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

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 (Δ_max) 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, Δ_max in the region allowed by the data, with similar behaviour as functions of the Higgs, gluino, stop mass or SUSY scale (m_susy = (m_t̄1 m_t̄2)^1/2) 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, Δ_q or vice versa. For all models, minimal fine tuning is achieved for Mhiggs near 115 GeV with aΔ_q ≈ Δ_max ≈ 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 ≈ Δ_max increase to between 500 and 1000. Possible corrections to these values are briefly discussed.