# Radiatively Generated Leptogenesis inS4 Flavor Symmetry Models

Authors: T. Phong Nguyen; P. V. Dong
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

We study how leptogenesis can be implemented in the seesaw models withS4 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 |〈mee〉| 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 |〈mee〉| through a high-energy CP phase ϕ, which is correlated with the low-energy
Majorana CP phases. It is shown that the predictions of |〈mee〉| for some fixed parameters of the
high-energy physics can be constrained by the current observation of baryon asymmetry.


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## Body

## 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(1.1)UTB=(23130-1613-12-161312)
can be regarded as the PMNS matrix UPMNS≡UTBPν [3–6], where Pν 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 A4 [7–16], T′ [17–21], and S4 [22–36]. The common feature of these models is that they are naturally realized at a very-high-energy scale Λ 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 (0ν2β) decays [45–48]. The exact TBM pattern forbids at low energy the CP violation in neutrino oscillations, due to Ue3=0. Therefore, any observation of the leptonic CP violation, for instance, in the 0ν2β 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 |〈mee〉| and examine a link between observable low-energy 0ν2β 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 SU(2)L×U(1)Y×S4 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 0ν2β decay through the seesaw mechanism.The rest of this work is organized as follows. In Section2, we present the low-energy observables in two variants of supersymmetric seesaw model based on flavor symmetry S4. We especially focus on the effective neutrino mass governing the 0ν2β 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. TwoS4 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 0ν2β decay, leptogenesis, and phenomenological difficulties associated with the models to be solved.


### 2.1. Bazzocchi-Merlo-Morisi Model

In this model the flavor symmetry isS4 accompanied with cyclic group Z5 and Froggatt-Nielsen symmetry U(1)FN [49], that is, Gf=S4×Z5×U(1)FN. The matter fields and flavons are given in Table 1. The superpotential for the lepton sector reads(2.1)wl=∑i=14θΛye,iΛ3ec(lXi)12hd+yμΛ2μc(lψη)12hd+yτΛτc(lψ)11hd+h.c.+⋅⋅⋅,wν=x(νcl)11hu+xd(νcνcφ)11+xt(νcνcΔ)11+h.c.+⋅⋅⋅,
where Xi=ψψη,ψηη,ΔΔξ′,Δφξ′ and the dots denote higher-order contributions.Table 1
Transformation properties of the lepton sector and all the flavons of the BMM model whereω=ei2π/3.
Fieldlecμcτcνchu,dθψηΔφξ′S43112121131111131231212Z5ω41ω2ω4ω11ω2ω2ω3ω31U(1)FN010000−100000The VEV alignment of flavons is(2.2)〈ψ〉=(010)Tυψ,〈Δ〉=(111)TυΔ,〈η〉=(01)Tυη,〈φ〉=(11)Tυφ,〈ξ′〉=υξ′,
where all the VEVs are of the same order of magnitude and for this reason being parameterized as VEVs/Λ=u. The remaining VEV which originates from a different mechanism is υθ, denoted by υθ/Λ=t. It is shown in [35] that u and t belong to a well-determined range 0.01<u,t<0.05.The mass matrix for the charged leptons is given by(2.3)ml=(ye(1)u2tye(2)u2tye(2)u2t0yμu000yτ)uυd.
The Dirac and RH-Majorana neutrino mass matrices are, respectively, obtained as(2.4)mνd=(100001010)xυu,MR=Beiα1(2reiϕ1-reiϕ1-reiϕ1-reiϕ1+2reiϕ-reiϕ1-reiϕ-reiϕ1+2reiϕ),
