#  EPHOU-21-014 CTPU-PTC-21-39 KEK-TH-2369 HUPD-2113 Modular symmetry in the
SMEFT

Tatsuo Kobayashi 1, Hajime Otsuka 2,3, Morimitsu Tanimoto 4, Kei Yamamoto 5

1Department of Physics, Hokkaido University, Sapporo 060-0810, Japan 2KEK
Theory Center, Institute of Particle and Nuclear Studies, KEK, 1-1 Oho,
Tsukuba, Ibaraki 305-0801, Japan
3Center for Theoretical Physics of the Universe, Institute for Basic Science,
Dajeon 34126, Korea
4Department of Physics, Niigata University, Niigata 950-2181, Japan 5Core of
Research for the Energetic Universe, Hiroshima University, Higashi-Hiroshima
739-8526, Japan

( Abstract
We study the modular symmetric standard-model effective field theory. We
employ the stringy Ansatz on coupling structure that 4-point couplings
$y^{(4)}$ of matter fields are written by a product of 3-point couplings
$y^{(3)}$ of matter fields, i.e., $y^{(4)}=y^{(3)}y^{(3)}$. In this framework,
we discuss the flavor structure of bilinear fermion operators and 4-fermion
operators, where the holomorphic and anti-holomorphic modular forms appear.
From the Ansatz, the $A_{4}$ modular-invariant semileptonic four-fermion
operator $[\,\bar{E}_{R}\Gamma E_{R}][\,\bar{D}_{R}\Gamma D_{R}\,]$ does not
lead to the flavor changing (FC) processes since this operator would be
constructed in terms of gauge couplings $g$ as $y^{(3)}\sim g$. The chirality
flipped bilinear operator $[\,\bar{D}_{R}\Gamma D_{L}\,]$ also does not lead
FC if the mediated mode corresponds to the Higgs boson $H_{d}$. In this case,
the flavor structure of this operator is the exactly same as the mass matrix.
On the other hand, if the flavor structure of the operator is not the exactly
same as the mass matrix, the situation would change drastically. Then, we
obtain the non-trivial relations of FC transitions at nearby fixed points
$\tau=i,\,\omega\,,i\infty$, which are testable in the future. As an
application, we discuss the relations of the lepton flavor violation processes
$\mu\to e\gamma,\,\tau\to\mu\gamma$ and $\tau\to e\gamma$ at nearby
$\tau_{e}=i$, where the successful lepton mass matrix was obtained. We also
study the flavor changing 4-quark operators in the $A_{4}$ modular symmetry of
quarks. As a result, the minimal flavor violation could be realized by taking
relevant specific parameter sets of order one. )

###### Contents

  1. 1 Introduction
  2. 2 $A_{4}$ modular symmetry and flavor of quarks and leptons
    1. 2.1 Modular flavor symmetry
    2. 2.2 Modular forms
    3. 2.3 Representation of down-type quarks and charged leptons
  3. 3 SMEFT 4-fermion operators in $A_{4}$ modular symmetry
  4. 4 Bilinear fermion operators $[\,\bar{D}_{R}\Gamma D_{L}\,]$ and $[\,\bar{D}_{L}\Gamma D_{R}\,]$
    1. 4.1 $[\,\bar{D}_{R}\Gamma D_{L}\,]$ and $[\,\bar{D}_{L}\Gamma D_{R}\,]$ bilinears in the flavor space
    2. 4.2 Diagonal matrix $M_{E}^{\dagger}M_{E}$ and $M_{q}^{\dagger}M_{q}$ at fixed points
      1. 4.2.1 Mass matrix and operators $\bar{D}_{L}\Gamma D_{R}$, $\bar{D}_{R}\Gamma D_{L}$ at the fixed point $\tau=i$
  5. B A model of mass matrices in the $A_{4}$ modular symmetry
  6. C Unitary transformation of $S$ and $ST$ and mass matrix
    1. C.1 Diagonal base of $S$ for $A_{4}$ triplet
    2. C.2 Diagonal bases of $ST$ and $T$ for $A_{4}$ triplet
    3. C.3 Mass matrix in the new bases of generators $S$ and $ST$
  7. D Mass matrix at nearby $\tau=i$
  8. E Decomposition of products of triplets
  9. F $U(2)$ flavor symmetry

##  1 Introduction

The Standard Model (SM) has been regarded as the low-energy limit of an full
theory at the high energy scale. After the first years of running of the LHC,
no new physics (NP) has been discovered. That is, there is a mass gap between
the SM spectrum and these hypothetical additional degrees of freedom such as
new particles.

Describing possible physics beyond the SM in general terms gets increasingly
important, and the systematic study goes under the name of the SMEFT: the SM
Effective Field Theory (EFT) based on the $SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}$ gauge symmetry, and the SM field content as dynamical degrees of
freedom below a cut-off scale $\mu>G^{-1/2}_{F}$. Several years after the
pioneering analysis in Ref.[1], the first complete list of non-redundant SMEFT
Lagrangian terms up to dimension-six has been presented in Ref.[2].

When all the possible flavor structures are taken into account in absence of
any flavor symmetry, a large proliferation in the number of independent
coefficients in the SMEFT occurs: there are 1350 CP-even and 1149 CP-odd
independent coefficients for the dimension-six operators [3]. The flavor
symmetry is a challenging hypothesis to reduce the number of independent
parameters of the flavor sector. Above all, the flavor symmetries $U(3)^{5}$
and $U(2)^{5}$ have been successfully applied to the SMEFT [4]. The $U(3)^{5}$
flavor symmetry is the maximal flavor symmetry allowed by the SM gauge sector,
while $U(2)^{5}$ is the corresponding subgroup acting only on the first two
(light) families. The $U(3)^{5}$ allows us to apply the Minimal Flavor
Violation (MFV) hypothesis [5, 6], which is the most restrictive hypothesis
consistent with the SMEFT, and suppress non-standard contributions to flavor-
violating observables [6]. In the $U(2)^{5}$ symmetry [7, 8, 9], it retains
most of the MFV virtues and allows us to have a much richer structure as far
as the dynamics of third family is concerned.

In the $U(3)^{5}$ and $U(2)^{5}$ flavor symmetric scenario, the Yukawa
couplings are understood as spurions (symmetry-breaking terms) which have non-
trivial representations under the symmetry. Assuming that flavor- and CP-
violating effects in NP is also controlled by the Yukawa, the flavor structure
of higher-dimensional operators are expressed by the spurions. Then, by
counting different order in the breaking terms of these symmetries, we can
classify the number of independent dimension-six operators, as studied in Ref.
[4]. These flavor symmetries are not the only options to efficiently suppress
flavor-violating observables in the SMEFT, and the non-Abelian discrete
symmetry would be one of the alternative choice.

The non-Abelian discrete groups have been discussed to challenge the flavor
problem of quarks and leptons [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].
Indeed, supersymmetric (SUSY) modular invariant theories give us an attractive
framework to address the flavor symmetry of quarks and leptons with non-
Abelian discrete groups [20]. In this approach, the quark and lepton mass
matrices are written in terms of modular forms which are holomorphic functions
of the modulus $\tau$. The arbitrary symmetry breaking sector of the
conventional models based on flavor symmetries is replaced by the moduli
space, and then Yukawa couplings are given by modular forms.

The well-known finite groups $S_{3}$, $A_{4}$, $S_{4}$ and $A_{5}$ are
isomorphic to the finite modular groups $\Gamma_{N}$ for $N=2,3,4,5$,
respectively[21]. The lepton mass matrices have been given successfully in
terms of $A_{4}$ modular forms [20]. Modular invariant flavor models have been
also proposed on the $\Gamma_{2}\simeq S_{3}$ [22], $\Gamma_{4}\simeq S_{4}$
[23] and $\Gamma_{5}\simeq A_{5}$ [24]. Based on these modular forms, the
flavor mixing of quarks and leptons has been discussed intensively in these
years. Phenomenological studies of the lepton flavors have been done based on
$A_{4}$ [25, 26, 27], $S_{4}$ [28, 29, 30] and $A_{5}$ [31]. A clear
prediction of the neutrino mixing angles and the Dirac CP phase was given in
the simple lepton mass matrices with the $A_{4}$ modular symmetry [26]. The
Double Covering groups $\rm T^{\prime}$ [32, 33] and $S_{4}^{\prime}$ [34, 35]
were also realized in the modular symmetry. Furthermore, phenomenological
studies have been developed in many works [36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,
83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101].

Superstring theory is a promising candidate for the unified theory including
gravity. Various string compactifications lead to four-dimensional low energy
field theories with the specific structure, where 4-point couplings
$y^{(4)}_{ijk\ell}$ of matter fields can be written by a product of 3-point
couplings $y^{(3)}_{ijm}$ of matter fields,

$y^{(4)}_{ijk\ell}=\sum_{m}y^{(3)}_{ijm}y^{(3)}_{mk\ell}\,,$ (1.1)

up to an overall factor, where the modes corresponding to $m$ may be light or
heavy modes. Furthermore, $n$-point couplings $y^{(n)}$ can also be written by
products of 3-point couplings $y^{(3)}$, i.e., $y^{(n)}=(y^{(3)})^{n-2}$.
Thus, the symmetries in 3-point couplings are still symmetries even for
higher-dimensional operators, and the flavor structures of higher-dimensional
operators are controlled by 3-point couplings. This structure in the string-
derived low-energy effective field theory meets the MFV hypothesis [102]. Note
that the string EFTs satisfy the relation (1.1) at the compactification scale
or the string scale, but it holds at the low-energy scale. This is because new
operators appearing through the vacuum expectation value (VEV) of scalar
fields and integrating out heavy states keep the relation (1.1).

In addition, these couplings in the string-derived effective field theory
depend on moduli, which represent geometrical characters of string compact
spaces such as size and shape. When we ignore the dynamic of moduli fields,
these moduli-dependent couplings behave as spurions. Then, the geometrical
symmetry, under which moduli transform non-trivially, would be important from
the viewpoint of the MFV, although Yukawa spurions transform $({\bf
3},{\bf\bar{3}},1,1,1)$, $({\bf 3},1,{\bf\bar{3}},1,1)$, and $(1,1,1,{\bf
3},{\bf\bar{3}})$ in the $U(3)^{5}=U(3)_{Q}\otimes U(3)_{U}\otimes
U(3)_{D}\otimes U(3)_{L}\otimes U(3)_{E}$ flavor symmetric MFV scenario, where
$\\{Q,U,D,L,E\\}$ denote the five independent types of the SM fermions. The
$U(2)^{5}$ flavor symmetric scenario can also be realized in string models due
to the fact that matter Yukawa couplings have rank 1 at the leading
order[103]. The modular symmetry is the geometrical symmetry, which changes
the basis of cycles of the torus $T^{2}$ as well as the orbifold
$T^{2}/\mathbb{Z}_{2}$.111The possible discrete modular symmetries on higher-
dimensional toroidal orbifolds were classified in the context of Type IIB
string theory[104].. Moreover, zero-modes transform non-trivially under the
modular symmetry (see, e.g., for heterotic string theory on orbifolds Ref.
[105] and for magnetized brane models Ref. [106]). That is, the modular
symmetry corresponds to the flavor symmetry in the low-energy effective field
theory. Yukawa couplings as well as other couplings transform non-trivially
under the modular symmetry, because these couplings depend on the modulus.
Calabi-Yau threefolds have many moduli, and their geometrical symmetries are
symplectic modular symmetries [107, 108]. Their phenomenological implications
were studied in Refs. [109, 110]. Hence, these observations strongly support
that flavor structures of higher-dimensional operators as well as Yukawa
couplings in the string EFTs are determined by the modular flavor symmetry.

A drawback of the MFV hypothesis is that it does not allow us to define a
clear power-counting in the SMEFT. This is because one of the breaking term,
namely Yukawa coupling $Y_{t}$, is large. It is therefore not obvious why one
should not consider more powers of $Y_{t}$ in the counting of independent
operators. On the other hand, it defines a clear power counting in the modular
symmetry due to the modular weights.

In this paper, we study the SMEFT with the $\Gamma_{N}$ modular flavor
symmetry. The modular flavor symmetry is regarded as a remnant of the
geometrical symmetry of the extra-dimensional space. Constraints on higher-
dimensional operators only by $\Gamma_{N}$ would be weak, in particular for
$\Gamma_{N}$ singlets and many parameters would be allowed. Hence, we employ
the relation (1.1) as Ansatz in the SMEFT. Through this Ansatz, higher-
dimensional operators are related with 3-point couplings, although the $m$
mode in Eq. (1.1) may be known or unknown.

