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Chiral Charge Erasure via Thermal Fluctuations of Magnetic Helicity

We consider a relativistic plasma of fermions coupled to an Abelian gauge field and carrying a chiral charge asymmetry, which might arise in the early Universe through baryogenesis. It is known that on large length scales, $λ\gtrsim 1/(αμ_5)$, the chiral anomaly opens an instability toward the erasure of chiral charge and growth of magnetic helicity. Here the chemical potential $μ_{5}$ parametrizes the chiral asymmetry and $α$ is the fine-structure constant. We study the process of chiral charge erasure through the thermal fluctuations of magnetic helicity and contrast with the well-studied phenomenon of Chern-Simons number diffusion. Through the fluctuation-dissipation theorem we estimate the amplitude and time scale of helicity fluctuations on the length scale $λ$, finding $δ\mathcal{H} \sim λT$ and $τ\sim αλ^3 T^2$ for a relativistic plasma at temperature $T$. We argue that the presence of a chiral asymmetry allows the helicity to grow diffusively for a time $t \sim T^3/(α^5 μ_5^4)$ until it reaches an equilibrium value $\mathcal{H} \sim μ_{5} T^2 / α$, and the chiral asymmetry is partially erased. If the chiral asymmetry is small, $μ_5 < T/α$, this avenue for chiral charge erasure is found to be slower than the chiral magnetic effect for which $t \sim T / (α^3 μ_{5}^2)$. This mechanism for chiral charge erasure can be important for the hypercharge sector of the Standard Model as well as extensions including ${\rm U}(1)$ gauge interactions, such as asymmetric dark matter models.