Quark-hadron duality in the ’t Hooft model for meson weak decays: Different quark diagram topologies

We compare the effects of different quark diagram topologies on the weak hadronic width of heavy-light mesons in the large $N_c$ limit. We enumerate the various topologies and show that the only one dominant (or even comparable) in powers of $N_c$ to the noninteracting spectator “tree” diagram is the “annihilation” diagram, in which the valence quark-antiquark pair annihilates weakly. We compute the amplitude for this diagram in the ’t Hooft model (QCD in $1+1$ spacetime dimensions with a large number of colors $N_c)$ at the hadronic level and compare to the Born term partonic level. We find that quark-hadron duality is not well satisfied, even after the application of a smearing procedure to the hadronic result. A number of interesting subtleties absent from the tree diagram case arise in the annihilation diagram case, and are described in detail.