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# Dirac Radiative Neutrino Mass with Modular Symmetry and Leptogenesis

Arnab Dasgupta<EMAIL_ADDRESS>School of Liberal Arts, Seoul-Tech,
Seoul 139-743, Korea  PITT PACC, Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, PA 15260, USA  Takaaki Nomura
<EMAIL_ADDRESS>College of Physics, Sichuan University, Chengdu 610065,
China  Hiroshi Okada<EMAIL_ADDRESS>Asia Pacific Center for
Theoretical Physics, Pohang 37673, Republic of Korea  Department of Physics,
Pohang University of Science and Technology, Pohang 37673, Republic of Korea
Oleg Popov<EMAIL_ADDRESS>Institute of Convergence Fundamental Studies,
Seoul National University of Science and Technology, Seoul 139-743, Korea
Department of Physics, Korea Advanced Institute of Science and Technology, 291
Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea  Scuola Normale
Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy  Morimitsu Tanimoto
<EMAIL_ADDRESS>Department of Physics, Niigata University,
Niigata 950-2181, Japan

###### Abstract

Minimalistic Dirac radiative neutrino mass model based on modular symmetry is
proposed. We predict maximum number of observables possible including neutrino
mass splittings, neutrino mass scale, lepton mixing angles, and Dirac phases
in the leptonic sector with minimum number of input parameters possible. Model
is capable of accommodating multicomponent dark matter, thanks to the
$\emph{R}-$parity and accidental scotogenic $\mathbb{Z}_{2}$ discrete
symmetry. Furthermore, even-though neutrinos are Dirac in our model, matter-
antimatter asymmetry of the Universe is achieved via neutrinogenesis
mechanism. Phenomenology of the dark sector including various dark matter
candidates is briefly discussed.

neutrino mass, modular symmetry, flavor symmetry, dark matter

###### pacs:

14.60.Pq, 95.35.+d, 12.60.-i, 14.60.St

††preprint: APCTP Pre2021 - 029, CTP-SCU/2021033††preprint: Prepared for
submission to Phys. Rev. D

###### Contents

  1. I Introduction
  2. II Model
    1. Boson sector
    2. Charged-lepton masses
  3. III Leptogenesis
  4. IV Neutrino masses
    1. Heavy neutral masses
    2. Neutrino mass generation
  5. V Neutrino analysis and discussion
  6. VI Discussion
  7. VII Conclusion

##  I Introduction

Physics beyond the standard model (SM) is required in explaining some issues
such as non-zero neutrino masses, existence of dark matter (DM) and matter-
antimatter asymmetry of the universe. In extending the SM, a new symmetry
plays important roles to restrict structure of new physics which can realize,
for example, stability of DM, neutrino mass generation at loop level
forbidding tree level mass and origin of structure for neutrino mass matrix.
Thus it is interesting to find a symmetry providing such properties with high
predictability.

In controlling flavor structure, attractive framework of symmetries is
proposed by papers Feruglio (2019); de Adelhart Toorop  _et al._ (2012), in
2017, where they applied modular non-Abelian discrete flavor symmetries to
quark and lepton sectors. Remarkably this framework has advantage that any
dimensionless couplings can also be transformed as non-trivial representations
under those symmetries. As a result, we do not need copious scalars to find a
predictive mass matrix. Furthermore we have a modular weight from the modular
origin that can play a role in stabilizing DM when appropriate charge
assignments are assigned to each of the fields in models. Along the line of
this idea, many approaches have appeared in the literature, e.g., based on
modular $A_{4}$ Feruglio (2019); Criado and Feruglio (2018); Kobayashi  _et
al._ (2018a); Okada and Tanimoto (2019a); Nomura and Okada (2019); Okada and
Tanimoto (2019b); de Anda  _et al._ (2018); Novichkov  _et al._ (2019a);
Nomura and Okada (2021a); Okada and Orikasa (2019a); Ding  _et al._ (2019a);
Nomura  _et al._ (2020); Kobayashi  _et al._ (2019a); Asaka  _et al._ (2020a);
Zhang (2019); Ding  _et al._ (2019b); Kobayashi  _et al._ (2020a); Nomura  _et
al._ (2021a); Wang (2020); Okada and Shoji (2020); Okada and Tanimoto (2020);
Behera  _et al._ (2020a, b); Nomura and Okada (2020a, b); Asaka  _et al._
(2020b); Okada and Tanimoto (2021a); Nagao and Okada (2020); Okada and
Tanimoto (2021b); Yao  _et al._ (2021a); Chen  _et al._ (2021); Kashav and
Verma (2021); Okada  _et al._ (2021); de Medeiros Varzielas and Lourenço
(2021); Nomura  _et al._ (2021b); Hutauruk  _et al._ (2020); Ding  _et al._
(2021a); Nagao and Okada (2021); Okada and Qi (2021), $S_{3}$ Kobayashi  _et
al._ (2018b, 2019b, 2019c); Okada and Orikasa (2019b); Mishra (2020); Du and
Wang (2021), $S_{4}$ Penedo and Petcov (2019); Novichkov  _et al._ (2019b);
Kobayashi  _et al._ (2019d); King and Zhou (2019); Okada and Orikasa (2019c);
Criado  _et al._ (2020); Wang and Zhou (2020); Zhao and Zhang (2021); King and
Zhou (2021); Ding  _et al._ (2021b); Zhang and Zhou (2021); Qu  _et al._
(2021); Nomura and Okada (2021b), $A_{5}$ Novichkov  _et al._ (2019c); Ding
_et al._ (2019c); Criado  _et al._ (2020), double covering of $A_{5}$ Wang
_et al._ (2021); Yao  _et al._ (2021b); Wang and Zhou (2021); Behera and
Mohanta (2021), larger groups Baur  _et al._ (2019a), multiple modular
symmetries De Medeiros Varzielas  _et al._ (2019), and double covering of
$A_{4}$ Liu and Ding (2019); Chen  _et al._ (2020a); Li  _et al._ (2021),
$S_{4}$ Novichkov  _et al._ (2021a); Liu  _et al._ (2021), and the other types
of groups Kikuchi  _et al._ (2020); Almumin  _et al._ (2021); Ding  _et al._
(2021c); Feruglio  _et al._ (2021); Kikuchi  _et al._ (2021); Novichkov  _et
al._ (2021b) in which masses, mixing, and CP phases for the quark and/or
lepton have been predicted 111For interested readers, some literature reviews
would be useful to understand the non-Abelian group and its applications to
flavor structure Altarelli and Feruglio (2010); Ishimori  _et al._ (2010,
2012); Hernandez and Smirnov (2012); King and Luhn (2013); King  _et al._
(2014); King (2017); Petcov (2018).. Majorana neutrino mass matrix with two
texture zeros can be also realized applying modular $\mathcal{A}_{4}$ symmetry
Zhang (2019). In addition to the lepton sector, stability of DM can be
realized at fixed points under the modular $A_{4}$ symmetry Kobayashi  _et
al._ (2021).

Further researches are found; a systematic approach to understand the origin
of CP transformations has been discussed in Ref. Baur  _et al._ (2019b),
CP/flavor violation in models with modular symmetry was discussed in Refs.
Kobayashi  _et al._ (2020b); Novichkov  _et al._ (2019d), a possible
correction from Kähler potential was discussed in Ref. Chen  _et al._ (2020b),
and systematic analysis of the fixed points (stabilizers) has been discussed
in Ref. de Medeiros Varzielas  _et al._ (2020).

In this paper, we construct a model realizing Dirac neutrino mass based on
framework of modular $A_{4}$ symmetry with supersymmetry in which we try to
find minimal contents for new particles and modular forms. The manuscript is
organized as follows: section II describes the model at hand, generation of
particle masses, including bosons and fermions, section III describes the
generation of the matter-anti-matter asymmetry of the universe with relevant
constraints on the model, generation of the neutrino masses and other heavy
neutral states is described in section IV, analysis and predictions for
leptonic sector is executed in section V, section VI discusses possible dark
matter candidates and various phenomenology of the model, section VII
concludes the work.

##  II Model

In this section we introduce our model that is SuperSymmetric (SUSY) and
applies modular $A_{4}$ symmetry. Model contains no flavon fields and is build
on Minimal SuperSymmetric Model (MSSM) by appending it with
$\bar{\nu},\eta,\eta^{\prime},\chi,$ and $N$ superfields. The particle content
is given in Tab. 3 where we summarize representations under modular $A_{4}$,
modular-weight $k$ and $3(B-L)$ value of each superfield. While $\bar{\nu}$ is
added to make neutrinos Dirac, $\eta,\chi$, and $N$ superfields are needed for
scotogenic (radiative) neutrino mass mechanism, where scotogenic symmetry is
modular $\mathcal{A}_{4}$ symmetry together with $\mathcal{R}-$parity. Here
$\eta^{\prime}$ is added to cancel gauge anomaly since our model is
SuperSymmetric. Model Lagrangian, _aka_ superpotential, is given as

$\displaystyle\mathcal{W}$
$\displaystyle=\bar{u}\boldsymbol{y_{u}}QH_{u}-\bar{d}\boldsymbol{y_{d}}QH_{d}+\mu
H_{u}H_{d}$ (1)
$\displaystyle-\delta_{1}L_{1}(\boldsymbol{y_{e}}\bar{e})_{1}H_{d}-\delta_{2}L_{1^{\prime}}(\boldsymbol{y_{e}}\bar{e})_{1^{\prime\prime}}H_{d}-\delta_{3}L_{1^{\prime\prime}}(\boldsymbol{y_{e}}\bar{e})_{1^{\prime}}H_{d}$
$\displaystyle+\alpha_{1}L_{1}(\boldsymbol{y_{l}}N)_{1}\eta+\alpha_{2}L_{1^{\prime}}(\boldsymbol{y_{l}}N)_{1^{\prime\prime}}\eta+\alpha_{3}L_{1^{\prime\prime}}(\boldsymbol{y_{l}}N)_{1^{\prime}}\eta$
$\displaystyle+\beta_{1}(N\boldsymbol{y_{\nu}})_{1}\bar{\nu}_{1}\chi_{1}+\beta_{2}(N\boldsymbol{y_{\nu}})_{1^{\prime}}\bar{\nu}_{1^{\prime\prime}}\chi_{1}+\beta_{3}(N\boldsymbol{y_{\nu}})_{1^{\prime\prime}}\bar{\nu}_{1^{\prime}}\chi_{1}$
$\displaystyle+\tilde{\beta}_{1}(N\boldsymbol{y_{\nu}})_{1}\bar{\nu}_{1}\chi_{2}+\tilde{\beta}_{2}(N\boldsymbol{y_{\nu}})_{1^{\prime}}\bar{\nu}_{1^{\prime\prime}}\chi_{2}+\tilde{\beta}_{3}(N\boldsymbol{y_{\nu}})_{1^{\prime\prime}}\bar{\nu}_{1^{\prime}}\chi_{2}$
$\displaystyle+\gamma_{h}\boldsymbol{y_{3}}\eta\chi_{1}H_{d}+\tilde{\gamma}_{h}\boldsymbol{y_{3}}\eta\chi_{2}H_{d}+\gamma_{N}\Lambda(N\boldsymbol{y_{n}}N)_{1}+\sum_{ij}\epsilon^{ij}\mu_{\chi}\boldsymbol{y_{\chi}}\chi_{i}\chi_{j}+\text{H.c.}\,,$

where we are using a two component notation, following Dreiner  _et al._
(2010). Here $\boldsymbol{y_{X}}(\boldsymbol{X}=e,l,\nu,3,\chi,\chi^{\prime})$
denotes modular forms whose representations and corresponding modular weight
are summarized in Table 2. We write modular forms
$\boldsymbol{y_{3}^{(2)}}=(y_{1},y_{2},y_{3})^{T}$,
$\boldsymbol{y_{1}^{(4)}}=y^{2}_{1}+2y_{2}y_{3}$ and
$\boldsymbol{y_{3}^{(4)}}=(y^{2}_{1}-y_{2}y_{3},y^{2}_{3}-y_{1}y_{2},y^{2}_{2}-y_{1}y_{3})^{T}$,
where $y_{i}$ is given by Dedekind eta$-$function $\eta(\tau)$ of modulus
$\tau$ and its derivative $\eta^{\prime}(\tau)$, as given in ref. Feruglio
(2019) ($y_{i}$ is written as $Y_{i}$ in the reference). On the other hand
$\\{\delta_{a},\alpha_{a},\beta_{a},\tilde{\beta}_{a},\gamma_{h},\tilde{\gamma}_{h},\epsilon^{ij}\\}$
are coupling constants. Terms that are forbidden by various symmetries are

$\displaystyle\mathcal{W}_{\not{P_{R}}}$
$\displaystyle=\mathcal{W}_{\not{P_{R}}}^{\text{MSSM}}+LH_{u}+\eta
H_{d}+\bar{\nu}N+N\chi+\boldsymbol{y_{\chi}^{\prime}}\chi\chi\chi,$ (2a)
$\displaystyle\mathcal{W}_{\not{\mathcal{A}_{4}}}$
$\displaystyle=y_{1}^{(2)}L\bar{\nu}H_{u}+\Lambda
y_{1}^{(2)}\bar{\nu}\bar{\nu}+y_{1,1^{\prime},1^{\prime\prime}}^{(3)}\Lambda\eta
L+y_{1}^{(3)}\Lambda\bar{\nu}\chi+y_{3}^{(1)}H_{u}H_{d}N,$ (2b)
$\displaystyle\mathcal{W}_{\not{P_{R}}\&\not{\mathcal{A}_{4}}}$
$\displaystyle=y_{1}^{(2)}\chi H_{u}H_{d},$ (2c)

where matter parity and $\mathcal{R}-$parity are defined as
$P_{M}=(-1)^{3(B-L)}$ and $P_{\mathcal{R}}=(-1)^{3(B-L)+2s}$ with $s$ being
spin of the particle, respectively. Parameters and observables of the leptonic
sector are listed in Tab. 3.

