Empirical determinations of Feynman integrals using integer relation algorithms

TL;DR

This paper demonstrates how integer relation algorithms like PSLQ and LLL can convert numerical Feynman integral results into exact forms, revealing reductions and relations to special functions and modular forms.

Contribution

It introduces the application of PSLQ and LLL algorithms to Feynman integrals, enabling exact evaluations and uncovering quadratic relations among integrals.

Findings

Reduced Feynman integrals to multiple polylogarithms

Discovered quadratic relations between Feynman integrals

Applied algorithms to integrals related to black hole physics

Abstract

Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modular forms. Moreover, these algorithms have revealed quadratic relations between Feynman integrals. A recent application concerning black holes involves quadratic relations between combinations of Feynman integrals with algebraic coefficients.