<%BANNER%>

Model-Independent Mass and Spin Determination for a Sequential Decay with a Jet and Two Leptons

Permanent Link: http://ufdc.ufl.edu/UFE0024843/00001

Material Information

Title: Model-Independent Mass and Spin Determination for a Sequential Decay with a Jet and Two Leptons
Physical Description: 1 online resource (124 p.)
Language: english
Creator: Burns, Michael
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: cascade, decay, independent, mass, measurement, model, sequential, spin
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We reconsidered existing techniques for the determination of the masses and spins of the four narrow resonances, $A$, $B$, $C$, and $D$, that might occur in a sequential decay, $D\rightarrow{}jC\rightarrow{}j\ell^{\pm}B\rightarrow{}j\ell^{\pm}\ell^{\mp}A$, where $j$ is a quark jet, $\ell^{\pm}\ell^{\mp}$ is a charged lepton (electron or muon) pair, $A$ is undetectable, and the energy and longitudinal momentum of $D$ are unknown. We found that these existing techniques rely on new physics model assumptions beyond the mere existence of the decay chain. We propose model-independent generalizations in order to improve these mass and spin determination techniques. We identified a two-fold ambiguity in the method of kinematic endpoints for mass determination that uses extreme kinematic values of $m_{ll}^2$, $m_{jl(lo)}^2$, $m_{jl(hi)}^2$, and $m_{jll}^2$ as observables, namely $m_{ll}^{2\max}$, $m_{jl(lo)}^{2\max}$, $m_{jl(hi)}^{2\max}$, $m_{jll}^{2\max}$, and $m_{jll(\theta > \pi/2)}^{2\min}$. While similar ambiguities have already been recognized, these ambiguities have not been well-understood, and the existing resolutions rely on model-dependent features of the single-variable invariant mass distributions, such as their shapes. In order to obtain a model-independent resolution to the two-fold ambiguity, we generalized from kinematic endpoints of single-variable distributions to kinematic boundary lines of double-variable distributions. In particular, we demonstrated a resolution of the ambiguity by using the double-variable invariant mass distributions in ($m_{ll}^2$,$m_{jll}^2$) and ($m_{jl(lo)}^2$,$m_{jl(hi)}^2$). Furthermore, our technique automatically determines if $m_B > m_C$. For spin determination, we reverted back to the single-variable invariant mass distributions in $m_{ll}^2$ and $m_{jl^{\pm}}^2$. Existing spin determination techniques that use these distributions assume fixed chiral projections for each decay vertex, where these chiral projections are determined by the assumed new physics model. These fixed chiral projections can artificially enhance the distinction between two different spin assignments. In order to isolate the spin determination, we derived basis functions that allow the spin-dependence of the invariant mass distributions to be separated from the dependence on the chiral projections of the vertices. Then, we demonstrated our spin determination technique by fitting simulated invariant mass distributions to our basis functions with floating chiral projection coefficients. While the floating chiral projections can weaken the distinction between two different spin assignments, our new basis functions clearly show that an intermediate vector particle will \emph{always} result in a characteristic deviation from pure phase space, regardless of the chiral projections. A fortunate byproduct of our spin determination technique is a set of constraints on the chiral projection coefficients, as a result of the fitting procedure, which may further distinguish a particular model of new physics beyond the distinction of the spin assignments alone.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Michael Burns.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Matchev, Konstantin T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024843:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024843/00001

Material Information

Title: Model-Independent Mass and Spin Determination for a Sequential Decay with a Jet and Two Leptons
Physical Description: 1 online resource (124 p.)
Language: english
Creator: Burns, Michael
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: cascade, decay, independent, mass, measurement, model, sequential, spin
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We reconsidered existing techniques for the determination of the masses and spins of the four narrow resonances, $A$, $B$, $C$, and $D$, that might occur in a sequential decay, $D\rightarrow{}jC\rightarrow{}j\ell^{\pm}B\rightarrow{}j\ell^{\pm}\ell^{\mp}A$, where $j$ is a quark jet, $\ell^{\pm}\ell^{\mp}$ is a charged lepton (electron or muon) pair, $A$ is undetectable, and the energy and longitudinal momentum of $D$ are unknown. We found that these existing techniques rely on new physics model assumptions beyond the mere existence of the decay chain. We propose model-independent generalizations in order to improve these mass and spin determination techniques. We identified a two-fold ambiguity in the method of kinematic endpoints for mass determination that uses extreme kinematic values of $m_{ll}^2$, $m_{jl(lo)}^2$, $m_{jl(hi)}^2$, and $m_{jll}^2$ as observables, namely $m_{ll}^{2\max}$, $m_{jl(lo)}^{2\max}$, $m_{jl(hi)}^{2\max}$, $m_{jll}^{2\max}$, and $m_{jll(\theta > \pi/2)}^{2\min}$. While similar ambiguities have already been recognized, these ambiguities have not been well-understood, and the existing resolutions rely on model-dependent features of the single-variable invariant mass distributions, such as their shapes. In order to obtain a model-independent resolution to the two-fold ambiguity, we generalized from kinematic endpoints of single-variable distributions to kinematic boundary lines of double-variable distributions. In particular, we demonstrated a resolution of the ambiguity by using the double-variable invariant mass distributions in ($m_{ll}^2$,$m_{jll}^2$) and ($m_{jl(lo)}^2$,$m_{jl(hi)}^2$). Furthermore, our technique automatically determines if $m_B > m_C$. For spin determination, we reverted back to the single-variable invariant mass distributions in $m_{ll}^2$ and $m_{jl^{\pm}}^2$. Existing spin determination techniques that use these distributions assume fixed chiral projections for each decay vertex, where these chiral projections are determined by the assumed new physics model. These fixed chiral projections can artificially enhance the distinction between two different spin assignments. In order to isolate the spin determination, we derived basis functions that allow the spin-dependence of the invariant mass distributions to be separated from the dependence on the chiral projections of the vertices. Then, we demonstrated our spin determination technique by fitting simulated invariant mass distributions to our basis functions with floating chiral projection coefficients. While the floating chiral projections can weaken the distinction between two different spin assignments, our new basis functions clearly show that an intermediate vector particle will \emph{always} result in a characteristic deviation from pure phase space, regardless of the chiral projections. A fortunate byproduct of our spin determination technique is a set of constraints on the chiral projection coefficients, as a result of the fitting procedure, which may further distinguish a particular model of new physics beyond the distinction of the spin assignments alone.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Michael Burns.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Matchev, Konstantin T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024843:00001


This item has the following downloads:


Full Text

PAGE 1

MODEL-INDEPENDENTMASSANDSPINDETERMINATIONFORASEQUEN TIAL DECAYWITHAJETANDTWOLEPTONS By MICHAELE.BURNS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

PAGE 2

c r 2009MichaelE.Burns 2

PAGE 3

\Ipledgeallegiancetoeectiveeldtheory,andtothereno rmalizationforwhichit stands;onetheoryunderGUTwithrunningparametersforall ." 3

PAGE 4

ACKNOWLEDGMENTS IowetremendousgratitudetoKyoungchulKong,aformerstud entofmyadvisor, andMyeonghunPark,afellowstudent,forcountlesscalcula tions,simulations,helpful discussions,andhelpwithmundanetype-setting.Iamalsog ratefulforhavingsuch awonderfuloce-mateandcompanion,Sung-SooKim,whosefr eshperspectivesand enthusiasmremindedmeeverydaythatIwasactuallystillin terestedinphysics.Ithank DarleneLatimer,NathanWilliams,andPamMarlinfortheirp atienceandhelptaking careoftheadministrativeB.S.(andIdon'tmeanBachelorof Science).Ofcourse,I wouldneverhavecompletedthisdissertationwithouttheen duringsupportofmyadvisor, KonstantinMatchev,whosomehowmanagedtoputupwithmefor 4years. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................7 LISTOFFIGURES ....................................8 ABSTRACT ........................................9 CHAPTER 1INTRODUCTION ..................................11 1.1TheBasicExperimentalSituation .......................11 1.2IntermediateResonancesoftheStandardModel ...............12 1.3Denitions ....................................14 1.4ExperimentalChallengesPertainingtoourAnalysis .............21 2MASSDETERMINATION .............................23 2.1TheMethodofKinematicEndpoints .....................23 2.1.1ForwardFormulas ............................25 2.1.2InversionFormulas ...........................27 2.2DuplicationAnalysis ..............................29 2.3KinematicBoundaryLinesforthe( m 2jl ( lo ) m 2jl ( hi ) )Distribution .......39 2.4KinematicBoundaryLinesforthe( m 2ll m 2jll )Distribution ..........49 3SPINDETERMINATION ..............................56 3.1ClassicationofHelicityCombinations ....................61 3.1.1HelicityBasisFunctions F IJ ......................63 3.1.2HelicityCoecients K IJ ........................66 3.2ObservableDistributions ............................68 3.2.1InvariantMassDistributionsintheHelicityBasis fF IJ g .......68 3.2.2InvariantMassDistributionsintheObservableBasis fF ; F ; F r ; F g 71 3.3DeterminationofModelParameters f ;;r g ................75 3.4TwinSpinScenariosFSFS/FSFVandFVFS/FVFV ............76 4CONCLUSIONS ...................................79 4.1KinematicBoundaries .............................79 4.2SpinCorrelations ................................81 4.3GeneralRemarks ................................83 APPENDIX AABSMEXAMPLE:SUSY .............................84 5

PAGE 6

BEXAMPLEINVERSIONFORMULASFOR N jl =1 ; 2 ; 3 .............87 CHELICITYBASISFUNCTIONS fF IJ g ......................89 DOBSERVABLESPINBASISFUNCTIONS fF ; F ; F r ; F g ...........95 ESIMPLESPINFITTINGPROCEDURE ......................100 FSPINDETERMINATIONEXAMPLES ......................103 F.1TheSPS1aStudyPoint ............................103 F.2InputfromSFSF( S =1) ............................107 F.3InputfromFSFS( S =2)andFSFV( S =3) .................109 F.4InputfromFVFS( S =4)andFVFV( S =5) ................111 F.5InputfromSFVF( S =6) ...........................113 F.6RemarksonSpinDeterminationattheTevatron ...............114 REFERENCES .......................................116 BIOGRAPHICALSKETCH ................................123 6

PAGE 7

LISTOFTABLES Table page 1-1Somerepresentativecomparisonsofournotation, R ij ...............17 1-2Thesixspinassignments. ..............................20 2-1Examplesofmassambiguities. ............................30 2-2Twoexamplesof exact duplication. .........................37 3-1Shorthandnotationforthesixdierentspinassignment s. .............57 3-2Classicationofmodelparametersaccordingtotheirco ntributionto K IJ ...64 3-3Availablemeasurementsofthemodelparameters and r ..........77 4-1Expectedoutcomesfromourspindiscriminationanalysi s. ............82 C-1Helicitybasisfunctionsforthedilepton. ......................94 C-2Helicitybasisfunctionsfor j` n ...........................94 D-1Observablebasisfunctionsforthedilepton. ....................99 D-2Observablebasisfunctionsfor j` n .........................99 7

