aainstitutetext: DESM (Physics), Regional Institute of Education (NCERT),
Bhubaneswar 751022, Indiabbinstitutetext: Department of Physics, Indian
Institute of Technology Bhilai, Raipur 492015, Indiaccinstitutetext:
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai
400076, India

# Effect of large light-heavy neutrino mixing and natural type-II seesaw
dominance to lepton flavor violation and neutrinoless double beta decay

Nitali Dash b Sudhanwa Patra c Prativa Pritimita c Urjit A. Yajnik
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We derive the lower bound on absolute scale of lightest neutrino mass for
normal hierarchy and inverted hierarchy pattern of light neutrinos by studying
the new physics contributions to charged lepton flavour violations in the
framework of a TeV scale left-right symmetric model. In the model the fermion
sector comprises of the usual quarks and leptons plus a fermion singlet per
generation and the scalar sector consists of isospin doublets, triplets and a
bidoublet. The framework allows large light-heavy neutrino mixing where the
light neutrino mass formula is governed by natural type-II seesaw mechanism
unlike the generic type-II seesaw dominance which assumes suppressed light-
heavy neutrino mixing. We demonstrate how sizeable loop-induced contribution
to light neutrino mass is kept under control such that light neutrino mass
formula is dominantly explained by type-II seesaw mechanism. We examine the
heavy neutrino contributions with large light-heavy neutrino mixing to charged
lepton flavour violating processes like $\mu\to e\gamma$, $\mu\to 3e$ and
$\mu\to e$ conversion inside a nuclei. We present a complementary study
between neutrinoless double beta decay and charged lepton flavour violation
taking into account single beta decay bound, double beta decay bound and
cosmology bounds on neutrino mass sum.

###### Keywords:

Seesaw mechanism, lepton flavour violation, left-right theories

##  1 Introduction

Neutrino oscillation data clearly indicate that neutrinos have small but non-
zero masses Olive:2016xmw . A simple theoretical paradigm for the origin of
neutrino mass is the seesaw mechanism, which predicts the Majorana nature of
neutrinos; for a review, see e.g. Ref. Mohapatra:2005wg . The existence of
right-handed (RH) neutrinos, as required for the type-I seesaw mechanism
Minkowski:1977sc ; Mohapatra:1979ia ; Yanagida:1979as ; GellMann:1980vs , or
the triplet scalars, as required for the type-II seesaw mechanism Magg:1980ut
; Schechter:1980gr ; Cheng:1980qt ; Lazarides:1980nt ; Mohapatra:1980yp , can
both be naturally motivated in ultraviolet-complete models of neutrino mass.
One such example is the left-right symmetric model (LRSM) Pati:1974yy ;
Mohapatra:1974gc ; Senjanovic:1975rk . In particular, TeV-scale models of
left-right symmetry breaking have a number of testable consequences for
collider signals in the gauge Keung:1983uu ; Ferrari:2000sp ; Schmaltz:2010xr
; Nemevsek:2011hz ; Chen:2011hc ; Chakrabortty:2012pp ; Das:2012ii ;
AguilarSaavedra:2012gf ; Han:2012vk ; Chen:2013fna ; Rizzo:2014xma ;
Deppisch:2014zta ; Deppisch:2015qwa ; Gluza:2015goa ; Ng:2015hba ;
Patra:2015bga ; Dobrescu:2015qna ; Brehmer:2015cia ; Dev:2015pga ;
Coloma:2015una ; Deppisch:2015cua ; Dev:2015kca ; Mondal:2015zba ; Aguilar-
Saavedra:2015iew ; Lindner:2016lpp ; Lindner:2016lxq ; Mitra:2016kov ;
Anamiati:2016uxp ; Khachatryan:2014dka ; Aad:2015xaa ; Khachatryan:2016jqo  as
well as Higgs sector Gunion:1989in ; Deshpande:1990ip ; Polak:1991vf ;
Barenboim:2001vu ; Azuelos:2004mwa ; Jung:2008pz ; Bambhaniya:2013wza ;
Dutta:2014dba ; Bambhaniya:2014cia ; Bambhaniya:2015wna ; Dev:2016dja ;
ATLAS:2014kca ; CMS:2016cpz ; ATLAS:2016pbt , neutrinoless double beta decay
($0\nu\beta\beta$) Mohapatra:1980yp ; Mohapatra:1981pm ; Picciotto:1982qe ;
Hirsch:1996qw ; Arnold:2010tu ; Tello:2010am ; Chakrabortty:2012mh ;
Nemevsek:2012iq ; Patra:2012ur ; Awasthi:2013ff ; Barry:2013xxa ; Dev:2013vxa
; Huang:2013kma ; Dev:2014xea ; Ge:2015yqa ; Borah:2015ufa ; Awasthi:2015ota ;
Horoi:2015gdv ; Bambhaniya:2015ipg ; Gu:2015uek ; Borah:2016iqd ;
Awasthi:2016kbk , low-energy charged lepton flavor violation (cLFV)
Riazuddin:1981hz ; Pal:1983bf ; Mohapatra:1992uu ; Cirigliano:2004mv ;
Cirigliano:2004tc ; Bajc:2009ft ; Tello:2010am ; Das:2012ii ; Barry:2013xxa ;
Dev:2013oxa ; Borah:2013lva ; Chakrabortty:2015zpm ; Awasthi:2015ota ;
Bambhaniya:2015ipg ; Borah:2016iqd ; Lindner:2016bgg ; Bonilla:2016fqd  and
electric dipole moment (EDM) Ecker:1983dj ; Nieves:1986uk ; Frere:1991jt ;
Nemevsek:2012iq ; Dev:2014xea ; Maiezza:2014ala , all of which could together
shed light on some of the unresolved issues in neutrino physics, such as the
Dirac vs. Majorana nature, mass hierarchy and absolute mass of the left-handed
(LH) neutrinos, and the leptonic $CP$ violation. These results will have far-
reaching implications for beyond the Standard Model (SM) physics in general.

In the conventional LRSM, where symmetry breaking is implemented with scalar
bidoublets and triplets, the light neutrino mass is governed by both type-I
Minkowski:1977sc ; Mohapatra:1979ia ; Yanagida:1979as ; GellMann:1980vs  and
type-II Magg:1980ut ; Schechter:1980gr ; Cheng:1980qt ; Lazarides:1980nt ;
Mohapatra:1980yp  seesaw contributions:

$\displaystyle M_{\nu}\ =\ -M_{D}M_{R}^{-1}M_{D}^{T}+M_{L}\ \equiv\
M_{\nu}^{\rm I}+M_{\nu}^{\rm II}\,.$ (1)

Here $M_{D}$ is the Dirac neutrino mass induced by the bidoublet vacuum
expectation value (VEV), while $M_{R}$ and $M_{L}$ are the Majorana masses of
the right and left-handed neutrinos respectively, induced by the triplet VEVs.
For phenomenological purposes, it is usually assumed that only one of the
contributions is dominant for the low-scale LRSM, with observable
ramifications for different experiments. For instance, in the type-I seesaw
dominance, we assume $M_{L}\to 0$ and the light neutrino mass crucially
depends on the Dirac mass matrix $M_{D}$, or the light-heavy neutrino mixing.
In fact, for exact left-right symmetry, $M_{D}$ can be expressed in terms of
$M_{\nu}$ and $M_{R}$ Nemevsek:2012iq , and it turns out that all the light-
heavy neutrino mixing effects are suppressed for TeV-scale parity restoration.
On the other hand, the type-II seesaw dominance can be realized with either
$M_{D}\to 0$ or with very high scale of parity restoration. The advantage here
is that the light and heavy neutrino mass matrices are directly proportional
to each other, thus leading to a more predictive scenario. In Ref.
Goswami:2020loc , the authors have considered these two types of dominance
separately to constrain lightest neutrino mass scale and heavy neutrino masses
from neutrinoless double beta decay and LFV in a TeV scale LRSM. But this
comes with the cost of losing all the light-heavy neutrino mixing effects on
the lepton number and/or flavor violating observables.

In this paper, we explore an extended scenario for low-scale LRSM
Pritimita:2016fgr , whose particle content is such that the light neutrino
mass generation is governed by a natural type-II seesaw mechanism, while still
allowing for observable light-heavy neutrino mixing effects. This has
important and non-trivial phenomenological consequences. The beautiful aspect
of this model is that the dominant new physics contributions to cLFV,
$0\nu\beta\beta$ and electron EDM processes can be expressed in terms of the
observed light neutrino oscillation parameters and lightest neutrino mass. As
a result of this, we can derive constraints on the lightest neutrino mass from
the non-observation of these rare processes. We also make a complementary
study between the low-energy cLFV and $0\nu\beta\beta$ processes within this
scenario. However, in order to highlight the contributions of right-handed
heavy neutrino and sterile neutrino to LFV decays and $0\nu\beta\beta$ decay,
we have focussed only on the diagrams mediated by them and ignored other
possible channels.

The paper is organized as follows; In Sec-2, we recapitulate the basic model
framework of left-right symmetric theory followed by a discussion on type-II
seesaw dominance with large light-heavy neutrino mixing and the condition to
achieve it. In Sec-3, we discuss new physics contributions to relevant LFV
processes due to this mixing and in Sec-4, we demonstrate how these
contributions to LFV constrain light neutrino masses. In Sec-5, we study how
the light neutrino masses are constrained by new physics contributions to
$0\nu\beta\beta$ decay and in Sec-6 we do a complementarity study between LNV
and LFV decays. We also show the variation of dipole moment of electron with
lightest neutrino mass and PMNS phase in that section. In Sec-7 we comment on
the recent muon (g-2) anomaly results and summarize our results in Sec-8.

##  2 The model framework of natural type-II seesaw dominance

###  2.1 Natural type-II seesaw dominance and large light-heavy neutrino
mixing

The left-right symmetric theory is based on the gauge group
$\mathcal{G}_{LR}\equiv SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times
U(1)_{B-L}$ Mohapatra:1974gc ; Pati:1974yy ; Senjanovic:1975rk . The fermion
sector comprises of quarks and leptons as follows,

$\displaystyle q_{L}=\begin{pmatrix}u_{L}\\\
d_{L}\end{pmatrix}\equiv[3,2,1,{\frac{1}{3}}]\quad,\quad
q_{R}=\begin{pmatrix}u_{R}\\\
d_{R}\end{pmatrix}\equiv[3,1,2,{\frac{1}{3}}]\,,$
$\displaystyle\ell_{L}=\begin{pmatrix}\nu_{L}\\\
e_{L}\end{pmatrix}\equiv[1,2,1,-1]\quad,\quad\ell_{R}=\begin{pmatrix}\nu_{R}\\\
e_{R}\end{pmatrix}\equiv[1,1,2,-1]\,$ (2)

The electric charge of individual components are related to the third
component of $SU(2)_{L,R}$ gauge groups and the difference between baryon and
lepton number.

$Q=T_{3L}+T_{3R}+\frac{B-L}{2}$

The fermion mass generation including light neutrino masses crucially depends
on how the left-right symmetry breaking happens. The left-right symmetry can
be spontaneously broken down to SM gauge group $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$ either by assigning VEV to a scalar doublet $H_{R}$
or scalar triplet $\Delta_{R}$, or with the help of both. In case the
spontaneous symmetry breaking is done with the help of doublet $H_{R}$ which
is the minimal scenario, Majorana masses can’t be generated for neutrinos and
thus it becomes less interesting from a phenomenology point of view. However
left-right symmetry breaking through $\Delta_{R}$ generates Majorana masses
for both light and heavy neutrinos thereby allowing lepton number violation
which can be probed by same-sign dilepton signatures at colliders as well as
$0\nu\beta\beta$ decay at low-energy experiments. In this case the neutrino
mass generation is governed by type-I plus type-II seesaw mechanism but it
gives negligible contribution to left-right mixing. The final step of symmetry
breaking occurs with the help of bidoublet $\Phi$ which breaks the electroweak
gauge group $SU(2)_{L}\times U(1)_{Y}$ to $U(1)_{\rm em}$ theory.

We briefly discuss below how the addition of a sterile neutrino
$S_{L}(1,1,1,0)$ per generation to this type-I plus type-II seesaw scheme
results in large left-right mixing and makes the scenario more interesting
phenomenologically. The neutral lepton sector of generic left-right symmetric
theories contains three active left-handed neutrinos ($\nu_{L}$) and three
right-handed neutrinos ($N_{R}$) which are their $SU(2)_{R}$ counterparts. We
extend the theory only by adding three sterile neutrinos ($S_{L}$), for the
purpose of generating light neutrino mass through natural type-II seesaw term,
as the type-I seesaw term gets exactly cancelled out in the presence of
$S_{L}$. More importantly this scenario may lead to new non-standard
contributions to neutrinoless double beta decay, lepton flavour violation and
the $T$ and $CP$-violating electric dipole moment (EDM) of charged leptons
because of the light-heavy neutrino mixing. Even though type-II seesaw
dominance is assumed in many left-right models Tello:2010am ;
Chakrabortty:2012mh ; Nemevsek:2012iq ; Barry:2013xxa ; Dev:2013vxa ;
Ge:2015yqa  in the context of low energy phenomenology, the light-heavy
neutrino mixing is very much suppressed in such cases. Our model differs from
these frameworks by naturally getting type-II seesaw term instead of assuming
it, along with large light-heavy neutrino mixing. The detailed derivation of
neutrino masses and mixing within this natural type-II seesaw and the
implications of large light-heavy neutrino mixing to $0\nu\beta\beta$ decay
can be found in ref Pritimita:2016fgr .

