# Scalar Triplet Leptogenesis with a CP violating phase

Sreerupa Chongdar<EMAIL_ADDRESS>Department of Physics and Astronomy,
National Institute of Technology Rourkela, Sundargarh, Odisha, India, $769008$
Sasmita Mishra<EMAIL_ADDRESS>Department of Physics and Astronomy,
National Institute of Technology Rourkela, Sundargarh, Odisha, India, $769008$

###### Abstract

We study baryogenesis through leptogenesis via decay of triplet scalars
embedded in the Standard Model. We consider two triplets scenario where the
vacuum expectation value developed by one of the triplets is complex. The
coupling of the triplet Higgs with the leptons and the Standard Model doublet
Higgs allows two decay channels, with branching ratios $B_{L}$ and $B_{\phi}$
respectively. It is known that the hierarchy between the two branching ratios
is sensitive to the generation of adequate CP violation and the efficiency of
leptogenesis. Requiring the observed baryon asymmetry in the observed range we
find the mass of the lightest triplet can be as low as $2\times 10^{10}$ GeV.
This motivates to study leptogenesis considering flavour effect. We study
flavour effects in leptogenesis and it is observed that adequate baryon
asymmetry can not be obtained for purely flavoured leptogenesis for the
hierarchical branching ratios, $B_{L}\gg B_{\phi}$. But the hierarchy in the
other way $B_{\phi}\gg B_{L}$ still allows successful baryogenesis through
leptogenesis. We also study resonant leptogenesis in the light of TeV scale
physics.

##  I Introduction

The origin of baryon asymmetry of the Universe (BAU) continues to be one of
the most important questions of fundamental physics. The observations from big
bang nucleosynthesis and cosmic microwave background (CMB) quantify the
asymmetry Spergel et al. (2003); Komatsu et al. (2009); Hinshaw et al. (2009);
Dunkley et al. (2009); Ade et al. (2016); Aghanim et al. (2020a),
$\eta_{B}=\frac{(n_{B}-n_{\overline{B}})}{n_{\gamma}}$ 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. The
Sakharov conditions Sakharov (1967) provide necessary ingredients for
dynamical generation of BAU viz; baryon number (B) violation, $C$ (charge
conjugation) and $CP$ (charge conjugation and parity) violation and out of
equilibrium dynamics in the processes involving baryons. Although these
ingredients are all present in the Standard Model (SM), the $CP$ violation is
not adequate and for the departure from thermal equilibrium, the electroweak
phase transition is not strongly first order. So one needs to go beyond the SM
to introduce new sources of $CP$ violation and provide a departure from
thermal equilibrium. Apart from BAU, there are other avenues where one needs
to contemplate beyond the SM. One such case is the results from neutrino
oscillation experiments that are the clearest evidence of the existence of new
physics. The experiments based on the neutrino oscillation phenomenon have
established the fact that this abundant particle in the Universe has non-zero
mass Fukuda et al. (1998); Ahmad et al. (2002a, b); Bahcall and Pena-Garay
(2004). The allowed ranges of neutrino oscillation parameters; three mixing
angles, $\theta_{12}$, $\theta_{23}$, $\theta_{13}$, two mass squared
differences $\Delta m^{2}_{21}$, $|\Delta m_{31}^{2}|$ ($\Delta
m^{2}_{ij}=m_{i}^{2}-m_{j}^{2}$) and one Dirac CP-violating phase $\delta$ by
latest neutrino oscillation experiments are given in table-1 Esteban et al.
(2020). Also the sum of neutrino masses are bounded below by recent Planck
bound $\sum m_{i}<0.12$ eV Aghanim et al. (2020b). This is in contrast to the
prediction of the SM, where neutrinos are massless. Subsequently one relies on
theories beyond the SM to circumvent the problem. There are attractive
neutrino mass models where neutrino masses arise via seesaw mechanism by
adding heavy states; right-handed neutrinos (type-I) Minkowski (1977);
Yanagida (1979); Mohapatra and Senjanovic (1980), scalar triplets (type-II) or
fermion triplets (type-III) and many more variants. In seesaw models, the
interaction of heavy states with the SM states originates neutrino mass along
with lepton number ($L$) asymmetry owing to decays of heavy states to the SM
lepton(s). The out of equilibrium decays of heavy states generate BAU through
leptogenesis mechanism Fukugita and Yanagida (1986). The lepton asymmetry is
converted to baryon asymmetry by $B-L$ conserving but $B+L$ violating
sphaleron process. In this framework, the Sakharov conditions are successfully
met. Thus, leptogenesis realizes a non-trivial link between two completely
independent experimental observations: BAU and neutrino mixing and hence mass.
In the present work we study baryogenesis through leptogenesis in the type-II
seesaw model where the SM is augmented with two scalar triplets, with one of
the triplets acquiring complex vacuum expectation value (vev).

The type-II seesaw mechanism is special from other variants in two ways.
First, the one-to-one relation between neutrino mass matrix and triplet Yukawa
coupling makes it easy to detect the flavor structure of the theory directly
from the neutrino mass matrix, which can be reconstructed from low energy
data. Second, only one triplet scalar can generate three light neutrino masses
though with this minimal set up there is a drawback with leptogenesis. In the
presence of complex couplings of the triplet with the SM Higgs and the
leptons, the leptonic CP asymmetry is generated only at higher loops and is
highly suppressed. Therefore, new sources (addition of heavy right handed
neutrino/ triplet scalar) for neutrino mass are required for successful
leptogenesis in the type-II framework. The addition of another triplet is
supported by another ground. Although substantial evidence has been collected
from experiments in support of neutrino oscillation, a lot remains to be known
including absolute values of masses, mass generation mechanism, and nature of
neutrinos: Dirac or Majorana. In the absence of a clear theoretical
understanding, various texture zeros in the neutrino mass matrix have been
investigated. The zero entries are often backed by some underlying flavor
symmetry in the lepton sector. Also, the reduced number of free parameters in
the neutrino mass matrix enhances the predictivity. In this context, two-zero
textures have gathered considerable attention. Two zero texture Frampton et
al. (2002); Fritzsch et al. (2011) in the lepton mass matrix arising from some
underlying flavor symmetry render useful studies. One such two zero texture
mass matrix represented as $B_{2}$ where $e-\mu$ and $\tau-\tau$ elements are
zero. This texture naturally emerges by adding two triplet scalars in the SM
and imposing a horizontal symmetry group ${\mathcal{Z}_{3}}$ Grimus and
Lavoura (2005); Fritzsch et al. (2011). The references Meloni et al. (2014);
Singh (2020) study the viability of two zero textures in the recent neutrino
oscillation experimental data and Planck Aghanim et al. (2020a) data and show
that the texture $B_{2}$ is consistent with data in the normal ordering of
neutrino masses.

In models where neutrino mass is generated through type-II seesaw the relevant
part of the Lagrangian related to generation on neutrino mass is the coupling
of the triplet field to the SM lepton fields ($L$) through Yukawa type
coupling ($Y_{\Delta}-L-\Delta-L$) and that to the SM Higgs fields ($\phi$)
through scalar coupling ($\mu-\phi-\Delta-\phi$). We call them $\Delta-L$ and
$\Delta-\phi$ terms respectively. If the latter type of coupling is absent
then the lepton number is violated spontaneously by the vev, $\omega$ of the
neutral component of the scalar triplet (as $\Delta$ carries $2$ units of
lepton number). This leads to the existence of a massless triplet scalar
Majoron, which would contribute to the invisible width of the $Z$-boson. So
this model is excluded by CERN LEP. Inclusion of the $\Delta-\phi$ coupling
term leads to breaking of the lepton number explicitly and eliminates Majoron.
The vev of the scalar triple, $\omega$ gives rise to the mass matrix of the
neutrinos $m_{ij}=\sqrt{2}Y_{\Delta_{ij}}\omega$;
$\omega\approx\mu\upsilon^{2}/2M^{2}_{\Delta}$, where $\mu$ is the coupling
strength of $\Delta-\phi$ term and $\upsilon$ and $M_{\Delta}$ are the vev of
the SM Higgs and mass of the triplet scalar respectively. An upper limit on
$\omega$ can be obtained from $\rho$-parameter constraint:
($\rho=M_{W}^{2}/M_{Z}^{2}\cos^{2}\theta_{w}$); $\omega/\upsilon\leq 0.03$ or
$\omega<8$ GeV.

A scalar triplet unlike a fermion singlet undergoes SM gauge interactions. The
latter can keep the triplets close to thermal equilibrium at temperatures
$T\lesssim 10^{15}$ GeV. This can result in reduced efficiency and hence seem
difficult to fulfill the out-of-equilibrium decay condition. However, a more
precise calculation by solving the full set of Boltzmann equations show that
leptogenesis is efficient even at much lower temperature Hambye and Senjanovic
(2004); Hambye et al. (2006, 2001). The presence of both $\Delta-L$ and
$\Delta-\phi$ terms in the Lagrangian allows two different decays of triplet
scalar and lepton number is violated only if both types of decays co-exist.
The interplay of respective branching ratios $B_{L}$ and $B_{\phi}$ play an
important role in successful baryogenesis through leptogenesis. The amount of
$CP$ violation, $\epsilon$, and efficiency, $\eta$ 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 ($\epsilon\propto\sqrt{B_{L}B_{\phi}}$). Solving a
full set of Boltzmann equations it was shown in Hambye et al. (2006) that even
for extremely hierarchical branching ratios it is possible to achieve the
required BAU provided the mass of the lightest triplet is bounded below
$2.8\times 10^{11}$ GeV. In this work, we show due to the interplay of complex
phases of triplet vev and elements of the Yukawa matrix it is possible to
relax the bound on lightest triplet mass by little more than an order of
magnitude while maintaining the hierarchical branching ratios and hence not
negotiating on efficiency. Also, we study flavour effects in leptogenesis and
find that in the hierarchical branching ratio cases (i) $B_{\phi}\gg B_{L}$
and (ii) $B_{L}\gg B_{\phi}$ give different results. We find in the latter
case which is known as the purely flavoured leptogenesis (PFL), the $CP$
violation parameter comes out to be extremely small, and hence producing the
required BAU may not be viable. Nevertheless, the other hierarchical limit in
(i) stands viable. For completeness, we also study TeV scale leptogenesis
taking resonant effect and Sommerfeld correction into account.

The possibility of explaining BAU, neutrino mass, and inflation simultaneously
are studied in Barrie et al. (2021) by adding a triplet Higgs to the SM.
Triplet leptogenesis in the Left-right symmetric model is explored in Hallgren
et al. (2008); Rink et al. (2021). The authors of Parida et al. (2020), Parida
and Samantaray (2021) and Chakraborty et al. (2020) study purely triplet
leptogenesis along with dark matter and vacuum stability issues in the SM
extended by triplets, in the context of $SU(5)$ unification and in the minimal
$SO(10)$ breaking respectively. The SM extended by gauged $U(1)_{B-L}$ where
neutrino mass is generated by triplet Higgs and simultaneous explanation of
dark matter has been studied in Mahapatra et al. (2020). The study of triplet
scalar leptogenesis in presence of right-handed neutrinos is studied in
Aristizabal Sierra (2013) and Mishra and Giri (2020) where the latter
incorporates $S_{3}$ and $Z_{2}$ symmetry on the extended SM. A study of dark
matter, generation of tiny neutrino mass, and BAU are carried out in Lu and Gu
(2017) in presence of mixed inert scalar triplets. By adding a TeV scale
scalar Higgs triplet and a second Higgs doublet and a fermion triplet to the
SM, the authors of Hati and Sarkar (2016) study dark energy and BAU.

In section section (II) we discuss the light neutrino mass model with the SM
extended with two triplets and determine the unknown parameters from neutrino
oscillation data. In section (IV) and (V) we study the unflavoured flavoured
leptogenesis respectively. The discussion on TeV scale leptogenesis is done in
section (VI). And we outline our conclusions and future perspective in section
(VII).

