Tetraquark and two-meson states at large $N_{\mathrm{c}}$

Considering four-point correlation functions of color-singlet quark bilinears, we investigate, in the large-$N_{\mathrm{c}}$ limit of QCD, the subleading diagrams that involve, in the $s$-channel of meson-meson scattering amplitudes, two-quark--two-antiquark intermediate states. The latter contribute, together with gluon exchanges, to the formation, at the hadronic level, of two-meson and tetraquark intermediate states. It is shown that the two-meson contributions, which are predictable, in general, from leading-order $N_{\mathrm{c}}$-behaviors, consistently satisfy the constraints resulting from the $1/N_{\mathrm{c}}$ expansion procedure and thus provide a firm basis for the extraction of tetraquark properties from $N_{\mathrm{c}}$-subleading diagrams. We find that, in general, tetraquarks, if they exist in compact form, should have narrow decay widths, of the order of $N_{\mathrm{c}}^{-2}$. For the particular case of exotic tetraquarks, involving four different quark flavors, two different types of tetraquark are needed, each having a preferred decay channel, to satisfy the consistency constraints.