Quantum Mayer graphs for Coulomb systems and the analog of the Debye potential

Within the Feynman-Kac path integral representation, the equilibrium quantities of a quantum plasma can be represented by Mayer graphs. The well known Coulomb divergencies that appear in these series are eliminated by partial resummations. In this paper, we propose a resummation scheme based on the introduction of a single effective potential phi that is the quantum analog of the Debye potential. A low density analysis of phi shows that it reduces, at short distances, to the bare Coulomb interaction between the charges (which is able to lead to bound states). At scale of the order of the Debye screening length kappa(D)(-1), phi approaches the classical Debye potential and, at large distances, it decays as a dipolar potential (this large distance behaviour is due to the quantum nature of the particles). The prototype graphs that result from the resummation obey the same diagrammatical rules as the classical graphs of the Abe-Meeron series. We give several applications that show the usefulness of phi to account for Coulombic effects at all distances in a coherent way.