# Soft supersymmetry breaking as the sole origin of neutrino masses and lepton
number violation

Anjan S. Joshipura<EMAIL_ADDRESS>Theoretical Physics Division,
Physical Research Laboratory, Navarangpura, Ahmedabad-380009, India  Ketan M.
Patel<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research
Laboratory, Navarangpura, Ahmedabad-380009, India

###### Abstract

We discuss a scenario in which the supergravity induced soft terms,
conventionally used for breaking supersymmetry, also lead to non-zero Majorana
neutrino masses. The soft terms lead to the spontaneous violation of the
lepton number at the gravitino mass scale $m_{3/2}$ which in turn leads to (i)
the Majorana masses of ${\cal O}(m_{3/2})$ for the right-handed neutrinos and
(ii) the $R$-parity breaking at the same scale. The former contributes to
light neutrino masses through the type I seesaw mechanism, while the latter
adds to it through neutrino-neutralino mixing. Both contributions can scale
inversely with respect to $m_{3/2}$ given that gaugino and Higgsino masses are
also of order $m_{3/2}$. Together, these two contributions adequately explain
observed neutrino masses and mixing. One realization of the scenario also
naturally leads to a $\mu$ parameter of ${\cal O}(m_{3/2})$. Despite the
lepton number symmetry breaking close to the weak scale, the Majoron in the
model exhibits very weak coupling to leptons, satisfying existing constraints
on Majoron-lepton interactions. The right-handed neutrinos in the model have a
large coupling to Higgsinos. This coupling and the relatively large heavy-
light neutrino mixing induced through the seesaw mechanism can lead to the
observable signals at colliders in terms of displaced vertices.

##  I Introduction

Violation of the lepton number, which otherwise is an automatic classical
symmetry of the standard model (SM), provides an appealing avenue to account
for small but non-vanishing neutrino masses Weinberg (1979). It enables
neutrinos to become Majorona particles and associates their masses to a new
scale in the theory. The exact relation between the neutrino mass scale and a
new scale depends on the nature of new physics introduced to violate the
lepton number (LN) Mohapatra  _et al._ (2007). For example, the most popular
among these is the type I seesaw mechanism Minkowski (1977); Yanagida (1979);
Mohapatra and Senjanovic (1980); Schechter and Valle (1980) in which two or
more right-handed (RH) neutrinos are introduced and coupled to the SM
neutrinos through Yukawa interactions. A Majorana mass $M_{N}$ of RH neutrino
$N$ denotes the scale of LN violation in this case. It can be put in by hand
or may arise from the spontaneous breaking of a gauge symmetry that includes
the LN Fritzsch and Minkowski (1975); Gell-Mann  _et al._ (1979). From the
perspective of the light neutrino masses, the RH neutrino mass scale can be as
high as $10^{16}$ GeV if they strongly couple to the SM. On the other hand,
$M_{N}$ can be as light as of the order of the electroweak scale provided that
their couplings with the SM leptons are small.

The other compelling reason to extend the SM comes from the gauge hierarchy
problem. The electroweak scale can be stabilised in the presence of some large
scale in the theory when the SM is supersymmetrised by replacing it with the
minimal supersymmetric standard model (MSSM), see Martin (1998) for a
comprehensive review. An efficient implementation of this scheme requires the
mass scale of the MSSM fields not very different from the electroweak scale.
The existence of all Dimopoulos and Georgi (1981); Dimopoulos  _et al._
(1981); Marciano and Senjanovic (1982) or at least some Arkani-Hamed and
Dimopoulos (2005); Giudice and Romanino (2004) of the super-partners around
the TeV scale also improves the unification of gauge couplings. Theoretically,
the mass scale of the MSSM is set by the $\mu$ parameter and the soft
supersymmetry (SUSY) breaking scale. The latter is linked to the mass
$m_{3/2}$ of the gravitino in theories based on supergravity (see Lahanas and
Nanopoulos (1987) for a review). Like in the case of SM, the MSSM can also be
extended to incorporate LN violation as both are independent extensions of the
SM. In this case, the scale of LN violation can coexist in the theory
independent of $\mu$ and $m_{3/2}$.

This paper aims to point out an interesting scenario in which the soft terms
generated in supergravity theory not only break the SUSY but also
spontaneously break the LN thereby linking the LN violation scale to the
gravitino mass $m_{3/2}$. The amalgamation of the otherwise two distinct
scales provides a compelling reason to expect a low-scale LN violation if the
SUSY is to resolve the gauge hierarchy problem or it is responsible for the
precision unification of the strong and electroweak forces. Soft SUSY breaking
terms proportional to $m_{3/2}$ exclusively generate Majorana masses for both
the RH and the light neutrinos. The masses of the first are ${\cal
O}(m_{3/2})$ while those of the latter are induced through the seesaw
mechanism. We propose a concrete and minimal realisation of the above scenario
in this paper. This minimal version is shown to be complete in the sense that
it leads to a consistent and predictive description of the neutrino masses and
mixing.

It is noteworthy that a connection between the mass of RH neutrino and the
SUSY breaking scale can also be established through a very different mechanism
Chun  _et al._ (1995, 1996); Chun (1999); Choi  _et al._ (2001); Lavignac and
Medina (2021); Joshipura and Patel (2021). In this case, the scale of the LN
violation is independent of the SUSY breaking scale. However, one of the RH
neutrinos is arranged to be a fermionic partner of the Goldstone boson of the
global LN symmetry and, therefore, this pair remains massless at the scale of
LN violation. Subsequently, the fermionic partner gets a mass of ${\cal
O}(m_{3/2})$ when SUSY is broken through supergravity. Even though there is no
in-principle connection between the LN breaking scale and $m_{3/2}$ in this
kind of setup, the viable neutrino masses in the simplest versions of this
framework force these two scales to stay close to each other Joshipura and
Patel (2021). In the framework we propose here, these two scales are
inherently connected leading to a more predictive and realistic setup for the
light neutrino masses.

The remainder of the paper is organised as follows. We discuss the basic
mechanism leading to a connection between the SUSY breaking and LN violation
in the next section. A comprehensive analysis of the light neutrino masses and
mixing is presented in section III. We compute the couplings of Majoron with
the light neutrinos and charged leptons in sections IV and V, respectively,
and discuss the phenomenological constraints. In section VI, we discuss the RH
neutrino mass spectrum and the possibility of probing them in direct search
experiments. The study is concluded in section VII.

##  II Lepton number violation from soft SUSY breaking

In the following, we discuss two different types of frameworks. Both lead to
violation of the LN through soft SUSY breaking. However, they are
characteristically different in their implication for the neutrino masses. The
first scenario assumes a conformal superpotential without any mass scale and
the other scenario allows for an explicit $\mu$ parameter.

###  II.1 Conformal $W$

We consider the MSSM extended by three RH neutrino superfields
$\hat{N}_{\alpha}$, ($\alpha=e,\mu,\tau$). To achieve the required scenario,
we demand that the superpotential $W$ is conformal, i.e. it does not contain
any mass scales. This can be done for example by imposing an $R$ symmetry on
$W$ under which all the chiral superfields carry charge $2/3$. This leads to
only cubic terms in $W$. All the standard terms present in the MSSM
superpotential, except for the $\mu\hat{H}_{1}\hat{H}_{2}$ term, are allowed
in this case. The superpotential for the RH neutrinos is completely fixed by
imposing an additional global LN-like symmetry under which
$(\hat{N}_{e},\hat{N}_{\mu},\hat{N}_{\tau})$ carry charges $(-1,1,0)$. This
leads to the following most general superpotential:

$W_{S}=\lambda\hat{N}_{e}\hat{N}_{\mu}\hat{N}_{\tau}+\frac{\kappa}{3}\hat{N}_{\tau}^{3}\,.$
(1)

Couplings between the MSSM sector and $\hat{N}_{\alpha}$ also get restricted
by the LN symmetry under which the leptonic doublets
$\hat{L}_{\alpha}^{\prime}$ carry charge $+1$. These are given by

$W_{LN}=\lambda_{\alpha}\,\hat{L}_{\alpha}^{\prime}\hat{H}_{2}\hat{N}_{e}+\lambda_{\alpha}^{\prime}\,\hat{L}_{\alpha}^{\prime}\hat{H}_{1}^{\prime}\hat{\alpha}^{c}+\eta\,\hat{N}_{\tau}\hat{H}_{1}^{\prime}\hat{H}_{2}\,,$
(2)

where $\hat{\alpha}^{c}$ are the weak singlet charged lepton superfields with
lepton number $-1$. The couplings between $\hat{L}_{\alpha}^{\prime}$ and
$\hat{\alpha}^{c}$ are chosen flavour diagonal without loss of generality.

Eqs. (1,2) admit a supersymmetric minimum in which scalar components of all
the superfields have a vanishing minimum. All fermions including
$\nu_{L\alpha}$, $N_{\alpha}$ are massless in this limit. These masses are
induced by the soft breaking of SUSY. We introduce the following soft terms
corresponding to $W_{S}$:

$V_{\rm
soft}=m_{3/2}\left(A_{\lambda}\lambda\tilde{N}_{e}\tilde{N}_{\mu}\tilde{N}_{\tau}+\frac{1}{3}A_{\kappa}\kappa\tilde{N}_{\tau}^{3}+{\rm
c.c.}\right)+m_{3/2}^{2}\sum_{\alpha}a_{N_{\alpha}}^{2}|\tilde{N}_{\alpha}|^{2}\,,$
(3)

where $\tilde{S}$ denotes a Lorentz scalar component of a superfield
$\hat{S}$. The neutrino mass generation through these soft SUSY-breaking terms
proceeds as follows. Minimization of the scalar potential with these soft
terms leads to non-zero vacuum expectation values (VEVs) for
$\tilde{N}_{\alpha}$. These generate the RH neutrino masses which in turn lead
to the masses for the left-handed neutrinos through the weak scale type I
seesaw mechanism. The non-zero VEVs also spontaneously break $R$-parity along
with the LN, generating additional R-violating contributions to the light
neutrino masses. Both these contributions together describe the observed
neutrino masses and mixing. We discuss some of the salient features of this
neutrino mass generation here and differ the detailed discussion to the next
section.

Minimisation of the scalar potential derived from $W_{S}$ and $V_{\rm soft}$
is carried out and discussed in Appendix A. It leads to the following VEVs:

$\langle\tilde{N_{\tau}}\rangle=\frac{C_{\tau}}{\lambda}m_{3/2}\,,~{}~{}\langle\tilde{N_{e}}\rangle=\langle\tilde{N_{\mu}}\rangle=\frac{C_{e}}{\lambda}m_{3/2}\,,$
(4)

where $C_{\tau}$ and $C_{e}$ are dimensionless parameters determined in
general by $\kappa$, $\lambda$ and the soft parameters. The equality in the
VEVs of $\tilde{N_{e}}$ and $\tilde{N_{\mu}}$ follows from an assumption
$a^{2}_{N_{e}}=a^{2}_{N_{\mu}}$ which makes the full scalar potential
invariant under the interchange of $\tilde{N}_{e}$ and $\tilde{N}_{\mu}$. In
absence of such an assumption, one finds
$\langle\tilde{N_{e}}\rangle\neq\langle\tilde{N_{\mu}}\rangle$ but with a
constraint,
$\langle\tilde{N_{e}}\rangle^{2}+\langle\tilde{N_{\mu}}\rangle^{2}=C_{e}^{2}m_{3/2}^{2}/\lambda^{2}$.
It is noteworthy that all three sneutrinos obtain VEV of the order of the
gravitino mass. In this way, the scale of the LN violation is linked to the
SUSY breaking scale.

Notably, the $\langle\tilde{N_{\tau}}\rangle$ also generates the $\mu$
parameter as in the NMSSM Ellwanger  _et al._ (2010). But unlike in the NMSSM
now the magnitude of $\mu$ is also naturally tied to the gravitino mass:

$\mu=-\eta\langle\tilde{N_{\tau}}\rangle=-\frac{\eta
C_{\tau}}{\lambda}m_{3/2}\,.$ (5)

The VEV $\langle\tilde{N_{e}}\rangle$ also breaks spontaneously the $R$-parity
Masiero and Valle (1990); Barbier  _et al._ (2005) and generates the bilinear
R-violation from Eq. (2), described by three parameters:

$\epsilon_{\alpha}\equiv-\lambda_{\alpha}\langle\tilde{N_{e}}\rangle=-\frac{\lambda_{\alpha}C_{e}}{\lambda}m_{3/2}\,.$
(6)

The RH neutrino mass matrix following from Eqs. (1) and (4) is given, in the
basis $(N_{e},N_{\mu},N_{\tau})$, by

$M_{R}=m_{3/2}\left(\begin{array}[]{ccc}0&C_{\tau}&C_{e}\\\
C_{\tau}&0&C_{e}\\\ C_{e}&C_{e}&\tilde{C}_{\tau}\\\ \end{array}\right)\,,$ (7)

where $\tilde{C}_{\tau}=\frac{\kappa}{\lambda}C_{\tau}$. This leads to the
three RH neutrino masses as

$\displaystyle M_{N_{1}}$ $\displaystyle=$ $\displaystyle C_{\tau}m_{3/2}\,,$
$\displaystyle M_{N_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{\tau}m_{3/2}\,\left(1+\frac{\kappa}{\lambda}-\left(\frac{\kappa}{\lambda}-1\right)\sec
2\theta\right)\,,$ $\displaystyle M_{N_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{\tau}m_{3/2}\,\left(1+\frac{\kappa}{\lambda}+\left(\frac{\kappa}{\lambda}-1\right)\sec
2\theta\right)\,,$ (8)

where

$\tan 2\theta=\frac{-2\sqrt{2}\,C_{e}}{C_{\tau}(1-\kappa/\lambda)}\,.$ (9)

One also finds

${\rm
Det}M_{R}=m_{3/2}^{3}\,C_{\tau}\left(2C_{e}^{2}-\frac{\kappa}{\lambda}C_{\tau}^{2}\right)\,.$
(10)

The geometrical mean of three RH neutrino masses is given by $({\rm
Det}M_{R})^{1/3}$. Typically, ${\rm Det}M_{R}\sim m_{3/2}^{3}$ when all the
three RH neutrinos have masses of ${\cal O}(m_{3/2})$. ${\rm Det}M_{R}\ll
m_{3/2}^{3}$ implies one or more RH neutrinos with a mass smaller than
$m_{3/2}$.