We take the level 3 finite modular groups, $\Gamma_{3}$ for the flavor
symmetry since the property of $A_{4}$ flavor symmetry has been well known
[111, 112, 113, 114, 115, 116, 117]. Based on the tensor decomposition of
$A_{4}$ modular group, we discuss the bilinear and 4-fermion operators with
flavor changing (FC) process at nearby fixed points. As an application, we
discuss the lepton flavor violation (LFV).

The paper is organized as follows. In section 2, we review on the $A_{4}$
modular flavor symmetry for flavors. In section 3, we discuss the 4-fermion
operators of the SMEFT in $A_{4}$ modular symmetry. In sections 4 and
LABEL:sec:LL, we study the flavor structure of the bilinear fermion operators
with chirality flip and chirality conserve, respectively. In section
LABEL:sec:qqqq, we discuss the 4-quark operator with $\Delta F=2$. Section
LABEL:sec:summary is devoted to the summary. In Appendix LABEL:App:SMEFT, we
summarizes the SMEFT operators. In Appendix B, we give a brief review on an
explicit $A_{4}$ modular flavor model. The $S$ and $ST$ transformations of its
mass matrix are given in Appendix C. In addition, the mass matrix at nearby
$\tau=i$ is given in Appendix D. In Appendix E, we present tensor products of
$A_{4}$ including modular forms. Appendix F summarizes briefly the $U(2)$
flavor symmetry of the quark sector.

##  2 $A_{4}$ modular symmetry and flavor of quarks and leptons

In this section, we briefly review the models with $A_{4}$ modular symmetry.

###  2.1 Modular flavor symmetry

The modular group $\bar{\Gamma}$ is the group of linear fractional
transformations $\gamma$ acting on the modulus $\tau$, belonging to the upper-
half complex plane as:

$\tau\longrightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d}\ ,~{}~{}{\rm
where}~{}~{}a,b,c,d\in\mathbb{Z}~{}~{}{\rm and}~{}~{}ad-bc=1,~{}~{}{\rm
Im}[\tau]>0~{},$ (2.1)

which is isomorphic to $PSL(2,\mathbb{Z})=SL(2,\mathbb{Z})/\\{\rm I,-I\\}$
transformation. This modular transformation is generated by $S$ and $T$,

$\displaystyle S:\tau\longrightarrow-\frac{1}{\tau}\ ,\qquad\qquad
T:\tau\longrightarrow\tau+1\ ,$ (2.2)

which satisfy the following algebraic relations,

$S^{2}={\rm I}\ ,\qquad(ST)^{3}={\rm I}\ .$ (2.3)

We introduce the series of groups $\Gamma(N)$, called principal congruence
subgroups, where $N$ is the level $1,2,3,\dots$. These groups are defined by

$\displaystyle\begin{aligned} \Gamma(N)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z})~{},~{}~{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}~{}~{}({\rm
mod}N)\right\\}\end{aligned}.$ (2.4)

For $N=2$, we define $\bar{\Gamma}(2)\equiv\Gamma(2)/\\{\rm I,-I\\}$. Since
the element $\rm-I$ does not belong to $\Gamma(N)$ for $N>2$, we have
$\bar{\Gamma}(N)=\Gamma(N)$. The quotient groups defined as
$\Gamma_{N}\equiv\bar{\Gamma}/\bar{\Gamma}(N)$ are finite modular groups. In
these finite groups $\Gamma_{N}$, $T^{N}={\rm I}$ is imposed. The groups
$\Gamma_{N}$ with $N=2,3,4,5$ are isomorphic to $S_{3}$, $A_{4}$, $S_{4}$ and
$A_{5}$, respectively [21].

Modular forms $f_{i}(\tau)$ of weight $k$ are the holomorphic functions of
$\tau$ and transform as

$f_{i}(\tau)\longrightarrow(c\tau+d)^{k}\rho(\gamma)_{ij}f_{j}(\tau)\,,\quad\gamma\in\bar{\Gamma}\,,$
(2.5)

under the modular symmetry, where $\rho(\gamma)_{ij}$ is a unitary matrix
under $\Gamma_{N}$.

Under the modular transformation of Eq. (2.1), chiral superfields $\psi_{i}$
($i$ denotes flavors) with weight $-k$ transform as [118]

$\psi_{i}\longrightarrow(c\tau+d)^{-k}\rho(\gamma)_{ij}\psi_{j}\,.$ (2.6)

We study global SUSY models. The superpotential which is built from matter
fields and modular forms is assumed to be modular invariant, i.e., to have a
vanishing modular weight. For given modular forms, this can be achieved by
assigning appropriate weights to the matter superfields.

The kinetic terms are derived from a Kähler potential. The Kähler potential of
chiral matter fields $\psi_{i}$ with the modular weight $-k$ is given simply
by

$\frac{1}{[i(\bar{\tau}-\tau)]^{k}}\sum_{i}|\psi_{i}|^{2},$ (2.7)

where the superfield and its scalar component are denoted by the same letter,
and $\bar{\tau}=\tau^{*}$ after taking VEV of $\tau$. The canonical form of
the kinetic terms is obtained by changing the normalization of parameters
[26]. The general Kähler potential consistent with the modular symmetry
possibly contains additional terms [119]. However, we consider only the
simplest form of the Kähler potential.

For $\Gamma_{3}\simeq\rm A_{4}$, the dimension of the linear space ${\cal
M}_{k}(\Gamma{(3)})$ of modular forms of weight $k$ is $k+1$ [120, 121, 122],
i.e., there are three linearly independent modular forms of the lowest non-
trivial weight $2$, which form a triplet of the $A_{4}$ group. These modular
forms have been explicitly given [20] in the symmetric base of the $A_{4}$
generators $S$ and $T$ for the triplet representation as shown in the next
subsection.

###  2.2 Modular forms

The holomorphic and anti-holomorphic modular forms with weight 2 compose the
$A_{4}$ triplet as: In the representation of the generators $S$ and $T$ for
$A_{4}$ triplet:

$\displaystyle\begin{aligned} S=\frac{1}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix},\end{aligned}\qquad\quad\begin{aligned}
T=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\end{aligned}$ (2.8)

where $\omega=e^{i\frac{2}{3}\pi}$, modular forms are given explicitly in
terms of the Dedekind eta function $\eta(\tau)$ and its derivative [20]:

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

Those are also expressed in the $q$ expansions, $q=\exp(2i\pi\tau)$:

$\displaystyle\begin{pmatrix}Y_{1}(\tau)\\\ Y_{2}(\tau)\\\
Y_{3}(\tau)\end{pmatrix}=\begin{pmatrix}1+12q+36q^{2}+12q^{3}+\dots\\\
-6q^{1/3}(1+7q+8q^{2}+\dots)\\\ -18q^{2/3}(1+2q+5q^{2}+\dots)\end{pmatrix}\,.$
(2.10)

###  2.3 Representation of down-type quarks and charged leptons

Assign the left-handed down-type quarks to $A_{4}$ triplets $\bf 3$ and the
three right-handed ones to $A_{4}$ three different singlets. Then, those are
expressed as follows:

$\displaystyle\begin{aligned} D_{L}=\begin{pmatrix}d_{L}\\\ s_{L}\\\
b_{L}\end{pmatrix}\,,\quad\bar{D}_{L}=\begin{pmatrix}\bar{d}_{L}\\\
\bar{b}_{L}\\\
\bar{s}_{L}\end{pmatrix}\,,\quad(d^{c}_{R},\,s^{c}_{R},\,b^{c}_{R})=(1,\,1^{\prime\prime},\,1^{\prime})\,,\quad(d_{R},\,s_{R},\,b_{R})=(1,\,1^{\prime},\,1^{\prime\prime})\,.\end{aligned}$
(2.11)

It is noticed that quarks of second and third families are exchanged each
other in $\bar{D}_{L}$. The weight of $D_{L}$ and $\bar{D}_{L}$, $k$ are
assigned to $2$ and $-2$, respectively. On the other hand, $k=0$ for singlets
$d^{c}_{R}$, $d_{R}$, etc..

The charged leptons are like down-type quarks as:

$\displaystyle\begin{aligned} E_{L}=\begin{pmatrix}e_{L}\\\ \mu_{L}\\\
\tau_{L}\end{pmatrix}\,,\quad\bar{E}_{L}=\begin{pmatrix}\bar{e}_{L}\\\
\bar{\tau}_{L}\\\
\bar{\mu}_{L}\end{pmatrix}\,,\quad(e^{c}_{R},\,\mu^{c}_{R},\,\tau^{c}_{R})=(1,\,1^{\prime\prime},\,1^{\prime})\,,\quad(e_{R},\,\mu_{R},\,\tau_{R})=(1,\,1^{\prime},\,1^{\prime\prime})\,.\end{aligned}$
(2.12)

The weight of $E_{L}$ and $\bar{E}_{L}$, $k$, are also $2$ and and $-2$,
respectively. On the other hand, $k=0$ for $e^{c}_{R}$, $e_{R}$, etc..

Most of modular flavor models, which have been studied, are supersymmetric
models. In the following sections, we study models below the supersymmetry
breaking scale. We assume that the light modes are exactly the same as the SM
with two doublet Higgs models.

##  3 SMEFT 4-fermion operators in $A_{4}$ modular symmetry

We write down 4-fermion operators as well as dipole operators in terms of
modular forms $Y(\tau)$. We also follow the Ansatz (1.1) that those higher-
dimensional operators are related with 3-point couplings, e.g., Yukawa
couplings with Higgs fields. Here, the higher-dimensional operators are
supposed to be generated by integrating out heavy superpartners, massive gauge
bosons and stringy modes. We have already many modular flavor symmetric
models, which lead to realistic quark and lepton mass matrices separately.
However, when we use the common value of the modulus $\tau$ for both quark and
lepton sectors, the models are severely constrained and very difficult to
realize all the experimental values of quark and lepton masses and their
mixing angles at the same time. In order to cover many modular flavor models,
we assume that the $A_{4}$ modular flavor symmetry in the lepton sector is
independent of the $A_{4}$ symmetry in the quark sector, i.e.,
$A^{E}_{4}\otimes A^{Q}_{4}$ symmetry. They have two independent moduli,
$\tau_{q}$ and $\tau_{e}$ for the quark sector and the lepton sector,
respectively. Such a setup can be realized through the compactification, that
the compact space includes $T^{2}\times T^{2}$, and the flavor structure in
the quark sector originates from one $T^{2}$, while the lepton flavor
structure originates from the other $T^{2}$. Indeed, a similar setup was
studied e.g., in Ref. [45]. Using this setup and Ansatz, we study their
implications on flavor changing processes.

As examples, consider the semileptonic flavor changing neutral processes,

$\displaystyle b\rightarrow s\,\bar{\mu}\mu\ (s\,\bar{e}e)\,,\qquad
b\rightarrow d\,\bar{\mu}\mu\ (d\,\bar{e}e)\,,\qquad s\rightarrow
d\,\bar{\mu}\mu\ (d\,\bar{e}e)\,,$ (3.1)

which are caused by the flavor changing $\Delta F=1$ operator. Impose the
modular $A_{4}$ symmetry on quarks and leptons, respectively, that is
$A^{E}_{4}\otimes A^{Q}_{4}$. The triplet modular forms with weight $2$ are
denoted as $Y(\tau_{q})$ and $Y(\tau_{e})$, which are different for quarks and
charged leptons because $\tau_{q}$ and $\tau_{e}$ are different. In order to
discuss relevant operators, we take a $A_{4}$ modular model, which leads to
the successful fermion mass matrices. Suppose that three left-handed quark and
lepton doublets are of a triplet of the $A_{4}$ group. The three right-handed
quarks and charged leptons are different singlets of $A_{4}$. On the other
hand, the Higgs doublets are supposed to be singlets of $A_{4}$. The generic
assignments of representations and modular weights to the fields are presented
in Table 1, where right-handed up-type quarks are omitted since those are not
necessary in following discussions.

| $Q_{L}$ | $(d^{c}_{R},\,s^{c}_{R},\,b^{c}_{R})$ | $L_{L}$ | $(e^{c}_{R},\,\mu^{c}_{R},\,\tau^{c}_{R})$ | $H_{d}$ |  $Y(\tau_{q})$, | $Y(\tau_{e})$
---|---|---|---|---|---|---|---
$SU(2)$ | $\bf 2$ | $\bf 1$ | $\bf 2$ | $\bf 1$ | $\bf 2$ | $\bf 1$
$A_{4}$ | $\bf 3$ | (1, 1′′, 1′) | $\bf 3$ | (1, 1′′, 1′) | $\bf 1$ | $\bf 3$
$k$ | $2$ | $(0,\ 0,\ 0)$ | $2$ | $(0,\,0,\,0)$ | 0 | $2$
Table 1: The assignment of $A_{4}$ representations and weights $k$ for down-
type quarks, charged leptons, down-type Higgs doublet and the modular forms.

