Missing energy look-alikes with 100 pb-1 at the LHC
Jay Hubisz, Joseph Lykken, Maurizio Pierini, Maria Spiropulu
aa r X i v : . [ h e p - ph ] J u l FERMILAB-PUB-08-012-TANL-HEP-PR-08-30
Missing energy look-alikes with 100 pb − at the LHC Jay Hubisz ∗ High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
Joseph Lykken † Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
Maurizio Pierini ‡ and Maria Spiropulu § Physics Department, CERN, CH 1211 Geneva 23, Switzerland (Dated: October 29, 2018)A missing energy discovery is possible at the LHC with the first 100 pb − of understood data. Wepresent a realistic strategy to rapidly narrow the list of candidate theories at, or close to, the momentof discovery. The strategy is based on robust ratios of inclusive counts of simple physics objects. Westudy specific cases showing discrimination of look-alike models in simulated data sets that are atleast 10 to 100 times smaller than used in previous studies. We discriminate supersymmetry modelsfrom non-supersymmetric look-alikes with only 100 pb − of simulated data, using combinations ofobservables that trace back to differences in spin. PACS numbers: 11.30.Pb,14.80.Ly,12.60.Jv
I. INTRODUCTIONA. Twenty questions at the LHC
Many well-motivated theoretical frameworks makedramatic predictions for the experiments at the LargeHadron Collider. These frameworks are generally basedupon assumptions about new symmetries, as is thecase for supersymmetry (SUSY) [1, 2] and little Higgs(LH)[3, 4], or upon assumptions about new degrees offreedom such as extra large [5] or warped [6] spatial di-mensions. Within each successful framework, one canconstruct a large number of qualitatively different mod-els consistent with all current data. Collectively thesemodels populate the “theory space” of possible physicsbeyond the Standard Model. The BSM theory space ismany dimensional, and the number of distinct modelswithin it is formally infinite. Since the data will notprovide a distinction between models that differ by suffi-ciently tiny or experimentally irrelevant details, infinity,in practice, becomes some large finite number N . Themapping of these N models into their experimental sig-natures at the LHC, though still incomplete, has beenexplored in great detail.As soon as discoveries are made at the LHC, physicistswill face the LHC Inverse Problem: given a finite setof measurements with finite resolutions, how does onemap back [7]-[9] to the underlying theory responsible forthe new phenomena? So far, not enough progress hasbeen made on this problem, especially as it relates to the ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] immediate follow-up of an early LHC discovery.When N is large, it is not a viable strategy to discrim-inate between N alternative explanations by performing N tests. However, as the game “twenty questions” illus-trates, a well-designed series of simple tests can identifythe correct alternative in of order log( N ) steps, proceed-ing along a decision tree such that, at each branching, oforder half of the remaining alternatives are eliminated.Addressing the LHC Inverse Problem implies designingand implementing this series of simple tests in the LHCexperiments, so that with high confidence a significantfraction of the remaining theory space is ruled out ateach step. The results of the first few tests will shape therequirements for future tests, so the immediate need is todevelop the strategy for the early tests. In this paper weprovide this strategy for the case of an LHC discovery inthe inclusive missing energy signature. B. Missing energy at the LHC
The existence of dark matter provides a powerful mo-tivation to explore missing energy signatures at the LHC,under the assumption that a significant fraction of darkmatter may consist of weakly interacting thermal relics.Missing energy at the LHC is experimentally challenging.Most of the energy of 14 TeV pp collisions is carried offby undetected remnants of the underlying event, so miss-ing energy searches actually look for missing transverseenergy ( E miss T ) of the partonic subprocess. E miss T searchesare plagued by instrumental and spurious backgrounds,including cosmic rays, scattering off beam halo and jetmismeasurement. Standard Model processes create anirreducible E miss T background from processes such as the Z boson decay to neutrinos and t ¯ t production followedby semileptonic decays of the top.In many theoretical frameworks with dark matter can-didates, there are heavy strongly interacting particleswith the same conserved charge or parity that makes thedark matter particle stable. These colored particles willbe pair-produced at the LHC with cross sections roughlyin the range 0.1 to 100 pb. Their subsequent decays willproduce Standard Model particles along with a pair ofundetected dark matter particles. Thus the generic ex-perimental signature is both simple and inclusive: large E miss T accompanied by multiple energetic jets. A detailedstrategy for early discovery with the inclusive E miss T sig-nature was presented in the CMS Physics Technical De-sign Report [10]-[12] and studied with full simulation ofthe CMS detector. After a series of cleanup and analysiscuts on a simulated E miss T trigger sample targeting thereduction of the instrumental and physics backgrounds,the signal efficiency remained as high as 25%. These re-sults indicate that, for signal cross sections as low as afew pb, an E miss T discovery could be made with the first100 pb − of understood LHC data . In our study we as-sume as the starting point that a greater than 5 σ excessof events will be seen in a 14 TeV LHC data sample of100 pb − with an inclusive missing energy analysis. Forinvariability and comparability we effectively adopt thefull analysis path and requirements used in [10]. C. Look-alikes at the moment of discovery
At the moment of discovery a large number of theorymodels will be immediately ruled out because, withinconservative errors, they give the wrong excess. Howevera large number of models will remain as missing energylook-alikes, defined as models that predict the same in-clusive missing energy excess, within some tolerance, inthe same analysis in the same detector, for a given in-tegrated luminosity. The immediate challenge is then tobegin discriminating the look-alikes.The look-alike problem was studied in [9] as it mightapply to a later mature phase of the LHC experiments.Even restricted to the slice of theory space populated by apartial scan of the MSSM, ignoring SM backgrounds andsystematic errors, and applying an uncorrelated χ -likestatistical analysis to 1808 correlated observables, thisstudy found that a large number of look-alikes remainedunresolved in a simulation equivalent to 10 fb − . A morerecent analysis [13] attempts to resolve these look-alikesin a simulation of a future linear collider.At the moment of an early discovery the look-alikeproblem will be qualitatively different. The data sam-ples will be much smaller, with a limited palette of ro-bust reconstructed physics objects. For example, τ or The first 100 pb − of understood LHC data will not be the first100 pb − of data written to tape. The 10 TeV data collected inthe early running will be used for calibrations and understandingof benchmark Standard Model processes. b tagging in multijet final states will be in developmentduring the 100 pb − era. In many small data samplespeaks and edges in invariant mass distributions may notbe visible, and most observables related to detailed fea-tures of the events will be rate limited. The observablesthat are available to discriminate the look-alikes in thevery early running will be strongly correlated by physicsand systematics making it imprudent to combine themin a multivariate analysis. D. Is it SUSY?
By focusing on the discrimination of look-alikes, we arepursuing a strategy of simple binary choices: is Model Aa significantly better explanation of the discovery dataset than Model B? Each answer carries with it a fewbits of important fundamental information about the newphysics process responsible for missing energy. Obviouslywe will need to make many distinct look-alike compar-isons before we can hope to build up a clear picture fromthese individual bits.Consider how this strategy might play out for answer-ing the basic question “is it SUSY?” It may not be pos-sible to answer this question conclusively during the 100pb − era. Our strategy will consist of asking a series ofmore modest questions, some of them of the form: “doesSUSY Model A give a significantly better explanation ofthe discovery data set than non-SUSY Model B?” Noneof these individual bits of information by itself is equiva-lent to answering “is it SUSY?” However we demonstratethat we can build up a picture from the data that con-nects back to features of the underlying theory.Furthermore, we demonstrate a concrete method to ob-tain indirect information about the spin of the new par-ticles. We establish how to discriminate between a non-SUSY model and its SUSY look-alikes. Even though wecannot measure the spins of the exotic particles directly,spin has significant effects on production cross sections,kinematic distributions and signal efficiencies. We arethus able to discriminate SUSY from non-SUSY usingcombinations of observables that trace back to differencesin spin. Our study shows that in favorable cases this canbe accomplished with data sets as small as 100 pb − . E. Outline
In section II we review in detail the missing energy dis-covery path, including the experimental issues and sys-tematics that limit our ability to fully reconstruct eventsfrom the discovery data set. We explain how the miss-ing energy signals are simulated, and the uncertaintiesassociated with these simulations. In section III we dis-cuss the problem of populating the parts of the theoryspace relevant to a particular missing energy discovery.In section IV we introduce two groups of look-alike mod-els relative to two different missing energy signals. Formodels differing only by spins, we discuss how cross sec-tions, kinematic distributions and efficiencies can be usedto distinguish them, drawing from formulae developedin the Appendix. In section V we define all of the ro-bust observables that we use to discriminate among thelook-alikes, and in section VI we describe the look-alikeanalysis itself and how we compute the significance of thediscriminations. Sections VII and VIII give a summary ofour results, with details relegated to further Appendices.Finally section IX describes the steps we are following toimprove this analysis for use with real data.
II. DISCOVERY ANALYSIS FOR MISSINGENERGY
Missing energy hadron collider data has been used pre-viously for successful measurements of Standard Modelprocesses with energetic neutrinos; these include the Z boson invisible decay rates, the top quark cross section,searches for the Higgs boson [14] and a precise extractionof the W mass from the reconstruction of the W trans-verse mass [15]. Pioneering searches for new phenomenain missing energy data sets at the Tevatron [16]-[19] ledto the development and understanding of the basic tech-niques that will be used in missing energy searches at theLHC.In an ideal detector, with hermetic 4 π solid angle cov-erage and excellent calorimeter resolution, the measure-ment of missing energy is the measurement of the neu-trino energy and the energy of any other neutral weaklyinteracting particles. In a real detector it is also a mea-surement of the energy that escapes detection due touninstrumented regions and other detector effects suchas imperfect calorimeter response. Muons are sourcesof missing energy since a muon typically deposits onlyof order a few GeV of its total energy in the calorime-ters . QCD jets produce real E miss T from semileptonicdecays of heavy flavor, and fake E miss T from detector-induced mismeasurements. Thus the E miss T distributionof a pure QCD multijet sample has a long tail relatedto non-Gaussian tails in the detector response. Thisgives rise to an important background to missing energysearches that is difficult to estimate prior to data. At theTevatron it has been shown that this background can bebrought under control by exploiting the fact that the fake E miss T from jet mismeasurements is highly correlated withthe azimuthal directions of the leading jets [16].There are other important sources of fake E miss T athadron colliders, including beam halo induced E miss T , cos-mic ray muons, noise in the data acquisition system, The energy loss of muons is mostly due to ionization up to muonenergies of 100 GeV. Above 100 GeV bremsstrahlung and nuclearlosses can cause a single “catastrophic” energy loss comparableto the total muon energy. and misreconstruction of the primary vertex. Eliminat-ing these sources requires unbiased filters based on cleandefinitions of event quality.To design a missing energy analysis, we need to havesome idea of the source of the E miss T in the signal. Thepossibilities include: • The E miss T is entirely from neutrinos. This couldarise from the direct decay of new heavy particlesto neutrinos, or decays of new heavy particles totop, W ’s, Z ’s or τ ’s. One appropriate discoverystrategy for this case is to look for anomalies in theenergetic tails of data sets with reconstructed top, W ’s or Z ’s. • The E miss T originates from a single weakly interact-ing exotic particle in the final state. An exampleof this possibility is graviton production in modelswith large extra dimensions [20]. If strong produc-tion occurs, the signal will consist predominately ofmonojets and large E miss T . Successful analyses forthis case were carried out at the Tevatron [21, 22].Other signals that fit this case arise from unpar-ticle models [23] and from models with s -channelresonances that have invisible decays. • The E miss T originates from many weakly interact-ing exotic particles. This can be the case in hiddenvalley models [24], where the weakly interacting ex-otics are light pions of the hidden sector. This caseis experimentally challenging. • The E miss T originates from two weakly interactingexotic particles in the final state. This is the casefor supersymmetry models with conserved R par-ity, where the weakly interacting particles are neu-tralino LSPs. It also applies for more generic mod-els with WIMP dark matter candidates.We focus on a discovery analysis developed for the lastcase. Thus we are interested in signal events with twoheavy WIMPs in the final state. For early discoveryat the LHC, the signal events should have strong pro-duction cross sections; we will assume that each WIMParises from the decay of a strongly interacting heavy par-ent particle. The most generic signature is therefore large E miss T in association with at least two high E T jets. Therewill be additional jets if the WIMP is not produced in a2-body decay of the parent particle. Furthermore, thereis a significant probability of an extra jet from QCD ra-diation, due to the large phase space. Thus it is onlyslightly less generic to design an inclusive analysis forlarge E miss T in association with three or more energeticjets. We will refer to this as the inclusive missing energysignature . The requirement of a third energetic jet greatly reduces the sizeand complexity of the Standard Model backgrounds. Thus while
In the basic 2 → p T roughly comparable to their mass m p . The WIMPs of mass m dm resulting from the parentparticle decays will fail to deposit energy ≥ m dm in thecalorimeters; if the WIMPS have fairly large p T , a sig-nificant fraction of this energy contributes to the E miss T .Thus either large m p or large m dm leads to large E miss T .Note the azimuthal directions of the WIMPS are anti-correlated, a feature inherited from their parents, so themagnitude of the total E miss T tends to be less than themagnitude of the largest single contribution.At the LHC, the most important Standard Modelsources of large real E miss T will be t ¯ t , single top, W and Z plus jets associated production, dibosons and heavyflavor decays. Most of these processes produce a hardlepton in association with the E miss T from an energeticneutrino. The exception is Z → ν ¯ ν . Even with a per-fect detector, Z → ν ¯ ν plus jets is an irreducible physicsbackground. A. Analysis path
In the real data this search will be performed start-ing from a primary data set that includes requirementsof missing energy, jets and general calorimetric activityat the trigger path; the trigger efficiency should be mea-sured in other data samples.For the offline analysis, we will adopt the inclusivemissing energy benchmark analysis studied with the fulldetector simulation for the CMS Physics Technical De-sign Report [10, 11].The first phase is a preselection based on the eventquality. The purpose of this primary cleanup is to discardevents with fake E miss T from sources such as beam halo,data acquisition noise and cosmic ray muons. To elimi-nate these types of backgrounds the benchmark analysisuses jet variables, averages them over the event to de-fine corresponding event variables, and uses these to dis-criminate real E miss T + multijet events from spurious back-grounds. The event electromagnetic fraction (EEMF) isdefined to be the E T weighted jet electromagnetic frac-tion. We define an event charged fraction (ECHF) asthe event average of the jet charged fraction (defined asthe ratio of the P p T of the tracks associated with a jetover the total calorimetric jet E T ). The preselection alsohas a quality requirement for the reconstructed primaryvertex.Events that are accepted by the preselection require-ments proceed through the analysis path if they have mature LHC analyses will explore the fully inclusive E miss T sig-nature, we assume here that an early discovery will be based ona multijet + E miss T data sample. missing transverse energy E miss T ≥
200 GeV and at leastthree jets with E T ≥
30 GeV within pseudorapidity | η | <
3. These requirements directly define the missing en-ergy signal signature. In addition the leading jet is re-quired to be within the central tracker fiducial volume i.e. | η | < .
7. Everywhere in this paper “jets” meansuncorrected (raw) jets with E T >
30 GeV and | η | < η − φ space. The missing energy is uncorrectedfor the presence of muons in the event.The rest of the analysis path is designed based onelimination of the major backgrounds. The QCD back-ground from mismeasured jets is reduced by rejectingevents where the E miss T is too closely correlated with theazimuthal directions of the jets. To reduce the large back-ground from W ( → ℓν )+jets, Z ( → ℓℓ )+jets and t ¯ t pro-duction an indirect lepton veto (ILV) scheme is designedthat uses the tracker and the calorimeters. The ILV re-tains a large signal efficiency while achieving a factor oftwo rejection of the remaining W and t ¯ t backgrounds.The veto is indirect because we don’t identify leptons–instead events are rejected if the electromagnetic fractionof one of the two leading jets is too large, or if the highest p T track of the event is isolated. The signals we are inter-ested in are characterized by highly energetic jets whileleptons in the signal originate from cascade decays of theparents or semileptonic B decays in the jets; thus evenwhen a signal event has leptons it is relatively unlikelyto be rejected by the ILV. For the models in our study,approximately 85% of all signal events and 70% of signalevents with muons or taus pass the ILV cut.The final selections require that the leading jet has E T >
180 GeV, and that the second jet has E T > H T >
500 GeV, where H T = X i =2 E iT + E missT , (1)where the E T is summed over the second, third andfourth (if present) leading jets. These cuts select forhighly energetic events, greatly favoring events with newheavy particles over the Standard Model backgrounds.Table I summarizes the benchmark analysis path. Thetable lists the cumulative efficiencies after each selectionfor a benchmark signal model and the Standard Modelbackgrounds. The signal model is the CMS supersym-metry benchmark model LM1, which has a gluino withmass 611 GeV and squarks with masses around 560 GeV.The last line of the table shows the expected number ofevents that survive the selection in a data set correspond-ing to 1000 pb − of integrated luminosity. For the QCDbackground and the single top background, which are notshown in the table, the estimated number of remainingevents is 107 and 3, respectively. Thus the total esti-mated Standard Model background after all selections is245 events per 1000 pb − . TABLE I: Cumulative selection efficiency after each requirement in the E miss T + multijets analysis path for a low mass SUSYsignal and the major Standard Model backgrounds (EWK refers to W/Z , W W/ZZ/ZW ), see [10, 11]).Cut/Sample Signal t ¯ t Z ( → ν ¯ ν )+ jets EWK + jetsAll (%) 100 100 100 100Trigger 92 40 99 57 E miss T >
200 GeV 54 0.57 54 0.9PV 53.8 0.56 53 0.9 N j ≥ | η j d | ≥ . ≥ ≥ Iso lead trk = 0 23 0.09 2.3 0.02
EMF ( j ,EMF ( j ≥ . E T, >
180 GeV, E T, >
110 GeV 14 0.015 0.5 0.003 H T >
500 GeV 13 0.01 0.4 0.002events remaining per 1000 pb − B. Triggers and “boxes”
Having established a benchmark analysis path, we alsoneed to define benchmark data samples. With the realLHC data these will correspond to data streams and datapaths from various triggers. For the inclusive missingenergy signature relevant triggers are the E miss T and jettriggers. A single lepton trigger is also of interest, sincemany models produce energetic leptons in associationwith large E miss T . For our study we have chosen simplebut reasonable [25, 26] parametrizations of the triggerefficiencies defining our four benchmark triggers : • The MET trigger is a pure inclusive E miss T trigger.It is 50% efficient for E miss T >
80 GeV, as seen inFigure 1. • The DiJet trigger requires two very high E T jets. Itis 50% efficient for uncorrected jet E T >
340 GeV,as seen in Figure 2. • The TriJet trigger requires three high E T jets. Itis 50% efficient for uncorrected jet E T >
210 GeV,as seen in Figure 3. These are made-up triggers for the purposes of our study. Theguidance on our parametrizations is from the published triggerand physics reports of the CMS experiment. We expect that thetrigger tables of the LHC experiments will include correspondingtrigger paths, richer and better in terms of the physics capture. • The Muon20 trigger requires an energetic muonthat is not necessarily isolated. The trigger is 88%efficient for muons with p T = 20 GeV/ c , asymptot-ing to 95% as seen in Figure 4.After applying the selection requirements, these fourtriggers define four potential discovery data sets. In oursimulation the DiJet, TriJet, and Muon20 data sets, afterthe inclusive missing energy analysis path is applied, areall subsets of the MET sample, apart from one or twoevents per 1000 pb − . Thus the MET is the largest,most inclusive sample. We perform one complete analy-sis based on the MET trigger. The other three triggersare then treated as defining three more boxes , i.e. experi-mentally well-defined subsets of the MET discovery dataset. The simplest physics observables are the counts ofevents in each box. C. Backgrounds and systematics
In the CMS study the total number of Standard Modelbackground events remaining after all selections is 245per 1000 pb − for an E miss T trigger sample. The erroron this estimate is dominated by i) the uncertainty inhow well the detector simulation software simulates the A perfectly designed trigger table will give rise to overlaps amongdatasets from different trigger paths due to both physics andslow/non-sharp trigger efficiency turn-ons (resolution). (GeV) missT
E0 50 100 150 200 250 300 T r i gg er e ff i c i e n c y FIG. 1: The E miss T trigger efficiency. (GeV) T E300 310 320 330 340 350 360 370 380 390 400 T r i gg er e ff i c i e n c y FIG. 2: The DiJet trigger efficiency. response of the actual CMS detector, and ii) the uncer-tainty on how well the Standard Model event generatorsemulate QCD, top production, and
W/Z plus jets pro-duction. Detailed studies of the real LHC data will berequired in order to produce reliable estimates of theseuncertainties.Prior to data we assign conservative error bars on thesebackground projections. We have checked that 100 pb − of data in the MET trigger sample is sufficient for a 5 σ discovery for the eight models in our study, even if wetriple the backgrounds quoted above and include a 15% (GeV) T E100 150 200 250 300 350 T r i gg er e ff i c i e n c y FIG. 3: The TriJet trigger efficiency. (GeV/c) T p0 10 20 30 40 50 60 T r i gg er e ff i c i e n c y FIG. 4: The Muon20 trigger efficiency. overall systematic error. The look-alike analysis will bedegraded, however, in the event that the Standard Modelbackgrounds turn out to be much larger than currentestimates.Prior to data, it is also difficult to make a reliable esti-mate of the main systematic uncertainties that will affectthe inclusive missing energy analysis. Systematic uncer-tainties will decrease over time, as the detectors are bet-ter understood, calibration studies are performed, andStandard Model physics is analyzed with the LHC data.For our study we have assumed that, at the moment ofdiscovery, the dominant systematic errors in the full dis-covery data set will come from three sources: • Luminosity uncertainty: it affects the counting ofevents. This systematic uncertainty is process in-dependent. • Detector simulation uncertainty: it mainly affectscalorimetry-related variables in our study, in par-ticular jet counting and the missing energy. Thissystematic is partially process dependent. • QCD uncertainty: it includes the uncertaintiesfrom the parton distribution functions, higher ordermatrix elements, and large logarithms. This uncer-tainty affects event counting, jet counting and theshapes of kinematic distributions. It is partiallyprocess dependent.Note that, since we use uncorrected jets, we do nothave a systematic from the jet energy scale. This istraded for a portion of the detector simulation uncer-tainty, i.e. how well we can map signal events into un-corrected jets as would be measured in the real detector.
