Standard Model Contributions to the Neutrino Index of Refraction in the Early Universe

With the standard electroweak interactions, the lowest-order coherent forward scattering amplitudes of neutrinos in a CP symmetric medium (such as the early universe) are zero, and the index of refraction of a propagating neutrino can only arise from the expansion of gauge boson propagators, from radiative corrections, and from new physics interactions. Motivated by nucleosynthesis constraints on a possible sterile neutrino (suggested by the solar neutrino deficit and a possible $17\ keV$ neutrino), we calculate the standard model contributions to the neutrino index of refraction in the early universe, focusing on the period when the temperature was of the order of a few $MeV$. We find sizable radiative corrections to the tree level result obtained by the expansion of the gauge boson propagator. For $\nu_e+e(\bar{e})\to \nu_e+e(\bar{e})$ the leading log correction is about $+10\%$, while for $\nu_e+\nu_e(\bar{\nu}_e)\to \nu_e+\nu_e(\bar{\nu}_e)$ the correction is about $+20\%$. Depending on the family mixing (if any), effects from different family scattering can be dominated by radiative corrections. The result for $\nu+\gamma\to\nu+\gamma$ is zero at one-loop level, even if neutrinos are massive. The cancellation of infrared divergence in a coherent process is also discussed.