#### Abstract

We study how leptogenesis can be implemented in the seesaw models with flavor symmetry, which lead to the tri-bimaximal neutrino mixing matrix. By considering renormalization group
evolution from a high-energy scale of flavor symmetry breaking (the GUT scale is assumed) to the
low-energy scale of relevant phenomena, the off-diagonal terms in a combination of Dirac Yukawa-coupling matrix can be generated and the degeneracy of heavy right-handed neutrino Majorana
masses can be lifted. As a result, the flavored leptogenesis is successfully realized. We also investigate
how the effective light neutrino mass associated with neutrinoless double beta decay can be
predicted along with the neutrino mass hierarchies by imposing the experimental data on the low-energy observables. We find a link between the leptogenesis and the neutrinoless double beta decay
characterized by through a high-energy CP phase *ϕ*, which is correlated with the low-energy
Majorana CP phases. It is shown that the predictions of for some fixed parameters of the
high-energy physics can be constrained by the current observation of baryon asymmetry.

#### 1. Introduction

The neutrino experimental data can provide an important clue for elucidating the origin of observed hierarchies in the mass matrices of quarks and leptons. The recent experiments of neutrino oscillation have gone into a new phase of precise determination of the mixing angles and squared mass differences [1, 2], which indicate that the tri-bimaximal mixing (TBM) for the three flavors of leptons can be regarded as the PMNS matrix [3–6], where is a diagonal matrix of CP phases. However, properties related to the leptonic CP violation have not been completely known yet. The large mixing angles, which may be suggestive of a flavor symmetry, are completely different from the quark mixing ones. Therefore, it is very important to find a model that naturally leads to those mixing patterns of quarks and leptons with a good accuracy. In recent years there have been a lot of efforts in searching for models which result in the TBM pattern naturally and a fascinating way seems to be the use of some discrete non-Abelian flavor groups added to the gauge groups of the standard model. There is a series of proposals based on groups [7–16], [17–21], and [22–36]. The common feature of these models is that they are naturally realized at a very-high-energy scale and the groups are spontaneously broken due to a set of scalar multiplets, the flavons.

In addition to the explanation of smallness of observed neutrino masses, the seesaw mechanism [37–39] has another appearing feature so-called leptogenesis mechanism for generation of observed baryon asymmetry of the Universe (BAU), through the decay of heavy right-handed (RH) Majorana neutrinos [40–44]. If this BAU was made via the leptogenesis, then the CP violation in leptonic sector is required. For the Majorana neutrinos of three flavors there are one Dirac-type phase and two Majorana-type phases, one (or a combination) of which in principle can be measured through neutrinoless double beta () decays [45–48]. The exact TBM pattern forbids at low energy the CP violation in neutrino oscillations, due to . Therefore, any observation of the leptonic CP violation, for instance, in the decay, can strengthen our belief in the leptogenesis by demonstrating that the CP is not a symmetry of leptons. It is interesting to explore this existence of the CP violation due to the Majorana CP-violating phases by measuring and examine a link between observable low-energy decay and the BAU. The authors in [35, 36] have shown that the TBM pattern can be generated naturally in the framework of the seesaw mechanism with symmetry. The textures of mass matrices as given in [35, 36] also could not generate a lepton asymmetry which is essential for the baryogenesis. In this paper, we investigate possibility of radiative leptogenesis when renormalization group (RG) effects are taken into account. We will show that the leptogenesis can be linked to the decay through the seesaw mechanism.

The rest of this work is organized as follows. In Section 2, we present the low-energy observables in two variants of supersymmetric seesaw model based on flavor symmetry . We especially focus on the effective neutrino mass governing the decay. In Section 3, we study RG effects on the Yukawa couplings and heavy Majorana neutrino mass matrices so that the leptogenesis becomes available. This leptogenesis in the two models due to the RG effects is studied in detail in Section 4. Finally, Section 5 is devoted to our conclusions.

#### 2. Two Models

In this section we give a review of the main features of Bazzocchi-Merlo-Morisi (BMM) model [35] and Ding model [36]. We simultaneously discuss the decay, leptogenesis, and phenomenological difficulties associated with the models to be solved.

##### 2.1. Bazzocchi-Merlo-Morisi Model

In this model the flavor symmetry is accompanied with cyclic group and Froggatt-Nielsen symmetry [49], that is, . The matter fields and flavons are given in Table 1. The superpotential for the lepton sector reads where and the dots denote higher-order contributions.

The VEV alignment of flavons is where all the VEVs are of the same order of magnitude and for this reason being parameterized as . The remaining VEV which originates from a different mechanism is , denoted by . It is shown in [35] that and belong to a well-determined range .

