Decays $\overline{B}\to D^{**}\pi$ and the Isgur-Wise functions ${\tau }_{1/2}(w)$, ${\tau }_{3/2}(w)$

We perform a phenomenological analysis of the decays $\overline{B}\to D^{**}\pi$, where $D^{**}$ is a $P$-wave excited meson with total angular momentum $j=\frac{1}{2}$ or $\frac{3}{2}$ for the light cloud, recently measured by the Belle Collaboration in the modes ${\overline{B}}^0\to D^{**+}{\pi }^{-}$ (Class I) and $B^{-}\to D^{**0}{\pi }^{-}$ (Class III). Making the reasonable assumption of naive factorization, that we test in $B\to D(D^{*})\pi$ decays, Class I decays allow to extract the Isgur-Wise form factors ${\tau }_{1/2}(w)$, ${\tau }_{3/2}(w)$ at $w\cong w_{\max }$ ($q^2\cong 0$). We obtain ${\tau }_{1/2}(w_{\max })<0.20$, ${\tau }_{3/2}(w_{\max })=0.31\pm 0.12$. We discuss the question of the $w$ dependence of these IW functions. We find agreement with the Bakamjian-Thomas quark model of form factors and, extrapolating at $w=1$, with Bjorken and Uraltsev sum rules. We discuss also Class III decays, where the $D^{**0}$ $(j=\frac{1}{2})$ emission diagram contributes. We extract the corresponding $f_{D_{1/2}}$ decay constant, that is in agreement with theoretical estimates at finite mass. Finally, we must warn that $1/m_Q$ corrections could be large and upset the results of the present stage of this analysis. On the other hand, we confront present data on the semileptonic rate of $B$ mesons to excited states with theoretical expectations.