Fine Tuning in General Gauge Mediation

We study the fine-tuning problem in the context of general gauge mediation. Numerical analyses toward for relaxing fine-tuning are presented. We analyse the problem in typical three cases of the messenger scale, that is, GUT ($2\times10^{16}$ GeV), intermediate ($10^{10}$ GeV), and relatively low energy ($10^6$ GeV) scales. In each messenger scale, the parameter space reducing the degree of tuning as around 10% is found. Certain ratios among gluino mass, wino mass and soft scalar masses are favorable. It is shown that the favorable region becomes narrow as the messenger scale becomes lower, and tachyonic initial conditions of stop masses at the messenger scale are favored to relax the fine-tuning problem for the relatively low energy messenger scale. Our spectra would also be important from the viewpoint of the $\mu-B$ problem.


Introduction
Low-energy supersymmetric extension of the standard model is one of promising candidates for a new physics at a TeV scale. The supersymmetry (SUSY) can stabilize the huge hierarchy between the weak scale and the Planck scale. That is a motivation for the low-energy SUSY.
In addition, the three gauge couplings are unified at the grand unified theory (GUT) scale, 2×10 16 GeV, in the minimal supersymmetric standard model (MSSM). Also, supersymmetric standard models have candidates for the dark matter.
Although low-energy SUSY solves the (huge) hierarchy problem between the weak scale and Planck/GUT scale, a few percent of fine-tuning is required in the MSSM as follows. The lightest CP-even Higgs mass m h is predicted as m h M Z at the tree level in the MSSM, but that is smaller than the experimental bound m h 114.4 GeV. However, the Higgs mass receives a large radiative correction depending on the averaged stop mass mt [1,2]. The experimental bound m h 114.4 GeV requires mt 1 TeV when |A t |/mt 1.0, where A t is the so-called A-term corresponding to the top Yukawa coupling. On the other hand, the stop mass also has a renormalization group (RG) effect on the soft scalar mass m Hu of the up-sector Higgs field as [3,4] ∆m 2 Hu ∼ − where y t is the top Yukawa coupling and Λ denotes a cut-off scale of the MSSM such as the Planck scale or GUT scale. This RG effect |∆m 2 Hu | would be comparable to the stop mass with a negative sign. Furthermore, the successful electroweak (EW) symmetry breaking requires where µ denotes the supersymmetric mass of the up-sector Higgs field H u and the downsector Higgs field H d . If m 2 Hu ∼ −m 2 t and mt = O(1) TeV, one needs a few percent of fine-tuning between µ 2 and m 2 Hu in order to derive the correct value of M Z . That is the socalled little hierarchy problem [5]. Several works have been done to address this issue [6]- [24].

Some of them include extensions of the MSSM.
In the bottom-up approach [25], it is found that non-universal gaugino masses with a certain ratio are favorable to improve fine-tuning in the MSSM when the messenger scale of SUSY breaking is the Planck/GUT scale. Such a favorable ratio of gaugino masses can be realized in the TeV scale mirage mediation [26,27,13,14] and gravity mediation, e.g. moduli mediation [28,22] and the SUSY breaking scenario, where F-components of gauge non-singlets are sizable [29,30,21] 1 . On the other hand, the spectrum of the constrained MSSM with the universal gaugino mass would be unfavorable. It is also pointed out that a negative value of the stop mass squared at the Planck/GUT scale would also be favorable [18,19].
Since the minimal gauge mediation [33] leads to the universal gaugino mass, that would be unfavorable from the viewpoint of fine-tuning [11,32]. Recently, Meade, Seiberg and Shih have extended the gauge mediation to general gauge mediation (GGM) [34]. (See also [35]- [46].) That leads to non-universal gaugino and soft scalar masses. Thus, it is important to study fine-tuning in the GGM. That is our purpose. 2 The important difference of the gauge mediation (including GGM) from other mediation scenarios such as gravity mediation is that the messenger scale can vary from the GUT scale to a TeV scale and predicted Aterms are very small in most of models. These would also lead to an important difference in the fine-tuning behavior.
This paper is organized as follows. In section 2, we briefly review on the fine-tuning problem in the MSSM. Section 3 is also a brief review on the GGM. In section 4, we analyse numerically on fine-tuning in the GGM. In section 5, we give a comment on the µ − B problem. Section 6 is devoted to conclusion.

