Boltzmann hierarchy for interacting neutrinos I: formalism

TL;DR

This paper derives a detailed Boltzmann hierarchy for interacting neutrinos and scalars from first principles, accounting for energy transfer and momentum dependence, with implications for models of neutrino interactions.

Contribution

It provides the first formal derivation of neutrino/scalar Boltzmann hierarchies including momentum-dependent collision terms from fundamental equations.

Findings

Derived Boltzmann hierarchy with energy transfer effects.

Identified no formal link between hierarchy and $(c_{eff}^2,c_{vis}^2)$ parameterisation.

Applicable to various ultrarelativistic fermion-scalar interaction scenarios.

Abstract

Starting from the collisional Boltzmann equation, we derive for the first time and from first principles the Boltzmann hierarchy for neutrinos including interactions with a scalar particle. Such interactions appear, for example, in majoron-like models of neutrino mass generation. We study two limits of the scalar mass: (i) An extremely massive scalar whose only role is to mediate an effective 4-fermion neutrino-neutrino interaction, and (ii) a massless scalar that can be produced in abundance and thus demands its own Boltzmann hierarchy. In contrast to, e.g., the first-order Boltzmann hierarchy for Thomson-scattering photons, our interacting neutrino/scalar Boltzmann hierarchies contain additional momentum-dependent collision terms arising from a non-negligible energy transfer in the neutrino-neutrino and neutrino-scalar interactions. This necessitates that we track each momentum mode of the phase space distributions individually, even if the particles were massless. Comparing our hierarchy with the commonly used $(c_{\rm eff}^2,c_{\rm vis}^2)$-parameterisation, we find no formal correspondence between the two approaches, which raises the question of whether the latter parameterisation even has an interpretation in terms of particle scattering. Lastly, although we have invoked majoron-like models as a motivation for our study, our treatment is in fact generally applicable to all scenarios in which the neutrino and/or other ultrarelativistic fermions interact with scalar particles.