An easy way to solve two-loop vertex integrals

TL;DR

This paper introduces a simplified method using negative dimensional integration to evaluate complex two-loop vertex integrals, making calculations more straightforward through polynomial forms and analytic continuation.

Contribution

It presents a novel approach employing negative dimensional integration for calculating two-loop vertex integrals with arbitrary propagator exponents and dimensions.

Findings

Successfully computes four two-loop three-point vertex diagrams.

Recovers known special cases with unit propagator exponents.

Demonstrates the method's simplicity and effectiveness.

Abstract

Negative dimensional integration is a step further dimensional regularization ideas. In this approach, based on the principle of analytic continuation, Feynman integrals are polynomial ones and for this reason very simple to handle, contrary to the usual parametric ones. The result of the integral worked out in $D<0$ must be analytically continued again --- of course --- to real physical world, $D>0$, and this step presents no difficulties. We consider four two-loop three-point vertex diagrams with arbitrary exponents of propagators and dimension. These original results give the correct well-known particular cases where the exponents of propagators are equal to unity.