Liouville term for neutrinos: Flavor structure and wave interpretation

TL;DR

This paper derives the complete flavor-dependent velocity structure of the neutrino kinetic equation's Liouville term, clarifying its wave interpretation and addressing recent questions about simplified models in astrophysical neutrino transport.

Contribution

It provides a detailed derivation of the flavor-dependent velocities in the neutrino kinetic equation and interprets the equation as a first-order wave equation at the oscillation scale.

Findings

Derived the full flavor-dependent velocity structure of the Liouville operator.

Showed that the simplified velocity expression is an approximation with higher-order corrections.

Argued that the kinetic equation can be viewed as a first-order wave equation at the neutrino oscillation scale.

Abstract

Neutrino production, absorption, transport, and flavor evolution in astrophysical environments is described by a kinetic equation $D\varrho=-i[{\sf H},\varrho]+{\cal C}[\varrho]$. Its basic elements are generalized occupation numbers $\varrho$, matrices in flavor space, that depend on time $t$, space $\bf x$, and momentum $\bf p$. The commutator expression encodes flavor conversion in terms of a matrix $\sf H$ of oscillation frequencies, whereas ${\cal C}[\varrho]$ represents source and sink terms as well as collisions. The Liouville operator on the left hand side involves linear derivatives in $t$, $\bf x$ and $\bf p$. The simplified expression $D=\partial_t+\hat{\bf p}\cdot{\partial}_{\bf x}$ for ultra-relativistic neutrinos was recently questioned in that flavor-dependent velocities should appear instead of the unit vector $\hat{\bf p}$. 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.