Big-bang nucleosynthesis and Leptogenesis in CMSSM

We have investigated the constrained minimal supersymmetric standard model with three right-handed Majorana neutrinos whether there still is a parameter region which is consistent with all existing experimental data/limits such as Leptogenesis and the dark matter abundance and we also can solve the Lithium problem. Using Casas-Ibarra parameterization, we have found that a very narrow parameter space of the complex orthogonal matrix elements where the lightest slepton can have a long lifetime, that is necessary for solving the Lithium problem. Further, under this condition, there is a parameter region that can give an explanation for the experimental observations. We have studied three cases of the right-handed neutrino mass ratio \mbox{\em (i)} $M_{2}=2 \times M_{1}$, \mbox{\em (ii)} $M_{2}=4 \times M_{1}$, \mbox{\em (iii)} $M_{2}=10 \times M_{1}$ while $M_{3}=40 \times M_{1}$ is fixed. We have obtained the mass range of the lightest right-handed neutrino mass that lies between $10^9$ GeV and $10^{11}$ GeV. The important result is that its upper limit is derived by solving the Lithium problem and the lower limit comes from Leptogenesis. Calculated low-energy observables of these parameter sets such as BR($\mu \to e \gamma$) is not yet restricted by experiments and will be verified in the near future.


I. INTRODUCTION
The standard models (SMs) of particle physics and cosmology have been successful to understand most of experimental and observational results obtained so far. Nonetheless, there are several phenomena which cannot be explained by these models. Among such phenomena, the mass and mixing of neutrinos, the Baryon asymmetry of the universe (BAU), the existence of the dark matter (DM), so-called Lithium (Li) problems are compelling evidences that require new physics laws for explanations. If all of these phenomena are addressed in particle physics, the new physics laws should be incorporated in a unified picture beyond the SM of particle physics.
Neutrino oscillation experiments (see Ref. [1] for recent review and global fit analysis) and cosmological observations [2,3] revealed that the masses of neutrinos are much lighter than those of other known SM particles. To generate such tiny masses many mechanisms have been proposed, among which most well-studied and the simplest mechanism is (type-I) seesaw mechanism [4][5][6][7][8]. In this mechanism, the heavy Majorana right-handed (RH) neutrinos are introduced and thus the Yukawa interactions of left-handed (LH) and RH neutrinos can be formed with the Higgs scalar, that gives rise to the flavor mixings in the neutrino sector. After integrating out the RH neutrinos, the LH neutrino masses become very light due to the suppression factor which is proportional to the inverse of the Majorana mass scale. Thus when we make use of the seesaw mechanics we can successfully generate the phenomenologically required masses and mixings of low-energy LH neutrinos.
Furthermore, the seesaw mechanism has another virtue, generating the baryon asymmetry [2] through Leptogenesis [9]. At the early stage of the universe, the RH Majorana neutrinos are produced in the thermal bath. As temperature decreases to their mass scale these neutrinos go out-of-thermal equilibrium, and at that time they decay into lepton with Higgs or anti-lepton with anti-Higgs. If CP symmetry is violated in the neutrino Yukawa coupling, the decay rates into lepton and anti-lepton are obviously different. That means that the lepton number asymmetry is generated through the decays of the heavy Majorana RH neutrinos, and then the lepton number asymmetry is converted to the baryon asymmetry by sphaleron process [10,11]: the seesaw mechanism explains two phenomena simultaneously. (see e.g. Refs. [12][13][14][15][16][17][18]) The existence of DM is also problem [19]. The dark matter must be a massive and stable or very long-lived particle compared with the age of the universe and do not carry electric neither color charges. Neutrino is only possible candidate for the DM within the SM, however, this possibility has been already ruled out because neutrino masses are too light. Thus, one should extend the SM so that the DM is incorporated. Supersymmetry (SUSY) with R parity is one of the attractive extensions in this regard, where the lightest SUSY particle (LSP) becomes absolutely stable. In many SUSY models, the LSP is the lightest neutralino that is a linear combination of neutral components of gauginos and higgsinos that are SUSY partners of electroweak gauge bosons and the Higgses, respectively. Therefore, the lightest neutralino LSP is a good candidate for the DM, and in fact the abundance of the neutralino LSP can be consistent with observational one of the DM [19] in specific parameter regions. In particular, the so-called coannihilation region is very interesting, in which the neutralino DM and the lighter stau, SUSY partner of tau lepton, as the next-LSP (NLSP) are degenerate in mass [20]. When the mass difference of the neutralino LSP and the stau NLSP is smaller than O(100) MeV, the stau NLSP becomes long-lived so that it can survive during the Big-Bang nucleosynthesis (BBN) proceeds [21][22][23]. Thus, the existence of the stau NLSP affects the primordial abundance of light elements. One can expect to find evidences of the stau NLSP in primordial abundance of light elements.
It has been reported that there are disagreements on the primordial abundances of 7 Li and 6 Li between the standard BBN prediction and observations. The prediction of the 7 Li abundance is about 3 times larger than the observational one (1.6±0.3)×10 −10 [24][25][26]. This discrepancy hardly seems to be solved in the standard BBN with the measurement errors. This is called the 7 Li problem. The 6 Li abundance is also disagreed with the observations. The predicted abundance is about 10 3 smaller than the observational abundance 6 Li/ 7 Li 5×10 −2 [27]. Although this disagreement is less robust because of uncertainties of theoretical prediction, it is called the 6 Li problem.
Since the disagreements cannot attribute to nuclear physics in the BBN [28], one needs to modify the standard BBN reactions. In Ref. [29], the authors have shown in the minimal SUSY standard model (MSSM) that negatively charged stau can form bound states with light nuclei, and immediately destroy the nuclei through the internal conversion processes during the BBN. Further, a detail analysis [30] has showed that in the coannihilation region where the lightest neutralino LSP is the DM and the stau NLSP has lifetime of O(10 3 ) sec., Li and Beryllium (Be) nuclei are effectively destroyed. The primordial density of 7 Li is reduced, while such a stau can promote to produce 6 Li density [31]. It turns out that both densities become the observational values. This is a solution of the dark matter and the Li problems in the MSSM scenario. It should be noted that the SUSY spectrum is highly predictive in this parameter region. In Ref. [32], the authors also showed whole SUSY spectrum in which the lightest neutralino mass is between 350 GeV and 420 GeV in the constrained MSSM (CMSSM). This result is consistent with non-observation of SUSY particle at the LHC experiment so far. However it is in the reach of the LHC Run-II.
In this article, we consider the CMSSM with the type I seesaw mechanism as a unified picture which successfully explains all phenomena as we have mentioned above. We aim to examine this model through searches of the long-lived charged particles at the LHC and lepton flavor violation (see e.g. Refs. [14,[33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]) at MEG-II, Mu3e and Belle-II experiments. This paper is organized as follows. In section II, we review the CMSSM with the heavy RH Majorana neutrinos. In section III, we show cosmological constraints such as dark matter, BBN and BAU, which we require to the model in our analysis. Then, we present the parameter sets of the CMSSM and RH Yukawa coupling which satisfies all requirements in section IV. Predictions on lepton flavor violating decays are shown in Section V. The last section is devoted to summary and discussion.

