Neutrino masses, mixing, and leptogenesis in an S3 model
Arturo Alvarez Cruz Myriam Mondragón

Cutoff-free Circuit Quantum Electrodynamics
    Hakan E. Türeci
    December 30, 2023
===========================================



  In this work we use previous results on the masses and mixing of
  neutrinos of an S3 model with three right-handed Majorana neutrinos
  and three Higgs doublets, to reduce one parameter in the case when
  two of the right-handed neutrinos are mass degenerate. We derive a new
  parameterization for the V_PMNS mixing matrix, with a new set of
  parameters, in the more general case where the right-handed
  neutrino masses are different.  With these results, we calculate
  leptogenesis and the associated baryogenesis in the model in the two
  different scenarios.  We show that it is possible to have enough
  leptogenesis to explain the baryonic asymmetry with right-handed
  neutrino masses above 10^6 GeV.






§ INTRODUCTION

The Standard Model (SM) is extremely successful, nevertheless the
discovery of neutrino masses and mixing in neutrino oscillation
experiments in 1998 <cit.>, presented evidence that is
necessary to go beyond it. Even before this discovery, the amount of
free parameters and the hierarchy problem, among others, have prompted
attempts to find a more fundamental theory, of which the SM is the
low-energy limit <cit.>.
Some of the goals of these new models are to understand the large
differences in the Yukawa couplings of the different fermions, the
hierarchy between the fundamental particles, and the amount of CP
violation and the structure of the CKM matrix <cit.>.
A popular way to approach these problems is to build models with
Non-Abelian flavor symmetries, often supplemented with extra Higgs
doublets. Common symmetries in flavor theories are, among many others,
A4, Q6 or S3
<cit.>.
The reason is that these models achieve in a natural way the Nearest
Neighbour Interaction textures in the fermion mass matrices
<cit.>.  The S3 extension of the SM
with three Higgs doublets (S3-3H)
<cit.> is a model in which a
symmetry on the permutation of three objects is imposed, which in
additon to the SM particles has another two Higgs doublets, as well as
three right-handed Majorana neutrinos, which are related to the left
ones through the seesaw mechanism (type I).

There has been a lot of work done on various S3 models (see for instance
<cit.>),
some of this work reproduces the CKM and PMNS matrices in agreement
with the current experimental data
<cit.>,
and there have been also studies of leptogenesis in a soft breaking S3
model <cit.>.  Nevertheless, most of this work has been
done in the case where two right-handed neutrinos are degenerate.  In
this way, it is an interesting question to extend the model and see
the possible new results with a generalization, taking into account
both degenerate and non-degenerate right-handed neutrino masses.
Following the idea of previous work <cit.>, we extend the analysis on the generalization of the S3-3H model.


Another question that the  SM  fails to explain is the observed baryon asymmetry.
It is well know that there are more baryons than antibaryons in the Universe. Nucleosynthesis is a solid and  consistent 
model of the creation of the nuclei in the early Universe, which
predicts a baryonic density of, 

    η=η_b-η_b/η_γ=η=(2.6-6.2)×10^-10.

Measurements of the Cosmic Background Radiation  <cit.> show a density of   

    η=(6.1±0.3)×10^-10,

in full agreement with the baryon density of the Nucleosynthesis<cit.>.

The idea to explain the baryon asymmetry through a dynamically process
was proposed by Sakharov in 1967 <cit.>.   The present 
cosmological observations favour the idea  that the matter-antimatter asymmetry
of the Universe may  be explained in terms of a dynamical generation
mechanism, called baryogenesis.  Also, it has been realized that a
successful model of baryogenesis cannot occur within the Standard
Model (SM).


Leptogenesis is a mechanism which generates baryon asymmetry by
creating a leptonic asymmetry through B + L violating electroweak
sphaleron transitions <cit.>.


Several things are needed for the occurrence of leptogenesis:

  
  * Heavy right handed neutrinos.
  
  * Majorana type neutrinos.
  
  * Decay of the right handed neutrinos to the left ones.

According to the original proposal of Fukugita and Yanagida
<cit.>, this mechanism also satisfies all the
Sakharov’s conditions <cit.> in order to produce a net
baryon asymmetry (for reviews see for instance <cit.>).

In this paper we explore the possibility of leptogenesis in the S3-3H
model, with degenerate and non-degenerate right-handed neutrino
masses, and calculate the associated baryogenesis.  We first study the
case where two of the right-handed neutrino masses are degenerate, and
then the more general case where all the right-handed neutrino masses
are different.  We scan the parameter space to find the leptogenesis
and associated baryogenesis dependence on the free parameters of the
model.  We find that there is a region of parameter space where enough
baryogenesis is produced through leptogenesis to explain the
baryon asymmetry of the Universe.

The outline of the paper is organized as follows: In section 2, the S3
model is introduced as well as some of its most important results. In
section 3 it is shown how to produce leptogenesis in the S3-3H model, and
the resultant baryogenesis is also computed. At the end, in section 4,
we conclude summarizing our main results.



§ S3-3H MODEL


In the Standard Model analogous fermions in different generations have
identical couplings to all gauge bosons of the strong, weak, and
electromagnetic interactions <cit.>. 




The group S3 consists of the six possible
permutations of three objects (f_1,f_2,f_3), and is the smallest
discrete non-abelian group. It has one 2-dimension irreducible
representation (irrep) and two of 1-dimension

    F_s=(f_1,f_2,f_3) ,    F_d1= - f_1 - f_2  + 2f_3 ,


    f_d2= f_2 - f_3 .

