# Quasi two-zero texture in Type-II seesaw at fixed points from modular
$A_{4}$ symmetry

Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Hiroshi Okada<EMAIL_ADDRESS>Department of
Physics, Henan Normal University, Xinxiang 453007, China

###### Abstract

We study a quasi two-zero neutrino texture based on type-II seesaw model in a
modular $A_{4}$ symmetry to evade the cosmological bound on the sum of
neutrino mass while keeping some predictability in neutrino sector. Working on
three fixed points for modulus, we discuss predictions of the model and show
the allowed points satisfying the cosmological bound on neutrino mass from
both CMB and CMB+BAO data.

##  I Introduction

Neutrino masses and their mixing patterns could be determined by a framework
of beyond the standard model (BSM) physics. One of the attractive
possibilities to restrict neutrino mass and mixing is to realize two zero
textures in the neutrino mass matrix with the diagonal charged-lepton mass
matrix. Currently it is found that only seven textures Fritzsch et al. (2011)
satisfy the latest neutrino oscillation data Esteban et al. (2020) as shown
below:

$\displaystyle\left[\begin{array}[]{ccc}0&0&\times\\\ 0&\times&\times\\\
\times&\times&\times\\\
\end{array}\right],\quad\left[\begin{array}[]{ccc}0&\times&0\\\
\times&\times&\times\\\ 0&\times&\times\\\
\end{array}\right],\quad\left[\begin{array}[]{ccc}\times&\times&0\\\
\times&0&\times\\\ 0&\times&\times\\\
\end{array}\right],\quad\left[\begin{array}[]{ccc}\times&0&\times\\\
0&\times&\times\\\ \times&\times&0\\\ \end{array}\right],$ (13)
$\displaystyle\left[\begin{array}[]{ccc}\times&0&\times\\\ 0&0&\times\\\
\times&\times&\times\\\
\end{array}\right],\quad\left[\begin{array}[]{ccc}\times&\times&0\\\
\times&\times&\times\\\ 0&\times&0\\\
\end{array}\right],\quad\left[\begin{array}[]{ccc}\times&\times&\times\\\
\times&0&\times\\\ \times&\times&0\\\ \end{array}\right].$ (23)

There are several ways to lead these textures introducing a symmetry regarding
lepton flavor. For example, a gauged $U(1)_{L_{\mu}-L_{\tau}}$ symmetry is
promising candidate to get some of these textures in addition to explanation
of muon anomalous magnetic dipole moment Ma et al. (2002); Baek et al. (2001);
Nagao et al. (2022); Lou and Nomura (2024); In fact local
$U(1)_{L_{i}-L_{j}}(i,j=e,\mu,\tau)$ symmetry and combination of them are
useful to restrict neutrino mass structure Asai et al. (2019); Asai (2020);
Nomura and Okada (2024). Two zero texture scenarios are attractive since it
provide us predictions for neutrino mixing angles, masses, CP violating phases
and neutrinoless double beta decay. However, these textures possess a
shortcoming that they cannot satisfy upper bounds on the sum of neutrino
masses $\sum D_{\nu}$ derived from analysis of cosmic microwave background
(CMB) and baryon acoustic oscillation (BAO) based on standard $\Lambda$CDM
cosmology; the upper limit is 120 meV from CMB with Planck data Vagnozzi et
al. (2017); Aghanim et al. (2020) and recently more stringent limit of $\sum
D_{\nu}<72$ meV is derived by combining Planck CMB data and BAO data from Dark
Energy Spectroscopic Instrument (DESI) Adame et al. (2024). Thus we would like
to realize a scenario in which neutrino mass structure has quasi two-zero
texture to obtain predictions for neutrino observables while satisfying
constraints on $\sum D_{\nu}$ at the same time.

In order to realize quasi two-zero textures, we apply modular $A_{4}$ flavor
symmetry for the type-II seesaw model. Here we consider $A_{4}$ since it is
known as the minimal non-Abelian group with irreducible triplet representation
Feruglio (2019); Criado and Feruglio (2018); Kobayashi et al. (2018); Okada
and Tanimoto (2019); Nomura and Okada (2019); Okada and Tanimoto (2021a); de
Anda et al. (2020); Novichkov et al. (2019); Nomura and Okada (2021); Okada
and Orikasa (2019); Ding et al. (2019a); Nomura et al. (2020); Kobayashi et
al. (2019); Asaka et al. (2020a); Zhang (2020); Ding et al. (2019b); Kobayashi
et al. (2020); Nomura et al. (2021); Wang (2020); Okada and Shoji (2020);
Okada and Tanimoto (2023); Behera et al. (2022a, b); Nomura and Okada (2022,
2020); Asaka et al. (2020b); Okada and Tanimoto (2021b); Nagao and Okada
(2022); Okada and Tanimoto (2021c); Kang et al. (2022); Qu and Ding (2024);
Ding et al. (2024); Ding and King (2023); Nomura et al. (2024); Kobayashi et
al. (2024); Petcov and Tanimoto (2024); Kobayashi and Tanimoto (2023). 111See
for a different approach to realize texture zeros in quark and lepton mass
matrices adopting the modular symmetries Lu et al. (2020); Kikuchi et al.
(2023); Ding et al. (2023); Devi (2023). We obtain the third type of two-zero
texture in Eq.(1) by assigning $A_{4}$ singlets for three left-handed leptons,
while the charged-lepton mass matrix is given by a non-trivial but predictable
mass matrix by assigning the right-handed charged-lepton fields to be $A_{4}$
triplet. Furthermore, we work on three fixed points of $\tau$; $i$, $\omega$
and $i\infty$, preserving $Z_{2}$, $Z_{3}$, and $Z_{2}$ symmetry,
respectively. These fixed points are statistically favored in the flux
compactification of Type IIB string theory Kobayashi et al. (2022) and
important in phenomenological point of views due to enhancement of
predictions.