D. Simulation of the signals
A realistic study of look-alikes requires full detectorsimulation. For the initial phase of this work a genera-tor level analysis is attractive, being computationally lessintensive and providing a clear link between observablesand the underlying theory models .In a generator level analysis, jets are reconstructed byapplying a standard algorithm to particles rather thanto calorimeter towers. This obviously does not capturethe effects of a realistic calorimeter response, calorimetersegmentation, and energy losses due to material in thetracker as well as magnetic field effects.A compromise between the full simulation and a gener-ator level analysis is a parameterized detector simulation.For the LHC the publicly available software packages in-clude AcerDET [27], and
PGS [28]. In such a simulation,electrons, muons and photons can be reconstructed us-ing parameterized efficiencies and resolutions based onabstract but educated rules-of-thumb for modern mul-tipurpose detectors. Jets are reconstructed in a virtualcalorimeter, from particle energies deposited in cells thatroughly mimic the segmentation of a real calorimeter.Calorimeter response is approximated by performing aGaussian smearing on these energy deposits. The E miss T isreconstructed from the smeared energies in these virtualtowers. The full
GEANT4 -based simulation is too slow to adequately sam-ple the entire theory space. Having completed the first ex-ploratory phase of this work, we are repeating the analysis tovalidate these results with the full experimental simulation.
We performed a preliminary study by comparing
PGS results to the full simulation results reported for theSUSY benchmark model LM1 [10, 11]. We found that
PGS jets are not a good approximation of uncorrectedjets in the full simulation, even for the most basic prop-erties such as the E T spectrum. Varying the parametersand adding simple improvements, such as taking into ac-count the 4 Tesla field in the barrel, did not change thisconclusion. PGS jets have a behavior, not surprisingly,that is intermediate between generator level jets and un-corrected full simulation jets.We developed a modified simulation called
PGSCMS with the geometry and approximate magnetic field ofthe CMS detector. The
PGS
Gaussian smearing anduninstrumented effects in the calorimeters are turned off.Electrons, muons and photons are extracted at generatorlevel, and
PGS tau reconstruction is not used. Track infor-mation is extracted as in the standard
PGS . The calorime-ter output improves on a generator level analysis in thatwe include approximations to the effects of segmentationand the 4 Tesla field, as well as an η correction derivedfrom the z value of the primary vertex. We parameterizedthe detector response in a limited set of look-up tablesas a function of the generator-level quantities.At the analysis level we apply parameterized correc-tions and reconstruction efficiencies inspired by the pub-lished CMS detector performance [29]. For the jets, weapply an E T and η dependent rescaling of their E T , tunedto reproduce the full simulation LM1 results in [10, 11].This rescaling makes the jets softer ( i.e. takes into ac-count the detector reconstruction): a 50 GeV generatorlevel jet becomes an approximately 30 GeV raw jet in ouranalysis.The E miss T reconstructed from PGSCMS is essentiallyidentical, modulo small calorimeter segmentation effects,to a a generator level analysis, i.e. our E miss T is virtuallyindistinguishable from the Monte Carlo truth E miss T ob-tained from minus the vector sum of the E T of neutrinos,muons and the other weakly interacting particles (suchas the LSP). We did not attempt to rescale the E miss T ;this is a complicated task since E miss T is a vector andin general energy losses, calorimeter response and mis-measurements tend to decrease the real large E miss T tailswhile increasing the E miss T tails in the distribution of non-real E miss T events. Instead of attempting to rescale the E miss T event by event, we raised the E miss T cut in ourbenchmark analysis to 220 GeV .Because of the limitations of our fast simulation, wealso simplified parts of the benchmark analysis. The firstphase primary cleanup is dropped since it is related tosupression of spurious processes that we do not simulate,and it is nearly 100% efficient for the signal. We also dropthe jet electromagnetic fraction cuts of the ILV, because In a realistic full simulation study with the first jet data in hand,our E miss T analysis will avoid such compromises. they are nearly 100% efficient for the signal.The resulting performance of our parameterized fastsimulation for the SUSY benchmark model LM1 is shownin Table II. The agreement with the full simulation studyis very good. The largest single cut discrepancy is 2%;this occurs for the QCD angular cuts, reflecting the ex-pected fact that our fast simulation does not accuratelyreproduce jet mismeasurement effects. Since the finalefficiencies agree to within 7%, it is plausible that look-alikes defined in our fast simulation study will remainlook-alikes in our upcoming full simulation study.It is important to note that this fast simulation doesnot reproduce the Standard Model background efficien-cies shown in Table I. In fact the discrepancies in thetotal efficiencies can approach an order of magnitude.This is to be expected. We are cutting very hard onthe Standard Model events, thus the events that pass arevery atypical. This is in contrast to the signal events,where the fraction that pass are still fairly generic, andtheir E T and E miss T spectra near the cuts are less steeplyfalling than those of the background. Since SM back-grounds cannot be estimated from a PGS level analysis,we take our backgrounds from the state-of-the-art analy-sis in [10]; this approach only works because we have alsomatched the analysis path used in [10].The full software chains we use in our study are sum-marized in Table III. All of the simulated data sets in-clude an average of 5 pileup events added to each sig-nal event, corresponding to low luminosity LHC running( ∼ cm − s − ). III. POPULATING THE THEORY SPACE
In Section II we gave a partial classification of BSMmodels according to how many new weakly interactingparticles appear in a typical final state. Our benchmark E miss T analysis is optimized for the case of two heavyweakly interacting particles per event, as applies to SUSYmodels with conserved R parity, little Higgs models withconserved T parity and Universal Extra Dimensions mod-els with conserved KK parity. This study is a first at-tempt at constructing groups of look-alike models drawnfrom this rather large fraction of the BSM theory space,and developing strategies to discriminate them shortlyafter an initial discovery.One caveat is that models from other corners of thetheory space may also be look-alikes of the ones consid-ered here. For example, models with strong production ofheavy particles that decay to boosted top quarks can pro-duce higher E T jets and larger E miss T from neutrinos thandoes Standard Model top production. Such look-alikepossibilities also require study, but they are not a majorworry since our results show that we have some ability todiscriminate heavy WIMPS from neutrinos even in smalldata sets. A. SUSY
In a large class of supersymmetry models with con-served R parity, not necessarily restricted to the MSSM,the LSP is either the lightest neutralino or a right-handedsneutrino .In addition, if the NLSP is a neutralino or sneu-trino and the LSP is a gravitino, the E miss T signatureis the same. Models based on gravity-mediated, gauge-mediated or anomaly-mediated SUSY breaking all pro-vide many candidate models.Because this relevant portion of SUSY theory space isalready so vast, there is a temptation to reduce the scopeof the LHC Inverse Problem by making explicit or im-plicit theoretical assumptions. To take an extreme, onecould approach an early LHC discovery in the E miss T chan-nel having already made the assumptions that (i) the sig-nal is SUSY, (ii) it has a minimal Higgs sector (MSSM), (iii) it has gravity-mediated SUSY breaking (SUGRA), (iv) the breaking is minimal (mSUGRA) and (v) test theoretical hypothesesin the LHC discovery data set combined with other mea-surements.For SUSY we have the benefit of more than one spec-trum calculator that can handle general models, morethan one matrix element calculator and event generationscheme, and a standardized interface via the SUSY LesHouches Accord (SLHA) [32]. There are still a few bugsin this grand edifice, but the existing functionality com-bined with the ability to perform multiple cross-checksputs us within sight of where we need to be when thedata arrives. B. Little Higgs
Little Higgs models are a promising alternative to weakscale supersymmetry [33]-[37]. In little Higgs models, theHiggs is an approximate Goldstone boson, with globalsymmetries protecting its mass (which originates from aquantum level breaking of these symmetries) from largeradiative corrections. Many of these LH models requirean approximate T parity discrete symmetry to reconcileLH with electroweak precision data. This symmetry issimilar to R parity in SUSY models. The new LH par-ticles that would be produced at the LHC would be oddunder this symmetry, enforcing the stability of the light-est particle that is odd under T parity. This new particleis weakly interacting and would manifest itself as missing Recent analyses [30, 31] have argued for the phenomenologicalviability of sneutrino dark matter.
TABLE II: Comparison of cut-by-cut selection efficiencies for our E miss T analysis applied to the SUSY benchmark model LM1.“Full” refers to the full simulation study [10, 11]; “Fast” is what we obtain from our parameterized fast simulation.Cut/Software Full FastTrigger and E miss T >
200 GeV 53.9% 54.5% N j ≥ | η j d | ≥ . Iso lead trk = 0 85.3% 85.5% E T, >
180 GeV, E T, >
110 GeV 63.0% 63.0% H T >
500 GeV 92.8% 93.9%Total efficiency 12.9% 13.8%TABLE III: Summary of software chains used in this study. The little Higgs spectrum is based on [38].
PGSCMS is a variationof
PGS v4 [28]. Software/Models Group 1 models Group 2 modelsSpectrum generator
Isajet v7.69 [39] or private little Higgsor
SUSY-HIT v1.1 [40] or
SuSpect v2.34 [41]Matrix element calculator
Pythia v6.4 [42]
MadGraph v4 [43]Event generator
Pythia v6.4 MadEvent v4 [44]with
BRIDGE [45]Showering and hadronization
Pythia v6.4 Pythia v6.4
Detector simulation
PGSCMS v1.2.5 PGSCMS v1.2.5 plus parameterized plus parameterizedcorrections corrections energy at the LHC .Just as in SUSY, new colored particles are the dom-inant production modes. These particles subsequentlygenerate high multiplicity final states through decaychains that end with the lightest T odd particle. In LHmodels, the strongly coupled particles are T odd quarks(TOQ’s), analogous to the squarks of SUSY. The weaklycoupled analogues of the gauginos are T odd spin one vec-tor bosons (TOV’s). In the models considered to date,there is no analog of the gluino: this is an important con-sideration in constructing supersymmetric look-alikes ofLH models.In this study, we work with a minimal implementationof a little Higgs model with T parity that is known as thelittlest Higgs model with T parity. This model is basedon a SU (5) /SO (5) pattern of global symmetry breaking.Each SM particle except the gluon has an associated LHpartner odd under T parity. There is also an extra pairof top partners, one T odd and the other T even, as well This symmetry may be inexact, or violated by anomalies [46].Such possibilities are model dependent [47, 48]. as singlets. The lightest T odd particle in this model welabel A H . It is a heavy gauge boson that is an admixtureof a heavy copy of the hypercharge gauge boson and aheavy W boson.For event generation, we use a private implementationof the littlest Higgs model within MadGraph . There is aneed to generalize this to a wider class of models.
C. Universal extra dimensions
Universal extra dimensions models are based on orb-ifolds of one or two TeV − size extra spatial dimen-sions [49]-[56]. The five-dimensional version of UED isthe simplest. At the first level of Kaluza-Klein ( KK )excitations, each Standard Model boson has an associ-ated partner particle, and each Standard Model fermionhas two associated partner particles (i.e. a vector-likepair). These KK partners are odd under a KK parity,the remnant of the broken translational invariance alongthe fifth dimension. This parity is assumed to be an ex-act symmetry. After taking into account mass splittingsdue to Standard Model radiative corrections, one findsthat the lightest KK odd partner is naturally the weakly0interacting partner of the hypercharge gauge boson. Awide variety of spectra for the KK odd partners can beobtained by introducing additional interactions that arelocalized at the orbifold fixed points; these choices dis-tinguish generic UED from the original minimal modelof [49]. These models resemble SUSY.A public event generation code based on a modifica-tion of Pythia is available for generic 5-dimensional UEDmodels [57]. There is a need to generalize this to a widerclass of models, e.g. 6-dimensional UED. In our studywe have not used any UED examples, but we will includethem in the future.
IV. DESCRIPTION OF THE MODELSA. Group 1
The five look-alike models of Group 1 are all MSSMmodels. Two of them (LM5 and LM8) are CMS SUSYbenchmark models, while another (LM2p) is a slight vari-ation of a CMS benchmark. It is a sobering coincidencethat these are look-alikes of the E miss T analysis, since thebenchmarks were developed by CMS to cover different ex-perimental signatures, not produce look-alikes. To roundout Group 1 we found two other MSSM look-alikes whosespectra and decay chains are as different from each otherand from the three CMS benchmarks as we could makethem.The models are consistent with all current experimen-tal constraints, but do not all give the “correct” relicdensity of dark matter. Any comparison of relic densi-ties to the so-called WMAP constraints assumes at leastthree facts not yet in evidence: (i) that dark matter is athermal relic, (ii) that there is only one significant speciesof dark matter and (iii) that cosmological evolution wasentirely radiation-dominated from the time of dark mat-ter decoupling until the time of Big Bang nucleosynthe-sis. A missing energy discovery at the LHC will help ustest whether these assumptions have any validity. Forexample, model LM8 produces a relic density an orderof magnitude larger than the WMAP upper bound; thusdiscriminating LM8 as a more likely explanation of anearly missing energy discovery would call into question[58, 59] assumptions (i) and (iii), or could be a hint thatthe lightest neutralino is not absolutely stable.LM2p, LM5 and LM8 are minimal supergravity mod-els [60]-[62]. They are specified by the usual high scalemSUGRA input parameters as shown in Table IV; be-cause the resulting superpartner spectra depend stronglyon RGE running from the high scale, a complete specifi-cation of the models also requires fixing the top quarkmass and the particular spectrum generator programused. We have used m top = 175 GeV and the ISAJETv7.69 generator [39], in order to maintain compatibilitywith the CMS Physics TDR [10]. Models LM5 and LM8are then identical to the mSUGRA benchmark models ofthe CMS Physics TDR, while LM2p is almost identical to benchmark model LM2; LM2p has a slightly larger valueof m / (360 versus 350 GeV) than LM2, which makes it more of a look-alike of the other Group 1 models. TABLE IV: Input parameters for the mSUGRA modelsLM2p, LM5 and LM8. The notation comforms to [39]. Themass parameters and trilinear A parameter have units ofGeV. LM2p LM5 LM8 m
185 230 500 m /
360 360 300 A β
35 10 10sign( µ ) + + +TABLE V: Input parameters for the MSSM models CS4d andCS6. The notation conforms to [40, 41]. The mass parametersand trilinear A parameters have units of GeV.CS4d CS6 M
620 400 M
930 600 M
310 200 A τ , A t , A b , A e , A u , A d -400 -300 M Q L , M t R , M b R
340 2000 M q u , M u R , M d R
340 2000 M τ L , M τ R , M e L , M e R
340 340 M h u , M h d β
10 10sign( µ ) + + The Group 1 models CS4d and CS6 are not minimalsupergravity; they are more general high scale MSSMmodels based on the compressed supersymmetry idea ofMartin [63, 64]. The high scale input parameters areshown in Table V. We have used m top = 175 GeVand the spectrum generator combination SuSpect v2.34 with
SUSY-HIT v1.1 [40, 41]. Model CS4d is in factpart of the compressed SUSY model line defined in [63].Model CS6 is a modification of compressed SUSY whereall of the squarks have been made very heavy, > ∼ ] m a ss [ G e V / c LM2p LM5 LM8 CS4d CS6 c~ t~ R l~ c~ – c~ t~ R d~ R u~ L u~ L d~g~ c~ t~ R l~ c~ – c~ t~ R d~ R u~ L u~ L d~g~ c~ c~ – c~ t~ R l~ t~g~ R d~ R u~ L u~ L d~ c~ t~ – c~ c~ t~ R l~ R d~ R u~g~ L u~ L d~ c~ t~ R l~ – c~ c~ g~ FIG. 5: The mass spectra of the MSSM models LM2p, LM5, LM8, CS4d and CS6. Only the most relevant particles are shown:the lighter gauginos ˜ χ , ˜ χ and ˜ χ ± , the lightest stau ˜ τ , the right-smuon and selectron denoted collectively as ˜ ℓ R , the lighteststop ˜ t , the gluino, and the left/right up and down squarks ˜ u L , ˜ u R , ˜ d L and ˜ d R . The very heavy ≃ Group 1 models both before and after our event selec-tion. The production fractions are much more similarafter the event selection than before it; this is expectedbecause the selection shapes the kinematics of the surviv-ing sample. Gluino pair production dominates for modelCS6, while squark-gluino and squark-squark productiondominate for the other four models. Pair production ofthe lightest stop is important for model CS4d before theselection cuts, but after the event selection very few ofthese events remain.Table VII shows the most relevant superpartner de-cay branching fractions. For models LM2p and LM5,gluino decay is predominantly to quark+squark; forLM8 and CS4d it is dominantly to top and the light-est stop, and gluinos decay in CS6 mostly through the three-body mode qq ˜ χ . For models LM2p, LM5 andLM8, left-squarks cascade through quark+chargino orquark+second neutralino; right-squarks have a two-bodydecay to quark+LSP; right-squarks in model LM8 alsohave a large branching to quark+gluino. In model CS4dleft-squarks decay almost entirely to quark+gluino, whileright-squarks decay almost entirely to quark+LSP; forCS6 all squarks except the stop decay dominantly toquark+gluino.In models LM2p, LM5 and LM8 the decays of thelightest stop split between b +chargino and top+LSP; forCS4d ˜ t decays 100% via the three-body mode bW + ˜ χ ,while for CS6 almost all of the decays are to top+gluino.Chargino decay is dominated by decays to the lighteststau and a neutrino for models LM2p and CS6, and by2 TABLE VI: Summary of LHC superpartner production for the Group 1 MSSM models LM2p, LM5, LM8, CS4d and CS6. Therelative percentages are shown for each model, both before and after the event selection. The squark–squark percentages shownare excluding the contributions from pair production of the lightest stops, which are shown separately. Note that squark–chargino includes the production of either chargino, and squark-neutralino includes all of the four neutralinos. The category“other” includes weak production as well as the associated production of gluinos with charginos or neutralinos. The total NLOcross sections are from
Prospino2 [65]. LM2p LM5 LM8 CS4d CS6NLO cross section (pb) 8.6 8.1 12.7 14.5 12.6before after before after before after before after before aftercuts cuts cuts cuts cuts cuts cuts cuts cuts cutssquark–squark 33% 36% 32% 38% 22% 33% 19% 34% 0.1% 0.1%squark–gluino 45% 55% 46% 52% 48% 54% 41% 55% 3.7% 7.4%gluino–gluino 7.2% 6.4% 7.4% 6.4% 14% 8.3% 11% 8% 95% 92%stop–stop 2.1% 1.1% 2.1% 0.9% 2.6% 1.5% 26% 1.4% - -squark–chargino 2.1% 0.5% 2.1% 0.7% 1.4% 0.7% 0.2% 0.2% - -squark–neutralino 1.7% 0.4% 1.8% 0.4% 1.2% 0.6% 0.6% 0.2% - -other 9.5% 0.7% 9.3% 0.8% 11% 0.8% 1.9% 0.3% 1.1% 0.1%TABLE VII: Summary of most relevant superpartner decaysfor the MSSM models LM2p, LM5, LM8, CS4d and CS6.LM2p LM5 LM8 CS4d CS6˜ g → ˜ qq
45% 45% - - - → ˜ b b
25% 20% 14% 2% - → ˜ t t
16% 23% 81% 94% - → q ¯ q ˜ χ - - 5% - 75%˜ u L → d ˜ χ ±
64% 64% 55% - - → u ˜ χ
32% 32% 27% - - → u ˜ g - - - 83% 85%˜ u R → u ˜ χ
99% 99% 62% 92% - → u ˜ g - - 38% - 85%˜ b → t ˜ χ −
42% 36% 35% 20% 9% → b ˜ χ
29% 23% 22% 14% 5% → b ˜ χ