##  II Type-II Seesaw Mechanism with two triplet scalar

In this section we consider the SM with one complex Higgs doublet, $\phi$ is
extended with two triplet scalars, $\Delta_{1},\Delta_{2}$ with hypercharge
${\mathcal{Y}}=2$ Bhupal Dev et al. (2013); Chaudhuri and Mukhopadhyaya
(2016); Chaudhuri et al. (2014);
$\Delta_{i}=(\delta_{i}^{++},\delta_{i}^{+},\delta_{i}^{0})$ with $i=1,2$. The
fields can be written in matrix form given by

$\phi=\begin{pmatrix}\phi^{+}\\\
\phi^{0}\end{pmatrix},\quad\Delta_{1}=\begin{pmatrix}\frac{\delta^{+}_{1}}{\sqrt{2}}&\delta^{++}_{1}\\\
\delta^{0}_{1}&-\frac{\delta^{+}_{1}}{\sqrt{2}}\end{pmatrix},\quad\Delta_{2}=\begin{pmatrix}\frac{\delta^{+}_{2}}{\sqrt{2}}&\delta^{++}_{2}\\\
\delta^{0}_{2}&-\frac{\delta^{+}_{2}}{\sqrt{2}}\end{pmatrix}.$ (1)

We can write the scalar potential in case of two triplet scalars, as

$\displaystyle V(\phi,\Delta_{1},\Delta_{2})$ $\displaystyle=$ $\displaystyle
a\phi^{{\dagger}}\phi+c(\phi^{{\dagger}}\phi)^{2}+\frac{b_{kl}}{2}{\rm
Tr}(\Delta^{{\dagger}}_{k}\Delta_{l})+\frac{d_{kl}}{4}({\rm
Tr}(\Delta^{{\dagger}}_{k}\Delta_{l}))^{2}+(e_{kl}-h_{kl})\phi^{{\dagger}}\phi{\rm
Tr}(\Delta^{{\dagger}}_{k}\Delta_{l})$ (2) $\displaystyle+$
$\displaystyle\frac{f_{kl}}{4}{\rm
Tr}(\Delta^{\dagger}_{k}\Delta^{\dagger}_{l}){\rm
Tr}(\Delta_{k}\Delta_{l})+h_{kl}\phi^{\dagger}\Delta^{\dagger}_{k}\Delta_{l}\phi+g{\rm
Tr}(\Delta^{{\dagger}}_{1}\Delta_{2})Tr(\Delta^{{\dagger}}_{2}\Delta_{1})$
$\displaystyle+$ $\displaystyle g^{\prime}{\rm
Tr}(\Delta^{{\dagger}}_{1}\Delta_{1}){\rm
Tr}(\Delta^{{\dagger}}_{2}\Delta_{2})+(\mu_{k}\phi^{{\dagger}}\Delta_{k}i\sigma_{2}\phi^{*}+h.c.),$

where $k,l=1,2$, and $\sigma$ is the Pauli matrix. The vacuum expectation
values (vev)s of the two triplet scalars and the Higgs doublet are given by

$<\delta_{1}^{0}>=\omega_{1},\quad<\delta_{2}^{0}>=\omega_{2},\quad<\phi^{0}>=\frac{v}{\sqrt{2}},$
(3)

where $v=246$ GeV. We have taken the vevs; $\omega_{1},\omega_{2}\ll v$,
keeping in mind the $\rho$-parameter constraint. There are two ways in which
one can achieve this requirement theoretically. One can assume $|\mu_{k}|\ll
v$ Grimus et al. (2000); Chaudhuri and Mukhopadhyaya (2016) or $b\gg v^{2}$ Ma
and Sarkar (1998). For considering two triplet scalars, there lies an extra
possibility of getting at least one CP-violating phase, which arises via a
relative phase between the triplets Ma and Sarkar (1998). For that purpose,
any of the two scalar triplets is made to have a complex vev. Let it be the
scalar triplet, $\Delta_{1}$. Consequently, the coefficient of the
corresponding trilinear term in the scalar potential is also considered to be
complex,

$\mu_{1}=|\mu_{1}|e^{i\beta},\quad\omega_{1}=|\omega_{1}|e^{i\alpha}.$ (4)

Now, $\mu$ and $\omega$ can be written in terms of real (unprimed) and
imaginary (primed) components,

$\mu=\begin{pmatrix}|\mu_{1}|cos\beta\\\
\mu_{2}\end{pmatrix},\quad\mu^{{}^{\prime}}=\begin{pmatrix}|\mu_{1}|sin\beta\\\
0\end{pmatrix},\quad\omega=\begin{pmatrix}|\omega_{1}|cos\alpha\\\
\omega_{2}\end{pmatrix},\quad\omega^{{}^{\prime}}=\begin{pmatrix}|\omega_{1}|sin\alpha\\\
0\end{pmatrix}.$ (5)

For calculation,

$\phi^{\dagger}\phi=\frac{1}{2}v^{2},$ (6)
$Tr(\Delta^{\dagger}_{k}\Delta_{l})=\omega^{T}_{k}\omega_{l}+\omega^{{}^{\prime}T}_{k}\omega^{{}^{\prime}}_{l},\quad\Delta^{\dagger}_{k}\Delta^{\dagger}_{l}=0,\quad\Delta_{k}\Delta_{l}=0,$
(7)
$\phi^{\dagger}\Delta_{k}\Delta^{{\dagger}}_{l}\phi=\frac{1}{2}\omega_{k}\omega^{*}_{l}v^{2},$
(8) $\phi^{\dagger}\Delta^{\dagger}_{l}\Delta_{k}\phi=0,$ (9)
$\mu_{k}\phi^{\dagger}\Delta_{k}i\sigma_{2}\phi^{*}=\frac{1}{2}v^{2}[\mu.\omega-\mu^{{}^{\prime}}.\omega^{{}^{\prime}}].$
(10)

As we have considered $\omega$’s to have small values, we neglect the quartic
terms, $Tr(\Delta_{1}^{\dagger}\Delta_{2})Tr(\Delta^{\dagger}_{2}\Delta_{1})$
and $Tr(\Delta_{1}^{\dagger}\Delta_{1})Tr(\Delta^{\dagger}_{2}\Delta_{2})$.
Now, minimising the potential we get the following relations among vevs and
phases as ,

$\omega=-v^{2}\left(B+\frac{1}{2}v^{2}(E-H)\right)^{-1}\mu,$ (11)
$sin\beta=\frac{\left(b_{11}+\frac{v^{2}}{2}(e_{11}-h_{11})\right)|\omega_{1}|sin\alpha}{v^{2}|\mu_{1}|},$
(12)

where $B=(b_{kl}),E=(e_{kl})$ and $H=(h_{kl})$. The equation (12) implies when
angle $\beta$ is absent, angle $\alpha$ has to be $n\pi$, $n=0,1,2,3$. This
also shows that the phases of $|\mu_{1}|$ and $|\omega_{1}|$ are related to
each other. The $\Delta L=2$ Yukawa interactions of the triplets are,

$\mathcal{L}_{Y}=\frac{(Y_{\Delta})^{k}_{ij}L^{T}_{i}Ci\sigma_{2}\Delta_{k}L_{j}}{\sqrt{2}}+h.c.,$
(13)

where $Y_{\Delta}$ is Yukawa coupling matrix and $C$ is the charge conjugation
matrix, $L_{i}$ denotes the left-handed lepton doublets, $i,j$ are the
summation indices over the three neutrino flavours, and $k=1,2$. The neutrino
mass matrix can be written in terms of the Yukawa coupling constants and the
real component of the triplet’s vev as Chaudhuri and Mukhopadhyaya (2016)

$(m_{\nu})_{ij}=Y^{(1)}_{\Delta_{ij}}|\omega_{1}|cos\alpha+Y^{(2)}_{\Delta_{ij}}\omega_{2}.$
(14)

The values of $\omega_{1}$, $\omega_{2}$ can be determined by the values of
the other parameters in the scalar potential. But the Yukawa coupling
constants are still undetermined. The Yukawa coupling matrix carries most of
the unknown parameters, thus the theory lacks in making a definite prediction
about fermion masses and flavour mixing.

##  III Neutrino mass model with two scalar triplets

In this section, we make a quantitative analysis of the elements of the
neutrino mass matrix given in eq. (14). We check allowed range of elements of
the mass matrix in light of the recent experimental data related to neutrino
mixing. To realize this, we conjecture the triplet Yukawa matrices to take the
form in such a manner that the neutrino mass matrix can be represented as

$\begin{pmatrix}Ae^{i\phi_{A}}&Be^{i\phi_{B}}&Ce^{i\phi_{C}}\\\
Be^{i\phi_{B}}&De^{i\phi_{D}}&Me^{i\phi_{M}}\\\
Ce^{i\phi_{C}}&Me^{i\phi_{M}}&Ne^{i\phi_{N}}\end{pmatrix}=\begin{pmatrix}ae^{i\phi_{a}}&be^{i\phi_{b}}&ce^{i\phi_{c}}\\\
be^{i\phi_{b}}&de^{i\phi_{d}}&me^{i\phi_{m}}\\\
ce^{i\phi_{c}}&me^{i\phi_{m}}&ne^{i\phi_{n}}\end{pmatrix}|\omega_{1}|cos\alpha+\begin{pmatrix}ae^{i\phi_{a^{\prime}}}&be^{i\phi_{b^{\prime}}}&ce^{i\phi_{c^{\prime}}}\\\
be^{i\phi_{b^{\prime}}}&de^{i\phi_{d^{\prime}}}&me^{i\phi_{m^{\prime}}}\\\
ce^{i\phi_{c^{\prime}}}&me^{i\phi_{m^{\prime}}}&ne^{i\phi_{n^{\prime}}}\end{pmatrix}\omega_{2},$
(15)

such that

$A^{2}=a^{2}[|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{a}-\phi_{a^{\prime}})],$
$\phi_{A}=tan^{-1}[\frac{|\omega_{1}|cos\alpha
sin\phi_{a}+\omega_{2}sin\phi_{a^{\prime}}}{|\omega_{1}|cos\alpha
cos\phi_{a}+\omega_{2}cos\phi_{a^{\prime}}}],$
$B^{2}=b^{2}[|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{b}-\phi_{b^{\prime}})],$
$\phi_{B}=tan^{-1}[\frac{|\omega_{1}|cos\alpha
sin\phi_{b}+\omega_{2}sin\phi_{b^{\prime}}}{|\omega_{1}|cos\alpha
cos\phi_{b}+\omega_{2}cos\phi_{b^{\prime}}}],$
$C^{2}=c^{2}[|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{c}-\phi_{c^{\prime}})],$
$\phi_{C}=tan^{-1}[\frac{|\omega_{1}|cos\alpha
sin\phi_{c}+\omega_{2}sin\phi_{c^{\prime}}}{|\omega_{1}|cos\alpha
cos\phi_{c}+\omega_{2}cos\phi_{c^{\prime}}}],$
$D^{2}=d^{2}[|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{d}-\phi_{d^{\prime}})],$
$\phi_{D}=tan^{-1}[\frac{|\omega_{1}|cos\alpha
sin\phi_{d}+\omega_{2}sin\phi_{d^{\prime}}}{|\omega_{1}|cos\alpha
cos\phi_{d}+\omega_{2}cos\phi_{d^{\prime}}}].$
$M^{2}=m^{2}[|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{m}-\phi_{m^{\prime}})],$
$\phi_{M}=tan^{-1}[\frac{|\omega_{1}|cos\alpha
sin\phi_{m}+\omega_{2}sin\phi_{m^{\prime}}}{|\omega_{1}|cos\alpha
cos\phi_{m}+\omega_{2}cos\phi_{m^{\prime}}}].$
$N^{2}=n^{2}[|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{n}-\phi_{n^{\prime}})],$
$\phi_{N}=tan^{-1}[\frac{|\omega_{1}|cos\alpha
sin\phi_{n}+\omega_{2}sin\phi_{n^{\prime}}}{|\omega_{1}|cos\alpha
cos\phi_{n}+\omega_{2}cos\phi_{n^{\prime}}}].$ (16)

In case of Majorana neutrinos, the neutrino mass matrix follows the
diagonalization

