Are there narrow flavor-exotic tetraquarks in large-$N_c$ QCD?

Understand

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.