Modular symmetry in the SMEFT

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