Tetraquarks in the Born-Oppenheimer approximation

The conventional loosely bound molecule interpretation of the $X(3872)$ is not compatible with the recent LHCb experimental measurement of the ratio of branching fractions $\mathcal{R}=\text{Br}(X\to\psi^\prime\gamma)/\text{Br}(X\to\psi\gamma)$. We systematically determine the entire tetraquark spectrum for $J=0,1,2$ and refine the calculation of $\mathcal{R}$ in an improved Born-Oppenheimer description of the $X(3872)$ compact tetraquark. This refinement yields a significantly better agreement with experimental data on ${\cal R}$ and on the spectroscopy of the states themselves. Extending the diquark-antidiquark paradigm to encompass tetraquarks that are linear superposition of open charm singlets and color octets, we discover that these exotic resonances manifest as compact shallow bound states of quarks in color force potentials.