$T_{cc}$ from finite volume energy levels: the left-hand cut problem and its solution

Lattice QCD has become a crucial tool for studying hadron-hadron interactions from first principles. However, significant challenges arise when extracting infinite-volume scattering parameters from finite-volume energy levels using the conventional L\"uscher method, particularly due to the presence of left-hand cuts induced by long-range interactions such as the one-pion exchange. To address these limitations, we propose a novel framework that combines chiral effective field theory and the plane-wave expansion with the Hamiltonian approach. By solving a Schr\"odinger-like equation in a finite volume, this method establishes a connection between finite-volume energy spectra and infinite-volume physical quantities, while effectively handling issues caused by left-hand cuts. Furthermore, the adoption of a plane-wave basis helps mitigating complexities associated with partial-wave mixing. Our preliminary numerical results at $m_\pi \approx 280$ MeV confirm that this approach efficiently overcomes the shortcomings of the L\"uscher method and indicate a resonant interpretation of the $T_{cc}(3875)$ state--in contrast to the virtual state suggested in conventional analyses.