. [ 16] only dealt with a pure Yang-Mills Es hidden sector. This was chosen for the purpose of illustration of the method but has several drawbacks from the point of view of phenomenology. First, the gauge coupling blows up very close to the unification scale and therefore does not allow for a large hierarchy. Second, there are no moduli-dependent threshold corrections and therefore this cannot be used to fix the vacuum expectation values of moduli fields, using for example T-duality arguments. A more realistic situation which would involve' moduli-dependent threshold corrections, would be the case of a hidden sector gauge group being a product of simple groups: C = Ha Ca- One immediate difficulty is the following: since we want to describe several gaugino condensates Ua "~ Tr(W"W,,)a, we need to introduce several vector superfields V~. However, since the theory has a single dilaton g, it must be identified with the lowest component of V = ~ Va. What should we do with the other components ga = V~[0=~=0? We will see that, in our description, these are non-propagating degrees of freedom which actually do not appear in the Lagrangian. Similarly only one antisymmetric tensor field (also associated with V = ~ a V~) is dynamical. This allows us to generalize our approach to the case of multicondensates. Let us stress that the goal is very different from the so-called "race-track" ideas [20] where going to the multicondensate case is necessary in order to get supersymmetry breaking. Here supersymmetry is broken already for a single condensate. Indeed, we will see that the picture which emerges from the multicondensate case (complete with threshold corrections and Green-Schwarz mechanism) is very different from the standard "race-track" description: indeed, the scalar potential is largely dominated by the condensate with the largest one-loop beta-function coefficient, To be more precise, we generalize in this paper the Lagrangian of Ref. [ 16] to models with arbitrary hidden sector gauge groups and with three untwisted (1,1) moduli T 1. We take the Kahler potential for the effective theory at the condensation scale to be n

K=k(V) +~gl,

gl = _ l n ( T z + 7~l),

1

V= ~V~,

(1.2)

a=l

where the Va are real vector supermultiplets and n is the number of (asymptotically free) non-Abelian gauge groups U~ in the hidden sector, /7

(1.3)

Chidden = H Ca @ U ( 1 ) m. a=l

We will take Chidden to be a subgroup of Es. We introduce both gaugino condensate superfields Ua and matter condensate superfelds H a that are non-propagating, A

Ua ~ Tr(W'~W~)a,

H'~ --~ I - [ (~/,A)"~ , A

(1.4)

30

P. Bindtruy et aL/Nuclear Physics B 493 (1997) 27-55

where ]A)a and (pa a r e the gauge and matter chiral superfields, respectively. The condensate H a is a chiral superfield of K~ihler chiral weight w = 0, while the condensate Uo associated with Go is a chiral superfield of weight w = 2, and is identified with the chiral projection of Va,

Uo = -(7)aD a -- 8R) Va,

Uo = -(D~73~ -

8Rt)Va.

(1.5)

We are thus introducing n scalar fields go = V~[0=~=-o. However, only one of these is physical, namely g = ~-]~aga;the others do not appear in the effective component Lagrangian constructed below. The effective Lagrangian for gaugino condensation is constructed and analyzed in Sections 2-5. In Appendix A we discuss a parallel construction using the chiral supermultiplet formulation for the dilaton and unconstrained chiral supermultiplets for the gaugino condensates, in order to illustrate the differences between the two approaches. In Section 6 we summarize our results and comment on their implications for gauge coupling unification.

2. Construction of the effective Lagrangian We adopt the following superfield Lagrangian:

ff-'eff= £~KE "}- £GS -']- Eth -]- £VY "]- £pot,

(2.1)

where

£KE= f d4OE[--2+ f(V)],

k(V) =lnV+g(V),

(2.2)

is the kinetic energy term for the dilaton, chiral and gravity superfields. The functions f(V), g(V) parameterize non-perturbative string effects. They are related by the condition V

dg(V) df(V) = -V +f, dV dV

(2.3)

which ensures that the Einstein term has the canonical form [ 16]. In the classical limit g = f = 0; we therefore impose the weak coupling boundary condition

g(V = 0) = 0

and

f ( V = 0) = 0.

(2.4)

Two counter terms are introduced to cancel the modular anomaly, namely the GreenSchwarz term [9], /~GS = b /

d40 EV ~ g l , l

b =

C 8¢r2,

and the term induced by string loop corrections [7],

(2.5)

P Bingtruy et aL/Nuclear Physics B 493 (1997) 27-55

31

d40 Ualn~]2(T ~') +h.c.

(2.6)

a,]

The parameters

bl=C-Ca+Z(1-2qa)

C A,

C=CEs,

(2.7)

A

vanish for orbifold compaetifications with no N = 2 supersymmetry sector [ 8]. Here Ca and C A are quadratic Casimir operators in the adjoint and matter representations, respectively, and q/a are the modular weights of the matter superfields @a of the underlying hidden sector theory. The term EVy = ~

l f

d4oE---Ua[b'aln(e-K/2Ua/tZ3) + ~-"b°~lnH /~ a R '~1, + h.c.,

a

(2.8)

a

where /z is a mass parameter of order one in reduced Planck units (that we will set to unity hereafter), is the generalization to supergravity [2,3] of the VenezianoYankielowicz superpotential term, including [ 21] the gauge invariant composite matter fields H a introduced in Eq. (1.4) (one can also take linear combinations of such gauge invariant monomials that have the same modular weight). Finally, .~pot =

1

/,

J d4OgeK/2w ( H a , T 1) -Jr-h.c.

(2.9)

is a superpotential for the matter condensates that respects the symmetries of the superpotential W(d~ A, T 1) of the underlying field theory. The coefficients b in (2.8) are dictated by the chiral and conformal anomalies of the underlying field theory. Under modular transformations, we have

aT 1 - ib TI---~ icT I + d'

ad-bc=l,

gl _+ Ft + Fq '

a,b,c, d c Z ,

Fl = ln( icT1+ d),

qDA __+e - "¢"~ 2.~l F I qla ~ A ,

A,, --+ e -½ ~-]', Imyl/~a,

X A --~ e ½~'(iImFI--2qAF') xA,

Ua __.+ e - i ~ - ~ q

1U ~

lmelua '

e- ~l

Fl q~ Hot '

0 -k+\e-½ ~ , ImFto,

A A qla = ~-~n,~ql" A

(2.10) \

The field theory loop correction to the effective Yang-Mills Lagrangian from orbifold compactification has been determined [22,23 ] using supersymmetric regularization procedures that ensure a supersymmetric form for the modular anomaly. Matching the variation under (2.10) of that contribution to the Yang-Mills Lagrangian with the variation of the effective Lagrangian (2.8) we require

8/2vy -

I ~1fd4oEuo[c~--~cA(1--2qA)]U+h.c., A,I

647r2

,

(2.11)

32

P. Bingtruy et al./Nuclear Physics B 493 (1997) 27-55

which implies

Oanaq' = ~

Ca-

CA ( 1 - 2q

VI.

