Semileptonic inclusive heavy meson decay: Duality in a nonrelativistic potential model in the Shifman-Voloshin limit

The quark-hadron duality in the inclusive semileptonic decay $\overrightarrow{B}{X}_{c}l\nu$ in the Shifman-Voloshin limit $\Lambda \ll \delta {m=m}_b-m_c\ll m_b{,m}_c$ is studied within a nonrelativistic potential model. The integrated semileptonic decay rate is calculated in two ways: first, by constructing the operator product expansion, and second by a direct summation of the exclusive channels. Sum rules (Bjorken, Voloshin, etc.) for the potential model are derived, providing a possibility to compare the two representations for $\Gamma \left(\overrightarrow{B}{X}_{c}l\nu \right).\mathrm{}$ An explicit difference between them referred to as the duality-violation effect is found. The origin of this effect is related to higher charm resonances which are kinematically forbidden in the decay process but are nevertheless picked up by the OPE. Within the considered ${1/m}_c^2$ order the OPE and the sum over exclusive channels match each other, up to the contributions of higher resonances, by virtue of the sum rules. In particular this is true for the terms of order $\delta m^2{/m}_c^2$ and $\Lambda \delta {m/m}_c^2$ which are present in each of the decay channels and cancel in the sum of these channels due to the Bjorken and Voloshin sum rules, respectively. The size of the duality violation effects is estimated to be of the order $O\left({\Lambda }^{2+b}{/m}_c^2\delta m^b\right)$ with $b>\mathrm{}0\mathrm{}$ depending on the details of the potential. Constraints for a better accuracy are discussed.