aainstitutetext: Department of Basic Sciences, Faculty of Computers and
Artificial Intelligence, Cairo University, Giza 12613, Egypt.

# No-scale gauge non-singlet inflation inducing TeV scale inverse seesaw
mechanism

Ahmad Moursy<EMAIL_ADDRESS>
###### Abstract

We develop a model of sneutrino inflation that is charged under $U(1)_{B-L}$
gauge symmetry, in no-scale supergravity framework. The model provides an
interesting modification of tribrid inflation. We impose $U(1)_{R}$ symmetry
on the renormalizable level while allow Planck suppressed non-renormalizable
operators that break R-symmetry. This plays a crucial role in realizing a
Starobinsly-like inflation scenario from one hand. On the other hand it plays
an essential role, as well as SUSY breaking effects, in deriving the tiny
neutrino masses via TeV inverse seesaw mechanism. Thus, we provide an
interpretation for the extremely small value of the $\mu_{S}$ mass parameter
required for inverse seesaw mechanism. We discuss a reheating scenario and
possible constraints on the model parameter space in connection to neutrino
masses.

##  1 Introduction

Remarkable studies over the past few decades were devoted to explore the
connection and interplay between different hierarchical scales in nature.
Finding such connections helps in understanding the fundamental origin of
these scales. This issue has attracted careful studies on the huge hierarchy
between the scales at the low energy observable sector, such as electroweak
and neutrino mass scales from one hand, and on the other hand the early
universe scales such as Planck, GUT, SUSY breaking and inflation scales.

Inflation paradigm is supported by CMB observations. Planck collaboration
Akrami:2018odb  has confirmed the values of the spectral index of the scalar
fluctuations $n_{s}=0.955-0.974$, up to 2 sigma exclusion limits, and the
tensor to scalar ratio $r<0.08$. One of the appealing models of inflation is
the Starobinsky model Starobinsky:1980te  whose predictions are consistent
with Planck observations. Starobinsky-like potentials can be implemented in
supergravity framework such as no-scale supergravity models Cremmer:1983bf ;
Ellis:1984bm ; Ellis:2013xoa ; Ellis:2013nxa ; Ellis:2017jcp ; Khalil:2018iip
; Moursy:2020sit . The latter models have a striking feature of solving
naturally the $\eta$-problem without any need to preserve R-symmetry that is
required in supersymmetric hybrid inflation models Dvali:1994ms ;
Antusch:2004hd .

It is tempting to study the link between inflation scale and neutrino mass
scale. The detected neutrino oscillations Super-Kamiokande:1998kpq ;
LSND:2001aii ; K2K:2002icj ; T2K:2011ypd ; DayaBay:2013yxg  implied that
neutrinos have tiny masses. The latter hinted at new physics beyond the
standard model that accounts for seesaw mechanisms Minkowski:1977sc ;
Mohapatra:1979ia ; Yanagida:1979as ; Schechter:1980gr . Interestingly right-
handed neutrino supermultiplets can be employed to play a double role in
realizing seesaw type I mechanism using its fermionic components, while the
scalar component (right-handed sneutrino) can be assigned as an inflaton, with
mass scale $\sim 10^{13}$ GeV. Chaotic inflation with a singlet right-handed
sneutrino was proposed in Murayama:1992ua . In Antusch:2004hd  sneutrino
hybrid inflation was introduced in supergravity framework with exact
R-symmetry while Heisenberg symmetry was imposed in Antusch:2008pn ;
Antusch:2009vg , in order to solve the $\eta$-problem. Non-singlet sneutrino
inflation was explored in Antusch:2010va ; Gonzalo:2016gey , where the
$\eta$-problem was avoided by either Heisenberg symmetry or shift symmetry.

In this work we consider a non-singlet inflaton under $U(1)_{B-L}$ gauge
symmetry that is assigned to right-handed sneutrino, in no-scale supergravity.
One remarkable feature of the present model is that the $U(1)_{B-L}$ gauge
symmetry is broken during inflation by the inflaton hence topological defects
such as cosmic strings are absent. After the inflation ends, the gauge
symmetry is broken in the true minimum at GUT scale which is a derived scale
from Planck scale $M_{P}$ and inflationary mass scale $\mu$. Moreover, the
inflation fields play a crucial rule in generating tiny neutrino masses via a
TeV inverse seesaw mechanism Mohapatra:1986aw ; Mohapatra:1986bd ; Gonzalez-
Garcia:1988okv  that allows non-small values of the neutrino Yukawa couplings
$Y_{\nu}$, without unnatural fine tuning. We introduce $U(1)_{R}$ symmetry
(R-symmetry) on the renormalizable level while we allow non-renormaizable
operators suppressed by Planck mass to break R-symmetry Civiletti:2013cra ;
Khalil:2018iip .

The paper is organized as follows. In section 2 we construct the model
discussing the role of each supermultiplet. In section 3 we explore the
inflation scenario. Also, we analyze the inflation effective potential,
trajectory and observables. We discuss consequences on post inflation era in
section 4. We point out an interesting feature of SUSY breaking in models of
hybrid inflation and exist in our model. We investigate constraints that arise
from reheating and then generate neutrino masses from inverse seesaw
mechanism. Finally, we draw our conclusions in section 5.

##  2 The model

We consider a no-scale supergravity model in which both the superpotential and
the Kähler potential are invariant under $U(1)_{B-L}$ gauge group. Moreover,
on the renormalizable level, the Lagrangian respects the $U(1)_{R}$ symmetry,
with $R[W]=2$ while non-renormalizable operators suppressed by Planck mass are
allowed to break R-symmetry Civiletti:2013cra ; Khalil:2018iip . A $Z_{3}$
discrete symmetry is imposed in order to guarantee a successful inflation and
TeV inverse seesaw mechanism. The field content of the model is listed as
follows.

  * •

The supermultiplets
$S_{1}=(\tilde{S}_{1},S_{1}),\,S_{2}=(\tilde{S}_{2},S_{2})$ are gauge non-
singlets under $U(1)_{B-L}$. Their scalar components are corresponding to the
inflaton while the fermionic components mix with the right-handed neutrino $N$
in the neutrino mass matrix.

  * •

The waterfall supermultiplets
$\phi_{1}=(\phi_{1},\tilde{\phi_{1}})\,,\,\phi_{2}=(\phi_{2},\tilde{\phi_{2}})$
whose scalar components are frozen at the origin during the inflation and
after the inflation ends they break the $B-L$ gauge symmetry at the true
minimum.

  * •

The singlet $S=(S,\tilde{S})$ plays a five-fold role: First, it is a driving
field where its F-term generates the potential of the waterfall fields, (the
scalar components of $\phi_{1},{\phi_{2}}$), hence they acquire vevs to break
the $B-L$ symmetry at GUT scale $M\sim\sqrt{\mu M_{P}}$. Second, the non-
renormalizable coupling to $S_{1},S_{2}$ will play important role in
flattening the inflation potential causing wider region of the parameter space
of a successful inflation as we will demonstrate in the following section.
Third, its decay contributes to reheating the universe. On the other hand SUSY
breaking effects generate a tadpole term in the scalar $S$ shifting its
minimum from the origin, hence the fourth role is realized in generating the
mixing term $\mu_{S}S_{1}S_{2}$ after the end of inflation and can contribute
to the neutrino mass matrix. Finally non-zero vev of $S$ can be employed to
solve the $\mu$-problem of the TeV scale MSSM by generating the TeV scale
mixing term $\mu\,H_{u}\,H_{d}$ that is needed to trigger the electroweak
symmetry breaking Dvali:1997uq .

