Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms

TL;DR

The paper introduces traintracks, a family of finite $L$-loop Feynman integrals in massless $\varphi^4$ theory whose transcendental content is governed by weight-$(L+1)$ hyperlogarithms integrated over $(L-1)$-dimensional elliptically fibered varieties, conjectured to be Calabi–Yau. A dual-conformal Feynman-parameterization is developed for $\frak{T}^{(L)}$, revealing a structure that yields a codimension-$(L+1)$ residue and hence an elliptic fiber in the residual integral; for $L=3$ the fiber space is a K3 surface, and for $L=4$ it is a Calabi–Yau threefold, with the CY property conjectured to persist at higher loops. The geometry is made explicit: the elliptic fibration $y^2=4x^3 - g_2(\vec{z})x - g_3(\vec{z})$ over $\mathbb{P}^{L-2}$ produces a K3 at $L=3$ and a CY$(L-1)$-fold at higher $L$, with concrete kinematic dependence affecting discriminants, $j$-invariants, and Frame shapes. Remarkably, traintracks also capture the entire leading contribution to a specific component amplitude in planar $\mathcal{N}=4$ SYM via a fishnet-deformed limit, illustrating the broad relevance of these geometries to perturbative QFT. The work argues for developing new tools beyond polylogarithm-symbol methods to study these Calabi–Yau-valued amplitudes and outlines compelling all-orders questions about the CY sequence in perturbation theory.

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.