The above $M_{R}$ together with the Dirac mass matrix following from Eq. (2)
leads to the seesaw contribution to the light neutrino masses as

$({\cal M}^{\rm
SS}_{\nu})_{\alpha\beta}=\tilde{m}_{0}\,\lambda_{\alpha}\lambda_{\beta}~{}.$
(11)

where

$\tilde{m}_{0}=\frac{v_{2}^{2}}{m_{3/2}}\frac{\lambda
C_{e}^{2}}{C_{\tau}(2\lambda C_{e}^{2}-\kappa
C_{\tau}^{2})}=\frac{v_{2}^{2}C_{e}^{2}}{m_{3/2}}\left(\frac{m_{3/2}^{3}}{{\rm
Det}M_{R}}\right)\,.$ (12)

The neutrino masses are determined by the Dirac neutrino Yukawa couplings and
the inverse of $m_{3/2}$. Small (high) values of the former (latter) can
result in the required suppression in the neutrino masses as it happens in the
usual seesaw case. The neutrino mass thus would be automatically suppressed in
models with high-scale SUSY breaking.

Interestingly, Eqs. (11,12) offer an alternative way of lowering the seesaw
contribution to neutrino masses even in low-scale SUSY models. This can be
done if the parameter $\lambda/\kappa$ in Eq. (1) is taken very small111This
can be done in a technically natural way since the limit $\lambda\rightarrow
0$ increases the symmetry of $W_{S}$.. This breaks LN at a scale higher than
$m_{3/2}$, see Eq. (4), and results in a small $\tilde{m}_{0}$. This way of
suppressing neutrino masses can be interpreted in terms of the inverse seesaw
mechanism often used for suppressing neutrino masses in the weak or TeV scale
seesaw models Mohapatra and Valle (1986); Mohapatra (1986); Nandi and Sarkar
(1986). It follows from Eq. (II.1) that one of the RH neutrinos, namely
$M_{N_{3}}$, becomes heavier compared to the other two if $\lambda/\kappa\ll
1$.

Integrating out $N_{\tau}$ results in a $5\times 5$ mass matrix, written in
the basis $(\nu^{\prime}_{\alpha},N_{e},N_{\mu})$,

${\cal M}_{\nu}^{\rm IS}=\left(\begin{array}[]{ccc}0_{3\times 3}&m_{D}&0\\\
m_{D}^{T}&\mu_{S}&M_{D}\\\ 0&M_{D}&\mu_{S}\\\ \end{array}\right)~{},$ (13)

where $M_{D}=m_{3/2}C_{\tau}$ and $m_{D}$ is $3\times 1$ matrix of the Yukawa
couplings $\lambda_{\alpha}$ multiplied by the Higgs VEV $v_{2}$. The matrix
${\cal M}_{\nu}^{\rm IS}$ has a form typically used in the inverse seesaw with

$\mu_{S}\equiv-\frac{\lambda}{\kappa}\frac{C_{e}^{2}m_{3/2}}{C_{\tau}}\,,$
(14)

representing the small LN breaking parameter suppressed by $\lambda/\kappa$.
For $M_{D}\gg m_{D}$, Eq. (13) leads to the light neutrino mass matrix

$\frac{\mu_{S}v_{2}^{2}}{m_{3/2}^{2}C_{\tau}^{2}}\,\lambda_{\alpha}\lambda_{\beta}\,$
(15)

which coincides with the seesaw result, Eq. (11), for $\lambda/\kappa\ll 1$.

It follows from Eq. (2) that only one combination of the neutrino fields
$\lambda_{\alpha}\nu_{\alpha}^{\prime}$ couples to $N_{e}$ and obtains its
mass through the canonical seesaw mechanism. An independent linear combination
of the neutrino fields obtains its mass from the $R$-parity violation induced
by the soft terms. The $R$-violation is characterised by $\epsilon_{\alpha}$
given in Eq. (6). They give rise to the VEV for the left-handed sneutrino
leading to the mixing of neutrinos with neutralinos. This mixing generates
mass for the other combination of neutrinos. The $R$-parity violating
parameters $\epsilon_{\alpha}$ increase with the gravitino mass. So naively,
one would expect that this contribution to the neutrino masses would increase
with $m_{3/2}$ in contrast to the seesaw contribution. However, this does not
happen and both contributions can be inversely proportional to $m_{3/2}$ in a
typical situation as we show below.

The $R$-parity violating contribution is specified in the rotated basis in
which bilinear $R$-parity violation is rotated away from the superpotential
Joshipura  _et al._ (2002). These unprimed bases are defined as

$\displaystyle\hat{H}_{1}$ $\displaystyle=$
$\displaystyle\hat{H}_{1}^{\prime}+\frac{\epsilon_{\alpha}}{\mu}\hat{L}_{\alpha}^{\prime}+{\cal
O}\left(\frac{\epsilon_{\alpha}^{2}}{\mu^{2}}\right)\,,$
$\displaystyle\hat{L}_{\alpha}$ $\displaystyle=$
$\displaystyle\hat{L}_{\alpha}^{\prime}-\frac{\epsilon_{\alpha}}{\mu}\hat{H}_{1}^{\prime}+{\cal
O}\left(\frac{\epsilon_{\alpha}^{2}}{\mu^{2}}\right)\,,$ (16)

where $\hat{L}_{\alpha}^{\prime}$ and $\hat{H}_{1}^{\prime}$ are the original
fields present in the superpotential Eq. (2). The $R$-parity violating light
neutrino mass matrix is obtained as (see the next section for the derivation)

$\left({\cal
M}_{\nu}^{R}\right)_{\alpha\beta}=A_{0}\,\omega_{\alpha}\omega_{\beta}\,,$
(17)

with

$A_{0}=-\frac{\mu(g^{2}M_{1}+g^{\prime 2}M_{2})}{2(M_{1}M_{2}\mu-
v_{1}v_{2}(g^{2}M_{1}+g^{\prime 2}M_{2}))}\,.$ (18)

Here $M_{1,2}$ are gaugino masses and $v_{1,2}$ are the VEVs of the Higgs
fields $\tilde{H}_{1,2}$.

In Eq. (17), $\omega_{\alpha}\equiv\langle\nu_{\alpha}\rangle$ denote the VEVs
of the scalar superpartners of the electrically neutral component of
$\hat{L}_{\alpha}$. They are determined in terms of the soft parameters by
minimising the relevant scalar potential. Writing

$\omega_{\alpha}=k_{\alpha}\epsilon_{\alpha}\,,$ (19)

we find in this case

$k_{\alpha}\simeq\frac{v_{1}}{\mu}\left(\frac{a_{H_{1}}^{2}-a_{L_{\alpha}}^{2}}{a_{L_{\alpha}}^{2}}\right)+\frac{v_{1}\tan\beta}{m_{3/2}}\left(\frac{A_{\eta}-A_{\alpha}}{a_{L_{\alpha}}^{2}}\right)\,,$
(20)

where $\tan\beta=v_{2}/v_{1}$. The details of the derivation of the above can
be found in Appendix B. If one assumes universal boundary conditions at a high
scale then the differences in soft parameters appearing in $k_{\alpha}$ are
generated at the low scale through renormalization effects and are small. In
particular, the first term which gives a dominant contribution to $k_{\alpha}$
is typically given by

$\frac{a_{H_{1}}^{2}-a_{L_{\alpha}}^{2}}{a_{L_{\alpha}}^{2}}\sim\frac{3y_{b}^{2}}{4\pi^{2}}\,\log\frac{M_{X}}{M_{Z}}\sim{\cal
O}\left(10^{-3}\right)\,.$ (21)

Here, $y_{b}$ is the bottom quark Yukawa coupling and $M_{X}$ is the scale
where the universality of the soft terms is imposed.

It is useful to eliminate Yukawa couplings and consider the ratio of the
contributions ${\cal M}_{\nu}^{R}$ and ${\cal M}_{\nu}^{\rm SS}$. We find

$\xi_{0}\equiv\frac{({\cal M}_{\nu}^{R})_{33}}{({\cal M}^{\rm
SS}_{\nu})_{33}}=\frac{A_{0}k_{\tau}^{2}m_{3/2}^{3}}{\lambda^{2}v_{2}^{2}}\,\left(\frac{{\rm
Det}M_{R}}{m_{3/2}^{3}}\right)\,.$ (22)

Following Eqs. (20,21), $k_{\tau}$ can be approximated as

$k_{\tau}\approx 10^{-3}\,\frac{v_{1}}{\mu}~{}.$ (23)

This implies

$\xi_{0}\approx\frac{A_{0}m_{3/2}^{3}}{\mu^{2}\tan^{2}\beta}\left(\frac{10^{-6}}{\lambda^{2}}\right)\,\left(\frac{{\rm
Det}M_{R}}{m_{3/2}^{3}}\right)\,.$ (24)

Since $\xi_{0}$ is related to the two non-zero neutrino masses, it cannot be
too small or large. The detailed fit to neutrino masses and mixing presented
in the next section show that the magnitude of $\xi_{0}$ is of order unity.
Requiring $\xi_{0}\sim 1$ has implications for the allowed $m_{3/2}$ and the
SUSY spectrum. Considering $M_{N_{i}}\sim m_{3/2}$, we note that:

  * •

If gaugino masses and the $\mu$ parameter are similar to $m_{3/2}$ then
$\xi_{0}$ is independent of $m_{3/2}$ since $A_{0}$ is inversely proportional
to the gaugino masses. This implies that like the seesaw contribution, the
$R$-parity violating contribution to the neutrino masses is also inversely
proportional to $m_{3/2}$. Therefore, the neutrino masses can be suppressed
without requiring too small Yukawa couplings as in the high-scale seesaw
models. This scenario would conflict with the gauge coupling unification which
requires gauginos at the electroweak scale.

  * •

If $\mu$ is held fixed around electroweak scale by choosing $\eta$
appropriately and the gaugino masses are also assumed to be of the order of
$\mu$ ($m_{3/2}$) then the $\xi_{0}$ is proportional to $m_{3/2}^{3}/\mu^{3}$
($m_{3/2}^{2}/\mu^{2}$). Then the requirement, $\xi_{0}\sim 1$, implies an
upper bound on $m_{3/2}/\mu$. We will explore this point more quantitatively
in the next section.

  * •

$\lambda$ cannot be arbitrarily small and can be around $10^{-2}$ if $\mu$
parameter and the gaugino masses are similar to $m_{3/2}$. Thus, $\lambda$ can
provide only a mild suppression in the neutrino masses through inverse seesaw.

Some of the above features change when $W$ is not conformal and contains an
independent scale as a bare $\mu$ term. We discuss this below.

###  II.2 $W$ with a bare $\mu$-term

An explicit $\mu$ parameter can exist if $R$ symmetry used to obtain cubic
potential is not insisted upon. This has a significant effect on neutrino
masses which we now discuss. Consider the same superpotential with an added
bare $\mu$-term, namely

$\displaystyle W$ $\displaystyle=$ $\displaystyle
W_{S}+W_{LN}-\mu_{0}\,\hat{H}^{\prime}_{1}\hat{H}_{2}\,.$ (25)

The effective $\mu$ parameter is now obtained as

$\mu=-\eta\langle\tilde{N}_{\tau}\rangle+\mu_{0}\,.$ (26)

We can altogether omit the $\eta$-dependent term. In this case, $\mu=\mu_{0}$
is not tied to the gravitino mass and represents an independent scale. This
can be technically done by imposing a discrete $Z_{3}$ symmetry under which
$\hat{\alpha}^{c}\rightarrow\omega^{2}\hat{\alpha}^{c}$,
$\hat{N}_{\alpha}\rightarrow\omega^{2}\hat{N}_{\alpha}$ and
$\hat{L}^{\prime}_{\alpha}\rightarrow\omega\hat{L}^{\prime}_{\alpha}$ with
$\omega^{3}=1$.

The $\mu_{0}$ term leads to an additional soft term
$B_{\mu_{0}}\mu_{0}\tilde{H}^{\prime}_{1}\tilde{H}_{2}$ in the scalar
potential. The $k_{\alpha}$ of Eq. (20) is now replaced by a more general
expression

$k_{\alpha}\simeq\frac{v_{1}}{\mu}\left(\frac{a_{H_{1}}^{2}-a_{L_{\alpha}}^{2}}{a_{L_{\alpha}}^{2}}\right)+\frac{v_{1}\tan\beta}{m_{3/2}}\left(\frac{A_{\eta}-A_{\alpha}}{a_{L_{\alpha}}^{2}}\right)+\frac{\mu_{0}}{\mu}\frac{v_{1}\tan\beta}{m_{3/2}}\left(\frac{B_{\mu_{0}}-A_{\alpha}}{a_{L_{\alpha}}^{2}}\right)\,.$
(27)

The above is obtained using Eq. (144) derived explicitly in Appendix B. Unlike
in the previous case, $k_{\alpha}$ is non-zero even at a high scale since
generally $B$ and $A$ parameters are not equal in supergravity-induced soft
terms. The last term, therefore, dominates in Eq. (27). This increases the
$R$-parity violating contribution by a factor of around $10^{6}$. Since the
last term in $k_{\alpha}$ dominates,

$k_{\tau}\approx\frac{\mu_{0}}{\mu}\frac{v_{2}}{m_{3/2}}\,.$ (28)

The ratio $\xi_{0}$ defined in Eq. (22) is now given by

$\xi_{0}\approx
A_{0}m_{3/2}\left(\frac{\mu_{0}}{\mu}\right)^{2}\left(\frac{1}{\lambda^{2}}\right)\,\left(\frac{{\rm
Det}M_{R}}{m_{3/2}^{3}}\right)\,.$ (29)

This suggests that $\lambda$ cannot be chosen small ruling out the possibility
of an inverse seesaw mechanism entirely in this case. Also since $A_{0}$ is
inversely proportional to the gaugino mass, one requires this mass to be
${\cal O}(m_{3/2})$ and $\mu_{0}\sim\mu$ to have $\xi_{0}$ of order one.