7% 2% 1% 50% - → b ˜ g - - - - 85%˜ t → b ˜ χ +1
45% 43% 42% - - → t ˜ χ
22% 25% 30% - 4% → t ˜ g - - - - 96% → bW + ˜ χ - - - 100% -˜ χ ± → W ± ˜ χ
5% 97% 100% 100% 2% → ˜ τ ± ν τ
95% - - - 77%˜ χ → Z ˜ χ
1% 11% 100% 100% - → h ˜ χ
3% 85% - - 2% → ˜ τ τ
96% 3% - - 77%˜ τ → τ ˜ χ decays to W +LSP for models LM5, LM8 and CS4d. Thesecond neutralino ˜ χ decays almost entirely to τ +staufor models LM2p and CS6, and goes 100% to Z +LSP inmodels LM8 and CS4d. The LM5 model has the distinct feature that 85% of ˜ χ decays are to Higgs+LSP.Table VIII shows the most significant inclusive finalstates for the Group 1 models. By final state we meanthat all unstable superpartners have decayed, while Stan-dard Model particles are left undecayed. We use q to de-note any first or second generation quark or antiquark,but list bottom and top quarks separately. The percent-age frequency of each final state is with respect to theevents passing our selection. The final states are inclu-sive, thus e.g. the events in the qqq ˜ χ ˜ χ final state area subset of those in the qq ˜ χ ˜ χ final state, and the totalpercentages in each column exceed 100% . By the sametoken, most exclusive final states actually have more par-tons than are listed for the corresponding inclusive entriesin Table VIII, so even at leading order parton level theyproduce more jets.For models LM2p, LM5 and CS6, the dominant inclu-sive final state is qq ˜ χ ˜ χ + X , i.e. multijets plus missingenergy from the two LSPs. This is the motivation behindthe design of our analysis. For model CS4d, the mostlikely production is squark-gluino followed by squark de-cay to quark+LSP; the gluino then decays to top+stop,with the stop decaying via the three-body mode bW + ˜ χ .The most popular exclusive final state is thus btW q ˜ χ ˜ χ .Similarly, for LM8 the most popular exclusive final statesare btW q ˜ χ ˜ χ and ttq ˜ χ ˜ χ , from squark-gluino produc-tion followed by gluino decay to top+stop.Final states with W ’s are prevalent in models LM5,LM8 and CS4d. The LM2p model stands out becauseof the high probability of taus in the final state. ModelLM5 produces a significant number of light Higgses fromsuperpartner decays. Model LM8 has a large fraction ofevents with Z bosons in the final state. Model CS4d is en-riched in final states with multiple tops and W ’s, of whichone representative example is shown: bbttW W ˜ χ ˜ χ .Summarizing this discussion, we list the most signifi-cant features of each model in Group 1:3 TABLE VIII: Summary of significant inclusive partonic fi-nal states for the Group 1 MSSM models LM2p, LM5, LM8,CS4d and CS6. By final state we mean that all unstablesuperpartners have decayed, while Standard Model particlesare left undecayed. Here q denotes any first or second genera-tion quark or antiquark, and more generally the notation doesnot distinguish particles from antiparticles. The percentagefrequency of each final state is with respect to the events pass-ing our selection.The final states are inclusive, thus e.g. theevents in the qqq ˜ χ ˜ χ final state are a subset of those in the qq ˜ χ ˜ χ final state, and the total percentages in each columnexceed 100% . LM2p LM5 LM8 CS4d CS6 qq ˜ χ ˜ χ
57% 61% 34% 38% 98% qqq ˜ χ ˜ χ
20% 19% 3% 4% 79% qqqq ˜ χ ˜ χ
1% 1% 1% 1% 77% τ ν τ q ˜ χ ˜ χ
39% 1% - - 1% τ τ q ˜ χ ˜ χ
25% 1% - - 1% b q ˜ χ ˜ χ
30% 25% 33% 69% 19% b t W q ˜ χ ˜ χ
10% 19% 31% 67% -
W q ˜ χ ˜ χ
25% 52% 56% 93% - h q ˜ χ ˜ χ
3% 20% - - - tt q ˜ χ ˜ χ
9% 4% 40% 11% 2%
Z q ˜ χ ˜ χ
10% 8% 35% 11% -
Z W q ˜ χ ˜ χ
2% 6% 23% 6% - bb tt W W ˜ χ ˜ χ - - 2% 18% -TABLE IX: Estimated number of events passing our selectionper 100 pb − of integrated luminosity, for the Group 1 modelsLM2p, LM5, LM8, CS4d and CS6. These estimates use NLOcross sections and the CTEQ5L pdfs.LM2p LM5 LM8 CS4d CS6211 200 195 195 212
Model LM2p:
800 GeV squarks are slightly lighterthan the gluino, and there is a 155 GeV stau. Dom-inant production is squark-gluino and squark-squark.Left-squarks decay about two-thirds of the time toquark+chargino, and one-third to quark+LSP; right-squarks decay to quark+LSP. Gluino decay is mostly toquark+squark. Charginos decay to the light stau plus aneutrino, while the second neutralino decays to τ +stau.Two-thirds of the final states after event selection haveat least one τ . Model LM5:
800 GeV squarks are slightly lighterthan the gluino. Dominant production is squark-gluinoand squark pairs. Left-squark decays about two-thirdsto quark+chargino, and one-third to quark+LSP; right-squarks decay to quark+LSP. Gluino decay is mostly toquark+squark. Charginos decay to a W and an LSP,while the second neutralino decays to a light Higgs andan LSP. After selection more than half of final states havea W boson, and a fifth have a Higgs. Model LM8:
The 745 GeV gluino is slightly lighterthan all of the squarks except ˜ b and ˜ t . Dominant pro-duction is squark-gluino and squark pairs. Left-squarksdecay about two-thirds to quark+chargino, and one-third to quark+LSP; right-squarks decay two-thirds toquark+LSP and one-third to quark+gluino. Gluino de-cay is dominantly to top and a stop; the 548 GeV stopsdecay mostly to b +chargino or top+LSP. Charginos de-cay to W +LSP, and the second neutralino decays to Z +LSP. After selection 40% of final states have two tops,which may or may not have the same sign. More thanhalf of the final states have a W , more than a third havea Z , and a quarter have both a W and a Z . Model CS4d:
The 753 GeV gluino is in between theright-squark and left-squark masses. The LSP is rela-tively heavy, 251 GeV, and the ratio of the gluino to LSPmass is small compared to mSUGRA models. Dominantproduction is squark-gluino and squark-squark. Left-squarks decay to quark+gluino, and right-squarks decayto quark+LSP. Gluinos decay to top and a stop; the 352GeV stops decay 100% to bW + ˜ χ . Two-thirds of the fi-nal states contain btW q ˜ χ ˜ χ , and a significant fractionof these contain more b ’s, t ’s and W ’s. Model CS6:
The 589 GeV gluino is much lighter thanthe 2 TeV squarks, and the ratio of the gluino to LSPmass is small compared to mSUGRA models. Productionis 92% gluino-gluino, and gluinos decay predominantlyvia the three-body mode qq ˜ χ . The final states consistalmost entirely of three or four quarks plus two LSPs,with a proportionate amount of the final state quarksbeing b ’s. TABLE X: Parameter choices defining the little Higgs modelLH2. We choose our conventions to agree with those foundin [38]: f is the symmetry-breaking scale, κ iq is the T -oddquark Yukawa coupling, κ il is the T -odd lepton Yukawa cou-pling and sin α is a mixing angle. CKM mixing has beensuppressed for our analysis.. f
700 GeV κ iq κ il α B. Group 2
Group 2 consists of three look-alike models: LH2, NM4and CS7, and a comparison model NM6. LH2 is a lit-tlest Higgs model with conserved T parity. The param-eter choices defining this model are shown in Table X.The mass spectrum of the lighter partners is shown inFigure 6; not shown are the heavier top partners T + , T − with tuned masses 3083 and 3169 GeV respectively, thecharged lepton partners ℓ H , ℓ H , ℓ H with mass 2522 GeVand the neutrino partners ν H , ν H , ν H with mass 25464 ] m a ss [ G e V / c LH2 NM6 NM4 CS7 H A H W H Z Hi d Hi u c~ – c~ c~ R d~ L u~ R u~ L d~ c~ – c~ c~ R u~ R d~ L u~ t~ L d~ R l~ c~ t~ R l~ – c~ c~ g~ FIG. 6: The mass spectra of the models LH2, NM6, NM4 and CS7. Only the most relevant partners are shown: the lightergauginos ˜ χ , ˜ χ and ˜ χ ± , the lightest stau ˜ τ , the right-smuon and selectron denoted collectively as ˜ ℓ R , the gluino, and theleft/right up and down squarks ˜ u L , ˜ u R , ˜ d L and ˜ d R . For the little Higgs model LH2, the relevant quark and vector partners areshown: the gauge boson partners A H , Z H , W H , and the three generations of quark partners u iH , d iH , i = 1 , , GeV. Model LH2 is consistent with all current experi-mental constraints [36, 38].NM6, NM4 and CS7 are all MSSM models. Thehigh scale input parameters are listed in Table XI. Wehave used m top = 175 GeV and the spectrum generator SuSpect v2.34 . The mass spectra are shown in Figure 6;not shown are the heavy gluinos of NM6 and NM4 withmasses 2000 and 1536 GeV respectively, and the > ∼ not look-alikes ofour benchmark inclusive missing energy analysis. ModelsNM4 and CS7, by contrast, are SUSY look-alikes of thelittle Higgs model LH2. The superpartner spectrum ofNM4 is roughly similar to the partner spectrum of LH2,but the superpartners are lighter. The spectrum of CS7has no similarity to that of LH2.The relative frequency of various LHC little Higgs part-ner production processes are shown in Table XII, for theLH2 model both before and after our event selection. ForLH2 the predominant process is gg or q ¯ q partons initiat-ing QCD production of a heavy partner quark-antiquarkpair; this process is completely equivalent to t ¯ t produc-tion at the LHC. The most striking feature of Table XII5 TABLE XI: Input parameters for the MSSM models NM6,NM4 and CS7. The notation conforms to [40, 41]. The massparameters and trilinear A parameters have units of GeV.NM6 NM4 CS7 M
138 105 428 M
735 466 642 M A t A b A τ , A t , A b , A e , A u , A d M Q L
755 590 2000 M t R
760 580 2000 M q u
770 590 2000 M u R , M b R
770 580 2000 M d R
765 580 2000 M τ L , M τ R , M e L , M e R M h u , M h d β
10 10 10sign( µ ) + + + is that nearly half of the total production involves weakinteractions. For example the second largest productionmechanism, 14% of the total, has two valence quarks inthe initial state producing a pair of first generation heavypartner quarks; at tree-level this is from s -channel anni-hilation into a W and t -channel exchange of a Z H or A H partner.The superpartner production at the LHC for the SUSYmodels NM6, NM4 and CS7 is summarized in Table XIII.For NM6 and NM4 a major contribution is from gg or q ¯ q partons initiating QCD production of a squark-antisquark pair. Production of a first generation squarkpair from two initial state valence quarks is also impor-tant; in contrast to the LH2 non-SUSY analog this isa QCD process with t -channel exchange of the heavygluino. For model CS7, which has a light gluino and veryheavy squarks, 96% of the production is gluino pairs.The primary decay modes for the lighter LH2 partnersare shown in Table XIV, while those for the SUSY mod-els are summarized in Table XV. Tables XVI and XVIIdisplay the most significant inclusive partonic final statesfor the Group 2 models.For LH2, a large fraction of heavy partner quarks havea direct 2-body decay to a quark and an A H WIMP. Theother heavy partner quark decay mode is a two stagecascade decay via the W H and Z H partner bosons. Sincethe W H decays 100% to W A H while the Z H decays 100%to hA H , a large fraction of events have a W or a Higgsin the final state.Analogous statements apply to the SUSY models NM6and NM4. We see that 100% of right-squarks and a sig-nificant fraction of left-squarks undergo a direct 2-bodydecay to quark+LSP. The rest have mostly a two stage cascade via the lightest chargino ˜ χ ± or the second neu-tralino ˜ χ . Since ˜ χ ± decays 100% to W +LSP, while the˜ χ decays dominantly to a Higgs+LSP, a significant frac-tion of events have a W or a Higgs in the final state.For the remaining SUSY model CS7, gluino pair pro-duction is followed by 3-body decays of each gluino toa quark-antiquark pair + LSP. As can be seen in Ta-ble XVII, this leads to high jet multiplicity but nothingelse of note besides a proportionate number of b ¯ b and t ¯ t pairs. TABLE XII: Production channels for little Higgs partners inthe LH2 model, both before and after the event selection.Here Q stands for any of the quark partners u iH , d iH , i =1 , ,
3. The total LO cross section as reported by
MadEvent is6.5 pb. before cuts after cuts Q i ¯ Q i
55% 64% u iH d iH , u iH u iH , d iH d iH
14% 16% d iH W + H , u iH W − H
12% 7% u iH Z H , u iH A H , d iH Z H , d iH A H
9% 5% Q i ¯ Q j , i = j
3% 3%other 7% 5%
C. Comparison of models differing only by spin
We have already noted that SUSY model NM6 hasa superpartner spectrum almost identical to the heavypartner spectrum of the non-SUSY little Higgs modelLH2. The only relevant difference, other than the spinsof the partners, is that model NM6 has a very heavy 2TeV gluino that has no analog in LH2. Despite being veryheavy, the gluino does make a significant contribution tosquark-squark production via t -channel exchange.This pair of models provides the opportunity for a com-parison of realistic models that within a good approxima-tion differ only by the spins of the partner particles. Ifit turned out that these two models were look-alikes inour benchmark inclusive missing energy analysis, thendiscriminating them would be physically equivalent todetermining the spins of at least some of the heavy part-ners.It is an ancient observation (see e.g. [66]) that modelsdiffering only by the spins of the new heavy exotics havesignificant differences in total cross section. The mostfamiliar example is the comparison of pair productionof heavy leptons near threshold with pair production ofspinless sleptons. For mass m and total energy √ s , thelepton cross section is proportional to β , β ≡ q − m s , (2)while the slepton cross section is proportional to β .Thus slepton production is suppressed near threshold6 TABLE XIII: Summary of LHC superpartner production for the Group 2 MSSM models NM6, NM4 and CS7. The relativepercentages are shown for each model, both before and after the event selection. Here ˜ q i denotes any of the three generationsof left and right squarks. Note that squark–chargino includes the production of either chargino, and squark-neutralino includesall of the four neutralinos. The LO total cross sections are as reported by MadEvent .NM6 NM4 CS7LO cross section (pb) 2.3 10.3 5.0before after before after before aftercuts cuts cuts cuts cuts cuts˜ q i ¯˜ q i
31% 29% 34% 26% - -˜ u ˜ d , ˜ u ˜ u , ˜ d ˜ d
32% 28% 29 23% - -squark–gluino 3% 10% 5% 23% 4% 8%gluino–gluino - - - - 96% 91%squark–chargino 2% 2% 3% 1% - -squark–neutralino 4% 1% 4% - - -˜ q i ¯˜ q j , i = j
15% 17% 17% 14% - -other 13% 13% 8% 13% - -TABLE XIV: Decay modes for the lighter little Higgs partnersof model LH2. d iH , i = 1 , → u i W H → d i Z H → d i A H d H → bZ H → bA H u iH , i = 1 , → d i W H → u i Z H → u i A H u H → bW H → tA H W H → W A H Z H → hA H compared to production of heavy leptons with the samemass. A somewhat less familiar fact is that the sleptonsnever catch up: even if we introduce both left and rightsleptons, to match the degrees of freedom of a Dirac lep-ton, the total cross section for left+right slepton pairsis one-half that of Dirac lepton pairs in the high energylimit β → TABLE XV: Summary of most relevant superpartner decaysfor the MSSM models NM6, NM4 and CS7.NM6 NM4 CS7˜ g → ˜ qq
66% 67% - → ˜ bb
17% 17% - → ˜ t t
17% 16% - → q ¯ q ˜ χ - - 99%˜ u L → d ˜ χ ±
39% 59% 12% → u ˜ χ
44% 12% -˜ u R → u ˜ χ b → b ˜ χ
24% 14% 6% → b ˜ χ
70% 86% -˜ t → b ˜ χ +1
40% 27% - → t ˜ χ
60% 73% 5%˜ χ ± → W ± ˜ χ → ˜ τν τ - - 35% → ˜ νℓ - - 28% → ˜ ντ - - 17%˜ χ → Z ˜ χ
22% 19% - → h ˜ χ
78% 81% - → ˜ τ τ - - 39% → ˜ νν - - 45% → ˜ ℓℓ - - 16% twins at the LHC using total cross section was first sug-gested by Datta, Kane and Toharia [67], and studied inmore detail in [68]. To implement this idea, we must alsocompare the relative efficiencies of the SUSY and non-SUSY twins in a real analysis, since what is measured inan experiment is not total cross section but rather crosssection times efficiency.An important observation is that the p T distributions,7 TABLE XVI: Significant inclusive partonic final states for thelittle Higgs model LH2. The percentage frequency of eachfinal state is with respect to the events passing our selection. qq A H A H W qq A H A H h qq A H A H bb A H A H W W qq A H A H hh bb A H A H hh qq A H A H tt A H A H qq ˜ χ ˜ χ
84% 83% 100% qqq ˜ χ ˜ χ
8% 16% 100% qqqq ˜ χ ˜ χ - - 95% bb q ˜ χ ˜ χ
2% 5% 11%
W qq ˜ χ ˜ χ
26% 35% - h q ˜ χ ˜ χ
14% 19% - tt q ˜ χ ˜ χ
1% 1% 11%
Z q ˜ χ ˜ χ
4% 5% -
W W q ˜ χ ˜ χ
4% 9% - in addition to the total cross sections, have large dif-ferences due solely to differences in spin. As an exam-ple, consider the LHC production of a pair of 500 GeVheavy quarks, versus the production of a pair of 500 GeVsquarks. We can compare the p T distributions by com-puting d log σdp T = 1 σ dσdp T (3)where we factor out the difference in the total cross sec-tions. Using the analytic formulae reviewed in the Ap-pendix, we have computed (3) for two relevant partonicsubprocesses. The first is gluon-gluon initiated produc-tion, for which the fully differential cross sections aregiven in (C33) and (C34); at leading order this arises TABLE XVIII: Estimated number of events passing our se-lection per 100 pb − of integrated luminosity, for the Group2 models LH2, NM6, NM4 and CS7. These estimates useleading order cross sections and the CTEQ5L pdfs.LH2 NM6 NM4 CS794 43 97 91 from an s -channel annihilation diagram, a gluon seag-ull for the squark case, and t and u channel exchangesof either the spin 1/2 heavy quark or the spin 0 squark.The second example is quark-antiquark initiated produc-tion, in the simplest case where the quark flavor does notmatch the quark/squark partner flavor; at leading orderthere is only one diagram: s -channel annihilation. Thefully differential cross sections are given in (C19) and(C20).For this simple example, we have integrated the fullydifferential cross sections over the parton fluxes, usingthe CTEQ5L parton distribution functions. The resultingnormalized p T distributions are shown in Figures 7 and8. For the gg initiated production the SUSY case has asignificantly softer p T distribution, while for the q ¯ q initi-ated production the SUSY case has a significantly harder p T distribution.We see similar differences in the complete models LH2and NM6. Figure 9 shows a comparison of the p T dis-tributions for heavy quark partner production from LH2and squark production for NM6. All of the leading orderpartonic subprocesses are combined in the plot, and noevent selection has been performed. The p T distributionfor the SUSY model is harder than for the non-SUSYmodel. Part of this net effect is due to intrinsic spin dif-ferences e.g. as depicted in Figure 8, and part is dueto SUSY diagrams with virtual gluino exchange. Onewould expect SUSY events to have a higher efficiency topass our missing energy selection than non-SUSY events.Indeed this is the case: 19% of NM6 events overall passthe selection, whereas only 14% of LH2 events do. Thehigher efficiency of SUSY NM6 events in passing the se-lection somewhat compensates for the smaller total crosssection compared to the non-SUSY LH2.The event counts can be obtained by multiplying eachtotal cross section times the total efficiency times theintegrated luminosity. For a 100 pb − sample, the totalsignal count is 94 events for model LH2 and 43 eventsfor model NM6. The net result is that although LH2and NM6 are twins in the sense of their spectra, they are not missing energy look-alikes in our benchmark analysis.Thus the good news is that, for models that differ only(or almost only) by spin, the event count in the discoverydata sample is already good enough to discriminate them.This is one of the important conclusions of our study.However this is also something of an academic exercise,since in the real experiment we will need to discriminatea large class of SUSY models from a large class of non-SUSY models. In this comparison a SUSY model canbe a look-alike of a non-SUSY model even though thespectra of partner particles don’t match. This is whathappens with SUSY models NM4 and CS7, which areboth look-alikes of LH2. Model NM4 looks particularlychallenging, since its superpartner spectrum is basicallyjust a lighter version of NM6. Compared to NM6, thetotal cross section of NM4 is more than 4 times larger(10.3 pb) while the efficiency to pass our missing energyselection is only half as good (9%). This gives a total8count of 97 events for 100 pb − , making NM4 a look-alike of LH2. (GeV/c) T p0 100 200 300 400 500 600 700 800 900 1000 - ( G e V / c ) T / dp s d l og FIG. 7: Comparison of the normalized p T distributions forleading order gg initiated production of a pair of 500 GeVparticles. The solid (red) line corresponds to quark-antiquarkpair; the dot-dashed (blue) line to a squark-antisquark pair.The distributions have been integrated over the parton fluxesusing the CTEQ5L pdfs.