PAGE 8

LISTOFFIGURES Figure page 1-1ExampledecaysintheSM. .............................13 1-2FourbasicdecayscenariosoftheHiggsboson. ...................14 1-3Weconsideredthesetwo\model-independent"decaydiag rams. .........15 1-4Arepresentationofthe11Regionsofinputmassparamete rspace. ........18 1-5Decisiontreeforchoosingthespinassignments. ..................21 2-1Anarticialsingle-variablehistogramtodemonstrate kinematicendpoints. ...26 2-2Themaps T 13 : R 1 7!R 3 and T 23 : R 2 7!R 3 ..................32 2-3Themap T 21 : R 2 7!R 1 ..............................33 2-4Theminimumvalue R min CD ( R BC ;R AB )requiredforduplication. ..........36 2-5Thegenericshape ONPF ..............................40 2-6Obtainingtheshapeofthe( m 2jl ( lo ) m 2jl ( hi ) )bivariatedistributionbyfolding. ...43 2-7Thegenericshapeofthebivariatedistribution( m 2jl ( lo ) m 2jl ( hi ) ) ..........45 2-8Scatterplotsof( m 2jl ( lo ) m 2jl ( hi ) ) ............................48 2-9Thegenericshape OVUS ..............................50 2-10Scatterplotsof( m 2ll m 2jll ). ..............................55 3-1The8dierenthelicitycombinations. ........................63 3-2Acontourplotofcos~ c asafunctionofcos c and f ..............69 A-1AnexampleofFig.1-3AfrommSUGRA. .....................86 E-1Contourplotsof 2 ( ;;r ). .............................101 F-1Dileptoninvariantmassdistributions, L + S ....................104 F-2Sumof j` + and j` invariantmassdistributions, S + ..............105 F-3Dierenceof j` + and j` invariantmassdistributions, D + ...........106 F-4DecaydiagramsforFSFSandFSFV. ........................110 F-5DecaydiagramsforFVFSandFVFV. .......................112 F-6DecaydiagramforSFVF. ..............................113 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MODEL-INDEPENDENTMASSANDSPINDETERMINATIONFORASEQUEN TIAL DECAYWITHAJETANDTWOLEPTONS By MichaelE.Burns August2009 Chair:KonstantinMatchevMajor:Physics Wereconsideredexistingtechniquesforthedetermination ofthemassesandspinsof thefournarrowresonances, A B C ,and D ,thatmightoccurinasequentialdecay, D jC j` B j` ` A ,where j isaquarkjet, ` ` isachargedlepton(electronormuon) pair, A isundetectable,andtheenergyandlongitudinalmomentumo f D areunknown. Wefoundthattheseexistingtechniquesrelyonnewphysicsm odelassumptionsbeyond themereexistenceofthedecaychain.Weproposemodel-inde pendentgeneralizationsin ordertoimprovethesemassandspindeterminationtechniqu es. Weidentiedatwo-foldambiguityinthemethodofkinematic endpointsfor massdeterminationthatusesextremekinematicvaluesof m 2ll m 2jl ( lo ) m 2jl ( hi ) ,and m 2jll asobservables,namely m 2maxll m 2maxjl ( lo ) m 2maxjl ( hi ) m 2maxjll ,and m 2minjll ( >= 2) .While similarambiguitieshavealreadybeenrecognized,theseam biguitieshavenotbeen well-understood,andtheexistingresolutionsrelyonmode l-dependentfeaturesofthe single-variableinvariantmassdistributions,suchasthe irshapes.Inordertoobtaina model-independentresolutiontothetwo-foldambiguity,w egeneralizedfromkinematic endpointsofsingle-variabledistributionstokinematicb oundarylinesofdouble-variable distributions.Inparticular,wedemonstratedaresolutio noftheambiguitybyusingthe double-variableinvariantmassdistributionsin( m 2ll m 2jll )and( m 2jl ( lo ) m 2jl ( hi ) ).Furthermore, ourtechniqueautomaticallydeterminesif m B >m C 9

PAGE 10

Forspindetermination,werevertedbacktothesingle-vari ableinvariantmass distributionsin m 2ll and m 2jl .Existingspindeterminationtechniquesthatusethese distributionsassumexedchiralprojectionsforeachdeca yvertex,wherethesechiral projectionsaredeterminedbytheassumednewphysicsmodel .Thesexedchiral projectionscanarticiallyenhancethedistinctionbetwe entwodierentspinassignments. Inordertoisolatethespindetermination,wederivedbasis functionsthatallowthe spin-dependenceoftheinvariantmassdistributionstobes eparatedfromthedependence onthechiralprojectionsofthevertices.Then,wedemonstr atedourspindetermination techniquebyttingsimulatedinvariantmassdistribution stoourbasisfunctionswith roatingchiralprojectioncoecients.Whiletheroatingch iralprojectionscanweakenthe distinctionbetweentwodierentspinassignments,ournew basisfunctionsclearlyshow thatanintermediatevectorparticlewill always resultinacharacteristicdeviationfrom purephasespace,regardlessofthechiralprojections.Afo rtunatebyproductofourspin determinationtechniqueisasetofconstraintsonthechira lprojectioncoecients,asa resultofthettingprocedure,whichmayfurtherdistingui shaparticularmodelofnew physicsbeyondthedistinctionofthespinassignmentsalon e. 10

PAGE 11

CHAPTER1 INTRODUCTION 1.1TheBasicExperimentalSituation ParticlephysicistsexpectthattheHiggseld,alongwitha ssociatedphenomena beyondtheStandardModel(BSM),shouldexhibitinvariantm assresonancesinthe rangefromabout100GeVuptoafewTeV[1,2].Thelowerendoft hismassrangeis justwithinreachoftheTevatronatFermilab,andtheyconti nuetosearchforevidence oftheHiggsboson[1,3,4].TheformerLargeElectron-Posit ronCollider(LEP)atThe EuropeanOrganizationforNuclearResearch(CERN)alsopro bedthelowerendofthis massrange[3,5],andthenewlycommissionedLargeHadronCo llider(LHC)atCERN isdesignedtoprobeevenhighermassesthanTevatronorLEP[ 3].Therehasbeensome collider-basedobservationofthedecayofeveryunstablef undamentalparticleinthe StandardModelofparticlephysics(SM),excepttheHiggsbo son.So,amajorgoalofthe LHCis,eithertoproduceHiggsbosonsandobservetheirdeca ys,ortostronglyruleout theHiggsmechanism.Bothofthesescenariosareexciting.M anyparticlephysicistsalso wanttoseeexperimentalevidenceforanextensionoftheSM, evenifaHiggsbosonis observed[1,6].So,thegreatexpectationisthatasignalwi llbediscoveredintheLHC datathatcannotbeexplainedbyanyparticleorinteraction thathaseverbeenobserved. However,thissignalmuststillbeinterpreted,whichprese ntsseveralchallenges. Inordertounderstandthedatathatareobtainedfromaparti clecollider,thedata mustbeanalyzedandinterpretedonvariouslevels.Therawd ataaresimplythephysical excitationsoftheparticledetector,whichisacomplexand carefullyplannedarrangement ofmaterials.Theserawdataareltered,quitenontriviall y,intoanumericalformat,which istheninterpretedintoreconstructedcolliderobjects.T hepossiblereconstructedobjects forahadroncollidersuchastheLHCare[7]: photon lepton:eitherelectronormuon 11

PAGE 12

jet:eitheruntagged,b-tagged,ortau missingtransversemomentum Somemodelisassumedateachlevelofdatainterpretation.D etectorexcitations aremodeledintermsoftheelectrostaticandnuclearforces thatbindmatter[7,8].The reconstructionofcolliderobjectsismodeledintermsofth esetofelectromagneticand nuclearpropertiesofeachparticlethatcouldbeproducedi nacollision.Thepossibility andfrequencyofcombinationsofspeciesandphysicalprope rties(i.e.the\signatures")of thesereconstructedobjectsaremodeledintermsoftheunde rlyingphysics.Wefocusedon aspecicsignature,namelyajetandtwoleptons,takingall otherlevelsofinterpretation forgranted.Theactualappearanceofanyspecicsignature dependsontheunderlying physics,andouranalysisappliesonlytodatainwhichthesi gnatureappears. 1.2IntermediateResonancesoftheStandardModel Weapproximatedthereconstructedelectrons,muons,andje tsasmassless.Inthis contexttheonlyessentiallymassiveparticlesoftheSMare [9]: theweakbosons, Z and W thetopquark, t theHiggsboson, h So,weapproximatethataSMcollisionproductcanbe,either anessentiallymassless particlethatdoesnotdecayintoanydistinctreconstructe dobjects,oroneofthemassive objectslistedabove. Thetwolightestparticlesinthelistthatdecaytodistinct reconstructedobjectsare theweakbosons,bothofwhichundergoasimpletwo-bodydeca y(Fig. 1-1 A&B).The topquarkundergoesatwo-bodydecaytoa W bosonwithabottomquark,andthen the W bosondecaystoatwo-particlestate(Fig. 1-1 C).Thisisanexampleofwhatwe calla\sequentialdecay".Asequentialdecayisadecaythat producesan intermediate on-shellresonance (aresonancethatiskinematicallyallowedtobeonitsmasss hell), whichthenitselfdecays,eithertoalighterintermediateo n-shellresonance,orgenerically 12

PAGE 13

W f f 0 Z 0 f f f f 0 b t W ABC Figure1-1.ExampledecaysintheSM.Inallthreediagrams, f f;f 0 g canbe,eithera leptonandneutrino,orapairofquarks,and f f; f g isageneric fermion-antifermionpair.A)andB)depictthegenericSMde caysoftheweak bosons,whicharebothkinematicallysimpletwo-bodydecay s.C)depictsthe SMdecayofthetopquark,whichisatwo-stagesequentialdec ay. tosomenalstate.Inthecaseofthetopquarkdecay,thereis onlyoneintermediate on-shellresonance,the W boson,sothetopquarkexhibitsatwo-stagesequentialdeca y. Ingeneral,asequentialdecayhastwoormorestages,onemor ethanthenumberof intermediateon-shellresonances. TherelevantdecayschemeoftheHiggsbosondependsonitsma ss. 1 A)IfthemassoftheHiggsbosonisatthelowerendofitsallow edrange(around 120GeV),thenitprimarilyundergoesatwo-bodydecaytoabo ttom-ravored quark-antiquarkpair(Fig. 1-2 A)[3]. B)Ifthemassisinitslowerintermediaterange(lessthanab out160GeV),thenit primarilyundergoesathree-bodydecaytoastatecontainin ga single intermediate weakboson(Fig. 1-2 B)[3].ThisprovidesanotherSMexampleofatwo-stage sequentialdecay,inwhichtherststageisathree-bodydec ay,andthelaststageis atwo-bodydecay. C)Ifthemassisinitsupperintermediaterange(betweenabo ut160GeVand350 GeV),thenitprimarilyundergoesatwo-bodydecaytoaweakb osonpair[3],each ofwhichundergoesatwo-bodydecay(Fig. 1-2 C).Thisscenariodoes not produce asequentialdecay,becausetherststageofthedecayprodu ces two intermediate on-shellresonancesinsteadofonlyone. 1 Atthetimeofthisresearch,themassoftheHiggsbosonisunk nown,butitis expectedtobearound120GeV[3]. 13

PAGE 14

A h b b B h f 000 f 00 V f 0 f D h t t b W + f f 0 b W f 00 f 000 C h V V f f 0 f 00 f 000 Figure1-2.FourbasicdecayscenariosoftheHiggsboson,de pendingonitsmass[3,10]. A)isperhapsthemostexpectedscenario,sincetheHiggsbos onisexpectedto beinthelightrange.B)isthesequentialdecayscenariofor theHiggsboson, producingafour-bodynalstate,anditisdominantinthelo werintermediate massrange.C)issimplyrelatedtoB)byreplacingtheo-she ll V inB)(not shown)withanon-shell V inC). V canbeanyof f W ;Z 0 g ,butthen V must bethecorresponding f W ;Z 0 g ,respectively.D)providesasignicantdecay channelintheheavymassrange. D)Ifthemassisinitshighrange(around500GeV),thentheup per-intermediate decayscheme(Fig. 1-2 C)remainsdominant,butatwo-bodydecaytoatop quark-antiquarkpair(Fig. 1-2 D)alsocontributessignicantlytothedecay[3], whichhasadistinctsignature.However,thisscenariodoes not resultinasequential decayoftheHiggsboson,againbecausetherststageproduc es two intermediate on-shellresonances. So,intheSM,therearepossiblytwotypesofsequentialdeca y,bothofwhichare two-stagedecays.Thisimpliesthatathree-stagesequenti aldecayisevidencefor underlyingphysicsBSM. 1.3Denitions Agivensignaturehasadvantagesanddisadvantages.Forexa mple,thechoiceof includingjetsandleptonsinasignaturedemonstratesthis trade-o.Weassumethata 14

PAGE 15

DCBA j ` ` DC A j ` ` AB Figure1-3.Weconsideredthesetwo\model-independent"de caydiagrams. A B C and D representsomeheavyresonanceswithunknownmassesandspi ns.Decay scenarioB)isanalternativetodecayscenarioA)inwhich m B >m C .We appliedouranalysistothejet( j ),\near"lepton( l n ),and\far"lepton( l f ), andweallowedtheverticestohavearbitrarycouplingstoth edierent chiralitiesofthesethreenal-stateparticles.Werestri ctedourspinanalysisto A)withaspin-1/2jet. 2-to-2subprocessofahadroniccollisionismostlikelytoc ontainQCD-chargedobjects inthenalstate.Soasignaturethatincludesjetscanprovi dehigherluminosity(i.e. morestatistics)comparedtoapurelyleptonicsignature.H owever,duetothecomposite structureofhadronsandtherelativestrengthoftheQCDcou pling,asignicantamount ofhadronicbackgroundradiationisexpectedinahadroncol lision,makingtheseparation betweenbackgroundjetsandsignaturejetsambiguous.Also ,thekinematicsofthe underlyingparton(i.e.quark)thatwewouldliketouseinou ranalysismaybequite dierentfromthekinematicsofthejetthatweassociatewit hthatparton.So,purely leptonicsignaturesaretypicallymorereliableandeasier tointerpretthansignatures thatincludejets.WecompromisedbyassumingthattheQCD-c hargefromthehadronic collisionresultsinasingleinitialhadronicjetandaQCDneutralintermediateresonance. Then,weselectedeventsinwhichtheQCD-neutralstatedeca ystotwoleptonsandsome otherQCD-neutralparticlethatweignored(e.g.aparticle ofdarkmatter,DM).So,in termsofFig. 1-3 D isQCD-charged,and C B ,and A areallQCD-neutral. Weimaginedtheexistenceofasequentialdecayofsomeheavy particle, D ,that producesasinglemasslessjetandtwomasslessleptons(Fig 1-3 ).Weconsideredajet fromtherstdecayinthedecaychainbecauseweexpectthatQ CDhardprocessesare 15

