Study on the structure of the four-quark states in terms of the Born-Oppenheimer approximation

TL;DR

This paper applies the Born-Oppenheimer approximation to study four-quark states, suggesting both molecular and tetraquark configurations can be stable and may mix, predicting partner states for observed exotic mesons.

Contribution

It introduces a novel application of the Born-Oppenheimer approximation to analyze the stability and structure of four-quark states, including potential mixing of molecular and tetraquark configurations.

Findings

Both molecular and tetraquark states can have energy minima, indicating possible stability.

Both structures can coexist and mix, leading to observable partner states.

Predicted partner exotic states should be detectable experimentally.

Abstract

In this work, we use the Born-Oppenheimer approximation where the potential between atoms can be approximated as a function of distance between the two nuclei to study the four-quark bound states. By the approximation, Heitler and London calculated the spectrum of hydrogen molecule which includes two protons (heavy) and two electrons (light). Generally, the observed exotic mesons $Z_b(10610)$, $Z_b(10650)$, $Z_c(3900)$ and $Z_c(4020)$($Z_c(4025)$) may be molecular states made of two physical mesons and/or in diquark-anti-diquark structures. In analog to the Heitler-London method for calculating the mass of hydrogen molecule, we investigate whether there exist energy minima for these two structures. By contrary to the hydrogen molecule case where only the spin-triplet possesses an energy minimum, there exist minima for both of them. It implies that both molecule and tetraquark states can be stable objects. But since they have the same quantum numbers, the two states may mix to result in the physical states. A consequence would be that partner exotic states co-existing with $Z_b(10610)$, $Z_b(10650)$, $Z_c(3900)$ and $Z_c(4020)$($Z_c(4025)$) are predicted and should be experimentally observed.