KANAZAWA-22-05

November, 2022

Subcritical regime of hybrid inflation
with modular $A_{4}$ symmetry

Yoshihiro Gunji,(a) Koji Ishiwata,(a) Takahiro Yoshida${}^{\it(b),\,(c)}$

(a)Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192,
Japan

(b)Department of Information, Kaishi Professional University, Niigata
950-0916, Japan

(c)Department of Physics, Niigata University, Niigata 950-2181, Japan

##  1 Introduction

Inflation, a hypothesis of accelerated expansion of the Universe, is supported
by the observation of the cosmic microwave background (CMB) radiation. It is
driven by the vacuum energy at the early state of the Universe and it is
realized by a slowly-rolling scalar field, called inflaton. According to
decades of the observations and analysis of the CMB, some properties of
inflation have been revealed. The latest results by Planck collaboration [1]
report that the amplitude $A_{s}$ and spectral index $n_{s}$ of the scalar
perturbation and the ratio $r$ of the tensor mode to the scalar mode are given
by $A_{s}=(2.0989\pm 0.029)\times 10^{-9}~{}~{}(68\%\,{\rm CL})$,
$n_{s}=0.9649\pm 0.0042~{}~{}(68\%\,{\rm CL})$, and $r<0.056~{}~{}(95\%\,{\rm
CL})$. The results already exclude some of inflation models and future
experiments are expected to measure the observables with more precision.

Besides the accelerated expansion of the Universe, the inflaton field plays
another important role, i.e., reheating to create the radiation dominated
Universe. Then the production of dark matter and baryons follows. Thus, it is
tempting to consider an inflation model that provide the sequence of the
thermal history after inflation. Moreover, the model would be motivated if it
is controlled by an underlying symmetry. The modular symmetry and
supersymmetry are the promising candidates.

The modular symmetry is the discrete symmetry associated with the
compactification of the extra dimensions inspired by superstring theory. It
has recently caught attention from the phenomenological point of view since
Ref. [2] pointed out that it gives a consistent pattern of the neutrino mixing
data [3, 4, 5, 6, 7, 8]. Variety types of symmetries have been studied so far;
for instance, the modular $S_{3}$ [9], $A_{4}$ [2, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], $A_{5}$
[53, 54, 55], and other modular groups [56, 57, 58, 59]. Quark masses and
mixings are investigated in Refs. [60, 61, 62, 63]. In the modular symmetric
framework, soft supersymmetry breaking terms [64] and modular stabilization
[65, 66, 67, 68, 69] are also discussed. For other phenomenological
applications, for example, the application of modular symmetry to grand-
unified theories is discussed in Refs. [70, 71, 72, 73, 74, 75, 76, 77, 78].
Moreover, in cosmology, models with dark matter candidates [79, 80, 81, 82,
83, 84, 85, 86] have been considered and Refs. [87, 88, 89, 90, 91, 92, 93,
94, 95] have applied it to leptogenesis.

In the present study we apply the modular $A_{4}$ symmetry to inflation.
Introducing three right-handed (s)neutrinos, we consider one of right-handed
sneutrinos plays the role of the inflaton field. Since the typical model of
the right-handed sneutrino inflation tends to be chaotic inflation due to the
quadratic term, which is already excluded by the CMB observations, we extend
the model to the $D$-term hybrid inflation by introducing a new Yukawa term.
These days variety of new $D$-term hybrid inflation models have been proposed,
depending on the symmetry of the Kähler potential; $R^{2}$ Starobinsky model
[96, 97] appears from the superconformal symmetry [98], the chaotic regime
emerges from the shift symmetry [99, 100], and $\alpha$-attractor [101] comes
from their combination, which is called superconformal subcritical hybrid
inflation [102]. A generalized version of the inflation model proposed in Ref.
[102] is intensitively analyzed in Ref. [103]. In our paper we study the
neutrino mixing and the dynamics of inflation based on the model in Ref.
[103]. In the framework, an unconventional neutrino mixing pattern and the CP
phases are obtained. We find that only the inverted hierarchy is consistent
with the data and in the valid parameters the upper limit for the scale of the
Majorana mass is obtained for successful inflation.

This paper is organized as follows. In Sec. 2, we introduce the supergravity
model to study. Then neutrino masses and mixing pattern are discussed in Sec.
3. Here we show the predicted CP phases and the effective neutrino mass for
the neutrinoless double beta decay. In Sec. 4, the dynamics of inflation is
studied both analytically and numerically, especially focusing on the impact
of the Majorana mass terms. Sec. 5 contains our conclusion.

##  2 The model

###  2.1 Brief review of modular symmetry

The modular symmetry is the geometric symmetry of the two dimensional torus.
The two dimensional torus $T^{2}$ is defined by $\mathbb{C}/\Lambda$, i.e.,
the complex plane $\mathbb{C}$ divided by the two dimensional lattice
$\Lambda=\\{\sum_{i=1}^{2}n_{i}\omega_{i}\,|\,n_{i}\in\mathbb{Z}\\}$ with
basis vectors $\omega_{i}\in\mathbb{C}$. Here, the basis vectors are given by
$\omega_{1}=2\pi R$ and $\omega_{2}=2\pi R\tau$ with $R\in\mathbb{R}$, and
$\tau=\omega_{2}/\omega_{1}$ is the modulus defined in the upper half plane
$\mathcal{H}=\\{\tau\in\mathbb{C}\,|\,\mathrm{Im}\tau>0\\}$. The same lattice
is constructed using the basis vectors transformed as

$\displaystyle\begin{pmatrix}\omega_{2}^{\prime}\\\
\omega_{1}^{\prime}\end{pmatrix}$
$\displaystyle=\gamma\begin{pmatrix}\omega_{2}\\\
\omega_{1}\end{pmatrix}\,,\qquad\gamma\in SL(2,\mathbb{Z})\,,$ (2.1)

where

$\displaystyle\Gamma\equiv
SL(2,\mathbb{Z})\equiv\biggl{\\{}\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\,\biggl{|}\biggr{.}\,\,a,b,c,d\in\mathbb{Z},\ ad-
bc=1\biggr{\\}}\,.$ (2.2)

Then, the modulus is transformed as

$\displaystyle\tau\to\gamma\tau=\frac{a\tau+b}{c\tau+d}\,.$ (2.3)

Eq. (2.3) is called modular transformation. It is seen that the transformation
law of $\tau$ is the same for $\gamma$ and $-\gamma$. Then, the group of
modular transformation is isomorphic to $PSL(2,\mathbb{Z})\equiv
SL(2,\mathbb{Z})/\pm\mathbb{I}\equiv\bar{\Gamma}$, which is called modular
group. The modular group is generated by two generators,

$\displaystyle S$ $\displaystyle=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad
T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\,,$ (2.4)

satisfying

$\displaystyle(ST)^{3}=S^{2}=\mathbb{I}\,.$ (2.5)

The modulus $\tau$ is transformed by $S$ and $T$ as

$\displaystyle\begin{split}S&:\,\tau\to-1/\tau\,,\\\
T&:\,\tau\to\tau+1\,.\end{split}$ (2.6)

For a positive integer $N$, the principal congruence subgroup of level $N$ is
defined by

$\displaystyle\Gamma(N)$ $\displaystyle=\biggl{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\ \biggl{|}\biggr{.}\,\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\,(\mathrm{mod}\,N)\biggr{\\}}\,.$ (2.7)

Note that $\Gamma(1)=\Gamma$ and $-\mathbb{I}\not\in\Gamma(N)$ for $N>2$.
Then, we introduce the groups $\bar{\Gamma}(N)$ as
$\bar{\Gamma}(2)=\Gamma(2)/\pm\mathbb{I}$ and
$\bar{\Gamma}(N)=\Gamma(N)\,(N>2)$. The quotient groups
$\Gamma_{N}\equiv\bar{\Gamma}/\bar{\Gamma}(N)$ are called finite modular
groups and the generators of $\Gamma_{N}$ have the additional relation,

$\displaystyle T^{N}=\mathbb{I}\,.$ (2.8)

It is known that the finite modular groups for $N=2,3,4,5$ are isomorphic to
the non-Abelian discrete groups, $\Gamma_{2}\simeq S_{3}$, $\Gamma_{3}\simeq
A_{4}$, $\Gamma_{4}\simeq S_{4}$, $\Gamma_{5}\simeq A_{5}$, respectively[104].
Under the modular transformation, the holomorphic functions of $\tau$, called
the modular forms are transformed as

$\displaystyle f(\tau)$ $\displaystyle\to
f(\gamma\tau)=(c\tau+d)^{k}\rho(\gamma)f(\tau)\,.$ (2.9)

Here $\rho(\gamma)$ is a unitary transformation of $\Gamma_{N}$ $(N>2)$ and
$k$ is non-negative even integer, called modular weight.

In this study, we focus on $\Gamma_{3}$, i.e., $A_{4}$. Then, the modular
forms $Y=(Y_{1},Y_{2},Y_{3})^{T}$ with representation of $A_{4}$ triplet and
modular weight $2$ are given by [2]

$\displaystyle Y_{1}$
$\displaystyle=\frac{i}{2\pi}\biggl{[}\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}-\frac{27\eta^{\prime}(3\tau)}{\eta(3\tau)}\biggr{]}\,,$
$\displaystyle Y_{2}$
$\displaystyle=\frac{-i}{\pi}\biggl{[}\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\omega\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\biggr{]}\,,$
(2.10) $\displaystyle Y_{3}$
$\displaystyle=\frac{-i}{\pi}\biggl{[}\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\biggr{]}\,,$

where $\omega=e^{2\pi i/3}$ and $\eta(\tau)$ is the Dedekind eta-function
defined by

$\displaystyle\eta(\tau)$
$\displaystyle=q^{1/24}\prod_{i=1}^{\infty}(1-q^{n}),\quad q=e^{2\pi
i\tau}\,.$ (2.11)

The modular forms $Y_{i}\,(i=1\text{--}3)$ satisfy the relation,

$\displaystyle Y_{2}^{2}+2Y_{1}Y_{3}=0\,.$ (2.12)

They are transformed under $T$ and $S$ transformations as

$\displaystyle\begin{split}S&:\,Y(-1/\tau)=\tau^{2}\rho(S)Y\,,\\\
T&:\,Y(\tau+1)=\rho(T)Y\,,\end{split}$ (2.13)

where

$\displaystyle\rho(S)$ $\displaystyle=\frac{1}{3}\begin{pmatrix}-1&2&2\\\
2&-1&2\\\ 2&2&-1\end{pmatrix}\,,\quad\rho(T)=\begin{pmatrix}1&&\\\ &\omega&\\\
&&\omega^{2}\end{pmatrix}\,.$ (2.14)

It is assumed that the superfields, denoted as $Z^{I}$, are also transformed
under $A_{4}$ as

$\displaystyle Z^{I}\to(c\tau+d)^{-k_{I}}\rho^{(I)}(\gamma)Z^{I}\,,$ (2.15)

where the modular weight $k_{I}$ will be determined later.

