The Fake Factor Method - University Of Pennsylvania
Chapter 8The Fake Factor MethodMisidentification is an important source of background for physics analyses using particle-level identification criteria. In the case of the di-lepton analyses presented in this thesis, this background arisesfrom W jet events in which a jet is misidentified as a lepton. It is important to measure this typeof background from data as the rate of misidentification may not be accurately modeled in the MC.The “fake factor” method is a data-driven procedure for modeling background from particle misidentification. This procedure is used both in the W W cross section measurement presented in Chapter 9and in the H W W ( ) search presented in Chapter ?. The fake factor method is presented in thischapter.The remainder of this chapter is organized as follows: Section 8.1 introduces background frommisidentification and the fake factor method, Section 8.2 describes the fake factor method, Section 8.3describes the fake factor method as applied in a di-lepton analysis, Section 8.3.3 describes systematicsassociated with the method, Section 8.3.5 describes the validation of the fact factor predictions,Section 8.4 describes a procedure to extend the method to account for misidentified leptons fromheavy-flavor decays.8.1IntroductionOne of the primary motivations for using physics signatures with leptons in the final state is the largebackground rejection provided by the lepton identification of the ATLAS detector. The vast majorityof QCD multi-jets can be suppressed by efficient lepton identification criteria. In ATLAS, the jetsuppression is at the level of 10 5 ; only jets in the tails of the detector response are misidentified asleptons. Despite the small lepton fake rates, a significant level of background from misidentificationcan be present due to the large production cross section of QCD jets at the LHC. Figure 8.1 compares151
W 8. JetBackground.The Fake Factor Method152vents can give rise to background to WW.epton and real MeT from Ws-IDed as LeptonW jet cross section gives significantontribution despite small lepton fake rate.WW jet(20 GeV)Rely on MCtion would have to get W jet physics right.tion would have to get the Jet ! Lepton piece right.ons / Conversions/ Heavy Flavors precise modeling of tails)WWHww(125)tor Method DataDrivenTechniqueFigure8.1: Productioncross-sections in 7 TeV. The W jet production cross section is contrastedagainst the W W and H W W ( ) cross sections. Add measured di-jet cross sections and 84TeVpredictions (Check x-sections and branching ratios)the di-jet and W jet production cross sections to those of standard model W W lνlν productionand H W W ( ) lνlν production. The sources of potential background from misidentification areproduced at rates orders of magnitude higher than the signal processes. These large cross sectionscan lead to a significant amount of background from misidentification which needs to be properlyestimated.There are several different sources of lepton misidentification depending on lepton type. In thefollowing misidentified leptons are referred to as “fake leptons”, or “fakes”. For electrons, fakes canarise from charged hadrons, photon conversions, or semi-leptonic heavy-flavor decays. In the case ofphoton conversions and semi-leptonic heavy-flavor decays, an actual electron is present in the finalstate. These electrons are still considered fake in the sense that they are not produced in isolationas part of the prompt decay of a particle of interest. In the following, the term fake applies to bothhadrons misidentified as leptons and to leptons from non-prompt sources. Prompt leptons produced inisolation, e.g. from the decays of W or Z bosons, are referred to as “real” or “true” leptons. Figure 8.2shows the contribution of the various sources of fake electrons passing a loose18 electron identificationcriteria. The contribution from true electrons is also shown as indicated by “W/Z/γ e”. The fakecomponent is sharply peaked at lower pT . At this level of selection, fake electrons are dominated by18In this figure a subset of the IsEM medium cuts are used. The cuts on Rhad and Rη are not applied.
