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Model Independent Particle Mass Measurements in Missing Energy Events at Hadron Colliders

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Title:
Model Independent Particle Mass Measurements in Missing Energy Events at Hadron Colliders
Creator:
PARK,MYEONGHUN
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (244 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Matchev, Konstantin T
Committee Members:
Korytov, Andrey
Field, Richard D
Ramond, Pierre
Tan, Jonathan
Graduation Date:
4/30/2011

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Subjects / Keywords:
Average linear density ( jstor )
Kinematics ( jstor )
Leptons ( jstor )
Mass ( jstor )
Mass spectra ( jstor )
Mathematical variables ( jstor )
Momentum ( jstor )
Particle decay ( jstor )
Particle mass ( jstor )
Topology ( jstor )
BEYOND -- HADRON -- SUPERSYMMETRY
Physics -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
This dissertation describes several new kinematic methods to measure the masses of new particles in events with missing transverse energy at hadron colliders. Each method relies on the measurement of some feature (a peak or an endpoint) in the distribution of a suitable kinematic variable. The first method makes use of the "Gator" variable smin, whose peak provides a global and fully inclusive measure of the production scale of the new particles. In the early stage of the LHC, this variable can be used both as an estimator and a discriminator for new physics over the standard model backgrounds. The next method studies the invariant mass distributions of the visible decay products from a cascade decay chain and the shapes and endpoints of those distributions. Given a sufficient number of endpoint measurements, one could in principle attempt to invert and solve for the mass spectrum. However, the non-linear character of the relevant coupled quadratic equations often leads to multiple solutions. In addition, there is a combinatorial ambiguity related to the ordering of the decay products from the cascade decay chain. We propose a new set of invariant mass variables which are less sensitive to these problems. We demonstrate how the new particle mass spectrum can be extracted from the measurement of their kinematic endpoints. The remaining methods described in the dissertation are based on "transverse" invariant mass variables like the "Cambridge" transverse mass MT2, the "Sheffield" contrasverse mass MCT and their corresponding one-dimensional projections MT2perp, MT2parallel, MCTperp, and MCTparallel with respect to the upstream transverse momentum UT. The main advantage of all those methods is that they can be applied to very short (single-stage) decay topologies, as well as to a subsystem of the observed event. The methods can also be generalized to the case of non-identical missing particles, as demonstrated in Chapter 7. A complete set of analytical results for the calculation of the relevant variables in each event, as well as the dependence of their endpoints on the underlying mass spectrum is given for each case. In some circumstances, the whole shape of the differential distribution can be theoretically predicted as well. The methods are illustrated with examples from supersymmetry and from top quark production in the standard model. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Matchev, Konstantin T.
Statement of Responsibility:
by MYEONGHUN PARK.

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Copyright PARK,MYEONGHUN. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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MODELINDEPENDENTPARTICLEMASSMEASUREMENTSINMISSINGENERGYEVENTSATHADRONCOLLIDERSByMYEONGHUNPARKADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011MyeonghunPark 2

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IdedicatethistomygrandmotherSunAeSeoandmywifeKellyChung. 3

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ACKNOWLEDGMENTS Firstandforemost,Iamdeeplyindebtedtomyadvisor,Prof.KonstantinMatchev,forhistime,patience,encouragement,muchstimulatingadvice.Tome,heisafatherwhokeepstryingtoteachhissonabouteverythingthatasonneedstoknowtobepreparedforaworldinfrontofhim.Inadditionthat,I'vealwaysbeenhappytoplaygameswithhisrstson,Anton.I'velearnedfrommyadvisoraboutmanythingsbeyondphysics,forexampleIlearnedhowIcanbeanicehusbandeventhoughIneedtospendlotsoftimeinmyofce.IamalsoextremelygratefultomycollaboratorDr.K.C.Kongforadviceandusefuldiscussions.IwouldliketothankDr.ParthaKonarforinterestingdiscussionsandcollaboration.Allofthemarereallynicelikeaselderbrothers.LikeasIlovemyfamily,Ilovemygroupmembers.Duringmyjourneytophysics,itwasagreatpleasuretoattendProf.PierreRamond'sclasses.Becauseofhim,Iopenedmyeyestothebeautyofsymmetriesandrealizedhowsymmetrycanguideustodeepquestionsandhandleproblems.IalsowanttothankProf.RickFieldforsupportingmetohavevariousexperiences.IwouldliketothankUFCMSgroupforallowingmetojoinCMScollaboration.ItisreallyfantastictohandlerealdatafromtheLHC.IamgratefultoDr.JosephLykkenandDr.StephenMrenna.Tohaveopportunitiestoworkwiththemwasreallyinterestingtome.Therearelotsofinterestingtasksthatweneedtodo,andthisisalwaysmypleasure.Ishouldmentionaboutseniorphysicistswhohavehelpedmealotandgaveadvicestoo.VisitingPusanandTokyoleadmeintoworldwidecollaborations.AllofthesecouldnotbedonewithoutsupportsfromProfessorKonstantinMatchev,ProfessorDeog-KiHongandProfessorMihokoM.Nojiri.IwouldliketothankDr.ChristopherLester,ProfessorMichaelPeskinandDr.JayWackerwhogavemeusefulcommentsandadvice,amongmanyotherphysiciststhat 4

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Ihavemetatworkshopsandseminarvisits.IwouldliketoshowmygratitudetoProf.PaulW.Chun'sadvicesandgreatcareonUFKoreanstudents.IdonotknowhowIcanshowmygratitudetomyparentsandfamily.Astherstson,Iknowhowmuchtheymissme.InfactIreallymissthemtoo.Duringmystudyabroad,theveryreasonthatIhavenotsufferedfromanyhomesicknessisbecauseIhaveafamilyheretoo.Ireallylovemyparentsinlawandappreciatetheirendlesssupport.Now,Iwillleavehereandgotoanotherplace.IdonotworryaboutmyfuturesinceIamwithmywife.Mywifeisthebestfriendallthetime.Ireallyappreciatemywife'sunderstandingsothatIcouldspendlotsoftimeinotherplacesforprofessionalvisits.Allofmyworkshereweremadepossiblebecauseofmybelovedfathers,myfatherinKorea,myfatherinlaw,myadvisorandmyLord. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 14 CHAPTER 1INTRODUCTION ................................... 16 2GENERALANALYSISWITHOUTANYASSUMPTIONS ............. 21 2.1TheNeedforaUniversal,GlobalandInclusiveMassVariable ....... 21 2.1.1Denitionofp smin ........................... 23 2.1.2p sminandtheUnderlyingEventProblem ............... 24 2.2DenitionoftheRECOlevelVariablep s(reco)min ................ 25 2.3DenitionoftheSubsystemVariablep s(sub)min ................. 29 2.4SMexample:DileptonEventsfromttproduction .............. 33 2.4.1EventSimulationDetails ........................ 34 2.4.2p s(reco)minVariable ............................. 34 2.4.3p s(sub)minVariable ............................. 43 2.5AnExclusiveSUSYExample:MultijetEventsFromGluinoProduction .. 47 2.6AnInclusiveSUSYExample:GMSBStudyPointGM1b .......... 53 2.7ComparisontoOtherInclusiveColliderVariables .............. 58 3INVARIANTMASSENDPOINTSMETHOD .................... 64 3.1ThreeGenericProblemsinInvariant-massEndpointMethods ....... 67 3.1.1Near-farLeptonAmbiguity ....................... 67 3.1.2InsufcientNumberofMeasurements. ................ 69 3.1.3ParameterSpaceRegionAmbiguity ................. 71 3.1.4PosingTheProblem .......................... 72 3.2NewVariables ................................. 73 3.2.1TheUnionm2j`n[m2j`f .......................... 74 3.2.2TheProductmj`nmj`f ........................ 75 3.2.3TheSumsm2j`n+m2j`f .......................... 75 3.2.3.1KinematicEndpointsofm2j`(s)()with1 ........ 76 3.2.3.2KinematicEndpointsofm2j`(s)()with<1and6=0 .. 78 3.2.4TheDifferencejm2j`n)]TJ /F3 11.955 Tf 11.96 0 Td[(m2j`fj ...................... 78 3.3TheoreticalAnalysis .............................. 80 3.3.1OurMethodAndTheSolutionForTheMassSpectrum ....... 80 3.3.2DisambiguationOfTheTwoSolutionsFormB ............ 81 6

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3.3.2.1Invariantmassendpointmethod .............. 81 3.3.2.2Invariantmasscorrelations ................. 83 3.4NumericalExamples .............................. 88 3.4.1MassMeasurementsAtStudyPointsLM1andLM6 ......... 88 3.4.2EliminatingTheFakeSolutionformB ................. 93 4SUBSYSTEMMT2METHOD ............................ 98 4.1AShortDecayChainX2!X1!X0 ..................... 101 4.1.1TheSubsystemVariableM(1,1,0)T2 ................... 102 4.1.2TheSubsystemVariableM(2,2,1)T2 ................... 111 4.1.3TheSubsystemVariableM(2,2,0)T2 ................... 113 4.1.4TheSubsystemVariableM(2,1,0)T2 ................... 121 4.2MT2-basedMassMeasurementmethods .................. 125 4.2.1PureMT2EndpointMethod ...................... 127 4.2.2MT2EndpointShapesAndKinks ................... 129 4.2.3HybridMethod:MT2EndpointsPlusAnInvariantMassEndpoint 131 5ONEDIMENSIONALPROJECTIONMETHOD .................. 134 5.1DetailedStudyOnMT2'sCharacteristics ................... 134 5.2UsingthePToftheUpstreamJetwithMT2Method ............ 143 5.3UsingFullPhaseSpaceInformationWithMCT ............... 152 6ASYMMETRICEVENTTOPOLOGY ........................ 160 6.1GeneralizingMT2ToAsymmetricEventTopologies ............. 161 6.2TheConventionalSymmetricMT2 ...................... 166 6.2.1Denition ................................ 166 6.2.2Computation .............................. 167 6.2.3Properties ................................ 170 6.2.3.1PropertyI:KnowledgeOfMpAsAFunctionofMc .... 170 6.2.3.2PropertyII:KinkInMT2(max)AtTheTrueMc ....... 172 6.2.3.3PropertyIII:PUTMInvarianceOfMT2(max)AtTheTrueMc 173 6.3TheGeneralizedAsymmetricMT2 ...................... 174 6.3.1Denition ................................ 175 6.3.2Computation .............................. 175 6.3.3Properties ................................ 178 6.3.3.1PropertyI:KnowledgeOfMpAsAFunctionOfM(a)cAndM(b)c .............................. 178 6.3.3.2PropertyII:RidgeInMT2(max)ThroughTheTrueM(a)cAndM(b)c ........................... 179 6.3.3.3PropertyIII:PUTMInvarianceOfMT2(max)AtTheTrueM(a)cAndM(b)c ........................ 180 6.3.4Examples ................................ 181 6.3.5CombinatorialIssues .......................... 183 7

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6.4TheSimplestEventTopology:OneStandardModelParticleOnEachSide 186 6.4.1AsymmetricCase ............................ 186 6.4.2SymmetricCase ............................ 193 6.4.3MixedCase ............................... 197 6.5AMoreComplexEventTopology:TwoVisibleParticlesOnEachSide .. 200 6.5.1Off-shellIntermediateParticle ..................... 200 6.5.2On-shellIntermediateParticle ..................... 207 6.6ApplicationToMoreGeneralCases ..................... 209 7CONCLUSIONS ................................... 211 7.1p smin ...................................... 211 7.2InvariantMassEndpointMethod ....................... 212 7.3SubsystemMT2Method ............................ 213 7.4OneDimensionalProjectionMethod ..................... 214 7.5AsymmetricEventTopology .......................... 214 APPENDIX AANALYTICALEXPRESSIONSFORTHESHAPESOFTHEINVARIANTMASSDISTRIBUTIONS ................................... 216 A.1DileptonMassDistributionm2`` ........................ 216 A.2CombinedJet-leptonMassDistributionm2j`(u) ................ 217 A.3Distributionofthesumm2j`(s)(=1) ..................... 217 A.4DistributionOfTheDifferencem2j`(d)(=1) ................. 219 A.5DistributionOfTheProductm2j`(p) ....................... 221 BANALYTICALEXPRESSIONSFORM(n,p,c)T2,max(~Mc,pT) ............... 222 B.1TheSubsystemVariableM(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT) ................ 223 B.2TheSubsystemVariableM(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT) ................ 224 B.3TheSubsystemVariableM(n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT) ............... 227 CTHESYMMETRICMT2INTHELIMITOFINFINITEPUTM ............ 229 REFERENCES ....................................... 236 BIOGRAPHICALSKETCH ................................ 244 8

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LISTOFTABLES Table page 2-1Selectedp squantitiesforeventsinFigure 2-4 2-5 and 2-12 .......... 40 2-2Masses(inGeV)oftheSUSYparticlesattheGM1bstudypoint. ........ 53 2-3Cross-sections(inpb)andparentmassthresholds(inGeV)forthedominantproductionprocessesattheGM1bstudypoint. .................. 55 3-1TherelevantpartoftheSUSYmassspectrumfortheLM1andLM6studypoints. ......................................... 87 5-1Selectedsparticlemasses(inGeV)atpointLM6. ................. 135 6-1MassspectraforthetwoexamplesstudiedinSections 6.4.1 and 6.4.2 .... 186 9

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LISTOFFIGURES Figure page 1-1Aschematicdescriptionofthemass-determinationmethodsdescribedinthisdissertation. ..................................... 20 2-1Thegenericeventtopologyusedtodenethep smin ............... 22 2-2Eventtopologyofasubsystem ........................... 30 2-3Distributionsofvariousp sminforthedileptontt ................. 35 2-4PGScalorimetermapoftheenergydepositsforadileptontteventwithonlytworeconstructedjets ................................ 38 2-5PGScalorimetermapoftheenergydepositsforadileptontteventwithmorethantworeconstructedjets ............................. 39 2-6Distributionofvariousp sminforW+W)]TJ /F1 11.955 Tf 10.4 -4.33 Td[(subsystemofttwithtworeconstructedleptons ........................................ 44 2-7Unit-normalizeddistributionofjetmultiplicityindileptonttevents. ....... 45 2-8Distributionsofvariousp sminforthedileptonttsampleforsubsystems. .... 45 2-9Unit-normalizeddistributionofjetmultiplicityingluinopairproductionevents 50 2-10Distributionofvariousp sminwithaSUSYexampleofgluinopairproduction,witheachgluinodecayingtofourjetsanda~01 .................. 51 2-11Distributionofvariousp sminwithaSUSYexampleofgluinopairproduction,witheachgluinodecayingtotwojetsanda~01LSPasin( 2 ). ........ 51 2-12PGScalorimetermapoftheenergydepositforaSUSYeventofgluinopairproduction,witheachgluinoforcedtodecayto4jetsandtheLSP ....... 52 2-13Distributionofthep s(cal)minandp s(reco)minvariablesininclusiveSUSYproductionfortheGMSBGM1abenchmarkstudypoint ................... 53 2-14Distributionsofvariousp sminfortheGMSBSUSYexample ........... 57 2-15Comparisonvariousp sminwithothertransversevariablesforttproduction .. 60 2-16Comparisonvariousp sminwithothertransversevariablesforthegluinopairproduction,witheachgluinodecayingto4jets. .................. 63 2-17Comparisonvariousp sminwithothertransversevariablesforgluinopairproductionwitheachgluinodecayingto2jets. ......................... 63 3-1ThetypicalcascadedecaychainofaBSMparticle,Ncascade=3 ......... 65 10

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3-2Comparisonofthepredictionsforthekinematicendpointsmmaxj`(s)()oftherealandfakesolutions .................................. 82 3-3Predictedscatterplotsofmj`(u)versusm``,forthecaseoftherealandfakesolutionsforeachofthetwostudypointsLM1andLM6 ............. 84 3-4One-dimensionalinvariantmassdistributionsforthecaseofLM1andLM1'spectra. ........................................ 90 3-5One-dimensionalinvariantmassdistributionsfortheLM6massspectrumandtheLM6'massspectrum ............................... 91 3-6Someotherone-dimensionalinvariantmassdistributionsofinterestforthecaseoftheLM1massspectrumandLM1' .................... 96 3-7Someotherone-dimensionalinvariantmassdistributionsofinterestfortheLM6massspectrumandtheLM6'massspectrum ................ 97 4-1IllustrationofthesubsystemM(n,p,c)T2variable ................... 99 4-2ThesubsystemM(n,p,c)T2variableswhichareavailableforn=1andn=2events. ........................................ 102 4-3DependenceoftheM(1,1,0)T2,maxupperkinematicendpointonthevalueofpT .... 107 4-4DependenceoftheM(2,2,0)T2,maxandM(2,1,0)T2,maxupperkinematicendpointsonthevalueofthetestmass~M0 .............................. 115 4-5Theamountofkinkasafunctionofthemassratiosp yandp z. ........ 116 4-6Unit-normalizeddistributionsofM(n,p,c)T2variablesindileptoneventsfromW+W)]TJ /F1 11.955 Tf -416.27 -18.79 Td[(pairproductionandttpairproduction. ...................... 126 4-7Unit-normalizedmb`invariantmass-squareddistributionsindileptonttevents. 133 5-1ThetypicalSUSYeventtopologyproducingtwoisolatedsame-signleptonsatpointLM6. ....................................... 135 5-2ThetwospecialmomentumcongurationsdenedinEquations( 5 5 ). .. 137 5-3MmaxT2versusthetestmass~Mc ........................... 140 5-4Scalingfactorsrelatingtheerror~MpintheextractionoftheMT2endpoint. .. 142 5-5Thegenericeventtopologyforone-dimensionalprojection. ........... 144 5-6Decompositionoftheobservedtransversemomentumvectors ......... 145 5-7Theunit-normalizedMT2?distribution. ....................... 147 5-8ObservableMT2?distributionafterhardcutsfor100fb)]TJ /F5 7.97 Tf 6.58 0 Td[(1ofLHCdata. ..... 148 11

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5-9Thefunction^N(~Mc) ................................. 150 5-10Zero-binsubtractedMCT?distributionaftercuts,forttdileptonevents. ..... 155 5-11ScatterplotsofMCT?versusMCTkandMCT?versusMCT ............ 156 5-12DCTdistributionsforfourdifferentvaluesofMp .................. 158 5-13FittedvaluesofDminCTasafunctionofMp. ..................... 159 6-1Thegenericeventtopologywithdifferentmissingparticles ............ 162 6-2MT2(max)(~Mc,PUTM)andMT2(max)(~Mc,PUTM)forseveralxedvaluesofPUTM 171 6-3Thethreedifferentevent-topologiesforvariousdecayingtypes. ......... 181 6-4Unit-normalizedMT2distributionsfortheeventtopologyofFigure 6-3 (b). ... 185 6-5MT2(max)asafunctionofthetwotestchildrenmasses,~M(a)cand~M(b)c,fortheeventtopologyofFigure 6-3 (a). ........................... 188 6-6MT2(max)fortheeventtopologyofFigure 6-3 (a)withxedupstreammomentumofPUTM=1TeV. ................................... 190 6-7MT2ridgestructurewhentwomissingparticlesaredifferent. .......... 191 6-8MT2(max)fortheeventtopologyofFigure 6-3 (a)withthesymmetricmassspectrumIIfromTable 6-1 withoutupstreammomentum. .................. 194 6-9MT2(max)fortheeventtopologyofFigure 6-3 (a)withthesymmetricmassspectrumIIfromTable 6-1 withxedupstreammomentum. ................. 194 6-10MT2ridgestructurewhentwomissingparticleshavesamemassspectrum. .. 195 6-11Unit-normalized,zero-binsubtractedMT2distributionforthefullmixedeventsample. ........................................ 198 6-12Thefourregionsinthe(~M(a)c,~M(b)c)parameterplaneleadingtothefourdifferenttypesofsolutionsfortheMT2endpointfortheoffshellscenarioillustratedinFigure 6-3 (b). .................................... 201 6-13MT2(max)asafunctionofthetwotestchildrenmasses,~M(a)cand~M(b)c,fortheeventtopologyofFigure 6-3 (b). ........................... 205 6-14Thefourregionsinthe(~M(a)c,~M(b)c)parameterplaneleadingtothefourdifferenttypesofsolutionsfortheMT2endpointfortheonshellscenarioillustratedinFigure 6-3 (c). .................................... 207 6-15[MT2(max)asafunctionofthetwotestchildrenmasses,~M(a)cand~M(b)c,fortheeventtopologyofFigure 6-3 (c). ........................... 209 12

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6-16Eventtopologyfortheeffectivelydifferentmissingpartices. ........... 210 A-1ComparisonofthenumericallyobtaineddifferentialinvariantmassdistributionsforstudypointLM1withtheanalyticalresults. ................... 218 C-1Theparameterplaneoftestchildrenmassessquared,dividedintothefourdifferentregionsRiusedtodenetheMT2endpointfunction. .......... 230 C-2MT2(max)fortheeventtopologyofFigure 6-3 (a)withxedupstreammomentumofPUTM=!1. ................................... 232 C-3MT2(max)fortheeventtopologyofFigure 6-3 (a)withthesymmetricmassspectrumIIfromTable 6-1 withupstreammomentumPUTM!1. ............. 232 C-4AstudyofthesharpnessoftheMT2ridgeforthecasewhenmissingparticlesaredifferent. ..................................... 233 C-5AstudyofthesharpnessoftheMT2ridgeforthecasewhenmissingparticlesarethesame. ..................................... 233 13

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELINDEPENDENTPARTICLEMASSMEASUREMENTSINMISSINGENERGYEVENTSATHADRONCOLLIDERSByMyeonghunParkMay2011Chair:KonstantinT.MatchevMajor:PhysicsThisdissertationdescribesseveralnewkinematicmethodstomeasurethemassesofnewparticlesineventswithmissingtransverseenergyathadroncolliders.Eachmethodreliesonthemeasurementofsomefeature(apeakoranendpoint)inthedistributionofasuitablekinematicvariable.TherstmethodmakesuseoftheGatorvariablep smin,whosepeakprovidesaglobalandfullyinclusivemeasureoftheproductionscaleofthenewparticles.IntheearlystageoftheLHC,thisvariablecanbeusedbothasanestimatorandadiscriminatorfornewphysicsoverthestandardmodelbackgrounds.Thenextmethodstudiestheinvariantmassdistributionsofthevisibledecayproductsfromacascadedecaychainandtheshapesandendpointsofthosedistributions.Givenasufcientnumberofendpointmeasurements,onecouldinprincipleattempttoinvertandsolveforthemassspectrum.However,thenon-linearcharacteroftherelevantcoupledquadraticequationsoftenleadstomultiplesolutions.Inaddition,thereisacombinatorialambiguityrelatedtotheorderingofthedecayproductsfromthecascadedecaychain.Weproposeanewsetofinvariantmassvariableswhicharelesssensitivetotheseproblems.Wedemonstratehowthenewparticlemassspectrumcanbeextractedfromthemeasurementoftheirkinematicendpoints.TheremainingmethodsdescribedinthedissertationarebasedontransverseinvariantmassvariablesliketheCambridgetransversemassMT2,theShefeld 14

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contrasversemassMCTandtheircorrespondingone-dimensionalprojectionsMT2?,MT2k,MCT?,andMCTkwithrespecttotheupstreamtransversemomentum~UT.Themainadvantageofallthosemethodsisthattheycanbeappliedtoveryshort(single-stage)decaytopologies,aswellastoasubsystemoftheobservedevent.Themethodscanalsobegeneralizedtothecaseofnon-identicalmissingparticles,asdemonstratedinChapter7.Acompletesetofanalyticalresultsforthecalculationoftherelevantvariablesineachevent,aswellasthedependenceoftheirendpointsontheunderlyingmassspectrumisgivenforeachcase.Insomecircumstances,thewholeshapeofthedifferentialdistributioncanbetheoreticallypredictedaswell.Themethodsareillustratedwithexamplesfromsupersymmetryandfromtopquarkproductioninthestandardmodel. 15

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CHAPTER1INTRODUCTIONTheLargeHadronCollider(LHC)atCERNhasbegunitslongawaitedexplorationoftheTeVscaleandreachedintegratedluminosityofL40pb)]TJ /F5 7.97 Tf 6.58 0 Td[(1at7TeV.StartingwiththehuntforstandardmodelHiggsparticle,weexpecttoseeBeyondtheStandardModel(BSM)phenomenaattheLHCwhichmayholdthekeytoourunderstandingofsomeverybasicquestionsaboutouruniverse:Whatisthedarkmatter?WhatarethefundamentalsymmetriesofNature?Arethereanyhiddendimensionsofspace?ApotentialdiscoveryofamissingenergysignalattheLHCmayrelatetoallthreeofthesequestions.PerhapsthemostcompellingphenomenologicalevidenceforBSMparticlesandinteractionsattheTeVscaleisprovidedbythedarkmatterproblem[ 1 ].Itisatantalizingcoincidencethataneutral,weaklyinteractingmassiveparticle(WIMP)intheTeVrangecanexplainalloftheobserveddarkmatterintheUniverse.AtypicalWIMPdoesnotinteractinthedetectorandcanonlymanifestitselfasmissingenergy.TheWIMPideathereforegreatlymotivatesthestudyofmissingenergysignaturesattheTevatronandtheLHC[ 2 ].ThelonglifetimeofthedarkmatterWIMPsistypicallyensuredbysomenewexactsymmetry,e.g.R-parityinsupersymmetry[ 3 ],KKparityinmodelswithextradimensions[ 4 ],T-parityinLittleHiggsmodels[ 5 6 ]etc.TheparticlesoftheStandardModel(SM)arenotchargedunderthisnewsymmetry,butthenewparticlesare,andthelightestamongthemisthedarkmatterWIMP.ThissetupguaranteesthattheWIMPcannotdecay,andmoreimportantly,thatWIMPsarealwayspair-producedatcolliders.Thecross-sectionsfordirectproductionofWIMPs(taggedwithajetoraphotonfrominitialstateradiation)athadroncollidersaretypicallytoosmalltoallowobservationabovetheSMbackgrounds[ 7 ].Thereforeonetypicallyconcentratesonthepairproductionoftheother,heavierparticles(e.g.superpartners,KK-partners,orT-partners),whichalsocarrynontrivialnewquantumnumbersjustliketheWIMPs.Onceproduced,those 16

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heavierpartnerswillcascadedecaydown,emittingSMparticleswhichareinprincipleobservableinthedetector.However,eachsuchcascadealsoinevitablyendsupwithaninvisibleWIMP,whoseenergyandmomentumareunknown.Sincetheheavypartnersarebeingpair-produced,therearetwosuchcascadesineachevent,andtherefore,twounknownWIMPmomenta.Inaddition,athadroncollidersthetotalpartonlevelenergyandmomentuminthecenterofmassframearealsounknown,andthustheexactreconstructionofthedecaychainsonaneventbyeventbasisisaverychallengingtask1.Inthisdissertation,wepresenthowwecanresolvethesedifcultiesmoresystematically.OurapproachisdescribedschematicallyintheFigure 1-1 SystematicApproachesforMassDeterminations.Theveryrstquestionthatwemaywanttoaskwouldbewhatisthescaleofthenewphysics.Toanswerthisquestion,itwouldbebesttoavoidanyassumptionsabouttheevent-topology.Westudiedvariableswhichmimictrueproductionenergyp ^sofnewparticles.Thereference[ 50 ]providesp sminthatisaminimizationofp ^swithonlyoneconditionfrommissingenergyconstraint.Itturnedoutthatthepeakofp sminprovidesaglobalandfullyinclusivemeasureoftheproductionscaleofthenewparticles.IntheearlystageoftheLHC,thisvariablecanbeusedbothasanestimatorandadiscriminatorfornewphysicsoverthestandardmodelbackgrounds.Moredetailedstudiesonp sminwillbeprovidedinChapter2.Sincewehavenotspeciedontheevent-topology,p sminitselfisnotpreciseenoughtodeterminethefullmassspectrumofthenewparticles.Inadditiontop sminonecanalsostudyinvariantmassofvariousvisibleparticles.Aswewillpointout,itispossibletoreconstructtheintermediateparticles'masseswhenthecascadedecaychainislongenough.Theproblemofusinginvariant-massisthatweneed 1SeeDarkmatterandcolliderphenomenologyofuniversalextradimensions(Phys.Rept.453,29(2007))[ 8 ]forarecentreview. 17

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tospecifywhichvisibleparticlecomesfromwhichintermediateparticle,namelywesufferfromcombinatorialproblems,thusifNcascade(thelengthofdecayingchain)islarge,itbecomesmoredifculttochoosetherightsetofvisibleparticlestoformtheinvariantmass.2Ontopofthisproblem,whenweinverttheinvariantmassendpointstosolveforthemassspectra,therewillbemultiplesolutionscomingfromthenon-linearcharacteristicsofcoupledquadraticequations[ 9 ].Thosedifcultiesinitiateourprojectsoninvariantmassmethods,anddetailedstudieswillbepresentedinChapter3.Whileinvariantmassmethodsrelyonasinglecascadedecaymodeofanewparticle,usuallynewphysicswillcomewithpairproducedparticlesduetothenewexactsymmetry.Thusifweusethisextraconditionaboutproducedparticles,wecanreconstructnewparticles'massevenforshortdecayingmodeslikeasNcascade=2whichisnotpossiblewiththeinvariantmassendpointsmethod.Wedevelopedasubsystemconceptwhichwecanapplytospecicdecaymodes.WeappliedthesubsystemtoaCambridgetransversevariablecalledMT2[ 10 ]inChapter4.WhileasubsystemMT2inprincipleworksforNcascade1,whenNcascade=1theresolutionofthemassdeterminationisnotsogood.Thisisbecausethemassdeterminationdependsonthehardnessoftheupstreammomentum,whichisnotbigenoughwhenupstreamobjectscomefromtheinitialstateradiation.WeproposedorthogonaldecompositionsofknownkinematicvariablessuchasMCTandMT2ontothatspecialtransversedirection[ 11 12 ].WerealizedthatthedoublytransversequantitieslikeMCT?andMT2?areparticularlyuseful,sincetheirkinematicendpointsareindependentof~UTandmakeitpossibletousethewholestructureofphasespacesothatwecandeterminethemassspectraofrelatedparticlesinaveryshortdecaychain.WeprovidethesemethodsinChapter5. 2p smindoesnotsufferfromthiskindofcombinatorialproblem. 18

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Givenourutterignoranceaboutthestructureofthedarkmattersector,wesetouttodevelopthenecessaryformalismforcarryingoutmissingenergystudiesathadroncollidersinaverygeneralandmodel-independentway,withoutrelyingonanyassumptionsaboutthenatureofthemissingparticles.Inparticular,wedidnotassumethatthetwomissingparticlesineacheventarethesame.WegeneralizedtheMT2ideatoasymmetriceventswithdifferentmissingparticlesinChapter6. 19

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Figure1-1. Aschematicdescriptionofthemass-determinationmethodsdescribedinthisdissertation. 20

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CHAPTER2GENERALANALYSISWITHOUTANYASSUMPTIONS 2.1TheNeedforaUniversal,GlobalandInclusiveMassVariableMostmethodsaremodel-dependentinthesensethateachmethodcruciallyreliesontheassumptionofaveryspeciceventtopology.OnecommonawofallmethodsonthemarketisthattheyusuallydonotallowanySMneutrinostoenterthetargetedeventtopology,andthemissingenergyistypicallyassumedtoariseonlyasaresultoftheproductionof(two)newdarkmatterparticles.Furthermore,eachmethodhasitsownlimitations.Forexample,thetraditionalinvariantmassendpointmethods[ 9 13 22 ]requiretheidenticationofasufcientlylongcascadedecaychain,withatleastthreesuccessivetwo-bodydecays[ 23 ].Thepolynomialmethods[ 24 31 ]alsorequiresuchlongdecaychainsandfurthermore,theeventsmustbesymmetric,i.e.musthavetwoidenticaldecaychainsperevent,orelsethedecaychainmustbeevenlonger[ 23 ].TherecentlypopularMT2methods[ 10 12 32 39 ]donotrequirelongdecaychains[ 23 ],buttypicallyassumethattheparentparticlesarethesameanddecaytotwoidenticalinvisibleparticles1.ThelimitationsoftheMCTmethods[ 11 42 43 ]arerathersimilar.Thekinematiccuspmethod[ 44 ]islimitedtothesocalledantlereventtopology,whichcontainstwosymmetricone-stepdecaychainsoriginatingfromasingles-channelresonance.Inlightofallthesevariousassumptions,itiscertainlydesirabletohaveauniversalmethodwhichcanbeappliedtoanyeventtopology.Thep sminvariableisdenedintermsofthetotalenergyEand3-momentum~Pobservedintheevent,andthusdoesnotmakeanyreferencetotheactualeventtopology.Itiscompletelygeneral,universalandfullyinclusive,andtothefullestextentmakesuseoftheavailableexperimentalinformation. 1SeeDarkMatterParticleSpectroscopyattheLHC:GeneralizingMT2toAsymmetricEventTopologies(JHEP1004,086(2010))[ 41 ]foramoregeneralapproachwhichavoidsthisassumption. 21

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Figure2-1. Thegenericeventtopologyusedtodenethep sminvariableinp smin:AGlobalinclusivevariablefordeterminingthemassscaleofnewphysicsineventswithmissingenergyathadroncolliders,(JHEP0903,085(2009))[ 50 ].Black(red)linescorrespondtoSM(BSM)particles.ThesolidlinesdenoteSMparticlesXi,i=1,2,...,nvis,whicharevisibleinthedetector,e.g.jets,electrons,muonsandphotons.TheSMparticlesmayoriginateeitherfrominitialstateradiation(ISR),orfromthehardscatteringandsubsequentcascadedecays(indicatedwiththegreen-shadedellipse).Thedashedlinesdenoteneutralstableparticlesi,i=1,2,...,ninv,whichareinvisibleinthedetector.Ingeneral,thesetofinvisibleparticlesconsistsofsomenumbernofBSMparticles(indicatedwiththereddashedlines),aswellassomenumbern=ninv)]TJ /F3 11.955 Tf 11.96 0 Td[(nofSMneutrinos(denotedwiththeblackdashedlines).TheidentitiesandthemassesmioftheBSMinvisibleparticlesi,(i=1,2,...,n)donotnecessarilyhavetobeallthesame,i.e.weallowforthesimultaneousproductionofseveraldifferentspeciesofdarkmatterparticles.Theglobaleventvariablesdescribingthevisibleparticlesare:thetotalenergyE,thetransversecomponentsPxandPyandthelongitudinalcomponentPzofthetotalvisiblemomentum~P.Theonlyexperimentallyavailableinformationregardingtheinvisibleparticlesisthemissingtransversemomentum~=PT. 22

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2.1.1Denitionofp sminConsiderthemostgenericmissingenergyeventtopologyshowninFigure 2-1 .Indeningp smin,oneimaginesacompletelygeneralsetupeacheventcontainssomenumbernvisofStandardModel(SM)particlesXi,i=1,2,...,nvis,whicharevisibleinthedetector,i.e.theirenergiesandmomentaareinprinciplemeasured.ExamplesofsuchvisibleSMparticlesarethebasicreconstructedobjects,e.g.jets,photons,electronsandmuons.ThevisibleparticlesXiaredenotedinFigure 2-1 withsolidblacklinesandmayoriginateeitherfromISR,orfromthehardscatteringandsubsequentcascadedecays(indicatedwiththegreen-shadedellipse).Inturn,themissingtransversemomentum~=PTarisesfromacertainnumberninvofstableneutralparticlesi,i=1,2,...,ninv,whichareinvisibleinthedetector.Ingeneral,thesetofinvisibleparticlesconsistsofsomenumbernofBSMparticles(indicatedwiththereddashedlines),aswellassomenumbern=ninv)]TJ /F3 11.955 Tf 12.65 0 Td[(nofSMneutrinos(denotedwiththeblackdashedlines).Asalreadymentionedearlier,the~=PTmeasurementalonedoesnotrevealthenumberninvofmissingparticles,norhowmanyofthemareneutrinosandhowmanyareBSM(darkmatter)particles.ThisgeneralsetupalsoallowstheidentitiesandthemassesmioftheBSMinvisibleparticlesi,(i=1,2,...,n)inprincipletobedifferent,asinmodelswithseveraldifferentspeciesofdarkmatterparticles[ 45 49 ].Ofcourse,theneutrinomassescanbesafelytakentobezero mi=0,fori=n+1,n+2,...,ninv.(2)Giventhisverygeneralsetup,ifwetrytominimizetheparton-levelMandelstaminvariantmassvariablep swhichisconsistentwiththeobservedvisible4-momentumvectorP(E,~P),wewillgetthefollowingminimumofp sas p smin(=M)p E2)]TJ /F3 11.955 Tf 11.95 0 Td[(P2z+q =M2+=PT2,(2) 23

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wherethemassparameter=Misnothingbutthetotalmassofallinvisibleparticlesintheevent: =MninvXi=1mi=nXi=1mi,(2)andthesecondequalityfollowsfromtheassumptionofvanishingneutrinomasses( 2 ).Theresult( 2 )canbeequivalentlyrewritteninamoresymmetricform p smin(=M)=q M2+P2T+q =M2+=PT2(2)intermsofthetotalvisibleinvariantmassMdenedas M2E2)]TJ /F3 11.955 Tf 11.95 0 Td[(P2x)]TJ /F3 11.955 Tf 11.95 0 Td[(P2y)]TJ /F3 11.955 Tf 11.96 0 Td[(P2zE2)]TJ /F3 11.955 Tf 11.96 0 Td[(P2T)]TJ /F3 11.955 Tf 11.96 0 Td[(P2z.(2)Noticethatinspiteofthecompletearbitrarinessoftheinvisibleparticlesectoratthispoint,thedenitionofp smindependsonasingleunknownparameter=M-thesumofallthemassesoftheinvisibleparticlesintheevent.Forfuturereference,oneshouldkeepinmindthattransversemomentumconservationatthispointimpliesthat ~PT+~=PT=0.(2)Themainresultfromp smin:AGlobalinclusivevariablefordeterminingthemassscaleofnewphysicsineventswithmissingenergyathadroncolliders,(JHEP0903,085(2009))[ 50 ]wasthatintheabsenceofISRandMPI,thepeakinthep smindistributionnicelycorrelateswiththemassthresholdofthenewlyproducedparticles.Thisobservationprovidesonegenericrelationbetweenthetotalmassoftheproducedparticlesandthetotalmass=Moftheinvisibleparticles. 2.1.2p sminandtheUnderlyingEventProblemAtthesametime,itwasalsorecognizedthateffectsfromtheunderlyingevent(UE),mostnotablyISRandMPI,severelyjeopardizethismeasurement.TheproblemisthatinthepresenceoftheUE,thep sminvariablewouldbemeasuringthetotalenergyofthefullsystemshowninFigure 2-1 ,whileforstudyinganynewphysicswearemostly 24

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interestedintheenergyofthehardscattering,asrepresentedbythegreen-shadedellipseinFigure 2-1 .TheinclusionoftheUEcausesadrasticshiftofthepeakofthep smindistributiontohighervalues,oftenbyasmuchasafewTeV[ 50 52 ].Asaresult,itappearedthatunlesseffectsfromtheunderlyingeventcouldsomehowbecompensatedfor,theproposedmeasurementofthep sminpeakwouldbeofnopracticalvalue.Themainpurposeofthischapteristoproposetwofreshnewapproachestodealingwiththeunderlyingeventproblemwhichhasplaguedthep sminvariableandpreventeditsmorewidespreaduseinhadroncolliderphysicsapplications.Weproposetwonewvariantsofthep sminvariable,whichwelabelp s(reco)minandp s(sub)minanddeneinSections 2.2 and 2.3 ,correspondingly.WeillustratethepropertiesofthesetwovariableswithseveralexamplesinSections 2.4 2.6 .Theseexampleswillshowthatbothp s(reco)minandp s(sub)minareunharmedbytheeffectsfromtheunderlyingevent,thusresurrectingtheoriginalideaofp sminproposedinReference[ 50 ]tousethepeakinthep smindistributionasarst,quick,model-independentestimateofthenewphysicsmassscale.InSection 2.7 wecomparetheperformanceofp sminagainstsomeotherinclusivevariableswhicharecommonlyusedinhadroncolliderphysicsforthepurposeofestimatingthenewphysicsmassscale. 2.2DenitionoftheRECOlevelVariablep s(reco)minIntherstapproach,weshallnotmodifytheoriginaldenitionofp sminandwillcontinuetousetheEquation( 2 )(oritsequivalentEquation( 2 )),preservingthedesireduniversal,globalandinclusivecharacterofthep sminvariable.Thenweshallconcentrateonthequestion,howshouldonecalculatetheobservablequantitiesE,~Pand=PTenteringthedeningEquations( 2 )and( 2 ).Thepreviousp sminstudies[ 50 52 ]usedcalorimeter-basedmeasurementsofthetotalvisibleenergyEandmomentum~Pasfollows.Thetotalvisibleenergyinthe 25

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calorimeterE(cal)issimplyascalarsumoverallcalorimeterdeposits E(cal)XE,(2)wheretheindexlabelsthecalorimetertowers,andEistheenergydepositinthetower.Asusual,sincemuonsdonotdepositsignicantlyinthecalorimeters,themeasuredEshouldrstbecorrectedfortheenergyofanymuonswhichmightbepresentintheeventandhappentopassthroughthecorrespondingtower.Thethreecomponentsofthetotalvisiblemomentum~Pwerealsomeasuredfromthecalorimetersas Px(cal)=XEsincos', (2) Py(cal)=XEsinsin', (2) Pz(cal)=XEcos, (2) whereand'arecorrespondinglythepolarandazimuthalangularcoordinatesofthecalorimetertower.Themissingtransversemomentumcansimilarlybemeasuredfromthecalorimeteras(Equation( 2 )) 6~PT(cal))]TJ /F8 11.955 Tf 24.48 3.16 Td[(~PT(cal).(2)Usingthesecalorimeter-basedmeasurements( 2 2 ),onecanmaketheidentication EE(cal), (2) ~P~P(cal), (2) ~=PT6~PT(cal) (2) 26

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inthedenition( 2 )andconstructthecorrespondingcalorimeter-basedp sminvariableas p s(cal)min(=M)q E2(cal))]TJ /F3 11.955 Tf 11.96 0 Td[(P2z(cal)+q =M2+6P2T(cal).(2)Thiswaspreciselythequantitywhichwasstudiedin[ 50 52 ]andshowntoexhibitextremesensitivitytothephysicsoftheunderlyingevent.HereweproposetoevaluatethevisiblequantitiesEand~PattheRECOlevel,i.e.intermsofthereconstructedobjects,namelyjets,muons,electronsandphotons.Tobeprecise,lettherebeNobjreconstructedobjectsintheevent,withenergiesEiand3-momenta~Pi,i=1,2,...,Nobj,correspondingly.TheninplaceofEquations( 2 2 ),letusinsteadidentify EE(reco)NobjXi=1Ei, (2) ~P~P(reco)NobjXi=1~Pi, (2) ~=PT6~PT(reco)=)]TJ /F8 11.955 Tf 9.87 3.15 Td[(~PT(reco), (2) andcorrespondinglydeneaRECO-levelp sminvariableas p s(reco)min(=M)q E2(reco))]TJ /F3 11.955 Tf 11.96 0 Td[(P2z(reco)+q =M2+6P2T(reco),(2)whichcanalsoberewritteninanalogytoEquation( 2 )as p s(reco)min(=M)q M2(reco)+P2T(reco)+q =M2+6P2T(reco),(2)where6PT(reco)andPT(reco)arerelatedasinEquation( 2 )andtheRECO-leveltotalvisiblemassM(reco)isdenedby M2(reco)E2(reco))]TJ /F8 11.955 Tf 12.53 3.15 Td[(~P2(reco).(2) 27

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WhatarethebenetsfromthenewRECO-levelp smindenedasinEquations( 2 2 )incomparisontotheoldcalorimeter-basedp smindenitioninanEquation( 2 )?Inordertounderstandthebasicidea,itisworthcomparingthecalorimeter-basedmissingtransversemomentum=PT(whichintheliteratureiscommonlyreferredtoasmissingtransverseenergy=ET)andtheanalogousRECO-levelvariable=HT,themissingHT.The~=HTvectorisdenedasthenegativeofthevectorsumofthetransversemomentaofallreconstructedobjectsintheevent: ~=HT)]TJ /F4 7.97 Tf 24.9 16.28 Td[(NobjXi=1~PTi.(2)Thenitisclearthatintermsofournotationhere,=HTisnothingbut6PT(reco).Itisknownthat=HTperformsbetterthan=ET[ 53 ].First,=HTislessaffectedbyanumberofadverseinstrumentalfactorssuchas:electronicnoise,faultycalorimetercells,pile-up,etc.Theseeffectstendtopopulatethecalorimeteruniformlywithunclusteredenergy,whichwilllaterfailthebasicqualitycutsduringobjectreconstruction.Incontrast,thetruemissingmomentumisdominatedbyclusteredenergy,whichwillbesuccessfullycapturedduringreconstruction.Anotheradvantageof=HTisthatonecaneasilyapplytheknownjetenergycorrectionstoaccountforthenonlineardetectorresponse.Forbothofthesereasons,CMSisnowusing=HTatboththetriggerlevelandofine[ 53 ].Nowrealizethatp s(cal)minisanalogoustothecalorimeter-based=ET,whileournewvariablep s(reco)minisanalogoustotheRECO-level=HT.Thuswemayalreadyexpectthatp s(reco)minwillinherittheadvantagesof=HTandwillbebettersuitedfordeterminingthenewphysicsmassscalethanthecalorimeter-basedquantityp s(cal)min.ThisexpectationisconrmedintheexplicitexamplesstudiedbelowinSections 2.4 and 2.5 .Apartfromthealreadymentionedinstrumentalissues,themostimportantadvantageofp s(reco)minfromthephysicspointofviewisthatitismuchlesssensitivetotheeffectsfromtheunderlyingevent,whichhaddoomeditscalorimeter-basedp s(cal)mincousin. 28

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Strictlyspeaking,theideaofp s(reco)mindoesnotsolvetheunderlyingeventproblemcompletelyandasamatterofprinciple.Everynowandthentheunderlyingeventwillstillproduceawell-denedjet,whichwillhavetobeincludedinthecalculationofp s(reco)min.Becauseofthiseffect,wecannotanymoreguaranteethatp s(reco)minprovidesalowerboundonthetruevaluep strueofthecenter-of-massenergyofthehardscatteringtheadditionaljetsformedoutofISR,pile-up,andsoon,willsometimescausep s(reco)mintoexceedp strue.Neverthelesswendthatthiseffectmodiesonlytheshapeofthep s(reco)mindistribution,butleavesthelocationofitspeaklargelyintact.Totheextentthatoneismostlyinterestedinthepeaklocation,p s(reco)minshouldalreadybegoodenoughforallpracticalpurposes. 2.3DenitionoftheSubsystemVariablep s(sub)minInthissectionweproposeanalternativemodicationoftheoriginalp sminvariable,whichsolvestheunderlyingeventproblemcompletelyandasamatterofprinciple.Thedownsideofthisapproachisthatitisnotasgeneralanduniversalastheonediscussedintheprevioussection,andcanbeappliedonlyincaseswhereonecanunambiguouslyidentifyasubsystemoftheoriginaleventtopologywhichisuntouchedbytheunderlyingevent.ThebasicideaisschematicallyillustratedinFigure 2-2 ,whichisnothingbutaslightrearrangementofFigure 2-1 exhibitingawelldenedsubsystem(delineatedbytheblackrectangle).TheoriginalnvisvisibleparticleXifromFigure 2-1 havenowbeendividedintotwogroupsasfollows: 1. TherearensubvisibleparticlesX1,...,Xnsuboriginatingfromwithinthesubsystem.TheirtotalenergyandtotalmomentumaredenotedbyE(sub)and~P(sub).Thesubsystemparticlesarechosensothattoguaranteethattheycouldnothavecomefromtheunderlyingevent. 2. Theremainingnvis)]TJ /F3 11.955 Tf 12.66 0 Td[(nsubvisibleparticlesXnsub+1,...,Xnvisarecreatedupstream(outsidethesubsystem)andhavetotalenergyE(up)andtotalmomentum~P(up).Theupstreamparticlesmayoriginatefromtheunderlyingeventorfromdecaysofheavierparticlesupstreamthisdistinctionisinconsequentialatthispoint. 29

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Figure2-2. ArearrangementofFigure 2-1 intoaneventtopologyexhibitingawelldenedsubsystem(delineatedbytheblackrectangle)withtotalinvariantmassp s(sub).TherearensubvisibleparticlesXi,i=1,2,...,nsub,originatingfromwithinthesubsystem,whiletheremainingnvis)]TJ /F3 11.955 Tf 11.95 0 Td[(nsubvisibleparticlesXnsub+1,...,Xnvisarecreatedupstream,outsidethesubsystem.ThesubsystemresultsfromtheproductionanddecaysofacertainnumberofparentparticlesPj,j=1,2,...,np,(someof)whichmaydecaysemi-invisibly.Allinvisibleparticles1,...,ninvarethenassumedtooriginatefromwithinthesubsystem. Wealsoassumethatallinvisibleparticles1,...,ninvoriginatefromwithinthesubsystem,i.e.thatnoinvisibleparticlesarecreatedupstream.Ineffect,allwehavedoneinFigure 2-2 istopartitiontheoriginalmeasuredvaluesofthetotalvisibleenergyEand3-momentum~PfromFigure 2-1 intotwoseparatecomponentsas E=E(up)+E(sub), (2) ~P=~P(up)+~P(sub). (2) 30

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Noticethatnowthemissingtransversemomentumisdenedas ~=PT)]TJ /F8 11.955 Tf 22.49 3.16 Td[(~PT(up))]TJ /F8 11.955 Tf 12.53 3.16 Td[(~PT(sub),(2)whilethetotalvisibleinvariantmassM(sub)ofthesubsystemisgivenby M2(sub)=E2(sub))]TJ /F8 11.955 Tf 12.53 3.15 Td[(~P2(sub).(2)TherewouldbeambiguitiesincategorizingagivenvisibleparticleXiasasubsystemoranupstreamparticle.Sinceourgoalistoidentifyasubsystemwhichisshieldedfromtheeffectsoftheunderlyingevent,thesafestwaytodothepartitionofthevisibleparticlesistorequirethatallQCDjetsbelongtotheupstreamparticles,whilethesubsystemparticlesconsistofobjectswhichareunlikelytocomefromtheunderlyingevent,suchasisolatedelectrons,photonsandmuons(andpossiblyidentied-jetsand,toalesserextent,taggedb-jets).Withthosepreliminaries,wearenowreadytoasktheusualp sminquestion:GiventhemeasuredvaluesofE(up),E(sub),~P(up)and~P(sub),whatistheminimumvaluep s(sub)minofthesubsystemMandelstaminvariantmassvariablep s(sub),whichisconsistentwiththosemeasurements?Proceedingasin[ 50 ],onceagainwendaverysimpleuniversalanswer,which,withthehelpofEquations( 2 )and( 2 ),canbeequivalentlywritteninseveraldifferentwaysasfollows: p s(sub)min(=M)=(q E2(sub))]TJ /F3 11.955 Tf 11.96 0 Td[(P2z(sub)+q =M2+=PT22)]TJ /F3 11.955 Tf 11.95 0 Td[(P2T(up))1 2 (2) =(q M2(sub)+P2T(sub)+q =M2+=PT22)]TJ /F3 11.955 Tf 11.96 0 Td[(P2T(up))1 2 (2) =(q M2(sub)+P2T(sub)+q =M2+=PT22)]TJ /F6 11.955 Tf 11.96 0 Td[((~PT(sub)+~=PT)2)1 2 (2) =jjpT(sub)+6pTjj, (2) 31

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whereinthelastlinewehaveintroducedtheLorentz1+2vectors pT(sub)q M2(sub)+P2T(sub),~PT(sub); (2) 6pTq =M2+=PT2,~=PT. (2) Asusual,thelengthofa1+2vectoriscomputedasjjpjj=p pp=p p20)]TJ /F3 11.955 Tf 11.96 0 Td[(p21)]TJ /F3 11.955 Tf 11.96 0 Td[(p22.Beforeweproceedtotheexamplesofthenextfewsections,asasanitycheckoftheobtainedresultitisusefultoconsidersomelimitingcases.First,bytakingtheupstreamvisibleparticlestobeanemptyset,i.e.~PT(up)!0,werecovertheusualexpressionforp smingiveninEquations( 2 2 ).Next,consideracasewithnoinvisibleparticles,i.e.=M=0andcorrespondingly,~=PT=0.Inthatcaseweobtainthatp s(sub)min=M(sub),whichisofcoursethecorrectresult.Finally,supposethattherearenovisiblesubsystemparticles,i.e.E(sub)=~P(sub)=M(sub)=0.Inthatcaseweobtainp s(sub)min==M,whichisalsothecorrectanswer.Asweshallsee,thesubsystemconceptofFigure 2-2 willbemostusefulwhenthesubsystemresultsfromtheproductionanddecaysofacertainnumbernpofparentparticlesPjwithmassesMPj,j=1,2,...,np,correspondingly.Thenthetotalcombinedmassofallparentparticlesisgivenby MpnpXj=1MPj.(2)BytheconjectureofReference[ 50 ],thelocationofthepeakofthep s(sub)min(=M)distributionwillprovideanapproximatemeasurementofMpasafunctionoftheunknownparameter=M.Byconstruction,theobtainedrelationshipMp(=M)willthenbecompletelyinsensitivetotheeffectsfromtheunderlyingevent.AtthispointitmayseemthatbyexcludingallQCDjetsfromthesubsystem,wehavesignicantlynarroweddownthenumberofpotentialapplicationsofthep s(sub)minvariable.Furthermore,wehaveapparentlyreintroducedacertainamountof 32