We can associate the particles in the model to doublets or to singlets with the following rules.
The direct product of two doublets p_D^T = (p_D1 , p_D2 ) and q_D^T = (q_D1 , q_D2 ) may be decomposed
into the direct sum of two singlets r_s and r_s' , and one doublet r_D^T where

    r_s=p_D1q_D1+p_D2q_D2   r_s'=p_D1q_D2-p_D2q_D1


    r_D^T=(r_D1,r_D2)=(p_D1q_D2+p_D2q_D1,p_D1q_D1+p_D2q_D2).

Since the Standard Model has only one Higgs SU(2)_L doublet, which can only be an S_3
singlet, it gives mass to the particles in the S_3 singlet representation.
To give mass to the rest of the particles we extend the Higgs sector
of the theory, by adding two more Higgs doublets. 
The quark and Higgs fields are

    Q^T=(u_L,d_L),u_r,d_r,


    L^t=(ν_L,e_L),e_R,ν_R   and   H.

 All of the fields have three species, and we assume that each one
forms a reducible representation 1S ⊕ 2. The first two
generations will be assigned to the doublet S3 irrep, and the third
generation to the singlet.  This applies to quarks, leptons, Higgs
fields, 
and right-handed neutrinos. The doublets carry capital indices I and J, which
run from 1 to 2, and the singlets are denoted by Q_3 , u_3R , d_3R , L_3 , e_3R , ν_3R and H_S.
 The subscript 3 denotes the singlet representation and not the third generation. 
The most
general renormalizable Yukawa interactions of this model are given by <cit.>

    L_Y=L_Y_D+L_Y_U+L_Y_E+L_Y_ν


where

    L_Y_D=     -Y^d_lQ_IH_S d_IR - Y^d_3Q_3H_S d_3r
        -Y^d_2[Q_Iκ_IJ H_l d_JR - Q_Iη _IJ H_2 d_JR]
        -Y^d_4Q_3H _Id _IR   -   Y^d_5Q_IH _ID _3R  + h.c.




    L_Y_U=   -Y^u_1Q_I(iσ _2 H^∗ _Su _IR)   -   Y^u_3Q_3(iσ _2 H^∗ _Su _3R)
       -Y^u_2[Q_Iκ_IJ(iσ _2 H^∗ _1u _JR)  -  Q_Iη_IJ(iσ _2 H^∗ _2u _JR)]
       -Y^u_4Q_3(iσ _2 H^∗ _Iu _IR)   -   Y^u_5Q_I(iσ _2 H^∗ _Iu _3R)  + h.c.




    L_Y_U=   -Y^u_1Q_I(iσ _2 H^∗ _Su _IR)   -   Y^u_3Q_3(iσ _2 H^∗ _Su _3R)
       -Y^u_2[Q_Iκ_IJ(iσ _2 H^∗ _1u _JR)  -  Q_Iη_IJ(iσ _2 H^∗ _2u _JR)]
       -Y^u_4Q_3(iσ _2 H^∗ _Iu _IR)   -   Y^u_5Q_I(iσ _2 H^∗ _Iu _3R)  + h.c.




    L_Y_E=   -Y^e_1L_IH _Se _IR)   -   Y^e_3L_3H _Se _3R)
       -Y^e_2[L_Iκ_IJH_1e _JR  -  L_Iη_IJH _2e _JR)]
       -Y^e_4L_3H _Ie _IR   -   Y^e_5L_IH _ID _3R  + h.c.




    L_Y_ν=   -Y^ν_1L_I(iσ _2 H^∗ _Sν _IR)   -   Y^ν_3L_3(iσ _2 H^∗ _Sν _3R)
       -Y^ν_2[L_Iκ_IJ(iσ _2 H^∗ _1ν _JR)  -  L_Iη_IJ(iσ _2 H^∗ _2ν _JR)]
       -Y^ν_4L_3(iσ _2 H^∗ _Iν _IR)   -   Y^ν_5L_I(iσ _2 H^∗ _Iν _3R)  + h.c.,

with

    κ=[ 0 1; 1 0;   ]η= [  1  0;  0 -1;    ].

Furthermore, we add to the Lagrangian the Majorana mass terms for the right-handed neutrinos

    L_M = -M_1ν ^T_1RCν_1R -M_2ν ^T_2RCν_2R -M_3ν ^T_3RCν_3R.

Due to the presence of three Higgs fields, the Higgs potential V_H (H_S , H_D ) is more 
complicated than that of the Standard Model <cit.>. In addition to the S3
symmetry, under certain conditions  the Higgs potential
exhibits a permutational symmetry
Z2 : H1 ↔ H2, which is not a subgroup of the flavor
group S3 <cit.>. The model has as well  an Abelian discrete symmetry
 that we will use as selection rules for the Yukawa couplings in the
 leptonic sector. In this paper, we will assume
 that the vacuum respects the accidental Z2 symmetry of
the Higgs potential and that

    <H_1>=<H_2>.

With these assumptions, the Yukawa interactions, eqs. (<ref>)-(<ref>) yield mass matrices, for
all fermions in the theory, of the general form

    M=
    [ μ_1+μ_2     μ_2     μ_5;     μ_2 μ_1-μ_2     μ_5;     μ_4     μ_4     μ_3;         ].

