Heavy-quark spin symmetry partners of the X(3872) revisited

We revisit the consequences of the heavy-quark spin symmetry for the possible spin partners of the $X(3872)$. We confirm that, if the $X(3872)$ were a DD̄^* molecular state with the quantum numbers JPC=1⁺⁺, then in the strict heavy-quark limit there should exist three more hadronic molecules degenerate with the $X(3872)$, with the quantum numbers 0⁺⁺, 1⁺⁻, and 2⁺⁺ in line with previous results reported in the literature. We demonstrate that this result is robust with respect to the inclusion of the one-pion exchange interaction between the D mesons. However, this is true only if all relevant partial waves as well as particle channels which are coupled via the pion-exchange potential are taken into account. Otherwise, the heavy-quark symmetry is destroyed even in the heavy-quark limit. Finally, we solve the coupled-channel problem in the 2⁺⁺ channel with nonperturbative pions beyond the heavy-quark limit and, contrary to the findings of previous calculations with perturbative pions, find for the spin-2 partner of the $X(3872)$ a significant shift of the mass as well as a width of the order of 50 MeV.