Hierarchy, Criticality and Vacuum Energy

A hierarchically small weak scale does not in general correspond to a point of enhanced symmetry, but it can still define a special configuration with respect to vacuum energy. When the vacuum energy is considered as a function of the Higgs mass, the non-analytic behavior at the point of electroweak symmetry breaking can potentially generate a local maximum at small Higgs vev. If the Higgs mass parameter is promoted to a dynamical field, or otherwise scans, then our universe could be localized to this critical point without fine-tuning via several mechanisms. For example, in certain simple models the only stable vacua reside near the critical point. Another possibility is that anthropic selection of the cosmological constant within a landscape prefers to maximize the Higgs-mass-dependent component of the vacuum energy. Independent of the mechanism, the principle of living a a critical point of the vacuum energy predicts that the electroweak scale squared should be two loop factors down from the scale of heavy particles giving loop corrections to the Higgs mass. We demonstrate this for the case of the MSSM, where we show that theories with heavy Higgsinos, stops, etc. as required by the LHC are not necessarily tuned in this framework, and obtain predictive constraints on the SUSY parameters.