(2.12)

eg,A

In the flat space limit where the reduced Planck mass mp -+ ~ , under a canonical scale transformation

A --+ e3~A,

U ---+e3CrU,

qbA ---+ e~q5A,

I I °' ~

e ~ A na°'IIa '

0 --+ e-l~o, we have the standard trace anomaly as determined by the/S-functions,

a£eff =

1

6 - T ~ 2 o"

~a i d 4 O E U a ( 3 C a - ~ A

c Aa ) + h . c . + O ( m ~ l

),

(2.13)

which requires

bana=~-~2

3Ca i

a

-kO(mpl) •

(2.14)

ot,A

Eqs. (2.12) and (2.14) are solved by [21] (up to O(rn-~ 1) corrections)

..='( a'a) Ca--

a

,

Z.c~ A A Oanaql = Z a,A A

b>a: y; cA o~,A

A

CA va qA, 47r 2 l

(2.15)

4~r2"

Note that the above arguments do not completely fix Z;~ff since we can a priori add chiral and modular invariant terms of the form A£. =

Z i btaa

d40EValnIe ~ , qsg " IH a l l --a > .

(2.16)

a,oL

For specific choices of the b ~ the matter condensates H '~ can be eliminated from the effective Lagrangian. However, the resulting component Lagrangian has a linear dependence on the unphysical scalar fields g.a - g.b, and their equations of motion impose physically unacceptable constraints on the moduli supermultiplets. To ensure that AE contains the fields ga only through the physical combination }-~aga, we have to impose b~a, = b2 independent of a. If these terms were added, the last condition in (2.15) would become A ~-~ bana Ca ot A + ~-~ h' n z = ~ (2.17) -a a 4¢r2ce,A

A

A

We shall not include such terms here.

P Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

33

! - ~ baq ot ot Combining (2.7) with (2.15) gives b~ = 8~r2(b - b~ , ). Superspace partial integration gives, for X any chiral superfield of zero KS_hler chiral weight,

+i'oP.

lnX ÷ h.c. = i

d40 EVa ln(XX)

-O,,, ( f d40~-DaVaE ~" ÷

h.c.),

(2.18)

where E am is an dement of the supervielbein, and the total derivative on the righthand side contains the chiral anomaly (~ araBm ~- FanFT) of the F-term on the left-hand side. Then combining the terms (2.2)-(2.9), the "Yang-Mills" part of the Lagrangian (2.1) can be expressed - up to a total derivatives that we drop in the subsequent analysis - as a modular invariant D-term,

+ ~-~ b~ In (H~/)~) o.

-

z W". InI/'-'-') '+""l

(2.19)

where H ~ = 1-I(cr~°) n~ = e}-~*q~gt/2Ha,

qT)A = e ~ # qAg'/2~ A

(2.20)

A

is a modular invariant field composed of elementary fields that are canonically normalized in the vacuum. The interpretation of this result in terms of renormalization group running will be discussed below. We have implicitly assumed affine level-one compactification. The generalization to higher affine levels is trivial. The construction of the component field Lagrangian obtained from (2.19) parallels that given in Ref. [ 16] for the case U = Es. Since the superfield Lagrangian is a sum of F-terms that contain only spinorial derivatives of the superfield Va, and the GreenSchwarz and kinetic terms that contain V~ only through the sum V, the unphysical scalars ga appear in the component Lagrangian only through the physical dilaton g. The result for the bosonic Lagrangian is 1/Z B 1 1 e = - ~ 7 - £ - (1 + b g ) ~-'~ (t, + T,)2

(Om~'OmtZ-PtF')

I

1

16e2 (eg,, + 1)[4(OmeOme- B'B,,) + ~u-4e"/'e (W~ + ueO] +l(gg(1)-2) [lf4M--bmbm--3{29I(~b'bUb--4WeK/2)+h.c.

}]

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

34

+-~

~

+ b'a In (e2-KFtaUa) + ~" ba a ln(qra'h ) a

+~1 [ b g l - 4 ~ l n l ~ 7 ( t l ) l i ] } ( f a - U a l f / l + h . c . ) 1 16g

bla(g.g,,, + l ) f t U a - - 4 g U a \ ~

+h.c. l + ~i ~ [ b ~ a l n ( u a~)

b Om Im t ! 2 ~--~ ~ - ~ B m + ~

~¢r~+(b;-b)2---R~et l

+ Z b a l n C r r °~~-) ]

bl

VmB~n

[;(t')(2iB~'Vmt'-uaF')+h.c.]

l,a

1

+eK/2 I ~ F I ( W I + K I W ) + ~ F a W a + h ' c "

'

(2.21)

where ~'(t)

-

oo

1 a~7(t)

- ~7(t)

- - , at

=

I-i (1 m= 1

g VIo=o_-o, =

1

2

m

8 °'=

~%B.~ = ~ [7;,,~. ~ga] v~ Io=o=o + ggao-,~,~b,,,,

a

u~ = U~lo=o=o= _(732 _ 8R) V~]o=O=o,

U = ~Ua, a

0,~ = Oa]o=O=o= _ ( ~ 2 _ 8Rt)V~[o=O=o,

= ~--~ t~a, a

-4F

- 4 J e~ = 7320~1o=0=0,

~ = D2Oalo=O=o ,

Fv = ~ F

~,

a

~-1- ~

rt°'lo=o=o,

¢r°'

=

B~lo=o__o,

- 4 F ~ = 7)2IIa[o=O__O, - 4 F ~ = 732Da]e=O__o, t I = TIIo=~=o, - 4 F I = D2TI]o=~____o, ~= T/Io=0=o, - 4 F I = 732T/10=0=o,

(2.22)

bm and M = (aT/)t = -6RIo=~=o are auxiliary components of the supergravity multiplet [ 13]. For n = 1, Ua = u, etc., (2.21) reduces to the result of Ref. [ 16]. The equations of motion for the auxiliary fields bin, M, F I, F a + F ~ and F ~ give, respectively, bm= 0,

M = ~ a

35

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

2(1 +

--

be)

aa (b - b~) +

((fl)Ret l

4eK/2 (2RetIVVl -- 17V)},

aaUa g--eg-(f +l)/b'j-~, bJ /S#b" H -

-- e2

Ir/(t/)Ib'°/2~b" H

1

( qr~r~'~r)-vUb"

a

qTr = H r 10=0=0'

0 =Z b°~ua+ 4¢r%K/2W~ Va.

(2.23)

a

Using these, the Lagrangian (2.17) reduces to l c~ = -in

e

-

2

om~' o ' t '

(l+bg)~-~(tz+?~)2

1

4g2

(g&, + 1) (OmgOmg- BmBm)

I

( ~a ) b~t Omlmtl - ~a b'awa+ b~¢~ VmB"a'- ~ ------TBm Re t +i Z bI [C(t')B'~'VmtI - h . c . ]

- V,

l,a

V= (gg';~6g+21) {rtu+g [gt (~a ffaUa-4eK/2W) + h.c.] } 1

~ 16(1+be)~

~ (

bI

)

Uo b--ffa+'~-~Z;(tl)Ret !

-4eK/2(2RetlWt--W) 2 + N1( g g , , - 2 )

~ b'bUb--4WeK/22 , (2.24)

where we have introduced the notation Ua = pa ei~°o ,

(2.25)

77"a = 71aei¢ ~ ,

and

2¢ 4 = - i l n \

~

abt

/

-

if w~ ~ o.

