PHYSICAL REVIEW D 97, 115013 (2018)
Big-bang nucleosynthesis and leptogenesis in the CMSSM
† ‡ ∥ Munehiro Kubo,1,* Joe Sato,1, Takashi Shimomura,2, Yasutaka Takanishi,1,§ and Masato Yamanaka3, 1Department of Physics, Saitama University, Shimo-Okubo 255, 338-8570 Saitama Sakura-ku, Japan 2Faculty of Education, University of Miyazaki, Gakuen-Kibanadai-Nishi 1-1, 889-2192 Miyazaki, Japan 3Maskawa Institute, Kyoto Sangyo University, Kyoto 603-8555, Japan
(Received 21 March 2018; published 8 June 2018)
We have studied the constrained minimal supersymmetric standard model with three right-handed neutrinos, and investigated whether there still is a parameter region consistent with all experimental data/ limits such as the baryon asymmetry of the Universe, the dark matter abundance and the lithium primordial abundance. Using Casas-Ibarra parametrization, we have found a very narrow parameter space of the complex orthogonal matrix elements where the lightest slepton can have a long lifetime, which is necessary for solving the lithium problem. We have studied three cases of the right-handed neutrino mass ratio (i) M2 ¼ 2 × M1, (ii) M2 ¼ 4 × M1, (iii) M2 ¼ 10 × M1, while M3 ¼ 40 × M1 is fixed. We have obtained the mass range of the lightest right-handed neutrino that lies between 109 and 1011 GeV. The important result is that its upper limit is derived by solving the lithium problem and the lower limit comes from leptogenesis. Lepton flavor violating decays such as μ → eγ in our scenario are in the reach of MEG-II and Mu3e.
DOI: 10.1103/PhysRevD.97.115013
I. INTRODUCTION type-I seesaw mechanism [4–8]. In this mechanism, the heavy Majorana right-handed (RH) neutrinos are introduced The standard models (SMs) of particle physics and and, thus, the Yukawa interactions of left-handed (LH) and cosmology have been successful at understanding most RH neutrinos can be formed with the Higgs scalar, which of the experimental and observational results obtained so gives rise to the flavor mixings in the neutrino sector. After far. Nonetheless, there are several phenomena which cannot integrating out the RH neutrinos, the LH neutrino masses be explained by these models. Among such phenomena, become very light due to the suppression factor which is the mass and mixing of neutrinos, the baryon asymmetry of proportional to the inverse of the Majorana mass scale. Thus, the Universe (BAU), the existence of the dark matter (DM), when we make use of the seesaw mechanism, we can and the so-called lithium (Li) problems are compelling successfully generate the phenomenologically required evidence that new physics laws are required. If all of these masses and mixings of LH neutrinos. phenomena are addressed in particle physics, the new Furthermore, the seesaw mechanism has another physics laws should be incorporated in a unified picture virtue, generating the baryon asymmetry [2] through beyond the SM of particle physics. leptogenesis [9]. At the early stage of the Universe, the Neutrino oscillation experiments (see Ref. [1] for recent RH Majorana neutrinos are produced in the thermal bath. review and global fit analysis) and cosmological observa- As the temperature decreases to their mass scale, these tions [2,3] revealed that the masses of neutrinos are much neutrinos go to out-of-thermal equilibrium, and at that lighter than those of other known SM particles. To generate time they decay into leptons with Higgs or antileptons such tiny masses, many mechanisms have been proposed, with anti-Higgs. If CP symmetry is violated in the among which the most well-studied and simplest is the neutrino Yukawa coupling, the decay rates into lepton and antilepton are obviously different. That means that the *[email protected] † lepton number asymmetry is generated through the decays [email protected] ‡ of the heavy Majorana RH neutrinos, and then the lepton [email protected] §[email protected] number asymmetry is converted to the baryon asymmetry ∥ [email protected] by the sphaleron process [10,11]: the seesaw mechanism explains two phenomena simultaneously. (see, e.g., Published by the American Physical Society under the terms of Refs. [12–18]) the Creative Commons Attribution 4.0 International license. The existence of DM is also a problem [19]. The dark Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, matter must be a massive and stable or a very long-lived and DOI. Funded by SCOAP3. particle compared with the age of the Universe and not
2470-0010=2018=97(11)=115013(15) 115013-1 Published by the American Physical Society MUNEHIRO KUBO et al. PHYS. REV. D 97, 115013 (2018) carry electric or color charges. LH neutrino is the only between 350 and 420 GeV in the constrained MSSM possible candidate for the DM within the SM; however, this (CMSSM). This result is consistent with nonobservation possibility already has been ruled out because neutrino of SUSY particle at the LHC experiment so far. However, it masses are too light. Thus, one should extend the SM so is in the reach of the LHC Run-II. that the DM is incorporated. Supersymmetry (SUSY) with In this article, we consider the CMSSM with the type I R parity is one of the attractive extensions in this regard, seesaw mechanism as a unified picture which successfully where the lightest SUSY particle (LSP) becomes absolutely explains all phenomena as we have mentioned above. We stable. In many SUSY models, the LSP is the lightest aim to examine this model through searches of the long- neutralino that is a linear combination of neutral compo- lived charged particles at the LHC and lepton flavor nents of gauginos and Higgsinos that are SUSY partners of violation (see, e.g., Refs. [14,33–47]) at MEG-II, Mu3e electroweak gauge bosons and the Higgses, respectively. and Belle-II experiments. This paper is organized as Therefore, the lightest neutralino LSP is a good candidate follows. In Sec. II, we review the CMSSM with the heavy for the DM, and in fact the abundance of the neutralino LSP RH Majorana neutrinos. In Sec. III, we show cosmological can be consistent with observational ones of the DM [19] constraints such as dark matter, BBN and BAU, which we in specific parameter regions. In particular, the so-called require to the model in our analysis. Then, we present the coannihilation region is very interesting, in which the parameter sets of the CMSSM and RH Yukawa coupling neutralino DM and the lighter stau, SUSY partner of the which satisfies all requirements in Sec. IV. Predictions on tau lepton, as the next-LSP (NLSP) are degenerate in mass lepton flavor violating decays are shown in Sec. V. The last [20]. When the mass difference of the neutralino LSP and section is devoted to a summary and discussion. the stau NLSP is smaller than Oð100Þ MeV, the stau NLSP becomes long-lived so that it can survive during the big- II. MODEL AND NOTATION bang nucleosynthesis (BBN) [21–23]. Thus, the existence of the stau NLSP affects the primordial abundance of light We consider the MSSM with RH Majorana neutrinos elements. One can expect to find evidence of the stau NLSP (MSSMRN). The superpotential for the lepton sector is in the primordial abundance of the light elements. given by It has been reported that there are disagreements on the 1 7 6 W ˆ c ˆ ˆ λ ˆ ˆ ˆ c − ˆ c ˆ c primordial abundance of Li and Li between the standard l ¼ EαðYEÞαβLβ · Hd þ βiLβ · HuNi 2 ðMNÞijNi Nj : BBN prediction and observations. The prediction of the 7Li abundance is about 3 times larger than the observational ð1Þ −10 one 1.6 0.3 × 10 [24–26]. This discrepancy hardly ˆ ˆ c ð Þ Here, Lα and Eα (α ¼ e; μ; τ; i; j ¼ 1, 2, 3) are the chiral seems to be solved in the standard BBN with the meas- 2 7 6 supermultiplets, respectively, of the SUð ÞL doublet lepton urement errors. This is called the Li problem. The Li 2 and of the SUð ÞL singlet charged lepton in the flavor basis abundance also disagrees with the observations. The which is given as the mass eigenstate of the charged lepton, 103 predicted abundance is about smaller than the obser- that is, the eigenstate of Y and hence implicitly ðY Þαβ ¼ 6 7 −2 E E vational abundance Li= Li ≃ 5 × 10 [27]. Although this ˆ c yαδαβ is assumed. Similarly N ði ¼ 1; 2; 3Þ is that of the disagreement is less robust because of uncertainties in the i RH neutrino and the indices denote the mass eigenstate, observations, it is called the 6Li problem. that is, the eigenstate of M , and implicitly M ¼ M δ Since the disagreements cannot be attributed to nuclear N Nij i ij is assumed, and the superscript C denotes the charge physics in the BBN [28], one needs to modify the standard ˆ ˆ BBN reactions. In Ref. [29], the authors have shown in the conjugation. Hu and Hd are the supermultiplets of the minimal SUSY standard model (MSSM) that negatively two Higgs doublet fields Hu and Hd. charged stau can form bound states with light nuclei and Below the lightest RH seesaw mass scale, the singlet ˆ c immediately destroy the nuclei through internal conversion supermultiplets Ni containing the RH neutrino fields are processes during the BBN. Further, a detailed analysis [30] integrated out, the Majorana mass term for the LH has shown that in the coannihilation region, where the neutrinos in the flavor basis is obtained, lightest neutralino LSP is the DM and the stau NLSP has a 1 lifetime of O 103 sec, Li and beryllium (Be) nuclei are Lν − ν ν ð Þ m ¼ 2 LαðmνÞαβ Lβ þ H:c:; ð2Þ effectively destroyed. The primordial abundance of 7Li is 6 2 λ −1 λ reduced, while such a stau can promote the production of Li ðmνÞαβ ¼ vuð νÞαiMi ð νÞiβ; ð3Þ [31]. It turns out that both densities become the observa- tional values. This is a solution of the dark matter and the where Mi ¼ðM1;M2;M3Þ and vu is vacuum expectation β Li problems in the MSSM scenario. It should be noted that value (VEV) of up-type Higgs field Hu, vu ¼ v sin the SUSY spectrum is highly predictive in this parameter with v ¼ 174 GeV. The matrix ðmνÞαβ can be diagonalized region. In Ref. [32], the authors also showed the whole by a single unitary matrix—Maki-Nakagawa-Sakata- — SUSY spectrum in which the lightest neutralino mass is matrix UMNS as
115013-2 BIG-BANG NUCLEOSYNTHESIS AND LEPTOGENESIS IN … PHYS. REV. D 97, 115013 (2018) pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi † 2 2 mν U D U ; 4 ≃ Δ ≃ Δ ð Þ¼ MNS mν MNS ð Þ thus mν3 matm and mν2 m⊙ and also that all mixing angles lie in the interval 0 < θ12; θ23; where Dmν ¼ diagðmν1 ;mν2 ;mν3 Þ. θ13 < π=2. Furthermore, the lightest LH neutrino mass is The solar, atmospheric, and reactor neutrino experiments fixed, for our main result, to be have shown at 3σ level that [48] 0 001 2 −5 2 mν1 ¼ . ðeVÞ; ð6Þ Δm12 ¼ð6.93 − 7.96Þ × 10 ðeV Þ; 2 −3 2 Δm23 ¼ð2.42 − 2.66Þ × 10 ðeV Þ; as we will see that we have no solution of the degener- 2 ate case. sin θ12 ¼ð0.250 − 0.354Þ; We will use the standard parametrization of the MNS 2 sin θ23 ¼ð0.381 − 0.615Þ; matrix, 2 sin θ13 ¼ð0.0190 − 0.0240Þ: ð5Þ ˆ 1 iα iβ UMNS ¼ Udiagð ;e ;e Þ; ð7Þ Note that in this articlewewill assume that the mass spectrum ≪ ≪ of light neutrinos is hierarchical ðmν1 mν2 mν3 Þ and with
0 1 −iδ B c13c12 c13s12 s13e C ˆ B iδ iδ C U ¼ @ −c23s12 − s23s13c12e c23c12 − s23s13s12e s23c13 A; ð8Þ iδ iδ s23s12 − c23s13c12e −s23c12 − c23s13s12e c23c13