If scalar $N$ somehow gets a non$-$zero VEV (it is $\mathcal{R}-$parity even),
then neutrinos would get a tree$-$level (still Dirac) mass via Dirac seesaw
(as explained in model$-$I tree level scenario of Ma and Popov (2017)). This
does not happen in our case because $y_{3}^{(1)}H_{u}H_{d}N$(eq. 2b) term is
forbidden due to modular invariance of the super$-$potential.
$y_{3}^{(1)}H_{u}H_{d}N$ is the only possible source term that can induce
$\left\langle N\right\rangle\neq 0$. In other words, the VEV of $N$ is not
induced by VEVs of $H_{u,d}$ and since $\mathcal{R}-$parity does not protect
$\left\langle N\right\rangle=0$, it indicates that there must be another
induced or accidental symmetry in the Lagrangian related to the fact that
$\left\langle N\right\rangle=0$.
The extra symmetry is $\mathbb{Z}_{2}$, which can be seen from the $\Lambda
N\boldsymbol{y_{n}}N$ term of eq. 1. The $\mathbb{Z}_{2}-$odd particles under
this accidental symmetry(this accidental symmetry is present because of
modular $\mathcal{A}_{4}$ symmetry invariance of the superpotential) are
$\hat{N},\hat{\chi},\hat{\eta}$. From this we can conclude that the lightest
of these $P_{\mathcal{R}}-$even states is a dark matter (DM) candidate which
makes our model a multi-particle dark matter model.

S$-$Field | SU(3)c | SU(2)L | U(1)Y | $\mathcal{A}_{4}$ | $-k$ | $3(B-L)$
---|---|---|---|---|---|---
Q | 3 | 2 | $\frac{1}{6}$ | $1$ | $0$ | $1$
$\bar{u}$ | $\boldsymbol{\bar{3}}$ | 1 | $-\frac{2}{3}$ | $1$ | $0$ | $-1$
$\bar{d}$ | $\boldsymbol{\bar{3}}$ | 1 | $\frac{1}{3}$ | $1$ | $0$ | $-1$
L | 1 | 2 | $-\frac{1}{2}$ | $\boldsymbol{1,1^{\prime},1^{\prime\prime}}$ | $-1$ | $-3$
$\bar{e}$ | 1 | 1 | $1$ | $\boldsymbol{3}$ | $-1$ | $3$
$\bar{\nu}$ | 1 | 1 | $0$ | $\boldsymbol{1,1^{\prime},1^{\prime\prime}}$ | $-1$ | $3$
$N$ | 1 | 1 | $0$ | $\boldsymbol{3}$ | $-1$ | $0$
$H_{u}$ | 1 | 2 | $\frac{1}{2}$ | $\boldsymbol{1}$ | $0$ | $0$
$H_{d}$ | 1 | 2 | $-\frac{1}{2}$ | $\boldsymbol{1}$ | $0$ | $0$
$\eta$ | 1 | 2 | $\frac{1}{2}$ | $\boldsymbol{1}$ | $-2$ | $3$
$\eta^{\prime}$ | 1 | 2 | $-\frac{1}{2}$ | $\boldsymbol{1}$ | $-2$ | $3$
$\chi$ | 1 | 1 | $0$ | $\boldsymbol{1}$ | $-2$ | $-3$
Table 1: Model particle content.333The matter$-$parity $(3(B-L))$ twin ($-$,$+$) particles are $(L,H_{d}),(\eta,H_{u}),(\bar{\nu}\chi,N)$. $k$ is the modular$-$weight. Field | $\mathcal{A}_{4}$ | $-k$
---|---|---
$\boldsymbol{y_{e}}=\boldsymbol{y_{3}^{(2)}}$ | $\boldsymbol{3}$ | $2$
$\boldsymbol{y_{l}}=\boldsymbol{y_{3}^{(4)}}$ | $\boldsymbol{3}$ | $4$
$\boldsymbol{y_{\nu}}=\boldsymbol{y_{3}^{(4)}}$ | $\boldsymbol{3}$ | $4$
$\boldsymbol{y_{3}}=\boldsymbol{y_{1}^{(4)}}$ | $\boldsymbol{1}$ | $4$
$\boldsymbol{y_{n}}=\boldsymbol{y_{3}^{(2)}}$ | $\boldsymbol{3}$ | $2$
$\boldsymbol{y_{\chi}}=\boldsymbol{y_{1}^{(4)}}$ | $\boldsymbol{1}$ | $4$
$\boldsymbol{y_{\chi}^{\prime}}=\boldsymbol{y_{1}^{(6)}}=6y_{1}y_{2}y_{3}$ | $\boldsymbol{1}$ | $6$
Table 2: Modular transformations of Yukawas and dimensionfull parameters of the model. Observable | Predicted/Input/Constrained
---|---
$\Delta m^{2}_{\text{sol}}$ | P
$\Delta m^{2}_{\text{atm}}$ | I
$m_{1}$ | C
$\sin^{2}\theta_{12}$ | P
$\sin^{2}\theta_{23}$ | P
$\sin^{2}\theta_{13}$ | P
$\delta_{CP}$ | P
$m_{ee}$ | P/C
$\sum m_{i}$ | P
$m_{e},m_{\mu},m_{\tau}$ | I
Model parameter | Constrained by/Free
$\boldsymbol{y_{e}}$ | $\alpha_{i},\tau$
$\boldsymbol{y_{\nu}}$ | $\alpha_{i},\tau$
$\tau$ | Scan
$\mu_{H}$ | $m_{\nu}$
$\mu$ | $G_{F},m_{h},\frac{\partial V}{\partial H_{u}^{0}}=0,\frac{\partial V}{\partial H_{d}^{0}}=0$
$v_{u},v_{d}\leftrightarrow v,\tan\beta$ | $m_{e},m_{\mu},m_{\tau},G_{F}$
Table 3: Parameters and observables of the leptonic sector.

### Boson sector

Here we discuss mass eigenstates and mixings of neutral scalar bosons which
are R-parity odd. Because of the term
$\gamma_{h}y_{3}\eta\chi_{1}H_{d}+\tilde{\gamma}_{h}y_{3}\eta\chi_{2}H_{d}$,
the neutral components of inert bosons $\eta$ and $\chi_{1,2}$ mix with each
other. We formulate their mixings as

$\displaystyle\left(\begin{matrix}\chi_{1}^{0}\\\ \chi_{2}^{0}\\\ \eta_{0}\\\
\end{matrix}\right)$ $\displaystyle=\begin{pmatrix}1&0&0\\\
0&c_{H_{23}}&-s_{H_{23}}\\\ 0&s_{H_{23}}&c_{H_{23}}\\\
\end{pmatrix}\begin{pmatrix}c_{H_{13}}&0&-s_{H_{13}}\\\ 0&1&0\\\
s_{H_{13}}&0&c_{H_{13}}\\\
\end{pmatrix}\begin{pmatrix}c_{H_{12}}&-s_{H_{12}}&0\\\
s_{H_{12}}&c_{H_{12}}&0\\\ 0&0&1\\\ \end{pmatrix}\left(\begin{matrix}H_{1}\\\
H_{2}\\\ H_{3}\\\ \end{matrix}\right),$ $\displaystyle\equiv
U_{H}\left(\begin{matrix}H_{1}\\\ H_{2}\\\ H_{3}\\\ \end{matrix}\right),$ (3)

where we consider mixing angles
$s_{H_{ij}}\equiv\sin\theta_{H_{ij}}$,$c_{H_{ij}}\equiv\cos\theta_{H_{ij}}$,
and mass eigenvalues $m_{H_{1,2,3}}$ as free parameters. In this paper, we do
not discuss charged scalar bosons since they are irrelevant for leptogenesis
and neutrino mass generation. Higgs sector in our model is the same as that of
MSSM and we do not discuss here.

### Charged-lepton masses

In this subsection we discuss charged lepton masses. Charged-lepton mass
matrix is given through the following Lagrangian:

$\displaystyle{\cal L}_{\ell}$
$\displaystyle=\delta_{1}(\bar{e}_{1}y_{1}+\bar{e}_{2}y_{3}+\bar{e}_{3}y_{2})H_{d}L_{1}+\delta_{2}(\bar{e}_{2}y_{2}+\bar{e}_{1}y_{3}+\bar{e}_{3}y_{1})H_{d}L_{2}$
$\displaystyle+\delta_{3}(\bar{e}_{3}y_{3}+\bar{e}_{1}y_{2}+\bar{e}_{2}y_{1})H_{d}L_{3}+{\rm
h.c.},$ (4)

where $\boldsymbol{y_{3}^{(2)}}\equiv[y_{1},y_{2},y_{3}]^{T}$ is applied for
terms for second line in Eq. 1. After the EW spontaneously breaking, we find

$\displaystyle
m_{\ell}=\left[\frac{v_{d}}{\sqrt{2}}\left(\begin{matrix}y_{1}&y_{3}&y_{2}\\\
y_{3}&y_{2}&y_{1}\\\ y_{2}&y_{1}&y_{3}\\\
\end{matrix}\right)\left(\begin{matrix}\delta_{1}&0&0\\\ 0&\delta_{2}&0\\\
0&0&\delta_{3}\\\ \end{matrix}\right)\right]_{\bar{e}L}.$ (5)

Then, the mass matrix is diagonalized by $D_{\ell}\equiv
V_{e_{R}}^{\dagger}m_{\ell}V_{e_{L}}$;
$|D_{\ell}|^{2}=V_{e_{L}}^{\dagger}m^{\dagger}_{\ell}m_{\ell}V_{e_{L}}$.

##  III Leptogenesis

The resultant leptonic asymmetry is realised by satisfying the Sakharov’s
condition Sakharov (1967). Now, in order to get a non-zero $CP$ violation one
needs to satisfy the Nanopolous-Weinberg theorem Nanopoulos and Weinberg
(1979) and along with that the condition pointed out by Adhikari-Rangarajan
Adhikari and Rangarajan (2002). The first condition specifies atleast how many
$B/L$ violating couplings on needs and the second condition tells us precisely
where to keep place such couplings.

In our scenario the only particle which breakes the lepton-number is $\chi$
through it’s mass term. The decay of $\chi$ will create an asymmetry in the
right$-$handed sector. Then, eventually the asymmetry from the right$-$handed
sector goes to the left$-$handed sector through the process
$\bar{\nu}\widetilde{\chi}\rightarrow\nu^{\dagger}\widetilde{\eta}^{\dagger}$.
The processes responsible for the generation of an asymmetry are given as
follows:

$\chi$$N$$\bar{\nu}$ | $\chi$$N$$\bar{\nu}$$\bar{\nu}$$\chi$$N$ | $\chi$$N$$\bar{\nu}$$\bar{\nu}$$\chi$$N$
---|---|---
Figure 1: The above figure shows the processes responsible for giving the $CP$
violation.

The diagram shown in fig.1 does gives a non-zero $CP$ which would eventually
give an asymmetry in the right handed sector. The asymmetry then would be
communicated to the left-handed sector through the following channel shown in
fig 2.

$\tilde{\eta}^{0}$$\nu$$\widetilde{\chi}$$\bar{\nu}$$\widetilde{N}\quad\widetilde{N}$
Figure 2: The above process is responsible for transferring the asymmetry from
the right handed sector to the left handed sector.

Although the above process would act as a source term and as well as the major
wash-out channel. So, in order to get the most asymmetry one would require
this wash-out factor to go out-of-equilibrium at the earlier time. Which boils
down in satisfying the following relation

$\displaystyle\frac{\gamma^{eq}_{Scatt}(\widetilde{\chi}\bar{\nu}\rightarrow\tilde{\eta}^{0}\nu)}{Hs}<1.$
(6)

In order to get the resultant asymmetry we need to solve the coupled boltzmann
equations.

