Approximating Feynman Integrals Using Complete Monotonicity and Stieltjes Properties

TL;DR

This work introduces two complementary numerical strategies for Feynman integrals: a complete monotonicity (CM) bootstrap that leverages positivity of all derivatives and linear differential equations to tightly bound integrals, and a Stieltjes-function approach that enables robust Padé representations for analytic continuation. The CM bootstrap yields tight bounds and feasible runtimes, demonstrated on massive bubble and equal-mass banana integrals up to four loops, with competitive performance against established tools. The Stieltjes framework proves scalar Feynman integrals belong to a class amenable to Padé approximants with strong convergence guarantees, enabling efficient high-precision evaluations even in analytically continued kinematics, as illustrated by a 20-loop banana example. Together, these methods provide a versatile, analytically anchored toolkit for high-loop numerical evaluations, with promising extensions to multivariate cases and ε-expansions in dimensional regularization.

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 Padé 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.

Figures (10)

• Figure 1: The one-loop massive bubble Feynman integral.
• Figure 2: Constraints obtained from the interplay of differential equations and complete monotonicity for the massive bubble integral. The dashed yellow and dashed blue lines correspond to upper and lower bounds, respectively. The white regions and gray regions indicate allowed and forbidden regions, respectively. The solid black line corresponds to the exact function value.
• Figure 3: Average runtime of the CM bootstrap for the 'banana' integrals at different loop orders with different precision goals.
• Figure 4: The domain $\mathcal{D}^+(\Delta)$ where Padé approximations are well-behaved. The convergence properties are discussed in detail in the main text. $R_{\text{max}}>0$ can take any value. $\Delta>0$ quantifies the distance from the cut on the negative real axis. Figure adapted from ref. BGM.
• Figure 5: Examples of Feynman integrals that are Stieltjes. (a) 'Banana' integrals in $D=2$. (b) Zig-zag integrals in $D=4$. (c) Ladder integrals in $D=6.$