KKilling the cMSSM softly
Philip Bechtle a,1 , Jos´e Eliel Camargo-Molina b,2 , Klaus Desch c,1 , Herbert K. Dreiner d,1,3 ,Matthias Hamer e,4 , Michael Kr¨amer f,5 , Ben O’Leary g,6 , Werner Porod h,6 ,Bj¨orn Sarrazin i,1 , Tim Stefaniak j,7 , Mathias Uhlenbrock k,1 , Peter Wienemann l,1 Physikalisches Institut, University of Bonn, Germany Department of Astronomy and Theoretical Physics, Lund University, SE 223-62 Lund, Sweden Bethe Center for Theoretical Physics, University of Bonn, Germany Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen, Germany Institut f¨ur Theoretische Physik und Astrophysik, University of W¨urzburg, Germany Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USAAugust 26, 2015
Abstract
We investigate the constrained Minimal Super-symmetric Standard Model (cMSSM) in the light of con-straining experimental and observational data from pre-cision measurements, astrophysics, direct supersymmetrysearches at the LHC and measurements of the propertiesof the Higgs boson, by means of a global fit using the pro-gram F
ITTINO . As in previous studies, we find rather pooragreement of the best fit point with the global data. We alsoinvestigate the stability of the electro-weak vacuum in thepreferred region of parameter space around the best fit point.We find that the vacuum is metastable, with a lifetime signif-icantly longer than the age of the Universe. For the first timein a global fit of supersymmetry, we employ a consistentmethodology to evaluate the goodness-of-fit of the cMSSMin a frequentist approach by deriving p -values from largesets of toy experiments. We analyse analytically and quan-titatively the impact of the choice of the observable set onthe p -value, and in particular its dilution when confrontingthe model with a large number of barely constraining mea-surements. Finally, for the preferred sets of observables, weobtain p -values for the cMSSM below 10%, i.e. we excludethe cMSSM as a model at the 90% confidence level. a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] e e-mail: [email protected] f e-mail: [email protected] g e-mail: [email protected] h e-mail: [email protected] i e-mail: [email protected] j e-mail: [email protected] k e-mail: [email protected] l e-mail: [email protected] Supersymmetric theories [1, 2] offer a unique extensionof the external symmetries of the Standard Model (SM)with spinorial generators [3]. Due to the experimental con-straints on the supersymmetric masses, supersymmetry mustbe broken. Supersymmetry allows for the unification of theelectromagnetic, weak and strong gauge couplings [4–6].Through radiative symmetry breaking [7, 8], it allows fora dynamical connection between supersymmetry breakingand the breaking of SU(2) × U(1), and thus a connection be-tween the unification scale and the electroweak scale. Fur-thermore, supersymmetry provides a solution to the fine–tuning problem of the SM [9, 10], if at least some of the su-persymmetric particles have masses below or near the TeVscale [11]. Furthermore, in supersymmetric models with R -parity conservation [12, 13], the lightest supersymmetricparticle (LSP) is a promising candidate for the dark matterin the universe [14, 15].Of all the implementations of supersymmetry, there isone which has stood out throughout in phenomenologicaland experimental studies: The constrained Minimal Super-symmetric Standard Model (cMSSM) [16, 17]. As we willshow in this paper, albeit it is a simple model with a greatset of benefits over the SM, it has come under severe ex-perimental pressure. To explain and – for the first time – toquantify this pressure is the aim of this paper.The earliest phenomenological work on supersymme-try was performed almost 40 years ago [12, 13, 18–20] inthe framework of global supersymmetry. Due to the masssum rule [21], realistic models require local supersymme-try, or supergravity [16, 22–24]. The cMSSM is an effectiveparametrisation motivated by realistic supergravity models.Since we wish to critically investigate the viability of the a r X i v : . [ h e p - ph ] A ug cMSSM in detail here, it is maybe in order to briefly recountsome of its history.The cMSSM as we know it was first formulated in [25].However, it is based on a longer development in the con-struction of realistic supergravity models. A globally super-symmetric model with explicit soft supersymmetry break-ing [26] added by hand was first introduced in [27]. It isformulated as an SU(5) gauge theory, but is otherwise al-ready very similar to the cMSSM, as we study it at collid-ers. It was however not motivated by a fundamental super-gravity theory. A first attempt at a realistic model of sponta-neously breaking local supersymmetry and communicatingit with gravity mediation is given in [28]. At tree-level, itincluded only the soft breaking gaugino masses. The softscalar masses were generated radiatively. The soft breakingmasses for the scalars were first included in [29, 30]. Hereboth the gauge symmetry and supersymmetry are brokenspontaneously [24]. In [30] the first locally supersymmetricgrand unified model was constructed. Connecting the break-ing of SU(2) × U(1) to supersymmetry breaking was first pre-sented in [7], this included for the first time the bi- and tri-linear soft-breaking B and A terms. Radiative electroweaksymmetry breaking was given in [8]. A systematic presenta-tion of the low–energy effects of the spontaneous breakingof local supersymmetry, which is communicated to the ob-servable sector via gravity mediation is given in [31, 32].Thus all the ingredients of the cMSSM, the five pa-rameters M , M / , tan β , sgn ( µ ) , A were present and un-derstood in early 1982. Here M and M / are the com-mon scalar and gaugino masses, respectively, and A is acommon trilinear coupling, all defined at the grand unifiedscale. The ratio of the two Higgs vacuum expectation val-ues is denoted by tan β , and µ is the superpotential Higgsmass parameter. Depending on the model of supersymme-try breaking there were various relations between these pa-rameters. By the time of [25], no obvious simple model ofsupersymmetry breaking had been found, and it was moreappropriate to parametrise the possibilities for phenomeno-logical studies, in terms of these five parameters. In manypapers the minimal supergravity model (mSUGRA) is oftendeemed synonymous with the cMSSM. However, more pre-cisely mSUGRA contains an additional relation between A and M reducing the number of parameters [33].The cMSSM is a very well-motivated, realistic and con-cise supersymmetric extension of the SM. Despite the smallnumber of parameters, it can incorporate a wide range ofphenomena. To find or to exclude this model has been themajor quest for the majority of the experimental and phe-nomenological community working on supersymmetry overthe last 25 years.In a series of F ITTINO analyses [34–37] we have con-fronted the cMSSM to precision observables, including inparticular the anomalous magnetic moment of the muon, ( g − ) µ , astrophysical observations like the direct dark mat-ter detection bounds and the dark matter relic density, andcollider constraints, in particular from the LHC experiments,including the searches for supersymmetric particles and themass of the Higgs boson.Amongst the previous work on understanding thecMSSM in terms of global analyses, there are both thoseapplying frequentist statistics [38–58] and Bayesian statis-tics [59–70]. While the exact positions of the minima de-pend on the statistical interpretation, they agree on the over-all scale of the preferred parameter region.We found that the cMSSM does not provide a good de-scription of all observables. In particular, our best fit pre-dicted supersymmetric particle masses in the TeV range orabove, i.e. possibly beyond the reach of current and futureLHC searches. The precision observables like ( g − ) µ orthe branching ratio of B meson decay into muons, BR ( B s → µ µ ) , were predicted very close to their SM value, and nosignal for dark matter in direct and indirect searches was ex-pected in experiments conducted at present or in the nearfuture.According to our analyses, the Higgs sector in thecMSSM consists of a light scalar Higgs boson with SM-like properties, and heavy scalar, pseudoscalar and chargedHiggs bosons beyond the reach of current and future LHCsearches. We also found that the LHC limits on supersym-metry and the large value of the light scalar Higgs massdrives the cMSSM into a region of parameter space withlarge fine tuning. See also [71–75] on fine-tuning. We thusconcluded that the cMSSM has become rather unattrac-tive and dull, providing a bad description of experimen-tal observables like ( g − ) µ and predicting grim prospectsfor a discovery of supersymmetric particles in the near fu-ture [76].While our conclusions so far were based on a poor agree-ment of the best fit points with data, as expressed in a ratherhigh ratio of the global χ to the number of degrees of free-dom, there has been no successful quantitative evaluation ofthe ”poor agreement” in terms of a confidence level . Thus,the cMSSM could not be excluded in terms of frequentiststatistics due to the lack of appropriate methods or the nu-merical applicability.Traditionally, a hypothesis test between two alternativehypotheses, based on a likelihood ratio, would be employedfor such a task. An example for this is e.g. the search for theHiggs boson, where the SM without a kinematically acces-sible Higgs as a “null hypothesis” is compared to an alterna-tive hypothesis of a SM with a given accessible Higgs Bosonmass. However, in the case employed here, there is a signif-icant problem with this approach: The SM does not have adark matter candidate and thus is highly penalised by theobserved cold dark matter content in the universe. (It is ac-tually excluded.) Thus, the likelihood ratio test will always prefer the supersymmetric model with dark matter againstthe SM, no matter how bad the actual goodness-of-fit mightbe. Thus, in the absence of a viable null hypothesis withoutsupersymmetry, in this paper we address this question bycalculating the p -value from repeated fits to randomly gen-erated pseudo-measurements. The idea to do this has existedbefore (see e.g. [77]), but due to the very high demand inCPU power, specific techniques for the re-interpretation ofthe parameter scan had to be developed to make such a resultfeasible for the first time. In addition to the previously em-ployed observables, here we included the measured Higgsboson signal strengths in detail. We find that the observed p -value depends sensitively on the precise choice of the setof observables.The calculation of a p -value allows us to quantitativelyaddress the question, whether a non-trivial cMSSM can bedistinguished from a cMSSM which, due to the decouplingnature of SUSY, effectively resembles the SM plus genericdark matter.The paper is organised as follows. In Sec. 2 we describethe method of determining the p -value from pseudo mea-surements. The set of experimental observables included inthe fit is presented in Sec. 3. The results of various fits withdifferent sets of Higgs observables are discussed in Sec. 4.Amongst the results presented here are also predictions fordirect detection experiments of dark matter, and a first studyof the vacuum stability of the cMSSM in the full area pre-ferred by the global fit. We conclude in Sec. 5. In this section, we describe the statistical methods employedin the fit. These include the scan of the parameter space,as well as the determination of the p -value. Both are non-trivial, because of the need for O ( ) theoretically validscan points in the cMSSM parameter space, where eachpoint uses about 10 to 20 seconds of CPU time. Therefore,in this paper optimised scanning techniques are used, anda technique to re-interpret existing scans in pseudo experi-ments (or “toy studies”) is developed specifically for the taskof determining the frequentist p -value of a SUSY model forthe first time.2.1 Performing and Interpreting the Scan of the ParameterSpaceIn this section, the specific Markov Chain Monte Carlo(MCMC) method used in the scan, the figure-of-merit usedfor the sampling, and the (re-)interpretation of the cMSSMparameter points in the scan is explained. The parameter space is sampled using a MCMC methodbased on the Metropolis-Hastings algorithm [78–80]. Ateach tested point in the parameter space the model pre-dictions for all observables are calculated and compared tothe measurements. The level of agreement between predic-tions and measurements is quantified by means of a total χ ,which in this case corresponds to the “Large Set” of observ-ables introduced in Section 3.1 : χ = (cid:0) O meas − O pred (cid:1) T cov − (cid:0) O meas − O pred (cid:1) + χ , (1)where O meas is the vector of measurements, O pred the cor-responding vector of predictions, cov the covariance matrixincluding theoretical uncertainties and χ the sum of all χ contributions from the employed limits, i.e. the quantitiesfor which bounds, but no measurements are applied.After the calculation of the total χ at the n th point inthe Markov chain, a new point is determined by throwinga random number according to a probability density calledproposal density. We use Gaussian proposal densities, wherefor each parameter the mean is given by the current param-eter value and the width is regularly adjusted as discussedbelow.The χ for the ( n + ) th point is then calculated and com-pared to the χ for the n th point. If the new point showsbetter or equal agreement between predictions and measure-ments, χ n + ≤ χ n , (2)it is accepted. If the ( n + ) th point shows worse agreementbetween the predictions and measurements, it is acceptedwith probability ρ = exp (cid:32) − χ n + − χ n (cid:33) , (3)and rejected with probability 1 − ρ . If the ( n + ) th pointis rejected, new parameter values are generated based on the n th point again. If the ( n + ) th point is accepted, new param-eter values are generated based on the ( n + ) th point. Sincethe primary goal of using the MCMC method is the accuratedetermination of the best fit point and a high sampling den-sity around that point in the region of ∆ χ ≤
