Upper critical dimension, dynamic exponent and scaling functions in the mode-coupling theory for the Kardar-Parisi-Zhang equation

We study the mode-coupling approximation for the KPZ equation in the strong coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension d_c=4, and the expansion z=2-(d-4)/4+O((4-d)^2) around d_c. We find the exact z=3/2 value in d=1, and estimate the values 1.62, 1.78 for z, in d=2,3. The result d_c=4 and the expansion around d_c are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2.