$\displaystyle\frac{dY_{\chi}}{dz}$
$\displaystyle=\frac{-1}{zsH}\left[\left(\frac{Y_{\chi}}{Y^{eq}_{\chi}}-1\right)\gamma_{D}(\chi\rightarrow\nu_{R}\widetilde{N})+\left(\frac{Y^{2}_{\chi}}{(Y^{eq}_{\chi})^{2}}-1\right)\gamma^{eq}_{Scatt}(\chi\chi\rightarrow
all)\right],$ (7a) $\displaystyle\frac{dY_{\Delta R}}{dz}$
$\displaystyle=\frac{1}{zsH}\left[\left(\frac{Y_{\chi}}{Y^{eq}_{\chi}}-1\right)\epsilon\gamma_{D}(\chi\rightarrow\nu_{R}\widetilde{N})-\frac{Y_{\Delta
R}}{Y^{eq}_{l}}\gamma_{D}(\chi\rightarrow\nu_{R}\widetilde{N})\right.$ (7b)
$\displaystyle-\left.2\frac{Y_{\Delta
R}}{Y^{eq}_{l}}\left[\gamma^{eq}_{scatt}(\nu_{R}\widetilde{N}\rightarrow\bar{\nu}_{R}\widetilde{N})+\gamma^{eq}_{scatt}(\nu_{R}\nu_{R}\rightarrow\widetilde{N}\widetilde{N})\right]\right.$
$\displaystyle+\left.\left(\frac{Y_{\Delta L}-Y_{\Delta
R}}{Y^{eq}_{l}}\right)\gamma^{eq}_{Scatt}(\widetilde{\chi}\nu_{R}\rightarrow
H\nu_{L})\right],$ $\displaystyle\frac{dY_{\Delta L}}{dz}$
$\displaystyle=\frac{1}{zsH}\left[\left(\frac{Y_{\Delta R}-Y_{\Delta
L}}{Y^{eq}_{l}}\right)\gamma^{eq}_{Scatt}(\widetilde{\chi}\nu_{R}\rightarrow
H\nu_{L})\right],$ (7c)

where $z=M_{\chi}/T$, $\gamma_{D}(i_{1}\rightarrow f_{1}+f_{2}+\cdots)$ and
$\gamma^{eq}_{scatt}(i_{1}i_{2}\rightarrow f_{1}+f_{2}+\cdots)$ is given as,

$\displaystyle\gamma_{D}(i_{1}\rightarrow f_{1}+f_{2}+\cdots)$
$\displaystyle=\frac{g_{i}}{2\pi^{2}}m^{2}_{i_{1}}TK_{1}(m_{i_{1}}/T)\Gamma(i_{1}\rightarrow
f_{1}+f_{2}+\cdots),$ (8a)
$\displaystyle\gamma^{eq}_{Scatt}(i_{1}i_{2}\rightarrow f_{1}+f_{2}+\cdots)$
$\displaystyle=\frac{g_{i_{1}}g_{i_{2}}T}{8\pi^{4}}\int^{\infty}_{s_{in}}ds\frac{p_{in}p_{out}}{\sqrt{s}}|\mathcal{M}(i_{1}i_{2}\rightarrow
f_{1}+f_{2}+\cdots)|^{2}K_{1}(\sqrt{s}/T),$ (8b)

in which $K_{1}$ is the modified Bessel function. As to get the estimate of
the asymmetry we start by calculating the CP asymmetry parameter $\varepsilon$
for $\chi\rightarrow N\nu_{R}$ decays which is given as

$\displaystyle\varepsilon_{i}$
$\displaystyle=\frac{1}{8\pi(Y_{\nu}^{\dagger}Y_{\nu})_{ii}}\Im[(Y^{\dagger}_{\nu}Y_{\nu})^{2}_{ij}]\frac{1}{\sqrt{x_{ji}}}\mathcal{F}(x_{ji}),$
(9a) $\displaystyle\mathcal{F}(x_{ji})$
$\displaystyle=\sqrt{x_{ji}}\left[f(x_{ji})-\frac{\sqrt{x_{ji}}}{x_{ji}-1}\right],$
(9b) $\displaystyle f(x_{ji})$
$\displaystyle=\sqrt{x_{ji}}\left[1+(1+x_{ji})\ln\left(\frac{x_{ji}}{x_{ji}+1}\right)\right],$
(9c)

with $x_{ji}=M^{2}_{j}/M^{2}_{i}$ and
$(Y_{\nu})_{1(2)}\equiv\beta_{a}\boldsymbol{y_{\nu}}(\tilde{\beta}_{a}\boldsymbol{y_{\nu}})$
omitting flavor indices. Furthermore, the decay $\Gamma_{i}$ is given as

$\displaystyle\Gamma_{i}$
$\displaystyle=\frac{M_{\chi_{i}}}{8\pi}(Y^{\dagger}_{\nu}Y_{\nu})_{ii}.$ (10)

Now assuming only the resonant case and taking the mass differences of the
decaying $\chi$ masses to be $M_{\chi_{j}}-M_{\chi_{i}}=\Gamma_{i}/2$. Which
simplifies the total asymmetry as follows

$\displaystyle\varepsilon$
$\displaystyle=\sum_{i}\varepsilon_{i}=\sin(2\phi).$ (11)

If we demand the out-of-equilibrium to occur around $T\sim M_{\chi}$ we can
safely put a constraint on $y_{\chi}$’s by the following relation

$\displaystyle\frac{\Gamma_{\chi}}{H}$
$\displaystyle=\left(\frac{8\pi^{3}g_{*}}{90}\right)^{-1/2}\frac{M_{pl}}{M_{\chi}}\frac{(Y^{\dagger}_{\nu}Y_{\nu})_{11}}{8\pi}=1\quad\textrm{assuming}\quad
M_{\chi_{1}}>M_{\chi_{2}},$ (12a) $\displaystyle(Y_{\nu})^{2}_{1}$
$\displaystyle=\frac{M_{\chi_{1}}}{M_{pl}}\sqrt{\frac{8\pi^{3}g_{*}}{90}},$
(12b) $\displaystyle(Y_{\nu})^{2}_{1}$ $\displaystyle=1.43\times
10^{-15}\left(\frac{M_{\chi_{1}}}{1{\rm TeV}}\right)=(Y_{\nu})^{2}_{2},$ (12c)

Now, in order to get the estimate we assume the processes leading to left-
right equilibration include
$\widetilde{\eta}\nu_{L}\rightarrow\widetilde{\chi}\nu_{R}$ mediated with
s-channel exchange of an $\widetilde{N}$. Approximately, at high temperatures
these processes have a rate

$\displaystyle\Gamma_{L-R}$ $\displaystyle\sim
2\frac{|(Y_{\nu})_{1}|^{2}|Y_{l}|^{2}}{M^{4}_{\widetilde{N}}}T^{5},$ (13)

where $Y_{l}\equiv\alpha_{a}\boldsymbol{y_{l}}$. This should be compared with
the Hubble rate

$\displaystyle H$
$\displaystyle=\sqrt{\frac{8\pi^{3}g_{*}}{90}}\frac{T^{2}}{M_{pl}}.$ (14)

The considerable constraint will be coming from the highest temperatures when
$T\simeq M_{\chi}$, i.e. those at which the asymmetry is generated

$\displaystyle
2\frac{|(Y_{\nu})_{1}|^{2}|Y_{l}|^{2}}{M_{\widetilde{N}}}M^{3}_{\chi_{1}}$
$\displaystyle\lesssim\frac{1}{M_{pl}}\sqrt{\frac{8\pi^{3}g_{*}}{90}}.$ (15)

The ratio boils down to

$\displaystyle|Y_{l}|^{2}$
$\displaystyle\lesssim\frac{M^{4}_{\widetilde{N}}}{2M^{4}_{\chi_{1}}}|Y_{l}|^{2}\leq
4\pi,$ (16a) $\displaystyle M_{\widetilde{N}}$
$\displaystyle=(8\pi)^{1/4}M_{\chi_{1}},$ (16b)

which basically tells us that the ratio $M_{\widetilde{N}}\sim M_{\chi_{1}}$.
Hence, the final asymmetry can be given as

$\displaystyle\eta_{B}$ $\displaystyle=a_{sph}\frac{86}{2387}\varepsilon
Y^{eq}_{\chi_{1}}(z=1),$ (17a) $\displaystyle\eta_{B}$
$\displaystyle=4.479\times 10^{-5}\sin(2\phi),$ (17b)

where $a_{sph}=28/79$ and the observed baryonic asymmetry
$\eta^{obs}_{B}=6\times 10^{-10}$ which can be transformed to

$\displaystyle\sin(2\phi)$ $\displaystyle=1.34\times
10^{-5}\quad\sim\phi=6.7\times 10^{-6}.$ (18)

##  IV Neutrino masses

Here generation of the Dirac neutrino masses is discussed, but we start with
the masses of the heavy neutral fermions.

### Heavy neutral masses

Before formulating the active neutrino mass matrix, let us formulate the
heavier Majorana neutral fermion $N$. The explicit form of Lagrangian is found
as

$\displaystyle{\cal L}_{N}$
$\displaystyle=M_{0}\left[y_{1}(2N_{1}N_{1}-N_{2}N_{3}-N_{3}N_{2})+y_{2}(2N_{2}N_{2}-N_{1}N_{3}-N_{3}N_{1})\right.$
$\displaystyle\left.+y_{3}(2N_{3}N_{3}-N_{1}N_{2}-N_{2}N_{1})\right]+{\rm
h.c.}$ (19)

Thus, the Majorana mass matrix is give by

$\displaystyle M_{N}=M_{0}\left(\begin{matrix}2y_{1}&-y_{3}&-y_{2}\\\
-y_{3}&2y_{2}&-y_{1}\\\ -y_{2}&-y_{1}&2y_{3}\\\ \end{matrix}\right).$ (20)

Then, this is diagonalized by $D_{N}\equiv U^{T}M_{N}U$; $|D_{N}|^{2}\equiv
U^{\dagger}M^{\dagger}_{N}M_{N}U$, furthermore $N\equiv U\psi$, where $\psi$
is mass eigenstate of $N$.

### Neutrino mass generation

In our model tree level Dirac neutrino mass term is fobidden by modular
invariance of the superpotential, while scotogenic Dirac neutrino mass term is
allowed due to presence of Majorana fermion $N$. The relevant interactions to
generate Dirac neutrino mass matrix is given by

$\displaystyle{\cal L}_{\nu}$
$\displaystyle=\alpha_{1}(N_{1}^{T}y^{\prime}_{1}+N_{2}^{T}y^{\prime}_{3}+N_{3}^{T}y^{\prime}_{2})\eta
L_{1}+\alpha_{2}(N_{2}^{T}y^{\prime}_{2}+N_{1}^{T}y^{\prime}_{3}+N_{3}^{T}y^{\prime}_{1})\eta
L_{2}+\alpha_{3}(N_{3}^{T}y^{\prime}_{3}+N_{1}^{T}y^{\prime}_{2}+N_{2}^{T}y^{\prime}_{1})\eta
L_{3}$
$\displaystyle+\beta_{1}\bar{\nu}_{1}(N_{1}y^{\prime}_{1}+N_{2}y^{\prime}_{3}+N_{3}y^{\prime}_{2})\chi_{1}+\beta_{2}\bar{\nu}_{2}(N_{2}y^{\prime}_{2}+N_{1}y^{\prime}_{3}+N_{3}y^{\prime}_{1})\chi_{1}+\beta_{3}\bar{\nu}_{3}(N_{3}y^{\prime}_{3}+N_{1}y^{\prime}_{2}+N_{2}y^{\prime}_{1})\chi_{1}+{\rm
h.c.}$
$\displaystyle+\tilde{\beta}_{1}\bar{\nu}_{1}(N_{1}y^{\prime}_{1}+N_{2}y^{\prime}_{3}+N_{3}y^{\prime}_{2})\chi_{2}+\tilde{\beta}_{2}\bar{\nu}_{2}(N_{2}y^{\prime}_{2}+N_{1}y^{\prime}_{3}+N_{3}y^{\prime}_{1})\chi_{2}+\tilde{\beta}_{3}\bar{\nu}_{3}(N_{3}y^{\prime}_{3}+N_{1}y^{\prime}_{2}+N_{2}y^{\prime}_{1})\chi_{2}+{\rm
h.c.},$ $\displaystyle\supset
N^{T}y_{\eta}\nu\eta^{0}+\bar{\nu}y_{\chi}N\chi^{0}_{1}+\bar{\nu}\tilde{y}_{\chi}N\chi^{0}_{2}+{\rm
h.c.},$
$\displaystyle=\psi^{T}U^{T}y_{\eta}\nu((U_{H})_{31}H_{1}+(U_{H})_{32}H_{2}+(U_{H})_{33}H_{3})+\bar{\nu}y_{\chi}U\psi((U_{H})_{11}H_{1}+(U_{H})_{12}H_{2}+(U_{H})_{13}H_{3})$
$\displaystyle+\bar{\nu}\tilde{y}_{\chi}U\psi((U_{H})_{21}H_{1}+(U_{H})_{22}H_{2}+(U_{H})_{23}H_{3})+{\rm
h.c.},$ (21)

where ${y_{3}^{(4)}}\equiv[y_{1}^{\prime},y_{2}^{\prime},y_{3}^{\prime}]^{T}$,
and we rewrite the interaction with mass eigenvector in the last line. Then,
we find each of Yukawa matrix to be

