Symbols of One-Loop Integrals From Mixed Tate Motives

TL;DR

The paper links the problem of computing symbols of one-loop 2m-gon integrals in 2m dimensions to the motivic structure of mixed Tate motives, using Goncharov's symbol framework to read the symbol directly from the Feynman parameterization. It develops an auto-motive recursive method for symbols, seeded by four-dimensional box integrals, and provides explicit recursion S_m(Q) = (1/2) sum_{i<j} S_{m-1}(Q_minus_ij) ⊗ R_ij. The authors validate the approach with concrete examples including the four-mass box and a three-mass hexagon in six dimensions, showing regulator independence and agreement with known polylogarithmic representations. This work offers a practical, motivically grounded algorithm for obtaining symbols of loop integrals, potentially guiding higher-loop analyses and the broader understanding of amplitudes via motivic and polylogarithmic structures.

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.