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

TL;DR

This paper investigates the $X(6900)$ tetraquark state as a $cc$-$ar car c$ bound state using Bethe-Salpeter equations, predicting spectra that align with experimental observations and exploring similar $bb$-$ar bar b$ states.

Contribution

The study develops a Bethe-Salpeter framework for diquark-antidiquark systems, providing numerical spectra that match the $X(6900)$ mass and proposing a method to determine the effective interaction.

Findings

Spectra of excited $cc$-$ar car c$ states can match $X(6900)$ mass.

Interaction kernel based on meson analogy used for diquark-antidiquark.

Brief spectra prediction for $bb$-$ar bar b$ states.

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.