We discuss the semileptonic 4-fermion operators, which are categorized as

$\displaystyle[\,\bar{E}_{L}\Gamma E_{L}\,][\,\bar{D}_{L}\Gamma D_{L}\,]$
$\displaystyle:Q_{\ell q}^{(1)},Q_{\ell q}^{(3)}\,,$
$\displaystyle[\,\bar{E}_{R}\Gamma E_{R}\,][\,\bar{D}_{R}\Gamma D_{R}\,]$
$\displaystyle:Q_{ed}\,,$ $\displaystyle[\,\bar{E}_{L}\Gamma
E_{L}][\,\bar{D}_{R}\Gamma D_{R}\,]$ $\displaystyle:Q_{\ell d}\,,$ (3.6)
$\displaystyle[\,\bar{E}_{R}\Gamma E_{R}\,][\,\bar{D}_{L}\Gamma D_{L}\,]$
$\displaystyle:Q_{qe}\,,$ $\displaystyle[\,\bar{E}_{L}\Gamma
E_{R}\,][\,\bar{D}_{R}\Gamma D_{L}\,]$ $\displaystyle:Q_{\ell edq}\,,$

where $L$ and $R$ denote the left-handed and the right-handed fields, and
$\Gamma$ denotes a generic combination of Dirac matrices, color and
$SU(2)_{L}$ generators, which play no role as far as the flavor structure is
concerned. Corresponding SMEFT operators $Q$ of which explicit expression are
shown in Appendix LABEL:App:SMEFT, are also listed.

First of all, let us construct the modular $A_{4}$-invariant 4-fermion
operators $[\,\bar{E}_{L}\Gamma E_{R}\,][\,\bar{D}_{R}\Gamma D_{L}\,]$ by
using the holomorphic and anti-holomorphic modular forms:

$\displaystyle[\,\bar{E}_{L}\Gamma E_{R}\,][\,\bar{D}_{R}\Gamma
D_{L}\,]\Rightarrow[\,\bar{E}_{L}Y^{*}(\tau_{e})\Gamma
E_{R}\,]_{1}[\,\bar{d}_{R}\Gamma Y(\tau_{q})d_{L}\,]_{1}\,,$ (3.7)

where the subscript $1$ denotes the $A_{4}$ trivial singlet. After the
decomposition of $A_{4}$ tensor product, we obtain coefficients for each
operator, which are written explicitly in the following sections. From the
viewpoint of the Ansatz, $y^{(4)}=y^{(3)}y^{(3)}$, these $[\,\bar{E}_{L}\Gamma
E_{R}\,][\,\bar{D}_{R}\Gamma D_{L}\,]$ operators would be constructed in terms
of Yukawa couplings with the Higgs fields. Thus, we assume that these
operators include the same coefficients as the Yukawa couplings, i.e. mass
matrices, as written explicitly later.

Next, the $A_{4}$ modular-invariant semileptonic 4-fermion operators
$[\,\bar{E}_{R}\Gamma E_{R}][\,\bar{D}_{R}\Gamma D_{R}\,]$ can be written by

$\displaystyle[\,\bar{E}_{R}\Gamma E_{R}\,][\,\bar{D}_{R}\Gamma
D_{R}\,]\Rightarrow\left[\,\sum_{i=1}^{3}r_{ei}\ \bar{e}_{iR}\Gamma
e_{iR}\,\right]\left[\,\sum_{i=1}^{3}r_{qi}\ \bar{d}_{iR}\Gamma
d_{iR}\,\right]\,,$ (3.8)

where $r_{ei}$ and $r_{qi}$ are arbitrary real constants. From the viewpoint
of the Ansatz (1.1), $y^{(4)}=y^{(3)}y^{(3)}$, these operators would be
constructed in terms of gauge couplings $g$ as $y^{(3)}\sim g$. Therefore, one
expects $r_{e(q)1}=r_{e(q)2}=r_{e(q)3}$, which do not lead to the FC
processes. Similarly, for the other operators, one of possible modular
$A_{4}$-invariant operators in our Ansatz (1.1) would be the type like
4-fermion operators mediated by gauge bosons. 222Note that our kinetic terms
(2.7) for left-handed fermions are not canonical. They couple with gauge
bosons as $\frac{1}{(\tau_{e}-\bar{\tau}_{e})^{2}}A^{\mu}[\,\bar{E}_{L}\Gamma
E_{L}\,]_{1}$ and
$\frac{1}{(\tau_{q}-\bar{\tau}_{q})^{2}}A^{\mu}[\,\bar{D}_{L}\Gamma
D_{L}\,]_{1}$. However, they do not lead to the FC processes similar to the
$[\,\bar{E}_{R}\Gamma E_{R}\,][\,\bar{D}_{R}\Gamma D_{R}\,]$ operator.
Operators including holomorphic and anti-holomorphic modular forms would lead
to the FC processes. Hence, another possibility for $A_{4}$ modular-invariant
semileptonic 4-fermion operators are constructed by using the holomorphic and
anti-holomorphic modular forms:

$\displaystyle[\,\bar{E}_{L}\Gamma E_{L}\,][\,\bar{D}_{L}\Gamma
D_{L}\,]\Rightarrow[\,\bar{E}_{L}Y^{*}(\bar{\tau_{e}})\Gamma
Y(\tau_{e})E_{L}\,]_{1}[\,\bar{D}_{L}Y^{*}(\bar{\tau_{q}})\Gamma
Y(\tau_{q})D_{L}\,]_{1}\,,$ (3.9) $\displaystyle[\,\bar{E}_{L}\Gamma
E_{L}][\,\bar{D}_{R}\Gamma
D_{R}\,]\Rightarrow[\,\bar{E}_{L}Y^{*}(\bar{\tau_{e}})\Gamma
Y(\tau_{e})E_{L}\,]_{1}\left[\,\sum_{i=1}^{3}r_{qi}\,\bar{d}_{iR}\Gamma
d_{iR}\right]\,,$ (3.10) $\displaystyle[\,\bar{E}_{R}\Gamma
E_{R}\,][\,\bar{D}_{L}\Gamma
D_{L}\,]\Rightarrow\left[\,\sum_{e_{i}=e,\,\mu,\,\tau}r_{e_{i}}\bar{e}_{iR}\Gamma
e_{iR}\,\right]\left[\,\bar{D}_{L}Y^{*}(\bar{\tau_{q}})\Gamma
Y(\tau_{q})D_{L}\,\right]_{1}\,.$ (3.11)

After decomposition of $A_{4}$ tensor products, we will give coefficients
explicitly in the following sections. These operators could be consistent with
the Ansatz, but the mode $m$ in Eq. (1.1) is unknown. Then, many free
parameters, which are not related with couplings in the renormalizable SM
Lagrangian, appear in these operators. At any rate, these operators are
$A_{4}$ modular invariant. The above $[\,\bar{E}_{L}\Gamma
E_{L}][\,\bar{D}_{R}\Gamma D_{R}\,]$ and $[\,\bar{E}_{R}\Gamma
E_{R}][\,\bar{D}_{L}\Gamma D_{L}\,]$ operators do not lead to the FC processes
in the quark sector and lepton sector, respectively. In the following
sections, we concentrate on these $[\,\bar{E}_{L}\Gamma
E_{R}][\,\bar{D}_{R}\Gamma D_{L}\,]$, $[\,\bar{E}_{R}\Gamma
E_{L}][\,\bar{D}_{L}\Gamma D_{R}\,]$ and $[\,\bar{E}_{L}\Gamma
E_{L}][\,\bar{D}_{L}\Gamma D_{L}\,]$ operators leading to the FC processes as
well as dipole operators. Moreover, 4-quark operators are also constructed in
a similar way.

##  4 Bilinear fermion operators $[\,\bar{D}_{R}\Gamma D_{L}\,]$ and
$[\,\bar{D}_{L}\Gamma D_{R}\,]$

The 4-fermion scalar operator $[\,\bar{E}_{L}E_{R}\,][\,\bar{D}_{R}D_{L}\,]$
does not appear at the tree level in the standard model (SM). Indeed, it is
not allowed by the exact $U(3)$ flavor symmetry. On the other hand, it appears
in three-point couplings of two fermions and modular forms in the modular
flavor symmetry. Moreover, due to the modular weights, they define a clear
power counting of modular forms.

Since the 4-fermion operators are given by the products of bilinear fermion
operators, interesting features from Ansatz (1.1) appear in the bilinear
fermion operators. For example, the 3-point couplings are realized in terms of
the modular forms typically in $[\,\bar{D}_{R}\Gamma D_{L}\,]$ operators.

In this section, we discuss the bilinear operators of quarks,
$[\,\bar{D}_{R}\Gamma D_{L}\,]$ and $[\,\bar{D}_{L}\Gamma D_{R}\,]$, and
corresponding ones of charged leptons. Those correspond the SMEFT operators
$Q$ of which explicit expression are shown in Appendix LABEL:App:SMEFT as
follows:

$\displaystyle[\,\bar{D}_{L}\Gamma D_{R}\,]$
$\displaystyle:Q_{dH},Q_{G},Q_{dW},Q_{dB}\,,$
$\displaystyle[\,\bar{D}_{R}\Gamma D_{L}\,]$
$\displaystyle:Q_{dH}^{\dagger},Q_{G}^{\dagger},Q_{dW}^{\dagger},Q_{dB}^{\dagger}\,,$
$\displaystyle[\,\bar{E}_{L}\Gamma E_{R}\,]$
$\displaystyle:Q_{eH},Q_{eW},Q_{eB}\,,$ $\displaystyle[\,\bar{E}_{R}\Gamma
E_{L}\,]$
$\displaystyle:Q_{eH}^{\dagger},Q_{eW}^{\dagger},Q_{eB}^{\dagger}\,.$

###  4.1 $[\,\bar{D}_{R}\Gamma D_{L}\,]$ and $[\,\bar{D}_{L}\Gamma D_{R}\,]$
bilinears in the flavor space

At first, let us begin with discussing the holomorphic operator
$[\,\bar{D}_{R}\Gamma D_{L}\,]$ and anti- holomorphic operator
$[\,\bar{D}_{L}\Gamma D_{R}\,]$ in the flavor space. The magnitudes of $LR$
couplings are proportional to modular forms. Taking account of
$\bar{D}_{R}=(d^{c},s^{c},b^{c})$, we can decompose the operator in the base
of Eq. (2.8) for $S$ and $T$ as:

$\displaystyle[\,\bar{D}_{R}\Gamma D_{L}\,]$
$\displaystyle\Rightarrow[\,\bar{D}_{R}\Gamma Y({\tau_{q}})D_{L}\,]_{\bf 1}=$
$\displaystyle[\,{\alpha_{d}\,\bar{d}_{R}}\Gamma(Y_{1}({\tau_{q}})d_{L}+Y_{2}({\tau_{q}})b_{L}+Y_{3}({\tau_{q}})s_{L})_{1}+{\beta_{d}\,\bar{s}_{R}}\Gamma(Y_{3}({\tau_{q}})b_{L}+Y_{1}({\tau_{q}})s_{L}+Y_{2}({\tau_{q}})d_{L})_{\bf
1^{\prime}}$
$\displaystyle+{\gamma_{d}\,\bar{b}_{R}}\Gamma(Y_{2}({\tau_{q}})s_{L}+Y_{1}({\tau_{q}})b_{L}+Y_{3}({\tau_{q}})d_{L})_{\bf
1^{\prime\prime}}]\,,$ $\displaystyle[\,\bar{D}_{L}\Gamma D_{R}\,]$
$\displaystyle\Rightarrow[\,\bar{D}_{L}Y^{*}({\tau_{q}})\Gamma D_{R}\,]_{\bf
1}=$
$\displaystyle[\,{\alpha_{d}^{*}\,d_{R}}\Gamma(Y_{1}^{*}({\tau_{q}})\bar{d}_{L}+Y_{2}^{*}({\tau_{q}})\bar{b}_{L}+Y_{3}^{*}({\tau_{q}})\bar{s}_{L})_{\bf
1}+{\beta_{d}^{*}\,s_{R}}\Gamma(Y_{3}^{*}({\tau_{q}})\bar{b}_{L}+Y_{1}^{*}({\tau_{q}})\bar{s}_{L}+Y_{2}^{*}({\tau_{q}})\bar{d}_{L})_{\bf
1^{\prime\prime}}$
$\displaystyle+{\gamma_{d}^{*}\,b_{R}}\Gamma(Y_{2}^{*}({\tau_{q}})\bar{s}_{L}+Y_{1}^{*}({\tau_{q}})\bar{b}_{L}+Y_{3}^{*}({\tau_{q}})\bar{d}_{L})_{\bf
1^{\prime}}]\,,$ (4.1)

where the subscript $1$, $1^{\prime}$, $1^{\prime\prime}$ denote the $A_{4}$
singlets, respectively. The parameters $\alpha_{d}$, $\beta_{d}$ and
$\gamma_{d}$ are constants. These expressions are written in the matrix
representation as

