#  Leptogenesis in $SO(10)$ Models with $A_{4}$ Modular Symmetry

Gui-Jun Ding1111E-mail<EMAIL_ADDRESS>Stephen F. King2222E-mail:
<EMAIL_ADDRESS>Jun-Nan Lu1333E-mail<EMAIL_ADDRESS>Bu-Yao
Qu1444E-mail<EMAIL_ADDRESS>
1Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China
2Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, U.K.

###### Abstract

We study the prediction for leptogenesis in two renormalizable supersymmetric
$SO(10)\times A_{4}$ modular models in which the neutrino mass is dominantly
generated by the type I seesaw mechanism. The evolution of the lepton
asymmetries are described in terms of the three-flavored density matrix
equations for three heavy Majorana neutrinos, where both vanishing initial
condition and thermal initial condition of the right-handed neutrinos are
considered. We also present an analytical approximation based on the Boltzmann
equations. We find regions of parameter space compatible with the measured
fermion masses and mixing parameters as well as the baryon asymmetry of the
Universe. The predictions for the light neutrino masses, the effective mass in
neutrinoless doble beta decay and the leptonic CP violation phases are
discussed.

##  1 Introduction

The origin of neutrino mass and mixing remains one of the most important
questions in particle physics, along with other unresolved questions such as
the matter-antimatter asymmetry of the Universe, and the possible unification
of the strong and electroweak interactions.

The excess of matter over antimatter in the Universe has been firmly
established. Although the standard model (SM) qualitatively satisfies the
Sakharov conditions [1], the CP violation in SM is too small to explain the
baryon asymmetry. A dynamical generation of baryon asymmetry requires new
physics beyond SM. The discovery of neutrino masses and mixing in neutrino
oscillation experiments is a great progress in particle physics. The smallness
of neutrino masses can be naturally explained in the type I seesaw mechanism
in which a number of heavy Majorana neutrinos are added to the SM, and a
direct cosmological implication is leptogenesis [2]. The observed baryon
asymmetry can be produced by the out-of equilibrium, CP and $B-L$ violating
decays of the right-handed neutrinos, see Refs. [3, 4] for recent review
papers. In the type I seesaw model with three right-handed neutrinos, 18
additional parameters are introduced and the leptogenesis can successfully
generate the observed baryon asymmetry if the neutrino Yukawa couplings and
right-handed neutrino masses are freely chosen. It is difficult to test
leptogenesis due to the large number of free parameters of in a generic seesaw
model.

Masses and mixing angles of elementary fermions are known with good precision
at present. The mass eigenvalues of quarks and leptons have a large hierarchy,
the flavor mixings in quark and lepton sectors are drastically different from
each other: small quark mixing angles versus large lepton mixing angles.
Understanding the origin of the flavor structure of SM is another of the
greatest challenges in particle physics. The fundamental principal behind the
observed patterns of fermion masses and mixing is still elusive, and the
approach of flavour symmetry has been extensively studied in the last several
decades [5, 6]. Many candidate groups for flavor symmetry have been discussed,
and many models which can describe some pieces of the flavor puzzle were
constructed. The flavor symmetry relates three generations of matter fields so
that the resulting quark and lepton mass matrices have certain structure and
the total number of free parameters in fermion mass matrices are reduced.

The flavor symmetry can not be exact symmetry and it has to be broken
explicitly or spontaneously to accommodate the non-trivial mixing patterns of
quarks and leptons. As a consequence, the fermion mass matrices are usually
expanded as power series of the symmetry breaking parameters. The breaking of
flavour symmetries typically relies on several scalar multiplets called
flavons whose vacuum expectation values should be oriented along certain
directions in flavor space. In addition, extra cyclic group generally is
necessary to forbid the unwanted operators and realize the correct vacuum
alignment [6]. Thus the construction of flavon potential is intricate and make
the flavor symmetry models rather complicated.

The modular invariance as flavor symmetry is recently proposed in [6]. From
the top-down perspective, modular invariance can naturally arise from the
compactification of a higher dimensional theory on a torus or an orbifold. In
minimal schemes based on modular invariance, flavons are not necessary and the
complex modulus $\tau$ is the unique symmetry breaking parameter, consequently
the vacuum alignment problem is considerably simplified. The action of modular
symmetry on the matter fields is characterized by the modular weights and
their transformations under the finite modular group
$\Gamma_{N}\equiv\Gamma/\Gamma(N)$, where $\Gamma(N)$ is the principal
congruence subgroup of level $N$ of the modular group $\Gamma\equiv PSL(2,Z)$
[6]. Modular invariance constrains the Yukawa couplings to be level $N$
modular forms which are specific holomorphic functions of $\tau$. To be more
general, one can use any normal subgroup of $\Gamma$ rather than $\Gamma(N)$
such that the Yukawa couplings are vector-valued modular forms of $SL(2,Z)$
[7]. The modular invariance approach to the flavor puzzle, especially for the
lepton flavor structure have been extensively exploited, and various modular
invariant models have been constructed in past years with the groups
$\Gamma_{2}\cong S_{3}$ [8, 9, 10, 11], $\Gamma_{3}\cong A_{4}$ [6, 12, 8, 13,
14, 15, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38], $\Gamma_{4}\cong S_{4}$ [39, 40, 41, 42, 43, 44, 45,
25, 46, 47], $\Gamma_{5}\cong A_{5}$ [48, 49, 44] and $\Gamma_{7}\cong
PSL(2,Z_{7})$ [50]. Similar attempts have been made to construct modular
invariant models for quarks and quark-lepton unification models based on the
modular groups $A_{4}$ [15, 18, 32, 36], $S_{4}$ [47], $T^{\prime}$ [51],
$S^{\prime}_{4}$ [52] and $A^{\prime}_{5}$ [53]. The formalism of modular
invariance has been extended to include odd weight modular forms [54],
fractional weight modular forms [55, 53] and it can involve several moduli
[41, 56]. The interplay of modular symmetry and generalized CP symmetry has
been studied [57, 58, 59, 60]. Moreover, modular invariance has been
intensively discussed from a top-down perspective [58, 59, 61, 62, 63, 64].

The origin of all quark and lepton masses and mixing may be related with each
other. This suggests to combine the Grand unified theories (GUTs) with flavor
symmetry. The electromagnetic, weak, and strong forces are merged into a
single force at high energies in GUTs. The quark and lepton fields are
embedded into one or few gauge multiplets, thus GUTs specifically predict
relations among the fermion masses. Inclusion of flavor symmetry allows to
predict the mixing parameters of quark and leptons. However, combining
traditional flavor symmetry with GUTs also requires the introduction of
flavons and vacuum alignment in order to break the flavor symmetry, see Ref.
[65] for a review. Therefore there is a strong motivation for introducing
modular symmetry in the context of GUTs such that the complication of flavor
symmetry breaking can be removed and the predictive power of the GUTs models
can be improved considerably. The possible combinations of $SU(5)$ GUTs and
modular symmetry groups have been discussed in the literature, and several
modular $SU(5)$ GUT models have been built at level 2 [10, 66], level 3 [14,
67, 68] and level 4 [69, 70, 71].

Besides the simple $SU(5)$ GUT, the $SO(10)$ GUT is another prototype of grand
unified theory. The unification of matter is even more complete in $SO(10)$
GUT, since the 16-dimensional spinor representation of $SO(10)$ can
accommodate all the known SM chiral fermions as well as the right handed
neutrino. This makes $SO(10)$ GUTs particularly attractive candidates for
explaining the origin of neutrino mass and mixing, the unification of the
strong and electroweak interactions, and the matter-antimatter asymmetry of
the universe, simultaneously. It is notable that both the right-handed
neutrino mass matrix and neutrino Yukawa coupling are constrained by the
$SO(10)$ GUT symmetry, and they are related with the quark and lepton mass
matrices. The $SO(10)$ GUTs employing an conventional flavor symmetry has been
studied so far. For example, models with $SO(10)\times A_{4}$ [72, 73, 74, 75]
and $SO(10)\times S_{4}$ [76, 77, 78, 79, 80] have already been investigated.
As in $SU(5)$ GUTs, the cumbersome vacuum alignment required in conventional
flavor symmetry can be eliminated by the use of modular symmetry.

Following this line of reasoning, SUSY $SO(10)\times A_{4}$ modular models
were constructed in [81]. We found that the models involving in addition to
the Higgs fields in the ${\mathbf{10}}$ and the ${\mathbf{\overline{126}}}$,
also a Higgs field in the ${\mathbf{120}}$, proved to give a successful
description of quark and lepton (including neutrino) masses and mixing, with
many such models being found with sums of modular weights of up to 8 or less.
The neutrino masses were generated by the type-I and/or type II seesaw
mechanisms [82, 83, 84, 85, 86] arising from the $SU(2)_{L}$ singlet and/or
triplet components of the Higgs in the ${\mathbf{\overline{126}}}$
representation. However the question of leptogenesis was not addressed for
these models.

In this paper, we study the prediction for leptogenesis in two of the
renormalizable SUSY $SO(10)\times A_{4}$ modular models from [81] in which the
neutrino mass is dominantly generated by the type I seesaw mechanism 555The
study of such models involving a mixture of type-I and type-II seesaw models
is much more complicated and left for a future project.. The evolution of the
lepton asymmetries are described in terms of the three-flavored density matrix
equations for three heavy Majorana neutrinos, where both vanishing initial
condition and thermal initial condition of the right-handed neutrinos are
considered. We also present an analytical approximation based on the Boltzmann
equations. We find regions of parameter space compatible with the measured
fermion masses and mixing parameters as well as the baryon asymmetry of the
Universe. The predictions for the light neutrino masses, the effective mass in
neutrinoless doble beta decay and the leptonic CP violation phases are
discussed.

This paper is organized as follows. In section 2 we briefly review the modular
invariant $SO(10)$ models and the predictions for the fermion mass matrices.
In section 3 we present the formalism of density matrix to describe the
evolution of the lepton asymmetries and the number densities of right-handed
neutrinos. Moreover, the decay and washout of three right-handed neutrinos are
considered in turns in the framework of Boltzmann equations, and an
approximation formula for the final lepton asymmetry is given. In section 4 we
present our numerical analysis for the concerned benchmark models, the results
of the fit and the predictions of the models are given. Section 5 concludes
the paper. In the appendix A we give two examples where the analytical
approximations disagree with the numerical results.

##  2 Fermion masses in $SO(10)$ GUTs with $A_{4}$ modular symmetry

It is known that all the fermionic multiplets of each SM generation, plus one
right-handed neutrino singlet, fit exactly into the 16 dimensional spinorial
representation of the $SO(10)$ grand unification group. As a consequence, in
renormalizable $SO(10)$ models, the Higgs fields that contribute to fermion
masses are in the $SO(10)$ representations $\mathbf{10}$,
$\overline{\mathbf{126}}$ and $\mathbf{120}$ which are denoted by $H$,
$\overline{\Delta}$ and $\Sigma$ respectively. The most general form of the
Yukawa superpotential for renormalizable $SO(10)$ is given by

$\mathcal{W}_{Y}=\mathcal{Y}_{ij}^{10}\psi_{i}\psi_{j}H+\mathcal{Y}_{ij}^{\overline{126}}\psi_{i}\psi_{j}\overline{\Delta}+\mathcal{Y}_{ij}^{120}\psi_{i}\psi_{j}\Sigma\,,$
(1)

with $i=1,2,3$, and $\psi_{i}$ stand for the three generations of matter
fields which are in the 16 dimensional representation of $SO(10)$. The Yukawa
coupling matrices $\mathcal{Y}^{10}$ and $\mathcal{Y}^{\overline{126}}$ are
symmetric in generation space, whereas $\mathcal{Y}_{ij}^{120}$ are
antisymmetric. Their explicit forms can be constrained by the flavor symmetry
which relates the three generations of fermions. The GUT Higgs $H$ and
$\overline{\Delta}$ contain one pair of up-type and down-type Higgs doublets
while $\Sigma$ has two pairs of such doublets. It is assumed that only one
linear combination of up-type and one of down-type Higgs doublets are light,
and the remaining combinations acquire GUT scale masses. The vacuum
expectation value (VEV) of the light Higgs doublets breaks the electroweak
symmetry and generate the fermion masses. The mass matrices of quarks and
charged leptons can be written as [87, 88, 89, 90, 91, 81]

$\displaystyle M_{u}$ $\displaystyle=$
$\displaystyle\left(\mathcal{Y}^{10}+r_{2}\mathcal{Y}^{\overline{126}}+r_{3}\mathcal{Y}^{120}\right)v_{u},$
$\displaystyle M_{d}$ $\displaystyle=$ $\displaystyle
r_{1}\left(\mathcal{Y}^{10}+\mathcal{Y}^{\overline{126}}+\mathcal{Y}^{120}\right)v_{d}\,,$
$\displaystyle M_{l}$ $\displaystyle=$ $\displaystyle
r_{1}\left(\mathcal{Y}^{10}-3\mathcal{Y}^{\overline{126}}+c_{e}\mathcal{Y}^{120}\right)v_{d}\,,$
(2)

where $v_{u}$ and $v_{d}$ refer to the VEVs of the MSSM Higgs fields $H_{u}$
and $H_{d}$ respectively, and some ratios of VEVs have been absorbed into the
Yukawa matrices $\mathcal{Y}^{10}$, $\mathcal{Y}^{\overline{126}}$ and
$\mathcal{Y}^{120}$ [81]. Moreover, the Higgs multiplet $\overline{\Delta}$
also contains the Pati-Salam multiplets
$(\overline{\mathbf{10}},\mathbf{3},\mathbf{1})$ and
$(\mathbf{10},\mathbf{1},\mathbf{3})$ which can generate the Majorana masses
for the left-handed and right-handed neutrinos. Hence generally the light
neutrino mass receives contributions from both type I and type II seesaw
mechanism,

$M_{\nu}=M_{L}-M_{D}M^{-1}_{R}M^{T}_{D}\,,$ (3)

with

$M_{D}=\left(\mathcal{Y}^{10}-3r_{2}\mathcal{Y}^{\overline{126}}+c_{\nu}\mathcal{Y}^{120}\right)v_{u},~{}~{}~{}M_{R}=v_{R}\mathcal{Y}^{\overline{126}},~{}~{}~{}M_{L}=v_{L}\mathcal{Y}^{\overline{126}}\,.$
(4)

The dimensionless parameters $r_{1,2,3}$ and $c_{e,\nu}$ in Eqs. (2,4) are
determined by the Clebsch–Gordan coefficients of $SO(10)$ and the mixing among
the Higgs fields. In the present work, we consider the scenario of type I
seesaw dominance which is the limit of $v_{L}\rightarrow 0$.