The Majorana mass for the left handed neutrinos ν_L is generated by the see-saw mechanism.
The corresponding mass matrix is given by

    M_ν=M_ν DM^-1(M_ν D)^T,

where  M = diag (M_1 , M_1 , M_3 ).
In principle, all entries in the mass matrices can be complex since there is no restriction coming
from the S3 flavor symmetry.
The mass matrices are diagonalized by bi-unitary transformations as

    U^†_d(u,e)LM_d(u,e)U_d(u,e)R=diag(m_d(u,e),m_s(u,e),m_b(u,e))


    U_ν^T M_ν U_ν =diag(m_ν 1,m_ν 2,m_ν 3).

The entries in this matrix are complex numbers, so the physical masses are their absolute
values.
The mixing matrices are, by definition,

    V_CKM=U^†_uLU_dL,   V_PMNS=U^†_eLU_ν K.

Where K is defined as the matrix that take out the phases of the diagonal  mass matrix,

    diag(m_ν 1,m_ν 2,m_ν 3)=K^†diag(|m_ν 1|,|m_ν 2|,|m_ν 3|)K^†.

A further reduction of the number of parameters in the leptonic sector may be achieved by
means of an Abelian Z_2 symmetry. A possible set of charge assignments of Z_2, compatible with
the experimental data on masses and mixings in the leptonic sector is given in Table I.

The Z_2 assignments forbid the following Yukawa couplings

    Y_1^e=Y_3^e=Y_1^ν=Y_5^ν.
 
Therefore, the corresponding entries in the mass matrices vanish.



 §.§ Mass matrix for the charged leptons

Under these assumptions, the mass matrix of the charged leptons takes the form 

    M_e=m_τ[  μ̃_̃2̃  μ̃_̃2̃  μ̃_̃5̃;  μ̃_̃2̃ -μ̃_̃2̃  μ̃_̃5̃;  μ̃_̃4̃  μ̃_̃4̃       0;         ].
 
The unitary matrix U_eL that enters in the definition of the mixing matrix, V_PMNS, is calculated from
 
    U^†_eLM_e M_e^† U_eL=diag(m^2_e,m^2_μ,m^2_τ),
 
where  m_3,m_μ and m_τ are the masses of the charged leptons, and 

    M_e M_e^τ=m^2_τ[               2|μ̃_̃2̃|^2                |μ̃_̃5̃|^2 2|μ̃_̃2̃||μ̃_̃4̃|e^-i δ_e;                |μ̃_̃5̃|^2               2|μ̃_̃2̃|^2                         0;  2|μ̃_̃2̃||μ̃_̃4̃|e^i δ_e                         0               2|μ̃_̃4̃|^2;                           ].
 
Notice that this matrix only has one phase factor. The parameters |μ̃_̃2̃|, |μ̃_̃4̃| and |μ̃_̃5̃| may readily be expressed in terms of the charged lepton masses. From the invariants of M_e M^†_e, we get the set of equations <cit.> 

    Tr(M_e M^†_e)=m^2_e+m^2_μ+m^2_τ=m^2_τ[4|μ̃_̃2̃|^2+2(|μ̃_̃4̃|^2+|μ̃_̃5̃|^2)]


    ξ(M_e M^†_e)=   m^2_τ(m^2_e+m^2_μ)+m_e^2 m_μ^2
    
    =   4m^4_τ[|μ̃_̃2̃|^2+|μ̃_̃2̃|^2(|μ̃_̃4̃|^2+|μ̃_̃5̃|^2)+|μ̃_̃4̃|^2|μ̃_̃5̃|^2]


    (M_eM_e^†)=m_e^2m_μ^2m_τ^2=4 m^6_τ|μ̃_̃2̃|^2|μ̃_̃4̃|^2|μ̃_̃5̃|^2,

where ξ(M_eM_e^†)=1/2[Tr(*M_eM_e^†))^2-Tr((M_eM_e^†)^2)].
Solving these equations for |μ̃_̃2̃|,|μ̃_̃4̃| and |μ̃_̃5̃|, we obtain

    |μ̃_̃2̃|^2=1/2m_e^2+m^2_μ/m^2_τ-m_e^2m_μ^2/m^2_τ(m^2_e+m^2_μ)+β.

In this expression, β is the smallest solution of the equation

    β^3-1/2(1-2y+6x/y)β^2-1/4(y-y^2-4z/y+7z-12z^2/y^2)β-
    1/8yz-1/2z^2/y^2+3/4z^2/y-z^3/y^3=0

where y=(m_e^2+m_μ^2)/ m_τ and z=μ^2_μμ^2_e / μ^4_τ.

An estimation of β at good order of magnitude is obtained from <cit.>

    β≃ -m_μ^2 m_e^2/2m^2_τ(m^2_τ-(m^2_τ+m^2_e)).

