On the modular invariance of mass eigenstates and CP violation

TL;DR

This paper explores how modular invariance affects mass eigenstates and CP violation in heterotic orbifolds, revealing that physical quantities can be modular invariant despite noninvariant components, with implications for CP violation and baryogenesis.

Contribution

It demonstrates that physical invariance can be achieved through patching noninvariant functions and clarifies the conditions under which CP violation can occur in modular invariant theories.

Findings

Mass eigenstates are inherently modular invariant.

Coupling constants are smooth invariant functions of T.

CP violation on the unit circle is generally suppressed in realistic models.

Abstract

We investigate the modular transformation properties of observable (light) fields in heterotic orbifolds, in the light of recent calculations of CP-violating quantities. Measurable quantities must be modular invariant functions of string moduli, even if the light fields are noninvariant. We show that physical invariance may arise by patching smooth functions that are separately noninvariant. CP violation for <T> on the unit circle, which requires light and heavy states to mix under transformation, is allowed in principle, although the Jarlskog parameter J_CP(T) must be amended relative to previous results. However, a toy model of modular invariant mass terms indicates that the assumption underlying these results is unrealistic. In general the mass eigenstate basis is manifestly modular invariant and coupling constants are smooth invariant functions of T, thus CP is unbroken on the unit circle. We also discuss the status of CP-odd quantities when CP is a discrete gauge symmetry, and point out a link with baryogenesis.