A simple expression for the four-point scalar function from Gaussian integrals and Fourier transform

TL;DR

This paper derives a simple, explicit one-dimensional expression for the four-point scalar function in quantum field theory using Gaussian integrals and Fourier transforms, simplifying calculations and clarifying singularity structures.

Contribution

It introduces a new explicit one-dimensional formula for the four-point scalar function, avoiding probabilistic methods and simplifying computational approaches.

Findings

Derived a one-dimensional integral expression involving square root and arcsine functions.

Disentangled singularities in the complex plane, isolating contributions as two-point functions.

Provided a simplified approach to compute four-point functions in quantum field theory.

Abstract

Recasting the $N$-point one loop scalar integral from Feynman to Schwinger parameters gives an integrand with a Gaussian form. By application of a Fourier transform, it is easy to derive explicit expressions for the two, three and four-point functions. The Fourier transformation disentangles singularities in the complex plane and extract their contribution as two-point functions in two dimensions. We explicitly derive a one dimensional expression for the (4D) four-point function whose integrand involves only square root and arcsine functions. This report is a condensed version of the approach developed in \cite{Benhaddou2016} which does not make use of probabilistic jargon.