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Within a two-loop leading-log approximation, we review the prediction for thelightest Higgs mass (m_h) in the framework of constrained MSSM (CMSSM), derivedfrom the naturalness requirement of minimal fine-tuning (Delta) of theelectroweak scale and dark matter consistency. As a result, the Higgs mass ispredicted to be just above the LEP2 bound, m_h=115.9\pm 2 GeV, corresponding toa 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 exponentiallyon either side of the above value of m_h, which stresses the relevance of thisprediction. A value m_h>121 (126) GeV cannot be accommodated within the CMSSMunless one accepts a fine-tuning cost worse than Delta>100 (1000),respectively. We review how the above prediction for m_h and Delta changesunder the addition of new physics beyond the MSSM Higgs sector, parametrized byeffective operators of dimensions d=5 and d=6. For d=5 operators, one canobtain values m_h\leq 130 GeV for Delta<10. The size of the supersymmetriccorrection that each individual operator of d=6 brings to the value of m_h forpoints with Delta<100, is found to be small, of few (<4) GeV for M=8 TeV, whereM is the scale of new physics. This value decreases (increases) byapproximately 1 GeV for a 1 TeV increase (decrease) of the scale M. Therelation of these results to the Atlas/CMS supersymmetry exclusion limits ispresented together with their impact for the CMSSM regions of lowestfine-tuning.

A review of naturalness and dark matter prediction for the Higgs mass in MSSM and beyond