where B=2|xd|υφ, C=2|xt|υΔ, and r=C/B are real and positive. The phases α1 and α2 are the arguments of xd,t, and ϕ≡α2-α1 being the only physical phase remained in MR. Notice that the MR can be exactly diagonalized by the TBM matrix:(2.5)MRD=VRTMRVR=Diag.(M1,M2,M3),M1=B|3reiϕ-1|,M2=2B,M1=B|3reiϕ+1|,VR=UTBVP,VP=Diag.(eiγ1/2,1,eiγ2/2),γ1,2=-arg(3reiϕ∓1).Integrating out the heavy degrees of freedom, we get the effective light neutrino mass matrix, which is given by the seesaw relation,meff=-(mνd)TMR-1mνd [37–39], and diagonalized by the TBM matrix:(2.6)UνTmeffUν=Diag.(m1,m2,m3)=-Diag.(x2υu2M1,x2υu2M2,x2υu2M3),Uν=UTBDiag.(e-iγ1/2,1,e-iγ2/2).In order to find the lepton mixing matrix we need to diagonalize the charged-lepton mass matrix:(2.7)mlD=Ulc†mlUl=Diag.(yeu2t,yμu,yτ)uυd,
where Ul is unity matrix. Therefore we get(2.8)UPMNS=Ul†Uν≡Uν=e-iγ1/2UTBDiag.(1,eiβ1,eiβ2),
where β1=γ1/2 and β2=(γ1-γ2)/2 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 mνd, 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 Gf symmetry group, predicting TBM and in turn no Dirac CP-violating phase. The remaining five physical parameters, β1,β2, m1, m2, and m3, are determined by the five real parameters B,C, υu, x, and ϕ.The light neutrino mass spectrum can have both normal or inverted hierarchy depending on the sign ofcosϕ. If cosϕ<0, one has normal hierarchy (NH), whereas if  cosϕ>0, 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 1σ [1, 2]:(2.9)|Δm312|=(2.29-2.52)×10-3eV2,Δm212=(7.45-7.88)×10-5eV2.
(Hereafter, we always use the experimental data at 1σ for our numerical calculations of low-energy observables.) The correlations between r and cosϕ for the NH spectrum (red (light) plot) and IH one [blue (dark) plot) are, respectively, presented in Figure 1.Figure 1
Allowed parameter region by the1σ experimental constraints (2.9) for the ratio r=C/B as a function of cosϕ. The blue (dark) and red (light) curves correspond to the IH and NH spectra.Because there is no Dirac CP-violating phase as mentioned, the only contribution from the Majorana phases to the0ν2β decay comes from β1. The effective neutrino mass governing the 0ν2β decay is given by(2.10)|〈mee〉|=13|2m1+m2e2iβ1|=m03(1-6rcos⁡ϕ+9r2)8.5+13.5r2+20.25r4-3r(13+12r2)cos⁡ϕ+9r2cos⁡2ϕ,
where m0=x2υu2/B. The behavior of |〈mee〉| is plotted in Figure 2 as a function of ϕ. The horizontal line (0.2 eV) is the current lower bound sensitivity [50–53] while the dashed line (10-2 eV) is a future sensitivity [54, 55].Figure 2
Prediction of the effective neutrino mass|〈mee〉| responsible for 0ν2β decay as a function of ϕ by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH spectra.Using (10) we can obtain the explicit relation betweenϕ and β1:(2.11)sin⁡2β1=-3rsin⁡ϕ1-6rcos⁡ϕ+9r2.Figure3 represents this relation corresponding to the NH spectrum [red (light) plot] and IH one [blue (dark) plot].Figure 3
The Majorana CP phaseβ1 as a function of ϕ plotted by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH spectra.In a basis where the charged current is flavor diagonal and the heavy neutrino mass matrixMR is diagonal and real, the Dirac mass matrix mνd gets modified to(2.12)mνd⟶Yνυu=VRTmνd,
where υu=υsinβ,  υ=176 GeV, and the coupling of Ni with leptons and scalar, Yν, is given by(2.13)Yν=xeiγ1/2(23-16-16e-iβ13e-iβ13e-iβ130e-iβ22-e-iβ22).
Concerned with the CP violation, we notice that the CP phase ϕ originating from mνd obviously takes part at the low-energy CP violation as the Majorana phases β1 and β2. On the other hand, the leptogenesis is associated with both the Yukawa coupling Yν and its combination:(2.14)H≡YνYν†=x2⋅Diag.(1,1,1).
This directly indicates that all off-diagonal Hij vanish, so the CP asymmetry could not be generated and neither leptogenesis. For the leptogenesis to be viable, the off-diagonalHijhave to be generated.


