Focus point in dark matter selected high-scale supersymmetry

In this paper, we explore conditions for focus point in the high-scale supersymmetry with the weak-scale gaugino masses. In this context the tension between the naturalness and LHC 2013 data about supersymmetry as well as the cold dark matter candidate are addressed simultaneously. It is shown that the observed Higgs mass can be satisfied in a wide classes of new models, which are realized by employing the non-minimal gauge mediation.


Introduction
The standard model (SM)-like Higgs scalar with mass around 125 GeV [1,2] discovered at the LHC needs some mechanism for stabilizing it against high-energy scale quantum correction. Among the well known candidates which achieve this naturally low-scale supersymmetry (SUSY) is expected to play an important role at the TeV scale. However, the first run of LHC has not observed any signal of new physics yet, and pushes the SUSY particle masses into multi-TeVs region [3,4]. So the absence of SUSY particles near the weak scale v, together with the observed Higgs mass, severely challenge the low-scale SUSY.
In the context of minimal supersymmetric standard model (MSSM) stop masses far above the weak scale is required by the observed Higgs mass when the mixing effect is weak. Given SUSY mass spectrum far above the weak scale, the naturalness is spoiled naively. However, this statement can be relaxed in some specific situations. SUSY with focusing phenomenon [5,6], which is named as focus point SUSY, is few of such examples. In focus point SUSY the sensitivity of up-type Higgs mass squared to the mass scale of SUSY mass spectrum is suppressed because of cancellation among the large renormalization group (RG) corrections. This phenomenon leads to a dramatical reduction of fine tuning associated with electroweak symmetry breaking (EWSB). As a result, it provides us an alternative choice of natural SUSY consistent with the LHC data and the observed Higgs mass.
Unfortunately, focus point SUSY can not be realized in the minimal setup from the viewpoint of model building such as conventional supergravity [7] and the minimal gauge mediation (GM) [8]. However, they are expected to be achieved in some subtle cases. For recent examples in the context of GM, see, e.g., [9][10][11][12]. These examples are restricted to special SUSY-breaking mediation scale M .

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Based on our earlier work [9], we will take an arbitrary M and generalize results above to general focus point. Consider the fact that the cold dark matter demands some electroweakino mass around the weak scale, we will focus on the high-scale SUSY in which gaugino masses are light in compared with other SUSY particle masses. 1 The paper is organized as follows. Firstly, we determine the conditions for focus point in SUSY with arbitrary M in section 2. Then, we analyze the prediction for the Higgs mass in focus point SUSY in section 3. We find that the observed Higgs mass requires the input value for m 2 Hu of order ∼ multi-TeVs. With such magnitude of m 2 Hu the observed Higgs mass can be explained in a wide classes of high-scale SUSY models. The second part of this paper is devoted to the realization of focus point in high-scale SUSY. In section 4 we will construct concrete and complete examples by employing non-minimal GM. We will show that for the case of small A t term focus point SUSY is viable for a wide range of M , which generalizes previous results in the literature. For the case of large A t term, we find that focus point SUSY is viable in a large classes of GM with direct Yukawa coupling between the messengers and the MSSM singlet.
Finally we discuss our results in section 5. The calculation of soft masses is included in appendix A.

Conditions for focus point
We begin with the conditions for focus point in SUSY. Given light gauginos, in compared with soft masses m 2Q Hu and A t term squared, the one-loop renormalization group equations (RGEs) for them are simply given by, where t ≡ ln Q M , with Q the RG running scale and M the SUSY-breaking mediation scale. Here y t denotes the top Yukawa coupling. All other SM Yukawa couplings will be ignored in our discussion. The correlation for these soft masses between scale M and the weak scale v = 174 GeV is obtained in terms of solving the RGEs in eq. (2.1). In particular, we have

