KANAZAWA-21-06

May, 2021

Inflation connected to the origin of $CP$ violation

Tsuyoshi Hashimotoa111e-mail<EMAIL_ADDRESS>Norma Sidik
Risdiantoa,b222e-mail<EMAIL_ADDRESS>
and Daijiro Suematsua333e-mail<EMAIL_ADDRESS>
aInstitute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192,
Japan

bDepartment of Physics Education, Universitas Islam Negeri Sunan Kalijaga, Jl.
Marsda Adisucipto, 55280, Yogyakarta, Indonesia

Abstract
We consider a simple extension of the standard model, which could give a
solution for its $CP$ issues such as the origin of both CKM and PMNS phases
and the strong $CP$ problem. The model is extended with singlet scalars which
allow the introduction of the Peccei-Quinn symmetry and could cause
spontaneous $CP$ violation to result in these phases at low energy regions.
The singlet scalars could give a good inflaton candidate if they have a
suitable nonminimal coupling with the Ricci scalar. $CP$ issues and inflation
could be closely related through these singlet scalars in a natural way. In a
case where inflaton is a mixture of the singlet scalars, we study reheating
and leptogenesis as notable phenomena affected by the fields introduced in
this extension.

##  1 Introduction

$CP$ symmetry is a fundamental discrete symmetry which plays an important role
in particle physics. In the standard model (SM), it is considered to be
violated explicitly through complex Yukawa coupling constants [1] and a
$\theta$ parameter in the QCD sector [2]. The former is known to explain very
well $CP$ violating phenomena in $B$ meson systems and so on [3]. The latter
is severely constrained through an experimental search of a neutron electric
dipole moment [4] and causes the notorious strong $CP$ problem [5]. Peccei-
Quinn (PQ) symmetry has been proposed to solve it [6]. If we assume that the
$CP$ symmetry is an original symmetry of nature, the complex phases of Yukawa
coupling constants have to be spontaneously induced through some mechanism at
high energy regions. It may be compactification dynamics in string theory near
at the Planck scale [7]. In that case, since nonzero $\theta$ could be caused
through radiative effects after the $CP$ violation, the PQ symmetry is
required to solve the strong $CP$ problem again.

As an alternative scenario for the realization of $CP$ symmetry, we may
consider it to be exact so that all coupling constants including Yukawa
couplings are real and also $\theta=0$ is kept to some scale much lower than
the Planck scale. In that case, the $CP$ symmetry is supposed to be
spontaneously broken and this violation can be expected to be transformed to a
complex phase in the CKM matrix effectively. If nonzero $\theta$ is not
brought about in this process, it is favorable for the strong $CP$ problem.
The Nelson-Barr (NB) mechanism [8] has been proposed as such a concrete
example. Unfortunately, radiative effects could cause nonzero $\theta$ with a
magnitude which contradicts the experimental constraints [9].aaa Introduction
of the PQ symmetry could solve this fault of the model. We consider such a
possibility in the extension of the model. However, the scenario is
interesting since it can present an explanation for the origin of the $CP$
violation at a much lower energy region than the Planck scale. As a
realization of the NB mechanism, a simple model has been proposed in [10]. The
model is extended in [11] to the lepton sector where the existence of a $CP$
violating phase in the PMNS matrix [12] is suggested through recent neutrino
oscillation experiments [13].

Observations of the CMB fluctuation [14, 15] suggest the existence of the
exponential expansion of the universe called inflation. Inflation is usually
considered to be induced by some slowly rolling scalar field called inflaton
[16]. It is a crucial problem to identify its candidate from a viewpoint of
the extension of the SM. Although the Higgs scalar has been studied as an only
promising candidate in the SM [17] under an assumption that it has a
nonminimal coupling with the Ricci scalar curvature [18], several problems
have been pointed out [19, 20, 21]. In this situation, it is interesting to
find an alternative candidate for inflaton in a certain extension of the SM
which could solve several problems in the SM. In this sense, the model
extended from a viewpoint of the $CP$ symmetry as described above could give
such a promising candidate. It contains singlet scalars which cause the
spontaneous $CP$ violation and allow the introduction of the PQ symmetry as a
solution for the strong $CP$ problem. If they couple with the scalar curvature
nonminimally, it could cause slow-roll inflation successfully. In this paper,
we discuss such a possibility that the inflation of the universe could be
related to the $CP$ violation in the SM. We study reheating and leptogenesis
as its phenomenology caused by extra fields introduced in the model to solve
the $CP$ issues.

Remaining parts of the paper are organized as follows. In section 2, we
describe the model studied in this paper and discuss both phases in the CKM
and PMNS matrices which are derived as a result of the spontaneous $CP$
violation. In section 3, we discuss the inflation brought about by the singlet
scalars which are related to the $CP$ issues and the reheating. After that, we
describe leptogenesis which could show a distinguishable feature from the
usual leptogenesis in the seesaw model. The paper is summarized in section 4.

##  2 Origin of $CP$ violation

###  2.1 An extended model

Our model is an $CP$ invariant extension of the SM with global $U(1)\times
Z_{4}$ symmetry and several additional fields. As fermions, we introduce a
pair of vector-like down-type quarks $(D_{L},D_{R})$, a pair of vector-like
charged leptons $(E_{L},E_{R})$, and three right-handed singlet fermions
$N_{j}~{}(j=1,2,3)$.bbbSimilar models with vector-like extra fermions have
been considered under different symmetry structures [11, 22].  We also
introduce an additional doublet scalar $\eta$ and two singlet complex scalars
$\sigma$ and $S$. Their representation and charge under the symmetry
$[SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}]\times U(1)\times Z_{4}$ are
summarized in Table 1.ccc $Z_{4}$ is imposed by hand to control the couplings
of the new fields to the SM contents. The SM contents are assumed to have no
charge of the global symmetry. Since this global $U(1)$ has color anomaly in
the same way as the KSVZ model [23] for a strong $CP$ problem, it can play the
role of the PQ symmetry. The present charge assignment for colored fermions
guarantees the domain wall number to be one ($N_{\rm DW}=1$) so that the model
can escape the domain wall problem [24, 25].

| $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)$ | $Z_{4}$ |  | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)$ | $Z_{4}$
---|---|---|---|---|---|---|---|---|---|---|---
$D_{L}$ | 3 | 1 | $-\frac{1}{3}$ | 0 | 2 | $D_{R}$ | 3 | 1 | $-\frac{1}{3}$ | 2 | 0
$E_{L}$ | 1 | 1 | $-1$ | 0 | 2 | $E_{R}$ | 1 | 1 | $-1$ | 2 | 0
$\sigma$ | 1 | 1 | 0 | $-2$ | 2 | $S$ | 1 | 1 | $0$ | 0 | 2
$N_{k}$ | 1 | 1 | 0 | $1$ | 1 | $\eta$ | 1 | 2 | $-\frac{1}{2}$ | $-1$ | $-1$

Table 1 New fields added to the SM and their representation and charge under
$[SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}]\times U(1)\times Z_{4}$.

The model is characterized by new Yukawa terms and scalar potential which are
invariant under the imposed symmetry

$\displaystyle-{\cal L}_{Y}$ $\displaystyle=$ $\displaystyle
y_{D}\sigma\bar{D}_{L}D_{R}+y_{E}\sigma\bar{E}_{L}E_{R}+\sum_{j=1}^{3}\Big{(}\frac{y_{N_{j}}}{2}\sigma\bar{N}_{j}^{c}N_{j}+y_{d_{j}}S\bar{D}_{L}d_{R_{j}}+\tilde{y}_{d_{j}}S^{\dagger}\bar{D}_{L}d_{R_{j}}$
$\displaystyle+$ $\displaystyle
y_{e_{j}}S\bar{E}_{L}e_{R_{j}}+\tilde{y}_{e_{j}}S^{\dagger}\bar{E}_{L}e_{R_{j}}+\sum_{\alpha=1}^{3}h_{\alpha
j}^{\ast}\eta\bar{\ell}_{\alpha}N_{j}\Big{)}+\sum_{\alpha,\beta=1}^{3}\frac{y_{\alpha\beta}}{M_{\ast}}(\bar{\ell}_{\alpha}\phi)(\bar{\ell}_{\beta}\phi)+{\rm
h.c.},$ $\displaystyle V$ $\displaystyle=$
$\displaystyle\lambda_{1}(\phi^{\dagger}\phi)^{2}+\lambda_{2}(\eta^{\dagger}\eta)^{2}+\lambda_{3}(\phi^{\dagger}\phi)(\eta^{\dagger}\eta)+\lambda_{4}(\phi^{\dagger}\eta)(\eta^{\dagger}\phi)+\frac{\lambda_{5}}{2M_{\ast}}[\sigma(\eta^{\dagger}\phi)^{2}+{\rm
h.c.}]$ (1) $\displaystyle+$
$\displaystyle\kappa_{\sigma}(\sigma^{\dagger}\sigma)^{2}+\kappa_{S}(S^{\dagger}S)^{2}+(\kappa_{\phi\sigma}\phi^{\dagger}\phi+\kappa_{\eta\sigma}\eta^{\dagger}\eta)(\sigma^{\dagger}\sigma)+(\kappa_{\phi
S}\phi^{\dagger}\phi+\kappa_{\eta S}\eta^{\dagger}\eta)(S^{\dagger}S)$
$\displaystyle+$ $\displaystyle\kappa_{\sigma
S}(\sigma^{\dagger}\sigma)(S^{\dagger}S)+m_{\phi}^{2}\phi^{\dagger}\phi+m_{\eta}^{2}\eta^{\dagger}\eta+m_{\sigma}^{2}\sigma^{\dagger}\sigma+m_{S}^{2}S^{\dagger}S+V_{b},$

where $d_{R_{j}}$ and $e_{R_{j}}$ are the SM down-type quarks and charged
leptons, respectively. $\ell_{\alpha}$ is a doublet lepton and $\phi$ is an
ordinary doublet Higgs scalar. Since $CP$ invariance is assumed, parameters in
Lagrangian are considered to be all real. In eq. (1), we list dominant terms
up to dimension five and $M_{\ast}$ is a cut-off scale of the model. Other
invariant terms are higher order and can be safely neglected in comparison
with the listed ones. $V_{b}$ is composed of terms which are invariant under
the global symmetry but violate the $S$ number.

For a while, we focus on a part of field space where the field values of
$\sigma$ and $S$ are much larger than both $\phi$ and $\eta$ to study the
potential composed of $\sigma$ and $S$ only. In the present study, we assume
that $V_{b}$ takes a following form:ddd Imposed symmetry allows terms
$m_{S}^{\prime 2}(S^{2}+S^{\dagger 2})$ in $V_{b}$. However, we assume that
their contribution is negligible since we focus our attention on a potential
valley where $\tilde{\sigma}\propto\tilde{S}$ is satisfied and then $\cos
2\rho$ could be a constant at the minimum of $V_{b}$ in that case.

$V_{b}=\alpha(S^{4}+S^{\dagger
4})+\beta\sigma^{\dagger}\sigma(S^{2}+S^{\dagger
2})=\frac{1}{2}\tilde{S}^{2}(\alpha\tilde{S}^{2}\cos
4\rho+\beta\tilde{\sigma}^{2}\cos 2\rho),$ (2)

where we define $\sigma=\frac{\tilde{\sigma}}{\sqrt{2}}e^{i\theta}$ and
$S=\frac{\tilde{S}}{\sqrt{2}}e^{i\rho}$. Along the minimum of $V_{b}$ for
$\rho$ which is fixed by $\frac{\partial V_{b}}{\partial\rho}=0$, the
potential of $\tilde{\sigma}$ and $\tilde{S}$ can be written as

$V(\tilde{\sigma},\tilde{S})=\frac{\tilde{\kappa}_{\sigma}}{4}(\tilde{\sigma}^{2}-w^{2})^{2}+\frac{\tilde{\kappa}_{S}}{4}(\tilde{S}^{2}-u^{2})^{2}+\frac{\kappa_{\sigma
S}}{4}(\tilde{\sigma}^{2}-w^{2})(\tilde{S}^{2}-u^{2}),$ (3)

where $\tilde{\kappa}_{\sigma}$ and $\tilde{\kappa}_{S}$ are defined as

$\tilde{\kappa}_{\sigma}=\kappa_{\sigma}-\frac{\beta^{2}}{4\alpha},\qquad\tilde{\kappa}_{S}=\kappa_{S}-2\alpha,$
(4)

and $w$ and $u$ are the vacuum expectation values (VEVs) of $\tilde{\sigma}$
and $\tilde{S}$. They are supposed to be much larger than the weak scale. They
keep the gauge symmetry but break down the global symmetry $U(1)\times Z_{4}$
into its diagonal subgroup $Z_{2}$.eeeIt guarantees the stability of the
lightest $Z_{2}$ odd field, which could be a dark matter (DM) candidate as
discussed later. Since the minimum of $V_{b}$ can be determined by using these
VEVs as $\cos 2\rho=-\frac{\beta}{4\alpha}\frac{w^{2}}{u^{2}}$ as long as
$\left|\frac{\beta}{4\alpha}\frac{w^{2}}{u^{2}}\right|\leq 1$ is satisfied,
the $CP$ symmetry is spontaneously broken to result in a low energy effective
model with the $CP$ violation. On the other hand, $\theta=0$ is satisfied
because of the global $U(1)$ symmetry relevant to $\sigma$ [26]. The stability
condition for the potential (3) can be given as

$\tilde{\kappa}_{\sigma},\tilde{\kappa}_{S}>0,\qquad
4\tilde{\kappa}_{\sigma}\tilde{\kappa}_{S}>\kappa_{\sigma S}^{2}.$ (5)

If we consider the fluctuation of $\tilde{\sigma}$ and $\tilde{S}$ around the
vacua $\langle\tilde{\sigma}\rangle$ and $\langle\tilde{S}\rangle$, mass
eigenstates are the mixture of them in general. If we take account of the
stability condition (5), mass eigenvalues can be approximately expressed as

$\displaystyle m^{2}_{\tilde{S}}\simeq
2\left(\tilde{\kappa}_{S}-\frac{\kappa_{\sigma
S}^{2}}{4\tilde{\kappa}_{\sigma}}\right)u^{2}\equiv
2\hat{\kappa}_{S}u^{2},\quad m^{2}_{\tilde{\sigma}}\simeq
2\tilde{\kappa}_{\sigma}w^{2}\qquad{\rm
for}~{}~{}\tilde{\kappa}_{\sigma}^{2}w^{2}\gg\tilde{\kappa}_{S}^{2}u^{2},$
$\displaystyle m_{\tilde{S}}^{2}\simeq 2\tilde{\kappa}_{S}u^{2},\quad
m^{2}_{\tilde{\sigma}}\simeq
2\left(\tilde{\kappa}_{\sigma}-\frac{\kappa_{\sigma
S}^{2}}{4\tilde{\kappa}_{S}}\right)w^{2}\equiv
2\hat{\kappa}_{\sigma}w^{2}\qquad{\rm
for}~{}~{}\tilde{\kappa}_{S}^{2}u^{2}\gg\tilde{\kappa}_{\sigma}^{2}w^{2}.$ (6)

Although they have a tiny subcomponent in these cases, a dominant component of
their eigenstates is $\tilde{S}$ and $\tilde{\sigma}$, respectively. The mass
of an orthogonal component to $\tilde{S}$ is found to be
$m^{2}_{S_{\perp}}=8\alpha u^{2}\left(1-\cos^{2}2\rho\right)$. Since the
global $U(1)$ symmetry works as the PQ symmetry mentioned above, and the axion
decay constant is given as $f_{a}=w$, the VEV $w$ should satisfy the following
condition [3, 27]:

$4\times 10^{8}~{}{\rm
GeV}~{}{{}^{<}_{\sim}}~{}w~{}{{}^{<}_{\sim}}~{}10^{11}{\rm GeV}.$ (7)

The NG-boson caused by the spontaneous breaking of this $U(1)$ becomes axion
[28] which is characterized by a coupling with photons [29]

$g_{a\gamma\gamma}=\frac{1.51}{10^{10}~{}{\rm GeV}}\left(\frac{m_{a}}{\rm
eV}\right).$ (8)

In the next part, we show that the effective model after the symmetry breaking
can have $CP$ phases in the CKM and PMNS matrices. They are induced by the
mass matrices for the down type quarks and the charged leptons through a
similar mechanism which has been discussed in [10] as a simple realization of
the NB mechanism [8] for the strong $CP$ problem.

###  2.2 $CP$ violating phases in CKM and PMNS matrices

Yukawa couplings of down-type quarks and charged leptons given in eq. (1)
derive mass terms as

$(\bar{f}_{Li},\bar{F}_{L}){\cal M}_{f}\left(\begin{array}[]{c}f_{R_{j}}\\\
F_{R}\\\ \end{array}\right)+{\rm h.c.},\qquad{\cal
M}_{f}=\left(\begin{array}[]{cc}m_{f_{ij}}&0\\\ {\cal F}_{f_{j}}&\mu_{F}\\\
\end{array}\right),$ (9)

where $f$ and $F$ represent $f=d,e$ and $F=D,E$ for down-type quarks and
charged leptons and ${\cal M}_{f}$ is a $4\times 4$ matrix. Each component of
${\cal M}_{f}$ is expressed as
$m_{f_{ij}}=h_{f_{ij}}\langle\tilde{\phi}\rangle$, ${\cal
F}_{f_{j}}=(y_{f_{j}}ue^{i\rho}+\tilde{y}_{f_{j}}ue^{-i\rho})$ and
$\mu_{F}=y_{F}w$. This mass matrix is found to have the same form proposed in
[10]. Since the global $U(1)$ symmetry works as the PQ symmetry and all
parameters in the model are assumed to be real, ${\rm arg}({\rm det}{\cal
M}_{f})=0$ is satisfied even if radiative effects are taken into account after
the spontaneous breaking of the $CP$ symmetry [11].

We consider the diagonalization of a matrix ${\cal M}_{f}{\cal
M}_{f}^{\dagger}$ by a unitary matrix

$\left(\begin{array}[]{cc}A_{f}&B_{f}\\\ C_{f}&D_{f}\\\
\end{array}\right)\left(\begin{array}[]{cc}m_{f}m_{f}^{\dagger}&m_{f}{\cal
F}_{f}^{\dagger}\\\ {\cal F}_{f}m_{f}^{\dagger}&\mu_{F}^{2}+{\cal F}_{f}{\cal
F}_{f}^{\dagger}\\\
\end{array}\right)\left(\begin{array}[]{cc}A_{f}^{\dagger}&C_{f}^{\dagger}\\\
B_{f}^{\dagger}&D_{f}^{\dagger}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\tilde{m}^{2}_{f}&0\\\
0&\tilde{M}^{2}_{F}\\\ \end{array}\right),$ (10)

where $\tilde{m}^{2}_{f}$ is a $3\times 3$ diagonal matrix in which generation
indices are not explicitly written. Eq. (10) requires

$\displaystyle
m_{f}m_{f}^{\dagger}=A_{f}^{\dagger}\tilde{m}^{2}_{f}A_{f}+C_{f}^{\dagger}\tilde{M}^{2}_{F}C_{f},\qquad{\cal
F}_{f}m_{f}^{\dagger}=B_{f}^{\dagger}m^{2}A_{f}+D_{f}^{\dagger}\tilde{M}^{2}_{F}C_{f},$
$\displaystyle\mu_{F}^{2}+{\cal F}_{f}{\cal
F}_{f}^{\dagger}=B_{f}^{\dagger}\tilde{m}^{2}_{f}B_{f}+D_{f}^{\dagger}\tilde{M}^{2}_{F}D_{f}.$
(11)

If $\mu_{F}^{2}+{\cal F}_{f}{\cal F}_{f}^{\dagger}$ is much larger than each
component of ${\cal F}_{f}m_{f}^{\dagger}$, which can be realized in the case
$u,w\gg\langle\phi\rangle$, we find that $B_{f},C_{f}$, and $D_{f}$ are
approximately given as

$B_{f}\simeq-\frac{A_{f}m_{f}{\cal F}_{f}^{\dagger}}{\mu_{F}^{2}+{\cal
F}_{f}{\cal F}_{f}^{\dagger}},\qquad C_{f}\simeq\frac{{\cal
F}_{f}m_{f}^{\dagger}}{\mu_{F}^{2}+{\cal F}_{f}{\cal F}_{f}^{\dagger}},\qquad
D_{f}\simeq 1.$ (12)

These guarantee the unitarity of the matrix $A_{f}$ within the present
experimental bound [3] since $|B_{f}|,|C_{f}|\sim\frac{|m_{f}|}{|{\cal
F}_{f}|}<O(10^{-7})$ is satisfied in each component. In such a case, it is
easy to find

$A_{f}^{-1}\tilde{m}_{f}^{2}A_{f}\simeq
m_{f}m_{f}^{\dagger}-\frac{1}{\mu_{F}^{2}+{\cal F}_{f}{\cal
F}_{f}^{\dagger}}(m_{f}{\cal F}_{f}^{\dagger})({\cal
F}_{f}m_{f}^{\dagger}),\quad\tilde{M}_{F}^{2}\simeq\mu_{F}^{2}+{\cal
F}_{f}{\cal F}_{f}^{\dagger}.$ (13)

The right-hand side of the first equation corresponds to an effective squared
mass matrix of the ordinary fermions $f$. It is derived through the mixing
with the extra heavy fermions $F$. Since its second term can have complex
phases in off-diagonal components as long as $y_{f_{i}}\not=\tilde{y}_{f_{i}}$
is satisfied, the matrix $A_{f}$ could be complex. Moreover, if
$\mu_{F}^{2}~{}{{}^{<}_{\sim}}~{}{\cal F}_{f}{\cal F}_{f}^{\dagger}$ is
realized, the complex phase of $A_{f}$ in eq. (13) could have a substantial
magnitude because the second term in the right-hand side has a comparable
magnitude with the first one. Although it can be realized for various
parameter settings, we consider a rather simple situation here,fffLeptogenesis
could depend on the strength of these couplings heavily in this model as
studied later.

$\langle\phi\rangle\ll w<u,\qquad y_{f_{j}}\sim\tilde{y}_{f_{j}}<y_{F}.$ (14)

Since the masses of vector-like fermions are expected to be of $O(10^{8})$ GeV
or larger in this case, they decouple from the SM and do not contribute to
phenomena at TeV regions.

The CKM matrix is determined as $V_{\rm CKM}={O_{u}}^{T}A_{d}$, where $O_{u}$
is an orthogonal matrix used for the diagonalization of a mass matrix for up-
type quarks. Thus, the $CP$ phase of $V_{\rm CKM}$ is caused by the one of
$A_{d}$. The same argument is applied to the leptonic sector and the PMNS
matrix is derived as $V_{\rm PMNS}=A_{e}^{\dagger}U_{\nu}$, where $U_{\nu}$ is
an orthogonal matrix used for the diagonalization of a neutrino mass matrix.
The Dirac $CP$ phase in the CKM matrix and the PMNS matrix can be induced from
the same origin of $CP$ violation. A concrete example of $A_{d}$ is given for
a simple case in Appendix A.

###  2.3 Effective model at a lower energy region

An effective model at lower energy regions than $w$ and $u$ can be obtained by
integrating out the heavy fields. It is reduced to the SM with a lepton sector
extended as the scotogenic neutrino mass model [30], which is characterized by
the terms invariant under the remaining $Z_{2}$ symmetry

$\displaystyle-{\cal L}_{\rm scot}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{3}\left[\sum_{j=1}^{3}\left(\tilde{h}_{\alpha
j}^{\ast}\bar{\ell}_{\alpha}\eta
N_{j}+\frac{M_{N_{j}}}{2}\bar{N}_{j}^{c}N_{j}\right)+\sum_{\beta=1}^{3}\frac{y_{\alpha\beta}}{M_{\ast}}(\bar{\ell}_{\alpha}\phi)(\bar{\ell}_{\beta}\phi)+{\rm
h.c.}\right]$ (15) $\displaystyle+$
$\displaystyle\tilde{m}_{\phi}^{2}\phi^{\dagger}\phi+\tilde{m}_{\eta}^{2}\eta^{\dagger}\eta+\tilde{\lambda}_{1}(\phi^{\dagger}\phi)^{2}+\tilde{\lambda}_{2}(\eta^{\dagger}\eta)^{2}+\tilde{\lambda}_{3}(\phi^{\dagger}\phi)(\eta^{\dagger}\eta)+\lambda_{4}(\phi^{\dagger}\eta)(\eta^{\dagger}\phi)$
$\displaystyle+$
$\displaystyle\frac{\tilde{\lambda}_{5}}{2}\left[(\phi^{\dagger}\eta)^{2}+{\rm
h.c.}\right],$

where neutrino Yukawa couplings $\tilde{h}_{\alpha j}$ are defined on the
basis for which the mass matrix of the charged leptons is diagonalized as
discussed in the previous part. Thus, they are complex now. After the
spontaneous breaking due to the VEVs of $\tilde{\sigma}$ and $\tilde{S}$,
coupling constants in eq. (15) are related to the ones contained in eq. (1) as

$\displaystyle\tilde{\lambda}_{1}=\lambda_{1}-\frac{\kappa_{\phi\sigma}^{2}}{4\tilde{\kappa}_{\sigma}}-\frac{\kappa_{\phi
S}^{2}}{4\tilde{\kappa}_{S}}+\frac{\kappa_{\sigma
S}\kappa_{\phi\sigma}\kappa_{\phi
S}}{4\tilde{\kappa}_{\sigma}\tilde{\kappa}_{S}},\qquad\tilde{\lambda}_{2}=\lambda_{2}-\frac{\kappa_{\eta\sigma}^{2}}{4\tilde{\kappa}_{\sigma}}-\frac{\kappa_{\eta
S}^{2}}{4\tilde{\kappa}_{S}}+\frac{\kappa_{\sigma
S}\kappa_{\eta\sigma}\kappa_{\eta
S}}{4\tilde{\kappa}_{\sigma}\tilde{\kappa}_{S}},$
$\displaystyle\tilde{\lambda}_{3}=\lambda_{3}-\frac{\kappa_{\phi\sigma}\kappa_{\eta\sigma}}{2\tilde{\kappa}_{\sigma}}-\frac{\kappa_{\phi
S}\kappa_{\eta S}}{2\tilde{\kappa}_{S}}+\frac{\kappa_{\sigma
S}\kappa_{\phi\sigma}\kappa_{\eta S}+\kappa_{\sigma
S}\kappa_{\eta\sigma}\kappa_{\phi
S}}{4\tilde{\kappa}_{\sigma}\tilde{\kappa}_{S}},\qquad\tilde{\lambda}_{5}=\lambda_{5}\frac{w}{M_{\ast}}.$
(16)

These connection conditions should be imposed at a certain scale $\bar{M}$,
which is taken to be $\bar{M}=\tilde{M}_{F}$ in the present study. Stability
of the potential (15) requires the following conditions to be satisfied
through scales $\mu<\bar{M}$:

$\tilde{\lambda}_{1},~{}\tilde{\lambda}_{2}>0,\qquad\tilde{\lambda}_{3},~{}\tilde{\lambda}_{3}+\lambda_{4}-|\tilde{\lambda}_{5}|>-2\sqrt{\tilde{\lambda}_{1}\tilde{\lambda}_{2}}.$
(17)

Potential stability (5) and (17) and perturbativity of the model from the weak
scale to the Planck scale can be examined by using renormalization group
equations (RGEs) for the coupling constants. Relevant RGEs at $\mu>\bar{M}$
are given in Appendix B. The mass parameters in eq. (15) are represented as

$\displaystyle M_{N_{j}}=y_{N_{j}}w,$
$\displaystyle\tilde{m}_{\phi}^{2}=m_{\phi}^{2}+\left(\kappa_{\phi\sigma}+\frac{\kappa_{\phi
S}\kappa_{\sigma S}}{2\tilde{\kappa}_{S}}\right)w^{2}+\left(\kappa_{\phi
S}+\frac{\kappa_{\phi\sigma}\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}\right)u^{2},$
$\displaystyle\tilde{m}_{\eta}^{2}=m_{\eta}^{2}+\left(\kappa_{\eta\sigma}+\frac{\kappa_{\eta
S}\kappa_{\sigma S}}{2\tilde{\kappa}_{S}}\right)w^{2}+\left(\kappa_{\eta
S}+\frac{\kappa_{\eta\sigma}\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}\right)u^{2}.$ (18)

If $\tilde{m}_{\eta}^{2}>0$ is satisfied and $\eta$ has no VEV, $Z_{2}$ is
kept as an exact symmetry of the model. In this model, we assume both
$|\tilde{m}_{\phi}|$ and $\tilde{m}_{\eta}$ have values of $O(1)$ TeV. Since
it has to be realized under the contributions from the VEVs $w$ and $u$,
parameter tunings are required.gggFor parameter values assumed in this
analysis, the order of required tuning is estimated as $O(10^{-10})$.

Fig. 1 Left: contours of $\Omega h^{2}=0.12$ are plotted in the
$(\lambda_{+},\tilde{\lambda}_{3})$ plane by a solid line for
$M_{\eta_{R}}=0.9$ (green), 1 (red) and 1.1 (blue) in a TeV unit. Since
$\eta_{R}$ should be lighter than the charged components, $\lambda_{4}<0$
should be satisfied which corresponds to a region above the diagonal black
solid line. Direct search bound from Xenon1T [34] is shown by the same color
dashed line for each $M_{\eta_{R}}$. Stability conditions (17) restrict the
allowed region to the area marked in the upper right quadrant (marked off by
the dot-dashed lines), for which $\tilde{\lambda}_{2}=0.1$ is assumed. Right:
an example of the running of coupling constants $\tilde{\lambda}_{i}$ for
$M=10^{\mu}~{}{\rm GeV}$. Initial values of $\tilde{\lambda}_{3}$ and
$\lambda_{4}$ are fixed as $\tilde{\lambda}_{3}=0.445$ and
$\lambda_{4}=-0.545$, which are included in the allowed region of the left
panel for $M_{\eta_{R}}=1$ TeV.
$\tilde{\lambda}_{3}+2\sqrt{\tilde{\lambda}_{1}\tilde{\lambda}_{2}}$ and
$\lambda_{+}+2\sqrt{\tilde{\lambda}_{1}\tilde{\lambda}_{2}}$ are also plotted
by red and blue lines, respectively.

Phenomenology on neutrinos and DM could be the same as the one which has been
studied extensively in various studies [31, 32] unless the axion is a dominant
component of DM. If the lightest neutral component of $\eta$ is DM which is
identified as its real component $\eta_{R}$, both DM relic abundance and DM
direct search constrain the parameters $\tilde{\lambda}_{3}$ and $\lambda_{4}$
[22, 33]. As a reference, in Fig. 1 we show their required values in the
$(\lambda_{+},\tilde{\lambda}_{3})$ plane for the cases $M_{\eta_{R}}=$0.9, 1
and 1.1 TeV where
$\lambda_{+}=\tilde{\lambda}_{3}+\lambda_{4}+\tilde{\lambda}_{5}$ and
$M_{\eta_{R}}^{2}=m_{\phi}^{2}+\lambda_{+}\langle\phi\rangle^{2}$. They should
be also consistent with the stability condition (17). The figure shows that
these could be satisfied with rather restricted values of
$\tilde{\lambda}_{3}$ and $\lambda_{4}$. A perturbativity requirement at
$\mu>\bar{M}$ also constrains the model strongly since DM relic abundance
requires $\tilde{\lambda}_{3}$ and $|\lambda_{4}|$ to take rather large
values. We have to take account of it to consider the model at high energy
regions. As an example, we show the result of RGE study in the right panel of
Fig. 1. Details of the assumed initial values are given in Appendix B. If the
initial values of $\tilde{\lambda}_{3}$ and $\lambda_{4}$ are chosen at a
larger $\tilde{\lambda}_{3}$ region in the left panel, the perturbatibity
condition is violated at a much lower region than the Planck scale. We do not
plot the behavior of $\kappa_{i}$ in this figure since their values are fixed
to be very small and they do not change their values substantially. If they
satisfy the condition (5), it can be confirmed to be kept up to the Planck
scale.

