Determine the spin ordering of the two heavier narrow pentaquark states

Determine whether the $P_c(4440)$ and $P_c(4457)$ states are assigned according to spin ordering I, $J[P_c(4440)]=1/2$ and $J[P_c(4457)]=3/2$, or the reverse ordering, $J[P_c(4440)]=3/2$ and $J[P_c(4457)]=1/2$, thereby resolving the model dependence not fixed by their peak masses.

Background

The paper treats $P_c(4440)$ and $P_c(4457)$ as possible $S$-wave $Sigma_c\bar D^*$ molecules with either $J^P=1/2^-$ or $J^P=3/2^-$. Because the measured peak masses do not establish which spin belongs to which state, the authors calculate both assignments and identify distinct $Lambda_c\bar D^*$ decay patterns for the two possible orderings.

Within the quark-interchange calculation, the $Lambda_c\bar D^*$ width is consistently larger for the $J^P=1/2^-$ assignment than for $J^P=3/2^-$ at a fixed peak mass. Comparing the widths of the two peaks could therefore help resolve the ordering experimentally, but the ordering itself remains unresolved in the paper.