Metastable supersymmetry breaking and supergravity at finite temperature

Lilia Anguelova* and Steven Thomas†

Department of Physics, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom

Riccardo Ricci‡

Theoretical Physics Group, Blackett Laboratory, Imperial College, London, SW7 2AZ, UKand The Institute for Mathematical Sciences, Imperial College, London, SW7 2PG, United Kingdom

(Received 10 October 2007; published 31 January 2008)

We study how coupling to supergravity affects the phase structure of a system exhibiting dynamicalsupersymmetry breaking in a meta-stable vacuum. More precisely, we consider the Seiberg dual of SQCDcoupled to supergravity at finite temperature. We show that the gravitational interactions decrease thecritical temperature for the second order phase transition in the quark direction, that is also present in theglobal case. Furthermore, we find that, due to supergravity, a new second order phase transition occurs inthe meson direction, whenever there is a nonvanishing constant term in the superpotential. Notably, thisphase transition is a necessary condition for the fields to roll, as the system cools down, towards the meta-stable susy breaking vacuum, because of the supergravity-induced shift of the meta-stable minimum awayfrom zero meson vevs. Finally, we comment on the phase structure of the KKLT model with upliftingsector given by the Seiberg dual of SQCD.

DOI: 10.1103/PhysRevD.77.025036 PACS numbers: 11.10.Wx, 12.60.Jv

I. INTRODUCTION

Understanding how supersymmetry breaking occurs is amajor problem on the road to connecting the underlyingsupersymmetric theory with the observed world.1 The idea,that supersymmetry can be broken due to dynamical effects[2], has long been considered phenomenologically verypromising, since it naturally leads to a large hierarchybetween the Planck and the susy breaking scales. How-ever, dynamical supersymmetry breaking has turned out tobe quite difficult to implement in a supersymmetric gaugetheory. The reason is that only rather complicated ex-amples [3] satisfy the strict conditions, necessary for theabsence of a global supersymmetric vacuum.

It was realized recently [4] that the situation changesdramatically, if one abandons the prejudice that the phe-nomenologically relevant vacuum has to be a global, andnot just local, minimum of the effective potential. In thiscase, one can relax the requirement that the theory lacks aglobal supersymmetric vacuum and search for models withmeta-stable, sufficiently long-lived, susy breaking vacua.From this new point of view, models with nonzero Wittenindex and without a conserved U�1� R-symmetry can beconsidered phenomenologically viable for supersymmetrybreaking. The spectrum of susy breaking theories is thensignificantly enriched. In particular, as shown in [4], meta-stable dynamical supersymmetry breaking occurs even inN � 1 SQCD with SU�Nc� gauge group and Nf massivefundamental flavors. This can be established by going tothe Seiberg (magnetic) dual description of this theory,

where supersymmetry is broken at tree level.2 For conve-nience, we will call this the ISS model from now on.During the last year, many more examples of meta-stabledynamical susy breaking were found in various phenom-enologically appealing settings [5]. Progress was alsomade on understanding the embedding of those field-theoretic models in string/M theory [6].

However, once we consider phenomenology in a local,instead of a global, minimum of the zero-temperatureeffective potential, the following question arises. Hownatural is it for the high-temperature system, that is theearly Universe, to end up in the meta-stable state aftercooling down? To address this question, the recent works[7,8] studied the ISS model at finite temperature. Theyfound that the meta-stable vacuum is thermodynamicallypreferable compared to the supersymmetric global ones.3

Although their conclusions agree, their approaches aredifferent and, in a sense, complimentary. In [7], they con-sider a path in field space, which extrapolates between thesusy breaking vacuum and a global vacuum, and constructthe effective potential along this path. Using this they showthat, even if at high temperature the system starts at a susyvacuum, it will end up in the meta-stable one as it coolsdown. On the other hand, [8] studies in great detail thephase structure around the origin of field space, which is alocal minimum of the nonzero temperature potential. Theyassume that at high temperature the quark and meson fieldsof the ISS model are localized near this point, which isreasonable since the number of light degrees of freedom atthe origin is largest and hence this state maximizes theentropy. With this starting point, [8] investigates the phase

*l.anguelova@qmul.ac.uk†s.thomas@qmul.ac.uk‡r.ricci@imperial.ac.uk1For a recent review of supersymmetry breaking see [1].

2We will review more details on that in the next section.3There are Nc of them as we review in Sec. II.

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structure of the free-energy as the temperature decreasesand finds out that there is a critical temperature, TQc , for asecond order phase transition in the quark direction (that is,the direction towards the meta-stable minimum). On theother hand, in the meson direction (i.e., the directiontowards a global susy vacuum) they find that only a firstorder phase transition can occur, at temperature smallerthan TQc , and that it is quite suppressed by a high potentialbarrier.4

We follow the approach of [8] for the ISS model coupledto supergravity. This is certainly necessary for more real-istic cosmological applications of the idea of dynamicalsupersymmetry breaking in a meta-stable vacuum. We willcompute the one-loop effective potential at finite tempera-ture by using the results of [10,11] for chiral multipletscoupled to supergravity. The nonrenormalizability of thelatter theory is not an issue as it is supposed to be viewed asan effective low-energy description only, not as a funda-mental theory. For more details on one-loop (albeit at T �0) calculations in supergravity coupled with various mattermultiplets, see [12]. However, we should note that theconsiderations of [10,11] treat the zero-temperature clas-sical supergravity contribution V0 to the T � 0 effectivepotential as an effective potential itself. In other words, theT � 0 loop corrections are viewed as already taken intoaccount in the standard sugra potential V0. So it mightseem conceivable that these results may be affected (de-spite susy being broken) by regularization subtleties simi-lar to those at T � 0. In the latter case, a regularizationcompatible with supersymmetry was developed in the lastfive references of [12] and was shown rather recently in[13] to sometimes have an impact on quantities of phe-nomenological interest, like the flavor-changing neutralcurrents. For the trivial Kahler potential, that we willneed, this is not the case. Nevertheless, it is important,although going well beyond the scope of our paper, toaddress this issue in full generality at T � 0. Anotherremark is due. Every gravitational system exhibits insta-bility under long-wave length gravitational perturbations[14]; this Jeans instability occurs also at finite temperature[15]. While it is certainly of great importance for structureformation in the early Universe, it is a subleading effect oncosmological scales on which the Universe is well approxi-mated by a homogeneous fluid. So we will limit ourselvesto considering the leading effect, by using the formulas of[10,11] for the effective potential, and will not address herethe Jeans instability.

We will show that the supergravity corrections decreasethe critical temperature for a second order phase transitionin the quark direction, TQc , for any Nc and Nf. While this isonly a small quantitative difference with the rigid case, inthe meson branch a significant qualitative difference can

occur. The reason is that in the relevant field-space regionthere are no contributions to the tree-level meson masses inthe rigid limit and so the supergravity corrections are theleading ones. As a result, it turns out that, whenever thesuperpotential contains a nonvanishing constant piece W0,there is a second order phase transition in the mesondirection at a temperature T’c , smaller than TQc . However,this is not a phase transition towards any of the globalsupersymmetric minima, as it occurs at nonvanishingquark vevs. Notably though, our new phase transition isprecisely what is needed for the system to roll towards themeta-stable vacuum since, due to the supergravity inter-actions, the latter is shifted away from the origin of themeson direction whenever W0 � 0, as shown in [16].

Considering the ISS model plus supergravity is, in fact,the first step towards a full investigation of the phasestructure of the KKLT scenario [17] with ISS upliftingsector at finite temperature. It was already argued in[16,18] that meta-stable susy breaking provides a naturalway of lifting the AdS KKLT vacuum to a de Sitter one,avoiding the problems encountered previously in the lit-erature. Recall that the original proposal was to introduceanti-D3 branes, which break supersymmetry explicitly,whereas the later idea to use nonvanishing D-terms [19]turned out to be quite hard to realize [20]. Studying thephase structure of the KKLT-ISS system is a big part of ourmotivation. However, in this case the computations be-come much more technically challenging. We make theinitial step by showing that the origin of the ISS field-spaceis no longer a local minimum of the temperature-dependentpart of the effective potential. The shift of the high-temperature minimum away from the ISS origin is relatedto the vev of the KKLT volume modulus.

This paper is organized as follows. In Sec. II we reviewnecessary background material about the ISS model. InSec. III we compute the one-loop temperature-dependentcontribution to the effective potential or, equivalently, thefree energy of the ISS model coupled to supergravity. Toachieve that, in Subsection III A we derive the mass matri-ces for both quark and meson vevs nonzero, that arecoming from the F-terms; in Subsection III B we takeinto account the D-terms. In Sec. IV we expand the generalresults of Sec. III in terms of the small parameter M�1

P ,whereMP is the Planck mass. This allows us to read off theleading supergravity corrections to the rigid theory, con-sidered in [8]. In Sec. V we compute the critical tempera-ture for a second order phase transition in the quarkdirection to O�M�2

P �. In Sec. VI we show that there isalso a second order phase transition in the meson directionand estimate the critical temperature for it. In Sec. VII weconsider the KKLT model with ISS uplifting sector andargue that the origin of the ISS field space is no longer alocal minimum of the high temperature effective potential.The shift of the minimum away from this origin is deter-mined by the vev of the KKLT volume modulus. In

4Reference [9] discusses in detail the suppression of thistransition in a class of O’Raireataigh models.

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Sec. VIII we discuss the implications of our results for thephase structure of the ISS model coupled to supergravityand for the end point of this system’s evolution at lowtemperature. We also outline open problems. Finally, inAppendix A we give some useful formulas for mass ma-trices near the origin of field space and in Appendix B weshow that no new supersymmetric minima appear in asmall neighborhood of the origin in the ISS model coupledto supergravity, in the field directions of interest.

