Leading RG logs in $\phi^4$ theory

TL;DR

This paper computes the leading renormalization group logs in $^4$ theory for any four-point Feynman diagram using both direct integral calculation and RG invariance via the Connes-Kreimer Lie algebra of graphs.

Contribution

It introduces two methods for calculating leading RG logs in $^4$ theory, including a novel use of the Connes-Kreimer Lie algebra approach.

Findings

Derived explicit formulas for leading RG logs in $^4$ theory.

Compared integral-based and algebraic methods for consistency.

Discussed non-RG logs like $( s/t)^n$ in the context of Feynman diagrams.

Abstract

We find the leading RG logs in $\phi^4$ theory for any Feynman diagram with 4 external edges. We obtain the result in two ways. The first way is to calculate the relevant terms in Feynman integrals. The second way is to use the RG invariance based on the Lie algebra of graphs introduced by Connes and Kreimer. The non-RG logs, such as $(\ln s/t)^n$, are discussed.