Ward identities and vanishing of the beta function for d=1 interacting Fermi systems

We give a self consistent and simplified proof of the ( asymptotic) vanishing of the Beta function in d = 1 interacting Fermi systems as a consequence of a few properties deduced from the exact solution of the Luttinger model. Moreover, since the vanishing of the Beta function is usually "proved'' in the physical literature through heuristic arguments based on Ward identities, we briefly discuss here also the possibility of exploiting this idea in a rigorous approach, by using a suitable Dyson equation. We show that there are serious difficulties, related to the presence of corrections ( for which we get careful bounds), which are usually neglected.