On the Perturbation Expansion of the KPZ Equation

We present a simple argument to show that the β-function of the d-dimensional KPZ equation (d≥2) is to all orders in perturbation theory given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\beta (g_R ) = (d - 2)g_R - [2/(8\pi )^{d/2} ]{\text{ }}\Gamma (2 - d/2)g_R^2 $$ \end{document} Neither the dynamical exponent z nor the roughness exponent ζ have any correction in any order of perturbation theory. This shows that standard perturbation theory cannot attain the strong-coupling regime and in addition breaks down at d = 4. We also calculate a class of correlation functions exactly.