The HAL QCD potential in $I=1$ $\pi \pi$ system with the $\rho$ meson bound state

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$\mathrm{/}$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$\mathrm{/}$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.