Towards the anomalous dimension toO(ΛQCD/mb)for phase space restrictedB¯→Xuℓν¯andB¯→Xsγ

We examine the anomalous dimension matrix appropriate for the phase space restricted $\overline{B}\to X_u\ell \overline{\nu }$ and $\overline{B}\to X_s\gamma$ decay spectra to subleading nonperturbative order. The time ordered products of the HQET Lagrangian with the leading order shape function operator are calculated, as are the anomalous dimensions of subleading operators. We establish the renormalizability and closure of a subset of the nonlocal operator basis, a requirement for the establishment of factorization theorems at this order. Operator mixing is found between the operators which occur to subleading order, requiring the subleading operator basis be extended. We comment on the requirement for new shape functions to be introduced to characterize the matrix elements of these new operators, and the phenomenological consequences for extractions of $|V_{ub}|$.