Finite- and infinite-volume study of $DD\pi$ scattering

We develop a comprehensive framework for extracting the pole position and properties of the doubly-charmed tetraquark $T_{\rm cc}^+(3875)$ from lattice QCD data using the relativistic three-particle formalism. This approach incorporates the effect of the one-pion exchange diagram in $DD\pi$ and $DD^*$ scattering, making it applicable at energies coinciding with the left-hand cut in the partial-wave projected $DD^*$ amplitude. We present an example application of this framework to existing lattice QCD data at $m_\pi = 280$ MeV. We solve the integral equations describing the $DD\pi$ reaction, use LSZ reduction to determine the corresponding $DD^*$ amplitude, and find the values of the infinite-volume two- and three-body $K$ matrices that lead to agreement with lattice $DD^*$ phase shifts within their uncertainties. Using these $K$ matrices in the three-particle quantization condition, we describe the finite-volume $DD^*$ spectrum and find good agreement with the lattice QCD energies. Our results suggest that, at this pion mass, the tetraquark appears as a pair of subthreshold complex poles whose precise location strongly depends on the value of the $DD\pi$ three-particle $K$ matrix.