###  2.2 The superpotential and Kähler potential

| $L$ | $E^{c}=(e^{c},\,\mu^{c},\,\tau^{c})$ | $N^{c}$ | $H_{u}$ | $H_{d}$ | $S_{\pm}$
---|---|---|---|---|---|---
U(1) | $0$ | $0$ | $0$ | $0$ | $0$ | $\pm q$
$A_{4}$ | $\boldsymbol{3}$ | $(\boldsymbol{1},\,\boldsymbol{1}^{\prime\prime},\,\boldsymbol{1}^{\prime})$ | $\boldsymbol{3}$ | $\boldsymbol{1}$ | $\boldsymbol{1}$ | $\boldsymbol{1}$
weight | $-k_{L}$ | $(-k_{e^{c}},-k_{\mu^{c}},-k_{\tau^{c}})$ | $-k_{N^{c}}$ | $-k_{H_{u}}$ | $-k_{H_{d}}$ | $-k_{S_{\pm}}$
Table 1: Field contents and their representations and modular weights.

We consider a supergravity model of inflation inspired by modular $A_{4}$
symmetry. In addition to the fields in the minimal supersymmetric standard
model (MSSM), we introduce three right-handed neutrinos
$N^{c}_{i}\,\,(i=1\text{--}3)$ and two new fields $S_{\pm}$. Here $S_{\pm}$
are gauge singlets under the MSSM gauge but are charged under a local U(1)
with charge $\pm q\,(q>0)$. They play important roles during inflation.

The superpotential allowed under $A_{4}$ symmetry is given by

$\displaystyle W$ $\displaystyle\supset W_{E}+W_{N}+W_{\rm hyb}\,,$ (2.16)

where

$\displaystyle W_{E}$
$\displaystyle=\alpha_{1}e^{c}H_{d}(LY)_{\boldsymbol{1}}+\alpha_{2}\mu^{c}H_{d}(LY)_{\boldsymbol{1}^{\prime}}+\alpha_{3}\tau^{c}H_{d}(LY)_{\boldsymbol{1}^{\prime\prime}}\,,$
(2.17) $\displaystyle W_{N}$
$\displaystyle=g_{1}(N^{c}H_{u}(LY)_{\boldsymbol{3}\mathrm{s}})_{\boldsymbol{1}}+g_{2}(N^{c}H_{u}(LY)_{\boldsymbol{3}\mathrm{a}})_{\boldsymbol{1}}\,,$
(2.18) $\displaystyle W_{\rm hyb}$ $\displaystyle=\lambda
S_{+}S_{-}(N^{c}Y)_{\boldsymbol{1}}+\Lambda(N^{c}N^{c}Y)_{\boldsymbol{1}}\,.$
(2.19)

The superpotential is similar to one considered in Ref. [105], but it is
extended to accommodate the modular $A_{4}$ symmetry. The contents of the
fields are listed in Table 1. The representation and the modular weights of
the fields are also shown in the table, which is based on the model considered
in Ref. [2, 11]; the right-handed neutrinos and the left-handed lepton
doublets $L_{i}=(\nu_{Li},l_{Li})^{T}\,\,(i=e,\mu,\tau)$ are the $A_{4}$
triplets and the others are the singlets. Here the three different singlet
representations $(\mathbf{1},\mathbf{1^{\prime\prime}},\mathbf{1^{\prime}})$
are assigned to the right-handed charged leptons
$E^{c}=(e^{c},\mu^{c},\tau^{c})$, respectively. We will discuss the assignment
of the modular weights soon later. In the superpotential,
$\alpha_{i}\,\,(i=1\text{--}3)$, $g_{i}\,\,(i=1,2)$, and $\lambda$ are Yukawa
coupling constants, and $\Lambda$ determines the mass scale of the right-
handed neutrinos. By redefinition of the fields, the parameters $\alpha_{i}$,
$\lambda$, $\Lambda$, and $g_{1}$ can be taken to be real without the loss of
generality.

For Kähler potential, we adopt a class of the canonical superconformal
supergravity model [106]. This type of model can be extended to the model with
a parameter $\alpha$, which corresponds to the parameter of the superconformal
$\alpha$ attractor model [101]. Recently the dynamics of inflation has been
analyzed in a generalized version of the model [103]. In order to consider the
similar inflation model, we consider the Kähler potential based on Ref.
[103]:#1#1#1We adopt the Planck unit, where $M_{\mathrm{pl}}=1$ for the Planck
mass $M_{\mathrm{pl}}$.

$\displaystyle K$
$\displaystyle=-3\log\Bigl(-\frac{\mathcal{N}}{3}\Bigr{missing})\,,$ (2.20)

where

$\displaystyle\mathcal{N}$
$\displaystyle=-|Z^{0}|^{n}\Bigl{(}-\frac{\Phi}{3}\Bigr{)}^{\alpha}\,,\quad-\frac{\Phi}{3}=1-\sum_{I}\frac{|Z^{I}|^{2}}{|Z^{0}|^{k_{I}}}\,.$
(2.21)

Here we have introduced a positive parameter $n$ in order to discuss the
modular weights of the field generically. We take $Z^{0}$ as
$Z^{0}=\bar{Z^{0}}=2\,\mathrm{Im}\,\tau$, In general, $\Phi$ includes
additional terms, such as $(YN^{c}N^{c})_{\mathbf{1}}$ or
$(\bar{Y}\bar{L}YL)_{\mathbf{1}}$ [2, 19]. In the current study we ignore them
to focus on the simple setting.#2#2#2This is also motivated by the results
given by Ref. [103]. In the literature it is shown such a simple Käher
potential gives a consistent result with the CMB observations.

The Kähler potential is further divided into the modular and matter parts as
$K=K^{\tau}+K^{\mathrm{m}}$, where

$\displaystyle K^{\tau}$ $\displaystyle=-3n\log(2\,\mathrm{Im}\,\tau)\,,\quad
K^{\mathrm{m}}=-3\alpha\log\Bigl(-\frac{\Phi}{3}\Bigr{missing})\,.$ (2.22)

The matter part is constructed to be modular invariant. Then, under the
modular transformation (2.3), the Kähler potential is transformed as

$\displaystyle K$ $\displaystyle\to
K+3n\bigl{[}\log(c\tau+d)+\log(c\bar{\tau}+d)\bigr{]}\,.$ (2.23)

In the supergravity the combination of the Kähler potential and
superpotential, which is defined by

$\displaystyle\mathcal{G}$ $\displaystyle=K+\log{W}+\log{\overline{W}}\,,$
(2.24)

should be modular invariant. Due to the invariance, the transformation of the
superpotential is determined as,

$\displaystyle\begin{split}W&\to e^{i\alpha(\gamma)}(c\tau+d)^{-3n}W\,,\\\
\overline{W}&\to
e^{-i\alpha(\gamma)}(c\bar{\tau}+d)^{-3n}\overline{W}\,.\end{split}$ (2.25)

Namely, its modular weight is $-3n$. Consequently, the weight of fields should
satisfy

$\displaystyle\begin{split}3n&=k_{H_{u}}+k_{H_{d}}=k_{S_{+}}+k_{S_{-}}+k_{N^{c}}-2=2k_{N^{c}}-2\\\
&=k_{N^{c}}+k_{L}+k_{H_{u}}-2=k_{E^{c}}+k_{L}+k_{H_{d}}-2\,,\end{split}$
(2.26)

where $k_{e^{c}}=k_{\mu^{c}}=k_{\tau^{c}}=k_{E^{c}}$. Those are the generic
conditions for the modular weights in the present model. For instance, if we
further impose the following conditions:

$\displaystyle k_{H_{u}}=k_{H_{d}}\equiv k_{H}\,,\quad
k_{S_{+}}=k_{S_{-}}\equiv k_{S}\,,$ (2.27)

the modular weights are uniquely determined for a given $n$ as

$\displaystyle k_{L}=1\,,\quad k_{N^{c}}=k_{E^{c}}=2k_{S}=k_{H}+1=(3n+2)/2\,.$
(2.28)

The value of $n$ is a free parameter and it may restrict possible terms in the
Kähler potential. However, we do not consider this direction seriously in the
current study.

