Landscape of pentaquark bound states with neural-network Variational Monte Carlo
We present a systematic study of the ground states of pentaquark systems using the neural network variational Monte Carlo method within the constituent quark potential model. We investigate all manifestly exotic pentaquark systems with negative parity, which cannot mix with the conventional baryons through the creation and annihilation of light or strange quark pair. We exhaust 387 color, spin, and isospin configurations and identify 21 pentaquark bound state candidates with binding energies ranging from $2$ to $15$ MeV. A striking observation is that all of the bound states are shallowly bound molecular states, and a deeply bound pentaquark state does not exist. The $QQqq\bar{Q}$, $ssqq\bar{Q}$, $Qsqq\bar{Q}$ and $Qqqq\bar{Q}$ bound states are composed of a heavy vector meson and a baryon, while the $QQqq\bar{s}$ and $Qqqq\bar{s}$ bound states are composed of a $K$ meson and a heavy baryon. As a cross-validation, the ground-state energies of the unbound systems converge to the baryon--meson thresholds from above, which are consistent with the S-wave scattering states and provide a reliable baseline for the bound-state criterion. Especially, we reproduce the lowest scattering states $η_c N$ and $η_c Λ$ in the $qqqc\bar{c}$ and $qqsc\bar{c}$ channels respectively, where the hidden-charm pentaquark resonances were observed experimentally. We also find that all fully heavy pentaquark states are unbound. Our results provide valuable guidance for future pentaquark searches at LHCb, Belle II, and other experiments.