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Supersymmetry Breaking and the Supersymmetric Flavour Problem: An Analysis of Decoupling the First Two Generation Scalars

The supersymmetric contributions to the Flavor Changing Neutral Current processes may be suppressed by decoupling the scalars of the first and second generations. It is known, however, that the heavy scalars drive the stop mass squareds negative through the two-loop Renormalization Group evolution. This tension is studied in detail. Two new items are included in this analysis: the effect of the top quark Yukawa coupling and the QCD corrections to the supersymmetric contributions to $Δm_K$. Even with Cabibbo-like degeneracy between the squarks of the first two generations, these squarks must be heavier than $\sim 40$ TeV to suppress $Δm_K$. This implies, in the case of a high scale of supersymmetry breaking, that the boundary value of the stop mass has to be greater than $\sim 7$ TeV to keep the stop mass squared positive at the weak scale. Low-energy supersymmetry breaking at a scale that is of the same order as the mass of the heavy scalars is also considered. In this case the finite parts of the two-loop diagrams are computed to estimate the contribution of the heavy scalar masses to the stop mass squared. It is found that for Cabibbo-like mixing between the squarks, the stop mass at the boundary needs to be larger than $\sim 2$ TeV. Thus, for both cases, the large boundary value of the stop masses leads to an unnatural amount of fine tuning to obtain the correct $Z$ mass.