Implications of Initial LHC Searches for Supersymmetry
Abstract
The CMS and ATLAS Collaborations have recently published the results of initial direct LHC searches for supersymmetry analyzing /pb of data taken at 7 TeV in the centre of mass. We incorporate these results into a frequentist analysis of the probable ranges of parameters of simple versions of the minimal supersymmetric extension of the Standard Model (MSSM), namely the constrained MSSM (CMSSM), a model with common nonuniversal Higgs masses (NUHM1), the very constrained MSSM (VCMSSM) and minimal supergravity (mSUGRA). We present updated predictions for the gluino mass, , the light Higgs boson mass, , and the spinindependent dark matter scattering cross section, . The CMS and ATLAS data make inroads into the CMSSM, NUHM1 and VCMSSM (but not mSUGRA) parameter spaces, thereby strengthening previous lower limits on sparticle masses and upper limits on in the CMSSM and VCMSSM. The favoured ranges of in the CMSSM, VCMSSM and mSUGRA are close to the Standard Model, but considerably larger values of are possible in the NUHM1. Applying the CMS and ATLAS constraints improves the consistency of the model predictions for with the LEP exclusion limits.
CERNPHTH/2010321, DCPT1102, DESY 10249, IPPP1101,
FTPIMINN10/39, KCLPHTH/201104, UMNTH2933/10
The results of experiments at the LHC will be makeorbreak for supersymmetry. Multiple analyses have shown that the ATLAS and CMS experiments at the LHC have excellent chances of discovering supersymmetry (SUSY) [1, 2] if it provides the astrophysical cold dark matter [3], and/or if sparticles are light enough to render natural the electroweak mass scale [4], and/or SUSY explains the apparent discrepancy between experimental measurement and the Standard Model (SM) prediction for the anomalous magnetic moment of the muon, [5, 6, 7]. In parallel, the CDF, DØ, ATLAS and CMS experiments should be able to establish or disprove the existence of a SMlike Higgs boson weighing less than about , which is the upper bound predicted by the minimal supersymmetric extension of the Standard Model (MSSM) [8, 9], provided that SUSY is realized at the TeV scale.
In anticipation of the LHC startup, many groups have ventured estimates of the possible masses of supersymmetric particles in variants of the MSSM [10]. The unconstrained MSSM contains too many parameters for a full exploration of its parameter space to be possible using present data. Therefore, we have focused on making estimates within the constrained MSSM (CMSSM) [11, 12] in which soft SUSYbreaking mass parameters are assumed to be universal at the GUT scale, in the simplest generalization of this model in which the universality is relaxed to allow nonuniversal Higgs masses (NUHM1) [13, 14, 15], in a very constrained model in which a supplementary relation is imposed on trilinear and bilinear soft SUSYbreaking masses (VCMSSM) [16], and in minimal supergravity (mSUGRA) in which, in addition, the gravitino mass is set equal to the common soft SUSYbreaking scalar mass before renormalization [16, 17].
Our estimates [12, 14, 15, 17, 18] have been made in a frequentist approach, in which we construct a global likelihood function with contributions from precision electroweak observables, physics observables, and the astrophysical cold dark matter density as well as the limits from the direct searches for Higgs bosons and sparticles at LEP. Our best fits in the CMSSM, NUHM1, VCMSSM and mSUGRA all suggested that sparticles should be relatively light, perhaps even within the reach of early runs of the LHC.
The first results of initial direct searches for SUSY in data recorded by the CMS and ATLAS detectors analyzing /pb of integrated luminosity of collisions at 7 TeV in the centre of mass taken in 2010 have now been published [19, 20]. In this paper we combine these newlypublished results with our previous analyses to give updated predictions for the preferred regions of parameter space, gluino and Higgs masses, and the spinindependent dark matter scattering cross section in the CMSSM, NUHM1, VCMSSM and mSUGRA frameworks.