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where the real coefficient I[M ] is defined as, For the case with negligble A t , eq. (2.2) reduces to the well known result [5,8], (2.4) As well known, for soft mass spectrum far above the weak scale in eq. (2.2) there is a large fine tuning naively. But this can be avoided if there is significant cancellation among large contributions to up-Higgs soft mass squared in eq. (2.2). Generally speaking, this cancellation does not happen in the minimal setup of model building. However, if it indeed happens in such as focus point SUSY, with the cancellation referred as focusing phenomenon, one can derive the focus condition, regarding eq. (2.2).
Contrary to the case with small A t term, where only two real parameters are needed, we define m 2 Hu [M ] ≡ m 2 0 and introduce three dimensionless real parameters x, y and z, . (2.5) In terms of eq. (2.5) soft masses at scale M can be simply parameterized by x, y and z, 6) and their values at the weak scale are obtained through the RGEs in eq. (2.1). Note that we have imposed the "focusing" condition m 2 Hu [v] = 0, which leads to a constraint on the input masses from eq. (2.2), Moreover, from eq. (2.6) one reads the allowed ranges for the parameters as, In figure 1 we show the "focusing" line in the plane of x and y for M = {10 6 , 10 9 , 10 12 , 10 15 } GeV. Each point on the line gives rise to the focusing phenomenon. Earlier works in [6,10] discussed focusing in GUT-scale model without A term. In these cases I[M GUT ] 1/3, so they correspond to the focus point (0, 2) in figure 1. Work in [7] addressed similar situation but with sizable A term, it corresponds to the gray line. Work in [9]

Higgs mass in focus point SUSY
In this section we discuss the prediction for the Higgs mass in high-scale SUSY with focus point. In particular, we would like to show which points in the focus lines in figure 1 can explain the observed Higgs mass m H = 125.5 GeV [1,2] at the LHC. It is well known that the tree-level contribution to the Higgs mass is up bounded by the Z boson mass, so loop correction must be taken into account for the Higgs mass fit. Moreover, contributions to the Higgs mass higher than the two-loop order should be considered when the squark masses are larger than ∼ 3 TeV. It has been shown in [14] that the three-loop contribution gives rise to ∼ 0.5-3 GeV uplifting of the Higgs mass. For this order approximation one can either use the numerical program in [15], or as we choose in this paper follow the three-loop analytic formula for the Higgs mass in ref. [16].
Explicitly we use the updated top quark mass M t = 173.3 GeV [17], QCD coupling structure constant α 3 = 0.1184, and adopt SUSY mass parameter µ = 200 GeV and gluino mass mg = 2 TeV for the fit. We choose the renormalization scale Q in [16] as the the By using eq. (2.6) we have, We show in figure 2 the contours of three-loop Higgs mass as projected to the twoparameter plane of (x, y) for tan β = 20 and M = simultaneously. It is shown that M s [v] 3 TeV for small mixing effect (x 0), which is consistent with the three-loop result in ref. [14] for the case of degenerate squark masses.
In each panel, one observes the sensitivity of Higgs mass to the input mass parameter m 0 , which clearly indicates that the observed Higgs mass constrains m 0 in the range 1.5 TeV ≤ m 0 ≤ 3.0 TeV. This range for m 0 is subject to the choice on parameter tan β, as tan β determines the tree-level contribution. We show the correlation between the range and tan β in figure 3, which suggests that for tan β = 5 we have 3.0 TeV ≤ m 0 ≤ 4.0 TeV instead. If one takes smaller tan β, it is expected that larger m 0 is needed.  Here a few comments are in order, regarding the fit to the observed Higgs mass in figure 2 and figure 3. iii), For m 0 with such order of magnitude, the Higgs couplings of Higgs sector are similar to those in the decoupling region. The average stop mass is above ∼ 3 TeV for arbitrary M , which has no conflict with the LHC 2013 data. So it is probably impossible to probe so heavy stops at the second run of LHC, and light electroweakinos will be the smoking gun of such high-scale SUSY with focus point.
In summary, the combination of figure 2 and figure 3 implies that given a focusing line referring to M most of focus points are consistent with the observed Higgs mass at the LHC by adjusting the underlying and overall energy scale m 0 in the soft mass spectrum. Typically, m 0 is in the range of [1.5, 3.0] TeV and [3.0, 4.0] TeV for tan β = 20 and tan β = 5, respectively. In the next section, we will proceed to construct high-scale SUSY with focus point which can explain the observed Higgs mass.

