SemileptonicBandΛbdecays and local duality in QCD

The inclusive and exclusive semileptonic decay distributions for $b\to c$ decay are computed in the Shifman-Voloshin limit. The inclusive decay distributions (computed using an operator product expansion) depend on quark masses, and the exclusive decay distributions depend on hadron masses. Nevertheless, we show explicitly how the first two terms in the $\frac{1}{m}$ expansion match between the inclusive and exclusive decays. Agreement between the inclusive and exclusive decay rates requires a minimum smearing region of size ${\Lambda }_{\mathrm{QCD}}$ before local duality holds in QCD. The ${\alpha }_s$ corrections to the inclusive and exclusive decay rates are also shown to agree to order $\frac{\left(\ln m\right)}{m^2}$. The $\frac{{\alpha }_s}{m^2}$ corrections are used to obtain the ${\alpha }_s$ correction to Bjorken's inequality on the slope of the Isgur-Wise function.