Leptonic and semi-leptonic neutrino interactions with muons in proto-neutron star cooling

Abstract

Abstract It is known that muons are scarce just after the birth of a proto-neutron star via a supernova explosion, but get more abundant as the proto-neutron star cools via neutrino emissions on the Kelvin–Helmholtz timescale. We evaluate all the relevant rates of the neutrino interactions with muons at different times in the proto-neutron star cooling. We are particularly interested in the late phase ($t \gtrsim 10 \operatorname{s}$), which will be accessible in the next Galactic supernova but has not been studied well so far. We calculate both leptonic and semi-leptonic processes, for the latter of which we also pay attention to the form factors with their dependence on the transferred momentum as well as to the modification of the dispersion relations for nucleons on the mean field level. We find that the flavor-exchange reactions νe + μ− → νμ + e− and $\bar{\nu }_{\mu } + \mu ^- \rightarrow \bar{\nu }_e + e^-$ can be dominant, particularly at low energies, over the capture of νe on neutron and the scatterings of $\bar{\nu }_{\mu }$ on nucleons as the opacity sources for these species, and that the inverse muon decay $\bar{\nu }_e + \nu _{\mu } + e^- \leftrightarrows \mu ^-$ can overwhelm the scatterings of $\bar{\nu }_e$ and νμ on nucleons again at low energies. At high energies, on the other hand, the corrections in the semi-leptonic processes mentioned above are more important. We also show the non-trivial energy and angular dependencies of the flavor-exchange reactions and the inverse muon decay. In the study of the diffusion coefficients from these reactions, we find that $\bar{\nu }_{\mu }$ is most affected. These pieces of information are indispensable for numerical computations and the interpretation of their results for proto-neutron star cooling, particularly at the very late phase.