The $B \to K^\ast \psi$ Polarization Puzzle

We point out that current estimates of form factors fail to explain the non-leptonic decays $B \to \psi K(K^{\ast})$ and that the combination of data on the semi-leptonic decays $D \to K(K^{\ast})\ell \nu$ and on the non-leptonic decays $B \to \psi K(K^{\ast})$ (in particular recent po\-la\-ri\-za\-tion data) severely constrain the form (normalization and $q^2$ dependence) of the heavy-to-light meson form factors, if we assume the factorization hypothesis for the latter. From a simultaneous fit to \bpsi and \dk data we find that strict heavy quark limit scaling laws applied to the form factors do not hold when going from $D$ to $B$ and must have large corrections that make softer the dependence on the masses. This is in contrast with the matrix elements themselves which are found to need smaller $1/m_Q$ corrections to the asymptotic heavy quark scaling laws. We also find that $A_1(q^2)$ should increase slower with \qq than $A_2, V, f_+$. We propose a simple parametrization of these corrections based on a quark model or on an extension of the \hhs laws to the \hl case, complemented with an approximately constant $A_1(q^2)$. This model may be viewed as assuming a precocious validity of strict heavy quark scaling laws for the current matrix elements. Although this model reproduces qualitatively the wanted features for mass and $q^2$ dependence, and thus reduces the discrepancy with data, it is insufficient to reach a full agreement with the experiment. In our opinion the puzzle is still there.