R-parity Violating Right-Handed Neutrino in Gravitino Dark Matter Scenario

A decay of the gravitino dark matter is an attractive candidate to explain the current excesses of the PAMELA/ATIC cosmic-ray data. However, R-parity violations are required to be very tiny in low-energy scale. We suggest a R-parity violation in the right-handed neutrino sector. The violation is suppressed by a see-saw mechanism. Although a reheating temperature is constrained from above, the thermal leptogenesis is found to work successfully with a help of the R-parity violating right-handed neutrino.


I. INTRODUCTION
Observations of comic rays have been greatly developed. After a series of cosmic-ray measurements, PAMELA recently published the first result of the positron fraction, which show an excess compared to the background at energies above 10GeV [1]. Interestingly, the ATIC collaboration also reported an excess of the electron plus position flux in a range of O(100)GeV [2]. Although the experimental data and the background estimations still involve ambiguities, the excesses may be a sign of the dark matter (DM). In this letter, we focus on decaying DM scenarios. The DM is not always required to be stable as long as its life time is long enough for the DM to survive until today. Then, its decay products are a possible source of the energetic cosmic rays.
It is non-trivial to realize such a long-lived particle. Actually, decay operators must be incredibly suppressed for the models to be viable. A well-motivated candidate of the DM is known to be the gravitino with a broken R-parity. This is because its decay is doubly suppressed by the Planck scale and R-parity violations [3]. Even with the Planck suppression, the R-parity violation is limited to be very tiny and looks unnatural unless some mechanisms are taken into account. As a solution of the fine-tuning, we propose a R-parity violation introduced in the right-handed neutrino sector. Then, the violation is suppressed in the low-energy scale by a see-saw mechanism. Actually, since the violation appears in low-energy phenomena though the right-handed neutrino, its effect becomes suppressed by the right-handed neutrino mass scale. We will show that the PAMELA/ATIC excesses are explained for M N ∼ 10 9 GeV without a parameter tuning.
This type of the R-parity violation does not only explain the PAMELA/ATIC anomalies, but also opens a window for the successful thermal leptogenesis [4] especially in a large gravitino mass region. Indeed, the mechanism is an attractive and elegant solution to the mystery of the baryon asymmetry of the universe, though it usually requires a relatively high reheating temperature, T R > ∼ 10 9 GeV (see e.g. [5]). In such a high reheating temperature, the severest cosmological constraints are provided by the big-bang nucleosynthesis (BBN) and the overclosure of the universe. The former constraint is due to a prolonged decay of the next-to-lightest superparticle (NLSP) into the gravitino lightest superparticle (LSP). In contrast, the NLSP can decay into the Standard Model (SM) particles much more efficiently through the R-parity violating operator. Thus, the BBN bound becomes ameliorated [6].
The overclosure constraint then puts a constraint on the reheating temperature. In particular, the thermal gravitino production is enhanced in a larger soft mass region. According to PAMELA/ATIC, the cosmic-ray excesses imply a heavy gravitino, namely large superparticle masses. We will see that the leptogenesis temperature is unlikely to be satisfied for m 3/2 > ∼ 1TeV unless the gluino mass parameter is relatively small at the GUT scale.
Nevertheless, since the R-parity violation in this letter contributes to a CP-asymmetric right-handed neutrino decay [7], the thermal leptogenesis will be shown to work to explain the observed baryon asymmetry easily. Thus, we will study the R-parity violating scenario in the right-handed neutrino sector as an attractive framework of the decaying gravitino DM.

II. R-PARITY VIOLATION IN RIGHT-HANDED NEUTRINO SECTOR
Let us consider the gravitino DM scenario with a decay due to R-parity violations. A spectrum of the cosmic rays from its decay products are determined by the type of the R-parity violating operator, the gravitino mass and its life time. In a case of the bilinear R-parity violation, the dominant decay channels are ψ 3/2 → W ℓ, Zν, γν and hν [8,9,10].
