MSSM with mh=125\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\mathrm{h}=125$$\end{document} GeV in high-scale gauge mediation

After the discovery of an SM-like Higgs with mh=125\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\mathrm{h}=125$$\end{document} 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 the gluino mass suggests that the messenger scale M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} is probably large if the magnitude of Λ∼100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \sim 100$$\end{document} TeV. In this paper, we study the 5¯+5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbf {5}}+ \mathbf {5}$$\end{document} model with M∼108-1012\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\sim 10^{8}{-}10^{12}$$\end{document} GeV and Λ≃100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \simeq 100$$\end{document} TeV. For moderate Higgs–messenger coupling, a viable model will be shown with moderate fine tuning. In this model, μ∼800\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \sim 800$$\end{document} GeV, and Bμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\mu }$$\end{document} nearly vanishes at the input scale, which can be constructed in a microscopic model.


Introduction
A SM-like Higgs boson with mass m h = 125 GeV [1,2] has been reported by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC). Its implications in the context of the minimal supersymmetric standard model (MSSM) have been extensively explored, see, e.g, [3][4][5][6][7][8][9][10][11][12][13][14][15]. Specifically, a large loop-induced correction to m h is needed, and this requires either a large stop mass, ∼ several TeV for A t = 0, or ∼1 TeV for maximal mixing. Since it is not promising to examine the former case at the LHC, we focus on the latter case, in which there is a large or at least moderate value for the A t term: ∼1 TeV.
In the scenario of gauge-mediated supersymmetry (SUSY) breaking [17][18][19][20][21][22][23] (for a recent review, see, e.g., [24]), the A term vanishes at one-loop level in models of minimal gauge mediation (GM). It is impossible to obtain a large A t in virtue of the renormalization group equations (RGEs) for low-scale GM, except that it receives a non-vanishing input value at the messenger scale M. According to the one-loop calculation of A t [25,26], it is required that we must add new Yukawa couplings between the Higgs sector and the messengers in the superpotential [27,28]. a e-mail: sibozheng.zju@gmail.com 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., a one-loop A term accompanied with a large, two-loop, positive m 2 H . A 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-loop A t , Yukawa couplings between (some of) the messengers and the doublet H u are added to the superpotential. In terms of imposing an additional global symmetry, the coupling of the doublet H d to the messengers can be forbidden. Together with a one-loop, negative, ( /M) 2  In this paper, we continue to address GM with messenger scale M > 10 3 TeV. There are three main motivations for this study. First, for a large messenger scale it is easy to accommodate the lower bound on the gluino mass. Second, it is also possible to achieve an A term as required in terms of large RGE, thus providing a 125 GeV Higgs boson. Finally, we will find that instead of the negative, one-loop, ( /M) 2 -suppressed contribution, the large RG corrections to m 2 H u due to the high messenger scale take over, thus providing EWSB. We will show that for a moderate value of the Higgs-messenger coupling there indeed exists a viable parameter space.
On the other hand, in contrast to SUSY models such as NMSSM, a large value of tan β is required in order to induce m h = 125 GeV in the MSSM. In the large tan β limit the four conditions of EWSB reduce to two simple requirements: a one-loop magnitude of the μ term and vanishing B μ (at least at two-loop level) at the input scale, which can be constructed in a microscopic model [32]. Alternatively, this type of μ-B μ combination is a consequence of adding the μ term by hand, i.e., a μ term exists in the SUSY limit.
The paper is organized as follows. In Sect. 2, we explore the model in detail. First, we find the parameter space composed of N , M, and λ u (with = 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. Putting 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 λ u ∼ 0.02−0.04. Finally, we conclude in Sect. 3.

