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

TL;DR

This paper calculates the volume and mean entropy of the set of unistochastic matrices of order 3, analyzes the distribution of the Jarlskog invariant, and discusses implications for particle physics matrices.

Contribution

It provides the first analytical computation of the volume and mean entropy of U_3, and derives the probability distribution of the Jarlskog invariant under different measures.

Findings

The volume of the unistochastic set U_3 is computed analytically.

The probability of small Jarlskog invariant values is low under both measures.

The results have implications for understanding CP violation in particle physics.

Abstract

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 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 U_3 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 CKM matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the MNS 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.