Upper Bounds on Gluino, Squark and Higgisino Masses in the Focus Point Gaugino Mediation with a Mild Fine Tuning $\Delta \le 100$

We show that upper bounds on the masses for gluino, squarks and higgsino are $m_{gluino} \le 5.5$ TeV, $m_{squark} \le 4.7$ TeV and $m_{higgsino} \le 650$ GeV in a focus point gaugino mediation. Here, we impose a mild fine tuning $\Delta \le 100$. This result shows that it is very challenging for the LHC to exclude the focus point gaugino mediation with the mild fine tuning. However, the ILC may have a potential for excluding the focus point gaugino mediation with such a mild fine tuning. It is also shown that vector-like matters reduce the required masses of the squark (stop) and gluino to explain the observed Higgs boson mass and enhance the testability of the model at the LHC. The fine-tuning is still kept mild.


Introduction
Gaugino dominated supersymmetry (SUSY) breaking mediation, so-called gaugino mediation, had been proposed as a solution to the flavor-changing neutral current (FCNC) problem [1,2], long time ago. In the gaugino mediation model, masses of squarks and sleptons are assumed to be suppressed compared to the gaugino masses at the high energy scale. The scalar masses at the weak scale are generated from gaugino loop contributions.
Since the gaugino contributions to masses of squarks and sleptons are always flavor independent, SUSY contributions to FCNC processes such as K meson mixing and µ → eγ are suppressed. 1 These flavor changing processes are serious obstacles to the low-energy SUSY.
If the SUSY is a solution to the hierarchy problem, focus point scenarios [6,7,8] are now attractive. This is because the relatively heavy Higgs boson of around 125 GeV suggests that SUSY particles are heavier than a few TeV [9], along with the nonobservations of the SUSY particles at the LHC. In the focus point scenarios, the EWSB can be explained naturally even if the SUSY particles are much heavier than the EWSB scale. It had been known in the general framework of gravity mediation that gaugino contributions to the Higgs potential have a focus point behavior at the electroweak scale if gaugino masses are non universal at the GUT scale [7]. Motivated by those considerations above, we proposed, recently, a gaugino dominated SUSY breaking scenario with non-universal gaugino masses called as "Focus Point Gaugino Mediation" [10,11]. 2 We showed that we can obtain the correct electroweak symmetry breaking with a much mild fine tuning even though soft masses of SUSY particles are in a region of a several TeV, thanks to the presence of a focus point. We also show that  (3) for the definition of ∆).
The purpose of this letter is to give upper bounds on SUSY particle masses in the FPGM requiring a mild fine tuning less than 1% (∆ ≤ 100) and discuss discovery or exclusion potential of the model at LHC and/or ILC.

Focus point gaugino mediation
In the focus point gaugino mediation, the EWSB scale becomes relatively insensitive to the gaugino mass parameter at the GUT scale, provided that the ratios of the wino mass M 2 to the gluino mass M 3 is M 3 /M 2 ∼ 8/3. The gaugino mass ratio is assumed to be determined by more fundamental physics; non-universal gaugino masses with fixed ratios arise as results of a product group unification model [10], an anomaly of a discrete Rsymmetry [11], and so on. (See Refs. [14] for the details of the product group unification models.) The Higgs soft SUSY breaking masses as well as the squark and sleptons masses are generated by the gaugino loops. Therefore the EWSB scale is determined by only the gaugino mass parameters and µ parameter. The vacuum expectation values of the up-type Higgs and down-type Higgs and their ratio are determined by following two conditions: where is the vacuum expectation value of the up type (down type) Higgs and ∆V is the one-loop corrections to the Higgs potential. Here, The soft masses of the up-type and down-type Higgs are denoted by m Hu and m H d , respectively. The Higgsino mass parameter is denoted by µ. The EWSB scale should satisfies the experimental value as mẐ ≃ 91.2 GeV [15].
In order to estimate the sensitivity of the EWSB scale with respect to the gaugino mass parameter, we adapt the following fine-tuning measure [16]: where µ 0 and B 0 are the Higgsino mass parameter and the Higgs B-parameter at the GUT scale, respectively. We assume that the ratios of the gaugino mass parameters are fixed at the GUT scale.
where M 1 is the bino mass at the GUT scale. Since the focus point behavior is insensitive to M 1 , we take r 1 = 1 in our numerical calculations.
In the universal gaugino mass case r 1 = r 3 = 1, the Higgs boson mass of m h = 125 GeV is explained with ∆ ≃ 1300; the required tuning is more than 0.1 % level. However, in the non-universal case, the required fine-tuning is reduced significantly. In Fig. 1, the contours of the Higgs boson mass (green) and ∆ (red) are shown. The Higgs boson mass is calculated using FeynHiggs 2.10.0 [17,18], which includes higher order corrections beyond 2-loop level [18]. We use SuSpect [19] to evaluate a SUSY mass spectrum and 2-loop renormalization group evolutions. In the focus point gaugino mediation, m h ≃ 125 GeV is explained with ∆ ∼ 50, when the gaugino mass ratios are set to be r 3 ∼ 3/8, r 1 = 1. Here, we take µ < 0, since it can be consistent with B 0 = 0 for tan β = O (10). Notice that the gaugino mediation model with B 0 = 0 is completely free from the SUSY CP problem.