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model-dependencewhichtheoriginalp sminapproachwastryingsohardtoavoid.Thoseareinprinciplevalidobjections,whichcanbeovercomebyusingthep s(reco)minvariableintroducedintheprevioussection.Nevertheless,wefeelthatthep s(sub)minvariablecanprovetobeusefulinitsownright,andinawidevarietyofcontexts.Toseethis,notethatthetypicalhadroncollidersignaturesofthemostpopularnewphysicsmodels(supersymmetry,extradimensions,LittleHiggs,etc.)arepreciselyoftheformexhibitedinFigure 2-2 .Onetypicallyconsidersproductionofcoloredparticles(squarks,gluinos,KK-quarks,etc.)whosecross-sectionsdominate.Inturn,thesecoloredparticlesshedtheircolorchargebyemittingjetsanddecayingtolighter,uncoloredparticlesinanelectroweaksector.Thedecaysofthelatterofteninvolveelectromagneticobjects,whichcouldbetargetedforselectioninthesubsystem.Thep s(sub)minvariablewouldthenbetheperfecttoolforstudyingthemassscalesintheelectroweaksector(inthecontextofsupersymmetry,forexample,theelectroweaksectoriscomposedofthecharginos,neutralinosandsleptons).Beforewemoveontosomespecicexamplesillustratingtheseideas,onelastcommentisinorder.Onemaywonderwhetherthep s(sub)minvariableshouldbecomputedattheRECO-levelorfromthecalorimeter.Sincethesubsystemwillusuallybedenedintermsofreconstructedobjects,themorelogicaloptionistocalculatep s(sub)minattheRECO-levelandlabelitasp s(sub,reco)min.However,tostreamlineournotation,inwhatfollowsweshallalwaysomittherecopartofthesuperscriptandwillalwaysimplicitlyassumethatp s(sub)miniscomputedatRECO-level. 2.4SMexample:DileptonEventsfromttproductionInthisandthenexttwosectionsweillustratethepropertiesofthenewvariablesp s(reco)minandp s(sub)minwithsomespecicexamples.InthissectionwediscussanexampletakenfromtheStandardModel,whichisguaranteedtobeavailableforearlystudiesattheLHC.Weconsiderdileptoneventsfromttpairproduction,wherebothW'sdecayleptonically.Inthiseventtopology,therearetwomissingparticles(twoneutrinos). 33

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Therefore,theseeventsverycloselyresemblethetypicalSUSY-likeevents,inwhichtherearetwomissingdarkmatterparticles.Inthenexttwosections,weshallalsoconsidersomeSUSYexamples.Inallcases,weperformdetailedeventsimulation,includingtheeffectsfromtheunderlyingeventanddetectorresolution. 2.4.1EventSimulationDetailsEventsaregeneratedwithPYTHIA[ 54 ](usingitsdefaultmodeloftheunderlyingevent)atanLHCof14TeV,andthenreconstructedwiththePGSdetectorsimulationpackage[ 55 ].WehavemadecertainmodicationsinthepubliclyavailableversionofPGStobettermatchittotheCMSdetector.Forexample,wetakethehadroniccalorimeterresolutiontobe[ 56 ] E=120% p E,(2)whiletheelectromagneticcalorimeterresolutionis[ 56 ] E2=S p E2+N E2+C2,(2)wheretheenergyEismeasuredinGeV,S=3.63%isthestochasticterm,N=0.124isthenoiseandC=0.26%istheconstantterm.Muonsarereconstructedwithinjj<2.4,andweusethemuonglobalreconstructionefciencyquotedin[ 56 ].WeusedefaultpTcutsonthereconstructedobjectsasfollows:3GeVformuons,10GeVforelectronsandphotons,and15GeVforjets.Forthettexamplepresentedinthissection,weusetheapproximatenext-to-next-to-leadingorderttcross-sectionoftt=8944+73+12)]TJ /F5 7.97 Tf 6.59 0 Td[(46)]TJ /F5 7.97 Tf 6.59 0 Td[(12pbatatopmassofmt=175GeV[ 57 ].FortheSUSYexamplesinthenexttwosectionsweuseleadingordercross-sections. 2.4.2p s(reco)minVariableWerstconsiderSUSY-likemissingenergyeventsarisingfromttproduction,whereeachW-bosonisforcedtodecayleptonically(toanelectronoramuon). 34

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Figure2-3. Distributionsofvariousp sminquantitiesdiscussedinthetext,forthedileptonttsampleattheLHCwith14TeVCMenergyand0.5fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofdata.Thedotted(yellow-shaded)histogramgivesthetruep sdistributionofthettpair.Thebluehistogramisthedistributionofthecalorimeter-basedp s(cal)minvariableintheidealcasewhenalleffectsfromtheunderlyingeventareturnedoff.Theredhistogramshowsthecorrespondingresultforp s(cal)mininthepresenceoftheunderlyingevent.Theblackhistogramisthedistributionofthep s(reco)minvariableintroducedinSection 2.2 .Allp smindistributionsareshownfor=M=0. Wedonotimposeanytriggerorofinerequirements,andsimplyplotdirectlytheoutputfromPGS2.Weshowvariousp squantitiesofinterestinFigure 2-3 ,setting=M=0,sinceinthiscasethemissingparticlesareneutrinosandaremassless.Thedotted(yellow-shaded)histogramrepresentsthetruep sdistributionofthettpair.Itquicklyrisesatthettmassthreshold Mp2mt=350GeV(2) 2Therefore,ourplotsinthissubsectionarenormalizedtoatotalnumberofeventsequaltottBR(W!e,)2. 35

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andtheneventuallyfallsoffatlargep sduetothepartondensityfunctionsuppression.Becausethetopquarksaretypicallyproducedwithsomeboost,thep struedistributioninFigure 2-3 peaksalittlebitabovethreshold: )]TJ /F2 11.955 Tf 5.48 -0.24 Td[(p struepeak>Mp.(2)Itisclearthatifonecoulddirectlymeasurethep struedistribution,oratleastitsonset,thettmassscalewillbeeasilyrevealed.Unfortunately,theescapingneutrinosmakesuchameasurementimpossible,unlessoneiswillingtomakeadditionalmodel-dependentassumptions3.Figure 2-3 alsoshowstwoversionsofthecalorimeter-basedp s(cal)minvariable:theblue(red)histogramisobtainedbyswitchingoff(on)theunderlyingevent(ISRandMPI).Thesecurvesrevealtwoveryinterestingphenomena.First,withouttheUE,thepeakofthep s(cal)mindistribution(bluehistogram)isveryclosetotheparentmassthreshold[ 50 ]: noUE=)p s(cal)minpeakMp.(2)ThemainobservationofParthaetal.[ 50 ]wasthatthiscorrelationoffersanalternative,fullyinclusiveandmodel-independent,methodofestimatingthemassscaleMpoftheparentparticles,evenwhensomeoftheirdecayproductsareinvisibleandnotseeninthedetector. 3Forexample,onecanusetheknownvaluesoftheneutrino,Wandtopmassestosolvefortheneutrinokinematics(uptodiscreteambiguities).However,thismethodassumesthatthefullmassspectrumisalreadyknown,andfurthermore,usestheknowledgeofthetopdecaytopologytoperfectlysolvethecombinatoricsproblemdiscussedintheIntroduction.Asanexample,consideracasewheretheleptonisproducedrstandtheb-quarksecond,i.e.whenthetoprstdecaystoaleptonandaleptoquark,whichinturndecaystoaneutrinoandab-quark.Thekinematicmethodwouldthenbeusingthewrongon-shellconditions.Theadvantageofthep sminapproachisthatitisfullyinclusiveanddoesnotmakeanyreferencetotheactualdecaytopology. 36

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Unfortunately,thenoUElimitofEquation( 2 )isunphysical,andthecorrespondingp s(cal)mindistribution(bluehistogramininFigure 2-3 )isunobservable.Whatisworse,whenonetriestomeasurethep s(cal)mindistributioninthepresenceoftheUE(redhistograminFigure 2-3 ),theresultingpeakisveryfarfromthephysicalthreshold: withUE=)p s(cal)minpeakMp.(2)InthettexampleofFigure 2-3 ,theshiftisontheorderof1TeV!Itappearsthereforethatinpracticethep s(cal)minpeakwouldbeuncorrelatedwithanyphysicalmassscale,andinsteadwouldbecompletelydeterminedbythe(uninteresting)physicsoftheunderlyingevent.Oncethenicemodel-independentcorrelationofEquation( 2 )isdestroyedbytheUE,itbecomesofonlyacademicvalue[ 8 50 52 58 ].However,Figure 2-3 alsosuggeststhesolutiontothisdifcultproblem.Ifwelookatthedistributionofthep s(reco)minvariable(blacksolidhistogram),weseethatitspeakhasreturnedtothedesiredvalue: p s(reco)minpeakMp,(2)Inordertomeasurephysicalmassthresholds,onesimplyneedstoinvestigatethedistributionoftheinclusivep s(reco)minvariable,whichiscalculatedatRECO-level.Eachpeakinthatdistributionsignalstheopeningofanewchannel,andfromEquation( 2 )thelocationofthepeakprovidesanimmediateestimateofthetotalmassofallparticlesinvolvedintheproduction.OurrstmainresultisthereforenicelysummarizedinFigure 2-3 ,whichshowsatotalof4distributions,3ofwhichareeitherunphysical(thebluehistogramofp s(cal)minintheabsenceoftheUE),unobservable(theyellow-shadedhistogramofp strue),oruseless(theredhistogramofp s(cal)mininthepresenceoftheUE).TheonlydistributioninFigure 2-3 whichisphysical,observableandusefulatthesametime,isthedistributionofp s(reco)min(solidblackhistogram). 37

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Figure2-4. PGScalorimetermapoftheenergydeposits,asafunctionofpseudorapidityandazimuthalangle,foradileptontteventwithonlytworeconstructedjets.Atthepartonlevel,thisparticulareventhastwob-quarksandtwoelectrons.Thelocationofab-quark(electron,muon)ismarkedwiththeletterq(e,).Agreycircledelineates(theconeof)areconstructedjet,whileagreendottedcircledenotesareconstructedlepton.IntheuppertwoplotsthecalorimeterislledatthepartonleveldirectlyfromPYTHIA,whilethelowertwoplotscontainresultsafterPGSsimulation.TheleftplotsshowabsoluteenergydepositsE,whileintherightplotstheenergyineachtowerisshownprojectedonthetransverseplaneasEcos. 38

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Figure2-5. PGScalorimetermapoftheenergydepositsforadileptontteventwithmorethantworeconstructedjets 39

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Table2-1. Selectedp squantities(inGeV)fortheeventsshowninFigures 2-4 2-5 and 2-12 .Thesecondcolumnshowsthetrueinvariantmassp strueoftheparentsystem:topquarkpairincaseofFigures 2-4 and 2-5 ,orgluinopairincaseofFigure 2-12 .Thethirdcolumnshowsthevalueofthep s(cal)minvariable( 2 )calculatedatthepartonlevel,withoutanyPGSdetectorsimulation,butwiththefulldetectoracceptancecutofjj<4.1.Thefourthcolumnliststhevalueofp s(cal)minobtainedafterPGSdetectorsimulation,whilethelastcolumnshowsthevalueofthep s(reco)minvariabledenedinEquation( 2 ). EventtypePYTHIApartonlevelafterPGSsimulationp struep s(cal)minp s(cal)minp s(reco)min tteventinFigure 2-4 42711101179363tteventinFigure 2-5 63825962761736SUSYeventinFigure 2-12 1954353935092085 Beforeconcludingthissubsection,weexplainthereasonfortheimprovedperformanceofthep s(reco)minvariableincomparisontothep s(cal)minversion.AsalreadyanticipatedinSection 2.2 ,thebasicideaisthatenergydepositswhichareduetohardparticlesoriginatingfromthehardscattering,tendtobeclustered,whiletheenergydepositsduetotheUEtendtobemoreuniformlyspreadthroughoutthedetector.Inordertoseethispictorially,inFigures 2-4 and 2-5 weshowaseriesofcalorimetermapsofthecombinedECAL+HCALenergydepositsasafunctionofthepseudorapidityandazimuthalangle.SincethecalorimeterinPGSissegmentedincellsof(,)=(0.1,0.1),eachcalorimetertowerisrepresentedbyasquarepixel,whichiscolor-codedaccordingtotheamountofenergypresentinthetower.Wehavechosenthecolorschemesothatlargerdepositscorrespondtodarkercolors.EachcalorimetermapinFigures 2-4 and 2-5 hasfourpanels.IntheuppertwopanelsthecalorimeterislledatthepartonleveldirectlyfromPYTHIA.Thiscorrespondstoaperfectdetector,whereweignoreanysmearingeffectsduetotheniteenergyresolution.ThelowertwoplotsinFigures 2-4 and 2-5 showthecorrespondingresultsafterPGSsimulation.Thusbycomparingtheplotsintheupperrowtothoseinthebottomrow,onecanseetheeffectofthedetectorresolution.Whilethenitedetectorresolutiondoesplaysomerole, 40

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wendthatitisofnoparticularimportanceforunderstandingthereasonbehindthebigswingsinthep sminpeaksobservedinFigure 2-3 .Letusinsteadconcentrateoncomparingtheplotsintheleftcolumnversusthoseintherightcolumn.TheleftplotsshowtheabsoluteenergydepositEinthecalorimetertower,whileintherightplotsthisenergyisshownprojectedonthetransverseplaneasEcos.Thedifferencebetweentheleftandtherightplotsisquitestriking.Theplotsontheleftexhibitlotsofenergy,whichisdepositedmostlyintheforwardcalorimetercells(atlargejj)[ 50 ].Theplotsontheright,ontheotherhand,showonlyafewclustersofenergy,concentratedmostlyinthecentralpartofthedetector.Thoseenergyclustersgiverisetotheobjects(jets,electronsandphotons)whicharereconstructedfromthecalorimeter.Furthermore,eachenergyclustercanbeeasilyidentiedwithaparton-levelparticleinthetopdecaychain.Inordertoexhibitthiscorrelation,inFigures 2-4 and 2-5 weusethefollowingnotationfortheparton-levelparticles:ab-quark(electron,muon)ismarkedwiththeletterq(e,).Agreycircledelineates(theconeof)areconstructedjet,whileagreendottedcirclemarksareconstructedlepton(electronormuon).Theleptonisolationrequirementimpliesthatgreencirclesshouldbevoidoflargeenergydepositsoff-center,andindeedweobservethistobethecase.Inparticular,Figure 2-4 showsabare-bonedileptontteventwithjusttworeconstructedjetsandtworeconstructedleptons(whichhappentobebothelectrons).AsseenintheFigure 2-4 ,thetwojetscanbeeasilytracedbacktothetwob-quarksatthepartonlevel,andtherearenoadditionalreconstructedjetsduetotheUEactivity.Becausetheeventissocleanandsimple,onemightexpecttoobtainareasonablevalueforp smin,i.e.closetothettthreshold.However,thisisnotthecase,ifweusethecalorimeter-basedmeasurementp s(cal)min.AsseeninTable 2-1 ,themeasuredvalueofp s(cal)minisveryfaroffontheorderof1TeV,eveninthecaseofaperfectdetector. 41

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ThereasonforthisdiscrepancyisnoweasytounderstandfromFigure 2-4 .Recallthatp s(cal)minisdenedintermsofthetotalenergyE(cal)inthecalorimeter,whichinturnisdominatedbythelargedepositsintheforwardregion,whichcamefromtheunderlyingevent.Moreimportantly,thosecontributionsaremoreorlessequallyspreadovertheforwardandbackwardregionofthedetector,leadingtocancellationsinthecalculationofthecorrespondinglongitudinalPz(cal)momentumcomponent.Asaresult,therstterminEquation( 2 )becomescompletelydominatedbytheUEcontributions[ 51 ].Letusnowseehowthecalculationofp s(reco)minisaffectedbytheUE.Sinceobjectreconstructionisdonewiththehelpofminimumtransversecuts(forclusteringandobjectid),therelevantcalorimeterplotsarethemapsontherightsideinFigure 2-4 .Weseethatthelargeforwardenergydepositswhichwerecausingthelargeshiftinp s(cal)minarenotincorporatedintoanyreconstructedobjects,andthusdonotcontributetothep s(reco)mincalculationatall.Ineffect,theRECO-levelprescriptionforcalculatingp sminisleavingoutpreciselytheunwantedcontributionsfromtheUE,whilekeepingtherelevantcontributionsfromthehardscattering.AsseenfromTable 2-1 ,thecalculatedvalueofp s(reco)minforthateventis363GeV,whichisindeedveryclosetothettthreshold.Itisalsosmallerthanthetruep svalueof427GeVinthatevent,whichistobeexpected,sincebydesignp sminp s,andthiseventdoesnothaveanyextraISRjetstospoilthisrelation.Itisinstructivetoconsideranother,morecomplexttdileptonevent,suchastheoneshowninFigure 2-5 .Thecorrespondingcalculatedvaluesforp s(cal)minandp s(reco)minareshowninthesecondrowofTable 2-1 .AsseeninFigure 2-5 ,thiseventhasadditionaljetsandalotmoreUEactivity.Asaresult,thecalculatedvalueofp s(cal)minisshiftedbyalmost2TeVfromthenominalp struevalue.Nevertheless,theRECO-levelprescriptionnicelycompensatesforthiseffect,andthecalculatedp s(reco)minvalueisonly736GeV,whichiswithin100GeVofthenominalp strue=638GeV.Noticethatinthisexample 42

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weendupwithasituationwherep s(reco)min>p strue.Figure 2-3 indicatesthatthishappensquiteoftenthetailofthep s(reco)mindistributionismorepopulatedthanthe(yellow-shaded)p struedistribution.Thisshouldbenocauseforconcern.Firstofall,weareonlyinterestedinthepeakofthep s(reco)mindistribution,andwedonotneedtomakeanycomparisonsbetweenp s(reco)minandp strue.Second,anysuchcomparisonwouldbemeaningless,sincethevalueofp strueisaprioriunknown,andunobservable. 2.4.3p s(sub)minVariableBeforeconcludingthissection,weshallusethettexampletoalsoillustratetheideaofthesubsystemp s(sub)minvariabledevelopedinSection 2.3 .Dileptontteventsareaperfecttestinggroundforthisidea,sincetheWWsubsystemdecaysleptonically,withoutanyjetactivity.WethereforedenethesubsystemasthetwohardisolatedleptonsresultingfromthedecaysoftheW-bosons.Correspondingly,werequiretworeconstructedleptons(electronsormuons)atthePGSlevel4,andplotthedistributionoftheleptonicsubsystemp s(sub)minvariableinFigure 2-6 .Asbefore,thedotted(yellow-shaded)histogramrepresentsthetruep sdistributionoftheW+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(pair.Asexpected,itquicklyrisesattheWWthreshold(denotedbytheverticalarrow),thenfallsoffatlargep s.Sincethep s(WW)truedistributionisunobservable,thebestwecandoistostudythecorrespondingp s(sub)mindistributionshownwiththesolidblackhistogram.Inthissubsystemexample,allUEactivityislumpedtogetherwiththeupstreamb-jetsfromthetopquarksdecays,andthushasnobearingonthepropertiesoftheleptonicp s(sub)min.Inparticular,wendthatthevalueofp s(sub)minisalwayssmallerthanthetruep s(WW)true.Moreimportantly,Figure 2-6 demonstratesthatthepeakinthep s(sub)mindistributionisfoundpreciselyatthemassthresholdoftheparticles(inthiscasethetwoWbosons)whichinitiatedthesubsystem. 4Theselectionefciencyforthetwoleptonsisontheorderof60%,whichexplainsthedifferentnormalizationofthedistributionsinFigures 2-3 and 2-6 43

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Figure2-6. [Distributionofvariousp sminforthedileptonsubsystemindileptontteventswithtworeconstructedleptonsinPGS.Thedotted(yellow-shaded)histogramgivesthetruep sdistributionoftheW+W)]TJ /F1 11.955 Tf 10.4 -4.33 Td[(pairinthoseevents.Theblackhistogramshowsthedistributionofthe(leptonic)subsystemvariablep s(sub)mindenedinSection 2.3 .Inthiscase,thesubsystemisdenedbythetwoisolatedleptons,whilealljetsaretreatedasupstreamparticles.TheverticalarrowmarkstheW+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(massthreshold. Inanalogyto( 2 )wecanalsowrite p s(sub)minpeakM(sub)p,(2)whereM(sub)pisthecombinedmassofalltheparentsinitiatingthesubsystem.Figure 2-6 showsthatinthettexamplejustconsidered,thisrelationholdstoaveryhighdegreeofaccuracy.Thisexampleshouldnotleavethereaderwiththeimpressionthathadronicjetsareneverallowedtobepartofthesubsystem.Onthecontrarythesubsystemmayverywellincludereconstructedjetsaswell.Thettcaseconsideredhereinfactprovidesaperfectexampletoillustratetheidea. 44

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Figure2-7. Unit-normalizeddistributionofjetmultiplicityindileptonttevents. Figure2-8. Distributionsofvariousp sminforthedileptonttsample,inadditiontothetwoleptons,thesubsystemnowalsoincludes:exactlytwob-taggedjets(blackhistogram);thetwohighestpTjets(bluehistogram);oralljets(redhistogram).Thedotted(yellow-shaded)histogramgivesthetruep sdistributionofthettpair. 45

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Letusreconsiderthettdileptonsample,andredenethesubsystemsothatwenowtargetthetwotopquarksastheparentsinitiatingthesubsystem.Correspondingly,inadditiontothetwoleptons,letusallowthesubsystemtoincludetwojets,presumablycomingfromthetwotopquarkdecays.Unfortunately,indoingso,wemustfaceavariantofthepartitioning5combinatorialproblemdiscussedintheintroduction:asseeninFigure 2-7 ,thetypicaljetmultiplicityintheeventsisrelativelyhigh,andwemustthereforespecifytheexactprocedurehowtoselectthetwojetswhichwouldenterthesubsystem.Weshallconsiderthreedifferentapproaches. B-tagging.Wecanusethefactthatthejetsfromtopquarkdecayareb-jets,whilethejetsfromISRaretypicallylightavorjets.Therefore,byrequiringexactlytwob-tags,andincludingonlythetwob-taggedjetsaspartofthesubsystem,wecansignicantlyincreasetheprobabilityofselectingthecorrectjets.Ofcourse,ISRwillsometimesalsocontributeb-taggedjetsfromgluonsplitting,butthathappensratherrarelyandthecorrespondingcontributioncanbesuppressedbyafurtherinvariantmasscutonthetwob-jets.Theresultingp s(sub)mindistributionforthesubsystemof2leptonsand2b-taggedjetsisshowninFigure 2-8 withtheblackhistogram.Weseethat,asexpected,thedistributionpeaksatthettthresholdandthistimeprovidesameasurementofthetopquarkmass: p s(sub)minpeakM(sub)p=2mt=350GeV.(2)Thedisadvantageofthismethodisthelossinstatistics:comparethenormalizationoftheblackhistograminFigure 2-8 afterapplyingthetwob-tags,tothedotted(yellow-shaded)distributionofthetruettdistributionintheselectedinclusivedileptonsample(withoutb-tags). SelectionbyjetpT.HereonecanusethefactthatthejetsfromtopdecaysareonaverageharderthanthejetsfromISR.Correspondingly,bychoosingthetwohighestpTjets(regardlessofb-tagging),onealsoincreasestheprobabilitytoselectthecorrectjetpair.ThecorrespondingdistributionisshowninFigure 2-8 withthebluehistogram,andisalsoseentopeakatthettthreshold.Animportant 5Byconstruction,thep sminandp s(sub)minvariablesneverhavetofacetheorderingcombinatorialproblem. 46

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advantageofthismethodisthatonedoesnothavetopaythepriceofreducedstatisticsduetothetwoadditionalb-tags. Noselection.Themostconservativeapproachwouldbetoapplynoselectioncriteriaonthejets,andincludeallreconstructedjetsinthesubsystem.Thenthesubsystemp s(sub)minvariableessentiallyrevertsbacktotheRECO-levelinclusivevariablep s(reco)minalreadydiscussedintheprevioussubsection.Notsurprisingly,wendthepeakofitsdistribution(redhistograminFigure 2-8 )nearthettthresholdaswell.Allthreeoftheseexamplesshowthatjetscanalsobeusefullyincorporatedintothesubsystem.Theonlyquestioniswhetheronecanndareliablewayofpreferentiallyselectingjetswhicharemorelikelytooriginatefromwithintheintendedsubsystem,asopposedtofromtheoutside.AsweseeinFigure 2-8 ,inthettcasethisisquitepossible,althoughingeneralitmaybedifcultinothersettings,liketheSUSYexamplesdiscussedinthenextsection. 2.5AnExclusiveSUSYExample:MultijetEventsFromGluinoProductionSincep sminisafullyinclusivevariable,arguablyitsbiggestadvantageisthatitcanbeappliedtopurelyjettyeventswithlargejetmultiplicities,wherenoothermethodonthemarketwouldseemtowork.Inordertosimulatesuchachallengingcase,weconsidergluinopairproductioninsupersymmetry,witheachgluinoforcedtoundergoacascadedecaychaininvolvingonlyQCDjetsandnothingelse.Inthissection,twodifferentpossibilitiesforthegluinodecayswereconsidered: Inonescenario,thegluino~gisforcedtoundergoatwo-stagecascadedecaytotheLSP.Intherststage,thegluinodecaystothesecond-lightestneutralino~02andtwoquarkjets:~g!qq~02.Inturn,~02itselfisthenforcedtodecayviaa3-bodydecayto2quarkjetsandtheLSP:~02!qq~01.Theresultinggluinosignatureis4jetsplusmissingenergy: ~g!jj~02!jjjj~01.(2)Therefore,gluinopairproductionwillnominallyresultin8jetevents.Ofcourse,asshowninFigure 2-9 ,theactualnumberofreconstructedjetsinsucheventsisevenhigher,duetotheeffectsofISR,FSRand/orstringfragmentation.AsinFigure 2-9 ,eachsucheventhasonaverage10jets,presentingaformidable 47

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combinatoricsproblem.Wesuspectthatall6massreconstructionmethodsonthemarketaredoomediftheyweretofacesuchascenario.Itisthereforeofparticularinteresttoseehowwellthep sminmethod(whichisadvertizedasuniversallyapplicable)wouldfareundersuchdirecircumstances. Inthesecondscenario,thegluinodecaysdirectlytotheLSPviaathree-bodydecay ~g!jj~01,(2)sothatgluinopair-productioneventswouldnominallyhave4jetsandmissingenergy.Forconcreteness,ineachscenariowexthemassspectrumaswasdonein[ 50 ]:weusetheapproximategauginounicationrelationstorelatethegauginoandneutralinomassesas m~g=3m~02=6m~01.(2)Wecanthenvaryoneofthesemasses,andchoosetheothertwoinaccordwiththeserelations.Sinceweassumethree-bodydecaysinEquations( 2 )and( 2 ),wedonotneedtospecifytheSUSYscalarmassparameters,whichcanbetakentobeverylarge.Inaddition,asimpliedbyEquation( 2 ),weimaginethatthelightesttwoneutralinosaregaugino-like,sothatwedonothavetospecifythehiggsinomassparametereither,anditcanbetakentobeverylargeaswell.Afterthesepreliminaries,ourresultsforthesetwoscenariosareshowninFigures 2-10 and 2-11 ,correspondingly.InFigure 2-10 (Figure 2-11 )weconsiderthe8-jetsignaturearisingfrom( 2 )(the4-jetsignaturearisingfrom( 2 )).Panels(a)correspondtoalightmassspectrumm~g=600GeV,m~02=200GeVandm~01=100GeV;whilepanels(b)correspondtoaheavymassspectrumm~g=2400GeV,m~02=800GeVandm~01=400GeV.Eachplotshowsthesamefourdistributions 6WiththepossibleexceptionoftheMTgenmethodofreference,C.LesterandA.Barr,MTGEN:Massscalemeasurementsinpair-productionatcolliders,(JHEP0712,102(2007)).[ 33 ],seeSection 2.7 below. 48

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asinFigure 2-3 .Thep smindistributionsareallplottedforthecorrectvalueofthemissingmassparameter,namely=M=2m~01.Overall,theresultsseeninFigures 2-10 and 2-11 arenottoodifferentfromwhatwealreadywitnessedinFigure 2-3 forthettexample.The(unobservable)distributionp strueshownwiththedottedyellow-shadedhistogramhasasharpturn-onatthephysicalmassthresholdMp=2m~g.IftheeffectsoftheUEareignored,thepositionofthisthresholdisgivenratherwellbythepeakofthep s(cal)mindistribution(bluehistogram).Unfortunately,theUEshiftsthepeakinp s(cal)minby1-2TeV(redhistogram).Fortunately,thedistributionoftheRECO-levelvariablep s(reco)min(blackhistogram)isstableagainstUEcontamination,anditspeakisstillintherightplace(nearMp).Havingalreadyseenasimilarbehaviorinthettexampleoftheprevioussection,theseresultsmaynotseemveryimpressive,untilonerealizesjusthowcomplicatedthoseeventsare.Forillustration,Figure 2-12 showsthepreviouslydiscussedcalorimetermapsforoneparticularjetevent.Thiseventhappenstohave11reconstructedjets,whichisconsistentwiththetypicaljetmultiplicityseeninFigure 2-9 .Thevaluesofthep squantitiesofinterestforthiseventarelistedinTable 2-1 .WeseethattheRECOprescriptionforcalculatingp sminisabletocompensateforashiftinp sofmorethan1.5TeV!AcasuallookatFigure 2-12 shouldbeenoughtoconvincethereaderjusthowdauntingthetaskofmassreconstructioninsucheventsis.Inthissense,theeasewithwhichthep sminmethodrevealsthegluinomassscaleinFigures 2-10 and 2-11 isquiteimpressive. 49

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Figure2-9. Unit-normalizeddistributionofjetmultiplicityingluinopairproductionevents,witheachgluinodecayingtofourjetsanda~01LSPasin( 2 ). 50

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Figure2-10. Distributionofvariousp sminwithaSUSYexampleofgluinopairproduction,witheachgluinodecayingtofourjetsanda~01LSPasindicatedin( 2 ).Themassspectrumischosenas:(a)m~g=600GeV,m~02=200GeVandm~01=100GeV;or(b)m~g=2400GeV,m~02=800GeVandm~01=400GeV.Allthreep smindistributionsareplottedforthecorrectvalueofthemissingmassparameter,inthiscase=M=2m~01. Figure2-11. Distributionofvariousp sminwithaSUSYexampleofgluinopairproduction,witheachgluinodecayingtotwojetsanda~01LSPasin( 2 ). 51

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Figure2-12. PGScalorimetermapoftheenergydepositforaSUSYeventofgluinopairproduction,witheachgluinoforcedtodecayto4jetsandtheLSPasin( 2 ).TheSUSYmassspectrumisasinFigures 2-10 (a)and 2-11 (a):m~g=600GeV,m~02=200GeVandm~01=100GeV.AsinFigures 2-4 and 2-5 ,thecirclesdenotejetsreconstructedinPGS,andhereqmarksthelocationofaquarkfromagluinodecaychain.Therefore,acirclewithoutaqinsidecorrespondstoajetresultingfromISRorFSR,whilealetterqwithoutanaccompanyingcirclerepresentsaquarkinthegluinodecaychainwhichwasnotsubsequentlyreconstructedasajet. 52

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Figure2-13. Distributionofthep s(cal)min(dottedred)andp s(reco)min(solidblack)variablesininclusiveSUSYproductionfortheGMSBGM1abenchmarkstudypointwithparameters=80TeV,Mmes=160TeV,Nmes=1,tan=15and>0.Thedottedyellow-shadedhistogramisthetruep sdistributionoftheparentpairofSUSYparticlesproducedatthetopofeachdecaychain(theidentityoftheparentparticlesvariesfromeventtoevent).Thep smindistributionsareshownfor=M=0andarenormalizedto1fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofdata.TheverticalarrowsmarkthemassthresholdsforafewdominantSUSYpair-productionprocesses. Table2-2. Masses(inGeV)oftheSUSYparticlesattheGM1bstudypoint.Here~uand~d(~`and~`)standforeitherofthersttwogenerationssquarks(sleptons). ~uL~dL~uR~dR~`L~`~`R~2~04~03~g908911872870289278145371371348690 ~t1~b1~t2~b2~2~~1~1~02~01~G8068638958782902771382062061060 2.6AnInclusiveSUSYExample:GMSBStudyPointGM1bIntheIntroductionwealreadymentionedthatp sminisafullyinclusivevariable.Herewewouldliketopointoutthattherearetwodifferentaspectsoftheinclusivitypropertyofp smin: Object-wiseinclusivity:p sminisinclusivewithregardstothetypeofreconstructedobjects.Thedenitionofp s(reco)mindoesnotdistinguishbetweenthedifferenttypes 53

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ofreconstructedobjects(andp s(cal)minmakesnoreferencetoanyreconstructedobjectsatall).Thismakesp sminaveryconvenientvariabletouseinthosecaseswherethenewlyproducedparticleshavemanypossibledecaymodes,andrestrictingoneselftoasingleexclusivesignaturewouldcauselossinstatistics.Forillustration,considerthegluinopairproductionexamplefromtheprevioussection.Eventhoughwearealwaysproducingthesametypeofparentparticles(twogluinos),ingeneraltheycanhaveseveraldifferentdecaymodes,leadingtoaverydiversesampleofeventswithvaryingnumberofjetsandleptons.Nevertheless,thep s(reco)mindistribution,plottedoverthiswholesignalsample,willstillbeabletopinpointthegluinomassscale,asexplainedinSection 2.5 Event-wiseinclusivity:p sminisinclusivealsowithregardstothetypeofevents,i.e.thetypeofnewparticleproduction.Forsimplicity,inourpreviousexampleswehavebeenconsideringonlyoneproductionmechanismatatime,butthisisnotreallynecessaryp smincanalsobeappliedinthecaseofseveralsimultaneousproductionmechanisms.Inordertoillustratethelastpoint,inthissectionweshallconsiderthesimultaneousproductionofthefullspectrumofSUSYparticlesataparticularbenchmarkpoint.WechosetheGM1bCMSstudypoint[ 59 ],whichisnothingbutaminimalgauge-mediatedSUSY-breaking(GMSB)scenarioontheSPS8Snowmassslope[ 81 ].Theinputparametersare=80TeV,Mmes=160TeV,Nmes=1,tan=15and>0.ThephysicalmassspectrumisgiveninTable 2-2 .PointGM1bischaracterizedbyaneutralinoNLSP,whichpromptlydecays(predominantly)toaphotonandagravitino.Therefore,atypicaleventhastwohardphotonsandmissingenergy,whichprovidegoodhandlesforsuppressingtheSMbackgrounds.WenowconsiderinclusiveproductionofallSUSYsubprocessesandplotthep smindistributionsofinterestinFigure 2-13 .Asusual,thedottedyellow-shadedhistogramisthetruep sdistributionoftheparentpairofSUSYparticlesproducedatthetopofeachdecaychain.Sincewedonotxtheproductionsubprocess,theidentityoftheparentparticlesvariesfromeventtoevent.Naturally,themostcommonparentparticlesaretheoneswiththehighestproductioncross-sections.ForpointGM1b,ata14TeVLHC,strongSUSYproductiondominates,andis87%ofthetotalcross-section.Afewofthedominantsubprocessesandtheircross-sectionsarelistedinTable 2-3 54

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Table2-3. Cross-sections(inpb)andparentmassthresholds(inGeV)forthedominantproductionprocessesattheGM1bstudypoint.Thelistedsquarkcross-sectionsaresummedoverthelightsquarkavorsandconjugatestates.ThetotalSUSYcross-sectionatpointGM1bis9.4pb. Process~1~02~+1~)]TJ /F5 7.97 Tf 0 -7.97 Td[(1~g~g~g~qR~g~qL~qR~qR~qL~qR~qL~qL (pb)0.830.432.032.171.900.360.500.28Mp(GeV)412412138015601600174017801820 Thetruep sdistributioninFigure 2-13 exhibitsaninterestingdouble-peakstructure,whichiseasytounderstandasfollows.AswehaveseenintheexclusiveexamplesfromSections 2.4 and 2.5 ,athadroncolliderstheparticlestendtobeproducedwithp sclosetotheirmassthreshold.AsseeninTable 2-2 ,theparticlespectrumoftheGM1bpointcanbebroadlydivided(accordingtomass)intotwogroupsofsuperpartners:electroweaksector(thelightestchargino~1,second-to-lightestneutralino~02andsleptons)withamassscaleontheorderof200GeVandastrongsector(squarksandgluino)withmassesoforder700)]TJ /F6 11.955 Tf 12.52 0 Td[(900GeV.Therstpeakinthetruep sdistribution(nearp s500GeV)arisesfromthepairproductionoftwoparticlesfromtheelectroweaksector,whilethesecond,broaderpeakintherangeofp s1500)]TJ /F6 11.955 Tf 12.36 0 Td[(2300GeVisduetothepairproductionoftwocoloredsuperpartners7.Eachoneofthosepeaksismadeupofseveralcontributionsfromdifferentindividualsubprocesses,butbecausetheirmassthresholds8aresoclose,theycannotbeindividuallyresolved,andappearasasinglebump.Ifonecouldsomehowdirectlyobservethetruep sSUSYdistribution(thedottedyellow-shadedhistograminFigure 2-13 ),thiswouldleadtosomeveryinterestingconclusions.First,fromthepresenceoftwoseparatepeaksonewouldknowimmediately 7Theattentivereadermayalsonoticetwobarelyvisiblebumps(near950GeVand1150GeV)reectingtheassociatedproductionofonecoloredandoneuncoloredparticle:~g~1,~g~02and~q~1,~q~02,correspondingly.8AfewindividualmassthresholdsareindicatedbyverticalarrowsinFigure 2-13 55

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thattherearetwowidelyseparatedscalesintheproblem.Second,thenormalizationofeachpeakwouldindicatetherelativesizeofthetotalinclusivecross-sections(inthisexample,oftheparticlesintheelectroweaksectorversusthoseinthestrongsector).Finally,thebroadnessofeachpeakisindicativeofthetotalnumberofcontributingsubprocesses,aswellasthetypicalmasssplittingsoftheparticleswithineachsector.Itmayappearsurprisingthatoneisabletodrawsomanyconclusionsfromasingledistributionofaninclusivevariable,butthisjustcomestoshowtheimportanceofp sasoneofthefundamentalcolliderphysicsvariables.Unfortunately,becauseofthemissingenergyduetotheescapinginvisibleparticles,thetruep sdistributioncannotbeobserved,andthebestonecandotoapproximateitistolookatthedistributionsofourinclusivep sminvariablesdiscussedinSection 2.2 :thecalorimeter-basedp s(cal)minvariable(dottedredhistograminFigure 2-13 )andtheRECO-levelp s(reco)minvariable(solidblackhistograminFigure 2-13 ).Firstletusconcentrateonthecalorimeter-basedversionp s(cal)min(dottedredhistogram).WecanimmediatelyseethedetrimentaleffectsoftheUE:rst,theelectroweakproductionpeakhasbeenalmostcompletelysmearedout,whilethestrongproductionpeakhasbeenshiftedupwardsbymorethanaTeV!Thisbehaviorisnottoosurprising,sincethesameeffectwasalreadyencounteredinourpreviousexamplesinSections 2.4 and 2.5 .Fortunately,wenowalsoknowthesolutiontothisproblem:oneneedstoconsidertheRECO-levelvariablep s(reco)mininstead,whichtracksthetruep sdistributionmuchbetter.WecanseeevidenceofthisinFigure 2-13 aswell.Inparticular,p s(reco)mindoesshowtwoseparatepeaks(indicatingthatSUSYproductiontakesplaceattwodifferentmassscales),thepeaksareintheirproperlocations(relativetothemissingmassscale=M),andhavethecorrectrelativewidth,hintingatthesizeofthemasssplittingsineachsector.Wethusconcludethatalloftheinterestingphysicsconclusionsthatonewouldbeabletoreachfromlookingatthetruep sdistributions,canstillbemadebasedontheinclusivedistributionofourRECO-levelp s(reco)minvariable. 56

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Figure2-14. Distributionsofvariousp sminfortheGMSBSUSYexampleconsideredinFigure 2-13 .Herethesubsystemisdenedintermsofthetwohardphotonsresultingfromthetwo~01!~G+decays.Theverticalarrowmarkstheonsetforinclusive~01~01production. Beforeconcludingthissection,weshalltaketheopportunitytousetheGM1bexampletoalsoillustratethep s(sub)minvariableproposedinSection 2.3 .Asalreadymentioned,theGM1bstudypointcorrespondstoaGMSBscenariowithapromptlydecayingBino-like~01NLSP.Mosteventsthereforecontaintwohardphotonsfromthetwo~01decaystogravitinos.ThenitisquitenaturaltodenetheexclusivesubsysteminFigure 2-2 intermsofthesetwophotons.Thecorrespondingp s(sub)mindistributionisshowninFigure 2-14 withtheblacksolidhistogram.Forcompleteness,wealsoshowthetruep sdistributionofthe~01pair(dottedyellow-shadedhistogram).Theverticalarrowmarksthelocationofthe~01~01massthreshold.Wenoticethatthepeakofthep s(sub)mindistributionnicelyrevealsthelocationoftheneutralinomassthreshold,andfromtheretheneutralinomassitself.Weseethatthemethodofp s(sub)minprovidesaverysimplewayofmeasuringtheNLSPmassinsuchGMSBscenarios(foranalternativeapproachbasedonMT2,see[ 61 ]). 57

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2.7ComparisontoOtherInclusiveColliderVariablesHavingdiscussedthenewlyproposedvariablesp s(reco)minandp s(sub)mininvarioussettingsinSections 2.4 2.6 ,weshallnowcomparethemtosomeotherglobalinclusivevariableswhichhavebeendiscussedintheliteratureinrelationtodeterminingamassscaleofthenewphysics.Forsimplicityhereweshallconcentrateonlyonthemostmodel-independentvariables,whichdonotsufferfromthetopologicalandcombinatorialambiguitiesmentionedintheIntroduction.Atthemoment,thereareonlyahandfulofsuchvariables.Dependingonthetreatmentoftheunknownmassesoftheinvisibleparticles,theycanbeclassiedintooneofthefollowingtwocategories: Variableswhichdonotdependonanunknowninvisiblemassparameter.ThemostpopularmembersofthisclassarethemissingHTvariable =HT)]TJ /F4 7.97 Tf 12.28 16.29 Td[(NobjXi=1~PTi,(2)whichissimplythemagnitudeofthe~=HTvectorfromEquation( 2 ),andthescalarHTvariable HT=HT+NobjXi=1PTi.(2)HerewefollowthenotationfromSection 2.2 ,where~PTiisthemeasuredtransversemomentumofthei-threconstructedobjectintheevent(i=1,2,...,Nobj).Themainadvantageof=HTandHTistheirsimplicity:bothareverygeneral,andaredenedpurelyintermsofobservedquantities,withoutanyunknownmassparameters.Thedownsideof=HTandHTisthattheycannotbedirectlycorrelatedwithanyphysicalmassscaleinamodel-independentway9. 9SomeearlystudiesofHT-likevariablesfoundinterestinglinearcorrelationsbetweenthepeakintheHTdistributionandasuitablydenedSUSYmassscaleinthecontextofspecicSUSYmodels,e.g.minimalsupergravity(MSUGRA)[ 13 63 64 ],minimalGMSB[ 63 ],ormixedmoduli-mediation[ 65 ].However,anysuchcorrelationsdonotsurvivefurtherscrutinyinmoregenericSUSYscenarios,seee.g.[ 66 ]. 58

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Variableswhichexhibitdependenceononeormoreinvisiblemassparameters.AstworepresentativesfromthisclassweshallconsiderMTgenfromReference[ 33 ]andp s(reco)minfromSection 2.2 here.WeshallnotrepeatthetechnicaldenitionofMTgen,andinsteadrefertheuninitiatedreadertotheoriginalpaper[ 33 ].SufceittosaythatthemethodofMTgenstartsoutbyassumingexactlytwodecaychainsineachevent.Thearisingcombinatorialproblemisthensolvedbybruteforcebyconsideringallpossiblepartitionsoftheeventintotwosides,computingMT2foreachsuchpartition,andtakingtheminimumvalueofMT2foundintheprocess.BothMTgenandp s(reco)minintroduceaprioriunknownparametersrelatedtothemassscaleofthemissingparticlesproducedintheevent.Inthecaseofp s(reco)min,thisissimplythesingleparameter=M,measuringthetotalinvisiblemass(inthesenseofascalarsumasdenedinEquation( 2 )).TheMTgenvariable,ontheotherhand,mustinprincipleintroducetwoseparatemissingmassparameters=M1and=M2(oneforeachsideoftheevent).However,theexistingapplicationsofMTgenintheliteraturehavetypicallymadetheassumptionthat=M1==M2,althoughthisisnotreallynecessaryandonecouldjustaseasilyworkintermsoftwoseparateinputs=M1and=M2[ 40 41 ].TheinconvenienceofhavingtodealwithunknownmassparametersinthecaseofMTgenandp s(reco)minisgreatlycompensatedbytheluxuryofbeingabletorelatecertainfeaturesoftheirdistributionstoafundamentalphysicalmassscaleinarobust,model-independentway.Inparticular,theupperendpointM(max)TgenoftheMTgendistributiongivesthelargerofthetwoparentmassesmaxfMP1,MP2g[ 62 ].Therefore,ifthetwoparentmassesarethesame,i.e.MP1=MP2,thentheparentmassthresholdMp=MP1+MP2issimplygivenby Mp=2M(max)Tgen.(2)Ontheotherhand,aswehavealreadyseeninSections 2.4 2.6 ,thepeakofthep s(reco)minissimilarlycorrelatedwiththeparentmassthreshold,seeEquation( 2 ).Inprinciple,allfour10ofthesevariablesareinclusivebothobject-wiseandevent-wise.Itisthereforeofinteresttocomparethemwithrespectto: 1. ThedegreeofcorrelationwiththenewphysicsmassscaleMp. 2. StabilityofthiscorrelationagainstthedetrimentaleffectsoftheUE.Figures 2-15 2-16 and 2-17 allowforsuchcomparisons. 10WecautionthereaderthatHTisoftendenedinamorenarrowsensethanEquation( 2 ).Forexample,sometimesthe=HTtermisomitted,sometimesthesuminEquation( 2 )islimitedtothereconstructedjetsonly;ortothefourhighestpTjetsonly;ortoalljets,butstartingfromthesecond-highestpTone. 59

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Figure2-15. Comparisonvariousp sminwithothertransversevariablesforttproduction.Inadditiontothetruep s(yellowshaded)andp s(reco)min(black)distribution,wealsoplotthedistributionsof2MTgen(reddots),2MTTgen(magentadots),HT(greendots)and=HT(bluedots),allcalculatedattheRECO-level.Allresultsincludethefullsimulationoftheunderlyingevent.Forplottingconvenience,the=HTdistributionisshownscaleddownbyafactorof2.TheverticaldottedlinemarksthettmassthresholdMp=2mt=350GeV. InFigure 2-15 werstrevisitthecaseofthedileptonttsamplediscussedinSection 2.4 .Inadditiontothetruep s(yellowshaded)andp s(reco)min(black)distributionalreadyappearinginFigure 2-3 ,wenowalsoplotthedistributionsof2MTgen(reddots),HT(greendots)and=HT(bluedots),allcalculatedattheRECO-level.Forcompleteness,inFigure 2-15 wealsoshowavariantofMTgen,calledMTTgen(magentadots),whereallvisibleparticlemomentaarerstprojectedonthetransverseplane,beforecomputingMTgenintheusualway[ 33 ].Allresultsincludethefullsimulationoftheunderlyingevent.Forplottingconvenience,the=HTdistributionisshownscaleddownbyafactorof2.BasedontheresultsfromFigure 2-15 ,wecannowaddressthequestion,whichinclusivedistributionshowsthebestcorrelationwiththeparentmassscale(inthiscasetheparentmassscaleisthettmassthresholdMp=2mt=350GeVmarked 60

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bytheverticaldottedlineinFigure 2-15 ).Letusbeginwiththetwovariables,=HTandHT,whichdonotdependonanyunknownmassparameters.Figure 2-15 revealsthatthe=HTdistributionpeaksveryfarfromthreshold,andthereforedoesnotrevealmuchinformationaboutthenewphysicsmassscale.Consequently,anyattemptatextractingnewphysicsparametersoutofthemissingenergydistributionalone,mustmakesomeadditionalmodel-dependentassumptions[ 2 ].Ontheotherhand,theHTdistributionappearstocorrelatebetterwithMp,sinceitspeakisrelativelyclosetothettthreshold.However,thisrelationshipispurelyempirical,anditisdifculttoknowwhatistheassociatedsystematicerror.Movingontothevariableswhichcarryadependenceonamissingmassparameter,p s(reco)min,2MTgenand2MTTgen,weseethatallthreeareaffectedtosomeextentbythepresenceoftheUE.Inparticular,thedistributionsof2MTgenand2MTTgenarenowsmearedandextendsignicantlybeyondtheirexpectedendpoint( 2 ).Notsurprisingly,theUEhasalargerimpacton2MTgenthanon2MTTgen.Ineithercase,thereisnoobviousendpoint.Nevertheless,onecouldinprincipletrytoextractanendpointthroughastraight-linet,forexample,butitisclearthattheobtainedvaluewillbewrongbyacertainamount(dependingonthechosenregionforttingandontheassociatedbackgrounds).Allthesedifcultieswith2MTgenand2MTTgenaresimplyareectionofthechallengeofmeasuringamassscalefromanendpointasin( 2 ),insteadoffromapeakasin( 2 ).Bycomparison,thedeterminationofthenewphysicsmassscalefromthep s(reco)mindistributionismuchmorerobust.AsshowninFigure 2-15 ,thep s(reco)minpeakisbarelyaffectedbytheUE,andisstillfoundpreciselyintherightlocation.AlloftheabovediscussioncanbedirectlyappliedtotheSUSYexamplesconsideredinSection 2.5 aswell.Asanillustration,Figures 2-16 and 2-17 revisittwoofthegluinoexamplesfromSection 2.5 .Weconsidergluinopair-productionwithalightSUSYspectrum(m~01=100GeV,m~02=200GeVandm~g=600GeV).TheninFigure 2-16 eachgluinodecaysto4jetsasinEquation( 2 ),whileinFigure 2-17 61

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eachgluinodecaysto2jetsasinEquation( 2 ).(ThusFigure 2-16 istheanalogueofFigure 2-10 (a),whileFigure 2-17 istheanalogueofFigure 2-11 (a).)TheconclusionsfromFigures 2-16 and 2-17 areverysimilar.Theseresultsconrmthat=HTisnotveryhelpfulindeterminingthegluinomassscaleMp=2m~g=1200GeV(indicatedbytheverticaldottedline).TheHTdistribution,ontheotherhand,hasanicewell-denedpeak,butthelocationoftheHTpeakalwaysunderestimatesthegluinomassscale(byabout250GeVineachcase).Figures 2-16 and 2-17 alsoconrmtheeffectalreadyseeninFigure 2-15 :thattheunderlyingeventcausesthe2MTgenand2MTTgendistributionstoextendwellbeyondtheirupperkinematicendpoint,thusviolating( 2 )andmakingthecorrespondingextractionofMpratherproblematic.Infact,justbylookingatFigures 2-16 and 2-17 ,onemightbetemptedtodeducethat,ifanything,itisthepeakin2MTgenthatperhapsmightindicatethevalueofthenewphysicsmassscaleandnotthe2MTgenendpoint.Finally,thep s(reco)mindistributionalsofeelstosomeextenttheeffectsfromtheUE,butalwayshasitspeakinthenearvicinityofMp.Therefore,amongtheveinclusivevariablesunderconsiderationhere,p s(reco)minappearstoprovidethebestestimateofthenewphysicsmassscale.ThecorrelationofEquation( 2 )isseentoholdverywellinFigure 2-17 andreasonablywellinFigure 2-16 62

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Figure2-16. Comparisonvariousp sminwithothertransversevariablesforthegluinopairproductionexamplefromSection 2.5 ,witheachgluinodecayingto4jetsasin( 2 ).WeusethelightSUSYmassspectrumfromFigure 2-10 (a).Theverticaldottedlinenowshowsthe~g~gmassthresholdMp=2m~g=1200GeV. Figure2-17. Comparisonvariousp sminwithothertransversevariablesforgluinopairproductionwitheachgluinodecayingto2jetsasin( 2 ).ComparetoFigure 2-11 (a). 63

