Double Inflation in Classically Conformal $B-L$ Model

Anish<EMAIL_ADDRESS>Nobuchika
<EMAIL_ADDRESS>Arnab
<EMAIL_ADDRESS>and Digesh
<EMAIL_ADDRESS>
a Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland,
b University of Alabama, Tuscaloosa, Alabama 35487, USA
c Centre for Strings, Gravitation and Cosmology, Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India
d Department of Physics and Materials Science, St. Mary’s College of Maryland,
St. Mary’s City, Maryland 20686, USA

It has recently been shown in Ref. [1] that the double-inflation scenario
based on the Coleman-Weinberg potential can successfully generate primordial
black holes (PBHs) with the inflationary predictions consistent with the
Planck measurements. These PBHs can play the role of dark matter in our
universe. In this paper, we propose the classically conformal minimal $B-L$
model as an ultra-violet (UV) completion of the scenario. In our model, the
$B-L$ Higgs field is identified with the inflaton and the electroweak symmetry
breaking is triggered by the radiative $B-L$ symmetry breaking with the
Coleman-Weinberg potential. We show that this UV completion leads to a viable
cosmological history after the double inflaton: the universe is reheated via
inflaton decay into right-handed neutrinos whose mass is determined
consistently by a relation between the number of e-folds and reheating
temperature. Using the general parameterization for neutrino Dirac Yukawa
couplings through the seesaw mechanism and the neutrino oscillation data, we
also show that the observed baryon asymmetry of the universe is successfully
reproduced by either resonant leptogenesis or non-thermal leptogenesis. Based
on the scalar power spectrum shown in Ref. [1], we evaluate the scalar induced
gravitational wave spectrum, which can be tested by various proposed
gravitational wave observatories like BBO, DECIGO etc.

##  1 Introduction

Cosmic inflation, which is the accelerated expansion of the universe at early
time, solves the horizon and the flatness problem as well as provides the
primordial density fluctuations essential for the large scale structure
formation [2, 3, 4, 5]. Rapidly increasing precision and cosmological data
from various observations, in particular, Cosmic Microwave Background (CMB)
measurement [6] has provided severe constraints on inflationary potential.

The inflaton field generates density fluctuation not only at the CMB scale,
${\rm k_{CMB}/{\rm Mpc^{-1}}}\sim 10^{-2}-10^{-1}$, but also over a wide range
of scales with ${\rm k\gg k_{CMB}}$. If the density fluctuations at some scale
are significantly enhanced and the scalar power spectrum
$P_{\zeta}\gtrsim{\cal O}(10^{-3})$, which is much greater than the power
spectrum $P_{\zeta}\gtrsim{\cal O}(10^{-9})$ corresponding to the CMB scales,
primordial black hole (PBH) can be formed. It is particularly interesting that
such PBHs can be a dark matter candidate in our Universe. See [7] for a recent
review.

In the context of a single-field inflation scenario, it has been shown that
the scalar power spectrum can be enhanced if the potential exhibits an
approximate inflection point. If the inflaton passes through this region with
a suitably small velocity, termed as the ultra-slow roll, the density
perturbations can be significantly enhanced to form PBHs [8, 9, 10, 11, 12,
13, 14]. See [15, 16, 17] for a recent review. Such an enhancement generally
requires fine-tuning of model parameters, with which the predicted value of
the spectral index mostly does not match the CMB data [6]. Successful PBH
formation has also been discussed in the multi-field inflation scenarios [18,
19], in hybrid inflation [20, 21], due to dynamics of spectator fields [22] or
others [23, 24, 25, 26].

Another possibility for the single field inflation is the so-called double
inflation scenario [1] in which the inflaton has a double-well potential. The
first inflation takes place when the inflaton rolls down from a large field
value, which produces density fluctuations responsible for the CMB spectrum.
After the first inflation, the field passes through a potential minimum and
climbs up the potential hill towards the potential origin. When the inflaton
almost comes to a stop in the vicinity of the potential origin, the second
inflation occurs. If the inflaton potential is very flat near the origin, the
ultra-slow role condition is satisfied to enhance the density fluctuation
during the second inflation, enough to form the PBHs.

Recently Ref. [1] analyzed the double-inflation scenario based on the Coleman-
Weinberg potential [27] to show that (i) the inflationary predictions from the
first inflation are consistent with the Planck data [6] and (ii) the density
fluctuation is sufficiently enhanced during the second inflation for
production of PBH. Note that the Coleman-Weinberg potential is ideal for
realizing this scenario since the mass-squared (the second derivative of the
potential) of the field at the origin is zero. In this paper, we propose
classically conformal $U(1)$ $B-L$ (Baryon number minus Lepton number)
extended Standard Model (SM) as an ultra-violet (UV) completion of the double
inflation scenario analyzed in Ref. [1].

From the particle physics viewpoint, the classically conformal $U(1)_{B-L}$
extended SM [28, 29] is particularly interesting since it can address several
outstanding problems in the SM. The $B-L$ symmetry is the unique anomaly-free
global symmetry in the SM. The so-called minimal $B-L$ model [30, 31, 32, 33,
34, 35] is the minimal framework for extending the SM to gauge the $B-L$
symmetry. The particle content of the SM is extended to include three right-
handed neutrinos to cancel $B-L$ related gauge anomalies and the $B-L$ Higgs
field to break the $B-L$ symmetry. Since the $B-L$ symmetry breaking generates
Majorana masses for the right-handed neutrinos, the Type-I seesaw mechanism
[36, 37, 38, 39, 40, 41] is implemented to explain the origin of the light
neutrino masses. The model can also explain the origin of baryon-asymmetry of
the universe through leptogenesis[42]. We consider the classically conformal
extension of the minimal $B-L$ model [28, 29]555See Refs. [43, 44, 45, 46] for
various cosmological implications of conformal B-L scenarios., where the $B-L$
gauge symmetry is radiatively broken by the Coleman-Weinberg mechanism. This
also triggers the breaking of the electroweak symmetry, and therefore, the
electroweak symmetry breaking originates from the radiative $B-L$ symmetry
breaking.