II. MODEL AND NOTATION
We consider the MSSM with RH Majorana neutrinos (MSSMRN). The superpotential for the lepton sector is given by Here L α and E c α (α = e, µ, τ ; i, j = 1, 2, 3), are the chiral supermultiplets respectively of the SU (2) L doublet lepton and of the SU (2) L singlet charged lepton in the flavor basis which is given as the mass eigenstate of the charged lepton, that is, the eigenstate of Y E and hence implicitly (Y E ) αβ = y α δ αβ is assumed. Similarly N c i (i = 1, 2, 3) is that of RH neutrino and indices denote the mass eigenstate, that is, the eigenstate of M N and implicitly M N ij = M i δ ij is assumed and the superscript C denotes the charge conjugation. H u and H d are the supermultiplets of the two Higgs doublet fields H u and H d .
Below the lightest RH seesaw mass scale, the singlet supermultiplets N c i containing the RH neutrino fields are integrated out, the Majorana mass term for the LH neutrinos in the flavor basis is obtained where M i = (M 1 , M 2 , M 3 ) and v u is vacuum expectation value (VEV) of up-type Higgs field H u , v u = v sin β with v = 174 GeV. The matrix (m ν ) αβ can be diagonalized by a single unitary matrix -Maki-Nakagawa-Sakata-matrix -U MNS as where D mν = diag(m ν 1 , m ν 2 , m ν 3 ). The solar, atmospheric and reactor neutrino experiments have shown at 3 σ level that [48]  Note that in this article we will assume that the mass spectrum of light neutrinos is hierarchical (m ν 1 m ν 2 m ν 3 ) and thus m ν 3 ∆m 2 atm and m ν 2 ∆m 2 and also that all mixing angles lie in the interval 0 < θ 12 , θ 23 , θ 13 < π/2. Furthermore, the lightest LH neutrino mass is fixed , for our main result, to be as we will see that we have no solution of degenerate case. We will use the standard parametrization of the MNS matrix where c ij = cos θ ij , s ij = sin θ ij , and δ is the Dirac CP-violating phase and α and β are two Majorana CP-violation phases. The input values of the angles and three CP-violation phases at GUT scale are set respectively by In addition, we parameterize the matrix of neutrino Yukawa couplings a la Casas-Ibarra [35] We adopt that R is a complex orthogonal matrix, R T R = 1, so that c ij = cos z ij and s ij = sin z ij with z ij = x ij + √ −1 y ij because we will calculate CP-violating process such as Leptogensis.

