Radiatively Generated Leptogenesis in 𝑆_𝟒 Flavor Symmetry Models

Abstract

We study how leptogenesis can be implemented in the seesaw models with 𝑆₄ 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 |⟨𝑚_𝑒𝑒⟩| 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 |⟨𝑚_𝑒𝑒⟩| through a high-energy CP phase ϕ, which is correlated with the low-energy Majorana CP phases. It is shown that the predictions of |⟨𝑚_𝑒𝑒⟩| 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 leptons𝑈_TB=√2√31√30−1√61√3−1√2−1√61√31√2 (1.1) can be regarded as the PMNS matrix 𝑈_PMNS≡𝑈_TB𝑃_𝜈 [3–6], where 𝑃_𝜈 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 𝑈_𝑒3=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 |⟨𝑚_ee⟩| 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 Section 2, 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. Two 𝑆₄ 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 is 𝑆₄ accompanied with cyclic group 𝑍₅ and Froggatt-Nielsen symmetry 𝑈(1)_FN [49], that is, 𝐺_𝑓=𝑆₄×𝑍₅×𝑈(1)_FN. The matter fields and flavons are given in Table 1. The superpotential for the lepton sector reads𝑤_𝑙=⁴_𝑖=1𝜃Λ𝑦_𝑒,𝑖Λ³𝑒^𝑐𝑙𝑋_𝑖12ℎ_𝑑+𝑦_𝜇Λ²𝜇^𝑐𝑙𝜓𝜂₁₂ℎ_𝑑+𝑦_𝜏Λ𝜏^𝑐𝑙𝜓₁₁ℎ_𝑑𝑤+ℎ.𝑐.+⋅⋅⋅,_𝜈=𝑥𝜈^𝑐𝑙₁₁ℎ_𝑢+𝑥_𝑑𝜈^𝑐𝜈^𝑐𝜑₁₁+𝑥_𝑡𝜈^𝑐𝜈^𝑐Δ₁₁+ℎ.𝑐.+⋅⋅⋅, (2.1) where 𝑋_𝑖=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 where 𝜔=𝑒^𝑖2𝜋/3.

The VEV alignment of flavons is𝜓=010^𝑇𝜐_𝜓,⟨Δ⟩=111^𝑇𝜐_Δ,𝜂=01^𝑇𝜐_𝜂,𝜑=11^𝑇𝜐_𝜑,𝜉^′=𝜐_𝜉′, (2.2) 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 𝜐_𝜃, denoted by 𝜐_𝜃/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 by𝑚_𝑙=𝑦_𝑒⁽¹⁾𝑢²𝑡𝑦_𝑒⁽²⁾𝑢²𝑡𝑦_𝑒⁽²⁾𝑢²𝑡0𝑦_𝜇𝑢000𝑦_𝜏𝑢𝜐_𝑑. (2.3) The Dirac and RH-Majorana neutrino mass matrices are, respectively, obtained as𝑚^𝑑_𝜈=100001010𝑥𝜐_𝑢,𝑀_𝑅=𝐵𝑒^𝑖𝛼12𝑟𝑒^𝑖𝜙1−𝑟𝑒^𝑖𝜙1−𝑟𝑒^𝑖𝜙1−𝑟𝑒^𝑖𝜙1+2𝑟𝑒^𝑖𝜙−𝑟𝑒^𝑖𝜙1−𝑟𝑒^𝑖𝜙−𝑟𝑒^𝑖𝜙1+2𝑟𝑒^𝑖𝜙, (2.4) where 𝐵=2|𝑥_𝑑|𝜐_𝜑, 𝐶=2|𝑥_𝑡|𝜐_Δ, and 𝑟=𝐶/𝐵 are real and positive. The phases 𝛼₁ and 𝛼₂ are the arguments of 𝑥_𝑑,𝑡, and 𝜙≡𝛼₂−𝛼₁ being the only physical phase remained in 𝑀_𝑅. Notice that the 𝑀_𝑅 can be exactly diagonalized by the TBM matrix:𝑀^𝐷_𝑅=𝑉^𝑇_𝑅𝑀_𝑅𝑉_𝑅=Diag.𝑀₁,𝑀₂,𝑀₃,𝑀₁=𝐵3𝑟𝑒^𝑖𝜙−1,𝑀₂=2𝐵,𝑀₁=𝐵3𝑟𝑒^𝑖𝜙,𝑉+1_𝑅=𝑈_𝑇𝐵𝑉_𝑃,𝑉_𝑃=Diag.𝑒^𝑖𝛾1/2,1,𝑒^𝑖𝛾2/2,𝛾_1,2=−arg3𝑟𝑒^𝑖𝜙∓1. (2.5)

Integrating out the heavy degrees of freedom, we get the effective light neutrino mass matrix, which is given by the seesaw relation, 𝑚_eff=−(𝑚^𝑑_𝜈)^𝑇𝑀_𝑅⁻¹𝑚^𝑑_𝜈 [37–39], and diagonalized by the TBM matrix:𝑈^𝑇_𝜈𝑚_eff𝑈_𝜈=Diag.𝑚₁,𝑚₂,𝑚₃=−Diag.𝑥²𝜐²_𝑢𝑀₁,𝑥²𝜐²_𝑢𝑀₂,𝑥²𝜐²_𝑢𝑀₃,𝑈_𝜈=𝑈_TBDiag.𝑒^{−𝑖𝛾1/2},1,𝑒^{−𝑖𝛾2/2}. (2.6)

In order to find the lepton mixing matrix we need to diagonalize the charged-lepton mass matrix:𝑚^𝐷_𝑙=𝑈^†_𝑙𝑐𝑚_𝑙𝑈_𝑙=Diag.𝑦_𝑒𝑢²𝑡,𝑦_𝜇𝑢,𝑦_𝜏𝑢𝜐_𝑑, (2.7) where 𝑈_𝑙 is unity matrix. Therefore we get𝑈_PMNS=𝑈^†_𝑙𝑈_𝜈≡𝑈_𝜈=𝑒^{−𝑖𝛾1/2}𝑈_TBDiag.1,𝑒^𝑖𝛽1,𝑒^𝑖𝛽2, (2.8) 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 𝑚^𝑑_𝜈, 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 𝐺_𝑓 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, 𝜐_𝑢, 𝑥, and 𝜙.

The light neutrino mass spectrum can have both normal or inverted hierarchy depending on the sign of cos𝜙. 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]:Δ𝑚²₃₁=(2.29−2.52)×10⁻³eV²,Δ𝑚²₂₁=(7.45−7.88)×10⁻⁵eV². (2.9) (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 the 1𝜎 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 the 0𝜈2𝛽 decay comes from 𝛽₁. The effective neutrino mass governing the 0𝜈2𝛽 decay is given by𝑚_ee=132𝑚₁+𝑚₂𝑒^2𝑖𝛽1=𝑚₀31−6𝑟cos𝜙+9𝑟²8.5+13.5𝑟²+20.25𝑟⁴−3𝑟13+12𝑟²cos𝜙+9𝑟²cos²𝜙, (2.10) where 𝑚₀=𝑥²𝜐²_𝑢/𝐵. The behavior of |⟨𝑚_ee⟩| 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 between 𝜙 and 𝛽₁:sin2𝛽₁=−3𝑟sin𝜙1−6𝑟cos𝜙+9𝑟². (2.11)

