#  $A_{4}$ modular invariance and the strong CP problem

S. T. Petcov 1,2111Also at: Institute of Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria. and M. Tanimoto 3
1INFN/SISSA, Via Bonomea 265, 34136 Trieste, Italy 2Kavli IPMU (WPI), UTIAS,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 3Department of
Physics, Niigata University, Ikarashi 2, Niigata 950-2181, Japan

( Abstract
We present simple effective theory of quark masses, mixing and CP violation
with level $N=3$ ($A_{4}$) modular symmetry, which provides solution to the
strong CP problem without the need for an axion. The vanishing of the strong
CP-violating phase $\bar{\theta}$ is ensured by assuming CP to be a
fundamental symmetry of the Lagrangian of the theory. The CP symmetry is
broken spontaneously by the vacuum expectation value (VEV) of the modulus
$\tau$. This provides the requisite large value of the CKM CP-violating phase
while the strong CP phase $\bar{\theta}$ remains zero or is tiny. Within the
considered framework we discuss phenomenologically viable quark mass matrices
with three types of texture zeros, which are realized by assigning both the
left-handed and right-handed quark fields to $A_{4}$ singlets ${\bf 1}$, ${\bf
1^{\prime}}$ and ${\bf 1^{\prime\prime}}$ with appropriate weights. The VEV of
$\tau$ is restricted to reproduce the observed CKM parameters. We discuss, in
particular, cases in which the modulus VEV is close to the fixed points $i$,
$\omega$ and $i\infty$, which are broadly consistent with the results on
modulus stabilisation. )

##  1 Introduction

The strong CP problem has been a puzzle in particle physics since the QCD
Lagrangian violates CP due to instanton effects [1, 2, 3, 4]. The CP violation
in QCD is described by the strong CP phase

$\displaystyle\bar{\theta}=\theta_{\rm QCD}+{\rm arg}\,{\rm
det}\,[M_{U}M_{D}]\,,$ (1)

where $M_{U}$ and $M_{D}$ denote the mass matrices of the up-type and the
down-type quarks, respectively, and $\theta_{\rm QCD}$ is the coefficient of
the topological charge term in the QCD Lagrangian:

$\displaystyle{\cal L}_{QCD}=\bar{Q}(i\ \cancel{D}-M_{Q})Q-\frac{1}{4}{\rm
Tr}\,G^{2}+\theta_{\rm QCD}\frac{g_{3}^{2}}{32\pi^{2}}{\rm Tr}\,G\tilde{G}\,.$
(2)

Here $G$ is the gluon field stress tensor and $Q$ is the multiplet of quark
fields with definite masses. While $\theta_{\rm QCD}$ and ${\rm arg}\,{\rm
det}\,[M_{U}M_{D}]$ are transformed into each other via chiral transformation,
the sum $\bar{\theta}$ is invariant.

The upper-bound of $\bar{\theta}$ is derived from the experimental upper bound
on the electric dipole moment (edm) of the neutron as $|\bar{\theta}|\lesssim
10^{-10}$ [5]. This bound is much smaller than the weak CP violation, for
example, the Jarlskog rephasing invariant $\sim 10^{-5}$ [6, 7]. Therefore,
the strong CP problem is the problem of understanding why $\bar{\theta}$ is so
small.

The most well known solution is the axion solution [8] (see also, e.g., [9]),
where $\bar{\theta}$ is a dynamical degree of freedom which is set to a small
value by a scalar potential. However, there have been no experimental hints
for the existence of the axion so far.

Another well known solution is provided by the Nelson-Barr model [10, 11],
where the CP symmetry is violated only by the mixings of Standard model (SM)
quarks with hypothetical extra heavy quarks, and the extended quark mass
matrix has a special structure with vanishing entries such that CP-violating
terms do not contribute to its determinant. Specific models, in which quark
mass matrices with real determinant and $\theta_{\rm QCD}=0$ still generate
the weak CP-violating (CPV) phase in the CKM quark mixing matrix have been
proposed in [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] as well.

The modular invariance opened up a new promising approach to the flavour
problem of quarks and leptons [24] (see also Refs. [25, 26, 27]). The strong
CP problem has been also discussed recently within the modular invariance
approach in Ref.[28]. In this study a simple effective theory of flavour and
CP symmetry breaking with vanishing $\bar{\theta}$ without the need for an
axion has been proposed. The analysis was done using the level $N=1$ full
modular group SL$(2,\mathbb{Z})$. A numerical example has shown that the
modular symmetry allows to solve the strong CP problem and to reproduce
correctly the quark masses and the CKM mixing matrix.

In this article we present alternative effective quark flavour models with
finite modular symmetry of level $N=3$ ($A_{4}$), which provide axion-less
solution to the strong CP problem. The $A_{4}$ modular symmetry has been
extensively used for understanding the origins of the quark and lepton
flavours [24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].
Finite groups including finite modular groups have been widely employed in
flavour model building (see, e.g., [25, 26, 44, 27, 45, 46, 47, 48, 49, 50,
51, 52] 222A rather complete list of articles on the modular invariance
approach to lepton and quark flavour problems is given in [52].  and the
reviews [53, 54, 55, 57, 58, 59, 60, 61, 62, 56, 63, 64]).

If CP is a fundamental symmetry of the Lagrangian, the QCD and the strong CPV
phases $\theta_{\rm QCD}$ and $\bar{\theta}$ will vanish. However, in order to
explain CP violation in weak interactions, the spontaneous breaking of the CP
symmetry has to generate the large measured value of the CPV phase in the CKM
matrix, while at the same time the strong CPV phase $\bar{\theta}$ should
still be zero or be tiny enough to be in agreement with experimental data on
the edm of the neutron. In other words, we have to look for a texture with
${\rm arg}\,{\rm det}\,[M_{U}M_{D}]=0$ with a realistic value for the CKM CPV
phase. Furthermore, as far as there is no accidental cancellation between
phases in the up-type and down-type mass matrices, ${\rm det}\,[M_{U}]$ and
${\rm det}\,[M_{D}]$ should be real, and positive by themselves.

Since constraints of $\bar{\theta}$ are very stringent, we need to take into
account the corrections to this parameter. The most important corrections are
333 The corrections to $\bar{\theta}$ from SM are known to be negligible [65,
66] [18, 28]:

  * •

Higher dimensional operators that spoil the structure of the mass matrices.

  * •

Corrections which are induced from supersymmetry (SUSY) breaking terms.

The first point has been addressed by introducing the modular symmetry within
the modular invariance approach to the quark (lepton) flavour problem [24], in
which the elements of the quark (charged lepton and neutrino) mass matrices
are modular forms of certain levels and weights that are holomorphic functions
of the modulus $\tau$. The modular forms are essentially non-perturbative
ones. Once the weight of the modular form is fixed there are no the higher
dimensional operator contributions. The symmetry is sharp.

The second point is the correction due to the SUSY breaking terms. This
correction has been discussed in the modular symmetry of flavours [28]. Since
the modular symmetry is in the framework of SUSY, the SUSY breaking could
contribute to $\bar{\theta}$ significantly. As far as SUSY is unbroken,
$\bar{\theta}$ cannot be generated radiatively. On the other hand, the SUSY
breaking sector, in principle, can introduce new sources of CP violation which
are model dependent. Assuming that SUSY is broken via gauge-mediation or
anomaly mediation below the mass scale of the modulus $\tau$, the
renomalization group and threshold corrections due to the SUSY breaking have
same flavour and CP structure as the SM ones [13]. Therefore, the corrections
to $\bar{\theta}$ are safely under control.

In the case of $A_{4}$ modular symmetry considered by us relevant texture
zeros of the quark mass matrices are realized if both left-handed (LH) and
right-handed (RH) quark fields are assigned to be $A_{4}$ singlets [67]. The
CP violation is generated only by the vacuum expectation value (VEV) of the
modulus $\tau$ [68]. By performing statistical analysis we obtain the values
of the VEV of $\tau$ that allow to reproduce the observed CP violation in the
quark sector.

It is known that the VEV of the modulus $\tau$ could be obtained by the
modulus stabilisation analysing relatively simple (supergravity-motivated)
modular- and CP-invariant potentials for the modulus $\tau$. In the modulus
stabilisation studies values of the VEV of $\tau$ close to the fixed points
[44] $i$, $\omega$ and $i\infty$ were found [69, 70, 71, 72, 73, 74, 75, 76,
77]. In view of this we search in this work for examples of $A_{4}$ modular
solutions to the strong CP problem for values of the VEV of $\tau$ close to
the fixed points.

The paper is organised as follows. In Section 2, we present quark mass
matrices with three types of texture zeros, which satisfy ${\rm arg}\,{\rm
det}\,[M_{Q}]=0$. In Section 3, we propose quark mass matrices with the
considered texture zeros in the case of $A_{4}$ modular symmetry and discuss
their CP violating phase structure. In Section 4, we show numerical examples
which allow to reproduce the observed quark masses, CKM mixing angles and CP-
violating phase for the VEV of $\tau$ close to the fixed points $i$, $\omega$
and $i\infty$. Section 5 is devoted to the summary and discussions. In
Appendix A the $A_{4}$ modular forms of weights up to 16 are presented. In
Appendix B we list the observed values of the quark masses, the CKM elements
and CPV phase, which serve as input for our numerical analyses. In Appendix C
we present numerically three examples of phenomenologically viable quark mass
matrices with texture zeros that allow to describe the data on quark masses,
mixing and CP violation.

##  2 Texture zeros and Real $\bf{\bf det}\,[M_{Q}]$

The texture zeros approach has a long history. In Ref.[79] Weinberg, assuming
the existence of two families of quarks, considered a mass matrix for the
down-type quark sector with zero (1,1) entry in the basis in which the up-type
quark mass matrix is diagonal. He further supposed that the down-type quark
mass matrix is symmetric. In this case the number of free parameters is
reduced to only two and hence he succeeded to predict the Cabibbo angle to be
$\sqrt{m_{d}/m_{s}}$, which is the so-called Gatto, Sartori, Tonin relation
[80]. Fritzsch extended the above approach to the three family case [81, 82].
Ramond, Roberts and Ross presented a systematic analysis with four or five
zeros for symmetric or hermitian quark mass matrices [83]. Their textures are
not viable today since they cannot describe the current rather precise data on
the CKM quark mixing matrix. However, the texture zero approach to the quark
mass matrices is still promising [85, 84] 444By making a suitable weak basis
transformation, one can obtain some sets of zeros of the quark mass matrices.
This issue was discussed in Refs. [86, 87, 88].

A systematic study of texture zeros has been presented for the down-type quark
mass matrix in the basis of diagonal up-type quark mass matrix in Ref.[89]
from the standpoint of “Occam’s Razor approach” [90], in which a minimum
number of parameters is allowed. The down-type quark mass matrix was arranged
to have the minimum number of parameters by setting three of its elements to
zero, while at the same time requiring that it describes successfully the CKM
mixing and CP violation without assuming it to be symmetric or hermitian. Some
of the texture zeros considered in Ref. [89] lead to real ${\rm
det}\,[M_{Q}]$. For the purpose of our study, where the modular invariance of
the quark mass matrices determines the texture zeros, we discuss three sets of
texture zeros for both the down-type and up-type mass matrices, $M_{Q}$, which
are texture zeros of the corresponding quark Yukawa couplings $Y_{Q}$ 555There
are thirteen available textures with three zeros, of which six textures lead
to real ${\rm det}\,[M_{Q}]$ [89]. They give three different relations between
the CKM angles and the quark masses as seen in Table 1 of Ref.[89]. The three
textures in Eq. (3) are representatives of the indicated six textures. :
$M_{Q}=v_{Q}Y_{Q}$, $Q=D,U$, $v_{D,U}$ being the VEVs of the down-type and up-
type doublet Higgs fields. We cannot take different texture zeros for the
down-type and up-type quark mass matrices in the modular invariant approach to
the strong CP problem as seen in the next section. In the right-left (RL)
convention for the mass matrices $(M_{Q})_{RL}=v_{Q}(Y_{Q})_{RL}$, which we
are going employ in our analysis, the three sets of textures we were referring
to above are:

$\displaystyle(1):\quad Y_{Q}=v^{-1}_{Q}\,M_{Q}=\begin{pmatrix}0&0&a_{Q}\\\
0&b_{Q}&c_{Q}\\\ d_{Q}&e_{Q}&f_{Q}\,e^{i\varphi_{Q}}\end{pmatrix}_{RL},$
$\displaystyle(2):\quad Y_{Q}=v^{-1}_{Q}\,M_{Q}==\begin{pmatrix}0&0&a_{Q}\\\
b_{Q}&0&c_{Q}\\\ e_{Q}&d_{Q}&f_{Q}\,e^{i\varphi_{Q}}\end{pmatrix}_{RL},$
$\displaystyle(3):\quad Y_{Q}=v^{-1}_{Q}\,M_{Q}=\begin{pmatrix}a_{Q}&0&0\\\
c_{Q}&b_{Q}&0\\\ f_{Q}\,e^{i\varphi_{Q}}&e_{Q}&d_{Q}\end{pmatrix}_{RL},$ (3)

where $Q=D,\,U$, and $a_{Q},\,b_{Q},\,c_{Q},\,d_{Q},\,e_{Q},\,f_{Q}$ are real
coefficients. The texture (1) is used in Ref. [28] for the discussion of the
strong CP problem. The textures (2) and (3) are obtained by exchange of
columns $(1\leftrightarrow 2)$ and $(1\leftrightarrow 3)$ of the matrix (1),
respectively. At the same time, the physics (the masses and the CKM matrix) is
not changed by the exchange of the rows of (1). The CPV phases $\varphi_{D}$
and $\varphi_{U}$ are assigned to specific entries so that the mass matrices
in Eq. (3) coincide in form with those obtained from modular invariance in our
further analysis. As we will see, the modular invariance constraints the down-
type and up-type quark mass matrices we will consider to have each only one
CPV phase originating from the VEV of the modulus $\tau$. The determinants of
the considered mass matrices are real:

$\displaystyle{\rm det}\,[M_{Q}]=\pm a_{Q}b_{Q}d_{Q}\,.$ (4)

We show typical examples of numerical fits for the three textures of Eq. (3)
in Appendix C, for which the input data are given in Appendix B.

