Scattering from production in 2d

TL;DR

This paper implements and tests numerical iterative schemes to reconstruct 2D scattering amplitudes from production probabilities, revealing convergence properties and a fractal structure related to CDD ambiguities, with implications for higher-dimensional S-matrix bootstrap.

Contribution

It applies Atkinson's constructive methods to 2D S-matrices using numerical algorithms, exploring convergence and uncovering a CDD fractal structure.

Findings

Algorithms converge within a specific amplitude region.

Discovered a CDD fractal structure linked to ambiguities.

Connections made to coupling maximization in S-matrix bootstrap.

Abstract

In 1968, Atkinson proved the existence of functions that satisfy all S-matrix axioms in four spacetime dimensions. His proof is constructive and to our knowledge it is the only result of this type. Remarkably, the methods to construct such functions used in the proof were never implemented in practice. In the present paper, we test the applicability of those methods in the simpler setting of two-dimensional S-matrices. We solve the problem of reconstructing the scattering amplitude starting from a given particle production probability. We do this by implementing two numerical iterative schemes (fixed-point iteration and Newton's method), which, by iterating unitarity and dispersion relations, converge to solutions to the S-matrix axioms. We characterize the region in the amplitude-space in which our algorithms converge, and discover a fractal structure connected to the so-called CDD ambiguities which we call "CDD fractal". To our surprise, the question of convergence naturally connects to the recent study of the coupling maximization in the two-dimensional S-matrix bootstrap. The methods exposed here pave the way for applications to higher dimensions, and expose some of the potential challenges that will have to be overcome.