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CHAPTER3INVARIANTMASSENDPOINTSMETHODInthischapterweconcentrateontheclassicmethodofkinematicalendpoints[ 13 ].Wesetouttoredesignthisstandardalgorithmforperformingthesestudies,bypursuingtwomainobjectives: ImprovingontheexperimentalprecisionoftheSUSYmassdetermination.Forexample,werequiredthatouranalysisbebasedexclusivelyonupperinvariantmassendpoints,whichareexpectedtobemeasuredwithagreaterprecisionthanthecorrespondinglowerendpoints(a.k.a.thresholds).Consequently,wedidnotmakeuseofthethresholdmeasurementmminj``(> 2),whichhasbeenanintegralpartofmostSUSYstudiessinceReference[ 16 ].Inthesamevein,wealsodemandedthatweshouldnotrelyonanyfeaturesobservedinatwo-orathree-dimensionalinvariantmassdistributionsuchmeasurementsareexpectedtobelessprecisethanthe(upper)endpointsextractedfromsimpleone-dimensionalhistograms. Avoidinganyparameterspaceregionambiguities.Itiswellknownthatsomeoftheinvariantmassendpointsusedintheconventionalanalysesarepiecewise-denedfunctions.ThisfeaturemaysometimesleadtomultiplesolutionsfortheSUSYmassspectrumintheLHCinverseproblem[ 9 17 67 69 ].Inordertosafeguardagainstthispossibility,weconservativelydemandedfromtheoutsetthatnoneofourendpointmeasurementsbegivenbypiecewisedenedfunctions.Thisratherstrictrequirementrulesoutthreeofthestandardendpointmeasurementsmmaxj``,mmaxj`(lo),andmmaxj`(hi).Inordertomeettheseobjectives,inSection 3.2 weproposedasetofnewinvariantmassvariableswhoseupperkinematicendpointscanbealternativelyusedforSUSYmassreconstructionstudies.TheninSection 3.3 weoutlinedasimpleanalysiswhichwasbasedontheparticularsetoffourinvariantmassvariables( 3 ),allofwhichsatisfyourrequirements.InSection 3.3.1 weprovidedsimpleanalyticalformulasfortheSUSYmassspectrumintermsofthefourmeasuredendpointsinEquation( 3 ).Oursolutionsrevealedasurprise:inspiteofthetwo-foldambiguityasinEquations( 3 3 )intheinterpretationoftwoofourendpointsMmaxj`(u)andmmaxj`(u),theanswerforthree(mD,mCandmA)outofthefourSUSYmassesisunique! 64

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Figure3-1. Thetypicalcascadedecaychainunderconsiderationinthischapter.HereD,C,BandAarenewBSMparticles,whilethecorrespondingSMdecayproductsare:aQCDjetj,anearlepton`nandafarlepton`f.ThischainisquitecommoninSUSY,withtheidenticationD=~q,C=~02,B=~`andA=~01,where~qisasquark,~`isaslepton,and~01(~02)istherst(second)lightestneutralino.InwhatfollowsweshallquoteourresultsintermsoftheDmassmDandthethreedimensionlesssquaredmassratiosRCD,RBCandRABdenedinEquation( 3 ). Thefourthmass(mB)isalsoknown,uptothetwo-foldambiguityasinEquation( 3 ),whichcanbeeasilyresolvedbyavarietyofmethodsdiscussedandillustratedinSections 3.3.2 and 3.4.2 .InSection 3.4 weappliedourtechniquetotwospecicexamplestheLM1andLM6CMSstudypoints.FollowingthepreviousSUSYstudies,forillustrationofourresultsweshallusethegenericdecaychainD!jC!j`nB!j`n`fAshowninFigure 3-1 .HereD,C,BandAarenewBSMparticleswithmassesmD,mC,mBandmA.TheircorrespondingSMdecayproductsare:aQCDjetj,anearlepton`nandafarlepton`f.ThisdecaychainisquitecommoninSUSY,withtheidenticationD=~q,C=~02,B=~`andA=~01,where~qisasquark,~`isaslepton,and~01(~02)istherst(second)lightestneutralino.However,ouranalysisisnotlimitedtoSUSYonly,sincethechaininFigure 3-1 alsoappearsinotherBSMscenarios,e.g.UniversalExtraDimensions[ 71 ].Forconcreteness,weshallassumethatallthreedecaysexhibitedinFigure 3-1 aretwo-body,i.e.weshallconsiderthemasshierarchy mD>mC>mB>mA>0.(3) 65

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Thispresentsthemostchallengingcase,inwhichonehastodetermineallfourmassesmD,mC,mBandmA.Theideaofthekinematicendpointmethodisverysimple.GiventheSMdecayproductsj,`nand`fexhibitedinFigure 3-1 ,formtheinvariantmass1ofeverypossiblecombination,m``,mj`n,mj`f,andmj``,plottheresultingdistributionsandmeasurethecorrespondingupperkinematicendpoints[ 13 16 17 ] (mmax``)2=m2DRCD(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RBC)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB); (3) )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`n2=m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RBC); (3) )]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`f2=m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB); (3) )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj``2=8>>>>>>>>>>>><>>>>>>>>>>>>:m2D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAC),forRCD
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NotethatthereareonlythreeindependentratiosinEquation( 3 ).WeshalltakethosetobeRAB,RBC,andRCD(seeFigure 3-1 ),andtheirdenitiondomainwillbetheinterval(0,1).2 3.1ThreeGenericProblemsinInvariant-massEndpointMethodsInspiteoftheirtransparenttheoreticalmeaning,thesetoffourendpointsinEquations( 3 through 3 )bythemselveshave(justiably)neverbeenusedasthesolebasisforaSUSYmassdeterminationanalysis.Thisisduetothreegenericproblems,whichareallverywellknown,andareseparatelyreviewedinthenextthreesubsections 3.1.1 3.1.2 and 3.1.3 .Ournewapproachtoresolvingthesethreeproblems,andtheoutlineoftherestofthepaperarepresentedinSection 3.1.4 3.1.1Near-farLeptonAmbiguityTherstproblemisthatonecannotdifferentiatebetweenthenearandfarleptons`nand`fonanevent-by-eventbasis.SincealldecaysinFigure 3-1 areprompt,bothleptonspointbacktotheprimaryinteractionvertexandthereisnowaytotellwhichcamerstandwhichcamesecond.Consequently,onecannotseparatelyconstructtheindividualmj`nandmj`finvariantmassdistributions,whoseupperendpointswouldbegivenbyEquations( 3 )and( 3 ).Thisproblemhasmotivatedmostofthepreviousinvariantmassstudiesintheliterature,beginningwith[ 16 ],tointroduceanalternativedenitionofthetwoj`distributions,simplybyorderingthetwomj`entriesineacheventbyinvariantmassasfollows mj`(lo)minfmj`n,mj`fg, (3) mj`(hi)maxfmj`n,mj`fg. (3) 2AsseeninEquation( 3 ),attimesweshallalsoutilizeoneormoreoftheotherthreeratios,RAC,RADandRBD,wheneverthiswillleadtoasimplicationoftheformulas.Ofcourse,thelatterthreeratiosarerelatedtoourpreferredsetfRAB,RBC,RCDgduetothetransitivitypropertyRijRjk=Rik. 67

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Bothofthenewlydenedquantitiesmj`(lo)andmj`(hi)alsoexhibitupperkinematicendpoints(mmaxj`(lo)andmmaxj`(hi),correspondingly).Sincetheindividualmj`(lo)andmj`(hi)distributionsareobservable,theirendpointsareexperimentallymeasurableandcanberelatedtotheunderlyingSUSYmassspectrumasfollows[ 16 17 ] )]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`(lo)2=8>>>>>>><>>>>>>>:)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`n2,for(2)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB))]TJ /F5 7.97 Tf 6.59 0 Td[(1>>>>>><>>>>>>>:)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`f2,for(2)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB))]TJ /F5 7.97 Tf 6.59 0 Td[(1
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mj`fshouldbeinvariantunderthesymmetry mj`n$mj`f.(3)Theadvantagesofourapproachmaynotbeimmediatelyobviousatthispoint,butwillbecomeclearintheprocessofourmassdeterminationanalysisinSection 3.3 below. 3.1.2InsufcientNumberofMeasurements.ThesecondproblemassociatedwiththeoriginalsetoffourmeasurementsinEquations( 3 3 ),aswellasthealternativesetinEquation( 3 ),isthatthemeasuredendpointsmaynotallbeindependentfromeachother.Indeed,therearecertainregionsofparameterspacewhereonendsthefollowingcorrelation[ 17 ] )]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj``2=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`(hi)2+(mmax``)2.(3)Inthiscase,thefourmeasurementsinEquation( 3 )areclearlyinsufcienttopindownallfourindependentinputparametersmD,mC,mBandmA.Therefore,onehastomeasureanadditionalindependentendpoint.Tothisend,ithasbeensuggestedtoconsidertheconstraineddistributionmj``(> 2),whichexhibitsausefullowerkinematicendpointmminj``(> 2)[ 16 ] mminj``(> 2)2=1 4m2D((1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC)(1+RCD) (3) +2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAC)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD))]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)p (1+RAB)2(1+RBC)2)]TJ /F6 11.955 Tf 11.96 0 Td[(16RAC).Thedistributionmj``(> 2)isnothingbuttheusualmj``distributionoverasubsetoftheoriginalevents,subjecttotheadditionaldileptonmassconstraint mmax`` p 2
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IntherestframeofparticleB,thiscutimpliesthefollowingrestrictionontheopeninganglebetweenthetwoleptons[ 70 ] > 2,(3)thusjustifyingthenotationformj``(> 2).Theadvantageofthethresholdendpointmeasurement( 3 )isthatitisalwaysindependentoftheotherfourmeasurementsin( 3 ).Asaresult,itwouldappearthattheenlargedsetofvekinematicendpointmeasurements mmax``,mmaxj``,mmaxj`(lo),mmaxj`(hi),mminj``(> 2)(3)shouldbeinprinciplesufcienttodetermineallfourunknownmasses(see,howeverSection 3.1.3 below).Unfortunately,thethreshold( 3 )alsosuffersfromcertaindisadvantages,whicharemostlyofexperimentalnature.Itisgenerallyexpectedthattheexperimentalprecisiononthedeterminationofthelowerkinematicendpoint( 3 )willberatherinferiorcomparedtotheprecisionontheotherfourupperkinematicendpoints( 3 )[ 17 ].Thereareseveralgenericreasonsforsuchapessimisticattitude.First,theregioninthemj``(> 2)distributionnearitslowerendpoint( 3 )israthersparselypopulated,resultinginashallowedgeandsizablestatisticalerrors.Tomakemattersworse,themj``(> 2)distributionnearitsloweredgeisaconvexfunction[ 72 ],whichmakesitevenmoredifculttotellwherethesignalendsandthetailsfromvarioussourcesbegin[ 17 ].Finally,thelowmassregionofalmostanyinvariantmassdistributioninSUSYisgenerallyassociatedwithlargerSM(aswellasSUSYcombinatorial)backgroundscomparedtoitshighmasscounterpart.OverallwendallthesedisadvantagessufcientlyconvincingsothatwewilldropthemeasurementinEquation( 3 )altogetherandwillneveruseitinthecourseofouranalysisinSection 3.3 below.Wewillbejustiedindoingso,sincethelinear 70

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dependenceproblem( 3 ),whichhasplaguedpreviousstudiesandwastheprimemotivationforintroducingthemminj``(> 2)measurementintherstplace,willhavenoeffectonouranalysis.Infact,wewillnotbeusingtheendpointmeasurementmmaxj`(hi)(forthereasonsgivenintheprevioussubsection 3.1.1 )andwewillnotbeusingtheendpointmeasurementmmaxj``(forthereasonsgiveninthefollowingsubsection 3.1.3 ).Oncethesetwoproblematicmeasurementsareremovedfromconsideration,thelineardependenceproblem( 3 )doesnotarise,andthethresholdmeasurement( 3 )isnotcentraltotheanalysisanymore. 3.1.3ParameterSpaceRegionAmbiguityThethirdproblemwiththeconventionalsetofmeasurements( 3 )isimmediatelyobviousfromthedeningEquations( 3 ),( 3 )and( 3 )forthekinematicendpointsmmaxj``,mmaxj`(lo),andmmaxj`(hi),correspondingly.Onecanseethattherelevantexpressionsarepiecewise-denedfunctions,i.e.theydependonthevaluesoftheindependentvariablesmA,mB,mCandmD.Forexample,therearefourdifferentcasesformmaxj``,andthreedifferentcasesforthepairof(mmaxj`(lo),mmaxj`(hi)).Altogether,thesegiveriseto9differentcases3whichmustbeseparatelyconsidered[ 9 17 ].Ofcourse,thisrepresentsaproblem,sincethemassesareaprioriunknown,anditisnotclearwhichcaseistherelevantone.Barringanymodel-dependentassumptions,oneisforcedtoconsiderallpossibilities,obtainasolutionforthespectrum,andonlyattheveryend,testwhetherthesolutionfallswithintheparameterspaceapplicableforthecaseathand.Thisproceduremayoftenresultinseveralalternativesolutions[ 9 17 67 69 73 74 ].Infact,werecentlyprovedthatthereexistsasizableparameterspaceregioninwhicheventhefullsetofmeasurements( 3 )wouldalwaysyieldtwoalternativesolutions,evenunderidealexperimentalconditions[ 9 ].Theproblemisfurtherexacerbatedbytheinevitable 3Theremaining3casesarealwaysunphysical[ 17 ]. 71

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experimentalerrorsonthemeasurements( 3 ),whichwouldallowforanevenlargernumberoffakeorduplicatesolutions[ 9 68 69 ].Havingidentiedtherootoftheduplicationproblemasthepiecewisedenitionofthemathematicalformulasin( 3 3 3 ),oursolutiontotheproblemwillbeagainverysimpleandconservative.Wewillsimplyavoidusinganykinematicendpointswhicharegivenintermsofpiecewise-denedexpressions.Thisrequirementautomaticallyeliminatesfromconsiderationthethreeconventionalendpointsmmaxj``,mmaxj`(lo),andmmaxj`(hi).Sincewealreadygaveuponmminj``(> 2)intheprevioussubsection,thisleavesmmax``astheonlymeasurementoutoftheconventionalset( 3 )thatweshalluseinouranalysis.Thisisperhapsthemostdrasticdifferencebetweenourapproachandallpreviousstudiesintheliterature. 3.1.4PosingTheProblemInthepreviousthreesubsectionswediscussedeachofthethreegenerictheoretical4problemswiththepreviousapplicationsofthekinematicendpointmethodformassdetermination.Wearenowreadytoexplicitlyformulateourmaingoalinthispaper.WeaimtodesignamethodformeasuringthemassesoftheparticlesinthedecaychainofFigure 3-1 ,whichisbasedonkinematicendpointinformation,andsatisesthefollowingrequirements: 4Inaddition,thereareproblemswhichareofexperimentalnature,e.g.identifyingthecorrectjetandthecorrectleptonpairresultingfromthedecaychaininFigure 3-1 .Thereexistsasetofstandardexperimentaltechniqueswhichareaimedatovercomingtheseproblems,e.g.theoppositeavorsubtractionforthetwoleptonsandthemixedeventsubtractionforthejet[ 75 ].Wrong``andj`pairingscanalsobeidentiedandaposterioriremovedwheneveraninvariantmassentryform``,mj`ormj``exceedsthecorrespondingkinematicendpointmmax``,mmaxj`(hi)ormmaxj``.Inwhatfollowsweshallassumethatthosepreliminarystepshavealreadybeendoneandthesampleswearedealingwithhavealreadybeenappropriatelysubtractedtoremovethecombinatorialbackground. 72

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Itdoesnotmakeuseofanykinematicendpointswhoseinterpretationisambiguous,i.e.whoseexpressionsintermsofthephysicalmassesarepiecewise-denedfunctions. Itdoesnotmakeuseofanylowerkinematicendpointssuchasthethresholdmminj``(> 2),duetotheexperimentalchallengeswithsuchmeasurements. Itreliessolelyon1-dimensionaldistributions,unlikethemethodsrecentlyadvertisedin[ 9 21 73 74 76 ],whichutilize2-dimensionalcorrelationplots.Whilethelatterdoprovideawealthofvaluableinformation,theyalsotypicallyrequiremoredatainordertoobtaingoodenoughstatisticsfordrawinganyrobustconclusionsfromthem.Incontrast,theone-dimensionaldistributionsshouldbeavailableratherearlyon,andwithsufcientstatisticsforendpointmeasurements.Asalreadyalludedtointheprevioussubsections,thersttworequirementsalreadyeliminatefouroutoftheveconventionalinputs( 3 ).Obviously,wewillneedtondawaytoreplacethosewithanalternativesetofkinematicendpointmeasurementswhichneverthelesssatisfytheaboverequirements.InSection 3.2 weintroduceandinvestigateanewsetofinvariantmassvariableswhoseupperendpointscanbeusefulforouranalysis.TheninSection 3.3 weoutlineourbasicmethod,whichmakesuseofsomeofthesenewvariables.WeillustrateourdiscussioninSection 3.4 withtwonumericalexamples:theLM1andLM6CMSstudypoints.InAppendix A wesupplytheanalyticexpressionsfortheshapesofthe1-dimensionalinvariantmassdistributionsusedinourmainanalysisinSection 3.3.1 .Thoseresultscanbeusefulinimprovingtheprecisionontheextractionofthekinematicalendpoints. 3.2NewVariablesInthissectionweproposeanewsetofinvariantmass(squared)variables.AsalreadyexplainedintheIntroduction,ourvariablesshouldbecomposedofm2j`nandm2j`finasymmetricway,inaccordancewith( 3 ).Consequently,anyplottingmanipulationsormathematicaloperationsinvolvingm2j`nandm2j`fshouldobeythesymmetryimpliedbyEquation( 3 ). 73

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3.2.1TheUnionm2j`n[m2j`fWebeginwiththesimplestcase,wherewepostponeapplyinganymathematicaloperationstom2j`nandm2j`f,andinsteadsimplyplotthem.TherequirementofEquation( 3 )impliesthattheonlypossibilityistoplacebothofthemtogetheronthesameplot,inessenceformingtheunion m2jl(u)m2j`n[m2j`f(3)oftheindividualm2j`nandm2j`fdistributions.Sinceeachindividualdistributionissmoothandhasakinematicendpoint,thesametwokinematicendpointsshouldbevisibleonthecombineddistributionm2jl(u)aswell5.Weshalldenotethelargerofthetwoendpointswith )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Mmaxjl(u)2maxn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`n2,)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`f2o(3)andthesmallerofthetwoendpointswith )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxjl(u)2minn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`n2,)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`f2o.(3)ThenewlyintroducedquantitiesMmaxjl(u)andmmaxjl(u)arenothingbuttheusualkinematicendpointsmmaxj`nandmmaxj`f,givenby( 3 )and( 3 ),correspondingly.Ofcourse,atthispointwedonotknowwhichiswhich,andwehaveanapparenttwo-foldambiguity:wecanhaveeither Mmaxjl(u)=mmaxj`n,mmaxjl(u)=mmaxj`f,ifRABRBC,(3)or Mmaxjl(u)=mmaxj`f,mmaxjl(u)=mmaxj`n,ifRABRBC.(3)Noticethatboth( 3 )and( 3 )areofciallyupperkinematicendpoints,andthussatisfyourbasicrequirements. 5Forspecicnumericalexamples,refertoSection 3.4 74

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Thebenetsofouralternativetreatment( 3 )inresponsetothenear-farleptonambiguityproblemofSection 3.1.1 ,arenowstartingtoemerge.Withtheconventionalordering( 3 3 )onehastodealwithathree-foldambiguityintheinterpretationoftheendpointsmmaxj`(lo)andmmaxj`(hi),asseeninEquations( 3 3 ).Instead,thesimpleunion( 3 )leadsonlytothetwo-foldambiguityofEquations( 3 3 ).Moreimportantly,theanalysisofSection 3.3.1 belowwillrevealthatinspiteoftheremainingtwo-foldambiguityinEquations( 3 3 ),onecanneverthelessuniquelydetermineallthreeofthemassesmD,mCandmA!Weconsiderthistobeoneoftheimportantresultsofthispaper. 3.2.2TheProductmj`nmj`fIntheremainderofthissection,weshallconstructnewinvariantmasssquaredvariablesoutofthetwoentriesm2j`nandm2j`f,simplybyapplyingvariousmathematicaloperationsontheminasymmetricfashion.Webeginwiththeproduct m2j`(p)mj`nmj`f(3)whoseendpointisgivenby )]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`(p)28>><>>:1 2m2D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB,forRBC0.5,m2D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)p RBC(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RBC)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB),forRBC0.5.(3)Unfortunately,thisendpointalsoturnsouttobepiecewise-dened,thusfailingoneofourbasicrequirementsfromtheIntroduction.Thereforeweshallnotusethisendpointinthecourseofouranalysis. 3.2.3TheSumsm2j`n+m2j`fAnotherpossibilityistoconsidervarioussums,forexamplem2j`n+m2j`for(mj`n+mj`f)2,asoriginallyproposedin[ 76 ].Herewegeneralizethediscussionin[ 76 ]andintroduceawholesetofnewvariables,m2j`(s)(),labelledbythecontinuousparameter 75

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,whicharedenedas m2j`(s)())]TJ /F3 11.955 Tf 5.48 -9.68 Td[(m2j`n+m2j`f1 .(3)Sinceisacontinuousparameter,inprinciplethereareinnitelymanymj`(s)variables!Noticethattheconventionalvariablesm2j`(lo)andm2j`(hi)from( 3 )and( 3 )arealsoincludedinourset,andaresimplygivenby m2j`(lo)m2j`(s)(), (3) m2j`(hi)m2j`(s)(1). (3) Weseethatournewset( 3 )isaverybroadgeneralizationoftheconventionaldenitions( 3 )and( 3 ),whichjustcorrespondtothetwoextremecases=.Ofcourse,theuserisfreetochooseatwill,andanynitevalueofwillleadtoanewvariablem2j`(s)().Inordertomakethenewvariablesm2j`(s)()usefulformassspectrumstudies,weneedtoprovidetheformulasfortheirkinematicendpoints(mmaxj`(s)())2.Theseformulasareeasytoderive,usingtheresultsfrom[ 9 ],andwepresenttheminthenexttwosubsections,whereitisconvenienttoconsiderseparatelythefollowingtwocases:1(inSection 3.2.3.1 )and<1,but6=0(inSection 3.2.3.2 ). 3.2.3.1KinematicEndpointsofm2j`(s)()with1Whenonechoosesavalueof1,them2j`(s)()endpointisgivenbythefollowingexpression )]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`(s)(1)28>><>>:)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`f2,RAB1)]TJ /F6 11.955 Tf 11.96 -0.17 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC))]TJ /F5 5.978 Tf 8.52 3.26 Td[(1 ,)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`()2,RAB1)]TJ /F6 11.955 Tf 11.96 -0.17 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC))]TJ /F5 5.978 Tf 8.52 3.26 Td[(1 ,(3)wheremmaxj`fwasalreadydenedinEquation( 3 ),andmmaxj`()isanewlydened,-dependentquantity )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`()2m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)hRBC(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RBC)i1 .(3) 76

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Asacross-check,onecanverifythatinthelimit!1theexpressioninEquation( 3 )reducestoEquation( 3 ),inagreementwithEquation( 3 ).Inthatcase,theupperlineinEquation( 3 )correspondstooptions()]TJ /F6 11.955 Tf 9.3 0 Td[(,1)and()]TJ /F6 11.955 Tf 9.3 0 Td[(,2)inEquation( 3 ),wheremmaxj`(hi)=mmaxj`f,whilethelowerlineinEquation( 3 )correspondstooption()]TJ /F6 11.955 Tf 9.3 0 Td[(,3)inEquation( 3 ),wheremmaxj`(hi)=mmaxj`n.Unfortunately,justliketheproductendpointofEquation( 3 ),theendpointofEquation( 3 )isingeneralpiecewise-dened,anddoesnotmeetourcriteria.However,thereisoneimportantexception,namelythecaseof=1,inwhichwedogetasinglydenedfunction.AccordingtothegeneraldenitioninEquation( 3 ),m2j`(s)(=1)issimplythesumofthetwom2j`entriesineachevent: m2j`(s)(=1)m2j`n+m2j`f.(3)Usingtheidentity m2j``=m2j`n+m2j`f+m2``,(3)Equation( 3 )canbeequivalentlyrewrittenas m2j`(s)(=1)m2j``)]TJ /F3 11.955 Tf 11.95 0 Td[(m2``.(3)Tondtheexpressionforitsendpoint,onecanset=1inEquation( 3 ),andthenrealizethatthelogicalconditionforexecutingtheupperlinebecomesRAB0,whichisimpossible,sincethemassratiosRijinEquation( 3 )arealwayspositivedenite.Therefore,theendpointmmaxj`(s)(=1)isalwayscalculatedaccordingtothelowerlineinEquation( 3 ),whichresultsin[ 76 ] )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`(s)(1)2m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAC).(3)NotethatthisendpointisperfectforourpurposessincetheEquation( 3 )isalwaysunique,i.e.itisindependentoftheparameterspaceregion.Thevariablem2j`(s)(=1)willthusplayacrucialroleinouranalysisbelow. 77

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3.2.3.2KinematicEndpointsofm2j`(s)()with<1and6=0Finally,inthecasewhen<1,but6=0,them2j`(s)()endpointisgivenbythefollowingexpression )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`(s)(<1)28>><>>:)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`()2,RBCh1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB) )]TJ /F5 5.978 Tf 5.76 0 Td[(1i)]TJ /F5 7.97 Tf 6.59 0 Td[(1,m2D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)h1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB) 1)]TJ /F13 5.978 Tf 5.75 0 Td[(i1)]TJ /F13 5.978 Tf 5.76 0 Td[( ,RBCh1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB) )]TJ /F5 5.978 Tf 5.76 0 Td[(1i)]TJ /F5 7.97 Tf 6.59 0 Td[(1,(3)wheremmaxj`()wasalreadydenedinEquation( 3 ).Againasacross-check,onecanverifythatinthelimit!theexpressioninEquation( 3 )reducestoEquation( 3 ),inagreementwithEquation( 3 ).Inthe!case,theupperlineinEquation( 3 )correspondstooption()]TJ /F6 11.955 Tf 9.3 0 Td[(,1)inEquation( 3 ),wheremmaxj`(lo)=mmaxj`n,whilethelowerlineinEquation( 3 )correspondstooptions()]TJ /F6 11.955 Tf 9.3 0 Td[(,2)and()]TJ /F6 11.955 Tf 9.3 0 Td[(,3)inEquation( 3 ),wheremmaxj`(lo)=mmaxj`(eq).Unfortunately,theendpointfunctioninEquation( 3 )isagainpiecewise-dened,anddoesnotmeetoneofourbasiccriteriaspelledoutintheintroduction.Inpassing,wenotethatthespecialcaseof=1 2,whichinvolvesthelinearsumofthetwomasses m2j`(s)(=1 2)(mj`n+mj`f)2,(3)waspreviouslyexploredin[ 76 77 ].Inthatcase,fromEquation( 3 )wendforitsendpoint mmaxj`(s)(1 2)28>><>>:m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)p RBC(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB)+p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(RBC2,RBC1)]TJ /F4 7.97 Tf 6.59 0 Td[(RAB 2)]TJ /F4 7.97 Tf 6.59 0 Td[(RAB,m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(2)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB),RBC1)]TJ /F4 7.97 Tf 6.59 0 Td[(RAB 2)]TJ /F4 7.97 Tf 6.59 0 Td[(RAB.(3) 3.2.4TheDifferencejm2j`n)]TJ /F3 11.955 Tf 11.96 0 Td[(m2j`fjFinally,onecanalsoconsiderasetofvariableswhichinvolvetheabsolutevalueofdifferencesbetweenm2j`nandm2j`f.InanalogywithEquation( 3 ),wecandene 78

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anotherinnitesetofvariables m2j`(d)()m2j`n)]TJ /F3 11.955 Tf 11.95 0 Td[(m2j`f1 .(3)Onceagain,theuserisfreetoconsiderarbitraryvaluesof.However,thisfreedomisredundant,whenitcomestotheissueofthekinematicendpointsofthevariablesinEquation( 3 ).Itisnotdifculttoseethattheendpointsofm2j`(d)()arealwaysgivenby )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`(d)()2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Mmaxjl(u)2(3)andareinfactindependentof!Therefore,forthepurposesofourdiscussion,itissufcienttoconsiderjustoneparticularvalueof.Inthefollowingweshallonlyuse=1: m2j`(d)(=1)m2j`n)]TJ /F3 11.955 Tf 11.95 0 Td[(m2j`f,(3)whichistheanalogueofm2j`(s)(=1)denedinEquation( 3 ).TheresultofEquation( 3 )impliesthattheendpointinEquation( 3 )doesnotcontainanynewamountofinformation,whichwasnotalreadypresentinthetwokinematicendpointsMmaxjl(u)andmmaxjl(u)discussedinSection 3.2.1 .Nevertheless,theindependentmeasurementof(mmaxjl(d)(1))2canstillbeveryuseful,sinceitwillmarkthelocationof(Mmaxjl(u))2onthem2jl(u)distribution.Thenonewillbelookingforthesecondendpoint(mmaxjl(u))2totheleft,i.e.intheregionofsmallerm2jl(u)values.Thiscompletesourdiscussionofthenewinvariantmassvariablesandtheirkinematicendpoints.Forourbasicproof-of-principlemeasurementtechniquepresentedinthenextSection 3.3.1 ,weshalluseonlythreeofthem,namelyMmaxj`(u),mmaxj`(u),andmmaxj`(s)(=1).However,theremainingvariablesareinprinciplejustasgood,theironlydisadvantagebeingthattheyfailedourarbitrarilyimposedconditionatthebeginningthattheendpointfunctionsshouldallberegionindependent.Ofcourse,onecould,andinfactshould,usealloftheavailablekinematicendpointinformation,whichina 79

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globaltanalysiscanonlyincreasetheexperimentalprecisionofthesparticlemassdetermination. 3.3TheoreticalAnalysis 3.3.1OurMethodAndTheSolutionForTheMassSpectrumOurstartingpointisthesetoffourmeasurements mmax``,Mmaxj`(u),mmaxj`(u),mmaxj`(s)(=1)(3)inplaceoftheconventionalsetinEquation( 3 ).ItiseasytoverifythatthemeasurementsinEquation( 3 )arealwaysindependentofeachother,andthusneversufferfromthelineardependenceproblemdiscussedinSection 3.1.2 .GiventhesetoffourmeasurementsinEquation( 3 ),itiseasytosolveforthemassspectrum.Tosimplifythenotation,weintroducethefollowingshorthandnotationfortheendpointsofthemasssquareddistributions L(mmax``)2,M)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Mmaxj`(u)2,m)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`(u)2,S)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`(s)(=1)2(3)Thesolutionforthemassspectrumisthengivenby m2D=Mm(L+M+m)]TJ /F3 11.955 Tf 11.95 0 Td[(S) (M+m)]TJ /F3 11.955 Tf 11.95 0 Td[(S)2; (3) m2C=MmL (M+m)]TJ /F3 11.955 Tf 11.96 0 Td[(S)2; (3) m2B=8>><>>:ML(S)]TJ /F4 7.97 Tf 6.59 0 Td[(M) (M+m)]TJ /F4 7.97 Tf 6.58 0 Td[(S)2,ifRABRBC,mL(S)]TJ /F4 7.97 Tf 6.59 0 Td[(m) (M+m)]TJ /F4 7.97 Tf 6.58 0 Td[(S)2,ifRABRBC; (3) m2A=L(S)]TJ /F3 11.955 Tf 11.95 0 Td[(m)(S)]TJ /F3 11.955 Tf 11.96 0 Td[(M) (M+m)]TJ /F3 11.955 Tf 11.95 0 Td[(S)2. (3) Itiseasytoverifythattheright-handsideexpressionsintheseequationsarealwayspositivedenite,sothatonecansafelytakethesquarerootandcomputethelinearmassesmD,mC,mBandmA.Noticethatinspiteofthetwo-foldambiguityinEquations 80

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( 3 3 ),thesolutionformD,mCandmAisunique!Indeed,theexpressionsformD,mCandmAaresymmetricundertheinterchangeM$m.Theremainingtwo-foldambiguityformBispreciselytheresultoftheambiguousinterpretationinEquations( 3 and 3 )ofthetwom2j`(u)endpoints,andisrelatedtothesymmetryunderEquation( 3 ),orequivalently,undertheinterchange RAB$RBC.(3)Inthenextsubsectionwediscussseveralwaysinwhichonecanlifttheremainingtwo-folddegeneracyformBwhichisduetoEquation( 3 ).Noticethegreatsimplicityofthismethod.TheexpressionsinEquations( 3 ),( 3 )and( 3 )areregionindependentandthereforeonedoesnothavetogothroughthestandardtrialanderrorprocedureinvolvingthe9parameterspaceregions(Nj``,Nj`)[ 9 17 ]associatedwiththevariousinterpretationsoftheendpointsmmaxj``,mmaxj`(lo)andmmaxj`(hi). 3.3.2DisambiguationOfTheTwoSolutionsFormBThemethodoutlinedinSection 3.3.1 allowedustondthetruemassesofparticlesA,CandD,butyieldstwoseparatepossiblesolutionsforthemassmBofparticleB.Weshallnowdiscussseveralwaysofliftingtheremainingtwo-folddegeneracyformB. 3.3.2.1InvariantmassendpointmethodOnepossibilityistouseanadditionalmeasurementofaninvariantmassendpoint.Indeed,asshowninSection 3.2 ,therearestillquiteafewone-dimensionalinvariantmassdistributionsatourdisposal,whichwehavenotusedsofar.Thoseincludetheconventionaldistributionsofm2j``,m2j`(lo)andm2j`(hi),aswellasthenewdistributionsm2j`(p),m2j`(s)()andm2j`(d)(1)whichweintroducedinSection 3.2 .Whichofthemcanbeusedforourpurposes?NotethattheduplicationinEquation( 3 )aroseduetothesymmetryinEquation( 3 ),sothatanykinematicendpointwhichviolatesthissymmetrywillbeabletodistinguishbetweenthetwosolutions. 81

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Figure3-2. Comparisonofthepredictionsforthekinematicendpointsmmaxj`(s)()oftherealandfakesolutions,asafunctionofarctan(inunitsof),forthetwoexamplesdiscussedindetailinSection 3.4 :(a)theLM1CMSstudypointand(b)theLM6CMSstudypoint.Ineachpanel,thepredictionofthereal(fake)solutionisplottedinred(blue).Theverticaldottedlineindicatesthecaseof= 4(=1),forwhichthetwosolutionsgiveanidenticalanswer,markedwithagreendot.Thehorizontaldottedlinesshowthecorrespondingasymptoticvaluesmmaxj`(hi)andmmaxj`(lo),obtainedat!(! 2). Letusbeginwiththeconventionaldistributionsm2j``,m2j`(lo),m2j`(hi)andm2j``(> 2),whoseendpointswedidnotuseinouranalysissofar.Itiseasytocheckthatmmaxj``,mmaxj`(hi)andmminj``(> 2)areinvariantundertheinterchangeasinEquation( 3 )andcannotbeusedfordiscrimination.However,mmaxj`(lo)isnotsymmetricunderEquation( 3 )andcandothejob.Infact,onecanshowthatthetwoduplicatesolutionsformBalways6givedifferentpredictionsformmaxj`(lo).Moreimportantly,manyofournewvariablesfromSection 3.2 canprovideanindependentcross-checkonthecorrectchoiceforthesolution.Forexample,thekinematicendpointinEquation( 3 )oftheproductvariablem2j`(p),alsoviolatesthesymmetryofEquation( 3 )anddistinguishesamongthetwosolutions.Theinnitesetofvariablesm2j`(s)()canalsobeused,andforalmostthewholerangeof<1.Tosee 6TheonlyexceptionisthetrivialcaseofRAB=RBC,butthenthetwosolutionsformBcoincide,andmBisagainuniquelydetermined. 82

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this,inFigure 3-2 wecomparethepredictionsforthekinematicendpointsmmaxj`(s)()oftherealandfakesolutions,forthetwoexamplesdiscussedindetailinSection 3.4 :(a)theLM1CMSstudypointand(b)theLM6CMSstudypoint.ThecorrespondingmassspectraarelistedinTable 3-1 below.Forconvenience,weplotversustheparameter arctan,(3)whichallowsustomapthewholedenitiondomain(,1)forintotheniteregion()]TJ /F7 7.97 Tf 10.5 4.7 Td[( 2, 2)for.Figure 3-2 showsthatformostoftheallowedrange,thetwosolutionspredictdifferentvaluesforthekinematicendpointsmmaxj`(s)().Infact,for< 4,thetwopredictionsarealwaysdifferent,apartfromthetrivialcaseof=0(=0).Evenfor> 4,therestillexistsarangeof,forwhich,atleasttheoretically,adiscriminationcanbemade.Thepredictionsareguaranteedtocoincideonlyfor= 4(=1)(astheyshould,seeEquation( 3 )),andforacertainrangeofthelargestpossiblevaluesof. 3.3.2.2InvariantmasscorrelationsAnotherwaytoresolvethetwofoldambiguityinoursolutioninEquation( 3 )istosimplygobacktotheoriginalmeasurementsofMmaxjl(u)andmmaxjl(u)andalreadyatthatpointtrytodecidewhichofthetwomeasuredmjl(u)endpointsismmaxj`nandwhichoneismmaxj`f.Asalreadydiscussedin[ 21 76 ],thisidenticationisinprinciplepossible,ifoneconsidersthecorrelationswhicharepresentinthetwo-dimensionaldistributionm2jl(u)versusm2ll.ThebasicideaisillustratedinFigure 3-3 ,whereweshowscatterplotsofmj`(u)versusm``,forthetwoexamplesusedinFigure 3-2 anddiscussedindetaillaterinSection 3.4 .Figure 3-3 (a)(Figure 3-3 (b))showstheresultforthereal(fake)solutioncorrespondingtotheLM1studypoint,whileFigures 3-3 (c)and 3-3 (d)showtheanalogousresultsfortheLM6studypoint.Ineachplotweused10,000entries,whichroughlycorrespondsto20fb)]TJ /F5 7.97 Tf 6.58 0 Td[(1(200fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1)ofdatafortheactualLM1(LM6)SUSYstudypoint.Hereandbelowweshowtheidealcasewhereweneglectsmearingeffectsduetothenitedetectorresolution,niteparticlewidthsandcombinatorialbackgrounds. 83

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Figure3-3. Predictedscatterplotsofmj`(u)versusm``,forthecaseoftherealandfakesolutionsforeachofthetwostudypointsLM1andLM6:(a)therealsolutionLM1;(b)thefakesolutionLM1';(c)therealsolutionLM6;and(d)thefakesolutionLM6'.Theredsolidhorizontal(bluedashedinclined)lineindicatestheconditionalmaximummmaxj`n(m``)(mmaxj`f(m``))givenbyEquation( 3 )(Equation( 3 )).Eachpanelcontains10,000entries.Theresultsshownhereareidealizedinthesensethatweneglectsmearingeffectsduetothenitedetectorresolution,niteparticlewidthsandcombinatorialbackgrounds.Noticetheuseofquadraticpowerscaleonthetwoaxes,whichpreservesthesimpleshapesofthescatterplots,evenwhenplottedversusthelinearmassesmj`(u)andm``. Allofourplotsareatthepartonlevel(usingourownMonte-Carlophasespacegenerator)andwithoutanycuts.Noticethatinordertoavoiddealingwiththelargenumericalvaluesofthesquaredmasses,weuseaquadraticpowerscaleonbothaxes,whichallowsustopreservethesimpleshapesofthescatterplotswhenplottingversusthelinearmassesthemselves. 84

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Figure 3-3 showsthatthecombineddistributionm2jl(u)issimplycomposedofthetwoseparatedistributionsm2j`nandm2j`f,buttheyarecorrelateddifferentlywiththedileptondistributionm2``.Inparticular,letusconcentrateontheconditionalmaximammaxj`n(m``)andmmaxj`f(m``),i.e.themaximumallowedvaluesofmj`nandmj`f,respectively,foragivenxedvalueofm``[ 21 76 ].AcloseinspectionofFigure 3-3 showsthatthevaluesofm2j`nandm2``areuncorrelated,andasaresult,theconditionalmaximummmaxj`n(m``)doesnotdependonm``.Inturn,thisimpliesthattheendpointvalue(mmaxj`n)2giveninEquation( 3 )canbeobtainedforanym2``: n)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`n2=mmaxj`n(m``)2=m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RBC),8m``2[0,mmax``].(3)BecauseofEquation( 3 ),theshapeofthem2j`nversusm2``scatterplotisasimplerectangle[ 21 76 ].ThisisconrmedbytheplotsinFigure 3-3 ,wherethe(red)horizontalsolidlineindicatestheconstantvalueasinEquation( 3 )fortheconditionalmaximummmaxj`n(m``).Incontrast,thevaluesofm2j`fandm2``arecorrelated.Theconditionalmaximummmaxj`f(m``)doesdependonthevalueofm``asfollows: )]TJ /F3 11.955 Tf 5.48 -9.69 Td[(mmaxj`f(m``)2=p+f)]TJ /F3 11.955 Tf 11.96 0 Td[(p Lm2``,(3)whereweintroducetheshorthandnotationusedin[ 9 ] f)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`f2=m2D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB), (3) pRBCf=m2D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RCD)RBC(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB). (3) Theabsolutemaximumofm2j`f,whichisgivenbyEquation( 3 )anddenotedherebyf,canonlybeobtainedwhenm2``itselfisatamaximum[ 21 76 ]: fmmaxj`f(mmax``)2.(3) 85

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Ontheotherhand,theconditionalmaximummmaxj`f(m``)obtainsitsminimumvalueatm2``=0andcorrespondsto[ 21 76 ] pmmaxj`f(0)2f.(3)Equations( 3 and 3 )implythattheshapeofthem2j`fversusm2``scatterplotisaright-angletrapezoid.ThisisconrmedbytheplotsinFigure 3-3 ,wherewemarkwitha(blue)dashedlinetheconditionalmaximuminEquation( 3 ).Withsufcientstatistics,thisdifferenceinthekinematicboundariesmaybeobservable,andwouldrevealtheidentityofmmaxj`nandmmaxj`f[ 21 76 ].Oncetheindividualmmaxj`nandmmaxj`fareknown,thesolutionforthemassspectrumisuniqueseee.g.AppendixAin[ 9 ].Ofcourse,incaseswherepf,namelyRBC1,itmaybedifcultinpracticetotellwhichofthetwoboundariesinthescatterplotisinclinedandwhichoneishorizontal7.OneexampleofthissortisofferedbypointLM6,whichhasRBC=0.91andleadstoaratheratmmaxj`f(m``)function,asseeninFigure 3-3 (c).Analternativeandsomewhatrelatedmethodwillbetoinvestigatetheshapesoftheone-dimensionaldistributionsthemselves[ 78 ].InAppendix A weprovidetheanalyticalexpressionsfortheshapesofthefourinvariantmassdistributionsm2``,m2j`(u),m2j`(s)(1)andm2j`(d)(1)usedinourbasicanalysisfromSection 3.3.1 .GivenwhatwehavealreadyseeninFigure 3-3 ,itisnotsurprisingthatthetrueandthefakesolutionspredictdifferentshapesfortheone-dimensionaldistributionsaswell.IntheLM1andLM6examplesconsideredbelowinSection 3.4 ,thisdifferenceisparticularlynoticeableforthem2j`(u)andm2j`(d)(1)distributions(seeFigures 3-4 (b), 3-4 (d), 3-5 (b)and 3-5 (d)),andcanbetestedexperimentally. 7Aseparateproblem,whicharisesinthecaseofpf,willbediscussedbelowinSection 3.4.1 86

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Table3-1. TherelevantpartoftheSUSYmassspectrumfortheLM1andLM6studypoints.ThecorrespondingduplicatedsolutionsLM1'andLM6'areobtainedbyinterchangingRBC$RABasinEquation( 3 ).Inthetablewealsolistthecorrespondingvaluesforvariousinvariantmassendpoints.TherstfourofthoserepresentourbasicsetofmeasurementsofEquation( 3 )discussedindetailinSection 3.4.1 ,whilethelasttwo(mmaxj`nandmmaxj`f)arenotdirectlyobservable.TheremaininginvariantmassendpointsareconsideredinSection 3.4.2 .Inthecaseofmmaxj`(s)(),weshowseveralrepresentativevaluesfor.Forthecompletevariation,refertoFigure 3-2 .Recallthatmmaxj`(s)(+1)=mmaxj`(hi)andmmaxj`(s)()=mmaxj`(lo). VariableLM1LM1'LM6LM6' mA(GeV)94.9158.15mB(GeV)118.9143.35291.0165.65mC(GeV)179.6304.8mD(GeV)561.6861.9RAB0.63700.43830.29540.9115RBC0.43830.63700.91150.2954RCD0.10230.1251mmax``(GeV)81.1076.12Mmaxj`(u)(GeV)398.8676.8mmaxj`(u)(GeV)320.6239.8mmaxj`(s)(=1)(GeV)451.8689.2mmaxj``(GeV)451.8689.2mminj``(> 2)(GeV)215.2176.4mmaxj`(hi)(GeV)398.8676.8mmaxj`(s)(=2)(GeV)406.6398.8676.8677.0mmaxj`(s)(=1.5)(GeV)417.9402.5676.8678.4mmaxj`(s)(=0.5)(GeV)611.0638.9886.0807.1mmaxj`(s)(=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.5)(GeV)142.9159.7174.9138.0mmaxj`(s)(=)]TJ /F6 11.955 Tf 9.3 0 Td[(1)(GeV)200.1225.9224.8184.8mmaxj`(lo)(GeV)274.6319.1239.8229.9mmaxj`(p)(GeV)292.0319.4393.7310.9mmaxj`n(GeV)398.8320.6239.8676.8mmaxj`f(GeV)320.6398.8676.8239.8 87

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3.4NumericalExamplesWeshallnowillustratetheideasoftheprevioussectionwithtwospecicnumericalexamples:theLM1andLM6SUSYstudypointsinCMS[ 75 ].ThemassspectraatLM1andLM6arelistedinTable 3-1 .PointLM1issimilartobenchmarkpointA(A')inReference[ 79 ](Reference[ 80 ])andtobenchmarkpointSPS1ainReference[ 81 ].PointLM6issimilartobenchmarkpointC(C')inReference[ 79 ](Reference[ 80 ]).TheTable 3-1 alsoliststhecorrespondingduplicatesolutionsLM1'andLM6',whichareobtainedbyinterchangingRBC$RAB,orequivalently,byreplacingthemassofBvia mB!m0B=mAmC mB.(3)ItisinterestingtonotethatLM1andLM6representbothsidesoftheambiguityinEquation( 3 ):atLM1,wehaveRAB>RBCandcorrespondingly,mmaxj`n>mmaxj`fandEquation( 3 )applies.Ontheotherhand,atLM6wehaveRAB
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inmindhere),particleBisascalar,whichautomaticallywashesoutanyspineffectsinthem2``andm2j`fdistributions.Furthermore,ifparticlesDandtheirantiparticlesDareproducedinequalnumbers,aswouldbethecaseifthedominantproductionisfromggand/orqqinitialstate,anyspincorrelationsinthem2j`ndistributionarealsowashedout.Underthosecircumstances,therefore,thepurephasespacedistributionsshownhereareinfactthecorrectanswer.Webeginourdiscussionwiththefourinvariantmassdistributionsm2``,m2j`(u),m2j`(s)(=1)andm2j`(d)(=1),whichformthebasisofourmethodoutlinedinSection 3.3.1 .Figure 3-4 (Figure 3-5 )showsthosefourdistributionsforthecaseofstudypointLM1(LM6).Ineachpanel,thered(solid)histogramcorrespondstothenominalspectrum(LM1orLM6),whiletheblue(dotted)histogramcorrespondstothefakesolution(LM1'orLM6'),whichisobtainedthroughthereplacement( 3 ).Forallguresinthissection,weusethesame4samplesof10,000eventseach,whichwerealreadyusedtomakeFigure 3-3 .Noticeoursomewhatunconventionalwayofllingandthenplottingthehistogramsinthissection.First,weshowdifferentialdistributionsinthecorrespondingmasssquared,i.e.dN=dm2.ThisisdoneinordertopreservetheconnectiontotheanalyticalresultsinAppendix A ,whicharewrittenthesameway.Moreimportantly,theshapesoftheone-dimensionalhistogramsaremuchsimplerinthecaseofdN=dm2asopposedtodN=dm[ 82 84 ].Inthenextstep,however,wechoosetoplotthethusobtainedhistogramversusthemassitselfratherthanthemasssquared.ThisallowsonetoreadoffimmediatelythecorrespondingendpointandcomparedirectlytothevalueslistedinTable 3-1 .Italsokeepsthex-axisrangewithinamanageablerange.However,sincethehistogramswerebinnedonamasssquaredscale,ifweweretousealinearscaleonthex-axis,wewouldgetbinswithvaryingsize.Thiswouldberatherinconvenientandmoreimportantly,woulddistortthenicesimpleshapesofthedN=dm2distributions.Therefore,weuseaquadraticscaleonthex-axis,whichpreservestheniceshapesandleadstoaconstantbinsizeoneachplot. 89

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Figure3-4. One-dimensionalinvariantmassdistributionsforthecaseofLM1(redsolidlines)andLM1'(bluedottedlines)spectra.ThekinematicendpointsinEquation( 3 )usedinouranalysisinSection 3.3.1 canbeobservedfromthesedistributionsasfollows:mmax``istheupperkinematicendpointofthem``distributioninpanel(a);Mmaxj`(u)istheabsoluteupperkinematicendpointseeninboththecombinedmj`(u)distributioninpanel(b),orthedifferencedistributionmj`(d)(1)inpanel(d);mmaxj`(u)istheintermediatekinematicendpointseeninpanel(b);andmmaxj`(s)(=1)istheupperkinematicendpointofthemj`(s)(=1)distributioninpanel(c). 90

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Figure3-5. One-dimensionalinvariantmassdistributionsfortheLM6massspectrum(redsolidlines)andtheLM6'massspectrum(bluedottedlines). 91

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Figures 3-4 and 3-5 illustratehoweachoneofthemeasurementsinEquation( 3 )canbeobtained.Forexample,mmax``istheclassicupperkinematicendpointofthem``distributionsinFigures 3-4 (a)and 3-5 (a).Thisendpointisverysharpandshouldbeeasilyobservable.Mmaxj`(u)istheabsoluteupperkinematicendpointseeninthecombinedmj`(u)distributioninFigures 3-4 (b)and 3-5 (b).Noticethatthesameendpointcanindependentlyalsobeobservedastheabsoluteupperkinematiclimitofthedifferencedistributionsmj`(d)(1)showninFigures 3-4 (d)and 3-5 (d).ThefactthattherearetwoindependentwaysofgettingtotheendpointMmaxj`(u)shouldallowforareasonableaccuracyofitsmeasurement.Uponcloserinspectionofthecombinedmj`(u)distributioninFigures 3-4 (b)and 3-5 (b),wealsonoticetheintermediatekinematicendpointmmaxj`(u)seenaround320GeVinFigure 3-4 (b)andaround240GeVinFigure 3-5 (b).Finally,mmaxj`(s)(=1)istheupperkinematicendpointofthemj`(s)(=1)distributionshowninFigures 3-4 (c)and 3-5 (c).Itisalsoratherwelldened,andshouldbewellmeasuredintherealdata.Atthispointwewouldliketocommentononepotentialproblemwhichisnotimmediatelyobvious,butneverthelesshasbeenencounteredinpracticalapplicationsoftheinvariantmasstechniqueforSUSYmassdeterminations[ 78 ].Ithasbeennotedthatinthecaseofpf(Equations( 3 3 )),thenumericaltforthemassspectrumbecomesratherunstable.GivenouranalyticalresultsinSection 3.3.1 ,wearenowabletotracetherootoftheproblem.NoticethatpfimpliesthatRBC1.Inthislimit,fromEquations( 3 ),( 3 ),( 3 )and( 3 )wend limRBC!1(L)=0,limRBC!1(n)=0,limRBC!1(M+m)]TJ /F3 11.955 Tf 11.95 0 Td[(S)=0.(3)ThismeansthatthefunctionsinEquations( 3 through 3 )givingthesolutionforthemassspectrumwillallbehaveas02 02,and,giventhestatisticaluctuationsinanactualanalysis,willhaveverypoorconvergenceproperties.Wenotethatthisproblemisnotlimitedtoourpreferredsetofmeasurements( 3 )andisrathergeneric,buthasbeenmissedinmostpreviousstudiessimplybecausethecaseofRBC1was 92

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rarelyconsidered.Figures 3-4 and 3-5 revealthat,asexpected,thereal(redsolidlines)andfake(bluedottedlines)solutionsalwaysgiveidenticalresultsforourbasicsetoffourendpointmeasurementsinEquation( 3 ).Thisisbydesign,andinordertodiscriminateamongtherealandthefakesolution,weneedadditionalexperimentalinput,asdiscussedinSection 3.3.2 .Beforeweproceedwiththedisambiguationanalysisinthenextsubsection,weshouldstressonceagainthattherealandfakesolutionsagreeon75%oftherelevantmassspectrum,i.e.theygivethesamevaluesforthemassesofparticlesD,CandA(Table 3-1 ).Theonlyquestionmarkatthispointis,whatisthemassofparticleB.Thisissueisaddressedinthefollowingsubsection. 3.4.2EliminatingTheFakeSolutionformBAsalreadydiscussedinSection 3.3.2 ,thereareseveralhandleswhichcoulddiscriminateamongthetwoalternativevaluesofmBintherealandthefakesolution.OnepossibilityistouseadditionalindependentmeasurementsofMT2kinematicendpoints.Anotherpossibility,discussedinSection 3.3.2.2 anddemonstratedexplicitlywithFigure 3-3 ,istousethedifferentcorrelationsinthe2-dimensionalinvariantmassdistributions(m2``,m2j`n)and(m2``,m2j`f).Thenear-farleptonambiguityisavoidedbystudyingthescatterplotof(m2``,m2j`(u)),showninFigure 3-3 ,whichshouldbeinprinciplesufcienttodiscriminateamongthetwoalternatives.Inkeepingwiththemainthemeofthispaper,inthissubsectionweshallconcentrateonthethirdpossibility,alreadysuggestedinSection 3.3.2.1 .Weshallsimplyexploreadditionalinvariantmassendpointmeasurements,whichwouldhopefullydiscriminateamongthetwosolutionsformB.Figures 3-6 and 3-7 showseveralinvariantmassdistributionswhichhavealreadybeenmentionedatonepointoranotherinthecourseofourpreviousdiscussion.Figure 3-6 showsthefollowing6distributions:(a)m2j``;(b)m2j`(hi);(c)m2j`(p);(d)m2j`(lo);(e)m2j`(s)(=)]TJ /F6 11.955 Tf 9.3 0 Td[(1)and(f)m2j`(s)(=1 2),fortheLM1massspectrum(redsolidlines)anditsLM1'counterpart(bluedottedlines).Figure 3-7 showsthesame6distributions,butfortheLM6andLM6'massspectra.InFigures 3-6 and 3-7 93

