Measures on Mixing Angles

TL;DR

This paper investigates measures on the space of quark mixing matrices to understand the smallness of CP violation, finding that incorporating quark mass hierarchy makes the observed CP violation appear natural rather than fine-tuned.

Contribution

It introduces a measure on the space of mixing matrices that accounts for quark mass hierarchy, providing a more natural explanation for the observed CP violation magnitude.

Findings

Different measures make the observed CP violation seem fine-tuned.

Incorporating quark mass hierarchy increases the likelihood of observed CP violation values.

The choice of mixing angles is linked to the quark mass hierarchy.

Abstract

We address the problem of the apparently very small magnitude of CP violation in the standard model, measured by the Jarlskog invariant J. In order to make statements about probabilities for certain values of J, we seek to find a natural measure on the space of Kobayashi-Maskawa matrices, the double quotient U(1)^2\SU(3)/U(1)^2. We review several possible, geometrically motivated choices of the measure, and compute expectation values for powers of J for these measures. We find that different choices of the measure generically make the observed magnitude of CP violation appear finely tuned. Since the quark masses and the mixing angles are determined by the same set of Yukawa couplings, we then do a second calculation in which we take the known quark mass hierarchy into account. We construct the simplest measure on the space of 3 x 3 Hermitian matrices which reproduces this known hierarchy. Calculating expectation values for powers of J in this second approach, we find that values of J close to the observed value are now rather likely, and there does not seem to be any fine tuning. Our results suggest that the choice of Kobayashi-Maskawa angles is closely linked to the observed mass hierarchy. We close by discussing the corresponding case of neutrinos.