where cij ¼ cos θij, sij ¼ sin θij, and δ is the Dirac CP- III. COSMOLOGICAL CONSTRAINT violating phase, and α and β are two Majorana CP- For our analysis we take into account three types of violation phases. The input values of the angles and three cosmological observables; (i) dark matter abundance CP-violation phases at GUT scale are set, respectively, so (ii) light element abundances (iii) baryon asymmetry of that at low energy the following values are realized, by the Universe. We show our strategy to find favored pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi parameter space from a standpoint of these observables s23 ¼ 0.441;s13 ¼ 0.02166;s12 ¼ 0.306; in the CMSSM with seesaw mechanism. α ¼ 0; β ¼ 0; δ ¼ 261°. ð9Þ A. Number densities of dark matter and of long-lived slepton In addition, we parametrize the matrix of neutrino Yukawa couplings ala` Casas-Ibarra [35] We consider the neutralino-slepton coannihilation sce- nario in the framework of CMSSM wherein the LSP is the χ˜0 1 pffiffiffiffiffiffiffiffi pffiffiffiffiffi Bino-like neutralino 1 and the NLSP is the lightest slepton λν ¼ U D R M; ð10Þ l˜ v MNS mν 1 that almost consists of RH stau including tiny flavor u mixing, X where ˜ ˜ l1 ¼ Cff; ð12Þ 0 1 f¼e;μ;τ c˜13c˜12 c˜13s˜12 s˜13 B C 2 2 2 1 B C where Ce þ Cμ þ Cτ ¼ , and each interaction state is R ¼ @ −c˜23s˜12 c˜23c˜12 − s˜23s˜13s˜12 s˜23c˜13 A: s˜23s˜12 − c˜23s˜13c˜12 −s˜23c˜12 − c˜23s˜13s˜12 c˜23c˜13 ˜ ˜ ˜ f ¼ cos θffL þ sin θffR: ð13Þ ð11Þ The flavor mixing Cf and left-right mixing angle θf are determined by solving RG equations with neutrino T 1 We adopt that R is a complex orthogonal matrix, R R ¼ , Yukawa. In our scenario Cτ ∼ 1 ≫ Ce;Cμ and sin θτ ∼ 1. so that c˜ ¼ cos z and s˜ ¼ sin z with z ¼ χ˜0 pffiffiffiffiffiffi ij ij ij ij ij The standard calculation for relic density of the 1 leads xij þ −1yij because we will calculate CP-violating to an overabundant dark matter density. A tight mass ˜ 0 process such as Leptogensis. degeneracy between l1 and χ˜1 assists in maintaining the
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X chemical equilibrium of SUSY particles with the SM sector dYn −s eq eq ¼ hσvi → ½Y Y − Y Y ; ð18Þ and can reduce the relic density below the Planck bound dz Hz ij SM i j i j χ˜0 l˜ [3]. This is called the coannihilation mechanism [20]. i;j¼ 1; 1 In a unique parameter space where the neutralino-slepton where z ¼ mχ˜0 =T, Y ¼ n =s is the number density of a coannihilation works well, we focus on the parameter space 1 i i 0 ˜ species i normalized to the entropy density s, and where the mass difference between χ˜1 and l1 is smaller than the tau mass, Yn ¼ n=s, respectively. Here H denotes the Hubble expan- σ sion rate, h viij→SM represents thermally averaged cross section for an annihilation channel ij → SM particles. δm ≡ ml˜ − mχ˜0 ˜ 0 2. Number density of long-lived slepton l1 → χ˜1l ðl ∋ e; μÞ: ð16Þ Even after the chemical decoupling, although the total In fact, the longevity depends on the degeneracy in mass density remains the current dark matter density, the ratio of 0 ˜ − ˜ þ and also on the magnitude of lepton flavor violation each number density of χ˜1, l1 , and l1 continues to evolve. [49,50]. As we will see in Sec. III B, we have to assume As long as the kinetic equilibrium with the SM sector is ˜ 0 δm 115013-4 BIG-BANG NUCLEOSYNTHESIS AND LEPTOGENESIS IN … PHYS. REV. D 97, 115013 (2018) 0 hσ vil˜ ↔χ˜0γne 1. Nonstandard nuclear reactions 1e 1 ≃ 1 08 109 2 0 ð . × ÞCe; ð22Þ in stau-nucleus bound state hσ vi˜ ˜0 nτ l1τ↔χ1γ We have constraints for the parameters at low energy so that BBN with the long-lived slepton works well. To see it, 0 hσ vil˜ μ↔χ˜0γnμ we have to take into account the following: 1 1 ≃ 9 93 107 2 0 ð . × ÞCμ: ð23Þ (1) Number density of the slepton at the BBN era hσ vil˜ τ↔χ˜0γnτ 1 1 (2) Nonstandard BBN process (a) Internal conversion [29,64] Here σ0 represents the cross section of relevant processes (b) Spallation [55] −5 (c) Slepton catalyzed fusion [31] for kinetic equilibrium. As long as Ce ≳ 3.2 × 10 and −4 Number density is calculated by numerically solving Cμ ≳ 1.0 × 10 , flavor changing processes maintain the Eqs. (24) and (25) if the lifetime is long enough. From kinetic equilibrium, and hence reduce nl˜ − . This means that −5 1 this requirement we obtain a constraint Cμ < Oð10 Þ and such a small flavor mixing can decrease nl˜ − significantly. −7 1 Ce < Oð10 Þ with the assumption δm X δ 3 5 10−9 θ 0 6 χ˜ 1 Ce m< . × MeV for sin e ¼ . : ð26Þ dY 0 eq eq − s σ v χ˜0 ↔ Yχ˜ Y − Y Y ¼ f h i 1X iY½ X i Y dz Hz ˜0 i≠χ1 eq eq 2. Nonstandard nuclear interactions Γ Y ˜0 sY sY … − Y ; 24 þh i½ χ1 ð X Þð X Þ i g ð Þ Internal conversion.—In a relatively early stage of the 7 X BBN, the long-lived slepton forms a bound state with Be dYl˜ 1 1 0 eq eq 7 − s σ v ˜ Y ˜ Y − Y Y and Li nucleus, respectively. These bound states give rise ¼ f h il1 X↔iY ½ l1 X i Y dz Hz ˜ i≠l1 to internal conversion processes [29], eq eq Γ − 0 … þh i½Yl˜ Yχ˜ ðsYX ÞðsYX Þ g: ð25Þ 7 ˜ − 0 7 1 1 ð Bel1 Þ → χ˜ 1 þ ντ þ Li; ð27aÞ ˜ 7 l˜ − → χ˜0 ν 7 Here Γ represents l1 decay rate of channels in Eqs. (15) ð Li 1 Þ 1 þ τ þ He: ð27bÞ and (16). The daughter 7Li nucleus in the process Eq. (27a) is destructed either by an energetic proton or the process B. Big-bang nucleosynthesis (27b) while the daughter 7He nucleus in the process To solve the lithium problem(s), we need a long-lived Eq. (27b) immediately decays into 6He nucleus and particle so that it survives until BBN starts, more precisely neutron, then rapid spallation processes by the background synthesis of 7Be begins. Fortunately, our model does have 6 ˜ particles convert the produced He into harmless nuclei, such a long-lived particle, i.e., l1. This slepton can e.g., 3He, 4He, etc. Hence, the nonstandard chain reactions 7 7 effectively destruct Be which would be Li just after the by the long-lived slepton could yield smaller 7Be and 7Li 7 7 BBN era. Since at the BBN era would-be Li exists as Be, abundances than those in the standard BBN scenario, 7 7 destructing Be effectively means reducing Li primordial which is precisely the requirement for solving the 7Li abundance. This long-lived slepton with degenerate mass problem. This is the scenario we proposed. 