LHC and ILC
The focus point gaugino mediation may be difficult to be excluded at the LHC, since the squarks and gluino are too heavy even when the mild fine-tuning ∆ = 50 − 100 is On the other hand, the Higgsino lighter than 690 GeV may be excluded at the ILC, by measuring the cross section σ(e − e + → µ − µ + ) very precisely. As shown in Fig. 1, the Higgsino mass is bounded from above as µ < 450 (650) GeV for ∆ < 50 (100). With this mass of the Higgsino, the gauge couplings change at O(0.1%) level as where g 2 (q 2 ) W H (g 2 (q 2 )) is the gauge coupling with (without) the Higgsino loop correction (see Appendix A). Taking q 2 = 500 GeV (1000 GeV), the Higgsino with the mass of µ ≃ 340 GeV (690 GeV) changes the coupling by 0.1%. Similarly, we have where g 1 (q 2 ) is the GUT normalized U(1) Y gauge coupling. This gives 0.03% change in the U(1) Y gauge coupling at the weak scale. Therefore, if the ILC with √ s = 1 TeV can measure σ(e − e + → µ − µ + ) at 0.1% level using polarized beams, the Higgsino mass up to 690 GeV can be excluded, even if the Higgsino is not produced directly at the ILC.
Finally let us comment on the case where vector-like matters are added to the MSSM.
The presence of the additional vector-like matters is motivated by, for instance, the existence of the non-anomalous discrete R-symmetry [20]. With the vector-like matters, the gluino and squarks become light compared to those in MSSM. In this case, the squarks and gluino can be discovered at the LHC.
We introduce N 5 pairs of the vector-like matters which are 5 and5 representation of the SU (5) GUT gauge group. The Yukawa couplings between vector-like matters and MSSM matters are assumed to be suppressed. Due to the presence of the vector-like matters, the renormalization group equations (RGEs), especially for gauge couplings and gaugino masses, change (see Appendix B). These changes lead to the significant changes in the SUSY mass spectrum, and the squark (stop) and gluino mass are reduced for a given Higgs boson mass [21]. The fine-tuning measure is defined with inclusion of the mass of the vector-like multiplets M 5 . ∆ = max(|∆ a |), ∆ a = ln mẐ ln µ 0 , Note that the sensitivity of mẐ with respect to M 5 is rather weak as ∆ 10 in the parameter space of interest. In

Conclusion and discussion
We have shown that the upper bounds on the gluino and squark masses are mg < 5.5 TeV and mq < 4.7 TeV (mg < 4.0 TeV and mq < 3.5 TeV) in the focus point gaugino mediation model with a mild fine-tuning, ∆ < 100 (50). These upper bounds show that it is difficult to exclude the FPGM model satisfying a mild fine-tuning at the LHC with √ s = 14 TeV.
On the other hand, the ILC may have a potential to exclude the FPGM model. A squared of a running gauge coupling changes by O(0.1%) level with radiative corrections from the Higgsinos. This change of the gauge couplings reflects a deviation in a cross section σ(e + e − → µ + µ − ) from the standard model prediction. If the ILC can measure this cross section precisely as 0.1% level, the Higgsino with the mass less than 650 GeV corresponding to ∆ < 100 can be excluded.
We have also shown that if the vector-like matters exist at the TeV or at an intermediate scale, the gluino and squark become light as mg ∼ 2.5 TeV and mq ∼ 2.5 TeV, and they can be in the region accessible to the LHC. We find that the fine-tuning is still kept mild even with the presence of those extra-matters.

Acknowledgment
We thank Shigeki Matsumoto for useful discussions. This work was supported by JSPS

A Running gauge coupling
The existence of the chiral fermion, the running gauge coupling is given by where b = (2/3)T (R) and T (R) is the Dynkin index of the representation R. The mass of the fermion is denoted by m. As for the Higgsinos, one-loop corrections give b = 2/5 and 2/3 for the GUT normalized U(1) Y couplings and SU(2) L , respectively. In the short distance limit −q 2 ≫ m 2 , we have with C = exp(5/3).
The ratio of the gauge coupling constants are given by

B The renormalization group equations with vectorlike matters
In this appendix, we give two-loop renormalization group equations in DR scheme with vector-like multiplets. Here, we define the renormalization scale as t = ln Q. The vector-like matters are introduced as 5 = (L ′ , D ′ ) and5 = (L ′ ,D ′ ) representations in SU (5) GUT gauge group. At the one-loop level, the renormalization group equations (RGEs) of a model with N 5 pairs of the vector-like multiplets change from those in the MSSM: where Q i is a hyper-charge of the chiral matter multiplet. We denote the gauge coupling, gaugino mass and the scalar mass squared as g a , M a and m 2 i . In gaugino mediation, [23], the RGEs of the gauge couplings at the two-loop level are given by With this B ab 2 , the RGEs of the gaugino masses are written as The new part of the RGE of the top Yukawa coupling through the change of anomalous dimensions is given by and that of the corresponding scalar trilinear coupling is The scalar masses receive negative corrections from the vector-like multiplets. The two-loop renormalization group equations for the scalar masses change as [23] dm 2 where Here, we have given only terms which arise from N 5 pairs of the vector-like matter multiplets. In gaugino mediation, δS ′ ≃ 0.
The Higgsino mass parameter also receives corrections: