Radiatively Generated Leptogenesis in S 4 Flavor Symmetry Models

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 |〈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.


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 Advances in High Energy Physics 1.1 can be regarded as the PMNS matrix U PMNS ≡ U TB P 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 A 4 7-16 , T 17-21 , and S 4 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 × S 4 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 S 4 .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.

Two S 4 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.

Bazzocchi-Merlo-Morisi Model
In this model the flavor symmetry is S 4 accompanied with cyclic group Z 5 and Froggatt-Nielsen symmetry The matter fields and flavons are given in Table 1.The superpotential for the lepton sector reads where X i ψψη, ψηη, ΔΔξ , Δϕξ and the dots denote higher-order contributions.
The VEV alignment of flavons is where B 2|x d |υ ϕ , C 2|x t |υ Δ , and r C/B are real and positive.The phases α 1 and α 2 are the arguments of x d,t , and φ ≡ α 2 − α 1 being the only physical phase remained in M R .Notice that the M R can be exactly diagonalized by the TBM matrix:

2.5
Integrating out the heavy degrees of freedom, we get the effective light neutrino mass matrix, which is given by the seesaw relation, R m d ν 37-39 , and diagonalized by the TBM matrix:

2.6
In order to find the lepton mixing matrix we need to diagonalize the charged-lepton mass matrix: where U l is unity matrix.Therefore we get 2.8 where β 1 γ 1 /2 and β 2 γ 1 − γ 2 /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, β 1 , β 2 , m 1 , m 2 , and m 3 , 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 cos φ.If cos φ < 0, one has normal hierarchy NH , whereas if cos φ > 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
Hereafter, we always use the experimental data at 1σ for our numerical calculations of lowenergy observables.The correlations between r and cos φ 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 0ν2β decay comes from β 1 .The effective neutrino mass governing the 0ν2β decay is given by   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 to Field where υ u υ sin β, υ 176 GeV, and the coupling of N i with leptons and scalar, Y ν , is given by 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 β 1 and β 2 .On the other hand, the leptogenesis is associated with both the Yukawa coupling Y ν and its combination:

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.

Ding Model
Ding model, proposed in 36 , possesses flavor symmetry group G f S 4 × Z 3 × Z 4 , where the three factors play different roles.The S 4 controls the mixing angles, the Z 3 guarantees the misalignment in flavor space between neutrino and charged-lepton eigenstates, and the Z 4 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.
The superpotential for the lepton sector reads Advances in High Energy Physics 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 M is also supposed to be real and positive.The phase φ ≡ α 2 − α 1 , where α 1 , α 2 are denoted as the arguments of y ν1 , y ν2 , respectively, is the only physical phase survived because the global phase α 1 can be rotated away.The real and positive components a and b are defined as After seesawing, the effective light neutrino mass matrix is obtained from which can be diagonalized by the TBM matrix: where with m 0 υ 2 u a 2 /M and r b/a.The lepton mixing matrix is given by where

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 CPviolating phase.There remain only five physical quantities, β 1 , β 2 , m 1 , m 2 , and m 3 , 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 cos φ: the NH corresponding to cos φ > 0 while the IH to cos φ < 0, which contrast with the previous model.The relation between r and cos φ 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 β 1 .The relevant effectiveneutrino mass is given by where m 0 a 2 υ 2 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 β 1 can be obtained from 2.   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: where

2.27
In this basis, the Dirac mass matrix m d ν gets the form where υ u υ • sin β, υ 176 GeV and the coupling of N i with leptons and scalar, Y ν , is given by Again, the CP phase φ which comes from m d ν also takes part at the low-energy CP violation as the Majorana phases β 1 and β 2 .On the other hand, the leptogenesis is associated with both the Yukawa coupling Y ν and its combination: 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.

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 offdiagonal terms of H Y ν Y † 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 : where t 1/16π 2 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 ∼ 10 16 GeV.
The RG equation for the Dirac neutrino Yukawa coupling can be written as 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:

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 as where A is an anti-Hermitian matrix A † −A due to the unitary of V R .Differentiating 3.3 we obtain

3.5
Absorbing the unitary factor into the Dirac Yukawa coupling Y ν ≡ V T R Y ν , the real diagonal part of 3.5 becomes

3.6
Advances in High Energy Physics 13 The RG equation for Y ν in the basis of diagonal M R is given by Finally, we obtain the RG equation for H responsible for the leptogenesis:

3.8
The heavy Majorana mass splitting generated through the relevant RG evolution is thus given by where H is defined in 2.30 .Neglecting the RG evolution of Y ν and its combination H Y ν Y † 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

Radiatively Induced Flavored Leptogenesis
As already noticed, the leptogenesis cannot be realized in the S 4 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 tan 2 β × 10 12 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 tan 2 β × 10 12 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 where Y ν and H Y ν Y † ν are in the basis where M R is real and diagonal.Here the loop function 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 , where Γ j H jj M j /8π is the decay width of jth RH neutrino and δ ij N 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 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 by where Γ α i is the partial decay width of N i into the lepton flavors and Higgs scalars, H M i 4π 3 g * /45 1/2 M 2 i /M Pl with the Planck mass M Pl 1.22 × 10 19 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 :  4.8

Bazzocchi-Merlo-Morisi Model
In this model, the RH neutrino masses are strongly hierarchical.For the NH case, the lightest RH neutrino mass is M 3 , then the leptogenesis is governed by the decay of M 3 neutrino.The explicit form of flavored CP asymmetries ε α 3 is given from 2.13 , 2.14 , 3.10 , and 4.1 : The corresponding washout parameters are

4.10
For the IH case, the lightest RH neutrino is of M 1 , then the leptogenesis is governed by the decay of M 1 neutrino.The flavored CP asymmetries ε α 1 are obtained as with corresponding washout parameters 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 10 13 GeV for the NH case, B 10 12 GeV for the IH case, and tan β 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 β 1 via the parameter φ.

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 N 3 to lepton asymmetries ε α i can be negligible due to H 13  ε τ 2 a 2 r sin φ 32π 3 − 2r cos φ 3r 2 t .

4.13
Here the mass slitting parameter δ 12 N which can be calculated from 2.30 and 3.9 ,

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 10 3 GeV and tan β 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 β 1 via the parameter φ.

Conclusions
We have studied the S 4 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 ν Y † 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 ν Y † 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.

Figure 1 :
Figure 1: Allowed parameter region by the 1σ experimental constraints 2.9 for the ratio r C/B as a function of cos φ.The blue dark and red light curves correspond to the IH and NH spectra.

Figure 2 :
Figure 2: Prediction of the effective neutrino mass | m ee | 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.

Figure 3 : 1 −3r sin φ 1 − 11 Figure 3
Figure 3: The Majorana CP phase β 1 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.

23 as sin 2β 1 6r sin φ 1 − 3r cos φ 1 −Figure 4 :
Figure 4: Allowed parameter region by the 1σ experimental constraints 2.9 for the ratio r b/a as a function of cos φ.The blue dark and red light curves correspond to the IH and NH ordering.

Figure 5 :
Figure 5: Prediction of the effective mass | m ee | 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 :
Figure6: Relation between the phase β 1 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.

Figure 7 :
Figure 7: Prediction of η B as a function of | m ee | 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.

Table 1 :
Transformation properties of the lepton sector and all the flavons of the BMM model where ω e i2π/3 .

Table 2 :
Representations of the matter fields of lepton sector and flavons under S 4 × Z 3 × Z 4 .
provided that the scale of heavy RH neutrino masses is about M ≤ 1 tan 2 β × 10 9 GeV where the μ and τ Yukawa couplings are in equilibrium and all the flavors are to be treated separately.And •10 9 GeV ≤ M i ≤ 1 tan 2 β •10 12 GeV where only the τ Yukawa coupling is in equilibrium and treated separately while the e and μ flavors are indistinguishable.Here ε 2 31H23 320.Actually, this is also correct if we take into account the RG effects on the H matrix since the radiative generation of H 13 31 , H 23 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 as