APCTP Pre2020-033 Spontaneous CP violation by modulus $\tau$
in $A_{4}$ model of lepton flavors

Let us start with discussing the generalised $CP$ symmetry [92, 99]. The CP transformation is non-trivial if the non-Abelian discrete flavor symmetry $G$ is set in the Yukawa sector of a Lagrangian. Let us consider the chiral superfields. The CP is a discrete symmetry which involves both Hermitian conjugation of a chiral superfield $\psi(x)$ and inversion of spatial coordinates,

$$\psi(x)\rightarrow{\bf X}_{\bf r}\overline{\psi}(x_{P})\ ,$$

(1)

where $x_{P}=(t,-{\bf x})$ and ${\bf X_{r}}$ is a unitary transformations of $\psi(x)$ in the irreducible representation $\bf r$ of the discrete flavor symmetry $G$. If ${\bf X_{r}}$ is the unit matrix, the $CP$ transformation is the trivial one. This is the case for the continuous flavor symmetry [99]. However, in the framework of the non-Abelian discrete family symmetry, non-trivial choices of ${\bf X_{r}}$ are possible. The unbroken $CP$ transformations of ${\bf X_{r}}$ form the group $H_{CP}$. Then, ${\bf X_{r}}$ must be consistent with the flavor symmetry transformation,

$$\psi(x)\rightarrow{\rho}_{\bf r}(g)\psi(x)\ ,\quad g\in G\ ,$$

(2)

where ${\rho}_{\bf{r}}(g)$ is the representation matrix for $g$ in the irreducible representation $\bf{r}$.

The consistent condition is obtained as follows. At first, perform a $CP$ transformation $\psi(x)\rightarrow{\bf X}_{\bf r}\overline{\psi}(x_{P})$, then apply a flavor symmetry transformation, $\overline{\psi}(x_{P})\rightarrow{\rho}_{\bf r}^{*}(g)\overline{\psi}(x_{P})$, and finally perform an inverse CP transformation. The whole transformation is written as $\psi(x)\rightarrow{\bf X}_{\bf r}\rho^{*}(g){\bf X}^{-1}_{\bf r}\psi(x)$, which must be equivalent to some flavor symmetry $\psi(x)\rightarrow{\rho}_{\bf r}(g^{\prime})\psi(x)$. Thus, one obtains [100]

$${\bf X}_{\bf r}\rho_{\bf r}^{*}(g){\bf X}^{-1}_{\bf r}={\rho}_{\bf r}(g^{\prime})\ ,\qquad g,\,g^{\prime}\in G\ .$$

(3)

This equation defines the consistency condition, which has to be respected for consistent implementation of a generalized CP symmetry along with a flavor symmetry [102, 101]. This chain $CP\rightarrow g\rightarrow CP^{-1}$ maps the group element $g$ onto $g^{\prime}$ and preserves the flavor symmetry group structure. That is a homomorphism $v(g)=g^{\prime}$ of $G$. Assuming the presence of faithful representations $\bf r$, Eq. (3) defines a unique mapping of $G$ to itself. In this case, $v(g)$ is an automorphism of $G$ [101].

It has been also shown that the full symmetry group is isomorphic to a semi-direct product of $G$ and $H_{CP}$, that is $G\rtimes H_{CP}$, where $H_{CP}\simeq\mathbb{Z}_{2}^{CP}$, is the group generated by the generalised $CP$ transformation under the assumption of $\bf X_{r}$ being a symmetric matrix [102].

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

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

(4)

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

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

(5)

which satisfy the following algebraic relations,

$$S^{2}=\mathbb{1}\ ,\qquad(ST)^{3}=\mathbb{1}\ .$$

(6)

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

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

(7)

For $N=2$, we define $\bar{\Gamma}(2)\equiv\Gamma(2)/\{\rm I,-I\}$. Since the element $\rm-I$ does not belong to $\Gamma(N)$ for $N>2$, we have $\bar{\Gamma}(N)=\Gamma(N)$. The quotient groups defined as $\Gamma_{N}\equiv\bar{\Gamma}/\bar{\Gamma}(N)$ are finite modular groups. In these finite groups $\Gamma_{N}$, $T^{N}=\mathbb{1}$ is imposed. The groups $\Gamma_{N}$ with $N=2,3,4,5$ are isomorphic to $S_{3}$, $A_{4}$, $S_{4}$ and $A_{5}$, respectively [23].

