Natural Boundaries for Scattering Amplitudes

TL;DR

This paper reveals that scattering amplitudes can have natural boundaries of analyticity, which act as barriers to analytic continuation, due to dense accumulations of Landau singularities in gapped theories.

Contribution

It introduces the concept of natural boundaries in scattering amplitudes and demonstrates their presence in 2-to-2 amplitudes with a mass gap, extending understanding of their analytic structure.

Findings

Natural boundaries occur on the second sheet of the lightest threshold cut.

Infinite ladder-type Landau singularities densely accumulate on the real axis.

Natural boundaries are likely common in higher-multiplicity gapped theories.

Abstract

Singularities, such as poles and branch points, play a crucial role in investigating the analytic properties of scattering amplitudes that inform new computational techniques. In this note, we point out that scattering amplitudes can also have another class of singularities called natural boundaries of analyticity. They create a barrier beyond which analytic continuation cannot be performed. More concretely, we use unitarity to show that $2 \to 2$ scattering amplitudes in theories with a mass gap can have a natural boundary on the second sheet of the lightest threshold cut. There, an infinite number of ladder-type Landau singularities densely accumulates on the real axis in the center-of-mass energy plane. We argue that natural boundaries are generic features of higher-multiplicity scattering amplitudes in gapped theories.