$m_{\nu}=Um^{diag}_{\nu}U^{T},$ (17)

where

$m^{diag}_{\nu}=\begin{pmatrix}m_{1}&0&0\\\ 0&m_{2}&0\\\
0&0&m_{3}\end{pmatrix},$ (18)

where $m_{1},m_{2}$ and $m_{3}$ are three neutrino mass eigen values. And $U$
is the PMNS matrix given by

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{pmatrix}{\rm
diag}(1,e^{i\alpha^{\prime}},e^{i(\beta^{\prime}+\delta)}),$ (19)

where $s_{ij}(c_{ij})\equiv\sin\theta_{ij}(\cos\theta_{ij})$ and
$\alpha^{\prime}$ and $\beta^{\prime}$ are Majorana phases. Now the neutrino
mass matrix elements can be expressed as:

$m_{ee}=c^{2}_{13}(m_{1}c^{2}_{12}+m_{2}e^{2i\alpha^{\prime}}s^{2}_{12})+m_{3}e^{2i\beta^{\prime}}s^{2}_{13}$
(20)
$m_{e\mu}=c_{13}((e^{2i\alpha^{\prime}}m_{2}-m_{1})s_{12}c_{12}c_{23}+e^{i\delta}(e^{2i\beta^{\prime}}m_{3}-e^{2i\alpha^{\prime}}m_{2}s^{2}_{12}-m_{1}c^{2}_{12})s_{23}s_{13})$
(21)
$m_{e\tau}=c_{13}((m_{1}-e^{2i\alpha^{\prime}}m_{2})s_{12}c_{12}s_{23}+e^{i\delta}(e^{2i\beta^{\prime}}m_{3}-e^{2i\alpha^{\prime}}m_{2}s^{2}_{12}-m_{1}c^{2}_{12})c_{23}s_{13})$
(22)
$m_{\mu\mu}=m_{1}(s_{12}c_{23}+c_{12}s_{23}s_{13}e^{i\delta})^{2}+m_{2}(c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta})^{2}e^{2i\alpha^{\prime}}+m_{3}s^{2}_{23}c^{2}_{13}e^{2i(\beta^{\prime}+\delta)}$
(23) $\displaystyle m_{\mu\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2}sin2\theta_{12}cos2\theta_{23}s_{13}e^{i\delta}(m_{1}-m_{2}e^{2i\alpha^{\prime}}){}$
$\displaystyle{}-\frac{1}{2}sin2\theta_{23}(m_{2}c^{2}_{12}e^{2i\alpha^{\prime}}+m_{1}s^{2}_{12}-e^{2i\delta}(m_{3}c^{2}_{13}e^{2i\beta^{\prime}}+s^{2}_{13}(m_{1}c^{2}_{12}+m_{2}e^{2i\alpha^{\prime}}s^{2}_{12}))){}$
$m_{\tau\tau}=m_{1}(c_{12}c_{23}s_{13}e^{i\delta}-s_{12}s_{23})^{2}+m_{2}(c_{12}s_{23}+s_{12}c_{23}s_{13}e^{i\delta})^{2}e^{2i\alpha^{\prime}}+m_{3}c^{2}_{23}c^{2}_{13}e^{2i(\beta^{\prime}+\delta)}$
(25)

NuFIT 5.0 (2020)
---
Neutrino Oscillation parameters |  Normal ordering 1$\sigma$ |  Normal ordering 3$\sigma$ |  Inverted ordering 1$\sigma$ |  Inverted ordering 3$\sigma$
$\theta_{12}/^{\circ}$ |  $33.44^{+0.77}_{-0.74}$ |  $31.27\rightarrow 35.86$ |  $33.45^{+0.78}_{-0.75}$ |  $31.27\rightarrow 35.87$
$sin^{2}\theta_{12}$ |  $0.304^{+0.012}_{-0.012}$ |  $0.269\rightarrow 0.343$ |  $0.304^{+0.013}_{-0.012}$ |  $0.269\rightarrow 0.343$
$\theta_{23}/^{\circ}$ |  $49.2^{+0.9}_{-1.2}$ |  $40.1\rightarrow 51.7$ |  $49.3^{+0.9}_{-1.1}$ |  $40.3\rightarrow 51.8$
$sin^{2}\theta_{23}$ |  $0.573^{+0.016}_{-0.020}$ |  $0.415\rightarrow 0.616$ |  $0.575^{+0.016}_{-0.019}$ |  $0.419\rightarrow 0.617$
$\theta_{13}/^{\circ}$ |  $8.57^{+0.12}_{-0.12}$ |  $8.20\rightarrow 8.93$ |  $8.60^{+0.12}_{-0.12}$ |  $8.24\rightarrow 8.96$
$sin^{2}\theta_{13}$ |  $0.02219^{+0.00062}_{-0.00063}$ |  $0.02032\rightarrow 0.02410$ |  $0.02238^{+0.00063}_{-0.00062}$ |  $0.02052\rightarrow 0.02428$
$\delta/^{\circ}$ |  $197^{+27}_{-24}$ |  $120\rightarrow 369$ |  $282^{+26}_{-30}$ |  $193\rightarrow 352$
$\frac{\Delta m^{2}_{21}}{10^{-5}eV^{2}}$ |  $7.42^{+0.21}_{-0.20}$ |  $6.82\rightarrow 8.04$ |  $7.42^{+0.21}_{-0.20}$ |  $6.82\rightarrow 8.04$
$\frac{\Delta m^{2}_{3l}}{10^{-3}eV^{2}}$ |  $+2.517^{+0.026}_{-0.028}$ |  $+2.435\rightarrow+2.598$ |  $-2.498^{+0.028}_{-0.028}$ |  $-2.581\rightarrow-2.414$
Table 1: Allowed ranges of neutrino oscillation parameters for normal and
inverted mass ordering

Comparing eq.(15) with eq.s (20 \- 25) we can see
$A=|m_{ee}|,B=|m_{e\mu}|,C=|m_{e\tau}|,D=|m_{\mu\mu}|,M=|m_{\mu\tau}|$ and
$N=|m_{\tau\tau}|$. To determine the values of the parameters ($A,B,C,D,M,N$)
we take the allowed ranges of neutrino oscillation parameters (three mixing
angles, $\theta_{12}$, $\theta_{23}$, $\theta_{13}$, two mass squared
differences $\Delta m^{2}_{21}$, $\Delta m^{2}_{3l}$ and one Dirac CP-
violating phase $\delta$) by latest neutrino oscillation experiments given in
table-1. In our study we assume normal ordering of mass eigen values. The
allowed values of the parameters as a function lightest neutrino mass $m_{1}$
consistent with experimental data are shown in fig.(11). A similar analysis on
elements of neutrino mass matrix is done in Merle and Rodejohann (2006). In
the next section we calculate CP asymmetry using these values of the
parameters that naturally arise in the formulation.

##  IV Baryogenesis through unflavoured Leptogenesis

In this section, we study flavour independent leptogenesis. We assume that
there was no lepton asymmetry when the temperature of the Universe was much
higher than the masses of triplets. As the temperature reached the mass scale
of one of the heavier triplets, lepton asymmetry was created due to the decay
of the heavier triplet, $\Delta_{1}$ in our case. The lepton asymmetry
produced in the $\Delta_{1}$ decays would be washed out by the lepton number
nonconserving interactions of $\Delta_{2}$. Thus, the effects of lepton number
violating processes due to $\Delta_{1}$, can be safely neglected. Only the
lepton number asymmetry arising from the interactions of $\Delta_{2}$ would
evolve with time. From now onwards we represent $\Delta_{2}$ as $\Delta$ with
mass $M_{\Delta}$ for our study.

###  IV.1 CP asymmetry parameter

The gauge invariance of $SU(2)_{L}\times U(1)_{\mathcal{Y}}$ at an energy
scale above electroweak symmetry breaking, ensures that we can pick up any one
of the three components of the triplet under consideration and the results are
valid for the other two. By choosing $\delta^{++}$ the decay channels possible
are,

$\delta^{++}\rightarrow\begin{cases}L_{i}^{+}\,L_{j}^{+}&(L=-2)\\\
\phi^{+}\,\phi^{+}&(L=0)\end{cases}$

with branching ratios

$\Gamma(\Delta\rightarrow LL)=\frac{M_{\Delta}}{16\pi}{\rm
Tr}(Y_{\Delta}Y_{\Delta}^{\dagger})=B_{L}\Gamma_{\Delta},\quad\Gamma(\Delta\rightarrow\phi\,\phi)=\frac{M_{\Delta}}{16\pi}\mu\mu^{\dagger}=B_{\phi}\Gamma_{\Delta}$
(26)

where $\Gamma_{\Delta}$ is the total decay width and $B_{L}$ and $B_{\phi}$
are the tree level branching ratios to leptons and SM Higgs doublets
respectively with $B_{L}+B_{\phi}=1$. The simultaneous existence of the above
two final states indicates that the lepton number is violated by two units.
However, the lepton asymmetry generated by $\delta^{++}$ would be neutralized
by the decays of $\delta^{--}$ unless CP conservation is also violated and the
decays are out of thermal equilibrium in the early Universe.

In the presence of complex couplings of the triplets with the SM Higgs and the
leptons, the interference of the tree-level and one-loop decay amplitudes
account for the required CP asymmetry. The tree-level and one-loop Feynman
diagrams are shown in fig.s (1) and (2), respectively. For a single triplet
scalar, the CP asymmetry is generated only at higher loops. So the addition of
heavier states (in our case another triplet scalar) contributes not only to CP
asymmetry but also to the neutrino masses. In this framework, vertex
correction contribution to the leptogenesis CP asymmetry is not considered at
the one-loop level. The average lepton number produced per decay gives the
expression of CP asymmetry,

$\epsilon=\Delta L\frac{\Gamma(\bar{\Delta}\rightarrow
LL)-\Gamma(\Delta\rightarrow\bar{L}\bar{L})}{\Gamma_{\Delta}+\Gamma_{\bar{\Delta}}}.$
(27)

Figure 1: Tree-level diagrams of scalar triplet decay to leptons and to Higgs
doublets respectively. Figure 2: One-loop diagram of scalar triplet decay to
leptons.

Here, $\Gamma$’s signify the decay/inverse decay rates. As each triplet decay
$\bar{\Delta}\rightarrow LL$ generates two leptons, so the factor $\Delta
L(=2)$ is multiplied. The CP asymmetry parameter in this case is given by
Branco et al. (2012)

$\epsilon=\frac{2M_{\Delta_{i}}}{4\pi
v^{2}}\sqrt{B_{L}B_{\phi}}\frac{Im[Tr(m^{(i)}_{\nu}m^{{\dagger}}_{\nu})]}{[Tr(m^{(i){\dagger}}_{\nu}m^{(i)}_{\nu})]^{\frac{1}{2}}},$
(28)

where neutrino mass $m_{\nu}=m^{(1)}_{\nu}+m^{(2)}_{\nu}$, $m^{(1)}_{\nu}$ is
the neutrino mass contributed by the heavier triplet scalar $\Delta_{1}$ and
$m^{(2)}_{\nu}$ is the neutrino mass contributed by the lighter triplet scalar
$\Delta_{2}$. The decay of the lighter triplet $\Delta_{2}$ gives rise to the
CP asymmetry, while the effect of heavier triplet $\Delta_{1}$ gets washed
out. Considering this fact, the CP asymmetry parameter is given by

$\epsilon=\frac{2M_{\Delta}}{4\pi
v^{2}}\sqrt{B_{L}B_{\phi}}\frac{Im[Tr(m^{(2)}_{\nu}m^{{\dagger}}_{\nu})]}{[Tr(m^{(2){\dagger}}_{\nu}m^{(2)}_{\nu})]^{\frac{1}{2}}},$
(29)

where $M_{\Delta}$ is the mass of the lighter triplet scalar.