7 can offer the solution to the Li problem [29,30,32,53–61]. We find that the time scale of the reaction is much shorter In addition, several articles [27,62,63] report that there are than the BBN time scale as long as δm is larger than several significant amount of 6Li though the standard BBN cannnot MeV. A parent nucleus is converted into another nucleus predict 6Li abundance. immediately once the bound state is formed. The bound Since we add the RH Majorana neutrinos, these Yukawa state formation makes the interaction between the slepton couplings are the seed of LFV, we have another constraint and a nucleus more efficient by two reasons: First, the to impose the longevity of the lifetime. To ensure the overlap of wave functions of the slepton and a nucleus longevity of the lifetime, only a very tiny electron and becomes large since these are confined in the small space. muon flavor can mix in the NLSP [23,53]. With keeping Second, the short distance between the slepton and a these facts in our mind, here we briefly recapitulate how to nucleus allows virtual exchange of the hadronic current solve the lithium problem(s). even if δm 115013-5 MUNEHIRO KUBO et al. PHYS. REV. D 97, 115013 (2018) ˜ Nonstandard process with bound helium.—The slepton and its superpartner N1. Typical parameters for solving the 4 7 6 10 −3 forms a bound state with He as well. This fact causes two Li and Li problems are M1 ∼ 10 GeV and jλα1j ∼ 10 . ≡ Γ ∼ nonstandard processes. One of these processes is the Further, the decay parameter should be K N1 =HðM1Þ 4 spallation process of the He nucleus [55], Oð1Þ and Kα ≡ K ·BRðN1 → lαϕÞ ∼ Oð0.1Þðα ∋ e; μ; τÞ. Here, HðM1Þ is the Hubble parameter at the temperature 4 ˜ − ˜0 ð Hel1 Þ → χ1 þ ντ þ t þ n; ð28aÞ T ¼ M1. In cases where the leptogenesis in the strong 12 washout regime takes place at T ≲ 10 GeV and Kα are 4 ˜ − 0 ð Hel1 Þ → χ˜1 þ ντ þ d þ n þ n; ð28bÞ comparable with each other, the lepton number of each flavor separately evolves, and it gives rise to Oð1Þ 4 ˜ − 0 corrections to the final lepton asymmetry with respect to ð Hel1 Þ → χ˜ þ ντ þ p þ n þ n þ n: ð28cÞ 1 where the flavor effects are ignored [67,68]. As studied in Refs. [69,70], the correction could be significant in the and the other channel is called slepton-catalyzed fusion SUSY flavored case. [31]; The lepton asymmetry is calculated by a set of the 4 ˜ − ˜ − 6 coupled evolution equations of the number densities of ð Hel1 Þþd → l1 þ Li: ð29Þ ˜ N1, N1, and the lepton numbers of each flavor. Since the Since the LFV coupling and δm determines which light superequilibration is maintained throughout the temper- elements are overproduced by these nonstandard reactions, ature range we consider [71], the equality of asymmetries 4 of each lepton and its scalar partner is also maintained, we need careful study of the evolution of the slepton- He 2 3 − and YB−L ¼ ×ðYΔ þYΔμ þYΔτ Þ with YΔα ¼ B= Lα. bound state for the parameter space of Cα ’s and δm.In e general, the spallation process is disastrous. In order to In the superequilibration regime, the primary piece of the suppress it, δm<30 MeV must be fulfilled. coupled equations are given as follows [72] The catalyzed fusion process enhances the 6Li produc- dY −z Y tion [31]. The thermal averaged cross section of the N1 N1 − 1 γ γs1 ¼ eq ½ N1 þ N1 ; ð32Þ catalyzed fusion is precisely calculated in Refs. [65,66], dz sHðM1Þ YN1 6 which is much larger than that of the Li production in the 4 6 dY ˜ − Y ˜ standard BBN, He þ d → Li þ γ, by 6 to 7 orders of Nþ z Nþ s1 ¼ − 2 ½γ ˜ þ γ ˜ ; ð33Þ 6 eq N1 N1 magnitude. The overproduction of the Li nucleus by the dz sHðM1Þ Y ˜ N1 catalyzed fusion process leads stringent constraints on 2 2 ðδmÞ Ce from below to make the slepton lifetime shorter dYΔ − YΔ N˜ z N˜ s2 ¼ ½γ ˜ þ γ ˜ than 5000 s [53]. With the lower bound on the lifetime eq N1 N1 dz sHðM1Þ Y ˜ N1 1700 s, in addition to the upper bound on Ce, Eq. (26), δ 10 YΔ we have a lower bound on it. For m ¼ MeV and YΔl s3 Hu s4 − ½γ ˜ − ½γ ˜ ; ð34Þ eq N1 eq N1 sin θe ¼ 0.6, Y Y l Hu −10 −10 1700 s ≤ τl˜ ≤ 5000 s ⇔ 2.0 × 10 ≤ C ≤ 3.5 × 10 dYΔ −z Y e i ε N1 − 1 γ γs1 ¼ i eq ½ N1 þ N1 dz sHðM1Þ Y ð30Þ N1 Y ˜ Nþ s1 þ ε − 2 ½γ ˜ þ γ ˜ is required. i eq N1 N1 Y ˜ Furthermore, there are several reports [27] that insist N1 6 1 there are significant amounts of Li. If we take these YΔl i s2 i s5 − eq γ þ γ þðγ ˜ þ γ ˜ Þ seriously, we can make use of the catalyzed fusion here Y 2 N1 N1 N1 N1 l and, in this case, the slepton lifetime must be between YΔ 1 Hu i s3 i s6 3500 s and 5000 s which corresponds to the requirement − eq γ þ γ þðγ ˜ þ γ ˜ Þ : ð35Þ Y 2 N1 N1 N1 N1 Hu −10 −10 3500 s ≤ τl˜ ≤ 5000 s ⇔ 2.0 × 10 ≤ Ce ≤ 2.5 × 10 : Here, z ¼ M1=T. We introduced transformed yield values ˜ ð31Þ for N1, Y ˜ ≡ Y ˜ þ Y ˜ , and YΔ ≡ Y ˜ − Y ˜ . γ and Nþ N1 N1 N˜ N1 N1 N1 γ ˜ ˜ N1 are thermally averaged decay rates of N1 and N1, C. Leptogenesis γsn 1 2 3 … respectively. X ðn ¼ ; ; ; Þ symbolizes a combination We calculate the lepton asymmetry assuming the RH of thermally averaged cross sections, and the explicit one is neutrinos