The parameters |μ̃_̃4̃|^2 and |μ̃_̃5̃|^2 are in terms of |μ̃_̃2̃|^2,

    |μ̃_̃4̃,̃5̃|^2=   1/4(1-m^2_μ+m^2_e/m_τ^2+4m_e^2m^2_μ/m_τ^2(m_e^2+m^2_μ)-β)
       ±1/4(√((1-m^2_μ+m^2_e/m_τ^2+4m_e^2m^2_μ/m_τ^2(m_e^2+m^2_μ)-β)^2)-m_μ^2 m_e^2/m^4_τ1/|μ̃_̃2̃|^2)

Once M_eM_e^† has been reparametrized in terms of the charged
lepton masses, it is straightforward to compute U_eL also as a
function of the lepton masses. Here  we will write the result to order (m_μ m_e /m^2_τ)^2
and x^4=(m_e/m_τ)^4

    M_e≃ m_τ[                   1/√(2)m̃_̃μ̃/√(1+x^2)                   1/√(2)m̃_̃μ̃/√(1+x^2)      1/√(2)√(1+x^2-m̃_̃μ̃^̃2̃)/√(1+x^2);                   1/√(2)m̃_̃μ̃/√(1+x^2)                  -1/√(2)m̃_̃μ̃/√(1+x^2)      1/√(2)√(1+x^2-m̃_̃μ̃^̃2̃)/√(1+x^2); m̃_̃ẽ(1+x^2)/√(1+x^2-m̃_̃μ̃^̃2̃)e^iδ _e m̃_̃ẽ(1+x^2)/√(1+x^2-m̃_̃μ̃^̃2̃)e^iδ _e                                       0;                                         ].


The unitary matrix U_eL that diagonalizes M_eM_e^† and
enters in the definition of the neutrino mixing matrix V_VPMNS,
equation (<ref>), is

    U_eL≃[      1      0      0;      0      1      0;      0      0 e^iδ_e;        ][  O_11 -O_12  O_13; -O_21  O_22  O_23; -O_31 -O_32  O_33;       ],

where 

    U_eL≃[  O_11 -O_12  O_13; -O_21  O_22  O_23; -O_31 -O_32  O_33;       ]=


    [             1/√(2)x(1+2m̃^̃2̃_̃μ̃+x^2+m̃^̃4̃_̃μ̃+2m̃^̃2̃_e)/√(1+m̃_̃μ̃^̃2̃+5x^2-m̃_̃μ̃^̃4̃-m̃^̃6̃_̃μ̃+m̃_̃ẽ^̃1̃2̃+12x^4)                     -1/√(2)(1-2m̃^̃2̃_̃μ̃+m̃^̃4̃_̃μ̃-2m̃^̃2̃_e)/√(1-m̃_̃μ̃^̃2̃+x^2+6m̃_̃μ̃^̃4̃-4m̃^̃6̃_̃μ̃+5m̃_̃ẽ^̃1̃2̃)                                                                                                                   1/√(2);                       -1/√(2)x(1+4x^2-m̃^̃4̃_̃μ̃-2m̃^̃2̃_e)/√(1+m̃_̃μ̃^̃2̃+5x^2-m̃_̃μ̃^̃4̃-m̃^̃6̃_̃μ̃+m̃_̃ẽ^̃1̃2̃+12x^4)                                1/√(2)(1-2m̃^̃2̃_̃μ̃+m̃^̃4̃_̃μ̃)/√(1-m̃_̃μ̃^̃2̃+x^2+6m̃_̃μ̃^̃4̃-4m̃^̃6̃_̃μ̃+5m̃_̃ẽ^̃1̃2̃)                                                                                                                   1/√(2); -√(1+2x^2-m̃^̃2̃_̃μ̃-m̃^̃2̃_̃ẽ)(1+m̃^̃2̃_̃μ̃+x^2-2m̃^̃2̃_e)/√(1+m̃_̃μ̃^̃2̃+5x^2-m̃_̃μ̃^̃4̃-m̃^̃6̃_̃μ̃+m̃_̃ẽ^̃1̃2̃+12x^4)    -x√(1+2x^2-m̃_̃μ̃^̃2̃-m̃_̃ẽ^̃2̃)(1+x^2-m̃^̃2̃_̃μ̃-2m̃^̃2̃_e)/√(1-m̃_̃μ̃^̃2̃+x^2+6m̃_̃μ̃^̃4̃-4m̃^̃6̃_̃μ̃+5m̃_̃ẽ^̃1̃2̃)                                                                                  m̃_̃ẽm̃_̃μ̃√(1+x^2)/√(1+x^2-m̃^̃2̃_̃μ̃);                                                                                                                          ]=

and where m̃_̃μ̃=m_μ/m_τ,m̃_̃ẽ=m_e/m_τ and
x=m_e/m_μ.



 §.§ The mass matrix of the neutrinos


With the Z_2 selection rule (Table <ref>), the mass matrix of the Dirac neutrinos takes the form

    M_ν D=
    [  μ^ν_2  μ^ν_2      0;  μ^ν_2 -μ^ν_2      0;  μ^ν_4  μ^ν_4  μ^ν_3 ]

Then, the mass matrix for the left-handed Majorana neutrinos is obtained from the see-saw mechanism,

    M_ν=M_ν DM̃^-1 (M_ν D)^T=
    [  (1/M_1+1/M_2)μ_2^2  (1/M_1-1/M_2)μ_2^2 (1/M_1+1/M_2)μ_2μ_4;  (1/M_1-1/M_2)μ_2^2  (1/M_1+1/M_2)μ_2^2 (1/M_1-1/M_2)μ_2μ_4; (1/M_1+1/M_2)μ_2μ_4 (1/M_1-1/M_2)μ_2μ_4 μ_4^2/M_2+μ_3^2/M_3;                     ] ,

where M_i are the right handed neutrino masses appearing in eq. (<ref>).