After the modulus parameter $\tau$ is fixed, we redefine the chiral
superfields as

$\displaystyle Z^{I}\to\check{Z}^{I}=Z^{I}\sqrt{3/|Z^{0}|^{k_{I}}}\,.$ (2.29)

Accordingly it is convenient to reparametrize the Yukawa couplings
($\alpha_{i}$, $g_{i}$, $\lambda$) and $\Lambda$ as ($\check{\alpha}_{i}$,
$\check{g}_{i}$, $\check{\lambda}$) and $\check{\Lambda}$ to give rise to the
same form of $W_{E}+W_{N}+W_{\rm hyb}$. Hereafter, we will write the model in
this field basis and omit the ‘check’ symbol for a simple notation unless
otherwise noticed. Namely, we use the superpotential given in Eqs. (2.17) –
(2.19) and the Kähler potential $K^{\rm m}$ of the matter part with the
function $\Phi$

$\displaystyle-\frac{\Phi}{3}=1-\frac{1}{3}\sum_{I}|Z^{I}|^{2}\,.$ (2.30)

For later calculation, we rewrite $W_{\rm hyb}$ as

$\displaystyle W_{\rm
hyb}=\lambda_{i}S_{+}S_{-}N^{c}_{i}+\frac{1}{2}M_{ij}N^{c}_{i}N^{c}_{j}\,,$
(2.31)

where $(\lambda_{1},\lambda_{2},\lambda_{3})=\lambda(Y_{1},Y_{3},Y_{2})$ and

$\displaystyle M$ $\displaystyle=\Lambda\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\
-Y_{3}&2Y_{2}&-Y_{1}\\\ -Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}\,.$ (2.32)

##  3 Neutrino masses and mixing pattern

In this section, we discuss the lepton sector, especially focusing on the
neutrino mixing pattern. From the superpotential, the light neutrinos acquire
masses in the seesaw mechanism [107, 108, 109, 110, 111, 112]. In our model,
however, $S_{+}$ obtains a vacuum expectation value (VEV), denoted as $\langle
S_{+}\rangle$, at the global minimum, which leads to an unconventional mass
matrix for the light neutrinos [105]. In addition, the components of the mass
matrices are given in a limited number of parameters due to $A_{4}$ symmetry,
which results in the characteristic pattern of the neutrino mixing and CP
phases in the lepton sector.

###  3.1 Neutrino masses and PMNS matrices

To give the mass matrices of the leptons, it is convenient to adopt
canonically normalized field basis. The canonically normalized field
$\hat{Z}^{I}$ is obtained by

$\displaystyle\hat{Z}^{I}=\sqrt{\alpha}Z^{I}\,.$ (3.1)

Here we have used $-\Phi/3\simeq 1$ at the global minimum. The validity of the
approximation is guaranteed by $\expectationvalue{S_{+}}\ll 1$, which will be
shown in Sec. 4. Accordingly we absorb the factor $\sqrt{\alpha}$ by
introducing

$\displaystyle(\hat{\alpha}_{i},\hat{g}_{i},\hat{\lambda})=\alpha^{-3/2}(\alpha_{i},g_{i},\lambda)\,,~{}~{}~{}\hat{\Lambda}=\alpha^{-1}\Lambda\,,$
(3.2)

to give the same form of the superpotential.

Let us see the neutrino mass matrix. We rewrite Eq. (2.19) as

$\displaystyle W_{\rm
hyb}=\hat{\lambda}_{i}\hat{S}_{+}\hat{S}_{-}\hat{N}^{c}_{i}+\frac{1}{2}\hat{M}_{ij}\hat{N}^{c}_{i}\hat{N}^{c}_{j}\,.$
(3.3)

where
$(\hat{\lambda}_{1},\hat{\lambda}_{2},\hat{\lambda}_{3})=\hat{\lambda}(Y_{1},Y_{3},Y_{2})$
and

$\displaystyle\hat{M}$
$\displaystyle=\hat{\Lambda}\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\
-Y_{3}&2Y_{2}&-Y_{1}\\\ -Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}\,.$ (3.4)

Then, the mass matrix of the neutrinos in the basis of
$(N^{c}_{1},N^{c}_{2},N^{c}_{3},\tilde{S}_{-},\nu_{Le},\nu_{L\mu},\nu_{L\tau})$#3#3#3We
use the same notation for the fermionic part as the chiral superfield for
$N^{c}_{i}$ and the leptons in the MSSM while $\tilde{S}_{-}$ is the fermionic
part of $S_{-}$. is given by

$\displaystyle\mathcal{M}$
$\displaystyle=\begin{pmatrix}\tilde{M}&\tilde{M}_{D}\\\
\tilde{M}_{D}^{T}&O\end{pmatrix}\,.$ (3.5)

Here $\tilde{M}$ and $\tilde{M}_{D}$ are $4\times 4$ and $4\times 3$ matrices,
respectively, written by

$\displaystyle\tilde{M}$
$\displaystyle=\hat{\lambda}\langle\hat{S}_{+}\rangle\left(\begin{array}[]{@{}c:c@{}}\hat{M}/\hat{\lambda}\langle\hat{S}_{+}\rangle&\begin{matrix}Y_{1}\\\
Y_{3}\\\ Y_{2}\end{matrix}\\\
\hdashline\begin{matrix}Y_{1}&Y_{3}&Y_{2}\end{matrix}&0\end{array}\right)\,,$
(3.8) $\displaystyle\tilde{M}_{D}$
$\displaystyle=\langle\hat{H}_{u}^{0}\rangle\begin{pmatrix}2\hat{g}_{1}Y_{1}&(-\hat{g}_{1}+\hat{g}_{2})Y_{3}&(-\hat{g}_{1}-\hat{g}_{2})Y_{2}\\\
(-\hat{g}_{1}-\hat{g}_{2})Y_{3}&2\hat{g}_{1}Y_{2}&(-\hat{g}_{1}+\hat{g}_{2})Y_{1}\\\
(-\hat{g}_{1}+\hat{g}_{2})Y_{2}&(-\hat{g}_{1}-\hat{g}_{2})Y_{1}&2\hat{g}_{1}Y_{3}\\\
0&0&0\end{pmatrix}\,,$ (3.9)

where $\langle\hat{H}_{u}^{0}\rangle$ is the VEV of up-type neutral Higgs. We
note that $\hat{\lambda}\expectationvalue*{\hat{S}_{+}}$ corresponds to the
scale of the inflaton mass, which will be shown later. Consequently, the light
neutrinos mass matrix is obtained by the seesaw mechanism as

$\displaystyle M_{\nu}$
$\displaystyle=-\tilde{M}_{D}^{T}\tilde{M}^{-1}\tilde{M}_{D}\,.$ (3.10)

Using Eq. (2.12), it is straightforward to obtain the mass matrix $M_{\nu}$ as

$\displaystyle\frac{M_{\nu 11}}{m_{\nu 0}}$
$\displaystyle=12\hat{Y}_{2}^{2}\bigl{[}-2(3+5\hat{g})+(-3+\hat{g})\hat{Y}_{2}^{3}\bigr{]}\,,$
$\displaystyle\frac{M_{\nu 22}}{m_{\nu 0}}$
$\displaystyle=\frac{4(2+\hat{Y}_{2}^{3})\bigl{[}(1+\hat{g})\hat{Y}_{2}^{6}+16\hat{Y}_{2}^{3}+8(-1+\hat{g})\bigr{]}}{\hat{Y}_{2}}\,,$
$\displaystyle\frac{M_{\nu 12}}{m_{\nu 0}}$ $\displaystyle=\frac{M_{\nu
21}}{m_{\nu 0}}=-\frac{M_{\nu 33}}{2m_{\nu
0}}=(-4+\hat{Y}_{2}^{3})\bigl{[}4(3+\hat{g})+(-3+5\hat{g})\hat{Y}_{2}^{3}\bigr{]}\,,$
(3.11) $\displaystyle\frac{M_{\nu 13}}{m_{\nu 0}}$ $\displaystyle=\frac{M_{\nu
31}}{m_{\nu
0}}=\hat{Y}_{2}\bigl{[}(-3+\hat{g})(16+\hat{Y}_{2}^{6})+4(6-11\hat{g})\hat{Y}_{2}^{3}\bigr{]}\,,$
$\displaystyle\frac{M_{\nu 23}}{m_{\nu 0}}$ $\displaystyle=\frac{M_{\nu
32}}{m_{\nu
0}}=\frac{-64(1+\hat{g})-48(2+\hat{g})\hat{Y}_{2}^{3}-24(-1+\hat{g})\hat{Y}_{2}^{6}+(1+\hat{g})\hat{Y}_{2}^{9}}{2\hat{Y}_{2}}\,,$

where

$\displaystyle\begin{split}&~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{\nu
0}\equiv\frac{(-1+\hat{g})}{(8+\hat{Y}_{2}^{3})^{2}}\frac{\hat{g}_{1}^{2}Y_{3}\langle\hat{H}_{u}\rangle^{2}}{\hat{\Lambda}}\,,\\\
\hat{Y}_{1}&\equiv Y_{1}/Y_{3}\,,\quad\hat{Y}_{2}\equiv
Y_{2}/Y_{3}\,,\quad\hat{g}\equiv\hat{g}_{2}/\hat{g}_{1}\,.\end{split}$ (3.12)

It is worth notifying that the mass matrix is independent of
$\hat{\lambda}\expectationvalue*{\hat{S}_{+}}$. Therefore, the scale of the
inflaton mass is not affected by the observations of the neutrino sector.
Finally $M_{\nu}$ is diagonalized by a unitary matrix $U_{\nu}$ as

$\displaystyle\mathrm{diag}(m_{1},m_{2},m_{3})=U_{\nu}^{T}M_{\nu}U_{\nu}\,.$
(3.13)

The obtained neutrino masses are then compared with the observed values shown
in Table 2. We notice that the lightest neutrino mass becomes zero since rank
of $M_{\nu}$ is two [105]. Therefore, by imposing the condition in which the
neutrino mass squared differences are within $3\sigma$ range of the observed
values, the cosmological upper bound on the sum of light neutrino masses [113,
114]

$\displaystyle\sum_{i}m_{i}\leq 120\,\mathrm{meV}\,,$ (3.14)

is automatically satisfied in our model for both the normal hierarchy (NH) and
inverted hierarchy (IH).

| Normal Hierarchy ($3\sigma$ range) | Inverted Hierarchy ($3\sigma$ range)
---|---|---
$\sin^{2}\theta_{12}$ | $0.269\text{\,--\,}0.343$ | $0.269\text{\,--\,}0.343$
$\sin^{2}\theta_{23}$ | $0.415\text{\,--\,}0.616$ | $0.419\text{\,--\,}0.617$
$\sin^{2}\theta_{13}$ | $0.02032\text{\,--\,}0.02410$ | $0.02052\text{\,--\,}0.02428$
$\displaystyle\Delta m_{21}^{2}/{10^{-5}\,\mathrm{eV}^{2}}$ | $6.82\text{\,--\,}8.04$ | $6.82\text{\,--\,}8.04$
$\displaystyle\Delta m_{3l}^{2}/{10^{-3}\,\mathrm{eV}^{2}}$ | $2.435\text{\,--\,}2.598$ | $-2.581\text{\,--\,}-2.414$
Table 2: Observed data of the neutrino mass squared differences and the
neutrino mixing angles from NuFIT 5.0 [115]. Here, $\Delta m_{3l}^{2}=\Delta
m_{31}^{2}>0$ (NH) and $\Delta m_{3l}^{2}=\Delta m_{32}^{2}<0$ (IH).

