Derivation of the Functional Renormalization Group β-function at order 1/N for manifolds pinned by disorder

In an earlier publication, we have introduced a method to obtain, at large N , the effective action for d-dimensional manifolds in a N -dimensional disordered environment. This allowed to obtain the Functional Renormalization Group (FRG) equation for N = ∞ and was shown to reproduce, with no need for ultrametric replica symmetry breaking, the predictions of the Mézard-Parisi solution. Here we compute the corrections at order 1/N . We introduce two novel complementary methods, a diagrammatic and an algebraic one, to perform the complicated resummation of an infinite number of loops, and derive the β-function of the theory to order 1/N . We present both the effective action and the corresponding functional renormalization group equations. The aim is to explain the conceptual basis and give a detailed account of the novel aspects of such calculations. The analysis of the FRG flow, comparison with other studies, and applications, e.g. to the strong-coupling phase of the Kardar-Parisi-Zhang equation are examined in a subsequent publication.