A Cutoff Procedure and Counterterms for Differential Renormalization

TL;DR

This paper investigates a real space cutoff approach in differential renormalization for massless 4 theory, demonstrating its consistency with unitarity and previous renormalization results up to three loops.

Contribution

It introduces a systematic cutoff procedure for differential renormalization and shows its compatibility with established renormalization group functions and unitarity.

Findings

Cutoff bare amplitudes match previous renormalized amplitudes plus cancelable singular terms.

Counterterms can be added to the Lagrangian to cancel surface term divergences.

Renormalization group functions (g) and (g) agree with earlier findings.

Abstract

Explicit divergences and counterterms do not appear in the differential renormalization method, but they are concealed in the neglected surface terms in the formal partial integration procedure used. A systematic real space cutoff procedure for massless $\phi^4$ theory is therefore studied in order to test the method and its compatibility with unitarity. Through 3-loop order, it is found that cutoff bare amplitudes are equal to the renormalized amplitudes previously obtained using the formal procedure plus singular terms which can be consistently cancelled by adding conventional counterterms to the Lagrangian. Renormalization group functions $\beta (g)$ and $\gamma (g)$ obtained in the cutoff theory also agree with previous results.