The structure of mass matrix in the basis $(\nu_{L},N^{c}_{R},S_{L})$ leading
to natural type-II seesaw dominance is given as,

$\displaystyle\mathcal{M}=\begin{split}\left[\begin{array}[]{c | c}\begin{array}[]{c c}{\color[rgb]{0,0,1}M_{L}}&M_{D}\\\ M^{T}_{D}&{\color[rgb]{0,0,1}M_{R}}\end{array}&\begin{array}[]{c}{\color[rgb]{1,0,0}\bf 0}\\\ M\end{array}\\\ \hline\cr\begin{array}[]{c c}{\color[rgb]{1,0,0}\bf 0}&\quad M^{T}\end{array}&\begin{array}[]{c}{\color[rgb]{0,0,1}\bf 0}\end{array}\\\ \end{array}\right]\end{split}\mathop{\xrightarrow{\hskip 71.13188pt}}^{{\color[rgb]{0,0,1}M_{R}\gg M>M_{D}\gg M_{L}}}\left\\{\begin{array}[]{c}m_{\nu}=M_{L}\quad\mbox{(type-II seesaw)}\\\\[5.69046pt] \mbox{light-heavy mixing}\simeq M_{D}/M\\\\[5.69046pt] m_{S}\simeq MM^{-1}_{R}M^{T}\,,\quad m_{N}=M_{R}\end{array}\right.$ (6)

where $M_{D}$ is the Dirac neutrino mass matrix connecting $\nu$ and $N_{R}$,
$M$ is the mixing matrix in the $\nu-S_{L}$ sector, $M_{L}$ $(M_{R})$ is the
Majorana mass matrix for left-handed (right-handed) neutrinos.

| Fields | $SU(3)_{c}$ | $SU(2)_{L}$ | $SU(2)_{R}$ | $B-L$
---|---|---|---|---|---
Fermions | $q_{L}$ | 3 | 2 | 1 | 1/3
$q_{R}$ | 3 | 1 | 2 | 1/3
$\ell_{L}$ | 1 | 2 | 1 | $-1$
$\ell_{R}$ | 1 | 1 | 2 | $-1$
$S$ | 1 | 1 | 1 | 0
Scalars | $\Phi$ | 1 | 2 | 2 | 0
$H_{L}$ | 1 | 2 | 1 | $-1$
$H_{R}$ | 1 | 1 | 2 | $-1$
$\Delta_{L}$ | 1 | 3 | 1 | 2
$\Delta_{R}$ | 1 | 1 | 3 | 2
Table 1: Particle content of left-right theories with type-II seesaw
dominance.

The symmetry breaking steps in our model and the subsequent mass generation
for fermions and bosons can be summed up as follows. We use both scalar
doublets and scalar triplets for left-right symmetry breaking in order to
obtain natural type-II seesaw mechanism even with non-negligible $M_{D}$. The
first step of symmetry breaking happens by assigning VEVs to both Higgs
doublet $H_{R}$ and Higgs triplet $\Delta_{R}$. As an immediate result of this
symmetry breaking the new gauge bosons $W^{\pm}_{R}$, $Z_{R}$ and right-handed
Majorana neutrinos get mass. The next step of symmetry breaking is done with
the help of SM Higgs doublet contained in the bidoublet $\Phi$, at the scale
$M_{Z}$. The SM fermions and gauge bosons $W_{L}$ and $Z$ get their mass at
this stage of symmetry breaking. The complete particle spectrum of the model
is given in Table 1. For fermions we have suppressed the family index, which
runs from 1 to 3.

With these fermions and scalars, the interaction lagrangian for leptons can be
written as,

$\displaystyle-\mathcal{L}_{Yuk}$ $\displaystyle=$
$\displaystyle\,\overline{\ell_{L}}\left[Y_{3}\Phi+Y_{4}\widetilde{\Phi}\right]\ell_{R}+f\,\left[\overline{(\ell_{L})^{c}}\ell_{L}\Delta_{L}+\overline{(\ell_{R})^{c}}\ell_{R}\Delta_{R}\right]\,$
(7)
$\displaystyle+F\,\overline{(\ell_{R})}H_{R}S^{c}_{L}+F^{\prime}\,\overline{(\ell_{L})}H_{L}S_{L}+\mu_{S}\overline{S^{c}_{L}}S_{L}\
+\mbox{h.c.}\,.$

The scalars take VEVs as follows,

$\displaystyle\langle\Phi\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}v_{1}&0\\\
0&v_{2}\end{pmatrix}\,,\quad\langle\Delta_{R}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\
v_{R}&0\end{pmatrix}\,,\quad\langle\Delta_{L}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\
v_{L}&0\end{pmatrix}\,,$ $\displaystyle\langle
H_{R}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
u_{R}\end{pmatrix}\,,\quad\langle
H_{L}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 0\end{pmatrix}\,.\quad$ (8)

After spontaneous symmetry breaking with the above assignment of VEVs to Higgs
scalars, the resulting mass matrix for neutral leptons in the basis
$\left(\nu_{L},N^{c}_{R},S_{L}\right)$ can be written as,

$\mathcal{M}_{\nu}=\left(\begin{array}[]{ccc}M_{L}&M_{D}&0\\\
M^{T}_{D}&M_{R}&M^{T}\\\ 0&M&0\end{array}\right)\,,$ (9)

where $M_{D}$ is the Dirac neutrino mass matrix connecting $\nu_{L}$ and
$N_{R}$, $M$ is the mixing matrix in the $N_{R}-S_{L}$ sector, $M_{L}$
$(M_{R})$ is the Majorana mass matrix for left-handed (right-handed) active
neutrinos generated dynamically by non-zero VEV of scalar triplet $\Delta_{L}$
$(\Delta_{R})$.

###  2.2 Relation between neutrino masses and mixings

One of the elegant features of this framework is that it connects heavy
neutrinos with light neutrinos by expressing heavy neutrino masses in terms of
oscillation parameters. The light and heavy neutrino masses,
$M_{L}=f_{L}\langle\Delta_{L}\rangle=fv_{L}$
$\left(M_{R}=f_{R}\langle\Delta_{R}\rangle=fv_{R}\right)$ are related as,

$m_{\nu}=M_{L}\propto M_{R}\,.$ (10)

Also in this set up the right-handed neutrino mixing is fully determined by
its left-handed counterpart and thus the right-handed and left-handed PMNS
matrices are of the same form,

$V^{PMNS}_{R}=V^{PMNS}_{L}\,.$ (11)

Our prime goal is to derive a bound on the lightest neutrino mass from new
physics contributions to lepton flavour violating prcoesses and the $T$ and
$CP$-violating electric dipole moment (EDM) of charged leptons. Before
estimating different new physics contributions to lepton flavor violation and
lepton number violation like neutrinoless double beta decay, we fix here the
involved input model parameters.

  * •

Masses and mixing of light neutrinos:- We consider absolute value of lightest
neutrino mass as a free parameter and express other light neutrino masses in
terms of lightest neutrino mass. For normal hierarchy of light neutrinos
($m_{1}\sim m_{2}<<m_{3}$), different light neutrino masses are related as,

$\displaystyle m_{1}=\mbox{lightest neutrino mass}\;\qquad$ $\displaystyle
m_{2}=\sqrt{m_{1}^{2}+\Delta m_{\rm sol}^{2}}\;\qquad$ $\displaystyle
m_{3}=\sqrt{m_{1}^{2}+\Delta m_{\rm atm}^{2}+\Delta m_{\rm sol}^{2}}\;.$ (12)

Similarly for inverted hierarchy ($m_{3}<<m_{1}\sim m_{2}$), different light
neutrino masses are related as,

$\displaystyle m_{3}=\mbox{lightest neutrino mass}\;\qquad$ $\displaystyle
m_{1}=\sqrt{m_{3}^{2}+\Delta m_{\rm atm}^{2}}\;\qquad$ $\displaystyle
m_{2}=\sqrt{m_{1}^{2}+\Delta m_{\rm sol}^{2}+\Delta M_{\rm atm}^{2}}\;.$ (13)

The leptonic PMNS mixing matrix is parametrized in terms of neutrino mixing
angles and phases as,

$\displaystyle U_{\rm{PMNS}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}c_{13}c_{12}&c_{13}s_{12}&s_{13}e^{-i\delta}\\\
-c_{23}s_{12}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\end{pmatrix}\cdot\mbox{P},$
(14)

where the mixing angles are denoted by $s_{ij}=\sin\theta_{ij}$,
$c_{ij}=\cos\theta_{ij}$ and the diagonal phase matrix carrying Majorana
phases $\alpha$ and $\beta$ is denoted by
$\mbox{P}=\mbox{diag}\left(1,e^{i\alpha},e^{i\beta}\right)$. We vary the
Majorana phases from $0\to\pi$. The experimental values of different
oscillation parameters for both NH and IH patterns of light neutrinos are
presented in Table.2. The light neutrino masses are in general diagonalised in
terms of unitary mixing matrix $U\equiv U_{\rm PMNS}$ in a basis where charged
lepton are already diagonal.

$m^{\rm diag}_{\nu}=U^{\dagger}_{\rm PMNS}m_{\nu}U^{*}_{\rm
PMNS}=\mbox{diag}\left(m_{1},m_{2},m_{3}\right)\,,$

and the physical masses are related to the mass matrix in flavour basis as,

$m_{\nu}=U_{\rm PMNS}m^{\rm diag}_{\nu}U^{T}_{\rm PMNS}\,.$

Oscillation Parameters | Within 3$\sigma$ range | within 3$\sigma$ range
---|---|---
| (Schwetz et al.GonzalezGarcia:2012sz ) | Gonzalez-Garcia et al (Gonzalez-Garcia:2014bfa )
$\Delta m^{2}_{\rm{21}}[10^{-5}\mbox{eV}^{2}]$ | 7.00-8.09 | 7.02 - 8.09
$|\Delta m^{2}_{\rm{31}}(\mbox{NH})|[10^{-3}\mbox{eV}^{2}]$ | 2.27-2.69 | 2.317 - 2.607
$|\Delta m^{2}_{\rm{31}}(\mbox{IH})|[10^{-3}\mbox{eV}^{2}]$ | 2.24-2.65 | 2.307 - 2.590
$\sin^{2}\theta_{s}$ | 0.27-0.34 | 0.270 - 0.344
$\sin^{2}\theta_{a}$ | 0.34-0.67 | 0.382 - 0.643
$\sin^{2}\theta_{r}$ | 0.016-0.030 | 0.0186 - 0.0250
Table 2: Neutrino oscillation parameters in 3$\sigma$ range.

  * •

Masses of heavy right-handed neutrinos:- Under the type-II seesaw dominance
scheme the light and heavy neutrino masses can be written as
$m_{\nu}=f\langle\Delta_{L}\rangle$ and
$M_{N}=f\langle\Delta_{R}\rangle=(v_{R}/v_{L})m_{\nu}$ with $f_{L}=f_{R}=f$.
Since $v_{L}$ and $v_{R}$ are constants, the light left-handed and heavy
right-handed neutrino masses are diagonalized by the same unitary mixing
matrix, i.e $U_{\rm PMNS}$. Thus the physical masses for right-handed
neutrinos $M_{i}$ are related to light neutrino mass eigenvalues $m_{i}$ as
$M_{i}\propto m_{i}$. This relation implies that if the light neutrinos are
normal hierarchical then the heavy right-handed neutrinos would also be
hierarchical in the same manner, i.e. if $m_{1}<m_{2}<<m_{3}$ then
$M_{N_{1}}<M_{N_{2}}<<M_{N_{3}}$.

Thus, if we fix the largest mass eigenvalue of heavy right-handed neutrino as
$M_{N}=M_{N_{3}}$ then the other two mass eigenvalues of right-handed
neutrinos can be expressed in terms of normal hierarchy (NH) pattern of light
neutrino masses as,

$\displaystyle M_{N_{1}}$ $\displaystyle=\frac{m_{1}}{m_{3}}M_{N},\text{ NH},$
(15a) $\displaystyle M_{N_{2}}$ $\displaystyle=\frac{m_{2}}{m_{3}}M_{N},\text{
NH}.$ (15b)

and when the largest mass eigenvalue of right-handed neutrino is fixed as
$M_{N}=M_{N_{2}}$ then the other physical masses for heavy right-handed
neutrinos would be related to inverted hierarchy (IH) pattern of light
neutrino masses as

$\displaystyle M_{N_{1}}$ $\displaystyle=\frac{m_{1}}{m_{2}}M_{N},\text{ IH},$
(16a) $\displaystyle M_{N_{3}}$ $\displaystyle=\frac{m_{3}}{m_{2}}M_{N},\text{
IH}.$ (16b)

  * •

Masses of sterile neutrinos:- The approximate seesaw block diagonalization
scheme for type-II seesaw dominance gives mass formulas for sterile neutrinos
as $M_{S}=-MM^{-1}_{R}M^{T}$. Assuming the matrix $M$ proportional to identity
matrix $M=m_{S}\mbox{diag}\\{1,1,1\\}$, the physical masses are inversely
proportional to heavy right-handed neutrinos and therefore inversely
proportional to light neutrino masses, i.e, $M_{S_{i}}\propto
1/M_{N_{i}}\propto 1/m_{i}$. As a result of this relation, when the light
neutrinos are normal hierarchical, then the sterile neutrinos would be
hierarchical in the inverse way, i.e. when $m_{1}<m_{2}<<m_{3}$ then
$M_{S_{3}}<M_{S_{2}}<<M_{S_{1}}$. Either we can fix the value of $m_{S}$ or
take the largest sterile neutrino mass as constant value and express other
sterile neutrino masses in terms of light neutrino mass eigenvalues. We fix
the largest sterile neutrino mass eigen value as $M_{S}=M_{S_{1}}$ and the
other sterile neutrino mass eigenvalues can be expressed in terms of normal
hierarchy (NH) pattern of light neutrino masses as,

$\displaystyle M_{S_{2}}$ $\displaystyle=\frac{m_{1}}{m_{2}}M_{S},\text{ NH},$
(17a) $\displaystyle M_{S_{3}}$ $\displaystyle=\frac{m_{1}}{m_{3}}M_{S},\text{
NH},$ (17b)

Similarly by fixing the largest sterile neutrino mass as $M_{S}=M_{S_{3}}$ the
physical masses of other sterile neutrinos can be expressed in terms of
inverted hierarchy (IH) pattern of light neutrino masses as,

$\displaystyle M_{S_{1}}$ $\displaystyle=\frac{m_{3}}{m_{1}}M_{S},\text{ IH},$
(18a) $\displaystyle M_{S_{2}}$ $\displaystyle=\frac{m_{3}}{m_{2}}M_{S},\text{
IH}.$ (18b)

  * •

Neutrino Mixings:- The flavor states $\nu_{L}$, $N_{R}$ and $S_{L}$ are
related to their mass eigenstates in the following way

$\displaystyle\begin{pmatrix}\nu_{L}\\\ S_{L}\\\
N^{c}_{R}\end{pmatrix}_{\alpha}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}{\mbox{V}}^{\nu\nu}&{\mbox{V}}^{\nu{S}}&{\mbox{V}}^{\nu{N}}\\\
{\mbox{V}}^{S\nu}&{\mbox{V}}^{SS}&{\mbox{V}}^{SN}\\\
{\mbox{V}}^{N\nu}&{\mbox{V}}^{NS}&{\mbox{V}}^{NN}\end{pmatrix}_{\alpha
i}\begin{pmatrix}\nu_{i}\\\ S_{i}\\\ N_{i}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}U_{\rm PMNS}&\frac{1}{m_{S}}M_{D}U^{*}_{\rm
PMNS}&\frac{v_{L}}{v_{R}}M_{D}U^{-1}_{\rm PMNS}{m^{\rm diag.}_{\nu}}^{-1}\\\
\frac{1}{m_{S}}M^{\dagger}_{D}U_{\rm PMNS}&U^{*}_{\rm
PMNS}&\frac{v_{L}}{v_{R}}m_{S}U^{-1}_{\rm PMNS}{m^{\rm diag.}_{\nu}}^{-1}\\\
\mathcal{O}&\frac{v_{L}}{v_{R}}m_{S}U^{-1}_{\rm PMNS}{m^{\rm
diag.}_{\nu}}^{-1}&U_{\rm PMNS}\end{pmatrix}_{\alpha
i}\begin{pmatrix}\nu_{i}\\\ S_{i}\\\ N_{i}\end{pmatrix}.$