III. COSMOLOGICAL CONSTRAINT
For our analysis we take into account three types of cosmological observables; (i) dark matter abundance (ii) light element abundances (iii) baryon asymmetry of the universe. We show our strategy to find favored parameter space from a standpoint of these observables in the CMSSM with seesaw mechanism.

A. Number densities of dark matter and of long-lived slepton
We consider the neutralino-slepton coannihilation scenario in the framework of CMSSM wherein the LSP is the Bino-like neutralino χ 0 1 and the NLSP is the lightest slepton 1 that almost consists of RH stau including tiny flavor mixing, where C 2 e + C 2 µ + C 2 τ = 1, and each interaction state is The flavor mixing C f and left-right mixing angle θ f are determined by solving RG equations with neutrino Yukawa. In our scenario C τ ∼ 1 C e , C µ and sin θ τ ∼ 1. The standard calculation for relic density of the χ 0 1 leads to an over-abundant dark matter density. A tight mass degeneracy between 1 and χ 0 1 assists to maintain the chemical equilibrium of SUSY particles with SM sector, and can reduce the relic density below the Planck bound [3]. This is called coannihilation mechanism [20].
In a unique parameter space for the neutralino-slepton coannihilation to work well, we focus on the space where the mass difference between χ 0 1 and 1 is smaller than tau mass Assuming flavor conservation i.e. 1 is purely RH stau, open decay channels of 1 are where π, a 1 and ρ are light mesons. Due to the phase space suppression and higher order coupling the 1 becomes a long-lived particle [21,23]. If the lepton flavor is violated, the following 2-body decays are allowed, In fact the longevity depends on the degeneracy in mass and also on the magnitude of lepton flavor violation [49,50]. As we will see in section III B, we have to assume δm < m µ , so the main decay mode is the 2-body decay 1 → χ 0 1 e and therefore the lifetime of the slepton τl is given by up to leading order of (δm) 2 , where g is the gauge coupling of SU (2) and θ W is the Weinberg angle, respectively.
The long-lived 1 has significant effect on light element abundances through exotic nuclear processes in the BBN era. To quantitatively determine this effect, we evaluate the number density of 1 on the era. As we will see, it is closely related with the relic density of χ 0 1 and it depends on not only δm but also on the magnitude of lepton flavor violation. Here we take decoupling limit of SUSY particles except for χ 0 1 and 1 .

Dark matter relic density
After SUSY particles ( χ 0 1 and 1 ) are chemically decoupled from SM sectors, their total density, n = n χ 0 , will be frozen. Since all of SUSY particles eventually decays into the LSP χ 0 1 , so that the dark matter relic density is indeed the total density. We find Boltzmann equation of the total density by adding each one of n χ 0 1 and n ± 1 [20,51], where z = m χ 0 1 /T , Y i = n i /s is the number density of a species i normalized to the entropy density s, and Y n = n/s, respectively. Here H denotes the Hubble expansion rate, σv ij→SM represents thermally averaged cross-section for an annihilation channel ij → SM particles. Relevant processes and the cross-sections are given in Ref. [52]. We search for favored parameters by numerically solving the equation to fit n to the observed dark matter density [48] where h = 0.678 is the Hubble constant normalized to H 0 = 100 km s −1 Mpc −1 , and ρ c = 1.054 × 10 −5 GeV cm −3 is the critical density of the universe.

