# Weak scale right-handed neutrino as pseudo-Goldstone fermion of
spontaneously broken $U(1)_{L_{\mu}-L_{\tau}}$

Anjan S. Joshipura<EMAIL_ADDRESS>Ketan M. Patel
<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research Laboratory,
Navarangpura, Ahmedabad-380 009, India.

###### Abstract

Possibility of a Right-Handed (RH) neutrino being a Goldstone fermion of a
spontaneously broken global $U(1)$ symmetry in a supersymmetric theory is
considered. This fermion obtains mass from the supergravity effects leading to
a RH neutrino at the electroweak scale with a mass similar to the gravitino
mass. A prototype model realizing this scenario contains just three gauge
singlet superfields needed for the type I seesaw mechanism. Masses of the
other two neutrinos are determined by the $U(1)$ breaking scale which too can
be around the electroweak scale. Light neutrinos obtain their masses in this
scenario through (a) mixing with the RH neutrinos (type I seesaw), (b) mixing
with neutralinos ($R$-parity breaking), (c) indirectly through mixing of the
RH neutrinos with neutralinos, and (d) radiative corrections. All these
contributions are described by the same set of a small number of underlying
parameters and provide a very constrained and predictive framework for the
neutrino masses which is investigated in detail for various choices of $U(1)$
symmetries. It is found that flavour independent $U(1)$ symmetries cannot
describe neutrino masses if the soft supersymmetry breaking terms are flavour
universal and one needs to consider flavour dependent symmetries. Considering
a particular example of $L_{\mu}-L_{\tau}$ symmetry, it is shown that viable
neutrino masses and mixing can be obtained without introducing any flavour
violation in the soft sector. The leptonic couplings of Majoron are worked out
in the model and shown to be consistent with various laboratory, astrophysical
and cosmological constraints. The neutrino data allows sizeable couplings
between the RH neutrinos and Higgsinos which can be used to probe the pseudo-
Goldstone fermion at colliders through its displaced decay vertex.

##  I Introduction

Extensions of the Standard Model (SM) by one or more Right-Handed (RH) singlet
fermions provide the most popular mechanism for obtaining light neutrino
masses Minkowski (1977); Yanagida (1979); Mohapatra and Senjanovic (1980);
Schechter and Valle (1980). The scale $M_{N}$ of the masses of these neutrinos
is set by different underlying physics. $M_{N}$ can be close to the Planck
scale in the absence of any symmetry protecting it. Grand unified theories
based on $SO(10)$ group can lead to $M_{N}$ in the range $10^{10}-10^{16}$ GeV
Fritzsch and Minkowski (1975); Gell-Mann  _et al._ (1979). Smaller $M_{N}$
close to TeV scale can be linked to theories with a low scale gauge symmetry
breaking such as $U(1)_{B-L}$ or the left right symmetry. The supersymmetric
theories with a spontaneously broken global symmetry provide an alternate
mechanism for protecting the RH neutrino masses upto TeV scale or less Chun
_et al._ (1995, 1996). If the global symmetry is broken by singlet
superfield(s) and the supersymmetry remains unbroken in the process then the
fermionic partner of the Goldstone Boson - the Goldstone Fermion - would also
remain massless. Once the supersymmetry is broken, say by supergravity
effects, this fermion can pick up a mass around the gravitino mass
$m_{3/2}\sim$ TeV or less. This pseudo-Goldstone Fermion (pGF) can be
identified with one of the sterile neutrinos. Implementation of this mechanism
generically requires more than one singlet superfields which may be identified
with the additional RH neutrinos participating in the seesaw mechanism.

The minimal choice of the global symmetry is $U(1)$. The basic assumption is
that the underlying theory is supersymmetric and invariant under $U(1)$. The
$U(1)$ is assumed to break spontaneously keeping supersymmetry intact. A
prototype model satisfying these requirements contains just three singlet
chiral superfields $\hat{N}_{\alpha}\equiv(\hat{Y},\hat{N},\hat{N}^{\prime})$
with $U(1)$ charges $(0,-1,1)$ and the superpotential

$W_{N}=\lambda\,(\hat{N}\hat{N}^{\prime}-f^{2})\,\hat{Y}\,.$ (1)

As usual, a chiral superfield $\hat{N}_{\alpha}$ contains a complex scalar
$\tilde{N}_{\alpha}$ and its fermionic partner which we denote as
$N_{\alpha}$. We identify the fermionic parts of $\hat{N}_{\alpha}$ with the
three RH neutrinos. Then Eq. (1), apart from breaking $U(1)$ also determines
the zeroeth order structure of the RH neutrino mass matrix which gets
corrected by the supergravity effects. The Dirac mass terms required to
complete the seesaw mechanism arise from the following superpotetial

$W_{D}=\lambda_{\alpha\beta}\,\hat{L}_{\alpha}\hat{N}_{\beta}\hat{H}_{2}\,.$
(2)

The structure of $\lambda_{\alpha\beta}$ is determined by the choice of the
quantum numbers of $(L_{\alpha},H_{2})$ under $U(1)$ symmetry and we will
consider various choices.

Eqs. (1) and (2) contribute to neutrino masses in two different ways. One
obtains non-zero Vacuum Expectation Values (VEVs) for scalars of $\hat{N}$,
$\hat{N}^{\prime}$ at the global supersymmetric minimum from Eq. (1). These
VEVs generate the masses for two of the RH neutrinos. The third RH neutrino
obtains its mass from the supersymmetry breaking effects and the three
neutrinos together generate the seesaw masses for the light neutrinos. VEVs
for $\hat{N}$, $\hat{N}^{\prime}$ also lead to spontaneous bilinear $R$-parity
violation through Eq. (2) which leads to additional contribution to neutrino
masses through their mixing with neutralinos (see for example Barbier  _et
al._ (2005) and references therein). These two entirely different
contributions to neutrino masses are governed by the same set of parameters
appearing in Eqs. (1,2) and provide a very constrained and predictive
framework for neutrino masses. Moreover, a simple requirement that the
$R$-parity violating contribution to neutrino masses is smaller or of the
order of the neutrino masses lead to a an upper bound on the RH sneutrino VEVs
since the bilinear $R$-parity breaking parameters are proportional to these
VEVs. This bound is around TeV but is model dependent as the the standard
seesaw also contributes to the neutrino masses. Since the RH sneutrino VEVs
control the masses of the other two RH neutrinos, they also have masses around
TeV. As a result, the masses of all three RH neutrinos are around TeV or less
and one obtains a viable scenario for the TeV scale RH masses without invoking
any additional gauge symmetry, see Deppisch  _et al._ (2015) for a review on
weak scale RH neutrinos.

A phenomenological motivation to consider seesaw models with TeV scale RH
neutrinos is the possibility of producing and detecting them at colliders.
Disadvantage of these models on the other hand is the smallness of the active
sterile mixing required to obtain the correct light neutrino masses. Since
this mixing governs the production and decay of the RH neutrino in the minimal
seesaw model it is not possible to observe such RH neutrino within the minimum
set up111There exist special textures of the Dirac and Majorana neutrino mass
matrices which lead to small neutrino masses and large active sterile mixing
even within the minimal seesaw, see for example Kersten and Smirnov (2007);
Lee  _et al._ (2013); Chattopadhyay and Patel (2017).. This changes if one or
more RH neutrinos are pGF of the type considered above. In this case, the
symmetry breaking VEV of the RH sneutrino directly couples the pGF to
Higgsinos and their production does not require suppressed mixing as is the
case with the standard seesaw model. Relatively small mixing
$|V_{N\nu_{i}}|\sim 10^{-6}$ of the pGF with neutrinos make them long lived
and pGF can lead to displaced vertex signature at in the collider experiments.
This possibility has been studied extensively recently Lavignac and Medina
(2021) and can also be realized within the scenario presented in this paper.

We introduce the basic framework in the next section choosing the above
mentioned case of the spontaneously broken $U(1)$ symmetry. We then discuss
structure of neutrino masses in section III assuming $U(1)$ to be flavour
independent. It is shown that models in this category cannot reproduce the
neutrino masses correctly if the soft supersymmetry (SUSY) breaking terms are
assumed flavour universal. We then discuss the case of the
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry in section IV which allows correct
description of the observed neutrino spectrum even with the universal soft
terms. The framework implies an existence of Goldstone Boson (GB) of
sponteneously broken $U(1)_{L_{\mu}-L_{\tau}}$ symmetry, namely the Majoron
Chikashige  _et al._ (1981). In section V, we derive the couplings of Majoron
with the charged and neutral leptons and discuss their phenomenological
implications. We also discuss about the possibility to detect the lightest RH
neutrino in the direct search experiment in section VI. The work is summarized
in section VII.

##  II Basic framework

The starting point of the models we discuss is the renormalizable
superpotential given in Eq. (1). The other terms in $W_{N}$, such as $Y^{2}$
or $Y^{3}$, can be forbidden by an additional symmetry like $R$ symmetry. The
minimization of the potential $V_{N}$ obtained from $W_{N}$ leads to a global
SUSY preserving minimum corresponding to

$\langle\tilde{N}\rangle=Us_{\phi}\,,~{}~{}\langle\tilde{N}^{\prime}\rangle=Uc_{\phi}\,,~{}~{}\langle\tilde{Y}\rangle=0\,,$
(3)

where $s_{\phi}=\sin\phi$, $c_{\phi}=\cos\phi$ and

$U=\frac{f}{\sqrt{s_{\phi}c_{\phi}}}\,,$ (4)

represents $U(1)$ breaking scale. Even in the presence of terms like $Y^{2}$
and $Y^{3}$ in $W_{N}$, Eq. (3) describes the global minimum of the potential.

The spontaneous breaking of global $U(1)$ symmetry implies a massless
Goldstone boson. Writing the scalar fields as

$\tilde{N}_{\alpha}=\langle\tilde{N}_{\alpha}\rangle+\frac{1}{\sqrt{2}}\left(\tilde{N}_{R\alpha}+i\tilde{N}_{I\alpha}\right)\,,$
(5)

at the minimum of the potential, we obtain the Goldstone boson as

$G_{B}=c_{\phi}\tilde{N}^{\prime}_{I}-s_{\phi}\tilde{N}_{I}\,.$ (6)

Its fermionic partner also remains massless in the SUSY preserving limit and
it is given by a similar linear combination

$G_{F}=c_{\phi}N^{\prime}-s_{\phi}N\,.$ (7)

This can also be seen explicitly by constructing singlet fermion mass matrix
from Eq. (1). It is obtained as

$M_{N}^{0}=\lambda U\left(\begin{array}[]{ccc}0&c_{\phi}&s_{\phi}\\\
c_{\phi}&0&0\\\ s_{\phi}&0&0\\\ \end{array}\right)\,.$ (8)

The diagonalization of the above matrix leads to a massless state given by Eq.
(7) and two massive and degenerate states with masses $|\lambda U|$.

The fermion mass matrix in Eq. (8) changes in the presence of the local SUSY
breaking terms. These terms shift the VEVs of $\tilde{N},\tilde{N}^{\prime}$
by ${\cal O}(m_{3/2})$ and also generate a nonzero VEV for $\tilde{Y}$ field
Goto and Yamaguchi (1992). The later makes the Goldstone fermion massive.
These shifts follow from the minimisation of the potential with the soft SUSY
terms Chun  _et al._ (1992); Chun and Lukas (1995). If the kinetic terms in
Kahler potential $K$ are minimal and hidden sector is assumed to have Polonyi
form then the shifts are given by Lahanas and Nanopoulos (1987)

$\delta\langle\tilde{N}\rangle=\delta\langle\tilde{N}^{\prime}\rangle=\frac{m_{3/2}}{\lambda}\left(1+\frac{f^{2}\lambda^{2}}{m_{3/2}^{2}}\right)^{\frac{1}{2}}\,,~{}~{}~{}\delta\langle\tilde{Y}\rangle=\frac{m_{3/2}}{\lambda}\,.$
(9)

Shifts for general soft terms are calculated in Chun and Lukas (1995). The
fermion mass matrix in SUGRA is given by

$M_{ab}=M_{P}\,e^{\frac{G}{2M_{P}^{2}}}\,\left(G_{ab}+\frac{1}{3}\frac{G_{a}G_{b}}{M_{P}^{2}}-G_{c}\,(G^{-1})^{c}_{d}\,G^{d}_{ab}\right)\,.$
(10)

with

$G=K+M_{P}^{2}~{}\ln|W/M_{P}^{3}|^{2}\,,$ (11)

$G_{a}=\frac{\delta G}{\delta\phi^{a}}$, $\phi^{a}=\phi_{a}^{*}$ is antichiral
superfield and

$W=g(z)+W_{N}(N_{\alpha})\,.$ (12)

Here, $g(z)$ describes the hidden sector potential and $W_{N}$ is given by Eq.
(1). We evaluate the fermion mass matrix by substituting $\langle
z\rangle=bM_{P}$, $\langle W\rangle\sim\langle
g(z)\rangle\sim\frac{m_{3/2}M_{P}^{2}}{c}$ where $c=e^{b^{2}/2}$ is a
numerical constant and remembering that now
$\lambda\langle\tilde{Y}\rangle\approx{\cal O}(m_{3/2})$. The gravitino mass
is defined as $m_{3/2}=M_{P}\,\exp(\langle G\rangle/2M_{P}^{2})$. The fermion
mass matrix following from Eq. (10) is then given by

$M_{N}^{\rm loc}=c\,m_{3/2}\left(\begin{array}[]{ccc}{\cal
O}\left(\frac{U^{2}}{M_{P}^{2}}\right)&\frac{\lambda
Uc_{\phi}}{m_{3/2}}&\frac{\lambda Us_{\phi}}{m_{3/2}}\\\ \frac{\lambda
Uc_{\phi}}{m_{3/2}}&{\cal
O}\left(\frac{U^{2}}{M_{P}^{2}}\right)&\frac{\lambda\langle\hat{Y}\rangle}{m_{3/2}}\\\
\frac{\lambda
Us_{\phi}}{m_{3/2}}&\frac{\lambda\langle\hat{Y}\rangle}{m_{3/2}}&{\cal
O}\left(\frac{U^{2}}{M_{P}^{2}}\right)\\\ \end{array}\right)\,.$ (13)

The above mass matrix approximately coincides with the $M_{N}^{0}$ in Eq. (8)
if one effectively assumes nonzero VEV for $\tilde{Y}$ in the latter. The
other corrections induced by SUSY breaking are ${\cal O}(U^{2}/M_{P}^{2})$ and
they are very small. Therefore, we approximate the singlet fermion mass matrix
in the presence of local SUSY breaking as

$M_{N}\simeq\lambda U\left(\begin{array}[]{ccc}0&c_{\phi}&s_{\phi}\\\
c_{\phi}&0&\epsilon\\\ s_{\phi}&\epsilon&0\\\ \end{array}\right)\,,$ (14)

where $\epsilon=\langle\hat{Y}\rangle/U<1$ and we have absorbed $c$ in
redefining $\lambda$. Note that $U$ denotes now a shifted VEV and it does not
strictly satisfy Eq. (4). The SUSY breaking generates the mass for the
lightest RH neutrino and split the masses of the other two singlet neutrinos
by ${\cal O}(m_{3/2})$. Explicitly,

$M_{N_{1}}\approx|2\epsilon\lambda
Uc_{\phi}s_{\phi}|\,,~{}~{}M_{N_{2,3}}\approx|\lambda U(1\pm\epsilon
c_{\phi}s_{\phi})|\,.$ (15)

The lightest RH neutrino is now a pseudo-Goldstone fermion which can be
written as

$N_{1}\simeq\cos\phi\,N^{\prime}-\sin\phi\,N-\epsilon\cos 2\phi\,Y\,,$ (16)

at the leading order in $\epsilon$.