(2.26)

To go further we have to be more specific. Assume 4 that for fixed ol, b~ v~ 0 for only one value of a. For example, we allow no representations (n, m) with both n and m v~ 1 under ~a ® ~b- Then Ua = 0 unless W~ 4~ 0 for every c~ with b~ v~ 0. We therefore assume that b~ 4~ 0 only if W~ v~ 0. 4For e.g. G = E6 ® SU(3), we t a k e / / _ ~ (27) 3 of E6 or (3) 3 of SU(3).

36

P. Bindtruy et aL/Nuclear Physics B 493 (1997) 27-55

Since t h e / / ~ are gauge invariant operators, we may take W linear i n / 7 ,

W(II, T) = ~ W ~ ( T ) I I %

W~(T) =c~H[rl(Tt)]2(qT-1),

o~

(2.27)

1

where r/(T) is the Dedekind function. If there are gauge singlets M i with modular weights q}, then the constants c~ are replaced by modular invariant functions,

c~ --+ w,~(M, T) = Co~H ( M i ) n7 H [ T ] ( T I ) ] 2nrq} . i

In addition, if s o m e the gauge group

M i

I

have gauge invariant couplings to vector-like representations of ]2(qA+q~+q)),

W ( ¢ , T, M) ~ CiABMi(I)a~Tf)B H [ 7 ] ( T / ) I

one has to introduce condensates H AB -~ q~a~f9B of dimension two, and corresponding terms in the effective superpotential,

W( II, T, M) ~ CiABMiII AB 1-I[v(T 1)

] 2(qAwq/B+~).

1 Since the M i are unconfined, they cannot be absorbed into the composite fields//. The case with only vector-like representations has been considered in Ref. [21 ]. To simplify the present discussion, we ignore this type of coupling and assume that the composite operators that are invariant under the gauge symmetry (as well as possible discrete global symmetries) are at least trilinear in the non-singlets under the confined gauge group. We further assume that there are no continuous global symmetries - such as a flavor SU(n)R®SU(n)L whose anomaly structure has to be considered [21]. With these assumptions the equations of motion (2.23) give, using ~-]~ b~q~[ + b~a/8rre = b - b~a, p2 = e--2b~o/b.eK e--(l+ f)/b~g--b ~-~.*gt/bo

H It/(t*)[4(b-b")/b~ H I

OZ

~q'r =

_e_~[k+~,(l_q,)g ] 4waUa, b~ 1

a

I

[b'~/4c,~[ -2b:/b°,

a

ba =--bat + ~-~b~.

(2.28)

og

Note that promoting the second equation above to a superfield relation, and substituting the expression on the right-hand side f o r / 7 in (2.19) gives

--

ba In e ~ , g ('=qT)[4Wa/ba] 2 £e

-

~,

b~

in

[(TI+~,)Irl2(T1)I2]} ) -[-~pot.

(2.29)

37

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

It is instructive to compare this result with the effective Yang-Mills Lagrangian, found [ 22,23 ] by matching field theory and string loop calculations. Making the identifications V -+ L, Ua --+ Tr(l'V~14;~)a, the effective Lagrangian at scale /z obtained from those results can be written as ~e~(~)=

I (

-2+ f(V)+

d40E

Va ~

C.-~

a

/*6gs4 ] 1 L)~6g~-3-~-nj - a-~2 ZCAln[ gs2/3Za(m)/g2/3(")zA(~)]

×ln r

A

~

-

In [(r* + f*)Irl2(r/) 12]

,

(2.3O)

i

with/xs2 ,-~ g~ ,-~ g in the string perturbative limit, f ( V ) = g(V) = 0. The first term in brackets in (2.29) can be identified with the corresponding term (2.30), provided

~-~ b: =

1

A,

A

bo=

Ca-- 5

a

I

In fact, this constraint follows from (2.15) if t h e / P are all of dimension three, which is consistent with the fact that only dimension-three operators survive in the superpotential in the limit mp -~ oo. Then ba is proportional to the fl-function for Ga, and Pa ~(IAaaal) has the expected exponential suppression factor for small coupling. In the absence of non-perturbative corrections to the K~hler potential ( f ( V ) = g ( V ) = 0), (2V[e=~__0) = (2g) = g2 = /~2 is the string scale in reduced Planck units and also the gauge coupling at that scale [ 22,23 ]. Therefore the argument of the log, =

-~/3 6s

-

~ I~sgs

,

(2.32)

gives the exact two-loop result for the coefficient of Ca in the renormalization group running from the string scale to the appropriate condensate scale [ 4,22,23 ]. The relation between (~-~) and (Ua), and hence the appearance of the gaugino condensate as the effective infrared cut-off for n~assless matter loops, is related to the Konishi anomaly [24]. The matter loop contr~butlons have additional two-loop corrections involving matter wave function renormalization [ 25-27 ], .

c9In ( /In/.12 Z Za) c ~

-

-

[

.

32~-21t [ , geg ~ e ~ l g l ( 1 - q a - q f - q C ) z A l ( ] ' L ) Z B l ( [ Z ) Z c l ( ] Z ) B C

×IWAscI2--4~--~(tz)C~(RA)

-+-O(g4) -[-O(qO2),

(2.33)

a

where C ~ ( R A ) = ( d i m G a / d i m R A ) C A, RA is the representation of Ga on @a. The boundary condition on ZA is [22] Za(IXs) = (1 -pag)-~, where PA is the coefficient of

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

38

e x p ( ~ l qagl)iq~a [2 in the Green-Schwarz counter term in the underlying field theory: V = ~ , g l + pAexp(~,qagt)[q~Zl2 + O(iq~A]4). The second line of (2.29) can be interpreted as a rough parameterization of the second line of (2.30). In the following analysis, we retain only dimension-three operators in the superpotential, and do not include any unconfined matter superfields in the effective condensate Lagrangian. The potential takes the form V= 16g 1 2 ~PaPbCOSOOabRab(tl),

Wab = COa-- rob,

a,b

Rab = (g&n + 1) (1 + bag) (1 + bbg) -- 3gZbabb + (1 +g2bg.~ ~ 1 da(tl)db(fl)' bz da(t t) = b - b " + 2--~((t/) Re t I - ~

b°~ [l - 4(q 7 - 1)Re t'g(tz)]

o/

= (b - ba) (1 +

4((tZ)RetZ).

(2.34)

Note that da(t l) o< F l vanishes at the self-dual point t I = 1,~'(t 1) = - 1 / 4 , r l ( t 1) 0.77. For Ret I ~ 1 we have, to a very good approximation, ( ( t I) ~ -¢r/12, rl(t* ) e -rrt/12. Note that also Pa - and hence the potential V - vanishes in the limits of large and small radii; from (2.28) we have lim p2 ~., ( 2 R e t 1) (b-b.)/boe-Cr(b-bo)RetZ/3ba ' tl__+oo

lim p2 ,.~ (2 Re t l ) (b"-b)/b"e-Cr(b-b")/3b" Retl,

(2.35)

tt--+0

where the second line follows from the first by the duality invariance of pa2. So there is potentially a "runaway moduli problem". However, as shown in Section 4, the moduli are stabilized at a physically acceptable vacuum, namely the self-dual point.

