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Leptogenesis in a model with Friedberg-Lee symmetry

We study the matter-antimatter asymmetry through the leptogenesis mechanism in a specific model with the Friedberg-Lee (FL) symmetry.We relate the leptogenesis with the CP violating Dirac and Majorana phases in the Maki-Nakagawa-Sakata leptonic mixing matrix and illustrate the net baryon asymmetry of the universe in terms of these phases.


I. INTRODUCTION
Neutrino oscillation experiments have indicated nonzero neutrino masses and mixings although they are not expected in the standard model (SM). One of the most plausible extensions of the SM to generate neutrino masses is the (type-I) seesaw mechanisms in which heavy right-handed Majorana neutrinos are introduced. This mechanism can explain not only the small neutrino masses but also the baryon asymmetry of the universe (BAU) via the leptogenesis mechanism [1,2]. In the lepton sector, the leptogenesis is related to the CP violating Dirac and Majorana phases in the Maki-Nakagawa-Sakata (MNS) leptonic mixing matrix as well as some possible high energy phases [3]. It is clear that there is no leptogenesis if all phases vanish. However, since there are many high energy phases [4] 1 , it is hard to make an explicit connection between the leptogenesis and the phases in the MNS matrix. In order to establish a simple relation between them, we need to reduce as many complex parameters as possible in the model. In Ref. [6], a family symmetry is used to minimize the number of arbitrary parameters in the Yukawa sector. Another possibility along this direction is to consider the so-called two-righthanded neutrino (2RHN) seesaw model [7], in which the number of parameters is less numerous than the ordinary seesaw model. In addition, spontaneous [8] and dynamical [9] CP violating approaches have been proposed.
In this Letter, we explore the leptogenesis mechanism in the model with the Friedberg-Lee (FL) symmetry [10] and directly relate it to the CP violating Dirac and Majorana phases in the MNS matrix. The FL symmetry is a translational hidden family symmetry for fermion mass terms. Several possible origins of the FL symmetry have been discussed in Ref. [11]. More detailed analyses of the symmetry have been given in Ref. [12]. The FL symmetry combined with a rotational symmetry has also been studied in Ref. [13].
As pointed out in Ref. [14], the introduction of the FL symmetry to the right-handed Majorana neutrinos suggests that there exists one massless right-handed neutrino with the absence of the corresponding column of the Dirac mass matrix in the basis of the diagonal right-handed Majorana mass matrix. As a result, the model can be regarded as the 2RHN seesaw model. This, in fact, motivates us to examine the leptogenesis in the context of the FL symmetry to see if it provides us with a testable seesaw and leptogenesis model. This Letter is organized as follows. In Sec. II, we propose a model with the FL symmetry on both the right-and left-handed neutrinos. We also examine the allowed parameters based on the present neutrino oscillation data. In Sec. III, we consider leptogenesis and estimate the net baryon asymmetry of the universe (BAU) as a function of the Dirac and Majorana phases. We give the conclusion in Sec. IV.

