Really Computing Non-perturbative Real Time Correlation Functions

TL;DR

This work addresses the nonperturbative infrared behavior of real-time correlation functions that govern baryon-number violation at high temperature, a regime where naive classical simulations are UV-sensitive. It proposes a modified GR-like algorithm that introduces an intermediate scale μ with $gT \ll μ \ll T$, integrates out hard modes to produce a hard thermal loop (HTL) Hamiltonian for soft modes, and then evolves the soft sector classically while averaging over a thermal initial ensemble to extract $<Q(t) Q(0)>_T$. The formal development connects Grigoriev-Rubakov's approach to HTL-based effective theories, using a toy $\lambda\phi^4$ model and discussing hot gauge theories via Braaten-Pisarski-Nair structures, with the aim of obtaining a cutoff-independent infrared prediction for the baryon-violation rate $\Gamma \sim \kappa (\alpha_W T)^4$. The paper emphasizes both the potential payoff of an infrared, nonperturbative real-time calculation and the significant theoretical and practical hurdles, notably in maintaining gauge invariance and achieving a robust, μ-independent algorithm in gauge theories.

Abstract

It has been argued by Grigoriev and Rubakov that one can simulate real time processes involving baryon number non-conservation at high temperature using real time evolution of classical equations, and summing over initial conditions with a classical thermal weight. It is known that such a naive algorithm is plagued by ultraviolet divergences. In quantum theory the divergences are regularized, but the corresponding graphs involve the contributions from the hard momentum region and also the new scale $\sim gT$ comes into play. We propose a modified algorithm which involves solving the classical equations of motion for the effective hard thermal loop Hamiltonian with an ultraviolet cutoff $μ\gg gT$ and integrating over initial conditions with a proper thermal weight. Such an algorithm should provide a determination of the infrared behavior of real time correlation function $<Q(t) Q(0)>_T$ determining the baryon violation rate. Hopefully, the results obtained in this modified algorithm would be cutoff-independent.