Another observable is the PMNS matrix, which is defined as $U\equiv
U_{l}^{\dagger}U_{\nu}$. Here $U_{l}$ is a unitary matrix that diagonalize the
charged lepton mass matrix $M_{E}$. $M_{E}$ is the same as one studied in Ref.
[11]. The result is parametrized in terms of three mixing angles
$\theta_{12}$, $\theta_{13}$, $\theta_{23}$, a Dirac phase
$\delta_{\mathrm{CP}}$, and a Majorana phase $\alpha_{21}$;

$\displaystyle U$
$\displaystyle=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\mathrm{CP}}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\mathrm{CP}}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\mathrm{CP}}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\mathrm{CP}}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\mathrm{CP}}}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}1&&\\\
&e^{i\frac{\alpha_{21}}{2}}&\\\ &&1\end{pmatrix}\,,$ (3.15)

where $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. The mixing angles
are determined by the observations and they are summarized in Table 2. We note
that there is only one Majorana phase since one of the light neutrinos is
massless. Then, the invariant quantities regarding the CP phases are given by
[116, 117, 118, 119, 120]

$\displaystyle J_{CP}$ $\displaystyle=\mathrm{Im}\,\bigl{[}U_{e1}U_{\mu
2}U_{e2}^{*}U_{\mu
1}^{*}\bigr{]}=s_{23}c_{23}s_{12}c_{12}s_{13}c_{13}^{2}\sin\delta_{CP}\,,$
(3.16) $\displaystyle I_{1}$
$\displaystyle=\mathrm{Im}\,\bigl{[}U_{e1}^{*}U_{e2}\bigr{]}=s_{12}c_{12}c_{13}^{2}\sin(\alpha_{21}/2)\,,$
(3.17) $\displaystyle I_{2}$
$\displaystyle=\mathrm{Im}\,\bigl{[}U_{e1}^{*}U_{e3}\bigr{]}=c_{12}s_{13}c_{13}\sin(-\delta_{CP})\,.$
(3.18)

In addition, the following relations are useful to determine the CP phases
[91]:

$\displaystyle\cos\delta_{CP}$ $\displaystyle=\frac{|U_{\tau
1}|^{2}-s_{12}^{2}s_{23}^{2}-c_{12}^{2}c_{23}^{2}s_{13}^{2}}{-2c_{12}s_{12}c_{23}s_{23}s_{13}}\,,$
(3.19) $\displaystyle\mathrm{Re}\,\bigl{[}U_{e1}^{*}U_{e2}\bigr{]}$
$\displaystyle=c_{12}s_{12}c_{13}^{2}\cos(\alpha_{21}/2)\,,$ (3.20)
$\displaystyle\mathrm{Re}\,\bigl{[}U_{e1}^{*}U_{e3}\bigr{]}$
$\displaystyle=c_{12}s_{13}c_{13}\cos(-\delta_{CP})\,.$ (3.21)

###  3.2 Observational consequences

Based on the arguments in the previous subsection, we compute the mixing
angles and CP phases for a given set of parameters. The relevant parameters
are $\tau$ and $\hat{g}$ and we parametrize them as

$\displaystyle\tau=\mathrm{Re}\,\tau+i\,\mathrm{Im}\,\tau\,,\quad\hat{g}=ge^{i\phi_{g}}\,,$
(3.22)

where $g=|\hat{g}|$ and $\phi_{g}$ is argument of $\hat{g}$. Considering the
fundamental domain $\mathcal{F}$ of $\bar{\Gamma}$ for $\tau$,

$\displaystyle\mathcal{F}$
$\displaystyle=\Bigl{\\{}\tau\in\mathcal{H}\,\,\Bigl{|}\Bigr{.}\,\,|\mathrm{Re}\,\tau|\leq\frac{1}{2},\,\,|\tau|\geq
1\Bigr{\\}}\,,$ (3.23)

we scan the following parameter ranges,

$\displaystyle\begin{split}&|\tau|\geq
1,\quad|\mathrm{Re}\,\tau|\leq\frac{1}{2},\quad\mathrm{Im}\,\tau\leq
1.65\,,\\\ &0\leq g\leq 6.5,\quad|\phi_{g}|\leq 180^{\circ}\,.\end{split}$
(3.24)

We use the 3$\sigma$ data of the mixing angles and mass squared differences
from NuFIT 5.0 [115], which are listed in Table 2. In the analysis, we take
the VEVs of the up- and down-type Higgs bosons as $v/2$ $(v\simeq
246.7\,\mathrm{GeV})$ by considering so-called high-scale SUSY to give
$125\,\mathrm{GeV}$ Higgs mass [121, 122].

First of all, we have found no allowed region for the NH case. Fig. 1 shows
the allowed regions for the $\tau$ and $\hat{g}$ for the IH case.

Figure 1: Allowed region of $\tau$ (left) and $\hat{g}=ge^{i\phi_{g}}$ (right)
for inverted hierarchy. Each dot indicates the value that is consistent with
the $3\sigma$ data of the neutrino oscillation experiments [115] summarized in
Table 2.

We found a specific pattern of the allowed values of $\tau$ and $\hat{g}$ is
seen. The allowed value of $\tau$ is limited in
$1.0\lesssim\mathrm{Im}\,\tau\lesssim 1.1$ and $|\mathrm{Re}\,\tau|\lesssim
0.1$, or $\mathrm{Im}\,\tau\sim 1.2$ and $|\mathrm{Re}\,\tau|\sim 0.5$. This
is a different feature compared to the previous work, e.g., Ref. [11], where
the allowed values of $\tau$ is distributed more widely. This difference comes
from the unconventional pattern of the active neutrino mass matrix (3.10)
given by Eqs. (3.8) and (3.9), which comes from additional Yukawa couplings
between the right-handed neutrinos and $S_{+}$ in Eq. (2.31) and the VEV of
$S_{+}$. Regarding $\hat{g}$, $g$ is distributed as $0.4\lesssim g\lesssim 5$.
The phase $\phi_{g}$ can take any value in $[-180^{\circ},180^{\circ}]$, but
it is given by a smooth function of $g$. Such behavior is in contrast to the
results in Ref. [11]; in the literature, the allowed region for both $g$ and
$\phi_{g}$ are more restricted.

Figure 2: Predicted CP phases $\delta_{\rm CP}$ and $\alpha_{21}$ from the
allowed parameter sets given in Fig. 1. The correlations between
$\sin^{2}\theta_{23}$ and the CP phases (top-left and top-right) and the
correlation between CP phases (bottom) are shown. The gray solid lines show
the boundaries of the $3\sigma$ region of the experimental data.

The discovered sets of the parameters lead to a new prediction for the CP
phases, which is shown in Fig. 2. In the plot we show the correlation between
$\sin^{2}\theta_{23}$ and the CP phases. For $\delta_{\rm CP}$, we found
$|\delta_{\rm CP}|\lesssim 120^{\circ}$ for any value of $\sin^{2}\theta_{23}$
in the 3$\sigma$ range. If $\sin^{2}\theta_{23}\simeq 0.55$, on the other
hand, then $\delta_{\rm CP}$ can take any values. Regarding $\alpha_{21}$, we
found $|\alpha_{21}|\sim 120\text{--}180^{\circ}$ for any value of
$\sin^{2}\theta_{23}$ in 3$\sigma$ range. An exception is $|\alpha_{21}|\sim
35^{\circ}$ for $\sin^{2}\theta_{23}\simeq 0.42\text{--}0.43$. Finally, we
found no specific correlation between $\delta_{\rm CP}$ and $\alpha_{21}$,
i.e., $|\alpha_{21}|\sim 120\text{--}180^{\circ}$ is predicted while various
value of $\delta_{\rm CP}$ is possible. To summarize, only the IH is allowed
and Fig. 2 is the prediction for the CP phases in our model.

Figure 3: Predicted effective neutrino mass of neutrinoless double beta decay
as a function of the Majorana CP phase (left) and the Dirac CP phase (right),
calculated from the results given in Fig. 2

Another observable consequence of this model is the neutrinoless double beta
decay $(A,Z)\to(A,Z+2)+2e^{-}$, which is a lepton number violating process in
low-energy phenomena. See, e.g., Ref. [123] for review. The decay rate of the
process is proportional to the effective neutrino mass-squared, which is given
by

$\displaystyle m_{\rm eff}=\Bigl{|}\sum_{i}m_{i}U_{ei}^{2}\Bigr{|}\,.$ (3.25)

The current upper limit on the effective mass of neutrinoless double beta
decay from KamLAND-Zen is 36–156 meV [124]. Note that the experimental upper
bound on the effective mass is affected by the uncertainty of the nuclear
matrix elements in the decay process. In the present model, one of the light
neutrinos is massless and we have seen that the IH is allowed. Therefore, the
effective neutrino mass is given by [125]

$\displaystyle m_{\rm
eff}^{2}=c_{13}^{4}(m_{1}^{2}c_{12}^{4}+m_{2}^{2}s_{12}^{4}+2m_{1}m_{2}c_{12}^{2}s_{12}^{2}\cos\alpha_{21})\,.$
(3.26)

The result is shown in Fig. 3. We found $14~{}{\rm meV}\lesssim m_{\rm
eff}\lesssim 28~{}{\rm meV}$ for the most parameter sets, where
$\alpha_{21}\sim\pm 180^{\circ}$. One can see that it becomes as large as
around 45–47 meV for $\alpha_{21}\sim\pm 35^{\circ}$, which can be understood
from Eq. (3.26). Namely the term proportional to $\cos\alpha_{21}$ is
constructive in this case. The resultant value $\order{10}~{}{\rm meV}$ of
$m_{\rm eff}$ is the same as one shown in Ref. [125]. Since the model predicts
a relatively large effective mass, neutrinoless double beta decay events may
be observed in future experiments. If the CP phases are constrained, for
instance, by considering the leptogenesis [126] in the present framework, then
$m_{\rm eff}$ may be more restrictive. That would be another future work to
pursue (see also the discussion in Sec. 4.5).

##  4 Inflation

$W_{\rm hyb}$ induces the $D$-term hybrid inflation. Motivated by the previous
studies [102, 103], we consider the subcritical regime of hybrid inflation,
where one of right-handed sneutrinos plays the role of the inflaton field. In
the present case, however, the additional Majorana mass terms may disturb the
dynamics during inflation. We will derive the inflaton potential and see how
the additional terms affect the dynamics. In this section, we examine the
impact of the heavy Majorana masses on the subcritical hybrid inflation
scenario. To be concrete, we assume

$\displaystyle|M_{ij}|\ll|\lambda_{i}\langle S_{+}\rangle|\,.$ (4.1)

Under the assumption, we expect that the same inflation model is obtained as
Ref. [103]. We will derive the upper bound for $M_{ij}$ quantitatively.

###  4.1 New field basis

We start with the basis given in the last paragraph of Sec. 2.2. At the global
minimum, the mass matrix squared $M_{\mathrm{sc}}^{2}$ of the scalar sector in
the basis
$(\tilde{N}^{c}_{1},\tilde{N}^{c}_{2},\tilde{N}^{c}_{3},S_{-})$#4#4#4$\tilde{N}^{c}_{i}$
and $S_{\pm}$ are scalar components of the chiral superfield $N^{c}_{i}$ and
$S_{\pm}$, respectively. is obtained as

$\displaystyle M_{\mathrm{sc}}^{2}$
$\displaystyle=\tilde{M}^{\dagger}\tilde{M}\,,$ (4.2)

where $\tilde{M}$ is equal to Eq. (3.8) without ‘hat’ symbol. Under the
condition (4.1), the mass matrix is approximated as

$\displaystyle M_{\mathrm{sc}}^{2}$ $\displaystyle=\langle
S_{+}\rangle^{2}\begin{pmatrix}|\lambda_{1}|^{2}&\lambda_{2}\lambda_{1}^{*}&\lambda_{3}\lambda_{1}^{*}&0\\\
\lambda_{1}\lambda_{2}^{*}&|\lambda_{2}|^{2}&\lambda_{3}\lambda_{2}^{*}&0\\\
\lambda_{1}\lambda_{3}^{*}&\lambda_{2}\lambda_{3}^{*}&|\lambda_{3}|^{2}&0\\\
0&0&0&\tilde{\lambda}^{2}\end{pmatrix}+\mathcal{O}(M_{ij}\lambda_{k}\langle
S_{+}\rangle)\,,$ (4.3)

where

$\displaystyle\tilde{\lambda}\equiv\sqrt{\sum_{i}|\lambda_{i}|^{2}}$ (4.4)

and it can be approximately diagonalized by a unitary matrix $V$ as

$\displaystyle V^{\dagger}M_{\mathrm{sc}}^{2}V$ $\displaystyle=\langle
S_{+}\rangle^{2}\begin{pmatrix}0&&&\\\ &0&&\\\ &&\tilde{\lambda}^{2}&\\\
&&&\tilde{\lambda}^{2}\end{pmatrix}+\mathcal{O}(M_{ij}\lambda_{k}\langle
S_{+}\rangle)\,,$ (4.5) $\displaystyle V$
$\displaystyle\equiv\left(\begin{array}[]{@{}c:c@{}}W&\begin{matrix}0\\\ 0\\\
0\end{matrix}\\\
\hdashline\begin{matrix}0&0&0\end{matrix}&\frac{\lambda_{3}^{*}}{|\lambda_{3}|}\end{array}\right)\,,\quad
W\equiv\begin{pmatrix}\frac{-\lambda_{3}|\lambda_{1}|}{\lambda_{1}\sqrt{\sum_{j\neq
2}|\lambda_{j}|^{2}}}&\frac{-\lambda_{2}\lambda_{1}^{*}}{\tilde{\lambda}\sqrt{\sum_{j\neq
2}|\lambda_{j}|^{2}|}}&\frac{|\lambda_{3}|\lambda_{1}^{*}}{\lambda_{3}^{*}\tilde{\lambda}}\\\
0&\sqrt{\frac{\sum_{j\neq
2}|\lambda_{j}|^{2}}{\tilde{\lambda}^{2}}}&\frac{|\lambda_{3}|\lambda_{2}^{*}}{\lambda_{3}^{*}\tilde{\lambda}}\\\
\frac{|\lambda_{1}|}{\sqrt{\sum_{j\neq
2}|\lambda_{j}|^{2}}}&\frac{-\lambda_{2}\lambda_{3}^{*}}{\tilde{\lambda}\sqrt{\sum_{j\neq
2}|\lambda_{j}|^{2}}}&|\lambda_{3}|/\tilde{\lambda}\end{pmatrix}\,.$ (4.8)

Therefore, it is legitimate to use a new basis of $(N^{\prime c}_{1},N^{\prime
c}_{2},N^{\prime c}_{3},S^{\prime}_{-})^{T}\equiv
V^{\dagger}(N^{c}_{1},N^{c}_{2},N^{c}_{3},S_{-})^{T}$. In this basis, it is
straightforward to find that $S^{\prime}_{\pm}$ only couple to $N^{\prime
c}_{3}$. Then, the superpotential (2.31) is written as

$\displaystyle W_{\rm
hyb}=\tilde{\lambda}S^{\prime}_{+}S^{\prime}_{-}N^{\prime
c}_{3}+\frac{1}{2}M^{\prime}_{ij}N^{\prime c}_{i}N^{\prime c}_{j}\,,$ (4.9)

where $M^{\prime}\equiv W^{T}MW$ is a $3\times 3$ matrix of order $|M_{ij}|$.
Therefore, $\phi\equiv\sqrt{2}\mathrm{Re}\,\tilde{N}^{\prime c}_{3}$ becomes
the inflaton field [105, 103].

The parameter $\alpha$ can be taken to various values and different
predictions for $n_{s}$ and $r$ are obtained accordingly, which is intensively
analyzed in Ref. [103]. In the current study, we take

$\displaystyle\alpha$
$\displaystyle=2/3\,,~{}~{}~{}\tilde{\lambda}=0.959\times
10^{-3}\,,\quad\xi=(0.604\times 10^{16}\,\mathrm{GeV})^{2}\,,$ (4.10)

and $q=y=1$. Then the best-fit value of observed spectral index $n_{s}$, along
with the tensor-to-scalar ratio $r\simeq 6\times 10^{-4}$, is obtained at the
last 60 $e$-folds of the subcritical regime of the hybrid inflation [103].
With the parameters, the condition (4.1) gives

$\displaystyle|M^{\prime}_{ij}|\ll 10^{13}\,{\rm GeV}\,,$ (4.11)

where $\expectationvalue{S^{\prime}_{+}}=(\xi/(2/3)q(1+\tilde{\xi}))^{1/2}$
and $\tilde{\xi}\equiv\xi/2q$ has been used, which will be shown in the next
subsection. In the following subsections, we will quantitatively examine the
condition (4.11) to keep the inflaton dynamics unchanged. To keep the readable
analytic expressions, we leave $\tilde{\lambda}$, $\xi$, $q$ and $y$ as they
are.

###  4.2 The scalar potential

The scalar potential is given by [103],

$\displaystyle V$ $\displaystyle=V_{F}+V_{D}\,,$ (4.12) $\displaystyle V_{F}$
$\displaystyle=\Bigl{(}-\frac{\Phi}{3}\Bigr{)}^{-1}\frac{3}{2}\bigl{[}\delta^{I\bar{J}}W_{I}\overline{W}_{\bar{J}}+\frac{1}{\Delta}\bigl{|}\delta^{I\bar{J}}W_{I}\Phi_{\bar{J}}-2W\bigr{|}^{2}+\frac{2}{\Phi}|W|^{2}\bigr{]}\,,$
(4.13) $\displaystyle V_{D}$
$\displaystyle=\frac{y^{2}}{2}\Bigl{[}\Bigl{(}-\frac{\Phi}{3}\Bigr{)}^{-1}\frac{2q}{3}\bigl{(}|S^{\prime}_{+}|^{2}-|S^{\prime}_{-}|^{2}\bigr{)}-\xi\Bigr{]}^{2}\,,$
(4.14)

where $V_{F}$ and $V_{D}$ are $F$ and $D$-term potentials, respectively. Here
$W_{I}\equiv\partial W/\partial z^{I}\,,\,\Phi_{I}\equiv\partial\Phi/\partial
z^{I}\,$, and $\Delta\equiv\Phi-\delta^{I\bar{J}}\Phi_{I}\Phi_{\bar{J}}$, and
$z^{I}$ shows the scalar component of $Z^{I}$. In the $D$-term potential, $y$
and $\xi\,(>0)$ are the gauge coupling and the Fayet-Iliopoulos (FI) term
related to the local U(1). Due to the FI-term, $S^{\prime}_{+}$ acquires a VEV
as $\langle S^{\prime}_{+}\rangle=(\xi/(2/3)q(1+\tilde{\xi}))^{1/2}$ at the
global minimum.#5#5#5From Eq. (4.10), it is clear that
$\expectationvalue*{S_{+}}\ll 1$.  Then, $s\equiv\sqrt{2}|S^{\prime}_{+}|$ is
identified with the waterfall field.

During inflation, we expect that the fields except for the inflaton and
waterfall fields are stabilized at the origin. Thus the scalar potential
during inflation is given as

$\displaystyle V(\phi,\,s)$ $\displaystyle\equiv
V|_{\sqrt{2}\mathrm{Re}\tilde{N}^{\prime
c}_{3}=\phi,\,\sqrt{2}|S^{\prime}_{+}|=s,\,\text{the others}=0}$
$\displaystyle=\Bigl{(}-\frac{\Phi(\phi,\,s)}{3}\Bigr{)}^{-1}\frac{3\phi^{2}}{4}\Bigl{[}\frac{\tilde{\lambda}^{2}}{2}s^{2}+\varDelta
M^{2}(\phi,\,s)\Bigr{]}$
$\displaystyle\quad+\frac{y^{2}}{8}\Bigl{[}\Bigl{(}-\frac{\Phi(\phi,\,s)}{3}\Bigr{)}^{-1}\frac{2q}{3}s^{2}-2\xi\Bigr{]}^{2}\,,$
(4.15)

where $\Phi(\phi,\,s)\equiv-3+(s^{2}+\phi^{2})/2$. $\Delta M^{2}(\phi,s)$ is
the term that originates in the Majorana mass term, which is given by

$\displaystyle\varDelta M^{2}(\phi,\,s)$
$\displaystyle\equiv\sum_{i=1}^{3}|M^{\prime}_{i3}|^{2}-|M^{\prime}_{33}|^{2}\frac{\phi^{2}}{12}\Bigl{(}-\frac{\Phi(\phi,\,s)}{3}\Bigr{)}^{-1}\,.$
(4.16)

###  4.3 Critical point

As in the literature [99, 100, 102, 105, 103], we focus on the dynamics of the
subcritical regime of the hybrid inflation. First of all, the Majorana mass
term shifts the critical point value of the hybrid inflation. The critical
point value is determined by the mass squared of the waterfall field, which is
given by

$\displaystyle m_{+}^{2}=m_{+,0}^{2}+\varDelta m_{+}^{2}\,,$ (4.17)

where

$\displaystyle m_{+,0}^{2}$ $\displaystyle\equiv
qy^{2}\xi\bigl{(}\Psi(\phi)-1\bigr{)}\,,$ (4.18) $\displaystyle\varDelta
m_{+}^{2}$ $\displaystyle\equiv
qy^{2}\xi\Psi(\phi)\frac{1}{3\tilde{\lambda}^{2}}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}$
$\displaystyle\quad\times\Bigl{[}\varDelta
M^{2}(\phi,0)+|M^{\prime}_{33}|^{2}\frac{\phi^{2}}{12}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}\Bigr{]}\,,$
(4.19) $\displaystyle\Psi(\phi)$
$\displaystyle\equiv\frac{\tilde{\lambda}^{2}}{2(2/3)^{2}qy^{2}\xi}\phi^{2}\,,\quad\Phi_{0}\equiv\Phi(\phi,0)\,.$
(4.20)

Here we have given the mass in a canonically normalized basis in accordance
with Ref. [103]. Consequently, the critical point value $\phi_{c}$ is obtained
by

$\displaystyle\phi_{c}^{2}$
$\displaystyle=\phi_{c,0}^{2}(1-\delta)+\mathcal{O}(M_{i3}^{\prime 4})\,,$
(4.21)

where $\phi_{c,0}^{2}={2(2/3)^{2}qy^{2}\xi}/{\tilde{\lambda}^{2}}$ [103] and

$\displaystyle\delta$
$\displaystyle\equiv\frac{2(2/3)^{2}}{\tilde{\lambda}^{2}}\frac{\varDelta
m_{+}^{2}(\phi_{c,0})}{\phi_{c,0}^{2}}\,.$ (4.22)

Imposing $|\delta|\ll 1$, the upper bound on $|M^{\prime}_{i3}|$ is obtained
as

$\displaystyle\sum_{i=1}^{2}|M^{\prime}_{i3}|^{2}$ $\displaystyle\ll
3\tilde{\lambda}^{2}-4qy^{2}\xi/9\simeq 3\times 10^{-7}\sim(1\times
10^{15}\,\mathrm{GeV})^{2}\,,$ (4.23) $\displaystyle|M^{\prime}_{33}|^{2}$
$\displaystyle\ll\frac{(3\tilde{\lambda}^{2}-4qy^{2}\xi/9)^{2}}{3\tilde{\lambda}^{2}(3\tilde{\lambda}^{2}-8qy^{2}\xi/9)}\simeq
4\times 10^{-8}\sim(5\times 10^{14}\,\mathrm{GeV})^{2}\,.$ (4.24)

Those are weaker bounds compared to Eq. (4.11). Therefore, the critical point
value is merely affected by the Majorana mass term.

###  4.4 Subcritical regime

Below the critical point, the waterfall field grows due to the tachyonic
instability [127, 99] and soon relaxes to the local minimum $s_{\mathrm{min}}$
of the classical path.#6#6#6 We have confirmed that the one-loop potential
dominates over the Majorana mass terms near the critical point under the
condition (4.11). Thus, the dynamics around the critical point is the same as
one studied in Ref. [103]. $s_{\mathrm{min}}$ is obtained by $\partial
V(\phi,s)/\partial s^{2}=0$ as

$\displaystyle s_{\mathrm{min}}^{2}$
$\displaystyle=\frac{3\xi}{q}\frac{(-\Phi_{0}/3)}{1+\tilde{\xi}(1-\Psi(\phi))}$
$\displaystyle\quad\times\Biggl{[}1-\Psi(\phi)\Biggl{\\{}1-\frac{\tilde{\xi}\bigl{[}1-\Psi(\phi)\bigr{]}(\Delta_{+}-\Delta_{-})-\Delta_{-}}{1+\tilde{\xi}\bigl{[}1-\Psi(\phi)(1+\Delta_{+})\bigr{]}}\Biggr{\\}}\Biggr{]}\,,$
(4.25)

where

$\displaystyle\Delta_{\pm}$
$\displaystyle\equiv\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}\frac{1}{3\tilde{\lambda}^{2}}\Bigl{[}\varDelta
M^{2}(\phi,0)\pm|M^{\prime}_{33}|^{2}\frac{\phi^{2}}{12}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}\Bigr{]}\,.$
(4.26)

Consequently the potential in the subcritical regime can be expressed by a
single field effective potential

$\displaystyle V_{\mathrm{sub}}(\phi)$ $\displaystyle\equiv
V(\phi,\,s_{\mathrm{min}}(\phi))$
$\displaystyle=y^{2}\xi^{2}\Psi(\phi)\Biggl{[}1-\frac{\Psi(\phi)}{2}+\frac{1+\tilde{\xi}\\{1-\Psi(\phi)\\}}{3\tilde{\lambda}^{2}\tilde{\xi}}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}$
$\displaystyle\qquad\qquad\times\Bigl{[}\varDelta
M^{2}(\phi,0)-|M^{\prime}_{33}|^{2}\frac{\phi^{2}}{12}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}\tilde{\xi}\\{1-\Psi(\phi)\\}\Bigr{]}\Biggr{]}+\mathcal{O}(M_{i3}^{\prime
4})$
$\displaystyle=y^{2}\xi^{2}\Psi(\phi)\Biggl{[}1-\frac{\Psi(\phi)}{2}+\frac{1}{3\tilde{\lambda}^{2}\tilde{\xi}}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{-1}\varDelta
M^{2}(\phi,0)\Biggr{]}+\mathcal{O}(M_{i3}^{\prime 4},\xi^{3})\,.$ (4.27)

In the last step, we have used $\xi\ll 1$. Therefore, $V_{\mathrm{sub}}$
reduces to the inflaton potential in Ref. [103] when

$\displaystyle|\varDelta M^{2}(\phi,0)|$ $\displaystyle\ll
3\tilde{\lambda}^{2}\tilde{\xi}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}\Bigl{[}1-\frac{\Psi(\phi)}{2}\Bigr{]}\,,$
(4.28)

is satisfied. This condition gives rise to upper bounds on $|M^{\prime}_{i3}|$
as

$\displaystyle\sum_{i=1}^{2}|M^{\prime}_{i3}|^{2}$ $\displaystyle\ll
3\tilde{\xi}\tilde{\lambda}^{2}(1+\phi_{c,0}^{2}/6)/2\sim(2\times
10^{12}\,\mathrm{GeV})^{2}\,,$ (4.29) $\displaystyle|M^{\prime}_{33}|^{2}$
$\displaystyle\ll
2\tilde{\xi}\tilde{\lambda}^{2}{(1-\phi_{c,0}^{2}/6)^{2}}/{|4-\phi_{c,0}^{2}|}\sim(9\times
10^{11}\,\mathrm{GeV})^{2}\,.$ (4.30)

Those are a bit tighter than the condition (4.11). When the above conditions
are satisfied, $s_{\mathrm{min}}$ is also in approximate agreement with the
local minimum value of the waterfall field in Ref. [103],

$\displaystyle s_{\mathrm{min}}^{2}$
$\displaystyle=\frac{3\xi}{q}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}\Bigl{[}1-\Psi(\phi)\Bigr{]}+\mathcal{O}(\xi\Delta_{-})\,.$
(4.31)

To summarize the inflaton dynamics is not affected if

$\displaystyle|M^{\prime}_{i3}|\ll 10^{12}\,\mathrm{GeV}\,,$ (4.32)

is satisfied.#7#7#7We have checked that the stability of the
$\tilde{L}_{i}H_{u}$ direction, pointed out by Ref. [128], is guaranteed if
$|M^{\prime}_{ij}|\lesssim 10^{18}\,{\rm GeV}$.  In the next subsection, we
will confirm the condition numerically.

###  4.5 Dynamics of scalar fields: numerical study

Figure 4: The time evolution of $s$, $\phi_{i}$, and $\tau_{i}$
$(i=1\text{--}3)$ as function of $e$-folds before the end of inflation. The
parameters are given in Eq. (4.39) and $\hat{\Lambda}=10^{10}\,\mathrm{GeV}$.
(Top-left) The waterfall field value $s$ (solid) given by the numerical
solution and that of the analytical approximation $s_{\mathrm{min}}$ (dotted)
are shown. (Top-right) The inflaton $\phi_{3}$ (solid) and $\tau_{3}$ (dashed)
are given. (Bottom) $\phi_{i}$ (solid) and $\tau_{i}$ (dashed) for $i=1,2$ are
shown.

The upper bound (4.32) is obtained under the assumption that the fields except
for the inflaton and waterfall fields are stabilized at the origin during
inflation. However, we need to investigate the validity of this assumption
since the values of $\tilde{N}^{\prime c}_{1,2}$ and
$\mathrm{Im}\,\tilde{N}^{\prime c}_{3}$ may grow to affect the dynamics of
inflation depending on the value of the Majorana masses. We will check the
stability of $\tilde{N}^{\prime c}_{1,2}$ and $\mathrm{Im}\,\tilde{N}^{\prime
c}_{3}$ by solving the equations of motion of $\tilde{N}^{\prime c}_{i}$ and
$S^{\prime}_{+}$ numerically and examine the condition (4.32) more
quantitatively.

To this end, we define the relevant scalar fields as
$\\{\phi_{i},\,\tau_{i},\,s\\}\equiv\varphi^{A}$ where#8#8#8Namely,
$\phi_{3}=\phi$ is the inflaton field in the notation of this subsection.

$\displaystyle\phi_{i}\equiv\sqrt{2}\mathrm{Re}\,\tilde{N}^{\prime
c}_{i}\,,\,\tau_{i}\equiv\sqrt{2}\mathrm{Im}\,\tilde{N}^{\prime c}_{i}\,.\,$
(4.33)

Then, the metric in terms of the field space of $\varphi^{A}$ is given by

$\displaystyle G_{AB}=\delta_{AB}\left\\{\begin{array}[]{ll}K^{\rm
m}_{N_{i}\bar{N}_{i}}&{\rm for}~{}\varphi^{A}=\phi_{i}\,,\tau_{i}\\\ K^{\rm
m}_{S^{+}\bar{S}^{+}}&{\rm for}~{}\varphi^{A}=s\end{array}\right.\,,$ (4.36)

where $K_{Z^{I}\bar{Z}^{J}}=\partialderivative*{K}{z^{I}}{\bar{z}^{J}}$. The
equations of motion of $\varphi^{A}$ are

$\displaystyle\ddot{\varphi}^{A}+3H\dot{\varphi}^{A}+G^{AB}\frac{\partial
V}{\partial\varphi^{B}}+\Gamma^{A}_{BC}\dot{\varphi}^{B}\dot{\varphi}^{C}=0\,.$
(4.37)

Here, dot denotes the time derivative and $H$ is the Hubble parameter that
depends on $\varphi^{A}$ and $\dot{\varphi}^{A}$. $G^{AB}$ is the inverse of
$G_{AB}$ and $\Gamma^{A}_{BC}$ is the connection defined by

$\displaystyle\Gamma^{A}_{BC}\equiv\frac{1}{2}G^{AD}(G_{DB,C}+G_{DC,B}-G_{BC,D})\,.$
(4.38)

For the numerical analysis, we take a benchmark point from the allowed region
of $\tau$ and $\hat{g}$, which is given by

$\displaystyle\begin{split}\mathrm{Re}\,\tau&=8.88\times
10^{-4}\,,\quad\mathrm{Im}\,\tau=1.08\,,\\\
g&=4.56\,,\quad\phi_{g}=-170^{\circ}\,.\end{split}$ (4.39)

With the parameters the corresponding neutrino mixing parameters are

$\displaystyle\begin{split}\sin^{2}\theta_{12}&=0.305\,,\quad\sin^{2}\theta_{13}=0.0224\,,\quad\sin^{2}\theta_{23}=0.572\,,\\\
\delta_{\mathrm{CP}}&=-160^{\circ}\,,\quad\alpha_{21}=-163^{\circ}\,,\end{split}$
(4.40)

and $|M^{\prime}_{ij}|$ is obtained as

$\displaystyle|M^{\prime}_{ij}|$
$\displaystyle=\begin{pmatrix}5.59&3.56&9.75\\\ 3.56&6.52&6.57\\\
9.75&6.57&3.53\end{pmatrix}\times
10^{9}\,\mathrm{GeV}\,\Bigl{(}\frac{\hat{\Lambda}}{10^{10}\,\mathrm{GeV}}\Bigr{)}\,.$
(4.41)

Since we are considering the subcritical regime, the initial values of $\phi$
and $s$ are set to $\phi^{2}_{\mathrm{init}}\equiv 0.98\phi_{c,0}^{2}$ and
$s_{\mathrm{init}}\equiv s_{\mathrm{min}}(\phi_{\mathrm{init}})$,
respectively, at the time $t=0$, while the other initial field values are set
to zero. We have confirmed that similar trajectories of the scalar fields are
obtained for slightly different the initial values of the inflaton and
waterfall fields.#9#9#9The waterfall field value is much smaller than the
inflaton one during inflation and the trajectory is almost straight along the
inflaton field direction. Therefore, the trajectory of $\phi_{3}$-$s$ system
has no additional impact on the curvature perturbation, which is already
studied in Ref. [103].

Figure 5:  The maximum values of $\phi_{i}\,\,(i=1,2)$ (left) and
$\tau_{i}\,\,(i=1\text{--}3)$ (right) during inflation as functions of
$\Lambda$. Red, blue, and green points correspond to $i=1$, $2$, and $3$,
respectively.

Fig. 4 shows the time evolution of $\varphi^{A}$ before the end of inflation
as a function of the $e$-folds $N_{e}$ defined by

$\displaystyle N_{e}(t)=\int^{t_{\rm end}}_{t}dt^{\prime}H\,,$ (4.42)

where $t_{\rm end}$ is the time at the end of inflation. In the plot,
$\hat{\Lambda}=10^{10}\,\mathrm{GeV}$ is taken to satisfy the condition (4.32)
at a percent level. We confirmed that the trajectory of the waterfall field
well agrees with the local minimum $s_{\rm min}$, given in Eq. (4.31). In
addition, we found that the $\tau_{i}$ $(i=1\text{--}3)$ merely move during
the inflation. On the other hand, $\phi_{1}$ and $\phi_{2}$ grow as large as
$\order{\expectationvalue{s}}$ as seen in the figure.#10#10#10$\phi_{i}$
($i=1,2$) are kicked by the term proportional to
$\sum_{j}\real(M_{ij}^{*}M_{j3})\phi_{i}\phi_{3}$. The direction that
$\phi_{i}$ is heading for depends on the sign of
$\sum_{j}\real(M_{ij}^{*}M_{j3})$. They are, however, still subdominant
components in the scalar potential and they do not have any impact on the
waterfall-inflaton dynamics. For smaller value of $\hat{\Lambda}$ we found
that the field values of $\phi_{1}$ and $\phi_{2}$ reduce almost linearly,
which is summarized in Fig. 5. Having confirmed the condition (4.32), we
derived more quantitative bound,

$\displaystyle\hat{\Lambda}\lesssim 10^{10}\,\mathrm{GeV}\,,$ (4.43)

or equivalently

$\displaystyle|M^{\prime}_{i3}|\lesssim 10^{10}\,\mathrm{GeV}\,.$ (4.44)

Figure 6: Time evolution of $\phi_{1}$ and $\phi_{2}$ as function of the
cosmic time $t$. Here $H_{c}\equiv H|_{\phi=\phi_{c}}\simeq 6.1\times
10^{12}\,\mathrm{GeV}$. The parameters are the same as Fig. 4.

In the bottom panels of Fig. 4, the field growths of $\phi_{1}$ and $\phi_{2}$
get accelerated. This is because the Hubble parameter begins to decrease after
the end of inflation since the inflaton field oscillates around the minimum to
behave as a matter component. After that, the $\phi_{1}$ and $\phi_{2}$ start
to oscillate with an angular frequency $\order{|M_{ij}|}$. We checked this
behavior numerically, which is shown in Fig. 6. Here we have assumed that the
inflaton field keeps the coherent oscillation. Realistically the inflaton
field decays to leptons and Higgsinos or sleptons and Higgses and it reheats
the universe. Then the radiation domination follows. Although $\phi_{1}$ and
$\phi_{2}$ continue the coherent oscillation, we expect that they do not
become the dominant component of the universe because their energy density is
highly suppressed. In that case, non-thermal leptogenesis by the inflaton
field works and it is expected to provide a sufficient number of lepton
asymmetry [126, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 105].
The details depend on the model parameters and they are beyond the scope of
our current study. We leave it for our future study.

##  5 Conclusion

We consider a supergravity model that has the modular $A_{4}$ symmetry. This
model accommodates the MSSM augmented by three right-handed neutrino fields
that have the Majorana masses. Additionally, two fields that are charged under
a gauged U(1) are introduced. They couple to the right-handed neutrinos via
the Yukawa interaction and consequently one of the scalar components plays a
role of the waterfall field during inflation and acquires a VEV at the global
minimum. With the extension, the pattern of the light neutrino mass, generated
by the seesaw mechanism, changes and one of the light neutrinos becomes
massless. On top of that, the modular $A_{4}$ symmetry restricts the mixing
pattern. Comparing with the current observations regarding the neutrino
mixings, we have found that only the IH case is allowed. The predicted
Majorana phase is around $\pm 35^{\circ}$, the Dirac phase is
$[-110^{\circ},-90^{\circ}],~{}[90^{\circ},110^{\circ}]$ and
$0.42\lesssim\sin^{2}\theta_{23}\lesssim 0.43$, or the Majorana phase is
$[-180^{\circ},-120^{\circ}],~{}[120^{\circ},180^{\circ}]$, the Dirac phase
takes values in $[-180^{\circ},180^{\circ}]$ and
$0.42\lesssim\sin^{2}\theta_{23}\lesssim 0.62$. The effective neutrino mass,
which determines the decay rate of the neutrinoless double beta decay, is
found to be around 14–28 meV and 45–47 meV. Such a relatively large effective
mass of $\mathcal{O}(10)$ meV will be explored in future experiments.

The supergravity model we consider is based on a hybrid inflation model that
has the subcritical inflation regime. Namely, inflation continues below the
critical point. In the present model, the three right-handed sneutrinos are
the candidates for the inflaton field. Compared to the inflation model studied
in Ref. [103], the existence of the Majorana mass terms is a crucial
difference, which may affect the inflation dynamics. Imposing the Majorana
mass not to affect the inflation dynamics, we have revealed that only one of
the right-handed sneutrinos turns out to couple to the waterfall field, then
we have derived the upper bound for the Majorana mass scale analytically. We
have confirmed the results numerically by solving the equations of motion for
scalar fields. It is found the other right-handed sneutrinos grow as large as
the VEV of the waterfall field. They are, however, negligible in the total
energy density during the inflation if the Majorana mass scale is smaller than
$\order{10^{10}}~{}{\rm GeV}$. Though their field values are much suppressed
during inflation, the scalar fields continue to oscillate after inflation,
which may affect the subsequent thermal history. For instance, they may
contribute to the generation of the lepton asymmetry. The details depend on
the parameter and we leave it for the future work.

