Focus Point in Gaugino Mediation ~ Reconsideration of the Fine-tuning Problem ~

We reconsider the fine-tuning problem in SUSY models, motivated by the recent observation of the relatively heavy Higgs boson and non-observation of the SUSY particles at the LHC. Based on this thought, we demonstrate a focus point-like behavior in a gaugino mediation model, and show that the fine-tuning is indeed reduced to about 2 percent level if the ratio of the gluino mass to wino mass is about 0.4 at the GUT scale. We show that such a mass ratio may arise naturally in a product group unification model without the doublet-triplet splitting problem. This fact suggests that the fine-tuning problem crucially depends on the physics at the high energy scale.


Introduction
The Higgs boson mass is a good probe of the supersymmetry (SUSY) breaking scale in the minimal SUSY standard model (MSSM) [1]. The observed Higgs boson mass of around 125 GeV [2,3] suggests, together with non-discovery of SUSY particles at the LHC, that the SUSY breaking scale is considerably higher than the electroweak scale. This already raises doubt of the low scale SUSY as a solution to the hierarchy problem. In fact, we need a fine-tuning at the level of 0.1 − 0.01% to reproduce the correct electroweak symmetry breaking scale if the squark and gluino masses are of order a few TeV.
The purpose of this paper is to argue that the issue of fine-tuning crucially depends on physics at a high energy, say GUT scale. A famous example is so called "Focus Point SUSY" [4] (see also [5,6] for recent discussions) in gravity mediation models. In this scenario, small gaugino masses and certain relations among stop masses, the up-type Higgs soft mass and the trilinear coupling of the stop, A t , are assumed. As a result, the Higgs boson mass of around 125 GeV can be accommodated within about 1% tuning.
Although the essential point of "Focus Point SUSY" is attractive, the relations among the scalar squared masses and A 2 t seem not so simple; the Kahler potential should be carefully chosen in order to reduce the fine-tuning to 1% level.
In this paper, we point out that the focus point like behavior also occurs in gaugino mediation models [7,8] with one simple relation; the required fine-tuning is indeed reduced significantly, depending on a gaugino mass ratio M 3 /M 2 at the GUT scale. Here, M 3 and M 2 are masses of gluino and wino at the GUT scale, respectively. It may be interesting that the mass ratio could be a parameter independent of SUSY breaking scale. We stress that the unnatural looking SUSY is a consequence of physics at high energy scale.

