# Low Energy Constraints From Absolute Neutrino Mass Observables and Lepton
Flavor Violation in Left-Right Symmetric Model

Srubabati Goswami<EMAIL_ADDRESS>Theoretical Physics Division, Physical
Research Laboratory, Ahmedabad - 380009, India  Vishnudath K. N
<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research
Laboratory, Ahmedabad - 380009, India

###### Abstract

We have studied the correlations among the three absolute neutrino mass
observables - the effective Majorana mass ($m_{ee}$) which can be obtained
from neutrinoless double beta decay, the electron neutrino mass ($m_{\beta}$)
which is measured in single beta decay experiments and the sum of the light
neutrino masses ($\Sigma$) which is constrained from cosmological
observations, in the context of minimal left-right symmetric model. Two
phenomenologically interesting cases of type-I seesaw dominance as well as
type-II seesaw dominance have been considered. We have taken into account the
independent constraints coming from lepton flavor violation, single $\beta$
decay, cosmology and neutrinoless double beta decay and have determined the
combined allowed parameter space that can be probed in the future experiments.
We have also analyzed the correlations and tensions between the different mass
variables. In addition, the constraints on the masses of the heavy particles
coming from lepton flavor violation and the bounds on three absolute neutrino
mass observables are also determined. We show that these constraints can rule
out some of the parameter space which are not probed by the collider
experiments.

##  I Introduction

The Standard Model (SM) of particle physics, despite being a highly successful
theory in many aspects, has certain limitations, which motivates one to think
of scenarios beyond SM. One of the main limitations is the absence of the
neutrino mass in SM. The observation of neutrino oscillations has shown that
neutrinos can convert from one flavour to another implying non-zero masses.
The global analysis of the neutrino oscillation data from various solar,
atmospheric, reactor and accelerator experiments has determined the values of
two mass squared differences ($\Delta m_{sol}^{2}\sim 7.4\times
10^{-5}~{}eV^{2}$ and $\Delta m_{atm}^{2}\sim 2.5\times 10^{-3}~{}eV^{2}$),
and three neutrino mixing angles ($\theta_{12}\sim 34^{\circ}$,
$\theta_{23}\sim 48^{\circ}$ and $\theta_{13}\sim 7.4^{\circ}$
)Capozzi:2017ipn ; deSalas:2017kay ; Esteban:2018azc .

In addition, we also do not know if neutrinos are Dirac particles or lepton
number violating Majorana particles. This can be probed in the neutrinoless
double beta decay ($0\nu\beta\beta$) experiments, which if observed, would
prove that neutrinos are Majorana particles Furry:1939qr ; Schechter:1980gr .
Also, the $0\nu\beta\beta$ experiments are sensitive to the absolute neutrino
masses through the dependence on the effective Majorana mass $m_{ee}$, unlike
the oscillation experiments which tell us only about the mass squared
differences. The combined constraints from KamLAND-Zen and GERDA experiments
put an upper limit on $m_{ee}$ in the range $0.071-0.161$ eV depending on the
values of the Nuclear Matrix Elements (NMEs) Agostini:2018tnm . Other than
$0\nu\beta\beta$, the non-oscillation data from single $\beta$ decay and the
cosmological observations put independent constraints on the absolute neutrino
masses. The rate of single $\beta$ decay depends on the electron neutrino mass
$m_{\beta}$ and this is constrained to be less than 1.1 eV by the KATRIN
experiment Aker:2019uuj  and the Planck-2018 data constrains the sum of active
light neutrino masses, $\Sigma$ to be less that 0.12 eV Aghanim:2018eyx . It
is important to study the combined constraints coming from these mutually
exclusive experiments and make predictions that can be tested in the future
experiments Fogli:2004as ; Pascoli:2005zb ; Fogli:2008ig ; DiIura:2016zsx ;
Xing:2016ymd ; Caldwell:2017mqu ; Agostini:2017jim .

One of the most popular and elegant way of generating neutrino masses is the
seesaw mechanism. According to this, neutrinos are Majorana particles whose
masses are generated by the dimension-5 Weinberg operator $\kappa
l_{L}l_{L}\Phi\Phi$ weinberg  through electroweak symmetry breaking. Depending
on the way in which the Weinberg operator is generated, there are three types
of seesaw mechanisms : (i) type-I seesaw mediated by heavy right-handed
neutrinos Minkowski:1977sc ; seesaw1 ; seesaw2 ; Mohapatra:1979ia , (ii) type-
II seesaw mediated by heavy scalar triplet Schechter:1980gr ; Schechter:1981cv
; Lazarides:1980nt ; Mohapatra:1980yp  and (iii) type-III seesaw mediated by
heavy fermionic triplets Foot:1988aq . There are many gauge extensions of the
SM that incorporate seesaw mechanism. Among these, the left-right symmetric
models (LRSMs) based on the gauge group $SU(3)_{C}\times SU(2)_{R}\times
SU(2)_{L}\times U(1)_{B-L}$ are of special significance Pati:1974yy ;
Mohapatra:1974gc ; Senjanovic:1975rk ; Mohapatra:1980qe . Here, the origin of
parity violation in SM is understood in a simple way such that the parity
symmetry is restored at a higher energy scale. Also, these models contain
right-handed neutrinos naturally and hence can generate neutrino masses by
type-I seesaw mechanism. Such models can be embedded in grand unified theories
based on the gauge group $SO(10)$ and the presence of the $SU(2)_{R}$ gauge
bosons and additional scalars makes their study interesting from an
experimental point of view. There are two most popular variants of the LRSM
which differ from each other in their Higgs sector. One of them contains two
Higgs doublets in addition to the Higgs bidoublet needed for the SM symmetry
breaking whereas the other variant has Higgs triplets instead of the Higgs
doublets. The model with Higgs triplets can also contribute to neutrino masses
via type-II seesaw mechanism. In addition, the associated charged scalars can
have interesting signatures in the collider experiments Gunion:1995mq ;
Huitu:1996su ; Huitu:1997vh ; Dev:2016dja ; Aaboud:2017qph ; Borah:2018yxd .
In this paper, we consider the LRSM with triplet scalars and from here
onwards, we refer to this model as the minimal left-right symmetric model
(MLRSM).

Another important feature of the MLRSM is that there can be a number of new
physics contributions to $0\nu\beta\beta$, coming from right-handed neutrinos
as well as the Higgs triplets, especially when these particles are at the TeV
scale Hirsch:1996qw ; Tello:2010am ; Chakrabortty:2012mh ; Barry:2013xxa ;
Dev:2014xea ; Awasthi:2015ota ; Bambhaniya:2015ipg ; Bonilla:2016fqd ;
Awasthi:2016kbk ; Deppisch:2017vne ; Borah:2017ldt ; Borgohain:2017akh . These
diagrams correspond to the short-range mechanism, unlike the long range
mechanism due to the standard light neutrino exchange. In such cases, there
can be an enhancement or suppression of the rate of $0\nu\beta\beta$ due to
the interference of different diagrams. One can study various limiting cases
where the new physics contributions can be expressed in terms of a few free
parameters along with the light neutrino masses and the Pontecorvo-Maki-
Nakagawa-Sakata (PMNS) mixing matrix Tello:2010am ; Chakrabortty:2012mh ;
Bambhaniya:2015ipg . The implications of TeV scale MLRSM for $0\nu\beta\beta$
under various limits and approximations have been studied in Tello:2010am ;
Chakrabortty:2012mh ; Barry:2013xxa ; Dev:2013vxa ; Dev:2014xea ;
Awasthi:2015ota ; Bambhaniya:2015ipg ; Awasthi:2016kbk ; Bonilla:2016fqd ;
Deppisch:2017vne ; Borah:2017ldt ; Borgohain:2017akh . In particular, the
authors of Bambhaniya:2015ipg ; Bonilla:2016fqd  have studied the combined
constraints from $0\nu\beta\beta$ and lepton flavour violation (LFV) in the
context of MLRSM by taking into account the scalar triplet contributions. It
has been observed that the LFV bounds put constraints on the absolute neutrino
masses and, it will be interesting to see how these constraints are reflected
in the allowed regions in the planes of the three mass variables,
$m_{\beta},~{}m_{ee}$ and $\Sigma$.

In this paper, we study the correlations among the three observables,
$m_{ee}$, $m_{\beta}$ and $\Sigma$ which depend on the absolute neutrino
masses, in MLRSM, for both type-I as well as type-II seesaw dominant cases.
Even though many such studies have been done in the case of standard three
neutrino picture Fogli:2004as ; Pascoli:2005zb ; Fogli:2008ig ; DiIura:2016zsx
; Xing:2016ymd ; Caldwell:2017mqu ; Agostini:2017jim , no detailed analysis
has been done for MLRSM. In particular, we have taken the independent
constraints coming from LFV, cosmology and $0\nu\beta\beta$ decay into account
and obtained the allowed parameter spaces that can be probed in the future
experiments. In addition, we study how the LFV constraints and the bounds on
$m_{ee}$, $m_{\beta}$ and $\Sigma$ restrict the masses of the heavy neutrinos,
the charged triplet scalar as well as the $W_{R}$ gauge boson. In fact, we
find that the LFV constraints along with the bounds on absolute neutrino mass
observables can rule out some of the parameter space which are not probed by
the collider experiments, thus providing a complimentary way to study the new
physics parameters.

The rest of the paper is organized as follows. In section-II, we describe the
MLRSM, it’s particle content and the mechanism of neutrino mass generation.
The rates of the LFV decays in this model and the constraints coming from them
are discussed in section-III. In section-IV, the predictions for
$0\nu\beta\beta$ in MLRSM are discussed and in section-V, we have discussed
the correlations among $m_{ee}$, $m_{\beta}$ and $\Sigma$. Constrains on the
new physics parameters from LFV and neutrino mass observables are discussed in
section-VI. Finally, we summarize our findings in section-VII.

##  II Minimal left-right symmetric model

The MLRSM is based on the gauge group $SU(3)_{C}\times SU(2)_{R}\times
SU(2)_{L}\times U(1)_{B-L}$. The matter multiplets for quarks and leptons in
this model are given by,

$Q_{L}\,\,\sim\,\,(2,1,\frac{1}{3})\,\,\,\,;\,\,Q_{R}\,\,\sim\,\,(1,2,\frac{1}{3}),$
$l_{L}\,\,\sim\,\,(2,1,-1)\,\,\,\,;\,\,l_{R}\,\,\sim\,\,(1,2,-1).$

Thus, the existence of right handed neutrinos as part of the $SU(2)_{R}$
doublet $l_{R}$ implies that the active light neutrino masses can be generated
via a type-I seesaw mechanism. The Higgs Sector consists of,

$\Phi\,\,\sim\,\,(2,2,0)\,\,\,\,;\,\,\,\,\Delta_{R}\,\,\sim\,\,(1,3,2)\,\,\,\,;\,\,\,\,\Delta_{L}\,\,\sim\,\,(3,1,2).$

Here, $\Delta_{R}$ is the $SU(2)_{R}$ triplet that breaks the $SU(2)_{R}\times
SU(2)_{L}\times U(1)_{B-L}$ gauge group to the SM gauge group $SU(2)_{L}\times
U(1)_{Y}$. The bidoublet $\Phi$ is responsible for breaking the SM symmetry
group to $U(1)_{em}$. The presence of the $SU(2)_{L}$ triplet Higgs is
required by left-right symmetry and it gives rise to type-II seesaw mechanism.
Thus, in general, MLRSM can have type- I+II seesaw mechanism for neutrino mass
generation. The Higgs fields acquire vevs as,

$\langle\Phi\rangle\,=\,\begin{pmatrix}v&0\\\
0&v^{\prime}\end{pmatrix}\,\,\,;\,\,\,\langle\Delta_{L}\rangle\,=\,v_{L}\,\,\,;\,\,\,\langle\Delta_{R}\rangle\,=\,v_{R}.$
(1)

The part of the Lagrangian that is relevant for the generation of neutrino
mass is,

$L_{l}=f_{L}\,\bar{l}_{L}^{c}\,\Delta_{L}\,l_{L}\,+\,f_{R}\,\bar{l}_{R}^{c}\,\Delta_{R}\,l_{R}\,\,+\,\bar{l}_{R}(Y_{D}\Phi\,+\,Y_{L}\tilde{\Phi})l_{L}\,\,+\textrm{h.c.}.$
(2)

Note that here, the generation indices have been suppressed and that $Y_{D}$,
$f_{L}$ and $f_{R}$ are $3\times 3$ matrices. Depending upon whether we take
the symmetry between the left and right sectors to be parity or charge
conjugation, the Yukawa coupling matrices for the left- and the right-handed
leptons with the corresponding scalar triplets will be related as,

$f_{L}=f_{R}\,\,\,\textrm{or}\,\,\,f_{L}=f_{R}^{*},$ (3)

respectively Maiezza:2010ic . In our work, we take parity as the symmetry
between left and right sectors. Once the Higgs fields acquire vevs, we will
get the neutrino mass matrix in the basis $(\nu_{L}^{c}\,,\,N_{R})$ as,

$\begin{pmatrix}f_{L}v_{L}&Y_{D}v\\\ Y_{D}^{T}v&f_{R}v_{R}\end{pmatrix}.$ (4)

The neutrino mass is generated through seesaw mechanism $(f_{R}v_{R}>>Y_{D}v)$
and we get light and heavy neutrino mass matrices as,

$m_{\nu}=f_{L}v_{L}-\frac{v^{2}}{v_{R}}Y_{D}^{T}\,f_{R}^{-1}Y_{D}$ (5)

and

$\,\,M_{R}\,\,=\,\,f_{R}v_{R}$ (6)

respectively. Now we can consider two interesting limits Tello:2010am ;
Chakrabortty:2012mh ; Bambhaniya:2015ipg  :

### Type-I Dominance

: In this case, $f_{L}v_{L}<<\frac{v^{2}}{v_{R}}Y_{D}^{T}\,f_{R}^{-1}Y_{D}$
and hence, the light neutrino mass matrix is given as,

