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Lecture Notes on Positivity Properties of Scattering Amplitudes

Prashanth Raman

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.

Lecture Notes on Positivity Properties of Scattering Amplitudes

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 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.

Paper Structure

This paper contains 46 sections, 19 theorems, 143 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Completely monotone and Stieltjes functions
  3. Completely monotone functions in one variable
  4. Definition and basic properties
  5. Examples.
  6. Basic properties.
  7. Convexity properties
  8. Interpolation of sequences via completely monotone functions
  9. Completely monotone functions in several variables
  10. Examples.
  11. Completely monotone functions on projective space
  12. Polar Duals and Projective Duality
  13. Stieltjes functions in one variable
  14. Defintion and basic properties
  15. Generalized Stieltjes Functions
  16. ...and 31 more sections

Key Result

Theorem 1

Bernstein's little theorem. If $f$ is completely monotone on $(a,b)$, then it admits an analytic continuation to a disk of radius $(b-a)$ centered at $x=b$ (after suitable translation of the interval).

Figures (9)

  • Figure 1: Plot of the angle-dependent cusp anomalous dimension for QCD and QED based on available perturbative data from ref henn2025positivitypropertiesscatteringamplitudes. As we will see in Section \ref{['CAD']} that this quantity is expected to be CM on $(0,1)$.
  • Figure 2: Wilson line containing a cusp with opening angle $\phi$.
  • Figure 3: Some families of Feynman integrals that are Stieltjes (a) Banana integrals in $D=2$ (b) box intgerals in $D=6$ (c) zig-zag integrals in $D=4$.
  • Figure 4: Regions inside the integration domain $\Delta$ where eq.(113),(114) hold fot $L=2$.
  • Figure 5: Behavior of ${\cal E}^{(L)}(u,u,u)$ in the region $0<u<1/4$. From bottom to top: strong coupling and $L=1,2,3,4$ loop results. The curves are positive, monotonically decreasing, and convex, as expected for completely monotone functions.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Definition
  • Theorem
  • Theorem
  • Theorem : Müntz theorem for CM functions, 5b5486ec-e7c3-38ae-810e-8390daa67e61
  • Theorem : Theorem 3.1 in LambyMarichalZenaidi2017
  • Definition
  • Definition : Extremal ray
  • Definition
  • Theorem : Bernstein--Hausdorff--Widder--Choquet
  • Definition
  • ...and 20 more