Production of heavy sterile neutrinos from vector boson decay at electroweak temperatures

In the standard model extended with a seesaw mass matrix, we study the production of sterile neutrinos from the decay of vector bosons at $T\simeq M_{W,Z}$. We derive a general quantum kinetic equation for the production of sterile neutrinos and their effective mixing angles valid in a wide range of temperature, to all orders in interactions of the standard model, and to leading order mixing angle $\theta \ll 1$. Production rates and effective mixing angles depend sensitively on helicity. Positive helicity states interact more weakly with the medium and their effective mixing angle is not modified significantly whereas the mixing angle for negative helicity is strongly suppressed by the medium. If $M_s \lesssim 8.35\,\mathrm{MeV}$, there are fewer states with negative helicity produced than those with positive helicity. There is an MSW resonance in the absence of lepton asymmetry, but is screened by the damping rate, without production enhancement. Negative helicity states freeze-out at $T^-_f\simeq 5\,\mathrm{GeV}$ and positive helicity states freeze-out at $T^+_f \simeq 8\,\mathrm{GeV}$, both distributions are far from thermal. Negative helicity states feature a broader momentum distribution than that for those with positive helicity. Sterile neutrinos produced via vector boson decay do not satisfy the abundance, lifetime and cosmological constraints to be the sole dark matter component in the universe but might solve the $^{7}Li$ problem, albeit at the very edge of the possible parameter space. A heavy sterile neutrino with a mass of a few MeV could decay into light sterile neutrinos, of a few keV in mass, that contribute to warm dark matter. We argue that heavy sterile neutrinos with lifetime $\leq 1/H_0$ reach local thermodynamic equilibrium.