The non-Hermitian, complex, symmetric neutrino mass matrix M_ν may be brought to a
diagonal form by a bi-unitary transformation, as

    U^T_ν M_ν U_ν=diag(m_ν 1 e^i ϕ _1,m_ν 2 e^i ϕ _2,m_ν 3 e^i ϕ _3)

Where U_ν is the matrix that diagonalizes the matrix M_ν.



  §.§.§ Neutrino matrix with degenerate masses.


In the case where M_1=M_2 the mass matrix is reduced to <cit.>

    M_ν=M_ν DM̃^-1 (M_ν D)^T=
    [  (1/M_1+1/M_1)μ_2^2                   0 (1/M_1+1/M_1)μ_2μ_4;                   0  (1/M_1+1/M_1)μ_2^2                   0; (1/M_1+1/M_1)μ_2μ_4                   0   μ_4^2/M1+μ_3^2/M3;                     ].

With this texture is easy to calculate the U_ν matrix that
diagonalizes M_ν^† M_ν, 

    M^†_ν M_ν=
    [ |A|^2+|B|^2           0   A^*B+B^*D;           0       |A^2|           0;   AB^*+BD^*           0 |B|^2+|D|^2 ]

with A=μ_2 ^2 / M_1 , B=2 μ_2 μ_4/M_1and D=2 μ_1 μ_2 / M_1+μ_3 ^2 /M_2, this matrix is diagonalized by 

    U_ν=
    [      1      0      0;      0      1      0;      0      0 e^iδ_ν;        ][  cosη  sinη     0;     0     0     1; -sinη  cosη     0;       ].

If we require that the defining equation (<ref>) be satisfied as an identity, we get the following set
of equations:

    2(μ_2^ν)^2/M1=   m_ν3 ,
    
    2(μ_2^ν)^2/M1=   m_ν1 cos^2η+m_ν2 sin^2η,
    
    2(μ_2^ν)(μ_4^ν)/M1=   sinηcosη(m_ν2-m_ν1)e^iδ_nu,
    
    2(μ_2^ν)^2/M1=   m_ν3 ,
    
    2(μ_2^ν)(μ_4^ν)/M1=   sinηcosη(m_ν2-m_ν1)e^iδ_nu,
    
    2(μ_4^ν)^2/M1+2(μ_3^ν)^2/M3=   (m_ν1sin^2η+m_ν2cos^2η)e^-2iδ_nu.

Solving these equations for sinη and cosη, we find

    sin^2η=m_ν 3-m_ν 1/m_ν 2-m_ν 1  cos^2η=m_ν 2-m_ν 3/m_ν 2-m_ν 1.

The unitarity of U_ν constrains sinη to be real and thus |sinη| ≤ 1, this condition fixes the phases ϕ_1 and ϕ_2 as

    |m_ν 1|sinϕ_1=|m_ν 2|sinϕ_2=|m_ν 3|sinϕ_3.

The real phase δ_ν appearing in eq. (<ref>) is not constrained by the unitarity of U_ν.
Therefore the U_ν matrix is,

    U_ν=
    [       1       0       0;       0       1       0;       0       0 e^iδ_nu;         ][  √(m_ν2-m_ν3/m_ν2-m_ν1)  √(m_ν3-m_ν1/m_ν2-m_ν1)                       0;                       0                       0                       1; -√(m_ν3-m_ν1/m_ν2-m_ν1)  √(m_ν2-m_ν3/m_ν2-m_ν1)                       0;                         ].

Now, the mass matrix of the Majorana neutrinos, M_ν, may be written in terms of the neutrino
masses; from (<ref>) and (<ref>,<ref>,<ref>), we get

    M_ν=
    [                             m_ν3                                0 √((m_ν3-m_ν1)(m_ν2-m_ν3))e^-iδ_ν;                                0                             m_ν3                                0; √((m_ν3-m_ν1)(m_ν2-m_ν3))e^-iδ_ν                                0          (m_ν1+m_ν2-m_ν3)e^-2δ_ν;                                  ]

The only free parameters in these matrices, other than the neutrino
masses, are the phase ϕ_ν, implicit in m_ν1,m_ν2 and m_ν3, and the Dirac phase δ_ν.

Therefore, the theoretical mixing matrix V_PMNS , is
given by

    V^th_PMNS=[  O_11cosη + O_31sinη e^iδ  O_11sinη - O_31cosη e^iδ                     -O_21; -O_12cosη + O_32sinη e^iδ -O_12sinη - O_32cosη e^iδ                      O_22;  O_13cosη - O_33sinη e^iδ  O_13sinη + O_33cosη e^iδ                      O_23;                           ]× K.

To obtain the expressions for the mixing angles we need to match the
theoretical and PDG expressions for the V_PMNS matrix 

    |V_PMNS^th|=|V_PMNS^PDG|

meaning |V_ij^th|=|V_ij^PDG|.
The standard parametrization of the Particle Data Group is 

    V_PMNS=
    [                       c_12c_13                       s_12c_13                  s_13e^-i δ_CP; -s_12c_23-c_12s_23s_13e^i δ_CP  c_12c_23-s_12s_23s_13e^i δ_CP                       s_23c_13;  s_12s_23-c_12c_23s_13e^i δ_CP -c_12s_23-s_12c_23s_13e^i δ_CP                       c_23c_13;                                ].