Neutrino mass is forbidden at tree level due to this $Z_{2}$ symmetry except
for the ones generated through dimension five Weinberg operators in eq. (15).
They could give a substantial contribution to the neutrino masses depending on
the cut-off scale $M_{\ast}$ and coupling constants $y_{\alpha\beta}$. In the
present study, however, we assume that their contribution is negligible and
the dominant contribution comes from one-loop diagrams with $\eta$ and $N_{j}$
in internal lines. Its formula is given as

${\cal M}^{\nu}_{\alpha\beta}\simeq\sum_{j=1}^{3}\tilde{h}_{\alpha
j}\tilde{h}_{\beta
j}\tilde{\lambda}_{5}\Lambda_{j},\qquad\Lambda_{j}=\frac{\langle\phi\rangle^{2}}{8\pi^{2}}\frac{1}{M_{N_{j}}}\ln\frac{M_{N_{j}}^{2}}{M_{\eta}^{2}},$
(19)

where
$M_{\eta}^{2}=\tilde{m}_{\eta}^{2}+(\tilde{\lambda}_{3}+\lambda_{4})\langle\phi\rangle^{2}$
and $M_{N_{j}}\gg M_{\eta}$ is supposed. As an example, one may assume a
simple flavour structure for neutrino Yukawa couplings [35]

$\tilde{h}_{ei}=0,~{}\tilde{h}_{\mu i}=\tilde{h}_{\tau i}\equiv
h_{i}~{}~{}(i=1,~{}2);\quad\tilde{h}_{e3}=\tilde{h}_{\mu 3}=-\tilde{h}_{\tau
3}\equiv h_{3}.$ (20)

This realizes a tri-bimaximal mixing which gives a simple and rather good 0-th
order approximation for the analysis of neutrino oscillation data and
leptogenesis [32]. If we impose the mass eigenvalues obtained from eq. (19)
for the case $|h_{1}|\ll|h_{2}|,|h_{3}|$ to satisfy the squared mass
differences required by the neutrino oscillation data, we find

$|h_{2}^{2}\tilde{\lambda}_{5}|\Lambda_{2}=\frac{1}{2}\sqrt{\Delta
m_{32}^{2}},\qquad|h_{3}^{2}\tilde{\lambda}_{5}|\Lambda_{3}=\frac{1}{3}\sqrt{\Delta
m_{21}^{2}}.$ (21)

Since we have $\Lambda_{2,3}\sim 7\times 10^{5}$ eV for $M_{2,3}\sim 10^{7}$
GeV and $M_{\eta}\sim 10^{3}$ GeV, the neutrino oscillation data [3] can be
explained by taking as an example

$y_{N_{j}}\sim 10^{-2},\quad|h_{2}|\sim 6\times 10^{-3},\quad|h_{3}|\sim
2\times 10^{-3},\quad|\tilde{\lambda}_{5}|\sim 10^{-3}.$ (22)

Even if we impose the neutrino oscillation data, $h_{1}$ can take a very small
value compared with $h_{2,3}$ [32]. It can play a crucial role for low scale
leptogenesis as seen later.

##  3 Inflation due to singlet scalars

###  3.1 Inflation

It is well known that a scalar field which couples nonminimally with the Ricci
scalar can cause inflation of the universe and the idea has been applied to
the Higgs scalar in the SM [17] and its singlet scalar extensions [36, 37]. If
the singlet scalars $S$ and $\sigma$, which are related to the $CP$ issues in
the SM couple with the Ricci scalar, it can play the role of inflaton in this
model. The action relevant to the inflation is given in the Jordan frame as

$\displaystyle S_{J}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}M_{\rm
pl}^{2}R-\xi_{\sigma}\sigma^{\dagger}\sigma
R-\xi_{S1}S^{\dagger}SR-\frac{\xi_{S2}}{2}(S^{2}+S^{\dagger 2})R\right.$ (23)
$\displaystyle+$
$\displaystyle\left.\partial^{\mu}\sigma^{\dagger}\partial_{\mu}\sigma+\partial^{\mu}S^{\dagger}\partial_{\mu}S-V(\sigma,S)\right],$

where $M_{\rm pl}$ is the reduced Planck mass and the coupling of $S$ is
controlled by the $Z_{4}$ symmetry. $V(\sigma,S)$ stands for the corresponding
part in the potential (1). Since inflation follows very complicated dynamics
if multi scalars contribute to it, we confine our study to the inflation in a
potential valley. Moreover, we assume $\xi_{S1}=-\xi_{S2}$ is satisfied for
simplicity. In that case, the coupling of $S$ with the Ricci scalar is reduced
to $\frac{1}{2}\xi_{S}S_{I}^{2}R$, where $S=\frac{1}{\sqrt{2}}(S_{R}+iS_{I})$
and $\xi_{S}=\xi_{S1}-\xi_{S2}$. If $S$ is supposed to evolve along a constant
$\rho$, which is determined as a potential minimum $\frac{\partial
V_{b}}{\partial\rho}=0$, the radial component $\tilde{S}$ couples with the
Ricci scalar as $\frac{1}{2}\tilde{\xi}_{S}\tilde{S}^{2}R$, where
$\tilde{\xi}_{S}$ is defined as $\tilde{\xi}_{S}=\xi_{S}\sin^{2}\rho$ and the
potential $V(\sigma,S)$ is expressed by eq.(3). Here we consider cases such
that both $\xi_{\sigma}$ and $\tilde{\xi}_{S}$ are positive only. Stability of
this potential requires the condition given in eq. (5). We neglect the VEVs
$w$ and $u$ for a while since they are much smaller than $O(M_{\rm pl})$,
which is the field values of $\tilde{\sigma}$ and $\tilde{S}$ during the
inflation. We also suppose that other scalars have much smaller values than
them.

We consider the conformal transformation for a metric tensor in the Jordan
frame

$\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu},\qquad\Omega^{2}=1+\frac{(\xi_{\sigma}\tilde{\sigma}^{2}+\tilde{\xi}_{S}\tilde{S}^{2})}{M_{\rm
pl}^{2}}.$ (24)

After this transformation to the Einstein frame where the Ricci scalar term
takes a canonical form, the action can be written as

$\displaystyle S_{E}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-\tilde{g}}\Big{[}-\frac{1}{2}M_{\rm
pl}^{2}\tilde{R}+\frac{1}{2}\partial^{\mu}\chi_{\sigma}\partial_{\mu}\chi_{\sigma}+\frac{1}{2}\partial^{\mu}\chi_{S}\partial_{\mu}\chi_{S}$
(25) $\displaystyle+$
$\displaystyle\frac{6\xi_{\sigma}\tilde{\xi}_{S}\frac{\tilde{\sigma}\tilde{S}}{M_{\rm
pl}^{2}}}{\left[(\Omega^{2}+\frac{6\xi_{\sigma}^{2}}{M_{\rm
pl}^{2}}\tilde{\sigma}^{2})(\Omega^{2}+\frac{6\tilde{\xi}_{S}^{2}}{M_{\rm
pl}^{2}}\tilde{S}^{2})\right]^{1/2}}\partial^{\mu}\chi_{\sigma}\partial_{\mu}\chi_{S}-\frac{1}{\Omega^{4}}V(\tilde{\sigma},\tilde{S})\Big{]},$

where $\chi_{\sigma}$ and $\chi_{S}$ are defined as [36]

$\frac{\partial\chi_{\sigma}}{\partial\tilde{\sigma}}=\frac{1}{\Omega^{2}}\sqrt{\Omega^{2}+6\xi_{\sigma}^{2}\frac{\tilde{\sigma}^{2}}{M_{\rm
pl}^{2}}},\qquad\frac{\partial\chi_{S}}{\partial\tilde{S}}=\frac{1}{\Omega^{2}}\sqrt{\Omega^{2}+6\tilde{\xi}_{S}^{2}\frac{\tilde{S}^{2}}{M_{\rm
pl}^{2}}}.$ (26)

If we introduce variables $\tilde{\chi},~{}\varphi$ to express
$\tilde{\sigma}$ and $\tilde{S}$ as
$\tilde{\sigma}=\tilde{\chi}\cos\varphi,~{}\tilde{S}=\tilde{\chi}\sin\varphi$,
the potential in the Einstein frame at the large field regions such as
$\xi_{\sigma}\tilde{\sigma}^{2}+\tilde{\xi}_{S}\tilde{S}^{2}\gg M_{\rm
pl}^{2}$ can be written as

$V(\tilde{\chi},\varphi)=\frac{M_{\rm
pl}^{4}}{4}\frac{\tilde{\kappa}_{S}\sin^{4}\varphi+\tilde{\kappa}_{\sigma}\cos^{4}\varphi+\kappa_{\sigma
S}\sin^{2}\varphi\cos^{2}\varphi}{(\xi_{\sigma}\cos^{2}\varphi+\tilde{\xi}_{S}\sin^{2}\varphi)^{2}}.$
(27)

We find that there are three types of valley along the minimum in the
$\varphi$ direction of this potential. They realize different types of
inflaton. Two of them are

${\rm(i)}~{}\varphi=0~{}~{}{\rm
for}~{}~{}2\tilde{\kappa}_{\sigma}\tilde{\xi}_{S}<\kappa_{\sigma
S}\xi_{\sigma},\qquad{\rm(ii)}~{}\varphi=\frac{\pi}{2}~{}~{}{\rm
for}~{}~{}2\tilde{\kappa}_{S}\xi_{\sigma}<\kappa_{\sigma S}\tilde{\xi}_{S}.$
(28)

In each case, a kinetic term mixing between $\chi_{\sigma}$ and $\chi_{S}$
disappears and inflaton can be identified with $\chi_{\sigma}$ for (i) and
$\chi_{S}$ for (ii), respectively.hhhIn different context, the inflaton
dominated by $\tilde{\sigma}$ and the $\tilde{S}$ inflaton have been discussed
in [38] and [11], respectively.

Another valley which is studied in this paper is realized at

$\sin^{2}\varphi=\frac{2\tilde{\kappa}_{\sigma}\tilde{\xi}_{S}-\kappa_{\sigma
S}\xi_{\sigma}}{(2\tilde{\kappa}_{S}\xi_{\sigma}-\kappa_{\sigma
S}\tilde{\xi}_{S})+(2\tilde{\kappa}_{\sigma}\tilde{\xi}_{S}-\kappa_{\sigma
S}\xi_{\sigma})},$ (29)

under the conditions

$2\tilde{\kappa}_{\sigma}\tilde{\xi}_{S}>\kappa_{\sigma S}\xi_{\sigma},\quad
2\tilde{\kappa}_{S}\xi_{\sigma}>\kappa_{\sigma S}\tilde{\xi}_{S},$ (30)

where we note that these are automatically satisfied for $\kappa_{\sigma S}<0$
since eq. (5) is imposed, and $\tilde{\xi}_{S}$ and $\xi_{\sigma}$ are assumed
to be positive. In this case the inflaton $\tilde{\chi}$ is a mixture of
$\tilde{\sigma}$ and $\tilde{S}$ with a constant value of
$\tilde{\sigma}/\tilde{S}$. Although the kinetic term mixing cannot be
neglected for a general $\sin\varphi$, it can be safely neglected if we
restrict it to the one in which the inflaton is dominated by
$\tilde{S}~{}(\sin^{2}\varphi\simeq 1)$ or
$\tilde{\sigma}~{}(\sin^{2}\varphi\simeq 0)$. We focus our study on the former
case where $\tilde{\chi}\gg\tilde{\sigma}$ is always satisfied during
inflation. If we additionally impose $\tilde{\xi}_{S}\gg\xi_{\sigma}$ on eq.
(29) and assume that the relevant couplings satisfyiiiIf we assume
$\tilde{\kappa}_{S}>|\kappa_{\sigma S}|$, $\sin\varphi$ is differently
expressed as
$\sin^{2}\varphi=1-\frac{\tilde{\kappa}_{S}\xi_{\sigma}}{\tilde{\kappa}_{\sigma}\tilde{\xi}_{S}}$.
However, we do not consider such a case in this paper.

$\kappa_{\sigma S}<0,\qquad\tilde{\kappa}_{S}<|\kappa_{\sigma
S}|<\tilde{\kappa}_{\sigma},$ (31)

$\sin\varphi$ is expressed as $\sin^{2}\varphi=1+\frac{\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}$. In this case, by using
$\hat{\kappa}_{S}=\tilde{\kappa}_{S}-\frac{\kappa_{\sigma
S}^{2}}{4\tilde{\kappa}_{\sigma}}$ potential can be expressed as
$V(\tilde{S})=\frac{1}{4\Omega^{4}}\hat{\kappa}_{S}\tilde{S}^{4}$ at the
bottom of the valley and $\cos 2\rho=-\frac{\beta}{4\alpha}\cot^{2}\varphi$
realizes the minimum of $V_{b}$ if
$\tan^{2}\varphi>\left|\frac{\beta}{4\alpha}\right|$ is satisfied. Nature of
the inflaton $\tilde{\chi}$ is fixed by the parameters $\tilde{\kappa}_{S}$,
$\tilde{\kappa}_{\sigma}$ and $\kappa_{\sigma S}$.

Squared mass of the orthogonal component to $\tilde{\chi}$ during inflation
can be estimated as $m_{\tilde{\chi}_{\perp}}^{2}=\frac{|\kappa_{\sigma
S}|M_{\rm pl}^{2}}{2\tilde{\xi}_{S}^{2}}$. Since the Hubble parameter $H_{I}$
satisfies $H_{I}^{2}=\frac{\hat{\kappa}_{S}M_{\rm
pl}^{2}}{12\tilde{\xi}_{S}^{2}}$ at the same period,
$H_{I}<m_{\tilde{\chi}_{\perp}}$ is satisfied under the condition (31).jjj
Mass of another orthogonal component $S_{I}$ is estimated as
$m_{S_{I}}^{2}\sim\frac{8\alpha M_{\rm pl}^{2}}{\tilde{\xi}_{S}^{2}}$ for a
case $\cos 2\rho\sim 0$, which is interested in the present study and then
$96\alpha>\hat{\kappa}_{S}$ is required for $m_{S_{I}}>H_{I}$. Although
$\alpha\ll\hat{\kappa}_{S}$ is assumed from the vacuum determination, it can
be satisfied for suitable values of $\alpha$. Thus, the inflation $\chi$
starts rolling along the valley within a few Hubble time independently of an
initial value of the inflaton. It justifies our analyzing the model as a
single field inflation model. On the other hand, since
$\tilde{\sigma}(=\sqrt{\frac{|\kappa_{\sigma
S}|}{2\tilde{\kappa}_{\sigma}}}\tilde{\chi})>\frac{H_{I}}{\sqrt{2\pi}}$ is
satisfied generally,kkkAlthough it may be escapable for
$\tilde{\xi}_{S}\ll\sqrt{|\kappa_{\sigma S}|}$, it is not the case in the
present model. the global $U(1)$ is spontaneously broken during inflation and
isocurvature fluctuation could be problematic [38]. However, even in that case
it is escapable since the axion needs not to be a dominant component of DM in
the present model. This problem is discussed later.

The canonically normalized inflaton $\chi$ can be expressed as [38]

$\Omega^{2}\frac{d\chi}{d\tilde{S}}=\sqrt{\gamma\Omega^{2}+6\tilde{\xi}_{S}^{2}\frac{\tilde{S}^{2}}{M_{\rm
pl}^{2}}},$ (32)

where $\gamma$ can be approximated along the valley as

$\gamma=1-\frac{\kappa_{\sigma S}}{2\tilde{\kappa}_{\sigma}}.$ (33)

If we use $\gamma\simeq 1$, the potential of $\chi$ obtained through
$V(\chi)=\frac{1}{\Omega^{4}}V(\tilde{\sigma},\tilde{S})$ can be derived by
using the solution of eq. (32) which is given as

$\frac{\chi}{M_{\rm pl}}=-\sqrt{6}~{}{\rm
arcsinh}\left(\frac{\sqrt{\frac{6}{\gamma}}\frac{\tilde{\xi}_{S}\tilde{S}}{M_{\rm
pl}}}{\sqrt{1+\frac{\tilde{\xi}_{S}}{M_{\rm
pl}^{2}}\tilde{S}^{2}}}\right)+\sqrt{\frac{\gamma+6\tilde{\xi}_{S}}{\tilde{\xi}_{S}}}{\rm
arcsinh}\left(\frac{\sqrt{\tilde{\xi}_{S}(1+\frac{6}{\gamma}\tilde{\xi}_{S})}\tilde{S}}{M_{\rm
pl}}\right).$ (34)

We derive the potential of $\chi$ through numerical calculation for a typical
value of $\tilde{\xi}_{S}$ by using eq. (34). Such examples of $V(\chi)$ are
shown in Fig. 2. It can be approximated as

$\displaystyle
V(\chi)=\left\\{\begin{array}[]{ll}\frac{\hat{\kappa}_{S}}{4\tilde{\xi}_{S}^{2}}M_{\rm
pl}^{4}&\quad\chi>M_{\rm pl},\\\
\frac{\hat{\kappa}_{S}}{6\tilde{\xi}_{S}^{2}}M_{\rm
pl}^{2}\chi^{2}&\quad\frac{M_{\rm pl}}{\tilde{\xi}_{S}}<\chi<M_{\rm pl},\\\
\frac{\hat{\kappa}_{S}}{4}\chi^{4}&\quad\chi<\frac{M_{\rm
pl}}{\tilde{\xi}_{S}}.\end{array}\right.$ (38)

The inflation ends at $\chi_{\rm end}\simeq M_{\rm pl}$. After the end of
inflation, there is a substantial region where the potential behaves as a
quadratic form before it is reduced to a quartic form at low energy regions
for the case $\tilde{\xi}_{S}\gg 1$ as in the Higgs inflation. However, such a
region can be neglected for the case $\tilde{\xi}_{S}<10$, which is the case
considered in this study. Since the inflaton oscillating in the quartic
potential behaves as radiation as shown later, the radiation domination starts
soon after the end of inflation in that case.

