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.
Highlights
The established nonperturbative approach to understanding the emergent phenomena of the relativistic quantum field theory of the strong interactions, quantum chromodynamics (QCD), is the numerical approach of lattice QCD
While experiment probes QCD through infinitevolume scattering observables such as the phase shift and inelasticity, the finite-volume and Euclidean-time aspects of the lattice formulation render the accessible quantity to be the spectrum of states in the finite-volume lattice
A formalism based on Hamiltonian effective field theory has been developed to address partial-wave mixing in a periodic finite volume
Summary
The established nonperturbative approach to understanding the emergent phenomena of the relativistic quantum field theory of the strong interactions, quantum chromodynamics (QCD), is the numerical approach of lattice QCD. In the low-lying nucleon spectrum, scattering states excited by lattice QCD interpolating fields are described within the Hamiltonian effective field theory in terms of the same hadronic scattering degrees of freedom [6,7,8,9]. Once the mixing of higher partial waves is taken into account, one must abandon the use of C3ðNÞ and work with higher dimension matrices The focus of this investigation is to create an optimal set of rest-frame cubic-group basis states, maximally reducing the dimension of the Hamiltonian and enabling the determination of several partial-wave scattering parameters simultaneously.
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