  * •

The right-handed neutrino supermultiplet $N=(\tilde{N},N)$ which contributes
to the neutrino mass matrix and has no role in the inflation scenario.

| $\,\,\,\,S_{1}\,\,\,\,$ | $\,\,\,\,S_{2}\,\,\,\,$ | $\,\,\,\,\phi_{1}\,\,\,\,$ | $\,\,\,\,\phi_{2}\,\,\,\,$ | $\,\,\,\,S\,\,\,\,$ | $\,\,\,\,N\,\,\,\,$
---|---|---|---|---|---|---
$U(1)_{B-L}$ | 1 | -1 | 2 | -2 | 0 | -1
$R$ | 1 | 1 | 0 | 0 | 2 | -1
$Z_{3}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega^{2}$ | 1 | $\omega$
Table 1: $U(1)_{B-L}$ and $R$ charge assignments for different superfields
$S_{1},S_{2},\phi_{1},\phi_{2},S,N$ in the inflation sector.

The superfields constituting the model, which are SM singlets, as well as the
respective ${B-L}$, the R-symmetry and $Z_{3}$ charges are given in Table 1.
In this respect the superpotential and the Kähler potential for the inflation
sector are given by

$\displaystyle\\!\\!\\!\\!\\!\\!W_{\text{i}nf}=\kappa_{1}S\left(\phi_{1}\phi_{2}-\mu
M_{P}\right)+\kappa_{2}\,S_{1}S_{2}\left(\dfrac{\phi_{1}\phi_{2}}{M_{P}}-\mu\right)+\dfrac{\lambda_{1}}{M_{P}}(S_{1}S_{2})^{2}+\dfrac{\lambda_{2}}{M_{P}}S^{2}S_{1}S_{2},$
(1) $\displaystyle
K=-3M_{P}^{2}\log\left[\dfrac{T+\overline{T}}{M_{P}}-\frac{|S|^{2}}{3M_{P}^{2}}-\frac{|S_{1}|^{2}}{3M_{P}^{2}}-\frac{|S_{2}|^{2}}{3M_{P}^{2}}-\frac{|\phi_{1}|^{2}}{3M_{P}^{2}}-\frac{|\phi_{2}|^{2}}{3M_{P}^{2}}\right],$
(2)

The superpotential (1) is a development of the tribrid inflation model
Antusch:2004hd ; Antusch:2008pn ; Antusch:2009vg ; Antusch:2010va ;
Antusch:2012jc ; Antusch:2013eca . As will be detailed later, the difference
is at two main points: The first is that the mass term of right-handed
neutrinos $S_{1}\,S_{2}$ vanishes at the end of inflation. The second is that
the vev of $S$ is not zero during the inflation. The parameter $\mu$
determines the scale of the inflation and $\kappa_{i},\lambda_{i}$ are
dimensionless couplings where $\lambda_{i}$ are responsible for the R-symmetry
breaking in the superpotential and $M_{P}=2.4\times 10^{18}$ GeV is the
reduced Planck mass.111Worthy to mention that the symmetry allows the Planck
suppressed non-renormalizable operators such as $S^{4}$,
$S^{2}\phi_{1}\,\phi_{2}$ and $(\phi_{1}\,\phi_{2})^{2}$. However they are
undesirable hence we can set the corresponding couplings to zero or we can
argue that they are forbidden in the UV sector by $R$-symmetry. We assume that
only $S,S_{i},N$ as well as the MSSM superfields carry R-charges. We are not
going to give more details on the UV sector as it is beyond the scope of the
paper. In this regard, the total scalar potential consists of the F-term and
D-term parts $V=V_{F}+V_{D}$. The F-term scalar potential is given by

$\displaystyle
V_{F}=e^{K/M_{P}^{2}}\left[D_{I}K^{I\bar{J}}D_{\bar{J}}\overline{W}-3\dfrac{|W|^{2}}{M_{P}^{2}}\right],$
(3)

where $I,J$ run over $(T,S,S_{1},S_{2},\phi_{1},\phi_{2})$, $K^{I\bar{J}}$ is
the inverse of the Kähler metric $K_{I\bar{J}}=\dfrac{\partial K}{\partial
Z^{I}\partial Z^{\bar{J}}}$, and $D_{I}$ is the Kähler derivative defined by
$D_{I}=\dfrac{\partial}{\partial Z^{I}}+\dfrac{1}{M_{P}^{2}}\dfrac{\partial
K}{\partial Z^{I}}$. We use the lower case letters $i,j$ to run over the
inflation sector fields $S,S_{1},S_{2},\phi_{1},\phi_{2}$. The D-term
potential is given by

$\displaystyle V_{D}=\frac{g^{2}}{2}{\text{R}e}f^{-1}_{AB}D^{A}D^{B},$ (4)

where $f_{AB}$ is the gauge Kinetic function and the indices $A,B$ are
corresponding to a representation of the gauge group under which $Z^{i}$ are
charged. The D-term $D^{A}$ is given by

$\displaystyle D^{A}=\frac{\partial K}{\partial
Z^{i}}\,\left(T^{A}\right)^{i}_{j}Z^{j},$ (5)

with $T^{A}$ are generators of the gauge group in the appropriate
representation. We will work in the D-flat direction, hence the total
potential will be given by F-term scalar potential as

$V=\dfrac{1}{\Omega^{2}}\,{{{\sum}}}_{i=1}^{5}\left|\frac{\partial
W}{\partial Z_{i}}\right|^{2},\hskip
28.45274pt\Omega=\frac{T+\bar{T}}{M_{P}}-\frac{|S|^{2}}{3M_{P}^{2}}-\frac{|\tilde{S}_{1}|^{2}}{3M_{P}^{2}}-\frac{|\tilde{S}_{2}|^{2}}{3M_{P}^{2}}-\frac{|\phi_{1}|^{2}}{3M_{P}^{2}}-\frac{|\phi_{2}|^{2}}{3M_{P}^{2}}.$
(6)

After calculating the scalar potential, we consider the modulus $T$ to be
stabilized at high scale by any of the mechanisms of Kachru:2003aw ;
Balasubramanian:2005zx ; Ellis:2013nxa , where
$\langle{\text{R}e}(T)\rangle=\tau_{0}$, $\langle{\text{I}m}(T)\rangle=0$. It
is clear that, the total scalar potential is positive semidefinite. Therefore
it has a global SUSY minimum which is Minkowskian, and is located at

$\displaystyle\langle
S\rangle=\langle\tilde{S}_{1}\rangle=\langle\tilde{S}_{2}\rangle=0\,\,\&\,\,\,\langle\phi_{1}\phi_{2}\rangle=\mu
M_{P}$ (7)

We will work in the D-flat direction associated with the $B-L$ symmetry,

$\displaystyle|\tilde{S}_{1}|^{2}-|\tilde{S}_{2}|^{2}+2|\phi_{1}|^{2}-2|\phi_{2}|^{2}=0.$
(8)

Without loss of generality we choose for simplicity the direction
$|\tilde{S}_{1}|=|\tilde{S}_{2}|=|\Sigma|$ and $|\phi_{1}|=|\phi_{2}|=|\phi|$,
that cancels two real degrees of freedom. Therefore, the scalar potential
simplifies to the following form