Fig. 2 Potential of the inflaton $\chi$ for $\tilde{\xi}_{S}=1$ (left panel)
and $\tilde{\xi}_{S}=10$ (right panel). In both panels,
$\tilde{\xi}_{S}/\xi_{\sigma}=20$ and $\tilde{\kappa}_{S}/|\kappa_{\sigma
S}|=|\kappa_{\sigma S}|/\tilde{\kappa}_{\sigma}=0.1$ are assumed and
$\tilde{\kappa}_{S}$ is fixed by using eq. (42) for ${\cal N}_{k}=55$. As
references, we also plot approximated potential
$\frac{\hat{\kappa}_{S}}{6\tilde{\xi}_{S}^{2}}M_{\rm pl}^{2}\chi^{2}$ and
$\frac{\hat{\kappa}_{S}}{4}\chi^{4}$ in eq. (38) as $\chi^{2}$ and $\chi^{4}$.
In these plots, a Planck unit $(M_{\rm pl}=1)$ is used.

The slow-roll parameters in this model can be estimated by using eq. (32) as
[16]

$\epsilon\equiv\frac{M_{\rm
pl}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}=\frac{8M_{\rm
pl}^{4}}{\gamma\tilde{\xi}_{S}\left(1+\frac{6}{\gamma}\tilde{\xi}_{S}\right)\tilde{S}^{4}},\qquad\eta\equiv
M_{\rm pl}^{2}\frac{V^{\prime\prime}}{V}=-\frac{8M_{\rm
pl}^{2}}{\gamma\left(1+\frac{6}{\gamma}\tilde{\xi}_{S}\right)\tilde{S}^{2}}.$
(39)

The $e$-foldings number ${\cal N}_{k}$ from the time when the scale $k$ exits
the horizon to the end of inflation is estimated by using eq. (32) as

$\displaystyle{\cal N}_{k}=\frac{1}{M_{\rm pl}^{2}}\int^{\chi_{k}}_{\chi_{\rm
end}}\frac{V}{V^{\prime}}d\chi=\frac{1}{8M_{\rm
pl}^{2}}(\gamma+6\tilde{\xi}_{S})(\tilde{S}_{k}^{2}-\tilde{S}_{\rm
end}^{2})-\frac{3}{4}\ln\frac{M_{\rm
pl}^{2}+\tilde{\xi}_{S}\tilde{S}_{k}^{2}}{M_{\rm
pl}^{2}+\tilde{\xi}_{S}\tilde{S}_{\rm end}^{2}}.$ (40)

Taking account of these, the slow-roll parameters in this inflation scenario
is found to be approximated as $\epsilon\simeq\frac{3}{4{\cal N}_{k}^{2}}$ and
$\eta\simeq-\frac{1}{{\cal N}_{k}}$. The field value of inflaton during the
inflation is found to be expressed as $\chi_{k}=\frac{\sqrt{6}}{2}M_{\rm
pl}\ln(32\tilde{\xi}_{S}{\cal N}_{k})$ by using eqs. (34) and (40), and its
potential $V_{k}(\equiv V(\chi_{k}))$ takes a constant value as shown in eq.
(38). On the other hand, if we use $\epsilon=1$ at the end of inflation, the
inflaton potential is estimated as $V_{\rm end}(\equiv V(\chi_{\rm
end}))\simeq 0.072\frac{\hat{\kappa}_{S}}{\tilde{\xi}_{S}^{2}}M_{\rm pl}^{4}$
which is found to be a good approximation from Fig. 2.

The spectrum of density perturbation predicted by the inflation is known to be
expressed as [16]

${\cal P}(k)=A_{s}\left(\frac{k}{k_{\ast}}\right)^{n_{s}-1},\qquad
A_{s}=\frac{V}{24\pi^{2}M_{\rm pl}^{4}\epsilon}\Big{|}_{k_{\ast}}.$ (41)

If we use the Planck data $A_{s}=(2.101^{+0.031}_{-0.034})\times 10^{-9}$ at
$k_{\ast}=0.05~{}{\rm Mpc}^{-1}$ [15], we find the Hubble parameter during the
inflation to be $H_{I}=1.4\times 10^{13}\left(\frac{60}{{\cal
N}_{k_{\ast}}}\right)$ GeV and the relation

$\hat{\kappa}_{S}\simeq 4.13\times
10^{-10}\tilde{\xi}_{S}^{2}\left(\frac{60}{{\cal N}_{k_{\ast}}}\right)^{2},$
(42)

which should be satisfied at the horizon exit time of the scale $k_{\ast}$. We
confine our study to the case $\tilde{\xi}_{S}<10$.

Fig. 3 Predicted values of the scalar spectral index $n_{s}$ and the tensor-
to-scalar ratio $r$ in this model. They are read off as the values at
intersection points of two lines with a fixed value of $\tilde{\xi}_{S}$ or
${\cal N}_{k}$. The coupling constant $\hat{\kappa}_{S}$ is varied in a range
from $10^{-7}$ to $10^{-10}$.

In Fig. 3, we plot predicted values for the scalar spectral index $n_{s}$ and
the tensor-to-scalar ratio $r$ in the present model. Since the quartic
coupling $\hat{\kappa}_{S}$ is a free parameter of the model under the
constraint (42), we vary $\hat{\kappa}_{S}$ in the range
$10^{-10}\leq\hat{\kappa}_{S}\leq 10^{-7}$ for fixed values of
$\tilde{\xi}_{S}$ or ${\cal N}_{k}$. The CMB constraint (42) is satisfied at
intersection points of the lines with a fixed value of $\tilde{\xi}_{S}$ or
${\cal N}_{k}$. The figure shows that the constraints of the observed CMB data
[15] are satisfied for the supposed parameters.

After the end of inflation, the inflaton $\chi$ starts oscillation in the
potential $V(\chi)$. At this stage, the description by $\chi$ is no longer
justified, especially, at the small field regions. $\varphi$ is not constant
in general there. $\tilde{S}$ and $\tilde{\sigma}$ should be treated
independently. In the following study, however, we confine our study to a
special inflaton trajectory and estimate the reheating phenomena by using
$\chi$ to give a rough evaluation of reheating temperature under the
assumption that inflaton follows a constant $\varphi$ trajectory.lll During
the first several oscillations, both $\varphi$ and $\rho$ can be numerically
confirmed to take constant values. This suggests that single field treatment
is rather good during the first few oscillations at least. In this case,
inflaton oscillation is described by the equation

$\frac{d^{2}\chi}{dt^{2}}+3H\frac{d\chi}{dt}+V^{\prime}(\chi)=0.$ (43)

Since the amplitude of $\chi$ evolves approximately as
$\Phi(t)=\frac{\tilde{\xi}_{S}}{\sqrt{\pi\hat{\kappa}_{S}}t}$ in the quadratic
potential after the end of inflation, the inflaton $\chi$ oscillates
$\frac{1}{2\pi\sqrt{3\pi}}(\tilde{\xi}_{S}-1)$ times before the potential (38)
changes from a quadratic form to a quartic one. This means that preheating
under the quadratic potential could play no substantial role for the case
$\tilde{\xi}_{S}<10$. In such a case, we need to consider the preheating in
the quartic potential only. The model with the quartic potential
$V(\chi)=\frac{\hat{\kappa}_{S}}{4}\chi^{4}$ becomes conformally invariant
[39].

If we introduce dimensionless conformal time $\tau$, which is defined by using
a scale factor $a$ as $ad\tau=\sqrt{\hat{\kappa}_{S}}\chi_{\rm end}dt$ and
also a rescaled field $f=\frac{a\chi}{\chi_{\rm end}}$, eq. (43) can be
rewritten as

$\frac{d^{2}f}{d\tau^{2}}+f^{3}=0.$ (44)

The solution of this equation which describes the inflaton oscillation is
known to be given by a Jacobi elliptic function $f(\tau)={\rm
cn}\left(\tau-\tau_{i},\frac{1}{\sqrt{2}}\right)$.mmmIf we take
$\tau_{i}\simeq 2.44$, $f(\tau)$ can be approximated by
$\cos\left(\frac{2\pi}{\tau_{0}}\tau\right)$, where $\tau_{0}$ is expressed by
using the complete elliptic integral of the first kind $K$ as
$\tau_{0}=4K(\frac{1}{\sqrt{2}})$ [39]. From the Friedman equation for this
inflaton oscillation, we find

$a(\tau)=\frac{\chi_{\rm end}}{2\sqrt{3}M_{\rm
pl}}\tau,\qquad\tau=2(3\hat{\kappa}_{S}M_{\rm pl}^{2})^{1/4}\sqrt{t}.$ (45)

Since $H=1/2t$ is satisfied, this oscillation era is radiation dominated. If
we take into account such a feature of the model that radiation domination
starts just after the end of inflation, the $e$-foldings number ${\cal N}_{k}$
can be expressed by noting a relation $k=a_{k}H_{k}$ as

${\cal N}_{k}=56.7-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\ln\left(\frac{V_{\rm
end}^{1/4}}{10^{14}~{}{\rm GeV}}\right)+2\ln\left(\frac{V_{k}^{1/4}}{V_{\rm
end}^{1/4}}\right),$ (46)

where $H_{k}^{2}=\frac{V_{k}}{3M_{\rm pl}^{2}}$ and the suffix 0 stands for
the present value of each quantity. Reheating temperature dependence of ${\cal
N}_{k}$ is weak or lost differently from the usual case [16] where substantial
matter domination is assumed to follow the inflation era.nnnIf reheating
occurs through a perturbative process at $\chi~{}{{}^{<}_{\sim}}~{}u$ where
matter domination is realized, its effect on ${\cal N}_{k}$ could also be
negligible as long as $\Gamma>H$ is satisfied at that stage where $\Gamma$ is
inflaton decay width. In the next part, we discuss the reheating temperature
expected to be realized in the present model.

###  3.2 Preheating and reheating

Before proceeding to the study of particle production under the background
oscillation of inflaton, we need to know the mass of the relevant particles
which is induced through the interaction with inflaton. Such interactions are
given as

$\displaystyle\Big{[}-\frac{y_{D}}{\sqrt{2}}\frac{\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}\chi\bar{D}_{L}D_{R}-\frac{y_{E}}{\sqrt{2}}\frac{\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}\chi\bar{E}_{L}E_{R}+\sum_{j=1}^{3}\Big{\\{}\frac{1}{\sqrt{2}}\left(y_{d_{j}}e^{i\rho}+\tilde{y}_{d_{j}}e^{-i\rho}\right)\chi\bar{D}_{L}d_{R_{j}}$
$\displaystyle+\frac{1}{\sqrt{2}}\left(y_{e_{j}}e^{i\rho}+\tilde{y}_{e_{j}}e^{-i\rho}\right)\chi\bar{E}_{L}e_{R_{j}}-\frac{y_{N_{j}}}{2\sqrt{2}}\frac{\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}\chi\bar{N}_{j}^{c}N_{j}\Big{\\}}+{\rm
h.c.}\Big{]}$ $\displaystyle+\frac{1}{2}(\kappa_{\phi
S}\phi^{\dagger}\phi+\kappa_{\eta
S}\eta^{\dagger}\eta)\chi^{2}-\frac{\kappa_{\sigma
S}}{2\tilde{\kappa}_{\sigma}}(\kappa_{\phi\sigma}\phi^{\dagger}\phi+\kappa_{\phi\sigma}\eta^{\dagger}\eta)\chi^{2}.$
(47)

The particles interacting with the inflaton $\chi$ have mass varying with the
oscillation of $\chi$ and their mass can be read off from eq. (47) as

$\displaystyle M_{N_{j}}\simeq\frac{y_{N_{j}}}{\sqrt{2}}\frac{|\kappa_{\sigma
S}|}{2\tilde{\kappa}_{\sigma}}\chi,\quad\tilde{M}_{F}\simeq\frac{\chi}{\sqrt{2}}\left[\sum_{j=1}^{3}(y_{f_{j}}^{2}+\tilde{y}_{f_{j}}^{2})+y_{F}^{2}\frac{\kappa_{\sigma
S}^{2}}{4\tilde{\kappa}_{\sigma}^{2}}\right]^{\frac{1}{2}},$ $\displaystyle
m_{\phi}^{2}\simeq\frac{1}{2}\left(\kappa_{\phi S}+\frac{|\kappa_{\sigma
S}|}{\tilde{\kappa}_{\sigma}}\kappa_{\phi\sigma}\right)\chi^{2},\quad
m_{\eta}^{2}\simeq\frac{1}{2}\left(\kappa_{\eta S}+\frac{|\kappa_{\sigma
S}|}{\tilde{\kappa}_{\sigma}}\kappa_{\eta\sigma}\right)\chi^{2},$ (48)

where $F=D$ or $E$ should be understood for $f=d$ or $e$, respectively. Since
the effect of nonminimal coupling is negligible during this oscillation
period, it is convenient to use the components of $\sigma$ and $S$, which are
parallel and orthogonal to the inflaton $\chi$ to describe their interactions.
If we indicate each of them as $\sigma_{\parallel}$, $\sigma_{\perp}$,
$S_{\parallel}$, and $S_{\perp}$, then their interactions are expressed as

$\displaystyle\frac{\tilde{\kappa}_{S}}{4}(S_{\parallel}^{2}+S_{\perp}^{2}-u^{2})^{2}+\frac{\kappa_{\sigma
S}}{4}(S_{\parallel}^{2}+S_{\perp}^{2}-u^{2})(\sigma_{\parallel}^{2}+\sigma_{\perp}^{2}-w^{2})+\frac{\tilde{\kappa}_{\sigma}}{4}\left(\sigma_{\parallel}^{2}+\sigma_{\perp}^{2}-w^{2}\right)^{2}.$

By combining these interactions with the composition of $\chi$, their masses
are found to be given byoooIt should be noted that the mass of $\sigma$ could
have another non-negligible contribution which is induced by explicit breaking
of the global $U(1)$ symmetry brought about by the quantum gravitational
effects. We do not take account of it in the present study.

$\displaystyle
m^{2}_{S_{\parallel}}\simeq\left(3\hat{\kappa}_{S}+\frac{\kappa_{\sigma
S}^{2}}{2\tilde{\kappa}_{\sigma}}\right)\chi^{2},\qquad
m_{\sigma_{\parallel}}^{2}\simeq\left(|\kappa_{\sigma S}|+\frac{\kappa_{\sigma
S}^{2}}{4\tilde{\kappa}_{\sigma}}\right)\chi^{2},$ $\displaystyle
m^{2}_{S_{\perp}}\simeq\hat{\kappa}_{S}\chi^{2},\qquad
m_{\sigma_{\perp}}^{2}\simeq\frac{\kappa_{\sigma
S}^{2}}{4\tilde{\kappa}_{\sigma}}\chi^{2}.$ (50)

The coupling constants relevant to these masses are restricted through the
assumed inflaton composition and the realization of the $CP$ phases in the CKM
and PMNS matrices. The discussion in the previous sections shows that such
requirements are satisfied for

$\hat{\kappa}_{S}<|\kappa_{\sigma
S}|<\tilde{\kappa}_{\sigma},\qquad\hat{\kappa}_{S}<y_{N_{j}},y_{f_{j}},\tilde{y}_{f_{j}}.$
(51)

We assume additionally

$\hat{\kappa}_{S}<g_{\phi}\equiv\kappa_{\phi S}+\frac{|\kappa_{\sigma
S}|}{\tilde{\kappa}_{\sigma}}\kappa_{\phi\sigma},\qquad\hat{\kappa}_{S}<g_{\eta}\equiv\kappa_{\eta
S}+\frac{|\kappa_{\sigma S}|}{\tilde{\kappa}_{\sigma}}\kappa_{\eta\sigma}.$
(52)

Since the oscillation frequency of the inflaton is
$\sim\sqrt{\hat{\kappa}_{S}}\chi$, decays or annihilations of the inflaton are
kinematically forbidden except for the one to $\sigma_{\perp}$ as found from
eqs. (48) and (50). In $\sigma_{\perp}$ case, the inflaton reaction rate to it
is much smaller than the Hubble parameter at this period because of the
smallness of its coupling with the inflaton, energy drain from the inflaton to
$\sigma_{\perp}$ is ineffective to be neglected. As a result, the energy
transfer from the inflaton oscillation to excited particles is expected to
occur at the time when the inflaton crosses the zero where the resonant
particle production is possible.