### 2.2. Ding Model

Ding model, proposed in [36], possesses flavor symmetry group Gf=S4×Z3×Z4, where the three factors play different roles. The S4 controls the mixing angles, the Z3 guarantees the misalignment in flavor space between neutrino and charged-lepton eigenstates, and the  Z4 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 Gf are assigned as in Table 2.Table 2
Representations of the matter fields of lepton sector and flavons underS4×Z3×Z4.
Fieldlecμcτcνchu,dφχϑηϕΔS431111211311131321223112Z3ωω2ω2ω211111ω2ω2ω2Z41i−1-i11ii111−1The superpotential for the lepton sector reads(2.15)wl=ye1Λ3ec(lφ)11(φφ)11hd+ye2Λ3ec((lφ)2(φφ)2)11hd+ye3Λ3ec((lφ)31(φφ)31)11hd+ye4Λ3ec((lχ)2(χχ)2)11hd+ye5Λ3ec((lχ)31(χχ)31)11hd+ye6Λ3ec(lφ)11(χχ)11hd+ye7Λ3ec((lφ)2(χχ)2)11hd+ye8Λ3ec((lφ)31(χχ)31)11hd+ye9Λ3ec((lχ)2(φφ)2)11hd+ye10Λ3ec((lχ)31(φφ)31)11hd+yμ2Λ2μc(l(φχ)32)12hd+yτΛτc(lφ)11hd+⋅⋅⋅,wν=yν1Λ((νcl)2η)11hu+yν2Λ((νcl)31ϕ)11hu+12M(νcνc)11+⋅⋅⋅,
where the dots denote higher-order contributions.The VEV alignment of flavons are assumed as follows:(2.16)〈φ〉=(0,υφ,0),〈χ〉=(0,υχ,0),〈ϑ〉=υϑ,〈η〉=(υη,υη),〈ϕ〉=(υϕ,υϕ,υϕ),〈Δ〉=υΔ.
The charged-lepton mass matrix is obtained by(2.17)ml=Diag.(yeυφ3Λ3,yμυφυχΛ2,yτυφΛ)υd,
where all the components are assumed to be real. The neutrino sector gives rise to the following Dirac and RH-Majorana mass matrices(2.18)mνd=eiα1(2beiϕa-beiϕa-beiϕa-beiϕa+2beiϕ-beiϕa-beiϕ-beiϕa+2beiϕ)υu,MR=(M0000M0M0),
where the quantity M is also supposed to be real and positive. The phase ϕ≡α2-α1, where α1,α2 are denoted as the arguments of yν1,yν2, respectively, is the only physical phase survived because the global phase α1 can be rotated away. The real and positive components a and b are defined as(2.19)a=|yν1|υηΛ,b=|yν2|υϕΛ.After seesawing, the effective light neutrino mass matrix is obtained frommeff=-(mνd)TMR-1mνd, which can be diagonalized by the TBM matrix:(2.20)UνTmeffUν=Diag.(m1,m2,m3),
where(2.21)m1=m0(1+9r2-6rcos⁡ϕ),m2=4m0,m3=m0(1+9r2+6rcos⁡ϕ),
with m0=υu2a2/M and r=b/a. The lepton mixing matrix is given by(2.22)UPMNS=Uν=e-iγ1/2UTBDiag.(1,eiβ1,eiβ2),
where β1=γ1/2,β2=(γ1-γ2)/2 are Majorana CP-violating phases with(2.23)γ1=arg{(a-3beiϕ)2},γ2=arg{-(a+3beiϕ)2}.
It is clear that the phase factored out to the left has no physical meaning. Moreover, the mixing angles are entirely fixed by the Gf symmetry, predicting TBM and in turn no Dirac CP-violating phase. There remain only five physical quantities, β1,β2, m1, m2, and m3, completely determined by the five parameters M, υu, a, b, and ϕ.There are two possible orderings in the masses of effective light neutrinos depending on the sign ofcosϕ: the NH corresponding to cosϕ>0 while the IH to cosϕ<0, which contrast with the previous model. The relation between r and cosϕ 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 0ν2β decay entirely comes from the Majorana phase β1. The relevant effective-neutrino mass is given by(2.24)|〈mee〉|=13|2m1+m2e2iβ1|=m01-4rcos⁡ϕ+2r2(2+3cos⁡2ϕ)-12r3cos⁡ϕ+9r4,
where m0=a2υu2/M. The behavior of |〈mee〉| as a function of ϕ 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 ϕ and β1 can be obtained from (2.23) as(2.25)sin⁡2β1=6rsin⁡ϕ(1-3rcos⁡ϕ)1-6rcos⁡ϕ+9r2,
which is presented in Figure 6 corresponding to the NH ordering [red (light) plot] and IH one [blue (dark) plot].Figure 4
Allowed parameter region by the1σ experimental constraints (2.9) for the ratio r=b/a as a function of cosϕ. The blue (dark) and red (light) curves correspond to the IH and NH ordering.Figure 5
Prediction of the effective mass|〈mee〉| responsible for 0ν2β as a function of ϕ by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH ordering.Figure 6
Relation between the phaseβ1 and ϕ as given by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH ordering.In a basis where the charged current is flavor diagonal, we diagonalizeMR in order to go into the physical mass basis of the RH neutrinos:(2.26)VRTMRVR=Diag.(M,M,-M),
where(2.27)VR=(100012-1201212).In this basis, the Dirac mass matrixmνd gets the form(2.28)mνd⟶Yνυu=VRTmνd,
where υu=υ·sinβ, υ=176 GeV and the coupling of Ni with leptons and scalar, Yν, is given by(2.29)Yν=(2beiϕa-beiϕa-beiϕ2(a-beiϕ)a+beiϕ2a+beiϕ20-(a+3beiϕ)2(a+3beiϕ)2).
Again, the CP phase ϕ which comes from mνd also takes part at the low-energy CP violation as the Majorana phases β1 and β2. On the other hand, the leptogenesis is associated with both the Yukawa coupling Yν and its combination:(2.30)H≡YνYν†=(2a2+6b2-4abcos⁡ϕ2(a2-3b2+2abcos⁡ϕ)02(a2-3b2+2abcos⁡ϕ)3a2+3b2-2abcos⁡ϕ000a2+9b2+6abcos⁡ϕ),
which directly indicates that all Im[Hij] 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.