$\displaystyle
V=\dfrac{1}{\Omega^{2}}\,\left[\left|\kappa_{1}\left(\phi^{2}-\mu
M_{P}\right)+\dfrac{2\lambda_{2}}{M_{P}}S\,\Sigma^{2}\right|^{2}+2\left|\phi\right|^{2}\left|\kappa_{1}S+\dfrac{\kappa_{2}}{M_{P}}\Sigma^{2}\right|^{2}\right.$
$\displaystyle\left.+2\left|\Sigma\right|^{2}\left|\dfrac{\kappa_{2}}{M_{P}}\left(\phi^{2}-\mu
M_{P}\right)+\dfrac{\lambda_{1}}{M_{P}}\Sigma^{2}+\dfrac{\lambda_{2}}{M_{P}}S^{2}\right|^{2}\right].$
(9)

The previous direction is not unique as one can choose another D-flat
direction in which $|\phi_{1}|^{2}-|\phi_{2}|^{2}=\gamma\neq 0$, which implies
that $|\tilde{S}_{1}|^{2}={|\tilde{S}_{2}|^{2}-2\gamma}$, hence only one
degree of freedom cancels. However, as we will show in the next section, all
the four real degrees of freedom $\alpha_{1,2},\beta_{1,2}$ are fixed to zero
during inflation, where $\phi_{j}=\dfrac{\alpha_{j}+i\beta_{j}}{\sqrt{2}}$.
Hence, during the inflation and at the global minimum $\gamma=0$, and we
return to the simple case of $|\tilde{S}_{1}|=|\tilde{S}_{2}|=|\Sigma|$. In
this regard we express the complex fields $S,\Sigma,\phi$ in terms of their
real components as follows

$\displaystyle
S=\frac{s+i\sigma}{\sqrt{2}}\,\,,\,\,\,\phi=\dfrac{\alpha+i\beta}{\sqrt{2}}\,\,,\,\,\,\Sigma=\dfrac{\chi+i\psi}{\sqrt{2}}.$
(10)

Accordingly, the global minimum of the potential is located at

$\displaystyle s=\sigma=\chi=\psi=\beta=0\,\,\&\,\,\,\alpha=\sqrt{2\mu
M_{P}}.$ (11)

It turns out that $\chi$ and $\psi$ are massless at the true minimum, since
the second term of (1) vanishes when the higgs fields acquire non-zero vev
$\sim\sqrt{\mu M_{P}}$. However, as we will demonstrate later, SUSY breaking
effects will generate mass terms for $\tilde{S}_{i}$. On the other hand the
squared masses of the other real components are given by

$\displaystyle
m_{s}^{2}=m_{\sigma}^{2}=\frac{3\,\kappa_{1}^{2}\,\mu\,M_{P}^{2}}{3\,\tau_{0}-\mu}\,\,\,\,\text{and}\,\,\,\,m_{\alpha}^{2}=m_{\beta}^{2}=\frac{\kappa_{1}^{2}\,\mu\,M_{P}^{2}}{\tau_{0}}\,.$
(12)

##  3 Inflation scenario and observables

###  3.1 Inflation trajectory and effective potential

Figure 1: The scalar potential of $x,\alpha$ near the SUSY vacuum (left panel)
and for large values of $x$ (right panel), with $\beta=y=\sigma=0$ and
$\lambda_{1}=0.3\times 10^{-5},\mu=\times
10^{-5},\kappa_{1}=10^{-4},\kappa_{2}\sim 1$. All values are given in the
units where $M_{P}=1$.

Here we study the trajectory of inflation and the resulting effective
inflation potential. The scalar potential is minimized at
$\phi_{1}=\phi_{2}=0$ for large values of $S_{1},S_{2}$. Along the inflation
trajectory $\phi_{1}=\phi_{2}=0$ and the effective inflation potential
receives a dominant contribution from the F-terms $F_{S_{i}}$ as well as a
contribution from $F_{S}$, that will be constrained from the inflation
observables, while the contribution of $F_{\phi_{i}}$ vanishes. On the other
hand, the inflation energy shifts the vev of $s$ from zero during inflation to
a small value, due to the coupling
$\dfrac{\lambda_{2}}{M_{P}}S^{2}S_{1}S_{2}$. We depict the shape of the scalar
potential of $(x,\alpha)$ during the inflation and near the global SUSY
minimum after the inflation ends in Fig. 1.

We first investigate the form of the scalar potential in the limit
$\lambda_{2}=0$ (the potential is minimized at $S=0$), then we expand the
scalar potential for small $s$ when $\lambda_{2}\neq 0$. In this respect, the
inflation potential, when $\lambda_{2}=0$, has the form

$\displaystyle\\!\\!\\!\\!\\!V(\\!S\\!=\\!0\\!)=\frac{9M_{P}^{2}\left[8\lambda_{1}^{2}|\Sigma|^{6}+\kappa_{1}^{2}\mu^{2}M_{P}^{4}+2\kappa_{2}^{2}\mu^{2}M_{P}^{2}|\Sigma|^{2}-4\kappa_{2}\lambda_{1}\mu
M_{P}\left(\overline{\Sigma}^{4}+\Sigma^{4}\right)\right]}{4\left(|\Sigma|^{2}-3M_{P}\tau_{0}\right)^{2}}.$
(13)

In order to have canonical kinetic terms we use the following field
redefinitions

$\displaystyle\Sigma=\sqrt{3M_{P}\tau_{0}}\,\tanh\left(\dfrac{x+iy}{\sqrt{6}M_{P}}\right),$
(14)

where $x,y$ are real scalar fields. As will be demonstrated latter, the field
$y$ acquires a mass of order Hubble scale during the inflation hence it is
stabilized at zero Moursy:2020sit . Therefore, the scalar potential takes the
form

$\displaystyle\\!\\!\\!\\!\\!\\!V=A\cosh^{4}\left(\frac{x}{\sqrt{6}}\right)\left[B^{2}\tanh^{6}\left(\frac{x}{\sqrt{6}}\right)-2B\tanh^{4}\left(\frac{x}{\sqrt{6}}\right)+\tanh^{2}\left(\frac{x}{\sqrt{6}}\right)+f\right]\\!,$
(15)

where,

$\displaystyle A=\dfrac{3\mu^{2}\kappa_{2}^{2}}{2\tau_{0}M_{P}}\,,\hskip
14.22636ptB=\dfrac{6\lambda_{1}\tau_{0}}{\kappa_{2}\,\mu}\,,\hskip
14.22636ptf=\dfrac{\kappa_{1}^{2}M_{P}}{6\kappa_{2}^{2}\tau_{0}},$ (16)

are dimensionless parameters that will be determined by slow-roll conditions
and inflation observables. We will use the units in which $M_{P}=1$ throughout
this section. When $f=0$ (or equivalently $\kappa_{1}=0$), we obtain the
potential studied in Khalil:2018iip . Therefore, in the limit where $B=1$ and
$f=0$, the potential (15) is asymptotically flat

$\displaystyle V_{0}=A\tanh^{2}\left(\dfrac{x}{\sqrt{6}}\right).$ (17)

Now we return to consider the effect of the coupling
$\dfrac{\lambda_{2}}{M_{P}}S^{2}S_{1}S_{2}$. It gives rise to a linear term in
$s$ in the $F_{S}$ contribution to the potential (2), hence $s$ acquires small
vev during the inflation and it is given by

$\displaystyle s=\frac{\sqrt{2Af}}{3\lambda_{2}},$ (18)

where we have expanded for large $x$ during the inflation and taken the
leading order.222This will not affect the canonical kinetic term of the
inflaton as $s$ is much small during inflation.

