Mode-coupling and renormalization group results for the noisy Burgers equation

We investigate the noisy Burgers equation (Kardar-Parisi-Zhang equation in 1 + 1 dimensions) using the dynamical renormalization group (to two-loop order) and mode-coupling techniques. The roughness and dynamic exponent are fixed by Galilean invariance and a fluctuation-dissipation theorem. The fact that there are no singular two-loop contributions to the two-point vertex functions supports the mode-coupling approach, which can be understood as a self-consistent one-loop theory where vertex corrections are neglected. Therefore, the numerical solution of the mode-coupling equations yields very accurate results for the scaling functions. In addition, finite-size effects can be studied. Furthermore, the results from exact Ward identities, as well as from second-order perturbation theory, permit the quantitative evaluation of the vertex corrections, and thus provide a quantitative test for the mode-coupling approach. It is found that the vertex corrections themselves are of the order 1. Surprisingly, however, their effect on the correlation function is substantially smaller.