Probing negative dimensional integration: two-loop covariant vertex and one-loop light-cone integrals

TL;DR

This paper explores the negative dimensional integration method (NDIM) for evaluating complex Feynman diagrams, demonstrating its effectiveness in covariant and non-covariant gauges, and comparing results with traditional techniques.

Contribution

It applies NDIM to two-loop and one-loop integrals in various gauges, showing its consistency and advantages over standard methods in complex quantum field theory calculations.

Findings

NDIM successfully evaluates two-loop three-point vertex integrals.

Results in light-cone gauge agree with traditional regularization methods.

NDIM simplifies the evaluation of Feynman diagrams in different regions.

Abstract

Negative dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method gives solutions in different regions of external momenta simultaneously. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals in the standard approach are transferred to a simpler solving of a system of linear algebraic equations, thanks to the polynomial character of the relevant integrands. We employ this method to evaluate a scalar integral for a massless two-loop three-point vertex with all the external legs off-shell, and consider several special cases for it, yielding results, even for distinct simpler diagrams. We also consider the possibility of NDIM in non-covariant gauges such as the light-cone gauge and do some illustrative calculations, showing that for one-degree violation of covariance (i.e., one external, gauge-breaking, light-like vector $n_{\mu}$) the ensuing results are concordant with the ones obtained via either the usual dimensional regularization technique, or the use of principal value prescription for the gauge dependent pole, while for two-degree violation of covariance --- i.e., two external, light-like vectors $n_{\mu}$, the gauge-breaking one, and (its dual) $n^{\ast}_{\mu}$ --- the ensuing results are concordant with the ones obtained via causal constraints or the use of the so-called generalized Mandelstam-Leibbrandt prescription.