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Lepton mass matrix from double covering of A4 modular flavor symmetry

  • We study a double covering of modular A4 flavor symmetry. To this end, we construct lepton models for canonical and radiative seesaw scenarios. Using irreducible doublet representations, heavier Majorana fermion masses are characterized by one free parameter that would differentiate from A4 symmetry. symmetry. Through χ square numerical analysis, we demonstrate that both scenarios produce some predictions in case of normal hierarchy reproducing neutrino oscillation data. However, no solution satisfies the neutrino oscillation data in case of radiative seesaw of inverted hierarchy.
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Hiroshi Okada and Yuta Orikasa. Lepton mass matrix from double covering of A4 modular flavor symmetry[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac92d8
Hiroshi Okada and Yuta Orikasa. Lepton mass matrix from double covering of A4 modular flavor symmetry[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac92d8 shu
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Lepton mass matrix from double covering of A4 modular flavor symmetry

  • 1. Asia Pacific Center for Theoretical Physics (APCTP) - Headquarters San 31, Hyoja-dong, Nam-gu, Pohang 790-784, Korea
  • 2. Department of Physics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea
  • 3. Institute of Experimental and Applied Physics, Czech Technical University in Prague, Husova 240/5, 110 00 Prague 1, Czech Republic

Abstract: We study a double covering of modular A4 flavor symmetry. To this end, we construct lepton models for canonical and radiative seesaw scenarios. Using irreducible doublet representations, heavier Majorana fermion masses are characterized by one free parameter that would differentiate from A4 symmetry. symmetry. Through χ square numerical analysis, we demonstrate that both scenarios produce some predictions in case of normal hierarchy reproducing neutrino oscillation data. However, no solution satisfies the neutrino oscillation data in case of radiative seesaw of inverted hierarchy.

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    I.   INTRODUCTION
    • Even after the discovery of the standard model (SM) Higgs, we wonder whether there is a way to accommodate neutral particles such as the active tiny neutrinos and dark matter (DM) candidates. To describe the observed neutrino sector, we might need a prescription about how to determine three mixings and two mass square differences in addition to CP phases, namely Majorana and Dirac phases, which are not accurately measured yet. Modular flavor symmetries are one of the most promising candidates to obtain predictive scenarios in the neutrino sector, given that these symmetries do not require many neutral bosons owing to a new degree of freedom: "modular weight." Moreover, DM can be considered stable by applying this degree of freedom. In fact, a great number of scientific papers on this research line have been published after the original paper [1]. For example, the modular A4 flavor symmetry has been discussed in Refs. [1,352], in addition to S3 in Refs. [5358], S4 in Refs. [5971], A5 in Refs. [64, 72, 73], double covering of A4 in Refs. [7476], double covering of S4 in Refs. [77, 78], and double covering of A5 in Refs. [7982]. Reference [83] discusses the CP phase of quark mass matrices in modular flavor symmetric models at the fixed point of τ. Soft-breaking terms on modular symmetry are discussed in Ref. [84]. Other types of modular symmetries have also been proposed to understand masses, mixings, and phases of the SM in Refs. [8594]. Different applications to various physics fields such as dark matter and the origin of CP are found in Refs. [6, 7, 11, 14, 63, 104112]. Mathematical studies such as possible correction from Kähler potential, systematic analysis of the fixed points, and moduli stabilization are discussed in Refs. [94, 113116]. Recently, the authors of Ref. [117] proposed a scenario to derive four-dimensional modular flavor symmetric models from higher-dimensional theory on extra-dimensional spaces with modular symmetry. It constrains modular weights and representations of fields and modular couplings in the four-dimensional effective field theory. Higher-dimensional operators in the SM effective field theory are also constrained in the higher-dimensional theory, in particular, the string theory [118]. Non-perturbative effects relevant to neutrino masses have been studied in the context of modular symmetry anomaly [119].

      In this study, we apply double covering of modular A4 symmetry T to the lepton sector and show several predictions in cases of canonical seesaw and radiative seesaw scenarios [120]. Interestingly, T has three has three irreducible doublet representations. Using these representations, heavier Majorana fermion masses are described by one free parameter (except τ) that would differentiate from A4 symmetry. Note symmetry. Note that the mathematical part of ath T has been thoroughly s has been thoroughly studied in Ref. [74], where the authors demonstrate a prediction in case of inverted hierarchy based on the canonical seesaw model.

