Liouville term for neutrinos: flavor structure and wave interpretation

Neutrino production, absorption, transport, and flavor evolution in astrophysical environments is described by a kinetic equation D ϱ=− i [ H ,ϱ]+𝒞[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 𝒞[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 + ⋅∂ x for ultra-relativistic neutrinos was recently questioned in that flavor-dependent velocities should appear instead of the unit vector . 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.