Within a two-loop leading-log approximation, we review the prediction for the lightest Higgs mass (m_h) in the framework of constrained MSSM (CMSSM), derived from the naturalness requirement of minimal fine-tuning (Delta) of the electroweak scale and dark matter consistency. As a result, the Higgs mass is predicted to be just above the LEP2 bound, m_h=115.9\pm 2 GeV, corresponding to a minimal Delta=17.8, value obtained from consistency with electroweak and WMAP (3\sigma) constraints, but without the LEP2 bound. Due to quantum corrections (largely QCD ones for m_h above LEP2 bound), Delta grows \approx exponentially on either side of the above value of m_h, which stresses the relevance of this prediction. A value m_h>121 (126) GeV cannot be accommodated within the CMSSM unless one accepts a fine-tuning cost worse than Delta>100 (1000), respectively. We review how the above prediction for m_h and Delta changes under the addition of new physics beyond the MSSM Higgs sector, parametrized by effective operators of dimensions d=5 and d=6. For d=5 operators, one can obtain values m_h\leq 130 GeV for Delta<10. The size of the supersymmetric correction that each individual operator of d=6 brings to the value of m_h for points with Delta<100, is found to be small, of few (<4) GeV for M=8 TeV, where M is the scale of new physics. This value decreases (increases) by approximately 1 GeV for a 1 TeV increase (decrease) of the scale M. The relation of these results to the Atlas/CMS supersymmetry exclusion limits is presented together with their impact for the CMSSM regions of lowest fine-tuning.Comment: 27 pages, 19 figures; (new figures and references added; improved presentation
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