Heavy flavor decays, OPE, and duality in the two-dimensional ’t Hooft model

The ’t Hooft model (two-dimensional QCD in the limit of a large number of colors) is used as a laboratory for exploring various aspects of heavy quark expansion in the nonleptonic and semileptonic decays of heavy flavors. We perform a complete operator analysis and construct the operator product expansion (OPE) up to terms $\mathcal{O}{(1/m}_Q^4),$ inclusively. The OPE-based predictions for the inclusive widths are then confronted with the “phenomenological” results, obtained by summation of all open exclusive decay channels, one by one. The summation is carried out analytically, by virtue of the ’t Hooft equation. The two alternative expressions for the total widths match. We comment on the recent claim in the literature of a ${1/m}_Q$ correction to the total width which would be in clear conflict with the OPE result. The issue of duality violations both in the simplified setting of the ’t Hooft model and in actual QCD is discussed. The amplitude of oscillating terms is estimated.