The decay is parametrized by the sneutrino vacuum expectation values (VEV). With the where L and H are the left-handed lepton and up-type Higgs, respectively. The parameters are the higgsino mass µ, the Higgs VEV v ≃ 174GeV, the Higgs VEV ratio tan β = H / H , and the sneutrino soft mass V soft = m 2 νν * ν . Then, the gravitino life time is estimated as for m 3/2 ≫ 100GeV, where η is defined as η ≡ | ν |/v. According to the current PAMELA/ATIC data, the mass and life time of the gravitino is favored to be which leads to the R-parity violation parameter as η ∼ 10 −10 m 3/2 1TeV This η satisfies with the other cosmological constraints [9]. The gravitino survives until today as η ≪ 10 −(5−6) for m 3/2 = O(100)GeV, and the parameter is small enough to avoid the wash-out of the lepton/baryon asymmetry since η < 10 −7 is satisfied. Furthermore, the BBN constraint is solved for η > ∼ 10 −(11−12) by virtue of the R-parity violation, and the Rparity violating contribution with η < 10 −7 is negligible for the neutrino mass. We assume that the neutrino mass is obtained by the see-saw mechanism in this letter.
However, from the naturalness point of view, the violation is too tiny and must be finetuned. We propose a simple framework to realize this tiny R-parity violation. We introduce the following R-parity violating operator in the right-handed neutrino sector, Accompanied by the right-handed neutrino mass and the Yukawa terms, W = M N NN/2 + Y N NLH, after decoupling the heavy right-handed neutrino, we obtain the effective R-parity violating operators in low-energy scale: It is noticed that the R-parity violating effects appear with 1/M N because they contribute through the right-handed neutrino. Thus, the R-parity violating effects are naturally suppressed by the see-saw mechanism.
Taking the Higgs VEVs, (6) behaves as a bilinear R-parity violating operator. Actually, combined with the µ term and the sneutrino soft mass, the sneutrino acquires VEV as Then, η is estimated to be wherem ν is a typical scale of the light neutrino masses defined as Here, we omitted flavor indices. As a result, we find that the gravitino life time τ 3/2 ∼ 10 26 sec required from PAMELA/ATIC is realized for λ ∼ 1 and M N ∼ 10 9 GeV. We want to emphasize that there is no fine-tuning in the R-parity violating parameters.
The gravitino decay is almost the same as that by the bilinear R-parity violation. In fact, it is dominantly induced by the sneutrino VEV (7), while the other decaying channels from (6) are subdominant. Although W = −λ 2 (HH) 2 /2M N is also derived from (5), the resultant decay operators are negligible for the cosmic ray spectra.
According to the current cosmic-ray data, the positron spectrum has a steep rise in highenergy region. This behavior prefers the gravitino DM decaying direct into the positron or anti-muon [11,12]. The decay product is determined by flavour structure of the sneutrino VEV. Denoting the flavour indexes explicitly in (7), Let us comment on radiative corrections with the R-parity violation (5). They induce a linear term of the right-handed neutrino in the Kähler potential. This then leads to the right-handed sneutrino VEV in the framework of the supergravity. Although this can give a similar contribution to the effective R-parity violating operators (6), the results in this letter do not change qualitatively. Thus, we will neglect it hereafter.
We may also have the R-parity violating operators in the right-handed neutrino sector other than (5) such as the superpotential terms, N and N 3 . However, their couplings are required to be suppressed since they can induce too large R-parity violations through the right-handed sneutrino VEV within the supergravity. Thus, we consider that the operators are forbidden by (discrete) symmetries.
In the above, we focused on the SUSY invariant operators for the R-parity violations.
Apart from them, we may have the violations in the SUSY breaking term. It can be noticed that the analysis is similar to the above, and the result is almost the same.

