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Nuclear Physics B 808 (2009) 237-259

www.elsevier.com/locate/nuclphysb

A light neutralino in hybrid models of supersymmetry breaking

Emilian Dudasab, Stéphane Lavignacc'*, Jeanne Parmentiera c

a Centre de Physique Théorique Ecole Polytechnique, F-91128 Palaiseau, France b Laboratoire de Physique Théorique 2, Université de Paris-Sud, Bât. 210, F-91405 Orsay, France c Institut de Physique Théorique 3, CEA-Saclay, F-91191 Gif-sur-Yvette cedex, France

Received 8 September 2008; accepted 17 September 2008

Available online 23 September 2008

Abstract

We show that in gauge mediation models where heavy messenger masses are provided by the adjoint Higgs field of an underlying SU(5) theory, a generalized gauge mediation spectrum arises with the characteristic feature of having a neutralino LSP much lighter than in the standard gauge or gravity mediation schemes. This naturally fits in a hybrid scenario where gravity mediation, while subdominant with respect to gauge mediation, provides f and B f parameters of the appropriate size for electroweak symmetry breaking.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction and motivations

Supersymmetry (SUSY) breaking is the central open question in supersymmetric extensions of the Standard Model. There are two major transmission mechanisms, each having its own advantages and disadvantages:

* Corresponding author.

E-mail address: Stephane.Lavignac@cea.fr (S. Lavignac).

1 Unité mixte du CNRS (UMR 7644).

2 Unité mixte du CNRS (UMR 8627).

3 Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique et Unité de Recherche associée au CNRS (URA 2306).

0550-3213/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2008.09.023

- gravity mediation [1] easily generates all soft terms needed at low energy in the Minimal Supersymmetric Standard Model (MSSM), including the / and Bj terms of the Higgs sector [2], all being of the order of the gravitino mass at high energy. A traditional problem is that the flavor universality needed in order to avoid flavor changing neutral current (FCNC) transitions is not automatic. The lightest supersymmetric particle (LSP) in gravity mediation is generically the lightest neutralino.

- gauge mediation (GMSB) [3-5] uses Standard Model gauge loops, and therefore successfully addresses the flavor problem of supersymmetric models. The soft terms are typically of the order of a scale determined by the SUSY breaking times a loop factor, which we call MGM in the following. There is however a serious problem in generating / and Bj of the right size [6]. The gravitino, whose mass m3/2 is much smaller than MGM, is the LSP. Its lightness is the main signature of gauge mediation.

An obvious way of combining the advantages and possibly reducing the disadvantages of both mechanisms is to assume

m3/2 ~ (0.01-0.1)Mgm, Mgm ~ 1 TeV. (1)

In this case, the FCNC amplitudes induced by the non-universal gravity contributions to soft scalar masses are suppressed by a factor of order m^/2/M^M. Concerning the j/Bj problem, an option would be to generate / ~ B ~ m3/2, through the Giudice-Masiero mechanism [2]. However, since MGM > m3/2, the squark and gluino masses are much larger than m3/2, and therefore electroweak symmetry breaking requires / > m3/2. As we will see explicitly later on, there is a way of generating / ~ MGM in the scenario considered in this paper, namely through Planck-suppressed non-renormalizable operators.

Combining the gauge and gravity mediation mechanisms is an obvious possibility, which has been considered in the past or more recently from various perspectives [7]. It is easy to see that such a hybrid scenario arises for messenger masses close to the GUT scale. Indeed, consider a set of messenger fields generically denoted by (&, &) coupling to a set of SUSY breaking fields, generically denoted by X:

Wm = &(kxX + m)&, (2)

with (X) = X0 + FX92. The gauge-mediated contributions to the MSSM soft terms are proportional to

g2 -XFX

Mgm = -M-, (3)

where M = -XX0 + m, and g2/16n2 is the loop suppression of gauge mediation. Since the gravitino mass is given by m3/2 ~ FX/MP (numerical factors are omitted in this introductory part), the ratio of the gauge to the gravity contribution reads

MGM g2 MP

Mgm (4)

m3/2 16n2 M

which shows that gravity mediation is subdominant for M < y6n_-XMP ~ -XMGUT, but not completely negligible if M lies within a few orders of magnitude of -XMGUT.

In the case where messengers come into vector-like pairs of complete SU(5) multiplets, such as (5, 5) or (10,10), and ignoring for simplicity a possible "flavor" structure in the messenger indices, the messenger mass matrix can be written as

M(X) = XxX + m, m = mol + ^ (S } + •••, (5)

where S is the SU(5) adjoint Higgs field. Indeed, any vector-like pair of complete SU(5) multiplets, besides having an SU(5) symmetric mass m0, can also couple to S and get an SU(5)-breaking mass term from its vev. Depending on the messenger representation, m could also receive contributions from other operators, denoted by dots in Eq. (5): operators involving other SU(5) Higgs representations than S, or higher-dimensional operators such as &S2<P /MP.

From a model-building perspective, the main novelty of the present paper4 is to consider the case where the messenger mass matrix is mostly given by the second term in m, i.e. we assume

M(X) = kxX + ^ (S}, with XxXo « ks (S}. (6)

As we shall see in Section 3.2, the latter condition is naturally satisfied when X is identified with the SUSY breaking field of a hidden sector, e.g. when X is the meson field of the ISS model [9]. Since5 (S} = 6vY, where v & 1016 GeV and Y is the hypercharge generator embedded in SU(5), Eq. (6) implies

M = 6XsVY, (7)

up to small corrections of order XXX0. Eq. (7) has a significant impact on the structure of the GMSB-induced soft terms in the visible (MSSM) sector. Most notably, since the gaugino masses Ma (where a refers to the SM gauge group factor Ga = SU(3)C, SU(2)L or U(\)Y) are proportional to Tr(^2/M), where the Qa's stand for the charges of the messenger fields under Ga, it is readily seen that the gauge-mediated contribution to the bino mass vanishes in the limit X0 = 0:

Mi |gmsb,X0=0 a Tr(Y2M-1) a Tr Y = 0. (8)

This result holds independently of the SU(5) representation of the messengers. A nonzero bino mass is generated from gravity mediation, from X0 = 0 and from possible other terms in m, but it is expected to be much smaller than the other gaugino masses, which are of order MGM. The resulting mass hierarchy,

M1 « M2 ~ M3 ~ f, (9)

leads to a light mostly-bino neutralino, which is therefore the LSP (unless M1 > 2m3/2 at the messenger scale, in which case the LSP is the gravitino). In addition to being theoretically well motivated, this scheme provides a natural realization of the light neutralino scenarios occasionally considered in the literature [10-14], and invoked more recently [15] in connection with the new DAMA/LIBRA data [16].

The plan of the paper is the following. In Section 2, we present the MSSM soft terms induced by the messenger mass matrix (6), which breaks the SU(5) symmetry in a well-defined manner. In Section 3, we couple the messenger sector to an explicit (ISS) supersymmetry breaking sector. We study the stability of the phenomenologically viable vacuum after including quantum corrections, and discuss the generation of the f and Bf terms by Planck-suppressed operators. In Section 4, we discuss the low-energy phenomenology of the scenario, paying particular attention to the dark matter constraint. Finally, we present our conclusions in Section 5. Appendices A-B

4 Preliminary results of this paper were reported at several conferences [8].

5 In the following, we define the SU(5) breaking vev v by (S} = v Diag(2, 2, 2, —3, —3). By identifying the mass of the superheavy SU(5) gauge bosons with the scale MGUT at which gauge couplings unify, we obtain v = V2/25MGUT/gGUT & 1016 GeV.

contain technical details about the computation of the MSSM soft terms and the quantum corrections to the scalar potential.

