  A salient feature shared by all tetraquark candidates observed in experiment
is the absence of flavor-exotic states of the type $\bar a b\bar c d$, with
four different quark flavors. This phenomenon may be understood from the
properties of large-$N_c$ QCD: On the one hand, consistency conditions for
flavor-exotic Green functions, potentially containing these tetraquark poles,
require the existence of two tetraquarks $T_A$ and $T_B$: each of them should
decay dominantly via a single two-meson channel, $T_A\to M_{\bar a b}M_{\bar c
d}$ and $T_B\to M_{\bar a d}M_{\bar c b}$, with suppressed rates $T_A\to
M_{\bar a d}M_{\bar c b}$ and $T_B\to M_{\bar a b}M_{\bar c d}$. On the other
hand, we have at hand only one diquark-antidiquark flavor structure $(\bar a
\bar c)(b d)$ that might produce a compact tetraquark bound state. Taking into
account that the diquark-antidiquark structure is the only viable candidate for
a compact tetraquark state, one concludes that it is impossible to have two
different narrow tetraquarks decaying dominantly into different two-meson
channels. This contradiction suggests that large-$N_c$ QCD does not support the
existence of narrow flavor-exotic tetraquarks. This argument does not rule out
the possible existence of broad molecular-type flavor-exotic states, or of
molecular-type bound states lying very close to the two-meson thresholds.
