$P_c(4380)$ in a constituent quark model

The constituent quark model with color-spin hyperfine potential is used to investigate the property of a compact pentaquark configuration with $J^p$=$3/2^-$ and isospin=1/2, which is the most likely quantum number of one of the recently observed exotic baryon states at LHCb. Starting from the characterization of the isospin, color, and spin states for the pentaquark configuration, we construct the total wave function composed of the spatial wave function, which we take to be symmetric and in S-wave, and the four orthogonal isospin $\otimes$ color $\otimes$ spin states that satisfy the Pauli principle. We then use the variational method to find a compact stable configuration. While there are compact configurations where the hyperfine potential is more attractive than the sum of $p$ and $J/\psi$ hyperfine potentials, we find that the ground state is the isolated $p$ and $J/\psi$ state. Furthermore, the mass of the excited state lies far above the observed pentaquark state leading us to conclude that the observed states can not be a compact multiquark configuration with $J^p$=$3/2^-$.