Dimensional renormalization: ladders to rainbows

TL;DR

This paper develops a method within dimensional renormalization to derive renormalization factors and Green functions for rainbow graphs and scattering diagrams at zero momentum transfer, ensuring correct anomalous dimensions in four dimensions.

Contribution

It provides an all-orders perturbative derivation of renormalization factors and Green functions for specific diagrams in dimensional renormalization, emphasizing the massless limit.

Findings

Derived renormalization factors to all orders for rainbow graphs.

Proved anomalous dimensions emerge correctly as D approaches 4.

Established a consistent framework for zero momentum transfer processes.

Abstract

Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive factors and renormalized Green functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering diagrams) at zero momentum transfer, in the context of dimensional renormalization, and we prove that the correct anomalous dimensions for those processes emerge in the limit D -> 4.