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

TL;DR

This paper investigates the decay properties of the $Z_c(3900)$ particle by constructing tetraquark currents and applying Fierz rearrangements, comparing compact tetraquark and molecular state interpretations.

Contribution

It systematically constructs tetraquark currents and derives their relations, providing a framework to analyze the $Z_c(3900)$ as either a tetraquark or molecular state.

Findings

Decay properties differ between tetraquark and molecular interpretations.

Fierz rearrangements relate different quark configurations.

Framework aids in understanding exotic hadron structures.

Abstract

We systematically construct all the tetraquark currents/operators of $J^{PC} = 1^{+-}$ with the quark configurations $[cq][\bar c \bar q]$, $[\bar c q][\bar q c]$, 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 $[\bar c q][\bar q c]$, we study decay properties of the $Z_c(3900)$ 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.