In this paper, we investigate the HAL QCD potential in the $I=1$ $\pi \pi$ scattering using the hybrid method for all-to-all propagators, in which a propagator is approximated by low-eigenmodes and the remaining high-eigenmode part is stochastically estimated. To verify the applicability of the hybrid method to systems containing quark creation$/$annihilation contributions such as the $\rho$ meson, we calculate the $I=1$ $\pi\pi$ potential with the 2+1 flavor gauge configurations on $16^3 \times 32$ lattice at the lattice spacing $a \approx 0.12$ fm and $(m_{\pi},m_{\rho}) \approx (870, 1230)$ MeV, in which the $\rho$ meson appears as a deeply-bound state. While we find that the naive stochastic evaluations for quark creation$/$annihilation contributions lead to extremely large statistical fluctuations, additional noise reduction methods enable us to obtain a sufficiently precise potential, which shows a strong attractive force. We also confirm that the binding energy and $k^3 \cot \delta$ obtained from our potential are roughly consistent with an existing $\rho$ meson bound state, within a large systematic error associated with our calculation, whose possible origin is also discussed.

中文翻译：

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具有 ρ 介子束缚态的 I = 1 π π 系统中的 HAL QCD 势

在本文中，我们使用全对全传播器的混合方法研究了 $I=1$$\pi\pi$ 散射中的 HAL QCD 势，其中传播器由低本征模近似，其余高-特征模部分是随机估计的。为了验证混合方法对包含夸克产生$/$湮灭贡献的系统的适用性，例如 $\rho$ 介子，我们使用 2+1 风味规配置计算了 $I=1$$\pi\pi$ 势在 $16^3 \times 32$ 格子上，格间距 $a \approx 0.12$ fm 和 $(m_{\pi},m_{\rho}) \approx (870, 1230)$ MeV，其中 $\ rho$ 介子表现为深度束缚态。虽然我们发现对夸克创造$/$湮灭贡献的幼稚随机评估导致了极大的统计波动，额外的降噪方法使我们能够获得足够精确的势，这显示出强大的吸引力。我们还确认从我们的势能中获得的结合能和 $k^3 \cot \delta$ 与现有的 $\rho$ 介子束缚态大致一致，在与我们的计算相关的大系统误差范围内，其可能的起源也是讨论。