V. OBSERVABLES
Having in mind an early discovery at the LHC, e.g.in the first 100 pb − of understood data, we have madeconservative assumptions about the physics objects thatwill be sufficiently well-understood for use in our look-alike analysis of a missing energy discovery.We assume that we can reconstruct and count high E T jets and hard tracks, as is required for our bench-mark missing energy selection. We do not assume thatvalidated jet corrections for multijet topologies will beavailable. We assume it will be possible to use the uncor-rected (raw) E miss T (without subtracting the momentum ofmuons or correcting for other calorimetric effects).We assume the ability to reconstruct and count high p T muons; a study of Z → µ + µ − events is a nec-essary precursor to understanding the Standard Model E miss T backgrounds. It will also be possible to count high E T electrons, however we are not yet including electronsin our study because of the high “fake” rate expected atstart-up. Multiflavor multilepton signatures are of greatimportance as model discriminators, though challengingwith small data sets; this is worthy of a separate dedi-cated study [69, 70].In our study instead of applying sophisticated b and τ tagging algorithms we isolate enriched samples of b quarks and hadronic τ ’s, by defining simple variables sim-ilar to the typical components of the complete tagging al- (GeV/c) T p0 100 200 300 400 500 600 700 800 900 1000 - ( G e V / c ) T / dp s d l og FIG. 8: Comparison of the normalized p T distributions forleading order q ¯ q initiated production of a pair of 500 GeV par-ticles, for the case that the initial parton flavor does not matchthe final parton flavor. The solid (red) line corresponds toquark-antiquark pair; the dot-dashed (blue) line to a squark-antisquark pair. The distributions have been integrated overthe parton fluxes using the CTEQ5L pdfs. gorithms: leptons in jets, track counting, and impact pa-rameter of charged tracks, to mention a few. This “poorman’s” tagging is not sufficient to obtain pure samplesof b ’s or τ ’s, but allows the discrimination of look-alikemodels with large differences in the b or τ multiplicity. A. Inclusive counts and ratios
We build our look-alike analysis strategy using simpleingredients. We start with the four trigger boxes definedin section II B: MET, DiJet, TriJet and Muon20. Forthe simulated data samples corresponding to each box,we compute the following inclusive counts of jets andmuons: • N, the number of events in a given box after ourbenchmark selection. • N(nj), the number of events with at least n jets(n=3,4,5). Note that N(3j) = N because of ourselection. • N(m µ -nj), the number of events with at least n jetsand m muons (n=3,4 and m=1,2). • N(ss µ ), the number of events with at least twosame-sign muons. • N(os µ ), the number of events with at least twoopposite-sign muons.9 (GeV/c) T p0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s FIG. 9: Comparison of the p T distributions for heavy quarksfrom the little Higgs model LH2 and squarks from the “twin”SUSY model NM6. The solid (red) line corresponds to heavyquark partners from model LH2; the dashed (blue) line tosquarks from model NM6. For both models 100,000 eventswere generated using MadGraph and the
CTEQ5L pdfs; no se-lection was applied.
In these counts a muon implies a reconstructed muonwith p T >
20 GeV/ c and | η | < . • r(nj)(3j) ≡ N(nj)/N(3j), with n=4,5, a measure ofjet multiplicity. • r(2 µ -nj)(1 µ -nj) ≡ N(2 µ -nj)/N(1 µ -nj), with n=3,4, ameasure of muon multiplicity.In appropriately chosen ratios of inclusive counts, im-portant systematic effects cancel partially or completely.For example, if the detector simulation does not preciselyreproduce the E T spectrum of signal jets as seen in thereal detector, this introduces a systematic error in theN(nj) counts, since jets are only counted if they have E T >
30 GeV. However we expect partial cancellation ofthis in the ratio of inclusive counts r(4j)(3j). Anotherlarge systematic in the jet counts, the pdf uncertainty,also cancels partially in the ratios. The luminosity un-certainty cancels completely in the jet ratios. As we dis-cuss below, ratios of correlated observables are also lesssensitive to statistical fluctuations.In order to enhance the robustness and realism of ourstudy, we have cast all of our physical observables intothe form of inclusive counts and ratios thereof, and ourlook-alike analysis only uses the ratios. This has theadded advantage of allowing us to compare different dis-criminating variables on a more even footing. In the next five subsections we explain how this casting into countsand ratios is done.
B. Kinematic observables
As noted in Section II, the distribution of the miss-ing transverse energy in the signal events is related to m dm , the mass of the WIMP, as well as to m p , the massof the parent particles produced in the original 2 → E miss T selection,we used the kinematic variable H T , as well as the E T ofthe leading and second leading jets. The distributions ofthese kinematic variables are also related to the underly-ing mass spectrum of the heavy partners.We have employed two other kinematic variables in ourstudy. The first is M , the total invariant mass of all of thereconstructed jets and muons in the event. The secondis M eff , the scalar sum of the E miss T with the E T of allthe jets in the event.Figures 10-13 show a comparison of the M eff , H T , M and E miss T distributions, after selection, for models LM2p,CS4d and CS6. These models, though look-alikes of our E miss T analysis, have a large spread in their superpartnerspectra, as is evident from Figure 5. For sufficiently largedata samples, this leads to kinematic differences that areapparent, as seen in Figures 10-13.All of the distributions exhibit broad peaks and longtails. In principle one could use the shapes of these distri-butions as a discrimination handle. This would require adeep understanding of the detector or a very conservativesystematic error related to the knowledge of the shapes.The location of the peaks is correlated with the mass m p of the parent particles in the events, but there is no prac-tical mapping from one to the other. By the same token,this implies that these kinematic distributions are highlycorrelated, and it is not at all clear how to combine theinformation from these plots.Our approach to kinematic observables in small datasets (low luminosity) is to define inclusive counts based onlarge bins. The dependence on the details of the detectorsimulation is strongly reduced by limiting the number ofbins and using a bin width much larger than the expecteddetector resolution.For M eff we define two bins and one new inclusivecount for the kinematic distributions in each box: • N(Meff1400) the number of events after selectionwith M eff > c .For H T we also define two bins and one new inclusivecount: • N(HT900) the number of events after selection with H T >
900 GeV/ c .Recall that the E miss T selection already required H T >
500 GeV. For the invariant mass M we define three binsand two new inclusive counts:0 ) (GeV/c eff M0 200 400 600 800 1000 1200 1400 1600 1800 2000 nu m b er o f e v e n t s FIG. 10: Comparison of the M eff distributions for Group1 MSSM models LM2p (solid red line), CS4d (dashed blueline) and CS6 (dotted magenta line). For each model 100,000events were generated then rescaled to 1000 pb − . (GeV) T H0 200 400 600 800 1000 1200 1400 1600 nu m b er o f e v e n t s FIG. 11: Comparison of the H T distributions for Group 1MSSM models LM2p (solid red line), CS4d (dashed blueline) and CS6 (dotted magenta line). For each model 100,000events were generated then rescaled to 1000 pb − . • N(M1400) the number of events after selection with
M > c ; • N(M1800) the number of events after selection with
M > c ;For E miss T we define four bins and three new inclusivecounts: • N(MET320), the number of events after selectionhaving E miss T >
320 GeV. ) total invariant mass (GeV/c0 200 400 600 800 1000 1200 1400 1600 1800 2000 nu m b er o f e v e n t s FIG. 12: Comparison of the distributions of the total invari-ant mass of jets and muons per event for Group 1 MSSMmodels LM2p (solid red line), CS4d (dashed blue line) andCS6 (dotted magenta line). For each model 100,000 eventswere generated then rescaled to 1000 pb − . (GeV) missT E0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s FIG. 13: Comparison of the E miss T distributions for Group1 MSSM models LM2p (solid red line), CS4d (dashed blueline) and CS6 (dotted magenta line). For each model 100,000events were generated then rescaled to 1000 pb − . • N(MET420), the number of events after selectionhaving E miss T >
420 GeV. • N(MET520), the number of events after selectionhaving E miss T >
520 GeV.Note that the E miss T selection already required E miss T >
220 GeV.1
C. Kinematic peaks and edges
With large signal samples, kinematic edges involvingleptons will be a powerful tool for model discriminationand to eventually extract the mass spectrum of the heavypartners. With small samples, in the range of 100 pb − to1000 pb − considered in our study, this will only be truein favorable cases. In fact for the 8 models studied here,we find no discrimination at all based on kinematic edgeswith leptons. This is due mostly to the small number ofhigh p T muons in our signal samples , as well as a lackof especially favorable decay chains.Although we do not observe any dimuon edges, we dosee a dimuon peak for model LM8. This is shown inFigure (14), for signal events after our missing energyselection rescaled to 1000 pb − of integrated luminosity.A Z peak is clearly visible, arising from squark decays toquark + ˜ χ , with the ˜ χ decaying 100% to Z + LSP. ) OS dimuon invariant mass (GeV/c0 20 40 60 80 100 120 140 160 180 200 nu m b er o f e v e n t s FIG. 14: The invariant mass distribution for opposite signdimuon pairs passing the missing energy selection, fromMSSM model LM8. The plot is from 100,000 generated eventsrescaled to 1000 pb − . D. Event shapes, hemispheres and cones
Event shapes are a way to extract information aboutthe underlying partonic subprocess and the resulting de-cay chains. This information is not uncorrelated with Low p T lepton and dilepton trigger paths as well as cross-triggerscombining leptons with jets and leptons with missing energy re-quirements are needed for Standard Model background calibra-tion and understanding; these will be important for signal ap-pearance and edge/threshold studies even at start-up. The LHCexperiments are preparing rich trigger tables along these lines[71]. kinematics, but it does have the potential to provide qual-itatively new characteristics and properties of the eventtopology. For our signal the partonic subprocess con-sists of the production of two heavy partner particles, soeach event has a natural separation into two halves thatwe will call “hemispheres”, consisting of all of the decayproducts of each partner. Associated to each hemisphereof the event should be one WIMP plus some number ofreconstructed jets and muons. A perfect hemisphere sep-aration is thus an associative map of each reconstructedobject into one of the two constituent decay chains.We can define two hemisphere axes in η - φ as the di-rections of the original parent particles; these axes arenot back-to-back because of the longitudinal (and trans-verse) boosts from the subprocess center-of-mass frameback to the lab frame. Even if we knew event by eventhow to boost to the center-of-mass frame, the hemisphereseparation would still not be purely geometrical, since insome events a decay product of one parent will end up inthe geometrical hemisphere of the other parent.Having defined a perfect hemisphere separation, weneed a practical algorithm to define reconstructed hemi-spheres. We will follow a study of 6 hemisphere algo-rithms presented in the CMS Physics TDR [10]. Thesealgorithms are geometrical and kinematic, based on thelarge though imperfect correlation between the η - φ vec-tor of the initial parent and the reconstructed objectsfrom this parent’s decay chain. The algorithms consistof two steps: a seeding method that estimates the twohemisphere axes, and an association method that usesthese two axes to associate each reconstructed object toone hemisphere.For the Group 1 model LM5, the CMS study foundthat jets were correctly associated to parent squarks 87%of the time, and to parent gluinos 70% of the time. Thedifferences in the performance of the 6 algorithms weresmall; the best-performing hemisphere algorithm com-bines the following methods: • Seeding method: The two hemisphere axes are cho-sen as the η - φ directions of the pair of reconstructedobjects with the largest invariant mass. The hard-est of these objects defines the leading hemisphere. • Association method: A reconstructed object is as-signed to the hemisphere that minimizes the Lunddistance [10].Figure 15 shows how well the seeding method produceshemisphere axes that match the actual axes defined bythe two parent particles. The separation is shown in unitsof ∆ R , defined as∆ R ≡ p (∆ η ) + (∆ φ ) . (4)The agreement is not overwhelmingly good, and is sub-stantially worse than that obtained for t ¯ t events, asshown in Figure 16. We have checked that all 6 hemi-sphere algorithms produce very similar results. Since theagreement is better for the leading hemisphere, all of our2single hemisphere derived observables are based just onthe leading hemisphere.We define three inclusive counts based on comparingthe two reconstructed hemispheres: • N(Hem j ) the number of events for which the recon-structed object multiplicity (jets + muons) in thetwo hemispheres differs by at least j, j=1,2,3.Once the two hemispheres are identified, we can breakthe degeneracy among the models by looking at thetopology in the events. For a given mass of the parentparticles, the events will look more jet-like rather thanisotropic if the decays are two-body rather than multi-body cascades. In the case of jet-like events, the projec-tion of the observed track trajectories on the transverseplane will cluster along the transverse boost of the par-ticles generating the cascade.In order to quantify this behaviour, we start from thecentral axes of each hemisphere and draw slices in thetransverse plane with increasing opening angles in φ sym-metric around the hemisphere axis. We refer to the slicesas cones , reminiscent of the cones used in CLEO analy-ses [72] to discriminate between jet-like QCD backgroundand isotropic decays of B meson pairs.We build five cones of opening angle 2 α ( α = 30 ◦ , 45 ◦ ,60 ◦ , 75 ◦ and 90 ◦ ) in each hemisphere. In terms of thesecones we define variables: • N(nt-c α ), the number of events having at least ntracks (n=10,20,30,40) in the leading hemispherecone of opening angle 2 α . • N(ntdiff-c α ), the number of events having a differ-ence of at least n tracks (n=10,20,30,40) betweenthe cones of opening angle 2 α in each hemisphere.Tracks are counted if they have p T > c and | η | < .
4. Since the cone of opening angle 2 α includes the oneof opening angle 2 β for α > β , these variables have aninclusive nature. E. The stransverse mass m T A potentially powerful observable for model discrimi-nation and mass extraction is the stransverse mass vari-able m T [73]-[75]. Let us briefly review how this is sup-posed to work for our missing energy signal. Ignoringevents with neutrinos, our signal events have two heavyparent particles of mass m p , each of which contributes tothe final state a WIMP of mass m dm plus some numberof visible particles. Supposing also that we have a per-fect hemisphere separation, we can reconstruct each setof visible particles into a 4-vector p Xµ . If we also knewthe mass and the p T of each WIMP, we could reconstructa transverse mass for each hemisphere from the formula m T = m X + m + 2( E XT E dm T − p XT · p dm T ) . (5) R D nu m b er o f e v e n t s FIG. 15: The distribution of the ∆ R separation between the η - φ direction of the parent superpartner and the reconstructedhemisphere axis. This is from 24,667 events of model LM5passing our selection. The solid red line is for the leadinghemisphere, while the dashed blue line is for the second hemi-sphere. R D nu m b er o f e v e n t s · FIG. 16: The distribution of the ∆ R separation between the η - φ direction of the parent top quark and the reconstructedhemisphere axis. This is from 3,000,000 Pythia t ¯ t events withno selection. The solid red line is for the leading hemisphere,while the dashed blue line is for the second hemisphere. This transverse mass is always less than or equal to themass m p of the parent particle. Thus the largest of thetwo transverse masses per event is also a lower bound on m p .Of course we do not know the p T of each WIMP, onlythe combined E miss T . Let p (1) T and p (2) T denote a possibledecomposition of the total p miss T into two azimuthal vec-tors, one for each WIMP. Note that this decomposition3ignores initial state radiation, the underlying event anddetector effects. Then we can define the stransverse massof an event as m T ( m dm ) ≡ (6)min p (1) T + p (2) T = p miss T h max h m T ( m dm ; p (1) T ) , m T ( m dm ; p (2) T ) ii . Since (with the caveats above) one of these partitions isin fact the correct one, this quantity is also a lower boundon the parent mass m p .For a large enough data sample, with the caveats aboveand ignoring finite decay widths, the upper endpoint ofthe stransverse mass distribution saturates at the par-ent mass m p , provided we somehow manage to input thecorrect value of the WIMP mass m dm . In the approxi-mation that the invariant mass m X of the visible decayproducts is small, the lower endpoint of the stransversemass distribution is at m dm .These impressive results seem to require that we know a priori the correct input value for m dm . However it hasbeen shown [76]-[79] that in principle there is a kink inthe plot of the upper endpoint value of m T as a functionof the assumed m dm , precisely when the input value of m dm equals its true value. Thus it may be possible toextract both m p and m dm simultaneously.Summarizing the remaining caveats, the stransversemass method requires a large data sample, and is pollutedby effects from incorrect hemisphere separation, ISR, theunderlying event, finite decay widths and detector effects.We can compare this to the precision extraction of the W mass at CDF [80, 81], using the transverse mass dis-tribution of the charged lepton and neutrino from the W decay. Here the WIMP mass is essentially zero, thedata samples are huge, and hemisphere separation is notapplicable since there is only one WIMP. In the CDFanalysis the ISR uncertainty was traded for an FSR un-certainty, by interpreting the vector sum of all the calori-metric E T not associated with the charged lepton as com-ing from ISR plus the underlying event; this then pollutesthe reconstructed neutrino p T with final state radiationfrom the lepton. The measured transverse mass distribu-tion has a considerable tail extending above the putativeendpoint at m W , but a precision mass (and width) ex-traction is still possible by modeling the distribution.In [82] it was shown that an imperfect hemisphere sep-aration greatly degrades the m T distribution for simu-lated LHC SUSY events. Two approaches to solve thisproblem have been suggested. The first, used in [82],is to reject events when the total invariant mass of thereconstructed objects in either hemisphere exceeds somevalue. This strategy is based on the fact that for a correcthemisphere separation with m T near the endpoint thehemisphere invariant mass is small, while incorrect hemi-sphere assignments naturally lead to large hemisphere in-variant masses. The second approach, used in [83], is toreplace m T with a new variable m T Gen . The new vari-able minimizes not only over all partitions of the p miss T ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s -1 -1
100 pb -1 FIG. 17: Comparison of the stransverse mass m T distribu-tions for model CS6, varying the cut on the maximum hemi-sphere invariant mass. The solid red line shows the m T dis-tribution when no cut is applied. The dashed blue line showsthe m T distribution rejecting events when the total recon-structed invariant mass of either hemisphere exceeds 300 GeV.The dotted magenta line shows the m T distribution whenthis cut is lowered to 200 GeV. In each case 100,000 eventswere generated then rescaled to 1000 pb − . The dotted hor-izontal lines are to guide the eye for where the distributioncuts off for 100 pb − and 1000 pb − . into p (1) T and p (2) T , but also over all possible hemisphereassignments. Since one of the hemisphere separations isin fact the correct one, m T Gen has the same endpoint as m T would have with a perfect hemisphere separation.Figure 17 shows the degradation of the m T distribu-tion for our model CS6. The m T endpoint should beat 589 GeV, the value of the gluino mass in CS6. How-ever the solid red line shows that a large fraction of eventsare above this endpoint, due to the imperfect hemisphereseparation. Applying the strategy of [82], the dotted ma-genta line shows that we regain the correct endpoint bymaking a hard cut of 200 GeV on the maximum recon-structed hemisphere invariant mass. However with sucha high requirement we take a big hit in statistics. Thedashed blue line shows that we still do pretty well witha 300 GeV requirement, while gaining a lot in statistics.For this study we have used the 300 GeV requirementin all of our m T analysis. The value is unoptimizedbut also unbiased, since it was determined by asking forapproximately the most stringent cut that retains rea-sonable statistics for the m T distributions of the entireset of models considered.Figure 18 shows a comparison of the m T distributionsfor two of our Group 1 look-alike models, LM2p and CS6.For LM2p, the parents are gluinos of mass 856 GeV andsquarks of mass approximately 800 GeV; for model CS6,by contrast, the parents are gluinos of mass 589 GeV. In4 ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s -1 -1
100 pb -1 FIG. 18: Comparison of the stransverse mass m T distribu-tions for look-alike models LM2p (solid red line) and CS6(dashed blue line). For each model 100,000 events were gen-erated then rescaled to 1000 pb − . The dotted horizontallines are to guide the eye for where the distribution cuts offfor 100 pb − and 1000 pb − . each case we have input the correct LSP mass, 142 GeVfor LM2p and 171 GeV for CS6.Each plot is rescaled to 1000 pb − . With this manyevents notice that we are just starting to saturate theappropriate endpoints at m T = m p , and notice the onsetof tails above the endpoints. The dotted lines in thefigure guide the eye to where the distributions cut off fordata samples of 100 pb − and 1000 pb − . Obviously for100 pb − we are not close to populating the endpoints.However even for 100 pb − there are significant differ-ences between the m T distributions of the two models.These differences only become larger if we use the sameinput mass for the LSP. Thus m T is at least as interest-ing for look-alike discrimination as the more traditionalkinematic variables discussed above. Furthermore, evenif we are not close to populating the endpoint, it mightbe possible to extract a direct estimate of m p by fittingor extrapolating the distributions.For our study we define five bins and four new inclusivecounts from m T : • N(mT2-300) the number of events after selectionwith m T >
300 GeV/ c , • N(mT2-400) the number of events after selectionwith m T >
400 GeV/ c , • N(mT2-500) the number of events after selectionwith m T >
500 GeV/ c . • N(mT2-600) the number of events after selectionwith m T >
600 GeV/ c . When comparing a model M1, playing the role of thedata, with a model M2, playing the role of the model totest, we will use the mass of the WIMP in model M2 asthe input mass in calculating m T for both models. F. Flavor enrichment
In order to have some model discrimination based onthe τ or b content, we need simple algorithms to createsubsamples enriched with b quarks and τ ’s. We refer tothese algorithms as “tagging”, despite the fact that thetagging efficiencies and the purity of the subsamples arerather poor.Without attempting any detailed optimizations, wehave designed two very simple tagging algorithms. Weexpect these algorithms to be robust, since they only re-quire a knowledge of uncorrected high E T jets, high p T muons, and basic counting of high p T tracks inside jets. τ enrichment: For each jet we define a 0.375 conecentered around the jet axis. Inside this cone we countall reconstructed charged tracks with p T > c . Ifonly one such track is found, and if this track has p T > c , we tag the jet as a τ jet.The τ algorithm is based on single-prong hadronic τ decays, which as their name implies produce a singlecharged track. In addition, leptonic decays of a τ toan electron and two neutrinos can be tagged, since somefraction of electrons reconstruct as jets. Soft tracks with p T ≤ c are not counted, a fact that makes thealgorithm much more robust. The p T >