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wefollowthesameplottingconventionsasinFigures 3-4 and 3-5 :weformthemasssquareddistributiondN=dm2,andthenplotversusthecorrespondinglinearmassmusingaquadraticscaleonthex-axis.Noticethatthesumofthem2j`(hi)distributioninFigure 3-6 (b)(Figure 3-7 (b))andthem2j`(lo)distributioninFigure 3-6 (d)(Figure 3-7 (d))preciselyequalsthecombineddistributionm2j`(u)inFigure 3-4 (b)(Figure 3-5 (b)).Inordertobeabletoseethisbythenakedeye,wehavekeptthesamexandyrangesonthecorrespondingplots.AsseeninFigures 3-6 and 3-7 ,notalloftheremaininginvariantmassdistributionsareabletodiscriminateamongthetwomBsolutions.AsexplainedinSection 3.3.2.1 ,thesuitabledistributionsarethosewhoseendpointsviolatethesymmetryinEquation( 3 ),whichcausedthemBambiguityintherstplace.Forexample,Figures 3-6 (a)and 3-7 (a)showthattheendpointofthem2j``distributionisthesamefortherealandthefakesolution.Thisistobeexpected,sincethedeningEquation( 3 )formmaxj``issymmetricunderEquation( 3 ).Figures 3-6 (a)and 3-7 (a)alsoshowthateventheshapesofthem2j``distributionsfortherealandfakesolutionareverysimilar.Inspiteofthis,theobservationofthem2j``endpointcanstillbeveryuseful,e.g.inreducingtheexperimentalerroronthemassdetermination.Similarcommentsapplytothem2j`(hi)distributionsshowninFigures 3-6 (b)and 3-7 (b).HereagaintheendpointisasymmetricfunctionofRABandRBC,andtherealandfakesolutionspredictidenticalendpoints.However,whiletheendpointsarethesame,thistimetheshapesarenot.TheshapedifferenceismorepronouncedinthecaseofLM1showninFigure 3-6 (b),andlessvisibleinthecaseofLM6showninFigure 3-7 (b).TheremainingfourdistributionsshowninFigures 3-6 (c-f)and 3-7 (c-f)alreadyhavedifferentendpointsandcanthusbeusedfordiscriminationamongtherealandfakesolutionformB.AlloftheendpointsinFigures 3-6 (c-f)and 3-7 (c-f)arerelativelysharpandshouldbemeasuredratherwell.OneshouldnotforgetthatinFigures 3-6 and 3-7 94

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weshowm2j`(s)()distributionsforonlythreerepresentativevaluesof:=inpanels(d),=)]TJ /F6 11.955 Tf 9.3 0 Td[(1inpanels(e),and=0.5inpanels(f).AsseeninFigure 3-2 ,thereareinnitelymanyotherchoicesfor,whichwouldstillexhibitdifferentendpointsfortherealandfakemBsolutions.OurconclusionisthatthroughasuitablecombinationofadditionalendpointmeasurementsonewouldbeabletotellaparttherealsolutionformBfromitsfakecousin. 95

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Figure3-6. Someotherone-dimensionalinvariantmassdistributionsofinterest,forthecaseoftheLM1massspectrum(redsolidlines)andLM1'massspectrum(bluedottedlines):(a)m2j``distribution;(b)m2j`(hi)distribution;(c)m2j`(p)distribution;(d)m2j`(lo)distribution;(e)m2j`(s)(=)]TJ /F6 11.955 Tf 9.3 0 Td[(1)distribution;(f)m2j`(s)(=1 2)distribution.Alldistributionsarethenplottedversusthecorrespondingmass,onaquadraticscaleforthex-axis. 96

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Figure3-7. Someotherone-dimensionalinvariantmassdistributionsofinterestfortheLM6massspectrum(redsolidlines)andtheLM6'massspectrum(bluedottedlines). 97

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CHAPTER4SUBSYSTEMMT2METHODTheideaforasubsystemMT2wasrstdiscussedin[ 85 ]andappliedin[ 86 ]foraspecicsupersymmetryexample(associatedsquark-gluinoproductionanddecay).Hereweshallgeneralizethatconceptforacompletelygeneraldecaychain.ThesubsystemMT2variablewillbedenedforthesubchaininsidetheblue(yellow-shaded)boxinFigure 4-1 .BeforewegiveaformaldenitionofthesubsystemMT2variables,letusrstintroducesometerminologyfortheBSMparticlesappearinginthedecaychain.WeshallnditconvenienttodistinguishthefollowingtypesofBSMparticles: Grandparents.ThosearethetwoBSMparticlesXnattheverytopofthedecaychainsinFigure 4-1 .Sincewehaveassumedsymmetricevents,thetwograndparentsineacheventareidentical,andcarrythesameindexn.Ofcourse,onemayrelaxthisassumption,andconsiderasymmetricevents,aswasdonein[ 86 87 ].Then,thetwograndparentswillbedifferent,andonewouldsimplyneedtokeeptrackoftwoseparategrandparentindicesn(1)andn(2). Parents.ThosearethetwoBSMparticlesXpatthetopofthesubchainusedtodenethesubsystemMT2variable.InFigure 4-1 thissubchainisidentiedbytheblue(yellow-shaded)rectangularbox.TheideabehindthesubsystemMT2issimplytoapplytheusualMT2denitionforthesubchaininsidethisbox.NoticethattheMT2conceptusuallyrequirestheparentstobeidentical,thereforeherewewillcharacterizethembyasingleparentindexp. Children.ThosearethetwoBSMparticlesXcattheveryendofthesubchainusedtodenethesubsystemMT2variable,asindicatedbytheblue(yellow-shaded)rectangularboxinFigure 4-1 .Thechildrenarealsocharacterizedbyasingleindexc.Ingeneral,thetruemassMcofthetwochildrenisunknown.Asusual,whencalculatingthevalueoftheMT2variable,oneneedstochooseachildtestmass,whichweshalldenotewithatilde,~Mc,inordertodistinguishitfromthetruemassMcofXc. Darkmattercandidates.ThosearethetwostableneutralparticlesX0appearingattheveryendofthecascadechain.Weseethatwhilethosearetheparticlesresponsibleforthemeasuredmissingmomentumintheevent,theyarerelevantforMT2onlyinthespecialcaseofc=0. 98

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Figure4-1. IllustrationofthesubsystemM(n,p,c)T2variabledenedinEquation( 4 ). Withthosedenitions,wearenowreadytogeneralizetheconventionalMT2denition[ 10 32 ].FromFigure 4-1 weseethatanysubchainisspeciedbytheparentindexpandthechildindexc,whilethetotallengthofthewholechain(andthusthetypeofevent)isgivenbythegrandparentindexn.Therefore,thesubsystemMT2variablewillhavetocarrythosethreeindicesaswell,andweshallusethenotationM(n,p,c)T2.InthefollowingweshallrefertothisgeneralizedquantityaseithersubsystemorsubchainMT2.Itisclearthatthesetofthreeindices(n,p,c)mustbeorderedasfollows: np>c0.(4)WeshallnowgiveaformaldenitionofthequantityM(n,p,c)T2,generalizingtheoriginalideaofMT2[ 10 32 ].Theparentandchildindicespandcuniquelydeneasubchain,withinwhichonecanformthetransversemassesM(1)TandM(2)Tofthetwoparents: M(k)T(p(k)p,p(k)p)]TJ /F5 7.97 Tf 6.58 0 Td[(1,...,p(k)c+1,~P(k)cT;~Mc),k=1,2.(4)Herep(k)i,c+1ip,arethemeasured4-momentaoftheSMparticleswithinthesubchain,~P(k)cTaretheunknowntransversemomentaofthechildren,while~Mcistheirunknown(test)mass.Then,thesubsystemM(n,p,c)T2isdenedbyminimizingthelargerofthetwotransversemassesasinEquation( 4 )overtheallowedvaluesofthechildren's 99

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transversemomenta~P(k)cT: M(n,p,c)T2(~Mc)=minP2k=1~P(k)cT=)]TJ /F15 7.97 Tf 7.99 5.98 Td[(P2k=1Pnj=c+1~p(k)jT)]TJ /F7 7.97 Tf 6.18 0.33 Td[(~pTnmaxnM(1)T,M(2)Too,(4)where~pTindicatesanyadditionaltransversemomentumduetoinitialstateradiation(ISR)(Figures 4-1 ).Noticethatinthisdenition,thedependenceonthegrandparentindexnentersonlythroughtherestrictiononthechildren'stransversemomenta~P(k)cT.Usingmomentumconservationinthetransverseplane 2Xk=1~P(k)0T+2Xk=1nXj=1~p(k)jT+~pT=0,(4)Furthermore,themeasurementofthemissingtransversemomentum~pT,missprovidestwoadditionalconstraints ~P(1)0T+~P(2)0T=~pT,miss(4)ontheunknowntransversemomentumcomponents~P(k)0TNowwecanrewritetherestrictiononthechildren'stransversemomenta~P(k)cTas 2Xk=1~P(k)cT=2Xk=1~P(k)0T+2Xk=1cXj=1~p(k)jT=~pT,miss+2Xk=1cXj=1~p(k)jT,(4)whereinthelaststepweusedEquation( 4 ).Equation( 4 )allowsustorewritethesubsystemM(n,p,c)T2denitionofEquation( 4 )inaformwhichdoesnotmanifestlydependonthegrandparentindexn: M(n,p,c)T2(~Mc)=minP2k=1~P(k)cT=~pT,miss+P2k=1Pcj=1~p(k)jTnmaxnM(1)T,M(2)Too.(4)However,thegrandparentindexnisstillimplicitlypresentthroughtheglobalquantity~pT,miss,whichknowsaboutthewholeevent.Weshallseebelowthattheinterpretationoftheexperimentallyobservableendpoints,kinks,etc.,forthesodenedsubsystemM(n,p,c)T2quantity,doesdependonthegrandparentindexn,whichjustiesournotation. 100

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WearenowinapositiontocompareoursubsystemM(n,p,c)T2quantitytotheconventionalMT2variable.Thelatterisnothingbutthespecialcaseofn=pandc=0: MT2M(n,n,0)T2,(4)i.e.theconventionalMT2issimplycharacterizedbyasingleintegern,whichindicatesthelengthofthedecaychain.WeseethatwearegeneralizingtheconventionalMT2variableintwodifferentaspects:rst,weareallowingtheparentsXptobedifferentfromtheparticlesXnoriginallyproducedintheevent(thegrandparents),andsecond,weareallowingthechildrenXctobedifferentfromthedarkmatterparticlesX0appearingattheendofthecascadechainandresponsibleforthemissingenergy.Thebenetsofthisgeneralizationwillbecomeapparentinthenextsection,whereweshalldiscusstheavailablemeasurementsfromthedifferentsubsystemM(n,p,c)T2variables. 4.1AShortDecayChainX2!X1!X0Aarelativelylong(n3)newphysicsdecaychaincanbehandledbyavarietyofmassmeasurementmethods,andinprincipleacompletedeterminationofthemassspectruminthatcaseispossibleatahadroncollider.Wealsoshowedthatarelativelyshort(n=1orn=2)decaychainwouldpresentamajorchallenge,andacompletemassdeterminationmightbepossibleonlythroughMT2methods.Fromnowonweshallthereforeconcentrateonlyonthismostproblematiccaseofn2.FirstletussummarizewhattypesofsubsystemM(n,p,c)T2measurementsareavailableinthecaseofn2.AsillustratedinFigure 4-2 ,thereexistatotalof4differentM(n,p,c)T2quantities.EachM(n,p,c)T2distributionwouldexhibitanupperendpointM(n,p,c)T2,max,whosemeasurementwouldprovideoneconstraintonthephysicalmasses.Inordertobeabletoinvertandsolveforthemassesofthenewparticlesintermsofthemeasuredendpoints,weneedtoknowtheanalyticalexpressionsrelatingtheendpointsM(n,p,c)T2,maxtothephysicalmassesMi. 101

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Figure4-2. ThesubsystemM(n,p,c)T2variableswhichareavailableforn=1andn=2events. InthissectionwesummarizethoserelationsforeachM(n,p,c)T2quantitywithn2.Someoftheseresults(e.g.portionsofSections 4.1.1 andSections 4.1.3 )havealreadyappearedintheliterature,andweincludethemhereforcompleteness.ThediscussioninSections 4.1.2 andSections 4.1.4 ,ontheotherhand,isnew.Inallcases,weshallallowforthepresenceofanarbitrarytransversemomentumpTduetoISR.Thisrepresentsageneralizationofallexistingresultsintheliterature,whichhavebeenderivedinthetwospecialcasespT=0[ 37 ]orpT=1[ 36 ].WeshallnditconvenienttowritetheformulasfortheendpointsM(n,p,c)T2,maxnotintermsoftheactualmasses,butintermsofthemassparameters (n,p,c)Mn 21)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M2c M2p.(4)Theadvantageofusingthisshorthandnotationwillbecomeapparentveryshortly. 4.1.1TheSubsystemVariableM(1,1,0)T2Westartwiththesimplestcaseofn=1showninFigure 4-2 (a).HereM(1,1,0)T2istheonlypossibility,anditcoincideswiththeconventionalMT2variable,asindicatedbyEquation( 4 ).Therefore,thepreviousresultsintheliteraturewhichhavebeenderived 102

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fortheconventionalMT2variableinEquation( 4 ),wouldstillapply.Inparticular,inthelimitofpT=0,theupperendpointM(1,1,0)T2,maxdependsonthetestmass~M0asfollows[ 37 ] M(1,1,0)T2,max(~M0,pT=0)=(1,1,0)+q 2(1,1,0)+~M20,(4)wheretheparameter(1,1,0)isdenedintermsofthephysicalmassesM1andM0accordingtoEquation( 4 ): (1,1,0)M1 21)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M20 M21=M21)]TJ /F3 11.955 Tf 11.96 0 Td[(M20 2M1.(4)Asusual,theendpointinEquation( 4 )canbeinterpretedasthemassM1oftheparentparticleX1,sothatEquation( 4 )providesarelationbetweenthemassesofX0andX1.IntheearlyliteratureonMT2,thisrelationhadtobederivednumerically,bybuildingtheMT2distributionsfordifferentvaluesofthetestmass~M0,andreadingofftheirendpoints.Nowadays,withtheworkofChoetal.[Measuringsuperparticlemassesathadroncolliderusingthetransversemasskink,JHEP0802,035(2008)][ 37 ],therelationisknownanalytically,and,asseenfromEquation( 4 ),isparameterizedbyasingleparameter(1,1,0).Therefore,inordertoextractthevalueofthisparameter,weonlyneedtoperformasinglemeasurement,i.e.weonlyneedtostudytheMT2distributionforoneparticularchoiceofthetestmass~M0.Weshallnditconvenienttochoose~M0=0,inwhichcaseEquations( 4 )and( 4 )give M(1,1,0)T2,max(~M0=0,pT=0)=2(1,1,0)=M21)]TJ /F3 11.955 Tf 11.95 0 Td[(M20 M1,(4)providingtherequiredmeasurementoftheparameter(1,1,0)demonstratestheusefulnessoftheMT2conceptjustasinglemeasurementoftheendpointoftheMT2distributionforasinglexedvalueofthetestmass~M0issufcienttoprovideuswithoneconstraintamongtheunknownmasses(M1andM0inthiscase).Unfortunately,onesinglemeasurementinEquation( 4 )isnotenoughtopindowntwodifferentmasses.InordertomeasurebothM0andM1,withoutany 103

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theoreticalassumptionsorprejudice,weobviouslyneedadditionalexperimentalinput.FromthegeneralexpressionasEquation( 4 )itisclearthatmeasuringotherM(1,1,0)T2,maxendpoints,fordifferentvaluesofthetestmass~M0,willnothelp,sincewewillsimplybemeasuringthesamecombinationofmasses(1,1,0)overandoveragain,obtainingnonewinformation.Anotherpossibilitymightbetoconsidereventswiththenextlongestdecaychain(n=2),which,asadvertisedintheIntroductionandshownbelowinSection 4.2 ,willbeabletoprovideenoughinformationforacompletemassdeterminationofallparticlesX0,X1andX2.However,theexistenceandtheobservationofthen=2decaychainiscertainlynotguaranteedtobeginwith,theparticlesX2maynotexist,ortheymayhavetoolowcross-sections.Itisthereforeofparticularimportancetoaskthequestionwhetherthen=1processinFigure 4-2 (a)alonecanallowadeterminationofbothM0andM1.Theanswertothisquestion,atleastinprinciple,isYes[ 36 ],andwhatismore,onecanachievethisusingtheverysameMT2variableM(1,1,0)T2.Thekeyistorealizethatinrealityatanycollider,andespeciallyathadroncollidersliketheTevatronandtheLHC,therewillbesizablecontributionsfrominitialstateradiation(ISR)withnonzeropT,whereoneormorejetsareradiatedofftheinitialstate,beforethehardscatteringinteraction.(InFigures 4-1 and 4-2 thegreenellipserepresentsthehardscattering,whileISRstandsforagenericISRjet.).ThiseffectleadstoadrasticchangeinthebehavioroftheM(1,1,0)T2,max(~M0,pT)function,whichstartstoexhibitakinkatthetruelocationofthechildmass~M0=M0: @M(1,1,0)T2,max(~M0,pT) @~M0!~M0=M0)]TJ /F7 7.97 Tf 6.59 0 Td[(6= @M(1,1,0)T2,max(~M0,pT) @~M0!~M0=M0+,(4)andfurthermore,thevalueofM(1,1,0)T2,maxatthatpointrevealsthetruemassoftheparentaswell: M(1,1,0)T2,max(~M0=M0,pT)=M1.(4) 104

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ThiskinkfeatureinEquations( 4 4 )wasobservedandillustratedinA.J.Barretal.[WeighingWimpswithKinksatColliders:InvisibleParticleMassMeasurementsfromEndpoints,JHEP0802,014(2008)]Section4.4[ 36 ].Wendthatitcanalsobeunderstoodanalytically,bygeneralizingtheresultofEquation( 4 )toaccountfortheadditionalISRtransversemomentum~pT.RecallthatEquation( 4 )wasderivedinRef.[ 37 ]undertheassumptionthatthemissingtransversemomentumduetothetwoescapingparticlesX0isexactlybalancedbythetransversemomentaofthetwovisibleparticlesx1usedtoformM(1,1,0)T2: ~P(1)0T+~P(2)0T+~p(1)1T+~p(2)1T=0.(4)Wemaysometimesrefertothissituationasabalancedmomentumconguration1.InthepresenceofISRwithsomenon-zerotransversemomentum~pT,Equation( 4 )ingeneralceasestobevalid,andismodiedto ~P(1)0T+~P(2)0T+~p(1)1T+~p(2)1T=)]TJ /F8 11.955 Tf 8.73 0.49 Td[(~pT,(4)inaccordancewith( 4 ).IncludingtheISReffects,wendthattheexpression( 4 )fortheM(1,1,0)T2,maxendpointsplitsintotwobranches M(1,1,0)T2,max(~M0,pT)=8><>:F(1,1,0)L(~M0,pT),if~M0M0,F(1,1,0)R(~M0,pT),if~M0M0,(4) 1ThisshouldnotbeconfusedwiththetermbalancedusedfortheanalyticMT2solutionsdiscussedin[ 33 37 ]. 105

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where F(1,1,0)L(~M0,pT)=8<:"(1,1,0)(pT)+r (1,1,0)(pT)+pT 22+~M20#2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(p2T 49=;1 2, (4) F(1,1,0)R(~M0,pT)=8<:"(1,1,0)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT)+r (1,1,0)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT))]TJ /F3 11.955 Tf 13.15 8.09 Td[(pT 22+~M20#2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(p2T 49=;1 2, (4) andthepT-dependentparameter(1,1,0)(pT)isdenedas (1,1,0)(pT)=(1,1,0)0@s 1+pT 2M12)]TJ /F3 11.955 Tf 17.11 8.09 Td[(pT 2M11A.(4)Bothbranchescorrespondtoextrememomentumcongurationsinwhichallthreetransversevectors~p(1)1T,~p(2)1Tand~pTarecollinear.ThedifferenceisthattheleftbranchF(1,1,0)Lcorrespondstotheconguration~p(1)1T""~p(2)1T""~pT,whiletherightbranchF(1,1,0)Rcorrespondsto~p(1)1T""~p(2)1T"#~pT.Therefore,thetwobranchesaresimplyrelatedas F(1,1,0)R(~M0,pT)=F(1,1,0)L(~M0,)]TJ /F3 11.955 Tf 9.3 0 Td[(pT).(4)ItiseasytoverifythatintheabsenceofISR,(i.e.forpT=0)ourgeneralresult( 4 )reducestothepreviousformula( 4 ).Ourresult( 4 )fortheM(1,1,0)T2,maxupperkinematicendpointasafunctionofthetestmass~M0isillustratedinFigure 4-3 (a).WeconsiderasingleISRjetandshowresultsforseveraldifferentvaluesofitstransversemomentumpT,startingfrompT=0(thegreensolidline)andincreasingthevalueofpTinincrementsofpT=100GeV.TheuppermostsolidlinecorrespondstothelimitingcasePT!1.Thetruevalueoftheparent(child)massismarkedbythehorizontal(vertical)dottedline.Thered(blue)linescorrespondtothefunctionF(1,1,0)L(F(1,1,0)R).ThesolidportionsofthoselinescorrespondtothetrueM(1,1,0)T2,maxendpoint,whilethedashedsegmentsaresimplytheextensionofF(1,1,0)LandF(1,1,0)Rintothewrongregionfor~M0,givingafalseendpoint. 106

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Figure4-3. (a)DependenceoftheM(1,1,0)T2,maxupperkinematicendpoint(solidlines)onthevalueofthetestmass~M0,forM1=300GeV,andM0=100GeV,andfordifferentvaluesofthetransversemomentumpToftheISRjet,startingfrompT=0(greenline),andincreasinguptopT=3TeVinincrementsofpT=100GeV,frombottomtotop.TheuppermostlinecorrespondstothelimitingcasepT!1.Thehorizontal(vertical)dottedlinedenotesthetruevalueoftheparent(child)mass.Solid(dashed)linesindicatetrue(false)endpoints.TheredlinescorrespondtothefunctionF(1,1,0)LdenedinEquation( 4 ),whilethebluelinescorrespondtothefunctionF(1,1,0)RdenedinEquation( 4 ).(b)Thevalueofthekink(1,1,0)denedin( 4 ),asafunctionofthedimensionlessratiospT M1andM0 M1. Figure 4-3 (a)revealsthatthetwobranchesinEquations( 4 )and( 4 )alwayscrossatthepoint(M0,M1),inagreementwithEquation( 4 ).Interestingly,thesharpnessoftheresultingkinkat~M0=M0dependsonthehardnessoftheISRjet,ascanbeseendirectlyfrom( 4 ).ForsmallpT,thekinkisbarelyvisible,andinthelimitpT!0weobtaintheoldresult( 4 )forthebalancedmomentumconguration,shownwiththegreensolidline,whichdoesnotexhibitanykink.Intheotherextreme,atverylargepT,weseeapronouncedkink,whichhasawell-denedlimitaspT!1.TheM(1,1,0)T2,maxkinkexhibitedinEquation( 4 )andinFigure 4-3 (a)isourrst,butnotlast,encounterwithakinkfeatureinanM(n,p,c)T2variable.BelowweshallseethattheMT2kinksarerathercommonphenomena,andweshallencounteratleasttwootherkinktypesbytheendofSection 4.1 .Therefore,wenditconvenienttoquantifythe 107

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sharpnessofanysuchkinkasfollows.ConsideragenericsubsystemM(n,p,c)T2variablewhoseendpointM(n,p,c)T2,max(~Mc,pT)exhibitsakink: M(n,p,c)T2,max(~Mc,pT)=8><>:F(n,p,c)L(~Mc,pT),if~McMc,F(n,p,c)R(~Mc,pT),if~McMc.(4)ThekinkappearsbecauseM(n,p,c)T2,max(~Mc,pT)isnotgivenbyasinglefunction,buthastwoseparatebranches.Therst(low)branchappliesfor~McMc,andisgivenbysomefunctionF(n,p,c)L(~Mc,pT),whilethesecond(high)branchisvalidfor~McMc,andisgivenbyadifferentfunction,F(n,p,c)R(~Mc,pT).ThefunctionM(n,p,c)T2,max(~Mc,pT)itselfiscontinuousandthetwobranchescoincideat~Mc=Mc: F(n,p,c)L(Mc,pT)=F(n,p,c)R(Mc,pT),(4)buttheirderivativesdonotmatch: @F(n,p,c)L @~Mc!~Mc=Mc6= @F(n,p,c)R @~Mc!~Mc=Mc,(4)leadingtotheappearanceofthekink.LetusdenetheleftandrightslopeoftheM(n,p,c)T2,max(~Mc,pT)functionat~Mc=Mcintermsoftwoangles(n,p,c)Land(n,p,c)R,correspondingly: tan(n,p,c)L @F(n,p,c)L(~Mc) @~Mc!~Mc=Mc, (4) tan(n,p,c)R @F(n,p,c)R(~Mc) @~Mc!~Mc=Mc. (4) Nowweshalldenetheamountofkinkastheangulardifference(n,p,c)betweenthetwobranches: (n,p,c)(n,p,c)R)]TJ /F6 11.955 Tf 11.95 0 Td[((n,p,c)L=arctan tan(n,p,c)R)]TJ /F6 11.955 Tf 11.96 0 Td[(tan(n,p,c)L 1+tan(n,p,c)Rtan(n,p,c)L!.(4) 108

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Alargevalueof(n,p,c)impliesthattherelativeanglebetweenthelowandhighbranchesatthepointoftheirjunction~Mc=Mcisalsolarge,andinthatsensethekinkwouldbemorepronouncedandrelativelyeasiertosee.ThisdenitioncanbeimmediatelyappliedtotheM(1,1,0)T2,maxkinkthatwejustdiscussed.Substitutingtheformulas( 4 )and( 4 )forthetwobranchesF(1,1,0)LandF(1,1,0)Rintothedenitions( 4 4 )andsubsequentlyinto( 4 ),weobtainanexpressionforthesize(1,1,0)oftheM(1,1,0)T2,maxkink: (1,1,0)=arctan M0(M21)]TJ /F3 11.955 Tf 11.95 0 Td[(M20)pTp 4M21+p2T M1(M21)]TJ /F3 11.955 Tf 11.96 0 Td[(M20)2+2M20M1(4M21+p2T)!.(4)Theresult( 4 )isillustratednumericallyinFigure 4-3 (b).Ascanbeseenfrom( 4 ),(1,1,0)dependsonthetwomassesM0andM1,aswellasthesizeoftheISRpT.However,since(1,1,0)isadimensionlessquantity,itsdependenceonthosethreeparameterscanbesimplyillustratedintermsofthedimensionlessratiospT M1andM0 M1.ThisiswhyinFigure 4-3 (b)weplot(1,1,0)(indegrees)asafunctionofpT M1andM0 M1.Figure 4-3 (b)conrmsthatthekinkdevelopsatlargepT,andiscompletelyabsentatpT=0,aresultwhichmayhavealreadybeenanticipatedonthebasisofFigure 4-3 (a).ForanygivenmassratioM0 M1,thekinkislargestforthehardestpossiblepT.InthelimitpT!1weobtain limpT!1(1,1,0)=arctanM21)]TJ /F3 11.955 Tf 11.96 0 Td[(M20 2M0M1.(4)FromFigure 4-3 (b)onecanseethatatsufcientlylargepT,the(1,1,0)contoursbecomealmosthorizontal,i.e.thesizeofthekink(1,1,0)becomesveryweaklydependentonpT.AcarefulexaminationofFigure 4-3 (b)revealsthattheasymptoticbehavioratpT!1isinagreementwiththeanalyticalresult( 4 ).Noticethatthemaximumpossiblevalueofanykinkofthetype( 4 )is(n,p,c)max=90.AccordingtoFigure 4-3 (b)andEquation( 4 ),inthecaseof(1,1,0)theabsolutemaximumcanbeobtainedonlyinthepT!1andM0!0limit.Theformerconditionwillneverbe 109

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realizedinarealisticexperiment,whilethelatterconditionmakestheobservationofthekinkratherproblematic,sincethelowbranchFLoftheM(1,1,0)T2,max(~M0,pT)functionistooshorttobeobservedexperimentally.Therefore,underrealisticcircumstances,wewouldexpectthesizeofthekink(1,1,0)tobeonlyontheorderofafewtensofdegrees,whicharethemoretypicalvaluesseeninFigure 4-3 (b).AccordingtoFigure 4-3 (b),foragivenxedpT,thesharpnessofthe(1,1,0)kinkdependsonthemasshierarchyoftheparticlesX1andX0.Whentheyarerelativelydegenerate,i.e.theirmassratioM0 M1islarge,thekinkisrelativelysmall.Conversely,whenX0ismuchlighterthanX1,thekinkismorepronounced.TheoptimummassratioM0 M1whichmaximizesthekinkforagivenpT,isratherweaklydependentonthepT,andforpT!1eventuallygoestozero,inagreementwithEquation( 4 ).However,formorereasonablevaluesofpTastheonesshownonthelefthalfoftheplot,theoptimalratioM0 M1variesbetween0.3(atpT0)to0.1(atpT5M1).Inthissense,thevalueofM0 M1=1 3whichwaschosenfortheillustrationinFigure 4-3 (a).Inconclusionofthissubsection,itisworthsummarizingthemainpointsfromit.Thegoodnewsisthatthe(1,1,0)kinkinprincipleoffersasecond,independentpieceofinformationaboutthemassesoftheparticlesX0andX1.WhentakentogetherwiththeM(1,1,0)T2,maxendpointmeasurement( 4 ),itwillallowustodeterminebothmassesM0andM1,inacompletelymodel-independentway.Ouranalyticalresultsregardingthe(1,1,0)kinkcomplementthestudyofA.J.Barretal.[WeighingWimpswithKinksatColliders:InvisibleParticleMassMeasurementsfromEndpoints,JHEP0802,014(2008)].[ 36 ],wherethiskinkwasrstdiscovered.However,onthedownside,weshouldmentionthatmuchofourdiscussionregardingthe(1,1,0)kinkmaybeoflimitedpracticalinterest,forseveralreasons.First,asseeninFigure 4-3 ,thekinkbecomesvisibleonlyforsufcientlylargevaluesofthepT.SincetheISRpTspectrumisfallingrathersteeply,onewouldneedtocollectrelativelylargeamountsofdata,inordertoguaranteethepresenceofeventswithsufcientlyhardISRjets.Eventhen, 110

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thecollectedeventsmaynotcontainthemomentumcongurationrequiredtogivethemaximumvalueofM(1,1,0)T2.AnalternativeapproachtomakeuseofthekinkstructurewouldbetomeasuretheendpointfunctionM(1,1,0)T2,max(~M0,pT)forseveraldifferentpTranges,andthentittotheanalyticalformula( 4 ).Whetherandhowwellthiscanworkinpractice,remainstobeseen,buttheresultsof[ 36 ]fromatoyexerciseintheabsenceofanybackgroundsanddetectorresolutioneffectsdonotappearveryencouraging.Nevertheless,whilethekinkstructure(1,1,0)maybedifculttoobserve,themeasurement( 4 )oftheendpointM(1,1,0)T2,max(~M0=0,pT=0)shouldberelativelystraightforward.InSections 4.2.1 and 4.2.3 weshallseethattheadditionalMT2informationfromeventswithn=2decaychainswilleventuallyallowustodeterminealltheunknownmasses. 4.1.2TheSubsystemVariableM(2,2,1)T2ThesubsystemvariableM(2,2,1)T2isillustratedinFigure 4-2 (b),whereweusethesubchainwithinthesmallerrectangleontheleft.M(2,2,1)T2isagenuinesubchainvariableinthesensethatweonlyusetheSMdecayproductsx2,andignoreanyremainingobjectsarisingfromthetwox1's.IntheabsenceofISR(pT=0)onecanadapttheresultsfrom[ 37 ]andshowthattheformulafortheM(2,2,1)T2endpointis M(2,2,1)T2,max(~M1,pT=0)=(2,2,1)+q 2(2,2,1)+~M21,(4)wheretheparameter(2,2,1)wasdenedinEquation( 4 ): (2,2,1)M2 21)]TJ /F3 11.955 Tf 13.15 8.08 Td[(M21 M22=M22)]TJ /F3 11.955 Tf 11.96 0 Td[(M21 2M2.(4)AlmostallofourdiscussionfromthepreviousSection 4.1.1 canbedirectlyappliedhereaswell.Forexample,inordertomeasuretheparameter(2,2,1),weonlyneedtoextracttheendpointofasingledistribution,forasinglexedvalueofthetestmass~M1.As 111

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before,wechoosetouse~M1=0.Theresultingendpointmeasurement M(2,2,1)T2,max(~M1=0,pT=0)=2(2,2,1)=M22)]TJ /F3 11.955 Tf 11.96 0 Td[(M21 M2(4)providestherequiredmeasurementoftheparameter(2,2,1)appearinginEquation( 4 ),aswellasoneconstraintonthemassesM1andM2involvedintheproblem.Moreimportantly,thenewconstraintinEquation( 4 )isindependentoftherelationinEquation( 4 )foundpreviouslyinSection 4.1.1 .ThenewvariableM(2,2,1)T2willalsoexhibitakinkintheplotofitsendpointM(2,2,1)T2,maxasafunctionofthetestmass~M1.Thisisthesametypeofkinkastheonediscussedintheprevioussubsection,thereforeallofourpreviousresultswouldapplyhereaswell.Inparticular,theanalyticalexpressionforthekinkisgivenby M(2,2,1)T2,max(~M1,pT)=8><>:F(2,2,1)L(~M1,pT),if~M1M1,F(2,2,1)R(~M1,pT),if~M1M1,(4)where F(2,2,1)L(~M1,pT)=8<:"(2,2,1)(pT)+r (2,2,1)(pT)+pT 22+~M21#2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(p2T 49=;1 2, (4) F(2,2,1)R(~M1,pT)=8<:"(2,2,1)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT)+r (2,2,1)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT))]TJ /F3 11.955 Tf 13.15 8.08 Td[(pT 22+~M21#2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(p2T 49=;1 2, (4) andthepT-dependentparameter(2,2,1)(pT)isdenedinanalogytoEquation( 4 ) (2,2,1)(pT)=(2,2,1)0@s 1+pT 2M22)]TJ /F3 11.955 Tf 17.11 8.08 Td[(pT 2M21A.(4)Thesizeofthenewkink(2,2,1)canbeeasilyreadofffromEquation( 4 ),whereoneshouldmaketheobviousreplacementsM0!M1andM1!M2. 112

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Wecannowgeneralizethetwoexamplesdiscussedsofar(M(1,1,0)T2andM(2,2,1)T2)tothecaseofanarbitrarygrandparentindexn,withp=nandc=n)]TJ /F6 11.955 Tf 11.95 0 Td[(1.Weget M(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)T2,max(~Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,pT)=8><>:F(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)L(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT),if~Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,F(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)R(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT),if~Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,(4)where F(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)L(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT)=8<:"(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)+r (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)+pT 22+~M2n)]TJ /F5 7.97 Tf 6.58 0 Td[(1#2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(p2T 49=;1 2,(4) F(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)R(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT)=8<:"(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT)+r (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT))]TJ /F3 11.955 Tf 13.15 8.09 Td[(pT 22+~M2n)]TJ /F5 7.97 Tf 6.58 0 Td[(1#2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(p2T 49=;1 2,(4)andthepT-dependentparameter(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)issimplythegeneralizationofEquations( 4 )and( 4 ): (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)=(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)0@s 1+pT 2Mn2)]TJ /F3 11.955 Tf 17.17 8.09 Td[(pT 2Mn1A.(4)Forn=1orn=2,thegeneralformulaofEquation( 4 )reproducesourpreviousresultsinEauations( 4 )and( 4 ),correspondingly. 4.1.3TheSubsystemVariableM(2,2,0)T2ThevariableM(2,2,0)T2isillustratedinFigure 4-2 (b),whereweusethewholechainwithinthelargerrectangle.AslongasweignoretheeffectsofanyISR,wehaveabalanced2momentumcongurationandtheanalyticalresultsfromRef.[ 37 ]wouldapply.Inparticular,theendpointM(2,2,0)T2,max(~M0,pT=0)isgivenby[ 37 ] M(2,2,0)T2,max(~M0,pT=0)=8><>:F(2,2,0)L(~M0,pT=0),if~M0M0,F(2,2,0)R(~M0,pT=0),if~M0M0,(4) 2InthesenseofEquation( 4 ).SeethediscussionfollowingEquation( 4 ). 113

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where F(2,2,0)L(~M0,pT=0)=(2,2,0)+q 2(2,2,0)+~M20, (4) F(2,2,0)R(~M0,pT=0)=(2,2,1)+(2,1,0)+q )]TJ /F8 11.955 Tf 5.48 -9.68 Td[((2,2,1))]TJ /F8 11.955 Tf 11.95 0 Td[((2,1,0)2+~M20, (4) andthevariousparameters(n,p,c)aredenedin( 4 ).NoticethattheseexpressionsarevalidonlyforpT=0.WehavealsoderivedthecorrespondinggeneralizedexpressionforM(2,2,0)T2,max(~M0,pT)forarbitraryvaluesofpT,whichwelistinAppendix B .Themoststrikingfeatureoftheendpointfunction( 4 )isthatitwillalsoexhibitakink(2,2,0)atthetruevalueofthetestmass~M0=M0.However,asemphasizedin[ 36 ],thephysicaloriginofthiskinkisdifferentfromthekinks(1,1,0)and(2,2,1)whichweencounteredpreviouslyinSections 4.1.1 and 4.1.2 .ThisiseasytounderstandinSections 4.1.1 and 4.1.2 wesawthatthekinks(1,1,0)and(2,2,1)ariseduetoISReffects,whileEquation( 4 )holdsintheabsenceofanyISR.Theexplanationforthe(2,2,0)kinkhasactuallyalreadybeenprovidedin[ 37 ].Inessence,onecantreattheSMdecayproductsx1andx2ineachchainasacompositeparticleofvariablemass,andthetwobranchesF(2,2,0)LandF(2,2,0)Rcorrespondtothetwoextremevaluesforthemassofthiscompositeparticle.Inspiteofitsdifferentorigin,thekinkinthefunction( 4 )sharesmanyofthesameproperties.Letususeaspecicexampleasanillustration.Considerapopularexamplefromsupersymmetry,suchasgluinopair-production,followedbysequentialtwo-bodydecaystosquarksandthelightestneutralinos.Thisispreciselyacascadeofthetypen=2,inwhichX2isthegluino~g,X1isasquark~q,andX0isthelightestneutralino~01.LetuschoosethesuperpartnermassesaccordingtotheSPS1amassspectrum,whichwasalsousedin[ 37 ]: M2=613GeV,M1=525GeV,M0=99GeV.(4) 114

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Figure4-4. DependenceoftheM(2,2,0)T2,maxandM(2,1,0)T2,maxupperkinematicendpointsonthevalueofthetestmass~M0,for(a)theSPS1aparameterpointinMSUGRA:M2=613GeV,M1=525GeV,andM0=99GeV;or(b)asplitspectrumM2=2000GeV,M1=200GeV,andM0=100GeV.Thehorizontal(vertical)dottedlinesdenotethetruevalueoftheparent(child)massforeachcase.Solid(dashed)linesindicatetrue(false)endpoints,whilered(blue)linescorrespondtoF(n,p,c)L(F(n,p,c)R)branches. TheresultingfunctionM(2,2,0)T2,max(~M0,pT=0)isplottedinFigure 4-4 (a)withtheuppersetoflines.ThereareseveralnoteworthyfeaturesofM(2,2,0)T2,max(~M0,pT=0)whichareevidentfromFigure 4-4 (a).First,whenthetestmass~M0isequaltothetruechildmassM0,theMT2endpointyieldsthetrueparentmass,inthiscaseM2: M(2,2,0)T2,max(~M0=M0,pT=0)=M2.(4)ThispropertyofMT2istruebydesign,andisconrmedbythedottedlinesinFigure 4-4 (a).Second,asseenfromEquation( 4 ),M(2,2,0)T2,max(~M0,pT=0)isnotgivenbyasinglefunction,buthastwoseparatebranches.Therst(low)branchF(2,2,0)Lappliesfor~M0M0,andisshowninFigure 4-4 (a)withredlines.Thesecond(high)branchF(2,2,0)Risvalidfor~M0M0andisshowninblueinFigure 4-4 (a).Whilethetwobranchescoincideat~M0=M0: F(2,2,0)L(M0,pT=0)=F(2,2,0)R(M0,pT=0).(4) 115

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Figure4-5. Theamountofkink:(a)(2,2,0)and(b)(2,1,0)indegrees,asafunctionofthemassratiosp yandp z.Thewhitedotandthewhiteasteriskdenotethelocationsinthis(p y,p z)parameterspaceofthetwosamplespectra( 4 )and( 4 )usedforFigures 4-4 (a)and 4-4 (b),correspondingly. Theirderivativesdonotmatch: @F(2,2,0)L @~M0!~M0=M06= @F(2,2,0)R @~M0!~M0=M0,(4)leadingtoakink(2,2,0)inthefunctionM(2,2,0)T2,max(~M0,pT=0)[ 34 37 ].Applyingthegeneraldenition( 4 ),weobtainthesizeofthiskinkquantitatively, (2,2,0)=arctan2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)p yz (y+z)(1+yz)+4yz,(4)wherewehavedenedthesquaredmassratios yM21 M22,zM20 M21.(4)Theresult( 4 )isplottedinFigure 4-5 (a)asafunctionofthemassratiosp yandp z.Figure 4-5 (a)demonstratesthatasbothyandzbecomesmall,thekink(2,2,0)getsmorepronounced.Figure 4-5 (a)alsoshowsthatthekink(2,2,0)isasymmetricfunctionofyandz,ascanalsobeseendirectlyfromEquation( 4 ).Therefore,thekink(2,2,0)willbebestobservableinthosecaseswhereyandzarebothsmall,andinaddition,themassspectrumhappenstoobeytherelationy=z,i.e.M1= 116

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p M0M2.ForthisspecialvalueofM1=p M0M2,theupperendpointoftheinvariantmassdistributionMx1x2isthesameasinthecasewhentheintermediateparticleX1isoff-shell,i.e.whenM1>M2.ThenwendthattheM(2,2,0)T2,maxformulasandcorrespondingkinkstructuresareidenticalintheon-shellandoff-shellcases.WeprovidemoredetailsinAppendix B .Unfortunately,theSPS1astudypointisratherfarfromthiscategorythespectrum( 4 )correspondstothevaluesp y=0.856andp z=0.189,whichareindicatedinFigure 4-5 (a)byawhitedot.ThisconclusionisalsosupportedbyFigure 4-4 (a),whichshowsarathermildkinkintheSPS1acase.Weshallberatherambivalentinourattitudetowardthe(2,2,0)kinkaswell.Whiletheinterpretationofthekinkisstraightforward,itsobservationintheactualexperimentisagainanopenissue.Ontheonehand,theexperimentalprecisionwoulddependontheparticularsignature,i.e.thetypeoftheSMparticlesx1andx2.Ifthoseareleptons,their4-momentap(k)1andp(k)2willbemeasuredrelativelywellandthekinkmightbeobservable.However,whenx1andx2arejets,theexperimentalresolutionmaynotbesufcient.Secondly,asseeninFigure 4-4 (a),thekinkitselfmaynotbeverypronounced,anditsobservabilitywillinfactdependontheparticularmassspectrum.Themainlessonfromtheabovediscussionisthatwhiletheexistenceofthekinkiswithoutadoubt,itsactualobservationisbynomeansguaranteed.Therefore,ourmainmassmeasurementmethod,describedlaterinSection 4.2.1 ,willnotuseanyinformationrelatedtothekink.InfactinSection 4.2.1 weshallshowthatonecancompletelyreconstructthemassspectrumofthenewparticles,usingjustmeasurementsofMT2endpoints,eachdoneatasinglexedvalueofthecorrespondingtestmass.Itisworthnotingthat,ingeneral,anendpointinaspectrumisasharperfeaturethanakinkofthetype( 4 ).Therefore,wewouldexpectthattheexperimentalprecisionontheextractedendpointswillbemuchbetterthanthecorrespondingprecisiononthekinklocation.Thekinkwillalsonotplayanyroleinourhybridmethod, 117

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describedinSection 4.2.3 .OnlyforthemethoddescribedinSection 4.2.2 ,weshalltrytomakeuseofthekinkinformation.LetusnowreturntoouroriginaldiscussionoftheM(2,2,0)T2endpoint( 4 ).FollowingourpreviousapproachfromSections 4.1.1 and 4.1.2 ,wewouldchooseaxedvalueofthetestmass~M0andmeasurethecorrespondingMT2endpoint.However,thepresenceoftwobranchesinEquations( 4 )and( 4 )leadstoaslightcomplication:forarandomlychosenvalueof~M0,wewillnotknowwhetherweshoulduseEquations( 4 )or( 4 )wheninterpretingtheendpointmeasurement.Thisrequiresustomakeveryspecialchoicesforthexedvalueof~M0,whichwouldremovethisambiguity.Itiseasytoseethatbychoosing~M0=0,wecanensurethattheendpointisalwaysdescribedbythelowbranchinEquation( 4 ),andtheM(2,2,0)T2,maxmeasurementcanthenbeuniquelyinterpretedas M(2,2,0)T2,max(~M0=0,pT=0)=2(2,2,0)=M22)]TJ /F3 11.955 Tf 11.95 0 Td[(M20 M2.(4)However,wecouldalsodesignaspecialchoiceof~M0,whichwouldselectthehighbranchinEquation( 4 )andagainuniquelyremovethebranchambiguity.Forthispurpose,wemustchooseavalueforthetestmass~M0whichissufcientlylarge,inordertosafelyguaranteethatitiswellbeyondthetruemassM0.SincethetruemassM0canneverexceedthebeamenergyEb,oneobvioussafeandratherconservativechoicefor~M0couldbe~M0=Eb,inwhichcasefrom( 4 )weget M(2,2,0)T2,max(~M0=Eb,pT=0)=(2,2,1)+(2,1,0)+q )]TJ /F8 11.955 Tf 5.48 -9.68 Td[((2,2,1))]TJ /F8 11.955 Tf 11.95 0 Td[((2,1,0)2+E2b (4) =M2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M2 2M21 M22+M20 M21+s M22 4M21 M22)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M20 M212+E2b.NoticethatthehighbranchfunctionF(2,2,0)RinEquation( 4 )isratheruniqueinoneveryimportantaspect:itdependsnotjustonone,butontwomassparameters,namelythecombinations(2,2,1)+(2,1,0)and(2,2,1))]TJ /F8 11.955 Tf 12.79 0 Td[((2,1,0).Incontrast,thelowbranchF(2,2,0)L,aswellasthepreviouslydiscussedendpointfunctionsM(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T2,max(~M0,pT=0), 118

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eachcontainedasingleparameter.Asaresult,inthosecaseswedidnotbenetfromanyextrameasurementsfordifferentvaluesofthetestmass~M0hadwedonethat,wewouldhavebeenmeasuringthesameparameteroverandoveragain.However,thesituationwithF(2,2,0)Risdifferent,andherewewillbenetfromanadditionalmeasurementforadifferentvalueof~M0.Forexample,letuschoose~M0=~Eb,with~Eb>Eb,whichwillstillkeepusonthehighbranch.Weobtainanotherconstraint M(2,2,0)T2,max(~M0=~Eb,pT=0)=(2,2,1)+(2,1,0)+q )]TJ /F8 11.955 Tf 5.48 -9.69 Td[((2,2,1))]TJ /F8 11.955 Tf 11.95 0 Td[((2,1,0)2+~E2b (4) =M2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(M2 2M21 M22+M20 M21+s M22 4M21 M22)]TJ /F3 11.955 Tf 13.15 8.08 Td[(M20 M212+~E2b.ItiseasytocheckthattheconstraintsinEquations( 4 through 4 )areallindependent,thusprovidingthreeindependentequations3forthethreeunknownmassesM0,M1andM2.ThesethreeEquations( 4 through 4 )canbesolvedrathereasily4,andoneobtainsthepropersolutionforthemassesM0,M1andM2,uptoatwo-foldambiguity: M2!M2,M1!M0 M1M2,M0!M0,(4)whichisnothingbuttheinterchangey$zataxedM2.Theambiguityarisesbecausetheexpression( 4 )fortheendpointM(2,2,0)T2,max(andconsequently,thesetofconstraints 3Inpractice,insteadofrelyingonindividualendpointmeasurementsforthreedifferentvaluesof~M0,onemayprefertousetheexperimentalinformationforthewholefunctionM(2,2,0)T2,max(~M0,pT=0)andsimplyttoittheanalyticalexpression( 4 )forthethreeoatingparametersM0,M1andM2,aswasdone[ 37 ].Asweshallseeshortly,thismethoddoesnotleadtoanynewinformation,andmayonlyimprovethestatisticalerroronthemassdetermination.Therefore,tokeepourdiscussionassimpleaspossible,weprefertotalkaboutthethreeindividualmeasurementsinEquations( 4 through 4 )asopposedtottingthewholedistribution( 4 ).4ThegeneralsolutionforM2,M1andM0intermsofthemeasuredendpoints( 4 4 )israthermessyandnotveryilluminating,thereforewedonotlistithere. 119

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( 4 4 ))isinvariantunderthetransformation( 4 ).Becauseofthisambiguity,inadditiontotheoriginalSPS1ainputvalues( 4 )forthemassspectrum,weobtainasecondsolution M2=613GeV,M1=115.6GeV,M0=99GeV.(4)ThissecondsolutionwasmissedintheanalysisofRef.[ 37 ].Itiseasytocheckthatthealternativemassspectrum( 4 )givesanidenticalM(2,2,0)T2,max(~M0,pT=0)distributionastheoneshowninFigure 4-4 (a),sothatitisimpossibletoruleitoutonthebasisofM(2,2,0)T2,maxmeasurementsalone.Thepreviousdiscussionrevealsanimportantandsomewhatoverlookedbenetfromtheexistenceofthekinkonecanmakenotone,nottwo,butthreeindependentendpointmeasurementsfromasingleM(n,p,c)T2distribution!Infact,weshallarguethatthethreemeasurementsinEquations( 4 through 4 )aremuchmorerobustthanthekinkmeasurement( 4 ).Forexample,whenthechildmassisrelativelysmall,thelowerbranchF(2,2,0)Lisrelativelyshortandthekinkwillbedifculttosee,evenunderidealexperimentalconditions.AnextremeexampleofthissortispresentedinSection 4.2 ,wherewediscusstopquarkevents,inwhichthechild(neutrino)massM0ispracticallyzeroandthekinkcannotbeseenatall.However,evenunderthosecircumstances,theendpointmeasurementsinEquations( 4 through 4 )arestillavailable.Moreimportantly,theconstraintsinEquations( 4 through 4 )areindependentofthepreviouslyfoundrelationsinEquations( 4 )and( 4 ),sothatthelattercanbeusedtoresolvethetwo-foldambiguityinEquation( 4 ).BeforewemoveontoadiscussionofthelastremainingsubsystemMT2quantityinthenextSection 4.1.4 ,letusrecapourmainresultderivedinthissubsection.WeshowedthattheM(2,2,0)T2variableyieldsthreeindependentendpointmeasurementsinEquations( 4 through 4 ),andpossiblyakinkmeasurementinEquation( 4 ).TheM(2,2,0)T2endpointmeasurementsbythemselvesaresufcienttodetermineallthree 120

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massesM0,M1andM2,uptothetwo-foldambiguityinEquation( 4 ).ThisrepresentsapureMT2-basedmassmeasurementmethod,whichdoesnotuseanyanykinkorinvariantmassinformation. 4.1.4TheSubsystemVariableM(2,1,0)T2ThevariableM(2,1,0)T2isillustratedinFigure 4-2 (b),whereweusethesubchainwithinthesmallerrectangleontheright.Thisisanothergenuinesubsystemquantity,sinceweonlyusetheSMdecayproductsx1andignoretheupstreamobjectsx2.However,theupstreamobjectsx2areimportantinthesensethattheyhavesomenon-zerotransversemomentum,andasaresult,thesumofthetransversemomenta~P(k)0TofthechildrenX0isnotbalancedbythesumofthetransversemomentaoftheSMobjectsx1usedintheMT2calculation: ~P(1)0T+~P(2)0T+~p(1)1T+~p(2)1T=)]TJ /F8 11.955 Tf 8.74 0.5 Td[(~p(1)2T)]TJ /F8 11.955 Tf 11.4 0.5 Td[(~p(2)2T)]TJ /F8 11.955 Tf 11.39 0.5 Td[(~pT6=0.(4)NoticethatevenintheabsenceofanyISRpT,thisisstillanunbalancedconguration,duetothetransversemomenta~p(1)2Tand~p(2)2Toftheupstreamobjectsx2.Therefore,wecannotusetheexistinganalyticalresultsonMT2,sincepreviousstudiesalwaysassumedthattheright-handsideofEquation( 4 )isexactlyzero,duetothelackofanyparticlesupstream.WethereforeneedtogeneralizetheprevioustreatmentsofMT2andobtainthecorrespondingendpointformulasforournewsubsystemM(2,1,0)T2variable.Inparticular,intheabsenceofanyintrinsicISR(i.e.,forpT=0),wendthattheendpointoftheM(2,1,0)T2distributionisgivenby M(2,1,0)T2,max(~M0,pT=0)=8><>:F(2,1,0)L(~M0,pT=0),if~M0M0,F(2,1,0)R(~M0,pT=0),if~M0M0,(4) 121