3. T h e a x i o n content o f the effective theory

Next we consider the axion states of the effective theory. If all W,, =# 0, the equations of motion for (-O a obtained from (2.24) read OF-. t m a 1 ( b~ua "~ m b og Ow--'Ta=-baV Bm-'2~-'~b'{,~,b \ 2 c b'Zuc +h.c. ] V Bm -- -=0.eOJa -

(3.l)

These give, in particular, aoV'B

off, a

= 0.

(3.2)

a

The 1-forms B ma are a priori dual to 3-forms,

1

(1._~Fnp q

B~ = ~em,,pq \ 3 ! 4

+

)

Onbl~aq

,

(3.3)

39

P. Bindtruy et al./Nuclear Physics B 493 (1997) 2 7 - 5 5

where F npq and bPaq a r e 3-form and 2-form potentials, respectively; (3.3) assures the constraints (1.1) for Tr()/V~)/V~) --+ Ua; explicitly, 2i

(D'~Do~ - 24Rt)Ua - (D,~D e~ - 24R)(]a = -2i*q~a = - - -3! - ' : ~mnpq~"amr'1/Pq * a = -16ivmBa,.

(3.4)

We obtain ,.

1

-ba q~a--2 Z b ~

(b~ua._[_h.c.).qbb=80V ~cb~u~c Oto--Ta'

o~,b

~-~ba*'Oa=O.

(3.5)

a

O, b pq can be removed by a gauge transformation F npq --+ F 1/pq + O [ n A pq] •

If F 1/pq ~

Thus

1 1 - E B'a* = 2--~a"~mnl'qO1/~q + -3!8 nmpqF npq a '

baF~,pq = O, a

bab~aq,

~Pq = ~ o

(3.6) ~--~

and we have the additional equations of motion, ff-.B = O,

-qO67-p ~-.B = O,

oc,,

\e(v~,~)) '

c= - -~

(3.7)

which are equivalent, respectively, to

e,,mpq~ ~V ~C.B =O,

1

1

6B~

t3 ~ £B = 0 ,

/

(3.8)

bb ~B~~

with

1 6 £

et3-Ba*

B=

(gg,,, + 1) Bm+ 2g2

b,oOn,w~ +

~

2a,b

b~

E c b°duc + h.c. Omwb

e4"~)] +Zbao [e.,ee4"~(emt, / -~-+,~\ 77-+h'c" +i~ ~b~ [((t')a"t'-h.c.] _~b amImt'Ret' a,l

(3.9)

1

Combining these with (3.1) and the equations of motion for g and t 1, one can eliminate B1/~, to obtain the equations of motion for an equivalent scalar-axion Lagrangian, with a massless axion dual to bm1/Again, these equations simplify considerably if we assume that for fixed ce, b~ 4= 0 for only one value of a. In this case (3.1) reduces to Ill

10V

a

x7 Bm

-

ba OtOa'

(3.10)

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

40 and we have

Oq~---~= O, Og

Oqb" - i ( ( t ' ) (q~ - 1) at 1

(3.11)

if we restrict the potential to terms of dimension three with no gauge singlets M i. Using }--],~b'~ ( q'[ - 1) + bl /Srr 2 = b - ba gives

l a_a_£._ (gg<,,+ e aB,,a

l) Bm_k_baOma) a

2g 2

+i~1 {om" [<(tl)(b-ba)-}-~] -h.c.} (gg<,, +l)Bm+bac~mo.ja+~-~gml m[(b_ba)'rr t, 6 2--------~ b ] 292 i Re t ' (3.12) where the last line corresponds to the approximation ( ( t 1) ~ - ¢ r / 1 2 . In the following we illustrate these equations using specific cases.

3.1. Single condensate As in Ref. [16] there is an axion ~o = o)a + (rr/6)(b/ba - 1) y ] s l m t I that has no potential, and, setting ! Emnpq~nbpq --

2

2g 2 (baOm m __ b S-" 0 m I m t I (ggm + 1) \ 2 z_.,l R e t - 7 - J '

(3.13)

the equations of motion derived from (2.24) are equivalent m those of the effective scalar Lagrangian,

l £n = - l T¢ e

2

- (1 -k- be)

F

omgl amtt

(tl + ~-,)2

(gg,,,g2+1) ( b a O m w _ b~~

!

452 (~-g,l,-I-1)3mgOmg-V(g,t', fl)

OmlmtI~ (baOmo)_b

~.~-~7)

Omimtl~

2F Reed)' (3.14)

3.2. Two condensates: bl 4= b2 Making the approximation r/(t) ~ e -eft~12, the Lagrangian (2.24) can be written as

1£8= e

1~ ~ 2 - (1 + be)

om?*Omt* ( d -}- ~l)2

1 4g2 (gg,,, + 1) (3mgamg - BmBm)

1

b

- m ~ m B m - °)'VmBm - "2 F

Om Im t l

Ret-i---7--Bm- V,

(3.15)

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

41

where 09--

b] °91-- b2

-67rZ Im

fl_ bl - 2 bib2 '

cot_

w l 2 + ~ - ~ i m/ t / ' / 3

B" = ~ b a B m.

(3.16)

a

We have

¢r

tl

bzr ZIm,' ) + gl ( o)'--g-

I

I

OV OV OV Owl = --0o)2 - 3o)12"

(3.17)

Then taking co, tot and t I as independent variables, the equations of motion for w, cot are

Bm=

~ m B m = O,

1

2 emnpqC)nDpq'

og VmBn~ = ~1. q~ --- t3 O9O)12 '

Bm = -3 !-18e

mnpq

F npq•

(3.18)

Substituting the first of these into the Lagrangian (3.15), we see that the axion co and the 3-form /~m drop out because they appear only linearly in the Lagrangian; hence they play the role of Lagrange multipliers. The equation of motion for bmn implies the constraint on the phase w, VmOn~o) = 0.

(3.19)

The equations of motion for Im t I and Fmnp read

[

OmImd

O=~'m (l+bg) 2 ~ Z + 2 R e t

O- (eg,~, + 1) Bin + 8row, 2g2

b~

-

-

2

b---~Bm] , ] -i(~tl-

h.c.)-~*(b,

-

8"' Im t I - -Re - - -t7 - '

(3.20)

and the equivalent scalar Lagrangian is 1

c)mtlOm tl

123=---~7-~-- (1 + b g ) ~

( t / + 7l)2

1

462 (gg(, + 1)OmgOmg

I

62 --g ( g, t I, 7l, o)12).

(3.21)

P. Bingtruy et aL/Nuclear Physics B 493 (1997) 27-55

42

As in Section 3.1, there is a single dynamical axion w ~ - or, via a duality transformation, %b - but there is now a potential for the axion.

3.3. General case We introduce n linearly independent vectors/~m, Bm, B,,, ^i i = 1 . . . n - 2 , and decompose the B~ ~ as i

B~'=aaBm+caB'+ZdaBi,

B^ im =~-~eaBnaZ.

