  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.

High Energy Physics - Phenomenology