##  III Flavour independent $U(1)$ symmetries and neutrino masses

We now discuss the interaction of the RH neutrinos with the SM leptons, Higgs
and their superpartners. This depends on the choice of the symmetry $U(1)$.
There are four possible choices. $U(1)$ is (a) a “hidden” symmetry acting only
on the RH fields $N_{\alpha}$, or (b) a leptonic symmetry acting on
$L_{\alpha}$ and $N_{\alpha}$, or (c) a Peccei Quinn like symmetry acting on
the Higgs doublet(s), on some of the right-handed quarks and may also be on
the leptonic fields, or (d) a leptonic family symmetry acting only on leptons
and distinguishing among them. We discuss the case (d) in detail in the next
section. The cases (a-c) can all be described by a general superpotential

$W=\lambda_{\alpha}\,\hat{L}_{\alpha}\hat{H}_{2}\hat{S}_{1}\,+\,\kappa\,\hat{H}_{1}\hat{H}_{2}\hat{S_{2}}\,$
(17)

where we have only shown the part relevant for the neutrino masses. Here
$\hat{S}_{1,2}$ can be any of the three RH fields $\hat{N}_{\alpha}$ depending
on the choice of the $U(1)$ symmetry. For the case (a), one has
$\hat{S}_{1}=\hat{S}_{2}=\hat{Y}$. The case (b) corresponds to $\kappa=0$ and
$\hat{S}_{1}=\hat{N}$ if $L_{\alpha}$ have $U(1)$ charge is +1. $W$ in this
case contains an additional $\mu$ term. The choice of $\hat{S}_{1,2}$ in case
(c) depends on the $U(1)$ charges of $\hat{H}_{1,2}$.

The cases (a-c) are similar and we consider the case (c) with
$\hat{S}_{1}=\hat{N}$ and $\hat{S}_{2}=\hat{N}^{\prime}$ for definiteness.
Without loss of generality, the parameters $\lambda_{\alpha}$ and $\kappa$ can
be chosen real. We shift the fields $\hat{N},\hat{N}^{\prime}$ by the VEV of
their scalar components. This leads to the following superpotetial

$W=\lambda_{\alpha}\,\hat{L}_{\alpha}\hat{H}_{2}\hat{N}\,+\,\kappa\,\hat{H}_{1}\hat{H}_{2}\hat{N^{\prime}}\,-\mu\hat{H}_{1}\hat{H}_{2}-\epsilon_{\alpha}\hat{L}_{\alpha}\hat{H}_{2}\,$
(18)

with

$\epsilon_{\alpha}=-\lambda_{\alpha}Us_{\phi}~{}~{}~{},~{}~{}~{}\mu=-\kappa
Uc_{\phi}~{}.$ (19)

The model contains ten neutral fermion fields which mix with each other
through the gauge and superpotential interactions. The corresponding mass
Lagrangian in the basis
$\psi_{0}=(\nu_{\alpha},N_{\alpha},-i\lambda_{1},-i\lambda_{2},\tilde{h}_{1}^{0},\tilde{h}_{2}^{0})^{T}$
is given by

$-{\cal L}_{\cal N}=\frac{1}{2}\,\psi_{0}^{T}\,{\cal M}_{\cal
N}\,\psi_{0}\,+\,{\rm h.c.}\,,$ (20)

where ${\cal M}_{\cal N}$ is a $10\times 10$ mass matrix which can be
parametrized as

${\cal M}_{\cal N}=\left(\begin{array}[]{ccc}0&m_{D}&m_{\nu\chi}\\\
m_{D}^{T}&M_{N}&m_{N\chi}\\\
m_{\nu\chi}^{T}&m_{N\chi}^{T}&M_{\chi}\end{array}\right)\,.$ (21)

Here, $M_{N}$ is singlet neutrino mass matrix as given by Eq. (14). $m_{D}$ is
$3\times 3$ Dirac neutrino mass matrix as can be read-off from Eq. (18),

$m_{D}=v_{2}\,\left(\begin{array}[]{ccc}0&\lambda_{e}&0\\\
0&\lambda_{\mu}&0\\\ 0&\lambda_{\tau}&0\\\ \end{array}\right)\,,$ (22)

with $v_{1,2}\equiv\langle H_{1,2}\rangle$ and
$\sqrt{v_{1}^{2}+v_{2}^{2}}\equiv v=174\,{\rm GeV}$. $m_{\nu\chi}$ and
$m_{N\chi}$ are $3\times 4$ mass matrices. They can be determined from Eq.
(18) and the gauge interactions as

$m_{\nu\chi}=\left(\begin{array}[]{cccc}-\frac{1}{\sqrt{2}}g^{\prime}\omega^{\prime}_{e}&\frac{1}{\sqrt{2}}g\omega^{\prime}_{e}&0&-\epsilon_{e}\\\
-\frac{1}{\sqrt{2}}g^{\prime}\omega^{\prime}_{\mu}&\frac{1}{\sqrt{2}}g\omega^{\prime}_{\mu}&0&-\epsilon_{\mu}\\\
-\frac{1}{\sqrt{2}}g^{\prime}\omega^{\prime}_{\tau}&\frac{1}{\sqrt{2}}g\omega^{\prime}_{\tau}&0&-\epsilon_{\tau}\end{array}\right)\,,~{}~{}m_{N\chi}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&\lambda_{\alpha}\omega_{\alpha}^{\prime}\\\ 0&0&\kappa v_{2}&\kappa
v_{1}\\\ \end{array}\right)\,,$ (23)

where $\omega_{\alpha}^{\prime}\equiv\langle\tilde{\nu}_{\alpha}\rangle$. The
explicit form of the neutralino mass matrix is

$M_{\chi}=\left(\begin{array}[]{cccc}M_{1}&0&-\frac{1}{\sqrt{2}}g^{\prime}v_{1}&\frac{1}{\sqrt{2}}g^{\prime}v_{2}\\\
0&M_{2}&\frac{1}{\sqrt{2}}gv_{1}&-\frac{1}{\sqrt{2}}gv_{2}\\\
-\frac{1}{\sqrt{2}}g^{\prime}v_{1}&\frac{1}{\sqrt{2}}gv_{1}&0&-\mu\\\
\frac{1}{\sqrt{2}}g^{\prime}v_{2}&-\frac{1}{\sqrt{2}}gv_{2}&-\mu&0\\\
\end{array}\right)\,.$ (24)

The tree level light neutrino masses can be obtained from Eq. (21). For
$m_{\nu\chi},m_{D}\ll M_{\chi},M_{N}$, the neutrino mass matrix at the leading
order is given by

$m_{\nu}=-m_{3\times 7}\,{\cal M}_{7\times 7}^{-1}\,(m_{3\times 7})^{T}\,.$
(25)

Here, $m_{3\times 7}\equiv(m_{D},m_{\nu\chi})$ is a $3\times 7$ analogue of
the conventional Dirac matrix in familiar type I seesaw mechanism. ${\cal
M}_{7\times 7}$ is a mass matrix of heavier states corresponding to the lower
$7\times 7$ block of Eq. (21).

Neglecting $\lambda_{\alpha}\omega_{\alpha}^{\prime}$ term222It is shown later
in this section that the SM neutrino mass scale requires $\lambda_{\alpha}\ll
1$ in this framework. Therefore, $\kappa
v_{1,2}\gg\lambda_{\alpha}\omega^{\prime}_{\alpha}$ unless $\kappa$ is
vanishingly small. in $m_{N\chi}$, an explicit evaluation of Eq. (25) gives

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle\tilde{m}_{0}\left[\left(1-\frac{\kappa
c_{\phi}U}{\mu}\left(2-\frac{\kappa
c_{\phi}U}{\mu}\right)\right)\,s_{\phi}^{2}\,\tilde{\cal
A}+\left(\frac{A_{0}M_{N_{1}}}{v_{2}^{2}}-\frac{2\kappa^{2}c_{\phi}^{2}A_{0}v_{1}}{\mu
v_{2}}\right)\,\tilde{{\cal B}}\right.$ (26) $\displaystyle-$
$\displaystyle\left.\frac{\kappa
A_{0}c_{\phi}(v_{1}^{2}-v_{2}^{2})}{v_{2}\mu}\left(1-\frac{\kappa
c_{\phi}U}{\mu}\right)\,s_{\phi}\,\tilde{{\cal H}}\right]\,,$

where,

$\tilde{m}_{0}=\frac{D_{4}v_{2}^{2}U^{2}\lambda^{2}}{\tilde{D}_{7}}=\frac{v_{2}^{2}}{M_{N_{1}}-\kappa^{2}c_{\phi}^{2}\zeta}\,,$
(27)

$D_{4}$ and $\tilde{D}_{7}$ are the determinants of $M_{\chi}$ and ${\cal
M}_{7\times 7}$, respectively. Explicit expression of $D_{4}$ is

$D_{4}={\rm Det}\,M_{\chi}=-\mu(M_{1}M_{2}\mu-(g^{2}M_{1}+g^{\prime
2}M_{2})v_{1}v_{2})\,.$ (28)

$\tilde{D}_{7}$ can be obtained using the second equality in Eq. (27) in terms
of the other known parameters where

$M_{N_{1}}\equiv 2\epsilon\lambda Us_{\phi}c_{\phi}\,,$ (29)

is the mass of the pGF and

$\zeta=\frac{v^{4}(g^{2}M_{1}+g^{\prime 2}M_{2})-4v_{1}v_{2}\mu
M_{1}M_{2}}{2D_{4}}\,.$ (30)

Further,

$A_{0}=-\frac{\mu(g^{2}M_{1}+g^{\prime 2}M_{2})}{2(M_{1}M_{2}\mu-
v_{1}v_{2}(g^{2}M_{1}+g^{\prime 2}M_{2}))}\,.$ (31)

The matrices in Eq. (26) are obtained as

$\tilde{{\cal
A}}=\left(\begin{array}[]{ccc}\lambda_{e}^{2}&\lambda_{e}\lambda_{\mu}&\lambda_{e}\lambda_{\tau}\\\
\lambda_{e}\lambda_{\mu}&\lambda_{\mu}^{2}&\lambda_{\mu}\lambda_{\tau}\\\
\lambda_{e}\lambda_{\tau}&\lambda_{\mu}\lambda_{\tau}&\lambda_{\tau}^{2}\\\
\end{array}\right)\,,~{}\tilde{{\cal
B}}=\left(\begin{array}[]{ccc}\omega_{e}^{2}&\omega_{e}\omega_{\mu}&\omega_{e}\omega_{\tau}\\\
\omega_{e}\omega_{\mu}&\omega_{\mu}^{2}&\omega_{\mu}\omega_{\tau}\\\
\omega_{e}\omega_{\tau}&\omega_{\mu}\omega_{\tau}&\omega_{\tau}^{2}\\\
\end{array}\right)\,,$ (32) $\tilde{{\cal
H}}=\left(\begin{array}[]{ccc}2\lambda_{e}\omega_{e}&\lambda_{e}\omega_{\mu}+\lambda_{\mu}\omega_{e}&\lambda_{e}\omega_{\tau}+\lambda_{\tau}\omega_{e}\\\
\lambda_{e}\omega_{\mu}+\lambda_{\mu}\omega_{e}&2\lambda_{\mu}\omega_{\mu}&\lambda_{\mu}\omega_{\tau}+\lambda_{\tau}\omega_{\mu}\\\
\lambda_{e}\omega_{\tau}+\lambda_{\tau}\omega_{e}&\lambda_{\mu}\omega_{\tau}+\lambda_{\tau}\omega_{\mu}&2\lambda_{\tau}\omega_{\tau}\\\
\end{array}\right)\,.$ (33)

Here, $\omega_{\alpha}$ denote the VEVs of sneutrino in the basis in which the
bilinear term is rotated away from the superpotential Joshipura  _et al._
(2002). Explicitly,

$\omega_{\alpha}=\omega^{\prime}_{\alpha}-\frac{v_{1}}{\mu}\epsilon_{\alpha}\,.$
(34)

If one neglects mixing $m_{N\chi}$ between the RH neutrino and neutralino
induced by $\kappa$, then the ${\cal M}_{7\times 7}$ becomes block diagonal
and $m_{\nu}$ becomes sum of two independent seesaw contributions
corresponding to the first two terms of Eq. (26) with $\kappa=0$. The first
term is the normal seesaw contribution while the second is the standard
$R$-parity violating contribution arising due to neutrino-neutralino mixing.
Both these contributions are of rank one and each can give mass to only one
neutrino. As seen from Eq. (26), the leading terms of the conventional seesaw
contribution is solely governed by the mass of the pGF.

