Can we understand the decay width of the $T_{cc}^+$ state?

Inspired by the recent discovery of a doubly charmed tetraquark state $T_{cc}^+$ by the LHCb Collaboration, we employ the effective Lagrangian approach to investigate the decay width of $T_{cc}^{+}\to D^{+} D^{0}\pi^{0}/D^{0} D^{0}\pi^{+}$ and $T_{cc}^{+}\to D^{0}D^{+}\gamma$ with the assumption that $T_{cc}^{+}$ is an isoscalar $DD^{\ast}$ molecule. We show that both the $T_{cc}\to D D\pi$ and $T_{cc}\to DD\gamma$ modes contribute to the decay width of $T_{cc}$, with the former being dominant. The resulting total decay width of about $\Gamma=63$ keV is smaller than the experimental decay width obtained from the Breit-Wigner fit of the LHCb data, $\Gamma=410\pm 165\pm 43^{+18}_{-38}$ keV, while close to the number obtained from the alternative unitary analysis, $\Gamma=48\pm 2^{+0}_{-14}$ keV, which supports the molecular nature of $T_{cc}$.