153107Data 2010 ( s 7 TeV)Monte CarloHadronsConversionsb"ec"eW/Z/ # * "e(a)106105104103EntriesEntries / GeV8. The Fake Factor Method(b) E T 7-26 GeV500004000030000ATL21010110-1020000ATLAS10000! Ldt 1.3 pb-110203040506070800090 1000.10.20.3E T [GeV]250200Monte CarloHadronsConversionsb/c/W/Z/ # * "eEntriesEntries [ 10 3 ]Figure 8.2: ET distribution for reconstructed electrons passing a loose identification criteria.6 The10data is shown along with300the different sources of electrons. The electrons are required to pass aData 2010 ( s 7 TeV)(c) E T 7-26 GeV(d) E T 7-26 GeVmodified selection similar to medium but without the Rhad and Rη requirements [?].510104hadrons and conversions. With tighter identification criteria the contributions from all three sources150are similar.ATLAS1032For muons the situation10heavy100is simpler. Almost all fake muons come from either semi-leptonicflavor decays or meson decays in flight. As above, these muons are referred to as fake despite the5010fact that an actual muon is present in the final state. The relative contribution after a loose muon1selection is shown in Figure08.3.The fakeis sharplypeaked01 component234 at lower5 pT . The fake muons01nBL selection, requiringare dominated by the heavy flavor contribution above 10 GeV. A tighter muon2345the muon to be well isolated and have a similar pT measurement in the inner detector and muonspectrometer, suppresses the decay-in-flight fraction even further. Unlike electrons, most analysesrequiring strict muon identification criteria only have misidentification from one source: semi-leptonicFigure 1: (a) Distribution of cluster transverse energy, ET , for thheavy-flavor decays.dates. The simulation uses PYTHIA with the W and Z/γ componto their NNLO total cross-sections and the heavy-flavour, conversiaccurate prediction of the fake background would require correctly simulating the particles that areto Onlythe atotalexpectationmisidentified, andcomponentsa precise model ofthenthe ratenormalisedof misidentification.small fractionof jets fake from the datsimulationsofwouldtherequiredistributionsof discriminatingused toleptons. Modelingthis rate correctlyan accurate modelingof the non-Gaussianvariablestails tron heavy-flavourW /Z/γsignalcomparedwouldto data: (b) thof the detector responseto jets. In addition, forpluselectrons,several sourcesof misidentificationall need to be properlypredicted.This level of ofdetailedmodeling is not expectedMC. TRTIt istweenthe numberhigh-thresholdhits fromandtheallhits on ththus necessary to(c)measureof backgrounddue to nmisidentificationdirectlywith data.thesourcesnumberof hits,electrontrack in the pixelBL , on theratio, E/p, between cluster energy and track momentum.Background from misidentification is not expected to be accurately modeled by the MC. Anthose expected in the simulation for heavy-flavour electrons.ficiencies are measured to be between 92.1% and 100.0%, wi
8. The Fake Factor Method154Figure 8.3: pT distribution of reconstructed muons after a loose muon selection. The data is shownalong with the different sources of “fake” muons.The fake factor method is a data-driven procedure for modeling background arising from misidentification. The method provides a measurement of the yield and the kinematic distributions of fakebackground. It is a general technique, applicable to any physics analysis in which particle-level selection criteria are used to suppress background. The fake factor method can be used with any numberof final state particles and is independent of the event selection. In the following it is presented in thecontext of modeling the background to misidentified electrons and muons, referred to as “leptons”,but the general discussion and techniques described are applicable to the background modeling of anyparticle with identification criteria: photons, hadronic-taus, heavy-flavor jets, or more exotic objectssuch as lepton-jets.The remainder of this chapter, presents the fake factor method in the context of a di-lepton missETanalysis. This is motivated by the use of method in the W W lνlν cross section measurementand the search for H W W ( ) lνlν presented elsewhere in this thesis: Chapters 9 and 10. In themissdi-lepton ETanalysis, referred to generically in the following as the “W W -analysis”, the primarysource of background from misidentification is W jet. QCD multi-jet background is also present ata much smaller level. Events in which W bosons are produced in association with jets give rise to
