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Upper bounds on supersymmetry breaking from gauge coupling unification

I derive conservative upper bounds on the supersymmetry breaking parameter $m_{1/2}$ as a function of the strong coupling in the Standard Supersymmetric Model (SSM) using gauge coupling unification. I find that over more than $99\%$ of the parameter space, $α_3>0.120$ implies that $m_{1/2}$ is below $10\TeV$ and $α_3>0.129$ implies that $m_{1/2}$ is below $1\TeV$. I express the variation of these bounds over the SSM parameter space with a numerical coefficient, $c$. I also find that in the SSM, a reasonable value of $50\GeV<m_{1/2}<1\TeV$ requires $α_3>0.119$ over the whole parameter space. These bounds are particularly sensitive to the value of $\sin^2θ_W=0.2317\pm0.0005$ used in the calculation. In more realistic models, heavy thresholds and gravitational effects will modify this result. Although these effects are theoretically calculable in specific models, more realistic models contain many unknown parameters in practice. I illustrate this point with minimal supersymmetric $SU(5)$ where the combined constraints of gauge coupling unification and proton decay require $α_3>0.119$ for $m_{1/2}<1\TeV$ and the upper bound on the supersymmetry breaking scale is greatly relaxed.