$\displaystyle[\,\bar{D}_{R}\Gamma Y({\tau_{q}})D_{L}\,]_{\bf 1}$
$\displaystyle=(\bar{d}_{R},\bar{s}_{R},\bar{b}_{R})\Gamma\begin{pmatrix}\alpha_{d}&0&0\\\
0&\beta_{d}&0\\\
0&0&\gamma_{d}\end{pmatrix}\begin{pmatrix}Y_{1}(\tau_{q})&Y_{3}(\tau_{q})&Y_{2}(\tau_{q})\\\
Y_{2}(\tau_{q})&Y_{1}(\tau_{q})&Y_{3}(\tau_{q})\\\
Y_{3}(\tau_{q})&Y_{2}(\tau_{q})&Y_{1}(\tau_{q})\end{pmatrix}\begin{pmatrix}d_{L}\\\
s_{L}\\\ b_{L}\end{pmatrix}\,,$ (4.2)
$\displaystyle[\,\bar{D}_{L}Y^{*}({\tau_{q}})\Gamma D_{R}\,]_{\bf 1}$
$\displaystyle=(\bar{d}_{L},\bar{s}_{L},\bar{b}_{L})\Gamma\begin{pmatrix}Y^{*}_{1}(\tau_{q})&Y^{*}_{2}(\tau_{q})&Y^{*}_{3}(\tau_{q})\\\
Y^{*}_{3}(\tau_{q})&Y^{*}_{1}(\tau_{q})&Y^{*}_{2}(\tau_{q})\\\
Y^{*}_{2}(\tau_{q})&Y^{*}_{3}(\tau_{q})&Y^{*}_{1}(\tau_{q})\end{pmatrix}\begin{pmatrix}\alpha_{d}^{*}&0&0\\\
0&\beta_{d}^{*}&0\\\ 0&0&\gamma_{d}^{*}\end{pmatrix}\begin{pmatrix}d_{R}\\\
s_{R}\\\ b_{R}\end{pmatrix}\,.$ (4.3)

It is useful to compare them with the down-type quark mass matrix $M_{d}$ in
the assignment of Table 1. The mass matrix is given in terms of weight 2
modular forms as:

$\displaystyle M_{d}$ $\displaystyle=v_{d}\begin{pmatrix}\alpha_{d(m)}&0&0\\\
0&\beta_{d(m)}&0\\\
0&0&\gamma_{d(m)}\end{pmatrix}\begin{pmatrix}Y_{1}(\tau_{q})&Y_{3}(\tau_{q})&Y_{2}(\tau_{q})\\\
Y_{2}(\tau_{q})&Y_{1}(\tau_{q})&Y_{3}(\tau_{q})\\\
Y_{3}(\tau_{q})&Y_{2}(\tau_{q})&Y_{1}(\tau_{q})\end{pmatrix}_{RL}\,,$ (4.4)

where the VEV of the Higgs field $H_{d}$ is denoted by $v_{d}$. Parameters
$\alpha_{d(m)}$, $\beta_{d(m)}$, $\gamma_{d(m)}$ can be taken to be real
constants. Since the bilinear operators appear in four-field operators, it is
reasonable to assume

$\displaystyle\alpha_{d}=c\alpha_{d(m)},\qquad\beta_{d}=c\beta_{d(m)},\qquad\gamma_{d}=c\gamma_{d(m)},$
(4.5)

from the viewpoint of the Ansatz Eq. (1.1), where the mode $m$ may correspond
to $H_{d}$. Here, $c$ is a common constant. Hereafter , we set $c=1$ for
simplicity. In this case, the matrix structure of bilinear operators
$[\,\bar{D}_{R}\Gamma D_{L}\,]$ appearing four-field operators is exactly the
same as the mass matrix. Obviously, the bilinear operator matrix is diagonal
in the basis for mass eigenstates. The FC processes such $b\to s$, $b\to d$,
$s\to d$ never happen. Hence, we obtain the very clear results in the modular
symmetric SMEFT with the Ansatz Eq. (1.1).

If the relation is violated, the situation would change drastically. When such
violations are small such as

$\displaystyle\alpha_{d}-\alpha_{d(m)}\ll\alpha_{d},\qquad\beta_{d}-\beta_{d(m)}\ll\beta_{d},\qquad\gamma_{d}-\gamma_{d(m)}\ll\gamma_{d},$
(4.6)

FC processes are still suppressed.

In what follows, we study larger violations such that $\alpha_{d}$,
$\beta_{d}$, and $\gamma_{d}$ are of ${\cal O}(\alpha_{d(m)})$, ${\cal
O}(\beta_{d(m)})$, and ${\cal O}(\gamma_{d(m)})$, respectively, but they are
different by factors from $\alpha_{d(m)}$, $\beta_{d(m)}$, and
$\gamma_{d(m)}$, i.e.

$\displaystyle\alpha_{d}-\alpha_{d(m)}\sim\alpha_{d},\qquad\beta_{d}-\beta_{d(m)}\sim\beta_{d},\qquad\gamma_{d}-\gamma_{d(m)}\sim\gamma_{d}.$
(4.7)

Unknown modes $m$ in Eq. (1.1) may contribute to such violations.

For the charged lepton operators $[\,\bar{E}_{R}\Gamma E_{L}\,]$ and
$[\,\bar{E}_{L}\Gamma E_{R}\,]$, we obtain the decompositions by replacing
$\tau_{q}$, $\alpha_{d}$, $\beta_{d}$ and $\gamma_{d}$ with $\tau_{e}$,
$\alpha_{e}$, $\beta_{e}$ and $\gamma_{e}$ in Eq. (4.1). As in the down-sector
quarks, the FC processes such as $\mu\to e$, $\tau\to e$, and $\tau\to\mu$
never occur when we assume $\alpha_{e}=\alpha_{e(m)}$,
$\beta_{e}=\beta_{e(m)}$, and $\gamma_{e}=\gamma_{e(m)}$, where
$\alpha_{e(m)}$, $\beta_{e(m)}$, and $\gamma_{e(m)}$ are parameters in the
charged lepton mass matrix as shown in Appendix B. This is the clear result in
the modular symmetric SMEFT with the Ansatz Eq. (1.1).

On the other hand, violations of the above parameter relation may lead to the
FC processes. In what follows, we study such violations such as

$\displaystyle\alpha_{e}-\alpha_{e(m)}\sim\alpha_{e},\qquad\beta_{e}-\beta_{e(m)}\sim\beta_{e},\qquad\gamma_{e}-\gamma_{e(m)}\sim\gamma_{e}.$
(4.8)

The $A_{4}$ flavor coefficients are given in Table 2 for relevant bilinear
operators of down-type quarks and charged leptons, where the overall strength
of the NP effect is not included. 333The overall strength of the NP effect is
omitted in coefficients of other Tables. Hereafter, without specifying them,
we denote $\alpha_{d,e(m)}$, $\beta_{d,e(m)}$, and $\gamma_{d,e(m)}$ by
$\alpha_{d,e}$, $\beta_{d,e}$, and $\gamma_{d,e}$, too, because they are the
same orders.

$\begin{matrix}{\bar{R}L}\\\ {\bar{L}R}\end{matrix}$ | $\begin{matrix}\bar{s}_{R}\Gamma b_{L}\\\ \bar{s}_{L}\Gamma b_{R}\end{matrix}$ | $\begin{matrix}\bar{d}_{R}\Gamma b_{L}\\\ \bar{d}_{L}\Gamma b_{R}\end{matrix}$ | $\begin{matrix}\bar{d}_{R}\Gamma s_{L}\\\ \bar{d}_{L}\Gamma s_{R}\end{matrix}$ | $\begin{matrix}\bar{\mu}_{R}\Gamma\tau_{L}\\\ \bar{\mu}_{L}\Gamma\tau_{R}\end{matrix}$ | $\begin{matrix}\bar{e}_{R}\Gamma\tau_{L}\\\ \bar{e}_{L}\Gamma\tau_{R}\end{matrix}$ | $\begin{matrix}\bar{e}_{R}\Gamma\mu_{L}\\\ \bar{e}_{L}\Gamma\mu_{R}\end{matrix}$ | $\begin{matrix}\bar{e}_{R}\Gamma e_{L}\\\ \bar{e}_{L}\Gamma e_{R}\end{matrix}$ | $\begin{matrix}\bar{\mu}_{R}\Gamma\mu_{L}\\\ \bar{\mu}_{L}\Gamma\mu_{R}\end{matrix}$
---|---|---|---|---|---|---|---|---
$\begin{matrix}{\rm Coeff.}\end{matrix}$ | $\begin{matrix}{\beta_{d}\,Y_{3}({\tau_{q}})}\\\ {\gamma_{d}\,Y_{2}^{*}({\tau_{q}})}\end{matrix}$ | $\begin{matrix}{\alpha_{d}\,Y_{2}({\tau_{q}})}\\\ {\gamma_{d}\,Y_{3}^{*}({\tau_{q}})}\end{matrix}$ | $\begin{matrix}{\alpha_{d}\,Y_{3}({\tau_{q}})}\\\ {\tilde{\beta}_{d}\,Y_{2}^{*}({\tau_{q}})}\end{matrix}$ | $\begin{matrix}{\beta_{e}\,Y_{3}({\tau_{e}})}\\\ {\gamma_{e}\,Y_{2}^{*}({\tau_{e}})}\end{matrix}$ | $\begin{matrix}{\alpha_{e}\,Y_{2}({\tau_{e}})}\\\ {\gamma_{e}\,Y_{3}^{*}({\tau_{e}})}\end{matrix}$ | $\begin{matrix}{\alpha_{e}\,Y_{3}({\tau_{e}})}\\\ {\beta_{e}\,Y_{2}^{*}({\tau_{e}})}\end{matrix}$ | $\begin{matrix}{\alpha_{e}\,Y_{1}({\tau_{e}})}\\\ {\alpha_{e}\,Y_{1}^{*}({\tau_{e}})}\end{matrix}$ | $\begin{matrix}{\beta_{e}\,Y_{1}({\tau_{e}})}\\\ {\beta_{e}\,Y_{1}^{*}({\tau_{e}})}\end{matrix}$
Table 2: $A_{4}$ flavor coefficients of the bilinear operators of down-type
quarks and charged leptons.

Therefore, the flavor structure of the these operators is predicted if the
modulus $\tau_{q,e}$ is fixed. It is noticed that above operators are given in
the flavor base. In order to move the mass eigenstate of the left-handed
quarks and leptons, we must fix the modulus $\tau_{q,e}$. The value of
$\tau_{q,e}$ depends on models, for example, Eqs. (B.5) and (B.10). The
interesting value of $\tau_{q,e}$ is fixed points of the modulus in the
fundamental domain of $SL(2,\mathbb{Z})$ since the moduli stabilization is
realized in a controlled way at nearby fixed points[123, 124]. Furthermore,
the fixed points are statistically favored in the string landscape [125]. We
discuss the phenomenology at nearby fixed points in the next subsection.