$m_{\nu}=-\frac{v^{2}}{v_{R}}Y_{D}^{T}\,f_{R}^{-1}Y_{D}.$ (7)

Assuming $Y_{D}\propto I$, where $I$ is the $3\times 3$ identity matrix, one
can see that the unitary matrices diagonalizing the light and the heavy
neutrino mass matrices are related as,

$V=U^{*},$ (8)

where $V$ is the unitary matrix diagonalizing the heavy neutrino mass matrix
$M_{R}$ and $U$ is the UPMNS matrix that diagonalizes $m_{\nu}$. Also, the
heavy neutrino masses are inversely proportional to the light neutrino masses,
i.e.,

$M_{N_{i}}\propto\frac{1}{m_{i}}.$ (9)

### Type-II Dominance

: In this case, $f_{L}v_{L}>>\frac{v^{2}}{v_{R}}Y_{D}^{T}\,f_{R}^{-1}Y_{D}$
and hence, the light neutrino mass matrix is given as,

$m_{\nu}=f_{L}v_{L}=f_{R}v_{L},$ (10)

where the last equality follows from left-right symmetry. Thus, the light and
the heavy neutrino mass matrices are diagonalized by the same unitary matrix,
i.e.,

$V=U.$ (11)

But now, the heavy neutrino masses are directly proportional to the light
neutrino masses, i.e.,

$M_{N_{i}}\propto m_{i}.$ (12)

In our work, we denote the mass of the heaviest right handed neutrino as
$M_{N}$.

Another assumption that we will be making is in the scalar sector. In scalar
sector, there are 20 real degrees of freedom (8 from the bidoublet, 6 from
$\Delta_{L}$ and 6 from $\Delta_{R}$), out of which, 6 will become Goldstone
bosons and one will be the physical SM Higgs particle with mas $\sim 125$ GeV.
The remaining physical scalars are the doublets
$H_{\phi}^{0},A_{\phi}^{0},H_{\phi}^{\pm}$, the left triplets
$H_{L}^{0},A_{L}^{0},\Delta_{L}^{\pm},\Delta_{L}^{\pm\pm}$ and the right
triplets $H_{R}^{0},A_{R}^{0},\Delta_{R}^{\pm},\Delta_{R}^{\pm\pm}$. All of
these particles are heavy and we assume that the masses of the doubly charged
scalars to be the same in left and right sectors. We define the parameter
$M_{\Delta}$ in terms of the masses of the two doubly charged scalars,

$\frac{1}{M_{\Delta}^{2}}=\frac{1}{{M_{\Delta_{L}^{\pm\pm}}}^{2}}+\frac{1}{{M_{\Delta_{R}^{\pm\pm}}}^{2}},$
(13)

and take $r=M_{N}/M_{\Delta}$, where $r$ is a fixed proportionality constant.
Thus we have,

$M_{\Delta_{R}^{++}}=\sqrt{2}M_{N}/r.$ (14)

##  III Constraints from Lepton Flavour Violation

The presence of the $SU(2)_{R}$ gauge bosons, triplet scalars and right handed
neutrinos can give rise to observable rates of charged LFV decays that can be
tested in future experiments. Ignoring the light-heavy mixing in the neutrino
sector as well as the left-right gauge boson mixing, the branching ratio for
the LFV decay $\mu\rightarrow e\gamma$ in MLRSM is given as Cirigliano:2004mv
; Barry:2013xxa ,

$BR(\mu\rightarrow
e\gamma)=\frac{3\alpha_{em}}{2\pi}\Big{(}|G_{L}^{\gamma}|^{2}+|G_{R}^{\gamma}|^{2}\Big{)}.$
(15)

In the above equation,

$G_{R}^{\gamma}=V_{\mu
i}V_{ei}^{*}~{}\Big{(}~{}\frac{M_{W_{L}}^{2}}{M_{W_{R}}^{2}}G_{1}^{\gamma}(b_{i})+\frac{2b_{i}}{3}\frac{M_{W_{L}}^{2}}{{M_{\Delta_{R}^{++}}}^{2}}~{}\Big{)}$
(16)

and,

$G_{L}^{\gamma}=V_{\mu
i}V_{ei}^{*}~{}b_{i}\Big{(}~{}\frac{2}{3}\frac{M_{W_{L}}^{2}}{{M_{\Delta_{L}^{++}}}^{2}}+\frac{1}{12}\frac{M_{W_{L}}^{2}}{{M_{\Delta_{L}^{+}}}^{2}}~{}\Big{)},$
(17)

where $a_{i}=\Big{(}\frac{M_{N_{i}}}{M_{W_{L}}}\Big{)}^{2}$,
$b_{i}=\Big{(}\frac{M_{N_{i}}}{M_{W_{R}}}\Big{)}^{2}$, $\alpha_{em}\approx
1/137$, $M_{W_{L}}$ and $M_{W_{R}}$ are the masses of left- and right- gauge
bosons respectively, $V$ is the unitary matrix that diagonalizes the right-
handed neutrino mass matrix, $M_{N_{i}}$ are the masses of heavy right-handed
neutrinos, and,

$G_{1}^{\gamma}(a)=\frac{-2a^{3}+5a^{2}-a}{4(1-a)^{3}}-\frac{3a^{3}}{2(1-a)^{4}}~{}\textrm{ln}a.$
(18)

The current upper bound on the branching ratio for $(\mu\rightarrow e\gamma)$
is TheMEG:2016wtm ,

$BR(\mu\,\rightarrow e\,\gamma)\,\,<\,4.2\times 10^{-13}.$ (19)

The upper bound on the branching ratio for the decay $\mu\rightarrow 3e$ also
puts very strong constraints on the masses of the heavy particles in low scale
seesaw models asBertl:1985mw ,

$\textrm{Br}(\mu\,\rightarrow 3e)\,\,<\,1.0\times 10^{-12}.$ (20)

The expression for the branching ratio of $\mu\rightarrow 3e$ in MLRSM is
given as Cirigliano:2004mv ; Leontaris:1985qc ; Swartz:1989qz ;
Cirigliano:2004tc ,

$BR(\mu\rightarrow 3e)=\frac{1}{2}|h_{\mu
e}h_{ee}^{*}|^{2}\Big{(}\frac{M_{W_{L}}^{4}}{{M_{\Delta_{L}^{++}}}^{4}}+\frac{M_{W_{L}}^{4}}{{M_{\Delta_{R}^{++}}}^{4}}\Big{)},$
(21)

where,

$h_{\alpha\beta}=\sum_{i=1}^{3}V_{\alpha i}V_{\beta
i}\frac{M_{N_{i}}}{M_{W_{R}}}.$ (22)

##  IV $0\nu\beta\beta$ in MLRSM

There are several extra contributions to $0\nu\beta\beta$ in MLRSM, coming
from right-handed currents, left-right mixing and scalar triplets. All these
will give extra contributions to the effective Majorana mass which can be
directly probed in $0\nu\beta\beta$ experiments. Assuming that there are no
large light-heavy neutrino as well as left-right gauge boson mixings, the half
life for $0\nu\beta\beta$ is given as Tello:2010am ,