Preheating under the background inflaton oscillation can generate the
excitations of $\chi$ itself and other scalars $\psi$ which couple with $\chi$
at its zero crossing [40]. In a quartic potential case [39], the model becomes
conformally invariant and the time evolution equations of $\chi_{k}(\simeq
S_{k})$ and $\psi_{k}$, which are the comoving modes with a momentum $k$, can
be transformed to the simple ones by rescaling them to the dimensionless
quantities in the same way as eq. (44). They are given as

$\displaystyle\frac{d^{2}}{d\tau^{2}}X_{k}+\omega_{k}^{2}X_{k}=0,\qquad\omega_{k}^{2}=\bar{k}^{2}+3f(\tau)^{2},$
$\displaystyle\frac{d}{d\tau^{2}}F_{k}+\tilde{\omega}_{k}^{2}F_{k}=0,\qquad\tilde{\omega}_{k}^{2}=\bar{k}^{2}+\frac{g_{\psi}}{\hat{\kappa}_{S}}f(\tau)^{2},$
(53)

where the rescaled variables are defined as

$\displaystyle X_{k}=\frac{a\chi_{k}}{\chi_{\rm end}},\qquad
F_{k}=\frac{a\psi_{k}}{\chi_{\rm end}},\qquad\bar{k}=\frac{ak}{\chi_{\rm
end}\sqrt{\hat{\kappa}_{S}}}.$ (54)

Function $f(\tau)$ is the solution of eq. (44) and $g_{\psi}$ stands for a
coupling constant of the relevant particle $\psi(=\sigma,S_{\perp},\phi,\eta)$
with the inflaton $\chi$, which can read off from eqs. (48) and (50).
Amplitudes $X_{k}$ and $F_{k}$ are known to show the exponential behavior
$\propto e^{\mu_{k}\tau}$ with a characteristic exponent $\mu_{k}$, which is
determined by a parameter $g_{\psi}/\hat{\kappa}_{S}$. Using the solutions of
eq. (53), the number density of the produced particle $\psi$ can be calculated
as

$n_{k}^{\psi}=\frac{\tilde{\omega}_{k}}{2\hat{\kappa}_{S}}\left(\frac{|F_{k}^{\prime}|^{2}}{\tilde{\omega}_{k}^{2}}+|F_{k}|^{2}\right)-\frac{1}{2}.$
(55)

Particle production based on eq. (53) at the inflaton zero crossing has been
studied in [39] and it is shown to be characterized by the parameter
$g_{\psi}/\hat{\kappa}_{S}$. We classify the relevant couplings into five
groups

${\rm(A)}~{}\frac{g_{\sigma_{\parallel}}}{\hat{\kappa}_{S}}\gg
1,\quad{\rm(B)}~{}\frac{g_{S_{\parallel}}}{\hat{\kappa}_{S}}=3,\quad{\rm(C)}~{}\frac{g_{S_{\perp}}}{\hat{\kappa}_{S}}=1,\quad{\rm(D)}~{}\frac{g_{\sigma_{\perp}}}{\hat{\kappa}_{S}}\ll
1,\quad{\rm(E)}~{}\frac{g_{\phi}}{\hat{\kappa}_{S}},~{}\frac{g_{\eta}}{\hat{\kappa}_{S}}>1,$
(56)

where we note that couplings in (A)-(D) are fixed by the present inflaton
composition but the ones in (E) are not constrained. Now we consider the
resonant particle production in each group. A maximum value of characteristic
exponent in (D) is very small so that it plays no effective role also in
preheating. In (B) and (C), both the fluctuations of $S_{\parallel}$ and
$S_{\perp}$ are produced fast, but it stops as soon as
$\langle|S_{\parallel}|^{2}\rangle$ and $\langle|S_{\perp}|^{2}\rangle$ reach
a certain value such as $0.5\chi_{\rm end}^{2}/a^{2}$. Although a maximum
value $\mu_{\max}$ of the characteristic exponent of (B) is much smaller than
the one of (C) and also the resonance band of (B) is much narrower than (C),
the interaction $S_{\parallel}^{2}S_{\perp}^{2}$ accelerates the production of
fluctuations of $S_{\parallel}$ through rescattering and they reach the
similar value [38, 41]. Since the backreaction of these fluctuations to the
inflaton oscillation restructures the resonance band, the resonant particle
production stops before causing much more conversion of the inflaton
oscillation energy to particle excitations. Moreover, since the decay of
excitations produced through these processes are also closed kinematically,
these could not play an efficient role in reheating. In (A), since
$\sigma_{\parallel}$ also couples to the inflaton directly, the resonant
production of its excitation stops at a certain stage due to the same reason
as (B) and (C). Even if the excited particles are allowed to decay to fermions
$F$ and $N_{j}$ kinematically, the decay width is much smaller than the Hubble
parameters to be neglected. As a result, if the process due to (E) is not
effective, preheating cannot play any role for reheating and reheating
proceeds through perturbative processes after the amplitude of inflaton is
smaller than the VEV $u$.

Here, we have to note that there is a possibility in (E) where the energy
transfer from the inflaton oscillation to radiation proceeds through
preheating since the produced excitations can decay to relativistic particles
differently from (B) and (C). In this case, $\phi$ and $\eta$ are produced as
excitations at the zero crossing of the inflaton where an adiabaticity
condition $\tilde{\omega}_{k}^{\prime}<\tilde{\omega}_{k}^{2}$ could be
violated for certain values of $\bar{k}$. By using the analytic solution of
eq. (53) derived in [39], the momentum distribution $n_{k}^{\psi}$ of the
produced particle $\psi$ through one zero crossing of the inflaton can be
estimated as

$n_{\bar{k}}^{\psi}=e^{2\mu_{k}\frac{\tau_{0}}{2}}=e^{-\left(\bar{k}/\bar{k}_{c}\right)^{2}},\qquad\bar{k}_{c}^{2}=\sqrt{\frac{g_{\psi}}{2\pi^{2}\hat{\kappa}_{S}}},$
(57)

where $\tau_{0}$ is an inflaton oscillation period and $\tau_{0}=7.416$. The
resonance is efficient for $\bar{k}<\bar{k}_{c}$. Thus, the particle number
density produced during one zero crossing of the inflaton is

$\bar{n}^{\psi}=\int\frac{d^{3}\bar{k}}{(2\pi)^{3}}n_{\bar{k}}^{\psi}=\int\frac{d^{3}\bar{k}}{(2\pi)^{3}}e^{-(\bar{k}/\bar{k}_{c})^{2}}=\frac{\bar{k}_{c}^{3}}{8\pi^{3/2}}.$
(58)

The energy transfer from the inflaton oscillation to relativistic particles is
caused through the decay of the produced particles $\psi(=\phi,\eta)$ and
thermalization proceeds. They can decay to light fermions through
$\phi\rightarrow\bar{q}t$ with a top Yukawa coupling $h_{t}$ and
$\eta\rightarrow\bar{\ell}N$ with neutrino Yukawa couplings $h_{j}$,
respectively. Here, we should note that $\eta$ can be heavier than $N_{j}$ at
this stage even if $\eta$ is the lightest one with $Z_{2}$ odd parity at the
weak scale. It is caused by the inflaton composition in the present model as
found from eq. (48). Their decay widths in the comoving frame are given by
using the conformally rescaled unit as

$\displaystyle\bar{\Gamma}_{\psi}=\frac{c_{\psi}y_{\psi}^{2}}{8\pi}\bar{m}_{\psi},\qquad\bar{m}_{\psi}=\frac{am_{\psi}}{\chi_{\rm
end}\sqrt{\hat{\kappa}_{S}}}=\sqrt{\frac{g_{\psi}}{\hat{\kappa}_{S}}}f(\tau),$
(59)

where $\psi=\phi,~{}\eta$, and $c_{\psi}$ are internal degrees of freedom
$c_{\phi}=3$ and $c_{\eta}=1$. The Yukawa coupling $y_{\psi}$ represents
$y_{\phi}=h_{t}$ and $y_{\eta}=h_{j}$. Since
$\bar{\Gamma}_{\psi}^{-1}<\tau_{0}/2$ is satisfied for $g_{\psi}>4\times
10^{-7}\left(\frac{\hat{\kappa}_{S}}{10^{-8}}\right)$, the produced $\psi$
decays to the light fermions completely before the next inflaton zero crossing
[42] and then it is not accumulated in such cases. We fix $\tau=0$ at the
first inflaton zero crossing so that $f(\tau)$ can be expressed approximately
as $f(\tau)=f_{0}\sin(cf_{0}\tau)$. Transferred energy density through the
$\psi$ decay during a half period of oscillation can be estimated
asppp$\bar{\rho}$ is defined as the energy density in the comoving frame by
using the conformally rescaled variables.

$\delta\bar{\rho}_{r}=\int^{\tau_{0}/2}_{0}d\tau\bar{\Gamma}_{\psi}\bar{m}_{\psi}\bar{n}_{\psi}e^{-\int_{0}^{\tau}\bar{\Gamma}_{\psi}\tau^{\prime}}=\frac{1}{8\pi^{3/2}(2\pi^{2})^{3/4}}\left(\frac{g_{\psi}}{\hat{\kappa}_{S}}\right)^{5/4}Y(f_{0},\gamma_{\psi}),$
(60)

where $\gamma_{\psi}$ and $Y(f_{0},\gamma_{\psi})$ are defined by using
$c=2\pi/\tau_{0}$ as

$\gamma_{\psi}=\frac{c_{\psi}y_{\psi}^{2}}{8\pi
c}\sqrt{\frac{g_{\psi}}{\hat{\kappa}_{S}}},\qquad
Y(f_{0},\gamma_{\psi})=c\gamma_{\psi}\int^{\tau_{0}/2}_{0}d\tau
f_{0}^{2}\sin^{2}(cf_{0}\tau)e^{-2\gamma_{\psi}\sin^{2}(\frac{cf_{0}\tau}{2})}.$
(61)

The energy density transferred to the light particles is accumulated at each
inflaton zero crossing linearly and its averaged value for $\tau$ is estimated
as

$\bar{\rho}_{r}(\tau)=\frac{2\tau}{\tau_{0}}\delta\bar{\rho}_{r}=6.5\times
10^{-4}\left(\frac{g_{\psi}}{\hat{\kappa}_{S}}\right)^{5/4}Y(f_{0},\gamma_{\psi})\tau,$
(62)

where the substantial change of $f_{0}$ is assumed to be negligible during
$\tau$. Since the total energy density of the inflaton oscillation energy
$\bar{\rho}_{\chi}$ and the transferred energy $\bar{\rho}_{r}$ to light
particles is conserved, reheating temperature realized through this process
can be estimated from $\bar{\rho}_{\chi_{\rm end}}=\bar{\rho}_{r}$. It can be
written by transferring it to the physical unit as

$\frac{1}{4\hat{\kappa}_{S}}\left(\frac{\sqrt{\hat{\kappa}_{S}}\chi_{\rm
end}}{a}\right)^{4}=\frac{\pi^{2}}{30}g_{\ast}T_{R}^{4},$ (63)

where we use $\bar{\rho}_{\chi_{\rm end}}=\frac{1}{4\hat{\kappa}_{S}}$ and
$g_{\ast}=130$. By applying eqs. (45) and (62) to this formula, we findqqqThe
same result can be obtained by using the relation $H=\frac{1}{2t}$ in the
radiation dominated era together with eqs. (45) and (62).

$T_{R}=5.9\times 10^{15}g_{\psi}^{5/4}Y(f_{0},\gamma_{\psi})~{}{\rm GeV}.$
(64)

Since $h_{t}\gg h_{j}$ is satisfied, reheating temperature is expected to be
determined by the produced $\phi$ as long as $\phi$ is dominantly produced.

If preheating cannot produce relativistic particles effectively, the dominant
energy is still kept in the inflaton oscillation. When the oscillation
amplitude of $\chi$ decreases to be $O(u)$, the inflaton starts decaying to
the light particles through the perturbative processes. Since the mass pattern
is expected under the present assumption for the coupling constants in (51) to
be

$2\tilde{m}_{\eta}<m_{\chi}<\tilde{M}_{D},\tilde{M}_{E},$ (65)

the inflaton decay is expected to occur mainly through
$\chi\rightarrow\eta^{\dagger}\eta$ and $\chi\rightarrow\phi^{\dagger}\phi$ at
tree level. The decay width of $\psi(=\phi,~{}\eta)$ is estimated as

$\displaystyle\Gamma_{\psi}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{\psi}^{2}}{16\pi\hat{\kappa}_{S}}m_{\chi},$ (66)

where $g_{\psi}$ is defined in eq. (52). After the inflaton decays to
$\eta^{\dagger}\eta$ and $\phi^{\dagger}\phi$, the SM contents are expected to
be thermalized through gauge interactions with $\eta$ and $\phi$ immediately.
Since $\Gamma_{\psi}>H$ is satisfied for
$g_{\psi}>10^{-7.1}\left(\frac{\hat{\kappa}_{S}}{10^{-8}}\right)^{1/2}\left(\frac{u}{10^{11}~{}{\rm
GeV}}\right)^{1/2}$ at $\chi\simeq u$, reheating temperature in such a case
can be estimated through
$\frac{1}{4}\hat{\kappa}_{S}u^{4}=\frac{\pi^{2}}{30}g_{\ast}T_{R}^{4}$ asrrr
Although a larger value of $u$ can make the reheating temperature much higher,
its upper bound exits. Since a larger $g_{\psi}$ is required in that case, it
could violate the perturbativity of the model and cause the upper bound for
it. For example, if we consider the case with $y_{F}=10^{-1.2}$, the
perturbativity is violated for $g_{\psi}>10^{-4.4}$. As a result, the
reheating temperature due to the perturbative process is bounded as
$T_{R}<6.3\times 10^{13}$ GeV.

$T_{R}\simeq 2.8\times
10^{8}\left(\frac{\hat{\kappa}_{S}}{10^{-8}}\right)^{1/4}\left(\frac{u}{10^{11}~{}{\rm
GeV}}\right)~{}{\rm GeV},$ (67)

which is independent of $g_{\psi}$. However, if $\Gamma_{\psi}>H$ is not
satisfied because of a small $g_{\psi}$, the reheating temperature is expected
to be determined through $\Gamma_{\psi}=H$ and then becomes smaller
proportionally to $g_{\psi}$.

Fig. 4 Left : reheating temperature $T_{R}$ (GeV) predicted for both the
preheating (a) and the perturbative process (b). They are plotted as a
function of $g_{\phi}$. In the case (a), the decay $\phi\rightarrow\bar{q}t$
is assumed. Right : contribution to the reheating temperature due to a
component $\sigma$ of the inflaton in a case $\kappa_{\phi S}=0$. The
reheating temperature $T_{R}$ (GeV) is plotted as a function of
$\kappa_{\phi\sigma}$. Each line represents $T_{R}$ for several values of
$(\tilde{\kappa}_{S},|\kappa_{\sigma S}|/\tilde{\kappa}_{\sigma})$, where
$\tilde{\kappa}_{\sigma}$ is fixed at $10^{-4.5}$ and $10^{-5.3}$ for
$\tilde{\kappa}_{S}=10^{-8}$ and $10^{-9}$, respectively.

In the left panel of Fig. 4, for a case $\psi=\phi$, the expected reheating
temperature through both processes is plotted as a function $g_{\phi}$ in a
case $\tilde{\kappa}_{S}=10^{-8}$, $\tilde{\kappa}_{\sigma}=10^{-4.5}$,
$|\kappa_{\sigma S}|/\tilde{\kappa}_{\sigma}=10^{-1.2}$, and $u=10^{11}$ GeV.
It shows that the reheating temperature is determined by the perturbative
process at $g_{\phi}<10^{-6}$. We also found from the figure that the
reheating at $g_{\phi}>10^{-6}$ proceeds through the preheating. Even if the
dominant component $S$ of the inflaton has no coupling with $\phi$ so that
$g_{\phi}\simeq|\kappa_{\sigma
S}|\kappa_{\phi\sigma}/\tilde{\kappa}_{\sigma}$, the preheating is caused by
the component $\sigma$. It is shown in the right panel where the reheating
temperature is plotted for $\kappa_{\phi\sigma}$ by varying $|\kappa_{\sigma
S}|/\tilde{\kappa}_{\sigma}$.sssThe condition
$\bar{\Gamma}_{\psi}^{-1}<\tau_{0}/2$ can be confirmed for the parameters used
here.  These figures show that $T_{R}>2.3\times 10^{8}$ GeV can be realized if
$g_{\phi}>4\times 10^{-8}$ is satisfied. However, since the perturbativity of
the model is found to be violated at $g_{\phi}>10^{-4.4}$ as mentioned in the
previous footnote, $\kappa_{\phi\sigma}<10^{-4.4}$ and
$\kappa_{\phi\sigma}<10^{-2.6}$ should be satisfied, and then the reheating
temperature cannot be higher than $\sim 10^{10}$ GeV as found from the figure.
Since the decay of $\phi$ is so effective, it decays soon after their
production and much before the inflaton amplitude becomes large during the
oscillation. This makes the energy transfer in the preheating inefficient.

In the usual leptogenesis in the seesaw scenario, the right-handed neutrinos
are supposed to be thermalized only through the neutrino Yukawa couplings
$h_{j}$. In the present model, neutrino mass eigenvalues obtained from eq.
(19) require $h_{2,3}=O(10^{-3})$ to explain the neutrino oscillation data as
discussed at a part of eq. (22). On the other hand, reheating temperature is
found to satisfy $T_{R}~{}{{}^{>}_{\sim}}~{}10^{8}$ GeV from the above
discussion. Since the decay width $\Gamma_{N_{2,3}}$ of $N_{2,3}$ and the
reheating temperature $T_{R}$ satisfy $\Gamma_{N_{2,3}}>H(T_{R})$ and
$T_{R}>M_{N_{2,3}}$, $N_{2,3}$ are also expected to be in the thermal
equilibrium through the inverse decay simultaneously at the reheating period.
In the case of $N_{1}$, however, it depends on the magnitude of its Yukawa
coupling $h_{1}$ which can be much smaller than others. We should note that
$N_{1}$ could be effectively generated in the thermal bath, even if $h_{1}$ is
extremely small, through the scattering of extra fermions which are expected
in the thermal equilibrium through gauge interactions in the case
$\tilde{M}_{D},\tilde{M}_{E}<T_{R}$. It is a noticeable feature of the present
model which opens a window for low scale leptogenesis.

