A Minimal System Including Weak Sphalerons for Investigating the
  Evolution of Matter Asymmetries and Hypermagnetic Fields
  We study simultaneous evolution of large scale hypermagnetic fields and the
asymmetries of quarks, leptons and Higgs boson in the temperature range from
10TeV to 100GeV. Above 10TeV, we identify all of the major fast interactions
and use the associated conservation laws as constraints on the initial
conditions at 10TeV. Below 10TeV, we identify the major processes which fall
out of equilibrium or emerge as non-negligible processes and derive the
relevant evolution equations. These include the Abelian anomalies which violate
fermion numbers, direct and inverse Higgs decays that change the chiralities of
fermions, and weak sphalerons which violate the left-handed fermion numbers. We
also consider the contributions of all fermionic chemical potentials to the
UY(1) Chern-Simons term which affects the evolution through the AMHD equations.
Thus, we present a minimal set of self-consistent initial conditions and
evolution equations which respect all constraints coming from conservation
laws, fast processes and charge neutrality of the plasma. We solve the coupled
evolution equations and find that initial large hypermagnetic field can produce
matter asymmetries starting from zero initial value, and vice versa provided an
initial seed of hypermagnetic field is present and the rate of the electron
Yukawa processes is lower. We find that our model yields acceptable values for
baryon asymmetry and magnetic field. However, the scale of the magnetic field
obtained is much smaller than the observational data, even when the turbulence
driven inverse cascade mechanism in the broken phase is taken into account.