We can straightforwardly read the equation for the mixing angles with

    |sin_θ_13|=|O_21|≃1/√(2)x1+4x^2-m̃_̃μ̃^̃4̃/√(1+m̃_̃μ̃^̃2̃+5x^2-m̃_̃μ̃^̃4̃) ,


    |sin_θ_23|=|O_22|/√(1-O^2_21)≃1/√(2)1-2m̃_̃μ̃^̃2̃+m̃_̃μ̃^̃4̃/√(1-4m̃_̃μ̃^̃2̃+x^2+6m̃_̃μ̃^̃4̃) ,

and

    tan_θ_12=O_11sin_η-O_31cos_η/O_31sin_η+O_11cos_η


    ≃ -√(m_ν2-m_ν3/m_ν3-m_ν1)× (√(1+2x^2-m̃^̃2̃_̃μ̃)(1+m̃_̃μ̃^̃2̃+x^2)-1/√(2)x(1+2m̃_̃μ̃^̃2̃+4x^2)√(m_ν3-m_ν1/m_ν2-m_ν3)/√(1+2x^2-m̃^̃2̃_̃μ̃)(1+m̃_̃μ̃^̃2̃+x^2)+1/√(2)x(1+2m̃_̃μ̃^̃2̃+4x^2)√(m_ν2-m_ν3/m_ν3-m_ν1)).

We can express tanθ_12 in terms of the differences of the
square of the masses as 

    tan^2θ_12=(Δ m^2_12+Δ m^2_13+|m_ν 3|^2 cos^2ϕ_nu)^1/2-|m_ν 3||cosϕ_nu|/(Δ m^2_13+|m_ν 3|^2 cos^2ϕ_nu)^1/2+|m_ν 3||cosϕ_nu|

where Δ m^2_ij=m_ν i^2-m_ν j^2.


We can use the experimental values of the masses of the charged leptons and the differences of the square of the masses to fit the mixing angles, 

    (sin^2θ_13)^th=1.1×10^-5,   (sin^2θ_13)^xp=2.19^+0.12_-0.12× 10^-2,

and

    (sin^2θ_23)^th=.499,   (sin^2θ_23)^xp=.5^+0.05_-0.05.

From expression (<ref>), we may readily derive expressions for
the neutrino masses in terms of tanθ_12, ϕ_ν and the differences of the squared masses,





    |m_3|=√(Δ m_13^2)/2tanθ_12cosϕ_ν1-tan^4θ_12+r^2/(1+tan^2θ_12)(1+tan^2θ_12+r^2) ,


    |m_1|=√(|m_ν 3|^2+Δ m_13^2) ,


    |m_2|=√(|m_ν 3|^2+Δ m_13^2(1+r^2)) ,



here r^2 = Δ m^2_12 / Δ m^2_13≈ 3 × 10^-2. This implies an inverted neutrino mass spectrum |m_ν 3 |
< |m_ν 1 | < |m_ν 2 |. 
As r^ 2 << 1, the sum of the neutrino masses is

    ∑^3_i=1 | m_ν_i | ≈Δ m^2_13/2cosϕ_νtanθ_12 (1 + 2 √(1 + 2 tan^2 θ_12 (2 cos^2 ϕ_ν - 1) + tan^4 θ_12) - tan^2 θ_12) .

The most restrictive cosmological upper bound <cit.> for this sum is 

    ∑|m_ν | ≤ 0.23eV .

This upper bound and the experimentally determined values of tanθ_12 and Δ m^2_i,j, give a lower bound for 

    cosϕ_ν≥ 0.55

or 0 ≤ϕ_ν≤ 57^∘.
We can use again equation (<ref>) to set the best value of ϕ, we find that with ϕ=50^∘ we get, 

    tanθ_12=0.665288

Hence, setting ϕ_ν = 50^∘ in
our formula, we find

    m_ν 1 = 0.052  eV,     m_ν 2 = 0.053  eV,   m_ν 3 = 0.019  eV.
                                                
The computed sum of the neutrino masses is

    (∑^3_i=1 |m_ν i |)^th      = 0.168508  eV,
 
below the cosmological upper bound given in eq. (<ref>), as
expected.

The above value of ϕ is in  agreement with the requirements for
leptogenesis, as we will show in section 3.
One of the successes of the S3-3H model  has been to predict  an angle
θ_13 different from zero, as well a very accurate angles
θ_12 and θ_23.  Nevertheless new experimental
results have shown that the angle θ_13 is greater than the
model predicts with degenerate right-handed neutrino masses. This is
the major reason to extend the model further, to the non-degenerate
case <cit.>,  and  where the angles fit the experimental value.  



  §.§.§ The mass matrix of the neutrinos without degeneration

In a more extensive analysis than <cit.>, we continue to study the case where the RHN masses are non-degenerate.
The effective neutrino mass matrix
m_ν is,

    M_ν=M_ν DM̃^-1 (M_ν D)^T=
    [  (1/M_1+1/M_2)μ_2^2  (1/M_1-1/M_2)μ_2^2 (1/M_1+1/M_2)μ_2μ_4;  (1/M_1-1/M_2)μ_2^2  (1/M_1+1/M_2)μ_2^2 (1/M_1-1/M_2)μ_2μ_4; (1/M_1+1/M_2)μ_2μ_4 (1/M_1-1/M_2)μ_2μ_4   μ_4^2/M2+μ_3^2/M3;                     ].