are hierarchical in mass that is generated by the shown in the Appendix in Ref. [72]. Relevant cross sections l H CP asymmetric reactions of the lightest RH neutrino N1 are given in Ref. [73]. Coefficient Cαβ (Cβ ) is a conversion 115013-6 BIG-BANG NUCLEOSYNTHESIS AND LEPTOGENESIS IN … PHYS. REV. D 97, 115013 (2018) l l factor from the asymmetryP of α (H) to that of β, eq l eq ðnl − nl¯ Þ=n ¼ − βC ðYΔ =Y Þ and ðn − n ¯ Þ= α Pα lα αβ β l H H eq − H eq nH ¼ βCβ ðYΔβ =Yl Þ. The entries are determined by constraints among the chemical potentials enforced by the equilibrium reactions at the stage where the asymmetries 10 are generated, T ∼ M1. In our scenario, M1 ∼ 10 GeV, l H and Cαβ and Cβ are [72] 0 1 906 −120 −120 1 B C Cl ¼ @ −75 688 −28 A; αβ 3 × 2148 −75 −28 688 1 H 37 52 52 C ¼ 2148 ð Þ: ð36Þ FIG. 1. Evolutions of jYB−Lj and each lepton asymmetry jYΔi j The CP asymmetry receives contributions from not only for a typical parameter in this paper. Horizontal band (gray) Ynon-f the RH neutrinos but also its scalar partner. The flavor corresponds to the observed baryon asymmetry. j B−L j shows the lepton asymmetry in the absence of flavor effect. dependent CP asymmetry for the channel Ni → lαϕ is defined as the importance of flavor effect, we also plot the non- ¯ † Γ N → lαϕ − Γ N → lαϕ flavored result with thin solid line. We find Oð1Þ correction εi ≡ ð i Þ ð i Þ α ¯ † ð37Þ to the final lepton asymmetry depending on the presence ΓðNi → lαϕÞþΓðNi → lαϕ Þ of the flavor effect. Since this correction is introduced λ and is obtained as [74] into the expected relation between M1 and αi, the flavor effects are critical ingredients to understand the correlation i i i εα ¼ εαðvertexÞþεαðwaveÞ; ð38Þ among the BBN, the BAU, and the charged LFV in our scenario. X 2 † 1 M M ℑ½ðλ λÞ λβ λα εi − j 1 i ji i i IV. ANALYSIS αðvertexÞ¼ 8π log þ 2 λ†λ ; Mi Mj ð Þii j A. Parameter space ð39Þ L Soft SUSY breaking term in the Lagrangian soft contains more than one hundred parameters in general. 2 X M εi − i In order to perform a phenomenological study, we make an αðwaveÞ¼ 8π 2 − 2 j Mj Mi assumption that three gauge couplings are unified at the ℑ λ†λ λ†λ λ λ GUT scale. At that scale, we presume that there exists a f½Mjð Þji þ Mið Þij βi αig : universal gaugino mass, m1 2. Besides, the scalar soft × λ†λ ð40Þ = ð Þii breaking part of the Lagrangian depends only on a common scalar mass m0 and trilinear coupling A0, in addition on the → ˜ χ˜ The CP asymmetries for other channels, Ni lα , ratio of VEVs, tan β. After fixing a sign ambiguity in the ˜ → χ˜ ˜ → ˜ ϕ Ni lα , and Ni lα , are defined similarly and given Higgsino mixing parameter μ, we complete five SUSY as the same results with Eqs. (38), (39), and (40). parameter space of the CMSSM: The lepton and slepton asymmetry converts to the baryon asymmetry, and the conversion factor in MSSM scenarios m1=2;m0;A0; tan β; signðμÞ: ð42Þ is YB ¼ð8=23ÞYB−L [75]. The required lepton asymmetry in 3 sigma range is Note that we have demonstrated our numerical analyses only in the signðμÞ > 0 case. −10 −10 2.414 × 10 ≲ jYB−Lj ≲ 2.561 × 10 ð41Þ In the neutrino Yukawa couplings, Eq. (10), there are 18 parameters since the matrix is 3 × 3 complex matrix. 2 for the observed baryon number Ωbh ¼ 0.0223 0.0002 We use the low-energy observed quantifies (i) three LH 1σ ( ) [48]. neutrino masses mν1 , mν2 , mν3 , (ii) three mixing angles θ θ θ Figure 1 shows the evolution of lepton number for a sin 23, sin 13, sin 12 in UMNS [Eq. (8)], and (iii) three CP- typical parameter obtained in this study. Numerical com- violating phases α, β, δ [Eq. (9)] as input parameters. They putations in this work are performed by using the complete are given in Sec. II. There are nine model parameters, set of coupled Boltzmann equations. For illustrating which we express in terms of three RH Majorana neutrino 115013-7 MUNEHIRO KUBO et al. PHYS. REV. D 97, 115013 (2018) masses M1, M2, M3 at GUT scale and the remaining three m1=2 ¼ 887.0 ðGeVÞ;A0 ¼ −3090 ðGeVÞ; complex angles in the R matrix. Thus, there are a total of tan β ¼ 25: ð43Þ nine free parameters and nine experimentally “observed” data in the Dirac-Yukawa couplings. At this moment, four SUSY parameters have been fixed The low-energy SUSY spectra and the low-energy including the sign of the μ term. The remaining parameter is flavor observables were computed by means of the the universal scalar mass, m0, which must lie at SPheno-3.3.8 [76,77] using two-loop beta functions with an option of the precision as quadrupole because the slepton m0 ≈ ½707.3; 707.4 ðGeVÞ; ð44Þ flavor mixing is required to be 10−12 order or even smaller. During these computations, we apply the set of constraints depending on the mass hierarchy structure of the RH displayed in Table I. We generate SLHA format files and send Majorana neutrino sector for fixing the value – them to micrOMEGAs_4.3.5 [78 80] which computes the of δm ¼ 0.01 GeV. Ω 2 neutralino relic density h and the spin-independent scat- Note that we have taken into account the logalismical tering cross section with nucleons, as we will briefly corrections of the corresponding scales and also the slepton mention below. mass running effect which are caused by the Dirac-Yukawa beta function. However, these effects are negligible for the B. Determining input parameters dark matter relic density calculations. In this subsection, we discuss in detail how we have It is important to mention that with the above given investigated a very wide range of parameter space. In values of SUSY parameters we obtain the SM-like Higgs mass about 125 GeV, i.e., the “right” combination of the principle, we must