# Significance of broken $\mu-\tau$ Symmetry in correlating $\delta_{CP}$,
$\theta_{13}$, Lightest neutrino Mass and neutrinoless double beta decay
$0\nu\beta\beta$

Gayatri Ghosh<EMAIL_ADDRESS>Department of Physics, Gauhati
University, Jalukbari, Assam-781015, India  Department of Physics, Pandit
Deendayal Upadhayay Mahavidyalaya, Karimganj, Assam-788720, India

###### Abstract

Leptonic CP Violating Phase $\delta_{CP}$ in the light neutrino sector and
leptogenesis via present matter antimatter asymmetry of the Universe entails
each other. Probing CP violation in light neutrino oscillation is one of the
challenging tasks today. The reactor mixing angle $\theta_{13}$ measured in
reactor experiments, LBL, DUNE with high precision in neutrino experiments
indicates towards the vast dimension of scope to detect $\delta_{CP}$. The
correlation between leptonic Dirac CPV phase $\delta_{CP}$, reactor mixing
angle $\theta_{13}$, lightest neutrino mass $m_{1}$ and matter antimatter
asymmetry of the Universe within the framework of $\mu-\tau$ symmetry breaking
assuming the type I seesaw dominance is extensively studied here. Small tiny
breaking of the $\mu-\tau$ symmetry allows a large Dirac CP violating phase in
neutrino oscillation which in turn is characterised by awareness of measured
value of $\theta_{13}$ and to provide a hint towards a better understanding of
the experimentally observed near maximal value of $\nu_{\mu}-\nu_{\tau}$
mixing angle $\theta_{23}\simeq\frac{\pi}{4}$. Precise breaking of the
$\mu-\tau$ symmetry is achieved by adding a 120 plet Higgs to the 10 $+$
$\bar{126}$ dimensional representation of Higgs. The estimated three
dimensional density parameter space of lightest neutrino mass $m_{1}$,
$\delta_{CP}$, reactor mixing angle $\theta_{13}$, is constrained here for the
requirement of producing the observed value of baryon asymmetry of the
Universe through the mechanism of leptogenesis. Carrying out numerical
analysis the allowed parameter space of $m_{1}$, $\delta_{CP}$, $\theta_{13}$,
is found out which can produce the observed baryon to photon density ratio of
the Universe.

##  I Introduction

In 1950, Brune Pontecorvo for the first time emphasized the idea of neutrino
oscillations which resembled $K^{0}-\bar{K^{0}}$ oscillations. In neutrino
oscillations, a neutrino originated with a definite flavor,
($\nu_{e},\nu_{\mu},\nu_{\tau}$) oscillates to a distinct contrasting lepton
flavor. neutrino oscillation reveals that each of the three states of neutrino
$\nu_{\alpha}$ in flavor basis is a superposition of three mass eigen states
($m_{1},m_{2},m_{3}$) 1 . neutrinos are massive and they mix with each other.
The massive neutrinos are formed in their gauge eigen states $(\nu_{\alpha})$
which is linked to their mass eigen states $\nu_{i}$. Gauge eigen states
participate in gauge interactions as

$\mid\nu_{\alpha}>=\sum U_{\alpha_{i}}\mid\nu_{i}>$ (1)

where, $\alpha=e,\mu,\tau$ , $\nu_{i}$ is the neutrino of distinct mass
$m_{i}$.U is parameterised as

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e_{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\\\
\end{pmatrix}$ (2)

where, $\theta_{12}=33^{0},\theta_{23}=38^{0}-53^{0},\theta_{13}=8^{0}$ 2  are
the solar, atmospheric and reactor angles according to the global fits
respectively. The Majorana phases $\alpha$, $\beta$ dwells in P, where

$P=\text{diag}\begin{pmatrix}1&e^{i\alpha}&e^{i(\beta+\delta)}\end{pmatrix}$
(3)

$U*P$ is known as the Pontecorvo-Maki-Nakagawa-Sakata $U_{PMNS}$ matrix 3 .
Since a $\nu$ of a given flavor $\alpha$ is a mixed state of atleast three
$\nu$ with distinct masses, this three generation mixing could result into the
flavor mixing mass matrix or PMNS matrix possessing an irreducible imaginary
component. This irreducible imaginary component is responsible for CP
asymmetry. CP violation interchanges every particle into its antiparticle.
$\delta_{CP}$ in PMNS matrix can induce CP violation. CP asymmetry can be
observed in neutrino oscillations. $\delta_{CP}$ phase measures the amount of
asymmetries between lepton oscillations and antilepton oscillations. neutrinos
being massive and they mix with each other. This may be a source of CP
violation if Sin$\delta_{CP}\neq 0$. The amount of $\delta_{CP}$ violation
phase in this case is estimated by the Jarkslog invariant 4 .

$J_{CP}=\frac{1}{8}Cos\theta_{13}Sin2\theta_{12}Sin2\theta_{23}Sin2\theta_{13}Sin\delta_{CP}$
(4)

