-
Next, we review our model in order to obtain the neutrino mass matrix. In addition to the minimal supersymmetric SM (MSSM), we introduce matter superfields including two right-handed neutral fermions
$ N^c_{1,2} $ that belong to doublet under the modular$ T^\prime $ group with modular weight$ -1 $ . We also add three chiral superfields$ \{\hat\chi, \hat{\eta}_1, \hat{\eta}_2 \} $ including two bosons$ \{\chi, \eta_1, \eta_2\} $ where there are superfields that are true singlets under the$ T^\prime $ group with$ \{-1,-1,-3\} $ modular weight. χ only plays a role in generating the neutrino mass matrix at one-loop level; therefore,$ \eta_{1,2} $ are inert bosons in addition to χ. Left-handed lepton doublets$ \{L_e,L_\mu,L_\tau\} $ are assigned to be triplet with$ -1 $ modular weight, while the right-handed ones$ \{e^c,\mu^c,\tau^c\} $ are set to be$ \{1,1'',1'\} $ with$ -1 $ modular weight. Two Higgs doublet$ H_{1,2} $ are invariant under the modular$ T^\prime $ symmetry. All the fields and their assignments are summarized in Table 1. Under these symmetries, the renormalizable superpotential is expressed as follows ④:Chiral superfields $\{{\hat{L}_{e}},{\hat{L}_{\mu}},{\hat{L}_{\tau}}\}$ $\{\hat{e}^c,\hat{\mu}^c,\hat{\tau}^c\}$ $\{\hat{N}^c_{1},\hat{N}^c_{2}\}$ $\hat{H}_1$ $\hat{H}_2$ $\hat{\eta}_1$ $\hat{\eta}_2$ $\hat{\chi}$ $SU(2)_L$ ${\bf{2}}$ ${\bf{1}}$ ${\bf{1}}$ ${\bf{2}}$ ${\bf{2}}$ ${\bf{2}}$ ${\bf{2}}$ ${\bf{1}}$ $U(1)_Y$ $-\frac12$ $1$ $0$ $\frac12$ $-\frac12$ $\frac12$ $-\frac12$ $0$ $T'$ $3$ $\{1,1'',1'\}$ $2$ $1$ $1$ $1$ $1$ $1$ $-k$ ${-1}$ $-1$ $-1$ $0$ $0$ $-1$ $-1$ $-3$ Table 1. Field contents of matter chiral superfields and their charge assignments under
$S U(2)_L\times U(1)_Y\times A_{4}$ in the lepton and boson sectors;$-k_I$ is the number of modular weight and the quark sector is the same as that of the SM.$ \begin{aligned}[b] {\cal W} =& \alpha_e [Y^{(2)}_3 \otimes\hat{e}^c\otimes \hat{L} \otimes\hat{H}_2] +\beta_e [Y^{(2)}_3 \otimes \hat{\mu}^c\otimes \hat{L} \otimes\hat{H}_2] +\gamma_e [Y^{(2)}_3 \otimes \hat{\tau}^c\otimes \hat{L}\otimes \hat{H}_2] \\ &+\alpha_\eta [Y^{(3)}_2 \otimes\hat{N}^c\otimes \hat{L} \otimes\hat{\eta}_1] +\beta_\eta [Y^{(3)}_{2''} \otimes\hat{N}^c\otimes \hat{L} \otimes\hat{\eta}_1] + M_{0} [Y^{(2)}_3 \otimes\hat{N}^c\otimes\hat{N}^c] \\ &+\mu_H \hat{H}_1 \hat{H}_2+\mu_\chi Y^{(6)}_1 \hat{\chi} \hat{\chi} + a Y^{(4)}_1 \hat{H}_1 \hat{\eta}_2 \hat{\chi} + b Y^{(4)}_1 \hat{H}_2 \hat{\eta}_1 \hat{\chi}, \end{aligned} $ (1) where R-parity is implicitly imposed in the above superpotential,
$ Y^{(2)}_3\equiv(f_1,f_2,f_3)^T $ is$ T' $ triplet with modular weight$ 2 $ , and$ Y^{(3)}_{2^{(\prime\prime)}} \equiv(y^{(\prime\prime)}_1,y^{(\prime\prime)}_2)^T $ is$ T^\prime $ doublet with modular weight$ 3 $ .⑤ The first line in Eq. (1) corresponds to the charged-lepton sector, while the second and third lines are related to the neutrino sector. The third line is particularly important if the neutrino mass matrix is induced at one-loop level as dominant contribution.After the electroweak spontaneous symmetry breaking, the charged-lepton mass matrix is given by
