  The approximated partial wave decomposition method to the discrete data on a
cubic lattice, developed by C. W. 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
$\Lambda_c N$ system calculated in the $(2+1)$-flavor QCD on a
$(32a~\mathrm{fm})^3$ lattice at the lattice spacing $a\simeq0.0907$ fm and
$m_\pi \simeq 700$ MeV. We find that the $l=0$ component can be successfully
extracted by Misner's method from the NBS wave function projected to $A_1^+$
representation of the cubic group, which contains small $l\ge 4$ 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, two methods give almost identical results on the central values and
on the 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.