###  2.1 $SO(10)$ models with $A_{4}$ modular symmetry

The $A_{4}$ group has twelve elements and it can be generated by two
generators $S$ and $T$ obeying the relations

$S^{2}=(ST)^{3}=T^{3}=1\,.$ (5)

It has four non-equivalent irreducible representations: one triplet
$\mathbf{3}$ and three singlets $\mathbf{1}$, $\mathbf{1^{\prime}}$ and
$\mathbf{1^{\prime\prime}}$. The tensor product of two triplets is
$\mathbf{3}\otimes\mathbf{3}=\mathbf{1}\oplus\mathbf{1^{\prime}}\oplus\mathbf{1^{\prime\prime}}\oplus\mathbf{3}_{S}\oplus\mathbf{3}_{A}$,
where $\mathbf{3}_{S}$ and $\mathbf{3}_{A}$ denote the symmetric and the
antisymmetric triplet combinations respectively. In terms of the components of
the two triplets $\alpha=\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)^{T}$
and $\beta=\left(\beta_{1},\beta_{2},\beta_{3}\right)^{T}$, we has

$\displaystyle\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\sim\mathbf{1}\,,$
$\displaystyle\alpha_{3}\beta_{3}+\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\sim\mathbf{1^{\prime}}\,,$
$\displaystyle\alpha_{2}\beta_{2}+\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}\sim\mathbf{1^{\prime\prime}}\,,$
$\displaystyle\begin{pmatrix}2\alpha_{1}\beta_{1}-\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
2\alpha_{3}\beta_{3}-\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
2\alpha_{2}\beta_{2}-\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\end{pmatrix}\sim\mathbf{3}_{S}\,,$
$\displaystyle\begin{pmatrix}\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}\end{pmatrix}\sim\mathbf{3}_{A}\,.$ (6)

There are three linearly independent modular forms of weight 2 at level 3 [6],
they transform in the triplet representation of $A_{4}$ up to the automorphy
factor $(c\tau+d)^{2}$, and they can be explicitly constructed in terms of the
Dedekind eta-function $\eta(\tau)$ as follow [54]:

$Y^{(2)}_{\mathbf{3}}(\tau)=\begin{pmatrix}\varepsilon^{2}(\tau)\\\
\sqrt{2}\,\vartheta(\tau)\varepsilon(\tau)\\\
-\vartheta^{2}(\tau)\end{pmatrix}\equiv\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\\\ Y_{3}(\tau)\end{pmatrix}\,,$ (7)

with

$\vartheta(\tau)=3\sqrt{2}\,\frac{\eta^{3}(3\tau)}{\eta(\tau)},\qquad\varepsilon(\tau)=-\frac{3\eta^{3}(3\tau)+\eta^{3}(\tau/3)}{\eta(\tau)}\,,$
(8)

and

$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^{n}\right),\qquad
q=e^{i2\pi\tau}\,.$ (9)

Then we can read out the $q$-expansion of the modular forms $Y_{1,2,3}(\tau)$
as

$\displaystyle
Y_{1}(\tau)=1+12q+36q^{2}+12q^{3}+84q^{4}+72q^{5}+36q^{6}+96q^{7}+180q^{8}+12q^{9}+216q^{10}+\dots\,,$
$\displaystyle
Y_{2}(\tau)=-6q^{1/3}\left(1+7q+8q^{2}+18q^{3}+14q^{4}+31q^{5}+20q^{6}+36q^{7}+31q^{8}+56q^{9}+32q^{10}+\dots\right)\,,$
$\displaystyle
Y_{3}(\tau)=-18q^{2/3}\left(1+2q+5q^{2}+4q^{3}+8q^{4}+6q^{5}+14q^{6}+8q^{7}+14q^{8}+10q^{9}+21q^{10}+\dots\right)\,.$
(10)

The higher weight modular forms of level 3 can be constructed from the tensor
product of $Y^{(2)}_{\mathbf{3}}$. There are five linearly independent weight
4 modular forms which can be arrange into the $A_{4}$ multiplets $\mathbf{1}$,
$\mathbf{1^{\prime}}$ and $\mathbf{3}$, i.e.

$\displaystyle Y^{(4)}_{\mathbf{1}}$ $\displaystyle=$
$\displaystyle(Y^{(2)}_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}=Y_{1}^{2}+2Y_{2}Y_{3}\,,$
$\displaystyle Y^{(4)}_{\mathbf{1}^{\prime}}$ $\displaystyle=$
$\displaystyle(Y^{(2)}_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}^{\prime}}=Y_{3}^{2}+2Y_{1}Y_{2}\,,$
$\displaystyle Y^{(4)}_{\mathbf{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Y^{(2)}_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{3}_{S}}=\begin{pmatrix}Y_{1}^{2}-Y_{2}Y_{3}\\\
Y_{3}^{2}-Y_{1}Y_{2}\\\ Y_{2}^{2}-Y_{1}Y_{3}\end{pmatrix}\,.$ (11)

At weight 6, we have three modular multiplets which transform in $\mathbf{1}$
and $\mathbf{3}$ of $A_{4}$,

$\displaystyle Y^{(6)}_{\mathbf{1}}$ $\displaystyle=$
$\displaystyle(Y^{(2)}_{\mathbf{3}}Y^{(4)}_{\mathbf{3}})_{\mathbf{1}}=Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3}\,,$
$\displaystyle Y^{(6)}_{\mathbf{3}I}$ $\displaystyle=$ $\displaystyle
Y^{(2)}_{\mathbf{3}}Y^{(4)}_{\mathbf{1}}=(Y_{1}^{2}+2Y_{2}Y_{3})\begin{pmatrix}Y_{1}\\\
Y_{2}\\\ Y_{3}\end{pmatrix}\,,$ $\displaystyle Y^{(6)}_{\mathbf{3}II}$
$\displaystyle=$ $\displaystyle
Y^{(2)}_{\mathbf{3}}Y^{(4)}_{\mathbf{1}^{\prime}}=(Y_{3}^{2}+2Y_{1}Y_{2})\begin{pmatrix}Y_{3}\\\
Y_{1}\\\ Y_{2}\end{pmatrix}\,.$ (12)

| $\psi_{1,2,3}$ | $H$ | $\Sigma$ | $\overline{\Delta}$
---|---|---|---|---
$SO(10)$ | $\mathbf{16}$ | $\mathbf{10}$ | $\mathbf{120}$ | $\overline{\mathbf{126}}$
$A_{4}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$k_{I}$ | $k_{F}$ | $k_{10}$ | $k_{120}$ | $k_{\overline{126}}$
Table 1: The transformation properties and modular weights of the $SO(10)$
matter fields and Higgs multiplets.

Although the $SO(10)$ GUT symmetry unifies all the fermions of each generation
into a single representation $\mathbf{16}$, the fermions in different
generations are not related so that the Yukawa coupling matrices
$\mathcal{Y}^{10}$, $\mathcal{Y}^{\overline{126}}$ and $\mathcal{Y}^{120}$ in
Eq. (1) can be arbitrary complex symmetric and antisymmetric matrices
respectively. In order to understand the observed flavor structure of quarks
and leptons, the $A_{4}$ modular symmetry is imposed in the $SO(10)$ GUT. We
assign the three generations of matter fields $\psi_{1,2,3}$ to a $A_{4}$
triplet since the flavor symmetry would effectively be the $Z_{3}$ subgroup
generated by $T$ rather than $A_{4}$ for the singlet assignment. The GUT Higgs
multiplets $H$, $\overline{\Delta}$ and $\Sigma$ all transform as
$\mathbf{1}$. Taking into account the $A_{4}$ modular symmetry, the $SO(10)$
Yukawa coupling of Eq. (1) becomes

$\displaystyle\mathcal{W}_{Y}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{r}_{a}}\alpha_{a}\left(\psi\psi
Y_{\mathbf{r}_{a}}^{(2k_{F}+k_{10})}(\tau)\right)_{\mathbf{1}}H+\sum_{\mathbf{r}_{c}}\beta_{b}\left(\psi\psi
Y_{\mathbf{r}_{b}}^{(2k_{F}+k_{120})}(\tau)\right)_{\mathbf{1}}\Sigma$ (13)
$\displaystyle~{}~{}~{}+\sum_{\mathbf{r}_{c}}\gamma_{c}\left(\psi\psi
Y_{\mathbf{r}_{c}}^{(2k_{F}+k_{\overline{126}})}(\tau)\right)_{\mathbf{1}}\overline{\Delta}\,,$

where $k_{F}$, $k_{10}$, $k_{\overline{126}}$ and $k_{120}$ are the modular
weights of $\psi$, $H$, $\overline{\Delta}$ and $\Sigma$ respectively. Notice
that one has to sum over the contributions of all independent modular
multiplets at the relevant weights. Using the multiplication rules in Eq. (6),
the Yukawa matrices $\mathcal{Y}^{10}$ can be read off as follows,

$\displaystyle\mathcal{Y}^{10}\Big{|}_{k=2k_{F}+k_{10}}=\alpha_{1}Y_{\mathbf{1}}^{(k)}(\tau)\left(\begin{matrix}1~{}&0~{}&0\\\
0~{}&0~{}&1\\\ 0~{}&1~{}&0\\\
\end{matrix}\right)+\alpha_{2}Y_{\mathbf{1}^{\prime}}^{(k)}(\tau)\left(\begin{matrix}0~{}&0~{}&1\\\
0~{}&1~{}&0\\\ 1~{}&0~{}&0\\\ \end{matrix}\right)$
$\displaystyle\qquad+\alpha_{3}Y_{\mathbf{1}^{\prime\prime}}^{(k)}(\tau)\left(\begin{matrix}0~{}&1~{}&0\\\
1~{}&0~{}&0\\\ 0~{}&0~{}&1\\\
\end{matrix}\right)+\alpha_{4}\left(\begin{matrix}2Y^{(k)}_{{{\mathbf{3}}},1}(\tau)~{}&~{}-Y^{(k)}_{{{\mathbf{3}}},3}(\tau)~{}&~{}-Y^{(k)}_{{{\mathbf{3}}},2}(\tau)\\\
-Y^{(k)}_{{{\mathbf{3}}},3}(\tau)~{}&~{}2Y^{(k)}_{{{\mathbf{3}}},2}(\tau)~{}&~{}-Y^{(k)}_{{{\mathbf{3}}},1}(\tau)\\\
-Y^{(k)}_{{{\mathbf{3}}},2}(\tau)~{}&~{}-Y^{(k)}_{{{\mathbf{3}}},1}(\tau)~{}&~{}2Y^{(k)}_{{{\mathbf{3}}},3}(\tau)\end{matrix}\right)\,,$
(14)

and $\mathcal{Y}^{\overline{126}}$ takes a similar form with $\alpha_{i}$
replaced by $\gamma_{i}$. Because the coupling with $\Sigma$ is antisymmetric
in the generation indices, only the antisymmetric triplet contraction
$(\psi\psi)_{\mathbf{3}_{A}}$ contributes to the Yukawa matrix
$\mathcal{Y}^{120}$,