6, while allow-ing the MCMC method to escape from local minima in the χ landscape, it is mandatory to neglect rejected points inthe progression of the Markov chain. However, the rejectedpoints may well be used in the frequentist interpretation ofthe Markov chain and for the determination of the p -value.Thus, we store them as well in order to increase the overallsampling density.An automated optimisation procedure was employed todetermine the width of the Gaussian proposal densities for Since the allowed region for all observable sets tested in Section 4differ only marginally, it does not matter significantly which observableset is chosen for the initial scan, as long as it efficiently samples therelevant parameter space each parameter for different targets of the acceptance rate ofproposed points. Since the frequentist interpretation of theMarkov chain does not make direct use of the point density,we can employ chains, where the proposal densities varyduring their evolution and in different regions of the param-eter space. We update the widths of the proposal densitiesbased on the variance of the last O ( ) accepted points inthe Markov chain. Also, different ratios of proposal densitiesto the variance of accepted points are used for chains startedin different parts of the parameter space, to optimally scanthe widely different topologies of the χ surface at differentSUSY mass scales. These differences stem from the varyingdegree of correlations between different parameters requiredto stay in agreement with the data, and from non-linearitiesbetween the parameters and observables. They are also themain reason for the excessive amount of points needed for atypical SUSY scan, as compared to more nicely behaved pa-rameter spaces. It has been ensured that a sufficient numberof statistically independent chains yield similar scan resultsover the full parameter space. For the final interpretation, allstatistically independent chains are added together.A total of 850 million valid points have been tested. Thepoint with the lowest overall χ = χ is identified as thebest fit point. In addition to the determination of the best fit point it is alsoof interest to set limits in the cMSSM parameter space. Forthe Frequentist interpretation the measure ∆ χ = χ − χ (4)is used to determine the regions of the parameter spacewhich are excluded at various confidence levels. For thisstudy the one dimensional 1 σ region ( ∆ χ <
1) and the twodimensional 2 σ region ( ∆ χ <
6) are used. In a Gaussianmodel, where all observables depend linearly on all parame-ters and where all uncertainties are Gaussian, this would cor-respond to the 1-dimensional 68% and 2-dimensional 95%Confidence Level (CL) regions. The level of observed de-viation from this pure Gaussian approximation shall be dis-cussed together with the results of the toy fits, which are anideal tool to resolve these differences.2.2 Determining the p -valueIn all previous instances of SUSY fits, no true frequentist p -value for the fit is calculated. Instead, usually the χ min / ndf iscalculated, from which for a linear model with Gaussian ob-servables a p -value can easily derived. It has been observedthat the χ min / ndf of constrained SUSY model fits such asthe cMSSM have been degrading while the direct limitson the sparticle mass scales from the LHC got stronger(see e.g. [34–36]). Thus, there is the widespread opinion that the cMSSM is obsolete. However, as the cMSSM isa highly non-linear model and the observable set includesnon-Gaussian observables, such as one-sided limits and theATLAS 0-lepton search, it is not obvious that the Gaussian χ -distribution for ndf degrees of freedom can be used tocalculate an accurate p -value for this model. Hence the mainquestion in this paper is: How obsolete is the cMSSM, ex-actly? To answer this, a machinery to re-interpret the scandescribed above had to be developed, since re-scanning theparameter space for each individual toy observable set iscomputationally prohibitive at present. Because during thisre-interpretation of the original scan a multitude of differentcMSSM points might be chosen as optima of the toy fits,such a procedure sets high demands on the scan density alsoover the entire approximate 2 to 3 sigma region around theobserved optimum.
After determining the parameter values that provide the bestcombined description of the observables suitable to con-strain the model, the question of the p -value for that modelremains: Under the assumption that the tested model at thebest fit point is the true description of nature, what is theprobability p to get a minimum χ as bad as, or worse than,the actual minimum χ ?For a set of observables with Gaussian uncertainties,this probability is calculated by means of the χ - distri-bution and is given by the regularised Gamma function, p = P (cid:16) n , χ (cid:17) . Here, n is the number of degrees of free-dom of the fit, which equals the number of observables mi-nus the number of free parameters of the model.In some cases, however, this function does not describethe true distribution of the χ . Reasons for a deviation in-clude non-linear relations between parameters and observ-ables (as evident in the cMSSM, where a strong variation ofthe observables with the parameters at low parameter scalesis observed, while complete decoupling of the observablesfrom the parameters occurs at high scales), non-Gaussianuncertainties as well as one-sided constraints, that in ad-dition might constrain the model only very weakly. Also,counting the number of relevant observables n might be non-trivial: For instance, after the discovery of the Higgs bo-son at the LHC, the limits on different Higgs masses set bythe LEP experiments are expected to contribute only veryweakly (if at all) to the total χ in a fit of the cMSSM. Thisis because the measurements at the LHC indicate that thelightest Higgs Boson has a mass significantly higher thanthe lower mass limit set by LEP. In such a situation, it is notclear how much such a one-sided limit actually is expectedto contribute to the distribution of χ values.For the above reasons, the accurate determination of the p -values for the fits presented in this paper requires the con- sideration of pseudo experiments or “toy observable sets”.Under the assumption that a particular best fit point providesan accurate description of nature, pseudo measurements aregenerated for each observable. Each pseudo measurementis based on the best fit prediction for the respective observ-able, taking into account both the absolute uncertainty onthe best fit point, as well as the shape of the underlyingprobability density function. For one unique set of pseudomeasurements, the fit is repeated, and a new best fit point isdetermined with a new minimum χ , i .This procedure is repeated n toy times, and the number n p of fits using pseudo measurements with χ , i ≥ χ isdetermined. The p -value is then given by the fraction p = n p n toy . (5)This procedure requires a considerable amount of CPU time;the number of sets of pseudo measurements is thus limitedand the resulting p -value is subject to a statistical uncer-tainty. Given the true p -value, p ∞ = lim n toy → ∞ p , (6) n p varies according to a binomial distribution B ( n p | p ∞ , n toy ) , which in a rough approximation givesan uncertainty of ∆ p = (cid:115) p · ( − p ) n toy (7)on the p -value. In the present fit of the cMSSM a few different classes of ob-servables have been used and the pseudo experiments havebeen generated accordingly. In this work we distinguish dif-ferent smearing procedures for the observables:a) For a Gaussian observable with best fit prediction O BFi and an absolute uncertainty σ BFi at the best fit point,pseudo measurements have been generated by throwinga random number according to the probability densityfunction P (cid:16) O toy i (cid:17) = √ πσ BFi · exp − (cid:16) O toy i − O BFi (cid:17) σ BFi . (8)b) For the measurements of the Higgs signal strengths andthe Higgs mass, the smearing has been performed bymeans of the covariance matrix at the best fit point.c) For the ATLAS 0-Lepton search [81] (see Section 3.1),the number of observed events has been smeared accord-ing to a Poisson distribution. The expectation value ofthe Poisson distribution has been generated for each toyby taking into account the nominal value and the sys-tematic uncertainty on both the background and signalexpectation at the best fit point. The systematic uncer-tainties are assumed to be Gaussian. d) The best fit point for each set of observables features alightest Higgs boson with a mass well above the LEPlimit. Assuming the best fit point, the number of ex-pected Higgs events in the LEP searches is thereforenegligible and has been ignored. For this reason, theLEP limit has been smeared directly assuming a Gaus-sian probability density function. Due to the enormous amount of CPU time needed to accu-rately sample the parameter space of the cMSSM and calcu-late a set of predictions at each point, a complete resamplingfor each set of pseudo measurements is prohibitive.For this reason the pseudo fits have been performed us-ing only the points included in the original Markov chain,for which all necessary predictions have been calculated inthe original scan.In addition, an upper cut on the χ (calculated with re-spect to the real measurements) of ∆ χ ≤
15 has been ap-plied to further reduce CPU time consumption. The cut ismotivated by the fact, that in order to find a toy best fitpoint that far from the original minimum, the outcome of thepseudo measurements would have to be extremely unlikely.While this may potentially prevent a pseudo fit from findingthe true minimum, tests with completely Gaussian toy mod-els have shown that the resulting χ distributions perfectlymatch the expected χ distribution for all tested numbers ofdegrees of freedom.As will be shown in Section 4.3, in general we observea trend towards less pseudo data fits with high χ valuesin the upper tail of the distribution than expected from thenaive gaussian case. This further justifies that the ∆ χ ≤ p -value calculated using the describedprocedure may be regarded as conservative in the sense thatthe true p -value may very well be even lower. Hence, if it isfound below a certain threshold of e.g. 5%, it is not expectedthat there is a bias that the true p -value for infinite statisticsis found at larger values. If for a particular toy fit the truebest fit point is not included in the original Markov chain,the minimum χ for that pseudo fit will be larger than thetrue minimum for that pseudo fit, which artificially increasesthe p -value. The parameters of the cMSSM are constrained by preci-sion observables, like ( g − ) µ , astrophysical observationsincluding in particular direct dark matter detection limitsand the dark matter relic density, by collider searches forsupersymmetric particles and by the properties of the Higgs Table 1
Precision observables used in the fit. a µ − a SM µ ( . ± . ) × − [82, 83]sin θ eff . ± . m t ( . ± . ± . ) GeV [85] m W ( . ± . ) GeV [86] ∆ m s ( . ± . ± . ) ps − [87] B ( B s → µµ ) ( . ± . ) × − [88] B ( b → s γ ) ( . ± . ± . ) × − [89] B ( B → τν ) ( . ± . ) × − [87] Table 2
Standard Model parameters that have been fixed. Please notethat m b and m c are MS masses at their respective mass scale, while forall other particles on-shell masses are used.1 / α em .