^ m

i

(3.22)

a

Then

baw a + ( b - ba ) -~

Im t 1 V mB~' = OoV mB"~ + w ' V mBm + ~-~ W~V mB'~', i

,oa = o~ + g~ Z l m ~ tl+~--~a

w ' - - -g-

Imd

+

' ' Va,0,

(3.23)

and the Lagrangian can be written as in (3.15) with an additional term,

-'-->

e

v m

e

i

(3.24)

"

i

The equations o f motion for the phases o) are -

VmBm =

OV

OoJ -

OV = O,

~

&o---~ a

3V

10V

1~

OV

1. ~ab

a

VmB~' •

OV =-Ow

e~' OV

i =-~-~baOwaa

1

~

ba - bb - babb '

(3.25)

8"~i'

i p = 8emnpq~qi give omw i = 0. Hence and the equations for Finn - -~-

Im t l

+ Rab,

Oab = constant.

(3.26)

Thus as in the two-condensate case o f Section 3.2, there is one dynamical axion with a potential. The dual scalar Lagrangian is the same as (3.21), with V = V(g, t 1, ?/, ~oa#).

4. The effective potential The potential (2.34) can be written in the form

P. Bindtruyet aL/NuclearPhysicsB 493 (1997)27-55

v= 16gl__2__(Vl -

43

v2 + v3) ,

vl = ( l + gg(, )

~a

( l + bae)

Ua 2

g2

v3-(l+bg) ~

,

v2 = 3g 2

2 ~a d a ( t l ) u a

~a baua 2' FI 2

=4g2(l+bg)~-~l

RetI

"

(4.1)

In the strong coupling limit lim V = (gg~l) - 2)

~a baua 2,

(4.2)

g---+oo

giving the same condition on the functions f, g as in Ref. [ 16] to assure boundedness of the potential. Note however that if Vl = v3 = 0 has a solution with v2 4= 0, the vacuum energy is always negative, u3 = 0 is solved by t ! = 1, i.e. the self-dual point. As explained below this is the only non-trivial minimum if the cosmological constant is fine-tuned to vanish. In the case of two condensates, there is no solution to u1 = 0, u2 va 0, for f ~> 0, and the cosmological constant can be fine-tuned to vanish, as will be illustrated below in a toy example. More generally, the potential is dominated by the condensate with the largest one-loop B-function coefficient, so the general case is qualitatively very similar to the single condensate case, and it appears that positivity of the potential can always be imposed. Otherwise, one would have to appeal to another source of supersymmetry breaking to cancel the cosmological constant, such as a fundamental 3form potential [28] whose field strength is dual to the constant that has been previously introduced in the superpotential [ 29 ], and/or an anomalous U ( 1 ) gauge symmetry [ 30]. In the following we study Z3-inspired toy models with E6 and/or SU(3) gauge groups in the hidden sector, and 3Nu matter superfields [ 31 ] in the fundamental representation f . Asymptotic freedom requires N27 ~< 3 and N3 ~< 5. For a true Z3 orbifold there are no moduli-dependent threshold corrections: b / = 0. In this case universal anomaly cancelation determines the average value of the matter modular weights in these toy models as: (2q~ 7 - 1) = 2/N27, (2q 3 - 1) = 18/N3. In some models Wilson line breaking of the hidden sector E8 generates vector-like representations that could acquire masses above the condensation scale, so that the universal anomaly cancelation sum rule is not saturated by light states alone. In this case the q~ no longer drop out of the equations, so some o f the above formulae would be slightly modified. In addition, one would have to include threshold effects [23,32], unless the masses of the heavy states are pushed to the string scale. Here, we assume for simplicity that the sum rule is saturated by the light states. Denoting the fundamental matter fields by (b)'~, ce = 1 . . . . . N f , the matter condensates can be constructed as 3

H~ =

l-I

)~ cI) ,

3 b~6 _ 4¢r2 ,

~ 1 bsu(3 ) = 8qr2 ,

l=l

where gauge indices have been suppressed.

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

44

In the analysis o f the models described below, we assume - for obvious phenomenological reasons - that the vacuum energy vanishes at the minimum (V) = 0. Thus we solve the equations

V= OV=o, c~x

x = g , t l,ma.

(4.3)

For x = g, t l, we have

Op~ 1 ( A x + 1 B ) Ox - 2 -~a x Pa, Be =

(1 + gg(1))

g2

(

'

--OX OV = A x - -~xg

=

b [1 + 4 ( ( t l ) R e t I] BI - 2~ Re t

1

(

V q- ] - ~ Eab PaPb COS 09ab ~aBXRab q- -~Xc9Rab

1 ~-~paPbCOSO)ab

3¢aR~b + -~xR~b

1692 ~b

+

(

Ax--[~xe+--n

)

(4.4)

-~a V,

where/3ab is defined in (3.25). By assumption, the last term in (4.4) vanishes in the vacuum. Note that the self-dual point, da(t l) = BI = O, t I = 1, is always a solution to the minimization equations for t 1. It is the only solution for the single condensate case. For the multicondensate case, if we restrict our analysis to the (relatively) weak coupling region, g < l / b _ , where b_ is the smallest/3-function constant, the potential is dominated by the condensate with the largest/3-function coefficient b+: V ~ p2R++/1692. Moreover, since ~rb/3ba > 1, the potential is always dominated by this term for Re t I > 1 (c.f. Eq. ( 2 . 3 5 ) ) , so the only minimum for Ret l > 1 is R e t I ---+c~, Pa --+ O. By duality the only minimum for Re t I < 1 is Re t I ---+0, Pa ---+0, so the self-dual point is the only non-trivial solution. Since our potential is always dominated by one condensate, the picture is very different from the "race-track" models studied previously [20]. At a self-dual point with V = 0, we have

~92V 1 ~ paPb c o s 09ab a(ti)2 ~, - ~ ab ×

(lq_b~.~(b-ba)(b-bb)--~nE3caRab C

3---2

-(] ~ - ~ )

6ne2 E tic+R++

.

(4.5)

C

Positivity o f the potential requires R++ >~ 0, and/3c+ ~< 0 by definition, so the extremum at a self-dual point with V = 0, p+ ~ 0 is a true minimum. In practice, the last term is negligible, and the normalized moduli squared mass is

P. Bindtruy et aL/Nuclear Physics B 493 (1997) 27-55

45

(b - b+)

(4.6)