##  3 Realization of texture zeros in $A_{4}$ modular symmetry

We will present next modular invariant mass matrices with level $N=3$
($A_{4}$) modular symmetry. They have the texture zeros of the matrices given
in Eq. (3). In order to realize the texture zeros, all quarks should be
assigned to $A_{4}$ singlets. However, the modular forms representing trivial
singlets 1 are just equivalent to the Eisenstein series, which correspond to
the modular forms of level $N=1$ modular symmetry, as shown in Eq. (52) of
Appendix A. An example of modular invariant mass matrix with level $N=1$
modular symmetry was presented in Ref. [28]. In this work, we employ modular
forms that are non-trivial singlets $\bf 1^{\prime}$ and $\bf
1^{\prime\prime}$ in addition to forms which are trivial singlets 1. In order
to reproduce the observed CP violation of the quark sector, two singlet
modular forms with same weight furnishing the same singlet representation are
required to avoid the vanishing of the CPV phase due to cancellation between
the contributions for the down-type and up-type quark mass matrices. The
smallest weight $k_{Y}$ at which there exists two $A_{4}$ modular forms of the
same singlet representation is $k_{Y}=12$, but the forms are trivial 1
singlets. Two non-trivial $\bf 1^{\prime}$ singlet modular forms of the same
weight exist at $k_{Y}=16$ and at larger weights, as seen in Appendix A. In
this section, using two ${\bf 1^{\prime}}$ singlet modular forms with weight
$k=16$ together with additional singlet modular forms, we construct quark mass
matrices with $A_{4}$ modular symmetry, which have the general forms given in
Eq. (3).

###  3.1 Quark mass matrix of model (1) and its CPV phase structure

We assign the $A_{4}$ representations and the weights for the relevant chiral
superfields as follows:

  * •

quark doublet (left-handed) $Q_{1}=(d,u)_{L},Q_{2}=(s,c)_{L},Q_{3}=(b,t)_{L}$:
$A_{4}$ singlets $(1,\,1,\,1^{\prime\prime})$ with weight $(-8,-4,8)$.

  * •

quark singlets (righ-handed) $(d^{c},s^{c},b^{c})$ and $(u^{c},c^{c},t^{c})$ :
$A_{4}$ singlets $(1^{\prime},\,1,\,1)$ with weight $(-8,4,8)$.

  * •

Higgs fields of down-type and up-type quark sectors $H_{U,D}$: $A_{4}$ singlet
$1$ with weight 0.

These assignments are summarized in Table 1.

| $(d,u)_{L},(s,c)_{L},(b,t)_{L}$ | $(d^{c},s^{c},b^{c}),\,(u^{c},c^{c},t^{c})$ | $H_{U}$ | $H_{D}$
---|---|---|---|---
$SU(2)$ | 2 | 1 | 2 | 2
$A_{4}$ | $(1,\,1\,,1^{\prime\prime})$ | $(1^{\prime},\,1\,,1)$ | $1$ | $1$
$k$ | $(-8,-4,\ 8)$ | $(-8,4,\ 8)$ | 0 | 0
Table 1: Assignments of $A_{4}$ representations and weights in model (1). We
note that the exponent in the automorphy factor in the modular transformations
of the quark superfields we work with is $(-\,k)$.

Taking account of the following tensor products of $A_{4}$ singlets [54, 55,
56],

$\displaystyle{\bf 1}\otimes{\bf 1}={\bf 1}\,,\qquad{\bf
1^{\prime}}\otimes{\bf 1^{\prime}}={\bf 1^{\prime\prime}}\,,\qquad{\bf
1^{\prime\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1^{\prime}}\,,\qquad{\bf
1^{\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1}\,,$ (5)

the superpotential terms $w_{D}$ and $w_{U}$ of the down-type and up-type
quark superfields with weights as given in Table 1 read:

$\displaystyle
w_{D}=\left[a_{D}d^{c}Q_{3}+b_{D}s^{c}Q_{2}+c_{D}s^{c}Q_{3}Y_{\bf
1^{\prime}}^{(12)}+d_{D}b^{c}Q_{1}+e_{D}b^{c}Q_{2}Y_{\bf
1}^{(4)}+f_{D}b^{c}Q_{3}\,(g_{D}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf
1^{\prime}{\rm B}}^{(16)})\right]H_{D}\,,$ $\displaystyle
w_{U}=\left[a_{U}u^{c}Q_{3}+b_{U}c^{c}Q_{2}+c_{U}c^{c}Q_{3}Y_{\bf
1^{\prime}}^{(12)}+d_{U}t^{c}Q_{1}+e_{U}t^{c}Q_{2}Y_{\bf
1}^{(4)}+f_{U}t^{c}Q_{3}(g_{U}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf
1^{\prime}{\rm B}}^{(16)})\right]H_{U}\,,$

where $Y_{\bf r}^{(k_{Y})}$ are modular forms of ${\bf r=1,1^{\prime}}$ with
weight $k_{Y}$ as seen in Appendix A. Parameters
$a_{D},\,b_{D},\,c_{D},\,d_{D},\,e_{D},\,f_{D},\,g_{D}$ and
$a_{U},\,b_{U},\,c_{U},\,d_{U},\,e_{U},\,f_{U},\,g_{U}$ are constants. These
constants are real as a consequence of the imposed CP invariance of the
Lagrangian of the theory. We note that $w_{D}$ and $w_{U}$ involve three
modular forms that are $\bf 1^{\prime}$ singlets, $Y_{\bf 1^{\prime}}^{(12)}$,
$Y_{\bf 1^{\prime}{\rm A}}^{(16)}$ and $Y_{\bf 1^{\prime}{\rm B}}^{(16)}$, in
addition to the trivial singlet one $Y_{\bf 1}^{(4)}$.

The Yukawa matrices of the down-type and up-type quarks follow from the
expressions for $w_{D}$ and $w_{U}$ and are given by:

$\displaystyle Y_{D}=\begin{pmatrix}0&0&a_{D}\\\ 0&b_{D}&c_{D}Y_{\bf
1^{\prime}}^{(12)}\\\ d_{D}&e_{D}Y_{\bf 1}^{(4)}&f_{D}(g_{D}Y_{\bf
1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm
B}}^{(16)})\end{pmatrix}_{RL}\,,\quad Y_{U}=\begin{pmatrix}0&0&a_{U}\\\
0&b_{U}&c_{U}Y_{\bf 1^{\prime}}^{(12)}\\\ d_{U}&e_{U}Y_{\bf
1}^{(4)}&f_{U}(g_{U}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm
B}}^{(16)})\end{pmatrix}_{RL}\,.$ (7)

We take the modular invariant kinetic terms simply as 666Possible non-minimal
additions to the Kähler potential, compatible with the modular symmetry, may
jeopardise the predictive power of the approach [91]. This problem is the
subject of ongoing research.

$\sum_{I}\frac{|\partial_{\mu}\psi^{(I)}|^{2}}{\langle-i\tau+i\bar{\tau}\rangle^{k_{I}}}~{},$
(8)

where $\psi^{(I)}$ denotes a chiral superfield with weight $k_{I}$, and
$\bar{\tau}$ is the anti-holomorphic modulus. The (anti-)holomorphic modulus
$\bar{\tau}$ is a dynamical field. It becomes a complex number after the
modulus $\tau$ takes a VEV. Then, one can set $\bar{\tau}=\tau^{*}$.

It is important to address the transformation needed to get the kinetic terms
of matter superfields in canonical form because the terms in Eq. (8) are not
canonical. The canonical form is obtained by an overall re-normalization of
the quark superfields of interest:

$\displaystyle\psi^{(I)}\rightarrow\sqrt{(2{\rm
Im}\tau_{q})^{k_{I}}}\,\psi^{(I)}\,.$ (9)

This changes some of the constant parameters in the quark mass matrices in the
following way:

$\displaystyle c_{Q}\rightarrow\hat{c}_{Q}=c_{Q}\,\sqrt{(2{\rm
Im}\tau)^{12}}=c_{Q}\,(2{\rm Im}\tau)^{6},\,$ $\displaystyle
e_{Q}\rightarrow\hat{e}_{Q}=e_{Q}\,\sqrt{(2{\rm Im}\tau)^{4}}=e_{Q}\,(2{\rm
Im}\tau)^{2},\,$ $\displaystyle
f_{Q}\rightarrow\hat{f}_{Q}=f_{Q}\,\sqrt{(2{\rm Im}\tau)^{16}}=f_{Q}\,(2{\rm
Im}\tau)^{8}\,,\qquad Q=D,\,U\,.$ (10)

The constants $a_{Q}$, $b_{Q}$, $d_{Q}$ remain unchanged. Thus, the mass
matrices of the down-type and up-type quarks take the form:

$\displaystyle M_{D}=v_{D}\begin{pmatrix}0&0&a_{D}\\\ 0&b_{D}&c_{D}(2{\rm
Im}\tau)^{6}Y_{\bf 1^{\prime}}^{(12)}\\\ d_{D}&e_{D}(2{\rm Im}\tau)^{2}Y_{\bf
1}^{(4)}&f_{D}(2{\rm Im}\tau)^{8}(g_{D}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf
1^{\prime}{\rm B}}^{(16)})\end{pmatrix}_{RL}\,,$ $\displaystyle
M_{U}=v_{U}\begin{pmatrix}0&0&a_{U}\\\ 0&b_{U}&c_{U}(2{\rm Im}\tau)^{6}Y_{\bf
1^{\prime}}^{(12)}\\\ d_{U}&e_{U}(2{\rm Im}\tau)^{2}Y_{\bf
1}^{(4)}&f_{U}(2{\rm Im}\tau)^{8}(g_{U}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf
1^{\prime}{\rm B}}^{(16)})\end{pmatrix}_{RL}\,.$ (11)

The determinants of the quark mass matrices $M_{D}$ and $M_{U}$ given in the
preceding equation are real:

$\displaystyle{\rm det}\,[M_{D}]=-a_{D}b_{D}d_{D}\,,\qquad\qquad{\rm
det}\,[M_{U}]=-a_{U}b_{U}d_{U}\,.$ (12)

The phase structure of the matrices in Eq. (11) is rather simple. The CP
violating phases originate from the VEV of the modulus $\tau$ via the modular
forms $Y_{\bf 1}^{(4)}$, $Y_{\bf 1^{\prime}}^{(12)}$, $Y_{\bf 1^{\prime}{\rm
A}}^{(16)}$ and $Y_{\bf 1^{\prime}{\rm B}}^{(16)}$. The phases of $Y_{\bf
1}^{(4)}$ and $Y_{\bf 1^{\prime}}^{(12)}$ in the $(2,3)$ and $(3,2)$ elements
can be factorised as follows:

