# Triplet scalar flavored leptogenesis with spontaneous CP violation

Sreerupa Chongdar  Sasmita Mishra 11footnotetext: Corresponding author.

###### Abstract

The inclusion of two triplet scalars in the Standard Model (SM) enables to
accommodate neutrino mass generation as well as baryogenesis through
leptogenesis. One of the essential ingredients of leptogenesis is the
violation of charge conjugation and parity (CP) symmetry in lepton number
violating decays of the triplet scalars. We work on the promising sector of
spontaneous CP violation (SCPV) which is manifested by the involvement of one
scalar singlet and two scalar fields, added to the SM. The predictive aspect
of the model is accomplished by imposing $A_{4}\times Z_{4}$ symmetry which
results in the traditional tribimaximal mixing pattern. With updated data on
neutrino oscillation, we study the parameter space of the model. The phase of
the complex vacuum expectation value (VEV) of the singlet scalar acts as the
common source of CP violation in both low and high energy sectors. Due to the
flavor symmetry of the model, required baryon asymmetry cannot be accomplished
via unflavored leptogenesis. In the temperature regime,
$\left[10^{9},10^{12}\right]$ GeV when flavor effects become important in the
study of leptogenesis, it is shown that baryogenesis is achievable. The rich
flavor interplay is explored through the study of the density matrix
equations. We also study the interplay of hierarchical branching ratios of the
decay of the triplet scalars and SCPV phase to accommodate the required CP
asymmetry to account for the final baryon asymmetry in the observational
range. Considering all possible mass hierarchies among the triplet scalars,
the flavor structure of the triplet Yukawa couplings results in different
scales of leptogenesis.

##  1 Introduction

The observations from Big Bang nucleosynthesis (BBN) and cosmic microwave
background (CMB) quantify the baryon asymmetry of the Universe (BAU) [46, 35,
32, 23, 5, 6], $\eta_{B}=\frac{(n_{B}-n_{\overline{B}})}{n_{\gamma}}$ as
$4.7\times 10^{-10}\leq\eta_{B}\leq 6.5\times 10^{-10}$, where $n_{B}$,
$n_{\overline{B}}$, and $n_{\gamma}=\frac{2\zeta(3)}{\pi^{2}}T^{3}$ denote the
number densities of baryons, antibaryons and gamma photons, respectively.
Baryogenesis through leptogenesis [25] has become a popular theory to explain
the matter-antimatter asymmetry of the Universe. Concurrently, thermal
leptogenesis could be an important consequence of neutrino mass generation via
the seesaw mechanism, since the heavy beyond Standard Model (BSM) particles
introduced through the seesaw mechanism could be promising candidates for
generating lepton asymmetry. This is how a seesaw framework could explain tiny
non-zero neutrino mass and the consequent leptogenesis leading to adequate
baryon asymmetry. Apart from the widely studied type-I seesaw mechanism with
heavy right-handed neutrinos, type-II and type-III seesaw mechanism with
scalar and fermion triplets, respectively, arise naturally from Weinberg’s
dimension-5 operator of neutrino mass generation. In this paper, we study
baryogenesis through flavored leptogenesis from the decay of triplet Higgs
scalar with spontaneous CP violation along with light neutrino mass generated
via type-II seesaw mechanism.

Neutrino mass generation through type-II seesaw mechanism [40, 37, 41, 19] and
leptogenesis have widely been studied for their unique features coming from
the out-of-equilibrium decay of the heavy triplet scalars [30, 39, 28, 18].
Although the addition of just one triplet scalar to the SM can effectively
explain the generation of neutrino mass, for producing non-vanishing CP
asymmetry for leptogenesis only one triplet scalar is not enough. While a
minimal purely triplet leptogenesis can be implemented through two heavy
triplet scalars [31, 26, 12], there are more interesting schemes like
type-I+II seesaw leptogenesis with one additional right-handed neutrino along
with a triplet scalar [27]. In the case of triplet leptogenesis, the flavor
interplay comes to be very crucial [36] due to the unique decay channels of
the scalar triplets. While flavored leptogenesis has already been studied in
the framework of explicit CP violation [20], it is equally appealing to
accommodate flavored leptogenesis through SCPV [38, 34, 48, 17, 42]. In this
work, we explore the role of the SCPV via the complex VEV of the scalar sector
in flavored leptogenesis.

Baryogenesis through leptogenesis from the decay of triplet Higgs has some
unique attributes as compared to the other variants of the leptogenesis
mechanism. They can be realized from the requirement of CP violation and out-
of-equilibrium conditions. One important ingredient of leptogenesis, CP
violation, in the case of scalar triplet leptogenesis, can come from two
possible sources: (i) Explicit CP violation sourced from the complex Yukawa
couplings of the triplet Higgs with leptons and complex scalar coupling to the
SM Higgs, (ii) Spontaneous CP violation sourced from the complex VEV of a
scalar. The latter one is more economical in terms of parameter counting and
attractive from a theoretical point of view [17]. In this case, the CP
violation necessary for leptogenesis and potentially observable low-energy
leptonic CP violation come from a common source. Also, in models of
spontaneously broken CP symmetry, extended by triplet scalars, specific to our
case, CP symmetry is imposed at the Lagrangian level, making the Yukawa
couplings real. In this case, spontaneous CP violation can be realized through
the complex VEVs developed by the triplet scalars of the model. The light
neutrino mass matrix being determined by the complex VEV of the triplets and
real Yukawa matrices, CP violation can be established at the low scale (such
as neutrino oscillation experiments). The high energy CP violation required
for the leptogenesis mechanism cannot be accommodated, in this case. However,
the model can be extended by singlet scalars to account for both low and high
energy CP violation [17]. Some recent studies by adding scalar singlets to the
SM that can give rise to spontaneous CP violation and light neutrino mass have
been performed in Ref. [16] in the context of dark matter and in Ref. [15], in
the context of leptogenesis in vanilla type-I seesaw.

The coupling of the triplet scalars to the leptons and the SM Higgs scalar
leads to two different decay channels of the triplet scalar and the lepton
number is violated only if both types of decays co-exist. The interplay of the
strength of the respective branching ratios, $B^{L}$ and $B^{\phi}$ plays an
important role in successful baryogenesis through leptogenesis, where
$B^{L}(B^{\phi})$ corresponds to the branching ratios of the triplet scalar to
the leptons (SM Higgs scalar). The amount of CP asymmetry, $\epsilon$, and the
efficiency of leptogenesis depend on $B^{L}$ and $B^{\phi}$ in a different
manner. The efficiency is maximal when $B^{L}\gg B^{\phi}$ or $B^{\phi}\gg
B^{L}$, whereas this condition corresponds to suppressed CP violation (as
$\epsilon\propto\sqrt{B^{L}B^{\phi}}$). Solving a full set of Boltzmann
equations, in a model-independent way, it was shown in Ref. [31] that even for
extremely hierarchical branching ratios, it is possible to achieve the
required BAU. In this work, we focus on leptogenesis from the decay of triplet
scalars with hierarchical branching ratios and CP violation originating from
the VEV of a scalar singlet, embedded in the SM along with with two triplet
scalars. Also to make the model more predictive we impose $A_{4}$ symmetry at
the Lagrangian level. Since in our case, adequate baryon asymmetry is obtained
for the triplet scalar mass around $\sim 10^{10}$ GeV, flavor effects become
important in this regime. Also, considering hierarchical branching ratios of
triplet scalar decays, washout effects play an important role in generating
adequate lepton/baryon asymmetry. With the improved computation of the kinetic
evolution of flavored lepton asymmetries, we use Density Matrix Equations
(DME) for the evaluation of final lepton asymmetries. We also study, with all
possible mass hierarchies among the triplet scalars, the flavor structure of
the model predicts different scales of leptogenesis.

The paper is organized as follows. The model is introduced in section (2).
Section (3) is devoted to pointing out the important features of flavored CP
asymmetry and DME for the analysis. Finally, the results are presented in
section (4) with the summary. The conclusion is given in section (5). Three
appendices (A), (B), (C) are added containing some of the detailed
calculations.

Field |  $L$ |  $e_{R},\mu_{R},\tau_{R}$ |  $\Delta_{1}$ |  $\Delta_{2}$ |  $\phi$ |  $S$ |  $\Phi$ |  $\psi$
---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ |  $2$ |  $1$ |  $3$ |  $3$ |  $2$ |  $1$ |  $1$ |  $1$
$U(1)_{Y}$ |  $-1/2$ |  $-1$ |  $1$ |  $1$ |  $1/2$ |  $0$ |  $0$ |  $0$
$A_{4}$ |  $3$ |  $1,1^{\prime\prime},1^{\prime}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $3$ |  $3$
$Z_{4}$ |  $i$ |  $-i$ |  $1$ |  $-1$ |  $i$ |  $-1$ |  $i$ |  $1$
Table 1: Field representations under $SU(2)_{L}$, $U(1)_{Y},A_{4}$ and $Z_{4}$
symmetries.

##  2 Model Description

In this section, we study an extension of the SM, developed in Ref. [17] to
account for SCPV. With updated data from neutrino oscillation and cosmology we
explore the parameter space of the model. In this model, the SM is extended
with one complex scalar singlet $S$ and two Higgs triplet scalars
$\Delta_{a}(a=1,2)$. To generate a realistic lepton mixing pattern, two scalar
fields, $\Phi$ and $\psi$ are added by imposing $A_{4}$ flavor symmetry and
$Z_{4}$ discrete symmetry. The appropriate charge assignments to all the
scalar and fermion fields are summarized in table (1). The two triplet Higgs
fields can be represented in $SU(2)$ representation as,

$\Delta_{a}=\begin{pmatrix}\Delta^{0}_{a}&-\frac{\Delta^{+}_{a}}{\sqrt{2}}\\\
-\frac{\Delta^{+}_{a}}{\sqrt{2}}&\Delta^{++}_{a}\end{pmatrix},$ (2.1)

The product rule to construct singlets from different products of irreducible
representations of $A_{4}$ in Altarelli- Feruglio basis [10, 9] is given as

$(a\otimes b)_{1}=a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2},$ $(a\otimes
b)_{1^{\prime}}=a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1},$ $(a\otimes
b)_{1^{\prime\prime}}=a_{2}b_{2}+a_{1}b_{3}+a_{3}b_{1},$ (2.2) $(a\otimes
b)_{3_{s}}=\frac{1}{3}(2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2},2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1},2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}),$
$(a\otimes
b)_{3_{a}}=\frac{1}{2}(a_{2}b_{3}-a_{3}b_{2},a_{1}b_{2}-a_{2}b_{1},a_{1}b_{3}-a_{3}b_{1}),$
$\displaystyle((a\otimes b)_{3_{s}}\otimes c)_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}(2a_{1}b_{1}c_{1}+2a_{2}b_{2}c_{2}+2a_{3}b_{3}c_{3}-a_{1}b_{2}c_{3}-a_{1}b_{3}c_{2}$
(2.3) $\displaystyle-$ $\displaystyle
a_{3}b_{1}c_{2}-a_{3}b_{2}c_{1}-a_{2}b_{1}c_{3}-a_{2}b_{3}c_{1}).$

CP invariance is imposed at the Lagrangian level and hence, all the parameters
are perceived to be real. The CP asymmetry is broken at high energy by the
complex VEV of the scalar singlet. Also $A_{4}\times Z_{4}$ symmetry is broken
at a high-scale with a specific vacuum configuration leading to the
tribimaximal (TBM) lepton mixing. Below the cutoff scale, $\Lambda$, the
effective Yukawa Lagrangian containing the lowest-order term
($\mathcal{O}(1/\Lambda)$) can be written as,