when $Sin\delta_{CP}\neq 0$ In leptogenesis, lepton-antilepton asymmetry is
explained if there are complex imaginary irreducible terms in the yukawa
couplings of lepton mass matrices. The lepton number generation of the early
Universe can be estimated by the complex CPV phase term in the fermion mass
matrices. The ongoing T2K experiment have reported that CP violating phase,
$\delta_{CP}$ excludes the value $\delta_{CP}=0,\pi$ 5  at the 2$\sigma$
confidence interval for either of the mass orderings, Normal ordering and
Inverted ordering. The value of Dirac CPV phase, $\delta_{CP}=276.5^{0}$ is
preferred in 6 . The neutrino mass matrix is invariant under $\mu-\tau$
exchange symmetry. In a basis where the charged leptons are mass eigen states.
Under the $\mu-\tau$ exchange symmetry, the 2-3 mixing is maximal, i.e,
$\theta_{23}=\frac{\pi}{4}$ and the 1-3 mixing is zero, i.e, $\theta_{13}=0$.
The deviation of $\delta\theta_{23}$ from the maximal angle $\theta_{23}$, the
explanation of reactor angle $\theta_{13}$ and the existence of CP violating
phase neccessitates of the spontaneous breaking of the $\mu-\tau$ exchange
symmetry in the neutrino sector. Here, in this work, an explicit form of the
Dirac neutrino mass matrix in broken $\mu-\tau$ 7  symmetry framework in type
I Seesaw mechanism is used in our calculation for generating baryon asymmetry
of the Universe via leptogenesis. This scenario is characterised by small
divergence of $\delta\theta_{23}$ from the maximal angle $\theta_{23}$, which
is consistent with a liberal size of $\theta_{13}\sim 8^{0}-9^{0}$ and a large
$\delta_{CP}$ phase in the neutrino sector. The renormalisable Dirac neutrino
yukawa couplings of the Dirac mass matrices is determined from the fermion
yukawa couplings to the 10, $\bar{126}$ and 120 dimensional fields of Higgs
multiplet in the SO(10) group. Higgs field under the 10 and $\bar{126}$
representations are symmetrical under the generalised $\mu-\tau$ symmetry,
while the 120 dimensional representation changes sign. This spontaneously
breaks the $\mu-\tau$ invariant symmetry, which in turn allows a generalised
$\delta_{CP}$ phase in the PMNS matrix. Here we made an effort for correlating
or constraining the values of $\delta_{CP}$ phase, non zero reactor angle
$\theta_{13}$ and lightest neutrino mass space for both the hierarchies in the
context of leptogenesis and current ratio of baryon to photon density of the
Universe. Both CPV phase $\delta_{CP}$ and reactor angle $\theta_{13}$ have
good vibes between each other. Precise value of $\theta_{13}$ plays an
imperative role in its CP violation phase measurements. On the basis of this
fact non zero values of $\theta_{13}$ is predicted here in consistency with
$\delta_{CP}$ phase. Taking into account constraints from the global fit
values of $\nu$ oscillation parameters and cosmology a density plot of the
favourable values of $\delta_{CP}$ phase, lightest neutrino mass,
$\theta_{13}$ is being initiated, which is compatible with the contraints on
the sum of the absolute neutrino masses, $\sum_{i}m(\nu_{i)}<0.23\hskip
2.84544pt\text{eV}$ from CMB, Planck 2015 data (CMB15+ LRG+ lensing + $H_{0}$)
8 . Constraints fom the leptonic asymmetry of the Universe is also considered
for further restricting $\delta_{CPV}$ phase space and lightest neutrino mass.
The leptonic CP asymmetry is being deliberated via leptogenesis in terms of
baryon density to photon density ratio of the Universe $\eta_{B}$ accessible
as $5.8\times 10^{-10}<\eta_{B}<6.6\times 10^{-10}$ 9 . We also calculate the
effective mass spectrum for neutrinoless double beta decay, $0\nu\beta\beta$
decay given by

$m_{ee}=|m_{1}c^{2}_{12}c^{2}_{13}+m_{2}s^{2}_{12}c^{2}_{13}e^{i\alpha_{21}}+m_{3}s^{2}_{13}e^{i(\alpha_{31}-2\delta_{CP})}|$
(5)

for favourable values of $\delta_{CP}$ phase and lightest $\nu$ mass explored
here in this work. In this paper, we apply the broken $\mu-\tau$ symmetry to
the Dirac neutrino Yukawa Couplings in type-I seesaw mechanism in SO(10) model
in predicting favourable values of $\delta_{CP}$ phase, lightest neutrino mass
and $\theta_{13}$. We then scan free parameters in these models and search for
the allowed region in which the neutrino oscillation data can be fitted. For
the allowed parameter sets, we show the predictions of observables like
$\delta_{CP}$ phase, lightest neutrino mass and $\theta_{13}$ in the neutrino
sector. Finally we show our predictions for the the effective mass spectrum
for neutrinoless double beta, $0\nu\beta\beta$ decay for favourable values of
$\delta_{CP}$ phase.

The paper is organized as follows. In section 2, we introduce our broken
$\mu-\tau$ symmetry models. In section 3, we perform parameter scan to fit
neutrino oscillation data and provide some informations in predicting
observables like $\delta_{CP}$ phase in the neutrino sector. Also we show our
predictions for the the effective mass spectrum for neutrinoless double beta,
$0\nu\beta\beta$ decay for favourable values of $\delta_{CP}$ phase. Section 4
is our results and conclusions for the Yukawa interactions associated with
broken $\mu-\tau$ symmetry model discussed here.

##  II Broken $\mu-\tau$ symmetry with type I Seesaw Mechanism

The Lagrangian of the type I Seesaw model is 10

$L_{N}^{M}=\frac{i}{2}\bar{N^{R}_{I}}\delta
N_{I}^{R}-y_{I\alpha}\bar{N}_{I}^{R}\tilde{\phi^{\dagger}}L_{\alpha}-\frac{1}{2}\bar{N^{R}_{I}}M_{IJ}(N_{J}^{R})^{C}+h.c$
(6)

Here, $y_{I\alpha}$ is the complex yukawa coupling
matrix,$L_{\alpha}=(\nu_{\alpha}^{L},L_{\alpha}^{l})^{T}$ is the standard
model left handed lepton doublet of flavor $\alpha$, when $\alpha=e,\mu,\tau$
and $\tilde{\phi}$ is the hypercharge conjugated Higgs Doublet,
$(\phi^{*}_{0},-\phi_{-})^{T}$.

The Lagrangian describes scenario of generation of $\nu$ masses via Higgs
mechanism. Electroweak symmetry breaking process allows neutral part of the
Higgs field to acquire a VEV, $v=246GeV$, $\sqrt{2}<\phi_{0}>=v$ so that the
left handed and right handed neutrinos forms massive Dirac fermions.