8. The Fake Factor Method155background to W W events when a jet is misidentified as a lepton. These events contain a real leptonand real missing energy from the W decay. With the jet misidentified as a lepton, the W jet eventshave two identified leptons, missing energy, and no other significant event characteristics. As a result,the W jet events cannot be readily suppressed by event selection. This background is particularlyimportant at low pT and is thus critical for the low mass Higgs search.8.2Fake Factor MethodThe fundamental idea of the fake factor method is simple: select a control sample of events enrichedin the background being estimated, and then use an extrapolation factor to relate these events to thebackground in the signal region. The method is data-driven provided the control sample is selectedin data, and the extrapolation factor is measured with data. For background arising from particlemisidentification, the extrapolation is done in particle identification space. The control sample isdefined using alternative particle selection criteria that are chosen such that the rate of misidentification is increased. The extrapolation factor relates background misidentified with this criteria, tobackground misidentified as passing the full particle selection of the signal region. The extrapolationfactor is referred to as the “fake factor”. The fake factor is measured and applied under the assumption that it is a local property of the particles being misidentified and is independent of the event-levelquantities. The fact that the extrapolation is done in an abstract particle identification space can beconceptually challenging, but the underlying procedure is straightforward.The control region is defined in order to select the background being estimated. The type ofbackground considered with the fake factor method arises from particle misidentification. To collectthis type background more efficiently, the particle selection in the signal region is replaced witha particle selection for which the misidentification rate is higher. This alternative particle selectioncriteria is referred to as the “denominator selection” or the “denominator definition”; particles passingthis criteria are referred to as “denominator objects” or simply “denominators”. The control regionis then defined to be the same as the signal region, except a denominator object is required in placeof the full particle selection in the signal region. For example, in the W W analysis, the control regionis defined to select W jet events in which the jet is misidentified as a lepton. A lepton denominatordefinition is chosen to enhance the misidentification rate from jets. The control region is then definedas events that contain one fully identified lepton, to select the real lepton from the W decay, and onedenominator object, to select the fake lepton from the jet. These events are required to pass the fullW W event selection, where the denominator is treated as if it were a fully identified lepton.For analyses where there are multiple sources of fake background, multiple control regions are
8. The Fake Factor Method156used. In the W W analysis, final states with both electrons and muons are considered: ee, eµ, andµµ. W jet background can arise from misidentification of either an electron or a muon. To accountfor this, separate electron and muon denominator selections are defined, and separate control regionsare used to predict the background from misidentification of the different lepton flavors.Events in the control region are related to the background in the signal region by the fake factor.The fake factor relates background which is misidentified as denominators, to background which ismisidentified as passing the full particle selection in the signal region. The full particle selectionin the signal region is referred to as the “numerator selection”; particles passing this criteria arereferred to as “numerator objects”. The fake factor extrapolates from background misidentified asdenominators, to background misidentified as numerators. It is important that the fake factor bemeasured in data. The fake factor measurement can be made in data using a pure sample of theobjects being misidentified. For the case of the W jet background, a pure sample of jets is needed.The fake factor can be measured in this sample by taking the ratio of the number of reconstructednumerators to the number of reconstructed denominators:f NNumerator.NDenominator(8.1)Because the sample consists of background, the reconstructed numerators and denominators in thesample are due to misidentification. The ratio of the object yields measures the ratio of the misidentification rates. This is the quantity needed to relate the background in the control region to thebackground in the signal region. For the W jet background in the W W analysis, the fake factor isdefined separately for electrons and muons, and measures the ratio of the rate at which jets pass thethe full lepton identification requirement to the rate at which they pass the denominator requirement.These fake factors are measured in data using a sample of di-jet events.The sample used to measure the fake factor cannot be the same as the control region used toselect the background being estimated. Events with numerators in the control region correspond tothe events in the signal region. Attempting to measure the extrapolation factor into the signal region,from the signal region is circular. The amount of background in the signal region would need to beknown in order to extract the fake factor, which is used to predict the amount of background in thesignal region. The fake factor method requires two separate control regions in data: the control regionused to select the background from which the extrapolation is made, and a control region used tomeasure the fake factor. In the following, the first region is referred to as “the background controlregion”, or in the case of the W jet background, as “the W jet control region”. The region usedto measure the fake factors is referred to as “the fake factor control region”, or in the case of theW jet background, as “the di-jet control region”. The event selection used to define the background