$\displaystyle\mathcal{Y}^{120}\Big{|}_{k=2k_{F}+k_{120}}=\beta_{1}\left(\begin{matrix}0~{}&~{}Y^{(k)}_{{{\mathbf{3}}},3}(\tau)~{}&~{}-Y^{(k)}_{{{\mathbf{3}}},2}(\tau)\\\
-Y^{(k)}_{{{\mathbf{3}}},3}(\tau)~{}&~{}0~{}&~{}Y^{(k)}_{{{\mathbf{3}}},1}(\tau)\\\
Y^{(k)}_{{{\mathbf{3}}},2}(\tau)~{}&~{}-Y^{(k)}_{{{\mathbf{3}}},1}(\tau)~{}&~{}0\end{matrix}\right)\,.$
(15)

###  2.2  Benchmark models

In the following, we present two typical $SO(10)\times A_{4}$ modular models
given in our previous work [81], then study the predictions for leptogenesis.
The first benchmark model denoted as BM1 is characterized by the modular
weights
$(2k_{F}+k_{10}\,,2k_{F}+k_{120}\,,2k_{F}+k_{\overline{126}})=(4,2,4)$, and it
is the non-minimal model 2 of [81]. Thus the modular invariant superpotential
is given by

$\displaystyle\mathcal{W}_{Y}$ $\displaystyle=$
$\displaystyle\alpha_{1}Y_{\mathbf{1}}^{(4)}\psi\psi
H+\alpha_{2}Y_{\mathbf{1}^{\prime}}^{(4)}\psi\psi
H+\alpha_{3}Y_{\mathbf{3}}^{(4)}\psi\psi
H+\beta_{1}Y_{\mathbf{3}}^{(2)}\psi\psi\Sigma$ (16)
$\displaystyle+\gamma_{1}Y_{\mathbf{1}}^{(4)}\psi\psi\overline{\Delta}+\gamma_{2}Y_{\mathbf{1}^{\prime}}^{(4)}\psi\psi\overline{\Delta}+\gamma_{3}Y_{\mathbf{3}}^{(4)}\psi\psi\overline{\Delta}\,.$

The second model denoted as BM2 is specified by the modular weights
$(2k_{F}+k_{10}\,,2k_{F}+k_{120}\,,2k_{F}+k_{\overline{126}})=(4,6,4)$, and it
is the non-minimal model 3 of [81]. We can straightforwardly read out the
superpotential as follows,

$\displaystyle\mathcal{W}_{Y}$ $\displaystyle=$
$\displaystyle\alpha_{1}Y_{\mathbf{1}}^{(4)}\psi\psi
H+\alpha_{2}Y_{\mathbf{1}^{\prime}}^{(4)}\psi\psi
H+\alpha_{3}Y_{\mathbf{3}}^{(4)}\psi\psi
H+\beta_{1}Y_{\mathbf{3}I}^{(6)}\psi\psi\Sigma$
$\displaystyle+\beta_{2}Y_{\mathbf{3}II}^{(6)}\psi\psi\Sigma+\gamma_{1}Y_{\mathbf{1}}^{(4)}\psi\psi\overline{\Delta}+\gamma_{2}Y_{\mathbf{1}^{\prime}}^{(4)}\psi\psi\overline{\Delta}+\gamma_{3}Y_{\mathbf{3}}^{(4)}\psi\psi\overline{\Delta}\,.$
(17)

In the case that the light neutrino mass is dominated by the type-I seesaw
contribution, the best fit values of the free parameters for the above two
models are listed in table 3 and table 4, and we see that the experimental
data can be accommodated very well.

##  3 Prediction for baryon asymmetry via leptogenesis

The baryon asymmetry of the Universe is a puzzle in particle physics, and its
value can be inferred from observations in two independent ways. The first is
from the big bang nucleosynthesis. With the assumption of only three light
neutrinos, the predictions for the abundances of the light elements D,
${}^{3}{\rm He}$, ${}^{4}{\rm He}$, and 7Li depend on a single parameter, that
is the difference between the number of baryons and anti-baryons normalized to
the number of photons:

$\eta\equiv\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}{\Big{|}}_{0},$ (18)

where the subscript $0$ means “at present time”. The range of $\eta$ which is
consistent with the abundances of D and ${}^{4}{\rm He}$ is determined to be
[92]

$\eta=(6.143\pm 0.190)\times 10^{-10}\,,$ (19)

at $1\sigma$ confidence level. This range corresponds to

$Y_{\Delta B}=(8.72\pm 0.27)\times 10^{-11}\,.$ (20)

Notice that $Y_{\Delta B}$ is defined as $Y_{\Delta
B}\equiv\frac{n_{B}-n_{\bar{B}}}{s}{\Big{|}}_{0}$ and it is related to the
parameter $\eta$ through $Y_{\Delta B}\simeq\eta/7.04$. Furthermore the baryon
asymmetry produces acoustic oscillations in the power spectrum of the cosmic
microwave background [93]. Observing these oscillations gives an even tighter
bound on the value of baryon asymmetry,

$Y_{\Delta B}=\left(8.703\pm
0.113\right)\times~{}10^{-11}~{}~{}~{}(95\%,\text{TT,TE,EE+lowE})$ (21)

It is impressive that the above two unrelated measurement approaches give
consistent range of baryon asymmetry. In this work, we will use the
conservative value of Eq. (20) from the big bang nucleosynthesis.

Generally speaking, both type I and type II seesaw mechanisms contribute to
the light neutrino mass in $SO(10)$ GUT, the contribution of type I seesaw is
assumed to be dominant over that of type II in the present work. Then the CP
violating out-of-equilibrium decay of the heavy Majorana neutrinos can
generate a lepton asymmetry which is subsequently partially converted to a
baryon asymmetry through the $B+L$ violating sphaleron processes [2]. In this
section, we shall study whether the measured value of the baryon asymmetry of
the Universe $Y_{\Delta B}=(8.72\pm 0.27)\times 10^{-11}$ in Eq. (20) can be
correctly generated through thermal leptogenesis in the above benchmark
$SO(10)\times A_{4}$ modular models. The evolution of the lepton asymmetry is
described by the density matrix equation.

###  3.1 Density matrix equation

In the simplest scenario of leptogenesis, the lepton and anti-lepton final
states from the decays of the heavy neutrino $N_{i}$ are in coherent
superposition of flavor,

$|\ell_{i}\rangle=\sum_{\alpha}C_{i\alpha}|\ell_{\alpha}\rangle\,,~{}~{}~{}~{}|\bar{\ell}_{i}\rangle=\sum_{\alpha}\bar{C}_{i\alpha}|\bar{\ell}_{\alpha}\rangle\,,$
(22)

where $i=1,2,3$ and $\alpha=e,\mu,\tau$, and the projection coefficients at
tree-level are given by

$C_{i\alpha}=\bar{C}_{i\alpha}=\dfrac{(\lambda_{\nu})_{\alpha
i}}{\sqrt{(\lambda_{\nu}^{\dagger}\lambda_{\nu})_{ii}}}\,.$ (23)

Notice that we work in the basis where both charged lepton and right-handed
neutrino mass matrices are diagonal. This is a good approximation for high
temperature $T\gg 10^{12}$ GeV when the charged lepton Yukawa interactions are
negligible. However, the coherent evolution of the states $|\ell_{i}\rangle$
and $|\bar{\ell}_{i}\rangle$ would be broken down and different lepton flavors
are distinguishable when these interactions are in thermal equilibrium. Then
the left-handed leptons can be rapidly converted to right-handed components
through scattering with the Higgs doublet. To be more specific, the $\tau$ and
$\mu$ Yukawa interactions come into thermal equilibrium at the temperature
$T\sim 10^{12}$ GeV and $T\sim 10^{9}$ GeV respectively.

It has been shown that the density matrix equation can provide an accurate
description for the time evolution of the lepton asymmetry [94, 95, 96, 97,
98, 99, 100]. The approach of the density matrix accounts for the quantum
decoherence effect and it allows to quantitatively describe the transitions
between the one-flavored, two-flavored and three-flavored regimes.
Comprehensive analysis of leptogenesis has been performed in the framework of
density matrix [101, 102, 103]. The formalism of the density matrix for
leptogenesis has been given in Ref. [100], we shall follow closely the
notation of [100] in the following. For three decaying heavy Majorana
neutrinos, the most general form of the density matrix equations are given by

$\displaystyle\dfrac{\mathrm{d}N_{N_{i}}}{\mathrm{d}z}$ $\displaystyle=$
$\displaystyle-D_{i}(N_{N_{i}}-N^{\rm eq}_{N_{i}})\,,$
$\displaystyle\dfrac{\mathrm{d}N^{B-L}_{\alpha\beta}}{\mathrm{d}z}$
$\displaystyle=$
$\displaystyle\sum_{i}\left[\varepsilon^{(i)}_{\alpha\beta}D_{i}(N_{N_{i}}-N^{\rm
eq}_{N_{i}})-\dfrac{1}{2}W_{i}\left\\{P^{0(i)},N^{B-L}\right\\}_{\alpha\beta}\right]$
(24) $\displaystyle-$
$\displaystyle\dfrac{\Im(\Lambda_{\tau})}{Hz}\left(\delta_{\alpha\tau}N^{B-L}_{\tau\beta}+\delta_{\beta\tau}N^{B-L}_{\alpha\tau}-2\delta_{\alpha\tau}\delta_{\beta\tau}N^{B-L}_{\tau\tau}\right)$
$\displaystyle-$
$\displaystyle\dfrac{\Im(\Lambda_{\mu})}{Hz}\left(\delta_{\alpha\mu}N^{B-L}_{\mu\beta}+\delta_{\beta\mu}N^{B-L}_{\alpha\mu}-2\delta_{\alpha\mu}\delta_{\beta\mu}N^{B-L}_{\mu\mu}\right)\,,$

with $z=M_{1}/T$. The projection matrices are

$P^{0(i)}_{\alpha\beta}=C_{i\alpha}C^{*}_{i\beta}\,,$ (25)

which describe how a given flavor of lepton is washed out. The quantity
$N_{N_{i}}$ is the number of the heavy neutrino $N_{i}$ in a portion of
comoving volume which contains one photon at the temperature $z\simeq 0$.
Hence the equilibrium number density $N^{\rm eq}_{N_{i}}$ for the Boltzmann
statistics is given by

$N^{\rm eq}_{N_{i}}(z)=\dfrac{3}{8}x_{i}z^{2}\mathcal{K}_{2}(z_{i})\,,$ (26)

where $x_{i}=M_{i}^{2}/M_{1}^{2}$, $z_{i}=\sqrt{x_{i}}z$, and
$\mathcal{K}_{2}(z)$ the modified Bessel functions of the second kind.
Consequently we have $N^{\rm eq}_{N_{i}}(z\simeq 0)=3/4$. The diagonal entries
$N^{B-L}_{\alpha\alpha}$ are the known number densities for the
$B/3-L_{\alpha}$ asymmetry, and the off-diagonal elements
$N^{B-L}_{\alpha\beta}$ describe the degree of coherence between the flavor
states. The rescaled decay rate $D_{i}$ of the right-handed neutrino $N_{i}$
is given by

$D_{i}(z)=K_{i}x_{i}z\dfrac{\mathcal{K}_{1}(z_{i})}{\mathcal{K}_{2}(z_{i})}\,,~{}~{}~{}K_{i}=\dfrac{M_{i}(\lambda_{\nu}^{\dagger}\lambda_{\nu})_{ii}}{8\pi
H(T=M_{i})}\,,$ (27)

where $H$ is the Hubble expansion rate. The washout parameters $W_{i}$ read

$W_{i}(z)=\frac{1}{4}K_{i}\sqrt{x_{i}}\,z_{i}^{3}\,{\cal K}_{1}(z_{i})\,.$
(28)

Furthermore, the last two terms in Eq. (24) describe the effect of charged
lepton Yukawa interactions which can induce the transition of left-handed
leptons to right-handed leptons. They describe decoherence via the damping of
the off-diagonal terms of the $B-L$ asymmetry matrix $N^{B-L}$ and they are
determined as

