Should we still believe in constrained supersymmetry?

Abstract

We calculate partial Bayes factors to quantify how the feasibility of the constrained minimal supersymmetric standard model (CMSSM) has changed in the light of a series of observations. This is done in the Bayesian spirit where probability reflects a degree of belief in a proposition and Bayes’ theorem tells us how to update it after acquiring new information. Our experimental baseline is the approximate knowledge that was available before LEP, and our comparison model is the Standard Model with a simple dark matter candidate. To quantify the amount by which experiments have altered our relative belief in the CMSSM since the baseline data we compute the partial Bayes factors that arise from learning in sequence the LEP Higgs constraints, the XENON100 dark matter constraints, the 2011 LHC supersymmetry search results, and the early 2012 LHC Higgs search results. We find that LEP and the LHC strongly shatter our trust in the CMSSM (with M ₀ and M _1/2 below 2 TeV), reducing its posterior odds by approximately two orders of magnitude. This reduction is largely due to substantial Occam factors induced by the LEP and LHC Higgs searches.

Appendices

Appendix A: Fast approximation to combined CLs limits for correlated likelihoods

In this appendix we offer a brief justification of the simplified method used to combine the ATLAS CL _s limits on sparticle production in our analysis. In this method an approximate combined confidence limit is obtained for a specified model point by simply taking the most powerful observed (lowest CL _s value) limit from one of several signal regions, or search channels. Our aim is to demonstrate a set of minimal conditions under which this procedure is conservative. This will be done by demonstrating conditions under which the following inequality holds:

$$ \min(CL_{s_1},CL_{s_2} ) \ge CL_{s_{1,2}} $$

(35)

where $CL_{s_{1}}$ is the value of the CL _s statistic for some signal model, derived from a dataset which we may call ‘channel 1’; $CL_{s_{2}}$ is the value of CL _s under the same signal model but derived from a correlated dataset ‘channel 2’; and $CL_{s_{1,2}}$ is the value of CL _s for this signal model derived from the full combination of the two datasets, accounting rigorously for correlations between datasets. This inequality does not hold in general, but if the experimental situation is such that it does hold, it means that the combined dataset results in a more powerful limit than either of the individual datasets alone, or conversely that considering only the most constraining of the two individual dataset limits is conservative. In the course of this exercise we will make use of the asymptotic results obtained in Ref. [105].

We remind the reader that the CL _s statistic is defined as

$$ CL_{s} = \frac{p_{s+b}}{1-p_b} $$

(36)

where p _s+b and p _b are p-values derived using the null hypotheses ‘s+b’ and ‘b’, respectively. s+b is the hypothesis that the data is generated from the nominal signal plus background model, while b supposes that the data contains background events only. In the CL _s method these p-values are computed using the likelihood ratio statistic

$$ q = -2\ln\frac{L_{s+b}}{L_b}=-2\ln\frac{L(\mu =1,\hat{\theta}(1))}{L(\mu=0,\hat{\theta}(0))}, $$

(37)

where L _s+b and L _b are the likelihoods of the ‘s+b’ and ‘b’ models, respectively. The second equality defines the background model as one which can be obtained by scaling the signal model by an appropriate ‘signal strength’ parameter μ, which is set to zero. $\hat{\theta}(1)$ and $\hat{\theta}(0)$ are the profiled values of any nuisance parameters. In the asymptotic limit (which requires sufficiently many candidate events) this statistic is given by the Wald approximation, with μ as the parameter of interest, as

$$ q = \frac{(\hat{\mu}-1)^2}{\sigma^2} - \frac{\hat {\mu}^2}{\sigma^2} = \frac{1-2\hat{\mu}}{\sigma^2}, $$