## Acknowledgments

We thank Tatsuo Kobayashi for valuable discussion at the early stage of this
project and Takehiko Asaka for useful comments on the preprint. This work is
supported by JST SPRING, Grant No. JPMJSP2135 (YG), JSPS KAKENHI Grant No.
JP18H05542, JP20H01894, JSPS Core-to-Core Program Grant No. JPJSCCA20200002
(KI), and JSPS KAKENHI Grant No. 20H01898 (TY).

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# I see you: A Vehicle-Pedestrian Interaction Dataset from Traffic
Surveillance Cameras

han4n  Hanan Quispe1 Jorshinno Sumire111footnotemark: 1 Patricia Condori2
Edwin Alvarez3
Harley Vera1
1Universidad Nacional de San Antonio Abad del Cusco 2Business on Engineering
and Technology - BE Tech 3Pontificia Universidad Católica del Perú
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>These authors contributed equally to this work

###### Abstract

The development of autonomous vehicles arises new challenges in urban traffic
scenarios where vehicle-pedestrian interactions are frequent e.g. vehicle
yields to pedestrians, pedestrian slows down due approaching to the vehicle.
Over the last years, several datasets have been developed to model these
interactions. However, available datasets do not cover near-accident scenarios
that our dataset covers. We introduce _I see you_ , a new vehicle-pedestrian
interaction dataset that tackles the lack of trajectory data in near-accident
scenarios using YOLOv5 and camera calibration methods. _I see you_ consist of
170 near-accident occurrences in seven intersections in Cusco-Peru. This new
dataset and pipeline code are available on
GitHub111https://github.com/hvzzzz/Vehicle_Trajectory_Dataset.

##  1 Introduction

The problem of predicting pedestrian paths is crucial for the development of
autonomous vehicles(AVs) that have to handle complex interactions with
pedestrians in urban traffic environments e.g. pedestrians recklessly crossing
the road, and vehicles speeding up to overtake a pedestrian. To address this
problem multiple datasets have been developed [1, 2, 3, 4]. Even though
existing datasets cover diverse vehicle-pedestrian interaction scenarios,
vehicle-pedestrian interaction in near accident scenarios is especially
important to avoid situations that can lead to safety hazards and is
challenging because pedestrians exhibit unpredictable maneuvers with sudden
speed changes [5]. Those sudden behavioral changes are hard to predict by an
AV as verified by [6] where the authors argued that safely maneuvering through
jaywalkers is a complex phenomenon that needs attention.

Near accident, data is difficult to collect [7]. This has led to the
development of multiple data-collecting approaches. Wu et al. [8] examined the
usage of a roadside LIDAR sensor to collect trajectories of road users and
identify vehicle-pedestrian near-crash events. Lee et al. [9] conducted a
study to collect physiological data from night-shift workers in near-crash
scenarios where 43.8% of the drives were terminated early for safety reasons.
Nasernejad et al. [10] developed an agent-based framework to model pedestrian
behavior in near misses and used surveillance camera footage to validate their
method. Anik [11] collected jaywalkers’ trajectories from surveillance camera
footage and proposed an artificial neural network to predict jaywalkers’
trajectory on a mid-block location in Bangladesh-India. To train or calibrate
such models is important to have publicly available ground truth trajectory
data. Nevertheless to the best of the author’s knowledge, this is the first
publicly available trajectory dataset that covers near-accident scenarios in
Latin America.

_I see you_ captures vehicle-pedestrian avoidance behaviors in dangerous
situations and scenarios where vehicle-pedestrian are very close but do not
represent a dangerous situation (Figure 1) as this would provide valuable
negative feedback. For each case, we provide processed vehicle and pedestrian
trajectories in GPS coordinates.

Figure 1: Vehicle-Pedestrian interactions captured by our dataset. Example
interaction 1 shows a dangerous situation. Example Interaction 2 shows a non-
dangerous situation where vehicle-pedestrian were very close.

##  2 _I see you_ Dataset

We developed a pipeline to collect _I see you_ (Figure 2). The object
detection task was performed using a YOLOv5 [12], for tracking we used
StrongSORT [13] then we used a linear Kalman filter [1] to remove noise from
the trajectories and finally for transformation to GPS coordinates we used
Perspective-n-Point [14].

Figure 2: Pipeline for the proposed system.

Video footage was collected from publicly available traffic surveillance
videos at seven signalized intersections in Cusco-Peru (Figure 3). The seven
intersections were selected near schools (Figure 3(a)), colleges (Figure
3(b)), churches (Figure 3(c)), and hospitals (Figure 3(d)) to ensure a diverse
distribution in pedestrian characteristics (genre, age). These intersections
also exhibit the transit from particular vehicles, taxis, and public transport
buses where the latter show, particularly aggressive driving behaviors.

Each video is 18 hours long taken from 6 am to 11:59 pm on February 5, 2022.
Every video has daytime, sunset and nighttime scenes these variations in
illumination lead us to fine-tune the YOLOv5 models used for detection. For
each category (vehicle, pedestrian) we used a different pre-trained YOLOv5
model, the vehicle category is detected by the model used in [15] and the
pedestrian category is detected by a model pre-trained in the COCO dataset.

(a) Servicentro

(b) Univesitaria

(c) Santa Ursula

(d) Haya de la Torre

Figure 3: Illumination variations of the collected footage in four
intersections.

To capture all illumination variations we sampled the 18-hour videos saving a
frame every 6 minutes resulting in 4246 images in total. Employing a $90\%$
train, $10\%$ test distribution, we obtained the results described in Table 1
for object detection in the test subset for each category.

Table 1: YOLOv5 Training Results Category | P | R |<EMAIL_ADDRESS>| mAP@0.5:0.95
---|---|---|---|---
Vehicle | $0.934$ | $0.916$ | $0.974$ | $0.76$
Pedestrian | $0.864$ | $0.853$ | $0.91$ | $0.595$

Since the YOLOv5 models had already been trained on the scenes every 6 minutes
we had to make sure that these same scenes were not presented to the detector
on the tracking step, this required a new sampling approach. We noted that if
a frame was taken when the traffic light was red, the scene remained almost
the same until the traffic light turned green. In the worst-case scenario a
frame could have been taken at the same time that the traffic light turned to
red, this implies that a scene could remain almost the same for up to 90
seconds (maximum duration of red light). To overcome that problem we separated
the 6 minutes interval between frames into 3 intervals 2 minutes long each,
then we saved the intermediate 2-minute interval (Figure 4) for the tracking
step.

Figure 4: Sampling approach used to save each 2-minute clip

The tracking step was conducted using StrongSORT [13], this tracker algorithm
used the YOLOv5 models previously fine-tuned to detect the road users in the
2-minute clips and extract the trajectories of each category assigning them id
numbers that were used in the manual labeling step. The resulting trajectories
were noisy with spikes indicating unreal accelerations and speeds, this fact
lead us to use a linear Kalman filter using the approach developed by Yang et
al. [1].

The filtered trajectories were manually labeled with the id generated by the
tracker of the vehicle and pedestrian involved in the near-accident scenario.
The labeled trajectories were converted from pixel coordinates to GPS
coordinates using semi-automatic camera calibration based on Perspective-n-
Point (PnP) [14] . Finally, we compare our dataset with other trajectory
datasets used for pedestrian trajectory prediction in various cases (Table 2).

Table 2: Comparison with Other Public Available Trajectory Datasets Dataset | Scenarios | Interaction | Agents | Method of | FPS | Amount
---|---|---|---|---|---|---
|  | Type |  | Annotation |  | of Trajec-
|  |  |  |  |  | tories
DUT [1] | Campus | Vehicle-Crowd | Car, Pedestrian | CSRT tracker | 23.98 | 1793
|  | in shared spaces |  |  |  |
_I see you_ | Signalized | Vehicle-Pedestrian | Car, Pedestrian | manual | 30 | 340
| Intersections | in near-accident |  |  |  |
|  | scenarios |  |  |  |
ETH [16] | Campus, | Pedestrians in | Pedestrian | manual | 2.5 | 650
| Urban Street | busy scenarios |  |  |  |
UCY [17] | Urban Street, | multi-human | Pedestrian | manual | 2.5 | 909
| Campus, Park | interaction scenarios |  |  |  |

##  3 Statistics

Speed is preferred to train pedestrian trajectory prediction models rather
than absolute position [18, 19]. This is because it does not depend on the
reference system and therefore it can be used as a more general way to
describe motion. In Figure 5 speed distributions were calculated for vehicles
and pedestrians. Both distributions show peaks at $0\frac{km}{h}$ this is
because vehicles and pedestrians are stopped by traffic lights at signalized
intersections. Also, each distribution has a second peak that shows the mean
speed of vehicles and pedestrians during the near-accident scenarios.

Figure 5: Speed distributions for vehicle and pedestrian categories in _I see
you_

##  4 Conclusions and Future Work

In this work, we addressed the need for trajectory data in near-accident
scenarios for which we have developed a pipeline for collecting the
trajectories of pedestrians and vehicles. During this process, we have
assessed the limitations of our pipeline in which manual labeling is a process
that can be automated using clusters to select the trajectories that
correspond to interactions in near accident scenarios and also could be
expanded to other types of interactions by selecting the appropriate clusters
for each type of interaction. Pedestrian risk-seeking behaviors could be
similar in other cities, however, to use the developed pipeline in other
locations it would be necessary to fine-tune the YOLOv5 models in those
locations to get good detection results.

## Acknowledgements

We would like to thank the Data Analysis Laboratory(LAAD) for providing the
GPU used for this project and Rodolfo Quispe for his continuous support and
feedback.

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