Light Higgsino and Gluino in $R$-invariant Direct Gauge Mediation

We provide a simple solution to the $\mu$-$B_\mu$ problem in the"$R$-invariant direct gauge mediation model". With the solution, the Higgsino and gluino are predicted to be light as $\mathcal{O}(100)$GeV and $\mathcal{O}(1)$TeV, respectively. Those gluino and Higgsino can be accessible at the LHC and future collider experiments. Moreover, dangerous dimension five operators inducing rapid proton decays are naturally suppressed by the $R$-symmetry.


Introduction
Models of gauge mediated supersymmetry breaking (GMSB) [1][2][3] 1 are very attractive, since dangerous flavor violating processes are naturally suppressed: soft supersymmetry (SUSY) breaking masses of sleptons and squarks are generated via gauge interactions, and hence, they are flavor-blind.
Among GMSB models, "R-invariant direct gauge mediation model" constructed in Refs. [9,10] (see also [11,12] for recent discussions) is highly successful, since the SUSY breaking minimum is stable. The model has an (spontaneously broken) R-symmetry, which may suppress dangerous proton decay operators. This model is also interesting from the view point of phenomenology. The gaugino masses are suppressed compared to sfermion masses even though the R-symmetry is spontaneously broken. These relatively light gauginos can be seen at future large hadron collider (LHC) experiments. On the other hand, squark masses mq including stop masses are O(10) TeV and the observed Higgs boson mass of 125 GeV [13,14] is easily explained with large radiative corrections from heavy stops [15][16][17][18][19].
The important remaining issue in this model is the µ -B µ problem [20][21][22][23][24][25][26][27][28][29]: if µ and B µ terms are generated dynamically, it usually predicts µ 2 B µ ∼ m 2 H u,d , where m Hu (m H d ) is a soft SUSY breaking mass for the up-type (down-type) Higgs. With the hierarchy of µ 2 and B µ , it has been considered to be difficult to realize the correct electroweak symmetry breaking (EWSB) for mq = O(0.1 -1) TeV. The situation changes for mq 10 TeV, since the hierarchy itself may not be a problem anymore.
The bare µ-term needs to be prohibited. If the bare µ-term is allowed by the Rsymmetry, the dimension five proton decay operators are also allowed by the symmetry under the assumption that the grand unified theory (GUT) exists. 2 These dimension five operators cause unacceptably rapid proton decays unless the soft SUSY breaking mass scale is extremely high as ∼ 10 10 GeV [30]. The µ -B µ problem might be related to the rapid proton decay problem.
In the minimal GMSB model, it has been shown that the µ -B µ problem is solved in a simple and naive way with a slight modification of the GUT relation among messenger masses for µ ∼ 100 GeV and |B µ | ∼ mq ∼ 10 TeV [31]. 3 In this letter, we point out that the µ -B µ problem is also solved in the R-invariant direct gauge mediation model in this way. With the solution, the Higgsino as well as the gluino is predicted to be light, which has a large impact on LHC and International linear collider (ILC) SUSY searches. We also point out that the violation of the GUT relation is not needed for the solution in this model. First, let us briefly review the R-invariant direct gauge mediation model. The model has a spontaneously broken R-symmetry, which suppresses gaugino masses compared to sfermion masses. The superpotential of the messenger sector is where Ψ and Ψ (Ψ andΨ ) are the messenger fields transformed as 5 (5) in SU (5) GUT gauge group. The above superpotential is invariant under U (1) R symmetry with the Rcharges of Q(Z) = Q(Ψ ) = Q(Ψ ) = 2 and Q(Ψ) = Q(Ψ) = 0. We assume Z has vacuum expectation values, which breaks R-symmetry and SUSY as where F Z = µ 2 Z . The R-symmetry is spontaneously broken by φ Z = 0. Such spontaneous breaking of the R-symmetry can be achieved in O'Raifeartaigh like models at tree-level [33][34][35] or one-loop level [36,37] if there exists a field with R-charge other than 0 or 2. Also, the spontaneously breaking can occur at the higher loop level [38][39][40][41]. In this paper, we do not specify the origin of the spontaneous R-symmetry breaking and take φ Z as a free parameter.
The messenger superfields, Ψ, Ψ ,Ψ andΨ , are decomposed as Ψ = Ψ D + ΨL,Ψ = ΨD + Ψ L , where Ψ where all parameters are taken to be real positive without loss of generality. For simplicity, further, we take M 1L = M 2L ≡ M L and M 1D = M 2D ≡ M D in the following discussions. Accordingly, the messenger sector are parametrized by the following five parameters: where In the case that c L = c D and M L = M D are satisfied at the GUT scale, r L and R L are fixed as r L ≈ R L ≈ 1/1.4 [10].
After integrating out the messenger fields, gauginos and sfermions obtain soft SUSY breaking masses. The gaugino masses are estimated as where F D and F L are numerical coefficient of O(0.1) (see [10] for complete formulae). Note that the gaugino masses are suppressed by factors, (Λ SUSY /M mess ) 2 and F L,D . On the other hand, the sfermion masses are not suppressed by the factor, and they are approximately given bỹ where