$\displaystyle y_{\eta}=\alpha_{1}\tilde{y}_{\eta}$
$\displaystyle=\alpha_{1}\left[\left(\begin{matrix}y_{1}^{\prime}&y_{3}^{\prime}&y_{2}^{\prime}\\\
y_{3}^{\prime}&y_{2}^{\prime}&y_{1}^{\prime}\\\
y_{2}^{\prime}&y_{1}^{\prime}&y_{3}^{\prime}\\\
\end{matrix}\right)\left(\begin{matrix}1&0&0\\\ 0&\tilde{\alpha}_{2}&0\\\
0&0&\tilde{\alpha}_{3}\\\ \end{matrix}\right)\right]_{N^{T}L},$ (22)
$\displaystyle y_{\chi}(\tilde{y}_{\chi})$
$\displaystyle=\left[\left(\begin{matrix}y_{1}^{\prime}&y_{3}^{\prime}&y_{2}^{\prime}\\\
y_{3}^{\prime}&y_{2}^{\prime}&y_{1}^{\prime}\\\
y_{2}^{\prime}&y_{1}^{\prime}&y_{3}^{\prime}\\\
\end{matrix}\right)\left(\begin{matrix}\beta_{1}(\tilde{\beta}_{1})&0&0\\\
0&\beta_{2}(\tilde{\beta}_{2})&0\\\ 0&0&\beta_{3}(\tilde{\beta}_{3})\\\
\end{matrix}\right)\right]_{\bar{\nu}N},$ (23)

where $\tilde{\alpha}_{2,3}\equiv\alpha_{2,3}/\alpha_{1}$ and $\alpha_{1}$ is
factored out for convinience in numerical analysis. In terms of these
interactions Dirac scotogenic neutrino mass diagram is given in Fig. 3.
Analytic form of the mass matrix is estimated to be

$\displaystyle
m_{\nu_{ij}}=-\frac{\alpha_{1}}{(4\pi)^{2}}\sum_{a=1}^{3}\sum_{A=1}^{3}(U^{T}\tilde{y}_{\eta})_{ia}D_{N_{a}}\left[(y_{\chi}U)_{aj}(U_{H})_{3A}(U_{H})_{1A}+(\tilde{y}_{\chi}U)_{aj}(U_{H})_{3A}(U_{H})_{2A}\right]f(r^{a}_{A}),$
(24a) $\displaystyle f(r^{a}_{A})=\frac{r^{a}_{A}\ln r^{a}_{A}}{1-r_{A}^{a}},$
(24b)

where $r^{a}_{A}\equiv\frac{m_{H_{A}}^{2}}{D_{a}^{2}}$. Here, we redefine
$\tilde{m}_{\nu}\equiv\frac{m_{\nu}}{\alpha_{1}}$. Then the neutrino mass
matrix is diagonalized by
$U_{\nu}^{T}\tilde{m}_{\nu}U_{\nu}\equiv$diag.($\tilde{m}_{1},\tilde{m}_{2},\tilde{m}_{3}$).
Finally, we find

$\displaystyle\alpha_{1}^{2}=\frac{\Delta m^{2}_{\rm
atm}}{\tilde{m}^{2}_{3}-\tilde{m}^{2}_{1}},\quad\Delta m^{2}_{\rm
sol}=\frac{\tilde{m}^{2}_{2}-\tilde{m}^{2}_{1}}{\tilde{m}^{2}_{3}-\tilde{m}^{2}_{1}}\Delta
m^{2}_{\rm atm},\quad({\rm NH}),$ (25a)
$\displaystyle\alpha_{1}^{2}=\frac{\Delta m^{2}_{\rm
atm}}{\tilde{m}^{2}_{2}-\tilde{m}^{2}_{3}},\quad\Delta m^{2}_{\rm
sol}=\frac{\tilde{m}^{2}_{2}-\tilde{m}^{2}_{1}}{\tilde{m}^{2}_{2}-\tilde{m}^{2}_{3}}\Delta
m^{2}_{\rm atm},\quad({\rm IH}),$ (25b)

where we require $\alpha_{1}^{2}\leq 4\pi$ to guarantee perturbativity of the
Yukawa coupling. Then, one finds $U_{PMNS}=V^{\dagger}_{eL}U_{\nu}$. Each of
mixing is given in terms of the component of $U_{MNS}$ as follows:

$\displaystyle\sin^{2}\theta_{13}=|(U_{PMNS})_{13}|^{2},\quad\sin^{2}\theta_{23}=\frac{|(U_{PMNS})_{23}|^{2}}{1-|(U_{PMNS})_{13}|^{2}},\quad\sin^{2}\theta_{12}=\frac{|(U_{PMNS})_{12}|^{2}}{1-|(U_{PMNS})_{13}|^{2}}.$
(26)

Figure 3: Dirac scotogenic neutrino mass diagram

Neutral fermion mass matrices are given by

$\displaystyle\left(\begin{matrix}\begin{matrix}0&m_{d}\\\
m_{d}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}\end{matrix}&\Large{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{0}}_{\mathcal{\not{A}}_{4}}}\\\
\Large{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{0}}_{\mathcal{\not{A}}_{4}}}&\begin{matrix}0&\boldsymbol{y_{3}}v_{d}\\\
\boldsymbol{y_{3}}v_{d}&\varepsilon\mu_{\chi}\boldsymbol{y_{\chi}}\end{matrix}\end{matrix}\right)$
$\displaystyle\left(\begin{matrix}\begin{matrix}0&\mu\\\
\mu&0\end{matrix}&\Large{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{0}}_{\mathcal{\not{A}}_{4}}}\\\
\Large{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{0}}_{\mathcal{\not{A}}_{4}}}&\gamma_{N}\Lambda\boldsymbol{y_{n}}\end{matrix}\right)$
(27)

in the $(\nu_{L},\bar{\nu}_{L},\eta_{L},\chi_{L})(P_{\mathcal{R}}=+)$ and
$(\tilde{h}_{uL}^{0},\tilde{h}_{dL}^{0},\tilde{N}_{L})(P_{\mathcal{R}}=-)$
basis, respectively, where bar over $\nu$ is notational like in Martin (1997).
Entries indicated in dark blue are forbidden by $\mathcal{A}_{4}$ modular
invariance. Therefore, we conclude that neutrinos are of Dirac type and do not
mix with other neutral fermions due to modular $\mathcal{A}_{4}$ symmetry and
$P_{\mathcal{R}}$ parity conservation.

Considering the constraints from eq.(12b) for $(Y_{\nu})_{1,2}$ and also
assuming $M_{\widetilde{N}}=M_{\chi_{1}}$ we have $\widetilde{y}_{l}\lesssim
1/\sqrt{2}$ from eq. (16a). Now, assuming the mass of $\chi$’s to be
$\mathcal{O}(1)$ TeV, $Y_{\nu}\simeq 3.78\times 10^{-8}$ to obtain observed
baryon asymmetry. We thus chose small $\beta_{a}$ and $\tilde{\beta}_{a}$
values in our numerical calculation below to achieve condition for
$(Y_{\nu})_{1(2)}\equiv\beta_{a}\boldsymbol{y_{\nu}}(\tilde{\beta}_{a}\boldsymbol{y_{\nu}})$.

##  V Neutrino analysis and discussion

In this section, we perform numerical $\Delta\chi^{2}$ analysis searching for
allowed region, satisfying neutrino oscillation data and LFVs. Also we show
our predictions, where we apply the best fit values for charged-lepton masses.
Here, we concentrate on the NH case, since IH is disfavored which would be
clarified by analytical estimation as can be seen in the previous section(if
possible).

In our numerical analysis, we randomly scan free parameters in the following
ranges

$\displaystyle\\{\tilde{\alpha}_{2},\tilde{\alpha}_{3}\\}\in[10^{-4},1.0],\quad\\{\beta_{1,2,3},\tilde{\beta}_{1,2,3}\\}\in[10^{-10},10^{-6}],\quad\sin\theta_{H_{12,13,23}}\in[-0.5,0.5],$
$\displaystyle\\{m_{H_{1}},M_{0}\\}\in[10^{3},10^{4}]\ {\rm GeV},\quad
m_{H_{2}}\in[1.0,1.1]M_{0},\quad m_{H_{3}}\in[1.1,1.2]M_{0},$ (28)

where $\tau$ runs over the fundamental region. Here we choose small
$\beta_{a}(\tilde{\beta}_{a})$ values that are required to realize Baryon
assymmetry indicated by Eq. 12c. Then, we perform numerical analysis and
discuss below. In Fig. 4, we shows the allowed region between real part of
$\tau$ and imaginary part of $\tau$, where the blue points are allowed within
2, green ones within 3, and red one within 5 of $\sqrt{\Delta\chi^{2}}$ for
five accurately known observables $\Delta m^{2}_{\rm atm},\Delta m^{2}_{\rm
sol},s_{12}^{2},s_{23}^{2},s_{13}^{2}$ in Nufit 5.0 Esteban  _et al._ (2019,
2020, ). The real part of $\tau$ runs whole the range in the fundamental
region, while the imaginary one runs over the region of $[1.0-1.7]$.

Figure 4: Allowed region of modulus $\tau$, where the blue points are allowed
within 2, green ones within 3, and red one within 5 of $\Delta\chi^{2}$
analysis.

Fig. 5 demonstrates the correlation between the sum of neutrino mass
eigenvalues ($\sum m_{i}$ eV) and Dirac CP phase $\delta_{CP}$. The legend is
the same as Fig.1. Dirac CP runs whole the ranges, while $\sum m_{i}$ tends to
be localized at around $0.06$ eV. It implies that the lightest neutrino mass
is very small compared to the other two masses.

Figure 5: Correlation between the sum of neutrino mass eigenvalues ($\sum
m_{i}$ eV) and Dirac CP phase $\delta_{CP}$, where the legend is the same as
the case of Fig.1.

Figs. 6 show relations between LFVs and $s^{2}_{23}$. All the three BR’s of
LFVs are less than $10^{-19}$, which are much smaller than the current upper
experimental bounds.

Figure 6: Relations between LFVs and $s^{2}_{23}$, where the legend is the
same as the case of Fig.1.

Finally, we show a benchmark in table 4, where we select it so that
$\sqrt{\Delta\chi^{2}}$ is minimum. The mass matrices for dimensionless
neutrino and charged-lepton are found as

$\displaystyle V_{e_{L}}$
$\displaystyle=\left[\begin{array}[]{ccc}-0.200&0.959&0.200\\\
-0.0596+0.0129i&0.187-0.0407i&-0.957+0.208i\\\
-0.955-0.211i&-0.203-0.0450i&0.0197+0.00429i\\\ \end{array}\right],$ (32)
$\displaystyle U_{\nu}$
$\displaystyle=\left[\begin{array}[]{ccc}0.338&0.730&-0.594\\\
-0.206-0.132i&-0.355-0.412i&-0.554-0.582i\\\
-0.906-0.0648i&0.408+0.0710i&-0.0147+0.0505i\\\ \end{array}\right].$ (36)

$\tau$ | $0.102244+1.61566i$
---|---
$[s_{H_{12}},s_{H_{23}},s_{{31}}]$ | $[0.393,0.0975,0.00692]$
$[\alpha_{\ell},\beta_{\ell},\gamma_{\ell}]$ | $[0.0596,0.000298,0.979]$
$[\alpha_{1},\tilde{\alpha}_{2},\tilde{\alpha}_{3}]$ | $[6.64\times 10^{-7},0.00966+0.000711i,0.00272+0.758i]$
$[\beta_{1},\beta_{2},\beta_{3}]$ | $[(17.9-3.90i)\times 10^{-10},(16.1-2.82i)\times 10^{-9},(208+2.83i)\times 10^{-9}]$
$[\tilde{\beta}_{1},\tilde{\beta}_{2},\tilde{\beta}_{3}]$ | $[(7.42+5.70i)\times 10^{-10},(175-2.74i)\times 10^{-9},(769+9.91i)\times 10^{-10}]$
$[m_{H_{1}},m_{H_{2}},m_{H_{3}}]$ | $[4661,4871,5493]\ {\rm GeV}$
$[M_{0},D_{N_{1}},D_{N_{2}},D_{N_{3}}]$ | $[8100,6340,10062,16402]\ {\rm GeV}$
$\Delta m^{2}_{\rm atm}$ | $2.52\times 10^{-3}{\rm eV}^{2}$
$\Delta m^{2}_{\rm sol}$ | $7.53\times 10^{-5}{\rm eV}^{2}$
$\sin^{2}\theta_{12}$ | $0.295$
$\sin^{2}\theta_{23}$ | $0.451$
$\sin^{2}\theta_{13}$ | $0.0219$
$\delta_{CP}$ | $18.0^{\circ}$
$\sum m_{i}$ |  $59.0$ meV
$\sqrt{\Delta\chi^{2}}$ | $1.32$
Table 4: A benchmark point of our input parameters and observables, where we
select it so that $\sqrt{\Delta\chi^{2}}$ is minimum.