952 [83] G F ( . × − ) GeV − [87] α s . m Z . m b .
18 GeV [87] m τ . m c .
275 GeV [87] boson. In this section we describe the observables that enterour fits. The measurements are given in Section 3.1 whilethe codes used to obtain the corresponding model predic-tions are described in Section 3.2.3.1 Measurements and exclusion limitsWe employ the same set of precision observables as in ourprevious analysis Ref. [36], but with updated measurementsas listed in Tab. 1. They include the anomalous magneticmoment of the muon ( g − ) µ ≡ a µ , the effective weak mix-ing angle sin θ eff , the masses of the top quark and W boson,the B s oscillation frequency ∆ m s , as well as the branchingratios B ( B s → µ µ ) , B ( B → τν ) , and B ( b → s γ ) . The Stan-dard Model parameters that have been fixed are collected inTab. 2. Note that the top quark mass m t is used both as anobservable, as well as a floating parameter in the fit, since ithas a significant correlation especially with the light Higgsboson mass.Dark matter is provided by the lightest supersymmetricparticle, which we require to be solely made up of the neu-tralino. We use the dark matter relic density Ω h = . ± . m ˜ χ ± > . χ extension provided by H IGGS -B OUNDS
IGGS S IGNALS IG - GS S IGNALS is a general tool which allows the test of anymodel with Higgs-like particles against the measurementsat the LHC and the Tevatron. Therefore, its default observ-able set of Higgs rate measurements is very extensive. Asis discussed in detail in Section 4.3, this provides maximalflexibility and sensitivity on the constraints of the allowedparameter ranges, but is not necessarily ideally tailored forgoodness-of-fit tests. There, it is important to combine ob-servables which the model on theoretical grounds cannotvary independently. In order to take this effect into account,in our analysis we compare five different Higgs observablesets:
Set 1 (Large Observable Set):
This set is the default ob-servable set provided with H
IGGS S IGNALS h → γγ and h → ZZ ( ∗ ) → (cid:96) channels. Itis used as a cross-check for the derived observable setsdescribed below. Set 2 (Medium Observable Set):
This set contains 10 in-clusive rate measurements, performed in the channels h → γγ , h → ZZ , h → WW , V h → V ( bb ) ( V = W , Z ),and h → ττ by ATLAS and CMS, listed in Tab. 3. As in Set 1 , 4 Higgs mass measurements are included.
Set 3 (Small Observable Set):
In this set, the h → γγ , h → ZZ and h → WW channels are combined toa measurement of a universal signal rate, denoted h → γγ , ZZ , WW in the following. Together with the V h → V ( bb ) and h → ττ from Set 2 , we have in total6 rate measurements. Furthermore, in each LHC exper-iment the Higgs mass measurements are combined. Theobservables are listed in Tab. 4.
Table 3
Higgs boson mass and rate observables of
Set 2 (Medium Ob-servable Set).Experiment, Channel observed µ observed m h ATLAS, h → WW → (cid:96) ν (cid:96) ν [100] 0 . + . − . -ATLAS, h → ZZ → (cid:96) [100] 1 . + . − . ( . ± . ) GeVATLAS, h → γγ [100] 1 . + . − . ( . ± . ) GeVATLAS, h → ττ [103] 1 . + . − . -ATLAS, V h → V ( bb ) [104] 0 . + . − . -CMS, h → WW → (cid:96) ν (cid:96) ν [105] 0 . + . − . -CMS, h → ZZ → (cid:96) [101] 0 . + . − . ( . ± . ) GeVCMS, h → γγ [102] 0 . + . − . ( . ± . ) GeVCMS, h → ττ [106] 0 . + . − . -CMS, V h → V ( bb ) [106] 1 . + . − . - Set 4 (Combined Observable Set):
In this set we furtherreduce the number of Higgs observables by combin-ing the ATLAS and CMS measurements for the Higgsdecays to electroweak vector bosons ( V = W , Z ), pho-tons, b -quarks and τ -leptons. These combinations areperformed by fitting a universal rate scale factor µ tothe relevant observables from within Set 1 . Furthermore,we perform a combined fit to the Higgs mass observ-ables of
Set 1 , yielding m h = ( . ± . ) GeV. Theobservables of this set are listed in Tab. 5.
Set 5 (Higgs mass only):
Here, we do not use any Higgssignal rate measurements. We only use one combinedHiggs mass observable, which in our case is m h =( . ± . ) GeV (see above).3.2 Model predictionsWe use the following public codes to calculate the predic-tions for the relevant observables: The spectrum is calcu-lated with SP
HENO Q = √ m ˜ t m ˜ t . At this scale the complete one-loop correc-tions to all masses of supersymmetric particles are calcu-lated to obtain the on-shell masses from the underlying DRparameters [111]. A measure of the theory uncertainty dueto the missing higher-order corrections is given by varying Note that the computing time needed for creating the pseudo-data fitspresented in Section 4.3 means that the fits were starting to be per-formed significantly before the combined measurement of the Higgsboson mass m h comb = . ± .
21 GeV by the ATLAS and CMS col-laborations was published [99]. We therefore performed our own com-bination, based on earlier results as published in [100–102]. Given theapplied theory uncertainty on the Higgs mass prediction of ∆ m h theo = .
64 GeV in the Higgs boson mass has a very smalleffect of ∆ χ ≈ O ( . / ) = . Table 4
Higgs boson mass and rate observables of
Set 3 (Small Ob-servable Set).Experiment, Channel observed µ observed m h ATLAS, h → WW , ZZ , γγ [100] 1 . + . − . ( . ± . ) GeVATLAS, h → ττ [103] 1 . + . − . -ATLAS, V h → V ( bb ) [104] 0 . + . − . -CMS, h → WW , ZZ , γγ † . + . − . ( . ± . ) GeVCMS, h → ττ [106] 0 . + . − . -CMS, V h → V ( bb ) [106] 1 . + . − . - † The combination of the CMS h → WW , ZZ , γγ channels has been per-formed with H IGGS S IGNALS using results from Ref. [101,105,107].The combined mass measurement for CMS is taken from Ref. [102].