We express below only those input model parameters in terms of neutrino
oscillation parameters that are required for estimating branching ratios for
LFV decays. The individual mixing matrices are expressed in terms of Dirac
neutrino mass matrix $M_{D}$, mixing term $M$ and right-handed Majorana mass
term $M_{R}$ as,

$\displaystyle{\mbox{V}}^{\nu\nu}=U_{\rm
PMNS}\,,\quad{\mbox{V}}^{\nu{S}}=\frac{1}{m_{S}}M_{D}U^{*}_{\rm
PMNS}\,,\quad{\mbox{V}}^{\nu{N}}=\frac{v_{L}}{v_{R}}M_{D}U^{-1}_{\rm
PMNS}{\widehat{m}_{\nu}}^{-1}\,,$
$\displaystyle{\mbox{V}}^{S\nu}=\frac{1}{m_{S}}M^{\dagger}_{D}U_{\rm
PMNS}\,,\quad{\mbox{V}}^{SS}=U^{*}_{\rm
PMNS}\,,\quad{\mbox{V}}^{SN}=\frac{v_{L}}{v_{R}}m_{S}U^{-1}_{\rm
PMNS}{\widehat{m}_{\nu}}^{-1}\,,$
$\displaystyle{\mbox{V}}^{N\nu}=\mathcal{O}\,,\quad{\mbox{V}}^{NS}=\frac{v_{L}}{v_{R}}m_{S}U^{-1}_{\rm
PMNS}{\widehat{m}_{\nu}}^{-1}\,,\quad{\mbox{V}}^{NN}=U_{\rm PMNS}\,.$ (20)

For simplification we have considered $M$ to be diagonal and degenerate. In
general the Dirac neutrino mass matrix $M_{D}$ is either of up-type quark mass
matrix or charged lepton mass matrix. However we have considered an $SO(10)$
GUT motivated structure for $M_{D}$ including RGE effects as,

$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0.0111&0.0384-0.0103\,i&0.038-0.4433\,i\\\
0.0384+0.0103\,i&0.29281&0.8623+0.0002\,i\\\
0.038+0.4433\,i&0.8623-0.0002\,i&77.7573\end{array}\right)\text{GeV}\,.$ (24)

  * •

Other model parameters:- The spontaneous symmetry breaking of left-right
symmetric model to SM is done by assigning a non-zero VEV to scalar triplet
$\Delta_{R}$ denoted by $v_{R}$. The value of $v_{R}$ decides the masses of
right-handed charged gauge bosons $W_{R}^{\pm}$, doubly charged scalar triplet
$\Delta^{++}$, right-handed neutrinos and others. Considering the bound on
$v_{R}>6$ TeV Bertolini:2014sua ; Maiezza:2010ic ; Zhang:2007da ; Beall:1981ze
; Bambhaniya:2014cia , we present below the other input model parameters.

$\displaystyle v_{R}\geq\mbox{15\,TeV}\,,\quad
M_{W_{R}}\geq\mbox{10\,TeV}\,,\quad
M_{\Delta^{++}}\simeq\mbox{10\,TeV}\,,\quad M_{N}\simeq\mbox{1\,TeV}\,.$ (25)

The parameters are chosen in such a way that they not only provide the plot
for natural type-II seesaw domiance but also ensure that the contributions
from charged scalar triplets and right-handed charged gauge boson $W_{R}$ are
negligible.

##  3 Lepton flavour violation in left-right symmetric model

A flavour violating process involving charged leptons has not been observed
yet. Many new physics models that discuss lepton flavour violation (LFV) are
constrained by muon decay experiments since the current limits on $\tau$
observables are less stringent. For the decay $l_{\alpha}\rightarrow
3l_{\beta}$, the SINDRUM experiment has set a limit of
$\text{BR}(l_{\alpha}\rightarrow 3l_{\beta})<10^{-12}$ Bellgardt:1987du  since
a long time, which is expected to improve significantly by the future Mu3e
experiment. Similarly for the decay $l_{\alpha}\rightarrow l_{\beta}\gamma$,
an impressive bound on its branching ratio $\text{BR}(l_{\alpha}\rightarrow
l_{\beta}\gamma)<4.2\times 10^{-13}$ is provided by the MEG collaboration
Adam:2013mnn  which will be improved by the upgraded MEG-II. Experiments like
Mu2e, COMET, PRIME focus on $\mu\rightarrow e$ conversion that will have a
sensitivity ranging from $10^{-14}$ to $10^{-18}$. The present bound and
future sensitivity of these lepton flavour violating processes are given in
Table.3.

LFV Decays | Present Bound | Near Future Sensitivity
---|---|---
(with Branching Ratios) |  | at ongoing search experiments
$\mbox{Br}\left(\mu\to{e\gamma}\right)$ | $5.7\times 10^{-13}$ | $6\times 10^{-14}$
$\mbox{Br}\left(\tau\to{e\gamma}\right)$ | $3.3\times 10^{-8}$ | $3\times 10^{-9}$
$\mbox{Br}\left(\tau\to{\mu\gamma}\right)$ | $4.4\times 10^{-8}$ | $3\times 10^{-9}$
$\mbox{Br}\left(\mu\to\mbox{3e}\right)$ | $1.0\times 10^{-12}$ | $10^{-16}$
$\mbox{Br}\left(\tau\to\mbox{eee}\right)$ | $3.0\times 10^{-8}$ | $10^{-9}$
$\mbox{Br}\left(\tau\to{\mu\mu\mu}\right)$ | $2.0\times 10^{-8}$ | $3\times 10^{-9}$
Table 3: Branching ratios for different LFV processes and their present
experimental bound and future sensitivity values taken from refs Adam:2013mnn
; Baldini:2013ke ; Aubert:2009ag ; Aushev:2010bq ; Bellgardt:1987du ;
Blondel:2013ia ; Hayasaka:2010np .

Within canonical seesaw models LFV can be induced via light neutrino mixing.
However in such models the branching ratios are found to be very much
suppressed which are well below the present and planned experimental
sensitivity. In left-right symmetric models dominant new contributions to LFV
arise when symmetry breaking occurs at few TeV scale. A detailed discussion on
LFV within manifest left-right symmetric model can be found in Barry:2013xxa ;
Cirigliano:2004mv . The relevant interaction terms which can mediate the
processes in a LRSM are given below.

  * •

Charged-current interactions in the lepton sector:
The charged-current interactions in the lepton sector within the present model
where neutral leptons comprising of $\nu_{L},N_{R},S_{L}$ are given by

$\displaystyle\mathcal{L}^{\rm\ell}_{\rm CC}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=e,\mu,\tau}\bigg{[}\frac{g_{L}}{\sqrt{2}}\,\overline{\ell}_{\alpha\,L}\,\gamma_{\mu}{\nu}_{\alpha\,L}\,W^{\mu}_{L}+\frac{g_{R}}{\sqrt{2}}\,\overline{\ell}_{\alpha\,R}\,\gamma_{\mu}{N}_{\alpha\,R}\,W^{\mu}_{R}\bigg{]}+\text{h.c.}$

Using masses and mixing relation for neutral leptons,

$\displaystyle\nu_{eL}=\mbox{V}^{\nu\nu}_{e\,i}\,\nu_{i}+\mbox{V}^{\nu\,S}_{e\,i}\,S_{i}+\mbox{V}^{\nu\,N}_{e\,i}\,N_{i},$
$\displaystyle
N_{eR}=0\times\,\nu_{i}+\mbox{V}^{N\,S}_{e\,i}\,S_{i}+\mbox{V}^{NN}_{e\,i}\,N_{i}$
(26)

the above CC interaction Lagrangian modifies as

$\displaystyle\mathcal{L}^{\rm\ell,m}_{\rm CC}$ $\displaystyle=$
$\displaystyle\frac{g_{L}}{\sqrt{2}}\,\bigg{[}\overline{e}_{\,L}\,\gamma_{\mu}\\{\mbox{V}^{\nu\nu}_{e\,i}\,\nu_{i}+\mbox{V}^{\nu\,S}_{e\,i}\,S_{i}+\mbox{V}^{\nu\,N}_{e\,i}\,N_{i}\\}\,W^{\mu}_{L}\bigg{]}+\mbox{h.c.}$
(27)
$\displaystyle+\frac{g_{R}}{\sqrt{2}}\,\bigg{[}\overline{e}_{\,R}\,\gamma_{\mu}\\{\mbox{V}^{N\,S}_{e\,i}\,S_{i}+\mbox{V}^{NN}_{e\,i}\,N_{i}\\}\,W^{\mu}_{R}\bigg{]}+\mbox{h.c.}$

  * •

Charged-current interactions in the quark sector:
The relevant charge-current interaction for quarks are

$\displaystyle\mathcal{L}^{\rm q}_{\rm CC}$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{g_{L}}{\sqrt{2}}\,\overline{u}_{\,L}\,\gamma_{\mu}{d}_{\,L}\,W^{\mu}_{L}+\frac{g_{R}}{\sqrt{2}}\,\overline{u}_{R}\,\gamma_{\mu}d_{R}\,W^{\mu}_{R}\bigg{]}+\text{h.c.}$
(28)

  * •

Scalar triplet interactions:
The other relevant terms involving scalar triplets are given by

$\displaystyle{\cal L}_{\Delta^{\pm}_{L}}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{L}^{+}}{\sqrt{2}}\left[\overline{{\nu_{L}}^{c}}f\ell_{L}+\overline{{\ell_{L}}^{c}}f\nu_{L}\right]+{\rm
h.c.}\,,$ (29) $\displaystyle{\cal L}_{\Delta^{\pm\pm}_{L,R}}$
$\displaystyle=$
$\displaystyle\Delta_{L,R}^{++}\overline{{\ell}^{c}}fP_{L,R}\ell+\Delta_{L,R}^{--}\overline{\ell}f^{\dagger}P_{R,L}{\ell}^{c}\,.$

In our model, LFV decays can be mediated by heavy right-handed neutrinos
$N_{R}$, extra sterile neutrinos $S_{L}$, charged scalar triplets
$\Delta^{\pm\pm}_{L,R},\Delta^{\pm}_{L,R}$ and gauge bosons $W_{L,R}$ which we
classify as follows,

  * •

due to purely left-handed currents (LL) arising from i) exchange of light
neutrinos, ii) exchange of heavy right-handed neutrinos and extra sterile
neutrinos with the involvement of light-heavy neutrino mixing, iii) exchange
of scalar triplets;

  * •

due to purely right-handed currents (RR) via exchange of right-handed charged
gauge boson, right-handed scalar triplet and heavy neutrinos;

  * •

due to involvement of left-handed as well as right-handed currents (LR).

However we focus here only on those contributions which involve large active-
sterile neutrino mixing, i.e. due to the neutrinos $N_{R}$ and $S_{L}$ in
order to constrain light neutrino masses from LFV decays. We ignore the other
possible contributions by imposing the following limiting conditions;

  * •

The mass of right-handed gauge boson $W_{R}$ is assumed to be much heavier
than the SM gauge boson $W_{L}$, i.e, $M_{W_{R}}\gg M_{W_{L}}$. The mass
eigenvalues for charged gauge bosons are given by

$\displaystyle M_{W_{1}}^{2}\approx\frac{1}{4}g_{L}^{2}v^{2}\,,\quad
M_{W_{2}}^{2}\approx\frac{1}{2}g_{R}^{2}v^{2}_{R}\,.$ (30)

With $v_{R}\simeq 15$ TeV and $g_{L}=g_{R}\simeq 0.623$, the $W_{L}-W_{R}$
mixing is found to be

$|\tan\,2\xi|\simeq\frac{v_{1}v_{2}}{v^{2}_{R}}\propto\frac{M^{2}_{W_{L}}}{M^{2}_{W_{R}}}\leq
10^{-4}\,.$ (31)

The mass of $W_{R}$ is found to be around $7~{}$TeV leading to negligible
contributions to LFV processes via right-handed currents and mixed currents.

  * •

The masses of scalar triplets and other scalars contained in the bidoublet are
assumed to be larger than heavy neutrinos, i.e, $M_{\Delta_{L,R}},M_{H_{1}}\gg
M_{N,S}$.

Figure 1: Feynman diagram for lepton flavor violating process $\mu\to e\gamma$
due to the exchange of mass eigenstates of heavy neutrinos $N_{i}$ and
$S_{i}$.

##  4 Constraints on light neutrino masses from LFV

In this section we derive constraints on absolute scale of lightest neutrino
mass inluding both normal hierarchy (NH) and inverted hierarchy (IH) patterns
from charged lepton flavour violating processes like $\mu\to e\gamma$, $\mu\to
3e$ and $\mu\to e$ conversion inside a nucleus. We do so by considering the
new contributions to these decays due to large light-heavy neutrino mixing in
our model.

The general expression for such a decay can be written as,

$\displaystyle\Gamma^{(0)}_{\mu}\equiv\Gamma_{\nu}(\mu^{+}\to
e^{+}\nu_{e}\overline{\nu_{\mu}})$ $\displaystyle\Gamma^{\rm Z}_{\rm
capt.}\equiv\Gamma\left(\mu^{-}+A(Z,N)\rightarrow\nu_{\mu}+A(Z-1,N+1)\right)$

and the expressions for branching ratios as,

$\displaystyle\text{Br}_{\mu\rightarrow
e\gamma}\equiv\frac{\Gamma(\mu\rightarrow e\gamma)}{\Gamma^{(0)}_{\mu}}$ (32)
$\displaystyle\text{R}_{\mu\rightarrow
e}^{A}\equiv\frac{\Gamma\left(\mu+A(N,Z)\rightarrow
e+A(N,Z)\right)}{\Gamma^{\rm Z}_{\rm capt.}}$ (33)
$\displaystyle\text{Br}_{\mu\rightarrow 3e}\equiv\frac{\Gamma(\mu\rightarrow
3e)}{\Gamma^{(0)}_{\mu}}\,.$ (34)

###  4.1 Bound on light neutrino mass from $\mu\to e\gamma$

The effective Lagrangian relevant for the lepton flavor violating process
$\mu\to e\gamma$ in our present work can be expressed as Barry:2013xxa ;
Cirigliano:2004mv ,

$\begin{split}{\cal L}_{\mu\to
e}=&-\frac{eg^{2}}{4(4\pi)^{2}m_{W_{L}}^{2}}m_{\mu}\overline{e}\sigma_{\mu\nu}(\mathcal{G}^{\gamma}_{L}P_{L}+\mathcal{G}^{\gamma}_{R}P_{R})\mu
F^{\mu\nu}\\\
&-\frac{\alpha_{W}^{2}}{2m_{W_{L}}^{2}}\sum_{q}\left\\{\overline{e}\gamma_{\mu}\left[\mathcal{W}^{q}_{L}P_{L}+\mathcal{W}^{q}_{R}P_{R}\right]\mu\;\overline{q}\gamma^{\mu}q\right\\}+{\rm
h.c.},\end{split}$ (35)

where $\sigma_{\mu\nu}\equiv\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$ and the
form factors are $\mathcal{G}^{\gamma}_{L,R}$ and $\mathcal{W}^{u,d}_{L,R}$.
The relevant contributions to $\mu\to e\gamma$ is given by

$\begin{split}i{\cal M}(\mu\to e\gamma)&=\frac{e\alpha_{W}}{8\pi
m_{W_{L}}^{2}}\epsilon_{\gamma}^{\mu}\overline{e}\left[\left(q^{2}\gamma_{\mu}-q_{\mu}\not{q}\right)\left(\mathcal{F}^{\gamma}_{L}P_{L}+\mathcal{F}^{\gamma}_{R}P_{R}\right)\right.\\\
&\left.-im_{\mu}\sigma_{\mu\nu}q^{\nu}\left(\mathcal{G}^{\gamma}_{L}P_{L}+\mathcal{G}^{\gamma}_{R}P_{R}\right)\right]\mu,\end{split}$
(36)

where the anapole and dipole form factors $\mathcal{F}^{\gamma}_{L,R}$ and
$\mathcal{G}^{\gamma}_{L,R}$ will be defined in subsequent paragraph.

Figure 2: Branching ratio of the lepton flavor violating process, $\mu\to
e\gamma$ as a function of lightest neutrino mass $m_{1}$ for NH and $m_{3}$
for IH. The blue (NH) and red (IH) coloured regions display the allow due to
the exchange of heavy right handed neutrino ($N_{R}$) (left-panel) and heavy
sterile neutrino ($S_{L}$) (right-panel).

The analytic expression for the branching ratio for the lepton flavor
violating process $\mu\to e\gamma$ shown in Feynman diagram in Fig.1 due to
mediation of heavy neutrinos 111We have neglected the other contributions to
LFV processes due to right-handed gauge boson $W_{R}$ and $W_{L}-W_{R}$ mixing
so that we can quantify the effect of light-heavy neutrino mixing on
$\text{Br}_{\mu\to e\gamma}$. Even though there are sizable contributions from
scalar triplets Cirigliano:2004mv ; Leontaris:1985qc ; Swartz:1989qz ;
Cirigliano:2004tc  we have not included them in our analysis by keeping very
large mass scalar triplets. ($N$ and $S$) is given by

$\text{Br}_{\mu\to
e\gamma}=\frac{\alpha_{W}^{3}s_{W}^{2}}{256\pi^{2}}\frac{m_{\mu}^{4}}{M_{W_{L}}^{4}}\frac{m_{\mu}}{\Gamma_{\mu}}|\mathcal{G}_{\gamma}^{\mu
e}|^{2}\,,$ (37)

where $\alpha_{W}=1/29.0$ as weak fine struture constant, $m_{\mu}=105$ MeV
being the muon mass, $M_{W_{L}}$ is the SM $W$-boson mass,
$s_{W}\equiv\sin\theta_{W}$ in which $\theta_{W}$ is the weak mixing angle and
$\Gamma_{\mu}=2.996\times 10^{-19}$ GeV Olive:2016xmw  is the total decay
width of the muon. The important parameter,$G_{\gamma}^{\mu e}$, for deriving
constraints on light neutrino masses, is of the following form,

$\displaystyle G_{\gamma}^{\mu e}$ $\displaystyle=$
$\displaystyle\bigg{|}\sum_{i=1}^{3}\bigg{\\{}{\mbox{V}^{\nu N}_{\mu
i}}^{*}{\mbox{V}^{\nu
N}_{ei}}\mathcal{G}_{\gamma}\left(x_{N_{i}}\right)+{\mbox{V}^{\nu S}_{\mu
i}}^{*}{\mbox{V}^{\nu
S}_{ei}}\mathcal{G}_{\gamma}\left(x_{S_{i}}\right)\bigg{\\}}\bigg{|}^{2},$
(38)

where $x_{N_{i}}=m_{N_{i}}^{2}/M_{W_{L}}^{2}$,
$x_{S_{i}}=m_{S_{i}}^{2}/M_{W_{L}}^{2}$ and the form of loop function is given
by

$\displaystyle G_{\gamma}(x)$ $\displaystyle=$
$\displaystyle-\frac{x(2x^{2}+5x-1)}{4(1-x)^{3}}-\frac{3x^{3}}{2(1-x)^{4}}\ln
x\,.$ (39)

The form of loop function $\mathcal{G}_{\gamma}(x)$ and its dependance with
change of lightest neutrino mass $m_{1}$ (for NH) and $m_{3}$ (for IH) is
presented in appendix- B. The other parameters $\mbox{V}^{\nu N}$ and
$\mbox{V}^{\nu S}$ are mixing matrices representing mixing between light-
active neutrinos with $N_{R}$ and $S_{L}$, respectively.

The variation of $\text{Br}_{\mu\to e\gamma}$ as a function of lightest
neutrino mass is displayed in Fig.2 with contributions coming from purely
$N_{R}$ is presented in left-panel and for $S_{L}$ contributions is shown
right-panel while the combine contributing is presented in Fig.3. The blue
color (NH) and red color (IH) regions are model prediction on $\mu\to e\gamma$
within $3-\sigma$ allowed range of neutrino mixing angles as well as for mass
squared differences. In x-axis, $m_{1}$ for NH ($m_{3}$ for IH) represents
absolute value of light neutrino mass. As given in Table.3, the current
experimental limit and future sensitivity by MEG ($\text{Br}_{\mu\to
e\gamma}\leq 4.2\times 10^{-13}$), MEG upgrade ($\text{Br}_{\mu\to
e\gamma}\leq 5.0\times 10^{-14}$) and PRISM/PRIME ($\text{Br}_{\mu\to
e\gamma}\leq 1.0\times 10^{-16}$) on branching ratio for $\mu\to e\gamma$ are
presented in dashed horizontal lines. The vertical shaded regions are PLANCK
bound on lightest neutrino mass with 95$\%$ C.L..

The estimated values of $\text{Br}_{\mu\to e\gamma}$ is mostly depend on the
sum of the heavy-light neutrino mixing parameters like ${\mbox{V}^{\nu S}_{\mu
i}}^{*}{\mbox{V}^{\nu S}_{ei}}$ for $S_{L}$ mediated contribution and
${\mbox{V}^{\nu N}_{\mu i}}^{*}{\mbox{V}^{\nu N}_{ei}}$ for $N_{R}$ mediated
contribution. The band strucure of NH and IH scenarios are coming since we
have taken $3\sigma$ range of oscillation parameters presented in Table.2 as
well as vary the phases; $\delta$ between [$0,2\pi$] and $\alpha,\beta$
between [$0,\pi$]. As presented in Table.3, the current experimental limit on
$\text{Br}_{\mu\to e\gamma}$ from MEG Adam:2013mnn  ($<4.2\times 10^{-13}$)
and the future experimental sensitivity from MEG Upgrade Baldini:2013ke
($<5.0\times 10^{-14}\,$) $\&$ PRISM/PRIME Kuno:2005mm  ($<1.0\times
10^{-16}\,$) are satisfied by the predicted branching ratio and one can derive
bound on lightest neutrino masses for NH as well as IH case.

  * •

Saturating the current experimental MEG bound ($\text{Br}_{\mu\to e\gamma}\leq
4.2\times 10^{-13}$), one can derive the limit on lightest neutrino mass in
the range of meV scale in IH case while most of the parameters are ruled out
for NH case.

  * •

The projected experimental sensitivity from MEG Upgrade Baldini:2013ke
($<5.0\times 10^{-14}\,$) limits lightest neutrino mass less than few meV for
NH while $4$ meV for IH case. However, most of the paremeter spaces satisfying
PRISM/PRIME Kuno:2005mm  ($<1.0\times 10^{-16}\,$) bound are lying in the
quasi-degenerate (QD) pattern of light neutrino masses which already ruled out
by PLANCK data.

The predicted branching ratio is allowed by the future sensitivity for
lightest neutrino mass $m_{1}\leq 10^{-3}$ eV in case of $N_{R}$ exchange
(left-panel Fig.2) and for interference term (Fig.3). But in case of $S_{L}$
exchange (right-panel Fig.2) small range of lightest neutrino mass
($m_{1},m_{3}$) is allowed by future limit.

Figure 3: Branching ratio of the process $\mu\to e\gamma$ vs lightest neutrino
mass, $m_{1}$ ($m_{3}$) for NH (IH) due to the combined effect of heavy right
handed and sterile neutrino.

###  4.2 Bound on light neutrino mass from $\mu\to 3e$ and $\mu\to e$
conversion

Under the assumptions that $M_{W_{R}}\gg M_{W_{L}}$, $\tan\xi\to 0$ and for
heavy scalar masses, the only relevant terms contributing to $\mu\to 3e$ as a
result of light-heavy neutrino mixing is given by

$\displaystyle\text{Br}_{\mu\to 3e}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{W}^{4}m_{\mu}^{5}}{24576\pi^{3}m_{W_{L}}^{4}\Gamma_{\mu}}\bigg{\\{}2\bigg{[}\bigg{|}\frac{1}{2}\mathcal{B}^{\mu
3e}_{LL}+\mathcal{F}^{Z_{1}}_{L}-2s_{W}^{2}(\mathcal{F}^{Z_{1}}_{L}-\mathcal{F}^{\gamma}_{L})\bigg{|}^{2}+\bigg{|}\frac{1}{2}\mathcal{B}^{\mu
3e}_{RR}\bigg{|}^{2}\bigg{]}$ (40) $\displaystyle\hskip
56.9055pt+\bigg{|}2s_{W}^{2}(\mathcal{F}^{Z_{1}}_{L}-\mathcal{F}^{\gamma}_{L})\bigg{|}^{2}+8s_{W}^{2}\mbox{Re}\left(2\mathcal{F}^{Z_{1}}_{L}+\mathcal{B}^{\mu
3e}_{LL}\right){\mathcal{G}^{\gamma}_{R}}^{*}$ $\displaystyle\hskip
56.9055pt-48s_{W}^{2}\mbox{Re}\left(\mathcal{F}^{Z_{1}}_{L}-\mathcal{F}^{\gamma}_{L}\right){\mathcal{G}^{\gamma}_{R}}^{*}+32s_{W}^{4}|\mathcal{G}^{\gamma}_{R}|^{2}\left[\ln\frac{m^{2}_{\mu}}{m^{2}_{e}}-\frac{11}{4}\right]\bigg{\\}}\,$

where $m_{e}(m_{\mu})$ is mass of electron (muon),
$s_{W}^{2}=\sin^{2}\theta_{W}$ and other loop factors are presented in
appendix-B. The estimated branching ratio for the process $\mu\to 3e$ is
satisfied by the experimental limit SINDRUM Bellgardt:1987du  ($<10^{-12}$)
for a broad range of lightest neutrino mass $m_{1},m_{3}\leq 0.03$ eV.

Figure 4: Branching ratio of the process $\mu\to 3e$ as a function of lightest
neutrino mass, $m_{1}$ ($m_{3}$) for NH (IH) pattern.

Fig.4 shows the variation of branching ratio of the process $\mu\to 3e$ with
respect to the lightest neutrino mass where NH and IH pattern of light neutrno
mass are represented by blue and red bands respectively. The bounds on the
branching ratio of this process are given by SINDRUM ($10^{-12}$) and Mu3e
($10^{-16}$) and the plot shows that both the bounds are satisfied by the
model’s predictions on this decay (due to light-heavy mixing) for both
patterns of light neutrino mass.

Similarly, the bound on light neutrino mass from $\mu\to e$ conversion rate
with the mediation of heavy neutrino N and sterile neutrino S can be found as
follows.

${\rm R}^{A(N,Z)}_{\mu\to e}=\frac{\alpha_{\rm
em}^{3}\alpha_{W}^{4}m_{\mu}^{5}}{16\pi^{2}m_{W_{L}}^{4}\Gamma_{\rm
capt}}\frac{Z_{\rm
eff}^{4}}{Z}\left|\mathcal{F}(-m_{\mu}^{2})\right|^{2}\left(|\mathcal{Q}_{L}^{W}|^{2}+|\mathcal{Q}_{R}^{W}|^{2}\right),$
(41)

where the relevant parameters are given in appendix C.

In Fig.5 we have plotted the variation of $\mu\to e$ conversion against the
lightest neutrino mass where the red curve represents the IH pattern of light
neutrino mass and blue band represents the NH pattern of light neutrino mass.
The experimental sensitivity of $\mu\to e$ conversion is represented by the
horizontal dashed line. The plot shows that the model’s predictions on decay
rate of this conversion (due to contributions from heavy neutrino $N_{R}$ and
sterile neutrino $S_{L}$) are not sensitive to the experimental bounds, for
both NH and IH pattern of light neutrino mass.

Figure 5: The rate for $\mu\to e$ conversion in Au nucleus as a function of
the lightest neutrino mass, $m_{1}$ ($m_{3}$) for NH (IH) due to light-heavy
mixing.

##  5 Constraints from neutrinoless double beta decay

We discuss here how the light-heavy neutrino mixing in left-right theories
with type-II seesaw dominance leads to sizable new contributions to
neutrinoless double beta decay. We give emphasis on left-handed current
effects due to the exchange of heavy neutrinos $N_{R}$ and $S_{L}$. To ignore
the effects of right-handed currents and the contributions of doubly charged
scalar triplets to $0\nu\beta\beta$ transition we have assumed $M_{W_{R}}\gg
M_{W_{L}}$ and large masses for scalar triplets. For a detailed discussion on
various new physics contributions to neutrinoless double beta decay in TeV
scale left-right symmetric model with large light-heavy neutrino mixing
through type-II seesaw dominance one may refer Pritimita:2016fgr .

Figure 6: Feynman diagrams for $0\,\nu\,\beta\beta$ transition due to purely
left-handed currents with the exchange of virtual Majorana neutrinos
$\nu_{i}$, $N_{j}$ and $S_{k}$.

###  5.1 Half-life and effective Majorana mass

The relevant contributions arising from purely left-handed currents with the
exchange of light active neutrinos (standard mechanism) and heavy neutrinos
$N$ and $S$ are shown in Fig.6. The inverse half-life of $0\nu\beta\beta$
transition for a given isotope is,

$\displaystyle\left[T_{1/2}^{0\nu}\right]^{-1}$ $\displaystyle=$
$\displaystyle G^{0\nu}_{01}\bigg{[}|{\cal
M}^{0\nu}_{\nu}\cdot\eta_{\nu}|^{2}+|{\cal
M}^{0\nu}_{N}\cdot\left(\eta_{N}+\eta_{S}\right)\big{|}^{2}\bigg{]}$ (42)

where the dimensionless lepton number violating parameters are given as,

$\displaystyle|\mathcal{\eta}_{\nu}|=\sum_{i=1,2,3}\frac{{\mbox{V}^{\nu\nu}_{ei}}^{2}\,m_{\nu_{i}}}{m_{e}}\,,|\mathcal{\eta}_{N}|=m_{p}\sum_{i=1,2,3}\frac{{\mbox{V}^{\nu
N}_{ei}}^{2}}{M_{N_{i}}}\,,|\mathcal{\eta}_{S}|=m_{p}\sum_{i=1,2,3}\frac{{\mbox{V}^{\nu
S}_{ei}}^{2}}{M_{S_{i}}}\,.$ (43)

In order to derive the bound on lightest neutrino mass by satuaring the
current experimental bounds on half-life of a given isotope, one has to
rewrite the inverse half-life in terms of a particle physics paremeter called
Effective Majorana mass that contains lepton number violating information in
it.

$\displaystyle\left[T_{1/2}^{0\nu}\right]^{-1}$ $\displaystyle=$
$\displaystyle G^{0\nu}_{01}\bigg{|}\frac{{\cal
M}^{0\nu}_{\nu}}{m_{e}}\bigg{|}^{2}\bigg{[}\big{|}m^{\nu}_{ee}\big{|}^{2}+\big{|}m^{N}_{ee}\big{|}^{2}+\big{|}m^{S}_{ee}\big{|}^{2}\bigg{]}$
(44) $\displaystyle=$ $\displaystyle G^{0\nu}_{01}\bigg{(}\frac{{\cal
M}^{0\nu}_{\nu}}{m_{e}}\bigg{)}^{2}\cdot|m^{\rm eff}_{\beta\beta}|^{2}\,.$

Here $|m^{\rm eff}_{\beta\beta}|^{2}$ is the sum of contributions from light
active neutrinos $\nu_{L}$, heavy right-handed neutrinos $N$ and sterile
neutrinos $S$. Thus the half-life of neutrinoless double beta decay process is
estimated by three kinds of contributions;

  * •

Phase-space factor $G^{0\nu}$ which is responsible for detailed kinematics of
the neutrinoless double beta decay process and is highly energy dependent,

  * •

Nuclear Matrix Elements (NMEs), $\mathcal{M}^{0\nu}_{\nu}$ and
$\mathcal{M}^{0\nu}_{N}$ for light and heavy neutrinos that take care of the
transition of the nucleus into daughter nuclei,

  * •

Particle physics parameter called Effective Majorana mass $m^{\rm
eff}_{\beta\beta}$ of the transition $2d\rightarrow 2u+2e^{-}$ inside the
involved nucleons.

Isotope |  $G^{0\nu}_{01}$ $[{\rm yrs}^{-1}]$ | ${\cal M}^{0\nu}_{\nu}$ | ${\cal M}^{0\nu}_{N}$
---|---|---|---
76Ge | $5.77\times 10^{-15}$ | 2.58–6.64 | 233–412
136Xe | $3.56\times 10^{-14}$ | 1.57–3.85 | 164–172
Table 4: $G^{0\nu}_{01}$ and NMEs Meroni:2012qf

The values of $G^{0\nu}$ and NMEs are different for different isotopes as
presented in Table.4 and the effective Majorana mass parameter $m^{\rm
eff}_{\beta\beta}$ is expressed in terms half-life of a given isotope as

$\displaystyle\left[T_{1/2}^{0\nu}\right]^{-1}=G^{0\nu}_{01}|\mathcal{M}^{0\nu}(A)|^{2}\cdot\bigg{(}\frac{m^{\rm
eff}_{\beta\beta}}{m_{e}}\bigg{)}^{2}\,$ $\displaystyle m^{\rm
eff}_{\beta\beta}=\frac{m_{e}}{\sqrt{G^{0\nu}_{01}T^{0\nu}_{1/2}}}\,.$ (45)

Using Table.4 and Table.5, one can derive theoretical limits on effective
Majorana mass which we use in our numerical estimations to derive limits on
absolute value of light neutrino masees.

Table 5: Limits on the half-life of $0\nu\beta\beta$. Experiment | Limit
---|---
HM | $1.9\times 10^{25}\ {\rm yrs}$
GERDA | $2.1\times 10^{25}\ {\rm yrs}$
Combined 76Ge | $3.0\times 10^{25}\ {\rm yrs}$
GERDA Phase II | $5.2\times 10^{25}\ {\rm yrs}$
EXO | $1.6\times 10^{25}\ {\rm yrs}$
KamLAND-Zen | $1.9\times 10^{25}\ {\rm yrs}$
Combined 136Xe | $3.4\times 10^{25}\ {\rm yrs}$
nEXO | $6.6\times 10^{27}\ {\rm yrs}$

###  5.2 Bound on lightest neutrino mass from $0\nu\beta\beta$

Standard mechanism: $m^{\nu}_{ee}$ from light neutrinos $\nu$ :- The standard
contribution to effective Majorana mass due to exchange of light neutrinos is
given by,

$m^{\nu}_{ee}=\sum_{i=1,2,3}\left(\mbox{V}^{\nu\nu}\right)^{2}_{ei}\,m_{i}\,m_{i}\simeq\sum_{i=1}^{3}(U_{\text{PMNS}})_{ei}^{2}\,m_{i}\,.$

where $V^{\nu\nu}$ is the mixing matrix containing non-unitarity effects which
is approximated to be $U_{\rm PMNS}$.

Figure 7: Left panel: Effective Majorana mass parameter ($|m_{ee}|$) due to
standard mechanism as a function of lightest neutrino mass $m_{1}$ ($m_{3}$)
for NH (IH). Right panel: SM contribution to the half-life of $0\nu\beta\beta$
transition as a function of lightest neutrino mass. The blue horizontal bands
show the limits on effective Majorana mass and half-life from GERDA and
EXO+KamLAND-Zen experiments. The vertical purple bands are for the constraints
on sum of light neutrino masses from cosmology data (PLANCK 2018) and KATRIN
detector.

New physics contribution: $m^{N,S}_{ee}$ from heavy neutrinos $N_{R}$ and
$S_{L}$ :-

Figure 8: Left panel: New physics contribution to the plot of effective
Majorana mass as a function of lightest neutrino mass, $m_{1}$ ($m_{3}$) for
NH (IH) via $W_{L}-W_{L}$ channel with the exchange of heavy neutrino $N_{R}$
and sterile neutrino $S_{L}$. Right panel: Contributions of $N_{R}$ and
$S_{L}$ to the plot of half-life vs lightest neutrino mass.

The effective Majorana masses due to the contributions of heavy neutrinos
$N_{R}$ and $S_{L}$ to $0\nu\beta\beta$ decay (represented by the second and
third Feynman diagrams in Fig.6) are as follows,

$\displaystyle|m^{N}_{ee}|=\langle
p^{2}\rangle\sum_{i=1,2,3}\frac{\left(\mbox{V}^{\nu
N}\right)^{2}_{ei}}{M_{N_{i}}}\,,$ $\displaystyle|m^{S}_{ee}|=\langle
p^{2}\rangle\sum_{i=1,2,3}\frac{\left(\mbox{V}^{\nu
S}\right)^{2}_{ei}}{M_{S_{i}}}\,.$ (46)

Here $M_{N_{i}}(M_{S_{i}})$ are the mass eigenvalues of heavy right-handed
(sterile) neutrinos. It is to be noted here that the type-II seesaw dominance
scheme in our model establishes a relationship between these mass eigenvalues
and the light neutrino masses. $\mbox{V}^{\nu N}$ and $\mbox{V}^{\nu S}$ are
the mixing matrices which represent the mixing between light-heavy neutrinos (
$\nu$ and $N_{R}$) and active-sterile neutrinos ($\nu$ and $S_{L}$)
respectively. The neutrino virtual momentum $\langle p^{2}\rangle$ plays an
important role as the half-life formula for $0\nu\beta\beta$ transition
differs for $m_{i}(M_{i})\ll\langle p^{2}\rangle$ or $m_{i}(M_{i})\gg\langle
p^{2}\rangle$. The typical expression for neutrino virtual momentum $\langle
p^{2}\rangle$ is written as,

$\displaystyle\langle p^{2}\rangle=-m_{e}\,m_{p}\,\frac{{\cal
M}^{0\nu}_{N}}{{\cal M}^{0\nu}_{\nu}}\simeq(\mbox{200\, MeV})^{2}\,.$ (47)

Fig.7 shows the effective Majorana mass parameter in the left panel and half-
life in the right panel due to the standard mechanism as a function of
lightest neutrino mass $m_{1}$(NH) and $m_{3}$(IH). Both the plots show that
the quasi-degenerate (QD) pattern of light neutrinos, i.e. $m_{1}\simeq
m_{2}\simeq m_{3}$ is disfavoured by cosmology data on sum of light neutrino
masses whereas the NH(green band) and IH(red band) patterns may not be probed
even by the next generation experiments. This propels the idea of exploring
possible new physics contributions to neutrinoless double beta decay which
might give a hint on mass hierarchy and lightest neutrino mass. Such
possibility occurs when the contributions of heavy neutrino $N_{R}$ and
sterile neutrino $S_{L}$ are considered. The same is shown in Fig.8. The
contributions of $N_{R}$ and $S_{L}$ to effective Majorana mass parameter and
half-life predictions makes both NH(blue band) and IH(orange band) patterns of
light neutrino masses sensitive to the current experimental bounds. In Fig.9
the contributions of all three types of neutrinos ($\nu,N_{R},S_{L}$) are
summed up and plotted against lightest neutrino mass. In this case also the
both NH(green band) and IH(red band) patterns saturate the current
experimental bounds on effective Majorana mass and half-life. In all the three
figures the horizontal blue bands stand for the improved limits on
$0\nu\beta\beta$ decay on effective Majorana mass and half-life with the
combined results of GERDA and KamLAND-Zen experiments. The vertial magenta
region is disfavoured by Planck-2018 Tian:2020tur  and KATRIN data
Aker:2019uuj  on sum of light neutrino masses. In these plots the largest
value of right-handed neutrino mass is fixed at $M_{N_{R}}=1$ TeV while
keeping $M_{W_{R}},M_{\Delta^{++}}>>M_{N_{R}}$. The Majorana phases and Dirac
CP-phase are varied between $0\to\pi$ for all the plots while other neutrino
oscillation parameters are taken in their allowed $3\sigma$ range.

The predictions of our model on lightest neutrino mass due to heavy and
sterile neutrino contributions to neutrinoless double beta decay are as
follows,

  * •

When the contributions of only $N_{R}$ and $S_{L}$ (excluding standard
contribution) are considered, the allowed values of lightest neutrino mass are
found to be in the range of $10-25$ meV for $m_{1}$(NH) and $25-40$ meV for
$m_{3}$(IH). This is done by saturating the effective Majorana mass
$m^{N+S}_{ee}$ with the current GERDA and KamLAND-Zen bounds. It remains the
same for half-life.

  * •

When the contributions of all three types of neutrinos ($\nu,N_{R},S_{L}$) are
considered the predicted values of lightest neutrino mass lies in the range of
$2-15$ meV for $m_{1}$(NH) and $15-25$ meV for $m_{3}$(IH) by sturating
$m^{\nu+N+S}_{ee}$ with the current experimental limits.

Using these predicted values of lightest neutrino mass other light neutrino
mass eigenvalues and heavy neutrino masses can be derived.

Figure 9: Effective Majorana mass (left panel) and half-life (right panel) as
a function of lightest neutrino mass $m_{1}$ ($m_{3}$) for NH (IH) due to the
combined contributions of standard mechanism, $N_{R}$ and $S_{L}$ mediated
diagrams.

##  6 Complementary studies between LNV and LFV Decays

###  6.1 Correlation between $m_{\beta}$ and sum of light neutrino masses
$\sum m_{i}$

In this subsection we examine how the combined constraints from single beta
decay and cosmology can limit absolute scale of light neutrino mass. The
important parameter for single $\beta$ decay sensitive to the electron
neutrino mass is defined as follows.

$m_{\beta}=\sqrt{\sum_{i}\mid U^{2}_{ei}m^{2}_{i}\mid}$ (48)

In the present model $U^{2}_{ei}$ is replaced by
$\mathcal{V}^{\nu\nu^{2}}_{ei}$ due to sizeable non-unitarity effects. The
current limit on $m_{\beta}$ from KATRIN experiment is $m_{\beta}<1.1\,$eV
while an improved bound of $0.2$ eV Aker:2019uuj  is expected in the future.
The sum of light neutrino masses is tightly constrained by Planck 2018 data
Tian:2020tur  which is $\sum m_{i}=m_{1}+m_{2}+m_{3}<0.12\,$ eV. Fig.10 shows
the variation of $\sum m_{i}$ and $m_{\beta}$ with lightest neutrino mass.

Figure 10: Plot showing the variation of $\sum m_{i}$ and $m_{\beta}$ with
lightest neutrino mass ($m_{1}$ for NH, $m_{3}$ for IH).

###  6.2 Correlation between $m_{ee}$ and sum of light neutrino masses $\sum
m_{i}$

Here we discuss the model predictions on effective Majorana mass by changing
the sum of light neutrino masses $\Sigma m_{i}$. We use the following limits
on sum of light neutrino masses Seljak:2004xh ; Costanzi:2014tna ; Palanque-
Delabrouille:2014jca  for displaying the allowed region of $m_{ee}$ in Fig.11.

$\displaystyle m_{\Sigma}<\mbox{84\,meV}\quad\quad~{}(1\sigma~{}\mbox{C.L.})$
$\displaystyle m_{\Sigma}<\mbox{146\,meV}\quad\quad(2\sigma~{}\mbox{C.L.})$
$\displaystyle m_{\Sigma}<\mbox{208\,meV}\quad\quad(3\sigma~{}\mbox{C.L.})$
(49)

In both the plots of Fig.11 the horizontal bands represents the experimental
bounds on effective Majorana mass $m_{ee}$ by GERDA and EXO+KamLAND-Zen while
the vertical dashed line shows the bound on sum of light neutrino masses from
cosmology. The plot in the left-panel shows the variation of effective
Majorana mass parameter due to standard mechanism with sum of light neutrino
masses. It shows that both NH (green band) and IH (red band) patterns of light
neutrino masses are not sensitive to the current experimetal bounds on
$m_{ee}$ and also disfavoured by cosmology when only light neutrino ($\nu$)
contribution is considered. The plot in the right-panel shows the variation of
effective Majorana mass parameter due to new physics contributions
(contributions of $N_{R}$ and $S_{L}$) with sum of light neutrino masses. It
shows that NH (purple band) pattern of light neutrino masses saturates the
experimental bound on $m_{ee}$ as well as the cosmology bound on sum of light
neutrino masses whereas IH (blue band) pattern is disfavoured by cosmology.
Thus the model with type-II seesaw dominance gives an important result on the
hierarchy of light neutrino masses when the contributions of heavy and sterile
neutrinos are considered.