$ \begin{align} m_\ell&= \frac {v_2}{\sqrt{2}} \left[\begin{array}{ccc} \alpha_e & 0 & 0 \\ 0 &\beta_e & 0 \\ 0 & 0 &\gamma_e \\ \end{array}\right] \left[\begin{array}{ccc} f_1 &f_3 & f_2 \\ f_2 &f_1 & f_3 \\ f_3 &f_2 & f_1 \\ \end{array}\right], \end{align} $
(2) where
$ \langle H_2\rangle\equiv [v_2/\sqrt2,0]^T $ . Then the charged-lepton mass eigenstate is found as$ {\rm diag}( |m_e|^2, |m_\mu|^2, |m_\tau|^2)\equiv V_{e_L}^\dagger m^\dagger_\ell m_\ell V_{e_L} $ . In our numerical analysis, we fix the free parameters$ \alpha_e,\beta_e,\gamma_e $ inserting the observed three charged-lepton masses by applying the following relations:$ \begin{align} &{\rm Tr}[m_\ell {m_\ell}^\dagger] = |m_e|^2 + |m_\mu|^2 + |m_\tau|^2, \end{align} $
(3) $ \begin{align} &{\rm Det}[m_\ell {m_\ell}^\dagger] = |m_e|^2 |m_\mu|^2 |m_\tau|^2, \end{align} $
(4) $ \begin{aligned}[b] &({\rm Tr}[m_\ell {m_\ell}^\dagger])^2 -{\rm Tr}[(m_\ell {m_\ell}^\dagger)^2]\\ =&2( |m_e|^2 |m_\mu|^2 + |m_\mu|^2 |m_\tau|^2+ |m_e|^2 |m_\tau|^2 ). \end{aligned} $
(5) The Dirac matrix consists of
$ \alpha_\eta $ and$ \beta_\eta $ ;$ N^c y_\eta L\eta_1 $ is given by$ \begin{align} y_\eta &= \left[\begin{array}{ccc} \dfrac{\beta_\eta}{\sqrt2} {\rm e}^{\frac{7\pi}{12}i}y^{\prime\prime}_2 & \alpha_\eta y_1 & \dfrac{\alpha_\eta}{\sqrt2} {\rm e}^{\frac{7\pi}{12}i}y_2 + \beta_\eta y^{\prime\prime}_1 \\ \dfrac{\beta_\eta}{\sqrt2} {\rm e}^{\frac{7\pi}{12}i}y^{\prime\prime}_1 + \alpha_\eta {\rm e}^{\frac{\pi}{6}i} y_2 & \beta_\eta {\rm e}^{\frac{\pi}{6}i} y^{\prime\prime}_2 & \dfrac{\alpha_\eta}{\sqrt2} {\rm e}^{\frac{7\pi}{12}i}y_1 \\ \end{array}\right]. \end{align} $
(6) The heavier Majorana mass matrix is given by
$ \begin{align} M_N &= M_0 \left[\begin{array}{ccc} f_2 & \dfrac1{\sqrt2} {\rm e}^{\frac{7\pi}{12}i} f_3 \\ \dfrac1{\sqrt2} {\rm e}^{\frac{7\pi}{12}i} f_3 & {\rm e}^{\frac{\pi}{6}i} f_1 \\ \end{array}\right] = M_0 {\tilde M}. \end{align} $
(7) The heavy Majorana mass matrix is diagonalized by a unitary matrix
$ V_N $ as follows:$ D_N\equiv V_N M_N V_N^T $ , where$ N^c\equiv \psi^cV_N^T $ ,$ \psi^c $ is the mass eigenstate. -
If all the bosons have nonzero VEVs, the neutrino mass matrix is generated via tree-level as follows:
$ \begin{align} m_\nu = \frac{v_{\eta_1}^2}{2 M_0} y_\eta^T \tilde M^{-1} y_\eta \equiv \kappa \tilde m_\nu, \end{align} $
(8) where
$ \kappa\equiv \dfrac{v_{\eta_1}^2}{2 M_0} $ and$ \langle\eta_1\rangle \equiv [0,v_{\eta_1}/\sqrt2]^T $ .$ m_\nu $ is diagonalized by a unitary matrix$ V_{\nu} $ ;$ D_\nu=|\kappa| \tilde D_\nu= V_{\nu}^T m_\nu V_{\nu}= |\kappa| V_{\nu}^T \tilde m_\nu V_{\nu} $ . Then,$ |\kappa| $ is determined by$ \begin{align} (\mathrm{NH}):\ |\kappa|^2= \frac{|\Delta m_{\rm atm}^2|}{\tilde D_{\nu_3}^2}, \quad (\mathrm{IH}):\ |\kappa|^2= \frac{|\Delta m_{\rm atm}^2|}{\tilde D_{\nu_2}^2}, \end{align} $
