Lower limit on the gravitino mass in low-scale gauge mediation with $m_H\simeq 125$GeV

We revisit low-scale gauge mediation models in light of recent observations of CMB Lensing and Cosmic Shear which put a severe upper limit on the gravitino mass, $m_{3/2} \lesssim 4.7$eV. With such a stringent constraint, many models of low-scale gauge mediation are excluded when the squark masses are required to be rather large to explain the observed Higgs boson mass. In this note, we discuss a type of low-scale gauge mediation models which satisfy both the observed Higgs boson mass and the upper limit on the gravitino mass. We also show that the gravitino mass cannot be smaller than about 1eV even in such models, which may be tested in future observations of 21 cm line fluctuations.


I. INTRODUCTION
Low-scale gauge mediation models with a light gravitino mass, m 3/2 < O(10) eV, is very attractive, since the gravitino with a mass in this range does not cause astrophysical nor cosmological problems [1][2][3]. In particular, such a light gravitino is consistent with high reheating temperature which is essential for many baryogenesis scenarios as typified in thermal leptogenesis [4] [see 5-7, for review]. A small gravitino mass is also motivated since it may require a milder fine-tuning of the cosmological constant due to a smaller supersymmetry (SUSY) breaking scale.
Recently, a severe upper limit on the mass of the light stable gravitino, m 3/2 4.7 eV (95% C.L.), has been put from CMB Lensing and Cosmic Shear [8]. With such a stringent constraint, many models of low-scale gauge mediation are excluded when the squark masses are required to be rather large to explain the observed Higgs boson mass, m H 125 GeV [9].
For example, we immediately find that the above upper limit on the gravitino mass excludes models in which the messenger fields couple to supersymmetry breaking sector perturbatively [10].
In this short note, we point out that the low-scale gauge mediation model can explain the observed Higgs boson mass even for m 3/2 4.7 eV when the messenger fields strongly couple to the SUSY breaking sector. We also show that the gravitino mass cannot be smaller than about 1 eV even in such models, which can be tested by future observations of 21 cm line fluctuations [11].

II. MODELS WITH LOW-SCALE GAUGE MEDIATION
A. Low-scale gauge mediation and Higgs boson mass In this note, we are interested in models with a gravitino mass in the eV range. For such a light gravitino mass, the SUSY breaking scale must be low as, Now, let us suppose that there are N m pairs of Ψ andΨ which are in the fundamental and The messenger fields couple to a SUSY breaking sector via a superpotential, Here, the SUSY breaking sector is encapsulated in Z whose vacuum expectation value (VEV) is assumed to be Z = θ 2 F . The coupling between the messenger fields and the SUSY breaking sector is given by the term proportional to y. In this simple setup, the mass splitting between the messenger scalars and the fermions is given by It should be noted that the mass splitting is required to be smaller than the messenger mass scale, M m , i.e., to avoid the tachyonic messenger fields.
Below the messenger scale, the masses of superparticles are given by, Here, C 2 is the quadratic Casimir invariant of representations of each sfermion, and g represents gauge coupling constant of the minimal SUSY Standard Model (MSSM). To satisfy the cosmological constraint on the gravitino mass, m 3/2 < 4.7 eV, the SUSY breaking scale is required to be √ F 140 TeV (see Eq. (1)). By combined with the non-tachyonic messenger condition Eq. (4), we find that the soft terms are limited from above; m gaugino N m g 2 16π 2 (yF ) 1/2 0.9 TeV × N m y 1/2 g 2 m 3/2 4.7 eV at the messenger scale. Let us discuss whether the above soft masses can be consistent with the observed Higgs boson mass, m H 125 GeV. In the MSSM, the Higgs boson mass is constrained as m H m Z at the tree level, which is enhanced by the top Yukawa radiative corrections [12]. Then, the observed Higgs boson mass, m H 125 GeV, requires the squark masses (in particular the stop masses) in multi-TeV range, which is in tension with the squark masses in Eq. (7) for In Fig. 1, we show the parameter region which is consistent with the observed Higgs boson mass, m H = 125.09 ± 0.21 ± 0.11 GeV [9]. In the figure, the Higgs boson mass is consistently explained at the 2σ level in the blue and red shaded regions for y = 1 and y = 4π, respectively. 1 Here, we take N m = 4, so that the resultant squark masses are as large as possible for given M m and F while keeping the perturbativity of the gauge coupling constants in the MSSM up to the scale of the grand unification. In our numerical analysis, we use softsusy-3.7.3 [13] to solve the renormalization group evolution of the 1 In our analysis, we define the χ 2 estimator, where δm H denotes the theoretical uncertainty.
take scheme dependences of the Higgs mass estimations into account, we also estimated the Higgs boson mass by using another code susyHD [15]. The corresponding parameter regions are shaded by darker blue/red. The figures show that the results obtained by using susyHD require slightly higher SUSY breaking scales (and hence heavier squark masses) to achieve the observed Higgs boson mass.
As the left panel shows, the region which is consistent with the observed Higgs boson mass is excluded by the cosmological constraint on the gravitino mass when the messenger fields couple to the SUSY breaking sector perturbatively. 2 When the messenger couple to the SUSY breaking sector strongly, i.e. y = O(4π), on the other hand, the observed Higgs boson mass can be explained even for m 3/2 4.7 eV. The figure also shows that there is a lower limit on the gravitino mass from the observed Higgs boson mass, According to Ref. [11], the gravitino dark matter in this range, i.e. Omniscope [18,19].
In the figures, the green shaded regions are excluded by SUSY searches at the LHC. By remembering that the next-to-lightest SUSY particle (NLSP) is the tau slepton for N m = 4, we show the lower limit on the gluino mass, m gluino 2.2-2.3 TeV, from the null results of searches for the tau slepton [20]. 3 The figure shows that the cosmological constraints are more stringent compared with the constraints put by direct searches at the LHC.

