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Neutrino damping in a fermion and scalar background

José F. Nieves^*

Sarira Sahu^†

Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, Río Piedras, San Juan 00936, Puerto Rico

Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Circuito Exterior, C. U., A. Postal 70-543, 04510 Mexico DF, Mexico

^*nieves@ltp.uprrp.edu
^†sarira@nucleares.unam.mx

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Phys. Rev. D 99, 095013 – Published 16 May, 2019

DOI: https://doi.org/10.1103/PhysRevD.99.095013

Abstract

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 $λ {\bar{f}}_{R} ν_{L} ϕ$ . 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 $λ^{(ν ν ϕ)} {\bar{ν}}_{R}^{c} ν_{L} ϕ$ 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 $ν ϕ$ 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 $ϕ$ -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.

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This is strictly true in the massless neutrino limit that we are considering, as we have already stated in Sec. 2a. In practice it is a valid approximation in the limit that the neutrino mass can be neglected in the calculation of the diagrams involved.
Notice that the antineutrino damping term is not defined with the additional minus sign as it is the case for the real part of the dispersion relation.
To prove Eq. (4.5) we rewrite the condition $E_{F} > ε_{\min}^{(I I)}$ in the form $\frac{1}{4 x} + x < \frac{E_{F}}{m_{f}},$ (A36)where $x \equiv \frac{κ m_{f}}{m_{ϕ}^{2}} .$ (A37)Now let us solve the equation $\frac{1}{4 x} + x = a,$ (A38)where $a$ is, at the moment, unspecified. The solutions are $x_{\pm} (a) = \frac{a}{2} \pm \frac{1}{2} \sqrt{a^{2} - 1} .$ (A39)The functions $x_{\pm} (a)$ are equal at $a = 1$ , and as $a$ increases $x_{+}$ increases while $x_{-}$ decreases. Since these functions, by construction, satisfy Eq. (a38), then their value at any $a$ (subject to $a > 1$ ) satisfies Eq. (a36) for all $a < E_{F} / m_{f}$ . It then follows that all the values of $x$ that lie between $x_{-} (E_{F} / m_{f})$ and $x_{+} (E_{F} / m_{f})$ satisfy Eq. (a36), while the values outside that range correspond to $a > E_{F} / m_{f}$ and therefore will violate it. Using the fact that $x_{\pm} (E_{F} / m_{f}) = \frac{1}{2} (\frac{E_{F}}{m_{f}}) \pm \frac{1}{2} \sqrt{{(\frac{E_{F}}{m_{f}})}^{2} - 1} = \frac{1}{2} (\frac{E_{F}}{m_{f}}) \pm \frac{1}{2} \frac{p_{F}}{m_{f}},$ (A40)proves Eq. (4.5).

Phys. Rev. D 99, 095013– Published 16 May, 2019

Vol. 99, Iss. 9 — 1 May 2019

Received 2 January 2019

DOI: https://doi.org/10.1103/PhysRevD.99.095013

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Neutrino damping in a fermion and scalar background

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