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 nonperturbative 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 $D{D}^{*}\to D{D}^{*}$ 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 quasibound pole derived from the complex scaling method within the Schrödinger 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 $D{D}^{*}$ 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.

Figure 1

The cut structures of the half-on-shell scattering amplitudes from the OPE potential involving the three-body dynamics under different cases. The red, green, orange and purple singularities correspond to $\kappa =0.01$, 0.03, 0.07 and 0.1 GeV, respectively and the dashed line denotes the branch cuts.

Figure 2

The evolution of the branch points in the OPE potential with the three-body dynamics using complex scaling transformations: $p=\kappa \to \kappa e^{-i\varphi }$ (upper plot) and $p=\kappa \to \kappa e^{i\varphi }$ (lower plot). The red (yellow) and green points correspond to $\kappa =0.01$ and 0.1 GeV, respectively, and points with the cross mark denotes $\varphi =0°$.

Figure 3

The physical on-shell $T$ matrix of the isoscalar $D{D}^{*}$ scattering below the threshold.

Figure 4

The physical on-shell $T$ matrix of the isoscalar $D{D}^{*}$ scattering above the threshold.