Analysis of the $ \boldsymbol{e^+ e^- \to J/\psi D\bar D}$ reaction close to the threshold concerning claims of a $\chi_{c0}(2P)$ state

We analyze the $D\bar{D}$ mass distribution from a recent Belle experiment on the $e^+e^- \to J/\psi D\bar{D}$ reaction, and show that the mass distribution divided by phase space does not have a clear peak above the $D\bar{D}$ threshold that justifies the experimental claim of a $\chi_{c0}(2P)$ state from those data. Then we use a unitary formalism with coupled channels $D^+D^-$, $D^0\bar{D}^0$, $D_s\bar{D}_s$, and $\eta\eta$, 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 $D\bar{D}$ amplitudes, some of them supporting a $D\bar{D}$ bound state and others not. The main conclusion is that the claim for the $\chi_{c0}(2P)$ state, already included in the PDG, is premature, but refined data can provide very valuable information on the $D\bar{D}$ scattering amplitude. As side effects, we warn about the use of a Breit-Wigner amplitude parameterization close to threshold, and show that the $D_s\bar{D}_s$ channel plays an important role in this reaction.