$\displaystyle\frac{\Im(\Lambda_{\mu})}{H\,z}$ $\displaystyle=$
$\displaystyle\frac{8\times 10^{-3}\,y_{\mu}^{2}\,T}{Hz}=1.72\times
10^{-10}\,\frac{M_{\text{Pl}}}{M_{1}}\,,$
$\displaystyle\frac{\Im(\Lambda_{\tau})}{H\,z}$ $\displaystyle=$
$\displaystyle\frac{8\times 10^{-3}\,y_{\tau}^{2}\,T}{Hz}=4.87\times
10^{-8}\,\frac{M_{\text{Pl}}}{M_{1}}\,,$ (29)

where $M_{\text{Pl}}\simeq 1.22\times 10^{19}$ GeV is the Planck mass, the
Yukawa couplings $y_{\mu}\simeq 6.07\times 10^{-4}$ and $y_{\tau}\simeq
1.02\times 10^{-2}$ are fixed by the $\mu$ and $\tau$ masses $m_{\mu}\simeq
105.66$ MeV and $m_{\tau}\simeq 1776.86$ MeV respectively. When the
temperature goes below $10^{12}$ GeV, the $\tau$ Yukawa interactions come into
thermal equilibrium and leads to decoherence of $\tau$ lepton states. This
implies the transition from an unflavoured to two-flavoured regime. When the
temperature drops below $10^{9}$ GeV the similar effects arise from the $\mu$
Yukawa interactions. Analogously, the electron Yukawa dependent damping term
should be considered for $M_{1}<10^{5}$ GeV. Finally the CP asymmetry
$\varepsilon_{\alpha\beta}^{(i)}$ generated by $N_{i}$ decay is of the
following form [104, 98, 95, 105, 100]

$\displaystyle\varepsilon_{\alpha\beta}^{(i)}$ $\displaystyle=$
$\displaystyle\frac{3}{32\pi\left(\lambda^{\dagger}_{\nu}\lambda_{\nu}\right)_{ii}}\,\sum_{j\neq
i}\bigg{\\{}i\left[(\lambda_{\nu})_{\alpha i}(\lambda^{*}_{\nu})_{\beta
j}(\lambda^{\dagger}\lambda)_{ji}-(\lambda^{*}_{\nu})_{\beta
i}(\lambda_{\nu})_{\alpha
j}(\lambda^{\dagger}_{\nu}\lambda_{\nu})_{ij}\right]f_{1}\left(\frac{x_{j}}{x_{i}}\right)$
(30) $\displaystyle+$ $\displaystyle i\left[(\lambda_{\nu})_{\alpha
i}(\lambda^{*}_{\nu})_{\beta
j}(\lambda^{\dagger}_{\nu}\lambda_{\nu})_{ij}-(\lambda^{*}_{\nu})_{\beta
i}(\lambda_{\nu})_{\alpha
j}(\lambda^{\dagger}_{\nu}\lambda_{\nu})_{ji}\right]f_{2}\left(\frac{x_{j}}{x_{i}}\right)\bigg{\\}}\,,$

with

$f_{1}(x)=\frac{2\sqrt{x}}{3}\left[(1+x)\,\ln\left(\frac{1+x}{x}\right)-\frac{2-x}{1-x}\right],~{}~{}~{}f_{2}(x)=\frac{2}{3(x-1)}\,.$
(31)

The diagonal components $\varepsilon^{(i)}_{\alpha\alpha}$ of the CP asymmetry
matrix are exactly the usual flavored CP asymmetries,

$\varepsilon_{\alpha\alpha}^{(i)}=\frac{3}{16\pi\left(\lambda^{\dagger}_{\nu}\lambda_{\nu}\right)_{ii}}\,\sum_{j\neq
i}\bigg{\\{}\Im\left[(\lambda^{*}_{\nu})_{\alpha i}(\lambda_{\nu})_{\alpha
j}(\lambda^{\dagger}_{\nu}\lambda_{\nu})_{ij}\right]f_{1}\left(\frac{x_{j}}{x_{i}}\right)+\Im\left[(\lambda^{*}_{\nu})_{\alpha
i}(\lambda_{\nu})_{\alpha
j}(\lambda^{\dagger}_{\nu}\lambda_{\nu})_{ji}\right]f_{2}\left(\frac{x_{j}}{x_{i}}\right)\bigg{\\}}\,,$
(32)

while the off-diagonal components
$\varepsilon^{(j)}_{\alpha\beta}=(\varepsilon^{(j)}_{\beta\alpha})^{*}$ and
they are not necessarily real [100]. The trace of the matrix
$N^{B-L}_{\alpha\beta}$ gives the total lepton asymmetry

$N^{\rm f}_{B-L}=\sum_{\alpha}N^{B-L}_{\alpha\alpha}\,,$ (33)

which is then converted into the baryon asymmetry of the Universe,

$\eta=\dfrac{28}{79}\dfrac{1}{27}\,N^{\rm f}_{B-L}\simeq 0.013\,N^{\rm
f}_{B-L}\,,$ (34)

where the factor $28/79$ accounts for the partial conversion of the $B-L$
asymmetry into the baryon asymmetry by sphaleron process, and the second
factor $1/27$ describes the dilution of the baryon asymmetry due to the change
of the photon density between leptogenesis and recombination. In the following
numerical analysis, we will use the Python package ULYSSES [106] to solve the
coupled density matrix equations shown in Eq. (24).

###  3.2 Analytic approximation within the Boltzmann equations

In our benchmark models, the right-handed neutrino masses are hierarchical
with $M_{3}\gg M_{2}\gtrsim 10^{12}\,{\rm GeV}$ while $10^{9}\,{\rm GeV}\ll
M_{1}\ll 10^{12}\,{\rm GeV}$. Therefore the decays of $N_{3}$ and $N_{2}$ are
in the single-flavored regime while $N_{1}$ decay is in the two-flavored
regime. In the following, we shall analyze the sequential decay of the three
heavy Majorana neutrinos and evolution of lepton symmetry in terms of
Boltzmann equations. In the first stage around the temperature $T\sim M_{3}$,
a $B-L$ asymmetry is generated from the $N_{3}$ decay. In the second (third)
stage at $T\sim M_{2}(M_{1})$ another $B-L$ asymmetry is generated from the
$N_{2}(N_{1})$ decays, while the inverse decay process of $N_{2}(N_{1})$
become effective and washout the asymmetry of $N_{3}$ decay to some level. In
each stage we can use the Boltzmann equations to obtain an approximate
analytical expression for the $B-L$ asymmetry.

In the first stage for $M_{3}\gtrsim T\gtrsim T_{B_{3}}$, where
$T_{B_{3}}\simeq M_{3}/z_{B_{3}}$ is the freeze-out temperature of the $N_{3}$
inverse decay and $z_{B_{3}}\simeq
2+4K_{3}^{0.13}\,e^{-2.5/K_{3}}=\mathcal{O}(1-10)$ [107], the rates of the
$\tau$ and $\mu$ Yukawa interactions are much smaller than the expansion rate
of the Universe. Consequently the charged lepton lepton state produced from
$N_{3}$ decay is in coherent superposition and the flavor states are
indistinguishable. In the single flavor approximation, the time evolution of
the number densities of $N_{3}$ and $B-L$ can be described by the following
semi-classical Boltzmann equations:

$\displaystyle\dfrac{{\rm d}N_{N_{3}}}{{\rm d}z_{3}}$ $\displaystyle=$
$\displaystyle-D_{3}(N_{N_{3}}-N^{\rm eq}_{N_{3}})\,,$
$\displaystyle\dfrac{{\rm d}N_{\Delta_{3}}}{{\rm d}z_{3}}$ $\displaystyle=$
$\displaystyle\varepsilon_{3}D_{3}(N_{N_{3}}-N^{\rm
eq}_{N_{3}})-W_{3}N_{\Delta_{3}}\,.$ (35)

The solution to the above equations is given by [108]

$\displaystyle N_{\Delta_{3}}({T\simeq
T_{B_{3}}})=\dfrac{3}{4}\varepsilon_{3}\kappa(K_{3})\,.$ (36)

where $\kappa(K_{3})$ is the so-called efficiency factor and its value depends
on the initial condition of decaying heavy neutrinos. For thermal initial
abundance of right-handed neutrinos, it is well approximated by [108],

$\displaystyle\kappa(x)=\dfrac{2}{xz_{B}(x)}\left[1-\exp\left(-\dfrac{1}{2}xz_{B}(x)\right)\right]\,.$
(37)

For the vanishing initial abundance of right-handed neutrinos, the
approximation formula for the efficiency factor is [108]

$\displaystyle\kappa(K)$ $\displaystyle=$
$\displaystyle\kappa^{-}(K)+\kappa^{+}(K)\,,$ $\displaystyle\kappa^{-}(K)$
$\displaystyle=$
$\displaystyle-2\,e^{-\frac{2}{3}N(K)}\left[e^{\frac{2}{3}\overline{N}(K)}-1\right]\,,$
$\displaystyle\kappa^{+}(K)$ $\displaystyle=$
$\displaystyle\dfrac{2}{z_{B}(K)K}\left[1-e^{-\frac{2}{3}z_{B}(K)K\overline{N}(K)}\right]\,,$
(38)

with

$\displaystyle N(K)=\dfrac{9\pi
K}{16}\,,~{}~{}~{}~{}~{}\overline{N}(K)=\dfrac{N(K)}{\left(1+\sqrt{\frac{N(K)}{3/4}}\right)^{2}}\,.$
(39)

The lepton state $|\ell_{3}\rangle$ produced by $N_{3}$ decay can be regarded
as coherent superposition of a $\ell_{2}$ parallel component and of a
$\ell_{2}$ orthogonal component which is denoted as
$|\ell_{\tilde{2}}\rangle$. To be more concrete, $|\ell_{\tilde{2}}\rangle$ is
the projection of $|\ell_{3}\rangle$ on the plane orthogonal to
$|\ell_{2}\rangle$ such that $\langle\ell_{\tilde{2}}|\ell_{2}\rangle=0$, and
its explicit expression is

$\displaystyle|\ell_{\tilde{2}}\rangle=\dfrac{1}{\sqrt{1-p_{32}}}(|\ell_{3}\rangle-\langle\ell_{2}|\ell_{3}\rangle|\ell_{2}\rangle)\,,$
(40)

where the probabilities $p_{32}\equiv|\langle\ell_{2}|\ell_{3}\rangle|^{2}$
and $p_{3\tilde{2}}\equiv|\langle\ell_{\tilde{2}}|\ell_{3}\rangle|^{2}$ with
$p_{32}+p_{3\tilde{2}}=1$. From Eq. (22), we can read out the general
expression of $p_{ij}$ as

$\displaystyle
p_{ij}=\Big{|}\sum_{\alpha}C_{i\alpha}C_{j\alpha}^{*}\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\dfrac{|(\lambda_{\nu}^{\dagger}\lambda_{\nu})_{ij}|^{2}}{(\lambda_{\nu}^{\dagger}\lambda_{\nu})_{ii}(\lambda_{\nu}^{\dagger}\lambda_{\nu})_{jj}}\,.$
(41)

Correspondingly the same decomposition can be made for the asymmetry
$N_{\Delta_{3}}$,

$\displaystyle N_{\Delta_{2}}({T\simeq T_{B_{3}}})$ $\displaystyle=$
$\displaystyle p_{32}N_{\Delta_{3}}({T\simeq
T_{B_{3}}})=\dfrac{3}{4}p_{32}\varepsilon_{3}\kappa(K_{3})\,,$ $\displaystyle
N_{\Delta_{\tilde{2}}}({T\simeq T_{B_{3}}})$ $\displaystyle=$ $\displaystyle
p_{3\tilde{2}}N_{\Delta_{3}}({T\simeq
T_{B_{3}}})=\dfrac{3}{4}(1-p_{32})\varepsilon_{3}\kappa(K_{3})\,.$ (42)

These two asymmetries can be used as initial condition at the beginning of the
$N_{2}$ production and decay. When the temperature goes down to $T\sim M_{2}$,
the $N_{2}$ decay and inverse processes break the coherent evolution of
$|\ell_{3}\rangle$ which becomes an incoherent mixture of $|\ell_{2}\rangle$
and $|\ell_{\tilde{2}}\rangle$. The $|\ell_{2}\rangle$ component undergoes the
washout from $N_{2}$ while the orthogonal component $|\ell_{\tilde{2}}\rangle$
is unwashed. The kinetic evolution equations are given by

$\displaystyle\dfrac{{\rm d}N_{N_{2}}}{{\rm d}z_{2}}$ $\displaystyle=$
$\displaystyle-D_{2}(N_{N_{2}}-N^{\rm eq}_{N_{2}})\,,$
$\displaystyle\dfrac{{\rm d}N_{\Delta_{2}}}{{\rm d}z_{2}}$ $\displaystyle=$
$\displaystyle\varepsilon_{2}D_{2}(N_{N_{2}}-N^{\rm
eq}_{N_{2}})-W_{2}N_{\Delta_{2}}\,,$ $\displaystyle\dfrac{{\rm
d}N_{\Delta_{\tilde{2}}}}{{\rm d}z_{2}}$ $\displaystyle=$ $\displaystyle 0\,.$
(43)

Hence the asymmetries at the temperature $T\simeq T_{B_{2}}$ take the form,

$\displaystyle N_{\Delta_{2}}({T\simeq T_{B_{2}}})$ $\displaystyle=$
$\displaystyle\dfrac{3}{4}\varepsilon_{2}\kappa(K_{2})+N_{\Delta_{2}}({T\simeq
T_{B_{3}}})\,e^{-\frac{3\pi}{8}K_{2}}\,,$ $\displaystyle
N_{\Delta_{\tilde{2}}}({T\simeq T_{B_{2}}})$ $\displaystyle=$ $\displaystyle
N_{\Delta_{\tilde{2}}}({T\simeq T_{B_{3}}})\,,$ (44)

