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

TL;DR

This paper investigates the heavy-quark spin symmetry partners of the $X(3872)$, confirming their existence under certain conditions and analyzing the effects of pion exchange and coupled channels on their properties.

Contribution

It provides a robust analysis of the heavy-quark spin symmetry partners of the $X(3872)$, including nonperturbative pion effects and coupled-channel dynamics beyond the heavy-quark limit.

Findings

Degenerate partners with specific quantum numbers exist in the heavy-quark limit.

Inclusion of all relevant partial waves and channels preserves heavy-quark symmetry.

The $2^{++}$ partner shows a significant mass shift and a ~50 MeV width beyond the heavy-quark limit.

Abstract

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 $D\bar{D}^*$ molecular state with the quantum numbers $J^{PC}=1^{++}$, 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.