Tuning supersymmetric models at the LHC: a comparative analysis at two-loop level.

Abstract

Abstract We provide a comparative study of the fine tuning amount (Δ) at the two-loop leading log level in supersymmetric models commonly used in SUSY searches at the LHC. These are the constrained MSSM (CMSSM), non-universal Higgs masses models (NUHM1, NUHM2), non-universal gaugino masses model (NUGM) and GUT related gaugino masses models (NUGMd). Two definitions of the fine tuning are used, the first (Δmax) measures maximal fine-tuning w.r.t. individual parameters while the second (Δ q ) adds their contribution in “quadrature”. As a direct consequence of two theoretical constraints (the EW minimum conditions), fine tuning (Δ q ) emerges at the mathematical level as a suppressing factor (effective prior) of the averaged likelihood ( $ L $ ) under the priors, under the integral of the global probability of measuring the data (Bayesian evidence p(D)). For each model, there is little difference between Δ q , Δmax in the region allowed by the data, with similar behaviour as functions of the Higgs, gluino, stop mass or SUSY scale ( $ {m_{\text{SUSY}}} = {\left( {{m_{{\overline t 1}}}{m_{{\overline t 2}}}} \right)^{{{{1} \left/ {2} \right.}}}} $ ) or dark matter and g − 2 constraints. The analysis has the advantage that by replacing any of these mass scales or constraints by their latest bounds one easily infers for each model the value of Δ q , Δmax or vice versa. For all models, minimal fine tuning is achieved for M higgs near 115 GeV with a Δ q  ≈ Δmax ≈ 10 to 100 depending on the model, and in the CMSSM this is actually a global minimum. Due to a strong (≈ exponential) dependence of Δ on M higgs, for a Higgs mass near 125 GeV, the above values of Δ q  ≈ Δmax increase to between 500 and 1000. Possible corrections to these values are briefly discussed.