PAGE 16

dominantattheLHC,sothat D carriescolor.Weconsideredthesubsequentdecayof C totwoopposite-sign-same-ravor(OSSF)leptonsinorderto copewithbackground. Indeed,ourrestrictiontoOSSFwasamodelassumption.Howe ver,thisplausiblechoice wasmadeonlyforconcretenessofthediscussionanddoes not representafundamental limitationofourmethods;thebasicideaappliestothegene ralcasewherethevisible particlesareany3SMfermions.Ofcourse,theoccurrenceof suchadecayprocessisnot guaranteed,butwebelievethatitprovidesagoodcombinati onofsimplicity,plausibility, anddistinctiveness. Ingeneral,welimitedouranalysistothetwoindependentma sshierarchiesshownin Inequalities 1{1 m D >m C >m A and m B >m A (1{1) Wealwaysassumedpositivemasses,andthisassumptionisim plicitinallofour expressions.Inthesehierarchies, A isalwaysthelightestresonanceinthechain. Kinematically,thesemasshierarchiesalwaysallowthedec ayfrom D to C ,andthe subsequentdecayfrom C to A ;however,theintermediatedecayfrom C to B isnot necessarilyallowed.Werefertoanoccurrenceofthecascad edecayinwhichthedecay from C to B does occurasa\three-stage"or\on-shell"scenario(Fig. 1-3 A).Werefer toanoccurrenceofthecascadedecayinwhichthedecayfrom C to B does not occur asa\two-stage"or\o-shell"scenario(Fig. 1-3 B).Thereferencetothe\shell"refersto themassshellof B :weassumedthat m B m C wouldallowonlyatwo-stagescenario(in which B iso-shell).Whilewerecognizethatitismoremeaningfult oreservethenames \three-stage"and\two-stage"torefertothedierentkind sof decaytopology andto reservethenames\on-shell"and\o-shell"torefertothed ierentkindsof massspectrum ,weusedthenamesforboththetopologyandthespectruminte rchangeably,and wemadenoattemptataconsistentusage.Indeed,whileit is physicallymeaningfulto consideran\o-shell" spectrum thatexhibitsacascadedecaywiththe\three-stage" 16

PAGE 17

Table1-1.Somerepresentativecomparisonsofournotation R ij ,withtwoother notationsforsquaredmassratiosthatappearintheliterat ure[11,12].Weused the R ij notationforthemassdetermination,andweusedthe x y z notationfor thespinanalysis. basicexpression R ij Ref.[11]Ref.[12] m 2A =m 2B R AB R A z m 2B =m 2C R BC R B y m 2C =m 2D R CD R C x m A =m C p R AC p R A R B p yz m B =m D p R BD p R B R C p xy m A =m D p R AD p R A R B R C p xyz topology (i.e.thereexistsaFeynmandiagramwithan o-shell intermediateparticle),we ignoredsuchcasesinordertoachieveamoremodel-independ entanalysis. Wefoundaconvenientmassparametrization(Equation 1{2 )forthemassdetermination (Chapter 2 )partofouranalysis,whichisessentiallyageneralizatio nofthenotationused in[11]. R ij m 2i m 2j where i;j 2f A;B;C;D g (1{2) Weadoptedasimpleralternativenotation(Equation 1{3 )forourspindetermination (Chapter 3 ),asusedin[12]. x m 2C m 2D y m 2B m 2C z m 2A m 2B (1{3) Table 1-1 showsthecomparisonofournotationtothatof[11,12]. Forourmassdeterminationanalysis(Chapter 2 ),wedened11Regionsofthe inputmassparameterspacebyextendingthedenitionsin[1 1]toincludetheo-shell scenario.Aslicethroughparameterspaceataconstant m C and m D (Fig. 1-4 )shows theseRegionsintermsof R AB and R BC (or,equivalently,asliceatconstant x showsthese Regionsintermsof y and z ).ThebordersoftheseRegionsdependon m C and m D only through R CD .WenametheseRegionswithanorderedpairofwholenumbers, ( N jll ;N jl ), whosemeaningisbasedonthemethodofkinematicendpoints. SimilarlytoRef.[11], weassignedthesenumbersarbitrarilyaccordingtoalgebra icdenitions(Equations 17

PAGE 18

Figure1-4.Arepresentationofthe11Regionsofinputmassp arameterspace,( N jll N jl ), ataxedvalueof R CD =0 : 3.TheseRegionsrepresenttheconditionson Equations 2{7 through 2{11 ,andtheyarecolor-codedtomatchFigs. 2-2 2-3 2-5 2-7 2-8 ,and 2-10 .TheshapesofthefourcoloredRegions, N jl ,are independentof R CD 1{4 and 1{5 ),whichpartitionthemasshierarchy(Equation 1{1 )intomanydierent sub-hierarchies.Thegeneralhierarchy(Equation 1{1 )isimpliedinthesedenitions (Equations 1{4 and 1{5 ). N jll = 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: 1 ; p m A m D >m C >m B 2 ;m C q m A m D >m B 3 ;m C >m B > p m A m D 4 ;m C > p m A m D >m B >m C q m A m D 5 ;m B >m C >m A > m 2C m D 6 ;m B >m C > p m A m D (1{4) N jl = 8>>>>>>><>>>>>>>: 1 ;m 2B m 2A >m 2C m 2B > 0 2 ;m 2B m 2A > m 2A m 2B ( m 2C m 2B ) > m 2A m 2B ( m 2B m 2A ) 3 ; p m A m C >m B 4 ;m B >m C (1{5) 18

PAGE 19

Intermsofthesquaredmassratios(Equation 1{2 ),theseconditionscanbewrittenas Equations 1{6 and 1{7 N jll = 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: 1 ;R AB >R AC >R CD 2 ; p R AD >R BC 3 ;R CD >R BD >R AB 4 ;R BC > p R AD \ R AB >R BD \ R CD >R AC 5 ;R BC > 1 >R AC >R CD 6 ;R BC > 1 >R CD >R AC (1{6) N jl = 8>>>>>>><>>>>>>>: 1 ; 1 >R BC > (2 R AB ) 1 > 0 2 ; 1 > (2 R AB ) 1 >R BC >R AB 3 ; 1 >R AB >R BC 4 ;R BC > 1 >R AC (1{7) Theon-shellscenariooccursif R BC < 1,andthisoccursinthe N jl =1 ; 2 ; 3Regions(the green,magenta,andcyanRegionsinFig. 1-4 ,respectively),whicharecollectivelythe sameasthe N jll =1 ; 2 ; 3 ; 4Regions,andthesearetheRegionsconsideredin[11].The o-shellscenariooccursif R BC > 1,andthisoccursinthe N jl =4Region(theyellow RegioninFig. 1-4 ),whichiscollectivelythesameasthe N jll =5 ; 6Regions[13]. Restrictingthespinanalysistothethree-stagescenario( Fig. 1-3 A),weidentiedthe generalformofthetwotypesofBSMvertexfunctions,showni nEquations 1{8 and 1{9 F S f = g L P L + g R P R (1{8) F V f = r ( g L P L + g R P R )(1{9) f representsamasslessSMfermion(jetorlepton), F representsamassivefermion, S representsamassivescalarboson,and V representsamassivevectorboson. P L and P R 19

PAGE 20

Table1-2.Thesixspinassignmentsoftheheavyparticles D C B and A inthedecay chainofFig. 1-3 A.ThelastcolumngivesonetypicalSUSYorUEDexample. SpinsDCBAExample SFSFS calar F ermion S calar F ermion~ q ~ 02 ~ ` ~ 01 SFVFS calar F ermion V ector F ermion ? FSFSF ermion S calar F ermion S calar q 1 Z H ` 1 r H FSFVF ermion S calar F ermion V ector q 1 Z H ` 1 r 1 FVFSF ermion V ector F ermion S calar q 1 Z 1 ` 1 r H FVFVF ermion V ector F ermion V ector q 1 Z 1 ` 1 r 1 representprojectionoperatorsforleft-handedandrighthandedfermions,respectively, andtheyactontheSMfermionwave-functiontotheright.For example,if f isaDirac fermion,then P L = 1 2 (1 r 5 ).ThevertexfunctionsmaybeHermitian-conjugated, dependingonthematrixelementofthephysicalprocess. g L and g R representthedetails ofthevertexfactor,andsothese\couplingconstants"esse ntiallyparametrizeour ignoranceofthedynamicsoftheparticularphysicsmodel.S pecically,wename g c for thedecayvertexfrom D to C g b forthedecayvertexfrom C to B ,and g a forthe decayvertexfrom B to A SincetheSMparticlesinFig. 1-3 Aareallspin-1/2fermions,theparticles A B C and D mustalternatebetweenfermions( F )andbosons( S or V )inordertomaintain Lorentzinvariance.Table 1-2 listsbynamethe6spinassignmentsthatwestudiedforour spindeterminationtechnique,whichwerealsoconsideredi n[14,15].Figure 1-5 showsa decisiontreerepresentingthe6spinassignmentsthatweco nsidered.Oursetupfollows closelytheconventionsofRefs.[12,14{18]. So,ourstartingpointwastoessentiallyparametrizeourig noranceofthematrix elementofthedecaychainintermsoffourmasses,six(gener allycomplex-valued) couplingconstants,andthevariouswaystoassignspinstot heheavyresonances.We emphasizethattheseparametersrepresent eective parameters;theydo not necessarily representparametersthatwouldappearinsomeBSMLagrangi an. 20

PAGE 21

? D : SF C : FSV B : SVFF A : FFSVSV Figure1-5.Decisiontreeforchoosingthespinassignments ofparticles D C B ,and A aslabeledinTable 1-2 S impliesspin-0, F impliesspin-1/2,and V implies spin-1.Thepossibilitiesaredeterminedbythesumrulesfo rangular momentum,assumingthateachofthethreenal-stateSMpart iclesare spin-1/2,andrestrictingthespintothethreepossibiliti es: S F ,and V .We didnotconsiderthepossibilityofavector D ,whichwouldhaveaddedtwo morespinassignments,VFSFandVFVF,toTab. 1-2 1.4ExperimentalChallengesPertainingtoourAnalysis Thereareanumberofexperimentalchallengestouniquelyid entifytheparticles comingfromthecascadeofFig. 1-3 E1 JetCombinatorics. Collisioneventsatahadroncolliderareexpectedtocontai na numberof\extra"jets.Someofthesejetsmaycomefrominiti alstateradiation, othersmayoriginatefromtheoppositecascadeinthesameev ent,andtheremay alsobejetsappearingfromthedecaysofheavierparticlesi ntoparticle D .This posesacombinatoricsproblem:whichoneofthemanyjetstha tareobservedinthe eventcorrespondstothejetinFig. 1-3 ?Thejetmaybeselectedasoneofthetwo hardestjetsintheeventif D and C havealargemasssplitting,butunfortunately thejetinquestionmustalsobeusedinordertodetermineift hisisindeedthecase. Fortunately,theredoesexistatechniquetoaddressthispr oblem:themixedevent technique.Themixedeventtechniquestatisticallyremove sthewrongjets[19]from thedata.Ref.[20]successfullyappliedthemixedeventtec hniquetodemonstrate SUSYmassmeasurementfortheSPS1astudypoint.Asubtracti onbyamixed eventtechniqueisparticularlywell-suitedforourpurpos es,sinceouranalysisrelies ononlythecumulativedistributions,sothatthetheidenti cationofthecorrectjet inasingleeventisnotcritical. 21

