The HAL QCD potential in the I=1 ππ system with the ρ meson bound state

We investigate the HAL QCD potential in 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. meson, we calculate the I = 1 pi pi potential with the (2 + 1)-flavor gauge configurations on a 16(3) x 32 lattice with lattice spacing a approximate to 0.12 fm and (m(pi), m(rho)) approximate to (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 the large systematic error associated with our calculation, whose possible origin is also discussed.