15 GeV/ c re-quirement on the single track reduces the backgroundfrom non- τ jets. Increasing the cone size decreases theefficiency to tag genuine τ ’s, because stray tracks aremore likely to be inside the cone; decreasing the conesize increases the fake rate. A genuine optimization ofthis algorithm can only be done with the real data.Table XIX shows the results of applying our τ taggingalgorithm to simulations of the Group 1 models LM2p,LM5, LM8, CS4d and CS6. The efficiency, defined as thenumber of τ tags divided by the number of generator-level τ ’s that end up reconstructed as jets, varies between 12%and 21%. The efficiency is lowest for models LM8 andCS4d, models where τ ’s come entirely from W and Z decays. The efficiency is highest for model LM2p, whichhas a large final state multiplicity of τ ’s from decays ofcharginos, second neutralinos and staus.The purity, defined as the fraction of τ tagged jets thatactually correspond to generator-level τ ’s, is quite low formodels LM8 and CS4d, and is only 8% for model CS6,which contains very few τ ’s. We obtain a reasonably highpurity of 55% for LM2p, the model with by far the largest τ multiplicity.We conclude that it is possible to obtain significantlyenriched samples of τ ’s from our simple algorithm, butonly for models that do have a high multiplicity of ener-getic τ ’s to begin with. From the counts in Table XIX, itis clear that this tagging method is not viable with 1005pb − of integrated luminosity. TABLE XIX: Results of our τ tagging algorithm appliedto the Group 1 models LM2p, LM5, LM8, CS4d and CS6.Counts are rescaled to 1000 pb − from 100,000 events permodel. The listing for τ jets counts generator level τ ’s thatare reconstructed as jets in events that pass our selection.LM2p LM5 LM8 CS4d CS6 τ jets per fb −
409 144 171 112 34tags per fb −
157 110 122 102 59correcttags per fb −
86 25 21 14 5efficiency 21% 18% 12% 13% 16%purity 55% 23% 17% 14% 8% b enrichment:
For each jet we search for a recon-structed muon inside the jet (recall that our muons have p T >
20 GeV/ c and | η | < . R < . b jet.This b algorithm is based on tagging muons fromsemileptonic B decays inside the b jet. This is inspired bythe “soft muon” tagging that was used in the top quarkdiscovery at the Tevatron [84, 85]. In our case “soft” isa misnomer, since in fact we only count reconstructedmuons with p T >
20 GeV/ c . This requirement makesthe tagging algorithm more robust, but reduces the effi-ciency.Table XX shows the results of applying our b taggingalgorithm to simulations of the Group 1 models LM2p,LM5, LM8, CS4d and CS6. The tagging efficiency is de-fined as the number of b tags divided by the number ofgenerator-level b ’s that are within ∆ R < . b ’s, the tagging ef-ficiency is poor: only about 5% for all models. Howeverthe purity of the samples is rather good: above 70% forevery model except CS6.We conclude that it is possible to obtain significantlyenriched samples of b ’s from our simple algorithm, butwith low efficiency. From the counts in Table XX, itis clear that this tagging method is not viable with 100pb − of integrated luminosity, but should become usefulas we approach 1000 pb − .In our study, discrimination based on τ ’s and b ’s is ob-tained from ratios that involve the following two inclusivecounts: • N( τ -tag), the number of events after selection hav-ing at least one τ tag. • N( b -tag), the number of events after selection hav-ing at least one soft muon b tag. VI. THE LOOK-ALIKE ANALYSIS
The look-alike analysis proceeds in four steps:
TABLE XX: Results of our b tagging algorithm applied to theGroup 1 models LM2p, LM5, LM8, CS4d and CS6. Countsare rescaled to 1000 pb − from 100,000 events per model. Thelisting for b jets counts generator level b quarks matched toreconstructed jets that pass our selection.LM2p LM5 LM8 CS4d CS6 b jets per fb − −
115 112 148 105 106correcttags per fb −
82 81 112 75 41efficiency 5% 5% 5% 5% 5%purity 72% 72% 75% 71% 39%
1. We choose one of the models to play the part ofthe data. We run the inclusive E miss T +jets analysis onthe MET trigger and verify that the predicted yield es-tablishes an excess (at > σ ) above the SM backgroundwith 100 pb − . We call the number of events selectedin this way the observed yield N data . In what follows,we assume that a subtraction of the residual StandardModel background has already been performed. We as-sume large signal over background ratios for the modelsconsidered so that the statistical error on the backgroundhas a small impact on the total error.2. We identify a set of models giving a predicted yield N compatible with N data . The compatibility is estab-lished if the difference in the two counts is less than twicethe total error, i.e if the pull | N data − N | σ ( N ) (7)is smaller that two. In the formula σ ( N ) represents theerror associated to the expected number of events N . Wecalculate it as the sum in quadrature of several contribu-tions: • A Poissonian error which takes into account thestatistical fluctuations associated to the event pro-duction (statistical component of the experimentalerror). • An error associated to the detector effects (system-atic component of the experimental error). • Theoretical error on the predicted number of events N (including a statistical and a systematic compo-nent).We discuss the origin of each contribution below.3. For each additional observable N i previously listed,we consider the value on the data ( N i data ) and the pre-dicted value N ij for the model j . We calculate the pullas in eqn. 7 and we identify the variable with the largestpull as the best discriminating counting variable. We ig-nore all the variables for which both the model and thedata give a yield below a fixed threshold N min . We use6 N min = 10, i.e. we require a minimum yield that is morethan three times its Poisson error √ N i ; for the data thiscorresponds to excluding at 3 σ the possibility that theobserved yield is generated by a fluctuation of the back-ground.4. We form ratios of some of the observables usedabove and we repeat the procedure of step 3. Since partof the uncertainties cancel out in the ratio, these variablesallow a better discrimination than the counting variables.In addition, provided that the two variables defining theratio are above the threshold N min , the ratios of two cor-related variables (such as N (4 j ) /N (3 j )) are less sensitiveto the statistical fluctuations. Details on the calculationof the errors on the ratios are given below.In each of the four trigger boxes we define the followingratios of correlated inclusive counts: • r( n j)(3j), with n=4,5 • r(MET320) • r(MET420) • r(MET520) • r(HT900) • r(Meff1400) • r(M1400) • r(M1800) • r(Hem j ) with j =1,2,3 • r(2 µ -nj)(1 µ -nj) with n=3,4 • r( τ -tag) • r( b -tag) • r(mT2-300) with the theory LSP mass • r(mT2-400) with the theory LSP mass • r(mT2-500) with the theory LSP mass • r(mT2-600) with the theory LSP mass • r(mT2-400/300) with the theory LSP mass • r(mT2-500/300) with the theory LSP mass • r(mT2-600/300) with the theory LSP mass • r(nt-c α ) for n=10,20,30,40 and α = 30 ◦ ,45 ◦ , 60 ◦ ,75 ◦ , 90 ◦ • r(ntdiff-c α ) for for n=10,20,30,40 and α = 30 ◦ , 45 ◦ ,60 ◦ , 75 ◦ , 90 ◦ For most of these ratios the numerator is the correspond-ing count defined in Section V and the denominator is thetotal event count in the trigger box. The exceptions arer( n j)(3j)=N( n j)/N(3j), r(2 µ -nj)(1 µ -nj)=N(2 µ -nj)/N(1 µ -nj), and r(mT2- n /300)=N(mT2- n )/N(mT2-300), n =400 , , • r(DiJet) • r(TriJet) • r(Muon20)As mentioned previously, it turns out that the DiJet,TriJet and Muon20 boxes are subsamples of the METbox to an excellent approximation, thus these ratios arealso ratios of inclusive counts.Finally we iterate and perform the transpose compar-isons (the model that was considered as data takes therole of the model). A. Theoretical uncertainty
We take into account several sources of uncertainty.First of all, there is an error associated to the knowledgeof the parton probability density functions (pdfs) thatare used to generate the event samples. In order to eval-uate this error, we produce and analyze all samples withthree different sets of pdfs:
CTEQ5L [86],
CTEQ6M [86], and
MRST2004nlo [87] or
MRST2002nlo [87] for Group 1 andGroup 2 respectively. We quote as central value the aver-age of the three values; for the pdf uncertainty we crudelyestimate it by taking half the spread of the three values.This uncertainty, as we will show, has important effectson the results.An additional error is given by the relative QCD scaleuncertainty when we compare different look-alike mod-els. This is an overall systematic on the relative crosssections that we take to be 5%. It is actually larger thanthis in our study, at least for the Group 2 models wherewe use LO cross sections, but we are assuming some im-provement by the time of the real discovery.There is an additional uncertainty for each observablefrom the missing higher order matrix elements. It is notincluded in the analysis shown here. It could be includedcrudely by running
Pythia with different values of theISR scale controlled by MSTP(68), similar to how weevaluate the pdf uncertainties. A better way is to include,for the signals, the higher order matrix elements for theemission of extra hard jets. The ideal approach would bea full NLO generator for the signals.The sum in quadrature of all these effects gives thesystematic error associated to the theoretical prediction.In the case of ratios, the error on the cross section cancelsout. In a similar way, the correlated error on the pdfscancels out by calculating the ratios for the three sets ofpdfs and then averaging them.7In the case of mSUGRA models, the result of the sim-ulation also depends on which RGE evolution code weuse to go from the parameters at the high scale to theSUSY spectrum at the Terascale. Rather than includingan error associated to such differences we take one of thecodes ( Isajet v7.69 or SuSpect v2.34 ) as part of thedefinition of the theory model we are considering.The theory predictions are also affected by a statisticalerror, related to the fact that the value of each observ-able is evaluated on a sample of limited size. Generatingthe same sample with a different Monte Carlo seed oneobtains differences on the predicted values of the observ-able. The differences, related to statistical fluctuations,are smaller for larger generated data sets. Consideringthat each number of events N ji for observable i and model j can be written as N ji = ǫ ji × σ j and that the error on σ j is already accounted for in the systematic contribution tothe theoretical error, the efficiency ǫ ji has an associatedbinomial error: σ ( ǫ i ) = s ǫ ji × (1 − ǫ ji ) N GEN (8)where N GEN is the size of the generated sample beforeany selection requirement. This error can be made negli-gible by generating data sets with large values of N GEN .We include the contribution of the statistical error sum-ming it in quadrature to the systematic error.When the variables defining the ratio are uncorrelated,the error on the ratio is obtained by propagating theerrors on the numerator and denominator, according tothe relation σ ( r ) = s(cid:18) σ ( N num ) N den (cid:19) + (cid:18) N num σ ( N den ) N (cid:19) (9)where r = N num /N den .This is not the correct formula in our case, since allof the counts on our ratios are correlated. For instance, N (4 j ) and N (3 j ) are correlated, since all the events withat least four jets have also three jets. Only a fractionof the events defining N (3 j ) will satisfy the requirementof an additional jet, i.e. applying the requirement of anadditional jet on the ≥ r (4 j )(3 j ) = N (4 j ) /N (3 j )the associated efficiency. The error on r is then givenby eqn. 8, replacing ǫ ji with the r and N GEN with N (4 j ).The same consideration applies to all the ratios built fromcorrelated variables. In order to use eqn. 8 for the error,we always define the ratios such that they are in the range[0,1]. For the CMS benchmark models, we used
Isajet v7.69 but com-pared the spectra results with
SuSpect v2.34 + SUSY-HIT v.1.1 and
SoftSusy v.2.0.14 [88]. The differences in the computedspectra led to differences in our observed yield of 3 to 10%.
B. Statistical uncertainty
The production of events of a given kind in a detectoris a process ruled by Poisson statistics. The error on acounting variable N is given by √ N .In analogy to the statistical error on the theoreticalpredictions, the statistical error on a ratio of two corre-lated variables is calculated according to eqn. 8, replacing ǫ ji with the r and N GEN with the value of the denomi-nator variable for the reference data luminosity. Unlikethe case of the theoretical error, this error is associatedto the statistics of the data set and not to the size of thegenerated sample; this error is intrinsic to the experimen-tal scenario we are considering and cannot be decreasedby generating larger Monte Carlo samples.
C. Systematic uncertainty
We consider two main sources of systematic error, theknowledge of the collected luminosity and the detector ef-fects on the counting variables. Estimating the two con-tributions to be of the order of 10%, we assign a globalsystematic error of 15% to the counting variables. Whencalculating the ratios, we expect this error to be stronglyreduced. On the other hand the cancellation is not ex-act, and it will be less effective at the start-up, becauseof potentially poorly-understood features related to thereconstruction. Hence we associate a residual systematicerror of 5% to the ratios.
VII. SUMMARY OF GROUP 1 RESULTS
Table XXI summarizes the results from the five MSSMmodels of Group 1. There are 20 pairwise model com-parisons. One model is taken as the simulated data, withthe observed yield scaled to either the 100 pb − discoverydata set or a 1000 pb − follow-up. The actual numberof events generated in each case was 100,000, thus the“data” has smaller fluctuations than would be present inthe real experiment; we are interested in identifying thebest discriminators and their approximate significance,not simulating the real experiment. Note, however, thatin our analysis we include the correct statistical uncer-tainty arising from the assumed 100 pb − or 1000 pb − ofintegrated luminosity in the “data” sample, as describedin Section VI.B. When the rescaled counts are below aminimum value, the corresponding ratio is not used inour analysis, as described in step 3 of analysis procedureoutlined in Section VI.Given a “data” model, we can compare it to four the-ory models. We want to understand to what extent wecan reject each theory model, based on discrepancies inour discriminating ratios. With the real LHC data, thetest needs to be performed as follows: given Model A andModel B, we will ask if Model A is a better description of8 TABLE XXI: Summary of the best discriminating ratios for model comparisons in Group 1. The models listed in rows are takenas simulated data, with either 100 or 1000 pb − of integrated luminosity assumed, and uncertainties as described in the text.The models listed in columns are then compared pairwise with the “data”. In each case, the three best distinct discriminatingratios are shown, with the estimated significance. By distinct we mean that we only list the best ratio of each type; thus ifr(5j)(4j) is listed, then r(4j)(3j) is not, etc. The asterix on the ratio r( b -tag) indicates that it is defined in the Muon20 box; allother ratios are defined in the MET box, and r(Muon20) denotes the ratio of the number of events in the Muon20 box to thenumber in the MET box. The m T ratios are computed using the LSP mass of the relevant “theory” model, not the “data”model. LM2p LM5 LM8 CS4d CS6LM2p100 r(5j)(3j) 1.6 σ r(5j)(3j) 4.4 σ r(MET520) 4.1 σ r(mT2-600/300) 11.4 σ r(mT2-300) 1.4 σ r(MET520) 3.7 σ r(HT900) 3.6 σ r(MET520) 10.6 σ r( τ -tag) 1.2 σ r(10t-c45) 2.9 σ r(Meff1400) 3.0 σ r(HT900) 6.8 σ τ -tag) 3.1 σ r(MET520) 8.2 σ r(MET520) 9.4 σ r(mT2-600/300) 33.0 σ r(5j)(3j) 2.8 σ r(mT2-500) 6.7 σ r(HT900) 6.4 σ r(MET520) 26.6 σ r(mT2-400) 2.6 σ r(5j)(3j) 6.5 σ r(mT2-600) 6.0 σ r(HT900) 14.6 σ LM5100 r(5j(3j) 1.8 σ r(5j)(3j) 2.9 σ r(HT900) 3.6 σ r(mT2-600/300) 11.6 σ r(mT2-300) 1.5 σ r(MET520) 2.7 σ r(Meff1400) 3.2 σ r(MET520) 9.2 σ r(10t-c30) 1.4 σ r(Muon20) 2.5 σ r(MET520) 3.1 σ r(HT900) 6.8 σ σ r(MET520) 6.0 σ r(MET520) 7.1 σ r(mT2-600/300) 33.7 σ r( τ -tag) 2.7 σ r(Muon20) 4.9 σ r(HT900) 6.4 σ r(MET520) 22.9 σ r(mT2-400) 2.6 σ r(5j)(3j) 4.3 σ r(mT2-600/400) 6.1 σ r(HT900) 14.6 σ LM8100 r(5j)(3j) 5.5 σ r(5j)(3j) 3.3 σ r(5j)(3j) 3.1 σ r(Muon20) 10.1 σ r(10t-c30) 3.7 σ r(Muon20) 3.1 σ r(mT2-400) 2.2 σ r(mT2500/300) 5.2 σ r(Muon20) 3.6 σ r(MET520) 2.4 σ r(20t-c45) 2.1 σ r(Hem3) 4.1 σ σ r(Muon20) 7.2 σ r(5j)(3j) 5.4 σ r(Muon20) 25.8 σ r(Muon20) 8.0 σ r(Hem3) 5.7 σ r(Hem3) 5.3 σ r(mT2-600/300) 20.1 σ r(Hem3) 7.3 σ r(5j)(3j) 5.6 σ r(Muon20) 4.1 σ r(Hem3) 14.2 σ CS4d100 r(MET520) 3.5 σ r(HT900) 3.0 σ r(5j)(3j) 2.8 σ r(Muon20) 6.8 σ r(HT900) 3.2 σ r(MET520) 2.7 σ r(mT2-300) 2.1 σ r(MET420) 5.5 σ r(Meff1400) 2.6 σ r(Meff1400) 2.6 σ r(10t-c30) 1.9 σ r(mT2-500/300) 5.2 σ σ r(MET520) 5.1 σ r(5j)(3j) 4.2 σ r(Muon20) 17.3 σ r(mT2-600) 5.3 σ r(mT2-600/400) 4.8 σ r(10tdiff-c30) 3.6 σ r(mT2-500) 12.8 σ r(HT900) 5.2 σ r(HT900) 4.5 σ r(Hem3) 3.6 σ r(MET520) 11.5 σ CS6100 r(MET420) 7.0 σ r(MET420) 6.0 σ r( b -tag) ∗ σ r(MET420) 4.3 σ r(mT2-500/300) 5.1 σ r(mT2-500/300) 4.6 σ r(Muon20) 5.2 σ r(Muon20) 4.0 σ r(HT900) 4.8 σ r(HT900) 4.5 σ r(MET420) 4.0 σ r(mT2-500/300) 2.9 σ σ r( b -tag) ∗ σ r( b -tag) ∗ σ r( b -tag) ∗ σ r( b -tag) ∗ σ r(MET520) 10.3 σ r(Muon20) 10.2 σ r(Muon20) 8.4 σ r(mT2-500) 10.2 σ r(mT2-500) 9.2 σ r(MET520) 7.6 σ r(MET420) 7.6 σ the data than Model B; this is a properly posed hypothe-sis test. Every time we reject a theory model as an expla-nation of the “data”, we learn something about the trueproperties of the model underlying the “data”. Whenseveral discriminating ratios have high significance, wemay learn more than one qualitative feature of the un-derlying model from a single pairwise comparison; this is not always the case however, since many ratios probevery similar features of the models and are thus highlycorrelated.In general the discriminating power of the robust ratiosis quite impressive. For 8 of the pairwise comparisonsat least one ratio discriminates at better than 5 σ with9only 100 pb − of simulated data . In only three cases(discussed in more detail below) do we fail to discriminateby at least 5 σ with 1000 pb − . In 14 out of 20 cases 1000pb − gives > σ discrimination with 3 or more differentratios, giving multiple clues about the underlying datamodel. ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s -1 FIG. 19: Comparison of the stransverse mass m T distribu-tions for look-alike models LM5 (solid red line) and CS4d(dashed blue line). For each model 100,000 events were gen-erated then rescaled to 1000 pb − . A. LM5 vs CS4d
We begin with one of the simplest pairwise compar-isons: LM5 is treated as the data and compared to the-ory model CS4d. Averaging over the three pdfs used inour study, we find that LM5 would produce 1951 sig-nal events in the 100 pb − discovery sample. This isonly 7% more than the 1817 events predicted by theorymodel CS4d, so the data model and theory model areindeed look-alikes. If we peek at the features of the twomodels we see that they have a number of phenomeno-logical similarities. Both models have about the sameproportion of squark-gluino and squark-squark produc-tion. In CS4d gluinos decay to stop-top, followed bythe three-body stop decay ˜ t → bW + ˜ χ ; this resemblesLM5 gluinos decaying to left-squark-quark, followed byleft-squark cascade to chargino-quark, with the charginodecaying to W ˜ χ . Both models have a large fraction ofevents with W ’s.At the moment of discovery CS4d is excluded asan explanation of the LM5 “data” by more than 3 σ As is common practice in high energy physics, we take 5 σ and3 σ as reference values for discovery and evidence, repsectively. in three kinematic ratios: r(HT900), r(Meff1400) andr(MET520). These ratios discriminate based on the pro-portion of highly energetic events; their values are about50% larger for LM5 than for CS4d. This indicates thatthe LM5 signal arises from production of heavier par-ent particles. From the superpartner spectra in Figure 5we see that indeed the gluino mass is about 100 GeVheavier in LM5 than in CS4d, and the lightest squarksare also somewhat heavier. Note that LM5 has a harder E miss T distribution even though its LSP mass is ∼ m T endpoint is a direct measure of the(largest) parent particle mass, we would expect the m T ratios to be good discriminators. However as can beseen in Figure 19 with 100 pb − we are hampered bypoor statistics near the endpoint. The best m T ratiois r(mT2-600/300) in the MET box, computed with theLSP mass of CS4d; it is defined as the number of eventsin the MET box with m T >
600 GeV divided by thenumber of events with m T >
300 GeV. This ratio has2.8 σ significance with 100 pb − .Making the same comparison at 1000 pb − , the sig-nificance of r(MET520), r(HT900) and r(Meff1400) asdiscriminators improves to 7.1, 6.4 and 5.9 σ respec-tively. More importantly, two of the m T ratios, r(mT2-600/300) and r(mT2-600/400), now discriminate at bet-ter than 6 σ . We have not attempted to perform a directextraction of the endpoint, but it is clear that the m T ratios can compete with the kinematic ratios as discrimi-nators while simultaneously providing more direct infor-mation about the underlying 2 → b jets, b tags and tagged b jets are all about thesame. Lacking an explicit reconstruction of tops or Higgs,we do not discriminate these models based on these fea-tures. B. LM2p vs LM5