## 2.1. Bazzocchi-Merlo-Morisi Model

In this model the flavor symmetry isS4 accompanied with cyclic group Z5 and Froggatt-Nielsen symmetry U(1)FN [49], that is, Gf=S4×Z5×U(1)FN. The matter fields and flavons are given in Table 1. The superpotential for the lepton sector reads(2.1)wl=∑i=14θΛye,iΛ3ec(lXi)12hd+yμΛ2μc(lψη)12hd+yτΛτc(lψ)11hd+h.c.+⋅⋅⋅,wν=x(νcl)11hu+xd(νcνcφ)11+xt(νcνcΔ)11+h.c.+⋅⋅⋅,
where Xi=ψψη,ψηη,ΔΔξ′,Δφξ′ and the dots denote higher-order contributions.Table 1
Transformation properties of the lepton sector and all the flavons of the BMM model whereω=ei2π/3.
Fieldlecμcτcνchu,dθψηΔφξ′S43112121131111131231212Z5ω41ω2ω4ω11ω2ω2ω3ω31U(1)FN010000−100000The VEV alignment of flavons is(2.2)〈ψ〉=(010)Tυψ,〈Δ〉=(111)TυΔ,〈η〉=(01)Tυη,〈φ〉=(11)Tυφ,〈ξ′〉=υξ′,
where all the VEVs are of the same order of magnitude and for this reason being parameterized as VEVs/Λ=u. The remaining VEV which originates from a different mechanism is υθ, denoted by υθ/Λ=t. It is shown in [35] that u and t belong to a well-determined range 0.01<u,t<0.05.The mass matrix for the charged leptons is given by(2.3)ml=(ye(1)u2tye(2)u2tye(2)u2t0yμu000yτ)uυd.
The Dirac and RH-Majorana neutrino mass matrices are, respectively, obtained as(2.4)mνd=(100001010)xυu,MR=Beiα1(2reiϕ1-reiϕ1-reiϕ1-reiϕ1+2reiϕ-reiϕ1-reiϕ-reiϕ1+2reiϕ),
where B=2|xd|υφ, C=2|xt|υΔ, and r=C/B are real and positive. The phases α1 and α2 are the arguments of xd,t, and ϕ≡α2-α1 being the only physical phase remained in MR. Notice that the MR can be exactly diagonalized by the TBM matrix:(2.5)MRD=VRTMRVR=Diag.(M1,M2,M3),M1=B|3reiϕ-1|,M2=2B,M1=B|3reiϕ+1|,VR=UTBVP,VP=Diag.(eiγ1/2,1,eiγ2/2),γ1,2=-arg(3reiϕ∓1).Integrating out the heavy degrees of freedom, we get the effective light neutrino mass matrix, which is given by the seesaw relation,meff=-(mνd)TMR-1mνd [37–39], and diagonalized by the TBM matrix:(2.6)UνTmeffUν=Diag.(m1,m2,m3)=-Diag.(x2υu2M1,x2υu2M2,x2υu2M3),Uν=UTBDiag.(e-iγ1/2,1,e-iγ2/2).In order to find the lepton mixing matrix we need to diagonalize the charged-lepton mass matrix:(2.7)mlD=Ulc†mlUl=Diag.(yeu2t,yμu,yτ)uυd,
where Ul is unity matrix. Therefore we get(2.8)UPMNS=Ul†Uν≡Uν=e-iγ1/2UTBDiag.(1,eiβ1,eiβ2),
where β1=γ1/2 and β2=(γ1-γ2)/2 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 mνd, 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 Gf symmetry group, predicting TBM and in turn no Dirac CP-violating phase. The remaining five physical parameters, β1,β2, m1, m2, and m3, are determined by the five real parameters B,C, υu, x, and ϕ.The light neutrino mass spectrum can have both normal or inverted hierarchy depending on the sign ofcosϕ. If cosϕ<0, one has normal hierarchy (NH), whereas if  cosϕ>0, 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 1σ [1, 2]:(2.9)|Δm312|=(2.29-2.52)×10-3eV2,Δm212=(7.45-7.88)×10-5eV2.
(Hereafter, we always use the experimental data at 1σ for our numerical calculations of low-energy observables.) The correlations between r and cosϕ for the NH spectrum (red (light) plot) and IH one [blue (dark) plot) are, respectively, presented in Figure 1.Figure 1
Allowed parameter region by the1σ experimental constraints (2.9) for the ratio r=C/B as a function of cosϕ. The blue (dark) and red (light) curves correspond to the IH and NH spectra.Because there is no Dirac CP-violating phase as mentioned, the only contribution from the Majorana phases to the0ν2β decay comes from β1. The effective neutrino mass governing the 0ν2β decay is given by(2.10)|〈mee〉|=13|2m1+m2e2iβ1|=m03(1-6rcos⁡ϕ+9r2)8.5+13.5r2+20.25r4-3r(13+12r2)cos⁡ϕ+9r2cos⁡2ϕ,
where m0=x2υu2/B. The behavior of |〈mee〉| is plotted in Figure 2 as a function of ϕ. The horizontal line (0.2 eV) is the current lower bound sensitivity [50–53] while the dashed line (10-2 eV) is a future sensitivity [54, 55].Figure 2
Prediction of the effective neutrino mass|〈mee〉| responsible for 0ν2β decay as a function of ϕ by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH spectra.Using (10) we can obtain the explicit relation betweenϕ and β1:(2.11)sin⁡2β1=-3rsin⁡ϕ1-6rcos⁡ϕ+9r2.Figure3 represents this relation corresponding to the NH spectrum [red (light) plot] and IH one [blue (dark) plot].Figure 3
The Majorana CP phaseβ1 as a function of ϕ plotted by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH spectra.In a basis where the charged current is flavor diagonal and the heavy neutrino mass matrixMR is diagonal and real, the Dirac mass matrix mνd gets modified to(2.12)mνd⟶Yνυu=VRTmνd,
where υu=υsinβ,  υ=176 GeV, and the coupling of Ni with leptons and scalar, Yν, is given by(2.13)Yν=xeiγ1/2(23-16-16e-iβ13e-iβ13e-iβ130e-iβ22-e-iβ22).
Concerned with the CP violation, we notice that the CP phase ϕ originating from mνd obviously takes part at the low-energy CP violation as the Majorana phases β1 and β2. On the other hand, the leptogenesis is associated with both the Yukawa coupling Yν and its combination:(2.14)H≡YνYν†=x2⋅Diag.(1,1,1).
This directly indicates that all off-diagonal Hij vanish, so the CP asymmetry could not be generated and neither leptogenesis. For the leptogenesis to be viable, the off-diagonalHijhave to be generated.


