Comparing implicit, differential, dimensional, and Bogolubov-Parasiuk-Hepp-Zimmermann renormalization

Abstract

We compare a momentum space implicit regularization (IR) framework with other renormalization methods which may be applied to dimension specific theories, namely differential renormalization (DfR) and the Bogolubov-Parasiuk-Hepp-Zimmermann (BPHZ) formalism. In particular, we define what is meant by minimal subtraction in IR in connection with DfR and dimensional renormalization. We illustrate with the calculation of the gluon self-energy a procedure by which a constrained version of IR automatically ensures gauge invariance at the one-loop level and handles infrared divergences in a straightforward fashion. Moreover, using the ${\ensuremath{\varphi}}_{4}^{4}$ 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 a renormalization group scale for the Callan-Symanzik equation. Finally we argue that constrained IR, constrained DfR, and dimensional reduction are equivalent within one-loop order.