B. Strongly interacting models
Here, let us illustrate how the strong coupling between the messenger sector and the SUSY breaking sector, i.e. y = O(4π) is achieved. When these two sectors are strongly interacting, the vacuum structure of the SUSY breaking sector is inevitably affected by the coupling to the messenger fields, and in most cases, SUSY breaking vacua are destabilized.
To avoid this problem, we need to assume that the messenger fields couple to a secondary SUSY breaking as realized in models of "cascade SUSY breaking" [22] (see also [23][24][25] for earlier works). There, a secondary SUSY breaking field S couples to the primary SUSY breaking field Z with Z = F θ 2 only through the Kähler potential, where Λ is the dynamical scale of the SUSY breaking sector (see [22] for more details). By using the Naive Dimensional Analysis (NDA) [26,27], the coefficient κ is expected to be of O((4π) 2 ) when both Z and S take part in the strong dynamics with the dynamical scales of O(Λ). By using the NDA, the primary SUSY breaking scale is also estimated to be 4 As a result, the term proportional to κ leads to a soft SUSY breaking mass of S, 5 Now, let us suppose that S and the messenger fields are composite states of some dynamics so that they couple in the superpotential with n ≥ 3. 6 By the NDA again, we expect k = O((4π) n−2 ) and λ = O((4π) n−2 ). Then, the scalar potential of S is roughly given by, 3 This limit is put by assuming N m = 3 in [20]. For N m = 4, the constraint might become slightly more stringent due to a relative smallness of the squark mass for N m = 3. When, the NLSP is the lightest neutralino, the lower limit on the gluino mass is slightly tighter, m gluino 1.6-1.7 TeV, which has been put by the null results of searches for the photons with missing energy [21]. 4 If we assume IYIT SUSY breaking model [28,29], this is achieved when the coupling between the gauge singlet and SP (N c ) fundamental quarks are of O(4π). 5 Here, we assume κ > 0 for simplicity, although a model in [22] is viable even for κ < 0. 6 A model with n = 5 is achieved in [22]. which leads to the VEVs of S, Putting these VEVs into the superpotential in Eq. (14), the messenger fields obtain their masses and the mass splittings, As another example to enhance the Higgs boson mass, it is also possible to introduce vector-like matter fields coupling to the Higgs doublets [32][33][34][35][36][37][38][39][40]. In those extensions, however, the more the vector-like matters are added, the severer upper limit on the messenger number 7 See also [30] for another strongly interacting messenger model. 8 Here, we assume that the NMSSM respects the Z 2 symmetry. If we allow Z 3 violating terms, it is possible to explain m H 125 GeV without having the Landau pole problem. In such cases, however, we generically suffer from tadpole problem and fine tuning problems.
from the perturbativity of the MSSM gauge coupling constants is put. With fewer messenger fields, the sparticle masses are difficult to be above the LHC constraints for m 3/2 1 eV.
One may also consider the extension of the MSSM with an additional U (1) gauge symmetry. In fact, the Higgs boson can be enhanced by the associated D-term potential of the new U (1) gauge interaction when the Higgs fields are charged under the symmetry [41, for review]. For that purpose, however, we need to require that the soft SUSY breaking masses of U (1) breaking fields should be of the order of the VEV of U (1). In view of stringent constraints on Z gauge bosons put by the LHC searches [42,43], the required SUSY breaking mass is at least in a few TeV range. Since we are assuming gauge mediation, the soft masses of U (1) breaking fields should also be provided by gauge mediation. As a result, for Let us also comment on the models with gauge mediation where the Higgs doublets and the messenger fields have small mixings [44][45][46][47][48]. In this class of models, a rather large A-terms are generated which enhances the Higgs boson mass. As a result, the observed Higgs boson mass can be explained for the sparticles masses in a few TeV range [45]. In such models, however, they generically suffer from instability problem of the SUSY breaking vacuum when the messenger scale is close to the SUSY breaking scale [49]. 9

