††institutetext:  1 Graduate University for Advanced Studies (Sokendai), 1-1
Oho, Tsukuba, Ibaraki 305-0801, Japan
2Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Republic
of Korea
3Department of Physics, Pohang University of Science and Technology, Pohang
37673, Republic of Korea
4 Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka
819-0395, Japan

# Residual flavor symmetry breaking in the landscape of modular flavor models

Keiya Ishiguro1 Hiroshi Okada2,3 Hajime Otsuka4<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a symmetry breaking of residual flavor symmetries realized at fixed
points of the moduli space. In the supersymmetric modular invariant theories,
a small departure of the modulus from fixed points is required to realize
fermion mass hierarchies and sizable CP-breaking effects. We investigate
whether one can dynamically fix the moduli values in the vicinity of the fixed
points in the context of Type IIB string theory. It is found that the string
landscape prefers $|\delta\tau|\simeq 10^{-5}$ for the deviation of the
complex structure modulus from all fixed points and the CP-breaking vacuum is
statistically favored. To illustrate phenomenological implications of
distributions of moduli values around fixed points, we analyze the lepton
sector on a concrete $A_{4}$ modular flavor model.

††preprint:

##  1 Introduction

The flavor symmetry is a powerful approach to understand the flavor structure
of quarks and leptons, and in addition, it provides the bridge between bottom-
up and top-down approaches of model building. Indeed, when the flavor symmetry
is embedded into a geometric symmetry of an extra-dimensional space, subgroups
of the geometric symmetry would control the flavor structure of matter zero-
modes. For instance, the $PSL(2,\mathbb{Z})$ modular symmetry incorporates the
phenomenologically interesting non-Abelian discrete symmetries such as
$S_{3},S_{4},A_{4}$ and $A_{5}$ in the principal subgroups
deAdelhartToorop:2011re . From the viewpoint of ultraviolet physics, it was
known that the $SL(2,\mathbb{Z})$ modular group and its subgroups appearing in
toroidal compactifications are connected to the flavor symmetries of matter
zero-modes in heterotic orbifold models Ferrara:1989qb ; Lerche:1989cs ;
Lauer:1990tm ; Baur:2019kwi ; Baur:2019iai  and Type IIB superstring theories
with magnetized D-branes Kobayashi:2018rad ; Kobayashi:2018bff ; Ohki:2020bpo
; Kikuchi:2020frp ; Kikuchi:2020nxn ; Kikuchi:2021ogn ; Almumin:2021fbk . Such
flavor symmetries are called modular flavor symmetries. The multi moduli cases
such as $Sp(2h,\mathbb{Z})$ symplectic modular symmetry are also discussed in
the context of heterotic string theory on toroidal orbifolds Baur:2020yjl  and
Calabi-Yau manifolds Strominger:1990pd ; Candelas:1990pi ; Ishiguro:2020nuf ;
Ishiguro:2021ccl .

From the phenomenological point of view, the modular flavor symmetries are
attractive for predicting the masses and mixing angles of quarks and leptons
under a certain value of the moduli fields Feruglio:2017spp ;
Kobayashi:2018vbk ; Penedo:2018nmg ; Novichkov:2018nkm ; Ding:2019xna ;
Liu:2019khw ; Chen:2020udk ; Novichkov:2020eep ; Liu:2020akv ; Wang:2020lxk ;
Yao:2020zml ; Ding:2020msi . The higher-dimensional operators in the Standard
Model effective field theory are also controlled by the modular symmetries
Kobayashi:2021pav ; Kobayashi:2022jvy , taking into account the selection rule
of the string theory Kobayashi:2021uam . The flavor symmetry of
quarks/leptons, as well as CP symmetry, is broken only by the modulus $\tau$
parametrizing the shape of the torus. Note that the CP transformation is
regarded as an outer automorphism of the modular group for the single modulus
Baur:2019kwi ; Novichkov:2019sqv  and multi moduli cases Ishiguro:2021ccl .
Once the modulus is fixed, there is no flavor symmetry in a generic moduli
space. However, there exist so-called fixed points in the fundamental region
of the $PSL(2,\mathbb{Z})$: $\tau=i,w,i\infty$ with
$w=\frac{-1+i\sqrt{3}}{2}$, preserving $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$ and
$\mathbb{Z}_{2}$ symmetries, respectively. Such fixed points play an important
role for several phenomenological applications of the lepton sector
Novichkov:2018ovf ; Novichkov:2018yse ; Novichkov:2018nkm ; Ding:2019gof ;
Okada:2019uoy ; King:2019vhv ; Okada:2020rjb ; Okada:2020ukr ; Okada:2020brs ;
Feruglio:2021dte ; Kobayashi:2021pav ; Kobayashi:2022jvy  as well as
controlling the effective action such as the dark matter (DM) stability
Kobayashi:2021ajl . To dynamically fix the moduli values gives a strong
prediction on proposed modular flavor models. These attempts were performed in
Refs. Kobayashi:2019xvz ; Kobayashi:2019uyt ; Kobayashi:2020uaj ;
Ishiguro:2020tmo ; Novichkov:2022wvg . However, in most of modular flavor
models, one requires a slight difference in moduli values from fixed points to
explain the observed masses and mixing angles of quarks/leptons as recently
discussed in Ref. Novichkov:2022wvg .

In this paper, we adopt a top-down approach to dynamically fix the moduli
values around the fixed points of the moduli space. In the string theory,
background fluxes can stabilize the moduli fields such that subgroups of
$SL(2,\mathbb{Z})$ are realized Kobayashi:2020hoc  and also the CP symmetry is
spontaneously broken Kobayashi:2020uaj ; Ishiguro:2020nuf . In addition, the
flux landscape prefers the stabilization of moduli fields at fixed points with
enhanced symmetries DeWolfe:2004ns ; Ishiguro:2020tmo . The purpose of this
paper is to investigate the stabilization of moduli values at nearby fixed
points and discuss the phenomenological implications. For concreteness, we
focus on Type IIB string theory on toroidal orientifolds, where the complex
structure moduli determine the flavor structure of quarks and leptons. Turning
on background three-form fluxes, these complex structure moduli, as well as
the dilaton, will be stabilized at statistically-favored symmetric points. To
break enhanced symmetries in the complex structure moduli space, we
incorporate non-perturbative effects whose existence is motivated by the
stabilization of remaining volume moduli associated with the metric of extra-
dimensional space. It is then expected that these non-perturbative effects can
slightly shift the values of complex structure moduli from fixed points.
Indeed, our systematic analysis of flux vacua with non-perturbative effects
reveals that the complex structure moduli are stabilized at nearby fixed
points whose magnitudes are controlled by non-perturbative effects.

Furthermore, we also incorporate the uplifting potential to obtain the de
Sitter (dS) vacuum as discussed in the Kachru-Kallosh-Linde-Trivedi (KKLT)
scenario Kachru:2003aw . Such a supersymmetry (SUSY) breaking source also
shift the value of the complex structure moduli from fixed points. 111Soft
SUSY breaking terms will keep the modular invariance in the moduli-mediated
SUSY breaking scenario Kikuchi:2022pkd , and their phenomenological aspects
are discussed in Refs. Du:2020ylx ; Kobayashi:2021jqu ; Otsuka:2022rak . It
turns out that the string landscape prefers $|\delta\tau|\simeq 10^{-5}$ for
the deviation of the complex structure modulus from fixed points
$\langle\tau\rangle=i,w,2i$, respectively.222Here, $\tau=i\infty$ is
approximated as $\tau=2i$. It corresponds to a specific SUSY breaking scale.
In addition, the CP-breaking vacua are statistically favored due to the
existence of non-perturbative effects as well as the uplifting source,
although the number of CP-breaking vacua is statistically small in the finite
number of flux vacua Ishiguro:2020tmo . These moduli values are well fitted
with observed masses and mixing angles in the lepton sector on a concrete
$A_{4}$ modular flavor model. Furthermore, a quasi-stable dark matter (DM)
would be realized due to the softly broken residual flavor symmetry at fixed
points.

This paper is organized as follows. After reviewing the structure of Type IIB
flux vacua on $T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$
orientifolds in section 2.1, we incorporate the non-perturbative effects to
stabilize the volume moduli in section 2.2. We numerically estimate the
deviations of the complex structure modulus $\tau$ from fixed points in
section 2.3, taking into account SUSY breaking effects. These effects slightly
break the enhanced symmetries in the moduli space of toroidal orientifolds.
Given these moduli values, we study the concrete $A_{4}$ modular flavor model
in section 3 with an emphasis on the lepton sector. The distributions of
$A_{4}$ model and the string landscape are compared. We summarize the paper in
section 4. In Appendix A, we list the $A_{4}$ modular forms used in this
paper.

##  2 Moduli distributions in the string landscape

In section 2.1, we first review the flux vacua in Type IIB string theory on
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$ orientifolds with an
emphasis on the enhanced symmetry in the complex structure moduli space. Next,
we focus on non-perturbative effects, which slightly break the enhanced
symmetries in moduli space of toroidal orbifolds as discussed in section 2.2.
Finally, we plot the deviation of the complex structure modulus from fixed
points and the typical SUSY breaking scale in section 2.3.

###  2.1 Flux vacua with enhanced symmetries

In Type IIB string theory on
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$ orientifolds, the moduli
Kähler potential and the flux-induced superpotential are given by333We follow
the convention of Ref. Ishiguro:2020tmo .