###  4.2 Diagonal matrix $M_{E}^{\dagger}M_{E}$ and $M_{q}^{\dagger}M_{q}$ at
fixed points

Residual symmetries arise whenever the VEV of the modulus $\tau$ breaks the
modular group $\overline{\Gamma}$ only partially. Here and in what follows, we
denote $\tau=\tau_{q,e}$ unless we specify it. Fixed points of modulus are the
case. There are only 2 inequivalent finite points in the fundamental domain of
$\overline{\Gamma}$, namely, $\tau=i$ and $\tau=\omega=-1/2+i\sqrt{3}/2$. The
first point is invariant under the $S$ transformation $\tau=-1/\tau$. In the
case of $A_{4}$ symmetry, the subgroup $\mathbb{Z}_{2}^{S}=\\{I,S\\}$ is
preserved at $\tau=i$. The second point is the left cusp in the fundamental
domain of the modular group, which is invariant under the $ST$ transformation
$\tau=-1/(\tau+1)$. Indeed, $\mathbb{Z}_{3}^{ST}=\\{I,ST,(ST)^{2}\\}$ is one
of subgroups of $A_{4}$ group. The right cusp at
$\tau=-\omega^{2}=1/2+i\sqrt{3}/2$ is related to $\tau=\omega$ by the $T$
transformation. There is also infinite point $\tau=i\infty$, in which the
subgroup $\mathbb{Z}^{T}_{3}=\\{I,T,T^{2}\\}$ of $A_{4}$ is preserved. We
summarize at three cases of the transformation:

$\displaystyle S\ {\rm invariant}\ :\ \tau=i\,,\qquad ST\ {\rm invariant}\,:\
\tau=\omega\,,\quad T\ {\rm invariant}\ :\ \tau=i\infty\,.$ (4.11)

If a residual symmetry of $S$ and $T$ in $A_{4}$ is preserved in mass matrices
of leptons and quarks, we have commutation relations between the mass matrices
and the generator $G\equiv S,T,ST$ as:

$\displaystyle[M_{RL}^{\dagger}M_{RL},\,{G}]=0\,,$ (4.12)

where $M_{RL}$ denotes the mass matrix of charged leptons and quarks, $M_{E}$
and $M_{q}\,(q=u,d)$.

Then the mass matrices $M_{E}^{\dagger}M_{E}$ and $M_{q}^{\dagger}M_{q}$ could
be diagonal in the diagonal basis of G at the fixed points. Therefore, the
hierarchical structures of flavor mixing are easily realized near those fixed
points.

####  4.2.1 Mass matrix and operators $\bar{D}_{L}\Gamma D_{R}$,
$\bar{D}_{R}\Gamma D_{L}$ at the fixed point $\tau=i$

At $\tau=i$, holomorphic and anti-holomorphic modular forms of weight $2$ are
given as:  7: $\psi^{2}H^{2}D$ $(\bar{L}L)$ $(\bar{R}R)$ $(\bar{R}R^{\prime})$
lepton $Q_{H\ell}^{(1)}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{\ell}_{p}\gamma^{\mu}\ell_{r})$
$Q_{He}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{e}_{p}\gamma^{\mu}e_{r})$
$Q_{H\ell}^{(3)}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}^{I}H)(\bar{\ell}_{p}\tau^{I}\gamma^{\mu}\ell_{r})$
quark $Q_{Hq}^{(1)}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{q}_{p}\gamma^{\mu}q_{r})$
$Q_{Hu}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{u}_{p}\gamma^{\mu}u_{r})$
$Q_{Hud}$ \+ h.c.
$i(\tilde{H}^{\dagger}D_{\mu}H)(\bar{u}_{p}\gamma^{\mu}d_{r})$ $Q_{Hq}^{(3)}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}^{I}H)(\bar{q}_{p}\tau^{I}\gamma^{\mu}q_{r})$
$Q_{Hd}$
$(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{d}_{p}\gamma^{\mu}d_{r})$
Class 8: Fermion Quadrilinears $(\bar{L}L)(\bar{L}L)$ $(\bar{R}R)(\bar{R}R)$
$(\bar{L}L)(\bar{R}R)$ semileptonic $Q_{\ell q}^{(1)}$
$(\bar{\ell}_{p}\gamma_{\mu}\ell_{r})(\bar{q}_{s}\gamma^{\mu}q_{t})$ $Q_{eu}$
$(\bar{e}_{p}\gamma_{\mu}e_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$ $Q_{\ell u}$
$(\bar{\ell}_{p}\gamma_{\mu}\ell_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$ $Q_{\ell
q}^{(3)}$
$(\bar{\ell}_{p}\gamma_{\mu}\tau^{I}\ell_{r})(\bar{q}_{s}\gamma^{\mu}\tau^{I}q_{t})$
$Q_{ed}$ $(\bar{e}_{p}\gamma_{\mu}e_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$
$Q_{\ell d}$
$(\bar{\ell}_{p}\gamma_{\mu}\ell_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$ $Q_{qe}$
$(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{e}_{s}\gamma^{\mu}e_{t})$ 4-quark
$Q_{qq}^{(1)}$ $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{q}_{s}\gamma^{\mu}q_{t})$
$Q_{uu}$ $(\bar{u}_{p}\gamma_{\mu}u_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$
$Q_{qu}^{(1)}$ $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$
$Q_{qq}^{(3)}$
$(\bar{q}_{p}\gamma_{\mu}\tau^{I}q_{r})(\bar{q}_{s}\gamma^{\mu}\tau^{I}q_{t})$
$Q_{dd}$ $(\bar{d}_{p}\gamma_{\mu}d_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$
$Q_{qu}^{(8)}$
$(\bar{q}_{p}\gamma_{\mu}T^{A}q_{r})(\bar{u}_{s}\gamma^{\mu}T^{A}u_{t})$
$Q_{ud}^{(1)}$ $(\bar{u}_{p}\gamma_{\mu}u_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$
$Q_{qd}^{(1)}$ $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$
$Q_{ud}^{(8)}$
$(\bar{u}_{p}\gamma_{\mu}T^{A}u_{r})(\bar{d}_{s}\gamma^{\mu}T^{A}d_{t})$
$Q_{qd}^{(8)}$
$(\bar{q}_{p}\gamma_{\mu}T^{A}q_{r})(\bar{d}_{s}\gamma^{\mu}T^{A}d_{t})$
4-lepton $Q_{\ell\ell}$
$(\bar{\ell}_{p}\gamma_{\mu}\ell_{r})(\bar{\ell}_{s}\gamma^{\mu}\ell_{t})$
$Q_{ee}$ $(\bar{e}_{p}\gamma_{\mu}e_{r})(\bar{e}_{s}\gamma^{\mu}e_{t})$
$Q_{\ell e}$
$(\bar{\ell}_{p}\gamma_{\mu}\ell_{r})(\bar{e}_{s}\gamma^{\mu}e_{t})$
$(\bar{L}R)(\bar{R}L)$ \+ h.c. $(\bar{L}R)(\bar{L}R)$ \+ h.c. semi-leptonic
$Q_{\ell edq}$ $(\bar{\ell}_{p}^{j}e_{r})(\bar{d}_{s}q_{tj})$ $Q_{\ell
equ}^{(1)}$ $(\bar{\ell}_{p}^{j}e_{r})\epsilon_{jk}(\bar{q}_{s}^{k}u_{t})$
4-quark $Q_{quqd}^{(1)}$
$(\bar{q}_{p}^{j}u_{r})\epsilon_{jk}(\bar{q}_{s}^{k}d_{t})$ $Q_{quqd}^{(8)}$
$(\bar{q}_{p}^{j}T^{A}u_{r})\epsilon_{jk}(\bar{q}_{s}^{k}T^{A}d_{t})$ $Q_{\ell
equ}^{(3)}$
$(\bar{\ell}_{p}^{j}\sigma_{\mu\nu}e_{r})\epsilon_{jk}(\bar{q}_{s}^{k}\sigma^{\mu\nu}u_{t})$
Table 12: List of all fermionic SMEFT operators in the Warsaw basis [2]. The
division in classes is adopted from Ref. [3]. The $p,r,s,t$ are flavor index,
and $j,k$ stand for SU(2) index. The operator classes 1–4 without fermion
fields are irrelevant in this paper, and not listed here.

##  Appendix B A model of mass matrices in the $A_{4}$ modular symmetry

We present a viable model for quarks [71]. Suppose that three left-handed
quark doublets are of a triplet of the $A_{4}$ group. The three right-handed
quarks are three different singlets of $A_{4}$. On the other hand, the Higgs
doublets are supposed to be singlets of $A_{4}$. The generic assignments of
representations and modular weights to the minimal supersymmetric standard
model (MSSM) fields are presented in Table 13, where weight 2 and 6 modular
forms are presented.

| $Q_{L}$ | $(d^{c}_{R},\,s^{c}_{R},\,b^{c}_{R})$ | $(u^{c}_{R},\,c^{c}_{R},\,t^{c}_{R})$ | $H_{q}$ | $Y_{\bf 3}^{(2)}$ | $Y_{\bf 3}^{(6)}$ | $Y_{\bf 3^{\prime}}^{(6)}$
---|---|---|---|---|---|---|---
$SU(2)$ | $\bf 2$ | $\bf 1$ | $\bf 1$ | $\bf 2$ | $\bf 1$
$A_{4}$ | $\bf 3$ | (1, 1′′, 1′) | (1, 1′′, 1′) | $\bf 1$ | $\bf 3$
$k$ | $2$ | $(0,\ 0,\ 0)$ | $(4,4,4)$ | 0 | $2$ | $6$ | $6$
Table 13: An example of assignments of weights $k$ for quarks and modular
forms.

Then, up-type $M_{u}$ and down-type $M_{d}$ quark mass matrices are given,
respectively as:

$\displaystyle M_{u}$
$\displaystyle=v_{u}\left[\begin{pmatrix}\alpha_{u(m)}&0&0\\\
0&\beta_{u(m)}&0\\\
0&0&\gamma_{u(m)}\end{pmatrix}\begin{pmatrix}Y_{1}^{(6)}&Y_{3}^{(6)}&Y_{2}^{(6)}\\\
Y_{2}^{(6)}&Y_{1}^{(6)}&Y_{3}^{(6)}\\\
Y_{3}^{(6)}&Y_{2}^{(6)}&Y_{1}^{(6)}\end{pmatrix}+\begin{pmatrix}\alpha^{\prime}_{u(m)}&0&0\\\
0&\beta^{\prime}_{u(m)}&0\\\
0&0&\gamma^{\prime}_{u(m)}\end{pmatrix}\begin{pmatrix}Y_{1}^{{}^{\prime}(6)}&Y_{3}^{{}^{\prime}(6)}&Y_{2}^{{}^{\prime}(6)}\\\
Y_{2}^{{}^{\prime}(6)}&Y_{1}^{{}^{\prime}(6)}&Y_{3}^{{}^{\prime}(6)}\\\
Y_{3}^{{}^{\prime}(6)}&Y_{2}^{{}^{\prime}(6)}&Y_{1}^{{}^{\prime}(6)}\end{pmatrix}\right]_{RL}$
$\displaystyle=v_{u}\begin{pmatrix}\alpha_{u(m)}&0&0\\\ 0&\beta_{u(m)}&0\\\
0&0&\gamma_{u(m)}\end{pmatrix}\begin{pmatrix}\tilde{Y}_{1}^{(6)}&\tilde{Y}_{3}^{(6)}&\tilde{Y}_{2}^{(6)}\\\
\tilde{Y}_{2}^{(6)}&\tilde{Y}_{1}^{(6)}&\tilde{Y}_{3}^{(6)}\\\
\tilde{Y}_{3}^{(6)}&\tilde{Y}_{2}^{(6)}&\tilde{Y}_{1}^{(6)}\end{pmatrix}_{RL}\,,$
$\displaystyle M_{d}$ $\displaystyle=v_{d}\begin{pmatrix}\alpha_{d(m)}&0&0\\\
0&\beta_{d(m)}&0\\\
0&0&\gamma_{d(m)}\end{pmatrix}\begin{pmatrix}Y_{1}&Y_{3}&Y_{2}\\\
Y_{2}&Y_{1}&Y_{3}\\\ Y_{3}&Y_{2}&Y_{1}\end{pmatrix}_{RL}\,,$ (B.5)

where $\tilde{Y}_{i}^{(6)}\equiv Y_{i}^{(6)}+g_{i}Y_{i}^{{}^{\prime}(6)}$ with
$g_{u1}=\alpha^{\prime}_{u(m)}/\alpha_{u(m)}$,
$g_{u2}=\beta^{\prime}_{u(m)}/\beta_{u(m)}$,
$g_{u3}=\gamma^{\prime}_{u(m)}/\gamma_{u(m)}$ and
$g_{q}\equiv\alpha_{q(m)}^{\prime}/\alpha_{q(m)}$. The VEV of the Higgs field
$H_{q}$ is denoted by $v_{q}$. Parameters $\alpha_{q}$, $\beta_{q}$,
$\gamma_{q}$ can be taken to be real, on the other hand, $g_{u1}$, $g_{u2}$,
$g_{u3}$ and $g_{u}$ are complex parameters.