The $\kappa$-dependent terms in Eq. (26) make significant contribution. This
can be seen by eliminating $\kappa$ using the second in Eq. (19) which leads
to

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle\tilde{m}_{0}\left[4s_{\phi}^{2}\,\tilde{\cal
A}+\frac{A_{0}M_{N_{1}}}{v_{2}^{2}}\left(1-\frac{2\mu
v_{1}v_{2}}{U^{2}M_{N_{1}}}\right)\,\tilde{{\cal
B}}+\frac{2A_{0}(v_{1}^{2}-v_{2}^{2})}{v_{2}U}\,s_{\phi}\,\tilde{{\cal
H}}\right]\,.$ (35)

The snuetrino VEVs arise from the parameters $\epsilon_{\alpha}$ and one can
parametrize them as

$\omega_{\alpha}\equiv
k_{\alpha}\epsilon_{\alpha}=-\lambda_{\alpha}k_{\alpha}Us_{\phi}\,,$ (36)

where we have used Eq. (19). The elements of matrix $m_{\nu}$ in Eq. (35) can
then be rewritten as

$\left(m_{\nu}\right)_{\alpha\beta}\
=\tilde{m}_{0}\,\lambda_{\alpha}\lambda_{\beta}\,s_{\phi}^{2}\left[4+\frac{A_{0}M_{N_{1}}U^{2}}{v_{2}^{2}}\left(1-\frac{2\mu
v_{1}v_{2}}{U^{2}M_{N_{1}}}\right)\,k_{\alpha}k_{\beta}-\frac{2A_{0}(v_{1}^{2}-v_{2}^{2})}{v_{2}}\,(k_{\alpha}+k_{\beta})\right]\,.$
(37)

Assuming the second and third terms are $\lesssim{\cal O}(1)$, one finds
typically

$|\lambda_{\alpha}|^{2}\sim{\cal
O}\left(\frac{m_{\nu}M_{N_{1}}}{v_{2}^{2}}\right)\sim{\cal
O}\left(10^{-13}\right)\times\left(\frac{M_{N_{1}}}{100\,{\rm GeV}}\right)\,.$
(38)

Therefore, $|\lambda_{\alpha}|\ll 1$ and neglecting $\lambda_{\alpha}$
dependent terms in $m_{N\chi}$ is justified. The third term in Eq. (37) is
smaller than the seesaw contribution for $k_{\alpha}\leq 1$. The second term
however increases with $k_{\alpha}U$ and can compete with the seesaw
contribution. Requiring that this term is at most of the order of the seesaw
term implies

$U^{2}|k_{\alpha}|^{2}\lesssim{\cal
O}\left(\frac{v_{2}^{2}}{|A_{0}|M_{N_{1}}}\right)\sim{\cal
O}\left(10^{6}\right)\,{\rm GeV}^{2}\times\left(\frac{2.8\times 10^{-4}\,{\rm
GeV}^{-1}}{|A_{0}|}\right)\times\left(\frac{100\,{\rm
GeV}}{M_{N_{1}}}\right)\,.$ (39)

The parameters $k_{\alpha}$ can be determined from the minimization of the
scalar potential of the model, see Appendix A for details. Generically,
$k_{\alpha}$ would be ${\cal O}(1)$. However, they can be very small if the
soft SUSY breaking terms are correlated (see, Appendix A). For $k_{\alpha}\sim
1$, Eq. (39) implies a strong upper bound on the symmetry breaking scale which
weakens with the decrease in the values of $k_{\alpha}$.

The magnitude of $k_{\alpha}$ depends on the trilinear soft terms and
sneutrino soft masses along with the other parameters and in general it can be
different for different $\alpha$. However, $k_{\alpha}$ can be identical in
minimal SUGRA type theories because of flavour universal soft parameters at
the scale of SUSY breaking mediation. A splitting among the values of
$k_{\alpha}$ then get induced by renormalization group effects and it is
expected to be small. Thus it is natural to assume flavour universal
$k_{\alpha}$ in a large class of models. In such a situation, Eq. (37) can
generate mass for only one neutrino and all the models in this category are
ruled out in spite of the presence of various contributions to neutrino
masses333This is not true if $\omega_{\alpha}$ and $\lambda_{\alpha}$ are
regarded as independent. One can obtain correct masses and mixing in this
case, see for example, model in Lavignac and Medina (2021). However, this
amounts to assuming flavour dependent $k_{\alpha}$..

These models become viable only if sizeable flavour violations in $k_{\alpha}$
are introduced. Even with the most general $k_{\alpha}$, Eq. (37) leads to a
massless neutrino. We find that to get large enough solar angle and viable
ratio of the solar to atmospheric squared mass differences, one requires
$(k_{2}-k_{3})/k_{3}\simeq{\cal O}(1)$. Such a sizeable difference between
$k_{\alpha}$ cannot come from the RG induced small corrections and one needs
to assume large flavour violation in the soft terms at the SUSY breaking
mediation scale. The need for such large flavour violation in case of the
purely bilinear $R$-parity violation was noticed before Joshipura  _et al._
(2002); Romao  _et al._ (2000); Hirsch  _et al._ (2000). This is an unwelcome
feature which can be cured if the flavour independent symmetry considered here
is replaced by a flavour dependent leptonic symmetry. It is possible in this
case to retain flavour universality of $k_{\alpha}$ and reproduce all the
neutrino data as we discuss in the next section.

##  IV $U(1)_{L_{\mu}-L_{\tau}}$ symmetry and neutrino masses

The example we consider here is based on the spontaneously broken
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry. The three RH neutrinos
$N_{\alpha}=(\hat{Y},\hat{N},\hat{N}^{\prime})$ retain the same charges as
before and we appropriately relabel them as a
$(\hat{N_{e}},\hat{N_{\mu}},\hat{N_{\tau}})$. The $U(1)$ charges for the three
generations of the lepton doublet superfields, $\hat{L}_{\alpha}$ are
$(0,1,-1)$ while the $SU(2)$ singlet charged lepton superfields,
$\hat{E}^{c}_{\alpha}$, have charges $(0,-1,1)$. The MSSM Higgs superfields
are neutral under $U(1)$. A renormalizable superpotential involving these
superfields is written as

$\displaystyle
W=\lambda_{\alpha}\,\hat{L}_{\alpha}\hat{H}_{2}\hat{N}_{\alpha}\,+\,\lambda_{\alpha}^{\prime}\,\hat{L}_{\alpha}\hat{H}_{1}\hat{E}^{c}_{\alpha}\,-\mu_{0}\hat{H}_{1}\hat{H}_{2}\,+\kappa\hat{N}_{e}\,\hat{H}_{1}\hat{H}_{2}\,$
(40)

The above along with $W_{N}$ gives the complete superpotential characterizing
the leptonic interactions in the model. Without loss of generality, the
parameters $\lambda_{\alpha}$, $\lambda^{\prime}_{\alpha}$ and $\mu_{0}$ can
be chosen real. The bilinear $R$-parity violation and the $\mu$ parameter
generated through the VEVs of scalar components of $\hat{N}_{\alpha}$ are
given by

$\epsilon_{\alpha}=-\lambda_{\alpha}\langle\tilde{N}_{\alpha}\rangle\,,~{}~{}\mu=\mu_{0}-\kappa\langle\tilde{N}_{e}\rangle\,$
(41)

The spontaneous breaking of $R$-parity also induces nonzero VEVs for the
sneutrino fields residing in $\hat{\nu}_{\alpha}$ at the minimum of the scalar
potential.

The masses of all the ten neutral fermions are given by the matrix in Eq.
(21). The matrices $M_{N}$, $M_{\nu\chi}$ and $M_{\chi}$ are the same as given
before by Eqs. (14,23,24), respectively. $m_{D}$ and $M_{N\chi}$ are different
compared to the flavour independent cases and are given by

$m_{D}=v_{2}\,{\rm
Diag.}\left(\lambda_{e},\lambda_{\mu},\lambda_{\tau}\right)\,,~{}~{}~{}~{}m_{N\chi}=\left(\begin{array}[]{cccc}0&0&\kappa
v_{2}&\kappa v_{1}+\lambda_{e}\omega_{e}^{\prime}\\\
0&0&0&\lambda_{\mu}\omega_{\mu}^{\prime}\\\
0&0&0&\lambda_{\tau}\omega_{\tau}^{\prime}\\\ \end{array}\right)\,.$ (42)

The tree level light neutrino mass matrix after the seesaw is still given by
Eq. (25). Neglecting $\lambda_{\alpha}\omega^{\prime}_{\alpha}$ terms in
$M_{N\chi}$, an explicit evaluation of $m_{\nu}^{(0)}$ gives

$\displaystyle m_{\nu}^{(0)}$ $\displaystyle=$ $\displaystyle m_{0}\left[{\cal
A}+\frac{A_{0}M_{N_{1}}}{v_{2}^{2}}\,{\cal B}+\frac{2\kappa\epsilon
U}{\mu}\,{\cal C}+\frac{\kappa
A_{0}\epsilon(v_{1}^{2}-v_{2}^{2})}{v_{2}\mu}\,{\cal D}\right.$ (43)
$\displaystyle+$ $\displaystyle\left.\frac{\kappa^{2}\epsilon\zeta}{\lambda
U}\,{\cal F}+\left(\frac{\kappa\epsilon U}{\mu}\right)^{2}\,{\cal
G}-\frac{2\kappa^{2}\epsilon^{2}A_{0}v_{1}}{\mu v_{2}}\,{\cal
B}+\frac{\kappa^{2}A_{0}\epsilon^{2}(v_{1}^{2}-v_{2}^{2})U}{v_{2}\mu^{2}}\,{\cal
H}\right]\,,$

where we have used relations Eqs. (34,41). Further,

$m_{0}=\frac{D_{4}v_{2}^{2}U^{2}\lambda^{2}}{D_{7}}=\frac{v_{2}^{2}}{M_{N_{1}}-\kappa^{2}\epsilon^{2}\zeta}\,,$
(44)

and $D_{4}$, $\zeta$ and $A_{0}$ are as defined earlier in Eqs. (28,30,31)
respectively. $D_{7}$ is the determinant of the new ${\cal M}_{7\times 7}$ and
it is different from $\tilde{D}_{7}$ defined earlier unless $\kappa=0$. Also,

${\cal A}=\left(\begin{array}[]{ccc}\epsilon^{2}\lambda_{e}^{2}&-\epsilon
s_{\phi}\lambda_{e}\lambda_{\mu}&-\epsilon
c_{\phi}\lambda_{e}\lambda_{\tau}\\\ -\epsilon
s_{\phi}\lambda_{e}\lambda_{\mu}&s_{\phi}^{2}\lambda_{\mu}^{2}&-s_{\phi}c_{\phi}\lambda_{\mu}\lambda_{\tau}\\\
-\epsilon
c_{\phi}\lambda_{e}\lambda_{\tau}&-s_{\phi}c_{\phi}\lambda_{\mu}\lambda_{\tau}&c_{\phi}^{2}\lambda_{\tau}^{2}\end{array}\right)\,,~{}{\cal
B}=\left(\begin{array}[]{ccc}\omega_{e}^{2}&\omega_{e}\omega_{\mu}&\omega_{e}\omega_{\tau}\\\
\omega_{e}\omega_{\mu}&\omega_{\mu}^{2}&\omega_{\mu}\omega_{\tau}\\\
\omega_{e}\omega_{\tau}&\omega_{\mu}\omega_{\tau}&\omega_{\tau}^{2}\\\
\end{array}\right)\,,$ (45) ${\cal
C}=\left(\begin{array}[]{ccc}\epsilon^{2}\lambda_{e}^{2}&0&0\\\
0&-s_{\phi}^{2}\lambda_{\mu}^{2}&-s_{\phi}c_{\phi}\lambda_{\mu}\lambda_{\tau}\\\
0&-s_{\phi}c_{\phi}\lambda_{\mu}\lambda_{\tau}&-c_{\phi}^{2}\lambda_{\tau}^{2}\end{array}\right)\,,$
(46) ${\cal
D}=\left(\begin{array}[]{ccc}2\epsilon\lambda_{e}\omega_{e}&\epsilon\lambda_{e}\omega_{\mu}-s_{\phi}\lambda_{\mu}\omega_{e}&\epsilon\lambda_{e}\omega_{\tau}-c_{\phi}\lambda_{\tau}\omega_{e}\\\
\epsilon\lambda_{e}\omega_{\mu}-s_{\phi}\lambda_{\mu}\omega_{e}&-2s_{\phi}\lambda_{\mu}\omega_{\mu}&-s_{\phi}\lambda_{\mu}\omega_{\tau}-c_{\phi}\lambda_{\tau}\omega_{\mu}\\\
\epsilon\lambda_{e}\omega_{\tau}-c_{\phi}\lambda_{\tau}\omega_{e}&-s_{\phi}\lambda_{\mu}\omega_{\tau}-c_{\phi}\lambda_{\tau}\omega_{\mu}&-2c_{\phi}\lambda_{\tau}\omega_{\tau}\\\
\end{array}\right)\,,$ (47) ${\cal F}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&\lambda_{\mu}\lambda_{\tau}\\\
0&\lambda_{\mu}\lambda_{\tau}&0\end{array}\right)\,,~{}~{}{\cal
G}=\left(\begin{array}[]{ccc}\epsilon^{2}\lambda_{e}^{2}&\epsilon
s_{\phi}\lambda_{e}\lambda_{\mu}&\epsilon c_{\phi}\lambda_{e}\lambda_{\tau}\\\
\epsilon
s_{\phi}\lambda_{e}\lambda_{\mu}&s_{\phi}^{2}\lambda_{\mu}^{2}&s_{\phi}c_{\phi}\lambda_{\mu}\lambda_{\tau}\\\
\epsilon
c_{\phi}\lambda_{e}\lambda_{\tau}&s_{\phi}c_{\phi}\lambda_{\mu}\lambda_{\tau}&c_{\phi}^{2}\lambda_{\tau}^{2}\end{array}\right)\,,$
(48) ${\cal
H}=\left(\begin{array}[]{ccc}2\epsilon\lambda_{e}\omega_{e}&\epsilon\lambda_{e}\omega_{\mu}+s_{\phi}\lambda_{\mu}\omega_{e}&\epsilon\lambda_{e}\omega_{\tau}+c_{\phi}\lambda_{\tau}\omega_{e}\\\
\epsilon\lambda_{e}\omega_{\mu}+s_{\phi}\lambda_{\mu}\omega_{e}&2s_{\phi}\lambda_{\mu}\omega_{\mu}&s_{\phi}\lambda_{\mu}\omega_{\tau}+c_{\phi}\lambda_{\tau}\omega_{\mu}\\\
\epsilon\lambda_{e}\omega_{\tau}+c_{\phi}\lambda_{\tau}\omega_{e}&s_{\phi}\lambda_{\mu}\omega_{\tau}+c_{\phi}\lambda_{\tau}\omega_{\mu}&2c_{\phi}\lambda_{\tau}\omega_{\tau}\\\
\end{array}\right)\,.$ (49)