$\mathcal{L}=\frac{y^{l}_{e}}{\Lambda}(\overline{L}\Phi)_{1}\phi
e_{R}+\frac{y^{l}_{\mu}}{\Lambda}(\overline{L}\Phi)_{1^{\prime}}\phi\mu_{R}+\frac{y^{l}_{\tau}}{\Lambda}(\overline{L}\Phi)_{1^{\prime\prime}}\phi\tau_{R}+\frac{y_{2}}{\Lambda}\Delta_{2}(L^{T}L\psi)_{1}+\frac{1}{\Lambda^{\prime}}\Delta_{1}(L^{T}L)_{1}(y_{1}S+y^{\prime}_{1}S^{*})+{\rm
h.c.}.$ (2.4)

The heavy scalar fields develop VEVs along the required directions as

$\langle\Phi\rangle=(r,0,0),\quad\langle\psi\rangle=(s,s,s),\quad\langle
S\rangle=v_{S}e^{i\alpha}.$ (2.5)

Now the Lagrangian can be written as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{y^{l}_{e}r}{\Lambda}\overline{L}_{e}\phi
e_{R}+\frac{y^{l}_{\mu}r}{\Lambda}\overline{L}_{\mu}\phi\mu_{R}+\frac{y^{l}_{\tau}r}{\Lambda}\overline{L}_{\tau}\phi\tau_{R}+\frac{y_{2}s}{3\Lambda}\Delta_{2}(2L^{T}_{e}L_{e}+2L^{T}_{\mu}L_{\mu}+2L^{T}_{\tau}L_{\tau}-L^{T}_{e}L_{\mu}$
(2.6) $\displaystyle-$ $\displaystyle
L^{T}_{e}L_{\tau}-L^{T}_{\mu}L_{e}-L^{T}_{\mu}L_{\tau}-L^{T}_{\tau}L_{e}-L^{T}_{\tau}L_{\mu})$
$\displaystyle+$
$\displaystyle\frac{v_{S}}{\Lambda^{\prime}}\Delta_{1}(L^{T}_{e}L_{e}+L^{T}_{\mu}L_{\tau}+L^{T}_{\tau}L_{\mu})(y_{1}e^{i\alpha}+y^{\prime}_{1}e^{-i\alpha})+{\rm
h.c.}$ $\displaystyle=$ $\displaystyle
Y^{l}_{\alpha\beta}\overline{L}_{\alpha}\phi
e_{R\beta}+Y^{\Delta_{1}}_{\alpha\beta}L^{T}_{\alpha}C\Delta_{1}L_{\beta}+Y^{\Delta_{2}}_{\alpha\beta}L^{T}_{\alpha}C\Delta_{2}L_{\beta}+{\rm
h.c.},$

where the Yukawa matrices are

$Y^{l}=\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}&0\\\
0&0&y_{\tau}\end{pmatrix},\quad
Y^{\Delta_{1}}=y_{\Delta_{1}}\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\quad
Y^{\Delta_{2}}=\frac{y_{\Delta_{2}}}{3}\begin{pmatrix}2&-1&-1\\\ -1&2&-1\\\
-1&-1&2\end{pmatrix},$ (2.7)

and

$y_{e,\mu,\tau}=\frac{r}{\Lambda}y^{l}_{e,\mu,\tau},\quad
y_{\Delta_{1}}=\frac{v_{S}}{\Lambda^{\prime}}(y_{1}e^{i\alpha}+y^{\prime}_{1}e^{-i\alpha}),\quad
y_{\Delta_{2}}=\frac{y_{2}}{\Lambda}s.$ (2.8)

It can be seen that the Yukawa coupling matrices $Y^{\Delta_{1}}$ and
$Y^{\Delta_{2}}$ exhibit $\mu-\tau$ symmetry and magic symmetry, respectively.
We assume the singlet scalar $S$ to be very heavy and to have decoupled at a
higher energy scale than electroweak (EW) and mass scales of the Higgs
triplets. The complex scalar $S$ plays a vital role in this model since the CP
symmetry, otherwise conserved in the Lagrangian level, is spontaneously broken
by the complex VEV of the scalar singlet. Hence, it is crucial to analyze the
scalar potential for $S$. The detail of the analysis can be found in Appendix
(A). The minimization of the potential leads to the following conditions,

$v^{2}_{S}=\frac{-2\lambda^{\prime}_{S}m^{2}_{S}+\lambda^{\prime\prime}_{S}\mu^{2}_{S}}{4\lambda_{S}\lambda^{\prime}_{S}-8\lambda^{\prime}_{S}-\lambda^{\prime\prime}_{S}},\quad\cos(2\alpha)=-\frac{\mu^{2}_{S}+\lambda^{\prime\prime}_{S}v^{2}_{S}}{4\lambda^{\prime}_{S}v^{2}_{S}}.$
(2.9)

To show that the last solution, which also leads in this case to the
spontaneous breaking of the CP symmetry, corresponds to the global minimum of
the potential, for a particular choice of parameter space such as
$m^{2}_{S}<0$, $\lambda^{\prime\prime}_{S}\simeq 0$, $\mu_{S}\simeq 0$ and
$\lambda_{S}>2\lambda^{\prime}_{S}>0$ and we obtain from Eq.(A.13), for case-3

$v^{2}_{S}\simeq-\frac{m^{2}_{S}}{2(\lambda_{S}-2\lambda^{\prime}_{S})},\quad\alpha\simeq\pm\frac{\pi}{4},$
(2.10)

which leads to

$V_{0}\simeq-\frac{m^{4}_{S}}{4(\lambda_{S}-2\lambda^{\prime}_{S})}$ (2.11)

which is the absolute minimum of the potential.

The triplet Higgs scalars contribute to the neutrino mass generation through
the type-II seesaw mechanism as

$m_{\nu}=m^{(1)}_{\nu}+m^{(2)}_{\nu},\quad
m^{(a)}_{\nu}=2u_{a}Y^{\Delta_{a}},$ (2.12)

where $m^{(1)}_{\nu}$ and $m^{(2)}_{\nu}$ are the contributions coming from
the triplet scalars $\Delta_{1}$ and $\Delta_{2}$, respectively. Here, $u_{a}$
is the VEV of the neutral components of the triplet scalars $\Delta_{a}$, and
it is given by

$u_{a}=\mu^{*}_{a}v^{2}/M_{a},\quad\mu_{1}=(\lambda_{1}e^{i\alpha}+\lambda^{\prime}_{1}e^{-i\alpha})v_{S}/M_{1}$
(2.13)

where $v=\langle\phi_{0}\rangle=174$ GeV. To diagonalize the neutrino mass
matrix, we define a unitary matrix $U$ as

$m_{\nu}=U^{*}d_{\nu}U^{\dagger},\quad
d_{\nu}=\rm{diag}(|z_{1}e^{i\beta}+z_{2}|,z_{1},|z_{1}e^{i\beta}-z_{2}|)\equiv
diag(m_{1},m_{2},m_{3}),$ (2.14)

where

$z_{a}=2|u_{a}y_{\Delta_{a}}|,\quad\beta={\rm arg}(u_{1}y_{\Delta_{1}})={\rm
arctan}\left(\frac{\lambda^{\prime}_{1}-\lambda_{1}}{\lambda^{\prime}_{1}+\lambda_{1}}{\rm
tan}\alpha\right)+{\rm
arctan}\left(\frac{y^{\prime}_{1}-y_{1}}{y^{\prime}_{1}+y_{1}}{\rm
tan}\alpha\right).$ (2.15)

In this model, where the neutrino mass matrix can be decomposed with matrices
with $\mu-\tau$ symmetry and magic symmetry, the mixing matrix takes the form

$U=e^{-i\sigma_{1}/2}U_{\rm{TBM}}K,$ (2.16)

where $U_{\rm{TBM}}$ is the tribimaximal mixing matrix and is given by,

$U_{\rm{TBM}}=\begin{pmatrix}\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{pmatrix},$
(2.17)

and the Majorana phases are the entries in the diagonal matrix $K$, given by,

$K=\rm{diag}(1,e^{i\gamma_{1}},e^{i\gamma_{2}}),$ (2.18)

with

$\gamma_{1}=(\sigma_{1}-\beta)/2,\quad\gamma_{2}=(\sigma_{1}-\sigma_{2})/2,\quad\sigma_{1,2}=\rm{arg}(z_{2}\pm
z_{1}e^{i\beta}).$ (2.19)

TBM mixing matrix has gathered considerable attention because of its proximity
with the observation of neutrino masses and mixing. However the absence of
Dirac-type CP violating phase asks for possible modifications of the model,
which we discuss in the next subsection. Considering the two cases of neutrino
mass hierarchy, we can have two sets of constraints on $z_{1}$ and $z_{2}$.
From Eq.(2.14), for normal hierarchy (NH), i.e. $m_{3}>m_{2}>m_{1}$, we obtain

$z_{2}+2z_{1}\cos\beta<0,\quad z_{2}-2z_{1}\cos\beta>0,$ (2.20)

and, for inverted hierarchy (IH) we have

$z_{2}+2z_{1}\cos\beta<0,\quad z_{2}-2z_{1}\cos\beta<0.$ (2.21)

Now the constraints in Eq.(2.21) cannot be satisfied simultaneously, which
indicates that the model under consideration does not qualify for the IH of
neutrino mass. In terms of the phase $\beta$ and two mass squared differences,
the parameters $z_{1}$ and $z_{2}$ can be written as,

$z_{1}=-\frac{1}{2\cos\beta}\frac{\Delta m^{2}_{31}}{\sqrt{2(\Delta
m^{2}_{31}-2\Delta m^{2}_{21})}}\simeq-\frac{1}{2\cos\beta}\sqrt{\frac{\Delta
m^{2}_{31}}{2}},$ (2.22)

and

$z_{2}=\sqrt{\frac{\Delta m^{2}_{31}-2\Delta
m^{2}_{21}}{2}}\simeq\sqrt{\frac{\Delta m^{2}_{31}}{2}}.$ (2.23)

Hence, the neutrino mass eigenvalues can be expressed as

$m_{1}=\sqrt{z^{2}_{1}-\Delta m^{2}_{21}},\quad m_{2}=z_{1},\quad
m_{3}=\sqrt{z^{2}_{1}+\Delta m^{2}_{31}-\Delta m^{2}_{21}}.$ (2.24)

###  2.1 Non-zero $\theta_{13}$ from perturbative corrections in the flavon
VEV

The measurement of non-zero $\theta_{13}$ [1, 4, 11] requires to modify the
TBM scenario. It can be achieved by exploiting the small perturbations around
the TBM vacuum alignment conditions. More specifically, by considering a small
perturbation around the flavon VEV of the form
$\langle\Phi\rangle=r(1,\varepsilon_{1},\varepsilon_{2})$ with
$|\varepsilon_{1,2}|\ll 1$ we obtain the transformed charged lepton Yukawa
matrix,

$Y^{l}=\begin{pmatrix}y_{e}&y_{\mu}\varepsilon_{1}&y_{\tau}\varepsilon_{2}\\\
y_{e}\varepsilon_{2}&y_{\mu}&y_{\tau}\varepsilon_{1}\\\
y_{e}\varepsilon_{1}&y_{\mu}\varepsilon_{2}&y_{\tau}\end{pmatrix},$ (2.25)

while the lepton mixing coming from the neutrino sector remains invariably of
TBM type, since $\langle\psi\rangle$ is not perturbed. The emergence of a non-
diagonal charged lepton Yukawa matrix can be diagonalized by biunitary
transformation $Y^{l}=U_{l}~{}{\rm Diagonal}(y_{e},y_{\mu},y_{\tau})~{}I$,
where $I$ is the identity matrix and $U_{l}$ is given by

$U_{l}=\begin{pmatrix}1&\varepsilon_{1}&\varepsilon_{2}\\\
\varepsilon_{2}&1&\varepsilon_{1}\\\
\varepsilon_{1}&\varepsilon_{2}&1\end{pmatrix}.$ (2.26)

Further, the lepton mixing matrix gets modified as

$U=e^{-i\sigma_{1}/2}U^{\dagger}_{l}U_{TBM}K.$ (2.27)

The $U$ matrix obtained in Eq. (2.27) can be identified as Pontecorvo-Maki-
Nakagawa-Sakata (PMNS) mixing matrix, $U_{{\rm PMNS}}$.

