Modular symmetry in the SMEFT

Abstract

We study the modular symmetric standard-model effective field theory. We employ the stringy Ansatz on coupling structure that 4-point couplings $y^{(4)}$ of matter fields are written by a product of 3-point couplings $y^{(3)}$ of matter fields, i.e., $y^{(4)}=y^{(3)}{y}^{(3)}$. In this framework, we discuss the flavor structure of bilinear fermion operators and 4-fermion operators, where the holomorphic and anti-holomorphic modular forms appear. From the Ansatz, the $A_4$ modular-invariant semileptonic four-fermion operator $[{\bar{E}}_R\Gamma E_R][{\bar{D}}_R\Gamma D_R]$ does not lead to the flavor changing (FC) processes since this operator would be constructed in terms of gauge couplings $g$ as $y^{(3)}\sim g$. The chirality flipped bilinear operator $[{\bar{D}}_R\Gamma D_L]$ also does not lead FC if the mediated mode corresponds to the Higgs boson $H_d$. In this case, the flavor structure of this operator is the exactly same as the mass matrix. On the other hand, if the flavor structure of the operator is not the exactly same as the mass matrix, the situation would change drastically. Then, we obtain the non-trivial relations of FC transitions at nearby fixed points $\tau =i,\omega ,i\infty$, which are testable in the future. As an application, we discuss the relations of the lepton flavor violation processes $\mu \to e\gamma ,\tau \to \mu \gamma$ and $\tau \to e\gamma$ at nearby ${\tau }_e=i$, where the successful lepton mass matrix was obtained. We also study the flavor changing 4-quark operators in the $A_4$ modular symmetry of quarks. As a result, the minimal flavor violation could be realized by taking relevant specific parameter sets of order one.

arXiv comment48 pages, 14 tables, revised discussion in Sec.4, several tables modified

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• Exploring the flavor structure of quarks and leptons with reinforcement learning
• Confronting the prediction of leptonic Dirac CP-violating phase with experiments
• Froggatt-Nielsen Meets the SMEFT
• Generalized Matter Parities from Finite Modular Symmetries
• Minimal Modular Flavor Symmetry and Lepton Textures Near Fixed Points
• Quasi two-zero texture in Type-II seesaw at fixed points from modular $A_4$ symmetry