2. Gauge mediation with GUT-induced messenger mass splitting

The main difference between minimal gauge mediation and the scenario considered in this paper6 lies in the messenger mass matrix (6). The messenger mass splitting depends on the SU(5) representation of the messenger fields. Denoting by (<i,<(i) the component messenger fields belonging to definite SM gauge representations and by Yi their hypercharge, one has

Tr(0 {£ )0) = 6v

yielding a mass Mi = 6XsvYi for (<i,<i) (again X0 = 0 is assumed). In the cases of (5, 5) and (10,10) messengers, the component fields and their masses are, respectively,

0(5) = {<3,1,1/3,(1,2,-1/2}, 0(10) = {(3,2,1/6 ,<3,1,-2/3,<1,1,1},

M = {2Xsv, -3Xsv}, M = {Xsv, -4Xsv, 6Xsv},

(11) (12)

where the subscripts denote the SU(3)C x SU(2)L x U(1)Y quantum numbers, and the components (i of 0 are in the complex conjugate representations.

The one-loop GMSB-induced gaugino masses are given by (see Appendix A)

Ma(^) =

aa(v-) 4n

J22Ta(Ri)

dln(detMi) FX

d ln X X

where the sum runs over the component messenger fields (<i,<(i), and Ta(Ri) is the Dynkin index of the representation Ri of (i. As noted in the introduction, with the messenger mass matrix (6), the gauge-mediated contribution to the bino mass vanishes irrespective of the SU(5) representation of the messengers, up to a correction proportional to XXX0 which will turn out to be negligible (see Section 3.2). Then M1 is mainly of gravitational origin:

M1 ~ m3/2 ■

As the messenger masses are not SU(5) symmetric, the running between the different messenger scales should be taken into account in the computation of the soft scalar masses. The corresponding formulae are given in Appendix A. For simplicity, we write below the simpler expressions obtained when the effect of this running is neglected. The two-loop MSSM soft scalar mass parameter m2x, induced by N1 messengers of mass M1 and N2 messengers of mass M2 and evaluated at the messenger scale, reads

ml=2T C

2N2Ta(R2)

d ln M2

d ln X

+ 2N1Ta(R1)

d ln M1 21 Fx

d ln X J X

In Eq. (15), Cax are the second Casimir coefficients for the superfield x, normalized to C(N) = (N2 - 1)/2N for the fundamental representation of SU(N) and to C\ = 3Y\/5 for U(1), and Ta (Ri) are the Dynkin indices for the messenger fields.

6 For recent analyses of general messenger masses, see e.g. Ref. [17].

While the vanishing of the GMSB contribution to the bino mass is a simple consequence of the underlying hypercharge embedding in a simple gauge group and of the structure of the mass matrix (6) (i.e. it is independent of the representation of the messengers), the ratios of the other superpartner masses, including the ratio of the gluino to wino masses M3/M2, do depend on the representation of the messengers. This is to be compared with minimal gauge mediation [5], in which the ratios of gaugino masses (namely, M1: M2 : M3 = a1 : a2 : a3) as well as the ratios of the different scalar masses is independent of the representation of the messengers [18]. Leaving a more extensive discussion of the mass spectrum to Section 4, we exemplify this point below with the computation of the gaugino and scalar masses in the cases of (5, 5) and (10, 10) messengers:

(i) (5, 5) messenger pairs

In this case the gluino and SU(2)L gaugino masses are given by

.. 1 a3 XxFX A/f 1 a2 XXFX

M3 = - Nm-----, M2 = —- Nm-----, (16)

2 4n XSv 3 4n Xsv

where Nm is the number of messenger pairs, leading to the ratio |M3/M2| = 3a3/2a2 (& 4 at f = 1 TeV). The complete expressions for the scalar masses can be found in Appendix A. For illustration, we give below the sfermion soft masses at a messenger scale of 1013 GeV, neglecting the running between the different messenger mass scales as in Eq. (15):

m2Q : m2Uc : m2DC : m2L : m2EC & 0.79 : 0.70 : 0.68 : 0.14 : 0.08, (17)

in units of NmMGM, with MGM = (a3/4n)(XXFX/XSv). InEq. (17) as well as inEq. (19) below, we used (a1/a3)(1013 GeV) = 0.65 and (a2/a3)(1013 GeV) = 0.85.

(ii) (10,10) messenger pairs

In this case the gluino and SU(2)L gaugino masses are given by

M3 = 7 Nma3 XXFX, M2 = 3Nma2 XXFX, (18)

4 4n Xsv 4n Xsv

leading to the ratio M3/M2 = 7a3/12a2 (& 1.5 at f = 1 TeV). In this case too, we give the sfermion soft masses at a messenger scale of 1013 GeV for illustration:

m2Q : mUc : m2Dc : m2L : m2Ec & 8.8 : 5.6 : 5.5 : 3.3 : 0.17, (19)

again in units of NmM^M. For the Higgs soft masses, one has mHtfu — m?iid — mL irrespective of the messenger representation.

In contrast to minimal gauge mediation with SU(5) symmetric messenger masses, in which the ratios of gaugino masses are independent of the messenger representation, in our scenario the gaugino mass ratios and more generally the detailed MSSM mass spectrum are representation dependent. There is however one clear-cut prediction, which distinguishes it from both minimal gauge mediation and minimal gravity mediation, namely the vanishing of the one-loop GMSB contribution to the bino mass. Notice also the lightness of the scalar partners of the right-handed leptons for (10, 10) messengers, which arises from the correlation between the hypercharge and the mass of the different component messenger fields (the lightest components have the smallest hypercharge). Finally, we would like to point out that, due to the fact that messengers carrying different SM gauge quantum numbers have different masses, gauge coupling unification is slightly modified compared to the MSSM. Since the messengers are heavy and their mass splitting is not very important, however, this effect is numerically small.

In the above discussion, the higher-dimensional operator X'Z0Z20/MP was assumed to be absent. Before closing this section, let us briefly discuss what relaxing this assumption would imply. If X'Z = 0, the messenger mass matrix (7) receives an additional contribution, which affects the gauge-mediated MSSM soft terms. In particular, M1 no longer vanishes:

6 2 X'Tv a1

M1|gmsb = -- d Tr Y 2-Z---1 NmMGM, (20)

5 XzMp a3

where d is the dimension of the messenger representation, and the trace is taken over the representation. Eq. (20) was derived under the assumption that the X'Z -induced corrections to the messenger masses are small, so that to a good approximation, the scalar and electroweak gaugino masses are still given by Eqs. (16)—(19). It is easy to show that this implies

M1IGMSB « 0.2NmMGM, (21)

for both (5, 5) and (10,10) messengers. In the rest of the paper, we shall therefore neglect the contribution of X'Z = 0 and assume that M1 is generated by gravity mediation.

3. A complete model

The computation of the MSSM soft terms performed in the previous section is to a large extent insensitive to the details of the sector that breaks supersymmetry. The generation of the / and Bj terms, on the other hand, depends on its details. The goal of the present section is to consider an explicit SUSY breaking sector, to couple it to the messenger sector, and to check that the following constraints are satisfied: (i) nonperturbative instabilities towards possible color-breaking vacua are sufficiently suppressed; (ii) / and Bj parameters of the appropriate size can be generated.