We recall that the CMSSM [11] has four input parameters: the universal soft SUSYbreaking scalar and gaugino masses , a universal trilinear soft SUSYbreaking parameter and the ratio of Higgs v.e.v.’s , as well as the sign of (the magnitude of and the bilinear SUSYbreaking parameter are fixed by the electroweak vacuum conditions). Concerning the latter, the results on [5, 6, 7] strongly favor a positive sign. In the NUHM1 [13], a common soft SUSYbreaking contribution to the masses of the two Higgs doublets is allowed to vary independently, so there are five independent parameters. On the other hand, the VCMSSM [16] imposes the supplementary constraint on the CMSSM, thereby removing as a free input and leaving three parameters, on which the further constraint in mSUGRA [21] imposes a severe restriction.
Our analysis has been performed using the MasterCode [12, 14, 15, 17, 18, 22]. The model parameter spaces are sampled using a Markov Chain Monte Carlo (MCMC) technique similar to that used in our previous papers. Our MCMC samplings of the CMSSM and NUHM1 parameter spaces each comprise some 25,000,000 points, whereas those of the VCMSSM and mSUGRA include some 30,000,000 and 17,000,000 points, respectively. The constraints are also treated similarly to our previous analyses, the significant changes being an updated value of the top quark mass, [23] and the use of the new determination of the SM contribution to [7], as described in [17] ^{1}^{1}1We have not updated our evaluation of [15], though higher values have been reported in [24, 25]. All results are compatible within the errors: the main differences in the central values are related to the values of and . We have checked that using the value given in [25], corresponding to less conservative treatments of and , would have negligible effects on our fits, except to increase globally by . Also we note that a large nonSM contribution to is not supported by data [26].. The numerical evaluation within the MasterCode [12, 14, 15, 17, 18, 22], combines SoftSUSY [27], FeynHiggs [9, 28, 29, 30], SuFla [31, 32], SuperIso [33, 34], a code providing supersymmetric predictions for electroweak observables based on [35, 36], MicrOMEGAs [37, 38, 39] and DarkSUSY [40, 41], making extensive use of the SUSY Les Houches Accord [42, 43].
The MasterCode is designed in such a way that the constraints from new observables can be taken into account and incorporated quickly and easily into the global likelihood function as ‘afterburners’ (i.e., as addons to the global function), provided that the contribution to the likelihood function from the new observable is available. The new ingredients in this analysis are the contributions from the direct SUSY searches performed by CMS and ATLAS [19, 20], which are incorporated as just such ‘afterburners’, via the procedures described below.
Previous studies [1, 2] had indicated that /pb at would provide sufficient sensitivity to probe regions of the planes favoured previously at the 95% and 68% confidence levels (CL), close to the previous bestfit points in the NUHM1 and CMSSM, though not as far as the bestfit points in the VCMSSM and mSUGRA. Therefore, the possibility that the actual experimental exclusion would be less than the expected sensitivity could not be excluded. However, this amount of luminosity could not be expected to lead to a 5 discovery of anomalous missingtransverseenergy events beyond those expected in the SM, and the interpretation of any excess would be ambiguous, since one could not expect to be able to discriminate between SUSY and other potential explanations. On the other hand, it was to be expected that even nonobservation of SUSY with this event sample would make an important contribution to the global likelihood function and possibly alter significantly the results of the fits.
The CMS result [19] is based on a search for multijet + events without accompanying leptons. The 13 events found in the signal region were compatible with the expected from SM backgrounds with a probability value of 30% The observed result allowed CMS to set a 95% CL (i.e., ) upper limit of 13.4 signal events. This would correspond to events for any possible signal, yielding for large sparticle masses. The central CMS result, shown in Fig. 5 of [19], is a 95% CL exclusion contour in the plane of the CMSSM for the particular values and . However, the sensitivity of a search for multijet + events is largely independent of these additional parameters within the CMSSM [19], and can also be taken over to the NUHM1, VCMSSM and mSUGRA models, which have similar signatures in these search channels.
Fig. 5 of [19] also presents a contour for the 95% CL exclusion expected in the absence of any signal, corresponding to events. This contour would correspond to an apparent significance of and hence . The observed 95% CL contour, on the other hand, corresponds to . We approximate the impact of the new CMS constraint by (where ) for each ray in the plane, fitting the parameters by requiring on the expected and observed 95% exclusion contours shown in Fig. 5 of [19] ^{2}^{2}2We have checked that this procedure gives a value for at the LM1 point in Fig. 5 of [19] that is consistent with the 19.2 events expected. We note that the functional form adopted for is approximate, and our later estimates of systematic errors in our results are based on studies of alternative forms..