We have neglected in the above discussions one more contribution to the light
neutrino masses which arise from the mixing of the RH neutrinos with Higgsino
originating from Eq. (2). This gives a much smaller contribution compared to
the ones already discussed. Inclusion of this requires consideration of the
full mass matrix of neutral fermions in the model which we discuss in the
following section.

##  III Light neutrino masses and mixing

The model has 10 neutral fermions denoted as
$\psi_{0}=(\nu_{\alpha}^{\prime},N_{\alpha},-i\lambda_{1},-i\lambda_{2},\tilde{h}_{1}^{0},\tilde{h}_{2}^{0})^{T}$.
The RH VEVs generated through soft terms cause mixing among them as mentioned
in the earlier section. The full mixing generated by the superpotential in Eq.
(25) and gauge interactions can be written as:

$-{\cal L}_{\cal N}=\frac{1}{2}\,\psi_{0}^{T}\,{\cal M}_{\cal
N}\,\psi_{0}\,+\,{\rm h.c.}\,,$ (30)

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

${\cal M}_{\cal N}=\left(\begin{array}[]{ccc}(0)_{3\times 3}&(m_{\nu
N})_{3\times 3}&(m_{\nu\chi})_{3\times 4}\\\ m_{\nu N}^{T}&(M_{R})_{3\times
3}&(m_{N\chi})_{3\times 4}\\\
m_{\nu\chi}^{T}&m_{N\chi}^{T}&(M_{\chi})_{4\times 4}\end{array}\right)\,.$
(31)

Here,

$\displaystyle m_{\nu N}$ $\displaystyle=$ $\displaystyle
v_{2}\left(\begin{array}[]{ccc}\lambda_{e}&0&0\\\ \lambda_{\mu}&0&0\\\
\lambda_{\tau}&0&0\end{array}\right)\,,~{}~{}m_{\nu\chi}=\left(\begin{array}[]{cccc}-\frac{1}{\sqrt{2}}g^{\prime}\omega^{\prime}_{e}&\frac{1}{\sqrt{2}}g\omega^{\prime}_{e}&0&-\epsilon_{e}\\\
-\frac{1}{\sqrt{2}}g^{\prime}\omega^{\prime}_{\mu}&\frac{1}{\sqrt{2}}g\omega^{\prime}_{\mu}&0&-\epsilon_{\mu}\\\
-\frac{1}{\sqrt{2}}g^{\prime}\omega^{\prime}_{\tau}&\frac{1}{\sqrt{2}}g\omega^{\prime}_{\tau}&0&-\epsilon_{\tau}\end{array}\right)\,.$
(38)

The $M_{R}$ is already given in Eq. (7) while

$\displaystyle m_{N\chi}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&0&\lambda_{\alpha}\omega_{\alpha}^{\prime}\\\
0&0&0&0\\\ 0&0&\eta v_{2}&\eta
v_{1}\end{array}\right)\,,~{}~{}M_{\chi}=\left(\begin{array}[]{cccc}M_{1}&0&-\frac{1}{\sqrt{2}}g^{\prime}v_{1}&\frac{1}{\sqrt{2}}g^{\prime}v_{2}\\\
0&M_{2}&\frac{1}{\sqrt{2}}gv_{1}&-\frac{1}{\sqrt{2}}gv_{2}\\\
-\frac{1}{\sqrt{2}}g^{\prime}v_{1}&\frac{1}{\sqrt{2}}gv_{1}&0&-\mu\\\
\frac{1}{\sqrt{2}}g^{\prime}v_{2}&-\frac{1}{\sqrt{2}}gv_{2}&-\mu&0\\\
\end{array}\right)\,.$ (46)

$M_{\chi}$ is the standard neutralino mass matrix of the MSSM Martin (1998).
$R$ parity-violating couplings mix the neutralinos with the SM neutrinos. This
mixing is denoted as $m_{\nu\chi}$. Specifically, Higgsino-neutrino mixing is
given by $\epsilon_{\alpha}$ and gaugino mixing is determined by the sneutrino
VEVs $\omega_{\alpha}^{\prime}\equiv\langle\nu_{\alpha}^{\prime}\rangle$.
$m_{N\chi}$ denotes the mixing of the singlet states $N_{\alpha}$ with
neutralinos following from Eq. (25). $m_{\nu N}$ represents $U(1)$ invariant
Dirac mass term between neutrinos and singlet states. The above ${\cal
M}_{\cal N}$ describes the general case with explicit $\mu_{0}$ term present.
The conformal case is obtained by putting $\mu_{0}=0$ in $\mu$ appearing in
${\cal M}_{\cal N}$.

###  III.1 Analytical form of effective neutrino mass matrix

The tree-level light neutrino mass matrix can be obtained from Eq. (31) in the
seesaw approximation as

$m_{\nu}=-m_{3\times 7}\,{\cal M}_{7\times 7}^{-1}\,(m_{3\times 7})^{T}\,,$
(47)

where $m_{3\times 7}=(m_{\nu N},m_{\nu N},m_{\nu\chi})$ and ${\cal M}_{7\times
7}$ is the lower-right $7\times 7$ block of the full matrix given in Eq. (31).
Neglecting $\lambda_{\alpha}\omega_{\alpha}^{\prime}$ term in $m_{N\chi}$ and
performing some algebraic simplification, we get

$\displaystyle m_{\nu}$ $\displaystyle=$ $\displaystyle\tilde{m}\left(a\,{\cal
A}+b\,{\cal B}+c\,{\cal C}\right)\,.$ (48)

Here,

$\tilde{m}=\frac{D_{4}v_{2}^{2}C_{e}^{2}m_{3/2}^{2}}{D_{7}}\,,$ (49)

with

$D_{4}={\rm Det}\,M_{\chi}=-\mu(M_{1}M_{2}\mu-(g^{2}M_{1}+g^{\prime
2}M_{2})v_{1}v_{2})\,,$ (50)

and

$D_{7}=D_{4}C_{\tau}m_{3/2}^{3}\left(\frac{2\lambda C_{e}^{2}-\kappa
C_{\tau}^{2}}{\lambda}-\frac{\eta^{2}C_{\tau}\zeta}{m_{3/2}}\right)\,,$ (51)

is determinant of $7\times 7$ matrix ${\cal M}_{7\times 7}$ with

$\zeta=\frac{v^{4}(g^{2}M_{1}+g^{\prime 2}M_{2})-4v_{1}v_{2}\mu
M_{1}M_{2}}{2D_{4}}\,.$ (52)

Substituting Eq. (51) in Eq. (49) and using Eq. (12), we find

$\tilde{m}=\frac{\tilde{m}_{0}}{1-\frac{\lambda\eta^{2}C_{\tau}}{(2\lambda
C_{e}^{2}-\kappa C_{\tau}^{2})}\frac{\zeta}{m_{3/2}}}\,.$ (53)

Since $\zeta/m_{3/2}\ll 1$ for $v_{1,2}\ll m_{3/2}$, the denominator provides
only a small correction to the equality, $\tilde{m}=\tilde{m}_{0}$.

The matrices in Eq. (48) are obtained as

${\cal
A}=\left(\begin{array}[]{ccc}\lambda_{e}^{2}&\lambda_{e}\lambda_{\mu}&\lambda_{e}\lambda_{\tau}\\\
\lambda_{e}\lambda_{\mu}&\lambda_{\mu}^{2}&\lambda_{\mu}\lambda_{\tau}\\\
\lambda_{e}\lambda_{\tau}&\lambda_{\mu}\lambda_{\tau}&\lambda_{\tau}^{2}\\\
\end{array}\right)\,,~{}{\cal
B}=\left(\begin{array}[]{ccc}\omega_{e}^{2}&\omega_{e}\omega_{\mu}&\omega_{e}\omega_{\tau}\\\
\omega_{e}\omega_{\mu}&\omega_{\mu}^{2}&\omega_{\mu}\omega_{\tau}\\\
\omega_{e}\omega_{\tau}&\omega_{\mu}\omega_{\tau}&\omega_{\tau}^{2}\\\
\end{array}\right)\,,$ (54) ${\cal
C}=\left(\begin{array}[]{ccc}2\lambda_{e}\omega_{e}&\lambda_{e}\omega_{\mu}+\lambda_{\mu}\omega_{e}&\lambda_{e}\omega_{\tau}+\lambda_{\tau}\omega_{e}\\\
\lambda_{e}\omega_{\mu}+\lambda_{\mu}\omega_{e}&2\lambda_{\mu}\omega_{\mu}&\lambda_{\mu}\omega_{\tau}+\lambda_{\tau}\omega_{\mu}\\\
\lambda_{e}\omega_{\tau}+\lambda_{\tau}\omega_{e}&\lambda_{\mu}\omega_{\tau}+\lambda_{\tau}\omega_{\mu}&2\lambda_{\tau}\omega_{\tau}\\\
\end{array}\right)\,.$ (55)

In the above,

$\omega_{\alpha}=\omega^{\prime}_{\alpha}-\frac{\epsilon_{\alpha}}{\mu}v_{1}\,,$
(56)

denote the VEVs of sneutrino in the basis in which the bilinear term is
rotated away from the superpotential, see Eq. (II.1). Finally, the
dimensionless coefficients in Eq. (48) are obtained as

$\displaystyle a$ $\displaystyle=$ $\displaystyle\left(1-\frac{\eta
C_{\tau}m_{3/2}}{\lambda\mu}\right)^{2}=\left(2-\frac{\mu_{0}}{\mu}\right)^{2}\,,$
$\displaystyle b$ $\displaystyle=$
$\displaystyle-\frac{A_{0}C_{\tau}^{2}\tilde{C}_{\tau}m_{3/2}}{C_{e}^{2}v_{2}^{2}}\left(1-2\frac{C_{e}^{2}}{C_{\tau}\tilde{C}_{\tau}}+2\frac{\eta^{2}v_{1}v_{2}}{\tilde{C}_{\tau}\mu
m_{3/2}}\right)\,,$ $\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{A_{0}\eta
C_{\tau}(v_{2}^{2}-v_{1}^{2})}{C_{e}v_{2}\mu}\left(1-\frac{\eta
C_{\tau}m_{3/2}}{\lambda\mu}\right)=\frac{A_{0}\lambda(v_{2}-v_{1}^{2}/v_{2})}{C_{e}m_{3/2}}\left(\frac{\mu_{0}}{\mu}-1\right)\left(2-\frac{\mu_{0}}{\mu}\right)\,,$
(57)

and $A_{0}$ is defined in Eq. (18).

The structures of the first two terms in Eq. (48) coincide with the seesaw and
$R$ parity violating terms obtained in the earlier section. The last term is
generated by the mixing among the RH neutrino $N_{\tau}$ and Higgsino coming
from Eq. (25). This mixing also modifies the overall coefficients of the
seesaw and $R$-violating interactions compared to the simplified case
considered earlier. Using $\omega_{\alpha}=k_{\alpha}\epsilon_{\alpha}$, the
neutrino mass matrix can be further simplified to the following form

$m_{\nu}=m\left(\begin{array}[]{ccc}r_{e}^{2}\left(1+\xi s_{e}^{2}+2\delta
s_{e}\right)&r_{e}r_{\mu}\left(1+\delta s_{\mu}+s_{e}\left(\delta+\xi
s_{\mu}\right)\right)&r_{e}\left(1+\delta+(\delta+\xi)s_{e}\right)\\\
r_{e}r_{\mu}\left(1+\delta s_{\mu}+s_{e}\left(\delta+\xi
s_{\mu}\right)\right)&r_{\mu}^{2}\left(1+\xi s_{\mu}^{2}+2\delta
s_{\mu}\right)&r_{\mu}\left(1+\delta+(\delta+\xi)s_{\mu}\right)\\\
r_{e}\left(1+\delta+(\delta+\xi)s_{e}\right)&r_{\mu}\left(1+\delta+(\delta+\xi)s_{\mu}\right)&1+\xi+2\delta\\\
\end{array}\right),$ (58)

where we have define $r_{e,\mu}=\frac{\lambda_{e,\mu}}{\lambda_{\tau}}$,
$k_{\tau}\equiv k$ and $s_{e,\mu}=\frac{k_{e,\mu}}{k}$. The parameters
$k_{\alpha}$ are given by Eqs. (20) or Eq. (27) depending on the choice of
$\mu$-term in the model. Also,

$\displaystyle m$ $\displaystyle=$
$\displaystyle\tilde{m}\lambda_{\tau}^{2}a=\tilde{m}_{0}\lambda_{\tau}^{2}a\left(1-\frac{\lambda\eta^{2}C_{\tau}}{(2\lambda
C_{e}^{2}-\kappa C_{\tau}^{2})}\frac{\zeta}{m_{3/2}}\right)^{-1}\,,$
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{\xi_{0}}{a}\left(1-\frac{2\lambda\eta^{2}v_{1}v_{2}C_{\tau}}{m_{3/2}\mu(2\lambda
C_{e}^{2}-\kappa C_{\tau}^{2})}\right)\,,$ $\displaystyle\delta$
$\displaystyle=$ $\displaystyle-\frac{c}{a}\frac{kC_{e}m_{3/2}}{\lambda}\,.$
(59)

Here, $\tilde{m}_{0}$ and $\xi_{0}$ are defined in Eqs. (12,22), respectively.
$a\tilde{m}_{0}$ denotes the overall scale of the seesaw contribution. The
$\xi$ corresponds to the ratio of the $R$-parity violating and seesaw terms.
$\tilde{m}$ and $\xi$ respectively reduce to $\tilde{m}_{0}$ and $\xi_{0}$ of
the earalier section when $\eta=0$. In the following, we shall only be working
with the tree-level neutrino mass matrix and will neglect the other sub-
dominant radiative contributions. These include corrections to the seesaw mass
matrix Grimus and Löschner (2018) and corrections generated by the effective
trilinear R parity violating terms present in the rotated basis Barbier  _et
al._ (2005). The tree-level mass matrix considered here is by itself
sufficient to reproduce the neutrino mass spectrum and the radiative
corrections are not expected to change the results significantly.

