Assignments of the $X(4140)$, $X(4500)$, $X(4630)$ and $X(4685)$ based on the QCD sum rules

TL;DR

This paper uses QCD sum rules to assign specific tetraquark states to several observed particles, including the new $X(4685)$, and confirms the tetraquark nature of these states through mass calculations and scattering state analysis.

Contribution

The study extends previous QCD sum rule analyses to include the $X(4685)$, providing mass predictions and clarifying the role of scattering states versus tetraquark contributions.

Findings

The $X(4685)$ is consistent with the first radial excitation of $X(4140)$.

Mass of $X(4685)$ predicted as 4.70 GeV, matching experimental data.

Tetraquark states are essential in saturating the QCD sum rules, beyond two-meson scattering states.

Abstract

In this article, we take into account our previous calculations based on the QCD sum rules, and tentatively assign the $X(4630)$ as the $D_s^*\bar{D}_{s1}-D_{s1}\bar{D}_s^*$ tetraquark molecular state or $[cs]_P[\bar{c}\bar{s}]_A+[cs]_A[\bar{c}\bar{s}]_P$ tetraquark state with the $J^{PC}=1^{-+}$, and assign the $X(3915)$ and $X(4500)$ as the 1S and 2S $[cs]_A[\bar{c}\bar{s}]_A$ tetraquark states respectively with the $J^{PC}=0^{++}$. Then we extend our previous works to investigate the LHCb's new tetraquark candidate $X(4685)$ as the first radial excited state of the $X(4140)$ with the QCD sum rules, and obtain the mass $M_{X}=4.70\pm0.12\,\rm{GeV}$, which is in very good agreement with the experimental value $4684 \pm 7 {}^{+13}_{-16}\,\rm{MeV}$. Furthermore, we investigate the two-meson scattering state contributions in details, and observe that the two-meson scattering states alone cannot saturate the QCD sum rules, the contributions of the tetraquark states play an un-substitutable role, we can saturate the QCD sum rules with or without the two-meson scattering states.