Functions Beyond Multiple Polylogarithms for Precision Collider Physics

TL;DR

This white paper surveys Feynman integrals that lie beyond multiple polylogarithms, highlighting elliptic and Calabi–Yau geometries that arise at two loops and beyond. It catalogs key non-polylogarithmic examples (sunrise/banana, traintracks, tardigrades/paramecia/amoebas, kite and ttbar double boxes) and reviews the current technology for handling them, including elliptic polylogarithms, Calabi–Yau periods, and numerical methods. The authors discuss canonical differential equations, rationalization strategies, and numerical schemes (AMFlow, sector decomposition, Mellin-Barnes), and they outline major open questions and directions, such as higher-genus function spaces, geometry cataloging, and generalized unitarity. The work emphasizes the importance of advancing both analytic and numerical tools to enable precision SM predictions and to deepen the mathematical understanding of perturbative quantum field theory at higher complexity.

Abstract

Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms -- whose properties as special functions are well understood -- more complex diagrams often involve integrals over complicated algebraic manifolds. Such diagrams already contribute at NNLO to the self-energy of the electron, $t \bar{t}$ production, $γγ$ production, and Higgs decay, and appear at two loops in the planar limit of maximally supersymmetric Yang-Mills theory. This makes the study of these more complicated types of integrals of phenomenological as well as conceptual importance. In this white paper contribution to the Snowmass community planning exercise, we provide an overview of the state of research on Feynman diagrams that involve special functions beyond multiple polylogarithms, and highlight a number of research directions that constitute essential avenues for future investigation.

Figures (11)

• Figure 1: The two-loop banana (or sunrise) diagram with generic internal masses.
• Figure 2: The $L$-loop banana integral, whose internal masses are all distinct, and which involves integrals over $(L{-}1)$-dimensional Calabi-Yau manifolds.
• Figure 3: The massless ten-point double box integral, which involves integrals over an elliptic curve.
• Figure 4: The traintrack class of Feynman integrals, where all internal propagators are massless. The dots in the middle loop represent $2-L$ additional boxes with a single massless external lines emerging from each new vertex. At $L$ loops, this diagram involves integrals over an $(L{-}1)$-dimensional Calabi-Yau manifold.
• Figure 5: The tardigrade diagrams, which exist for even numbers of loops and involve integrals over $(2L{-}2)$-dimensional Calabi-Yau manifolds at $L$ loops.