Towards a precision calculation of $N_{\mathrm{eff}}$ in the Standard Model III: Improved estimate of NLO contributions to the collision integral

Abstract

We compute the dominant QED correction to the neutrino-electron interaction rate in the vicinity of neutrino decoupling in the early universe, and estimate its impact on the effective number of neutrino species $N_{\mathrm{eff}}$ in cosmic microwave background anisotropy observations. We find that the correction to the interaction rate is at the sub-percent level, consistent with a recent estimate by Jackson and Laine. Relative to that work we include the electron mass in our computations, but restrict our analysis to the enhanced $t$-channel contributions. The fractional change in $N_{\mathrm{eff}}^{\mathrm{SM}}$ due to the rate correction is of order $10^{-5}$ or below, i.e., about a factor of 30 smaller than that recently claimed by Cielo {\it et al.}, and below the nominal computational uncertainties of the current benchmark value of $N_{\mathrm{eff}}^{\mathrm{SM}}=3.0440\pm 0.0002$. We therefore conclude that aforementioned number remains to be the state-of-the-art benchmark for $N_{\mathrm{eff}}^{\mathrm{SM}}$ in the standard model of particle physics.

34 pages, 8 figures, 2 tables; v2: version accepted by JCAP

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