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

Figure 3 

The Majorana CP phase 𝛽1 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 matrix 𝑀_𝑅 is diagonal and real, the Dirac mass matrix 𝑚^𝑑_𝜈 gets modified to𝑚^𝑑_𝜈⟶𝑌_𝜈𝜐_𝑢=𝑉^𝑇_𝑅𝑚^𝑑_𝜈, (2.12) where 𝜐_𝑢=𝜐sin𝛽, 𝜐=176 GeV, and the coupling of 𝑁_𝑖 with leptons and scalar, 𝑌_𝜈, is given by𝑌_𝜈=𝑥𝑒^𝑖𝛾1/223−1√6−1√6𝑒^−𝑖𝛽1√3𝑒^−𝑖𝛽1√3𝑒^−𝑖𝛽1√30𝑒^−𝑖𝛽2√2−𝑒^−𝑖𝛽2√2. (2.13) Concerned with the CP violation, we notice that the CP phase 𝜙 originating from 𝑚^𝑑_𝜈 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 𝑌_𝜈 and its combination:𝐻≡𝑌_𝜈𝑌^†_𝜈=𝑥²⋅Diag.(1,1,1). (2.14) This directly indicates that all off-diagonal 𝐻_𝑖𝑗 vanish, so the CP asymmetry could not be generated and neither leptogenesis. For the leptogenesis to be viable, the off-diagonal 𝐻_𝑖𝑗 have to be generated.

2.2. Ding Model

Ding model, proposed in [36], possesses flavor symmetry group 𝐺_𝑓=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 𝐺_𝑓 are assigned as in Table 2.

Table 2 

Representations of the matter fields of lepton sector and flavons under 𝑆₄×𝑍₃×𝑍₄.

The superpotential for the lepton sector reads𝑤_𝑙=𝑦_𝑒1Λ³𝑒^𝑐𝑙𝜑₁₁𝜑𝜑₁₁ℎ_𝑑+𝑦_𝑒2Λ³𝑒^𝑐𝑙𝜑₂𝜑𝜑₂₁₁ℎ_𝑑+𝑦_𝑒3Λ³𝑒^𝑐𝑙𝜑₃₁𝜑𝜑₃₁₁₁ℎ_𝑑+𝑦_𝑒4Λ³𝑒^𝑐𝑙𝜒₂𝜒𝜒₂₁₁ℎ_𝑑+𝑦_𝑒5Λ³𝑒^𝑐𝑙𝜒₃₁𝜒𝜒₃₁₁₁ℎ_𝑑+𝑦_𝑒6Λ³𝑒^𝑐𝑙𝜑₁₁𝜒𝜒₁₁ℎ_𝑑+𝑦_𝑒7Λ³𝑒^𝑐𝑙𝜑₂𝜒𝜒₂₁₁ℎ_𝑑+𝑦_𝑒8Λ³𝑒^𝑐𝑙𝜑₃₁𝜒𝜒₃₁₁₁ℎ_𝑑+𝑦_𝑒9Λ³𝑒^𝑐𝑙𝜒₂𝜑𝜑₂₁₁ℎ_𝑑+𝑦_𝑒10Λ³𝑒^𝑐𝑙𝜒₃₁𝜑𝜑₃₁₁₁ℎ_𝑑+𝑦_𝜇2Λ²𝜇^𝑐𝑙𝜑𝜒₃₂₁₂ℎ_𝑑+𝑦_𝜏Λ𝜏^𝑐𝑙𝜑₁₁ℎ_𝑑𝑤+⋅⋅⋅,_𝜈=𝑦_𝜈1Λ𝜈^𝑐𝑙₂𝜂₁₁ℎ_𝑢+𝑦_𝜈2Λ𝜈^𝑐𝑙₃₁𝜙₁₁ℎ_𝑢+12𝑀𝜈^𝑐𝜈^𝑐₁₁+⋅⋅⋅, (2.15) where the dots denote higher-order contributions.

