Lecture Notes on Positivity Properties of Scattering Amplitudes

TL;DR

This paper reviews positivity properties of scattering amplitudes, focusing on completely monotone and Stieltjes functions, and explores their mathematical structure, physical origins, and applications in quantum field theory and positive geometry.

Contribution

It introduces the role of CM and Stieltjes functions in QFT, connecting classical analysis with modern amplitude and geometric methods.

Findings

Identifies positivity constraints in scalar Feynman integrals and Coulomb branch amplitudes.

Establishes links between these functions and positive geometry interpretations.

Provides evidence for geometric volume interpretations of these functions.

Abstract

We review completely monotone (CM) and Stieltjes functions, which are classes of functions obeying an infinite hierarchy of positivity constraints. While these are classical concepts in analysis, such properties have recently been shown to arise in many fundamental building blocks and observables of quantum field theory (QFT), including scalar Feynman integrals in the Euclidean region and Coulomb branch amplitudes in $\mathcal{N}=4$ SYM. After reviewing their mathematical structure, we discuss the physical and geometric origins of these properties, ranging from unitarity and analyticity in scattering amplitudes to the structure of parametric representations of Feynman integrals. We then survey a number of applications, including constraints on the analytic S-matrix, implications for numerical bootstrap methods, and connections to positive geometry, where we present evidence for a close relation between these functions and geometric volume interpretations. These notes are based on an extended series of lectures delivered at the \emph{Positive Geometry in Scattering Amplitudes and Cosmological Correlators} workshop, held at the International Centre for Theoretical Sciences (ICTS), Bengaluru, in February 2025.