$\displaystyle K$ $\displaystyle=-\ln(-i(S-\bar{S}))-2\ln{\cal
V}(T)-3\ln\left(i(\tau-\bar{\tau})\right),$ (2.1)

where $S,T,\tau$ denote the dilaton, Kähler moduli (volume moduli) and the
complex structure modulus, respectively. Here and in what follows, we adopt
the reduced Planck mass unit unless we specify it, and we consider the
isotropic torus $\tau=\tau_{1}=\tau_{2}=\tau_{3}$ to simplify our analysis. In
Type IIB flux compactifications, one can consider the so-called Gukov-Vafa-
Witten type superpotential Gukov:1999ya  induced by background three-form
fluxes:

$\displaystyle W_{\rm flux}$ $\displaystyle=a^{0}\tau^{3}-3a\tau^{2}-3b\tau-
b_{0}-S\left(c^{0}\tau^{3}-3c\tau^{2}-3d\tau-d_{0}\right),$ (2.2)

where $\\{a^{0},a,b,b_{0},c^{0},c,d,d_{0}\\}$ represent three-form flux quanta
with the notation of Ref. Ishiguro:2020tmo . These integers are now quantized
in multiple of 8 on $T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$
geometry. In this paper, we analyze the SUSY minima:

$\displaystyle\partial_{S}W=\partial_{\tau}W=W=0\,,$ (2.3)

at which the energy of scalar potential vanishes
$V=e^{K}(K^{I\bar{J}}D_{I}W\overline{D_{J}W}-3|W|^{2})=0$ with
$D_{I}W=\partial_{I}W+W\partial_{I}K$. Here, we use the so-called no-scale
structure for the Kähler moduli: $K^{i\bar{j}}K_{i}K_{\bar{j}}=3$ with
$K_{i}=\partial_{T^{i}}K$ and $K^{i\bar{j}}$ being the inverse of Kähler
metric. The SUSY conditions can be analytically solved by redefining the
superpotential as

$\displaystyle W_{\rm RR}=a^{0}\tau^{3}-3a\tau^{2}-3b\tau-
b_{0}=(r\tau+s)P(\tau)\,,$ (2.4) $\displaystyle W_{\rm
NS}=c^{0}\tau^{3}-3c\tau^{2}-3d\tau-d_{0}=(u\tau+v)P(\tau)\,.$ (2.5)

Here, we denote a quadratic (integer-coefficient) polynomial $P(\tau)$ with
respect to $\tau$, and the minimum of $\tau$ is found by solving $P(\tau)=0$.
Following Ref. Betzler:2019kon , we write

$\displaystyle P(\tau)=l\tau^{2}+m\tau+n\,,$ (2.6)

under $m^{2}-4ln<0$, whose expression leads to the vacuum expectation value of
$\tau$:

$\displaystyle\langle\tau\rangle$
$\displaystyle=\frac{-m+\sqrt{m^{2}-4ln}}{2l}\quad(l,n>0)\,,$
$\displaystyle\langle\tau\rangle$
$\displaystyle=\frac{-m-\sqrt{m^{2}-4ln}}{2l}\quad(l,n<0)\,.$ (2.7)

The vacuum expectation value of the dilaton field is obtained by solving
$\partial_{\tau}W=0$, i.e.,

$\displaystyle
P(\tau)\partial_{\tau}\\{(r\tau+s)-S(u\tau+v)\\}+\\{(r\tau+s)-S(u\tau+v)\\}\partial_{\tau}P(\tau)=0\,.$
(2.8)

Since the $\tau$ is now stabilized at Eq. (2.7) determined by $P(\tau)=0$, we
require

$\displaystyle\langle S\rangle=\frac{r\tau+s}{u\tau+v}\,.$ (2.9)

Note that the condition $\partial_{\tau}P(\tau)=0$ gives rise to a real $\tau$
and the dilaton cannot be stabilized anymore. At this stage, one cannot
stabilize the Kähler moduli and requires additional sources such as non-
perturbative effects.

Before going into the detail of the volume moduli stabilization, we also
review the structure of flux vacua on the toroidal orientifold. Remarkably,
the background three-form fluxes induce a net D3-brane charge:

$\displaystyle N_{\rm flux}$ $\displaystyle=\frac{1}{l_{s}^{4}}\int
H_{3}\wedge
F_{3}=c^{0}b_{0}-d_{0}a^{0}+\sum_{i=1}^{3}(c^{i}b_{i}-d_{i}a^{i})=c^{0}b_{0}-d_{0}a^{0}+3(cb-
da)\,,$ (2.10)

with the string length $l_{s}$, which should be canceled on a compact
manifold. Taking into the contributions of D3/D7-branes and O3/O7-planes, the
flux-induced D3-brane charge is constrained as

$\displaystyle 0\leq N_{\rm flux}\leq N_{\rm flux}^{\rm max}={\cal
O}(10^{5})\,.$ (2.11)

Here, we admit the F-theory extension of the Type IIB orientifolds where a
largest value of O3-plane contribution is given by $1820448$ Candelas:1997eh ;
Taylor:2015xtz . Furthermore, $N_{\rm flux}$ should be in multiple of 192 due
to the fact that $\\{a^{0},a,b,b_{0},c^{0},c,d,d_{0}\\}\in 8\mathbb{Z}$.

For concreteness, we focus on the vacuum structure of $\tau$ whose fixed
points in the moduli space are $\tau=i,w=\frac{-1+i\sqrt{3}}{2},i\infty$, each
of which corresponds to the $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$,
$\mathbb{Z}_{2}$ fixed points, respectively. The $\tau=i,w$ fixed points are
statistically favored in the flux landscape, as seen in Fig. 1, where the
higher the degeneracy of vacua, the darker the color is. Note that one cannot
realize $\tau=i\infty$ requiring the infinite value of flux quanta, and it is
inconsistent with the charge cancellation condition of D3-brane charge (2.11),
namely the tadpole cancellation condition. In this respect, we adopt $\tau=2i$
as an approximation of $\tau=i\infty$. Such an approximation will often be
used in the phenomenological analysis of modular flavor models.

Figure 1: The numbers of stable vacua on the fundamental domain of $\tau$ in
the case of $N_{\rm flux}^{\rm max}=192\times 1000$ Ishiguro:2020tmo .

###  2.2 Stabilization of volume moduli by non-perturbative effects

In this section, we analyze the stabilization of volume moduli along the line
of KKLT scenario Kachru:2003aw . The stabilization of volume moduli is
performed by the following non-perturbative effects:

$\displaystyle W=W_{\rm flux}(\tau,S)+W_{\rm np}(S,T)\,,$ (2.12)

where

$\displaystyle W_{\rm np}=\sum_{m}C_{m}e^{ia_{m}T+ib_{m}S}$ (2.13)

is supposed to be generated by D-brane instanton effects with
$a_{m},b_{m}=2\pi$ or strong dynamics on D7-branes wrapping the rigid cycle
with $a_{m}=2\pi/N$ and $N$ being the rank of the gauge group. Here and in
what follows, we consider a simple setup where the volume of internal manifold
is determined by a single Kähler modulus $T$ whose Kähler potential is given
by $K=-3\ln(i(\bar{T}-T))$.

In the KKLT construction, the dilaton and the complex structure moduli are
determined in the context of Type IIB flux compactifications as discussed in
section 2.1. Note that the vacuum expectation value of flux superpotential
vanishes in our analysis in the previous section; thereby the dilaton-
dependent non-perturbative effects would induce the constant term in the
effective superpotential:

$\displaystyle W_{\rm eff}=W_{\rm np}(\langle S\rangle,T)\,,$ (2.14)

which includes the following terms required in the KKLT construction:

$\displaystyle W_{\rm eff}\simeq\langle e^{ibS}\rangle+Ce^{iaT}\,.$ (2.15)

Thus, the overall Kähler modulus is stabilized at $T=T_{0}$ satisfying

$\displaystyle D_{T}W_{\rm eff}=\partial_{T}W_{\rm eff}+K_{T}W_{\rm eff}=0\,,$
(2.16)

at which the minimum value of $T$ is estimated as

$\displaystyle a\langle T\rangle\approx\ln(C/w_{0})\,,$ (2.17)

with $w_{0}=\langle e^{ibS}\rangle$. Here, the origin of small superpotential
$w_{0}$ relies on the dilaton-dependent non-perturbative effects. It is also
possible to realize the small flux superpotential in Type IIB/F-theory flux
compactifications (see, Refs. Demirtas:2019sip ; Honma:2021klo , for the large
complex structure regime). In what follows, the prefactor $C$ is assumed to be
a constant, in particular, 1.

So far, we have assumed that the dilaton and the complex structure moduli are
stabilized in flux compactifications. However, the presence of non-
perturbative effects slightly shifts their values. Indeed, the true vacuum is
found by solving

$\displaystyle D_{I}W=0\,,$ (2.18)

which changes the moduli values obtained in the previous section. To find the
slight difference from the fixed points of complex structure modulus, we
utilize the perturbation method; the non-perturbative superpotential $W_{\rm
np}$ causes the shift of the minima:

$\displaystyle\tau$ $\displaystyle=\langle\tau\rangle+\delta\tau\,,$
$\displaystyle S$ $\displaystyle=\langle S\rangle+\delta S\,,$ $\displaystyle
T$ $\displaystyle=\langle T\rangle+\delta T\,,$ (2.19)

where the reference points $\\{\langle\tau\rangle,\langle S\rangle,\langle
T\rangle\\}$ are given in Eqs. (2.7), (2.9) and (2.17), respectively.

Following Ref.Abe:2006xi , we estimate the deviation up to a linear order. Let
us consider the Kähler-invariant quantity $G=K+\ln|W|^{2}$ satisfying
$G_{A}=\partial_{A}G=0$ at the SUSY minima. Here and in what follows, the
index $A$ denotes both the holomorphic and anti-holomorphic fields:
$\\{S,T,\tau,\bar{S},\bar{T},\bar{\tau}\\}$. From the expansion (2.19),
$G_{A}$ is expanded as

$\displaystyle
G_{A}=G_{A}\bigl{|}_{\langle\rangle}+\delta\phi^{B}G_{AB}\bigl{|}_{\langle\rangle}+({\cal
O}(\delta\phi)^{2})\,,$ (2.20)

where $\bigl{|}_{\langle\rangle}$ means
$\bigl{|}_{\tau=\langle\tau\rangle,S=\langle S\rangle,T=\langle T\rangle}$.
Under the assumption $a,b>1$, we obtain

$\displaystyle G_{IJ},G_{\bar{I}\bar{J}}\gg G_{I\bar{J}},G_{\bar{I}J}\,,$
(2.21)

which implies that $G_{AB}$ and $G^{AB}$ are diagonalized only by the
holomorphic and anti-holomorphic parts, respectively. As a result, we obtain

$\displaystyle\delta\phi^{I}=G^{IJ}G_{J}\bigl{|}_{\langle\rangle}+({\cal
O}(\delta\phi)^{2})\,,$ (2.22)

whose explicit form is written by

$\displaystyle\delta\tau$ $\displaystyle=W_{\rm
eff}\left(-\frac{G_{S}}{W_{S\tau}}\right)\biggl{|}_{\langle\rangle}+{\cal
O}(W_{\rm eff}^{2})\,,$ $\displaystyle\delta S$ $\displaystyle=\frac{W_{\rm
eff}}{W_{S\tau}}\left(\frac{W_{\tau\tau}}{W_{S\tau}}G_{S}-G_{\tau}\right)\biggl{|}_{\langle\rangle}+{\cal
O}(W_{\rm eff}^{2})\,,$ $\displaystyle\delta T$
$\displaystyle=\left(-\frac{G_{ST}}{G_{TT}}\right)\biggl{|}_{\langle\rangle}\delta
S\,.$ (2.23)

Note that the internal volume should be larger than the string length,

$\displaystyle{\rm Im}(T)\gg 1\,,$ (2.24)

and the weak string coupling ${\rm Im}(S)>1$; thereby the magnitude of the
flux superpotential is exponentially small:

$\displaystyle\langle W_{\rm eff}\rangle\simeq w_{0}+e^{iaT}<10^{-3}\,.$
(2.25)

Here and in the following numerical calculations, we take $a=b=2\pi$ for
concreteness.