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where F(2,1,0)L(~M0,pT=0)=((2,2,0))]TJ /F8 11.955 Tf 11.95 0 Td[((2,2,1)+q 2(2,2,0)+~M202)]TJ /F8 11.955 Tf 11.96 0 Td[(2(2,2,1))1 2, (4) F(2,1,0)R(~M0,pT=0)=((2,1,0)+q )]TJ /F8 11.955 Tf 5.48 -9.68 Td[((2,2,1))]TJ /F8 11.955 Tf 11.95 0 Td[((2,1,0)2+~M202)]TJ /F8 11.955 Tf 11.96 0 Td[(2(2,2,1))1 2, (4) andthevariousparameters(n,p,c)aredenedin( 4 ).ThecorrespondingexpressionsforgeneralpT(i.e.,arbitraryintrinsicISR)arelistedinAppendix B .FromEquation( 4 )weseethat,onceagain,theendpointfunctionM(2,1,0)T2,max(~M0,pT=0)wouldexhibitakink(2,1,0)atthetruevalueofthetestmass~M0=M0: @F(2,1,0)L @~M0!~M0=M06= @F(2,1,0)R @~M0!~M0=M0.(4)TheexistenceofthiskinkshouldcomeasnosurpriseRef.[ 36 ]showed(inthepT!1limit)thatanytypeofupstreammomentumwillgenerateakinkinanotherwisesmoothMT2,maxfunction.Asbefore,thevalueoftheMT2endpointM(2,1,0)T2,maxatthekinklocationrevealsthetruemassoftheparent: M(2,1,0)T2,max(~M0=M0,pT=0)=M1.(4)Atthesametime,thephysicaloriginofthiskinkisdifferentfromeitherofthetwokinktypes((1,1,0)and(2,2,0))discussedearlier.Clearly,thenewkinkisdifferentfrom(2,2,0),whichwasduetothevaryinginvariantmassofthefx1,x2gsystem.HereweareusingasingleSMparticlex1whosemassisconstant.Furthermore,thenewkink(2,1,0)cannotbeduetoanyISRlikeinthecaseof(1,1,0),sinceEquation( 4 )doesnotaccountforanyISReffects.Therealreasonforthisnew(2,1,0)kinkisathirdone,namely,thekinematicalrestrictionsplacedbythedecaysoftheupstreamparticles(inthiscase,thegrandparentsX2).Wenowproceedtoinvestigatethenewkink(2,1,0)quantitatively.Usingthesameexampleofgluinopair-productionfortheSPS1amassspectrum( 4 ),weplot 122

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thefunction( 4 )inFigure 4-4 (a).ComparingthelowerandtheuppersetoflinesinFigure 4-4 (a),wenoticethattheM(2,1,0)T2,maxandM(2,2,0)T2,maxvariablesshareseveralcommoncharacteristics.Theybothexhibitakinkatthetruelocationofthechildmass~M0=M0,whiletheirvaluesatthatpointrevealthetrueparentmassineachcase:M1forM(2,1,0)T2,maxandM2forM(2,2,0)T2,max.Usingthedenition( 4 ),wendthatthesizeofthe(2,1,0)kinkisgivenby (2,1,0)=arctan(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y2)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)p z 2z(1+y2)+y(1+z2)+2yz,(4)wheretheparametersyandzwerealreadydenedin( 4 ).Thekink(2,1,0)isplottedinFigure 4-5 (b)asafunctionofp yandp z.Wenoticethatthekinkstructurebecomesmorepronouncedforrelativelysmallyandz.ComparingFigures 4-5 (a)and 4-5 (b),weseethatforanygivensetofvaluesforyandz,the(2,1,0)kinkdiscussedhereismorepronouncedthanthe(2,2,0)kinkfromtheprevioussubsection5.Thedifferenceisparticularlynoticeableintheregionofp y0andp z0.2.TheSPS1amassspectrum( 4 )inourpreviousexamplewasratherfarawayfromthisregion,asindicatedbythewhitedotsinFigure 4-5 .Nowletuschooseadifferentmassspectrum,whichisclosertotheregionwherethedifferencebetweenthetwokinksbecomesmorenoticeable,forexample M2=2000GeV,M1=200GeV,M0=100GeV,(4)correspondingtothepointmarkedwiththewhiteasteriskinFigures 4-5 (a)and 4-5 (b).TheresultingendpointfunctionsM(2,2,0)T2,maxandM(2,1,0)T2,maxareplottedinFigure 4-4 (b).IndeedweseethatwiththisnewspectrumthekinkintheM(2,1,0)T2,maxfunctionismuchmorenoticeablethanthekinkintheM(2,2,0)T2,maxfunction.Therefore,ourrstconclusionregardingtheM(2,1,0)T2variableisthatitskinkisingeneralsharperandappearsto 5Thisstatementcanalsobeveriedusingtheanalyticalformulas( 4 )and( 4 ). 123

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bemorepromisingthanthepreviouslydiscussedkinkintheM(2,2,0)T2variablefromSection 4.1.3 .Followingourpreviousstrategy,weshallnotdwelltoolongonthekink,butinsteadweshalldiscusstheavailableendpointmeasurementsforvariousvaluesof~M0.Again,thepresenceoftwobranchesinEquation( 4 )canbeusedtoouradvantage.AsinSection 4.1.3 ,werstchooseatestmassvalue~M0=0,whichwouldselectthelowbranch( 4 )andresultinanendpointmeasurement M(2,1,0)T2,max(~M0=0,pT=0)=2q (2,2,0)((2,2,0))]TJ /F8 11.955 Tf 11.96 0 Td[((2,2,1)). (4) Justasbefore,wecouldalsochoosearatherlargevaluefor~M0=Eb,whichwouldselectthehighbranch( 4 )andresultinthemeasurement M(2,1,0)T2,max(~M0=Eb,pT=0)=((2,1,0)+q )]TJ /F8 11.955 Tf 5.48 -9.69 Td[((2,2,1))]TJ /F8 11.955 Tf 11.96 0 Td[((2,1,0)2+E2b2)]TJ /F8 11.955 Tf 11.95 0 Td[(2(2,2,1))1 2.(4)Athirdchoice,~M0=~Eb,with~Eb>Eb,wouldyieldyetanotherendpointmeasurement M(2,1,0)T2,max(~M0=~Eb,pT=0)=((2,1,0)+q )]TJ /F8 11.955 Tf 5.48 -9.68 Td[((2,2,1))]TJ /F8 11.955 Tf 11.95 0 Td[((2,1,0)2+~E2b2)]TJ /F8 11.955 Tf 11.96 0 Td[(2(2,2,1))1 2.(4)AgainweobtainedthreeEquations( 4 through 4 )forthethreeunknown-parameters(2,2,0),(2,2,1)and(2,1,0),orequivalently,forthethreeunknownmassesM0,M1andM2.Equations( 4 through 4 )areallindependentandcanbeeasilysolved,givingatotaloffoursolutions.However,threeofthesolutionsarealwaysunphysical,sothatweendupwithasingleuniquesolution.ThisrepresentsanimportantadvantageoftheM(2,1,0)T2,maxvariableincomparisonwiththeM(2,2,0)T2,maxvariablediscussedinSection 4.1.3 .TherewefoundthatM(2,2,0)T2,maxalwaysgivesrisetwoatwo-foldambiguityinthemassspectrum,whilenowweseethatM(2,1,0)T2,maxdoesnotsufferfromthisproblemandalreadybyitselfallowsforacompleteandunambiguousdeterminationofthemassspectrum. 124

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4.2MT2-basedMassMeasurementmethodsInthissectionweusetheanalyticalresultsderivedintheprevioussectiontoproposethreedifferentstrategiesfordeterminingthemassesinn2decaychains.Weshallillustrateeachofourmethodswithaspecicexample,forwhichwechoosetoconsiderthedileptonsamplesfromW+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(andttevents.Theformerisanexampleofthen=1decaychainexhibitedinFigure 4-2 (a),whilethelatterisanexampleofthen=2decaychaininFigure 4-2 (b).Mostimportantly,thesesamplesalreadyexistintheTevatrondataandwillalsobeamongthersttobestudiedattheLHC.Correspondingly,throughoutthissectionweshallusethefollowingmassspectrum M2=mt=173GeV,M1=mW=80GeV, (4) M0=m=0GeV.Beforewebegin,letusreviewthefourdifferentM(n,p,c)T2variableswhichareinprincipleavailableinthatcase.EachoneofthemisplottedinFigure 4-6 forvedifferentvaluesofthecorrespondingtestmass(0,100,200,300and400GeV).InFigure 4-6 (a)weshowtheM(1,1,0)T2variablefromW+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(pairproductionevents,whileinFigures 4-6 (b-d)wecorrespondinglyshowtheM(2,2,1)T2,M(2,2,0)T2andM(2,1,0)T2variablesfromttevents.WeusedPYTHIA[ 54 ]foreventgenerationanddidnotimposeanyselectioncuts,sincetheywillnotaffectthelocationoftheMT2endpoint6.TheplotsaremadefortheTevatron(appcolliderwitha2TeVcenter-of-massenergy),wheretherelevantdataisalreadyavailable.ThecorrespondinganalysisfortheLHCisverysimilar.AllofourplotsinthisSectionhavethefullISReffects. 6Thecutswouldhaveanimpactontheoverallacceptanceandefciency.Thiseffectisnotrelevanthere,sinceweareshowingunit-normalizeddistributions.Thecutsmayalsodistorttheshapeofeachdistribution,butshouldpreservethelocationoftheupperendpoint. 125

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Figure4-6. Unit-normalizeddistributionsofM(n,p,c)T2variablesindileptoneventsfrom(a)W+W)]TJ /F14 11.955 Tf 10.41 -4.34 Td[(pairproductionand(b-d)ttpairproduction.Eachpanelshowsresultsforvedifferentvalues(0,100,200,300and400GeV)ofthecorrespondingtestmass.ThemethodsofSections 4.2.1 and 4.2.3 onlymakeuseoftheMT2endpointatzerotestmass,M(n,p,c)T2,max(~Mc=0),whichisindicatedbytheverticalredarrow.Inpanel(c),thetwodottedlineM(2,2,0)T2distributionscorrespondtothecorrectandthewrongpairingofthetwob-jetswiththeleptons,whilethesolidlinedistributionistheaverageofthesetwo. AsdiscussedinSection 4.1 ,thepresenceofISRwithnonzeropTwillincreasethenominalM(n,p,c)T2endpoints: M(n,p,c)T2,max(~Mc,pT)M(n,p,c)T2,max(~Mc,0),(4)wheretheequalityisobtainedonlywhen~Mc=Mc.ISRwillthereforeintroducesomesystematicerrorwhenoneistryingtomeasureM(n,p,c)T2,max(~Mc,0).Thesizeofthis 126

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errordependsontheISRpTspectrum,whichinturndependsonthetypeofcollider(TevatronorLHC).AttheTevatron,thiswillnotbesuchaseriousissue,asevidencedfromFigure 4-6 ,wheretheobservedendpointsinthepresenceofISRmatchprettywellwiththeirexpectedvaluesforthepT=0case.Ontheotherhand,attheLHCthismaybecomeaproblem,whichcanbehandledinoneoftwoways.First,dependingontheparticularsignature,onemaybeabletoselectasamplewithpT0(atacertaincostinstatistics),byimposingasuitablydesignedjetvetotoremovejetsfromISR.Alternatively,onecanusethefulleventsample(whichwouldincludeISRjets),andmakeuseofourgeneralformulasinAppendix B ,whichcontaintheexplicitpTdependenceofM(n,p,c)T2,max.InthepreviousSection 4.1 wederivedthatinthecaseofn2cascades,thereare8differentMT2endpointmeasurements:oneforM(1,1,0)T2,max(seeEquation( 4 )andSection 4.1.1 ),oneforM(2,2,1)T2,max(seeEquation( 4 )andSection 4.1.2 ),threeforM(2,2,0)T2,max(seeEquations( 4 through 4 )andSection 4.1.3 ),andthreeforM(2,1,0)T2,max(seeEquations( 4 through 4 )andSection 4.1.4 ).GiventhatwearetryingtodetermineonlythreemassesM0,M1andM2,itisclearthatthese8measurementsshouldbesufcienttocompletelydeterminethespectrum.ThenumberofavailablemeasurementsisinfactmuchlargerthanthenumberofM(n,p,c)T2,maxvariables.Indeed,asshowninSections 4.1.3 and 4.1.4 ,therearecaseswherewemightbeabletoobtainmorethanonemassconstraintfromagivenM(n,p,c)T2,maxvariable.Ofcourse,the8measurementscannotallbeindependentamongthemselves,astheyonlydependonthreeparameters.Ourthreemethodsbelowwillbedistinguishedbasedonwhichsubsetofthesemeasurementsweareusing. 4.2.1PureMT2EndpointMethodWiththismethod,weuseMT2endpointmeasurementsEnpcatasinglexedvalueofthetestmass,whichforconveniencewetaketobe~Mc=0: EnpcM(n,p,c)T2,max(~Mc=0,pT=0).(4) 127

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ThecorrespondingformulasinterpretingthosemeasurementsintermsofthephysicalmassesM0,M1andM2werederivedinSection 4.1 : E110M(1,1,0)T2,max(0,0)=M21)]TJ /F3 11.955 Tf 11.96 0 Td[(M20 M1=M2p y(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z), (4) E221M(2,2,1)T2,max(0,0)=M22)]TJ /F3 11.955 Tf 11.96 0 Td[(M21 M2=M2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y), (4) E220M(2,2,0)T2,max(0,0)=M22)]TJ /F3 11.955 Tf 11.96 0 Td[(M20 M2=M2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(yz), (4) E210M(2,1,0)T2,max(0,0)=1 M2q (M22)]TJ /F3 11.955 Tf 11.95 0 Td[(M20)(M21)]TJ /F3 11.955 Tf 11.95 0 Td[(M20)=M2p y(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(yz). (4) Usingthemassspectrum( 4 ),thepredictedlocationsofthesefourMT2endpointsare E110=80GeV, (4) E221=136GeV, (4) E220=173GeV, (4) E210=80GeV, (4) whicharemarkedwiththeverticalredarrowsinFigure 4-6 .Giventhatwehavefourmeasurements( 4 through 4 )foronlythreeparametersM0,M1andM2,oneshouldbeabletouniquelydetermineallthreeoftheunknownparameters.Naively,itseemsthatusingjustthreeofthemeasurements( 4 through 4 )shouldbesufcientforthispurpose,andfurthermore,thatanythreeofthemeasurements( 4 through 4 )willdothejob.However,oneshouldexercisecaution,sincenotallfourmeasurements( 4 through 4 )areindependent.ItiseasytocheckthatE221,E220andE210obeythefollowingrelation E2210=E220(E220)]TJ /F3 11.955 Tf 11.95 0 Td[(E221).(4)ThismeansthatinordertobeabletosolveforthemassesfromEquations( 4 through 4 ),wemustalwaysmakeuseoftheE110measurementinEquation( 4 ), 128

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andthenwehavethefreedomtochooseanytwooutoftheremainingthreemeasurements( 4 through 4 ).Forexample,usingthesetofthreemeasurementsfE110,E221,E220g(i.e.Equations( 4 through 4 )),themassesareuniquelydeterminedas M0=E110E221(E220)]TJ /F3 11.955 Tf 11.96 0 Td[(E221)E220(E220)]TJ /F3 11.955 Tf 11.95 0 Td[(E221))]TJ /F3 11.955 Tf 11.96 0 Td[(E21101 2 E2110)]TJ /F6 11.955 Tf 11.95 0 Td[((E220)]TJ /F3 11.955 Tf 11.96 0 Td[(E221)2, (4) M1=E110E221(E220)]TJ /F3 11.955 Tf 11.95 0 Td[(E221) E2110)]TJ /F6 11.955 Tf 11.95 0 Td[((E220)]TJ /F3 11.955 Tf 11.96 0 Td[(E221)2, (4) M2=E2110E221 E2110)]TJ /F6 11.955 Tf 11.96 0 Td[((E220)]TJ /F3 11.955 Tf 11.95 0 Td[(E221)2. (4) Similarly,onecansolveforM0,M1andM2usingthesetofmeasurementsfE110,E220,E210g,oralternatively,thesetofmeasurementsfE110,E221,E210g.Ineachcase,theremainingfourthunusedmeasurementprovidesausefulconsistencycheckonthemassdetermination. 4.2.2MT2EndpointShapesAndKinksThemethodproposedinSection 4.2.1 usesthemeasuredendpointsfromseveraldifferentM(n,p,c)T2variables.NowwediscussanalternativemethodwhichmakesuseofasingleM(n,p,c)T2variable.Letusbeginwiththesimplestcaseofn=1asshowninFigure 4-2 (a).Inthatcase,wehaveonlyoneMT2variableatourdisposal,namelyM(1,1,0)T2.ItspropertieswerediscussedinSection 4.1.1 ,whereweshowedthatitsendpointM(1,1,0)T2,maxcanallowthedeterminationofbothmassesM0andM1,atleastasamatterofprinciple.Indeed,theendpointmeasurement( 4 )atzerotestmassprovidesonerelationamongM0andM1.ThekeyobservationinSection 4.1.1 (whichwasrstdonein[ 36 ])wasthatwiththeinclusionofISReffects,theendpointfunctionM(1,1,0)T2,max(~M0,pT)exhibitsakinkat~M0=M0,whichcanthenbeusedtodeterminebothmassesM0andM1.ThemethodcanbereadilyappliedtotheexistingdileptoneventsamplefromW+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(pairproduction,whichwillallowanindependentmeasurementoftheWmassmWandtheneutrinomassm.Whiletheprecisionofthismeasurementwillnotbecompetitive 129

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withexistingWandneutrinomassdeterminations,itisneverthelessusefultotesttheviabilityofthisapproachwithrealdata.Nowletusdiscussthemorecomplicatedcaseofn=2,whichinourexamplecorrespondstottpairproductionwithbothtopsdecayingleptonically.AsdiscussedinSections 4.1.2 4.1.3 and 4.1.4 ,herewehaveachoiceofthreedifferentMT2variables:M(2,2,1)T2,M(2,2,0)T2,andM(2,1,0)T2.Becauseofthelargerttcross-section,weexpectthatthestatisticalprecisiononeachoneofthosethreevariableswillbebetterthantheM(1,1,0)T2variableofthen=1case.AsshowninSections 4.1.3 and 4.1.4 ,eachofthetwovariablesM(2,2,0)T2andM(2,1,0)T2exhibitsakinkinitsendpointM(n,p,c)T2,maxwhenconsideredasafunctionofthetestmass~M0,evenwhenthetransversemomentumoftheintrinsicISRintheeventiszero,pT=0.Then,whichofthesetwovariablesisbettersuitedforamassdetermination?ThecaseofM(2,2,0)T2,max(~M0,pT=0)wasalreadydiscussedin[ 34 36 37 ].HerewewouldliketoproposethealternativemeasurementofM(2,1,0)T2,max(~M0,pT=0).Whatismore,wewouldliketoemphasizethatourfunctionM(2,1,0)T2,max(~M0,pT=0)offersseveraluniqueadvantagesoverthepreviouslyconsideredcaseofM(2,2,0)T2,max(~M0,pT=0): 1. ThesubsystemvariableM(2,1,0)T2doesnotsufferfromthecombinatoricsproblemwhichispresentforM(2,2,0)T2.Indeed,whenconstructingtheM(2,2,0)T2distribution,onehastodecidehowtopairuptheb-jetswiththetwoleptons.Becauseitisdifculttodistinguishbetweenabandab,thereisatwo-foldambiguitywhichisquitedifculttoresolvebyothermeans.Incontrast,oursubsystemvariableM(2,1,0)T2doesnotmakedirectuseoftheb-jets,andisthereforefreeofsuchcombinatoricsissues. 2. AswealreadysawinSection 4.1.3 ,evenunderperfectexperimentalconditions,thettotheM(2,2,0)T2,maxendpointresultsintwoseparatesolutionsforthemassspectrum:onesolution(Equation( 4 ))isgivenbythetruevaluesoftheinputmasses,whilethesecondsolution(Equation( 4 ))isobtainedbythetransformationinEquation( 4 ).UsingM(2,2,0)T2,maxalone,thereisnowaytotellthedifferencebetweenthesetwomassspectra.Incontrast,ourvariableM(2,1,0)T2doesnotsufferfromthisambiguity,andaccordingtoourresultsfromSection 4.1.4 thesolutionisalwaysunique. 130

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3. ThethirdadvantageofthesubsystemvariableM(2,1,0)T2isrelatedtotheexpectedprecisiononthedeterminationofthemasses.AswepointedoutinSection 4.1.4 andillustratedexplicitlyinFigure 4-5 ,thekink(2,1,0)intheM(2,1,0)T2,max(~M0,pT=0)functionismuchsharperthanthecorrespondingkink(2,2,0)intheM(2,2,0)T2,max(~M0,pT=0)function.ThiscanalsobeseenexplicitlyfromthetwoexamplesshowninFigure 4-4 .Asaresult,weexpectthatthekinkstructurecanbebetteridentiedinthecaseofM(2,1,0)T2,max,whichwouldleadtosmallererrorsonthemassdetermination.Ofcourse,onecould(andinfactshould)usetheexperimentalinformationfrombothM(2,2,0)T2,maxandM(2,1,0)T2,max,ifavailable.OurmaingoalhereissimplytopointouttheobviousadvantagesofthesubsystemvariableM(2,1,0)T2,whichsofarhasnotbeenusedintheliterature. 4.2.3HybridMethod:MT2EndpointsPlusAnInvariantMassEndpointAnycascadewithn2willprovideacertainnumberofmeasurementslikeasinvariantmassendpointsinadditiontotheMT2measurementsdiscussedsofar.Inparticular,forthen=2exampleconsideredhere,therewillbeonemeasurementoftheendpointoftheMx1x2invariantmassdistribution.TheformulafortheendpointMx1x2,maxintermsoftheunknownphysicalmassesM0,M1andM2isingeneralgivenby EimMx1x2,max=1 M1q (M22)]TJ /F3 11.955 Tf 11.96 0 Td[(M21)(M21)]TJ /F3 11.955 Tf 11.95 0 Td[(M20)=M2p (1)]TJ /F3 11.955 Tf 11.96 0 Td[(y)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(z).(4)Inthecaseoftteventsconsideredhere,thisissimplytheendpointoftheinvariantmassdistributionmb`ofeachleptonanditscorrespondingb-jet.Thisdistribution(unit-normalized)isshowninFigure 4-7 .Unfortunately,hereoneisfacingthesamecombinatorialproblemaswiththeM(2,2,0)T2variablewecannoteasilytellthechargeoftheb-jet,thereforeaprioriitisnotclearwhichb-jetgoeswithwhichlepton.Fortunately,thereareonlytwopossibilities:theresultfromthecorrect(wrong)pairingisshowninFigure 4-7 withthegreen(blue)dottedline.Weseethatthegreenhistogramwiththecorrectpairinghasanendpointattheexpectedlocation Eim=s (m2t)]TJ /F3 11.955 Tf 11.96 0 Td[(m2W)(m2W)]TJ /F3 11.955 Tf 11.95 0 Td[(m2) m2W=153.4GeV,(4) 131

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witharelativelysmalltailduetothenitewidtheffects.Moreimportantly,the(blue)distributionfromthewrongpairingsisrelativelysmooth,andasaresulttheendpointasinEquation( 4 )ispreservedintheexperimentallyobservable(red)distribution,whichincludesallpossibleb`pairings.Nowwecanaddthenewmeasurement( 4 )tothepreviouslydiscussedsetofmeasurementsinEquations( 4 through 4 ).WeobtainatotalofvemeasurementsforthethreeunderlyingparametersM0,M1andM2,thereforethereexisttworelationsamongthemeasurements.TherstrelationisalreadygivenbyEquation( 4 )anddoesnotinvolvetheinvariantmassendpointinEquation( 4 ).Thesecondrelationisgivenby E2im=E221E2110 E220)]TJ /F3 11.955 Tf 11.96 0 Td[(E221.(4)Wecannowconsiderahybridmethod,whichwouldmakeuseoftheinvariantmassendpoint( 4 ),plusanytwooftheMT2measurementsinEquations( 4 through 4 ).Inprinciple,oneagainneedstobecarefulandmakesurethatthethreeusedmeasurementsareindependent.Fortunately,asseenfromEquations( 4 4 ),theinvariantmassendpointEimisindependentfromanypairofMT2measurements.Thereare6possiblepairsamongtheMT2measurements( 4 through 4 ),andinprincipleeachonecanbeusedincombinationwiththeinvariantmassendpoint( 4 ).Whatisthebestchoice?Wendthatinall6ofthosecasesoneobtainsauniquesolutionforthemassesM0,M1andM2.Therefore,theoptimalchoiceisdictatedbytheexperimentalprecisiononeachofthemeasurements( 4 through 4 ).Weexpectthatthemeasurement( 4 )ofM(1,1,0)T2,maxwillbelesspreciseduetothesmallercross-sectionforW+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(production.Similarly,M(2,2,0)T2,maxsuffersfromthecombinatorialproblemalreadymentionedearlier.ThereforeforourillustrationofthehybridmethodwechoosetousetheM(2,2,1)T2endpoint( 4 ),theM(2,1,0)T2endpoint( 4 ),andtheinvariantmassendpoint( 4 ). 132

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Figure4-7. Unit-normalizedmb`invariantmass-squareddistributionsindileptonttevents.Thegreen(blue)dottedlinecorrespondstothecorrect(wrong)pairingoftheleptonsandtheb-jets,whiletheredsolidlineistheaverageofthosetwodistributions.Theendpoint( 4 )ofthemb`distributionismarkedbytheverticalredarrow. Thesolutionforthemassesintermsofthosethreemeasurementsisgivenby M0=p 2E221Eim2E221E2210+E221E2im)]TJ /F3 11.955 Tf 11.95 0 Td[(E2imp E2221+4E22101 2 E2221+2E2im)]TJ /F3 11.955 Tf 11.96 0 Td[(E221p E2221+4E2210, (4) M1=p 2E221EimE221p E2221+4E2210)]TJ /F3 11.955 Tf 11.95 0 Td[(E22211 2 E2221+2E2im)]TJ /F3 11.955 Tf 11.96 0 Td[(E221p E2221+4E2210, (4) M2=2E221E2im E2221+2E2im)]TJ /F3 11.955 Tf 11.95 0 Td[(E221p E2221+4E2210. (4) ItiseasytocheckthatsubstitutingthemeasuredvaluesoftheendpointsE221,E210andEimfromEquations( 4 ),( 4 )and( 4 ),intoEquations( 4 )aboveyieldsthevaluesfortheneutrino,Wandtopquarkmass,correspondingly. 133

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CHAPTER5ONEDIMENSIONALPROJECTIONMETHODMostofmassreconstructionmethodsrelyonexclusivechannelswhereasufcientlylongdecaychaincanbeproperlyidentied.Unfortunately,thisalmostinevitablyrequirestheuseofhadronicjetsinsomeformintheanalysisinmostSUSYmodels,themainLHCsignalisduetothestrongproductionofcoloredsuperpartners,whosecascadedecaystotheneutralLSPnecessarilyinvolvehadronicjets.Formanyreasons,jetsarenotoriouslydifculttodealwith,especiallyinahadroncolliderenvironment.BecauseofthehighjetmultiplicityinSUSYsignalevents,anyjet-basedanalysisisboundtofaceaseverecombinatorialproblemandisunlikelytoachieveanygoodprecision.Thusitisimperativetohavealternativemethodswhichavoidthedirectuseofjetsandinsteadrelyonlyonthewellmeasuredmomentaofany(isolated)leptonsintheevent.Inthischapter,wedescribemethods,whicharefreeofthejetcombinatorialproblem.Inthersttwosection,wewilluseMT2andthelastsection,wewillillustratehowtousefullphasespaceinformationwithcontravsevariableMCTasanexample. 5.1DetailedStudyOnMT2'sCharacteristicsForillustrationintwosections,weshallusethestandardexampleofR-parityconservingsupersymmetrywitha~01LSP.Itscollidersignatureshavebeenextensivelystudied,andtypicallyinvolvejets,leptonsandmissingtransverseenergy[ 89 ].Amongthose,theinclusivesame-signdileptonchannelhasalreadybeenidentiedasauniqueopportunityforanearlySUSYdiscoveryattheLHC[ 75 90 ].Thetwoleptonsofthesamechargecanbeeasilytriggeredon,andprovideagoodhandleforsuppressingtheSMbackground.InouranalysisweusetheLM6CMSstudypoint[ 75 ],whoserelevantmassspectrumisgiveninTable 5-1 .AtpointLM6,signaleventswithtwoisolatedsame-signleptonstypicallyarisefromtheSUSYeventtopologyinFigure 5-1 134

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Table5-1. Selectedsparticlemasses(inGeV)atpointLM6.Welisttheaverage~qLmassM~qL=1 2(M~uL+M~dL). M~gM~qLM~+1M~`LM~`M~01 939.8862305.3291.0275.7158.1 Figure5-1. ThetypicalSUSYeventtopologyproducingtwoisolatedsame-signleptonsatpointLM6(seetextfordetails).Thediagramforapairofnegativelychargedleptons`)]TJ /F8 11.955 Tf 7.09 -4.34 Td[(`)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(isanalogous. Considertheinclusiveproductionofsame-signcharginos,whichdecayleptonicallyasshownintheyellow-shadedboxinFigure 5-1 .Theresultingsneutrino(~`)couldbetheLSPitself,or,asinthecaseofLM6,mayfurtherdecayinvisiblytoaneutrinoandthetrueLSP~01.Suchsame-signcharginopairstypicallyresultfromsquarkdecays,asindicatedinFigure 5-1 .Inturn,thesquarksmaybeproduceddirectlythroughat-channelgluinoexchange,orindirectlyingluinodecays.Notethatthetwosame-signleptonsinFigure 5-1 areaccompaniedbyanumberofupstreamobjects(typicallyjets)whichmayoriginatefromvarioussources,e.g.initialstateradiation,squarkdecays,ordecaysofevenheavierparticlesupthedecaychain.Inordertostayclearofjetcombinatorialissues,weshalladoptafullyinclusiveapproachtothesame-signdileptonsignature,bytreatingalltheupstreamobjectswithintheblackrectangularframeinFigure 5-1 asasingleentityoftotaltransversemomentum~PT.Giventhisverygeneralsetup,wenowposethefollowingquestion:assumingthataSUSYdiscoveryismadeintheinclusivesame-signdileptonchannel,isitpossibletomeasuretheindividualsparticlemassesMpandMcinvolvedintheleptonicdecays 135

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ofFigure 5-1 ,usingonlythetransversemomentaofthetwoleptons~p(1)`Tand~p(2)`T,andthetotalupstreamtransversemomentum~PT?Althoughitmayappearthatthosethreevectorsdonotprovidealotofinformationtogoon,weshallshowthatthisispossible.Wediscussthreedifferentapproaches.MethodI.Letusconcentratedirectlyontheobservedleptonmomenta~p(i)`T.ConsiderthetwocollinearmomentumcongurationsillustratedinFigure 5-2 anddenedasfollows.Ineachconguration,theleptonmomentaarethesame:~p(1)`T=~p(2)`T;andthentheycanbeeitherparalleloranti-paralleltothemeasuredupstream~PT: s=+1)~p(1)`T=~p(2)`T""~PT; (5) s=)]TJ /F6 11.955 Tf 9.3 0 Td[(1)~p(1)`T=~p(2)`T"#~PT. (5) Inwhatfollowsweshallusetheintegers=+1(s=)]TJ /F6 11.955 Tf 9.3 0 Td[(1)torefertotheparallel(anti-parallel)conguration:scos(~p(1)`T,~PT)=cos(~p(2)`T,~PT).Nowletusmeasurethemaximumleptonmomentumineachconguration: p`T(sPT)max~p(1)`T=~p(2)`T^cos(~p(1)`T,~PT)=snp(i)`To.(5)Observethatbothp`T(+PT)andp`T()]TJ /F3 11.955 Tf 9.3 0 Td[(PT)canbedirectlymeasuredfromtheleptonpTdistributions.Forexample,constructa2Dscatterplotfx,ygof x=cos(~p(1)`T+~p(2)`T,~PT),y=j~p(1)`T+~p(2)`Tj,(5)withthecutj~p(1)`T)]TJ /F8 11.955 Tf 11.4 0.5 Td[(~p(2)`Tj<(0),andtakethelimit p`T(sPT)=limx!sy 2.(5)Armedwiththetwomeasurementsp`T(+PT)andp`T()]TJ /F3 11.955 Tf 9.3 0 Td[(PT),wecannowdirectlysolveforthemassesMpandMc.Theformulaforp`T(sPT)is p`T(sPT)=M2p)]TJ /F3 11.955 Tf 11.95 0 Td[(M2c 4M2pq 4M2p+(sPT)2)]TJ /F3 11.955 Tf 11.96 0 Td[(sPT.(5) 136

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Figure5-2. ThetwospecialmomentumcongurationsdenedinEquations( 5 5 ). InvertingEquation( 5 ),weget Mp=p p`T()]TJ /F3 11.955 Tf 9.3 0 Td[(PT)p`T(+PT) p`T()]TJ /F3 11.955 Tf 9.3 0 Td[(PT))]TJ /F3 11.955 Tf 11.95 0 Td[(p`T(+PT)PT,(5)thusxingtheabsolutemassscaleintheproblem.OncetheparentmassMpisknown,thechildmassMcis Mc=Mps 1)]TJ /F6 11.955 Tf 11.95 0 Td[(2p`T()]TJ /F3 11.955 Tf 9.3 0 Td[(PT))]TJ /F3 11.955 Tf 11.95 0 Td[(p`T(+PT) PT.(5)ThuswefoundthetruesparticlemassesMpandMcdirectlyintermsofthemeasuredleptonmomentap`T(PT)andupstreammomentumPT.NotethatthechoiceofthevalueforPTinEquations( 5 )and( 5 )isarbitrary,whichcanbeusedtoouradvantage,e.g.toselectthemostpopulatedPTbin,minimizingthestatisticalerror.MethodII.Inourpreviousmethod,theleptonmomentap`T(PT)weremeasureddirectlyfromthedataasimpliedbyEquation( 5 ).Alternatively,wecanobtainthemindirectlyfromtheendpointoftheCambridgeMT2variable.Tobemoreprecise,weapplythesubsystemMT2variableintroducedin[ 23 ]tothepurelyleptonicsubsystemintheyellow-shadedboxofFigure 5-1 .FollowingthegenericnotationofReference[ 23 ],wedenotetheinput(test)massofthesneutrinochildas~Mc.ThesubsystemMT2variableisnowdenedasfollows.FirstformthetransversemassMTforeach(chargino)parent M(i)Ts ~M2c+2j~p(i)`Tjq ~M2c+j~p(i)cTj2)]TJ /F8 11.955 Tf 11.39 0.5 Td[(~p(i)`T~p(i)cT 137

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intermsoftheassumedtestmass~Mcandtransversemomentum~p(i)cTforeach(sneutrino)child.JustlikethetraditionalMT2,theleptonicsubsystemMT2variable[ 23 ]isdenedthroughaminimizationprocedureoverallpossiblepartitionsoftheunknownchildrenmomenta~p(k)cT,consistentwithtransversemomentumconservationPk(~p(k)cT+~p(k)`T)+~PT=0 MT2(~Mc,~PT,~p(i)`T)minnmaxnM(1)T,M(2)Too.(5)TheMT2distributionhasanupperkinematicendpoint MmaxT2(~Mc,PT)maxalleventsnMT2(~Mc,~PT,~p(i)`T)o,(5)whichcanbeexperimentallymeasuredandsubsequentlyinterpretedasthecorrespondingparentmass~Mp ~Mp(~Mc,PT)MmaxT2(~Mc,PT),(5)providingonefunctionalrelationshipamong~Mpand~Mc,butleavingtheindividualmassesstilltobedetermined.ForustheimportanceoftheMT2variable( 5 )isthatthemomentumcongurationsinFigure 5-2 arepreciselytheoneswhichdetermineitsendpointMmaxT2.ThecompleteanalyticaldependenceoftheMT2endpoint~Mp(~Mc,PT)onbothofitsarguments~McandPTisnowknown[ 23 ]: ~Mp(~Mc,PT)=8><>:~Mp(~Mc,+PT),if~McMc,~Mp(~Mc,)]TJ /F3 11.955 Tf 9.3 0 Td[(PT),if~McMc,(5)where ~Mp(~Mc,sPT)=p`T(sPT)+s p`T(sPT)+sPT 22+~M2c2)]TJ /F6 11.955 Tf 13.15 8.09 Td[((sPT)2 41 2.(5)ThuswecanalternativelyobtainthesparticlemassesbymeasuringjusttwoMT2kinematicendpoints,witharbitrarychoicesforthetestmass~Mcandtheupstream 138

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PT.Forconcreteness,letuspicksomexed~M0candP0T,formthecorrespondingMT2distribution( 5 )andmeasureitsendpoint~M0p,alsomakinganoteofthecongurations0: n~M0c,P0Tomeasure)166(!n~M0p,s0o.(5)Nowperformasecondsuchmeasurement n~M00c,P00Tomeasure)166(!n~M00p,s00o.(5)ByinvertingEquation( 5 ),thesetwomeasurementsallowtheexperimentaldeterminationof p`T(s0P0T)=~M02p)]TJ /F6 11.955 Tf 14.57 2.65 Td[(~M02c 4~M02pq 4~M02p+(s0P0T)2)]TJ /F3 11.955 Tf 11.95 0 Td[(s0P0T(5)andsimilarlyforp`T(s00P00T).Nowtakingtheratio rp`T(s0P0T) p`T(s00P00T)=p 4M2p+(s0P0T)2)]TJ /F3 11.955 Tf 11.95 0 Td[(s0P0T p 4M2p+(s00P00T)2)]TJ /F3 11.955 Tf 11.95 0 Td[(s00P00T,(5)whereinthesecondstepweusedEquation( 5 ),wecansolveEquation( 5 )forthetrueparentmassMpintermsofmeasuredquantities: Mp=)]TJ /F3 11.955 Tf 9.3 0 Td[(rs0P0Ts00P00T (1)]TJ /F3 11.955 Tf 11.95 0 Td[(r2)2r)]TJ /F3 11.955 Tf 14.3 8.09 Td[(s0P0T s00P00Tr)]TJ /F3 11.955 Tf 13.15 8.09 Td[(s00P00T s0P0T1 2,(5)andthenndthetruechildmassMcfromEquation( 5 )as Mc=Mp241)]TJ /F12 11.955 Tf 11.95 20.44 Td[( 1)]TJ /F6 11.955 Tf 15.76 10.74 Td[(~M02c ~M02p!q 4~M02p+(s0P0T)2)]TJ /F3 11.955 Tf 11.95 0 Td[(s0P0T p 4M2p+(s0P0T)2)]TJ /F3 11.955 Tf 11.96 0 Td[(s0P0T351 2(5)withMpalreadygivenbyEquation( 5 ).Notethaninthismethod,thevaluesof~M0c,~M00c,P0TandP00Tcanbechosenatwill,allowingforrepeatedmeasurementsofMpandMc.MethodIII.ThethirdandnalmethodforextractingthetwomassesMpandMcwillmakeuseofthecelebratedkinkintheMT2endpointfunction( 5 ). 139

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Figure5-3. MmaxT2versusthetestmass~Mc,asobtainedinoursimulations(datapoints)fromasamplewithPT=42050GeV,ortheoreticallyfromEquation( 5 )(bluesolidline),aswellastheirdifference(lowerpanel). Since~Mp(~Mc,+PT)and~Mp(~Mc,)]TJ /F3 11.955 Tf 9.3 0 Td[(PT)havedifferentslopesatthecrossoverpoint~Mc=Mc,thefunction~Mp(~Mc,PT)hasaslopediscontinuitypreciselyatthecorrectvalueMcofthechildmass,providinganalternativemeasurementoftheabsolutemassscale.TheprocedureisillustratedinFigure 5-3 fortheLM6studypointofTable 5-1 .ThebluesolidlineshowsthetheoreticallyexpectedshapefromEquation( 5 ),forPT=420GeV,whichisroughlythemeanofthePTdistributionatpointLM6.IntheLM6casethekinkisverymild,only3.3[ 23 ].Inordertotesttheprecisionofthethreemethods,weperformeventsimulationsusingthePYTHIAeventgenerator[ 54 ]andPGSdetectorsimulation[ 55 ].WeconsidertheLHCatitsnominalenergyof14TeVand100fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofdata.Toensurediscovery,weusestandardCMScutsasfollows[ 75 91 ]:exactlytwoisolatedleptonswithpT>10GeV,atleastthreejetswithpT>(175,130,55)GeV,=ET>200GeVandavetoontaujets.Withthosecuts,inthedimuonchannelalone,theremainingSMbackgroundcross-sectionisrathernegligible(0.15fb),whiletheSUSYsignalis14fb,already 140

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leadingtoa22discoverywithjust10fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofdata[ 75 91 ].InordertocomparetothetheoreticalresultinFigure 5-3 ,weselecta50GeVPTbinaroundPT=420GeVandconstructaseriesofMT2distributions,fordifferentinputvaluesof~Mc.Foreachcase,weincludeallSMandSUSYcombinatorialbackgrounds,andextracttheMmaxT2endpointbyalinearunbinnedmaximumlikelihoodt,obtainingthedatapointsshowninFigure 5-3 .WeseethattheMT2endpointcanbedeterminedratherwell(~Mp<3GeV),butonlyontherightbranch~McMc.Incontrast,theMT2endpointsontheleftbranch~McMcareconsiderablyunderestimated,washingouttheexpectedkink.Therearetwoseparatereasonsbehindthiseffect.RecallthattheMT2endpointontheleftbranchisobtainedinthecongurations=+1ofFigure 5-2 ,whichrequirestheleptontobeemittedinthebackwarddirection.Asaresult,theparentboostfavorscongurationswiths')]TJ /F6 11.955 Tf 23.93 0 Td[(1overs'+1.Anotherconsequenceisthatleptonswiths'+1aresofterandmoreeasilyrejectedbytheofinepTcuts.WeconcludethatMmaxT2measurementsontheleftbranchareingeneralnotveryreliable,andtendtojeopardizethetraditionalkinkmethod.Forexample,usingMethodIIItotthedatainFigure 5-3 (greendottedline),wendbesttvaluesofonlyMp(t)=212GeVandMc(t)=188GeV.MethodIhasasimilarproblem,sincep`T(+PT)ismeasuredfromeventsinthes=+1conguration.Usingthe~MpmeasurementsfromFigure 5-3 at~Mc=0and~Mc=1TeV,wendfromEquation( 5 )thatp`T(+420GeV)=8.8GeVandp`T()]TJ /F6 11.955 Tf 9.3 0 Td[(420GeV)=50.6GeV(comparetothenominalvaluesof14.8GeVand53.6GeV,correspondingly).TheresultingmassdeterminationviaEquations( 5 5 )isMp(t)=212GeVandMc(t)=190GeV.WeseethatinbothMethodIandMethodIII,themassesareunderestimatedduetothesystematicunderestimationoftheleftMmaxT2branchinFigure 5-3 .Itisthereforeofgreatinteresttohaveanalternativemethod,whichreliesontherightMmaxT2branchalone. 141

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Figure5-4. Scalingfactorsrelatingtheerror~MpintheextractionoftheMT2endpointtotheresultinguncertaintiesMpandMcontheparentandchildmassescalculatedfrom( 5 )and( 5 ),asafunctionofthetrueinputmassesMcandMp. ThisiswheretheavailablefreedominMethodIIcomesintoplay,sincebothtestmasses~M0cand~M00ccanbechosenontherightbranch.TakingP0T=35050GeVandP00T=50050GeVandrepeatingourearlieranalysis,wendthat~Mpontherightbranchisstillontheorderof3GeV,asinFigure 5-3 .TheresultingerrorMp(Mc)onthemeasuredparent(child)masscanbeeasilypropagatedfromEquations( 5 5 ).ThetworatiosMp=~MpandMc=~MpareshowninFigure 5-4 ,whereforconcretenesswehavetaken~M0c=~M00c=1000GeV.Figure 5-4 revealsthattheLM6inputvaluesofMcandMpareratherunlucky,sincetheerror~MpontheMT2endpointisthenampliedbyafactorofalmost70.However,ifMcandMphappenedtobedifferent,withtherestofthespectrumthesame,theprecisionquicklyimproves.Forexample,with~Mp=3GeV,themassescanbedeterminedtowithin30GeV(75GeV)withintheyellow(orange)region.Oneshouldkeepinmindthatthedominant 142

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uncertaintyon~MpisduetotheSUSYcombinatorialbackground.Wehaveveriedthatintheabsenceofsuchcombinatorialbackground,~Mp<1GeVandthetypicalprecisiononMpandMcfromFigure 5-4 isthenatthelevelof10%. 5.2UsingthePToftheUpstreamJetwithMT2MethodUnfortunately,inordertoapplypreviousmethod,onemustworkwithasubsetofeventswithinarelativelynarrowxedPTrangeofupstreamobjects(includingjets),incurringsomelossinstatistics.Tobemoregeneral,wetreatallupstreamparticlesintoasinglesectorasupstreamobjectU,anddenoteitstotaltransversemomentum~UTasinFigure 5-5 .Inthissection,weproposeanewmethodwhichusesthefulldataset.Asintheprevioussection,wedenetheendpointMmaxT2ofMT2distributionastheparentmass~Mpwiththetrialchildmass~McforagivenUT,: ~Mp(~Mc,UT)MmaxT2(~Mc,UT).(5)Hereweproposetoobtainarelationbyusingthepropertythatthefunction~Mp(~Mc,UT)isindependentofUTatthetruechildmassMc: ~Mp(Mc,UT+UT))]TJ /F6 11.955 Tf 14.56 2.66 Td[(~Mp(Mc,UT)=0,8UT,(5)whichwecanrewritemoreinformativelyas ~Mp(~Mc,UT))]TJ /F6 11.955 Tf 14.57 2.66 Td[(~Mp(~Mc,0)0,(5)withequalitybeingachievedonlyfor~Mc=Mc.Equation( 5 )impliesthat,foranygiven~Mc,therewillalwaysbeacertainnumberofeventswhoseMT2valueswillexceedthereferencevalue~Mp(~Mc,0),unlessthetrialmass~MchappenstocoincidewiththetruechildmassMc.Inordertoquantifythiseffect,wedenethefunction N(~Mc)XalleventsHMT2)]TJ /F6 11.955 Tf 14.57 2.66 Td[(~Mp(~Mc,0).(5) 143

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Figure5-5. Thegenericeventtopologyunderconsideration.Allparticlesvisibleinthedetectorareclusteredintothreegroups:upstreamobjectsUwithtotaltransversemomentum~UT,andtwocompositevisibleparticlesVi(i=1,2),eachwithinvariantmassmiandtotaltransversemomentum~piT. H(x)istheHeavisidestepfunction.FromthedenitionofN(~Mc)itisclearthatitisminimizedat~Mc=Mc,whereintheorywewouldexpect NminminfN(~Mc)g=N(Mc)=0.(5)Inreality,thevalueofNminwillbeliftedfrom0,duetoniteparticlewidtheffects,detectorresolution,etc.NeverthelessweexpectthatthelocationoftheN(~Mc)minimumwillstillbeat~Mc=Mc,allowingadirectmeasurementofthechildmassMc: Mc=n~McjN(~Mc)=Nmino,(5)whichisourrstmainresult.OncethechildmassMcisfoundfromEquation( 5 ),thetrueparentmassMpisobtainedasusualfromEquation( 5 )asMp=~Mp(Mc,UT).Atthispointitisnotclearwhetherwehavegainedanythingstatistics-wise,sincethereferencequantity~Mp(~Mc,0)appearinginthedenitionofEquation( 5 )hastobemeasuredataxedUT=0anyway.Oursecondmainresultisthat~Mp(~Mc,0)caninfactbemeasuredfromthefulldatasetwithnolossinstatisticsasfollows. 144

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Figure5-6. DecompositionoftheobservedtransversemomentumvectorsfromFigure 5-5 inthetransverseplane. StepI.Orthogonaldecompositionoftheobservedtransversemomentawithrespecttothe~UTdirection.TheTevatronandLHCcollaborationscurrentlyusexedaxescoordinatesystemstodescribetheirdata.Instead,weproposetorotatethecoordinatesystemfromoneeventtoanother,sothatthetransverseaxesarealwaysalignedwiththedirectionTkselectedbythemeasuredupstreamtransversemomentumvector~UTandthedirectionT?orthogonaltoit(Figure 5-6 ).ThevisibletransversemomentumvectorsfromFigure 5-5 arethendecomposedas ~piTk1 U2T~piT~UT~UT, (5) ~piT?~piT)]TJ /F8 11.955 Tf 11.39 0.49 Td[(~piTk=1 U2T~UT~piT~UT. (5) StepII.ConstructingthetransverseandlongitudinalcontransversemassesMT2?andMT2k.Nowwedene1DMT2decompositionsincompleteanalogywiththestandardMT2denitionofEquation( 5 ): MT2kmin~pc1Tk+~pc2Tk=~=PTknmaxnM1Tk,M2Tkoo, (5) MT2?min~pc1T?+~pc2T?=~=PT?fmaxfM1T?,M2T?gg. (5) 145

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Thesedecompositionsareextremelyuseful.Foronce,the1DvariablesinEquations( 5 5 )canbecalculatedviasimpleanalyticexpressionsasshownbelow.Incontrast,ageneralformulafortheoriginalMT2variableinEquation( 5 )inthepresenceofarbitraryUTisunknownandonestillhastocomputeMT2numerically[ 38 ].Moreimportantly,MT2?allowsustomeasurethereferencequantity~Mp(~Mc,0)inEquation( 5 )fromthefulldataset,usingeventswithanyvalueofUT.Tounderstandthebasicidea,itissufcienttoconsiderthesimplest,yetmostchallengingcaseofasinglestepdecaychain.LetVibeasingle,(approximately)masslessSMparticle:m1=m2=0.(Thediscussionforthemassivecaseproceedsanalogously.)Inwhatfollows,forillustrationweshallusethesame-signdileptonchannelinsupersymmetry,whereeachViisaleptonresultingfromacharginodecaytoasneutrino[ 39 ].Thecharginosthemselvesareproducedindirectlyinthedecaysofsquarksandgluinos.ForconcretenessweshalluseaSUSYspectrumgivenbytheLM6CMSstudypointasinTable 5-1 .InoursimulationsweusethePYTHIAeventgenerator[ 54 ]andthePGSdetectorsimulationprogram[ 55 ].ThevariableMT2?hasseveraluniqueproperties.Eventwise,itcanbecalculatedanalyticallyas MT2?=p AT?+q AT?+~M2c, (5) AT?1 2(j~p1T?jj~p2T?j+~p1T?~p2T?).TheendpointoftheMT2?distributionisgivenby MmaxT2?(~Mc)=+q 2+~M2c,(5)intermsoftheparameterintroducedin[ 23 ] Mp 21)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M2c M2p.(5) 146

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Figure5-7. Theunit-normalizedMT2?distribution( 5 )forthesame-signdileptonchannelinaSUSYmodelwithLM6CMSmassspectrumandachoiceoftestmass~Mc=100GeV.Theyellowshadeddistributionshowsthetheoreticallypredictedshape( 5 ),matchingverywellthepartonlevelresultfromPYTHIAwithnocuts(redhistogram).Thegreen(blue)histogramisthecorrespondingresultafterPGSdetectorsimulationwithmild(hard)cutsasexplainedinthetext.TheendpointexpectedfromEquation( 5 )is132.1GeVandismarkedwiththeverticalarrow. Equation( 5 )revealsperhapsthemostimportantfeatureoftheMT2?variable:itsendpointisindependentoftheupstreamPTandcanthusbemeasuredwiththewholedatasample.Wecanevenpredictanalyticallytheshapeofthe(unit-normalized)differentialMT2?distribution dN dMT2?=N0?(MT2?)]TJ /F6 11.955 Tf 14.57 2.65 Td[(~Mc)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(N0?)dN dMT2?,(5)whereN0?isthefractionofeventsinthelowest~McbinMT2?=~Mc,whiletheshapeoftheremaining(unit-normalized)MT2?distributionisgivenby(Figure 5-7 ) dN dMT2?=M4T2?)]TJ /F6 11.955 Tf 14.56 2.66 Td[(~M4c 2M3T2?ln 2MT2? M2T2?)]TJ /F6 11.955 Tf 14.56 2.66 Td[(~M2c!.(5) 147

