Measures on mixing angles

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 3x3 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.