The determinant of $m_{\nu}$ is vanishing. It predicts a massless neutrino.
The eigenvector corresponding to the massless state can be determined using
$\sum_{j}(m_{\nu})_{ij}\,u_{j}=0$. This gives

$\left(u_{1},u_{2},u_{3}\right)=\pm\frac{1}{\sqrt{N}}\,\left(r_{\mu}(s_{\mu}-1),\,-r_{e}(s_{e}-1),\,r_{e}r_{\mu}(s_{e}-s_{\mu})\right)\,.$
(60)

In the diagonal basis of the charged leptons, the above is identified with the
first (third) column of the leptonic mixing matrix for normal (inverted)
hierarchy in the neutrino masses. The observed neutrino mixing angles then
necessarily require $s_{e}\neq 1$, $s_{\mu}\neq 1$ as well as $s_{e}\neq
s_{\mu}$. This requires sizeable flavour violations in the soft terms, a
feature which was noticed also earlier in Joshipura  _et al._ (2002); Romao
_et al._ (2000); Hirsch  _et al._ (2000) in the context of different
frameworks.

###  III.2 Example solutions

We now demonstrate that the effective neutrino mass matrix, $m_{\nu}$ as
obtained in Eq. (58), can account for the realistic neutrino masses and mixing
observables. As mentioned earlier, the charged lepton Yukawa couplings can be
chosen real and diagonal without loss of generality. Moreover, the parameters
$\lambda$, $\lambda_{\alpha}$, $\eta$ and $\mu_{0}$ appearing in the
superpotential can be made real through redefinitions of various superfields.
This along with an assumption for the real VEVs of $N_{\alpha}$ and
$\nu_{\alpha}$ lead to real $r_{e,\mu}$, $s_{e,\mu}$ and $\delta$ in Eq. (58).
The phase of $m$ is also unphysical. Altogether, these six real parameters and
complex $\xi$ are required to reproduce the realistic values of six
observables, namely the solar and atmospheric neutrino masses, three mixing
angles and the Dirac CP phase.

Parameters |  NH1 |  NH2 |  NH3 |  NH4
---|---|---|---|---
$m$ [eV] | 1.0 | 1.0 | 0.01 | 0.01
$\xi$ | -0.9831 | 1.0216 | -2.6937 | -1.7155
$\delta$ | 0 | -0.9983 | 0 | -0.4891
$r_{e}$ | 0.0782 | -0.1196 | 1.387 | -1.3511
$r_{\mu}$ | -0.5964 | 0.9562 | -3.2619 | 3.1967
$s_{e}$ | 1.1216 | 0.5015 | 0.5859 | 0.501
$s_{\mu}$ | 0.971 | 0.8866 | 0.6797 | 0.6163
Parameters |  NH5 |  NH6 |  NH7 |  NH8
$m$ [eV] | 1.0 | 1.0 | 0.01 | 0.01
$|\xi|$ | 0.9841 | 0.5857 | 2.704 | 1.8462
${\rm Arg}[\xi]$ | -3.1349 | -0.01 | -3.096 | -3.0176
$\delta$ | 0 | -0.7847 | 0 | -0.4372
$r_{e}$ | -0.0814 | 0.1684 | 1.3628 | -1.3123
$r_{\mu}$ | -0.6049 | -0.4458 | 3.2386 | -3.1601
$s_{e}$ | 1.1158 | 1.3175 | 0.5813 | 0.5004
$s_{\mu}$ | 0.9714 | 0.7802 | 0.678 | 0.6197
Table 1: Example solutions of the parameters of the effective neutrino mass matrix which reproduces the neutrino masses and mixing parameters within $1\sigma$ assuming normal hierarchy. The first four solutions correspond to the CP conserving case. Parameters |  IH1 |  IH2 |  IH3 |  IH4
---|---|---|---|---
$m$ [eV] | 1.0 | 1.0 | 0.01 | 0.01
$\xi$ | -0.982 | -0.5409 | -0.4514 | -0.347
$\delta$ | 0 | -0.2272 | 0 | -0.1066
$r_{e}$ | -0.2517 | -0.2393 | -11.2255 | 14.6181
$r_{\mu}$ | -0.9077 | 0.8048 | 3.0287 | 3.7629
$s_{e}$ | 1.1493 | 1.2026 | 1.5 | 1.4252
$s_{\mu}$ | 1.0082 | 0.9881 | 1.3626 | 1.3282
Parameters |  IH5 |  IH6 |  IH7 |  IH8
$m$ [eV] | 1.0 | 1.0 | 0.01 | 0.01
$|\xi|$ | 0.9826 | 0.0836 | 3.0314 | 2.8087
${\rm Arg}[\xi]$ | -3.1412 | -0.0039 | -3.1386 | -3.1384
$\delta$ | 0 | -0.5321 | 0 | -0.0918
$r_{e}$ | -0.5623 | -0.6534 | 4.3557 | 4.4436
$r_{\mu}$ | -0.743 | 0.725 | -1.7175 | 1.735
$s_{e}$ | 1.0669 | 1.1278 | 0.5083 | 0.5
$s_{\mu}$ | 0.99 | 0.9772 | 0.7529 | 0.7463
Table 2: Four example solutions of the parameters of the effective neutrino
mass matrix which reproduces the neutrino masses and mixing parameters within
$1\sigma$ assuming inverted hierarchy. The first four solutions correspond to
the CP conserving case.

Following the usual $\chi^{2}$ optimization technique Joshipura and Patel
(2021), we find several example solutions for the parameters in $m_{\nu}$
which lead to the viable neutrino spectrum. First, we consider real $\xi$
which leads to the vanishing CP phase, $\delta_{\rm CP}$. The latter is not
directly observed and the present fits to the neutrino oscillation data
Capozzi  _et al._ (2017); de Salas  _et al._ (2021); Esteban  _et al._ (2020)
allows $\delta_{\rm CP}=0$ within $\pm 3\sigma$. We find solutions for two
example values of $m$ and also consider a possibility in which the parameter
$\delta$ can be zero. The solutions obtained in this way are displayed as the
first four solutions in Table 1 (2) for normal (inverted) hierarchy in the
neutrino masses. Subsequently, we also consider complex $\xi$ to account for
the Dirac CP phase. These solutions are listed as NH5 to NH8 in the case of
normal and IH5 to IH8 in the case of inverted hierarchy in the respective
tables. All the NH (IH) solutions give $\Delta m_{21}^{2}=7.41\times
10^{-5}\,{\rm eV}^{2}$, $\Delta m_{31}^{2}=2.511\times 10^{-3}\,{\rm eV}^{2}$
($\Delta m_{23}^{2}=2.498\times 10^{-3}\,{\rm eV}^{2}$),
$\sin^{2}\theta_{12}=0.303$, $\sin^{2}\theta_{23}=0.572$
($\sin^{2}\theta_{23}=0.578$) and $\sin^{2}\theta_{13}=0.02203$
($\sin^{2}\theta_{13}=0.02219$). Solutions with complex $\xi$ give
$\sin\delta_{\rm CP}=-0.3$ ($\sin\delta_{\rm CP}=-0.96$) for the NH (IH) case.
All these values correspond to the central values as obtained by the latest
fit to the neutrino oscillation data (see, NuFIT 5.2) in Esteban  _et al._
(2020).

The noteworthy feature of these fits is that $|\xi|\sim 1$ implying that the
scales associated with both the $R$-violating contribution and the seesaw
terms are similar in magnitudes. This can be used to draw important
conclusions on the basic parameters of the model. The exact expression of
$\xi$, using Eqs. (III.1,10), can be rewritten as

$\xi=\frac{A_{0}k^{2}}{\lambda^{2}v_{2}^{2}}{\rm
Det}M_{R}\left(2-\frac{\mu_{0}}{\mu}\right)^{-2}\left(1-\frac{2\lambda^{2}v_{1}v_{2}\mu}{{\rm
Det}M_{R}}\left(1-\frac{\mu_{0}}{\mu}\right)^{2}\right)\,.$ (61)

Scenarios in which the common gaugino mass scale, $M_{1}\simeq
M_{2}\simeq\mu\simeq\mu_{0}\equiv M_{g}$, is arranged to stay close to the
electroweak scale and $m_{3/2}\gg v$, then leads to

$\xi\simeq\frac{m_{3/2}}{2M_{g}}\,\frac{1}{\lambda^{2}}\,\left(\frac{{\rm
Det}M_{R}}{m_{3/2}^{3}}\right)\,.$ (62)

While deriving the above, we have used the expression Eq. (28) for the $k$
parameter. The fact that the value of $|\xi|$ cannot be arbitrarily large, as
indicated by the fits, implies that the gaugino mass scale cannot be much
smaller than $m_{3/2}$ if ${\rm Det}M_{R}\sim m_{3/2}^{3}$. In other words, it
is not possible to arrange gauginos at the electroweak scale and all the RH
neutrinos at an arbitrarily high scale. The ratio $\frac{{\rm
Det}M_{R}}{m_{3/2}^{3}}$ is determined by the soft SUSY breaking parameters
and $\frac{\kappa}{\lambda}$, see Eq. (10). In supergravity models, the soft
parameters are naturally of order unity and hence $\frac{{\rm
Det}M_{R}}{m_{3/2}^{3}}\sim{\cal O}(1)$. However, it can be suppressed in
other models of supersymmetry breaking. For example, the parameters $A$ and
$B$ vanish at a high scale in the gauge mediated models Giudice and Rattazzi
(1999) with the minimal messenger sector and get generated purely radiatively.
In such scenarios, $\frac{{\rm Det}M_{R}}{m_{3/2}^{3}}$ can be naturally small
offering the possibility of raising the value of $m_{3/2}$ compared to the
gaugino masses.

One can also use the fitted value of $m$ and $\xi$ to determine the Yukawa
coupling $\lambda_{\tau}^{2}$. Using Eq. (III.1) and neglecting small $\zeta$
dependent term, one finds

$\displaystyle\lambda_{\tau}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{m\xi\lambda^{2}}{A_{0}k^{2}C_{e}^{2}m_{3/2}^{2}}=\frac{\mu^{2}}{\mu_{0}^{2}}\frac{\lambda^{2}}{C_{e}^{2}}\,\frac{m\xi}{A_{0}v_{2}^{2}}\,,$
(63)

where the second equality is obtained using the value of $k$ from Eq. (28). It
is to be noted that $\lambda_{\tau}^{2}$ as given above is determined by the
$R$ parity violating contribution $m\xi$ rather than by the seesaw
contribution $m$. For $\mu\sim\mu_{0}$, $A_{0}=(2M_{g})^{-1}$ and
$C_{e}=\lambda=1$, we get

$\displaystyle\lambda_{\tau}^{2}$ $\displaystyle\sim$ $\displaystyle 2\times
10^{-11}\,\left(\frac{m\xi}{0.1\,{\rm eV}}\right)\,\left(\frac{100\,{\rm
GeV}}{v_{2}}\right)^{2}\,\left(\frac{M_{g}}{1\,{\rm TeV}}\right)\,.$ (64)

This is to be compared with the expectation based on the seesaw,
$\lambda_{\tau}^{2}\sim\frac{m_{\nu}M_{R}}{v^{2}}\sim 10^{-11}$, for a TeV
scale $M_{R}$ and $m_{\nu}\sim 0.1$ eV. The similarity of the magnitudes of
$\lambda_{\tau}^{2}$ obtained through these two different approaches is not a
coincidence since the model requires both these contributions to be of similar
order for the realistic neutrino mass spectrum.

The above observations can also be summarized by comparing the overall scales
of the seesaw and $R$-parity contributions to the light neutrino masses. The
first is proportional to the parameter $m$ given in Eq. (III.1) which goes as
$\lambda_{\tau}^{2}v_{2}^{2}/m_{3/2}$. The second is determined as $m\xi$ and
it is independent of $m_{3/2}$ for the fixed gaugino mass scale $M_{g}$ as can
be observed from the expression of $\xi$ in Eq. (62). The requirement that $m$
and $m\xi$ are of a similar order, therefore, puts an upper bound on $m_{3/2}$
as it is shown in Fig. 1. Further, if one chooses $M_{g}\simeq m_{3/2}$, both
the contributions scale as $\lambda_{\tau}^{2}v_{2}^{2}/m_{3/2}$ leading to a
situation similar to the one obtained in the case of conformal $W$ discussed
in section II.

Figure 1: Contours of $m=0.01$ eV (red dashed), $m=1$ eV (red solid),
$m\xi=0.01$ eV (green dashed), $m=1$ eV (green solid) for various values of
$m_{3/2}$ and $\lambda_{\tau}^{2}$ and for gaugino mass $M_{g}=10^{3}$ GeV
(left panel) and $M_{g}=10^{5}$ GeV (right panel). Only in the overlapping
region, a realistic light neutrino mass spectrum can be reproduced.