Fine tuning in the MSSM
Here, we briefly review the fine-tuning problem in the MSSM by showing explicitly equations.
In our analysis, we neglect the Yukawa couplings except the top Yukawa coupling y t . Then, the Higgs sector in the MSSM is described as the following superpotential, where Q 3 , and U 3 are the chiral superfields corresponding to the left-and right-handed top quarks, respectively. The Higgs fields and top-stop multiplets as well as the gaugino fields play an important role in the fine-tuning problem. Thus, we concentrate on these fields.
Their soft SUSY breaking terms are given by where m X (X = H u,d , Q 3 , U 3 ) are the soft scalar masses for X, respectively, µB is the SUSY breaking mass, i.e. the so-called B-term. Note that we utilize the same notation for denoting a chiral superfield and its lowest scalar component.
The soft SUSY breaking mass for the up-type Higgs m Hu is subject to relatively large logarithmic radiative correction (1) from mainly stop loops. The radiative correction ∆m 2 can be obtained in gauge mediated SUSY breaking model [34]. A careful definition of gauge mediation mechanism has been given in the work, that is, in the limit that the MSSM gauge couplings α i → 0, the theory decouples into the MSSM and a separate hidden sector which breaks SUSY. Following the convention, we label the gauge groups, SU(3), SU(2) and U(1) of the MSSM by a = 3, 2, 1, respectively. Within the framework of the GGM, the three gaugino masses M a (a = 1, 2, 3) of the MSSM are given at the messenger scale M as, In general, B a (a = 1, 2, 3) are three independent complex parameters. If CP phases of B a are not aligned each other, that would lead to a serious CP problem. Thus, we use B a as three real parameters. The soft scalar masses squared are also given in the GGM as at M, where c 2 (f ; a) is the quadratic Casimir of the representation of fermion f under the gauge group corresponding to the label a. Here, A a (a = 1, 2, 3) are three independent real parameters. Hereafter, we concentrate on the models with ζ = 0. 3 In this case, there are the mass relations at M where and m E f denote soft scalar masses for the f -th generation of the left-handed squarks, up-sector right-handed squarks, down-sector right-handed squarks, left-handed sleptons and right-handed sleptons. Thus, the U(1) Y D-term S, i.e., vanishes at the messenger scale M. Furthermore, its RG equation is given as where t ≡ 2 log(M Z /μ),μ is an arbitrary energy scale, and b 1 = 33/5 (and b 2 = 1, b 3 = −3 for references). Thus, when S is vanishing at M, it vanishes at any scale. For concreteness, we show explicitly the initial conditions of the soft scalar masses, m Q 3 , m U 3 , m Hu and m H d as for convenience. Similarly, we define the ratios of gaugino masses to the gluino mass, The initial condition of the A-term in the GGM is given as at M. Thus, the A-term A t at the weak scale is given only by the RG effect between the weak scale and the messenger scale M. This initial condition is important because the stop mixing A t /mt at the weak scale has a significant effect on the Higgs mass (7). By utilizing these gaugino and sfermion masses given in the GGM, we numerically analyze the fine-tuning problem in the next section.