$\frac{1}{T_{1/2}^{0\nu}}=G_{01}^{0\nu}\Big{(}|M_{\nu}^{0\nu}\eta_{\nu}|^{2}+|M_{N}^{0\nu}\eta_{R}|^{2}\Big{)}.$
(23)

Here, $M_{\nu}^{0\nu}$ and $M_{N}^{0\nu}$ are the nuclear matrix elements for
the light- and heavy-neutrino exchanges respectively and $G_{01}^{0\nu}$ is
the phase space factor. $\eta_{\nu}$ and $\eta_{R}$ represent the left-handed
and right-handed amplitudes respectively, and are given as,

$\eta_{\nu}=\frac{1}{m_{e}}\sum_{i}U_{ei}^{2}m_{i}\,\,\,\,;\,\,\,\,\eta_{R}=m_{p}\Big{(}\frac{M_{W_{L}}}{M_{W_{R}}}\Big{)}^{4}\Big{(}\sum_{i}\frac{V^{2}_{ei}}{M_{N_{i}}}+\sum_{i}\frac{V^{2}_{ei}M_{N_{i}}}{M_{\Delta_{R}^{++}}^{2}}\Big{)}.$
(24)

Here, $m_{e}$ is the electron mass and $m_{p}$ is the proton mass. The
corresponding effective Majorana neutrino mass is given as,

$m_{ee}=\sqrt{\Big{|}\sum_{i}U_{ei}^{2}m_{i}\Big{|}^{2}+\Big{|}\langle
p^{2}\rangle\Big{(}\frac{M_{W_{L}}}{M_{W_{R}}}\Big{)}^{4}\Big{(}\sum_{i}\frac{V^{2}_{ei}}{M_{N_{i}}}+\sum_{i}\frac{V^{2}_{ei}M_{N_{i}}}{M_{\Delta_{R}^{++}}^{2}}\Big{)}\Big{|}^{2}}.$
(25)

Here, $\,\,\langle p^{2}\rangle=|m_{e}m_{p}M_{N}^{0\nu}/M_{\nu}^{0\nu}|$
encapsulates the contribution due to the nuclear matrix element. Its value is
$\sim(153-184~{}\textrm{MeV})^{2}$ for ${}^{76}Ge$ isotope Meroni:2012qf .
Note that we have not included the contribution from the $W_{L}-W_{R}$ mixing
diagram in the above expression. Recently, in reference Li:2020flq , the
authors have emphasized the importance of long-range pion exchange diagrams.
Then, the contribution from $W_{L}-W_{R}$ mixing can be significant and can
dominate over all the other contributions to $0\nu\beta\beta$. In particular,
the authors of reference Li:2020flq  considered charge conjugation as the
left-right symmetry in which case, the $W_{L}-W_{R}$ mixing (which is
proportional to $\textrm{sin}2\beta$ where $\textrm{tan}\beta=v^{\prime}/v$)
can be large. In our case, we consider parity as the left-right symmetry for
which $\textrm{sin}2\beta$ is restricted Senjanovic:2014pva ;
Senjanovic:2015yea  and the $W_{L}-W_{R}$ mixing diagram can be neglected when
$\beta$ is very small.

---
Figure 1: Predictions for $m_{ee}$ in MLRSM with type-I dominance for
$M_{N}=500$ GeV. The left (right) panel is for NH (IH) and the upper (lower)
panel is for $r=0.01$ ($r=0.5$). We have fixed $M_{W_{R}}=7$ TeV. The Green
region is without including the LFV constraints and the orange region
satisfies the LFV constraints. The blue dashed lines correspond to the
predictions in SM with three light Majorana neutrinos. The yellow region is
disfavored by Planck-2018 Aghanim:2018eyx  and the region above the brown band
is disfavored by the combined constraints from KamLAND-Zen and GERDA
experiments Agostini:2018tnm . The horizontal dotted black line corresponds to
the future sensitivity of nEXO Kharusi:2018eqi  and the solid black line
corresponds to the future sensitivity of LEGEND-1k Abgrall:2017syy ;
Agostini:2017jim .
---
Figure 2: Same as Fig.1, but for type-II dominance.

In Fig.1, we have shown the predictions for $m_{ee}$ in MLRSM with type-I
dominance. The left and the right panels are for normal hierarchy (NH) and
inverted hierarchy (IH) of the light neutrino masses respectively, and the
upper and the lower panels are for $r=0.01$ and $r=0.5$ respectively. In all
the plots in this paper, we have kept $M_{W_{R}}=7$ TeV and $M_{N}=500$ GeV
unless otherwise mentioned. This implies that the upper and the lower panels
are for $M_{\Delta_{R}^{++}}\sim 70$ TeV and $M_{\Delta_{R}^{++}}\sim 1.4$ TeV
respectively. The ATLAS search for same- sign dileptonic new physics signals
has set lower bounds on the masses of $\Delta_{L}^{++}$ and
$\Delta_{R}^{++}$Aaboud:2017qph . These are given as
$M_{\Delta_{L}^{++}}>770-870$ GeV and $M_{\Delta_{R}^{++}}>660-760$ GeV,
assuming $BR(\Delta^{\pm\pm}\rightarrow l^{\pm}l^{\pm})=100\%$. These chosen
values are also allowed by the constraints from perturbativity as well as
collider experimens as given in reference Maiezza:2016ybz ; Chauhan:2018uuy .
We have varied the light neutrino mass squared differences and the mixing
angles in their $3\sigma$ ranges Esteban:2018azc  and the Majorana phases have
been varied in the range $0-\pi$ for all the plots in this paper. The yellow
region is disfavored by Planck-2018 data ($\Sigma=m_{1}+m_{2}+m_{3}<0.12$ eV)
Aghanim:2018eyx  and the region above the brown band ($0.071-0.161$ eV) is
disfavored by the combined constraints from KamLAND-Zen and GERDA experiments
Agostini:2018tnm . This is a band because of the NME uncertainty Engel:2016xgb
; Agostini:2018tnm ; Kotila:2012zza .

The green regions in Fig.1 are the predictions for the MLRSM with out
including LFV bounds whereas the orange regions satisfy the LFV bounds. The
blue dashed lines give the predictions for the SM case with three light
Majorana neutrinos. First let us consider the case of NH with type-I dominance
(left panels). In the upper panel, the orange and the green regions almost
overlap implying that there are no additional constraints from LFV. However,
for $r=0.5$, values of $m_{1}<0.0045$ eV are disfavored by LFV constraints and
only a narrow region above is allowed by both LFV and $0\nu\beta\beta$ since
the Majorana phases corresponding to the region below the orange band give
large values of LFV decay rates Bambhaniya:2015ipg . A part of the parameter
space is also disfavored by the Planck data. We can see that the cancellation
region for NH in MLRSM with type-I dominance gets shifted slightly to the low
$m_{1}$ region as compared to the standard three neutrino scenario. Also, for
large values of $m_{1}$ (the quasi-degenerate region), the predictions for
$m_{ee}$ in both the cases are same whereas for smaller values of $m_{1}$,
MLRSM with type-I dominance gives higher values as compared to the three
neutrino picture Chakrabortty:2012mh ; Bambhaniya:2015ipg .