II. TWISTED FRIEDBERG-LEE SYMMETRIC SEESAW MODEL
A. framework of model We start with the conventional (type-I) seesaw framework with three right-handed Majorana neutrinos. The relevant Lagrangian is given by where we have omitted family indeces. We assume the diagonal charged lepton mass matrix and impose the twisted FL symmetry [13] only on neutrinos where ξ is a non-local Grassmann parameter, η is a column vector of c-numbers, and S is the permutation matrix between the second and third families, given by Here, we have adopted the specific combination η ∝ (−2, 1, 1) T discussed in Ref. [13] so that the resulting neutrino mixing is tribimaximal. Due to the symmetry, the Majorana mass matrix takes the form In addition to the twisted FL symmetry in Eq. (2), we consider a Z 2 symmetry for the lepton doublet and charged singlet of the first family. Consequently, the Dirac mass matrix is given by We note that without the Z 2 symmetry, Eq. (5) has the same form as Eq. (4). As a result, the model cannot reproduce a realistic neutrino mass spectrum because we focus on a scenario in which one of three light neutrinos is massless in this Letter.
The Majorana mass matrix in Eq. (4) can be diagonalized by the tribimaximal matrix so that where P = diag(1, e iφ R /2 , 1) is a diagonal phase matrix of the right-handed Majorana neutrinos. In this basis, the Dirac mass matrix in Eq. (5) becomes Note that α can be always real by suitable redefinitions of the left-handed leptons. As pointed out in Ref. [14], in this basis the right-handed neutrino of the first family can be regarded as a non-interacting massless neutrino. By omitting this field, we can move to 3 × 2 dimensional Dirac mass matrix basis and rewrite Eq. (8) as where D R = diag(M 2 , M 3 ). The mass matrix of the light neutrinos is as follows This matrix can be diagonalized by the tribimaximal matrix and has only one non-zero eigenvalue, m 3 . Thus, there are two interacting and one non-interacting massless neutrinos and no CP violating phase in the MNS matrix. Clearly, it is inconsistent with the experimental data of existing at least two massive light neutrinos.
In order to obtain a realistic model, we need to introduce symmetry breaking terms in Eq. (5), given by Note that the breaking terms violate both the permutation symmetry in Eq. (2) and the Z 2 symmetry, but preserve the translational symmetry so that the first family light neutrino remains massless. Note also that although we could introduce breaking terms for the Majorana mass matrix as well, we only focus on the effect from the Dirac mass matrix in the following discussions.
In the diagonal basis of the right-handed neutrinos, the Dirac mass matrix can be still regarded as an 3 × 2 dimensional matrix 2 and becomes In what follows, we consider the basis where α is real but ∆ i are complex and for simplicity assume ∆ 1 ≡ ∆ ≡ |∆|e iφ ∆ and ∆ 2 = ∆ 3 = ∆ 4 = 0. The mass matrix of the light neutrinos is given by where the second and third terms are responsible for the deviations from the tribimaximal mixing with while ∆ and φ R generate CP violation in the MNS matrix. Here, we define the MNS matrix as where s θ = sin θ (c θ = cos θ) with δ is the Dirac phase which has to satisfy to guarantee the right hand side of Eq. (16) to be real, and Ω = diag(1, e iγ/2 , 1) is a diagonal Majorana phase matrix. The mixing angles are given by We note that our definitions of the Dirac and Majorana phases (δ and γ) are different from δ pdg and γ pdg of the standard parametrization proposed by the Particle Data Group [16]. The relations between them are given by cos δ pdg = c 2 12 c 2 23 + s 2 12 s 2 23 s 2 13 − 1/3(1 − 3s 2 12 ) − 3/2s 2 13 − 2(1 − 3s 2 13 ) s 13 cos δ 2s 12 c 12 s 23 c 23 s 13 , where s ij (c ij ) means sin θ ij (cos θ ij ), respectively. The mass matrix in Eq. (13) is diagonalized by Eq. (15), leading to the masses of the light neutrinos to be The Majorana phase is given by where as input parameters. The remaining two quantities are the masses of the heavy neutrinos, M 2 and M 3 . As we will discuss later, in order to account for the measured value of the BAU, they should be O(10 10∼11 ) GeV, corresponding to α ∼ 0.004 and |∆| ∼ 0.002. If From Eqs. (18) and (20), one can see that the Dirac phase δ can be described by sin θ 23 and sin θ 13 . Furthermore, from Eqs. (19) and (29), one obtains that 0.0073 < sin 2 θ 13 . Finally, we would like to briefly remark on the possibility to test our model. As our model predicts the novel relation sin 2 θ 13 ≃ 1/3 − sin 2 θ 12 based on Eqs. (18) and (19), more precise determinations of mixing angles would provide us a chance to rule out or confirm the model in future. For instance, the smaller value of sin 2 θ 12 , which is sin 2 θ 12 = 0.304 +0.000 −0.016 , results in sin 2 θ 13 = 0.0293 ∼ 0.0453 which is beyond the 1σ range given in Ref. [17]. On the other hand, the larger value sin 2 θ 12 = 0.304 +0.022 −0.000 corresponding to sin 2 θ 13 = 0.0073 ∼ 0.0293 is well coincident with [17] and a recently proposed global analysis of the neutrino oscillation data [18].

III. LEPTOGENESIS
As discussed in the previous section, our model results in nonzero values of δ and sin θ 13 as shown in Eqs. (30) and (31). This means that the CP symmetry is always violated in the lepton sector at 1σ level even if there is no Majorana phase γ. In this section, we consider the unflavored leptogenesis mechanism 3 via the out-of-equilibrium decays of the heavy right-handed neutrinos. The CP violating parameter in the leptogenesis due to the i-th heavy neutrino decays is written as [2] where i, j = 2 or 3, F (x) is given by Y R is the Dirac mass matrix in the diagonal basis of the right-handed neutrinos and charged leptons, given in Eq. (12), with the first (second) column referred as Y R j2 (Y R j3 ).
The dilution factor κ i is approximately given by [21] κ where with M pl = 1.22 × 10 19 GeV and g * = 106.75. The net BAU is found to be where ω = 28/79.
We note that, in general, the CP asymmetry depends on both φ R and φ ∆ which are responsible for the Dirac and Majorana phases. In the followings, we first examine two extreme cases of (A) φ R = 0 and φ ∆ = 0 and (B) φ R = 0 and φ ∆ = 0, and then study the general case of (C) φ R = 0 and φ ∆ = 0. Since ε 2 ≫ ε 3 , we will only concentrate on ε 2 .
A. φ R = 0 and φ ∆ = 0 For φ R = 0 and φ ∆ = 0, the CP asymmetry in Eq. (33) can be simplified as 3 The importance of the flavor effects is discussed in Ref. [20]. Similarly, for φ R = 0 and φ ∆ = 0, the CP asymmetry is given by where the phase φ R is responsible for both Dirac and Majorana phases. In Fig. 4, we display η B as a function of δ (upper) or γ (lower) similar to Fig are continuous and broad.

IV. CONCLUSION
We have studied the BAU through the leptogenesis mechanism in the model with the FL symmetry. We have tried to make a connection between the leptogenesis and the CP violating Dirac and Majorana phases in the MNS matrix. In particular, we have demonstrated that there exists a wide range of these phases to achieve the measure BAU, allowed by the neutrino data.