$\displaystyle M_{D}=v_{D}P_{R}^{*}\begin{pmatrix}0&0&a_{D}\\\
0&b_{D}&c_{D}(2{\rm Im}\tau)^{6}|Y_{\bf 1^{\prime}}^{(12)}|\\\
d_{D}&e_{D}(2{\rm Im}\tau)^{2}|Y_{\bf 1}^{(4)}|&f_{D}(2{\rm
Im}\tau)^{8}(g_{D}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm
B}}^{(16)})e^{-i(\varphi_{4}+\varphi_{12})}\end{pmatrix}P_{L}\,,$
$\displaystyle M_{U}=v_{U}P_{R}^{*}\begin{pmatrix}0&0&a_{U}\\\
0&b_{U}&c_{U}(2{\rm Im}\tau)^{6}|Y_{\bf 1^{\prime}}^{(12)}|\\\
d_{U}&e_{U}(2{\rm Im}\tau)^{2}|Y_{\bf 1}^{(4)}|&f_{U}(2{\rm
Im}\tau)^{8}(g_{U}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm
B}}^{(16)})e^{-i(\varphi_{4}+\varphi_{12})}\end{pmatrix}P_{L}\,.$ (13)

The phase matrices $P_{R}$ and $P_{L}$ are given by:

$\displaystyle P_{R}=\begin{pmatrix}e^{i\varphi_{12}}&0&0\\\ 0&1&0\\\
0&0&e^{-i\varphi_{4}}\end{pmatrix}\,,\qquad
P_{L}=\begin{pmatrix}e^{-i\varphi_{4}}&0&0\\\ 0&1&0\\\
0&0&e^{i\varphi_{12}}\end{pmatrix}\,,$ (14)

where

$\displaystyle\varphi_{4}\equiv\arg\,Y_{\bf
1}^{(4)}\,,\qquad\varphi_{12}\equiv\arg\,Y_{\bf 1^{\prime}}^{(12)}\,.$ (15)

The matrix $P_{R}$ does not appear in the CKM mixing matrix. Since $P_{L}$ is
common in down-type quark and up-type quark mass matrices, the phase matrix
$P_{L}$ is cancelled out in the CKM matrix due to $P_{L}^{*}P_{L}=1$. Thus,
the CP violation originates from the phases of the $(3,3)$ elements of $M_{D}$
and $M_{U}$ in Eq. (13):

$\displaystyle\arg\ [(g_{D}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf
1^{\prime}{\rm B}}^{(16)})\
e^{-i(\varphi_{4}+\varphi_{12})}]\equiv\Phi^{(1)}_{D}\,,\qquad\arg\
[(g_{U}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm B}}^{(16)})\
e^{-i(\varphi_{4}+\varphi_{12})}]\equiv\Phi^{(1)}_{U}\,.$ (16)

These phases are fixed by the values of the VEV of modulus $\tau$ and of the
real parameters $g_{D}$ and $g_{U}$. Due to the two ${\bf 1^{\prime}}$ singlet
weight 16 modular forms, non-trivial CP-violating phases appear in down-type
and up-type quark mass matrices as well as in the CKM matrix.

In Appendix C.1 we show that the mass matrices $M_{Q}$, $Q=D,U$, given in Eq.
(13), which correspond to the first (upper) matrix of Yukawa couplings in Eq.
(3), are completely consistent with the observed quark masses and CKM quark
mixing and CP violation when their elements have the numerical values reported
in Eq. (76) of Appendix C.1 and the CPV phases in the (3,3) elements in
$M_{D}$ and $M_{U}$ have the values $24.36^{\circ}$ and $-\,150.26^{\circ}$,
respectively. This implies that the quark matrices in Eq. (13), obtained using
the $A_{4}$ modular invariance, could be consistent with the data on quark
masses, mixing and CP violation if the phases $\Phi^{(1)}_{D}$ and
$\Phi^{(1)}_{U}$ in Eq. (16) have the indicated values. If we will be able to
adjust the VEV of the modulus $\tau$ and the values of the real constants
$g_{D}$ and $g_{U}$ so that the phases $\Phi^{(1)}_{D}$ and $\Phi^{(1)}_{U}$
get these values, we will have viable quark mass matrices since the constants
$a_{D},\,b_{D},\,c_{D},\,d_{D},\,e_{D},\,f_{D}$ and
$a_{U},\,b_{U},\,c_{U},\,d_{U},\,e_{U},\,f_{U}$ can be used to reproduce the
numerical values in the matrices in Eq. (76). These numerical values are
derived by fitting the quark masses, the CKM matrix elements and the Jarlskog
rephasing invariant $J_{CP}$ quoted in Eqs. (68), (70) and (73) in Appendix B.

We plot in Fig. 2 the values of the modulus $\tau$ 777By “values of the
modulus $\tau$” here and in what follows we mean “values of the VEV of the
modulus $\tau$”. , which allow to reproduce the quoted numerical values of the
CPV phases $\Phi^{(1)}_{D}=24.36^{\circ}$ and
$\Phi^{(1)}_{U}=-\,150.26^{\circ}$ within $1\%$ deviation by scanning $g_{D}$
and $g_{U}$ in the ranges of $|g_{D}|,|g_{U}|=0-0.5$ (blue points),
$|g_{D}|,|g_{U}|=0.5-2$ (red points) and $|g_{D}|,|g_{U}|=2-10$ (magenta
points). These ranges are chosen in order to illustrate the $g_{D}$ and
$g_{U}$ dependences. As is indicated by the figure, there is a viable region
of values of $\tau$ up to ${\rm Im}\,\tau\simeq 2$. We find a region near the
fixed points $\tau=\omega$ that is also viable. Indeed, in subsection 4.1 we
easily obtain an example of a parameter set near the fixed points
$\tau=\omega$ which provides a good description of the quark data. There is
scarcely a viable point close to $\tau=i$.

In Fig. 2 we show the corresponding region in the $g_{D}-{\rm Im}\,\tau$ plane
for values of $|g_{D}|,|g_{U}|<10$. The imaginary part of $\tau$ can have
relatively large values ${\rm Im}\,\tau\cong(1.5-2.0)$ in the region of
$|g_{D}|<0.5$, the maximal value ${\rm Im}\,\tau\cong 2.0$ being reached for
$|g_{D}|\ll 1$. We do not show the analogous plot in the $g_{U}-{\rm
Im}\,\tau$ plane since it is almost the same as the one in Fig. 2.

We will present in subsection 4.1 the results of the fits of the the quark
masses, CKM elements and the Jarlskog rephasing invariant $J_{CP}$ quoted in
Eqs. (68), (70) and (73) of Appendix B.

Figure 1: The region in ${\rm Re}\,\tau$-${\rm Im}\,\tau$ plane consistent
with with observed CP phase of CKM matrix in model (1). Blue, red and magenta
points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and $2$-$10$,
respectively. Figure 2: The region in $g_{D}$-${\rm Im}\,\tau$ plane
consistent with observed CP phase of CKM matrix in model (1). Blue, red and
magenta points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and
$2$-$10$, respectively.
---

Let us compare the allowed region obtained by us in the $A_{4}$ modular model
with quark mass matrices given in Eq. (13) with that in the model proposed in
Ref. [28], in which modular forms of level $N=1$ modular symmetry are used.
These modular forms are SL$(2,\mathbb{Z})$ singlets - the Eisenstein series
with weight $4$, $8$ and $12$, $E_{4}$, $E_{8}$ and $E_{12A}$, $E_{12B}$. They
coincide with the singlet ${\bf 1}$ modular forms of the $A_{4}$ modular group
of the same weights: $E_{4}\equiv Y^{(4)}_{\bf 1}$, $E_{8}\equiv Y^{(8)}_{\bf
1}=E^{2}_{4}$, $E_{12A}=E^{3}_{4}\equiv Y^{(12)}_{{\bf 1}A}$,
$E_{12B}=E^{2}_{6}-E^{3}_{4}\equiv Y^{(12)}_{{\bf 1}B}$, where $E_{6}\equiv
Y^{(6)}_{\bf 1}$ and $Y^{(4)}_{\bf 1}$, $Y^{(6)}_{\bf 1}$, $Y^{(8)}_{\bf 1}$,
$Y^{(12)}_{{\bf 1}A}$ and $Y^{(12)}_{{\bf 1}B}$ are given in Appendix A. The
parameters $g_{D}$ and $g_{U}$ appear in the quark mass matrices of the model
in the same way they appear in the Yukawa matrices in Eq. (7):

$\displaystyle M_{Q}=v_{Q}\,\begin{pmatrix}0&0&a_{Q}\\\ 0&b_{Q}&c_{Q}(2{\rm
Im}\tau)^{4}Y_{\bf 1}^{(8)}\\\ d_{Q}&e_{Q}(2{\rm Im}\tau)^{2}Y_{\bf
1}^{(4)}&f_{Q}(2{\rm Im}\tau)^{6}(g_{Q}Y_{\bf 1{\rm A}}^{(12)}+Y_{\bf 1{\rm
B}}^{(12)})\end{pmatrix}_{RL}\,,\quad Q=D,U\,.$ (17)

The results of this analysis performed by us are shown in Fig. 4 and Fig. 4.
The distributions of $\tau$ and $g_{D}$ are almost the same as those shown in
Figs. 2 and 2 because the textures of $M_{D}$ and $M_{U}$ are the same in the
two models. We will see later that the distributions of $\tau$ and $g_{D}$
depend significantly on the textures of the quark mass matrices.

Figure 3: The region in ${\rm Re}\,\tau$-${\rm Im}\,\tau$ plane consistent
with the observed CPV phase of CKM matrix in the case of $N=1$ modular
summetry of the quark mass matrices, where Eisenstein series are used as
modular forms [28]. Blue, red and magenta points correspond to
$|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and $2$-$10$, respectively. See the text
for further details. Figure 4: The region in $g_{D}$-${\rm Im}\,\tau$ plane
consistent with observed CP phase of CKM matrix in the case of $N=1$. Blue,
red and magenta points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and
$2$-$10$, respectively.
---

###  3.2 Quark mass matrix of model (2) and its CPV phase structure

We present the second model (2), in which the assignments of the weights for
the relevant chiral superfields are as follows:

  * •

quark doublet (left-handed) $Q_{1}=(d,u)_{L},Q_{2}=(s,c)_{L},Q_{3}=(b,t)_{L}$:
$A_{4}$ singlets $(1,\,1,\,1^{\prime\prime})$ with weight $(-4,-8,8)$.

  * •

quark singlets (righ-handed) $(d^{c},s^{c},b^{c})$ and $(u^{c},c^{c},t^{c})$ :
$A_{4}$ singlets $(1^{\prime},\,1,\,1)$ with weight $(-8,4,8)$.

  * •

Higgs fields of down-type and up-type quark sectors $H_{U,D}$: $A_{4}$ singlet
$1$ with weight 0.

These assignments are summarized in Table 2.

| $(d,u)_{L},(s,c)_{L},(b,t)_{L}$ | $(d^{c},s^{c},b^{c}),\,(u^{c},c^{c},t^{c})$ | $H_{U}$ | $H_{D}$
---|---|---|---|---
$SU(2)$ | 2 | 1 | 2 | 2
$A_{4}$ | $(1,\,1\,,1^{\prime\prime})$ | $(1^{\prime},\,1\,,1)$ | $1$ | $1$
$k$ | $(-4,\ -8,8)$ | $(-8,4,\ 8)$ | 0 | 0
Table 2: Assignments of $A_{4}$ representations and weights in model (2).