In the framework of the classically conformal $B-L$ model, we identify the
$B-L$ Higgs field as the inflaton in the double-inflation scenario [1].
Focusing on a benchmark parameter set to realize a successful double
inflation, we show the successful thermal history of the universe after the
double inflation, namely, reheating after the double inflation, leptogenesis,
electroweak symmetry breaking and neutrino mass generation. We also show that
enhanced density fluctuations needed for the PBH production produce (scalar-
induced) gravitational waves, which can be tested by various planned
observations in the future [47].

The paper is organized as follows: in section 2, we introduce classically
conformal $B-L$ model with radiative symmetry breaking. In section 3, we
discuss the double inflation scenario with the $B-L$ Higgs field identified as
inflation. Successful reheating after the double inflation and thermal
leptogenesis are discussed in Section 4. Section 5 is devoted to conclusions
and discussions. In this section, we also discuss the scalar-induced
gravitational wave spectrum associated with the enhanced power spectrum to
form the PBHs.

##  2 Classically Conformal Minimal $B-L$ Model

| SU(3)c | SU(2)L | U(1)Y | U(1)B-L
---|---|---|---|---
$q_{L}^{i}$ | 3 | 2 | $+1/6$ | $+1/3$
$u_{R}^{i}$ | 3 | 1 | $+2/3$ | $+1/3$
$d_{R}^{i}$ | 3 | 1 | $-1/3$ | $+1/3$
$\ell^{i}_{L}$ | 1 | 2 | $-1/2$ | $-1$
$N_{R}^{i}$ | 1 | 1 | $0$ | $-1$
$e_{R}^{i}$ | 1 | 1 | $-1$ | $-1$
$H$ | 1 | 2 | $-1/2$ | $0$
$\Phi$ | 1 | 1 | $0$ | $+2$
Table 1: Particle content of the minimal $B-L$ extended SM. Besides the SM
particles, it includes three right-handed neutrinos $N_{R}^{i}$ (where
$i=1,2,3$ denotes the generation index) and a complex scalar $\Phi$.

The particle content of the minimal anomaly-free extension of the SM with
gauged $B-L$ (baryon minus lepton number) symmetry is listed in Table 1, where
$N_{R}^{i}$ ($i=1,2,3$) are three generations of right-hand neutrinos (RHNs),
$\Phi$ is the U(1)B-L Higgs field, and the rest are the SM particles.

The Yukawa sector involving the new particles is given by

$\displaystyle{\cal
L}\supset\left(-\sum_{i,j=1}^{3}Y_{D}^{ij}\overline{N_{R}^{i}}\;H^{\dagger}\ell_{L}^{j}-\frac{1}{2}\sum_{i=1}^{3}Y_{M}^{i}\Phi\overline{N_{R}^{i~{}C}}N^{i}_{R}\right)+{\rm
h.c.},$ (2.1)

where the first and the second terms are, respectively, the Dirac and Majorana
Yukawa couplings, and we have chosen the flavor diagonal basis for the
Majorana Yukawa couplings.

The renormalizable scalar potential consistent with classically conformal
invariance does not include the mass-squared terms at the tree level and is
expressed as

$\displaystyle
V=\lambda_{H}\left(H^{\dagger}H\right)^{2}+\lambda_{\Phi}\left(\Phi^{\dagger}\Phi\right)^{2}-\lambda_{\rm
mix}\left(H^{\dagger}H\right)\left(\Phi^{\dagger}\Phi\right),$ (2.2)

where the couplings $\lambda_{H},~{}\lambda_{\Phi}$ and $\lambda_{\rm mix}$
are taken to be positive. Assuming $\lambda_{\rm mix}$ is negligibly small, we
can analyze the Higgs potentials for $\Phi$ and $H$ separately as a good
approximation. This assumption will be justified later. Let us first analyze
the $B-L$ Higgs sector. By choosing the renormalization scale to be the vacuum
expectation value (VEV) of $\Phi$, the Coleman-Weinbeg potential [27] at the
one-loop level is given by

$\displaystyle
V(\phi)=\frac{\lambda_{\Phi}}{4}\phi^{4}+\frac{\beta_{\Phi}}{8}\phi^{4}\left(\ln\left[\frac{\phi^{2}}{v_{\phi}^{2}}\right]-\frac{25}{6}\right)+V_{0},$
(2.3)

where $\phi/\sqrt{2}=\Re[\Phi]$ is a real scalar,
$v_{\phi}=\langle\phi\rangle$ is the VEV of $\Phi$, and
$V_{0}=\beta_{\Phi}v_{\phi}^{4}/16$ ensures $V(v_{\phi})=0$. Imposing the
stationary condition $\left.dV/d\phi\right|_{\phi=v_{\phi}}=0$ we get a
relation between the renormalized self-coupling,

$\displaystyle\lambda_{\Phi}=\frac{11}{6}\beta_{\Phi},$ (2.4)

where

$\displaystyle\lambda_{\Phi}=\frac{1}{3!}\left.\frac{d^{4}V(\phi)}{d\phi^{4}}\right|_{\phi=v_{\phi}},$
(2.5)

and the coefficient of the one-loop corrections,

$\displaystyle\beta_{\Phi}=\frac{1}{16\pi^{2}}\left(20\lambda_{\Phi}^{2}+96g^{4}-\sum_{i=1}^{3}(Y_{M}^{i})^{4}\right)\simeq\frac{1}{16\pi^{2}}\left(96g^{4}-\sum_{i=1}^{3}(Y_{M}^{i})^{4}\right).$
(2.6)

Here, $g$ is the $B-L$ gauge coupling, and we have used $\lambda_{\Phi}^{2}\ll
g^{4}$ indicated by Eq. (2.4).

The mass of the $B-L$ Higgs boson is given by

$\displaystyle m_{\phi}^{2}$ $\displaystyle=$
$\displaystyle\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi=v_{\phi}}=\beta_{\Phi}v_{\phi}^{2}\simeq\frac{1}{16\pi^{2}}\left(96g^{4}-\sum_{i=1}^{3}(Y_{M}^{i})^{4}\right)v_{\phi}^{2}$
(2.7) $\displaystyle=$
$\displaystyle\frac{3}{2\pi^{2}}g^{2}m_{Z^{\prime}}^{2}\left(1-\frac{2}{3}\sum_{i=1}^{3}\left(\frac{m_{N^{i}}}{m_{Z^{\prime}}}\right)^{4}\right),$

where

$\displaystyle
m_{Z^{\prime}}=2gv_{\phi},\;\;m_{N^{i}}=\frac{Y_{M}^{i}}{\sqrt{2}}v_{\phi},$
(2.8)

are the $B-L$ gauge boson ($Z^{\prime}$ boson) and the Majorana RHN masses
that are generated by the breaking of $B-L$ symmetry. The vacuum stability
condition $m_{\phi}^{2}>0$ demands
$m_{Z^{\prime}}>\left(\frac{2}{3}\right)^{1/4}\sum_{i=1}^{3}m_{N^{i}}$, or
equivalently, $\sum_{i=1}^{3}Y_{M}^{i}<(96)^{1/4}g$.