## 2.2. Ding Model

Ding model, proposed in [36], possesses flavor symmetry group Gf=S4×Z3×Z4, where the three factors play different roles. The S4 controls the mixing angles, the Z3 guarantees the misalignment in flavor space between neutrino and charged-lepton eigenstates, and the  Z4 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 Gf are assigned as in Table 2.Table 2
Representations of the matter fields of lepton sector and flavons underS4×Z3×Z4.
Fieldlecμcτcνchu,dφχϑηϕΔS431111211311131321223112Z3ωω2ω2ω211111ω2ω2ω2Z41i−1-i11ii111−1The superpotential for the lepton sector reads(2.15)wl=ye1Λ3ec(lφ)11(φφ)11hd+ye2Λ3ec((lφ)2(φφ)2)11hd+ye3Λ3ec((lφ)31(φφ)31)11hd+ye4Λ3ec((lχ)2(χχ)2)11hd+ye5Λ3ec((lχ)31(χχ)31)11hd+ye6Λ3ec(lφ)11(χχ)11hd+ye7Λ3ec((lφ)2(χχ)2)11hd+ye8Λ3ec((lφ)31(χχ)31)11hd+ye9Λ3ec((lχ)2(φφ)2)11hd+ye10Λ3ec((lχ)31(φφ)31)11hd+yμ2Λ2μc(l(φχ)32)12hd+yτΛτc(lφ)11hd+⋅⋅⋅,wν=yν1Λ((νcl)2η)11hu+yν2Λ((νcl)31ϕ)11hu+12M(νcνc)11+⋅⋅⋅,
where the dots denote higher-order contributions.The VEV alignment of flavons are assumed as follows:(2.16)〈φ〉=(0,υφ,0),〈χ〉=(0,υχ,0),〈ϑ〉=υϑ,〈η〉=(υη,υη),〈ϕ〉=(υϕ,υϕ,υϕ),〈Δ〉=υΔ.
The charged-lepton mass matrix is obtained by(2.17)ml=Diag.(yeυφ3Λ3,yμυφυχΛ2,yτυφΛ)υd,
where all the components are assumed to be real. The neutrino sector gives rise to the following Dirac and RH-Majorana mass matrices(2.18)mνd=eiα1(2beiϕa-beiϕa-beiϕa-beiϕa+2beiϕ-beiϕa-beiϕ-beiϕa+2beiϕ)υu,MR=(M0000M0M0),
where the quantity M is also supposed to be real and positive. The phase ϕ≡α2-α1, where α1,α2 are denoted as the arguments of yν1,yν2, respectively, is the only physical phase survived because the global phase α1 can be rotated away. The real and positive components a and b are defined as(2.19)a=|yν1|υηΛ,b=|yν2|υϕΛ.After seesawing, the effective light neutrino mass matrix is obtained frommeff=-(mνd)TMR-1mνd, which can be diagonalized by the TBM matrix:(2.20)UνTmeffUν=Diag.(m1,m2,m3),
where(2.21)m1=m0(1+9r2-6rcos⁡ϕ),m2=4m0,m3=m0(1+9r2+6rcos⁡ϕ),
with m0=υu2a2/M and r=b/a. The lepton mixing matrix is given by(2.22)UPMNS=Uν=e-iγ1/2UTBDiag.(1,eiβ1,eiβ2),
where β1=γ1/2,β2=(γ1-γ2)/2 are Majorana CP-violating phases with(2.23)γ1=arg{(a-3beiϕ)2},γ2=arg{-(a+3beiϕ)2}.
It is clear that the phase factored out to the left has no physical meaning. Moreover, the mixing angles are entirely fixed by the Gf symmetry, predicting TBM and in turn no Dirac CP-violating phase. There remain only five physical quantities, β1,β2, m1, m2, and m3, completely determined by the five parameters M, υu, a, b, and ϕ.There are two possible orderings in the masses of effective light neutrinos depending on the sign ofcosϕ: the NH corresponding to cosϕ>0 while the IH to cosϕ<0, which contrast with the previous model. The relation between r and cosϕ 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 0ν2β decay entirely comes from the Majorana phase β1. The relevant effective-neutrino mass is given by(2.24)|〈mee〉|=13|2m1+m2e2iβ1|=m01-4rcos⁡ϕ+2r2(2+3cos⁡2ϕ)-12r3cos⁡ϕ+9r4,
where m0=a2υu2/M. The behavior of |〈mee〉| as a function of ϕ 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 ϕ and β1 can be obtained from (2.23) as(2.25)sin⁡2β1=6rsin⁡ϕ(1-3rcos⁡ϕ)1-6rcos⁡ϕ+9r2,
which is presented in Figure 6 corresponding to the NH ordering [red (light) plot] and IH one [blue (dark) plot].Figure 4
Allowed parameter region by the1σ experimental constraints (2.9) for the ratio r=b/a as a function of cosϕ. The blue (dark) and red (light) curves correspond to the IH and NH ordering.Figure 5
Prediction of the effective mass|〈mee〉| responsible for 0ν2β as a function of ϕ by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH ordering.Figure 6
Relation between the phaseβ1 and ϕ as given by the 1σ experimental constraints (2.9). The blue (dark) and red (light) curves correspond to the IH and NH ordering.In a basis where the charged current is flavor diagonal, we diagonalizeMR in order to go into the physical mass basis of the RH neutrinos:(2.26)VRTMRVR=Diag.(M,M,-M),
where(2.27)VR=(100012-1201212).In this basis, the Dirac mass matrixmνd gets the form(2.28)mνd⟶Yνυu=VRTmνd,
where υu=υ·sinβ, υ=176 GeV and the coupling of Ni with leptons and scalar, Yν, is given by(2.29)Yν=(2beiϕa-beiϕa-beiϕ2(a-beiϕ)a+beiϕ2a+beiϕ20-(a+3beiϕ)2(a+3beiϕ)2).
Again, the CP phase ϕ which comes from mνd also takes part at the low-energy CP violation as the Majorana phases β1 and β2. On the other hand, the leptogenesis is associated with both the Yukawa coupling Yν and its combination:(2.30)H≡YνYν†=(2a2+6b2-4abcos⁡ϕ2(a2-3b2+2abcos⁡ϕ)02(a2-3b2+2abcos⁡ϕ)3a2+3b2-2abcos⁡ϕ000a2+9b2+6abcos⁡ϕ),
which directly indicates that all Im[Hij] 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 ofH=YνYν† (for BMM model), which are necessary ingredients for a successful leptogenesis mechanism.The radiative behavior of heavy RH-Majorana mass matrixMR is dictated by the following RG equation [56–60]:(3.1)dMRdt=2[(YνYν†)MR+MR(YνYν†)T],