The mass matrices of the down-type and up-type quarks read:

$\displaystyle M_{Q}=v_{Q}\begin{pmatrix}0&0&{a_{Q}}\\\
{b_{Q}}&0&{c_{Q}}(2{\rm Im}\tau)^{6}Y_{\bf 1^{\prime}}^{(12)}\\\ {e_{Q}}(2{\rm
Im}\tau)^{2}Y_{\bf 1}^{(4)}&{d_{Q}}&{f_{Q}}(2{\rm Im}\tau)^{8}(g_{Q}Y_{\bf
1^{\prime}_{\rm A}}^{(16)}+Y_{\bf 1^{\prime}_{\rm
B}}^{(16)})\end{pmatrix}_{RL}\,,\qquad Q=D,U\ .$ (18)

The CP violating phases come from the modulus $\tau$ in modular forms of
$Y_{\bf 1}^{(4)}$, $Y_{\bf 1^{\prime}}^{(12)}$, $Y_{\bf 1^{\prime}_{\rm
A}}^{(16)}$ and $Y_{\bf 1^{\prime}_{\rm B}}^{(16)}$. The phases of $Y_{\bf
1}^{(4)}$ and $Y_{\bf 1^{\prime}}^{(12)}$ in the $(2,3)$ and $(3,1)$ elements
can be factorised as follows:

$\displaystyle M_{Q}=v_{Q}P_{R}^{*}\begin{pmatrix}0&0&{a_{Q}}\\\
{b_{Q}}&0&{c_{Q}}(2{\rm Im}\tau)^{6}|Y_{\bf 1^{\prime}}^{(12)}|\\\
{e_{Q}}(2{\rm Im}\tau)^{2}|Y_{\bf 1}^{(4)}|&{d_{Q}}&{f_{Q}}(2{\rm
Im}\tau)^{8}(g_{Q}Y_{\bf 1^{\prime}_{\rm A}}^{(16)}+Y_{\bf 1^{\prime}_{\rm
B}}^{(16)})e^{-i(\varphi_{4}+\varphi_{12})}\end{pmatrix}_{RL}\hskip-8.5359ptP_{L}\,,\
\ Q=D,U\ .$ (19)

The phase matrices $P_{R}$ and $P_{L}$ are given by:

$\displaystyle P_{R}=\begin{pmatrix}e^{i\varphi_{12}}&0&0\\\ 0&1&0\\\
0&0&e^{-i\varphi_{4}}\end{pmatrix}\,,\qquad P_{L}=\begin{pmatrix}1&0&0\\\
0&e^{-i\varphi_{4}}&0\\\ 0&0&e^{i\varphi_{12}}\end{pmatrix}\,,$ (20)

where $\varphi_{4}$ and $\varphi_{12}$ are defined in Eq.(15). As in the model
(1), the matrix $P_{R}$ does not contribute to the CKM mixing matrix, while
$P_{L}$, being common to down-type quark and up-type quark mass matrices, is
cancelled out in the CKM matrix. The CP violation is generated by the phases
of the $(3,3)$ elements of $M_{D}$ and $M_{U}$ in Eq. (19),

$\displaystyle\arg\ [(g_{D}Y_{\bf 1{\rm A}}^{(16)}+Y_{\bf 1{\rm B}}^{(16)})\
e^{-i(\varphi_{4}+\varphi_{12})}]\equiv\Phi^{(2)}_{D}\,,\qquad\arg\
[(g_{U}Y_{\bf 1{\rm A}}^{(16)}+Y_{\bf 1{\rm B}}^{(16)})\
e^{-i(\varphi_{4}+\varphi_{12})}]\equiv\Phi^{(2)}_{U}\,,$ (21)

which are determined by the VEV of the modulus $\tau$ and the real parameters
$g_{D}$ and $g_{U}$.

The analysis which follows is analogous to that performed for the model (1) in
the preceding section. The second (middle) matrix of Yukawa couplings in Eq.
(3) generates the quark mass matrices reported in Eq. (81) of Appendix C.2,
which have the same structure as the the mass matrices in Eq. (19), obtained
by employing the $A_{4}$ modular symmetry. It is shown in C.2 that these mass
matrices successfully describe the data on the quark masses, mixing and CP
violation if their elements have the numerical values given in Eq. (83) and
the CPV phases in the (3,3) elements in $M_{D}$ and $M_{U}$ posses the values
$104.84^{\circ}$ and $56.63^{\circ}$, respectively. Reproducing the real
numerical values of the elements of $M_{D}$ and $M_{U}$ reported in Eq. (83)
using the constants $a_{D},\,b_{D},\,c_{D},\,d_{D},\,e_{D},\,f_{D}$ and
$a_{U},\,b_{U},\,c_{U},\,d_{U},\,e_{U},\,f_{U},$ does not pose a problems.
Thus, if we find values of the VEV of $\tau$ and of the real constants $g_{D}$
and $g_{U}$ that generate CPV phases $\Phi^{(2)}_{D}=104.84^{\circ}$ and
$\Phi^{(2)}_{U}=56.63^{\circ}$, then the $A_{4}$ modular matrices $M_{D}$ and
$M_{U}$ in Eq. (19) will be phenomenologically viable for the so found values
of the modulus VEV and $g_{D}$ and $g_{U}$.

We have performed the relevant analysis and show in Fig. 6 the region of
values of the VEV of $\tau$ for which the phases
$\Phi^{(2)}_{D}=104.84^{\circ}$ and $\Phi^{(2)}_{U}=56.63^{\circ}$ are
reproduced within $1\%$ deviation by scanning $g_{D}$ and $g_{U}$ in the
ranges of $|g_{D}|,|g_{U}|=0-0.5$ (blue points), $|g_{D}|,|g_{U}|=0.5-2$ (red
points) and $|g_{D}|,|g_{U}|=2-10$ (magenta points). The allowed points of
$\tau$ are considerably reduced as compared with those in Fig. 2. However,
there is still a viable region of values of $\tau$ close to ${\rm
Im}\,\tau=1.9$. We also find a region near the fixed points $\tau=\omega$ that
is also viable. Indeed, in subsection 4.2 we present an example of a parameter
set near the fixed points $\tau=\omega$ as well as the ones close to $\tau=i$
and at the large ${\rm Im}\,\tau$.

In Fig. 6 we show the corresponding region in the $g_{D}-{\rm Im}\,\tau$ plane
for values of $|g_{D}|,|g_{U}|<10$. The imaginary part of $\tau$ reaches the
value of $1.9$ for $|g_{D}|\ll 1$. The analogous plot of $g_{U}-{\rm
Im}\,\tau$ plane is almost the same one as the one in Fig. 6 and we do not
show it here.

We will present in subsection 4.2 the results of the fits of the the quark
masses, CKM elements and the Jarlskog rephasing invariant $J_{CP}$ quoted in
Eqs. (68), (70) and (73) of Appendix B.

Figure 5: The region in ${\rm Re}\,\tau$-${\rm Im}\,\tau$ plane consistent
with with observed CP phase of CKM matrix in model (2). Blue, red and magenta
points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and $2$-$10$,
respectively.  Figure 6: The region in $g_{D}$-${\rm Im}\,\tau$ plane
consistent with observed CP phase of CKM matrix in model (2). Blue, red and
magenta points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and
$2$-$10$, respectively.
---

###  3.3 Quark mass matrix of model (3) and its CPV phase structure

For the model (3), the assignments of the weights for the relevant chiral
superfields is given as follows:

  * •

quark doublet (left-handed) $Q_{1}=(d,u)_{L},Q_{2}=(s,c)_{L},Q_{3}=(b,t)_{L}$:
$A_{4}$ singlets $(1^{\prime\prime},\,1,\,1)$ with weight $(8,-4,-8)$.

  * •

quark singlets (righ-handed) $(d^{c},s^{c},b^{c})$ and $(u^{c},c^{c},t^{c})$ :
$A_{4}$ singlets $(1^{\prime},\,1,\,1)$ with weight $(-8,4,8)$.

  * •

Higgs fields of down-type and up-type quark sectors $H_{U,D}$: $A_{4}$ singlet
$1$ with weight 0.

These assignments are summarized in Table 3.

| $(d,u)_{L},(s,c)_{L},(b,t)_{L}$ | $(d^{c},s^{c},b^{c}),\,(u^{c},c^{c},t^{c})$ | $H_{U}$ | $H_{D}$
---|---|---|---|---
$SU(2)$ | 2 | 1 | 2 | 2
$A_{4}$ | $(1^{\prime\prime},\,1\,,1)$ | $(1^{\prime},\,1\,,1)$ | $1$ | $1$
$k$ | $(8,\ -4,\ -8)$ | $(-8,4,\ 8)$ | 0 | 0
Table 3: Assignments of $A_{4}$ representations and weights in model (3).

The mass matrices of the down-type and up-type quarks have the form:

$\displaystyle M_{Q}=v_{Q}\begin{pmatrix}{a_{Q}}&0&0\\\ {c_{Q}}(2{\rm
Im}\tau)^{6}Y_{\bf 1^{\prime}}^{(12)}&{b_{Q}}&0\\\ {f_{Q}}(2{\rm
Im}\tau)^{8}(g_{Q}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm
B}}^{(16)})&{e_{Q}}(2{\rm Im}\tau)^{2}Y_{\bf
1}^{(4)}&{d_{Q}}\end{pmatrix}_{RL}\,,\qquad Q=D,U\ .$ (22)

Since the number of parameters are 14, four parameters are redundant for
reproducing the quark masses and CKM elements (10 observables). In this case,
following Ref.[89], we set $c_{U}=e_{U}=f_{U}=0$ and we investigate whether
one can describe successfully the quark data with this additional
phenomenological assumption. As a consequence of setting the constants
$c_{U}$, $e_{U}$ and $f_{U}$ to zero the up-type quark mass matrix is
diagonal.

The CP violating phases originate from the modulus $\tau$ and appear in the
down-type quark mass matrix $M_{D}$ via the modular forms $Y_{\bf 1}^{(4)}$,
$Y_{\bf 1^{\prime}}^{(12)}$, $Y_{\bf 1^{\prime}_{\rm A}}^{(16)}$ and $Y_{\bf
1^{\prime}_{\rm B}}^{(16)}$. The phases of $Y_{\bf 1}^{(4)}$ and $Y_{\bf
1^{\prime}}^{(12)}$ in the $(2,3)$ and $(3,1)$ elements can be factored out as
follows:

$\displaystyle M_{D}=v_{D}P_{R}^{*}\begin{pmatrix}{a_{D}}&0&0\\\ {c_{D}}(2{\rm
Im}\tau)^{6}|Y_{\bf 1^{\prime}}^{(12)}|&{b_{D}}&0\\\ {f_{D}}(2{\rm
Im}\tau)^{8}(g_{D}Y_{\bf 1^{\prime}_{\rm A}}^{(16)}+Y_{\bf 1^{\prime}_{\rm
B}}^{(16)})e^{-i(\varphi_{4}+\varphi_{12})}&{e_{D}}(2{\rm Im}\tau)^{2}|Y_{\bf
1}^{(4)}|&{d_{D}}\end{pmatrix}_{RL}P_{L}\,.$ (23)

The phase matrices $P_{R}$ and $P_{L}$ are

$\displaystyle P_{R}=P_{L}=\begin{pmatrix}1&0&0\\\ 0&e^{-i\varphi_{12}}&0\\\
0&0&e^{-i(\varphi_{4}+\varphi_{12})}\end{pmatrix}\,,$ (24)

where $\varphi_{4}$ and $\varphi_{12}$ are given in Eq. (15).

On the other hand, the up-type quark mass matrix is diagonal. Therefore,
$P_{L}$ and $P_{R}$ do not contribute to the CKM matrix. Thus, the CP
violation is generated by the phase of the $(3,1)$ element of $M_{D}$ in Eq.
(22),

$\displaystyle\arg\ [(g_{D}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf
1^{\prime}{\rm B}}^{(16)})\
e^{-i(\varphi_{4}+\varphi_{12})}]\equiv\Phi^{(3)}_{D}\,,$ (25)

which is determined by the VEV of $\tau$ and the real parameter $g_{D}$.

As shown in Eq. (88) of Appendix C.3, the quark mass matrices in Eq. (22)
could be completely consistent with observed masses and the CKM matrix if the
phase of Eq. (25) has the value $\Phi^{(3)}_{D}=66.36^{\circ}$. By adjusting
the value of the VEV of the modulus $\tau$ and of the real constant $g_{D}$ to
get $\Phi^{(3)}_{D}=66.36^{\circ}$, we will obtain a phenomenologically viable
down-quark mass matrix $M_{D}$. The real constants
$a_{D},\,b_{D},\,c_{D},\,d_{D},\,e_{D},\,f_{D}$ can be used to reproduce the
real numerical values in $M_{D}$ in Eq. (90).

We plot in Fig. 8 the values of the VEV of $\tau$ for which one can generate
$\Phi^{(3)}_{D}=66.36^{\circ}$ within $1\%$ deviation by scanning $g_{D}$ and
$g_{U}$ again in the ranges of $|g_{D}|,|g_{U}|=0-0.5$ (blue points),
$|g_{D}|,|g_{U}|=0.5-2$ (red points) and $|g_{D}|,|g_{U}|=2-10$ (magenta
points). In the considered case it is possible to reproduce the observed CP
violation in the quark sector even for $\tau$ close to $\omega$ and $\tau$
“close” to $i\infty$. Indeed, we present examples of viable parameter sets for
$\tau\simeq\omega$ and ${\rm Im}\,\tau\simeq 1.7$ in subsection 4.3.