set all the parameters simultaneously so values of tan β, A0 and stop mass generated by the universal that all the requirements are fulfilled. However, concep- scalar mass m0 are selected in our calculation processes. tually, we can set the parameters step by step with the small We show here an example parameter in the top panel of correction from the following steps. Table II. With this parameter set, the flavor mixing Ce, Cμ, and the mixing angle sin θe, and the lifetime of slepton are 1. The CMSSM parameters calculated. The results are listed in the middle-panel of Let us start with the constraints on the lightest neutralino Table II. It is clear that our model with these parameters 6 7 mass from relic abundance. For our analysis, we take into solve also both Li and Li problems. Furthermore, we have account cosmological data—dark matter abundance—that calculated the observed quantities, the relic density of dark arise from the Planck satellite analysis [2]. In this article, matter, mass of dark matter, and the mass difference the neutralino relic density, Ωh2, must satisfy the 3 sigma between the NLSP and LSP, δm, with same parameters. range: Ωh2 ∈ 0.1126; 0.1246 [48]. In CMSSM type The results are displayed in the bottom-panel of Table II.As ½ 2 theory, the lightest neutralino mass will be of order of has been noted, our result, Ωh ¼ 0.1154, satisfies the relic 400 GeV. In the framework of MSSMRN which we density obtained from the Planck satellite analysis [2]. consider, the lightest neutralino mass becomes about 380 GeV. What is more, we fix the mass difference 2. The Yukawa coupling δm ¼ 0.01 GeV as already studied [53]; furthermore, we In order to find a set of parameters with which our decide to use tan β ¼ 25 because with this value we can model—MSSMRH with boundary condition at the GUT easily obtain the right amount of the relic density which scale—we have performed parameter scan in the following must be within the 3 sigma rage of cosmological data. “systematic” way. Essentially we do not scan all mass range Accordingly, the three SUSY parameters, m1 2, A0, and = of the RH Majorana neutrinos, but we fix the mass ratio of β tan , are set at the following values: 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 M3=M1 ¼ 40, and TABLE I. The experimental constraints. we investigate the following three scenarios in this article. Namely, Quantity Reference (1) M2 ¼ 2 × M1, M3 ¼ 40 × M1 Ωh2 [0.1126, 0.1246] [48] (2) M2 ¼ 4 × M1, M3 ¼ 40 × M1 10 40 mh (124.4, 125.8) GeV [48] (3) M2 ¼ × M1, M3 ¼ × M1; −4 BRðB → sγÞ ½2.82; 3.29 × 10 [81] i.e., only the ratios of M2=M1 are different in each setup. → μþμ− þ2.1 −9 Fixing the mass of the lightest RH Majorana neutrino and BRðBs Þ 2.8−1 8 × 10 [82] . arranging the elements of the complex orthogonal matrix BRðBu → τν¯Þ 0.52 115013-8 BIG-BANG NUCLEOSYNTHESIS AND LEPTOGENESIS IN … PHYS. REV. D 97, 115013 (2018) TABLE II. The example of input parameters and output parameters. Input parameters Value m0 707 (GeV) m1=2 887 (GeV) A0 −3089 ðGeVÞ tan β 25 μ=jμjþ1 10−3 mν1L ðeVÞ 4 04 10−3 mν2L . × ðeVÞ 1 18 10−2 mν3L . × ðeVÞ 10 M1 2.0 × 10 ðGeVÞ 10 M2 8.0 × 10 ðGeVÞ 11 FIG. 2. The lightest slepton lifetime as a function of x23. M3 8 0 10 . × ðGeVÞ The blue and green band corresponds to the lifetime required to α 0 7 7 6 solve the Li problem only and both the Li and Li problems, β 0 respectively. δ 261 x12 2.28948 x13 3.56000 −5 When x23 differs from this range, Ce is Oð10 Þ and hence x23 4.80532 the slepton lifetime becomes much shorter. One can see that y12 1.02 the real part x23 is determined almost uniquely to solve the Li y13 0.1 problems. No need to say that allowed regions of the real part y23 0.1 of the complex angles are also depend on the mass structure Output parameters Value of the RH Majorana neutrinos thus we have to seek an other tiny parameter space when we change the value of M1. −10 Ce 3.28 × 10 −6 Furthermore we check the parameters obtained in this way Cμ 2.94 × 10 whether they reproduce the right amount of baryon asym- sin θe 0.188 τ metry as explained in Sec. III C. l˜ 4217 (s) Output parameters Value 3. The allowed mass region of the lightest Ωh2 0.115 right-handed Majorana neutrino mχ˜0 379.6 (GeV) 1 We describe our main results in this subsection. First, we −2 δm 1.01 × 10 ðGeVÞ 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 calculate the baryon asymmetry. For simplicity, we fix the hierarchical structure. With respect to the lithium problem, three cases which we have investigated are listed here: values of y23 ¼ y13 ¼ 0.1 and vary only y12 in the complex 2 40 part of the mixing angles. The real part of complex angles, (1) case of M2 ¼ × M1, M3 ¼ × M1 (a) Taking into account the 6Li and 7Li problem x12, x13, x23, are obtained through the electron mixing in slepton mass matrix, C . To have a enough lifetime of e 7 8 108 ≤ ≤ 7 0 1010 slepton for solving the lithium problem, only extremely . × M1 . × ðGeVÞ: ð45Þ −5 narrow ranges of x12, x13, x23—of the order of 10 —are 7 −10 (b) Taking into account only the Li problem allowed because of Ce being of the order of 10 .To illustrate how the real parts of the flavor mixing are 8 11 7.8 × 10 ≤ M1 ≤ 1.0 × 10 ðGeVÞ: ð46Þ determined, we show the lifetime of the slepton in terms of x23 in Fig. 2. The RH Majorana mass is taken to (2) case of M2 ¼ 4 × M1, M3 ¼ 40 × M1 2 0 1010 6 7 M1 ¼ . × GeV in case 2. The blue and green bands (a) Taking into account the Li and Li problem represent the slepton lifetime required to solve only the 7Li 7 6 9 10 problem, Eq. (30), and both Li and Li problems, Eq. (31), 1.9 × 10 ≤ M1 ≤ 7.0 × 10 ðGeVÞ: ð47Þ respectively. The lifetime changes 2 orders of magnitude for −5 7 the narrow range of x23 of order 10 . This is because Ce ∼ (b) Taking into account only the Li problem 10−10 is realized due to the fine-tuned cancellation among 9 11 the LFV terms in renormalization group equation running. 