$P$-wave charmonium contribution to hidden-charm states from reanalysis of lattice QCD data

We reanalyze, considering the contribution of $P$-wave charmonia, lattice data for the $D \bar{D}$-$D_s\bar{D}_s$ coupled-channel of S. Prelovsek et al. [JHEP 06, 035 (2021)] and $D\bar{D}^*$ systems of S. Prelovsek et al. [Phys. Rev. Lett. 111, 192001 (2013)] with $m_{\pi}\simeq 280$ and $266$ MeV, and $L=24a/32a$ ($a\simeq 0.09$ fm) and $L=16a$ ($a\simeq0.1239(13)$ fm), respectively. The hidden-charm states with $J^{PC}=0^{++}$, $1^{++}$, and $2^{++}$ quantum numbers are then searched for. For $0^{++}$, the analysis reveals three poles in the $D\bar{D}$-$D_s\bar{D}_s$ coupled-channel amplitude, corresponding to three states. Two of these poles, located near the $D\bar{D}$ and $D_s\bar{D}_s$ thresholds, can be interpreted as mostly molecular states. A third pole above the $D_s\bar{D}_s$ threshold is originated from the $P$-wave $\chi_{c0}(2P)$ charmonium state. The number of poles found in the $D\bar D$-$D_s \bar D_s$ system is the same as that found in the original lattice analysis though the position of the third pole changes sizeably. In the $1^{++}$ sector, we find two poles in the complex energy plane. The first one is related to the molecular $X(3872)$ state, with a compositeness exceeding $90\%$, while the second one, stemming from the $\chi_{c1}(2P)$ charmonium, appears above the $D\bar{D}^*$ threshold and it likely corresponds to the recently discovered $\chi_{c1}(4010)$ state. In the $2^{++}$ sector, we also report two poles and find that the dressed $\chi_{c2}(2P)$ is lighter than the $D^*\bar{D}^*$ molecular state, with the dynamics of the latter closely related to that of the heavy-quark spin-symmetry partner of the $X(3872)$. Our exploratory study of the $1^{++}$ and $2^{++}$ sectors offers valuable insights into their dynamics, but given that the fits that we carry out are underconstrained, more lattice data are required to draw robust conclusions.