1 Introduction

UT-12-37

IPMU12-0208

Enhanced Higgs Mass in a Gaugino Mediation Model

without the Polonyi Problem

Takeo Moroi, Tsutomu T. Yanagida and Norimi Yokozaki

Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

Kavli IPMU, University of Tokyo, Kashiwa, 277-8583, Japan

## 1 Introduction

The presence of so called Polonyi field is an inevitable ingredient in the gravity mediation when the gravitino mass is  TeV, otherwise we have vanishing gaugino masses at the tree level and the contributions from the anomaly mediation are too small [1, 2].111 In scenarios with TeV gravitino, the Polonyi field is not required since the gaugino masses of TeV are induced at one-loop level. Without assuming a particular form of the Kahler potential, ”Pure Gravity Mediation” scenario [3, 4] was proposed where the scalar masses are  TeV and the gaugino masses are  TeV. (A similar model was proposed in Ref. [5].) The scenario can naturally explain the Higgs mass of around 125 GeV with heavy stops [6]. However, this Polonyi field causes serious cosmological problems. In particular, its decay in the early universe produces too much entropy, resulting in a huge dilution of the primordial baryon-number asymmetry. Furthermore, its decay occurs during/after the Big-Bang Nucleosynthesis (BBN) and destroys light elements produced by the BBN [7]. From a cosmological view point, the Polonyi problem severely restricts the supersymmetry (SUSY) breaking scenarios.

In a series of recent works, it has been pointed out [8, 9] that the serious Polonyi problem can be solved if the Polonyi field has enhanced couplings to inflaton. This observation is based on the adiabatic evolution of the Polonyi field following the inflaton potential, originally suggested by Linde [10]. The integration of inflaton field induces also enhanced self couplings of the Polonyi field; the Polonyi mass becomes much heavier than the gravitino mass. In addition, enhanced couplings of the Polonyi field to the gauge kinetic function are preferred in order to solve the Polonyi problem. This is because the enhanced couplings make the decay of faster and the cosmological constraint becomes weaker [11]. Consequently, relatively high reheating temperature GeV is allowed [9]. With this reheating temperature, non-thermal leptogenesis works for baryogenesis [12].

Such a setup suggests a gaugino mediated SUSY breaking scenario [13].222If we assume a separation between the Polonyi field and quark, lepton and Higgs multiplets in the conformal frame, we have vanishing soft masses for sfermions [13]. Based on this observation, the gaugino mediation scenario was first proposed in [13] (see also  [14]). (See Refs. [15] for a scenario in the framework of extra dimension.) Here, we consider the case where the interactions between the gauge multiplets and the Polonyi field are enhanced while those between chiral multiplets in the minimal supersymmetic standard model (MSSM) and the Polonyi field are not. We study its phenomenological consequences paying particular attention to the Higgs mass. As well as the cosmological advantages mentioned above, such a scenario is also favored because it can solve the serious SUSY FCNC problem [13]. As we discuss below, in our setup, the gravitino mass is much smaller than the gaugino and the Polonyi masses because of the enhanced couplings.333A similar setup was considered in [16]. The sfermions masses and the scalar trilinear couplings (so-called -terms) are also of the order of the gravitino mass at the tree-level,444If we assume the sequestered form of the Kahler potential, sfermion masses and -terms vanish at the tree level [13, 1]. and dominantly arise from the gaugino masses through the renormalization group evolution. Therefore, the SUSY CP and flavor problems are relaxed.

In this letter, motivated by the recent discovery of the Higgs-like boson at the ATLAS and CMS experiments [17], we investigate the Higgs boson mass in the gaugino mediated SUSY breaking scenario. If the particle content of the MSSM is assumed up to the cut-off scale (which will be taken to be at the GUT scale), the Higgs mass of around 125 GeV is realized only in the region out of the reach of the LHC experiment [18]. However, we point out that the existence of the extra matters can drastically enhance the trilinear coupling of the stop, even if there are no direct couplings to the MSSM matter fields. As a result, the Higgs mass can be as large as GeV with large -term [19] in a region within the reach of the LHC experiment; gluino mass is around 12 TeV. The existence of the extra matters at the scale much lower than the Planck scale is expected in many models, for example, like grand unified theory, axion models, and so on.

This paper is organized as follows. In section 2, we explain the setup of our “gaugino dominated SUSY breaking” scenario and show that the Higgs mass of 125 GeV can be consistent only with the gluino mass heavier than about 4 TeV with the particle content of the MSSM [18]. In section 3, we show that the trilinear coupling of the stop is enhanced if the extra matters exist; the Higgs mass can be explained with the gluino mass of TeV. The final section is devoted to conclusion and discussion.

