  The strong decays of the pseudoscalar and scalar double-charmed tetraquarks $
T_{cc;\overline{u}\overline{d}}^{+}$ and $\widetilde{T}_{cc;\overline{u}
\overline{d}}^{+}$ are investigated in the framework of the QCD sum rule
method. The mass and coupling of these exotic four-quark mesons are calculated
in the framework of the QCD two-point sum rule approach by taking into account
vacuum condensates of the quark, gluon, and mixed local operators up to
dimension 10. Our results for masses $m_{T}=(4130~\pm 170)~ \mathrm{MeV} $ and
$m_{\widetilde{T}}=(3845~\pm 175)~\mathrm{MeV}$ demonstrate that these
tetraquarks are strong-interaction unstable resonances and decay to
conventional mesons through the channels $T_{cc; \overline{u}\overline{d}}^{+}
\to D^{+}D^{\ast }(2007)^{0},~D^{0}D^{\ast }(2010)^{+}$ and
$\widetilde{T}_{cc;\overline{u}\overline{d}}^{+}\to D^{+}D^{0}$. Key quantities
necessary to compute the partial width of these decay modes, i.e., the strong
couplings of two $D$ mesons and a corresponding tetraquark $g_i,~i=1,2$, and
$G$ are extracted from the QCD three-point sum rules. The full width $\Gamma
_{T}=(129.9\pm 23.5)~\mathrm{ MeV}$ demonstrates that the tetraquark
$T_{cc;\overline{u}\overline{d}}^{+}$ is a broad resonance, whereas the scalar
exotic meson with $\Gamma _{ \widetilde{T}}=(12.4\pm 3.1)~\mathrm{MeV}$ can be
classified as a relatively narrow state.
