Modular flavor symmetric models ¹¹1The contribution to a special book dedicated to the memory of Professor Harald Fritzsch

One of important mysteries in particle physics is the origin of the flavor structure, i.e., fermion mass hierarchies, their mixing angles, and CP violation. Various studies have been done to understand their origin. One of traditional approaches is the texture zeros proposed by Weinberg and Fritzsch, where zero entries are put in the fermion mass matrices [1, 2]. Indeed, the Fritzsch Ansatz [3, 4] gave significant prediction power for the flavor mixing although the origin of zeros are unclear. This approach leads to the texture zero analysis where some elements of mass matrices are required to be zero to reduce the degrees of freedom in mass matrices. Some famous works have been made in the texture zeros [5, 6, 7, 8, 9]. Those zeros can be related with certain symmetries.

Flavor symmetries are interesting approaches to attack the origin of fermion mass hierarchies and mixing angles. Froggatt-Nielsen have taken the $U(1)$ symmetry to explain the observed masses and mixing angles for quarks [10]. Furthermore, the $S_{3}$ symmetry was used for quark mass matrices [11, 12]. It was also discussed to understand the large mixing angle [13] in the oscillation of atmospheric neutrinos [14]. For the last twenty years, non-Abelian discrete flavor symmetries have been developed, that is motivated by the precise observation of flavor mixing angles of leptons [15, 16, 17, 19, 20, 21, 22, 23, 24, 18, 25].

The standard model (SM) is the low-energy effective field theory from the viewpoint of underlying theory, and it is referred to as the SM effective field theory (SMEFT) [26, 27, 28]. The SMEFT includes many higher dimensional operators, and they contribute to flavor changing processes and muon $(g-2)$. Flavor symmetries are useful not only to derive realistic fermion masses and their mixing angles, but to control higher dimensional operators in the SMEFT. Indeed, the $U(3)^{5}$ and $U(2)^{5}$ symmetries control the SMEFT operators [29]. The $U(3)^{5}$ symmetry [30] allows us to apply the Minimal Flavor Violation (MFV) hypothesis [31, 32], which is the most restrictive hypothesis consistent with the SMEFT. In the $U(2)^{5}$ symmetry [33, 34, 35], it retains most of the MFV virtues and allows us to have a much richer structure as far as the dynamics of third family is concerned. Thus, flavor symmetries are useful to connect between the low-energy physics and high-energy physics such as superstring theory.

Superstring theory is a promising candidate for unified theory of all the interactions including gravity and matters such as quarks and leptons, and Higgs modes. Superstring theory must have six-dimensional (6D) compact space in addition to four-dimensional (4D) space times. Geometrical symmetries of 6D compact space control 4D effective field theory. For example, in certain compactifications there appears non-Abelian discrete symmetries such as $D_{4}$ and $\Delta(54)$ [36, 37, 38, 39, 40].

The modular symmetry is a geometrical symmetry of $T^{2}$ and $T^{2}/Z_{2}$, and corresponds to change of their cycle basis. Matter modes transform non-trivially under the modular symmetry. (See for hetetrotic string theory on orbifolds Refs. [41, 42, 43, 44] and magnetized D-brane models Refs. [45, 46, 47, 48, 49, 50, 51] . ²²2Calai-Yau manifolds have larger symplectic modular symmetries of many moduli [52, 53, 54, 55] .) That is, the modular symmetry is a flavor symmetry. Indeed, finite modular groups include $S_{3}$, $A_{4}$, $S_{4}$, $A_{5}$, which have been used in 4D flavor models so far, while $\Delta(98)$ and $\Delta(384)$ are also included as subgroups.

The well-known finite groups $S_{3}$, $A_{4}$, $S_{4}$ and $A_{5}$ are isomorphic to the finite modular groups $\Gamma_{N}$ for $N=2,3,4,5$, respectively[56]. The lepton mass matrices have been presented in terms of $\Gamma_{3}\simeq A_{4}$ modular forms [57]. Modular forms have also been obtained for $\Gamma_{2}\simeq S_{3}$ [58], $\Gamma_{4}\simeq S_{4}$ [59] and $\Gamma_{5}\simeq A_{5}$ [60, 61], respectively. By using them, the viable lepton mass matrices have been obtained for $\Gamma_{4}\simeq S_{4}$ [59], and then $\Gamma_{5}\simeq A_{5}$ [60, 61].

The 4D CP symmetry can be embedded into a proper Lorentz symmetry in higher dimensional theory such as superstring theory [62, 63, 64, 65, 66, 67]. From this viewpoint, CP violation in 4D effective field theory would originate from the compactification, that is, the moduli stabilization. (See for early studies on the CP violation through the moduli stabilization Refs. [68, 69, 70, 71] .) Recently, the spontaneous breaking of the CP symmetry was studied through the moduli stabilization due to 3-form fluxes Refs. [72, 54]. In modular flavor symmetric models, the CP symmetry is combined with the modular symmetry as well as other symmetries, and is enlarged [73, 74, 75, 76, 77, 55] ³³3See for the CP symmetry in the Calabi-Yau compactification Refs. [54, 55, 78] .. The CP-invariant vacua and CP-preserving modulus values increase by the modular symmetry. It is important to study the CP violation in such models with the enlarged symmetry [79, 80].

