On messengers couplings in extended GMSB models

We discuss, for the first time, the role of the superpotential couplings of three messenger fields in a GMSB-type unification model in which messenger sector consists of both 5+5bar and 10+10bar of SU(5). It turns out that these interactions are relevant when coexist with appropriate messenger-MSSM couplings. Then they induce extra contributions to 2-loop soft masses. In the present model, we derive all such soft terms and discuss constraints which have to be satisfied to avoid rapid proton decay and mu/B_mu problem. As an example, it is shown how superpotential couplings of three messengers influence mass spectra when the model is restricted by additional global U(1)_q symmetry. We find that masses of the lightest sleptons are the most sensitive to those new interactions, what in some cases results in the NLSP/NNLSP pattern in which stau or selectron is lighter than the lightest neutralino.


Introduction
The recent measurement of the Higgs-like boson mass at the LHC [1,2] triggers a lot of questions about its consequences for models of SUSY breaking mediation [3]. It is well-known that m h 0 ≈ 125 GeV can be achieved in the MSSM by adequate left-right stops mixing, which in turn originates e.g. from large A-terms at the EWSB scale [4]. It is hard to accommodate them in the standard Gauge-Mediated Supersymmetry Breaking (GMSB) models, but they naturally arise at 1-loop level in so-called extended 1 GMSB models [5,6,7,8]. In that class of frameworks, messengers interact with themselves and with MSSM matter fields via renormalizable superpotential couplings (herein called marginal couplings). 2 One usually considers two types of them: matter-matter-messenger and matter-messenger-messenger. That topic has been widely investigated from quite a long time in various settings, and it is well known that some of those couplings significantly change phenomenology of the usual GMSB models [9] because they generate additional 1-and 2-loop soft terms. Depending on the context, the focus was on messenger-Higgs [10,11], messenger-quark [12] or messenger-lepton [13] interactions. Selection rules are usually delivered by R-parity, or some 1 Models which include marginal couplings of messengers are also called Yukawa-Deflected, More Generic or Flavoured Gauge Mediation models. 2 We adopt terminology used in [8].
global U (1) symmetry, which in some cases is related to Froggatt-Nielsen mechanism. For simplicity, it is often assumed that only couplings to the third family of the matter are relevant (i.e. some hierarchy of interactions is assumed, akin to the one in MSSM Yukawas). However, there are attempts to justify the structure of the couplings and address the issue of FCNC in the case when messengers interact also with the light families of MSSM [13,14,15,16]. The upshot being that relatively small hierarchy (much milder than the one in MSSM Yukawas) renders predictions of those models perfectly consistent with low-energy observables. Recently, all matter-matter-messenger and matter-messenger-messenger couplings have been analysed in the case when messengers are in SU (5) representations of low dimension (singlet, fundamental, antisymmetric and adjoint) [8,17].
Moreover, the wave-function renormalization method, which is relevant for computing soft terms, was substantially improved [8]. Finally, it is worthwhile to mention that the attractive feature of extended GMSB models is that they not only accommodate for large A-terms, but also in some cases realize non-standard NLSP/NNLSP mass patterns (e.g. stop/bino) with NLSP mass in the range that may be probed at the LHC [18].
In this letter we analyse the possible marginal couplings of messengers in GMSB-type model which contains one pair of messengers in representation 5 + 5 and one in 10 + 10 of SU (5). The novel elements are superpotential couplings of the form: messenger-messenger-messenger. We show that they are relevant for the phenomenology when coexist with appropriate messenger-messengermatter or messenger-matter-matter interactions. Then they contribute to 2-loop soft masses. At the same time, they do not generate A-terms nor give contributions to 1-loop soft masses. The latter feature is expected to be important for low-scale GMSB models.
One can check that proper phenomenology put several constraints on the discussed couplings.
Some of them must be highly suppressed to avoid inducing operators which would lead to rapid proton decay, or those which make it difficult to realize EWSB [19]. In the model under consideration, we analyse the issue of baryon/lepton number violation and generation of µ and B µ terms at 1-loop. To elude mentioned problems extra global U (1) q symmetry is introduced. Detailed analysis of the phenomenology of the models restricted by that symmetry shows that, even in such simplified frameworks, superpotential couplings of three messengers influence mass spectrum significantly (e.g. by changing NLSP/NNLSP pattern).
The paper is organized as follows. In Section 2 we recall details of the extended GMSB model, M GU T , and in the theory below M GU T they are absent from the spectrum. However, it is very convenient to use SU (5) notation though -all couplings involving Higgs fields H 5,5 are understood to be projected onto doublet components of 5, 5.
We assume that the SUSY breaking in the hidden sector can be parametrized by gauge singlet chiral superfield X (so-called spurion) which lowest component and F -term spontaneously get vev i.e. X = M + θ 2 F . As usual, the following superpotential couplings between spurion and messengers provide mass M for the latter. We choose M to be of order 10 14 GeV. For the simplicity, it is assumed that in (1) the spurion coupling to both pairs of messengers is the same. SUSY breaking effects are transmitted from the hidden sector to MSSM via the messengers. In the extended Gauge Mediation models, messengers interactions which are relevant for the mediation mechanism are not only couplings to gauge fields and to the spurion, but also to MSSM fields. Therefore, we consider all marginal superpotential couplings of messengers and MSSM matter that are allowed by gauge symmetry (including couplings of three messengers). They can be organized in terms of SU (5) invariants. It is easy to check that in the model under consideration such terms are of the form: 4 5 10 10, 5 10 10, 5 5 10 and 5 5 10. We assume that they are hierarchical i.e. only coupling to the heaviest family of the MSSM is of order one -the interactions with other two families are assumed to be small enough not to induce large FCNC effects. Taking into account full flavour structure of the Yukawas and messenger couplings to the first and second family is left for the future work.