This is the most difficult pair of look-alikes in ourstudy. From Figure 5 we see that the superpartner spec-tra are almost identical; the only significant difference isthat LM2p has a much lighter stau. As a result, LM2pevents are much more likely to contain τ ’s, while LM5events are much more likely to contain W ’s (mostly fromchargino decays).As Table XXI shows, at the moment of discovery LM5cannot be ruled out as the explanation of LM2p “data”.Without τ tagging, we would not have 3 σ discriminationeven with 1000 pb − ; the best we could do is the jetmultiplicity ratio r(5j)(3j) with 2.8 σ : LM5 produces morehigh multiplicity jet events after selection, because we canget two jets from a W decay in LM5 compared to onlyone hadronic τ from a stau decay in LM2p.0 TABLE XXII: Summary of the best discriminating ratios for model comparisons in Group 2. The models listed in rows aretaken as simulated data, with either 100 or 1000 pb − of integrated luminosity assumed, and uncertainties as described in thetext. The models listed in columns are then compared pairwise with the “data”. In each case, the three(five) best distinctdiscriminating ratios for 100(1000) pb − are shown, with the estimated significance. By distinct we mean that we only list thebest ratio of each type; thus if r(5j)(4j) is listed, then r(4j)(3j) is not, etc. Square brackets denote ratios defined in the DiJet,TriJet or Muon20 boxes; all other ratios are defined in the MET box, and r(DiJet), r(TriJet) denotes the ratio of the numberof events in the DiJet/TriJet boxes to the number in the MET box. The m T ratios are computed using the LSP mass of therelevant “theory” model, not the “data” model.LH2 NM4 CS7LH2100 r(mT2-500) 4.9 σ r(mT2-500) 6.7 σ r(Meff1400) 3.0 σ r(MET420) 6.5 σ r(M1400) 2.7 σ r(4j)(3j) 4.0 σ σ r(mT2-500) 18.9 σ r(mT2-300) [TriJet] 11.0 σ r(MET420) 16.7 σ r(mT2-400) [DiJjet] 7.9 σ r(mT2-500) [TriJet] 8.8 σ r(Meff1400) 7.2 σ r(4j)(3j) [DiJet] 7.3 σ r(M1400) 6.6 σ r(mT2-300) [DiJet] 6.7 σ NM4100 r(Meff1400) 4.2 σ r(Meff1400) 4.3 σ r(M1400) 4.0 σ r(DiJet) 4.1 σ r(mT2-400) 3.8 σ r(MET420) 4.0 σ σ r(Meff1400) 11.2 σ r(TriJet) 10.4 σ r(MET520) 10.6 σ r(M1400) 9.8 σ r(DiJet) 10.6 σ r(DiJet 8.2 σ r(HT900) 9.0 σ r(HT900) 8.0 σ r(4j)(3j) 6.1 σ CS7100 r(MET420) 4.9 σ r(4j)(3j) 4.4 σ r(4j)(3j) 4.6 σ r(MET420) 3.3 σ r(mT2-400) 4.1 σ r(Hem1) 3.2 σ σ r(4j)(3j) 9.4 σ r(TriJet) 10.4 σ r(5j)(3j) [DiJet] 7.4 σ r(MET420) 9.6 σ r(Meff1400) 7.4 σ r(4j)(3j) 9.5 σ r(DiJet) 6.9 σ r(mT2-500) 8.3 σ r(HT900) 6.2 σ With our crude τ tagging algorithm we manage to dis-criminate LM2p and LM5 at 3.1 σ with 1000 pb − . Asseen in Table XIX, the LM2p sample has almost threetimes as many hadronic τ ’s that reconstruct as jets thandoes the LM5 sample. This results in four times as manyjets being correctly tagged as hadronic τ ’s, and almost50% more total τ tags for LM2p versus LM5.The only other ratios that give even 2 σ discrimina-tion are based on the stransverse mass m T . The ratior(mT2-400) counts the number of events with m T > m T is computed using the LSP mass of the theorymodel LM5. Figure 20 compares the m T distributionsof LM2p and LM5. The shapes of the distributions arevery similar; the only obvious difference is that there arefewer events in the LM5 bins. It is not surprising that the shapes are similar, sinceLM2p and LM5 are very similar models with nearly iden-tical gluino and squark masses. Thus ratios like r(mT2-600/400) should not be good discriminators, and indeedthey are not. The ratio r(mT2-400), which is a gooddiscriminator, is obviously just picking up the fact thatthere are fewer events in the LM5 bins than in the LM2pbins.Figure 21 compares the m T distributions of LM2psubsamples with fixed multiplicity of jets+muons. Wesee that events with higher multiplicity are significantlymore likely to fail the 300 GeV hemisphere invariant massupper bound that we imposed to make up for the effecton m T of imperfect hemisphere separation. This makessense since events with higher multiplicity are more likelyto have mistakes in the hemisphere assignments.1 ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s -1 FIG. 20: Comparison of the stransverse mass m T distribu-tions for look-alike models LM2p (solid red line) and LM5(dashed blue line). For each model 100,000 events were gen-erated then rescaled to 1000 pb − . ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s -1 FIG. 21: Comparison of the stransverse mass m T distribu-tions for model LM2p with a fixed number of reconstructedobjects (jets or muons): 3 objects (solid red line), 4 objects(dashed blue line) and 5 objects (dotted magenta line). Foreach case 100,000 events were generated then rescaled to 1000pb − . Thus the discriminating power of r(mT2-400) in thiscase is correlated with r(5j)(3j), not with kinematic ratioslike r(HT900) and r(Meff1400).It is important to note that the m T ratios have someability to discriminate based on neutrinos in the finalstate: Figure 22 shows a comparison of the m T dis-tributions for LM2p events containing neutrinos versusthose without neutrinos. The events with neutrinos have ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s -1 FIG. 22: Comparison of the stransverse mass m T distribu-tions for two subsamples of model LM2p: events with neu-trinos (solid red line) and events without neutrinos (dashedblue line). A total of 100,000 events were generated, rescaledto 1000 pb − , then sorted into the two subsamples. a softer m T distribution, i.e. the subsample with neutri-nos is less efficient at populating the m T upper endpoint.Models LM2p and LM5 differ greatly in the proportionof events after selection that have neutrinos: about 50%for LM2p but only about 10% for LM5. The neutrinocontent effect on the m T distributions actually reducesthe discrimination of LM2p versus LM5, because the neu-trino effect works in the opposite direction from the dom-inant effect of jet multiplicity.This example shows that the interpretation of the m T ratios requires a comparison with other discrimi-nators. If the m T ratios r(mT2- xxx/yyy ) have a highsignificance positively correlated with e.g. r(HT900) andr(Meff1400), then the m T ratios are predominantly in-dicating kinematics. If the m T ratios r(mt2- xxx ) havea high significance but r(mT2- xxx/yyy ) do not (as oc-curred here), we expect they will be positively corre-lated with the jet ratios, indicating a difference in themultiplicity of reconstructed objects. If the m T ratiosr(mT2- xxx/yyy ) have a high significance uncorrelated ornegatively correlated with either kinematics or jet mul-tiplicity, this could signal the presence of three unseenparticles (e.g. two LSPs and a neutrino) in the final stateof a large fraction of events. C. CS4d vs LM8
This is the second most difficult pair of look-alikes inour study. From Figure 5 we see that the gluino andsquark superpartner spectra are roughly similar. Thegluino masses agree to within 10 GeV; in LM8 the leftand right squarks are nearly degenerate and slightly heav-2ier than the gluino, while in CS4d the right squarks areslightly lighter than the gluino and the left-squarks areabout 180 GeV heavier.Both models have about the same fractions of squarkpair, squark-gluino and gluino pair production after se-lection. In both models the gluino decays predominatelyto top-stop. Both models produce many tops, b ’s and W ’s. LM8 also produces a lot of Z ’s from ˜ χ decays.The main difference between these models is that CS4dhas a much lighter stop and a much heavier LSP. With m ˜ t =352 GeV and m ˜ χ =251 GeV a two-body light stopdecay cannot occur, and this stop can just barely managethe three-body decay ˜ t → bW + ˜ χ .Consider the case that we perform our look-alike anal-ysis taking CS4d as the “data” model and LM8 as thetheory model. At the moment of discovery, LM8 willexplain not only the overall size of the signal but alsoits kinematics: with 100 pb − we find no kinematic ob-servable that discriminates better than r(Meff1400) with1.7 σ , and even with 1000 pb − no kinematic observablesdiscriminate at even the 3 σ level. Given that LM8 isan mSUGRA benchmark used at the LHC experimentsit may be tempting to falsely conclude that LM8 is theprobable explanation of the discovery! Since LM8 has230 GeV charginos, even a preliminary result in this di-rection could be used, for example, as a justification tostart building a 500 GeV linear collider. Since the actualchargino mass of the underlying compressed SUSY modelis 352 GeV, this would lead to embarrassment.Fortunately our look-alike analysis gives additional dis-criminating handles: • The ratio r(Muon20) has a 3.4 σ significance with1000 pb − , reflecting a larger fraction of recon-structed muons in LM8 events over CS4d. • With 1000 pb − there are enough dimuons to re-construct the Z peak as shown in Figure 14 forLM8, while no peak appears for CS4d. • The ratio r(5j)(3j) in the MET box differs by 2.8 σ at 100 pb − , increasing to 4.2 σ with 1000 pb − ,reflecting a larger jet multiplicity in LM8 eventsversus CS4d. • With 1000 pb − we also see discrepancies in theevent shape variables. One of these, r(Hem3), isthe fraction of events where the object counts inthe two hemispheres differ by at least 3; the other,r(10tdiff-c300), is the fraction of events where thetrack count in a 30 ◦ cone around each hemisphereaxis differs by at least 10. With 1000 pb − boththese ratios have a significance of 3.6 σ , reflectingmore symmetrical object counts in CS4d eventsthan for LM8.From Table XX we would have hoped for a significantdifference in r( b -tag) between LM8 and CS4d. Howevereven with 1000 pb − we obtain only a 2.4 σ significancewith this ratio; this is due to the combination of low efficiency in the b tagging and bad luck in that this ratiohappens to have a rather large uncertainty from the pdfspread.These handles exclude LM8 with reasonable confidenceas the explanation of the “data”. They also give clueson how to modify LM8 (still within the hypothesis ofSUSY) to better fit the data. The Z peak in LM8 comesfrom left-squarks decaying to ˜ χ ; making the left-squarksheavier will cause them to decay instead to quark-gluino.To keep the parent kinematics and observed yield con-stant, this also suggests lowering the right-squark masses.This has the added benefit of favoring the 2-body squarkdecay ˜ q R → q ˜ χ over the decay ˜ q R → q ˜ g , which lowersthe jet multiplicity. The large hemisphere asymmetriesin LM8 are derived from gluino cascades resulting in twotop quarks, and thus up to six jets, in a single hemi-sphere. The obvious way to reduce this without dras-tically changing the model is to squeeze out the phasespace for the 2-body stop decay ˜ t → t ˜ χ , and furthersqueeze the 3-body stop decay. By thus reducing theamount of visible energy reconstructed in the events, thisreduces the hemisphere asymmetries, the jet multiplicity,and the muon counts.Thus with 1000 pb − we might not only exclude LM8but also come close to guessing CS4d from this simplelook-alike comparison. Without the benefit of additionalmodel comparisons this guess would be relying on thestrong assumptions that the “data” was SUSY (an as-sumption that we will relax in the Group 2 analysis) andthat the data was full of b ’s and tops despite lacking ex-plicit confirmation from b tagging or top reconstruction. r ( M E T ) r ( M E T ) r ( M E T ) r ( H e m ) r ( H e m ) r ( H T ) ) m )( m r ( -t a g ) t r ( r ( m T - ) r ( m T - ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 23: Breakdown of estimated uncertainties for discrimi-nating ratios with 1000 pb − , in the comparison of look-alikemodels CS6 and CS4d, with CS6 treated as the “data”. r ( j )( j ) r ( M E T ) r ( M E T ) r ( M E T ) r ( H e m ) r ( H e m ) r ( M ) r ( H T ) r ( M e ff ) ) m )( m r ( E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 24: Breakdown of estimated uncertainties for discrimi-nating ratios with 1000 pb − , in the comparison of look-alikemodels CS6 and LM8, with CS6 treated as the “data”. r ( M E T ) r ( M E T ) r ( M E T ) r ( H e m ) r ( H e m ) r ( H T ) ) m )( m r ( -t a g ) t r ( r ( m T - ) r ( m T - ) [ % ] X ) / X ( X - -15-10-5051015 CTEQ5l1CTEQ6mMRST2004nlo
FIG. 25: Pdf spreads for discriminating ratios with 1000 pb − ,in the comparison of look-alike models CS6 and CS4d, withCS6 treated as the “data”. D. CS6 vs LM2p, LM5, LM8 and CS4d
This is a complete example of how we could deduce thecorrect model, under the assumption of SUSY, based onhow it fails to match with four incorrect SUSY models.We take CS6 as the “data” and compare it to LM2p,LM5, LM8 and CS4d.With 100 pb − , the ratio r(mT2-500/300) is muchsmaller for the “data” than for LM2p, LM5, LM8 andCS4d, with significance 5.1, 4.6, 3.3 and 2.9 σ respectively.This indicates that the parent particle mass in the datamodel is quite a bit lighter than in model CS4d, whichhas the lightest squarks and gluinos of the four theory r ( j )( j ) r ( M E T ) r ( M E T ) r ( M E T ) r ( H e m ) r ( H e m ) r ( M ) r ( H T ) r ( M e ff ) ) m )( m r ( [ % ] X ) / X ( X - -15-10-5051015 CTEQ5l1CTEQ6mMRST2004nlo
FIG. 26: Pdf spreads for discriminating ratios with 1000 pb − ,in the comparison of look-alike models CS6 and LM8, withCS6 treated as the “data”. models.To keep the overall data yield constant, we can con-template two possible modifications of the spectrum inCS4d: either make the gluino lighter and the squarksheavier, or vice-versa.The other striking result with 100 pb − is that the“data” has only 1/4 as many events in the Muon20 boxas the theory model LM8, and 1/3 as many as the theorymodel CS4d. At the same time, the “data” has nearlythree times more b tags in the Muon20 box than doesLM8, a 6.5 σ discrepancy. With a little more data thesame puzzling discrepancy turns up in the comparisonwith CS4d.How to explain this? Recall that both models LM8 andCS4d produce a large number of b ’s and W ’s, and LM8also has large numbers of Z ’s. Events ending up in theMuon20 box have hard muons from either W / Z decaysor from energetic cascades with semileptonic B decays injets. If we removed all of the W ’s and Z ’s we would havemany fewer events in the Muon20 box. However preciselyin this case all of the events in the Muon20 box will havehard muons inside b jets. This would then explain a high b tagging rate in this box combined with a smaller overallcount.Already with 100 pb − we have strong clues that thedata model has a light gluino, heavy squarks, and thatthe gluino decays do not involve W ’s, Z ’s or sleptons.Table XIX shows that the “data” has only 59 τ tagsper 1000 pb − ; this gives a 6 σ deficiency in τ tags forthe data relative to model LM2p after 1000 pb − . Thelack of τ tags indicates we are not making a lot of τ ’s,but if we are not making either W ’s or τ ’s then we areprobably not making charginos from gluino decays. Thissuggests that the mass splitting between the gluino andthe chargino is relatively small, and that the gluino has4a three-body decay. A three-body decay mediated by avirtual chargino would imply more muons, so we a ledto the three-body decay ˜ g → q ¯ q ˜ χ mediated by a virtualsquark.Putting it all together (within the hypothesis of SUSY)leads to a model like CS6, with heavy squarks, a lightgluino, and a compressed gaugino spectrum. Productionis dominated by gluino pairs, and gluino decays are dom-inated by the three-body mode to q ¯ q ˜ χ . The only muonsare from semileptonic B decays.This scenario also makes predictions that can be testedwith more data. For example, the hemisphere countsshould be quite symmetrical, since we are almost alwaysproducing a pair of the same particles with the samedecays. Indeed this prediction is borne out with 1000pb − , where both r(Hem2) and r(Hem3) are > σ smallerfor CS6 than for LM8.These hemisphere ratios also demonstrate the impor-tance of the pdf uncertainties. Figures 23 and 24 showthe breakdown of the uncertainties for some of the dis-criminating ratios with 1000 pb − in the comparisons ofCS6 “data” with models CS4d and LM8. For CS4d ther(Hem2) ratio, which discriminates at 4.6 σ , is systemat-ics limited, while the r(Hem3) ratio still has a ratherlarge statistical uncertainty. For LM8 we notice thatthe theory systematic is the largest uncertainty for bothr(Hem2) and r(Hem3).These differences are explained by Figures 25 and 26,which show the spread in the values of the ratios as wevary the parton distribution functions used in the simula-tion. Note that the pdf spreads are model dependent: thespreads for the hemisphere ratios r(Hem2) and r(Hem3)are twice as large for LM8 as they are for CS4d. Thisexplains why the total theory systematic for these ratiosis larger for LM8 than for CS4d. However because ofbetter statistics the hemisphere ratios appear to discrim-inate better for CS6 vs LM8 than they do for CS6 vsCS4d. The caveat is that the validity of this statementdepends crucially on whether our estimates of the pdfuncertainties are at least roughly accurate. VIII. SUMMARY OF GROUP 2 RESULTS
Table XXII summarizes the results from the models ofGroup 2. There are 6 pairwise model comparisons. Onemodel is taken as the simulated data, with the numberof signal events scaled to either the 100 pb − discoverydata set or a 1000 pb − follow-up. In our analysis weinclude the correct statistical uncertainty arising from theassumed 100 pb − or 1000 pb − of integrated luminosityin the “data” sample, as described in Section VI.B.The most remarkable feature of these results is thatwe achieve greater than 4 σ discrimination of non-SUSYmodel LH2 from its SUSY look-alikes NM4 and CS7, al-ready at the moment of discovery . With the larger 1000pb − data set, we achieve > σ discrimination in everycase, for more than five distinct ratios per comparison. Thus even with small data sets we can both distinguishSUSY and non-SUSY explanations of the same excessand have multiple handles to inform us about key prop-erties of the true model behind the data.To see how this works in detail, let us take LH2 as our“data”. Suppose that we lived in a world where particletheorists believed that any missing energy signal has tobe explained by SUSY (until recently we did in fact live insuch a world). Then clever theorists might construct theSUSY model NM4 shown in Figure 6 as an explanationof the missing energy discovery.Applying our look-alike analysis, however, one detectsa problem already with 100 pb − . The m T ratio r(mT2-500) (computed using the NM4 LSP mass) is three timeslarger for the “data” than for the SUSY model NM4,a nearly 5 σ discrepancy. As seen in Table XXIII, this ispositively correlated with the m T ratio r(mT2-500/300),which is more than twice as large for the “data” as forNM4, a 3 σ discrepancy, and uncorrelated with the jetmultiplicity, which shows no significant difference.These results strongly suggest that the true model un-derlying the “data” has heavier parent particles thandoes SUSY model NM4. However the SUSY enthusiastwill be quite confused by this conclusion, since the kine-matic ratio r(Meff1400) is more than two times greater for NM4 than it is for the “data”, a 3 σ discrepancy thatseems to directly contradict our previous conclusion. Aconservative SUSY enthusiast might choose to wait formore data, but this will only reinforce the confusion; with1000 pb − the ratio r(mT2-500/300) indicates with 8.5 σ significance that the NM4 squarks are too light, whiler(Meff1400), r(M1400) and r(HT900) all indicate at > σ that SUSY NM4 events are too energetic!At some point our SUSY enthusiast may decide to re-place SUSY model NM4 with SUSY model CS7. This isa bright idea since the parent production in CS7 is allgluinos, instead of all squarks as in NM4, and the gluinokinematics are naturally softer then squark kinematics,for comparable masses. Thus, as seen in Table XXIV,CS7 matches both the kinematics and the overall yieldof the “data” much better than NM4: even with 1000pb − r(Meff1400), r(M1400) and r(HT900) have no dis-crepancies as large as 3 σ .However our SUSY enthusiast still has serious prob-lems. Table XXIV shows that now the E miss T distri-butions are way off: even with 100 pb − the ratior(MET420) is more than twice as big for the “data” asfor CS7, while r(MET520) is three times as big; these areboth > σ discrepancies. Furthermore the jet multiplic-ities don’t match: the ratio r(4j)(3j) is almost twice aslarge for CS7 as for the “data”, a 4 σ discrepancy with100 pb − .Figures 27 and 28 demonstrate the robustness of theseresults, by showing the breakdown of the experimentaland theoretical uncertainties for the relevant ratios. Withthe exception of r(4j)(3j), the uncertainties on all of theratios that we have been discussing are completely dom-inated by the low statistics of our small “data” sample.5Thus, for example, doubling the pdf uncertainties wouldnot alter any of the conclusions reached above.It is not obvious that our SUSY diehard can fix up aSUSY candidate to falsely explain the non-SUSY “data”,while surviving the scrutiny of our look-alike analysis.This applies even for small data sets on the order of afew hundred inverse picobarns. The key observation isthat although SUSY models have many adjustable pa-rameters, the number of adjustable parameters relevant to this look-alike analysis is small compared to the num-ber of robust discriminators. LH2 vs. NM4 [100 pb − ]Variable LH2 NM4 SeparationMETr(mT2-500) 0.16 0.05 4.87r(mT2-400) 0.44 0.21 4.84r(mT2-300) 0.75 0.54 3.49r(Meff1400) 0.11 0.25 2.99r(mT2-500/300) 0.21 0.09 2.98r(M1400) 0.07 0.19 2.69r(mT2-400/300) 0.58 0.40 2.48r(HT900) 0.13 0.24 2.34r(MET420) 0.48 0.37 2.00r(mT2-500/400) 0.36 0.22 1.47TABLE XXIII: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of LH2vs. NM4, taking LH2 as the “data”, assuming an integratedluminosity of 100 pb − .LH2 vs. CS7 [100 pb − ]Variable LH2 CS7 SeparationMETr(mT2-500) 0.27 0.08 6.68r(MET420) 0.48 0.20 6.49r(MET520) 0.21 0.07 5.06r(MET320) 0.78 0.53 4.29r(mT2-500/300) 0.32 0.12 4.24r(4j)(3j) 0.36 0.61 4.04r(mT2-400) 0.63 0.40 4.00r(mT2-300) 0.85 0.62 3.55r(mT2-500/400) 0.43 0.19 3.52r(Hem1) 0.79 0.63 2.59TABLE XXIV: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of LH2vs. CS7, taking LH2 as the “data”, assuming an integratedluminosity of 100 pb − . r ( M E T ) r ( M E T ) r ( M ) r ( H T ) r ( M e ff ) r ( m T - ) r ( m T - ) r ( m T - ) r ( m T - / ) r ( m T - / ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 27: Breakdown of estimated uncertainties for discrimi-nating ratios with 100 pb − , in the comparison of look-alikemodels LH2 and NM4, with LH2 treated as the “data”. r ( j )( j ) r ( j )( j ) r ( M E T ) r ( M E T ) r ( M E T ) r ( H e m ) r ( m T - ) r ( m T - ) r ( m T - ) r ( m T - / ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 28: Breakdown of estimated uncertainties for discrimi-nating ratios with 100 pb − , in the comparison of look-alikemodels LH2 and CS7, with LH2 treated as the “data”. IX. DISCUSSION AND OUTLOOK
We have presented a concrete strategy for determiningthe underlying theory model of an early missing energydiscovery at the LHC. Applying this look-alike analysis toa realistic simulation, we were able to distinguish a non-SUSY model from its SUSY look-alikes essentially at themoment of discovery, with little more than 100 pb − ofintegrated luminosity. In 23 of 26 pairwise comparisons,mostly SUSY with SUSY, we were able to discriminatelook-alikes at better than 5 σ significance with at least onerobust observable and 1000 pb − or less of integrated lu-minosity. Even in the three cases with the worst discrim-6ination we found strong hints of the key properties of theunderlying model; these would be confirmed with moredata and/or by our improving the look-alike analysis.One surprise of our study (at least to us) was the sen-sitivity and robustness of the ratios based on the strans-verse mass m T . Keep in mind that we did not applythe m T distributions to their originally intended use i.e.extracting masses from endpoints and kinks, and we ap-plied our m T ratios to data sets 100 times smaller thanused in previous studies. Nevertheless we found that the m T ratios are among our best discriminators. One of themost important features of the m T ratios is that to firstapproximation they do not depend on the spins of theparent particles. Since ratios based on more traditionalkinematic distributions like H T and M eff have a large de-pendence on the spins of the parent particles, comparing m T ratios to these ratios is a powerful discriminator forspin.Our main goal in this study was to develop a look-alike analysis for missing energy that can be successfullyapplied to the LHC data in the first year of physics run-ning. The crucial properties of such an analysis are re-alism, robustness, validation and sensitivity. We brieflysummarize where we stand with respect to establishingthese properties. Realism
We have employed state-of-the-art event generation forthe missing energy signals, but only at leading order ineach subprocess. In the next phase we include the possi-bility of extra high E T jets at the matrix element level.We are performing this study within the CMS collabora-tion and hence replace our fast simulation with the fullCMS detector simulation. This also allows us to includethe Standard Model backgrounds in the full analysis, re-placing the background subtraction assumed here. Robustness
Our analysis is already quite robust against disappoint-ments in the performance of the LHC detectors duringthe first physics run. We have assumed a minimal paletteof reconstructed objects and triggers. Because our analy-sis uses only ratios of correlated inclusive counts, there isa large cancellation of theoretical and experimental un-certainties, and we can make simple apples–with–applescomparisons between different observables.Despite the fact that we are considering small datasets, we have not employed any multivariate statisticalmethods. As mentioned earlier such methods are left forthe era of demonstrated understanding of the correlationsbetween observables. By dispensing with these methodswe lose sensitivity but gain a cleaner more robust analy-sis. We gain additional robustness from the physics redun-dancy built into our choice of correlated observables. Forexample, jet multiplicity is correlated with track countsin the cones, and sometimes with the ratios of m T countsto total number of events in the box. The hemisphereratios are correlated with the difference counts in thecones. Muon counting is correlated with b tagging. Thefour trigger boxes provide us with four complete sets ofratios, allowing further comparisons and cross-checks.Our main deficiency in robustness is the limited num-ber of theory models simulated for this study. In the nextphase we are including a much larger number and varietyof models.A possible approach to expanding this analysis is to ap-ply the idea of OSETs [89], as a strategy for effectivelysampling the entire theory space. A disadvantage of thisapproach is that, by definition, we give up our spin sen-sitivity and more generally any discrimination based ondetails of the matrix elements. Validation
No studies performed to date of the LHC phenomenol-ogy of any BSM theory have adequately validated un-certainties. The experimental uncertainties cannot besufficiently validated until we have LHC data. The the-oretical uncertainties could be validated sooner, but thiswill require many more detailed studies adhering to atleast the degree of realism attempted here.In the next phase of this analysis (currently being per-formed within the CMS collaboration) we are validatingthe Standard Model backgrounds of the inclusive missingenergy signature using the CMS full simulation frame-work. This, among other things, is allowing us to com-pute the backgrounds in each of our trigger boxes andstudy the effect of varying the selection criteria, e.g. re-laxing or modifying the ILV.The full detector simulation itself needs to be validatedagainst real LHC data, using these same Standard Modelprocesses as benchmarks. Similar comments apply tothe parton distribution functions. In both cases we needto develop a more sophisticated parametrization of theuncertainties.The event generation chains used in our study, whilestate-of-the-art, have not been adequately validated; wehave performed several cross-validation checks and ob-tained significant discrepancies. This is an importanttask for the entire LHC theory community.