###  2.2 Parameter space study

To check the compatibility of the model with the latest experimental results,
it is important to study the parameter space of the model constrained by the
neutrino oscillation data. The latest global-fit of the observational data are
summarized in table (2) [24]. Also, the limits for effective mass, $|m_{ee}|$
appearing in the half-life of neutrinoless double beta ($0\nu\beta\beta$)
decay are constrained by GERDA as $<(79-180)$ meV [7] and KamLAND-Zen as
$<(36-156)$ meV [2]. The future sensitivity of $|m_{ee}|$ from the future ton-
scale experiments like CUPID [14, 13], LEGEND [3], and nEXO [8, 33] are going
to probe $|m_{ee}|\leq 0.01$ eV.

NuFIT 5.0 [24]
---
Neutrino Oscillation parameters |  Normal ordering 1$\sigma$ |  Normal ordering 3$\sigma$
$\sin^{2}\theta_{12}$ |  $0.303^{+0.012}_{-0.012}$ |  $0.270\rightarrow 0.341$
$\sin^{2}\theta_{23}$ |  $0.451^{+0.019}_{-0.016}$ |  $0.408\rightarrow 0.603$
$\sin^{2}\theta_{13}$ |  $0.02225^{+0.00056}_{-0.00059}$ |  $0.02052\rightarrow 0.02398$
$\delta/^{\circ}$ |  $232^{+36}_{-26}$ |  $144\rightarrow 350$
$\frac{\Delta m^{2}_{21}}{10^{-5}{\rm eV}^{2}}$ |  $7.41^{+0.21}_{-0.20}$ |  $6.82\rightarrow 8.03$
$\frac{\Delta m^{2}_{3l}}{10^{-3}{\rm eV}^{2}}$ |  $+2.507^{+0.026}_{-0.027}$ |  $+2.427\rightarrow+2.590$
Table 2: Allowed ranges of neutrino oscillation parameters for normal mass
ordering.

The three mixing angles, $\theta_{12}$, $\theta_{23}$ and $\theta_{13}$ can be
extracted from the standard parameterization of the $U_{\rm PMNS}$ matrix as

$\sin^{2}\theta_{12}=\frac{|U_{12}|^{2}}{1-|U_{13}|^{2}},\quad\sin^{2}\theta_{23}=\frac{|U_{23}|^{2}}{1-|U_{13}|^{2}},\quad\sin^{2}\theta_{13}=|U_{13}|^{2}.$
(2.28)

From equations (2.27) and (2.28), three mixing angles, $\theta_{12}$,
$\theta_{23}$ and $\theta_{13}$ are related to the small perturbations
$\varepsilon_{1}$ and $\varepsilon_{2}$ through the $U$ matrix, given in Eq.
(2.27). The explicit relations relating the three mixing angles and the small
perturbations are shown below:

$\sin^{2}\theta_{12}\simeq\frac{1}{3}[1+2(\varepsilon_{1}+\varepsilon_{2})],\quad\sin^{2}\theta_{23}\simeq\frac{1}{2}[1-2\varepsilon_{2}],\quad\sin^{2}\theta_{13}\simeq\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2})^{2},$
(2.29)

which further satisfies the relation,

$\sin^{2}\theta_{13}\simeq\frac{(4\sin^{2}\theta_{23}-3\cos^{2}\theta_{12})^{2}}{8}.$
(2.30)

(a)

(b)

Figure 1: Left: The plot shows the parameter space for the perturbations
$\varepsilon_{1}$ and $\varepsilon_{2}$ constrained by the $3\sigma$ data of
three mixing angles. Right: The plot shows the dependence of $|m_{ee}|$ (eV)
on the phase $\beta$ (rad) using the bounds of the $3\sigma$ data of three
mixing angles and the two mass squared differences. The experimental limits on
$|m_{ee}|$ are given by GERDA (cyan) and KamLAND-Zen (magenta). The orange
line corresponds to the future sensitivity of $|m_{ee}|$ from the future ton-
scale experiments like CUPID, LEGEND, and nEXO.

By imposing the experimentally obtained constraints on the mixing angle, as
given in table (2), the allowed parameter space for the perturbations
$[\varepsilon_{1},\varepsilon_{2}]$ are obtained as shown in Fig. (1(a)). We
use the same set of mixing angle constraints along with the bounds on two mass
squared differences to assess the effective neutrino mass, which is given by

$|m_{ee}|=\frac{1}{3}\left|2m_{1}(1-\varepsilon_{1}-\varepsilon_{2})+m_{2}(1+2\varepsilon_{1}+2\varepsilon_{2})e^{2i\gamma_{1}}\right|,$
(2.31)

where the phase $\gamma_{1}$ is given in Eq. (2.19). Through the phase,
$\gamma_{1}$, it is possible to draw a link between the effective mass and its
dependence on the phase $\beta$, which appears in the expressions of neutrino
masses as well as in the high-scale CP violation, as it will be elaborated
later on. In Fig. (1(b)) we show the dependence of $|m_{ee}|$ on the phase
$\beta$. The plot shows that the allowed effective mass in turn has imposed a
bound on the phase $\beta\leq 270^{\circ}$. Apart from the bound on the
effective neutrino mass $|m_{ee}|\lesssim 0.15$ eV, the bound on the total
neutrino mass $\Sigma=m_{1}+m_{2}+m_{3}<0.12$ eV [6] puts a more stringent
restriction on the parameter space of phase $\beta$ as
$\sim[120^{\circ},240^{\circ}]$. We have used this restricted parameter space
of the phase $\beta$ to calculate the flavored CP asymmetries in section (4).

##  3 Baryogenesis through leptogenesis

The Sakharov conditions [44] provide necessary ingredients for dynamical
generation of BAU viz; baryon number violation, C and CP violation and out-of-
equilibrium dynamics in the processes involving baryons. In the framework of
baryogenesis through leptogenesis, the necessary CP violation required for
successful leptogenesis is provided by the decay of heavy particles added to
the SM. In the type-II seesaw scenario, the decays of the triplet scalars
$\Delta_{1}$ and $\Delta_{2}$ serve the purpose of generating adequate CP
asymmetry.

The flavor-independent CP asymmetry, as shown in Eq. (3.9), crucially depends
on the factor ${\rm Tr}(m^{(a)}_{\nu}m^{(b)\dagger}_{\nu})$ that comes from
the neutrino mass matrices given in Eq. (2.12). In the framework under
consideration, this factor vanishes due to the specific flavor structure of
the Yukawa matrices $Y^{\Delta_{a}}$, causing the flavor-independent CP
asymmetry to diminish. Hence, the flavor-independent analysis of leptogenesis
is not feasible in our chosen framework. Moreover, since scalar triplets
generally couple to lepton pairs from different flavor generations, it is more
appropriate to consider the flavor effects while studying triplet scalar
leptogenesis.

###  3.1 CP violation

A triplet scalar can decay into two leptons or two SM Higgs doublets at tree
level. Lepton asymmetry can be produced only if the two decay channels co-
exist. The non-vanishing lepton asymmetry can be produced for each triplet
component from the decay of triplet scalar $\Delta_{a}$ into two leptons and
the CP asymmetry can be estimated through,

$\epsilon_{a}=\Delta L\sum_{\alpha\beta}\frac{\Gamma(\Delta^{*}_{a}\rightarrow
L_{\alpha}+L_{\beta})-\Gamma(\Delta_{a}\rightarrow\overline{L}_{\alpha}+\overline{L}_{\beta})}{\Gamma_{\Delta_{a}}+\Gamma_{\Delta^{*}_{a}}},$
(3.1)

where $\Delta L=2$ since the lepton number is violated by two units. The tree
level branching ratios $B^{L}_{a}$ and $B^{\phi}_{a}$,

$B^{L}_{a}\Gamma_{\Delta_{a}}\equiv\sum_{\alpha,\beta}\Gamma(\Delta^{*}_{a}\rightarrow
L_{\alpha}+L_{\beta})=\frac{M_{a}}{8\pi}{\rm
Tr}(Y^{\Delta_{a}\dagger}Y^{\Delta_{a}}),$ (3.2)
$B^{\phi}_{a}\Gamma_{\Delta_{a}}\equiv\Gamma(\Delta^{*}_{a}\rightarrow\phi+\phi)=\frac{M_{a}}{8\pi}|\mu_{a}|^{2},$
(3.3)

satisfy the relation $B^{L}_{a}+B^{\phi}_{a}=1$, provided

$\Gamma_{\Delta_{a}}=\frac{M_{a}}{8\pi}\left[{\rm
Tr}(Y^{\Delta_{a}\dagger}Y^{\Delta_{a}})+|\mu_{a}|^{2}\right],$ (3.4)

which is the total triplet decay width. In the case of a lepton flavor
violating interaction, the interference of tree level (Fig. 2) and one-loop
(Fig. 3) diagrams give rise to a non-vanishing CP asymmetry, given by

$\epsilon^{\alpha\beta}_{a}\simeq-\frac{g(x_{b})}{2\pi}\frac{c_{\alpha\beta}{\rm
Im}[\mu^{*}_{a}\mu_{b}Y^{\Delta_{a}}_{\alpha\beta}Y^{\Delta_{b}*}_{\alpha\beta}]}{{\rm
Tr}(Y^{\Delta_{a}\dagger}Y^{\Delta_{a}})+|\mu_{a}|^{2}},$ (3.5)

where $x_{b}=\frac{M^{2}_{b}}{M^{2}_{a}}$, and the on-loop self energy
function,

$g(x_{b})=\frac{\sqrt{x_{b}}(1-x_{b})}{(x_{b}-1)^{2}+(\Gamma_{\Delta_{b}}/M_{a})^{2}},$
(3.6)

and

$\displaystyle c_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
2-\delta_{\alpha\beta}\quad\rm{for}\quad\Delta^{0}_{a},\Delta^{++}_{a},$
$\displaystyle=$ $\displaystyle 1\quad\rm{for}\quad\Delta^{+}_{a}.$

In the present scenario, we can rewrite Eq. (3.5) as

$\epsilon^{\alpha\beta}_{a}\simeq-\frac{g(x_{b})}{4\pi}\frac{M_{b}(B^{L}_{a}B^{\phi}_{a})^{\frac{1}{2}}}{v^{2}}\frac{c_{\alpha\beta}{\rm
Im}[m^{(a)}_{\nu,\alpha\beta}m^{*}_{\nu,\alpha\beta}]}{[{\rm
Tr}(m^{(a)\dagger}_{\nu}m^{(a)}_{\nu})]^{1/2}}.$ (3.7)

If we consider the triplet scalar masses to be hierarchical, satisfying
$M_{a}\ll M_{b}$ then

$\epsilon^{\alpha\beta}_{a}\simeq\frac{M_{a}(B^{L}_{a}B^{\phi}_{a})^{\frac{1}{2}}}{v^{2}}\frac{c_{\alpha\beta}{\rm
Im}[m^{(a)}_{\nu,\alpha\beta}m^{*}_{\nu,\alpha\beta}]}{[{\rm
Tr}(m^{(a)\dagger}_{\nu}m^{(a)}_{\nu})]^{1/2}}.$ (3.8)