##  VI Discussion

Our model is capable of accommodating multiparticle dark matter (DM) due to
presence of $P_{\mathcal{R}}$ and modular $\mathcal{A}_{4}$ symmetry
invariances. A $P_{\mathcal{R}}$ _odd_ candidate
($\widetilde{N}_{L},\widetilde{\eta}_{s},\widetilde{\chi}_{s}$) is the
canonical SUSY WIMP, which in our model is connected to the radiative neutrino
mass generation via scotogenic mechanism. Second component of the
multiparticle DM is guaranteed by the accidental $\mathbb{Z}_{2}$ discrete
symmetry which is induced by the modular $\mathcal{A}_{4}$ invariance of the
model. Here possible DM candidates are odd under this $\mathbb{Z}_{2}$
symmetry and even under $P_{\mathcal{R}}$, which are
$N_{s},\eta^{0}_{L},\chi_{L}$ superfields. Furthermore, in our scenario we are
assuming $\chi_{L}$ to be heavier in order to generate asymmetry in the right-
handed sector. The mixture of $\eta^{0}_{L},\chi_{L}$ (as shown in eq. 27) is
not singlet under SM gauge symmetry therefore $N_{s}$, which is a SM singlet,
is the best DM candidate odd under accidental $\mathbb{Z}_{2}$ symmetry.
Further details on dark matter abundance and direct detection are left for
elsewhere.

##  VII Conclusion

In this work a Dirac radiative neutrino mass model based on modular
$\mathcal{A}_{4}$ symmetry was presented. Being a scotogenic neutrino mass
model, it demonstrates a natural connection between naturally small Dirac
neutrino mass origin and existence of dark matter. Modular $\mathcal{A}_{4}$
symmetry in this work achieves simultaneously three main goals: Firstly, the
Diracness of the neutrinos, _aka_ forbids all majorana neutrino mass terms,
secondly it builds a connection of neutrino with dark matter through the well
know scotogenic mechanism, and finally, it predicts and reproduces Dirac phase
as well as neutrino mass splittings in the leptonic sector. This model favors
normal neutrino mass hierarchy, while disfavors the inverted one. Heavy
neutral, dark, fermions are of the order of $\mathcal{O}(1-10)$ TeV, dark
neutral scalars are of the order $\mathcal{O}(1-5)$ TeV, whereas light
neutrino masses are of the order of $0.1$ eV, and the sum of neutrino masses
is around $60$ meV. Even though, neutrinos are Dirac and the lepton number is
conserved in our model, we achieve the matter asymmetry of the universe by
means of the lepton number violation in the right$-$handed neutrino sector via
a mechanism known as _neutrinogenesis_. The required phase in the combination
of the lepton sector yukawa couplings is of the order $\mathcal{O}(10^{-6})$.
Model is able to predict preferred Dirac phase in the leptonic sector, as well
as neutrino mass splittings and mass order, accommodate multicomponent dark
matter, thanks to the _R_ $-$parity and accidental scotogenic $\mathbb{Z}_{2}$
symmetry with minimum set of input parameters. This is the first model of
Dirac scotogenic neutrino mass based on modular symmetries.

###### Acknowledgements.

This research was supported by an appointment to the JRG Program at the APCTP
through the Science and Technology Promotion Fund and Lottery Fund of the
Korean Government. This was also supported by the Korean Local Governments -
Gyeongsangbuk-do Province and Pohang City (H.O.). A.D. and O.P. are supported
by the National Research Foundation of Korea Grants No. 2017K1A3A7A09016430
and No. 2017R1A2B4006338. OP is also supported by the Samsung Science and
Technology Foundation under Grant No. SSTF-BA1602-04 and National Research
Foundation of Korea under Grant Number 2018R1A2B6007000. Feynman diagrams were
created using Ti _k_ Z-Feynman package Ellis (2017).