The model can be described by a superpotential of the form:

W = wmssm + wsb(x, ■■■) + Wm(0, 0, X, Z) + Wgut(Z), (22)

where WSB(X, ■■■) describes the SUSY breaking sector, Wm(0,0,X,Z) the couplings of the messengers fields (0, 0) to the SUSY breaking fields X and to the SU(5) adjoint Higgs field Z, and WGUT(Z) describes the breaking of the unified gauge symmetry, SU(5) ^ SU(3)C x SU(2)l x U(1)Y. In this paper, we consider the case where Wm(0, 0, X, Z) = 0(XXX + X zZ)0. The details of the GUT sector are irrelevant for our purposes and will not be further discussed in the following. The implicit assumption here is that the SUSY breaking sector and the GUT sector only couple via gravity and via the messenger fields. It is therefore reasonable to expect that they do not influence significantly their respective dynamics.

3.1. The SUSY breaking sector

A generic dynamical supersymmetry breaking sector [19] coupled to the messenger sector is enough for our purposes. For concreteness and simplicity, we consider here the ISS model [9], namely N = 1 SUSY QCD with Nf quark flavors and gauge group SU(Nc) in the regime Nc < Nf < 3 Nc. In the IR, the theory is strongly coupled, giving rise to a low-energy physics that is better described by a dual "magnetic" theory with gauge group SU(Nf - Nc), Nf flavors of quarks q'a and antiquarks ¿¡a, and meson (gauge singlet) fields Xj (i, j = 1,---, Nf, a = {^■^N, with N = Nf - Nc). The magnetic theory is IR free and can be analyzed perturbatively.

The superpotential of the magnetic theory,

Wiss = hqixjq* — hf2 Tr X, (23)

leads to supersymmetry breaking a la O'Raifeartaigh, since the auxiliary fields (-FX)j = hq'aqa — hf 2Sj cannot all be set to zero. Indeed, the matrix q'aqa is at most of rank N, whereas the second term hf 2Si has rank Nf > N. The supersymmetry-breaking ISS vacuum is defined by (q'a} = (qa} = , (X} = 0. At tree level, there are flat directions along which the components i, j = (N + 1),...,Nf of Xj are non-vanishing; quantum corrections lift them and impose (X} =0 [9]. This means that the R-symmetry under which X is charged is not spontaneously broken, which in turn implies that no gaugino masses are generated in the minimal ISS model. Another important feature of the ISS vacuum is that it is metastable. Indeed, according to the Witten index, the theory possesses Nc supersymmetric vacua. These vacua are obtained in the magnetic description by going along the branch with nonzero meson vev's, (X} = 0, where magnetic quarks become massive and decouple, so that the low-energy theory becomes strongly coupled again. In order to ensure that the lifetime of the ISS vacuum is larger than the age of the universe, one requires f « Am, where Am is the scale above which the magnetic theory is strongly coupled.

3.2. Coupling the SUSY breaking sector to messengers

Let us now couple the SUSY breaking sector to the messenger sector by switching on the superpotential term XX@X<P, and address the following two questions:

• How is the vacuum structure of the model affected, in particular is the ISS vacuum still metastable and long lived?

• Is it possible to generate f and Bf of the appropriate size?

The first question has been investigated in several works [20] in the case of SU(5) symmetric messenger masses. We reanalyze it in our scenario and come to a similar conclusion: the messenger fields induce a lower minimum which breaks the SM gauge symmetries, a rather common feature of gauge mediation models. To our knowledge, the solution we propose for the second issue has not been discussed in the literature.7 We now proceed to address the above two questions in detail.

3.2.1. Stability of the phenomenologically viable vacuum

It is well known that coupling a SUSY breaking sector to a messenger sector generally introduces lower minima in which the messenger fields have nonzero vev's. Since the messengers carry SM gauge quantum numbers, these vacua are phenomenologically unacceptable. Such minima also appear in our scenario. Summarizing the analysis done in Appendix B, we indeed find two types of local supersymmetry-breaking minima at tree level:

• the ISS vacuum with no messenger vev's and energy V($4> = 0) = (Nf - N)h2f4;

7 For recent approaches to the f/Bf problem of gauge mediation, see Ref. [21].

lower minima with messenger vev's = ^-i N+1 x,i ni—2 and energy

-Hi=N+1 X'x,ihf2 i,j)f{i=j=1,...,N} \kX,j I2

V(0<P = 0) = h2f 4 Nf - N -

i=N +1

(i,M{i=j = 1,...,N} )

Transitions from the phenomenologically viable ISS minimum to the second class of minima, in which the SM gauge symmetry is broken by the messenger vev's, must be suppressed. An estimate of the lifetime of the ISS vacuum in the triangular approximation gives t - exp((A0)4/AV), with

AV T. l2 - & (24)

A)4 ^ 1 XJ

( (i,j)i[i=j = 1,...,N}

The lifetime of the phenomenologically viable vacuum is therefore proportional to e1/X2. To ensure that it is larger than the age of the universe, it is enough to have X2 < 10-3.

We conclude that, as anticipated, the superpotential coupling Xx@X<P induces new minima with a lower energy than the ISS vacuum, in which the messenger fields acquire vev's that break the SM gauge symmetry. In order to ensure that the ISS vacuum is sufficiently long lived, the coupling between the ISS sector and the messenger sector, XX, has to be small. We believe that this result is quite generic.

Let us now discuss the stability of the phenomenologically viable vacuum under quantum corrections. As shown in Ref. [9], the ISS model possesses tree-level flat directions that are lifted by quantum corrections. The novelty of our analysis with respect to Ref. [9] is that we include messenger loops in the computation of the one-loop effective potential, and we find that these corrections result in a nonzero vev for X. The detailed analysis is given in Appendix B; here we just notice that since the messenger fields do not respect the R-symmetry of the ISS sector, it is not surprising that coupling the two sectors induces a nonzero vev for X (which otherwise would be forbidden by the R-symmetry). Indeed, the one-loop effective potential for the meson fields reads, keeping only the leading terms relevant for the minimization procedure (see Appendix B for details):

2 2 2 1 4 2

Vi-loop(X0,^0) = 2Nh f2\Yo\2 + 64^2 (8h4/2(ln4 - 1)N(Nf - N)|Xo|2

10Nmh2f 4\Tr'X|2 r , ,, ,1

+-m/-L (Tr' X)X0 + (Tr" X)Y0 + h.c. , (25)

3Xsv JJ

~ ~ Nf

where we have set XX = X01Nf-N, Y = Y01N and defined Tr' X = J2i=N+1 X'Xi, Tr'' X = J2N=1 XlX i. In Eq. (25), the first line contains the tree-level potential for X and the one-loop corrections computed in Ref. [9], whereas the linear terms in the second line are generated by messenger loops. The latter induce vev's for the meson fields:

5Nm\Tr' X\2(Tr' X)* f2

<X0> ~--m-^-)--—, (26)

12(ln4 - 1)h2N(Nf - N)Xsv

5Nm \Tr' X\2(Tr'' X)* f2 <Y0>" l92n2(V fV (27)

Notice that, due to (Y0} = 0, magnetic quarks (and antiquarks) do contribute to supersymmetry breaking: Fq ~ qX = 0 (Fq ~ qX = 0), while Fq = F-q = 0 in the ISS model as a consequence of the R-symmetry. Here instead, the R-symmetry is broken by the coupling of the ISS sector to the messengers fields, and the F-terms of the magnetic (anti-)quarks no longer vanish. We have checked that, in the messenger direction, < = < = 0 is still a local minimum. We have also checked that the nonzero vev's (26) and (27) resulting from quantum corrections do not affect the discussion about the lifetime of the phenomenologically viable vacuum. Notice that these vev's also appear in the standard case where messenger masses are SU(5) symmetric.

