Construction of general $N$-body lattice operators with arbitrary momenta

TL;DR

The paper develops a universal, projection-based method to construct lattice QCD operators for arbitrary multi-hadron systems with any momentum, spin, and internal quantum numbers, directly projecting continuum $|J M angle$ structures onto lattice irreps. It provides explicit operator bases for one- and two-hadron sectors and demonstrates multi-hadron constructions up to four bodies, with moving-frame extensions and internal-symmetry projections (isospin, $C$-, $G$-, and $P$-parity). The construction is implemented in the open-source OpTion toolkit, enabling automated generation of complete operator bases and facilitating robust finite-volume spectroscopy and scattering analyses. This framework supports systematic studies of increasingly complex hadronic systems in lattice QCD, including multi-body interactions and coupled-channel analyses, across multiple symmetry sectors and frames.

Abstract

We present a systematic method for constructing lattice QCD operators for systems of an arbitrary number of particles with arbitrary momentum, spin, and internal quantum numbers. Explicit constructions are provided for one-, two-, three-, and four-hadron operators, covering all irreducible representations of the relevant lattice symmetry groups in both rest and moving frames. The construction procedure has been implemented in the open-source package \texttt{OpTion} (Operator construcTion), available at https://github.com/wittscien/OpTion. The paper and the package are designed to serve as a practical and extensible dictionary for future lattice QCD studies, as lattice calculations advance towards increasingly complex hadronic systems.