Again,

$Im[Tr(m^{(2)}_{\nu}m^{{\dagger}}_{\nu})]=Im[Tr(m^{(2)}_{\nu}m^{(1){\dagger}}_{\nu})],$
(30)

as $Im[Tr(m^{(2)}_{\nu}m^{(2){\dagger}}_{\nu})]=0$. So

$\epsilon=\frac{2M_{\Delta}}{4\pi
v^{2}}\sqrt{B_{L}B_{\phi}}\frac{Im[Tr(m^{(2)}_{\nu}m^{(1){\dagger}}_{\nu})]}{[Tr(m^{(2){\dagger}}_{\nu}m^{(2)}_{\nu})]^{\frac{1}{2}}}.$
(31)

If we consider only the lighter triplet scalar contributing to light neutrino
mass matrix, then $m_{\nu}=m^{(2)}_{\nu}$ and using the expression of the CP
asymmetry parameter given in eq.(29) we get $\epsilon=0.$ Hence, only one
triplet scalar is not enough to obtain the lepton asymmetry.

###  IV.2 CP asymmetry parameter: Formulation

In case of triplet leptogenesis, for hierarchical light neutrinos, the upper
bound on CP asymmetry parameter is given by,

$|\epsilon|\lesssim
10^{-6}\times(B_{L}B_{\phi})^{\frac{1}{2}}\frac{M_{\Delta}}{10^{10}GeV}\frac{m_{atm}}{0.05eV}.$
(32)

Taking, $B_{L}=0.0005$, $B_{\phi}=0.9995$, $M_{\Delta}=10^{10}$ GeV,
$m_{atm}\equiv\sqrt{\Delta m^{2}_{atm}}\approx 0.05$ eV, the upper bound on
the CP asymmetry parameter can be obtained as

$|\epsilon|\lesssim 0.022\times 10^{-6}.$

For a realistic calculation in our case, by using eq.s (31) and (III) we
obtain,

$\epsilon=\frac{2M_{\Delta}}{4\pi
v^{2}}\sqrt{B_{L}B_{\phi}}F(\alpha,\phi,\omega_{i})\sqrt{A^{2}+2B^{2}+2C^{2}+D^{2}+2M^{2}+N^{2}},$
(33)

where

$F(\alpha,\phi,\omega_{i})=\frac{|\omega_{1}|cos\alpha
sin\phi}{\sqrt{|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos\phi}},$ (34)

where $A,B,C,D,M,N$ are determined from neutrino oscillation data as described
in section (II). Here we make an assumption that if the phase differences,
$\phi_{a^{\prime}}-\phi_{a}\approx\phi_{b^{\prime}}-\phi_{b}\approx\phi_{c^{\prime}}-\phi_{c}\approx\phi_{d^{\prime}}-\phi_{d}\approx\phi_{m^{\prime}}-\phi_{m}\approx\phi_{n^{\prime}}-\phi_{n}$,
conspire to be equal then we can write,
$|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos(\phi_{a,b,c,d,m,n}-\phi_{a^{\prime},b^{\prime},c^{\prime},d^{\prime},m^{\prime},n^{\prime}})=|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos\phi=f(|\omega_{1}|,\omega_{2},\alpha,\phi).$ So

$A^{2}=a^{2}f(|\omega_{1}|,\omega_{2},\alpha,\phi),$
$B^{2}=b^{2}f(|\omega_{1}|,\omega_{2},\alpha,\phi),$
$C^{2}=c^{2}f(|\omega_{1}|,\omega_{2},\alpha,\phi),$
$D^{2}=d^{2}f(|\omega_{1}|,\omega_{2},\alpha,\phi).$
$M^{2}=m^{2}f(|\omega_{1}|,\omega_{2},\alpha,\phi).$
$N^{2}=n^{2}f(|\omega_{1}|,\omega_{2},\alpha,\phi).$ (35)

Figure 3: Isoplots showing correlation among CP asymmetry parameter, triplet
mass $M_{\Delta}$ and the phase $\alpha$.

The CP-violation parameter depends on three parameters $M_{\Delta}$, and the
two phases $\alpha$ and $\phi$ apart from branching ratios $B_{L}$ and
$B_{\phi}$. To get baryon asymmetry $\sim\mathcal{O}(10^{-10})$ the $\epsilon$
parameter should be of the order $\mathcal{O}(10^{-6}-10^{-8})$ requiring
efficiency, $\eta$ of the order $\mathcal{O}(10^{-3}-10^{-2})$. We plot
different correlation plots among these three variables in the fig.s (3), (4)
and (5). We have taken $B_{L}=0.995$, $\omega_{1}=2.229$ GeV and
$\omega_{2}=0.002$ GeV. Figure (3) shows that adequate CP violation parameter
$\epsilon$ can be produced for triplet mass as low as $2\times 10^{10}$ GeV.
This result is less than by an order of magnitude as reported by Hambye et al.
(2006). In fig.(4) we can see $\epsilon$ of order $10^{-7}$ can be achieved
for $\phi=1.57$ for $M_{\Delta}\simeq 10^{10}$ GeV. In fig.(5) shows the
contours of $\epsilon$ in the plane of $\alpha$ and $\phi$ for
$M_{\Delta}=5\times 10^{10}$ GeV. This range of triplet mass gives a scope to
study flavoured leptogenesis in this framework. We take this exercise in the
upcoming sections.

Figure 4: Isoplots showing correlation among CP asymmetry parameter, triplet
mass $M_{\Delta}$ and the phase $\phi$.  Figure 5: Isoplots showing
correlation among CP asymmetry parameter and the phases $\alpha$ and $\phi$
for $M_{\Delta}=5\times 10^{10}$ GeV.

###  IV.3 Boltzmann equations

The reactions carried out by all the relevant particles in the early Universe
can be classified based on their rates in comparison to the Hubble expansion
rate of the Universe, as follows:

  * •

If the reactions are much faster than the Hubble expansion rate at a
temperature $T_{0}$, then the particles involved are said to have attained
thermal equilibrium. These reactions are to be considered in the study of
particle asymmetries.

  * •

If the reactions have comparable rates as the Hubble expansion, then those
reactions cannot attain thermodynamic equilibrium. Those are to be accounted
for via Boltzmann equations which describe the evolution of all the
asymmetries in the heat bath.

  * •

If the reactions are much slower than the Hubble expansion rate at a
temperature $T_{0}$, then those reactions usually have no relevance in the
study of the evolution of particle asymmetries. Though utmost care should be
taken before dropping these reactions terms falling under this category, from
the Boltzmann equations, especially in the case of Higgs triplet scalars.
Because, particularly in purely flavoured leptogenesis, very slow triplet
interactions play a vital role in dominating the baryogenesis process. Hence,
in this case, an extra Boltzmann equation corresponding to the scalar triplet
asymmetry is included.

The particles involved in the case of scalar triplet leptogenesis are lepton
doublets (L), Higgs doublets, ($\phi$) and scalar triplets ($\Delta$). Here,
we consider the case of a single flavoured approximation, without spectator
processes for simplicity. However, it is observed that the relaxation of
single flavour approximation in case of scalar triplet Aristizabal Sierra et
al. (2014); Lavignac and Schmauch (2015) and in fermion singlet Nardi et al.
(2006); Abada et al. (2006) leptogenesis shows some novel features in purely
flavoured case . We take up this study in the upcoming sections. The number
density of particle of species $p$ is denoted as $n_{p}$ and by
$n_{\overline{p}}$, the number density of the corresponding anti-particle is
symbolized. Here, the Boltzmann equations are expressed as the function of
$z=\frac{M_{\Delta}}{T}$. The asymmetry is denoted as $\Delta_{p}$, where $p$
is the species of particles is given by

$\Delta_{p}=\frac{(n_{p}-n_{\overline{p}})}{s},$ (36)

where $p\equiv{\Delta,\phi,L}$ in the case under consideration. Total
$\Delta,\overline{\Delta}$ density is given by

$\Sigma_{\Delta}=\frac{(n_{\Delta}+n_{\overline{\Delta}})}{s}.$ (37)

The Boltzmann equations are given by Hallgren et al. (2008); Hambye et al.
(2006),

$sHz\frac{d\Sigma_{\Delta}}{dz}=-\left(\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1\right)\gamma_{D}-2\left(\frac{\Sigma^{2}_{\Delta}}{\Sigma^{2eq}_{\Delta}}-1\right)\gamma_{A},$
(38)
$sHz\frac{d\Delta_{L}}{dz}=X-2\gamma_{D}B_{L}\left(\frac{\Delta_{L}}{Y^{eq}_{L}}+\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}\right),$
(39)
$sHz\frac{d\Delta_{\phi}}{dz}=X-2\gamma_{D}B_{\phi}\left(\frac{\Delta_{\phi}}{Y^{eq}_{\phi}}-\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}\right),$
(40)
$sHz\frac{d\Delta_{\Delta}}{dz}=-\gamma_{D}\left(\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}+B_{L}\frac{\Delta_{L}}{Y^{eq}_{L}}-B_{\phi}\frac{\Delta_{\phi}}{Y^{eq}_{\phi}}\right),$
(41)

where

$X=\gamma_{D}\epsilon\left(\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1\right).$
(42)

Here, $\gamma_{D}$ is the total decay space-time density of triplet scalar
$\Delta$ and its anti-particle $\overline{\Delta}$, $\gamma_{A}$ is the space-
time density of $SU(2)_{L}\otimes U(1)_{\mathcal{Y}}$ gauge scattering (shown
in figure 6), given by

$\gamma_{D}=s\Gamma_{\Delta}\Sigma^{eq}_{\Delta}\frac{K_{1}(z)}{K_{2}(z)},$
(43)
$\gamma_{A}=\frac{M_{\Delta}T^{3}e^{-\frac{2M_{\Delta}}{T}}}{64\pi^{4}}(9g^{4}+12g^{2}g^{2}_{Y}+3g^{4}_{Y})\left(1+\frac{3T}{4M_{\Delta}}\right)$
(44)

Figure 6: Feynman diagrams that contribute to the interaction rate
$\gamma_{A}$

where $\Gamma_{\Delta}=\frac{M_{\Delta}^{2}\tilde{m}_{\Delta}}{16\pi
v^{2}\sqrt{B_{L}B_{\phi}}}$, $\tilde{m}_{\Delta}={\rm
Tr}(m^{(2)\dagger}_{\nu}m^{(2)}_{\nu})$,
$\Sigma^{eq}_{\Delta}=2Y^{eq}_{\Delta}=2\times\frac{45g_{\Delta}}{4\pi^{4}g_{*}}z^{2}K_{2}(z)$
and $g,g_{\mathcal{Y}}$ are the coupling constants corresponding to
$SU(2)_{L}$ and $U(1)_{\mathcal{Y}}$ interactions, respectively, having
values, $g=\frac{e}{sin\theta_{W}}=0.651742$ and
$g_{Y}=\frac{e}{cos\theta_{W}}=0.461388$. Again, $g_{\Delta}=3$,
$g_{*}=106.75$ are the triplet scalar and SM particle-degrees of freedom,
respectively Gonzalez Felipe et al. (2013). The other reaction densities,
$\gamma$’s are to be calculated using the integration given below,

$\gamma=\frac{T}{64\pi^{4}}\int^{\infty}_{\textbf{s}_{min}}d\textbf{s}~{}\textbf{s}^{\frac{1}{2}}K_{1}\left(\frac{\sqrt{\textbf{s}}}{T}\right)\hat{\sigma}(\textbf{s}).$
(45)

Here, s is the Mandelstam variable, square of center of mass energy Halzen and
Martin (1984), $\hat{\sigma}$ is the reduced cross section of $2\rightarrow 2$
scatterings. In this case, $\textbf{s}_{min}=4M^{2}_{\Delta}$. There are some
important differences to be noted between the standard leptogenesis scenario
involving heavy Majorana neutrinos and the scalar triplets Hambye and
Senjanovic (2004, 2004); Ma and Sarkar (1998).

  * •

Triplet scalar has two types of decays, into Higgs bosons and into anti-
leptons.

  * •

Scalar triplet is not a self-conjugate state, unlike heavy Majorana neutrinos.
So a nonvanishing asymmetry arises between the triplet ($\Delta$) and anti-
triplet ($\overline{\Delta}$) abundances. It definitely affects the
leptogenesis dynamics differently. That is why, one extra Boltzmann equation,
corresponding to $\Delta_{\Delta}$, has appeared in this scenario.

