Partial wave decomposition on the lattice and its applications to the HAL QCD method

Abstract

The approximated partial-wave decomposition method for discrete data on a cubic lattice, developed by Misner, is applied to the calculation of S-wave hadron-hadron scatterings by the HAL QCD method in lattice QCD. We consider the Nambu-Bethe-Salpeter (NBS) wave function for the spin-singlet ΛcN system calculated in (2+1)-flavor QCD on a (32a fm)3 lattice with lattice spacing a≃0.0907 fm and mπ≃700 MeV. We find that by Misner's method the l=0 component can be successfully extracted from the NBS wave function projected to the A1+ representation of the cubic group, which contains small l≥4 components. Furthermore, while the higher partial-wave components are enhanced so as to produce significant comb-like structures in the conventional HAL QCD potential if the Laplacian approximated by the usual second-order difference is applied to the NBS wave function, such structures are found to be absent in the potential extracted by Misner's method, where the Laplacian can be evaluated analytically for each partial-wave component. Despite the difference in the potentials, the two methods give almost identical results on the central values and magnitude of statistical errors for the fits of the potentials, and consequently on the scattering phase shifts. This indicates not only that Misner's method works well in lattice QCD with the HAL QCD method, but also that the contaminations from higher partial waves in the study of S-wave scatterings are well under control even in the conventional HAL QCD method. It will be of interest to study interactions in higher partial-wave channels in the HAL QCD method with Misner's decomposition, where the utility of this new technique may become clearer.