PAGE 22

E2 Leptoncombinatorics. Theremaybeadditionalisolatedleptonsintheevent,so onemightconsiderselectingeventsexclusivelywithexact lytwoleptons.However, eventhen,itisnotguaranteedthatthosetwoleptonsarecom ingfromthesame decaychain.Fortunately,thereagainexistsatechniqueto addressthisproblem: oppositeleptonravorsubtraction(OLFS).Thisisagainast atisticaltechnique, similarinideatothemixedeventtechnique.Foroursignatu re,OLFSsolvesthe leptoncombinatoricsproblem[19]byformingthelinearcom binationof f e + e g + f + gf e + gf + e g ,inwhichtheeectsoftheuncorrelatedleptonsinthe signal(includingSMbackgroundsinvolvingtopquarks,b-j ets, W bosons,andtau jets,andevenuncorrelatedBSMbackgrounds)cancelout. E3 Quark-antiquarkjetambiguity. Assumingthat A isneutral,thechargeofthe jetinFig. 1-3 isxedbythechargeof D .However,sincethechargeofajetis diculttodetermine 2 ,weassumedthatwecouldnotdeterminethechargeof D Soourinvariantmassdistributionsactuallyrepresentthe sumoftheindividual contributionsfromboth D 'sandantiD 's. Weassumedthatthejetandleptoncombinatoricshavebeenap propriatelyresolved. Weincorporatedthe\quark-antiquark"ambiguityintooura nalysis,showingthatthe distinctionisunnecessaryformassdeterminationandcana lsobeaccountedasan additionalmodelparameterforspindetermination. 2 If q isaheavyravor,thenthedistinction can bemade(statistically).Weignoredthis possibilityinordertodemonstratethatourmethodworksev enintheworstcasescenario ofjet-chargeambiguity. 22

PAGE 23

CHAPTER2 MASSDETERMINATION 2.1TheMethodofKinematicEndpoints WiththeimminenceofdatafromtheLHCtherehavebeennumero usdevelopments ofmassdeterminationtechniques[11,13,19,21{61].Ourco ntributiontomassdetermination appliestothetwodierentscenariosshowninFig. 1-3 .Ourgoalwastoimprovethe methodofkinematicendpointstodeterminetheunknownmass es,either f m D ;m C ;m B ;m A g intheon-shellscenario,or f m D ;m C ;m A g intheo-shellscenario.Itmaybepossibleto determine m B evenintheo-shellscenario(Fig. 1-3 B)[33].However,anumberof inruences,suchasthespinsoftheheavyresonancesandtheb ackgrounds,plaguethis determination,andwepreferredouranalysistobeinsensit ivetotheseinruencesby relyingononlyabsolutekinematiceects. Weappliedthemethodofkinematicendpointstovariousrepr esentativespectra, usingtheinvariantmassesofthefourdierentcombination softhejetandtwoleptons f m ll ;m jll ;m jl n ;m jl f g .Weassumedthattwooftheinvariantmasses, m ll and m jll ,are unambiguouslyobservable,buttheothertwocombinationsa reambiguousbecausethe decayverticescorrespondingto l n and l f cannotbedeterminedonanevent-by-event basis.Onepossibilityistodistinguishthetwoleptonsbas edontheircharges,which leadstoinvariantmassdistributionsthatareparticularl ywell-suitedtospinstudies [12,14,62,63].However,formassdeterminationweusedthe populareventvariables denedinEquations 2{1 and 2{2 [11,26]. m jl ( lo ) smallerof m jl n ;m jl f foreachevent(2{1) m jl ( hi ) largerof m jl n ;m jl f foreachevent(2{2) Themethodofkinematicendpoints,whenappliedtoourdecay chain,beginswiththe setoffourobservableslistedinExpression 2{3 m maxll ;m maxjll ;m maxjl ( lo ) ;m maxjl ( hi ) (2{3) 23

PAGE 24

Inusingthemethodofkinematicendpoints,oneassumesthat thesefourobservables (endpoints)dependonthemasses, m A m B m C and m D ,andthenexpectsthatthe expressionsfortheendpointsintermsofthemassescan,inp rinciple,beinverted[64]. 1 Unfortunately,theinversionoftheexpressionsforthefou rendpointsalonecanbe ambiguous[31,64{71],andwecategorizevariousreasonsfo rtheambiguity. 1.Theremaynotbefourindependentexpressionsforthefour measurements (Expression 2{3 )intermsofthefourmasses.Indeed,inRegions(2,3),(3,2) and(3,1)ofFig. 1-4 ,theseexpressionsexhibitthedependenceshowninEquatio n 2{4 [31]. m 2maxjll = m 2maxjl ( hi ) + m 2maxll (2{4) Inthiscase,thefourmeasurements(Equation 2{3 ),whenthoughtofasfour constraintsonthe4-dimensionalmassspace,allsimultane ouslyintersectalongan extendedcurveratherthanatadiscretepointsolution[37] .Soonemustsupplement theseconstraintswithanadditionalmeasurementconstrai ntthatwillselecta particularpointalongthiscurve.Thedistributionof m jll subjecttotheconstraint inEquation 2{5 providesa lower kinematicendpoint(i.e.\threshold"), m minjll ( > 2 ) thathasbeensuggestedforthispurpose[26]. m minjll ( > 2 ) m minjll s.t. m 2ll > 1 2 m 2maxll (2{5) ( istheanglebetweenthetwoleptonsintherestframeof B .)Wesimplyprecluded theambiguityinEquation 2{4 byextendingtheoriginalsetof4measurements (Expression 2{3 )tothesetofvemeasurementslistedinExpression 2{6 m maxll ;m maxjll ;m maxjl ( lo ) ;m maxjl ( hi ) ;m minjll ( > 2 ) (2{6) 2.Thevemeasurements(Expression 2{6 )inevitablycomewithsomeexperimental errors,sothatwithinthoseexperimentaluncertainties,t woormoresolutionsare possible[31,32,65].Evenwith300fb 1 ofdataattheLHC,theexperimental uncertainties(nitedetectorresolution,statisticalan dsystematicerrors,etc.) willstillallowtwosolutions[72].Weshowanexampleofthi sambiguityinthe SPS1acolumnsofTab. 2-1 .Theexperimentalresolutionshouldimprovewith timesothatthisambiguitymayeventuallyberesolved.Fore xample,increasing 1 Incidentally,wealsorecentlydevelopedamassdeterminat iontechniquethatwe callsubm T 2 [59].Thesubm T 2 techniquecanbeusedtodeterminemassesinaneven shorterdecaychain,undercertainassumptionsinaddition totheonesthatwehavemade regardingthemethodofkinematicendpoints. 24

PAGE 25

theintegratedluminosityreducesthestatisticalerror.I tisalsoworthnotingthat Ref.[31]conservativelyassignedaratherlargesystemati cerrorforthe m minjll ( > 2 ) measurement,sincetheanalyticalshapeofitsedgewasunkn ownatthetime.The shapewasafterwardderivedin[36],sothatnowallvemeasu rements(Expression 2{6 )canbeconsideredonequalfooting. 3.Discreteambiguitiesariseduetotheverynatureofthema thematicalproblem. Theexpressionsfortheendpointsintermsofthemassesare piecewise-dened functions,andthemethodofkinematicendpointsprovidesn ocriteriatodetermine whichdenitionistherelevantone.Themethodofkinematic endpointsrequires onetoconsiderallpossibilities,obtaineachsolution,an dtestforconsistency attheveryend.So,evenintheidealcaseofaperfectexperim ent,whichwould yieldresultsforallvemeasurements(Expression 2{6 )withvanishingerrorbars, theremaystillbemultiplesolutionstotheinversion.Weid entiedthespecic circumstanceswhenthistakesplace.Weemphasizethatthes ediscreteambiguities arisedueto mathematicallyidentical valuesfor allve observablesformorethan onediscretesetofinputmasses.Therefore,noimprovement sintheexperimental resolutionareabletoresolvethisduplication.Twoexampl esofasimilarkindof ambiguityaregivenin[66],buttheiranalysisusesonlyfou routoftheveavailable measurements(Expression 2{3 ).Table 2-1 showsthattheinclusionof m minjll ( > 2 ) resolvestheambiguitiespresentedin[66].Incontrast,we useallvemeasurements (Expression 2{6 )andwestillndsomepartsofthemassparameterspaceinwhi ch theambiguityoccurs,namelyinRegions(2,3),(3,2),and(3 ,1)ofFig. 1-4 ,precisely theRegionsinwhich m minjll ( > 2 ) isrequiredtoresolvetheambiguityduetoEquation 2{4 Numerical examplesofaduplicationsimilartoourshavepreviouslybe en presentedin[64],andour analytical formulashelptounderstandthereasonforthe duplicationpresentedthere. 2.1.1ForwardFormulas Forconvenience,weusethelowercaseletters a b c d ,and e torepresentthe kinematicendpointvalues(nottobeconfusedwiththeupper caseletters A B C and D thatlabeltheunknownresonances).Theendpointsaregiven intermsoftheinputmasses accordingtoknownexpressions(Equations 2{7 through 2{11 )[13,19,22,24,29,31]. N jll and N jl areRegionlabelsthataredeterminedbythespectrum(Equat ions 1{4 and 1{5 ). a m 2maxll = 8><>: m 2D R CD (1 R BC )(1 R AB ) ;N jl =1 ; 2 ; 3 m 2D R CD (1 p R AC ) 2 ;N jl =4 (2{7) 25

PAGE 26

m v eventcountperbinm v p E 2 v j ~p v j 2 m v =0 m minv ( c ) m maxv Figure2-1.Anarticialsingle-variablehistogramtodemo nstratethemeaningofa kinematicendpoint, m maxv ,andthreshold, m minv ( c ) ,neglectingexperimental ambiguities,suchasbackground,thatwouldobscuretheloc ationsofthese points. v standsforagenericcombinationofobservableparticles,i .e. v 2f ``;j``;j` ( lo ) ;j` ( hi ) g .( c )standsforagenericrestrictiononthesample (e.g.Equation 2{5 ).Themethodofkinematicendpointsusestheextreme variables m maxv and m minv ( c ) ,ignoringtheshapeofthedistributionat intermediatevaluesof m v b m 2maxjll = 8>>>>>>><>>>>>>>: m 2D (1 R CD )(1 R AC ) ;N jll =1 ; 5 m 2D (1 R BC )(1 R AB R CD ) ;N jll =2 m 2D (1 R AB )(1 R BD ) ;N jll =3 m 2D 1 p R AD 2 ;N jll =4 ; 6 (2{8) c m 2maxjl ( lo ) = 8>>>><>>>>: m 2D (1 R CD )(1 R BC ) ;N jl =1 m 2D (1 R CD )(1 R AB )(2 R AB ) 1 ;N jl =2 ; 3 1 2 m 2D (1 R CD )(1 R AC ) ;N jl =4 (2{9) d m 2maxjl ( hi ) = 8>>>><>>>>: m 2D (1 R CD )(1 R AB ) ;N jl =1 ; 2 m 2D (1 R CD )(1 R BC ) ;N jl =3 m 2D (1 R CD )(1 R AC ) ;N jl =4 (2{10) 26

PAGE 27

e m 2minjll ( > 2 ) = 8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>: 1 4 m 2D ( (1 R AB )(1 R BC )(1+ R CD )+2(1 R AC )(1 R CD ) (1 R CD ) p (1+ R AB ) 2 (1+ R BC ) 2 16 R AC ) if N jl =1 ; 2 ; 3 1 4 m 2D (1 p R AC ) 2 R CD (1 p R AC ) +(1 R CD ) 3+ p R AC p 1+ R AC +6 p R AC if N jl =4 (2{11) Equations 2{7 through 2{11 arepiecewise-dened:theexpressionsfortheendpoints dependontheRegionofmassparameterspace(Fig. 1-4 )inwhichthespectrumoccurs. UsingEquations 2{7 2{8 and 2{10 ,itiseasytoverifyEquation 2{4 (writtenin simpliednotationas b = a + d )inRegions(3,1),(3,2)and(2,3).Therefore,onemustrely ontheadditionalmeasurementof e (Equation 2{11 )inordertoobtainadiscretesolution intheseRegions. Noticetheabsenceofthe\ B "indexintheexpressionsfortheendpointsif N jl =4 or N jll =5 ; 6.Thissimplyimpliesthat,if B iso-shell,thentheexpressionsforthe endpointsareindependentof m B 2.1.2InversionFormulas Wederivedtheexactanalyticalinversionformulasfor all oftherelevantparameter spaceRegionsofFig. 1-4 .Untilnow,inversionformulaehadbeenderivedforonly6of the 11regions,namely(1,1),(1,2),(1,3),(4,1),(4,2)and(4, 3),andeventhesedidnotinclude the e measurement[31].Notethat,ingeneral,thesystemappears tobeover-constrained sincetherearefourunknowns( m A m B m C and m D )intermsofvemeasurements( a b c d and e ). TheapproachofRef.[31](whichconsideredonlytheon-shel lcaseinFig. 1-3 A)was touseonly a b c ,and d .Onereasontoneglect e wasthatthe\forward"expressionfor e 27