In Eq. (6), $M_{IJ}$ is a symmetric matrix of left handed violating Majorana
masses, where $M_{IJ}=M_{I}\delta_{IJ}$ is real and diagonal. Here right
handed neutrino masses is larger than the Electroweak Scale. The $\nu$ masses
is then suppresed by right handed neutrino yukawa couplings, and also by

$\frac{v}{M_{I}}<1$ (7)

In type I Seesaw, the baryon asymmetry of the Universe (BAU) occurs via
leptogenesis mechanism via out of equilibrium decay of heavy RH Majorana
neutrinos in the early Universe via electroweak sphaleron processes 11 . The
resulting Majorana mass matrix of light SM neutrinos is

$M_{\alpha\beta}=-(m_{D}^{T})_{\alpha I}(M_{IJ}^{-1})(m_{D})_{J\beta}$ (8)

where $(m_{D})_{\alpha I}$ is the Dirac mass matrix. Eq. 8 shows that in type
I seesaw mechanism, SM $\nu$ masses are suppressed by the combination of small
yukawa couplings and large RH $\nu$ masses. neutrino mass matrix on
diagonalision, gives two eigen values $-$ light neutrino $\sim$
$\frac{m^{2}_{D}}{M_{R}}$ and a heavy neutrino state $\sim$ $M_{R}$. This is
known as type I seesaw mechanism.

In SO(10), handed heavy Majorana neutrino couples to the left handed $\nu$ via
Dirac mass matrix $m_{D}$. Out of the decay of the lightest of the RH Majorana
neutrinos, $M_{1}$, i.e $M_{3},M_{2}\gg M_{1}$ will contribute to CP asymmetry
6 ; 12  (for leptogenesis), i.e $\epsilon^{CP}_{l}$ and leptogenesis SO1 ; SO2
; SO3 ; SO4 ; SO5 ; SO6 ; SOF .

At the end of inflation 13 , a certain number density of right-handed
neutrinos, $n_{\nu_{R}}$, were created, which is linked to the present
cosmological scenario. Right-handed neutrinos decayed, with a decay rate that
reads, at tree level

$\Gamma_{D_{i}}=\Gamma(\nu_{R_{i}\rightarrow
l_{i}+H_{u}}+\Gamma(\nu_{R_{i}\rightarrow\tilde{l_{i}}+\tilde{H_{u}}})=\frac{1}{8}(Y_{\nu}Y_{\nu}^{\dagger})_{ii}M_{i}$
(9)

It is convenient to work in a basis of right-handed neutrinos where RH $\nu$
mass matrix is diagonal, the type I contribution to $\epsilon^{CP}_{l}$ is
given by decay of $M_{1}$ or the CP violating parameter is given as,

$\epsilon^{CP}_{l}=\frac{\Gamma_{D_{i}}-\bar{\Gamma_{D_{i}}}}{\Gamma_{D_{i}}+\bar{\Gamma_{D_{i}}}},$
(10)

where

$\Gamma_{D_{i}}=\Gamma(\nu_{R_{i}}\rightarrow
l_{i}H_{u})+\Gamma(\nu_{R_{i}}\rightarrow\tilde{L_{i}}\tilde{h_{u}})=\frac{1}{8}(Y_{\nu}Y_{\nu}^{\dagger})M_{i}$
(11)

where $\Gamma(\nu_{R_{i}}\rightarrow l_{i}H_{u})$ means decay rate of heavy
Majorana RH $\nu$ of mass $M_{1}$ to a lepton and Higgs. In electroweak
sphaleron process, asymmetries produced by the out of equilibrium decay of
$M_{2}$ and $M_{3}$ gets washed out by lepton number violating interactions
after $\nu_{R}$ or $M_{1}$ decay. In lepton number violating interactions
(decays, inverse decays and scatterings) must be out of equilibrium when the
right-handed neutrinos decay. In the basis where RH $\nu$ mass matrix is
diagonal, the type I contribution to $\epsilon^{CP}_{l}$ is given by Mh ,

$\epsilon_{CP}=-\frac{3M_{1}}{8\pi}\frac{Im[\Delta
m^{2}_{\odot}Q^{2}_{12}+\Delta
m^{2}_{A}Q^{2}_{13}]}{\upsilon^{2}\sum|Q_{1j}|^{2}m_{j}}.$ (12)

where $\upsilon$ is the Higg’s vev. $Q$ is a complex unitary orthogonal matrix
where $Q$ is parameterized as Osc
:$Q=D_{\sqrt{M^{-1}}}Y_{\nu}UD_{\sqrt{K^{-1}}}$ where $Y_{\nu}$ is the Dirac
neutrino Yukawa couplings. To reproduce the physical, low-energy, parameters,
i.e. the light neutrino masses (contained in $D_{K}$) and mixing angles and CP
phases (contained in $U_{PMNS}$), we have taken the most general Dirac
neutrino mass matrix in broken $\mu-\tau$ symmetry framework as kod

$Y_{\nu}=\frac{M_{\nu}}{246\hskip 5.69046ptGeV}=\frac{1}{246\hskip
5.69046ptGeV}\begin{pmatrix}11353.7&-11193.7+12692.2i&-11193.7-12692.2i\\\
-11193.7-12692.2i&62440.4&62279.4-14582.3i\\\
-11193.7+12692.2i&62279.4+14582.3i&62021.3\\\ \end{pmatrix}$ (13)

In the flavor basis, where the charged-lepton Yukawa matrix, $Y_{e}$ and gauge
interactions are flavour-diagonal,

$D_{K}=U^{T}Y_{\nu}^{T}D_{M}^{-1}Y_{\nu}U$ (14)

In terms of user defined Dirac neutrino mass matrices, kod

$K=Y_{\nu}^{T}M_{R}^{-1}Y_{\nu}$ (15)