Number density of long-lived slepton
Even after the chemical decoupling, although the total density remains the current dark matter density, the ratio of each number density of χ 0 1 , − 1 , and + 1 continues to evolve. As long as the kinetic equilibrium with the SM sector is maintained, 1 and χ 0 1 follow the Boltzmann distribution, and hence − 1 number density until the kinetic decoupling is We focus on the parameter space where δm < m µ , m µ being the muon mass. Then the lifetime of 1 is long enough, and we are able to solve the 7 Li and 6 Li problems [30,53]. Processes maintaining the kinetic equilibrium in the space are 1 Even for a tiny lepton flavor violation (LFV), flavor changing processes are relevant due to much larger densities of e and µ compared with that of τ for the universe temperature 1 Note that the process ± 1 γ ↔ χ 0 1 e ± must not be included. The process should be incorporated into a corrective part of the decay (inverse decay) ± 1 ↔ χ 0 1 e ± . Similarly, if the decay ± 1 ↔ χ 0 1 µ ± is open, the process ± 1 γ ↔ χ 0 1 µ ± also must not be taken into account.
smaller than m τ . For example, for a reference universe temperature T = 70 MeV, reaction rates of these processes are σ v Here σ represents the cross-section of relevant processes for kinetic equilibrium. As long as C e 3.2 × 10 −5 and C µ 1.0 × 10 −4 , flavor changing processes maintain the kinetic equilibrium, and hence reduce n − 1 . This means that such a small flavor mixing can decrease n − 1 significantly.

B. Big-Bang Nucleosynthesis
To solve the Lithium problem(s), we need a long-lived particle so that it survives until BBN starts, more precisely synthesis of 7 Be begins. Fortunately, our model does have such a long-lived particle, i.e., 1 . This slepton can effectively destruct 7 Be which would be 7 Li just after the BBN era. Since at the BBN era would-be 7 Li exists as 7 Be, destructing 7 Be effectively means reducing 7 Li primordial abundance. This long-lived slepton with degenerate mass can offer the solution to the 7 Li problem [29,30,32,[53][54][55][56][57][58][59][60][61]. In addition, several articles [27,62,63] report that there are significant amount of 6 Li though the standard BBN cannnot predict 6 Li abundance.
Since we add the RH Majorana neutrinos, these Yukawa couplings are the seed of LFV, we have another constraint to impose the longevity of the lifetime. To ensure the longevity of the lifetime, only a very tiny electron and muon flavor can mix in the NLSP [23,53]. With keeping these facts in our mind, here we briefly recapitulate how to solve the Lithium problem(s).

Non-standard nuclear reactions in stau-nucleus bound state
We have constraints for the parameters at low energy so that BBN with the long-lived slepton works well. To see it we have to take into account the followings: (1) Number density of the slepton at the BBN era (2) Non-standard BBN process (a) Internal Conversion [29,64] (b) Spallation [55] (c) Slepton catalyzed fusion [31] Number density is calculated by numerically solving Eqs. (24) and (25) if the lifetime is long enough. From this requirement we obtain a constraint C µ < O(10 −5 ) and C e < O(10 −7 ) with the assumption δm < m µ [53].
In addition, since its lifetime must be long enough (≥ 1700 s) there is more stringent constraint on C e with δm as has pointed out in Ref. [53].
2. Non-standard nuclear interactions a. Internal Conversion: In a relatively early stage of the BBN, the long-lived slepton forms a bound state with 7 Be and 7 Li nucleus respectively. These bound states give rise to internal conversion processes [29], The daughter 7 Li nucleus in the process Eq. (27a) is destructed either by an energetic proton or the process (27b) while the daughter 7 He nucleus in the process Eq. (27b) immediately decays into 6 He nucleus and neutron, then rapid spallation processes by the background particles convert the produced 6 He into harmless nuclei, e.g. 3 He, 4 He etc. Hence the nonstandard chain reactions by the long-lived slepton could yield smaller 7 Be and 7 Li abundances than those in the standard BBN scenario, that is precisely the requirement for solving the 7 Li problem. This is the scenario we proposed.
We find that the time scale of the reaction is much shorter than the BBN time scale as long as δm is larger than several MeV. A parent nucleus is converted into another nucleus immediately once the bound state is formed. The bound state formation makes the interaction between the slepton and a nucleus more efficient by two reasons: First, the overlap of wave functions of the slepton and a nucleus becomes large since these are confined in the small space. Second, the short distance between the slepton and a nucleus allows virtual exchange of the hadronic current even if δm < m π .
b. Non-standard process with bound Helium: The slepton forms a bound state with 4 He as well. This fact causes two non-standard processes. One of these processes is the spallation process of the 4 He nucleus [55], and the other channel is called slepton-catalyzed fusion [31]; Since the LFV coupling and δm determines which light elements are over-produced by these non-standard reactions, we need careful study of the evolution of the slepton-4 He bound state for the parameter space of C α 's and δm. In general the spallation process is disastrous. In order to suppress it δm < 30 MeV must be fulfilled. The catalyzed fusion process enhances the 6 Li production [31]. Thermal averaged crosssection of the catalyzed fusion is precisely calculated in Refs. [65,66], which is much larger than that of the 6 Li production in the Standard BBN, 4 He + d → 6 Li + γ, by 6-7 orders of magnitude. The over-production of 6 Li nucleus by the catalyzed fusion process leads stringent constraints on (δm) 2 C 2 e from below to make the slepton lifetime shorter than 5000 s [53]. With the lower bound on the lifetime 1700 s, in addition to the upper bound on C e , Eq. (26) we have lower bound on it. For δm = 10 MeV and sin θ e = 0.6, 1700 s ≤ τ ≤ 5000 s ⇔ 2.0 × 10 −10 ≤ C e ≤ 3.5 × 10 −10 is required. Furthermore there are several reports [27] that insists there are significant amount of 6 Li. If we take it seriously, we can make use of the catalyzed fusion here and in this case the slepton lifetime must be between 3500 s and 5000 s and it corresponds to the requirement 3500 s ≤ τ ≤ 5000 s ⇔ 2.0 × 10 −10 ≤ C e ≤ 2.5 × 10 −10 . (31)

