${\chi }_{c1}(4010)$ as a $D^{\ast \pm }{D}^{\mp }$ Molecular State: Mass, Decay Constant, and Width

The charmonium-like state ${\chi }_{c1}(4010)$, recently observed by the LHCb Collaboration in $B^{+}\to D^{\ast \pm }{D}^{\mp }{K}^{+}$ decays, is investigated as an axial-vector $D^{\ast \pm }{D}^{\mp }$ molecular state within the QCD sum rule approach. A two-point sum rule incorporating quark, gluon, and mixed vacuum condensates up to dimension $10$ yields the mass and decay constant, $m_{{\chi }_{c1}(4010)}=(4005.89\pm 47.15)\text{ MeV}$ and $f_{{\chi }_{c1}(4010)}=(2.87\pm 0.55)\times {10}^{-3}{\text{GeV}}^4$, in agreement with the LHCb measurement. Since this mass lies above the $D^{\ast \pm }{D}^{\mp }$ threshold, the strong coupling constants at the ${\chi }_{c1}(4010)D^{\ast \pm }{D}^{\mp }$ vertices are extracted from three-point sum rules and used to compute the partial widths of the $D^{\ast -}{D}^{+}$ and $D^{\ast +}{D}^{-}$ channels. The resulting total width, ${\mathrm{\Gamma }}_{{\chi }_{c1}(4010)}=(57.64\pm 9.64)\text{ MeV}$, is compatible with experiment within uncertainties. These results support a $D^{\ast \pm }{D}^{\mp }$ molecular interpretation of ${\chi }_{c1}(4010)$ and offer sum-rule predictions for comparison with future measurements at BESIII, LHCb, and Belle~II.