$U$ is the PMNS matrix and $M_{R}$ is the RH neutrino Majorana scale. We can
always choose to work in a basis of right neutrinos where M is diagonal
$D_{M}=Diag(M_{1},M_{2},M_{3})$ where $M_{3},M_{2}\gg M_{1}$. Equation (12)
expresses $\epsilon_{CP}$ in terms of both the solar ($\Delta m^{2}_{21}$) and
atmospheric ($\Delta m^{2}_{A}$) mass squared differences, eq.(12) also
reveals that CP asymmetry is linked to Dirac CPV phase. Here we utilse this
fact to generate the allowed region of $\delta_{CP}$ phase in context of
laptogenesis. As has been discussed in Mh , the lepton $–$ antilepton
asymmetry gets connected to both the solar and the atmospheric mass difference
square. The transformation of the lepton asymmetry into a baryon asymmetry by
non-perturbative $B+L$ violating (sphaleron) processes is discussed in 6 .

neutrino masses and mixings are connected with the atmospheric and solar
neutrino fluxes, this is suitable to explain flavor changing neutral current
processes, FCNC processes, like $\mu\rightarrow e+\gamma$ processes. In
supersymmetric theories like cMSSM, NUHM, NUGM and NUSM where the origin of
the $\nu$ masses is via see-saw mechanism, it can be shown that the prediction
for $BR(\mu\rightarrow e,\gamma)$, $BR(\tau\rightarrow\mu,\gamma)$ and
$BR(\tau\rightarrow e,\gamma)$ is in general larger than the experimental
upper MEG bound GG ; GG1 . Also some studies on decays of b flavoured hadrons
in the context of cMSSM/mSUGRA models is being done in bfh .

Let H, F (G) be any complex symmetric (anti) matrices in general, which are a
measure of the fermion yukawa couplings to the 10, $\bar{126}$ and 120 Higgs
field respectively. Here,

$H=\begin{pmatrix}h_{11}&h_{12}&h_{12}\\\ h_{12}&h_{22}&h_{23}\\\
h_{12}&h_{23}&h_{22}\\\ \\\ \end{pmatrix}$ (16)

Similarly

$F=\begin{pmatrix}f_{11}&f_{12}&f_{12}\\\ f_{12}&f_{22}&f_{23}\\\
f_{12}&f_{23}&f_{22}\\\ \\\ \end{pmatrix}$ (17)

Similarly

$G=\begin{pmatrix}0&g_{12}&-g_{12}\\\ -g_{12}&0&g_{23}\\\ g_{12}&-g_{23}&0\\\
\\\ \end{pmatrix}$ (18)

h,f,g are all real. $\mu-\tau$ symmetry is invariant under the exchange of
second and third generation fermions. When $\mu-\tau$ symmetry is added with
SO(10) grand unified theory then a general symmetry results which satisfies

$S^{T}(H,F,G)S=(H,F,-G)$ (19)

where

$S=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\ \\\ \end{pmatrix}$ (20)

A small explicit breaking of $\mu-\tau$ symmetry is put by hand, by inheriting
the property,

$h_{22}\neq h_{33}$ (21)

This introduces CP violation in PMNS matrices and results in $\theta_{13}$
being non zero. The amount of breaking used here for generating non zero CP
asymmetry producing a measurable CP violating phase via Dirac neutrino Yukawa
Couplings used from kod  is

$\frac{h_{22}-h_{33}}{h_{22}+h_{33}}=0.0045$ (22)

Exact $\mu-\tau$ symmetry imposes $Sin^{2}\theta_{23}=0.$ One can break the
symmetry spontaneously through the vev of the 120 plet and by use of Eq. (22).
A required amount of CP violating phase $\delta_{CP}$ is generated by
explicitly breaking $\mu-\tau$ symmetry. This assumption by using Eq.(22)
leads to $Sin^{2}\theta_{23}\sim 0.42-0.63$ and $Sin^{2}\theta_{13}>0.006$.
All these remarks enables one to use Dirac neutrino Yukawa Couplings mass
matrices reproduced by explicit use of broken $\mu-\tau$ symmetry embedded in
Eq. (22).

##  III Numerical Analysis

In this section, numerical analysis has been carried out. Firstly, the free
parameter called the lightest $\nu$ mass, the $\nu$ oscillation parameters
like reactor angle $\theta_{13}$ Dirac CPV phase, $\delta_{CP}$, Majorana
phases $\alpha_{21}$, $\alpha_{31}$ in broken $\mu-\tau$ symmetry, type I
Seesaw model are scanned to search for dependance of $\delta_{CP}$ phase on
$\theta_{13}$, $m_{1}$ ($m_{3}$), Jarkslog invariant $J_{CP}$, effective mass
for $)\nu\beta\beta$ decay in case of normal ordering (inverted ordering) in
the context of producing correct baryon asyymetry of the Universe 9 . We use
best fit values of $\nu$ oscillation parameters. The two mass square
differences $\Delta m^{2}_{12}$, $\Delta m^{2}_{13}$ are embedded in neutrino
mixing matrix so we are left out with lightest $\nu$ mass as only free
parameter in this model. In the chargd lepton basis, we parameterize the PMNS
matrix $U_{PMNS}$, by diagonalizing the neutrino mass matrix $m_{\nu}$ in
terms of three mixing angles $\theta_{ij}$ ($i,j=1,2,3$; $i<j$), one CP
violating Dirac CPV phase $\delta_{CP}$, and two Majorana phases
($\alpha_{21}$ and $\alpha_{31}$) as follows:

$U^{\ast}P^{\ast}m_{\nu}P^{\dagger}U^{\dagger}=m_{\nu}^{D}$ (23)

The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is UP, where U is

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e_{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\\\
\end{pmatrix}$ (24)

where,
$\theta_{12}=33.82^{0},\theta_{23}=48.3^{0}(48.6^{0}),\theta_{13}=8.61^{0}(8.65^{0})$
pdg  in case of normal hierarchy (inverted hierarchy) are the solar,
atmospheric and reactor angles respectively. The Majorana phases reside in P,
where

$P=\text{diag}\begin{pmatrix}1&e^{i\alpha_{21}}&e^{i(\alpha_{31}+\delta)}\end{pmatrix}$
(25)

We have taken complex and orthogonal matrix $R=U_{PMNS}$, in terms of user
defined Dirac neutrino Yukawa Couplings defined in Eq. (13) in order to
produce correct baryon asymmetry of the Universe.

