The decays $\bar{B}\to D^{\ast \ast }\pi$ and the Isgur-Wise functions ${\tau }_{1\mathrm{/}2}(w)$, ${\tau }_{3\mathrm{/}2}(w)$

We perform a phenomenological analysis of the decays $B\to D^{\ast \ast }\pi$, where $D^{\ast \ast }$ is a $P$-wave excited meson with total angular momentum $j = {1 \over 2}$or${3 \over 2}$ for the light cloud, recently measured by the Belle Collaboration in the modes ${\bar{B}}^0\to D^{\ast \ast +}{\pi }^{-}$ (Class I) and $B^- \to D^{**0}\pi^-$ (Class III). Making the reasonable assumption of naive factorization, that we test in $B\to D(D^{\ast })\pi$ decays, Class I decays allow to extract the Isgur-Wise form factors ${\tau }_{1\mathrm{/}2}(w)$, ${\tau }_{3\mathrm{/}2}(w)$ at $w \cong w_{max}$($q^2 \cong 0$)\mathrm{.}Weobtain$\tau_{1/2}(w_{max}) < 0.20$,$\tau_{3/2}(w_{max}) = 0.31 \pm 0.12$\mathrm{.}Wediscussthequestionofthe$w$ 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 $D^{\ast \ast 0}$ $(j = {1 \over 2})$emissiondiagramcontributes\mathrm{.}Weextractthecorresponding$f_{D_{1/2}}$ decay constant, that is in agreement with theoretical estimates at finite mass. Finally, we must warn that $1\mathrm{/}{m}_Q$ 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.