Figure 2: The ratio of the squared mass of the real scalar fields
$\alpha_{i},\beta_{i},y$ and $\sigma$, to the squared Hubble scale, with
$A=3\times 10^{-10},f=3.33\times 10^{-11}\,,\,\lambda_{2}=0.005,\,B=1$. All
values are given in the units where $M_{P}=1$.

Now, we consider for completeness, that all real components of $\phi_{1,2}$
that can account for possible D-flat directions, $\alpha_{i},\beta_{i}$. We
show that they are fixed at zero during inflation as well as the real degrees
of freedom $y,\sigma$. As a matter of fact, all real degrees of freedom, apart
from the inflaton $x$, acquire masses larger than the Hubble scale $H$ during
inflation Moursy:2020sit . We present the ratio of field dependent squared
masses to the squared Hubble scale during inflation in Fig. 2.

Returning to the scalar $\alpha$ that accounts for the waterfall phase, the
scalar potential (2) is minimized at $\alpha=0$ during the inflation, for $x$
is rolling at large values, since the field dependent squared mass
$m^{2}_{\alpha}$ is positive and is of order squared Hubble scale during
inflation. When $x$ reaches a critical value $x_{c}$ after the inflation ends,
$m^{2}_{\alpha}$ flips it sign to negative values which triggers the waterfall
phase Moursy:2020sit . Expanding for small values of the inflaton $x$, we have

$\displaystyle
m_{\alpha}^{2}\simeq\frac{\kappa_{1}^{2}\,\mu\left(\mu-3\tau_{0}\right)}{6\tau_{0}^{2}},$
(19)

which is negative as long as $\mu<3\tau_{0}$. The critical value of the
inflaton $x_{c}$ will be given as

$\displaystyle
x_{c}=\frac{2\kappa_{2}\mu}{\tau_{0}\,\left(\kappa_{2}+4\lambda_{1}\right)}.$
(20)

Figure 3: The scalar potential $\dfrac{V_{\text{inf}}}{A}$ of the inflaton $x$
in the two cases when $s$ is zero (solid curves) and non-zero (dashed curves),
with $A=3\times 10^{-10}\,,\,\lambda_{2}=0.5,\,B=1$ and we took two values for
$f=3.33\times 10^{-9}$ and $3.33\times 10^{-5}$. All values are given in the
units where $M_{P}=1$.

We expand the scalar potential (2) to second order in $s$ with the other
fields being fixed at the origin, and consider the limit where $B=1$, then we
obtain the inflation effective potential as follows

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!V_{\text{inf}}$ $\displaystyle=$
$\displaystyle
V_{0}+A\,f\cosh^{4}\left(\frac{x}{\sqrt{6}}\right)-6\,\lambda_{2}\,\sqrt{\dfrac{f}{2\,A}}\,s\,V_{0}\,\cosh^{4}\left(\frac{x}{\sqrt{6}}\right)+\frac{s^{2}\cosh^{2}\left(\frac{x}{\sqrt{6}}\right)}{6A\tau_{0}}\times$
(21)
$\displaystyle\times\left[A\,f\cosh^{4}\left(\frac{x}{\sqrt{6}}\right)+V_{0}\left(-3A\,\lambda_{2}\sqrt{\frac{6\tau_{0}}{A}}+A+27\lambda_{2}^{2}\,\tau_{0}\,\sinh^{2}\left(\frac{x}{\sqrt{6}}\right)\right)\right]\\!\\!.$

Figure 4: The inflation potential $\dfrac{V_{\text{inf}}}{A}$ with $A=3\times
10^{-10},\lambda_{2}=0.5$, $s\neq 0$ and $s=0$ for the solid and dashed curves
respectively, and we took two values for $f=3.33\times 10^{-9}$ and
$3.33\times 10^{-5}$. The left panel corresponds to $B=1+10^{-5}$ and the
right panel corresponds to $B=1-10^{-5}$. All values are given in the units
where $M_{P}=1$.

If $\lambda_{2}=0$, then $s$ will be stabilized at zero during the inflation,
hence the third and fourth terms will be absent in the inflation potential
(21), and we return to the potential (15). It is clear that the second term
proportional to $f$ spoils the asymptotic flatness of the potential at large
values of the inflaton $x$, hence $f$ should be small enough in order to
guarantee the flatness of the potential during the inflation. For small values
of $s$, the third term dominates over the fourth term during the inflation. As
a consequence, the third term in (21) has a desirable effect in cancelling the
dangerous effect of the second term hence flattening the potential during the
inflation. In Fig. 3 we show the inflation potential for $B=1$ when $s$ is
stabilized at zero and non-zero values during inflation where it was indicated
that the potential is flattened for $s\neq 0$ during the inflation as depicted
in the solid curves. When $B\neq 1$, we define $B(\xi\neq 0)=1+\xi$, hence
extra terms proportional to $\xi$ are generated and this may spoil the
flatness during the inflation. Accordingly the constraint $|\xi|\lesssim{\cal
O}(10^{-4})$ should be imposed Khalil:2018iip . Fig. 4 illustrates the
inflation potential for $B\neq 1$ ($\xi\neq 0$) for different values of $f$.
The solid curves corresponding to non-zero values of $s$, which coincide for
different values of $f$.

###  3.2 Inflation observables

Now we move to discuss the model predictions of inflation observables and
explore the possible constraints on the different parameters. The inflation
observables, tensor-to-scalar ratio $r$, spectral tilt $n_{s}$ and the scalar
amplitude $A_{s}$ will be investigated. In terms of the slow-roll parameters
$\epsilon$ and $\eta$, the observables are defined as follows

$\displaystyle r$ $\displaystyle=$ $\displaystyle 16\epsilon,$ $\displaystyle
n_{s}$ $\displaystyle=$ $\displaystyle 1-6\epsilon+2\eta,$ $\displaystyle
A_{s}$ $\displaystyle=$ $\displaystyle\frac{V}{24\pi^{2}\epsilon},$

where the above observables are computed at the crossing horizon value of the
inflaton field $x_{*}$. The number of e-folds is given by

$\displaystyle N=\int_{x_{e}}^{x_{*}}\frac{1}{\sqrt{2\epsilon}}\,dx,$ (22)

where $x_{e}$ is the value of the inflaton at the end of inflation. The
observed value of the scalar amplitude $A_{s}\simeq 1.95896\pm 0.10576\times
10^{-9}$ at 68% CL Akrami:2018odb , which fixes $A\sim 10^{-10}$.

Figure 5: A logarithmic plot for the model predictions in the $n_{s}-r$ plane.
We have fixed $A\sim 10^{-10}$. The solid (dash-dotted) blue curves correspond
to $N=60$ ($N=50$) respectively where we scanned over $B=1+\xi$ with
$\xi\in[-10^{-4},10^{-4}]$, with $f=\lambda_{2}=0$. The other segments
correspond to fixed $B=1\,(\xi=0)$ with different values of $f,\,\lambda_{2}$
and we have scanned over $N=50-60$ (small-big) dots, for each segment. The
dark and light red regions correspond to the 1 and 2 sigma exclusion limits
released by the Planck collaboration (2018) (TT+ TE+ EE + LowE + Lensing +
BK14)Akrami:2018odb .

