The recent discovery of a 125 GeV Higgs-like resonance at LHC, coupled with the lack of evidence for weak scale supersymmetry (SUSY), has severely constrained SUSY models such as minimal supergravity (mSUGRA)/CMSSM. As LHC probes deeper into SUSY model parameter space, the little hierarchy problem—how to reconcile the $Z$ and Higgs boson mass scale with the scale of SUSY breaking—will become increasingly exacerbated unless a sparticle signal is found. We evaluate two different measures of fine-tuning in the mSUGRA/CMSSM model. The more stringent of these, ${\Delta }_{\mathrm{HS}}$, includes effects that arise from the high-scale origin of the mSUGRA parameters while the second measure, ${\Delta }_{\mathrm{EW}}$, is determined only by weak scale parameters: hence, it is universal to any model with the same particle spectrum and couplings. Our results incorporate the latest constraints from LHC7 sparticle searches, LHCb limits from $B_s\to {\mu }^{+}{\mu }^{-}$ and also require a light Higgs scalar with $m_h\sim 123–127\mathrm{GeV}$. We present fine-tuning contours in the $m_0$ vs $m_{1/2}$ plane for several sets of $A_0$ and $\tan \beta$ values. We also present results for ${\Delta }_{\mathrm{HS}}$ and ${\Delta }_{\mathrm{EW}}$ from a scan over the entire viable model parameter space. We find a ${\Delta }_{\mathrm{HS}}\gtrsim {10}^3$, or at best 0.1%, fine-tuning. For the less stringent electroweak fine-tuning, we find ${\Delta }_{\mathrm{EW}}\gtrsim {10}^2$, or at best 1%, fine-tuning. Two benchmark points are presented that have the lowest values of ${\Delta }_{\mathrm{HS}}$ and ${\Delta }_{\mathrm{EW}}$. Our results provide a quantitative measure for ascertaining whether or not the remaining mSUGRA/CMSSM model parameter space is excessively fine-tuned and so could provide impetus for considering alternative SUSY models.

• Received 26 October 2012

DOI:https://doi.org/10.1103/PhysRevD.87.035017