Numerical Evaluation of Feynman Integrals Using Positivity Properties

Positivity properties in quantum field theory provide powerful constraints on observables, often reflecting underlying principles such as unitarity and analyticity. In this talk, we explore two related notions of positivity: complete monotonicity (CM), where an observable and all its signed derivatives remain positive, and its refinement, the Stieltjes property. These positivity properties impose infinitely many constraints on functions and have recently been shown to arise in a wide range of quantum field theory observables, including scalar Feynman integrals in the Euclidean region, tree-level string amplitudes, and 2–2 scattering amplitudes satisfying dispersion relations.

After introducing these positivity properties, we will focus on scalar Feynman integrals, discussing in detail how these properties follow from parametric representations and under what conditions they hold. We will then present two novel approaches to leveraging these properties for the numerical evaluation of multi-loop Feynman integrals. The first approach combines the differential equations satisfied by Feynman integrals with CM constraints to formulate a convex optimization problem. Implemented as a linear program, this allows us to numerically bootstrap integral values with high precision in the Euclidean region. The second approach uses the Stieltjes property to construct efficient rational approximations via Padé approximants, which converge across the cut complex plane and enable high-precision evaluation in physical kinematics. Finally, we will present examples and, time permitting, discuss how this framework leads to rational approximations of certain transcendental functions.