Interacting Fermi liquid at finite temperature: Part I: Convergent Attributions

Using the method of continuous renormalization group around the Fermi surface, we prove that a two-dimensional jellium interacting system of Fermions at low temperature T is a Fermi liquid (analytic in the coupling constant g) for g < const/|logT|, and satisfying uniform bounds on the first and second derivatives of the selfenergy. This bound is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermions systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof.