Real invariant matrices and flavour-symmetric mixing variables with emphasis on neutrino oscillations

In fermion mixing phenomenology, the matrix of moduli squared, P=(|U|2), is well known to carry essentially the same information as the complex mixing matrix U itself, but with the advantage of being phase-convention independent. The matrix K, formed from the real parts of the mixing-matrix “plaquette” products, is similarly invariant. In this Letter, the P and K matrices are shown to be entirely equivalent, both being directly related (in the leptonic case) to the observable, locally L/E-averaged transition probabilities in neutrino oscillations. We study an (over-)complete set of flavour-symmetric Jarlskog-invariant functions of mass-matrix commutators, rewriting them simply as moment-transforms of such (real) invariant matrices.