The Khuri-Jones Threshold Factor as an Automorphic Function

TL;DR

This paper explores the mathematical structure of the Khuri-Jones threshold correction in scattering amplitudes, revealing its automorphic function properties and how group transformations differentiate bound states from resonances.

Contribution

It provides a novel automorphic function framework for the Khuri-Jones correction and links group symmetry changes to physical state distinctions in scattering theory.

Findings

The correction is an automorphic function for a dihedron.

Transformation of symmetry groups distinguishes bound states from resonances.

Expression for partial wave amplitude at the pole is derived.

Abstract

The Khuri-Jones correction to the partial wave scattering amplitude at threshold is an automorphic function for a dihedron. An expression for the partial wave amplitude is obtained at the pole which the upper half-plane maps on to the interior of semi-infinite strip. The Lehmann ellipse exists below threshold for bound states. As the system goes from below to above threshold, the discrete dihedral (elliptic) group of Type 1 transforms into a Type 3 group, whose loxodromic elements leave the fixed points 0 and $\infty$ invariant. The transformation of the indifferent fixed points from -1 and +1 to the source-sink fixed points 0 and $\infty$ is the result of a finite resonance width in the imaginary component of the angular momentum. The change in symmetry of the groups, and consequently their tessellations, can be used to distinguish bound states from resonances.