The ATLAS result [20] is based on a search for multijet + events with one accompanying electron or muon. The 2 events found in the signal region were compatible with the expected from SM backgrounds with a probability value of 16%. The central ATLAS result, shown in Fig. 2 of [20], is again a 95% CL exclusion contour in the plane of the CMSSM for the particular values and , which is also only moderately dependent on these additional parameters within the CMSSM [20], and can also be taken over to the NUHM1, VCMSSM and mSUGRA models ^{3}^{3}3This would not be the case for multilepton + jets + searches, which have a stronger dependence on , leading in particular to weaker bounds for large .. The observed result allowed ATLAS to set a 95% CL (i.e., ) upper limit on sparticle production that corresponds to 4.8 signal events and . This would correspond to a downward fluctuation of events for any possible signal, yielding an estimate of for large sparticle masses.
In order to estimate the ATLAS sensitivity to sparticle masses in the parameter region beyond the observed 95% CL, we follow [44], based on [45], in which the integrated luminosities required for discovering sparticles of various masses were estimated. There it was found empirically that , where denotes a (similar) squark and gluino mass. This suggests that for equal squark and gluino masses the effective event rate . Using this as a guide, we assume that the effective numbers of events expected for relevant points in the plane scale as . We then calculate the corresponding significances assuming the observed signal of estimated above, to estimate the corresponding values of . As an approximate analytic interpolating form, we use , fitting and along rays in the plane. Our analysis is based exclusively on published material [19, 20, 44, 45], and we consider it prudent to assign a larger systematic error to our implementation of the ATLAS constraint.
As shown in Fig. 5 of [19] and Fig. 2 of [20], the direct physics reaches of the CMS and ATLAS data in the models studied are much greater than those of the earlier CDF, DØ and LEP2 searches for sparticles, so the latter do not make significant contributions to the global function in the regions of interest for our analysis. On the other hand, the LEP2 Higgs search [46, 47] had an indirect reach in the plane that is comparable to those of the direct CMS and ATLAS constraints in the models studied, as we see explicitly later.
Conservatively, we do not attempt to combine the CMS and ATLAS constraints in the following discussion, which would require detailed modelling of their likelihood functions and a better understanding of correlations between observables in the CMS and ATLAS searches. The functions obtained by combining the CMS and ATLAS constraints separately with previous constraints in the planes for the CMSSM, NUHM1, VCMSSM and mSUGRA models are shown in Fig. 1. In each panel, the new 68% and 95% CL contours incorporating the CMS (ATLAS) results are shown as red and blue dashed (solid) lines, and the corresponding previous contours are shown as dotted lines. (It should be noted that the updated values of and and some further technical improvements have changed slightly the best fits and likelihood contours with respect to our earlier publications [14, 15, 17].) We also indicate as open and full green stars the new bestfit points including the CMS and ATLAS results, respectively, and indicating the preLHC bestfit points by green ‘snowflakes’ ^{4}^{4}4We recall that in mSUGRA there is a secondary minimum of the likelihood function with that is located at [17], which is not visible in Fig. 1 and lies well beyond the initial LHC reach.. In panels where only one green star is visible, the CMS and ATLAS bestfit points lie on top of each other, within the accuracy adopted for our numbers. The jagged boundaries of the 95% CL CMSSM and NUHM1 contours at large and reflect the uncertainties in sampling the slow variations in their likelihood functions.
