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Neutrino effective potential and damping in a fermion and scalar background in the resonance region

We consider the propagation of a neutrino or an antineutrino in a medium composed of fermions $f$ and scalars $ϕ$ interacting via a Yukawa-type coupling of the form $\bar fνϕ$, for neutrino energies at which the processes like $ν+ ϕ\leftrightarrow f$ or $ν+ \bar f \leftrightarrow \barϕ$, and the corresponding ones for the antineutrino, are kinematically accessible. The relevant energy values are around $|m^2_ϕ- m^2_f|/2m_ϕ$ or $|m^2_ϕ- m^2_f|/2m_f$, where $m_ϕ$ and $m_f$ are the masses of $ϕ$ and $f$, respectively. We refer to either one of these regions as a resonance energy range. Near these points, the one-loop formula for the neutrino self-energy has a singularity. From a technical point of view, that feature is indicative that the self-energy acquires an imaginary part, which is associated with damping effects and cannot be neglected, while the integral formula for the real part must be evaluated using the principal value of the integral. We carry out the calculations explicitly for some cases that allow us to give analytic results. Writing the dispersion relation in the form $ω= κ+ V_{\text{eff}} - iγ/2$, we give the explicit formula for $V_{\text{eff}}$ and $γ$ for the cases considered. When the neutrino energy is either much larger or much smaller than the resonance energy, $V_{\text{eff}}$ reduces to the effective potential that has been already determined in the literature in the high or low momentum regime, respectively. The virtue of the formula we give for $V_{\text{eff}}$ is that it is valid also in the \emph{resonance energy range}, which is outside the two limits mentioned. As a guide to possible applications we give the relevant formulas for $V_{\text{eff}}$ and $γ$, and consider the solution to the oscillation equations including the damping term, in a simple two-generation case.