For the Normally ordered light $\nu$ masses, we have

$M_{R}^{diag}=\text{diag}(M_{1},M_{2},M_{3})=M_{1}\text{diag}(1,\frac{M_{2}}{M_{1}},\frac{M_{3}}{M_{1}})=M_{1}\text{diag}(1,\frac{m_{1}}{m_{2}},\frac{m_{1}}{m_{3}})$
(26)

With $m_{1}\in[10^{-6}eV,10^{-1}eV],and,m_{2}^{2}-m_{1}^{2}=7.39\times
10^{-5}eV^{2},m_{3}^{2}-m_{1}^{2}=2.48\times 10^{-3}eV^{2}$ as is evident from
the $\nu$ oscillation data pdg , $m_{1}$ being the lightest of three $\nu$
masses. For the inverted ordered light $\nu$ masses, we have

$M_{R}^{diag}=\text{diag}(M_{1},M_{2},M_{3})=M_{1}\text{diag}(1,\frac{M_{2}}{M_{1}},\frac{M_{3}}{M_{1}})=M_{1}\text{diag}(1,\frac{m_{1}*m_{3}}{m_{2}^{2}},\frac{m_{1}}{m_{2}})$
(27)

with $m_{3}$ being the lightest of three $\nu$ masses. Here we take $M_{1}\sim
10^{12}$ GeV. For normal ordering, the choices of lightest neutrino mass is
$m_{1}=0.07118eV$ whereas for inverted ordering, the choice of lightest
neutrino mass is $m_{3}=0.0657eV$. This sustainable allowance of
$m_{1}(m_{3})=0.07(0.065)eV$ signifies a neutrino mass spectrum where the sum
of absolute neutrino masses lies below the cosmological upper bound,
$\sum_{i}m(\nu_{i})<0.23\text{eV}$ An . Next random scan of the $\nu$ mixing
matrix parameter space for NH, IH in order to produce correct baryon asymmetry
of the Universe $5.8\times 10^{-10}<Y_{B}<6.6\times 10^{-10}$ is performed in
the following 3 $\sigma$ranges of $\delta_{CP}$ with respect to the tabulated
$\chi^{2}$ map of the SuperKamiokande analysis of the data within
$\Delta\chi^{2}=6.2$ pdg :

$m_{1}(m_{3})\in[10^{-6}\hskip 2.84544pteV,0.1\hskip 2.84544pteV]\hskip
2.84544pt([10^{-6}\hskip 2.84544pteV,0.1\hskip 2.84544pteV])$
$\delta_{CP}\in[141,370]\hskip 5.69046pt\text{for Normal ordering}$
$\delta_{CP}\in[205,354]\hskip 5.69046pt\text{for inverted ordering}$
$\theta_{13}\in[7.9,8.9]\hskip 5.69046pt\text{for Normal ordering for
tabulated $\Delta\chi^{2}=9.5$}$ $\theta_{13}\in[8.0,9.0]\hskip
5.69046pt\text{for inverted ordering for tabulated $\Delta\chi^{2}=9.5$}$
$\theta_{23}\in[40.8,51.3]\hskip 5.69046pt\text{for normal ordering for
tabulated $\Delta\chi^{2}=6.2$}$

While doing parameter scan, we find favoured values of lightest $\nu$ mass,
dirac CPV phase $\delta_{CP}$, for producing correct baryon asymmetry of the
Universe, $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$.

The lepton flavor effects are significant if the lightest right handed
Majorana neutrino mass $M_{\nu_{R1}}$ is below $10^{12}$ GeV. Here
$M_{1}=10^{12}$ GeV. In the type I seesaw mechanism one can always find the
right handed neutrino mass matrix as;

$M_{\alpha\beta}=-(m_{D}^{T})_{\alpha I}(M_{IJ}^{-1})(m_{D})_{J\beta}$ (28)

where $(m_{D})_{\alpha I}$ is the Dirac mass matrix. We consider a Dirac
neutrino mass matrix defined in Eq. (13). Here when we fix
$m_{1}(m_{3}),Y_{\nu},M_{R}$ the remaining free parameter in the neutrino
sector within our broken $\mu-\tau$ framework is the leptonic CPV phase
$\delta_{CP}$. When we vary the CPV phase $\delta_{CP}$ we compute the
favoured regions of $m_{1}(m_{3})$. The variations of leptonic CPV phase
$\delta_{CP}$ with $m_{1}(m_{3})$, $\theta_{13}$, $J_{CP}$, $m_{ee}$ for
$0\nu\beta\beta$ are shown in figures below.

For global fit values of $\nu$ oscillation parameters, we compute the Jarlskog
invariant, $\delta_{CP}$ given by PMNS matrix elements $U_{\alpha i}$. We also
compute the Jarkslog invariant for allowed values of $\delta_{CP}$ phase,
$\theta_{13}$ and lightest $\nu$ mass explored here in this work for both
normal ordering and inverted ordering.

$J_{CP}=Im[U_{e1}U_{\mu 2}U^{*}_{e2}U^{*}_{\mu
1}]=s_{23}c_{23}s_{12}c_{12}s_{13}c^{2}_{13}sin\delta_{CP}$ (29)

We also calculate the favourable space of the effective mass for
$0\nu\beta\beta$ decay for favourable values of $\delta_{CP}$ phase,
$\theta_{13}$ and lightest $\nu$ mass given by,

$m_{ee}=|m_{1}c^{2}_{12}c^{2}_{13}+m_{2}s^{2}_{12}c^{2}_{13}e^{i\alpha_{21}}+m_{3}s^{2}_{13}e^{i({\alpha_{31}-2\delta_{CP}})}|$
(30)

Figure 1: Predictions in broken $\mu-\tau$ symmetry model for Normal ordering.
The left panel: predicted favoured values of $({m_{1},\theta_{13}})$ plane for
best fit values of $\delta_{CP}=222^{0}$ with $\Delta\chi^{2}=6.2$ w/o SK-ATM
pdg  ( allowed by updated values of correct baryon asymmetry of the Universe)
as a result of contribution of type I Seesaw mechanism to neutrino mass
matrix. Similarly, The right panel: predicted favoured values
$({m_{1},\delta_{CP}})$ plane, for best fit value of $\theta_{13}=8.41$ with
$\Delta\chi^{2}=9.5$pdg

.