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

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

(8)

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

Superstring theory on the torus $T^{2}$ or orbifold $T^{2}/Z_{N}$ has the modular symmetry [103, 104, 105, 106, 107, 108]. Its low energy effective field theory is described in terms of supergravity theory, and string-derived supergravity theory has also the modular symmetry. Under the modular transformation of Eq. (4), chiral superfields $\psi_{i}$ ($i$ denotes flavors) transform as [109],

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

(9)

We study global supersymmetric models, e.g., minimal supersymmetric extensions of the Standard Model (MSSM). 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

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

(10)

where the superfield and its scalar component are denoted by the same letter, and $\bar{\tau}=\tau^{*}$ after taking VEV of $\tau$. Therefore, the canonical form of the kinetic terms is obtained by changing the normalization of parameters [28]. The general Kähler potential consistent with the modular symmetry possibly contains additional terms [110]. However, we consider only the simplest form of the Kähler potential.

For $\Gamma_{3}\simeq A_{4}$, the dimension of the linear space ${\cal M}_{k}(\Gamma{(3)})$ of modular forms of weight $k$ is $k+1$ [111, 112, 113], i.e., there are three linearly independent modular forms of the lowest non-trivial weight $2$, which form a triplet of the $A_{4}$ group, ${\bf Y^{(\rm 2)}_{\bf 3}}(\tau)=(Y_{1}(\tau),\,Y_{2}(\tau),\,Y_{3}(\tau))^{T}$. As shown in Appendix A, these modular forms have been explicitly obtained [22] in the symmetric base of the $A_{4}$ generators $S$ and $T$ for the triplet representation:

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

(11)

where $\omega=\exp(i\frac{2}{3}\pi)$ .

The CP transformation in the modular symmetry was given by using the generalized CP symmetry [93]. We summarize the discussion in Ref.[93] briefly. Consider the CP and modular transformation $\gamma$ of the chiral superfield $\psi(x)$ assigned to an irreducible unitary representation $\bf r$ of $\Gamma_{N}$. The chain $CP\rightarrow\gamma\rightarrow CP^{-1}=\gamma^{\prime}\in\bar{\Gamma}$ is expressed as:

	$\displaystyle\psi(x)$	$\displaystyle\xrightarrow{\,CP\,}{\bf X}_{\bf r}\overline{\psi}(x_{P})\xrightarrow{\ \gamma\ }(c\tau^{*}+d)^{-k}{\bf X}_{\bf r}\,{\rho}_{\bf r}^{*}(\gamma)\overline{\psi}(x_{P})$
		$\displaystyle\xrightarrow{\,CP^{-1}\,}(c\tau^{*}_{CP^{-1}}+d)^{-k}{\bf X}_{\bf r}\,{\rho}_{\bf r}^{*}(\gamma){\bf X}_{\bf r}^{-1}\psi(x)\,,$		(12)

where $\tau_{CP^{-1}}$ is the operation of $CP^{-1}$ on $\tau$. The result of this chain transformation should be equivalent to a modular transformation $\gamma^{\prime}$ which maps $\psi(x)$ to $(c^{\prime}\tau+d^{\prime})^{-k}{\rho}_{\bf r}(\gamma^{\prime})\psi(x)$. Therefore, one obtains

$\displaystyle{\bf X}_{\bf r}\rho_{\bf r}^{*}(\gamma){\bf X}^{-1}_{\bf r}=\left(\frac{c^{\prime}\tau+d^{\prime}}{c\tau^{*}_{CP^{-1}}+d}\right)^{-k}{\rho}_{\bf r}(\gamma^{\prime})\,.$

(13)

Since ${\bf X}_{\bf r}$, $\rho_{\bf r}$ and $\rho_{\bf r^{\prime}}$ are independent of $\tau$, the overall coefficient on the right-hand side of Eq. (13) has to be a constant (complex) for non-zero weight $k$:

$\displaystyle\frac{c^{\prime}\tau+d^{\prime}}{c\tau^{*}_{CP^{-1}}+d}=\frac{1}{\lambda^{*}}\,,$

(14)

where $|\lambda|=1$ due to the unitarity of $\rho_{\bf r}$ and $\rho_{\bf r^{\prime}}$. The values of $\lambda$, $c^{\prime}$ and $d^{\prime}$ depend on $\gamma$.

Taking $\gamma=S$ ($c=1$, $d=0$) , and denoting $c^{\prime}(S)=C$, $d^{\prime}(S)=D$ while keeping $\lambda(S)=\lambda$, we find $\tau=(\lambda\tau^{*}_{CP^{-1}}-D)/C$ from Eq. (14), and consequently,

$\displaystyle\tau\xrightarrow{\,CP^{-1}\,}\tau_{CP^{-1}}=\lambda(C\tau^{*}+D)\,,\qquad\tau\xrightarrow{\,CP\,}\tau_{CP}=\frac{1}{C}(\lambda\tau^{*}-D)\,.$

(15)

Let us act with chain $CP\rightarrow T\rightarrow CP^{-1}$ on the mudular $\tau$ itself:

$\displaystyle\tau\xrightarrow{\,CP\,}\tau_{CP}=\frac{1}{C}(\lambda\tau^{*}-D)\xrightarrow{\ T\ }\frac{1}{C}(\lambda(\tau^{*}+1)-D)\xrightarrow{\,CP^{-1}\,}\tau+\frac{\lambda}{C}\,.$

(16)

The resulting transformation has to be a modular transformation, therefore $\lambda/C$ is an integer. Since $|\lambda|=1$, we find $|C|=1$ and $\lambda=\pm 1$. After choosing the sign of $C$ as $C=\mp 1$ so that ${\rm Im}[\tau_{CP}]>0$, the CP transformation of Eq. (15) turns to

$\displaystyle\tau\xrightarrow{\,CP\,}n-\tau^{*}\,,$

(17)

where $n$ is an integer. The chain $CP\rightarrow S\rightarrow CP^{-1}=\gamma^{\prime}(S)$ imposes no furher restrictions on $\tau_{CP}$. It is always possible to redefine the CP transformation in such a way that $n=0$ by using the freedom of $T$ transformation. Therefore, we define that the modulus $\tau$ transforms under CP as

$\displaystyle\tau\xrightarrow{\,CP\,}-\tau^{*}\,,$

(18)

without loss of generality.

The same transformation of $\tau$ was also derived from the higher dimensional theories [94]. The four-dimensional CP symmetry can be embedded into $(4+d)$ dimensions as higher dimensional proper Lorentz symmetry with positive determinant. That is, one can combine the four-dimensional CP transformation and $d$-dimensional transformation with negative determinant so as to obtain $(4+d)$ dimensional proper Lorentz transformation. For example in six-dimensional theory, we denote the two extra coordinates by a complex coordinate $z$. The four-dimensional CP symmetry with $z\rightarrow z^{*}$ or $z\rightarrow-z^{*}$ is a six-dimensional proper Lorentz symmetry. Note that $z=x+\tau y$, where $x$ and $y$ are real coordinates. The latter transformation $z\rightarrow-z^{*}$ maps the upper half plane ${\rm Im}[\tau]>0$ to the same half plane. Hence, we consider the transformation $z\rightarrow-z^{*}$ $(\tau\rightarrow-\tau^{*})$ as the CP symmetry.

Chiral superfields and modular forms transform in Eqs. (8) and (9), respectively, under a modular transformation. Chiral superfields also transform in Eq. (1) under the CP transformation. The CP transformation of modular forms were given in Ref.[93] as follows. Define a modular multiplet of the irreducible representation $\bf r$ of $\Gamma_{N}$ with weight $k$ as $\bf Y^{\rm(k)}_{\bf r}(\tau)$, which is transformed as:

$\displaystyle\bf Y^{\rm(k)}_{\bf r}(\tau)\xrightarrow{\,{\rm CP}\,}Y^{\rm(k)}_{\bf r}(-\tau^{*})\,,$

(19)

under the CP transformation. The complex conjugated CP transformed modular forms $\bf Y^{\rm(k)*}_{\bf r}(-\tau^{*})$ transform almost like the original multiplets $\bf Y^{\rm(k)}_{\bf r}(\tau)$ under a modular transformation, namely:

$\displaystyle\bf Y^{\rm(k)*}_{\bf r}(-\tau^{*})\xrightarrow{\ \gamma\ }Y^{\rm(k)*}_{\bf r}(-(\gamma\tau)^{*})={\rm(c\tau+d)^{k}}\rho_{\bf{r}}^{*}({\rm u}(\gamma))Y^{\rm(k)*}_{\bf r}(-\tau^{*})\,,$

(20)

where $u(\gamma)\equiv CP\gamma CP^{-1}$. Using the consistency condition of Eq. (3), we obtain

$\displaystyle\bf X_{r}^{T}Y^{\rm(k)*}_{\bf r}(-\tau^{*})\xrightarrow{\ \gamma\ }{\rm(c\tau+d)^{k}}\rho_{\bf{r}}(\gamma)X_{r}^{T}Y^{\rm(k)*}_{\bf r}(-\tau^{*})\,.$

(21)

Therefore, if there exist a unique modular multiplet at a level $N$, weight $k$ and representation $\bf r$, which is satisfied for $N=2$–$5$ with weight $2$, we can express the modular form $\bf Y^{\rm(k)}_{\bf r}(\tau)$ as:

$\displaystyle\bf Y^{\rm(k)}_{\bf r}(\tau)={\rm\kappa}X_{r}^{T}Y^{\rm(k)*}_{\bf r}(-\tau^{*})\,,$

(22)

where $\kappa$ is a proportional coefficient. Since $\bf Y^{\rm(k)}_{\bf r}(-(-\tau^{*})^{*})=\bf Y^{\rm(k)}_{\bf r}(\tau)$, Eq. (22) gives $\bf X_{r}^{*}X_{r}={\rm|\kappa|^{2}}\mathbb{1}_{r}$. Therefore, the matrix $\bf X_{r}$ is symmetric one, and $\kappa=e^{i\phi}$ is a phase, which can be absorbed in the normalization of modular forms. In conclusion, the CP transformation of modular forms is given as:

$\displaystyle\bf Y^{\rm(k)}_{\bf r}(\tau)\xrightarrow{\,{\rm CP}\,}Y^{\rm(k)}_{\bf r}(-\tau^{*})=X_{r}Y^{\rm(k)*}_{\bf r}(\tau)\,.$

(23)

It is also emphasized that $\bf X_{r}=\mathbb{1}_{r}$ satisfies the consistency condition Eq. (3) in a basis that generators of $S$ and $T$ of $\Gamma_{N}$ are represented by symmetric matrices because of $\rho^{*}_{\bf{r}}(S)=\rho^{\dagger}_{\bf{r}}(S)=\rho_{\bf{r}}(S^{-1})=\rho_{\bf{r}}(S)$ and $\rho^{*}_{\bf{r}}(T)=\rho^{\dagger}_{\bf{r}}(T)=\rho_{\bf{r}}(T^{-1})$.

The CP transformations of chiral superfields and modular multiplets are summalized as follows:

$\displaystyle\tau\xrightarrow{\,{\rm CP}\,}-\tau^{*}\,,\qquad\psi(x)\xrightarrow{\,{\rm CP}\,}X_{r}\overline{\psi}(x_{P})\,,\qquad\bf Y^{\rm(k)}_{\bf r}(\tau)\xrightarrow{\,{\rm CP}\,}Y^{\rm(k)}_{\bf r}(-\tau^{*})=X_{r}Y^{\rm(k)*}_{\bf r}(\tau)\,,$

(24)

where $\bf X_{r}=\mathbb{1}_{r}$ can be taken in the base of symmetric generators of $S$ and $T$. We use this CP transformation of modular forms to construct the CP invariant mass matrices in the next section.

Let us discuss numerical results for NH of neutrino masses. The ratios $\alpha_{e}/\gamma_{e}$ and $\beta_{e}/\gamma_{e}$ are given after fixing charged lepton masses and $\tau$ as shown in Appendix C. However, in practice, we scan $\alpha_{e}/\gamma_{e}$ and $\beta_{e}/\gamma_{e}$ to obtain the observed charged lepton mass ratio and include them in $\chi^{2}$ fit as well as three mixing angles and $\Delta m_{\rm atm}^{2}/\Delta m_{\rm sol}^{2}$.