III. UPPER BOUND ON REHEATING TEMPERATURE
In the decaying gravitino DM scenario, the cosmological constraint comes from the overclosure of the universe. Actually, the gravitino is thermally produced in the hot plasma. At the leading order, the relic abundance is evaluated as where the definition of the parameters are found in [13]. In the numerical analysis, we include the electroweak contributions, which can be sizable especially when the gluino mass parameter is rather small compared to the Bino and Wino ones at the GUT (namely T R ) scale. It should be mentioned that the gravitino production rate includes an O(1) uncertainty from unknown higher order contributions and nonperturbative effects [14]. In addition, resummation of thermal masses potentially increases the rate by about a factor of two a [15]. Since the massive gravitino behaves as a cold dark matter, its abundance is constrained from the measurements. According to the WMAP 5-year data, the abundance is required to satisfy [17], at the 2σ level. Thus, provided the gaugino and gravitino masses, we obtain an upper bound on the reheating temperature.
It is interesting to study the overclosure bound in terms of the superparticle masses at the weak scale, which are expected to be measured by LHC. From (9) we notice that the gravitino production is evaluated with the parameters at the reheating temperature scale.  [14] (see also the discussion in [18]). In the following numerical analysis, we evolve the renormalization group running at the 2-loop accuracy and include the 1-loop threshold corrections by means of SOFTSUSY 2.0.18 [19].
In Fig. 1, we plot the maximal reheating temperature allowed by the overclosure bound with varying the gluino mass M 3 compared to the gaugino mass M 1/2 at the GUT scale.
Here, the Bino and Wino masses are set to be equal M 1 = M 2 ≡ M 1/2 and realize the NLSP mass m NLSP on each solid lines. Since the universal soft scalar mass m 0 except for the a Apart from the thermal production, we also have nonthermal contributions to the gravitino abundance, particularly from inflaton decay [16]. Since they are model dependent, we neglect them for simplicity.
Higgs ones is chosen to be m 0 = M 1/2 , the Bino-like neutralino is the NLSP, while the Higgs mass squareds are assumed to be m 2H = m 2 0 and m 2 H = −m 2 0 for the electroweak symmetry breaking to give arise especially in a large m 0 region. The scalar trilinear couplings, a 0 , and tan β are a 0 = 0 and tan β = 30, though they are irrelevant for the maximal reheating temperature.
The gravitino mass is relevant for the upper bound on the reheating temperature. We assumed that it is equal to the NLSP mass. This turns out to maximize T R for M 3 /M 1/2 > 0.5. Although the possible maximal temperature is obtained by a smaller gravitino mass for a lighter gluino region, the difference of T R is less than 10% even for M 3 /M 1/2 = 0.2. It is commented that the gluino can become the NLSP for M 3 /M 1/2 < 0.2.
We find that the maximal reheating temperature can be as large as O(10 8 )GeV in a large parameter region, while T R ≥ 10 9 GeV is realized for a smaller NLSP mass, e.g.

m NLSP
< ∼ 600GeV for the universal gaugino mass M 3 = M 1/2 . This is because the thermal gravitino abundance (9) increases as the soft mass scale is larger. From Fig. 1, we see that T R > 10 9 GeV leads to m NLSP < 1.5TeV for M 3 /M 1/2 > 0.2 (see [25] for a hierarchical gaugino mass case). According to PAMELA, the gravitino mass is indicated as Then, it is easy to obtain the leptogenesis temperature in a wide class of models (see [18], in which discussions of superparticle mass spectrum are also explored).
Thus, the SUSY breaking models become specified to realize the leptogenesis temperature.
