$B$ decays to radially excited $D$ mesons in heavy quark effective theory

Semileptonic transitions $\overline{B}\to D^{(n)}\ell \overline{\nu }$, where $D^{(n)}(n\ne 0)$ is a radially excited meson, have recently attracted much attention as a way to understand some puzzles between theory and data. In the heavy quark limit the inelastic Isgur-Wise function vanishes at zero recoil, ${\xi }^{(n)}(1)=0(n\ne 0)$. We consider here the $1/m_Q$ corrections within heavy quark effective theory. We find simple formulas that involve the derivative of the inelastic IW function at $w=1$, ${\xi }^{(n{)}^{\prime }}(1)(n\ne 0)$. We propose a number of ways of isolating this important quantity in practice, with emphasis on lattice QCD. We formulate also a generalization to the inelastic case of Luke’s theorem.