Explicit quark-hadron duality in heavy-light meson weak decays in the ’t Hooft model

We compute the nonleptonic weak decay width of a heavy-light meson in 1+1 spacetime dimensions with a large number of QCD colors (the ’t Hooft model) as a function of the heavy quark mass. In this limit, QCD is exactly soluble, and decay modes are dominated by two-particle final states. We compare the results to the tree-level partonic decay width of the heavy quark in order to test quark-hadron duality in this universe. We find that this duality is surprisingly well satisfied in the heavy quark limit, in that the difference between the sum of exclusive partial widths and the tree-level partonic width approaches a constant as $\overrightarrow{M}\infty$, and the deviation is well-fit by a small $1/M$ correction. We comment on the meaning of this conclusion and its implications for the use of quark-hadron duality in hadronic physics.