$b\to sγ$ Constraints on the Minimal Supergravity Model with Large $\tanβ$

In the minimal supergravity model (mSUGRA), as the parameter $\tan\beta$ increases, the charged Higgs boson and light bottom squark masses decrease, which can potentially increase contributions from $tH^\pm$, $\tg\tb_j$ and $\tz_i\tb_j$ loops in the decay $b\to s\gamma$. We update a previous QCD improved $b\to s\gamma$ decay calculation to include in addition the effects of gluino and neutralino loops. We find that in the mSUGRA model, loops involving charginos also increase, and dominate over $tW$, $tH^\pm$, $\tg\tq$ and $\tz_i\tq$ contributions for $\tan\beta\agt 5-10$. We find for large values of $\tan\beta \sim 35$ that most of the parameter space of the mSUGRA model for $\mu 0$ and large $\tan\beta$, most of parameter space is allowed, although the regions with the least fine-tuning (low $m_0$ and $m_{1/2}$) are ruled out due to too low a value of $B(b\to s\gamma)$. We compare the constraints from $b\to s\gamma$ to constraints from the neutralino relic density, and to expectations for sparticle discovery at LEP2 and the Fermilab Tevatron $p\bar p$ colliders. Finally, we show that non-universal GUT scale soft breaking squark mass terms can enhance gluino loop contributions to $b\to s\gamma$ decay rate even if these are diagonal.