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.