8. The Fake Factor Method157control region is dictated by the signal selection. There are no constraints on the event selection usedto define the fake factor control region other than that it be dominated by background and distinctfrom the background control region.After the control region is defined, and the is fake factor measured, the background in the signalregion is calculated by weighting the event yield in the control region by the fake factor:NBackground f NBackground Control .(8.2)The event yield in the control region measures the amount of background passing the event selectionbut with a misidentified denominator instead of a misidentified numerator. This is related to thebackground passing the event selection with a misidentified numerator, i.e the background in thesignal region, by the ratio of the misidentification rates, i.e. the fake factor. This is expressed,colloquially, in equation form as:NBkg.X N f NBkg.X D NN NBkg.X D ,ND(8.3)where: N represents a numerator object, D a denominator object, and X stands for any object or eventselection unrelated to the misidentification in question. In the background calculation, the rate of thebackground misidentification in the fake factor control region is assumed to be the same as the rateof background misidentification in the background control region. A systematic uncertainty needs tobe included to account for this assumption. This uncertainty is referred to as “sample dependence”,and is often the dominant uncertainty on the background prediction. For the W jet background, thefake factor is measured in di-jet events and is applied to events in the W jet control region. Thesample dependence uncertainty is the leading uncertainty in the W jet background prediction.The fake factor often has a dependence on the kinematics of the misidentified objects. This canbe accounted for by measuring the fake factor separately in bins of the relevant kinematic variableand applying it based on the kinematics of the denominator in the background control sample. Thetotal background yield is then calculated as:NBkg.X N if i Ni,Bkg.X D ,(8.4)where i labels the different kinematic bins. In the case of the W jet background, the fake factor ismeasured in bins of lepton pT . The W jet background is then calculated as: jetNWNumerator Numerator i jetf i Ni,WNumerator Denominator ,(8.5)
8. The Fake Factor Method158where i labels the pT bin of the fake factor and the denominator object in the W jet control region.The fake factor method can model the event kinematics of the background due to misidentification.This is done by binning the background control region in the kinematic variable of interest. Thecorresponding background distribution in the signal region is obtained by scaling with the fake factor,bin-by-bin:j,Bkg.Nj,Bkg.X N f NX D ,where: j labels the bins of the kinematic distribution being modeled. A similar extension can beapplied to Equation 8.4, in the case of a fake factor kinematic dependence:Nj,Bkg.X N if i Ni,j,Bkg.X D ,(8.6)where i labels the kinematic dependence of the fake factor and the denominator object, and j labelsthe kinematic bins of the distribution being modeled.In the discussion thus far the control regions have been assumed to consist purely of background. Inpractice, both the background control region and the fake factor control region will have contaminationfrom sources other than the background of interest. To an extent, this can be mitigated by carefulchoice of denominator definition. For example, in the case of lepton misidentification from jets,the denominator definition can be chosen to explicitly exclude selection criteria that is efficient fortrue leptons. This reduces the contamination of true leptons in the denominator selection and thusthe control regions. The residual sources of contamination have to be subtracted from the controlregions. In many cases this subtraction can be done from MC, by running the control region selectionon the contaminating samples. In some cases the contamination in the control region arises frommisidentification, in which case the fake factor method can be applied twice: once to predict thecontamination in the control region, and once to predict the background in the signal region. Examplesof these corrections for the W W analysis will be discussed in Section 8.3.This concludes the introduction of the basic idea and methodology of the fake factor procedure.The following section motivates the fake factor method from another point of view. The rest of thechapter provides examples of the fake factor method in use. Subtleties that can arises in practiceare discussed, systematic uncertainties associated with the method are described, and data-drivenmethods to validated the background fake factor procedure are presented. Finally, an extension tothe basic method that simultaneously accounts for several sources of background from misidentificationis presented.