## 2 Gaugino mediation without Polonyi problem

Let us discuss the setup of our gaugino mediation scenario. We assume that the Polonyi field strongly couples to the inflaton field and the gauge kinetic functions, while the couplings of to the other fields (matters and Higgs) are suppressed. The relevant part of the Kahler potential is given by

 K=−c2I|Z|2|I|2M2P−c2Z|Z|44M2P, (1)

where and are the Polonyi field and the inflaton field, respectively, and is the reduced Plank scale. The coefficient is required to be as large as so that the Polonyi abundance is sufficiently suppressed by the adiabatic suppression mechanism [10, 8, 9]. The second term arises by the radiative corrections from the inflaton loops and is also expected to be . As a result, the Polonyi mass is enhanced compared to the gravitino mass:

 mZ=cZFZMP≃cZ√3m3/2, (2)

where is the gravitino mass. Here we assume that the SUSY is dominantly broken by the field so that . The couplings of to the gauge kinetic functions are also assumed to be enhanced:

 L∋−cg∫d2θZWαWαMP+h.c., (3)

where . (Here we take the basis in which the gauge kinetic function is canonically normalized.) With Eq. (3), the gaugino mass is given by

 Mλ=2cgFZMP≃2√3cgm3/2. (4)

Since we consider the scenario that , the gravitino is the LSP and candidate for a dark matter. The Polonyi field decays into SM gauge bosons through the operator (3), and its decay width is given by

 Γ(Z→2 gaugebosons)≃c2g3m3Z2πM2P≃1.2sec−1(cg100)2(mZ103GeV)3. (5)

With the suppression of the Polonyi abundance and the relatively short lifetime (less than 1 second), the BBN constraint can be avoided relatively easily [11] and the reheating temperature  GeV is allowed for  GeV and  [9]. With such a relatively high reheating temperature, enough baryon asymmetry can be generated by the non-thermal leptogenesis [12].

In our setup, the gravitino mass as well as the scalar masses are suppressed compared to the gaugino masses at the cut-off scale. Then the scalar masses dominantly arise from the renormalization group effect between the cut-off scale and the SUSY scale (which is the mass scale of the MSSM SUSY particles) of  TeV. We calculate the low-energy SUSY parameters by numerically solving two-loop renormalization group equations (RGEs). Here, we use SuSpect [20] to evaluate the spectrum of the SUSY particles. The boundary condition is taken such that all the scalar masses and trilinear coupling constants vanish and the gaugino masses are universal at the cut-off scale (which is taken to be the GUT scale). Then, we calculate the lightest Higgs boson mass using FeynHiggs [21]. In fig. 1, the contours of the constant Higgs mass are shown. In the same figure, we also plot the contours of constant -parameter at the GUT scale, where the -parameter is defined as

 V∋BμHuHd+h.c., (6)

with () being the up-type (down-type) Higgs and being the Higgsino mass parameter in the superpotential. (Here, is assumed to be a free parameter, and is added by hand.) The GUT-scale value of the -parameter (which is denoted as ) is expected to be of the same order of the gravitino mass. As we have mentioned,  GeV is required for non-thermal leptogenesis with avoiding the BBN constraints. So, we take  GeV as representative values.

Adapting the uncertainty in the Higgs mass calculation of GeV [22, 23], Higgs mass as large as GeV may be realized if the gluino mass is heavier than about  TeV [18]. Unfortunately, such a heavy gluino (and squarks) is out of the reach of the LHC experiments.

## 3 Enhanced A-term from extra matters

So far, we have seen that, if we adopt the particle content of the MSSM, the Higgs mass of is hardly realized in gaugino mediation model with gluino and squarks which are within the reach of the LHC experiment. Now, we show that such a conclusion is altered if there exist extra vector-like multiplets at around TeV. Many models such as grand unified theory and axion models predict the existence of the extra matters at the scale much lower than the Planck scale.

The existence of extra matters may enhance the Higgs mass via two effects. First, if the extra matters have sizable Yukawa interaction with the Higgs fields, the radiative correction below the SUSY scale may significantly enhance the lightest Higgs mass [24]. This is the case with extra matters, for example. Second, the presence of the vector-like multiplets changes the beta-functions of the gauge couplings and the gauginos. Consequently, the trilinear coupling of the stop becomes larger than the case without extra matters, resulting in the enhancement of the Higgs mass. The first effect has been intensively studied in recent works [25], so we concentrate on the second one, assuming that the extra matters have no Yukawa interaction with the Higgs fields.