II. ISS MODEL

It was argued in [21] that SQCD with SU�Nc� gaugegroup and SU�Nf�L � SU�Nf�R flavor symmetry has adual (magnetic) description in terms of an SU�Nf � Nc�gauge theory coupled to certain matter fields. When thecondition Nf < 3Nc=2 is satisfied the magnetic theory isIR free. The matter content of the dual theory comprisestwo chiral superfields q and ~q, that are transforming in the( �Nf; 1) and (1;Nf) representations of the flavor symmetrygroup, respectively, and are in the fundamental and anti-fundamental representations of SU�Nf � Nc� respectively(and so are called quarks), and a gauge-singlet chiralmeson superfield � in the (Nf; �Nf) flavor group represen-tation. Hence the index structure of the magnetic quark

and meson fields is the following: qai , ~q�ja and �i

�j with i �

1; . . . ; Nf and a � 1; . . . ; N, where N � Nf � Nc. For eas-ier comparison with the literature, we will also use thenotation Nm � N and Nc � Ne, implying, in particular,that Nf � Nm � Ne. In terms of this notation the abovecondition for IR free dual theory is Ne > 2Nm. In thefollowing we will only consider this case.

The magnetic theory has the following tree-level super-potential:

W � hTrq�~q� h�2Tr�: (1)

The second term breaks the flavor group to its diagonalsubgroup and corresponds to a quark mass term in themicroscopic theory (i.e., the original SU�Nc� gauge the-ory). The Kahler potential is the canonical one

K � Trqyq� Tr~qy~q� Tr�y�: (2)

The magnetic description can be used to prove theexistence of a meta-stable vacuum, which breaks super-symmetry at tree-level [4]. Indeed, it is immediate to seethat the F-term condition

F�ji� h�qai ~qja ��2�ji � � 0 (3)

cannot be satisfied as the matrix qai ~qja has at most rank Nmwhile �ji has rank Nf. The moduli space of meta-stablevacua can be parameterized as

q �Q0

� �; ~qT �

~Q0

!; � �

0 00 ’

� �; (4)

where’ is an �Nf � N� � �Nf � N�matrix whileQ and ~Q

are N � N matrices satisfying the condition Q ~Q ��1N�N . The point of maximum global symmetry is at

hq1i � h~qT1 i � �1N; hq2i � h~q2i � 0; h�i � 0;

(5)

where we have denoted: qT � �q1; q2� with q1 and q2

being N � N and N � �Nf � N� matrices, respectively. Itwill also be useful for the future to introduce the followingnotation for the generic components of the Nf � Nf matrix�:

� ��11 �12

�21 �22

� �; (6)

where �11 is Nm � Nm, �12 is Nm � Ne, �21 is Ne � Nm,and finally �22 is an Ne � Ne matrix.

The value of the scalar potential in each meta-stableminimum in (4) is

Vmin � �Nf � N�h2�4: (7)

Usually, when supersymmetry is spontaneously broken,the moduli space of classical vacua is not protected againstquantum corrections. As a result, the quantum modulispace is typically smaller and one may wonder whetherany of the meta-stable vacua survive in it. In this regard, itwas shown in [4] that the classically flat directions aroundthe maximally symmetric vacuum (5) acquire positivemasses at one-loop through the supersymmetricColeman-Weinberg potential [22]. This meta-stable mini-mum is therefore tachionic-free and from now on we willalways mean (5), when we talk about a supersymmetrybreaking vacuum.

In addition to the perturbative corrections that we justdiscussed, there are also nonperturbative ones. Namely,gaugino condensation in the magnetic gauge groupSU�N� induces the Affleck-Dine-Seiberg (ADS) superpo-tential [23]

WADS � N�hNf

det�

�Nf�3N

�1=N; (8)

where � is the UV cutoff of the magnetic theory, i.e., thescale above which the magnetic description becomesstrongly coupled and hence not well-defined. Adding thisdynamically generated contribution to the classical ISSsuperpotential leads to Nc supersymmetric vacua, charac-terized by nonvanishing meson vevs5:

hh�i�ji � ���3N�Nf�=Nc�i�j; hqi � h~qTi � 0; (9)

in agreement with the Witten index [24] of the microscopictheory. The meta-stable vacuum can be made long-lived bytaking � parametrically small as in that case the tunnellingto the supersymmetric vacuum is strongly suppressed.6

5We denote by � the quantity ���1.6Recall that Nf > 3N and hence for very small � the meson

vev, h�i, in (9) becomes very large.

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Since the ADS superpotential is suppressed by powers ofthe UV cutoff, for small meson fields it is completelynegligible compared to the tree-level one, (1). So in thefollowing we will drop WADS from our considerations, aswe will study the finite temperature effective potential onlyin a neighborhood of the origin of field space.

III. ONE-LOOP EFFECTIVE POTENTIAL ATNONZERO T

In the present section we compute the one-loop effectivepotential at finite temperature for the ISS model coupled tosupergravity. Its analysis in subsequent sections will enableus to deduce the phase structure of this theory near theorigin.

Let us start by recalling some generalities about thepath-integral derivation of the effective potential in a the-ory with a set of fields f�Ig. An essential step in that is toshift �I by a constant background �I. Equivalently, weexpand the Lagrangian around a nonzero background, f�Ig,for the fields. Using this expansion, one can derive withfunctional methods a formula for the effective potential.The original derivation of [25] was only for zero-temperature renormalizable field theory. The same kindof considerations apply also for finite temperature and upto one loop give [26]

Veff��� � Vtree��� � V�1�0 ��� � V

�1�T ���; (10)

where Vtree is the classical potential, V�1�0 is the zero-temperature one-loop contribution, encoded in theColeman-Weinberg formula, and finally the temperature-dependent correction is

V�1�T ��� � ��2T4

90

�gB �

7

8gF

�

�T2

24�TrM2

s ��� � 3 TrM2v��� � TrM2

f���

�O�T�: (11)

For convenience, from now on we will denote this lastexpression simply by VT . Here gB and gF are the totalnumbers of bosonic and fermionic degrees of freedomrespectively; 7 TrM2

s , TrM2v, and TrM2

f are the coefficientsof the quadratic terms of scalar, vector, and fermion8 fieldscomputed from the shifted classical potential or, in otherwords, the mass matrices of those fields in the classicalbackground f�Ig. The expansion (11) is valid in the high-temperature regime, more precisely when all masses aremuch smaller than the energy scale set by the temperature.

The above result for the one-loop effective potential atfinite temperature was shown in [10] to also hold forcoupling to supergravity.

We turn now to computing TrM2s , TrM2

v, and TrM2f for

our case. The classical background � around which we willbe expanding is

hq1i � h~q1i � Q1Nm�Nm; h�11i � ’11Nm�Nm;

h�22i � ’21Ne�Ne ;(12)

with zero vevs for all remaining fields and with Q, ’1, ’2

all being real.

A. F-terms

In this subsection we consider the contribution from theF-terms. The D-terms will be taken into account in the nextone. For convenience, from now on we denote collectivelyall components of the fields q, ~q, and � simply by �I.

1. Preliminaries

Recall that the classical F-term supergravity potentialis9:

V � eKfKI �JDIWD �J�W � 3jWj2g; (13)

where I, J run over all scalar fields in the theory, KI �J is theinverse of the Kahler metric and the Kahler covariantderivative is

DIW � @IW � @IKW: (14)

The supergravity Lagrangian is invariant under Kahlertransformations

K��I; ��I� ! K��I; ��I� � F��I� � �F� ��I�;

W��I� ! e�F��I�W��I�:

(15)

One can exploit this invariance,10, by taking F��I� �logW��I�, in order to show [28] that the scalar potentialdepends only on the combination

G � K � lnjWj2; (16)

but not onW andK separately. In terms of this function, wecan rewrite (13) as follows:

7Note that this is different from the number of fields. Forexample, for NB chiral superfields the number of scalar degreesof freedom is gB � 2NB. In fact, below we will denote by NB thenumber of complex scalars.

8In (11) the quantity TrM2f is computed summing over Weyl

fermions.

9In this section we setMP � 1. The explicit dependence on thePlanck mass will be reinserted later when needed.

10In principle, at the one-loop level the invariance under Kahlertransformations may be broken. The resulting Kahler anomalyhas been studied in great detail in [27] and references therein.For us, however, this will not be an issue, since the Kahlerpotential we consider is gauge invariant and there are no Fayet-Iliopoulos terms, as will be evident in Sec. III B.

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V � eG�GIGI � 3� (17)

where

GI �@G@�I

; GJ �@G@ ��J

: (18)

This notation utilizes the fact that for us the Kahler poten-tial is canonical, i.e., KI �J � �I �J, see (2), and so ��J �KJ �L �� �L.

For such a Kahler potential, the expressions for thescalar and fermionic mass matrices are [10]11:

TrM2s � h2eGf�GIJ �GIGJ��GIJ �GIGJ�

� �NB � 1�GIGI � 2NBgi (19)

and

TrM2f � TrM2

1=2 � TrM23=2

� heGf�GIJ �GIGJ��GIJ �GIGJ� � 2gi; (20)

respectively. Here NB is the number of complex scalars.Using (17), it is easy to show that (19) follows from

TrM2s � 2

@2V@�J@ ��J

: (21)

The derivation of the fermion mass squared is a bit moreinvolved since one has to disentangle the mixing betweenthe gravitino and the Goldstino in the supergravityLagrangian. After having dealt with that, one findsTrM2

3=2 � �2eG. The anomalous negative sign in the grav-itino contribution is due to the fact that in TrM2

1=2 we havesummed over the physical matter fermions and theGoldstino.