Marginal couplings of messengers and MSSM matter
The part of the superpotential which contains marginal couplings of messengers Y = {Y 5 , Y 5 , Y 10 , Y 10 } is of the following form where W YYY are novel interactions of three messengers. They are crucial for the further discussion, and they will be discussed in details in the next section. On the other hand, the other two terms 3 Higgs fields are denoted by H5 and H 5 , while φ 5 and φ10 stand for the superfields containing quarks Q, U, D and leptons L, E of the third family and their superpartners. Subscripts indicate SU (5) representations. All matter superfields are collectively denoted by Φ. Abusing notation, we denote matter superfields with the same symbol as their Standard Model components. 4 All such terms which involve messengers can a priori appear while in the visible sector one needs only couplings: H5φ10φ10 and H 5 φ 5 φ10. The issue of dangerous operator φ 5 φ 5 φ10 shall be discussed in the Section 2.3.
in (2) include messenger couplings to matter fields Φ = {H 5 , H 5 , φ 5 , φ 10 } i.e. 5 are matter-matter-messenger couplings, and are matter-messenger-messenger couplings. We shall denote coupling constants h A collectively by h. Let us recall that in the extended GMSB models one usually considers only interactions of messengers Y and MSSM matter fields Φ of those two types listed above: (3) and (4). 6 Both of them, in fact, change predictions of the standard GMSB models in several ways e.g. by enhancing left-right squarks mixing, or allowing various non-standard types of NLSP/NNLSP patterns [20].
Below messenger scale M , one gets MSSM with soft terms. Gaugino masses M (r) λ are of the same form as in the GMSB models i.e. they arise at 1-loop where r = 1, 2, 3 corresponds to the gauge group U (1) Y , SU (2) L and SU ( [6,7,8]. One can find that A-terms generated by (3) and (4) are of the The numerical coefficients C A,B are given in the Table 1. The A-terms given in (6) are related to the trilinear terms in the scalar potential V in the following way: where y t,b,τ denote MSSM Yukawa couplings of the third family. The scalars Φ ∈ {H u , H d , L, E, Q, U , D} receive 2-loop corrections to soft SUSY breaking mass terms from three sources what can be written 5 For the simplicity, we assume that all superpotential coupling constants are real. 6 W ΦΦY and W ΦYY are sometimes called, respectively, type II and type I messenger couplings [8]. 7 In the discussed model gravitino is the LSP with mass m 3/2 ∼ 1 GeV. 8 The 1-loop soft masses generated by hA are negligible because of ξ/M ≪ 1 . 9 When messengers couplings to the first and second family of the MSSM matter are relevant then also R-parity violating A-terms L i L j E k and U i U j D k can appear.  Table 1: as