##  IV Majoron-neutrino couplings

In the present framework, there exists a Goldstone boson of the spontaneously
broken LN symmetry, namely the Majoron Chikashige  _et al._ (1981). It can be
identified as

$J\simeq\frac{1}{\sqrt{2}}\left(\tilde{N}_{I\mu}-\tilde{N}_{Ie}\right)+\frac{\omega^{\prime}_{\alpha}}{U}\,\tilde{\nu}_{I\alpha}\,,$
(65)

where
$U=\sqrt{\langle\tilde{N_{e}}\rangle^{2}+\langle\tilde{N_{\mu}}\rangle^{2}}$
is the scale of the $U(1)_{L}$ breaking and the subscript $I$ denotes the
imaginary part of the corresponding scalar field. The Majoron is an $SU(2)$
singlet and does not directly couple to the SM neutrinos. However, this
coupling is generated through (a) light-heavy neutrino mixing and (b)
neutrino-neutralino mixing. This coupling can be parametrized as

${\cal L}_{\nu J}=-\frac{i}{2}g_{ij}\,\nu^{T}_{Li}C^{-1}\nu_{Lj}\,J\,+{\rm
h.c.}\,,$ (66)

in the physical neutrino basis.

The strongest limits on $g_{ij}$ arise from the precision measurement of the
acoustic peak of the CMB spectrum which requires that neutrinos must be free
streaming at the time of photon decoupling and therefore cannot couple
strongly to a nearly massless Majoron. This requirement translates into
Hannestad and Raffelt (2005); Raffelt and Weiss (1995); Ayala  _et al._ (2014)

$\displaystyle|g_{ii}|$ $\displaystyle\lesssim$ $\displaystyle 10^{-7}\,,$
$\displaystyle|g_{ij}|$ $\displaystyle\lesssim$ $\displaystyle 0.61\times
10^{-11}\,\left(\frac{0.05\,{\rm eV}}{m_{\nu_{j}}}\right)^{2}\,,$ (67)

for $i\neq j$ and $m_{\nu_{j}}>m_{\nu_{i}}$. We now show that these limits are
satisfied in the model in spite of relatively low LN breaking scale $\sim
m_{3/2}$ without conflicting with constraints from neutrino masses and mixing.

Contribution to $g_{ij}$ arising from the light-heavy neutrino mixing in the
present model can be estimated as follows. Eq. (25) gives rise to the
following interaction between the heavy neutrinos and the Majoron:

${\cal L}^{(a)}_{\nu
J}=-\frac{i\lambda}{\sqrt{2}}\,\left(N_{e}^{T}C^{-1}N_{\tau}\,\tilde{N_{\mu}}_{I}+N^{T}_{\mu}C^{-1}N_{\tau}\,\tilde{N_{e}}_{I}\right)+{\rm
h.c.}\,.$ (68)

Subsequently, the couplings with light neutrinos can be estimated using the
diagonalization of the relevant blocks of ${\cal M}_{\cal N}$. The latter can
be block diagonalized at the leading order by a unitary matrix,

$U_{\cal N}=\left(\begin{array}[]{ccc}1&\rho_{\nu N}&\rho_{\nu\chi}\\\
-\rho_{\nu N}^{\dagger}&1&\rho_{N\chi}\\\
-\rho_{\nu\chi}^{\dagger}&-\rho_{N\chi}^{\dagger}&1\end{array}\right)\,+\,{\cal
O}(\rho^{2})\,,$ (69)

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

$\rho_{\nu N}\simeq(m_{\nu
N}M_{R}^{-1})^{*}\,,~{}~{}\rho_{\nu\chi}\simeq(m_{\nu\chi}M_{\chi}^{-1})^{*}\,,~{}~{}\rho_{N\chi}\simeq(m_{N\chi}M_{\chi}^{-1})^{*}\,.$
(70)

The neutral fermions in the block-diagonal basis are given by $U_{\cal
N}^{\dagger}\psi^{0}$. Writing the light neutrinos in block diagonal basis as
$\nu_{\alpha}^{\prime}$ and using Eq. (70), one finds

$N_{e}=-(\rho_{\nu
N}^{\dagger})_{1\alpha}\,\nu_{L\alpha}^{\prime}\,,~{}~{}N_{\mu}=-(\rho_{\nu
N}^{\dagger})_{2\alpha}\,\nu_{L\alpha}^{\prime}\,,~{}~{}N_{\tau}=-(\rho_{\nu
N}^{\dagger})_{3\alpha}\,\nu_{L\alpha}^{\prime}\,.$ (71)

Substituting the above in Eq. (68) and using the definition of $J$ in Eq.
(65), we find the light neutrino-Majoron interaction as

${\cal L}^{(a)}_{\nu
J}=-\frac{i}{2}g_{ij}^{(a)}\,\nu^{T}_{Li}C^{-1}\nu_{Lj}\,J\,+{\rm h.c.}\,,$
(72)

with

$\displaystyle g^{(a)}_{ij}$ $\displaystyle=$
$\displaystyle\lambda\left((\rho_{\nu N}^{\dagger})_{1\alpha}-(\rho_{\nu
N}^{\dagger})_{2\alpha}\right)(\rho_{\nu N}^{\dagger})_{3\beta}\,U_{\alpha
i}U_{\beta j}\,$ (73) $\displaystyle=$ $\displaystyle\frac{\lambda
C_{e}v_{2}^{2}}{C_{\tau}m_{3/2}^{2}(C_{\tau}\tilde{C}_{\tau}-2C_{e}^{2})}\,\lambda_{\alpha}\lambda_{\beta}\,U_{\alpha
i}U_{\beta j}\,$ $\displaystyle=$
$\displaystyle-\frac{\lambda\tilde{m}_{0}}{C_{e}m_{3/2}}\,\lambda_{\alpha}\lambda_{\beta}\,U_{\alpha
i}U_{\beta j}\,,$

where we use Eq. (12) to arrive at the last equality. In the above, we have
used $\nu_{L\alpha}^{\prime}=U_{\alpha i}\nu_{Li}$ where $\nu_{Li}$ are the
mass eigenstates which are identified as physical light neutrinos. The
couplings $g^{(a)}_{ij}$ are suppressed by $\tilde{m}_{0}/m_{3/2}$ as a
consequence of heavy-light mixing.

The second contribution arises from the Yukawa interaction in Eq. (25) and
through gauge interactions quantified by the following terms:

$\displaystyle{\cal L}^{(b)}_{\nu J}$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{2}}\lambda_{\alpha}\,\nu_{L\alpha}^{\prime
T}\tilde{N_{e}}_{I}C^{-1}\tilde{h}_{2}^{0}-\frac{i}{2}(g^{\prime}\tilde{B}^{T}-g\tilde{W}^{0T})C^{-1}\nu^{\prime}_{L\beta}\,\tilde{\nu}_{I\beta}+{\rm
h.c.}\,,$ (74)

where $-i\lambda_{1}\equiv\tilde{B}$ and $-i\lambda_{2}\equiv\tilde{W}^{0}$.
The Higgsino, Bino and Wino contain the light neutrinos through mixing. From
the block diagonalization, one finds

$\displaystyle\tilde{h}_{2}^{0}$ $\displaystyle=$
$\displaystyle-(\rho_{\nu\chi}^{\dagger})_{4\alpha}\,\nu_{L\alpha}^{\prime}=\frac{v_{1}}{\mu}A_{0}\,\omega_{\alpha}\,U_{\alpha
i}\,\nu_{Li}\,,$ $\displaystyle\tilde{B}$ $\displaystyle=$
$\displaystyle-(\rho_{\nu\chi}^{\dagger})_{1\alpha}\,\nu_{L\alpha}^{\prime}=-\frac{\sqrt{2}g^{\prime}M_{2}}{g^{2}M_{1}+g^{\prime
2}M_{2}}\,A_{0}\,\omega_{\alpha}\,U_{\alpha i}\,\nu_{Li}\,,$
$\displaystyle\tilde{W}^{0}$ $\displaystyle=$
$\displaystyle-(\rho_{\nu\chi}^{\dagger})_{2\alpha}\,\nu_{L\alpha}^{\prime}=\frac{\sqrt{2}gM_{1}}{g^{2}M_{1}+g^{\prime
2}M_{2}}\,A_{0}\,\omega_{\alpha}\,U_{\alpha i}\,\nu_{Li}\,,$ (75)

Substituting the above in Eq. (74) and using Eq. (65), we find

${\cal L}^{(b)}_{\nu
J}=-\frac{i}{2}g_{ij}^{(b)}\,\nu^{T}_{Li}C^{-1}\nu_{Lj}\,J\,+{\rm h.c.}\,,$
(76)

where

$g^{(b)}_{ij}=-\frac{A_{0}\lambda}{C_{e}m_{3/2}}\omega_{\alpha}\omega_{\beta}\,U_{\alpha
i}U_{\beta j}\,,$ (77)

where $A_{0}$ is as defined in Eq. (18).

Combining the two contributions and using the definition Eq. (66), the
$3\times 3$ light neutrino-Majoron coupling matrix can be written as

$g=-\frac{\lambda}{C_{e}m_{3/2}}\,\,U^{T}\,\left(\tilde{m}_{0}{\cal
A}+A_{0}{\cal B}\right)\,U\,.$ (78)

Note that the quantity in the bracket above is the effective neutrino mass
matrix $m_{\nu}$ in the limit $\eta\to 0$. Therefore, using the simplified
expression of $m_{\nu}$ obtained in Eq. (58) with $\eta=0$, we can express

$g=g_{0}\,\,U^{T}\,\left(\begin{array}[]{ccc}r_{e}^{2}\left(1+\xi_{0}s_{e}^{2}\right)&r_{e}r_{\mu}\left(1+\xi_{0}s_{e}s_{\mu}\right)&r_{e}\left(1+\xi_{0}s_{e}\right)\\\
r_{e}r_{\mu}\left(1+\xi_{0}s_{e}s_{\mu}\right)&r_{\mu}^{2}\left(1+\xi_{0}s_{\mu}^{2}\right)&r_{\mu}\left(1+\xi_{0}s_{\mu}\right)\\\
r_{e}\left(1+\xi_{0}s_{e}\right)&r_{\mu}\left(1+\xi_{0}s_{\mu}\right)&1+\xi_{0}\\\
\end{array}\right)\,U\,,$ (79)

with

$g_{0}=-\frac{\tilde{m}_{0}\lambda_{\tau}^{2}}{\langle\tilde{N}_{e}\rangle}\,,$
(80)

and $\xi_{0}$ as defined in Eq. (22). When $\eta=0$, the matrix in the bracket
in Eq. (79) is identified with $m_{\nu}$ and $g$ becomes a diagonal matrix
leading to only $g_{ii}$ nonzero. This case is relatively poorly constrained
as seen from the limits in Eq. (IV).

Numerically, all the dimensionless parameters appearing in the matrix shown in
Eq. (79) are of order one as can be seen from the numbers in Tables 1 and 2.
The overall scale of neutrino-majoron coupling, therefore, is given by
$g_{0}$. Also noting that the numerator in Eq. (80) is $m$ in the limit
$\eta\to 0$, one can estimate an overall size of $g_{0}$ as

$\displaystyle|g_{0}|$ $\displaystyle\simeq$
$\displaystyle\frac{m}{|\langle\tilde{N}_{e}\rangle|}=10^{-11}\,\left(\frac{m}{1\,{\rm
eV}}\right)\left(\frac{100\,{\rm
GeV}}{|\langle\tilde{N}_{e}\rangle|}\right)\,.$ (81)

It is seen that $|g_{0}|$ remains suppressed for a large range in the
parameter space. The matrix appearing in Eq. (78) is also completely
determined by the fits of the table since $\xi_{0}\approx\xi$. For example,
for solution NH8 listed in Table 1, we get

$|g|\simeq 10^{-13}\,\left(\frac{100\,{\rm
GeV}}{|\langle\tilde{N}_{e}\rangle|}\right)\,\left(\begin{array}[]{ccc}0&0&0\\\
0&1.79624&2.55013\\\ 0&2.55013&1.52902\\\ \end{array}\right)\,.$ (82)

We find that all the solutions listed in Tables 1 and 2 satisfy the
constraints, Eq. (IV), for $|\langle\tilde{N}_{e}\rangle|\gtrsim 100$ GeV.

##  V Majoron-Charged lepton couplings

As in the case of the neutrinos, the charged leptons $e_{\alpha}$ do not
directly couple to Majoron but these couplings arise from their mixing with
the charginos. In general, the Majoron-charged lepton couplings can be
parametrized as

${\cal L}_{ej}=i\,h_{\alpha\beta}\,e_{\alpha}e^{c}_{\beta}J\,+\,{\rm h.c.}\,.$
(83)

Majoron has both the diagonal and the off-diagonal couplings. The latter is
constrained from the flavour violating decays such as $e_{\alpha}\rightarrow
e_{\beta}\,J$ Albrecht  _et al._ (1995); Jodidio  _et al._ (1986). Non-
observation of these implies Raffelt and Weiss (1995); Ayala  _et al._ (2014):

$|h_{e\mu}|<1.9\times 10^{-11}\,,~{}~{}|h_{\tau e}|,|h_{\tau\mu}|<1.1\times
10^{-7}\,.$ (84)

The requirement that the red giant stars do not lose energy through Majoron
emission from electrons constrain $|h_{ee}|$, namely

$|h_{ee}|\leq 2.57\times 10^{-13}\,.$ (85)

Despite such strong constraints, a Majoron with $U(1)_{L}$ breaking scale as
low as $\sim 100$ GeV can satisfy the above limits as we discuss in detail
below.

