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

Journal Article · Tue Dec 15 04:00:00 EST 2009 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3272543· OSTI ID:21294530

Dunkl, Charles ^[1]; Zyczkowski, Karol ^[2]

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1. Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137 (United States)
2. Institute of Physics, Jagiellonian University, 30-059 Cracow (Poland) and Center for Theoretical Physics, Polish Academy of Sciences, 00-668 Warsaw (Poland)

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that B{sub ij}=|U{sub ij}|{sup 2} for i,j=1,...,N. The set U{sub 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{sub 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 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.

• OSTI ID: 21294530
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 50; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
CP INVARIANCE
ENTROPY
KOBAYASHI-MASKAWA MATRIX
NEUTRINO OSCILLATION
PARITY
PROBABILITY
SET THEORY
STOCHASTIC PROCESSES