Modular symmetry in the SMEFT

TL;DR

This paper explores the implications of modular symmetry in the Standard Model effective field theory, analyzing flavor structures and flavor-changing processes, and proposing testable relations at specific modular fixed points.

Contribution

It introduces a string-inspired Ansatz for coupling structures in SMEFT with modular symmetry, deriving novel relations for flavor-changing processes and minimal flavor violation scenarios.

Findings

Flavor-changing processes are suppressed or related to mass matrices under the Ansatz.

Testable relations for FC transitions are derived at fixed points like τ=i and τ=ω.

Minimal flavor violation can be achieved with specific parameter choices.

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.