In the limit $\kappa\to 0$, the first term in Eq. (43) quantifies the standard
seesaw contribution:

$m_{\nu}^{S}\equiv-
m_{D}\,M_{N}^{-1}\,m_{D}^{T}=\frac{v_{2}^{2}}{M_{N_{1}}}\,{\cal A}\,.$ (50)

Contrary to the flavour independent cases, this contribution leads to non-zero
masses for all three neutrinos. The leading mass

$m_{\nu_{3}}\simeq\frac{v_{2}^{2}}{|M_{N_{1}}|}\left(c_{\phi}^{2}\lambda_{\tau}^{2}+s_{\phi}^{2}\lambda_{\mu}^{2}\right)\,,$
(51)

is controlled by the mass of the pGF. The lower $2\times 2$ block of the
matrix ${\cal A}$ along with this mass can account for the atmospheric
neutrino oscillations. The other two masses are degenerate when the terms of
${\cal O}(\epsilon^{2})$ are neglected in ${\cal A}$. Degeneracy is lifted by
the latter but by itself it is not sufficient to account for the solar scale
and angle. The seesaw term in Eq. (43) however gets significant contribution
from the $\kappa$ dependent terms ${\cal C}$, ${\cal F}$ and ${\cal G}$ when
$\left|\kappa U\epsilon/\mu\right|\sim{\cal O}(1)$. These terms by themselves
are sufficient to reproduce the neutrino spectrum as we will discuss. For
$\kappa=0$, the second term denotes the usual contribution to the neutrino
masses from the sneutrino VEVs induced by $R$-parity violation:

$m_{\nu}^{\cancel{R}}\equiv-
m_{\nu\chi}\,M_{\chi}^{-1}\,m_{\nu\chi}^{T}=A_{0}\,{\cal B}\,.$ (52)

This contribution also gets corrected by the $\kappa$ dependent terms as shown
in Eq. (43).

As mentioned earlier, the parameters $\omega_{\alpha}$ denote the VEVs of
sneutrino in the basis in which the bilinear term is shifted away Joshipura
_et al._ (2002). They can be parametrized as
$\omega_{\alpha}=k_{\alpha}\epsilon_{\alpha}$ and $k_{\alpha}$ can be
determined from minimization of the scalar potential of the model, see
Appendix A for details. The magnitude of $k_{\alpha}$ depends on the trilinear
soft terms and sneutrino soft masses along with the other parameters and in
general their value can be different for different $\alpha$. However,
$k_{\alpha}$ are expected to be universal in minimal SUGRA type theories
because of flavour universal soft parameters at the scale of SUSY breaking
mediation. A splitting among the values of $k_{\alpha}$ then get induced by
renormalization group effects. We assume that these effects are small and
$k_{\alpha}$ are approximately universal at the electroweak scale as well. For
$\omega_{\alpha}=k\epsilon_{\alpha}$, we find

${\cal B}=k^{2}U^{2}\,{\cal G}\,,~{}~{}{\cal D}=-2kU\,{\cal C}\,,~{}~{}{\cal
H}=-2kU\,{\cal G}\,.$ (53)

Substituting these in Eq. (43) and defining
$r_{e,\mu}=\lambda_{e,\mu}/\lambda_{\tau}$, we obtain the neutrino mass matrix
after combining the different contributions as

$m^{(0)}_{\nu}=m_{0}\lambda_{\tau}^{2}\left(\begin{array}[]{ccc}\epsilon^{2}r_{e}^{2}(1+\eta+\eta^{\prime})&-\epsilon
r_{e}r_{\mu}s_{\phi}(1-\eta)&-\epsilon r_{e}c_{\phi}(1-\eta)\\\ -\epsilon
r_{e}r_{\mu}s_{\phi}(1-\eta)&r_{\mu}^{2}s_{\phi}^{2}(1+\eta-\eta^{\prime})&-r_{\mu}s_{\phi}c_{\phi}(1-\eta+\eta^{\prime}-\zeta^{\prime})\\\
-\epsilon
r_{e}c_{\phi}(1-\eta)&-r_{\mu}s_{\phi}c_{\phi}(1-\eta+\eta^{\prime}-\zeta^{\prime})&c_{\phi}^{2}(1+\eta-\eta^{\prime})\\\
\end{array}\right),$ (54)

where,

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{A_{0}k^{2}U^{2}M_{N_{1}}}{v_{2}^{2}}+\left(\frac{\kappa\epsilon
U}{\mu}\right)^{2}\left(1-\frac{2A_{0}k(kv_{1}\mu+v_{1}^{2}-v_{2}^{2})}{v_{2}}\right)\,,$
$\displaystyle\eta^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2\kappa\epsilon
U}{\mu}\left(1-\frac{A_{0}k(v_{1}^{2}-v_{2}^{2})}{v_{2}}\right)\,,$
$\displaystyle\zeta^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2\kappa^{2}\epsilon^{2}\zeta}{M_{N_{1}}}\,.$ (55)

If the induced sneutrino VEVs $\omega_{\alpha}$ are small then only the seesaw
terms contribute to Eq. (54). This contribution by itself is sufficient to
describe neutrino masses and mixing parameters including the Dirac CP phase.
However, it requires relatively low value of the $U(1)$ breaking scale $U$.
This is quantified as follows. One obtains a relation
$\eta=\frac{1}{4}\eta^{\prime 2}$ from Eq. (IV) in the purely seesaw limit.
Eq. (53) is then described in terms of five real parameters
$m_{0}\lambda_{\tau}^{2}$, $\eta^{\prime}$, $\phi$, $r_{e}\epsilon$, $r_{\mu}$
and a complex $\zeta^{\prime}$ which control neutrino masses and mixing in
this case. Explicitly,

$\displaystyle r_{e}\epsilon$ $\displaystyle=$
$\displaystyle-0.649\,,~{}~{}r_{\mu}=-13.23\,,~{}~{}\eta^{\prime}=0.714\,,~{}~{}\phi=-0.09\,,$
$\displaystyle\zeta^{\prime}$ $\displaystyle=$ $\displaystyle
0.516+0.022\,i\,,~{}~{}m_{0}\lambda_{\tau}^{2}=0.0367\,{\rm eV}\,,$ (56)

reproduce all the neutrino squared mass differences and mixing angles within
less than 1$\sigma$ and the Dirac CP phase within 1.4$\sigma$ ranges given by
the latest global fit results, NuFIT v5.0 Esteban  _et al._ (2020). The
implied value of $\eta^{\prime}=\frac{2\kappa\epsilon
U}{\mu}=\frac{2\kappa\langle\tilde{N}_{e}\rangle}{\mu}$ determines the VEV of
$\tilde{N}_{e}$ induced by the supergravity effect. The definitions of
$\zeta^{\prime}$ and $M_{N_{1}}$ can be used to obtain a relation

$U^{2}=\left|\frac{\kappa\eta^{\prime}\mu\zeta}{\lambda\zeta^{\prime}\sin
2\phi}\right|\,.$ (57)

The values in Eq. (IV) then imply

$U=432\,{\rm
GeV}\,\times\sqrt{\left|\frac{\kappa}{\lambda}\right|}\,\times\left(\frac{\mu}{\rm
TeV}\right)^{1/2}\times\,\left(\frac{\zeta}{24.13\,{\rm GeV}}\right)^{1/2}\,.$
(58)

corresponding to a relatively low breaking of the $U(1)$ symmetry. The pGF
mass is determined as

$|M_{N_{1}}|=\left|\frac{\lambda}{\kappa}\,\eta^{\prime}\mu\,s_{\phi}c_{\phi}\right|=64\,{\rm
GeV}\,\times\left|\frac{\lambda}{\kappa}\right|\times\left(\frac{\mu}{\rm
TeV}\right)\,,$ (59)

while the heavier two neutrinos have masses $432\sqrt{|\lambda\kappa|}\,{\rm
GeV}\mp\frac{1}{2}|M_{N_{1}}|$.

The $R$-parity violating contribution to the seesaw model can be significant
with increasing $U$ and need to be included. Also, Eq. (54) is a neutrino mas
matrix as predicted by the model at the tree level. The neutrino masses also
receive higher order corrections and for the masses of RH neutrinos of ${\cal
O}(100)$ GeV, these corrections are non-negligible in general. We, therefore,
include 1-loop correction to determine the viability of this model in
reproducing the neutrino masses and mixing parameters and to derive its
reliable predictions. There are several sources of 1-loop correction to the
neutrino masses in this model. The dominant contribution arises from the self-
energy correction mediated by the lightest Higgs $H$ and $Z$ bosons Grimus and
Lavoura (2002); Aristizabal Sierra and Yaguna (2011). These are evaluated in
Grimus and Lavoura (2002) for multi Higgs doublets. Using this we obtain in
our case,

$\delta m_{\nu}^{(1)}=\frac{1}{32\pi^{2}v^{2}}\,m_{D}\,U_{N}^{*}\,{\rm
Diag.}\left(g(M_{N_{1}}),g(M_{N_{2}}),g(M_{N_{3}})\right)\,U_{N}^{\dagger}\,m_{D}^{T}\,,$
(60)

where $U_{N}$ is a unitary matrix which diagonalizes $M_{N}$ such that

$M_{N}=U_{N}{\rm Diag.}(M_{N_{1}},M_{N_{2}},M_{N_{3}})U_{N}^{T}\,,$ (61)

and

$g(M_{i})=M_{i}\left(\frac{1}{\sin^{2}\beta}\,\frac{\log\frac{M_{i}^{2}}{m_{H}^{2}}}{\frac{M_{i}^{2}}{m_{H}^{2}}-1}\,+\,3\,\frac{\log\frac{M_{i}^{2}}{m_{Z}^{2}}}{\frac{M_{i}^{2}}{m_{Z}^{2}}-1}\right)\,.$
(62)

Similar corrections at 1-loop also arise from supersymmetric particles in the
loop. We list them in Appendix B. Complete determination of the magnitude of
these correction in general requires specification of SUSY breaking sector and
depending on the later their contribution can be sizeable or sub-dominant. For
a range of parameters of our interest, we find that these contributions remain
small and $\delta m_{\nu}^{(1)}$ given in Eq. (60) captures the most relevant
and dominant 1-loop correction.

The 1-loop corrected neutrino mass matrix is therefore given by

$m_{\nu}=m^{(0)}_{\nu}+\delta m_{\nu}^{(1)}\,.$ (63)

The charged lepton mass matrix is dominantly given by $3\times 3$ matrix
$m_{l}$ which is diagonal and real. Hence, the lepton mixing matrix can be
determined from diagonalization of $m_{\nu}$ only. Note that
$\lambda_{\alpha}$, $\epsilon$ and $U$ can be chosen real without loss of
generality. $A_{0}$ and $k$ are also real if no CP violation is introduced in
the soft SUSY breaking sector. The CP violation in the lepton sector,
therefore, arises only from a parameter $\lambda$. Considering the present
generic limits on the masses of neutralinos and charginos Zyla  _et al._
(2020), we take $M_{1}\simeq M_{2}\simeq\mu=1$ TeV to evaluate the neutrino
mass spectrum. We also fix $\tan\beta\equiv v_{2}/v_{1}=2$. This implies
$\zeta=24.13$ GeV from Eq. (30) and $A_{0}=-2.8\times 10^{-4}\,{\rm GeV^{-1}}$
from Eq. (31). We fix $\kappa=1$ and choose four sample values for $U$ and
determine the remaining dimensionless eight real parameters (real $r_{e}$,
$r_{\mu}$, $\epsilon$, $\phi$, $\lambda_{\tau}$, $k$ and complex $\lambda$)
using $\chi^{2}$ minimization. The $\chi^{2}$ function includes two neutrino
squared mass differences, three mixing angles and a Dirac CP phase. We use the
values of these observables from the latest global fit results (NuFIT v5.0)
Esteban  _et al._ (2020). We also assume normal ordering in neutrino masses.

The results for four example values of $U$ are displayed in Table 1.