Table 5
Higgs boson mass and rate observables of
Set 4 (CombinedObservable Set).Experiment, Channel observed µ observed m h ATLAS+CMS, h → WW , ZZ . + . − . ( . ± . ) GeVATLAS+CMS, h → γγ . + . − . ATLAS+CMS, h → ττ . + . − . -ATLAS+CMS, V h , tth → bb . + . − . - the scale Q between Q / Q . We find that the uncer-tainty on the strongly interacting particles is about 1-2%,whereas for the electroweakly interacting particles it is oforder a few per mille [109].Properties of the Higgs bosons as well as a µ , ∆ m s , sin θ eff and m W are obtained with F EYN H IGGS
EYN H IGGS U - PER I SO HENO agreewithin the theoretical uncertainties, see also [118] for a com-parison with other codes.For the astrophysical observables we use M I - CR OMEGA S ARK
SUSY 5.0.5 [121] via theinterface program A
STRO F IT [122] for the direct detectioncross section.For the calculation of the expected number of signalevents in the ATLAS jets plus missing transverse momen-tum search, we use the Monte Carlo event generator H ER - WIG ++ [123] and a carefully tuned and validated versionof the fast parametric detector simulation D
ELPHES [124].For tan β =
10 and A =
0, a fine grid has been producedin M and M / . In addition, several smaller, coarse gridshave been defined in A and tan β for fixed values of M and Fig. 1
Comparison of the emulation of the ATLAS 0-Lepton searchwith the published ATLAS result. In red dots we show the Atlas me-dian expected limit; the red lines denote the corresponding 1 σ uncer-tainty. The central black line is the result of the F ITTINO implemen-tation described in the text. The upper and lower black curves are thecorresponding 1 σ uncertainty. The yellow dots are the observed Atlaslimit. M / along the expected ATLAS exclusion curve to correctthe signal expectation for arbitrary values of A and tan β .We assume a systematic uncertainty of 10% on the expectednumber of signal events. In Fig. 1 we compare the expectedand observed limit as published by the ATLAS collaborationto the result of our emulation. The figure shows that the pro-cedure works reasonably well and is able to reproduce withsufficient precision the expected exclusion curve, includingthe ± σ variations.We reweight the events depending on their productionchannel according to NLO cross sections obtained fromP ROSPINO [125–127]. Renormalisation and factorisationscales have been chosen such that the NLO+NLL resummedcross section normalisations [128–132] are reproduced forsquark and gluino production.For all predictions we take theoretical uncertainties intoaccount, most of which are parameter dependent and reeval-uated at every point in the MCMC. For the precision ob-servables, they are given in Tab. 6. For the dark matter relicdensity we assume a theoretical uncertainty of 10%, for theneutralino-nucleon cross section entering the direct detec-tion limits we assign 50% uncertainty (see Ref. [36] for adiscussion of this uncertainty arinsing from pion-nuclueonform factors), for the Higgs boson mass prediction 2 . Table 6
Theoretical uncertainties of the precision observables used inthe fit. a µ − a SM µ θ eff . m t m W . ∆ m s B ( B s → µµ ) B ( b → s γ ) B ( B → τν ) always numerically implemented in the same way, or thatSM loop contributions might be missing from the SUSYcalculation. In most cases these differences are within thetheory uncertainty, or can be used to estimate those. Onesuch case of interest for this fit occurs in the program F EYN -H IGGS , which does not exactly reproduce the SM Higgs de-coupling limit [134] as used by the LHC Higgs cross-sectionworking group [133]. To compensate this, we rescale theHiggs production cross sections and partial widths of theSM-like Higgs boson. We determine the scaling factors bythe following procedure [134]: We fix tan β =
10. We setall mass parameters of the MSSM (including the param-eters µ and m A of the Higgs sector) to a common value m SUSY . We require all sfermion mixing parameters A f to beequal. We vary them by varying the ratio X t / m SUSY , where X t = A t − µ / tan β . The mass of the Higgs boson becomesmaximal for values of this ratio of about ±
2. We scan theratio between these values. In this way we find for each m SUSY two parameter points which give a Higgs boson massof about 125.5 GeV. One of these has negative X t , the otherpositive X t . We then determine the scaling factor by requir-ing that for m SUSY = X t the productioncross sections and partial widths of the SM-like Higgs bo-son are the same as for a SM Higgs boson with the samemass of 125.5 GeV. We then determine the uncertainty onthis scale factor by comparing the result with scale factorswhich we would have gotten by choosing m SUSY = m SUSY = X t . This additional un-certainty is taking into account in the χ computation. Bythis procedure we derive scale factors between 0 .
95 and 1 . . In Section 4.1, we show results based on the simplistic andcommon profile likelihood technique, which all frequentistfits, including us, have hitherto been employing. In Sec-tion 4.2 a full scan of the allowed parameter space for astable vacuum is shown, before moving on to novel resultsfrom toy fits in Section 4.3. (GeV) M0 2000 4000 6000 8000 10000 ( G e V ) / M
2D 95% CL 1D 68% CL best fit (a) 1 σ and 2 σ contour plot in in the ( M , M / )–plane for theSmall Observable Set. (GeV) M0 2000 4000 6000 8000 10000 ( G e V ) / M
2D 95% CL 1D 68% CL best fit (b) 1 σ and 2 σ contour plot in in the ( M , M / )–plane for theMedium Observable Set. (GeV) M0 2000 4000 6000 8000 10000 ( G e V ) / M
2D 95% CL 1D 68% CL best fit (c) 1 σ and 2 σ contour plot in the ( M , M / )–plane for theLarge Observable Set. (GeV) A 10000 5000 0 5000 10000 β t an
2D 95% CL 1D 68% CL best fit (d) 1 σ and 2 σ contour plot in the ( A , tan β )–plane for theMedium Observable Set. Fig. 2 σ and 2 σ contour plots for different projections and different observable sets. It can be seen that the preferred parameter region does notdepend on the specific observable set. ev-ery cMSSM parameter point in the scan, we rely here onthe profile likelihood technique. This means, we show vari-ous projections of the 1D-1 σ /1D-2 σ /2D-2 σ regions definedas regions which satisfy ∆ χ < / / .
99 respectively. Inthis context, profile likelihood means that out of the 5 phys-ical parameters in the scan, the parameters not shown in aplot are profiled, i.e. a scan over the hidden dimensions isperformed and for each selected visible parameter point thelowest χ value in the hidden dimensions is chosen. Ob-viously, no systematics nuisance parameters are involved,since all systematic uncertainties are given by relative orabsolute Gaussian uncertainties, as discussed in Section 2.One should keep in mind that this correspondence is actu- ally only exact when the observed distribution of χ valuesin a set of toy fits is truly χ -distributed, which – as dis-cussed in Section 4.3 – is not the case. Nevertheless, sincethe exact method is not computationally feasible, this stan-dard method, as used in the literature in all previous frequen-tist results, gives a reasonable estimation of the allowed pa-rameter space. In Section 4.3 more comparisons between thesets of toy fit results and the profile likelihood result will bediscussed.Note that for the discussion in this and the next section,we treat the region around the best fit point as “allowed”,even though, depending on the observable set, an exclusionof the complete model will be discussed in Section 4.3.All five Higgs input parameterisations introduced inSection 3 lead to very similar results when interpreted withthe profile likelihood technique. As an example, Figures 2(a)- 2(c) show the ( M , M / )–projection of the best fit point,1D-1 σ and 2D-2 σ regions for the Small, the Large and theMedium Observable Set. Thus, in the remainder of this sec- Table 7
Central values and 1 σ uncertainties of the free model param-eters at the global and secondary minimum when using the MediumObservable SetParameter global minimum secondary minimum M (GeV) 387 . + . − . . + . − . M / (GeV) 918 . + . − . . + . − . A (GeV) − . + . − . . + . − . tan β . + . − . . + . − . m t (GeV) 174 . + . − . . + . − . (GeV) M0 2000 4000 6000 8000 10000 ( G e V ) / M Focus Point region Coannihilation region τ∼ Coannihilation region ± χ∼ A Funnel region region σ
1D 1
Fig. 3 σ region in the ( M , M / )–plane for the Medium ObservableSet. Regions with different dark matter annihilation mechanisms areindicated. The enclosed red areas denote the best fit regions shown inFig. 2(b). tion, we concentrate on results from the Medium ObservableSet.The ( M , M / )–projection in Fig. 2(b) shows two dis-joint regions. While in the region of the global χ minimum,values of less than 900 GeV for M and less than 1300 GeVfor M / are preferred at 1 σ , in the region of the secondaryminimum values of more than 7900 GeV for M and morethan 2100 GeV for M / are favored (see also Tab. 7).The different regions are characterised by different dom-inant dark matter annihilation mechanisms as shown inFig. 3. Here we define the different regions similarly toRef. [46] by the following kinematical conditions, which wehave adapted such that each point of the 2D-2 σ region be-longs to at least one region:- ˜ τ coannihilation: m ˜ τ / m ˜ χ − < .
15- ˜ t coannihilation: m ˜ t / m ˜ χ − < .
2- ˜ χ ± coannihilation: m ˜ χ ± / m ˜ χ − < . A / H funnel: | m A / m ˜ χ − | < .
