$B$ decays to radially excited $D$ mesons in Heavy Quark Effective Theory

Semileptonic transitions $\overline{B} \to D^{(n)} \ell \overline{\nu}$, where $D^{(n)} (n \not = 0)$ is a radially excited meson, have recently attracted much attention as a way to understand some puzzles between theory and data. Following closely the formalism of Falk and Neubert for the elastic case, we study the $1/m_Q$ corrections to the heavy quark limit, in which the inelastic Isgur-Wise function vanishes at zero recoil, $\xi^{(n)}(1) = 0\ (n \not = 0)$. We find simple formulas that involve the derivative $\xi^{(n)'}(1)$, and we propose a number of ways of isolating this quantity in practice. We formulate also a generalization to the inelastic case of Luke's theorem. On the other hand, although some $1/m_Q$ HQET results are satisfied in the Bakamjian-Thomas relativistic quark model, we emphasize the problems concerning these corrections in this scheme.