Model  Minimum  Probability  (no LEP)  

(GeV)  (GeV)  (GeV)  (GeV)  
CMSSM  (21.3)  (32%)  (320)  (60)  (170)  (11)  (107.9) 
with CMS  22.0  29%  370  80  340  14  112.6 
with ATLAS  24.9  16%  400  100  430  16  112.8 
NUHM1  (19.3)  (31%)  (260)  (110)  (1010)  (8)  (121.9) 
with CMS  20.9  28%  380  90  70  14  113.5 
with ATLAS  23.3  18%  490  110  630  25  116.5 
VCMSSM  (22.5)  (31%)  (300)  (60)  (30)  (9)  (109.3) 
with CMS  23.8  25%  340  70  50  9  115.5 
with ATLAS  27.1  13%  390  90  70  11  117.0 
mSUGRA  (29.4)  (6.1%)  (550)  (230)  (430)  (28)  (107.8) 
with CMS  29.4  6.1%  550  230  430  28  121.2 
with ATLAS  30.9  5.7%  550  230  430  28  121.2 
In Table 1 we compare the bestfit points found in this paper incorporating the CMS and ATLAS constraints with preLHC results [14, 15, 17] in the CMSSM, NUHM1 and VCMSSM (the fits in mSUGRA are essentially unchanged when LHC data are included in the fits, only the best coannihilation fit is reported). In addition to the minimum value of and the fit probability in each scenario, we include the values of and at all the bestfit points. We estimate systematic errors of in the values for quoted in Table 1 for the bestfit points in the CMS analyses of the CMSSM, NUHM1 and VCMSSM, associated with the ambiguities in the implementations of the LHC constraints and the slow variations in the functions. In the cases of the ATLAS analyses, we have an additional systematic uncertainty in the implementation of the constraint, and estimate a somewhat larger error . Table 1 also shows the values of that would be estimated in each model if the LEP Higgs constraint were neglected.
The absences of supersymmetric signals in the CMS and ATLAS data [19, 20] invalidate portions of the CMSSM, NUHM1 and VCMSSM parameter spaces at low that were previously allowed at the 95% and 68% CLs ^{5}^{5}5Strictly speaking, the and 2.30 contours that we plot may not always correspond exactly to these CLs, but we ignore any possible differences here since previous studies [14, 15, 17] showed acceptable coverage, which should not have deteriorated in the presence of additional constraints. For recent studies in a related context, see [48]., but do not impinge significantly on the corresponding regions for mSUGRA. In the cases of the CMSSM and VCMSSM, the LHC data disfavour the low tips of the coannihilation regions and increase significantly the bestfit values of , as seen in Fig. 1 and Table 1. However, it should be kept in mind that is quite a shallow function of near the bestfit points, particularly in the NUHM1. In the case of the NUHM1, the CMS and ATLAS data disfavour a slice of parameter space at low and extending from the coannihilation region towards the lightHiggs funnel discussed in [17]. The CMS and particularly ATLAS data extend the 95% CL regions to larger and , particularly in the NUHM1.
While the CMS constraint, as seen in Table 1, leads only to small increases in the global of for each model, the inclusion of the ATLAS constraint results in increases in by . This indicates that there is no tension between the CMS data and previous constraints, whereas some tension may arise from the ATLAS constraint. Correspondingly, the fit probabilities in the different models are reduced by including the CMS and ATLAS constraints, but generally not to unacceptable levels. It should be recalled, though, that our ATLAS implementation is more uncertain, and our implementations of the constraints should, conservatively, each be assigned an uncertainty at least as large as the value of for CMS and ATLAS, respectively. On the other hand, the absence of a supersymmetric signal at the LHC with a luminosity of /fb at a centreofmass energy TeV would increase the global minimum of sufficiently to put severe pressure on these models.
Fig. 2 displays the effects of the CMS and ATLAS constraints on the planes in the CMSSM, NUHM1, VCMSSM and mSUGRA ^{6}^{6}6These planes are truncated at , so that perturbative RGE calculations remain reliable.. We see that the ranges of preferred at the 68% CL are extended to significantly larger values in the CMSSM and (particularly) the NUHM1, to a lesser extent in the VCMSSM, and not at all in mSUGRA. The bestfit values of are also increased in the CMSSM and NUHM1, but not significantly in the VCMSSM and mSUGRA. The increases are mainly due to the constraint: larger sparticle masses lead to a smaller supersymmetric contribution that can be compensated by a larger value of .
In the regions of parameter space of interest to the CMSSM, NUHM1, VCMSSM and mSUGRA, the direct reach of the LHC for supersymmetry is strongly influenced by the gluino mass, . Accordingly, we display in Fig. 3 the oneparameter functions for relative to the minima in all these models. In each case, we display the new likelihood functions incorporating CMS and ATLAS data as dashed and solid lines, respectively, and those given by the preLHC fits as dotted lines. The plots display the contributions in each model relative to the bestfit points in that model. In this and subsequent figures, the oneparameter functions for mSUGRA are essentially unchanged when the LHC data are included but are shown for comparison purposes.