Figure 11: Allowed region of effective Majorana mass parameter ($|m_{ee}|$) as
a function of sum of light neutrino masses ($\Sigma m_{i}$) for standard
mechanism (left-panel) and new physics contributions from $N_{R}$ and $S_{L}$
mediated diagrams (right-panel).

We also do a comparative analysis between neutrinoless double beta decay and
LFV processes to examine how light-heavy neutrino mixing is constrained from a
combined study of LNV and LFV.

Figure 12: Correlation plot between LFV processes $\mu\to e\gamma$ and LNV
$0\nu\beta\beta$ decay for 76Ge isotope.

###  6.3 Electric dipole moment (EDM) of charged leptons

The dipole moment of electron is an important process whose experimental
observation will reveal violation of Parity $P$ and Time reversal $T$ symmetry
(or violation of CP invariance) in nature. The electric dipole moment formula
derived for charged leptons $\ell_{\alpha}(\alpha=e,\mu,\tau)$ is given by
Nieves:1986uk ; Nemevsek:2012iq ; Tello:2012qda ; Barry:2013xxa

$\displaystyle d_{\alpha}=\frac{e\alpha_{\rm\small W}}{8\pi
M^{2}_{W_{L}}}\mbox{\Large Im}\bigg{[}\sum^{3}_{i=1}{\mbox{V}}^{\nu N}_{\alpha
i}{\mbox{V}}^{NN}_{i\alpha}\,\xi\,\mathcal{G}^{\gamma}_{2}(x_{N_{i}})M_{N_{i}}\bigg{]}\,,$
(50)

where $\alpha_{W}\simeq 1/30$ is weak fine structure constant,
$M_{W_{L}}=80.3$ GeV, $\xi\leq 10^{-4}$ is the $W_{L}-W_{R}$ mixing and
$M_{N}$ is the mass matrix for heavy neutrinos. The mixing matrices are
$\mbox{V}^{\nu N}=(v_{L}/v_{R})M_{D}U_{\rm PMNS}m^{-1}_{\nu}$,
$\mbox{V}^{NN}=U_{\rm PMNS}$. The resulting dipole moment for electron can be
expressed as a function of lightest neutrino mass and one can derive similar
bound on absolute scale lightest neutrino mass by saturating the experimental
bound.

In the present work, we can go to a basis where both Dirac neutrino mass
matrix $M_{D}$ connecting $\nu_{L}-N_{R}$ and heavy Dirac mass $M$ connecting
$N_{R}-S_{L}$ can be diagonal simultaneously. Taking $M_{D}\simeq
80\mbox{diag}(m_{e},m_{\mu},m_{\tau})$ GeV and $M_{D}\simeq
500\mbox{diag}(1,1,1)$ GeV, we found that the dipole moment only depends upon
PMNS phases contributing to the imaginary part. Considering vanishing Majorana
phases, the variation of electron dipole moment with Dirac CP-phase
$\delta_{\rm CP}$ is shown in Fig.13 The horizontal line represents the
current bound on electron EDM set by ACME Baron:2013eja .

Figure 13: Electric dipole moment, $d_{e}$ as a function of Dirac CP-phase,
$\delta_{\rm CP}$ for NH and IH patterns of light neutrino masses considering
diagonal structures for Dirac neutrino mass matrix $M_{D}$ and $N_{R}-S_{L}$
mixing matrix $M$.

##  7 Comments on muon (g-2) anomaly

An intensive activity is going on to explain the recent results declared by
Fermi National Accelerator Laboratory (FNAL) on the measurement of muon
magnetic dipole moment, $a_{\mu}=\frac{g_{\mu}-2}{2}$ that just confirmed the
existing anomaly in the muon sector. The theoretical value of $a_{\mu}$
estimated by Standard Model Tanabashi:2018oca ; Blum:2013xva ; Bennett:2006fi
; Abi:2021gix , the measured value by Brookhaven National Laboratoty (BNL)
Bennett:2006fi  and the recent result by FNAL Abi:2021gix  are as follows.

$\displaystyle a_{\mu}^{\text{SM}}=(11659181.0\pm 4.3)\times 10^{-10}$ (51)
$\displaystyle a_{\mu}^{\text{BNL}}=(11659208.0\pm 6.3)\times 10^{-10}$ (52)
$\displaystyle a_{\mu}^{\text{FNAL}}=(11659204.0\pm 5.4)\times 10^{-10}$ (53)

Combining both the results of BNL and FNAL, the new world average now stands
at,

$a_{\mu}^{\text{2021}}=(11659206.1\pm 4.1)\times 10^{-10}\,.\\\ $ (54)

Previously there existed a 3.7 $\sigma$ deviation in the SM’s estimated value
and the measured value by BNL which has now sharpened to 4.2 $\sigma$
considering the FNAL result with a precise deviation of,

$\Delta
a_{\mu}^{\text{2021}}=(a_{\mu}^{\text{BNL}}+a_{\mu}^{\text{FNAL}})-a_{\mu}^{\text{SM}}=(25.1\pm
5.9)\times 10^{-10}.\\\ $ (55)

Many new physics scenarios have been proposed so far to explain the anomaly,
for an incomplete list of which one may refer Jegerlehner:2009ry ;
Lindner:2016bgg ; Majumdar:2020xws ; Patra:2016shz  (and references therein) .
In this framework new contributions to $a_{\mu}$ arise from the interactions
of sterile neutrino $S_{L}$ due to large mixing with muons. This contribution
depends on sterile neutrino mass and its mixing with muons, which are related
to light neutrino masses and oscillation parameters in type-II seesaw
dominance scheme as discussed earlier. Thus, this scenario opens the
possibility of constraning light neutrino masses and mass hierarchy from FNAL
results on $a_{\mu}$. Other sizeable contributions can arise from $W_{R}$
mediated channels and scalar exchange if one considers left-right symmetry
breaking at TeV scale. Such cases are studied in one of our recent works
Majumdar:2020xws , where we have extensively calculated the contributions of
neutral fermions, gauge bosons and scalars to $a_{\mu}$ in a TeV scale
extended Left-right model.

##  8 Conclusion

We considered a new mechanism of natural type-II seesaw dominance that allows
large light-heavy neutrino mixing within a framework of left-right symmetric
model. The fermion sector of the model contains usual quarks and leptons plus
an extra sterile neutrino per generation while its scalar sector consists of
Higgs doublets, triplets and bidoublet. The extra particles help in generating
large light-heavy neutrino mixing that gives new contributions to various LFV
processes like $\mu\to e\gamma$, $\mu\to 3e$ and $\mu\to e$ conversion inside
nuclei. As a result of these new contributions the new branching ratios can be
accessible to present as well as planned experiments. We have demonstrated how
the model parameters are suitably adjusted to make the contribution of inverse
seesaw and loop induced light neutrino masses sub-dominant. Rather we have
generated the light neutrino masses through type-II seesaw mechanism.

All the physical masses and mixing of neutral leptons are expressed in terms
of light neutrino masses and PMNS mixing matrix. Thus the new physics
contributions to various LFV processes arising from heavy and sterile
neutrinos depend upon light neutrino mass. As a result of this, the bound on
absolute scale of light neutrino mass can be derived by saturating the
experimental limits on LFV processes, which we find to be in the meV range. We
have plotted branching ratios for LFV processes such as $\mu\to e\gamma$,
$\mu\to 3e$ and $\mu\to e$ conversion inside nuclei as a function of the
lightest neutrino mass for NH as well as IH pattern of light neutrinos in
order to derive the bound on light neutrino mass. We have also plotted dipole
moment of electron as a function of light neutrino mass and found that the
result depends only upon PMNS phase. Thus in our model the dipole moment of
electron can uniquely say about the relative phases of PMNS mixing matrix. We
have also studied the new contributions to neutrinoless double beta decay
arising from purely left-handed current due to exchange of heavy Majorana
neutrinos in our model. It is found that these new contributions can saturate
the experimental bound for $m_{1}>0.001$eV for NH. We have shown the
correlation between LFV and LNV by focusing on $\text{Br}_{\mu\rightarrow
e\gamma}\mbox{vs.}T_{1/2}^{0\nu}[^{76}\mbox{Ge}]$ in Fig. 12.

## Acknowledgements

The authors would like to thank P.S. Bhupal Dev for the useful discussions
during the early stage of this work.

##  9 Appendix

##  Appendix A Derivation of neutrino masses and mixings in extended left-
right seesaw model (ELRSM)

###  A.1 ELRSM mass matrix and form of unitary mixing matrix

We discuss here the implementation of extended seesaw mechanism for neutrino
masses and mixing and derivation of type-II seesaw dominance within left-right
symmetric models. Apart from usual quarks and leptons, the fermion sector is
extended with an extra sterile neutrino $\nu_{S_{L}}$ (or $\nu_{S}$ in short-
hand notation) while scalar sector consists of usual scalar bidoublet $\Phi$,
doublets ($H_{R},H_{L}$) and triplets ($\Delta_{R},\Delta_{L}$). The left-
right symmetry is broken down to SM theory through spontaneous symmetry
breaking when the scalars $H_{R}$ or $\Delta_{R},\Delta_{L}$ take VEV. The
scalar $H_{L}$ doesn’t play any role here, rather it is present due to the
left-right symmetry. The SM theory further breaks down to low energy theory
with the help of bidoblet $\Phi$. We call this scheme the extended left-right
seesaw model (ELRSM). The neutral fermions neeeded for ELRSM are active left-
handed neutrinos, $\nu_{L}$, active right-handed neutrinos, $\nu_{R}$ and
sterile neutrinos, $\nu_{S}$. The relevant mass terms within ELRSM are given
by

$\displaystyle\mathcal{L}_{\rm ELRSM}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{M_{D}}+\mathcal{L}_{M}+\mathcal{L}_{M_{L}}+\mathcal{L}_{M_{R}}+\mathcal{L}_{\mu_{S}}$
$\displaystyle\mathcal{L}_{M_{D}}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha,\beta}\overline{\nu_{\alpha
L}}[M_{D}]_{\alpha\beta}\nu_{\beta R}\mbox{+ h.c.}$
$\displaystyle\mathcal{L}_{M}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha,\beta}\overline{\nu_{\alpha
S}}[M]_{\alpha\beta}\nu_{\beta R}\mbox{+ h.c.}$
$\displaystyle\mathcal{L}_{M_{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{\alpha,\beta}\overline{\nu^{c}_{\alpha
L}}[M_{L}]_{\alpha\beta}\nu_{\beta L}\mbox{+ h.c.}$
$\displaystyle\mathcal{L}_{M_{R}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{\alpha,\beta}\overline{\nu^{c}_{\alpha
R}}[M_{R}]_{\alpha\beta}\nu_{\beta R}\mbox{+ h.c.}$
$\displaystyle\mathcal{L}_{\mu_{S}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{\alpha,\beta}\overline{\nu^{c}_{\alpha
S}}[\mu_{S}]_{\alpha\beta}\nu_{\beta S}\mbox{+ h.c.}$ (56)

The flavour states for active left-handed neutrinos $\nu_{\alpha L}$, sterile
neutrinos $\nu_{\beta S}$ and right-handed neutrinos $\nu_{\gamma R}$ are
defined as follows

$\displaystyle\nu_{\alpha L}=\begin{pmatrix}\nu_{eL}\\\ \nu_{\mu L}\\\
\nu_{\tau L}\end{pmatrix}\,,\quad\nu_{\beta S}=\begin{pmatrix}\nu_{S_{1L}}\\\
\nu_{S_{2L}}\\\ \nu_{S_{3L}}\end{pmatrix}\,,\quad\nu_{\gamma
R}=\begin{pmatrix}\nu_{N_{1R}}\\\ \nu_{N_{2R}}\\\
\nu_{N_{3R}}\end{pmatrix}\,.$ (57)

Similarly, the mass states for neutral fermions are given by

$\displaystyle\nu_{iL}=\begin{pmatrix}\nu_{1L}\\\ \nu_{2L}\\\
\nu_{3L}\end{pmatrix}\,,\quad S_{jL}=\begin{pmatrix}S_{1L}\\\ S_{2L}\\\
S_{3L}\end{pmatrix}\,,\quad N^{c}_{kR}=\begin{pmatrix}N^{c}_{1R}\\\
N^{c}_{2R}\\\ N^{c}_{3R}\end{pmatrix}\,.$ (58)

In the flavour basis $\left(\nu_{L},\nu_{S},\nu^{c}_{R}\right)$ the resulting
$9\times 9$ neutral lepton mass matrix within ELRSM is given by,

$\displaystyle\mathcal{M}_{\rm ELRSM}=\begin{pmatrix}M_{L}&0&M_{D}\\\
0&\mu_{S}&M\\\ M^{T}_{D}&M^{T}&M_{R}\end{pmatrix}$ (59)

Here $M_{D}$ is the Dirac mass term connecting $\nu_{L}-\nu_{R}$, $M$ is the
mixing term between $\nu_{R}-\nu_{S}$ while $M_{L}$, $M_{R}$, $\mu_{S}$ are
Majorana mass terms for $\nu_{L}$, $\nu_{R}$ and $\nu_{S}$, respectively. The
mass hierarchy is given by

$\displaystyle M_{R}\gg M>M_{D}>\mu_{S}\gg M_{L}\,.$ (60)

The diagonalisation of $\mathcal{M_{\rm ELRSM}}$ after changing it from
flavour basis to mass basis is done by a generalized unitary transformation
as,

$\displaystyle\mid\Psi\rangle_{\rm flavor}=\mathcal{V}\mid\Psi\rangle_{\rm
mass}$ (61) or, $\displaystyle\begin{pmatrix}\nu_{\alpha L}\\\ \nu_{\beta
S}\\\ \nu^{c}_{\gamma R}\end{pmatrix}=\begin{pmatrix}\mathcal{V}_{\alpha
i}^{\nu\nu}&\mathcal{V}_{\alpha j}^{\nu S}&\mathcal{V}_{\alpha k}^{\nu N}\\\
\mathcal{V}_{\beta i}^{S\nu}&\mathcal{V}_{\beta j}^{SS}&\mathcal{V}_{\beta
k}^{SN}\\\ \mathcal{V}_{\gamma i}^{N\nu}&\mathcal{V}_{\gamma
j}^{NS}&\mathcal{V}_{\gamma k}^{NN}\end{pmatrix}\begin{pmatrix}\nu_{i}\\\
S_{j}\\\ N^{c}_{k}\end{pmatrix}\,$ (63)
$\displaystyle\mathcal{V}^{\dagger}\mathcal{M_{\rm
ELRSM}}\mathcal{V}^{*}=\mathcal{\widehat{M}}_{\rm ELRSM}$ $\displaystyle\hskip
62.59596pt=\mbox{diag}\left(m_{i},m_{S_{j}},m_{N_{k}}\right)$
$\displaystyle\hskip
62.59596pt=\mbox{diag}\left(m_{1},m_{2},m_{3},m_{S_{1}},m_{S_{2}},m_{S_{3}},m_{N_{1}},m_{N_{2}},m_{N_{3}}\right)$

Here the indices $\alpha,\beta,\gamma$ run over three generations of light
left-handed neutrinos, sterile neutrinos and heavy right-handed neutrinos in
flavor basis respectively, whereas the indices $i,j,k$ run over corresponding
mass states.