4.1. Single condensate with matter In this c a s e a

Ot l

flab

= 0, and the minimization equations for

t 1 require

[1 + 4 ( ( t / ) R e t ' [ 2 = 0 ,

which is solved by 1 + 4 s r ( t t ) R e t t = 0, t I = 1. Then v3 = F t = 0, and the potential is qualitatively the same as in the E8 case [ 16] - except for the fact that the moduli are fixed. (Note however that if flab = 0 one can choose the b~, in (2.16) such that the matter composites drop out o f the effective Lagrangian; then Raa is independent of the moduli which remain undetermined.) The quantitative difference from the E8 case is the value of the fl-function coefficient: be6 = (12 -3N27)/87"r 2, bsu(3) = ( 6 - N 3 ) / 1 6 ~ -2. As in Ref. [ 16] we take the non-perturbative contribution to the dilaton Kahler potential to be o f the form [ 17] f = Ae -B/e or [ 18] f = Ae -8lye, and fine tune the constant A to get a vanishing cosmological constant. Attention has been drawn to the leading correction for small coupling that is of the form f = Ae -~/vW. If we restrict f to this form we have to require a rather large value for the coefficient: A ~-- 40 to cancel the cosmological constant. On the other hand, the important feature of f here is its behavior in the strong coupling limit; if f contains terms of the form Ae -BW'/2 the strong coupling limit will be dominated by the term with the largest value of n. In the numerical analysis we take f = Ae-B/v; adding to this a term o f the form f = Ate -B'/v~ will not significantly affect the analysis. We find that the vev of g is insensitive to the content of the hidden sector; it is completely determined by string non-perturbative effects, provided a potential for g is generated by the strongly coupled hidden Yang-Mills sector. More specifically, taking f = Ae -~/v we find that (V) = 0 requires A ~ e 2 ~ 7.4, and the dilaton is stabilized at a value (g) ~ B/2. Taking B = 1 gives (g) ~ 0.5, ( f ( g ) ) ~ 1, and the squared gauge coupling at the string scale is gs2 = (2g/(1 + f ) ) ~ 0.5. If instead we use f = ae -B/v~, the corresponding numbers are A ~ 2e 3 ~ 40, (g) ~ B2/9, gZs ~ 2B2/27. From now on we take f = Ae -1/v. The potential for Ga = E6, N27 = 1, is plotted in Figs. 1-3. Fig. 1 shows the potential in the g, In t plane, where we have set t I = t, Im t = 0; with this choice of variables the t-duality invariance of the potential is manifest. Fig. 2 shows the potential for g at the self-dual point t I = 1, and Fig. 3 shows the potential for In t with g fixed at its vev. The qualitative features of the potential are independent of the content of the hidden sector. Fixing A in each case by the condition (V) = 0, we find for Ca = E6

A=

7.359 , 7.381

(g) =

0.501 ~ ~ , 0.500

for N27 =

{1 2 . 3

(4.7)

46

P. Bin~truy et aL/Nuclear Physics B 493 (1997) 27-55

V

-±

Fig. 1. The scalar potential V (in reduced Planck units) is plotted versus g and Int.

x 10 m

V 0.6

•

0.4

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 2. The scalar potential V (in reduced Planck units) is plotted versus g with t l = 1 (the self-dual point).

-16

xlO

V

-1.5

-i

-0.5

0.5

1

llS

lnt Fig. 3. The scalar potential V (in reduced Planck units) is plotted versus lnt with g = (g).

For Ca = S U ( 3 ) , N3 = 1, w e find A = 7.383, = 0.500 ~ g2. A s discussed in Section 5, the scale o f s u p e r s y m m e t r y b r e a k i n g in this case is far too small, and decreases w i t h i n c r e a s i n g N3.

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

47

4.2. Two condensates We have

OV OV . . . . . Owl 0w2

plp2R12 sin w12,

t~caPaPbgabCOS09ab=/~21 (p2Rll - p2R22) •

(4.8)

abc

Minimization with respect to wl requires either (sin co12) = 0 or (R12) = 0. Identifying bt = b+, b2 = b - , positivity of the potential requires Rll >~ 0, which in turn implies R12 > 0, SO the extrema in w are at sin wa2 = 0, with cosw12 = - i ( 4 1 ) corresponding to minima (maxima)

0O922

-plp2R12coswl2,

mo,~2

)2PlP2R12

•

(4.9)

There is also a small Im tl-wl2 mixing. Note that while in contrast to the single condensate case, the dynamical axion is no longer massless, its mass is exponentially suppressed relative to the gravitino mass by a factor N v / ~ / p l . We do not expect this feature to persist when kinetic terms are introduced for the condensate fields. For G = E6 ® S U ( 3 ) , the potential is dominated by the E6 condensate, and the results are the s a m e as in (4.7). The only other gauge groups in the restricted set considered here that are subgroups of Es are G = [ S U ( 3 ) ] " , n ~< 4; these cannot generate sufficient supersymmetry breaking.

5. Supersymmetry breaking The pattern and scale of supersymmetry breaking are determined by the vevs of the F-components of the chiral superfields. From the equations of motion for 7r~ and Pa we obtain, at the self-dual point (F l) = 0:

(F~) - (gg(~, + 1) .,~ ~+g~"~bb~b 4g2ba

~ -~a'~r~u+(1 +gb+) -1,

b

b7 sO, 1 (Fa+P~)=4--C-~(gg,~,+l)(l+gba)

u~ a + g

3b2+ 1 + gba (Uafi+ + FtaU+) 4b~ 1 + gb+

bbab + h . c .

] (5.1)

where the approximations on the right-hand sides are exact for a single condensate. The dominant contribution is from the condensate with the largest fl-function coefficient,

3p2b+ (F+ + V+) -

2

(5.2)

P Bin(truy et aL/Nuclear Physics B 493 (1997) 27-55

48

The fact that the F ~ vanish in the vacuum is a desirable feature for phenomenology. Nonuniversal squark and slepton masses that could induce unacceptably large flavor-changing neutral currents are thereby avoided. However, this feature might be modified in the presence of moduli-dependent threshold effects ~ ln(/x 2) w h e r e ] z 2 = ( e x p ( ~ l qilgI)tMi[ 2) is a modular invariant squared mass and M i is a gauge singlet with modular weights ql. Another important parameter for soft supersymmetry breaking in the observable sector is the gravitino mass. The derivation of the gravitino part of the Lagrangian again parallels the construction in Ref. [16]. The gravitino mass m 0 is determined by the term £mass(q~) = - - g O O'mn~I

Lta +

+ b'a ln(e2-XFtaU~) + Ol

eK/2~vomormnO n + h.c.,

In [z/(

(5.3)

giving, when the equations of motion (2.23) are imposed,

m0 =

1

(IMI) = ~([

~-~b'aua - 4 e X / 2 W I ) =

1 1 ~([ ~-~b~ua[) ~ "~b+(p+).

a

(5.4)

a

The scale of supersymmetry breaking is governed by the vev (2.28) of the condensate with the largest/3-function Coefficient. This includes the usual suppression factor (pa) c~ e -Ub"gz~, where g~ = (2/?/(1 + f ) ) is the effective squared coupling constant at the string scale. However, there are other important parameters that determine the scale of the hierarchy between the supersymmetry breaking scale and the Planck scale. The dependence on the moduli provides a second exponential suppression factor,

(Pa) (X ( H Iw(fl)12(b-b°~/b°) = 1~7(1)16(b-b°~/b° ~ e -~r(b-b")/2ba.

(5.5)

l

On the other hand, the numerical factor I-[, lb'~/4c,~[ -bUt'° gives an exponential enhancement if c~ ~ 1. This is the largest numerical uncertainty in our analysis. A priori, c, is related to the Yukawa couplings for matter in the hidden sector. However, there is an arbitrary normalization factor in the definition of H% If the hidden sector Yukawa couplings were known, it might be possible to estimate c,~ by a matching condition for the vevs of the second lines of (2.29) and (2.30). In our numerical analysis we have set ca = 1. Then, if the hidden gauge group with the largest condensate is G+ = E6 with 3N27 matter chiral superfields in the fundamental representation, we obtain

m0 =

1.1 × 10 -9 3.3 × 10 -11 1.65 × 10 -15

for N27 =

{ 1 2 , 3

(5.6)

in reduced Planck units. For G+ = SU(3) with three matter chiral fields in the fundamental representation, we obtain an unacceptably large gauge hierarchy: m 0 = 2.2 × 10-32; rnd decreases rapidly as N3 increases, i.e. as the/3-function coefficient decreases.