In Fig. 8, we show the values of ${\rm Im}\,\tau$ leading (within $1\%$
deviation) to the requisite value of $\Phi^{(3)}_{D}$ versus $|g_{D}|<10$. The
magnitude of ${\rm Im}\,\tau$ increases rapidly when $|g_{D}|$ decreases
towards $0$.

Figure 7: The region in ${\rm Re}\,\tau$-${\rm Im}\,\tau$ plane consistent
with with observed CP phase of CKM matrix in model (3). Blue, red and magenta
points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and $2$-$10$,
respectively.  = Figure 8: The region in $g_{D}$-${\rm Im}\,\tau$ plane
consistent with observed CP phase of CKM matrix in model (3). Blue, red and
magenta points correspond to $|g_{D}|,|g_{U}|=0$-$0.5$, $0.5$-$2$ and
$2$-$10$, respectively.
---

Let us emphasise that in the case of the discussed model (3) we have set the
redundant parameters $c_{U}=e_{U}=f_{U}=0$ “by hand” to get diagonal mass
matrix for the up-type quarks. This set-up is not guaranteed within the
framework of the modular invariance approach without additional (symmetry)
assumptions.

##  4 Reproducing quark masses and CKM parameters

We present next numerical examples of successfully reproducing the observed
quark masses, the CKM mixing angles and CPV phase in the cases of the three
models considered by us.

###  4.1 Fitting model (1)

We present three examples of model (1) corresponding to the $\tau$ being
relatively close to the fixed points $\tau=i$, $\omega$ and $\infty$,
respectively.

####  4.1.1 $\tau$ close to $i$

The first one is an example of model (1), where $\tau$ is rather close to $i$.
This case is comparable to the example in Ref.[28]. We show the numerical
values of parameters obtained in the fit of the quark masses, CKM mixing
angles, the CPV phase $\delta_{CP}$ and of the Jarlskog rephasing invariant
$J_{CP}$ [6], as defined in [7] and quoted in Eqs. (68), (70) and (73) of
Appendix B:

$\displaystyle\frac{b_{D}}{a_{D}}=-0.724,\quad\frac{c_{D}}{a_{D}}=0.0430,\quad\frac{d_{D}}{a_{D}}=0.0864,\quad\frac{e_{D}}{a_{D}}=0.135,\quad\frac{f_{D}}{a_{D}}=0.00750,$
$\displaystyle\frac{b_{U}}{a_{U}}=0.972,\quad\frac{c_{U}}{a_{U}}=0.361,\quad\frac{d_{U}}{a_{U}}=0.631,\quad\frac{e_{U}}{a_{U}}=1.49,\quad\frac{f_{U}}{a_{U}}=0.154,$
$\displaystyle g_{D}=-0.749,\quad g_{U}=-1.10,\quad\tau=0.1811+i\,1.1583,\quad
N\sigma=0.52,$ (26)

where $N\sigma$ denotes a measure of goodness of the fit. By employing the sum
of one-dimensional $\Delta\chi^{2}$ for eight observable quantities
$m_{d}/m_{b},\,m_{s}/m_{b},\,m_{u}/m_{t},\,m_{c}/m_{t},\,|V_{us}|,\,|V_{cb}|,\,|V_{ub}|,\,\delta_{CP}$,
it is defined as $N\sigma\equiv\sqrt{\Delta\chi^{2}}$. The result of the fit
of the quark mass ratios, three CKM elements $|V_{us}|$, $|V_{cb}|$,
$|V_{ub}|$, the phase $\delta_{CP}$ and of the $J_{CP}$ factor are collected
in Table 4.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $8.84$ | $2.80$ | $5.89$ | $0.2251$ | $0.0401$ | $0.00352$ | $2.84\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.8^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 4: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

We comment of the values of parameters in Eq. (26). The parameters $a_{Q}$,
$b_{Q}$, $d_{Q}$ ($Q=D,U$) are real constants that do not couple to modular
forms. On the other hand, $c_{Q}$, $e_{Q}$, $f_{Q}$ ($Q=D,U$) are constants
multiplying the modular forms. Since the normalizations of each of the three
modular forms present in the expressions for the quark mass matrices $M_{D}$
and $M_{U}$ are arbitrary, the magnitudes of the parameters $a_{Q}$, $b_{Q}$,
$d_{Q}$ and respectively those of $c_{Q}$, $e_{Q}$, $f_{Q}$ should be compared
separately among themselves. The constants multiplying the modular forms are
of the same order in magnitude:

$\displaystyle\left|\frac{c_{D}}{f_{D}}\right|\simeq
5.7\,,\qquad\left|\frac{e_{D}}{f_{D}}\right|\simeq
18\,,\qquad\left|\frac{c_{U}}{f_{U}}\right|\simeq
2.3\,,\qquad\left|\frac{e_{U}}{f_{U}}\right|\simeq 9.7\,.$ (27)

In what concerns the constants, $a_{Q}$, $b_{Q}$, $d_{Q}$, only the ration
$|d_{D}/a_{D}|$ is somewhat smaller than the other ratios $|b_{D}/a_{D}|$,
$|b_{U}/a_{U}|$ and $|d_{U}/a_{U}|$:

$\displaystyle\left|\frac{b_{D}}{a_{D}}\right|\simeq
0.72\,,\qquad\left|\frac{d_{D}}{a_{D}}\right|\simeq
0.086\,,\qquad\left|\frac{b_{U}}{a_{U}}\right|\simeq
0.97\,,\qquad\left|\frac{d_{U}}{a_{U}}\right|\simeq 0.63\,.$ (28)

####  4.1.2 $\tau$ close to $\omega$

We present the second example of successful fit of the quark data, in which
the VEV of the modulus $\tau$ is close to the fixed point
$\omega=-\,0.5+i\,\sqrt{3}/2$ (the left cusp). The numerical values of
parameters read:

$\displaystyle\frac{b_{D}}{a_{D}}=0.895,\quad\frac{c_{D}}{a_{D}}=0.816,\quad\frac{d_{D}}{a_{D}}=0.0830,\quad\frac{e_{D}}{a_{D}}=0.524,\quad\frac{f_{D}}{a_{D}}=0.00973,$
$\displaystyle\frac{b_{U}}{a_{U}}=1.26,\quad\frac{c_{U}}{a_{U}}=5.90,\quad\frac{d_{U}}{a_{U}}=0.667,\quad\frac{e_{U}}{a_{U}}=7.53,\quad\frac{f_{U}}{a_{U}}=0.264,$
$\displaystyle g_{D}=-0.809,\ \ g_{U}=3.33,\ \ \tau=-0.4900+i\,0.9493,\ \
\tau-\omega=0.0093+i\,0.0832\,,\ N\sigma=0.87.$ (29)

In Table 5 we present the results of the fit of the quark mass ratios, the
three CKM elements $|V_{us}|$, $|V_{cb}|$, $|V_{ub}|$, of the CPV phase
$\delta_{CP}$ and of the $J_{CP}$ factor.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.88$ | $8.73$ | $2.85$ | $5.57$ | $0.2252$ | $0.0398$ | $0.00349$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.9^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 5: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

The ratios of the constants multiplying the modular forms are of the same
order of magnitude:

$\displaystyle\left|\frac{c_{D}}{f_{D}}\right|\simeq
83.8\,,\qquad\left|\frac{e_{D}}{f_{D}}\right|\simeq
53.9\,,\qquad\left|\frac{c_{U}}{f_{U}}\right|\simeq
22.3\,,\qquad\left|\frac{e_{U}}{f_{U}}\right|\simeq 28.5\,.$ (30)

In what concerns the other constants, only the ratio of $|d_{D}/a_{D}|$ is
again somewhat smaller than the other ratios:

$\displaystyle\left|\frac{b_{D}}{a_{D}}\right|\simeq
0.90\,,\qquad\left|\frac{d_{D}}{a_{D}}\right|\simeq
0.083\,,\qquad\left|\frac{b_{U}}{a_{U}}\right|\simeq
1.26\,,\qquad\left|\frac{d_{U}}{a_{U}}\right|\simeq 0.67\,.$ (31)

####  4.1.3 $\tau$ “close” to $i\infty$

We present the third example of successful fit of quark data, in which the VEV
of the modulus $\tau$ has a relatively large imaginary part “close” to the
fixed point $i\infty$. The numerical values of parameters read:

$\displaystyle\frac{b_{D}}{a_{D}}=-0.469,\quad\frac{c_{D}}{a_{D}}=0.0138,\quad\frac{d_{D}}{a_{D}}=0.0581,\quad\frac{e_{D}}{a_{D}}=0.0296,\quad\frac{f_{D}}{a_{D}}=0.0497,$
$\displaystyle\frac{b_{U}}{a_{U}}=1.585,\quad\frac{c_{U}}{a_{U}}=0.0867,\quad\frac{d_{U}}{a_{U}}=2.504,\quad\frac{e_{U}}{a_{U}}=0.601,\quad\frac{f_{U}}{a_{U}}=4.282,$
$\displaystyle g_{D}=0.000130,\ \ g_{U}=-0.00316,\ \ \tau=0.1097+i\,1.8213,\ \
\ N\sigma=1.31.$ (32)

In Table 6 we present the results of the fit of the quark mass ratios, the
three CKM elements $|V_{us}|$, $|V_{cb}|$, $|V_{ub}|$, of the CPV phase
$\delta_{CP}$ and of the $J_{CP}$ factor.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.88$ | $9.71$ | $2.82$ | $3.44$ | $0.2248$ | $0.0401$ | $0.00353$ | $2.84\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.6^{\circ}$
Exp | $1.88$ | $8.72$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 6: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

The constants multiplying the modular forms present in $M_{D}$ and in $M_{U}$
are of the same order of magnitude except for $|c_{U}/f_{U}|$, which is
somewhat smaller than $|e_{U}/f_{U}|$:

$\displaystyle\left|\frac{c_{D}}{f_{D}}\right|\simeq
0.28\,,\qquad\left|\frac{e_{D}}{f_{D}}\right|\simeq
0.59\,,\qquad\left|\frac{c_{U}}{f_{U}}\right|\simeq
0.02\,,\qquad\left|\frac{e_{U}}{f_{U}}\right|\simeq 0.14\,.$ (33)

In what concerns the other constants in $M_{D}$ and in $M_{U}$, the ratio of
$|d_{D}/a_{D}|$ is somewhat smaller than $|b_{D}/a_{D}|$, the other ratios
$|b_{U}/a_{U}|$ and $|d_{U}/a_{U}|$ being of the same order of magnitude:

$\displaystyle\left|\frac{b_{D}}{a_{D}}\right|\simeq
0.47\,,\qquad\left|\frac{d_{D}}{a_{D}}\right|\simeq
0.058\,,\qquad\left|\frac{b_{U}}{a_{U}}\right|\simeq
1.59\,,\qquad\left|\frac{d_{U}}{a_{U}}\right|\simeq 2.50\,.$ (34)

###  4.2 Fitting model (2)

We present three examples of model (2) corresponding to the $\tau$ being
relatively close to the fixed points $\tau=i$, $\omega$ and $\infty$,
respectively.