Model building
In this section we consider the realization of focusing phenomenon in high-scale SUSY. We restrict us to GM [18] for this purpose.

Models without A t term
Firstly we discuss gauge mediated focus point in high-scale SUSY with small A t term. In the context of GM, there are two important observations for our discussion. (1), the boundary value A t [M ] disappears at the one-loop level when there are no direct Yukawa-like couplings between the messenger field(s) and MSSM matter field(s) in the superpotential.
(2), the gaugino masses vanish at the same order when the mass matrix for messenger fields M satisfies det M = const. (3), the scalar soft masses do not disappear at this order generally.
As shown in [19], points (1)-(3) can be satisfied in a simple model. In this model the messenger fields are a set of chiral and bi-fundamental supermultiplets q + q ,q +q , l + l , l +l that transforme under SU

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The renormalizable superpotential consistent with SM gauge symmetry in this model reads as, where SUSY-breaking sector X = θ 2 F , Yukawa couplings λs and tree-level masses ms are assumed to be real for simplicity. It can be easily verified that both the mass matrixes Ms for messenger vector (q, q ) and (l, l ) both satisfy det M = const. So the gaugino masses indeed vanish at the one-loop order of O(F/M ) as desired. The soft scalar masses differ from those of the minimal GM, which are given by, respectively, Here A = 1 8π 2 F 2 M 2 , which determines the overall magnitude of soft masses above, and s 1,2,3 are given by For the case of small A t term the focusing condition in eq. (2.4) reduces to R act = R req , with , In figure 4 the top line shows the ratio R req /R act as function of M , which reproduces the well known result that in the minimal GM m 2 Hu is too small [8] to satisfy the focus condition, which holds for the whole range of M . However, the ratio changes when the soft mass spectrum deviates from the minimal GM. The soft mass spectrum in eq. (4.3) depends on parameter s 2 relative to s 3 . We fix s 2 = 1 in the following discussion, and show that R req /R act is suppressed to be near unity by decreasing soft mass mQ 3

Models with A t term
Now we consider focus point in high-scale SUSY with large A t term. In the model discussed in this subsection, the messenger fields are the same as previously studied in eq. (4.1), except that we add a singlet of SM gauge group S and its bi-fundamental fieldS into the messenger sector. These messengers are coupled to the SUSY-breaking spurion field X = θ 2 F through superpotential, W = X ll + qq +SS + m l l l + ll + m q q q + qq . (4.7) We will generalize the number of messenger fields to n pairs, and simply take universal couplings between messengers and X fields and universal masses m q m l M for our analysis. Similar to our previous observations, gaugino masses vanish at the one-loop level of order O(F/M ) in our setup.
For our purpose we further deform the model defined in eq. (4.7) by adding a direct Yukawa coupling between the lepton-like messengerl and H u , δW = λH u Sl. (4.8) This new superpotential can be argued to be natural by imposing hidden parity. As mentioned in [9], eq. (4.8) can be protected in terms of either imposing some hidden matter JHEP03(2015)098 with A = n While the later one reads as, (4.10)

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Here a few comments are in order, regarding eq. (4.10). The contribution to soft scalar mass spectrum due to Yukawa-type interaction has been previously studied in [22][23][24]. However, they can not be directly applied to our case. Because the SUSY-breaking spurion superfield X differs from the conventional situation in which X stand = X + F θ 2 , with X = 0. We present the derivation of eq. (4.10) in appendix A. Focusing phenomenon in this model has been discussed in [9] for intermediate scale M ∼ 10 8 GeV. Here, our discussions are more general, with scale M unfixed.
In this model the input parameters related to focusing conditions eq. (2.7) and eq. In summary, focus point high scale SUSY can be realized in non-minimal GM. But only large M is allowed for the case with large A t term. In the next subsection, we address the problem of small gaugino mass in general focus point SUSY, which can be reconciled with the LHC 2013 data in terms of reasonable modification to the original messenger sector.

