Partial Wave Mixing in Hamiltonian Effective Field Theory

TL;DR

This paper develops a formalism within Hamiltonian effective field theory to relate finite-volume lattice QCD spectra to partial-wave scattering, accounting for cubic symmetry mixing, and demonstrates it with $ ext{I}=2$ $ ext{ extpi} ext{ extpi}$ scattering data.

Contribution

The paper introduces a formalism that maps partial-wave scattering potentials to finite-volume lattice spectra within Hamiltonian effective field theory, accounting for symmetry-induced mixing.

Findings

Successfully applied to lattice QCD data for $ ext{I}=2$ $ ext{ extpi} ext{ extpi}$ scattering.

Extracted $s$, $d$, and $g$ partial-wave phase shifts from finite-volume spectra.

Provides a basis reduction method for Hamiltonian calculations in finite volume.

Abstract

The spectrum of excited states observed in the finite volume of lattice QCD is governed by the discrete symmetries of the cubic group. This finite group permits the mixing of orbital angular momentum quanta in the finite volume. As experimental results refer to specific angular momentum in a partial-wave decomposition, a formalism mapping the partial-wave scattering potentials to the finite volume is required. This formalism is developed herein for Hamiltonian effective field theory, an extension of chiral effective field theory incorporating the L\"uscher relation linking the energy levels observed in finite volume to the scattering phase shift. The formalism provides an optimal set of rest-frame basis states maximally reducing the dimension of the Hamiltonian, and it should work in any Hamiltonian formalism. As a first example of the formalism's implementation, lattice QCD results for the spectrum of an isospin-2 $\pi\pi$ scattering system are analyzed to determine the $s$, $d$, and $g$ partial-wave scattering information.