We obtain the flavor independent or unflavored CP asymmetry from Eq. (3.8), by
summing over all lepton flavors,

$\epsilon_{a}\simeq\frac{M_{a}(B^{L}_{a}B^{\phi}_{a})^{\frac{1}{2}}}{v^{2}}\frac{{\rm
Im}[{\rm Tr}(m^{(a)}_{\nu}m^{\dagger}_{\nu})]}{[{\rm
Tr}(m^{(a)\dagger}_{\nu}m^{(a)}_{\nu})]^{1/2}}.$ (3.9)

This CP asymmetry has an upper bound as can be seen from the expression,

$|\epsilon_{a}|\leq\frac{M_{a}(B^{L}_{a}B^{\phi}_{a})^{\frac{1}{2}}}{v^{2}}[{\rm
Tr}(m^{(a)}_{\nu}m^{\dagger}_{\nu})]^{1/2}=\frac{M_{a}(B^{L}_{a}B^{\phi}_{a})^{\frac{1}{2}}}{v^{2}}\left(\sum_{k}m^{2}_{k}\right)^{1/2}.$
(3.10)

Further, for hierarchical light neutrinos, one finds

$|\epsilon_{a}|\lesssim
10^{-6}(B^{L}_{a}B^{\phi}_{a})^{\frac{1}{2}}\left(\frac{M_{a}}{10^{10}GeV}\right)\left(\frac{\sqrt{\Delta
m^{2}_{31}}}{0.05eV}\right)$ (3.11)

which reaches its absolute maxima at $B^{L}_{a}=B^{\phi}_{a}=\frac{1}{2}$.

Figure 2: Tree-level decay diagram: Triplet scalar decays into lepton and SM
Higgs, respectively. Figure 3: One-loop decay diagram: Triplet scalar decays
into leptons through Higgs loop and lepton loop, respectively.

###  3.2 Triplet leptogenesis and flavor covariance

Flavor effects can emerge from the misalignment of flavor and mass eigenbasis,
which can persist in the off-diagonal entries of charged lepton Yukawa
matrices. The phenomenon of particle mixing with primordial plasma as a
background medium also contributes to flavor effects in the early universe.
Such effects can be addressed through quantum statistical mechanics, precisely
considering the non-vanishing off-diagonal entries of particle number
densities of particle species tagged with flavor quantum numbers. In the case
of thermal leptogenesis, the charged lepton flavor coherence affects the
washout process, thereby affecting the generation of baryon asymmetry. Hence
it is necessary to develop theoretical frameworks to assess the flavor
effects. We will use a formalism, namely closed time path (CTP) formalism, in
which particle densities are replaced by Green’s functions to describe non-
equilibrium phenomena like thermal leptogenesis [36, 22, 47, 43, 45]. The
flavor covariance could be conveniently studied in a density matrix formalism
[36] through flavor-covariant Boltzmann equations. The temperature range
$[10^{9},10^{12}]$ GeV appears more interesting for the study of such flavor
covariance since this is an intermediate temperature range $\lesssim 10^{12}$
GeV, when the flavor coherence between $\tau$-leptons and the other two
leptons $(e,\mu)$ is broken, yet before the complete disintegration of the
flavor coherence that happens further below $10^{9}$ GeV.

Hence, we will study the leptogenesis through a set of flavor-covariant
Boltzmann equations and perceive the asymmetries in terms of a density matrix
$(\Delta_{l})_{\alpha\beta}$. In general, it is a $3\times 3$ matrix in lepton
flavor space. The lepton doublet asymmetries are stored as diagonal entries,
whereas, the off-diagonal entries symbolise the quantum correlation between
the different flavors. In the most general way, we can write such a flavor-
covariant equation to express the evolution of flavored lepton asymmetries, as

$sHz\frac{d(\Delta_{l})_{\alpha\beta}}{dz}=\rm{Source}+\rm{Washout}.$ (3.12)

The source term on the right-hand side is proportional to the flavor-covariant
CP asymmetry matrix $\epsilon^{\alpha\beta}$. Different lepton number
violating processes like inverse triplet and anti-triplet decays washout the
lepton asymmetry generated through decay processes. Similarly, $\Delta L=2$
scatterings via s - and t - channel triplet exchange, 2 - lepton - 2 - lepton
scatterings, as well as lepton flavor violating processes also contribute to
affect the total lepton asymmetry. So, they come under the class of washout
processes. To avoid complexity, we will only stick to the washout associated
with inverse decays. Apart from the inverse decay and scattering processes,
there are different SM reactions, such as strong and EW sphaleron process and
Yukawa couplings that indirectly affect the final asymmetry. They are
considered spectator processes. For a specific temperature range, these
spectator processes in thermal equilibrium can impose some relations among
chemical potentials and thereby among the asymmetries. As a consequence,
different asymmetries appearing in the Boltzmann equation can be expressed in
terms of asymmetries conserved by all active SM processes.

We write the evolution of the asymmetries of the particle species under
consideration in a set of flavor-covariant Boltzmann equations, with respect
to a factor $z=\frac{M_{a}}{T}$, $T$ is the temperature of the Universe,
$H(z)$ is the Hubble rate of expansion of the Universe,
$H(z)=\frac{H_{0}(M_{a})}{z^{2}}$,
$H_{0}(T)=\sqrt{\frac{4\pi^{3}}{45}g_{*}}\frac{T^{2}}{M_{P}}$,
$M_{P}=1.22\times 10^{19}$ GeV is the Planck mass,
$s=\left(\frac{2\pi^{2}}{45}\right)g_{*}T^{3}$ is the total entropy density
and $g_{*}=106.75$. The asymmetries are defined as
$\Delta_{x}=\frac{n_{x}-n_{\overline{x}}}{s}$ where $n_{x}(n_{\bar{x}})$ is
the number density of the species, $x(\bar{x})$, and
$\Sigma_{\Delta}=\frac{n_{\Delta}+n_{\overline{\Delta}}}{s}$ is the total
triplet number density. Also, $Y_{x}=\frac{n_{x}}{s}$ is the comoving number
density,

$Y^{\rm eq}_{\Delta}=\frac{45g_{\Delta}}{4\pi^{4}g_{*}}z^{2}K_{2}(z),\quad
Y^{\rm eq}_{L}=\frac{3}{4}\frac{45\varsigma(3)}{2\pi^{4}g_{*}}g_{L},\quad
Y^{\rm eq}_{\phi}=\frac{45\varsigma(3)}{2\pi^{4}g_{*}}g_{\phi},$

are the equilibrium values, with $g_{\Delta}=1$ for each triplet component,
$g_{L}=2$ and $g_{\phi}=2$, $\varsigma(3)\simeq 1.202$, $K_{2}(z)$ is the
modified Bessel function. Different reaction densities are elaborated in
Appendix (B).

We are interested to study the evolution of the asymmetries $\Delta_{\phi}$,
$\Delta_{\Delta}$ and $(\Delta_{l})_{\alpha\beta}$ through a set of flavor-
covariant Boltzmann equations. Within the temperature range
$T\subset[10^{9},10^{12}]$ GeV, the $\tau$-Yukawa couplings enter equilibrium
and consequently, the $(e,\tau)$, $(\mu,\tau)$, $(\tau,e)$, $(\mu,e)$ entries
of the $3\times 3$ density matrix $(\Delta_{l})_{\alpha\beta}$ come to be
zero. So it is convenient to assign $2\times 2$ matrix
$(\Delta^{0}_{l})_{\alpha\beta}$ $(\alpha,\beta=e,\mu)$ and
$\Delta_{l_{\tau}}$ instead of the $3\times 3$ density matrix
$(\Delta_{l})_{\alpha\beta}$ with $(\alpha,\beta=e,\mu,\tau)$ to express the
asymmetries and quantum correlations stored in $e-\mu$ flavored lepton
subspace and the asymmetries stored in $\tau$-lepton, respectively. Among all
the asymmetries, $\Delta_{\phi}$, $\Delta_{\Delta}$, $\Delta_{l_{\tau}}$,
$(\Delta^{0}_{l})_{\alpha\beta}$, only $\Delta_{\Delta}$ is conserved by all
the SM interactions. Hence, it is necessary to define conserved asymmetries
$\Delta_{\tau}$ and $\Delta^{0}_{\alpha\beta}$ (instead of $\Delta_{l_{\tau}}$
and $(\Delta^{0}_{l})_{\alpha\beta}$) along with $\Delta_{\Delta}$ to be used
in the Boltzmann equations. The Higgs and flavored lepton asymmetries can be
expressed as functions of these conserved asymmetries ($\Delta_{\Delta}$,
$\Delta_{\tau}$, $\Delta^{0}_{\alpha\beta}$) using suitable chemical potential
relations (discussed in equations (B.4 \- B.10)). For the temperature range
$T\subset[10^{9},10^{12}]$ GeV, we obtain the relations as [36]

$(\Delta^{0}_{l})_{\alpha\beta}=\left(\frac{86}{589}{\rm
Tr}(\Delta^{0}_{\alpha\beta})+\frac{60}{589}\Delta_{\tau}+\frac{8}{589}\Delta_{\Delta}\right)\delta_{\alpha\beta}-\Delta^{0}_{\alpha\beta},$
(3.13) $\Delta_{l_{\tau}}=\frac{30}{589}{\rm
Tr}(\Delta^{0}_{\alpha\beta})-\frac{390}{589}\Delta_{\tau}-\frac{52}{589}\Delta_{\Delta},$
(3.14)

and

$\Delta_{\phi}=-\frac{164}{589}{\rm
Tr}\left(\Delta^{0}_{\alpha\beta}\right)-\frac{224}{589}\Delta_{\tau}-\frac{344}{589}\Delta_{\Delta},$
(3.15)

where $\alpha$, $\beta$ describe any two orthogonal in the $(l_{e},l_{\mu})$
flavor subspace now. Therefore, within the temperature range
$T\subset[10^{9},10^{12}]$ GeV, the flavor-covariant Boltzmann equations are
given by

$sHz\frac{d\Sigma_{\Delta}}{dz}=-\left(\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1\right)\gamma_{D}-2\left(\left(\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}\right)^{2}-1\right)\gamma_{A},$
(3.16)
$sHz\frac{d\Delta^{0}_{\alpha\beta}}{dz}=-\left(\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1\right)\gamma_{D}\epsilon^{\alpha\beta}+\tilde{\mathcal{W}}^{D}_{\alpha\beta},$
(3.17)
$sHz\frac{d\Delta_{\tau}}{dz}=-\left(\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1\right)\gamma_{D}\epsilon^{\tau\tau}+\tilde{\mathcal{W}}^{D}_{\tau},$
(3.18)
$sHz\frac{d\Delta_{\Delta}}{dz}=-\frac{1}{2}\tilde{\mathcal{W}}^{D}_{\Delta},$
(3.19)

where, $\tilde{\mathcal{W}}^{D}$ represent different washout terms associated
with inverse decay processes. In order to avoid complexity of presentation, we
represent $Y^{\Delta_{a}}\equiv Y$ and

$\tilde{\mathcal{W}}^{D}_{\alpha\beta}=\frac{2B^{L}}{\lambda^{2}_{l}}\left[(YY^{\dagger})_{\alpha\beta}\frac{\Delta_{\Delta}}{\Sigma_{\Delta}}+\frac{1}{4Y^{eq}_{L}}\left(2Y(\Delta^{0}_{l})^{T}Y^{\dagger}+YY^{\dagger}\Delta^{0}_{l}+\Delta^{0}_{l}YY^{\dagger}\right)_{\alpha\beta}+\frac{1}{2Y^{eq}_{L}}Y_{\alpha\tau}Y^{*}_{\beta\tau}\Delta_{l_{\tau}}\right]\gamma_{D},$
(3.20)
$\tilde{\mathcal{W}}^{D}_{\Delta}=Tr(\tilde{\mathcal{W}}^{D})+\tilde{\mathcal{W}}^{D}_{\tau}-W^{D}_{\phi},$
(3.21)
$\tilde{\mathcal{W}}^{D}_{\tau}=\frac{2B^{L}}{\lambda^{2}_{l}}\left[\left(YY^{\dagger}\right)_{\tau\tau}\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}+\frac{1}{2Y^{eq}_{L}}\left(\left(Y(\Delta^{0}_{l})^{T}Y^{\dagger}\right)_{\tau\tau}+\left(\left(YY^{\dagger}\right)_{\tau\tau}+|Y_{\tau\tau}|^{2}\right)\Delta_{l_{\tau}}\right)\right]\gamma_{D},$
(3.22)

and

$W^{D}_{\phi}=2B^{\phi}\left(\frac{\Delta_{\phi}}{Y^{eq}_{\phi}}-\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}\right)\gamma_{D},$
(3.23)

where $(\Delta^{0}_{l})_{\alpha\beta}$, $\Delta_{l_{\tau}}$ and
$\Delta_{\phi}$ are given by equations (3.13 \- 3.15).