####  4.2.1 $\tau$ close to $i$

The first example is close to $\tau=i$. For the numerical values of parameters
resulting from the fit, we obtain in this case:

$\displaystyle{\frac{b_{D}}{a_{D}}=-0.0541},\quad{\frac{c_{D}}{a_{D}}=0.0212},\quad{\frac{d_{D}}{a_{D}}=0.871},\quad{\frac{e_{D}}{a_{D}}=0.0114},\quad{\frac{f_{D}}{a_{D}}=0.0157},$
$\displaystyle{\frac{b_{U}}{a_{U}}=5.58\times
10^{-3}},\quad{\frac{c_{U}}{a_{U}}=5.76\times
10^{-2}},\quad{\frac{d_{U}}{a_{U}}=1.63},\quad{\frac{e_{U}}{a_{U}}=5.10\times
10^{-4}},\quad{\frac{f_{U}}{a_{U}}=1.11\times 10^{-1}},$ $\displaystyle
g_{D}=1.363,\quad g_{U}=1.151,\quad\tau=0.1706+i\,1.1417,\quad N\sigma=0.82.$
(35)

The results of the fit of the observables are shown in Table 7.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $9.10$ | $2.80$ | $6.47$ | $0.2250$ | $0.0400$ | $0.00360$ | $2.86\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $65.6^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 7: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

The constants multiplying the modular forms are of the same order of magnitude
except for $e_{U}$, which is significantly smaller:

$\displaystyle\left|\frac{c_{D}}{f_{D}}\right|\simeq
1.35\,,\qquad\left|\frac{e_{D}}{f_{D}}\right|\simeq
0.73\,,\qquad\left|\frac{c_{U}}{f_{U}}\right|\simeq
0.52\,,\qquad\left|\frac{e_{U}}{f_{U}}\right|\simeq 0.0046\,.$ (36)

The other constants are hierarchical:

$\displaystyle\left|\frac{b_{D}}{a_{D}}\right|\simeq
0.054\,,\qquad\left|\frac{d_{D}}{a_{D}}\right|\simeq
0.87\,,\qquad\left|\frac{b_{U}}{a_{U}}\right|\simeq
0.0056\,,\qquad\left|\frac{d_{U}}{a_{U}}\right|\simeq 1.63\,.$ (37)

####  4.2.2 $\tau$ close to $\omega$

We present the second example, in which the VEV of the modulus $\tau$ is close
to the fixed point $\omega$. The numerical values of parameters obtained in
the fit are:

$\displaystyle{\frac{b_{D}}{a_{D}}=0.0505},\quad{\frac{c_{D}}{a_{D}}=-0.278},\quad{\frac{d_{D}}{a_{D}}=0.839},\quad{\frac{e_{D}}{a_{D}}=0.0317},\quad{\frac{f_{D}}{a_{D}}=-0.00963},$
$\displaystyle{\frac{b_{U}}{a_{U}}=4.75\times
10^{-3}},\quad{\frac{c_{U}}{a_{U}}=0.889},\quad{\frac{d_{U}}{a_{U}}=1.72},\quad{\frac{e_{U}}{a_{U}}=-1.89\times
10^{-3}},\quad{\frac{f_{U}}{a_{U}}=-6.34\times 10^{-2}},$ $\displaystyle
g_{D}=1.64,\ \,g_{U}=-0.840,\ \,\tau=-0.4474+i\,0.9357,\
\,\tau-\omega=0.0526+i\,0.0697,\ \,N\sigma=0.96.$ (38)

The results of the fit of the observables are shown in Table 8.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.89$ | $9.27$ | $2.86$ | $4.48$ | $0.2252$ | $0.0400$ | $0.00349$ | $2.76\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $64.7^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 8: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

The constants multiplying the modular forms are of the same order of magnitude
except for $e_{U}$, which is significantly smaller also in this case:

$\displaystyle\left|\frac{c_{D}}{f_{D}}\right|\simeq
28\,,\qquad\left|\frac{e_{D}}{f_{D}}\right|\simeq
3.3\,,\qquad\left|\frac{c_{U}}{f_{U}}\right|\simeq
14\,,\qquad\left|\frac{e_{U}}{f_{U}}\right|\simeq 0.030\,.$ (39)

The other constants are hierarchical:

$\displaystyle\left|\frac{b_{D}}{a_{D}}\right|\simeq
0.051\,,\qquad\left|\frac{d_{D}}{a_{D}}\right|\simeq
0.84\,,\qquad\left|\frac{b_{U}}{a_{U}}\right|\simeq
0.0048\,,\qquad\left|\frac{d_{U}}{a_{U}}\right|\simeq 1.72\,.$ (40)

####  4.2.3 $\tau$ ”close” to $i\infty$

We present the third example at the VEV of $\tau$ having a relatively large
imaginary part “close” to $i\infty$. The numerical values of parameters
obtained in the fit are:

$\displaystyle{\frac{b_{D}}{a_{D}}=1.07\times 10^{-2}},\ \
{\frac{c_{D}}{a_{D}}=-4.96\times 10^{-3}},\ \ {\frac{d_{D}}{a_{D}}=-0.166},\ \
{\frac{e_{D}}{a_{D}}=2.35\times 10^{-3}},\ \ {\frac{f_{D}}{a_{D}}=5.38\times
10^{-3}},$ $\displaystyle{\frac{b_{U}}{a_{U}}=1.58\times 10^{-4}},\ \
{\frac{c_{U}}{a_{U}}=2.99\times 10^{-3}},\ \ {\frac{d_{U}}{a_{U}}=0.465},\ \
{\frac{e_{U}}{a_{U}}=-6.98\times 10^{-6}},\ \ {\frac{f_{U}}{a_{U}}=-4.11\times
10^{-2}},$ $\displaystyle g_{D}=0.124,\quad
g_{U}=-0.0709,\quad\tau=0.3102+i\,1.5672,\quad N\sigma=0.774.$ (41)

The results of the fit of the observables are shown in Table 9.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $9.79$ | $2.79$ | $5.96$ | $0.2250$ | $0.0402$ | $0.00352$ | $2.82\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $65.3^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 9: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

The constants multiplying the modular forms are of the same order of magnitude
for the down-type-quarks, but hierarchical for up-type quarks:

$\displaystyle\left|\frac{c_{D}}{f_{D}}\right|\simeq
0.92\,,\qquad\left|\frac{e_{D}}{f_{D}}\right|\simeq
0.44\,,\qquad\left|\frac{c_{U}}{f_{U}}\right|\simeq
0.073\,,\qquad\left|\frac{e_{U}}{f_{U}}\right|\simeq 0.00017\,.$ (42)

The other constants are hierarchical:

$\displaystyle\left|\frac{b_{D}}{a_{D}}\right|\simeq
0.011\,,\qquad\left|\frac{d_{D}}{a_{D}}\right|\simeq
0.17\,,\qquad\left|\frac{b_{U}}{a_{U}}\right|\simeq
0.00016\,,\qquad\left|\frac{d_{U}}{a_{U}}\right|\simeq 0.47\,.$ (43)

###  4.3 Fitting model (3)

We present three examples of model (3) corresponding to the $\tau$ being
relatively close to the fixed points $\tau=i$, $\omega$ and $\infty$,
respectively.

####  4.3.1 $\tau$ close to $i$

The first one is an example of model (3), where $\tau$ is rather close to $i$.
For the numerical values of parameters resulting from the fit, we obtain in
this case:

$\displaystyle{\frac{a_{D}}{d_{D}}=-9.38\times
10^{-4}},\quad{\frac{b_{D}}{d_{D}}=-1.83\times
10^{-2}},\quad{\frac{c_{D}}{d_{D}}=2.66\times
10^{-5}},\quad{\frac{e_{D}}{d_{D}}=7.65\times 10^{-3}},$
$\displaystyle{\frac{f_{D}}{d_{D}}=3.58\times
10^{-6}},\quad{\frac{a_{U}}{d_{U}}=5.39\times
10^{-6}},\quad{\frac{b_{U}}{d_{U}}=2.80\times
10^{-3}},\quad{\frac{c_{U}}{d_{U}}=0},\quad{\frac{e_{U}}{d_{U}}=0},\quad{\frac{f_{U}}{d_{U}}=0},$
$\displaystyle g_{D}=1.199,\quad\tau=0.2505+i\,1.1356,\quad N\sigma=0.15.$
(44)

Both the constants multiplying the modular forms present in $M_{D}$ and the
other constants in $M_{D}$ and $M_{U}$ have very different values with the
their relevant ratios exhibiting strong hierarchies. The results of the fit of
the observables are shown in Table 10.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $9.13$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.84\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.5^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 10: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

####  4.3.2 $\tau$ close to $\omega$

We present the second example, in which the VEV of the modulus $\tau$ is close
to the fixed point $\omega$. The numerical values of parameters obtained in
the fit are:

$\displaystyle{\frac{a_{D}}{d_{D}}=9.68\times
10^{-4}},\quad{\frac{b_{D}}{d_{D}}=1.82\times
10^{-2}},\quad{\frac{c_{D}}{d_{D}}=4.00\times
10^{-4}},\quad{\frac{e_{D}}{d_{D}}=-3.04\times 10^{-2}},$
$\displaystyle{\frac{f_{D}}{d_{D}}=-2.65\times
10^{-6}},\quad{\frac{a_{U}}{d_{U}}=5.39\times
10^{-6}},\quad{\frac{b_{U}}{d_{U}}=2.80\times
10^{-3}},\quad{\frac{c_{U}}{d_{U}}=0},\quad{\frac{e_{U}}{d_{U}}=0},\quad{\frac{f_{U}}{d_{U}}=0},$
$\displaystyle
g_{D}=-0.869,\quad\tau=-0.4706+i\,0.9359,\quad\tau-\omega=0.0295+i\,0.0699,\quad
N\sigma=0.24.$ (45)

As in the previous examples, there is a strong hierarchy among both the
constants multiplying the modular forms present in $M_{D}$ and the other
constants in $M_{D}$ and $M_{U}$.

The results of the fit of the observables are shown in Table 11.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.86$ | $9.43$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00354$ | $2.84\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.4^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 11: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

####  4.3.3 $\tau$ ”close” to $i\infty$

Finally, we present an example at the VEV of $\tau$ having a relatively large
imaginary part “close” to $i\infty$. The numerical values of parameters
obtained in the fit are:

$\displaystyle{\frac{a_{D}}{d_{D}}=9.45\times
10^{-4}},\quad{\frac{b_{D}}{d_{D}}=-1.83\times
10^{-2}},\quad{\frac{c_{D}}{d_{D}}=-8.22\times
10^{-6}},\quad{\frac{e_{D}}{d_{D}}=-3.54\times 10^{-3}},$
$\displaystyle{\frac{f_{D}}{d_{D}}=-1.28\times
10^{-5}},\quad{\frac{a_{U}}{d_{U}}=5.39\times
10^{-6}},\quad{\frac{b_{U}}{d_{U}}=2.80\times
10^{-3}},\quad{\frac{c_{U}}{d_{U}}=0},\quad{\frac{e_{U}}{d_{U}}=0},\quad{\frac{f_{U}}{d_{U}}=0},$
$\displaystyle g_{D}=0.00925,\quad\tau=-0.288076+i\,1.68188,\quad
N\sigma=0.097.$ (46)

In this example, there is also a strong hierarchy among both the constants
multiplying the modular forms present in $M_{D}$ and the other constants in
$M_{D}$ and $M_{U}$.

The results of the fit of the observables are shown in Table 12.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $9.24$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00352$ | $2.83\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.4^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 12: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

##  5 Summary and discussions

We have discussed the strong CP problem within the modular invariance approach
to flavour. Working with $A_{4}$ ($N=3$) modular symmetry we have constructed
simple models which provide solution to the strong CP problem without the need
for an axion. In these models it is assumed that CP is a fundamental symmetry
of the Lagrangian. As a consequence, the strong CPV phase $\bar{\theta}=0$.
The CP symmetry is broken spontaneously by the VEV of the modulus $\tau$
($\tau_{\rm VEV}$), so the large CPV phases in the CKM matrix is generated and
at the same time $\bar{\theta}$ remains zero or gets a tiny value compatible
with the existing stringent experimental limit $\bar{\theta}<10^{-10}$.

To be more specific, we have considered three models, i.e., three types of
down-type and up-type mass matrices $M_{D}$ and $M_{U}$, which have, as a
consequence of the $A_{4}$ modular symmetry, three zero elements, or three
texture zeros, each (Eqs. (11), (18) and (22)). The position of the zeros and
the requirement of CP invariance ensures that ${\rm det}M_{D}$ and ${\rm
det}M_{U}$ are real quantities. The quark mass matrices $M_{D}$ and $M_{U}$
contain three $A_{4}$ modular forms of weights 12 and 16 which are ${\bf
1^{\prime}}$ singlets, $Y^{(12)}_{1^{\prime}}$, $Y^{(16)}_{1^{\prime}A}$ and
$Y^{(16)}_{1^{\prime}B}$, and one modular form of weight 4 which is ${\bf 1}$
singlet, $Y^{(4)}_{1}$. The presence of two modular forms of the same weight
furnishing the same singlet representation of $A_{4}$ is required in order to
describe correctly the observed CP violation in the quark sector when the CP
symmetry is broken spontaneously by $\tau_{\rm VEV}$. In the case of model (3)
we have considered phenomenologically the presence of three additional zero
elements in $M_{U}$ to reduce the number of redundant constant parameters in
the model. This set-up, in which $M_{U}$ is a diagonal matrix, is not
guaranteed by modular invariance without additional (symmetry) assumptions.