In this way, the deviation of the vacuum values $\\{\delta\tau,\delta S,\delta
T\\}$ are determined by the non-perturbative effects, implying that the
deviation is naturally suppressed with respect to the volume modulus. From the
phenomenological point of view, such a small deviation of $\tau$ is useful for
predicting the masses and mixing angles of quarks and leptons, as discussed in
detail in section 3. Before going into a phenomenological analysis, we discuss
the supersymmetry breaking effects in the next section.

###  2.3 Moduli values at nearby fixed points

So far, we have analyzed the stabilization of the complex structure modulus,
dilaton and Kähler moduli at SUSY minima. However, the obtained vacuum energy
is negative, i.e., anti-de Sitter (AdS) vacuum. To realize a dS vacuum, it is
required to uplift the AdS vacuum to the dS one. Among several proposals for
the uplifting scenarios, we focus on the anti D3-brane as originally utilized
in the KKLT scenario Kachru:2003aw . The anti D3-brane induces the positive
vacuum energy due to its SUSY breaking effect,

$\displaystyle V_{\rm up}=\frac{D}{(i(\bar{T}-T))^{3}}\,,$ (2.26)

where a constant $D$ is chosen to realize the present vacuum energy. Then, the
effective scalar potential is described as

$\displaystyle
V=e^{K}\left(K^{I\bar{J}}D_{I}W\overline{D_{J}W}-3|W|^{2}\right)+V_{\rm
up}\,,$ (2.27)

indicating that the uplifting source further causes the shift of the moduli
values obtained in the previous section.

To see the deviation of complex structure moduli values from fixed points, we
numerically calculate the deviation of $\tau$ from $\langle\tau\rangle=i,w,2i$
by utilizing a finite number of flux vacua with $N_{\rm flux}^{\rm max}=1000$.
By calculating the minimization condition of the full scalar potential
$\partial_{I}V=0$ for $I=\tau,S,T$, we find deviations of the complex
structure modulus from fixed points $\delta\tau\equiv\tau-\langle\tau\rangle$
as shown in Figs. 2, 3 and 4. It turns out that the flux landscape prefers
$|\delta\tau|\simeq 10^{-5}$ from fixed points $\langle\tau\rangle=i,w,2i$,
but there is no sizable difference in the phase direction. It means that the
CP symmetry parametrized by $\tau\rightarrow-\bar{\tau}$ is broken in a
generic moduli space. Furthermore, we plot the typical SUSY breaking scale,
i.e., the gravitino mass $m_{3/2}=e^{K/2}W$, in the same figures. At the
statistically favored moduli values $|\delta\tau|={\cal O}(10^{-5})$, the
gravitino mass is $m_{3/2}={\cal O}(10^{-5})$ in the reduced Planck mass unit.
Note that the small $\delta\tau$ is originating from non-perturbative effects
and the uplifting source, both of which are the same order.

Figure 2: The number of vacua at nearby $\langle\tau\rangle=i$ as a function
of $|\delta\tau|$ in the left panel and ${\rm Arg}(\delta\tau)$ in the right
panel, respectively. In the left panel, the absolute value of gravitino mass
is plotted as a function of $|\delta\tau|$.

Figure 3: The number of vacua at nearby $\langle\tau\rangle=w$ as a function
of $|\delta\tau|$ in the left panel and ${\rm Arg}(\delta\tau)$ in the right
panel, respectively. In the left panel, the absolute value of gravitino mass
is plotted as a function of $|\delta\tau|$.

Figure 4: The number of vacua at nearby $\langle\tau\rangle=2i$ as a function
of $|\delta\tau|$ in the left panel and ${\rm Arg}(\delta\tau)$ in the right
panel, respectively. In the left panel, the absolute value of gravitino mass
is plotted as a function of $|\delta\tau|$.

##  3 $A_{4}$ modular flavor model

To illustrate implications of distributions of moduli fields around fixed
points, we study the phenomenology of lepton sector on a concrete $A_{4}$
modular flavor model.

###  3.1 Setup

|  $L$ |  $\\{e^{c},\mu^{c},\tau^{c}\\}$ |  $N^{c}$ |  $H_{u}$ |  $H_{d}$
---|---|---|---|---|---
$SU(2)_{L}$ | $\bf{2}$ | $\bf{1}$ | $\bf{1}$ | $\bf{2}$ | $\bf{2}$
$U(1)_{Y}$ | $-\frac{1}{2}$ | $1$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$
$A_{4}$ | ${\bf 3}$ | $\\{\bf 1,1^{\prime},1^{\prime}\\}$ | ${\bf 3}$ | ${\bf 1}$ | ${\bf 1}$
$-k_{I}$ | $-2$ | $\\{-2,-4,-4\\}$ | $-2$ | $0$ | $0$
Table 1: Charge assignments under $SU(2)_{L}\times U(1)_{Y}\times A_{4}$ in
the lepton and Higgs sectors, where $k_{I}$ denotes the modular weight of
matter superfields $\Phi_{I}$.

For concreteness, we specify charge assignments under $SU(2)_{L}\times
U(1)_{Y}\times A_{4}$ for the lepton and Higgs sectors as summarized in Tab.
1. Here, the $A_{4}$ group belongs to the $SL(2,\mathbb{Z})$ modular group
parametrized by the modulus $\tau$. The Yukawa couplings are constructed in a
modular invariant way. (For more details, see, Appendix A.) Then, we can write
down the modular invariant superpotential:

$\displaystyle W$ $\displaystyle=y_{e}(Y_{\bf
3}^{(4)}L)_{1}H_{d}e^{c}+\sum_{\bf r=3,3^{\prime}}y_{\mu}^{({\bf r})}(Y_{\bf
r}^{(6)}L)_{1}H_{d}\mu^{c}+\sum_{\bf r=3,3^{\prime}}y_{\tau}^{({\bf
r})}(Y_{\bf r}^{(6)}L)_{1}H_{d}\tau^{c}$ $\displaystyle+\sum_{\bf
r=1,1^{\prime}}y_{d}^{({\bf r})}(Y_{\bf r}^{(4)}LH_{u}N^{c})_{1}+y_{d}^{({\bf
3_{S}})}(Y_{\bf 3}^{(4)}H_{u}(LN^{c})_{\bf 3_{S}})_{1}+y_{d}^{({\bf
3_{A}})}(Y_{\bf 3}^{(4)}H_{u}(LN^{c})_{\bf 3_{A}})_{1}$
$\displaystyle+\sum_{\bf r=1,1^{\prime},3}M^{({\bf r})}(Y_{\bf
r}^{(4)}N^{c}N^{c})_{1},$ (3.1)

where $Y_{\bf r}^{(k)}$ denotes the holomorphic modular form with weight $k$
for ${\bf r}$ representations under the $A_{4}$ group, and
$\\{y_{e},y_{\mu}^{({\bf r})},y_{\tau}^{({\bf r})},y_{d}^{({\bf r})}\\}$ are
parameters. Here, we introduce the Majorana mass terms to realize small
neutrino masses.

In the following, we enumerate the mass matrix of the lepton sector.

  1. 1.

Charged-lepton mass matrix

After the electroweak symmetry breaking, charged-lepton mass matrix is written
as

$\displaystyle(m_{l})_{LR}=\frac{v_{d}}{\sqrt{2}}\begin{pmatrix}Y_{1}^{(4)}&Y_{3}^{(6)}+\epsilon_{\mu}Y_{3^{\prime}}^{(6)}&Y_{3}^{(6)}+\epsilon_{\tau}Y_{3^{\prime}}^{(6)}\\\
Y_{3}^{(4)}&Y_{2}^{(6)}+\epsilon_{\mu}Y_{2^{\prime}}^{(6)}&Y_{2}^{(6)}+\epsilon_{\tau}Y_{2^{\prime}}^{(6)}\\\
Y_{2}^{(4)}&Y_{1}^{(6)}+\epsilon_{\mu}Y_{1^{\prime}}^{(6)}&Y_{1}^{(6)}+\epsilon_{\tau}Y_{1^{\prime}}^{(6)}\\\
\end{pmatrix}\times\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}^{(\bf 3)}&0\\\
0&0&y_{\tau}^{(\bf 3)}\\\ \end{pmatrix},$ (3.2)

where we introduce

$\displaystyle\langle
H_{d}\rangle=v_{d},\quad\epsilon_{\mu}=\frac{y_{\mu}^{(\bf
3^{\prime})}}{y_{\mu}^{({\bf 3})}},\quad\epsilon_{\tau}=\frac{y_{\tau}^{(\bf
3^{\prime})}}{y_{\tau}^{({\bf 3})}},\quad Y_{\bf
3}^{(k)}=\begin{pmatrix}Y_{1}^{(k)}\\\ Y_{2}^{(k)}\\\ Y_{3}^{(k)}\\\
\end{pmatrix},\quad Y_{\bf
3^{\prime}}^{(k)}=\begin{pmatrix}Y_{1^{\prime}}^{(k)}\\\
Y_{2^{\prime}}^{(k)}\\\ Y_{3^{\prime}}^{(k)}\\\ \end{pmatrix}.$ (3.3)

The explicit modular forms are listed in Appendix A. Then the charged-lepton
mass square eigenstate can be found by ${\rm
diag}(|m_{e}|^{2},|m_{\mu}|^{2},|m_{\tau}|^{2})\equiv
V^{\dagger}_{l_{L}}m_{l}^{\dagger}m_{l}V_{l_{L}}$. We numerically determine
the three parameters $y_{e},y_{\mu}^{(\bf 3)},y_{\tau}^{(\bf 3)}$ to fit the
three charged-lepton masses by applying the relations:

$\displaystyle{\rm
Tr}[m_{l}^{\dagger}m_{l}]=|m_{e}|^{2}+|m_{\mu}|^{2}+|m_{\tau}|^{2},$ (3.4)
$\displaystyle{\rm
Det}[m_{l}^{\dagger}m_{l}]=|m_{e}|^{2}|m_{\mu}|^{2}|m_{\tau}|^{2},$ (3.5)
$\displaystyle({\rm Tr}[m_{l}^{\dagger}m_{l}])^{2}-{\rm
Tr}[(m_{l}^{\dagger}m_{l})^{2}]=2(|m_{e}|^{2}|m_{\nu}|^{2}+|m_{\mu}|^{2}|m_{\tau}|^{2}+|m_{e}|^{2}|m_{\tau}|^{2}).$
(3.6)

Therefore, input parameters are $\epsilon_{\mu},\ \epsilon_{\tau}$ in the
charged-lepton sector.

