$T_{cc}^{+}$ and $X(3872)$ with the complex scaling method and $DD(\bar{D})\pi$ three-body effect

Abstract

We use the leading order (LO) contact interactions and OPE potentials to investigate the newly observed double-charm state $T_{cc}^{+}$. The $DD\pi$ three-body effect is important in this system since the intermediate states can go on shell. We keep the dependence of the pion propagators on the center-of-mass energy, which results in a unitary cut of the OPE potential at the $DD\pi$ three-body threshold. By solving the complex scaled Schrödinger equation, we find a pole corresponding to the $T_{cc}^{+}$ on the physical Riemann sheet. Its width is around 80 keV and nearly independent of the choice of the cutoff. Assuming the $D\bar{D}\pi$ and $D{\bar{D}}^{\ast }$ channels as the main decay channels, we apply the similar calculations to the $X(3872)$, and find its width is even smaller. Besides, the isospin breaking effect is significant for the $X(3872)$ while its impact on the $T_{cc}^{+}$ is relatively small.

Comments: 25 pages, 10 figures, 6 tables