On the Contributions to the $\bf U_Y(1)$ Chern-Simons Term and the Evolution of Fermionic Asymmetries and Hypermagnetic Fields

We study simultaneous evolution of electron, neutrino and quark asymmetries, and large scale hypermagnetic fields in the symmetric phase of the electroweak plasma in the temperature range $100$GeV$\leq T\leq 10$TeV, taking into account the chirality flip processes via inverse Higgs decays and fermion number violation due to Abelian anomalies. We present a derivation of the coefficient of the Chern-Simons term for the hypercharge gauge field, showing that the left-handed and right-handed components of each fermion species contribute with opposite sign. This is in contrast to the results presented in some of the previous works. The $\textrm{U}_{\textrm{Y}}(1)$ Chern-Simons term affects the resulting anomalous magnetohydrodynamic (AMHD) equations. We solve the resulting coupled evolution equations for the lepton and baryon asymmetries, as well as the hypermagnetic field to obtain their time evolution along with their values at the electroweak phase transition ($T_{EW} \sim 100$GeV) for a variety of critical ranges for their initial values at $T=10$TeV. We first investigate the results of this sign change, by directly comparing our results with those obtained in one of the previous works and find that matter asymmetry generation increases considerably in the presence of a strong hypermagnetic field. Furthermore, we find that a strong hypermagnetic field can generate matter asymmetry starting from absolutely zero asymmetry, while matter asymmetry can generate a hypermagnetic field provided the initial value of the latter is nonzero.