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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 apropagator is approximated by low-eigenmodes and the remaining high-eigenmodepart is stochastically estimated. To verify the applicability of the hybridmethod to systems containing quark creation$/$annihilation contributions suchas the $\rho$ meson, we calculate the $I=1$ $\pi\pi$ potential with the 2+1flavor 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 whichthe $\rho$ meson appears as a deeply-bound state. While we find that the naivestochastic evaluations for quark creation$/$annihilation contributions lead toextremely large statistical fluctuations, additional noise reduction methodsenable us to obtain a sufficiently precise potential, which shows a strongattractive 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 ourcalculation, whose possible origin is also discussed.

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