Solving the flavor-covariant Boltzmann equations numerically, we obtain
flavored lepton asymmetries and we estimate the final baryon asymmetry as [36]

$\eta_{B}=7.04\times\frac{12}{37}\times\sum_{\alpha}\Delta_{\alpha},$ (3.24)

where $\Delta_{\alpha}=\Delta_{B/3-L_{\alpha}}$.

##  4 Baryogenesis through flavored leptogenesis: Results

We obtain the final baryon asymmetry given in Eq.(3.24) via lepton asymmetries
$\Delta_{\alpha}$ by solving the full set of DMEs given in equations (3.16 \-
3.19), for two cases. The cases depend upon the mass hierarchy among the two
triplet scalars; (I) $M_{1}\ll M_{2}$ (II) $M_{2}\ll M_{1}$. In thermal
leptogenesis, the Yukawa couplings of the corresponding triplet scalars in
both cases play an important role in the generation of adequate CP asymmetries
and washout effects. We further study, in each case, the role of hierarchical
branching ratios $(B^{L}_{a}\sim 0.0005,0.005,0.05)$ in generating adequate
baryon asymmetry as compared to comparable branching ratios $(B^{L}_{a}\sim
0.5)$. The CP-violating phase $\alpha/\beta$ plays an important role in
compensating the suppression in the CP asymmetry, which can manifest due to
hierarchy in the branching ratios. We also take different sets of triplet
$\Delta_{a}$ masses $M_{a}\sim 10^{10}$ GeV, flavored CP asymmetries
$\epsilon^{\alpha\beta}_{a}$ and calculate the baryon asymmetry for different
hierarchy of branching ratios.

###  4.1 Formulation of flavored CP asymmetries

The expression of flavored CP asymmetry from triplet $\Delta_{1}$ decay

$\epsilon^{\alpha\beta}_{1}=c_{\alpha\beta}\frac{M^{2}_{1}|u_{1}|^{2}}{2\pi\left(3z^{2}_{1}v^{4}+4M^{4}_{1}|u_{1}|^{4}\right)}\times\rm{Im}\left[m^{(1)}_{\nu,\alpha\beta}m^{*}_{\nu,\alpha\beta}\right],$
(4.1)

and the same originating from triplet $\Delta_{2}$ decay

$\epsilon^{\alpha\beta}_{2}=c_{\alpha\beta}\frac{M^{2}_{2}|u_{2}|^{2}}{4\pi\left(z^{2}_{2}v^{4}(1-2\varepsilon_{1}-2\varepsilon_{2})+2M^{4}_{2}|u_{2}|^{4}\right)}\times\rm{Im}\left[m^{(2)}_{\nu,\alpha\beta}m^{*}_{\nu,\alpha\beta}\right],$
(4.2)

get generated from Eq. (3.8) using the expressions of branching ratios (Eq.
(3.3)) and total triplet decay width (Eq. (3.4)). The total neutrino mass
matrix expression (Eq. (2.14)) is obtained by taking the modified
diagonalising matrix $U$ given in Eq. (2.27). The $\gamma_{1,2}$, $\beta$
phase relations are taken from Eq. (2.19). Using the neutrino mass matrix
relation

$m_{\nu}=U^{*}d_{\nu}U^{\dagger}=U^{*}d^{(1)}_{\nu}U^{\dagger}+U^{*}d^{(2)}_{\nu}U^{\dagger},$
(4.3)

we can obtain the neutrino mass matrix contribution from triplet $\Delta_{1}$,
as

$m^{(1)}_{\nu}=U^{*}d^{(1)}_{\nu}U^{\dagger}=z_{1}e^{i\beta}\begin{pmatrix}1&2\varepsilon_{1}&2\varepsilon_{2}\\\
2\varepsilon_{1}&2\varepsilon_{2}&1\\\
2\varepsilon_{2}&1&2\varepsilon_{1}\end{pmatrix}$ (4.4)

where

$d^{(1)}_{\nu}=\rm
diag\left(z_{1}e^{i(\beta-\sigma_{1})},z_{1},-z_{1}e^{i(\beta-\sigma_{2})}\right).$
(4.5)

and we can similarly obtain the neutrino mass matrix contribution from triplet
$\Delta_{2}$, as

$m^{(2)}_{\nu}=U^{*}d^{(2)}_{\nu}U^{\dagger}=\frac{1}{3}z_{2}(1-\varepsilon_{1}-\varepsilon_{2})\begin{pmatrix}2&-1&-1\\\
-1&2&-1\\\ -1&-1&2\end{pmatrix}$ (4.6) $d^{(2)}_{\nu}=\rm
diag\left(z_{2},0,z_{2}\right).$ (4.7)

Within the energy range $[10^{9},10^{12}]$ GeV, the flavored CP asymmetries
can be expressed as $\epsilon^{\alpha\beta}_{a}(\alpha,\beta=e,\mu)$ and
$\epsilon^{\tau\tau}_{a}$, where $a=1,2$ is associated with triplet
$\Delta_{1,2}$. Using equations (4.3), (4.4) and (4.6) we calculate, the
flavored CP asymmetries. The specific expressions are given in equations (C.8
\- C.11) and (C.12 \- C.15) in Appendix (C). The dependence of $\beta$ on the
CP asymmetries are plotted in the figures (8) and (9) for different values
$B^{L}_{1(2)}=0.5,0.05,0.005,0.0005$. The triplet VEV $|u_{1}|$ ($|u_{2}|$)
are taken in the range $[0.1-100]$ eV. To estimate the CP asymmetry from the
model, we have chosen the small perturbation parameters from the allowed
parameter space shown in Fig.(1(a)), as $\varepsilon_{1}\sim-0.125$ and
$\varepsilon_{2}\sim 0.075$. We have varied the phase $\beta$ within the range
$[120^{\circ},240^{\circ}]$ so that it also remains consistent with the latest
cosmological bound on the sum of the neutrino masses and the effective
neutrino mass $|m_{ee}|\lesssim 0.15$ eV. The phase $\sigma_{1}$ is taken in
such a way that it satisfies the relations given in Eq.(2.19). The viable
values of the parameters $z_{1}$ and $z_{2}$ are calculated as functions of
two mass squared differences and phase $\beta$, from the expressions equations
(2.22) and (2.23) by making sure that it is consistent with the bound on total
neutrino mass $\Sigma=m_{1}+m_{2}+m_{3}<0.12$ eV through the expression in
Eq.(2.24). It is observed that CP asymmetries of the order
$\mathcal{O}(10^{-7})$ can be produced for triplet masses $\sim 10^{10}$ GeV.
The figures (8) and (9) are generated by taking $M_{1}$ ($M_{2}$) $=2\times
10^{10}$ GeV and taking the mass squared differences in $3\sigma$ range. The
scale is suitable for studying two-flavor leptogenesis which we take up in the
upcoming sections.

###  4.2 Baryogenesis from $\Delta_{1}$ decay

First, we consider the case -I, $M_{1}\ll M_{2}$, where the lepton asymmetry
is generated by the decay processes of triplet $\Delta_{1}$. We obtain the
final baryon asymmetry given in Eq. (3.24) via lepton asymmetries
$\Delta_{\alpha}$ by solving the set of DMEs given in equations (3.16 \-
3.19), for different sets of triplet mass $M_{1}$ and the flavored CP
asymmetries $\epsilon^{\alpha\beta}_{1}$. The results of baryon asymmetry for
some of the benchmark points are briefed in table (3). In table (3), we show
the interplay of hierarchical branching ratios and the SCPV phase $\beta$ on
the flavored CP asymmetry parameters. For example, for $M_{1}=3\times 10^{10}$
GeV, using Eq. (3.2), we calculate the branching ratios, $B_{1}^{L}$ using the
allowed values of the parameters of the model. Using the same values of the
parameters we also calculate the corresponding CP asymmetries as shown in the
table. While the hierarchy in $B_{1}^{L}$ increases, it should have resulted
in suppression of the magnitude of the CP asymmetries (as can be seen in Eq.
(3.8)). But the contribution of the SCPV phase $\beta$ in
$\epsilon^{\alpha\beta}_{1}$ (equations (C.8 \- C.11)) compensates the
suppression arising from the increase in the hierarchy of the branching
ratios. It should be noted here that $B_{1}^{L}$ is independent of the phase
$\beta$. This feature also enhances the efficiency of leptogenesis which can
be seen from the increase in the final baryon asymmetry as shown in the table.

$M_{1}({\rm GeV})$ |  $B^{L}_{1}$ |  $\beta(^{\circ})$ |  $\epsilon^{ee}_{1}/10^{-8}$ |  $\epsilon^{e\mu}_{1}/10^{-9}$ |  $\epsilon^{\mu\mu}_{1}/10^{-9}$ |  $\epsilon^{\tau\tau}_{1}/10^{-9}$ |  $|\eta_{B}|/10^{-10}$
---|---|---|---|---|---|---|---
|  $0.48$ |  $178$ |  $0.70$ |  $0.84$ |  $1.01$ |  $-1.68$ |  $0.04$
$3\times 10^{10}$ |  $0.06$ |  $175$ |  $2.06$ |  $2.46$ |  $2.94$ |  $-4.91$ |  $0.29$
|  $0.005$ |  $165$ |  $2.04$ |  $2.43$ |  $2.91$ |  $-4.86$ |  $0.73$
|  $0.0005$ |  $126$ |  $2.06$ |  $2.46$ |  $2.94$ |  $-4.91$ |  $1.74$
|  $0.34$ |  $178$ |  $2.79$ |  $3.32$ |  $3.98$ |  $-6.63$ |  $0.30$
$4\times 10^{10}$ |  $0.05$ |  $175$ |  $2.82$ |  $3.36$ |  $4.03$ |  $-6.71$ |  $0.44$
|  $0.005$ |  $164$ |  $2.82$ |  $3.36$ |  $4.03$ |  $-6.71$ |  $1.07$
|  $0.0006$ |  $126$ |  $2.83$ |  $3.37$ |  $4.04$ |  $-6.74$ |  $2.50$
Table 3: Baryon asymmetries through flavored leptogenesis for the triplet mass
hierarchy case-1, $M_{1}\ll M_{2}$ by solving DME in the energy range
$[10^{9},10^{12}]$ GeV for different values of $M_{1}\sim 10^{10}$ GeV,
$\epsilon^{\alpha\beta}_{1}$, and hierarchical $(B^{L}_{1}\sim
0.0005,0.005,0.05)$ as well as comparable $(B^{L}_{1}\sim 0.5)$ branching
ratio $B^{L}_{1}$.