This paper is organized as follows. In Sec. II, we show the field contents and
their assignments, and derive two zero textures in the neutrino mass matrix
and specific mass matrix for the charged-lepton. We then analytically
formulate their mixings and mass eigenvalues. In Sec. III, we give numerical
analyses in each fixed points with normal hierarchy and inverted one and show
some predictions for each case. Finally, we summarize and conclude in Sec. IV.

##  II Model setup

|  $\hat{L}$ |  $\hat{\overline{\ell}}$ |  $\hat{H}_{u}$ |  $\hat{H}_{d}$ |  $\hat{\Delta}_{u}$ |  $\hat{\Delta}_{d}$
---|---|---|---|---|---|---
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{2}$ | $\bm{2}$ | $\bm{3}$ | $\bm{3}$
$U(1)_{Y}$ | $-\frac{1}{2}$ | $+1$ | $+\frac{1}{2}$ | $-\frac{1}{2}$ | $+1$ | $-1$
$A_{4}$ | $\\{\bm{1}\\}$ | $\bm{3}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$
$-k_{I}$ | $-5$ | $-1$ | $0$ | $0$ | $0$ | $0$
Table 1: Charge assignments of the SM lepton and new superfields under
$SU(2)_{L}\otimes U(1)_{Y}\otimes A_{4}$ where $-k_{I}$ is the number of
modular weight. Here $\\{\bm{1}\\}=\\{1,1^{\prime\prime},1^{\prime}\\}$
indicates assignment of $A_{4}$ singlets.

The Type-II seesaw mechanism requests an isospin triplet scalar field $\Delta$
to generate neutrino mass where we adopt a supersymmetric scenario to forbid
infinitely possible terms. Therefore we introduce $\Delta_{u}$ and
$\Delta_{d}$ having $+1$ and $-1$ hypercharge, respectively, to cancel gauge
anomaly. 222In analysis, we do not consider superpartners assuming they are
sufficiently heavy without affecting phenomenology. The superfields are
denoted by $\hat{\psi}$ where $\psi$ indicates any field. In the model we
assign three types of $A_{4}$ singlets
$\\{\bm{1}\\}=\\{1,1^{\prime\prime},1^{\prime}\\}$ to the left-handed leptons
$L$ with $-5$ modular weight, and $A_{4}$ triplet to the right-handed charged-
leptons $\bar{\ell}$ with $-1$ modular weight. The other fields are trivial
$A_{4}$ representations with zero modular weight. Relevant fields and their
assignments are summarized in Table 1. The renormalizable superpotential which
are invariant under the modular $A_{4}$ is found as

$\displaystyle W_{\ell}=$
$\displaystyle[Y^{(6)}_{3,3^{\prime}}\hat{\overline{\ell}}\hat{L}_{\ell}]\hat{H}_{d}+{[Y^{(10)}_{1,1^{\prime}}\hat{L}\hat{\Delta}_{u}\hat{L}]}+\mu_{H}\hat{H}_{u}\hat{H}_{d}+\mu_{\Delta}{\rm
Tr}(\hat{\Delta}_{u}\hat{\Delta}_{d})+\lambda_{u}\hat{H}_{u}\hat{\Delta}_{d}\hat{H}_{u}+\lambda_{d}\hat{H}_{d}\hat{\Delta}_{u}\hat{H}_{d},$
(24)

where $Y_{A}^{(k_{Y})}$ indicates a modular form with $A_{4}$ representation
$A$ and modular weight $k_{Y}$, and $[\cdots]$ represents $A_{4}$ singlets
constructed by the $A_{4}$ representations associated with superfields and
modular forms. Note that superpotential is invariant under modular $A_{4}$
symmetry since sum of modular weights are zero for each term.

###  II.1 Charged-lepton mass matrix

The mass matrix of charged-lepton comes from the first term in Eq.(24). After
spontaneous electroweak symmetry breaking, the mass matrix is given by

$\displaystyle
m_{\ell}=\frac{v_{d}}{\sqrt{2}}\left(\begin{array}[]{ccc}y_{1}+py^{\prime}_{1}&y_{2}+qy^{\prime}_{2}&y_{3}+ry^{\prime}_{3}\\\
y_{3}+py^{\prime}_{3}&y_{1}+qy^{\prime}_{1}&y_{2}+ry^{\prime}_{2}\\\
y_{2}+py^{\prime}_{2}&y_{3}+qy^{\prime}_{3}&y_{1}+ry^{\prime}_{1}\end{array}\right)\left(\begin{array}[]{ccc}a_{e}&0&0\\\
0&b_{e}&0\\\ 0&0&c_{e}\end{array}\right),$ (31)