      This paper is organized as follows. In Sec. II, we review our model setup in the lepton sector, for deriving the superpotential, charged-lepton mass matrix, Dirac Yukawa matrix, and Majorana mass matrix. In Sec. III, we formulate the neutrino mass matrix and its observables in case of canonical seesaw. Then, we address the radiative seesaw model, showing the soft breaking terms that play a crucial role in generating the neutrino mass matrix at one-loop level. In Sec. IV, we perform the numerical χ square analysis and show predictive figures for normal and inverted hierarchies of the canonical and radiative seesaws. Finally, we conclude and summarize our model in Sec. V. In Appendix A, we summarize formulas on the double covering of modular ormula A4 symmetry.

    II.   MODEL
    • 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 Nc1,2 that belong to doublet under the modular T group with modular weight 1. We also add three chiral superfields {ˆχ,ˆη1,ˆη2} including two bosons {χ,η1,η2} where there are superfields that are true singlets under the T group with {1,1,3} modular weight. χ only plays a role in generating the neutrino mass matrix at one-loop level; therefore, η1,2 are inert bosons in addition to χ. Left-handed lepton doublets {Le,Lμ,Lτ} are assigned to be triplet with 1 modular weight, while the right-handed ones {ec,μc,τc} are set to be {1,1,1} with 1 modular weight. Two Higgs doublet H1,2 are invariant under the modular T symmetry. All the fields and their assignments are summarized in Table 1. Under these symmetries, the renormalizable superpotential is expressed as follows :

      Chiral superfields
      {ˆLe,ˆLμ,ˆLτ} {ˆec,ˆμc,ˆτc} {ˆNc1,ˆNc2} ˆH1 ˆH2 ˆη1 ˆη2 ˆχ
      SU(2)L21122221
      U(1)Y1210121212120
      T3{1,1,1}211111
      k11100113

      Table 1.  Field contents of matter chiral superfields and their charge assignments under SU(2)L×U(1)Y×A4 in the lepton and boson sectors; kI is the number of modular weight and the quark sector is the same as that of the SM.

      W=αe[Y(2)3ˆecˆLˆH2]+βe[Y(2)3ˆμcˆLˆH2]+γe[Y(2)3ˆτcˆLˆH2]+αη[Y(3)2ˆNcˆLˆη1]+βη[Y(3)2ˆNcˆLˆη1]+M0[Y(2)3ˆNcˆNc]+μHˆH1ˆH2+μχY(6)1ˆχˆχ+aY(4)1ˆH1ˆη2ˆχ+bY(4)1ˆH2ˆη1ˆχ,

      (1)

      where R-parity is implicitly imposed in the above superpotential, Y(2)3(f1,f2,f3)T is T triplet with modular weight 2, and Y(3)2()(y()1,y()2)T is T doublet with modular weight 3. 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

      m=v22[αe000βe000γe][f1f3f2f2f1f3f3f2f1],

      (2)

      where H2[v2/2,0]T. Then the charged-lepton mass eigenstate is found as diag(|me|2,|mμ|2,|mτ|2)VeLmmVeL. In our numerical analysis, we fix the free parameters αe,βe,γe inserting the observed three charged-lepton masses by applying the following relations:

      Tr[mm]=|me|2+|mμ|2+|mτ|2,

      (3)

      Det[mm]=|me|2|mμ|2|mτ|2,

      (4)

      (Tr[mm])2Tr[(mm)2]=2(|me|2|mμ|2+|mμ|2|mτ|2+|me|2|mτ|2).

      (5)

      The Dirac matrix consists of αη and βη; NcyηLη1 is given by

      yη=[βη2e7π12iy2αηy1αη2e7π12iy2+βηy1βη2e7π12iy1+αηeπ6iy2βηeπ6iy2αη2e7π12iy1].

      (6)

      The heavier Majorana mass matrix is given by

      MN=M0[f212e7π12if312e7π12if3eπ6if1]=M0˜M.

      (7)

      The heavy Majorana mass matrix is diagonalized by a unitary matrix VN as follows: DNVNMNVTN, where NcψcVTN, ψc is the mass eigenstate.