(9) where
$ \Delta m_{\rm atm}^2 $ denotes atmospheric neutrino mass difference squares, and NH and IH represent the normal hierarchy and inverted hierarchy, respectively. Subsequently, the solar mass different squares can be written in terms of$ |\kappa| $ as follows:$ \begin{align} (\mathrm{NH}):\ \Delta m_{\rm sol}^2= |\kappa|^2 \tilde D_{\nu_2}^2, \quad(\mathrm{IH}):\ \Delta m_{\rm sol}^2= |\kappa|^2 ({\tilde D_{\nu_2}^2-\tilde D_{\nu_1}^2}), \end{align} $
(10) which can be compared to the observed value. The observed mixing matrix is defined by
$ U=V^\dagger_L V_\nu $ [121], where it is parametrized by three mixing angles, i.e., i.e.,$ \theta_{ij} (i,j=1,2,3; i < j) $ , one CP violating Dirac phase$ \delta_{CP} $ , and one Majorana phase$ \alpha_{21} $ as follows:$ \begin{equation} U = \begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} {\rm e}^{-{\rm i} \delta_{CP}} \\ -s_{12} c_{23} - c_{12} s_{23} s_{13} {\rm e}^{{\rm i} \delta_{CP}} & c_{12} c_{23} - s_{12} s_{23} s_{13} {\rm e}^{{\rm i} \delta_{CP}} & s_{23} c_{13} \\ s_{12} s_{23} - c_{12} c_{23} s_{13} {\rm e}^{{\rm i} \delta_{CP}} & -c_{12} s_{23} - s_{12} c_{23} s_{13} {\rm e}^{{\rm i} \delta_{CP}} & c_{23} c_{13} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & {\rm e}^{{\rm i} \frac{\alpha_{21}}{2}} & 0 \\ 0 & 0 & 1 \end{pmatrix}, \end{equation} $ (11) where
$ c_{ij} $ and$ s_{ij} $ stand for$ \cos \theta_{ij} $ and$ \sin \theta_{ij} $ , respectively. Then, each mixing is expressed in terms of the component of U as follows:$ \begin{aligned}[b]& \sin^2\theta_{13}=|U_{e3}|^2,\quad \sin^2\theta_{23}=\frac{|U_{\mu3}|^2}{1-|U_{e3}|^2},\\& \sin^2\theta_{12}=\frac{|U_{e2}|^2}{1-|U_{e3}|^2}, \end{aligned} $
(12) and the Majorana phase
$ \alpha_{21} $ and Dirac phase$ \delta_{CP} $ are expressed in terms of the following relations:$ \begin{aligned}[b] & {\rm{Im}}[U^*_{e1} U_{e2}] = c_{12} s_{12} c_{13}^2 \sin \left( \frac{\alpha_{21}}{2} \right), \\& {\rm{Im}}[U^*_{e1} U_{e3}] = - c_{12} s_{13} c_{13} \sin \delta_{CP} , \end{aligned} $
(13) $ \begin{aligned}[b] & {\rm{Re}}[U^*_{e1} U_{e2}] = c_{12} s_{12} c_{13}^2 \cos \left( \frac{\alpha_{21}}{2} \right), \\& {\rm{Re}}[U^*_{e1} U_{e3}] = c_{12} s_{13} c_{13} \cos \delta_{CP} , \end{aligned} $
(14) where
$ \alpha_{21}/2,\ \delta_{CP} $ are subtracted from π when$ \cos(\alpha_{21}/2),\ \cos\delta_{CP} $ are negative. In addition, the effective mass for the neutrinoless double beta decay is given by$ \begin{aligned}[b] \langle m_{ee}\rangle=&|\kappa||\tilde D_{\nu_1} \cos^2\theta_{12} \cos^2\theta_{13}+\tilde D_{\nu_2} \sin^2\theta_{12} \cos^2\theta_{13}{\rm e}^{{\rm i}\alpha_{21}}\\&+\tilde D_{\nu_3} \sin^2\theta_{13}{\rm e}^{-2{\rm i}\delta_{CP}}|, \end{aligned} $
(15) where its observed value could be measured by KamLAND-Zen in future [122].