We have performed statistical analyses of the three models and have shown that
they are phenomenologically viable. We have shown that model (1) (Eq. (11)),
model (2) (Eq. (18) and model (3) (Eq. (22) and the related discussion) can
describe well the quark data for values of $\tau_{\rm VEV}$ close to $i$,
$\tau_{\rm VEV}$ close to $\omega=-\,0.5+i\,\sqrt{3}/2$ (the left cusp) and
$\tau_{\rm VEV}$ “close” to $i\infty$. The values of the VEV of the modulus
$\tau$, $\tau_{\rm VEV}=i$, $\tau_{\rm VEV}=\omega$ and $\tau_{\rm
VEV}=i\infty$, as is well known, are the only fixed points of the modular
group in its fundamental domain. Values of $\tau_{\rm VEV}$ close to three
fixed points have been found in studies of the modulus stabilisation (see,
e.g., [69, 70, 71, 72, 73, 74, 75, 76, 77]), in which $\tau_{\rm VEV}$ is
obtained by analysing the absolute minima of relatively simple (supergravity-
motivated) modular- and CP-invariant potentials for the modulus $\tau$. The
results for $\tau_{\rm VEV}$ we have found in the statistical analyses of the
three models considered by us are broadly consistent with the results on
modulus stabilisation.

Our work is an extension of the pioneering work of Ref.[28] where one pattern
of texture zeros is considered in the framework of the level $N=1$ modular
symmetry. We have presented phenomenologically viable models with finite
modular symmetry of level $N=3$ ($A_{4}$) by using three patterns of texture
zeros. We have studied the distributions of the VEV of the modulus $\tau$
allowing to reproduce the observed quark masses CKM mixing angles and CP
violation. And we have found examples of the models which are viable for
values of $\tau_{\rm VEV}$ close to the fixed points of the modular group,
which are broadly consistent with the results on modulus stabilisation.

Our work together with Ref.[28] promotes the modular invariance as a
successful approach and framework providing solutions not only to the quark
and lepton flavour problems but also to the strong CP problem.

## Acknowledgments

The work of S. T. P. was supported in part by the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 860881-HIDDeN, by the Italian INFN program on Theoretical
Astroparticle Physics and by the World Premier International Research Center
Initiative (WPI Initiative, MEXT), Japan. The authors would like to thank
Kavli IPMU, University of Tokyo, where part of this study was done, for the
kind hospitality.

## Appendix

##  Appendix A Modular forms of $A_{4}$ with higher weights

The lowest weight 2 triplet modular forms of $A_{4}$ are given as:

$\displaystyle{\bf Y^{(\rm 2)}_{3}}=\begin{pmatrix}Y_{1}\\\ Y_{2}\\\
Y_{3}\end{pmatrix}=\begin{pmatrix}1+12q+36q^{2}+12q^{3}+\dots\\\
-6q^{1/3}(1+7q+8q^{2}+\dots)\\\ -18q^{2/3}(1+2q+5q^{2}+\dots)\end{pmatrix}\,,$
(47)

where $q=\exp[2\pi i\tau]$. They satisfy also the constraint [24]:

$\displaystyle Y_{2}^{2}+2Y_{1}Y_{3}=0~{}.$ (48)

For weight 4, five modular forms are given as:

where ${\bf Y^{(\rm 4)}_{1^{\prime\prime}}}$ vanishes due to the constraint of
Eq. (48).

For weigh 6, there are seven modular forms as:

For weigh 8, there are nine modular forms as:

For weigh 10, there are eleven modular forms as:

$\displaystyle{\bf Y^{(\rm
10)}_{1}}=(Y_{1}^{2}+2Y_{2}Y_{3})(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3})=E_{10}=E_{4}E_{6}\,,$
$\displaystyle{\bf Y^{(\rm
10)}_{1^{\prime}}}=(Y_{3}^{2}+2Y_{1}Y_{2})(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3})\
,$ $\displaystyle{\bf Y^{(\rm 10)}_{3,{\rm
1}}}\equiv\begin{pmatrix}Y_{1,1}^{(10)}\\\ Y_{2,1}^{(10)}\\\
Y_{3,1}^{(10)}\end{pmatrix}=(Y_{1}^{2}+2Y_{2}Y_{3})^{2}\begin{pmatrix}Y_{1}\\\
Y_{2}\\\ Y_{3}\end{pmatrix}\,,$ $\displaystyle{\bf Y^{(\rm 10)}_{3,{\rm
2}}}\equiv\begin{pmatrix}Y_{1,2}^{(10)}\\\ Y_{2,2}^{(10)}\\\
Y_{3,2}^{(10)}\end{pmatrix}=(Y_{3}^{2}+2Y_{1}Y_{2})^{2}\begin{pmatrix}Y_{2}\\\
Y_{3}\\\ Y_{1}\end{pmatrix}\,,$ $\displaystyle{\bf Y^{(\rm 10)}_{3,{\rm
3}}}\equiv\begin{pmatrix}Y_{1,3}^{(10)}\\\ Y_{2,3}^{(10)}\\\
Y_{3,3}^{(10)}\end{pmatrix}=(Y_{1}^{2}+2Y_{2}Y_{3})(Y_{3}^{2}+2Y_{1}Y_{2})\begin{pmatrix}Y_{3}\\\
Y_{1}\\\ Y_{2}\end{pmatrix}\,.$ (49)

For weigh 12, there are thirteen modular forms as:

For weigh 14, there are fifteen modular forms as:

$\displaystyle{\bf Y^{(\rm
14)}_{1}}=(Y_{1}^{2}+2Y_{2}Y_{3})^{2}(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3})=E_{4}^{2}E_{6}\,,$
$\displaystyle{\bf Y^{(\rm
14)}_{1^{\prime}}}=(Y_{1}^{2}+2Y_{2}Y_{3})(Y_{3}^{2}+2Y_{1}Y_{2})(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3})\,,$
$\displaystyle{\bf Y^{(\rm
14)}_{1"}}=(Y_{3}^{2}+2Y_{1}Y_{2})^{2}(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3})\,.$
(50)

Four triplets are obtained by ${\bf Y^{(\rm 10)}_{3}}\otimes{\bf Y^{(\rm
4)}_{1}}$ and ${\bf Y^{(\rm 8)}_{3}}\otimes{\bf Y^{(\rm 6)}_{1}}$.

For weigh 16, there are seventeen modular forms as:

$\displaystyle{\bf Y^{(\rm 16)}_{1{\rm
A}}}=(Y_{1}^{2}+2Y_{2}Y_{3})^{4}=E_{4}^{4}\,,\qquad\qquad{\bf Y^{(\rm
16)}_{1{\rm
B}}}=(Y_{1}^{2}+2Y_{2}Y_{3})(Y_{3}^{2}+2Y_{1}Y_{2})^{3}=E_{4}(E_{6}^{2}-E_{4}^{3})\,,$
$\displaystyle{\bf Y^{(\rm 16)}_{1^{\prime}{\rm
A}}}=(Y_{1}^{2}+2Y_{2}Y_{3})^{3}(Y_{3}^{2}+2Y_{1}Y_{2})\,,\quad{\bf Y^{(\rm
16)}_{1^{\prime}{\rm B}}}=(Y_{3}^{2}+2Y_{1}Y_{2})^{4}\,,$ $\displaystyle{\bf
Y^{(\rm 16)}_{1"}}=(Y_{1}^{2}+2Y_{2}Y_{3})^{2}(Y_{3}^{2}+2Y_{1}Y_{2})^{2}\,.$
(51)

Four triplets are obtained by ${\bf Y^{(\rm 12)}_{3}}\otimes{\bf Y^{(\rm
4)}_{1}}$ and ${\bf Y^{(\rm 10)}_{3}}\otimes{\bf Y^{(\rm 6)}_{1}}$.

The modular form $E_{k}$ is the holomorphic normalized Eisenstein series with
weight $k$, which is given

$\displaystyle
E_{k}(\tau)=\frac{1}{{2\zeta(k)}}\sum_{(m,n)\not=(0.0)}\frac{1}{(m+n\tau)^{k}}\,,$
(52)

where $m$ and $n$ are integers.

We show the values of singlets of modular form at the fixed points, $i$,
$\omega$, $i\infty$ in Table 13.

$k$ | $\bf r$ | $\tau_{0}=i$ | $\tau_{0}=\omega$ | $\tau_{0}=i\infty$
---|---|---|---|---
$2$ | ${\bf 3}\,(Y_{1},Y_{2},Y_{3})$ | $Y_{0}\,(1,1-\sqrt{3},-2+\sqrt{3})$ | $Y_{0}\,(1,\omega,-\frac{1}{2}\omega^{2})$ | $Y_{0}^{2}\,(1,0,0)$
$4$ | $\\{\bf 1,\ 1^{\prime}\\}$ | $Y_{0}^{2}\,\\{(6\sqrt{3}-9),-(6\sqrt{3}-9)\\}$ | $Y_{0}^{2}\,(0,\ \frac{9}{4}\omega)$ | $Y_{0}^{2}\,\\{1,\ 0\\}$
$6$ | $\bf 1$ | 0 | $Y_{0}^{3}\,\frac{27}{8}$ | $Y_{0}^{3}$
$8$ | $\bf 1$ | $Y_{0}^{4}\,27\,(7-4\sqrt{3})$ | 0 | $Y_{0}^{4}$
| $\bf 1^{\prime}$ | $-Y_{0}^{4}\,27\,(7-4\sqrt{3})$ | 0 | 0
| $\bf 1^{\prime\prime}$ | $Y_{0}^{4}\,27\,(7-4\sqrt{3})$ | $Y_{0}^{4}\,\frac{81}{16}\omega^{2}$ | 0
$10$ | $\\{\bf 1,\ 1^{\prime}\\}$ | $Y_{0}^{5}\,\\{0,\ 0\\}$ | $Y_{0}^{5}\,(0,\ \frac{243}{32}\omega)$ | $Y_{0}^{5}\,\\{1,\ 0\\}$
$12$ | $\\{\bf 1_{\rm A},\ 1_{\rm B}\\}$ | $81Y_{0}^{6}\,\\{\,(26\sqrt{3}-45),-(26\sqrt{3}-45)\\}$ | $Y_{0}^{6}\,\\{0,\ \frac{729}{64}\\}$ | $Y_{0}^{6}\,\\{1,\ 0\\}$
| $\\{\bf 1^{\prime},\ 1^{\prime\prime}\\}$ | $-81Y_{0}^{6}\,\\{(26\sqrt{3}-45),-(26\sqrt{3}-45)\\}$ | {0, 0} | {0, 0}
$14$ | $\\{\bf 1,\ 1^{\prime},\ 1"\\}$ | $\\{0,\ 0,\ 0\\}$ | $Y_{0}^{7}\,\\{0,\ 0,\ \frac{2187}{128}\omega^{2}\\}$ | $Y_{0}^{7}\,\\{1,\ 0,\ 0\\}$
$16$ | $\\{\bf 1_{\rm A},\ 1_{\rm B}\\}$ | $729Y_{0}^{8}\,\\{-(56\sqrt{3}-97),(56\sqrt{3}-97)\\}$ | $\\{0,\ 0\\}$ | $Y_{0}^{8}\,(1,\ 0)$
| $\\{\bf 1^{\prime}_{A},\ 1^{\prime}_{B}\\}$ | $729Y_{0}^{8}\,\\{(56\sqrt{3}-97),-(56\sqrt{3}-97)\\}$ | $Y_{0}^{8}\,\\{0,\ \frac{6561}{256}\omega\\}$ | {0, 0}
| $1"$ | $-729Y_{0}^{8}\,(56\sqrt{3}-97)$ | 0 | 0
| $Y_{0}=Y_{1}(\tau_{0})$ | $Y_{1}(i)=1.0225...$ | $Y_{1}(\omega)=0.948...$ | $Y_{1}(i\infty)=1$
Table 13: Modular forms of singlets with weight
$k=4,\,6,\,8,\,10,\,12,\,14,\,16$, at the fixed point $\tau_{0}$.

