[Paper Review] Functions Beyond Multiple Polylogarithms for Precision Collider Physics

Jacob L. Bourjaily, Johannes Broedel+15 more|arXiv (Cornell University)|Mar 14, 2022

Particle physics theoretical and experimental studiesPhysics and Astronomy33 citations

TL;DR

This white paper surveys Feynman integrals beyond multiple polylogarithms, focusing on elliptic and Calabi–Yau geometries, and outlines current methods and future research directions for precision collider predictions.

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.

Motivation & Objective

• Motivate the study of Feynman integrals that extend beyond multiple polylogarithms for precision collider predictions.
• Catalogue well-studied examples where algebraic roots and elliptic/Calabi–Yau geometries arise in loop integrals.
• Summarize current techniques for handling elliptic and higher-dimensional integrals in perturbation theory.
• Highlight open questions and future research directions in the math-physics of non-polylogarithmic integrals.

Proposed method

• Describe how loop integrals reduce to Symanzik form and how Feynman parameterization leads to algebraic roots.
• Explain when integrals can be expressed in terms of multiple polylogarithms and when elliptic or Calabi–Yau structures appear.
• Discuss differential equations approaches for master integrals and the role of epsilon-expansion in canonical vs non-canonical forms.
• Survey the known classes of non-polylogarithmic integrals (sunrise/banana, traintrack, tardigrade/paramecium/amoeba) and their associated geometries.
• Outline current function spaces (elliptic polylogarithms, modular forms, Calabi–Yau periods) used to express these integrals.
• Address practical evaluation strategies and the conceptual implications of non-polylogarithmic function spaces.

Experimental results

Research questions

• RQ1 What classes of Feynman integrals require functions beyond multiple polylogarithms at two loops and beyond?
• RQ2 How do elliptic and Calabi–Yau geometries arise in known Feynman diagrams, and what are the current methods to evaluate them?
• RQ3 What are the limitations of canonical differential equations forms for non-polylogarithmic integrals, and what alternative strategies exist?
• RQ4 What open questions remain about the role of higher-dimensional varieties in scattering amplitudes and their physical implications?

Key findings

• Non-polylogarithmic integrals appear at two loops and in diagrams with multiple kinematic variables, necessitating elliptic and Calabi–Yau function spaces.
• Elliptic polylogarithms and iterated integrals over elliptic curves provide a framework for single-elliptic-curve cases, while higher-dimensional Calabi–Yau geometries arise in more complex families (e.g., banana/traintrack diagrams).
• Canonical dlog-based differential equations often fail for elliptic/Calabi–Yau cases, requiring non-algebraic changes of variables or linear-in-epsilon but non-homogeneous forms.
• Master integral strategies can still be pursued via Baikov representation and maximal cuts, enabling iterative solution construction when canonical forms are unavailable.
• Several well-studied diagram families (sunrise/banana, traintracks, tardigrades/paramecia/amoebas) illustrate the variety of geometries and function spaces encountered in perturbative amplitudes.
• There is active development of numerical and symbolic tools for elliptic and Calabi–Yau integrals, with ongoing work to connect geometry to physical principles and predictions.