Spectrum of the $uudc\bar{c}$ hidden charm pentaquark with an SU(4) flavor-spin hyperfine interaction

We study a few of the lowest states of the pentaquark $uudc\bar{c}$, of positive and negative parity, in a constituent quark model with an SU(4) flavor-spin hyperfine interaction. For positive parity we introduce space wave functions of appropriate permutation symmetry with one unit of orbital angular momentum in the internal motion of the four-quark subsystem or an orbital excitation between the antiquark and the four quark subsystem which remains in the ground state. We show that the lowest positive parity states $1\mathrm{/}{2}^{+},3\mathrm{/}{2}^{+}$ are provided by the first alternative and are located below the $1\mathrm{/}{2}^{-}$ and the $1\mathrm{/}{2}^{+}$ states with all quarks in the ground state. We compare our results with the LHCb three narrow pentaquark structures reported in 2019.