For the lepton sector, we also present mass matrices of a successful model
[77], where the neutrino mass matrix is given in terms of weight 4 modular
forms by using Weinberg operator. The assignments of representations and
modular weights to the lepton fields are presented in Table 14.

| $L_{L}$ | $(e^{c}_{R},\mu^{c}_{R},\tau^{c}_{R})$ | $H_{q}$ | $Y_{\bf r}^{\rm(2)}$ | $Y_{\bf r}^{\rm(4)}$
---|---|---|---|---|---
$SU(2)$ | $\bf 2$ | $\bf 1$ | $\bf 2$ | $\bf 1$
$A_{4}$ | $\bf 3$ | (1, 1′′, 1′) | $\bf 1$ | $\bf 3$ | $\bf\\{3,1,1^{\prime}\\}$
$k$ | $2$ | $(0,\ 0,\ 0)$ | 0 | $2$ | $4$
Table 14:  Representations and weights $k$ for MSSM fields and modular forms
of weight $2$ and $4$.

The charged lepton mass matrix and neutrino ones are given as:

$\displaystyle M_{E}=v_{d}\begin{pmatrix}\alpha_{e(m)}&0&0\\\
0&\beta_{e(m)}&0\\\
0&0&\gamma_{e(m)}\end{pmatrix}\begin{pmatrix}Y_{1}&Y_{3}&Y_{2}\\\
Y_{2}&Y_{1}&Y_{3}\\\ Y_{3}&Y_{2}&Y_{1}\end{pmatrix}_{RL},$ $\displaystyle
M_{\nu}=\frac{v_{u}^{2}}{\Lambda}\left[\begin{pmatrix}2Y_{1}^{(4)}&-Y_{3}^{(4)}&-Y_{2}^{(4)}\\\
-Y_{3}^{(4)}&2Y_{2}^{(4)}&-Y_{1}^{(4)}\\\
-Y_{2}^{(4)}&-Y_{1}^{(4)}&2Y_{3}^{(4)}\end{pmatrix}+g^{\nu}_{1}Y_{\bf
1}^{\rm(4)}\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}+g^{\nu}_{2}Y_{\bf
1^{\prime}}^{\rm(4)}\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\right]\,,$ (B.10)

respectively, where $\alpha_{e(m)}$, $\beta_{e(m)}$ and $\gamma_{e(m)}$ are
real parameters and $g_{1}^{\nu}$, $g_{2}^{\nu}$ are supposed to be real.

##  Appendix C Unitary transformation of $S$ and $ST$ and mass matrix

The mass matrix is transformed by the unitary transformation, which transforms
the generator $S$ and $ST$. We discuss details at the fixed points $\tau=i$
and $\tau=\omega$.

###  C.1 Diagonal base of $S$ for $A_{4}$ triplet

The generators of $A_{4}$ group for the triplet are:

$\displaystyle\begin{aligned} S=\frac{1}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix},\end{aligned}\qquad\begin{aligned}
T=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\end{aligned}$ (C.1)

where $\omega=e^{i\frac{2}{3}\pi}$ for a triplet. The eigenvalues of $S$ is
$(1\,,-1\,,-1)$. This is the diagonal base of $T$.

In order to present the mass matrices in the diagonal base of $S$, we move to
the diagonal base of $S$ as follows:

$\displaystyle\begin{aligned}
U_{S1}\,{S}\,U_{S1}^{\dagger}=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\
0&0&-1\end{pmatrix},\quad
U_{S2}\,{S}\,U_{S2}^{\dagger}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{pmatrix},\quad
U_{S3}\,{S}\,U_{S3}^{\dagger}=\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\
0&0&1\end{pmatrix},\end{aligned}$ (C.2)

where

$\displaystyle\begin{aligned} U_{Si}\equiv P_{i}\begin{pmatrix}\
\frac{2}{\sqrt{6}}&-\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{6}}\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\\
0&-\frac{1}{\sqrt{2}}&\ \frac{1}{\sqrt{2}}\end{pmatrix},\quad
P_{1}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\ \
P_{2}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix},\ \
P_{3}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}\,.\end{aligned}$
(C.3)

Then, the generator $T$ is not anymore diagonal. However, the eigenvalue $-1$
of $S$ is degenerated, there is a freedom of the rotation between
corresponding rows and between columns.

As seen in subsection C.3, the Dirac mass matrix $M_{RL}$ is transformed as:

$\displaystyle\hat{M}_{RL}=M_{RL}U_{Si}^{\dagger}\,,$ (C.4)

If there is a residual symmetry of $A_{4}$ in the Dirac mass matrix,
$\mathbb{Z}_{2}^{S}=\\{I,S\\}$, which is realized at the fixed point $\tau=i$
as presented in Eq. 4.11, the generator $S$ commutes with
$\hat{M}_{RL}^{\dagger}\hat{M}_{RL}$,

$\displaystyle\begin{aligned}
\left[\hat{M}_{RL}^{\dagger}\hat{M}_{RL}\,,\,S\right]=0\,.\end{aligned}$ (C.5)

Therefore, the mass matrix could be diagonal in the diagonal base of $S$.

###  C.2 Diagonal bases of $ST$ and $T$ for $A_{4}$ triplet

We can move $ST$ to the diagonal base by the six-type unitary transformations
$V_{ST}$ as follows:

$\displaystyle\begin{aligned}
U_{STi}\,{ST}\,U_{STi}^{\dagger}=P_{i}\begin{pmatrix}\omega^{2}&0&0\\\
0&\omega&0\\\ 0&0&1\end{pmatrix}P_{i}^{T},\end{aligned}$ (C.6)

where

$\displaystyle\begin{aligned}
U_{STi}=\frac{1}{3}P_{i}\begin{pmatrix}-2\omega^{2}&-2\omega&1\\\
-\omega^{2}&\ 2\omega&2\\\ \ 2\omega^{2}&-\omega&2\end{pmatrix},\ \
P_{4}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix},\ \
P_{5}=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{pmatrix},\ \
P_{6}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\end{pmatrix},\end{aligned}$ (C.7)

and $P_{i}(i=1,2,3)$ are given in Eq. (C.3). The order of eigenvalues of $ST$
depends on $P_{i}$. We have eigenvalues $\\{\omega^{2},\,\omega,\,1\\}$ for
$P_{1}$, $\\{\omega,\,\omega^{2},\,1\\}$ for $P_{2}$,
$\\{\omega^{2},\,1,\,\omega\\}$ for $P_{3}$, $\\{1,\,\omega,\,\omega^{2}\\}$
for $P_{4}$, $\\{1,\,\omega^{2},\,\omega\\}$ for $P_{5}$ and
$\\{\omega,\,1,\,\omega^{2}\\}$ for $P_{6}$, respectively. Thus, there are
independent six unitary transformations to move $ST$ to the diagonal base.

As seen in subsection C.3, the Dirac mass matrix $M_{RL}$ is transformed as:

$\displaystyle\hat{M}_{RL}=M_{RL}U_{STi}^{\dagger}\,.$ (C.8)

If there exists the residual symmetries of the $A_{4}$ group
$\mathbb{Z}_{3}^{ST}=\\{I,ST,(ST)^{2}\\}$ which is realized at the fixed point
$\tau=\omega$ as presented in Eq. (4.11), we have

$\displaystyle\begin{aligned}
\left[\hat{M}_{RL}^{\dagger}\hat{M}_{RL}\,,\,ST\right]=0\,,\end{aligned}$
(C.9)

which leads to the diagonal $\hat{M}_{RL}^{\dagger}\hat{M}_{RL}$ because $ST$
has three different eigenvalues.

On the other hand, the generator $T$ is already diagonal in the original base
of Eq. (C.1). If there exists the residual symmetries of the $A_{4}$ group
$\mathbb{Z}_{3}^{T}=\\{I,T,T^{2}\\}$, which is realized at $\tau=i\infty$,

$\displaystyle\begin{aligned}
\left[M_{RL}^{\dagger}M_{RL}\,,\,T\right]=0\,,\end{aligned}$ (C.10)

which also gives the diagonal $M_{RL}^{\dagger}M_{RL}$.

###  C.3 Mass matrix in the new bases of generators $S$ and $ST$

Define the new basis of generators, $\tilde{G}\ (G=S,ST)$ by a unitary
transformation $U$ as:

$\displaystyle\begin{aligned} \hat{G}=UGU^{\dagger}\ ,\end{aligned}$ (C.11)

where $\hat{G}$, $G$ and $U$ are $3\times 3$ matrices. Since the $A_{4}$
triplet transforms under the unitary transformation as:

$\displaystyle\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\end{pmatrix}_{\bf 3}\to
G\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\end{pmatrix}_{\bf
3}=U^{\dagger}\hat{G}U\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\end{pmatrix}_{\bf
3}\ .$ (C.12)

Thus, in the new basis, the $A_{4}$ triplet transforms as:

$\displaystyle\begin{pmatrix}\hat{a}_{1}\\\ \hat{a}_{2}\\\
\hat{a}_{3}\end{pmatrix}_{\bf 3}\to\hat{G}\ \begin{pmatrix}\hat{a}_{1}\\\
\hat{a}_{2}\\\ \hat{a}_{3}\end{pmatrix}_{\bf 3},$ (C.13)

where

$\displaystyle\begin{pmatrix}\hat{a}_{1}\\\ \hat{a}_{2}\\\
\hat{a}_{3}\end{pmatrix}_{\bf 3}=U\begin{pmatrix}a_{1}\\\ a_{2}\\\
a_{3}\end{pmatrix}_{\bf 3}.$ (C.14)

Let us rewrite the Dirac mass matrix $M_{RL}$ in the new base ($\hat{G}$) of
the triplet left-handed fields. Denoting $L$ and $\hat{L}$ to be triplets of
the left-handed fields in the base of $G$ and $\hat{G}$, respectively, and $R$
to be right-handed singlets, the Dirac mass matrix is written as:

$\displaystyle\begin{aligned}
\bar{R}M_{RL}L=\bar{R}M_{RL}U^{\dagger}\hat{L}\,,\end{aligned}$ (C.15)

where

$\displaystyle L=U^{\dagger}\hat{L}\,.$ (C.16)

Then, the Dirac mass matrix $\hat{M}_{RL}$ in the new base is given as:

$\displaystyle\hat{M}_{RL}=M_{RL}U^{\dagger}\,.$ (C.17)

##  Appendix D Mass matrix at nearby $\tau=i$

Both down-type quark mass matrix of Eq. (B.5) and the charged lepton mass
matrix of Eq. (B.10) are given in terms of weight $2$ modular forms.