where $v_{d}$ is the vacuum expectation value (VEV) of $H_{d}$, and
$\\{a_{e},b_{e},c_{e}\\}$ are real parameters without loss of generality while
$\\{p,q,r\\}$ are complex parameters. Furthermore,
$Y^{(6)}_{3}\equiv(y_{1},y_{2},y_{3})$ and
$Y^{(6)}_{3^{\prime}}\equiv(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3})$.
The mass eigenvalues for the charged-leptons are obtained via mixings matrices
$V_{R},V_{L}$ as diag.$(m_{e},m_{\mu},m_{\tau})\equiv
V_{R}^{\dagger}m_{\ell}V_{L}$. Therefore,
$V_{L}^{\dagger}m_{\ell}^{\dagger}m_{\ell}V_{L}={\rm
diag.}(|m_{e}|^{2},|m_{\mu}|^{2},|m_{\tau}|^{2})$. We numerically determine
the free parameters $a_{e},b_{e},c_{e}$ in order to fit the three observed
charged-lepton masses, applying the following three relations:

$\displaystyle{\rm
Tr}[m_{\ell}^{\dagger}m_{\ell}]=|m_{e}|^{2}+|m_{\mu}|^{2}+|m_{\tau}|^{2},\quad{\rm
Det}[m_{\ell}^{\dagger}m_{\ell}]=|m_{e}|^{2}|m_{\mu}|^{2}|m_{\tau}|^{2},$
$\displaystyle({\rm Tr}[m_{\ell}^{\dagger}m_{\ell})^{2}-{\rm
Tr}[(m_{e}^{\dagger}m_{e})^{2}]=2(|m_{e}|^{2}|m_{\mu}|^{2}+|m_{\mu}|^{2}|m_{\tau}|^{2}+|m_{e}|^{2}|m_{\tau}|^{2}).$
(32)

In the following, we estimate qualitative behavior of their mass eigenvalues
and mixing matrix on the three fixed points.

####  II.1.1 $\tau=i$

In case of $\tau=i$, we can simplify the modular forms, at the leading order,
as follows; $Y^{(6)}_{3}\sim(-3+2\sqrt{3},-9+5\sqrt{3},12-7\sqrt{3})$ and
$Y^{(6)}_{3^{\prime}}\sim(-12+7\sqrt{3},3-2\sqrt{3},9-5\sqrt{3})$. They lead
us to the one massless eigenvalue. Therefore, the electron mass is induced at
sub-leading order. The eigenvector in the massless eigenvalue is given by

$\displaystyle\frac{1}{A}\left[b_{e}c_{e}(1+q+qr),a_{e}c_{e}(1+r+pr),a_{e}b_{e}(1+p+pq)\right]^{T},$
(33)

where $A\equiv
1/{\sqrt{b_{e}^{2}c_{e}^{2}|1+q+qr|^{2}+a_{e}^{2}b_{e}^{2}|1+p+pq|^{2}+a_{e}^{2}c_{e}^{2}|1+r+pr|^{2}}}$.
Thus, in some limiting cases, we find the first column of $V_{L}$ is given by

$\displaystyle a_{e}\ll
b_{e},c_{e},\quad\left[1,\frac{a_{e}(1+r+pr)}{b_{e}(1+q+qr)},\frac{a_{e}(1+p+pq)}{c_{e}(1+q+qr)}\right]^{T}+{\cal
O}(a_{e}^{2}),$ (34) $\displaystyle b_{e}\ll
a_{e},c_{e},\quad\left[\frac{b_{e}(1+q+qr)}{a_{e}(1+r+pr)},1,\frac{b_{e}(1+p+pq)}{c_{e}(1+r+pr)}\right]^{T}+{\cal
O}(b_{e}^{2}),$ (35) $\displaystyle c_{e}\ll
a_{e},b_{e},\quad\left[\frac{c_{e}(1+q+qr)}{a_{e}(1+p+pq)},\frac{c_{e}(1+r+pr)}{b_{e}(1+p+pq)},1\right]^{T}+{\cal
O}(c_{e}^{2}).$ (36)

They indicate that we have some specific structure of $V_{L}$ when parameters
$\\{a_{e},b_{e},c_{e}\\}$ are hierarchical. In fact such hierarchical points
are obtained in our numerical analysis.

####  II.1.2 $\tau=\omega$

In case of $\tau=\omega$, we can simplify the modular forms, at the leading
order, such that $Y^{(6)}_{3}\sim(0,0,0)$ and
$Y^{(6)}_{3^{\prime}}\sim(-1,2\omega,2\omega^{2})$. They lead us to the unit
mixing matrix for $V_{L}$. Therefore, the neutrino oscillation data is
determined only the the structure of the neutrino matrix at the leading order.
The mass eigenvalues are given by

$\displaystyle\left(\begin{array}[]{ccc}m_{e}^{2}&0&0\\\ 0&m_{\mu}^{2}&0\\\
0&0&m_{\tau}^{2}\end{array}\right)\sim\left(\begin{array}[]{ccc}a_{e}^{2}|p|^{2}&0&0\\\
0&b_{e}^{2}|q|^{2}&0\\\ 0&0&c_{e}^{2}|r|^{2}\end{array}\right).$ (43)

In this case deviation from two-zero texture originates from difference of
$\tau$ from the fixed point $\omega$.