Generation of µ/B µ terms
Next, we introduce messenger-Higgs couplings to generate µ and B µ -terms. The relevant part of the superpotential is given by where S andS are gauge singlet superfields with the R-charge assignment, Q(S)+Q(S) = 0. Here, Q(H u ) + Q(H d ) = 4 and the bare µ term, µH u H d , is not allowed by U (1) R symmetry. 4 Also, a dangerous dimension five proton decay operator, 10 10 105, is prohibited by the symmetry. So far we have eight free parameters in this model: where Λ SUSY , M mess , R, r L and R L are defined in the previous subsection, and Integrating out the messenger fields, S andS, the µ-parameter and soft SUSY breaking mass parameters are generated as With a particular choice of R-charges, the seesaw mechanism can be incorporated.
for R = r L = R L = 1, R S = 7 and Λ SUSY /M mess = 0.95. The analytic forms of Eq. (10) can be found in Appendix A.
The above µ and B µ must satisfy conditions for the EWSB. The conditions are given by where m Z is the Z boson mass and tan β is a ratio of the VEVs, v u /v d ; ∆V is one-loop corrections to the Higgs potential. The Higgs soft masses and ∆V are evaluated at the stop mass scale, M stop . The µ-parameter is roughly estimated as where (m 2 H u,d ) GMSB are contributions from gauge mediation in Eq. (7), and (∆m 2 Hu ) rad contains radiative corrections from stop and gluino loops and is negative. Since |(m 2 Hu ) GMSB | |(∆m 2 Hu ) rad |, µ-parameter determined by the EWSB conditions is larger than O(0.1)M stop in usual GMSB models. However, in our model, the small µ-parameter is obtained with sizable δm 2 Hu , i.e. Eq.(10) and Eq.(11) are consistently satisfied.

Results
In this section, we discuss the mass spectra of SUSY particles and survey the parameter region where the mass of observed Higgs boson and the EWSB are correctly explained.

SM-like Higgs mass
First, we estimate the mass of the lightest CP-even neutral Higgs boson, m h 0 . Figure 1 shows the value of m h 0 on (Λ SUSY , tan β) plane with the other parameters fixed. Here we compute mass spectra of SUSY particles using softsusy-4.0.1 [44] with appropriate modifications and then m h 0 is estimated using SUSYHD [45]. In the left (right) figure, we

Electroweak symmetry breaking
We next check whether the EWSB conditions are correctly satisfied. For this purpose, we solve the EWSB conditions (Eq. (11)) using softsusy-4.0.1 and compare the solutions with µ and B µ in Eq. (10). Figure 3 shows the difference between our predictions (µ and , respectively. It should be noted that the difference between |µ| and |µ EWSB | is very sensitive to k u . In other words, we need a fine-tuning of k u to find the parameter region where the prediction of µ is consistent with the EWSB conditions.

Mass spectra in some benchmark points
Finally, we show the typical mass spectra in our model. Here, we pick up four benchmark points shown in Table 1 and O(1)TeV and they can be good targets for the forthcoming collider experiments.
In our model, the lightest SUSY particle (LSP) is always gravitino. Typical gravitino mass is estimated as where M pl 2.4 × 10 18 GeV denotes the reduced Planck mass. 5 With the above gravitino mass, the next to the lightest SUSY particle behaves as a stable particle in collider time scale.

Conclusion and discussion
We have provided a simple solution to the µ-B µ problem in R-invariant direct gauge mediation. In contrast to the case of minimal gauge mediation shown in Ref. [31], the solution works even when the GUT relations among the parameters in the messenger sector are satisfied. 5 Provided that the R-symmetry is explicitly broken by a constant term in the superpotential, the mass of the R-axion is given by  The Higgsino is predicted to be light as ∼ 100 -500 GeV with the solution. Since the gravitino is expected to be heavier than 10 -100 keV, the lightest neutralino, which is Higgsino-like, is stable inside a detector. This light Higgsino is a good target at the LHC [46][47][48][49][50][51] and ILC [52]. The gluino is also likely to be light as 2 -3 TeV, which can be tested at the future LHC experiment [53]. Moreover, the dangerous dimension five operators inducing rapid proton decays are naturally suppressed by the R-symmetry.

JP17H02875 (N.Y.), and 16H06490 (R.N.).
A Analytic formulae for µ/B µ -term, A-terms and m 2 H u,d In this appendix, we give analytic formulae for µ/B µ -term, A-terms and m 2 H u,d at one-loop level. The definition for these parameters is the same with that in Ref. [31].
To begin with, we summarize the mass eigenstates of the messenger fermions and sfermions. After the spontaneous SUSY and U (1) R symmetry breaking, the mass matrices for messenger lepton and slepton, m L and m 2 L , are given by These mass matrices are diagonalized by orthogonal matrices U , V and V as with m L i (i = 1, 2) and m 2 L i (i = 1, 2, 3, 4) being real and non-negative. The mass matrices for messenger quark/squark can be diagonalized in the same way. Now we are ready to calculate µ/B µ -term and soft SUSY breaking parameters. After integrating out messenger fields, S and S, we find where Here F , F and A denote the finite one-loop functions which are defined as A(m 2 ) = −m 2 ln m 2 , with m S = c S φ Z , m 2 S 1 = m 2 S − c L µ 2 Z , m 2 S 2 = m 2 S + c L µ 2 Z and