A Realistic Extension of Gauge-Mediated SUSY-Breaking Model with Superconformal Hidden Sector

The sequestering of supersymmetry (SUSY) breaking parameters, which is induced by superconformal hidden sector, is one of the solutions for the mu/B_mu problem in gauge-mediated SUSY-breaking scenario. However, it is found that the minimal messenger model does not derive the correct electroweak symmetry breaking. In this paper we present a model which has the coupling of the messengers with the SO(10) GUT-symmetry breaking Higgs fields. The model is one of the realistic extensions of the gauge mediation model with superconformal hidden sector. It is shown that the extension is applicable for a broad range of conformality breaking scale.


Introduction
Low-energy supersymmetry (SUSY) is a very attractive model of physics beyond the standard model (SM). In the minimal supersymmetric standard model (MSSM), however, general SUSY-breaking masses of squarks and sleptons induce too large FCNC and/or CP violation effects in low-energy observables. These SUSY FCNC and CP problems should be solved in realistic SUSY-breaking models.
Gauge-mediated SUSY breaking (GMSB) [1,2,3,4,5]  One of the difficulties in the model building of GMSB is the origin of the B µ term, which is the SUSY-breaking term corresponding to the supersymmetric mass of the MSSM Higgs doublets, µ. B µ has the mass dimension two. From viewpoints of naturalness and electroweak symmetry breaking, B µ and µ are required to be comparable to the other SUSY-breaking mass parameters in the MSSM. The correct size of µ is realized when µ is generated at one-loop level or even when the MSSM Higgs doublets are directly coupled with S in the superpotential with a small coupling (∼ 10 −(2−3) ). However, if B µ is simultaneously induced with µ, B µ is relatively enhanced by a one-loop factor. This problem is sometimes called as the µ/B µ problem. Several mechanisms are proposed for this problem [6,7,8,9].
It is pointed out in Refs. [8,9] that the µ/B µ problem is solved in the GMSB models with the superconformal hidden sector (SCHS). The conformal sequestering suppresses B µ , in addition to sfermion mass squareds m 2 f (f = q, u, d, l, e) [10], relative to the A parameters and gaugino masses. The SUSY-breaking parameters at the scale M X at which the conformality is broken are given as [9] m 2f = 0, (f = q, u, d, l, e), where In this paper we discuss the electroweak symmetry breaking under the boundary condition for the SUSY-breaking parameters given in Eq. (1). It is found that the electroweak symmetry breaking conditions have no physical solution when the messenger sector is minimal and the GUT relation among the gaugino masses is imposed. We propose an extension of the minimal messenger model in which the messenger multiplets are coupled with the GUT-symmetry breaking sector in order to avoid the introducing CP violation.
It is shown that this extension makes the model phenomenologically viable and that it is applied for arbitrary scale for M X .
The organization of the paper is as follows. In Section 2, we review the GMSB models with SCHS. We show that the minimal messenger model has no realistic vacuum with the electroweak symmetry broken. In Section 3, we propose an extension of the minimal model, which the messenger sector is coupled with the GUT-symmetry breaking Higgs VEV. Section 4 is devoted to conclusion.

