MSSM with $m_{h}=125$ GeV in High-Scale Gauge Mediation

After the discovery of SM-like Higgs with $m_{h}=125$ GeV, it is increasingly urgent to explore a solution to the hierarchy problem. In the context of MSSM from gauge-mediated SUSY breaking, the lower bound on gluino mass suggests that the messenger scale $M$ is probably large if the magnitude of $\Lambda\sim 100$ TeV. In this paper, we study $\overline{\mathbf{5}}+ \mathbf{5}$ model with $M\sim 10^{8}-10^{12}$ GeV and $\Lambda\simeq 100$ TeV. For moderate Higgs-messenger coupling, it will be shown a viable model with moderate fine tuning. In this model, $\mu\sim 800$ GeV and $B_{\mu}$ nearly vanishes at input scale, which can be constructed in microscopic model.


Introduction
A SM-like Higgs boson with mass m h = 125 GeV [1] has been reported by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC). Its implications in the context of minimal supersymmetric standard model (MSSM) have been extensively explored, see, e.g, [2][3][4][5][6][7][8][9][10][11][12][13][14]. Specifically, large loop-induced correction to m h is needed, this requires either large stop mass ∼ several TeV for A t = 0, or ∼ 1 TeV for maximal mixing. Since it isn't promising to examine the former case at the LHC, we focus on the later case, in which there is a large or at least moderate value of A t term ∼ 1 TeV.
In scenario of gauge-mediated supersymmetry (SUSY) breaking [16] (For a modern review, see, e.g., [17].), A term vanishes at one-loop level in models of minimal gauge mediation (GM). It is impossible to obtain large A t in virtue of renormalization group equations (RGEs) for low-scale GM, except that it receives non-vanishing input value at the messenger scale M. According to the one-loop calculation about A t [18,19], it requires that we must add new Yukawa couplings between the Higgs sector and the messengers in the superpotential [20,21].
However, a little hierarchy of A − m 2 H similar to µ − B µ is usually induced after adding new Yukawa couplings between the Higgs sector and the messengers, i.e, an one-loop A term accompanied with a large, two-loop, positive m 2 H . Large and positive m 2 H spoils the electroweak symmetry breaking (EWSB). In order to evade this problem, the messengers should couple to X in the same way as in the minimal GM. In order to generate one- suppressed contribution to m 2 Hu isn't significant. The EWSB seems impossible for this range of M.
In this paper, we continue to address GM with messenger scale M > 10 3 TeV. There are three main motivations for this study. At first, for large messenger scale it is easy to accommodate the lower bound on gluino mass. Second, it is also possible to achieve A term as required in terms of large RGE, thus providing 125 GeV Higgs boson. Finally, we will find that instead of the negative, one-loop, (Λ/M) 2 suppressed contribution, the large RG corrections to m 2 Hu due to high messenger scale take over, thus providing EWSB. We will show that for moderate value of Higgs-messenger coupling there indeed exists viable parameter space.
On the other hand, in contrast to SUSY models such as NMSSM, large value of tan β is required in order to induce m h = 125 GeV in the MSSM. In large tan β limit the four conditions of EWSB reduce to two simple requirements: an one-loop magnitude of µ term and vanishing B µ (at least at two-loop level) at the input scale, which can be constructed in microscopic model [24]. Alternatively, this type of µ − B µ is a consequence of adding µ term by hand, i.e, µ term exists in SUSY limit.
The paper is organized as follows. In section 2, we explore the model in detail. At fist, we find the parameter space composed of N, M and λ u (with Λ = 100 TeV fixed) that gives rise to m h = 125 GeV. Then we continue to discuss stringent constraints arising from EWSB, which finally sets the magnitude of µ ∼ 800 GeV. Put all these results together, we argue that the model of 5 + 5 that fills out SU(5) is viable in GM with M ∼ 10 8 − 10 12 GeV if α λu ∼ 0.02 − 0.04. Finally we conclude in section 3.