Now let us consider IH with type-I dominance (right panel of Fig.1). Here
also, the orange region completely overlaps with the green region for $r=0.01$
and hence all the points are allowed by LFV. But for $r=0.5$, only a narrow
region above is allowed by LFV for $m_{3}>0.04$ eV whereas all the points with
$m_{3}<0.04$ eV are allowed. The region $m_{3}>0.04$ eV is also disfavored by
Planck-2018 data. We can also see that there is no cancellation region as in
the standard three neutrino picture Chakrabortty:2012mh ; Bambhaniya:2015ipg .
In addition, we can see that the predictions for $m_{ee}$ in both the cases
(MLRSM and standard three generation) are same for large values of $m_{3}$
whereas for smaller values of $m_{3}$, MLRSM gives much higher values than
that in the three neutrino picture, a part of which is disfavored by the bound
from KamLand-Zen and GERDA. Thus for type-I dominance with larger values of
$r$ ($r=0.5$), lower values of $m_{lightest}$ are constrained by LFV in the
case of NH, whereas, higher values of $m_{lightest}$ are disfavored by LFV in
the case of IH.

The horizontal dotted black line in Fig.1 corresponds to the future
sensitivity of nEXO, which is a $0\nu\beta\beta$ experiment using
${}^{136}Xe$. The future $3\sigma$ sensitivity of the nEXO is
$T_{1/2}=5.7\times 10^{27}$ years Kharusi:2018eqi . This can be converted into
a band in $m_{ee}$ by including the NME uncertainties whereas we have given
the lowest value in this band ($m_{ee}=0.007$ eV) by using the highest value
of NME. Similarly, the horizontal solid black line corresponds to the future
sensitivity of LEGEND-1k Abgrall:2017syy ; Agostini:2017jim . LEGEND-1k is a
$0\nu\beta\beta$ experiment using ${}^{76}Ge$ and has a $3\sigma$ discovery
sensitivity of $T_{1/2}=4.5\times 10^{27}$ years. This translates into a band
in $m_{ee}$ with lowest value as $0.017$ eV. From the figure, we can see that
if the LGEND-1k does not observe a signal in the next run, it will rule out IH
completely for both the considered values of $r$ (or $M_{\Delta_{R}^{++}}$).
This is true for MLRSM as well as the standard three neutrino picture. Also,
LEGEND-1k and nEXO can rule out some part of the low $m_{1}$ region for MLRSM
with $r=0.01$ if no signal is observed.

In Fig.2, we have shown similar plots as in Fig.1, but for type-II dominance.
The interesting feature to note here is the absence of cancellation region for
NH (left panel). Here also, the green and the orange regions overlap
completely for $r=0.01$ (which is true for IH as well, as can be seen from the
right panel). For $r=0.5$, the values of $m_{ee}$ in the green region with
$m_{1}>0.048$ eV are constrained by LFV unlike in the type-I dominance case
where lower values of $m_{1}$ are disfavored for NH. This is also disfavored
by Planck 2018 Aghanim:2018eyx . On the other hand, the predictions for IH is
almost the same as in the three neutrino picture. This is because we have
taken $M_{W_{R}}=7$ TeV. For smaller values of $M_{W_{R}}$, the predictions
for type-II IH would have been different from the standard three generation
case Tello:2010am ; Chakrabortty:2012mh ; Bambhaniya:2015ipg . For IH with
$r=0.5$, the values of predicted $m_{ee}$ in the green region are disfavored
by LFV through the entire range of $m_{3}$ since the corresponding values of
Majorana phases are disfavored. From the left panel of Fig.2, we can see that
a part of the parameter space for low $m_{1}$ is already disfavored by the
current constraints on $0\nu\beta\beta$ for NH in MLRSM. Also, IH with $r=0.5$
will be disallowed if LEGEND-1k fails to observe a positive signal and NH with
$r=0.5$ will be be disallowed if nEXO fails to observe a positive signal.

##  V Correlations of $m_{ee}$ to $m_{\beta}$ and $\Sigma$

---
Figure 3: Correlation of $m_{\beta}$ against $\Sigma$. The left (right) panel
is for NH (IH). The yellow region is disfavored by Planck-2018 Aghanim:2018eyx
. The black solid and dotted lines correspond to the future sensitivity of
KATRIN Aker:2019uuj  and Project 8 Esfahani:2017dmu  respectively.
---
Figure 4: Lightest neutrino mass (Red), $m_{\beta}$ (blue) and $\Sigma$
(Magenta) plotted against the lightest neutrino mass. The left (right) panel
is for NH (IH).
---
Figure 5: Correlations of $m_{ee}$ against $m_{\beta}$ and $\Sigma$ for type-I
dominance with $M_{N}=500$ GeV, $r=0.01$ ($M_{\Delta_{R}^{++}}\sim 70$ TeV)
and $M_{W_{R}}=7$ TeV. The Green region is without demanding the LFV
constraints which is completely covered by the the orange region for which LFV
constraints are satisfied. The blue dashed lines correspond to the predictions
in standard three neutrino picture. The yellow region in the upper panels and
the region shaded with gray dashes in the lower panels are disfavored by
Planck-2018 and the region above the brown band is disfavored by the
constraints from KamLAND-Zen and GERDA Agostini:2018tnm . The purple region
corresponds to the future sensitivity of KATRIN Aker:2019uuj  and the vertical
dashed black line corresponds to the future sensitivity of Project 8
Esfahani:2017dmu . The horizontal dotted and solid black lines correspond to
the future sensitivities of nEXO Kharusi:2018eqi  and LEGEND-1k
Abgrall:2017syy ; Agostini:2017jim  respectively.
---
Figure 6: Same as Fig.5, but for type-I dominance with $r=0.5$
($M_{\Delta_{R}^{++}}\sim 1.4$ TeV).
---
Figure 7: Same as Fig.5, but for type-II dominance with $r=0.01$
($M_{\Delta_{R}^{++}}\sim 70$ TeV).
---
Figure 8: Same as Fig.5, but for type-II dominance with $r=0.5$
($M_{\Delta_{R}^{++}}\sim 1.4$ TeV).

As it was already mentioned, in addition to $0\nu\beta\beta$, the non-
oscillation data from single $\beta$ decay and cosmology put independent
constraints on the absolute neutrino masses. The single $\beta$ decay is
sensitive to the electron neutrino mass, $m_{\beta}$, which is defined as,

$m_{\beta}=\sqrt{\sum_{i}|U_{ei}^{2}m_{i}^{2}|}.$ (26)

The most stringent limit on $m_{\beta}$ has been obtained recently by KATRIN
(Karlsruhe Tritium Neutrino Experiment) according to which Aker:2019uuj ,

$m_{\beta}<1.1~{}\textrm{eV}.$ (27)

In future, KATRIN is expected to probe values of $m_{\beta}$ as low as 0.2 eV.
The Project 8 is another experiment which is under construction and it aims to
achieve a sensitivity of $m_{\beta}\sim 0.04$ eV Esfahani:2017dmu .

