Double-charm tetraquark under the complex scaling method

The LHCb Collaboration discovered a double-charm tetraquark $T_{cc}^{+}$ with a very small width. We investigate the $T_{cc}^{+}$ as a $DD^{*}$ molecule with $J^{P}=1^{+}$ in the framework of the one-boson-exchange potential model. The isospin breaking effect and $S-D$ wave coupling are taken into account carefully. We adopt the complex scaling method (CSM) to study the $DD^{*}$ system and obtain a quasibound state corresponding to the $T_{cc}^{+}$. Its binding energy relative to the $D^{0}D^{*+}$ and width are $-354$ keV and $61$ keV respectively. The isospin breaking effect is found to be enormous, and the $S-$wave $D^{0}D^{*+}$ and $D^{+}D^{*0}$ components give dominant contributions with the probabilities of $72.1\%$ and $27.1\%$ respectively. In addition, we do not find any resonances in the $DD^{*}$ system. As a by-product, we study the $X(3872)$ as a $(D\bar{D}^*-D^*\bar{D})/\sqrt{2}$ molecule with $J^{PC}=1^{++}$. We also find a quasibound state corresponding to the $X(3872)$. Its binding energy relative to the $D^{0}\bar{D}^{*0}$ threshold and width are $-111$ keV and $26$ keV respectively. The $S-$wave $(D^{0}\bar{D}^{*0}-D^{*0}\bar{D}^{0})/\sqrt{2}$ component dominates this state with the probability of $92.7\%$.