Narrow exotic tetraquark mesons in large-$N_c$ QCD

Discussing four-point Green functions of bilinear quark currents in large-$N_c$ QCD, we formulate rigorous criteria for selecting diagrams appropriate for the analysis of potential tetraquark poles. We find that both flavor-exotic and cryptoexotic (i.e., flavor-nonexotic) tetraquarks, if such poles exist, have a width of order $O(1/N_c^2)$, so they are parametrically narrower compared to the ordinary $\overline{q}q$ mesons, which have a width of order $O(1/N_c)$. Moreover, for flavor-exotic states, the consistency of the large-$N_c$ behavior of “direct” and “recombination” Green functions requires two narrow flavor-exotic states, each coupling dominantly to one specific meson-meson channel.

Figure 1

Perturbative expansion of the Green functions ${\mathrm{\Gamma }}_{\mathrm{I}}^{(\mathrm{dir})}$.

Figure 2

Diagrams for the recombination Green function ${\mathrm{\Gamma }}^{(\text{rec})}$.

Figure 3

Diagrams contributing to ${\mathrm{\Gamma }}_{\mathrm{I},T}^{(\mathrm{dir})}$. Crossed propagators denote on-shell particles contributing to the four-quark cut.

Figure 4

Diagrams contributing to the recombination Green function ${\mathrm{\Gamma }}_T^{(\mathrm{rec})}$.