Neutrino damping in a fermion and scalar background

We consider the propagation of a neutrino in a background composed of a scalar particle and a fermion using a simple model for the coupling of the form $\lambda\bar f_R\nu_L\phi$. In the presence of these interactions there can be damping terms in the neutrino effective potential and index of refraction. We calculate the imaginary part of the neutrino self-energy in this case, from which the damping terms are determined. The results are useful in the context of Dark Matter-neutrino interaction models in which the scalar and/or fermion constitute the dark-matter. The corresponding formulas for models in which the scalar particle couples to two neutrinos via a coupling of the form $\lambda^{(\nu\nu\phi)}\bar\nu^c_R\nu_L\phi$ are then obtained as a special case, which can be important also in the context of neutrino collective oscillations in a supernova and in the Early Universe hot plasma before neutrino decoupling. A particular feature of our results is that the damping term in a $\nu\phi$ background is independent of the antineutrino-neutrino asymmetry in the background. Therefore, the relative importance of the damping term may be more significant if the neutrino-antineutrino asymmetry in the background is small, because the leading $Z$-exchange and $\phi$-exchange contributions to the effective potential, which are proportional to the neutrino-antineutrino asymmetry, are suppressed in that case, while the damping term is not.