Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms

TL;DR

This paper introduces a new class of finite four-dimensional multi-loop Feynman integrals involving hyperlogarithms over elliptically fibered Calabi-Yau varieties, crucial for advanced amplitude calculations in various quantum field theories.

Contribution

It identifies and conjectures the Calabi-Yau nature of the varieties involved in these integrals, providing explicit examples at three and four loops and demonstrating their relevance across multiple theories.

Findings

Explicit identification of a K3 surface at three loops.

Evidence for a Calabi-Yau threefold at four loops.

Relevance of these integrals for amplitude representations in diverse theories.

Abstract

We describe a family of finite, four-dimensional, $L$-loop Feynman integrals that involve weight-$(L+1)$ hyperlogarithms integrated over $(L-1)$-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau. At three loops, we identify the relevant K3 explicitly; and we provide strong evidence that the four-loop integral involves a Calabi-Yau threefold. These integrals are necessary for the representation of amplitudes in many theories---from massless $\varphi^4$ theory to integrable theories including maximally supersymmetric Yang-Mills theory in the planar limit---a fact we demonstrate.