(38)

where $\hat{\mu}$ is the best fit value of μ given some dataset, and σ ² is the variance of $\hat{\mu}$ (which is normally distributed) under either the ‘s+b’ or the ‘b’ models, that is, σ takes the values σ _s+b and σ _b when the μ=1 and μ=0 models are assumed to be generating the data, respectively. Using so-called ‘Asimov’ datasets, which when observed cause $\hat{\mu}$ to adopt its true value (either 1 or 0; see Ref. [105]) we can obtain σ ² as

$$\begin{aligned} &\sigma^2 = \frac{1-2\mu'}{q_A}, \quad \mbox{so that}\quad \sigma^2_{s+b} = \frac{1}{\vert q_{A_{s+b}}\vert} \quad \mathrm{and} \end{aligned}$$

(39)

$$\begin{aligned} & \sigma^2_{b} = \frac{1}{\vert q_{A_b}\vert} \end{aligned}$$

(40)

where μ′ is the assumed true value of μ and q _A is the value of q obtained using the relevant Asimov dataset.

The asymptotic distribution f of the statistic q is normal with mean (1−2μ′)/σ ² and variance 4/σ ² so the p- in Eq. (36) can be computed by values

$$\begin{aligned} p_{s+b} = \int_{q_{\mathrm{obs}}}^{\infty} f(q|s+b) \,dq &= \ 1 - \varPhi\biggl(\frac{q_{\mathrm{obs}} + 1/\sigma^2_{s+b}}{2/\sigma_{s+b}} \biggr) \\ &= 1 - \varPhi\biggl(\frac{q_{\mathrm{obs}} - q_{A_{s+b}}}{2\sqrt {|q_{A_{s+b}}|}} \biggr) \end{aligned}$$

(41)

and

$$ \begin{aligned}[b] p_{b} = \int_{-\infty}^{q_{\mathrm{obs}}} f(q|b) \,dq &= \ \varPhi\biggl(\frac{q_{\mathrm{obs}} -1/\sigma^2_{b}}{2/\sigma _{b}} \biggr) \\ &= \varPhi\biggl(\frac{q_{\mathrm{obs}} - q_{A_b}}{2\sqrt {|q_{A_b}|}} \biggr). \end{aligned} $$

(42)

Let us now go to the case where the observed events in all channels are in accordance with the background hypothesis, such that $\hat{\mu}\sim0$. Then $q_{\mathrm{obs}}\sim q_{A_{b}}$. Furthermore, in this case the 95 % CL _s limit lies near model points which predict low signals, so we may further take σ∼σ _s+b ∼σ _b (since the distribution f under both s+b and b hypotheses will be very similar). Also note that in this limit $q_{A_{s+b}}=-q_{A_{b}}$. Our p-values can thus be simplified to

$$ p_{s+b} = 1 - \varPhi\bigl(\sqrt{|q_A|} \bigr) \quad \mathrm{and} \quad p_{b} = \varPhi(0 ) = \frac{1}{2} $$

(43)

(where we have also used the knowledge that $\mathrm {sign}(q_{A_{s+b}}) = -1$). We can thus write the inequality of Eq. (35) as:

$$\begin{aligned} &\frac{p_{s+b;1}}{1-p_{b;1}} \ge\frac{p_{s+b;1,2}}{1-p_{b;1,2}} \\ &\quad \rightarrow\quad 1 - \varPhi\bigl(\sqrt{|q_{1_{A}}|} \bigr) \ge 1 - \varPhi\bigl(\sqrt{|q_{1,2_{A}}|} \bigr) \end{aligned}$$

(44)

where we have assumed WLOG that $CL_{s_{1}}\le CL_{s_{2}}$. The function Φ(x) is monotonically increasing with x, so our inequality will hold if

$$ \vert q_{1_{A}}\vert\le\vert q_{1,2_{A}} \vert. $$

(45)

To determine when this is the case, we need to express $q_{1,2_{A}}$ in terms of the parameters describing $q_{1_{A}}$ and $q_{2_{A}}$. We can do this by obtaining the two parameter Wald expansion for the combined test statistic q _1,2 (i.e. taking a Taylor expansion of q about the best fit values of μ ₁ and μ ₂, up to second order):

$$\begin{aligned} q_{1,2}&= \frac{1}{1-\rho} \biggl(\frac{1-2\hat {\mu}_1}{\sigma_1^2} + \frac{1-2\hat{\mu}_2}{\sigma_2^2} \\ &\quad {}-2\rho\frac{1-\hat{\mu}_1-\hat{\mu}_2}{\sigma_1\sigma_2} \biggr), \end{aligned}$$