Since $10^{9}\,{\rm GeV}\ll M_{1}\ll 10^{12}\,{\rm GeV}$, the $\tau$ Yukawa
interaction becomes effective before the onset of $N_{1}$ washout processes.
Let us indicate with $M_{1}\ll T_{\star}\ll 10^{12}\,{\rm GeV}$ that value of
temperature below which one can approximate the coherence of $|\tau\rangle$
and the orthogonal component completely damped. Thus the lepton quantum state
can be treated as an incoherent mixture of three components: $|\tau\rangle$,
$|\tilde{\tau}_{2}\rangle$ and $|\tilde{\tau}_{\tilde{2}}\rangle$, where
$|\tilde{\tau}_{2}\rangle$ and $|\tilde{\tau}_{\tilde{2}}\rangle$ are
projections of $|\ell_{2}\rangle$ and $|\ell_{\tilde{2}}\rangle$ on the
$e-\mu$ plane, respectively. Hence the lepton states
$|\tilde{\tau}_{2}\rangle$ and $|\tilde{\tau}_{\tilde{2}}\rangle$ read as

$\displaystyle|\tilde{\tau}_{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{1-p_{2\tau}}}\left(|\ell_{2}\rangle-\langle\tau|\ell_{2}\rangle|\tau\rangle\right)=\dfrac{1}{\sqrt{1-p_{2\tau}}}\left(C_{2\mu}|\mu\rangle+C_{2e}|e\rangle\right)\,,$
$\displaystyle|\tilde{\tau}_{\tilde{2}}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{1-p_{\tilde{2}\tau}}}\left(|\ell_{\tilde{2}}\rangle-\langle\tau|\ell_{\tilde{2}}\rangle|\tau\rangle\right)\,.$
(45)

The corresponding probabilities are
$p_{2\tau}=|\langle\tau|\ell_{2}\rangle|^{2}=|C_{2\tau}|^{2}$,
$p_{2\tilde{\tau}_{2}}=|\langle\tilde{\tau}_{2}|\ell_{2}\rangle|^{2}$,
$p_{\tilde{2}\tau}=|\langle\tau|\ell_{\tilde{2}}\rangle|^{2}$ and
$p_{\tilde{2}\tilde{\tau}_{\tilde{2}}}=|\langle\tilde{\tau}_{\tilde{2}}|\ell_{\tilde{2}}\rangle|^{2}$
with

$\displaystyle p_{\tilde{2}\tau}$ $\displaystyle=$
$\displaystyle|\langle\tau|\ell_{\tilde{2}}\rangle|^{2}=\dfrac{1}{1-p_{32}}\Big{[}p_{3\tau}+p_{32}p_{2\tau}-2\Re\Big{(}\sum_{\alpha}C_{3\tau}C^{*}_{2\tau}C_{2\alpha}C^{*}_{3\alpha}\Big{)}\Big{]}\,,$
(46)
$\displaystyle\hskip-25.29494ptp_{2\tau}+p_{2\tilde{\tau}_{2}}=p_{\tilde{2}\tau}+p_{\tilde{2}\tilde{\tau}_{\tilde{2}}}=1\,.$

Thus the $B-L$ asymmetry before $N_{1}$ decay can be decomposed into the
following three parts:

$\displaystyle N_{\Delta_{\tau}}(T\simeq T_{\star})$ $\displaystyle=$
$\displaystyle p_{2\tau}N_{\Delta_{2}}(T\simeq
T_{B_{2}})+p_{\tilde{2}\tau}N_{\Delta_{\tilde{2}}}(T\simeq T_{B_{2}})\,,$
$\displaystyle N_{\Delta_{\tilde{\tau}_{2}}}(T\simeq T_{\star})$
$\displaystyle=$ $\displaystyle(1-p_{2\tau})N_{\Delta_{2}}(T\simeq
T_{B_{2}})\,,$ $\displaystyle N_{\Delta_{\tilde{\tau}_{\tilde{2}}}}(T\simeq
T_{\star})$ $\displaystyle=$
$\displaystyle(1-p_{\tilde{2}\tau})N_{\Delta_{\tilde{2}}}(T\simeq
T_{B_{2}})\,.$ (47)

The asymmetries in the tau flavor and the orthogonal components
$|\tilde{\tau}_{1}\rangle$ and $|\tilde{\tau}_{\tilde{1}}\rangle$ experience
different washout from inverse decay, The quantum state
$|\tilde{\tau}_{1}\rangle$ is the projection of $|\ell_{1}\rangle$ on the
$e-\mu$ plane and $|\tilde{\tau}_{\tilde{1}}\rangle$ is the quantum state
orthogonal to $|\tilde{\tau}_{1}\rangle$ in $e-\mu$ plane,

$\displaystyle|\tilde{\tau}_{1}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{1-p_{1\tau}}}\left(|\ell_{1}\rangle-\langle\tau|\ell_{1}\rangle|\tau\rangle\right)=\dfrac{1}{\sqrt{1-p_{1\tau}}}\left(C_{1\mu}|\mu\rangle+C_{1e}|e\rangle\right)\,,$
$\displaystyle|\tilde{\tau}_{\tilde{1}}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{1-p_{1\tau}}}\left(-C^{*}_{1e}|\mu\rangle+C^{*}_{1\mu}|e\rangle\right)\,,$
(48)

with $p_{1\tau}=|C_{1\tau}|^{2}$. One can use the two-flavored Boltzmann
equations to describe the time evolution of the asymmetries during $N_{1}$
leptogenesis,

$\displaystyle\dfrac{{\rm d}N_{N_{1}}}{{\rm d}z_{1}}$ $\displaystyle=$
$\displaystyle-D_{1}(N_{N_{1}}-N^{\rm eq}_{N_{1}})\,,$
$\displaystyle\dfrac{{\rm d}N_{\Delta_{\tau}}}{{\rm d}z_{1}}$ $\displaystyle=$
$\displaystyle\varepsilon_{1\tau}D_{1}(N_{N_{1}}-N^{\rm
eq}_{N_{1}})-p_{1\tau}W_{1}N_{\Delta_{\tau}}\,,$ $\displaystyle\dfrac{{\rm
d}N_{\Delta_{\tilde{\tau}_{1}}}}{{\rm d}z_{1}}$ $\displaystyle=$
$\displaystyle\varepsilon_{1\tilde{\tau}_{1}}D_{1}(N_{N_{1}}-N^{\rm
eq}_{N_{1}})-p_{1\tilde{\tau}_{1}}W_{1}N_{\Delta_{\tilde{\tau}_{1}}}\,,$
$\displaystyle\dfrac{{\rm d}N_{\Delta_{\tilde{\tau}_{\tilde{1}}}}}{{\rm
d}z_{1}}$ $\displaystyle=$ $\displaystyle 0\,,$ (49)

where
$\varepsilon_{1\tilde{\tau}_{1}}\equiv\varepsilon_{1e}+\varepsilon_{1\mu}$.
Thus the frozen values of the asymmetries at the end of $N_{1}$ washout are
given by

$\displaystyle N_{\Delta_{\tau}}(T\simeq T_{B_{1}})$ $\displaystyle=$
$\displaystyle\dfrac{3}{4}\varepsilon_{1\tau}\kappa(K_{1\tau})+N_{\Delta_{\tau}}(T\simeq
T_{\star})\,e^{-\frac{3\pi}{8}K_{1\tau}}\,,$ $\displaystyle
N_{\Delta_{\tilde{\tau}_{1}}}(T\simeq T_{B_{1}})$ $\displaystyle=$
$\displaystyle\dfrac{3}{4}\varepsilon_{1\tilde{\tau}_{1}}\kappa(K_{1\tilde{\tau}_{1}})+N_{\Delta_{\tilde{\tau}_{1}}}({T\simeq
T_{\star}})\,e^{-\frac{3\pi}{8}K_{1\tilde{\tau}_{1}}}\,,$ $\displaystyle
N_{\Delta_{\tilde{\tau}_{\tilde{1}}}}(T\simeq T_{B_{1}})$ $\displaystyle=$
$\displaystyle N_{\Delta_{\tilde{\tau}_{\tilde{1}}}}(T\simeq T_{\star})$ (50)

where $K_{1\tilde{\tau}_{1}}\equiv K_{1e}+K_{1\mu}$. Notice that decomposing
$N_{\Delta\tilde{\tau}_{2}}(T\simeq T_{\star})$ and
$N_{\Delta\tilde{\tau}_{\tilde{2}}}(T\simeq T_{\star})$ along the directions
of $|\tilde{\tau}_{1}\rangle$ and $|\tilde{\tau}_{\tilde{1}}\rangle$ gives
rise to $N_{\Delta_{\tilde{\tau}_{1}}}({T\simeq T_{\star}})$ and
$N_{\Delta_{\tilde{\tau}_{\tilde{1}}}}(T\simeq T_{\star})$ as follows,

$\displaystyle N_{\Delta_{\tilde{\tau}_{1}}}({T\simeq T_{\star}})$
$\displaystyle=$ $\displaystyle
p_{\tilde{\tau}_{2}\tilde{\tau}_{1}}N_{\Delta_{\tilde{\tau}_{2}}}(T\simeq
T_{\star})+p_{\tilde{\tau}_{\tilde{2}}\tilde{\tau}_{1}}N_{\Delta_{\tilde{\tau}_{\tilde{2}}}}(T\simeq
T_{\star})\,,$ $\displaystyle N_{\Delta_{\tilde{\tau}_{\tilde{1}}}}(T\simeq
T_{\star})$ $\displaystyle=$
$\displaystyle(1-p_{\tilde{\tau}_{2}\tilde{\tau}_{1}})N_{\Delta_{\tilde{\tau}_{2}}}(T\simeq
T_{\star})+(1-p_{\tilde{\tau}_{\tilde{2}}\tilde{\tau}_{1}})N_{\Delta_{\tilde{\tau}_{\tilde{2}}}}(T\simeq
T_{\star})\,,$ (51)

with

$\displaystyle
p_{\tilde{\tau}_{2}\tilde{\tau}_{1}}=|\langle\tilde{\tau}_{1}|\tilde{\tau}_{2}\rangle|^{2}=\dfrac{\left|C^{*}_{1\mu}C_{2\mu}+C^{*}_{1e}C_{2e}\right|^{2}}{(1-p_{2\tau})(1-p_{1\tau})}\,,$
$\displaystyle
p_{\tilde{\tau}_{\tilde{2}}\tilde{\tau}_{1}}=|\langle\tilde{\tau}_{1}|\tilde{\tau}_{\tilde{2}}\rangle|^{2}=\dfrac{\left|C^{*}_{1\mu}C_{3\mu}+C^{*}_{1e}C_{3e}-(C^{*}_{1\mu}C_{2\mu}+C^{*}_{1e}C_{2e})\sum_{\alpha}C^{*}_{2\alpha}C_{3\alpha}\right|^{2}}{(1-p_{1\tau})(1-p_{\tilde{2}\tau})(1-p_{32})}\,.$
(52)

We would like to mention that the factors $C_{1\alpha}$, $C_{2\alpha}$,
$C_{3\alpha}$ depend on the neutrino Yukawa, as shown in Eq. (23). The total
final asymmetry is the sum of the three terms

$\displaystyle N_{B-L}(T\simeq T_{B_{1}})=N_{\Delta_{\tau}}(T\simeq
T_{B_{1}})+N_{\Delta_{\tilde{\tau}_{1}}}(T\simeq
T_{B_{1}})+N_{\Delta_{\tilde{\tau}_{\tilde{1}}}}(T\simeq T_{B_{1}})\,.$ (53)

It turns out that our benchmark models are in the strong washout regime with
$K_{3},K_{2},K_{1\tau},K_{1\tilde{\tau}_{1}}\gg 1$, thus we neglect the
contributions which undergo the washout exponential suppression of at least
one right-handed neutrino. The final lepton asymmetry will be then dominated
by the unwashed terms and it is approximately given by

$\displaystyle N_{B-L}^{\rm f}$ $\displaystyle=$
$\displaystyle\dfrac{3}{4}\Big{[}\varepsilon_{1\tau}\kappa(K_{1\tau})+\varepsilon_{1\tilde{\tau}_{1}}\kappa(K_{1\tilde{\tau}})+(1-p_{\tilde{\tau}_{2}\tilde{\tau}_{1}})(1-p_{2\tau})\varepsilon_{2}\kappa(K_{2})$
(54)
$\displaystyle~{}~{}~{}+(1-p_{\tilde{\tau}_{\tilde{2}}\tilde{\tau}_{1}})(1-p_{\tilde{2}\tau})(1-p_{32})\varepsilon_{3}\kappa(K_{3})\Big{]}\,.$

Notice that this approximation formula is applicable to both thermal and
vanishing initial abundance of right-handed neutrinos, and the corresponding
efficiency factors are given in Eq. (37) and Eq. (38) respectively.