Higher dimensional operators can be computed within the framework of superstring theory. Allowed couplings are controlled by stringy symmetries and $n$-point couplings are written by products of 3-point couplings. The modular flavor symmetry also control these higher dimensional operators [81, 82, 83].

In addition to the above aspects, the modular flavor symmetries were recently extended to models for dark matter [84, 85], soft supersymmetry breaking terms [86, 87, 88], matter parity [89], the strong CP problem[90], etc.

The paper is organized as follows. In section 2, we give a brief review on modular symmetry. In section 3, we study modular flavor symmetric models. As an illustrating example, we explain $A_{4}$ modular symmetric models. In section 4, we study texture structure at the fixed point $\tau=\omega$. In section 5, we study CP violation in modular symmetric models. In section 6, we study SMEFT with modular flavor symmetry. Section 7 is devoted to conclusion. In Appendix A, we review modular forms of $A_{4}$. In Appendix B, the tensor product decomposition is given in the $A_{4}$ group.

The modular symmetry is a geometrical symmetry of the two-dimensional torus, $T^{2}$. The two-dimensional torus is constructed as division of the two-dimensional Euclidean space $R^{2}$ by a lattice $\Lambda$, $T^{2}=R^{2}/\Lambda$. Instead of $R^{2}$, one can use the one-dimensional complex plane. As shown in Fig. 1, the lattice is spanned by two basis vectors, $e_{1}$ and $e_{2}$ as $m_{1}e_{1}+m_{2}e_{2}$, where $m_{1}$ and $m_{2}$ are integer. Their ratio,

$\displaystyle\tau=\frac{e_{2}}{e_{1}},$

(1)

in the complex plane, represents the shape of $T^{2}$, and the parameter $\tau$ is called the modulus.

The same lattice can be spanned by other basis vectors such as

$\displaystyle\left(\begin{array}[]{c}e^{\prime}_{2}\\ e^{\prime}_{1}\end{array}\right)=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\left(\begin{array}[]{c}e_{2}\\ e_{1}\end{array}\right),$

(8)

where $a,b,c,d$ are integer satisfying $ad-bc=1$. That is the $SL(2,Z)$.

Under the above transformation, the modulus $\tau$ transforms as follows,

$\displaystyle\tau\longrightarrow\tau^{\prime}=\gamma\tau=\frac{a\tau+b}{c\tau+d}\,.$

(9)

That is the modular symmetry [91, 92, 93, 94]. For the element $-e$ in $SL(2,Z)$,

$\displaystyle-e=\left(\begin{array}[]{cc}-1&0\\ 0&-1\end{array}\right),$

(12)

the modulus $\tau$ is invariant, $\tau\longrightarrow\tau^{\prime}=(-\tau)/(-1)=\tau$. Thus, the modular group is $\bar{\Gamma}=PSL(2,Z)=SL(2,Z)/\{e,-e\}$. It is sometimes called the inhomogeneous modular group. On the other hand, the group, $\Gamma=SL(2,Z)$ is called the homogeneous modular group or the full modular group.

The generators of $\Gamma\simeq SL(2,Z)$ are written by $S$ and $T$,

$\displaystyle S=\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right),\qquad T=\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right).$

(17)

They satisfy the following algebraic relations,

$\displaystyle S^{4}=(ST)^{3}=e.$

(18)

Note that

$\displaystyle S^{2}=-e.$

(19)

On $\bar{\Gamma}=PSL(2,Z)$, they satisfy

$\displaystyle S^{2}=(ST)^{3}=e.$

(20)

These relations are also confirmed explicitly by the following transformations:

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

(21)

which are shown on the lattice $\Lambda$ in Figs. 3 and 3.

In addition to the above algebraic relations of $\bar{\Gamma}=PSL(2,Z)$, we can require $T^{N}=e$, i.e.

$\displaystyle S^{2}=(ST)^{3}=T^{N}=e.$

(22)

They can correspond to finite groups such as $S_{3},A_{4},S_{4},A_{5}$ for $N=2,3,4,5$. In practice, we define the principal congruence subgroup $\Gamma(N)$ as

$\displaystyle\Gamma(N)=\left\{\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in{\Gamma}~{}\right|\left.\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)=\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right)~{}~{}({\rm mod}~{}N)\right\}.$

(29)

It includes $T^{N}$, but not $S$ or $T$. Then, we define the quotient $\Gamma_{N}=\bar{\Gamma}/\bar{\Gamma}(N)$, where the above algebraic relations are satisfied. It is found that $\Gamma_{N}$ with $N=2,3,4,5$ are isomorphic to $S_{3},A_{4},S_{4},A_{5}$, respectively [56].