  * •

The triplet scalar is charged under $SU(2)_{L}\otimes U(1)_{\mathcal{Y}}$. So
the triplet and anti-triplet are able to annihilate before they take part in
decay. A significantly large lepton asymmetry can be generated only if the
decay rates are higher than the rate of gauge annihilations, which are
typically in thermal equilibrium. But, it seems to contradict Sakharov’s third
condition: the condition of being out of thermal equilibrium. But as triplet
has several decay channels, it is possible for some of those to be out of
equilibrium and that resolves the mentioned conflict. Again, the final lepton
asymmetry does not depend on the initial conditions, as the gauge scatterings
keep the triplet scalar abundance $\Sigma_{\Delta}$ close to thermal
equilibrium. Thereby, it can be observed that the Boltzmann equation
corresponding to $\Sigma_{\Delta}$ is dominated by $\gamma_{D}$, which puts
$\Sigma_{\Delta}$ close to $\Sigma^{eq}_{\Delta}$, whereas gauge scatterings
have a negligible effect on it.

Figure 7: In the left panel: Evolution of interaction rates
$\gamma_{D\phi},\gamma_{Dl},\gamma_{A}$ in units of $Hn_{\gamma}$ with $Z$ In
right panel: evolution of abundancies with $Z$ for
$B_{L}=0.9995,B_{\phi}=0.0005$.

###  IV.4 Discussion

We solve Boltzmann eq.s (38), (39), (40) and (41) and the relevant abundancies
are plotted in fig. (7) in units of $|\epsilon|$ against
$z=\frac{M_{\Delta}}{T}$ for heirarchical $B_{L}$ and $B_{\phi}$. In case of
triplet scalar leptogenesis, the efficiency factor can be defined as the ratio
of lepton asymmetry $\Delta_{L}$, in units of CP asymmetry parameter
$\epsilon$ and the total triplet density $\Sigma_{\Delta}$ for $z\ll 1$,

$\eta=\frac{\Delta_{L}/\epsilon}{\Sigma_{\Delta}(z\ll 1)}.$ (46)

In fig.(7), $\gamma_{Dl}=B_{L}\gamma_{D}$, $\gamma_{D\phi}=B_{\phi}\gamma_{D}$
and $\gamma_{A}$ is plotted against $M_{\Delta}/T$. For this,
$B_{\phi}=5\times 10^{-4}$, $B_{L}=1-B_{\phi}$ are used. The decay rates
$\gamma_{Dl}$, $\gamma_{D\phi}$ and $\gamma_{A}$ are shown for different
values of branching ratios, $B_{L}$, $B_{\phi}$ in fig.(12). The corresponding
abundancy plots are also given shown therein. It is to be noted that the
efficiency factor is proportional to the asymmetry $\Delta_{L}$, as expressed
in eq.(46), and it is observed from the fig.s (7) and (12) that, when
$B_{L}\gg B_{\phi}$ or $B_{\phi}\gg B_{L}$, the asymmetry $\Delta_{L}$ is
larger. Hence, the efficiency is found to be maximal. On the other hand, the
reduced asymmetry $\Delta_{L}$ makes efficiency minimal when
$B_{L}=B_{\phi}=\frac{1}{2}$.

For hierarchical branching ratios, the amount of $CP$ violation is suppressed
but this enhances the efficiency. Fig. (7) shows one of the case of
hierarchical branching ratios with $M_{\Delta}=5\times 10^{10}$ GeV, whereas
fig.s (12) show that the efficiency decreases significantly for equal
branching ratios. We present some of the instructive numerical estimates of
$CP$ violation parameter, efficiency and amount of BAU in table (2).

$M_{\Delta}(GeV)$ |  $B_{L}$ |  $\epsilon$ |  $\eta$ |  $\eta_{B}$
---|---|---|---|---
$2\times 10^{9}$ |  $0.995$ |  $\pm 1.5\times 10^{-8}$ |  $0.015357$ |  $6.6\times 10^{-12}$
$2\times 10^{9}$ |  $0.9995$ |  $\pm 6.0\times 10^{-9}$ |  $0.0721$ |  $1.25\times 10^{-11}$
$1\times 10^{10}$ |  $0.995$ |  $\pm 8.0\times 10^{-8}$ |  $0.021445$ |  $4.975\times 10^{-11}$
$1\times 10^{10}$ |  $0.9995$ |  $\pm 3.0\times 10^{-8}$ |  $0.066$ |  $5.742\times 10^{-11}$
$5\times 10^{10}$ |  $0.995$ |  $\pm 4.0\times 10^{-7}$ |  $0.0240$ |  $2.784\times 10^{-10}$
$5\times 10^{10}$ |  $0.9995$ |  $\pm 1.5\times 10^{-7}$ |  $0.096878$ |  $4.214\times 10^{-10}$
$1\times 10^{11}$ |  $0.995$ |  $\pm 8.0\times 10^{-7}$ |  $0.01675$ |  $3.886\times 10^{-10}$
$1\times 10^{11}$ |  $0.9995$ |  $\pm 3.0\times 10^{-7}$ |  $0.1$ |  $8.7\times 10^{-10}$
Table 2: Baryon asymmetries obtained for different values $M_{\Delta},B_{L}$
and $\epsilon$.

In our calculation of $CP$ violation triplet vevs are kept fixed at
$\omega_{1}=2.299$ GeV and $\omega_{2}=0.002$ GeV. The values of $\omega_{1}$
is subjected to suppression due to $\cos\alpha$ as can be seen in eq.(14). In
calculating the values of $\epsilon$ the following neutrino mass matrix
elements in eV is taken.

$M_{\nu}=\begin{pmatrix}0.028306&0.012501&0.015691\\\
0.012501&0.018767&0.017914\\\ 0.015691&0.017914&0.016304.\end{pmatrix}$ (47)

Such a value of the mass matrix is chosen from fig.(11) such that the values
of light neutrino masses for normal hierarchy satisfy the recent Planck data
Aghanim et al. (2020a) such that

$m_{1}=0.030000~{}eV,\quad m_{2}=0.031131~{}eV\quad m_{3}=0.058134~{}eV.$ (48)

The sum of neutrino masses,

$\Sigma=m_{1}+m_{2}+m_{3}=0.119266~{}eV,$ (49)

is bounded by Planck data. Here we have not taken the renormalization group
evolution of neutrino mass matrix in to account. As shown in Hambye et al.
(2006) the result is only $M_{\nu}(\rm High~{}scale)\sim(1.2/1.3)M_{\nu}$
Barbieri et al. (2000); Giudice et al. (2004).

We can summarise the outcomes of this section as follows. The $CP$ asymmetry
parameter increases with the increase in the triplet mass $M_{\Delta}$. It
decreases with the increase in the hierarchical branching ratios. In case of
unflavoured leptogenesis, the baryon asymmetry obtained seems to be increasing
with the increasing triplet mass and hierarchical branching ratio. But a
maximal $\epsilon$ value can be achieved for triplet mass as low as $2\times
10^{10}$ GeV. Such a value of triplet mass motivates to study baryogenesis
through leptogenesis considering the flavour effects which we take up in next
section.

##  V Flavoured leptogenesis (Two flavour approximation)

At temperature, $T\gtrsim 10^{15}$ GeV all the SM reactions are slow and not
in thermal equilibrium and only the interactions related to triplet are
relevant. Only top quark related Yukawa interactions enter thermal equilibrium
for $T\subset[10^{12},10^{15}]$ GeV. The Yukawa interactions related to
bottom, charm, and tau, electroweak sphalerons, and QCD instantons enter
thermal equilibrium for $T\subset[10^{9},10^{12}]$ GeV. The processes induced
by $\tau-$Yukawa are in thermal equilibrium and it breaks the coherence
between $\tau$-lepton and the other two leptons (e, $\mu$). Also, electroweak
sphalerons are in thermal equilibrium in this temperature regime baryon number
is no longer conserved while individual $B/3-L_{i}$ is conserved. So the
evolution $B-L$ asymmetry should be done by following the evolution of the
flavoured charge symmetries $B/3-L_{i}$, where $i=a,\tau$, $a$ being the
coherent superposition of $e$ and $\mu$ lepton flavours. The strange and muon
Yukawa interactions resume thermal equilibrium in the range
$T\subset[10^{5},10^{9}]$ GeV. In this regime, the lepton doublets completely
lose their quantum coherence. Finally all the SM interactions are in thermal
equilibrium for $T\lesssim 10^{5}$ GeV. The flavour decoherence requires
careful quantum treatment. However, a simplified treatment can be given
considering two relevant processes: (i) the SM lepton Yukawa interactions
($L-\phi$) and (ii) lepton related triplet inverse decays, ($L-\Delta$).
Consider the situation where at a certain time if the $L-\phi$ interaction
rate becomes faster than the Hubble rate but $L-\Delta$ processes are much
faster than (i). In this case, the coherent superposition of leptons produced
from the decay of $\Delta$ will inverse decay before it has the time to
undergo $L-\phi$ interaction. In this case flavour decoherence has to wait
until the reduction in the temperature of the hit bath when the $L-\Delta$
interaction in (ii) is Boltzmann suppressed.

Below $T=10^{12}$ GeV the Yukawa introduced processes like
$q_{3}L_{\tau}\rightarrow t_{R}\tau_{R}$ are in equilibrium, so the coherence
between $\tau$-leptons and other two flavours is broken. So the coherence of
$l_{a}=(Y_{1e}L_{e}+Y_{1\mu}L_{\mu})/\sqrt{|Y_{1e}|^{2}+|Y_{1\mu}|^{2}}$ is
preserved. In this temperature regime $T\subset[10^{9},10^{12}]$ GeV, the
progress of leptogenesis can be described by $2-$flavour approximation. The
lepton asymmetries generated in this region can be quantified in terms of
$\Delta_{l_{a}}$ and $\Delta_{l_{\tau}}$. However, the muon Yukawa coupling is
out of equilibrium for temperature $T>10^{9}$ GeV. Below this temperature
leptogenesis has to be studied treating the three flavours separately and the
study becomes fully flavoured. For our purpose, we only consider $2-$flavour
approximation.

###  V.1 Formulation of flavoured CP asymmetry parameter

In type-II seesaw mechanism with two triplet Higgs scalars (without any
fermion singlet) leptogenesis is generated when the triplet scalars decay into
two leptons. In the presence of complex couplings of the triplets with SM
Higgs doublet and the leptons, the interference of the tree level and one loop
decay amplitudes gives rise to CP asymmetries. Now, there can be two types of
one-loop decay, one is mediated by SM Higgs and the other is mediated by
leptons. They are shown in fig.s- (8(a)), (8(b)).