where t=(1/16π2)ln(M/Λ′) and M is an arbitrary renormalization scale. The cutoff scale Λ′ can be regarded as the Gf breaking scale Λ′=Λ and assumed to be in order of the GUT scale, Λ′~1016 GeV.The RG equation for the Dirac neutrino Yukawa coupling can be written as(3.2)dYνdt=Yν[(T-3g22-35g12)+Yl†Yl+3Yν†Yν],
where T=Tr(3Yu†Yu+Yν†Yν), Yu and Yl are the Yukawa couplings of up-type quarks and charged leptons, and g2,1 are the SU(2)L and U(1)Y gauge coupling constants, respectively.Let us first reformulate (3.1) in the basis where MR is diagonal. Since MR is symmetric, it can be diagonalized by a unitary matrix VR as mentioned:(3.3)VRTMRVR=Diag.(M1,M2,M3).
As the structure of MR changes with the evolution of the scale, the VR depends on the scale too. The RG evolution of VR(t) can be written as(3.4)ddtVR=VRA,
where A is an anti-Hermitian matrix A†=-A due to the unitary of VR. Differentiating (3.3) we obtain(3.5)dMiδijdt=AijTMj+MiAij+2{VRT[(YνYν†)MR+MR(YνYν†)T]VR}ij.
Absorbing the unitary factor into the Dirac Yukawa coupling Yν≡VRTYν, the real diagonal part of (3.5) becomes(3.6)dMidt=4Mi(YνYν†)ii.
The RG equation for Yν in the basis of diagonal MR is given by(3.7)dYνdt=Yν[(T-3g22-35g12)+Yl†Yl+3Yν†Yν]+ATYν.Finally, we obtain the RG equation forH responsible for the leptogenesis:(3.8)dHdt=2(T-3g22-35g12)H+2Yν(Yl†Yl)Yν†+6H2+ATH+HA*.
The heavy Majorana mass splitting generated through the relevant RG evolution is thus given by(3.9)δNij=4(Hii-Hjj)t,
where H is defined in (2.30). Neglecting the RG evolution of Yν and its combination H=YνYν†, all the necessary components for the flavored leptogenesis in Ding model are available. The flavored CP asymmetries ɛiα can be obtained from (2.29), (2.30), (3.9), and (4.3).Notice however that in BMM model a nonvanishing CP asymmetry requiresIm[Hij(Yν)iα(Yν)jα*]≠0 with Yν defined in (2.13). Therefore, to have a viable radiative leptogenesis we need to induce a nonvanishing Hij(i≠j) at the leptogenesis scale. Indeed, this is possible since the RG effects due to the τ-Yukawa coupling contribution imply at the leading order yields [56–60](3.10)Hij(t)=2yτ2(Yν)i3(Yν)j3*×t.
The flavored CP asymmetries ɛiα 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 theS4 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 aboveT~(1+tan2β)×1012 GeV, do not distinguish among lepton flavors, called conventional or unflavored leptogenesis. However, if the scale of heavy RH neutrino masses is about M≤(1+tan2β)×1012 GeV, we need to take into account lepton flavor effects, called flavored leptogenesis.In this case, the CP asymmetry as generated by the decay ofith heavy RH neutrino far from almost degenerate is given by [61–71](4.1)ɛiα=18πHii∑j≠iIm⁡[Hij(Yν)iα(Yν)jα*]g(Mj2Mi2),
where Yν and H=YνYν† are in the basis where MR is real and diagonal. Here the loop function g(Mj2/Mi2) is(4.2)g(Mj2Mi2)≡gij(x)=x[21-x-ln⁡1+xx].
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 ith heavy RH neutrino (Ni) when decaying into a lepton flavor α(=e,μ,τ) and dominated by the one-loop self-energy contributions [74],(4.3)ɛiα=∑j≠iIm⁡[Hij(Yν)iα(Yν)jα*]16πHiiδNij(1+Γj24MjδNij2),
where Γj=HjjMj/8π is the decay width of jth RH neutrino and δNij is mass splitting parameter defined as(4.4)δNij=1-MjMi.As reminded in the previous section, by properly taking into account the RG effects, the nonzero flavored CP asymmetriesɛiα as given above can be obtained.Once the initial values ofɛiα are fixed, the final result of BAU, ηB, 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 Kiα which are the wash-out factors due to the inverse decay of Majorana neutrino Ni into the lepton flavor α. The explicit form of Kiα is given by(4.5)Kiα=ΓiαH(Mi)=(Yν†)αi(Yν)iαυu2m*Mi,
where Γiα is the partial decay width of Ni into the lepton flavors and Higgs scalars, H(Mi)≃(4π3g*/45)1/2Mi2/MPl with the Planck mass MPl=1.22×1019 GeV and the effective number of degrees of freedom g*≃228.75 is the Hubble parameter at temperature T=Mi, and the equilibrium neutrino mass m*≃10-3. 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 flavorɛiα is weighted differently by the corresponding washout parameter Kiα, appearing with a different weight in the final formula for the baryon asymmetry [75, 76]:(4.6)ηB≃-10-2∑Ni[ɛieκ(93110Kie)+ɛiμκ(1930Kiμ)+ɛiτκ(1930Kiτ)],
provided that the scale of heavy RH neutrino masses is about M≤(1+tan2β)×109 GeV where the μ and τ Yukawa couplings are in equilibrium and all the flavors are to be treated separately. And(4.7)ηB≃-10-2∑Ni[ɛi2κ(541761Ki2)+ɛiτκ(494761Kiτ)]
is given if (1+tan2β)·109 GeV ≤Mi≤(1+tan2β)·1012 GeV where only the τ Yukawa coupling is in equilibrium and treated separately while the e and μ flavors are indistinguishable. Here ɛi2=ɛie+ɛiμ, Ki2=Kie+Kiμ.The wash-out factors are given by(4.8)κiα≃(8.25Kiα+(Kiα0.2)1.16)-1.


