Positivity properties of scattering amplitudes are often associated with fundamental principles such as unitarity and causality. Yet, in many cases, these properties arise from deeper mathematical structures. In this talk, we explore an infinite set of positivity constraints obeyed by certain amplitudes—and related observables—that go beyond conventional expectations.
Specifically, we highlight regions where these quantities and all of their signed derivatives are positive—a striking behavior known as complete monotonicity in mathematics. We begin by introducing completely monotone functions and their key properties, including their integral representations. We then demonstrate how fundamental objects in quantum field theory, such as scalar Feynman integrals, exhibit this property.
Building on this, we then present evidence that a variety of observables—such as Coulomb branch amplitudes, the cusp anomalous dimension, and remainder functions in planar N=4 super Yang-Mills theory—exhibit complete monotonicity across multiple perturbative orders. We conclude by outlining potential applications of these positivity constraints in the S-matrix bootstrap program, in improving numerical techniques for evaluating Feynman integrals, and in the ongoing study of positive geometries in quantum field theory.