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Figure5-8. ObservableMT2?distributionafterhardcutsfor100fb)]TJ /F5 7.97 Tf 6.58 0 Td[(1ofLHCdata.ThetotalstackeddistributionconsistsoftheSUSYsignal(red)andtheSMbackground(blue).Thesolidlineistheresultofasimplelineart,revealingendpointsat134.4GeVand172.4GeV. Noticethatthisshapedoesnotdependonanyunknownkinematicparameters,suchastheunknowncenter-of-massenergyorlongitudinalmomentumoftheinitialhardscattering.Itisalsoinsensitivetospincorrelationeffects,whenevertheupstreammomentumresultsfromproductionand/ordecayprocessesinvolvingscalarparticles(e.g.squarks)orvectorlikecouplings(e.g.theQCDgaugecoupling).ItisevenindependentoftheactualvalueoftheupstreammomentumPT.ThuswearenotrestrictedtoaparticularPTrangeandcanusethewholeeventsampleintheMT2?analysis.Foranychoiceof~Mc(inFigure 5-7 weused~Mc=100GeV),Equation( 5 )isaone-parametercurvewhichcanbettedtothedatatoobtaintheparameterandfromtheretheMT2?endpoint( 5 ).Asalways,therearepracticallimitationstotheuseofsuchshapetting.First,theshapeinEquation( 5 )ismodiedinthepresenceofmildcuts,whicharerequiredforleptonidenticationinPGS(greenhistograminFigure 5-7 ),andmoreimportantly, 148

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forthediscoveryofthesame-signdileptonSUSYsignalovertheSMbackgrounds.Toensurediscovery,weusehardcutsasfollows[ 75 91 ]:exactlytwoisolatedleptonswithpT>10GeV,atleastthreejetswithpT>(175,130,55)GeV,=PT>200GeVandavetoontaujets.Withthosecuts,inthedimuonchannelalone,theremainingSMbackgroundcross-sectionisdominatedbyttandisjust0.15fb,whiletheSUSYsignalis14fb,leadingtoa22discoverywithjust10fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofdata[ 75 91 ].ThedistortionoftheMT2?shapewiththesehardofinecutsisillustratedbytheblue(rightmost)histograminFigure 5-7 .TheactualMT2?distributionwhichweexpecttoobservewith100fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofdata,isshowninFigure 5-8 andiscomprisedofarelativelysmallSMbackgroundcomponent(blue)andadominantSUSYsignalcomponent(red).InspiteofthepresenceofasizableSUSYcombinatorialbackground,theMT2?endpointexpectedfromFigure 5-7 isclearlyvisibleanditslocationfromasimplelineartisobtainedas134.4GeV,whichisveryclosetothenominalvalueof132.1GeV.(Interestingly,thedatarevealsasecondendpointat172.4GeV,whichisduetoeventsinwhichonecharginodecaysthroughachargedslepton:~1!~`L!~01[ 39 ].Itsnominalvalueis169.2GeV.)Ournalkeyobservationisthat ~Mp(~Mc,0)=MmaxT2(~Mc,0)=MmaxT2?(~Mc),(5)whichallowstorewritethefunctionN(~Mc)ofEquation( 5 )as N(~Mc)XalleventsHMT2)]TJ /F3 11.955 Tf 11.96 0 Td[(MmaxT2?(~Mc).(5)TheMT2?analysisjustdescribedallowsaveryprecisemeasurementofthebenchmarkquantityMmaxT2?(~Mc)appearinginEquation( 5 ),sothatthefunctionN(~Mc)itselfcanbereliablyreconstructed,usingthewholeeventsampleallthewaythroughouttheanalysis,withoutanylossinstatistics. 149

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Figure5-9. Thefunction^N(~Mc)denedinEquation( 5 ).Theblue(red)setofmeasurementsarewith(without)SUSYcombinatorialbackground.Theerrorbarsshownarepurelystatistical. WeshowourresultinFigure 5-9 ,whereforconvenienceweunit-normalizethefunctionN(~Mc)as ^N(~Mc)=N(~Mc)=hN(~Mc)i,(5)wheretheaveragingisperformedovertheplottedrangeof~Mc.Asexpected,thefunction^N(~Mc)exhibitsaminimuminthevicinityofthetruesneutrinomass~Mc=Mc=275.7GeV.IgnoringtheSUSYcombinatorialbackground,thismeasurement(reddatapoints)isquiteprecise,atthelevelofafewpercent.Inordertoreducethecombinatorialbackground,weselecteventswith~Mc60GeV.TheresultingMcmeasurement(bluedatapoints)isatthelevelof10%.ThisprecisionisclearlysufcienttoexcludeSMneutrinosasthesourceofthemissingenergy,hintingatapotentialdarkmatterdiscoveryattheLHC.Notethatthetraditional 1ThemeasuredvalueofMmaxT2?inFigure 5-8 alreadyimpliesthatthemasssplittingMp)]TJ /F3 11.955 Tf 11.95 0 Td[(Mcisontheorderof30GeV,resultinginarathersoftleptonpTspectrum. 150

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kinkfortheLM6studypointisonlyafewdegreesandappearsdifculttoobserveexperimentally[ 39 ],unliketheresultinFigure 5-9 .Forcompleteness,wenowdiscussthepropertiesofthevariableMT2k.Inanygivenevent,itcanbecalculatedasfollows.If~p1Tk~p2Tk0,thensimplyMT2k=~Mc,andtheeventfallsinthelowest~Mcbin.Inthealternativecaseof~p1Tk~p2Tk>0,MT2kcanbefoundfrom M2T2k=j~p1Tkj+q ~M2c+j~p(1)cTkj22)]TJ /F12 11.955 Tf 11.96 13.75 Td[(~p1Tk+~p(1)cTk2,wherethechildtestmomentumis ~p(1)cTk=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2h~UT+(1)]TJ /F8 11.955 Tf 11.95 0 Td[()~p1Tk+(1+)~p2Tki,vuut ~M2c ~p1Tk~p2Tk+j~UT+~p1Tk+~p2Tkj2 j~p1Tk+~p2Tkj2.TheendpointMmaxT2koftheMT2kdistributionisidenticaltotheendpoint( 5 )oftheMT2distributionitself: MmaxT2k=MmaxT2(~Mc,UT)=~Mp(~Mc,UT),(5)andisthusknownasafunctionofUT[ 23 ] MmaxT2k=(sUT)+s (sUT)+sUT 22+~M2c2)]TJ /F6 11.955 Tf 13.15 8.08 Td[((sUT)2 41 2,(5)where (sUT)0@s 1+sUT 2Mp2)]TJ /F3 11.955 Tf 13.76 8.09 Td[(sUT 2Mp1A,ands=1isanintegerdenedasssgn(Mc)]TJ /F6 11.955 Tf 14.83 2.66 Td[(~Mc).Unfortunately,theshapeoftheMT2kdistributioncannotbepredictedinamodel-independentway,sinceitissensitivetotheunderlyingp ^s,aswellastothemeasuredUT.Nevertheless,onecanimagineseveralusefulapplicationsofMT2k.Forexample,onecanstudytheboundaryofthetwo-dimensionalfMT2?,MT2kgdistributionaswewilldowithonedimensionalprojectedvariablesofMCT.Onecouldalsoconsiderananalogueof( 5 ),denedintermsof 151

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MT2k: Nk(~Mc)XalleventsHMT2k)]TJ /F3 11.955 Tf 11.95 0 Td[(MmaxT2?(~Mc),(5)whoseminimumwillalsomarkthelocationofthetruechildmassMc. 5.3UsingFullPhaseSpaceInformationWithMCTInthissection,weshowhowtogetmoreinformationfromtheboundaryofphasespace,with1D-decompsedMCTvariable.ComparedtoMT2approaches,MCTmethodhastwoadvantages.First,itissimpleritusesonlytheobservedobjectsU,V1andV2intheeventandmakesnoreferencetothemissingparticlekinematics(ormass).Second,itismoreprecise,sinceitutilizesthewholekinematicboundaryoftherelevanttwo-dimensionaldistributionandnotjustthekinematicendpointofitsone-dimensionalprojection.StepI.ConstructingthetransverseandlongitudinalcontransversemassesMCT?andMCTk.Ourstartingpointistheoriginalcontransversemassvariable[ 42 ] MCT=q m21+m22+2(e1Te2T+~p1T~p2T),(5)whereeiTisthetransverseenergyofVi eiT=q m2i+j~piTj2.(5)ForeventswithUT=0,MCThasanupperendpointwhichisinsensitivetotheunknownp ^s,providingonerelationamongMpandMc[ 42 43 ] MmaxCT(UT=0)=q m21+m22+2m1m2cosh(1+2),(5)where sinhi1 2(M2p,M2c,m2i) 2Mpmi, (5) (x,y,z)x2+y2+z2)]TJ /F6 11.955 Tf 11.96 0 Td[(2xy)]TJ /F6 11.955 Tf 11.96 0 Td[(2xz)]TJ /F6 11.955 Tf 11.96 0 Td[(2yz. (5) 152

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Unfortunately,theUT=0limitisnotparticularlyinterestingathadroncolliders(especiallyforinclusivestudies),sinceasignicantamountofupstreamUTistypicallygeneratedbyISR(andother)jets.Onepossiblexistouseallevents,butmodifythedenition( 5 )toapproximatelycompensateforthetransverse~UTboost[ 43 ].OnethenrecoversadistributionwhoseendpointisstillgivenbyEquation( 5 ).Alternatively,onecouldsticktotheoriginalMCTvariable,andsimplyaccountfortheUTdependenceofitsendpointas MmaxCT(UT)=q m21+m22+2m1m2cosh(2+1+2)(5)whereiwerealreadydenedinEquation( 5 ),and sinhUT 2Mp,coshs 1+U2T 4M2p.(5)Ourapproachhereistoutilizetheone-dimensionalprojectionsfromEquations( 5 5 )andconstructone-dimensionalanaloguesoftheMCTvariable MCT?q m21+m22+2(e1T?e2T?+~p1T?~p2T?), (5) MCTkr m21+m22+2e1Tke2Tk+~p1Tk~p2Tk, (5) wherethecorrespondingtransverseenergiesare eiT?q m2i+j~piT?j2,eiTkq m2i+j~piTkj2.(5)ThebenetofthedecompositionasEquations( 5 5 )isthatonegetstwoforthepriceofone,i.e.twoindependentandcomplementaryvariablesinsteadofthesinglevariableinEquation( 5 ).ThevariableMCT?inparticularisveryusefulforourpurposes.Toillustratethebasicidea,itissufcienttoconsiderthemostcommoncase,whereViisapproximatelymassless(mi=0),astheleptonsinourttexample.AcrucialpropertyofMCT?isthat 153

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itsendpointisindependentofUT: MmaxCT?=M2p)]TJ /F3 11.955 Tf 11.95 0 Td[(M2c Mp,8UT.(5)InfactthewholeMCT?distributionisinsensitivetoUT: dN dMCT?=N0?(MCT?)+(Ntot)]TJ /F3 11.955 Tf 11.96 0 Td[(N0?)dN dMCT?,(5)whereN0?isthenumberofeventsinthezerobinMCT?=0.Usingphasespacekinematics,wendthattheshapeoftheremaining(unit-normalized)zero-bin-subtracteddistributionissimplygivenby dN d^MCT?)]TJ /F6 11.955 Tf 21.92 0 Td[(4^MCT?ln^MCT?(5)intermsoftheunit-normalizedMCT?variable ^MCT?MCT? MmaxCT?.(5)TheobservableMCT?distributionforourttexampleisshowninFigure 5-10 ,for10fb)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofLHCdataat7TeV.EventsweregeneratedwithPYTHIA[ 54 ]andprocessedwiththePGSdetectorsimulator[ 55 ].Weapplystandardbackgroundrejectioncutsasfollows[ 92 ]:werequiretwoisolated,oppositesignleptonswithpiT>20GeV,m`+`)]TJ /F8 11.955 Tf 10.69 -0.3 Td[(>12GeV,andpassingaZ-vetojm`+`)]TJ /F2 11.955 Tf 9.65 -0.3 Td[()]TJ /F3 11.955 Tf 12.2 0 Td[(MZj>15GeV;atleasttwocentraljetswithpT>30GeVandjj<2.4;anda=ETcutof=ET>30GeV(=ET>20GeV)foreventswithsameavor(oppositeavor)leptons.Wealsodemandatleasttwob-taggedjets,assumingaatb-taggingefciencyof60%.Withthosecuts,theSMbackgroundfromotherprocessesisnegligible[ 92 ].Figure 5-10 demonstratesthattheMCT?endpointcanbemeasuredquitewell.SincethetheoreticallypredictedshapeofEquation( 5 )isdistortedbythecuts,weusealinearslopewithGaussiansmearing,andtfortheendpointandtheresolutionparameter. 154

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Figure5-10. Zero-binsubtractedMCT?distributionaftercuts,forttdileptonevents.Theyellow(lower)portionisoursignal,whiletheblue(upper)portionshowsttcombinatorialbackgroundwithisolatedleptonsarisingfromorbdecays. WendMmaxCT?=80.9GeV(comparetothetruevalueMmaxCT?=80.4GeV),whichgivesoneconstraint( 5 )amongMpandMc.Atthispoint,asecond,independentconstraintcaninprinciplebeobtainedfromananalogousmeasurementoftheMmaxCTendpointinEquation( 5 )ataxedvalueofUT(resultinginlossinstatistics),afterwhichthetwomassescanbefoundfrom Mp=UTMmaxCT(UT)MmaxCT? (MmaxCT(UT))2)]TJ /F6 11.955 Tf 11.95 0 Td[((MmaxCT?)2, (5) Mc=q Mp)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Mp)]TJ /F3 11.955 Tf 11.95 0 Td[(MmaxCT?. (5) However,theorthogonaldecompositionasEquations( 5 5 )offersanotherapproach,whichwepursueinthelaststep.StepII.Fittingtokinematicboundarylines.Itisknownthattwo-dimensionalcorrelationplotsrevealalotmoreinformationthanone-dimensionalprojectedhistograms[ 22 ].Tothisend,considerthescatterplotofMCT?vsMCTkinFigure 5-11 (a),whereforillustrationweused10,000eventsatthepartonlevel. 155

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Figure5-11. Scatterplotsof(a)MCT?versusMCTkand(b)MCT?versusMCT,foraxedrepresentativevalueUT=75GeV.Thesolidlinesshowthecorrespondingboundariesdenedin( 5 )and( 5 ),forthecorrectvalueofMmaxCT?andseveraldifferentvaluesofMpasshown. ForagivenvalueofMCT?,theallowedvaluesofMCTkareboundedby M(lo)CTk(MCT?)MCTkM(hi)CTk(MCT?),(5)whereM(lo)CTk(MCT?)=0and M(hi)CTk(MCT?)=MmaxCT?q 1)]TJ /F6 11.955 Tf 14.56 2.65 Td[(^M2CT?cosh+sinh.(5)Figure 5-11 (a)revealsthattheendpointMmaxCTkoftheone-dimensionalMCTkdistributionisobtainedatMCT?=0 MmaxCTk=M(hi)CTk(0)=MmaxCT?(cosh+sinh)=1 21)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M2c M2pq 4M2p+U2T+UT. (5) NoticethateventsinthezerobinsMCT?=0andMCTk=0fallononeoftheaxesandcannotbedistinguishedontheplot. 156

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NowconsiderthescatterplotofMCT?vsMCTshowninFigure 5-11 (b).MCTissimilarlyboundedby M(lo)CT(MCT?)MCTM(hi)CT(MCT?),(5)wherethistimeM(lo)CT(MCT?)=MCT?and M(hi)CT(MCT?)=MmaxCT?cosh+q 1)]TJ /F6 11.955 Tf 14.56 2.66 Td[(^M2CT?sinh.(5)WeseethattheendpointMmaxCToftheone-dimensionalMCTdistributionisalsoobtainedforMCT?=0: MmaxCT=M(hi)CT(0)=MmaxCT?(cosh+sinh)=MmaxCTk.(5)Figure 5-11 revealsaconceptualproblemwithone-dimensionalprojections.WhileallpointsinthevicinityoftheboundarylinesinEquations( 5 )and( 5 )aresensitivetothemasses,theMmaxCT?endpointisextractedmostlyfromeventswithMCT?MmaxCT?,whiletheMmaxCTkandMmaxCTendpointsareextractedmostlyfromtheeventswithMCT?0.Theeventsneartheboundary,butwithintermediatevaluesofMCT?,willnotenterefcientlyeitheroneoftheseendpointdeterminations.Sohowcanonedobetter,giventheknowledgeoftheboundarylineofEquation( 5 )?Inthespiritof[ 93 ],wedenethesigneddistancetothecorrespondingboundary,e.g. DCT(Mp,Mc)M(hi)CT(MCT?,UT,Mp,Mc))]TJ /F3 11.955 Tf 11.96 0 Td[(MCTandsimilarlyforDCTk.ThekeypropertyofthisvariableisthatforthecorrectvaluesofMpandMc,itslowerendpointDminCTisexactlyzero(seeFigure 5-12 (b)): DminCT(Mp,Mc)=0.(5)InthatcasetheboundarylineprovidesaperfectlysnugttothescatterplotnoticethegreenboundarylinemarkedinFigure 5-11 (b). 157

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Figure5-12. DCTdistributionsforfourdifferentvaluesofMp(andMcgivenfrom( 5 )).Theyellow(lightshaded)histogramsuseonlyeventsinthezerobinMCT?=0.Theredsolidlinesshowlinearbinnedmaximumlikelihoodts. WhileingeneralEquation( 5 )representsatwo-dimensionalttoMpandMc,inpracticeonecanalreadyusetheMmaxCT?measurementtoreducetheproblemtoasingledegreeoffreedom,e.g.theparentmassMp,aspresentedinFigures 5-11 and 5-12 .WeseethatthecorrectparentmassMp=80GeVprovidesaperfectenvelope,forwhichDminCT=0.If,ontheotherhand,Mpistoolow,agapdevelopsbetweentheoutlyingpointsinthescatterplotandtheirexpectedboundary,whichresultsinDminCT>0. 158

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Figure5-13. FittedvaluesofDminCTasafunctionofMp. Conversely,ifMpistoohigh,someoftheoutlyingpointsfromthescatterplotfalloutsidetheboundaryandhaveDCT<0,leadingtoDminCT<0,asseeninFigure 5-12 (c,d).TheresultingtforDminCTasafunctionofMpfromourPGSdatasampleisshowninFigure 5-13 ,whichsuggeststhataWmassmeasurementatthelevelofafewpercentmightbeviable. 159

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CHAPTER6ASYMMETRICEVENTTOPOLOGYInthischapter,westudyasymmetriceventsbasedonfollowingreasons. Singledarkmattercomponent.Acommonassumptionthroughoutthecolliderphenomenologyliteratureisthatcollidersareprobingonlyonedarkmatterspeciesatatime,i.e.thatthemissingenergysignalatcollidersisduetotheproductionofoneandonlyonetypeofdarkmatterparticles.Ofcourse,thereisnoastrophysicalevidencethatthedarkmatterismadeupofasingleparticlespecies:itmayverywellbethatthedarkmatterworldhasarichstructure,justlikeours[ 95 ].Consequently,ifthereexistseveraltypesofdarkmatterparticles,eachcontributingsomefractiontothetotalrelicdensity,apriorithereisnoreasonwhytheycannotallbeproducedinhighenergycollisions.Theoreticalmodelswithmultipledarkmattercandidateshavealsobeenproposed[ 45 47 96 101 ]. Identicalmissingparticlesineachevent.Aseparateassumption,commontomostpreviousstudies,isthatthetwomissingparticlesineacheventareidentical.Thisassumptioncouldinprinciplebeviolatedaswell,evenifthesingledarkmattercomponenthypothesisistrue.Thepointisthatoneofthemissingparticlesintheeventmaynotbeadarkmatterparticle,butsimplysomeheaviercousinwhichdecaysinvisibly.Aninvisiblydecayingheavyneutralino(~0i!~01withi>1)andaninvisiblydecayingsneutrino(~!~01)aretwosuchexamplesfromsupersymmetry.Asfarastheeventkinematicsisconcerned,themassoftheheaviercousinisarelevantparameterandapproximatingitwiththemassofthedarkmatterparticlewillsimplygivenonsensicalresults.AnotherrelevantexampleisprovidedbymodelsinwhichtheSUSYcascademayterminateinanyoneofseverallightneutralparticles[ 102 ].Givenourutterignoranceaboutthestructureofthedarkmattersector,inthischapterwesetouttodevelopthenecessaryformalismforcarryingoutmissingenergystudiesathadroncollidersinaverygeneralandmodel-independentway,withoutrelyingonanyassumptionsaboutthenatureofthemissingparticles.Inparticular,weshallnotassumethatthetwomissingparticlesineacheventarethesame.Weshallalsoallowforthesimultaneousproductionofseveraldarkmatterspecies,oralternatively,fortheproductionofadarkmattercandidateinassociationwithaheavier,invisiblydecayingparticle.Undertheseverygeneralcircumstances,weshalltrytodevelopamethodformeasuringtheindividualmassesofallrelevantparticles-thevarious 160

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missingparticleswhichareresponsibleforthemissingenergy,aswellastheirparentswhichwereoriginallyproducedintheevent. 6.1GeneralizingMT2ToAsymmetricEventTopologiesIngeneral,bynowthereisawidevarietyoftechniquesavailableformassmeasurementsinSUSY-likemissingenergyevents.Sucheventsarecharacterizedbythepairproductionoftwonewparticles,eachofwhichundergoesasequenceofcascadedecaysendingupinaparticlewhichisinvisibleinthedetector.Eachtechniquehasitsownadvantagesanddisadvantages1.Forourpurposes,wechosetorevampthemethodoftheCambridgeMT2variable[ 10 ]andadaptittothemoregeneralcaseofanasymmetriceventtopologyshowninFigure 6-1 .Considertheinclusiveproductionoftwoidentical2parentsofmassMpasshowninFigure 6-1 .Theparentparticlesmaybeaccompaniedbyanynumberofupstreamobjects,suchasjetsfrominitialstateradiation[ 35 36 103 ],orvisibledecayproductsofevenheavier(grandparent)particles[ 23 ].Theexactoriginandnatureoftheupstreamobjectswillbeofnoparticularimportancetous,andtheonlyinformationaboutthemthatweshallusewillbetheirtotaltransversemomentum~PUTM.Inturn,eachparentparticleinitiatesadecaychain(showninred)whichproducesacertainnumbern()ofStandardModel(SM)particles(showningray)andanintermediatechildparticleofmassM()c.Throughoutthischapterweshallusetheindextoclassifyvariousobjectsasbelongingtotheupper(=a)orlower(=b)branchinFigure 6-1 .Thechildparticlemayormaynotbeadarkmattercandidate:ingeneral,itmaydecayfurtherasshownbythedashedlinesinFigure 6-1 1Foracomparativereviewofthethreemaintechniques,[ 23 ].2Inprinciple,theassumptionofidenticalparentscanalsoberelaxed,byasuitablegeneralizationoftheMT2variable,inwhichthemassratioofthetwoparentsistreatedasanadditionalinputparameter[ 40 ]. 161

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Figure6-1. Thegenericeventtopologyunderconsiderationinthischapter.Weconsidertheinclusivepair-productionoftwoparentparticleswithidenticalmassesMp.Theparentsmaybeaccompaniedbyupstreamobjects,e.g.jetsfrominitialstateradiation,visibledecayproductsofevenheavierparticles,etc.Thetransversemomentumofallupstreamobjectsismeasuredanddenotedby~PUTM.Inturn,eachparentparticleinitiatesadecaychain(showninred)whichproducesacertainnumbern()ofSMparticles(showningray)andanintermediatechildparticleofmassM()c,where=a(=b)forthebranchabove(below).Ingeneral,thechildparticledoesnothavetobethedarkmattercandidate,andmaydecayfurtherasshownbythedashedlines.TheMT2variableisdenedforthesubsysteminsidetheblueboxandisdenedintermsoftwoarbitrarychildrentestmasses~M(a)cand~M(b)c.Then()SMparticlesfromeachbranchformacompositeparticleoftransversemomentum~p()Tandinvariantmassm(),correspondingly.Thetrialtransversemomenta~q()Tofthechildrenobeythetransversemomentumconservationrelationshowninsidethegreenbox.Ingeneral,thenumbern(),aswellasthetypeofSMdecayproductsineachbranchdonothavetobethesame. 162

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WeshallapplythesubsystemMT2concept[ 23 85 ]tothesubsystemwithinthebluerectangularframe.TheSMparticlesfromeachbranchwithinthesubsystemformacompositeparticleofknown3transversemomentum~p()Tandinvariantmassm().SincethechildrenmassesM(a)candM(b)careaprioriunknown,thesubsystemMT2willbedenedintermsoftwotestmasses~M(a)cand~M(b)c.InFigure 6-1 ,~q()Tarethetrialtransversemomentaofthetwochildren.Theindividualmomenta~q()Tarealsoaprioriunknown,buttheyareconstrainedbytransversemomentumconservation: ~q(a)T+~q(b)T~Qtot=)]TJ /F6 11.955 Tf 9.3 0 Td[((~p(a)T+~p(b)T+~PUTM).(6)Giventhisverygeneralsetup,inSection 6.3 weshallconsiderageneralizationoftheusualMT2variablewhichcanapplytotheasymmetriceventtopologyofFigure 6-1 .Therewillbetwodifferentaspectsoftheasymmetry: Firstandforemost,weshallavoidthecommonassumptionthatthetwochildrenhavethesamemass.Thiswillbeimportantfortworeasons.Ontheonehand,itwillallowustostudyeventsinwhichthereareindeedtwodifferenttypesofmissingparticles.Weshallgiveseveralsuchexamplesinthesubsequentsections.Moreimportantly,theendpointoftheasymmetricMT2variablewillallowustomeasurethetwochildrenmassesseparately.Therefore,evenwhentheeventscontainidenticalmissingparticles,asisusuallyassumedthroughouttheliterature,onewouldbeabletoestablishthisfactexperimentallyfromthedata,insteadofrelyingonanadhoctheoreticalassumption. AscanbeseenfromFigure 6-1 ,ingeneral,thenumberaswellasthetypesofSMdecayproductsineachbranchmaybedifferentaswell.Onceweallowforthechildrentobedifferent,andgiventhefactthatwestartfromidenticalparents,thetwobranchesofthesubsystemwillnaturallyinvolvedifferentsetsofSMparticles.Inwhatfollows,whenreferringtothemoregeneralMT2variabledenedinSection 6.3 ,weshallinterchangeablyusethetermsasymmetricorgeneralizedMT2.Incontrast, 3WeassumethattherearenoneutrinosamongtheSMdecayproductsineachbranch. 163

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weshallusethetermsymmetricwhenreferringtothemoreconventionalMT2denitionwithidenticalchildren.ThetraditionalMT2approachassumesthatthechildrenhaveacommontestmass~Mc~M(a)c=~M(b)candthenproceedstondonefunctionalrelationbetweenthetruechildmassMcandthetrueparentmassMpasfollows[ 10 ].ConstructseveralMT2distributionsfordifferentinputvaluesofthetestchildrenmass~McandthenreadofftheirupperkinematicendpointsMT2(max)(~Mc).Theseendpointmeasurementsaretheninterpretedasanoutputparentmass~Mp,whichisafunctionoftheinputtestmass~Mc: ~Mp(~Mc)MT2(max)(~Mc).(6)TheimportanceofthisfunctionalrelationisthatitisautomaticallysatisedforthetruevaluesMpandMcoftheparentandchildmasses: Mp=MT2(max)(Mc).(6)Inotherwords,ifwecouldsomehowguessthecorrectvalueMcofthechildmass,thefunction( 6 )willprovidethecorrectvalueMpoftheparentmass.However,sincethetruechildmassMcisaprioriunknown,theindividualmassesMpandMcstillremainundeterminedandmustbeextractedbysomeothermeans.Atthispoint,itmayseemthatbyconsideringtheasymmetricMT2variablewithnon-identicalchildrenparticles,wehaveregressedtosomeextent.Indeed,weareintroducinganadditionaldegreeoffreedominEquation( 6 ),whichnowreads ~Mp(~M(a)c,~M(b)c)MT2(max)(~M(a)c,~M(b)c).(6)ThestandardMT2endpointmethodwillstillallowustondtheparentmass~Mp,butnowitisafunctionoftwoinputparameters~M(a)cand~M(b)cwhicharecompletelyunknown.Ofcourse,ifoneknewthecorrectvaluesofthetwochildrenmassesM(a)candM(b)centeringEquation( 6 ),thetrueparentmassMpwillbegiveninamanneranalogousto 164

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Equation( 6 ): Mp=MT2(max)(M(a)c,M(b)c).(6)OurmainresultisthatinspiteoftheapparentremainingarbitrarinessinEquation( 6 ),onecanneverthelessuniquelydetermineallthreemassesMp,M(a)candM(b)c,justbystudyingthebehaviorofthemeasuredfunction~Mp(~M(a)c,~M(b)c).Moreimportantly,thisdeterminationcanactuallybedoneintwodifferentways!OurrstmethodissimplyageneralizationoftheobservationmadeinReferences.[ 23 34 37 ]thatundercertaincircumstances(varyingm()ornonvanishingupstreammomentumPUTM),thefunction( 6 )developsakinkpreciselyatthecorrectvalueMcofthechildmass: @~Mp(~Mc) @~Mc!~Mc+)]TJ /F12 11.955 Tf 11.96 20.45 Td[( @~Mp(~Mc) @~Mc!~Mc)]TJ /F7 7.97 Tf 6.58 0 Td[(8><>:=0,if~Mc6=Mc,6=0,if~Mc=Mc.(6)Inotherwords,thefunction( 6 )iscontinuous,butnotdifferentiableatthepoint~Mc=Mc.IntheasymmetricMT2case,wendthatthefunction( 6 )issimilarlynon-differentiableatasetofpointsf(~M(a)c,~M(b)c)g,sothatthekinkofEquation( 6 )isgeneralizedtoaridgeonthe2-dimensionalhypersurfacedenedbyEquation( 6 )inthethree-dimensionalparameterspaceoff~M(a)c,~M(b)c,~Mpg.4Interestinglyenough,theridgeoften(albeitnotalways)exhibitsaspecialpointwhichmarkstheexactlocationofthetruevalues(M(a)c,M(b)c).Oursecondmethodfordeterminingthetwochildrenmasses~M(a)cand~M(b)cisevenmoregeneralandisapplicableunderanycircumstances.ThemainstartingpointisthatjustliketheendpointofthesymmetricMT2,theendpointoftheasymmetricMT2alsodependsonthevalueoftheupstreamtransversemomentumPUTM,sothat 4Reference[ 40 ]studiedtheorthogonalscenarioofdifferentparents(M(a)p6=M(b)p)andidenticalchildren(M(a)c=M(b)c)andfoundasimilarnon-differentiablefeature,calledacrease,onthecorrespondingtwo-dimensionalhypersurfacewithinthethree-dimensionalparameterspacef~Mc,~M(a)p,~M(b)pg. 165

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Equation( 6 )ismoreproperlywrittenas ~Mp(~M(a)c,~M(b)c,PUTM)=MT2(max)(~M(a)c,~M(b)c,PUTM).(6)NowwecanexplorethePUTMdependenceinEquation( 6 )andnotethatitisabsentforpreciselytherightvaluesof~M(a)cand~M(b)c: @MT2(max)(~M(a)c,~M(b)c,PUTM) @PUTM~M(a)c=M(a)c,~M(b)c=M(b)c=0.(6)Whilethispropertyhasbeenknown,itwasrarelyusedinthecaseofthesymmetricMT2,sinceitoffersredundantinformation:oncethecorrectchildmassMcisfoundthroughtheMT2kinkasEquation( 6 ),theparentmassMpisgivenbyEquation( 6 )andtherearenoremainingunknowns,thusthereisnoneedtofurtherinvestigatethePUTMdependence.InthecaseoftheasymmetricMT2,however,westartwithoneadditionalunknownparameter,whichcannotalwaysbedeterminedfromtheridgeinformationalone.Therefore,inordertopindownthecompletespectrum,weareforcedtomakeuseofEquation( 6 ).ThenicefeatureofthePUTMmethodisthatitalwaysallowsustodeterminebothchildrenmassesM(a)candM(b)c,withoutrelyingontheridgeinformationatall.Inthissense,ourtwomethodsarecomplementaryandeachcanbeusedtocross-checktheresultsobtainedbytheother. 6.2TheConventionalSymmetricMT2 6.2.1DenitionWebeginourdiscussionbyrevisitingtheconventionaldenitionofthesymmetricMT2variablewithidenticaldaughters,followingthegeneralnotationintroducedinFigure 6-1 .LetusconsidertheinclusiveproductionoftwoparentparticleswithcommonmassMp.Eachparentinitiatesadecaychainproducingacertainnumbern()ofSMparticles.Inthissectionweassumethatthetwochainsterminateinchildrenparticlesofthesamemass:M(a)c=M(b)c=Mc.(FromSection 6.3 onweshallremovethisassumption.)InmostapplicationsofMT2intheliterature,thechildrenparticles 166

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areidentiedwiththeverylastparticlesinthedecaychains,i.e.thedarkmattercandidates.However,thesymmetricMT2canalsobeusefullyappliedtoasubsystemoftheoriginaleventtopology,wherethechildrenaresomeotherpairof(identical)particlesappearingfurtherupthedecaychain[ 23 85 ].TheMT2variableisdenedintermsofthemeasuredinvariantmassm()andtransversemomentum~p()Tofthevisibleparticlesoneachside(Figure 6-1 ).Withtheassumptionofidenticalchildren,thetransversemassofeachparentis M()T)]TJ /F8 11.955 Tf 4.91 -9.19 Td[(~p()T;~q()T;m();~Mc=r m2()+~M2c+2e()~e())]TJ /F8 11.955 Tf 11.39 0.5 Td[(~p()T~q()T,(6)where~Mcisthecommontestmassforthechildren,whichisaninputtotheMT2calculation,while~q()Tistheunknowntransversemomentumofthechildparticleinthe-thchain.InEquation( 6 )wehavealsointroducedshorthandnotationforthetransverseenergyofthecompositeparticlemadefromthevisibleSMparticlesinthe-thchain e()=q m2()+~p()T~p()T(6)andforthetransverseenergyofthecorrespondingchildparticleinthe-thchain ~e()=q ~M2c+~q()T~q()T.(6)Thentheevent-by-eventsymmetricMT2variableisdenedthroughaminimizationprocedureoverallpossiblepartitionsofthetwochildrenmomenta~q()T[ 10 ] MT2)]TJ /F7 7.97 Tf 5.07 -9.36 Td[(~p(a)T,~p(b)T;m(a),m(b);~Mc,PUTM=min~q(a)T+~q(b)T=~Qtot"maxnM(a)T)]TJ /F7 7.97 Tf 5.08 -9.35 Td[(~p(a)T;~q(a)T;m(a);~Mc,M(b)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(b)T;~q(b)T;m(b);~Mco#, (6) consistentwiththemomentumconservationconstraint( 6 )inthetransverseplane. 6.2.2ComputationThestandarddenitionasinEquation( 6 )oftheMT2variableissufcienttocomputethevalueofMT2numerically,givenasetofinputvaluesforitsarguments. 167

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Theright-handsideofEquation( 6 )representsasimpleminimizationproblemintwovariables,whichcanbeeasilyhandledbyacomputer.Infact,therearepubliclyavailablecomputercodesforcomputingMT2[ 104 105 ].Thepubliccodeshaveevenbeenoptimizedforspeed[ 38 ]andgiveresultsconsistentwitheachother(aswellaswithourowncode)5.Nevertheless,itisusefultohaveananalyticalformulaforcalculatingtheevent-by-eventMT2forseveralreasons.First,ananalyticalformulaisextremelyvaluablewhenitcomestounderstandingthepropertiesandbehaviorofcomplexmathematicalfunctionslikeinEquation( 6 ).Second,computingMT2fromaformulawillbefasterthananynumericalscanningalgorithm.ThecomputingspeedbecomesanissueespeciallywhenoneconsidersvariationsofMT2likeMT2gen,whereinadditiononeneedstoscanoverallpossiblepartitionsofthevisibleobjectsintotwodecaychains[ 33 ].Thereforeweshallpayspecialattentiontotheavailabilityofanalyticalformulasandweshallquotesuchformulaswhenevertheyareavailable.Inthesymmetriccasewithidenticalchildren,ananalyticalformulafortheevent-by-eventMT2existsonlyinthespecialcasePUTM=0.Itwasderivedin[ 33 ]andweprovideithereforcompleteness.(Inthenextsectionweshallpresentitsgeneralizationfortheasymmetriccaseofdifferentchildren.)ThesymmetricMT2isknowntohavetwotypesofsolutions:balancedandunbalanced[ 32 33 ].ThebalancedsolutionisachievedwhentheminimizationprocedureinEquation( 6 )selectsamomentumcongurationfor~q()Tinwhichthetransversemassesofthetwoparentsarethesame:M(a)T=M(b)T.Inthatcase,typicallyneitherM(a)TnorM(b)Tisatitsglobal(unconstrained)minimum.Inwhatfollows,weshalluseasuperscriptBtorefertosuchbalanced-typesolutions.TheformulaforthebalancedsolutionMBT2ofthe 5Unfortunately,theassumptionofidenticalchildrenishardwiredinthepubliccodesandtheycannotbeusedtocalculatetheasymmetricMT2introducedbelowinSection 6.3 withoutadditionalhacking.WeshallreturntothispointinSection 6.3 168

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symmetricMT2variableisgivenby[ 33 37 ] hMBT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~Mci2=~M2c+AT+vuut 1+4~M2c 2AT)]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.95 0 Td[(m2(b)!A2T)]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a)m2(b),(6)whereATisaconvenientshorthandnotationintroducedin[ 37 ] AT=e(a)e(b)+~p(a)T~p(b)T(6)ande()wasalreadydenedinEquation( 6 ).Ontheotherhand,unbalancedsolutionsarisewhenoneofthetwoparenttransversemasses(M(a)TorM(b)T,asthecasemaybe)isatitsglobal(unconstrained)minimum.DenotingthetwounbalancedsolutionswithasuperscriptU,wehave[ 32 ] MUaT2)]TJ /F7 7.97 Tf 5.08 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~Mc=m(a)+~Mc, (6) MUbT2)]TJ /F7 7.97 Tf 5.08 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~Mc=m(b)+~Mc. (6) GiventhethreepossibleoptionsforMT2,Equations( 6 ),( 6 )and( 6 ),itremainstospecifywhichoneactuallytakesplaceforagivensetofvaluesfor~p(a)T,~p(b)T,m(a),m(b),~McandPUTM=0intheevent6.Thebalancedsolution( 6 )applieswhenthefollowingtwoconditionsaresimultaneouslysatised: M(b)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(b)T;~q(b)T=)]TJ /F7 7.97 Tf 6.24 0.33 Td[(~q(a)T(0)+~=PT;m(b);~McM(a)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T;~q(a)T=~q(a)T(0);m(a),~Mc=m(a)+~Mc, (6) M(a)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T;~q(a)T=)]TJ /F7 7.97 Tf 6.24 0.33 Td[(~q(b)T(0)+~=PT;m(a);~McM(b)T)]TJ /F7 7.97 Tf 5.08 -9.35 Td[(~p(b)T;~q(b)T=~q(b)T(0);m(b);~Mc=m(b)+~Mc, (6) where ~q()T(0)=~Mc m()~p()T,(=a,b),(6) 6RecallthatEquation( 6 )onlyappliesforPUTM=0. 169

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givestheglobal(unconstrained)minimumofthecorrespondingparenttransversemassM()T.TheunbalancedsolutionMUaT2applieswhenthecondition( 6 )isfalseandconditioninEquation( 6 )istrue,whiletheunbalancedsolutionMUbT2applieswhenthecondition( 6 )istrueandconditioninEquation( 6 )isfalse.ItiseasytoseethatconditionsinEquations( 6 )and( 6 )cannotbesimultaneouslyviolated,sothesethreecasesexhaustallpossibilities. 6.2.3PropertiesGivenitsdenition( 6 ),onecanreadilyformandstudythedifferentialMT2distribution.Althoughitsshapeingeneraldoescarrysomeinformationabouttheunderlyingprocess,ithasbecomecustomarytofocusontheupperendpointMT2(max),whichissimplythemaximumvalueofMT2foundintheeventsample: MT2(max)(~Mc,PUTM)=maxalleventshMT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~Mci. (6) Noticethatintheprocessofmaximizingoverallevents,thedependenceon~p(a)T,~p(b)T,m(a)andm(b)disappears,andMT2(max)dependsonlyontwoinputparameters:~McandPUTM,thelatterenteringthrough~QtotinthemomentumconservationconstraintofEquation( 6 ).ThemeasuredfunctioninEquation( 6 )isthestartingpointofanyMT2-basedmassdeterminationanalysis.Weshallnowreviewitsthreebasicpropertieswhichmakeitsuitableforsuchstudies[ 12 ]. 6.2.3.1PropertyI:KnowledgeOfMpAsAFunctionofMcThispropertywasalreadyidentiedintheoriginalpapersandservedasthemainmotivationforintroducingtheMT2variableintherstplace[ 10 32 ].Mathematicallyitcanbeexpressedas ~Mp(~Mc,PUTM)MT2(max)(~Mc,PUTM). (6) ThisisthesameasEquation( 6 ),butnowwehavebeencarefultoincludetheexplicitdependenceonPUTM,whichwillbeimportantinoursubsequentdiscussion. 170

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Figure6-2. Plotsof(a)theMT2endpointMT2(max)(~Mc,PUTM)denedinEquation( 6 ),and(b)thefunctionMT2(max)(~Mc,PUTM)denedinEquation( 6 )asafunctionofthetestchildmass~Mc,forseveralxedvaluesofPUTM:PUTM=0GeV(solid,green),PUTM=500GeV(dot-dashed,black),PUTM=1TeV(dashed,red),andPUTM=2TeV(dotted,blue).TheprocessunderconsiderationispairproductionofsleptonsofmassMp=300GeV,whichdecaydirectlytothelightestneutralino~01ofmassMc=100GeV. AsindicatedinEquation( 6 ),thefunction~Mp(~Mc,PUTM)canbeexperimentallymeasuredfromtheMT2endpointofEquation( 6 ).ThecrucialpointnowisthattherelationinEquation( 6 )issatisedbythetruevaluesMpandMcoftheparentandchildmass,correspondingly: Mp=MT2(max)(Mc,PUTM). (6) NoticethatEquation( 6 )holdsforanyvalueofPUTM,soinpracticalapplicationsofthismethodonecouldchoosethemostpopulatedPUTMbintoreducethestatisticalerror.Ontheotherhand,sinceaprioriwedonotknowthetruemassMcofthemissingparticle,Equation( 6 )givesonlyonerelationbetweenthemassesofthemotherandthechild.ThisisillustratedinFigure 6-2 (a),whereweconsiderthesimpleexampleof 171

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directsleptonpairproduction7,whereeachslepton(~`)decaystothelightestneutralino(~01)byemittingasinglelepton`:~`!`+~01.Herethesleptonistheparentandtheneutralinoisthechild.TheirmasseswerechosentobeMp=300GeVandMc=100GeV,correspondingly,asindicatedwiththeblackdottedlinesinFigure 6-2 (a).Inthisexample,theupstreamtransversemomentumPUTMisprovidedbyjetsfrominitialstateradiation.InFigure 6-2 (a)weplotthefunctioninEquation( 6 )versus~Mc,forseveralxedvaluesofPUTM.ThegreensolidlinerepresentsthecaseofnoupstreammomentumPUTM=0.InagreementwithEquation( 6 ),thislinepassesthroughthepoint(Mc,Mp)correspondingtothetruevaluesofthemassparameters.NoticethatthepropertyofEquation( 6 )continuestoholdforothervaluesofPUTM.Figure 6-2 (a)showsthreemorecases:PUTM=500GeV(dotdashedblackline),PUTM=1TeV(dashedredline)andPUTM=2TeV(dottedblueline).Allthosecurvesstillpassthroughthepoint(Mc,Mp)withthecorrectvaluesofthemasses,illustratingtherobustnessofthepropertyofEquation( 6 )withrespecttovariationsinPUTM. 6.2.3.2PropertyII:KinkInMT2(max)AtTheTrueMcThesecondimportantpropertyoftheMT2variablewasidentiedratherrecently[ 23 34 37 ].Interestingly,theMT2endpointMT2(max),whenconsideredasafunctionoftheunknowninputtestmass~Mc,oftendevelopsakinkasEquation( 6 )atpreciselythecorrectvalue~Mc=Mcofthechildmass.Theappearanceofthekinkisarathergeneralphenomenonandoccursundervariouscircumstances.Itwasoriginallynoticedineventtopologieswithcompositevisibleparticles,whoseinvariantmassm()isavariableparameter[ 34 37 ].Lateritwasrealisedthatakinkalsooccursinthepresenceofnon-zeroupstreammomentumPUTM[ 23 35 36 ],asintheexampleofFigure 6-2 (a),wherePUTMarisesduetoinitialstateradiation.AscanbeseeninFigure 6-2 (a),the 7ThecorrespondingeventtopologyisshowninFigure 6-3 (a)belowwithM(a)c=M(b)c=Mc. 172

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kinkisabsentforPUTM=0,butassoonasthereissomenon-vanishingPUTM,thekinkbecomesreadilyapparent.Asexpected,thekinklocation(markedbytheverticaldottedline)isatthetruechildmass(Mc=100GeV),wherethecorrespondingvalueofMT2(max)(markedbythehorizontaldottedline)isatthetrueparentmass(Mp=300GeV).Figure 6-2 (a)alsodemonstratesthatwiththeincreaseinPUTM,thekinkbecomesmorepronounced,thusthemostfavorablesituationsfortheobservationofthekinkarecaseswithlargePUTM,e.g.whentheupstreammomentumisduetothedecaysofheavier(grandparent)particles[ 23 ].InSection 6.3.3 weshallseehowthekinkfeature( 6 )ofthesymmetricMT2endpoint~Mp(~Mc)denedbyEquation( 6 )isgeneralizedtoaridgefeatureontheasymmetricMT2endpoint~Mp(~M(a)c,~M(b)c)denedinEquation( 6 ). 6.2.3.3PropertyIII:PUTMInvarianceOfMT2(max)AtTheTrueMcThispropertyistheonewhichhasbeenleastemphasizedintheliterature.NoticethattheMT2endpointfunctionofEquation( 6 )ingeneraldependsonthevalueofPUTM.However,therstpropertyofEquation( 6 )impliesthatthePUTMdependencedisappearsatthecorrectvalueMcofthechildmass: @MT2(max)(~Mc,PUTM) @PUTM~Mc=Mc=0.(6)Inordertoquantifythisfeature,letusdenethefunction MT2(max)(~Mc,PUTM)MT2(max)(~Mc,PUTM))]TJ /F3 11.955 Tf 11.96 0 Td[(MT2(max)(~Mc,0),(6)whichmeasurestheshiftoftheMT2endpointduetovariationsinPUTM.ThefunctionMT2(max)(~Mc,PUTM)canbemeasuredexperimentally:thersttermontheright-handsideofEquation( 6 )issimplytheMT2endpointobservedinasubsampleofeventswithagiven(preferablythemostcommon)valueofPUTM,whilethesecondtermontheright-handsideofEquation( 6 )containstheendpointM(max)T2?ofthe1-dimensional 173

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MT2?variableintroducedin[ 12 ]: MT2(max)(~Mc,0)=M(max)T2?(~Mc).(6)Giventhedenition( 6 ),thethirdpropertyofEquation( 6 )canberewrittenas MT2(max)(~Mc,PUTM)0,(6)wheretheequalityholdsonlyfor~Mc=Mc: MT2(max)(Mc,PUTM)=0,8PUTM.(6)Equations( 6 )and( 6 )provideanalternativewaytodeterminethetruechildmassMc:simplyndthevalueof~McwhichminimizesthefunctionMT2(max)(~Mc,PUTM).ThisprocedureisillustratedinFigure 6-2 (b),wherewerevisitthesleptonpairproductionexampleofFigure 6-2 (a)andplotthefunctionMT2(max)(~Mc,PUTM)denedinEquation( 6 )versusthetestmass~Mc,forthesamesetof(xed)valuesofPUTM.Clearly,thezeroofthefunctionasEquation( 6 )occursatthetruechildmass~Mc=Mc=100GeV,inagreementwithEquation( 6 ).InourstudiesoftheasymmetricMT2caseinthenextsections,weshallndthatthethirdpropertyinEquation( 6 )isextremelyimportant,sinceitwillalwaysallowusthecompletedeterminationofthemassspectrum,includingbothchildrenmassesM(a)candM(b)c. 6.3TheGeneralizedAsymmetricMT2AfterthisshortreviewofthebasicpropertiesoftheconventionalsymmetricMT2variableinEquation( 6 ),wenowturnourattentiontothelesstrivialcaseof~M(a)c6=~M(b)c.FollowingthelogicofSection 6.2 ,inSection 6.3.1 werstintroducetheasymmetricMT2variableandtheninSections 6.3.2 and 6.3.3 wediscussitscomputationandmathematicalproperties,correspondingly. 174

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6.3.1DenitionThegeneralizationoftheusualdenition( 6 )totheasymmetriccaseof~M(a)c6=~M(b)cisstraightforward[ 40 ].WecontinuetofollowtheconventionsandnotationofFigure 6-1 ,butnowwesimplyavoidtheassumptionthatthechildrenmassesareequal,andweleteachonebeanindependentinputparameter~M()c.Withoutlossofgenerality,inwhatfollowsweassumeM(b)cM(a)c.ThetransversemassofeachparentinEquation( 6 )isnowafunctionofthecorrespondingchildmass~M()c: M()T)]TJ /F8 11.955 Tf 4.92 -9.18 Td[(~p()T;~q()T;m();~M()c=r m2()+~M()c2+2e()~e())]TJ /F8 11.955 Tf 11.4 0.5 Td[(~p()T~q()T,(6)wherethetransverseenergye()ofthecompositeSMparticleonthe-thsideoftheeventwasalreadydenedinEquation( 6 ),whilethetransverseenergy~e()ofthechildisnowgeneralizedfromEquation( 6 )to ~e()=r ~M()c2+~q()T~q()T.(6)Theevent-by-eventasymmetricMT2variableisdenedinanalogytoEquation( 6 )andisgivenby[ 40 ] MT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)c,PUTM=min~q(a)T+~q(b)T=~=PT"maxnM(a)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T;~q(a)T;m(a);~M(a)c,M(b)T)]TJ /F7 7.97 Tf 5.07 -9.36 Td[(~p(b)T;~q(b)T;m(b);~M(b)co#, (6) whichisnowafunctionoftwoinputtestchildrenmasses~M(a)cand~M(b)c.Inthespecialcaseof~M(a)c=~M(b)c~Mc,theasymmetricMT2variabledenedinEquation( 6 )reducestotheconventionalsymmetricMT2variable( 6 ). 6.3.2ComputationInthissubsectionwegeneralizethediscussioninSection 6.2.2 andpresentananalyticalformulaforcomputingtheevent-by-eventasymmetricMT2variable( 6 ).Justliketheformula( 6 )forthesymmetriccase,ourformulawillholdonlyinthespecialcaseofPUTM=0.Asbefore,theasymmetricMT2variablehastwotypesof 175

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solutionsbalancedandunbalanced.Thebalancedsolutionoccurswhenthefollowingtwoconditionsaresimultaneouslysatised(comparetotheanalogousconditions(Equations 6 )and( 6 )forthesymmetriccase) M(b)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(b)T;~q(b)T=)]TJ /F7 7.97 Tf 6.24 0.33 Td[(~q(a)T(0)+~=PT;m(b);~M(b)cM(a)T)]TJ /F7 7.97 Tf 5.08 -9.35 Td[(~p(a)T;~q(a)T=~q(a)T(0);m(a),~M(a)c=m(a)+~M(a)c, (6) M(a)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T;~q(a)T=)]TJ /F7 7.97 Tf 6.23 0.33 Td[(~q(b)T(0)+~=PT;m(a);~M(a)cM(b)T)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(b)T;~q(b)T=~q(b)T(0);m(b);~M(b)c=m(b)+~M(b)c, (6) where,inanalogytoEquation( 6 ), ~q()T(0)=~M()c m()~p()T,(=a,b),(6)isthetestchildmomentumattheglobalunconstrainedminimumofM()T.ThebalancedsolutionforMT2isnowgivenby hMBT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)ci2=~M2++AT+ m2(b))]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a) 2AT)]TJ /F3 11.955 Tf 11.96 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.95 0 Td[(m2(b)!~M2)]TJ /F2 11.955 Tf -339.31 -43.11 Td[(vuut 1+4~M2+ 2AT)]TJ /F3 11.955 Tf 11.96 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.96 0 Td[(m2(b)+ 2~M2)]TJ ET q 0.478 w 222.35 -338.65 m 316.59 -338.65 l S Q BT /F6 11.955 Tf 222.35 -349.84 Td[(2AT)]TJ /F3 11.955 Tf 11.96 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.96 0 Td[(m2(b)!2q A2T)]TJ /F3 11.955 Tf 11.96 0 Td[(m2(a)m2(b), (6) whereATwasdenedinEquation( 6 ).Forconvenience,inEquation( 6 )wehaveintroducedtwoalternativemassparameters ~M2+1 2n)]TJ /F6 11.955 Tf 8.09 -7.02 Td[(~M(b)c2+)]TJ /F6 11.955 Tf 8.09 -7.02 Td[(~M(a)c2o, (6) ~M2)]TJ /F2 11.955 Tf 11.07 2.96 Td[(1 2n)]TJ /F6 11.955 Tf 8.09 -7.03 Td[(~M(b)c2)]TJ /F12 11.955 Tf 11.95 9.68 Td[()]TJ /F6 11.955 Tf 8.09 -7.03 Td[(~M(a)c2o, (6) inplaceoftheoriginaltrialmasses~M(a)cand~M(b)c.Thenewparameters~M+and~M)]TJ /F1 11.955 Tf -435.11 -22.11 Td[(aresimplyadifferentparametrizationofthetwodegreesoffreedomcorrespondingtotheunknownchildmasses~M(a)cand~M(b)centeringthedenitionoftheasymmetricMT2.Theparameters~M+and~M)]TJ /F1 11.955 Tf 10.4 1.79 Td[(allowustowriteformula( 6 )inamorecompactform.Moreimportantly,theyalsoallowtomakeeasycontactwiththeknownresultsfrom 176

