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.