Gaugino mass
The LHC 2013 data [3,4] has reported gluino mass bound about ∼ 1.3 TeV. This mass bound is not satisfied in the focus point SUSY discussed so far. For example, the gluino mass relative to m Hu at scale M in the case with large A t discussed in IV.B is given by, where function F(a, b) is defined in [19]. Together with the constraint on m 0 due to observed Higgs mass as shown in the section 3, one can estimate the boundary value of mg 3 [M ]. Eq. (4.12) implies that the gluino mass is far below the present lower bound.
Some reasonable modification should be taken into account in order to complete the discussions about model building. Now we re-examine the smallness of gaugino masses, which JHEP03(2015)098 attributes to the fact that det M = const and consequently mg r ∼ αr 4π F ∂ ln det M/∂ ln X ∼ 0. When we employ small tree-level mass terms for some of messengers such as 2 δW = m q q + m l l , (4.13) it will lead to the replacement in det M for quark and lepton messengers [25], (4.14) If so, the correction to soft scalar mass spectrum and gaugino masses is of order O(m 4 /M 4 ) and , respectively. Provided the former correction can be very weak so that the focusing still holds, but the later one can be large enough to reconcile with the LHC gaugino mass bound. For example, we choose m 0.06M 10 8 GeV. For m 0 ∼ 3 TeV we have √ F ∼ 6 × 10 7 GeV, and further mg 3 2 TeV from eq. (4.15). In this model both the bino and wino masses are around ∼ 1 TeV, so they are the main target at the 14-TeV LHC. The discussions about model building in high-scale SUSY with focus point are therefore completed.

Conclusion
In this paper we have explored focus point in the high-scale SUSY which has weak-scale electroweakino masses. We have derived conditions for the focusing phenomenon in general. We have analyzed in detail the prediction for the Higgs mass in such focus point SUSY. The observed Higgs mass at the LHC requires the input value m 2 Hu of order ∼ 2 TeV and ∼ 3 TeV for tan β = {20, 5}, respectively. We also address the model building of focus point SUSY by employing non-minimal GM. The main results include (a) for the case with small A t term, focus point is allowed for a wide range of M ; (b) for the case with large A t term, focus point is only permitted in high-scale SUSY.
One may worry about the stability of focus point discussed so far by following two facts. One fact is that the soft scalar mass spectrum is directly related and thus sensitive to the underlying mass scales F and M . The other one is that the focusing phenomenon imposes constraint on these soft masses as shown in eq. (2.2). So, the significant reduction due to focus point seems to be spoiled for a small change of either F or M . However, this is not true. Because only the overall magnitude of soft mass spectrum but not their relative ratios is determined by F/M . As shown in eq. (2.7) the focusing condition eq. (2.2) is actually not dependent on either F or M , and only dependent on M indirectly through RG effect. Once we fix M , for example in some GUT-scale SUSY models, the focus point induced by the accidental cancellation, is actually stable against to high mass scale.

[M ]
A · cos 2 θ · − with g r (r = 1, 2, 3) refer to the SM gauge couplings. For the model under study we have tan θ = 1 and d H = n, n being number of messenger pairs. C r = C r Hu + C r i + C r j is the sum of quadratic Casimirs of the fields which participate in the Higgs-messengermessenger Yukawa coupling, with i, j referring to messenger fields involved. In our case, C r = 3 10 , 3 2 , 0 .
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