####  II.1.3 $\tau=i\infty$

In case of $\tau=i\infty$, we can simplify the modular forms, at the leading
order, as $Y^{(6)}_{3}\sim(1,0,0)$ and $Y^{(6)}_{3^{\prime}}\sim(0,0,0)$. They
also lead us to the unit mixing matrix for $V_{L}$. Therefore, the neutrino
oscillation data is determined only the the structure of the neutrino matrix
at the leading order. The mass eigenvalues are given by

$\displaystyle\left(\begin{array}[]{ccc}m_{e}^{2}&0&0\\\ 0&m_{\mu}^{2}&0\\\
0&0&m_{\tau}^{2}\end{array}\right)\sim\left(\begin{array}[]{ccc}a_{e}^{2}&0&0\\\
0&b_{e}^{2}&0\\\ 0&0&c_{e}^{2}\end{array}\right).$ (50)

Again, we obtain deviation from two-zero texture due to difference of $\tau$
from fixed point $\tau=i\infty$.

###  II.2 Neutrino mass matrix

The mass matrix of neutrino comes from the second term in Eq.(24). After
spontaneous electroweak symmetry breaking with $\Delta_{u}$ VEV, the mass
matrix is given by

$\displaystyle
m_{\nu}=\frac{v_{\Delta_{u}}}{\sqrt{2}}\left(\begin{array}[]{ccc}a_{\nu}Y^{(10)}_{1}&b_{\nu}Y^{(10)}_{1^{\prime}}&0\\\
b_{\nu}Y^{(10)}_{1^{\prime}}&0&c_{\nu}Y^{(10)}_{1}\\\
0&c_{\nu}Y^{(10)}_{1}&Y^{(10)}_{1^{\prime}}\end{array}\right),$ (54)

where $\\{a_{\nu},b_{\nu},c_{\nu}\\}$ are complex free parameters. Here
$v_{\Delta_{u}}$ is VEV of $\Delta_{u}$ that is required to be
$v_{\Delta}\lesssim{\cal O}(1)$ GeV to satisfy the constraint on the $\rho$
parameter Zyla et al. (2020). The mass eigenvalues for the active neutrinos
$D_{\nu}=\\{D_{\nu_{1}},D_{\nu_{2}},D_{\nu_{3}}\\}$ are obtained via mixings
matrix $V_{\nu}$ as $D_{\nu}\equiv V_{\nu}^{T}m_{\nu}V_{\nu}$. Therefore,
$V_{\nu}^{\dagger}m_{\nu}^{\dagger}m_{\nu}V_{\nu}={\rm
diag.}(|D_{\nu_{1}}|^{2},|D_{\nu_{2}}|^{2},|D_{\nu_{3}}|^{2})$ where
$D_{\nu_{1,2,3}}$ are the neutrino mass eigenvalues. Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) mixing matrix $U(\equiv U_{PMNS})$ is defined by
$U=V_{L}^{\dagger}V_{\nu}$. Then, the above texture is diagonalized in terms
of $U$ and $V_{L}$ as follows:

$\displaystyle
m_{\nu}=V^{*}_{L}U^{*}\left[\begin{array}[]{ccc}D_{\nu_{1}}&0&0\\\
0&D_{\nu_{2}}&0\\\ 0&0&D_{\nu_{3}}\\\
\end{array}\right]U^{\dagger}V^{\dagger}_{L}.$ (58)

The matrix $U$ can be decomposed by $VP$. Here $V$ is the three by three
unitary matrix consisting of three mixing angles $\theta_{12,23,13}$ and one
Dirac CP-phase $\delta$:

$\displaystyle V=\left[\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\\\
\end{array}\right]\left[\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\\
0&1&0\\\ -s_{13}e^{i\delta_{CP}}&0&c_{13}\\\
\end{array}\right]\left[\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\ \end{array}\right],$ (68)

where $s_{ij},\ c_{ij}$ are respectively short-hand notations for
$\sin\theta_{ij},\ \cos\theta_{ij}$ ($ij=12,23,13$). $P$ is the phase matrix
given by $\rm{diag.}(1,e^{i\alpha/2},e^{i\beta/2})$ with $\\{\alpha,\
\beta\\}$ being Majorana phases.

Now we rewrite the neutrino mass matrix in terms of complex mass eigenvalues
and mixings as follows:

$\displaystyle m_{\nu}$
$\displaystyle=V^{*}_{L}V^{*}\left[\begin{array}[]{ccc}e^{-i\alpha/2}&0&0\\\
0&e^{i\beta/2}&0\\\ 0&0&1\\\
\end{array}\right]\left[\begin{array}[]{ccc}D_{\nu_{1}}&0&0\\\
0&D_{\nu_{2}}&0\\\ 0&0&D_{\nu_{3}}\\\
\end{array}\right]\left[\begin{array}[]{ccc}1&0&0\\\ 0&e^{-i\alpha/2}&0\\\
0&0&e^{i\beta/2}\\\ \end{array}\right]V^{\dagger}V^{\dagger}_{L}$ (78)
$\displaystyle\equiv
V^{*}_{L}V^{*}\left[\begin{array}[]{ccc}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\\\ \end{array}\right]V^{\dagger}V^{\dagger}_{L},$ (82)

where
$(\lambda_{1},\lambda_{2},\lambda_{3})\equiv(D_{\nu_{1}},D_{\nu_{2}}e^{-i\alpha},D_{\nu_{3}}e^{-i\beta})$.
Here, we redefine $U^{\prime}\equiv V^{*}_{L}V^{*}$. Therefore, we find