3.2.2. Generation of the f and Bf terms

As stressed in the introduction, due to the hierarchy of scales m3/2 « MGM, the Giudice-Masiero mechanism fails to generate a f term of the appropriate magnitude for radiative elec-troweak symmetry breaking. Fortunately, there are other sources for f and Bf in our scenario.

A crucial (but standard) hypothesis is the absence of a direct coupling between the hidden SUSY breaking sector and the observable sector (i.e. the MSSM). In particular, the coupling XHuHd should be absent from the superpotential. The fields of the ISS sector therefore couple to the MSSM fields only via non-renormalizable interactions and via the messengers. It is easy to check that non-renormalizable interactions involving the ISS and MSSM fields have a significant effect only on the f and Bf terms, whereas they induce negligible corrections to the MSSM soft terms and Yukawa couplings. The most natural operators mixing the two sectors, which are local both in the electric and in the magnetic phases of the ISS model, are the ones built from the mesons X. It turns out, however, that such operators do not generate f and Bf parameters of the appropriate magnitude.

Fortunately, a more interesting possibility arises in our scenario, thanks to the loop-induced vev of the meson fields discussed in the previous subsection. Indeed, the Planck-suppressed operator

X1TT~ HuHd, (28)

in spite of being of gravitational origin, yields a f term that can be parametrically larger than m3/2. This allows us to assume m3/2 « MGM, as needed to suppress the most dangerous FCNC transitions, consistently with electroweak symmetry breaking (which typically requires a f term of the order of the squark and gluino masses). More precisely, the operator (28) generates

f=h tN^ m3/2, (29)

, ,, 5Nm|Tr' X|2(Tr'' X) MP r-

B = —2hY) =--ml n 1 ) —V3 m3/2, (30)

^ 01 96n2N,JWc Xsv 3/2 v '

where we used m3/2 = NfN+1 F'xj I2/T>MP = TWchf2/TiMP. Using Eqs. (26) and (27), it is easy to convince oneself that one can obtain f ~ 1 TeV for e.g. m3/2 ~ (10-100) GeV, by taking a small enough ISS coupling h. As a numerical example, one can consider for instance m3/2 = 50 GeV, Nc = 5, Nf = 7 and X1/h = 10, in which case f = 775 GeV. As for the B parameter, it turns out to be somewhat smaller than m3/2. For instance, taking as above Nc = 5, Nf = 7 and assuming further Nm = 1, |Tr'X|2 = 10—3 and XSv = 1013 GeV, one obtains B = —0.49(Tr" X)m3/2. This will in general be too small for a proper electroweak symmetry breaking, even if Tr" X ~ 1 is possible in principle (contrary to Tr' X, Tr" X is not constrained by the lifetime

of the ISS vacuum). However, B¡ also receives a contribution from the non-renormalizable operator

X2 -VrHuHd, i31)

which gives a negligible contribution to x, but yields B¡ = -X2^/3Nc(X0)m3/2. Using Eq. (29), one then obtains

R . hN X2 5Nm|Tr' X|2(Tr' X)* Mp R

B = -X2~~ (X0) = 7 TTTTi -1woAr9 rrr\-m3/2, (32)

Xi N Xi 12(ln4 - 1)h2N2^/NC Xsv '

which is enhanced with respect to Eq. (30) by the absence of the loop factor and by the presence of h2 in the denominator. It is then easy to obtain the desired value of the B parameter. As an illustration, choosing the same parameter values as in the above numerical examples and taking h = 0.1, one obtains B/X2 = 7.9 TeV, while choosing Tr' X = 10-2 (instead of 10-3/2) gives B/X2 = 250 GeV.

We conclude that Planck-suppressed operators can generate x and B¡ parameters of the appropriate size in our scenario, thanks to the vev's of the meson fields induced by messenger loops, which are crucial for the generation of B¡. As mentioned in the previous subsection, these vev's appear independently of whether the messenger masses are split or not. Therefore, the x and B¡ terms can be generated in the same way in more standard gauge mediation models with SU(5) symmetric messenger masses.

Notice that there is a price to pay for the above solution to the x/B¡ problem: the interaction term (28), which is local in the magnetic ISS description, becomes non-local in the electric description, analogously to the qXq coupling of the magnetic Seiberg duals [22].

4. Low-energy phenomenology

The phenomenology of minimal gauge mediation has been investigated in detail in the past (see e.g. Ref. [18]). The main distinctive feature of our scenario with respect to standard gauge mediation is the presence of a light neutralino, with a mass of a few tens of GeV in the picture where M1 ~ m3/2 ~ (10-100) GeV. As is well known, such a light neutralino is not ruled out by LEP data: the usually quoted lower bound M-0 > 50 GeV assumes high-scale gaugino mass

unification, and can easily be evaded once this assumption is relaxed.8 The other features of the superpartner spectrum depend on the messenger representation. Particularly striking is the lightness of the lR with respect to other sfermions (including the lL) in the case of (10,10) messengers. The values of the soft terms at the reference messenger scale9 Mmess = 1013 GeV are given by Eqs. (16)-(19). One can derive approximate formulae for the gaugino and the first two generation sfermion masses at low energy by neglecting the Yukawa contributions in the

8 More precisely, for a mostly-bino neutralino (as in our scenario, where M\ ^ M2, i^i), there is no mass bound

from LEP if either M-0 + M-0 > 200 GeV or selectrons are very heavy [23]. The former constraint is satisfied by all

superpartner mass spectra considered in this section. Furthermore, a mostly-bino neutralino has a suppressed coupling to the Z boson and thus only gives a small contribution to its invisible decay width.

9 As explained in Appendix B.2, the requirement that our metastable vacuum is sufficiently long lived constrains the messenger scale Mmess = k^v to lie below 1014 GeV or so. Demanding Mgm/®3/2 ~ 10 further pushes it down to

1013 GeV.

one-loop renormalization group equations, as expressed by Eq. (A.4). At the scale / = 1 TeV, one thus obtains

M2 - 0.25NmMGu, M3 - NmMGu, (33)

m2Qia - (0.79 + 0.69Nm)NmMGM, m- (0.70 + 0.66Nm)NmM^M, (34)

m2Dc2 - (0.68 + 0.66Nm)NmM2GM, m2Li 2 - (0.14 + 0.03Nm)NmM^M, (35)

m2Ec2 - 0.08NmMGM + 0.12M2, (36) for (5, 5) messengers, and

M2 - 2.2NmMGM, M3 - 3.5NmMGM, (37)

mQi 2 - (8.8 + 10.4Nm)NmMGM, m^ - (5.6 + 8.1Nm)NmMGM, (38)

m2Dc2 - (5.5 + 8.1Nm)NmMGM, mL1,2 - (3.3 + 2.3Nm)NmM^M, (39)

mE^ - 0.17NmMGM + 0.12M2, (40)

for (10,10) messengers, where MGM = (a3(Mmess)/4n)(XXFX/Xsv). Furthermore, one has in both cases:

Mx0 « 0.5M1. (41)

