Mass matrices with CP phase in modular flavor symmetry

TL;DR

This paper investigates how CP violation can arise in quark mass matrices within modular flavor symmetric models, especially focusing on the effects of T-symmetry breaking at fixed points like tau = e^{2πi/3}, using magnetized orbifold models as examples.

Contribution

It explores the conditions under which CP violation emerges in modular flavor models, highlighting the role of T-symmetry breaking and the behavior of multiple Higgs modes.

Findings

CP violation can originate from T-symmetry breaking at fixed points.

The behavior of Higgs modes under T-transformation affects the CP phase.

An example model shows a non-zero CP phase at tau = e^{2πi/3}.

Abstract

We study the CP violation and the CP phase of quark mass matrices in modular flavor symmetric models. The CP symmetry remains at $\tau = e^{2 \pi i/3}$ by a combination of the $T$-symmetry of the modular symmetry. However, the $T$-symmetry breaking may lead to the CP violation at the fixed point $\tau = e^{2 \pi i/3}$. We study such a possibility in magnetized orbifold models as examples of modular flavor symmetric models. These models, in general, have more than one candidates for Higgs modes, while generic string compactifications also lead to several Higgs modes. These Higgs modes have different behaviors under the $T$-transformation. The light Higgs mode can be a linear combination of those modes so as to lead to realistic quark mass matrices. The CP phase of mass matrix does not appear in a certain case, which is determined by the $T$-transformation behavior. Deviation from it is important to realize the physical CP phase. We discuss an example leading to non-vanishing CP phase at the fixed point $\tau = e^{2 \pi i/3}$.