Before starting the actual computations, two remarks arein order. First, it may seem that it is more illuminating toperform the calculations as in [8], i.e., to compute sepa-rately the mass squareds of every field and then add them.However, for us this becomes rather cumbersome, whereasthe trace formulas above provide a very efficient way ofhandling things. And second, the formulation of the super-gravity Lagrangian in terms of the function G, (16), ap-pears to encounter a problem for vanishing superpotential,as W enters various terms in the denominator. That will bean issue for the D-terms in Subsection III B and we will usethere a more modern formulation that is equally valid forW � 0 and W � 0. Here we simply note that for the F-terms there is no problem, since the apparent negativepowers of W, coming from derivatives of G, are cancelledby the positive powers from eG. (This will be made moreexplicit in Subsection III A 2.) So the F-term results nevercontain division by zero. This is an important point as inlater sections we will be interested in the effective potential

at the origin of field space, where the ISS superpotentialvanishes.

2. Mass matrices

We are finally ready to find the F-term mass matricesTrM2

s and TrM2f for the ISS model coupled to supergravity.

To do so more efficiently, we note that, instead of findingseparate expressions for the various ingredients GIJGIJ,GIJGIGJ, and GIGJGIGJ which enter the mass formulas,it is computationally much more convenient to calculatethe whole combination �GIJ �GIGJ��GIJ �GIGJ� foreach particular choice of I and J. This avoids introducinga big number of terms that have to cancel at the end, as wenow explain. From the definition of G, i.e., G �K � lnW � ln �W, we have that12

GI � KI �WI

W; GIJ � �

WIWJ

W2 �WIJ

W: (22)

Therefore, in the expressions GIJGIJ, GIJGIGJ, andGIGJGIGJ one seems to obtain many terms proportionalto 1=W �W2. Taking into account the jWj2 factor comingfrom eG, one is left with many terms1=jWj2. However, itis clear that they have to cancel at the end, since the scalarpotential (13) does not include any negative powers of jWj2

and so if we were computing the masses of each fieldseparately and adding them (as in [8]) we could not pos-sibly obtain terms 1=jWj2. This cancellation can beincorporated from the start by using (22) to write

GIJ �GIGJ �WIJ

W�KIWJ � KJWI

W� KIKJ (23)

or equivalently

GIJ �GIGJ �WIJ

W�GIKJ � KIGJ � KIKJ: (24)

It is evident now that this expression does not contain any1=W2 terms. Incidentally, this also makes it obvious that,as expected, the expression eG�GIJ �GIGJ��GIJ �GIGJ�does not contain any powers of W in the denominator.

To illustrate how much the use of (23) or (24) simplifiesthe computation, let us look for instance at the followingterm:

R�11�22 � �G�11�22 �G�11G�22��G�11�22�G�11

G�22�:

(25)

To make use of (24), we note that

11We will consider a noncanonical Kahler potential in Sec. VII,when we address the KKLT setup with ISS uplifting sector.

12Recall that for us the Kahler potential is the canonical one andso, in terms of the notation introduced in Eq. (18), we have thatKJI � �JI and KIJ � 0.

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W�11�22� 0;

hK�11i � ’11Nm�Nm;

hK�22i � ’21Ne�Ne ;

hG��11�i�ji �

�’1 �

hW0�Q2 ��2�

��

�ji ;

hG��22�k�li �

�’2 �

h�2

W0

���lk:

(26)

Therefore, one immediately finds

hR�11�22i �

��’1 �

h�Q2 ��2�

W0

�’2 �

�’2 �

h�2

W0

�’1

� ’1’2

�2��

�ji�

�lk���

i�j�k�l�; (27)

where we denoted by W0 the value of the ISS potential in

the background (12). The last factor gives ���ji�

�lk���

i�j�k�l� �

�ii�kk � NmNe. So we obtain

hR�11�22i �

�’1’2 �

h�Q2 ��2�

W0’2 �

h�2

W0’1

�2NmNe:

(28)

Similarly, it is very easy to compute

hR�11�11i �’21

�’1� 2

h�Q2��2�

W0

�2N2m;

hR�22�22i �’22

�’2� 2

h�2

W0

�2N2e ;

hRq1�11i � hR~q1�11i �

�hQW0

�2N2m

�Q2

��h’1

W0� 1

�’1�

h�Q2��2�

W0

�2N2m

� 2hQ2

W0

��h’1

W0� 1

�’1�

h�Q2��2�

W0

�Nm;

hRq1�22i � hR~q1�22i �Q2

�’2�

hW0�’1’2��

2�

�2NmNe;

hRq2�21i � hR~q2�12i �

�hQW0

�2NmNe;

hRq1q1i � hR~q1 ~q1i �Q4

�1�

2hW0

’1

�2N2m;

hRq1 ~q1i �

�h’1

W0

�2N2m�Q4

�1�

2h’1

W0

�2N2m

�2h’1Q2

W0

�1�

2h’1

W0

�Nm;

hRq2 ~q2i �

�h’2

W0

�2NmNe: (29)

For all remaining pairs hR�I�J i � 0.

The last ingredient in (19), that we need to compute, ishGIGIi. The only nonvanishing components are for I ��11; �22; q1; ~q1 and we find

hGIGIi �

�’1 �

h�Q2 ��2�

W0

�2Nm �

�’2 �

h�2

W0

�2Ne

� 2Q2

�h’1

W0� 1

�2Nm: (30)

One can notice that all expressions above depend onlyon Q2, not on Q alone. Hence the temperature-dependentpart of the one-loop effective potential as a function ofQ isof the form13

VT � eQ2�AQ6 � BQ4 � CQ2 �D� (31)

for any values of the meson vevs ’1 and ’2. ThereforeQ � 0 is always an extremum. In the meson directionsthings are not so apparent, as there are odd powers of ’1

and ’2. However, one can see that all of them multiplyeither a power of Q2 or the first power of W0. Since W0 islinear in the meson vevs, for Q � 0 the dependence of VTon ’1 and ’2 is, in fact, at least quadratic (by that we alsomean mixed terms, i.e., with ’1’2). We will see inSec. III A that the point �Q;’1; ’2� � �0; 0; 0� is a localminimum of VT , as was also the case for vanishing super-gravity interactions [8].14 Before that, however, let us firstconsider the D-term contributions to the mass matrices ofthe various fields.

B. D-term masses

The D-terms for super Yang-Mills coupled to supergrav-ity were derived for the first time in [28]. However thatformulation, entirely in terms of the functionG of Eq. (16),is not convenient for our purposes since the D-terms have asingular dependence on the superpotential. For example,denoting by g the gauge coupling constant and by T� thegenerators of the gauge group, the D-term scalar potentialwas found to be

VD �1

2D�D�; (32)

with

D� � gGITI�J�

J �1

2gTI�J�

J DIWW

; (33)

where in the last equality we have substituted GI � KI �WI=W. Clearly, VD is not well-defined when the super-

13We drop from now on the piece that is T4, as it does notdepend on the vevs of the fields and so it does not contribute tothe derivatives of VT , which are the quantities that will be ofinterest for us.

14This is no longer true if one includes the KKLT sector, as weargue in Sec. VII.

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potential W vanishes. The same problem, i.e., division byW, appears also in the fermionic mass terms. So if we adoptthe formulation given in [28], TrM2

f seems to diverge forW � 0. That is problematic since we would like to studythe effective potential at the origin of field space, where theISS superpotential vanishes.

It has been noted long ago that the above formulation isnot suitable in the presence of a vanishing superpotentialand the latter case has to be studied separately, without theuse of the function G � K � lnjWj2. However, a carefulstudy of the supergravity Lagrangian that is valid both forW � 0 and W � 0 was performed only recently (to thebest of our knowledge) in [29]. What is relevant for us isthat VD can be written as

VD �g2

2�i�I�@IK � 3ir��

2; (34)

where �I� are the gauge transformations of the scalar fields,i.e., ���I � �I��f�

Jg�, and r� are functions determined bythe gauge variations of the superpotential

��W � �I�@IW � �3r�W; (35)

where the second equality is required for gauge invarianceof the action. These functions also characterize the gaugenoninvariance of the Kahler potential

��K��; ��� � 3�r���� � �r�� ����: (36)

In the case of nonvanishing superpotential one can use (35)to express r� in terms of W and @IW. Substituting theresult in (34), one finds (32) and (33) upon using �I� �iTI�J�

J. This, indeed, shows that the formulation of [29]reduces to the one in [28] when W � 0.

Since the ISS superpotential is gauge invariant, we haver� � 0 and

VD �g2

2

XN2m�1

��1

�Tr�qy1T�q1 � ~q1T�~qy1 ��2 (37)

as in [8]. We can then borrow the results, found in theglobal supersymmetry case, to obtain a 4g2Q2�N2

m � 1�contribution to TrM2

s . Also, the vector boson mass is thesame as in [8] and so it gives a 4g2Q2�N2

m � 1� contribu-tion to TrM2

v.Let us now come to the fermionic sector. The mass

matrices are [29]

MIJ �DIDJM (38)

MI� � �i�@IP � �

1

4�Ref��1P@If�

�(39)

M� � �1

4@ �If�K

�IJMJ; (40)

where f� are the gauge kinetic functions, the action of thecovariant derivative DI on M � eK=2W is

D I � @I �1

2@IK; (41)

MI �DIM and finally

P � � i�I�@IK: (42)

These expressions are clearly well-defined and nonsingulareven when W � 0, unlike the analogous formulas in [28].In our case, obviously the index I runs now only over thequark fields. For a flat Kahler potential, one can easilyverify from (38) that

MIJMIJ � eG�GIJ �GIGJ��GIJ �GIGJ�; (43)

so that we recover correctly the contribution from thematter fermions and the Goldstino in (20). Since for usf� � 1=g2 � const for all � and , M� � 0 while thecontribution to TrM2

f from the mixing of Gaugino and

hyperini can be found from (39) using P � � qy1T�q1 �

~q1T�~qy1 . It reads

h2MI�MI�i � 8g2Q2�N2

m � 1�: (44)

In the supergravity Lagrangian there is one more mixingbetween fermions, which could potentially add a term toTrM2

f, namely, the mixing between the gravitino andGaugino. In [29] it is of the form P ��

�. However, inour case hP�i � 0 and so this mixing does not contribute.To recapitulate, the D-terms give exactly the same contri-bution as in the rigid case considered in [8].