## References

  * Feruglio (2019) F. Feruglio, in  _From My Vast Repertoire …: Guido Altarelli’s Legacy_ , edited by A. Levy, S. Forte, and G. Ridolfi (2019) pp. 227–266, arXiv:1706.08749 [hep-ph] .
  * de Adelhart Toorop  _et al._ (2012) R. de Adelhart Toorop, F. Feruglio, and C. Hagedorn, Nucl. Phys. B858, 437 (2012), arXiv:1112.1340 [hep-ph] .
  * Criado and Feruglio (2018) J. C. Criado and F. Feruglio, SciPost Phys. 5, 042 (2018), arXiv:1807.01125 [hep-ph] .
  * Kobayashi  _et al._ (2018a) T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, JHEP 11, 196 (2018a), arXiv:1808.03012 [hep-ph] .
  * Okada and Tanimoto (2019a) H. Okada and M. Tanimoto, Phys. Lett. B791, 54 (2019a), arXiv:1812.09677 [hep-ph] .
  * Nomura and Okada (2019) T. Nomura and H. Okada, Phys. Lett. B 797, 134799 (2019), arXiv:1904.03937 [hep-ph] .
  * Okada and Tanimoto (2019b) H. Okada and M. Tanimoto, (2019b), arXiv:1905.13421 [hep-ph] .
  * de Anda  _et al._ (2018) F. J. de Anda, S. F. King, and E. Perdomo, (2018), arXiv:1812.05620 [hep-ph] .
  * Novichkov  _et al._ (2019a) P. P. Novichkov, S. T. Petcov, and M. Tanimoto, Phys. Lett. B793, 247 (2019a), arXiv:1812.11289 [hep-ph] .
  * Nomura and Okada (2021a) T. Nomura and H. Okada, Nucl. Phys. B 966, 115372 (2021a), arXiv:1906.03927 [hep-ph] .
  * Okada and Orikasa (2019a) H. Okada and Y. Orikasa, (2019a), arXiv:1907.13520 [hep-ph] .
  * Ding  _et al._ (2019a) G.-J. Ding, S. F. King, and X.-G. Liu, (2019a), arXiv:1907.11714 [hep-ph] .
  * Nomura  _et al._ (2020) T. Nomura, H. Okada, and O. Popov, Phys. Lett. B 803, 135294 (2020), arXiv:1908.07457 [hep-ph] .
  * Kobayashi  _et al._ (2019a) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, Phys. Rev. D 100, 115045 (2019a), [Erratum: Phys.Rev.D 101, 039904 (2020)], arXiv:1909.05139 [hep-ph] .
  * Asaka  _et al._ (2020a) T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida, JHEP 01, 144 (2020a), arXiv:1909.06520 [hep-ph] .
  * Zhang (2019) D. Zhang, (2019), arXiv:1910.07869 [hep-ph] .
  * Ding  _et al._ (2019b) G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, JHEP 12, 030 (2019b), arXiv:1910.03460 [hep-ph] .
  * Kobayashi  _et al._ (2020a) T. Kobayashi, T. Nomura, and T. Shimomura, Phys. Rev. D 102, 035019 (2020a), arXiv:1912.00637 [hep-ph] .
  * Nomura  _et al._ (2021a) T. Nomura, H. Okada, and S. Patra, Nucl. Phys. B 967, 115395 (2021a), arXiv:1912.00379 [hep-ph] .
  * Wang (2020) X. Wang, Nucl. Phys. B 957, 115105 (2020), arXiv:1912.13284 [hep-ph] .
  * Okada and Shoji (2020) H. Okada and Y. Shoji, Nucl. Phys. B 961, 115216 (2020), arXiv:2003.13219 [hep-ph] .
  * Okada and Tanimoto (2020) H. Okada and M. Tanimoto, (2020), arXiv:2005.00775 [hep-ph] .
  * Behera  _et al._ (2020a) M. K. Behera, S. Singirala, S. Mishra, and R. Mohanta, (2020a), arXiv:2009.01806 [hep-ph] .
  * Behera  _et al._ (2020b) M. K. Behera, S. Mishra, S. Singirala, and R. Mohanta, (2020b), arXiv:2007.00545 [hep-ph] .
  * Nomura and Okada (2020a) T. Nomura and H. Okada, (2020a), arXiv:2007.04801 [hep-ph] .
  * Nomura and Okada (2020b) T. Nomura and H. Okada, (2020b), arXiv:2007.15459 [hep-ph] .
  * Asaka  _et al._ (2020b) T. Asaka, Y. Heo, and T. Yoshida, Phys. Lett. B 811, 135956 (2020b), arXiv:2009.12120 [hep-ph] .
  * Okada and Tanimoto (2021a) H. Okada and M. Tanimoto, Phys. Rev. D 103, 015005 (2021a), arXiv:2009.14242 [hep-ph] .
  * Nagao and Okada (2020) K. I. Nagao and H. Okada, (2020), arXiv:2010.03348 [hep-ph] .
  * Okada and Tanimoto (2021b) H. Okada and M. Tanimoto, JHEP 03, 010 (2021b), arXiv:2012.01688 [hep-ph] .
  * Yao  _et al._ (2021a) C.-Y. Yao, J.-N. Lu, and G.-J. Ding, JHEP 05, 102 (2021a), arXiv:2012.13390 [hep-ph] .
  * Chen  _et al._ (2021) P. Chen, G.-J. Ding, and S. F. King, JHEP 04, 239 (2021), arXiv:2101.12724 [hep-ph] .
  * Kashav and Verma (2021) M. Kashav and S. Verma, JHEP 09, 100 (2021), arXiv:2103.07207 [hep-ph] .
  * Okada  _et al._ (2021) H. Okada, Y. Shimizu, M. Tanimoto, and T. Yoshida, JHEP 07, 184 (2021), arXiv:2105.14292 [hep-ph] .
  * de Medeiros Varzielas and Lourenço (2021) I. de Medeiros Varzielas and J. a. Lourenço, (2021), arXiv:2107.04042 [hep-ph] .
  * Nomura  _et al._ (2021b) T. Nomura, H. Okada, and Y. Orikasa, Eur. Phys. J. C 81, 947 (2021b), arXiv:2106.12375 [hep-ph] .
  * Hutauruk  _et al._ (2020) P. T. P. Hutauruk, D. W. Kang, J. Kim, and H. Okada, (2020), arXiv:2012.11156 [hep-ph] .
  * Ding  _et al._ (2021a) G.-J. Ding, S. F. King, and J.-N. Lu, (2021a), arXiv:2108.09655 [hep-ph] .
  * Nagao and Okada (2021) K. I. Nagao and H. Okada, (2021), arXiv:2108.09984 [hep-ph] .
  * Okada and Qi (2021) H. Okada and Y.-h. Qi, (2021), arXiv:2109.13779 [hep-ph] .
  * Kobayashi  _et al._ (2018b) T. Kobayashi, K. Tanaka, and T. H. Tatsuishi, Phys. Rev. D 98, 016004 (2018b), arXiv:1803.10391 [hep-ph] .
  * Kobayashi  _et al._ (2019b) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi, and H. Uchida, Phys. Lett. B 794, 114 (2019b), arXiv:1812.11072 [hep-ph] .
  * Kobayashi  _et al._ (2019c) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, (2019c), arXiv:1906.10341 [hep-ph] .
  * Okada and Orikasa (2019b) H. Okada and Y. Orikasa, Phys. Rev. D 100, 115037 (2019b), arXiv:1907.04716 [hep-ph] .
  * Mishra (2020) S. Mishra, (2020), arXiv:2008.02095 [hep-ph] .
  * Du and Wang (2021) X. Du and F. Wang, JHEP 02, 221 (2021), arXiv:2012.01397 [hep-ph] .
  * Penedo and Petcov (2019) J. T. Penedo and S. T. Petcov, Nucl. Phys. B939, 292 (2019), arXiv:1806.11040 [hep-ph] .
  * Novichkov  _et al._ (2019b) P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, JHEP 04, 005 (2019b), arXiv:1811.04933 [hep-ph] .
  * Kobayashi  _et al._ (2019d) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, (2019d), arXiv:1907.09141 [hep-ph] .
  * King and Zhou (2019) S. F. King and Y.-L. Zhou, (2019), arXiv:1908.02770 [hep-ph] .
  * Okada and Orikasa (2019c) H. Okada and Y. Orikasa, (2019c), arXiv:1908.08409 [hep-ph] .
  * Criado  _et al._ (2020) J. C. Criado, F. Feruglio, and S. J. D. King, JHEP 02, 001 (2020), arXiv:1908.11867 [hep-ph] .
  * Wang and Zhou (2020) X. Wang and S. Zhou, JHEP 05, 017 (2020), arXiv:1910.09473 [hep-ph] .
  * Zhao and Zhang (2021) Y. Zhao and H.-H. Zhang, JHEP 03, 002 (2021), arXiv:2101.02266 [hep-ph] .
  * King and Zhou (2021) S. F. King and Y.-L. Zhou, JHEP 04, 291 (2021), arXiv:2103.02633 [hep-ph] .
  * Ding  _et al._ (2021b) G.-J. Ding, S. F. King, and C.-Y. Yao, Phys. Rev. D 104, 055034 (2021b), arXiv:2103.16311 [hep-ph] .
  * Zhang and Zhou (2021) X. Zhang and S. Zhou, JCAP 09, 043 (2021), arXiv:2106.03433 [hep-ph] .
  * Qu  _et al._ (2021) B.-Y. Qu, X.-G. Liu, P.-T. Chen, and G.-J. Ding, Phys. Rev. D 104, 076001 (2021), arXiv:2106.11659 [hep-ph] .
  * Nomura and Okada (2021b) T. Nomura and H. Okada, (2021b), arXiv:2109.04157 [hep-ph] .
  * Novichkov  _et al._ (2019c) P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, JHEP 04, 174 (2019c), arXiv:1812.02158 [hep-ph] .
  * Ding  _et al._ (2019c) G.-J. Ding, S. F. King, and X.-G. Liu, (2019c), arXiv:1903.12588 [hep-ph] .
  * Wang  _et al._ (2021) X. Wang, B. Yu, and S. Zhou, Phys. Rev. D 103, 076005 (2021), arXiv:2010.10159 [hep-ph] .
  * Yao  _et al._ (2021b) C.-Y. Yao, X.-G. Liu, and G.-J. Ding, Phys. Rev. D 103, 095013 (2021b), arXiv:2011.03501 [hep-ph] .
  * Wang and Zhou (2021) X. Wang and S. Zhou, JHEP 07, 093 (2021), arXiv:2102.04358 [hep-ph] .
  * Behera and Mohanta (2021) M. K. Behera and R. Mohanta, (2021), arXiv:2108.01059 [hep-ph] .
  * Baur  _et al._ (2019a) A. Baur, H. P. Nilles, A. Trautner, and P. K. S. Vaudrevange, Phys. Lett. B 795, 7 (2019a), arXiv:1901.03251 [hep-th] .
  * De Medeiros Varzielas  _et al._ (2019) I. De Medeiros Varzielas, S. F. King, and Y.-L. Zhou, (2019), arXiv:1906.02208 [hep-ph] .
  * Liu and Ding (2019) X.-G. Liu and G.-J. Ding, JHEP 08, 134 (2019), arXiv:1907.01488 [hep-ph] .
  * Chen  _et al._ (2020a) P. Chen, G.-J. Ding, J.-N. Lu, and J. W. F. Valle, Phys. Rev. D 102, 095014 (2020a), arXiv:2003.02734 [hep-ph] .
  * Li  _et al._ (2021) C.-C. Li, X.-G. Liu, and G.-J. Ding, JHEP 10, 238 (2021), arXiv:2108.02181 [hep-ph] .
  * Novichkov  _et al._ (2021a) P. P. Novichkov, J. T. Penedo, and S. T. Petcov, Nucl. Phys. B 963, 115301 (2021a), arXiv:2006.03058 [hep-ph] .
  * Liu  _et al._ (2021) X.-G. Liu, C.-Y. Yao, and G.-J. Ding, Phys. Rev. D 103, 056013 (2021), arXiv:2006.10722 [hep-ph] .
  * Kikuchi  _et al._ (2020) S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada, and H. Uchida, JHEP 11, 101 (2020), arXiv:2007.06188 [hep-th] .
  * Almumin  _et al._ (2021) Y. Almumin, M.-C. Chen, V. Knapp-Pérez, S. Ramos-Sánchez, M. Ratz, and S. Shukla, JHEP 05, 078 (2021), arXiv:2102.11286 [hep-th] .
  * Ding  _et al._ (2021c) G.-J. Ding, F. Feruglio, and X.-G. Liu, SciPost Phys. 10, 133 (2021c), arXiv:2102.06716 [hep-ph] .
  * Feruglio  _et al._ (2021) F. Feruglio, V. Gherardi, A. Romanino, and A. Titov, JHEP 05, 242 (2021), arXiv:2101.08718 [hep-ph] .
  * Kikuchi  _et al._ (2021) S. Kikuchi, T. Kobayashi, and H. Uchida, Phys. Rev. D 104, 065008 (2021), arXiv:2101.00826 [hep-th] .
  * Novichkov  _et al._ (2021b) P. P. Novichkov, J. T. Penedo, and S. T. Petcov, JHEP 04, 206 (2021b), arXiv:2102.07488 [hep-ph] .
  * Note (1) For interested readers, some literature reviews would be useful to understand the non-Abelian group and its applications to flavor structure Altarelli and Feruglio (2010); Ishimori  _et al._ (2010, 2012); Hernandez and Smirnov (2012); King and Luhn (2013); King  _et al._ (2014); King (2017); Petcov (2018).
  * Kobayashi  _et al._ (2021) T. Kobayashi, H. Okada, and Y. Orikasa, (2021), arXiv:2111.05674 [hep-ph] .
  * Baur  _et al._ (2019b) A. Baur, H. P. Nilles, A. Trautner, and P. K. S. Vaudrevange, Nucl. Phys. B 947, 114737 (2019b), arXiv:1908.00805 [hep-th] .
  * Kobayashi  _et al._ (2020b) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi, and H. Uchida, Phys. Rev. D 101, 055046 (2020b), arXiv:1910.11553 [hep-ph] .
  * Novichkov  _et al._ (2019d) P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, JHEP 07, 165 (2019d), arXiv:1905.11970 [hep-ph] .
  * Chen  _et al._ (2020b) M.-C. Chen, S. Ramos-Sánchez, and M. Ratz, Phys. Lett. B 801, 135153 (2020b), arXiv:1909.06910 [hep-ph] .
  * de Medeiros Varzielas  _et al._ (2020) I. de Medeiros Varzielas, M. Levy, and Y.-L. Zhou, JHEP 11, 085 (2020), arXiv:2008.05329 [hep-ph] .
  * Dreiner  _et al._ (2010) H. K. Dreiner, H. E. Haber, and S. P. Martin, Phys. Rept. 494, 1 (2010), arXiv:0812.1594 [hep-ph] .
  * Ma and Popov (2017) E. Ma and O. Popov, Phys. Lett. B764, 142 (2017), arXiv:1609.02538 [hep-ph] .
  * Note (2) The matter$-$parity $(3(B-L))$ twin ($-$,$+$) particles are $(L,H_{d}),(\eta,H_{u}),(\bar{\nu}\chi,N)$.
  * Sakharov (1967) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967), [Usp. Fiz. Nauk161,no.5,61(1991)].
  * Nanopoulos and Weinberg (1979) D. V. Nanopoulos and S. Weinberg, Phys. Rev. D20, 2484 (1979).
  * Adhikari and Rangarajan (2002) R. Adhikari and R. Rangarajan, Phys. Rev. D65, 083504 (2002), arXiv:hep-ph/0110387 [hep-ph] .
  * Martin (1997) S. P. Martin,  , 1 (1997), [Adv. Ser. Direct. High Energy Phys.18,1(1998)], arXiv:hep-ph/9709356 [hep-ph] .
  * Esteban  _et al._ (2019) I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz, JHEP 01, 106 (2019), arXiv:1811.05487 [hep-ph] .
  * Esteban  _et al._ (2020) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, JHEP 09, 178 (2020), arXiv:2007.14792 [hep-ph] .
  * (95) I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz, NuFIT 5.0 (2029), www.nu-fit.org, (2020) .
  * Ellis (2017) J. Ellis, Comput. Phys. Commun. 210, 103 (2017), arXiv:1601.05437 [hep-ph] .
  * Altarelli and Feruglio (2010) G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82, 2701 (2010), arXiv:1002.0211 [hep-ph] .
  * Ishimori  _et al._ (2010) H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Prog. Theor. Phys. Suppl. 183, 1 (2010), arXiv:1003.3552 [hep-th] .
  * Ishimori  _et al._ (2012) H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, and M. Tanimoto,  _An introduction to non-Abelian discrete symmetries for particle physicists_ , Vol. 858 (2012).
  * Hernandez and Smirnov (2012) D. Hernandez and A. Y. Smirnov, Phys. Rev. D 86, 053014 (2012), arXiv:1204.0445 [hep-ph] .
  * King and Luhn (2013) S. F. King and C. Luhn, Rept. Prog. Phys. 76, 056201 (2013), arXiv:1301.1340 [hep-ph] .
  * King  _et al._ (2014) S. F. King, A. Merle, S. Morisi, Y. Shimizu, and M. Tanimoto, New J. Phys. 16, 045018 (2014), arXiv:1402.4271 [hep-ph] .
  * King (2017) S. F. King, Prog. Part. Nucl. Phys. 94, 217 (2017), arXiv:1701.04413 [hep-ph] .
  * Petcov (2018) S. T. Petcov, Eur. Phys. J. C 78, 709 (2018), arXiv:1711.10806 [hep-ph] .


# On the structure vector field in lightlike hypersurfaces

Samuel Ssekajja  School of Mathematics
University of the Witwatersrand
Private Bag 3, Wits 2050
South Africa<EMAIL_ADDRESS>
###### Abstract.

We study lightlike hypersurfaces of an indefinite almost contact metric-
manifold $\bar{M}$. We prove that there are only two types of such
hypersurfaces, known as ascreen and inascreen, with respect to the position of
the structure vector field of $\bar{M}$. We also show that the second class of
hypersurfaces naturally admits an almost Hermitian structure.

###### Key words and phrases:

Lightlike hypersurfaces, Ascreen hypersurfaces, Inascreen hypersurfaces

###### 2010 Mathematics Subject Classification:

Primary 53C25; Secondary 53C40, 53C50

##  1\. Introduction

Lightlike hypersurfaces, $M$, of indefinite almost contact metric manifolds,
$\bar{M}$, have been studied by many authors. Among those are the following
[1], [3], [4], [5], [6], [7], [8], [9] and [10]. A lot of research has been
done on such hypersurfaces to-date, and most of the work is on tangential
hypersurfaces, that is; those which are tangent to the structure vector field
of $\bar{M}$. Although it is relatively easy to study tangential lightlike
hypersurfaces, it has been shown that most of the well-known lightlike
hypersurfaces are non-existent in this case. For example, these hypersurfaces
can not be totally umbilical (although they have been assumed to be totally
umbilical in the paper [9]), totally screen umbilical or screen conformal.
This shows that the position of the structure vector field relative to each
hypersurface has a great impact on the underlying geometry. In an effort to
extend the study of lightlike hypersurfaces to non-tangential ones, D. H. Jin
introduced a new class of lightlike hypersurfaces and named it ascreen [5, 6,
7, 8]. In this class of hypersurfaces, the structure vector field is nowhere
tangent to the hypersurface. In fact, this vector field lies in the orthogonal
complement of the screen distribution, over $M$, in the ambient space. Some
results have been proved regarding this class of hypersurfaces. Some of those
results can be seen in the above articles. A natural question arises here;

######  Question 1.1.

Given a lightlike hypersurface $M$ of an indefinite almost contact metric-
manifold $\bar{M}$. Is it possible to precisely locate the position of the
structure vector field of $\bar{M}$ relative to $M$?

Our answer to this question is affirmative. In fact, we prove, in Theorem 3.5,
that there are only two types of such hypersurfaces, i.e. the ascreen and what
we have named inascreen. The second class (inascreen) also includes the well-
down tangential lightlike hypersurfaces. Finally, we prove that this class of
hypersurfaces admits an an almost Hermitian structure (see Theorem 4.5).

