  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.

High Energy Physics - Phenomenology;High Energy Physics - Lattice;Nuclear Theory