Numerical Analyses
We study the fine-tuning problem in the GGM and present numerical analyses. In gauge mediated SUSY breaking models, phenomenological consequences at the EW scale generally depend on the messenger scale M. We present our analyses for three typical messenger Firstly, we give the soft parameters at the EW scale by integrating the 1-loop RG equations [3]. The gaugino mass at the EW scale are In this analysis, we use the values of gauge couplings at the EW scale asα These couplings in the MSSM would be unified at the GUT scale within a good accuracy. In addition, we use the running top mass m t = 164.5 GeV at M Z and tan β = 10 for numerical analysis.
The scalar masses such as m Q 3 , m U 3 , m H u,d , and A t , which are important to discuss the fine-tuning problem, are given for each typical messenger scale as (ii) M = 10 10 GeV, (iii) M = 10 6 GeV, Here, we have used the initial conditions, A t (M) = S(M) = 0. The change of RG effects between the cases (ii) and (iii) is rather drastic compared with one between (i) and (ii).
If all soft parameters are taken as the same order, B a ∼ m X (M), the averaged top squark mass is approximated for each messenger scale as For a fixed value of |A t (M Z )/mt|, a large value of m 2 t would be favorable to realize the Higgs mass m h ≥ 114.4 GeV. That implies that a higher messenger scale would be favorable for a fixed value of the gluino mass, i.e. B 3 . In order to satisfy the experimental bound for the Higgs mass, the lower bound for B 3 is roughly estimated as .
A large value of |A t (M Z )/mt| would be favorable to realize the Higgs mass m h ≥ 114.4 GeV. That implies that a higher messenger scale would be favorable.
On the other hand, the dominant part of the RG effects in m 2 H d (22), (27) and (32) is due to the gluino mass, i.e. B Toward the numerical analyses of the fine-tuning problem, we introduce fine-tuning parameters [5], (ii) M = 10 10 GeV We have assumed that B a and A a are independent of each other. However, in a definite theory, they are not independent, but certain ratios are predicted in each theory. That is, in a definite theory there is one parameter, which determines the overall size of soft SUSY breaking terms. We choose B 3 as such a parameter and the ratios a a and b a are fixed in a theory. Then, we consider the fine-tuning only for B 3 , i.e. ∆ B 3 under fixed ratios of a a and b a . Varying a a and b a means that we compare different theories in the theory space of the GGM. Then, the fine-tuning parameter can be rewritten as (ii) M = 10 10 GeV Coefficients of b 1 and a 1 in the above equations are very small. Thus, those terms would not be important unless b 1 = O(10) or a 1 = O(100). Therefore, we concentrate on others and throughout our numerical analyses we take b 1 = a 1 = 1 as a typical value. It is found that the coefficients of a 2 and b 2 2 , which determines the wino mass, are negative. Hence, it would be favorable to cancel the dominant term by relatively large b 2 and/or a 2 . That is, models satisfying (ii) M = 10 10 GeV would be interesting in the theory space. For fixed values of a 2 and a 3 , a favorable value of b 2 is determined. That means a favorable ratio between the gluino and wino masses such as Ref. [25]. For a fixed value of b 2 , a linear correlation between a 2 and a 3 is required. On the other hand, for a fixed value of a 2 (a 3 ) a quadratic relation between b 2 and a 3 (a 2 ) is required.
Our results show that a certain ratio between the gluino mass and wino mass is favorable. Also, the tachyonic initial condition for stop masses at the messenger scale would be favorable, in particular in the low messenger scale scenario. For M < 10 6 GeV, the favorable region corresponds to only negative values of both a 2 and a 3 . The A-term A t plays a role in this result. Its initial value vanishes at M, i.e. A t (M) = 0, and its value at M Z is generated by RG effect as Eqs. (24), (29), (34), which are determined mainly by B 3 and B 2 . However, a value of |A t (M Z )| at M Z is smaller as the messenger scale becomes lower, because the RG effects become smaller. On the other hand, a large value of the stop mixing |A t /mt| is favorable to increase the Higgs mass, m h . Thus, if a value of |A t (M Z )| is small, we have to decrease a value mt to obtain a large stop mixing |A t /mt|. That can be realized by imposing the tachyonic initial condition of the stop mass at M. Figure  500 4.19 −45 < ∼ a 2 < ∼ 5 −10 < ∼ a 3 < ∼ 2 2 (a) 300 4.01 −40 < ∼ a 2 < ∼ 10 −9 < ∼ a 3 < ∼ 5 2 (b) 500 1 −10 < ∼ a 2 < ∼ 50 −15 < ∼ a 3 < ∼ 0 2 (c) 300 1 −10 < ∼ a 2 < ∼ 50 −12 < ∼ a 3 < ∼ 3 2 (d)  Figure  500 4.12 We also give the mass spectra of gluino, wino, and stop for typical parameters of the favorable regions in Table 4. We find that the smallest masses of wino and stop are realized in the case (i) with B 3 = 300 GeV, a 3 = −1, and a 2 = 30 as M 2 ≃ 517 GeV and mt ≃ 555 GeV. On the other hand, the largest masses of wino and stop are given in the case (iii) with B 3 = 10 3 GeV, a 3 = 1 and a 2 = −50 as M 2 ≃ 7150 GeV and mt ≃ 2420 GeV.

µ − B problem
Here, we comment on the µ-term and B-term. How to generate the µ-term and B-term is another important issue. Within the framework of the gauge mediation, a simple mechanism to generate the µ-term would lead to This ratio would cause a problem if When both (54) and (55) hold, we could not realize the successful EW symmetry breaking.
few TeV. Indeed, if we can obtain the following hierarchy, we can realize the successful EW symmetry breaking. It has been already pointed out in [48] that the above hierarchy would be favorable in the gauge mediation. Also, such a pattern has been studied within the framework of the TeV scale mirage scenario [14], i.e. the mass pattern II. This pattern of hierarchy can be realized in our analyses. A relatively large B 2 is favorable to obtain a large m H d seen as in (23), (28), and (33). For example, if we take M = 10 6 GeV, By using sin 2β = 2µB 2|µ| 2 + m 2 Hu + m 2 with tan β = 10, the above value of m 2 H d (M Z ) ≃ 2.89 2 TeV 2 determines the value of µB as µB ≃ 911 2 GeV 2 .
That is, we have µB/µ 2 = O(100) for µ ∼ 100 GeV. Such a ratio µB/µ 2 could be realized by a simple mechanism to generate the µ-term and B-term (54). 5 Therefore, this parameter set, which relaxes the fine-tuning problem, would also be favorable from the viewpoint of the µ − B problem.
conditions of scalar masses are favored, in particular in the relatively low messenger scale scenario. Furthermore, the type of spectra with µ ≈ 100 GeV and a few TeV of other SUSY breaking masses is also favorable from the viewpoint of the µ − B problem. Thus, it would be important to construct explicit models, which realize certain ratios among gaugino and scalar masses.
Note to be added While this paper was being completed, Ref. [49] appeared, where also fine tuning in the GGM was studied.