##  2\. Preliminaries

An odd-dimensional semi-Riemannian manifold $(\bar{M},\bar{g})$ is called an
almost contact metric-manifold [11, 12] if there are a $(1,1)$ tensor field
$\bar{\phi}$, a vector field $\zeta$ , called structure vector field, and a
1-form $\eta$ such that

$\displaystyle\eta(\zeta)=1\quad\mbox{and}\quad\bar{\phi}^{2}=-I+\eta\otimes\zeta,$
(2.1)

where $I$ denotes the identity transformation. From (2.1), we have

$\displaystyle\bar{\phi}\zeta=0\quad\mbox{and}\quad\eta\circ\bar{\phi}=0.$
(2.2)

Futhermore, on an almost contact manifold $\bar{M}$ the following holds

$\displaystyle\bar{g}(\bar{\phi}X,\bar{\phi}Y)$
$\displaystyle=\bar{g}(X,Y)-\eta(X)\eta(Y),$ (2.3)

for any $X$ and $Y$ tangent to $\bar{M}$. It follows from (2.1) and (2.3) that

$\displaystyle\bar{g}(X,\zeta)=\eta(X).$ (2.4)

We can, also, see from (2.1), (2.2) and (2.3) that $\bar{\phi}$ is skew-
symmetric with respect to $\bar{g}$, i.e.

$\displaystyle\bar{g}(\bar{\phi}X,Y)=-\bar{g}(X,\bar{\phi}Y),$ (2.5)

for any $X$ and $Y$ tangent to $\bar{M}$.

Let $(\bar{M},\bar{g})$ be a semi-Riemannian manifold, and let $(M,g)$ be a
hypersurface of $\bar{M}$, where $g=\bar{g}|_{M}$ is the induced metric tensor
on $M$. We call $M$ a lightlike hypersurface if the normal bundle $TM^{\perp}$
of $M$ is a vector subbundle of the tangent bundle $TM$, of rank 1. Moreover,
it is known [2, 3] that the complementary bundle to $TM^{\perp}$ in $TM$,
called the screen distribution and denoted by $S(TM)$, is non-degenerate and
the following decomposition holds;

$\displaystyle TM=S(TM)\perp TM^{\perp},$ (2.6)

where $\perp$ denotes the orthogonal direct sum. A lightlike hypersurface $M$
with a chosen screen distribution will be denoted by $M=(M,g,S(TM))$. Then,
there exists a unique vector bundle $\mathrm{tr}(TM)$, called the lightlike
transversal bundle of $M$ with respect to $S(TM)$, of rank 1 over $M$ such
that for any non-zero section $\xi$ of $TM^{\perp}$ on a coordinate
neighbourhood $\mathcal{U}\subset M$, there exists a unique section $N$ of
$\mathrm{tr}(TM)$ on $\mathcal{U}$ satisfying the conditions

$\displaystyle\bar{g}(\xi,N)=1\quad\mbox{and}\quad\bar{g}(N,N)=\bar{g}(N,Z)=0,$
(2.7)

for any $Z$ tangent to $S(TM)$. Consequently, we have the following
decomposition.

$\displaystyle T\bar{M}|_{M}$
$\displaystyle=S(TM)\perp\\{TM^{\perp}\oplus\mathrm{tr}(TM)\\}$ (2.8)
$\displaystyle=TM\oplus\mathrm{tr}(TM),$

where $\oplus$ denotes a direct sum, not necessarily orthogonal.

Let $\bar{\nabla}$ be the Levi-Civita connection of $\bar{M}$ and let $P$ be
the projection morphism of $TM$ onto $S(TM)$, with respect to (2.6). Then the
local Gauss-Weingarten equations of $M$ and $S(TM)$ are given by [2, 3].

$\displaystyle\bar{\nabla}_{X}Y=\nabla_{X}Y+B(X,Y)N,\quad\bar{\nabla}_{X}N=-A_{N}X+\tau(X)N,$
and
$\displaystyle\nabla_{X}PY=\nabla^{*}_{X}PY+C(X,PY)\xi,\quad\nabla_{X}\xi=-A^{*}_{\xi}X-\tau(X)\xi,$

respectively, for all $X$ and $Y$ tangent to $M$, $\xi$ tangent to
$TM^{\perp}$ and $N$ tangent to $\mathrm{tr}(TM)$. $\nabla$ and $\nabla^{*}$
are the induced connections on $TM$ and $S(TM)$, respectively. $B$ and $C$ are
the local second fundamental forms of $M$ and $S(TM)$, respectively.
Furthermore, $A_{N}$ and $A^{*}_{\xi}$ are the shape operators of $TM$ and
$S(TM)$ respectively, while $\tau$ is a 1-form on $TM$. Moreover,

$\displaystyle g(A^{*}_{\xi}X,Y)=B(X,Y)\quad\mbox{and}\quad
g(A_{N}X,PY)=C(X,PY),$

for any $X$ and $Y$ tangent to $M$. Although $\nabla^{*}$ is a metric
connection, $\nabla$ is generally not, and certisfy the relation

$\displaystyle(\nabla_{X}g)(Y,Z)=B(X,Y)\theta(Z)+B(X,Z)\theta(Y),$

where $\theta$ is 1-form given by $\theta(X)=\bar{g}(X,N)$, for all $X$, $Y$
and $Z$ tangent to $M$. For more details on lightlike hypersurfaces, we refer
the reader to the books [2, 3].

##  3\. A precise location of the structure vector field $\zeta$

Let $M$ be a lightlike hypersurface of an almost contact metric manifold
$\bar{M}$. Let $\xi$ and $N$ be the lightlike sections spanning the normal
bundle $TM^{\perp}$ and transversal bundle $\mathrm{tr}(TM)$ over $M$,
respectively. Since the structure vector field $\zeta$ is tangent to
$\bar{M}$, we decompose it according to (2.8) as

$\displaystyle\zeta=W+a\xi+bN,$ (3.1)

where $W$ is a smooth section of $S(TM)$, while $a$ and $b$ are smooth
functions on $M$. Using (2.5) and (2.4), we have

$\displaystyle a=\eta(N)\quad\mbox{and}\quad b=\eta(\xi).$ (3.2)

Furthermore, using the first relation in (2.1), together with (2.4) and (3.1),
we derive $g(W,W)+2ab=1$. We say that $M$ is tangent to $\zeta$ whenever
$b=\eta(\xi)=0$. In this case, C. Calin [1] has shown that $a=0$ too, i.e.
$\zeta$ belongs to $S(TM)$.

Let $M$ be a lightlike hypersurface of an indefinite almost contact metric
manifold $\bar{M}$. By virtue of relation (2.5), we have
$\bar{g}(\bar{\phi}\xi,\xi)=-\bar{g}(\xi,\bar{\phi}\xi)$, where $\xi$ is
tangent to $TM^{\perp}$. It follows that $\bar{g}(\bar{\phi}\xi,\xi)=0$ and
hence the $\bar{\phi}$ is always tangent to $M$. Let us choose a screen
distribution $S(TM)$ such that $\bar{\phi}\xi$ is tangent to it. Thus, by
(2.5), we have $\bar{g}(\bar{\phi}N,N)=0$ and
$\bar{g}(\bar{\phi}N,\xi)=-\bar{g}(N,\bar{\phi}\xi)=0$. These relations shows
that $\bar{\phi}N$ is also tangent to $M$, and in particular belonging to
$S(TM)$. Furthermore, we have $g(\bar{\phi}\xi,\bar{\phi}N)=1-ab$. Hence,
$\bar{\phi}TM^{\perp}$ and $\bar{\phi}\mathrm{tr}(TM)$ are vector subbundles
of $S(TM)$ of rank 1. Thus, there exists a non-degenerate distribution
$D^{\prime}$, such that

$\displaystyle
S(TM)=\\{\bar{\phi}TM^{\perp}\oplus\bar{\phi}\mathrm{tr}(TM)\\}\perp
D^{\prime}.$ (3.3)

From (2.6), (2.8) and (3.3), the decompositions of $TM$ and $T\bar{M}$ becomes

$\displaystyle TM$
$\displaystyle=\\{\bar{\phi}TM^{\perp}\oplus\bar{\phi}\mathrm{tr}(TM)\\}\perp
D^{\prime}\perp TM^{\perp};$ (3.4) $\displaystyle T\bar{M}_{|M}$
$\displaystyle=\\{\bar{\phi}TM^{\perp}\oplus\bar{\phi}\mathrm{tr}(TM)\\}\perp
D^{\prime}\perp\\{TM^{\perp}\oplus\mathrm{tr}(TM)\\}.$ (3.5)

Furthermore, from (3.3), we can decompose $W$ as

$\displaystyle W=W^{\prime}+f_{1}\bar{\phi}N+f_{2}\bar{\phi}\xi,$ (3.6)

where $W^{\prime}$ is a smooth section of $D^{\prime}$, while $f_{1}$ and
$f_{2}$ are smooth functions on $M$.

######  Proposition 3.1.

$\bar{\phi}D^{\prime}\subset S(TM)$.

###### Proof.

On one hand, using (2.5) and (3.3), we derive

$\displaystyle\bar{g}(\bar{\phi}X^{\prime},\xi)=-\bar{g}(X^{\prime},\bar{\phi}\xi)=0,$
(3.7)

for every $X^{\prime}$ tangent to $D^{\prime}$. Relation (3.7) shows that
$\bar{\phi}X^{\prime}$ is tangent to $M$. On the other hand, we have

$\displaystyle\bar{g}(\bar{\phi}X^{\prime},N)=-\bar{g}(X^{\prime},\bar{\phi}N)=0,$

which, indeed, shows that $\bar{\phi}X^{\prime}$ is tangent to $S(TM)$. ∎

######  Proposition 3.2.

$D^{\prime}$ is $\bar{\phi}$-invariant, i.e. $\bar{\phi}D^{\prime}\subseteq
D^{\prime}$, if and only if one of the following holds:

  1. (1)

$a=b=0$;

  2. (2)

$W^{\prime}=0$.

###### Proof.

In view of the first relation in (2.3), we have

$\displaystyle\bar{g}(\bar{\phi}X^{\prime},\bar{\phi}N)=-a\eta(X^{\prime})\quad\mbox{and}\quad\bar{g}(\bar{\phi}X^{\prime},\bar{\phi}\xi)=-b\eta(X^{\prime}),$
(3.8)

for every $X^{\prime}$ tangent to $D^{\prime}$. Now, if $D^{\prime}$ is
$\bar{\phi}$-invariant then (3.8) gives

$\displaystyle a\eta(X^{\prime})=b\eta(X^{\prime})=0.$ (3.9)

Relation (3.9) shows that either $a=b=0$ or
$\eta(X^{\prime})=g(X^{\prime},W^{\prime})=0$, that is $W^{\prime}=0$. The
converse is obvious. ∎

######  Definition 3.3.

A lightlike hypersurface $M$ of an indefinite almost contact metric-manifold
$\bar{M}$ is said to be an ascreen [5, 6, 7, 8] lightlike hypersurface of
$\bar{M}$ if the vector field $\zeta$ belongs to
$S(TM)^{\perp}=TM^{\perp}\oplus\mathrm{tr}(TM)$.

In any ascreen lightlike hypersurface, $W^{\prime}=0$ and $a,b\neq 0$. It
follows that $D^{\prime}$ is $\bar{\phi}$-invariant. Moreover, the following
result about ascreen hypersurfaces is known.

######  Theorem 3.4 ([5]).

Let $M$ be a lightlike hypersurface of an indefinite almost contact metric-
manifold $\bar{M}$. Then $M$ is an ascreen lightlike hypersurface of $\bar{M}$
if and only if $\bar{\phi}TM^{\perp}=\bar{\phi}\mathrm{tr}(TM)$.

In the next theorem, we show that there are only two types of lightlike
hypersurface $M$ of an indefinite almost contact manifold $\bar{M}$ according
to the position of the structure vector field $\zeta$.

######  Theorem 3.5.

Let $M$ be a lightlike hypersurface of an indefinite almost contact metric-
manifold $\bar{M}$. Then, $\zeta$ takes exactly one of the following forms:

  1. (1)

$\zeta=a\xi+bN$, i.e. $M$ is an ascreen hypersurface;

  2. (2)

$\zeta=W^{\prime}+a\xi+bN$, where $W^{\prime}$ is a non-zero section of
$D^{\prime}$.