Parameters |  BP 1 |  BP 2 |  BP 3 |  BP 4
---|---|---|---|---
$U$ [GeV] | $10^{3}$ | $10^{4}$ | $10^{5}$ | $10^{6}$
$\kappa$ | $1.0$ | $1.0$ | $1.0$ | $1.0$
$\lambda$ | $0.5591+0.1449\,i$ | $0.6148-0.2204\,i$ | $0.7513+0.0327\,i$ | $-0.8436-0.0283\,i$
$\epsilon$ | 0.1648 | -0.0955 | 0.0276 | 0.0035
$\phi$ | 3.0965 | -3.011 | -2.8527 | -0.7434
$\lambda_{\tau}$ | $9.73\times 10^{-8}$ | $2.321\times 10^{-7}$ | $-5.895\times 10^{-7}$ | $-1.2484\times 10^{-6}$
$r_{\mu}$ | 27.2463 | -8.3797 | 4.614 | -1.5014
$r_{e}$ | -5.2322 | -140.889 | -16.6385 | -55.4134
$k$ | 0 | 0.0252 | -0.006 | 0.0005
$\Delta m^{2}_{21}$ [$10^{-5}$ eV2] | 7.42 | 7.35 | 7.42 | 7.42
$\Delta m^{2}_{31}$ [$10^{-3}$ eV2] | 2.516 | 2.517 | 2.517 | 2.517
$\sin^{2}\theta_{12}$ | 0.302 | 0.299 | 0.304 | 0.304
$\sin^{2}\theta_{23}$ | 0.578 | 0.584 | 0.573 | 0.573
$\sin^{2}\theta_{13}$ | 0.0221 | 0.0221 | 0.0222 | 0.0222
$\sin\delta_{\rm CP}$ | -0.78 | -0.57 | -0.29 | -0.29
$\sum m_{\nu_{i}}$ [eV] | 0.146 | 0.086 | 0.09 | 0.092
$|m_{\beta\beta}|$ [eV] | 0.039 | 0.016 | 0.019 | 0.019
$M_{N_{1}}$ [GeV] | 8.34 | 159.7 | 1132 | 2912
$M_{N_{2}}$ [GeV] | 581.1 | 6479 | $7.466\times 10^{4}$ | $8.426\times 10^{5}$
$M_{N_{3}}$ [GeV] | 589.5 | 6639 | $7.579\times 10^{4}$ | $8.455\times 10^{5}$
Table 1: The best fit values of model parameters, corresponding results for
neutrino masses and mixing observables and relevant predictions for four
benchmark solutions obtained for $\kappa=1$ and different values of $U$.

The solution BP 1 represent the pure seesaw case discussed earlier in Eq. (IV)
but it now includes the radiatiove correction. This is shown here for specific
values of various parameters for comparison. The benchmark points BP 3 and BP
4 reproduce the central values of neutrino squared mass differences and mixing
parameters as obtained from the latest global fit results Esteban  _et al._
(2020). BP 1 and BP 2 reproduce all the observables within $1.3\sigma$. It is
seen that the 1-loop correction considered in Eq. (60) significantly modifies
one of the elements of $m_{\nu}$ from its tree level value. This is shown
explicitly in Appendix B where we also discuss the strengths of the other
1-loop corrections. We also repeat the same analysis but for $\kappa=0$ and
the corresponding results are discussed in Appendix C.

In Table 1, we also list predictions for the sum of the three light neutrino
masses, the effective mass parameter relevant for neutrinoless double beta
decay (i.e. $m_{\beta\beta}=U_{ei}^{2}\,m_{\nu_{i}}$), and mass spectrum of
the RH neutrinos. It can be seen that all the benchmark points predict
relatively degenerate mass spectrum for the light neutrinos. The obtained
value of $|m_{\beta\beta}|$ is smaller than the current upper limits from
various recent experiments seearching for neutrinoless double beta decay as
listed in Dolinski  _et al._ (2019). The strongest constraint on the neutrino
mass spectrum of the model comes from the cosmological observations. The
current limit from Planck collaboration is $\sum m_{\nu_{i}}<0.12$ eV Aghanim
_et al._ (2020) (see also Vagnozzi  _et al._ (2017)) which seems to be in
conflict with the prediction of BP 1. However, in deriving the constraint on
the sum of the neutrino masses from cosmology, it is assumed that the
neutrinos are stable or atleast their lifetime is greater than the age of the
universe. If neutrinos decay into invisible radiation on the time scales
shorter than the age of the universe, the above constraint can be replaced by
a much weaker limit Chacko  _et al._ (2020), as weak as $\sum m_{\nu_{i}}<0.9$
eV. In our model, neutrinos can decay into Majoron allowing such a
possibility. We compute the neutrino lifetime in the next section and show
that the solution BP 1 can evade the strongest limit444Alternatively, the
stricter limit on $\sum m_{\nu_{i}}$ can also be relaxed in models based on
dynamical dark energy with arbitrary equation of state Vagnozzi  _et al._
(2018)..

A pair of the heavier neutrinos is quasi-degenerate while the mass of the pGF
is predicted to be $<{\cal O}(200)$ GeV for some of the benchmark points. The
later is within the kinematic reach of the LHC. However, it’s production and
detection involves coupling with the SM particles which arise through active-
sterile neutrino mixing in the standard scenario and hence the current bounds
are sensitive to such mixing. We estimate this in section VI and obtain that
the magnitude of the underlying mixing is $\lesssim 10^{-6}$. For such a small
mixing, the current ATLAS Aad  _et al._ (2015) and CMS Khachatryan  _et al._
(2015, 2016) analysis do not put any limit on the RH neutrino masses.

##  V Majoron and its couplings

As a consequence of spontaneous breaking of $U(1)_{L_{\mu}-L_{\tau}}$, there
exists a massless Goldstone boson - Majoron - which was primarily given by the
linear combination Eq. (6). Since two of the three lepton doublets are also
charged under $U(1)_{L_{\mu}-L_{\tau}}$, this structure gets modified to

$J\simeq\cos\phi\,\tilde{N}_{I\tau}-\sin\phi\,\tilde{N}_{I\mu}+\frac{\omega^{\prime}_{\mu}}{U}\,\tilde{\nu}_{I\mu}-\frac{\omega^{\prime}_{\tau}}{U}\,\tilde{\nu}_{I\tau}\,.$
(64)

Majoron couples to the leptons at the tree level in this model while the
couplings with the other SM particles can arise through loop(s). The leptonic
Majoron couplings are strongly constrained from cosmology and astrophysics.

The strongest constraints on the neutrino-Majoron couplings, denoted by
$g_{ij}$ here, come from the observed CMB spectrum which requires that the
neutrinos should be free streaming at the time of the photon decoupling. This
restricts both the diagonal and the off-diagonal couplings Hannestad and
Raffelt (2005); Raffelt and Weiss (1995); Ayala  _et al._ (2014). For the
diagonal neutrino couplings with Majoron, the constraint reads
$|g_{ii}|\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10^{-7}$
while the off-diagonal couplings are required to be

$|g_{ij}|\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>0.61\times
10^{-11}\,\left(\frac{0.05\,{\rm eV}}{m_{\nu_{i}}}\right)^{2}$ (65)

for the hierarchical neutrino masses. The off-diagonal couplings
$h_{\alpha\beta}$ of Majoron to the charged leptons are constrained from the
non observation of the decay $e_{\alpha}\rightarrow e_{\beta}J\,.$ The
branching ratio of $\alpha$ lepton decaying into Majoron and $\beta$ lepton is
estimated as

${\rm
BR}[\alpha\to\beta\,J]=\frac{\Gamma[\alpha\to\beta\,J]}{\Gamma[\alpha\to\beta\,\nu_{\alpha}\,\bar{\nu}_{\beta}]}\simeq\frac{12\pi^{2}}{G_{F}^{2}m_{\alpha}^{4}}\,|h_{\alpha\beta}|^{2}\,,$
(66)

for $m_{\alpha}\gg m_{\beta},m_{J}$. The experimental limits ${\rm BR}[\mu\to
e\,J]\leq 2.6\times 10^{-6}$ Jodidio  _et al._ (1986) and ${\rm BR}[\tau\to
e/\mu\,J]\leq 10^{-3}$ Albrecht  _et al._ (1995) require

$|h_{\mu e}|<1.9\times 10^{-11}\,,~{}~{}|h_{\tau e}|,|h_{\tau\mu}|<1.1\times
10^{-7}\,.$ (67)

The electron coupling to Majoran is contained from the energy loss from red
giant and require Raffelt and Weiss (1995); Ayala  _et al._ (2014)

$|h_{ee}|<2.57\times 10^{-13}\,.$ (68)

We systematically evaluate leptonic couplings to Majoron below and show that
the above bounds are satisfied for the set of parameters which give the
correct neutrino spectrum. In particular, the Majoron coupling to electron is
absent in the model in the seesaw limit and $g_{ee}$ much smaller than the
value required above arise from higher order terms in the seesaw expansion.

###  V.1 Couplings with neutrinos

In the $U(1)_{L_{\mu}-L_{\tau}}$ based model, the dominant interaction of the
light neutrinos with Majoron arises from primarily two sources: (a) through
light-heavy neutrino mixing from $W_{N}$ in Eq. (1), and (b) through neutrino-
neutralino mixing from $W$ in Eq. (40) and gauge interactions.

From the F-term of $W_{N}$ and substitution given in Eq. (5), the relevant
interaction terms are

${\cal L}^{(a)}_{\nu
J}=-\frac{i\lambda}{\sqrt{2}}\,\left(N_{\tau}^{T}C^{-1}N_{e}\,\tilde{N}_{I\mu}+N^{T}_{\mu}C^{-1}N_{e}\,\tilde{N}_{I\tau}\right)+{\rm
h.c.}\,.$ (69)

The above gives rise to coupling between Majoron and the light neutrinos
through heavy-light neutrino mixing. The later can be computed using
diagonalization of the relevant blocks of neutral fermion mass matrix given in
Eq. (21) with $M_{N}$, $m_{\nu\chi}$, $M_{\chi}$ as written in Eqs. (14,23,24)
respectively and $m_{D}$, $m_{N\chi}$ given in Eq. (42). The full matrix
${\cal M}_{\cal N}$ is block diagonalized at the leading order using a unitary
matrix

$U_{\cal N}=\left(\begin{array}[]{ccc}1&\rho_{D}&\rho_{\nu\chi}\\\
-\rho_{D}^{\dagger}&1&\rho_{N\chi}\\\
-\rho_{\nu\chi}^{\dagger}&-\rho_{N\chi}^{\dagger}&1\end{array}\right)\,+\,{\cal
O}(\rho^{2})\,.$ (70)

such that $U_{\cal N}^{T}{\cal M}_{\cal N}U_{\cal N}$ is an approximate block
diagonal matrix. Here,

$\rho_{D}\simeq(m_{D}M_{N}^{-1})^{*}\,,~{}~{}\rho_{\nu\chi}\simeq(m_{\nu\chi}M_{\chi}^{-1})^{*}\,,~{}~{}\rho_{\chi
N}\simeq(m_{N\chi}M_{\chi}^{-1})^{*}\,.$ (71)

The neutral fermions in the block-diagonal basis are then obtained from the
original basis as $U_{\cal N}^{\dagger}\psi^{0}$. Using this and denoting the
light neutrinos in block diagonal basis as $\nu_{\alpha}^{\prime}$, we find

$\displaystyle N_{e}$ $\displaystyle=$
$\displaystyle-(\rho_{D}^{\dagger})_{1\alpha}\,\nu_{L\alpha}^{\prime}=(-1)^{1-\delta_{\alpha
1}}\,B_{0}\,\epsilon\,\epsilon_{\alpha}\,\nu_{L\alpha}^{\prime}\,,$
$\displaystyle N_{\mu}$ $\displaystyle=$
$\displaystyle-(\rho_{D}^{\dagger})_{2\alpha}\,\nu_{L\alpha}^{\prime}=(-1)^{1-\delta_{\alpha
2}}\,B_{0}\,\sin\phi\,\epsilon_{\alpha}\,\nu_{L\alpha}^{\prime}\,,$
$\displaystyle N_{\tau}$ $\displaystyle=$
$\displaystyle-(\rho_{D}^{\dagger})_{3\alpha}\,\nu_{L\alpha}^{\prime}=(-1)^{1-\delta_{\alpha
3}}\,B_{0}\,\cos\phi\,\epsilon_{\alpha}\,\nu_{L\alpha}^{\prime}\,,$ (72)

where

$B_{0}=\frac{v_{2}}{U^{2}\lambda\epsilon\sin
2\phi}=\frac{v_{2}}{M_{N_{1}}U}\,.$ (73)

Substituting Eq. (V.1) in Eq. (69) and use of Eq. (64) determine the couplings
of heavy and light neutrinos with Majoron. As we are interested in the latter,
the relevant part is obtained as

${\cal L}^{(a)}_{\nu
J}=-i\frac{B_{0}}{\sqrt{2}U}\,\frac{v_{2}}{U}\,(-1)^{(1-\delta_{\beta
1})}\,\epsilon_{\beta}\left(\epsilon_{\mu}\,\nu_{L\mu}^{\prime
T}-\epsilon_{\tau}\,\nu_{L\tau}^{\prime
T}\right)C^{-1}\nu_{L\beta}^{\prime}\,J\,+{\rm h.c.}\,.$ (74)

In the physical basis of the light neutrinos,
$\nu_{L\alpha}^{\prime}=U_{\alpha i}\,\nu_{Li}$ where $U_{\alpha i}$ are the
elements of the PMNS matrix, the above term is written as

${\cal L}^{(a)}_{\nu J}=-ig_{ij}^{(a)}\,\nu^{T}_{Li}C^{-1}\nu_{Lj}\,J\,+{\rm
h.c.}\,,$ (75)

with

$g^{(a)}_{ij}=\frac{B_{0}}{\sqrt{2}U}\,\frac{v_{2}}{U}\,\sum_{\beta}\,(-1)^{(1-\delta_{\beta
1})}\,\left(\epsilon_{\mu}\,U_{\mu i}-\epsilon_{\tau}\,U_{\tau
i}\right)\,\epsilon_{\beta}\,U_{\beta j}\,.$ (76)