In Eqs. (33)-(40), the unknown gravitational contribution to the soft terms is not taken into account, apart from Mi which is taken as an input (we neglected subdominant terms proportional to M2 in all sfermion masses but m2EC ). These formulae fit reasonably well the results

obtained by evolving the soft terms from Mmess = 1013 GeV down to / = 1 TeV with the code SUSPECT [24]. For the third generation sfermion masses, most notably for mQ3 and m2vc, the Yukawa couplings contribute sizeably to the running and the above formulae do not apply. The Higgs and neutralino/chargino spectrum also depend on tan j3 and on the values of the \x and B/j, parameters, which are determined from the requirement of proper radiative electroweak symmetry breaking. As for the lightest neutralino, Eq. (41) implies that M-0 < m3/2 as long as M1 < 2m3/2, a condition which is unlikely to be violated if M1 is of gravitational origin, and we can therefore safely assume that the lightest neutralino is the LSP. The gravitino is then the NLSP, and its late decays into /[V tend to spoil the successful predictions of Big Bang nucleosynthesis (BBN) if it is abundantly produced after inflation. This is the well-known gravitino problem [25], and it is especially severe for a gravitino mass in the few 10 GeV range, as in our scenario. We are therefore led to assume a low reheating temperature in order to reduce the gravitino abundance, typically TR < (105-106) GeV, which strongly disfavor baryogenesis mechanisms occurring at very high temperatures, such as (non-resonant) thermal leptogenesis.

While the lightness of is a welcome feature from the point of view of distinguishing the present scenario from other supersymmetric models (for recent studies of the collider signatures of a light neutralino, see e.g. Refs. [13,14]), it might be a problem for cosmology. Indeed, a neutralino with a mass below, say, 50 GeV will generally overclose the universe, unless some annihilation processes are very efficient [11-13]: (i) the annihilation into t+t-and bb via s-channel exchange of the CP-odd Higgs boson A, or (ii) the annihilation into a fermion-antifermion pair via t- and u-channel exchange of a light sfermion. The process

(i) can bring the relic neutralino abundance down to the observed dark matter level (namely, ^DMh2 = 0.1099 ± 0.0062 [26]) if A is light, tan j is large and xf contains a sizeable hig-gsino component (which requires \/\ ~ 100 GeV). More precisely, can be as light as 6 GeV for MA ~ 90 GeV and tan j > 30 [12,13], in the anti-decoupling regime for the lightest Higgs boson h. The process (ii) is more efficient for light sleptons (lR) and large values of tan j. In particular, in the large mA region where the process (i) is not relevant, X? can be as light as 18 GeV without exceeding the observed dark matter density if mT1 is close to its experimental bound of 86 GeV and tan j ~ 50 [11,13]. Note that experimental limits on superpartner masses and rare processes have been imposed in deriving these bounds.

We were not able to find values of MGM, Nm and tan j leading to a light A boson (say,

MA < 120 GeV); hence we must considerer Mx0 > 18 GeV in order to comply with the dark

matter constraint. In Table 1, we display 6 representative spectra with 20 GeV < Mx0 < 45 GeV

and light lR masses (apart from model 1), corresponding to different numbers and types of messengers, and different values of MGM and tan j. The superpartner masses were obtained by running the soft terms from Mmess = 1013 GeV down to low energy with the code SUSPECT. Apart from M1, which is taken as an input, the unknown subdominant gravitational contributions to the soft terms have not been included (we shall comment on this later). As is customary, f1 and f2 refer to the lighter and heavier f mass eigenstates; for the first two generations of sfermions, they practically coincide with fR and fL. We also indicated in Table 1 the bino and down higgsino components of the lightest neutralino, in the notation X0 = Z11B + Z12^3 + Z13H0 + Z14H°.

Let us now comment on these spectra. In the case of messengers in (5, 5) representations, taking into account the LEP lower bound on the lightest Higgs boson mass (mh > 114.4 GeV) and the experimental limits on the superpartner masses generally leads to relatively heavy lR (see model 1), although larger values of tan j yield a lighter T1 (for instance, shifting tan j from 30 to 50 in model 1 gives mT1 = 150 GeV). However, one can accommodate a lighter T1 if one assumes a large number of messengers, as exemplified by model 2. Light sleptons are more easily obtained with messengers in (10, 10) representations (models 3/3bis and 4), or in both (5, 5) and (10,10) representations (models 5 and 6). Note that both mT1 and m j1,e1 are close to their experimental limits in model 4. Apart from the mass of the lightest neutralino (and to a lesser extent of lR), the low-energy spectrum very weakly depends on the actual value of M1 (compare models 3 and 3bis, which only differ by the value of M1). In the last column of Table 1, we give the relic density of X0 computed by the code MicrOMEGAs [27,28]. One can see that, for MX0 ~ (20-25) GeV, QX0h2 lies above the observed dark matter density, even though lR

are light (models 1 to 3); this can be traced back to the small higgsino admixture of X0, which

suppresses the Z boson exchange contribution [13]. Larger values of MX0 enable the relic density

to fall in the 2a WMAP range (models 3bis to 6).

We conclude that the scenario of supersymmetry breaking considered in this paper can provide supersymmetric models with a light neutralino (MX0 ~ 40 GeV) accounting for the dark

matter of the universe. We have checked that the models of Table 1 are consistent with the negative results from direct dark matter detection experiments such as CDMS [29] and XENON [30]. Since the spin-independent (spin-dependent) neutralino-nucleon cross section is dominated by Higgs boson and squark exchange diagrams (Z boson and squark exchange diagrams), it is expected to be rather small in our scenario, in which squarks are heavy and the neutralino is mostly a bino. This is confirmed by a numerical computation with MicrOMEGAs, which gives typical

Table 1

Supersymmetric mass spectra obtained by running the soft terms from Mmess = 1013 GeV down to low energy with the code SUSPECT (all masses in GeV).

1 2 3 3bis 4 5 6

n(55) N — "(10,10) Mgm 1 0 1000 6 0 200 0 1 300 0 1 300 0 4 110 1 1 220 3 1 160

M1 50 50 50 85 80 85 85

tan ß 30 24 15 15 9 15 15

sign(ß) + + + + + + +

h 114.7 115.0 115.2 115.2 116.5 114.6 114.8

A 779.2 645.4 892.2 892.4 1015 735.8 662.7

H 0 779.2 645.5 892.4 892.6 1015 735.9 662.8

H ± 783.3 650.3 895.7 895.9 1018 740.1 667.5

*Î 259.4 305.0 560.2 560.3 676.7 408.0 223.9

747.8 636.8 693.9 694.0 970.4 590.4 597.5

x0 24.5 23.5 23.2 42.9 38.1 43.0 42.9

¿2° 259.4 305.0 560.1 560.3 677.1 408.0 223.9

¿3° 743.3 629.8 596.9 597.1 691.0 570.8 589.2

X 0 745.7 634.7 693.8 693.9 970.4 590.4 596.3

|ZU| 0.9982 0.9975 0.9971 0.9971 0.9978 0.9968 0.9969

|Z13i 0.0599 0.0708 0.0750 0.0755 0.0648 0.0792 0.0772

g 1064 1207 1097 1097 1527 1028 1063

h 984.6 927.3 861.7 861.6 1080 795.7 809.5

t2 1156 1074 1240 1240 1468 1058 1002

«1 j C1 1195 1087 1135 1135 1361 1006 987.9

Ù2,C2 1240 1115 1327 1327 1555 1118 1043

b1 1128 1040 1123 1123 1356 995.4 966.2

b_2 1169 1079 1224 1224 1451 1038 987.1

d1, 1184 1085 1134 1134 1360 1005 987.1

d2,S2 1243 1117 1329 1329 1557 1121 1046

t1 242.2 99.0 86.3 89.3 87.0 96.7 95.2

T2 420.3 289.4 696.2 696.3 753.1 498.6 349.8

294.4 150. 6 131.5 133.6 105.4 123.6 117.4

¿2,ß2 413.4 275.1 699.1 699.2 754.1 500.1 348.5

vt 396.6 260.5 691.4 691.5 749.0 491.4 337.6

405.8 263.6 694.8 694.9 750.1 493.9 339.5

2 h C5 6.40 0.428 0.279 0.122 0.124 0.118 0.116

values of 10-46-10-45 cm2 for the spin-independent cross-section, and of 10 -46-10 -45 cm2 for the spin-dependent cross-section.