GMSB Models with SCHS and Minimal Messenger Model
The gauge-mediated SUSY-breaking model with superconformal hidden sector has nontrivial prediction for the SUSY-breaking parameters at MSSM as in Eq. (1). Here, we review the derivation. See Ref. [9] for the detail.
We first discuss the gaugino and sfermion masses in the model as warming up. After decoupling of the messenger multiplets with the SM gauge quantum numbers, the following effective interactions for the gauge and matter multiplets in the MSSM with a singlet in the hidden sector S are generated, When S gets the F -term VEV, S θ 2 = F S , the first and second terms generate the gaugino and sfermion masses, respectively. The coefficients c a λ are at one-loop level while c f m 2 are at two-loop level. The explicit forms for them can be read off from formulae given in Ref. [11].
After the hidden sector enters into conformal regime at Λ ⋆ , above two terms receive huge radiative correction. At µ R (< Λ ⋆ ), the effective interactions are given as where Here, Z S is the wave function renormalization of S and R(S) is the R charge for S. When S is singlet under the hidden gauge groups, R(S) is larger than 2/3 so that Z S (µ R ) > 1.
The 1PI contribution to operators S † S is parametrized by α S in the above equation.
The gaugino masses M a at M X , at which the conformality is broken, are given as When α S > 0, the sfermion masses are suppressed, and the conformal sequestering is realized as [10] m 2 f = 0, (f = q, u, d, l, e).
Next, let us move to the Higgs sector. Here, the messenger sector is assumed to be minimal among models where the µ term is generated by one-loop diagrams. Then, the messenger multiplets are embedded in SU(5) 10 and 10 ⋆ -dimensional multiplets. 1 The messenger multiplets have an interaction with the Higgs doublets H u and H d in the superpotential, where Q m , U m , and E m (Q m ,Ū m , andĒ m ), which come from the SU(5) 10 (10 ⋆ ) multiplet, have SU(5) symmetric mass and interaction terms.
Integration of the messenger sector leads to the effective interactions of the Higgs doublets with S as Here, the coefficients of the operators, c µ , c Bµ , c Au , and c A d are generated at one-loop level, while c Hu m 2 and c H d m 2 are vanishing at one-loop level.
After the hidden sector enters into conformal regime, the effective interactions become The terms proportional to (Z |S| 2 −1) come from diagrams with the Higgs doublet exchange.
Since the tree-level diagrams with the Higgs exchange do not contribute to the effective Lagrangian, one is subtracted from Z |S| 2 there. Therefore, the SUSY-breaking terms in the Higgs sector at M X are In the derivation of Eq. (11), we redefined the Higgs doublets as Since we now derived the SUSY-breaking terms in the minimal messenger model, Eq. (12) satisfies the relationship |A Hu A H d | = |µ| 2 . In Appendix we give formulae for the SUSY-breaking terms of the Higgs sector in more general messenger cases. Now we discuss the electroweak symmetry breaking in the GMSB models with SCHS.
The minimization condition of the Higgs potential at tree level results in where m 2 1 ≡ (m 2 H d + µ 2 ) and m 2 2 ≡ (m 2 Hu + µ 2 ). In the GMSB models with SCHS, the Higgs boson mass squareds are zero at tree level even after including the supersymmetric mass µ. Thus, the electroweak symmetry breaking and the stability of the Higgs boson potential are sensitive to the radiative corrections to them.
In Fig. 1  It is found from Fig. 1 that m 2 A is always negative. This implies that the vacuum is not stabilized. In order to qualitatively understand this result, we derived approximation of the mass parameters in the Higgs potential from the renormalization-group equations (RGE) as where t SU SY = log(M X /m SU SY )/2π, α t (≡ y 2 t /4π) and α a (a = Y, 2, 3) are for the topquark Yukawa and gauge coupling constants, respectively. Here, we include the one-loop contributions due to the electroweak and top-quark Yukawa interactions and two-loop contributions due to the strong one. The later one is comparable to the one-loop terms when the gluino mass is larger than others, as in the GUT relation. These equations are semi-quantitatively valid when α a t SU SY , α t t SU SY ≪ 1. Even when α a t SU SY , α t t SU SY ∼ O(1), we can guess the qualitative behaviors, such as relative signs and sizes among the terms, using the equations.
It is found that A Hu/H d are negative in Eq. (12). This implies that the one-loop contributions to B µ /µ are constructive. Sizable values of B µ /µ lead to suppression of µ/M 3 from Eq. (13) for tan β > ∼ 1. In those cases the two-loop contribution, which enhanced by the gluino mass, derives m 2 A to be negative. When tan β ≃ 1, µ/M 3 ∼ 1 is possible. However, it is found from the figure that m 2 A is still negative. One of the solutions for the problem is introduction of non-zero B µ at M X . If Z |S| 2 (M X ) is accidentally around O(10 −(2−3) ), B µ keeps its sizable value at M X . However, its sign is positive relatively to µ since This is constructive to the RGE contribution to B µ , while the deconstructive interference is rather required for the electroweak symmetry breaking. This is also noticed in Ref. [12].
If the operator S † S is mixed with other operators whose D-component VEVs are nonvanishing, the sign of the contribution to B µ may be changed.
The second solution is extension of the messenger sector. When introducing multiple messengers with different supersymmetric masses and couplings with S, the deconstructive interference in B µ is possible. However, arbitrary introduction of the messengers leads to CP phases in the SUSY-breaking parameters. That is not favored from phenomenological viewpoints.

