On-shell representations of two-body transition amplitudes: single external current

TL;DR

This paper derives an exact analytic framework for two-body scattering amplitudes involving a single external current, enabling the extraction of resonance and bound-state form-factors from lattice QCD data.

Contribution

It provides a rigorous analytic representation for 2+current amplitudes and establishes a complete framework for their determination from lattice QCD.

Findings

Derived an exact analytic representation for 2+current amplitudes.

Defined resonance and bound-state form-factors rigorously.

Explored the narrow-width limit and Ward-Takahashi identity effects.

Abstract

This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, $2+\mathcal{J}\to 2$. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form-factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.