(46)

where ρ characterises linear correlations between the two channels, taking values in the domain (−1,1), and $\hat{\mu}_{1},\hat {\mu}_{2}$ and $\sigma_{1}^{2},\sigma_{2}^{2}$ are the best fit μ values and their variances, as obtained above for each individual channel. Again we use the Asimov dataset for the background hypothesis, which sets $\hat{\mu}_{1}=\hat{\mu}_{2}=0$, to find $q_{1,2_{A,b}}$:

$$ q_{1,2_{A,b}} = \frac{1}{1-\rho} \biggl( \frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2} -2\rho\frac{1}{\sigma _1\sigma_2} \biggr), $$

(47)

which, like $q_{1_{A,b}}$, is strictly positive. Using this expression together with Eq. (39) we can rewrite the inequality of Eq. (45) as

$$ \frac{1}{\sigma^2_1} \le\frac{1}{1-\rho} \biggl( \frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2} -2\rho\frac{1}{\sigma _1\sigma_2} \biggr). $$

(48)

One can readily see that Eq. (48) holds in the case ρ=0, i.e. when no correlations exist between channels. Knowing this, we may vary ρ from this point and see where the equality is achieved in order to check if the inequality may be violated. Setting the equality we solve for σ ₁, finding the two general solutions

$$ \sigma_1=\sigma_2 \bigl(\rho\pm \sqrt{(\rho-1)\rho} \bigr), $$

(49)

from which it is apparent that no real solutions exist for 0<ρ<1, while such solutions do exist for −1<ρ<0. We could convert this to a bound on the allowed values of σ ₁/σ ₂, since only the positive root solution can give a positive σ ₁, but negative correlations are not relevant for our signal regions, which are correlated due to shared events, so we are done.

We can thus conclude that if channel correlations are linear and positive, the observed event counts are not far from the expected background, the nominal signal hypothesis at the limit is small, and enough events are observed for asymptotic formulae to hold, then we can safely take the most powerful limit from among several channels as an estimate of the full combination, without overestimating the combined limit. Violations of these conditions may result in the target inequality of Eq. (35) being violated, with a particular concern being that this can occur as the observed events differ from the background expectation; however, it is difficult to determine the general conditions under which this happens. Certainly if one channel sees an excess above the background while another does not then in general the combined limit will be weaker than one obtained using only the more constraining (background-like) channel.

Nevertheless, in our special case we may be confident that our method remains approximately valid thanks to the procedure used by ATLAS to produce their official limits (in Ref. [102]), to which our approximate limits are fitted. ATLAS also do not attempt to rigorously account for the correlations between channels; they follow a similar procedure to us and, for each point in the CMSSM parameter space, take the limit from the channel with the best expected limit. We, on the other hand, take the channel with the best observed limit, which, following the discussion of this appendix, can be expected to less reliably approximate the rigorous combination.

We follow our more approximate procedure because ATLAS do not provide the expected limits on the mean signal for each channel; however, it is possible to estimate these using the asymptotic formulae discussed in this appendix, and so we use these estimates to gauge the seriousness of the difference between our method and the one used by ATLAS.

To do this a model of the likelihood in each signal region is needed. Taking the random variable to be the best fit signal strength $\hat {\mu}$, the simplest option is the normal limit of a Poisson likelihood, with standard deviation σ modified by convolution with normal signal and background systematics σ _s and σ _b . The mean and variance are then simply

$$\begin{aligned} &\mu= \frac{n-b}{s} \end{aligned}$$

(50)

$$\begin{aligned} &\sigma^2 = \bigl(\mu'\bigl(s + \sigma_s^2 \bigr) + \sigma_b^2\bigr)/s^2 \end{aligned}$$

(51)

where n=μ′s+b is the expected total number of events (and μ′=1 or 0 as before). ATLAS provide estimates of σ _b so we use these, however σ _s is not provided since it varies point to point. This variation would require a large effort to model so we simply fit a single value for σ _s for each channel, ensuring that the observed 95 % CL _s limits obtained from our simplified likelihood agree with ATLAS (we have also checked that varying this value has little effect on our results).

