Analytic solution of Hedin’s equations in zero dimensions

Abstract

Feynman diagrams for the many-body perturbational theory are enumerated by solving the system of Hedin’s equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B 71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.