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 $U(1{)}^2\setminus SU(3)/U(1{)}^2$. 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 $3\times 3$ 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.

Figure 1

Probability distribution for $\log |J|$ using the $SU(3)$-invariant flag measure, with fit to $p(|J|)\propto |J{|}^{\lambda }$.

Figure 2

Probability distribution for $\log |J|$ using the Kaluza-Klein measure, with fit to $p(|J|)\propto |J{|}^{\lambda }$.

Figure 3

Probability distribution for $\log |J|$ using a uniform distribution, with fit to $p(|J|)\propto |J{|}^{\lambda }$.