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Section 6.2 bytakingthesymmetriclimit~M(a)c=~M(b)c~Mcas ~M+!~Mc,~M)]TJ /F2 11.955 Tf 10.4 1.79 Td[(!0.(6)ItiseasytoseethatinthesymmetriclimitasEquation( 6 )ourbalancedsolutioninEquation( 6 )fortheasymmetricMT2reducestotheknownresultofEquation( 6 )forthesymmetricMT2.AninterestingfeatureoftheasymmetricbalancedsolutionistheappearanceofasignonthesecondlineofEquation( 6 ).Inprinciple,thissignambiguityispresentinthesymmetriccaseaswell,buttheretheminussignalwaysturnsouttobeunphysicalandthesignissuedoesnotarise[ 33 ].However,intheasymmetriccase,bothsignscanbephysicalsometimesandonemustmakethepropersignchoiceinEquation( 6 )asfollows.Forthegivensetoftestmasses(~M(a)c,~M(b)c),calculatethetransversecenter-of-massenergy p ^sT=e(a)+e(b)+2(e(b))]TJ /F3 11.955 Tf 11.96 0 Td[(e(a))~M2)]TJ ET q 0.478 w 135.2 -331.73 m 229.43 -331.73 l S Q BT /F6 11.955 Tf 135.2 -342.92 Td[(2AT)]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.95 0 Td[(m2(b)(e(b)+e(a))AT)]TJ /F6 11.955 Tf 11.96 0 Td[((e(b)m2(a)+e(a)m2(b)) q A2T)]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a)m2(b)vuut 1+4~M2+ 2AT)]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.96 0 Td[(m2(b)+ 2~M2)]TJ ET q 0.478 w 289.4 -383.54 m 383.64 -383.54 l S Q BT /F6 11.955 Tf 289.4 -394.73 Td[(2AT)]TJ /F3 11.955 Tf 11.95 0 Td[(m2(a))]TJ /F3 11.955 Tf 11.96 0 Td[(m2(b)!2, (6) correspondingtoeachsignchoiceinEquation( 6 ),andcomparetheresulttotheminimumallowedvalueofp ^sT p ^sT(min)=e(a)+e(b)+r Q2tot+~M(a)c+~M(b)c2. (6) TheminussigninEquation( 6 )takesprecedenceandapplieswheneveritisphysical,i.e.wheneverp ^s)]TJ /F4 5.978 Tf 0 -10.94 Td[(T>p ^sT(min).Intheremainingcaseswhenp ^s)]TJ /F4 5.978 Tf 0 -10.94 Td[(T


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anunbalancedsolution,inanalogytoEquations( 6 )and( 6 ): MUaT2)]TJ /F7 7.97 Tf 5.07 -9.36 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)c=m(a)+~M(a)c, (6) MUbT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)c=m(b)+~M(b)c. (6) TheunbalancedsolutionMUaT2ofEquation( 6 )applieswhenthecondition( 6 )isfalseandcondition( 6 )istrue,whiletheunbalancedsolutionMUbT2ofEquation( 6 )applieswhentheconditionofEquation( 6 )istrueandconditionofEquation( 6 )isfalse.Equations( 6 ),( 6 )and( 6 )representoneofourmainresults.TheygeneralizetheanalyticalresultsofReferences.[ 33 37 ]andallowthedirectcomputationoftheasymmetricMT2variablewithouttheneedforscanningandnumericalminimizations.Thisisanimportantbenet,sincetheexistingpubliccodesforMT2[ 104 105 ]onlyapplyinthesymmetriccaseM(a)c=M(b)c. 6.3.3PropertiesAllthreepropertiesofthesymmetricMT2discussedinSection 6.2.3 readilygeneralizetotheasymmetriccase. 6.3.3.1PropertyI:KnowledgeOfMpAsAFunctionOfM(a)cAndM(b)cIntheasymmetriccase,theendpointMT2(max)oftheMT2distributionstillgivesthemassoftheparent,onlythistimeitisafunctionoftwoinputtestmassesforthechildren: ~Mp(~M(a)c,~M(b)c,PUTM)=MT2(max)(~M(a)c,~M(b)c,PUTM).(6)Theimportantpropertyisthatthisrelationissatisedbythetruevaluesofthechildrenandparentmasses: Mp=MT2(max)(M(a)c,M(b)c,PUTM).(6)ThusthetrueparentmassMpwillbeknownoncewedeterminethetwochildrenmassesM(a)candM(b)c. 178

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6.3.3.2PropertyII:RidgeInMT2(max)ThroughTheTrueM(a)cAndM(b)cInthesymmetricMT2case,theendpointfunctioninEquation( 6 )isnotcontinuouslydifferentiableandhasakinkatthetruechildmass~Mc=Mc.IntheasymmetricMT2case,theendpointfunctioninEquation( 6 )issimilarlynon-differentiableatasetofpoints n~M(a)c(),~M(b)c()o(6)parametrizedbyasinglecontinuousparameter.ThegradientoftheendpointfunctioninEquation( 6 )suffersadiscontinuityaswecrossthecurvedenedbyEquation( 6 ).SinceEquation( 6 )representsahypersurfaceinthethree-dimensionalparameterspaceoff~M(a)c,~M(b)c,~Mpg,thegradientdiscontinuitywillappearasaridge(sometimesalsoreferredtoasacrease[ 40 ])onourthree-dimensionalplotsbelow.Theimportantpropertyoftheridgeisthatitpassesthroughthecorrectvaluesforthechildrenmasses,evenwhentheyaredifferent: M(a)c=~M(a)c(0), (6) M(b)c=~M(b)c(0), (6) forsome0.ThustheridgeinformationprovidesarelationamongthetwochildrenmassesandleavesuswithjustasingleunknowndegreeoffreedomtheparameterinEquation( 6 ).Interestingly,theshapeoftheridgeprovidesaquicktestwhetherthetwomissingparticlesareidenticalornot8.Iftheshapeoftheridgeinthe(~M(a)c,~M(b)c)planeissymmetricwithrespecttotheinterchange~M(a)c$~M(b)c,i.e.underamirrorreectionwithrespecttothe45line~M(a)c=~M(b)c,thenthetwomissingparticlesarethesame. 8Tobemoreprecise,theridgeshapetestswhetherthetwomissingparticleshavethesamemassornot. 179

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Conversely,whentheshapeoftheridgeisnotsymmetricunder~M(a)c$~M(b)c,themissingparticlesareingeneralexpectedtohavedifferentmasses. 6.3.3.3PropertyIII:PUTMInvarianceOfMT2(max)AtTheTrueM(a)cAndM(b)cThethirdMT2property,whichwasdiscussedinSection 6.2.3.3 ,isreadilygeneralizedtotheasymmetriccaseaswell.NotethatEquation( 6 )impliesthatthePUTMdependenceoftheasymmetricMT2endpointinEquation( 6 )disappearsatthetruevaluesofthechildrenmasses: @MT2(max)(~M(a)c,~M(b)c,PUTM) @PUTM~M(a)c=M(a)c,~M(b)c=M(b)c=0.(6)ThisequationistheasymmetricanalogueofEquation( 6 ).ProceedingasinSection 6.2.3.3 ,letusdenethefunction MT2(max)(~M(a)c,~M(b)c,PUTM)MT2(max)(~M(a)c,~M(b)c,PUTM))]TJ /F3 11.955 Tf 11.95 0 Td[(MT2(max)(~M(a)c,~M(b)c,0),(6)whichquantiestheshiftoftheasymmetricMT2endpointasEquation( 6 )inthepresenceofPUTM.Bydenition, MT2(max)(~M(a)c,~M(b)c,PUTM)0,(6)withequalitybeingachievedonlyforthecorrectvaluesofthechildrenmasses: MT2(max)(M(a)c,M(b)c,PUTM)=0,8PUTM.(6)TheEquation( 6 )revealsthepowerofthePUTMinvariancemethod.UnlikethekinkmethoddiscussedinSection 6.3.3.2 ,whichwasonlyabletondarelationbetweenthetwochildrenmassesM(a)candM(b)c,thePUTMinvarianceimpliedbyEquation( 6 )allowsustodetermineeachindividualchildrenmass,withoutanytheoreticalassumptions,andeveninthecasewhenthetwochildrenmasseshappentobedifferent(M(a)c6=M(b)c). 180

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Figure6-3. Thethreedifferentevent-topologiesunderconsiderationinthischapter.Ineachcase,twoparentswithmassMpareproducedonshellanddecayintotwodaughtersof(generallydifferent)massesM(a)candM(b)c.Case(a),whichisthesubjectofSection 6.4 ,hasasinglemasslessvisibleSMparticleineachlegandsomearbitraryupstreamtransversemomentum~PUTM.Intheremainingtwocases(b)and(c),whicharediscussedinSection 6.5 ,therearetwomasslessvisibleparticlesineachleg,whichformacompositevisibleparticlewithvaryinginvariantmassm().TheintermediateparticleofmassM()iis(b)heavyandoff-shell(M()i>Mp),or(c)on-shell(Mp>M()i>M()c).Forsimplicity,wedonotconsideranyupstreammomentumincases(b)and(c). 6.3.4ExamplesInthenexttwosectionsweshallillustratethethreepropertiesdiscussedsofarinSection 6.3.3 withsomeconcreteexamples.InsteadofthemostgeneraleventtopologydepictedFigure 6-1 ,herewelimitourselvestothethreesimpleexamplesshowninFigure 6-3 .Thesimplestpossiblecaseiswhenn()=1,i.e.wheneachcascadedecaycontainsasingleSMparticle,asinFigure 6-3 (a).Inthisexample,m()isconstant.Forsimplicity,weshalltakem()0,whichisthecaseforaleptonoralightavorjet.IftheSMparticleisaZ-bosonoratopquark,itsmasscannotbeneglected,andonemustkeepthepropervalueofm().This,however,isonlyatechnicaldetail,whichdoesnotaffectourmainconclusionsbelow.Inspiteofitssimplicity,thetopologyofFigure 6-3 (a)isactuallythemostchallengingcase,duetothelimitednumberofavailablemeasurements[ 23 ].Inordertobeabletodetermineallindividualmasses 181

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inthatcase,onemustconsidereventswithupstreammomentum~PUTM,asillustratedinFigure 6-3 (a).Thisisnotaparticularlyrestrictiveassumption,sincethereisalwaysacertainamountofPUTMintheevent(attheveryleast,frominitialstateradiation).InSection 6.4 thetopologyofFigure 6-3 (a)willbeextensivelystudied-rstfortheasymmetriccaseofM(a)c6=M(b)cinSection 6.4.1 ,andthenforthesymmetriccaseofM(a)c=M(b)cinSection 6.4.2 .AnothersimplesituationariseswhentherearetwomasslessvisibleSMparticlesineachleg,asillustratedinFigures 6-3 (b)and 6-3 (c).Ineithercase,theinvariantmassm()isnotconstantanymore,butvarieswithinacertainrangemmin()m()mmax(),wheremmin()=0,whilethevalueofmmax()dependsonthemassM()iofthecorrespondingintermediateparticle.InFigure 6-3 (b)weassumeM()i>Mp,sothattheintermediateparticleisoff-shelland mmax()=Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M()c.(6)Theoff-shellcaseofFigure 6-3 (b)willbediscussedinSection 6.5.1 .Incontrast,inFigure 6-3 (c)wetakeMp>M()i>M()c,inwhichcasetheintermediateparticleison-shellandtherangeform()isnowlimitedfromaboveby mmax()=Mpvuuut 241)]TJ /F12 11.955 Tf 11.96 20.45 Td[( M()i Mp!235241)]TJ /F12 11.955 Tf 11.96 20.45 Td[( M()c M()i!235.(6)Weshalldiscusstheon-shellcaseofFigure 6-3 (c)inSection 6.5.2 .IntheeventtopologiesofFigures 6-3 (b)and 6-3 (c),themassm()isvaryingandtheridgeofEquation( 6 )willappeareveniftherewerenoupstreamtransversemomentumintheevent.Therefore,inourdiscussionofFigures 6-3 (b)and 6-3 (c)inSection 6.5 belowweshallassumePUTM=0forsimplicity.Thepresenceofnon-zeroPUTMwillonlyadditionallyenhancetheridgefeature. 182

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6.3.5CombinatorialIssuesBeforegoingontotheactualexamplesinthenexttwosections,weneedtodiscussoneminorcomplication,whichisuniquetotheasymmetricMT2variableandwasnotpresentinthecaseofthesymmetricMT2variable.Thequestionis,howdoesoneassociatethevisibledecayproductsobservedinthedetectorwithaparticulardecaychain=aor=b.Thisistheusualcombinatoricsproblem,whichnowhastwodifferentaspects: Therstissueisalsopresentinthesymmetriccase,whereonehastodecidehowtopartitiontheSMparticlesobservedinthedetectorintotwodisjointsets,oneforeachcascade.Inthetraditionalapproach,wherethechildrenparticlesareassumedtobeidentical,thetwosetsareindistinguishableanditdoesnotmatterwhichoneisrstandwhichoneissecond.Thisparticularaspectofthecombinatorialproblemwillalsobepresentintheasymmetriccase. Intheasymmetriccase,however,thereisanadditionalaspecttothecombinatorialproblem:nowthetwocascadesaredistinguishable(bythemassesofthechildparticles),soevenifwecorrectlydividethevisibleobjectsintothepropersubsets,westilldonotknowwhichsubsetgoestogetherwithM(a)candthusgetsalabel=a,andwhichgoestogetherwithM(b)candgetslabelledby=b.Thisleadstoanadditionalcombinatorialfactorof2whichisabsentinthesymmetriccasewithidenticalchildren.Theseverityofthesetwocombinatorialproblemsdependsontheeventtopology,aswellasthetypeofsignatureobjects.Forexample,therearecaseswheretherstcombinatorialproblemiseasilyresolved,orevenabsentaltogether.ConsidertheeventtopologyofFigure 6-3 (a)withaleptonastheSMparticleoneachside.Inthiscase,thepartitionisunique,andtheupstreamobjectsarejets,whichcanbeeasilyidentied[ 39 ].NowconsidertheeventtopologiesofFigures 6-3 (b)and 6-3 (c),withtwooppositesign,sameavorleptonsoneachside.Sucheventsresultfrominclusivepairproductionofheavierneutralinosinsupersymmetry.Byselectingeventswithdifferentleptonavors:e+e)]TJ /F8 11.955 Tf 7.08 -4.34 Td[(+)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(,wecanovercometherstcombinatorialproblemaboveanduniquelyassociatethee+e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(pairwithonecascadeandthe+)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(pairwiththeother.However,thesecondcombinatorialproblemremains,aswestillhavetodecidewhichof 183

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thetwoleptonpairstoassociatewith=aandwhichtoassociatewith=b.Recallthatthelabels=aand=barealreadyattachedtothechildparticles,whicharedistinguishableintheasymmetriccase.Weusetheconventionthat=aisattachedtothelighterchildparticle: ~M(a)c~M(b)c,(6)whichalsoensuresthatthe~M)]TJ /F1 11.955 Tf 10.41 1.79 Td[(parameterdenedinEquation( 6 )isreal.Wecanputthisdiscussioninmoreformaltermsasfollows.Thecorrectassociationofthevisibleparticleswiththecorrespondingchildrenwillyield MT2)]TJ /F7 7.97 Tf 5.07 -9.36 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)c,(6)whiletheother,wrongassociationwillgivesimply MT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(b)c,~M(a)c.(6)BothofthesetwoMT2valuescanbecomputedfromthedata,butaprioriwedonotknowwhichonecorrespondstothecorrectassociation.Thesolutiontothisproblemishoweveralreadyknown[ 23 33 ]:onecanconservativelyusethesmallerofthetwo M(<)T2minMT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)c,MT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(b)c,~M(a)c(6)inordertopreservethelocationoftheupperMT2endpoint.ThisisillustratedinFigure 6-4 ,whereweshowresultsfortheeventtopologyofFigure 6-3 (b)withamassspectrumasfollows:M(a)c=100GeV,M(b)c=200GeVandMp=600GeV.Thetestchildrenmassesaretakentobethetruemasses:~M(a)c=M(a)cand~M(b)c=M(b)c.Thedottedblackdistributionistheunit-normalizedtrueMT2distribution,whereoneignoresthecombinatorialproblemandusestheMonteCarloinformationtomakethecorrectassociation.Theredhistogramshowstheunit-normalizeddistributionoftheM(<)T2variabledenedinEquation( 6 ). 184

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Figure6-4. Unit-normalizedMT2distributionsfortheeventtopologyofFigure 6-3 (b).ThemassspectrumischosenasM(a)c=100GeV,M(b)c=200GeVandMp=600GeV.Thetestchildrenmassesaretakentobethetruemasses:~M(a)c=M(a)cand~M(b)c=M(b)c.ThedottedblackdistributionisthetrueMT2distribution,ignoringthecombinatorialproblem.TheredhistogramshowsthedistributionoftheM(<)T2variabledenedinEquation( 6 )whilethebluehistogramshowsthedistributionoftheM(>)T2variabledenedinEquation( 6 ). WeseethatthedenitioninEquation( 6 )preservesthecorrespondingendpoint: M(<)T2(max)=MT2(max).(6)Ofcourse,wecanalsoconsiderthealternativecombination M(>)T2maxMT2)]TJ /F7 7.97 Tf 5.08 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(a)c,~M(b)c,MT2)]TJ /F7 7.97 Tf 5.07 -9.35 Td[(~p(a)T,~p(b)T;m(a),m(b);~M(b)c,~M(a)c,(6)whoseunit-normalizeddistributionisshowninFigure 6-4 withthebluehistogram.OnecanseethatsomeofthewrongcombinationentriesintheM(>)T2histogramviolatetheoriginalendpointMT2(max),yetthereisstillawelldenedM(>)T2endpoint M(>)T2(max)MT2(max).(6) 185

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Table6-1. MassspectraforthetwoexamplesstudiedinSections 6.4.1 and 6.4.2 .AllmassesaregiveninGeV. SpectrumCaseM(a)cM(b)cMp IDifferentchildren250500600IIIdenticalchildren100100300 Strictlyspeaking,inouranalysisinthenextsections,weonlyneedtostudytheM(<)T2endpointinEquation( 6 ),whichcontainstherelevantinformationaboutthephysicalMT2endpoint.Atthesametime,withourconventionasEquation( 6 )forthechildrenmasses,weonlyneedtoconcentrateontheupperhalf~M(b)c~M(a)cofthe(~M(a)c,~M(b)c)plane.However,forcompletenessweshallalsopresentresultsfortheM(>)T2endpointinEquation( 6 ),andweshallusethelower(~M(b)c<~M(a)c)halfofthe(~M(a)c,~M(b)c)planetoshowthose.ThustheMT2endpointshowninourplotsbelowshouldbeinterpretedasfollows MT2(max)=8><>:M(<)T2(max),if~M(a)c~M(b)c,M(>)T2(max),if~M(a)c>~M(b)c.(6) 6.4TheSimplestEventTopology:OneStandardModelParticleOnEachSideInthissection,weconsiderthesimplesttopologywithasinglevisibleparticleoneachsideoftheevent.WealreadyintroducedthisexampleinSection 6.3.4 ,alongwithitseventtopologyinFigure 6-3 (a).InSection 6.4.1 belowwerstdiscussanasymmetriccasewithdifferentchildren.LaterinSection 6.4.2 weconsiderasymmetricsituationwithidenticalchildrenmasses.ThemassspectraforthesetwostudypointsarelistedinTable 6-1 6.4.1AsymmetricCaseBeforewepresentournumericalresults,itwillbeusefultoderiveananalyticalexpressionfortheasymmetricMT2endpointinEquation( 6 )intermsofthecorrespondingphysicalspectrumofTable 6-1 andthetwotestchildrenmasses~M(a)cand~M(b)c.Ourresultwillgeneralizethecorrespondingformuladerivedin[ 37 ]forthe 186

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symmetriccaseof~M(a)c=~M(b)c~Mcandnoupstreammomentum(PUTM=0).FortheeventtopologyofFigure 6-3 (a)theMT2endpointisalwaysobtainedfromthebalancedsolutionandisgivenby[ 37 ] MT2(max)(~Mc,PUTM=0)=ppc+q 2ppc+~M2c.(6)Herewemadeuseoftheconvenientshorthandnotationintroducedin[ 23 ]fortherelevantcombinationofphysicalmasses npcMn 2(1)]TJ /F12 11.955 Tf 11.95 16.86 Td[(Mc Mp2).(6)TheparameterdenedinEquation( 6 )issimplythetransversemomentumofthe(massless)visibleparticleinthoseeventswhichgivethemaximumvalueofMT2[ 39 ].SquaringEquation( 6 ),wecanequivalentlyrewriteitas M2T2(max)(~Mc,PUTM=0)=22ppc+~M2c+q 42ppc(2ppc+~M2c).(6)NowletusderivetheanalogousexpressionsfortheasymmetriccaseM(a)c6=M(b)c.Justlikethesymmetriccase,theasymmetricendpointMT2(max)alsocomesfromabalancedsolutionandisgivenby M2T2(max)(~M(a)c,~M(b)c,PUTM=0)=22ppc+~M2++q 42ppc(2ppc+~M2+)+~M4)]TJ /F6 11.955 Tf 9.07 2.61 Td[(, (6) wheretheparameters~M2+and~M2)]TJ /F1 11.955 Tf 10.41 2.96 Td[(werealreadydenedinEquation( 6 )and( 6 ),whileppcisnowthegeometricaverageofthecorrespondingindividualppcparameters 2ppcppcappcb(M2p)]TJ /F12 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M(a)c2)(M2p)]TJ /F12 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M(b)c2) 4M2p.(6)Itiseasytocheckthatinthesymmetriclimit ~M(b)c!~M(a)c=)ppc!ppc,~M+!~Mc,~M)]TJ /F2 11.955 Tf 10.41 1.8 Td[(!0,(6)Equation( 6 )reducestoitssymmetriccounterpartofEquation( 6 ),asitshould. 187

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Figure6-5. MT2(max)asafunctionofthetwotestchildrenmasses,~M(a)cand~M(b)c,fortheeventtopologyofFigure 6-3 (a)withnoupstreammomentum(PUTM=0),andtheasymmetricmassspectrumIfromTable 6-1 :(M(a)c,M(b)c,Mp)=(250,500,600)GeV.Weshow(a)athreedimensionalviewand(b)contourplotprojectiononthe(~M(a)c,~M(b)c)plane(redcontourlines).Thegreendotmarksthetruevaluesofthechildrenmasses.Panel(b)alsoshowsagradientplot,wherelonger(shorter)arrowsimplysteeper(gentler)slope.Akinkstructureisabsentinthiscase.ThesymmetricendpointMT2(max)(~Mc)ofEquation( 6 )canbeobtainedbygoingalongthediagonalorangeline~M(b)c=~M(a)c. WearenowreadytopresentournumericalresultsfortheeventtopologyofFigure 6-3 (a).WersttaketheasymmetricmassspectrumIfromTable 6-1 andconsiderthecasewithnoupstreammomentum,whenformula( 6 )applies.Figure 6-5 showsthecorrespondingMT2endpointasafunctionofthetwotestchildrenmasses~M(a)cand~M(b)c.Inpanel(a)wepresentathreedimensionalview,whileinpanel(b)weshowacontourplotprojectiononthe(~M(a)c,~M(b)c)plane(redcontourlines).Oneitherpanel,thegreendotmarksthetruevaluesofthechildrenmasses,M(a)candM(b)c.Panel(b)alsoshowsagradientplot,wherelonger(shorter)arrowsimplysteeper(gentler)slope.ThesymmetricendpointMT2(max)(~Mc,PUTM=0)ofEquation( 6 )canbeobtainedbygoingalongthediagonalorangeline~M(b)c=~M(a)cinFigure 6-5 (b).WeremindthereaderthattheendpointMT2(max)plottedinFigure 6-5 shouldbeinterpretedasinEquation( 6 ). 188

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Figure 6-5 illustratestherstbasicpropertyoftheasymmetricMT2variable,whichwasdiscussedinSection 6.3.3.1 .TheMT2endpointallowsustondonerelationbetweenthetwochildrenmasses~M(a)cand~M(b)candtheparentmass~Mp=MT2(max),andinordertodoso,wedonothavetoassumeequalityofthechildrenmasses,asisalwaysdoneintheliterature.Thecrucialadvantageofourapproach,inwhichweallowthetwochildrenmassestobearbitrary,isitsgeneralityandmodel-independence.ItallowsustoextractthebasicinformationcontainedintheMT2endpoint,withoutmuddlingitupwithadditionaltheoretical(andunproven)assumptions.Unfortunately,togoanyfurtheranddetermineeachindividualmass,wemustmakeuseoftheadditionalpropertiesdiscussedinSections 6.3.3.2 and 6.3.3.3 .InthecaseofthesimplesteventtopologyofFigure 6-3 (a)consideredhere,theybothrequirethepresenceofsomeupstreammomentum[ 23 36 ].Asaproofofconcept,wenowreconsiderthesametypeofevents,butwithaxedupstreammomentumofPUTM=1TeV.(Theupstreammomentummaybeduetoinitialstateradiation,ordecaysofheavierparticlesupstream.)ThecorrespondingresultsareshowninFigure 6-6 .Figure 6-6 demonstratesthesecondbasicpropertyoftheasymmetricMT2variablediscussedinSection 6.3.3.2 .UnliketheresultshowninFigure 6-5 (a),whichwasperfectlysmooth,thistimetheMT2(max)functioninFigure 6-6 (a)showsaridge,correspondingtotheslopediscontinuitymarkedwiththeblacksolidlineinFigure 6-6 (b).Themostimportantfeatureoftheridgeisthefactthatitpassesthroughthegreendotmarkingthetruevaluesofthechildrenmasses.NoticethatapplyingthetraditionalsymmetricMT2approachinthiscasewillgiveacompletelywrongresult.Ifweweretoassumeequalchildrenmassesfromtheverybeginning,wewillbeconstrainedtothediagonalorangelineinFigure 6-6 (b).TheMT2endpointwillthenstillexhibitakink,butthekinkwillbeinthewronglocation.IntheexampleshowninFigure 6-6 (b),wewillunderestimatetheparentmass,whileforthechildmasswewillndavaluewhichissomewhereinbetweenthetwotruemassesM(a)candM(b)c. 189

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Figure6-6. MT2(max)fortheeventtopologyofFigure 6-3 (a)withxedupstreammomentumofPUTM=1TeV.Theridgestructure(shownastheblacksolidline)isrevealedbythesuddenincreaseintheslope(gradient)inpanel(b).Noticethattheridgegoesthroughthetruevaluesofthechildrenmassesmarkedbythegreendot. Usingtheridgeinformation,wenowknowanadditionalrelationamongthechildrenmasses,whichallowsustoexpressallthreemassesintermsofasingleunknownparameter,asillustratedinFigure 6-7 (a).Letuschoosetoparametrizetheridgebythepolarangleinthe(~M(a)c,~M(b)c)plane: =tan)]TJ /F5 7.97 Tf 6.59 0 Td[(1 ~M(b)c ~M(a)c!.(6)UsingtheridgeinformationfromFigure 6-6 ,wecanthenndallthreemassesasafunctionof.TheresultisshowninFigure 6-7 (a).Themass~M(a)cofthelighterchildisplottedinred,themass~M(b)coftheheavierchildisplottedinblue,whiletheparentmass~Mpisplottedinblack.Withourconvention( 6 )forthechildrenmasses,onlyvaluesof45arephysical,andthecorrespondingmassesareshownwithsolidlines.ThedottedlinesinFigure 6-7 (a)showtheextrapolationintotheunphysicalregion<45.Figure 6-7 (a)hassomeimportantandfarreachingimplications.Forexample,onemaynowstartaskingthequestion:Aretherereallyanymassiveinvisibleparticlesinthoseevents,oristhemissingenergysimplyduetoneutrinoproduction[ 94 ]? 190

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Figure6-7. (a)ParticlemassesobtainedalongtheMT2(max)ridgeseeninFigure 6-6 .TheridgeisparametrizedbytheangledenedinEquation( 6 ).Thetwochildrenmasses~M(a)c()(inred)and~M(b)c()(inblue)aswellastheparentmass~Mp(inblack)arethenplottedasafunctionof.Inourconvention( 6 )onlyvaluesof45arephysical,andthecorrespondingmassesareshownwithsolidlines.Dottedlinesshowtheextrapolationfor<45.(b)ContourplotofthequantityMT2(max)(~M(a)c,~M(b)c,PUTM=1TeV)denedinEquation( 6 ),inthe(~M(a)c,~M(b)c)plane.ThisplotisobtainedsimplybytakingthedifferencebetweenFigure 6-6 (a)andFigure 6-5 (a).ThesolidblackcurveindicatesthelocationoftheMT2(max)ridge.Onlythepointcorrespondingtothetruechildrenmasses(thegreendot)satisesthePUTMinvarianceconditionMT2(max)=0fromEquation( 6 ). TheridgeresultsshowninFigure 6-7 (a)begintoprovidetheanswertothatquitefundamentalquestion.AccordingtoFigure 6-7 (a),foranyvalueofthe(stillunknown)parameter,thetwochildrenparticlescannotbesimultaneouslymassless.Thismeansthatthemissingenergycannotbesimplyduetoneutrinos,i.e.thereisatleastonenew,massiveinvisibleparticleproducedinthemissingenergyevents.Atthispoint,wecannotbecertainthatthisisadarkmatterparticle,butestablishingtheproductionofaWIMPcandidateatacolliderisbyitselfatremendouslyimportantresult.Noticethatwhilewecannotbesureaboutthemassesofthechildren,theparentmassMpis 191

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determinedwithaverygoodprecisionfromFigure 6-7 (a):thefunction~Mp()isalmostatandratherinsensitivetotheparticularvalueof9.OncewehaveprovedthatsomekindofWIMPproductionisgoingon,thenextimmediatequestionis:howmanysuchWIMPparticlesarepresentinthedataoneortwo?Unfortunately,theridgeanalysisofFigure 6-7 (a)alonecannotprovidetheanswertothisquestion,sincethevalueofisstillundetermined.If=90,oneofthemissingparticlesismassless,whichisconsistentwithaSMneutrino.Therefore,ifwereindeed90,themostplausibleexplanationofthisscenariowouldbethatonlyoneofthemissingparticlesisagenuineWIMP,whiletheotherisaSMneutrino.Ontheotherhand,almostanyothervalueof<90wouldguaranteethattherearetwoWIMPcandidatesineachevent.Inthatcase,thenextimmediatequestionis:aretheythesameoraretheydifferent?Fortunately,ourasymmetricapproachwillallowansweringthisquestioninamodel-independentway.Ifisdeterminedtobe45,thetwoWIMPparticlesarethesame,i.e.weareproducingasinglespeciesofdarkmatter.Ontheotherhand,if45<<90,thenwecanbecertainthattherearenotone,buttwodifferentWIMPparticlesbeingproduced.Weseethatinordertocompletelyunderstandthephysicsbehindthemissingenergysignal,wemustdeterminethevalueof,i.e.wemustndtheexactlocationofthetruechildrenmassesalongtheridge.OneofourmainresultsinthischapteristhatthiscanbedonebyusingthethirdMT2propertydiscussedinSection 6.3.3.3 .TheideaisillustratedinFigure 6-7 (b),whereweshowacontourplotinthe(~M(a)c,~M(b)c)planeofthequantityMT2(max)(~M(a)c,~M(b)c,PUTM)denedinEquation( 6 ),foraxedPUTM=1TeV.Thisplotisobtainedsimplybytakingthedifferencebetween 9Interestingly,fortheexampleinFigure 6-7 (a),themaximumvalueof~Mp()happenstogivethetrueparentmassMp,butwehavecheckedthatthisisacoincidenceanddoesnotholdingeneralforotherexampleswhichwehavestudied. 192

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Figure 6-6 (a)andFigure 6-5 (a).(Amorepracticalmethodforobtainingthisinformationwasproposedin[ 12 ].)RecallthatthefunctionMT2(max)wasintroducedinordertoquantifythePUTMinvarianceoftheMT2endpoint,anditisexpectedthatMT2(max)vanishesatthecorrectvaluesofthechildrenmasses(seeEquation( 6 )).ThisexpectationisconrmedinFigure 6-7 (b),wherewendtheminimum(zero)oftheMT2(max)functionexactlyattherightspot(markedwiththegreendot)alongtheMT2(max)ridge.ThustheMT2(max)functioninFigure 6-7 (b)completelypinsdownthespectrum,andinthiscasewouldrevealthepresenceoftwodifferentWIMPparticles,withunequalmassesM(a)c6=M(b)c.OuranalysisthusshowsthatcolliderscannotonlyproduceaWIMPdarkmattercandidateandmeasureitsmass,asdiscussedintheexistingliterature,buttheycandoamuchmoreelaboratedarkmatterparticlespectroscopy,asadvertizedinthetitle.Inparticular,theycanprobethenumberandtypeofmissingparticles,includingparticlesfromsubdominantdarkmatterspecies,whichareotherwiseunlikelytobediscoveredexperimentallyintheusualdarkmattersearches. 6.4.2SymmetricCaseWhileinourapproachthetwochildrenmasses~M(a)cand~M(b)caretreatedasindependentinputs,this,ofcourse,doesnotmeanthattheapproachisonlyvalidincaseswhenthechildrenmassesaredifferenttobeginwith.Thetechniquesdiscussedintheprevioussubsectionremainapplicablealsointhemoreconventionalcasewhenthechildrenareidentical,i.e.whencollidersproduceasingledarkmattercomponent.Inordertoillustratehowourmethodworksinthatcase,weshallnowworkoutanexamplewithequalchildrenmasses.WestillconsiderthesimplesteventtopologyofFigure 6-3 (a),butwiththesymmetricmassspectrumIIfromTable 6-1 .WethenrepeattheanalysisdoneinFigures 6-5 6-6 ,and 6-7 andshowthecorrespondingresultsinFigures 6-8 6-9 and 6-10 193

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Figure6-8. MT2(max)fortheeventtopologyofFigure 6-3 (a)withnoupstreammomentum.ParticleshavethesymmetricmassspectrumIIfromTable 6-1 ,i.e.(M(a)c,M(b)c,Mp)=(100,100,300)GeV. Figure6-9. MT2(max)fortheeventtopologyofFigure 6-3 (a)withxedupstreammomentumPUTM=1TeV.ParticleshavethesymmetricmassspectrumIIfromTable 6-1 ,i.e.(M(a)c,M(b)c,Mp)=(100,100,300)GeV. TheconclusionsfromthisexerciseareverysimilartowhatwefoundearlierinSection 6.4.1 fortheasymmetriccase.TheMT2endpointstillprovidesonerelationamongthetwochildrenmasses~M(a)cand~M(b)candtheparentmass~Mp=MT2(max).ThisrelationisshowninFigure 6-8 (Figure 6-9 )forthecasewithout(with)upstreammomentumPUTM.AsseeninFigure 6-8 ,intheabsenceofanyupstreamPUTM,thefunction~Mp(~M(a)c,~M(b)c)issmoothandrevealsnothingaboutthechildrenmasses. 194

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Figure6-10. ThesameasinFigure 6-7 butforthesymmetricmassspectrumIIfromTable 6-1 ,i.e.(M(a)c,M(b)c,Mp)=(100,100,300)GeV.Noticethat,incontrasttoFigure 6-7 ,theminimumoftheMT2(max)functionisnowobtainedat~M(a)c=~M(b)c,implyingthatthetwomissingparticlesarethesame. However,thepresenceofupstreammomentumsignicantlychangesthepictureandthefunction~Mp(~M(a)c,~M(b)c)againdevelopsaridge,whichisclearlyvisible10inboththethree-dimensionalviewofFigure 6-9 (a),aswellasthegradientplotinFigure 6-9 (b).TheridgeinformationnowfurtherconstrainsthechildrenmassestotheblacksolidlineinFigure 6-9 (b),leavingonlyoneunknowndegreeoffreedom.Parametrizingitwiththepolarangleasin( 6 ),weobtainthespectrumasafunctionof,asshowninFigure 6-10 (a).Onceagainwendthefortuitousresultthatinspiteoftheremainingarbitrarinessinthevalueof,theparentmassMpisverywelldetermined,since~Mp()isaveryweaklyvaryingfunctionof.Furthermore,bothFigure 6-9 (a)andFigure 6-9 (b) 10Wecautionthereaderthatherewearepresentingonlyaproofofconcept.Intheactualanalysistheridgemayberatherdifculttosee,foravarietyofreasons-detectorresolution,nitestatistics,combinatorialandSMbackgrounds,etc.Nevertheless,weexpectthattheridgewillbejustaseasilyobservableasthetraditionalkinkinthesymmetricMT2endpoint.Ifthekinkcanbeseeninthedata,theridgecanbeseentoo,andthereisnoreasontomaketheassumptionofequalchildrenmasses.Conversely,ifthekinkistoodifculttosee,theridgewillremainhiddenaswell. 195

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exhibitahighdegreeofsymmetryunder~M(a)c$~M(b)c,whichisagoodhintthatthechildrenareinfactidentical.ThissuspicionisconrmedinFigure 6-10 (b),wherewendthatthePUTMdependencedisappearsatthesymmetricpoint~M(a)c=~M(b)c=100GeV,revealingthetruemassesofthetwochildren.InthetwoexamplesconsideredsofarinSections 6.4.1 and 6.4.2 ,weusedaxednitevalueoftheupstreamtransversemomentumPUTM=1TeV,whichisprobablyratherextremeinrealisticmodels,onemightexpecttypicalvaluesofPUTMontheorderofseveralhundredGeV.However,thingsbegintogetmuchmoreinterestingifoneweretoconsiderevenlargervaluesofPUTM.Ontheonehand,theridgefeaturebecomessharperandeasiertoobserve[ 23 ].Moreimportantly,theridgestructureitselfismodied,andasecondsetofridgelinesappears11atsufcientlylargePUTM.Allridgelinesintersectpreciselyatthepointmarkingthetruevaluesofthechildrenmasses,thusallowingthecompletedeterminationofthemassspectrumbytheridgemethodalone.ThisprocedurewasdemonstratedexplicitlyinReference.[ 40 ],whichinvestigatedtheextremecaseofPUTM=1forastudypointwithdifferentparentsandidenticalchildren.TheassumptionofPUTM=1justiedtheuseofadecouplingargument,inwhichthetwobranches=aand=baretreatedindependently,allowingthederivationofsimpleanalyticalexpressionsfortheMT2endpoint[ 40 ].InAppendix C wereproducetheanalogousanalyticalresultsatPUTM!1forthecaseofinteresthere(identicalparentsanddifferentchildren)andstudyindetailthePUTMdependenceoftheridgelines.Unfortunately,wendthatthevaluesofPUTMnecessarytorevealtheadditionalridgestructure,aretoolargetobeofanyinterestexperimentally.Onthepositiveside,thePUTMinvariancemethoddiscussedinSection 6.2.3.3 doesnotrequire 11AkeenobservermayhavealreadynoticedahintofthoseinFigures 6-7 (b)and 6-10 (b). 196

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suchextremelylargevaluesofPUTMandcaninprinciplebetestedinmorerealisticexperimentalconditions. 6.4.3MixedCaseForsimplicity,sofarinourdiscussionwehavebeenstudyingonlyonetypeofmissingenergyeventsatatime.Inreality,themissingenergysamplemaycontainseveraldifferenttypesofevents,andthecorrespondingMT2measurementswillrstneedtobedisentangledfromeachother.Forconcreteness,considertheinclusivepairproductionofsomeparentparticlep,whichcandecayeithertoachildparticleaofmassM(a)c,oradifferentchildparticlebofmassM(b)c.LetthecorrespondingbranchingfractionsbeBaandBb,i.e.BaB(p!a)andBbB(p!b).Furthermore,letbdecayinvisibly12toa.Suchasituationcanbeeasilyrealizedinsupersymmetry,forexample,withtheparentbeingasquark,aslepton,oragluino,theheavierchildbbeingaWino-likeneutralino~02andthelighterchildabeingaBino-likeneutralino~01.Theheavierneutralinohasalargeinvisibledecaymode~02!~01,ifitsmasshappenstofallbetweenthesneutrinomassandtheleft-handedsleptonmass:M~
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Figure6-11. Unit-normalized,zero-binsubtractedMT2distribution(blackhistogram)forthefullmixedeventsample,aswellastheindividualcomponentsaa(red),ab(blue)andbb(green).Wetookzerotestmassesforthechildren~M(a)c=~M(b)c=0andequalbranchingfractionfortheparentsBa=Bb=50%.ThemassspectrumistakenfromtheasymmetricstudypointIinTable 6-1 withM(a)c=250GeV,M(b)c=500GeVandMp=600GeV.Thethreearrowsindicatetheexpectedendpointsforeachindividualcomponentinthesample. Forthisplot,weusedtheasymmetricmassspectrumIfromTable 6-1 :M(a)c=250GeV,M(b)c=500GeVandMp=600GeV,andchosezerotestmassesforthechildren~M(a)c=~M(b)c=0.Fordeniteness,wexedequalbranchingfractionsBa=Bb=50%,sothattherelativenormalizationofthethreeindividualsamplesisNaa:Nbb:Nab=1:1:2.Figure 6-11 showsthattheobservableMT2distributionissimplyasuperpositionoftheMT2distributionsofthethreeindividualsamplesaa,abandbb,whichareshownwiththered,blueandgreenhistograms,correspondingly.EachindividualsampleexhibitsitsownMT2endpoint,markedwithaverticalarrow,whichcanalsobeseeninthecombinedMT2distribution.UsingEquation( 6 ),thethreeendpointsarefoundto 198

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be aa!M(aa)T2(max)(0,0,0)=Mp241)]TJ /F12 11.955 Tf 11.95 20.44 Td[( M(a)c Mp!235=496GeV, (6) ab!M(ab)T2(max)(0,0,0)=Mpvuuut 241)]TJ /F12 11.955 Tf 11.96 20.44 Td[( M(a)c Mp!235241)]TJ /F12 11.955 Tf 11.96 20.44 Td[( M(b)c Mp!235=301GeV, (6) bb!M(bb)T2(max)(0,0,0)=Mp241)]TJ /F12 11.955 Tf 11.95 20.44 Td[( M(b)c Mp!235=183GeV. (6) NowsupposethatallthreeendpointsasinEquations( 6 through 6 )areseeninthedata.Theirinterpretationisfarfromobvious,andinfact,therewillbedifferentcompetingexplanations.Ifoneinsistsonthesinglemissingparticlehypothesis,therecanbeonlyonetypeofchildparticle,andtheonlywaytogetthreedifferentendpointsinFigure 6-11 istohaveproductionofthreedifferentpairsofparentparticles,eachofwhichdecaysinexactlythesameway.Sincethethreeparentmassesareaprioriunrelated,onedoesnotexpectanyparticularcorrelationamongthethreeobservedendpointsinEquations( 6 through 6 ).Nowconsideranalternativeexplanationwhereweproduceasingletypeofparents,buthavetwodifferentchildrentypes.Thissituationalsogivesrisetothreedifferenteventtopologies,withthreedifferentMT2endpoints,aswejustdiscussed.However,nowthereisapredictedrelationamongthethreeMT2endpoints,whichfollowssimplyfromEquations( 6 through 6 ): M(ab)T2(max)(0,0,0)=q M(aa)T2(max)(0,0,0)M(bb)T2(max)(0,0,0).(6)Iftheparentsarethesameandthechildrenaredifferent,thisrelationmustbesatised.Iftheparentsaredifferentandthechildrenarethesame,apriorithereisnoreasonwhyEquation( 6 )shouldhold,andifitdoes,itmustbebypurecoincidence.ThepredictionofEquation( 6 )thereforeisadirecttestofthenumberofchildren 199

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particles.AnothertestcanbeperformedifwecouldestimatetheindividualeventcountsNaa,NabandNbb,althoughthisappearsratherdifcult,duetotheunknownshapeoftheMT2distributionsinFigure 6-11 .Intheasymmetricexamplediscussedhere,wehaveanotherprediction,namely Nab=2p NaaNbb,(6)whichisanothertestofthedifferentchildrenhypothesis.NoticethatEquation( 6 )holdsregardlessofthebranchingfractionsBaandBb,althoughifoneofthemdominates,thetwoendpointswhichrequiretheother(rare)decaymaybetoodifculttoobserve.Ofcourse,theultimatetestofthesinglemissingparticlehypothesisisthebehavioroftheintermediateMT2endpointinFigure 6-11 correspondingtotheasymmetriceventsoftypeab.ApplyingeitheroneofthetwomassdeterminationmethodsdiscussedearlierinFigures 6-7 and 6-10 ,weshouldndthatM(ab)T2(max)isaresultofasymmetricevents,indicatingthesimultaneouspresenceoftwodifferentinvisibleparticlesinthedata. 6.5AMoreComplexEventTopology:TwoVisibleParticlesOnEachSideInthissection,weconsidertwomoreexamples:theoff-shelleventtopologyofFigure 6-3 (b)isdiscussedinSection 6.5.1 ,whiletheon-shelleventtopologyofFigure 6-3 (c)isdiscussedinSection 6.5.2 .(Forsimplicity,wedonotconsideranyPUTMinthissection.)Nowtherearetwovisibleparticlesineachleg,whichformacompositevisibleparticleofvaryingmassm().Ingeneral,bystudyingtheinvariantmassdistributionofm(),oneshouldbeabletoobservetwodifferentinvariantmassendpoints,suggestingsometypeofanasymmetricscenario. 6.5.1Off-shellIntermediateParticleHereweconcentrateontheexampleofFigure 6-3 (b).Sincetheintermediateparticleisoffshell,themaximumkinematicallyallowedvalueform()isgivenbyEquation( 6 ). 200

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Figure6-12. Thefourregionsinthe(~M(a)c,~M(b)c)parameterplaneleadingtothefourdifferenttypesofsolutionsfortheMT2endpoint,fortheoff-shelleventtopologyofFigure 6-3 (b).Thegreendotmarksthetruelocationofthetwochildrenmasses.Withineachregion,weindicatetherelevantmomentumcongurationforthevisibleparticles(redarrows)andthechildrenparticles(bluearrows)ineachleg(aorb).Themomentaarequotedintheback-to-backboostedframe[ 37 ],inwhichthetwoparentsareatrest.Abluedotimpliesthatthecorrespondingdaughterisatrestandthereforethetwovisibleparticlesareemittedback-to-back.ThetwobalancedsolutionsaredenotedasBandB0,whilethetwounbalancedsolutionsareUaandUb.TheblacksolidlinesrepresentphasechangesbetweendifferentsolutiontypesanddelineatetheexpectedlocationsoftheridgesintheMT2(max)functionshowninFigure 6-13 RecallthatforthesimpletopologyofFigure 6-3 (a)discussedintheprevioussection,theMT2endpoint( 6 )alwayscorrespondedtoabalancedsolution.Moreprecisely,theMT2variablewasmaximizedforamomentumconguration~p()TinwhichMT2wasgivenbythebalancedsolution( 6 ).However,inthissectionweshallndthatforthemorecomplextopologiesofFigures 6-3 (b)and 6-3 (c),theMT2endpointmayresultfromoneoffourdifferentcasesaltogether:twodifferentbalancedsolutions,whichweshalllabelasBandB',ortheunbalancedsolutionsUaandUbdiscussedinSection 201

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6.3.2 .DependingonthetypeofsolutiongivingtheendpointMT2(max),the(~M(a)c,~M(b)c)parameterplanedividesintothethreeregions13showninFigure 6-12 .ThegreendotinFigure 6-12 denotesthetruechildrenmassesinthisparameterspace.Withineachregion,weshowtherelevantmomentumcongurationforthevisibleparticles(redarrows)andthechildrenparticles(bluearrows)ineachleg(aorb).Themomentaarequotedintheback-to-backboostedframe[ 37 ],inwhichthetwoparentsareatrest.Thelengthofanarrowisindicativeofthemagnitudeofthemomentum.Abluedotimpliesthatthecorrespondingdaughterisatrestandthereforethetwovisibleparticlesareemittedback-to-back.ThetwobalancedsolutionsaredenotedasBandB0,whilethetwounbalancedsolutionsareUaandUb.TheblacksolidlinesrepresentphasechangesbetweendifferentsolutiontypesanddelineatetheexpectedlocationsoftheridgesintheMT2(max)functionshowninFigure 6-13 below.PerhapsthemoststrikingfeatureofFigure 6-12 isthatthethree(infact,allfour)regionscometogetherpreciselyatthegreendotmarkingthetruevaluesofthetwochildrenmasses.TheboundariesoftheregionsshowninFigure 6-12 willmanifestthemselvesasthelocationsoftheridges(i.e.gradientdiscontinuities)intheMT2(max)function.Therefore,weexpectthatbystudyingtheridgestructureandndingitstriplepoint,onewillbeabletocompletelydeterminethemassspectrum.WeshallnowgiveanalyticalformulasfortheMT2endpointineachofthefourregionsofFigure 6-12 .WebeginwiththetwobalancedsolutionsBandB0,forwhichtheevent-by-eventbalancedsolutionforMT2isgivenbyEquation( 6 ).IntheparameterspaceregionofFigure 6-12 whichisadjacenttotheorigin,wendthebalancedcongurationB,inwhichallvisibleparticleshavethesamedirectioninthe 13ThefourthcaseoftheB0balancedsolutionhappenstocoincidewiththetwounbalancedsolutionsalongtheboundarybetweenUaandUb. 202

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back-to-backboostedframe.Asaresult,wehave m(a)=m(b)=0(6)and AT=(M2p)]TJ /F12 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M(a)c2)(M2p)]TJ /F12 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M(b)c2) 2M2p.(6)SubstitutingEquations( 6 )and( 6 )inthebalancedMT2solution( 6 ),whereweshouldtaketheplussign,weobtain hMBT2(max)(~M(a)c,~M(b)c)i2=22ppc+~M2++q 42ppc(2ppc+~M2+)+~M4)]TJ /F6 11.955 Tf 9.07 2.61 Td[(,(6)whichwerecognizeasthebalancedsolution( 6 )foundforthedecaytopologyofFigure 6-3 (a).MovingawayfromtheorigininFigure 6-12 ,wendasecondbalancedsolutionB'alongtheboundaryoftheunbalancedregionsUaandUb.Inthiscasethevisibleparticlesareback-to-back,andtheirinvariantmassismaximized: m()=Mp)]TJ /F3 11.955 Tf 11.95 0 Td[(M()c,(6)andcorrespondingly AT=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(a)c)]TJ /F3 11.955 Tf 12.95 -9.68 Td[(Mp)]TJ /F3 11.955 Tf 11.95 0 Td[(M(b)c.(6)SubstitutingEquations( 6 )and( 6 )inthebalancedMT2solution( 6 ),weobtaintheB'-typeMT2endpointas hMB0T2(max)(~M(a)c,~M(b)c)i2=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(a)c)]TJ /F3 11.955 Tf 12.95 -9.69 Td[(Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(b)c+~M2++2Mp)]TJ /F3 11.955 Tf 11.95 0 Td[(M(a)c)]TJ /F3 11.955 Tf 11.95 0 Td[(M(b)c M(b)c)]TJ /F3 11.955 Tf 11.96 0 Td[(M(a)c~M2)]TJ /F6 11.955 Tf 7.08 2.95 Td[(.(6)ThecorrespondingformulasfortheunbalancedcasesUaandUbareobtainedbytakingthemaximumvaluefortheinvariantmassofthevisibleparticlesinthe 203

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correspondingdecaychain: m(a)=mmax(a)=Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(a)cforregion(Ua), (6) m(b)=mmax(b)=Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(b)cforregion(Ub). (6) ThecorrespondingformulaforMT2(max)isthengivenby MUaT2(max)(~M(a)c)=Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(a)c+~M(a)c, (6) MUbT2(max)(~M(b)c)=Mp)]TJ /F3 11.955 Tf 11.96 0 Td[(M(b)c+~M(b)c. (6) OnecannowusetheanalyticalresultsofEquations( 6 ),( 6 ),( 6 )and( 6 )tounderstandtheridgestructureshowninFigure 6-12 .Forexample,theboundarybetweentheBandUaregionsisparametricallygivenbythecondition MBT2(max)(~M(a)c,~M(b)c)=MUaT2(max)(~M(a)c),(6)whiletheboundarybetweentheBandUbregionsisparametricallygivenby MBT2(max)(~M(a)c,~M(b)c)=MUbT2(max)(~M(b)c).(6)Ontheotherhand,theboundary MUaT2(max)(~M(a)c)=MUbT2(max)(~M(b)c)(6)betweenthetwounbalancedregionsUaandUbisquiteinteresting.Theparametricequation( 6 )isnothingbutastraightlineinthe(~M(a)c,~M(b)c)plane: ~M(b)c=M(b)c)]TJ /F3 11.955 Tf 11.95 0 Td[(M(a)c+~M(a)c,(6)asseeninFigure 6-12 204

