QCD Improved $b\to sγ$ Constraints on the Minimal Supergravity Model

Recent advances in the QCD corrections to $b\to s\gamma$ decay in the MSSM include {\it i.}) evaluation of the relevant operators, Wilson coefficients and anomalous dimension matrix elements for the various MSSM effective theories valid at scales beyond $Q =M_W$, {\it ii.}) calculations of most of the needed anomalous dimension matrix elements to next-to-leading order for scales $m_b\alt Q <M_W$, and {\it iii.}) calculations of ${\cal O}(\alpha_s)$ virtual and bremsstrahlung corrections to the $b\to s\gamma$ decay operators at scale $Q\sim m_b$. We assemble all these known results to gain an estimate of $B(b\to s\gamma )$ for the parameter space of the minimal supergravity model (mSUGRA). We find a much reduced scale dependence of our result compared to usual leading-log evaluations. Comparison with the latest CLEO results yields stringent constraints on parameter space. Much of mSUGRA parameter space is ruled out for $\mu <0$, especially for large $tan\beta$. We compare these results with other constraints from cosmology and non-standard vacua. Also, we compare with expectations for discovering mSUGRA at LEP2, the Tevatron and the CERN LHC.