Perform the unitary transformation of $D_{L}\to U_{S}D_{L}$, where
$U_{S}=U_{23}(15^{\circ})U_{S2}$ (see Appendix C.3)

$\displaystyle U_{S}=U_{23}(-15^{\circ})U_{S2}=\begin{pmatrix}1&0&0\\\ 0&\cos
15^{\circ}&\sin 15^{\circ}\\\ 0&-\sin 15^{\circ}&\cos 15^{\circ}\\\
\end{pmatrix}\,U_{S2}=\frac{1}{2\sqrt{3}}\begin{pmatrix}2&2&2\\\
\sqrt{3}+1&-2&\sqrt{3}-1\\\ \sqrt{3}-1&-2&\sqrt{3}+1\end{pmatrix}\,.$ (D.1)

Then, the down-type quark mass matrix at $\tau=i$ in Eq. (B.5) is simply given
as:

$\displaystyle
M_{d}\,=\frac{1}{2}v_{d}\begin{pmatrix}0&3(\sqrt{3}-1)\tilde{\alpha}_{d(m)}&-(3-\sqrt{3})\tilde{\alpha}_{d(m)}\\\
0&-3(\sqrt{3}-1)\tilde{\beta}_{d(m)}&-(3-\sqrt{3})\tilde{\beta}_{d(m)}\\\
0&0&2(3-\sqrt{3})\tilde{\gamma}_{d(m)}\end{pmatrix}_{RL}\,,$ $\displaystyle
M_{d}^{\dagger}M_{d}=\frac{1}{2}v_{d}^{2}\begin{pmatrix}0&0&0\\\
0&9(2-\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}+\tilde{\beta}_{d(m)}^{2})&3(3-2\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})\\\
0&3(3-2\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})&3(2-\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}+\tilde{\beta}_{d(m)}^{2}+4\tilde{\gamma}_{d(m)}^{2})\end{pmatrix}_{LL}\,,$
(D.2)

where $\tilde{\alpha}_{d(m)}=(6-3\sqrt{3})Y_{1}(i)\alpha_{d(m)}$,
$\tilde{\beta}_{d(m)}=(6-3\sqrt{3})Y_{1}(i)\beta_{d(m)}$ and
$\tilde{\gamma}_{d(m)}=(6-3\sqrt{3})Y_{1}(i)\gamma_{d(m)}$ and
$\tilde{\gamma}_{d(m)}$ is supposed to be much larger than
$\tilde{\alpha}_{d(m)}$ and $\tilde{\beta}_{d(m)}$. The rotation
$U_{23}(-15^{\circ})$ realizes that the (3,3) entry is much large than other
entries. Since two eigenvalues of $S$ are degenerate such as $(1,-1,-1)$,
there is still a freedom of the $2$–$3$ family rotation. Therefore,
$M_{d}^{\dagger}M_{d}$ could be diagonal after the small $2$–$3$ family
rotation of ${\cal
O}(\tilde{\alpha}_{d(m)}^{2}\tilde{/}\gamma_{d(m)}^{2},\,\tilde{\beta}_{d(m)}^{2}/\tilde{\gamma}_{d(m)}^{2})$.
The charged lepton mass matrix is the same one in Eq. (D.2) by replacing the
subscript $d$ with $e$.

Since the quark and lepton mass matrices cannot reproduce the observed CKM and
PMNS matrices at fixed points as discussed in Ref.[71]. Therefore, the
deviations from the fixed points are required to realize observed masses and
mixing angles.

Let us consider the small deviation from $\tau=i$. By using the approximate
modular forms of weight 2 at $\tau=i+\epsilon$, we present
$M_{d}^{\dagger}M_{d}$ including of order $\epsilon$. By the unitary
transformation $D_{L}\to U_{12}U_{S}D_{L}$, where $U_{12}$ is the first-second
family exchange,

$\displaystyle U_{12}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix}\,,$
(D.3)

the mass matrix of Eq. (D.2) is modified as:

$\displaystyle M_{d}^{\dagger}M_{d}\simeq
v_{d}^{2}\begin{pmatrix}\frac{9}{2}(2-\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}+\tilde{\beta}_{d(m)}^{2})&3(\sqrt{3}-2)(\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})\epsilon_{1}&\frac{3}{2}(3-2\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})\\\
3(\sqrt{3}-2)(\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})\epsilon_{1}^{*}&2(2-\sqrt{3})(\tilde{\gamma}_{d(m)}^{2}+\tilde{\alpha}_{d(m)}^{2}+\tilde{\beta}_{d(m)}^{2})|\epsilon_{1}|^{2}&(3-2\sqrt{3})(2\tilde{\gamma}_{d(m)}^{2}-\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})\epsilon_{1}^{*}\\\
\frac{3}{2}(3-2\sqrt{3})(\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})&(3-2\sqrt{3})(2\tilde{\gamma}_{d(m)}^{2}-\tilde{\alpha}_{d(m)}^{2}-\tilde{\beta}_{d(m)}^{2})\epsilon_{1}&\frac{3}{2}(2-\sqrt{3})(4\tilde{\gamma}_{d(m)}^{2}+\tilde{\alpha}_{d(m)}^{2}+\tilde{\beta}_{d(m)}^{2})\end{pmatrix}\,,$
(D.4)

where $\gamma_{d(m)}^{2}\gg\alpha_{d(m)}^{2},\beta_{d(m)}^{2}$ is taken and
$\epsilon_{2}=2\epsilon_{1}$ in Eq. (LABEL:epS12) is put. The extra
transformation $U_{12}$ is performed to keep the hierarchy among matrix
elements $(3,3)\gg(2,3)\gg(1,3)$ because the CKM mixing angles are
hierarchical.

Then, we can estimate the mixing matrix $U_{dm}$ of order ${\cal
O}(\epsilon_{1})$ to diagonarize $M_{d}^{\dagger}M_{d}$:

$\displaystyle U_{md}^{\dagger}M_{d}^{\dagger}M_{d}U_{md}={\rm
diag}(m_{d}^{2},m_{s}^{2},m_{b}^{2})\,,\qquad\quad
U_{md}\simeq\begin{pmatrix}1&s_{12}^{d}e^{i\eta}&0\\\
-s_{12}^{d}e^{-i\eta}&1&s_{23}^{d}\\\
s_{12}^{d}s_{23}^{d}e^{-i\eta}&-s_{23}^{d}&1\end{pmatrix}\,,$ (D.5)

where $\eta=2{\rm arg}\,[\epsilon_{1}]$. The mixing angle $s_{23}^{d}$ is of
order ${\cal O}(\epsilon_{1})$. On the other hand, $s_{12}^{d}$ depends on
$\tilde{\alpha}_{d(m)}^{2}/\tilde{\gamma}_{d(m)}^{2}$,
$\tilde{\beta}_{d(m)}^{2}/\tilde{\gamma}_{d(m)}^{2}$ and $\epsilon_{1}$. In
the numerical fit, $s_{12}^{d}\sim 0.1$ has been obtained [47].

On the other hand, the transformation of the charged leptons is still
$E_{L}\to U_{S}E_{L}$, where $U_{12}$ of Eq. (D.3) does not appear because the
large mixing angle between the second and third families is reproduced in the
neutrino mass matrix under this transformation, which is preferable for the
lepton mixing matrix. Then, at $\tau=i+\epsilon$, we have

$\displaystyle M_{E}^{\dagger}M_{E}\simeq
v_{d}^{2}\begin{pmatrix}2(2-\sqrt{3})(\tilde{\gamma}_{e(m)}^{2}+\tilde{\alpha}_{e(m)}^{2}+\tilde{\beta}_{e(m)}^{2})|\epsilon_{1}|^{2}&3(\sqrt{3}-2)(\tilde{\alpha}_{e(m)}^{2}-\tilde{\beta}_{e(m)}^{2})\epsilon_{1}^{*}&(3-2\sqrt{3})(2\tilde{\gamma}_{e(m)}^{2}-\tilde{\alpha}_{e(m)}^{2}-\tilde{\beta}_{e(m)}^{2})\epsilon_{1}^{*}\\\
3(\sqrt{3}-2)(\tilde{\alpha}_{e(m)}^{2}-\tilde{\beta}_{e(m)}^{2})\epsilon_{1}&\frac{9}{2}(2-\sqrt{3})(\tilde{\alpha}_{e(m)}^{2}+\tilde{\beta}_{e(m)}^{2})&\frac{3}{2}(3-2\sqrt{3})(\tilde{\alpha}_{e(m)}^{2}-\tilde{\beta}_{e(m)}^{2})\\\
(3-2\sqrt{3})(2\tilde{\gamma}_{e(m)}^{2}-\tilde{\alpha}_{e(m)}^{2}-\tilde{\beta}_{e(m)}^{2})\epsilon_{1}&\frac{3}{2}(3-2\sqrt{3})(\tilde{\alpha}_{e(m)}^{2}-\tilde{\beta}_{e(m)}^{2})&\frac{3}{2}(2-\sqrt{3})(4\tilde{\gamma}_{e(m)}^{2}+\tilde{\alpha}_{e(m)}^{2}+\tilde{\beta}_{e(m)}^{2})\end{pmatrix}\,,$
(D.6)

where $\gamma_{e(m)}^{2}\gg\alpha_{e(m)}^{2},\beta_{e(m)}^{2}$ is taken and
$\epsilon_{2}=2\epsilon_{1}$ in Eq. (LABEL:epS12) is put. It is remarked that
the phase of $\epsilon_{1}$ can be removed. That is a real matrix. Since
$\alpha_{e(m)}^{2}/\gamma_{e(m)}^{2}$ and $\beta_{e(m)}^{2}/\gamma_{e(m)}^{2}$
are smaller than $10^{-3}$, it is noticed that this matrix is a rank one
matrix in the limit of $\alpha_{e(m)}^{2}=\beta_{e(m)}^{2}=0$. Taking relevant
value of $\alpha_{e(m)}^{2}/\gamma_{e(m)}^{2}$,
$\beta_{e(m)}^{2}/\gamma_{e(m)}^{2}$ and $|\epsilon_{1}|$, desired lepton
masses could be obtained.

The mixing matrix $U_{me}$ to diagonarize $M_{E}^{\dagger}M_{E}$:

$\displaystyle U_{me}^{\dagger}M_{E}^{\dagger}M_{E}U_{me}={\rm
diag}(m_{e}^{2},m_{\mu}^{2},m_{\tau}^{2})\,,\qquad\quad
U_{me}\simeq\begin{pmatrix}1&s_{12}^{e}&s_{13}^{e}\\\ -s_{12}^{e}&1&0\\\
-s_{13}^{d}&0&1\end{pmatrix}\,,$ (D.7)

where $s_{13}^{e}$ is of order ${\cal O}(|\epsilon_{1}|)$. On the other hand,
$s_{12}^{e}$ depends on $\tilde{\alpha}_{e(m)}^{2}/\tilde{\gamma}_{e(m)}^{2}$,
$\tilde{\beta}_{e(m)}^{2}/\tilde{\gamma}_{e(m)}^{2}$ and $|\epsilon_{1}|$. In
the numerical fit, $s_{12}^{e}\sim 0.1$ has been obtained [77].

##  Appendix E Decomposition of products of triplets

We take the generators of $A_{4}$ group for the triplet as follows:

$\displaystyle\begin{aligned} S=\frac{1}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix},\end{aligned}\qquad\begin{aligned}
T=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\end{aligned}$ (E.1)

where $\omega=e^{i\frac{2}{3}\pi}$ for a triplet. In this base, the
multiplication rule is

$\displaystyle\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\end{pmatrix}_{\bf
3}\otimes\begin{pmatrix}b_{1}\\\ b_{2}\\\ b_{3}\end{pmatrix}_{\bf 3}$
$\displaystyle=\left(a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}\right)_{\bf
1}\oplus\left(a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}\right)_{{\bf 1}^{\prime}}$
$\displaystyle\oplus\left(a_{2}b_{2}+a_{1}b_{3}+a_{3}b_{1}\right)_{{\bf
1}^{\prime\prime}}$
$\displaystyle\oplus\frac{1}{3}\begin{pmatrix}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\end{pmatrix}_{{\bf
3}}\oplus\frac{1}{2}\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ a_{3}b_{1}-a_{1}b_{3}\end{pmatrix}_{{\bf 3}\ }\ ,$
$\displaystyle{\bf 1}\otimes{\bf 1}={\bf 1}\ ,\qquad$ $\displaystyle{\bf
1^{\prime}}\otimes{\bf 1^{\prime}}={\bf 1^{\prime\prime}}\ ,\qquad{\bf
1^{\prime\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1^{\prime}}\ ,\qquad{\bf
1^{\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1}\ ,$ (E.2)

where

$\displaystyle T({\bf 1^{\prime})}=\omega\,,\qquad T({\bf
1^{\prime\prime}})=\omega^{2}.$ (E.3)

More details are shown in the review [11, 12].

By using these multiplication rules, we have

$\displaystyle(YD_{L})_{\bf 1}$
$\displaystyle=Y_{1}d_{L1}+Y_{2}d_{3L}+Y_{3}d_{2L}\,,$
$\displaystyle(YD_{L})_{\bf 1^{\prime}}$
$\displaystyle=Y_{3}d_{3L}+Y_{1}d_{2L}+Y_{2}d^{L}_{1}\,,$
$\displaystyle(YD_{L})_{\bf 1^{\prime\prime}}$
$\displaystyle=Y_{2}d_{2L}+Y_{1}d_{3L}+Y_{3}d_{1L}\,,$ (E.4)
$\displaystyle(YD_{L})_{\bf 3s}=\left(\begin{array}[]{c}(YD_{L})_{3s1}\\\
(YD_{L})_{3s2}\\\
(YD_{L})_{3s3}\end{array}\right)=\frac{1}{3}\left(\begin{array}[]{c}2Y_{1}d_{1L}-Y_{2}d_{3L}-Y_{3}d_{2L}\\\
2Y_{3}d_{3L}-Y_{1}d_{2L}-Y_{2}d_{1L}\\\
2Y_{2}d_{2L}-Y_{1}d_{3L}-Y_{3}d_{1L}\end{array}\right)_{\bf 3s}\,,$ (E.11)
$\displaystyle(YD_{L})_{\bf 3a}=\left(\begin{array}[]{c}(YD_{L})_{3a1}\\\
(YD_{L})_{3a2}\\\
(YD_{L})_{3a3}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{c}Y_{2}d_{3L}-Y_{3}d_{2L}\\\
Y_{1}d_{2L}-Y_{2}d_{1L}\\\ Y_{3}d_{1L}-Y_{1}d_{3L}\end{array}\right)_{\bf
3a}\,,$ (E.18)

where the $A_{4}$ triplets $Y_{\bf 3}$ and $D_{L\bf 3}$ are abbreviated as $Y$
and $D_{L}$, respectively.

