Extension of the HAL QCD approach to inelastic and multi-particle scatterings in lattice QCD

TL;DR

This paper extends the HAL QCD method to handle inelastic and multi-particle scatterings in lattice QCD by deriving asymptotic wave functions and constructing energy-independent potentials above inelastic thresholds.

Contribution

The authors develop a framework to apply the HAL QCD approach to inelastic and multi-particle systems, including derivation of asymptotic behaviors and construction of coupled channel potentials.

Findings

Derived asymptotic behaviors of NBS wave functions for multi-particle systems.

Constructed energy-independent coupled channel potentials above inelastic thresholds.

Demonstrated the applicability of the extended HAL QCD method to inelastic scatterings.

Abstract

We extend the HAL QCD approach, with which potentials between two hadrons can be obtained in QCD at energy below inelastic thresholds, to inelastic and multi-particle scatterings. We first derive asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave function at large space separations for systems with more than 2 particles, in terms of the one-shell $T$-matrix consrainted by the unitarity of quantum field theories. We show that its asymptotic behavior contains phase shifts and mixing angles of $n$ particle scatterings. This property is one of the essential ingredients of the HAL QCD scheme to define "potential" from the NBS wave function in quantum field theories such as QCD. We next construct energy independent but non-local potentials above inelastic thresholds, in terms of these NBS wave functions. We demonstrate an existence of energy-independent coupled channel potentials with a non-relativistic approximation, where momenta of all particles are small compared with their own masses. Combining these two results, we can employ the HAL QCD approach also to investigate inelastic and multi-particle scatterings.