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Figure6-13. MT2(max)asafunctionofthetwotestchildrenmasses,~M(a)cand~M(b)c,fortheoff-shelleventtopologyofFigure 6-3 (b).WeusethemassspectrumfromtheexampleinFigure 6-4 :M(a)c=100GeV,M(b)c=200GeVandMp=600GeVandforsimplicityconsideronlyeventswithPUTM=0. ItisnoweasytounderstandthetriplepointstructureinFigure 6-12 .ThetriplepointisobtainedbythemergingofallthreeboundariesasinEquations( 6 ),( 6 )and( 6 ),i.e.when MBT2(max)(~M(a)c,~M(b)c)=MB0T2(max)(~M(a)c,~M(b)c)=MUaT2(max)(~M(a)c)=MUbT2(max)(~M(b)c).(6)Itiseasytocheckthat~M(a)c=M(a)cand~M(b)c=M(b)cidenticallysatisfytheseequations,therebyprovingthatthetripleintersectionoftheboundariesseeninFigure 6-12 indeedtakesplaceatthetruevaluesofthechildrenmasses.Theseresultsareconrmedinournumericalsimulations.InFigure 6-13 wepresent(a)athreedimensionalviewand(b)agradientplotoftheridgestructurefoundineventswiththeoff-shelltopologyofFigure 6-3 (b).ThemassspectrumforthisstudypointwasxedasinFigure 6-4 ,namelyM(a)c=100GeV,M(b)c=200GeVandMp=600GeV.Sincetheridgestructureforthistopologydoesnotrequirethepresenceofupstreammomentum,forsimplicityweconsideronlyeventswithPUTM=0.TheridgepatternisclearlyevidentinFigure 6-13 (a),whichshowsathree-dimensionalviewoftheMT2endpointfunctionMT2(max)(~M(a)c,~M(b)c).ItisevenmoreapparentinFigure 6-13 (b),whereonecanseea 205

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sharpgradientchangealongtheridgelines:inregionsUaandUb,thecorrespondinggradientvectorspointintrivialdirections(eitherhorizontallyorvertically),inaccordwithEquations( 6 )and( 6 ).Ontheotherhand,thegradientinregionBisverysmall,andtheMT2endpointfunctionisratherat.Thegreendotmarksthelocationofthetruechildrenmasses(M(a)c=100GeV,M(b)c=200GeV)andisindeedtheintersectionpointofthethreeridgelines.Asexpected,thecorrespondingMT2(max)atthatpointisthetrueparentparticlemassMp=600GeV.Atthispoint,itisinterestingtoaskthequestion,whatwouldbetheoutcomeofthisexerciseifoneweretomaketheusualassumptionofidenticalchildren,andapplythetraditionalsymmetricMT2tothissituation.TheanswercanbededucedfromFigure 6-13 (b),wherethediagonalorangedotdashedlinecorrespondstotheusualassumptionof~M(a)c=~M(b)c.Inthatcase,onestillndsakink,butatthewronglocation:inFigure 6-13 (b)theintersectionofthediagonalorangelineandthesolidblackridgelineoccursat~M(a)c=~M(b)c=65.3GeVandthecorrespondingparentmassis~Mp=565.3GeV.Therefore,thetraditionalkinkmethodcaneasilyleadtoawrongmassmeasurement.ThentheonlywaytoknowthattherewassomethingwrongwiththemeasurementwouldbetostudytheeffectoftheupstreammomentumandseethattheobservedkinkisnotinvariantunderPUTM.Weshouldnotethat,dependingontheactualmassspectrum,thetwo-dimensionalridgepatternseeninFigures 6-12 and 6-13 (b)maylookverydifferently.Forexample,thebalancedregionBmayormaynotincludetheorigin.Onecanshowthatif Mp
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Figure6-14. Thefourregionsinthe(~M(a)c,~M(b)c)parameterplaneleadingtothefourdifferenttypesofsolutionsfortheMT2endpointfortheonshellscenarioillustratedinFigure 6-3 (c). ThisexerciseteachesusthatthefailsafeapproachtomeasuringthemassesinmissingenergyeventsistoapplyfromtheverybeginningtheasymmetricMT2concept. 6.5.2On-shellIntermediateParticleOurnalexampleistheon-shelleventtopologyillustratedinFigure 6-3 (c).NowthereisanadditionalparameterwhichentersthegamethemassM()ioftheintermediateparticleinthe-thdecaychain.Asaresult,theallowedrangeofinvariantmassesforthevisibleparticlepaironeachsideislimitedfromabovebyEquation( 6 ).InthiscasewendthattheMT2endpointexhibitsasimilarphasestructureastheoneshowninFigure 6-12 .OneparticularpatternisillustratedinFigure 6-14 ,whichexhibitsthesamefourregionsB,B',UaandUbseeninFigure 6-12 .ThedifferencenowisthatregionB'isconsiderablyexpanded,andasaresult,regionBdoesnothaveacommonborderwithregionsUaandUbanymore.ThetriplepointofFigure 6-12 hasnowdisappearedandthecorrectvaluesofthechildrenmassesnowliesomewhereon 207

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theborderbetweenregionsBandB',buttheirexactlocationalongthisridgelineisatthispointunknown.Justlikewedidfortheoff-shellcaseinSection 6.5.1 ,weshallnowpresentanalyticalformulasfortheMT2endpointineachregionofFigure 6-14 .InthebalancedregionB,wendthesameresultsasEquations( 6 through 6 )asintheoff-shellcaseconsideredinthepreviousSection 6.5.1 .TheotherbalancedregionB'ischaracterizedby m()=mmax(),(6)wheremmax()isgivenbyEquation( 6 ),and AT=M2p 4242)]TJ /F12 11.955 Tf 11.95 20.45 Td[( M(a)i Mp!2)]TJ /F12 11.955 Tf 11.95 20.45 Td[( M(a)c M(a)i!235242)]TJ /F12 11.955 Tf 11.95 20.45 Td[( M(b)i Mp!2)]TJ /F12 11.955 Tf 11.96 20.45 Td[( M(b)c M(b)i!235+M2p 424 M(a)c M(a)i!2)]TJ /F12 11.955 Tf 11.96 20.44 Td[( M(a)i Mp!23524 M(b)c M(b)i!2)]TJ /F12 11.955 Tf 11.96 20.44 Td[( M(b)i Mp!235. (6) TheformulafortheendpointMB0T2(max)inregionB0isthensimplyobtainedbysubstitutingEquations( 6 )and( 6 )intothebalancedsolution( 6 ).Finally,theMT2endpointintheunbalancedregionsUaandUbisgivenby MUaT2(max)(~M(a)c)=mmax(a)+~M(a)c, (6) MUbT2(max)(~M(b)c)=mmax(b)+~M(b)c, (6) wheremmax(a)andmmax(b)aregivenbyEquation( 6 ).InFigure 6-15 wepresentournumericalresultsinthison-shellscenario.Themassspectrumisxedas:M(a)c=100GeV,M(b)c=200GeV,M(a)i=M(b)i=550GeVandMp=1TeV,andwestilldonotincludetheeffectsofanyupstreammomentum.Figure 6-15 (a)showsthethree-dimensionalviewoftheMT2endpointfunctionMT2(max)(~M(a)c,~M(b)c),whichexhibitsthreedifferentsetsofridges,whicharemoreeasilyseeninthegradientplotofFigure 6-15 (b). 208

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Figure6-15. MT2(max)asafunctionofthetwotestchildrenmasses,~M(a)cand~M(b)c,fortheeventtopologyofFigure 6-3 (c),withamassspectrumM(a)c=100GeV,M(b)c=200GeV,M(a)i=M(b)i=550GeVandMp=1TeV. Asusual,thegreendotmarksthetruechildrenmasses.Figure 6-15 (b)showsthattheridgelineseparatingthetwobalancedregionsBandB'doesgothroughthegreendotandthusrevealsarelationshipbetweenthetwochildrenmasses,leavingtheridgelineparameterastheonlyremainingunknowndegreeoffreedom.However,unliketheoff-shellcaseofSection 6.5.1 ,nowthereisnospecialpointonthisridgeline,andwecannotcompletelypindownthemassesbytheridgemethod.Thus,inordertodetermineallmassesintheproblem,onemustuseanadditionalpieceofinformation,forexamplethevisibleinvariantmassendpoint( 6 )orthePUTMinvariancemethodsuggestedinSection 6.3.3.3 6.6ApplicationToMoreGeneralCasesInthissectionwediscussafewotherpossibleapplicationsoftheasymmetricMT2idea,besidestheexamplesalreadyconsidered. 1. Invisibledecaysofthenext-to-lightestparticle.Mostnewphysicsmodelsintroducesomenewmassiveandneutralparticlewhichplaystheroleofadarkmattercandidate.Oftentheverysamemodelsalsocontainother,heavierparticles,whichforcolliderpurposesbehavejustlikeadarkmattercandidate:theydecayinvisiblyandresultinmissingenergyinthedetector.Forexample,insupersymmetryonemayndaninvisiblydecayingsneutrino~`!`~01,inUEDonendsaninvisibly 209

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Figure6-16. Eventtopologyfortheeffectivelydifferentmissingpartices.TheblacksolidlinesrepresentSMparticleswhicharevisibleinthedetectorwhileredsolidlinesrepresentparticlesatintermediatesages.Themissingparticlesaredenotedbydottedlines.(a)Squarkpairproductionwithdecaychainsterminatingintwodifferentinvisibleparticles(~01and~`,correspondingly).Inthiscase~`decaysinvisibly.(b)ThesubsystemMT2variableappliedtottevents.TheW-bosoninthelowerlegistreatedasachildparticleandcandecayeitherhadronicallyorleptonically. decayingKKneutrino1!1,etc.Thesescenarioscaneasilygenerateanasymmetriceventtopology.Forexample,considerthestrongproductionofasquark(~q)pair,asillustratedinFigure 6-16 (a).Oneofthesquarkssubsequentlydecaystothesecondlightestneutralino~02,whichinturndecaystothelightestneutralino~01byemittingtwoSMfermions~02!`+`)]TJ /F6 11.955 Tf 8.26 -4.34 Td[(~01(or~02!jj~01).Theothersquarkdecaystoachargino~1,whichthendecaystoasneutrinoas~1!`~`.Since~`canonlydecayinvisibly,weobtaintheasymmetriceventtopologyoutlinedwiththeblueboxinFigure 6-16 (a).Thetwosquarksaretheparents,thelightestneutralino~01istherstchild,andthesneutrino~`isthesecondchild. 2. ApplyingMT2toanasymmetricsubsystem.OnecanalsoapplytheMT2ideaeventoeventsinwhichthereisonlyone(orevenno)missingparticlestobeginwith.SuchanexampleisshowninFigure 6-16 (b),whereweconsiderttproductioninthedileptonorsemi-leptonicchannel.Intherstlegwecantakeb`asourvisiblesystemandtheneutrino`astheinvisibleparticle,whileintheotherlegwecantreattheb-jetasthevisiblesystemandtheW-bosonasthechildparticle.Inthiscase,therestillshouldbearidgestructurerevealingthetruet,Wandmasses. 3. Multi-componentdarkmatter.Ofcourse,themodelmaycontaintwo(ormore)differentgenuinedarkmatterparticles[ 45 47 96 101 ],whoseproductioninvariouscombinationswillinevitablyleadattimestoasymmetriceventtopologies. 210

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CHAPTER7CONCLUSIONSWehaveproposedmethodsformassmeasurementsinmissingenergyeventsathadroncolliders.Inthischapter,wewillsummarizeallproposedvariables. 7.1p sminWeproposedp s(reco)minandp s(sub)min,whichhaveaclearphysicalmeaning:theminimumCMenergyinthe(sub)system,whichisrequiredinordertoexplaintheobservedsignalinthedetector.Therstvariant,theRECO-levelvariablep s(reco)minisbasicallyamodicationoftheprescriptionforcomputingtheoriginalp sminvariable:insteadofusing(muon-corrected)calorimeterdeposits,aswasdonein[ 50 51 ],onecouldinsteadcalculatep sminwiththehelpofthereconstructedobjects(jetsandisolatedphotons,electronsandmuons).OurexamplesinSections 2.4 2.5 and 2.6 showedthatthisproceduretendstoautomaticallysubtractoutthebulkoftheUEcontributions,renderingthep s(reco)minvariablesafe.Oursecondsuggestionwastoapplyp smintoasubsystemoftheobservedevent,whichissuitablydenedsothatitdoesnotincludethecontributionsfromtheunderlyingevent.Theeasiestwaytodothisistovetojetsfromenteringthedenitionofthesubsystem.Inthiscase,thesubsystemvariablep s(sub)miniscompletelyunaffectedbytheunderlyingevent.However,dependingontheparticularscenario,inprincipleonecouldalsoallow(certainkindsof)jetstoenterthesubsystem.Aslongasthereisanefcientmethod(throughcuts)ofselectingjetswhich(mostlikely)didnotoriginatefromtheUE,thisshouldworkaswell,asdemonstratedinFig. 2-6 withourttexample.p s(reco)min(andtosomeextentp s(sub)min)isageneral,global,andinclusivevariable,whichcanbeappliedtoanytypeofevents,regardlessoftheeventtopology,numberortypeofreconstructedobjects,numberortypeofmissingparticles,etc.Forexample,allofthearbitrarinessassociatedwiththenumberandtypeofmissingparticlesisencodedbyasingleparameter=M. 211

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Themostimportantpropertyofbothp s(reco)minandp s(sub)ministhattheyexhibitapeakintheirdistributions,whichdirectlycorrelateswiththemassscaleMpoftheparentparticles.Comparedtoakinematicendpoint,apeakisafeaturewhichismucheasiertoobserveandsubsequentlymeasurepreciselyovertheSMbackgrounds. 7.2InvariantMassEndpointMethodWithournewproposedsetsofinvariantmassendpoint,theprecisionoftheBSMmassdeterminationisexpectedtoimprove.Weprovidetheanalyticalexpressionsforalldifferentialinvariantmassdistributionsusedinourbasicanalysis:m2``,m2j`(u)andm2j`(s)(1).Wealsoprovidethecorrespondingexpressionforthem2j`(d)(1)distribution,whoseupperendpointoffersanindependentmeasurementofMmaxj`(u)Finally,wealsolisttheformulaforthedifferentialdistributionofm2jl(p),whoseendpointcanbeusedforselectingthecorrectmBsolution.Theknowledgeoftheshapeofthewholedistributionisindispensableandgreatlyimprovestheaccuracyoftheendpointextraction.IntheabsenceofanyanalyticalresultslikethoseinAppendix A ,onewouldbeforcedtousesimplelinearextrapolations,whichwouldleadtoasignicantsystematicerror.Clearly,notallinvariantmassvariableswillhavetheirendpointsmeasuredwithexactlythesameprecisionsomeendpointswillbemeasuredbetterthanothers.Thisdifferencecanbeduetomanyfactors,e.g.theslopeofthedistributionneartheendpoint,theshape(convexversusconcave)ofthedistributionneartheendpoint,theactuallocationoftheendpoint,thelevelofSMandSUSYcombinatorialbackgroundneartheendpoint,etc.Weprovideanumberofavailablemeasurementstremendouslyexceedsthenumberofunknownmassparameters.Thus,wecanchoosethebestinvariantmassendpointvariablesforspecicapplication.Allofthenewvariablesexhibitmildersensitivitytotheparameterspaceregion,incomparisontotheconventionalendpointmmaxj``.Theendpointforeachofourvariablesisgivenbyatmosttwodifferentexpressions,asopposedtofourinthecaseofmmaxj``.Anotableexceptionisthevariablemj`(s)(1),whoseendpointisactuallyuniquelypredicted, 212

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andisindependentoftheparameterspaceregion.Wethereforestronglyencouragetheuseofmj`(s)(1)infutureanalysesofSUSYmassdeterminations.Wecanalreadyuniquelydeterminethreeoutofthefourmassesinvolvedintheproblem.Then,theadditionofafthmeasurement,asdiscussedinSections 3.3.2.1 and 3.4.2 ,issufcienttopindownallfouroftheBSMmasses.Incontrast,withtheconventionalapproach,onealsostartswithfourmeasurementsasin( 3 ),butintheworstcasescenariothisresultsininnitelymanysolutions,duetothelineardependenceproblem( 3 )discussedinSection 3.1.2 .Addingafthmeasurementasin( 3 )helps,butonceagain,theworstcasescenarioleadstotwoalternativesolutions[ 9 ].Inordertoresolvetheremainingduplication,andthusguaranteeuniquenessofthesolutionunderanycircumstances,oneneedsatleast6measurements. 7.3SubsystemMT2MethodWeshowedthattheMT2methodbyitselfissufcientforacompletemassspectrumdetermination,evenintheproblematiccasesofNcascade=1orNcasecade=2.Webackedourclaimwithtwoexplicitexamples:W+W)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(pairproduction,whichisanexampleofann=1chain,andttpairproduction,whichisanexampleofann=2chain.WeshowedthattheMT2methodinprincipleprovidesmorethanenoughmeasurementsfortheunambiguousdeterminationofthecompletemassspectrum.WhenapplyingtheMT2method,wegeneralizedtheconceptofMT2byintroducingvarioussubsystem(orsubchain)M(n,p,c)T2variables.ThelatteraredenedsimilarlytotheconventionalMT2variable,butarelabelledbythreeintegersn,p,andc,whosemeaningisasfollows.Theintegernlabelsthegrandparentparticleoriginallyproducedinthehardscatteringandinitiatingthedecaychain.WethenapplytheusualMT2concepttothesubchainstartingattheparentparticlelabelledbypandterminatingatthechildparticlelabelledbyc.Ingeneral,thechildparticledoesnothavetobetheverylast(i.e.themissing)particleinthedecaychain,justliketheparentparticledoesnothavetobetheveryrstparticleproducedintheevent.TheintroductionoftheM(n,p,c)T2 213

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subchainvariablesgreatlyproliferatesthenumberofavailableMT2-typemeasurements,andallowsustomakefulluseofthepoweroftheMT2concept. 7.4OneDimensionalProjectionMethodWeproposedonedimensionalprojectionsofthekinematicvariablesMT2andMCTwithrespectto~PTofupstreamobjects.Bydoingthisdecomposition,wecanmeasureBSMparticles'massspectrainaveryshortdecaywherepairproducedparticlesinahardcollisionaredecayingintomissingparticlesandonevisibleparticle(Ncascade=1).TotheextentthatthedenitionofMT2?reliesonlyonthedirectionandnotthemagnitudeoftheupstream~PT,ourmethodisinsensitivetothejetenergyscaleerror[ 11 ].Wehavealsoprovidedexactanalyticalformulasforthecomputationofthe1DdecomposedMT2,MT2kandtheshapeoftheMT2?distribution.Weshowhowtheperpendicularandtheparallelprojectedvariablesarerelatedwitheachother.Bystudyingthemaximumallowedboundaryofthisco-relation,wecanincreasestatisticsandcorrespondinglygetmoreprecisemeasurements. 7.5AsymmetricEventTopologyThedarkmattersignaturesatcollidersalwaysinvolvemissingtransverseenergy.Sucheventswillbequitechallengingtofullyreconstructand/orinterpret.Allpreviousstudieshavemade(eitherexplicitlyorimplicitly)theassumptionthateacheventhastwoidenticalmissingparticles.Ourmainpointisthatthisassumptionisunnecessary,andbysuitablemodicationsoftheexistinganalysistechniquesonecaninprincipletestboththenumberandthetypeofmissingparticlesinthedata.OurproposalherewastomodifytheCambridgeMT2variable[ 10 ]bytreatingeachchildrenmassasanindependentinputparameter.Inthisapproach,oneobtainstheMT2endpointMT2(max)asafunctionofthetwochildrenmasses~M(a)cand~M(b)c,andproceedstostudyitsproperties.ThefunctionMT2(max)(~M(a)c,~M(b)c)exhibitsaridgestructure(i.e.agradientdiscontinuity).Thepointcorrespondingtothecorrectchildrenmassesalwayslies 214

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onaridgeline,thustheridgelinesprovideamodel-independentconstraintamongthechildrenmasses,justliketheMT2endpointprovidesamodel-independentconstraintonthemassesofthechild(ren)andtheparent.Ingeneral,theMT2endpointfunctionalsodependsonthevalueoftheupstreamtransversemomentumintheevent.MT2(max)(~M(a)c,~M(b)c,PUTM).However,thePUTMdependencedisappearscompletelyforpreciselytherightvaluesofthechildrenmasses.Thisprovidesasecond,quitegeneralandmodel-independent,methodformeasuringtheindividualparticlemassesinsuchmissingenergyevents. 215

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APPENDIXAANALYTICALEXPRESSIONSFORTHESHAPESOFTHEINVARIANTMASSDISTRIBUTIONSThisappendixAprovidestheanalyticalexpressionsfortheshapesoftheinvariantmassdistributionsm2``,m2j`(u)m2j`n[m2j`f,m2j`(s)(1)m2j`n+m2j`f,m2j`(d)(1)jm2j`n)]TJ /F3 11.955 Tf 10.77 0 Td[(m2j`fj,andm2jl(p).Tosimplifytheexpressions,weintroducetheshorthandnotationforthecorrespondingendpoints,whichwasalreadyintroducedinEquations( 3 ),( 3 ),( 3 )and( 3 ): L(mmax``)2=m2DRCD(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RAB), (A) n)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`n2=m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RBC), (A) f)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(mmaxj`f2=m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB), (A) pRBCf=m2D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(RCD)RBC(1)]TJ /F3 11.955 Tf 11.96 0 Td[(RAB). (A) Inthisappendix,weshallignorespincorrelationsandconsideronlypurephasespacedecays.Generalresultsincludingspincorrelationsform2``,m2j`nandm2j`fexistandcanbefoundin[ 84 ].Weshallunit-normalizethem2``,m2j`(s),m2j`(d)andm2j`(p)distributions,towhicheacheventcontributesasingleentry.Incontrast,theuniondistributionm2j`(u)hastwoentriesperevent,soitwillbenormalizedto2instead.Itisalsoconvenienttowritethedistributionsintermsofmassessquaredinsteadoflinearmasses.Ofcourse,thetwoaretriviallyrelatedby dN dm=2mdN dm2.(A) A.1DileptonMassDistributionm2``Thedifferentialdileptoninvariantmassdistributionisgivenby dN dm2``=1 L,(A) 216

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whichisunit-normalized: ZL0dm2``dN dm2``=1.(A) A.2CombinedJet-leptonMassDistributionm2j`(u)Thedifferentialdistributionforum2j`(u)isgivenby dN du=(n)]TJ /F3 11.955 Tf 11.95 0 Td[(u)(u)1 n+(p)]TJ /F3 11.955 Tf 11.95 0 Td[(u)(u)ln(f=p) f)]TJ /F3 11.955 Tf 11.96 0 Td[(p+(f)]TJ /F3 11.955 Tf 11.96 0 Td[(u)(u)]TJ /F3 11.955 Tf 11.95 0 Td[(p)ln(f=u) f)]TJ /F3 11.955 Tf 11.96 0 Td[(p,(A)where(x)istheusualHeavisidestepfunction (x)8>><>>:1,x0,0,x<0.(A)Itiseasytoverifythenormalizationcondition ZM0dudN du=2,(A)whereM(Mmaxj`(u))2wasalreadydenedinEquation.( 3 ).InFigure A-1 (a)wecross-checkthepredictionofEquation.( A )(bluedashedline)withthenumericallyobtainedm2j`(u)distributioninFigure 3-4 (b)(redsolidline),forthecaseofstudypointLM1.Weseethatwithinthestatisticalerrors,ourformulaisinperfectagreementwiththenumericalresult. A.3Distributionofthesumm2j`(s)(=1)Thedifferentialdistributionform2j`(s)(=1)isgivenby dN d=1 f)]TJ /F3 11.955 Tf 11.96 0 Td[(p((m)]TJ /F8 11.955 Tf 11.96 0 Td[()()lnfn fn)]TJ /F8 11.955 Tf 11.96 0 Td[((f)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+(M)]TJ /F8 11.955 Tf 11.96 0 Td[()()]TJ /F3 11.955 Tf 11.96 0 Td[(m)lnM M)]TJ /F6 11.955 Tf 11.95 0 Td[((f)]TJ /F3 11.955 Tf 11.96 0 Td[(p)+(n+p)]TJ /F8 11.955 Tf 11.96 0 Td[()()]TJ /F3 11.955 Tf 11.95 0 Td[(M)lnfn)]TJ /F8 11.955 Tf 11.95 0 Td[((f)]TJ /F3 11.955 Tf 11.95 0 Td[(p) p(n+p)]TJ /F3 11.955 Tf 11.96 0 Td[(f)), (A) 217

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FigureA-1. ComparisonofthenumericallyobtaineddifferentialinvariantmassdistributionsforstudypointLM1(redsolidlines)withtheanalyticalresultspresentedinthisappendix(bluedashedlines):(a)thedistributionofthecombinedjet-leptonmassum2j`(u)fromFigure 3-4 (b)versustheanalyticalpredictionofEquation.( A );(b)thedistributionofthesumm2j`(s)(=1)fromFigure 3-4 (c)versustheanalyticalpredictionofEquation.( A );(c)thedistributionofthedifferencem2j`(d)(=1)fromFigure 3-4 (d)versustheanalyticalpredictionofEquations( A through A );(d)thedistributionoftheproductm2jl(p)fromFigure 3-6 (c)versustheanalyticalpredictionofEquations( A through A ). wherem(mmaxj`(u))2wasdenedin( 3 ),andn,fandpweredenedinEquations( A through A ).ThenormalizationconditionforEquation.( A )reads ZS0ddN d=1,(A)whereSisdenedinEquation.( 3 ).Asacross-check,Figure A-1 (b)showsthatouranalyticalformulainEquation.( A )agreeswiththenumericalresultfromFigure 3-4 (c)fortheLM1studypoint. 218

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A.4DistributionOfTheDifferencem2j`(d)(=1)Thedifferentialdistributionforthedifferencem2j`(d)(=1)dependsonthevaluesofRBCandRAB.Tosimplifythenotation,wedeneanantisymmetricfunction L(x,y)=)]TJ /F3 11.955 Tf 9.29 0 Td[(L(y,x)lnnf+x(f)]TJ /F3 11.955 Tf 11.95 0 Td[(p) nf+y(f)]TJ /F3 11.955 Tf 11.95 0 Td[(p),(A)whichweheavilyuseinwritingdowntheresultforthedifferentialdistribution.Noticethattherearevariousequivalentwaystowritedowntheseformulas,duetothetransitivityproperty L(x,y)+L(y,z)=L(x,z).(A)Form2j`(d)(=1)oneneedstoconsiderveseparatecases:If2 3)]TJ /F4 7.97 Tf 6.59 0 Td[(RABRBC<1,then dN d=1 f)]TJ /F3 11.955 Tf 11.96 0 Td[(p((n)]TJ /F6 11.955 Tf 11.95 0 Td[()()hL(0,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)+L()]TJ /F6 11.955 Tf 9.29 0 Td[(,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)i+(p)]TJ /F3 11.955 Tf 11.95 0 Td[(n)]TJ /F6 11.955 Tf 11.95 0 Td[()()]TJ /F3 11.955 Tf 11.96 0 Td[(n)L(0,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)+(f)]TJ /F6 11.955 Tf 11.96 0 Td[()()]TJ /F6 11.955 Tf 11.95 0 Td[((p)]TJ /F3 11.955 Tf 11.95 0 Td[(n))L(f,)). (A) If1 2)]TJ /F4 7.97 Tf 6.59 0 Td[(RABRBC<2 3)]TJ /F4 7.97 Tf 6.58 0 Td[(RAB,then dN d=1 f)]TJ /F3 11.955 Tf 11.95 0 Td[(p((p)]TJ /F3 11.955 Tf 11.96 0 Td[(n)]TJ /F6 11.955 Tf 11.96 0 Td[()()hL(0,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)+L()]TJ /F6 11.955 Tf 9.3 0 Td[(,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)i+(n)]TJ /F6 11.955 Tf 11.96 0 Td[()()]TJ /F6 11.955 Tf 11.95 0 Td[((p)]TJ /F3 11.955 Tf 11.95 0 Td[(n))hL(f,)+L()]TJ /F6 11.955 Tf 9.3 0 Td[(,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)i+(f)]TJ /F6 11.955 Tf 11.95 0 Td[()()]TJ /F3 11.955 Tf 11.96 0 Td[(n)L(f,)). (A) 219

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IfRABRBC<1 2)]TJ /F4 7.97 Tf 6.59 0 Td[(RAB,then dN d=1 f)]TJ /F3 11.955 Tf 11.95 0 Td[(p((n)]TJ /F3 11.955 Tf 11.96 0 Td[(p)]TJ /F6 11.955 Tf 11.96 0 Td[()()hL(f,)+L(f,0)i+(n)]TJ /F6 11.955 Tf 11.96 0 Td[()()]TJ /F6 11.955 Tf 11.95 0 Td[((n)]TJ /F3 11.955 Tf 11.96 0 Td[(p))hL(f,)+L()]TJ /F6 11.955 Tf 9.3 0 Td[(,)]TJ /F3 11.955 Tf 9.3 0 Td[(n)i+(f)]TJ /F6 11.955 Tf 11.95 0 Td[()()]TJ /F3 11.955 Tf 11.96 0 Td[(n)L(f,)). (A) IfRAB 2)]TJ /F4 7.97 Tf 6.59 0 Td[(RABRBC
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A.5DistributionOfTheProductm2j`(p)Finally,forcompletenesswealsolistthedifferentialdistributionfortheproductvariableinEquation.( 3 ),forwhichhereweshallusetheshorthandnotationm2jl(p).Tofurthersimplifythenotation,wedenethefunction X()p n 2(f)]TJ /F3 11.955 Tf 11.95 0 Td[(p)p nfp f2n+4(p)]TJ /F3 11.955 Tf 11.96 0 Td[(f)2,(A)wheren,fandparedenedasbeforeinEquations( A through A ).Therearetwoseparatecases:IfRBC0.5,thedistributionismadeupoftwobranchesjoiningat=p np(see,forexampletheLM1distributioninFigure 3-6 (c)andtheLM6'distributioninFigure 3-7 (c)) dN d=2 nf((p np)]TJ /F8 11.955 Tf 11.95 0 Td[()()hlnn p+2ln X)]TJ /F6 11.955 Tf 7.08 1.79 Td[(()i+fp n 2p f)]TJ /F3 11.955 Tf 11.96 0 Td[(p)]TJ /F8 11.955 Tf 11.96 0 Td[(()]TJ 11.95 8.45 Td[(p np)2lnX+() X)]TJ /F6 11.955 Tf 7.09 1.79 Td[(()). (A) IfRBC0.5,thereisasinglebranch,asillustratedbytheLM1'distributioninFigure 3-6 (c)andtheLM6distributioninFigure 3-7 (c): dN d=2 nf(p np)]TJ /F8 11.955 Tf 11.95 0 Td[()()(lnn p+2ln X)]TJ /F6 11.955 Tf 7.09 1.79 Td[(()).(A)Inbothofthosecases,thenormalizationconditionis Zmax0ddN d=1,(A)wheremaxisthecorrespondingm2jl(p)endpointdenedinEquation.( 3 ).Figure A-1 (d)demonstratesthatouranalyticalresultofEquation.( A )agreeswellwiththenumericallyderivedm2jl(p)distributioninFigure 3-6 (c)fortheLM1studypoint. 221

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APPENDIXBANALYTICALEXPRESSIONSFORM(N,P,C)T2,MAX(~MC,PT)ThepurposeofthisAppendixBistocollectinoneplaceallrelevantformulasforthevarioussubsystemMT2endpointsM(n,p,c)T2,max(~Mc,pT)inthepresenceofinitialstateradiation(ISR)witharbitrarytransversemomentumpT.Inallcases,wewillndthatM(n,p,c)T2,max(~Mc,pT)isgivenbytwobranches: M(n,p,c)T2,max(~Mc,pT)=8><>:F(n,p,c)L(~Mc,pT),if~McMc,F(n,p,c)R(~Mc,pT),if~McMc.(B)InwhatfollowsweshalllisttheanalyticexpressionsforeachbranchF(n,p,c)LandF(n,p,c)R,forallpossible(n,p,c)caseswithn)]TJ /F3 11.955 Tf 12.23 0 Td[(c2.ThegrandparentsXn,theparentsXpandthechildrenXcarealwaysassumedtobeon-shell.However,anyintermediateparticlesXmwithn>m>porp>m>cmayormaynotbeon-shell,andthetwocaseswillhavetobetreateddifferently.SuchanexampleisprovidedbytheendpointfunctionM(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)discussedbelowinSection B.2 .Forconvenience,ourresultswillbewrittenintermsofthemassparameters(n,p,c)denedinEquation( 4 ) (n,p,c)Mn 21)]TJ /F3 11.955 Tf 13.15 8.09 Td[(M2c M2p.(B)Theseparametersrepresentcertaincombinationsofthemassesofthegrandparents(Mn),parents(Mp)andchildren(Mc),anddonotcontainanydependenceontheISRtransversemomentumpT.AswediscussedinSections 4.1 and 4.2 ,thesearegenerallythequantitieswhicharedirectlymeasuredbyexperiment.Therefore,withtheMT2method,thegoalofanyexperimentwouldbetoperformasufcientnumberof-parametermeasurementsandthenfromthosetodeterminetheparticlemassesthemselves. 222

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Insomespecialcases,namelyn=p,weshallalsodenepT-dependentparameters,wherethepTdependenceisexplicitlyshownasanargument: (n,n,c)(pT)=(n,n,c)0@s 1+pT 2Mn2)]TJ /F3 11.955 Tf 17.17 8.09 Td[(pT 2Mn1A.(B)WhenpT=0,thepT-dependentparameters( B )simplyreducetothepT-independentones( B ): (n,n,c)(pT=0)=(n,n,c).(B)Wealsoremindthereaderthattestmassesforthechildrenaredenotedwithatilde:~Mc,whilethetruemassofanyparticledoesnotcarryatildesign. B.1TheSubsystemVariableM(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT)ThecorrespondingexpressionswerealreadygiveninEquations( 4 )and( 4 )andwelistthemhereforcompleteness: F(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)L(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT)==8<:"(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)+r (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)+pT 22+~M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(1#2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(p2T 49=;1 2, (B) F(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)R(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,pT)==8<:"(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT)+r (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)()]TJ /F3 11.955 Tf 9.3 0 Td[(pT))]TJ /F3 11.955 Tf 13.15 8.08 Td[(pT 22+~M2n)]TJ /F5 7.97 Tf 6.58 0 Td[(1#2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(p2T 49=;1 2, (B) wherethepT-dependentparameter(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)(pT)wasalreadydenedin( B ): (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(pT)=(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)0@s 1+pT 2Mn2)]TJ /F3 11.955 Tf 17.17 8.09 Td[(pT 2Mn1A.(B)AsalreadymentionedinSection 4.1.1 ,theleftbranchF(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)Lcorrespondstothemomentumconguration~p(1)nT""~p(2)nT""~pT,whiletherightbranchF(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)Rcorrespondsto~p(1)nT""~p(2)nT"#~pT. 223

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B.2TheSubsystemVariableM(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)InthiscasethereisanintermediateparticleXn)]TJ /F5 7.97 Tf 6.59 0 Td[(1betweentheparentXnandthechildXn)]TJ /F5 7.97 Tf 6.59 0 Td[(2(Figures 4-1 ).OurformulasbelowarewritteninsuchawaythattheycanbeappliedbothinthecasewhentheintermediateparticleXn)]TJ /F5 7.97 Tf 6.58 0 Td[(1isonshell(Mn>Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(1)andinthecasewhenXn)]TJ /F5 7.97 Tf 6.59 0 Td[(1isoff-shell(Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(1Mn).Inbothcases(off-shelloron-shell)wendthattheleftbranchofM(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)isgivenby F(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)L(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)=8<:"(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(pT)+r (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(pT)+pT 22+~M2n)]TJ /F5 7.97 Tf 6.58 0 Td[(2#2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(p2T 49=;1 2,(B)wherethepT-dependentparameter(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)(pT)wasalreadydenedin( B ): (n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(pT)=(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)0@s 1+pT 2Mn2)]TJ /F3 11.955 Tf 17.17 8.09 Td[(pT 2Mn1A.(B)TherightbranchF(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)Risgivenbythreedifferentexpressions,dependingonthemassspectrumandthesizeoftheISRpT: F(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)= (B) =8>>>>><>>>>>:F(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)L(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,)]TJ /F3 11.955 Tf 9.3 0 Td[(pT),ifpT>M2n)]TJ /F4 7.97 Tf 6.59 0 Td[(M2n)]TJ /F5 5.978 Tf 5.76 0 Td[(2 Mn)]TJ /F5 5.978 Tf 5.75 0 Td[(2,F(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R,o(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT),ifpTM2n)]TJ /F4 7.97 Tf 6.59 0 Td[(M2n)]TJ /F5 5.978 Tf 5.75 0 Td[(2 Mn)]TJ /F5 5.978 Tf 5.76 0 Td[(2andMn,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2(pT)Mxn)]TJ /F5 5.978 Tf 5.75 0 Td[(1xn,max,F(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R,on(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT),ifpTM2n)]TJ /F4 7.97 Tf 6.59 0 Td[(M2n)]TJ /F5 5.978 Tf 5.75 0 Td[(2 Mn)]TJ /F5 5.978 Tf 5.76 0 Td[(2andMn,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2(pT)Mxn)]TJ /F5 5.978 Tf 5.75 0 Td[(1xn,max.HereMn,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2(pT)isapT-dependentmassparameterdenedas Mn,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2(pT)8<:"r M2n+p2T 4)]TJ /F3 11.955 Tf 11.96 0 Td[(Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2#2)]TJ /F3 11.955 Tf 13.16 8.09 Td[(p2T 49=;1 2,(B)whichinthelimitpT!0reducesto Mn,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2(pT=0)=Mn)]TJ /F3 11.955 Tf 11.96 0 Td[(Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,(B) 224

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justifyingitsnotation.NoticethatMn,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2(pT)isalwayswell-dened,sinceitisonlyusedwhentheconditionpT(M2n)]TJ /F3 11.955 Tf 12.97 0 Td[(M2n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)=Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(2issatisedandtheexpressionunderthesquarerootinEquation( B )isnonnegative.TheothermassparameterappearinginEquation( B ),Mxn)]TJ /F5 5.978 Tf 5.76 0 Td[(1xn,max,isthefamiliarendpointoftheinvariantmassdistributionofthefxn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,xngSMparticlepair: Mxn)]TJ /F5 5.978 Tf 5.76 0 Td[(1xn,max8><>:1 Mn)]TJ /F5 5.978 Tf 5.75 0 Td[(1p (M2n)]TJ /F3 11.955 Tf 11.96 0 Td[(M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2),ifMn)]TJ /F5 7.97 Tf 6.59 0 Td[(1
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Finally,thequantitypvis(pT)appearinginEquation( B )isashorthandnotationforthetotaltransversemomentumofthevisibleparticlesxnandxn)]TJ /F5 7.97 Tf 6.58 0 Td[(1ineachleg:pvisj~p(k)nT+~p(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)Tj.InthecaserelevantforF(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R,on,thevalueofpvisisgivenby pvis(pT)((n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+(n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2))pT 2Mn+j(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F8 11.955 Tf 11.96 0 Td[((n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)js 1+p2T 4M2n.(B)ItiseasytocheckthatinthelimitofpT!0ourEquations( B )and( B )reducetotheknownresultsforthecaseofnoISR(Equations(70)and(74)inW.S.Choetal.[Measuringsuperparticlemassesathadroncolliderusingthetransversemasskink,JHEP0802,035(2008)][ 37 ]).TheleftbranchF(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)LinEquation( B )correspondstothemomentumconguration~p(k)nT+~p(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T""~pT,whiletherightbranchF(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)RinEquation( B )correspondsto~p(k)nT+~p(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T"#~pT.Inthelattercase,F(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)R,oisobtainedwhenXn)]TJ /F5 7.97 Tf 6.58 0 Td[(2isatrest:P(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)T=0,whileF(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R,oncorrespondstothecasewhenP(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)T=1 2pT)]TJ /F3 11.955 Tf 11.96 0 Td[(pvis(pT). 226

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B.3TheSubsystemVariableM(n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)T2,max(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)HerewegeneralizeourpT=0resultofEquation( 4 )fromSection 4.1.4 tothecaseofarbitraryISRpT: F(n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)L(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)==8<:24(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(^pT)+s (n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(^pT)+^pT 22+~M2n)]TJ /F5 7.97 Tf 6.58 0 Td[(2352)]TJ /F6 11.955 Tf 13.3 8.08 Td[(^p2T 49=;1 2, (B) F(n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R(~Mn)]TJ /F5 7.97 Tf 6.59 0 Td[(2,pT)==8<:24(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)()]TJ /F6 11.955 Tf 9.44 0 Td[(^pT)+s (n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)()]TJ /F6 11.955 Tf 9.44 0 Td[(^pT))]TJ /F6 11.955 Tf 13.3 8.09 Td[(^pT 22+~M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2352)]TJ /F6 11.955 Tf 13.3 8.09 Td[(^p2T 49=;1 2, (B) wherewehaveintroducedtheshorthandnotation ^pTpT+2(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)(pT).(B)Noticethatthesecondtermontheright-handsidecontainsthepT-dependentparameterdenedinEquation( B ).TheleftbranchF(n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)LinEquation( B )correspondstothemomentumconguration~p(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T""~p(k)nT""~pT,whiletherightbranchF(n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)RinEquation( B )correspondsto~p(k)(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T"#~p(k)nT""~pT.ItisworthcheckingthatourgeneralpT-dependentresultsinEquations( B )and( B )reducetoourpreviousformulasinEquations( 4 )and( 4 )inthepT!0limitandinthespecialcaseofn=2.FirsttakingthelimitpT!0fromEquations( B )and( B )weget limpT!0^pT=2(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1),(B) 227

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limpT!0(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(^pT)=(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)(2(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))=(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2))]TJ /F8 11.955 Tf 11.95 0 Td[((n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1),(B) limpT!0(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)()]TJ /F6 11.955 Tf 13.09 0 Td[(^pT)=(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)()]TJ /F6 11.955 Tf 9.3 0 Td[(2(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))=(n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2).(B)SubstitutingEquations( B through B )intoEquations( B )and( B ),weget F(n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)L(~Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(2,pT=0)==((n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2))]TJ /F8 11.955 Tf 11.96 0 Td[((n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)+q 2(n,n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(2)+~M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(22)]TJ /F8 11.955 Tf 11.95 0 Td[(2(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))1 2, (B) F(n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)R(~Mn)]TJ /F5 7.97 Tf 6.58 0 Td[(2,pT=0)==((n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)+q )]TJ /F8 11.955 Tf 5.48 -9.68 Td[((n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F8 11.955 Tf 11.96 0 Td[((n,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)2+~M2n)]TJ /F5 7.97 Tf 6.59 0 Td[(22)]TJ /F8 11.955 Tf 11.95 0 Td[(2(n,n,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))1 2, (B) whicharenothingbutthegeneralizationsofEquations( 4 )and( 4 )forarbitraryn. 228

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APPENDIXCTHESYMMETRICMT2INTHELIMITOFINFINITEPUTMInthisAppendixC,werevisitourprevioustwoexamplesfromSections 6.4.1 and 6.4.2 ,thistimeconsideringtheinnitelylargePUTMlimit[ 40 ].Whilethissituationisimpossibletoachieveinarealexperiment,itsadvantageisthatitcanbetreatedbyanalyticalmeans.InthePUTM!1limit,thedecouplingargumentholds[ 40 ],andonendsthefollowinganalyticalexpressionfortheMT2endpointasafunctionofthetwotestchildrenmasses~M(a)cand~M(b)c: MT2(max)(~M(a)c,~M(b)c,1)=8>>>>>>>>>>>><>>>>>>>>>>>>:q M2p)]TJ /F6 11.955 Tf 11.95 0 Td[((M(a)c)2+(~M(a)c)2,if(~M(a)c,~M(b)c)2R1,q M2p)]TJ /F6 11.955 Tf 11.96 0 Td[((M(b)c)2+(~M(b)c)2,if(~M(a)c,~M(b)c)2R2,~M(b)c M(b)cMp,if(~M(a)c,~M(b)c)2R3,~M(a)c M(a)cMp,if(~M(a)c,~M(b)c)2R4,(C)wherethefourdeningregionsRi,(i=1,...,4)areshowninFigure C-1 andaredenedasfollows: R1:~M(b)c
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FigureC-1. Theparameterplaneoftestchildrenmassessquared,dividedintothefourdifferentregionsRiusedtodenetheMT2endpointfunction( C ).TheircommonboundariesLijareparametricallydenedinEquations( C C ).Theblackdotcorrespondstothetruevaluesofthechildrenmasses. AsindicatedinFigure C-1 ,eachLijisastraightlineintheparameterspaceofthechildrentestmassessquaredandisgivenby L12:(~M(b)c)2=(M(b)c)2)]TJ /F6 11.955 Tf 11.95 0 Td[((M(a)c)2+(~M(a)c)2,~M(a)cM(a)c; (C) L23:~M(b)c=M(b)c,~M(a)cM(a)c; (C) L34:~M(b)c=M(b)c M(a)c~M(a)c,~M(a)cM(a)c; (C) L14:~M(a)c=M(a)c,~M(b)cM(b)c. (C) AsseeninFigure C-1 ,allfourlinesLijmeetatthetruechildrenmasspoint~M(a)c=M(a)c,~M(b)c=M(b)c,whereinturntheMT2endpointMT2(max)givesthetrueparentmassMp,inaccordancewithEquation( 6 ).Withthosepreliminaries,wearenowinapositiontorevisitourtwoexamplesfromSections 6.4.1 and 6.4.2 .Figures C-2 and C-3 arethecorrespondinganaloguesofFigures 6-6 and 6-9 inthecaseofinnitePUTM.Comparingwithourearlierresults,wenoticebothquantitativeandqualitativechangesintheridgestructure.First,thesmooth 230

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ridgeinFigure 6-6 (b)(Figure 6-9 (b))hasnowbeendeformedintotwostraightlinesegments,onehorizontal(L23)andtheothervertical(L14),whichmeetatanangleof90preciselyatthetruevaluesofthechildrenmasses.Moreimportantly,Figures C-2 and C-3 nowexhibitanotherpairofridgesL12andL34(plottedinredinFigures C-2 (b)and C-3 (b)),whichwereabsentfromtheearlierguresinSection 6.4 .ThesystemoffourridgesseeninFigures C-2 (a)and C-3 (a)isverysimilartothecreasestructureobservedinA.J.Barretal.[Transversemassesandkinematicconstraints:fromtheboundarytothecrease,JHEP0911,096(2009)][ 40 ].WethusconrmtheresultofReference.[ 40 ]thatintheinnitePUTMlimitthereexistfourdifferentridges,whosecommonintersectionpointrevealsthetruemassesoftheparentandchildrenparticles.Atthispointitisinstructivetocontrastthetwosetsofridgelines:L23andL14(showninFigures C-2 (b)and C-3 (b)inblack)versusL12andL34(showninFigures C-2 (b)and C-3 (b)inred).TheboundariesL23andL14separatetheunionofregionsR1andR2fromtheunionofregionsR3andR4.Alongthoseboundaries,weobserveatransitioninthecongurationofvisiblemomentawhichyieldsthemaximumpossiblevalueofMT2.Moreprecisely,inregionsR1andR2wendthatthevisiblemomenta~p()TforMT2(max)areparalleltothedirectionoftheupstreammomentum~PUTM,whileinregionsR3andR4wendthat~p()Tareanti-parallelto~PUTM.ThisfactremainstrueevenatnitevaluesofPUTM,whichiswhytheridgelinesL23andL14couldalsobeseenintheearlierplotsfromSection 6.4 atnitePUTM=1TeV.Ontheotherhand,theridgelinesL12andL34showninredinFigures C-2 (b)and C-3 (b)areduetothedecouplingargument[ 40 ],whichisstrictlyvalidonlyintheinnitePUTMlimit.ThisiswhytheseridgesbecomeapparentonlyatverylargevaluesofPUTM,andaregraduallysmearedoutatsmallerPUTM. 231

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FigureC-2. MT2(max)fortheeventtopologyofFigure 6-3 (a)withxedupstreammomentumofPUTM=!1. FigureC-3. MT2(max)fortheeventtopologyofFigure 6-3 (a)withthesymmetricmassspectrumIIfromTable 6-1 withupstreammomentumPUTM!1. 232

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FigureC-4. AstudyofthesharpnessoftheMT2ridgefortheexampleconsideredinSection 6.4.1 .TheeventtopologyisthatofFigure 6-3 (a)andthemassspectrumisM(a)c=250GeV,M(b)c=500GeVandMp=600GeV.WeplottheasymmetricMT2endpointMT2(max)(~M(a)c(),~M(b)c(),PUTM),asafunctionoftheangularvariableparameterizingthecircleofradiusRdenedinEquations( C C ).TheradiusRofthecircleistakentobeR=50GeVinpanel(a)andR=5GeVinpanel(b).WepresentresultsforfourdifferentchoicesoftheupstreammomentumPUTMaslabelledintheplot. FigureC-5. AstudyofthesharpnessoftheMT2ridgeforthecasewhenmissingparticlesarethesamewheretheinputmassspectrumisxedasM(a)c=100GeV,M(b)c=100GeVandMp=300GeV. 233

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TheevolutionoftheridgestructureasafunctionofPUTMisshowninFigures C-4 and C-5 .Inordertocomparethesharpnessofthefourridges,wechoosetovarythetestchildrenmasses~M(a)cand~M(b)calongacirclecenteredontheirtruevaluesandwithaxedradiusR.Suchacircleisguaranteedtocrossallfourridges,andcanbeparameterizedintermsofanangularcoordinateasfollows ~M(a)c()=M(a)c+Rcos, (C) ~M(b)c()=M(b)c+Rsin. (C) TheninFigure C-4 (Figure C-5 )weplottheasymmetricMT2endpointMT2(max)(~M(a)c(),~M(b)c(),PUTM),asafunctionoftheangularvariable,forthecaseofmassspectrumIstudiedinSection 6.4.1 (massspectrumIIstudiedinSection 6.4.2 ).TheradiusRistakentobeR=50GeVinpanels(a)andR=5GeVinpanels(b).Wepresentresultsforfourdifferentchoicesoftheupstreammomentum:PUTM=100GeV(blacklines),PUTM=1TeV(bluelines),PUTM=4TeV(magentalines),andPUTM=1(redlines).NoticethattheredlinesatPUTM=1inFigures C-4 and C-5 aredirectlycorrelatedtothethree-dimensionalplotsofFigures C-2 and C-3 ,whilethebluelinesatPUTM=1TeVinFigures C-4 and C-5 aredirectlycorrelatedtothethree-dimensionalplotsofFigures 6-6 and 6-9 .EachoneofthepreviouslydiscussedridgesmanifestsitselfasakinkinFigures C-4 and C-5 .Indeed,theredlinesforPUTM=1revealfourclearkinks,which(fromlefttoright)correspondtotheridgelinesL34,L23,L12,andL14.UsingEquations( C C ),itiseasytondtheexpectedlocationofeachkinkinthePUTM!1limit:=f63.4,180,204.9,270gforFigure C-4 (a),=f63.4,180,206.4,270gforFigure C-4 (b),and=f45,180,225,270gforFigures C-5 (a)and C-5 (b).However,astheupstreammomentumisloweredtomorerealisticvalues,thekinksgraduallywashout,albeittoadifferentdegree.Asanticipatedfromourearlierresults,thesmearingeffectisquitesevereforL34andL12,andbythetimewereachPUTM=1TeV,those 234

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twokinkshavecompletelydisappeared.Ontheotherhand,L23andL14areaffectedtoalesserdegreebythesmearingeffectandarestillvisibleatPUTM=1TeV,butbyPUTM=100GeVtheyareessentiallygoneaswell.NoticethatthevariationinPUTMaffectsnotonlythesharpnessofthekinks,butalsotheirlocation.Thiswastobeexpected,sincewealreadysawthattheshapeoftheridgeisdifferentatPUTM=1TeVandPUTM=1:comparetheblackridgelinesinFigures 6-6 (b)and 6-9 (b)tothoseinFigures C-2 (b)and C-3 (b).Finally,asacuriousfactwenoticethattheresultsshowninpanels(a)andpanels(b)ofFigures C-4 and C-5 areapproximatelyrelatedbyasimplescalingwithaconstantfactor. 235

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BIOGRAPHICALSKETCH MyeonghunParkwasborninChung-Ju,SouthKorea.Hehasoneoldersisterandtwoyoungerbrothers.HegrewupmostlyinJeon-JuCity,graduatingfromDong-AmHighSchoolin1995.HeearnedhisB.S.inPhysicsandMathematicsinKoreaAdvancedInstituteofScienceandTechnologyin2000.HenishedM.S.degreeinPhysicsfromSeoulNationalUniversityon2002.Aftergraduationhefullledmilitaryserviceasanavyofcerfrom2002to2005.Fortwoyears,hehadbeenaengineeringofcerincombatships,andspenthislastserviceyearinNavywarehouses.HereceivedhisPh.D.fromtheUniversityofFloridainthespringof2011.HewillgototheEuropeanOrganizationforNuclearResearch(CERN)inGenevaSwitzerlandforapostdoctoralfellowshipfrom2011to2013.MyeonghunParkmarriedKellyChungin2009. 244