The three charged lepton and two chargino fields can be represented as

$\psi_{-}=\left(\begin{array}[]{c}e_{\alpha}\\\
\chi_{m}\end{array}\right)\,,~{}~{}~{}\psi_{+}=\left(\begin{array}[]{c}e_{\alpha}^{c}\\\
\chi_{m}^{c}\end{array}\right)$ (86)

with

$\chi_{m}=\left(\begin{array}[]{c}-i\lambda^{-}\\\
h_{1}^{-}\end{array}\right)\,,~{}~{}~{}\chi_{m}^{c}=\left(\begin{array}[]{c}-i\lambda^{+}\\\
h_{2}^{+}\end{array}\right)$ (87)

The charged fermion mass term is parametrized by

$-{\cal L}_{c}=\psi_{-}^{T\,}{\cal M}_{c}\,\psi_{+}\,+\,{\rm h.c.}\,,$ (88)

where $5\times 5$ matrix ${\cal M}_{c}$ is defined as

${\cal M}_{c}=\left(\begin{array}[]{cc}m_{l}&m_{lc}\\\
m_{lc}^{\prime}&M_{c}\\\ \end{array}\right)~{}.$ (89)

Here, $m_{l}$ denotes the (diagonal) charged lepton mass matrix before mixing
with charginos. $m_{lc}$, $m_{lc}^{\prime}$ arise from the $R$-parity
violation and are given by

$m_{lc}=\left(\begin{array}[]{cc}g\omega_{e}^{\prime}&\epsilon_{e}\\\
g\omega_{\mu}^{\prime}&\epsilon_{\mu}\\\
g\omega_{\tau}^{\prime}&\epsilon_{\tau}\\\
\end{array}\right)\,,~{}~{}m_{lc}^{\prime}=\left(\begin{array}[]{ccc}0&0&0\\\
\lambda_{e}^{\prime}\omega_{e}^{\prime}&\lambda_{\mu}^{\prime}\omega_{\mu}^{\prime}&\lambda_{\tau}^{\prime}\omega_{\tau}^{\prime}\\\
\end{array}\right)\,.$ (90)

$M_{c}$ is the usual chargino mass matrix in the MSSM,

$M_{c}=\left(\begin{array}[]{cc}M_{2}&gv_{1}\\\ gv_{2}&\mu\\\
\end{array}\right)~{}.$ (91)

Next, we define a basis

$\psi_{\pm}^{\prime}=U_{\pm}^{\dagger}\,\psi_{\pm}\,,$ (92)

such that ${\cal M}_{c}$ assumes a block diagonal form in this basis.
Explicitly,

$U_{-}^{T}{\cal M}_{c}U_{+}=\left(\begin{array}[]{cc}m^{\rm
eff}_{l}&0_{3\times 2}\\\ 0_{2\times 3}&M^{\rm eff}_{c}\\\
\end{array}\right)\,.$ (93)

The $U_{\pm}$ can be parameterized as

$U_{\pm}=\left(\begin{array}[]{cc}\sqrt{1-B_{\pm}B_{\pm}^{\dagger}}&B_{\pm}\\\
-B_{\pm}^{\dagger}&\sqrt{1-B_{\pm}^{\dagger}B_{\pm}}\\\ \end{array}\right)\,.$
(94)

It is always possible Grimus and Lavoura (2000) to choose $3\times 2$ matrices
$B_{\pm}$ which puts ${\cal M}_{c}$ in the block diagonal form of Eq. (93).

The coupling between the Majoron and charged lepton can be directly obtained
from the basic Lagrangian as done in the case of neutrinos. Alternatively, one
can use the effective Lagrangian describing the coupling of Majoron to the
divergence of the broken $U(1)$ current $j^{\mu}$,

$-{\cal L}_{eJ}=-\frac{i}{U}\,J\,\partial_{\mu}j^{\mu}~{}.$ (95)

Here,

$j^{\mu}=-\bar{e}_{\alpha}\bar{\sigma}^{\mu}e_{\alpha}+\bar{e}^{c}_{\alpha}\bar{\sigma}^{\mu}e^{c}_{\alpha}~{}$
(96)

is the leptonic current in the flavour basis. $U$ denotes the $U(1)_{L}$
symmetry breaking scale as already defined in the previous section. The mixing
terms $m_{lc}$ and $m_{lc}^{\prime}$ violate $U(1)_{L}$ and hence contribute
to $\partial_{\mu}j^{\mu}$. This contribution is given by

$\partial_{\mu}j^{\mu}=i\,\psi^{T}_{-}\left(q{\cal M}_{c}+{\cal
M}_{c}q^{c}\right)\psi_{+}\,+\,{\rm h.c.}\,,$ (97)

where

$q=\left(\begin{array}[]{cc}{1}_{3\times 3}&{0}_{3\times 2}\\\ {0}_{2\times
3}&{0}_{2\times
2}\end{array}\right)~{}~{}~{},~{}~{}~{}q^{c}=\left(\begin{array}[]{cc}-{1}_{3\times
3}&{0}_{3\times 2}\\\ {0}_{2\times 3}&{0}_{2\times 2}\end{array}\right)\,$
(98)

are the diagonal charge matrices representing $U(1)_{L}$ charges of the
underlying charged leptons and charginos. Eq. (97) can be rewritten in the
block diagonal basis as

$\partial_{\mu}j^{\mu}=i\psi^{\prime T}_{-}U_{-}^{T}\left(q{\cal
M}_{c}+q^{c}{\cal M}_{c}\right)U_{+}\psi_{+}^{\prime}\,+\,{\rm h.c.}\,.$ (99)

Using Eqs. (93,94), the Majoron charged lepton coupling can be written in the
block diagonal basis as

$\displaystyle{\cal L}_{eJ}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}\langle\tilde{N}_{e}\rangle}\left(m^{\rm
eff}_{l}\,B_{+}B_{+}^{\dagger}-B_{-}^{*}B_{-}^{T}\,m^{\rm
eff}_{l}\right)_{\alpha\beta}\,e_{\alpha}e_{\beta}^{c}J\,+\,{\rm h.c.}\,.$
(100)

Comparing with the definition Eq. (83), we get

$h_{\alpha\beta}=\frac{1}{\sqrt{2}\langle\tilde{N}_{e}\rangle}\left(m^{\rm
eff}_{l}\,B_{+}B_{+}^{\dagger}-B_{-}^{*}B_{-}^{T}\,m^{\rm
eff}_{l}\right)_{\alpha\beta}\,.$ (101)

This equation is valid to all orders in the seesaw expansion. $m^{\rm
eff}_{l}$ is the effective charged lepton mass matrix whose eigenvalues
correspond to the masses of the charged leptons.

$B_{\pm}$ can be iteratively determined using Eq. (93) in the seesaw
approximation $m_{l}<m_{lc},m_{lc}^{\prime},\ll M_{C}$. At the leading order,
one finds Grimus and Lavoura (2000)

$B_{+}\approx(M_{c}^{-1}m_{lc}^{\prime})^{\dagger}\,,~{}~{}B_{-}\approx(m_{lc}M_{c}^{-1})^{*}\,,~{}~{}m_{l}^{\rm
eff}\approx m_{l}-m_{lc}M_{c}^{-1}m_{lc}^{\prime}\,.$ (102)

Substitution of the above in Eq. (101) indicates that the charged lepton-
Majoron couplings, at the leading order, are suppressed by two powers of
seesaw expansion parameter $m_{lc}M_{c}^{-1}$ or $m^{\prime}_{lc}M_{c}^{-1}$.
This is different from neutrino-Majoron coupling which goes as $\sim m_{\nu
N}M_{R}^{-1}$ at the leading order. The model, therefore, predicts
additionally suppressed coupling between the Majoron and the charged leptons.

Explicit computation of $h_{\alpha\beta}$ using the leading order expressions
of $B_{\pm}$ and neglecting the terms of ${\cal O}(v_{1,2}/\mu)$, we get

$\displaystyle
h_{\alpha\beta}=-\frac{1}{\sqrt{2}\langle\tilde{N}_{e}\rangle}\,\left(m_{\beta}\left(\frac{\epsilon_{\alpha}\epsilon_{\beta}}{\mu^{2}}+g^{2}\frac{\omega^{\prime}_{\alpha}\omega^{\prime}_{\beta}}{M_{2}^{2}}\right)-m_{\alpha}\frac{\lambda_{\alpha}^{\prime}\lambda_{\beta}^{\prime}\omega^{\prime}_{\alpha}\omega^{\prime}_{\beta}}{\mu^{2}}\right)\,.$
(103)

Using Eq. (II.1) and noting that $k_{\alpha}$ of Eqs. (20,27) are smaller than
one, the dominant contribution to $h_{\alpha\beta}$ comes from the term
proportional to $\epsilon_{\alpha}\epsilon_{\beta}$. This gives, for example,

$\displaystyle h_{ee}$ $\displaystyle\simeq$ $\displaystyle
r_{e}^{2}\lambda_{\tau}^{2}m_{e}\frac{\langle\tilde{N}_{e}\rangle}{\mu^{2}}~{},$
(104) $\displaystyle=$ $\displaystyle 4\times
10^{-16}\,\left(\frac{\lambda_{\tau}^{2}}{10^{-10}}\right)\left(\frac{\langle\tilde{N}_{e}\rangle}{100\,{\rm
GeV}}\right)\left(\frac{500\,{\rm GeV}}{\mu}\right)^{2}\,,$

where we take $r_{e}\equiv\frac{\lambda_{e}}{\lambda_{\tau}}=4.44$ which is
the largest value among all the solutions listed in Tables 1 and 2. The
obtained value of $h_{ee}$ is far below the constraint (85). Similarly, the
off-diagonal elements are also found much below the existing constraint, Eq.
(84).

##  VI Right-handed neutrinos and direct search prospects

The three RH neutrinos in the model have masses close to the electroweak scale
and they can be produced at colliders if their mixing with the other particles
is favourable. In this section, we discuss such possibilities in detail and
point out ranges of parameters of the model where the direct detection of RH
neutrinos is possible. There are two standard methods of detecting the RH
neutrinos, see Deppisch  _et al._ (2015) for a review. The RH neutrinos can
couple to the standard $W$ and $Z$ through their mixing with the active
neutrinos. The same mixing also causes the decay of the $N_{i}$ into charged
leptons. This leads to dilepton + 2 jet signals which are free from the SM
background. Such signals however depend on the fourth power of the RH mixing
$|V_{\alpha N_{i}}|$ and are therefore sensitive to relatively large mixing
only. In the $R$-parity violating supersymmetric models, the RH neutrinos can
also mix with the neutralinos and this mixing could be much larger than
$|V_{\alpha N_{i}}|$. In such a situation, the RH neutrinos are produced in
the decays of neutralinos and further decay through their mixing with the
active neutrinos. Relatively small mixing with active neutrinos can result in
the delayed decays of the RH neutrinos which lead to the distinctive displaced
vertex signatures as discussed in Lavignac and Medina (2021).

The RH neutrino mass eigenstates $N_{i}$ are related to the flavour states
$N_{\alpha}$ as

$N_{\alpha}=(U_{R})_{\alpha i}N_{i}\,,$ (105)

where $U_{R}$ diagonalizes the mas matrix $M_{R}$ given in Eq. (7).
Explicitly, $U_{R}$ is given by

$U_{R}\simeq\left(\begin{array}[]{ccc}\frac{i}{\sqrt{2}}&\frac{\cos\theta}{\sqrt{2}}&\frac{\sin\theta}{\sqrt{2}}\\\
-\frac{i}{\sqrt{2}}&\frac{\cos\theta}{\sqrt{2}}&\frac{\sin\theta}{\sqrt{2}}\\\
0&-\sin\theta&\cos\theta\\\ \end{array}\right)\,,$ (106)

Mixing of $N_{i}$ with the charged leptons follows from Eqs. (69,105).
Specifically, one can write

$\nu^{\prime}_{\alpha}=(\rho_{\nu D})_{\alpha\beta}N_{\beta}=(\rho_{\nu
D})_{\alpha\beta}(U_{R})_{\alpha i}N_{i}\equiv V_{\alpha N_{i}}N_{i}\,.$ (107)

Using the above relation, we find

$\displaystyle V_{\alpha N_{1}}$ $\displaystyle=$
$\displaystyle=\frac{-i\lambda_{\alpha}v_{2}}{\sqrt{2}M_{N_{1}}}=\frac{-ir_{\alpha}\lambda_{\tau}v_{2}}{\sqrt{2}M_{N_{1}}}~{},$
$\displaystyle V_{\alpha N_{2}}$ $\displaystyle=$
$\displaystyle\frac{c_{\theta}\lambda_{\alpha}v_{2}}{\sqrt{2}M_{N_{2}}}=\frac{c_{\theta}r_{\alpha}\lambda_{\tau}v_{2}}{\sqrt{2}M_{N_{2}}}~{},$
$\displaystyle V_{\alpha N_{3}}$ $\displaystyle=$
$\displaystyle\frac{s_{\theta}\lambda_{\alpha}v_{2}}{\sqrt{2}M_{N_{3}}}=\frac{s_{\theta}r_{\alpha}\lambda_{\tau}v_{2}}{\sqrt{2}M_{N_{3}}}\,.$
(108)

The predicted RH mixing, $V_{\alpha N_{i}}\sim\frac{(m_{\nu N}U_{R})_{\alpha
i}}{M_{N_{i}}}$ is in agreement with a typical seesaw scenario. For
$\lambda_{\tau}\sim 10^{-5}$ and $M_{N_{i}}\sim 100$ GeV, one gets $V_{\alpha
N_{i}}\sim r_{\alpha}\,10^{-7}$. This is much smaller than the current limit
from the direct searches, $|V_{\alpha N}|^{2}<10^{-5}$ Aad  _et al._ (2019);
Sirunyan  _et al._ (2018), see for example, the left panel in Fig. 2. One can
raise $V_{\alpha N_{i}}$ for a smaller $M_{N_{i}}$ and a larger
$\lambda_{\tau}$. For example, $\mu/\mu_{0}\sim 5$ and $M_{N_{1}}=100$ GeV in
Eq. (63) gives $\lambda_{\tau}\sim 2\times 10^{-3}$ and leads to
$V_{eN_{3}}\sim 3\times 10^{-3}\sin\theta$. While this is much larger than the
typical TeV scale seesaw prediction, this still falls short of the existing
limits from the collider searches. Nevertheless, it is possible to probe such
or even smaller mixing through the specific displaced vertex signature as
suggested in Lavignac and Medina (2021) provided that RH neutrinos are
produced through neutralinos.

