Improved actions, the perfect action, and scaling by perturbation theory in Wilsons renormalization group: the two dimensional $O(N)$-invariant non linear $σ$-model in the hierarchical approximation

We propose a method using perturbation theory in the running coupling constant and the idea of scaling to determine improved actions for lattice field theories combining Wilson's renormalization group with Symanzik's improvement program . The method is based on the analysis of a single renormalization group transformation. We test it on the hierarchical $O(N)$ invariant $\sigma$ model in two dimensions.