Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant

Abstract

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that Bij=|Uij|2 for i,j=1,…,N. The set U3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U3 with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant J, defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on U(3) and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding |J| smaller than the value observed for the Cabbibo–Kobayashi–Maskawa matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the Maki–Nakagawa–Sakata matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.