The mass matrix for the charged leptons is given by The Dirac and RH-Majorana neutrino mass matrices are, respectively, obtained as where , , and are real and positive. The phases and are the arguments of , and being the only physical phase remained in . Notice that the can be exactly diagonalized by the TBM matrix:

Integrating out the heavy degrees of freedom, we get the effective light neutrino mass matrix, which is given by the seesaw relation, [37–39], and diagonalized by the TBM matrix:

In order to find the lepton mixing matrix we need to diagonalize the charged-lepton mass matrix: where is unity matrix. Therefore we get where and are Majorana CP violating phases. The phase factored out to the left has no physical meaning, since it can be eliminated by a redefinition of the charged lepton fields. The light neutrino mass eigenvalues are simply the inverse of the heavy neutrino ones, a part from a minus sign and the global factor from , as can be seen in (2.6). There are nine physical parameters consisting of the three light neutrino masses, three mixing angles, and three CP-violating phases in general. The mixing angles are entirely fixed by the symmetry group, predicting TBM and in turn no Dirac CP-violating phase. The remaining five physical parameters, , , , and , are determined by the five real parameters , , , and .

The light neutrino mass spectrum can have both normal or inverted hierarchy depending on the sign of . If , one has normal hierarchy (NH), whereas if , one has inverted hierarchy (IH). In order to see how this correlation in the allowed parameter space is constrained, we consider the experimental data at [1, 2]: (Hereafter, we always use the experimental data at for our numerical calculations of low-energy observables.) The correlations between and for the NH spectrum (red (light) plot) and IH one [blue (dark) plot) are, respectively, presented in Figure 1.

Because there is no Dirac CP-violating phase as mentioned, the only contribution from the Majorana phases to the decay comes from . The effective neutrino mass governing the decay is given by where . The behavior of is plotted in Figure 2 as a function of . The horizontal line (0.2 eV) is the current lower bound sensitivity [50–53] while the dashed line ( eV) is a future sensitivity [54, 55].

Using (10) we can obtain the explicit relation between and :

Figure 3 represents this relation corresponding to the NH spectrum [red (light) plot] and IH one [blue (dark) plot].

In a basis where the charged current is flavor diagonal and the heavy neutrino mass matrix is diagonal and real, the Dirac mass matrix gets modified to
where , GeV, and the coupling of with leptons and scalar, , is given by
Concerned with the CP violation, we notice that the CP phase originating from obviously takes part at the low-energy CP violation as the Majorana phases and . On the other hand, the leptogenesis is associated with both the Yukawa coupling and its combination:
This directly indicates that all off-diagonal vanish, so the CP asymmetry could not be generated and neither leptogenesis. *For the leptogenesis to be viable, the off-diagonal ** have to be generated. *

##### 2.2. Ding Model

Ding model, proposed in [36], possesses flavor symmetry group , where the three factors play different roles. The controls the mixing angles, the guarantees the misalignment in flavor space between neutrino and charged-lepton eigenstates, and the is crucial to eliminating unwanted couplings and reproducing observed mass hierarchies. In this framework the mass hierarchies are controlled by spontaneously breaking of the flavor symmetry instead of the Froggatt-Nielsen mechanism [49]. The matter fields of lepton sector and flavons under are assigned as in Table 2.

The superpotential for the lepton sector reads where the dots denote higher-order contributions.

The VEV alignment of flavons are assumed as follows: The charged-lepton mass matrix is obtained by where all the components are assumed to be real. The neutrino sector gives rise to the following Dirac and RH-Majorana mass matrices where the quantity is also supposed to be real and positive. The phase , where are denoted as the arguments of , respectively, is the only physical phase survived because the global phase can be rotated away. The real and positive components and are defined as

After seesawing, the effective light neutrino mass matrix is obtained from , which can be diagonalized by the TBM matrix: where with and . The lepton mixing matrix is given by where are Majorana CP-violating phases with It is clear that the phase factored out to the left has no physical meaning. Moreover, the mixing angles are entirely fixed by the symmetry, predicting TBM and in turn no Dirac CP-violating phase. There remain only five physical quantities, , , , and , completely determined by the five parameters , , , , and .

There are two possible orderings in the masses of effective light neutrinos depending on the sign of : the NH corresponding to while the IH to , which contrast with the previous model. The relation between and for the NH spectrum (red plot) and IH one (blue plot) is included in Figure 4. Similarly to the previous model, the contribution to the decay entirely comes from the Majorana phase . The relevant effective-neutrino mass is given by where . The behavior of as a function of is plotted in Figure 5, where the horizontal line and dashed line are the current lower bound and the future one as mentioned. Moreover, the relation between and can be obtained from (2.23) as which is presented in Figure 6 corresponding to the NH ordering [red (light) plot] and IH one [blue (dark) plot].

