CMSSM, naturalness and the "fine-tuning price" of the Very Large Hadron Collider

The absence of supersymmetry or other new physics at the Large Hadron Collider (LHC) has lead many to question naturalness arguments. With Bayesian statistics, we argue that natural models are most probable and that naturalness is not merely an aesthetic principle. We calculate a probabilistic measure of naturalness, the Bayesian evidence, for the Standard Model (SM) with and without quadratic divergences, confirming that the SM with quadratic divergences is improbable. We calculate the Bayesian evidence for the Constrained Minimal Supersymmetric Standard Model (CMSSM) with naturalness priors in three cases: with only the $M_Z$ measurement; with the $M_Z$ measurement and LHC measurements; and with the $M_Z$ measurement, $m_h$ measurement and a hypothetical null result from a $\sqrt{s}=100\text{TeV}$ Very Large Hadron Collider (VLHC) with $3000\mathrm{/}\text{fb}$. The “fine-tuning price” of the VLHC given LHC results would be $\sim 400$, which is slightly less than that of the LHC results given the electroweak scale ($\sim 500$).