Lepton asymmetry growth in the symmetric phase of an electroweak plasma with hypermagnetic fields versus its washing out by sphalerons

We study lepton asymmetry evolution in plasma of the early Universe before the electroweak phase transition (EWPT) accounting for chirality flip processes via Higgs decays (inverse decays) entering equilibrium at temperatures below TRL10 TeV, TEW<T<TRL. We solve appropriate kinetic equations for leptons and Higgs bosons taking into account the lepton number violation due to Abelian anomalies for right and left electrons and neutrinos in the self-consistent hypercharge field obeying Maxwell equations modified by the contribution of the Standard Model of electroweak interactions. The violation of left lepton numbers and the corresponding violation of the baryon number due to sphaleron processes in symmetric phase is taken into account as well. Assuming the Chern-Simons wave configuration of the seed hypercharge field, we get the estimates of baryon and lepton asymmetries evolved from the primordial right electron asymmetry existing alone as partial asymmetry at T≥TRL. One finds a strong dependence of the asymmetries on the Chern-Simons wave number. We predict a nonzero chiral asymmetry Δμ=μ_eR-μ_eL≠0 in this scenario evolved down to the EWPT moment that can be used as an initial value for the Maxwellian field evolution after EWPT.