Hermitizing the HAL QCD potential in the derivative expansion

TL;DR

This paper presents a formalism to hermitize the non-hermitian HAL QCD potential in the derivative expansion, demonstrating that the hermitized potential improves the physical interpretation and comparison with phenomenological models.

Contribution

The authors develop a systematic method to hermitize the HAL QCD potential at all orders in the derivative expansion, including an exact hermitization at NLO, enhancing its physical applicability.

Findings

NLO potential corrections are small for the $ ext{Xi} ext{Xi}$ scattering case.

Hermitized NLO potential yields better local behavior than non-hermitian version.

The formalism facilitates comparison with phenomenological interactions and use in many-body systems.

Abstract

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 to 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.