##  4 Numerical results

Parameters | $\mu_{i}\pm 1\sigma$ | Parameters | $\mu_{i}\pm 1\sigma$
---|---|---|---
$m_{t}/\text{GeV}$ | $83.155\pm 3.465$ | $\theta_{12}^{q}$ | $0.229\pm 0.001$
$m_{b}/\text{GeV}$ | $0.884\pm 0.035$ | $\theta_{13}^{q}$ | $0.0037\pm 0.0004$
$m_{u}/m_{c}$ | $0.0027\pm 0.0006$ | $\theta_{23}^{q}$ | $0.0397\pm 0.0011$
$m_{c}/m_{t}$ | $0.0025\pm 0.0002$ | $\delta_{CP}^{q}/^{\circ}$ | $56.34\pm 7.89$
$m_{d}/m_{s}$ | $0.051\pm 0.007$ | $\sin^{2}\theta_{12}^{l}$ | $0.304\pm 0.012$
$m_{s}/m_{b}$ | $0.019\pm 0.002$ | $\sin^{2}\theta_{23}^{l}$ | $0.450_{-0.016}^{+0.019}$
$m_{e}/m_{\mu}$ | $0.0048\pm 0.0002$ | $\sin^{2}\theta_{13}^{l}$ | $0.02246\pm 0.00062$
$m_{\mu}/m_{\tau}$ | $0.059\pm 0.002$ | $\delta_{CP}^{l}/^{\circ}$ | $230_{-25}^{+36}$
$m_{b}/m_{\tau}$ | $0.73\pm 0.03$ | $r\equiv\Delta m_{21}^{2}/\Delta m_{31}^{2}$ | $0.02956\pm 0.00084$
$Y_{B}$ | $(8.72\pm 0.27)\times 10^{-11}$ | $\Delta m_{21}^{2}/\text{eV}^{2}$ | $7.42^{+0.21}_{-0.20}\times 10^{-5}$
Table 2:  The best fit values $\mu_{i}$ and $1\sigma$ uncertainties of the
quark and lepton parameters when evolved to the GUT scale as calculated in
[109], with the SUSY breaking scale $M_{\text{SUSY}}=500$ GeV and
$\tan\beta=10$, where the error widths represent $1\sigma$ intervals. The
values of lepton mixing angles, leptonic Dirac CP violation phases
$\delta^{l}_{CP}$ and the neutrino mass squared difference are taken from
NuFIT 5.1 [110] for normal ordering neutrino masses with Super-Kamiokande
data.

In this section, we shall perform a detailed numerical analysis to explore
whether the renormalizable $SO(10)\times A_{4}$ modular models of section 2.2
can explain the experimentally measured values of masses and mixing parameters
of both quarks and leptons as well as the matter-antimatter asymmetry of the
Universe. We perform a $\chi^{2}$ analysis to optimize the values of the free
parameters. The $\chi^{2}$ function is defined as

$\chi^{2}=\sum_{i}\left(\frac{P_{i}(x)-\mu_{i}}{\sigma_{i}}\right)^{2}\,.$
(55)

Here $\mu_{i}$ and $\sigma_{i}$ denote the experimental central values and the
$1\sigma$ uncertainties of the observables respectively, and their values are
obtained by evolving their low energy values to the GUT scale with the
renormalization group equations, as shown in table 2. For the neutrino
oscillation data, we assume normal ordering neutrino mass spectrum so that the
renormalization group induced corrections to the neutrino masses and mixing
parameters can be negligible for small $\tan\beta$, and we use the most recent
results from the NuFit collaboration [110].

The mechanism of modulus stabilization is still an open question, consequently
we treat the complex modulus $\tau$ as a free parameter in the fundamental
region
$\mathcal{D}=\left\\{\tau\in\mathbb{C}\Big{|}\text{Im}\tau>0,|\text{Re}\tau|\leq\frac{1}{2},|\tau|\geq
1\right\\}$, which can represent all possible values of $\tau$ in the upper
half complex plane up to a modular transformation. In the numerical analysis,
the magnitudes of all coupling constants are limited in the region
$[0,10^{4}]$ and their phases are freely varied in the range $[0,2\pi]$. The
overall scales $\alpha_{1}v_{u}$, $\alpha_{1}r_{1}v_{u}$ and
$\alpha^{2}_{1}v^{2}_{u}/v_{R}$ are fixed by the experimentally measured
values of top quark mass, bottom quark mass and solar neutrino mass squared
difference $\Delta m^{2}_{21}$. For each set of given values of the input
parameters, the quark and lepton mass matrices in Eqs. (2, 3) can be
straightforwardly diagonalized and subsequently mixing angles and CP violation
phases can be extracted in the usual way. We numerically minimize the
$\chi^{2}$ function by using the minimization algorithms TMinuit [111]
developed by CERN to determine the optimum values of the input parameters. In
the present work, we will focus the scenario of type I seesaw dominance, and
the out of equilibrium decay of right-handed neutrinos is used to generate the
baryon asymmetry of the Universe.

For each model, we present the point in parameter space which minimizes the
$\chi^{2}$, as a result of a numerical minimization procedure. We shall
consider two typical initial conditions of heavy Majorana neutrinos: thermal
initial abundance and vanishing initial abundance. We show the numerical
results in table 3 and table 4 for vanishing and thermal initial conditions
respectively. It is notable that the value of $\tau$ is close to the residual
modular symmetry fixed point $\tau_{ST}\equiv-1/2+i\sqrt{3}/2$. Since both
models at the best points are in the strong washout regime, the prediction for
the baryon asymmetry is not so sensitive to the initial condition of right-
handed neutrinos. Thus the values of each input parameter have little
difference for the concerned two initial conditions. We also present the
predictions for the mass ratios of charged fermions $m_{e}/m_{\mu}$,
$m_{\mu}/m_{\tau}$, $m_{u}/m_{c}$, $m_{c}/m_{t}$, $m_{d}/m_{s}$, $m_{s}/m_{b}$
and $m_{b}/m_{\tau}$, the mixing angles $\theta^{l}_{12}$, $\theta^{l}_{13}$,
$\theta^{l}_{23}$, $\theta^{q}_{12}$, $\theta^{q}_{13}$ and $\theta^{q}_{23}$
as well as the quark CP violation phase $\delta^{q}_{CP}$, which are
compatible with experimental data. Moreover, from the fitted values of the
parameters, we can derive the predictions for the still unmeasured observables
including the leptonic Dirac CP violation phase $\delta^{l}_{CP}$ and Majorana
CP phases $\alpha_{21}$ and $\alpha_{31}$, the light neutrino masses
$m_{1,2,3}$, the effective Majorana neutrino mass $m_{\beta\beta}$ for
neutrinoless double decay and the right-handed neutrino masses $M_{1,2,3}$ at
the best fit point. The right-handed neutrino masses have a hierarchical
pattern $M_{3}\gg M_{2}\gg M_{1}$ which are in the range of $10^{10}-10^{13}$
GeV. The effective Majorana mass $m_{\beta\beta}$ is determined to be $0.532$
(1.059) meV and 0.622 (0.550) meV in the models BM1 and BM2 respectively for
vanishing (thermal) initial condition, they are much below the current most
stringent bound $m_{\beta\beta}<(61-165)$ meV given by the KamLAND-Zen
Collaboration [112]. Future tonne-scale neutrinless double decay experiments
can reduce the sensitivity on $m_{\beta\beta}$ to few meV such as
$m_{\beta\beta}<(4.7-20.3)$ meV from nEXO [113] and $m_{\beta\beta}<(9-21)$
meV from LEGEND-1000 [114]. Consequently our predictions for $m_{\beta\beta}$
are even out of reach of tonne-scale detectors. As shown in tables 3 and 4, we
have split the $\chi^{2}$ function into four different contributions
$\chi^{2}_{total}=\chi^{2}_{l}+\chi^{2}_{q}+\chi^{2}_{b\tau}+\chi^{2}_{Y_{B}}$,
where $\chi^{2}_{l}$, $\chi^{2}_{q}$, $\chi^{2}_{b\tau}$ and
$\chi^{2}_{Y_{B}}$ stand for the contributions induced by the deviations of
the lepton sector observables, the quark sector observables, the mass ratio
$m_{b}/m_{\tau}$ and the baryon asymmetry $Y_{B}$ from their central values
respectively. We see that the largest deviation arises from the quark sector
in both models, and $\chi^{2}_{total}$ is dominated by $\chi^{2}_{q}$ although
the predictions for the quark masses and mixing parameters are in the
experimentally allowed region.

Regarding the baryon asymmetry, the concrete form of the neutrino Yukawa
coupling matrix and right-handed neutrino masses are fixed at the best fit
point, we numerically solve the density matrix equation of Eq. (24) with the
help of ULYSSES to obtain the frozen value of $B-L$ asymmetry. It is
remarkable that the numerical results agree well with the analytical
approximations. We see that the observed matter-antimatter asymmetry can be
accommodated. In general, the decays of three right-handed neutrinos
$N_{1,2,3}$ all contribute to the final baryon asymmetry, and asymmetries
generated by $N_{2,3}$ are partially washed out by the $N_{1}$ process. In the
model BM1 for both initial conditions and the BM2 model with vanishing initial
abundance, the leptogensis is dominated by $N_{1}$ decay. The contributions of
$N_{2}$ and $N_{3}$ are as important as the that of $N_{1}$ in the BM2 for
thermal initial abundance. Moreover, we plot the evolution of $Y_{B}$ with the
temperature $T$ for the two benchmark models in figure 1, and the
corresponding numerical results of Boltzmann equations are displayed for
comparison.

Furthermore, we comprehensively scan the parameter space around the best fit
points of table 3 and table 4, and we require all the fermion masses and
mixing parameters are in the experimentally preferred $3\sigma$ regions. The
experimentally allowed values of the complex modulus $\tau$ and the
correlations between the masses and mixing parameters are shown in figures 2,
4, 6 and 8 for the benchmark models with vanishing and thermal initial
conditions. The corresponding predictions for the baryon asymmetry $Y_{B}$
with respect to $\sin^{2}\theta^{l}_{23}$, $\delta^{l}_{CP}$, $\alpha_{21}$,
$\alpha_{31}$ are displayed in figures 3, 5, 7 and 9. We would like to mention
that the contribution of $Y_{B}$ to $\chi^{2}$ is not included in these
figures. We see that imposing successful leptogenesis yields specific
predictions for the fermion masses and mixing parameters.