Extension
One of the extensions of the GMSB with SCHS without introducing CP violation is introduction of coupling of the messengers with the GUT-symmetry breaking Higgs fields.
Let us consider following superpotential; Here, ψ andψ are the messengers and Σ is the GUT-symmetry breaking Higgs fields.
The messengers are 10 and 10 ⋆ -dimensional multiplets in the SU(5) GUTs, and 16 and 16 ⋆ in the SO(10) GUTs.
It is found that this extension does not work well in the SU(5) GUTs. When the SU (5) breaking Higgs field is a 24-dimensional multiplet, the messenger masses are proportional to their hypercharges so that the bino mass is zero at one-loop level. When the SU (5) breaking Higgs field is a 75-dimensional multiplet, the SU(2) doublet messenger quark and singlet messenger quark masses are degenerate with the opposite sign. Then, µ and A parameters are zero at one-loop level.
These problems are resolved when the messenger masses are generated by the higherdimensional operators with Σ. In those cases the colored messengers are relatively lighter so that the gluino becomes heavier. From the electroweak symmetry breaking condition in Eq. (14), which is reduced m 2 Z ≃ −2m 2 2 for tan β > ∼ 1, larger M 3 leads to larger µ. However, m 2 A (≃ m 2 1 − 1/2m 2 Z ) is likely to be tachyonic due to large µ. Thus, we consider the SO(10) GUTs. Here, we assume that Σ is a 45-dimensional multiplet. The messenger masses are given by hypercharge Q Y and (B − L) charge of the In the following, we consider a case where only the SU(5) 10 and 10 ⋆ -dimensional components of the 16 and 16 ⋆ -dimensional multiplets become effective in generation of the SUSY-breaking terms in the MSSM. This is only for simplicity, because when the SO(10) full multiplets contribute to SUSY-breaking mediation, the M Y /M B−L dependence of soft breaking parameters is more complicated. Actually, it is realized when an SO (10) 10-dimensional multiplet is introduced in the messenger sector. In that case, we can add following terms to the superpotential, where ψ H (ψ H ) are 16(16 ⋆ )-dimensional multiplets and φ is a 10-dimensional matter mul- viable. We studied the other regions. However, though we found points to be consistent with the electroweak symmetry breaking conditions, their spectrums are quite light so that they are experimentally excluded.
We show mass spectra and the branching ratio BR(b → sγ) at several points, M X = 10 8 , 10 11 , and 10 14 GeV, in Table 1 using SuSpect 2.41 [13] and SusyBSG 1.1.2 [14]. All of them are consistent with the Higgs boson mass bound, sparticle mass bounds [15] and branching ratio of b → sγ [16]; In all sample points, the right-handed slepton masses are very small compared with other sparticle masses. As we have seen in Eq. (1), the scalar fermion soft masses are nearly zero at M X . In addition, when y > 0, the bino is light compared to the wino and gluino. As a result, the right-handed slepton masses are such small in this model.
Using SuSpect 2.34, we also calculated supersymmetric contributions to the anomalous magnetic moment of the muon, a µ = (g − 2) µ /2. The comparison between the measurements [17] and the SM theoretical predictions [18] for a µ is The left-handed sleptons are so heavy that the SUSY contribution to a µ is suppressed.
When the deviation is confirmed in future, this model would be disfavored.

Conclusion
We have studied the electroweak symmetry breaking for the GMSB models with SCHS which solve the µ/B µ problem by the conformal sequestering. It is found that the correct electroweak symmetry breaking is not derived in the minimal messenger model with GUT relation among the gaugino masses.
In this paper we also propose an extension of the minimal model which has the coupling of the messengers with the SO (10)    The µ, B µ , and A terms are given as