Abstract
This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, $2+\mathcal{J}\ensuremath{\rightarrow}2$. 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.
Highlights
Resolving the hadronic spectrum has proven to be a significant challenge due to the nonperturbative nature of quantum chromodynamics (QCD)
IVA, we prove the well-known result that the on-shell partial-wave scattering amplitude can be written in the form, 3Although evident for two-body systems, this was proven for three-particle systems in Refs. [39,40], where it was shown that previous results describing three-body amplitudes obtained using all-orders perturbation theory [41] and unitarity constraints [42] were consistent
We have presented a model independent on-shell decomposition for transition amplitudes of two hadrons interacting with an external current
Summary
Resolving the hadronic spectrum has proven to be a significant challenge due to the nonperturbative nature of quantum chromodynamics (QCD). [30] by Lellouch and Lüscher laid the foundation to develop a general technique to match finite-volume matrix elements to 1 þ J → 2 transition processes [31,32], where J is some external local current, 1 refers to a state of just a QCD-stable hadron and 2 is an asymptotic state of two hadrons An application of this formalism was used to calculate the pion photoproduction in the π þ γ⋆ → ππ process, from which the π þ γ⋆ → ρ transition form factor was determined for heavier-than-physical pions by two distinct groups [33,34].
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