Figure 2: Predictions in broken $\mu-\tau$ symmetry model for Normal ordering:
Contour plot for predicted favoured values of $({m_{1},\theta_{13}})$ plane
for best fit values of $\delta_{CP}=222^{0}$ of $\Delta_{\chi^{2}}=6.2$ w/o
SK-ATM pdg  ( allowed by updated values of correct baryon asymmetry of the
Universe) as a result of contribution of type I Seesaw mechanism to neutrino
mass matrix.

Figure 3: Predictions in broken $\mu-\tau$ symmetry model for Normal ordering:
The left panel: three dimensional plot of preferred values of
$({m_{1},\delta_{CP}})$ plane for best fit values of $\theta_{13}=8.41$ of
$\Delta\chi^{2}=9.5$ w/o SK-ATM pdg  (in the light of recent ratio of the
baryon to photon density bounds $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$)
as a result of contribution of type I Seesaw mechanism to neutrino mass
matrix. Similarly, The right panel: three dimensional plot of favourable
values of $({m_{1},\theta_{13}})$ plane, for best fit value of
$\delta_{CP}=222^{0}$ of $\Delta\chi^{2}=6.2$ pdg  (in the light of recent
ratio of the baryon to photon density bounds $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$).

Figure 4:  Predictions in broken $\mu-\tau$ symmetry model for Normal
ordering: The left panel: preferred values of $\delta_{CP}$ for best fit
values of $\theta_{13}=8.41$ of $\Delta\chi^{2}=9.5$ w/o SK-ATM pdg  (in the
light of recent ratio of the baryon to photon density bounds $5.8\
times10^{-10}<\eta<6.6\times 10^{-10}$) as a result of contribution of type I
Seesaw mechanism to neutrino mass matrix. Similarly, the right panel: favoured
values of lightest neutrino mass, $m_{1}$, for best fit value of
$\delta_{CP}=222^{0}$ of $\Delta\chi^{2}=6.2$ pdg  (in the light of recent
ratio of the baryon to photon density bounds $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$).

Figure 5: Predictions in broken $\mu-\tau$ symmetry model for Normal ordering:
The right panel: preferred three dimensional regions of ($\delta_{CP}$,
$\theta_{23}$, $J_{CP}$) plane for best fit values of $\theta_{13}=8.41$ of
$\Delta\chi^{2}=9.5$ w/o SK-ATM pdg . The left panel: allowed two dimensional
space of ($\delta_{CP}$, $J_{CP}$) plane for best fit values of
$\theta_{13}=8.41$ of $\Delta\chi^{2}=9.5$ w/o SK-ATM pdg . Figure 6:
Predictions in broken $\mu-\tau$ symmetry model for Normal ordering: The right
panel: allowed two dimensional space of ($\delta_{CP}$, $J_{CP}$ ) plane for
absolute range of $\delta_{CP}\in[-180,+180]$.

Figure 7: Predictions in broken $\mu-\tau$ symmetry model for Normal ordering:
The right panel: preferred three dimensional regions of ($\delta_{CP}$,
$\theta_{23}$, $J_{CP}$) plane for favoured values of
$\delta_{CP}\in[303,308]$ (in the light of recent ratio of the baryon to
photon density bounds $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$) for best
fit values of $\theta_{13}=8.41$ of $\Delta\chi^{2}=9.5$ w/o SK-ATM pdg . The
left panel: allowed two dimensional space of ($\delta_{CP}$, $J_{CP}$) plane
for favoured values of $\delta_{CP}\in[303,308]$ (in the light of recent ratio
of the baryon to photon density bounds $5.8\times 10^{-10}<\eta<6.6\times
10^{-10}$) for best fit values of $\theta_{13}=8.41$ of $\Delta\chi^{2}=9.5$
w/o SK-ATM pdg  (in the light of recent ratio of the baryon to photon density
bounds $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$) .

Figure 8: Predictions in broken $\mu-\tau$ symmetry model for Normal ordering:
The right panel: preferred three dimensional regions of ($\theta_{13}$,
$\theta_{23}$, $J_{CP}$) plane for best fit values of $\delta_{CP}=222^{0}$ of
$\Delta\chi^{2}=6.2$ w/o SK-ATM pdg . The left panel: allowed two dimensional
space of ($\theta_{13}$, $J_{CP}$) plane for best fit values of
$\delta_{CP}=222^{0}$ of $\Delta\chi^{2}=6.2$ w/o SK-ATM pdg  Figure 9:
Predictions in broken $\mu-\tau$ symmetry model for Normal ordering: Variation
of $J_{CP}$ as a function of $\theta_{23}$, $\theta_{23}\in[40.8,51.3]$ for
normal ordering for tabulated $\Delta\chi^{2}=6.2.$ pdg .