(a) Tree level decay diagram: Triplet decaying into charged bi-lepton and SM
Higgs, respectively

(b) One-loop decay diagram: Triplet decaying into charged bi-lepton through
Higgs loop and lepton loop, respectively

Figure 8: Feynman diagram for triplet Higgs scalar decaying into charged bi-
lepton and SM Higgs

In the case of the Higgs loop, it accounts for both total lepton number
violation and lepton flavour violation. On the other hand, the one-loop
mediated by leptons only causes lepton flavour violation. Based on these two
contributions, the total CP asymmetry can be expressed as,

$\epsilon=\epsilon_{\cancel{L},\cancel{F}}+\epsilon_{\cancel{F}},$ (50)

where $\epsilon_{\cancel{L},\cancel{F}}$ and $\epsilon_{\cancel{F}}$
contributions come from the interference of tree-level diagrams with one loop
diagrams of the Higgs loop and the lepton loop respectively. The corresponding
expressions are given by,

$\epsilon^{l_{(i)}(\cancel{L},\cancel{F})}=\frac{1}{2\pi}\frac{Im[(Y^{\dagger}_{2}Y_{1})_{ii}\mu^{*}_{\Delta_{2}}\mu_{\Delta_{1}}]}{M^{2}_{\Delta_{2}}Tr[Y_{2}Y^{\dagger}_{2}]+|\mu_{\Delta_{2}}|^{2}}g\left(\frac{M^{2}_{\Delta_{2}}}{M^{2}_{\Delta_{1}}}\right)$
(51)
$\epsilon^{l_{(i)}(\cancel{F})}=\frac{1}{2\pi}M^{2}_{\Delta_{2}}\frac{Im[(Y^{\dagger}_{2}Y_{1})_{ii}Tr[Y_{2}Y^{\dagger}_{1}]]}{M^{2}_{\Delta_{2}}Tr[Y_{2}Y^{\dagger}_{2}]+|\mu_{\Delta_{2}}|^{2}}g\left(\frac{M^{2}_{\Delta_{2}}}{M^{2}_{\Delta_{1}}}\right)$
(52)

where

$g(x)=\frac{x(1-x)}{(1-x)^{2}+xy},\quad
x=\frac{M^{2}_{\Delta_{a}}}{M^{2}_{\Delta_{b}}},\quad
y=\left(\frac{\Gamma^{tot}_{\Delta_{b}}}{M_{\Delta_{b}}}\right)^{2}$

If one of the triplets couples dominantly with leptons, then
$\epsilon_{\cancel{L},\cancel{F}}\approx 0$, and
$\epsilon=\epsilon_{\cancel{F}}$ and the entire CP asymmetry gets contribution
mainly by the interference of tree level and one loop decay amplitudes
mediated by the leptons. This can be guaranteed by choosing the branching
ratios to be $B_{L}\gg B_{\phi}.$ Hence the leptogenesis evolving from this CP
asymmetry is called to be purely flavoured leptogenesis. In our study, the
determination of Yukawa parameters that are relevant for determination on
leptonic branching ratios are determined from neutrino mass matrix elements up
to phase factor. So the smallness of the couplings make the CP asymmetry
parameter extremely small. So in our study PFL is not viable. Nevertheless,
the case of $B_{\phi}\gg B_{L}$ remains still viable for flavoured
leptogenesis. The expression of the flavoured CP asymmetry parameters for
three lepton flavour generations ($L=e$, $\mu$, $\tau$), in case of two
triplet leptogenesis, is given byBranco et al. (2012)

$\epsilon^{\alpha\beta}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}}{4\pi
v^{2}}\times\frac{Im[(m^{(2)}_{\nu})_{\alpha\beta}(m^{(1)*}_{\nu})_{\alpha\beta}]}{[Tr(m^{(2)\dagger}_{\nu}m^{(2)}_{\nu})]^{\frac{1}{2}}},$
(53)

We further substitute, $\phi_{a^{\prime}}-\phi_{a}=\phi_{1}$,
$\phi_{b^{\prime}}-\phi_{b}=\phi_{2}$,$\phi_{c^{\prime}}-\phi_{c}=\phi_{3}$,
$\phi_{d^{\prime}}-\phi_{d}=\phi_{4}$, $\phi_{m^{\prime}}-\phi_{m}=\phi_{5}$,
$\phi_{n^{\prime}}-\phi_{n}=\phi_{6}$, then we construct a function
$f_{i}(|\omega_{1}|,\omega_{2},\alpha,\phi_{i})$ as given by

$f_{i}=|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos\phi_{i}$

The CP asymmetry parameter expressions get reduced to

$\epsilon^{ee}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}|\omega_{1}|cos\alpha}{4\pi
v^{2}}\times\frac{A^{2}sin\phi_{1}/f_{1}}{F},$ (54)
$\epsilon^{e\mu}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}|\omega_{1}|cos\alpha}{4\pi
v^{2}}\times\frac{B^{2}sin\phi_{2}/f_{2}}{F},$ (55)
$\epsilon^{e\tau}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}|\omega_{1}|cos\alpha}{4\pi
v^{2}}\times\frac{C^{2}sin\phi_{3}/f_{3}}{F},$ (56)
$\epsilon^{\mu\mu}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}|\omega_{1}|cos\alpha}{4\pi
v^{2}}\times\frac{D^{2}sin\phi_{4}/f_{4}}{F},$ (57)
$\epsilon^{\mu\tau}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}|\omega_{1}|cos\alpha}{4\pi
v^{2}}\times\frac{M^{2}sin\phi_{5}/f_{5}}{F},$ (58)
$\epsilon^{\tau\tau}=\frac{M_{\Delta_{2}}(B_{L}B_{\phi})^{\frac{1}{2}}|\omega_{1}|cos\alpha}{4\pi
v^{2}}\times\frac{N^{2}sin\phi_{6}/f_{6}}{F},$ (59)

where
$F=\sqrt{\frac{A^{2}}{f_{1}}+\frac{2B^{2}}{f_{2}}+\frac{2C^{2}}{f_{3}}+\frac{D^{2}}{f_{4}}+\frac{2M^{2}}{f_{5}}+\frac{N^{2}}{f_{6}}}$,
and
$f_{i}=|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos\phi_{i}$. And further

$\epsilon^{e}=\epsilon^{ee}+\epsilon^{e\mu}+\epsilon^{e\tau}$ (60)
$\epsilon^{\mu}=\epsilon^{\mu e}+\epsilon^{\mu\mu}+\epsilon^{\mu\tau}$ (61)
$\epsilon^{\tau}=\epsilon^{\tau e}+\epsilon^{\tau\mu}+\epsilon^{\tau\tau}$
(62) $\epsilon^{a}=\epsilon^{e}+\epsilon^{\mu}.$ (63)

###  V.2 Boltzmann equations

The out of equilibrium triplet decays give rise to $B/3-L_{i}$ and $\phi$
asymmetries. The $\phi$ asymmetry gets partially transferred to SM fermion
asymmetries mediated by Yukawa interactions. To analyse the evolution of
$\phi$ asymmetry no separate Boltzmann equation is needed as its evolution is
determined by the chemical equilibrium condition carried out by the reactions
in thermal equilibrium at a temperature $T_{0}$. The triplet abundancy relies
on the reactions that either generate or wash out the $B/3-L_{i}$ asymmetry in
the hot plasma. There are four classes of such interactions namely,

  * •

Yukawa and scalar induced decays and inverse decays

  * •

Lepton flavour violating and lepton number violating s and t channel
$2\leftrightarrow 2$ scatterings mediated by scalars and induced by Yukawa-
scalars

  * •

Lepton flavour violating s and t channel $2\leftrightarrow 2$ scatterings
mediated by scalars and induced by Yukawa-scalars

  * •

Gauge induced $2\leftrightarrow 2$ scatterings.

Among these interactions, considering only decays and inverse decays, gauge
induced reactions and the off-shell pieces of the s channel process can lead
to a quite accurate calculation of $B-L$ asymmetry. The Boltzmann equations
for purely flavoured leptogenesis are given by Aristizabal Sierra et al.
(2014)

$szH(z)\frac{d\Delta_{\Delta}}{dz}=-\gamma_{D}\left[\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}-\sum_{k}(\sum_{i}B_{l_{i}}C^{l}_{ik}-B_{\phi}C^{\phi}_{k})\frac{\Delta_{k}}{Y^{eq}_{l}}\right],$
(64)
$szH(z)\frac{d\Sigma_{\Delta}}{dz}=-\left[\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1]\gamma_{D}-2[\frac{\Sigma^{2}_{\Delta}}{\Sigma^{eq^{2}}_{\Delta}}-1\right]\gamma_{A},$
(65)
$szH(z)\frac{d\Delta_{B/3-L_{i}}}{dz}=-\left[\frac{\Sigma_{\Delta}}{\Sigma^{eq}_{\Delta}}-1\right]\epsilon_{l_{i}}\gamma_{D}+2\sum_{j}\left[\frac{\Delta_{\Delta}}{\Sigma^{eq}_{\Delta}}-\frac{1}{2}\sum_{k}C^{l}_{ijk}\frac{\Delta_{k}}{Y^{eq}_{l}}\right]B_{l_{ij}}\gamma_{D}$
(66)

where $z=\frac{M_{\Delta}}{T}$, and

$\Delta_{k}=\begin{pmatrix}\Delta_{\Delta}\\\ \Delta_{B/3-L_{a}}\\\
\Delta_{B/3-L_{\tau}}\end{pmatrix},\quad C^{l}_{ijk}=C^{l}_{ik}+C^{l}_{jk},$

$H(z)$ is the Hubble’s rate of expansion of the Universe,
$H(z)=\frac{H_{0}(M_{\Delta})}{z^{2}}$,
$H_{0}(T)=\sqrt{\frac{4\pi^{3}}{45}g_{*}}\frac{T^{2}}{m_{P}}$,
$m_{P}=1.22\times 10^{19}$ GeV is the Planck mass,
$s=(\frac{2\pi^{2}}{45})g_{*}T^{3}$ is the total entropy density at the
leptogenesis temperature scale, $g_{*}=106.75$,
$\Delta_{x}=\frac{n_{x}-n_{\overline{x}}}{s}$ is the asymmetry corresponding
to $x$ particle, and
$\Sigma_{\Delta}=\frac{n_{\Delta}+n_{\overline{\Delta}}}{s}$ is the total
triplet number density. Also, $Y=\frac{n_{x}}{s}$ is the comoving number
density,

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

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. The decay rate and gauge scattering rate is
expressed in section (IV.3). The tree level branching ratios of the triplet
scalar to leptons and SM Higgs respectively, are expressed as,

$B_{L}=\frac{M_{\Delta}}{8\pi\Gamma_{\Delta}}Tr(Y_{\Delta}^{\dagger}Y_{\Delta}),\quad
B_{\phi}=\frac{M_{\Delta}}{8\pi\Gamma_{\Delta}}|\mu|^{2},\quad
B_{L}+B_{\phi}=1,$ $B_{L}=\sum_{i,j=a,\tau}B_{l_{ij}}$
$B_{L}=\begin{pmatrix}B_{l_{aa}}&B_{l_{a\tau}}\\\ B_{l_{\tau
a}}&B_{l_{\tau\tau}}\end{pmatrix}$
$B_{l_{aa}}=\frac{M_{\Delta}}{8\pi\Gamma_{\Delta}}\sum_{i,j=1,2}|Y^{(2)}_{\Delta_{ij}}|^{2}$
$B_{l_{a\tau}}=\frac{M_{\Delta}}{8\pi\Gamma_{\Delta}}\sum_{i=1,2}|Y^{(2)}_{\Delta_{i3}}|^{2}$
$B_{l_{\tau
a}}=\frac{M_{\Delta}}{8\pi\Gamma_{\Delta}}\sum_{j=1,2}|Y^{(2)}_{\Delta_{3j}}|^{2}$
$B_{l_{\tau\tau}}=\frac{M_{\Delta}}{8\pi\Gamma_{\Delta}}|Y^{(2)}_{\Delta_{33}}|^{2}$

where $\lambda=\frac{\mu}{M_{\Delta}}$. Within the temperature range
$[10^{5},10^{9}]$ GeV the quantum coherence of the lepton doublets is
completely broken, and through the chemical equilibrium calculation $C^{l}$
and $C^{\phi}$ matrices are obtained as

$C^{l}=\begin{pmatrix}-6/359&307/718&-18/359\\\
39/359&-21/718&117/359\end{pmatrix}$
$C^{\phi}=\begin{pmatrix}258/359&41/359&56/359\end{pmatrix}$

The asymmetries in lepton flavours and SM Higgs can be obtained using the
expressions given below:

$\Delta_{l_{i}}=-\sum_{k}C^{l}_{ik}\Delta_{k},\quad\Delta_{\phi}=-\sum_{k}C^{\phi}_{k}\Delta_{k}.$
(67)

The final baryon asymmetry is estimated by $\eta_{B}=7.04~{}\Delta_{B}$, where

$\Delta_{B}=3\times\frac{12}{37}\sum_{i}\Delta_{B/3-L_{i}}.$ (68)

The Boltzmann equations are solved numerically to obtain the dynamics of the
asymmetries $\Delta_{x}$ with the factor $z=\frac{M_{\Delta}}{T}$. The initial
conditions used are given below,

$\Sigma_{\Delta}(z\ll 1)=\Sigma^{eq}_{\Delta}(z\ll
1),\quad\Delta_{\Delta}(z\ll 1)=0,\quad\Delta_{l_{\alpha}}(z\ll
1)=0,\quad\Delta_{\phi}(z\ll 1)=0.$

###  V.3 Discussion

We solve the flavoured Boltzmann equations as given in eq.s (64), (65) and
(66) for $M_{\Delta}=5\times 10^{10}$ GeV. The abundancies are plotted in
fig.(9). Various cases of triplet mass and $B_{\phi}$ are studied and are
summarises in table (3). However, we record two sets of calculation for
illustration for $M_{\Delta}=5\times 10^{10}$GeV, and $B_{L}=0.0005$. In both
cases the $CP$ violation parameter, $\epsilon$ is calculated by using eq.s (54
\- 63) for different choices of phase factors.

