Implicit Regularisation Technique: Calculation of the Two-loop $\phi^4_4$-theory $\beta$-function

TL;DR

This paper introduces an implicit regularisation scheme that avoids explicit regulators, allowing symmetry-preserving calculations of divergent amplitudes, demonstrated through two-loop $eta$-function calculations in QED and $^4_4$-theory.

Contribution

It presents a novel implicit regularisation method that separates divergences from finite parts without explicit regulators, preserving symmetries during renormalisation.

Findings

Successfully calculated the QED $eta$-function at one-loop order.

Computed the $^4_4$-theory $eta$-function at two-loop order.

Demonstrated the scheme's ability to handle divergences while maintaining symmetries.

Abstract

We propose an implicit regularisation scheme. The main advantage is that since no explicit use of a regulator is made, one can in principle avoid undesirable symmetry violations related to its choice. The divergent amplitudes are split into basic divergent integrals which depend only on the loop momenta and finite integrals. The former can be absorbed by a renormalisation procedure whereas the latter can be evaluated without restrictions. We illustrate with the calculation of the $QED$ and $\phi^4_4$-theory $\beta$-function to one and two-loop order, respectively.