# Derivative Expansion of Wave Function Equivalent Potentials

## TL;DR

This paper investigates the convergence properties of the derivative expansion of wave function equivalent potentials in a non-relativistic model, demonstrating improved accuracy through tuning expansion parameters.

## Contribution

It generalizes the derivative expansion method for non-local potentials and analyzes its convergence using analytic solutions, offering insights into optimizing potential models.

## Key findings

- Convergence of the expansion is confirmed via scattering phase shifts.
- Tuning interpolating fields or expansion scale improves convergence.
- The generalized derivative expansion converges better than previous methods.

## Abstract

Properties of the wave function equivalent potentials introduced by HAL QCD collaboration are studied in a non-relativistic coupled-channel model. The derivative expansion is generalized, and then applied to the energy-independent and non-local potentials. The expansion coefficients are determined from analytic solutions to the Nambu-Bethe-Salpeter wave functions. The scattering phase shifts computed from these potentials are compared with the exact values to examine the convergence of the expansion. It is confirmed that the generalized derivative expansion converges in terms of the scattering phase shift rather than the functional structure of the non-local potentials. It is also found that the convergence can be improved by tuning either the choice of interpolating fields or expansion scale in the generalized derivative expansion.