In a basis where the charged current is flavor diagonal, we diagonalize in order to go into the physical mass basis of the RH neutrinos: where

In this basis, the Dirac mass matrix gets the form
where , GeV and the coupling of with leptons and scalar, , is given by
Again, the CP phase which comes from also takes part at the low-energy CP violation as the Majorana phases and . On the other hand, the leptogenesis is associated with both the Yukawa coupling and its combination:
which directly indicates that all vanish and in turn unflavored leptogenesis could not take place. *However, flavored leptogenesis can work if the degeneracy of the heavy Majorana neutrino masses is lifted. *

#### 3. Relevant RG Equations

In both models, the CP asymmetries due to the decay of heavy RH Majorana neutrinos at leading order vanish; therefore the leptogenesis could not take place. The radiative effects due to RG running from a high to low scale can naturally lead not only to a degenerate splitting of heavy Majorana masses (for Ding model), but also to an enhancement in vanished off-diagonal terms of (for BMM model), which are necessary ingredients for a successful leptogenesis mechanism.

The radiative behavior of heavy RH-Majorana mass matrix is dictated by the following RG equation [56–60]: where and is an arbitrary renormalization scale. The cutoff scale can be regarded as the breaking scale and assumed to be in order of the GUT scale, GeV.

The RG equation for the Dirac neutrino Yukawa coupling can be written as where , and are the Yukawa couplings of up-type quarks and charged leptons, and are the and gauge coupling constants, respectively.

Let us first reformulate (3.1) in the basis where is diagonal. Since is symmetric, it can be diagonalized by a unitary matrix as mentioned: As the structure of changes with the evolution of the scale, the depends on the scale too. The RG evolution of can be written as where is an anti-Hermitian matrix due to the unitary of . Differentiating (3.3) we obtain Absorbing the unitary factor into the Dirac Yukawa coupling , the real diagonal part of (3.5) becomes The RG equation for in the basis of diagonal is given by

Finally, we obtain the RG equation for responsible for the leptogenesis: The heavy Majorana mass splitting generated through the relevant RG evolution is thus given by where is defined in (2.30). Neglecting the RG evolution of and its combination , all the necessary components for the flavored leptogenesis in Ding model are available. The flavored CP asymmetries can be obtained from (2.29), (2.30), (3.9), and (4.3).

Notice however that in BMM model a nonvanishing CP asymmetry requires with defined in (2.13). Therefore, to have a viable radiative leptogenesis we need to induce a nonvanishing at the leptogenesis scale. Indeed, this is possible since the RG effects due to the -Yukawa coupling contribution imply at the leading order yields [56–60] The flavored CP asymmetries can then be obtained from (2.13), (2.14), (3.10), and (4.1).

#### 4. Radiatively Induced Flavored Leptogenesis

As already noticed, the leptogenesis cannot be realized in the models at the leading order, so this section is devoted to study the flavored leptogenesis with the effects of RG evolution.

The lepton asymmetries, which are produced by out-of-equilibrium decays of heavy RH neutrinos in early Universe at temperatures above GeV, do not distinguish among lepton flavors, called conventional or unflavored leptogenesis. However, if the scale of heavy RH neutrino masses is about GeV, we need to take into account lepton flavor effects, called flavored leptogenesis.

In this case, the CP asymmetry as generated by the decay of th heavy RH neutrino far from almost degenerate is given by [61–71] where and are in the basis where is real and diagonal. Here the loop function is This function depends strongly on the hierarchy of light neutrino masses.

For an almost degenerate heavy Majorana mass spectrum, the leptogenesis can be naturally implemented through the resonant leptogenesis [72, 73]. In this case, the CP asymmetry is generated by the th heavy RH neutrino () when decaying into a lepton flavor and dominated by the one-loop self-energy contributions [74],
where is the decay width of *j*th RH neutrino and is mass splitting parameter defined as

As reminded in the previous section, by properly taking into account the RG effects, the nonzero flavored CP asymmetries as given above can be obtained.