We are going to assume that the phases of the μ_3 and μ_4
terms are aligned, therefore we can write M_ν in polar form
M_ν=P M_ν P, with M_ν, real and
P=diag(e^-i θ_μ 2,e^-i θ_μ 2,e^i(ca/2
  -θ_μ 4)) In this way M_ν can be
expressed in terms of a matrix with two texture zeros class I as:

    M_ν=μ_0 𝕀_3× 3+M_ν'

with μ_0=2 |μ_2|^2/M1.

Therefore, the U_ν matrix is P^† U_1 where U_1 is the matrix that diagonalizes M'_ν.
We can take a rotation u_π /4,

    u_π /4=1/√(2)[    1    1    0;   -1    1    0;    0    0 √(2) ],

to the M'_ν matrix, 

    u_π /4^T M'_ν u_π /4=na
    [             0             0             1;             0 √(2)ψ (ψ_2-1)           ψ_2;             1           ψ_2           μ_c ]

with na=M_2/(√(2)|μ_2||μ_4|), ψ=√(2)|μ_2|/|μ_4|, ψ_2=M_2/M_1 and μ_c=μ_4^2/M1+μ_3^2/M3-μ_0.
The Matrix that diagonalizes u_π /4^T M'_ν u_π /4 is 

    U_2=
    [                ψ_2 n_1                ψ_2 n_2                ψ_2 n_3; -(1+μ_c λ_1-λ^2_1) n_1 -(1+μ_c λ_2-λ^2_2) n_2 -(1+μ_c λ_3-λ^2_3) n_3;            ψ_2 λ_1 n_1            ψ_2 λ_2 n_2            ψ_2 λ_3 n_3 ].

where n_i is a normalization factor and λ_i is the i-th eigenvalue of  M'_ν.With U_1=u_π/4 U_2, therefore



    U_ν=
    [ O'_11 O'_12 O'_13; O'_21 O'_22 O'_23; O'_31 O'_32 O'_33;       ],


    [  n_1 (ψ_2+ f_1)  n_2 (ψ_2+ f_2)  n_3 (ψ_2+ f_3); n_1 (-ψ_2+ f_1) n_2 (-ψ_2+ f_2) n_3 (-ψ_2+ f_3);     ψ_2 n_1 λ_1     ψ_2 n_2 λ_2     ψ_2 n_3 λ_3;                 ],


where f_i=(-1-μ_c λ_i+λ^2_i). 
In the same way as in  the degenerate scenario we have

    |V_PMNS^th|=|V_PMNS^PDG| ,

in terms of the mixing angles with

    s_13=O'_13 ,    s_23=O'_23/√(1-O_13^'2) ,    s_12=O'_12/√(1-O_13^'2) .

In the non-degenerate scenario we have three free parameters (ψ,
  ψ_2,   μ_c) for the neutrino matrix. In this model the
PMNS matrix may be obtained numerically. We have used the following values for the masses given in <cit.>

    m_e =    0.5109989461 ± 0.0000000031  MeV,
    
    m_μ =    105.6583745 ± 0.0000024  MeV,
    
    m_τ =    1776.86 ± 0.12  MeV.

In order to obtain the numerical values for the three free parameters 
we perform a χ^2 analysis on the parameter space to find their
best fit points

    χ^2=(sin(θ_12)^2-sin(θ_12)^2)^2/σ^2_sinθ_12^2+(sin(θ_12)^2-sin(θ_12)^2)^2/σ^2_sinθ_12^2+(sin(θ_12)^2-sin(θ_12)^2)^2/σ^2_sinθ_12^2

where we have taken the following experimental values for the V_PMNS elements <cit.>

    sin^2_θ_12= 0.304 ± 0.014 ,  sin^2_θ_23 = 0.50 ± 0.05 ,  sin^2_θ_13= (2.19 ± 0.12) × 10^-2 .

The best values for the free parameters are thus found to be

    ψ_2 = 1.1431 ,  ψ = 1.3091 ,  μ_c = 1.6502 eV,

at one sigma C.L. with χ^2=3.74×10^-15 as the minimal value. These correspond to the following mixing angles

    sin(θ_12)^2=   0.3039,   sin(θ_23)^2=0.4999, 
       sin(θ_13)^2=0.0218.








§ LEPTOGENESIS IN AN S3-3H MODEL

The Yukawa couplings of the neutrinos allow the decay of the right-handed neutrinos into the left-handed ones.

    -Y^ν_1L_I(iσ_2H^∗_Sν_IR).

As shown in <cit.>, the asymmetry is defined to be

    ϵ _1=∑_αΓ (N_1→ℓ _α H)-Γ (N_1→ℓ _αH)/∑_αΓ (N_1→ℓ _α H) +Γ (N_1→ℓ _αH) .

where Γ is the decay rate, and N_1 is the decaying right-handed neutrino.

The possible decays up to tree level are shown in <ref>,



The asymmetry generated by these decays are,

    ϵ≃-3/8π1/(h_νh^†_ν)∑_i=2,3Im{ (h_νh^†_ν)^2_1i}[f(M^2_i/M^2_1)+g(M^2_i/M^2_1)].

and the self interactions are

    f(x)=√(x)[1-(1+x)ln(1+x/x)] ,


    g(x)=√(x)/1-x .

This function depends strongly on the hierarchy of the light neutrino masses. It can lead to
a strong enhancement of the CP asymmetries if the masses M_2 and M_3 are
nearly degenerate.


The relation between the lepton and baryon asymmetry is given through the sphaleron process <cit.>

    Y_B=aY_B-L=a/a-1Y_L  ,

where a is  a=(8N_f+4N_H)/(22N_f+13N_H), N_f is the number of families and N_H is the number of Higgs doublets.