Marginal couplings of three messengers
The part of the superpotential which contains interactions of three messenger fields can be written as where η i are coupling constants of order one, collectively denoted by η. In the discussed framework, (10) induce extra corrections to scalar soft masses m 2 Φ Φ † Φ. As in the case of h couplings, one can obtain them using wave-function renormalization method [6,7,8]. Those new terms can be written in the following form where (3) and (4). Finally, α f 's are related to MSSM Yukawa couplings y t,b,τ : Full list of coefficients C (Φ) can be found in the Appendix B. The 2-loop contributions to soft masses displayed in eq. (11) are new results for the discussed class of extended GMSB models. The consecutive components of the sum (11) arise from 2-loop diagrams with two h and two η vertices, three h and one η vertex and from diagram with two h, one η and one y f vertex respectively. Let us mention that beside (11) also mass term m 2 mixing H d and slepton doublet L is generated by η couplings. 11 The explicit form of m 2 can be found in the Appendix B.8.
The role of η will be further analysed in the Section 3, where we focus on the simplest examples of models restricted by additional global U (1) q symmetry. That symmetry is introduced in the next section to meet phenomenological bounds.

Operators generating proton decay and µ/B µ terms
Having specified all possible superpotential couplings of messengers, it is important to know what are obstructions in getting realistic low-energy phenomenology, and how they are related to the discussed couplings (2). The obstacles one can face are e.g. rapid proton decay, absence of proper EWSB (µ/B µ problem [19]), or R-parity violating soft terms in the Lagrangian [22]. In this Section we shall comment on such dangerous operators generated by (2) at tree-and 1-loop level.
When we discussed possible messenger couplings (2), gauge invariance and renormalizability were used as the only selection rules. Hence the tree-level dimension 4 operator φ 5 φ 5 φ 10 | θ 2 could as well be present in the Lagrangian of the visible sector. However, it is well-known fact that this term would lead to rapid proton decay. So, first of all, it is necessary to ensure that it cannot appear in the superpotential. The baryon/lepton number violation would also be induced by dimension 5 operators: . The latter appears after integrating out messengers. Its source in the model under consideration is the tree-level exchange of (Y 5 , Y 5 ) messengers. It is clear that if matter-matter-messenger couplings h 2 and h 6 (cf. (3)) occur simultaneously then they generate at the tree-level the following effective operator To meet experimental bounds, at least one of these couplings must be highly suppressed such that [23]. Integrating out messengers results also in corrections to the Kähler potential which may violate baryon/lepton number. In the discussed model, only the following operator of dimension 6 is relevant for the phenomenology: If it is not suppressed then it lead to rapid proton decay, mainly in the channel p → π 0 e + . To satisfy the lower limit on the proton lifetime, h 2 has to fulfill h 2 10 −16+t M [24]. 11 After EWSB it may induce non-zero vev for sneutrino [21].
The second serious issue is the µ/B µ problem [19]. We shall assume that there is no µH 5 H 5 term in the superpotential, and mass term for Higgs superfields is generated via the following correction to the Kähler potential when F -term of spurion superfield X gets vev (c µ is a coupling constant of order one). To avoid µ/B µ problem, we require that at the same time a term of the form X † XH 5 H 5 /M 2 GU T is absent. However, µ and B µ are also generated at 1-loop by messenger-matter couplings i.e. the following corrections to the Kähler potential appear after integrating out messengers: where , which lead to µ 2 ≪ B µ when F -term of X gets vev. Therefore, to get proper electroweak symmetry breaking with µ generated by (14), one has to suppress c ′ µ,Bµ [11]. To ensure those phenomenological constraints without fine-tuning parameters, it is necessary to impose additional selection rules which restrict structure of the Lagrangian. It seems that the most handy and economical way to achieve it is to add extra global U (1) q symmetry. Such solution is realized e.g. in F-theory GUT models [20,25,26,27], and in models which use Froggatt-Nielsen mechanism to address Yukawas hierarchy problem.
In the rest of the paper, we exploit that idea, and analyse what are necessary conditions to forbid dangerous operators which were discussed above. Then we shall examine what are the lowenergy predictions of the simplified model in which U (1) q symmetry dictates the structure of the superpotential.

