Liouville term for neutrinos: flavor structure and wave interpretation

Abstract

Neutrino production, absorption, transport, and flavor evolution in astrophysical environments is described by a kinetic equation Dϱ=−i[H,ϱ]+\U0001d49e[ϱ]. Its basic elements are generalized occupation numbers ϱ, matrices in flavor space, that depend on time t, space x, and momentum p. The commutator expression encodes flavor conversion in terms of a matrix H of oscillation frequencies, whereas \U0001d49e[ϱ] represents source and sink terms as well as collisions. The Liouville operator on the left hand side involves linear derivatives in t, x and p. The simplified expression D=∂t+p̂⋅∂x for ultra-relativistic neutrinos was recently questioned in that flavor-dependent velocities should appear instead of the unit vector p̂. Moreover, a new damping term was postulated as a result. We here derive the full flavor-dependent velocity structure of the Liouville term although it appears to cause only higher-order corrections. Moreover, we argue that on the scale of the neutrino oscillation length, the kinetic equation can be seen as a first-order wave equation.