Wilsonian Flow and Mass-Independent Renormalization

We derive the Gell-Mann and Low renormalization group equation in the Wilsonian approach to renormalization of massless $g\phi^4$ in four dimensions, as a particular case of a non-linear equation satisfied at any scale by the Wilsonian effective action. We give an exact expression for the $\beta$ and $\gamma_{\phi}$ functions in terms of the Wilsonian effective action at the Wilsonian renormalization scale $\L_R$; at the first two loops they are simply related to the gradient of the flow of the relevant couplings and have the standard values; beyond two loops this relation is spoilt by corrections due to irrelevant couplings. We generalize this analysis to the case of massive $g\phi^4$, introducing a mass-independent Wilsonian renormalization scheme; using the flow equation technique we prove renormalizability and we show that the limit of vanishing mass parameter exists. We derive the corresponding renormalization group equation, in which $\beta$ and $\gamma_{\phi}$ are the same as in the massless case; $\gamma_m$ is also mass-independent; at one loop it is the gradient of a relevant coupling and it has the expected value.