$\displaystyle m_{\nu}=U^{\prime}\left[\begin{array}[]{ccc}\lambda_{1}&0&0\\\
0&\lambda_{2}&0\\\ 0&0&\lambda_{3}\\\ \end{array}\right]U^{\prime T}.$ (86)

Then, applying the above neutrino mass matrix for two zero textures denoted by
$(m_{\nu})_{ab}=(m_{\nu})_{cd}=0$ ($ab\neq cd$), one finds the following
relations:

$\displaystyle\Lambda_{12}$
$\displaystyle\equiv\frac{\lambda_{1}}{\lambda_{2}}=\frac{U^{\prime}_{a3}V_{b3}U^{\prime}_{c2}U^{\prime}_{d2}-U^{\prime}_{a2}U^{\prime}_{b2}U^{\prime}_{c3}U^{\prime}_{d3}}{U^{\prime}_{a1}U^{\prime}_{b1}U^{\prime}_{c3}U^{\prime}_{d3}-U^{\prime}_{a3}U^{\prime}_{b3}U^{\prime}_{c1}U^{\prime}_{d1}},$
$\displaystyle\Lambda_{13}$
$\displaystyle\equiv\frac{\lambda_{1}}{\lambda_{3}}=\frac{U^{\prime}_{a3}V_{b3}U^{\prime}_{c2}U^{\prime}_{d2}-U^{\prime}_{a2}U^{\prime}_{b2}U^{\prime}_{c3}U^{\prime}_{d3}}{U^{\prime}_{a2}U^{\prime}_{b2}U^{\prime}_{c1}U^{\prime}_{d1}-U^{\prime}_{a1}U^{\prime}_{b1}U^{\prime}_{c2}U^{\prime}_{d2}}.$
(87)

Furthermore, one finds the following mass ratios and Majorana phases:

$\displaystyle\frac{D_{\nu_{1}}}{D_{\nu_{2}}}$
$\displaystyle=\left|\Lambda_{12}\right|,\
\frac{D_{\nu_{1}}}{D_{\nu_{3}}}=\left|\Lambda_{13}\right|,\ \alpha={\rm
arg}\left[\Lambda_{13}\right],\ \beta={\rm arg}\left[\Lambda_{12}\right].$
(88)

Two mass square differences $\Delta m^{2}_{\mathrm{sol}}\equiv
D_{\nu_{2}}^{2}-D_{\nu_{1}}^{2}$ and $\Delta m^{2}_{\mathrm{atm}}$ are given
by

$\displaystyle\Delta
m^{2}_{\mathrm{sol}}=D_{\nu_{1}}^{2}(\left|\Lambda_{12}\right|^{-2}-1),$ (89)
$\displaystyle({\bf NH}):~{}\Delta
m^{2}_{\mathrm{atm}}=D_{\nu_{1}}^{2}(\left|\Lambda_{13}\right|^{-2}-1),\quad({\bf
IH}):~{}\Delta
m^{2}_{\mathrm{atm}}=D_{\nu_{1}}^{2}(\left|\Lambda_{12}\right|^{-2}-\left|\Lambda_{13}\right|^{-2}).$
(90)

In addition, the effective mass for the neutrinoless double beta decay and sum
of neutrino masses are respectively given by

$\displaystyle\langle
m_{ee}\rangle=\left|\sum_{i=1}^{3}D_{\nu_{i}}U_{ei}^{2}\right|=\left|D_{\nu_{1}}c^{2}_{12}c^{2}_{13}+D_{\nu_{2}}s^{2}_{12}c^{2}_{13}e^{i\alpha}+D_{\nu_{3}}s^{2}_{13}e^{i(\beta-2\delta_{CP})}\right|$
$\displaystyle=D_{\nu_{1}}\left|c^{2}_{12}c^{2}_{13}+\left|\Lambda_{12}\right|^{-1}s^{2}_{12}c^{2}_{13}e^{i\alpha}+\left|\Lambda_{13}\right|^{-1}s^{2}_{13}e^{i(\beta-2\delta_{CP})}\right|,$
(91)
$\displaystyle\sum_{i=1}^{3}D_{\nu_{i}}=D_{\nu_{1}}\left(1+\left|\Lambda_{12}\right|^{-1}+\left|\Lambda_{13}\right|^{-1}\right),$
(92)

where its predicted value is constrained by the current KamLAND-Zen data and
could be measured in future Abe et al. (2024). The upper bound is found as
$\langle m_{ee}\rangle<(28-122)$ meV at 90 % confidence level where the range
of the bound comes from the use of different method estimating nuclear matrix
elements. Sum of neutrino masses is constrained by the minimal cosmological
model $\Lambda$CDM $+\sum D_{\nu_{i}}$ that provides the upper bound on $\sum
D_{\nu}\leq$ 120 meV Vagnozzi et al. (2017); Aghanim et al. (2020), although
it becomes weaker if the data are analyzed in the context of extended
cosmological models Olive et al. (2014). Recently, DESI and CMB data
combination provides more stringent upper bound on the sum of neutrino masses
$\sum D_{\nu}\leq$ 72 meV Adame et al. (2024). We note that $\sum
D_{\nu}\simeq 172$ meV with IH when the two-zero texture is exact Fritzsch et
al. (2011) and it is beyond the Planck CMB limit. Thus some deviation from the
two-zero texture is necessary to satisfy the constraints on $\sum D_{\nu}$.
The two observable $\langle m_{ee}\rangle$ and $\sum D_{\nu_{i}}$ are also
taken into account in the numerical analysis.

