Possible Avenues in Supersymmetry and Naturalness

The purpose of this thesis is to investigate a set of possible realizations of Supersymmetry and naturalness in beyond the Standard Model physics. We will mostly be interested in models that combine both aspects, but this will not prevent us from investigating models that are only supersymmetric without being natural or that are natural without being supersymmetric. More precisely, we will look at four different cases. First, we introduce our own variation of the Twin Higgs model, which we refer to as the spontaneous Z2 breaking Twin Higgs. The main feature of the model is that the Z2 breaking is spontaneous instead of explicit. In addition to the theoretical appeal of this, other advantages are that it reduces tuning and makes many mirror particles very heavy. Second, we study collider constraints on Mini-Split models of Supersymmetry. More precisely, we study the current constraints on anomaly and gauge mediation, as well as their prospects for Run-II of the LHC and a future 100 TeV proton collider. We assume the Higgsinos to be heavy and a slightly lighter third generation of squarks. Third, we study constraints coming from electroweak precision tests on models of U(1)R Supersymmetry where the R-charge is associated with the lepton number. Contributions from new physics are encoded in a set of operators which are used to calculate corrections to a set of electroweak observables. The model is then constrained by fitting with experimental inputs. Finally, we investigate whether supersymmetric models where the U(1)R symmetry is instead associated with the baryon number can lead to successful baryogenesis once this symmetry is broken. Indeed, the U(1)R symmetry is expected to be broken by either anomaly mediation or Planck scale suppressed operators, thereby breaking baryon number conservation. We find that it is possible to obtain the correct baryon density, but that a Mini-Split spectrum is required and that the U(1)R symmetry must be badly broken. We also find that the model does not require the Higgsinos to be as heavy as in the MSSM.