The VEV alignment of flavons are assumed as follows:𝜑=0,𝜐_𝜑,0,𝜒=0,𝜐_𝜒,0,⟨𝜗⟩=𝜐_𝜗,𝜂=𝜐_𝜂,𝜐_𝜂,𝜙=𝜐_𝜙,𝜐_𝜙,𝜐_𝜙,⟨Δ⟩=𝜐_Δ. (2.16) The charged-lepton mass matrix is obtained by𝑚_𝑙=Diag.𝑦_𝑒𝜐³_𝜑Λ³,𝑦_𝜇𝜐_𝜑𝜐_𝜒Λ²,𝑦_𝜏𝜐_𝜑Λ𝜐_𝑑, (2.17) where all the components are assumed to be real. The neutrino sector gives rise to the following Dirac and RH-Majorana mass matrices𝑚^𝑑_𝜈=𝑒^𝑖𝛼12𝑏𝑒^𝑖𝜙𝑎−𝑏𝑒^𝑖𝜙𝑎−𝑏𝑒^𝑖𝜙𝑎−𝑏𝑒^𝑖𝜙𝑎+2𝑏𝑒^𝑖𝜙−𝑏𝑒^𝑖𝜙𝑎−𝑏𝑒^𝑖𝜙−𝑏𝑒^𝑖𝜙𝑎+2𝑏𝑒^𝑖𝜙𝜐_𝑢,𝑀_𝑅=,𝑀0000𝑀0𝑀0 (2.18) 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 𝑦_𝜈1,𝑦_𝜈2, 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 as𝑎=𝑦_𝜈1𝜐_𝜂Λ,𝑏=𝑦_𝜈2𝜐_𝜙Λ. (2.19)

After seesawing, the effective light neutrino mass matrix is obtained from 𝑚_eff=−(𝑚^𝑑_𝜈)^𝑇𝑀_𝑅⁻¹𝑚^𝑑_𝜈, which can be diagonalized by the TBM matrix:𝑈^𝑇_𝜈𝑚_eff𝑈_𝜈=Diag.𝑚₁,𝑚₂,𝑚₃, (2.20) where𝑚₁=𝑚₀1+9𝑟²,𝑚−6𝑟cos𝜙₂=4𝑚₀,𝑚₃=𝑚₀1+9𝑟²+6𝑟cos𝜙, (2.21) with 𝑚₀=𝜐²_𝑢𝑎²/𝑀 and 𝑟=𝑏/𝑎. The lepton mixing matrix is given by𝑈_PMNS=𝑈_𝜈=𝑒^{−𝑖𝛾1/2}𝑈_TBDiag.1,𝑒^𝑖𝛽1,𝑒^𝑖𝛽2, (2.22) where 𝛽₁=𝛾₁/2,𝛽₂=(𝛾₁−𝛾₂)/2 are Majorana CP-violating phases with𝛾₁=arg𝑎−3𝑏𝑒^𝑖𝜙2,𝛾₂=arg−𝑎+3𝑏𝑒^𝑖𝜙2. (2.23) It is clear that the phase factored out to the left has no physical meaning. Moreover, the mixing angles are entirely fixed by the 𝐺_𝑓 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, 𝜐_𝑢, 𝑎, 𝑏, and 𝜙.

There are two possible orderings in the masses of effective light neutrinos depending on the sign of cos𝜙: 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 by𝑚_ee=132𝑚₁+𝑚₂𝑒^2𝑖𝛽1=𝑚₀1−4𝑟cos𝜙+2𝑟²2+3cos2𝜙−12𝑟³cos𝜙+9𝑟⁴, (2.24) where 𝑚₀=𝑎²𝜐²_𝑢/𝑀. The behavior of |⟨𝑚_ee⟩| 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) assin2𝛽₁=6𝑟sin𝜙1−3𝑟cos𝜙1−6𝑟cos𝜙+9𝑟², (2.25) 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 the 1𝜎 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 phase 𝛽1 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 diagonalize 𝑀_𝑅 in order to go into the physical mass basis of the RH neutrinos:𝑉^𝑇_𝑅𝑀_𝑅𝑉_𝑅=Diag.(𝑀,𝑀,−𝑀), (2.26) where𝑉_𝑅=01100√2−1√201√21√2. (2.27)