  * •

Set-I
In this set we choose
$\alpha=2.1349,\phi_{1}=1.0,\phi_{2}=2.0,\phi_{3}=6.0,\phi_{4}=5.0,\phi_{5}=3.699,\phi_{6}=1.4$
giving

$\epsilon_{a}=-2.10\times 10^{-8},\epsilon_{\tau}=-1.29\times 10^{-9}$ (69)

The abundancies are plotted in fig. (9) giving $\eta_{B}=6.37\times 10^{-10}$

Figure 9: Abundancies plot from two flavoured approximation for
$M_{\Delta}=5\times 10^{10}$ GeV, $B_{L}=0.0005$, $B_{\phi}=0.9995$ .

  * •

Set-II
In this set we choose
$\alpha=2.1349,\phi_{1}=\phi_{2}=\phi_{3}=1.0,\phi_{4}=5.0,\phi_{5}=4.699,\phi_{6}=2.399$
giving

$\epsilon_{a}=-2.67\times 10^{-8},\epsilon_{\tau}=-3.61\times 10^{-9}$ (70)

and the final BAU in this case is given by $\eta_{B}=8.678\times 10^{-10}$.

$M_{\Delta}(GeV)$ |  $B_{\phi}$ |  $\epsilon_{l_{a}}$ |  $\epsilon_{l_{\tau}}$ |  $\eta_{B}$
---|---|---|---|---
$5\times 10^{10}$ |  $0.9995$ |  $-2.67\times 10^{-8}$ |  $-3.61\times 10^{-9}$ |  $8.678\times 10^{-10}$
$5\times 10^{10}$ |  $0.9995$ |  $-2.10\times 10^{-8}$ |  $-1.29\times 10^{-9}$ |  $6.37\times 10^{-10}$
$5\times 10^{10}$ |  $0.995$ |  $-3.25\times 10^{-8}$ |  $-2.21\times 10^{-9}$ |  $5.937\times 10^{-10}$
$5\times 10^{10}$ |  $0.995$ |  $-3.686\times 10^{-8}$ |  $-3.924\times 10^{-9}$ |  $6.98\times 10^{-10}$
$5\times 10^{10}$ |  $0.95$ |  $-7.61\times 10^{-8}$ |  $-4.936\times 10^{-9}$ |  $7.288\times 10^{-10}$
$5\times 10^{10}$ |  $0.95$ |  $-7.065\times 10^{-8}$ |  $-3.771\times 10^{-9}$ |  $6.68\times 10^{-10}$
Table 3: Baryon asymmetries from $2$-flavoured leptogenesis for different
values of $M_{\Delta}$ and $B_{\phi}$.

##  VI Low scale leptogenesis

The leptonic asymmetry can be resonantly enhanced if the relation between
their respective masses and decay width satisfy
$|M_{\Delta_{j}}-M_{\Delta_{i}}|\sim\frac{1}{2}\Gamma_{\Delta_{j}}$. The CP
violation parameter can be given as after summing over the lepton flavours

$\epsilon_{i,res}\simeq\frac{\sqrt{B_{L}^{i}B_{\phi}^{i}}Im[(m^{(i)}_{\nu}m^{(j)\dagger}_{\nu})]}{[Tr(m^{(i)\dagger}_{\nu}m^{(i)}_{\nu})]^{\frac{1}{2}}[Tr(m^{(j)\dagger}_{\nu}m^{(j)}_{\nu})]^{\frac{1}{2}}}$
(71)

This is neither suppressed by light neutrino mass nor the scalar triplet
masses. It is bounded by unitarity constraint $|\epsilon_{i}|<2{\rm
min}(B_{L},B_{\phi})$. This feature opens up the possibility of triplet
leptogenesis at the TeV scale.

In this regime the triplets become non-relativistic and new features emerge in
their annihilation rate $\gamma_{A}$. The non-relativistic scatterings among
gauge particles are affected by non-perturbative electroweak Sommerfeld
corrections. This introduces new corrections in $\gamma_{A}$. The scatterings
are distorted by the Coulomb force as when the charged particles become non-
relativistic, their kinetic energy is lower than the electrostatic potential
energy. These dynamics decreases the efficiency by $30\%$. The efficiency
decreases at $M_{\Delta}\leq 3$ TeV because when the triplet decays produce
lepton asymmetry, sphalerons no longer convert it into baryon asymmetry. And
for temperature below the mass of the Higgs boson, the electroweak symmetry
starts to break suppressing the sphaleron transition rates. The baryogenesis
via leptogenesis is only possible at $M_{\Delta}\geq 1.6$ TeV Strumia (2009).
Referring to eq.(45) of the annihilation rate the reduced cross section of
$2\rightarrow 2$ scatterings can be represented as

$\hat{\sigma}_{A}(s)=c_{s}\beta+c_{p}\beta^{3}+\mathcal{O}(\beta^{5}),~{}\beta=\sqrt{1-4M_{\Delta}^{2}/s},$
(72)

where $\beta$ is the triplet velocity in $\Delta\Delta^{*}$ center of mass
frame. The annihilation rate is given by

$\gamma_{A}=\frac{M_{\Delta}T^{3}e^{-2M_{\Delta}/T}}{32\pi^{3}}\left[c_{s}+\frac{3T}{2M_{\Delta}}(c_{p}+\frac{c_{s}}{2})+\mathcal{O}(T/M_{\Delta})^{2}\right],~{}c_{s}=\frac{9g_{2}^{4}+12g_{2}^{2}g_{\mathcal{Y}}^{2}+3g_{\mathcal{Y}}^{2}}{2\pi}$
(73)

where the hypercharge ${\mathcal{Y}}$ for the leptons is $1/2$.

For a single abelian mass less vector with potential $V=\alpha/r$, the
Sommerfeld correction to $\sigma=S(x)\sigma_{\rm perturbative}$ where

$S(x)=\frac{-\pi x}{1-e^{\pi x}},\quad x=\alpha/\beta.$ (74)

The thermally averaged value of Sommerfeld correction is

$S_{T}(y)=\frac{\int_{0}\beta^{2}e^{-M_{\Delta}\beta^{2}/T}S(\alpha/\beta)d\beta}{\int_{0}\beta^{2}e^{-M_{\Delta}\beta^{2}/T}d\beta},~{}y\equiv\alpha/\beta_{T}$
(75)

where $\beta_{T}\equiv\sqrt{T/M_{\Delta}}$ is characteristic velocity of the
thermal bath. The function $S_{T}$ can be computed numerically and leads to
$S_{T}(x)\approx S(x)$. The generalization to massive vectors needs a list of
potential and annihilation matrices and has been discussed in Hisano et al.
(2007). For $SU(2)_{L}$ invariant limit ($M\gg M_{W,Z}$) the Sommerfeld
corrections was calculated in Strumia (2009). The isospin of two-body triplet
states $\Delta\Delta^{*}$ have total isospin $I={1,3,5}$, as it can be
decomposed as $3\bigotimes 3=1_{S}\bigoplus 3_{A}\bigoplus 5_{S}$ which can
result from total spin $s=0,1$ and total angular momentum $L$. The potentials
are $V=-2\alpha_{2}/r,-\alpha_{2}/r,\alpha_{2}/r$ for singlet, triplet and
quintuplet states respectively. Since in the energy regime we are interested
$s$-wave $(L=0)$ annihilations are important, the final two SM states can have
isospin $I={1,3,5}$. The correction to annihilation rate from Sommerfeld
corrections is given by,

$c_{s}=\frac{2g_{2}^{4}+3g_{Y}^{4}/2}{\pi}S_{T}(-(2\alpha_{2}+\alpha_{Y})\sqrt{z})+\frac{5g_{2}^{4}}{2\pi}S_{T}((\alpha_{2}-\alpha_{Y})\sqrt{z})+\frac{6g_{2}^{2}g_{Y}^{2}}{\pi}S_{T}(-(\alpha_{2}+\alpha_{Y})\sqrt{z}),$
(76)

where the first, second and third term on right hand side of the equation
correspond to final isospin state $I=1,2$ and $3$ respectively. In the limit
$S_{T}\rightarrow 1$ we get back the annihilation rate for given in eq. (44)
in section (IV.3). We solve Boltzmann equations in this regime as shown in
fig.(10).

Figure 10: Abundancies plot by solving Boltzmann equation for TeV scale
triplet mass, $M_{\Delta}=1600$ GeV, $B_{L}=B_{\phi}=\frac{1}{2}$. Abundancy,
$\Delta_{L}$ is also plotted in units of $\epsilon$.

###  VI.1 Discussion

We calculate the amount of $CP$ violation and efficiency parameter to quantify
BAU. Using eq.(71) the CP violation parameter can be expressed as

$\epsilon_{i,res}=\frac{(B_{L}B_{\phi})^{\frac{1}{2}}[a^{2}sin\phi_{1}+2b^{2}sin\phi_{2}+2c^{2}sin\phi_{3}+d^{2}sin\phi_{4}+2m^{2}sin\phi_{5}+n^{2}sin\phi_{6}]}{[a^{2}+2b^{2}+2c^{2}+d^{2}+2m^{2}+n^{2}]},$
(77)

where

$a=\frac{A}{f(|\omega_{1}|,\omega_{2},\alpha,\phi_{1})},~{}b=\frac{B}{f(|\omega_{1}|,\omega_{2},\alpha,\phi_{2})},~{}c=\frac{C}{f(|\omega_{1}|,\omega_{2},\alpha,\phi_{3})}$
(78)

$d=\frac{D}{f(|\omega_{1}|,\omega_{2},\alpha,\phi_{4})},~{}m=\frac{M}{f(|\omega_{1}|,\omega_{2},\alpha,\phi_{5})},~{}n=\frac{N}{f(|\omega_{1}|,\omega_{2},\alpha,\phi_{6})},~{}$
(79)

$f(|\omega_{1}|,\omega_{2},\alpha,\phi_{i})=|\omega_{1}|^{2}cos^{2}\alpha+\omega^{2}_{2}+2|\omega_{1}|\omega_{2}cos\alpha
cos\phi_{i}$ (80)

We take the numerical values of neutrino mass matrix elements from eq. (47).
The calculated CP asymmetry

$\epsilon=-2.970569\times 10^{-2},$

taking $B_{L}=B_{\phi}=\frac{1}{2}$. We have chosen, $\omega_{1}=2.299$GeV,
$\omega_{2}=0.002$ GeV. The phases are chosen to be $\alpha=3.99$,
$\phi_{1}\approx 0.994$, $\phi_{2}\approx 0.994$, $\phi_{3}\approx 3.0$,
$\phi_{4}\approx 4.99$, $\phi_{5}\approx 4.29$, and $\phi_{6}\approx 4.39$. By
solving the set of Boltzmann equations from (38 \- 41) we plot the abundancy,
$\Delta_{L}$ for $B_{L}=B_{\phi}=\frac{1}{2}$ and find $\eta=7\times 10^{-7}$
from fig.(10). And the baryon asymmetry is calculated to be,

$\eta_{B}=6.03025\times 10^{-10}.$

##  VII Conclusion

We have studied baryogenesis through leptogenesis from the decay of scalar
triplets. The coexistence of two decay processes of latter and hence two
branching ratios ($\Delta-\phi$ and $\Delta-L$) introduce new features in
leptogenesis as compared to other scenarios. The amount of baryon asymmetry
generation depends on the strength of $CP$ violation and efficiency of the
$CP$ violating decay. The $CP$ asymmetry parameter is determined by the
complex phases of the Yukawa matrix and vev of the triplet scalar through the
neutrino mass matrix, triplet mass and branching ratios of triplet scalars to
the doublet scalars and the SM leptons. It is known that strongly hierarchical
branching ratios enhance the efficiency but suppress the $CP$ violation. The
generation of baryon asymmetry then requires the mass of the triplets to be
between $\sim 10^{9}-10^{12}$ GeV. In our study, we find that for a sizable
amount of $CP$ asymmetry ($\sim 10^{-7}$) and efficiency ($\sim 10^{-2}$) the
triplet mass can be as low as $2\times 10^{10}$ GeV. In this case, the order
of magnitude of generated baryon asymmetry matches with the observed value.
The reported value of triplet mass is suitable for studying the flavour
effects in leptogenesis.