PAGE 28

(Equation 2{11 )appearstoocomplicatedtouseinananalyticalinversion. However,there aretwoproblemswiththisapproach: Itcannotproduceadiscretesolutioninthethreeon-shellR egions(3,1),(3,2)and (2,3),duetoEquation 2{4 ItleadstodiscreteambiguitieseveninotherRegions,exem pliedbySU1andSU3 inTab. 2-1 Therefore,inordertoobtaininverserelationsthatareval idforanyallowedspectrum (Equation 1{1 ),thethreshold m minjll ( > 2 ) must beused,andoneof a b or d ,canbe eliminated.Wedroppedthe b measurementfromourlist(Expression 2{6 ),leaving thefourmeasurementsshowninExpression 2{12 a m 2maxll ;c m 2maxjl ( lo ) ;d m 2maxjl ( hi ) ;e m 2minjll ( >= 2) (2{12) Sincewedidnotrelyonthe b measurementinourinversion,thelineardependence between a b and d (Equation 2{4 )wasneveranissue.Thesame4endpoints(Equation 2{12 )canbeusedinallparameterspaceRegions(Fig. 1-4 ).Also,weonlyneededto considerthe4Regionsaccordingto N jl (the4color-codedRegionsofFig. 1-4 )forour inversion.Theseregionscanbedenedintermsofonlytwoma ssparameters, R AB and R BC Wefoundaconvenientformfortheinversionformulaethatap pliestoall4Regions. m 2A = G N jl N jl 1 N jl 1 r N jl 1 (2{13) m 2B = G N jl N jl 1 N jl 1 r N jl (2{14) m 2C = G N jl N jl 1 N jl r N jl (2{15) m 2D = G N jl N jl N jl r N jl (2{16) Thisformprovidesmodularitytotheinversionformulae(wh ichisconvenientforcoding theinversionformulaeintoFORTRAN,forexample).Thequan tities G N jl N jl N jl and r N jl areparticularcombinationsofthemeasuredendpointsgive ninEquations 2{17 28

PAGE 29

through 2{20 .TheydependontheRegion( N jl ),justlikethe\forward"expressionsfor theendpointsintermsoftheinputmasses. G 1 g (2 d g ) 2 c ( d g ) g 1 a + G 1 G 1 1 d G 1 r 1 c G 1 (2{17) G 2 g (2 d g )( d c ) g ( d c )+2 c ( d g ) 2 a + G 2 G 2 2 d G 2 r 2 c d c (2{18) G 3 ( g (2 d g ) 2 c ( d g )) d gd +2 c ( d g ) 3 a + G 3 G 3 3 c ( d + G 3 ) dG 3 r 3 d G 3 (2{19) G 4 d + g + p (2 d g ) g 4 a + G 4 G 4 4 r 4 d + G 4 2 G 4 (2{20) Wesimpliedtheexpressionsfor G N jl byrealizingaparticularcombinationofendpoints, Equation 2{21 g 2 e a (2{21) Awordofcautionisinorderregardingtheo-shellscenario (Equation 2{20 ).Inthat case, m B isnotarelevantparameter,sothereareonlyreallythreeun knowns: m A m C and m D .Atthesametime,thereareonefewerindependentconstrain ts,sinceEquations 2{9 and 2{10 implyEquation 2{22 c = 1 2 d (2{22) Forthepurposeofinversion,intheo-shellcasewechoseto omit c andworkonlywith f a;d;g g ,whicharetheonlythreeendpointsappearinginEquation 2{20 .Theappearance ofthesquarerootinEquation 2{20 shouldnotbeaproblem,sinceInequality 2{23 is satisedintheo-shellscenario. 1 < d g < 2+ p 2(2{23) Thesetofanalyticalinversionformulae(Equation 2{13 through 2{16 with 2{17 through 2{21 )isourrstmainresult. 2.2DuplicationAnalysis Forconvenience,werefertothemassparameterspaceRegion forwhich N jl = i as \ R i ".Weconsideredthefourcolor-codedRegions(Fig. 1-4 ),andaskedthequestion:Is 29

PAGE 30

Table2-1.Examplesofmassambiguities[31,65,66].The\Tr ue"columnrepresentsthe solutionthatwasfoundinthesameRegionofmassparameters paceasthe \Input"spectrum.The\False"columnrepresentsasecondso lutionthatwas foundinadierentRegion.TheFalsesolutionforSPS1a( )producesve endpoints(Expression 2{6 )thataresimilartotheveendpointsproducedby theTruesolution,butonlywithintheexperimentaluncerta inties.TheFalse solutionsforSU1andSU3producefourendpoints(Expressio n 2{3 )that exactlymatchthefourendpointsproducedbytherespective Truesolutions, buttheyproduceconrictingvaluesof m minjll ( > 2 ) SPS1a( )[31,65]SU1[66]SU3[66] VariableInputTrueFalseTrueFalseTrueFalseRegion(1,1)(1,1)(1,2)(1,1)(1,3)(1,3)(1,1) m ~ 01 96.196.385.3137.0122.1118.0346.8 m ~ l R 143.0143.2130.4254.0127.5155.0411.1 m ~ 02 176.8177.0165.5264.0245.9219.0451.6 m ~ q L 537.2537.5523.5760.0743.6631.0899.9 m maxll 77.077.077.161.61.100.100. m maxjl ( lo ) 298.3298.3299.6194.194.322.322. m maxjl ( hi ) 375.6375.6375.7600.600.418.418. m maxjll 425.8425.8425.6609.609.499.499. m minjll ( > 2 ) 200.6200.6205.1143.148.247.214. itpossiblethatidenticallythesamevaluesoftheendpoint s f a;c;d;e g canbeobtained fromamassspectrumbelongingtoa dierent parameterspaceRegion R j 6 = R i ?Ifthe answerwas\yes",wethenaskedtwofollow-upquestions:rs t,exactlyinwhichpartsof R i and R j doesthisduplicationoccur,andsecond,cantheotherendpo intmeasurement, b ,resolvethistwo-foldambiguity? WeconsideredEquations 2{7 and 2{9 through 2{11 asamap F i ofthecorresponding parameterspaceregion R i ontothespaceofvaluesofthekinematicendpoints. R i F i 7!f a;c;d;e g (2{24) Similarly,Equations 2{13 through 2{21 provideaninversemap F 1 j fromthespaceof kinematicendpointsbackontothemassparameterspace. f a;c;d;e g F 1 j 7!R j (2{25) 30

PAGE 31

Thecompositionofthetwomapsisatransformationofmasspa rameterspaceintoitself. T ij F 1 j F i (2{26) R i T ij 7!R j (2{27) For i = j ,thecompositemapisabijectiveidentitymap.Forthethree on-shellcases, i =1 ; 2 ; 3, i 6 = j ,themappinghasthegenericformshowninEquations 2{28 through 2{31 R 0 AB = f AB ( R AB ;R BC )(2{28) R 0 BC = f BC ( R AB ;R BC )(2{29) R 0 CD = f CD ( R AB ;R BC ;R CD )(2{30) m 0D = m D f D ( R AB ;R BC ;R CD )(2{31) Oneimportantfeatureofthe T ij map(Equation 2{28 through 2{31 )isthatthetwo transformedvalues, f R 0 AB ;R 0 BC g ,dependononlythetwooriginalvalues, f R AB ;R BC g regardlessoftheothertwovalues, f m D ;R CD g .Noticethat R AB and R BC areprecisely theparametersdeningthefourregions R i (Fig. 1-4 ).Therefore,forthepurposesofour duplicationanalysisitwassucienttoconsiderthesimple rtransformationofEquations 2{28 and 2{29 only,insteadofthemoregeneralmappinggivenbyallfourEq uations 2{28 through 2{31 Foranontrivialmapping,ourdenitionsrequirethattheob tainedvalues, f R 0 AB ;R 0 BC g belongto R j ,whichisnotautomaticallyguaranteedbyEquations 2{7 2{9 through 2{11 and 2{13 through 2{21 .Inordertondallsuchoccurrences,weconsideredallposs ible transformations T ij with i 6 = j .Webeganwiththeon-shellcase( i;j =1 ; 2 ; 3),forwhich thereare6possiblemappings T ij .Forthepurposeofndingtheduplicatedportionof parameterspace,itwassucienttoconsideronly3ofthem,w hichwechoseas T 13 T 23 and T 21 .ThecorrespondingresultsareshowninFigs. 2-2 and 2-3 31

PAGE 32

Figure2-2.Themaps T 13 : R 1 7!R 3 (toptwopanels)and T 23 : R 2 7!R 3 (bottomtwo panels).Inbothcasesthetargetregion R 3 isshadedincyan.Under T 13 ,the green-shadedregion ABD inthetopleftpaneltransformsintothe green-hatchedregion A 0 B 0 D 0 ofthetoprightpanel.Under T 23 ,the magenta-shadedregion BCD inthebottomleftpaneltransformsintothe magenta-hatchedregion B 0 C 0 D 0 ofthebottomrightpanel.Inbothcases,the transformed(primed)regionfallscompletelywithinthebo undariesofthe intendedtarget( R 3 ),implyingduplication. Fig. 2-2 showstheimagesets T 13 ( R 1 )(toptwopanels)and T 23 ( R 2 )(bottomtwo panels),whileFig. 2-3 showstheimageset T 21 ( R 2 ).Thecolor-shadedregionsintheleft andrightpanelsexhibitthedomains R i andranges R j Theimagesetsoccurwherethe cross-hatchedregionsintherightpanelsoverlaptherange s.Forexample,inFig. 2-2 T 13 mapsthewholegreen-shadedregion ABD ontheleftintothegreen-hatchedregion A 0 B 0 D 0 ontheright,whileinFig. 2-2 (Fig. 2-3 ) T 23 ( T 21 )mapsthewholemagenta-shaded region BCD ontheleftintothemagenta-hatchedregion B 0 C 0 D 0 ontheright.Duplication occurswhenevertheimagesetisnonempty.Weseethatduplic ationoccursinthecase of T 13 and T 23 ,butnotfor T 21 .(Althoughimagepointsof T 21 ( R 2 )thatareonopposite 32

PAGE 33

Figure2-3.ThesameasFig. 2-2 ,butforthemap T 21 : R 2 7!R 1 ,wheretheintended targetisthegreen-shadedregion R 1 .Under T 21 ,themagenta-shadedregion BCD intheleftpaneltransformsintothemagenta-hatchedregio n B 0 C 0 D 0 of therightpanel.Theimage B 0 C 0 D 0 hasnooverlapwithitsintendedtarget R 1 exceptalongthe BD = B 0 D 0 boundary,whichisinvariantunderthe T 12 transformation. sides,butclosetotheboundaryline BD ,giverathersimilarvaluesforthemeasured kinematicendpoints f a;c;d;e g .) Atthisstageoftheanalysis,wewonderedwhethertheresult (Fig. 2-2 )would besucienttoprovetheexistenceofduplication.Indeed,F ig. 2-2 toldusnothing abouttheremainingtwoparameters R CD and m D andmorespecicallyabouttheir transformedvalues R 0 CD and m 0D underthemappings T 13 and T 23 .However,wefound thatallmappingsof R CD and m D leadtoallowedvaluesof R 0 CD and m 0D .Therefore,the duplicationexamplesshowninFig. 2-2 trulyrepresentaproblematictwo-foldambiguity inthemethodofkinematicendpoints.Wethenperformedasim ilaranalysisinvolvingthe o-shellregion R 4 andfoundnooccurrenceofduplication,whichwasnotsurpri sing,since theo-shellcaseismorerestricted,duetoEquation 2{22 .Wesummarizeourresultas follows: 33

