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

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.