We present a specific result corresponding to a set with $M_{1}=3\times
10^{10}$ GeV, and flavored CP asymmetries $\epsilon^{ee}_{1}\sim 10^{-7}$,
$\epsilon^{e\mu,~{}\mu e,~{}\mu\mu,~{}\tau\tau}_{1}\sim 10^{-8}$ as given in
table (4). We have plotted the obtained flavored lepton asymmetries and baryon
asymmetry in Fig. (4(a)) along with the evolution of different washout terms
in Fig. (4(b)). Although we have shown the results for final baryon asymmetry
for a few benchmark points, we have taken different data sets corresponding to
particular flavored CP asymmetries for $M_{1}\sim 10^{10}$ GeV and
hierarchical branching ratios and we have plotted the obtained baryon
asymmetry for each set in Fig. (6(a)).

$M_{1}({\rm GeV})$ |  $B^{L}_{1}$ |  $|u_{1}|(\rm eV)$ |  $\epsilon^{ee}_{1}/10^{-7}$ |  $\epsilon^{e\mu}_{1}/10^{-8}$ |  $\epsilon^{\mu\mu}_{1}/10^{-8}$ |  $\epsilon^{\tau\tau}_{1}/10^{-8}$ |  $|\eta_{B}|/10^{-10}$
---|---|---|---|---|---|---|---
$3\times 10^{10}$ Fig. (4) |  $0.58$ |  $4.80$ |  $4.47$ |  $5.32$ |  $6.38$ |  $-1.06$ |  $5.29$
Table 4: Baryon asymmetries through leptogenesis by solving DME in the energy
range $[10^{9},10^{12}]$ GeV for $M_{1}=3\times 10^{10}$ GeV.

(a)

(b)

Figure 4: Solution of DMEs for asymmetries and washouts for set with
$M_{1}=3\times 10^{10}$ GeV by using the flavored CP asymmetries from table
(4). Left: The color lines represent different asymmetries. Using the
numerically obtained asymmetries the final baryon asymmetry is calculated. The
green lines correspond to the observational value of the BAU. Right: The color
lines represent numerically obtained amount of washouts along different flavor
directions in units of $sHz$.

###  4.3 Baryogenesis from $\Delta_{2}$ decay

We further consider the triplet mass hierarchy case (II), with $M_{2}\ll
M_{1}$ and take different sets with triplet $\Delta_{2}$ mass $M_{2}\sim
10^{10}$ GeV, flavored CP asymmetries and hierarchical branching ratios. The
results are similar to case (I). The results are briefed in table (5). In this
case, for comparatively lower values of triplet mass, the model can result in
adequate baryon asymmetry. We present a specific result corresponding to a set
with $M_{2}=1.4\times 10^{10}$ GeV, and flavored CP asymmetries
$\epsilon^{ee}_{2}\sim 10^{-7}$, $\epsilon^{e\mu,~{}\mu e,~{}\mu\mu}_{2}\sim
10^{-8}$, $\epsilon^{\tau\tau}_{2}\sim 10^{-6}$ as given in table (6). We have
plotted the obtained flavored lepton asymmetries and baryon asymmetry in Fig.
(5(a)) along with the evolution of different washout terms in Fig. (5(b)).
Similar to the case -I, we have taken different data sets corresponding to
particular flavored CP asymmetries for $M_{2}\sim 10^{10}$ GeV and
hierarchical branching ratios and we have plotted the obtained baryon
asymmetry for each set in Fig. (6(b)).

(a)

(b)

Figure 5: Solution of DMEs for asymmetries and washouts for set with $M_{2}=1.4\times 10^{10}$ GeV by using the flavored CP asymmetries from table (6). Left: The color lines represent different asymmetries. Using the numerically obtained asymmetries the final baryon asymmetry is calculated. The green lines correspond to the observational value of the BAU. Right: The color lines represent numerically obtained amount of washouts along different flavor directions in units of $sHz$. $M_{2}({\rm GeV})$ |  $B^{L}_{2}$ |  $\beta(^{\circ})$ |  $\epsilon^{ee}_{2}/10^{-8}$ |  $\epsilon^{e\mu}_{2}/10^{-9}$ |  $\epsilon^{\mu\mu}_{2}/10^{-9}$ |  $\epsilon^{\tau\tau}_{2}/10^{-7}$ |  $|\eta_{B}|/10^{-10}$
---|---|---|---|---|---|---|---
|  $0.59$ |  $176$ |  $-0.66$ |  $-0.79$ |  $-0.95$ |  $0.75$ |  $0.04$
$1\times 10^{10}$ |  $0.06$ |  $172$ |  $-0.67$ |  $-0.80$ |  $-0.96$ |  $0.76$ |  $0.07$
|  $0.005$ |  $157$ |  $-0.67$ |  $-0.80$ |  $-0.96$ |  $0.75$ |  $0.86$
|  $0.0006$ |  $126$ |  $-0.73$ |  $-0.87$ |  $-1.05$ |  $0.75$ |  $3.26$
|  $0.53$ |  $177$ |  $-1.29$ |  $-1.53$ |  $-1.84$ |  $1.45$ |  $0.01$
$2\times 10^{10}$ |  $0.05$ |  $176$ |  $-1.42$ |  $-1.68$ |  $-2.02$ |  $1.59$ |  $0.03$
|  $0.006$ |  $156$ |  $-1.42$ |  $-1.69$ |  $-2.03$ |  $1.58$ |  $1.66$
|  $0.0005$ |  $126$ |  $-1.42$ |  $-1.69$ |  $-2.03$ |  $1.45$ |  $7.51$
Table 5: Baryon asymmetries through flavored leptogenesis for the triplet mass hierarchy case - II, $M_{2}\ll M_{1}$ by solving DME in the energy range $[10^{9},10^{12}]$ GeV for different values of $M_{2}\sim 10^{10}$ GeV, $\epsilon^{\alpha\beta}_{2}$, and hierarchical $(B^{L}_{2}\sim 0.0005,0.005,0.05)$ as well as comparable $(B^{L}_{2}\sim 0.5)$ branching ratio $B^{L}_{2}$. $M_{2}({\rm GeV})$ |  $B^{L}_{2}$ |  $|u_{2}|(\rm eV)$ |  $\epsilon^{ee}_{2}/10^{-8}$ |  $\epsilon^{e\mu}_{2}/10^{-9}$ |  $\epsilon^{\mu\mu}_{2}/10^{-9}$ |  $\epsilon^{\tau\tau}_{2}/10^{-7}$ |  $|\eta_{B}|/10^{-10}$
---|---|---|---|---|---|---|---
$1.4\times 10^{10}$ Fig. (5) |  $0.0006$ |  $48.1$ |  $-1.00$ |  $-1.20$ |  $-1.44$ |  $1.04$ |  $4.92$
Table 6: Baryon asymmetries through leptogenesis by solving DME in the energy
range $[10^{9},10^{12}]$ GeV for $M_{2}=1.4\times 10^{10}$ GeV.

(a)

(b)

Figure 6: Plot of obtained baryon asymmetries $|\eta_{B}|$ for different
masses of the triplet scalars in left (right) for $M_{1}(M_{2})$ in GeV. The
figures also show a comparative study of the efficiency of leptogenesis and
its dependence on the branching ratio hierarchy. The bound in green shows the
range of the baryon asymmetry from observations.

###  4.4 Summary

The results obtained in this section can be summarized as follows:

  1. 1.

The SCPV phase $\beta$ acts as a common link between the high energy CPV in
leptogenesis and low energy effect such as neutrinoless double beta decay as
shown in Fig. (1(b)).

  2. 2.

The hierarchical branching ratios result in different amounts of washout
effects along different flavor directions. The amount of washouts greatly
influences the efficiency of leptogenesis. Comparing figures (4(b))
($B_{1}^{L}=0.58$) and (5(b)) ($B_{2}^{L}=0.0006$), one can see that in the
first case, the washout effects along the flavor directions are more as
compared to the second case (except $ee$-flavor direction). This feature can
be attributed to the specific flavor structure of the triplet Yukawa coupling
matrices of the model. Although, in the second case, the extreme hierarchy in
the branching ratios can in principle suppress the flavored CP asymmetry
parameters, the presence of SCPV phase $\beta$ can neutralise the effect.

  3. 3.

The process of generation of baryon asymmetry from lepton asymmetry requires
that CP violation should be maximal. Observing Eq. (3.8), it is ideal to
generate maximal CP asymmetry for lower values of $M_{1}(M_{2})$. This would
require $B^{L}\sim B^{\phi}$. In our analysis, in bringing down the
$M_{1}(M_{2})$ scale as low as possible it can be observed that

    1. (a)

In case - I, $M_{1}\ll M_{2}$, (in Fig. (4), also Fig. (6(a))) the observable
range of baryon asymmetry can be achieved for triplet mass $M_{1}$ as low as
$\sim 3\times 10^{10}$ GeV for comparable branching ratios $B^{L}_{1}\sim
0.5$. Taking hierarchical branching ratios and requiring adequate flavored CP
asymmetries, baryon asymmetry can only be produced in the observable range for
a triplet mass $M_{1}>5\times 10^{10}$ GeV for a highly hierarchical branching
ratio $B^{L}_{1}\sim 0.0005$.

    2. (b)

In case - II, $M_{2}\ll M_{1}$, we find from Fig. 5 ( also Fig. (6(b))) the
observational range of baryon asymmetry can be achieved for triplet mass
$M_{2}$ as low as $\sim 1\times 10^{10}$ GeV for highly hierarchical branching
ratio $B^{L}_{2}\sim 0.0005$.

    3. (c)

From Fig. (6), for both cases - I and II, with hierarchical branching ratios
the baryon asymmetry increases for a fixed value of $M_{1}(M_{2})$. As shown
in tables (3) and (5), the suppression in CP asymmetry due to the hierarchical
branching ratio, can be compensated by the extra source of CP violation due to
the presence of the SCPV phase $\beta$ in the neutrino mass matrix. As a
result, even with the modified washout effect from hierarchical branching
ratios, it is possible to achieve maximal CP violation and baryon asymmetry
can be achieved in the observable range. The flavor structure of the model is
suitable to draw these conclusions.

##  5 Conclusion

Spontaneous violation of CP symmetry is very promising in explaining CP
violation in low-energy processes like neutrino oscillation and high-energy
process like leptogenesis. In general CP violation in these two sectors are
not related [21] and are model-dependent. In this paper, we have studied, in
an extended Standard model, neutrino mass generated via type-II seesaw
mechanism and leptogenesis from the decay of heavy scalar triplet with a
common source of CP violation. The model provides a viable explanation of
baryon asymmetry through thermal leptogenesis, by sourcing enough CP asymmetry
spontaneously broken due to the presence of a scalar singlet with complex VEV.
Moreover, the neutrino mass matrix exhibits TBM structure which generally
fails to address the occurrence of non-zero CHOOZ mixing angle, $\theta_{13}$.
By incorporating small perturbations around flavon VEV in the high-energy
scale, it can accommodate $\theta_{13}\neq 0$ in the model. With the latest
neutrino oscillation data, bound on total neutrino mass and the effective
neutrino mass, we have analyzed the parameter space of the model. Using the
constrained parameter space of the model we have studied flavored leptogenesis
from the decay of the triplet scalars. The flavor structure of the model
suppresses flavor-independent CP asymmetry and hence we explore the effect of
flavor in leptogenesis by using a set of flavor-covariant Boltzmann equations
considering a density matrix formalism. In our study, the scale of
leptogenesis is $\mathcal{O}(10^{10})$ GeV. We also study the role of
hierarchical branching ratios and the SCPV phase $\beta$ in generating maximal
CP asymmetry to generate baryon asymmetry in the observational range.
Requiring mass of triplet as low as possible, we find BAU in the observable
range can be produced for $M_{1}=3\times 10^{10}$ GeV for comparable branching
ratios. Whereas the same can be achieved for $M_{2}=1.4\times 10^{10}$ GeV for
hierarchical branching ratios. The flavor structure of the triplet Yukawa
couplings plays an important role in generating the washout effects which are
inevitable for thermal leptogenesis.

