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 $B_0 \gtrsim 10^{-14\sim16} \, {\rm G}$.