Relativistic correction to the binding energies of two-body hadronic molecular states

This study presents a systematic estimation of the relativistic correction to the binding energies of two-body hadronic molecular states by comparing the numerical solutions of the three-dimensional (3D) Schr{ö}dinger, 3D Salpeter, and fully relativistic four-dimensional (4D) Bethe-Salpeter (BS) equations derived from the same underlying interaction. The numerical results reveal a counter-intuitive property: for hadronic molecular states whose binding energies are in the MeV range, the relativistic correction is unexpectedly large. This finding contradicts the conventional expectation that a heavier exchanged mass in the interaction implies suppressed relativistic effects. Specifically, we first benchmark the results using the Wick-Cutkosky model with a one-boson-exchange (OBE) interaction of mass $m_{ex}$, and then extend the analysis to the physical $D\bar{D}$ system. We find within the $1\sim 50$ MeV binding energy region, the relativistic correction is substantial, amounting to $-90\% \sim -70\%$ of the non-relativistic result. Such a significant correction strongly suggests that analyses based solely on the 3D Schr{ö}dinger or 3D Salpeter equations for hadronic molecular states should be treated with caution.