  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 $T_{cc}^+\to
D^0D^0\pi^+$, $T_{cc}^+\to D^+D^0\pi^0$ and radiative decays $D^+D^0 \gamma$.
Our results show that the decay width of $T_{cc}^+\to D^0D^0\pi^+$ 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 $|D^{*+}D^0\rangle$ 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.
