Operators for scattering of particles with spin

TL;DR

This paper develops and compares methods for constructing lattice operators to simulate scattering of particles with spin, enabling better identification of continuum quantum numbers in lattice QCD simulations.

Contribution

It introduces three methods, including helicity and partial-wave approaches, to construct lattice operators with well-defined continuum quantum numbers for particle scattering.

Findings

Operators with good continuum quantum numbers are constructed.

Helicity and partial-wave methods clarify the relation between lattice operators and continuum states.

Guidelines for selecting operators to enhance desired state couplings are provided.

Abstract

Operators for simulating the scattering of two particles with spin are constructed. Three methods are shown to give the consistent lattice operators for PN, PV, VN and NN scattering, where P, V and N denote pseudoscalar meson, vector meson and nucleon. The projection method leads to one or several operators $O_{\Gamma,r,n}$ that transform according to a given irreducible representation $\Gamma$ and row r. However, it gives little guidance on which continuum quantum numbers of total J, spin S, orbital momentum L or single-particle helicities $\lambda_{1,2}$ will be related with a given operator. This is remedied with the helicity and partial-wave methods. There first the operators with good continuum quantum numbers $(J,P,\lambda_{1,2})$ or $(J,L,S)$ are constructed and then subduced to the irreps $\Gamma$ of the discrete lattice group. The results indicate which linear combinations $O_{\Gamma,r,n}$ of various n have to be employed in the simulations in order to enhance couplings to the states with desired continuum quantum numbers. The total momentum of two hadrons is restricted to zero since parity P is a good quantum number in this case.