Figure 10: Predictions in broken $\mu-\tau$ symmetry model for Normal
ordering: The left panel depicts predicted three dimensional space of
$(m_{1},\delta_{CP},\theta_{13})$ for $m_{ee}$ [eV], $0\nu\beta\beta$ decay
for favoured values of $m_{1}$,$\delta_{CP},\theta_{13}$ (in the light of
recent ratio of the baryon to photon density bounds, $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$). The right panel depicts predicted three
dimensional space of ($m_{1},\delta_{CP},\theta_{13}$) for $m_{ee}$ [eV],
$0\nu\beta\beta$ decay for values of $m_{1}$,$\delta_{CP},\theta_{13}$ in the
given three $\sigma$ range, corresponding to $\Delta\chi^{2}=6.2$ and
$\Delta\chi^{2}=9.5$ w/o SK-ATM pdg .
Figure 11: Predictions in broken $\mu-\tau$ symmetry model for Normal
ordering: depicts density plot of $(m_{1},\delta_{CP}$) for $m_{ee})$ [eV],
$0\nu\beta\beta$ decay for favoured values of
$m_{1}$,$\delta_{CP},\theta_{13}$ (in the light of recent ratio of the baryon
to photon density bounds, $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$). .

Figure 12: Predictions in broken $\mu-\tau$ symmetry model for Normal
ordering. Left panel depicts predicted three dimensional space of
$(m_{ee},\delta_{CP},m_{1})$ for $m_{ee}$ [eV], $0\nu\beta\beta$ decay for
favoured values of $m_{1}$,$\delta_{CP},\theta_{13}$ (in the light of recent
ratio of the baryon to photon density bounds, $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$). Left panel depicts predicted three
dimensional space of $(m_{ee},\delta_{CP},m_{1})$ for $m_{ee}$ [eV],
$0\nu\beta\beta$ decay for favoured values of $m_{1}$,$\delta_{CP},m_{ee}$ for
values of $\delta_{CP}$ in the given three $\sigma$ range, corresponding to
$\Delta\chi^{2}=9.5$ w/o SK-ATM pdg  Figure 13: The predicted three
dimensional space of $(m_{ee},\delta_{CP},\theta_{13})$ for $m_{ee}$ [eV],
$0\nu\beta\beta$ decay for favoured values of
$m_{1}$,$\delta_{CP},\theta_{13}$ (in the light of recent ratio of the baryon
to photon density bounds, $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$) for
lightest $\nu$ mass $m_{1}=0.07118$ eV.

Figure 14: Predictions in broken $\mu-\tau$ symmetry model for Inverted
ordering. The left panel: predicted favoured values of $\delta_{CP}$ (in the
light of recent ratio of the baryon to photon density bounds, $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$) for lightest $\nu$ mass $m_{3}=0.0657$ eV,
as a result of contribution of type I Seesaw mechanism to neutrino mass
matrix. Similarly, The right panel: predicted allowed three dimensional space
of $(\delta_{CP},\theta_{13},J_{CP})$ plane for allowed regions of Jarkslog
invariant, $J_{CP}$ values for for best fit value of $\theta_{23}=48.6^{0}$ of
$\Delta\chi^{2}=6.2$ pdg  as a result of contribution of type I Seesaw
mechanism to neutrino mass matrix. Figure 15: Predictions in broken $\mu-\tau$
symmetry model for Inverted ordering: Three dimensional plot for predicted
favoured values of $(m_{3},\delta_{CP},\eta)$ plane for best fit values of
$\theta_{13}=8.49^{0}$ of $\Delta\chi^{2}=9.5$ w/o SK-ATM pdg  (in the light
of recent ratio of the baryon to photon density bounds, $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$) as a result of contribution of type I
Seesaw mechanism to neutrino mass matrix.

Figure 16: Predictions in broken $\mu-\tau$ symmetry model for Inverted
ordering. The left panel: density plot of predicted favoured values of
$({m_{3},\theta_{13}})$ plane for best fit values of $\delta_{CP}=285^{0}$ of
$\Delta\chi^{2}=6.2$ w/o SK-ATM pdg  ( allowed by updated values of correct
baryon asymmetry of the Universe) as a result of contribution of type I Seesaw
mechanism to neutrino mass matrix. Similarly, The right panel: Three
dimensional plot for predicted favoured values of
$(\theta_{13},\delta_{CP},\eta)$ plane for lightest $\nu$ mass, $m_{3}=0.0657$
(in the light of recent ratio of the baryon to photon density bounds,
$5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$) as a result of contribution of
type I Seesaw mechanism to neutrino mass matrix

.

Figure 17: Predictions in broken $\mu-\tau$ symmetry model for Inverted
ordering. The left panel: density plot of predicted favoured values of
$({m_{3},\theta_{13}})$ plane for best fit values of $\delta_{CP}=285^{0}$ of
$\Delta\chi^{2}=6.2$ w/o SK-ATM pdg  ( allowed by updated values of correct
baryon asymmetry of the Universe) as a result of contribution of type I Seesaw
mechanism to neutrino mass matrix. Similarly, The right panel: The predicted
two dimensional space of $(m_{3},\theta_{13})$ for $m_{ee}$ [eV],
$0\nu\beta\beta$ decay, for best fit values of $\delta_{CP}=285^{0}$ of
$\Delta\chi^{2}=6.2$ w/o SK-ATM pdg

.

Figure 18: Predictions in broken $\mu-\tau$ symmetry model for Inverted
ordering: predicted allowed three dimensional space of
$(delta_{CP},\theta_{23},J_{CP})$ plane for allowed regions of Jarkslog
invariant, $J_{CP}$ values for for best fit value of $\theta_{13}=8.49$ of
$\Delta\chi^{2}=9.5$ pdg  as a result of contribution of type I Seesaw
mechanism to neutrino mass matrix

.

Figure 19: Predictions in broken $\mu-\tau$ symmetry model for Inverted
ordering. The left panel depicts predicted three dimensional space of
$(m_{ee},\delta_{CP},m_{3})$ for values of $m_{3}\in[10^{-6},0.1]$ eV,
$\delta_{CP}$ in the given 3 $\sigma$range, corresponding to
$\Delta\chi^{2}=6.2$ and best fit values of $\theta_{13}=8.49^{0}$
corresponding to $\Delta\chi^{2}=9.5$ w/o SK-ATM [27]. The right panel depicts
predicted three dimensional space of $(m_{ee},\delta_{CP},\theta_{13})$ for
values of lightest $\nu$ mass, $m_{3}=0.0657eV$, $\delta_{CP},\theta_{13}$ in
the given 3 $\sigma$range, corresponding to $\Delta\chi^{2}=6.2$ and
$\Delta\chi^{2}=9.5$ w/o SK-ATM [27]

.