    III.   ACTIVE NEUTRINO MASS MATRIX

      A.   Canonical seesaw

      1.   Mass matrix of mν
    • If all the bosons have nonzero VEVs, the neutrino mass matrix is generated via tree-level as follows:

      mν=v2η12M0yTη˜M1yηκ˜mν,

      (8)

      where κv2η12M0 and η1[0,vη1/2]T. mν is diagonalized by a unitary matrix Vν; Dν=|κ|˜Dν=VTνmνVν=|κ|VTν˜mνVν. Then, |κ| is determined by

      (NH): |κ|2=|Δm2atm|˜D2ν3,(IH): |κ|2=|Δm2atm|˜D2ν2,

      (9)

      where Δm2atm 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 |κ| as follows:

      (NH): Δm2sol=|κ|2˜D2ν2,(IH): Δm2sol=|κ|2(˜D2ν2˜D2ν1),

      (10)

      which can be compared to the observed value. The observed mixing matrix is defined by U=VLVν [121], where it is parametrized by three mixing angles, i.e., i.e., θij(i,j=1,2,3;i<j), one CP violating Dirac phase δCP, and one Majorana phase α21 as follows:

      U=(c12c13s12c13s13eiδCPs12c23c12s23s13eiδCPc12c23s12s23s13eiδCPs23c13s12s23c12c23s13eiδCPc12s23s12c23s13eiδCPc23c13)(1000eiα2120001),

      (11)

      where cij and sij stand for cosθij and sinθij, respectively. Then, each mixing is expressed in terms of the component of U as follows:

      sin2θ13=|Ue3|2,sin2θ23=|Uμ3|21|Ue3|2,sin2θ12=|Ue2|21|Ue3|2,

      (12)

      and the Majorana phase α21 and Dirac phase δCP are expressed in terms of the following relations:

      Im[Ue1Ue2]=c12s12c213sin(α212),Im[Ue1Ue3]=c12s13c13sinδCP,

      (13)

      Re[Ue1Ue2]=c12s12c213cos(α212),Re[Ue1Ue3]=c12s13c13cosδCP,

      (14)

      where α21/2, δCP are subtracted from π when cos(α21/2), cosδCP are negative. In addition, the effective mass for the neutrinoless double beta decay is given by

      mee=|κ||˜Dν1cos2θ12cos2θ13+˜Dν2sin2θ12cos2θ13eiα21+˜Dν3sin2θ13e2iδCP|,

      (15)

      where its observed value could be measured by KamLAND-Zen in future [122].

    • 2.   Neutrino masses at the fixed points
    • 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 τ=i,ω,i according to the string theory. At these fixed points, modular forms are obtained by simple forms. Table 2 shows modular forms at τ=i. 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 τ=ω. 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 τ=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.

    • B.   Radiative seesaw

    • 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}} .

    • 1.   Inert boson
    • 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.

    • 2.   Mass matrix of m_\nu
    • 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.

    IV.   NUMERICAL ANALYSIS
    • In this section, we show numerical \Delta \chi^2 analysis for each of the cases, fitting the four reliable experimental data; \Delta m_{\rm sol}^2, \sin^2\theta_{13},\sin^2\theta_{23},\sin^2\theta_{12} in Ref. [123], where \Delta m^2_{\rm atm} is supposed to be the input value. In case of IH for the radiative seesaw model, we would not find any allowed region within 5\sigma . Thus, we do not discuss this case hereafter. The dimensionful input parameters are randomly selected in the range of [ 10^{2}-10^7 ] GeV, while the dimensionless ones are selected in the range of [ 10^{-10}-10^{-1} ] except for τ.

    • A.   NH for the canonical seesaw model

    • Figure 1 represents NH for the canonical seesaw model. We show an allowed region of τ in the top left panel, \langle m_{ee}\rangle in terms of sum of neutrino masses \sum m_i in the top right one, Majorana phase \alpha_{21} and Dirac CP phase \delta_{CP} in the bottom left one, and Dirac CP phase \delta_{CP} versus sum of neutrino masses \sum m_i in the bottom right one, respectively. Each color corresponds to the range of \Delta\chi^2 values such that blue represents \Delta\chi^2 \leq 1 , green represents 1< \Delta\chi^2 \le 4 , yellow represents 4< \Delta\chi^2\le 9 , and red represents 9< \Delta\chi^2 \le 25 . These figures suggest that, within 5\sigma , 0.058 eV \lesssim \sum{ m_i }\lesssim 0.06 eV, 0.001 {\rm {eV}}\lesssim\langle m_{ee}\rangle\lesssim 0.004 eV, any value is possible for \alpha_{21} , and \delta_{CP} tends to be localized near 120 ^\circ and 240 ^\circ. As analytically estimated, we have found several solutions near \tau=i .