C. Leptogenesis
We calculate the lepton asymmetry assuming the RH neutrinos being hierarchical in mass that is generated by the CP asymmetric reactions of the lightest RH neutrino N 1 and its superpartnerÑ 1 . Typical parameters for solving the 7 Li and 6 Li problems are M 1 ∼ 10 10 GeV and |λ α1 | ∼ 10 −3 . Further, the decay parameter should be , µ, τ ). Here H(M 1 ) is the Hubble parameter at the temperature T = M 1 . In cases where the Leptogenesis in the strong washout regime takes place T 10 12 GeV and K α are comparable with each other, the lepton number of each flavor separately evolves, and it gives rise to O(1) corrections to the final lepton asymmetry with respect to where the flavor effects are ignored [67,68]. As studied in Refs. [69,70] the correction could be significant in SUSY flavored case.
The lepton asymmetry is calculated by a set of the coupled evolution equations of the number densities of N 1 , N 1 , and lepton numbers of each flavor. Since the super-equilibration is maintained throughout the temperature range we consider [71], the equality of asymmetries of each lepton and its scalar partner is also maintained, and Y B−L = 2× Y ∆e +Y ∆µ +Y ∆τ with Y ∆α = B/3 − L α . In the super-equilibration regime, the primary piece of the coupled equations are given as follows [72] Here . γ N 1 and γ N 1 are thermally averaged decay rates of N 1 and N 1 , respectively. γ s n X (n = 1, 2, 3, ...) symbolizes a combination of thermally averaged cross-sections, and the explicit one is shown in Appendix in Ref. [72]. Relevant cross-sections are given in Ref. [73]. Coefficient C αβ (C H β ) is a conversion factor from the asymmetry of α (H) to that of β , (n α − n¯ α ) /n eq α = − β C αβ Y ∆ β /Y eq and (n H − nH) /n eq The CP asymmetry receives contributions from not only the RH neutrinos but also its scalar partner. The flavor dependent CP asymmetry for the channel N i → α φ is defined as and is obtained as [74], The CP asymmetries for other channels, N i → l α χ, N i → l α χ, and N i → l α φ, are defined similarly, and given as the same results with Eqs. (38), (39) and (40). The lepton and slepton asymmetry converts to the baryon asymmetry, and the conversion factor in MSSM scenarios is Y B = (8/23)Y B−L [75]. The required lepton asymmetry in 3 sigma range is 2.414 × 10 −10 |Y B−L | 2.561 × 10 −10 for the observed baryon number Ω b h 2 = 0.0223 ± 0.0002 (1σ) [48]. Figure 1 shows the evolution of lepton number for a typical parameter obtained in this study. Numerical computations in this work are performed by using the complete set of coupled Boltzmann equations. For illustrating the importance of flavor effect, we also plot the non-flavored result with thin solid line. We find O(1) correction to the final lepton asymmetry depending on the presence of the flavor effect. Since this correction is introduced into the expected relation between M 1 and λ αi , the flavor effects are critical ingredients to understand the correlation among the BBN, the BAU, and the charged LFV in our scenario.