After the radiative breaking of the $B-L$ symmetry via the Coleman-Weinberg
mechanism, the mixing quartic coupling term in Eq. (2.2) generates the
negative mass squared term for the SM Higgs doublet,

$V\supset\frac{\lambda_{H}}{4}h^{4}-\frac{\lambda_{\rm
mix}}{4}v_{\phi}^{2}h^{2},$ (2.9)

where we have replaced $H$ by $H=1/\sqrt{2}\,(0\;\,h)^{T}$ in the unitary
gauge, so that the electroweak symmetry breaking is triggered. The SM Higgs
boson mass ($m_{h}$) is expressed as

$m_{h}^{2}=\lambda_{\rm mix}v_{\phi}^{2}=2\lambda_{H}v_{h}^{2},$ (2.10)

where $v_{h}=246$ GeV is the SM Higgs VEV. Considering the Higgs boson mass of
$m_{h}=125$ GeV and the LEP experimental constraint on the VEV
$v_{\phi}\gtrsim 10$ TeV [48, 49, 50, 51], we find $\lambda_{\rm mix}\lesssim
10^{-4}$ and the smallness of $\lambda_{\rm mix}$ is justified.

After the breaking of the electroweak and the $B-L$ gauge symmetries, the
Dirac and Majorana masses for the neutrinos are generated in Eq. (2.1):

$\displaystyle{\cal
L}\supset\left(-\sum_{i,j=1}^{3}m_{D}^{ij}\overline{N_{R}^{i}}\;\nu_{L}^{j}-\frac{1}{2}\sum_{i=1}^{3}M_{N^{i}}\overline{N_{R}^{i~{}C}}N^{i}_{R}\right)+{\rm
h.c.},$ (2.11)

The resulting neutrino mass matrix is given by

$\displaystyle M_{\nu}=\begin{pmatrix}0&m_{D}^{T}\\\
m_{D}&M_{N}\end{pmatrix},$ (2.12)

which, for $M_{N}\gg m_{D}$, leads to the seesaw formula for the light
neutrino mass matrix

$\displaystyle m_{\nu}\simeq m_{D}^{T}M_{N}^{-1}m_{D}.$ (2.13)

Using the Maki-Nakagawa-Sakata (MNS) matrix $U_{MNS}$ we can diagonalize
$m\nu$ as

$\displaystyle D_{\nu}=U_{MNS}^{T}\;m_{\nu}\;U_{MNS},$ (2.14)

where $D_{\nu}={\rm diag}(m_{1},m_{2},m_{3})$ with the mass eigenvalues
$m_{i}$. The explict form of the MNS matrix is given by

$\displaystyle U_{\rm
MNS}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)\left(\begin{array}[]{ccc}e^{i\sigma_{1}}&0&0\\\
0&e^{i\sigma_{2}}&0\\\ 0&0&1\end{array}\right),$ (2.21)

where $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$, and $\delta$ and
$\sigma_{i}$ are the CP-violating Dirac and Majorana phases, respectively.
Neutrino mass parameters are determined by the oscillation data [52]:

$\displaystyle\sin^{2}\theta_{12}=0.307,\quad\sin^{2}\theta_{13}=2.20\times
10^{-2},\quad\Delta m_{21}^{2}=7.53\times 10^{-5}\;{\rm
eV},\quad\delta=1.23\pi\;{\rm rad}$
$\displaystyle\sin^{2}\theta_{23}=0.534\;{\rm(NH)},\quad\Delta
m_{32}^{2}\equiv(m_{3}^{2}-m_{2}^{2})=2.437\times 10^{-3}\;{\rm
eV^{2}}\;{\rm(NH)},$
$\displaystyle\sin^{2}\theta_{23}=0.547\;{\rm(IH)},\quad\;\Delta
m_{32}^{2}\equiv(m_{2}^{2}-m_{3}^{2})=2.519\times 10^{-3}\;{\rm
eV^{2}}\;{\rm(IH)},\qquad$ (2.22)

Here, the Normal Hierarchy (NH) implies $m_{lightest}\equiv m_{1}<m_{2}<m_{3}$
while in the Inverted Hierarchy (IH) $m_{lightest}\equiv m_{3}<m_{1}<m_{2}$.
Hence, for the NH,

$\displaystyle m_{2}=\sqrt{m_{lightest}^{2}+\Delta m_{12}^{2}},\quad
m_{3}=\sqrt{m_{lightest}^{2}+\Delta m_{12}^{2}+\Delta m_{23}^{2}},$ (2.23)

while for the IH,

$\displaystyle m_{1}=\sqrt{m_{lightest}^{2}+\Delta m_{23}^{2}-\Delta
m_{12}^{2}},\quad m_{2}=\sqrt{m_{lightest}^{2}+\Delta m_{23}^{2}}.$ (2.24)

The neutrino mass matrix can be conveniently parametrized as [53, 54]

$\displaystyle m_{D}=\sqrt{M_{N}}O\sqrt{D_{\nu}}U_{\rm MNS}^{\dagger},$ (2.25)

where
$\sqrt{M_{N}}\equiv\mbox{diag}(\sqrt{m_{N^{1}}},\sqrt{m_{N^{2}}},\sqrt{m_{N^{3}}})$
and

$\displaystyle\sqrt{D_{\nu}}=\left\\{\begin{array}[]{ccc}\sqrt{D_{\nu}^{\rm
NH}}&=&{\rm
diag}\left(\sqrt{m_{lightest}},\sqrt{m_{2}},\sqrt{m_{3}}\right),\\\
\sqrt{D_{\nu}^{\rm IH}}&=&{\rm
diag}\left(\sqrt{m_{lightest}},\sqrt{m_{1}},\sqrt{m_{2}}\right).\end{array}\right.$
(2.28)

In (2.25), $O$ is a $3\times 3$ orthogonal matrix

$\displaystyle O=\left(\begin{array}[]{ccc}C_{12}C_{13}&S_{12}C_{13}&S_{13}\\\
-S_{12}C_{23}-C_{12}S_{23}S_{13}&C_{12}C_{23}-S_{12}S_{23}S_{13}&S_{23}C_{13}\\\
S_{12}S_{23}-C_{12}C_{23}S_{13}&-C_{12}S_{23}-S_{12}C_{23}S_{13}&C_{23}C_{13}\end{array}\right),$
(2.32)

where $S_{ij}=\sin\omega_{ij}$ and $c_{ij}=\cos\omega_{ij}$ with complex
numbers $\omega_{ij}$.