  2. 2.

Dirac Yukawa mass matrix

$\displaystyle(m_{D})_{LN}$
$\displaystyle=\frac{v_{u}}{\sqrt{2}}\Biggl{[}\frac{y_{d}^{({\bf
3_{S}})}}{3}\begin{pmatrix}2Y_{1}^{(4)}&-Y_{3}^{(4)}&-Y_{2}^{(4)}\\\
-Y_{3}^{(4)}&2Y_{2}^{(4)}&-Y_{1}^{(4)}\\\
-Y_{2}^{(4)}&-Y_{1}^{(4)}&2Y_{3}^{(4)}\\\ \end{pmatrix}+\frac{y_{d}^{({\bf
3_{A}})}}{2}\begin{pmatrix}0&Y_{3}^{(4)}&-Y_{2}^{(4)}\\\
-Y_{3}^{(4)}&0&Y_{1}^{(4)}\\\ Y_{2}^{(4)}&-Y_{1}^{(4)}&0\\\ \end{pmatrix}$
$\displaystyle\hskip 40.0pt+y_{d}^{({\bf 1})}Y_{\bf
1}^{(4)}\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\ \end{pmatrix}+y_{d}^{({\bf
1^{\prime}})}Y_{\bf 1^{\prime}}^{(4)}\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\
\end{pmatrix}\Biggl{]}$
$\displaystyle=m_{d_{0}}\Biggl{[}\begin{pmatrix}2Y_{1}^{(4)}&(-1+g_{D})Y_{3}^{(4)}&-(1+g_{D})Y_{2}^{(4)}\\\
-(1+g_{D})Y_{3}^{(4)}&2Y_{2}^{(4)}&(-1+g_{D})Y_{1}^{(4)}\\\
(-1+g_{D})Y_{2}^{(4)}&-(1+g_{D})Y_{1}^{(4)}&2Y_{3}^{(4)}\\\ \end{pmatrix}$
$\displaystyle\hskip 40.0pt+h_{1}\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\
\end{pmatrix}+h_{2}\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\
\end{pmatrix}\Biggl{]}$ $\displaystyle\equiv m_{d_{0}}\tilde{m}_{D},$ (3.7)

where we define

$\displaystyle\langle H_{u}\rangle=v_{u},\quad
m_{d_{0}}\equiv\frac{y_{d}^{(\bf 3_{S})}}{3\sqrt{2}}v_{u},\quad
g_{D}=\frac{3y_{d}^{(\bf 3_{A})}}{2y_{d}^{(\bf 3_{S})}},\quad
h_{1}=\frac{3y_{d}^{({\bf 1})}Y_{\bf 1}^{(4)}}{y_{d}^{(\bf 3_{S})}},\quad
h_{2}=\frac{3y_{d}^{({\bf 1^{\prime}})}Y_{\bf 1^{\prime}}^{(4)}}{y_{d}^{(\bf
3_{S})}}.$ (3.8)

  3. 3.

Majorana mass matrix

$\displaystyle M_{N}$
$\displaystyle=\frac{M_{1}}{3}\begin{pmatrix}2Y_{1}^{(4)}&-Y_{3}^{(4)}&-Y_{2}^{(4)}\\\
-Y_{3}^{(4)}&2Y_{2}^{(4)}&-Y_{1}^{(4)}\\\
-Y_{2}^{(4)}&-Y_{1}^{(4)}&2Y_{3}^{(4)}\\\ \end{pmatrix}+M_{2}Y_{\bf
1}^{(4)}\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\ \end{pmatrix}+M_{3}Y_{\bf
1^{\prime}}^{(4)}\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\ \end{pmatrix}$
$\displaystyle=M_{0}\Biggl{[}\begin{pmatrix}2Y_{1}^{(4)}&-Y_{3}^{(4)}&-Y_{2}^{(4)}\\\
-Y_{3}^{(4)}&2Y_{2}^{(4)}&-Y_{1}^{(4)}\\\
-Y_{2}^{(4)}&-Y_{1}^{(4)}&2Y_{3}^{(4)}\\\
\end{pmatrix}+f_{1}\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\
\end{pmatrix}+f_{2}\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\
\end{pmatrix}\Biggl{]}$ $\displaystyle\equiv M_{0}\tilde{M}_{N},$ (3.9)

where we define

$\displaystyle M_{0}\equiv\frac{M_{1}}{3},\quad f_{1}=\frac{3Y_{\bf
1}^{(4)}M_{2}}{M_{1}},\quad f_{2}=\frac{3Y_{\bf
1^{\prime}}^{(4)}M_{3}}{M_{1}}.$ (3.10)

Then, the active neutrino mass matrix is given by

$\displaystyle m_{\nu}$ $\displaystyle\approx-
m_{D}^{T}M_{N}^{-1}m_{D}=-\kappa\tilde{m}_{D}^{T}\tilde{M}_{N}^{-1}\tilde{m}_{D}=-\kappa\tilde{m}_{\nu},$
(3.11)

where the mass dimensional parameter $\kappa$ is defined by
$m_{d_{0}}^{2}/M_{0}$. $\tilde{m}_{\nu}$ is diagonalized by applying a unitary
matrix as
$V^{\dagger}_{\nu}(\tilde{m}_{\nu}^{\dagger}\tilde{m}_{\nu})V_{\nu}=(\tilde{m}_{1}^{2},\tilde{m}_{2}^{2},\tilde{m}_{3}^{2})$.
In this case, $\kappa$ is determined by

$\displaystyle({\rm NH}):\ \kappa^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{m}_{3}^{2}-\tilde{m}_{1}^{2}},\quad({\rm IH}):\
\kappa^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{m}_{2}^{2}-\tilde{m}_{3}^{2}},$ (3.12)

where $\Delta m_{\rm atm}^{2}$ is atmospheric neutrino mass square difference,
and NH and IH stand for normal and inverted hierarchies, respectively. The
solar mass square difference is also found in terms of $\kappa$ as follows:

$\displaystyle\Delta m_{\rm
sol}^{2}={\kappa^{2}}({\tilde{m}_{2}^{2}-\tilde{m}_{1}^{2}}).$ (3.13)

In our numerical analysis, we regard $\Delta m_{\rm atm}^{2}$ as an input
parameter from experiments so that $\Delta m_{\rm sol}^{2}$ be output
parameter giving numerical
$(\tilde{m}_{1}^{2},\tilde{m}_{2}^{2},\tilde{m}_{3}^{2})$. Then, one finds
$U_{\rm PMNS}=V^{\dagger}_{l_{L}}V_{\nu}$, and it is parametrized by three
mixing angles $\theta_{ij}(i,j=1,2,3;i<j)$, one CP violating Dirac phase
$\delta_{\rm CP}$, and two Majorana phases $\\{\alpha_{21},\alpha_{32}\\}$ as
follows:

$U_{\rm PMNS}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm
CP}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\text{CP}}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\text{CP}}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\text{CP}}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\text{CP}}}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&e^{i\frac{\alpha_{21}}{2}}&0\\\
0&0&e^{i\frac{\alpha_{31}}{2}}\end{pmatrix},$ (3.14)

where $c_{ij}$ and $s_{ij}$ stand for $\cos\theta_{ij}$ and $\sin\theta_{ij}$,
respectively. These mixings are rewritten in terms of the component of $U_{\rm
PMNS}$ as follows:

$\displaystyle\sin^{2}\theta_{13}=|(U_{\rm
PMNS})_{13}|^{2},\quad\sin^{2}\theta_{23}=\frac{|(U_{\rm
PMNS})_{23}|^{2}}{1-|(U_{\rm
PMNS})_{13}|^{2}},\quad\sin^{2}\theta_{12}=\frac{|(U_{\rm
PMNS})_{12}|^{2}}{1-|(U_{\rm PMNS})_{13}|^{2}}.$ (3.15)

In addition, we can compute the Jarlskog invariant, $\delta_{\text{CP}}$ from
PMNS matrix elements $(U_{\rm PMNS})_{\alpha i}\equiv U_{\alpha i}$:

$J_{\rm CP}=\text{Im}[U_{e1}U_{\mu 2}U_{e2}^{*}U_{\mu
1}^{*}]=s_{23}c_{23}s_{12}c_{12}s_{13}c^{2}_{13}\sin\delta_{\rm CP},$ (3.16)

and the Majorana phases are also estimated in terms of other invariants
$I_{1}$ and $I_{2}$ constructed by PMNS matrix elements:

$I_{1}=\text{Im}[U^{*}_{e1}U_{e2}]=c_{12}s_{12}c_{13}^{2}\sin\left(\frac{\alpha_{21}}{2}\right),\
I_{2}=\text{Im}[U^{*}_{e1}U_{e3}]=c_{12}s_{13}c_{13}\sin\left(\frac{\alpha_{31}}{2}-\delta_{\rm
CP}\right).$ (3.17)

Furthermore, the effective mass for the neutrinoless double beta decay is
given by

$\displaystyle\langle
m_{ee}\rangle=\kappa|\tilde{D}_{\nu_{1}}c^{2}_{12}c^{2}_{13}+\tilde{D}_{\nu_{2}}s^{2}_{12}c^{2}_{13}e^{i\alpha_{21}}+\tilde{D}_{\nu_{3}}s^{2}_{13}e^{i(\alpha_{31}-2\delta_{\rm
CP})}|\,,$ (3.18)

where its observed value could be measured by KamLAND-Zen experiment in future
KamLAND-Zen:2016pfg . In our numerical analysis below, we will do $\Delta\chi$
square analysis referring to Ref. Esteban:2020cvm .