8. The Fake Factor Method8.2.1159Motivation of Fake Factor MethodThis section motivates the fake factor method in another way. The fake factor method is introducedas an extension of a simpler, more intuitive, background calculation. With this approach, the meaningof the fake factor and the major source of its systematic uncertainty are made explicit. The method ispresented in the context of modeling misidentified leptons in W jet events, but as mentioned above,the discussion is more generally applicable.A simple, straightforward way to calculate the W jet background is to scale the number of eventswith a reconstructed W and a reconstructed jet by the rate at which jets fake leptons:W jetW jetN(Lepton Lepton) FLepton N(Lepton Jet),(8.7)W jetwhere: N(Lepton Lepton)represents the W jet background to di-lepton events passing a given eventW jetselection, FLepton is the jet fake rate, and N(Lepton Jet)is the number of W jet events with a leptonand a reconstructed jet passing the event selection. This method is simple: the number of reconstructed W jet events is counted in data, and the rate at which jets are misidentified as leptonsis used to predict the background in the signal region. The procedure would be fully data drivenprovided FLepton is determined from data.The problem with this naive method is that the systematic uncertainty associated with the extrapolation from reconstructed jet to misidentified lepton is large. One source of systematic uncertaintycomes from different jet types. There are a lot of different kinds of jets: quark jets, gluon jets,heavy-flavor jets, etc. Each of these different jet types will have a different fake rate. The fake rate,FLepton , is measured in a control sample with a particular mix of jet types and is only applicable forthat specific mixture. For example, di-jet events are dominated by gluon jets. Figure 8.4a shows theleading order Feynman diagram for di-jet production at the LHC. If these events are used to measureFLepton , the fake rate will be mainly applicable to gluon jets. However, the jets in W jet events tendto be quark-initiated jets. Figure 8.4b shows the leading order diagram for W jet production. Thegluon-jet fake rates may be substantially different from those of the quark jets they are used to model.Differences in composition between jets in the N(Lepton Jet) sample, and jets in the sample used tomeasure FLepton , is a large source of systematic uncertainty in Equation 8.7.Differences between the reconstructed jet energy, and the reconstructed energy of the misidentifiedlepton, lead to another source of systematic uncertainty in the naive method. When extrapolatingfrom reconstructed jets there are two relevant energy scales: the energy of the jet and the energy of themisidentified lepton. A jet with a given energy can be misidentified as lepton with a different energy.For example, 100 GeV jets can be misidentified as 20 GeV electrons, or they can be misidentified as
8. The Fake Factor Methodg160gqWq gg(a)gq (b)Figure 8.4: Leading order Feynman diagrams for (a) di-jet production and (b) W jet production.The jets in the di-jet sample are gluon initiated, whereas jets in the W jet sample are quark initiated.100 GeV electrons. In general, the rate at which jets are misidentified as leptons depends on both theenergy of the initial jet and the energy of the lepton it is misidentified as. The rate at which 100 GeVjets are misidentified as 20 GeV electrons will be different from the rate at which 100 GeV jets arereconstructed as 100 GeV electrons. This multidimensional kinematic dependence is not accountedfor in Equation 8.7 and leads to a source of further systematic uncertainty.A more precise calculation of the W jet background can be made by extending the simple procedure to explicitly account for the effects mentioned above. An updated calculation of the backgroundwould be written as:W jetN(Lepton Lepton) i,j,q /gi,jW jet jFLepton(q /g) N(Lepton Jet)(q /g),(8.8)W jeti,jwhere: N(Lepton Lepton)is the total W jet background, FLepton(q /g) is the jet fake rate, andW jet ji,jN(Lepton Jet)(q /g) is the number of lepton plus jet events. The fake rate, FLepton(q /g), is binned ac-cording to the pT of the reconstructed jet, denoted by the superscript j, and the pT of the misidentifiedlepton, denoted by the superscript i. The fake rate is a function of the different types of jet: quark jet,W jet jgluon jet, etc, denoted by q /g. The observed number of lepton plus jet events, N(Lepton Jet)(q /g),is also binned in jet pT , and is a function of the reconstructed jet type. Calculating the total background requires summing over the different jet types, the pT of the reconstructed jet, and the pT ofthe misidentified lepton.The updated W jet prediction in Equation 8.8 is precise but much more complicated. The fakerate is now a matrix. It maps reconstructed jets of pT j into the misidentified leptons of pT i. Thismatrix is awkward to work with in practice, and challenging to measure in data. The matrix elementsF ij can only be measured in events with a misidentified lepton, whereas, they are applied to jetsin the control region without a corresponding misidentified lepton in the event. The correspondencebetween the pT scale of jets misidentified as leptons and jets in the control region would have to beestablished and validated.
