Bayesian naturalness of the CMSSM and CNMSSM

The recent discovery of the 125.5 GeV Higgs boson at the LHC has fueled interest in the next-to-minimal supersymmetric standard model (NMSSM) as it may require less fine-tuning than the minimal model to accommodate such a heavy Higgs. To this end we present Bayesian naturalness priors to quantify fine-tuning in the (N)MSSM. These priors arise automatically as Occam razors in Bayesian model comparison and generalize the conventional Barbieri-Giudice measure. In this paper we show that the naturalness priors capture features of both the Barbieri-Giudice fine-tuning measure and a simple ratio measure that has been used in the literature. We also show that according to the naturalness prior the constrained version of the NMSSM is less tuned than the CMSSM.

Figure 1

Fine-tuning measures ${\mathrm{\Delta }}_{\mathrm{BG}}$ (top), ${\mathrm{\Delta }}_J$ (middle), ${\mathrm{\Delta }}_{\mathrm{EW}}$ (bottom) in the $M_0$ vs. $M_{1/2}$ plane for $A_0=-2.5\mathrm{TeV}$, $\tan \beta =10$ and $\mathrm{sgn}(\mu )=1$ in the CMSSM. The color code quantifies the value of ${\mathrm{\Delta }}_{\mathrm{EW}}$ and ${\mathrm{\Delta }}_J$. Since ${\mathrm{\Delta }}_{\mathrm{BG}}$ is dominated by the $\mu$ derivative it is low in the small $M_0$ and $M_{1/2}$ region. Although ${\mathrm{\Delta }}_{\mathrm{BG}}$, by definition, is formally part of ${\mathrm{\Delta }}_J$ the numerical behavior of the latter is similar to that of ${\mathrm{\Delta }}_{\mathrm{EW}}$. All massive parameters are in GeV unit. No experimental constraints applied except that the lightest supersymmetric particle is electrically neutral and the EWSB condition is satisfied.

Figure 2

Same as Fig. 1 except for the constrained NMSSM. $A_{0,\kappa ,\lambda }=-2.5\mathrm{TeV}$ and $\tan \beta =10$ are assumed. $\lambda$ is sampled from the range [0, 0.8].

Figure 3

Fine-tuning with respect to $m_{h^0}$ for the CMSSM (upper) and CNMSSM (lower). $A_0=-2.5\mathrm{TeV}$ and $\tan \beta =10$ for both models.

Figure 4

Fine-tuning with respect to $m_{h^0}$ for the CMSSM (upper) and CNMSSM (lower). $A_0=-2.5\mathrm{TeV}$ and $\tan \beta =3$ for both models.

Figure 5

Fits to various observables in the framework of the slightly relaxed CNMSSM for fixed values of $A_0=-2.5\mathrm{TeV}$ and $\tan \beta =10$. $A_{\lambda }$ and $A_{\kappa }$ are allowed to vary independently from $A_0$. Table 1 shows the experimental values of the observables that were used in this fit. The top frame is the fit to the relic density alone while the bottom is the two observable combined fit. The statistical significance with which each model point can be rejected is given in units of $\sigma s$. As the figure shows on the $A_0=-2.5\mathrm{TeV}$ and $\tan \beta =10$ hypersurface a good fit to both observables can be obtained for the low Jacobian tuning of ${\mathrm{\Delta }}_J\sim 1$.