The Planck 2018 results put very stringent limit on the maximum value of the
sum of active light neutrino masses as Aghanim:2018eyx ,

$\Sigma=m_{1}+m_{2}+m_{3}<0.12~{}\textrm{eV}.$ (28)

The value of $m_{\beta}$ in the MLRSM that we are considering will be the same
as in the standard three neutrino picture. This is because, we have considered
the mixing between the $W_{L}$ and $W_{R}$ gauge bosons to be negligible. In
Fig.3, we have plotted the correlation of $m_{\beta}$ against $\Sigma$. The
left and the right panels are for NH and IH respectively. The yellow region is
disfavored by Planck-2018 and the black solid and dotted lines correspond to
the future sensitivity of KATRIN and Project 8 respectively. As already
pointed out, these plots are independent of any other model parameters other
than the the UPMNS mixing matrix and light neutrino masses. From this figure,
we can see that the region of parameter space that KATRIN is going to probe is
in strong tension with cosmology. On the other hand, the Project 8 experiment
will have the potential to probe the remaining IH region that is allowed by
cosmology, thus enabling it to distinguish between NH and IH. Any measurement
of $m_{\beta}$ outside the range shown in these figures will definitely be a
signal of some other new physics.

In Fig.4, we have plotted the lightest neutrino mass (Red), $m_{\beta}$ (blue)
and $\Sigma$ (Magenta) against the lightest neutrino mass. The left panel is
for NH and the right panel is for IH. From this figure, we can see that the
values of the lightest neutrino mass and $m_{\beta}$ become almost same for
$m_{1}\geq 0.02$ eV for NH and $m_{3}\geq 0.1$ eV for IH. We can also see that
there exists a predicted minimum value for both $m_{\beta}$ as well as
$\Sigma$ from Fig.4. For NH, $m_{\beta~{}min}\sim$ 0.009 eV,
$\Sigma_{min}\sim$ 0.06 eV and for IH, $m_{\beta~{}min}\sim$ 0.05 eV and
$\Sigma_{min}\sim$ 0.1 eV.

It is also very instructive to plot the allowed intervals of $m_{ee}$ as a
function of $m_{\beta}$ and $\Sigma$. In Fig.5, we have plotted these for
type-I dominance with $r=0.01$ ($M_{\Delta_{R}^{++}}\sim 70$ TeV since $M_{N}$
= 500 GeV). The left and the right panels are for NH and IH respectively. The
yellow region in the upper panel and the region shaded with gray dashes in the
lower panel are disfavored by Planck-2018 and the region above the brown band
is disfavored by the combined constraints from KamLAND-Zen and GERDA
experiments. The horizontal black dotted and solid lines correspond to the
future sensitivity of nEXO and LEGEND-1k respectively. The purple region
corresponds to the future sensitivity of KATRIN and the vertical dashed black
line corresponds to the future sensitivity of Project 8 Esfahani:2017dmu . The
blue dashed lines give the predictions for SM with three Majorana neutrinos.
The orange region are the predictions for $m_{ee}$ in MLRSM with type-I
dominance after including the LFV constraints. We have seen in the previous
section that the LFV bounds do not put extra constraints for $r=0.01$. From
this figure, the values of $m_{\beta}$ and $\Sigma$ in MLRSM corresponding to
the cancellation region for NH (left panel) are around $\sim 0.01$ eV and
$\sim 0.06$ eV respectively and this is the same as the standard three
generation case. The major difference in the predictions for MLRSM with that
of the three generation case is the presence of an enhancement in $m_{ee}$
values for $m_{\beta}\sim 0.01$ eV and $\Sigma\sim 0.06$ eV. This new region
is allowed even after imposing the cosmological constraints and can be probed
in LEEGEND-1k and nEXO. There is an enhancement in the predicted values of
$m_{ee}$ in the case of IH as well (right panel) for $m_{\beta}\sim 0.07$ eV
and $\Sigma\sim 0.1$ eV, though here, most of the enhancement region is
disfavored by the current $0\nu\beta\beta$ bounds. Also, for IH, only a very
narrow region with $m_{\beta}\sim 0.044-0.052$ eV is allowed after imposing
the constraints from cosmology. The earlier mentioned tension between the
region KATRIN is going to probe and the constraints from cosmology can be seen
here as well. This is expected since the expressions for $m_{\beta}$ and
$\Sigma$ are the same in MLRSM as well as the standard three neutrino picture.
All the allowed parameter space for IH can be probed completely by LEGEND-1k
and Project 8 experiments.

In Fig.6, we have shown similar plots for type-I dominance with $r=0.5$
($M_{\Delta_{R}^{++}}\sim 1.4$ TeV). Here also, the green regions show the
predictions without including the LFV constraints whereas the orange region is
after including the LFV constraints. As seen earlier, the contribution of the
triplet scalar to LFV becomes important in this case and a major portion of
the predicted values of $m_{ee}$ are disallowed. In particular, the
cancellation region as well as the enhancement region for NH are disfavored by
LFV constraints and a very narrow region remains allowed, as can be seen from
the left panel of Fig.6. In the case of IH (right panel of Fig.6), the
enhancement region is allowed by LFV, but most of it is disfavored by Kam-
LAND-Zen + GERDA as in the previous figure. Also, since the regions that are
disfavored by LFV in IH are disfavored by cosmology as well, the net allowed
parameter space for type-I IH is the same for $r=0.01$ and $r=0.5$. In this
case, the small allowed parameter region (orange region) for NH can be probed
by nEXO whereas that for IH can be probed by LEGEND-1k and Project 8.

Figs.7 and 8 show the correlations of $m_{ee}$ against $\Sigma$ and
$m_{\beta}$ for $r=0.01$ and $r=0.5$ respectively in the case of type-II
dominance. Here, we can see that there is no region with complete cancellation
for $m_{ee}$ as we have already seen in the previous section. Also, there is
considerable enhancement in the predicted values of $m_{ee}$ for NH, part of
which is disfavored by $0\nu\beta\beta$ constraints. In addition, here also,
the regions that are disfavored by LFV are disfavored by cosmology as well,
making the allowed parameter space for type-II NH the same for $r=0.01$ and
$r=0.5$. For IH, the predictions overlap almost completely with that for the
standard three neutrino picture. Thus, for $r=0.01$ (Fig.7), since there are
no additional constraints from LFV, the allowed parameter space in the
$m_{\beta}-m_{ee}$ plane is almost the same for IH in type-II LR as well as
three neutrino picture. On the other hand, for $r=0.5$ (Fig.8), the LFV
constraints are severe in MLRSM and only a very small region around the point
($m_{\beta}\sim 0.045$, $m_{ee}\sim 0.047$) is allowed. For $r=0.01$, even
though most of the $m_{ee}$ parameter space for NH can be probed by nEXO,
there is still a small region which is beyond the reach of nEXO. For IH with
$r=0.01$ (Fig.7) as well as $r=0.5$ (Fig.8), all the allowed parameter space
for IH is well within the reach of LEGEND-1k and Project 8.

##  VI Constraining New Physics Parameters From LFV and Neutrino Mass
Observables

In the previous section, we discussed how the constraints from LFV affect the
predictions for correlations among the absolute neutrino mass observables. In
this section, we will see how the LFV constraints as well as the bounds on the
values of the three neutrino mass observables restrict the allowed values of
new physics parameters. Specifically, we will study the bounds on the heavy
neutrino mass $M_{N}$, the parameter $r$ (or equivalently, the mass of the
scalar $M_{\Delta_{R}}^{++}$) and the mass of the $SU(2)_{R}$ $W$ gauge boson,
$M_{W_{R}}$. This also enables us to see the impact of varying $r$ and $M_{N}$
on the absolute neutrino mass observables. For all the plots in this section,
oscillation parameters are varied in the $3\sigma$ ranges, Majorana phases are
varied in the range $0-\pi$,$m_{lightest}$ is varied in the range
$10^{-4}-0.1$ eV and $r$ is varied in the range $0-2$. Also, the upper and the
lower panels are for type-I and type-II seesaw dominances, and the left and
the right panels are for NH and IH, respectively in all the figures following.

---
Figure 9: The correlation of effective Majorana mass $m_{ee}$ against $r$. The
upper (lower) panel is for type-I (type-II) dominance and the left (right)
panel is for NH(IH). We have fixed $M_{W_{R}}=7$ TeV. The pink points satisfy
the LFV constraints whereas the blue points satisfy LFV constraints as well as
the current bounds on $m_{ee}$, $m_{\beta}$ and $\Sigma$. Oscillation
parameters are varied in the $3\sigma$ ranges, Majorana phases are varied in
the range $0-\pi$, $M_{N}$ is varied in the range $0.1-30$ TeV and
$m_{lightest}$ is varied in the range $10^{-4}-0.1$ eV. The horizontal black
line indicates the bound on $m_{ee}$ ($m_{ee}<0.161$ eV).

In Fig.9, we have plotted the correlation of the effective Majorana mass
$m_{ee}$ against $r$. Here, $M_{N}$ is varied in the range $0.1-30$ TeV. The
pink points satisfy the LFV constraints whereas the blue points satisfy LFV
constraints as well as the current bounds on $m_{ee}$, $m_{\beta}$ and
$\Sigma$. The bound on $m_{ee}$ comes from KamLAND-Zen and GERDA and is taken
as $m_{ee}<0.161$ eV (horizontal black line), which is the most liberal upper
value corresponding to the lowest value of nuclear matrix elements. From this
figure, we can see that there is an upper bound on $r$ (or correspondingly, a
lower bound on $M_{\Delta_{R}^{++}}$) due to the constraints that we have
imposed. These bounds for all the four cases are given in Table 1. Also, a
very important result which is of special mention is that the bounds on
$M_{\Delta_{R}^{++}}$ given in table-1 is more stringent than the collider
bounds ( $M_{\Delta_{R}^{++}}>660-760$ GeV) Aaboud:2017qph  in the regime
$M_{N}>760$ GeV for all the four cases.

We have seen earlier that for small $r$, LFV constraints are not that
important and the parameter space allowed by $0\nu\beta\beta$ is allowed by
LFV. In fact, from Fig.9, we can see that there are some parameters for small
r which are allowed by LFV but that gives 0nubb to be more than the current
upper bound. In the case of type-I NH, cancellation region as well as
predictions for $m_{ee}\sim 0.1$ eV are present only for the lower values of
$r$. Specifically, for $r>0.26$, ${m_{ee}}_{max}\sim 0.03$ eV and for
$r>0.45$, ${m_{ee}}_{min}\sim 0.006$ eV. The pink points below the black line
which are allowed by both LFV as well as $0\nu\beta\beta$ bounds are in fact
disfavored by the bound on $\Sigma$ since those large values of $m_{ee}$
correspond to the quasi-degenerate region with $\Sigma>0.12$ eV. For type-I
IH, all the allowed values of $r$ can give predictions for $m_{ee}$ in the
range $\sim 0.015-0.161$ eV. In type-II dominant case with NH, the predictions
for $m_{ee}$ can take values in the range $2\times 10^{-4}-0.161$ eV,
depending upon the values of other parameters. Note that in Fig.2, we saw that
the minimum predicted value for $m_{ee}$ when $M_{N}=500$ GeV for type-II NH
is $\sim 2\times 10^{-3}$ whereas in Fig.9, we can see that the predictions
for $m_{ee}$ in type-II NH can be as low as $2\times 10^{-4}$. This is because
we have varied $M_{N}$ in the range $0.1-30$ TeV. From Eqn. 25, the
contribution to $m_{ee}$ due to heavy neutrino exchange is inversely
proportional to $M_{N}$ and hence, for large $M_{N}$, the values of $m_{ee}$
can be smaller. In type-II dominance with IH, for $r>0.3$, the predicted
values of $m_{ee}$ are around $\sim 0.05$ eV whereas for $r<0.3$, $m_{ee}$
lies in the range $\sim 0.013-0.06$ eV. Thus, the maximum predicted values of
$m_{ee}$ that are allowed by LFV and bounds on absolute neutrino mass
observables lie considerably below the upper bound on $m_{ee}$ that we have
taken. This is because the values of $m_{ee}$ in the range 0.06-0.161 eV (pink
points below the black line) are disfavored by the Planck data since these
high values of $m_{ee}$ come from large values of $m_{lightest}$ giving
$\Sigma>0.12$ eV. Thus, from Fig.9, we see that MLRSM in different limits
gives slightly different upper limits on $r$ and lower limits on
$M_{\Delta_{R}^{++}}$ and how all the three bounds (bounds from LVF,
$0\nu\beta\beta$ and cosmology) conspire to give this upper limit, since had
it been only LFV, much larger values of $r$ would have been allowed as shown
by the pink points.

$\begin{array}[]{|c|c|c|}\hline\cr\textrm{Model}&\textrm{Bound on
}r&\textrm{Bound on
}M_{\Delta_{R}}^{++}(M_{\Delta_{R}^{++}}=\sqrt{2}M_{N}/r)\\\ \hline\cr Type-
INH&r<1.26&M_{\Delta_{R}^{++}}>1.12~{}M_{N}\\\ \hline\cr Type-
IIH&r<1.28&M_{\Delta_{R}^{++}}>1.10~{}M_{N}\\\ \hline\cr Type-
IINH&r<1.35&M_{\Delta_{R}^{++}}>1.05~{}M_{N}\\\ \hline\cr Type-
IIIH&r<1.08&M_{\Delta_{R}^{++}}>1.31~{}M_{N}\\\ \hline\cr\end{array}$ Table 1:
The upper bounds on $r$ and $M_{\Delta_{R}}^{++}$ (in terms of $M_{N}$) for
the four different cases that we have studied.
---
Figure 10: Parameter space allowed by various constraints in the
$M_{W_{R}}-M_{N}$ plane. $r$, $m_{lightest}$ and oscillation parameters are
varied in the same ranges as in the previous plots. The green points satisfy
the bound from $0\nu\beta\beta$ and the gray points satisfy the bounds on
$0\nu\beta\beta$, $m_{\beta}$ and $\Sigma$ as well as the constraints from
LFV. The region to the left of the blue and red lines are disfavored by the
ATLAS and CMS analysis respectively. The black line corresponds to
$M_{N}=M_{W_{R}}$. The upper (lower) panel is for type-I (type-II) dominance
and the left (right) panel is for NH (IH).