In this basis, the Dirac mass matrix 𝑚^𝑑_𝜈 gets the form𝑚^𝑑_𝜈⟶𝑌_𝜈𝜐_𝑢=𝑉^𝑇_𝑅𝑚^𝑑_𝜈, (2.28) where 𝜐_𝑢=𝜐⋅sin𝛽, 𝜐=176 GeV and the coupling of 𝑁_𝑖 with leptons and scalar, 𝑌_𝜈, is given by𝑌_𝜈=2𝑏𝑒^𝑖𝜙𝑎−𝑏𝑒^𝑖𝜙𝑎−𝑏𝑒^𝑖𝜙√2𝑎−𝑏𝑒^𝑖𝜙𝑎+𝑏𝑒^𝑖𝜙√2𝑎+𝑏𝑒^𝑖𝜙√20−𝑎+3𝑏𝑒^𝑖𝜙√2𝑎+3𝑏𝑒^𝑖𝜙√2. (2.29) Again, the CP phase 𝜙 which comes from 𝑚^𝑑_𝜈 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 𝑌_𝜈 and its combination:𝐻≡𝑌_𝜈𝑌^†_𝜈=2𝑎²+6𝑏²−4𝑎𝑏cos𝜙√2𝑎²−3𝑏²0√+2𝑎𝑏cos𝜙2𝑎²−3𝑏²+2𝑎𝑏cos𝜙3𝑎²+3𝑏²−2𝑎𝑏cos𝜙000𝑎²+9𝑏²,+6𝑎𝑏cos𝜙 (2.30) which directly indicates that all Im[𝐻_𝑖𝑗] 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 𝐻=𝑌_𝜈𝑌^†_𝜈 (for BMM model), which are necessary ingredients for a successful leptogenesis mechanism.

The radiative behavior of heavy RH-Majorana mass matrix 𝐌_𝑅 is dictated by the following RG equation [56–60]:𝑑𝐌_𝑅𝐘𝑑𝑡=2_𝜈𝐘^†_𝜈𝐌_𝑅+𝐌_𝑅𝐘_𝜈𝐘^†_𝜈^𝑇, (3.1) where 𝑡=(1/16𝜋²)ln(𝑀/Λ^′) and 𝑀 is an arbitrary renormalization scale. The cutoff scale Λ^′ can be regarded as the 𝐺_𝑓 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 as𝑑𝐘_𝜈𝑑𝑡=𝐘_𝜈𝑇−3𝑔²₂−35𝑔²₁+𝐘^†_𝑙𝐘_𝑙+3𝐘^†_𝜈𝐘_𝜈, (3.2) where 𝑇=Tr(3𝑌^†_𝑢𝑌_𝑢+𝐘^†_𝜈𝐘_𝜈), 𝑌_𝑢 and 𝐘_𝑙 are the Yukawa couplings of up-type quarks and charged leptons, and 𝑔_2,1 are the SU(2)_𝐿 and U(1)_𝑌 gauge coupling constants, respectively.

Let us first reformulate (3.1) in the basis where 𝐌_𝑅 is diagonal. Since 𝐌_𝑅 is symmetric, it can be diagonalized by a unitary matrix 𝑉_𝑅 as mentioned:𝑉^𝑇_𝑅𝐌_𝑅𝑉_𝑅=Diag.𝑀₁,𝑀₂,𝑀₃. (3.3) As the structure of 𝐌_𝑅 changes with the evolution of the scale, the 𝑉_𝑅 depends on the scale too. The RG evolution of 𝑉_𝑅(𝑡) can be written as𝑑𝑉𝑑𝑡^𝑅=𝑉_𝑅𝐴, (3.4) where 𝐴 is an anti-Hermitian matrix 𝐴^†=−𝐴 due to the unitary of 𝑉_𝑅. Differentiating (3.3) we obtain𝑑𝑀_𝑖𝛿_𝑖𝑗𝑑𝑡=𝐴^𝑇_𝑖𝑗𝑀_𝑗+𝑀_𝑖𝐴_𝑖𝑗+2𝑉^𝑇_𝑅𝐘_𝜈𝐘^†_𝜈𝐌_𝑅+𝐌_𝑅𝐘_𝜈𝐘^†_𝜈^𝑇𝑉_𝑅𝑖𝑗. (3.5) Absorbing the unitary factor into the Dirac Yukawa coupling 𝑌_𝜈≡𝑉^𝑇_𝑅𝐘_𝜈, the real diagonal part of (3.5) becomes𝑑𝑀_𝑖𝑑𝑡=4𝑀_𝑖𝑌_𝜈𝑌^†_𝜈𝑖𝑖. (3.6) The RG equation for 𝑌_𝜈 in the basis of diagonal 𝐌_𝑅 is given by𝑑𝑌_𝜈𝑑𝑡=𝑌_𝜈𝑇−3𝑔²₂−35𝑔²₁+𝐘^†_𝑙𝐘_𝑙+3𝑌^†_𝜈𝑌_𝜈+𝐴^𝑇𝑌_𝜈. (3.7)