Figure 1: The evolution of the baryon asymmetry $Y_{B}$ with temperature for the two benchmark models, the free parameters take the best fit values in table 3 and table 4. The red line and blue line are the numerical results obtained by solving the density matrix equation and Boltzmann equation of section 3.2 respectively. The top-left (right) panel is for the $\texttt{BM1}\,(\texttt{BM2})$ model with vanishing initial abundance. The bottom-left (right) panel for the BM1 (BM2) model with thermal initial abundance. parameters | BM1 | BM2
---|---|---
$\tau$ | $-0.4982+0.8862i$ | $-0.4796+0.8958i$
$r_{2}$ | $1.293~{}e^{i1.790\pi}$ | $1.572~{}e^{i1.219\pi}$
$r_{3}$ | $0.343~{}e^{i0.395\pi}$ | $1.374~{}e^{i1.921\pi}$
$c_{e}$ | $9.891~{}e^{i1.447\pi}$ | $1.591~{}e^{i0.731\pi}$
$c_{\nu}$ | $7.485~{}e^{i0.108\pi}$ | $0.952~{}e^{i1.644\pi}$
$\alpha_{2}/\alpha_{1}$ | $0.310~{}e^{i1.839\pi}$ | $0.600~{}e^{i1.025\pi}$
$\alpha_{3}/\alpha_{1}$ | $0.307~{}e^{i1.838\pi}$ | $0.597~{}e^{i1.034\pi}$
$\beta_{1}/\alpha_{1}$ | $0.093~{}e^{i0.862\pi}$ | $3.481~{}e^{i0.139\pi}$
$\beta_{2}/\alpha_{1}$ | — | $0.025~{}e^{i1.606\pi}$
$\gamma_{1}/\alpha_{1}$ | $0.295~{}e^{i1.400\pi}$ | $0.243~{}e^{i0.664\pi}$
$\gamma_{2}/\alpha_{1}$ | $0.297~{}e^{i0.094\pi}$ | $0.112~{}e^{i1.550\pi}$
$\gamma_{3}/\alpha_{1}$ | $0.300~{}e^{i0.095\pi}$ | $0.114~{}e^{i1.555\pi}$
$\alpha_{1}^{2}v_{u}^{2}/v_{R}({\rm meV})$ | $15.272$ | $7.053$
$\alpha_{1}v_{u}/{\rm GeV}$ | $20.673$ | $18.157$
$\alpha_{1}r_{1}v_{d}$/GeV | $0.272$ | $0.240$
$\sin^{2}\theta_{12}^{l}$ | $0.308$ | $0.294$
$\sin^{2}\theta_{13}^{l}$ | $0.02178$ | $0.02248$
$\sin^{2}\theta_{23}^{l}$ | $0.465$ | $0.455$
$\delta_{CP}^{l}/^{\circ}$ | $267.074$ | $333.240$
$\alpha_{21}/^{\circ}$ | $212.233$ | $222.547$
$\alpha_{31}/^{\circ}$ | $237.998$ | $31.091$
$m_{e}/m_{\mu}$ | $0.00488$ | $0.00484$
$m_{\mu}/m_{\tau}$ | $0.0584$ | $0.0586$
$m_{1}/{\rm meV}$ | $3.065$ | $2.645$
$m_{2}/{\rm meV}$ | $9.143$ | $9.011$
$m_{3}/{\rm meV}$ | $50.597$ | $50.375$
$m_{\beta\beta}/{\rm meV}$ | $0.532$ | $0.622$
$M_{1}/{\rm GeV}$ | $6.546\times 10^{10}$ | $7.447\times 10^{10}$
$M_{2}/{\rm GeV}$ | $1.716\times 10^{12}$ | $3.307\times 10^{12}$
$M_{3}/{\rm GeV}$ | $4.853\times 10^{13}$ | $3.012\times 10^{13}$
$v_{R}/{\rm GeV}$ | $2.798\times 10^{13}$ | $4.674\times 10^{13}$
$\theta_{12}^{q}$ | $0.230$ | $0.229$
$\theta_{13}^{q}$ | $0.00411$ | $0.00332$
$\theta_{23}^{q}$ | $0.0432$ | $0.0421$
$\delta_{CP}^{q}/^{\circ}$ | $83.606$ | $76.967$
$m_{u}/m_{c}$ | $0.00329$ | $0.00285$
$m_{c}/m_{t}$ | $0.00286$ | $0.00287$
$m_{d}/m_{s}$ | $0.0353$ | $0.0491$
$m_{s}/m_{b}$ | $0.0169$ | $0.0206$
$m_{b}/m_{\tau}$ | $0.715$ | $0.803$
$Y_{B}^{\rm num}$ | $8.541\times 10^{-11}$ | $8.516\times 10^{-11}$
$Y_{B}^{\rm ap}$ | $6.806\times 10^{-11}$ | $6.355\times 10^{-11}$
$\chi^{2}_{l}$ | $3.523$ | $9.202$
$\chi^{2}_{q}$ | $33.912$ | $16.555$
$\chi^{2}_{b\tau}$ | $0.264$ | $5.958$
$\chi^{2}_{Y_{B}}$ | $0.441$ | $0.573$
$\chi^{2}_{total}$ | $38.140$ | $31.715$
Table 3: The best fit values of the free parameters and the corresponding predictions for the masses and mixing parameters of leptons and quarks as well as the baryon asymmetry $Y_{B}$ in the two typical models BM1 and BM2. We assume vanishing initial abundance of the right-handed neutrinos, $Y_{B}^{\rm num}$ is the numerical result from density matrix equation, and $Y_{B}^{\rm ap}$ denotes the result of analytical approximation in section 3.2.  parameters | BM1 | BM2
---|---|---
$\tau$ | $-0.4949+0.8861i$ | $-0.4997+0.8975i$
$r_{2}$ | $1.414~{}e^{i1.811\pi}$ | $2.441~{}e^{i1.377\pi}$
$r_{3}$ | $0.315~{}e^{i0.268\pi}$ | $1.910~{}e^{i1.903\pi}$
$c_{e}$ | $10.425~{}e^{i1.365\pi}$ | $1.675~{}e^{i0.613\pi}$
$c_{\nu}$ | $7.276~{}e^{i0.149\pi}$ | $0.890~{}e^{i1.214\pi}$
$\alpha_{2}/\alpha_{1}$ | $0.286~{}e^{i1.912\pi}$ | $0.626~{}e^{i1.011\pi}$
$\alpha_{3}/\alpha_{1}$ | $0.283~{}e^{i1.912\pi}$ | $0.631~{}e^{i1.016\pi}$
$\beta_{1}/\alpha_{1}$ | $0.104~{}e^{i0.774\pi}$ | $3.590~{}e^{i0.104\pi}$
$\beta_{2}/\alpha_{1}$ | — | $0.016~{}e^{i1.247\pi}$
$\gamma_{1}/\alpha_{1}$ | $0.197~{}e^{i1.405\pi}$ | $0.333~{}e^{i0.639\pi}$
$\gamma_{2}/\alpha_{1}$ | $0.311~{}e^{i0.102\pi}$ | $0.140~{}e^{i1.600\pi}$
$\gamma_{3}/\alpha_{1}$ | $0.313~{}e^{i0.102\pi}$ | $0.141~{}e^{i1.597\pi}$
$\alpha_{1}^{2}v_{u}^{2}/v_{R}({\rm meV})$ | $12.574$ | $8.865$
$\alpha_{1}v_{u}/{\rm GeV}$ | $19.718$ | $13.894$
$\alpha_{1}r_{1}v_{d}$/GeV | $0.266$ | $0.242$
$\sin^{2}\theta_{12}^{l}$ | $0.321$ | $0.305$
$\sin^{2}\theta_{13}^{l}$ | $0.02258$ | $0.02235$
$\sin^{2}\theta_{23}^{l}$ | $0.438$ | $0.433$
$\delta_{CP}^{l}/^{\circ}$ | $329.478$ | $229.431$
$\alpha_{21}/^{\circ}$ | $175.644$ | $184.054$
$\alpha_{31}/^{\circ}$ | $240.818$ | $287.703$
$m_{e}/m_{\mu}$ | $0.00481$ | $0.00502$
$m_{\mu}/m_{\tau}$ | $0.0585$ | $0.0599$
$m_{1}/{\rm meV}$ | $2.156$ | $7.053$
$m_{2}/{\rm meV}$ | $8.880$ | $11.133$
$m_{3}/{\rm meV}$ | $49.719$ | $50.466$
$m_{\beta\beta}/{\rm meV}$ | $1.059$ | $0.550$
$M_{1}/{\rm GeV}$ | $7.083\times 10^{10}$ | $2.798\times 10^{10}$
$M_{2}/{\rm GeV}$ | $1.682\times 10^{12}$ | $1.793\times 10^{12}$
$M_{3}/{\rm GeV}$ | $5.608\times 10^{13}$ | $1.739\times 10^{13}$
$v_{R}/{\rm GeV}$ | $3.092\times 10^{13}$ | $2.177\times 10^{13}$
$\theta_{12}^{q}$ | $0.229$ | $0.230$
$\theta_{13}^{q}$ | $0.00359$ | $0.00416$
$\theta_{23}^{q}$ | $0.0421$ | $0.0440$
$\delta_{CP}^{q}/^{\circ}$ | $55.549$ | $86.564$
$m_{u}/m_{c}$ | $0.00278$ | $0.00285$
$m_{c}/m_{t}$ | $0.00274$ | $0.00260$
$m_{d}/m_{s}$ | $0.0302$ | $0.0446$
$m_{s}/m_{b}$ | $0.0173$ | $0.0156$
$m_{b}/m_{\tau}$ | $0.708$ | $0.702$
$Y_{B}^{\rm num}$ | $8.526\times 10^{-11}$ | $8.458\times 10^{-11}$
$Y_{B}^{\rm ap}$ | $7.800\times 10^{-11}$ | $5.215\times 10^{-11}$
$\chi^{2}_{l}$ | $10.694$ | $2.539$
$\chi^{2}_{q}$ | $16.122$ | $35.517$
$\chi^{2}_{b\tau}$ | $0.554$ | $0.847$
$\chi^{2}_{Y_{B}}$ | $0.518$ | $0.940$
$\chi^{2}_{total}$ | $27.888$ | $38.843$
Table 4:  Similar to table 3, but for thermal initial abundance of the right-
handed neutrinos.

##  5 Conclusion

In this paper we have studied the prediction for leptogenesis in two
renormalizable supersymmetric $SO(10)\times A_{4}$ modular models in which the
neutrino mass is dominantly generated by the type I seesaw mechanism. The
evolution of the lepton asymmetries are described in terms of the three-
flavored density matrix equations for three heavy Majorana neutrinos, where
both vanishing initial condition and thermal initial condition of the right-
handed neutrinos are considered. We also presented an analytical approximation
based on the Boltzmann equations. We found regions of parameter space
compatible with the measured fermion masses and mixing parameters as well as
the baryon asymmetry of the Universe. The predictions for the light neutrino
masses, the effective mass in neutrinoless doble beta decay and the leptonic
CP violation phases were discussed.

The right-handed neutrinos naturally appear in $SO(10)$ GUT, all the chiral
fermions of one generation plus one additional right-handed neutrino are
unified into a single 16 dimensional spinor representation of $SO(10)$. Thus
$SO(10)$ GUT provides a natural explanation for the tiny neutrino mass through
the type I seesaw mechanism. From the view of cosmology, it is well known that
the CP violating decay of the right-handed neutrinos can generate the matter-
antimatter asymmetry of the Universe, this is the so-called leptogenesis.
Moreover, the mass and Yukawa coupling of the right-handed neutrinos are
closely related to those of quarks and charged leptons, and their concrete
values can be determined by requirement that the measured masses, mixing
angles and CP violation phases of both quarks and leptons should be
accommodated. It is attractive that the fermion masses and mixing together
with baryon asymmetry of the Universe can be explained in a given $SO(10)$
model.

Modular symmetry is an important progress to address the flavor puzzle of the
SM, the vacuum alignment problem in conventional flavor symmetry models is
greatly simplified, and the complex modulus $\tau$ could be the unique source
of flavor symmetry breaking. As a consequence, the modular symmetry models
could be quite predictive. In the present work, we have analyzed the
prediction for leptogenesis in two benchmark renormalizable $SO(10)$ models
with $A_{4}$ modular symmetry where the neutrino masses dominantly arise from
type I seesaw mechanism. The three generations of matter fields are assigned
to triplet of $A_{4}$ modular symmetry, and the Higgs fields $H$, $\Sigma$,
$\overline{\Delta}$ in the 10, 120 and 16 dimensional representations of
$SO(10)$ are invariant under $A_{4}$. As a consequence, the structure of the
$SO(10)\times A_{4}$ modular models are fully specified by the modular weights
of the fermion fields and Higgs fields, as shown in table 1. The modular
weights are
$(2k_{F}+k_{10}\,,2k_{F}+k_{120}\,,2k_{F}+k_{\overline{126}})=(4,2,4)$ and
$(2k_{F}+k_{10}\,,2k_{F}+k_{120}\,,2k_{F}+k_{\overline{126}})=(4,6,4)$
respectively in the two concerned benchmark models.

We used the density matrix equations to describe the kinetic evolution of the
lepton asymmetry. In comparison with Boltzmann equations, the density matrix
equations give a quantitative description of lepton flavor effects in
leptogenesis, and allows to precisely calaulate the final lepton asymmetry for
arbitrary right-handed neutrino mass spectrum. The right-handed neutrino
masses are hierarchical with $M_{3}\gg M_{2}\gtrsim 10^{12}\,{\rm GeV}$ and
$10^{9}\,{\rm GeV}\ll M_{1}\ll 10^{12}\,{\rm GeV}$ in our benchmark models.
Hence the decays of $N_{3}$ and $N_{2}$ are in the single-flavored regime
while $N_{1}$ decay is in the two-flavored regime. In order to understand the
physical process of leptogenesis, we assume the three heavy Majorana neutrinos
decay in sequence. The lepton asymmetries generated from the decay of each
right-handed neutrino and the washout effects are governed by the Boltzmann
equation, the analytical approximate solutions are given. The approximation
agrees with the numerical results of density matrix equations well if the
interplay of the different right-handed neutrinos and the interference of
lepton flavor are insignificant.