###  3.2 Numerical analysis

In this section, we show the allowed region with $\chi$ square analysis to
satisfy the current neutrino oscillation data, where we randomly select within
the following ranges of input parameters,

$\displaystyle|\delta\tau|\in[10^{-20},0.1],\
\\{\epsilon_{\mu},\epsilon_{\tau},g_{D},f_{1},f_{2},h_{1},h_{2}\\}\in[10^{-4},2],$
(3.19)

where we assume all the parameters (except $\tau$) are real for simplicity. We
also take Yukawa couplings of the SM charged leptons at the GUT scale $2\times
10^{16}$ GeV and $\Delta m_{\rm atm}^{2}$ as input parameters, where
$\tan\beta=5$ is taken as a bench mark Bjorkeroth:2015ora :

$\displaystyle y_{e}=(1.97\pm 0.024)\times 10^{-6},\ y_{\mu}=(4.16\pm
0.050)\times 10^{-4},\ y_{\tau}=(7.07\pm 0.073)\times 10^{-3},$ (3.20)
$\displaystyle|\Delta m_{\rm atm}^{2}|=(2.431-2.598)\times 10^{-21}\ {\rm
eV^{2}}\,\,\ {\rm for}\ {\rm NH},$ (3.21) $\displaystyle|\Delta m_{\rm
atm}^{2}|=(2.412-2.583)\times 10^{-21}\ {\rm eV^{2}}\,\,\ {\rm for}\ {\rm
IH},$ (3.22)

where the charged-lepton masses are within 1$\sigma$ region, while $\Delta
m_{\rm atm}^{2}$ is within 3$\sigma$ region. Here, the lepton masses are
defined by $m_{\ell}=y_{\ell}v_{H}$ with $v_{H}=174$ GeV. Then, we pick the
output data up only when the $\chi$ square is within 5$\sigma$ considering
five measured neutrino oscillation data; $(\Delta m^{2}_{\rm sol},\
\sin^{2}\theta_{13},\ \sin^{2}\theta_{23},\ \sin^{2}\theta_{12}$)
Esteban:2018azc . Here, we do not include $\delta_{\rm CP}$ in the $\chi$
square analysis due to the large ambiguity of experimental results in
$3\sigma$ interval. In general, IH case is more difficult to accumulate more
data to satisfy the neutrino oscillation data, because the minimum $\chi$
square is $2.7$ in Nufit 5.0 Esteban:2018azc .

####  3.2.1 Nearby $\tau=i$

Figure 5: Each of color represents ${\rm blue}\leq 1\sigma$, $1\sigma<{\rm
green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$, $3\sigma<{\rm
red}\leq 5\sigma$.

In Fig. 5, we show our several allowed regions on $\tau$ at nearby $\tau=i$ in
case of NH, where each of color represents ${\rm blue}\leq 1\sigma$,
$1\sigma<{\rm green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$,
$3\sigma<{\rm red}\leq 5\sigma$. The up-left one represents the allowed region
of the imaginary part of $\tau$ in terms of the real part of $\tau$. The up-
right one demonstrates the allowed region of neutrinoless double beta decay
$\langle m_{ee}\rangle$ in terms of the lightest active neutrino mass $m_{1}$.
There are two linear correlations between them. Furthermore, the smaller
$\chi$ square is localized at nearby their smaller masses. The down-left one
shows the allowed region of Majorana phases $\alpha_{21}$ and $\alpha_{31}$.
Since we take all input parameters (except $\tau$) to be real values, both the
allowed regions are localized at nearby by $0^{\circ},\ 180^{\circ}$. The
down-right one depicts the allowed region of Dirac phase $\delta_{\text{CP}}$
in terms of the sum of neutrino masses $\sum m_{i}$. The vertical line is the
upper bound on cosmological constraint $\sum m_{i}<0.12$ eV Planck:2018vyg .
There is an intriguing tendency that allowed region of smaller $\chi$ square
is localized at smaller $\sum m_{i}$ that is within the cosmological bound.
Another feature is that the best fit value of Dirac CP phase $\sim
195^{\circ}$ would be reproduced when we allow up to $5\sigma$ interval.

Figure 6: $|\delta\tau|<10^{-15}$ for black,
$10^{-15}\leq|\delta\tau|<10^{-12}$ for gray,
$10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta.

In Fig. 6, we show the several figures in terms of deviation from $\tau=i$ in
the same case of Fig. 5 at 5 $\sigma$ interval, where each of color represents
$|\delta\tau|<10^{-15}$ for black, $10^{-15}\leq|\delta\tau|<10^{-12}$ for
gray, $10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta. The up-left one is the same as
the case of up-right one in Fig. 5. It implies that smaller deviations
$|\delta\tau|$ tend to be localized at nearby their smaller masses. The up-
right one is the same as the case of down-left one in Fig. 5. This figure
would show rather trivial. Therefore, the smaller deviation is localized at
$0^{\circ}$ and $180^{\circ}$, while the larger deviation deviates from these
two points. It directly follows from our phase source is $\tau$ only. The
down-left one is the same as the case of down-right one in Fig. 5. The smaller
deviation would be favored in the point of view of the bound on cosmological
constraint.

Figure 7: Each of color represents ${\rm blue}\leq 1\sigma$, $1\sigma<{\rm
green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$, $3\sigma<{\rm
red}\leq 5\sigma$.

In Fig. 7, we show our several allowed regions on $\tau$ at nearby $\tau=i$ in
case of IH, where color legends are the same as the one of Fig. 5. Therefore,
we have found only the allowed region of $3\sigma-5\sigma$. The up-left one
represents the allowed region of imaginary part of $\tau$ in terms of real
part of $\tau$. The up-right one demonstrates the allowed region of
neutrinoless double beta decay $\langle m_{ee}\rangle$ in terms of the
lightest active neutrino mass $m_{3}$. There are two correlations between
them; one is a linear line and another is a slightly curved one. The solutions
tend to be localized at nearby smaller mass of $m_{3}$ with $\langle
m_{ee}\rangle=0.015,\ 0.05$ eV. The down-left one shows the allowed region of
Majorana phases $\alpha_{21}$ and $\alpha_{31}$. Both the allowed regions are
localized at nearby by $0^{\circ},\ 180^{\circ}$ similar to the case of NH.
The down-right one depicts the allowed region of the sum of neutrino masses
$\sum m_{i}$ in terms of Dirac phase $\delta_{\text{CP}}$. The vertical line
is the upper bound on cosmological constraint. $\delta_{\rm CP}$ is allowed at
the points $0^{\circ}$ and $180^{\circ}$. While a large part of $\sum m_{i}$
would be ruled out by the cosmological bound. Therefore, we would predict a
narrow range of $0.1~{}{\rm eV}\leq\sum m_{i}\leq 0.12~{}{\rm eV}$ in this
case.

Figure 8: $|\delta\tau|<10^{-15}$ for black,
$10^{-15}\leq|\delta\tau|<10^{-12}$ for gray,
$10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta.

In Fig. 8, we show the several figures in terms of deviation from $\tau=i$
where the color legends are the same as the one in Fig. 6. The up-left one
corresponds to the case of up-right one in Fig. 7. It implies that smaller
deviations $|\delta\tau|$ tend to be localized at nearby their smaller masses
$m_{3}$. The up-right one corresponds to the case of down-left one in Fig. 7.
This figure would show rather trivial. Therefore, the smaller deviation is
localized at $0^{\circ}$ and $180^{\circ}$, while the larger deviation
deviates from these two points. It directly follows from the fact that our
phase source is $\tau$ only. The down-left one corresponds to the case of
down-right one in Fig. 7. The smaller deviation would be favored in the point
of view of the bound on cosmological constraint.

Finally, we discuss ratios of the number of solutions in a corresponding range
of $-{\rm Log}_{10}|\delta\tau|$ to the number of whole solutions for both the
string landscape in Fig. 2 and the $A_{4}$ model within $5\sigma$. Fig. 9
indicates both the distributions of $A_{4}$ model with NH and the moduli
fields in the string landscape peak around $|\delta\tau|={\cal O}(10^{-5})$,
but such a signal will not be found in the IH case.

Figure 9: Ratios of the number of solutions in a corresponding range of $-{\rm
Log}_{10}|\delta\tau|$ to the number of whole solutions for both the string
landscape in Fig. 2 and the $A_{4}$ model within $5\sigma$. We present the NH
and the IH in the left and right panels, respectively.

####  3.2.2 Nearby $\tau=\omega$

Figure 10: Each of color represents ${\rm blue}\leq 1\sigma$, $1\sigma<{\rm
green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$, $3\sigma<{\rm
red}\leq 5\sigma$.

In Fig. 10, we show our several allowed regions on $\tau$ at nearby
$\tau=\omega$ in case of NH, where the color legends are the same as the one
in Fig. 6. The up-left one represents the allowed region of the imaginary part
of $\tau$ in terms of the real part of $\tau$. The smaller $\chi$ square
denoted by blue color is closest to the fixed point of $\tau=\omega$, which
would be a good tendency. The up-right one demonstrates the allowed region of
neutrinoless double beta decay $\langle m_{ee}\rangle$ in terms of the
lightest active neutrino mass $m_{1}$. There is a linear correlation with
width between them. Furthermore, all the regions of $\chi$ square tend to run
the whole range. The down-left one shows the allowed region of Majorana phases
$\alpha_{21}$ and $\alpha_{31}$. Even though the whole region is allowed,
there exist two islands at around $-50^{\circ}\leq\alpha_{21},\alpha_{31}\leq
50^{\circ}$. The down-right one depicts the allowed region of Dirac phase
$\delta_{\text{CP}}$ in terms of the sum of neutrino masses $\sum m_{i}$. The
vertical line is the upper bound on cosmological constraint. Below this bound,
the whole region is allowed for $\delta_{\rm CP}$. At nearby this bound,
$\delta_{\rm CP}$ is allowed by $0^{\circ}-100^{\circ}$ and
$270^{\circ}-360^{\circ}$. Furthermore, the smaller $\chi$ square tends to be
localized at nearby the cosmological bound, and its testability would be
enhanced.

Figure 11: $|\delta\tau|<10^{-15}$ for black,
$10^{-15}\leq|\delta\tau|<10^{-12}$ for gray,
$10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta.

In Fig. 11, we show the several figures in terms of deviation from
$\tau=\omega$ where the color legends are the same as the one in Fig. 6. The
up-left one corresponds to the case of up-right one in Fig. 10. The up-right
one corresponds to the case of down-left one in Fig. 10. The down-left one
corresponds to the case of down-right one in Fig. 10. These figures show us
that larger deviation; $10^{-3}\leq|\delta\tau|<10^{-1}$, is requested when
the neutrino oscillations are satisfied. It is not favored by the theoretical
point of view as we already discussed in Sec. 2.

Figure 12: Each of color represents ${\rm blue}\leq 1\sigma$, $1\sigma<{\rm
green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$, $3\sigma<{\rm
red}\leq 5\sigma$.

