  Wilson's original formulation of the renormalization group is perturbative in
nature. We here present an alternative derivation of the infinitesimal momentum
shell RG, akin to the Wegner and Houghton scheme, that is a priori exact. We
show that the momentum-dependence of vertices is key to obtain a diagrammatic
framework that has the same one-loop structure as the vertex expansion of the
Wetterich equation. Momentum dependence leads to a delayed functional
differential equation in the cutoff parameter. Approximations are then made at
two points: truncation of the vertex expansion and approximating the functional
form of the momentum dependence by a momentum-scale expansion. We exemplify the
method on the scalar $\varphi^{4}$-theory, computing analytically the
Wilson-Fisher fixed point, its anomalous dimension $\eta(d)$ and the critical
exponent $\nu(d)$ non-perturbatively in $d\in[3,4]$ dimensions. The results are
in reasonable agreement with the known values, despite the simplicity of the
method.

Statistical Mechanics