Use and Misuse of QCD Sum Rules in Heavy-to-light Transitions: the Decay $B\to\rho e \nu$ Reexamined

The existing calculations of the form factors describing the decay $B\to\rho e \nu$ from QCD sum rules have yielded conflicting results at small values of the invariant mass squared of the lepton pair. We demonstrate that the disagreement originates from the failure of the short-distance expansion to describe the $\rho$ meson distribution amplitude in the region where almost the whole momentum is carried by one of the constituents. This limits the applicability of QCD sum rules based on the short-distance expansion of a three-point correlation function to heavy-to-light transitions and calls for an expansion around the light-cone, as realized in the light-cone sum rule approach. We derive and update light-cone sum rules for all the semileptonic form factors, using recent results on the $\rho$ meson distribution amplitudes. The results are presented in detail together with a careful analysis of the uncertainties, including estimates of higher-twist effects, and compared to lattice calculations and recent CLEO measurements. We also derive a set of ``improved'' three-point sum rules, in which some of the problems of the short-distance expansion are avoided and whose results agree to good accuracy with those from light-cone sum rules.