We then use this model likelihood to estimate the expected limits on the signal yield in each channel for each point in our training dataset, and obtain an estimate of the ATLAS combined observed limit by taking the observed CL _s value of each training data point to be the one obtained from the channel with the lowest expected CL _s value for that point (i.e. following ATLAS’s method). We find the difference between this estimate of the ATLAS limit and the one used in our analysis to be very small: of the 26491 training points there are 100 which are classified (into excluded/not excluded) differently by the two limits. We show these points in Fig. 10; they predominantly occur in a group clustered at low m ₀, and for most of them the observed strongest limit comes from R1, while we estimate that the expected strongest limit comes from R2.

Fig. 10

Classification of training data for the ATLAS 1 fb⁻¹ jets+MET search used in the main analysis. Two methods for combining the ATLAS limits for each search channel are used: the method used in this analysis uses the most constraining observed CL _s value from the set of channels at each training data point to determine its classification, while ATLAS use the observed CL _s value from the signal region with the most powerful expected exclusion. We have estimated the limit that would be obtained from the ATLAS method using asymptotic approximations for the signal likelihood. Training data model points which are excluded at 95 % CL _s by both limits are coloured red, while model points not excluded by either are coloured green. Points where conflict exists are coloured black. The official ATLAS limit is overlaid for comparison. Points are sampled from the full CMSSM parameter space as described in the text, but are projected onto the (m ₀,m _1/2) plane for visualisation (Color figure online)

Appendix B: Plots of CMSSM profile likelihoods and marginalised posteriors

This appendix contains the figures referred to in Sect. 7. We refer the reader to that section for further information.

Fig. 11

The evolution of the profile of the (log-)likelihood function from the “pre-LEP” situation (first row), to including the LEP Higgs search and XENON100 data (second row), to adding the 1 fb⁻¹ LHC sparticle searches (third row), to folding in the 2012 February Higgs search results. Contours containing 68 % and 95 % confidence regions are shown. The above results were obtained using the log prior. Results obtained using the CCR prior (not shown) show variations consistent with the different sampling density but are qualitatively similar

Fig. 12

The evolution of the profile of the (log-)likelihood function from the “pre-LEP” situation (first row), to including the LEP Higgs search and XENON100 data (second row), to adding the LHC sparticle searches (third row), to folding in the 2012 February Higgs search results. Contours containing 68 % and 95 % confidence regions are shown. The above results were obtained using the log prior and have been reweighted to estimate the effect of removing the δa _μ constraint. Significant deterioration of the sampling is seen due to the shift in the preferred regions away from the originally sampled regions, however the general impact of removing the δa _μ constraint can be seen in the motion of the preferred regions upwards in the mass parameters. Of particular note is the very strong shift to high A ₀ when the ATLAS Higgs search results are imposed, which is much less pronounced in Fig. 11, indicating very strong tension between the ATLAS Higgs search results and the δa _μ constraint. Results obtained using the CCR prior (not shown) show variations consistent with the different sampling density but are qualitatively similar

Fig. 13

The evolution of the CMSSM marginalised posterior probability distributions from the “pre-LEP” situation (first row), to including the LEP Higgs search and XENON100 data (second row), to adding the LHC sparticle searches (third row), to folding in the 2012 February Higgs search results. Log priors are used and 68 % and 95 % credible regions are shown

Fig. 14

The evolution of the CMSSM marginalised posterior probability distributions from the “pre-LEP” situation (first row), to including the LEP Higgs search and XENON100 data (second row), to adding the LHC sparticle searches (third row), to folding in the 2012 February Higgs search results. Natural (“CCR”) priors are used and 68 % and 95 % credible regions are shown. The natural prior can be seen to favour lower M ₀ and tanβ than the log prior