A lighter gluino mass parameter is favored at the GUT scale, and thus a mass spectrum of the superparticles tends to degenerate at the weak scale especially when the gravitino mass is heavier b . On the other hand, the universal gaugino mass models look inconsistent with the thermal leptogenesis in the light of the ATIC result, though the usual gauge-and many gravity-mediation models have this feature. Thus, we will revisit the thermal leptogenesis in the next section in the presence of the R-parity violation.
b This mass spectrum looks like that of the mirage mediation [26]. However, it is usually obtained by enhancing the anomaly mediation, namely with a large gravitino mass, e.g. m 3/2 > ∼ 100TeV. Thus, the gravitino is not the LSP. Before proceeding to the next section, let us comment on the other parameter dependence.
In the analysis, we take m 0 = M 1/2 at the GUT scale. For tiny m 0 , the stau can be the NLSP. Then, the upper constraint in Fig. 1 becomes severer for fixed m NLSP , since the gaugino masses increase. Thus, the model parameters discussed above are limited more severely.

IV. THERMAL LEPTOGENESIS REVISITED
We now revisit the thermal leptogenesis in the presence of the R-parity violating operator (5). In the above, we found that a gluino mass is bounded from above to realize the leptogenesis temperature T R > ∼ 10 9 GeV in a large gravitino mass region. In particular, the universal gaugino mass models are inconsistent with the temperature if we take the ATIC result.
On the other hand, we have introduced the R-parity violating operator in the right-handed neutrino sector. We will see that a CP asymmetry of the decay of the right-handed neutrino is enhanced by the violation [7], and thus the leptogenesis mechanism works successfully.
In the thermal leptogenesis, the baryon density relative to the photon density becomes (cf. [20]) n B n γ ≃ −1.04 × 10 −2 ǫ 1 κ, which is compared to the observation [17], Here and hereafter, we assume the hierarchical right-handed neutrinos and ignore the flavour effects which does not affect the lower bound on the reheating temperature. The efficiency factor κ represents effects of washout and scattering processes, which is obtained by solving the Boltzmann equations. For the case of zero initial abundance of the right-handed neutrinos, its maximal value is known to be κ ≃ 0.2 [5,21].
The CP asymmetry ǫ 1 in the decay of the lightest right-handed neutrino, N 1 → LH, is determined by the structure of the neutrino sector. With κ < ∼ 0.2 the observed baryon asymmetry requires In the R-parity preserved models, it is known that the |ǫ 1 | has an upper bound as [22,23,24] due to the see-saw relation. Here m i (m 1 < m 2 < m 3 ) are the mass eigenvalues of the light neutrinos, and M N 1 is the mass of the lightest right-handed neutrino N 1 . Using the atmospheric neutrino mass squared difference ∆m 2 atm ≃ (2.5 ± 0.2) × 10 −3 eV 2 , the lower bound on the right-handed neutrino mass becomes with the 3σ values of n B /n γ and ∆m 2 atm . Consequently, the corresponding lower bound on the reheating temperature is obtained as T R > ∼ 1 × 10 9 GeV.
The upper bound on |ǫ 1 | (14) can be relaxed by the R-parity violating operator (5). The operator contributes to the CP asymmetric decay, N 1 → LH, via the diagrams exchanging H andH (and N) in the loop. In addition, a decay, N 1 → HH, gives arise with the R-parity violating operator of N 1 . Consequently, we obatin [7] ǫ (RPV) 1 where the loop function is . We notice that the result depends on λ 1 and λ i (i = 1). In order to avoid a strong washout by a (inverse) decay, λ 1 is favored to satisfy 11 , while λ i is allowed to be O(1). Then, the R-parity violating processes can dominate the CP asymmetry of the right-handed neutrino decay.
Actually, it is estimated as for We can see that the result exceeds the upper bound (14) for λ i ∼ 1 and 11 , i.e. c 1 ∼ 1. In the last section, we found that T R = O(10 8 )GeV is obtained even in a heavy NLSP mass region from Fig. 1. On the other hand, we saw in this section that the CP asymmetry of the right-handed neutrino can be drastically enhanced by the R-parity violating operator (5), and the condition (13) is easily satisfied, e.g. for M 1 = 10 8 GeV. Since N 1 is expected to be produced sufficiently in the thermal bath once T R exceeds M N 1 , the thermal leptogenesis consequently works in a wide class of models with explaining the cosmic-ray anomalies from PAMELA/ATIC. To be explicit, let us give an example: λ i =1 ∼ 1 with the lightest right- One may be worried that such a large R-parity violation enhances washout effects. The (inverse) decay and ∆L = 1 processes indeed receive additional contributions from the R-parity violating operator. Considering that they are controlled by the washout mass parameter,m 1 = (Y N Y † N ) 11 v 2 /M N 1 , we notice that the extra effects just redefinem 1 as the new one given in the previous paragraph. On the other hand, the R-parity violation can enhance the ∆L = 2 washout. Actually, we obtain LH → HH (H means H orH) by exchanging a heavier right-handed neutrino in the diagram. Since the diagram is almost the same as the R-parity preserved one except for a coupling and corresponding field, the additional contribution is roughly estimated as where γ (sub) ∆L=2 represents the ∆L = 2 washout term with a subtraction of the resonance [21]. Thus, γ RPV becomes larger than γ ∆L=2 is negligibly small for this M N 1 , we can numerically check that the washout is dominated by the (inverse) decay and ∆L = 1 scattering even for λ i = O(1) (see e.g. [21] for the numerical estimations of the washout terms in the R-parity preserved case). Although γ (sub) ∆L=2 increase as M N 1 grows, γ RPV decreases at the same time. Thus, the R-parity violating contribution does not change the result.

V. CONCLUSIONS
The gravitino DM scenario is an attractive candidate to explain the PAMELA/ATIC excesses. However, the R-parity violation has to be tightly suppressed in the low-energy scale.
In this letter, we proposed a framework of the R-parity violation to solve the fine-tuning.
Introducing the violation in the right-handed neutrino sector, the effects are naturally suppressed in the low-energy scale by the see-saw mechanism. We found that the life time required by PAMELA/ATIC, τ 3/2 = O(10 26 )sec, is obtained by the R-parity violating operator W = λNHH with λ ∼ 1 for M N = O(10 9 )GeV.
The R-parity violation not only provides a well source of the cosmic rays to explain the PAMELA/ATIC anomalies, but also opens a window for the thermal leptogenesis. In the presence of the R-parity violation, the reheating temperature is bounded from above by the overclosure bound due to the thermal gravitino production. We found that T R > ∼ 10 9 GeV is accessible for m 3/2 < 600GeV, while m 3/2 > ∼ 1TeV favored by ATIC restricts the gluino mass to be suppressed for T R > ∼ 10 9 GeV. Nonetheless, the leptogenesis was shown to work, since the R-parity violating interactions in the right-handed neutrino sector enhances the CP asymmetric right-handed decay. Therefore, the gravitino LSP with the R-parity violation in the right-handed neutrino sector is a nice candidate of the decaying DM scenarios.
It is worthwhile to clarify the parameter dependence of the phenomena. The cosmic-ray spectra are sensitive to larger λ 2 and corresponding M N 2 but insensitive to M N 1 as long as λ 1 ≪ λ 2 , while the lightest right-handed neutrino determines the thermal leptogenesis, i.e.
the baryon asymmetry depends on M N 1 . From (17) it is possible to lower M N 1 as well as the reheating temperature required by the thermal leptogenesis, though too small M N 1 needs tiny λ 1 to avoid a strong washout.
The cosmic-ray spectra of the anti-matter and photon also depends on the R-parity violation. In this letter, the flavor structure is determined by the neutrino Yukawa coupling and λ i . By obtaining more cosmic-ray data in future such as FGST and AMS-02 and refining the astrophysical knowledge, we may distinguish the R-parity violating operators.
Furthermore, since the R-parity violating operator can be embedded in a high-scale models such as SU (5) GUT and heterotic string models, combining with the future collider data we might be able to study physics in high-energy scale.