2- focus point region: | µ / m ˜ χ − | < . σ region belongs either to the ˜ τ coannihilation or the focus point region. Additionally a subset of the pointsin the ˜ τ coannihilation region fulfills the criterion for the A / H -funnel, while some points of the focus point regionfulfill the criterion for χ ± coannihilation. No point in thepreferred 2D-2 σ region fulfills the criterion for ˜ t coannihi-lation, due to relatively large stop masses.At large M and low M / a thin strip of our preferred2D-2 σ region is excluded at 95% confidence level by AT-LAS jets plus missing transverse momentum searches re-quiring exactly one lepton [135] or at most one lepton andb-jets [136] in the final state. Therefore an inclusion of theseresults in the fit is expected to remove this small part of thefocus point region without changing any conclusion.Also note that the parameter space for values of M larger than 10 TeV was not scanned such that the preferred2D-2 σ focus point region is cut off at this value. Becausethe decoupling limit has already been reached at these largemass scales we do not expect significant changes in the pre-dicted observable values when going to larger values of M .Hence we also expect the 1D-1 σ region to extend to largervalues of M than visible in Fig. 2(b) due to a low samplingdensity directly at the 10 TeV boundary. For the same reasonthis cut is not expected to influence the result of the p -valuecalculation. If it does it would only lead to an overestimationof the p -value.In the ˜ τ coannihilation region negative values of A be-tween − − β between 6 and 35 are preferred, while in the focuspoint region large positive values of A above 3400 GeVand large values of tan β above 36 are favored. This can beseen in the ( A , tan β )–projection shown in Fig. 2(d) and inTab. 7.While the ˜ τ coannihilation region predicts a spin inde-pendent dark matter–nucleon scattering cross section whichis well below the limit set by the LUX experiment, this mea-surement has a significant impact on parts of the focus pointregion for lightest supersymmetric particle (LSP) masses be-tween 200 GeV and 1 TeV, as shown in Fig. 4. The plot alsoshows how the additional uncertainty of 50% on σ SI shiftsthe implemented limit compared to the original limit set byLUX. It can be seen that future improvements by about 2orders of magnitude in the sensitivity of direct detection ex-periments, as envisaged e.g. for the future of the XENON 1Texperiment [137], could at least significantly reduce the re-maining allowed parameter space even taking the systematicuncertainty into account, or finally discover SUSY dark mat-ter. The predicted mass spectrum of the Higgs bosons andsupersymmetric particles at the best fit point and in the one-dimensional 1 σ and 2 σ regions is shown in Fig. 5. Due tothe relatively shallow minima of the fit a wide ranges ofmasses is allowed at 2 σ for most of the particles. The massesof the coloured superpartners are predicted to lie above 1.5 (GeV) χ m ( pb ) S I σ − − − − − − best fit point1D 68% CL2D 95% CLLUX 90% CL = 1.64 LUX2 χ L L L S U SY S U SY SUSY
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Fig. 4 σ and 2D2 σ regions in m ˜ χ - σ SI for the Medium Observ-able Set. The LUX exclusion is shown in addition. The dashed lineindicates a χ contribution of 1.64 which corresponds to a 90% CLupper limit. This line does not match the LUX exclusion line becausewe use a theory uncertainty of 50% on σ SI . h A H + H χ χ χ χ χ χ R l~ L l~ τ∼ τ∼ R q~ L q~ b~ b~ t~ t~ g~ pa r t i c l e m a ss ( G e V ) environment σ
1 environment σ Fig. 5
The Higgs and supersymmetric particle mass spectrum as pre-dicted by our fit using the Medium set of Higgs observables.
TeV, but due to the focus point region also masses above 10TeV are allowed at 1 σ . Similarly the heavy Higgs bosonshave masses of about 1.5 TeV at the best fit point, but massesof about 6 TeV are preferred in the focus point region. Thesleptons, neutralinos and charginos on the other hand canstill have masses of a few hundred GeV.A lightest Higgs boson with a mass as measured by theATLAS and CMS collaborations can easily be accommo-dated, as shown in Fig. 6. As required by the signal strengthmeasurements, it is predicted to be SM-like. Fig. 7 shows acomparison of the Higgs production cross sections for dif-ferent production mechanisms in p-p collisions at a centre-of-mass energy of 14 TeV. These contain gluon-fusion (ggh),vector boson fusion (qqh), associated production (Wh, Zh),and production in assiciation with heavy quark flavours(bbh, tth). Compared to the SM prediction, the cMSSM pre- (GeV) h m118 120 122 124 126 128 130 γγ → CMS, h 4l → ZZ → CMS, h γγ → ATL, h 4l → ZZ → ATL, h best fit valuedata68 % CL95 % CL L L L S U S Y S U SY SUSY
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Fig. 6
Our predicted mass of the light Higgs boson, together with the1 σ and 2 σ ranges. The LHC measurements used in the fit are shown aswell. Note that the correlated theory uncertainty of ∆ m h theo = ∆ m h fit = . dicts a slightly smaller cross section in all channels exceptthe bbh channel. The predicted signal strengths µ in the dif-ferent final states for p-p collisions at a centre-of-mass en-ergy of 8 TeV is also slightly smaller than the SM prediction,as shown in Fig. 8. The current precision of these measure-ments does, however, not allow for a discrimination betweenthe SM and the cMSSM based solely on measurements ofHiggs boson properties.4.2 Vacuum stabilityThe scalar sector of the SM consists of just one complexHiggs doublet. In the cMSSM the scalar sector is dramat-ically expanded with an extra complex Higgs doublet, aswell as the sfermions ˜ e L , R , ˜ ν L , ˜ u L , R , ˜ d L , R of the first family,and correspondingly of the second and third families. Thusthere are 25 complex scalar fields. The corresponding com-plete scalar potential V cMSSM is fixed by the five parameters: ( M , M / , A , tan β , sgn ( µ )) . The minimal potential energyof the vacuum is obtained for constant scalar field valueseverywhere. Given a fixed set of these cMSSM parameters,it is a computational question to determine the minimum of V cMSSM . Ideally this minimum should lead to a Higgs vac-uum expectation such that SU(2) L × U(1) Y → U(1) EM , as inthe SM. However, it was observed early on in supersym-metric model building, that due to the extended scalar sec-tor, some sfermions could obtain non-vanishing vacuum ex- @ 14 TeV SM σ / σ tthbbhZhWhqqhggh best fit value68 % CL95 % CL L L L S U S Y S U SY SUSY
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Fig. 7
Predicted production cross sections at 14 TeV of the light Higgsboson relative to the SM value for a Higgs boson with the same mass pectation values (vevs). There could be additional minimaof the scalar potential which would break SU(3) c and/orU(1) EM and thus colour and/or charge [7, 138–140]. If theseminima are energetically higher than the conventional elec-troweak breaking minimum, then the latter is consideredstable. If any of these minima are lower than the conven-tional minimum, our universe could tunnel into them. Ifthe tunneling time is longer than the age of the Universeof 13.8 gigayears [90], we denote our favored vacuum asmetastable, otherwise it is unstable. However, this is onlya rough categorisation. Since even if the tunneling time isshorter than the age of the universe, there is a finite proba-bility, that it will have survived until today. When computingthis probability, we set a limit of 10% survival probability.We wish to explore here the vacuum stability of the preferredparameter ranges of our fits.The recent observation of the large Higgs boson massrequires within the cMSSM large stop masses and/or alarge stop mass splitting. Together with the tuning of thelighter stau mass to favor the stau co-annihilation region(for the low M fit region), this typically drives fits to fa-vor a very large value of | A | relative to | M | , cf. Tab. 7.(For alternative non-cMSSM models with a modified stopsector, see for example [141–144].) This is exactly the re- µ Vbb → CMS, Vh ττ → CMS, h γγ → CMS, h 4l → ZZ → CMS, h ν l ν l → WW → CMS, h Vbb → ATL, Vh ττ → ATL, h γγ → ATL, h 4l → ZZ → ATL, h ν l ν l → WW → ATL, h best fit valuedata68 % CL95 % CL L L L S U S Y S U SY SUSY
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Fig. 8
Our predicted µ values of the light Higgs boson relative to theSM value for a Higgs boson with the same mass. The measurementsused in the fit are shown as well. gion, which typically suffers from the SM-like vacuum be-ing only metastable, decaying to a charge- and/or colour-breaking (CCB) minimum of the potential [145, 146].For the purpose of a fit, in principle a likelihood valuefor the compatibility of the lifetime of the SM-like vacuumof a particular parameter point with the observation of theage of the Universe should be calculated and should be im-plemented as a one-sided limit. Unfortunately, the effort re-quired to compute all the minima of the full scalar potential and to compute the decay rates for every point in the MCMC and to implement this in the likelihood function is beyondpresent capabilities [145].Effectively, whether or not a parameter point has an un-acceptably short lifetime has a binary yes/no answer. There-fore, as a first step, and in the light of the results of the possi-ble exclusion of the cMSSM in Section 4.3, we overlay ourfit result from Section 4.1 over a scan of the lifetime of thecMSSM vacuum over the complete parameter space.The systematic analysis of whether a potential has min-ima which are deeper than the desired vacuum configura-tion has been automated in the program V EVACIOUS [147].When restricting the analysis to only a most likely relevantsubset of the scalar fields of the potential, i.e. not the full (GeV) M ( G e V ) / M L L L S U S Y S U SY SUSY
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Fig. 9
Preferred 1D-1 σ and 2D-2 σ regions in M - M / for theMedium Observable Set. The filled areas contain stable points, whilethe doted lines enclose points which are metastable but still might bevery long-lived. The whole preferred 2D-2 σ focus point region leadsto a stable vacuum, while the coannihilation region contains both sta-ble and metastable points. There are no stable points in the preferred1D-1 σ coannihilation region.