A. Parameter Space
Soft SUSY breaking term in the Lagrangian L soft contains more than one hundred parameters in general. In order to perform phenomenological study we make an assumption that three gauge couplings unified at GUT scale and further for reduction of the number of parameters. At that scale we presume that there exists a universal gaugino mass, m 1/2 . Besides, the scalar soft breaking part of the Lagrangian depends only on a common scalar mass m 0 and trilinear coupling A 0 , in addition on the ratio of VEVs, tan β. After fixing a sign-ambiguity in the higgsino mixing parameter µ we complete five SUSY parameter space of the CMSSM: m 1/2 , m 0 , A 0 , tan β, sign(µ) .
Note that we have demonstrated our numerical analyses only in the sign(µ) > 0 case. In the neutrino Yukawa couplings, Eq.(10), there are 18 parameters since the matrix is 3 × 3 complex matrix. We use the low-energy observed quantifies (i) three LH neutrino masses m ν 1 , m ν 2 , m ν 3 (ii) three mixing angles sin θ 23 , sin θ 13 , sin θ 12 in U MNS (Eq.(8)) and (iii) three CP-violating phases α, β, δ (Eq.(9)) as input parameters. They are given in Sec. II. There are 9 model parameters, which we express in terms of 3 RH Majorana neutrino masses M 1 , M 2 , M 3 at GUT scale and remain 3 complex angles in R matrix. Thus, there are total 9 free parameters and 9 experimentally "observed" data in the Dirac Yukawa couplings.
The low-energy SUSY spectra and the low-energy flavor observables were computed by means of the SPheno-3.3.8 [76,77] using two-loop beta functions with an option of the precision as quadrupole because the slepton flavor mixing is required to be 10 −12 order or even smaller. During these computation we apply the set of constraints displayed in Table I. We generate SLHA format files and send them to micrOMEGAs 4.3.5 [78][79][80] which computes the neutralino relic density Ωh 2 and the spin-independent scattering cross-section with nucleons, as we will briefly mention below.

B. Determining input parameters
In this subsection we discuss in detail how we have investigated very wide range of parameter space. In principle we must set all the parameters simultaneously so that all the requirement are fulfilled. However conceptually we can set the parameters step by step with the small correction from the following steps.

The CMSSM parameters
Let us start with the constraints on the lightest neutralino mass from relic abundance. For our analysis we take into account cosmological data -dark matter abundance -that arises from the Planck satellite analysis [2]. In this article we request the neutralino relic density, Ωh 2 , must satisfy the 3 sigma range: Ωh 2 ∈ [0.1126, 0.1246] [48]. In CMSSM type theory the lightest neutralino mass will be of order of 400 GeV. In the framework of MSSMRN which we consider the lightest neutralino mass becomes about 380 GeV. What is more we fix the mass difference δm = 0.01 GeV as already studied [53], furthermore we decide to use tan β = 25 because with this value we can easily obtain a right amount of the relic density which must be within 3 sigma rage of cosmological data. Accordingly, three of SUSY parameters, m 1/2 , A 0 and tan β, we set in the following values At this moment, four SUSY parameters are fixed including sign of the mu-term, the remaining parameter, the universal scalar mass, m 0 , must lie on depending on the mass hierarchy structure of the RH Majorana neutrino sector for fixing the value of δm = 0.01 GeV, not only due to the logalismical corrections of the corresponding scales but also the slepton mass running effect which are caused by the Dirac Yukawa betafunction. However, the effect of Dirac Yukawa runnings for calculation of the dark matter relic density is negligible and thus we can safely ignore this effect.
Input Parameters value m 0 707 (GeV)  It is important to mention that with the above given values of SUSY parameters we obtain the SM-like Higgs mass about 125 GeV, i.e. the "right" combination of the values of tan β, A 0 and stop mass generated by the universal scalar mass m 0 are selected in our calculation processes.
We show here an example parameter in the left-panel of Table II. With this parameter set, the flavor mixing C e , C µ , and the mixing angle sin θ e , and the lifetime of slepton are calculated. The results are listed in the right-top-panel of Table II. It is clear that our model with these parameters solve also both 6 Li and 7 Li problems. Furthermore, we have calculated the observed quantities, the relic density of dark matter, mass of dark matter, and the mass difference between the NLSP and LSP, δm, with same parameters. The results are displaced in the right-down-panel of Table II. As has been noted, our result, Ωh 2 = 0.1154, satisfies the relic density obtained from the Planck satellite analysis [2].

