Sequences of Trees and Higher-Order Renormalization Group Equations

TL;DR

This paper introduces higher-order renormalization group equations, characterizes sequences of trees satisfying these equations under Feynman rules, and discusses special cases requiring specific Feynman rule choices.

Contribution

It defines and characterizes higher-order renormalization group equations for sequences of trees, extending the understanding of renormalization in quantum field theory.

Findings

Characterization of sequences satisfying k-th order RG equations

Conditions under which Feynman rules produce Green functions obeying these equations

Comments on special cases with specific Feynman rule requirements

Abstract

We define a notion of higher order renormalization group equation and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a $k$th order renormalization group equation when applying any choice of Feynman rules results in a Green function satisfying a $k$th order renormalization group equation, and we characterize all such sequences of trees. We also make some comments on sequences of trees which require special choices of Feynman rules in order to satisfy a higher order renormalization group equation.