Boltzmann hierarchy for interacting neutrinos I: formalism

TL;DR

This work derives the first-principles Boltzmann hierarchy for ultrarelativistic neutrinos interacting with a scalar, covering two scalar-mass limits: a very heavy mediator and a massless scalar. It reveals momentum-dependent collision terms that prevent simple energy-integrated closures and shows that the common $(c_{ m eff}^2,c_{ m vis}^2)$ parameterisation cannot, in general, be interpreted as a particle-scattering description. The results provide explicit, implementable hierarchies for both neutrinos (and scalars in the massless limit) and highlight fundamental differences from Thomson-like photon hierarchies, with implications for CMB analyses of neutrino interactions. The framework is broadly applicable to any ultrarelativistic fermions interacting with scalars, and sets the stage for future numerical implementation in CAMB/CLASS-like codes to assess cosmological signatures of such interactions.

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.

Figures (5)

• Figure 1: Tree-level diagrams for the $2 \to 2$ scattering processes described by the Lagrangian \ref{['Lagrangian']} in the $s$, $t$, and $u$-channels. The neutrinos have been explicitly assumed to be Majorana particles.
• Figure 2: Integral kernels $K^{\rm m}_{0,1,2,3}$, defined in equation \ref{['eq:massivekernelP']} and plotted here in units of $T_{\nu,0}^4$, as functions of $y\equiv |\boldsymbol{q}|/ T_{\nu,0}$ and $y' \equiv |\boldsymbol{q}'|/ T_{\nu,0}$.
• Figure 3: The functions ${\cal X}^{\nu}(|\boldsymbol{q}|,\eta)$ (blue) and ${\cal X}^{\phi}(|\boldsymbol{q}|,\eta)$ (red), defined in equation \ref{['SumMassless']} and shown here in units of $T_{\nu,0}$, as functions of $y \equiv |\boldsymbol{q}|/T_{\nu,0}$, assuming a Maxwell-Boltzmann background distribution for both the neutrinos and scalar particles. Both functions have been evaluated at the time of recombination for a neutrino mass of $m_{\nu}=0.05$ eV.
• Figure 4: Integral kernels ${\cal K}^{\nu}_{0,1,2,3}$, defined in equation \ref{['SumMassless']} and shown here as $\log_{10}(|{\cal K}^{\nu}_{0,1,2,3}|)$, as functions of $y\equiv |\boldsymbol{q}|/ T_{\nu,0}$ and $y' \equiv |\boldsymbol{q}'|/ T_{\nu,0}$, assuming a Maxwell--Boltzmann background distribution for the neutrinos and scalar particles. The functions have been evaluated at the time of recombination for a neutrino mass of $m_{\nu}=0.05$ eV. For ${\cal K}^{\nu}_{0}$, ${\cal K}^{\nu}_{1}$ and ${\cal K}^{\nu}_{3}$ the two green--blue lines mark the points where the functions flip sign.
• Figure 5: Same as figure \ref{['Fig:MasslessKernel1']}, but for the kernels $\mathfrak{K}^{\nu}_{0,1,2,3}$.