Once the initial values of are fixed, the final result of BAU, , can be given by solving a set of flavor-dependent Boltzmann equations including the decay, inverse decay, and scattering processes as well as the nonperturbative sphaleron interaction. In order to estimate the washout effects, we introduce parameters which are the wash-out factors due to the inverse decay of Majorana neutrino into the lepton flavor . The explicit form of is given by where is the partial decay width of into the lepton flavors and Higgs scalars, with the Planck mass GeV and the effective number of degrees of freedom is the Hubble parameter at temperature , and the equilibrium neutrino mass . From (2.13), (2.29), and (4.5) we can obtain the washout parameters corresponding to the two models.

Each lepton asymmetry for a single flavor is weighted differently by the corresponding washout parameter , appearing with a different weight in the final formula for the baryon asymmetry [75, 76]: provided that the scale of heavy RH neutrino masses is about GeV where the and Yukawa couplings are in equilibrium and all the flavors are to be treated separately. And is given if GeV GeV where only the Yukawa coupling is in equilibrium and treated separately while the and flavors are indistinguishable. Here , .

The wash-out factors are given by

##### 4.1. Bazzocchi-Merlo-Morisi Model

In this model, the RH neutrino masses are strongly hierarchical. For the NH case, the lightest RH neutrino mass is , then the leptogenesis is governed by the decay of neutrino. The explicit form of flavored CP asymmetries is given from (2.13), (2.14), (3.10), and (4.1): The corresponding washout parameters are

For the IH case, the lightest RH neutrino is of , then the leptogenesis is governed by the decay of neutrino. The flavored CP asymmetries are obtained as with corresponding washout parameters

Applying (4.6), (4.7), and (4.8), the BAU for two cases are then obtained. Notice also that in the NH case the leptogenesis has no contribution from the electron flavor decay channel which makes the scale of the heavy RH neutrino mass for a successful leptogenesis higher than that of the IH case.

The prediction for as a function of is shown in Figure 7 where we have used GeV for the NH case, GeV for the IH case, and as inputs. The horizontal solid and dashed lines correspond to the central value of BAU experimental data [77–79] and phenomenologically allowed region , respectively. As shown in Figure 7, the current observation of can narrowly constrain the value of for the NH and IH spectrum, respectively. Combining the results presented in Figures 2 and 3 with those from the leptogenesis, we can pin down the Majorana CP phase via the parameter .

**(a)**

**(b)**

##### 4.2. Ding Model

In this model, all the heavy RH neutrinos are exactly degenerate. By considering the RG effects, their masses get a tiny splitting (almost degenerate), which lead to a resonant leptogenesis as contributed from all these heavy RH neutrinos. However, if we neglect the RG effects on the matrix, the contribution of to lepton asymmetries can be negligible due to . (Actually, this is also correct if we take into account the RG effects on the matrix since the radiative generation of is very small.) Combined with (2.29), (2.30), (3.9), and (4.3), the flavor-dependent CP asymmetries are obtained as Here the mass slitting parameter which can be calculated from (2.30) and (3.9), has been used. The explicit form of is found as With the help of (4.6), the BAU is obtained then.

The prediction for as a function of is shown in Figure 8 where we have used GeV and . The horizontal solid and dashed lines correspond to the central value of the BAU experiment result [77–79] and phenomenologically allowed region , respectively. As seen in Figure 8, the current observation of can narrowly constrain the value of for the NH and IH spectrum, respectively. Again, combining the results in Figures 5 and 6 with those from the leptogenesis, we can pin down the Majorana CP phase via the parameter .

**(a)**

**(b)**

#### 5. Conclusions

We have studied the models in the context of a supersymmetric seesaw model which naturally lead to the TBM form for the lepton mixing matrix. In BMM model, the combination is proportional to unity whereas in Ding model the heavy RH Majorana masses are exactly degenerate. This would forbid the desirable leptogenesis to occur in each model. Therefore, for a viable leptogenesis the off-diagonal terms of in BMM model have to be generated, while in Ding model the degeneracy of heavy RH Majorana masses has to be lifted. We have shown that these can be easily achieved by the RG effects from a high-energy scale to the low-energy scale which result in the successful leptogenesis.

We have also studied implications to the low-energy observables such as the decay. It gives the definite predictions for decay parameter . Interestingly we have found a link between the leptogenesis and amplitude of decay through a high-energy CP phase . We have shown how the high-energy CP phase is correlated to the low-energy Majorana CP phase and examined how the leptogenesis can be related with the decay. It is pointed out that the predictions of for the NH and IH spectra can be constrained by the current observation of the baryon asymmetry of the universe as .

#### Acknowledgments

T. P. Nguyen is grateful to Exchange Program for East Asia Young Researchers, JSPS for financial support. He thanks Professor Yoshimasa Kurihara and Members of Theory Division at KEK for support. This work is supported in part by National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under Grant no. 103.03-2011.35.