IV. EXPANSION IN M�1P AND RIGID SUSY LIMIT

In Sec. III, we computed all ingredients of the finitetemperature one-loop effective potential for the ISS modelcoupled to supergravity. Now we will make connectionwith the globally supersymmetric theory by inserting inthe relevant formulas the explicit dependence on thePlanck mass, MP, and expanding in powers of M�1

P . Forlater use, we will extract the leading supergravity correc-tions to the rigid results.

Let us start with the tree-level supergravity potential

V � M4Pe

G�M2PG

IGI � 3� �1

2DaDa (45)

with

G �K

M2P

� logjWj2

M6P

: (46)

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As discussed in the previous section, the D-term contribu-tion is the same as in the rigid limit. It is easy to see that theexpansion of (45) gives

V � WIWI �1

2DaDa �

1

M2P

�KWIWI

� 2Re�KIWIW� � 3jWj2� �O�M�4

P �; (47)

where obviously the first two terms are the standard globalsusy result. For future use, we note that taking I ��11; �22 in the last equation gives, to leading order in thesupergravity corrections, the following contribution to theclassical F-term potential for the quarks:

VF � �h2Trq1q

y1 ~qy1 ~q1 � h

2�2�Trq1 ~q1 � Trqy1 ~qy1 �

� h2�4�Nm � Ne���

1�Trq1q

y1 � Tr~q1 ~qy1M2P

�: (48)

Inserting the MP dependence in the thermal one-looppotential VT yields15:

VT �T2

24M2P

�eG�3M4

P

XIJ

RIJ � 2�NB � 1�M2PG

IGI

� 2�2NB � 1���� hVDi; (49)

where we have denoted by RIJ the quantity �GIJ �GIGJ��GIJ �GIGJ�, as before. To obtain the explicitpowers of MP in RIJ, we note that due to (46) Eq. (24)becomes

GIJ �GIGJ �WIJ

W�GIKJM2P

�GJKIM2P

�KIKJM4P

: (50)

Now, expanding (49) we find

VT �T2

24

��3WIJWIJ � 3

K

M2P

WIJWIJ

� 6jWj2

M2P

Re�WIJ

W2 �WIKJ � KIWJ�

�

� 2�NB � 1�

M2P

WIWI

��� hVDi �O�M�4

P �: (51)

Together, Eqs. (47) and (51) give the general expression forthe zeroth and first orders in theM�1

P expansion of the one-loop effective potential at finite temperature.

Let us now apply the above formulas for the background(12), but with vanishing meson vevs. To see what (51)leads to, note that the only contributions to the first twoterms come from hjWq1�11

j2i � hjW~q1�11j2i � h2Q2N2

m andhjWq2�21

j2i � hjW~q2�12j2i � h2Q2NmNe and that the only

nonzero hWIi are hW�11i � h�Q2 ��2� and hW�22i ��h�2. Hence we obtain

VT �T2

2h2Q2�N2

m � NmNe��1� 2

Q2NmM2P

�

� T2Q2g2�N2m � 1� �

T2

M2P

�h2Q2�Q2 ��2�Nm

�1

12�NB � 1��h2�Q2 ��2�2Nm � h

2�4Ne�

�O

�1

M4P

�: (52)

Taking MP ! 1, we find the global supersymmetry resultfor VT .16

Local minimum at the origin

At high temperature, the temperature-dependent contri-bution VT completely dominates the effective potential andso the minima of Veff are given by the minima of VT . Let usnow address the question whether the origin of field spaceis a local minimum of VT .

In principle, this could be a complicated problem, as wehave to consider a function of three variables,VT�Q;’1; ’2�. However, things are enormously simplifiedby the fact that, as we noted at the end of Sub-section III A 2, Q � 0 is an extremum for any ’1 and ’2.Since it is also a local minimum in the rigid limit, it willclearly remain such when taking into account the sublead-ing supergravity corrections in our case. So we are left withinvestigating a function of two variables, ’1 and ’2. Thepresence of terms linear in any of them could, potentially,shift the position of the minimum away from the point�’1; ’2� � �0; 0�.

17 However, such terms in the ISS modelcoupled to supergravity appear only multiplied by powersof Q2, as we noted below Eq. (31). So, given that�’1; ’2� � �0; 0� is a local minimum in the global super-symmetry case, it will remain such after coupling to su-pergravity, to all orders in the 1=MP expansion.

It is, nevertheless, instructive to write down explicitlythe expression for VT to leading order in the 1=MP correc-tions

16Our result is in agreement with that of [8], upon correcting atypo there. Namely, they have overcounted by a factor of 2 thenumber of Weyl fermions (in their notation) ��

11 and ��q1 �

�~q1�=

���2p

.17This is what happens for the KKLT-ISS model, as we will see

in Sec. VII.

15Recall that, as we mentioned in footnote 13, we drop forbrevity the constant T4 contribution to effective potential in(11), as it does not affect our considerations.

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VT�Q;’1; ’2� �T2

2h2Q2�N2

m � NeNm� � T2Q2g2�N2

m � 1� �T2

4h2�’2

1N2m � ’

22NmNe�

�T2

M2P

�h2Q2�2’2

1 �Q2 ��2�Nm �

�NB � 1�

12fh2��Q2 ��2�2 � 2’2

1�Nm � h2�4Neg

� �2Q2Nm � ’21Nm � ’

22Ne�

�1

2h2Q2�N2

m � NmNe� �1

4h2�’2

1N2m � ’2

2NmNe����O

�1

M4P

�: (53)

Clearly, this is consistent with (52). We have collected theterms that survive in the MP ! 1 limit on the first line.Obviously, the origin of field space �Q;’1; ’2� � �0; 0; 0�is a minimum. One can also notice that to this order VT is afunction of the form VT�Q2; ’2

1; ’22�, i.e., it does not depend

on odd powers of any of the vevs. One can verify that theterms linear in ’ and multiplying powers of Q2, that wementioned above, start appearing at O�1=M6

P�, while termslinear in ’1’2 appear at O�1=M4

P�.It will be useful for the next sections to extract from (53)

a couple of special cases. One case is the second derivativein the quark direction only

@2VT@Q2

��������Q�0;’1�0;’2�0� T2�h2�N2

m � NmNe�

� 2g2�N2m � 1�

�T2

M2P

1

3h2�2Nm�NB � 5�: (54)

The other interesting case is the potential in the mesondirections only

VT�0; ’1; ’2� �T2

4

�h2�’2

1N2m � ’2

2NmNe�

�h2

M2P

�’21Nm � ’

22Ne�

2Nm

�1

M2P

1

3h2�4�NB � 1��Nm � Ne�

�: (55)

Finally, let us note that if we want to study other minimaof the potential, which are away from the origin of fieldspace �Q;’1; ’2� � �0; 0; 0�, we would have to include inour considerations the nonperturbative superpotentialWADS, see Eq. (8).

V. PHASE TRANSITION IN QUARK DIRECTION

As we saw in the previous section, at high temperaturethe origin of field space is a local minimum of the effectivepotential. Lowering the temperature, one reaches a point atwhich this minimum becomes unstable and the fields startevolving towards new vacua. As recalled in Sec. II, at zerotemperature the ISS model possesses supersymmetric va-cua at nonzero meson vev and a meta-stable vacuum in thequark branch. Adding the supergravity interactions resultsin a slight shift of the positions of those minima. Inparticular, the meta-stable vacuum shifts to small vevs

for some of the meson directions [16]. Hence, in order toend up in it, the system must undergo a second order phasetransition in those same meson directions, as well as in thequark ones. In the next section we will see that this isindeed the case. In the present section, we will find thecritical temperature, TQc , for the onset of a second orderphase transition towards nonvanishing quark vevs, whilekeeping all meson vevs at zero.

Before turning to the supergravity corrections to TQc , letus recall how to compute the critical temperature in ageneric field theory [26]. Suppose that we have a theorywith a set of fields f�Ig. To find the effective potential, onehas to shift �I by constant background fields �I, as wereviewed in Sec. III. The effective potential is a function of�I only and we consider the case when it is of the formVeff��

2�.18 The location of the minima is then determinedby

@Veff��2�

@�I� 2�I

@Veff��2�

@�2 � 0: (56)

Clearly, �I � 0 satisfies this condition. Other minima, at�I � 0, can only occur when @Veff=@�

2 � 0. The criticaltemperature, at which rolling towards such minima begins,can be found by requiring [26]

@Veff

@�2

����������0�@V0

@�2

����������0�@VT@�2

����������0� 0; (57)

where we have split the one-loop effective potential intozero-temperature and temperature-dependant contribu-tions, V0 and VT respectively. This is equivalent to19

@VT@�2

����������0� �

m2

2; (58)

18This will be the case for us. However, even in principle thisassumption is not a restriction but merely a simplification, as weexplain in a subsequent footnote.

19The assumption that Veff depends only on �2, but not on �alone, guarantees that � � 0 is an extremum of the effectivepotential and thus simplifies the computations technically. In thegeneral case, instead of (58) one has to solve, symbolically, thefollowing system: V0eff��c; Tc� � 0; V00eff��c; Tc� � 0 in order tofind the critical temperature Tc, at which the phase transitionoccurs, and the vev �c, at which the relevant extremum of Veff issituated. Clearly, here we have denoted by 0 and 00 first andsecond derivatives with respect to �, respectively.