In Fig. 12, we show our several allowed regions on $\tau$ at nearby
$\tau=\omega$ in case of IH, where the color legends are the same as the one
in Fig. 6. The up-left one represents the allowed region of the imaginary part
of $\tau$ in terms of the real part of $\tau$. We have found only the allowed
region of $2\sigma-5\sigma$. The up-right one demonstrates the allowed region
of neutrinoless double beta decay $\langle m_{ee}\rangle$ in terms of the
lightest active neutrino mass $m_{3}$. There seems to be a linear correlation
between them, and 0.02 eV$\lesssim\langle m_{ee}\rangle\lesssim$ 0.06 eV up to
$5\sigma$, but the allowed regions are localized at nearby small masses up to
$2\sigma$. The down-left one shows the allowed region of Majorana phases
$\alpha_{21}$ and $\alpha_{31}$. We find the allowed regions
$100^{\circ}\lesssim\alpha_{21}\lesssim 280^{\circ}$ and
$50^{\circ}\lesssim\alpha_{31}\lesssim 340^{\circ}$. The down-right one
depicts the allowed region of Dirac phase $\delta_{\text{CP}}$ in terms of the
sum of neutrino masses $\sum m_{i}$. The allowed region at yellow plots; $\sum
m_{i}\simeq 0.11$ eV, is totally within the cosmological constraint. This
implies that $m_{3}$ is almost zero combined with the up-right figure.

Figure 13: $|\delta\tau|<10^{-15}$ for black,
$10^{-15}\leq|\delta\tau|<10^{-12}$ for gray,
$10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta.

In Fig. 13, we show the several figures in terms of deviation from
$\tau=\omega$ where the color legends are the same as the one in Fig. 8. The
up-left one corresponds to the case of up-right one in Fig. 12. The up-right
one corresponds to the case of down-left one in Fig. 12. The down-left one
corresponds to the case of down-right one in Fig. 12. These figures also show
us that larger deviation; $10^{-3}\leq|\delta\tau|<10^{-1}$, is requested when
the neutrino oscillations are satisfied. It is not favored by the theoretical
point of view as we already discussed in Sec. 2. In conclusion, in the case of
$\tau=\omega$, both the case of NH and IH would not be favored by the
theoretical viewpoint.

####  3.2.3 Nearby $\tau=2i$

Figure 14: Each of color represents ${\rm blue}\leq 1\sigma$, $1\sigma<{\rm
green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$, $3\sigma<{\rm
red}\leq 5\sigma$.

In Fig. 14, we show our several allowed regions on $\tau$ at nearby $\tau=2i$
in case of NH, where the color legends are the same as the one in Fig. 6. The
up-left one represents the allowed region of the imaginary part of $\tau$ in
terms of the real part of $\tau$. The smaller $\chi$ square denoted by blue
color is closest to the fixed point of $\tau=2i$, which would be a good
tendency. The up-right one demonstrates the allowed region of neutrinoless
double beta decay $\langle m_{ee}\rangle$ in terms of the lightest active
neutrino mass $m_{1}$. There is main linear correlation between them. We find
the allowed regions 0 eV$\leq m_{1}\leq$0.014 eV, and 0 eV$\leq\langle
m_{ee}\rangle\leq$0.013 eV. The down-left one shows the allowed region of
Majorana phases $\alpha_{21}$ and $\alpha_{31}$. Both the phases allow to be
$0^{\circ}$ or $180^{\circ}$. The down-right one depicts the allowed region of
Dirac phase $\delta_{\text{CP}}$ in terms of the sum of neutrino masses $\sum
m_{i}$. The whole allowed region of $\sum m_{i}$ is totally within the bound
on cosmological constraint; 0.058 eV$\leq\sum m_{i}\leq$ 0.082 eV, whereas the
allowed region of $\delta_{\rm CP}$ is the same as Majorana phases;
$0^{\circ}$ or $180^{\circ}$. Note that it is trivial that we find no phases
since the situation is similar to the case of $\tau=i$.

Figure 15: $|\delta\tau|<10^{-15}$ for black,
$10^{-15}\leq|\delta\tau|<10^{-12}$ for gray,
$10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta.

In Fig. 15, we show the several figures in terms of deviation from $\tau=2i$
in the same case of Fig. 5, where the color legends are the same as the one in
Fig. 6. The up-left one is the same as the case of up-right one in Fig. 14.
The up-right one is the same as the case of down-left one in Fig. 14. The
down-left one is the same as the case of down-right one in Fig. 14. These
figures suggest us that size of deviation almost run the whole ranges that are
allowed by the neutrino oscillation data.

Figure 16: Each of color represents ${\rm blue}\leq 1\sigma$, $1\sigma<{\rm
green}\leq 2\sigma$, $2\sigma<{\rm yellow}\leq 3\sigma$, $3\sigma<{\rm
red}\leq 5\sigma$.

In Fig. 16, we show our several allowed regions on $\tau$ at nearby $\tau=2i$
in case of IH, where color legends are the same as the one of Fig. 5.
Therefore, we have found only the allowed region of $2\sigma-5\sigma$. The up-
left one represents the allowed region of the imaginary part of $\tau$ in
terms of the real part of $\tau$. The up-right one demonstrates the allowed
region of neutrinoless double beta decay $\langle m_{ee}\rangle$ in terms of
the lightest active neutrino mass $m_{3}$. We find the allowed regions as
follows: 0 eV$\leq m_{3}\leq$0.03 eV and 0.014 eV$\leq\langle
m_{ee}\rangle\leq$0.04 eV up to $5\sigma$, but the allowed regions are
localized at nearby small masses at yellow plots. The down-left one shows the
allowed region of Majorana phases $\alpha_{21}$ and $\alpha_{31}$.
$\alpha_{21}$ is allowed by $100^{\circ}$ to $200^{\circ}$, while
$\alpha_{31}$ is wider region than $\alpha_{21}$. However the allowed regions
are localized at nearby $\alpha_{21}=180^{\circ}$ and $\alpha_{31}=0^{\circ}$
at yellow plots. The down-right one depicts the allowed region of Dirac phase
$\delta_{\text{CP}}$ in terms of the sum of neutrino masses $\sum m_{i}$. The
vertical line is the upper bound on cosmological constraint. $\delta_{\rm CP}$
is allowed at the points $0^{\circ}$ and $180^{\circ}$. On the other hand,
almost half the points of $\sum m_{i}$ would be ruled out by the cosmological
bound. Therefore, we would predict a narrow range of $0.1$eV $\leq\sum
m_{i}\leq 0.12$ eV that is almost the same as the one in case of $\tau=i$.

Figure 17: $|\delta\tau|<10^{-15}$ for black,
$10^{-15}\leq|\delta\tau|<10^{-12}$ for gray,
$10^{-12}\leq|\delta\tau|<10^{-10}$ for purple,
$10^{-10}\leq|\delta\tau|<10^{-7}$ for brown,
$10^{-7}\leq|\delta\tau|<10^{-5}$ for blue green,
$10^{-5}\leq|\delta\tau|<10^{-3}$ for orange, and
$10^{-3}\leq|\delta\tau|<10^{-1}$ for magenta.

In Fig. 17, we show the several figures in terms of deviation from $\tau=2i$
in the same case of Fig. 5, where the color legends are the same as the one in
Fig. 8. The up-left one is the same as the case of up-right one in Fig. 16.
The up-right one is the same as the case of down-left one in Fig. 16. The
down-left one is the same as the case of down-right one in Fig. 16. These
figures suggest us that size of deviation almost run the whole ranges that are
allowed by the neutrino oscillation data.

Figure 18: Ratios of the string landscape in Fig. 4 and the $A_{4}$ model
within $5\sigma$, where the ratios are defined as those of the number of
solutions in a corresponding range of $-{\rm Log}_{10}|\delta\tau|$ to the
number of whole solutions. We present the NH and the IH in the left and right
panels, respectively.

In a similar to $\tau\simeq i$, we plot ratios of the number of solutions in a
corresponding range of $-{\rm Log}_{10}|\delta\tau|$ to the number of whole
solutions for both the string landscape in Fig. 2 and the $A_{4}$ model within
$5\sigma$ in Fig. 18. It indicates both the distributions of $A_{4}$ model
with NH and the moduli fields in the string landscape peak around
$|\delta\tau|={\cal O}(10^{-5})$, but such a signal will not be found in the
IH case.

Here, we summarize our results where $\tau=\omega$ does not favor a
theoretical point of view from the string landscape. Thus, we focus on
$\tau=i$ and $\tau=2i$ only. In the case of $\tau=i$ with NH, there is an
intriguing tendency that the allowed region of smaller $\chi$ square is
localized at smaller $\sum m_{i}$ that is within the cosmological bound.
Another feature is that the best fit value of Dirac CP phase $\sim
195^{\circ}$ would be reproduced when we allow up to $5\sigma$ interval. It
implies that smaller deviations $|\delta\tau|$ tend to be localized at nearby
their smaller masses. In the case of $\tau=i$ with IH, There are two
correlations between them; one is a linear line, and another is a slightly
curved one. The solutions tend to be localized at nearby smaller mass of
$m_{3}$ with $\langle m_{ee}\rangle=0.015,\ 0.05$ eV. A large part of $\sum
m_{i}$ would be ruled out by the cosmological bound. Therefore, we would
predict a narrow range of $0.1~{}{\rm eV}\leq\sum m_{i}\leq 0.12~{}{\rm eV}$.
It implies that smaller deviations $|\delta\tau|$ tend to be localized at
nearby their smaller masses $m_{3}$. The smaller deviation would be favored in
the point of view of the bound on the cosmological constraint. Both the
distributions of $A_{4}$ model with NH and the moduli fields in the string
landscape peak around $|\delta\tau|={\cal O}(10^{-5})$, but such a signal is
not found. In the case of $\tau=2i$ with NH, the smaller $\chi$ square denoted
in blue is closest to the fixed point of $\tau=2i$, which would be a good
tendency. We find the allowed regions 0 eV$\leq m_{1}\leq$0.014 eV, and 0
eV$\leq\langle m_{ee}\rangle\leq$0.013 eV. The whole allowed region of $\sum
m_{i}$ is totally within the bound on cosmological constraint; 0.058
eV$\leq\sum m_{i}\leq$ 0.082 eV. The size of the deviation from $\tau=2i$
almost runs the whole ranges that are allowed by the neutrino oscillation
data. In the case of $\tau=2i$ with IH, we find the allowed regions as
follows: 0 eV$\leq m_{3}\leq$0.03 eV and 0.014 eV$\leq\langle
m_{ee}\rangle\leq$0.04 eV up to $5\sigma$, but the allowed regions are
localized at nearby small masses at yellow plots. $\alpha_{21}$ is allowed by
$100^{\circ}$ to $200^{\circ}$, while $\alpha_{31}$ is wider region than
$\alpha_{21}$. Almost half the points of $\sum m_{i}$ would be ruled out by
the cosmological bound. Therefore, we would predict a narrow range of $0.1$eV
$\leq\sum m_{i}\leq 0.12$ eV that is almost the same as the one in the case of
$\tau=i$. The size of the deviation from $\tau=2i$ almost runs the whole
ranges that are allowed by the neutrino oscillation data. Both the
distributions of $A_{4}$ model with NH and moduli fields in the string
landscape peak around $|\delta\tau|={\cal O}(10^{-5})$, but such a signal is
not found in the IH.

