We present in detail the implementation of the Blaizot-Méndez-Wschebor (BMW) approximation scheme of the nonperturbative renormalization group, which allows for the computation of the full momentum dependence of correlation functions. We discuss its signification and its relation with other schemes, in particular the derivative expansion. Quantitative results are presented for the testground of scalar O(N ) theories. Besides critical exponents which are zero-momentum quantities, we compute in three dimensions in the whole momentum range the two-point function at criticality and, in the high temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.

We demonstrate the power of a recently-proposed approximation scheme for the non-perturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtained within this scheme, and compute the two-point functions of the O(N ) theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and non-universal quantities at modest numerical cost.PACS numbers: 05.10. Cc,64.60.ae,11.10.Hi The renormalization group, in its non-perturbative version [1,2] (also referred to as the exact renormalization group), provides a general formalism giving access, for arbitrary coupling strength, to a whole set of physically important quantities, universal as well as non-universal [3,4], thermodynamic functions and momentum-dependent correlation functions, etc. However, most studies within this framework involve approximations that restrict their scope to the calculation of thermodynamical quantities, or correlation functions with vanishing external momenta. In order to access the full momentum dependence, Blaizot, Méndez-Galain, and Wschebor (BMW) have recently introduced an approximation scheme which overcomes this limitation [5]. In principle, this scheme allows to compute, in all dimensions, at and away from criticality, both universal and non-universal quantities, as well as momentumdependent properties from p = 0 up to the ultra-violet cut-off Λ (inverse lattice spacing).In this Letter, we present the first complete implementation of the leading order approximation of the BMW scheme, and demonstrate its power by using O(N ) models as a testbed. We compute the entire momentum dependence of the two-point functions in two and three dimensions and obtain excellent results for both universal and non-universal quantities.We start by a brief outline of the formalism. In order to simplify the presentation, we shall write only the equations corresponding to the case of a scalar field theory with quartic coupling, i.e., restrict the presentation to the case N = 1 (corresponding to the Ising model). The strategy of the renormalization group is to build a family of theories indexed by a momentum scale parameter k, such that fluctuations are smoothly taken into account as k is lowered from the microscopic scale Λ down to 0. In practice, this is achieved by adding to the original Euclidean action S a mass-like term of the formis chosen so that R k (q 2 ) ∼ k 2 for q k, which effectively suppresses the modes ϕ(q k), and so that it vanishes for q k, leaving the modes ϕ(q k) unaffected. One then defines a scale-dependent partition functionand a scale-dependent effective actionwith φ = δ ln Z k /δJ. where

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