-
In general, it is difficult to show analytical predictions in arbitrary τ. However, we might be able to conduct demonstrations in specific points of τ such as fixed points. There are three important points
$ \tau=i,\omega,i\infty $ according to the string theory. At these fixed points, modular forms are obtained by simple forms. Table 2 shows modular forms at$ \tau=i \infty $ . In this case, a triplet modular form becomes$ (1, 0, 0)^T $ and there is a massless right-handed neutrino. Table 3 shows modular forms at$ \tau=\omega $ . In this case, doublet modular forms become$ (0, 0)^T $ and all neutrinos are massless. These two cases are not suitable in our model. Table 4 shows modular forms at$ \tau=i $ . Using Eqs. (9) and (10), we obtain the following relations:k $\boldsymbol r$ $ \tau=i \infty $ $ 2n-1 $ $ \bf 2 $ $ (0, 1)^T $ $ \bf 2' $ $ (0, 0)^T $ $ \bf 2'' $ $ (1, 0)^T $ $ 2n $ $ \bf 1 $ $ 1 $ $ \bf 1' $ $ 0 $ $ \bf 1'' $ $ 0 $ $ \bf 3 $ $ (1, 0, 0)^T $ Table 2. Modular forms at
$ \tau=i \infty $ ; note that we ignore overall factors, and n is a positive integer.k $\boldsymbol r$ $ \tau=\omega $ $ 6n-5 $ $ \bf 2 $ ,$ \bf 2' $ ,$ \bf 2'' $ $ (0, 0)^T $ $ 6n-4 $ $ \bf 1 $ $ 0 $ $ \bf 1' $ $ 0 $ $ \bf 1'' $ $ 1 $ $ \bf 3 $ $ (1, \omega, -\frac12 \omega^2)^T $ $ 6n-3 $ $ \bf 2 $ ,$ \bf 2' $ ,$ \bf 2'' $ $ (0, 0)^T $ $ 6n-2 $ $ \bf 1 $ $ 0 $ $ \bf 1' $ $ 1 $ $ \bf 1'' $ $ 0 $ $ \bf 3 $ $ (1, -\frac12 \omega, \omega^2)^T $ $ 6n-1 $ $ \bf 2 $ ,$ \bf 2' $ ,$ \bf 2'' $ $ (0, 0)^T $ $ 6n $ $ \bf 1 $ $ 1 $ $ \bf 1' $ $ 0 $ $ \bf 1'' $ $ 0 $ $ \bf 3 $ $ (1, -2 \omega, -2\omega^2)^T $ Table 3. Modular forms at
$ \tau=\omega $ ; note that we ignore overall factors, and n is a positive integer.k $\boldsymbol r$ $ \tau=i $ $ 4n-3 $ $ \bf 2 $ ,$ \bf 2' $ ,$ \bf 2'' $ $ ( (-1)^{\frac{7}{12}} (1 + \sqrt{3}), -\sqrt{2})^T $ $ 4n-2 $ $ \bf 1 $ ,$ \bf 1' $ ,$ \bf 1'' $ $ 0 $ $ \bf 3 $ $ (1, 1 + \sqrt{3}, -2 - \sqrt{3})^T $ ,$ (1, -2 + \sqrt{3}, 1 - \sqrt{3})^T $ $ 4n-1 $ $ \bf 2 $ ,$ \bf 2' $ ,$ \bf 2'' $ $ (-1 + (-1)^{1/6}, 1)^T $ $ 4n $ $ \bf 1 $ ,$ \bf 1' $ ,$ \bf 1'' $ $ 1 $ $ \bf 3 $ $ (1, 1, 1)^T $ Table 4. Modular forms at
$ \tau=i $ ; note that we ignore overall factors, and n is a positive integer.$ \begin{align} (\mathrm{NH}):\ \frac{\Delta m_{\rm sol}^2}{|\Delta m_{\rm atm}^2|}= \frac{{\tilde D_{\nu_2}^2}}{\tilde D_{\nu_3}^2}, \quad (\mathrm{IH}):\ \frac{\Delta m_{\rm sol}^2}{|\Delta m_{\rm atm}^2|}= \frac{{\tilde D_{\nu_2}^2-\tilde D_{\nu_1}^2}}{\tilde D_{\nu_2}^2}. \end{align} $
(16) These relations are functions of
$ \beta_\eta/\alpha_\eta $ and the equations have solutions if and only if the neutrino mass ordering is NH case. In the next section, we numerically check whether these analytical estimations are reasonable or not. -
When