The model
We follow the formalism of spurion superfields, X = M + θ 2 F, which stores the information of the 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 that grand unification still viable. The superpotential is chosen to be As noted in the introduction and argued in [28], this choice, which can be realized by global symmetry, reconciles the A-m 2 H u problem. The total contributions to m 2 H u 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 ) is the Casimir invariant for H u , α r are the SM gauge couplings (r = 1, 2, 3), d H the effective number of messengers coupled to the Higgs, and C r = C r H u + C r i + C r j , i, j, referring to the messengers. The same Yukawa coupling also induces deviations to A t , etc., from that of minimal GM 2 , In the following, we consider a type of 5 The superpotential (2.1) explicitly reads (2.5) For this model, the number of messenger pairs is N = d H .
As shown in the last line of (2.2), the modifications to m 2 H u due to the Higgs-messenger coupling is controlled by d H and the magnitude of the Yukawa coupling λ u . Also, we note that the one-loop, negative, ( /M) 2 -suppressed contribution is tiny in comparison with those in (2.2) for /M 1/4π . For low-scale gauge mediation, the appearance of the oneloop negative ( /M) 2 -suppressed contribution guarantees that we obtain a negative m 2 H μ at the EW scale [28]. The authors of Ref. [27] focused on the possibility of driving m 2 H u negative by the α 3 α λ u term in (2.2). In our case, as we have emphasized in the previous section, we consider large RG running due to a high messenger scale. Therefore, the central point in our note is that the positive m 2 H μ at the input scale is driven negative at the EW scale by a large RG correction.
The parameter space is described by the following four parameters: (2.6) The first three determine the input values of the soft mass parameters at the messenger scale, while the last one controls the magnitudes of RG corrections when we run from M to the EW scale. By setting m h = 125 GeV, we can fix one parameter; let us choose . Note that to obtain a natural EWSB it is suggested that 100 TeV, while to generate a large gaugino mass which exceeds 1 TeV a lower bound on N is set. Thus, with a specific N , can be tightly constrained to be in a narrow range.
By imposing negative m 2 H u as favored by the EWSB, one can constrain the magnitude of λ u . If λ u is rather small, it will induce too small an A t term. Conversely, if it is rather large, it will be impossible to drive m 2 H u negative at the EW scale. With an estimate on the range of λ u at 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 H u at the scale M. This in turn determines the allowed range of M.
Putting all these observations together, we can examine the EWSB in the large-tan β limit in the parameter space of (2.6) favored by the above requirements.