##  3 Double Inflation in the Classically Conformal Minimal $B-L$ Model

Let us consider the following action in the Jordan frame with a non-minimal
gravitational coupling between the inflaton ($\phi$) and the Ricci scalar
curvature ($\cal R$) and non-minimal kinetic term for $\phi$:

$\displaystyle{\cal S}_{J}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}f(\phi){\cal
R}+\frac{1}{2}K(\phi)g^{\mu\nu}\left(\partial_{\mu}\phi\right)\left(\partial_{\nu}\phi\right)-V_{J}(\phi)\right].$
(3.1)

Following Ref. [1], we set $f(\phi)=1+\xi\phi^{2}$ with $\xi>0$ being the non-
minimal coupling parameter, the non-minimal kinetic term
$K(\phi)=(1-\phi^{2}/6\alpha)^{-n}$, and the inflaton potential $V_{J}(\phi)$
given by

$\displaystyle
V_{J}(\phi)=\frac{\lambda}{4}\left(\frac{1}{2}\phi^{4}\left(\ln\left[\frac{\phi^{2}}{v_{\phi}^{2}}\right]-\frac{1}{2}\right)+\frac{1}{4}v_{\phi}^{4}\right).$
(3.2)

This potential is essentially the same as the potential in Eq. (2.3) with the
identification of $\lambda=\frac{6}{11}\lambda_{\Phi}$ and
$\lambda=\beta_{\Phi}$. Throughout this paper we use the Planck unit by
setting the reduced Planck mass $M_{P}=2.44\times 10^{18}$ GeV to be 1.

After performing the conformal transformation, $g_{{\mu\nu}}\to
g_{E{\mu\nu}}=f(\phi)g_{\mu\nu}$, the action in the Einstein frame is given by

$\displaystyle S_{E}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g_{E}}\left[-\frac{1}{2}{\cal
R}_{E}+g_{E}^{\mu\nu}\left(\partial_{\mu}\sigma\right)\left(\partial_{\nu}\sigma\right)-V_{E}(\sigma)\right].$
(3.3)

Here, $V_{E}$ is the potential in the Einstein Frame,

$\displaystyle
V_{E}(\phi(\sigma))=\frac{V_{J}(\phi(\sigma))}{f(\phi(\sigma))^{2}},$ (3.4)

and $\sigma$ (inflaton field in Einstein Frame) is related to the original
field $\phi$ by

$\displaystyle\left(\frac{d\sigma}{d\phi}\right)^{2}=\left(\frac{K}{f}+\frac{3}{2}\left(\frac{f^{\prime}}{f}\right)^{2}\right),$
(3.5)

where a “$\it prime$” denotes a derivative with respect to $\phi$. Using this
relation, the slow-roll parameters can be expressed in terms of $\phi$ as
follows:

$\displaystyle\epsilon(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{V_{E}^{\prime}}{V_{E}\;\sigma^{\prime}}\right)^{2},$
$\displaystyle\eta(\phi)$ $\displaystyle=$
$\displaystyle\frac{V_{E}^{\prime\prime}}{V_{E}\;(\sigma^{\prime})^{2}}-\frac{V_{E}^{\prime}\;\sigma^{\prime\prime}}{V_{E}\;(\sigma^{\prime})^{3}},$
$\displaystyle\zeta(\phi)$ $\displaystyle=$
$\displaystyle\left(\frac{V_{E}^{\prime}}{V_{E}\;\sigma^{\prime}}\right)\left(\frac{V_{E}^{\prime\prime\prime}}{V_{E}\;(\sigma^{\prime})^{3}}-3\frac{V_{E}^{\prime\prime}\;\sigma^{\prime\prime}}{V_{E}\;(\sigma^{\prime})^{4}}+3\frac{V_{E}^{\prime}\;(\sigma^{\prime\prime})^{2}}{V_{E}\;(\sigma^{\prime})^{5}}-\frac{V_{E}^{\prime}\;\sigma^{\prime\prime\prime}}{V_{E}\;(\sigma^{\prime})^{4}}\right).$
(3.6)

The amplitude of the scalar curvature perturbation $\Delta_{\cal R}$ is given
by

$\Delta_{\cal R}^{2}=\left.\frac{V_{E}}{24\pi^{2}\epsilon}\right|_{k},$ (3.7)

where $k$ is the pivot scale. The Planck result sets the power spectrum
$\Delta_{\mathcal{R}}^{2}=2.195\times 10^{-9}$ [55] for $k=0.05~{}\rm
Mpc^{-1}$. The number of e-folds is given by

$\displaystyle N_{k}=\frac{1}{\sqrt{2}}\int_{\phi_{\rm
e}}^{\phi_{k}}d\phi\frac{\sigma^{\prime}}{\sqrt{\epsilon(\phi)}}$ (3.8)

where $\phi_{k}$ is the inflaton value at horizon exit of the scale
corresponding to $k$, and $\phi_{e}$ is the inflaton value at the end of
inflation, which is defined by $\epsilon(\phi_{e})=1$. To solve the horizon
and flatness problems, $N_{k}=50-60$ is necessary.

The predictions for the inflationary observables, namely, the scalar spectral
index $n_{s}$ and the tensor-to-scalar ratio $r$ are expressed in terms of
slow-roll parameters as

$\displaystyle n_{s}=1-6\epsilon+2\eta,\;\;\;\;\;r=16\epsilon,\;\;$ (3.9)

which are evaluated at $\phi=\phi_{k}$.