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where m2 is the unshifted tree level20 mass squared of thefield, whose nonzero vev characterizes the new vacuum.This last equation can only be solved if m2 < 0. In theglobal supersymmetry case, among the quark fields, whichhave nonzero vevs at the meta-stable minimum, the onlyones with negative tree-level mass squared are the compo-nents of Re�q1 � ~qt1�=

���2p� Req� [8]. Since for small vevs

the gravitational corrections are subleading, we are guar-anteed that this will be the case for us as well.

A. Mass matrix diagonalization

We will now compute the tree-level squared masses ofthe fields Req�, that are necessary for finding the criticaltemperature in the quark direction. Before considering thesupergravity corrections, it will be useful to give the deri-vation of these masses in the global supersymmetry limit.For more generality, we keep Q � 0 in this latter case,although we will take Q � 0 when we turn to the super-gravity contributions to m2 in (58).

From the tree-level scalar potential of the global theory,V � KI �J@IW@ �J

�W, one finds [8]

h2Q2�jq1j2 � j~q1j

2� � h2Q2 Tr�q1 ~qy1 � qy1 ~q1�

� h2�Q2 ��2��Tr~q1q1 � H:c:�: (59)

When we turn on gravitational interactions we will have asimilar expression but, generically, with different coeffi-cients. So it is of benefit to consider the general expression

A�jq1j2 � j~q1j

2� � B�Trq1 ~qy1 � Trqy1 ~q1�

� C�Tr~q1q1 � H:c:� (60)

with arbitrary A, B, and C. It can be diagonalized easily byintroducing the combinations

q� �q1 � ~qt1���

2p ; q� �

q1 � ~qt1���2p : (61)

Substituting the inverse transformation,

q1 �q� � q����

2p and ~qt1 �

q� � q����2p ; (62)

in (60), we find

A�jq�j2 � jq�j

2� � B�jq�j2 � jq�j

2�

� C1

2Tr�q2

� � q2� � q

y2� � q

y2� �: (63)

Finally, we decompose q� � Re�q�� � iIm�q�� and ob-tain the following mass terms:

�A� B� C��Re�q���2 � �A� B� C��Im�q���2

� �A� B� C��Re�q���2 � �A� B� C��Im�q���2:

(64)

Reading off the values of A, B, and C from (59), weobtain from (64)

m2req� � h2�3Q2 ��2�; m2

Imq�� h2�Q2 ��2�;

m2req� � h2��2 �Q2�; m2

Imq�� h2�Q2 ��2�:

(65)

We see that, as already mentioned above, only the compo-nents Req� of the field q� have negative mass squareds forzero shift Q.21 Therefore, only their masses should appearon the right-hand side of Eq. (58). We note that the massesfor the fields Imq� and Imq� differ from those given inTable 5 of [8]. Fortunately their typos cancel out in the totalTrM2

s .Let us now apply the result (64) for the diagonalization

of the expression (60) to the case of interest for us. Namely,we consider the ISS model coupled to supergravity and wewant to compute the following derivatives:

@2q1q

y1

V; @2q1 ~qy1

V; @2~q1q1

Vetc:; (66)

which give the coefficients A, B, and C. As before, V is thetree-level supergravity scalar potential. The only nonvan-ishing derivatives, upon setting Q � 0, are22

m2q1q

y1

� m2~q1 ~qy1�

1

M2P

h2�4�Nm � Ne�;

m2q1 ~q1� m2

qy1 ~qy1� �h2�2:

(67)

Hence, applying (64), we find that the tree-level mass ofeach of the N2

m real fields in Req� is

m2Req�

� �h2�2 �1

M2P

h2�4�Nm � Ne�: (68)

20In principle, m2 gets also contributions from the Coleman-Weinberg potential. However, in our case the origin of fieldspace is not a local minimum of the zero temperature potentialand so perturbation theory around it does not make sense. Hence,for us, m2 is purely classical. In fact, to be more precise, weshould note that at the origin the Coleman-Weinberg potentialbecomes complex. As shown in [30], the imaginary part of theeffective potential encodes the decay rate of a system with aperturbative instability. This decay process leads to the breakingup of the initial homogeneous field configuration into variousdomains, in each of which the fields are rolling toward a differentclassical solution, and should not be confused with a nonpertur-batively induced tunneling between different minima of thepotential. We will not dwell on that further in the present paper.

21Clearly, the negative mass squared of Imq� is irrelevant as inthe meta-stable supersymmetry breaking vacuum hq�i � 0.

22For more details see Appendix A. Note that (67) are exactexpressions, i.e., to all orders in 1=MP. However, since theyhappen to be at most of O� 1

M2P�, they can also be read off from the

part of the scalar potential (48), which contains only the relevantleading order supergravity corrections.

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The first term corresponds to the global supersymmetryresult, as can be seen from (65), while the second is acorrection due to the supergravity interactions.

Note that we did not need to include the D-terms in thecomputation of the unshifted masses. The reason is that, aswe have seen in Sec. III B, the contribution of these termsto the mass squared of the fields is always proportional toQ and so vanishes for zero shift.

B. Critical temperature

To find the critical temperature in the quark direction,recall that we shift the fields q1i

a and ~q1b�j by constant

matrices Q�ai and Q��jb respectively, see Eq. (12). In the

basis of q� and q�, see Eq. (61), the only fields which getshifted are theNm diagonal components of q�. From those,only the Nm fields Req� have negative tree-level masssquared as we saw above. Therefore we can find the criticaltemperature from23

@2VT@Q2

��������Q�0� �Nmm

2Req�

: (69)

Recall that, to leading order in the supergravity correc-tions, we have [see (54)]

@2VT@Q2

��������Q�0� T2�h2�N2

m � NmNe� � 2g2�N2m � 1�

�1

M2P

T2h2�2Nm�NB � 5�

3: (70)

Together with (68), this then leads to

�TQc �2 �h2�2Nm

h2�N2m � NmNe� � 2g2�N2

m � 1�

�1

M2P

h2�4�Nm � Ne�Nmh2�N2

m � NmNe� � 2g2�N2m � 1�

�1

M2P

h4�4N2m�NB � 5�

3�h2�N2m � NmNe� � 2g2�N2

m � 1�2

�O

�1

M4P

�: (71)

To see whether the order 1=M2P correction increases or

decreases the critical temperature, recall that Nf > 3N,i.e., Ne > 2Nm. Let us write

Ne � 2Nm � p; p 2 Z� (72)

and substitute this in the total numerator of the O� 1M2P� terms

in (71)

Numerator � �h4�4

�4N2

m �10

3Nmp�

2p2 � 5

3

�N2m

� 2h2�4g2�3N2m � pNm��N2

m � 1�: (73)

Clearly, for any Nm and for any p > 1 every term in theabove expression is negative definite. For p � 1 there is apositive contribution from the last term in the first bracket:h4�4N2

m. However, it is outweighed by the first two termsin that bracket for any value of Nm. So the conclusion isthat the supergravity interactions cause TQc to decreasecompared to the rigid case for any Nm and Ne.

VI. PHASE TRANSITION IN MESON DIRECTION

In the global supersymmetry case, a second order phasetransition is only possible in a field-space direction withnonzero squark vevs [8].24 However, in our case thingsmay be quite different due to the supergravity-inducedcontributions to the tree-level meson masses. Note that,in the approximation of neglecting Wdyn, the meson fieldshave zero classical mass in the global limit [8] and so thesupergravity corrections are the leading ones.25 Thus thepossibility occurs that in the region of small vevs, forwhich neglecting the dynamically generated superpotentialis well justified, some of the meson directions may developnegative tree-level mass squareds due to supergravity. Wewill see in the following when that happens.

Let us address this issue in a slightly more general setupas in [16], namely, by adding to the ISS superpotential aconstant piece. I.e., we consider W � W0 �WISS, whereWISS is as in (1) and W0 � const. This is also a usefulpreparation for the KKLT-ISS model that we will discussmore in the next subsection. It will turn out that we need tocompute the tree-level supergravity-induced meson massesnot only at the origin of field space but also along the quarkdirection, i.e., for Q � 0 and ’1; ’2 � 0.26 Also, we willhave to keep track of terms of up to O�1=M4

P� to see thenovel effect.

Turning to the computation, from (A6) we find

23Clearly, this is equivalent to (58) as @2VT=@Q2 �

2@VT=@�Q2�.

24Although a first order phase transition can still occur in themeson direction [8].

25Recall that we do not have to take into account contributionsfrom zero-temperature one-loop effects in the rigid theory as theorigin of field space is not a local minimum of the zero-temperature potential.

26For reasons that will become clear below, we will be inter-ested only in the range Q 2 �0; �. So we are still allowed toneglect Wdyn.

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m2��11�

i�j� ��11�

�lk

� 2h2Q2��ji�

k�l�

1

M2P

�ki �

�j�lh2��Q2 ��2�2Nm ��4Ne � �

�ji�

k�lh2��Q2 ��2��3Q2 ��2� � 4Q4Nm

�1

M4P

�ki �

�j�l��2W2

0 � 2h2Q2Nm��Q2 ��2�2Nm ��4Ne� � ��ji�

k�l4h2Q4Nm�2�Q2 ��2� �Q2Nm

;

m2��22�

m�n �

��22��pq�

1

M2P

f�qm� �n�ph

2��Q2 ��2�2Nm ��4Ne � h

2�4� �nm�

q�pg

�1

M4P

�qm� �n�pf�2W2

0 � 2h2Q2Nm��Q2 ��2�2Nm ��4Neg;

m2��11�

i�j� ��22�

�pq� �h2�2

Q2 ��2

M2P

�4Q4NmM4P

�

�ji�

q�p;

m2��22�

n�m�

��11��lk

� �h2�2

Q2 ��2

M2P

�4Q4NmM4P

� �nm�

k�l;

(74)

where clearly i; j; k; l � 1; . . . ; Nm and m; n; p; q �1; . . . ; Ne. Note that W0 appears only at order 1=M4

P. Allterms of the form m2

�11�11, m2

�22�22, and m2

�11�22vanish (see

Appendix A). To understand which mass matrix one has todiagonalize, let us recall that the meson fields are shifted asfollows: ��11�

i�j by ’1�

i�j and ��22�

m�n by ’2�

m�n , see Eq. (12).