###  A.2 Seesaw block diagonalization for ELRSM neutrino mass matrix

The block diagonaliztion is done in two steps. In the first step, the ELRSM
neutrino mass matrix $\mathcal{M_{\rm ELRSM}}$ is reduced to an ‘Intermediate
Block Diagonal’ form $\mathcal{M_{\rm IBD}}$ and in the second step this
$\mathcal{M_{\rm IBD}}$ is further diagonalised to $\mathcal{M_{\rm
BD}}=\mbox{diag}\left(m_{\nu},m_{S},m_{N}\right)$, from which we obtain the
mass formulae for three types of neutrinos $\nu_{L},\nu_{S},\nu_{R}$ as the
three diagonal elements. Finally, in order to get the physical masses for all
types of neutrinos we need unitary transformations for individual mass
matrices as,

$\mathcal{M_{\rm diag}}=\mathcal{\widehat{M}}_{\rm
ELRSM}=\mbox{diag}\left(m_{i},m_{S_{j}},m_{N_{k}}\right)=\mbox{diag}\left(m_{1},m_{2},m_{3},m_{S_{1}},m_{S_{2}},m_{S_{3}},m_{N_{1}},m_{N_{2}},m_{N_{3}}\right)$

####  A.2.1 Determination of $\mathcal{M_{\rm IBD}}$

Let us first rewrite the ELRSM mass matrix $\mathcal{M}_{\rm ELRSM}$ given in
eq.(59) to generic type-I+II seesaw Grimus:2000vj  as,

$\displaystyle\mathcal{M_{\rm ELRSM}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\mathcal{M}_{L}&\mathcal{M}^{T}_{D}\\\
\mathcal{M}_{D}&\mathcal{M}_{R}\end{pmatrix}\,,$ where,
$\displaystyle\mathcal{M}_{L}=\begin{pmatrix}M_{L}&0\\\
0&\mu_{S}\end{pmatrix}_{6\times
6}\,,\quad\mathcal{M}_{D}=\begin{pmatrix}M^{T}_{D}&M^{T}\end{pmatrix}_{3\times
6}\,,\quad\mathcal{M}_{R}=M_{R}\,.$ (64)

Using the seesaw approximations given in eq.(60), it can be shown that
$|\mathcal{M}_{R}|\gg\mathcal{M}_{D}\gg\mathcal{M}_{L}$. First
$\mathcal{M_{\rm ELRSM}}$ can be simplified to intermediate block diagonalized
form $\mathcal{M_{\rm IBD}}$ by integrating out the heaviest right-handed
neutrinos from other neutral states. Thus, the first block diagonalized
approximate unitary mixing matrix $\mathcal{W}_{1}$ gives

$\displaystyle\mathcal{W}^{T}_{1}\mathcal{M}_{\rm
ELRSM}\mathcal{W}_{1}=\mathcal{M}_{\rm IBD}$ $\displaystyle\text{where},$
$\displaystyle\mathcal{M}_{\rm IBD}=\begin{pmatrix}M^{\rm light}_{\rm
Eff}&{\bf 0}_{3\times 6}\\\ {\bf 0}_{3\times 6}&M^{\rm heavy}\end{pmatrix},$
and
$\displaystyle\mathcal{W}_{1}=\begin{pmatrix}\sqrt{1-\mathcal{B}\mathcal{B}^{\dagger}}&\mathcal{B}\\\
-\mathcal{B}^{\dagger}&\sqrt{1-\mathcal{B}^{\dagger}\mathcal{B}}\end{pmatrix}$
(65)

Now $\mathcal{M_{\rm IBD}}$ gives effective mass matrix in the modified basis
of left-handed active and sterile neutrinos as,

$\displaystyle M^{\rm light}_{\rm Eff}$ $\displaystyle\equiv$ $\displaystyle
M^{\rm
Eff}_{ELRSM}=\mathcal{M}_{L}-\mathcal{M}^{T}_{D}\mathcal{M}^{-1}_{R}\mathcal{M}_{D}$
(66) $\displaystyle=$ $\displaystyle\begin{pmatrix}M_{L}&0\\\
0&\mu_{S}\end{pmatrix}-\begin{pmatrix}M_{D}\\\
M\end{pmatrix}M^{-1}_{R}\begin{pmatrix}M^{T}_{D}&M^{T}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}M_{L}&0\\\
0&\mu_{S}\end{pmatrix}-\begin{pmatrix}M_{D}M^{-1}_{R}M^{T}_{D}&M_{D}M^{-1}_{R}M^{T}\\\
MM^{-1}_{R}M_{D}&MM^{-1}_{R}M^{T}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}M_{L}-M_{D}M^{-1}_{R}M^{T}_{D}&-M_{D}M^{-1}_{R}M^{T}\\\
-MM^{-1}_{R}M_{D}&\mu_{S}-MM^{-1}_{R}M^{T}\end{pmatrix}$

and block diagonalized mass formula for the integrated out heavy right-handed
neutrinos as,

$\displaystyle M^{\rm heavy}$ $\displaystyle\equiv$ $\displaystyle
m_{N}=M_{R}+\cdots$ (67)

Using the standard seesaw block diagonalization methodology, one can get

$\mathcal{B}^{\dagger}_{1}=\begin{pmatrix}M_{D}M^{-1}_{R}\\\
MM^{-1}_{R}\end{pmatrix}^{T}$

and the approximated intermediate block diagonalized mixing matrix up to the
order of $\mathcal{O}(1/M_{R})$ is given by

$\displaystyle\mathcal{W}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1-\frac{1}{2}ZZ^{\dagger}&-\frac{1}{2}ZY^{\dagger}&Z\\\
-\frac{1}{2}YZ^{\dagger}&1-\frac{1}{2}YY^{\dagger}&Y\\\
-Z^{\dagger}&-Y^{\dagger}&1-\frac{1}{2}\left(Z^{\dagger}Z+Y^{\dagger}Y\right)\end{pmatrix}$
(68)

where $Z=M_{D}M^{-1}_{R}$ and $Y=MM^{-1}_{R}$.

In this way the light active and sterile neutrinos contained in the effective
block diagonalized mass matrix $M^{\rm light}_{\rm Eff}$ get completely
decoupled from heaviest right-handed neutrinos. Thus the first seesaw
intermediate block diagonalization brings down a $9\times 9$ matrix into two
smaller matrices; a block diagonalized $6\times 6$ matrix for $\nu_{L}$ and
$\nu_{S}$ and a $3\times 3$ matrix for $\nu_{R}$. Now we have to repeat the
same seesaw block diagonalization procedure for $M^{\rm light}_{\rm Eff}$ to
further block diagonalize the light neutrino states.

####  A.2.2 Determination of $\mathcal{M_{\rm BD}}$

We require another approximated unitary mixing matrix $\mathcal{S}$ (and
$\mathcal{W}_{2}$ for accounting the integrated out right-handed neutrinos) to
further block diagonalize the effective light neutrino mass matrix $M^{\rm
light}_{\rm Eff}$ in order to get mass matrices for $\nu_{L}$ and $\nu_{S}$.
Let us write $M^{\rm light}_{\rm Eff}$ as given in eq.66 in a simpler form as,

$\displaystyle M^{\rm light}_{\rm Eff}$ $\displaystyle=$
$\displaystyle-\begin{pmatrix}M_{L}-M_{D}M^{-1}_{R}M^{T}_{D}&-M_{D}M^{-1}_{R}M^{T}\\\
-MM^{-1}_{R}M_{D}&\mu_{S}-MM^{-1}_{R}M^{T}\end{pmatrix}=\begin{pmatrix}\mathcal{M}^{\prime}_{L}&\mathcal{M}^{\prime
T}_{D}\\\ \mathcal{M}^{\prime}_{D}&\mathcal{M}^{\prime}_{R}\end{pmatrix}\,,$
where, $\displaystyle\mathcal{M}^{\prime}_{L}=M_{L}-M_{D}M^{-1}_{R}M^{T}_{D}$
(69) $\displaystyle\mathcal{M}^{\prime}_{D}=-MM^{-1}_{R}M^{T}_{D}$
$\displaystyle\mathcal{M}^{\prime}_{R}=\mu_{S}-MM^{-1}_{R}M^{T}$

Now repeating the same procedure of type-I+II seesaw block diagonalization
along with the mass hierarchy given in eq.(60), the diagonalized mass matrix
and mixing matrix look as follows,

$\displaystyle\mathcal{S}^{T}\mathcal{M}^{\rm
light}_{Eff}\mathcal{S}=\begin{pmatrix}m_{\nu}&0\\\ 0&m_{S}\end{pmatrix}$
$\displaystyle\mathcal{S}=\begin{pmatrix}\sqrt{1-\mathcal{A}\mathcal{A}^{\dagger}}&\mathcal{A}\\\
-\mathcal{A}^{\dagger}&\sqrt{1-\mathcal{A}^{\dagger}\mathcal{A}}\end{pmatrix}=\begin{pmatrix}1-\frac{1}{2}XX^{\dagger}&X\\\
-X^{\dagger}&1-\frac{1}{2}X^{\dagger}X\end{pmatrix}$ (70)

where active-sterile neutrino mixing is given by the matrix $X=M_{D}M^{-1}$.
Thus, we get the block diagonalized mass formulae for light active neutrinos,
sterile neutrinos and right-handed neutrinos as follows.

$\displaystyle\mathcal{W}^{T}_{2}\mathcal{M}^{IBD}_{\rm
ELRSM}\mathcal{W}_{2}=\mathcal{M}^{BD}_{\rm ELRSM}$
$\displaystyle\mathcal{M}^{\rm BD}_{\rm ELRSM}=\begin{pmatrix}m_{\nu}&0&0\\\
0&m_{S}&0\\\ 0&0&m_{N}\end{pmatrix}$ (71)

The approximated unitary mixing matrix $\mathcal{W}_{2}$ is given by,

$\displaystyle\mathcal{W}_{2}=\begin{pmatrix}\mathcal{S}&{\bf 0}_{6\times
3}\\\ {\bf 0}_{3\times 6}&{\bf 1}_{3\times 3}\end{pmatrix}$
$\displaystyle\hskip
28.45274pt=\begin{pmatrix}\sqrt{1-\mathcal{A}\mathcal{A}^{\dagger}}&\mathcal{A}&{\bf
0}_{3\times 3}\\\
-\mathcal{A}^{\dagger}&\sqrt{1-\mathcal{A}^{\dagger}\mathcal{A}}&{\bf
0}_{3\times 3}\\\ {\bf 0}_{3\times 3}&{\bf 0}_{3\times 3}&{\bf 1}_{3\times
3}\end{pmatrix}=\begin{pmatrix}1-\frac{1}{2}XX^{\dagger}&X&{\bf 0}_{3\times
3}\\\ -X^{\dagger}&1-\frac{1}{2}X^{\dagger}X&{\bf 0}_{3\times 3}\\\ {\bf
0}_{3\times 3}&{\bf 0}_{3\times 3}&{\bf 1}_{3\times 3}\end{pmatrix}$ (72)

####  A.2.3 Radiative contribution to light neutrino masses

The block diagonalized mass formulas for light active neutrinos, sterile
neutrinos and heavy neutrinos are given by

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\prime}_{L}-\mathcal{M}^{\prime
T}_{D}\mathcal{M}^{\prime^{-1}}_{R}\mathcal{M}^{\prime}_{D}$ (73)
$\displaystyle=$ $\displaystyle
M_{L}-M_{D}M^{-1}_{R}M^{T}_{D}-\big{(}-M_{D}M^{-1}_{R}M^{T}\big{)}\cdot\big{(}-MM^{-1}_{R}M^{T}\big{)}^{-1}\cdot\big{(}\mu_{S}-MM^{-1}_{R}M^{T}_{D}\big{)}$
$\displaystyle=$ $\displaystyle
M_{L}-M_{D}M^{-1}_{R}M^{T}_{D}+M_{D}M^{-1}_{R}M^{T}_{D}+\big{(}\frac{M_{D}}{M}\big{)}\mu_{S}\big{(}\frac{M_{D}}{M}\big{)}^{T}$
$\displaystyle=$ $\displaystyle m^{\rm II}+m^{\rm inv}$ $\displaystyle m_{S}$
$\displaystyle=$ $\displaystyle\mu_{S}-MM^{-1}_{R}M^{T}$ (74) $\displaystyle
m_{N}$ $\displaystyle=$ $\displaystyle M_{R}$ (75)

An interesting feature of approximated seesaw block diagonalization scheme
within ELRSM is that type-I seesaw contribution gets exactly cancelled out at
tree level. Thus at tree level light neutrinos get mass through type-II seesaw
and inverse seesaw. There is also a sizable contribution to light neutrino
masses at 1-loop level.