In our analysis, we consider the roll-back scenario discussed in Ref. [1]. In
this case, after the conclusion of the slow-roll inflation, which occurs at
$\phi>v_{\phi}$, the inflaton rolls down towards a potential minimum at
$\phi=v_{\phi}$ and passes through the potential minimum, and then climbs up
the potential hill towards the origin $\phi=0$. The inflaton comes to a stop
in the vicinity of $\phi=0$, rolls back towards $\phi=v_{\phi}$, and then
oscillates around the minimum until it decays to reheat the universe. There
are two phases of inflation along with the evolution of inflaton. The slow-
roll inflation takes place for $\phi>v_{\phi}$ which generates comsological
perturbations corresponding to the observed CMB anisotropy. The second
inflation occurs during the rolling back near $\phi=0$. Since the curvature of
the potential at $\phi=0$ is zero in the CW potential, the ultra-slow roll
condition is satisfied, and density fluctuations are significantly enhanced at
small scales to form PBHs.

It has been shown in Ref. [1] that with a suitable choice of the parameters,
namely, $\xi$, $n$, $\alpha$, $\lambda$, $v_{\phi}$, the double-inflation can
successfully produce inflationary predictions consistent with the CMB data and
also generate large fluctuations to form PBHs. For our analysis, we choose the
benchmark for the right panel of Fig. 2 in the paper [1]:
$\lambda=1.79222\times 10^{-8}$ and $v_{\phi}=0.3854781$, $n=4$, and
$\alpha=2v_{\phi}^{2}/3$. According to the paper, this benchmark
predicts666Although the paper [1] does not provide the explicit value for the
tensor-to-scalar ratio ($r$) for this benchmark, Table 1 showing the results
for various benchmark parameters indicates that the resultant inflationary
predictions for $\xi\ll 1$ are almost the same, $r\simeq 0.016$.
$n_{s}=0.96072$ and for the number of e-folds $N_{0}=N_{I}+N_{II}=55$, where
$N_{I}=41$ and $N_{II}=14$ are the number of e-folds generated during the
first and second phases of inflation, respectively. We note that
$\phi_{k}\simeq\sqrt{6\alpha}=2v_{\phi}$, which is close to the singular point
of the non-minimal kinetic function $K(\phi)$.

##  4 Reheating after Inflation

In the scenario we consider, after the end of the second inflation at
$t_{end}$, the inflaton oscillates around the potential minimum at
$\phi=v_{\phi}$ until it decays to reheat the universe. We use the sudden
decay approximation, and all the inflation energy is transmitted to the
radiation at $t=\tau_{\phi}=1/\Gamma_{\phi}$, where $\tau_{\phi}$ is the
lifetime of the inflaton and $\Gamma_{\phi}$ is its decay width. We assume
that the universe is instantly thermalized at $t=\tau_{\phi}$. This is the
beginning of the Standard Big Bang cosmology. This radiation-dominated era
continues until the matter-radiation equality is reached at time $t_{eq}$.
After that, the Universe stays matter-dominated until today $t_{0}$.

We now evaluate the number of e-folds based on the above picture. Denoting the
co-moving wave number of the CMB scale by $k$, the scale factor by $a_{k}$ and
the Hubble parameter by $H_{k}$, they are related by $k=a_{k}H_{k}$ at the
horizon exit scale. It follows that

$\frac{k}{a_{0}H_{0}}=\frac{a_{k}H_{k}}{a_{0}H_{0}}=\frac{a_{k}}{a_{end}}\frac{a_{end}}{a_{rh}}\frac{a_{rh}}{a_{eq}}\frac{a_{eq}}{a_{0}}\frac{H_{k}}{H_{0}},$
(4.1)

where $a_{end}$, $a_{rh}$, $a_{eq}$, and $a_{0}$ are the scale factors at the
end of inflation ($t_{end}$), at the reheating time ($\tau_{\phi}$), at
matter-radiation equality ($t_{eq}$), and at present ($t_{0}$). Also,
$H_{0}=100h$ km s-1 Mpc-1 with $h=0.674$ [55]. Each ratio in the right hand
side of Eq. (4.1) is evaluated as follows: $\frac{a_{k}}{a_{end}}=e^{N_{k}}$;
$\frac{a_{end}}{a_{rh}}=\left(\frac{\rho_{rh}}{\rho_{end}}\right)^{1/3}$,
where $\rho_{end}\simeq V_{E}(0)$ and $\rho_{rh}\simeq 3\Gamma_{\phi}^{2}$ are
the energy densities of inflaton at $t_{end}$ and $\tau_{\phi}$, respectively,
and we have used the fact that the during the oscillation between $t_{end}$
and $\tau_{\phi}$ the inflaton potential is approximately quadratic, hence the
equation-of-state $w=0$;
$\frac{a_{rh}}{a_{eq}}=\left(\frac{\rho_{eq}}{\rho_{rh}}\right)^{1/4}\left(g_{*}^{eq}/g_{*}^{rh}\right)^{1/12}$,
where $g_{*}^{eq}\simeq 43/11$, $g_{*}^{rh}\simeq 100$, $\rho_{eq}=1.3\times
10^{-110}$ is the energy density at the matter-radiation equality ($t_{eq}$),
and we have used $w=1/3$ for radiation; $a_{eq}=1/(1+z_{eq})$, where
$z_{eq}=3387$ [52] is the redshift of the matter-radiation equality;
$H_{k}\simeq\sqrt{V_{E}(\phi_{k})/3}$, where $V_{E}(\phi_{k})$ is the value of
the inflaton potential evaluated at the pivot scale. Plugging all the ratios
we obtain the formula for e-folding number $N_{k}$:

$N_{k}\equiv\ln\frac{a_{end}}{a_{k}}\simeq
66.9-\ln\frac{k}{a_{0}H_{0}}+\frac{1}{12}\ln\frac{\rho_{rh}}{\rho_{end}}+\frac{1}{4}\ln\frac{V_{E}(\phi_{k})}{\rho_{end}}+\frac{1}{4}\ln
V_{E}(\phi_{k})+\frac{1}{12}\left(\ln g_{*}^{eq}-\ln g_{*}^{rh}\right).$ (4.2)

For the fixed value of $N_{k}=55$ and the pivot scale $k$, this equation
determines $\Gamma_{\phi}$.