Finally, we obtain the RG equation for 𝐻 responsible for the leptogenesis:𝑑𝐻𝑑𝑡=2𝑇−3𝑔²₂−35𝑔²₁𝐻+2𝑌_𝜈𝐘^†_𝑙𝐘_𝑙𝑌^†_𝜈+6𝐻²+𝐴^𝑇𝐻+𝐻𝐴^∗. (3.8) The heavy Majorana mass splitting generated through the relevant RG evolution is thus given by𝛿_𝑁^𝑖𝑗=4𝐻_𝑖𝑖−𝐻_𝑗𝑗𝑡, (3.9) where 𝐻 is defined in (2.30). Neglecting the RG evolution of 𝑌_𝜈 and its combination 𝐻=𝑌_𝜈𝑌^†_𝜈, all the necessary components for the flavored leptogenesis in Ding model are available. The flavored CP asymmetries 𝜀^𝛼_𝑖 can be obtained from (2.29), (2.30), (3.9), and (4.3).

Notice however that in BMM model a nonvanishing CP asymmetry requires Im[𝐻_𝑖𝑗(𝑌_𝜈)_𝑖𝛼(𝑌_𝜈)^∗_𝑗𝛼]≠0 with 𝑌_𝜈 defined in (2.13). Therefore, to have a viable radiative leptogenesis we need to induce a nonvanishing 𝐻_𝑖𝑗(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]𝐻_𝑖𝑗(𝑡)=2𝑦²_𝜏𝑌_𝜈𝑖3𝑌_𝜈^∗_𝑗3×𝑡. (3.10) The flavored CP asymmetries 𝜀^𝛼_𝑖 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 𝑆₄ 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 𝑇∼(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 of 𝑖th heavy RH neutrino far from almost degenerate is given by [61–71]𝜀^𝛼_𝑖=18𝜋𝐻_{𝑖𝑖𝑗≠𝑖}Im𝐻_𝑖𝑗𝑌_𝜈𝑖𝛼𝑌_𝜈^∗_𝑗𝛼𝑔𝑀²_𝑗𝑀²_𝑖, (4.1) where 𝑌_𝜈 and 𝐻=𝑌_𝜈𝑌^†_𝜈 are in the basis where 𝑀_𝑅 is real and diagonal. Here the loop function 𝑔(𝑀²_𝑗/𝑀²_𝑖) is𝑔𝑀²_𝑗𝑀²_𝑖≡𝑔_𝑖𝑗(𝑥)=√𝑥21−𝑥−ln1+𝑥𝑥. (4.2) 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 (𝑁_𝑖) when decaying into a lepton flavor 𝛼(=𝑒,𝜇,𝜏) and dominated by the one-loop self-energy contributions [74],𝜀^𝛼_𝑖=_𝑗≠𝑖Im𝐻_𝑖𝑗𝑌_𝜈𝑖𝛼𝑌_𝜈^∗_𝑗𝛼16𝜋𝐻_𝑖𝑖𝛿_𝑁^𝑖𝑗Γ1+^2𝑗4𝑀_𝑗𝛿_𝑁^𝑖𝑗2, (4.3) where Γ_𝑗=𝐻_𝑗𝑗𝑀_𝑗/8𝜋 is the decay width of jth RH neutrino and 𝛿_𝑁^𝑖𝑗 is mass splitting parameter defined as𝛿_𝑁^𝑖𝑗𝑀=1−^𝑗𝑀_𝑖. (4.4)

