Probing with Penguins: A lattice calculation of the branching ratio for some of the exclusive modes of $b\to s\gamma$

We calculate the leading-order matrix element for exclusive decays of $b \to s\gamma$inthequenchedapproximationoflatticeQCDona$24^3\times48$latticeat$\beta=6.2$, using an O(a)-improved fermion action. The matrix element is used to extract the on-shell form factor $T_1(q^2=0)$ for $B \to K^\gamma$and$B_s \to \phi\gamma$,usingtwodifferentassumptionsaboutthe$q^2$dependenceoftheformfactorsforthesedecays\mathrm{.}For$B \to K^\gamma$,$T_1(q^2=0)$isdeterminedtobe$0.159^{+34}{-33}\pm 0.067$or$0.124^{+20}{-18} \pm 0.022$ in the two cases. We find the results to be consistent (in the Standard Model) with the CLEO experimental branching ratio of $BR(B\to K^{\ast }\gamma )=(4.5\pm 1.5\pm 0.9)\times {10}^{-5}$.