For each of the CMSSM, NUHM1 and VCMSSM, we see that the side of the likelihood function below the bestfit point is shifted to larger by similar amounts to . The bestfit values of in the CMSSM, NUHM1 and VCMSSM are now to , and the sides of the likelihood function beyond the bestfit points rise quite similarly in these models, though more slowly in the NUHM1. In mSUGRA the most likely values of are unchanged by either CMS or ATLAS, lying in the range to , with a secondary minimum in the light Higgs funnel region at and large [17] that lies beyond the present reach of the LHC ^{7}^{7}7A vestige of this region is visible in the VCMSSM at , see also the lower left panel of Fig. 2..
The experimental search for a SMlike Higgs boson is heating up, with interesting prospects for both the Tevatron collider [49] and the LHC [1, 2], see in particular [50]. Accordingly, we display in Fig. 4 the oneparameter functions for the lightest MSSM Higgs mass in the CMSSM, NUHM1, VCMSSM and mSUGRA. In this figure we do not include the direct limits from LEP [46, 47] or the Tevatron, so as to illustrate whether there is a conflict between these limits and the predictions of supersymmetric models. For each model we display the new likelihood functions incorporating the ATLAS data as solid lines, indicating the theoretical uncertainty in the calculation of of by red bands. We also show, as dashed lines without red bands, the central value of the prediction based on the CMS constraint, and as dotted lines without red bands the preCMS predictions for (all discarding the LEP constraint).
In the case of the CMSSM, we see that the CMS and ATLAS constraints increase the consistency of the model prediction with the direct LEP limit on , indicated by the yellow region: the bestfit value is found at after the inclusion of the CMS (ATLAS) constraints (but still neglecting the LEP Higgs searches, see also Table 1), with an estimated theoretical error of . In the case of the NUHM1, apart from somewhat lower bestfit values of after including the CMS (ATLAS) constraint (we recall that the function was very shallow in the direction of lower masses), we see that the main effect of the LHC data is to increase substantially the oneparameter function at low masses . (It should be remembered that in the NUHM1 the LEP constraint is weakened at low because the coupling may be reduced, which is not possible in the CMSSSM, VCMSSM and mSUGRA [51, 52]. The LHC data render a large reduction less likely.) Now most of the preferred region in the NUHM1 is indeed above . In the case of the VCMSSM, the LHC constraint strongly disfavours a Higgs boson below the LEP limit. In the case of the mSUGRA coannihilation region, the global minimum of the preLHC with the LEP constraint disregarded was found in an isolated region at low , resulting in . The inclusion of either the CMS or ATLAS data removes this isolated region, pushing the preferred range to larger compatible with the LEP constraint [46, 47]. The global minimum with either the CMS or ATLAS constraint included is very similar to the preCMS fit with the LEP constraint included. In general, these plots support the remark made earlier that the LHC data have an effect comparable to the LEP limit in constraining the models studied.
For this reason, the LHC data do not have a great impact on the amounts of finetuning of parameters required. We have evaluated the finetuning measure proposed in [53] for the best pre (post)LHC fits, finding in the CMSSM 100 (120) [140] before the LHC (with the CMS constraint) [with the ATLAS constraint]. The corresponding numbers in the other models are: NUHM 250 (230) [310], VCMSSM 130 (110) [140], mSUGRA 250 in all cases. ^{8}^{8}8See [54] for a contrasting view that does not take into account the LEP constraint on .
One of the other observables of potential interest for testing variants of the MSSM is , and we recall that the sensitivity of the LHCb experiment from the data recorded during the 2010 LHC run approaches that already achieved by the CDF and DØ experiments [55]. As seen in Fig. 5, we find that the effects of the LHC data on this observable are negligible in mSUGRA and very small in the VCMSSM. However, the larger values of now favoured in the CMSSM increase the likelihood of an enhancement in beyond the SM value. As was discussed previously [17], the best prospects for a deviation from the SM prediction for are in the NUHM1, where the bestfit value is significantly larger than in the SM and substantially larger values are now possible, thanks to the larger values of now preferred. On the other hand, the bestfit values for in the CMSSM, VCMSSM and mSUGRA are peaked within 10% of the SM value, indistinguishable from the SM given the theoretical and experimental uncertainties.
