The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) which was introduced in the eighties by van Beijren, Kutner and Spohn to describe, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with one conserved scalar quantity. Following a Physics heuristics, the non-linearity is usually chosen to be quadratic as this is the first term that cannot be removed via a Galileian transformation and should thus provide the first non-trivial contribution to the dynamics. As shown by Hairer and Quastel in dimension 1 in the so-called weakly asymmetric scaling, such derivation is not fully correct and, when considering a generic non-linearity higher order terms do contribute to the limit. In the present talk, we consider the critical dimension 2 and prove that, under a logarithmically superdiffusive scaling (no weak asymmetry is required), the same is true, meaning that the limiting contribution of the non-linearity comes in via its second order coefficient of its Hermite expansion. We conclude with some remarks in higher dimension, for which instead every term in the expansion does contribute to the limit. This is a joint work with Q. Moulard and T. Klose.