$ \eta_{1,2},\ \chi $ are inert bosons, the neutrino mass matrix is induced at one-loop level via mixings among neutral components of inert bosons. Before discussing the neutrino sector, we formulate the Higgs sector. The valid soft SUSY-breaking terms to construct the neutrino mass matrix are found as follows:$ \begin{aligned}[b] -{\cal L}_{\rm soft} =& \mu_{BH}^2 H_1 H_2 + \mu_{B\chi}^2 Y^{(6)}_1 \chi\chi + A_a Y^{(4)}_1 H_1\eta_2 \chi\\&+ A_b Y^{(4)}_1 H_2\eta_1 \chi +m^2_{H_1}|H_1|^2+m^2_{H_2}|H_2|^2 \\&+m^2_{\eta_1}|\eta_1|^2+m^2_{\eta_2}|\eta_2|^2+m^2_{\chi}|\chi|^2 + {\rm h.c.},\end{aligned} $
where
$ m^2_{\eta_{1,2}} $ and$ m^2_{\chi} $ include the invariant coefficients$ 1/(\tau^*-\tau)^{k_{\eta_{1,2},\chi}} $ . -
Inert bosons χ,
$ \eta_1 $ , and$ \eta_2 $ mix each other through the soft SUSY-breaking terms of$ A_{a,b} $ and$ \mu_{B\eta} $ after the spontaneous electroweak symmetry breaking. Here, we suppose that$ \mu_{B\eta},\ A_a<<A_b $ for simplicity. Then, the mixing dominantly comes from χ and$ \eta_1 $ only. This assumption does not affect the structure of the neutrino mass matrix. Thus, the mass eigenstate is defined by$ \begin{align} \left[\begin{array}{c} \chi_{R,I} \\ \eta_{1_{R,I}} \\ \end{array}\right]= \left[\begin{array}{cc} c_{\theta_{R,I}} & -s_{\theta_{R,I}} \\ s_{\theta_{R,I}} & c_{\theta_{R,I}} \\ \end{array}\right] \left[\begin{array}{c} \xi_{1_{R,I}} \\ \xi_{2_{R,I}} \\ \end{array}\right], \end{align} $
(17) where
$ c_{\theta_{R,I}}, s_{\theta_{R,I}} $ are the shorthand notations of$ \sin\theta_{R,I} $ and$ \cos\theta_{R,I} $ , respectively;$ \xi_{1,2} $ denotes the mass eigenstates for$ \chi,\eta_1 $ , and their mass eigenvalues are denoted by ed by$m_{i_{R,I}}\ (i=1,2)$ . Note that the mixing angle θ simultaneously diagonalizes the mass matrix of real and imaginary parts. -
The active neutrino mass matrix
$ m_\nu $ is induced at one-loop level as follows:$ \begin{aligned}[b] m_\nu =- \frac{1}{2(4\pi)^2} (y^T_\eta)_{i\alpha} (V_N)_{\alpha a} D_{N_a} (V_N^T)_{a\beta} (y_\eta)_{\beta j} \end{aligned} $
$ \begin{aligned}[b]\quad\quad &\times \Big[ s^2_{\theta_R} f(m_{\xi_{1_R}},D_{N_a}) +c^2_{\theta_R} f(m_{\xi_{2_R}},D_{N_a}) \\& -s^2_{\theta_I} f(m_{\xi_{1_I}},D_{N_a}) -c^2_{\theta_I} f(m_{\xi_{2_I}},D_{N_a}) \Big] , \end{aligned} $
(18) $ \begin{align} f(m_1,m_2)&=\int_0^1\ln\left[ x\left(\frac{m_1^2}{m_2^2}-1\right)+1 \right]. \end{align} $
(19) In order to fit the atmospheric mass square difference, we extract
$ \alpha_\eta $ from$ y_\eta $ and redefine$ m_\nu\equiv \alpha_\eta^2 \tilde m_\nu $ . Then, we can proceed with the discussion by following the same approach as in the case of canonical seesaw by regarding$ \alpha_\eta^2 $ as κ, where κ is a parameter in the canonical seesaw model. In case of radiative seesaw, one might find that two fixed points at$ \tau=\omega,i\infty $ are not favorable owing to absence of enough degrees of freedom of non-vanishing right-handed neutrino masses, analogous to the observation in the canonical seesaw case. However, it is not possible to realize analytical predictions because of the highly complicated loop function even for a fixed point of$ \tau=i $ . Thus, we cannot help relying on numerical analysis only. -