PAGE 34

Foreverypointwith R AB
PAGE 35

f a;b;c;d;e g canalsobeobtainedfromadierentparameterspacepointwi th R BC
PAGE 36

Figure2-4.Theminimumvalue R min CD ( R BC ;R AB )requiredforduplication,asafunction of R BC and R AB .Thewhiteasterisks(circles)marktheduplicatepairof points P 31 and P 23 ( P 32 and P 0 23 )inTable 2-2 .Duplicationdoesnotoccurin thesolidredregion. chose m D =500GeV(anotherniceroundnumber).Theresultingmasses m A m B and m C arereadilycomputedintermsof m D andthemassratios.Wenamedtheresulting spectrum\studypoint P 31 ",whichislistedinthesecondcolumnofTable 2-2 .Given P 31 weusedthetransformation T 13 toobtainthematchingspectrumin R 3 ,whichislistedin thethirdcolumnofTable 2-2 underthenameof\studypoint P 23 ".Thecorresponding pointismarkedwiththewhiteasteriskintheupperrightcor nerofFig. 2-4 .Inthecase ofEquation 2{33 ,wefollowedasimilarprocedure,exceptthatwestartedwit hapointin region R 2 (indicatedwithawhitecircleinFig. 2-4 )andthenusedthe T 23 transformation toobtainthecorrespondingpointin R 3 (alsoindicatedwithawhitecircleinFig. 2-4 ). Thetworesultingmassspectra(called P 32 and P 0 23 )aregiveninthefourthandfth columnsofTable 2-2 ,respectively. Additionalconstraintsareneededinordertoresolvetheam biguity.Oneoption istoconsideralongerdecaychain,whichwouldyieldsevera ladditionalendpoint measurements.Forexample,thedecaychainsinFig. 1-3 maybeginwithanevenheavier 36

PAGE 37

Table2-2.Twoexamplesof exact duplicationunderEquations 2{32 and 2{33 .Thepairs ofstudypoints P 31 and P 23 ,aswellas P 32 and P 0 23 ,exhibitidenticalvaluesfor allvekinematicendpoints(Expression 2{6 ).Point P 31 belongsto R 1 ,point P 32 belongsto R 2 ,andpoints P 23 and P 0 23 belongto R 3 .Thelasteightrows givetheendpointmeasurementsthatareavailablefromthet wo-dimensional distributions( m 2jl ( lo ) ;m 2jl ( hi ) )(Equations 2{41 through 2{43 )and( m 2ll ;m 2jll ) (Equations 2{62 through 2{65 and 2{67 ). R 1 $R 3 R 2 $R 3 (3,1)(2,3)(3,2)(2,3) Variable P 31 P 23 P 32 P 0 23 m A (GeV)236.643915.618126.491241.618 m B (GeV)374.166954.747282.843346.073 m C (GeV)418.331083.10447.214554.133 m D (GeV)500.001172.57500.00610.443 R AB 0.4000.9200.2000.487 R BC 0.8000.7770.4000.390 R CD 0.7000.8530.8000.824 R min CD 0.6860.8450.7740.800 m maxll (GeV) p a 145.145.310.310. m maxjll (GeV) p b 257.257.369.369. m maxjl ( lo ) (GeV) p c 122.122.149.149. m maxjl ( hi ) (GeV) p d 212.212.200.200. m minjll ( > 2 ) (GeV) p e 132.132.248.248. m maxjl f (GeV) p f 212.127.200.183. m ( p ) jl f (GeV) p p 190.112.126.115. m maxjl n (GeV) p n 122.212.173.200. m jll (+) (0)(GeV) p s 226.240.214.230. m jll (+) ( a t )(GeV) p t 263.257.374.369. m jll (+) ( a on )(GeV) p u 257.257.369.369. m jll ( ) ( a on )(GeV) p v 190.193.355.360. m jll (+) ( a o )(GeV) p w 256.243.372.367. particle(say, E ),attheexpenseofasinglenewparameter(themassofpartic le E )[32]. Alternatively,onemaysupplement(orevenreplace)Expres sions 2{6 withmeasurements fromafutureleptoncollider[65].Instead,weconcentrate donthequestion,\What additionalinformationthatisalreadypresentinthehadro ncolliderdatacanbeusedto resolvetheambiguity?" Inverygeneralterms,thekinematicsofthedecayinFig. 1-3 isgovernedbysome three-dimensionaldierentialdistribution,Expression 2{35 ,where and r aresome 37

PAGE 38

suitablychosenanglesspecifyingtheparticulardecaycon guration[12]. d 3 dddr (2{35) Throughachangeofvariables,theseanglescanbetradedfor threeinvariantmass combinationsoftheobserveddecayproductsinFig. 1-3 ,e.g. m ll ;m jl + ;m jl [15],butany genericsetofmasses f m 1 ;m 2 ;m 3 g ispossible.So,onecanreplaceExpression 2{35 with Expression 2{36 d 3 dm 1 dm 2 dm 3 (2{36) Weimaginethatthedistribution(Expression 2{36 )isobservableasathree-dimensional histogram.Itprovidesmuchmoreinformationaboutthedeca yinFig. 1-3 comparedto thesinglevariabledistributions.Onecansimplyintegrat eouttwoofthethreemassesin ordertoobtainthesinglevariabledistributionforsay, m 1 asdemonstratedinEquation 2{37 ,whichexhibitsakinematicendpoint, m max1 d dm 1 Z dm 2 dm 3 d 3 dm 1 dm 2 dm 3 (2{37) However,theintegrateddistributionignoressomeoftheor iginalinformationcontainedin Expression 2{36 .Theintegrationonasingleinvariantmassproducesatwo-d imensional (bivariate)distribution,e.g.( m 1 ;m 2 )asdemonstratedinEquation 2{38 ,andmaintains someoftheinformationthatisintegratedoutforthecorres pondingsinglevariable distributions,i.e. m 1 and m 2 separately. d 2 dm 1 dm 2 Z dm 3 d 3 dm 1 dm 2 dm 3 (2{38) Atwo-dimensionaldistributionexhibits boundarylines (orcurves)ratherthanasingle endpoint.Giventhatbivariateandtrivariatedistributio nsaremoreinformativethanthe simpleone-dimensionalhistograms,wenditsurprisingth attheyhavenotbeenused moreofteninthepreviousanalysesofmassdetermination. 38

PAGE 39

Wefoundthattheshapeofthebivariatedistribution( m 2jl ( lo ) ;m 2jl ( hi ) )canbeused toqualitativelyidentifytheRegion(Fig. 1-4 ),thusresolvingtheduplication[67,69{71]. Wealsofoundanalyticalformulasintermsoftheparameters R ij (Denition 1{2 )for theboundariesofthekinematicallyalloweddistributions ,whichcanbeusedtofurther quantitativelyimproveonthemassdetermination[71].Wei dentiedsomedenitive pointsontheboundarylines,forwhichweprovideanalytice xpressionsintermsof R ij Thesedenitivepointsaretypicallyhiddenasambiguousfe aturesoftheone-dimensional distributionsbutareeasytoseeandunderstandonthebivar iatedistributions. 2.3KinematicBoundaryLinesforthe( m 2jl ( lo ) m 2jl ( hi ) )Distribution Weexaminedthekinematicboundarylinesofthe two-variable invariantmass distributionin( m 2jl ( lo ) m 2jl ( hi ) ). d 2 dm 2jl ( lo ) dm 2jl ( hi ) (2{39) Wefoundthatthequalitativeshapeoftheboundarylinesoft hisdistributioncanbeused toresolvetheduplicationproblem,andalsothatextrameas urementscanbeobtained fromtheseboundarylines. Thevariables m jl ( lo ) and m jl ( hi ) (Equations 2{1 and 2{2 )dealwiththeambiguityin theidenticationofthe\near"and\far"leptons l n and l f inFig. 1-3 .Theshapeofthe ( m jl ( lo ) m jl ( hi ) )distributionisrelatedtothecorresponding( m 2jl n m 2jl f )distribution. d 2 dm 2jl n dm 2jl f (2{40) Inprinciple,bothdistributions(Equation 2{39 and 2{40 )dependnotonlyonthemass spectrum,butalsoonthespinsandonthechiralitiesofthec ouplingconstantsofthe particles A B C and D [14,17,63].However,thelocationandtheshapeofthebound ary linesaredeterminedbykinematicsalone,anddonotdependo nthespinandtypeof couplings.Totheextentthatwewereonlyinterestedinthes eboundarylines,itwas thereforesucienttoconsideronlypurephasespacedecays ,inwhichcasetheanalytical resultsforthedistributionswere,inprinciple,alreadya vailable[11].So,forourmass 39

PAGE 40

Figure2-5.Thegenericshape ONPF ofthebivariatedistributioninthe( m 2jl n ;m 2jl f ) plane. determinationanalysisweusetheterm\shape"torefertoth eshapeoftheboundary lines,andnottheprobabilitydistribution. Theshapeofthe( m 2jl n m 2jl f )distributionisasimpleright-angletrapezoid,illustra ted inFig. 2-5 .Point O issimplytheoriginofthecoordinatesystem.Point N (for\near") liesonthe m 2jl n axis,anditscoordinateisthemaximumpossiblevalueofthe jet-near leptoninvariantmass,givenintermsoftheinputspectruma ccordingtoEquation 2{41 n m maxjl n 2 = m 2D (1 R CD )(1 R BC )(2{41) Similarly,point F (for\far")liesonthe m 2jl f axis,anditscoordinateisthemaximum possiblevalueofthejet-farleptoninvariantmass,giveni ntermsoftheinputspectrum accordingtoEquation 2{42 f m maxjl f 2 = m 2D (1 R CD )(1 R AB )(2{42) 40

PAGE 41

Thecoordinatesofpoint P are( n;p ),where p isgivenintermsoftheinputspectrum accordingtoEquation 2{43 p m ( p ) jl f 2 fR BC = m 2D (1 R CD ) R BC (1 R AB )(2{43) Since N and P sharethesame m 2jl n coordinate n ,point P alwaysliesdirectlyabovepoint N .Atthesametime,thedenitionof p impliesEquation 2{44 ,sothatpoint P always lieslowerthanpoint F (Fig. 2-5 ). p
PAGE 42

ThecoloredregionsinFig. 2-5 showtheallowedlocationsofpoint P ,andare color-codedtomatchFig. 1-4 .ThetwowhiteareasinFig. 2-5 arenotaccessibletopoint P .Theregionwith m 2jl f >f isforbiddenaccordingtoEquation 2{44 .Similarly,thewhite triangularareaneartheorigin,denedbyEquation 2{51 isalsonotallowedaccordingto Inequality 2{52 followingfromEquations 2{41 through 2{43 m 2jl f
PAGE 43

Figure2-6.Obtainingtheshapeofthe( m 2jl ( lo ) m 2jl ( hi ) )bivariatedistributionbyfoldingthe ( m 2jl n m 2jl f )distributionacrosstheline m 2jl n = m 2jl f .Thisparticularexample appliesto R 3 .Fortheotherthreeregions,refertoFig. 2-7 stillberepresentedinFig. 2-5 ,andinfactthisrepresentationisunique:thereisasingle allowedlocationforpoint P at n = f and p =0.InFig. 2-5 thisuniquelocationis indicatedwithayellow-shadedcircle,whichcorrespondst othewholeyellow-shadedregion R 4 inFig. 1-4 .Inotherwords,intheo-shellcasewecanrandomlyassign\ near"and \far"labelstothetwoleptonsineachevent,andthenthesha pe ONF oftheresulting ( m 2jl n ;m 2jl f )distributionwillbeanisoscelesrighttriangle. Theboundariesofthedistribution(Expression 2{40 )containsomeusefulinformation: theirshapeuniquelyidentiestheparameterspaceregion( R i )andyieldsthemeasurements f n;f;p;q g giveninEquations 2{41 through 2{43 and 2{45 .Wefoundthatitisactually possibletopreserveandsubsequentlyextractthisadditio nalinformationfromthe observabledistribution(Expression 2{39 ),usingthesimpleintuitiveunderstandingofthe shapeexhibitedinFig. 2-5 .Thekeywastorealizethatthereorderingofthe( m 2jl n ;m 2jl f ) pairintothe( m 2jl ( lo ) ;m 2jl ( hi ) )pair(Equation 2{1 and 2{2 )simplycorrespondsto\folding" thetrapezoid ONPF inFig. 2-5 acrosstheline m 2jl n = m 2jl f .Thisprocedureisshown pictoriallyinFig. 2-6 ,whereforillustrationweusedanexamplefrom R 3 .PanelA showsthetrapezoidalshapeoftheoriginal( m 2jl n m 2jl f )distributionfromFig. 2-5 .The 43

PAGE 44

( m 2jl n ;m 2jl f )distributioncanbeconvertedintothe( m 2jl ( lo ) ;m 2jl ( hi ) )distributionsimplyby reinterpretingthe m 2jl n axisas m 2jl ( lo ) andthe m 2jl f axisas m 2jl ( hi ) .Fromthatpointofview, thetrapezoid ONPF inFig. 2-6 dividesintotwoadjacentregions: OQF (blue-shaded) and ONPQ (red-shaded).Withintheblue-shadedarea OQF wehave m 2jl n
PAGE 45

