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 $φ^{4}$-theory, computing analytically the Wilson-Fisher fixed point, its anomalous dimension $η(d)$ and the critical exponent $ν(d)$ non-perturbatively in $d\in[3,4]$ dimensions. The results are in reasonable agreement with the known values, despite the simplicity of the method.
Link between New Versions of the Hierarchical Reference Theory of Liquids and of the Non Perturbative Renormalization Group in Statistical Field Theory
我提出了一个新的版本的分层参考理论的液体。同时给出了巨正则系综和统计场论框架下的两种形式。在后者中，该理论是新版本的非微扰重整化群的化身（J. Phys. A：Math. Gen. \textbf{42}，225004（2009））。导出了Wilsonian作用量和Wetterich有效平均作用量的流，并建立了这两个泛函之间的简单关系。液体的标准分层参考理论（\textit{Adv. Phys.} \textbf{44}，211（1995））被恢复用于传播子的急剧红外截止