Let us add for completeness that models 1 to 3 can be made consistent with the observed dark matter density by assuming a small amount of R-parity violation [10]. In fact, in the presence

of R-parity violation, nothing prevents us from considering even smaller neutralino masses by lowering10 m3/2.

Some comments are in order regarding the subdominant supergravity contributions to the soft terms and their effects in flavor physics. First of all, these contributions will shift the values of the soft terms at Mmess by a small amount and correspondingly affect the spectra presented in Table 1. Since supergravity contributions are parametrically suppressed with respect to gauge contributions by a factor m3/2/(NmMGM) for gaugino masses, and by a factor m3/2/(VNmMGM) for scalar masses, we do not expect them to change the qualitative features of the spectra.11 Also, the gravity-mediated A-terms are suppressed by the small gravitino mass, and they should not affect the sfermion masses in a significant way. The most noticeable consequence of the supergravity contributions is actually to introduce flavor violation in the sfermion sector at the messenger scale:

{M2X ).. = m2xSlJ + (Xx)ijml/2 (x = Q,Uc,Dc,L,Ec ), (42)

where m2xSlj is the flavor-universal gauge-mediated contribution, and the coefficients (Xx)lj are at most of order one. As is well known, flavor-violating processes are controlled by the mass insertion parameters (here for the down squark sector):

(Mq)j isd ï _ (M2Dc)ÍJ fs d , _ (Aj 2 d

plu , (tu, (sua0.=», (43)

J m- J m- J m

where (MQ )ij, (MD c)ij and (Ad)ijvd are the off-diagonal entries of the soft scalar mass matrices renormalized at low energy and expressed in the basis of down quark mass eigenstates, and m¿ is an average down squark mass.

Neglecting the RG-induced flavor non-universalities, which are suppressed by a loop factor and by small CKM angles, the mass insertion parameters (&dMM)ij (M = L, R) arising from the non-universal supergravity contributions are suppressed by a factor my2/m2-, and possibly also by small coefficients (XQ^Dc)ij .For m3/2 = 85 GeV and m ¿ ~ 1 TeV as in the spectra displayed in Table 1, we find (&LL)ij ~ 7 x 10-3 (^Q)ij and (SRR)ij ~ 7 x 10-3 (kDc)ij, which is sufficient to cope with all experimental constraints (in the presence of large CP-violating phases, however, eK further requires y7(kQ)i2(kDc)n < 0.04, see e.g. Ref. [33]). As for the (&LR)ij, they are typically suppressed by m3/2mdi /m2 and are therefore harmless.

The situation is much more problematic in the slepton sector, where processes such as \x ^ ey and t ^ /xy put strong constraints on the (8eMN)ij, M,N = L,R (see e.g. Ref. [34]). Indeed, the leptonic S's are less suppressed than the hadronic ones, due to the smallness of the slepton masses: for m3/2 = 50 GeV and mLi ~ 500 GeV, mEc ~ 100 GeV, one e.g. finds ($LL)ij ~ 10-2(^L)i;' and (SeRR)ij ~ 0.3 (kEc)ij. To cope with the experimental constraints, which are particularly severe in the presence of a light neutralino and of light sleptons, we need to

10 Assuming M1 ~ m3/2, one can reach Mx^0 ~ 5 GeV by choosing m3/2 ~ 10 GeV. We refrain from considering much lower values of m3/2, which would render the generation of ¡â ~ Mgm less natural. However, we note that in recent models of moduli stabilization [31,32], gravity (moduli) contributions to gaugino masses are typically smaller than m3/2 by one order of magnitude.

11 For values of m3/2 as large as 80-85 GeV, however, the supergravity contribution to the R masses is expected to be comparable to the GMSB one. In this case the parameters of the models in Table 1 must be adjusted in order to keep the sleptons sufficiently light.

assume close to universal supergravity contributions to slepton soft masses, perhaps due to some flavor symmetry responsible for the Yukawa hierarchies. Possible other sources of lepton flavor violation, e.g. radiative corrections induced by heavy states, should also be suppressed. Let us stress that the same problem is likely to be present in any light neutralino scenario in which the neutralino annihilation dominantly proceeds through slepton exchange. Alternatively, in models where the relic density of /0 is controlled by a small amount of R-parity violation, all sleptons can be relatively heavy as in model 1, thus weakening the constraints from the non-observation of lepton flavor violating processes.

Throughout this paper, we assumed that the non-renormalizable operator @E2@/MP is absent from the superpotential and that M1 is purely of gravitational origin. Let us mention for completeness the alternative possibility that this operator is present and gives the dominant contribution to M1. In this case, the lightest neutralino mass is no longer tied up with the mass of the gravitino, which can be the LSP as in standard gauge mediation. This makes it possible to solve the lepton flavor problem by taking m3/2 < 10 GeV and considering a model with relatively heavy I R. Such a scenario is still characterized by a light neutralino, but it is no longer the LSP, and the dark matter abundance is no longer predicted in terms of parameters accessible at high-energy colliders. Furthermore, since the NLSP is the lightest neutralino, some amount of R-parity violation is needed to avoid the strong BBN constraints [25].

5. Conclusions

In this paper, we have shown that models in which supersymmetry breaking is predominantly transmitted by gauge interactions lead to a light neutralino if the messenger mass matrix is oriented with the hypercharge generator, M ~ vY. This arises naturally if the main contribution to messenger masses comes from a coupling to the adjoint Higgs field of an underlying SU(5) theory. In this case, the bino receives its mass from gravity mediation, leading to a light neu-tralino which is then the LSP. While from a model building perspective the gravitino, hence the neutralino, could be much lighter, we considered a typical neutralino mass in the 20-45 GeV range and worked out the corresponding low-energy superpartner spectrum. We noticed that, in the case of (10, 10) messengers or of a large number of (5, 5) messengers, the scalar partners of the right-handed leptons are much lighter than the other sfermions, making it possible for a neutralino with a mass around 40 GeV to be a viable dark matter candidate. However, such a SUSY spectrum also creates potential FCNC problems in the lepton sector, which asks for a high degree of universality or alignment in slepton masses.

In the hybrid models of supersymmetry breaking considered in this paper, the gravity-mediated contributions, although subdominant, are essential in generating the / and Bj terms through Planck-suppressed operators. We studied the case where the SUSY breaking sector is provided by the ISS model and found that, as expected, messenger loops induce a breaking of the R-symmetry in the ISS vacuum. The associated meson vev's happen to be of the appropriate size for generating the Bj term needed for electroweak symmetry breaking. We stress that this mechanism also works for more general messenger mass matrices than the one studied in this paper, in particular in the simpler case of SU(5) symmetric messenger masses.

While the vanishing of the GMSB contribution to the bino mass is a simple consequence of the messenger mass matrix (6) and of the embedding of the hypercharge into a simple gauge group, the other features of the superpartner spectrum depend on the representation of the messengers, in contrast to minimal gauge mediation. For example, the gluino to wino mass ratio is |M3/M2| = 3a3/2a2 for (5, 5) messengers and |M3/M2| =7a3/12a2 for (10,10) messengers.

The experimental evidence for one of these mass ratios at the LHC, together with the discovery of a light neutralino LSP, would be a clear signature of the hybrid models of supersymmetry breaking studied in this paper. In most high-energy scenarios, gaugino masses are assumed to be universal, leading to the hierarchy Mi: M2 : M3 = a\ : a2 : a3 at low energy. The possibility that non-universal gaugino masses be related to the lightness of the neutralino LSP by an underlying GUT structure appears to be appealing and deserves further investigation.

Acknowledgements

We thank Geneviève Bélanger, Marco Cirelli, Tony Gherghetta, Yann Mambrini, Mariano Quiros, Alberto Romagnoni and Carlos Savoy for useful discussions and comments. We are grateful to Yann Mambrini for providing us with an improved version of the code SUSPECT. This work has been supported in part by the ANR grants ANR-05-BLAN-0079-02, ANR-05-BLAN-0193-02, ANR-05-JCJC-0023, the RTN contracts MRTN-CT-2004-005104 and MRTN-CT-2004-503369, the CNRS PICS #2530 and 3747 and the European Union Excellence Grant MEXT-CT-2003-509661.

Appendix A. Gauge contributions to the MSSM gaugino and scalar masses

In this appendix, we compute the gauge-mediated contributions to the MSSM soft terms in the scenario with a GUT-induced messenger mass splitting considered in this paper. We use the method of Ref. [35], appropriately generalized to the case of several types of messengers with different masses.

A.1. General formulae

The gauge-mediated contributions to gaugino masses are encoded in the running of the gauge couplings [35]:

1 _ 1 b±. /^uv\ y- 2Ta(Ri) /^UV\ (A1)

g2(x) g2(Auv) 8n2 % / ) ^ 8n2 % Mi ), (.)

where ba = 3C2(Ga) R Ta(R) is the beta function coefficient of the gauge group factor Ga, and the sum runs over several types of messengers fa) with masses Mi (/ < Mi < ^UV) belonging to the SM gauge representations Ri. Ta(Ri) is the Dynkin index of the representation Ri, normalized to 1/2 for fundamental representations of SU(N). For U(1), we use the SU(5)

normalization ai = 3aY; correspondingly, Ti(Ri) should be understood as 3Yi /5, where the hypercharge Y is defined by Y = Q - T3 (so that YQ = 1 /6, YUc = -2/3, YDc = 1/3, YL = -1/2 and YEc = 1). The one-loop gaugino masses are then given by [35]

aaiv)^^ ln(detMi) Fx

Mate) = -Oï^Yl 2Ta(Ri)

4n ^ d ln X X

■ (A.2)

The gauge-mediated contributions to scalar masses are encoded in the wave-function renormal-ization of the MSSM chiral superfields x [35]:

ni aq(^uv) YC^2 ( aa(M2)\2C^ f aa(Ml)\2CX-b

where f < Mi < M2 < Auv, ba,i = ba - 2NiTa(Ri), ba,2 = ba,i - 2N2Ta(R2), and q are the quadratic Casimir coefficients for the superfield x, normalized to C(N) = (N2 - 1)/2N for the fundamental representation of SU(N) and to Cj = 3Y2/5 for ^(1). In Eq. (A.3), we considered for simplicity only 2 types of messengers, characterized by their masses M1i2 (which should not be confused with the bino and wino masses), SM gauge representations R1i2 and multiplicities N1j2. Following Ref. [35], we obtain for the soft mass parameter m2x:

= 2£ C

a I "a

aa(v-)

2N T (R )è2 J- (2N2Ta(R2))2 è2 è2 \ 2N2Ta(R2Ka,2 +--T-[?a, 1 - ?a,2>

+ 2NiTa(Ri)fai

d ln M1

d ln X

1 - è 2

1 èa, 1

d ln(det M)

d ln M2

d ln X

d ln X

where èa,i = aM d = 1,2) and det M = MN1 m2N2 . In Eq. (A.4), the first term in square brackets contains the contribution of the messengers of mass M2 renormalized at the scale M1, the second term represents the contribution of the messengers of mass M1, and the third term the running from the messenger scale M1 down to the low-energy scale n-.

A.2. (5, 5) and (10, 10) messengers with GUT-induced mass splitting

We are now in a position to evaluate the MSSM gaugino and scalar masses induced by Nm (5, 5) messenger pairs with a mass matrix M(X) given by Eq. (6). Inside each pair, the SU(3)C triplets have a mass 2X^ v, while the SU(2)L doublets have a mass —3X%v (we omit the contribution of X0 = 0, which as discussed in Section 3.2 turns out to be negligible). Applying Eq. (A.2), we obtain for the one-loop gaugino masses:

1 a3 XX FX M3 = - Nm —3 -X-X,

2 4n -xv

1 a2 -X —X

M2 =--Nm-2

3 4n -xv

M1 = 0.

In computing scalar masses, we neglect for simplicity the running of the gauge couplings between different messenger scales, which amounts to set aa(Mi) = aa(M2) = aa(Mmess) in Eq. (A.4), where Mmess is an average messenger mass. Summing up all gauge contributions, we can cast the scalar masses in the form

m2x(Mn

^ N — s) — Nm / ar\.

^ n 4n

-X—X

where aa = aa(Mmess) and the coefficients da are given in the following table:

da SU(3)C SU(2)L U(1)

Q 2/3 1/6 1/180

Uc 2/3 0 4/45

Dc 2/3 0 1/45

L 0 1/6 1/20

Ec 0 0 1/5

Hu,Hd 0 1/6 1/20

Consider now Nm (10,10) messenger pairs. Inside each pair, the (03,2,+i/6, $3 2 -1/6) fields have a mass k^v, ($3 1 -2/3, 03,1,+2/3) have a mass —4k%v, and (01,1,+1, 01,1,-1) have a mass 6k^v. Then the gaugino masses are given by

7 kxFx kxFx

M3 = - Nm-3 M2 = 3Nm-2 kX^, M1 = 0, (A.7)

and the scalar masses by Eq. (A.6), with coefficients dX given by:

dx SU(3)c SU(2)l U(1)

Q 11/2 9/2 1/90

Uc 11/2 0 8/45

Dc 11/2 0 2/45

L 0 9/2 1/10

Ec 0 0 2/5

Hu,Hd 0 9/2 1/10

Appendix B. Quantum corrections and metastability of the vacuum

B.1. Tree-level vacuum structure

We are searching for the minima of the scalar potential

V = \FX |2 + \Fbx |2 + \Fq|2 + \ Fq|2 + |F$|2 + F|2 + \FS |2, (B.1)

\Fx\2 = E- hf2 + k'xM2, i = 1

\Fbb\2 = E If)+kXM2,

(i,j№=j = 1,...,N}

|Fq |2 = E \hx j

a,i = 1,...,N

|Fq |2 = E \hqiX) a,j = 1,...,N

F |2 = \(kx x + kSS)$\2, F |2 =\$(kxX + k^)\2,

|Fs |2 =

d^GUT 2

We choose a basis in which qai qja is a rank N diagonal matrix:

qrn 0 0 0 0

0 0 0 0

0 0 qNqN 0 0

0 0 0 0 0

0 0 0 0 0

The potential (B.1) does not contain the supergravity contributions nor the corresponding soft terms, which are expected to have a negligible impact in the present discussion. WGUT(£) is the superpotential for the SU(5) adjoint Higgs field £, whose vev is responsible for the spontaneous breaking of SU(5).