25 complex scalar fields, and ignoring the calculation oflifetimes, this code runs sufficiently fast that we are ableto present an overlay of which parameter points have CCBminima deeper than the SM-like minimum in Figures 9 and10. However, only the stop and stau fields were allowed tohave non-zero values in determining the overlays, in addi-tion to the neutral components of the two complex scalarHiggs doublets. The ˜ τ L , R , ˜ t L , R are suspected to have thelargest effect [145]. The computation time when includingmore scalar fields which are allowed to vary increases expo-nentially. Thus the more detailed investigations below are re-stricted to a set of benchmark points. Note that the overlaysin Figures 9 and 10 only show whether metastable vacuamight occur at a given point, or whether the vacuum is in-stable at all. The actual lifetime is not yet considered in thisstep. See the further considerations below.There are analytical conditions in the literature forwhether MSSM parameter points could have dangerouslydeep CCB minima, see for example [7, 139, 140, 148–152].These can be checked numerically in a negligible amount ofCPU time, while performing a fit. However, these conditionshave been explicitly shown to be neither necessary nor suffi-cient [153]. In particular they have also been shown numer-ically to be neither necessary nor sufficient for the relevantregions of the cMSSM parameter space which we considerhere [145].Since the exact calculation of the lifetime of a metastableSM-like vacuum is so computationally intensive, we unfor-tunately must restrict this to just the best-fit points of thestau co-annihilation and focus point regions of our the fit, asdetermined in Section 4.1. As an indicator, though, the ex-tended ˜ τ co-annihilation region of the cMSSM investigated β tan / M A -15-10-505 2D 95% CL, stable1D 68% CL, stable2D 95% CL, metastable1D 68% CL, metastable L L L S U S Y S U SY SUSY
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Fig. 10
Preferred 1D-1 σ and 2D-2 σ regions in tan β - A / M for theMedium Observable Set. The filled areas contain stable points, whilethe doted lines enclose points which are metastable but still might bevery long-lived. Points leading to a metastable vacuum have usuallylarger negative values of A relative to M when compared to pointswith a stable vacuum at the same tan β . The part of the 1D-1 σ regionwhich belongs to the focus point region fulfills A / M ∼ A / M and is metastable. in [145] had SM-like vacuum lifetimes, which were all ac-ceptably long compared to the observed age of the Universe.The 1D1 σ best-fit points in Section 4.1 where checkedfor undesired minima, allowing, but not requiring, simulta-neously for all the following scalar fields to have non-zero,real VEVs: H d , H u , ˜ τ L , ˜ τ R , ˜ ν µ L , ˜ b L , ˜ b R , ˜ t L , ˜ t R . The focus pointregion best-fit point was found to have an absolutely stableSM-like minimum against tunneling to other minima, as nodeeper minimum of V cMSSM was found at the 1-loop level.The SM-like vacuum of the best-fit ˜ τ co-annihilation pointwas found to be metastable, with a deep CCB minimumwith non-zero stau and stop VEVs. Furthermore there wereunbounded-from-below directions with non-zero values forthe µ -sneutrino scalar field in combination with nonzerovalues for both staus, or both sbottoms, or both chiralitiesof both staus and sbottoms. This does not bode very wellfor the absolute best-fit point of our cMSSM fit. However,further effects must be considered.The parameter space of the MSSM which has directionsin field space, where the tree-level potential is unboundedfrom below was systematically investigated in Ref. [152].We confirmed the persistence of the runaway directionsat one loop with V EVACIOUS out to field values of theorder of twenty times the renormalisation scale. This isabout the limit of trustworthiness of a fixed-order, fixed-renormalisation-scale calculation [147]. However, this is notquite as alarming as it may seem. The appropriate renor-malisation scale for very large field values should be of theorder of the field values themselves, and for field values ofthe order of the GUT scale, the cMSSM soft SUSY-breaking mass-squared parameters by definition are positive. Thus thepotential at the GUT scale is bounded from below, as none ofthe conditions for unbounded-from-below directions givenin [152] can be satisfied without at least one negative mass-squared parameter. Note, even the Standard Model suffersfrom a potential which is unbounded from below at a fixedrenormalisation scale. Though in the case of the SM it onlyappears at the one-loop level. Nevertheless, RGEs show thatthe SM is bounded from below at high energies [154].Furthermore, the calculation of a tunneling time out ofa false minimum does not technically require that the Uni-verse tunnels into a deeper minimum . In fact, the state whichdominates tunneling is always a vacuum bubble, with a fieldconfiguration inside, which classically evolves to the truevacuum after quantum tunneling [155, 156]. Hence the life-time of the SM-like vacuum of the stau co-annihilation best-fit point could be calculated at one loop even though thepotential is unbounded from below at this level. The min-imal energy barrier through which the SM-like vacuum ofthis point can tunnel is associated with a final state withnon-zero values for the scalar fields H d , H u , ˜ τ L , ˜ τ R , and ˜ ν µ L .The lifetime was calculated by using the program V EVA - CIOUS through the program C
OSMO T RANSITIONS [157] tobe roughly e ∼ times the age of the Universe.Therefore, we consider the ˜ τ coannihilation region best-fitpoint as effectively stable.As well as asking whether a metastable vacuum has alifetime at least as long as the age of the Universe at zerotemperature, one can also ask whether the false vacuumwould survive a high-temperature period in the early Uni-verse. Such a calculation has been incorporated into V EVA - CIOUS [158]. In addition to the fact that the running of theLagrangian parameters ensures that the potential is boundedfrom below at the GUT scale, the effects of non-zero tem-perature serve to bound the potential from below, as well. Infact the CCB minima of V cMSSM evaluated at the parametersof the stau co-annihilation best-fit point are no longer deeperthan the configuration with all zero VEVs, which is assumedto evolve into the SM-like minimum as the Universe cools,for temperatures over about 2300 GeV. The probability oftunneling into the CCB state integrated over temperaturesfrom 2300 GeV down to 0 GeV was calculated to be roughlyexp ( e − ) . So while having a non-zero-temperature de-cay width about e − / e − = e + times larger thanthe zero-temperature decay width, the SM-like vacuum, orits high-temperature precursor, of the stau co-annihilationbest-fit point has a decay probability which is still utterlyinsignificant.4.3 Toy based resultsPseudo datasets have been generated for a total of 7 differ-ent minima based on 6 different observable sets. For the Medium, Small and Combined Observable Sets, roughly1000 sets of pseudo measurements have been taken into ac-count, as well as for the observable set without the Higgsrates. For the Medium Observable Set, in addition to thebest fit point, we also study the p -value of the local mini-mum in the focus point region. Due to relaxed requirementson the statistical uncertainty of a p -value in the range of O ( . ) , as compared to O ( . ) , we use only 125 pseudodatasets for the Large Observable Set. Finally, to study theimportance of (g-2) µ , a total of 500 pseudo datasets havebeen generated based on the best fit point for the MediumObservable Set without (g-2) µ . A summary of all p -valueswith their statistical uncertainties and a comparison to thenaive p -value according to the χ -distribution for Gaussiandistributed variables is shown in Tab. 8.Figure 11(a) shows the χ -distribution for the best fitpoint of the Medium Observable Set, from which we de-rive a p -value of (4 . ± . χ -distribution for the pseudo fits using the Com-bined Observable Set in figure 11(b). Both distributionsare significantly shifted towards smaller χ -values com-pared to the corresponding χ -distributions for Gaussiandistributed variables. Several observables are responsible forthe large deviation between the two distributions, as shownin Fig. 12(a), where the individual contributions of all ob-servables to the minimum χ of all pseudo best fit points areplotted.First, H IGGS B OUNDS does not contribute significantlyto the χ at any of the pseudo best-fit points, which is alsothe case for the original fit. The reason for this is, that for themajority of tested points, the χ contribution from H IGGS -B OUNDS reflects the amount of violation of the LEP limiton the lightest Higgs boson mass by the model. Since themeasurements of the Higgs mass by ATLAS and CMS liesignificantly above this limit, it is extremely unlikely thatin one of the pseudo datasets the Higgs mass is rolled suchthat the best fit point would receive a χ penalty due to theLEP limit. This effectively eliminates one degree of free-dom from the fit. In addition, the predicted masses of A , H and H ± lie in the decoupled regime of the allowed cMSSMparameter space. Thus there are no contributions from heavyHiggs or charged Higgs searches as implemented in H IGGS -B OUNDS .The same effect is observed slightly less pronounced forthe LHC and LUX limits, where the best fit points are muchcloser to the respective limits than in the case of H
IGGS -B OUNDS . Finally we observe that for each pseudo datasetthe cMSSM can very well describe the pseudo measurementof the dark matter relic density, which further reduced theeffective number of degrees of freedom.Figure 12(a) also shows that the level of disagreementbetween measurement and prediction for ( g − ) µ is smallerin each single pseudo dataset than in the original fit with the Table 8
Summary of p -valuesObservable Set χ /ndf naive p -value (%) toy p -value (%)Small 27.1/16 4.0 1 . ± . . ± . . ± . . ± . ± . ± . χ f r a c t i on s Toy Fits (NDF = 22) χ best fit point30.42 0.7) % ± p = ( 4.9 (a) Minimal χ values from toy fits using theMedium Observable Set. χ f r a c t i on s toy fits (NDF = 13) χ best fit point17.5 0.8) % ± p = ( 8.3 (b) Minimal χ values from toy fits using theCombined Observable Set. Fig. 11