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

Once the initial values of 𝜀^𝛼_𝑖 are fixed, the final result of BAU, 𝜂_𝐵, 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 𝐾^𝛼_𝑖 which are the wash-out factors due to the inverse decay of Majorana neutrino 𝑁_𝑖 into the lepton flavor 𝛼. The explicit form of 𝐾^𝛼_𝑖 is given by𝐾^𝛼_𝑖=Γ^𝛼_𝑖𝐻𝑀_𝑖=𝑌^†_𝜈𝛼𝑖𝑌_𝜈𝑖𝛼𝜐²_𝑢𝑚_∗𝑀_𝑖, (4.5) where Γ^𝛼_𝑖 is the partial decay width of 𝑁_𝑖 into the lepton flavors and Higgs scalars, 𝐻(𝑀_𝑖)≃(4𝜋³𝑔_∗/45)^1/2𝑀²_𝑖/𝑀_Pl with the Planck mass 𝑀_Pl=1.22×10¹⁹ GeV and the effective number of degrees of freedom 𝑔_∗≃228.75 is the Hubble parameter at temperature 𝑇=𝑀_𝑖, 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 flavor 𝜀^𝛼_𝑖 is weighted differently by the corresponding washout parameter 𝐾^𝛼_𝑖, appearing with a different weight in the final formula for the baryon asymmetry [75, 76]:𝜂_𝐵≃−10⁻²_𝑁𝑖𝜀^𝑒_𝑖𝜅93𝐾110^𝑒𝑖+𝜀^𝜇_𝑖𝜅19𝐾30^𝜇𝑖+𝜀^𝜏_𝑖𝜅19𝐾30^𝜏𝑖, (4.6) 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. And𝜂_𝐵≃−10⁻²_𝑁𝑖𝜀²_𝑖𝜅541𝐾761^2𝑖+𝜀^𝜏_𝑖𝜅494𝐾761^𝜏𝑖 (4.7) is given if (1+tan²𝛽)⋅10⁹ GeV ≤𝑀_𝑖≤(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 𝜀²_𝑖=𝜀^𝑒_𝑖+𝜀^𝜇_𝑖, 𝐾²_𝑖=𝐾^𝑒_𝑖+𝐾^𝜇_𝑖.

The wash-out factors are given by𝜅^𝛼_𝑖≃8.25𝐾^𝛼_𝑖+𝐾^𝛼_𝑖0.2^1.16−1. (4.8)

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 𝑀₃, then the leptogenesis is governed by the decay of 𝑀₃ neutrino. The explicit form of flavored CP asymmetries 𝜀^𝛼₃ is given from (2.13), (2.14), (3.10), and (4.1):𝜀^𝑒₃𝜀≃0,_𝜇3≃𝜀^𝜏₃≃𝑦²_𝜏𝑥²124𝜋2sin2𝛽₂⋅𝑔₃₁−sin2𝛽₁−𝛽₂⋅𝑔₃₂⋅𝑡. (4.9) The corresponding washout parameters are𝐾^𝑒₃=0,𝐾₃^𝜇,𝜏≃34𝐾^𝑒₁. (4.10)

