Multidimensional mSUGRA likelihood maps

We calculate the likelihood map in the full 7-dimensional parameter space of the minimal supersymmetric standard model assuming universal boundary conditions on the supersymmetry breaking terms. Simultaneous variations of $m_0$, $A_0$, $M_{1/2}$, $\tan \beta$, $m_t$, $m_b$ and ${\alpha }_s(M_Z)$ are applied using a Markov chain Monte Carlo algorithm. We use measurements of $b\to s\gamma$, $(g-2{)}_{\mu }$ and ${\Omega }_{DM}{h}^2$ in order to constrain the model. We present likelihood distributions for some of the sparticle masses, for the branching ratio of $B_s^0\to {\mu }^{+}{\mu }^{-}$ and for $m_{\tilde{\tau }}-m_{{\chi }_1^0}$. An upper limit of $2\times {10}^{-8}$ on this branching ratio might be achieved at the Tevatron, and would rule out $29\%$ of the currently allowed likelihood. If one allows for non-thermal-neutralino components of dark matter, this fraction becomes $35\%$. The mass ordering allows the important cascade decay ${\tilde{q}}_L\to {\chi }_2^0\to {\tilde{l}}_R\to {\chi }_1^0$ with a likelihood of $24\pm 4\%$. The stop-coannihilation region is highly disfavored, whereas the light Higgs region is marginally disfavored.

Figure 1

Estimate of potential scale reduction shown as a function of the number of Markov chain Monte Carlo steps. The upper bound we require for convergence is shown as a horizontal line.

Figure 2

Likelihood maps of mSUGRA parameter space. The graphs show the likelihood distributions sampled from 7D parameter space and marginalized down to two. The likelihood (relative to the likelihood in the highest bin) is displayed by reference to the bar on the right-hand side of each plot.

Figure 3

Likelihood distributions of masses in mSUGRA. The graphs show the likelihood distributions marginalized down to 2D. The likelihood (relative to the likelihood in the highest bin) is displayed by reference to the bar on the right-hand side of each plot.

Figure 4

(a) Selected sparticle mass likelihood distributions in mSUGRA. (b) The stau-neutralino mass difference likelihood distribution where the inset shows a blowup of the quasi-mass-degenerate region. (c) Branching ratio for the decay $B_s\to {\mu }^{+}{\mu }^{-}$. The Tevatron upper bound is displayed by a vertical line. (d) Likelihood density marginalized to the 2D plane $BR(B_s\to {\mu }^{+}{\mu }^{-})$ versus $M_{1/2}$. (e) Correlation between $BR(B_s\to {\mu }^{+}{\mu }^{-})$ and $(g-2{)}_{\mu }$. (f) Likelihood marginalized to the $\tan \beta$-$M_A$ plane.

Figure 5

A line through the stop-coannihilation region in mSUGRA: $\tan \beta =10$, $M_{1/2}=350\mathrm{GeV}$, $A_0=-2000\mathrm{GeV}$ and central standard model inputs. (a) Dark matter relic density, fractional stop-neutralino mass splitting $\Delta \equiv (m_{{\tilde{t}}_1}-m_{{\chi }_1^0})/m_{{\chi }_1^0}$ and light Higgs mass. The horizontal lines show the $1\sigma$ WMAP-derived limits upon ${\Omega }_{DM}{h}^2$ derived from Eq. (4), (b) muon magnetic moment and $BR[b\to s\gamma ]$ predictions.

Figure 6

Likelihood maps of mSUGRA parameter space including theoretical uncertainty. The graphs show the likelihood distributions sampled from 8D parameter space and marginalized down to two. The likelihood (relative to the likelihood in the highest bin) is displayed by reference to the bar on the right-hand side of each plot.

Figure 7

(a) $p(\lambda |m_{{\Omega }_{DM}{h}^2})$, the probability distribution assumed for $\lambda$, the ratio of predicted SUSY dark matter to total dark matter, given some measured value $m_{{\Omega }_{DM}{h}^2}$. (b) Comparison of the likelihood penalty ${\mathcal{L}}_{{\Omega }_{DM}{h}^2}$ paid for a prediction of SUSY dark matter $p_{{\Omega }_{DM}{h}^2}$ with (“extra DM allowed”) and without (“no extra DM allowed”) the allowance of non-thermal-neutralino dark matter.

Figure 8

Likelihood maps of mSUGRA parameter space allowing for non-thermal-neutralino contributions to the dark matter relic density. The graphs show the likelihood distributions sampled from 7D parameter space and marginalized down to two. The likelihood (relative to the likelihood in the highest bin) is displayed by reference to the bar on the right-hand side of each plot.

Figure 9

Investigation on the effects of allowing for a non-thermal-neutralino component of dark matter in the branching ratios for the decay $B_s\to {\mu }^{+}{\mu }^{-}$ in mSUGRA. The current Tevatron upper limit is displayed by a vertical line.

Figure 10

Binned samples of the double Gaussian distribution $p_{dGau}(x)$. The normalization is arbitrary and has no relevance here. Part (a) uses a Metropolis-Hastings algorithm and yields a good approximation whereas part (b) uses the adaptive algorithm.

Figure 11

Binned sampling of a Cauchy distribution $1/(1+x^2)$ (shown as the smooth solid curve) truncated at $x=\pm 500$. Part (a) shows the result of direct sampling and yields a good approximation whereas part (b) uses the adaptive algorithm.