Subleading shape function contributions to the hadronic invariant mass spectrum inB¯→Xulν¯ldecay

We study the $O\left({\Lambda }_{\mathrm{QCD}}{/m}_b\right)$ corrections to the singly and doubly differential hadronic invariant mass spectra $d\Gamma {/ds}_H$ and $d\Gamma {/ds}_H{\mathrm{dq}}^2$ in $\overline{B}\to X_{u}l{\overline{\nu }}_l$ decays, and discuss the implications for the extraction of the CKM matrix element $|V_{\mathrm{ub}}|.$ Using simple models for the subleading shape functions, the effects of subleading operators are estimated to be at the few percent level for experimentally relevant cuts. The subleading corrections proportional to the leading shape function are larger, but largely cancel in the relation between the hadronic invariant mass spectrum and the photon spectrum in $\overline{B}\to X_s\gamma .$ We also discuss the applicability of the usual prescription of convoluting the partonic level rate with the leading light-cone wave function of the b quark to subleading order.