U(1) q symmetry
As introduced in the previous section, to get rid of rapid proton decay and the µ/B µ problem we shall invoke extra global U (1) q symmetry and appropriately choose charges of the visible sector fields Φ, messengers (Y, Y ) and the spurion X . The requirements discussed in Section 2.3 can be rephrased as follows: (a) in the superpotential of the visible sector there is no φ 5 φ 5 φ 10 term, but the standard Yukawa couplings H 5 φ 10 φ 10 and H 5 φ 5 φ 10 are present, (b) supersymmetric mass term for the Higgses is forbidden, however they couple to X † in the Kähler potential. One of the ways to satisfy those conditions is to assign the following charges to the fields: where k and l are nonequal integers such that charge of the X field (i.e. q X ) is different from 0.
Moreover, extra condition comes from spurion-messenger couplings (1). They are allowed only if the following relations hold: where R = 5, 10 denotes representation of SU (5). Hence analysis of the possible ΦΦY, ΦYY and YYY couplings boils down to inspecting charges of Y 5 and Y 10 . Let us remark that if they are set as in (16) then operator φ 5 φ 3 10 /M GU T | θ 2 is ruled out. Furthermore, condition (a) is basically equivalent to the requirement that φ 5 and H 5 have different U (1) q charges, what prevents them from mixing in the kinetic terms. It can also be checked that when these fields have different charges then (2) does not lead to R-parity violating A-term (cf. Table 1). Moreover, it should be emphasized that operators (12) and (15) are also ruled out in spite of the fact that in the effective theory, below scale M , global U (1) q symmetry is spontaneously broken by the vev of the lowest component of X.
The reason is that they would arise from couplings φ 5 φ 3 10 /X and H 5 H 5 X † /X, respectively, which are forbidden in the parent theory provided (a) and (b) are satisfied. On the other hand, U (1) q symmetry protects the proton from decay via dimension 6 operator (13) only if Finally, mass term H 5 φ 5 in the superpotential which would cause appearance of φ 5 φ 5 φ 10 operator after redefinition of matter fields should also be forbidden. To ensure this additional condition, we require that 2k + 3l = 0.
3 Phenomenology of the simplest models with YYY In this Section we study how low-energy predictions of the discussed extended GMSB models depend on messenger couplings (10), taking into account restrictions imposed by U (1) q symmetry. We shall use discussed above set of constraints (16)- (19) to select the simplest models involving marginal interactions of messengers. Analysis of more complicated cases is straightforward, and will be given elsewhere. The cases with only η couplings allowed give at the leading order the same results as the standard GMSB model with the effective number of messengers equal to 4, and will not be investigated here. It is also straightforward to check that there is no charge assignment which allows Surprisingly, there is a lot of ways to assign charges which allow for two h and one η interaction but not all of them are relevant for phenomenology. We shall examine two of them which lead to the biggest A t -terms in that class of models (cf. Table 1). Namely, we analyse models in which U (1) q admits the following couplings: (III) (h 8 , h 11 , η 2 ) or (IV) (h 8 , h 11 , η 4 ).
Charges corresponding to these cases are displayed in the third (III) and fourth (IV) row of the For the cases (I)-(IV) listed above, we adopt the initial conditions for the soft SUSY breaking terms presented in Sections 2.1 and 2.2, and compute the low-energy spectrum and the electroweak symmetry breaking with an appropriately modified SuSpect code [28]. Approximation of vanishing Yukawa couplings of the first two generations of fermions is used, so the MSSM mass spectra we obtain are degenerate for these generations. In the following, we shall call sfermions of the first and second generations with the name of the first generation (see e.g. Figure 1).
A number of constraints is imposed on the obtained mass spectra. We require that the scalar potential is bounded from below, and that there are no low lying color or charge breaking minima.
All cases with tachyons in the spectrum, Higgs boson mass smaller than 123 GeV or bigger than 127 GeV and BR(b → sγ) lying outside the 2σ range (2.87 − 4.33) × 10 −4 are discarded during the analysis. We keep only models in which the squark and gluino masses lie within the allowed 95% CL range determined for a simplified setup in Ref. [29].
In the case (I) the part of the superpotential which contains marginal couplings of the messengers is of the following form (cf. (2)): Although in such setup the left-right stop mixing would be large enough (cf. (6) and Table 1) to get m h 0 ∼ 125 GeV even for ξ ∼ 10 5 GeV, this case is not phenomenologically satisfactory as here supersymmetric mass term M ′ φ 5 Y 5 is allowed. If M ′ ∼ M GU T , then the mass of the heavy combination of Y 5 and φ 5 is of order GUT scale. As a result, soft terms generated by gauge mediation mechanism are of comparable size as those induced by gravity. Hence this case will not be analysed further.
In the case (II) only couplings h 14 and η 2 are allowed by U (1) q symmetry i.e. (cf. (2)): To get the lightest Higgs mass m h 0 ≈ 125 GeV in this simplified model, one has to set ξ ≈ 1.6 × 10 5 GeV. The fact that m h 0 is not enhanced significantly by h 14 coupling can be traced back to relatively small value of C (t) 14 in the A t -term (6). The choice of ξ results in rather large values of sparticles masses. Bino/wino/gluino masses are about 0.9/1.6/4.0 TeV, and they hardly depend on η 2 . On the other hand, masses of other neutralinos and charginos, Higgses (beside the lightest one), squarks and heavier sleptons do vary, but stay above 1.5 TeV when η 2 changes in the range 0 − 1.4 and h 14 is fixed to 1.2. However, it turns out that η 2 coupling influences NLSP/NNLSP pattern. The lighter sleptons masses are sensitive to that messenger coupling, and can be as low as 400 GeV for h 14 , η 2 ∼ 1. Figure 1 displays how masses of sleptons τ 1 , e 1 depend on the value of η 2 . Increasing η 2 above 0.4 − 0.5 changes NLSP/NNLSP pattern from B/ τ 1 to τ 1 / B and then to τ 1 / e 1 (with nearly degenerated masses) for small tan β, and from τ 1 / B to τ 1 / e 1 for moderate and large values of tan β.

