Sterile neutrino rates for general $M$, $T$, $\mu$, $k$: review of a theoretical framework

Abstract

The temperature of sphaleron freeze-out, $T\approx 130$ GeV, sets a scale for mechanisms for generating lepton asymmetries and converting them to baryon asymmetry (leptogenesis). For Majorana masses $M\gg \pi T$, leptogenesis takes place in the non-relativistic regime, while for $M\ll \pi T$, the dynamics is ultrarelativistic. The intermediate case $M\sim \pi T$ is the most cumbersome, as no effective theory methods are available. We review the definitions of and provide integral representations for all rate coefficients and mass corrections of $\mathcal{O}(h_{\nu }^2)$ in this regime, such that the expressions extrapolate to known limits and are gauge independent to the order computed, both in the symmetric and the Higgs phase, for any helicity, momentum, and set of chemical potentials. Road signs towards a numerical evaluation are offered.

Comments: 36 pages. v2: minor fixes, published version