The superpotential of the Yukawa is strongly constrained by modular symmetry,
it involves 7 couplings $\alpha_{1,2,3}$, $\beta_{1}$ and $\gamma_{1,2,3}$ in
the BM1 model, and an additional parameter $\beta_{2}$ in the BM2 model.
Moreover, the quark and lepton mass matrices depend on 5 parameters
$r_{1,2,3}$, $c_{e,\nu}$ which describe the light Higgs combinations. Notice
that the Higgs sector of the renormalizable $SO(10)$ models with all 10, 120
and 126 dimensional Higgs multiplets are complicated and it reduces the
predictive power of $SO(10)$ models. The 126-plet Higgs $\overline{\Delta}$
contains a right-handed scalar triplet $\Delta_{R}$ whose vacuum expectation
value $\langle\Delta_{R}\rangle=v_{R}$ gives mass to the right-handed
neutrinos. The parameters $\alpha_{1}$, $r_{1}$ and $v_{R}$ can be taken real
without loss of generality. Including the complex modulus $\tau$, the BM1 and
BM2 models depend on 25 and 27 real parameters respectively. We fit all these
free parameters using the 20 observables in table 2. In particular, we take
into account the matter-antimatter asymmetry of the Universe through
leptogenesis besides the measured masses and mixing parameters of quarks and
leptons. As shown in table 3 and table 4, the experimental data can be well
accommodated and it is remarkable that all the coupling constants are order
one. Furthermore, we can obtain predictions for unmeasured quantities such as
the Majorana CP phases $\alpha_{21}$, $\alpha_{31}$, the effective mass
$m_{\beta\beta}$ of neutrinoless double decay and the right-handed neutrino
masses $M_{1,2,3}$. $m_{\beta\beta}$ is predicted in the range 0.5-1.1 meV, it
is far below the sensitivity of current and future experiments. Hence our
models would be ruled out if a positive signal of neutrinoless double beta
decay is reported in future.

## Acknowledgements

GJD, JNL and BYQ are supported by the National Natural Science Foundation of
China under Grant Nos. 11975224, 11835013 and the Key Research Program of the
Chinese Academy of Sciences under Grant NO. XDPB15. SFK acknowledges the STFC
Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research
and Innovation programme under Marie Skłodowska-Curie grant agreement HIDDeN
European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN).

## Appendix

##  Appendix A  Two examples of analytical approximation failure

The formalism of density matrix allows for a more general description of
leptogenesis than the semi-classical Boltzmann equations. The off-diagonal
elements of the density matrix describe the degree of coherence between the
flavour states, so that the decoherence effects are incorporated into the
equations. Moreover, the density matrix equations can describe the dynamical
process of the transition between flavour-regimes, and allows to calculate the
lepton asymmetry in the intermediate regimes where the single-flavored and
two-flavored treatments are inadequate. Hence it is found that the Boltzmann
equations fail to describe correctly the generation of baryon asymmetry in
many cases [101, 102, 103]. Although the analytical approximation agree well
with the numerical results of the density matrix equation in section 4, we
note that this not always true. In particular, when the effects of decoherence
cannot be neglected or there are non-trivial interplay between the decays and
the washout processes of the three right-handed neutrinos. We give two such
benchmark points in table 5 at which our analytical estimates fail to give the
correct value of baryon asymmetry, and even the sign of $Y_{B}$ is wrong.

For the model BM1 with the values of the free parameters given in table 5, the
decay parameters of $N_{2}$ and $N_{3}$ are determined to be $K_{2}=90$ and
$K_{3}=98$ respectively. Therefore the freeze-out temperature of $N_{3}$ is
approximately $T_{B_{3}}\simeq M_{3}/9\simeq 2M_{2}$, the effects of $N_{2}$
are not negligible before the freeze-out of $N_{3}$ inverse decay, as can be
seen from the left-top panel of figure 10. In addition, the asymmetries
generated by $N_{2}$ and $N_{3}$ are only partially washed out by the $N_{1}$
decay and inverse decay processes and in fact they dominate the final
asymmetry. Moreover, the mass of $N_{1}$ is $2.308\times 10^{11}\,{\rm GeV}$,
consequently it is in the transition region between the single- and two-
flavored leptogenesis. This might contribute to the failure of the analytical
estimates, since the transitions between the different flavour regimes are not
taken into account in the Boltzmann equations and the baryon asymmetry $\eta$
could change its sign during the transitions [103]. Analogously, for another
model BM2, there is also non-trivial interplay between the decay and the
washout processes of the heavy right-handed neutrinos. The contribution of
$N_{1}$ is found to be subdominant, and the $N_{1}$ mass is $8.367\times
10^{9}$ GeV which is close to the transition scale between the two-flavor and
three-flavor regimes. Furthermore, from figure 10 we see that the decoherence
of flavor states is not infinitely fast as assumed in the Boltzmann equations.
This can also lead to deviations of analytical approximation from the
numerical results of density matrix.

parameters | BM1 | BM2
---|---|---
$\tau$ | $-0.4863+0.8863i$ | $-0.5+0.9031i$
$r_{2}$ | $1.305~{}e^{i1.880\pi}$ | $2.067~{}e^{i1.488\pi}$
$r_{3}$ | $0.267~{}e^{i0.402\pi}$ | $1.731~{}e^{i1.902\pi}$
$c_{e}$ | $8.188~{}e^{i1.401\pi}$ | $1.247~{}e^{i0.785\pi}$
$c_{\nu}$ | $7.473~{}e^{i0.139\pi}$ | $0.890~{}e^{i0.999\pi}$
$\alpha_{2}/\alpha_{1}$ | $0.286~{}e^{i1.852\pi}$ | $0.555~{}e^{i0.948\pi}$
$\alpha_{3}/\alpha_{1}$ | $0.282~{}e^{i1.862\pi}$ | $0.554~{}e^{i0.948\pi}$
$\beta_{1}/\alpha_{1}$ | $0.082~{}e^{i0.801\pi}$ | $3.321~{}e^{i0.085\pi}$
$\beta_{2}/\alpha_{1}$ | — | $0.014~{}e^{i1.995\pi}$
$\gamma_{1}/\alpha_{1}$ | $0.299~{}e^{i1.180\pi}$ | $0.340~{}e^{i0.511\pi}$
$\gamma_{2}/\alpha_{1}$ | $0.247~{}e^{i0.129\pi}$ | $0.154~{}e^{i1.539\pi}$
$\gamma_{3}/\alpha_{1}$ | $0.247~{}e^{i0.120\pi}$ | $0.149~{}e^{i1.546\pi}$
$\alpha_{1}^{2}v_{u}^{2}/v_{R}({\rm meV})$ | $8.242$ | $12.535$
$\alpha_{1}v_{u}/{\rm GeV}$ | $24.249$ | $15.606$
$\alpha_{1}r_{1}v_{d}$/GeV | $0.313$ | $0.272$
$\sin^{2}\theta_{12}^{l}$ | $0.313$ | $0.309$
$\sin^{2}\theta_{13}^{l}$ | $0.02241$ | $0.02247$
$\sin^{2}\theta_{23}^{l}$ | $0.443$ | $0.430$
$\delta_{CP}^{l}/^{\circ}$ | $280.211$ | $318.757$
$\alpha_{21}/^{\circ}$ | $199.009$ | $163.502$
$\alpha_{31}/^{\circ}$ | $237.526$ | $205.183$
$m_{e}/m_{\mu}$ | $0.00467$ | $0.00471$
$m_{\mu}/m_{\tau}$ | $0.0592$ | $0.0571$
$m_{1}/{\rm meV}$ | $2.009$ | $3.206$
$m_{2}/{\rm meV}$ | $8.845$ | $9.191$
$m_{3}/{\rm meV}$ | $50.036$ | $50.132$
$m_{\beta\beta}/{\rm meV}$ | $0.374$ | $0.325$
$M_{1}/{\rm GeV}$ | $2.308\times 10^{11}$ | $8.367\times 10^{9}$
$M_{2}/{\rm GeV}$ | $5.582\times 10^{12}$ | $1.934\times 10^{12}$
$M_{3}/{\rm GeV}$ | $1.023\times 10^{14}$ | $1.641\times 10^{13}$
$v_{R}/{\rm GeV}$ | $7.134\times 10^{13}$ | $1.943\times 10^{13}$
$\theta_{12}^{q}$ | $0.229$ | $0.229$
$\theta_{13}^{q}$ | $0.00535$ | $0.00393$
$\theta_{23}^{q}$ | $0.0414$ | $0.0366$
$\delta_{CP}^{q}/^{\circ}$ | $86.767$ | $65.647$
$m_{u}/m_{c}$ | $0.00269$ | $0.00250$
$m_{c}/m_{t}$ | $0.00268$ | $0.00292$
$m_{d}/m_{s}$ | $0.0306$ | $0.0601$
$m_{s}/m_{b}$ | $0.0171$ | $0.0177$
$m_{b}/m_{\tau}$ | $0.767$ | $0.672$
$Y^{\rm num}_{B}$ | $8.681\times 10^{-11}$ | $8.454\times 10^{-11}$
$Y_{B}^{\rm ap}$ | $-6.378\times 10^{-11}$ | $-1.179\times 10^{-12}$
$\chi^{2}_{l}$ | $3.151$ | $8.969$
$\chi^{2}_{q}$ | $41.647$ | $16.306$
$\chi^{2}_{b\tau}$ | $1.540$ | $3.775$
$\chi^{2}_{Y_{B}}$ | $0.021$ | $0.971$
$\chi^{2}_{total}$ | $46.359$ | $30.021$
Table 5:  Benchmark points at which the analytical approximation of section
3.2 fails for the concerned models. The vanishing initial abundance of right-
handed neutrinos is assumed in BM1 model while thermal initial abundance is
assumed in BM2.

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Figure 2: The values of the complex modulus $\tau$ compatible with
experimental data and the correlations between the neutrino mixing angles, CP
violation phases, quark mass ratios and mixing parameters for the model BM1
with vanishing initial condition. The vertical and horizontal dashed lines are
the $3\sigma$ bounds taken from [110]. Note that the contribution of $Y_{B}$
to $\chi^{2}$ is not included here.  Figure 3: The numerical results for the
correlation between the baryon asymmetry $Y_{B}$ and
$\sin^{2}\theta^{l}_{23}$, $\delta^{l}_{CP}$, $\alpha_{21}$, $\alpha_{31}$ in
the BM1 model with vanishing initial condition. The horizontal band denotes
the experimentally allowed region of $Y_{B}$. The star “$\bigstar$” stands for
the best fitting point in table 3. Note that the contribution of $Y_{B}$ to
$\chi^{2}$ is not included here.  Figure 4: The values of the complex modulus
$\tau$ compatible with experimental data and the correlations between the
neutrino mixing angles, CP violation phases, quark mass ratios and mixing
parameters for the model BM2 with vanishing initial condition. The vertical
and horizontal dashed lines are the $3\sigma$ bounds taken from [110]. Note
that the contribution of $Y_{B}$ to $\chi^{2}$ is not included here.  Figure
5: The numerical results for the correlation between the baryon asymmetry
$Y_{B}$ and $\sin^{2}\theta^{l}_{23}$, $\delta^{l}_{CP}$, $\alpha_{21}$,
$\alpha_{31}$ in the BM2 model with vanishing initial condition. The
horizontal band denotes the experimentally allowed region of $Y_{B}$. The star
“$\bigstar$” stands for the best fitting point in table 3. Note that the
contribution of $Y_{B}$ to $\chi^{2}$ is not included here.  Figure 6: The
values of the complex modulus $\tau$ compatible with experimental data and the
correlations between the neutrino mixing angles, CP violation phases, quark
mass ratios and mixing parameters for the model BM1 with thermal initial
condition. The vertical and horizontal dashed lines are the $3\sigma$ bounds
taken from [110]. Note that the contribution of $Y_{B}$ to $\chi^{2}$ is not
included here.  Figure 7: The numerical results for the correlation between
the baryon asymmetry $Y_{B}$ and $\sin^{2}\theta^{l}_{23}$, $\delta^{l}_{CP}$,
$\alpha_{21}$, $\alpha_{31}$ in the BM1 model with thermal initial condition.
The horizontal band denotes the experimentally allowed region of $Y_{B}$. The
star “$\bigstar$” stands for the best fitting point in table 4. Note that the
contribution of $Y_{B}$ to $\chi^{2}$ is not included here.  Figure 8: The
values of the complex modulus $\tau$ compatible with experimental data and the
correlations between the neutrino mixing angles, CP violation phases, quark
mass ratios and mixing parameters for the model BM2 with thermal initial
condition. The vertical and horizontal dashed lines are the $3\sigma$ bounds
taken from [110]. Note that the contribution of $Y_{B}$ to $\chi^{2}$ is not
included here.  Figure 9: The numerical results for the correlation between
the baryon asymmetry $Y_{B}$ and $\sin^{2}\theta^{l}_{23}$, $\delta^{l}_{CP}$,
$\alpha_{21}$, $\alpha_{31}$ in the BM2 model with thermal initial condition.
The horizontal band denotes the experimentally allowed region of $Y_{B}$. The
star “$\bigstar$” stands for the best fitting point in table 4. Note that the
contribution of $Y_{B}$ to $\chi^{2}$ is not included here.

Figure 10: The evolution of the baryon asymmetry $Y_{B}$ with temperature for
the local minimum shown in table 5. The left and right panels are the BM1 with
vanishing initial abundance and the BM2 with thermal initial abundance
respectively. The evolution of the $B-L$ asymmetry number densities
$N^{B-L}_{\alpha\beta}$, $\alpha,\beta=e,\mu,\tau$ displayed in the lower
panel is described by the density matrix equations in Eq. (24).