Comparing Implicit, Differential, Dimensional and BPHZ Renormalisation

TL;DR

This paper compares various renormalisation methods, demonstrating how implicit regularisation can automatically ensure gauge invariance, handle divergences, and relate to other schemes like BPHZ and dimensional reduction at one loop.

Contribution

It introduces a constrained implicit regularisation framework that aligns with differential and dimensional renormalisation, providing explicit calculations and a parametrization for finite-infinite separation.

Findings

IR ensures gauge invariance at one loop

IR displays finite parts explicitly in calculations

Constrained IR, DfR, and dimensional reduction are equivalent at one loop

Abstract

We compare a momentum space implicit regularisation (IR) framework with other renormalisation methods which may be applied to dimension specific theories, namely Differential Renormalisation (DfR) and the BPHZ formalism. In particular, we define what is meant by minimal subtraction in IR in connection with DfR and dimensional renormalisation (DR) .We illustrate with the calculation of the gluon self energy a procedure by which a constrained version of IR automatically ensures gauge invariance at one loop level and handles infrared divergences in a straightforward fashion. Moreover, using the $\phi^4_4$ theory setting sun diagram as an example and comparing explicitly with the BPHZ framework, we show that IR directly displays the finite part of the amplitudes. We then construct a parametrization for the ambiguity in separating the infinite and finite parts whose parameter serves as renormalisation group scale for the Callan-Symanzik equation. Finally we argue that constrained IR, constrained DfR and dimensional reduction are equivalent within one loop order.