Quantitative Predictions for $B$ semileptonic decays into $D,D^*$ and the orbitally excited $D^{**}$ in quark models \`a la Bakamjian-Thomas

Once chosen the dynamics in one frame, the rest frame in this paper, the Bakamjian and Thomas method allows to define relativistic quark models in any frame. These models have been shown to provide, in the infinite quark mass limit, fully covariant current form factors as matrix elements of the quark current operator. In this paper we use the rest frame dynamics fitted from the meson spectrum by various authors, already shown to provide a reasonable value for $\rho^2$. From the general formulae for the scaling invariant form factors $\xi^{(n)}(w)$, $\tau_{1/2}^{(n)}(w)$ and $\tau_{3/2}^{(n)}(w)$, we predict quantitavely the $B$ semileptonic branching ratios to the ground state and orbitally excited charmed mesons $D, D^\ast$ and $D^{\ast\ast}$. We check Bjorken's sum rule and discuss the respective contributions to it. We find $\xi(w)\simeq (2/(1+w)) ^2$, resulting from the fact that the ground state wave function is Coulomb-like. We also find $\tau_{3/2}\simeq 0.5 (2/(1+w))^3$ and $\tau_{1/2}(w)\ll \tau_{3/2}(w)$. Very small branching ratios into $j=1/2$ orbitally excited $D$'s results. The overall agreement with experiment is rather good within the present accuracy which is poor for the orbitally excited charmed mesons. We predict a ratio $Br(B\to D^\ast_2 l \nu)/Br(B\to D_1 l \nu)=1.55\pm 0.15$ as a mere consequence of the heavy quark symmetry. If some faint experimental indications that $Br(B\to D_1 l \nu)\simeq Br(B\to D^\ast_2 l \nu)$ were confirmed, it would indicate a sizeable $O(1/m_c)$ correction.