The Model
We follow the formalism of spurion superfield X = M + θ 2 F , which stores the information of SUSY-breaking hidden sector. The visible sector is the ordinary MSSM, which communicates with hidden X-sector via messenger sector. The messengers φ i ,φ i (with number of pairs N) fill out either 10 + 10 or 5 + 5 of SU(5), which can ensure grand unification still viable. The superpotential is chosen to be, As noted in the introduction and argued in [21], this choice which can be realized by global symmetry reconciles the A − m 2 Hu problem.
The total contributions to m 2 Hu at input scale M are mainly composed of two parts: with the first line in (2.2) arising from the minimal GM, and the second line in (2.2) arising from the Yukawa coupling in (2.1). Here C i (H u ) being the Casimir for H u , α r being the SM gauge couplings (r = 1, 2, 3), d H being the effective number of messengers coupled to Higgs, and C r = C r Hu + C r i + C r j , i, j referring to the messengers. This same Yukawa coupling also induces deviations to A t , etc., from that of minimal GM 2 , In what follows, we consider a type of 5+5 model 3 , in which the SU(3)×SU(2)×U (1) representations of messengers take the form: The superpotential (2.1) reads explicitly, For this model, the number of messenger pairs N = d H .
As shown in the last line of (2.2), the modifications to m 2 Hu due to Higgs-messenger coupling is controlled by d H and the magnitude of Yukawa coupling λ u . Also note that  Hµ at EW scale [21]. The authors of Ref. [20] focused on the possibility of driving negative m 2 Hu by the α 3 α λu term in (2.2).
In our case, as we have emphasized in the previous section, we consider large RG running due to high messenger scale. Therefore, the central point in our note is that the positive m 2 Hµ at the input scale is driven to be negative at EW scale by large RG correction. The parameter space is described by the following four parameters, The first three determine the input values of soft mass parameters at the messenger scale, while the last one controls the magnitudes of RG corrections when we run from M to EW scale.
By setting m h = 125 GeV, we can fix one parameter, let us chose Λ. Note that to obtain a natural EWSB it suggests Λ 100 TeV, while to generate a large gaugino mass which exceeds 1 TeV sets a lower bound on NΛ. Thus, with a specific N, Λ can be tightly constrained to be in a narrow range.
By imposing negative m 2 Hu as favored by the EWSB, one can constrain the magnitude of λ u . If λ u is rather small, it will induce too small A t term. Conversely, if it is rather large, there will be impossible to drive the m 2 Hu to be negative at EW scale. With an estimate on the range of λ u in hand, the constraint on A t at the messenger scale can be explicitly derived. In virtue of RGE for A t , one builds the connection between the value of A t required by m h = 125 GeV at the EW scale and that required by negative m 2 Hu at the scale M. This in turn determines the allowed range of M.
Put all these observations together, we can examine the EWSB in large-tan β limit in the parameter space of (2.6) favored by above requirements.

Constraints from m h = 125 GeV
The two-loop mass of Higgs boson in the MSSM reads [6,15], where X t = A t − µ cot β ≃ A t in large-tan β limit, M S = √ mt 1 mt 2 being the average stop mass, and υ = 174 GeV.
We show in fig.1 the plots of m h = 125 ± 1 GeV as function of A t with fixed Λ = 10 2 TeV for M = 10 8 GeV. The three colors represent different numbers of messenger pairs 4 . Fig.1 shows that for N = 2, 3 | A t |∼ 1.0 − 1.5 TeV at EW scale can provide 125 GeV.
As M increases to ∼ 10 12 GeV, there is no significant modification to the value of A t as required. Hu too large at input scale to be driven negative at EW scale. Actually, the positivity of stop soft masses at input scale require α λu < 0.1. On the other hand, the region λ u < 0.01 provides | A t | term too small at input scale to accommodate required value of | A t | 1 TeV at EW scale. Fig. 1 and 2 show us the possible range for α λu as 0.01 α λu < 0.1 (2.8) 4 One can examine that for Λ = 10 2 TeV, the parameter space N 1 and M ∼ 10 8 − 10 12 GeV can give rise to RGE large enough for M 3 such that at EW scale its value is larger than mass bound ∼ 1 TeV  Corresponding the range of | A t | is from several hundred GeV to ∼ 1 TeV at the input scale. Similar plots can also be found for 10 + 10. For realistic EWSB, fig.2 shows that large RGE for m 2 Hu must take over in the most range of (2.8). Now we consider the connection between the values of A t at input and EW scale in virtue of fig.1. Roughly we need a correct RGE which ensures that A t runs from a correct value at messenger scale to ∼ −1 TeV at the EW scale. Recall the beta function β At below the messenger scale for A t [23], Since the sign of β At depends on the relative magnitude of | A t | to M 3 , or concretely the magnitude of α λu to α 3 , it can be either positive or negative in of the range of (2.8).
By using (2.9) we show the RGE for A t in the cases N = 1 ( fig.3 In summary, the parameter space for Λ = 10 2 TeV which can explain the 125 GeV Higgs boson is restricted to region N > 2 and α λu ∼ 0.03 − 0.05. In the next subsection, we will explore EWSB in this narrow region, determine soft breaking masses at EW scale, and measure the fine tuning in this type of model.