Distribution of minimal χ values from toy fits using two different sets of Higgs observables. A χ distribution for Gaussian distributedvariables is shown for comparison. real dataset. The 1-dimensional distribution of the pseudobest fit values of ( g − ) µ is shown in Fig. 13(a). The figureshows that under the assumption of our best fit point, not asingle pseudo dataset would yield a prediction of ( g − ) µ that is consistent with the actual measurement. As a com-parison, Fig. 13(b) shows the 1-dimensional distribution forthe dark matter relic density, where the actual measurementcan well be accommodated in any of the pseudo best fit sce-narios. To further study the impact of ( g − ) µ on the p -value, we repeat the toy fits without this observable and geta p -value of ( ± ) %. This shows that the relatively low p -value for our baseline fit is mainly due to the incompatibil-ity of the ( g − ) µ measurement with large sparticle masses,which are however required by the LHC results.Interestingly, under the assumption that the minimum inthe focus point region is the true description of nature, weget a slightly better p -value (7.8%) than we get with the ac-tual best fit point. Figure 12(b) shows the individual contri-butions to the pseudo best fit χ at the pseudo best fit pointsfor the toy fits performed around the local minimum in thefocus point region. There are two variables with higher av-erage contributions compared to the global minimum: m top and the LHC SUSY search. In particular for the LHC SUSYsearch, the LHC contribution to the total χ is, on average,significantly higher than for the pseudo best fit points for theglobal minimum. The number of expected signal events for the minimum in the focus point region is 0, while it is > τ -coannihilation region, tend to predict an ex-pected number of signal events larger than zero. Since forthe pseudo measurements based on the minimum in the fo-cus point region an expectation of 0 is assumed, this natu-rally leads to a larger χ contribution from the ATLAS 0- (cid:96) analysis. The effect on the distribution of the total χ isshown in figure 14. Another reason might be that the fo-cus point region is sampled more coarsely than the regionaround the global minimum. This increases the probabilitythat the fit of the pseudo dataset misses the actual best fitpoint, due to our procedure of using only the points in theoriginal MCMC. This effect should however only play a mi-nor role, since the parameter space is still finely scanned andonly a negligible fraction of scan points are chosen numer-ous times as best fit points in the pseudo data fits.To further verify that this effect is not only caused by thecoarser sampling in the focus point region, we performed an-other set of 500 pseudo fits based on the global minimum, re-ducing the point density in the ˜ τ -coannihilation region suchthat it corresponds to the point density around the local min-imum in the focus point region. We find that the resulting χ distribution is slightly shifted with respect to the χ distri-bution we get from the full MCMC. The shift is, however, χ contribution to 0 5 10 15 20 25 30 DM Ω LHC SUSY Search µ (g 2) top m W m ) γ s → BR ( b ) µ µ → s BR ( B ) ν τ → BR ( B LUX) eff θ ( sin s B m ∆ HiggsBounds h mHiggs signal strengths full rangelocal 95% Intervallocal 68% Intervaloriginal best fit point (a) Toy fits smeared around the global minimum in the˜ τ -coannihilation region. χ contribution to 0 5 10 15 20 25 30 DM Ω LHC SUSY Search µ (g 2) top m W m ) γ s → BR ( b ) µ µ → s BR ( B ) ν τ → BR ( B LUX) eff θ ( sin s B m ∆ HiggsBounds h mHiggs signal strengths full rangelocal 95% Intervallocal 68% Intervaloriginal best fit point (b) Toy fits smeared around the local minimum in the focus pointregion. Fig. 12
Individual χ contributions of all observables/observable sets at the best fit points of the toy fits using the Medium Set of Higgs observableswith observables smeared around the global and the local minimum of the observed χ contour. The predicted measurements at the best fit points ofthe individual pseudo data fits are used to derive the local CL intervals shown in the plots. These are compared with the individual χ contributionof each observable at the global or local minimum. Note the different scale shown on the top which is used for HiggsSignals, which contains 14observables. Also note that m h contains contributions from 4 measurements for this observable set. SM µ - a exp µ a-2 -1 0 1 2 3 4 -9 × f r a c t i on s toy fits σ σ -10 × -11 × + 8.8 -10 × - 3.5 cMSSM data -9 × ± (2.87 (a) Anomalous magnetic moment of the muon. h CDM Ω f r a c t i on s toy fits σ σ cMSSM data 0.012 ± (b) Dark matter relic density. Fig. 13
Distribution of the predictions of the best fit points of the pseudo data fits for two different observables used in the fit, compared with therespective measurements. too small to explain the difference between p -values we findfor the global minimum and the local minimum in the focuspoint region.As an additional test, we investigate a simple toy modelwith only Gaussian observables and a single one-sided limitcorresponding to the LHC SUSY search we use in our fitof the cMSSM. Also in this very simple model we find sig-nificantly different χ distributions for fits based on pointsin a region with/without a significant signal expectation forthe counting experiment. We thus conclude that the true p -value for the local minimum in the focus point region is infact higher than the true p -value for the global minimum ofour fit.In order to ensure that there are no more points with ahigher χ and a higher p -value than the local minimum in the focus point region, we generate 200 pseudo datasets fortwo more points in the focus point region. The two pointsare the points with the highest/lowest M in the local 1 σ region around the focus point minimum. We find that the χ distributions we get from these pseudo datasets are in goodagreement with each other and also with the χ distributionderived from the pseudo experiments around the focus pointminimum, and hence conclude that the local minimum inthe focus point region is the point with the highest p -valuein the cMSSM.To study the impact of the Higgs rates on the p -value,and in order to compare to the observable sets used by otherfit collaborations, which exclude the Higgs rate measure-ments from the fit on the basis that in the decoupling regime they do not play a vital role, we perform toy fits for the χ nu m be r o f p s eudo be s t f i t po i n t s global minimumfocus point minimum Fig. 14
Comparison of the χ distributions obtained from toy fits usingthe global minimum and the local minimum in the focus point regionof the Medium Observable Set. observable set without Higgs rates and derive a p -value of ( . ± . ) %. In the decoupling limit, the cMSSM predic-tions for the Higgs rates are very close to the SM, so thatthe LHC is not able to distinguish between the two modelsbased on Higgs rates measurements (see Fig. 8). Becauseof the overall good agreement between the Higgs rate mea-surements and the SM prediction, the inclusion of the Higgsrates in the fit improves the fit quality despite some tensionbetween the ATLAS and CMS measurements.As discussed in Section 2, it is important to understandthe impact of the parametrisation of the measurements onthe p -value. To do so, we compare our baseline fit with twomore extreme choices. First, we use the Small ObservableSet which combines h → γγ , h → ZZ , and h → WW mea-surements but keeps ATLAS and CMS measurements sepa-rately. We use this choice because an official ATLAS combi-nation is available. The equivalent corresponding CMS com-bination is produced independently by us. Using this observ-able set we get a p -value of ( . ± . ) %. Here the cMSSMreceives a χ penalty from the trend of the ATLAS signalstrength measurements to values µ ≥ µ ≤ h → VV channels.As a cross-check, we employ the Large Observable Set,which contains all available sub-channel measurements sep-arately. Using this observable set, we get a p -value fromthe pseudo data fits of ( . ± . ) %. As observed in Sec-tion 4.1, the Large Observable Set yields the same preferredparameter region as the Small, Medium and Combined Ob-servable sets. Yet, its p -value from the pseudo data fits sig-nificantly differs.To explain this interesting result we consider a simplifiedexample: For i = , . . . , N , let x i be Gaussian measurementswith uncertainties σ i and corresponding model predictions a i ( P ) for a given parameter point P . We assume that themeasurements from x n to x N are uncombined measurements of the same observable; then a i = a n for all i ≥ n . There arenow two obvious possibilities to compare measurements andpredictions: – We can compare each of the individual measurementswith the corresponding model predictions by calculating χ = N ∑ i = (cid:18) x i − a i σ i (cid:19) . This would correspond to an approach where the modelis confronted with all avaialable observables, irrespec-tive of the question whether they measure independentquantities in the model or not. One example for such asituation would be the Large Observable Set of Higgssignal strength measurements, where several observ-ables measure different detector effects, but the samephysics. – We can first combine the measurements x i , i ≥ n to ameasurement ¯ x which minimises the function f ( x ) = N ∑ i = n (cid:18) x i − x σ i (cid:19) (9)and has an uncertainty of ¯ σ and then use this combina-tion to calculate χ = n − ∑ i = (cid:18) x i − a i σ i (cid:19) + (cid:18) ¯ x − a n ¯ σ (cid:19) . (10)This situation now corresponds to first calculating onephysically meaningful quantity (e.g. a common signalstrength for h → γγ in all VBF categories, and all gg → h categories) and only then to confront the model to thecombined measurement.Plugging in the explicit expressions for ( ¯ x , ¯ σ ), using 1 / ¯ σ = ∑ Ni = n ( / σ i ) and defining χ = f ( ¯ x ) one finds χ = χ − χ . (11)Hence doing the combination of the measurements be-fore the fit is equivalent to using a χ -difference which inturn is equivalent to the usage of a log-likelihood ratio. Thenumerator of this ratio is given by the likelihood L model ofthe model under study, e.g. the cMSSM. The denominatoris given by the maximum of a phenomenological likelihood L pheno which depends directly on the model predictions a i .These possess an expression as functions a i ( P ) of the modelparameters P of L model . Note that in L pheno however, the a i are treated as n independent parameters. We now identify χ ≡ − L model and χ ≡ − L pheno . When insert-ing a i ( P ) , one is guaranteed to find L pheno ( a ( P ) , . . . , a n ( P )) = L model ( P ) . (12)Using these symbols, χ can be written as χ = − L model ( P ) L pheno ( ˆ a , . . . , ˆ a n ) , (13) /ndf χ a r b i t r a r y un i t s L L L S U S Y S U SY SUSY
FITTINO
Fig. 15
Numerical example showing the distribution of χ / ndf usingcombined and split measurements. Using split measurements smallervalues of χ / ndf are obtained. Because in this example all measure-ments are Gaussian, this is equivalent to larger p -values. We call theeffect of obtaining larger p -values when using split measurements thedilution of the p -value. where ˆ a , . . . , ˆ a n maximise L pheno . Note that in this formu-lation the model predictions a i do not necessarily need tocorrespond directly to measurements used in the fit, as it isthe case for our example. For instance the model predictions a i might contain cross sections and branching ratios whichare constrained by rate measurements.Using ndf split = N , ndf combined = n and ndf data = N − n Eq. (11) implies χ ndf split = χ ndf combined + N − n + χ ndf data + n . (14)The more uncombined measurements are used, the larger N − n gets and the less the p -value depends on the first termon the right hand side, which measures the agreement be-tween data and model. At the same time, the p -value de-pends more on the second term on the right hand which mea-sures the agreement within the data. Especially, for n fixedand N → ∞ : χ ndf split = χ ndf data . (15)Since in the case of purely statistical fluctuations of thesplit measurements around the combined value the agree-ment within the data is unlikely to be poor, the expectationis χ ndf data ≈ physical combined observables. So most of thetime the p -value will get larger when uncombined measure-ments are used, hiding deviations between model and data.As a numerical example Fig. 15 shows toy distributions of χ ndf combined and χ ndf split for one observable (n=1), ten measure-ments (N=10) and a 3 σ deviation between the true value andthe model prediction. We call this effect dilution of the p-value . It explains the large p -value for the Large ObservableSet by the overall good agreement between the uncombinedmeasurements.On the other hand if there is some tension within thedata, which might in this hypothetical example be causedpurely by statistical or experimental effects, the “innocent”model is punished for these internal inconsistencies of thedata. This is observed here for the Medium Observable Setand Small Observable Set. Hence, and in order to incorpo-rate our assumption that ATLAS and CMS measured thesame Higgs boson, we produce our own combination of cor-responding ATLAS and CMS Higgs mass and rate measure-ments. We also assume that custodial symmetry is preservedbut do not assume that h → γγ is connected to h → WW and h → ZZ as in the official ATLAS combination usedin Small Observable Set. We call the resulting observableset Combined Observable Set. Note that for simplicity wealso combine channels for which the cMSSM model predic-tions might differ due to different efficiencies for the differ-ent Higgs production channels. This could be improved ina more rigorous treatment. For instance the χ could be de-fined by Eq. (13) using a likelihood L pheno which containsboth the different Higgs production cross sections and thedifferent Higgs branching ratios as free parameters a i .Using the Combined Observable Set we get a p -value of ( . ± . ) %, which is significantly smaller than the diluted p -value of ( . ± . ) % for the Large Observable Set. Thegood agreement within the data now shows up in a small χ / ndf of 68 . /
65 for the combination but no longer affectsthe p -value of the model fit. On the other hand the p -valuefor the Combined Observable Set is larger than the one forthe Medium Observable Set, because the tension betweenthe ATLAS and CMS measurements is not included. Thistension can be quantified by producing an equivalent AT-LAS and CMS combination not from the Large ObservableSet but from the Medium Observable Set giving a relativelybad χ / ndf of 10 . / M and M / . The 68% and 95% CL re-gions according to the total pseudo best fit χ are shown.As expected by the non-Gaussian behaviour of our fit, somedifferences between the results obtained by the profile like-lihood technique and the pseudo fit results can be observed.For the pseudo fits, in both parameters M and M / , the95% CL range is slightly smaller compared to the allowed [GeV] M0 2000 4000 6000 8000 10000 f r a c t i on s toy fits σ σ cMSSM (a) [GeV] M0 500 1000 1500 2000 2500 3000 3500 f r a c t i on s toy fits σ σ cMSSM (b) Fig. 16
Distribution of the pseudo best fit values for (a) M and (b) M / . range according to the profile likelihood. Considering thefits of the pseudo datasets, M is limited to values < . M / is limited to values < . . p -value would have to be evaluated atevery single point in the parameter space. In this paper we present what we consider the final analysisof the cMSSM in light of the LHC Run 1 with the programF
ITTINO .In previous iterations of such a global analysis of thecMSSM, or any other more general SUSY model, the fo-cus was set on finding regions in parameter space that glob-ally agree with a certain set of measurements, either usingfrequentist or bayesian statistics. However, these analysesshow that a constrained model such as the cMSSM has be-come rather trivial: because of the decoupling behaviour atsufficiently high SUSY mass scales the phenomenology re-sembles that of the SM with dark matter. This does how-ever not disqualify the cMSSM as a valid model of Nature.In addition, there are undeniable fine-tuning challenges, butalso these do not statistically disqualify the model. There-fore, before abandoning the cMSSM, we apply one crucialtest, which it has not been performed before: we calculatethe p -value of the cMSSM through toy tests. A likelihood ratio test of the cMSSM against the SMwould be meaningless, since the SM cannot acommodatedark matter. Thus we apply a goodness-of-fit test of thecMSSM. As in every likelihood test (also in likelihood ra-tio tests), the sensitivity of the test towards the validity ofthe model depends on the number of degrees of freedomin the observable set, while the sensitivity towards the pre-ferred parameter range does not. Thus, when calculating the p -value of the cMSSM, it is important that the observableset is chosen carefully. First, only such observables shouldbe considered for which the cMSSM predictions are, in prin-ciple, sensitive to the choice of the model parameters, inde-pendent of the actually measured values of the observables.This excludes e.g. many LEP/SLD precision observables,for which the cMSSM by construction always predicts theSM value for any parameter value. Second, it is importantthat observables are combined whenever the correspondingcMSSM predictions are not independent. Otherwise the re-sulting p -value would be too large by construction. It shouldbe noted that the allowed parameter space for all observablesets studied here is virtually identical. It is only the impacton the p -value which varies.In order to study this dependence, several observablesets are studied. The main challenge arises from the Higgsrate measurements. Since the cMSSM Higgs rate predic-tions are, in principle, very sensitive to the choice of modelparameters, the corresponding measurements have to be in-cluded in a global fit. Using the preferred observable sets“combined” and “medium” (as described in Section 4.3 andTab. 8), we calculate a p -value of the cMSSM of 4.9%and 8.4%, respectively. In addition, the cMSSM is excludedat the 98.7% CL if Higgs rate measurements are omitted.The main reason for these low p -values is the tension be-tween the direct sparticle search limits from the LHC andthe measured value of the muon anomalous magnetic mo- ment ( g − ) µ . If e.g. ( g − ) µ is removed from the fit, the p -value of the cMSSM increases to about 50%. However,there is no justification for arbitrarily removing one variable a posteriori . On the other hand, the observable sets couldbe artificially chosen to be too detailed, such that there aremany measurements for which the model predictions cannotbe varied individually. This is the case for the Large Observ-able Set of Higgs rate observables in Tab. 8, the inclusion ofwhich does thus not represent a methodologically stringenttest of the p -value of the cMSSM.Thus, the main result of this analysis is that the cMSSMis excluded at least at the 90% CL for reasonable choices ofthe observable set.The best-fit point is in the ˜ τ -coannihilation region at M ≈
500 GeV, with a secondary minimum in the focus-point region at M / , M (cid:29) p -values of coannihilation and focus-point regions can serveas an estimate of a likelihood-ratio test between a cMSSMat M ≈
500 GeV which can be tested at the LHC, and a“SM with dark matter” with squark and gluino masses be-yond about 5 TeV. Since the focus point manifests a morelinear relation between observables and input parameters inthe toy fits, and thus a more χ -distribution like behaviour,it reaches a slightly higher p -value than the ˜ τ -coannihilationregion. This shows that even the best-fit region offers no sta-tistically relevant advantage over the “SM with dark matter”.Thus, we can conclude that the cMSSM is not only excludedat the 90% to 95% CL, but that it is also statistically mostlyindistinguishable from a hypothetical SM with dark matter.In addition to this main result, we apply the first com-plete scan of the possibility of the existence of chargeor colour-breaking minima within a global fit of thecMSSM. In addition, we calculate the lifetime of the bestfit points. We find that the focus-point best-fit-region is sta-ble, while the ˜ τ -coannihilation best-fit region is either stableor metastable, with a lifetime significantly longer than theage of the Universe.It is important to note that the exclusion of the cMSSMat the 90% CL or more does in general not apply to less re-stricted SUSY models. The combination of measurementsrequiring low slepton and gaugino mass scales, such as ( g − ) µ , and the high mass scales preferred by the SM-like Higgs and the non-observation of coloured sparticlesat the LHC puts the cMSSM under extreme tension. In thecMSSM these mass scales are connected. A more generalSUSY model where these scales are decoupled, and prefer-ably also with a complete decoupling of the third generationsleptons and squarks from the first and second generation,would easily circumvent this tension.Therefore, the future of SUSY searches at the LHCshould emphasize the coverage of any phenomenologicalscenario which allows sleptons, and preferably also thirdgeneration squarks, to remain light, while the other sparti- cles can become heavy. Many loopholes with light SUSYstates still exist, as analyses as in [159] show, and there existpotentially promising experimental anomalies which couldbe explained by more general SUSY models [160–162].On the other hand, the analysis presented here shows thatSUSY does not directly point towards a non-SM-like lightHiggs boson. The uncertainty on the predictions of ratios ofpartial decay widths and other observables at the LHC aresignificantly smaller than the direct uncertainty of the LHCHiggs rate measurements. This is because of the high SUSYmass scale, also for third generation squarks, imposed bythe combination of the cMSSM and the direct SUSY parti-cle search limits. These do not allow the model to vary thelight Higgs boson properties sufficiently to make use of theexperimental uncertainty in the Higgs rate measurements.This might change for a more general SUSY model, butthere is no direct hint in this direction. The predicted levelof deviation of the light Higgs boson properties from theSM prediction at the O ( ) level is not accessible even at ahigh-luminosity LHC and requires an e + e − collider.In summary, we find that the undeniable freedom inchoosing the observable set – before looking at the exper-imental values of the results – introduces a remaining soft-ness into the exclusion of the cMSSM. Therefore, while wemight have preferred to find SUSY experimentally, we findthat at least we can almost complete the second most reveredtask of a physics measurement: with the combination of as-trophysical, precision collider and energy frontier measure-ments in a global frequentist analysis we (softly) kill thecMSSM. Acknowledgements
We thank Sven Heinemeyer and Thomas Hahnfor very helpful discussions during the preparations of the Higgs bo-son decay rate calculations. This work was supported by the DeutscheForschungsgemeinschaft through the research grant HA 7178/1-1, bythe U.S. Department of Energy grant number DE-FG02-04ER41286,by the BMBF Theorieverbund and the BMBF-FSP 101 and in partby the Helmholtz Alliance “Physics at the Terascale”. T.S. is sup-ported in part by a Feodor-Lynen research fellowship sponsored by theAlexander von Humboldt Foundation. We also thank the HelmholtzAlliance and DESY for providing Computing Infrastructure at the Na-tional Analysis Facility.
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