In Fig. 5, we displayed the predictions of our model for $n_{s}$, $r$ in a
logarithmic plot versus the Planck limits. First, we fixed $f=\lambda_{2}=0$
and scanned over $B=1+\xi$ with $-10^{-4}\,\leq\,\xi\,\leq\,10^{-4}$ on the
solid (dash-dotted) blue curves for $N=60$ ($N=50$), respectively. On the
other segments, we scanned over $N=50-60$, and fix $B=1\,(\xi=0)$. The orange
segment illustrates the situation when the potential is asymptotically flat
(17) where $f=\lambda_{2}=0$. However, $f$ (or $\kappa_{1}$) should not be
zero in order to keep the potential terms responsible for stabilizing the
waterfall fields at the true minimum. Therefore fixing $\lambda_{2}=0$ (which
means that $s=0$ on the inflation trajectory), the inflation observables
constrain $f\lesssim 2\times 10^{-6}$ as indicated in the green dashed
segment. Turning on $\lambda_{2}$ to non-zero value will flatten the potential
causing successful inflation. The latter case is illustrated in the solid
green segment where we took the same value of $f$ on the dashed one and let
$\lambda_{2}\sim 10^{-6}$. Any greater value of $\lambda_{2}$ makes the
potential more flat at larger inflaton value. Therefore we investigate greater
values of $f\lesssim{\cal O}(1)$ that spoil the flatness of the potential
against greater values of $\lambda_{2}$. The Purple segment corresponds to
$\lambda_{2}\sim 0.05$ and $f\sim 1$. For larger value of $\lambda_{2}$ purple
segment moves towards the orange one. For $\lambda_{2}\gtrsim 0.5$, the purple
and orange segments coincide. For $B\neq 1$ and $\lambda_{2}\gtrsim 0.5$,
different potentials with different values of $f$ coincide as depicted in Fig.
(4). In order to have successful inflation, the constraint $|\xi|\lesssim{\cal
O}(10^{-4})$ should be satisfied.

$\,\,\,\,\lambda_{2}\,\,\,\,$ | $\,\,\,\,f\,\,\,\,$ | $\,\,\,\,\mu\,\,\,\,$ | $\,\,\,\,\kappa_{1}\,\,\,\,$ | $\,\,\,\,\kappa_{2}\,\,\,\,$ | $\,\,\,\,\lambda_{1}\,\,\,\,$ | $\,\,\,\,\tau_{0}\,\,\,\,$ | $\,\,\,\,M\,\,\,\,$
---|---|---|---|---|---|---|---
$0-10^{-6}$ | $1.7\times 10^{-6}$ |  | $2.8\times 10^{-6}$
---
$2.8\times 10^{-5}$
| $10^{-3}$
---
$10^{-2}$
$0.9$ |  | $4.3\times 10^{-6}$
---
$4.3\times 10^{-7}$
| $0.1$
---
$10$
| $1.6\times 10^{-3}$
---
$5.3\times 10^{-3}$
$0.05-0.9$ | $0.9$ |  | $4\times 10^{-6}$
---
$3.6\times 10^{-3}$
| $0.95$
---
$0.1$
| $0.9$
---
$0.01$
| $3\times 10^{-6}$
---
$3\times 10^{-7}$
| $0.2$
---
$20$
| $2\times 10^{-3}$
---
$6\times 10^{-2}$
Table 2: The numerical values of the parameters of the superpotential and the
modulus vev $\tau_{0}$ in the units $M_{P}=1$. We have considered $B(\xi=0)=1$
and $A\sim 10^{-10}$.

In Table 2, we have provided some specific values of the superpotential
parameters as well as the modulus vev $\tau_{0}$ corresponding to the values
of $A,f,\lambda_{2}$ and $B=1$, that were discussed above in the inflation
observables. We have focused on showing different regimes of small and large
values of $\kappa_{1}$ and $\kappa_{2}$, since they play important roles in
the inverse seesaw mechanism that we will demonstrate in the next section. We
emphasis that $f$ can’t be zero in order to account for the correct dynamics
of the waterfall fields.

###  3.3 Loop corrections

Here we discuss the effects from one-loop correction to the inflation
potential, and two loop correction contributing to the gauge $\eta$-problem
(or Dvali problem) mentioned in Dvali:1995fb . We will show that they don’t
have significant effect on the inflation potential.

  * •

The Coleman-Weinberg one-loop corrections Coleman:1973jx  are given by

$V_{\text{1-loop}}(\Sigma)=\frac{1}{64\pi^{2}}\text{Str}\left[m^{4}\left(\log\left(\frac{m^{2}}{Q^{2}}\right)-\frac{3}{2}\right)\right],$
(23)

where $\text{Str}\,m(\Sigma)^{4}=\,\sum_{i}(-1)^{2J_{i}}(2J_{i}+1)m_{i}^{4}$
is the supertrace taken over all inflaton dependent mass eigenvalues of states
with spin $J_{i}$. The stabilized fields during the inflation have $m_{i}\sim
H$. But $H^{2}M_{p}^{2}\sim V_{\text{inf}}\sim A\,M_{p}^{4}$, therefore
$H^{2}\sim 10^{-10}\,M_{P}^{2}$ during the inflation. Accordingly, the 1-loop
correction $V_{\text{1-loop}}\sim\frac{H^{4}}{64\pi^{2}}\sim{\cal
O}(10^{-22})\,M_{P}^{4}$, which is negligible compared to the tree level
potential for any relevant renormalization scale.

  * •

For the two loop Dvali problem, it may arises in the context of hybrid
inflation if there is a gauge non-singlet flat directions ($\Sigma$ in our
case) as well as a singlet $S$ coupled to the non-singlet waterfall fields.
This results in large vacuum energy by the F-term $F_{S}$ that breaks SUSY
during the inflation. Therefore a significant two loop contribution to
$\Sigma$ effective mass is estimated as Antusch:2010va

$\displaystyle\delta
m^{2}\sim\dfrac{g_{B-L}^{4}}{(4\pi)^{4}}\dfrac{|W_{S}|^{2}}{m_{\phi}^{2}},$
(24)

where $m_{\phi}$ is the SUSY preserving mass of the waterfall fields and
$g_{B-L}$ is the ${B-L}$ gauge coupling. If $\delta m\sim H$, this implies
$|\eta|\sim 1$ and the slow-roll conditions are violated causing an
$\eta$-problem called the "gauge $\eta$-problem" or "Dvali Problem". However
we will show that the two loop contributions are suppressed compared to the
tree level inflaton mass during inflation. Similar to the situation in
Gonzalo:2016gey , SUSY breaking scale $M_{SB}$ during inflation gets
contributions from $F_{S}=\kappa_{1}\mu M_{P}$ and $F_{S_{i}}$. However the
dominant contribution comes from $F_{S_{i}}$. According to the non-
renormalization theorem, corrections should be proportional to powers of
$M_{SB}\sim\langle F_{S_{i}}\rangle$. Therefore the corrections arise from the
following contributions Gonzalo:2016gey ; Antusch:2010va ,