##  4 Phenomenological signature of the model

###  4.1 Leptogenesis

The most interesting feature of this inflation scenario is that thermal
leptogenesis could generate sufficient baryon number asymmetry even for
$M_{N_{1}}<10^{9}$ GeV without relying on resonance effect. In the ordinary
seesaw framework, neutrino mass is generated as
$(m_{\nu})_{\alpha\beta}=\frac{h_{\alpha j}h_{\beta
j}\langle\phi\rangle^{2}}{M_{N_{j}}}$ through Yukawa interaction $h_{\alpha
j}\bar{\ell}_{\alpha}\phi N_{j}$. Baryon number asymmetry in the universe [43]
is expected to be generated by the same interaction through thermal
leptogenesis [44]. If we assume the sufficient lepton asymmetry is generated
through the out-of-equilibrium decay of the lightest right-handed neutrino,
which has been in the thermal equilibrium, then the reheating temperature
$T_{R}$ is required to be larger than its mass $T_{R}>M_{N_{1}}$. Moreover,
since it has to be produced sufficiently in the thermal bath, its Yukawa
coupling $h_{\alpha 1}$ should not be so small. On the other hand, the
neutrino mass formula gives a severer upper bound on $h_{\alpha 1}$ for a
smaller $M_{N_{1}}$ under the constraints of neutrino oscillation data. These
impose a lower bound for $M_{N_{1}}$ such as $10^{9}$ GeV [45]. This condition
for $M_{N_{1}}$ is not changed even if $T_{R}\gg 10^{9}$ GeV is satisfied. The
problem is caused by such a feature of the model that both the production and
the out-of-equilibrium decay of the right-handed neutrino have to be caused
only by the same neutrino Yukawa coupling. It does not change in the original
scotogenic model either [32]. In that model, the right-handed neutrino mass
can be much smaller than $10^{9}$ GeV keeping the neutrino Yukawa couplings to
be rather larger values by fixing $|\lambda_{5}|$ at a smaller value in a
consistent way with the neutrino oscillation data. However, the washout of the
generated lepton number due to the inverse decay of the right-handed neutrinos
becomes so effective in that case. As a result, successful leptogenesis cannot
be realized for a lighter right-handed neutrino than $10^{8}$ GeV.tttLow scale
leptogenesis in the scotogenic model has been studied intensively in [46].
However, the lightest right-handed neutrino is assumed to be in the thermal
equilibrium initially there. It is a notable aspect in the present model that
this situation can be changed by the particles which are introduced to explain
the $CP$ issues in the SM.

We note that the interaction between the right-handed neutrino $N_{1}$ and
extra vector-like fermions $F$ mediated by $\tilde{\sigma}$ could change the
situation.uuuThe similar mechanism has been discussed in models with a
different type of inflaton [33, 47]. The lightest right-handed neutrino
$N_{1}$ can be effectively produced in the thermal bath through the extra
fermions scattering $\bar{D}_{L}D_{R},\bar{E}_{L}E_{R}\rightarrow N_{1}N_{1}$
mediated by $\tilde{\sigma}$ if $D_{L,R}$ and/or $E_{L,R}$ are in the thermal
equilibrium at a certain temperature $T$. In that case, both conditions
$T>\tilde{M}_{F},M_{N_{1}}$ and $\Gamma_{FF}\simeq H(T)$ are required to be
satisfied, where $\Gamma_{FF}$ is the reaction rate of this scattering. Mass
of these fermions is determined by the VEVs $u$ and $w$ which should be larger
than the lower bound of PQ symmetry breaking scale. Since the rough estimation
of $\Gamma_{FF}\simeq H(T)$ for relativistic $F$ and $N_{1}$ gives

$T\simeq 5.8\times
10^{8}\left(\frac{y_{F}}{10^{-1.2}}\right)^{2}\left(\frac{y_{N_{1}}}{10^{-2}}\right)^{2}{\rm
GeV},$ (68)

we find that $T>\tilde{M}_{F},M_{N_{1}}$ could be satisfied for suitable
values of $y_{F}$ and $y_{N_{1}}$. It is crucial that this does not depend on
the magnitude of the $N_{1}$ Yukawa coupling $h_{1}$. If an extremely small
value is assumed for $h_{1}$, successful leptogenesis is allowed in a
consistent way with neutrino oscillation data even for $M_{N_{1}}<10^{9}$ GeV.

After $N_{1}$ is produced in the thermal bath through the scattering of the
extra fermions mediated by $\tilde{\sigma}$, it is expected to decay to
$\ell_{\alpha}\eta^{\dagger}$ by a strongly suppressed Yukawa coupling. Since
its substantial decay occurs after the washout processes are frozen out, the
generated lepton number asymmetry can be efficiently converted to the baryon
number asymmetry through sphaleron processes. This scenario can be checked by
solving Boltzmann equations for $Y_{N_{1}}$ and $Y_{L}(\equiv
Y_{\ell}-Y_{\bar{\ell}})$, where $Y_{\psi}$ is defined as
$Y_{\psi}=\frac{n_{\psi}}{s}$ by using the $\psi$ number density $n_{\psi}$
and the entropy density $s$. Boltzmann equations analyzed here are given as

$\displaystyle\frac{dY_{N_{1}}}{dz}=-\frac{z}{sH(M_{N_{1}})}\left(\frac{Y_{N_{1}}}{Y^{\rm
eq}_{N_{1}}}-1\right)\left[\gamma_{D}^{N_{1}}+\left(\frac{Y_{N_{1}}}{Y^{\rm
eq}_{N_{1}}}+1\right)\sum_{F=D,E}\gamma_{F}\right],$
$\displaystyle\frac{dY_{L}}{dz}=-\frac{z}{sH(M_{N_{1}})}\left[\varepsilon\left(\frac{Y_{N_{1}}}{Y^{\rm
eq}_{N_{1}}}-1\right)\gamma_{D}^{N_{1}}-\frac{2Y_{L}}{Y_{\ell}^{\rm
eq}}\sum_{j=1,2,3}\left(\frac{\gamma_{D}^{N_{j}}}{4}+\gamma_{N_{j}}\right)\right],$
(69)

where $z=\frac{M_{N_{1}}}{T}$ and an equilibrium value of $Y_{\psi}$ is
represented by $Y_{\psi}^{\rm eq}$. $H(T)$ is the Hubble parameter at
temperature $T$ and the $CP$ asymmetry $\varepsilon$ for the decay of $N_{1}$
is expressed as

$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\sum_{j=2,3}\frac{{\rm
Im}[\sum_{\alpha}(\tilde{h}_{\alpha 1}\tilde{h}_{\alpha
j}^{\ast})]^{2}}{\sum_{\alpha}\tilde{h}_{\alpha 1}\tilde{h}_{\alpha
1}^{\ast}}F\left(\frac{M_{N_{j}}^{2}}{M_{N_{1}}^{2}}\right)$ (70)
$\displaystyle=$
$\displaystyle\frac{1}{16\pi}\left[4|h_{2}|^{2}F\left(\frac{y_{N_{2}}^{2}}{y_{N_{1}}^{2}}\right)\sin
2(\theta_{1}-\theta_{2})+|h_{3}|^{2}F\left(\frac{y_{N_{3}}^{2}}{y_{N_{1}}^{2}}\right)\sin
2(\theta_{1}-\theta_{3})\right],$

where $h_{j}=|h_{j}|e^{i\theta_{j}}$ and
$F(x)=\sqrt{x}\left[1-(1+x)\ln\frac{1+x}{x}\right]$. A reaction density for
the decay $N_{j}\rightarrow\ell_{\alpha}\eta^{\dagger}$ and for the lepton
number violating scattering mediated by $N_{j}$ is expressed by
$\gamma_{D}^{N_{j}}$ and $\gamma_{N_{j}}$, respectively [32]. $\gamma_{F}$
represents a reaction density for the scattering
$\bar{D}_{L}D_{R},\bar{E}_{L}E_{R}\rightarrow N_{1}N_{1}$. We assume that
$(D_{L},D_{R})$ and $(E_{L},E_{R})$ are in the thermal equilibrium and
$Y_{N_{1}}=Y_{L}=0$ at $z=z_{R}(\equiv\frac{M_{N_{1}}}{T_{R}})$.

Now we fix the model parameters for numerical study of eq. (69) by taking
account of the discussion in the previous part. We consider two cases for the
VEVs of the singlet scalars such that

${\rm(I)}~{}w=10^{9}~{}{\rm GeV},~{}u=10^{11}~{}{\rm
GeV},\qquad{\rm(II)}~{}w=10^{11}~{}{\rm GeV},~{}u=10^{13}~{}{\rm GeV},$ (71)

where the axion could be a dominant DM in case (II). The parameters
$\tilde{\kappa}_{S},\tilde{\kappa}_{\sigma}$, and $\kappa_{\sigma S}$, which
characterize the inflaton $\chi$, are fixed to
$\tilde{\kappa}_{S}=10^{-8},\tilde{\kappa}_{\sigma}=10^{-4.5}$, and
$|\kappa_{\sigma S}|=10^{-6.1}$. These are used in the right panel of Fig. 4.
The condition ${\cal F}_{f}{\cal F}_{f}^{\dagger}>\mu_{F}^{2}$ for which the
$CP$ phases in the CKM and PMNS matrices can be generated is reformulated as
$\delta\equiv\tilde{M}_{F}/\mu_{F}>\sqrt{2}$. If we confine our study on a
case $y_{f}=(0,0,y)$ and $\tilde{y}_{f}=(0,\tilde{y},0)$ for simplicity,vvvIt
is considered as an example in Appendix A. we have a relation
$y^{2}+\tilde{y}^{2}=(\delta^{2}-1)\frac{w^{2}}{u^{2}}y_{F}^{2}$ among Yukawa
couplings of the extra fermions. We fix them as $\delta=\sqrt{3}$,
$\tilde{y}/y=0.5$ and $y_{D}=y_{E}=10^{-1.2}$ at the scale $\bar{M}$.
Parameters relevant to the neutrino mass generation are fixed as

$\displaystyle y_{N_{2}}=2\times 10^{-2},\quad y_{N_{3}}=4\times
10^{-2}\quad{\rm for~{}(I)~{}and~{}(II)},$ $\displaystyle y_{N_{1}}=7\times
10^{-3},\quad|h_{1}|=6\times 10^{-7}\quad|\tilde{\lambda}_{5}|=10^{-3},\quad
M_{\eta}=1~{}{\rm TeV}\quad{\rm for~{}(I)},$ $\displaystyle
y_{N_{1}}=10^{-3},\quad|h_{1}|=6\times
10^{-5},\quad|\tilde{\lambda}_{5}|=5\times 10^{-3},\quad M_{\eta}=0.9~{}{\rm
TeV}\quad{\rm for~{}(II)}.$ (72)

These parameters fix the mass of relevant particles as

$\displaystyle{\rm(I)}$ $\displaystyle M_{N_{1}}=7\times 10^{6}~{}{\rm
GeV},\quad M_{N_{2}}=2\times 10^{7}~{}{\rm GeV},\quad M_{N_{3}}=4\times
10^{7}~{}{\rm GeV},$ $\displaystyle\tilde{M}_{D}=\tilde{M}_{E}=1.1\times
10^{8}~{}{\rm GeV},\quad m_{\tilde{\sigma}}=8\times 10^{6}~{}{\rm GeV},$
$\displaystyle{\rm(II)}$ $\displaystyle M_{N_{1}}=10^{8}~{}{\rm GeV},\quad
M_{N_{2}}=2\times 10^{9}~{}{\rm GeV},\quad M_{N_{3}}=4\times 10^{9}~{}{\rm
GeV},$ (73) $\displaystyle\tilde{M}_{D}=\tilde{M}_{E}=1.1\times 10^{10}~{}{\rm
GeV}.\quad m_{\tilde{\sigma}}=8\times 10^{8}~{}{\rm GeV}.$

Although the mediator has a small component $\tilde{S}$, it can be safely
treated as $\tilde{\sigma}$. For these parameters, the $CP$ asymmetry
$\varepsilon$ in the $N_{1}$ decay takes a value of $O(10^{-6})$ in both cases
if the maximum $CP$ phase is assumed. DM is determined by the couplings
$\tilde{\lambda}_{3}$ and $\lambda_{4}$. Since they are fixed so as to realize
the correct DM abundance by the neutral component of $\eta$ in the case (I),
it cannot saturate the required DM abundance for the same
$\tilde{\lambda}_{3}$ and $\lambda_{4}$ in the case (II) as found from Fig. 1.
The axion could be a dominant component of DM in the case (II) since $w$ is
taken to be a sufficient value for it.

We give a remark on these couplings here. It is crucial to examine whether the
above parameters used in this analysis are consistent with the potential
stability conditions (5), (17) and also the perturbativity of the model under
constraints coming from the requirements for the DM relic abundance and the
reheating temperature. If DM relic abundance is realized by the neutral
component of $\eta$, both $\tilde{\lambda}_{3}$ and $\lambda_{4}$ should take
values shown in Fig. 1. On the other hand, the reheating temperature required
for sufficient leptogenesis can be realized for
$\kappa_{\phi\sigma}~{}{{}^{>}_{\sim}}~{}10^{-4}$ or $\kappa_{\phi
S}~{}{{}^{>}_{\sim}}~{}10^{-7}$ as found from the analysis of the reheating
temperature. Since they can give rather large contributions to the
$\beta$-functions of the scalar quartic couplings $\kappa_{\eta\sigma}$ and
$\kappa_{\eta S}$ for example, the perturbativity up to the inflation scale
could be violated. An upper bound on $g_{\phi}$ has to be imposed to escape it
and it results in an upper bound on the reheating temperature discussed
already. The parameter sets used here have been confirmed to satisfy these
conditions through the RGEs study. Details of the used parameters in the
analysis are addressed in Appendix B and an example of the results of this
study is presented in the right panel of Fig. 1.

Fig. 5 Evolution of $Y_{N_{1}}$ and $Y_{L}$ obtained as solutions of Boltzmann
equations. Results of the case (I) is shown in the left panel for
$\kappa_{\phi\sigma}=10^{-4}$ and $\kappa_{\phi S}=\kappa_{\eta S}=0$. Results
of the the case (II) is shown in the right panel for $\kappa_{\phi
S}=\kappa_{\eta S}=10^{-6}$ and $\kappa_{\phi\sigma}=\kappa_{\eta\sigma}=0$.
Other parameters in each case are given in the text. Initial values for them
are fixed as $Y_{N_{1}}=Y_{L}=0$ at $z=z_{R}$. $\rho_{N_{1}}/\rho_{\rm R}$
represents a ratio of the energy density of $N_{1}$ to the one of radiation.

Solutions of the Boltzmann equations in the cases (I) and (II) are are shown
in Fig. 5. The lightest right-handed neutrino mass in each case is
$M_{N_{1}}=7\times 10^{6}$ and $10^{8}$ GeV. In both cases, the sufficient
baryon number asymmetry is found to be produced. The figure for the case (I)
shows clearly that the present scenario works well. $Y_{N_{1}}$ reaches a
value near $Y_{N_{1}}^{\rm eq}$ through the scattering of the extra fermions
as expected. Substantial out-of-equilibrium decay occurs at $z>10$ to generate
the lepton number asymmetry. The delay of the decay due to the small $h_{1}$
could make the washout of the lepton number asymmetry ineffective. On the
other hand, we cannot definitely find a signature of the scenario in the case
(II), where the $N_{1}$ mass is near the bound for which the usual
leptogenesis can generate the required baryon number asymmetry in the original
scotogenic model [32]. The figure shows that additional contribution to the
$N_{1}$ production starts at $z\simeq 0.1$. It is considered to be brought
about by the $N_{1}$ inverse decay since it is expected to become effective
around $z\sim\left(\frac{6.3\times
10^{-5}}{h_{1}}\right)\left(\frac{M_{N_{1}}}{10^{8}~{}{\rm GeV}}\right)$. The
figure shows that it plays a main role for the $N_{1}$ production. As a
result, the leptogenesis in this case results in the ordinary one where the
lower mass bound of $N_{1}$ is $O(10^{8})$ GeV.