$\displaystyle m_{\nu}=m^{\rm II}+m^{\rm inv}+m^{rad}_{\nu}\,.$ (76)

The analytic expression for this one loop contribution to light neutrino mass
mediated by SM $W$ and $Z$-bosons is given by

$\displaystyle m^{\rm rad}_{\nu}\equiv\Delta M$ $\displaystyle\simeq$
$\displaystyle M_{D}\frac{\alpha_{W}}{16\pi
m_{W}^{2}}M_{R}\left[\frac{m_{H}^{2}}{M_{R}^{2}-m_{H}^{2}{\bf
1}_{3}}\ln\left(\frac{M_{R}^{2}}{m_{H}^{2}}\right)+\frac{3m_{Z}^{2}}{M_{R}^{2}-m_{Z}^{2}{\bf
1}_{3}}\ln\left(\frac{M_{R}^{2}}{m_{Z}^{2}}\right)\right]M^{T}_{D}$ (77)
$\displaystyle\simeq$ $\displaystyle M_{D}M_{R}^{-1}x_{R}\,f(x_{R})M_{D}^{T}$

where the one-loop function $f(x_{R})$ is defined as

$\displaystyle
f(x_{R})=\frac{\alpha_{W}}{16\pi}\left[\frac{x_{H}}{x_{R}-x_{H}}\ln\left(\frac{x_{R}}{x_{H}}\right)+\frac{3x_{Z}}{x_{R}-x_{Z}}\ln\left(\frac{x_{R}}{x_{Z}}\right)\right]$
(78)

with $x_{R}\equiv\hat{M}_{R}^{2}/m_{W}^{2}$, $x_{H}\equiv
m_{H}^{2}/m_{W}^{2}$, $x_{Z}\equiv m_{Z}^{2}/m_{W}^{2}$, $\hat{M}_{R}$ as
diagonal matrix.

####  A.2.4 Complete diagonalization and physical neutrino masses

After block diagonalization, the mass matrix for the three types of neutrinos
are further diagonalized by respective unitary mixing matrices as follows,

$\displaystyle\mathcal{U}=\begin{pmatrix}U_{\nu}&0&0\\\ 0&U_{S}&0\\\
0&0&U_{N}\end{pmatrix}$ (79)

resulting in physical masses for all the neutrinos.

$\displaystyle
U^{\dagger}_{\nu}m_{\nu}U^{*}_{\nu}=\hat{m}_{\nu}=\mbox{diag}\left(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}}\right)$
$\displaystyle
U^{\dagger}_{S}m_{S}U^{*}_{S}=\hat{m}_{S}=\mbox{diag}\left(m_{s_{1}},m_{s_{2}},m_{s_{3}}\right)$
$\displaystyle
U^{\dagger}_{N}m_{N}U^{*}_{N}=\hat{m}_{N}=\mbox{diag}\left(m_{N_{1}},m_{N_{2}},m_{N_{3}}\right)$
(80) $\displaystyle\mathcal{U}_{9\times 9}=\begin{pmatrix}{\bf
U_{\nu}}_{3\times 3}&{\bf 0}_{3\times 3}&{\bf 0}_{3\times 3}\\\ {\bf
0}_{3\times 3}&{U_{S}}_{3\times 3}&{\bf 0}_{3\times 3}\\\ {\bf 0}_{3\times
3}&{\bf 0}_{3\times 3}&{U_{N}}_{3\times 3}\end{pmatrix}$ (81)

The complete $9\times 9$ mixing matrix is derived as,

$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\mathcal{W}_{1}\cdot\mathcal{W}_{2}\cdot\mathcal{U}$ (82)
$\displaystyle=\begin{pmatrix}1-\frac{1}{2}ZZ^{\dagger}&-\frac{1}{2}ZY^{\dagger}&Z\\\
-\frac{1}{2}YZ^{\dagger}&1-\frac{1}{2}YY^{\dagger}&Y\\\
-Z^{\dagger}&-Y^{\dagger}&1-\frac{1}{2}\left(Z^{\dagger}Z+Y^{\dagger}Y\right)\end{pmatrix}\cdot\begin{pmatrix}1-\frac{1}{2}XX^{\dagger}&X&0\\\
-X^{\dagger}&1-\frac{1}{2}X^{\dagger}X&0\\\
0&0&1\end{pmatrix}\cdot\begin{pmatrix}U_{\nu}&0&0\\\ 0&U_{S}&0\\\
0&0&U_{N}\end{pmatrix}$ $\displaystyle\simeq$
$\displaystyle\begin{pmatrix}U_{\nu}\left(1-\frac{1}{2}XX^{\dagger}\right)&-U_{S}X&ZU_{N}\\\
-U_{\nu}X^{\dagger}&U_{S}\left(1-\frac{1}{2}X^{\dagger}X\right)&U_{N}Y\\\
\left(Z^{\dagger}X^{\dagger}X\right)U_{\nu}&-U_{S}Y^{\dagger}&U_{N}\left(1-\frac{1}{2}Y^{\dagger}Y\right)\end{pmatrix}$

Putting $X=M_{D}M^{-1}$, $Y=MM^{-1}_{R}$, $Z=M_{D}M^{-1}_{R}$ and fixing the
typical magnitudes for $M_{D}\simeq$ 10 GeV ,$M\simeq$ 100 GeV, $M_{R}\simeq$
1000 GeV - 10 TeV, we get $X\simeq 0.1$, $Y\simeq 0.01$, $Z\simeq 10^{-3}$.
Since $U_{\nu}$, $U_{N}$ and $U_{S}$ are of $\mathcal{O}(1)$, the matrix
elements of $\mathcal{V}$ are approximated to be

$\displaystyle\begin{pmatrix}\mathcal{V}_{\alpha
i}^{\nu\nu}&\mathcal{V}_{\alpha j}^{\nu S}&\mathcal{V}_{\alpha k}^{\nu N}\\\
\mathcal{V}_{\beta i}^{S\nu}&\mathcal{V}_{\beta j}^{SS}&\mathcal{V}_{\beta
k}^{SN}\\\ \mathcal{V}_{\gamma i}^{N\nu}&\mathcal{V}_{\gamma
j}^{NS}&\mathcal{V}_{\gamma
k}^{NN}\end{pmatrix}\simeq\begin{pmatrix}1&0.1&10^{-3}\\\ 0.1&0.91&0.01\\\
0&0.01&1.0\end{pmatrix}$ (83)

###  A.3 Achieving type-II seesaw dominance by including radiative
contributions

In addition to the generic Dirac neutrino mass $M_{D}$ and Majorana neutrino
mass $M_{R}$, there are two more terms; a mixing term $M$ that connects right-
handed neutrinos with extra serile neutrinos and a small Majorana mass term
$\mu_{S}$ for only sterile neutrinos. With a small but non-zero $\mu_{S}$, an
additional contribution to light neutrino masses arises through inverse seesaw
mechanism as $m^{\rm inv}_{\nu}=(M_{D}/M)\mu_{S}(M_{D}/M)^{T}$ apart from the
type-II seesaw term at tree level. This inverse seesaw term can be avoided and
type-II seesaw dominance can be achieved with $\mu_{S}\to 0$. However SM
radiative corrections involving the SM $Z$ and Higgs boson also contribute to
light neutrino masses $m^{\rm rad}_{\nu}$ as pointed out originally by
Pilaftsis:1991ug  and later by Dev:2012bd ; Dev:2012sg . The presence of
inverse seesaw and one loop radiative corrections to light neutrino mass might
spoil the plot for type-II seesaw dominance. We briefly discuss below how one
can obtain the type-II seesaw dominance.

  * •

By considering suppressed value of Dirac neutrino mass:-
The total contribution to light neutrino masses is the sum of type-II, inverse
and radiative seesaw mechanism, out of which the inverse and radiative seesaw
contributions mostly depend on Dirac neutrino masses. Thus a suppressed value
of Dirac neutrino mass will give negligible contribution to inverse as well as
radiative corrections to light neutrinos masses. In such case (i.e. in the
limit of suppressed or vanishing Dirac neutrino masses), the only scheme for
generating light neutrino masses will be type-II seesaw dominance.

  * •

Due to effective cancellation between inverse and radiative contributions:-
Alternatively type-II seesaw dominance can be achieved by allowing
exact/partial cancellation between inverse seesaw and radiative contribution
such that they marginally/sub-dominantly contribute to the light neutrino
masses. This is phenomenologically more interesting because it allows large
light-heavy neutrino mixing which can contribute to lepton number violation
and lepton flavour violation. We have followed this method in our present
model, i.e. we have allowed cancellation to make $m_{\nu}^{inv}+m_{\nu}^{\rm
rad}=0$.

##  Appendix B Loop functions involved in lepton flavour violating processes

The relevant loop functions arising in various lepton flavour violating (LFV)
processes like $\mu\to e\gamma$, $\mu\to 3e$ and $\mu\to e$ conversion inside
a nuclei used in our analysis are given by

$\displaystyle\begin{split}\mathcal{G}^{\gamma}_{1}(x)&=-\frac{2x^{3}+5x^{2}-x}{4(1-x)^{3}}-\frac{3x^{3}}{2(1-x)^{4}}\ln{x},\\\\[2.84526pt]
\mathcal{G}^{\gamma}_{2}(x)&=\frac{x^{2}-11x+4}{2(1-x)^{2}}-\frac{3x^{2}}{(1-x)^{3}}\ln{x},\\\\[2.84526pt]
\mathcal{F}_{\gamma}(x)&=\frac{7x^{3}-x^{2}-12x}{12(1-x)^{3}}-\frac{x^{4}-10x^{3}+12x^{2}}{6(1-x)^{4}}\ln{x},\\\\[2.84526pt]
\end{split}$ (84)

$\displaystyle\begin{split}\mathcal{F}_{\rm
box}(x,y)&=\left(4+\frac{xy}{4}\right)I_{2}(x,y,1)-2xyI_{1}(x,y,1),\\\\[2.84526pt]
\mathcal{F}_{\rm
Xbox}(x,y)&=-\left(1+\frac{xy}{4}\right)I_{2}(x,y,1)-2xyI_{1}(x,y,1),\\\\[2.84526pt]
\mathcal{G}_{\rm
box}(x,y,\eta)&=-\sqrt{xy}\left[(4+xy\eta)I_{1}(x,y,\eta)-(1+\eta)I_{2}(x,y,\eta)\right],\\\
\mathcal{G}_{Z}(0,x)&=-\frac{x\ln{x}}{2(1-x)}\,,\\\\[2.84526pt]
\mathcal{F}_{\rm
box}(0,x)&=\frac{4}{1-x}+\frac{4x}{(1-x)^{2}}\ln{x}\,,\\\\[2.84526pt]
\mathcal{F}_{\rm
Xbox}(0,x)&=-\frac{1}{1-x}-\frac{x\ln{x}}{(1-x)^{2}}\,,\\\\[2.84526pt]
\mathcal{F}_{Z}(x)&=-\frac{5x}{2(1-x)}-\frac{5x^{2}}{2(1-x)^{2}}\ln{x}\,,\\\\[2.84526pt]
\end{split}$ (85)

These loop functions involve heavy right-handed neutrino mass $m_{N}$ and
sterile neutrino mass $m_{S}$. In our model these masses $m_{N}$ and $m_{S}$
are proportional to light neutrino mass $m_{\nu}$. Thus the relevant loop
functions needed for various LFV processes can be expressed in terms of
lightest neutrino mass.

##  Appendix C Expression for $\mu\longrightarrow e$ conversion with the
mediation of neutrinos $N$ and $S$

${\rm R}^{A(N,Z)}_{\mu\to e}=\frac{\alpha_{\rm
em}^{3}\alpha_{W}^{4}m_{\mu}^{5}}{16\pi^{2}m_{W_{L}}^{4}\Gamma_{\rm
capt}}\frac{Z_{\rm
eff}^{4}}{Z}\left|\mathcal{F}(-m_{\mu}^{2})\right|^{2}\left(|\mathcal{Q}_{L}^{W}|^{2}+|\mathcal{Q}_{R}^{W}|^{2}\right),$
(86)

where, the relevant terms containing only light-heavy neutrino mixing are
given by

$\displaystyle\mathcal{Q}^{W}_{L}=(2Z+N)\left[\mathcal{W}^{u}_{L}-\frac{2}{3}s_{W}^{2}G^{\gamma}_{R}\right]+(Z+2N)\left[W^{d}_{L}+\frac{1}{3}s_{W}^{2}G^{\gamma}_{R}\right],$
$\displaystyle\mathcal{Q}^{W}_{R}=(2Z+N)\left[\mathcal{W}^{u}_{R}\right]+(Z+2N)\left[W^{d}_{R}\right],$
(87)

and the factor used here are expressed as,

$\displaystyle
W^{u}_{L}=\frac{2}{3}s_{W}^{2}\mathcal{F}^{\gamma}_{L}+\left(-\frac{1}{4}+\frac{2}{3}s_{W}^{2}\right)\mathcal{F}^{Z_{1}}_{L}+\frac{1}{4}\mathcal{B}^{\mu
euu}_{LL},$ $\displaystyle W^{u}_{R}\to 0,$ $\displaystyle
W^{d}_{L}=-\frac{1}{3}s^{2}_{W}\mathcal{F}^{\gamma}_{L}+\left(\frac{1}{4}-\frac{1}{3}s_{W}^{2}\right)\mathcal{F}^{Z_{1}}_{L}+\frac{1}{4}\mathcal{B}^{\mu
edd}_{LL},$ $\displaystyle W^{d}_{R}\to 0\,.$ (88)

$\frac{V^{(p)}}{\sqrt{Z}}=\frac{Z_{eff}^{2}F(-m_{\mu}^{2})\alpha_{\rm
em}^{\frac{3}{2}}}{4\pi}\,,$ (89)

and $V^{(p)}/Z\simeq V^{(n)}/N$. The key box diagram form factors are
expressed as

$\displaystyle\begin{split}\mathcal{B}^{\mu
euu}_{LL}&=\sum_{i=1}^{3}\bigg{\\{}{\mbox{V}^{\nu N}_{\mu
i}}^{*}{\mbox{V}^{\nu N}_{ei}}\left[\mathcal{F}_{\rm
box}(0,x_{i})-\mathcal{F}_{\rm box}(0,0)\right],\\\ \mathcal{B}^{\mu
edd}_{LL}&\simeq\sum_{i=1}^{3}{\mbox{V}^{\nu N}_{\mu i}}^{*}{\mbox{V}^{\nu
N}_{ei}}\left\\{\mathcal{F}_{\rm Xbox}(0,x_{i})-\mathcal{F}_{\rm
Xbox}(0,0)\right.\\\ &\left.+|V_{td}|^{2}\left[\mathcal{F}_{\rm
Xbox}(x_{t},x_{i})-\mathcal{F}_{\rm Xbox}(0,x_{i})-\mathcal{F}_{\rm
Xbox}(0,x_{t})+F_{\rm Xbox}(0,0)\right]\right\\},\\\ \mathcal{B}^{\mu
eqq}_{RR}&=\frac{m_{W_{L}}^{2}}{m_{W_{R}}^{2}}\mathcal{B}^{\mu
eqq}_{LL}(\mbox{V}^{\nu
N}\leftrightarrow{\mbox{V}^{NN}}^{*}\,;\,x^{N}_{i}\leftrightarrow
y^{N}_{i}\,;\,x_{t}\leftrightarrow y_{t})\to 0\,,\end{split}$ (90)

where $x_{t}=m_{t}^{2}/m_{W_{L}}^{2}$ and $y_{t}=m_{t}^{2}/m_{W_{R}}^{2}$.

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