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We investigate the noisy Burgers equation (Kardar--Parisi--Zhang equation in1+1 dimensions) using the dynamical renormalization group (to two--loop order)and mode--coupling techniques. The roughness and dynamic exponent are fixed byGalilean invariance and a fluctuation--dissipation theorem. The fact that thereare no singular two--loop contributions to the two--point vertex functionssupports the mode--coupling approach, which can be understood as aself--consistent one--loop theory where vertex corrections are neglected.Therefore, the numerical solution of the mode coupling equations yields veryaccurate results for the scaling functions. In addition, finite--size effectscan be studied. Furthermore, the results from exact Ward identities, as well asfrom second--order perturbation theory permit the quantitative evaluation ofthe vertex corrections, and thus provide a quantitative test for themode--coupling approach. It is found that the vertex corrections themselves areof the order one. Surprisingly, however, their effect on the correlationfunction is substantially smaller

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