EWSB
In the previous section we address the parameter space which provides m h = 125 GeV at the EW scale. In what follows, we continue to explore another question whether this parameter space induces the EWSB simultaneously. We begin with the conditions of EWSB involving soft parameters in the Higgs sector in the MSSM: which together with guarantee a stable vacuum. In the large tan β limit together with negative m 2 Hu and positive m 2 H d , (2.10) and (2.11) reduce to, The first constraint in (2.12) fixes the magnitude of µ at the input scale µ(M) through RGE for µ, and the last of which is the new constraint to be satisfied. In traditional MSSM without Yukawa coupling λ u , m 2 Hu is driven negative at EW scale through RGE. In our model, similar phenomenon appears for small α λu . Otherwise, the the input values significantly increase for m 2 Hu and decrease for stop masses squared due to large α λu , which would spoil EWSB. In fig.6 we show RGEs for m Hu and m H d for input scale M = 10 8 GeV (10 12 GeV) and α λu = 0.03 for the case N = 3. Larger value of α λu isn't viable. Following fig. 6 we obtain µ ∼ 800 GeV at EW scale. With this value of µ term, we can estimate the REG for B µ as follows. In our model, B µ ≃ 0 (at two-loop level) at the input scale, therefore the second condition of (2.12) can be trivially satisfied in virtue of REG for B µ ,

Conclusions
The discovery of SM-like Higgs with m h = 125 GeV verifies SM as a precise low-energy effective theory. It is increasingly urgent to find a solution to stabilize the mass of this scalar. At present status, SUSY is still in the short list of frameworks which can provide such a solution with some fine tunings. This paper is devoted to explore gauge-mediated SUSY with latest results reported by the LHC experiments.
The constraint of m h = 125 GeV and the lower bound on Λ needs an A t term ∼ 1 TeV. However, since the vanishing of A t soft term at input scale at one-loop level, it is impossible to obtain such large value in low-scale GM except we either introduce direct Higgs-messenger coupling or consider high messenger scale.
The situation is rather different in GM with high messenger scale (with Λ ∼ 100 TeV fixed). At first, the negative m 2 Hu required by EWSB must be realized due to large RG correction instead of the one-loop, negative, (Λ/M) 2 -suppressed contribution [21]. Second, the A t term can still be obtained with small or moderate Higgs-messenger coupling.
Finally, the large lower bound on gluino mass suggests that large messenger scale is favored. Therefore, it is necessary to explore viable GM with M > 10 7 GeV. Following these motivations, we find that a type of 5 + 5 model [21] is viable in GM with M ∼ 10 8 − 10 12 GeV and moderate value of Higgs-messenger coupling α λu . In this model, m 2 Hu is driven to be negative although it has positive input value ∼ O(1) TeV 2 . At EW scale we obtain EWSB with the magnitude of µ ∼ 800 GeV, m h = 125 GeV and M 3 > 1 TeV.
This magnitude of µ term can be either generated at one-loop level at input scale, with B µ vanishing at least at two-loop level, or considered as an input scale in SUSY limit. For the first case, we refer the reader to [24], where this type of µ and B µ terms can indeed be realized in terms of adding SM singlets to the MSSM. The latter choice makes the model complete, although it seems ad hoc to add a µ term by hand.