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