Hence there are Nm fields shifted by ’1 (let us denote themby �1) and Ne fields shifted by ’2 (let us denote them by�2). In the quark direction we could factor out an overallNm (since both q1 and ~q1 have the same number of com-ponents) and diagonalize the remaining mass matrix.27

Now things are somewhat more complicated as there aredifferent numbers of �1 and �2 fields. More precisely,from (74) we see that we have to diagonalize the expres-sion

M2� �

�2h2Q2N2

m �1

M2P

�NmC1 � N2mC2�

�1

M4P

�NmC3 � N2mC4�

��1

��1

�

�1

M2P

h2�Q2 ��2�2NmNe �1

M4P

NeC3

��2

��2

� h2�2NmNe

�Q2 ��2

M2P

�4Q4NmM4P

�� ��1

��2 � ��2�1�; (75)

where

C1 � h2��Q2 ��2�2Nm ��4Ne;

C2 � h2��Q2 ��2��3Q2 ��2� � 4Q4Nm;

C3 � �2W20 � 2Q2NmC1;

C4 � 4h2Q4Nm�2�Q2 ��2� �Q2Nm�:

(76)

Notice that W0 enters the above formulas only at order1=M4

P.Before proceeding further, let us make an important

remark about the position of the local minimum that isour starting point at high temperature. From (29) and (30),one can see that for W0 � 0 terms linear in ’1 and ’2 startappearing at order 1=M4

P. This implies that the position ofthe minimum is shifted to some point �Q;’1; ’2� ��0; ’ 1; ’

2� with ’ 1, ’ 2 � 0. However, one can easily

verify that both ’ 1 and ’ 2 are of O�1=M4P�.

28 Therefore,in all terms of (A1) with an explicit 1=MP dependence onecan take ’ 1, ’ 2 � 0 since we are working to order 1=M4

P.Hence the only place the nonzero ’ 1, ’ 2 can make adifference in is the zeroth order terms. This translates tothe zeroth order term in (A6). However, that term is inde-pendent of the meson vevs as the superpotential is linear inthose and the indices I, J are along meson directions. Thusthe results (74)–(76) are unchanged by the nonzero ’ 1, ’ 2.For convenience, in the following we will keep referring tothe local minimum at �0; ’ 1; ’

2� as ‘‘the minimum at the

origin of field space.’’Let us now go back to (75). Since the coefficients of

�1��1 and �2

��2 are not the same, we cannot diagonalizethis expression immediately by using the results ofSubsection VA. However, it is still useful to change basisto the real components of the fields: �1 � Re�1 � iIm�1

and �2 � Re�2 � iIm�2. Then M2� acquires the form

M2� �

X2

i�1

�M11x2i � 2M12xiyi �M22y2

i �; (77)

where �x1; y1� � �Re�1; Re�2� and �x2; y2� ��Im�2; Im�2�. Thus the problem is reduced to the diago-nalization of the 2� 2 matrix Mab with a; b � 1; 2 and sothe eigenvalues are given by

27This is why we ended up with m2 � Nmm2Req�

in (69), wherem2

Req�was the result of the diagonalization.

28More precisely, one finds ’ 1 �4�2W0

hNm1M4P�O� 1

M6P� and ’ 2 �

4�2W0

hNm1M4P�O� 1

M6P�. We have not looked at the subleading orders,

so we abstain from claiming that ’ 1 � ’ 2.

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m2� �

TrM��������������������������������������TrM�2 � 4 detM

p2

: (78)

Before proceeding further with the Q � 0 considerations,let us first take a look at what happens at the point�0; ’ 1; ’

2�, which for us, as explained above, is the same

as looking at the origin of field space.For Q � 0, the expressions (74) and (75) simplify sig-

nificantly. Namely, we have

m2��11�

i�j� ��11�

�lk

jOr �h2�4

M2P

��ki ��j�l�Nm � Ne� � �

�ji�

k�l

�2W2

0

M4P

�ki ��j�l;

m2��22�

m�n �

��22��pqjOr �

h2�4

M2P

��qm� �n�p�Nm � Ne� � �

�nm�

q�p

�2W2

0

M4P

�qm� �n�p;

m2��11�

i�j� ��22�

�pqjOr � �

h2�4

M2P

��ji�

q�p;

m2��22�

n�m�

��11��lk

jOr � �h2�4

M2P

� �nm�

k�l;

(79)

where jOr denotes evaluation at the origin, and therefore

M2�jOr �

�h2�4

M2P

NmNe �2W2

0

M4P

Nm

��1

��1

�

�h2�4

M2P

NmNe �2W2

0

M4P

Ne

��2

��2

�h2�4

M2P

NmNe��1��2 � ��1�2�: (80)

Applying (78), we find that the two eigenvalues are

m2� �

2h2�4

M2P

NmNe �W2

0

M4P

�Nm � Ne�;

m2� � �

W20

M4P

�Nm � Ne�:

(81)

So, clearly, for any Nm and Ne there is a negative mesonmass-squared direction. However, note that its value is oflower order in 1=MP than the leading term (which is ofzeroth order) in the negative squark mass squared that isdriving the quark phase transition; see (68). Hence evenwithout calculating the critical temperature T’c , that wouldcorrespond to m2

�, we can be sure that it would be muchsmaller than TQc of Subsection V B. Therefore, by the timethe temperature starts approaching T’c , the system wouldhave already undergone the second order phase transitionin the quark direction and would be rolling along the Qaxis. So let us get back to considering (78) with Q � 0.

One easily finds that, up to order 1=M4P, the two eigen-

values of the matrix M are

m2� � 2h2Q2N2

m �O

�1

M2P

�;

m2� �

h2�Q2 ��2�2NmNeM2P

�1

M4P

Ne2Q2 ��4Q2W2

0 � 4h2Q4�Q2 ��2�2N2m

� 4h2�4Q4NmNe � h2�4�Q2 ��2�2Ne: (82)

The expression for m2� seems divergent for Q � 0.

However, we just saw in the previous paragraph that atthe point Q � 0 things are completely regular.29 The rea-son for the apparent problem in (82) is the following. Toobtain the last formulas, we had to expand the square rootin (78) for small 1=MP. This is perfectly fine when Q is oforder�. However, whenQ! 0 one has to be more carefulas the ‘‘leading’’ zeroth order contribution, 2h2Q2, in theabove mass formulas becomes � �4=M2

P (keep in mindthat �2=MP may be small, but it is definitely finite). So toextract the correct behavior of (78) in the limit Q! 0, onehas to first expand in small Q and then expand this result insmall 1=MP to the desired order. Doing that, one recovers(81) at the zeroth order in the Q expansion.

The lesson we learn from the above considerations isthat (82) is valid in a neighborhood of the point Q � �, inwhich Q and � have comparable orders of magnitude. Inthat neighborhood both m2

� are positive definite except atQ � �, where one has 30

m2� � 2h2�2N2

m �O

�1

M2P

�;

m2� � �

2W20NeM4P

�2h2�6N2

e�Nm � 1�

M4P

:

(83)

So, in principle, the sign of m2�jQ�� depends on the

relative magnitudes of W0 and h�3, except for the caseNm � 1. However, if one wants the positive vacuum energydensity in the meta-stable state to be very small, then thefollowing relation should be satisfied [16,18]

W20

M2P

� h2�4Ne; (84)

where � means that the two sides are of the same order ofmagnitude. If one assumes this relation, then the first termin m2

� is dominant as by order of magnitude it ish2�4=M2

P and so m2�jQ�� < 0. Hence, when (84) holds,

the new effect due to W0 � 0 corrects the order 1=M2P

results instead of those at O�1=M4P�.

29This is, in fact, the main reason we performed that compu-tation explicitly; it was already clear from (74) that if there is acritical temperature T’c at Q � 0, then it must be that T’c � TQc .

30Strictly speaking, (83) is also valid in the small neighborhoodin which Q��! 0, i.e., when Q��� �2=MP. The argu-ment is analogous to the one below Eq. (82).

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Let us see what conclusions one can make for the sign ofm2� throughout the interval �0; � for the cases when the

W0 contribution is of order 1=M4P and of order 1=M2

P,respectively. In either case, the sign of the eigenvalue m2

�

is determined by the sign of det M, see (78).31 However, inthe first case, the leading order contribution is Q2�Q2 ��2�2 and so is positive definite except at Q � 0 and atQ � �.32 At these two points the subleading term (of order1=M4

P) determines the sign and it is negative due to thenegative W0 contribution. On the other hand, in thecase when (84) holds we should only look at the terms ofup to O�1=M2

P�. The leading contribution to det M, whichis first order in 1=M2

P, is now det M�1� Q2�Q2 ��2�2h2NmNe �Q2W2

0Ne=M2P, which upon using (84)

gives

det M�1� � Q2h2��Q2 ��2�2NmNe ��4N2e< 0 (85)

at any point in �0; � as Nm < Ne. To recapitulate: when(84) is imposed, we find that m2

� < 0 for any Q 2 �0; �,while without (84) m2

� is negative only in small neighbor-hoods ofQ � 0 andQ � �. In a similar way one can showthat m2

� is positive definite without (84), whereas with (84)it is negative only in a small neighborhood of the origin.