##  III Numerical analysis and phenomenology

In this section, we carry out numerical analysis to fit the neutrino data and
find some predictions of the models. Then some implications for phenomenology
are discussed. We randomly select our input parameters within the following
ranges

$\displaystyle(|p|,|q|,|r|)\in[10^{-3},10^{3}]\ {\rm eV}.$ (93)

Also, we allow $|\epsilon|=[10^{-5}-0.1]$ at each of the fixed point where
$\tau=\tau_{fixed}+\epsilon$. Note that Im[$\tau_{fixed}$]$=[2-10]$ in case of
$\tau=i\infty$.

Figure 1: Numerical analyses at nearby $\tau=i$. The left figure represents
the case of NH and the right one IH. Cyan points satisfy $\sum D_{\nu}\leq$
120 meV.

Figure 2: Numerical analyses at nearby $\tau=i$. The up and down figures
represent the NH and the IH cases, respectively. The left ones demonstrate
neutrinoless double beta decay $\langle m_{ee}\rangle$ in terms of the
lightest neutrino mass where red dashed horizontal lines indicate the current
upper bound from KamLAND-Zen with different estimation for nuclear matrix
elements.. The center ones shows Majorana phases $\alpha$ and $\beta$. The
right ones depict the Dirac CP phase in terms of the sum of neutrino masses
where the red(magenta) dashed vertical lines indicate the upper bound on $\sum
D_{\nu}$ from Planck CMB (+ DESI BAO) data.

###  III.1 $\tau=i$

In Fig. 1, we figure out the allowed range of $\tau$ at nearby $\tau=i$ where
the left figure represents the case of NH and the right one is for IH. It
implies that IH would have more points at nearby $\tau=i$. Cyan points satisfy
$\sum D_{\nu}\leq$ 120 meV. In Fig. 2, we show predictions in the case of
$\tau=i$. The up and down figures represent the NH and the IH cases,
respectively. The left ones demonstrate neutrinoless double beta decay
$\langle m_{ee}\rangle$ in terms of the lightest neutrino masses. The dotted
red horizontal lines are bounds for the current KamLAND-Zen as discussed in
the neutrino sector. In both the cases, most of the points are within this
bound and good testability would be expected. The center ones shows Majorana
phases $\alpha$ and $\beta$. The figures imply that points of NH case would be
localized at nearby $\alpha=0$ and $\beta=\pi$ although these tendency would
be the same. The right ones depict the Dirac CP phase in terms of the sum of
neutrino masses. The vertical red dashed line shows the upper bound of
cosmological limit $\sum D_{\nu}\leq 120$ meV while the vertical magenta
dashed line shows the upper bound of $\sum D_{\nu}\leq 72$ meV. For both NH
and IH case, we find the allowed region to satisfy $\sum D_{\nu}\leq 120$ meV
due to the deviation from two-zero texture originated from the modular
structure of charged-lepton mass matrix.

Figure 3: Numerical analyses at nearby $\tau=\omega$ where the legends of
figures are the same as Fig. 1.

Figure 4: Numerical analyses at nearby $\tau=\omega$. All the legends are the
same as Figs. 2.

###  III.2 $\tau=\omega$

In Fig. 3, we figure out the allowed range of $\tau$ at nearby $\tau=\omega$
where the left figure represents the case of NH and the right one is for IH.
These figures suggest that both the tendency are same. In Fig. 4, we show
predictions in the case of $\tau=\omega$. The up and down figures represent
the NH and the IH cases, respectively. The left ones demonstrate neutrinoless
double beta decay $\langle m_{ee}\rangle$ in terms of the lightest neutrino
masses where the dotted red horizontal lines are bounds from the current
KamLAND-Zen as given in the Fig. 2. In both the cases, most of the points are
within the bound and good testability would be expected. The center ones show
Majorana phases $\alpha$ and $\beta$, and the figures imply that points of NH
case would be localized at nearby $\alpha=0$ and $\beta=\pi$ although these
tendency would be the same. The right ones represent the Dirac CP phase in
terms of the sum of neutrino masses where the vertical red (magenta) dashed
line shows the upper bound of $\sum m_{\nu}$ the same as the Fig. 2. It would
show that IH case is better than the NH one to satisfy the bound of $\sum
m_{\nu}<120$ meV.

Figure 5: Numerical analyses at nearby $\tau=i\infty$ where the legends of
figures are the same as Fig. 1.

Figure 6: Numerical analyses at nearby $\tau=i\infty$. All the legends are the
same as Figs. 2.