We now evaluate the neutrino-Majoron coupling which arise from the neutrino-
neutralino mixing. The relevant Lagrangian can be read from Eq. (40) and gauge
interaction as

$\displaystyle{\cal L}^{(b)}_{\nu J}$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{2}}\left(\lambda_{\mu}\,\nu_{L\mu}^{T}\tilde{N}_{I\mu}+\lambda_{\tau}\,\nu_{L\tau}^{T}\tilde{N}_{I\mu}\right)C^{-1}\tilde{h}_{2}^{0}+\frac{i}{2}(g^{\prime}\tilde{B}^{T}-g\tilde{W}^{0T})C^{-1}\nu_{L\beta}\,\tilde{\nu}_{I\beta}+{\rm
h.c.}\,,$ (77)

where we define $-i\lambda_{1}\equiv\tilde{B}$ and
$-i\lambda_{2}\equiv\tilde{W}^{0}$. The neutral fermions $\tilde{h}_{2}^{0}$,
$\tilde{B}$ and $\tilde{W}^{0}$ in the above contain the light neutrinos when
${\cal M}_{\cal N}$ is block diagonalized. From Eqs. (70,71), we obtain

$\displaystyle\tilde{h}_{2}^{0}$ $\displaystyle=$
$\displaystyle-(\rho_{\nu\chi}^{\dagger})_{4\alpha}\,\nu_{L\alpha}^{\prime}=\frac{v_{1}}{\mu}A_{0}\,\omega_{\alpha}\,U_{\alpha
i}\,\nu_{Li}\,,$ $\displaystyle\tilde{B}$ $\displaystyle=$
$\displaystyle-(\rho_{\nu\chi}^{\dagger})_{1\alpha}\,\nu_{L\alpha}^{\prime}=-\frac{\sqrt{2}g^{\prime}M_{2}}{g^{2}M_{1}+g^{\prime
2}M_{2}}\,A_{0}\,\omega_{\alpha}\,U_{\alpha i}\,\nu_{Li}\,,$
$\displaystyle\tilde{W}^{0}$ $\displaystyle=$
$\displaystyle-(\rho_{\nu\chi}^{\dagger})_{2\alpha}\,\nu_{L\alpha}^{\prime}=\frac{\sqrt{2}gM_{1}}{g^{2}M_{1}+g^{\prime
2}M_{2}}\,A_{0}\,\omega_{\alpha}\,U_{\alpha i}\,\nu_{Li}\,,$ (78)

where $A_{0}$ and $\omega_{\alpha}$ are defined in Eqs. (31,34). Substituting
the above relations in Eq. (77) and use of Eq. (64) lead to the following
interaction between the SM neutrinos and Majoron:

${\cal L}^{(b)}_{\nu J}=-ig_{ij}^{(b)}\,\nu^{T}_{Li}C^{-1}\nu_{Lj}\,J\,+{\rm
h.c.}\,,$ (79)

with

$g^{(b)}_{ij}=\frac{A_{0}}{\sqrt{2}U}\,\sum_{\beta}\,\left(\omega_{\mu}\,U_{\mu
i}-\omega_{\tau}\,U_{\tau i}\right)\,\omega_{\beta}\,U_{\beta j}\,.$ (80)

Combining the two contributions, the light neutrino-Majoron coupling is
parametrized as

$-{\cal L}_{\nu J}=ig_{ij}\,\nu^{T}_{Li}C^{-1}\nu_{Lj}\,J\,+{\rm h.c.}\,,$
(81)

with

$g_{ij}=g_{ij}^{(a)}+g_{ij}^{(b)}=\frac{B_{0}v_{2}}{\sqrt{2}\,U^{2}}\,\sum_{\beta}\left((-1)^{(1-\delta_{\beta
1})}+\eta_{0}\right)\left(\epsilon_{\mu}\,U_{\mu i}-\epsilon_{\tau}\,U_{\tau
i}\right)\,\epsilon_{\beta}\,U_{\beta j}\,,$ (82)

where we have used $\omega_{\alpha}=k\epsilon_{\alpha}$ and

$\eta_{0}=\frac{A_{0}k^{2}U^{2}M_{N_{1}}}{v_{2}^{2}}\,.$ (83)

Note that for $\kappa\to 0$, $\eta_{0}$ denotes the relative strength of
$R$-parity violating contribution compared to the standard seesaw contribution
to the neutrino masses. For $|\eta_{0}|<1$, the neutrino-Majoron coupling is
dominantly determined by the light-heavy neutrino mixing while for
$|\eta_{0}|>1$ it mainly arises from the neutrino-neutralino mixing.

The interaction in Eq. (81) leads to neutrino decay into Majoron whenever it
is kinematically allowed. The decay width can be estimated as

$\Gamma[\nu_{j}\to\nu_{i}\,J]=\frac{\lambda^{1/2}[m_{\nu j}^{2},m_{\nu
i}^{2},m_{J}^{2}]}{16\pi\,m_{\nu j}^{3}}\,|g_{ji}|^{2}\,\left(m_{\nu
j}^{2}+m_{\nu i}^{2}-m_{J}^{2}-2m_{\nu i}m_{J}\right)\,,$ (84)

where $\lambda[x,y,z]=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$. Assuming massless
Majoron, we compute the total decay width of $\nu_{3}$ as
$\Gamma_{\nu}\equiv\Gamma[\nu_{3}\to\nu_{2}\,J]+\Gamma[\nu_{3}\to\nu_{1}\,J]$
and the lifetime $\tau_{\nu}=1/\Gamma_{\nu}$ for the benchmark solutions
listed in Table 1. The results are displayed in Table 2 where we also list the
obtained value of $|\eta_{0}|$ and the relevant off-diagonal and the largest
diagonal couplings $|g_{ij}|$.

Parameters |  BP 1 |  BP 2 |  BP 3 |  BP 4
---|---|---|---|---
$|\eta_{0}|$ | 0 | 0.118 | 4.79 | 7.67
${\rm Max.}\\{|g_{ii}|\\}$ | $1.2\times 10^{-14}$ | $5.8\times 10^{-15}$ | $1.8\times 10^{-16}$ | $2.8\times 10^{-17}$
$|g_{32}|$ | $3.3\times 10^{-14}$ | $8.9\times 10^{-16}$ | $4.5\times 10^{-17}$ | $6.2\times 10^{-18}$
$|g_{31}|$ | $4.0\times 10^{-14}$ | $1.4\times 10^{-15}$ | $4.1\times 10^{-17}$ | $4.9\times 10^{-18}$
$\tau_{\nu}$ [sec.] | $2.3\times 10^{14}$ | $2.4\times 10^{17}$ | $1.7\times 10^{20}$ | $1.0\times 10^{22}$
Table 2: Predictions for the largest diagonal coupling of neutrinos with
Majoron, off-diagonal couplings relevant for neutrino decays and the
corresponding neutrino lifetime for the benchmark solutions.

The neutrino lifetime turns out to be smaller than the age of the universe for
solution BP 1 and hence the stringent limit on the sum of neutrino masses does
not apply on this solutions as discussed earlier. The derived values of the
neutrino-Majoron couplings for all the benchmark solutions obey the limits Eq.
(65).

###  V.2 Couplings with charged leptons

The chargino spectrum of the model consists of three pairs of the charged
lepton fields and pairs of charged wino and Higgsino. In the basis
$\psi_{+}=\left(e^{c}_{\alpha},-i\lambda^{+},h_{2}^{+}\right)^{T}$ and
$\psi_{-}=\left(e_{\alpha},-i\lambda^{-},h_{1}^{-}\right)^{T}$, the charged
fermion masses are given by

$-{\cal L}_{\cal C}=\psi_{-}^{T}\,{\cal M}_{\cal C}\,\psi_{+}\,+\,{\rm
h.c.}\,,$ (85)

with

${\cal M}_{\cal C}=\left(\begin{array}[]{cc}m_{l}&m_{lC}\\\
m_{lC}^{\prime}&M_{C}\end{array}\right)\,,$ (86)

and

$m_{l}=v_{1}\,{\rm
Diag.}\left(\lambda_{e}^{\prime},\lambda_{\mu}^{\prime},\lambda_{\tau}^{\prime}\right)\,,~{}~{}M_{C}=\left(\begin{array}[]{cc}M_{2}&gv_{2}\\\
gv_{1}&\mu\end{array}\right)\,,$ (87)
$m_{lC}=\left(\begin{array}[]{cc}g\omega^{\prime}_{e}&\epsilon_{e}\\\
g\omega^{\prime}_{\mu}&\epsilon_{\mu}\\\
g\omega^{\prime}_{\tau}&\epsilon_{\tau}\end{array}\right)\,,~{}~{}m_{lC}^{\prime}=\left(\begin{array}[]{ccc}0&0&0\\\
-\omega^{\prime}_{e}\lambda^{\prime}_{e}&-\omega^{\prime}_{\mu}\lambda^{\prime}_{\mu}&-\omega^{\prime}_{\tau}\lambda^{\prime}_{\tau}\end{array}\right)\,.$
(88)

From $W$ in Eq. (40) and gauge interactions, we get the following interaction
term involving the charged fermions and Majoron

$\displaystyle{\cal L}_{lJ}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\,\left(\lambda_{\mu}\,\mu\tilde{N}_{I\mu}+\lambda_{\tau}\,\tau\tilde{N}_{I\tau}\right)\tilde{h}_{2}^{+}+\frac{i}{\sqrt{2}}\left(\lambda^{\prime}_{\mu}\,\tilde{\nu}_{I\mu}\mu^{c}+\lambda^{\prime}_{\tau}\,\tilde{\nu}_{I\tau}\tau^{c}\right)\tilde{h}_{1}^{-}\,$
(89) $\displaystyle-$
$\displaystyle\frac{i}{\sqrt{2}}\,g\left(\tilde{\nu}_{I\mu}\,\mu+\tilde{\nu}_{I\tau}\,\tau\right)(-i\lambda^{+})\,+\,{\rm
h.c.}\,.$

The chargino-charged lepton mixing can be evaluated by block diagonalizing the
matrix ${\cal M}_{\cal C}$ given in Eq. (86). For $m_{l}<m_{lC}<M_{C}$, this
is done using the following leading order unitary matrices

$U_{+}=\left(\begin{array}[]{cc}1&(M_{C}^{-1}m_{lC}^{\prime})^{\dagger}\\\
-M_{C}^{-1}m_{lC}^{\prime}&1\end{array}\right)\,,~{}~{}U_{-}=\left(\begin{array}[]{cc}1&(m_{lC}M_{C}^{-1})^{*}\\\
-(m_{lC}M_{C}^{-1})^{T}&1\end{array}\right)\,.$ (90)

such that

$U_{-}^{T}\,{\cal M}_{\cal
C}\,U_{+}\approx\left(\begin{array}[]{cc}m_{l}-m_{lC}\,M_{C}^{-1}\,m_{lC}^{\prime}&0\\\
0&M_{C}\end{array}\right)\,.$ (91)

The fields $\tilde{h}_{2}^{+}$, $\lambda^{+}$ and $\tilde{h}_{1}^{-}$ can then
be written in the block diagonal basis of the charged leptons,
$e^{\prime}_{\alpha}$ and $e^{c\prime}_{\alpha}$, as

$\displaystyle h_{2}^{+}$ $\displaystyle=$
$\displaystyle-\left(M_{C}^{-1}m_{lC}^{\prime}\right)_{5\alpha}\,e^{c\prime}_{\alpha}=\frac{M_{2}}{M_{2}\mu-g^{2}v_{1}v_{2}}\,\omega^{\prime}_{\alpha}\lambda^{\prime}_{\alpha}\,e^{c\prime}_{\alpha}\,,$
$\displaystyle-i\lambda^{+}$ $\displaystyle=$
$\displaystyle-\left(M_{C}^{-1}m_{lC}^{\prime}\right)_{4\alpha}\,e^{c\prime}_{\alpha}=-\frac{gv_{2}}{M_{2}\mu-g^{2}v_{1}v_{2}}\,\omega^{\prime}_{\alpha}\lambda^{\prime}_{\alpha}\,e^{c\prime}_{\alpha}\,\,,$
$\displaystyle h_{1}^{-}$ $\displaystyle=$
$\displaystyle-\left((M_{C}^{-1})^{T}m_{lC}^{T}\right)_{5\alpha}\,e^{\prime}_{\alpha}=-\frac{M_{2}\epsilon_{\alpha}-g^{2}v_{2}\omega_{\alpha}^{\prime}}{M_{2}\mu-g^{2}v_{1}v_{2}}\,e^{\prime}_{\alpha}\,\,.$
(92)

Substitution of Eqs. (V.2,64) in Eq. (89) leads to

$-{\cal
L}_{lJ}=\frac{i}{\sqrt{2}}\frac{1}{Uv_{1}}\,\left(\omega^{\prime}_{\alpha}m_{\alpha}\xi_{\mu}\,\mu^{\prime}\,e^{c\prime}_{\alpha}+\omega_{\mu}m_{\mu}\xi_{\alpha}\,e^{\prime}_{\alpha}\,\mu^{c\prime}-(\mu\to\tau)\right)\,J\,+\,{\rm
h.c.}\,,$ (93)

where $m_{\alpha}\simeq v_{1}\lambda^{\prime}_{\alpha}$ are the charged lepton
masses, and

$\xi_{\alpha}=\frac{M_{2}\epsilon_{\alpha}-g^{2}v_{2}\omega^{\prime}_{\alpha}}{M_{2}\mu-g^{2}v_{1}v_{2}}\,.$
(94)

As can be seen from Eq. (91) the charged lepton mass matrix,
$M_{l}=m_{l}-m_{lC}\,M_{C}^{-1}m_{lC}^{\prime}$, in the block diagonal basis
is approximately diagonal since $m_{l}$ is diagonal. The primed fields and
corresponding physical states are related with each other by a unitary matrix
very close to the identity matrix. Hence, the Majoron dominantly couples to
the second and third generation charged leptons and its coupling to a pair of
electrons arises only through $e-\alpha$ mixing $U_{e\alpha}\approx{\cal
O}\left(\frac{\epsilon_{e}\lambda^{\prime}_{\alpha}\omega^{\prime}_{\alpha}}{m_{e}\mu}\right)$
induced by the seesaw correction $m_{lC}M_{C}^{-1}m^{\prime}_{lC}$.