### 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 isM3, then the leptogenesis is governed by the decay of M3 neutrino. The explicit form of flavored CP asymmetries ɛ3α is given from (2.13), (2.14), (3.10), and (4.1):(4.9)ɛ3e≃0,ɛ3μ≃ɛ3τ≃yτ2x224π(12sin⁡2β2⋅g31-sin⁡2(β1-β2)⋅g32)⋅t.
The corresponding washout parameters are(4.10)K3e=0,K3μ,τ≃34K1e.For the IH case, the lightest RH neutrino is ofM1, then the leptogenesis is governed by the decay of M1 neutrino. The flavored CP asymmetries ɛ1α are obtained as(4.11)ɛ1e≃-yτ2x236πsin⁡2β1⋅g12⋅t,ɛ1μ≃yτ2x224π(13sin⁡2β1⋅g12-12sin⁡2β2⋅g13)⋅t,ɛ1τ≃yτ2x224π(13sin⁡2β1⋅g12+12sin⁡2β2⋅g13)⋅t,
with corresponding washout parameters(4.12)K1e≃2m03m*(1-6rcos⁡ϕ+9r2),K1μ,τ≃14K1e.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 forηB as a function of |〈mee〉| is shown in Figure 7 where we have used B=1013 GeV for the NH case, B=1012 GeV for the IH case, and tanβ=30 as inputs. The horizontal solid and dashed lines correspond to the central value of BAU experimental data ηBCMB=6.1×10-10 [77–79] and phenomenologically allowed region 2×10-10≤ηB≤10-9, respectively. As shown in Figure 7, the current observation of ηBCMB can narrowly constrain the value of |〈mee〉| 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 β1 via the parameter ϕ.Prediction ofηB as a function of |〈mee〉| for the NH case (a) and IH case (b). The horizontal solid and dashed lines correspond to the experimental central value and phenomenologically allowed region.
(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 theH matrix, the contribution of N3 to lepton asymmetries ɛiα can be negligible due to H13(31)=H23(32)=0. (Actually, this is also correct if we take into account the RG effects on the H matrix since the radiative generation of H13(31),H23(32) is very small.) Combined with (2.29), (2.30), (3.9), and (4.3), the flavor-dependent CP asymmetries ɛiα are obtained as(4.13)ɛ1e≃a2rsin⁡ϕ32π(1-2rcos⁡ϕ+3r2)t,ɛ1μ≃ɛ1τ≃-a2rsin⁡ϕ64π(1-2rcos⁡ϕ+3r2)t,ɛ2e≃-a2rsin⁡ϕ16π(3-2rcos⁡ϕ+3r2)t,ɛ2μ≃ɛ2τ≃a2rsin⁡ϕ32π(3-2rcos⁡ϕ+3r2)t.
Here the mass slitting parameter δN12 which can be calculated from (2.30) and (3.9),(4.14)δN12=-δN21=-4a2(1-3r2+2rcos⁡ϕ),
has been used. The explicit form of Kiα is found as(4.15)K1e≃4r2m0m*,m0=a2υu2M,K1μ,τ≃m0m*(1-2rcos⁡ϕ+r2),K2e≃2m0m*(1-2rcos⁡ϕ+r2),K2μ,τ≃m02m*(1+2rcos⁡ϕ+r2).
With the help of (4.6), the BAU is obtained then.The prediction forηB as a function of |〈mee〉| is shown in Figure 8 where we have used M=103 GeV and tanβ=1. The horizontal solid and dashed lines correspond to the central value of the BAU experiment result ηBCMB=6.1×10-10 [77–79] and phenomenologically allowed region 2×10-10≤ηB≤10-9, respectively. As seen in Figure 8, the current observation of ηBCMB can narrowly constrain the value of |〈mee〉| 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 β1 via the parameter ϕ.Prediction ofηB as a function of |〈mee〉| or the NH case (a) and IH case (b). The horizontal solid and dashed lines correspond to the experimental central value and phenomenologically allowed region.
(a)(b)