In Fig.10, we have plotted the allowed parameter space in the
$M_{W_{R}}-M_{N}$ plane. The oscillation parameters $r$ and $m_{lightest}$ are
varied in the same ranges as in the previous plot. Here, the upper (lower)
panel is for type-I (type-II) dominance and the left (right) panel is for NH
(IH). The green points satisfy the bound from $0\nu\beta\beta$ whereas the
gray points satisfy the bounds on $0\nu\beta\beta$, $m_{\beta}$ and $\Sigma$
as well as the constraints from LFV. Thus, only the gray points are allowed by
all the constraints. The black line corresponds to $M_{N}=M_{W_{R}}$ and the
collider experiments have given some bounds only below this line
($M_{N}<M_{W_{R}}$), where the signal with two leptons and two jets
($W_{R}\rightarrow N_{R}~{}l\rightarrow l~{}l~{}j~{}j$) in the final state is
kinematically observable Nemevsek:2011hz . The region to the left of the blue
line is disfavored by the ATLAS Aaboud:2019wfg  analysis done with the mass of
$N_{R}$ in the range $0.1-18$ TeV. The region to the left of the red line is
disfavored by the CMS Sirunyan:2018vhk  analysis.

Here, we can see that both $0\nu\beta\beta$ and LFV can give constraints on
the $M_{W_{R}}-M_{N}$ plane that are complimentary to the bounds from the
collider. The region disfavored by $0\nu\beta\beta$ are shown by the white
region in Fig.10, which corresponds to the low values of $M_{N}$ and
$M_{W_{R}}$. In reference Dev:2013vxa , the authors had studied the
constraints from $0\nu\beta\beta$ in the $M_{W_{R}}-M_{N}$ plane for type-II
dominance and they had observed that $0\nu\beta\beta$ provide constraints only
for NH. Here, from Fig.10, we can see that with the updated combined
constraints on $m_{ee}$ from KamLAND-Zen and GERDA experiments
Agostini:2018tnm , $0\nu\beta\beta$ puts constraints on IH case as well. Also,
it is to be noted that majority of the parameter space disfavored by
$0\nu\beta\beta$ are also disfavored by the bounds from ATLAS and CMS
experiments (Note that there is a small band corresponding to very small
values of $M_{W_{R}}$, with $M_{W_{R}}<0.5$ TeV where large values of
$M_{N}(\sim 10\textrm{ TeV})$ are disfavored by $0\nu\beta\beta$, which is not
shown in the figure). On the other hand, the combined constraints from LFV and
$\Sigma$ disfavor certain regions (the green points lying above the red and
blue lines) that are allowed by collider and $0\nu\beta\beta$. This is in the
low $M_{W_{R}}$ and high $M_{N}$ region. It is interesting to note that the
two cases with IH is relatively more constrained compared to the two NH cases.
For instance, when $M_{N}=10$ TeV, the lower bounds on $M_{W_{R}}$ are $\sim$
2.4 TeV, 3.8 TeV, 2.8 TeV and 4 TeV respectively for type-I NH, type-I IH,
type-II NH and type-II IH respectively. Thus LFV along with the absolute
neutrino mass observables provide strong constraints on the values of
$M_{W_{R}}$ in the regions that are not covered by the LHC analyses.

##  VII Conclusion

In this work we have studied the correlations among the three absolute
neutrino mass observables, $m_{ee}$, $m_{\beta}$ and $\Sigma$, in the context
of minimal left-right symmetric model(MLRSM). Two phenomenologically
interesting cases of type-I seesaw dominance as well as type-II seesaw
dominance were considered for both NH as well as IH. From the study of
correlations of absolute neutrino mass observables, we observed that the
predictions for MLRSM can be very different compared to the standard three
generation case. This is due to the contributions of the right handed current
and the doubly charged scalar to $0\nu\beta\beta$. In particular, we found
that there are new regions with considerable enhancement in the predicted
values for $m_{ee}$ in all the cases except for the type-II dominance case
with IH. Some of these newly allowed regions are already disfavored by the
current bounds on $0\nu\beta\beta$ and a large part of the parameter space can
be probed by the upcoming $0\nu\beta\beta$ experiments like LEGEND-1k and
nEXO. Also, there exists a tension in the region of $m_{\beta}$ that the
experiment KATRIN is going to probe and the bounds from cosmology. The planned
experiment Project 8 with a sensitivity of $m_{\beta}\sim 0.04$ eV has the
potential to probe the remaining IH region that is allowed by cosmology.

We have also taken into account the independent constraints coming from
charged lepton flavor violation and studied how these bounds are translated
into the planes of absolute neutrino mass observables. We found that the
constraints from LFV put very stringent limits on the prediction for absolute
neutrino mass observables, especially for low mass range of the triplet
scalar. We studied the effect of LFV for two different values of
$M_{\Delta_{R}^{++}}$, viz, 70 TeV and 1.4 TeV. For $M_{\Delta_{R}^{++}}\sim
70$ TeV, LFV constraints had no effect on the predictions for absolute
neutrino mass observables whereas for $M_{\Delta_{R}^{++}}\sim 1.4$ TeV, major
part of the allowed parameter space was disfavored by LFV bounds. The mutually
independent future measurements or null results from cosmology, single beta
decay and $0\nu\beta\beta$ decay can throw more light into the underlying
model.

We have also studied the constraints on the masses of the heavy particles
using the bounds from LFV and the three neutrino mass observables. In
particular, we found that different models give different lower limit on
$M_{\Delta_{R}}^{++}$ for a given value of $M_{N}$. As a general result, we
found $M_{\Delta_{R}}^{++}>M_{N}$ (Table 1) and this bound is more stringent
than the collider bounds on $M_{\Delta_{R}^{++}}$
($M_{\Delta_{R}^{++}}>660-760$ GeV) in the regime $M_{N}>760$ GeV. We have
also determined the allowed parameter space in the $M_{W_{R}}-M_{N}$ planes
and observed that LFV along with the absolute neutrino mass observables
provide strong constraints on the values of $M_{W_{R}}$ in the regions that
are not covered by the LHC analyses. The IH cases were found to be more
constrained compared to the corresponding NH cases. All these can compliment
the future searches for the heavy particles at the colliders. In addition,
these studies can also be complimented with the theoretical constraints on the
model parameter space considered, for instance, the constraints coming from
absolute stability of the electroweak vacuum, tree level unitarity and
perturbativity.

## Acknowledgement

The authors would like to thank Dr. Gulab Bambhania for his involvement during
the initial stages of the work and Dr. Namit Mahajan for useful discussions.

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