Effective potential for the massless KPZ equation

TL;DR

This paper applies a quantum field theory-inspired formalism to the massless KPZ equation, revealing its ultraviolet renormalization properties and symmetry-breaking behavior in low dimensions.

Contribution

It introduces a functional integral approach to analyze the massless KPZ equation, demonstrating one-loop ultraviolet renormalizability in 1-3 dimensions and symmetry breaking in 1 and 2 dimensions.

Findings

One-loop effective potential is UV renormalizable in 1-3 dimensions.

Massless KPZ exhibits Coleman-Weinberg type symmetry breaking in 1 and 2 dimensions.

Symmetry breaking does not occur in 3 dimensions.

Abstract

In previous work we have developed a general method for casting a classical field theory subject to Gaussian noise (that is, a stochastic partial differential equation--SPDE) into a functional integral formalism that exhibits many of the properties more commonly associated with quantum field theories (QFTs). In particular, we demonstrated how to derive the one-loop effective potential. In this paper we apply the formalism to a specific field theory of considerable interest, the massless KPZ equation (massless noisy vorticity-free Burgers equation), and analyze its behaviour in the ultraviolet (short-distance) regime. When this field theory is subject to white noise we can calculate the one-loop effective potential and show that it is one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, and fails to be ultraviolet renormalizable in higher dimensions. We show that the one-loop effective potential for the massless KPZ equation is closely related to that for lambda phi^4 QFT. In particular we prove that the massless KPZ equation exhibits one-loop dynamical symmetry breaking (via an analog of the Coleman-Weinberg mechanism) in 1 and 2 space dimensions, and that this behaviour does not persist in 3 space dimensions.