Constraints from m h = 125 GeV
The two-loop mass of the Higgs boson in the MSSM reads [7,16] where X t = A t −μ cot β A t in the 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 a function of A t with fixed = 10 2 TeV for M = 10 8 GeV. The three colors represent different numbers of messenger pairs. 4 Figure 1 shows that for N = 2, 3 | A t |∼ 1.0−1.5 TeV at the EW scale one is provided with 125 GeV. As M increases to ∼10 12 GeV, there is no significant modification to the value of A t , as required.
We show in Fig. 2 plots of m 2 H u at the input scale as functions of λ u with different N s for = 100 TeV and M = 10 8 GeV. A negative m 2 H u can be obtained in the range 0.02 α λ u 0.04 for N = 1. On increasing N the allowed range will shrink. In particular, only the neighborhood of α λ u 0.02 is possible for N = 2. Similar plots can be found for M = 10 12 GeV. Following Fig. 2 one observes that the region α λ u > 0.1 induces m 2 H u to be positive and too large at the input scale to be driven negative at the EW scale. Actually, the positivity of the stop soft masses at the input scale require α λ u < 0.1. On the other hand, the region λ u < 0.01 provides an | A t | term too small at the input scale to accommodate the required value of | A t | 1 TeV at the 4 One can examine that for = 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 the mass bound ∼1 TeV [29,30] reported by the LHC experiments. Hu can be obtained in the range 0.02 α λu 0.04 for N = 1. On increasing N the allowed range will shrink. In particular, only the neighborhood of α λu 0.02 is possible for N = 2. Similar plots can be found for different choices of M EW scale. Figures 1 and 2 show the possible range for α λ u as 0.01 α λ u < 0.1. (2.8) The corresponding 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 a large RGE for m 2 H u must be taken in most of the range of (2.8).
Now we consider the connection between the values of A t at input and the 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 A t below the messenger scale for A t [31], Since the sign of β A t depends on the magnitude of | A t | relative to M 3 , or concretely the magnitude of α λ u relative to α 3 , it can be either positive or negative in the range of (2.8). By using (2.9) we show the RGE for A t in the cases N = 1 (Fig. 3), N = 2 (Fig. 4), and N = 3 (Fig. 5), from which the dependence on α λ u and M is obvious. In particular N = 1 is excluded (as shown from Fig. 3) regardless of the values of M and α λ u . α λ u < 0.05 is excluded for the case of N = 2. In the case of N = 3, the parameter space allowed is possible for the whole range of M ∼ 10 8 −10 12 GeV and for α λ u ∼ 0.03−0.05. In summary, the parameter space for = 10 2 TeV which can explain the 125 GeV Higgs boson is restricted to the region N > 2 and α λ u ∼ 0.03−0.05. In the next subsection, we will explore EWSB in this narrow region, determine the soft breaking masses at EW scale, and measure the fine tuning in this type of model.   Fig. 6 we obtain μ ∼ 800 GeV at EW scale. With this value of the 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 the REG for B μ , For M = 10 8 (10 12 ) GeV, substituting μ ∼ 800 GeV (from Fig. 6) into (2.13) and using the REGs for A t (Fig. 5) and M 1,2 , one obtains the RGE for B μ term in each case 5 , as shown by the green plots in Fig. 6. One finds that for small input scale M = 10 8 GeV, B μ ∼ (10 GeV) 2 . As we adjust the value of M to 10 12 GeV, the RG correction to B μ increases to (500 GeV) 2 . Finally, we use the traditional definition c = max{c i } to measure fine tuning in the model, where c i = ∂ ln m 2 Z / ∂ ln m 2 i , m 2 i being the soft breaking mass squared. For the mass spectrum in Fig. 6, the main contribution to the large value of c arises from the stop masses, the gaugino masses, and the μ term. Typically we have c 150-200, which suggests that this type of model is moderately fine tuned.

Conclusions
The discovery of a SM-like Higgs boson with m h = 125 GeV verifies the SM as a precise low-energy effective theory. It is increasingly urgent to find a solution to stabilize the mass of this scalar. The present status is that SUSY is still in the short 5 It is crucial to note that the sign of B μ must be positive, even if its absolute value is rather small in comparison with the magnitude of (m 2 H d − m 2 Hu ).
list of frameworks which can provide such a solution with some fine tunings. This paper is devoted to an exploration of gauge-mediated SUSY with the latest results reported by the LHC experiments. The constraint of m h = 125 GeV and the lower bound on need an A t term ∼1 TeV. However, since the A t soft term at the input scale at one-loop level vanishes, it is impossible to obtain such a large value in low-scale GM except when we either introduce a direct Higgs-messenger coupling or consider a high messenger scale.
The situation is rather different in GM with a high messenger scale (with ∼ 100 TeV fixed). First, the negative m 2 H u required by EWSB must be realized due to the large RG correction instead of the one-loop, negative, ( /M) 2suppressed contribution [28]. Second, the A t term can still be obtained with a small or moderate Higgs-messenger coupling. Finally, the large lower bound on the gluino mass suggests that a 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 [28] is viable in GM with M ∼ 10 8 −10 12 GeV and with a moderate value of the Higgs-messenger coupling α λ u . In this model, m 2 H u is driven negative, although it has a positive input value, ∼ O(1) TeV 2 . At the EW scale we obtain EWSB with the magnitude of μ ∼ 800 GeV, m h = 125 GeV, and M 3 > 1 TeV.
This magnitude of the μ term can be either generated at one-loop level at the input scale, with B μ vanishing at least at two-loop level, or it can be considered as an input scale in the SUSY limit. For the first case, we refer the reader to [32], 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.