Anomalous non-conservation of fermion/chiral number in Abelian gauge theories at finite temperature

We discuss the non-conservation of fermion number (or chirality breaking, depending on the fermionic charge assignment) in Abelian gauge theories at finite temperature. We study different mechanisms of fermionic charge disappearance in the high temperature plasma, using both analytical estimates and real-time classical lattice numerical simulations. We investigate the random walk of the Chern-Simons number $Q \propto \int d^4x F_{\mu\nu}{\tilde F}^{\mu\nu}$, and show that it has a diffusive behaviour in the presence of an external magnetic field $B$. This indicates that the mechanism for fermionic number non-conservation for $B \neq 0$, is due to fluctuations of the gauge fields, similarly as in the case of non-Abelian gauge theories. We determine numerically, with lattice simulations, the rate $\Gamma$ of chirality non-conservation associated with this diffusion. We find the rate to be a factor $\mathcal{O}(10)$ larger compared to previous theoretical estimates, what calls for a revision of the implications of Abelian fermion number and chirality non-conservation for baryogenesis, magnetogenesis and chiral symmetry evolution. An erratum can be found at the end of the manuscript, following the results obtained in JHEP 10 (2019) 142 (1904.11892 [hep-th]), which reduce our original claim in the discrepancy between theory and numerics by a factor $\sim 5-6$.