On the modular invariance of mass eigenstates and CP violation

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.