Naive scaling at the level of the Lagrangian does not directly correspond to physical scaling because a cutoff is required to give a meaning to the field theory and the path integral. Changing the length scale also changes the number of degrees of freedom. This complication is taken into account by the renormalization group. The main result at the upper critical dimension is that scale invariance remains valid for large factors b , but with additional l n ( b ) factors in the scaling of the coordinates and fields.