###  III.3 $\tau=i\infty$

In Fig. 5, we figure out the allowed range of $\tau$ at nearby $\tau=i\infty$
where the left figure represents the case of NH and the right one is for IH.
The NH case allows the range of Im[$\tau$]$=[2-4]$ while the IH case
Im[$\tau$]$=[2-5]$. In Fig. 6, we show predictions in the case of
$\tau=i\infty$ where the up and down figures represent the NH and the IH
cases, respectively. The left ones demonstrate neutrinoless double beta decay
$\langle m_{ee}\rangle$ in terms of the lightest neutrino masses. The dashed
red horizontal lines are bounds from the current KamLAND-Zen as the Fig. 2. In
both the cases, most of the points are within this bound and would be tested
in near future. The center ones shows Majorana phases $\alpha$ and $\beta$
where the figures imply that both points would be localized at nearby
$\alpha=0$ and $\beta=\pi$. Furthermore, IH case would be more localized
compared to the NH case at the other regions. The right ones represent the
Dirac CP phase in terms of the sum of neutrino masses where the vertical red
(magenta) dashed line shows the upper bound of $\sum m_{\nu}$ the same as the
Fig. 2. It would show that NH case is better than the IH one to satisfy $\sum
m_{\nu}<120$ meV that is opposite behavior from the case of $\tau=\omega$.
Moreover, the only case of NH in $\tau=i\infty$ satisfies the bound of DESI
and CMB $\sum D_{\nu}\leq 72$ meV.

##  IV Summary and discussion

We have studied a model with quasi two-zero texture of neutrino mass matrix in
order to satisfy the cosmological bounds on neutrino mass while keeping
predictability of neutrino observables. The quasi two-zero texture is realized
by applying a modular $A_{4}$ symmetry for the type-II seesaw model. The two-
zero texture of Yukawa couplings among triplet scalar and neutrino is obtained
due to modular $A_{4}$ symmetry, and deviation from two-zero structure is
originated from charged lepton mass matrix which has non-trivial structure
determined by modular $A_{4}$.

We have worked on three fixed points of $\tau$ that are favored by the flux
compactification of type IIB string theory as well as phenomenological points
of views. We have demonstrated several predictions for each points and shown
that CMB bound can be satisfied for all the fixed points while CMB+BAO bound
can be satisfied only near the fixed point $\tau=i\infty$. In addition the
model would indicate specific decay patterns of doubly charged scalar boson
from triplet via Yukawa coupling. As the Yukawa coupling has the two-zero
texture we expect decay of doubly charged scalar into $\mu\mu$ and $e\tau$ are
suppressed. Therefore the model could be tested in various ways such as
neutrino observables, neutrinoless double beta decay, cosmological bound on
sum of neutrino mass and collider physics associated with scalar boson from
the triplet Higgs.

## Acknowledgments

The work was supported by the Fundamental Research Funds for the Central
Universities (T. N.).