Figure 1:  The plot shows $N_{k}$ as a function of $m_{N}$ for two pivot
scales, $k=0.05$ (diagonal solid line) and $k=0.002$ (diagonal dashed line).
The gray shaded region depicts $m_{N}>m_{\phi}/2$, which is excluded by
kinematics.

Let us discuss the decay modes for inflaton in the classically conformal
$U(1)_{B-L}$ model. Since the inflaton is identified with the $B-L$ Higgs
boson, there are two possible decay modes: inflaton decays to SM Higgs pair
through the mixed quartic coupling ($\lambda_{mix}$) in Eq. (2.2) and inflaton
decays to a pair of right-handed neutrinos through the Majorana Yukawa
couplings ($Y_{M}^{i}$) in Eq. (2.13). The partial decay width for $\phi\to
H^{\dagger}H$ is given by

$\displaystyle\Gamma_{\phi}(\phi\to H^{\dagger}H)\simeq\frac{\lambda_{\rm
mix}^{2}v_{\phi}^{2}}{8\pi
m_{\phi}}=\frac{m_{h}^{4}}{8\pi{\sqrt{\lambda}}v_{\phi}^{3}},$ (4.3)

where we have used $m_{\phi}^{2}=\lambda v_{\phi}^{2}$ and $\lambda_{\rm
mix}=m_{h}^{2}/v_{\phi}^{2}$. The partial decay width for $\phi\to
N_{R}^{i}N_{R}^{i}$ is given by

$\displaystyle\Gamma_{\phi}(\phi\to
N_{R}^{i}N_{R}^{i})\simeq\frac{(Y_{M}^{i})^{2}}{16\pi}m_{\phi}=\frac{\sqrt{\lambda}m_{N^{i}}^{2}}{8\pi
v_{\phi}},$ (4.4)

where we have used $m_{N^{i}}=Y_{M}^{i}v_{\phi}/\sqrt{2}$ and assumed
$m_{N^{i}}<m_{\phi}/2$ with $m_{\phi}\simeq 1.3\times 10^{14}$ GeV for our
benchmark777Note that the vacuum stability condition,
$m_{Z^{\prime}}>\left(\frac{2}{3}\right)^{1/4}\sum_{i=1}^{3}m_{N^{i}}$, sets a
lower bound on $Z^{\prime}$ boson mass, $m_{Z^{\prime}}\gtrsim 10^{11}$ GeV..
For a successful thermal leptogenesis we will discuss below, the lightest
right-handed neutrino mass $m_{N^{1}}\gtrsim 1$ TeV. We have assumed that
reheating occurs after the second inflation. This assumption leads to an upper
bound on the decay width, $\Gamma_{\phi}^{\rm max}<\sqrt{H(\phi\simeq
0)}=\sqrt{\frac{V_{E}(0)}{3}}$, which is satisfied for $m_{N^{i}}<m_{\phi}/2$.

Assuming the inflaton total decay width is dominated by its decay mode to a
single generation of right-handed neutrino pair, Eq. (4.2) relates $N_{k}$
with $m_{N^{i}}$ for a fixed pivot scale $k$. In Fig. 1, we show $N_{k}$ as a
function of $m_{N^{i}}$ for $k=0.05$ (diagonal solid line) and $k=0.002$
(diagonal dashed line). The gray shaded region depicting
$m_{N^{i}}>m_{\phi}/2$ is excluded. For the benchmark value $N_{k}=55$ (the
horizontal red line), we have found a solution $m_{N^{i}}=3.1\times 10^{11}$
GeV only for $k=0.002$, which corresponds to $\Gamma_{\phi}\simeq 0.55$ GeV.
The corresponding reheating temperature is estimated by
$T_{R}\simeq\left(\frac{90}{\pi^{2}g_{*}}\right)^{1/4}\sqrt{\Gamma_{\phi}(\phi\to
N_{R}^{i}N_{R}^{i})}$ which yields $T_{R}=6.4\times 10^{8}$ GeV. Although
$m_{N^{i}}>T_{R}$, this estimate is justified if $N_{R}^{i}$ decay width is
greater than the inflaton decay width, and created $N_{R}^{i}$’s instantly
decay to the SM particles. Here we calculate the partial decay width of
$N_{R}^{i}$ to the SM final states, $\Gamma(N_{R}^{i}\to H\ell_{L}^{j})$, and
its total total decay width is found to be

$\displaystyle\Gamma_{i}=\Sigma_{j}\Gamma(N_{R}^{i}\rightarrow\ell_{L}^{j}+H)=\frac{m_{N^{i}}}{8\pi}(Y_{D}Y_{D}^{\dagger})_{ii}\;=\frac{m_{N^{i}}^{2}}{4\pi
v_{h}^{2}}\left(OD_{\nu}O^{\dagger}\right)_{ii}\simeq\frac{1}{4\pi}\frac{m_{N_{i}}^{2}m_{i}}{v_{h}^{2}},$
(4.5)

where we have applied the seesaw formula Eq. (2.13) and assumed $O\simeq{\bf
1}_{3\times 3}$. Using $m_{N^{i}}=3.1\times 10^{11}$ GeV, we find that
$\Gamma_{i}/{\rm GeV}\simeq 1.3\times 10^{7}\left(\frac{m_{i}}{0.1\;{\rm
eV}}\right)\;{}\gg\Gamma_{\phi}/{\rm GeV}\simeq 0.55$.