##  Appendix B Input data of quark masses and CKM elements

The modulus $\tau$ breaks the modular invariance by obtaining a VEV at some
high mass scale. We assume this to be the GUT scale. Correspondingly, the
values of the quark masses and CKM parameters at the GUT scale play the role
of the observables that have to be reproduced by the considered quark flavour
models. They are obtained using the renormalisation group (RG) equations which
describe the “running” of the observables of interest from the electroweak
scale, where they are measured, to the GUT scale. In the analyses which follow
we adopt the numerical values of the quark Yukawa couplings at the GUT scale
$2\times 10^{16}$ GeV derived in the framework of the minimal SUSY breaking
scenarios with $\tan\beta=5$ [78]:

$\displaystyle\begin{aligned} &\frac{y_{d}}{y_{b}}=9.21\times 10^{-4}\ (1\pm
0.111)\,,\qquad\frac{y_{s}}{y_{b}}=1.82\times 10^{-2}\ (1\pm 0.055)\,,\\\
\rule[15.0pt]{0.0pt}{1.0pt}&\frac{y_{u}}{y_{t}}=5.39\times 10^{-6}\ (1\pm
0.311)\,,\qquad\frac{y_{c}}{y_{t}}=2.80\times 10^{-3}\ (1\pm 0.043)\,.\\\
\end{aligned}$ (68)

The quark masses are given as $m_{q}=y_{q}v_{H}$ with $v_{H}=174$ GeV. The
choice of relatively small value of $\tan\beta$ allows us to avoid relatively
large $\tan\beta$-enhanced threshold corrections in the RG running of the
Yukawa couplings. We set these corrections to zero.

The quark flavour mixing is given by the CKM matrix, which has three
independent mixing angles and one CP violating phase. These mixing angles are
given by the absolute values of the three CKM elements $|V_{us}|$, $|V_{cb}|$
and $|V_{ub}|$. We take the present data on the three CKM elements in Particle
Data Group (PDG) edition of Review of Particle Physics [7] as:

$\displaystyle\begin{aligned} |V_{us}|=0.22500\pm
0.00067\,,\quad|V_{cb}|=0.04182^{\pm
0.00085}_{-0.00074}\,,\quad|V_{ub}|=0.00369\pm 0.00011\,.\end{aligned}$ (69)

By using these values as input and $\tan\beta=5$ we obtain the CKM mixing
angles at the GUT scale of $2\times 10^{16}$ GeV [78]:

$\displaystyle|V_{us}|=0.2250\,(1\pm 0.0032)\,,\quad|V_{cb}|=0.0400\,(1\pm
0.020)\,,\quad|V_{ub}|=0.00353\,(1\pm 0.036)\,.$ (70)

The tree-level decays of $B\to D^{(*)}K^{(*)}$ are used as the standard
candles of the CP violation. The latest world average of the CP violating
phase is given in PDG2022 [7] as:

$\delta_{CP}={66.2^{\circ}}^{+3.4^{\circ}}_{-3.6^{\circ}}\,.$ (71)

Since the phase is almost independent of the evolution of RG equations, we
refer to this value in the numerical discussions. The rephasing invariant CP
violating measure $J_{\rm CP}$ [6] is also given in [7]:

$J_{\rm CP}=3.08^{+0.15}_{-0.13}\times 10^{-5}\,.$ (72)

Taking into account the RG effects on the mixing angles for $\tan\beta=5$, we
have at the GUT scale $2\times 10^{16}$ GeV:

$J_{\rm CP}=2.80^{+0.14}_{-0.12}\times 10^{-5}\,.$ (73)

##  Appendix C Quark mass matrices with texture zeros

In this Appendix, we present three numerical examples of the quark mass matrix
with the texture zeros, which are consistent with the observed masses and CKM
elements given in Appendix B.

###  C.1 Texture (1)

Consider the quark mass matrices with the texture zeros of model (1):

$\displaystyle M_{Q}=v_{Q}\begin{pmatrix}0&0&a_{Q}\\\ 0&b_{Q}&c_{Q}\\\
d_{Q}&e_{Q}&f_{Q}\,e^{i\varphi_{Q}}\end{pmatrix}_{RL},\qquad Q=D,U\ ,$ (74)

where
$a_{D},\,b_{D},\,c_{D},\,d_{D},\,e_{D},\,f_{D},\,a_{U},\,b_{U},\,c_{U},\,d_{U},\,e_{U},\,f_{U}$
are real constants, and $\phi_{D},\,\phi_{U}$ are CP violating (CPV) phases.
The determinants of the mass matrices $M_{D}$ and $M_{U}$ are real and are
given in terms of the real parameters $a_{Q}$, $b_{Q}$ and $d_{D}$:

$\displaystyle{\rm det}\,[M_{D}]=-a_{D}b_{D}d_{D}\,,\qquad\qquad{\rm
det}\,[M_{U}]=-a_{U}b_{U}d_{U}\,.$ (75)

If these mass matrices keep strictly their zero structures and describe
correctly the observed CP violation, they are candidates for solving the
strong CP problem.

Since the number of parameters in $M_{D}$ and $M_{U}$ are 14, four parameters
are redundant from the point of view of reproducing the six quark masses and
the four independent CKM mixing angles and CPV phase (10 observables ). We
present the mass matrices in the case of $|a_{Q}|\sim|b_{Q}|\sim|d_{Q}|$.
Using as input the observed quark masses and the CKM parameters and performing
a statistical analysis we show a typical numerical example of the mass
matrices which reproduce the quark data in Appendix B:

$\displaystyle M_{D}\sim\begin{pmatrix}0&0&1\\\ 0&0.901&-11.63\\\
0.105&-0.695&12.96\,e^{i24.36^{\circ}}\end{pmatrix},\ \
M_{U}\sim\begin{pmatrix}0&0&1\\\ 0&1.23&-80.80\\\
-0.586&9.63&386.81\,e^{-i150.26}\\\ \end{pmatrix},$ (76)

where $|a_{Q}|\sim|b_{Q}|\sim|d_{Q}|$ are approximately satisfied with only
$|d_{D}|$ being somewhat smaller than the other five constants 888The
magnitudes of the constants, including $c_{Q}$, $e_{Q}$ and $f_{Q}$, $Q=D,U$,
which in the modular $A_{4}$ model multiply the modular forms present in
$M_{D}$ and $M_{U}$, are discussed in section 4. . The matrices $M_{D}$ and
$M_{U}$ with the numerical values of the real constants and of the CPV phases
$\varphi_{D}=24.13^{\circ}$ and $\varphi_{U}=-150.26^{\circ}$ as given in Eq.
(76), provide a good quality of the fit of the quark mass ratios, the CKM
mixing angles and the CPV phase $\delta_{\rm CP}$ ($N\sigma=0.840$ C.L.), as
seen in Table 14.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.86$ | $9.52$ | $2.80$ | $4.18$ | $0.2249$ | $0.0400$ | $0.00355$ | $2.86\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $67.0^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 14: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

###  C.2 Texture (2)

Consider next the quark mass matrices with the texture zeros of model (2):

$\displaystyle M_{Q}=v_{Q}\begin{pmatrix}0&0&a_{Q}\\\ b_{Q}&0&c_{Q}\\\
e_{Q}&d_{Q}&f_{Q}\,e^{i\varphi_{Q}}\end{pmatrix}_{RL},\qquad Q=D,U\ ,$ (81)

where again the constants
$a_{D},\,b_{D},\,c_{D},\,d_{D},\,e_{D},\,f_{D},\,a_{U},\,b_{U},\,c_{U},\,d_{U},\,e_{U},\,f_{U}$
are real, and $\varphi_{D},\,\varphi_{U}$ are CPV phases. The determinants of
the mass matrices $M_{D}$ and $M_{U}$ are given in terms of
$a_{Q},\,b_{Q},\,d_{D}$ and are real:

$\displaystyle{\rm det}\,[M_{D}]=a_{D}b_{D}d_{D}\,,\qquad\qquad{\rm
det}\,[M_{U}]=a_{U}b_{U}d_{U}\,.$ (82)

If these mass matrices keep their zero elements strictly zero and describe the
the observed CP violation, they are candidates for solving the strong CP
problem.

Performing a statistical analysis we have found a typical numerical example of
the mass matrices $M_{D}$ and $M_{U}$ which describe the quark data in
Appendix B:

$\displaystyle M_{D}\sim\begin{pmatrix}0&0&1\\\ 0.0610&0&4.576\\\
0.0380&0.733&12.901\,e^{i\,104.836^{\circ}}\end{pmatrix},\ \
M_{U}\sim\begin{pmatrix}0&0&1\\\ 0.005752&0&12.44\\\
0.00290&1.625&83.89\,e^{i\,56.64}\\\ \end{pmatrix},$ (83)

The quark mass ratios and the CKM matrix obtained by diagonalising the
matrices $M_{D}$ and $M_{U}$ given in Eq. (83) are completely consistent with
observed one including the CP violating phase $\delta_{\rm CP}$
($N\sigma=0.217$ C.L.), as seen in Table 15.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $9.21$ | $2.82$ | $5.43$ | $0.2249$ | $0.0400$ | $0.00352$ | $2.82\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.3^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 15: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

###  C.3 Texture (3)

Let us analyse finally the quark mass matrices with the texture zeros of model
(3):

$\displaystyle M_{Q}=v_{Q}\begin{pmatrix}a_{Q}&0&0\\\ c_{Q}&b_{Q}&0\\\
f_{Q}\,e^{i\varphi_{Q}}&e_{Q}&d_{Q}\end{pmatrix}_{RL},\qquad Q=D,U\ ,$ (88)

where as in the previous two cases the constants $a_{D},\,b_{D}$,
$c_{D},\,d_{D}$, $e_{D},\,f_{D}$, $a_{U},\,b_{U}$, $c_{U},\,d_{U}$,
$e_{U},\,f_{U}$ are real, and $\varphi_{D},\,\varphi_{U}$ are CPV phases. Also
in this case the determinants of the quark mass matrices $M_{D}$ and $M_{U}$
are given in terms of real parameters, namely of $a_{Q},\,b_{Q},\,d_{D}$, and
are real:

$\displaystyle{\rm det}\,[M_{D}]=a_{D}b_{D}d_{D}\,,\qquad\qquad{\rm
det}\,[M_{U}]=a_{U}b_{U}d_{U}\,,$ (89)

As in the previous two cases we note that if the zero elements of these mass
matrices remain strictly zero, and the mass matrices describe correctly the
observed quark masses, and especially the quark mixing angles and CP violation
in the quark sector, they are a candidate for solving the strong CP problem.

We will present next a typical numerical example of the mass matrices which
describe the quark data. Since the number of parameters is 14, four parameters
are redundant for reproducing the quark masses and CKM elements (10
observables). In this case, following Ref.[89], we set $c_{U}=e_{Q}=f_{U}=0$
(also $\varphi_{U}=0$). Performing a statistical analysis we get a good
description of the quark data in Appendix B with:

$\displaystyle M_{D}\sim\begin{pmatrix}0.000948&0&0\\\ 0.004209&0.01828&0\\\
0.003519\,e^{i\,66.25^{\circ}}&0.0400&1\end{pmatrix},\ \
M_{U}\sim\begin{pmatrix}5.39\times 10^{-6}&0&0\\\ 0&2.80\times 10^{-3}&0\\\
0&0&1\\\ \end{pmatrix},$ (90)

The quark mass ratios, and especially the CKM mixing angles, the $J_{\rm CP}$
factor and the CPV phase $\delta_{\rm CP}$ obtained using the numerical
matrices given in Eq. (90) are completely consistent with observed one
($N\sigma=0.113$ C.L.), as seen in Table 16.

| $\frac{m_{s}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{2}$ | $\frac{m_{d}}{m_{b}}\hskip-2.84526pt\times\hskip-2.84526pt10^{4}$ | $\frac{m_{c}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{3}$ | $\frac{m_{u}}{m_{t}}\hskip-2.84526pt\times\hskip-2.84526pt10^{6}$ | $|V_{us}|$ | $|V_{cb}|$ | $|V_{ub}|$ | $|J_{\rm CP}|$ | $\delta_{\rm CP}$
---|---|---|---|---|---|---|---|---|---
Fit | $1.87$ | $9.23$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00352$ | $2.82\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
Exp | $1.82$ | $9.21$ | $2.80$ | $5.39$ | $0.2250$ | $0.0400$ | $0.00353$ | $2.80\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | $66.2^{\circ}$
$1\,\sigma$ | $\pm 0.10$ | $\pm 1.02$ | $\pm 0.12$ | $\pm 1.68$ | $\pm 0.0007$ | $\pm 0.0008$ | $\pm 0.00013$ | ${}^{+0.14}_{-0.12}\hskip-2.84526pt\times\hskip-2.84526pt10^{-5}$ | ${}^{+3.4^{\circ}}_{-3.6^{\circ}}$
Table 16: Results of the fits of the quark mass ratios, CKM mixing angles,
$J_{\rm CP}$ and $\delta_{\rm CP}$. ’Exp’ denotes the values of the
observables at the GUT scale, including $1\sigma$ error.

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