First, we show how -parameters, a squark mass and the ratio of the -parameter to the squark mass are enhanced with extra matters. For this purpose, we use one-loop RGEs (although our numerical calculations are performed at the two-loop level). With the presence of the extra matters, the beta-functions of the gauge coupling constants and gaugino masses are given by

where are the gauge couplings for , and , respectively, and are the coefficients of the beta-functions with the MSSM matter content. The number of the extra vector-like multiplets in units of fundamental and anti-fundamental representation of GUT gauge group, , is denoted as . For , and at the SUSY scale are smaller than those at the GUT scale.555Although the one-loop beta-function of vanishes for , inclusion of the higher order corrections leads to the positive beta-function. The change of these beta-functions dramatically alters the squark masses and trilinear couplings at the SUSY scale. In particular, the changes of the stop masses and stop trilinear coupling lead to the important consequences in the Higgs boson mass and the SUSY search at the LHC.

Neglecting Yukawa couplings, the RGE of an -parameter of a squark can be written as

 dAqdlnQ = 116π2(cag2aMa) (9) = 116π2[ca(8π2ba+N5)dg2adlnQ(Mag2a)].

Notice that is an RGE invariant quantity, i.e., constant. The coefficient is common to all . (Here, we neglect the effects of Yukawa coupling constants, which do not change the following discussion qualitatively. Our numerical calculation will be performed with the effects of Yukawa coupling constants.) Eq. (9) can be solved as

 A(Q) = −ca2(ba+N5)[g2a(MGUT)−g2a(Q)](Mag2a) (10) = −ca2(g4a(Q)8π2lnMGUTQ)(1−ba+N58π2g2a(Q)lnMGUTQ)−1(Mag2a).

Taking the renormalization scale as the mass of the extra matter, , i.e., the decoupling scale, larger results in a larger for the fixed value of , since does not depend on ; the trilinear couplings including are enhanced for the fixed gluino mass.

Similarly, the RGE of a squark mass can be written as

 dm2~qdlnQ = 116π2(−dag2aM2a) (11) = 116π2⎡⎣−da(4π2ba+N5)dg4adlnQ(Mag2a)2⎤⎦,

where is common to all squark mass. By solving Eq. (11), we obtain

 m2~q(Q) = 14(daba+N5)[g4a(MGUT)−g4a(Q)](Mag2a)2 (12) = dag4a(Q)4(g2a(Q)8π2lnMGUTQ)[(1−ba+N58π2g2a(Q)lnMGUTQ)−2 +(1−ba+N58π2g2a(Q)lnMGUTQ)−1](Mag2a)2.

Again, becomes larger as increases for the fixed value of . Thus by adding extra matters, the stop mass is expected to be larger with the fixed value of the gluino mass at the SUSY scale. The ratio is also enhanced. Neglecting the contributions from and , the ratio becomes

 (A(Q)m~q(Q))2 = c23d3(b3+N5)(1−g23(Q)/g23(MGUT))(1+g23(Q)/g23(MGUT))−1 (13) = (c23d3)(g2(Q)8π2lnMGUTQ)(2−b3+N58π2g2(Q)lnMGUTQ)−1.

Taking , we obtain the enhanced ratio for larger . Consequently, the Higgs boson mass is enhanced as the number of the extra matters increases, because of the larger ratio of and the larger .

The results of the numerical calculations are shown in fig. 2. The gluino mass is fixed to be 1.2 TeV and in both left and right panels. In the left panel, the Higgs mass as a function of is shown. Three curves correspond to from bottom to top. For comparison, the dashed line, which is evaluated in the MSSM is also drawn. Remarkably, the Higgs mass reaches 125 GeV with and TeV. In the right panel, we also show the normalized trilinear coupling , where and , with and being the lighter and heavier stop masses. Three curves correspond to from top to bottom. The enhancement of can be seen. The number of the vector-like multiplets is motivated by the grand unified theory, and can be regarded as a complete vector-like family.

The contours of the constant Higgs mass for and TeV are shown in fig. 3 (left panel). The regions near the lines GeV are consistent with the boundary condition of the gaugino mediated SUSY breaking scenario. The Higgs mass is calculated to be GeV for the gluino mass of TeV. Correspondingly, a squark mass is TeV (right panel). Notice that these gluino and squarks may be observed at the LHC. In this region, the mass of the (right-handed) slepton is about GeV and the Bino (Wino) mass is GeV ( GeV), provided the universal gaugino masses at the GUT scale. Such relatively light non-colored SUSY particles may be seen at future linear collider experiments.

We also show that contours of the constant Higgs mass for and TeV in fig. 4. The Higgs mass of about 125 GeV (or larger) is realized in a wide region where the gluino mass is larger than about TeV. Correspondingly, the Bino and Wino masses are larger than 380 GeV and 630 GeV, respectively. Notice that the squark mass is larger than 3 TeV, which is too heavy to be observed at the LHC. The sleptons are also heavier than about 1 TeV. In the calculation, we adapt the universal gaugino masses at the GUT scale and the Bino is the next-to-the lightest SUSY particle (NLSP) in the whole region. (For the case without the GUT relation, see the discussion below.)