We have already studied the lepton mass matrices in Eqs. (27) and (31) phenomenologically at the nearby fixed points of the modulus because the spontaneous CP violation in Type IIB string theory is possibly realized at nearby fixed points, where the moduli stabilization is performed in a controlled way [119, 120]. There are two fixed points in the fundamental domain of $PSL(2,\mathbb{Z})$, $\tau=i$ and $\tau=\omega$. Indeed, the viable $\tau$ of our lepton mass matrices is found around $\tau=i$ [75].

Based on this result of Ref. [75], we scan $\tau$ around $i$ while neutrino couplings $g^{\nu}_{1}$ and $g^{\nu}_{2}$ are scanned in the real space of $[-10,\,10]$. As a measure of good-fit, we adopt the sum of one-dimensional $\chi^{2}$ function for four accurately known dimensionless observables $\Delta m_{\rm atm}^{2}/\Delta m_{\rm sol}^{2}$, $\sin^{2}\theta_{12}$, $\sin^{2}\theta_{23}$ and $\sin^{2}\theta_{13}$ in NuFit 5.0 [116]. In addition, we employ Gaussian approximations for fitting $m_{e}/m_{\tau}$ and $m_{\mu}/m_{\tau}$ by using the data of PDG [121].

In Fig. 1 we show the allowed region on the ${\rm Re}\,[\tau]$ – ${\rm Im}\,[\tau]$ plane, where three mixing angles and $\Delta m_{\rm atm}^{2}/\Delta m_{\rm sol}^{2}$ are consistent with observed ones. The green, yellow and red regions correspond to $2\sigma$, $3\sigma$ and $5\sigma$ confidence levels, respectively.

The allowed region of $\tau$ is restricted in the narrow regions. This result is contrast to the previous one in Ref. [75], where non-trivial phases of $g^{\nu}_{1}$ and $g^{\nu}_{2}$ enlarged the allowed region of $\tau$. The predicted range of $\tau$ is in ${\rm Re}\,[\tau]=\pm[0.073,0.083]$ and ${\rm Im}\,[\tau]=[1.006,1.014]$ at $3\,\sigma$ confidence level (yellow), which are close to the fixed point $\tau=i$.

The allowed region of $g^{\nu}_{1}$ and $g^{\nu}_{2}$ is also shown in Fig. 2, where $g^{\nu}_{1}$ is in the rather wide region of $[-0.18,0.18]$ while $g^{\nu}_{2}$ is restricted in $[-0.87,-0.79]$ at $3\,\sigma$ confidence level (yellow).

Due to restricted ${\rm Re}\,[\tau]$, the CP violating Dirac phase $\delta_{CP}$, which is defined in Appendix D, is predicted clearly. In Fig. 3, we show prediction of $\delta_{CP}$ versus the sum of neutrino masses $\sum m_{i}$. It is remarked that $\delta_{CP}$ is almost independent of $\sum m_{i}$. The predicted ranges of $\delta_{CP}$ are narrow such as $[98^{\circ},110^{\circ}]$ and $[250^{\circ},262^{\circ}]$ at $3\,\sigma$ confidence level (yellow). The predicted ranges $[98^{\circ},110^{\circ}]$ and $[250^{\circ},262^{\circ}]$ correspond to ${\rm Re}\,[\tau]=(0.073$–$0.083)$ and ${\rm Re}\,[\tau]=-(0.073$–$0.083)$, respectively. The predicted $\sum m_{i}$ is in $[82,\,102]$ meV for $3\,\sigma$ confidence level (yellow). The minimal cosmological model, ${\rm\Lambda CDM}+\sum m_{i}$, provides the upper-bound $\sum m_{i}<120$ meV [117, 118]. Thus, our predicted sum of neutrino masses is consistent with the cosmological bound $120$  meV.

In Fig. 4, we show the allowed region on the $\sin^{2}\theta_{23}$ – $\sum m_{i}$ plane. Since $\sum m_{i}$ depends on the value of $\sin^{2}\theta_{23}$ significantly, the crucial test of our prediction will be available in the near future.