Sensitivity
We have demonstrated the importance of being able toobtain subsamples of the discovery data set enriched in τ ’s and b ’s. To do this successfully will require a differentapproach to flavor tagging in the early LHC running, anapproach that emphasizes robustness and fast validation7over efficiency and purity. It seems likely that a dedicatedeffort could achieve results as good as our preliminarystudy, and possibly much better.We have seen that one of the virtues of the m T ratiosis sensitivity to the number of weakly interacting par-ticles in the final state. It is important to study thisfurther, along with the potential to extract estimates ofthe masses of parent particles and the LSP from smalldata sets. Dark matter at 100 pb − We have demonstrated a concrete strategic solutionto the LHC Inverse Problem applicable under realisticconditions to early physics running at the LHC. Since themissing energy signature is motivated by the existenceof dark matter, we should also address what might becalled the Dark Matter Inverse Problem: given a missingenergy discovery at the LHC, what can we learn aboutdark matter and the cosmological events that producedit? This problem has not been addressed at all for the100 pb − era of LHC running. Given an early missingenergy discovery at the LHC, this problem will becomeone of the most interesting questions in particle physics,especially tied in to results from the ongoing direct andindirect dark matter searches. LM5 vs. CS4d [100 pb − ]Variable LM5 CS4d SeparationMETr(HT900) 0.38 0.24 3.59r(Meff1400) 0.32 0.21 3.15r(MET520) 0.28 0.19 3.12DiJetr(DiJet) 0.22 0.15 2.37r(MET520) 0.28 0.23 0.56r(Hem2) 0.27 0.32 0.51TriJetr(TriJet) 0.22 0.17 1.89r(Meff1400) 0.65 0.59 0.62r(HT900) 0.73 0.67 0.59Muon20r(HT900) 0.35 0.25 1.17r(mT2-300) 0.27 0.36 1.00r(mT2-400) 0.26 0.34 0.93TABLE XXV: Largest separation (in units of σ ) for the com-parison of LM5 vs. CS4d (error on CS4d) assuming an inte-grated luminosity of 100 pb − . LM5 vs. CS4d [100 pb − ]Variable LM5 CS4d SeparationMETr(HT900) 0.38 0.24 3.59r(Meff1400) 0.32 0.21 3.15r(MET520) 0.28 0.19 3.12r(mT2-600/300) 0.29 0.17 2.83r(mT2-600/400) 0.31 0.18 2.82r(mT2-600/500) 0.46 0.30 2.22r(MET420) 0.49 0.40 2.14r(mT2-300) 0.36 0.45 1.98r(mT2-400) 0.34 0.43 1.87r(mT2-600) 0.11 0.08 1.47TABLE XXVI: Largest separation (in units of σ ) for the com-parison of LM5 vs. CS4d (error on CS4d) assuming an inte-grated luminosity of 100 pb − .LM5 vs. CS4d [1000 pb − ]Variable LM5 CS4d SeparationMETr(MET520) 0.28 0.19 7.07r(HT900) 0.38 0.24 6.41r(mT2-600/400) 0.31 0.18 6.10DiJetr(DiJet) 0.22 0.15 4.80r(mT2-600/300) 0.29 0.15 2.69r(mT2-600/400) 0.30 0.16 2.69TriJetr(TriJet) 0.22 0.17 3.50r(MET520) 0.28 0.24 1.31r(Meff1400) 0.65 0.59 1.29Muon20r(mT2-600/300) 0.23 0.13 3.04r(mT2-600/400) 0.25 0.14 2.98r(HT900) 0.35 0.25 2.63TABLE XXVII: Best discriminating ratios in each trigger box,with separations in units of σ , for the comparison of LM5vs. CS4d, taking LM5 as the “data”, assuming an integratedluminosity of 1000 pb − . Acknowledgments
The authors are grateful to Keith Ellis, Gordon Kane,Filip Moortgat, Stephen Mrenna, Luc Pape, TilmanPlehn, Chris Quigg, Sezen Sekmen, Phillip Schuster, Pe-ter Skands and Natalia Toro for useful discussions. JLand MS acknowledge the Michigan Center for Theoret-ical Physics for hospitality during the concluding phaseof this work; JH similarly acknowledges the Kavli Insti-tute for Theoretical Physics. Fermilab is operated by the8
LM5 vs. CS4d [1000 pb − ]Variable LM5 CS4d SeparationMETr(MET520) 0.28 0.19 7.07r(HT900) 0.38 0.24 6.41r(mT2-600/400) 0.31 0.18 6.10r(mT2-600/300) 0.29 0.17 6.04r(Meff1400) 0.32 0.21 5.94r(mT2-600/500) 0.46 0.30 4.88r(MET420) 0.49 0.40 3.82r(mT2-600) 0.11 0.08 3.52r(mT2-300) 0.36 0.45 3.12r(mT2-400) 0.34 0.43 3.02TABLE XXVIII: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of LM5vs. CS4d, taking LM5 as the “data”, assuming an integratedluminosity of 1000 pb − .LM2p vs. LM5 [100 pb − ]Variable LM2p LM5 SeparationMETr(5j)(3j) 0.33 0.40 1.64r(mT2-300) 0.41 0.34 1.44r(mT2-400) 0.30 0.25 1.34r(4j)(3j) 0.64 0.69 1.18r( τ -tag) 0.07 0.05 1.17r(10t-c45) 0.30 0.36 1.14r(10t-c30) 0.16 0.20 1.13r(10t-c75) 0.48 0.53 1.07r(10t-c60) 0.41 0.47 1.07r(mT2-500) 0.16 0.13 1.05TABLE XXIX: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of LM2pvs. LM5, taking LM2p as the “data”, assuming an integratedluminosity of 100 pb − . Fermi Research Alliance LLC under contract DE-AC02-07CH11359 with the U.S. Dept. of Energy. Researchat Argonne National Laboratory Division is supportedby the U.S. Dept. of Energy under contract DE-AC02-06CH11357.
APPENDIX A: DETAILED RESULTS FORGROUP 11. LM5 vs CS4d
This comparison is described in section VII A. Wetreat model LM5 as the data and CS4d as the the-ory model. For 100 pb − of integrated luminosity, Ta- r ( j )( j ) r ( j )( j ) -t a g ) t r ( r ( m T - ) r ( m T - ) r ( m T - ) r ( t- c30 ) r ( t- c45 ) r ( t- c60 ) r ( t- c75 ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 29: Breakdown of estimated uncertainties for discrimi-nating ratios with 100 pb − , in the comparison of look-alikemodels LM2p and LM5, with LM2p treated as the “data”. r ( j )( j ) r ( j )( j ) -t a g ) t r ( r ( m T - ) r ( m T - ) r ( m T - ) r ( t- c30 ) r ( t- c45 ) r ( t- c60 ) r ( t- c75 ) [ % ] X ) / X ( X - -20-15-10-505101520 CTEQ5l1CTEQ6mMRST2004nlo
FIG. 30: Pdf spreads for discriminating ratios with 100 pb − ,in the comparison of look-alike models LM2p and LM5, withLM2p treated as the “data”. ble XXV shows the three best discriminating ratios asdefined in each of the trigger boxes: MET, DiJet, TriJetand Muon20. Table XXVI shows the ten best discrimi-nating ratios in the MET box. The separation in units of σ is computed from the total estimated theoretical andexperimental uncertainties as described in section VI.The same information for 1000 pb − is shown inTables XXVII and XXVIII. The notation r(DiJet),r(TriJet) and r(Muon20) denotes the ratio of the numberof events in that box with the number of events in theMET box. Since (plus or minus about one event in 1000pb − ) the DiJet, TriJet and Muon20 boxes are subsam-ples of the MET box, these are ratios of inclusive counts.9 s Number of -1
10 1 10 r(4j)(3j)r(5j)(3j)-tag) t r(r(mT2-300)r(mT2-400)r(mT2-500)r(10t-c30)r(10t-c45)r(10t-c60)r(10t-c75) LM2p vs. LM5 MET [100pb]
FIG. 31: Pulls of the best discriminating ratios with 100 pb − ,in the comparison of look-alike models LM2p and LM5, withLM2p treated as the “data”.LM2p vs. LM5 [1000 pb − ]Variable LM2p LM5 SeparationMETr( τ -tag) 0.07 0.05 3.14r(5j)(3j) 0.33 0.40 2.77r(mT2-400) 0.30 0.25 2.56r(mT2-500) 0.16 0.13 2.53r(mT2-300) 0.41 0.34 2.27r(10t-c30) 0.16 0.20 1.67r(MET520) 0.31 0.28 1.67r(20t-c45) 0.07 0.09 1.61r(10tdiff-c30) 0.14 0.16 1.61r(4j)(3j) 0.64 0.69 1.56TABLE XXX: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of LM2pvs. LM5, taking LM2p as the “data”, assuming an integratedluminosity of 1000 pb − .
2. LM2p vs LM5
This comparison is described in section VII B. Wetreat model LM2p as the data and LM5 as the theorymodel. For 100 pb − , the ten best discriminating ra-tios are listed in Table XXIX and the pulls are displayedin Figure 31. While the τ tag ratio is not discriminat-ing well, Figure 29 shows that this is due entirely to poorstatistics in this small simulated data sample: the experi-mental statistical uncertainty is 30%, compared to exper-imental and theory systematics both estimated at around5%. The small theory statistical uncertainty shown is theerror from the finite Monte Carlo statistics in simulatingthe theory model. Table XXX shows the improvement CS4d vs. LM8 [100 pb − ]Variable CS4d LM8 SeparationMETr(5j)(3j) 0.40 0.54 2.83r(mT2-300) 0.41 0.32 2.14r(4j)(3j) 0.70 0.81 2.11r(10t-c30) 0.20 0.28 1.93r(20t-c60) 0.16 0.23 1.78r(20t-c45) 0.08 0.13 1.73r(10t-c30)(10t-c90) 0.34 0.43 1.72r(10t-c45) 0.36 0.46 1.70r(30t-c90) 0.11 0.17 1.68r(Meff1400) 0.21 0.28 1.65TABLE XXXI: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of CS4dvs. LM8, taking CS4d as the “data”, assuming an integratedluminosity of 100 pb − .CS4d vs. LM8 [1000 pb − ]Variable CS4d LM8 SeparationMETr(5j)(3j) 0.40 0.54 4.21r(10tdiff-c30) 0.15 0.20 3.63r(Hem3) 0.07 0.11 3.58r(mT2-300) 0.41 0.32 3.43r(mT2-400) 0.25 0.20 3.24r(20t-c30) 0.02 0.04 3.09r(20tdiff-c45) 0.06 0.08 3.05r(20tdiff-c60) 0.12 0.16 3.04r(20t-c45) 0.08 0.13 3.03r(10t-c30)(10t-c90) 0.34 0.43 2.95TABLE XXXII: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of CS4dvs. LM8, taking CS4d as the “data”, assuming an integratedluminosity of 1000 pb − . in the discriminating power of r( τ -tag) with 1000 pb − ,due to the reduction in the experimental statistical un-certainty. Note that in this difficult case no other ratiodiscriminates with better than 3 σ significance.In Figure 29 one also notices large differences in therelative size of the theory systematics for the differentratios. These are due to large differences in the spreadof values when we vary the parton distribution functionsused in the simulation, as shown in Figure 30. Noticethat the pdf spreads vary from less than 5% for the jetmultiplicity ratio r(4j)(3j) to greater than 20% for thecone track count ratio r(10t-c30).This is an important generic feature of our results. Wefind the pdf uncertainties to be process dependent andthus model dependent. We find also that the relative pdf0 r ( j )( j ) r ( j )( j ) r ( H e m ) r ( H e m ) r ( M ) r ( M ) r ( H T ) r ( M e ff ) r ( m T - ) r ( m T - ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 32: Breakdown of estimated uncertainties for discrimi-nating ratios with 100 pb − , in the comparison of look-alikemodels CS4d and LM8, with CS4d treated as the “data”. r ( t- c90 ) r ( t- c90 ) r ( t- c90 ) r ( t d i ff- c30 ) r ( t d i ff- c45 ) r ( t d i ff- c45 ) r ( t d i ff- c60 ) r ( t d i ff- c60 ) r ( t d i ff- c75 ) r ( t d i ff- c75 ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 33: Breakdown of estimated uncertainties for discrimi-nating ratios with 100 pb − , in the comparison of look-alikemodels CS4d and LM8, with CS4d treated as the “data”. uncertainties for different ratios in the same model varyby factors as large as 4 or 5.
3. CS4d vs LM8
This comparison is described in section VII C. Wetreat model CS4d as the data and LM8 as the theorymodel. Tables XXXI and XXXII show the ten best dis-criminating ratios for 100 pb − and 1000 pb − respec-tively. We observe that the best discriminator with 100pb − , the jet multiplicity ratio r(5j)(3j), remains the bestwith 1000 pb − . However the second and third best dsi-criminators with 1000 pb − , r(10tdiff-c30) and r(Hem3),are not even among the top ten best with 100 pb − . CS6 vs. LM2p [100 pb − ]Variable CS6 LM2p SeparationMETr(MET420) 0.22 0.53 6.99r(MET520) 0.09 0.31 6.10r(mT2-500/300) 0.08 0.39 5.14DiJetr(DiJet) 0.11 0.21 3.04r(5j)(3j) 0.54 0.32 3.00r(4j)(3j) 0.83 0.62 2.56TriJetr(MET420) 0.29 0.53 2.96r(MET320) 0.55 0.77 2.84r(HT900) 0.57 0.75 2.46Muon20r( b -tag) 0.74 0.36 4.25r(Muon20) 0.06 0.14 3.14r(MET420) 0.23 0.48 2.55TABLE XXXIII: Best discriminating ratios in each triggerbox, with separations in units of σ , for the comparison of CS6vs. LM2p, taking CS6 as the “data”, assuming an integratedluminosity of 100 pb − .CS6 vs. LM2p [100 pb − ]Variable CS6 LM2p SeparationMETr(MET420) 0.22 0.53 6.99r(MET520) 0.09 0.31 6.10r(mT2-500/300) 0.08 0.39 5.14r(mT2-500/400) 0.17 0.54 5.11r(MET320) 0.54 0.78 4.93r(HT900) 0.18 0.38 4.80r(mT2-500) 0.03 0.16 4.56r(Meff1400) 0.16 0.32 3.91r(mT2-400/300) 0.49 0.73 3.88r(mT2-400/300) 0.49 0.73 3.88TABLE XXXIV: Best discriminating ratios in the MET box,with separations in units of σ , for the comparison of CS6vs. LM2p, taking CS6 as the “data”, assuming an integratedluminosity of 100 pb − . These again are generic features of our results. Fig-ures 32 and 33 show that with 100 pb − r(5j)(3j) alreadyhas a statistical uncertainty comparable to the estimatedsystematics, while many other ratios still have large 15 to20% statistical uncertainties. This includes r(10tdiff-c30)and r(Hem3), ratios that are sensitive to what fraction ofevents have large hemisphere differences in object countsor track counts. Thus qualitatively new features of theevents emerge automatically as good discriminators as1 r ( M e ff ) -t a g ) t r ( r ( m T - ) r ( m T - ) r ( m T - ) r ( m T - / ) r ( m T - / ) r ( m T - / ) r ( m T - / ) r ( m T - / ) E rr o r [ % ] Exp. Statistical ErrorExp. Systematic ErrorTeo. Statistical ErrorTeo. Systematic Error
FIG. 34: Breakdown of estimated uncertainties for discrimi-nating ratios with 100 pb − , in the comparison of look-alikemodels CS6 and LM2p, with CS6 treated as the “data”. ) (GeV/c T2 m0 100 200 300 400 500 600 700 800 900 1000 nu m b er o f e v e n t s FIG. 35: Comparison of the m T distribution of the CS6“data” (solid red line) to that of the theory model LM2p(dashed blue line) for 100 pb − . Here m T is computed usingthe LSP mass of the theory model LM2p. the integrated luminosity goes up, without changing thedesign of the look-alike analysis.
