Dictionary revision for mapping boundary data to bound states

The correspondence between gravitational observables derived from scattering processes and adiabatic invariants in bound orbits, within the framework of the Post-Minkowskian (PM) expansion, has garnered significant attention in the study of bound orbital systems. However, the existing dictionary for this correspondence, \textcolor{black}{characterized by the transformation $\beta\rightarrow i \beta$ and $b\rightarrow \pm i |b|$ with $\beta=arccosh \gamma$}, produces complex-valued quantities of bound orbits in 4PM calculations. These results contradict fundamental physical principles, thereby highlighting deficiencies in the existing dictionary. Our research identifies a critical issue: the Fourier transform of the scattering amplitude incorporates a factor of $(p_\infty^2)^{-n/2}$. This factor introduces singularities at $p_\infty^2 = 0$, thereby rendering the original dictionary become ineffective, as it assumes the possibility of connecting both scattering states and bound states at the singular point. We propose a rigorous modification by employing Hawking's method for black hole radiation, specifically analytical continuing $p_\infty^2 \rightarrow p_\infty^2 e^{-i \pi}$. We also evaluate the new dictionary by comparing the binding energy calculated using effective one-body theory with numerical relativity simulation data from the SXS collaboration. Our findings indicate a remarkable agreement between the two sets of results. This revised dictionary enhances the applicability of gravitational observables derived from scattering processes to bound orbits and effectively fulfills the objectives envisioned by the pioneers who proposed this correspondence.