The Yukawa coupling
In order to find a set of parameters with which our model -MSSMRH with boundary condition at the GUT scale -we have performed parameter scan in the following "systematic" way. Essentially we do not scan all mass range of the RH Majorana neutrinos, but we fix the mass ratio of these particles. It means that the second heaviest and the heaviest RH Majorana masses are given by a function of the lightest RH one, hereby we fix the ratio M 3 /M 1 = 40, and we investigate the following three scenarios in this article. Namely, i.e. only the ratios of M 2 /M 1 are different in each setup.
Fixing the mass of the lightest RH Majorana neutrino and arranging the elements of the complex orthogonal matrix R, i.e. real part of the complex angles (x 12 , x 13 , x 23 ) and complex part of ones, y 12 , y 13 , y 23 , we are now able to calculate the baryon asymmetry. For simplicity we fix the values of y 23 = y 13 = 0.1 and vary only y 12 in the complex part of the mixing angles. The real part of complex angles, x 12 , x 13 , x 23 , are obtained through the electron mixing in slepton mass matrix, C e . To have a enough lifetime of slepton for solving the Lithium problem, only extremely narrow ranges of x 12 , x 13 , x 23 , -of the order of 10 −5 -are allowed because of C e being of the order of 10 −10 . To illustrate how the real parts of the flavor mixing are determined, we show the lifetime of the slepton in terms of x 23 in Figure 2. The RH Majorana mass is taken to M 1 = 2.0 × 10 10 GeV in case 2. The blue and green bands represent the slepton lifetime required to solve only the 7 Li problem, Eq.(30), and both 7 Li and 6 Li problems, Eq.(31), respectively. The lifetime changes two orders of magnitude for the narrow range of x 23 of order 10 −5 . This is because C e ∼ 10 −10 is realized due to the fine-tuned cancellation among the LFV terms in renormalization group equation running. When x 23 differs from this range, C e is O(10 −5 ) and hence the slepton lifetime becomes much shorter. One can see that the real part x 23 is determined almost uniquely to solve the Li problems. No need to say that allowed regions of the real part of the complex angles are also depend on the mass structure of the RH Majorana neutrinos thus we have to seek an other tiny parameter space when we change the value of M 1 . Furthermore we check the parameters obtained in this way whether they reproduce the right amount of baryon asymmetry as explained in Sec. III C.

The allowed mass region of the lightest right-handed Majorana neutrino
We describe our main results in this subsection. First we discuss the allowed mass range of the lightest RH Majorana neutrino. We have found the upper-and lower-limit for mass of the lightest RH Majorana neutrino corresponding to its hierarchical structure. With respect to the Lithium problem, three cases which we have investigated are listed here: • Taking into account only 7 Li problem 1.9 × 10 9 ≤ M 1 ≤ 1.0 × 10 11 (GeV) .   One might be wonder why we do not write the upper limit of the lightest RH Majorana neutrino mass in the third case. Essentially we do not need to get the values that definitely exist because the region where the upper limit would be is already excluded by the current experiment date of BR(µ → eγ).
The upper limits of the lightest RH Majoarana neutrino in three different cases are obtained from the limits of C e and C µ , in fact we need to suppress both the slepton mixing C e and C µ . Naively, these flavor mixing are scaled to the Yukawa couplings, so that it is easy to understand why we let the absolute values of the Dirac Yukawa couplings be |λ αi | 1 to satisfy the experimental constraints of C e and C µ . At the same time, of course we must satisfy the low-energy neutrino experiment data namely ∆m 2 and three mixing angles according to Eq. (3) in which we do not consider an extreme fine-tuning in matrix multiplications. Therefore the eigenvalues of M R , or the lightest RH Majorana neutrino mass since we fix the mass ratios M 2 /M 1 and M 3 /M 1 , should be lighter than the case of |λ αi | ∼ 1. Further Yukawa coupling constants square is scaled to the RH Majorana neutrino masses, at a certain mass it becomes impossible for C's to be small enough. Thus we have the upper bound for M 1 . In Figures 3, we show the slepton lifetime as a function of x 12 , x 13 , x 23 to illustrate this explanation. The RH Majorana mass M 1 is taken to 1.2 × 10 11 GeV and 3.0×10 10 GeV in the left and right panels, respectively. In the left panels, the lifetime cannot reach 1700 s even when x 12,13,23 are fine-tuned. On the other hand, in the right panels where M 1 is taken to be smaller, the lifetime can be longer than 1700 s. Note that all of the real parts are determined in very narrow range as we explained in Fig. 2. The flavor mixing C e is more tightly constrained to solve the 6 Li problem (or evade 6 Li over production) and hence the upper bound is more stringent. It is worth noting that with similar reason, we cannot have a degenerate solution for left-handed neutrino mass since in this case also rather large Yukawa coupling is necessary.
On the other hand, to reproduce the matter-antimatter asymmetry generated by Leptogenesis so that the CP violating of Majorana decay processes ε i α (Eq.(37)) should be large than 10 −6 , if one does not take into account the flavor effects neither does not consider also an accidental fine-tuning cancellation in λ † ν λ ν . Thus we need a sufficiently large Yukawa couplings and large value of M 1 is required (as we know non-flavor Leptogenesis case, M 1 10 9 GeV [85].) As it is scaled to the RH Majorana neutrino mass at a certain point such a sufficiently large coupling can not be realized.
In concluding, we mention that we have found there exists only such tiny allowed parameter space for the lightest RH Majorana neutrino where all experimental data and constrains are fulfilled within 3 sigma range.

