Chiral Charge Erasure via Thermal Fluctuations of Magnetic Helicity

Abstract: 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, $\lambda \gtrsim 1/(\alpha \mu_5)$, the chiral anomaly opens an instability toward the erasure of chiral charge and growth of magnetic helicity. Here the chemical potential $\mu_{5}$ parametrizes the chiral asymmetry and $\alpha$ 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 $\lambda$, finding $\delta \mathcal{H} \sim \lambda T$ and $\tau \sim \alpha \lambda^3 T^2$ 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 $t \sim T^3/(\alpha^5 \mu_5^4)$ until it reaches an equilibrium value $\mathcal{H} \sim \mu_{5} T^2 / \alpha$, and the chiral asymmetry is partially erased. If the chiral asymmetry is small, $\mu_5 < T/\alpha$, this avenue for chiral charge erasure is found to be slower than the chiral magnetic effect for which $t \sim T / (\alpha^3 \mu_{5}^2)$. 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.