In this appendix, we summarize some formulas in the framework of
$ T^\prime $ modular symmetry belonging to the$ SL(2,\mathbb{Z}) $ modular symmetry. The$ SL(2,Z_3) $ modular symmetry corresponds to the$ T^\prime $ modular symmetry. The modulus τ transforms as$ \begin{align} & \tau \longrightarrow \gamma\tau= \frac{a\tau + b}{c \tau + d}, \end{align}\tag{A1} $
with
$ \{a,b,c,d\} \in Z_3 $ satisfying$ ad-bc=1 $ and$ {\rm Im} [\tau]>0 $ . The transformation of modular forms$ f(\tau) $ are given by$ \begin{align} & f(\gamma\tau)= (c\tau+d)^k f(\tau)\; , \; \; \gamma \in SL(2,Z_3)\; , \end{align}\tag{A2} $
where
$ f(\tau) $ denotes holomorphic functions of τ with the modular weight k.In a similar way, the modular transformation of a matter chiral superfield
$ \phi^{(I)} $ with the modular weight$ -k_I $ is given by$ \begin{equation} \phi^{(I)} \to (c\tau+d)^{-k_I}\rho^{(I)}(\gamma)\phi^{(I)}, \end{equation} \tag{A3}$
where
$ \rho^{(I)}(\gamma) $ stands for a unitary matrix corresponding to$ T^\prime $ transformation. Note that the superpotential is invariant when the sum of modular weight from fields and modular form is zero and the term is a singlet under the$ T^\prime $ symmetry. It restricts a form of the superpotential, as expressed in Eq. (1).Modular forms are constructed on the basis of weight 1 modular form,
$ Y^{(1)}_2=(Y_1, Y_2)^T $ , transforming as a doublet of$ T^\prime $ . Their explicit forms are written by the Dedekind eta-function$ \eta(\tau) $ with respect to τ [1, 74]:$ \begin{aligned}[b] Y_{1}(\tau) =& \sqrt{2}{\rm e}^{{\rm i}\frac{7\pi}{12}} \frac{\eta^3(3\tau)}{\eta(\tau)}, \\ Y_{1}(\tau) =& \sqrt{2}{\rm e}^{{\rm i}\frac{7\pi}{12}} \frac{\eta^3(3\tau)}{\eta(\tau)}. \end{aligned} $
Modular forms of higher weight can be obtained from tensor products of
$ Y^{(1)}_2 $ . We enumerate some modular forms used in our analysis:$ \begin{align} Y_1^{(4)} = -4 Y_1^3 Y_2 - (1-i) Y_2^4, \end{align} \tag{A4}$
$ \begin{align} Y^{(6)}_{\bf 1} &= (1-i){\rm e}^{{\rm i}\pi/6}Y_2^6 -(1+i){\rm e}^{{\rm i}\pi/6}Y_1^6 - 10 {\rm e}^{{\rm i}\pi/6}Y_1^3Y_2, \end{align}\tag{A5} $
$ \begin{align} Y^{(2)}_3 &\equiv(f_1,f_2,f_3)^T = ( {\rm e}^{{\rm i}\pi/6}Y_2^2, \sqrt{2} {\rm e}^{{\rm i}7\pi/12} Y_1 Y_2, Y_1^2 )^T, \end{align} \tag{A6}$
$ \begin{align} Y^{(3)}_{2} &\equiv(y_1,y_2)^T = ( 3{\rm e}^{{\rm i}\pi/6}Y_1 Y_2^2, \sqrt{2}{\rm e}^{{\rm i}5\pi/12}Y_1^3 - {\rm e}^{{\rm i}\pi/6}Y_2^3 )^T, \end{align}\tag{A7} $
$ \begin{align} Y^{(3)}_{2^{\prime\prime}} &\equiv (y^{\prime\prime}_1, y^{\prime\prime}_2)^T = ( Y_1^3 + (1-i) Y_2^3, -3 Y_1^2 Y_2 )^T. \end{align}\tag{A8} $
Lepton mass matrix from double covering of A4 modular flavor symmetry
- Received Date: 2022-06-11
- Available Online: 2022-12-15
Abstract: We study a double covering of modular