Such behaviour can be explained as follows.
In this simplified model influence of marginal coupling of three messengers η 2 on mass spectrum is rather moderate because η 2 affects only soft mass of left squarks doublet Q. One can check (cf. Appendix B.5) that η 2 contribution to (8) can be written as which is always non-negative. For natural choice of coupling constants (i.e. h, η ∼ 1) it is of the same order as the following correction to (8) induced by h 14 coupling and also of the same order as the standard GMSB contribution In the framework defined by the third (III) choice of U (1) q charges, superpotential (2) is of the following form and only corrections to H u mass are generated by messenger marginal couplings: Here we choose ξ = 10 5 GeV. Bino/wino/gluino masses are about 0.55/1.03/2.71 TeV, and, as previously, they hardly depend on η 2 . The first observation is that increasing η 2 gives smaller µ at the EWSB scale.
Clearly, m 2 L,h is amplified by large tan β. It can be seen in the Figure 2 that for large value of tan β and η 2 1.4 the NLSP is the lighter stau (which is mostly left-handed). The NNLSP is tau sneutrino and masses of both particles raise when η 2 increases. Simultaneously, the lighter selectron mass drops such that for η 2 ≈ 1.4 it becomes close to the stau mass, and eventually NLSP/NNLSP pattern changes from τ 1 / ν τ to e 1 / τ 1 .
In the case (IV) superpotential is of the form and coupling of three messengers induces the following contributions to soft masses of left squarks and right up squark Their influence on the mass spectrum can be described in the following way. Bino/wino/gluino masses are about 0.55/1.03/2.71 TeV, and, as before, they hardly depend on η 4 . On the other hand, behaviour of µ is different than in previous case. In this scenario µ increases when η 4 raises.
The same effect can be seen in the Higgs sector (beside the lightest Higgs scalar) and for heavier chargino and heavier neutralinos. It turns out that to fulfill phenomenological constraints it is necessary to impose extra selection rules on (2). Otherwise rapid proton decay or µ/B µ problem can occur. We deal with those issues by invoking additional global U (1) q symmetry. The charge assignments which lead to the smallest number of allowed interaction terms were found. Using derived corrections to the soft masses (11), we have performed analysis of the phenomenology of the models involving the smallest number of marginal couplings of three messengers. The main conclusion is that in those scenarios the lightest slepton masses are the most sensitive to η couplings, which alter them not directly but only via D-term contribution to RGE running. We have shown that due to η, even for small tan β (see Figure 1 and Figure 2, left plots), a stau or selectron can be lighter than the lightest neutralino, and have masses as low as 300 − 400 GeV, which is close to recent LHC exclusion limit. Such situation is not typical for the GMSB model.
It would be worthwhile to extend analysis of the parameter space of the presented model, especially to the low M region, and investigate whether marginal couplings of three messengers are relevant for realizing radiative EWSB. Moreover, one can check if there is any common NLSP/NNLSP pattern which emerge when one considers models with more h and η couplings allowed. Finally, while we assumed that messenger-matter couplings are hierarchical, it would be interesting to investigate full flavour structure of those interactions in the discussed model.

Acknowledgments
The author would like to thank J. Gluza, J. with two h A couplings and one η i coupling which accommodate the largest A t -terms in that class of models. In the table below, the charges of the fields are written as multiplicities of the smallest charge (denoted by q 1 , q 2 , q 3 and q 4 respectively).

B Numerical coefficients in 2-loop soft masses
In this Appendix we tabulate numerical values of the coefficients C can be written as follows: Coefficients C