An Algebraic Formula for Two Loop Renormalization of Scalar Quantum Field Theory

TL;DR

This paper derives a general algebraic formula for two-loop renormalization counterterms in scalar quantum field theories with up to two derivatives, extending previous one-loop results and applicable to higher derivative interactions.

Contribution

It provides a new algebraic method for calculating two-loop counterterms in scalar QFTs with derivatives, extending 't Hooft's one-loop formula and applicable to higher derivative theories.

Findings

Derived a general two-loop counterterm formula for scalar QFTs.

Identified that factorizable diagrams do not affect RG equations.

Method applicable to theories with higher derivatives, with future work on EFTs.

Abstract

We find a general formula for the two-loop renormalization counterterms of a scalar quantum field theory with interactions containing up to two derivatives, extending 't~Hooft's one-loop result. The method can also be used for theories with higher derivative interactions, as long as the terms in the Lagrangian have at most one derivative acting on each field. We show that diagrams with factorizable topologies do not contribute to the renormalization group equations. The results in this paper will be combined with the geometric method in a subsequent paper to obtain the counterterms and renormalization group equations for the scalar sector of effective field theories (EFT) to two-loop order.