Approximating Feynman Integrals Using Complete Monotonicity and Stieltjes Properties

TL;DR

This paper introduces two innovative numerical methods leveraging complete monotonicity and Stieltjes properties to efficiently approximate Feynman integrals, including multi-loop cases, with potential applications in complex kinematic regions.

Contribution

The paper presents new numerical approaches based on complete monotonicity and Stieltjes properties, enabling efficient and accurate computation of Feynman integrals across different kinematic regimes.

Findings

Successfully applied methods to multi-loop Feynman integrals with internal masses.

Demonstrated rational approximations for 20-loop banana-type integrals.

Validated the effectiveness of the approaches in Euclidean and physical scattering regions.

Abstract

We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives have a fixed sign, in the Euclidean kinematic region. This imposes strong constraints on the function space. Simultaneously, these integrals obey systems of linear differential equations with respect to kinematic parameters. By imposing that the solutions to these differential equations satisfy complete monotonicity across the Euclidean region, we develop an efficient and highly constraining numerical bootstrap method. We provide a proof of principle of the power of our approach by applying it to a class of multi-loop Feynman integrals with internal masses. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, such as dimension and propagator exponents, are not only CM but in fact Stieltjes functions. The latter can be described efficiently by Pad\'e approximants that are known to converge in the cut complex plane. This means that these representations are valid also in analytically continued kinematics, such as physical scattering regions. These insights allow us to obtain rational approximations to Feynman integrals from minimal information, such as a Taylor expansion about a soft limit. We demonstrate the effectiveness of this method by applying it to a 20-loop banana-type Feynman integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.