$B \rar K^* \ \gamma$} from Hybrid Sum Rule

Abstract

Using the {\it hybrid} moments-Laplace sum rule (HSR), which is well-defined for $M_b \rar \infty$, in contrast with the $popular$ double Borel (Laplace) sum rule (DLSR), which blows up in this limit when applied to the heavy-to-light processes, we show that the form factor of the $B \rar K^* \ \gamma$ radiative transition is dominated by the light-quark condensate for $M_b \rar \infty$ and behaves like $\sqrt M_b$. The form factor is found to be $ F^{B\rar K^*}_1(0) \simeq (30.8 \pm 1.3 \pm 3.6 \pm 0.6)\times 10^{-2}, $ where the errors come respectively from the procedure in the sum rule analysis, the errors in the input and in the $SU(3)_f$-breaking parameters. This result leads to $Br(B\rar K^* \ \gamma) \simeq (4.45 \pm 1.12) \times 10^{-5}$ in agreement with the recent CLEO data. Parametrization of the $M_b$-dependence of the form factor including the $SU(3)_f$-breaking effects is given in (26), which leads to $F^{B\rar K^*}_1(0)/ F^{B\rar \rho}_1(0) \simeq (1.14 \pm 0.02)$.