The above conclusions imply that at some temperatureT’c , below TQc of Sec. V, there will be another second orderphase transition, this time in the meson direction corre-sponding to m2

�. We will explain in Sec. VIII that thisphase transition is actually necessary in order for thesystem to roll towards the meta-stable vacuum, due to theshifting of the latter from �’1; ’2� � �0; 0� to �’1; ’2� ��2=MP;�

2=MP�. However, computing the critical tem-perature is now much more complicated than in the quarkcase. One reason is that the phase transition occurs at someQ � 0. [With (84) imposed, this could a priori be any pointin the interval �0; �, whereas without this constraint it hasto occur at Q � �.] Another, much more serious issue isthat for temperatures below TQc there are masses that aregreater than the temperature and so the high-temperatureapproximation (11) cannot be used. Instead, one shouldconsider the full integral expression

Veff��� � V0��� �T4

2�2

XI

� nI

��Z 1

0dxx2 ln�1� exp��

��������������������������������x2 �m2

I ���=T2

q��;

(86)

where nI are the numbers of degrees of freedom and theupper (lower) sign is for bosons (fermions).

Unfortunately, that means that we cannot obtain a simpleanalytic answer for the meson critical temperature T’c .However, we can make an estimate of its magnitude inthe vein of [7]. Namely, let us consider a path in field spaceconnecting the point along the Q axis, at which the rollingin the meson direction starts, with the point �Q;’1; ’2� �

�0; 0; 0�, where the meta-stable minimum is, and take intoaccount only fields whose masses change significantlyalong this path.33 Now, let us assume that all relevantmasses are either much smaller or much greater than T.We will see below that our results are consistent with thisassumption. Then for fields with m� T we can still use(11), whereas for fields with m� T we can utilize theapproximation [see (3.5) of [7]]

�Z 1

0dxx2 ln�1� exp��

�������������������������x2 �m2=T2

q��

T4

�m

2�T

�3=2

exp��mT

�: (87)

In principle, the full temperature-dependent part of theeffective potential is obtained by summing over all fields.Note however, that the exponential in (87) strongly sup-presses the contributions of the fields with massm� T. Inother words, as long as there is at least one field with m�T, one can neglect the heavy fields to leading order.

As we saw above, without imposing (84) the phasetransition occurs at Q � �. Let us assume that this istrue also with (84). Clearly, for the latter case this assump-tion will only give us a lower bound on T’c , but this is asgood as one can get in that case without studying the fulldynamical evolution of the system. At Q � �, the heavymasses in our system are m2

Req1�, m2

Imq1�, m2

Req2�, m2

Imq2�,

and m2�. All of them are h2�2 �O�1=M2

P� and to lead-ing order remains constant along the above field-spacepath. The light masses that determine T’c in this approxi-mation are m2

Req1�, m2

Imq1�, m2

Req2�, m2

Imq2�, and m2

�.34

Hence we can obtain an estimate for the critical tempera-ture by solving

@2’�V

lT � �m

2�;

@’�VlT � �@’�V0 at �Q;’1; ’2� � ��;’

1; ’

2�:

(88)

In this system of equations VlT denotes the temperature-dependent part of the effective potential, that results from(11) by summing only over the light fields, and ’� is thelinear combination of ’1 and ’2 that corresponds to themass-squared eigenvalue m2

�. One then finds that

31If det M< 0, then the expression under the square root isgreater than �TrM�2 and so m2

� < 0.32More precisely, except in very small neighborhoods of those

points; see the remark in footnote 30.

33Recall that the reason for this is that fields with (nearly)constant masses contribute only to the field-independent T4 termin the effective potential, which we are not interested in.

34All of them vary significantly (meaning that the magnitude oftheir change is comparable to the magnitude of their leadingorder) along the path of interest and so contribute essentially tothe effective potential.

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T’c W0

hM2P

��2

MP; (89)

where in the second equality we have used (84). Note thatthe heavy masses (let us denote them bymh with h runningover all heavy fields) are all of zeroth order in 1=MP,whereas the light masses (denote them byml with l runningover the light fields) are all multiplied by the small constanth compared to (89). So our assumption, that at tempera-tures near T’c we have mh � T � ml, is consistent withthe estimate in (89).

Before concluding this section, let us comment on thefirst order phase transition found in [8]. Its critical tem-perature is of order �T’c �2 h�2. When it is reached,tunnelling becomes possible between h�i � 0 and a mini-mum away from the origin in the meson direction h�i 1Nf , which at zero temperature becomes the supersymmet-

ric vacuum at h�i � h�1���3N�Nf�=�Nf�N�1Nf . On theother hand, our second order phase transition occurs inthe field direction ’� � ’2 �

NeNm

�2

M2P’1 �O� 1

M4P� and at

hq1i � h~q1i � 0.35 Which one of T’c and T’c is greaterdepends on the relative magnitudes of h and ��=MP�

2.However, regardless of that, the second order phase tran-sition is much more likely to take precedence since the firstorder one, as any tunnelling event, is exponentially sup-pressed. To gain a better understanding of the phase struc-ture in the meson direction and, in particular, to be able toestimate the supergravity corrections to the height of thepotential barrier relevant for the first order phase transition,we would need to take into account the nonperturbativedynamically generated contribution to the superpotential.We leave that for future research.

VII. TOWARDS KKLT-ISS AT FINITE T

The proposal of [17] is a significant progress towardsfinding dS vacua in string theory with all moduli stabilized.However, the uplifting of their AdS minimum to a de Sitterone has been rather difficult to implement in a controlledway. It was shown recently in [16,18], that this can beachieved easily by using the ISS model as the upliftingsector. They considered the following coupling:

W � WISS �WKKLT; K � KISS � KKKLT; (90)

where WISS and KISS are given by (1) and (2), whereas

WKKLT � ~W0 � ae�b� and KKKLT � �3 ln��� ���:

(91)

In the string context, the constant ~W0 is due to nonzerobackground fluxes. By tuning it suitably, one can achieve

an almost vanishing cosmological constant and a lightgravitino mass [18], which is an important improvementcompared to the models with D-term uplifting. Therefore,it would be quite interesting to investigate the phase struc-ture of the KKLT-ISS model (90) at finite temperature. Inthis section, we limit ourselves to a discussion of the fate ofthe �Q;’1; ’2� � �0; 0; 0� ISS minimum when the KKLTfield � is included.

In Subsection III A), we noted that the presence of termslinear in the meson vevs ’1 and ’2 can shift the minimumof the effective potential away from the origin of the ISSfield space. This, of course, refers to terms that are notmultiplying Q2 since Q � 0 is always a local minimum aslong as �� MP, see Eq. (54). However, in that sectionsuch linear terms were not appearing. Now the situation isvery different, as the inclusion of the KKLT sector intro-duces many terms linear in ’1 and ’2. One source of themcomes from the constant piece ~W0 in the KKLT super-potential, as one can easily convince oneself by looking atthe form of hGIGIi in (30).36 In addition, there are manymore contributions, linear in the meson vevs, from mixedterms between the KKLT and ISS sectors in the total TrM2.For example, the total fermionic TrM2

f is now

TrM2f � he

G�KA �BKC �D�rAGC �GAGC�

� ��r �BG �D �G �BG �D� � 2i; (92)

where A � f�; Ig and the index I runs over the ISS fields.Writing this out, we have

TrM2f � he

G�K� ��K� ���r�G� �G�G���r ��G �� �G ��G ���

� 2K� ��KI �J�r�GI �G�GI��r ��G �J �G ��G �J�

� �GIJ �GIGJ��GIJ �GIGJ� � 2i; (93)

where in the second line we have used thatr�GI � rIG�.37 The last line is the familiar ISS plussupergravity contribution, which for W � WISS �WKKLT

gives the linear term mentioned in the beginning of thisparagraph. Let us now take a more careful look at theremaining terms. One can verify that the second line con-tains, among many others, the terms

�3

��� ���3�WKI �WI � �WKIWI

�: (94)

It is easy to see that these also lead to contributions whichare linear in ’1 and ’2. Further linear terms come from the

36Recall that the full expression is heGGIGIi. Hence the mixedterms in (30) give contributions proportional to ’1

~W0 and to’2

~W0.37The relation r�GI � rIG� is due to @IG� � @�GI ��W�WI=W

2 together with the fact that the only nonvanishingChristoffel symbols are ���� and � ��

�� ��.

35This expression for ’� is valid with or without (84) as W0appears for first time at order 1=M6

P.

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first line of (93) and also from the mixed terms in the scalarpotential

V � eK�KA �BDAWD �B�W � 3jWj2�; (95)

which determines the bosonic TrM2b via [11]

TrM2b � 2

�KC �D @2V

@�C@ �� �D

�: (96)

Therefore, in the coupled KKLT-ISS model, generically,the origin of the ISS field space �Q;’1; ’2� � �0; 0; 0� isnot a local minimum anymore. The coefficients of theterms linear in the meson vevs are functions of the KKLTfield �. Hence h�i is related to the magnitude of the shift ofthe minimum in the ISS plane. Although very interesting,we leave further analysis of the KKLT-ISS system forfuture work.

VIII. DISCUSSION

We studied the effective potential at finite temperaturefor the ISS model coupled to supergravity. Assuming thatat high temperature the fields are at the origin of fieldspace, which is a local minimum, we investigated the phasestructure of the system as it cools down. In the quarkdirection, the situation is analogous to the rigid case [8].Namely, there is a second order phase transition at certaincritical temperature, TQc . The effect of the supergravitycorrections is to decrease TQc compared to its global super-symmetry counterpart. In the meson branch however, anew feature appears whenever the superpotential containsa nonvanishing constant piece W0. Recall that in the globaltheory all meson fields always had a local minimum at theorigin of the meson direction, with no tree-level contribu-tions to their masses squared. Now we find that, whenW0 � 0, for some of them this ceases to be true at sometemperature T’c , below which negative tree-level super-gravity corrections to their effective masses squared areoutweighing the positive one-loop temperature dependentcontributions. Hence, the supergravity interactions lead tothe occurrence of a new second order phase transitionwhenever W0 � 0.

