Critical Analysis of Theoretical Estimates for $B$ to Light Meson Form Factors and the $B \to ψK(K^{\ast})$ Data

Abstract

We point out that current estimates of form factors fail to explain the non-leptonic decays $B \to ψK(K^{\ast})$ and that the combination of data on the semi-leptonic decays $D \to K(K^{\ast})\ell ν$ and on the non-leptonic decays $B \to ψK(K^{\ast})$ (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 do not hold when going from $D$ to $B$ and must have large corrections that make softer the dependence on the masses. We 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 $A_1(q^2)$. We analyze in the light of these data and theoretical input various theoretical approaches (lattice calculations, QCD sum rules, quark models) and point out the origin of the difficulties encountered by most of these schemes. In particular we check the compatibility of several quark models with the heavy quark scaling relations.