Upper critical dimension of the Kardar-Parisi-Zhang equation

A recent mapping of the interface roughening problem to directed polymers by Lassig and Kinzelbech has shown that the upper critical dimension of the Kardar-Parisi-Zhang equation is less than or equal to four. By combining the mode-coupling technique with a small-cu (roughening exponent) expansion, we show that the upper critical dimension is four. The validity of this conclusion is obviously limited by the applicability of the mode-coupling technique to the strong-coupling regime.