1.9 × 10 ≤ M1 ≤ 1.0 × 10 ðGeVÞ: ð48Þ 115013-9 MUNEHIRO KUBO et al. PHYS. REV. D 97, 115013 (2018) 100000 100000 10000 10000 1000 1000 slepton lifetime (s) slepton lifetime (s) 100 100 10 10 2.906 2.908 2.91 2.912 2.914 2.1005 2.101 2.1015 2.102 2.1025 2.103 x12 x12 100000 100000 10000 10000 1000 1000 slepton lifetime (s) slepton lifetime (s) 100 100 10 10 3.51436 3.51438 3.5144 3.51442 3.51444 3.51446 3.51448 3.5145 3.51452 3.51454 3.51456 3.557 3.5575 3.558 3.5585 3.559 3.5595 x13 x13 100000 100000 10000 10000 1000 1000 slepton lifetime (s) slepton lifetime (s) 100 100 10 10 4.94185 4.94186 4.94187 4.94188 4.94189 4.94190 4.8342 4.83425 4.8343 4.83435 4.8344 4.83445 x23 x23 11 FIG. 3. The slepton lifetime in terms of x12, x13, x23 in case 2. In the left and right panel, M1 is taken to 1.2 × 10 GeV and 3.0 × 1010 GeV, respectively. (3) case of M2 ¼ 10 × M1, M3 ¼ 40 × M1 exist because the region where the upper limit would (a) Taking into account the 6Li and 7Li problem be is already excluded by the current experiment date of BRðμ → eγÞ. 9 2.35 × 10 ðGeVÞ ≤ M1: ð49Þ The upper limits of the lightest RH Majoarana neutrino in three different cases are obtained from the limits of Ce One might wonder why we do not write the upper limit and Cμ, in fact we need to suppress both the slepton mixing of the lightest RH Majorana neutrino mass in the third case. Ce and Cμ. Naively, these flavor mixing are scaled to the Essentially, we do not need to get the values that definitely Yukawa couplings, so that it is easy to understand why we 115013-10 BIG-BANG NUCLEOSYNTHESIS AND LEPTOGENESIS IN … PHYS. REV. D 97, 115013 (2018) let the absolute values of the Dirac-Yukawa couplings be 10−6, if one does not take into account the flavor effects neither jλαij ≪ 1 to satisfy the experimental constraints of Ce and does not consider also an accidental fine-tuning cancellation † Cμ. At the same time, of course we must satisfy the low- in λνλν. Thus we need a sufficiently large Yukawa couplings 2 energy neutrino experiment data, namely Δm , and three and large value of M1 is required (as we know in the nonflavor 9 mixing angles according to Eq. (3) in which we do not leptogenesis case, M1 ≳ 10 GeV [85].) As it is scaled to the consider an extreme fine-tuning in matrix multiplications. RH Majorana neutrino mass at a certain point such a Therefore, the eigenvalues of MR, or the lightest RH sufficiently large coupling can not be realized. Majorana neutrino mass since we fix the mass ratios In concluding, we note that there exists only a tiny M2=M1 and M3=M1, should be lighter than the case of allowed parameter space for the lightest RH Majorana λ ∼ 1 j αij . Further Yukawa coupling constants square is neutrino where all experimental data and constraints are scaled to the RH Majorana neutrino masses, at a certain fulfilled within the 3 sigma range. mass it becomes impossible for C’s to be small enough. Thus, we have the upper bound for M1. In Fig. 3, we show the slepton lifetime as a function of x12, x13, x23 to illustrate V. PREDICTIONS FROM PARAMETER SEARCH this explanation. The RH Majorana mass M1 is taken to A. Predictions mainly from CMSSM parameters 1.2 × 1011 GeV and 3.0 × 1010 GeV in the left and right As explained in Sec. IV B 1, CMSSM parameter is almost panels, respectively. In the left panels, the lifetime cannot determined uniquely from mχ˜0 , δm, and SM Higgs mass. reach 1700 s even when x12;13;23 are fine-tuned. On the 1 other hand, in the right panels where M1 is taken to be Therefore the dark matter relic abundance, SUSY mass smaller, the lifetime can be longer than 1700 s. Note that all spectrum, and the contribution to the muon magnetic anoma- − 2 of the real parts are determined in very narrow range as we lous moment g are more or less predicted uniquely. In our analysis, the relic abundance of the neutralino explained in Fig. 2. The flavor mixing Ce is more tightly constrained to solve the 6Li problem (or evade 6Li over- density is production) and hence the upper bound is more stringent. It 2 is worth noting that with similar reason, we cannot have a Ωh ¼ 0.115: ð50Þ degenerate solution for left-handed neutrino mass since in this case also rather large Yukawa coupling is necessary. For calculation of the spin-independent cross section On the other hand, to reproduce the matter-antimatter with the nucleon, we use the following values of the quark asymmetry generated by leptogenesis so that the CP violating form-factors in the nucleon which are the default values in i of Majorana decay processes εα [Eq. (37)] should be large than the MICROMEGAS code, TABLE III. SUSY particle masses. Particle Mass (GeV) Mixing ˜ 1 453 103 ˜ ˜ ˜ d1 . × d1 ≃ ð0.9910 − 0.0000iÞbL þð0.1289 − 0.0000iÞbR ˜ 1 696 103 ˜ ˜ ˜ d2 . × d2 ≃ ð0.9916 − 0.0000iÞbR þð−0.1286 þ 0.0000iÞbL ˜ 1 850 103 ˜ d3 . × d3 ≃ ð0.9997 þ 0.0189iÞs˜R þð0.0068 þ 0.0001iÞs˜L ˜ 1 851 103 ˜ ˜ ˜ d4 . × d4 ≃ ð−0.9263 − 0.3766iÞdR þð−0.0003 − 0.0001iÞdL ˜ 1 925 103 ˜ ˜ d5 . × d5 ≃ ð−0.9835 − 0.016iÞs˜L þð0.1664 − 0.0588iÞdL ˜ 1 926 103 ˜ ˜ d6 . × d6 ≃ ð0.8698 − 0.4605iÞdL þð0.1752 − 0.0229iÞs˜L 2 ˜ ˜ u˜ 1 8.775 × 10 u˜1 ≃ ð0.9604 − 0.0000iÞtR þð0.2749 − 0.0000iÞtL 3 ˜ ˜ u˜ 2 1.502 × 10 u˜2 ≃ ð−0.9603 þ 0.0000iÞtL þð0.2784 − 0.0000iÞtR ˜ 3 ˜ ˜ ˜ u3 1.858 × 10 u3 ≃ ð0.9999 − 0.0001iÞcR þð0.0103 þ 0.0000iÞcL ˜ 3 ˜ ˜ ˜ u4 1.858 × 10 u4 ≃ ð0.2862 þ 0.9581iÞuR þð0.0000 þ 0.0000iÞuL ˜ 3 ˜ ˜ ˜ u5 1.924 × 10 u5 ≃ ð0.9958 þ 0.0045iÞcL þð0.0659 þ 0.0618iÞuL 3 u˜ 6 1.924 × 10 u˜6 ≃ ð−0.7492 þ 0.6560iÞu˜L þð0.0092 − 0.0899iÞc˜L ˜ 3 796 102 ˜ l1 . × l1 ≃ ð−0.9852 þ 0.0000iÞτ˜R þð−0.1710 − 0.0000iÞτ˜L ˜ 7 806 102 ˜ l2 . × l2 ≃ ð−0.6766 − 0.7360iÞμ˜R þð−0.0141 − 0.0154iÞμ˜L ˜ 7 817 102 ˜ l3 . × l3 ≃ ð−0.6639 þ 0.7477iÞe˜R þð0.0000 þ 0.7605iÞe˜L ˜ 7 980 102 ˜ l4 . × l4 ≃ ð0.9852 þ 0.0000iÞτ˜L þð−0.1710 − 0.0000iÞτ˜R ˜ 9 215 102 ˜ l5 . × l5 ≃ ð0.6681 þ 0.7311iÞμ˜L þð0.1077 − 0.0835iÞe˜L ˜ 9 219 102 ˜ l6 . × l6 ≃ ð−0.7833 þ 0.6064iÞe˜L þð0.0919 þ 0.1006iÞμ˜L g˜ 1.986 × 103 115013-11 MUNEHIRO KUBO et al. PHYS. REV. D 97, 115013 (2018) 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 p=n to fs ) can lead to a shift by a factor of 2–6 in the spin- independent cross section [78]. Masses of supersymmetric particles are shown in Table III. Note that these spectrum is predicted just above the current experimental limits [48]. The interesting prediction of MSSMRN is a small contribution to the muon anomalous magnetic moment g − 2, δaμ: −10 δaμ ¼ 3.537 × 10 : ð53Þ With this contribution, the discrepancy of the theoretical value and the experimental one becomes within 3σ; i.e., our model satisfies a limit for aμ 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 a sizable charged LFV (CLFV). Figure 4 shows the branching ratio of LFV decays as a function of M1 in three different cases. Current bounds (gray region) and future sensitivity (dashed line) are summarized in Table IV. All of the reaction rates are crudely proportional to the second lightest Majorana neutrino mass M2. The dependence comes from the elements of the Dirac neutrino Yukawa matrix λν that have large absolute values for a fixed active neutrino parameter λ ∝ jð νÞi2j M2. All curves satisfy the requirement to solve the 7Li problem while the thick solid lines fulfill those for the 7Li and 6Li problems. The parameter of the RH neutrino is narrowed down to a small space to solve the 7Li and 6Li problems and to generate successfully large lepton asymmetry. The predic- μ → γ μ → 3 FIG. 4. BRðμ → eγÞ,BRðμ → 3eÞ,BRðτ → μγÞ, and tions for BRð e Þ and BRð eÞ lie in the range BRðτ → 3μÞ as a function of M1 for M2 ¼ 2 × M1, 4 × M1, where the recent and near future experiment can probe. Our 7 and 10 × M1. The Li problem is solved with parameters in each scenario can be precisely illuminated by combining LFV line, while both the 7Li and 6Li problems are solved only for observables and unique collider signals [94–100]. It should thick part. Gray region is excluded by MEG experiment, and the be emphasized that when we consider the 6Li=7Li problems horizontal lines show future sensitivity. in the constrained MSSMRN, it is no surprise that we have p 0 033 p 0 023 p 0 26 fd ¼ . ;fu ¼ . ;fs ¼ . ; TABLE IV. Current bound and future sensitivity of branching fn ¼ 0.042;fn ¼ 0.018;fn ¼ 0.26; ð51Þ d u s ratio of LFV decays. and we get Process Bound Sensitivity −13 −14 SI −47 2 μ → eγ 4.2 × 10 [87] 6 × 10 [88] σ ¼ 1.05 × 10 cm ; ð52Þ −12 −16 μ → 3e 1.0 × 10 [89] 1 × 10 [90] τ → μγ 4 4 10−8 1 10−9 so that our dark matter candidate satisfies easily the limit . × [91] × [92] τ → 3μ 2.1 × 10−8 [93] 1 × 10−9 [92] of the spin-independent cross section with the nucleon 115013-12 BIG-BANG NUCLEOSYNTHESIS AND LEPTOGENESIS IN … PHYS. REV. D 97, 115013 (2018) not yet observed CLFV. As a matter of fact, we will observe violation. We have also found that the degenerate mass CLFV processes in the near future. hierarchy of the active neutrinos is hardly realized in this region because rather large Yukawa couplings are necessary VI. SUMMARY AND DISCUSSION for the degenerate hierarchy. The flavor mixing among sleptons is significantly canceled through the renormaliza- We have investigated the parameter space of the con- strained minimal supersymmetric standard model with tion group equation running by adjusting the complex three RH Majorana neutrinos by requiring low-energy angles. For this reason, the lightest slepton becomes a long-enough-lived particle, and we thus are able to solve neutrino masses and mixings. At the same time, we have 7 6 applied experimental constraints such as dark matter the Li= Li problems. abundance, Li abundances, baryon asymmetry, and results Furthermore, we have calculated the branching ratios of from the LHC experiment, anomalous magnetic moment, the lepton flavor violating decay using the allowed param- eter sets. It is found that the upper bounds of BRðμ → eγÞ and flavor observations. We have scanned the parameter of −13 −15 the complex orthogonal matrix R in Eq. (10) assuming a and BRðμ → 3eÞ are Oð10 Þ and Oð10 Þ for M2 ¼ −12 −14 relation among the RH neutrino masses (see Sec. IV B) and 2 × M1 and Oð10 Þ and Oð10 Þ for M2 ¼ 4 × M1, have found that the allowed parameter sets really exist respectively. The LFV decays, μ → eγ and μ → 3e, are in where all of the phenomenological requirements are the reach of MEG-II and Mu3e. satisfied. As shown in Sec. IV, the range of the lightest RH Majorana 9 11 ACKNOWLEDGMENTS neutrino mass M1 is roughly 10 GeV ≤ M1 ≤ 10 GeV. The lower bound of M1 is determined to obtain a suffi- This work is supported by JSPS KAKENHI cient amount of matter-antimatter asymmetry, while the Grants No. 25105009 (J. S.), No. 15K17654 (T. S.), upper bound is determined to suppress large lepton flavor No. 16K05325, and No. 16K17693 (M. Y.) [1] P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola, and [18] S. Pascoli, S. T. Petcov, and A. Riotto, Nucl. Phys. B774,1 J. W. F. Valle, arXiv:1708.01186. (2007). [2] P. A. R. Ade et al. 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