Tetraquark state $X(6900)$ and the interaction between diquark and antidiquark

Abstract

Recently LHCb declared a new structure $X(6900)$ in the final state di-$J/\psi$ which is popularly regarded as a $cc$-$\bar{c}\bar{c}$ tetraquark state. %popularly. Within the Bethe-Salpeter (B-S) frame we study the possible $cc$-$\bar{c}\bar{c}$ bound states and the interaction between diquark ($cc$) and antidiquark ($\bar{c}\bar{c}$). In this work $cc$ ($\bar{c}\bar{c}$) is treated as a color anti-triplet (triplet) axial-vector so the quantum numbers of $cc$-$\bar{c}\bar{c}$ bound state are $0^{+}$, $1^{+}$ and $2^{+}$. Learning from the interaction in meson case and using the effective coupling we suggest the interaction kernel for diquark and antidiquark system. Then we deduce the B-S equations for different quantum numbers. Solving these equations numerically we find the spectra of some excited states can be close to the mass of $X(6900)$ when we assign appropriate values for parameter $\kappa$ introduced in the interaction (kernel).We also briefly calculate the spectra of $bb$-$\bar{b}\bar{b}$ bound states. Future measurement of $bb$-$\bar{b}\bar{b}$ state will help us to determine the exact form of effective interaction.

14 pages, two figures

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