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

With the assumptions that the $T_{cc}^+$ discovered at LHCb is a $D^{*}D$ hadronic molecule, using a nonrelativistic effective field theory we calculate the radiative partial widths of $T_{cc}^* \to D^*D\gamma$ with $T_{cc}^*$ being a $D^{*}D^{*}$ shallow bound state and the heavy-quark-spin partner of $T_{cc}^+$. The $I=0$ $D^*D$ rescattering effect with the $T_{cc}$ pole is taken into account. The results show that the isoscalar $D^{\ast} D$ rescattering can increase the tree-level decay width of $T_{cc}^{\ast +}\rightarrow D^{*+}D^0\gamma$ by about $50\%$, while decrease that of $T_{cc}^{\ast +}\rightarrow D^{*0}D^+\gamma$ by a similar amount. The two-body partial decay widths of the $T_{cc}^{*+}$ into $T_{cc}^+\gamma$ and $T_{cc}^+\pi^0$ are also calculated, and the results are about $6~\rm{keV}$ and $3~\rm{keV}$, respectively. Considering that the $D^*$ needs to be reconstructed from the $D\pi$ or $D\gamma$ final state in an experimental measurement, the four-body partial widths of the $T_{cc}^{*+}$ into $DD\gamma\gamma$ and $DD\pi\gamma$ are explicitly calculated, and we find that the interference effect between different intermediate $D^*D\gamma$ states is small. The total radiative decay width of the $T_{cc}^*$ is predicted to be about $24~\rm{keV}$. Adding the hadronic decay widths of $T_{cc}^* \to D^*D\pi$, the total width of the $T_{cc}^*$ is finally predicted to be $(65\pm2)$ keV.