8. The Fake Factor Method161Another complication arises from the different jet types. Separate fake rate matrices are neededfor each jet type. These are then applied based on the jet type seen in the control region. Associatingjet types to reconstructed jets is not straightforward. Reconstructed jet observables that correlate tojet type would have to be identified and validated. Uncertainties due to jet misclassification wouldneed to be assigned. A procedure for measuring the separate fake rate matrices would also haveto be established. Measuring the fake rate matrices and understanding the systematic uncertaintiesassociated with the complications described above is not practical.The fake factor method is designed to retain the precision of the updated W jet calculation,while avoiding the complicated calculation in Equation 8.8. By defining an additional lepton criteria,referred to as the denominator selection, Equation 8.8 can be trivially rewritten as:W jetN(Lepton Lepton) i,j,q /gi,jFLepton(q /g)i,jFDenominator(q /g)i,jW jet j FDenominator(q /g) N(Lepton Jet)(q /g),(8.9)i,jwhere FDenominator(q /g) represents the rate at which jets are misidentified as denominators. As forleptons, the fake rate for denominators will depend on jet type and will be represented by a matrix: jetsof a given pT can be misidentified as denominators of a different pT . Because the identification criteriafor leptons and denominators are different, the corresponding jet fake rates will also be different.In general, the differences between lepton and denominator fake rates will be complicated. Thesedifferences will depend on the jet type, the jet pT , and the misidentified lepton pT .The crux of the fake factor method is the assumption that the lepton and denominator fake ratesare related by a single number19 that is independent of all the other fake rate dependencies. Theassumption is that the lepton fake rates can be expressed in terms of the denominator fake rates as:i,ji,jFLepton(q /g) f FDenominator(q /g),(8.10)where f is a constant number, referred to as the “fake factor”. The assumption is that all the fake ratevariation due to the underlying jet physics is the same for leptons and denominators, up to a numericalconstant. This is an approximation. The degree to which the approximation is correct depends on thelepton and denominator definitions. In the fake factor method, a systematic uncertainty is needed toaccount for the extent to which this assumption is valid. This systematic uncertainty is the underlyingcause of sample dependence.19In practice the dependence on lepton pT is accounted for the the fake factors, but this detail is ignored for now.
8. The Fake Factor Method162With the fake factor assumption, the W jet background in Equation 8.9, can be written as:W jetN(Lepton Lepton) f F i,jDenominator (q /g)i,j,q /g i,j,q /gi,jFDenominator(q /g)i,jW jet j FDenominator(q /g) N(Lepton Jet)(q /g),i,jW jet jf FDenominator(q /g) N(Lepton Jet)(q /g).(8.11)Because the fake factor is assumed to be independent of jet type and pT , it can be factored out of thesum:W jetN(Lepton Lepton) f i,j,q /gi,jW jet jFDenominator(q /g) N(Lepton Jet)(q /g).(8.12)At first glance, the expression in Equation 8.12 is no simpler than the one started with in Equation 8.8.The denominator fake rate matrix has all the same complications as the lepton fake rate matrix. Thedependence on jet type and all the associated complexity involved with observing it, is still present.However, the upshot of Equation 8.12 is that the sum on the right-hand side is observable in data.It is simply the number of reconstructed events with a lepton and a denominator. Equation 8.12 canbe written as:W jetW jetN(Lepton Lepton) f N(Lepton Denominator).(8.13)where, N(Lepton Denominator) is the number of observed lepton-denominator events. In a sense, bygoing through the denominator objects, the detector performs the complicated sums in Equations 8.8and 8.12. In the fake factor method, the background extrapolation is made from reconstructeddenominators instead of reconstructed jets. This provides a W jet measurement with the precisionof Equation 8.8, without having to perform the complicated calculation. This simplification comes atthe cost of the added systematic uncertainty associated with the assumption in Equation 8.10.In order for the fake factor method to be data-driven, the fake factor as defined in Equation 8.10,needs to be measured in data. This can be done by measuring the ratio of reconstructed leptons todenominators in a di-jet control sample. Assuming a pure di-jet sample, all
8. The Fake Factor Method 154 Figure 8.3: p T distribution of reconstructed muons after a loose muon selection. The data is shown along with the different sources of "fake" muons. The fake factor method is a data-driven procedure for modeling background arising from misiden-
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