P. Bingtruy et aL/Nuclear Physics B 493 (1997) 27-55

49

6. Concluding remarks In the class of models studied here, the introduction of a parameterization for nonperturbative contributions to the KShler potential for the dilaton generically allows a stable vacuum at a non-trivial, phenomenologically acceptable point in the dilaton/moduli space. In particular, when we impose the constraint that the cosmological constant vanishes, we find that in the linear multiplet formulation, the moduli t 1 are stabilized at the self-dual point, and their associated auxiliary fields vanish in the vacuum, which implies the phenomenologically desirable feature of universal soft supersymmetry breaking parameters. As shown in Appendix A, these features do not survive in the parallel construction starting from the chiral multiplet formalism because of the explicit s-dependence of the superpotential. They may also be modified in the linear multiplet formalism in the presence of moduli-dependent intermediate-scale threshold effects. However, the case with no such threshold corrections serves to illustrate the difference between the two approaches. We have argued that the linear multiplet approach more faithfully respects the physics of the underlying strongly coupled Yang-Mills theory. A salient feature of our formalism is that there is little qualitative difference between a single condensate and a multi-condensate scenario. For several condensates with equal (or very similar) /3-functions, the potential reduces to that of the single condensate case, except that there may be fiat directions. If bl = b2 . . . . . bk, then at the selfdual point P a / P l = (a = constant and the potential vanishes identically in the direction ~ka= 1 ~a ei~°° = O, Pa>k = 0. This always has a solution if (a = 1, in which case the fiat direction preserves supersymmetry and there is no barrier between this solution and the interesting, supersymmetry breaking solution. For different t-functions, the potential is dominated by the condensate(s) with the largest fl-functon coefficient, and the result is essentially the same as in the single condensate case, except that a small mass is generated for the dynamical axion. In all cases non-perturbative corrections to the dilaton KS.hler potential are required to stabilize the dilaton. This picture is very different from previously studied "race-track" models [20] where dilaton stabilization is achieved through cancelations among different condensates with similar t-functions. The qualitative difference between an Es hidden sector and one with a product gauge group is the presence of matter; in the E8 case the potential is independent of the moduli, which therefore remain undetermined in the classical vacuum of the effective condensate theory. As discussed previously [ 11,12,16], kinetic energy terms for the condensate fields Pa, °)a, as well as an axion mass comparable to the condensation scale, can be generated by including a dependence of the K~ihler potential k (and correspondingly the function f ) on the variables Ua, Oa. Terms of the form V -2n ~a(Ua~Ja) n and V - 2 n ( U U ) n are generated both by classical string corrections [33] and by field theory loop corrections [27]. Note that once the condensate fields are integrated out these induce, by virtue of their vevs (2.14), "non-perturbative" corrections to the Kahler potential for the dilaton, of the type discussed by Banks and Dine [ i7]. However, in the single condensate case [34] it was found that these terms are insufficient to stabilize the dilaton,

50

P Bingtruy et al./Nuclear Physics B 493 (1997) 27-55

and one must appeal instead to string non-perturbative effects. 5 We expect the same conclusion to hold in the multicondensate case. If this is so, the interpretation of contributions to the K~.hler potential of the form f = Ae -B/v as arising from field theoretic corrections to our static model may be questionable. We therefore adopt the point of view that the unknown function f parameterizes string non-perturbative corrections. In the static models studied here, cancelation of the cosmological constant by string non-perturbative corrections alone requires that they are significant at the vacuum: ( f ( g ) ) ~ (2g) ~ 1. This has implications 6 for phenomenological analyses [36] of gauge coupling unification. Including non-perturbative corrections to the K~hler potential for the linear multiplet L, i.e. taking k ( L ) = lnL + g(L) with f ( L ) related to g(L) as in (2.3) with V ~ L, the two-loop boundary condition [22] on the MS gauge couplings now reads (for affine level one) ga 2(/xs) = g s 2 ÷ ~Ca 2 {g(e) ÷ In [ f ( g ) ÷ 1]

--

ln2}

1

167r2 ~

b / I n [(t' + / )

]r/2(tl)12] ,

I gs2 - f + 1 2g '

2 1 1 tZs =geg-l = ~eg- ( / + 1 ) ~ .

(6.1)

Note that the tree coupling of the effective field theory is now 2g~-2 = ( ( f + 1)/g), and integration over the condensate fields with vevs given by (2.28) gives corrections to the Kahler potential for g of the form [ 17] N e-"/bo~ = e-~Y+l>/b,,e, when kinetic terms for Ua, Ua are included. On the other hand, we expect string non-perturbative effects [18] to be N e -n=/v7 since the linear supermultiplet containing the 3-form d[nb pq] is the fundamental field in string compactifications (as opposed e.g. to 5-brane compactification [37], in which the dilaton is in a chiral multiplet and the moduli are in linear multiplets). If one performs a duality transformation in the usual way [ 14] via a Lagrange multiplier S + S,

l~.=

/

I

1

d40g -2÷ f(L)÷'~(L÷O)(SWS

,

where ~2 is the Chern-Simons superfield, L is unconstrained and S is chiral, the equations of motion for L give precisely S + S = ( f + 1)/L, so that Re s is always the tree-level inverse squared coupling constant in the chiral formulation of the effective field theory. Including the Green-Schwarz term and loop corrections in the chiral formulation [ 23 ] again gives (6.1).

5 It can be shown that the static model of Ref. [ 16] is indeed the low energy limit of the dynamicalmodel of Ref. [34]. 6 Other gauge-dependentthreshold corrections [35] have recently been found.

P Bingtruy et al./Nuclear Physics B 493 (1997) 27-55

51

Acknowledgements RB. and M.K.G. would like to acknowledge support from the Miller Institute for Basic Research in Science, and also the Aspen Center for Physics where part of this work was completed. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-95-14797.

Appendix A. Chiral multiplet formulation There has been interest in the question as to Whether the linear and chiral multiplet formulations are equivalent at the quantum level. They are presumably equivalent in the sense that we may perform a duality transformation at the superfield level on the Lagrangian (2.1) so as to recast it entirely in terms of chiral supermultiplets; the resulting effective Lagrangian is apt to be rather complicated. The more practical question that we address in this appendix is the extent to which the above results can be reproduced if one takes as a starting point the usual chiral supermultiplet formalism for the dilaton with the gaugino condensates represented by unconstrained chiral supermultiplets, and naively generalizes the methods commonly used in this context. In the chiral multiplet formulation, the Green-Schwarz term appears as a correction to the Kahler potential, which we take to be

K ( S , T I) = ln(L) + ~ ( L ) _ ] _ ~ g l , 1

L -1 = S + S -

beg',

(A.1)

t

where ~ is the correction from non-perturbative string effects. Modular invariance of the Yang-Mills Lagrangian at the quantum level is assured by the transformation property of S under (2.10),

S---+ S + b ~'~ F l,

(A.2)

I

and modular covariance of the Kahler potential (K -+ K + ~-]~I(F! -k- pl) ) requires that it depends on S only through the real superfield L. We introduce static condensate superfields H ~, Ua as before, but now the superfield

Ua = eX/2 H3a

(A.3)

does not satisfy the constraint (3.4) because Ha is taken to be an unconstrained chiral superfield. 7 We construct the superpotential in analogy to (2.1), using the standard Veneziano-Yankielowicz approach, Wtot -~- Wcond -t-

W(H),

(A.4)

7This is probably where the departure from the approach of Section 2 is the most sensitive. The correct procedure - which is not the one usually followed- wouldbe to use a 3-formsupermultiplet description [28].

