Derivative expansion in the HAL QCD method for a separable potential

TL;DR

This paper examines the effectiveness of the derivative expansion in the HAL QCD method for extracting physical observables from a solvable separable potential, demonstrating rapid convergence and clarifying its role in obtaining phase shifts and energies.

Contribution

It shows that the derivative expansion in the HAL QCD method accurately extracts physical observables and converges faster than the formal derivative expansion, clarifying its purpose.

Findings

Potentials give reasonable phase shifts at leading order.

Higher order expansion improves accuracy.

Derivative expansion converges faster than formal expansion.

Abstract

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 inputs to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schr\"odinger 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.