The $CP$ asymmetries arising because of the presence of two hierarchical
triplets introduce two one loop diagrams. The interference of tree level
process with that of one loop mediated by doublet Higgs and leptons. The
former one is both lepton number violating as well as lepton flavour violating
($\epsilon_{\not{F},\not{l}}$) and the latter one is only lepton flavour
violating ($\epsilon_{\not{F}}$). The purely flavoured, $\epsilon_{\not{F}}$
dominates over the $\epsilon_{\not{F},\not{l}}$ in the case where triplets
couple to leptons more than the doublet Higgs. This ensures the hierarchical
branching ratios, $B_{L}\gg B_{\phi}$. In our study, the different modes of
$B_{L}$ are determined from /corresponding elements of the Yukawa coupling
matrix which in turn are determined from neutrino mass matrix elements. So
their smallness and the dependence of branching ratios
($\sqrt{B_{L}B_{\phi}}$) amount in negligible amount of $CP$ violation. So
producing adequate BAU seems not attainable even by considering flavour
effects. However, the other choice of $B_{\phi}\gg B_{L}$ and flavour effects
show to meet the objective with proper choice of $CP$ violating phases of the
vev of the triplet Higgs scalar and other phases of neutrino mass matrix. We
also study the possibility of TeV scale leptogenesis taking resonant effects
into account where the two triplet scalars can be nearly degenerate in their
masses. This class of study introduces new corrections in the triplet
annihilation rate reduces the efficiency.

Triplet scalars arise naturally in many models beyond the SM such as left-
right symmetric model. They offer rich phenomenology in energy frontiers like
the Large Hadron Collider and processes like neutrinoless double beta decay
and lepton flavour violation. They have also been studied combined with
inflation Barrie et al. (2021) and strong CP violation Bertolini et al.
(2015). Adding triplets to the SM and augmenting with different symmetries
explain the origin of different textures in the neutrino mass matrix. It
offers to study combinations of above laid avenues in the future.

(a) $m_{ee}$ vs $m_{1}$

(b) $m_{e\mu}$ vs $m_{1}$

(c) $m_{e\tau}$ vs $m_{1}$

(d) $m_{\mu\mu}$ vs $m_{1}$

(e) $m_{\mu\tau}$ vs $m_{1}$

(f) $m_{\tau\tau}$ vs $m_{1}$

Figure 11: correlation plots for elements of neutrino mass matrix

(a) $B_{L}=0.0005,B_{\phi}=0.9995$

(b) $B_{L}=0.0005,B_{\phi}=0.9995$

(c) $B_{L}=0.5000,B_{\phi}=0.5000$

(d) $B_{L}=0.5000,B_{\phi}=0.5000$

Figure 12: Dependence of decay rates on $B_{L}$ and $B_{\phi}$ and
corresponding abundancy plots

On the left column, the dependence of decay rates on $B_{L}$ and $B_{\phi}$ is
studied; on the right column, corresponding dependence of abundancies on
$B_{L}$ and $B_{\phi}$ is studied, $M_{\Delta}=10^{10}$ GeV,
$\tilde{m}_{\Delta}=0.05$ eV is taken, abundancies ($\Delta_{L}$,
$\Delta_{\phi}$, $|\Delta_{\Delta}|$) are plotted in units of $\epsilon$.

## References

  * Spergel et al. (2003) D. N. Spergel et al. (WMAP), Astrophys. J. Suppl. 148, 175 (2003), eprint astro-ph/0302209.
  * Komatsu et al. (2009) E. Komatsu et al. (WMAP), Astrophys. J. Suppl. 180, 330 (2009), eprint 0803.0547.
  * Hinshaw et al. (2009) G. Hinshaw et al. (WMAP), Astrophys. J. Suppl. 180, 225 (2009), eprint 0803.0732.
  * Dunkley et al. (2009) J. Dunkley et al. (WMAP), Astrophys. J. Suppl. 180, 306 (2009), eprint 0803.0586.
  * Ade et al. (2016) P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A13 (2016), eprint 1502.01589.
  * Aghanim et al. (2020a) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020a), [Erratum: Astron.Astrophys. 652, C4 (2021)], eprint 1807.06209.
  * Sakharov (1967) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  * Fukuda et al. (1998) Y. Fukuda et al. (Super-Kamiokande), Phys. Rev. Lett. 81, 1562 (1998), eprint hep-ex/9807003.
  * Ahmad et al. (2002a) Q. R. Ahmad et al. (SNO), Phys. Rev. Lett. 89, 011301 (2002a), nucl-ex/0204008.
  * Ahmad et al. (2002b) Q. R. Ahmad et al. (SNO), Phys. Rev. Lett. 89, 011302 (2002b), nucl-ex/0204009.
  * Bahcall and Pena-Garay (2004) J. N. Bahcall and C. Pena-Garay, New J. Phys. 6, 63 (2004), hep-ph/0404061.
  * Esteban et al. (2020) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, JHEP 09, 178 (2020), eprint 2007.14792.
  * Aghanim et al. (2020b) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020b), eprint 1807.06209.
  * Minkowski (1977) P. Minkowski, Phys. Lett. B67, 421 (1977).
  * Yanagida (1979) T. Yanagida, in _Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe_ , edited by O. Sawada and A. Sugamoto (Tsukuba, Japan, 1979), p. 95.
  * Mohapatra and Senjanovic (1980) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
  * Fukugita and Yanagida (1986) M. Fukugita and T. Yanagida, Phys. Lett. B174, 45 (1986).
  * Frampton et al. (2002) P. H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B 536, 79 (2002), eprint hep-ph/0201008.
  * Fritzsch et al. (2011) H. Fritzsch, Z.-z. Xing, and S. Zhou, JHEP 09, 083 (2011), eprint 1108.4534.
  * Grimus and Lavoura (2005) W. Grimus and L. Lavoura, J. Phys. G 31, 693 (2005), eprint hep-ph/0412283.
  * Meloni et al. (2014) D. Meloni, A. Meroni, and E. Peinado, Phys. Rev. D 89, 053009 (2014), eprint 1401.3207.
  * Singh (2020) M. Singh, EPL 129, 1 (2020), eprint 1909.01552.
  * Hambye and Senjanovic (2004) T. Hambye and G. Senjanovic, Phys. Lett. B 582, 73 (2004), eprint hep-ph/0307237.
  * Hambye et al. (2006) T. Hambye, M. Raidal, and A. Strumia, Phys. Lett. B 632, 667 (2006), eprint hep-ph/0510008.
  * Hambye et al. (2001) T. Hambye, E. Ma, and U. Sarkar, Nucl. Phys. B 602, 23 (2001), eprint hep-ph/0011192.
  * Barrie et al. (2021) N. D. Barrie, C. Han, and H. Murayama (2021), eprint 2106.03381.
  * Hallgren et al. (2008) T. Hallgren, T. Konstandin, and T. Ohlsson, JCAP 01, 014 (2008), eprint 0710.2408.
  * Rink et al. (2021) T. Rink, W. Rodejohann, and K. Schmitz, Nucl. Phys. B 972, 115552 (2021), eprint 2006.03021.
  * Parida et al. (2020) M. K. Parida, M. Chakraborty, S. K. Nanda, and R. Samantaray, Nucl. Phys. B 960, 115203 (2020), eprint 2005.12077.
  * Parida and Samantaray (2021) M. K. Parida and R. Samantaray, Nucl. Phys. B 970, 115484 (2021), eprint 2102.01605.
  * Chakraborty et al. (2020) M. Chakraborty, M. K. Parida, and B. Sahoo, JCAP 01, 049 (2020), eprint 1906.05601.
  * Mahapatra et al. (2020) S. Mahapatra, N. Narendra, and N. Sahu (2020), eprint 2002.07000.
  * Aristizabal Sierra (2013) D. Aristizabal Sierra, Nucl. Phys. B Proc. Suppl. 237-238, 43 (2013), eprint 1212.3302.
  * Mishra and Giri (2020) S. Mishra and A. Giri, J. Phys. G 47, 055008 (2020), eprint 1909.12147.
  * Lu and Gu (2017) W.-B. Lu and P.-H. Gu, Nucl. Phys. B 924, 279 (2017), eprint 1611.02106.
  * Hati and Sarkar (2016) C. Hati and U. Sarkar, Eur. Phys. J. C 76, 236 (2016), eprint 1511.02874.
  * Bhupal Dev et al. (2013) P. Bhupal Dev, D. K. Ghosh, N. Okada, and I. Saha, JHEP 03, 150 (2013), [Erratum: JHEP 05, 049 (2013)], eprint 1301.3453.
  * Chaudhuri and Mukhopadhyaya (2016) A. Chaudhuri and B. Mukhopadhyaya, Phys. Rev. D 93, 093003 (2016), eprint 1602.07846.
  * Chaudhuri et al. (2014) A. Chaudhuri, W. Grimus, and B. Mukhopadhyaya, JHEP 02, 060 (2014), eprint 1305.5761.
  * Grimus et al. (2000) W. Grimus, R. Pfeiffer, and T. Schwetz, Eur. Phys. J. C 13, 125 (2000), eprint hep-ph/9905320.
  * Ma and Sarkar (1998) E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998), eprint hep-ph/9802445.
  * Merle and Rodejohann (2006) A. Merle and W. Rodejohann, Phys. Rev. D 73, 073012 (2006), eprint hep-ph/0603111.
  * Branco et al. (2012) G. Branco, R. Felipe, and F. Joaquim, Rev. Mod. Phys. 84, 515 (2012), eprint 1111.5332.
  * Aristizabal Sierra et al. (2014) D. Aristizabal Sierra, M. Dhen, and T. Hambye, JCAP 08, 003 (2014), eprint 1401.4347.
  * Lavignac and Schmauch (2015) S. Lavignac and B. Schmauch, JHEP 05, 124 (2015), eprint 1503.00629.
  * Nardi et al. (2006) E. Nardi, Y. Nir, E. Roulet, and J. Racker, JHEP 01, 164 (2006), eprint hep-ph/0601084.
  * Abada et al. (2006) A. Abada, S. Davidson, A. Ibarra, F. X. Josse-Michaux, M. Losada, and A. Riotto, JHEP 09, 010 (2006), eprint hep-ph/0605281.
  * Gonzalez Felipe et al. (2013) R. Gonzalez Felipe, F. Joaquim, and H. Serodio, Int. J. Mod. Phys. A 28, 1350165 (2013), eprint 1301.0288.
  * Halzen and Martin (1984) F. Halzen and A. D. Martin, _QUARKS AND LEPTONS: AN INTRODUCTORY COURSE IN MODERN PARTICLE PHYSICS_ (1984), ISBN 978-0-471-88741-6.
  * Barbieri et al. (2000) R. Barbieri, P. Creminelli, A. Strumia, and N. Tetradis, Nucl. Phys. B 575, 61 (2000), eprint hep-ph/9911315.
  * Giudice et al. (2004) G. Giudice, A. Notari, M. Raidal, A. Riotto, and A. Strumia, Nucl. Phys. B 685, 89 (2004), eprint hep-ph/0310123.
  * Strumia (2009) A. Strumia, Nucl. Phys. B 809, 308 (2009), eprint 0806.1630.
  * Hisano et al. (2007) J. Hisano, S. Matsumoto, M. Nagai, O. Saito, and M. Senami, Phys. Lett. B 646, 34 (2007), eprint hep-ph/0610249.
  * Bertolini et al. (2015) S. Bertolini, L. Di Luzio, H. Kolešová, and M. Malinský, Phys. Rev. D 91, 055014 (2015), eprint 1412.7105.