  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.
