On Finite Representation of Dimensionally Regularized One-loop Integrals

TL;DR

This paper explores finite representations of one-loop integrals in quantum field theory using dimensional regularization, proposing conditions and schemes to achieve finite, well-defined integral expressions that match traditional regularization results.

Contribution

It introduces a framework for formulating finite integral expressions corresponding to dimensional regularization, comparing Gaussian and cut-off schemes, and establishing conditions for their equivalence.

Findings

Finite integral expressions can be constructed matching dimensional regularization results.

Gaussian and cut-off regularization schemes can be effectively used to achieve finiteness.

A scheme involving an additional scale operator removes divergences and yields finite, local integral representations.

Abstract

Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond tree level, the actual integrals can be highly divergent, at least in a traditional sense. In particular, standard one-loop integrals can be expressed in terms of an explicit formula, which associates both ultraviolet and infrared divergent parameter values to analytically continued special function expressions. We aim to discuss the formulation of finite integral expressions corresponding to the analytically continued structures. Effectively, we wish to establish conditions which form an equivalence class for this analytical continuation, or rather form a proper set/conditions of regularization techniques leading to it. This is further demonstrated by considering both partially and fully successful strategies side-by-side, with major emphasis on the two simplest functioning schemes: Gaussian and cut-off regularization. By explicit computations we aim to associate these generalisations of the initial integrals with the results from dimensional regularization, considering both multiple mass (or momentum) scales as well as scaleless cases. We achieve the finiteness of the sought-after integrals by applying one of these suitable schemes along with an additional scheme related scale. This enables us to devise a proper representation of the dimensionally regularized expressions (or a local description) through an operator removing all excess terms with the additional scale(s).