These results show that the model with suitable parameters can generate a
sufficient amount of baryon number asymmetry through leptogenesis even if the
reheating temperature is lower than $10^{9}$ GeV as long as $M_{N_{1}}<T_{R}$
GeV is satisfied. In the present model, both the right-handed neutrino mass
and the extra fermion mass are determined by $M_{N_{j}}=y_{N_{j}}w$ and
$\tilde{M}_{F}=\delta y_{F}w$. Since $w$ is fixed as the $PQ$ symmetry
breaking scale and $\delta>1$ is imposed for the realization of the
substantial $CP$ phases in the CKM and PMNS matrices, their mass cannot be
arbitrarily smaller than $10^{9}$ GeV under the condition that the scattering
of the extra fermions to the right-handed neutrinos is effective. Because of
this reason, low scale leptogenesis, which can be a distinguishable feature of
the model, tends to be allowed only for the case where the PQ symmetry
breaking occurs at a neighborhood of its lower bound. Even in that case,
successful leptogenesis is expected to be realized only in the range
$M_{N_{1}}>4\times 10^{6}\left(\frac{y_{F}}{10^{-1.2}}\right)^{-1/2}$ GeV for
the case $T_{R}>10^{8}$ GeV since $T$ in eq. (68) should satisfy
$T>\tilde{M}_{F},M_{N_{1}}$ for the sufficient $N_{1}$ production.

Fig. 6 Left : baryon number asymmetry generated at the expected reheating
temperature $T_{R}$ (GeV) in the case (I). It is plotted by a red solid line
for $y_{D}=10^{-1.2}$ and by green crosses for $y_{D}=10^{-1.5}$. Right :
baryon number asymmetry generated at $T_{R}=10^{9}$ GeV for various values of
$\tilde{\lambda}_{5}$ and $y_{N_{1}}$ in the case (I). In both panels, other
parameters are fixed at the ones given in (72).

In Fig. 6, we show the baryon number asymmetry $Y_{B}$ generated in the case
(I) varying the values of relevant parameters. In the left panel, $Y_{B}$ is
plotted as a function of the reheating temperature, which is fixed by the
inflaton composition and its coupling with $\phi$ and $\eta$. Two values of
$y_{F}$ are used in this plot. The $N_{1}$ production in the present scenario
depends on the reheating temperature and the couplings $y_{F},~{}y_{N_{1}}$. A
red solid line representing $Y_{B}$ is expected for the parameters given in
eq.(72) for the case (I). It becomes larger and reaches an upper bound
$Y_{B}\simeq 2.5\times 10^{-10}$ when the reheating temperature increases to
$10^{10}$ GeV. This behavior can be understood if we take into account that
the equilibrium number density of extra fermions are suppressed by the
Boltzmann factor at lower reheating temperature and then the $N_{1}$
production due to the scattering of extra fermions is suppressed. We also plot
$Y_{B}$ for a smaller value of $y_{F}$ by green crosses at some typical
$T_{R}$. They show that $Y_{B}$ takes smaller values for a smaller $y_{F}$
since the $N_{1}$ production cross section is proportional to
$y_{F}^{2}$.wwwSince the effect of Boltzmann suppression caused by its mass
$\tilde{M}_{F}=\delta y_{F}w$ could be dominant at lower $T_{R}$ compared with
the effect on the cross section, the smaller $y_{F}$ gives a larger $Y_{B}$ at
$T_{R}<10^{8}$ GeV in this case. In the right panel, $Y_{B}$ is plotted by
varying $|\tilde{\lambda}_{5}|$ and $y_{N_{1}}$. A red solid line represents
it as a function of $|\tilde{\lambda}_{5}|$ for a fixed $y_{N_{1}}=5\times
10^{-3}$. Since the neutrino oscillation data have to be imposed on eq. (21),
Yukawa couplings $h_{2,3}$ are settled by $|\tilde{\lambda}_{5}|$, $y_{N_{2}}$
and $y_{N_{3}}$. The $CP$ asymmetry $\varepsilon$ and the washout of the
generated lepton number asymmetry are mainly determined by $h_{2,3}$ for the
fixed $y_{N_{2}}$ and $y_{N_{3}}$ as found from eq.(70). Since a smaller
$|\tilde{\lambda}_{5}|$ makes $h_{2,3}$ larger and then both $\varepsilon$ and
washout larger, $Y_{B}$ takes a maximum value for a certain
$|\tilde{\lambda}_{5}|$, which is found in the figure. We also plot $Y_{B}$ by
varying $y_{N_{1}}$ for a fixed $|\tilde{\lambda}_{5}|$ in the same panel. A
smaller $y_{N_{1}}$ makes the $N_{1}$ production less effective for a fixed
$y_{F}$ and then its lower bound is expected to appear for successful
leptogenesis. It gives the lower bound of $M_{N_{1}}$ as $\sim 4\times 10^{6}$
GeV as predicted above.

Although other parameters are fixed at the ones given in (72) in these
figures, it is useful to give remarks on their dependence here. If $\delta$
takes a larger value, the mass of extra fermions $\tilde{M}_{F}$ becomes
larger to suppress the reaction density $\gamma_{F}$ due to the Boltzmann
factor. As a result, the $N_{1}$ number density generated through the
scattering becomes smaller and the resulting $Y_{B}$ also becomes smaller. If
$h_{1}$ is much smaller, $N_{1}$ decay delays and the entropy produced through
the decay of relic $N_{1}$ might dilute the generated lepton number asymmetry.

###  4.2 Dark matter and isocurvature fluctuations

This model has two DM candidates. One is the lightest neutral component of
$\eta$ with $Z_{2}$ odd parity which is an indispensable ingredient of the
model. It is known to be a good DM candidate which does not cause any
contradiction with known experimental data as long as its mass is in the TeV
range where the coannihilation can be effective [49, 48, 32]. As found from
Fig. 1, both the DM abundance and the DM direct search bound can be satisfied
if the couplings $\tilde{\lambda}_{3}$ and $|\lambda_{4}|$ take suitable
values of $O(1)$. Although these parameters could affect the perturbativity of
the scalar quartic couplings through the radiative corrections, we can safely
escape such problems in certain parameter regions. The results obtained for
the case (I) in the previous part are derived by supposing that the required
DM is $\eta_{R}$.

Axion is another promising candidate in the model. However, the axion could be
a dominant component of DM only for $f_{a}\sim 10^{11}$ GeV although it
depends on the contribution from the axion string decay [50]. We consider the
case (II) as such an example. As described before, the PQ symmetry is
spontaneously broken during the inflation since the inflaton contains the
radial component of $\sigma$. As a result, the axion appears as the phase
$\theta$ of $\sigma$. Since the axion potential is flat during the inflation,
the axion gets a quantum fluctuation $\delta A=\left(H/2\pi\right)^{2}$ and it
can cause isocurvature fluctuation in the CMB amplitude [51, 52]. A
canonically normalized axion $A$ is defined by noting eq. (26) as

$\frac{\partial
A}{\partial\theta}=\frac{\tilde{\sigma}}{\Omega^{2}}\sqrt{\Omega^{2}+6\xi_{\sigma}\frac{\tilde{\sigma}^{2}}{M_{\rm
pl}^{2}}}\simeq\frac{\sqrt{\chi M_{\rm
pl}}}{\tilde{\xi}_{S}^{1/4}}\frac{|\kappa_{\sigma
S}|}{\tilde{\kappa}_{\sigma}}\equiv\chi_{\rm iso}.$ (74)

Since the axion interacts with other fields very weakly, it causes the
isocurvature fluctuation as the fluctuation of its number density $n_{A}$. The
amplitude of its power spectrum can be expressed as

${\cal P}_{i}(k)=\Big{\langle}\left|\frac{\delta
n_{A}}{n_{A}}\right|^{2}\Big{\rangle}=\frac{H_{k}^{2}}{\pi^{2}\chi_{\rm
iso}^{2}\langle\theta^{2}\rangle}.$ (75)

Since the axion is only a source of the isocurvature fluctuation in this
model, its fraction in the power spectrum is given as

$\alpha=\frac{R_{a}^{2}{\cal P}_{i}(k)}{R_{a}^{2}{\cal P}_{i}(k)+{\cal
P}_{s}(k)}\simeq 8~{}\epsilon~{}\tilde{\xi}_{S}^{1/2}\frac{M_{\rm
pl}}{\chi_{k}}\frac{R_{a}^{2}}{\langle\theta^{2}\rangle}\left(\frac{\tilde{\kappa}_{\sigma}}{\kappa_{\sigma
S}}\right)^{2},$ (76)

where ${\cal P}_{s}(k)=A_{s}$ which is given in eq.(41). $R_{a}$ is a fraction
of the axion energy density in the CDM and defined as
$R_{a}=\Omega_{a}/\Omega_{\rm CDM}$. If we use a relation [53]

$R_{a}=\frac{\langle\theta^{2}\rangle}{6\times
10^{-6}}\left(\frac{f_{a}}{10^{16}~{}{\rm GeV}}\right)^{7/6},$ (77)

we find

$\alpha=3.25\times 10^{-5}~{}\tilde{\xi}_{S}^{1/2}\frac{M_{\rm
pl}}{\chi_{k}}\left(\frac{55}{{\cal
N}_{k}}\right)^{2}\left(\frac{f_{a}}{10^{10}~{}{\rm
GeV}}\right)^{7/6}R_{a}\left(\frac{\tilde{\kappa}_{\sigma}}{\kappa_{\sigma
S}}\right)^{2}.$ (78)

Since the Planck data constrain $\alpha$ as $\alpha\leq 0.037$ at $k=0.05$
Mpc-1 [15], we have a condition for the model to be consistent with the
present observation of the isocurvature fluctuation in the CMB as

$R_{a}<\frac{67}{\tilde{\xi}_{S}^{1/2}}\frac{\chi_{k}}{M_{\rm
pl}}\left(\frac{{\cal N}_{k}}{55}\right)^{2}\left(\frac{10^{11}~{}{\rm
GeV}}{w}\right)^{7/6}\left(\frac{\kappa_{\sigma
S}}{\tilde{\kappa}_{\sigma}}\right)^{2},$ (79)

where $f_{a}=w$ is used. In the case (I), this gives no constraint and the
parameters used in the present study to estimate the reheating temperature in
Fig. 4 and the baryon number asymmetry are consistent with the observational
data. DM can be identified with the neutral component of $\eta$. On the other
hand, in the case (II), the isocurvature condition can be satisfied for
$R_{a}<0.21$ if $|\kappa_{\sigma S}|/\tilde{\kappa}_{\sigma}=10^{-1.6}$ is
assumed and for $R_{a}<0.034$ if $|\kappa_{\sigma
S}|/\tilde{\kappa}_{\sigma}=10^{-2}$ is assumed. The isocurvature constraint
forbids the axion to be a dominant DM component and the neutral component of
$\eta$ is required to play a role of the dominant DM in this case also.

##  5 Summary

We have proposed a model which could give an explanation for the origin of the
$CP$ phases in both the CKM and PMNS matrices and the strong $CP$ problem. It
is a simple extension of the SM with vector-like extra fermions and several
scalars. In order to control the couplings of new fields, global symmetry is
imposed. If the $CP$ symmetry is spontaneously broken in a singlet scalar
sector at an intermediate scale, it can be transformed to the CKM and PMNS
matrices through the mixing between the extra fermions and the ordinary quarks
or the charged leptons. On the other hand, since the colored extra fermions
play the same role as the ones in the KSVZ model for the strong $CP$ problem,
the strong $CP$ problem could be solved through the PQ mechanism. After the
symmetry breaking due to the singlet scalars, the leptonic sector of the model
is reduced to the scotogenic model, which can explain the small neutrino
masses and the DM abundance due to the remnant discrete symmetry of the
imposed symmetry. Singlet scalars introduced to explain the $CP$ issues can
play a role of inflaton if it has a nonminimal coupling with the Ricci scalar.
We suppose this coupling is of order one. In that case, although it gives the
similar prediction for the scalar spectral index and the tensor to scalar
ratio to the one of the Higgs inflation, reheating phenomena is different from
it since the radiation domination starts just after the end of inflation.

The model has a notable phenomenological feature in addition to these. The
extra fermions which are introduced for the $CP$ issues could make the thermal
leptogenesis generate the sufficient baryon number asymmetry even if the
lightest right-handed neutrino mass is much lower than $10^{9}$ GeV, which is
the well-known lower bound of the right-handed neutrino mass for successful
leptogenesis in the ordinary seesaw scenario. Although the model allows low
scale leptogenesis, it is difficult to distinguish it from other thermal
leptogenesis model experimentally. However, if we consider its supersymmetric
extension, it could give a possibility to escape the gravitino problem. The
model is constrained by the isocurvature fluctuation which is caused by the
spontaneous breaking of the PQ symmetry during the inflation. We find that its
present observation can be consistent with the model even if the DM relic
abundance is imposed on the model. Although the relic density of axion should
be a small fraction of the DM, there is a neutral component of the inert
doublet scalar as an alternative candidate of DM in the model. It can explain
the DM abundance just as in the scotogenic model without affecting other
predictions of the model. It is remarkable that the model has potentiality to
explain various issues in the SM although the model is rather simple. It may
deserve further study.

## Appendix A A simple example for $A_{f}$

In this Appendix, we present a simple example which could bring about a phase
in the CKM matrix. In this example, we assume $w=10^{9}$ GeV and $u=10^{11}$
GeV, and also the relevant Yukawa couplings $h_{d}$, $y_{d}$ and
$\tilde{y}_{d}$ to be written by using real constant parameters as

$h_{d}=c\left(\begin{array}[]{ccc}\epsilon^{4}&\epsilon^{3}&p_{1}\epsilon^{3}\\\
\epsilon^{3}&\epsilon^{2}&p_{2}\epsilon^{2}\\\ \epsilon^{2}&p_{3}&-p_{3}\\\
\end{array}\right),\quad y_{d}=(0,y,0),\quad\tilde{y}_{d}=(0,0,\tilde{y}).$
(80)

As long as $\epsilon$ satisfies $\epsilon<1$, the down type quark mass matrix
$m_{d}(\equiv h_{d}\langle\tilde{\phi}\rangle)$ has hierarchical mass
eigenvalues. Here, we introduce $X_{ij}$ and $Y_{ij}$ whose definition is
given as

$\displaystyle
X_{ij}=1+p_{i}p_{j}+\frac{y^{2}+\tilde{y}^{2}p_{i}p_{j}+y\tilde{y}(p_{i}+p_{j})\cos
2\rho}{y^{2}+\tilde{y}^{2}}\left(1-\frac{1}{\delta^{2}}\right),$
$\displaystyle Y_{ij}=\frac{y\tilde{y}(p_{i}-p_{j})\sin
2\rho}{y^{2}+\tilde{y}^{2}}\left(1-\frac{1}{\delta^{2}}\right),$ (81)

where $\delta$ is defined as $\delta=\tilde{M}_{F}/\mu_{F}$. If we define
$R_{ij}$ and $\theta_{ij}$ by using these quantities as

$R_{ij}=\sqrt{X_{ij}^{2}+Y_{ij}^{2}},\quad\tan\theta_{ij}=\frac{Y_{ij}}{X_{ij}},$
(82)

the component of eq. (13) is found to be expressed under the assumption
$\mu_{D}^{2}<{\cal F}_{d}{\cal F}_{d}^{\dagger}$ as

$(A_{d}^{-1}m^{2}A_{d})_{ij}=c^{2}\langle\tilde{\phi}\rangle^{2}\epsilon_{ij}R_{ij}e^{i\theta_{ij}},$
(83)

where $\epsilon_{ij}$ is defined as

$\epsilon_{11}=\epsilon^{6},\quad\epsilon_{22}=\epsilon^{4},\quad\epsilon_{33}=1,\quad\epsilon_{12}=\epsilon_{21}=\epsilon^{5},\quad\epsilon_{13}=\epsilon_{31}=\epsilon^{3},\quad\epsilon_{23}=\epsilon_{32}=\epsilon^{2}.$
(84)

By solving eq. (83), we find that $A_{d}$ is approximately written as

$A_{d}\simeq\left(\begin{array}[]{ccc}1&-\lambda&\lambda^{3}\left(\frac{X_{23}}{|\alpha|^{2}X_{33}}e^{i\vartheta}-\frac{X_{13}}{|\alpha|^{3}X_{33}}\right)\\\
\lambda&1&-\lambda^{2}\frac{X_{23}}{|\alpha|^{2}X_{33}}e^{i\vartheta}\\\
\lambda^{3}\frac{X_{13}}{|\alpha|^{3}X_{33}}&\lambda^{2}\frac{X_{23}}{|\alpha|^{2}X_{33}}e^{-i\vartheta}&1\\\
\end{array}\right),$ (85)

where the constants $\lambda$, $\alpha$, and $\vartheta$ are defined by

$\alpha=\frac{X_{12}X_{33}-X_{13}X_{23}e^{-i(\theta_{23}+\theta_{12}-\theta_{13})}}{X_{22}X_{33}-X_{23}^{2}},\quad\lambda=|\alpha|\epsilon,\quad\vartheta=\arg(\alpha)+\theta_{23}+\theta_{12}-\theta_{13}.$
(86)

This expression shows that $A_{d}$ could have a nontrivial phase which gives
the origin of the CKM phase. If the diagonalization matrix $O^{L}$ for a mass
matrix of the up type quarks takes an almost diagonal form, the CKM matrix
could be obtained as $V_{\rm CKM}\simeq A_{d}$. As an example, if we assume
$\cos\rho=\frac{\pi}{4}$ and fix other parameters as

$\displaystyle y_{F}=10^{-1.5},\quad\delta=\sqrt{3},\quad y=4\times
10^{-4},\quad\tilde{y}=2\times 10^{-4},$ $\displaystyle p_{1}=1.1,\quad
p_{2}=-0.9,\quad p_{3}=1,\quad\epsilon=0.2,\quad c=0.014,$ (87)

then we obtain $\lambda=0.22$ and the Jarlskog invariant [54] as $J(\equiv{\rm
Im}[A_{12}A_{13}^{\ast}A_{23}A_{22}^{\ast}])=-1.6\times 10^{-6}$. The mass
eigenvalues for the down type quarks are obtained as

$\displaystyle
m_{d}=\left\\{X_{11}-\frac{X_{13}^{2}}{X_{33}}+|\alpha|^{2}\left(X_{22}-\frac{X_{23}^{2}}{X_{33}}-2\right)\right\\}^{1/2}\epsilon^{3}c\langle\tilde{\phi}\rangle\simeq
3.3~{}{\rm MeV},$ $\displaystyle
m_{s}=\left(X_{22}-\frac{X_{23}^{2}}{X_{33}}\right)^{1/2}\epsilon^{2}c\langle\tilde{\phi}\rangle\simeq
138~{}{\rm MeV},$ $\displaystyle
m_{b}=X_{33}^{1/2}c\langle\tilde{\phi}\rangle\simeq 4.2~{}{\rm GeV}.$ (88)

Although a diagonalization matrix $A_{e}$ for the charged lepton sector may be
considered to take the same form as $A_{d}$, it is not favorable for a large
$CP$ phase in the PMNS matrix. In that case, since $A_{e}$ is the nearly
diagonal, large flavor mixing has to be caused only by the neutrino sector and
the Dirac $CP$ phase in the PMNS matrix becomes small as a result. A large
$CP$ phase in the PMNS matrix requires $A_{e}$ to have rather large off-
diagonal elements also.