We estimate the magnitude of the coupling appearing in Eq. (93) for the
benchmark solutions listed in Table 1. $\omega_{\alpha}^{\prime}$ can be
expressed in terms of the known parameters using Eqs. (34,36). For
$\mu,M_{2}\gg v$, one finds

$h_{\alpha\beta}\equiv\frac{\omega^{\prime}_{\alpha}m_{\alpha}\xi_{\beta}}{\sqrt{2}Uv_{1}}\simeq\left(k+\frac{v_{1}}{\mu}\right)\frac{m_{\alpha}\epsilon_{\alpha}\epsilon_{\beta}}{\sqrt{2}\,v_{1}\mu
U}\,.$ (95)

The magnitudes of the largest value of $h_{\alpha\beta}$ obtained for
different benchmark points are given in Table 3. These are consistent with the
experimental bounds given in Eq.(67).

Parameters |  BP 1 |  BP 2 |  BP 3 |  BP 4
---|---|---|---|---
${\rm Max.}\\{|h_{\alpha\beta}|\\}$ | $10^{-17}$ | $10^{-15}$ | $10^{-14}$ | $10^{-13}$
Table 3: Maximum value of charged leptons’ coupling with Majoron obtained for
the different benchmark solutions.

The strength of couplings is larger for solutions with the higher $U(1)$
breaking scale in contrast to neutrino-Majoron coupling. As mentioned earlier,
the coupling of Majoron with electrons will be further suppressed and will be
much smaller than $10^{-13}$ for all the solutions. This is in agreement with
the astrophysical bounds from red giant Raffelt and Weiss (1995); Ayala  _et
al._ (2014) and SN 1987A Keil  _et al._ (1997).

##  VI Direct searches of pseudo-Goldstone fermion

In the standard seesaw models, the direct searches of heavy neutrinos
typically depend on production mechanisms which involve light-heavy neutrino
mixing. Therefore, the production cross section is suppressed for small
mixings between the active and sterile neutrinos Pilaftsis (1992). This
picture changes significantly in the present framework as the lightest RH
neutrino, which is a pseudo-Goldstone fermion, can be produced from the decays
of neutralinos Lavignac and Medina (2021). The production of Higgsino-like
neutralino state is governed by electroweak interactions and it is not
suppressed in general. The RH neutrinos mix with neutralinos through mass
mixing term $m_{\chi N}$ in Eq. (21). If this mixing is large enough then
$N_{1}$ is produced in the decays of Higgsinos. For $M_{N_{1}}\ll
m_{\chi^{0}}$, the produced heavy neutrino is boosted and its subsequent
decays into $W$ and charged leptons or $Z$ and light neutrinos give rise to
displaced vertex signature. Such a signature has a better visibility due to
less SM background and hence it improves significantly the prospects of
detecting the pGF. Note that the decays of $N_{1}$ in the visible particles
are still governed by the light-heavy neutrino mixing. The details of the
displaced vertex signature of pGF of this kind are recently discussed in
Lavignac and Medina (2021).

The parameter of primary interest here is the mixing between RH neutrinos and
neutralino. This can be estimated following the same procedure outlined in
section IV-A. One finds from Eq. (70), $N_{\alpha}=(\rho_{N\chi})_{\alpha
m}\,\psi_{0,m}$ with $m=7,..,10$. This leads to the following leading order
mixing between the RH neutrinos and neutralinos in the block-diagonal basis:

$\displaystyle N_{e}$ $\displaystyle\simeq$ $\displaystyle\frac{2\kappa
A_{0}vM_{1}M_{2}}{\mu(g^{2}M_{1}+g^{\prime
2}M_{2})}\,\left(\cos\beta\,\tilde{h}^{0}_{1}+\sin\beta\,\tilde{h}^{0}_{2}+\frac{g^{\prime}v}{\sqrt{2}M_{1}}\cos
2\beta\,\tilde{B}-\frac{gv}{\sqrt{2}M_{2}}\cos 2\beta\,\tilde{W^{0}}\right).$
(96)

We have neglected $\lambda_{\alpha}$ dependent terms in $m_{N\chi}$ while
deriving the above. $N_{\mu,\tau}$ do not contain any neutralino state at the
leading order. Substituting the above in Eq. (16), one can determine the
mixing between the pGF $N_{1}$ and neutralinos. As it can be seen, $N_{1}$ has
dominant mixing with the Higgsino states and they are obtained as

$\displaystyle V_{{N_{1}}\tilde{h}^{0}_{1}}$ $\displaystyle=$
$\displaystyle-\left(\frac{2A_{0}vM_{1}M_{2}}{\mu(g^{2}M_{1}+g^{\prime
2}M_{2})}\right)\,\kappa\epsilon\cos
2\phi\,\cos\beta\,\simeq\frac{v}{\mu}\kappa\epsilon\cos 2\phi\,\cos\beta\,,$
$\displaystyle V_{{N_{1}}\tilde{h}^{0}_{2}}$ $\displaystyle\simeq$
$\displaystyle\tan\beta\,V_{{N_{1}}\tilde{h}^{0}_{1}}\,,$ (97)

for $v\ll M_{1,2}$. This can be compared with the $N_{1}$-$\nu_{i}$ mixing
which can be read-off from similar expressions, Eq. (V.1), as

$V_{{N_{1}}{\nu_{i}}}=V_{{N_{1}}{\nu_{\alpha}}}\,U_{\alpha i}\,,$ (98)

where

$\displaystyle V_{{N_{1}}{\nu_{e}}}$ $\displaystyle\simeq$ $\displaystyle-
B_{0}\,\epsilon_{e}\cos 2\phi\,(1+\epsilon^{2})\,,$ $\displaystyle
V_{{N_{1}}{\nu_{\mu}}}$ $\displaystyle\simeq$ $\displaystyle-
B_{0}\,\epsilon_{\mu}\,(1-\epsilon^{2}\cos 2\phi)\,,$ $\displaystyle
V_{{N_{1}}{\nu_{\tau}}}$ $\displaystyle\simeq$ $\displaystyle
B_{0}\,\epsilon_{\tau}\,(1+\epsilon^{2}\cos 2\phi)\,.$ (99)

For $\kappa\sim{\cal O}(1)$, one finds
$|V_{{N_{1}}\tilde{h}^{0}_{1,2}}|\gg|V_{{N_{1}}{\nu_{i}}}|$ as the later is
suppressed by Yukawa couplings $\lambda_{\alpha}$. The estimated values of the
above mixing for the different benchmark solutions are listed in Table 4.

Parameters |  BP 1 |  BP 2 |  BP 3 |  BP 4
---|---|---|---|---
$\left|V_{{N_{1}}\tilde{h}^{0}_{1}}\right|$ | 0.013 | 0.0072 | 0.0007 | 0.0011
$\left|V_{{N_{1}}{\nu_{1}}}\right|$ | $1.6\times 10^{-6}$ | $2.6\times 10^{-6}$ | $2.7\times 10^{-8}$ | $6.4\times 10^{-8}$
$\left|V_{{N_{1}}{\nu_{2}}}\right|$ | $1.0\times 10^{-6}$ | $1.3\times 10^{-6}$ | $3.2\times 10^{-7}$ | $3.9\times 10^{-7}$
$\left|V_{{N_{1}}{\nu_{3}}}\right|$ | $2.5\times 10^{-6}$ | $4.7\times 10^{-7}$ | $1.1\times 10^{-7}$ | $2.6\times 10^{-7}$
Table 4: Estimated magnitudes of the pseudo-Goldstone fermion’s mixing with
Higgsino and light neutrinos for the benchmark solutions.

For the first two solutions, the obtained values of mixings are such that the
pGF with mass less than $200$ GeV can be detected through displaced vertex
signature at the LHC Lavignac and Medina (2021).

##  VII Summary and Discussions

Possibility of a RH neutrino being a Goldstone fermion of a spontaneously
broken global $U(1)$ symmetry in the supersymmetric theories offers an
interesting opportunity to confirm or otherwise, the mechanism of neutrino
mass generation in direct search experiments. The singlet fermion is naturally
light with mass of the order of gravitino mass scale in minimal supergravity
like setup. Moreover, it can couple to some of the supersymmetric particles
with large enough couplings such that the former can be produced at colliders
through decays of the later. Investigating this general framework in detail,
we find that the observed neutrino mass spectrum severely restricts the nature
of $U(1)$ symmetry.

The light neutrino masses arise in the framework through the standard type I
seesaw mechanism as well as through the $R$-parity violating RH snuetrino
VEVs. Because of their common origin, these contributions are highly
correlated. It is shown that if all three families of the SM leptons have the
same $U(1)$ charge and the soft SUSY breaking terms are flavour universal then
two of the three light neutrinos remain massless despite the presence of more
than one sources of neutrino mass generation. One requires large flavour
violations in the soft sector to induce the solar neutrino mass scale with one
neutrino still remaining massless. We show that this picture changes
significantly when the underlying $U(1)$ is flavour dependent symmetry.
Considering a particular example of $L_{\mu}-L_{\tau}$ symmetry, it is shown
that viable neutrino masses and mixing can be obtained without introducing any
flavour violation in the soft sector.

The framework also contains a Majoron - the Goldstone boson of
$U(1)_{L_{\mu}-L_{\tau}}$. We derive its couplings with the SM leptons and
show that it couples weakly enough to evade the present constraints from
cosmology, the stellar cooling as well as the constraints from the flavour
violating decays of the charged leptons. Given this, and the fact that the
viable neutrino mass spectrum can be achieved without any explicit breaking of
$U(1)_{L_{\mu}-L_{\tau}}$, the Majoron can remain massless in the present
framework. Light neutrinos can also decay into Majoron in this case and we
find that the neutrino lifetime can be of the order of the age of universe or
smaller if the scale of $U(1)$ breaking is $\lesssim\,10^{4}$ GeV.

The nature of neutrino mass generation mechanism in this framework appears
similar to the one in $\mu\nu$SSM model Lopez-Fogliani and Munoz (2006);
Escudero  _et al._ (2008); Ghosh and Roy (2009); Fidalgo  _et al._ (2009). In
the later, the bilinear terms including the $\mu$ term are assumed to be
absent at the begining and they arise from the trilinear terms when RH
neutrinos take VEVs. Typically, the most general trilinear terms are
considered which explicitly violate the $U(1)$ lepton number. As a result,
both the Dirac and RH neutrino mass matrices take general form in $\mu\nu$SSM.
In contrast to this, the present framework uses a flavour dependent
combination of lepton number as a global symmetry which leads to highly
restricted forms of $m_{D}$ and $M_{N}$ and still gives viable explanation of
lepton spectrum. More importantly, the structure of the RH neutrino masses is
determined here from the combined requirement of the unbroken global
supersymmetry and the spontaneous violation of the $U(1)$ symmetry. The $U(1)$
symmetry protects the masses of a boson and its superpartner which is
identified as the lightest RH neutrino. The later receives mass from the local
SUSY breaking effects. A weak scale RH neutrino as pGF, therefore, naturally
emerge in this setup. Thus, the formalism presented here is more constrained
than the $\mu\nu$SSM. If $\mu_{0}=0$ set by some mechanism, this shift in VEVs
also generates a TeV scale $\mu$ parameter in the present framework in a
simlar way discussed recently in Dvali and Jankowsky (2020). In this case, one
typically finds $\mu\sim M_{N_{1}}\sim m_{3/2}$ and hence sub-TeV scale RH
neutrino may face strong constraints from non-observation of neutralinos
and/or charginos.

The existence of a pGF coupled to Higgsino and the charged leptons offers an
interesting possibility to search for this fermion at colliders through the
displaced vertex signal. It is shown that such a signal can be observed
Lavignac and Medina (2021) for the parameters of the model required to fit the
neutrino data.

## Acknowledgements

The work of KMP is partially supported by a research grant under INSPIRE
Faculty Award (DST/INSPIRE/04/2015/000508) from the Department of Science and
Technology, Government of India. The computational work reported in this paper
was performed on the High Performance Computing (HPC) resources (Vikram-100
HPC cluster) at the Physical Research Laboratory, Ahmedabad.