Since T’c < TQc , as we saw in Sec. VI, the fields first startrolling away from the origin in the quark direction. Whenthe temperature decreases enough, the same happens alsoin the meson direction. However, unlike in the rigid case,the second phase transition does not imply that the systemis moving away from the supersymmetry breaking vac-uum. The reason is that the coupling to supergravity leadsto slight shifting of the position of the meta-stable mini-mum [16].38 Whereas in the rigid theory it was given

by hq1i � h~q1i � �1Nm and h�i � 0, in the locallysupersymmetric case some of the meson vevs also acquirenonzero value: �hdiag�11i; hdiag�22i� � �’1; ’2� ��2=MP;�

2=MP� � hq1i. Hence the latter phase transi-tion is, in fact, a necessary condition for the system toevolve towards the meta-stable vacuum. Of course, tofollow with more precision the evolution of this systemas the temperature decreases, one has to study the fulleffective potential for both hqi � 0 and h�i � 0 awayfrom a small neighborhood of the origin.

In the above paragraph, we reached the conclusion thatthe final state of the system at T 0 is likely to be themeta-stable vacuum. However, one should be cautioussince, similarly to [8], our considerations assume thermalequilibrium. Hence, although suggestive, they are not com-pletely conclusive. As was pointed out in [8], for properunderstanding of the evolution of the system one shouldaddress also the dynamics of the fields at finite tempera-ture. Actually, even before worrying about dynamics, onemay be concerned that in our case the situation is compli-cated by the existence of new supergravity-induced super-symmetric minima.39 Indeed, it was shown in [18] thatsuch a minimum occurs in the KKLT-ISS model.40

However, this new vacuum only appears due to the inter-action with the KKLT sector; it is easy to see that the lastcondition in the solution for this minimum, Eq. (19) of[18], is only satisfied with an appropriate choice of (someof) the tunable KKLT parameters W0, a, and b. Still, onemay wonder whether there could be a solution to thesupersymmetry preserving equations for the ISS plus su-pergravity sector alone. In Appendix B we show that this isnot possible (at least in the field-space directions of inter-est, i.e., for an ansatz for the vevs that is of the same type asthe one in [18]).

The new solution of [18] is only one indication that theKKLT-ISS system is quite intricate to study. Another is thefact that, as we saw in Sec. VII, the interaction with theKKLT sector leads to shifting of the high-temperatureminimum of the effective potential away from the originof the ISS field space. Understanding the phase structure ofthis system is of great interest. However, the technicalcomplications involved are rather significant. Therefore,it may be beneficial to gain preliminary intuition about itby considering the recently proposed O’KKLT model [31],as the latter is much simpler while having quite similar

38This is not a trivial consequence of including supergravity, asin the ISS model coupled to sugra with W0 � 0 there is noshifting of the meta-stable vacuum compared to the global case.

39That is, susy minima other than those induced by the non-perturbative superpotential WADS, see Eq. (8). Recall that theWitten index gives only the number of global susy minima ofglobally supersymmetric theories. Hence, in the present contextit is not applicable and so one cannot immediately rule out thepresence of additional solutions.

40It is true that this minimum is much further out in field spacethan the meta-stable one, but its very existence raises thepossibility that it may be quite premature to make conclusionsabout the final state at low temperature, based solely on studyingthe immediate neighborhood of the origin.

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behavior. In addition, the O’KKLT model has a signifi-cance of its own, as it was argued in [31] to be of value instudies of cosmological inflation.41 We hope to address thisin the future.

ACKNOWLEDGMENTS

We would like to thank V. Khoze, R. Russo, and G.Travaglini for useful discussions. The work of L. A. issupported by the EC Marie Curie Research TrainingNetwork MRTN-CT-2004-512194 Superstrings.

APPENDIX A: USEFUL MASS MATRIXFORMULAS

By differentiating the F-term part of (45) with respect to�I and ��J one finds

@I@JV � eG�M6P�G

LJ �GLGJ��GLI �GLGI�

�M4PGIG

J � �JI �M4PG

LGL � 2M2P�: (A1)

Substituting eG � eK=M2P jWj2=M6

P and

GLI �GLGI �1

M2P

WLI � KLWI � KIWL

W�KLKIM4P

;

(A2)

we see that all powers of W in the denominator cancel out.We want to compute the value of the resulting expressionfor two cases: One is for zero background vevs of allscalars. And the other is for ’1; ’2 � 0 but Q � 0.

In the first case, i.e., for Q;’1; ’2 � 0, one has thathKi � 0 and hKIi � 0 for8I. Let us denote hWi � W0 andfor the moment consider W0 � 0 for more generality. Wefind that

h@I@JVi � hWIL

�WJLJLi �1

M2P

��JI hWL�WLi � hWI

�WJi�;

(A3)

where all higher orders in the 1=MP expansion vanish dueto hKi, hKIi � 0 regardless of the value of W0. Realizingthat the vevs of all double derivatives of W vanish at theorigin of field space, we finally obtain

h@I@JVi �1

M2P

��JI hWL�WLi � hWI

�WJi�: (A4)

Note that this result, apart from being exact to all orders, isalso completely independent of W0.

Applying (A4) to compute the mass matricesm2q �q, where

q is either q1 or ~q1 and �q is either �q1 or �~q1, one arrives athalf of the relations in (67). The other half, i.e., of types

m2qq andm2

�q �q, can be derived in a similar way. [It is perhapsmore convenient to differentiate the potential in the form(13).] One finds that at the origin of field space

h@K@LVi � hKI �JWKLI�W �Ji; (A5)

which is exactly the expression for the rigid case, since allsecond derivatives of the superpotential vanish for zerobackground fields. And again, this is the exact result to allorders in 1=MP.

In the case of ’1, ’2 � 0, but Q � 0, we will beinterested in the meson mass matrices. For I; J runningover the meson field components only, we still have thathKIi � 0, although now hKi � 0. Keeping W0 � 0 andexpanding (A1) up to order 1=M4

P, we find

h@I@JVi � hWIL�WJLi �

1

M2P

f�JI hWL�WLi � h �WIWJi

� hWIKL �WJL �WILKL �WJ � KWIL�WJLig

�1

M4P

�JI �hKWL

�WLi � 2W20 � �

�K2

2WIL

�WJL�

� hKihWIKL �WJL �WILKL �WJi

: (A6)

Obviously, for Q � 0 the above formula agrees with (A4).However, recall that the I; J indices in it run only over themeson fields, whereas L runs also over the squarks.

Finally, one can easily convince oneself that if K;L runonly over the mesons, whereas the squarks are the onlyfields with nonzero vevs, then h@K@LVi � 0 to all ordersand for any value of W0.

APPENDIX B: ON EXISTENCE OF NEW SUSYSOLUTIONS

We investigate here whether there are solutions to thesupersymmetry preserving equations for the ISS modelcoupled to supergravity, with the nonperturbative super-potential WADS of Eq. (8) still neglected.

The susy equations are

Dq1W � 0; D�11

W � 0; D~q1W � 0;

D�22W � 0:

(B1)

Similarly to [18], we consider the following ansatz for theexpectation values of the quark and meson fields in thesolutions we are seeking

hq1i � �11Nm; h�11i � 11Nm;

h~q1i � �21Nm; h�22i � 21Ne ;(B2)

where all vevs are real. Evaluating (B1) in this backgroundgives

41In fact, this statement only applies to a variant of O’KKLTconsidered in Sec. III there. This same variant is also the modelthat is most useful as an approximation of KKLT-ISS since it isthe one, whose supersymmetric global minimum is at finite fieldvalues.

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Dq1: h�2 1 ��1hWi � 0;

D~q1: h�1 1 ��2hWi � 0;

D�11: h��1�2 ��

2� � 1hWi � 0;

D�22: � h�2 � 2hWi � 0:

(B3)

The two equations on the first line of (B3) imply that �21 �

�22. On the other hand, the ones on the second line lead to

2 � ��2

�1�2 ��2 1: (B4)

Now, using this last relation and the equations for Dq1and

D�22, we find42

21 �

�1

�2��1�2 ��

2�;

2 � ��2

��1

�2��1�2 ��2�

�1=2:

(B5)

So far, we have expressed all vevs in terms of one of them,which could be either �1 or �2. Let us choose this to be�1. To obtain an independent equation for it, we need touse the explicit vev of the superpotential

hWi � h��1�2 1Nm ��2� 1Nm � 2Ne��: (B6)

Combining this with (B4) and theDq1equation in (B3), we

find

��2

�1�

1

M2P

��1�2Nm ��2

�Nm �

�2Ne��1�2 ��

2�

��;

(B7)

where we have reinserted the explicit dependence on MPthat comes from DIW � @IW � �KI=M2

P�W. Let us nowconsider first the case �2 � �1. Then (B7) becomes aquadratic equation for �2

1, whose solutions are

��1

MP

�2�

��MP

�2�

1

2Nm�

1

2Nm

��������������������������������������1� 4

��MP

�4NmNe

s:

(B8)

Since we would like both�1 � MP and �� MP in orderto have a reliable field theory description, only the plussign in (B8) is meaningful. Hence, we have

��1

MP

�2�

��MP

�2�

��MP

�4Ne �O

��8

M8P

�: (B9)

This implies that �21 ��

2 < 0, which is inconsistent with(B5) since the vevs 1 and 2 are real. If we take in turn�2 � ��1 and repeat the above steps, we end up with

��1

MP

�2� �

��MP

�2�O

��4

M4P

�; (B10)

which is again inconsistent for real vevs. So we concludethat coupling to supergravity does not increase the numberof vacua of the ISS model.

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