We define $SL(2,Z_{N})$ by

$\displaystyle SL(2,Z_{N})=\left.\left\{\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\right|a,b,c,d\in Z_{N},ad-bc=1\right\},$

(32)

where $Z_{N}$ denotes integers modulo $N$. The group $\Gamma_{N}$ is isomorphic to $PSL(2,Z_{N})=SL(2,Z_{N})/\{e,-e\}$ for $N>2$, while $\Gamma_{2}$ is isomorphic to $SL(2,Z_{2})$, because $e=-e$ in $SL(2,Z_{2})$.

Similar to $\Gamma_{N}$, we can define $\Gamma^{\prime}_{N}=SL(2,Z)/\Gamma(N)$, and it is the double cover of $\Gamma_{N}$. That is, the groups $\Gamma^{\prime}_{N}$ for $N=3,4,5$ are isomorphic to the double covering groups of $A_{4},S_{4},A_{5}$, i.e. $T^{\prime},S^{\prime}_{4},A^{\prime}_{5}$, respectively, although $\Gamma_{2}^{\prime}$ is isomorphic to $S_{3}$.

The upper half-plane of the modulus space $\tau$ is mapped onto itself. For example, $\Gamma$ does not include the basis change. $(e_{1},e_{2})\longrightarrow(e_{1},-e_{2})$, i.e. $\tau\to-\tau$. In practice, we find

$\displaystyle{\rm Im}(\gamma\tau)=|c\tau+d|^{-2}{\rm Im}(\tau).$

(33)

Thus, the modular group is represented on the upper half-plane of $\tau$. Obviously one can map any value of $\tau$ on the upper half-plane into the region, $-\frac{1}{2}\leq{\rm Re}(\tau)\leq\frac{1}{2}$ by $T^{n}$. Furthermore, by the modular transformation one can map any value of $\tau$ on the upper half-plane into the following region:

$\displaystyle-\frac{1}{2}\leq{\rm Re}(\tau)\leq\frac{1}{2},\qquad|\tau|>1,$

(34)

which is called the fundamental domain. Suppose that ${\rm Im}(\gamma\tau)$ is a maximum value among all of $\gamma$ for a fixed value of $\tau$. If $|\gamma\tau|<1$, we map it by $S$, and we find

$\displaystyle{\rm Im}(S\gamma\tau)=\frac{{\rm Im}(\gamma\tau)}{|\gamma\tau|^{2}}>{\rm Im}(\gamma\tau).$

(35)

That is inconsistent with the assumption that ${\rm Im}(\gamma\tau)$ is a maximum value among all of $\gamma$. That is, we find $|\gamma\tau|>1$. Thus, we can map $\tau$ on the upper half-plane into the fundamental region by the modular transformation. The point $\tau=i$ is the fixed point under $S$ because $S:i\to-\frac{1}{i}=i$, where $Z_{2}$ symmetry remains. Similarly, the point $\tau=e^{2\pi i/3}$ is the fixed point under $ST$, where $Z_{3}$ symmetry remains.

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

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

(36)

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

Under the modular transformation, chiral superfields $\psi_{i}$ ($i$ denotes flavors) with weight $-k$ transform as [95]

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

(37)

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

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

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

(38)

where the superfield and its scalar component are denoted by the same letter, and $\bar{\tau}=\tau^{*}$ after taking vacuum expectation value (VEV) of $\tau$. The canonical form of the kinetic terms is obtained by changing the normalization of parameters. The general Kähler potential consistent with the modular symmetry possibly contains additional terms [96]. However, we consider only the simplest form of the Kähler potential.

In this section, we discuss the flavor model of quark and lepton mass matrices. There is a difference between the modular symmetry and the usual flavor symmetry. Coupling constants such as Yukawa couplings also transform non-trivially under the modular symmetry and are written as functions of the modulus called modular forms, which are holomorphic functions of the modulus $\tau$. On the other hand, coupling constants are invariant under the traditional flavor symmetries.

The flavor model of lepton mass matrices have been proposed based on the finite modular group $\Gamma_{3}\simeq A_{4}$ [57]. This approach based on modular invariance opened up a new promising direction in the studies of the flavor physics and correspondingly in flavor model building.

Texture zeros of the fermion mass matrix provide an attractive approach to understand the flavor mixing. Those can be related with some flavor symmetries. Indeed, zeros of the mass matrix are discussed in the modular symmetry of flavors [162, 163, 121].

The flavor structure has been investigated in magnetized orbifold models with multi-Higgs modes[228, 229, 230], which are interesting compactification from higher dimensional theory such as superstring theory. They lead to a four-dimensional chiral theory, which has the modular symmetry [45, 46, 47, 48, 49, 50, 51]. In these models, quark and lepton masses and their mixing angles were discussed [231, 233, 232, 214, 234, 235, 236]. These magnetized orbifold models lead to multi-Higgs modes, while generic string compactification also leads more than one candidates for Higgs fields.

In this section, we show that texture zeros are generally realized at the fixed points $\tau=\omega$ in the modular flavor models with multi-Higgs [237].