$CP$ violation in modular invariant flavor models

TL;DR

This paper investigates how spontaneous CP violation can occur in modular invariant flavor models through the stabilization of the modulus tau, highlighting the role of T symmetry violation for CP violation.

Contribution

It demonstrates that the CP-invariant potential's minimum occurs at specific tau values, and CP violation requires T symmetry breaking within modular invariant flavor models.

Findings

CP is conserved at the potential minimum with Re[tau]=0 or 1/2

Physical CP phase vanishes in the studied models

T symmetry violation is necessary for spontaneous CP violation

Abstract

We study the spontaneous $CP$ violation through the stabilization of the modulus $\tau$ in modular invariant flavor models. The $CP$-invaraiant potentential has the minimum only at ${\rm Re}[\tau] = 0$ or 1/2. From this prediction, we study $CP$ violation in modular invariant flavor models. The physical $CP$ phase is vanishing. The important point for the $CP$ conservation is the $T$ transformation in the modular symmetry. One needs the violation of $T$ symmetry to realize the spontaneous $CP$ violation.