A. Predictions mainly from CMSSM parameters
As explained in Sec.IV B 1, CMSSM parameter is almost determined uniquely from m χ 0 1 , δm, and SM Higgs mass. Therefore the dark matter relic abundance, SUSY mass spectrum, and the contribution to the muon magnetic anomalous moment g − 2 are more or less predicted uniquely.
In our analysis, the relic abundance of the neutralino density is For calculation of the spin-independent cross-section with nucleon we use the following values of the quark form-factors in the nucleon which are the default values in the mi-crOMEGAs code and we get σ SI = 1.05 × 10 −47 cm 2 , so that our dark matter candidate satisfies easily the limit of the spin-independent crosssection with nucleon reported by the LUX collaboration [86], even including the main uncertainty from the strange quark coefficient. If we use another set of quark coefficients (the large corrections to f p/n s ) can lead to a shift by a factor 2-6 in the spin-independent cross-section [78].
Masses of supersymmetric particles are shown in Table III. Note that these spectrum are predicted just above the current experimental limits [48].
With this contribution, the discrepancy of the theoretical value and the experimental one becomes within 3σ, i.e. our model satisfies a limit for a µ at a 3 sigma level.

B. Predictions for Charged LFV
Since the slepton mixing is induced by the existence of the Dirac Yukawa couplings via the RGE effect, we have sizable charged LFV (CLFV). Figure 4 shows the branching ratio of LFV decays as a function of M 1 in three different cases. Current bound (gray region) and future sensitivity (dashed line) are summarized in Table IV. All of reaction rates are crudely proportional to the second lightest Majorana neutrino mass M 2 . The dependence comes from the elements of the Dirac neutrino Yukawa matrix λ ν that have large absolute values for a fixed active neutrino parameter |(λ ν ) i2 | ∝ M 2 . All curves satisfy the requirement to solve the 7 Li problem while the thick solid lines fullfil those for 7 Li and 6 Li problems.
The parameter of RH neutrino is narrowed down to a small space to solve the 7 Li and 6 Li problems and to generate successfully large lepton asymmetry. The predictions for    BR(µ → eγ) and BR(µ → 3e) lie in the range where the recent and near future experiment can probe. Our scenario can be precisely illuminated by combining LFV observables and unique collider signals [94][95][96][97][98][99][100]. It should be emphasized that when we consider the 6 Li/ 7 Li problems in the constrained MSSMRN, it is no surprise that we have not observed yet CLFV. As a matter of fact, we will observe CLFV processes in the near future.

VI. SUMMARY AND DISCUSSION
We have investigated the parameter space of the constrained minimal supersymmetric standard model with three RH Majorana neutrinos by requiring low-energy neutrino masses and mixings. At the same time we have applied experimental constraints such as dark matter abundance, Li abundances, baryon asymmetry, and results from the LHC experiment, anomalous magnetic moment and flavor observations. We have scanned the parameter of the complex orthogonal matrix R in Eq. (10) assuming a relation among the RH neutrino masses (see Sec. IV B), and have found that the allowed parameter sets really exist where all of the phenomenological requirements are satisfied.
As shown in Sec. IV, it is found that the range of the lightest RH Majorana neutrino mass M 1 is roughly 10 9 GeV ≤ M 1 ≤ 10 11 GeV. The lower bound of M 1 is determined to obtain sufficient amount of matter-anti-matter asymmetry while the upper bound is determined to suppress large lepton flavor violation. We have also found that the degenerate mass hierarchy of the active neutrinos are hardly realized in this region, because rather large Yukawa couplings are necessary for the degenerate hierarchy. The flavor mixing among sleptons are significantly canceled through the renormalization group equation running by adjusting the complex angles. For this reason the lightest slepton becomes long-enough-lived particle and we thus are able to solve the 7 Li/ 6 Li problems.