The massless higher-loop two-point function

TL;DR

This paper introduces a new analytical method for massless Feynman integrals, providing a criterion based on graph topology to determine when these integrals evaluate to multiple zeta values, and extends known results to higher loops.

Contribution

A novel method for computing massless Feynman integrals analytically, with a topology-based criterion for evaluating to multiple zeta values, applicable to higher-loop graphs.

Findings

Massless 2-loop 2-point functions are expressible in terms of multiple zeta values.

Planar graphs' coefficients evaluate to multiple zeta values.

Non-planar graphs with crossing number 1 may involve roots of unity.

Abstract

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph $K_{3,4}$ at one edge.