On Dark Matter Selected High-Scale Supersymmetry

The prediction for the Higgs mass in the dark matter selected high-scale SUSY is explored. We show the bounds on SUSY-breaking scale in models of SM $+\tilde{w}$ and SM $+\tilde{h}/\tilde{s}$ due to the observed Higgs mass at the LHC. We propose that effective theory below scale $\tilde{m}$ described by SM $+\tilde{w}$ is possibly realized in gauge mediation with multiple spurion fields that exhibit significant mass hierarchy, and that by SM $+\tilde{h}/\tilde{s}$ can be realized with direct singlet-messenger-messenger coupling for singlet Yukawa coupling $\lambda\sim(v/\tilde{m})^{1/2}g_{\text{SM}}$. Finally, the constraint on high-scale SUSY is investigated in the light of inflation physics if these two subjects are directly related.


Introduction
In particle physics the interest on supersymmetry (SUSY) is based on four main reasons: i) solution of the naturalness problem, ii) successful gauge coupling unification, iii) viable thermal dark matter (DM) candidate with mass near weak scale, iv) ingredients of string theory. The so called low-scale SUSY addresses i)-iii). However, the first run of LHC shows bad prospect for conventional low-scale SUSY. In contrast to low-scale SUSY, only iv) is addressed in the "minimal " high-scale SUSY, it is because of that in "minimal " high-scale SUSY all superpartner masses are far above the weak scale, and the connection between weak and SUSY breaking scale is lost. But iii) should be addressed in any realistic model, and this can be done in two classes of "non-minimal " high-scale SUSY 1 .
The first one is named as "split" SUSY [1], in which the scalar superpartners masses are far above the weak scale, but all fermionic superpartners including gauginog and higgsinosh u andh d are light due to the protection of R symmetry. In this class of high-scale SUSY DM is identified as the lightest supersymmetric particle. The observed Higgs mass in high precision [2], by virtue of two-loop RGEs and one-loop threshold corrections, suggests that the scale of SUSY breaking m ≤ 10 8 GeV [3,4] when scalar superpartner threshold corrections are not very large. Above the scalem the physical states are described as the minimal supersymmetric standard model (MSSM), while below it described as standard model (SM) +g +h.
The other class was firstly studied in [5], in which R-symmetry breaking isn't suppressed, and either some new parity instead of R symmetry keeps higgsino (h) and singlino (s) light or there exists a light winow DM due to environmental selection. In the former case, the singlino state is needed because of that pure higgsino DM isn't viable. So below the scalem the physical states are described as SM+h/s (SM+w) when DM is mixing state ofh ands (w-like). The observed Higgs mass, similar to the analysis performed in the first class, can be used to constrain the scale of SUSY breakingm. Since the region of model parameters discussed in [5] corresponds to Higgs mass of order 127 − 142 GeV in model SM+w and 141 − 210 GeV in model SM+h/s (see Table 4 therein) it is necessary and also interesting to revise the Higgs mass in such DM selected high-scale SUSY. This is the aim of this paper. In particular, instead of takingm = 10 14 GeV and large tan β limit as in [5] ,m will be considered as a free parameter in this paper, and region of small tan β will be covered also.
In section 2, similar to Split SUSY [1] we discuss the two-loop RGE for Higgs quartic coupling λ and one-loop RGEs for gauge couplings and other Yukawa couplings, and threshold correction to λ due to heavy SUSY particles will be parameterized for prediction for Higgs mass. In section 3, we estimate prediction for the Higgs mass M h in models of SM+w and SM+h/s, with uncertainty for M h due to uncertainty of top quark mass and threshold correction. Finally, we conclude in section 4. RGEs for parameters related to Higgs mass are presented in appendix A.

RGEs and Threshold Corrections
Inspired by the case of split SUSY [3], the two-loop RGEs for SUSY model parameters in SM+w and SM+h/s can be similarly derived, and the results are presented in appendix A. In appendix A, we show the results in SM+w, SM+h/s and split SUSY simultaneously, in order to illustrate the differences among them. A few comments are in order. i), As the Higgs mass is directly related to λ, we consider the RGE for λ at two-loop order but for the others only at one-loop order. ii), The one-loop beta functions for gauge couplings can be derived by either following [7,8] or taking the insights of Weinberg [9]. iii), An additional parameter g λ appears in SM+h/s (see Eq.(3.1)), in compared with split SUSY and SM+w, but it affects RGEs of SM gauge (Yukawa) couplings only at two-loop (one-loop) order. Hence, we include one-loop effects due to Yukawa g λ in RGE for SM Yukawa g t .
The value of Higgs quartic coupling at scalem, λ(m), is determined by the SUSY boundary condition, up to threshold correction δλ arising from heavy scalar and fermionic superpartners. Solving the RGE for λ, one obtains the electroweak (EW) scale Higgs mass M h = 2λv 2 , v = 174 GeV.
Threshold corrections related to our models at one-loop level have been considered in [4,6] and at the two-loop level in [6]. In the next section, we consider this model parameter in the range | δλ |≤ 0.03, which is sufficient to cover the uncertainty of theoretic value of δλ in high-scale SUSY. . It indicates that in split SUSY λ and the prediction for Higgs mass at EW scale is roughly the largest among the three models for same tan β and δλ, except that the correction due to RGE effects is large enough to violate above expectation.

Higgs Mass
In this section, we estimate the prediction for the Higgs mass. Similar to the case of split SUSY [4], we use the updated experimental values of top quark mass M t = 173.3 ± 0.76 GeV [11] and QCD coupling α 3 (M Z ) = 0.1184±0.0007 [12] for our analysis. The measured value for the Higgs mass, M h = 125.15 ± 0.25 GeV is obtained from a naive average of the ATLAS and CMS results [2]. As the Higgs mass is rather sensitive to top Yukawa, the dominant one-loop QCD corrections to top Yukawa δg t ≃ −0.065 [3,10] will be applied to the prediction.

SM+w
In model SM+w the model parameters are wino mass mw andm. Parameter mw is constrained by the DM relic abundance, from which mw ∼ 2 TeV [5]. Parameterm relates to the boundary value for the Higgs mass at high energy scale, and thus the measured Higgs mass is sensitive to it. Compare our prediction for Higgs mass in Fig. 2 with previous result in [5,6]. M h approaches to ∼ 140 GeV for large value of tan β, which is consistent with the prediction of M h ≃ 141 − 142 GeV in [5,6]. On the other hand, the prediction for Higgs mass should be similar to the minimal high-scale SUSY studied in [4], because the deviation from SM is smaller than split SUSY. shown that the uncertainty of M h is about ∼ 10 GeV for | δλ |≃ 0.03 in compared with Fig.2, and m ≤ 10 7 GeV is still allowed.

SM+h/s
In model SM+h/s three new parameters enter in the effective Lagrangian belowm [5], L = L SM (q, u, d, l, e, h) + µh uhd + m 2 2 s 2 + g λhds h + h.c. Ref. [5] has shown that the observed DM relic abundance can be explained in the wide range 0 < g λ < 0.9. As λ(m) is sensitive to g λ (M Z ), we choose g λ = 0.2 in the small g λ region (≤ 0.4) and g λ = 0.8 in the large g λ region (≥ 0.7) for comparison. Fig. 4 shows that the uncertainty of prediction for the Higgs mass at high energy scale is suppressed at EW scale, and there is only about ∼ 1 − 2 GeV uncertainty due to uncertainty of m t .
For g λ = 0.2 (left panel) it shows that the model is excluded form ≥ 10 13 GeV, while the model is excluded form ≥ 10 9 GeV instead for g λ = 0.8 (right panel). This obviously differs from the case for SM +w. It is because that the deviation from SM in this model is larger than in SM +w, especially in the large g λ region.

Conclusions
Inspired by the present LHC results on SUSY, the prediction for the Higgs mass in high-scale SUSY with weak interacting massive DM is explored in this paper. Similar to well known split SUSY, models of SM +w and SM +h/s, in which wino and mixing state of higgsino and singlino serves as DM respectively, are studied in detail. The main results in this study include: i), In model of SM +w, the SUSY-breaking scalem is allowed in the whole range of 10 4 GeV ≤m ≤ 10 16 GeV for vanishing threshold correction, andm ≤ 10 7 GeV is still allowed for δλ max = 0.03. ii), in model of SM +h/s with vanishing threshold correction,m ≤ 10 13 GeV (10 9 GeV ) is allowed in the small (large) g λ region. For threshold corrections δλ max = 0.03, the model is still allowed form ≤ 10 7 GeV.
Although high-scale SUSY loses its connection to EW scale and has no promising prospect for discovery at the LHC, the prediction for Higgs mass in this paper encourages relating it to cosmology of early universe, especially the inflation physics. It supports the idea [13] that inflation and SM superpartners probably share the same origin of SUSY breaking of order ∼ 10 16 GeV.

Acknowledgement
The author thanks Yang Bai for correspondence and Yang Ma for the help of numerical simulation.

A RGE
Given a coupling g i , its RGE can be written as, The relevant coupling in SM+w include three SM gauge couplings g i , the third-generation Yukawa couplings (g t ), and the Higgs quartic (λ), while an extra Yukawa g λ must also be included in SM+h/s. Compare with split SUSY, the three gaugino couplings defined in [3] disappear either in SM+w or SM+h/s. At one-loop order the RGEs for gauge couplings g i are given by, where the coefficients b i are shown in table 1.