  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.

High Energy Physics - Phenomenology