Effective non-perturbative real-time dynamics of soft modes in hot gauge theories
D. T. Son
TL;DR
The paper derives a gauge-invariant, non-perturbative Fokker-Planck equation for the real-time diffusion of soft gauge-field configurations in hot non-Abelian plasmas on scale $ (g^2T)^{-1}$ and time $ (g^4T)^{-1}$. Starting from the Schwinger-Keldysh formalism and the influence-functional method, it systematically integrates out hard modes to obtain a Langevin description with a damping term $j$ and a noise kernel $N$, then fixes the ambiguity by formulating the corresponding Fokker-Planck equation. The damping and noise are computed explicitly; the resulting FP equation is gauge invariant and consistent with the known thermodynamics of soft modes, reproducing the static Gibbs distribution in the equilibrium limit. The work provides a framework for using the FP (or the associated Langevin form) in numerical simulations of long-distance hot-gauge dynamics and for connecting dynamical soft-mode evolution to baryon-number violation rates in the electroweak theory.
Abstract
We derive, from first principle, the Fokker-Planck equation describing the non-perturbative real-time evolution of a gauge field at high temperatures on the spatial scale $(g^2T)^{-1}$ and the time scale $(g^4T)^{-1}$, where $g$ is the gauge coupling and $T$ is the temperature. The knowledge of the effective dynamics on such spatial and time scales is crucial for the understanding and quantitative calculation of the baryon number violation rate at high temperatures in the electroweak theory. The Fokker-Planck equation, which describes the diffusive motion of soft components of the gauge field in the space of field configurations, is shown to be gauge-invariant and consistent with the known thermodynamics of soft gauge modes. We discuss the ways the Fokker-Planck equation can be made useful for numerical simulations of long-distance dynamics of hot gauge theories.