We find that all the F-terms, except FX, can be set to zero. However, F$ = F$> = 0 has two types of solutions. More precisely, for values of X such that the matrix (acting on SU(5) gauge indices) XXX + X££ is

• invertible, then $ =$ = 0;

• non-invertible, then both $ and $ can have a nonzero vev.

Indeed, if XXX + X£ai = 0, where ai is an eigenvalue of £, the values of $ and $ are not fixed by the constraint F$ = F$> = 0. The equation F£ = 0 implies that they must be of the

form $ = (0,...,0,$a0, 0,..., 0) and $T = (0,..., 0, $a0, 0,..., 0). Indeed, one has F£p = f '(£)p - p&p + X£$a$p = 0, where a, p = 1,..., 5 are SU(5) indices and f(£) is defined by ^GUT(£) = f(£) - P Tr £ (the specific form of the function f is irrelevant here). Working in a SU(5) basis in which £p is diagonal, one concludes that at most one component in $ and $ can be nonzero, and it must be the same component. As for Fq and Fq, they can always be fixed to zero by choosing the matrix Xj to be symmetric, with the vectors q'a = qa (a = 1,...,N), solutions of hqlaqa - hf2 + klXi$$ = 0 (so as to satisfy the constraint \FX|2 = 0), belonging to its kernel. Note that the value of X is not completely determined at this level.

We have succeeded to set all F-terms but FXb to zero without completely fixing the value of X. For generic couplings X'X j, it is still possible to arrange for the matrix XXX + X££ to have a zero eigenvalue, in which case $ and $ can be nonzero. We can minimize \FX\2 in both cases ($$ = 0 versus $$ = 0), which yields two types of local supersymmetry-breaking minima:

• $$ = 0, with the ISS energy ¥0 = (Nf - N)h2 f4;

• $$ = - T^N+i hf2/Ea, jmi=j=1,..., N}\^X, j\2, with ¥0 = h2f4(Nf - N -\ Di=N+1 XX,i \2/H(i,j)/{i=j = 1,...,N} \^X,j \2).

B.2. Lifetime of the metastable vacuum

Following Ref. [9], we evaluate the lifetime of the metastable ISS vacuum in the triangle approximation. The decay rate is proportional to

( (A$)4\ AV ^ , .2 -2

exp--, with-j = y \XXi I = X2. (B.4)

FV AV ) (A$)4 1 X,jl

In order for the metastable vacuum to be sufficiently long lived, we require X2 < 10 3. The individual couplings X'X j must then typically be of order 10-2, except the ones corresponding to i = j = 1,...,N, which can in principle be larger. From Eq. (4) we see that, for Tr'X = EN=/N+1XX,i = 10-2, Mgm/^3/2 - in — ! - 1n13

which in turn requires Xs — 10-3.

10 corresponds to a messenger scale ksv ~ 10 GeV,

B.3. Quantum corrections to the scalar potential

As explained in Ref. [9], the ISS model has a tree-level flat direction along the i, j = (N + 1),...,N/ components of Xj. In the absence of messengers, quantum corrections enforce (X) = 0. In this section, we add the contribution of the messengers to the one-loop effective potential for X and study its behaviour around fa = fa = 0. Our aim is to determine whether the ISS vacuum remains metastable and long lived in our scenario after quantum corrections have been included.

We parametrize the quantum fluctuations in the following way:

SZf \ j, , s„ ^ -_( /e- + Sxf ^

sz X ),

q = (fe + Sx,Sp), q =

with X = X0 + SX and Y = Y0 + SY. The only F-term from the ISS sector that is relevant for the computation of the messenger contribution to the one-loop effective potential is the one of x:

-FX ; = h Tr Nc [SpSp ^ ff, - hf 2Sff,+ kf ' SfaSfa,

where /, /' = (N + 1),...,N/. The terms of the scalar potential that contribute to the scalar messenger mass matrix are:

\h Trnc (SpSp ^ ff, - hf Sff / + kXfff ' SfaSfa\2 + \(kXX + m)Sfa\2 + \Sfa(kXX + m)\2.

ff (B.7)

Around the vacuum with zero messenger vev's, fa = fa = 0, there is no quadratic mixing between the ISS and messenger fields. We can therefore compute separately the contributions of the ISS and messenger sectors to the effective potential.

Let us first consider the messenger sector. With the notations Mj = kXX + mj (where the index I refers to different components of the messenger fields in definite SM gauge representations, and mj = 6k^Yjv), Tr/ k = ifN +1kX i and t = hf 2Tr/ k, the scalar mass matrix reads:

(fat fat faj faj)

HMj |2

|Mj |2 -t

-t * |Mj |2

|Mj |2/

(faj\ faj

We then find the mass spectrum (which is non-tachyonic since |t | = |kXFX \^k2sv

m20j = |Mj |2 ±|t | = |kXX + mj |2 ± hf 2|Tr/ k|. The contribution of the messenger sector to the effective potential is then

■m2):

V(1)- =

1 4 (M2

-- StrM4lnl M

64n2 V A2

2 2 /det MtM\\

20 |t |2 + 211 |2lni a2 \\.

(B.10)

As for the contribution of the ISS sector, it is given by [9]:

Vss = ¿28h4/2(ln4 - 1)N(Nf - N)|Xo|2, (B.11)

where we have set X = X01Nf -N, Y = Y01N, and we have omitted a term proportional to | Y012, which is subleading with respect to the tree-level potential for Y0, VISS(Y0) = 2Nh2f2|Y0|2 (by contrast, the term proportional to |X0|2 in vSS is fully relevant, since there is no tree-level potential for X0). To VISS + VIS1S), we add the linearized field-dependent one-loop contribution of the messenger sector, using the fact that ^ |«k2sv2:

(1) _ NmT k\2h2f4

V( ) =

35 2 10

(IxX)2 +--kxX + h.c.

m|u2 3ksv

(B.12)

As will become clear after minimization of the full one-loop effective potential, the quadratic term in is suppressed with respect to the quadratic terms in VISS by (X) « k*v, and can

therefore be dropped. Minimizing VISS + v/SS + , one finds that the contribution of the messenger fields to the effective potential destabilizes the tree-level ISS vacuum and creates small tadpoles for the meson fields:

5Nm|Tr' k|2(Tr' k)* f2

(X0)~--—--)--— « ksv, (B.13)

12(ln4 - 1)h2N(Nf - N)ksv *

5N—Tr' k|2(Tr" k)* f2

(Y0)---—-Lr-- — « k*v, (B.14)

X 0 192n2N ksv

where Tr" k = J2N=ik'X i. The contribution of Eqs. (B.13) and (B.14) to the vacuum energy, being suppressed both by a loop factor and by (X0), (Y0) « k*v, is negligible compared with the ISS energy. Hence, we still have a metastable vacuum around (l) = (<j>) = 0, with a small tadpole induced for X. This plays an important role in generating \x and B/j, parameters of the appropriate size in the MSSM Higgs sector, as discussed in Section 3.

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