Figure 20: Predictions in broken $\mu-\tau$ symmetry model for Inverted
ordering: depicts predicted density plot of $(\delta_{CP},m_{3})$ for values
of $m_{3}\in[10^{-6},0.1]$ eV, $\delta_{CP}$ in the given 3 $\sigma$range,
corresponding to $\Delta\chi^{2}=6.2$ and best fit values of
$\theta_{13}=8.49^{0}$ corresponding to $\Delta\chi^{2}=9.5$ w/o SK-ATM [27].

##  IV Results and Conclusion

In this work, we have used user defined Dirac Neutrino Yukawa couplings for
the yukawa interactions associated with the broken $\mu-\tau$ symmetry model
for the generation of non-zero reactor mixing angle $\theta_{13}$, leptonic CP
phase $\delta_{CP}$ in type I seesaw Mechanism in the light of leptogenesis
and then the transformation of the lepton asymmetry into a baryon asymmetry by
non-perturbative $B+L$ violating (sphaleron, sakharov conditions) processes as
discussed in 6 . A small explicit breaking of $\mu-\tau$ symmetry kod
inherits the property of generating non zero CP violation in $U_{PMNS}$
matrices, $\delta_{CP}$ phase and results in $\theta_{13}$ being non zero.
Here we consider the type I seesaw as the main donor to neutrino mass. We also
take into account both inverted and normal ordering of neutrino mass spectrum
as well as two different types of lightest neutrino mass
$m_{1}(m_{3}=0.07118eV(0.0657eV))$ to visualise the results of hierarchical
$\nu$ mass spectrum. In case of normal ordering of $\nu$ masses, the
dependance of $\delta_{CP}$ phase on lightest $\nu$ mass is predicted in
figures 1,3,4 (in the light of recent ratio of the baryon to photon density
bounds, $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$ ). The favoured values of
$\delta_{CP}$ phase is found to lie between $\delta_{CP}\in[302^{0},304^{0}]$
for best fit values of $\theta_{13}=8.41$ corresponding to
$\Delta\chi^{2}=9.5$ w/o SK-ATM pdg  (in the light of recent ratio of the
baryon to photon density bounds $5.8\times 10^{-10}<\eta<6.6\times 10^{-10}$)
as a result of contribution of type I Seesaw mechanism to neutrino mass
matrix. The favoured values of lightest $\nu$ mass, $m_{1}$ in this case comes
out to be $\in[0.078,0.1]$ eV. In case of inverted hierarchy the variation of
$\delta_{CP}$ phase is found to be very intense with best fit valuesof
$\theta_{13}=8.49$ corresponding to $\Delta\chi^{2}=9.5$ w/o SK-ATM pdg . Here
only two values of $\delta_{CP}$ phase is favoured,
$\delta_{CP}=220^{0},223^{0},252^{0},268^{0},293^{0},309^{0},345^{0}$ (in the
light of recent ratio of the baryon to photon density bounds $5.8\times
10^{-10}<\eta<6.6\times 10^{-10}$) as is evident from figure 15. The allowed
spectrum of lightest $\nu$ mass is $m_{3}$, $\in[0.02,0.055]$ eV. We also plot
the allowed values of $|m_{ee}|$ eV for neutrinoless double beta decay and the
Jarkslog invariant, $J_{CP}$ in figures 5,6,7,8,9,10,11,12,13 for normal
ordering of $\nu$ masses. Prediction of future leptonic CP violation
experiments should be able to rule out or take into account some of the
results discussed in this work. If we abide by the best fit values of leptonic
CP phase $\delta_{CP}=\frac{3}{2}\pi$ discussed in the literature r  then our
scenario, $\delta_{CP}\in 268^{0}$ for inverted ordering of $\nu$ masses
favours the value of $\delta_{CP}=\frac{3}{2}\pi$ for T2K neutrino and
antineutrino appearance results. We show the variation of baryon asymmetry
with leptonic $\delta_{CP}$phase in table I.

$\mu-\tau$ broken symmetry model | Calculated leptonic CP phase $\delta_{CP}$
---|---
$m_{1}\in[0.078,0.1]$ eV (NH) | $\delta_{CP}\in[302^{0},304^{0}]$
$m_{3}\in[0.02,0.055]$ eV (IH) | $\delta_{CP}=220^{0},223^{0},252^{0},268^{0},293^{0},309^{0},345^{0}$
Table 1: Values of $\delta_{CP}$ phase giving correct updated values of baryon
asymmetry.

We also plot the allowed values of $|m_{ee}|$ eV for neutrinoless double beta
decay and the Jarkslog invariant, $J_{CP}$ in figures 14,17,18,19,20 for
inverted ordering of $\nu$ masses. Future LBL experiments, will hunt for the
leptonic CP phase and potentially will measure it with precision. Neutrinoless
double beta decay will indicate towards Majorana CPV phase. The fundamental
mysteries in the Universe are about the findings of the nature of the massive
neutrinos-Dirac or Majorana. This may be sorted out by the experiments like
GERDA, CUORE, KamLAND-Zen, EXO, LEGEND, nEXO etc. Determination of the status
of leptonic CP asymmetry (T2K, NO$\nu$A, T2HK, DUNE), determination of type of
neutrino mass ordering (T2K+NO$\nu$A, JUNO, PINGU, ORCA, T2HKK, DUNE) and
determination of the order of absolute neutrino mass scale (KATRIN, Cosmology)
are few of the most challenging tasks today. The ideas presented in this work
may definitely will rule in or rule out some of the favoured space in few of
the above experiments.

##  V Acknowledgement

I would like to thank my supervisor Prof. Kalpana Bora for useful discussion
on this topic.

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