Abstract
The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice. To this end, instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for particle-dimer scattering is expanded in the basis functions of different irreducible representations of the octahedral group. Such a projection is of particular importance for the three-body problem in the finite volume due to the occurrence of three-body singularities above breakup. Additionally, we study the numerical solution and properties of such a projected quantization condition in a simple model. It is shown that, for large volumes, these solutions allow for an instructive interpretation of the energy eigenvalues in terms of bound and scattering states.
Highlights
Lattice QCD calculations provide an ab initio access to hadronic processes
The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice
Instead of spherical harmonics, the kernel of the BetheSalpeter equation for particle-dimer scattering is expanded in the basis functions of different irreducible representations of the octahedral group
Summary
Lattice QCD calculations provide an ab initio access to hadronic processes. These calculations are usually performed in a small cubic volume with periodic boundary conditions and require an infinite-volume extrapolation for the comparison to experimental data. The finite-volume interaction has discrete poles from three-body intermediate states This averts the convergence of the partial-wave expansion, making the expansion in eigenfunctions of the octahedral symmetry group unavoidable in this energy region. We shall in particular demonstrate that, owing to the octahedral symmetry, the three-body quantization condition [3,4,5,6,10,11,12, 14,15,16,17] can be split into different independent equations, whose solutions determine the finite-volume energy spectrum in different irreducible representations (irreps) of the octahedral group This is extremely convenient, since the source and sink operators in lattice calculations are usually chosen to transform as irreducible tensor operators under the octahedral group, allowing one to determine the energy spectra in different irreps independently.
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