Very recently, the LHCb Collaboration reported the doubly charmed tetraquark state $T_{cc}^+$ below the $D^{*+}D^0$ threshold about $273$ keV. As a very near-threshold state, its long-distance structure is very important. In the molecular scheme, we relate the coupling constants of $T_{cc}^+$ with $D^{*0}D^+$ and $D^{*+}D^0$ to its binding energy and mixing angle of two components with a coupled-channel effective field theory. With the coupling constants, we investigate the kinetically allowed strong decays $T_{cc}^+to D^0D^0pi^+$, $T_{cc}^+to D^+D^0pi^0$ and radiative decays $D^+D^0 gamma$. Our results show that the decay width of $T_{cc}^+to D^0D^0pi^+$ is the largest one, which is just the experimental observation channel. Our theoretical total strong and radiative widths are in favor of the $T_{cc}^+$ as a $|D^{*+}D^0rangle$ dominated bound state. The total strong and radiative width in the single channel limit and isospin singlet limit are given as $59.7^{+4.6}_{-4.4} text{ keV}$ and $46.7^{+2.7}_{-2.9} text{ keV}$, respectively. Our calculation is cutoff-independent and without prior isospin assignment. The absolute partial widths and ratios of the different decay channels can be used to test the structure of $T_{cc}^+$ state when the updated experimental results are available.
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