Symbols of One-Loop Integrals From Mixed Tate Motives

TL;DR

This paper presents a method to directly determine the symbol of one-loop integrals in quantum field theory using mixed Tate motives, enabling recursive computation from known simpler cases.

Contribution

It introduces a recursive algorithm based on mixed Tate motives to compute symbols of one-loop integrals directly from Feynman parameterizations.

Findings

The symbol of a three-mass hexagon integral in six dimensions is explicitly derived.

The method simplifies the computation of one-loop integrals in higher dimensions.

A recursive approach is established from four-dimensional box integrals.

Abstract

We use a result on mixed Tate motives due to Goncharov (arXiv:alg-geom/9601021) to show that the symbol of an arbitrary one-loop 2m-gon integral in 2m dimensions may be read off directly from its Feynman parameterization. The algorithm proceeds via recursion in m seeded by the well-known box integrals in four dimensions. As a simple application of this method we write down the symbol of a three-mass hexagon integral in six dimensions.