$\displaystyle\delta
m_{\Sigma}^{2}\sim\,\,\,\dfrac{g_{B-L}^{4}}{(4\pi)^{4}}\dfrac{M_{SB}^{4}}{M_{Z^{\prime}}^{2}}\,,\hskip
14.22636pt\dfrac{g_{B-L}^{4}}{(4\pi)^{4}}\dfrac{M_{SB}^{4}H}{M_{Z^{\prime}}^{2}}\,,\hskip
14.22636pt\dfrac{g_{B-L}^{4}}{(4\pi)^{4}}\dfrac{M_{SB}^{4}H^{2}}{M_{Z^{\prime}}^{2}}\,,$
(25)

where $M_{Z^{\prime}}\sim g_{B-L}\langle\Sigma\rangle$ is the $U(1)_{B-L}$
gauge boson. Assuming that during inflation, $\Sigma\sim 1M_{P}$, $g_{B-L}\sim
0.1$, $\mu\sim 10^{-5}M_{P}$, $\kappa_{1}\leq 0.1$, $\kappa_{2}\sim{\cal
O}(1)$, $\lambda_{1}\sim 10^{-5}$, $\tau_{0}\sim 1M_{P}$, we find

$\displaystyle\delta m_{\Sigma}^{2}\sim\,\,\,10^{-14}\,M_{P}^{2}\,,\hskip
14.22636pt10^{-19}\,M_{P}^{2}\,,\hskip 14.22636pt10^{-24}\,M_{P}^{2}\,,$ (26)

which are negligible compared to the tree level mass squared of the inflaton
during the inflation $m_{\Sigma}^{2}\sim\mu^{2}\sim 10^{-10}M_{P}^{2}$. Hence
Dvali problem is absent in our scenario.

##  4 SUSY breaking, reheating and neutrino masses

In this section we investigate the link between the inflation sector and low
energy physics sectors. We study the stringent constraints arising from
reheating scenarios on the parameter space that is allowed by inflation
observables as demonstrated in the previous section. Also, we discuss the
neutrino masses which are one of the important motivations of our model. They
are connected to the inflation sector via $S,\,S_{1},\,S_{2}$. As we will see,
SUSY breaking and $R$-symmetry will play essential rule in providing a tiny
mass term for $S_{1}\,S_{2}$, hence TeV inverse seesaw mechanism can be
realized, with $Y_{\nu}$ not being small.

The assigned $B-L$ and $R$ charges in Table 1, forbid the terms $SNN$,
$\phi_{1}NN$ as well as higher order operators containing $NN$.333Here
operators such as $(\phi_{1}\phi_{2})^{n}\phi_{1}NN$ and
$(\phi_{1}\phi_{2})^{n}\,N\,S_{1}$ are not allowed by R-symmetry in the UV
sector as we indicated in Section 2. On the other hand the operator
$S^{2}\phi_{1}NN$ allowed by the symmetry will be suppressed by $M_{P}^{2}$
and has no effect in the neutrino mass matrix. Therefore the Majorana mass
term ${M_{N}NN}$ is suppressed. On the other hand $Z_{3}$ symmetry with the
assigned charges in Table 1, prevents terms such as $\phi_{1}S_{2}^{2}$ and
$\phi_{2}S_{1}^{2}$ hence any operator containing $S_{1}^{2}$ and $S_{2}^{2}$
is forbidden. The relevant terms in the superpotential to the neutrino masses
which are allowed by symmetry, are given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!W_{\nu}=Y_{\nu}LH_{u}N+Y_{S}\,S\,N\,S_{1}+\dfrac{\lambda_{3}}{M_{P}}\,S\,\phi_{1}\,N\,S_{2}+\left[\dfrac{\lambda_{2}}{M_{P}}S^{2}+\kappa_{2}\left(\dfrac{\phi_{1}\phi_{2}}{M_{P}}-\mu\right)\right]S_{1}S_{2},$
(27)

where the last term is common between the inflation and neutrino sector, hence
a clear connection between the inflation and low energy physics of neutrino
arises. SUSY breaking effects generate a tadpole term in the scalar potential
$-2\kappa_{1}M^{2}\,m_{3/2}\,S+h.c.$, causing a shift in the vev of
$S,\phi_{1},\phi_{2}$ in terms of the gravitino mass $m_{3/2}$, as indicated
in Dvali:1997uq ; Barbieri:1982eh ; Buchmuller:2000zm

$\displaystyle\langle S\rangle\simeq\dfrac{m_{3/2}}{\kappa_{1}}\,\,,\hskip
28.45274pt\langle|\phi_{1}|\rangle=\langle|\phi_{2}|\rangle\simeq
M\left(1-\dfrac{m_{3/2}^{2}}{\kappa_{1}^{2}M^{2}}\right).$ (28)

As advocated in the previous section, the term proportional to $S^{2}$ in the
superpotential (27) had an important role in flattening the inflation
potential, and here it appears the third and fourth roles of $S$ in reheating
and generating the tiny neutrino masses. Indeed a mass term
$\mu_{S}\,S_{1}\,S_{2}$ will be induced due to the shifts in the vevs of $S$
and $\phi_{i}$ in (28).

###  4.1 Constraints from reheating scenarios

At early time, after the inflation ended, the inflaton fields
$\tilde{S}_{1},\tilde{S}_{2}$ are massless. However, they acquire masses
$m_{S_{i}}\ll M$, due to SUSY breaking. Therefore reheating the universe
occurs via the decay of the heavy inflation sector fields $S$ and $\phi_{i}$
to $N(\tilde{N})$ and $S_{i}(\tilde{S}_{i})$, according to the interactions in
the superpotential (27). Moreover $\tilde{S}_{1},\tilde{S}_{2}$ can decay to
the SM particles and contribute to the reheating according to their masses. In
this respect the reheating temperature is given as Giudice:2003jh ;
Pradler:2006hh ; Ellis:2015jpg ; Ellis:2016spb ; Dudas:2017rpa ; Ellis:2020lnc

$\displaystyle
T_{R}=\left(\frac{10}{g_{s}}\right)^{1/4}\left(\frac{2\,\Gamma\,M_{P}}{\pi\,c}\right)^{1/2},$
(29)

where $\Gamma$ is the total decay width of the inflation sector fields, $c$ is
a constant of $\cal O$(1), and $g_{s}$ is the effective number of light
degrees of freedom in the thermal bath at $T_{R}$.

In models of supergravity inflation, an upper bound on the reheating
temperature arises due to the gravitino overproduction problem Ellis:1982yb ;
Khlopov:1984pf ; Moroi:1995fs ; Ellis:1984eq ; Ellis:1984er ; Moroi:1993mb ;
Kawasaki:2004yh ; Antusch:2010mv ; Kawasaki:2008qe ; Ferrantelli:2010as .
Gravitinos can be produced in the early universe thermally or non-thermally.
If gravitinos are the lightest supersymmetric particle (LSP), then they are
stable and can be a dark matter candidate. However, for high $T_{R}$, the
energy density of gravitinos can be larger than the present total energy
density of the universe hence overcloses the universe. Since the number
density of gravitinos is proportional to $T_{R}$, this leads to a bound on the
reheating temperature in terms of $m_{3/2}$. On the other hand if gravitinos
are not stable, they may decay into the LSP (which can be neutralinos) before
the Big Bang Nucleosynthesis (BBN), and hence the dark matter abundance
imposes a constraint on gravitino production. Moreover, they can decay during
and after the BBN. Constraints from BBN on both stable and unstable gravitinos
were discussed in Kawasaki:2008qe . If $m_{3/2}\lesssim 10$ TeV, most of
gravitinos decay after the start of BBN. Therefore, preserving the BBN
predictions, which is compatible with the observed abundances of the light
elements 3He, 4He, 6Li and D, requires small abundance of gravitinos. This
imposes an upper bound on $T_{R}$ depending on the MSSM mass spectrum and
$m_{3/2}$ as follows

$\displaystyle 10^{6}\,\text{GeV}\lesssim T_{R}\lesssim 2\times
10^{10}\,\text{GeV}\,,\hskip 14.22636pt{\text{for}}\hskip
14.22636pt300\,\text{GeV}\,\lesssim m_{3/2}\lesssim 3\times
10^{3}\,\text{GeV}\,.$ (30)

For stable gravitinos, the MSSM-LSP becomes the next to LSP (NLSP) that decays
to gravitinos. If their life time is long enough, this may affect the
abundances of the light elements. Therefore BBN constrains $m_{3/2}$.