## 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 isM3, then the leptogenesis is governed by the decay of M3 neutrino. The explicit form of flavored CP asymmetries ɛ3α is given from (2.13), (2.14), (3.10), and (4.1):(4.9)ɛ3e≃0,ɛ3μ≃ɛ3τ≃yτ2x224π(12sin⁡2β2⋅g31-sin⁡2(β1-β2)⋅g32)⋅t.
The corresponding washout parameters are(4.10)K3e=0,K3μ,τ≃34K1e.For the IH case, the lightest RH neutrino is ofM1, then the leptogenesis is governed by the decay of M1 neutrino. The flavored CP asymmetries ɛ1α are obtained as(4.11)ɛ1e≃-yτ2x236πsin⁡2β1⋅g12⋅t,ɛ1μ≃yτ2x224π(13sin⁡2β1⋅g12-12sin⁡2β2⋅g13)⋅t,ɛ1τ≃yτ2x224π(13sin⁡2β1⋅g12+12sin⁡2β2⋅g13)⋅t,
with corresponding washout parameters(4.12)K1e≃2m03m*(1-6rcos⁡ϕ+9r2),K1μ,τ≃14K1e.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 forηB as a function of |〈mee〉| is shown in Figure 7 where we have used B=1013 GeV for the NH case, B=1012 GeV for the IH case, and tanβ=30 as inputs. The horizontal solid and dashed lines correspond to the central value of BAU experimental data ηBCMB=6.1×10-10 [77–79] and phenomenologically allowed region 2×10-10≤ηB≤10-9, respectively. As shown in Figure 7, the current observation of ηBCMB can narrowly constrain the value of |〈mee〉| 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 β1 via the parameter ϕ.Prediction ofηB as a function of |〈mee〉| for the NH case (a) and IH case (b). The horizontal solid and dashed lines correspond to the experimental central value and phenomenologically allowed region.
(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 theH matrix, the contribution of N3 to lepton asymmetries ɛiα can be negligible due to H13(31)=H23(32)=0. (Actually, this is also correct if we take into account the RG effects on the H matrix since the radiative generation of H13(31),H23(32) is very small.) Combined with (2.29), (2.30), (3.9), and (4.3), the flavor-dependent CP asymmetries ɛiα are obtained as(4.13)ɛ1e≃a2rsin⁡ϕ32π(1-2rcos⁡ϕ+3r2)t,ɛ1μ≃ɛ1τ≃-a2rsin⁡ϕ64π(1-2rcos⁡ϕ+3r2)t,ɛ2e≃-a2rsin⁡ϕ16π(3-2rcos⁡ϕ+3r2)t,ɛ2μ≃ɛ2τ≃a2rsin⁡ϕ32π(3-2rcos⁡ϕ+3r2)t.
Here the mass slitting parameter δN12 which can be calculated from (2.30) and (3.9),(4.14)δN12=-δN21=-4a2(1-3r2+2rcos⁡ϕ),
has been used. The explicit form of Kiα is found as(4.15)K1e≃4r2m0m*,m0=a2υu2M,K1μ,τ≃m0m*(1-2rcos⁡ϕ+r2),K2e≃2m0m*(1-2rcos⁡ϕ+r2),K2μ,τ≃m02m*(1+2rcos⁡ϕ+r2).
With the help of (4.6), the BAU is obtained then.The prediction forηB as a function of |〈mee〉| is shown in Figure 8 where we have used M=103 GeV and tanβ=1. The horizontal solid and dashed lines correspond to the central value of the BAU experiment result ηBCMB=6.1×10-10 [77–79] and phenomenologically allowed region 2×10-10≤ηB≤10-9, respectively. As seen in Figure 8, the current observation of ηBCMB can narrowly constrain the value of |〈mee〉| 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 β1 via the parameter ϕ.Prediction ofηB as a function of |〈mee〉| or the NH case (a) and IH case (b). The horizontal solid and dashed lines correspond to the experimental central value and phenomenologically allowed region.
(a)(b)


## 5. Conclusions

We have studied theS4 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 YνYν† 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 YνYν† 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 the0νββ decay. It gives the definite predictions for 0ν2β decay parameter |〈mee〉|. Interestingly we have found a link between the leptogenesis and amplitude of 0ν2β decay |〈mee〉| through a high-energy CP phase ϕ. We have shown how the high-energy CP phase ϕ is correlated to the low-energy Majorana CP phase and examined how the leptogenesis can be related with the 0ν2β decay. It is pointed out that the predictions of |〈mee〉| for the NH and IH spectra can be constrained by the current observation of the baryon asymmetry of the universe as 6.1×10-10.



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*Source: 254093-2012-02-27.xml*