Soft Leptogenesis and Gravitino Dark Matter in Gauge Mediation

We study soft leptogenesis in gauge mediated supersymmetry breaking models with an enhanced A-term for the right-handed neutrino. We find that this scenario can explain the baryon asymmetry of the present universe, consistently with the gravitino dark matter for a wide range of gravitino mass m_{3/2}=O(MeV)-O(GeV). We also propose an explicit model which induces the necessary A-term for the right-handed neutrino.


Introduction
Gauge-mediated supersymmetry breaking (GMSB) [1,2,3,4,5,6,7,8,9,10] is an attractive way of communicating supersymmetry (SUSY) breaking effects to the supersymmetric standard model (SSM), since flavor-changing neutral currents (FCNC) and dangerous CP violating phases are naturally suppressed. In addition, the mass spectrum of the superparticles in the SUSY standard model sector is determined by only a few parameters, which may be tested at the LHC in the near future.
In GMSB, the gravitino is the lightest SUSY particle and stable with R-parity. Therefore the gravitino is a candidate for the dark matter. In fact, gravitinos are produced by the scattering processes of thermal particles after the inflation [11,12,13], and its abundance is given by where Ω 3/2 and m 3/2 are the density parameter and the mass of the gravitino, respectively, h ≃ 0.73 is the normalized Hubble parameter, mg is the gluino mass, and T R is the reheating temperature after the inflation. Therefore, the gravitino becomes a viable dark matter candidate for T R < ∼ O(10 7 ) GeV × (m 3/2 /0.1 GeV), or it can explain the observed cold dark matter density, Ω CDM h 2 ≃ 0.11 [14], if the reheating temperature saturates the bound. Furthermore, the notorious inflaton-induced gravitino problem [15], which excludes most of the inflation models in the gravity-mediated SUSY breaking scenario, can be avoided in GMSB models.
However, another big puzzle in cosmology, the origin of the matter anti-matter asymmetry of the universe, is not easy to solve in this framework: • In the standard thermal leptogenesis with heavy right-handed (RH) neutrinos [16], there is a lower bound on the mass of the RH neutrino, M N 2 × 10 9 GeV [17,18], which requires a high reheating temperature T R > O(10 9 ) GeV. This would lead to a too much gravitino abundance for m 3/2 < ∼ O(10) GeV [cf. Eq. (1)].
• Electroweak baryogenesis [25,26,27] seems also difficult because the necessary ingredients, a first order phase transition and sufficient CP phases, are absent in GMSB.
In this paper, we would like to propose a viable baryogenesis scenario in GMSB, which is consistent with the gravitino dark matter for a wide range of gravitino mass O(MeV)-O(GeV). 1 The framework is a simple GMSB model supplemented by an enhanced A-term for the RH neutrino, and the baryon asymmetry is produced by the soft leptogenesis [36,37].
Soft leptogenesis [36,37] is an attractive way of generating baryon asymmetry. The SUSY breaking terms introduce a mixing between the RH sneutrinos and their antiparticles. This induces significant CP violation in sneutrino decays in similar ways to B 0 -B 0 and K 0 -K 0 mixings. An attractive feature of the soft leptogenesis is that M N (and T R ) can be smaller than that in the standard leptogenesis and therefore there is a possibility of generating baryon asymmetry without generating too much gravitino dark matter.
Interestingly, a successful soft leptogenesis favors a small B-term for the RH neutrino, which is naturally realized in the framework of GMSB [38]. In Ref. [38], the authors investigated the soft leptogenesis in a minimal GMSB setup, and found a viable parameter region with very light gravitino m 3/2 16 eV. In the minimal setup, the RH neutrino Aterm is suppressed, and hence sufficient baryon asymmetry cannot be generated for m 3/2 O(100 eV) satisfying the gravitino constraint. We extend their study with an enhanced Aterm, and show that there is a viable region with with the gravitino dark matter. We also show an explicit model which generates an enhanced A-term through the coupling between the messenger and up-type Higgs, without introducing additional unwanted CP phases in the low energy.
This paper is organized as follows: In section 2, we briefly review soft leptogenesis, and then show that the baryon asymmetry can be explained in our scenario. In section 3, we introduce a concrete model which generates the necessary A-term through a coupling between the up-type Higgs and the messenger. Section 4 is devoted to summary and discussion.
Let us first briefly review the soft leptogenesis following Ref. [37]. We consider only the lightest RH neutrino and sneutrino for simplicity. The superpotential for the RH neutrino is given by where N, L i and H u are the chiral superfields for the RH neutrino, the lepton doublets, the up-type Higgs, respectively. The soft SUSY breaking terms containing RH sneutrinõ M N and Y ν,i are taken to be real by redefining the phases of the superfields, N and L i .
SUSY breaking terms introduce the mixing betweenÑ andÑ * , which induces lepton asymmetry in decays ofÑ andÑ * . Then, the generated lepton asymmetry is converted into the baron asymmetry through the sphaleron process [39]. The baryon to entropy ratio is given by [ where Γ is the width of the RH sneutrinos θ is the CP phase given by θ = arg(B ν A * ν ), and η is a factor which describes the effects caused by the inefficiency in the production of the RH sneutrinos, the wash-out effects, and v = 174 GeV is the vacuum expectation value of Higgs. Now let us estimate the baryon asymmetry by using Eq. (4) in our setup. For simplicity, we assume m 1 ≃ 10 −3 eV and η ≃ 0.1, 3 which leads to We also assume that B ν is dominated by the gravity-mediation contribution, |B ν | ∼ m 3/2 , 4 as discussed in Ref. [38]. Fig. 1 shows Now we estimate the required size of A ν . As discussed above, smaller M N is favored by the constraint from the gravitino abundance, which corresponds to Γ < |B ν | [cf. Eq. (7)].
By taking Γ ≪ |B ν |, we obtain Therefore in order to explain the observed value of n B /s, |A ν | ≃ (100 GeV − 10 TeV) is required for |B ν | ≃ (0.01 − 1)m 3/2 . However, such a large A-term is not generated in a minimal GMSB. In fact, it was found [38] that a successful soft leptogenesis and the gravitino constraint require an ultralight gravitino m 3/2 < ∼ 16 eV, as far as the A-term is generated through the renormalization-group evolutions. In the next section, we show an explicit model which can generate a large A-term for the RH neutrino.