##  Appendix A Sneutrino VEVs and $k$ factor

In this Appendix, we show the origin of Eq. (36) in the present framework. The
procedure is similar to the one carried out previously in Joshipura  _et al._
(2002); Giudice  _et al._ (1993). When the scalar components of
$\hat{N}_{\alpha}$ take VEVs, the bilinear terms in $W$ are given by

$W\supset-\,\epsilon_{\alpha}\,\hat{L}_{\alpha}\hat{H}_{2}\,-\,\mu\,\hat{H}_{1}\hat{H}_{2}\,.$
(100)

The $F$ and $D$ term potential evaluated from the above terms is

$\displaystyle V_{F,D}$ $\displaystyle=$
$\displaystyle\left|\mu\tilde{H}_{1}+\epsilon_{\alpha}\tilde{L}_{\alpha}\right|^{2}+\left(|\mu|^{2}+|\epsilon_{\alpha}|^{2}\right)|\tilde{H}_{2}|^{2}+\frac{1}{8}(g^{2}+g^{\prime
2})\left(|\tilde{H}_{1}|^{2}-|\tilde{H}_{2}|^{2}+|\tilde{L}_{\alpha}|^{2}\right)^{2},$
(101)

where index $\alpha$ is summed over. Further, the relevant soft terms can be
written as

$V_{\rm
soft}=m_{1}^{2}\,|\tilde{H}_{1}|^{2}+m_{2}^{2}\,|\tilde{H}_{2}|^{2}+m_{\alpha}^{2}\,|\tilde{L}_{\alpha}|^{2}+\left\\{B_{\mu}\,\mu\,\tilde{H}_{1}\tilde{H}_{2}+A_{\alpha}\,\epsilon_{\alpha}\,\tilde{L}_{\alpha}\tilde{H}_{2}+{\rm
h.c.}\right\\}\,.$ (102)

The term proportional to $\epsilon_{\alpha}$ form $W$ can be rotated away with
the following redefinitions of the fields

$\mu\hat{H}_{1}+\epsilon_{\alpha}\hat{L}_{\alpha}\to\mu\hat{H}_{1}\,,~{}~{}~{}~{}-\epsilon_{\alpha}\hat{H}_{1}+\mu\hat{L}_{\alpha}\to\mu\hat{L}_{\alpha}\,.$
(103)

In this new basis, $V_{F,D}$ does not contain terms involving
$\epsilon_{\alpha}$ while $V_{\rm soft}$ gets replaced by

$V_{\rm soft}\to V_{\rm
soft}\,-\,\left\\{\frac{\epsilon_{\alpha}}{\mu}(m_{1}^{2}-m_{\alpha}^{2})\,\tilde{H}_{1}^{\dagger}\tilde{L}_{\alpha}\,+\,\left(B_{\mu}-A_{\alpha}\right)\epsilon_{\alpha}\,\tilde{L}_{\alpha}\tilde{H}_{2}\,+{\rm
h.c.}\right\\}\,.$ (104)

In the new basis, the VEV of $\tilde{L}_{\alpha}$ is denoted by
$\omega_{\alpha}$. It is related to the VEV $\omega^{\prime}_{\alpha}$ in the
original basis by Eq. (34) as can be explicitly seen from Eq. (103). Using the
definition Eq. (36) and minimizing the full potential, we obtain

$k_{\alpha}\simeq\frac{v_{1}}{\mu}\frac{(m_{1}^{2}-m_{\alpha}^{2})-\mu\tan\beta\left(B_{\mu}-A_{\alpha}\right)}{m_{\alpha}^{2}+\frac{1}{2}M_{Z}^{2}\,\cos
2\beta}\,.$ (105)

The first term in $k_{\alpha}$ can be suppressed for universal soft masses,
however the difference $B_{\mu}-A_{\alpha}$ can be arbitrary and hence
$k_{\alpha}$ need not be small. For flavour universal soft terms,
$k_{\alpha}\equiv k$ can be universal at SUSY breaking mediation scale and
difference among $k_{\alpha}$ can arise through renormalization group
evolution.

##  Appendix B On the size of 1-loop contributions to the neutrino masses

The resulting values of the tree level neutrino mass matrix $m_{\nu}^{(0)}$
and 1-loop corrected seesaw contribution $\delta m_{\nu}^{(1)}$ as defined in
Eqs. (43) and (60) are obtained for example, in case of solution BP 4 as

$\displaystyle m_{\nu}^{(0)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}5.84\,-0.319i&-8.379+0.932i&-6.071+0.675i\\\
-8.379+0.932i&-21.508-2.723i&-34.962-1.322i\\\
-6.071+0.675i&-34.962-1.322i&-11.29-1.429i\\\ \end{array}\right)\,{\rm
meV}\,,$ (109) $\displaystyle\delta m_{\nu}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}11.491&-0.006&0.005\\\
-0.006&0.419&-0.298\\\ 0.005&-0.298&0.219\\\ \end{array}\right)\,{\rm meV}\,.$
(113)

The $(1,1)$ element of the neutrino mass matrix receives significant
correction from $\delta m_{\nu}^{(1)}$ while the other elements are dominantly
determined by tree level contribution. This characteristic has also been seen
in the other benchmark solutions listed in Table 1.

In the above, we considered 1-loop self-energy diagram of light neutrinos with
Higgs/$Z$ and RH neutrinos in the loop. In general, the neutrino masses also
receive corrections from SUSY particles in the loop. A comprehensive
discussion on such corrections is given in Dedes  _et al._ (2006). We outline
these contributions and discuss their magnitudes in the context of our model.

  1. 1.

Neutralino - $\tilde{\nu}_{R}$ loop: This contribution arises when both
Higgs/$Z$ and RH neutrinos are replaced by their SUSY partners in the self-
energy diagram. The contribution is $R$-parity conserving and it goes like

$(\delta
m_{\nu})_{\alpha\beta}\sim\frac{1}{16\pi^{2}}M_{N}\,\lambda_{\alpha}\lambda_{\beta}\,\left(B_{0}\left[m_{\nu}^{2},m_{\tilde{\nu}^{+}_{R}}^{2},M_{N}^{2}\right]-B_{0}\left[m_{\nu}^{2},m_{\tilde{\nu}^{-}_{R}}^{2},M_{N}^{2}\right]\right)\,,$
(114)

where $m_{\tilde{\nu}^{\pm}_{R}}$ are masses of CP even and odd RH sneutrinos.
We have used the general result derived in Dedes  _et al._ (2006) to evaluate
the above contribution. Apparently, the above contribution depends on the mass
spectrum of sneutrinos and hence its complete determination would require
specification of the SUSY breaking mechanism. The contribution vanishes when
the RH sneutrinos are degenerate as can be seen from Eq. (114).

  2. 2.

Neutralino-$H/Z$ loop: This arises when RH neutrino is replaced by neutralinos
in the loop considered initially. This contribution arises through R-parity
violating mixing between neutrino and neutralino and it goes as

$(\delta
m_{\nu})_{\alpha\beta}\sim\frac{1}{16\pi^{2}}M_{1}\,\left(\frac{\omega_{\alpha}\omega_{\beta}}{M_{1}^{2}}\right)\,\frac{m_{H/Z}^{2}}{M_{1}^{2}}\,\ln\frac{M_{1}^{2}}{m_{H/Z}^{2}}\,,$
(115)

where $\omega_{\alpha}=k\epsilon_{\alpha}$ as defined in Eq. (34) and $M_{1}$
is a common neutralino mass scale. With $M_{1}=1$ TeV and the values of
$\omega_{\alpha}$ obtained for solution BP 2, the largest magnitude of this
contribution is ${\cal O}(10^{-6})$ eV. Hence, it is sub-dominant in
comparison to loop corrected seesaw contribution given in Eq. (109).

  3. 3.

Neutralino - $\tilde{\nu}_{L}$ loop: This contribution also depends on the
details of soft SUSY breaking sector. It vanishes for the degenerate masses of
CP even and odd $\tilde{\nu}^{\pm}_{L}$. However, in the presence of
$R$-parity violating soft terms, $\tilde{\nu}^{\pm}_{L}$ mix with the other CP
even and odd neutral scalars and the mass splitting between
$\tilde{\nu}^{\pm}_{L}$ eigenstates gets induced. The contribution to the
neutrino mass is approximated as Dedes  _et al._ (2006)

$(\delta
m_{\nu})_{\alpha\beta}\sim\frac{1}{16\pi^{2}}M_{1}\,g^{2}\,\left(\frac{M_{1}}{m_{\tilde{\nu}_{L}}}\right)^{2}\,\frac{b_{\alpha}b_{\beta}\tan^{2}\beta}{(M_{N}^{2}-m_{\tilde{\nu}_{L}}^{2})^{2}}\,,$
(116)

where $b_{\alpha}=B\epsilon_{\alpha}$. For typical values of the parameters,
$M_{1}=1$ TeV, $m_{\tilde{\nu}_{L}}=2$ TeV, $\tan\beta=2$ and $b_{\alpha}=1.0$
GeV2, one obtains $\delta m_{\nu}\sim{\cal O}(10^{-5})$ eV.

  4. 4.

Chargino-slepton loop: This contribution is also computed in Dedes  _et al._
(2006) but assuming vanishing VEVs for sneutrinos. We estimate this
considering nonzero sneutrino VEVs and find that the dominant contribution is
given by

$(\delta
m_{\nu})_{\alpha\beta}\sim\frac{1}{16\pi^{2}}m_{l}\,\tan\beta\frac{\omega_{\alpha}}{M_{C}}\,\frac{b_{\beta}}{M_{H^{+}}^{2}}\,,$
(117)

where $m_{l}$ is charged lepton mass, $M_{2}$ is chargino mass. Another
similar contribution also arise when $b_{\beta}$ is replaced by
$\epsilon_{\beta}m_{l}$ in the above equation. These contributions are
suppressed for $\omega_{\alpha}\ll\epsilon_{\alpha}$. For typical values,
$M_{2}=1$ TeV, $M_{H^{+}}=2$ TeV, $\tan\beta=2$, $b_{\alpha}=1.0$ GeV2 and
$\omega_{\alpha}=k\epsilon_{\alpha}$, the largest contribution from Eq. (117)
for solution BP 2 is estimated as $\delta m_{\nu}\sim 10^{-8}$ eV which is
again negligibly small compared to the 1-loop seesaw corrected contribution.

Therefore, the 1-loop correction to the neutrino masses from the SUSY
particles in the loop is insignificant in comparison to the loop corrected
seesaw contribution that we have taken into consideration. The 2-loop
correction to the seesaw is expected to remain smaller than 1-loop
contribution as it is further suppressed by factor of
$\lambda^{2}_{\alpha}/(4\pi)^{2}$. Hence, the results and predictions for the
various observable parameters that we have derived using $m_{\nu}$ as given in
Eq. (25) are robust.

##  Appendix C Neutrino spectrum with $\kappa=0$

In this Appendix, we show the same results as displayed in Tables 1, 2 and 3
but for vanishingly small $\kappa$ and $\tan\beta=1.5$. These cases correspond
to vanishingly small mixing between neutralino and RH neutrinos. As it can be
seen, one can reproduce neutrino mass spectrum consistent with all the current
experimental bounds in this case as well. For BP 6 and BP 7, the predicted
value of the sum of light neutrino masses is larger than the current Planck
limit. However, the neutrino lifetime obtained in these cases are shorter than
the age of the universe as it can be seen from the corresponding values in
Table 6. Hence, these solutions can evade the Planck constraint Chacko  _et
al._ (2020) as discussed in section IV.

Parameters |  BP 5 |  BP 6 |  BP 7 |  BP 8
---|---|---|---|---
$U$ [GeV] | $10^{3}$ | $10^{4}$ | $10^{5}$ | $10^{6}$
$\kappa$ | $0$ | $0$ | $0$ | $0$
$\lambda$ | $-0.1+0.0004\,i$ | $0.1168-0.0013\,i$ | $-0.9998-0.3\,i$ | $0.2973+0.0891\,i$
$\epsilon$ | 0.0139 | -0.0377 | -0.0015 | -0.0007
$\phi$ | -0.7764 | 2.1609 | -2.1093 | 1.7723
$\lambda_{\tau}$ | $7.59\times 10^{-8}$ | $5.031\times 10^{-7}$ | $4.261\times 10^{-7}$ | $9.032\times 10^{-7}$
$r_{\mu}$ | 1.0749 | 0.7156 | -0.8108 | 0.2771
$r_{e}$ | -43.4421 | -11.7682 | 85.9164 | 64.9009
$k$ | 4.2802 | 0.082 | 0.0118 | 0.0014
$\Delta m^{2}_{21}$ [$10^{-5}$ eV2] | $7.43$ | 7.43 | 7.42 | 7.42
$\Delta m^{2}_{31}$ [$10^{-3}$ eV2] | $2.513$ | 2.515 | 2.517 | 2.517
$\sin^{2}\theta_{12}$ | 0.309 | 0.303 | 0.304 | 0.304
$\sin^{2}\theta_{23}$ | 0.6 | 0.596 | 0.573 | 0.573
$\sin^{2}\theta_{13}$ | 0.0221 | 0.0221 | 0.0222 | 0.0222
$\sin\delta_{\rm CP}$ | -0.14 | -0.42 | -0.29 | -0.29
$\sum m_{\nu_{i}}$ [eV] | 0.252 | 0.23 | 0.093 | 0.093
$|m_{\beta\beta}|$ [eV] | 0.079 | 0.071 | 0.02 | 0.02
$M_{N_{1}}$ [GeV] | 1.39 | 40.7 | 133.4 | 91.2
$M_{N_{2}}$ [GeV] | 99.31 | 1147 | 104317 | 310266
$M_{N_{3}}$ [GeV] | 100.7 | 1188 | 104451 | 310357
Table 5: Same as Table 1 but with $\kappa=0$. Parameters |  BP 5 |  BP 6 |  BP 7 |  BP 8
---|---|---|---|---
$|\eta_{0}|$ | 0.341 | 0.366 | 2.497 | 2.495
${\rm Max.}\\{|g_{ii}|\\}$ | $1.5\times 10^{-14}$ | $1.4\times 10^{-15}$ | $1.2\times 10^{-16}$ | $1.2\times 10^{-17}$
$|g_{32}|$ | $5.9\times 10^{-14}$ | $5.9\times 10^{-15}$ | $1.4\times 10^{-17}$ | $1.4\times 10^{-18}$
$|g_{31}|$ | $6.7\times 10^{-14}$ | $6.0\times 10^{-15}$ | $3.1\times 10^{-18}$ | $3.1\times 10^{-19}$
$\tau_{\nu}$ [sec.] | $9.1\times 10^{13}$ | $9.9\times 10^{15}$ | $3.2\times 10^{21}$ | $3.2\times 10^{23}$
Table 6: Same as Table 2 but with $\kappa=0$. Parameters |  BP 5 |  BP 6 |  BP 7 |  BP 8
---|---|---|---|---
${\rm Max.}\\{|h_{\alpha\beta}|\\}$ | $10^{-16}$ | $10^{-15}$ | $10^{-14}$ | $10^{-13}$
Table 7: Same as Table 3 but with $\kappa=0$.

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