##  4 Conclusions

The residual flavor symmetries appearing in fixed points of
$PSL(2,\mathbb{Z})$ moduli space are employed in a wide variety of modular
flavor models, but a small departure of the modulus from fixed points is
required to realize the observed masses and mixing angles of quarks/leptons
and CP-breaking effects in the bottom-up modular invariant theories. In this
paper, we have explicitly demonstrated the breaking of residual flavor
symmetry from the top-down approach.

Following Ref. Ishiguro:2020nuf , we have studied the moduli stabilization in
the context of Type IIB string theory on
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$ orientifold. In Type IIB
flux compactifications, it was known that $\mathbb{Z}_{2}$ and
$\mathbb{Z}_{3}$ fixed points on the fundamental domain of the complex
structure modulus space are statistically favored in the finite number of
vacua. However, the volume moduli have not been stabilized yet, and a present
stage of acceleration of the Universe should be realized. In this respect, we
have incorporated non-perturbative corrections to the superpotential as well
as uplifting sources to stabilize the volume moduli at the dS vacua. These
sources naturally shift the value of $\tau$ from fixed points by a small
amount. We find that the deviations of $\tau$ from fixed points
$\langle\tau\rangle=i,w,2i$ are statistically favored at $|\delta\tau|\simeq
10^{-5}$ and the CP symmetry $\tau\rightarrow-\bar{\tau}$ is broken in a
generic choice of background fluxes. Since the SUSY is broken by the existence
of uplifting source, the typical SUSY-breaking scale, i.e., the gravitino
mass, is estimated as of ${\cal O}(10^{13})$ GeV at the small departure
$|\delta\tau|\simeq 10^{-5}$. In this way, the top-down approach restricts
ourselves to the specific value of the modulus $\tau$ as well as the SUSY-
breaking scale.

To illustrate phenomenological implications, we analyze the concrete $A_{4}$
modular flavor model with an emphasis on the lepton sector. Under charge
assignments for the lepton and Higgs sectors in Tab. 1, we have presented
several predictions in the vicinity of three fixed points by a global
$\chi^{2}$ analysis in both the normal and inverted hierarchies of neutrinos.
It turns out that there exist many phenomenologically promising models around
$\langle\tau\rangle=i$ with the normal hierarchy, whose number is compared
with that of the string landscape in Fig. 9. It implies similar distributions
for string and $A_{4}$ models with respect to $\delta\tau$. Furthermore, there
is an intriguing tendency that allowed region of smaller $\chi$ square is
localized at smaller $\sum m_{i}$ that is within the cosmological bound.

Before closing our paper, it is worthwhile mentioning the quasi-stable DM
candidate due to the tiny deviation from the fixed points in $\tau=i,\ 2i$. In
Ref. Kobayashi:2021ajl , especially, DM and neutrino oscillation data can
simultaneously be explained at $\tau=i$ where DM is a Majorana heavy fermion
with modular weight $-2$. In this set up 444In order to identify DM, we would
need to assign a singlet under $A_{4}$ symmetry in order to avoid mixings
among Majorana fermions that spoil the stability of DM. Also, we might need to
construct a model with all the singlets under $A_{4}$ to get neutrino
oscillation data. In this sense, our model has to be modified when there is DM
in a theory., DM decays into leptons and Higgses via a Dirac term. Assuming
order one free parameter and the DM mass ($m_{X}$) is much heavier than the
leptons and Higgses, we estimate its lifetime ($\tau_{X}$) as follows:

$\displaystyle\tau_{X}\simeq 1.32\times
10^{-25}\times\left|Y^{(6)}_{1}\right|^{-2}\left(\frac{1\ {\rm
TeV}}{m_{X}}\right){\rm sec}.$ (4.1)

When $m_{X}=1$ TeV, the upper limit of $\left|Y^{(6)}_{1}\right|$ be less than
the order $10^{-21}$ in order $X$ to be a quasi-stable DM imposing
$10^{17}{\rm sec}\lesssim\tau_{X}$. Here, $10^{17}$ sec is the age of the
Universe. This constraint is equivalent to $|\delta\tau|\lesssim 5.57\times
10^{-9}$ that is within our valid parameter space.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers JP20K14477 (Hajime O.)
and JP22J12877 (K.I). The work of Hiroshi O. is supported by the Junior
Research Group (JRG) Program at the Asia-Pacific Center for Theoretical
Physics (APCTP) through the Science and Technology Promotion Fund and Lottery
Fund of the Korean Government and was supported by the Korean Local
Governments-Gyeongsangbuk-do Province and Pohang City. Hiroshi O. is sincerely
grateful for all the KIAS members.

##  Appendix A $A_{4}$ modular forms

Note that the modulus-dependent modular forms are constructed by the weight 2
modular form,

$\displaystyle Y_{\bf 3}^{(2)}=\begin{pmatrix}Y_{1}\\\ Y_{2}\\\ Y_{3}\\\
\end{pmatrix},$ (A.1)

with

$\displaystyle Y_{1}(\tau)$
$\displaystyle=\frac{i}{2\pi}\left(\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}-\frac{27\eta^{\prime}(3\tau)}{\eta(3\tau)}\right),$
(A.2) $\displaystyle Y_{2}(\tau)$
$\displaystyle=\frac{-i}{\pi}\left(\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\omega\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\right),$
(A.3) $\displaystyle Y_{3}(\tau)$
$\displaystyle=\frac{-i}{\pi}\left(\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\right)\,,$
(A.4)

where $\eta(\tau)$ denotes the Dedekind eta-function and $\omega=e^{2\pi
i/3}$. Recalling that the other modular forms are constructed by tensor
products of $Y_{\bf 3}^{(2)}$, we list the modular forms used in our analysis:

$\displaystyle{Y^{\rm(4)}_{\bf
3}}(\tau)=\begin{pmatrix}Y_{1}^{2}-Y_{2}Y_{3}\\\ Y_{3}^{2}-Y_{1}Y_{2}\\\
Y_{2}^{2}-Y_{1}Y_{3}\end{pmatrix}\,,$ $\displaystyle Y_{\bf
1}^{(4)}=Y_{1}^{2}+2Y_{2}Y_{3}\,,\qquad\qquad Y_{\bf
1^{\prime}}^{(4)}=Y_{3}^{2}+2Y_{1}Y_{2}\,.$ $\displaystyle{Y^{\rm(6)}_{\bf
3}}(\tau)=Y_{\bf 1}^{(4)}{Y^{\rm(2)}_{\bf
3}}(\tau)=(Y_{1}^{2}+2Y_{2}Y_{3})\begin{pmatrix}Y_{1}\\\ Y_{2}\\\
Y_{3}\end{pmatrix}\,,$ $\displaystyle{Y^{\rm(6)}_{\bf
3^{\prime}}}(\tau)=Y_{\bf 1^{\prime}}^{(4)}{Y^{\rm(2)}_{\bf
3}}(\tau)=(Y_{3}^{2}+2Y_{1}Y_{2})\begin{pmatrix}Y_{3}\\\ Y_{1}\\\
Y_{2}\end{pmatrix}\,.$ (A.5)

