The fine-tuning cost of the likelihood in SUSY models

In SUSY models, the fine-tuning of the electroweak (EW) scale with respect to their parameters γi={m0,m1/2,μ0,A0,B0,…}γi={m0,m1/2,μ0,A0,B0,…} and the maximal likelihood L to fit the experimental data are usually regarded as two different problems. We show that, if one regards the EW minimum conditions as constraints that fix   the EW scale, this commonly held view is not correct and that the likelihood contains all the information about fine-tuning. In this case we show that the corrected likelihood is equal to the ratio L/ΔL/Δ of the usual likelihood L and the traditional fine-tuning measure Δ   of the EW scale. A similar result is obtained for the integrated likelihood over the set {γi}{γi}, that can be written as a surface integral of the ratio L/ΔL/Δ, with the surface in γiγi space determined by the EW minimum constraints. As a result, a large likelihood actually demands a large ratio L/ΔL/Δ or equivalently, a small χnew2=χold2+2lnΔ. This shows the fine-tuning cost to the likelihood (χnew2) of the EW scale stability enforced by SUSY, that is ignored in data fits. A good χnew2/d.o.f.≈1 thus demands SUSY models have a fine-tuning amount Δ≪exp(d.o.f./2)Δ≪exp(d.o.f./2), which provides a model-independent criterion for acceptable fine-tuning. If this criterion is not met, one can thus rule out SUSY models without a further χ2χ2/d.o.f. analysis. Numerical methods to fit the data can easily be adapted to account for this effect.