We present results of a lattice study of the $S$-wave and $P$-wave $DD^*$ scattering phase shifts using Lüscher’s method under the twisted boundary conditions to investigate the doubly charmed tetraquark $T_{cc}^+$ observed by the LHCb collaboration. Although the scattering phase shift at zero momentum gives information about the number of bound states according to Levinson’s theorem, Lüscher’s method under the periodic boundary condition only accesses the scattering phase shifts at some discrete momenta and is not suitable for watching the signal of bound state formation.
On the other hand, the twisted boundary condition has the advantage that the scattering phase shift at any momentum can be calculated and that not only the $S$-wave scattering phase shift but also the $P$-wave scattering phase shift can be obtained simultaneously.
In this study, we perform the simulation for the $DD^*$ and $BB^*$ systems in the $I=0$ channel using 2+1 flavor PACS-CS gauge ensembles simulated at $m_\pi=295$ and 411 $\mathrm{MeV}$.