Probing supergravity models with the bto sγ microscope

We present a calculation of the branching ratio $\brbsg$ in two well motivated supersymmetric models: the minimal $SU(5)$ and the no-scale flipped $SU(5)$ supergravity models. We find that the improved CLEO upper bound ($\brbsg<5.4\times10^{-4}$ at 95\% CL) does not yet constrain the minimal $SU(5)$ supergravity model, where $\brbsg_{minimal}=(2.3-3.6)\times10^{-4}$. In the flipped $SU(5)$ model the CLEO bound is constraining, although still not very significantly, even for light charged Higgs masses. An improvement in sensitivity by a factor of two will probe all (more than half) of the parameter space of the minimal (flipped) $SU(5)$ supergravity model. This ``resolution" of the $\bsg$ microscope far surpasses that of present collider experiments. In the flipped model there exists a significant region of parameter space where $\brbsg$ is highly suppressed due to a new phenomenon involving a complicated cancellation against the QCD correction parameter.