Figure2-7.Thegenericshapeofthebivariatedistribution ( m 2jl ( lo ) m 2jl ( hi ) )foreachofthe fourparameterspaceregions:A) R 1 ,B) R 2 ,C) R 3 ,andD) R 4 .Eachpanel showsthetypicalshape(red-shaded)oftheresulting( m 2jl ( lo ) ;m 2jl ( hi ) ) distribution,afterthe\folding"inFig. 2-6 .Hatched(unhatched)areas correspondtodouble-density(single-density).Eachpane lalsoshowsthe originallocationofthepoint P inthe( m 2jl n ;m 2jl f )plot,aswellastheallowed positionsofpoint P ,followingthecolorconventionsofFigs. 1-4 and 2-5 45

PAGE 46

Theshapeofthe( m 2jl ( lo ) ;m 2jl ( hi ) )distributionallowsustouniquelydeterminetheregion ofmassparameterspace.Forexample,thetypicalshapeforR egion R 1 ,exhibitedin Fig. 2-7 A,consistsofaright-angletriangularregion OO 0 N 0 ofdoubledensityanda right-angletrapezoidalregion N 0 O 0 PF ofsingledensity.Inthiscase,point P isdirectly observable,anditscoordinatesimmediatelyyieldthequan tities n and p denedin Equations 2{41 and 2{43 .Inaddition,onecanalsomeasurethelocation f ofpoint F alongthe m 2jl ( hi ) axis,givenbyEquation 2{42 .Thisgivesatotalofthreeindependent measurements: n p and f ,whichshouldbeorderedas n
PAGE 47

situationcorrespondstotheon-shellcasesofRegions R 2 or R 3 ,withpoint P lyingvery closetotheyellowdotrepresentingRegion R 4 .Inspiteofhavingthesameshapeoftheir boundarylines,thetwodistributionswillbequitedieren t,astheywillexhibitadierent pointdensity.Inparticular,forallthreeon-shellcases, thepurephasespaceprobability distributionisgivenbyEquation 2{56 d 2 dm 2jl n dm 2jl f = 1 n m 2jl f ( m 2jl n ) = 1 fn ( f p ) m 2jl n (for R BC < 1)(2{56) Withinthekinematicallyallowedregion,thedensityisind ependentof m 2jl f .Inthelimit p 0,Expression 2{56 becomessingularwhen m 2jl n n .Thissingularityisregularized bythewidthofparticle B andthebranchingfractionforthe C B decay.Incontrast, thecorrespondingdensityintheo-shellcaseisquitedie rent,andinparticulardoesnot exhibitsuchsingularfeatures.Furthermore,wefoundthat the( m 2jll ;m 2ll )boundarylines wouldappearquitedierentforsuchanon-shellpointcompa redtoano-shellpoint. So,theshapeofthekinematicboundarylinesofthe( m 2jl ( lo ) ;m 2jl ( hi ) )distribution uniquelyidentiestheregion,asshowninFig. 2-7 .Sincetheduplicatesolutionsthatwe foundalwaysappearintwodierentregions,thisisinprinc iplesucienttoeliminatethe wrongsolution.Also,thescatterplotsoerthepossibilit yofadditionalmeasurements, andattheveryleastameasurementofthequantity p .AscanbeseenfromTable 2-2 ,the valueof p isalreadydierentforeachpairofduplicatespectra,and, providedthatitcan bemeasuredwithsucientprecision,canalsobeusedtoremo vetheambiguity. OurconclusionsareconrmedbyFig. 2-8 ,whichshows( m 2jl ( lo ) ;m 2jl ( hi ) )scatter plotsforthefourstudypointsfromTable 2-2 .Thegureindeedshowsthateachpair ofduplicatepointshasidenticalvaluesfortheendpointso ftheseparate single-variable distributions m 2jl ( lo ) and m 2jl ( hi ) .However,theshapesofthescatterplotshaveobvious dierences.Wethereforeconcludethatthetwo-variabledi stributionin( m 2jl ( lo ) ;m 2jl ( hi ) )in principlecanresolvethetwo-foldambiguity. 47

PAGE 48

Figure2-8.Scatterplotsof( m 2jl ( lo ) m 2jl ( hi ) )forthe4studypointsfromTable 2-2 .A quadraticscalewasusedonbothaxes.Thetheoreticalkinem aticboundary linesareoutlinedwiththecorrespondingcolorforeachReg ion,followingthe colorcodingconventionsofFigs. 1-4 and 2-5 .Eachplothas10,000data points.Weassumedthatallparticles A B C and D areexactlyon-shell,and thatall4-momentaareperfectlymeasured. 48

PAGE 49

Asanalremark,wepointoutthatwhenthetwo-dimensionals catterplotslike thoseinFig. 2-8 areprojectedontotheaxestoobtainthecorrespondingonedimensional distributionsofeither m 2jl ( lo ) or m 2jl ( hi ) ,thelatteroftenexhibitsomepeculiarfeaturesthat wereclassiedaseither\feet"or\drops"inRef.[11].Theo riginofthesefeaturesisnow easytounderstandintermsofthetwo-variabledistributio n.Forexample,thescatter plotsinFigs. 2-8 Band 2-8 D,whenprojectedontothe m 2jl ( hi ) axis,bothexhibita\drop" atthe m 2jl ( hi ) endpoint,whichissimplyduetotheratupperboundary P 0 N 0 inFig. 2-7 C. Similarly,theprojectionofthedistributioninFig. 2-8 Contothe m 2jl ( hi ) axisexhibitsa \foot"extendingfrom n to f .The\foot"canbeeasilyunderstoodintermsofthegeneric shapeofFig. 2-7 B,whereitarisesfromtheprojectionofthesingledensitya rea N 0 HF 2.4KinematicBoundaryLinesforthe( m 2ll m 2jll )Distribution Inadditiontothekinematicboundarylinesofthe( m 2jl ( lo ) m 2jl ( hi ) )distribution,we alsoexaminedthekinematicboundarylinesofthedistribut ioninExpression 2{57 ,whose genericshape OVUS isshowninFig. 2-9 d 2 dm 2jll dm 2ll (2{57) Theredarearepresentskinematicallyallowedvaluesforth epairofobservables( m 2ll m 2jll ). ThekinematicboundarylinesofExpression 2{57 generallyconsistof4segments.The upper( SU )andlower( OV )curvedboundariesaresegmentsofahyperboliccurve ( OWS ),whiletheleft( OS )andright( UV )boundariesarestraightlinesegments.We describetheshapeofthe( m 2ll ;m 2jll )boundarywithparametricequationsfortheupper andlowercurvedboundaries SW and OW ,togetherwiththelocationoftheverticalline UV .Wechose m 2ll asthelineparameterdescribingthecurvedboundaries.The upper boundaryline SUW isgivenbyEquation 2{58 whilethelowerboundaryline OVW is givenbyEquation 2{59 [13]. m 2jll (+) ( m 2ll )= 1+ R CD 2 m 2ll R CD + 1 2 m 2D (1 R CD )(1 R AC ) 49

PAGE 50

Figure2-9.Thegenericshape OVUS ofthebivariatedistribution(Expression 2{57 )in the( m 2ll ;m 2jll )plane. + 1 R CD 2 ( m 2ll R CD m 2D (1+ R AC ) 2 4 m 4D R AC ) 1 2 (2{58) m 2jll ( ) ( m 2ll )= 1+ R CD 2 m 2ll R CD + 1 2 m 2D (1 R CD )(1 R AC ) 1 R CD 2 ( m 2ll R CD m 2D (1+ R AC ) 2 4 m 4D R AC ) 1 2 (2{59) Theverticalstraightlinesegment UV isingenerallocatedat m 2ll = a ,where a isthevalue ofthedileptoninvariantmassendpoint(Equation 2{7 ). Sincetheexpressionfor a dependsonthemassshellconditionof B ,weintroduce separatenotationfortheendpoint a ineachofthesetwocases.Intheon-shellscenario (Fig. 1-3 A)weuse a on ,andintheo-shellscenario(Fig. 1-3 B)weuse a o .FromEquation 2{7 wedetermineEquation 2{60 ,whichisobviousonFig. 2-9 a on a o (2{60) 50

PAGE 51

EqualityinEquation 2{60 isachievedwhen R AB = R BC ,i.e.whentheon-shellspectrum happenstolieexactlyontheborderbetween R 2 and R 3 (Fig. 1-4 ).Inphysicaltermsthis meansthat m B isequaltothegeometricmeanof m A and m C ,showninEquation 2{61 m B = p m A m C ) a on = a o (2{61) Thisrepresentsanotherpotentialsourceofconfusioninex tractingthemassspectrum{ themeasurementofthedileptoninvariantmassendpoint a alonetellsusnothingabout whethertheintermediateparticle B ison-shelloro-shell[33].Inparticular,when Equation 2{61 holds,theentireareainside OWS iskinematicallyaccessible,andthe boundaryofthe( m 2ll m 2jll )distributionintheon-shellscenariocannotbedistingui shed fromtheboundaryofthe( m 2ll m 2jll )distributionintheo-shellscenario.However,we alreadydeterminedthattheshapeofthe( m 2jl ( lo ) ;m 2jl ( hi ) )distributiondistinguishesthe o-shellscenario(Fig. 2-7 D)fromtheon-shellscenario(Fig. 2-7 Athrough 2-7 C). Weidentiedseveralpointsalongthehyperbola OWS inFig. 2-9 .Point O issimply theorigin(0 ; 0)ofthe( m 2ll ;m 2jll )coordinatesystem.Point W isthetipofthehyperbola at m 2ll = a o ,wheretheupperbranch m 2jll (+) ( m 2ll )meetsthelowerbranch m 2jll ( ) ( m 2ll ).The m 2jll coordinateof W isgivenbyEquation 2{62 w m 2jll (+) ( a o ) m 2jll ( ) ( a o )= m 2D 1 R CD p R AC 1 p R AC (2{62) Point S isthe m 2jll -interceptoftheupperkinematicboundaryline m 2jll (+) ( m 2ll ).The m 2jll coordinateofpoint S isgivenbyEquation 2{63 s m 2jll (+) (0)= m 2D (1 R CD )(1 R AC )(2{63) Points U and V labeltheintersectionsoftheverticalboundary UV withtheupperand lowerhyperbolicbranches(Equations 2{58 and 2{59 ),respectively.Theysharethesame m 2ll coordinate a on ,whiletheir m 2jll coordinatesarecorrespondinglygivenbyEquations 51

PAGE 52

2{64 and 2{65 u m 2jll (+) ( a on ) (2{64) = 1 2 m 2D (1+ R CD )(1 R BC )(1 R AB )+(1 R CD )(1 R AC + j R BC R AB j ) v m 2jll ( ) ( a on ) (2{65) = 1 2 m 2D (1+ R CD )(1 R BC )(1 R AB )+(1 R CD )(1 R AC j R BC R AB j ) Point T occursatthemaximumoftheupperbranch(Equation 2{58 ).The m 2ll coordinate a t ofpoint T isgivenbyEquation 2{66 a t m 2D R CD p R AD 1 p R AD (2{66) The m 2jll coordinate t ofpoint T canbefoundbysubstitutingEquation 2{66 into Equation 2{58 ,andisgivenintermsoftheinputmassspectrumbyEquation 2{67 t m 2jll (+) ( a t )= m 2D 1 p R AD 2 (2{67) Point W isnotpartoftheactualboundaryexceptintheo-shellscen ario,orif Equation 2{60 holds.Point T maynotevenbedenedif a t < 0.Wecalculatetheslopeof theupperbranch m 2jll (+) ( m 2ll )atpoint S dm 2jll (+) dm 2ll m 2ll =0 = R CD R AC R CD (1 R AC ) (2{68) SincethedenominatorontheR.H.S.ofEquation 2{68 isalwayspositive,thesignof thederivativeisdeterminedbytherelativesizeof R CD and R AC .When R CD
PAGE 53

boundary OVUS .Intheo-shellcaseof N jll =6,point T liesontheboundary,andthe maximumvalueof m 2jll isgivenby t ,inagreementwithEquation 2{8 .However,inthe remainingthreeon-shellcases( N jll =2 ; 3 ; 4)point T isincludedonlyifitliestotheleftof the UV line(Equation 2{69 ). a t 0(2{70) Alternatively,point T willnotbeontheboundaryifEquation 2{71 issatised. ( R BC R AB R CD )( R AB R BD ) < 0(2{71) ThetwofactorsenteringEquations 2{70 and 2{71 cannotbesimultaneouslynegative:if thatwerethecase,wewouldhaveEquation 2{72 ,whichcontradictsourbasicassumption (Equation 1{1 ). R BC R AB R CD < 0 ) m 2B < m A m D m 2C R AB R BD < 0 ) m A m D =>; ) m D 0 ) ( R BC R AB R CD )( R AB R BD ) < 0 ) a t >a on (2{73) Inthiscase, T isnotpartoftheboundary.Then,themaximumvalueof m 2jll is obtainedatpoint U andisgivenbyEquation 2{64 .Sinceinthiscase R BC < R AB R CD
PAGE 54

