Persistent fine-tuning in supersymmetry and the NMSSM

We examine the use of modified Higgs sectors to address the little hierarchy problem of supersymmetry. Such models reduce weak-scale fine-tuning by allowing lighter stops, but they retain some fine-tuning independent of the Higgs. The modified Higgs sector can also introduce model-dependent tunings. We consider the model-independent constraints on naturalness from squark, gluino, chargino and neutralino searches, as well as the model-dependent tuning in the PQ- and R-symmetric limits of the NMSSM, where cascade decays or additional quartic interactions can hide a Higgs without relying on heavy stops. Obtaining viable electroweak symmetry breaking requires a tuning of $\sim 1-10%$ in the PQ limit. A large A-term is also necessary to make charginos sufficiently heavy, and this introduces an additional weak-scale tuning of $\sim 10%$. The R-symmetric limit requires large marginal couplings, a large singlet soft mass, and a tuning of $\sim 5-10%$ to break electroweak symmetry. Hiding MSSM-like Higgs states below $\approx 112$ GeV requires additional tunings of A-terms at the $\sim 1-10%$ level. Thus, although the NMSSM has rich discovery potential, it suffers from a unique fine-tuning problem of its own.