Nonetheless, the above derived upper bounds on reheating temperature can be
relaxed to $\sim{\cal O}(10^{14})$ GeV as indicated in Ferrara:2010in ;
Ellis:2016spb , where models of higgs inflation usually predict such a high
$T_{R}$. It was pointed out that the gravitino production problem can be
addressed if gravitinos are heavy enough $m_{3/2}>{\cal O}$(30 TeV) such that
they decay before the BBN, or very light as in gauge mediation SUSY breaking
models, $m_{3/2}\ll 1$ GeV where an upper bound on $T_{R}\lesssim{\cal
O}(10^{14}\,\text{GeV})$ is not problematic with the BBN observations.

Now we discuss the constraints from the reheating scenario imposed on the
model parameters. We extract the interactions of $S_{i}$, $S$ and $\phi_{i}$
leading to decay into fermions and scalars, that are not suppressed and
contribute to $T_{R}$, as follows

$\displaystyle{\cal
L}_{S_{i}}=\left(Y_{S}Y_{\nu}^{*}\frac{m_{3/2}}{\kappa_{1}}\,S_{1}+\lambda_{3}Y_{\nu}^{*}\,\frac{M\,m_{3/2}}{\kappa_{1}M_{P}}\,S_{2}\right)\,\tilde{L}^{*}H_{u}^{*}+\,h.c.$
(31) $\displaystyle{\cal L}_{\phi_{i}}$ $\displaystyle=$
$\displaystyle\kappa_{2}^{2}\,\mu\,\sqrt{\dfrac{\mu}{M_{P}}}\,(\phi_{1}+\phi_{2})(\tilde{S_{1}}\tilde{S}_{1}^{*}+\tilde{S_{2}}\tilde{S}_{2}^{*})+\kappa_{2}\sqrt{\dfrac{\mu}{M_{P}}}\left(\phi_{1}+\phi_{2}\right)\,\bar{S_{1}}^{c}\,S_{2}^{c}\,+\,h.c.$
(32) $\displaystyle{\cal L}_{S}$ $\displaystyle=$ $\displaystyle
S\Bigg{[}\dfrac{|Y_{S}|^{2}\,m_{3/2}}{\kappa_{1}}(\tilde{S_{2}}\tilde{S}_{2}^{*}+\tilde{N}\tilde{N}^{*})+(Y_{S}Y_{\nu}^{*}\tilde{S_{1}}\tilde{L}^{*}H_{u}^{*}+\lambda_{3}Y_{\nu}^{*}\sqrt{\dfrac{\mu}{M_{P}}}\tilde{S_{2}}\tilde{L}^{*}H_{u}^{*}+h.c.)$
(33)
$\displaystyle+2\kappa_{1}\,\kappa_{2}\,\mu\tilde{S}_{1}\tilde{S}_{2}\Bigg{]}+\left[S\left(Y_{S}\,\bar{N}^{c}\,S_{1}^{c}+\lambda_{3}\sqrt{\dfrac{\mu}{M_{P}}}\,\bar{N}^{c}\,S_{2}^{c}\right)+\,h.c.\right]$

\begin{overpic}[width=260.17464pt]{reheat_constraint1}
\put(55.0,2.0){$\displaystyle\lambda_{3}$} \put(-5.0,53.0){$\displaystyle
Y_{S}$} \end{overpic} Figure 6: The allowed regions in the plane
$(\lambda_{3},Y_{S})$, corresponding to constraints on reheating temperature.

In case of TeV scale gravitinos, $T_{R}\lesssim 2\times 10^{10}$ GeV hence
$\Gamma\lesssim 10^{3}$ GeV. Studying the branching ratios of different decay
channels with respect to the total decay width, one finds that for large
$Y_{S}$, the dominant decay channel is $S\to\,\bar{N}^{c}\,S_{1}^{c}$, hence a
constraint on $Y_{S}<10^{-4}$. In this case, the channel
$S\to\,\bar{N}^{c}\,S_{2}^{c}$, $\phi_{i}\to\tilde{S_{i}}\tilde{S}_{j}^{*}$
become the dominant. The latter introduces the constraints
$\lambda_{3}<10^{-3}$ and $\kappa_{2}<0.01$. As advocated above, these
constraints can be relaxed if gravitinos were light $m_{3/2}\lesssim{\cal
O}$(0.1 GeV) or very heavy $m_{3/2}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}{\cal O}$(100
TeV).

In Fig. 6, we show the allowed region under the curves, for the couplings
$Y_{S}$ and $\lambda_{3}$, where we have fixed $\mu$ from the scalar
perturbations observations and $Y_{\nu}\sim{\cal O}(1)$. The red curve
corresponds to $T_{R}\sim 2\times 10^{10}$ GeV with $\kappa_{1}\sim 10^{-5}$,
$\kappa_{2}\sim 10^{-3}$ and $\tau_{0}\sim 10^{-3}\,M_{P}$. The solid
blue(purple) curves correspond to the relaxation $T_{R}\sim 10^{13}$ GeV, when
gravitinos are too heavy or too light, with the respective values
$\kappa_{1}\sim 10^{-2}(10^{-3})$, $\kappa_{2}\sim 10^{-2}(10^{-1})$ and
$\tau_{0}\sim 1(10)\,M_{P}$. For the dashed we have changed $\tau_{0}\sim
0.1(0.01)\,M_{P}$ respectively.

###  4.2 Neutrino masses

The Lagrangian of the neutrino masses can be derived from the superpotential
(27), after electroweak symmetry breaking, as follows