      Figure 1.  (color online) NH for the canonical seesaw model: an allowed region of τ is shown in the top left panel, \langle m_{ee}\rangle in terms of sum of neutrino masses \sum m_i in the top right one, Majorana phase \alpha_{21} and Dirac CP phase \delta_{CP} in the bottom left one, and Dirac CP phase \delta_{CP} versus sum of neutrino masses \sum m_i in the bo in the bottom right one, respectively. Each color corresponds to the range of \Delta\chi^2 values such that blue represents \Delta\chi^2 \leq 1 , green represents 1< \Delta\chi^2 \le 4 , yellow represents 4< \Delta\chi^2 \le 9 , and red represents 9< \Delta\chi^2 \le 25 .

    • B.   IH for the canonical seesaw model

    • Figure 2 represents IH for the canonical seesaw model, where the legends and colors are the same as those of NH for the canonical seesaw case. These figures suggest that 0.098\ {\rm eV}\lesssim\sum m_i\lesssim0.102\ {\rm eV} , \langle m_{ee}\rangle \simeq 0.05 eV, \alpha_{21} \simeq 0^\circ , 50^\circ\lesssim\delta_{CP}\lesssim130^\circ , and 230^\circ\lesssim\delta_{CP}\lesssim310^\circ . In our analytical estimation, we have no solutions near \tau=i . However, according to this figure, there would exist solutions near \tau=i . Thus, we investigated the behavior of modular Yukawa functions in terms of τ and found that this function is rather sensitive to deviations from \tau=i .

      Figure 2.  (color online) IH for the canonical seesaw model; the legends and colors are the same as those of NH for the canonical seesaw case.

    • C.   NH for the radiative seesaw model

    • Figure 3 represents NH for the radiative seesaw model; the legends and colors are the same as those of NH for the canonical seesaw case. These figures suggest that 0.057 {\rm eV}\le(\sum m_i,\langle m_{ee}\rangle)\le0.06 eV, and any values are allowed for phases. In the radiative case, there are solutions near \tau=i only in case of NH. This is what we expect from our analytical estimation.

      Figure 3.  (color online) NH for the radiative seesaw model; the legends and colors are the same as those of NH for the canonical seesaw case.

    V.   CONCLUSION AND DISCUSSION
    • We studied a double covering of modular A_4 flavor symmetry in which we constructed lepton models in cases of canonical seesaw and radiative seesaw applying irreducible doublet representations to heavier Majorana fermions that do not have A_4 symmetry. Then, we have some predictions for both cases except for the IH of radiative seesaw. From our numerical analyses, we conclude the following:

      1. In case of NH in the canonical seesaw scenario, within hin 5\sigma , 0.058 {\rm eV}\lesssim\sum m_i\lesssim0.06 eV, 0.001 {\rm eV}\lesssim\langle m_{ee}\rangle\lesssim0.004 eV, any value is possible for \alpha_{21} , and \delta_{CP} tends to be localized near 120 ^\circ and 240 ^\circ .

      2. In case of IH in the canonical seesaw scenario, within hin 5\sigma , 0.098\ {\rm eV}\lesssim\sum m_i\lesssim0.102\ {\rm eV} , \langle m_{ee}\rangle \simeq 0.05 eV, \alpha_{21} \simeq 0^\circ , 50^\circ\lesssim\delta_{CP}\lesssim130^\circ , and 230^\circ\lesssim\delta_{CP}\lesssim310^\circ .

      3. In case of NH in the radiative seesaw scenario, within 5\sigma , 0.057 {\rm eV}\le(\sum m_i,\langle m_{ee}\rangle)\le0.06 eV, and any values are allowed for phases.

      As a future work, it would be interesting to apply this modular symmetry to both the quark and lepton sectors with the common modulus τ. We expect it to result in different predictions from A_4 modular symmetry due to especially irreducible doublet representations.

    APPENDIX A: FORMULAS IN MODULAR T^\prime FRAMEWORK
    • 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}

Reference (123)

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