Nonconservation of lepton current and asymmetry of relic neutrinos
  The neutrino asymmetry in the early universe plasma, $n_\nu - n_{\bar \nu}$,
is calculated both before and after the electroweak phase transition (EWPT).
The leptogenesis before EWPT within the standard model is well known to be
driven by the abelian anomaly in a massless hypercharge field. The generation
of the neutrino asymmetry in the Higgs phase after EWPT, in its turn, has not
been considered previously because of the absence of any quantum anomaly in an
external electromagnetic field for such electroneutral particles as neutrino,
unlike the Adler anomaly for charged left and right polarized massless
electrons in the same electromagnetic field. Using the neutrino Boltzmann
equation, modified by the Berry curvature term in the momentum space, we
establish the violation of the macroscopic neutrino current in plasma after
EWPT and exactly reproduce the nonconservation of the lepton current in the
symmetric phase before EWPT arising in quantum field theory due to the non-zero
lepton hypercharge and corresponding triangle anomaly in an external
hypercharge field. In the last case the violation of the lepton current is
derived through the kinetic approach without the computation of the
corresponding Feynman diagrams. Then the new kinetic equation is applied for
the calculation of the neutrino asymmetry accounting for the the Berry
curvature and the electroweak interaction with background fermions in the Higgs
phase, including the stage after the neutrino decoupling at the absence of
neutrino collisions in plasma. This asymmetry is found to be rather small to be
observed. Thus, a difference of the relic neutrino and antineutrino densities,
if exists, should be acquired mainly in the symmetric phase before EWPT.