$\displaystyle{\cal
L}_{\nu}=m_{D}\,\bar{\nu}_{L}\,N^{c}+M_{R_{1}}\,\bar{N}^{c}\,S_{1}^{c}+M_{R_{2}}\,\bar{N}^{c}\,S_{2}^{c}+\mu_{S}\,\bar{S_{1}}^{c}\,S_{2}^{c}+h.c.,$
(34) $\displaystyle{\text{with}}\,\hskip 14.22636ptm_{D}$ $\displaystyle=$
$\displaystyle Y_{\nu}\,v\sin(\beta)\,,\,\hskip
28.45274ptM_{R_{1}}=\dfrac{Y_{S}\,m_{3/2}}{\kappa_{1}}\,,\hskip
28.45274ptM_{R_{2}}=\dfrac{\lambda_{3}\,m_{3/2}\,M}{\kappa_{1}\,M_{P}}\,,$
$\displaystyle\mu_{S}$ $\displaystyle=$
$\displaystyle\dfrac{\lambda_{2}\,}{M_{P}}\,\langle
S\rangle^{2}\,+\,\dfrac{\kappa_{2}\,}{M_{P}}\left[\langle\phi_{1}\phi_{2}\rangle-M^{2}\right]$
(35) $\displaystyle=$
$\displaystyle\dfrac{m_{3/2}^{2}}{\kappa_{1}^{2}\,M_{P}}\left[\lambda_{2}\,-\,\kappa_{2}\,+\,\dfrac{\kappa_{2}\,m_{3/2}^{2}}{2\kappa_{1}^{2}\,M^{2}}\,\right]\,,$

where $v$ is the electroweak vev and $\tan\beta=\dfrac{\langle
H_{u}\rangle}{\langle H_{d}\rangle}$. The last term in $\mu_{S}$ is suppressed
by the super-heavy scale $M^{2}$ and dominant effects come from the first two
terms. The neutrino mass matrix, in the basis $(\nu_{L},N,S_{1},S_{2})$ (for
one flavor), is given by

\begin{overpic}[width=212.47617pt]{massnu1}
\put(55.0,2.0){$\displaystyle\lambda_{3}$} \put(1.0,50.0){$\displaystyle
Y_{S}$} \end{overpic}\begin{overpic}[width=212.47617pt]{massnu2}
\put(55.0,2.0){$\displaystyle\lambda_{3}$} \put(1.0,55.0){$\displaystyle
Y_{S}$} \end{overpic}\begin{overpic}[width=212.47617pt]{massnu3}
\put(55.0,2.0){$\displaystyle\lambda_{3}$} \put(1.0,55.0){$\displaystyle
Y_{S}$} \end{overpic}

Figure 7: The regions of parameter space in the plane $(\lambda_{3},Y_{S})$
allowed by neutrino masses and reheating temperature constraints. The solid
lines represent the lightest neutrino eigenstate $|m_{\nu_{l}}|\sim{\cal
O}(0.1)$ eV, while the dotted curve represent the constraints from reheating.

$\displaystyle{M}_{\nu}=\left(\begin{array}[]{cccc}0&{m_{D}}&0&0\\\
{m_{D}}&0&{M_{R_{1}}}&M_{R_{2}}\\\ 0&{M_{R_{1}}}&0&\mu_{S}\\\
0&M_{R_{2}}&\mu_{S}&0\\\ \end{array}\right).$ (40)

For the mass hierarchy $\mu_{S}\ll m_{D}\ll M_{R_{i}}$, the eigenvalues
consist of a light mass corresponding to the active neutrino, a relatively
light one corresponding a sterile neutrino and two heavy states as follows

$\displaystyle m_{\nu_{l}}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{D}^{2}\,\mu_{S}}{2M_{R_{1}}M_{R_{2}}}\,,$
$\displaystyle m_{\nu_{1}}$ $\displaystyle\simeq$
$\displaystyle\frac{\mu_{S}\left(-\frac{m_{D}^{2}}{2M_{R_{2}}}-2M_{R_{2}}\right)}{M_{R_{1}}}\,\sim\,\mu_{S}\,,$
$\displaystyle m_{\nu_{2,3}}$ $\displaystyle\simeq$
$\displaystyle\pm\left(M_{R_{1}}+\frac{m_{D}^{2}+M_{R_{2}}^{2}}{2M_{R_{1}}}\right)+\frac{M_{R_{2}}\mu_{S}}{M_{R_{1}}}\sim\pm\left(M_{R_{1}}+M_{R_{2}}\right).$
(41)

In Table 3, we list three points in the parameter space that give neutrino
mass of order ${\cal O}(0.1)$ eV, where upper bound on the sum over the
neutrino masses is given by $\sum_{\gamma}m_{\nu_{\gamma}}<0.15$ eV, at 95% CL
Giusarma:2016phn ; RoyChoudhury:2018gay . We have taken different value of
$m_{3/2}$ corresponding to different regimes in the reheating scenario and
fixed $\sin(\beta)\sim 1$. The value of $M$ can range between
$10^{15}-10^{17}$ GeV.

$\,\,\,\,m_{3/2}\,\,\,\,$ | $\,\,\,\,\lambda_{2}\,\,\,\,$ | $\,\,\,\,\lambda_{3}\,\,\,\,$ | $\,\,\,\,\kappa_{1}\,\,\,\,$ | $\,\,\,\,\kappa_{2}\,\,\,\,$ | $\,\,\,\,Y_{\nu}\,\,\,\,$ | $\,\,\,\,Y_{S}\,\,\,\,$ | $\,\,\,\,|m_{\nu_{1}}|\,\,\,\,$ | $\,\,\,\,|m_{\nu_{2,3}}|\,\,\,\,$
---|---|---|---|---|---|---|---|---
$10^{3}$ | 0.008 | $0.002$ | $10^{-5}$ | $10^{-3}$ | 0.05 | $3\times 10^{-5}$ | $3\times 10^{-5}$ | $4.4\times 10^{3}$
$0.1$ | 0.9 | 0.4 | $10^{-7}$ | 0.1 | 0.9 | 0.01 | $2.1\times 10^{-7}$ | $1.06\times 10^{4}$
$10^{5}$ | 0.1 | 0.01 | 0.01 | $0.01$ | 0.5 | 0.0032 | $6.6\times 10^{-7}$ | $3.21\times 10^{4}$
Table 3: Three benchmark points in the parameter space that give the active
neutrino masses $|m_{\nu_{l}}|\sim{\cal O}(0.1)$ eV and satisfy the
constraints from inflation observables and reheating. All mass scales are
given in GeV.

Fig. 7 depicts the allowed values of the parameters in the plane
$(\lambda_{3},Y_{S})$, while fixing the other parameters, where the solid
curves correspond to active neutrino mass value $|m_{\nu_{l}}|\sim{\cal
O}(0.1)$ eV. We have fixed $\sin(\beta)\sim 1$, and $Y_{\nu}\sim{\cal O}(1)$
in the case where $m_{3/2}=0.1,10^{5}$ GeV. We have imposed the constraints
from inflation observables and reheating temperature according to the cases
discussed in (4.1). Therefore the segments between the black dots give the
allowed values in the parameter space $(\lambda_{3},Y_{S})$ for fixed values
of the other parameters.

##  5 Conclusions

We have constructed an inflation model in which the non-singlet right-handed
sneutrinos $S_{1},S_{2}$ contain the inflaton component, in no-scale
supergravity. We have modified the tribrid inflation model to implement a
scenario in which the inflation sector is linked to a TeV inverse seesaw
mechanism. In this regard, we have explained the smallness of the $\mu_{S}$
mass parameter on a basis of the symmetry defined on the model and SUSY
breaking effects. We have discussed different regimes in the parameter space
that lead to successful inflation compatible with Planck observation. We found
that the reheating scenario imposes severe constraints on the parameter space
when gravitino mass is TeV scale. These constraints can be relaxed for very
heavy or ultra-light gravitinos in order to avoid the BBN constraints. We have
derived the neutrino masses taking into account the latter constraints. The
model may have interesting consequences on the low energy phenomenology.

###### Acknowledgements.

The author would like to thank Stefan Antusch and Waleed Abdallah for
encouragement and stimulating discussions. The work of A. M. is supported in
part by the Science, Technology and Innovation Funding Authority (STDF) under
grant No. (33495) YRG.

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