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Dispersive analysis of the $γγ\to D \bar{D}$ data and the confirmation of the $D \bar{D}$ bound state

In this paper, we present a data-driven analysis of the $γγ\to D^+D^-$ and $γγ\to D^0\bar{D}^0$ reactions from threshold up to 4.0 GeV in the $D\bar{D}$ invariant mass. For the $S$-wave contribution, we adopt a partial-wave dispersive representation, which is solved using the $N/D$ ansatz. The left-hand cuts are accounted for using the model-independent conformal expansion. The $D$-wave $χ_{c2}(3930)$ state is described as a Breit-Wigner resonance. The resulting fits are consistent with the data on the invariant mass distribution of the $e^+e^- \to J/ψD\bar{D}$ process. Performing an analytic continuation to the complex $s$-plane, we find no evidence of a pole corresponding to the broad resonance $X(3860)$ reported by the Belle Collaboration. Instead, we find a clear bound state below the $D\bar{D}$ threshold at $\sqrt{s_B} = 3695(4)$ MeV, confirming the previous phenomenological and lattice predictions.