Testing SUSY at the LHC: Electroweak and Dark matter fine tuning at two-loop order

In the framework of the Constrained Minimal Supersymmetric Standard Model (CMSSM) we evaluate the electroweak fine tuning measure that provides a quantitative test of supersymmetry as a solution to the hierarchy problem. Taking account of current experimental constraints we compute the fine tuning at two-loop order and determine the limits on the CMSSM parameter space and the measurements at the LHC most relevant in covering it. Without imposing the LEPII bound on the Higgs mass, it is shown that the fine tuning computed at two-loop has a minimum $\Delta=8.8$ corresponding to a Higgs mass $m_h=114\pm 2$ GeV. Adding the constraint that the SUSY dark matter relic density should be within present bounds we find $\Delta=15$ corresponding to $m_h=114.7\pm 2$ GeV and this rises to $\Delta=17.8$ ($m_h=115.9\pm 2$ GeV) for SUSY dark matter abundance within 3$\sigma$ of the WMAP constraint. We extend the analysis to include the contribution of dark matter fine tuning. In this case the overall fine tuning and Higgs mass are only marginally larger for the case SUSY dark matter is subdominant and rises to $\Delta=28.7$ ($m_h=116.98\pm 2$ GeV) for the case of SUSY dark matter saturates the WMAP bound. For a Higgs mass above these values, fine tuning rises exponentially fast. The CMSSM spectrum that corresponds to minimal fine tuning is computed and provides a benchmark for future searches. It is characterised by heavy squarks and sleptons a nd light neutralinos, charginos and gluinos.