HyperFORM -- a FORM package for parametric integration with hyperlogarithms

TL;DR

HyperFORM delivers a FORM-based framework for symbolically integrating hyperlogarithms multiplied by rational functions, extending parametric integration to cases with rational letters and arguments. By porting HyperInt functionality to FORM, it exploits FORM's efficiency and parallelism to tackle large Feynman-integral expressions, including multi-loop cases and zigzag topologies. The paper details implementation, regularization, fibration-basis conversion, and a suite of examples, alongside performance benchmarks that emphasize scalability and hardware effects. Together, these contributions broaden the toolkit for analytic Feynman integral calculations and chart a path toward handling more complex integrands and higher-loop computations.

Abstract

We present an implementation of algorithms for the symbolic integration of hyperlogarithms multiplied by rational functions in the computer algebra system FORM. This implementation encompasses cases where hyperlogarithms have rational letters or a rational argument. It complements the previous implementation, HyperInt, in MAPLE by Erik Panzer, utilizing the advantages of FORM in the efficient handling of large symbolic expressions. Among a wide range of applications, this approach enables the computation of many Feynman integrals.

Figures (7)

• Figure 1: The two-point function of the FA topology with external momentum $p^2 \neq 0$ and labels for all propagators.
• Figure 2: The six-loop zigzag diagram with vertices labeled by numbers and edges by Schwinger variables.
• Figure 3: Integration sequence for zigzag 5.
• Figure 4: CPU time as a function of the number of FORM cores. For details of the computing environment see the main text.
• Figure 5: Computation time as a function of CPU clock frequency for two different RAM frequencies, using FORM with 16 cores.