## Appendix B RGEs for coupling constants

In order to examine both the stability and the perturbativity of the model
from the weak scale to the Planck scale, we have to know the running of the
coupling constants in eq. (1). If we fix these coupling constants at an
intermediate scale and solve the RGEs to the Planck scale, we can find their
values throughout the scale. The one-loop RGEs of the relevant coupling
constants are given as

$\displaystyle
16\pi^{2}\mu\frac{\partial\lambda_{1}}{\partial\mu}=24\lambda_{1}^{2}+\lambda_{3}^{2}+(\lambda_{3}+\lambda_{4})^{2}+\kappa_{\phi\sigma}^{2}+\kappa_{\phi
S}^{2}+\frac{3}{8}\left(3g^{4}+g^{\prime 4}+2g^{2}g^{\prime 2}\right)$
$\displaystyle\hskip 56.9055pt-3\lambda_{1}\left(3g^{2}+g^{\prime
2}-4h_{t}^{2}\right)-6h_{t}^{4},$ $\displaystyle
16\pi^{2}\mu\frac{\partial\lambda_{2}}{\partial\mu}=24\lambda_{2}^{2}+\lambda_{3}^{2}+(\lambda_{3}+\lambda_{4})^{2}+\kappa_{\eta\sigma}^{2}+\kappa_{\eta
S}^{2}+\frac{3}{8}\left(3g^{4}+g^{\prime 4}+2g^{2}g^{\prime 2}\right)$
$\displaystyle\hskip 56.9055pt-3\lambda_{2}\left(3g^{2}+g^{\prime
2}\right)+4\lambda_{2}\left(2h_{2}^{2}+3h_{3}^{2}\right)-8h_{2}^{4}-18h_{3}^{4},$
$\displaystyle
16\pi^{2}\mu\frac{\partial\lambda_{3}}{\partial\mu}=2(\lambda_{1}+\lambda_{2})(6\lambda_{3}+2\lambda_{4})+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\kappa_{\phi\sigma}\kappa_{\eta\sigma}+2\kappa_{\phi
S}\kappa_{\eta S}$ $\displaystyle\hskip
56.9055pt+\frac{3}{4}\left(3g^{4}+g^{\prime 4}-2g^{2}g^{\prime
2}\right)-3\lambda_{3}\left(3g^{2}+g^{\prime
2}\right)+2\lambda_{3}\left(3h_{t}^{2}+2h_{2}^{2}+3h_{3}^{2}\right),$
$\displaystyle
16\pi^{2}\mu\frac{\partial\lambda_{4}}{\partial\mu}=4(\lambda_{1}+\lambda_{2})\lambda_{4}+8\lambda_{3}\lambda_{4}+4\lambda_{4}^{2}+3g^{2}g^{\prime
2}-3\lambda_{4}\left(3g^{2}+g^{\prime 2}\right)$ $\displaystyle\hskip
56.9055pt+2\lambda_{4}\left(3h_{t}^{2}+2h_{2}^{2}+3h_{3}^{2}\right),$
$\displaystyle
16\pi^{2}\mu\frac{\partial\kappa_{S}}{\partial\mu}=20\kappa_{S}^{2}+\kappa_{\sigma
S}^{2}+2(\kappa_{\phi S}^{2}+\kappa_{\eta
S}^{2})+4\kappa_{S}\left[3(y_{d}^{2}+\tilde{y}_{d}^{2})+y_{e}^{2}+\tilde{y}_{e}^{2}\right]$
$\displaystyle\hskip
56.9055pt-2\left[3(y_{d}^{2}+\tilde{y}_{d}^{2})^{2})+(y_{e}^{2}+\tilde{y}_{e}^{2})^{2}\right],$
$\displaystyle
16\pi^{2}\mu\frac{\partial\kappa_{\sigma}}{\partial\mu}=20\kappa_{\sigma}^{2}+\kappa_{\sigma
S}^{2}+2(\kappa_{\phi\sigma}^{2}+\kappa_{\eta\sigma}^{2})+4\kappa_{\sigma}\left(3y_{D}^{2}+y_{E}^{2}+\frac{1}{2}y_{N_{3}}^{2}\right)-2\left(3y_{D}^{4}+y_{E}^{4}+\frac{1}{2}y_{N_{3}}^{4}\right),$
$\displaystyle 16\pi^{2}\mu\frac{\partial\kappa_{\sigma
S}}{\partial\mu}=4\kappa_{\sigma
S}^{2}+8(\kappa_{S}+\kappa_{\sigma})\kappa_{\sigma S}+2(\kappa_{\phi
S}\kappa_{\phi\sigma}+\kappa_{\eta S}\kappa_{\eta\sigma})$
$\displaystyle\hskip 56.9055pt+2\kappa_{\sigma
S}\left[3(y_{d}^{2}+\tilde{y}_{d}^{2})+y_{e}^{2}+\tilde{y}_{e}^{2}+3y_{D}^{2}+y_{E}^{2}+\frac{1}{2}y_{N_{3}}^{2}\right]-4[3y_{D}^{2}(y_{d}^{2}+\tilde{y}_{d}^{2})+y_{E}^{2}(y_{e}^{2}+\tilde{y}_{e}^{2})],$
$\displaystyle
16\pi^{2}\mu\frac{\partial\kappa_{\phi\sigma}}{\partial\mu}=4\kappa_{\phi\sigma}^{2}+2\kappa_{\sigma
S}\kappa_{\phi
S}+2\kappa_{\eta\sigma}(2\lambda_{3}+\lambda_{4})+4\kappa_{\phi\sigma}(3\lambda_{1}+2\kappa_{\sigma})$
$\displaystyle\hskip
56.9055pt+2\kappa_{\phi\sigma}\left(3y_{D}^{2}+y_{E}^{2}+\frac{1}{2}y_{N_{3}}^{2}+3h_{t}^{2}-\frac{9}{4}g^{2}-\frac{3}{4}g^{\prime
2}\right),$ $\displaystyle
16\pi^{2}\mu\frac{\partial\kappa_{\eta\sigma}}{\partial\mu}=4\kappa_{\eta\sigma}+2\kappa_{\sigma
S}\kappa_{\eta
S}+2\kappa_{\phi\sigma}(2\lambda_{3}+\lambda_{4})+4\kappa_{\eta\sigma}(3\lambda_{2}+2\kappa_{\sigma})$
$\displaystyle\hskip
56.9055pt+2\kappa_{\eta\sigma}\left(3y_{D}^{2}+y_{E}^{2}+\frac{1}{2}y_{N_{3}}^{2}+2h_{2}^{2}+3h_{3}^{2}-\frac{9}{4}g^{2}-\frac{3}{4}g^{\prime
2}\right)-4(h_{2}^{2}+h_{3}^{2})y_{N_{3}}^{2}$ $\displaystyle
16\pi^{2}\mu\frac{\partial\kappa_{\phi S}}{\partial\mu}=4\kappa_{\phi
S}^{2}+2\kappa_{\sigma S}\kappa_{\phi\sigma}+2\kappa_{\eta
S}(2\lambda_{3}+\lambda_{4})+4\kappa_{\phi S}(3\lambda_{1}+2\kappa_{S})$
$\displaystyle\hskip 56.9055pt+2\kappa_{\phi
S}\left[3(y_{d}^{2}+\tilde{y}_{d}^{2})+y_{e}^{2}+\tilde{y}_{e}^{2}+3h_{t}^{2}-\frac{9}{4}g^{2}-\frac{3}{4}g^{\prime
2}\right],$ $\displaystyle 16\pi^{2}\mu\frac{\partial\kappa_{\eta
S}}{\partial\mu}=4\kappa_{\eta S}^{2}+2\kappa_{\sigma
S}\kappa_{\eta\sigma}+2\kappa_{\phi S}(2\lambda_{3}+\lambda_{4})+4\kappa_{\eta
S}(3\lambda_{2}+2\kappa_{S})$ $\displaystyle\hskip 56.9055pt+2\kappa_{\eta
S}\left[3(y_{d}^{2}+\tilde{y}_{d}^{2})+y_{e}^{2}+\tilde{y}_{e}^{2}+2h_{2}^{2}+3h_{3}^{2}-\frac{9}{4}g^{2}-\frac{3}{4}g^{\prime
2}\right],$ $\displaystyle 16\pi^{2}\mu\frac{\partial
y_{d}}{\partial\mu}=y_{d}\left[-8g_{s}^{2}-\frac{2}{3}g^{\prime
2}+\frac{1}{2}y_{D}^{2}+4y_{d}^{2}+3\tilde{y}_{d}^{2}+y_{e}^{2}+\tilde{y}_{e}^{2}\right],$
$\displaystyle
16\pi^{2}\mu\frac{\partial\tilde{y}_{d}}{\partial\mu}=\tilde{y}_{d}\left[-8g_{s}^{2}-\frac{2}{3}g^{\prime
2}+\frac{1}{2}y_{D}^{2}+3y_{d}^{2}+4\tilde{y}_{d}^{2}+y_{e}^{2}+\tilde{y}_{e}^{2}\right],$
(89) $\displaystyle 16\pi^{2}\mu\frac{\partial
y_{e}}{\partial\mu}=y_{e}\left[-6g^{\prime
2}+\frac{1}{2}y_{E}^{2}+3(y_{d}^{2}+\tilde{y}_{d}^{2})+2y_{e}^{2}+\tilde{y}_{e}^{2}\right],$
$\displaystyle
16\pi^{2}\mu\frac{\partial\tilde{y}_{e}}{\partial\mu}=\tilde{y}_{e}\left[-6g^{\prime
2}+\frac{1}{2}y_{E}^{2}+3(y_{d}^{2}+\tilde{y}_{d}^{2})+y_{e}^{2}+2\tilde{y}_{e}^{2}\right],$
$\displaystyle 16\pi^{2}\mu\frac{\partial
y_{D}}{\partial\mu}=y_{D}\left(-8g_{s}^{2}-\frac{2}{3}g^{\prime
2}+4y_{D}^{2}+y_{E}^{2}+\frac{1}{2}y_{d}^{2}+\frac{1}{2}\tilde{y}_{d}^{2}+\frac{1}{2}y_{N_{3}}^{2}\right),$
$\displaystyle 16\pi^{2}\mu\frac{\partial
y_{E}}{\partial\mu}=y_{E}\left(-6g^{\prime
2}+3y_{D}^{2}+2y_{E}^{2}+\frac{1}{2}y_{e}^{2}+\frac{1}{2}\tilde{y}_{e}^{2}+\frac{1}{2}y_{N_{3}}^{2}\right),$
$\displaystyle 16\pi^{2}\mu\frac{\partial
y_{N_{3}}}{\partial\mu}=y_{N_{3}}\left[3y_{D}^{2}+y_{E}^{2}+\frac{3}{2}y_{N_{3}}^{2}+2(h_{2}^{2}+h_{3}^{2})\right],$
$\displaystyle 16\pi^{2}\mu\frac{\partial
h_{2}}{\partial\mu}=h_{2}\left(-\frac{9}{4}g^{2}-\frac{3}{4}g^{\prime
2}+5h_{2}^{2}+3h_{3}^{2}\right),$ $\displaystyle 16\pi^{2}\mu\frac{\partial
h_{3}}{\partial\mu}=h_{3}\left(-\frac{9}{4}g^{2}-\frac{3}{4}g^{\prime
2}+2h_{2}^{2}+\frac{15}{2}h_{3}^{2}+\frac{1}{2}y_{N_{3}}^{2}\right),$
$\displaystyle 16\pi^{2}\mu\frac{\partial
h_{t}}{\partial\mu}=h_{t}\left(\frac{9}{2}h_{t}^{2}-8g_{s}^{2}-\frac{9}{4}g^{2}-\frac{17}{12}g^{\prime
2}\right),$ $\displaystyle 16\pi^{2}\mu\frac{\partial
g_{s}}{\partial\mu}=-\frac{19}{3}g_{s}^{3},\qquad 16\pi^{2}\mu\frac{\partial
g}{\partial\mu}=-3g^{3},\qquad 16\pi^{2}\mu\frac{\partial
g^{\prime}}{\partial\mu}=\frac{79}{9}g^{\prime 3},$

where $g_{s}$, $g$, and $g^{\prime}$ are the gauge coupling constants of the
SM. In these equations, we assume eq. (20) for neutrino Yukawa couplings and
eq. (80) for $y_{f}$ and $\tilde{y}_{f}$, and also only $y_{N_{3}}$ is taken
into account. Contributions from $V_{b}$ are also neglected in these RGEs.

Initial values of the coupling constants which are used in the RGE study of
the right panel of Fig. 1 are taken to be the same ones which are used in the
analysis of the leptogenesis for the case (I). A part of them are fixed at the
weak scale as

$\displaystyle\tilde{\lambda}_{1}=0.13,\quad\tilde{\lambda}_{2}=0.1,\quad\tilde{\lambda}_{3}=0.445,\quad\lambda_{4}=-0.545,\quad\lambda_{5}=-10^{-3},$
$\displaystyle h_{1}=6\times 10^{-7},\quad h_{2}=8.3\times 10^{-3},\quad
h_{3}=3.9\times 10^{-3},$ (90)

where $\tilde{\lambda}_{3}$ and $\lambda_{4}$ are fixed by the DM constraint
shown in the left panel of Fig. 1 and $h_{2,3}$ are fixed by using the
nautrino mass formula and the neutrino oscillation data. Remaining ones are
fixed at a scale $\bar{M}$ as

$\displaystyle\tilde{\kappa}_{S}=10^{-8},\quad\tilde{\kappa}_{\sigma}=10^{-4.5},\quad|\kappa_{\sigma
S}|=10^{-6.1},\quad\kappa_{\phi\sigma}=\kappa_{\eta\sigma}=10^{-4},\quad\kappa_{\phi
S}=\kappa_{\eta S}=10^{-7}$ $\displaystyle y_{N_{1}}=7\times 10^{-3}.\quad
y_{N_{2}}=2\times 10^{-2},\quad y_{N_{3}}=4\times 10^{-2},$ $\displaystyle
y_{D}=y_{E}=10^{-1.2},\quad y_{e}=y_{d}=8\times
10^{-4},\quad\tilde{y}_{e}=\tilde{y}_{d}=4\times 10^{-4}.$ (91)

These satisfy the imposed conditions (5), (14), (17), (31) and (42).

## Acknowledgements

This work is partially supported by a Grant-in-Aid for Scientific Research (C)
from Japan Society for Promotion of Science (Grant No. 18K03644). N. S. R is
supported by Program 5000 Doktor under Ministry of Religious Affairs (MORA),
Republic of Indonesia.

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