We can express the lepton asymmetry in terms of the CP asymmetry 

    Y_L=n_L-n_L̅/s=κϵ_i/g* ,


where g is 110, the number of relativistic degrees of freedom,
κ is obtained from solving the Boltzmann equations, and it can be
reparametrized in terms of K defined as the ratio of Γ_1, the
tree-level decay width of N_1 to H the Hubble parameter at
temperature M_1, where k=Γ_1/H<1 describes a process out of
thermal equilibrium and κ<1 describes the washout effect <cit.>:

    κ≈0.3/K(ln(K))^0.6  for   10<K<10^6 ,


    κ≈1/2√(K^2+9)  for   0<K<10 .

The decay width of N_1 by the Yukawa interaction at tree level and
Hubble parameter in terms of the temperature T and the Planck scale
M_pl are Γ_1=(m^†_D m_D)_11M_1/(8π v^2) and
H=1.66g^*1/2T^2/M_pl respectively. At temperature T=M_1 the
ratio K is

    K=M_pl/1.66√(g*)(8π v^2)(m^†_D m_D)_11/M_1 .








 §.§ Baryon asymmetry in the degenerate scheme


Putting all the above ingredients together, the asymmetry for the S3-3H model is

    ϵ=Im[e^2iδ^*M_2 m_3 √(M_2(m_2-m_3)(m_3-m_1))/√(m_3)](f[M_3^2/M_1^2]+g[M_3^2/M_1^2])/8π |M_2 m_3| .

The value of the baryon asymmetry has a dependence on ϕ and the
masses of the neutrinos |m_1|,M_1,|m_2|,M_2,|m_3|,M_3, where the
masses of the right-handed neutrinos are considered real.  We can
calculate the dependence of the baryon asymmetry on the phase
δ. As can be seen from eq. (<ref>) the asymmetry is a
periodic function of δ, where the  masses give the
scale of the baryon asymmetry.


The maximum value for the baryon asymmetry is on δ=3/4π.  As
figure <ref> shows, the leptogenesis is crucially
dependent on the phase.  The value of the baryon asymmetry is
determined by the masses of the light neutrinos and the ratios of
the right-handed neutrino masses. The see-saw mechanism relates the masses of
the right-handed neutrinos to the light ones, 
making the right-handed neutrino masses bigger than 10^12 GeV, in order to be in agreement with the experimental data.

We calculate the asymmetry generated in the best case scenario where
δ=3/4π. In this case we can see from fig. <ref> that the masses of the
right-handed neutrinos could be of order of 10^7 GeV to produce leptogenesis. The graph also
shows the region of resonant leptogenesis M_1-M_3≃1/2Γ_N_1,3, where the asymmetry increases above the
one observed in the Universe, lowering even more the possible mass of
the right-handed neutrinos or the δ phase.





 §.§ Non-degenerate scenario


The value of the baryon asymmetry in the non-degenerate case has
dependence on the angles of the M_d matrix μ_2a, μ_3a and
the real masses of the neutrinos M_1,M_2,M_3, where μ_ia is
the angle of μ_i=|μ_i|e^iμ_ia.  We can calculate the
dependence of the baryon asymmetry on the phases μ_2a
, μ_3a.  As in the degenerate case the magnitude or scale of the baryon
assymetry is given by the masses.



The maximum of the asymmetry is achieved in all the lines where
μ_2a-μ_3a=π /2 +nπ, where n is any integer, as can be
seen from fig. <ref>.  Again, this is independent of the
masses of the neutrinos, the masses only fix the scale of the
asymmetry.  Taking the best values of the angles we can see that the
scale of the masses of the right-handed neutrinos can be lower and
that the region of resonant leptogenesis is wider. Therefore, this
gives a wider region in parameter space fulfilling the baryon
asymmetry explanation.  In fig. <ref> we show the baryon
asymmetry dependence on the M_1 and M_3 masses. The darker shades
of green correspond to more asymmetry, whereas the red regions
correspond to an excess of baryon asymmetry as compared to the one
observed in the Universe, for the maximum value of the phases.




§ CONCLUSIONS


The minimal S3-3H extension of the SM acommodates well the masses and
mixings of quarks and leptons, and gives naturally a non-zero value
for the neutrino reactor mixing angle.  We re-derived previous results
on the neutrino sector with recent experimental data, taking into
account a new value of the phase ϕ to include the angle
θ_12 in the model.  In the non-degenerate right-handed
neutrino mass case, we find a new parametrization of the V_PMNS
matrix and use the experimental values in a χ^2 analysis to fit
the new parameters. We find thus a new region in parameter space where
the model predicts the mixing angles correctly.

We then calculated the leptogenesis and the associated baryogenesis in
this model in the case of two right-handed degenerate neutrino masses,
and in the more general case of non-degenerate masses.  We show that
there are regions in parameter space which allow leptogenesis as a
mechanism to solve the observed baryonic asymmetry with right-handed
neutrino masses starting from 10^6 GeV.



§ ACKNOWLEDGEMENTS
 We acknowledge useful discussions with J. Kersten and
  A. Mondragón.  This work is partially supported by a UNAM grant
  PAPIIT IN111115.



Frere:2006iz
Davidson:2008bu
Ishimori:2010au
Antusch:2003kp
h-physrev3

