Correlation function and the inverse problem in the $BD$ interaction

We carry a study of the correlation functions of the $B^0 D^+, B^+ D^0$ system, which develops a bound state by about $40\mev$, using input consistent with the $T_{cc}(3875)$ state. Then we face the inverse problem of starting from these correlation functions to determine scattering observables related to the system, including the existence of the bound state and its molecular nature. The important output of the approach is the uncertainty by which these observables can be obtained, assuming errors in the $B^0 D^+, B^+ D^0$ correlation functions typical of current ones in present correlation functions. We observe that it is possible to obtain scattering lengths and effective ranges with relative high precision and the existence of a bound state. While the pole position is obtained with errors of the order of $50 \%$ of the binding energy, the molecular probability of the state is obtained with a very small error of the order of $6\%$. All these findings can serve as motivation to perform such measurements in future runs of high energy hadron collisions.