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

In order to keep ${\rm det}\,[M_{D}M_{U}]$ to be real, the following condition of the weights are required [29]:

$\displaystyle\sum^{3}_{i=1}(2\,k_{Qi}+k_{d_{i}}^{c}+k_{u_{i}}^{c})=0\,,$

(5)

where $k_{Qi}$ and $k_{d_{i}}^{c}(k_{u_{i}}^{c})$ denote weights for the left-handed quarks and right-handed down (up)-type quarks, respectively. The weights of Higgs are set to be zero. The condition of Eq.(5) guarantees that the modular symmetry has no QCD anomaly.

We assign the $A_{4}$ representations and the weights for quarks and Higgs 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 $(-8,4,8)$, respectively.

• •

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

The assigned weights satisfy the condition in Eq.(5). These assignments are summarized in Table 1.

Taking account of the following tensor products of $A_{4}$ singlets [55, 56, 57],

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

(6)

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:

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

(8)

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 [92]. However, those do not affect $\arg\det M_{Q}$ as discussed in Ref.[29].

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

(9)

where $\psi^{(I)}$ denotes a chiral superfield with weight $k_{I}$, and $\bar{\tau}$ is the anti-holomorphic modulus. The (anti-)holomorphic modulus 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. (9) are not canonical. The canonical form is obtained by an overall re-normalization of the quark superfields of interest:

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

(10)

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

	$\displaystyle c_{Q}\rightarrow\hat{c}_{Q}=c_{Q}\,\sqrt{(2{\rm Im}\tau)^{12}}=c_{Q}\,(2{\rm Im}\tau)^{6},\,$
	$\displaystyle e_{Q}\rightarrow\hat{e}_{Q}=e_{Q}\,\sqrt{(2{\rm Im}\tau)^{4}}=e_{Q}\,(2{\rm Im}\tau)^{2},\,$
	$\displaystyle f_{Q}\rightarrow\hat{f}_{Q}=f_{Q}\,\sqrt{(2{\rm Im}\tau)^{16}}=f_{Q}\,(2{\rm Im}\tau)^{8}\,,\qquad Q=D,\,U\,.$
			(11)

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:

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

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

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

(13)

The phase structure of the matrices in Eq. (12) 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:

	$\displaystyle M_{D}=v_{D}P_{R}^{*}\begin{pmatrix}0&0&a_{D}\\ 0&b_{D}&c_{D}(2{\rm Im}\tau)^{6}|Y_{\bf 1^{\prime}}^{(12)}|\\ d_{D}&e_{D}(2{\rm Im}\tau)^{2}|Y_{\bf 1}^{(4)}|&f_{D}(2{\rm Im}\tau)^{8}(g_{D}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm B}}^{(16)})e^{-i(\varphi_{4}+\varphi_{12})}\end{pmatrix}P_{L}\,,$
	$\displaystyle M_{U}=v_{U}P_{R}^{*}\begin{pmatrix}0&0&a_{U}\\ 0&b_{U}&c_{U}(2{\rm Im}\tau)^{6}|Y_{\bf 1^{\prime}}^{(12)}|\\ d_{U}&e_{U}(2{\rm Im}\tau)^{2}|Y_{\bf 1}^{(4)}|&f_{U}(2{\rm Im}\tau)^{8}(g_{U}Y_{\bf 1^{\prime}{\rm A}}^{(16)}+Y_{\bf 1^{\prime}{\rm B}}^{(16)})e^{-i(\varphi_{4}+\varphi_{12})}\end{pmatrix}P_{L}\,.$		(14)

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

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

(15)

where

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

(16)

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. (14):

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

(17)

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. (14), 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. (81) 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. (14), 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. (17) 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. (81). These numerical values are derived by fitting the quark masses, the CKM matrix elements and the Jarlskog rephasing invariant $J_{CP}$ quoted in Eqs. (73), (75) and (78) 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. (73), (75) and (78) of Appendix B.

Let us compare the allowed region obtained by us in the $A_{4}$ modular model with quark mass matrices given in Eq. (14) with that in the model proposed in Ref. [29], 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. (8):

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

(18)

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.

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 satisfy the weight condition of Eq. (5). Those are summarized in Table 2.

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

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

(19)

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:

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

(20)

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

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

(21)

where $\varphi_{4}$ and $\varphi_{12}$ are defined in Eq.(16). 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. (20),

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

(22)

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. (86) of Appendix C.2, which have the same structure as the the mass matrices in Eq. (20), 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. (88) 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. (88) 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. (20) 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. (73), (75) and (78) of Appendix B.

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 satisfy the weight condition of Eq.(5). Those are summarized in Table 3.

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

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

(23)

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.[90], 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:

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

(24)

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

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

(25)

where $\varphi_{4}$ and $\varphi_{12}$ are given in Eq. (16).