Cut structures and an observable singularity in the three-body threshold dynamics: the $T_{cc}^+$ case

The three-body threshold effect, the distinctive and intriguing non-perturbative dynamics in the low-energy hadron-hadron scattering, has acquired compelling significance in the wake of the recent observation of the double-charm tetraquark $T_{cc}^+$. This dynamics is characterized by the emergence of singular points and branch cuts within the interaction potential, occurring when the on-shell condition of the mediated particle is satisfied. The presence of these potential singularities indicates that the system is no longer Hermitian and also poses intractable challenges in obtaining exact solutions for dynamical scattering amplitudes. In this work, we develop a complex scaled Lippmann-Schwinger equation as an operation of analytical continuation of the $T$ matrix to resolve this problem. Through a practical application to the $DD^* \to DD^*$ process, we reveal complicated cut structures of the three-body threshold dynamics in the complex plane, primarily stemming from the one-pion exchange. Notably, our methodology succeeds in reproducing the $T_{cc}^+$ structure, in alignment with the quasi-bound pole derived from the complex scaling method within the Schr\"odinger equation framework. More remarkably, after solving the on-shell $T$ matrix on the positive real axis of momentum plane, we find an extra new structure in the $DD^*$ mass spectrum, which arises from a right-hand cut at a physical pion mass and should be observable in Lattice QCD simulations and future high-energy experiments.