Focus point in gaugino mediation
The recent analyses [9] of the adiabatic solution [10] to the Polonyi problem [11] in gravity mediation scenario would suggest a small gravitino mass, m 3/2 , compared with the gaugino masses M 1/2 , that is, m 3/2 ≪ M 1/2 , and hence the gaugino mediation model [12] is very attractive. Furthermore, it is well known that the flavor changing neutral current (FCNC) problem is ameliorated substantially in the gaugino mediation models [7,8]. Motivated by those facts, we consider a gaugino mediation model throughout this paper and point out that the focus point-like behavior occurs with a suitable choice of the ratio of M 3 and M 2 ; if the ratio of M 3 /M 2 ∼ 0.4, the required fine-tuning can be reduced. 1 Note that the bino mass M 1 is not important, as shown later.
In our setup, among superfields in MSSM, only gauge kinetic functions have enhanced couplings to the Polonyi field which has a SUSY breaking F-term, and hence the scalar masses, the Higgs B-term and scalar trilinear couplings are much smaller than gaugino masses at the high energy scale [12]. The gravitino is the lightest SUSY particle (LSP) and candidate for a dark matter (see [12] for details). Let us parameterize the gaugino mediation model as where M 1 , M 2 and M 3 are the bino, wino and gluino mass at the GUT scale, respectively, and µ 0 denotes the Higgsino mass parameter at the GUT scale. Here, the scalar masses, the Higgs B-term as well as the scalar trilinear couplings are neglected for simplicity, and they are induced by renormalization group (RG) evolutions between the GUT scale and the SUSY scale. The universal gaugino mass corresponds to r 1 = r 3 = 1. Here and hereafter, we take r 1 , r 3 > 0.
The successful electroweak symmetry breaking occurs with a particular balance among the soft SUSY breaking mass of up-and down-type Higgs (H u and H d ), the Higgs B-term and the SUSY invariant mass µ. Including radiative corrections to the Higgs potential, the electroweak symmetry breaking scale is determined by the following condition: where v u and v d are the vacuum expectation values of H 0 u and H 0 d , respectively, and ∆V is the radiative correction to the Higgs potential. The soft mass squared of H u and H d are denoted by m 2 Hu and m 2 H d , respectively, and µ is the Higgsino mass parameter at the SUSY scale. The electroweak symmetry breaking scale is, in principle, determined by Eq. (2) although it is fixed to reproduce mẐ ≃ 91.2 GeV [14]. Neglecting ∆V and the terms suppressed by tan 2 β, Eq. (2) where the two-loop renormalization group equations [15] are used. We obtain m 2 0. This indicates that the fine-tuning can be reduced with a certain choice of r 3 , that is, the ratio of M 3 to M 2 . Notice that the coefficients of the terms proportional to M 1 are small in most of the viable region, 2 and hence, their contributions to m 2 Hu are not important as long as M 1 ∼ M 3 . In Fig. 1, we show the focus point-like behavior for different choice of r 3 (and r 1 ). The scale where m 2 Hu vanish is shifted to the low-scale as r 3 becomes small, and hence, by taking smaller value of r 3 , it is expected that the amount of the fine-tuning is reduced.
In order to evaluate the degrees of fine-tuning, we adapt the following fine-tuning where a is a parameter at the GUT scale and a = M 1/2 and µ 0 in our model. Notice that ∆ µ is always ∼ 2µ 2 /(91.2 GeV) 2 , since the SUSY mass parameter µ is almost unchanged during the RG evolution between the GUT scale to the SUSY scale, i.e., µ ≃ µ 0 , and hence a small ∆ µ simply means a small µ. On the other hand, roughly speaking, a small ∆ M 1/2 means a small change of m 2 Hu , and hence, a small µ does not always correspond to a small fine-tuning.
First, we show results of the universal gaugino mass case, i.e., r 1 = r 3 = 1.0 in In the case of non-universal gaugino masses, the fine-tuning is reduced significantly due to the focus point-like behavior. In Fig. 3, the Higgs boson mass as a function of M 1/2 is shown for different r 3 . The ratio r 1 is taken as r 1 = 0.4. The slight change of the ratio correspond to the change of the dominant contributions to ∆. In the region with small M 1/2 , ∆ is simply determined by the size of µ parameter. As M 1/2 gradually increases, |µ| becomes small. However, (∂ ln m 2 Z )/(∂ ln M 2 1/2 ) dominates ∆, and the fine-tuning becomes worse. This change is also reflected in the steep slope of |µ|; the small |µ| is necessary for small ∆ but it is not sufficient. It is noticed that the fine-tuning measure is reduced to ∆ ≃ 60 (123) for r 3 = 0.37 (0.39), where the gaugino mass is taken as M 1/2 ≃ 4100 (6200) GeV. The observed Higgs boson mass of around 125 GeV can be consistent with about 2 % tuning. The detailed mass spectra are shown in Table. 1. Since some of the squark masses can be smaller than 3 TeV, they may be observed at LHC with √ s = 14 TeV. In addition, the lightest stau, chargino and neutralino can be around 350 GeV, which may be target of future linear collider experiments.  Table 1: The mass spectrum and ∆. The scalar trilinear coupling of the stop is denoted by A t . Here, the gravitino is the LSP.

Conclusions and discussion
In this paper we have shown that the required fine tuning is substantially reduced at the level of ∼ 2% in a gaugino mediation model if the ratio of the gluino mass to the wino mass at the GUT scale is about 0.4. 5 The Higgs boson mass of around 125 GeV can be explained without a severe fine-tuning, even if the colored SUSY particles are as heavy as a few TeV.
The deviation of the universal gaugino mass is clearly inconsistent with the minimal SUSY GUT scenario. However, we show in this section that the required mass ratio, M 3 /M 2 ∼ 0.4, is even natural in one of the product group unification (PGU) models [19,20], which were proposed to solve the doublet-triplet splitting problem in the minimal SUSY GUT.
(M 1/2 = 6200 GeV, r 1 = 1.5 and r 3 = 0.39), which is below the current experimental bound, d e 10 −27 e cm [14]. As we have stated, the change of the bino mass does not affect the focus point-like behavior significantly, that is, the bino can be heavy without an increase of the fine-tuning. Therefore, the constraint from the EDM can be avoided relatively easily, but still the electron EDM is expected to be seen at feature experiments.