52

P. Bingtruy et al./Nuclear Physics B 493 (1997) 27-55

where W ( H ) is the same as in (2.27), and

Woo.d = Wc + Wvy + Wth,

Wc = ¼S ~[_ H~, a

Wvy= 1

H3a

3b,alnHa +

balnHO ~ , ot

1

bl

w~ = ~ ~ 8--~H3 lnD72(r') 1,

(A.5)

where W c represents the classical contribution. H~3 transforms in the same way as Ua under rigid chiral and conformal transformations, and the anomaly matching conditions give the same constraints on the b's as in Section 2. Then it is straightforward to check that under the modular transformation (2.10) with Ha --+ exp(-~--~I F t / 3 ) , we have Wcond ---+exp(--~--~4 F~/3)Wcond' as required for modular invariance of the Lagrangian. Summing the various contributions, the superpotential for Ha can be written in the form

Wcond = ~

baH ~ In

e s/bo

[ ~ ( T 1) ] -bJ47rZb°

( H " ) b"/b"

a

.

(A.6)

I

The bosonic Lagrangian takes the standard form £8=

_1~

2

_

M i l l + Ki~, ( F i E ~ - ~alxz i~l~ a z fft'~ )

+ e x/2 IF i (Wi + K i W ) -- a4W + h.c.] ,

(A.7)

where Z i = S, T1,Ha, H % z i = Zll0=9=0. In our static model K i o ~ , K i = 0 for z i , z m = H a , / I % and the equations of motion for F i give Wi = 0 for these fields. This determines the chiral superfields Ha, I1 '~ as holomorphic functions of S, T I. Making the same restrictions on W ( H ) and the b~ as in Section 2, we obtain n 3 = e(2n+l)irr(b'~--ba)/b.--b'Jbc, e--S/b.

H [rl(Zl ) ]2(b-ba)/b. I I I l

= - 4ca

,

b~/4c~l-bT/b"'

ce

b a 4= O.

(A.8)

I

As in (2.28), the correct dependence of the gaugino condensates on the gauge coupling constant ((Re s)-~/2), s = SI0=0=0, is recovered. Note however, that in contrast to (2.28) the gaugino condensate phases are quantized once Im s is fixed at its vev. Using these results gives 1 Wtot = W ( S , T 1) = - - ~ ~

baH 3.

(a.9)

a

The effective potential is determined in the standard way after eliminating the remaining auxiliary fields through their equations of motion,

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55 Pe~ = --eX/2K ie~ (Wi + KiW) ,

M = --3eK/2W,

V ( s , d , ? I) =e x [g i#' (Wi + KiW) (17V,~q- KrnlTV) -

53 Z i = S , T 1,

31w12].

(A.10)

The inverse Kahler metric for the KS_hler potential (A.1) is 4(Re tl) 2 81J, (1 - bKs)

Ktj

KI~ -

2b Re t 1 (1 - b K s ) '

KS~ = 1 - bKs + 3b2Ks~ Ks~( 1 - bKs)

(A.11)

and the potential reduces to

eK

V = 1 ---bKs

{

K ~ ' (1 - bKs + 3b2Ks~)IWs + KsWI 2 + 4 E

(Ret') 2

l

-3eXlWl 2.

(A.12)

We have

-2Re/(W, + K,W) = - ~

1 - bKs a

1 3 ws + x,w= Z a-~ ( 1 - K, bo) I-l~,

(A.13)

a

and the potential can be written in the form

V

eK

K-"

16(1 -- bgs) ~

(A.14)

]hahbla CoS WabRab,

ab

where now co~ is the phase of h 3 = H31o=o__o,~O~bis defined as before, and

I1 +

Rab = babbfab(g) + (b - ba) (b - bb) ~

4Retl£~(t 1) 12,

g = LI0=0=0,

1

fab(g) = ( 1 -

bKs) I ( l - b a K s ) ( 1 - baKs) - 3 ]

(A.15)

In the absence of non-perturbative effects Ks = - g , Ks~ = g2, fab --+ - 2 b g as g ---+ c~, and the potential is unstable in the strong coupling direction, as expected. A positive definite potential requires that f + + ( g ) be positive semi-definite where, as before, b+ is the largest ba. Note that the perturbative expression for faa (g) is negative for bag > 1.4, while in the linear multiplet formalism, the corresponding expression is negative only for bag > 2.4, so non-perturbative effects are required to be more important in the chiral multiplet formulation. If there is only one condensate, the self-dual point for the moduli

P. Bindtruy et al./Nuclear Physics B 493 (1997) 27-55

54

is again a minimum, but (F 1) v~ O. In the general case, the minimization equations for the moduli read

3t ~ - 16(1 --- bKs) ~ Ihahbl3 cos~°ab ab n

~

( ( t I) ~--~caRab-[- ~-~Ra c

V,

(A.16)

where /3ab is defined as in (3.25). Again imposing (V) = 0, the m i n i m u m is shifted slightly away from the self-dual point if some/gab 4= 0. The effective Lagrangian in the linear multiplet formalism - like the string and field theory loop-corrected Y a n g - M i l l s Lagrangian [22,23] - depends only on the variables t I and the modular invariant field g, so the Lagrangian is invariant under modular transformations on the t l alone. In contrast, the effective Lagrangian in the standard chiral multiplet approach has an explicit s-dependence which accounts for the fact that the self-dual point is not the minimum. The standard chiral construction forces a holomorphic coefficient for the interpolating superfield for the Yang Mills composite superfield U ~ Tr(~/V~W~), and hence cannot faithfully reflect non-holomorphic contributions from the G r e e n - S c h w a r z term and field theory toop corrections. The last point can be evaded by incorporating these renormalization effects in the K~flaler potential [38,15,32] rather than in the superpotential, in which case it is also possible to recover invariance under continuous infinitesimal S-duality rotations in the weak coupling limit. Again, this a property o f the Y a n g - M i l l s Lagrangian and the linear multiplet formulation o f condensation, but not of the chiral multiplet formulation. 8 However, in this last approach the relation o f the effective Lagrangian for condensation to the underlying Y a n g - M i l l s Lagrangian is much less transparent. We emphasize that we do not claim that there is no effective chiral Lagrangian dual to that o f Section 2, with the same physics. However, a straightforward approach based on the chiral multiplet formalism leads to different physics, in particular the non-vanishing o f the moduli F-terms in the vacuum, which has implications for flavor-changing neutral currents.

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