Analysis of the $e^{+}{e}^{-}$ → J/ψ$D\overline{D}$ reaction close to the threshold concerning claims of a χ${}_{c0}$(2P) state

We analyze the $D\overline{D}$ mass distribution from a recent Belle experiment on the $e^{+}{e}^{-}$ → J/ψ$D\overline{D}$ reaction, and show that the mass distribution divided by phase space does not have a clear peak above the $D\overline{D}$ threshold that justifies the experimental claim of a χ${}_{c0}$(2P) state from those data. Then we use a unitary formalism with coupled channels D${}^{+}$D${}^{\text{â}}\text{ˆ}\text{’}$, D${}^0$${\overline{D}}^0$, D${}_s$${\overline{D}}_s$, and ηη, with some of the interactions taken from a theoretical model, and use the data to fix other parameters. We then show that, given the poor quality of the data, we can get different fits leading to very different $D\overline{D}$ amplitudes, some of them supporting a $D\overline{D}$ bound state and others not. The main conclusion is that the claim for the χ${}_{c0}$(2P) state, already included in the PDG, is premature, but refined data can provide very valuable information on the $D\overline{D}$ scattering amplitude. As side effects, we warn about the use of a Breit-Wigner amplitude parameterization close to threshold, and show that the $D_s{\overline{D}}_s$ channel plays an important role in this reaction.