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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.

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Notes

  1. Here ‘more extreme’ can be defined in numerous ways.

  2. Strictly, some prior dependence remains due to the choice of parameter values considered possible by the prior, most often arising from the choice of scan range; however, this is the same kind of dependence that exists in a frequentist analysis. As well as this there exists the possibility that d 1 strictly forbids certain values of θ, and these too should be excluded from the computation of \(P(d_{2}|H_{i},\hat{\theta})\).

  3. The full volume of parameter space viable at this inference step, V total, is defined by the informative prior. If the likelihood function for the new data was constant in a region V and zero outside of it, then the fraction f=V/V total would be the Occam factor.

  4. I.e. in a generic event counting experiment we assume the expected number of signal events at the best-fit point to be close to zero.

  5. Note that a very small value for the evidence from learning some data implies a very large amount of information was gained about the model. This may sound like a good thing; however, it means that little was known about the model before this data arrived and so the model was not very useful for predicting what that data would be. PBFs penalise this failure; however, if the information gain was sufficiently large then the model may in fact become highly predictive about future data, and may thus fare much better in future PBF tests.

  6. The reader may protest that the SM+DM is not just a fine-tuning “puzzle”, it is a very extreme example of fine-tuning! However, this is only true if one considers it from a pre-‘electroweak data’ perspective. The SM+DM presumably suffers a very large PBF penalty for failing to predict the electroweak scale (and for this scale being observed very far from, say, the Planck scale, where a priori arguments based on the hierarchy problem may place it); however, these considerations enter before the ‘baseline’ data we choose for our inference sequence and so do not directly enter our PBFs. The complete assessment of which model best reflects reality should of course take these matters into account.

  7. Except for an early LHCb lower bound on BR(B s μμ).

  8. This is a small lie; we do compute marginalised posteriors for each update, which indeed correspond to the “informative” priors for the subsequent update. Nevertheless we do not explicitly use them in this fashion.

  9. The rest of the Standard Model parameters of course also enter explicitly, but we may reasonably consider priors over those to be statistically independent of the CMSSM parameters, such that measuring the values of these parameters results in PBFs of 1.

  10. Within the range of m h values compatible with d new, i.e. the dark sector theory is permitted to exclude values of m h which are also well excluded by d new.

  11. For a scale parameter this is the Jeffreys prior.

  12. These Δχ 2 curves are almost quadratic in logm h , implying a close to Gaussian likelihood function; however, we have digitised the most loose boundaries of the displayed curves to be conservative. As a result the likelihood function we reconstruct has a flat maximum from ∼80 GeV to ∼100 GeV.

  13. If, due to tuning arguments, we except m h to adopt a value on the largest allowed scale, rather than all scales being equally likely, then a flat prior cut off at this scale may indeed better represent this belief. The lack of sensitivity of the informative “pre-LEP” prior to this choice reflects the fact that before the “pre-LEP” update the Standard Model prediction for the Higgs mass is already quite well constrained.

  14. Aside from nested sampling and several variants of Markov Chain Monte Carlo methods, the list of techniques used to scan the CMSSM has expanded in recent years to also include genetic algorithms and neural networks [76, 77].

  15. The authors estimate these Bayes factors using both flat and log priors; here we refer to the log prior results only since we do not use flat priors. In addition, the δa μ constraint is shown to strongly drive the preference of μ>0 so if the validity of this constraint is questioned (we consider the effects of removing it in Sect. 7) then the impact of ignoring the μ<0 branch may also merit revisitation.

  16. We do this because P eff requires renormalisation group running to be evaluated, i.e. our spectrum generator needs to be run before we can evaluate P eff.

  17. In practice a larger scan volume will decrease the scan resolution and reduce the accuracy of results, so scan prior volume dependence would still exist in this indirect form.

  18. We ignore the variation of a with the predicted signal rate as it is small for small signal.

  19. Actually the CL s method is used so the limit is drawn where p=p s /(1−p b )=0.1 [106], but this correction weakens the limit so it is conservative to ignore it and in this case makes little difference anyway, given our other approximations.

  20. In Sect. 6.3 we construct the ATLAS Higgs search likelihood function using almost identical techniques, but argue that each fitted slice needs to be normalised relative to the others using the likelihood of the best-fit point on each slice. This occurs because the best-fit point of each slice lies a varying number of standard deviations from the zero signal point (zero cross section), which we know to have the same likelihood for every slice. A similar normalisation is in principle required to recover the true likelihood function computed by Xenon; however, the variance of the limit appears to be approximately Gaussian in the logarithm of the cross section, making extrapolation of the likelihood to the zero cross section point extremely unreliable. In addition, the reconstruction method we use for the ATLAS Higgs search likelihood relies on plots of the signal best fit against m X , whereas here we use a plot of the 90 % confidence limit. Performing the extraction using the limit curve requires more assumptions than a best-fit curve, so combined with the logarithmic difficulty we judge that this technique would produce poor results, and so we prefer to stick with the simpler technique described. The Xenon limit turns out to be of very minor importance to our final inferences anyway so we are not concerned with small errors in our reconstructed likelihood. In hindsight we expect that even simply applying a hard cut at the observed Xenon limit would negligibly affect our inferences.

  21. From Eq. (5), using the suggested σ s =50±8 MeV and σ l =47±9 MeV [112], with m s /m l =m s /(2(m u +m d ))=26±4 [86].

  22. We demonstrate this in Appendix A.

  23. The point was found during a wide ranging scan of the CMSSM parameter space, hence the esoteric choice of parameters.

  24. All other points in the test sample are “not excluded” and labelled as such, or “excluded” and labelled as such.

  25. The apparent ‘thinning’ of the likelihood toward higher M 0 and M 1/2 is a mere sampling artefact.

  26. We note that the new limits cut off much less than half of the posterior remaining in the “ATLAS-Higgs” dataset (shown in the last frame of Fig. 14), so the corresponding “additional” PBF is likewise much less than two.

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Acknowledgements

The authors are indebted to Sudhir Gupta and Doyoon Kim for their assistance with the calculation of Higgs boson production cross sections and decays. B.F. is thankful to Farhan Feroz for assistance with MultiNest. M.J.W. thanks Teng Jian Khoo and Ben Allanach for conversations regarding the calculation of ATLAS-based likelihoods for candidate SUSY models. This research was funded in part by the ARC Centre of Excellence for Particle Physics at the Tera-scale, and in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences Grant No. KJCX2.YW.W10. A.B. acknowledges the support of the Scottish Universities Physics Alliance. The use of Monash Sun Grid (MSG) and Edinburgh ECDF high-performance computing facilities is also gratefully acknowledged. Most numerical calculations were performed on the Australian National Computing Infrastructure (NCI) National Facility SGI XE cluster and Multi-modal Australian ScienceS Imaging and Visualisation Environment (MASSIVE) cluster.

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Correspondence to Benjamin Farmer.

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 σσ 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=μs+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
figure 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
figure 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
figure 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
figure 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
figure 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

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Balázs, C., Buckley, A., Carter, D. et al. Should we still believe in constrained supersymmetry?. Eur. Phys. J. C 73, 2563 (2013). https://doi.org/10.1140/epjc/s10052-013-2563-y

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  • DOI: https://doi.org/10.1140/epjc/s10052-013-2563-y

Keywords

  • Dark Matter
  • Higgs Mass
  • Higgs Search
  • Constrained Minimal Supersymmetric Standard Model
  • Muon Anomalous Magnetic Moment