Left-hand cut and the HAL QCD method

TL;DR

This paper examines the treatment of the left-hand cut in the HAL QCD method by analyzing non-relativistic quantum mechanics with potentials, showing how to correctly handle the LHC and IR cutoff effects to improve scattering calculations.

Contribution

It provides a detailed analysis of the LHC problem and offers guidance on how to properly incorporate it into the HAL QCD potential method based on quantum mechanics insights.

Findings

The phase shift with an IR cutoff converges to the analytic continuation result as the cutoff increases.

The binding momentum can be accurately obtained even with a large but finite IR cutoff.

The study clarifies the conditions under which the HAL QCD method correctly accounts for the LHC effects.

Abstract

We investigate how the left-hand cut (LHC) problem is treated in the HAL QCD method. For this purpose, we first consider the effect of the LHC to the scattering problem in non-relativistic quantum mechanics with potentials. We show that the $S$-matrix or the scattering phase shift obtained from the potential including the Yukawa term ($e^{- m_\pi r}/r$) with the infra-red (IR) cutoff $R$ is well-defined even for the complex momentum $k$ as long as $R$ is finite, and they are compared with those obtained by the analytic continuation without the IR cutoff. In the $R\to\infty$ limit, the phase shift approaches the result from the analytic continuation at ${\rm Im}\, k < m_\pi/2$, while they differ at ${\rm Im}\, k > m_\pi/2$, except $k= k_b$, where $k_b$ is the binding momentum. We also observe that $k_b$ can be correctly obtained even at finite but large $R$. Using knowledge obtained in the non-relativistic quantum mechanics, we present how we should treat the LHC in the HAL QCD potential method.