Decay properties of $P_c$ states through the Fierz rearrangement

We systematically study hidden-charm pentaquark currents with the quark configurations $[\bar{c}u][udc]$, $[\bar{c}d][uuc]$, and $[\bar c c][u u d]$. Some of their relations are derived using the Fierz rearrangement of the Dirac and color indices, and the obtained results are used to study strong decay properties of $P_c$ states as ${\bar{D}}^{(\ast )}{\mathrm{\Sigma }}_c$ hadronic molecules. We calculate their relative branching ratios for the $J\mathrm{/}\psi p$, ${\eta }_{c}p$, ${\chi }_{c0}p$, ${\chi }_{c1}p$, ${\bar{D}}^{(\ast )0}{\mathrm{\Lambda }}_c^{+}$, $\bar D^{0} \Sigma_c^{+}$,and$\bar D^{-} \Sigma_c^{++}$ decay channels. We propose to search for the $P_c(4312)$ in the ${\eta }_{c}p$ channel and the $P_c(4440)\mathrm{/}{P}_c(4457)$ in the ${\bar{D}}^0{\mathrm{\Lambda }}_c^{+}$ channel.