The mixing between heavy neutrinos $N_{i}$ and neutralinos, relevant for
displaced vertex signatures at the colliders, can be estimated using

$N_{\alpha}=\left(\rho_{N\chi}\right)_{\alpha m}\,(\psi_{0})_{m}\,.$ (109)

Using Eqs. (70,31) and ignoring the term
$\lambda_{\alpha}\omega^{\prime}_{\alpha}$ in $m_{N\chi}$, we find that only
the state $N_{\tau}$ has mixing with Higgsinos and gauginos at the leading
order. It is given by

$\displaystyle N_{\tau}$ $\displaystyle\simeq$ $\displaystyle\frac{2\eta
A_{0}vM_{1}M_{2}}{\mu(g^{2}M_{1}+g^{\prime
2}M_{2})}\left(\cos\beta\,\tilde{h}^{0}_{1}+\sin\beta\,\tilde{h}^{0}_{2}+\frac{g^{\prime}v}{\sqrt{2}M_{1}}\cos
2\beta\,\tilde{B}-\frac{gv}{\sqrt{2}M_{2}}\cos 2\beta\,\tilde{W^{0}}\right).$
(110)

Overall, the couplings with Higgsinos are suppressed by $v/\mu$ only while the
ones with Bino (Wino) are suppressed by additional power of $v/M_{1}$
($v/M_{2}$). Replacing $N_{\tau}$ by physical states through Eq. (106), we
find the heavy neutrino-Higgsino mixing as

$V_{N_{2}\tilde{h}_{1}^{0}}\simeq-\frac{2\eta
A_{0}vM_{1}M_{2}}{\mu(g^{2}M_{1}+g^{\prime
2}M_{2})}\,\cos\beta\sin\theta\,,~{}~{}~{}V_{N_{3}\tilde{h}_{1}^{0}}\simeq\frac{2\eta
A_{0}vM_{1}M_{2}}{\mu(g^{2}M_{1}+g^{\prime 2}M_{2})}\,\cos\beta\cos\theta\,.$
(111)

As it can be seen, $N_{1}$ does not mix with Higgssinos at the leading order.

The states $N_{2}$ and $N_{3}$ can be efficiently produced in the decays of
Higgsinos provided that $M_{N_{2,3}}$ are smaller than the Higgsino masses.
Motivated by this, we look for the values of parameters for which $N_{3}$ is
the lightest among all three RH neutrinos222The role of $N_{2}$ and $N_{3}$
can be interchanged by changing the sign of $\cos 2\theta$, see Eq. (II.1)..
As can be seen from Eq. (II.1), the ratios $M_{N_{3}}/M_{N_{1}}$ and
$M_{N_{3}}/M_{N_{2}}$ depend only on $\theta$ and $\kappa/\lambda$. By varying
the values of these parameters in reasonable ranges, we look for the region in
which $M_{N_{3}}$ is the lightest RH neutrino. The results are displayed in
Fig. 2 in which the regions shaded in orange correspond to
$M_{N_{3}}<M_{N_{1,2}}$.

Figure 2: Regions in $\theta$ and $\kappa/\lambda$ leading to
$M_{N_{3}}<M_{N_{1,2}}$. The black contours indicate the correlation, Eq.
(112), resulting from a specific choice of soft parameters.

In the same figure, we also show the contours corresponding to

$\tan^{2}2\theta=\frac{8}{1-\frac{\kappa}{\lambda}}\,.$ (112)

The above arises from a relation between $C_{e}$ and $C_{\tau}$ obtained as
Eq. (136) as a result of a specific choice of the soft parameters.

Next, for $M_{N_{3}}<M_{N_{1,2}}$, we estimate the values of mixing parameters
$V_{eN_{3}}$ and $V_{N_{3}\tilde{h}_{1}^{0}}$ given in Eq (VI) and (111),
respectively. This requires the specification of several parameters. Using
Eqs. (26,II.1), we set

$\eta=\lambda\,\frac{\mu_{0}-\mu}{M_{N_{1}}}=\lambda\,\frac{\mu_{0}-\mu}{2M_{N_{3}}}\,\left(1+\frac{\kappa}{\lambda}+\left(\frac{\kappa}{\lambda}-1\right)\sec
2\theta\right)\,.$ (113)

For $\lambda_{\tau}$, we use the expression obtained in Eq. (63) and replace
$C_{e}$ using Eq. (9). Also, $\lambda$ can be determined from Eq. (61) and
using the leading order expression of $\xi$ and also using Eqs. (II.1,10). It
leads to

$\lambda^{2}\simeq
2M_{N_{3}}\frac{A_{0}}{\xi}\left(\frac{\mu_{0}/\mu}{2-\mu_{0}/\mu}\right)^{2}\frac{2C_{e}^{2}-\frac{\kappa}{\lambda}C_{\tau}^{2}}{1+\frac{\kappa}{\lambda}+(\frac{\kappa}{\lambda}-1)\sec
2\theta}\,.$ (114)

For the MSSM parameters, we take $M_{1}=1$ TeV, $M_{2}=\mu_{0}=2$ TeV,
$\mu=500$ GeV, $\tan\beta=2$ and choose $C_{\tau}=0.1$ as a specific example.
With these, $V_{eN_{3}}$ and $V_{N_{3}\tilde{h}_{1}^{0}}$ are obtained as
functions of $r_{e}$, $m\xi$, $\kappa/\lambda$, $\theta$ and $M_{N_{3}}$. The
first two can be extracted directly from the neutrino mass fits while the next
two can be appropriately chosen from Fig. 1 such that $N_{3}$ is the lightest
RH neutrino. We chose $r_{e}$ and $m\xi$ from solution NH8 and randomly vary
$\kappa/\lambda$ ($\theta$) in range $0.1$-$5$ ($1.0$-$2.0$) and compute
$V_{eN_{3}}$ and $V_{N_{3}\tilde{h}_{1}^{0}}$ as function of $M_{N_{3}}$. The
results are displayed in Fig. 2. We also show the typical value of $|V_{\alpha
N}|^{2}$ expected naively from a seesaw relation, $|V_{\alpha
N}|^{2}=\frac{0.05\,{\rm eV}}{M_{N}}$, as a solid line in the left panel of
Fig. 3.

Figure 3: Variations of $|V_{eN_{3}}|^{2}$
($|V_{N_{3}\tilde{h}_{1}^{0}}|^{2}$) with $M_{N_{3}}$ in the left (right)
panel, for example, values of various parameters as described in the text. The
solid black line in the left panel depicts the values of $|V_{\alpha N}|^{2}$
expected from a typical seesaw relation, $|V_{\alpha N}|^{2}=\frac{0.05\,{\rm
eV}}{M_{N}}$. The grey region covered by dashed contour is excluded at $95\%$
C.L. by the direct searches of heavy neutral leptons Aad  _et al._ (2019);
Sirunyan  _et al._ (2018).

The predicted values of $|V_{eN_{3}}|$ can be larger than the typical
expectation from seesaw estimates by several orders of magnitude. Also, a
number of points for $M_{N_{3}}<100$ GeV fall either in the already excluded
region or are quite close to it. As seen from both the panels in Fig. 3, the
considered range of parameters give
$|V_{N_{3}\tilde{h}_{1}^{0}}|\gg|V_{eN_{3}}|$ for a given $M_{N_{3}}$ which is
ideally suitable for the displaced vertex signatures at the colliders. For
comparison, $|V_{N\tilde{\chi}_{1}^{0}}|^{2}\sim 10^{-4}$ and
$|V_{eN}|^{2}\sim 10^{-12}$ are shown to lead to observable signatures for
$M_{N}\sim 100$ GeV and for the similar values of $M_{1,2}$ and $\mu$ in
Lavignac and Medina (2021). The values of $|V_{N_{3}\tilde{h}_{1}^{0}}|$ and
$|V_{eN_{3}}|$ we get for $M_{N_{3}}\sim 100$ GeV, can be considerably larger
than these. Hence, the present model can provide a more favourable setup for
probing the RH neutrinos in the direct search experiments.

##  VII Discussions

Extension of the SM to include the LN as an explicit global or local symmetry
is primarily aimed at providing tiny but non-vanishing masses for the light
neutrinos through the seesaw mechanisms. On the other hand, extension with
enlarged space-time symmetries such as supersymmetry is aimed to ameliorate
some structural issues pertaining to the SM. In general, these are two
separate classes of new physics and they may be operating at two different
scales. They are often combined to play complementary roles. A more intriguing
possibility is that both these frameworks arise from a common mechanism and,
therefore, their scales are closely related. We have put forward one such
mechanism and discussed its most relevant phenomenological aspects in detail.

The mechanism uses a global LN symmetry and three SM singlet superfields, two
of which are charged under the $U(1)_{L}$. The Lorentz scalar components of
these superfields acquire non-zero vacuum expectation values and break
$U(1)_{L}$ spontaneously only when the soft terms in supergravity are switched
on. This relates the $U(1)_{L}$ breaking scale with the SUSY breaking scale
which is proportional to the gravitino mass $m_{3/2}$. The fermionic
components of the SM singlet superfields which play the role of RH neutrinos
also get masses of order $m_{3/2}$. The same scale also governs the neutrino-
neutralino mixing through $R$-parity violation effectively induced by the VEVs
of the SM singlet scalars. It is also possible to link the MSSM
$\mu$-parameter with the common scale $m_{3/2}$ utilising the vacuum structure
of the RH sneutrinos.

The identification of various scales with a common scale $m_{3/2}$ in the
underlying framework leads to the following interesting and noteworthy
observations:

  * •

The light neutrinos acquire masses through the usual seesaw mechanism as well
as from $R$-parity violation. The realistic neutrino mass spectrum requires
that both these contributions are of similar magnitude. In the case of
$\mu\sim m_{3/2}$ (i.e. conformal superpotential), both these contributions
are proportional to $m_{3/2}^{-1}$. Therefore, the usual scale of the seesaw
would also imply high-scale SUSY. On the other hand, keeping the SUSY scale
close to the electroweak scale necessarily leads to low scale seesaw mechanism
with RH neutrino masses of the order of a few hundred GeV.

  * •

If $\mu$ is kept independent of $m_{3/2}$ and held fixed close to the
electroweak scale, the viable light neutrino masses do not allow a large
hierarchy between $\mu$ and $m_{3/2}$. This again implies a low-scale seesaw
mechanism and a SUSY breaking scale not very far from the electroweak scale.
In particular, the gauginos and RH neutrinos are not allowed to be arbitrarily
heavy in this case.

  * •

The spontaneous breaking of $U(1)_{L}$ at the scale $\sim\,m_{3/2}$ leads to a
Majoron with coupling to the light neutrinos of ${\cal O}(m_{\nu}/m_{3/2})$.
This turns out to be small enough to satisfy all the present constraints even
for massless Majoron.

  * •

The existence of near weak-scale RH neutrinos can be probed through their
displaced vertex signature in which they are produced from Higgssinos at the
colliders and subsequently decay into the SM particles. It is seen that for a
range of parameters, the model predicts sizeable mixings between Higgssino and
RH neutrino and between the heavy and light neutrinos enhancing the
possibility for direct detection.

All along, we have assumed that supergravity is responsible for the generation
of the soft SUSY breaking. However, the underlying mechanism is applicable
even if the SUSY breaking arises from some other sources.

## Acknowledgements

This work is partially supported under the MATRICS project (MTR/2021/000049)
by the Science & Engineering Research Board (SERB), Department of Science and
Technology (DST), Government of India. KMP acknowledges support from the ICTP
through the Associates Programme (2023-2028) where part of this work was
completed.

##  Appendix A Potential minimization and VEVs of RH sneutrinos

In this Appendix, we provide a derivation of the vacuum expectation values
given in Eq. (4). The relevant scalar potential of the model derived from
$W_{S}$ in Eq. (1) along with the soft terms is obtained as

$V=|\lambda|^{2}|\tilde{N}_{\tau}|^{2}\left(|\tilde{N}_{e}|^{2}+{|\tilde{N}_{\mu}|^{2}}\right)+|\lambda\tilde{N}_{e}\tilde{N}_{\mu}+\kappa\tilde{N}_{\tau}^{2}|^{2}+V_{\rm
soft}\,,$ (115)

where $V_{\rm soft}$ is given in Eq. (3).

