Semileptonic $B\to D^{\ast \ast }$ decays in Lattice QCD : a feasibility study and first results

We compute the decays $B\to D_0^{\ast }$ and $B\to D_2^{\ast }$ with finite masses for the $b$ and $c$ quarks. We first discuss the spectral properties of both the $B$ meson as a function of its momentum and of the $D_0^{\ast }$ and $D_2^{\ast }$ at rest. We compute the theoretical formulae leading to the decay amplitudes from the three-point and two-point correlators. We then compute the amplitudes at zero recoil of $B\to D_0^{\ast }$ which turns out not to be vanishing contrary to what happens in the heavy quark limit. This opens a possibility to get a better agreement with experiment. To improve the continuum limit we have added a set of data with smaller lattice spacing. The ${B\to D^\ast_2}$ vanishes at zero recoil and we show a convincing signal but only slightly more than 1 sigma from 0. In order to reach quantitatively significant results, we plan to fully exploit smaller lattice spacings as well as another lattice regularization.