We derive new expressions for the Rayleigh-Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional renormalization in quantum field theory, involves the Birkhoff decomposition of some Laurent series built up out of explicit fully non-resonant terms present in the usual expression of the Rayleigh-Schrödinger series. More generally we prove that such a decomposition provides solutions of a universal " mould equation ", introduced by two of us in an earlier article, which solves general normal form problems in Lie algebras.