Finally, we recall that it is expected that the sensitivities of direct searches for the spinindependent scattering of dark matter particles on heavy nuclei will soon be increasing substantially [56]. Accordingly, we display in Fig. 6 the oneparameter functions for the spinindependent neutralino dark matter scattering cross section in the CMSSM, NUHM1, VCMSSM and mSUGRA. We display the new likelihood functions incorporating CMS and ATLAS data and calculated assuming the default value MeV [57] as dashed and solid lines, as previously, and those given by the previous fits as dotted lines. We see that in the CMSSM and VCMSSM the LHC constraint reduces substantially the likelihood of relatively large values of cm for , whereas the likelihood functions are little changed in mSUGRA and the NUHM1, due in the latter case to compensation between preferences for larger and larger . We have recalculated the values of at the bestfit points in the different models assuming , close to the lower end of the plausible range [57], and found reductions in by factors to 4 in each case.
Fig. 7 displays the regions of the planes favoured at the 68% and 95% CL in the CMSSM, NUHM1, VCMSSM and mSUGRA models, assuming , comparing the post and preLHC contours (dashed/solid lines and dotted lines, respectively), and showing the bestfit points as solid and open stars, respectively ^{9}^{9}9The region at small in the mSUGRA plot corresponds to the light Higgs funnel discussed in [17].. The models predict, in general, a negative correlation between and , which is why the stronger lower limit on and hence provided by CMS and ATLAS (see Fig. 3) corresponds to a stronger upper limit on , as seen in Fig. 6. Conversely, any future upper limit on (measurement of) would correspond to a lower limit on (preferred range of) and hence . However, this correlation is weakened by the uncertainty in , which should be taken into account in the interpretation of any future constraint on .
In summary, we have shown that the initial CMS and ATLAS searches for supersymmetry at the LHC at 7 TeV already have significant impacts on the regions of the CMSSM, NUHM1 and VCMSSM parameter spaces favoured at the 68 and 95% CL, shifting the bestfit points to somewhat larger values (with an estimated systematic uncertainty of (CMS) to 20% (ATLAS) in ). They are now located at , corresponding to , within the prospective reach of the LHC with 1/fb of analyzed data at a centreofmass energy of 7 TeV or more ^{10}^{10}10However, we recall that significantly larger values of are allowed at the 95% CL.. On the other hand, the CMS and ATLAS data have no impact on the mSUGRA fit. Increases in the preferred values of in the CMSSM and particularly the NUHM1 increase the likelihood that exceeds significantly the SM value. The model predictions for (discarding the direct searches by LEP) are now in better agreement with the direct search limits, because of the somewhat heavier mass spectrum required by CMS and ATLAS. Within the NUHM1, the new contribution decreases significantly the likelihood of a light Higgs boson with . Within the CMSSM and VCMSSM frameworks, the CMS and ATLAS results also diminish significantly the likelihood that cm, but have little impact in the NUHM1 and mSUGRA.
The search for supersymmetry is about to enter its critical stage, and further results from the LHC with tailored analyses and increased luminosity will soon provide crucial new information, as will dark matter search experiments.
Note added
During the completion of this work three papers addressing related topics appeared [58, 59, 60]. Ref. [58] studies the phenomenology of the mSUGRA rapidannihilation funnel region. Ref. [59] analyzes the CMS constraint within the CMSSM and reaches similar conclusions on the increases in and that it entails. Ref. [60] considers a more general model than those examined here.
This work was supported in part by the European Community’s MarieCurie Research Training Network under contracts MRTNCT2006035505 ‘Tools and Precision Calculations for Physics Discoveries at Colliders’ and MRTNCT2006035482 ‘FLAVIAnet’, and by the Spanish MEC and FEDER under grant FPA200501678. The work of S.H. was supported in part by CICYT (grant FPA 2007–66387 and FPA 2010–22163C0201), and the work of K.A.O. was supported in part by DOE grant DE–FG02–94ER–40823 at the University of Minnesota. K.A.O. also thanks SLAC (supported by the DOE under contract number DEAC0276SF00515) and the Stanford Institute for Theoretical Physics for their hospitality and support while this work was being finished.
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