## References

  * (1) R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, _Finite Modular Groups and Lepton Mixing_ , _Nucl. Phys. B_ 858 (2012) 437 [1112.1340].
  * (2) S. Ferrara, .D. Lust and S. Theisen, _Target Space Modular Invariance and Low-Energy Couplings in Orbifold Compactifications_ , _Phys. Lett. B_ 233 (1989) 147.
  * (3) W. Lerche, D. Lust and N.P. Warner, _Duality Symmetries in $N=2$ Landau-ginzburg Models_, _Phys. Lett. B_ 231 (1989) 417.
  * (4) J. Lauer, J. Mas and H.P. Nilles, _Twisted sector representations of discrete background symmetries for two-dimensional orbifolds_ , _Nucl. Phys. B_ 351 (1991) 353.
  * (5) A. Baur, H.P. Nilles, A. Trautner and P.K.S. Vaudrevange, _Unification of Flavor, CP, and Modular Symmetries_ , _Phys. Lett. B_ 795 (2019) 7 [1901.03251].
  * (6) A. Baur, H.P. Nilles, A. Trautner and P.K.S. Vaudrevange, _A String Theory of Flavor and $\mathscr{CP}$_, _Nucl. Phys. B_ 947 (2019) 114737 [1908.00805].
  * (7) T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba and T.H. Tatsuishi, _Modular symmetry and non-Abelian discrete flavor symmetries in string compactification_ , _Phys. Rev. D_ 97 (2018) 116002 [1804.06644].
  * (8) T. Kobayashi and S. Tamba, _Modular forms of finite modular subgroups from magnetized D-brane models_ , _Phys. Rev. D_ 99 (2019) 046001 [1811.11384].
  * (9) H. Ohki, S. Uemura and R. Watanabe, _Modular flavor symmetry on a magnetized torus_ , _Phys. Rev. D_ 102 (2020) 085008 [2003.04174].
  * (10) S. Kikuchi, T. Kobayashi, S. Takada, T.H. Tatsuishi and H. Uchida, _Revisiting modular symmetry in magnetized torus and orbifold compactifications_ , _Phys. Rev. D_ 102 (2020) 105010 [2005.12642].
  * (11) S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada and H. Uchida, _Modular symmetry by orbifolding magnetized $T^{2}\times T^{2}$: realization of double cover of $\Gamma_{N}$_, _JHEP_ 11 (2020) 101 [2007.06188].
  * (12) S. Kikuchi, T. Kobayashi and H. Uchida, _Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds_ , _Phys. Rev. D_ 104 (2021) 065008 [2101.00826].
  * (13) Y. Almumin, M.-C. Chen, V. Knapp-Pérez, S. Ramos-Sánchez, M. Ratz and S. Shukla, _Metaplectic Flavor Symmetries from Magnetized Tori_ , _JHEP_ 05 (2021) 078 [2102.11286].
  * (14) A. Baur, M. Kade, H.P. Nilles, S. Ramos-Sanchez and P.K.S. Vaudrevange, _Siegel modular flavor group and CP from string theory_ , _Phys. Lett. B_ 816 (2021) 136176 [2012.09586].
  * (15) A. Strominger, _SPECIAL GEOMETRY_ , _Commun. Math. Phys._ 133 (1990) 163.
  * (16) P. Candelas and X. de la Ossa, _Moduli Space of Calabi-Yau Manifolds_ , _Nucl. Phys. B_ 355 (1991) 455.
  * (17) K. Ishiguro, T. Kobayashi and H. Otsuka, _Spontaneous CP violation and symplectic modular symmetry in Calabi-Yau compactifications_ , _Nucl. Phys. B_ 973 (2021) 115598 [2010.10782].
  * (18) K. Ishiguro, T. Kobayashi and H. Otsuka, _Symplectic modular symmetry in heterotic string vacua: flavor, CP, and R-symmetries_ , _JHEP_ 01 (2022) 020 [2107.00487].
  * (19) F. Feruglio, _Are neutrino masses modular forms?_ , in _From My Vast Repertoire …: Guido Altarelli’s Legacy_ , A. Levy, S. Forte and G. Ridolfi, eds., pp. 227–266 (2019), DOI [1706.08749].
  * (20) T. Kobayashi, K. Tanaka and T.H. Tatsuishi, _Neutrino mixing from finite modular groups_ , _Phys. Rev. D_ 98 (2018) 016004 [1803.10391].
  * (21) J.T. Penedo and S.T. Petcov, _Lepton Masses and Mixing from Modular $S_{4}$ Symmetry_, _Nucl. Phys. B_ 939 (2019) 292 [1806.11040].
  * (22) P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, _Modular A 5 symmetry for flavour model building_, _JHEP_ 04 (2019) 174 [1812.02158].
  * (23) G.-J. Ding, S.F. King and X.-G. Liu, _Neutrino mass and mixing with $A_{5}$ modular symmetry_, _Phys. Rev. D_ 100 (2019) 115005 [1903.12588].
  * (24) X.-G. Liu and G.-J. Ding, _Neutrino Masses and Mixing from Double Covering of Finite Modular Groups_ , _JHEP_ 08 (2019) 134 [1907.01488].
  * (25) P. Chen, G.-J. Ding, J.-N. Lu and J.W.F. Valle, _Predictions from warped flavor dynamics based on the $T′$ family group_, _Phys. Rev. D_ 102 (2020) 095014 [2003.02734].
  * (26) P.P. Novichkov, J.T. Penedo and S.T. Petcov, _Double cover of modular $S_{4}$ for flavour model building_, _Nucl. Phys. B_ 963 (2021) 115301 [2006.03058].
  * (27) X.-G. Liu, C.-Y. Yao and G.-J. Ding, _Modular invariant quark and lepton models in double covering of $S_{4}$ modular group_, _Phys. Rev. D_ 103 (2021) 056013 [2006.10722].
  * (28) X. Wang, B. Yu and S. Zhou, _Double covering of the modular $A_{5}$ group and lepton flavor mixing in the minimal seesaw model_, _Phys. Rev. D_ 103 (2021) 076005 [2010.10159].
  * (29) C.-Y. Yao, X.-G. Liu and G.-J. Ding, _Fermion masses and mixing from the double cover and metaplectic cover of the $A_{5}$ modular group_, _Phys. Rev. D_ 103 (2021) 095013 [2011.03501].
  * (30) G.-J. Ding, S.F. King, C.-C. Li and Y.-L. Zhou, _Modular Invariant Models of Leptons at Level 7_ , _JHEP_ 08 (2020) 164 [2004.12662].
  * (31) T. Kobayashi, H. Otsuka, M. Tanimoto and K. Yamamoto, _Modular symmetry in the SMEFT_ , _Phys. Rev. D_ 105 (2022) 055022 [2112.00493].
  * (32) T. Kobayashi, H. Otsuka, M. Tanimoto and K. Yamamoto, _Lepton flavor violation, lepton $(g-2)_{\mu,\,e}$ and electron EDM in the modular symmetry_, 2204.12325.
  * (33) T. Kobayashi and H. Otsuka, _On stringy origin of minimal flavor violation_ , _Eur. Phys. J. C_ 82 (2022) 25 [2108.02700].
  * (34) P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, _Generalised CP Symmetry in Modular-Invariant Models of Flavour_ , _JHEP_ 07 (2019) 165 [1905.11970].
  * (35) P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, _Modular S 4 models of lepton masses and mixing_, _JHEP_ 04 (2019) 005 [1811.04933].
  * (36) P.P. Novichkov, S.T. Petcov and M. Tanimoto, _Trimaximal Neutrino Mixing from Modular A4 Invariance with Residual Symmetries_ , _Phys. Lett. B_ 793 (2019) 247 [1812.11289].
  * (37) G.-J. Ding, S.F. King, X.-G. Liu and J.-N. Lu, _Modular S 4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing_, _JHEP_ 12 (2019) 030 [1910.03460].
  * (38) H. Okada and M. Tanimoto, _Towards unification of quark and lepton flavors in $A_{4}$ modular invariance_, _Eur. Phys. J. C_ 81 (2021) 52 [1905.13421].
  * (39) S.F. King and Y.-L. Zhou, _Trimaximal TM 1 mixing with two modular $S_{4}$ groups_, _Phys. Rev. D_ 101 (2020) 015001 [1908.02770].
  * (40) H. Okada and M. Tanimoto, _Quark and lepton flavors with common modulus $\tau$ in $A_{4}$ modular symmetry_, 2005.00775.
  * (41) H. Okada and M. Tanimoto, _Modular invariant flavor model of $A_{4}$ and hierarchical structures at nearby fixed points_, _Phys. Rev. D_ 103 (2021) 015005 [2009.14242].
  * (42) H. Okada and M. Tanimoto, _Spontaneous CP violation by modulus $\tau$ in $A_{4}$ model of lepton flavors_, _JHEP_ 03 (2021) 010 [2012.01688].
  * (43) F. Feruglio, V. Gherardi, A. Romanino and A. Titov, _Modular invariant dynamics and fermion mass hierarchies around $\tau=i$_, _JHEP_ 05 (2021) 242 [2101.08718].
  * (44) T. Kobayashi, H. Okada and Y. Orikasa, _Dark matter stability at fixed points in a modular $A_{4}$ symmetry_, 2111.05674.
  * (45) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi, _$A_{4}$ lepton flavor model and modulus stabilization from $S_{4}$ modular symmetry_, _Phys. Rev. D_ 100 (2019) 115045 [1909.05139].
  * (46) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T.H. Tatsuishi and H. Uchida, _$CP$ violation in modular invariant flavor models_, _Phys. Rev. D_ 101 (2020) 055046 [1910.11553].
  * (47) T. Kobayashi and H. Otsuka, _Challenge for spontaneous $CP$ violation in Type IIB orientifolds with fluxes_, _Phys. Rev. D_ 102 (2020) 026004 [2004.04518].
  * (48) K. Ishiguro, T. Kobayashi and H. Otsuka, _Landscape of Modular Symmetric Flavor Models_ , _JHEP_ 03 (2021) 161 [2011.09154].
  * (49) P.P. Novichkov, J.T. Penedo and S.T. Petcov, _Modular Flavour Symmetries and Modulus Stabilisation_ , 2201.02020.
  * (50) T. Kobayashi and H. Otsuka, _Classification of discrete modular symmetries in Type IIB flux vacua_ , _Phys. Rev. D_ 101 (2020) 106017 [2001.07972].
  * (51) O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, _Enumerating flux vacua with enhanced symmetries_ , _JHEP_ 02 (2005) 037 [hep-th/0411061].
  * (52) S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, _De Sitter vacua in string theory_ , _Phys. Rev. D_ 68 (2003) 046005 [hep-th/0301240].
  * (53) S. Kikuchi, T. Kobayashi, K. Nasu, H. Otsuka, S. Takada and H. Uchida, _Modular symmetry of soft supersymmetry breaking terms_ , 2203.14667.
  * (54) X. Du and F. Wang, _SUSY breaking constraints on modular flavor $S_{3}$ invariant SU(5) GUT model_, _JHEP_ 02 (2021) 221 [2012.01397].
  * (55) T. Kobayashi, T. Shimomura and M. Tanimoto, _Soft supersymmetry breaking terms and lepton flavor violations in modular flavor models_ , _Phys. Lett. B_ 819 (2021) 136452 [2102.10425].
  * (56) H. Otsuka and H. Okada, _Radiative neutrino masses from modular $A_{4}$ symmetry and supersymmetry breaking_, 2202.10089.
  * (57) S. Gukov, C. Vafa and E. Witten, _CFT’s from Calabi-Yau four folds_ , _Nucl. Phys. B_ 584 (2000) 69 [hep-th/9906070].
  * (58) P. Betzler and E. Plauschinn, _Type IIB flux vacua and tadpole cancellation_ , _Fortsch. Phys._ 67 (2019) 1900065 [1905.08823].
  * (59) P. Candelas, E. Perevalov and G. Rajesh, _Toric geometry and enhanced gauge symmetry of F theory / heterotic vacua_ , _Nucl. Phys. B_ 507 (1997) 445 [hep-th/9704097].
  * (60) W. Taylor and Y.-N. Wang, _The F-theory geometry with most flux vacua_ , _JHEP_ 12 (2015) 164 [1511.03209].
  * (61) M. Demirtas, M. Kim, L. Mcallister and J. Moritz, _Vacua with Small Flux Superpotential_ , _Phys. Rev. Lett._ 124 (2020) 211603 [1912.10047].
  * (62) Y. Honma and H. Otsuka, _Small flux superpotential in F-theory compactifications_ , _Phys. Rev. D_ 103 (2021) 126022 [2103.03003].
  * (63) H. Abe, T. Higaki and T. Kobayashi, _Remark on integrating out heavy moduli in flux compactification_ , _Phys. Rev. D_ 74 (2006) 045012 [hep-th/0606095].
  * (64) KamLAND-Zen collaboration, _Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen_ , _Phys. Rev. Lett._ 117 (2016) 082503 [1605.02889].
  * (65) I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, _The fate of hints: updated global analysis of three-flavor neutrino oscillations_ , _JHEP_ 09 (2020) 178 [2007.14792].
  * (66) F. Björkeroth, F.J. de Anda, I. de Medeiros Varzielas and S.F. King, _Towards a complete A ${}_{4}\times$ SU(5) SUSY GUT_, _JHEP_ 06 (2015) 141 [1503.03306].
  * (67) I. Esteban, M.C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, _Global analysis of three-flavour neutrino oscillations: synergies and tensions in the determination of $\theta_{23}$, $\delta_{CP}$, and the mass ordering_, _JHEP_ 01 (2019) 106 [1811.05487].
  * (68) Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].