Focus points and naturalness in supersymmetry

We analyze focus points in supersymmetric theories, where a parameter’s renormalization group trajectories meet for a family of ultraviolet boundary conditions. We show that in a class of models including minimal supergravity, the up-type Higgs mass parameter has a focus point at the weak scale, where its value is highly insensitive to the universal scalar mass. As a result, scalar masses as large as 2 to 3 TeV are consistent with naturalness, and all squarks, sleptons and heavy Higgs scalars may be beyond the discovery reaches of the CERN Large Hadron Collider and proposed linear colliders. Gaugino and Higgsino masses are, however, still constrained to be near the weak scale. The focus point behavior is remarkably robust, holding for both moderate and large $\tan \beta ,$ any weak scale gaugino masses and A parameters, variations in the top quark mass within experimental bounds, and for large variations in the boundary condition scale.