Fixing the EW scale in supersymmetric models after the Higgs discovery

Abstract

TeV-scale supersymmetry was originally introduced to solve the hierarchy problem and therefore fix the electroweak (EW) scale in the presence of quantum corrections. Numerical methods testing the SUSY models often report a good likelihood L (or chi^2=-2ln L) to fit the data {\it including} the EW scale itself (m_Z^0) with a {\it simultaneously} large fine-tuning i.e. a large variation of this scale under a small variation of the SUSY parameters. We argue that this is inconsistent and we identify the origin of this problem. Our claim is that the likelihood (or chi^2) to fit the data that is usually reported in such models does not account for the chi^2 cost of fixing the EW scale. When this constraint is implemented, the likelihood (or chi^2) receives a significant correction (delta_chi^2) that worsens the current data fits of SUSY models. We estimate this correction for the models: constrained MSSM (CMSSM), models with non-universal gaugino masses (NUGM) or higgs soft masses (NUHM1, NUHM2), the NMSSM and the general NMSSM (GNMSSM). For a higgs mass m_h\approx 126 GeV, one finds that in these models (delta_chi^2)/ndf> 1.5 (approx 1 for GNMSSM), which violates the usual condition of a good fit (total chi^2/ndf approx 1) already before fitting observables other than the EW scale itself (ndf=number of degrees of freedom). This has (negative) implications for SUSY models and it is suggested that future data fits properly account for this effect, if one remains true to the original goal of SUSY. Since the expression of delta_chi^2 that emerges from our calculation depends on a familiar measure of fine-tuning, one concludes that EW fine-tuning is an intrinsic part of the likelihood to fit the data that includes the EW scale (m_Z^0).