## References

  * Fritzsch et al. (2011) H. Fritzsch, Z.-z. Xing, and S. Zhou, JHEP 09, 083 (2011), eprint 1108.4534.
  * Esteban et al. (2020) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, JHEP 09, 178 (2020), eprint 2007.14792.
  * Ma et al. (2002) E. Ma, D. P. Roy, and S. Roy, Phys. Lett. B 525, 101 (2002), eprint hep-ph/0110146.
  * Baek et al. (2001) S. Baek, N. G. Deshpande, X. G. He, and P. Ko, Phys. Rev. D 64, 055006 (2001), eprint hep-ph/0104141.
  * Nagao et al. (2022) K. I. Nagao, T. Nomura, H. Okada, and T. Shimomura (2022), eprint 2212.14528.
  * Lou and Nomura (2024) Y. Lou and T. Nomura (2024), eprint 2406.01030.
  * Asai et al. (2019) K. Asai, K. Hamaguchi, N. Nagata, S.-Y. Tseng, and K. Tsumura, Phys. Rev. D 99, 055029 (2019), eprint 1811.07571.
  * Asai (2020) K. Asai, Eur. Phys. J. C 80, 76 (2020), eprint 1907.04042.
  * Nomura and Okada (2024) T. Nomura and H. Okada, Phys. Lett. B 848, 138393 (2024), eprint 2304.03905.
  * Vagnozzi et al. (2017) S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho, and M. Lattanzi, Phys. Rev. D 96, 123503 (2017), eprint 1701.08172.
  * Aghanim et al. (2020) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], eprint 1807.06209.
  * Adame et al. (2024) A. G. Adame et al. (DESI) (2024), eprint 2404.03002.
  * Feruglio (2019) F. Feruglio, _Are neutrino masses modular forms?_ (2019), pp. 227–266, eprint 1706.08749.
  * Criado and Feruglio (2018) J. C. Criado and F. Feruglio, SciPost Phys. 5, 042 (2018), eprint 1807.01125.
  * Kobayashi et al. (2018) T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, JHEP 11, 196 (2018), eprint 1808.03012.
  * Okada and Tanimoto (2019) H. Okada and M. Tanimoto, Phys. Lett. B 791, 54 (2019), eprint 1812.09677.
  * Nomura and Okada (2019) T. Nomura and H. Okada, Phys. Lett. B 797, 134799 (2019), eprint 1904.03937.
  * Okada and Tanimoto (2021a) H. Okada and M. Tanimoto, Eur. Phys. J. C 81, 52 (2021a), eprint 1905.13421.
  * de Anda et al. (2020) F. J. de Anda, S. F. King, and E. Perdomo, Phys. Rev. D 101, 015028 (2020), eprint 1812.05620.
  * Novichkov et al. (2019) P. Novichkov, S. Petcov, and M. Tanimoto, Phys. Lett. B 793, 247 (2019), eprint 1812.11289.
  * Nomura and Okada (2021) T. Nomura and H. Okada, Nucl. Phys. B 966, 115372 (2021), eprint 1906.03927.
  * Okada and Orikasa (2019) H. Okada and Y. Orikasa (2019), eprint 1907.13520.
  * Ding et al. (2019a) G.-J. Ding, S. F. King, and X.-G. Liu, JHEP 09, 074 (2019a), eprint 1907.11714.
  * Nomura et al. (2020) T. Nomura, H. Okada, and O. Popov, Phys. Lett. B 803, 135294 (2020), eprint 1908.07457.
  * Kobayashi et al. (2019) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, Phys. Rev. D 100, 115045 (2019), [Erratum: Phys.Rev.D 101, 039904 (2020)], eprint 1909.05139.
  * Asaka et al. (2020a) T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida, JHEP 01, 144 (2020a), eprint 1909.06520.
  * Zhang (2020) D. Zhang, Nucl. Phys. B 952, 114935 (2020), eprint 1910.07869.
  * Ding et al. (2019b) G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, JHEP 12, 030 (2019b), eprint 1910.03460.
  * Kobayashi et al. (2020) T. Kobayashi, T. Nomura, and T. Shimomura, Phys. Rev. D 102, 035019 (2020), eprint 1912.00637.
  * Nomura et al. (2021) T. Nomura, H. Okada, and S. Patra, Nucl. Phys. B 967, 115395 (2021), eprint 1912.00379.
  * Wang (2020) X. Wang, Nucl. Phys. B 957, 115105 (2020), eprint 1912.13284.
  * Okada and Shoji (2020) H. Okada and Y. Shoji, Nucl. Phys. B 961, 115216 (2020), eprint 2003.13219.
  * Okada and Tanimoto (2023) H. Okada and M. Tanimoto, Phys. Dark Univ. 40, 101204 (2023), eprint 2005.00775.
  * Behera et al. (2022a) M. K. Behera, S. Singirala, S. Mishra, and R. Mohanta, J. Phys. G 49, 035002 (2022a), eprint 2009.01806.
  * Behera et al. (2022b) M. K. Behera, S. Mishra, S. Singirala, and R. Mohanta, Phys. Dark Univ. 36, 101027 (2022b), eprint 2007.00545.
  * Nomura and Okada (2022) T. Nomura and H. Okada, JCAP 09, 049 (2022), eprint 2007.04801.
  * Nomura and Okada (2020) T. Nomura and H. Okada (2020), eprint 2007.15459.
  * Asaka et al. (2020b) T. Asaka, Y. Heo, and T. Yoshida, Phys. Lett. B 811, 135956 (2020b), eprint 2009.12120.
  * Okada and Tanimoto (2021b) H. Okada and M. Tanimoto, Phys. Rev. D 103, 015005 (2021b), eprint 2009.14242.
  * Nagao and Okada (2022) K. I. Nagao and H. Okada, Nucl. Phys. B 980, 115841 (2022), eprint 2010.03348.
  * Okada and Tanimoto (2021c) H. Okada and M. Tanimoto, JHEP 03, 010 (2021c), eprint 2012.01688.
  * Kang et al. (2022) D. W. Kang, J. Kim, T. Nomura, and H. Okada, JHEP 07, 050 (2022), eprint 2205.08269.
  * Qu and Ding (2024) B.-Y. Qu and G.-J. Ding (2024), eprint 2406.02527.
  * Ding et al. (2024) G.-J. Ding, S.-Y. Jiang, S. F. King, J.-N. Lu, and B.-Y. Qu (2024), eprint 2404.06520.
  * Ding and King (2023) G.-J. Ding and S. F. King (2023), eprint 2311.09282.
  * Nomura et al. (2024) T. Nomura, H. Okada, and H. Otsuka, Nucl. Phys. B 1004, 116579 (2024), eprint 2309.13921.
  * Kobayashi et al. (2024) T. Kobayashi, T. Nomura, H. Okada, and H. Otsuka, JHEP 01, 121 (2024), eprint 2310.10091.
  * Petcov and Tanimoto (2024) S. T. Petcov and M. Tanimoto (2024), eprint 2404.00858.
  * Kobayashi and Tanimoto (2023) T. Kobayashi and M. Tanimoto (2023), eprint 2307.03384.
  * Lu et al. (2020) J.-N. Lu, X.-G. Liu, and G.-J. Ding, Phys. Rev. D 101, 115020 (2020), eprint 1912.07573.
  * Kikuchi et al. (2023) S. Kikuchi, T. Kobayashi, M. Tanimoto, and H. Uchida, Eur. Phys. J. C 83, 591 (2023), eprint 2207.04609.
  * Ding et al. (2023) G.-J. Ding, F. R. Joaquim, and J.-N. Lu, JHEP 03, 141 (2023), eprint 2211.08136.
  * Devi (2023) M. R. Devi (2023), eprint 2303.04900.
  * Kobayashi et al. (2022) T. Kobayashi, H. Otsuka, M. Tanimoto, and K. Yamamoto, Phys. Rev. D 105, 055022 (2022), eprint 2112.00493.
  * Zyla et al. (2020) P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Abe et al. (2024) S. Abe et al. (KamLAND-Zen) (2024), eprint 2406.11438.
  * Olive et al. (2014) K. A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014).