We parametrize the VEVs of the three scalar fields as

$\langle\tilde{N}_{\tau}\rangle=q\,,~{}~{}\langle\tilde{N}_{e}\rangle=p\cos\phi\,,~{}~{}\langle\tilde{N}_{\mu}\rangle=p\sin\phi\,.$
(116)

The parameters $p$, $q$ and $\phi$ are to be obtained by solving the
minimization conditions $\frac{\partial V}{\partial\tilde{N}_{\alpha}}=0$ at
the minimum for
$\tilde{N}_{\alpha}=\tilde{N}_{\tau},\tilde{N}_{e},\tilde{N}_{\mu}$. For
simplicity, we assume all the parameters in $V$ real and also require real
solutions for the VEVs. Using a linear combination of two of the minimization
conditions

$\cos\phi\,\frac{\partial V}{\partial\tilde{N}_{e}}-\sin\phi\,\frac{\partial
V}{\partial\tilde{N}_{\mu}}=0\,,$ (117)

we obtain

$\cos
2\phi=\frac{(a_{N_{\mu}}^{2}-a_{N_{e}}^{2})\,m_{3/2}^{2}}{(a_{N_{\mu}}^{2}+a_{N_{e}}^{2})\,m_{3/2}^{2}+2q^{2}\lambda^{2}}\,,$
(118)

at the minimum of the potential. Using the orthogonal linear combination

$\sin\phi\,\frac{\partial V}{\partial\tilde{N}_{e}}+\cos\phi\,\frac{\partial
V}{\partial\tilde{N}_{\mu}}=0\,,$ (119)

we solve for $p^{2}$ and obtain

$p^{2}=0\,~{}{\rm
or}\,~{}p^{2}=-\frac{1}{\lambda^{2}}\left((a_{N_{e}}^{2}+a_{N_{\mu}}^{2})m_{3/2}^{2}+2q^{2}\lambda^{2}+\frac{2q\lambda}{\sin
2\phi}(m_{3/2}A_{\lambda}+q\kappa)\right)\,.$ (120)

Next, we assume $a_{N_{e}}^{2}=a_{N_{\mu}}^{2}$ for simplification which leads
to $\cos 2\phi=0$ and degenerate VEVs for $\tilde{N}_{e}$ and
$\tilde{N}_{\mu}$. Using the remaining minimization condition, $\frac{\partial
V}{\partial\tilde{N}_{\tau}}=0$, and substituting the non-vanishing solution
for $p^{2}$ given in Eq. (120) along with $\cos 2\phi=0$, we find the
following equation:

$q^{3}+bq^{2}+cq+d=0\,,$ (121)

with

$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{3A_{\lambda}(\kappa+\lambda)-A_{\kappa}\kappa}{2\lambda(2\kappa+\lambda)}\,m_{3/2}\,,$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{(A_{\lambda}^{2}-a_{N_{\tau}}^{2})\lambda+2a_{N_{e}}^{2}(\lambda+\kappa)}{2\lambda^{2}(2\kappa+\lambda)}\,m_{3/2}^{2}\,,$
$\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{A_{\lambda}a_{N_{e}}^{2}}{2\lambda^{2}(2\kappa+\lambda)}\,m_{3/2}^{3}\,.$
(122)

The other solution, $p^{2}=0$, leads to $q=0$. Eq. (121) needs to be solved to
determine $q$ which then can be used to find the VEVs in terms of the
parameters of the scalar potential.

It is convenient to convert Eq. (121) into the so-called “Depressed cubic”
equation Dickson  by the following change of variable:

$q=u-\frac{b}{3}\,.$ (123)

In the new variable, the equation has no quadratic term and it is given by

$u^{3}+Pu+Q=0\,,$ (124)

where

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{3c-b^{2}}{3}\,,$
$\displaystyle Q$ $\displaystyle=$ $\displaystyle\frac{2b^{3}-9bc+27d}{27}\,.$
(125)

The discriminant of the Depressed cubic equation is given by

$\Delta=-(4P^{3}+27Q^{2})\,.$ (126)

The following possibilities exist for the solutions Dickson  of Eq. (124):

  * •

If $\Delta>0$, then the underlying polynomial has three distinct real roots.
They are explicitly given by

$u_{k}=2\sqrt{\frac{-P}{3}}\,\cos\left[\frac{1}{3}\cos^{-1}\left(\frac{3Q}{2P}\sqrt{\frac{-3}{P}}\right)-\frac{2\pi
k}{3}\right]\,,~{}~{}~{}k=0,1,2\,.$ (127)

  * •

For $\Delta=0$ and $P\neq 0$, the polynomial has three real roots out of which
two are degenerate. These are

$u_{1}=\frac{3Q}{P}\,,~{}~{}~{}~{}u_{2,3}=-\frac{3Q}{2P}\,.$ (128)

  * •

If $\Delta<0$ then the polynomial has one real and two complex roots. The real
root can be represented, for $P<0$, as

$u_{0}=-2\frac{|Q|}{Q}\,\sqrt{-\frac{P}{3}}\,\cosh\left[\frac{1}{3}\cosh^{-1}\left(-\frac{3|Q|}{2P}\sqrt{-\frac{3}{P}}\right)\right]\,,$
(129)

and, for $P>0$, as

$u_{0}=-2\sqrt{\frac{P}{3}}\,\sinh\left[\frac{1}{3}\sinh^{-1}\left(\frac{3Q}{2P}\sqrt{\frac{3}{P}}\right)\right]\,.$
(130)

Substituting $b$, $c$ and $d$ from Eq. (A) in Eq. (A), one can determine
$\Delta$ which turns out to be proportional to $(m_{3/2}/\lambda)^{6}$. With
an appropriate choice of the values of the soft parameters, $\Delta\geq 0$ can
be obtained leading to the real solutions for $q_{i}$.

As an explicit example, consider

$A_{\kappa}=A_{\lambda}\equiv
A\,,~{}~{}a_{N_{e}}^{2}=a_{N_{\tau}}^{2}=A^{2}/8\,.$ (131)

This leads to $\Delta=0$ for which the roots can be estimated from Eq. (128).
Substituting this choice of the soft parameters in Eq. (A) and using Eq. (128)
and (123), we find the real solutions $q_{i}$ as

$q_{1}=-\frac{Am_{3/2}}{2\kappa+\lambda}\,~{}~{}~{}{\rm
and}~{}~{}~{}q_{2}=q_{3}=-\frac{Am_{3/2}}{4\lambda}\,$ (132)

Substitution of the above in $p^{2}$ expression Eq. (120) leads to

$p_{1}^{2}=-\frac{A^{2}m_{3/2}^{2}(\lambda-2\kappa)^{2}}{4\lambda^{2}(\lambda+2\kappa)^{2}}\,~{}~{}~{}{\rm
and}~{}~{}~{}p_{2}^{2}=p_{3}^{2}=\frac{A^{2}m_{3/2}^{2}(\lambda-\kappa)}{8\lambda^{3}}\,.$
(133)

The first solution leads to an imaginary $p$ while the second corresponds to
real $p$ if $\lambda>\kappa$. Moreover, the solution corresponding to
$(q_{2},p_{2})$ is a global minimum.

In summary, for $a_{N_{\mu}}=a_{N_{e}}$, the scalar potential given in Eq.
(115) can lead to the following VEV configuration:

$\langle\tilde{N}_{\tau}\rangle=\frac{C_{\tau}}{\lambda}m_{3/2}\,,~{}~{}\langle\tilde{N}_{e}\rangle=\langle\tilde{N}_{\mu}\rangle=\frac{C_{e}}{\lambda}m_{3/2}\,,$
(134)

where $C_{\tau}$ and $C_{e}$ are dimensionless parameters determined by
$\lambda$, $\kappa$ and the soft parameters. The non-trivial relation between
the VEVs obtained in Eq. (120) implies the following correlation between
$C_{e}$ and $C_{\tau}$:

$\frac{C_{e}^{2}}{C_{\tau}^{2}}=-1-\frac{\kappa}{\lambda}-\frac{a_{N_{e}}^{2}}{C_{\tau}^{2}}-\frac{A_{\lambda}}{C_{\tau}}\,.$
(135)

For the specific choice of the soft terms used in Eq. (131) and the resulting
$q_{2}$ in Eq. (132), the above becomes

$\frac{C_{e}^{2}}{C_{\tau}^{2}}=1-\frac{\kappa}{\lambda}\,,$ (136)

which is completely determined in terms of the ratio $\kappa/\lambda$.
However, the desired value of $C_{\tau}$ and $C_{e}$ can be obtained by the
appropriate choice of $a_{N_{e}}$ and $A_{\lambda}$ in general. Therefore, the
VEVs in Eq. (134) are parametrized in a general way and considered for
phenomenological analysis.

##  Appendix B Calculation of $k_{\alpha}$

We derive expressions for $k_{\alpha}$ displayed in Eqs. (20,27) in this
Appendix following a similar calculation carried out earlier in Giudice  _et
al._ (1993); Joshipura  _et al._ (2002) for a different framework. When the
scalar components of $\hat{N}_{e,\tau}$ take VEVs, the bilinear terms in $W$
are given by

$W\supset-\,\epsilon_{\alpha}\,\hat{L}^{\prime}_{\alpha}\hat{H}_{2}\,-\,\mu\,\hat{H}^{\prime}_{1}\hat{H}_{2}\,$
(137)

Here, $\epsilon_{\alpha}=-\lambda_{\alpha}\langle\tilde{N}_{e}\rangle$ and we
consider a general case in which $\mu$ contains both the contributions such as
shown in Eq. (26). The scalar potential evaluated from the above is then given
by

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

where the last term in the first line arises from the $F$-term of
$\hat{N}_{\tau}$. Further, the relevant soft terms can be written as

$\displaystyle V_{\rm soft}$ $\displaystyle=$ $\displaystyle
m_{3/2}\left(A_{\alpha}\epsilon_{\alpha}\tilde{L}^{\prime}_{\alpha}\tilde{H}_{2}+B_{\mu}\mu\tilde{H}^{\prime}_{1}\tilde{H}_{2}+{\rm
c.c.}\right)\,$ (139) $\displaystyle+$ $\displaystyle
m_{3/2}^{2}\left(a_{H_{1}}^{2}\,|\tilde{H}^{\prime}_{1}|^{2}+a_{H_{2}}^{2}\,|\tilde{H}_{2}|^{2}+a_{L_{\alpha}}^{2}\,|\tilde{L}^{\prime}_{\alpha}|^{2}\right)\,,$

where we have defined

$B_{\mu}\mu=B_{\mu_{0}}\mu_{0}-A_{\eta}\eta\langle\tilde{N}_{\tau}\rangle\,.$
(140)

In Eqs. (138,139), the index $\alpha$ is summed over.

Rotating away the term proportional to $\epsilon_{\alpha}$ form Eq. (137)
using the redefinitions:

$\mu\hat{H}^{\prime}_{1}+\epsilon_{\alpha}\hat{L}_{\alpha}^{\prime}=\mu\hat{H}_{1}\,,~{}~{}-\epsilon_{\alpha}\hat{H}^{\prime}_{1}+\mu\hat{L}^{\prime}_{\alpha}=\mu\hat{L}_{\alpha}\,,$
(141)

one finds that $V_{F,D}$ and $V_{\rm soft}$ get modified to

$\displaystyle\tilde{V}_{F,D}$ $\displaystyle=$
$\displaystyle|\mu|^{2}\left(|\tilde{H}_{1}|^{2}+|\tilde{H}_{2}|^{2}\right)+|\eta|^{2}\left|\tilde{H}_{1}-\frac{\epsilon_{\alpha}}{\mu}\tilde{L}_{\alpha}\right|^{2}|\tilde{H}_{2}|^{2}$
$\displaystyle+$ $\displaystyle\frac{1}{8}(g^{2}+g^{\prime
2})\left(|\tilde{H}_{1}|^{2}-|\tilde{H}_{2}|^{2}+|\tilde{L}_{\alpha}|^{2}\right)^{2}\,,$
$\displaystyle\tilde{V}_{\rm soft}$ $\displaystyle=$ $\displaystyle
m_{3/2}\left((A_{\alpha}-B_{\mu})\epsilon_{\alpha}\tilde{L}_{\alpha}\tilde{H}_{2}+B_{\mu}\mu\tilde{H}_{1}\tilde{H}_{2}+{\rm
c.c.}\right)\,$ (142) $\displaystyle+$ $\displaystyle
m_{3/2}^{2}\left(a_{H_{1}}^{2}\,|\tilde{H}_{1}|^{2}+a_{H_{2}}^{2}\,|\tilde{H}_{2}|^{2}+a_{L_{\alpha}}^{2}\,|\tilde{L}_{\alpha}|^{2}-\left\\{\frac{\epsilon_{\alpha}}{\mu}(a_{H_{1}}^{2}-a_{L_{\alpha}}^{2})\,\tilde{H}_{1}^{*}\tilde{L}_{\alpha}+{\rm
c.c.}\right\\}\right)\,.$

In the new basis, the VEV of the electrically neutral components of the
$\tilde{L}_{\alpha}$ are denoted by $\omega_{\alpha}$. It is related to the
VEV $\omega^{\prime}_{\alpha}$ in the original basis by Eq. (141). Using the
defining relation, $\omega_{\alpha}=k_{\alpha}\epsilon_{\alpha}$, and
minimizing the potential $\tilde{V}_{F,D}+\tilde{V}_{\rm soft}$, we obtain

$k_{\alpha}\simeq\frac{v_{1}}{\mu}\frac{\left(a_{H_{1}}^{2}-a_{L_{\alpha}}^{2}+|\eta|^{2}\frac{v_{2}^{2}}{m_{3/2}^{2}}\right)+\frac{\mu}{m_{3/2}}(B_{\mu}-A_{\alpha})\tan\beta}{a_{L_{\alpha}}^{2}+\frac{1}{2}\frac{M_{Z}^{2}}{m_{3/2}^{2}}\,\cos
2\beta}\,.$ (143)

Neglecting the contributions of ${\cal O}(v^{2}/m_{3/2}^{2})$, the above can
be further simplified to

$k_{\alpha}\simeq\frac{v_{1}}{\mu}\left(\frac{a_{H_{1}}^{2}-a_{L_{\alpha}}^{2}}{a_{L_{\alpha}}^{2}}\right)+\frac{v_{1}}{m_{3/2}}\left(\frac{B_{\mu}-A_{\alpha}}{a_{L_{\alpha}}^{2}}\right)\tan\beta\,.$
(144)

The first term in $k_{\alpha}$ is suppressed for the universal soft masses.
Depending on the presence or absence of bare $\mu_{0}$ in the model, the
nature of the second term in $k_{\alpha}$ can be decided. For example, when
$\mu_{0}=B_{\mu_{0}}=0$, using Eq. (140) and the fact that
$\mu=-\eta\langle\tilde{N}_{\tau}\rangle$, Eq. (144) can be reduced to

$k_{\alpha}\simeq\frac{v_{1}}{\mu}\left(\frac{a_{H_{1}}^{2}-a_{L_{\alpha}}^{2}}{a_{L_{\alpha}}^{2}}\right)+\frac{v_{1}}{m_{3/2}}\left(\frac{A_{\eta}-A_{\alpha}}{a_{L_{\alpha}}^{2}}\right)\tan\beta\,.$
(145)

Both terms in $k_{\alpha}$ can then be suppressed for the universal soft
terms.

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