We investigate how the derivative expansion in the HAL QCD method works to extract physical observables, using a separable potential in quantum mechanics, which is solvable but highly non-local in the coordinate system. We consider three cases for in
puts to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schrodinger equation with some boundary conditions (3) time-dependent wave function made by a linear combination of finite number of eigenfunctions at low energy to mimic the finite volume effect. We have found that, for all three cases, the potentials provide reasonable scattering phase shifts even at the leading order of the derivative expansion, and they give more accurate results as the order of the expansion increases. By comparing the above results with those from the formal derivative expansion for the separable potential, we conclude that the derivative expansion is not a way to obtain the potential but a method to extract physical observables such as phase shifts and binding energies, and that the scattering phase shifts from the derivative expansion in the HAL QCD method converge to the exact ones much faster than those from the formal derivative expansion of the separable potential.
A formalism is given to hermitize the HAL QCD potential, which needs to be non-hermitian except the leading order (LO) local term in the derivative expansion as the Nambu-Bethe-Salpeter (NBS) wave functions for different energies are not orthogonal t
o each other. It is shown that the non-hermitian potential can be hermitized order by order to all orders in the derivative expansion. In particular, the next-to-leading order (NLO) potential can be exactly hermitized without approximation. The formalism is then applied to a simple case of $Xi Xi (^{1}S_{0}) $ scattering, for which the HAL QCD calculation is available to the NLO. The NLO term gives relatively small corrections to the scattering phase shift and the LO analysis seems justified in this case. We also observe that the local part of the hermitized NLO potential works better than that of the non-hermitian NLO potential. The hermitian version of the HAL QCD potential is desirable for comparing it with phenomenological interactions and also for using it as a two-body interaction in many body systems.
