On the Tuning Condition of Split Supersymmetry

Split Supersymmetry does not attempt to solve the hierarchy problem, but it assumes a tuning condition for the electroweak scale. We clarify the meaning of this condition and show how it is related to the underlying parameters. Simple assumptions on the structure of the soft terms lead to predictions on tan β and on the physical Higgs mass. Split Supersymmetry [1, 2, 3] does not attempt to give a natural explanation of the hierarchy problem but, as in the Standard Model, a parameter fine-tuning is assumed to obtain the correct breaking of electroweak symmetry. In this paper we want to show that, although the theory does not provide a dynamical explanation for the tuning, its existence leads to important information on the underlying parameters and on measurable physical quantities. Let us start by considering the potential in Split Supersymmetry for the real and neutral Higgs field h, valid at energies lower than the squark and slepton mass scale m̃, V = m 2 h + λ 8 h. (1) The Higgs mass parameter m satisfies to the RG equation 16πμ̄ d dμ̄ m = [ 6λ− 3 2 (