4. CS6 vs LM2p, LM5, LM8 and CS4d
This comparison is described in section VII D. Wetreat model CS6 as the data and compare to theory mod-els LM2p, LM5, LM8 and CS4d. Tables XXXIII andXXXIV show the best discriminating ratios for the com-parison of LM2p to the CS6 “data” with 100 pb − . We see that two E miss T ratios and two m T ratios already dis-criminate at better than 5 σ .Figure 34 shows the breakdown of the uncertainitesfor some of the ratios in the this comparison. Observethat the kinematic ratio r(Meff1400) has a smaller ex-perimental statistical uncertainty than do the m T ra-tios r(mT2-500/300) and r(mT2-500/400); neverthelessTable XXXIV shows that r(Meff1400) is a worse discrim-inator.This is a generic feature of our results. The distribu-tions in M eff are rather broad, whereas the m T distribu-tions are steeply falling as one approaches the endpointregion. As seen in Figure 35, when the parent particlemass differences are large, m T is an intrinsically gooddiscriminator even for quite small data samples withinvery modest resolutions.Tables XXXV and XXXVI show the best discriminat-ing ratios for the comparison of LM8 ans CS4d to theCS6 “data” with 100 pb − . These results show the im-portance of the ratios in the Muon20 box. CS6 vs. LM8 [100 pb − ]Variable CS6 LM8 SeparationMETr(MET420) 0.22 0.39 4.03r(Hem2) 0.19 0.34 3.60r(MET520) 0.09 0.20 3.45DiJetr(DiJet) 0.11 0.18 2.21r(Hem2) 0.28 0.35 0.87r(MET320) 0.59 0.66 0.86TriJetr(Meff1400) 0.50 0.64 1.58r(HT900) 0.57 0.68 1.35r(MET320) 0.55 0.66 1.21Muon20r( b -tag) 0.74 0.29 6.45r(Muon20) 0.06 0.24 5.19r(MET420) 0.23 0.36 1.78TABLE XXXV: Best discriminating ratios in each trigger box,with separations in units of σ , for the comparison of CS6vs. LM8, taking CS6 as the “data”, assuming an integratedluminosity of 100 pb − . APPENDIX B: DETAILED RESULTS FORGROUP 21. LH2 vs NM4 and CS7
This comparison is described in section VIII. We treatthe little Higgs model LH2 as the data and compare toSUSY models NM4 and CS7. Tables XXXVII-XL show2
CS6 vs. CS4d [100 pb − ]Variable CS6 CS4d SeparationMETr(MET420) 0.22 0.40 4.25r(MET320) 0.54 0.70 3.26r(MET520) 0.09 0.19 3.15DiJetr(5j)(3j) 0.54 0.39 1.53r(DiJet) 0.11 0.15 1.42r(4j)(3j) 0.83 0.69 1.41TriJetr(MET320) 0.55 0.70 1.63r(MET420) 0.29 0.43 1.53r(HT900) 0.57 0.67 1.17Muon20r(Muon20) 0.06 0.18 3.95r(MET320) 0.54 0.69 1.73r(MET420) 0.23 0.37 1.59TABLE XXXVI: Best discriminating ratios in each triggerbox, with separations in units of σ , for the comparison of CS6vs. CS4d, taking CS6 as the “data”, assuming an integratedluminosity of 100 pb − . the best discriminating ratios for the comparison of NM4and CS7 to the LH2 “data” with 1000 pb − .With the better statistics of 1000 pb − , Ta-ble XXXVIII illustrates even more clearly the conun-drum discussed in section VIII. The m T ratio r(mT2-500/300) shows unquestionably that the parent particlemasses in model NM4 are too small to fit the “data”.The E miss T distribution of NM4 also appears to be toosoft, with 4.3 σ significance. However the kinematic dis-tributions for NM4 represented by r(Meff1400), r(M1400)and r(HT900) are all too hard, with > σ significance.The other impressive feature of these tables is thatwith 1000 pb − we acquire several highly discriminatingratios in the DiJet and TriJet boxes. With real data thiswould provide an impressive redundancy of cross-checks,still within the original design of our look-alike analysis.The large number of independent highly discriminatingrobust ratios seen here provide a powerful tool to resolveSUSY look-alikes from non-SUSY look-alikes. APPENDIX C: COMPARISON OF SQUARKPRODUCTION WITH HEAVY QUARKPRODUCTION1. smuon production versus muon production
Let’s compare the QED processes e + e − → µ + µ − and e + e − → ˜ µ R ¯˜ µ R . We will use the conventions and notationof Peskin and Schroeder (PS) [90], and work in the ap- LH2 vs. NM4 [1000 pb − ]Variable LH2 NM4 SeparationMETr(mT2-500) 0.16 0.05 14.11r(mT2-400) 0.44 0.21 11.13r(mT2-500/300) 0.21 0.09 8.52DiJetr(mT2-400) 0.32 0.12 7.89r(mT2-300) 0.64 0.32 7.79r(DiJet) 0.11 0.22 5.94TriJetr(mT2-300) 0.62 0.19 10.96r(mT2-400) 0.34 0.07 10.91r(TriJet) 0.06 0.15 5.94Muon20r(mT2-400) 0.38 0.14 5.03r(mT2-300) 0.72 0.42 4.30r(Meff1400) 0.10 0.34 3.50TABLE XXXVII: Best discriminating ratios in each triggerbox, with separations in units of σ , for the comparison of LH2vs. NM4, taking LH2 as the “data”, assuming an integratedluminosity of 1000 pb − .LH2 vs. NM4 [1000 pb − ]Variable LH2 NM4 SeparationMETr(mT2-500) 0.16 0.05 14.11r(mT2-400) 0.44 0.21 11.13r(mT2-500/300) 0.21 0.09 8.52r(Meff1400) 0.11 0.25 7.24r(M1400) 0.07 0.19 6.57r(mT2-300) 0.75 0.54 6.26r(mT2-400/300) 0.58 0.40 5.77r(HT900) 0.13 0.24 5.67r(M1800) 0.02 0.07 4.82r(MET420) 0.48 0.37 4.32TABLE XXXVIII: Best discriminating ratios in the METbox, with separations in units of σ , for the comparison of LH2vs. NM4, taking LH2 as the “data”, assuming an integratedluminosity of 1000 pb − . proximation that the electron and positron are massless.In this notation p and p ′ denote the incoming 4-momentaof the electron and positron, while k and k ′ denote theoutgoing 4-momenta of the muons or smuons. The pho-ton 4-momentum is denoted by q = p + p ′ . We will use m interchangably to denote the mass of the muon or smuon,assuming them (in this pedagogical example) to be de-generate.The leading order QED matrix element for e + e − → LH2 vs. CS7 [1000 pb − ]Variable LH2 CS7 SeparationMETr(mT2-500) 0.27 0.08 18.87r(MET420) 0.48 0.20 16.73r(MET520) 0.21 0.07 14.49DiJetr(4j)(3j) 0.20 0.67 7.30r(mT2-300) 0.72 0.31 6.73r(mT2-400) 0.53 0.22 6.26TriJetr(mT2-500) 0.20 0.04 8.83r(mT2-300) 0.68 0.32 7.43r(mT2-400) 0.53 0.22 7.18Muon20r(mT2-300) 0.84 0.35 1.57r(mT2-400) 0.60 0.24 1.32TABLE XXXIX: Best discriminating ratios in each triggerbox, with separations in units of σ , for the comparison of LH2vs. CS7, taking LH2 as the “data”, assuming an integratedluminosity of 1000 pb − .LH2 vs. CS7 [1000 pb − ]Variable LH2 CS7 SeparationMETr(mT2-500) 0.27 0.08 18.87r(MET420) 0.48 0.20 16.73r(MET520) 0.21 0.07 14.49r(mT2-600) 0.05 0.01 14.11r(mT2-500/300) 0.32 0.12 11.17r(mT2-500/400) 0.43 0.19 9.77r(mT2-600/300) 0.06 0.01 9.77r(mT2-400) 0.63 0.40 8.46r(MET320) 0.78 0.53 8.17TABLE XL: Best discriminating ratios in the MET box, withseparations in units of σ , for the comparison of LH2 vs. CS7,taking LH2 as the “data”, assuming an integrated luminosityof 1000 pb − . µ + µ − is¯ v s ′ ( p ′ )( − ieγ µ ) u s ( p ) (cid:18) − iq (cid:19) ¯ u r ( k )( − ieγ µ ) v r ′ ( k ′ ) (C1)The corresponding matrix element for e + e − → ˜ µ R ¯˜ µ R is¯ v s ′ ( p ′ )( − ieγ µ ) u s ( p ) (cid:18) − iq (cid:19) ( − ie ( k µ − k ′ µ )) (C2)In each case, we compute the squared matrix element,averaging over the spins of the electrons. For e + e − → µ + µ − we also sum over the spins of the muons, thus P s,s ′ ,r,r ′ |M ( s, s ′ , r, r ′ ) | = e q tr [ p ′ γ µ p γ ν ] × tr (cid:2) ( k + m ) γ µ ( k ′ − m ) γ ν (cid:3) , (C3)while for e + e − → ˜ µ R ¯˜ µ R we have P s,s ′ |M ( s, s ′ ) | = e q tr (cid:2) p ′ γ µ ( k µ − k ′ µ ) p γ ν ( k ν − k ′ ν ) (cid:3) . (C4)From now on we will follow the convention of Ellis, Stir-ling and Webber (ESW) [91] and use a barred summationto denote the average over initial spins and sum over finalspins (if any). Thus performing the traces (C3) becomes P |M| = e q (cid:2) ( p · k )( p ′ · k ′ )+( p · k ′ )( p ′ · k )+ m ( p · p ′ ) (cid:3) (C5)while (C4) becomes P |M| = e q (cid:2) ( p · p ′ )( k · k ′ ) − ( p · k )( p ′ · k ′ ) − ( p · k ′ )( p ′ · k )+( p · k )( p ′ · k ) + ( p · k ′ )( p ′ · k ′ ) − m ( p · p ′ ) (cid:3) . (C6)The kinematics of both cases are identical; in the centerof mass frame: p · p ′ = s , k · k ′ = s β ) ; p · k = p ′ · k ′ = s − β cos θ ) ; (C7) p · k ′ = p ′ · k = s β cos θ ) , where θ is the polar angle of the final state muon orsmuon, and β ≡ q − m s . (C8)Substituting, (C5) becomes P |M| = e (cid:2) − β (1 − cos θ ) (cid:3) , (C9)and (C6) becomes P |M| = e β (1 − cos θ ) . (C10)Thus the differential cross section for e + e − → µ + µ − atleading order in QED is dσd (cos θ ) = πα s β (cid:2) − β (1 − cos θ ) (cid:3) , (C11)while for e + e − → ˜ µ R ¯˜ µ R we get dσd (cos θ ) = πα s β (1 − cos θ ) , (C12)agreeing with Farrar and Fayet [66].Since a Dirac muon has two complex scalar superpart-ners, ˜ µ R and ˜ µ L , from now on we will write the combined4cross section for e + e − → ˜ µ R ¯˜ µ R + ˜ µ L ¯˜ µ L , which is justtwice the expression in (C12). Thus we have the totalcross sections: σ ( e + e − → µ + µ − ) = 2 πα s β (3 − β ) ; σ ( e + e − → ˜ µ R ¯˜ µ R + ˜ µ L ¯˜ µ L ) = 2 πα s β . (C13)In the high energy limit β →
1, the leading order muonpair cross section is exactly twice the smuon pair crosssection, as noted for example in [92].
2. q¯q → Q ¯Q versus q¯q → ˜q¯˜q From these formulae it is easy to obtain the lead-ing order cross sections for hadroproduction of heavyquarks/squarks from light q ¯ q initial parton states. Firstwe introduce a kinematic notation more suitable forhadroproduction. The subprocess Mandelstam invari-ants are given byˆ t = m − ˆ s − β cos θ ) ;ˆ u = m − ˆ s β cos θ ) ; (C14)2 m = ˆ s + ˆ t + ˆ u . It is convenient to use the dimensionless variables definedby ESW: τ = m − ˆ t ˆ s ; τ = m − ˆ u ˆ s ; (C15) ρ = 1 − β ;1 = τ + τ . We change variables using d (cos θ ) d ˆ t = 2 β ˆ s . (C16)Thus (C11), the leading order QED differential cross sec-tion for e + e − → µ + µ − , becomes dσd ˆ t = 2 πα ˆ s h τ + τ + ρ i , (C17)while the e + e − → ˜ µ R ¯˜ µ R + ˜ µ L ¯˜ µ L cross section is dσd ˆ t = 2 πα ˆ s h − τ − τ − ρ i . (C18)To convert these formulae into cross sections for theleading order QCD subprocesses q ¯ q → Q ¯ Q (heavy quarkproduction) and q ¯ q → ˜ q R ¯˜ q R + ˜ q L ¯˜ q L (squark production), we replace α by α s , and insert a factor of 2 / s -channel diagram.Thus we get the fully differential cross section for q ¯ q → Q ¯ Q : d σdx dx d ˆ t = 4 πα s s f ( x ) f ( x ) h τ + τ + ρ i , (C19)and for q ¯ q → ˜ q R ¯˜ q R + ˜ q L ¯˜ q L we have d σdx dx d ˆ t = 4 πα s s f ( x ) f ( x ) h − τ − τ − ρ i (C20)where f ( x ) and f ( x ) are the pdfs for the initial statequark and antiquark.We can compare the heavy quark cross section (C19)to ESW by making the change of variables d σdy dy d p T = x x π × d σdx dx d ˆ t , (C21)where y , y are the lab frame rapidities of the heavyquarks, and p T is the transverse momentum. Thus d σdy dy d p T = 4 α s s x f x f h τ + τ + ρ i , (C22)agreeing with eqn. 10.51 of ESW.Similarly, we can compare the differential cross sectionfor squark production (C20) to the literature. The com-parison requires a couple of kinematic identities:ˆ u ˆ t − m = ˆ s h − τ − τ − ρ i ; (C23)ˆ u ˆ t − m = 14 (cid:2) ˆ s (ˆ s − m ) − (ˆ u − ˆ t ) (cid:3) . (C24)Using (C23), we see that the differential cross section forsquark production (C20) agrees with Dawson, Eichtenand Quigg (DEQ) [93], and (C24) shows that we agreewith Harrison and Llewellyn Smith [94].The total cross sections are obtained by integration: σ ( q ¯ q → Q ¯ Q ) = Z dx dx Z ˆ t max ˆ t min d ˆ t d σdx dx d ˆ t (C25)= 4 πα s Z dx dx f f ˆ s Z τ max τ min dτ (1 − τ + 2 τ + ρ . Using τ max = (1 + β ), τ min = (1 − β ), this becomes σ ( q ¯ q → Q ¯ Q ) = 8 πα s Z dx dx f f (ˆ s + 2 m )ˆ s β (C26)5which agrees with the result of Combridge [95]. The anal-ogous total cross section for squark production is σ ( q ¯ q → ˜ q R ¯˜ q R + ˜ q L ¯˜ q L ) =4 πα s Z dx dx f f (ˆ s − m )ˆ s β (C27)= 4 πα s Z dx dx f f s β , which agrees with Harrison and Llewellyn Smith [94].
3. gg → Q ¯Q versus gg → ˜q¯˜q For gluon fusion, we start with the leading order matrixelements as given by ESW and DEQ. At leading orderthere are three diagrams for each process, correspondingto the s , t and u channels, and an additional gluon seagulldiagram for the squark case. In the t and u channels weare not only producing particles of different spins but alsoexchanging particles of different spins. For gg → Q ¯ Q wehave (see Table 10.2 of [91]): P |M| = g s (cid:16) τ τ − (cid:17) (cid:16) τ + τ + ρ − ρ τ τ (cid:17) , (C28)while for gg → ˜ q R ¯˜ q R + ˜ q L ¯˜ q L we have (see eqn. 3.26 of[93]): P |M| = g s (cid:16) + u − ˆ t ) s (cid:17) × h m ˆ t (ˆ t − m ) + m ˆ u (ˆ u − m ) + m (ˆ t − m )(ˆ u − m ) i . (C29)Converting to the ESW kinematic variables and expand-ing, (C29) becomes P |M| = (C30) g s h + ρ − τ τ − ρ τ τ − ρ τ τ + ρ τ τ i . This can be refactored into P |M| = (C31) g s (cid:16) τ τ − (cid:17) (cid:16) − τ − τ − ρ + ρ τ τ (cid:17) . Notice that the sum of the squared matrix elements forheavy quark and squark production with the same masshas a very simple form: P |M| ( gg → Q ¯ Q + ˜ q R ¯˜ q R + ˜ q L ¯˜ q L ) = (C32) g s (cid:16) τ τ − (cid:17) . An analogous simplification also occurs in the q ¯ q initiatedproduction, as is obvious from comparing (C17) to (C18).It does not appear that these elegant SUSY relationshave ever been noticed in the literature! This may bebecause these are not, strictly speaking, MSSM relations.In the MSSM, electroweak symmetry breaking does not occur in the SUSY limit where the soft breaking termsare turned off. Since the MSSM fermions are massless inthe absence of EWSB, there are no MSSM cross sectionrelations between degenerate heavy quarks and squarks.In the simple processes that we considered above, theSUSY limit actually corresponds to some more genericvectorlike SUSY theory.The corresponding fully differential cross section for gg → Q ¯ Q is d σdx dx d ˆ t = πα s ˆ s f ( x ) f ( x ) (C33) × (cid:18) τ τ − (cid:19) (cid:18) τ + τ + ρ − ρ τ τ (cid:19) , and for gg → ˜ q R ¯˜ q R + ˜ q L ¯˜ q L we have d σdx dx d ˆ t = πα s ˆ s f ( x ) f ( x ) (C34) × (cid:18) τ τ − (cid:19) (cid:18) − τ − τ − ρ + ρ τ τ (cid:19) . The total cross sections are σ ( gg → Q ¯ Q ) = πα s Z dx dx f f ˆ s h − β +31 β + [32 − β + (1 − β ) ] ln 1 + β − β i ; (C35) σ ( gg → ˜ q R ¯˜ q R + ˜ q L ¯˜ q L ) = πα s Z dx dx f f ˆ s h β − β − [16(1 − β ) + (1 − β ) ] ln 1 + β − β i . (C36)
4. gq → QW versus gq → ˜q ˜W The process gq → QW contributes to single top pro-duction in the Standard Model via qb → tW − . At leadingorder there is both an s -channel and a t -channel diagram.This can be compared to the SUSY process gq → ˜ q ˜ W ,i.e., the associated production of a squark with a Wino.Again in the corresponding t -channel diagrams we areexchanging particles of different spins (a quark versus asquark).This process obviously cares about EWSB and the factthat we have chiral fermions rather than vectorlike ones.Thus as already explained above we do not expect anelegant SUSY limit relating the two processes. Indeedone observes intrinsic differences already at the level ofcomparing the W and Wino decay widths into Q ¯ q and˜ q ¯ q , respectively. Assuming that these decays were kine-matically allowed, we can extract the leading order ex-pressions from eqn. 5.15 of [96] and eqn. B.88a of [97],6in the limit that we neglect the light quark mass:Γ( W → Q ¯ q ) = 3 g W π m W (cid:20) m m W (cid:21)(cid:20) − m m W (cid:21) ;(C37)Γ( ˜ W → ˜ q ¯ q ) = 3 g W π m ˜ W (cid:20) (cid:21)" − m m W , (C38)where as before m denotes the heavy quark or squarkmass. These formulae coincide in the limit m = m W = m ˜ W , but this is a kinematic limit, not a SUSY limit.The fully differential cross section for gq → QW wascomputed at leading order by Halzen and Kim [98], whilethe leading order cross section for gq → ˜ q ˜ W is given inDEQ [93]. We convert these expressions to ESW nota-tion, introducing a new dimensionless variable δ : δ ≡ m W − m ˆ s , (C39)with m W replaced by m ˜ W in the SUSY case. We alsoreplace (C8) by the definition of β appropriate for twounequal mass final state particles: β ≡ s(cid:20) − ( m + m W ) ˆ s (cid:21) (cid:20) − ( m − m W ) ˆ s (cid:21) . (C40)Thus for gq → QW we obtain d σdx dx d ˆ t = g W α s s f ( x ) f ( x ) " − δρ + 4 δ (C41)+ (cid:20) − δρ + 4 δ (cid:21) (cid:20) δ + τ + 1 − δ (1 − δ ) τ + ρδ τ (cid:21) , and for gq → ˜ q ˜ W we have d σdx dx d ˆ t = g W α s s f ( x ) f ( x ) (C42) × " − δ − τ + 2 δ (1 − δ ) τ − ρδ τ . The total cross sections are σ ( gq → QW ) = g W α s Z dx dx f f ˆ s δ ) − β ] " β (cid:2) − − δ ) β + δ (31 + 13 δ + 21 δ ) (cid:3) (C43)+2 [1 − δ (1 − δ )] (cid:2) − β ) + δ (2 + 3 δ ) (cid:3) ln 1 + β − δ − β − δ ; σ ( gq → ˜ q ˜ W ) = g W α s Z dx dx f f ˆ s × (cid:20) β − δβ + 4 δ (1 − δ )ln 1 + β − δ − β − δ (cid:21) , (C44) where the integrals were performed using the kinematicrelations: τ max = 12 (1 + β − δ ) ; (C45) τ min = 12 (1 − β − δ ) . (C46)To compare the total cross sections, it is much simplerto consider the special limit m = m W = m ˜ W , i.e., thecase δ = 0. 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