Decay properties of the $Z_c(3900)$ through the Fierz rearrangement

We systematically construct all the tetraquark currents/operators of $J^{PC} = 1^{±}$withthequarkconfigurations$[cq][\bar c \bar q]$,$[\bar c q][\bar q c]$,and$[\bar c c][\bar q q]$($q=u/d$), and derive their relations through the Fierz rearrangement of the Dirac and color indices. Using the transformations of $[qc][\bar{q}\bar{c}]\to [\bar{c}c][\bar{q}q]$ and $[\bar c q][\bar q c]$,westudydecaypropertiesofthe$Z_c(3900)$ as a compact tetraquark state; while using the transformation of $[\bar c q][\bar q c] \to [\bar c c][\bar q q]$, we study its decay properties as a hadronic molecular state.