Analysis of the $Z(4430)$ as the first radial excitation of the $Z_c(3900)$

TL;DR

This paper uses QCD sum rules to analyze the $Z_c(3900)$ and $Z(4430)$, proposing they are the ground and first radial excited states of axial-vector tetraquarks, respectively, and supports this with mass calculations.

Contribution

It provides a consistent QCD sum rule analysis that supports the interpretation of $Z_c(3900)$ and $Z(4430)$ as related tetraquark states with specific excitation levels.

Findings

$Z_c(3900)$ is the ground state of the axial-vector tetraquark.

$Z(4430)$ is the first radial excitation of the same tetraquark family.

Mass calculations align with experimental observations.

Abstract

In this article, we take the $Z_c(3900)$ and $Z(4430)$ as the ground state and the first radial excited state of the axial-vector tetraquark states with $J^{PC}=1^{+-}$, respectively, and study their masses and pole residues with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-10 in a consistent way in the operator product expansion. The numerical result favors assigning the $Z_c(3900)$ and $Z(4430)$ as the ground state and first radial excited state of the axial-vector tetraquark states, respectively.