The fine-tuning cost of the likelihood in SUSY models

Abstract

In SUSY models, the fine tuning of the electroweak (EW) scale with respect to their parameters gamma_i={m_0, m_{1/2}, mu_0, A_0, B_0,...} 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/Delta of the usual likelihood L and the traditional fine tuning measure Delta of the EW scale. A similar result is obtained for the integrated likelihood over the set {gamma_i}, that can be written as a surface integral of the ratio L/Delta, with the surface in gamma_i space determined by the EW minimum constraints. As a result, a large likelihood actually demands a large ratio L/Delta or equivalently, a small chi^2_{new}=chi^2_{old}+2*ln(Delta). This shows the fine-tuning cost to the likelihood (chi^2_{new}) of the EW scale stability enforced by SUSY, that is ignored in data fits. A good chi^2_{new}/d.o.f.\approx 1 thus demands SUSY models have a fine tuning amount Delta<<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 chi^2/d.o.f. analysis. Numerical methods to fit the data can easily be adapted to account for this effect.