Hadronic decays of the heavy-quark-spin molecular partner of $T_{cc}^{+}$

Starting from the hypothesis that the $T_{cc}^{+}$ discovered at LHCb is a $D^{\ast +}{D}^0\mathrm{/}{D}^{\ast 0}{D}^{+}$ hadronic molecule, we consider the partial width of its heavy quark spin partner, the $T_{cc}^{\ast +}$ as a $D^{\ast +}{D}^{\ast 0}$ shallow bound state, decaying into the $D^{\ast }D\pi$ final states including the contributions of the $D^{\ast }D$ and $D^{\ast }\pi$ final state interaction by using a nonrelativistic effective field theory. Because of the existence of the $T_{cc}^{+}$ pole, the $I=0$ $D^{\ast }D$ rescattering can give a sizeable correction up to about $40\mathrm{\%}$ to the decay widths considering only the tree diagrams, and the $D^{\ast }\pi$ rescattering correction is about $10\mathrm{\%}$. The four-body partial widths of the $T_{cc}^{\ast +}$ into $DD\pi \pi$ are also explicitly calculated, and we find that the interference effect between different intermediate $D^{\ast }D\pi$ states is small. The total width of the $T_{cc}^{\ast +}$ is predicted to be about 41 keV.