##  Appendix A Scalar potential of the model

The scalar potential invariant under the symmetries as given in table (1), can
be generally realised as

$V=V_{S}+V_{\phi}+V_{\Delta}+V_{S\phi}+V_{S\Delta}+V_{\phi\Delta}+V_{S\phi\Delta},$
(A.1)

where

$\displaystyle V_{S}$
$\displaystyle=\mu^{2}_{S}(S^{2}+S^{*2})+m^{2}_{S}S^{*}S+\lambda^{\prime}_{S}(S^{4}+S^{*4})+\lambda^{\prime\prime}_{S}S^{*}S(S^{2}+S^{*2})+\lambda_{S}(S^{*}S)^{2},$
(A.2) $\displaystyle V_{\phi}$
$\displaystyle=m^{2}_{\phi}\phi^{\dagger}\phi+\lambda_{\phi}(\phi^{\dagger}\phi)^{2},$
(A.3) $\displaystyle V_{\Delta}$
$\displaystyle=\Sigma_{a}M^{2}_{a}Tr(\Delta^{\dagger}_{a}\Delta_{a})+\Sigma_{a,b}[(\lambda_{\Delta})_{ab}Tr(\Delta^{\dagger}_{a}\Delta_{a})Tr(\Delta^{\dagger}_{b}\Delta_{b})+(\lambda^{\prime}_{\Delta})_{ab}Tr(\Delta^{\dagger}_{a}\Delta_{a}\Delta^{\dagger}_{b}\Delta_{b})],$
(A.4)

$\displaystyle V_{S\phi}$
$\displaystyle=\eta_{S}(S^{*}S)(\phi^{\dagger}\phi)+\eta^{\prime}_{S}(S^{2}+S^{*2})(\phi^{\dagger}\phi),$
(A.5)

$\displaystyle V_{S\Delta}$
$\displaystyle=\Sigma_{a}Tr(\Delta^{\dagger}_{a}\Delta_{a})[\eta_{a}(S{2}+S^{*2})+\xi_{a}S^{*}S],$
(A.6)

$\displaystyle V_{\phi\Delta}$
$\displaystyle=\Sigma_{a}[\xi^{\prime}_{a}(\phi^{\dagger}\phi)Tr(\Delta^{\dagger}_{a}\Delta_{a})+\xi^{\prime\prime}_{a}(\phi^{\dagger}\Delta^{\dagger}_{a}\Delta_{a}\phi)]+(\mu_{2}M_{2}\tilde{\phi}^{T}\Delta_{2}\tilde{\phi}+{\rm
H.c.}),$ (A.7)

$\displaystyle V_{S\phi\Delta}$
$\displaystyle=\tilde{\phi}^{T}\Delta_{1}\tilde{\phi}(\lambda_{1}S+\lambda^{\prime}_{1}S^{*})+{\rm
H.c.},$ (A.8)

where $\tilde{\phi}=i\sigma_{2}\phi^{*}$. It is interesting to note that a
viable leptogenesis scenario in this context requires a term
$\mu_{1}M_{1}\tilde{\phi}^{T}\Delta_{1}\tilde{\phi}$ such a term is forbidden
due to the choice of $Z_{4}$ symmetry. However, the complex VEV acquired by
the scalar singlet $S$ enables this term to be generated from the scalar
potential as shown in Eq. (A.8). Furthermore, the addition of two heavy scalar
fields allows us to explore the spontaneous breaking of the discrete $A_{4}$
symmetry.

The potential can be written in tree level as

$V_{0}=m^{2}_{S}v^{2}_{S}+\lambda_{S}v^{4}_{S}+2(\mu^{2}_{S}+\lambda^{\prime\prime}_{S}v^{2}_{S})v^{2}_{S}\cos(2\alpha)+2\lambda^{\prime}_{S}v^{4}_{S}\cos(4\alpha).$
(A.9)

Minimizing the potential with respect to $v_{S}$ and $\alpha$ we obtain,

$\frac{\partial V_{0}}{\partial
v_{S}}=2v_{S}[m^{2}_{S}+2\lambda_{S}v^{2}_{S}+2(\mu^{2}_{S}+2\lambda^{\prime\prime}_{S}v^{2}_{S})\cos(2\alpha)+4\lambda^{\prime}_{S}v^{2}_{S}\cos(4\alpha)]=0,$
(A.10)

and

$\frac{\partial
V_{0}}{\partial\alpha}=-4v^{2}_{S}\sin(2\alpha)[(\mu^{2}_{S}+\lambda^{\prime\prime}_{S}v^{2}_{S})+4\lambda^{\prime}_{S}v^{2}_{S}\cos(2\alpha)]=0.$
(A.11)

Three possible solution to the above equations, excluding the trivial solution
$v_{S}=0$ that leads to $V_{0}=0$, are given by

  1. 1.

$v^{2}_{S}=-\frac{m^{2}_{S}+2\mu^{2}_{S}}{2(\lambda_{S}+2\lambda^{\prime}_{S}+2\lambda^{\prime\prime}_{S})},\quad\alpha=0,\pm\pi,$
(A.12)

  2. 2.

$v^{2}_{S}=\frac{-m^{2}_{S}+2\mu^{2}_{S}}{2(\lambda_{S}+2\lambda^{\prime}_{S}-2\lambda^{\prime\prime}_{S})},\quad\alpha=\pm\frac{\pi}{2},$
(A.13)

  3. 3.

$v^{2}_{S}=\frac{-2\lambda^{\prime}_{S}m^{2}_{S}+\lambda^{\prime\prime}_{S}\mu^{2}_{S}}{4\lambda_{S}\lambda^{\prime}_{S}-8\lambda^{\prime}_{S}-\lambda^{\prime\prime}_{S}},\quad\cos(2\alpha)=-\frac{\mu^{2}_{S}+\lambda^{\prime\prime}_{S}v^{2}_{S}}{4\lambda^{\prime}_{S}v^{2}_{S}}.$
(A.14)

To show that the last solution, which also leads in this case to the
spontaneous breaking of the CP symmetry, corresponds to the global minimum of
the potential, we consider $m^{2}_{S}<0$, $\lambda^{\prime\prime}_{S}\simeq
0$, $\mu_{S}\simeq 0$ and $\lambda_{S}>2\lambda^{\prime}_{S}>0$ and we obtain
from Eq. (A.13),

$v^{2}_{S}\simeq-\frac{m^{2}_{S}}{2(\lambda_{S}+2\lambda^{\prime}_{S})},\quad\alpha=0,\pm\frac{\pi}{2},\pm\pi.$
(A.15)

For cases - 1, and 2,

$V_{0}\simeq-\frac{m^{4}_{S}}{4(\lambda_{S}+2\lambda^{\prime}_{S})}.$ (A.16)

For case - 3

$v^{2}_{S}\simeq-\frac{m^{2}_{S}}{2(\lambda_{S}-2\lambda^{\prime}_{S})},\quad\alpha\simeq\pm\frac{\pi}{4},$
(A.17)

which leads to

$V_{0}\simeq-\frac{m^{4}_{S}}{4(\lambda_{S}-2\lambda^{\prime}_{S})}$ (A.18)

which is the absolute minimum of the potential.

##  Appendix B Reaction Densities

The space-time density of total decay of triplet scalar $\Delta$ and its anti-
particle $\overline{\Delta}$, $\gamma_{D}$ and that of $2\leftrightarrow 2$
scattering, $\gamma_{A}$ are given by,

$\gamma_{D}=s\Gamma_{\Delta}\Sigma^{eq}_{\Delta}\frac{K_{1}(z)}{K_{2}(z)},$
(B.1) $\gamma_{A}=\frac{M^{4}_{a}}{64\pi^{4}}\int^{\infty}_{x_{\rm
min}}dx\sqrt{x}\frac{K_{1}\left(z\sqrt{x}\right)\hat{\sigma}_{S}}{z}.$ (B.2)

Here, $x={\rm s}/M^{2}_{a}$, s is the Mandelstam variable, square of center of
mass energy [29], $\hat{\sigma}_{S}$ is the reduced cross-section of
$2\rightarrow 2$ scatterings (relevant diagrams in fig.(7)). For processes
mediated by gauge bosons $x_{\rm min}=4$ and for Yukawa-induced reactions
$x_{\rm min}=0$.

The corresponding reaction density of scattering is given by,

$\gamma_{A}=\frac{M_{a}T^{3}e^{-\frac{2M_{a}}{T}}}{64\pi^{4}}(9g^{4}+12g_{2}^{2}g^{2}_{\mathcal{Y}}+3g^{4}_{Y})\left(1+\frac{3T}{4M_{a}}\right).$
(B.3)

Figure 7: Feynman diagrams that contribute to the interaction rate
$\gamma_{A}$

Here we take $\Gamma_{\Delta}=\frac{M^{2}_{a}\tilde{m}_{\Delta}}{16\pi
v^{2}\sqrt{B^{L}B^{\phi}}}$, $\tilde{m}_{\Delta}={\rm
Tr}(m^{(a)\dagger}_{\nu}m^{(a)}_{\nu})$, $\Sigma^{\rm eq}_{\Delta}=2Y^{\rm
eq}_{\Delta}=2\times\frac{45g_{\Delta}}{4\pi^{4}g_{*}}z^{2}K_{2}(z)$ and
$g_{\Delta}=3$, $g_{*}=106.75$ are the triplet scalar and SM particle-degrees
of freedom, respectively [26]. $g_{2},g_{\mathcal{Y}}$ are the coupling
constants corresponding to $SU(2)_{L}$ and $U(1)_{\mathcal{Y}}$ interactions,
respectively, having values, $g_{2}=\frac{e}{\sin\theta_{W}}=0.651742$ and
$g_{\mathcal{Y}}=\frac{e}{\cos\theta_{W}}=0.461388$. The other reaction
densities, $\gamma$’s are to be calculated using the integration given in Eq.
(B.2).

