Quark-hadron duality for heavy meson mixings in the 't Hooft model

We study local quark-hadron duality and its violation for the $D^0-\bar{D}^0$, $B^0_d-\bar{B}^0_d$ and $B^0_s-\bar{B}^0_s$ mixings in the 't Hooft model, offering a laboratory to test QCD in two-dimensional spacetime together with the large-$N_c$ limit. With the 't Hooft equation being numerically solved, the width difference is calculated as an exclusive sum over two-body decays. The obtained rate is compared to inclusive one that arises from four-quark operators to check the validity of the heavy quark expansion (HQE). In view of the observation in four-dimensions that the HQE prediction for the width difference in the $D^0-\bar{D}^0$ mixing is four orders of magnitude smaller than the experimental data, in this work we investigate duality violation in the presence of the GIM mechanism. We show that the order of magnitude of the observable in the $D^0-\bar{D}^0$ mixing is enhanced in the exclusive analysis relative to the inclusive counterpart, when the 4D-like phase space function is used for the inclusive analysis. By contrast, it is shown that for the $B^0_d-\bar{B}^0_d$ and $B^0_s-\bar{B}^0_s$ mixings, small yet non-negligible corrections to the inclusive result emerge, which are still consistent with what is currently indicated in four-dimensions.