##  5 Leptogenesis

We now consider the leptogenesis scenario from out-of-equilibrium $N$ decay
which generates lepton asymmetry. The lepton asymmetry is then converted to
baryon asymmetry by the so-called Sphaleron process [56, 57, 58]. The observed
baryon number asymmetry $n_{B}/s\simeq 10^{-10}$ [55, 52] originates from the
lepton number asymmetry,

$\displaystyle\frac{n_{B}}{s}\simeq\frac{n_{L}}{s},$ (5.1)

where $n_{B}$ ($n_{L}$) is the baryon (lepton) number density, and $s$ is the
entropy density. In the leptogensis, $n_{L}$ is controlled by the CP-asymmetry
parameter $\varepsilon$ [59]:

$\displaystyle\varepsilon_{i}$ $\displaystyle\equiv$
$\displaystyle\frac{\sum_{j}\left[\Gamma(N_{R}^{i}\rightarrow\ell_{L}^{j}+H)-\Gamma(N_{R}^{i}\rightarrow\overline{\ell_{L}^{j}}+H^{\dagger})\right]}{\sum_{j}\left[\Gamma(N_{R}^{i}\rightarrow\ell_{L}^{j}+H)+\Gamma(N_{R}^{i}\rightarrow\overline{\ell_{L}^{j}}+H^{\dagger})\right]}$
(5.2) $\displaystyle=$ $\displaystyle-\sum_{j\neq i}\frac{{\rm
Im}[(m_{D}m_{D}^{\dagger})^{2}_{ij}]}{(m_{D}m_{D}^{\dagger})_{ii}(m_{D}m_{D}^{\dagger})_{jj}}\frac{m_{N^{i}}}{m_{N^{j}}}\frac{\Gamma_{j}}{m_{N^{j}}}\left(\frac{1}{2}V_{j}+S_{j}\right),$

where

$\displaystyle V_{j}$ $\displaystyle=$ $\displaystyle
2\frac{m_{N^{j}}^{2}}{m_{N^{i}}^{2}}\left[\left(1+\frac{m_{N^{j}}^{2}}{m_{N^{i}}^{2}}\right)\ln\left(1+\frac{m_{N^{i}}^{2}}{m_{N^{j}}^{2}}\right)-1\right],$
(5.3) $\displaystyle S_{j}$ $\displaystyle=$
$\displaystyle\frac{m_{N^{j}}^{2}\Delta M_{ji}^{2}}{(\Delta
M_{ji}^{2})^{2}+m_{N^{i}}^{2}\Gamma_{j}^{2}},$ (5.4)

are contributions from the decay including the vertex corrections and self-
energy corrections, respectively, with $\Delta
M_{ji}^{2}=m_{N^{j}}^{2}-m_{N^{i}}^{2}$. We can see that $V_{j}\simeq
S_{j}\simeq 1$ if the right-handed neutrino mass spectrum is hierarchical. If
two right-handed neutrinos $N_{i}$ and $N_{j}$ are degenerate in mass but
$\Delta M_{ji}^{2}\gtrsim m_{N^{i}}\Gamma_{j}$, then $S_{j}\simeq
m_{N^{j}}^{2}/\Delta M_{ji}^{2}=\frac{1}{1-(m_{N^{j}}/m_{N^{i}})^{2}}\gg 1$ is
enhanced. By using the Eq. (2.25), we rewrite Eq. (5.2) as

$\displaystyle\varepsilon_{i}=-\sum_{j\neq i}\frac{{\rm
Im}[(OD_{\nu}O^{\dagger})^{2}_{ij}]}{(OD_{\nu}O^{\dagger})_{ii}}\frac{\left(\frac{1}{2}V_{j}+S_{j}\right)}{4\pi
v_{h}^{2}}\;m_{N^{i}}.$ (5.5)

Note that $\varepsilon_{i}$ is independent of the MNS matrix and is determined
by the orthogonal matrix $O$, and both light and heavy neutrino mass spectra.

There are two possibilities depending on whether (i) the lightest right-handed
neutrino is thermalized in the SM particle plasma or (ii) it has never been
thermalized. In case (i), there is a lower bound on the lightest right-handed
neutrino mass $m_{N^{1}}>10^{9-10}$ GeV [60, 61] for the successful
leptogenesis if the right-handed neutrino mass spectrum is hierarchical. In
thermal leptogenesis, if the washing-out effects are negligible, the resultant
baryon-to-entropy ration is given by $n_{B}/s\sim
n_{L}/s\sim\varepsilon_{1}/g_{*}\simeq 0.01\times\varepsilon_{1}$, which gives
a lower bound on $\varepsilon_{1}\gtrsim 10^{-8}$ to reproduce the observed
value $n_{B}/s\simeq 10^{-10}$. This condition corresponds to the lower bound
on $m_{N^{1}}>10^{9-10}$ GeV. Since $N_{R}^{1}$ is thermalized before it
decays, $T_{R}\gtrsim m_{N^{1}}$, the lower bound on $m_{N^{1}}>10^{9-10}$ GeV
is not satisfied in our case. However, if the lighter of the two right-handed
neutrinos are almost degenerate in mass, $S_{j}$ is enhanced as we have
discussed before, and as a result the CP asymmetry parameter $\varepsilon$ is
enhanced. Hence, a successful leptogenesis can still be realized even for
$m_{N^{1}}<10^{9-10}$ GeV. This scenario is known as resonant leptogenesis
[62, 63]. We can apply resonant leptogenesis by setting $m_{N^{1}}\simeq
m_{N^{2}}<T_{R}=6.4\times 10^{8}$ GeV and identifying
$m_{N^{3}}=m_{N}=3.1\times 10^{11}$ GeV with the heaviest right-handed
neutrino mass.

In case (ii), we can identify $m_{N^{1}}=m_{N}=3.1\times 10^{11}$ GeV with the
lightest right-handed neutrino while we set $m_{N^{2,3}}>m_{\phi}/2$. Since
$m_{N^{1}}>T_{R}$, $N_{R}^{1}$ has never been in thermal equilibrium with the
SM plasma. The universe is thermalized by $N_{R}^{1}$ decay to SM particles as
discussed before, and this decay also generates lepton asymmetry of the
universe. This scenario is called non-thermal leptogenesis [64, 65]. The
generated lepton number $n_{L}$ is estimated as

$\displaystyle n_{L}\sim n_{\phi}\times\varepsilon,$ (5.6)

where $n_{\phi}\sim\frac{\rho_{rh}}{m_{\phi}}$ is the number density of
inflaton when it decays. Using the entropy density
$s\sim\frac{\rho_{rh}}{T_{R}}$, we obtain
$\frac{n_{B}}{s}\sim\frac{T_{R}}{m_{\phi}}\;\varepsilon=5.1\times
10^{-6}\;\varepsilon$. Note that since $N_{R}^{1}$ is never in thermal
equilibrium, there is no washing-out effect. To reproduce the observed baryon
asymmetry, $n_{B}/s\simeq 10^{-10}$, we adjust $\varepsilon\sim 10^{-5}$. For
the hierarchical right-handed neutrino mass spectrum,

$\displaystyle\varepsilon_{1}\simeq-\frac{3m_{N^{1}}}{8\pi
v_{h}^{2}}\;\sum_{j=2}^{3}\frac{{\rm
Im}[(OD_{\nu}O^{\dagger})^{2}_{1j}]}{(OD_{\nu}O^{\dagger})_{11}}.$ (5.7)

For example, we find $\varepsilon_{1}=10^{-5}$ with $m_{lightest}\simeq
10^{-3}$ eV is satisfied for NH with $\omega_{12}\simeq+1.85\;i$,
$\omega_{23}=0$ and $\omega_{13}=\frac{\pi}{4}-2\;i$, while
$\omega_{12}\simeq+0.87\;i$, $\omega_{23}=0$ and
$\omega_{13}=\frac{\pi}{4}+2\;i$ for IH.