Within the temperature range $10^{9}$ GeV $<T<10^{12}$ GeV, the chemical
potential relations are given by:

$\sum_{i=1,2,3}(\mu_{q_{i}}+2\mu_{u_{i}}-\mu_{d_{i}})-\sum_{\alpha=e,\mu,\tau}(\mu_{l_{\alpha}}+\mu_{e_{\alpha}})+2\mu_{\phi}6\mu_{\Delta}=0\quad(\Delta_{Y}=0)$
(B.4)
$\sum_{i=1,2,3}(2\mu_{q_{i}}+\mu_{u_{i}}+\mu_{d_{i}})=0\quad(\Delta_{B}=0)$
(B.5) $\sum_{i=1,2,3}(2\mu_{q_{i}}-\mu_{u_{i}}-\mu_{d_{i}})=0\quad(\text{QCD
sphaleron})$ (B.6)
$\sum_{i=1,2,3}2\mu_{q_{i}}+\sum_{\alpha=e,\mu,\tau}\mu_{l_{\alpha}}=0\quad(\text{EW
sphaleron})$ (B.7) $\mu_{q_{i}}-\mu_{u_{i}}+\mu_{\phi}=0\quad(\text{up-type
quark Yukawa})$ (B.8) $\mu_{q_{i}}-\mu_{d_{i}}-\mu_{\phi}=0\quad(\text{down-
type quark Yukawa})$ (B.9)
$\mu_{l_{\alpha}}-\mu_{e_{\alpha}}-\mu_{\phi}=0\quad(\text{charged lepton
Yukawa})$ (B.10)

##  Appendix C Estimation of flavored CP asymmetries

The important factors appearing in the flavored CP asymmetry parameters as
given in Eq. (3.8) are calculated as follows:

$\rm{Im}\left[m^{(1)}_{\nu,ee}m^{*}_{\nu,ee}\right]=\frac{2}{3}z_{1}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\left(1-\varepsilon_{1}-\varepsilon_{2}\right)\sin(\beta-\sigma_{1}),$
(C.1) $\displaystyle\rm{Im}\left[m^{(1)}_{\nu,e\mu}m^{*}_{\nu,e\mu}\right]$
$\displaystyle=$
$\displaystyle-\frac{2}{3}\varepsilon_{1}z_{1}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{1})$
(C.2) $\displaystyle=$ $\displaystyle\rm{Im}\left[m^{(1)}_{\nu,\mu
e}m^{*}_{\nu,\mu e}\right],$
$\displaystyle\rm{Im}\left[m^{(1)}_{\nu,\mu\mu}m^{*}_{\nu,\mu\mu}\right]$
$\displaystyle=$
$\displaystyle\varepsilon_{2}z_{1}\Big{[}\frac{1}{3}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{1})$
(C.3) $\displaystyle+$
$\displaystyle\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{2})\Big{]},$
$\displaystyle\rm{Im}\left[m^{(1)}_{\nu,\tau\tau}m^{*}_{\nu,\tau\tau}\right]$
$\displaystyle=$
$\displaystyle\varepsilon_{1}z_{1}\Big{[}\frac{1}{3}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{1})$
(C.4) $\displaystyle+$
$\displaystyle\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{2})\Big{]},$
$\displaystyle\rm{Im}\left[m^{(2)}_{\nu,ee}m^{*}_{\nu,ee}\right]$
$\displaystyle=$
$\displaystyle-\frac{2}{3}z_{2}\Big{[}\frac{2}{3}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}(1-2\varepsilon_{1}-2\varepsilon_{2})\sin\sigma_{1}$
(C.5) $\displaystyle+$
$\displaystyle\frac{1}{3}z_{1}(1+\varepsilon_{1}+\varepsilon_{2})\sin\beta\Big{]},$
$\displaystyle\rm{Im}\left[m^{(2)}_{\nu,e\mu}m^{*}_{\nu,e\mu}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{3}z_{2}\Big{[}(\frac{7}{6}\varepsilon_{1}+\frac{1}{6}\varepsilon_{2}-\frac{1}{3})\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{3}(1+\varepsilon_{1}+\varepsilon_{2})z_{1}\sin\beta$
$\displaystyle-$
$\displaystyle\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2})\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{2}\Big{]}$
$\displaystyle=$ $\displaystyle\rm{Im}\left[m^{(2)}_{\nu,\mu e}m^{*}_{\nu,\mu
e}\right],$
$\displaystyle\rm{Im}\left[m^{(2)}_{\nu,\mu\mu}m^{*}_{\nu,\mu\mu}\right]$
$\displaystyle=$
$\displaystyle-\frac{2}{3}z_{2}\Big{[}(-\frac{5}{6}\varepsilon_{1}+\frac{1}{6}\varepsilon_{2}-\frac{1}{6})\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{1}$
(C.6) $\displaystyle+$
$\displaystyle\frac{1}{3}(1+\varepsilon_{1}+\varepsilon_{2})z_{1}\sin\beta$
$\displaystyle+$
$\displaystyle\frac{1}{2}(1-\varepsilon_{1}-3\varepsilon_{2})\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{2}\Big{]},$

and

$\displaystyle\rm{Im}\left[m^{(2)}_{\nu,\tau\tau}m^{*}_{\nu,\tau\tau}\right]$
$\displaystyle=$
$\displaystyle-\frac{2}{3}z_{2}\Big{[}(\frac{1}{6}\varepsilon_{1}-\frac{5}{6}\varepsilon_{2}+\frac{1}{6})\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{1}$
(C.7) $\displaystyle+$
$\displaystyle\frac{1}{3}(1+\varepsilon_{1}+\varepsilon_{2})z_{1}\sin\beta$
$\displaystyle+$
$\displaystyle\frac{1}{2}(1-3\varepsilon_{1}-\varepsilon_{2})\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{2}\Big{]}.$

Finally from equations (4.1), (C.1), (C.2), (C.3) and (C.4), the flavored CP
asymmetries $\epsilon^{\alpha\beta}_{1}$ are formed as,

$\epsilon^{ee}_{1}=\frac{M^{2}_{1}|u_{1}|^{2}}{2\pi\left(3z^{2}_{1}v^{4}+4M^{4}_{1}|u_{1}|^{4}\right)}\Big{[}\frac{2}{3}z_{1}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\left(1-\varepsilon_{1}-\varepsilon_{2}\right)\sin(\beta-\sigma_{1})\Big{]},$
(C.8) $\displaystyle\epsilon^{e\mu}_{1}$ $\displaystyle\approx$
$\displaystyle\frac{M^{2}_{1}|u_{1}|^{2}}{2\pi\left(3z^{2}_{1}v^{4}+4M^{4}_{1}|u_{1}|^{4}\right)}\Big{[}-\frac{2}{3}\varepsilon_{1}z_{1}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{1})\Big{]}$
(C.9) $\displaystyle\approx$ $\displaystyle\epsilon^{\mu e}_{1},$
$\displaystyle\epsilon^{\mu\mu}_{1}$ $\displaystyle\approx$
$\displaystyle\frac{M^{2}_{1}|u_{1}|^{2}}{2\pi\left(3z^{2}_{1}v^{4}+4M^{4}_{1}|u_{1}|^{4}\right)}\times\varepsilon_{2}z_{1}\Big{[}\frac{1}{3}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{1})$
(C.10) $\displaystyle+$
$\displaystyle\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{2})\Big{]},$

and

$\displaystyle\epsilon^{\tau\tau}_{1}$ $\displaystyle\approx$
$\displaystyle\frac{M^{2}_{1}|u_{1}|^{2}}{2\pi\left(3z^{2}_{1}v^{4}+4M^{4}_{1}|u_{1}|^{4}\right)}\times\varepsilon_{1}z_{1}\Big{[}\frac{1}{3}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{1})$
(C.11) $\displaystyle+$
$\displaystyle\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin(\beta-\sigma_{2})\Big{]}.$

From equations (4.2), (C.5), (C), (C.6) and (C.7), the flavored CP asymmetries
$\epsilon^{\alpha\beta}_{2}$ are formed as,

$\displaystyle\epsilon^{ee}_{2}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{2}|u_{2}|^{2}}{4\pi\left(z^{2}_{2}v^{4}(1-2\varepsilon_{1}-2\varepsilon_{2})+2M^{4}_{2}|u_{2}|^{4}\right)}\times-\frac{2}{3}z_{2}$
(C.12)
$\displaystyle\Big{[}\frac{2}{3}\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}(1-2\varepsilon_{1}-2\varepsilon_{2})\sin\sigma_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{3}z_{1}(1+\varepsilon_{1}+\varepsilon_{2})\sin\beta\Big{]},$
$\displaystyle\epsilon^{e\mu}_{2}$ $\displaystyle\approx$
$\displaystyle\frac{M^{2}_{2}|u_{2}|^{2}}{4\pi\left(z^{2}_{2}v^{4}(1-2\varepsilon_{1}-2\varepsilon_{2})+2M^{4}_{2}|u_{2}|^{4}\right)}\times\frac{1}{3}z_{2}$
(C.13)
$\displaystyle\Big{[}(\frac{7}{6}\varepsilon_{1}+\frac{1}{6}\varepsilon_{2}-\frac{1}{3})\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{3}(1+\varepsilon_{1}+\varepsilon_{2})z_{1}\sin\beta$
$\displaystyle-$
$\displaystyle\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2})\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{2}\Big{]}$
$\displaystyle\approx$ $\displaystyle\epsilon^{\mu e}_{2},$
$\displaystyle\epsilon^{\mu\mu}_{2}$ $\displaystyle\approx$
$\displaystyle\frac{M^{2}_{2}|u_{2}|^{2}}{4\pi\left(z^{2}_{2}v^{4}(1-2\varepsilon_{1}-2\varepsilon_{2})+2M^{4}_{2}|u_{2}|^{4}\right)}\times-\frac{2}{3}z_{2}$
(C.14)
$\displaystyle\Big{[}(-\frac{5}{6}\varepsilon_{1}+\frac{1}{6}\varepsilon_{2}-\frac{1}{6})\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{3}(1+\varepsilon_{1}+\varepsilon_{2})z_{1}\sin\beta$
$\displaystyle+$
$\displaystyle\frac{1}{2}(1-\varepsilon_{1}-3\varepsilon_{2})\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{2}\Big{]},$

and

$\displaystyle\epsilon^{\tau\tau}_{2}$ $\displaystyle\approx$
$\displaystyle\frac{M^{2}_{2}|u_{2}|^{2}}{4\pi\left(z^{2}_{2}v^{4}(1-2\varepsilon_{1}-2\varepsilon_{2})+2M^{4}_{2}|u_{2}|^{4}\right)}\times-\frac{2}{3}z_{2}$
(C.15)
$\displaystyle\Big{[}(\frac{1}{6}\varepsilon_{1}-\frac{5}{6}\varepsilon_{2}+\frac{1}{6})\left(z^{2}_{1}+z^{2}_{2}+2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{3}(1+\varepsilon_{1}+\varepsilon_{2})z_{1}\sin\beta$
$\displaystyle+$
$\displaystyle\frac{1}{2}(1-3\varepsilon_{1}-\varepsilon_{2})\left(z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2}\cos\beta\right)^{\frac{1}{2}}\sin\sigma_{2}\Big{]}.$

(a) $\epsilon^{ee}_{1}$ as a function of $\beta$

(b) $\epsilon^{e\mu}_{1}$ as a function of $\beta$

(c) $\epsilon^{\mu\mu}_{1}$ as a function of $\beta$

(d) $\epsilon^{\tau\tau}_{1}$ as a function of $\beta$

Figure 8: Flavored CP asymmetries generated due to triplet $\Delta_{1}$
decays, as a function of phase $\beta$ for comparable and hierarchical
branching ratios. The triplet $\Delta_{1}$ mass $M_{1}$ is taken $2\times
10^{10}$ GeV.

(a) $\epsilon^{ee}_{2}$ as a function of $\beta$

(b) $\epsilon^{e\mu}_{2}$ as a function of $\beta$

(c) $\epsilon^{\mu\mu}_{2}$ as a function of $\beta$

(d) $\epsilon^{\tau\tau}_{2}$ as a function of $\beta$

Figure 9: Flavored CP asymmetries generated due to triplet $\Delta_{2}$
decays, as a function of phase $\beta$ for comparable and hierarchical
branching ratios. The triplet $\Delta_{2}$ mass $M_{2}$ is taken $2\times
10^{10}$ GeV.

## Acknowledgments

SC acknowledges the valuable discussions with Rohan Pramanick on Spontaneous
CP violation and leptogenesis.

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