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 $\mathrm{CP}$-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 ${\nu }_e+e\left(\overline{e}\right)\to {\nu }_e+e\left(\overline{e}\right)$ the leading-log correction is about +10%, while for ${\nu }_e+{\nu }_e\left({\overline{\nu }}_e\right)\to {\nu }_e+{\nu }_e\left({\overline{\nu }}_e\right)$ 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 the one-loop level, even if neutrinos are massive. The cancellation of infrared divergence in a coherent process is also discussed.