Emergence of the $ρ$ resonance from the HAL QCD potential in lattice QCD

Abstract

We investigate the $I=1$ $ππ$ interaction using the HAL QCD method in lattice QCD. We employ the (2+1)-flavor gauge configurations on $32^3 \times 64$ lattice at the lattice spacing $a \approx 0.0907$ fm and $m_π \approx 411$ MeV, in which the $ρ$ meson appears as a resonance state. We find that all-to-all propagators necessary in this calculation can be obtained with reasonable precision by a combination of three techniques, the one-end trick, the sequential propagator, and the covariant approximation averaging (CAA). The non-local $I=1$ $ππ$ potential is determined at the next-to-next-to-leading order (N$^2$LO) of the derivative expansion for the first time, and the resonance parameters of the $ρ$ meson are extracted. The obtained $ρ$ meson mass is found to be consistent with the value in the literature, while the value of the coupling $g_{ρππ}$ turns out to be somewhat larger. The latter observation is most likely attributed to the lack of low-energy information in our lattice setup with the center-of-mass frame. Such a limitation may appear in other P-wave resonant systems and we discuss possible improvement in future. With this caution in mind, we positively conclude that we can reasonably extract the N$^2$LO potential and resonance parameters even in the system requiring the all-to-all propagators in the HAL QCD method, which opens up new possibilities for the study of resonances in lattice QCD.