A Model
In this section, we give a concrete model which generates the enhanced A ν , through a new coupling between the messenger field and the up-type Higgs. We will also show that In GMSB, the messenger mass is given by the following superpotential: where Ψ andΨ are messenger superfields and transform as 5 and5 under the GUT SU(5), respectively. X is a superfield which has a scalar and an auxiliary vacuum expectation values, X = M + F θ 2 . We consider only one pair of messengers for simplicity, however an extension to the multi-messenger case is straightforward.
In order to generate the A-term, L = −A ν Y νL H uÑ + h.c., we consider the following superpotential [43,44]: whereΨ ℓ is the leptonic part of the messengerΨ and S is a gauge singlet. In order to forbid unwanted terms, such as NH uΨℓ and NH d Ψ ℓ , 6 we introduce a messenger parity Z ′ 2 in addition to the R-parity (equivalently matter parity). S, Ψ andΨ are odd and the others are even under Z ′ 2 . There is also another term, λ ′ SH d Ψ ℓ , which is allowed by the symmetry. However, this term is irrelevant to the generation of A ν , and therefore we neglect it in following discussion, for simplicity.
We take λ and the messenger mass M to be real by redefining the phases of Ψ and Ψ. We assume that the mass of S, M S satisfy the relation, M N < M S < M. The term in Eq.(12) decouples after integrating out the messenger superfields. The leading 6 If these term exist, the large B ν can be generated.
contribution to the A-term which is proportional to (F/M) is given by the one-loop diagram expressed in Fig. 2. This contribution is also extracted from the wave-function renormalization of H u by the analytic continuation method [43]. The leading term of A ν is given by which can be of the order of 100 GeV − 1 TeV.
The corresponding terms in the superpotential and soft breaking terms are defined by We assume that the Higgs µ term is generated above the messenger scale. Under this assumption, there is no physical phase from GMSB, since the phases of the soft breaking parameters are the same, arg(F/M) and we can remove them by the U(1) R transformation.
On the other hand, the neutrino B-term, B ν is generated by the gravity-mediation and the order of gravitino mass. Therefore its phase is expected to be completely different from arg(F/M). With the U(1) R transformation and a phase transformation of H u , the parameters transform as, If we choose 2θ R = −arg(F/M) and θ Hu = −arg(µ) − arg(F/M), only B ν is complex and its phase is arg(B ν F * /M). Therefore the new interaction term does not lead to large CP violation in low energy phenomena. 8 We considered soft leptogenesis in gauge mediated SUSY breaking scenario, including the simple interaction term which contains up-type Higgs and leptonic part of the messenger.
The interaction term generates A ν , A u , soft SUSY breaking mass for up-type Higgs and squarks, and Higgs B-term. With the large A ν soft leptogenesis works successfully, which is consistent with the gravitino dark matter for a wide range of gravitino mass. The phases of A ν , A u and Higgs B-term are aligned with those of other SUSY breaking terms from gauge mediated SUSY breaking. Therefore inclusion of the new interaction term does not lead to large CP violation in low energy phenomena. Interestingly, the additional contributions to the soft terms lead to a different spectrum pattern of SUSY particles from that of ordinary gauge mediation, which may be tested at LHC in the near future.