###### Proof.

From relations (3.1) and (3.6), $\zeta$ can be written as

$\displaystyle\zeta=W^{\prime}+f_{1}\bar{\phi}N+f_{2}\bar{\phi}\xi+a\xi+bN.$
(3.10)

Taking the inner product of (3.10) with $\bar{\phi}\xi$ and $\bar{\phi}N$ in
turns, and considering (2.3) and (2.2), we have

$\displaystyle b^{2}f_{2}=f_{1}(1-ab)\quad\mbox{and}\quad
a^{2}f_{1}=f_{2}(1-ab),$ (3.11)

respectively. It follows from (3.11), that

$\displaystyle(2ab-1)f_{1}=0\quad\mbox{and}\quad(2ab-1)f_{2}=0.$ (3.12)

From (3.12), we have the following cases:

  1. (1)

$2ab=1$: Taking the inner product on both sides of (3.10) with $\zeta$, we get
$g(W^{\prime},W^{\prime})+2ab=1$. Thus, we have $g(W^{\prime},W^{\prime})=0$,
which implies that $W^{\prime}=0$ by the fact that $D^{\prime}$ is non-
degenerate. Then $\zeta$ in (3.10) reduces to

$\displaystyle\zeta=f_{1}\bar{\phi}N+f_{2}\bar{\phi}\xi+a\xi+bN.$ (3.13)

Applying $\bar{\phi}$ to (3.13), and remembering that $\bar{\phi}\zeta=0$, we
get

$\displaystyle a\bar{\phi}\xi+b\bar{\phi}N-f_{2}\xi-f_{1}N+f\zeta=0,$ (3.14)

where $f=af_{1}+bf_{2}$. Then using (3.13) and (3.14), we get

$\displaystyle(ff_{2}+a)\bar{\phi}\xi+$
$\displaystyle(ff_{1}+b)\bar{\phi}N=0,$ (3.15) $\displaystyle af$
$\displaystyle=f_{2},$ (3.16) $\displaystyle bf$ $\displaystyle=f_{1}.$ (3.17)

Note that none of the functions $ff_{2}+a$ and $ff_{1}+b$ seen in (3.15)
vanishes. In fact, if one assumes that $ff_{2}+a=0$, then (3.16) leads to

$\displaystyle aff_{2}+a^{2}=f^{2}_{2}+a^{2}=0.$ (3.18)

Relation (3.18) shows that $f_{2}=a=0$. But this is impossible since $2ab=1$.
Also, if $ff_{1}+b=0$, relation (3.17) leads to $f_{1}=b=0$, which is again
impossible. Thus, we may rewrite (3.15) as

$\displaystyle\bar{\phi}\xi=\lambda\bar{\phi}N,$ (3.19)

where $\lambda=-\frac{ff_{1}+b}{ff_{2}+a}$ is a non-zero smooth function on
$M$. It is clear from (3.19) that
$\bar{\phi}TM^{\perp}=\bar{\phi}\mathrm{tr}(TM)$, and hence by Theorem 3.4,
$M$ is ascreen.

  2. (2)

$2ab\neq 1$: In this case, we see that $f_{1}=f_{2}=0$ and therefore
$\zeta=W^{\prime}+a\xi+bN$, from which we get
$g(W^{\prime},W^{\prime})+2ab=1$. Furthermore, $W^{\prime}\neq 0$ since if one
takes $W^{\prime}=0$, we get $2ab=1$, which is a contradiction,

which completes the proof. ∎

######  Corollary 3.6.

Every lightlike hypersurface of an indefinite almost contact metric-manifold
in which $D^{\prime}$ is $\bar{\phi}$-invariant, i.e.
$\bar{\phi}D^{\prime}\subseteq D^{\prime}$, is either ascreen or tangential
with $\zeta$ tangent to $D^{\prime}$.

Using part (2) of Theorem 3.5, we have the following definition.

######  Definition 3.7.

A lightlike hypersurface $M$ of an definite almost contact metric-manifold
$\bar{M}$ is called inascreen if $W^{\prime}\neq 0$. In addition, if $b\neq 0$
then $M$ will be called a proper inascreen lightlike hypersurface.

######  Example 3.8.

Any lightlike hypersurface of an indefinite almost contact metric-manifold
that is tangent to the structure vector field $\zeta$ is inascreen with
$a=b=0$.

In view of Proposition 3.2 and Definition 3.7, we hve the following.

######  Proposition 3.9.

The only inascreen lightlike hypersurfaces of an indefinite almost contact
metric-manifold with a $\bar{\phi}$-invariant $D^{\prime}$ are those tangent
to the structure vector field $\zeta$.

##  4\. Proper inascreen lightlike hypersurfaces

Since inascreen lightlike hypersurfaces with $b=0$ are the well-known
tangential lightlike hypersurfaces, in this section we shall focus only on the
proper ones, i.e. those in which $b\neq 0$.

######  Proposition 4.1.

Let $M$ be a proper inascreen lightlike hypersurface of an indefinite almost
contact metric-manifold $\bar{M}$ then, the following holds:

  1. (1)

$D^{\prime}$ is never a $\bar{\phi}$-invariant distribution;

  2. (2)

$\bar{\phi}N$ and $\bar{\phi}\xi$ are linearly independent vector fields;

  3. (3)

$\zeta$ is linearly independent with any $X$ tangent to $M$.

###### Proof.

In a proper inascreen lightlike hypersurface, we know that $W^{\prime}\neq 0$
and $b\neq 0$. It follows from Proposition 3.2 that $D^{\prime}$ is never
$\bar{\phi}$-invariant. Suppose that

$\displaystyle l_{1}\bar{\phi}N+l_{2}\bar{\phi}\xi=0,$ (4.1)

for some smooth functions $l_{1}$ and $l_{2}$ on $M$. Taking the inner product
of (4.1) with $\bar{\phi}N$ and $\bar{\phi}\xi$ by turns, we get

$\displaystyle a^{2}l_{1}=l_{2}(1-ab)\quad\mbox{and}\quad
b^{2}l_{2}=l_{1}(1-ab),$ (4.2)

respectively. The two relations in (4.2) leads to

$\displaystyle l_{1}(1-2ab)=0\quad\mbox{and}\quad l_{2}(1-2ab)=0.$ (4.3)

Since $2ab\neq 1$, (4.3) gives $l_{1}=l_{2}=0$. Finally, suppose that

$\displaystyle l_{3}X+l_{4}\zeta=0,$ (4.4)

for any tangent vector field $X$, where $l_{3}$ and $l_{4}$ are some smooth
functions on $M$. Then, taking the inner product of this relation with $\xi$
leads to $l_{4}\eta(\xi)=l_{4}b=0$. Since $b\neq 0$, we get $l_{4}=0$.
Consequently, $l_{3}X=0$, which gives $l_{3}=0$. ∎

Using part (2) of Proposition 4.1, we deduce the following.

######  Proposition 4.2.

Let $M$ be a proper inascreen lightlike hypersurface of an indefinite almost
contact metric-manifold $\bar{M}$ then, $\dim M\geq 4$ and $\dim\bar{M}\geq
5$.

As $\zeta$ is nowhere tangent to $M$, we put

$\displaystyle\bar{\phi}X=\phi X+\omega(X)\zeta,$ (4.5)

for any $X$ tangent to $M$. Here, $\phi X$ is the tangential part (with
respect to $\zeta$) of $\bar{\phi}X$ to $M$. It is easy to see that $\phi$ and
$\omega$ are tensor fields of type $(1,1)$ and $(0,1)$, respectively, on $M$.
We say that $M$ is invariant if $\omega=0$. Since $\bar{\phi}\xi$ is tangent
to $M$, it follows from (4.5) that

$\displaystyle\bar{\phi}\xi=\phi\xi\quad\mbox{and}\quad\omega(\xi)=0.$ (4.6)

Applying $\bar{\phi}$ to (4.5), and using (2.1) and (2.2), we have

$\displaystyle-X+\eta(X)\zeta=\phi^{2}X+\omega(\phi X)\zeta.$ (4.7)

Comparing components in (4.7) we get

$\displaystyle\phi^{2}X$ $\displaystyle=-X\quad\mbox{and}\quad\omega(\phi
X)=\eta(X),$ (4.8)

for any $X$ tangent to $M$. Thus, the tensor $\phi$ is an almost complex
structure on $M$.

######  Proposition 4.3.

There exist no invariant inascreen lightlike hypersurface of an indefinite
almost contact metric-manifold $\bar{M}$.

###### Proof.

$M$ is invariant whenever $\omega=0$. It follows from (4.8) that $\eta(X)=0$,
for any $X$ tangent to $M$. Taking $X=\xi$ in the last relation, we get
$0=\eta(\xi)=b$, which is a contradiction to $b\neq 0$. ∎

By a direct calculation, while using (2.3), (2.5) and (4.5), we derive

$\displaystyle g(\phi X,\phi Y)$
$\displaystyle=g(X,Y)-\eta(X)\eta(Y)+\omega(X)\omega(Y),$ (4.9) $\displaystyle
g(\phi X,Y)+\omega(X)\eta(Y)$ $\displaystyle=-g(X,\phi Y)-\omega(Y)\eta(X),$
(4.10)

for any $X$ and $Y$ tangent to $M$.

######  Proposition 4.4.

There exit no proper inascreen lightlike hypersurface of an indefinite almost
contact metric-manifold such that;

  1. (1)

$(g,\phi)$ is an almost Hermitian structure on $M$;

  2. (2)

$\phi$ is skew-symmetric with respect to $g$,

for any $X$ and $Y$ tangent to $M$.

###### Proof.

Suppose that $(g,\phi)$ is an almost Hermitian structure on $M$. Then $g$ must
satisfy the relation $g(\phi X,\phi Y)=g(X,Y)$, for any $X$ and $Y$ tangent to
$M$. In view of (4.9), this implies that

$\displaystyle\eta(X)\eta(Y)=\omega(X)\omega(Y),$ (4.11)

for any $X$ and $Y$ tangent to $M$. With $X=Y=\xi$ in (4.11), and considering
relation (4.6), we get

$\displaystyle b^{2}=\eta(\xi)\eta(\xi)=\omega(\xi)\omega(\xi)=0,$

which is impossible for a proper inascreen lightlike hypersurface. Next,
suppose that $\phi$ is skew-symmetric with respect to $g$. Then relation
(4.10) implies that

$\displaystyle\omega(X)\eta(Y)=-\omega(Y)\eta(X),$ (4.12)

for any $X$ and $Y$ tangent to $M$. Taking $Y=\xi$ in (4.12) and considering
(4.6), we have $\omega(X)b=0$. Since $b\neq 0$, we get $\omega=0$ meaning that
$M$ is invariant. However, this is not possible by Proposition 4.3. ∎

Let us set $\tilde{g}=g+\omega\otimes\omega$. This metric $\tilde{g}$ is
degenerate on $M$ since $\tilde{g}(X,\xi)=0$ and $\tilde{g}(X,\phi\xi)=0$, for
every $X$ tangent to $M$. Moreover, using (4.8) and (4.9) we have

$\displaystyle\tilde{g}(\phi X,\phi Y)$ $\displaystyle=g(\phi X,\phi
Y)+\omega(\phi X)\omega(\phi Y)$
$\displaystyle=g(X,Y)-\eta(X)\eta(Y)+\omega(X)\omega(Y)+\omega(\phi
X)\omega(\phi Y)$ $\displaystyle=g(X,Y)+\omega(X)\omega(Y)$
$\displaystyle=\tilde{g}(X,Y).$ (4.13)

Therefore, in view of (4), we have the following.

######  Theorem 4.5.

Every proper inascreen lightlike hypersurface of an indefinite almos contact
metric-manifold admits an almost Hermitian structure $(\tilde{g},\phi)$.

## References

  * [1] C. Calin, Contributions to geometry of CR-submanifold, Thesis, University of Iasi, Iasi, Romania, 1998.
  * [2] K . L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
  * [3] K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhauser Veriag AG Basel-Boston-Berlin, 2010.
  * [4] K. L. Duggal and B. Sahin, Generalized Cauchy-Riemann lightlike submanifolds of Kaehler manifolds, Acta Mathematica Hungarica, vol. 112, no. 1-2, pp. 107-130, 2006.
  * [5] D. H. Jin, Ascreen lightlike hypersurfaces of indefinite Sasakian manifold, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math, 20(1), pp. 25-35, 2013.
  * [6] D. H. Jin, Special Lightlike Hypersurfaces of Indefinite Kaehler Manifolds, Filomat 30:7, 1919-1930, 2016.
  * [7] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold, Indian J Pure Appl Math 41, pp. 569-581, 2010.
  * [8] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite cosymplectic manifold, Commun. Korean Math. Soc. 27, No. 1, pp. 185-195, 2012.
  * [9] T. H. Kang, S. D. Jung, B. H. Kim, H. K. Pak and J. S. Pak, Lightlike hypersurfaces of indefinite Sasakian manifolds, Indian Journal of Pure and Applied Mathematics, vol. 34, no. 9, pp. 1369-1380, 2003.
  * [10] F. Massamba, Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds, Kodai Math. J. 31(3), pp. 338-358, 2008.
  * [11] T. Takahashi, Sasakian manifold with pseudo-Riemannian metric, The Tohoku Mathematical Journal, vol. 21, pp. 271-290, 1969.
  * [12] S. Tanno, Sasakian manifolds with constant $\phi$-holomorphic sectional curvature, The Tohoku Mathematical Journal, vol. 21, pp. 501-507, 1969.