Sum rules and asymptotic behaviors of neutrino mixing in dense matter

Abstract

It has proved convenient to define the effective lepton flavor mixing matrix U˜ and neutrino mass-squared differences Δ˜ji≡m˜j2−m˜i2 (for i,j=1,2,3) to describe the phenomena of neutrino mixing and flavor oscillations in a medium, but the prerequisite is to establish direct and transparent relations between these effective quantities and their fundamental counterparts in vacuum. With the help of two sets of sum rules for U˜ and Δ˜ji, we derive new and exact formulas for moduli of the nine elements of U˜ and the sides of its three Dirac unitarity triangles in the complex plane. The asymptotic behaviors of |U˜αi|2 and Δ˜ji (for α=e,μ,τ and i,j=1,2,3) in very dense matter (namely, allowing the matter parameter A=22GFNeE to mathematically approach infinity) are analytically unraveled for the first time, and in this connection the confusion associated with the parameter redundancy of θ˜12, θ˜13, θ˜23 and δ˜ in the standard parametrization of U˜ is clarified.