$\boldsymbol{\chi_{c1}}(3872)$ and its Partners in the Diabatic Born-Oppenheimer Approximation for QCD

Abstract

In the Born-Oppenheimer approximation for QCD, the exotic hidden-charm tetraquark meson $\chi_{c1}(3872)$ is a near-threshold bound state in Born-Oppenheimer potentials associated with an isospin-0 adjoint meson. The $\chi_{c1}(3872)$ is the $1^{++}$ member of a heavy-quark spin-symmetry multiplet whose other members have $J^{PC}$ quantum numbers $0^{++}$, $1^{+-}$, and $2^{++}$. We introduce a simple model for the Born-Oppenheimer potentials that interpolates between the adjoint-meson potential at short distances and the triplet-meson-pair potential at large distances. We take into account the spin splittings of charm mesons nonperturbatively for the first time by solving the diabatic Schr\"odinger equation. We also take into account the spin splittings of the adjoint meson as well as a narrow avoided crossing with the quarkonium potential. We tune the energy of $\chi_{c1}(3872)$ to the $D^* \bar{D}$ threshold and then calculate the spin splittings of the other members of the multiplet and their decay widths into charm-meson pairs. We also calculate the energies and decay widths of the corresponding multiplet of hidden-bottom tetraquarks. These calculations provide a template for the quantitative analysis of all hidden-heavy hadrons using the Born-Oppenheimer approximation for QCD.