The similar formulae are obtained for$Y_{k}^{*}\bar{d}^{L}_{j}$, but it is
noticed that the exchange between $1^{\prime}$ and $1^{\prime\prime}$, and the
second entry and third one in the triplet representation, respectively,
through their complex conjugates. That is:

$\displaystyle(Y^{*}\bar{D}_{L})_{\bf 1}$
$\displaystyle=Y_{1}^{*}\bar{d}^{L}_{1}+Y_{2}^{*}\bar{d}^{L}_{3}+Y_{3}^{*}\bar{d}^{L}_{2}\,,$
$\displaystyle(Y^{*}\bar{D}_{L})_{\bf 1^{\prime\prime}}$
$\displaystyle=Y_{3}^{*}\bar{d}^{L}_{3}+Y_{1}^{*}\bar{d}^{L}_{2}+Y^{*}_{2}\bar{d}^{L}_{1}\,,$
$\displaystyle(Y^{*}\bar{D}_{L})_{\bf 1^{\prime}}$
$\displaystyle=Y_{2}^{*}\bar{d}^{L}_{2}+Y_{1}^{*}\bar{d}^{L}_{3}+Y_{3}^{*}\bar{d}^{L}_{1}\,,$
(E.19) $\displaystyle(Y^{*}\bar{D}_{L})_{\bf
3s}=\left(\begin{array}[]{c}(Y^{*}\bar{D}_{L})_{3s1}\\\
(Y^{*}\bar{D}_{L})_{3s2}\\\
(Y^{*}\bar{D}_{L})_{3s3}\end{array}\right)=\frac{1}{3}\left(\begin{array}[]{c}2Y_{1}^{*}\bar{d}_{1L}-Y_{2}^{*}\bar{d}_{3L}-Y_{3}^{*}\bar{d}_{2L}\\\
2Y_{2}^{*}\bar{d}_{2L}-Y_{1}^{*}\bar{d}_{3L}-Y_{3}^{*}\bar{d}_{1L}\\\
2Y_{3}^{*}\bar{d}_{3L}-Y_{1}^{*}\bar{d}_{2L}-Y_{2}^{*}\bar{d}_{1L}\end{array}\right)_{\bf
3s}\,,$ (E.26) $\displaystyle(Y^{*}\bar{D}_{L})_{\bf
3a}=\left(\begin{array}[]{c}(Y^{*}\bar{D}_{L})_{3a1}\\\
(Y^{*}\bar{d}_{L})_{3a2}\\\
(Y^{*}\bar{D}_{L})_{3a3}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{c}Y_{2}^{*}\bar{d}_{3L}-Y_{3}^{*}\bar{d}_{2L}\\\
Y_{3}^{*}\bar{d}_{1L}-Y_{1}^{*}\bar{d}_{3L}\\\
Y_{1}^{*}\bar{d}_{2L}-Y_{2}^{*}\bar{d}_{1L}\end{array}\right)_{\bf 3a}\,.$
(E.33)

By using these decompositions, we can calculate the product of following
triplets:

$\displaystyle[(Y^{*}\bar{D}_{L})_{\bf 3s}\otimes(YD_{L})_{\bf 3s}]_{\bf
3s}=\frac{1}{3}\left(\begin{array}[]{c}2(Y^{*}\bar{D}_{L})_{3s1}(YD_{L})_{3s1}-(Y^{*}\bar{D}_{L})_{3s2}(YD_{L})_{3s3}-(Y^{*}\bar{D}_{L})_{3s3}(YD_{L})_{3s2}\\\
2(Y^{*}\bar{D}_{L})_{3s3}(YD_{L})_{3s3}-(Y^{*}\bar{D}_{L})_{3s1}(YD_{L})_{3s2}-(Y^{*}\bar{D}_{L})_{3s2}(YD_{L})_{3s1}\\\
2(Y^{*}\bar{D}_{L})_{2s1}(YD_{L})_{2s1}-(Y^{*}\bar{D}_{L})_{3s1}(YD_{L})_{3s3}-(Y^{*}\bar{D}_{L})_{3s3}(YD_{L})_{3s1}\end{array}\right),$
(E.37) $\displaystyle[(Y^{*}\bar{D}_{L})_{\bf 3s}\otimes(YD_{L})_{\bf
3s}]_{\bf
3a}=\frac{1}{2}\left(\begin{array}[]{c}(Y^{*}\bar{D}_{L})_{3s2}(YD_{L})_{3s3}-(Y^{*}\bar{D}_{L})_{3s3}(YD_{L})_{3s2}\\\
(Y^{*}\bar{D}_{L})_{3s1}(YD_{L})_{3s2}-(Y^{*}\bar{D}_{L})_{3s2}(YD_{L})_{3s1}\\\
(Y^{*}\bar{D}_{L})_{3s3}(YD_{L})_{3s1}-(Y^{*}\bar{D}_{L})_{3s1}(YD_{L})_{3s3}\end{array}\right).$
(E.41)

Other six decompositions $[(Y^{*}\bar{D}_{L})_{\bf 3s}\otimes(YD_{L})_{\bf
3a}]_{\bf 3s}$, $[(Y^{*}\bar{D}_{L})_{\bf 3a}\otimes(YD_{L})_{\bf 3s}]_{\bf
3s}$, $[(Y^{*}\bar{D}_{L})_{\bf 3a}\otimes(YD_{L})_{\bf 3a}]_{\bf 3s}$,
$[(Y^{*}\bar{D}_{L})_{\bf 3s}\otimes(YD_{L})_{\bf 3a}]_{\bf 3a}$,
$[(Y^{*}\bar{D}_{L})_{\bf 3a}\otimes(YD_{L})_{\bf 3s}]_{\bf 3a}$ and
$[(Y^{*}\bar{D}_{L})_{\bf 3a}\otimes(YD_{L})_{\bf 3a}]_{\bf 3a}$ are written
similarly.

##  Appendix F $U(2)$ flavor symmetry

The $U(2)^{5}$ flavor symmetry is the subgroup of $U(3)^{5}$ and distinguishes
the first two families of fermions from the third one [7, 8, 9]. The flavor
symmetry is decomposed as

$\displaystyle U(2)^{5}=U(2)_{L}\otimes U(2)_{Q}\otimes U(2)_{E}\otimes
U(2)_{U}\otimes U(2)_{D}~{}.$ (F.1)

Under this symmetry, the first two families transform as doublets of the
$U(2)$ subgroups, whereas third family ones as a singlet. Then, the third-
family Yukawa couplings are allowed by the symmetry, and it provides a natural
explanation of why third-family Yukawa couplings are large. A minimal set of
$U(2)^{5}$ breaking terms (spurions) which reproduce the observed SM flavor
parameters, without tuning and with minimal size for the breaking terms, is
given by [7]

$\displaystyle V_{\ell}\sim\left(2,1,1,1,1\right)~{},\qquad
V_{q}\sim\left(1,2,1,1,1\right)~{},$
$\displaystyle\Delta_{e}\sim\left(2,1,\bar{2},1,1\right)~{},\qquad\Delta_{u}\sim\left(1,2,1,\bar{2},1\right)~{},\qquad\Delta_{d}\sim\left(1,2,1,1,\bar{2}\right)~{},$
(F.2)

where $V_{q,\ell}$ are complex two-vectors and $\Delta_{e,u,d}$ are complex
$2\times 2$ matrices. In terms of these spurions, the $3\times 3$ Yukawa
matrices can be decomposed as

$\displaystyle
Y_{e}=y_{\tau}\left(\begin{matrix}\Delta_{e}&x_{\tau}V_{\ell}\\\
0&1\end{matrix}\right),$ $\displaystyle
Y_{u}=y_{t}\left(\begin{matrix}\Delta_{u}&x_{t}V_{q}\\\
0&1\end{matrix}\right),$ $\displaystyle
Y_{d}=y_{b}\left(\begin{matrix}\Delta_{d}&x_{b}V_{q}\\\
0&1\end{matrix}\right),$ (F.3)

where $y_{\tau,t,b}$ and $x_{\tau,t,b}$ are free complex parameters, expected
to be of order $\mathcal{O}(1)$. Using the residual $U(2)^{5}$ invariance, we
can transform the spurions to the following explicit form (see Ref. [4])

$V_{q(\ell)}=e^{i\bar{\phi}_{q(\ell)}}\begin{pmatrix}0\\\
\epsilon_{q(l)}\end{pmatrix}~{},\quad\Delta_{e}=O_{e}^{\intercal}\,\begin{pmatrix}\delta^{\prime}_{e}&0\\\
0&\delta_{e}\end{pmatrix}~{},\quad\Delta_{u}=U_{u}^{\dagger}\begin{pmatrix}\delta^{\prime}_{u}&0\\\
0&\delta_{u}\end{pmatrix}~{},\quad\Delta_{d}=U_{d}^{\dagger}\begin{pmatrix}\delta^{\prime}_{d}&0\\\
0&\delta_{d}\end{pmatrix}~{},$ (F.4)

where $O$ and $U$ represent $2\times 2$ orthogonal and complex unitary
matrices, respectively

$\displaystyle O_{e}=\begin{pmatrix}c_{e}&s_{e}\\\
-s_{e}&c_{e}\end{pmatrix}\,,\qquad
U_{q}=\begin{pmatrix}c_{q}&s_{q}\,e^{i\alpha_{q}}\\\
-s_{q}\,e^{-i\alpha_{q}}&c_{q}\end{pmatrix}\,,$ (F.5)

with $s_{i}\equiv\sin\theta_{i}$ and $c_{i}\equiv\cos\theta_{i}$. The
$\epsilon_{i}$ and $\delta^{(\prime)}_{i}$ are small positive real parameters
controlling the overall size of the spurions. From the observed hierarchies of
the Yukawa couplings, it is expected that they have following order relation

$1\gg\epsilon_{i}\gg\delta_{i}\gg\delta^{\prime}_{i}>0$ (F.6)

where

$\displaystyle\epsilon_{i}=\mathcal{O}(y_{t}|V_{ts}|)=\mathcal{O}(10^{-1})~{},\delta_{i}=\mathcal{O}\left(\frac{y_{c}}{y_{t}},\frac{y_{s}}{y_{b}},\frac{y_{\mu}}{y_{\tau}}\right)=\mathcal{O}(10^{-2})~{},\delta^{\prime}_{i}=\mathcal{O}\left(\frac{y_{u}}{y_{t}},\frac{y_{d}}{y_{b}},\frac{y_{e}}{y_{\tau}}\right)=\mathcal{O}(10^{-3}).$
(F.7)

In terms of the spurion expressions discussed above, we can classify the
number of independent operators in the SMEFT with a $U(2)^{5}$ flavor
symmetry. Following the complete classification in Ref. [4], we focus on the
chirality-flipped $d_{i}\to d_{j}$ and $\ell_{i}\to\ell_{j}$ transitions in
this paper. Here, we classify the operators up to $\mathcal{O}(V^{3},\Delta
V)\sim\mathcal{O}(10^{-3})$, given the size of the spurions in Eq. (F.7). The
$b_{i}\to b_{j}$ transition allowed by different $U(2)$ breaking terms are
summarized in Table LABEL:tab:U2RLbs, where we follow the result of Table 4 in
Ref.[4]. The case of $\ell_{i}\to\ell_{j}$ transition is given in the same
way, and is summarised in Table LABEL:tab:leptonLRatU2. We denote with latin
(greek) letters the real (complex) couplings appearing in hermitian (non-
hermitian) structures. Terms with the same number of spurions are denoted by
the same latin or greek letter with different subscript.

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