##  6 Conclusions and Discussions

The classically conformal minimal $B-L$ model is particularly interesting
because it can address (i) the origin of the electroweak symmetry breaking
triggered by the radiative $B-L$ symmetry breaking with the Coleman-Weinberg
potential; (ii) the origin of the neutrino mass through the seesaw mechanism
with the heavy right-handed Majorana neutrinos whose masses are generated by
the $B-L$ symmetry breaking; (iii) the origin of the baryon-asymmetry of the
universe through leptogenesis with the CP-asymmetric out-of-equilibrium decay
of the heavy neutrinos.

It has recently been shown in Ref. [1] that the double-inflation scenario
based on the Coleman-Weinberg potential can successfully generate PBHs, which
can be a dark matter candidate of the universe, while the inflationary
predictions are consistent with the Planck measurements. In this paper, we
have proposed a UV completion of the scenario discussed in Ref. [1] by the
classically conformal minimal $B-L$ model with the identification of the $B-L$
Higgs field to be the inflaton. We have shown this UV completion leads to a
viable history of the early universe after the double inflaton: the universe
is reheated via inflaton decay into a pair of right-handed neutrinos, and the
reheating temperature is determined by its relation to the number of e-folds,
and then the right-handed neutrino mass is fixed. With the determined values
of reheating temperature and right-handed neutrino mass, we have considered
baryogenesis via leptogenesis, and found that either thermal or non-thermal
leptogenesis can successfully reproduce the observed baryon asymmetry in the
universe consistently with the neutrino oscillation data.

Figure 2: The scalar-induced GW spectrum obtained by using the power spectrum
shown in Fig. 2 of Ref. [1], along with the current and future reach of
various GW observations.

Finally, we discuss the scalar-induced GWs from the enhanced power spectrum
associated with PBH production. Although in the first-order perturbation, the
scalar and tensor modes do not mix, the scalar modes can source tensor modes
in the second order of perturbation [66, 67, 68, 69, 70] (see Ref. [47] for a
review). As shown in Fig. 2 of Ref. [1], the second inflation remarkably
enhances the scalar perturbations and the power spectrum (${\cal
P}_{\zeta}$)888$\Delta_{\cal R}^{2}$ in Eq. (3.7) is defined as ${\cal
P}_{\zeta}$ evaluated at the pivot scale $k$. exhibits a sharp peak at $k_{\rm
peak}={\cal O(}10^{14})$ Mpc-1. The temperature of the universe when this
scale re-entered the horizon scale ($T_{ren}$) is estimated as
$T_{ren}\sim\frac{k_{\rm peak}}{2\pi}\frac{M_{P}}{T_{0}}\sim 10^{6}$ GeV,
where $T_{0}\simeq 2.7$ K is the present CMB temperature. Hence, the
fluctuations re-entered the horizon during the radiation-dominated era after
the reheating at $T_{R}=6.4\times 10^{8}$ GeV. In this case, the scalar-
induced GW spectrum is calculated by [71]

$\displaystyle\Omega_{\rm GW}h^{2}\approx 2.14\times
10^{-3}\frac{g_{*}}{106.75}\left(\frac{g_{*,s}}{106.75}\right)^{-\frac{4}{3}}\\!\\!\int_{-1}^{1}\\!\\!{\rm
d}x\\!\\!\int_{1}^{\infty}\\!\\!{\rm
d}y\,\mathcal{P}_{\zeta}\left(\frac{y-x}{2}k\right)\mathcal{P}_{\zeta}\left(\frac{x+y}{2}k\right)F(x,y)\,,$
(6.1)

where

$\displaystyle F(x,y)$ $\displaystyle=$
$\displaystyle\frac{(x^{2}\\!+\\!y^{2}\\!-\\!6)^{2}(x^{2}-1)^{2}(y^{2}-1)^{2}}{(x-y)^{8}(x+y)^{8}}$
$\displaystyle\times\left\\{\left(x^{2}-y^{2}+\frac{x^{2}\\!+\\!y^{2}\\!-\\!6}{2}\ln\left|\frac{y^{2}-3}{x^{2}-3}\right|\right)^{\\!2}+\frac{\pi^{2}(x^{2}\\!+\\!y^{2}\\!-\\!6)^{2}}{4}\theta(y-\sqrt{3})\right\\}.$

The resultant power spectrum from the double inflation for our benchmark is
shown in the right panel of Fig. 2 in Ref. [1]. Employing their power spectrum
as an input to Eq. (6.1), we calculate the scalar-induced GW spectrum. Our
result is shown in Fig. 2. The sharp peak of $P_{\zeta}$ at $k_{\rm
peak}={\cal O(}10^{14})$ Mpc-1 induces the remarkable enhancement of the GW
spectrum at $f\sim 1$ Hz. Interestingly, the predicted scalar-induced GW
spectrum well-overlaps with the search reach of the proposed/planned GW
observations, like LISA, DECIGO, BBO, SKA and ET [72, 73, 74, 75, 76].

## Acknowledgement

The work of N.O. is supported in part by the United States Department of
Energy Grant No. DE-SC0012447 and DE-SC0023713. AP thanks Indian Association
for the Cultivation of Science, Kolkata for financial support through Research
Associateship, where part of the work was done. AP also wishes to thank the
Indo-French Centre for the Promotion of Advanced Research for supporting the
postdoctoral fellowship through the proposal 6704-4 under the Collaborative
Scientific Research Programme.

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