The effects of the U$_\textrm{Y}$(1) Chern-Simons term and its baryonic   contribution on matter asymmetries and hypermagnetic fields
  In this paper, we study the significance of the U$_\textrm{Y}$(1) Chern-Simons term in general, and its baryonic contribution in particular, for the evolution of the matter asymmetries and the hypermagnetic field in the temperature range $100$GeV$\leq T \leq 10$TeV. We show that an initial helical hypermagnetic field, denoted by $B_Y^{(0)}$, can grow matter asymmetries from zero initial value. However, the growth which is initially quadratic with respect to $B_Y^{(0)}$, saturates for values larger than a critical value. The inclusion of the baryonic contribution reduces this critical value, leading to smaller final matter asymmetries. Meanwhile, $B_Y(T_{EW})$ becomes slightly larger than $B_Y^{(0)}$. In the absence of the U$_\textrm{Y}$(1) Chern-Simons term, the final values of matter asymmetries grow without saturation. Conversely, we show that an initial matter asymmetry can grow an initial seed of hypermagnetic field, provided the Chern-Simons term is taken into account. The growth process saturates when the matter asymmetry drops abruptly. When the baryonic contribution is included, the saturation occurs at an earlier time, and $B_Y (T_{EW})$ becomes larger. We also show the results can be within the acceptable range of present day data, provided the inverse cascade process is also taken into account. 
S. Rostam Zadeh and S. S. Gousheh
hep-ph