For the IH case, the lightest RH neutrino is of 𝑀₁, then the leptogenesis is governed by the decay of 𝑀₁ neutrino. The flavored CP asymmetries 𝜀^𝛼₁ are obtained as𝜀^𝑒₁≃−𝑦²_𝜏𝑥²36𝜋sin2𝛽₁⋅𝑔₁₂𝜀⋅𝑡,_𝜇1≃𝑦²_𝜏𝑥²124𝜋3sin2𝛽₁⋅𝑔₁₂−12sin2𝛽₂⋅𝑔₁₃𝜀⋅𝑡,_𝜏1≃𝑦²_𝜏𝑥²124𝜋3sin2𝛽₁⋅𝑔₁₂+12sin2𝛽₂⋅𝑔₁₃⋅𝑡, (4.11) with corresponding washout parameters𝐾^𝑒₁≃2𝑚₀3𝑚_∗1−6𝑟cos𝜙+9𝑟²,𝐾₁^𝜇,𝜏≃14𝐾^𝑒₁. (4.12)

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 𝜂_𝐵 as a function of |⟨𝑚_ee⟩| 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 𝜂^CMB_𝐵=6.1×10⁻¹⁰ [77–79] and phenomenologically allowed region 2×10⁻¹⁰≤𝜂_𝐵≤10⁻⁹, respectively. As shown in Figure 7, the current observation of 𝜂^CMB_𝐵 can narrowly constrain the value of |⟨𝑚_ee⟩| 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.

(a)

(b)

Figure 7 

Prediction of 𝜂𝐵 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.

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 𝐻 matrix, the contribution of 𝑁₃ to lepton asymmetries 𝜀^𝛼_𝑖 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 𝜀^𝛼_𝑖 are obtained as𝜀^𝑒₁≃𝑎²𝑟sin𝜙32𝜋1−2𝑟cos𝜙+3𝑟²𝑡,𝜀^𝜇₁≃𝜀^𝜏₁≃−𝑎²𝑟sin𝜙64𝜋1−2𝑟cos𝜙+3𝑟²𝑡,𝜀^𝑒₂≃−𝑎²𝑟sin𝜙16𝜋3−2𝑟cos𝜙+3𝑟²𝑡,𝜀^𝜇₂≃𝜀^𝜏₂≃𝑎²𝑟sin𝜙32𝜋3−2𝑟cos𝜙+3𝑟²𝑡. (4.13) Here the mass slitting parameter 𝛿_𝑁¹² which can be calculated from (2.30) and (3.9),𝛿_𝑁¹²=−𝛿_𝑁²¹=−4𝑎²1−3𝑟²+2𝑟cos𝜙, (4.14) has been used. The explicit form of 𝐾^𝛼_𝑖 is found as𝐾^𝑒₁≃4𝑟²𝑚₀𝑚_∗,𝑚₀=𝑎²𝜐²_𝑢𝑀,𝐾₁^𝜇,𝜏≃𝑚₀𝑚_∗1−2𝑟cos𝜙+𝑟²,𝐾^𝑒₂≃2𝑚₀𝑚_∗1−2𝑟cos𝜙+𝑟²,𝐾₂^𝜇,𝜏≃𝑚₀2𝑚_∗1+2𝑟cos𝜙+𝑟². (4.15) With the help of (4.6), the BAU is obtained then.

The prediction for 𝜂_𝐵 as a function of |⟨𝑚_ee⟩| 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 𝜂^CMB_𝐵=6.1×10⁻¹⁰ [77–79] and phenomenologically allowed region 2×10⁻¹⁰≤𝜂_𝐵≤10⁻⁹, respectively. As seen in Figure 8, the current observation of 𝜂^CMB_𝐵 can narrowly constrain the value of |⟨𝑚_𝑒𝑒⟩| 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.

(a)

(b)

Figure 8 

Prediction of 𝜂𝐵 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.

5. Conclusions

We have studied the 𝑆₄ 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 𝑌_𝜈𝑌^†_𝜈 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 𝑌_𝜈𝑌^†_𝜈 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 0𝜈𝛽𝛽 decay. It gives the definite predictions for 0𝜈2𝛽 decay parameter |⟨𝑚_ee⟩|. Interestingly we have found a link between the leptogenesis and amplitude of 0𝜈2𝛽 decay |⟨𝑚_ee⟩| 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 |⟨𝑚_ee⟩| 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.

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.