$I=2$ $\pi\pi$ scattering phase shift from the HAL QCD method with the LapH smearing

Physical observables, such as the scattering phase shifts and the binding energies, calculated from the non-local HAL QCD potential do not depend on the sink operators used to define the potential. This is called the scheme independence of the HAL QCD method. In practical applications, the derivative expansion of the non-local potential is employed, so that physical observables may receive some scheme dependence at given order of the expansion. In this paper, we compare the $I=2$ $\pi\pi$ scattering phase shifts obtained in the point-sink scheme (the standard scheme in the HAL QCD method) and the smeared-sink scheme (the LapH smearing newly introduced in the HAL QCD method). Although potentials in different schemes have different forms as expected, we find that, for reasonably small smearing size, the resultant scattering phase shifts agree with each other if the next-to-leading order (NLO) term is taken into account. We also find that the HAL QCD potential in the point-sink scheme has negligible NLO term for wide range of energies, which implies a good convergence of the derivative expansion in this case, while the potential in the smeared-sink scheme has non-negligible NLO contribution. Implication of this observation to the future studies of resonance channels (such as the $I=0$ and $1$ $\pi\pi$ scatterings) with smeared all-to-all propagators is briefly discussed. All computations in this paper have been performed at the lattice spacing $a\simeq 0.12$ fm ($1/a \simeq 1.6$ GeV) on a $16^3\times 32$ lattice with the pion mass $m_\pi\simeq 870$ MeV.