So far, we have seen that the existence of extra matters at the SUSY scale enhances the Higgs mass in the gauge mediated SUSY breaking scenario. More enhancement may be realized if there exist additional extra matters at the intermediate scale. One of the motivations to consider such extra matters at the intermediate scale is SUSY axion model. As an example of the enhancement due to the extra matters at the intermediate scale, we consider the case where there exists one pair of extra matters at GeV 666The mass can be smaller than the Peccei-Quinn (PQ) symmetry breaking scale, i.e., for instance, Yukawa coupling of  PQ breaking scale. (as well as three pairs of extra matters at TeV scale). In fig. 5, contours of the Higgs mass (left panel) and squark mass (right panel) are shown.

Finally, we comment on cosmological implication of Bino-like neutralino as the NLSP. In the early universe, neutralinos are produced and may decay after the BBN epoch in particular when R-parity is conserved. Hadro- and photo-dissociation processes are caused by the decay of the neutralino, and the success of the BBN may be spoiled if the lifetime of the neutralino is longer than sec [11]. In the present case where , the parameter region which we are interested in mostly conflicts with the BBN constraints.777If stau is the NLSP, which may be the case in the gaugino mediation model without extra matters, the constraint is much weaker [11]. Assuming that stau decays into tau and gravitino via supercurrent interaction, the stau mass is required to be larger than 200 GeV for (with being the mass of stau), in order for successful BBN scenario. In addition, for , no constraint is obtained. Such a problem may be solved if the NLSP is Wino-like neutralino instead of Bino-like neutralino.888The GUT relation among gaugino masses may not hold even if the SM gauge couplings are unified. For example, in the product-group unification scenario [26], this is the case [27]. In such a case, the thermal relic abundance of the NLSP is significantly suppressed, and the BBN constraints are relaxed. Interestingly, if the Wino-like neutralino is the NLSP, the signal of the Wino production may be observed at the LHC [28, 4]. Another possibility to avoid the confliction with the BBN constraints is to introduce small R-parity violation.

## 4 Conclusion and discussion

In this letter, we have considered a gaugino mediated SUSY breaking scenario without the Polonyi problem. The gaugino mediation naturally occurs with the requirements for avoiding the serious Polonyi problem. With this setup, we have evaluated the Higgs boson mass and found that the Higgs mass of around 125 GeV may be realized with gluino mass heavier than about 4 TeV in the MSSM. Unfortunately, such heavy colored SUSY particles are out of reach of the LHC experiment.

However, if there exist a number of extra matters at TeV, the Higgs boson mass can be significantly enhanced with relatively small gluino mass of TeV. This is because the trilinear coupling of the scalar top is enhanced by the change of the RGEs of the gauge coupling constants and gaugino masses. With the uncertainty in the calculation of the Higgs boson mass ( GeV), the squark mass of around 2 TeV becomes consistent with observed value of the Higgs boson mass. Such gluino and squarks are within the reach of the LHC experiment. In addition, it is notable that the masses of other SUSY particles can be much below ; in the parameter region of our interest, the masses of slepton, Bino, and Wino are about GeV, GeV, and GeV, respectively. These non-colored SUSY particles can be targets of future linear collider experiments.

In this letter, we have concentrated on the scenario in which only the gauge multiplets (and the inflaton) have enhanced couplings to the Polonyi field. From the point of view of solving the SUSY FCNC problem, the Higgs fields may also strongly couple to the Polonyi field. If so, the behaviors of the SUSY breaking parameters may be significantly altered [18]. In particular, the cut-off-scale values of the -parameters and soft SUSY breaking Higgs masses are expected to become much larger than the gravitino mass. Such a scenario will be studied in a separate publication [29].

Our scenario is consistent with the cosmological observation. The baryon asymmetry can be from non-thermal leptogenesis with relatively high reheating temperature as GeV. The gravitino is the LSP and the candidate for dark matter. If the R-parity is conserved, the gravitino mass of , which is expected in the present scenario, may conflict with the BBN constraints in particular when the Bino-like neutralino is the NLSP. However, the BBN constraints can be avoided if the NLSP is Wino-like neutralino or if a small violation of R-parity is introduced. Even with a small R-parity violation, the gravitino can be long-lived enough to be a viable candidate for dark matter [30, 31] .

## Acknowledgment

This work is supported by the Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 22244021 (T.M. and T.T.Y.) No. 23104008 (T.M.), No. 60322997 (T.M.), and also by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The work of N.Y. is supported in part by JSPS Research Fellowships for Young Scientists.

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