III. CONCLUSIONS AND DISCUSSIONS
In this paper, we revisited low-scale gauge mediation models in light of recent observation of CMB Lensing and Cosmic Shear which put a severe upper limit on the gravitino mass, m 3/2 4.7 eV. Such a stringent constraint excludes wide range models of low-scale gauge mediation when the squark masses are required to be rather large to explain the observed Higgs boson mass. In this note, we pointed out that strongly interacting low-scale gauge mediation still survives even if we require that the models satisfy both the observed Higgs boson mass and the upper limit on the gravitino mass. We also show that the gravitino mass cannot be smaller than about 1 eV even when the messenger fields strongly couple to the SUSY breaking sector. 9 Details of this type of models in the context of the low-scale gauge mediation will be discussed elsewhere.
As an interesting aspect of the strongly coupled low-scale gauge mediation it may naturally provide dark matter candidate, the baryonic composite states in the SUSY breaking sector or the messenger sector [22,30,[50][51][52][53]. The baryonic composite states are given by higher dimensional operators, and hence, they couple to the SM particles very weakly. As a result, they are expected to be long lived. Furthermore, the annihilation cross section of the baryonic composite states via strong interaction can saturates the unitarity limit, which requries the dark matter mass of O(100) TeV so that the dark matter density can be explained by the thermal freeze-out [54]. 10 Therefore, the baryonic composite states of the strong dynamics at around O(100) TeV in the low-scale gauge mediation naturally explain the observed dark matter density.
Taking the thought one step further, this observation might provide an interesting perspective on the naturalness problem. Let us consider a distribution of the SUSY breaking scale in the ensemble of vacua (or theories) [56,57], which is expected to be biased towards a lower scale for a flat universe. When dark matter is provided as composite states of the strongly coupled low-scale gauge mediation, the dynamical scale cannot be much smaller than O(100) TeV to avoid the lack of dark matter due to a large annihilation cross section.
Thus, the final distribution should have a peak at around O(100) TeV, since the scale lower than O(100) TeV is not habitable. 11 Therefore, in this interpretation, the Higgs boson mass and rather heavy squark masses are outcomes of the cosmological selection on the dark matter density [30]. For some scenario, the effective annihilation cross section of baryonic composite states can exceed the unitarity limit on each partial wave modes [55]. 11 See e.g. [58,59] for more on habitable conditions. 12 See [60] for related discussion on the cosmological selection.