Evolution of the Baryon Asymmetry through the Electroweak Crossover in
  the Presence of a Helical Magnetic Field
  We elaborate upon the model of baryogenesis from decaying magnetic helicity
by focusing on the evolution of the baryon number and magnetic field through
the Standard Model electroweak crossover. The baryon asymmetry is determined by
a competition between the helical hypermagnetic field, which sources baryon
number, and the electroweak sphaleron, which tends to wash out baryon number.
At the electroweak crossover both of these processes become inactive; the
hypermagnetic field is converted into an electromagnetic field, which does not
source baryon number, and the weak gauge boson masses grow, suppressing the
electroweak sphaleron reaction. An accurate prediction of the relic baryon
asymmetry requires a careful treatment of the crossover. We extend our previous
study [K. Kamada and A. J. Long, Phys. Rev. D94, 065301 (2016),
arXiv:1606.08891[astro-ph.CO]], taking into account the gradual conversion of
the hypermagnetic into the electromagnetic field. If the conversion is not
completed by the time of sphaleron freeze-out, as both analytic and numerical
studies suggest, the relic baryon asymmetry is enhanced compared to previous
calculations. The observed baryon asymmetry of the Universe can be obtained for
a primordial magnetic field that has a present-day field strength and coherence
length of $B_0 \sim 10^{-17} \, {\rm G}$ and $\lambda_0 \sim 10^{-3} \, {\rm
pc}$ and a positive helicity. For larger $B_0$ the baryon asymmetry is
overproduced, which may be in conflict with blazar observations that provide
evidence for an intergalactic magnetic field of strength $B_0 \gtrsim
10^{-14\sim16} \, {\rm G}$.