N jll =3:Inthiscase,thesecondfactorinEquation 2{72 isnegative. R AB R BD < 0 ) R BC R AB R CD > 0 ) ( R BC R AB R CD )( R AB R BD ) < 0 ) a t >a on (2{74) Onceagain,point T doesnotbelongtotheboundary,andthemaximumvalueof m 2jll isobtainedatpoint U andisgivenbyEquation 2{64 .Thistime,however, R AB 0 R AB R BD > 0 ) ) ( R BC R AB R CD )( R AB R BD ) > 0 ) a t
2 ) ) 2 (Equation 2{11 ).Byrestrictingthe( m 2ll m 2jll )distributionstopointswith m 2ll > 1 2 a on ,i.e.totherightofthedashedline EE 0 ,thesingle-variable m 2jll distribution willexhibita lower endpoint,whosevalue e isgivenbythe m 2jll coordinateofpoint E in Fig. 2-9 .Intheon-shellcase, e isdenedbyEquation 2{76 ,and,intheo-shellcase, e is denedbyEquation 2{77 e m 2jll ( ) ( a on = 2)(2{76) e m 2jll ( ) ( a o = 2)(2{77) Inprinciple,thenewlyintroducedquantities s t u v ,and w canbedetermined fromtheobservableboundaryofthe( m 2ll m 2jll )distribution(Fig. 2-9 ),where t and w may requireextrapolation.Table 2-2 liststheirsquarerootvaluesforourfourstudypoints P 31 P 23 P 32 and P 0 23 .Asexpected,thevalueof u ismatchedidenticallyforeachpair. However,thetwootherdirectlyobservablequantities s and v dier,andinprinciple canbeusedtoresolvetheduplication.Thisisillustratedi nFig. 2-10 .UnlikeFig. 2-8 thedierencesbetweenthe( m 2ll m 2jll )scatterplotsforeachduplicatedpairareonly quantitative,andmaybediculttoobserveinpractice.Dup licationoccursonlyin 54

PAGE 55

Figure2-10.ThesameasFig. 2-8 ,butfor( m 2ll m 2jll ). regionswith N jll =2or N jll =3.Inbothcases,theshapeofthe( m 2ll ;m 2jll )scatterplotis rathersimilar:theslopeatpoint S ispositive,andtheupperboundary SU excludespoint T .Furthermore,theduplicationanalysisensuresthattheri ghtmostverticalboundary UV occursinthesamelocation a on .Anyway,wealreadydeterminedthatthe qualitative shape ofthekinematicboundaryofthe( m 2jl ( lo ) ;m 2jl ( hi ) )distributioncanresolvetheduplication. 55

PAGE 56

CHAPTER3 SPINDETERMINATION Afterconsideringthemassdeterminationofparticles A B C and D fromthe decayprocessrepresentedinFig. 1-3 A,wealsowantedtodevelopamodel-independent techniquetodeterminethespinsoftheseparticles.(Weass umedonlythethree-stage cascadedecayforthespindetermination.)Asinthemassdet ermination,weassumedthat wecouldnotobservetheenergyandmomentumof A ,sothatwecouldnotreconstruct theresonances.Forsimplicity,weassumedthatthemasses f m A ;m B ;m C ;m D g had alreadybeendetermined. Recentlytherehasbeenalotofeortondevelopingvarioust echniquesfordiscriminating amongdierentmodelscenarios[12,14{18,35,62,73{86].S inceweassumedthatthespin of A isunknown,thisgivesrisetoseveraldistinctpossibiliti esforthespinassignmentsof theheavyresonances.Evenifthespinof A wereknown,thisstillwouldnotcompletelyx thespinsoftheprecedingparticles B C and D ThelastcolumnofTable 1-2 givessometypicalexamplesinvolvingthesquarks~ q sleptons ~ ` andneutralinos~ 0i insupersymmetry,theKKquarks q 1 ,KKleptons ` 1 and KKgaugebosons Z 1 and r 1 in5D(or6D)UED[87],andthespinlessgaugebosons r H and Z H in6DUED[88].CaseSFVFwouldrequireeitherascalarleptoq uarkoragauge bosonthatcarriesleptonnumber,sowedonotprovideanexam ple[63].Nevertheless,we includedthiscaseinourstudyinordertocomparetotheresu ltsof[14,15].Weemphasize thatthesupersymmetryandUEDexamplesinTable 1-2 serveonlyasillustrations,and weneverrestrictedtheanalysistoanyparticularmodel.In particular,ouranalysisdid notrelyonanyfeaturesofthemassspectrumnorthecoupling sthatmightbeexpectedin SUSYorUED.Itisforthisreasonthatweclaimamodel-indepe ndentanalysis. Ourmaingoalwastoassessthepossibilityofdiscriminatin gbetweenthe6dierent alternativesinTable 1-2 ,usingobservableinvariantmassdistributionsofthevisi ble particles(thejetandthetwoleptons)inFig. 1-3 A.Ourmethodofspindetermination 56

PAGE 57

Table3-1.Shorthandnotationforthesixdierentspinassi gnmentslistedinTable 1-2 shorthand: S =123456 spinassignment:SFSFFSFSFSFVFVFSFVFVSFVF alsoprovidesanindependentmeasurementofcertaincombin ationsofcouplingsand mixinganglesoftheheavypartners.Forconvenience,wepro videashorthandnotation forthe6dierentspinassignments,identifyingeachonewi thanintegerfrom1to6,as showninTable 3-1 Sincewedonotknowwhetherthejetrepresentsaquarkoranan tiquark,thenwe alsodonotknowwhetherthecascadewasinitiatedbyapartic le D oritsantiparticle D .Ata p p collidersuchastheTevatron,weassumethatthefraction f of D particles producedinthedataisequaltothefraction f ofantiparticles D .However,ata pp collidersuchastheLHCweexpectanexcessofparticlesover antiparticles,butwithout knowledgeoftheprecisevalue.Therefore f ,thefractionofparticles(asopposedto antiparticles)producedfromthecollision,isinprincipl eanunknownparameter,which signicantlyaectstheobservable m jl + and m jl distributions.Previousstudiesofspin measurementshavexed f toaspecicvalueforsomecorrespondingstudypoint[ ? ]. However,wedecidedthatthiswasunjustied.Theinruenceo f f onthespinextraction wasconsideredin[77,79],where f wasleftasaroatingparameterandconsequently theextractionofthespinsbecamemuchmoredicult.Wefoll owedasimilarapproach, treating f asaroatinginputparameter. WeassumethatthethreeSMparticlesinoursignatureareall spin-1/2particles, whosecouplingstotheheavypartnersateachvertexareunkn own.Theobserved invariantmassdistributionsdependonthechiralityoftho secouplings,andthis presentsanambiguityindeterminingthespins.Agivenseto fmeasuredinvariant massdistributionscouldinprinciplebeexplainedbymoret hanonespinconguration, dependingonthechiralitiesforthefermioncouplings.Wem adethosecouplings completelyarbitrary,andparametrizedthemintermsofind ependentchiralitycoecients 57

PAGE 58

ateachvertex(Equations 1{8 and 1{9 ).Ingeneral,therearethreedierentsetsofvertex coecients f g L ;g R g ,oneateachvertex,denotedas f c L ;c R g f b L ;b R g and f a L ;a R g from lefttorightinFig. 1-3 A.Thecouplings f c L ;c R g areassociatedwiththe D C j vertex, thecouplings f b L ;b R g areassociatedwiththe C B ` n vertex,andthecouplings f a L ;a R g areassociatedwiththe B A ` f vertex.Noticethatthisparametrizationallowsthesame coecientstodescribeeitherkindofinteraction(Equatio n 1{8 or 1{9 )ateachvertex withoutdiscrimination.Thisallowedustoroattheseparam etersindependentlyofthe spinassignment.Sincewewerenotconcernedwiththeactual cross-section,butonly theshapeofthedistributions,wefounditconvenienttouni tnormalizethecouplings (Equations 3{1 ). j a L j 2 + j a R j 2 =1 j b L j 2 + j b R j 2 =1 j c L j 2 + j c R j 2 =1(3{1) Itturnsoutthatonlythe relative helicitiesofthenal-stateSMparticlesdetermine theshapesofthedistributions,soweparametrizedthe relative chiralityofeachvertex withasingletangent(Equations 3{2 ). tan a = j a R j j a L j tan b = j b R j j b L j tan c = j c R j j c L j (3{2) Since g L and g R areingeneralcomplexparameters,eachoftheSMfermionint eractionsis parametrizedbyfourrealvalues.Thenormalizationcondit ion(Equation 3{1 )eliminates onedegreeoffreedom,andEquation 3{2 identiestheinterestingdegreeoffreedom explicitlyastan .Theremainingtwodegreesoffreedom,thecomplexphasesof the couplingconstants,remainarbitraryandcannotbemeasure dfromtheinvariantmass distributionsthatweconsidered.Fortunately,thesecomp lexphaseshavenoinruenceon thespindetermination. Weemphasizethatthecouplings g L and g R inEquations 1{8 and 1{9 arethe couplingsofmasseigenstates.Therefore,whenevertherei smixingamongtheheavy partnerstates,ourcouplings g L and g R areingeneralmatrices,whicharerelated 58

PAGE 59

tothecouplings g (0) L and g (0) R oftheinteractioneigenstatesthroughrotationsbythe correspondingmixingangles.Duetothismixing,wedonotex pecttheexperimentally measuredcouplings g L and g R tobepurelychiral,eveninmodelswithpurelychiral couplings g (0) L and g (0) R .TheeectofheavyfermionmixinginaspecicUEDmodelwas previouslyconsideredin[80].Wegeneralizedtheanalysis toincludealsoarbitraryheavy bosonmixingandarbitrarycouplings.Oneofourmainresult swasidentifyingwhich particularcombinationsofthemodelparameterscaninprin ciplebemeasuredfromthe invariantmassdistributionsofthethreeSMfermions( j ` n ,and ` f ),andtoproposethe actualmethodformeasuringthem.Therearethreesuchcombi nations,whichwecalled and r .Eachoneofthemisobservableinprinciple,andrepresents somecombinationof couplingsandmixingangles.Itisinthissensethatourmeth odyieldsameasurementof thecouplingsandmixinganglesoftheheavypartners. Ourbasicassumptionisthattheshapesoftheinvariantmass distributionsdependon thespinsoftheheavyparticlesalongthedecaychain.Purep hasespacepredictssquared massdistributionsthatare,either ratthroughoutforthetwo\near-type"distributions( m 2ll and m 2jl n ),or composedofaratpieceinthelowrangejuxtaposedwithaloga rithmicpieceinthe highrangeofthe m 2jl f distribution[11]. Deviationfromthispurephasespacepredictionimpliessom ekindofspincorrelations [62].However,observingdistributionswhichareconsiste ntwiththepurephasespace predictiondoes notnecessarily implythatallparticlesinvolvedinthedecayarescalars{ spincorrelationsmayhavebeenpresentfortheindividualh elicities,butmayhavebeen washedoutwhenthedierenthelicitycontributionswereco mbinedtoformtheobservable distributions[80]. Thegeneralapproachinpreviousspinstudieshasbeentocom parethedatafrom agivenstudypointwithinonespecicmodeltothecorrespon dingdataobtainedfrom anothermodelalternativewithdierentchoiceofspinsfor theheavypartners.Acommon 59

PAGE 60

rawinallsuchstudieswasthatthecouplingsandparticle-a ntiparticlefractionwerexed tobe identical inthetwomodels,sothatanyremainingdierencecanbeinte rpretedas amanifestationofspins.Suchanapproachtospinmeasureme ntdependsonthespecic modelthatisassumed,andhaslittleusebeyondtheassumedm odel.Sincethechirality parameters a b and c andtheparticle-antiparticlefraction f arenotindependently measuredpriortothespindetermination,thereisnoreason torequiresamevaluesfor theseparametersforeachofthedierentspinconguration sunderstudy.Therefore,the properquestiontoaskinsteadis: Doesaparticularsetofspinassignmentstthedata forsomechoice ofthechiralityparameters a b and c ,and forsomechoice ofthe particle-antiparticleratio f ? Wedevelopedsomemathematicaltoolstoaddressthisquesti oninamodel-independent way.Ourtoolsdonotrequirethevalueofthefraction f ,northevaluesofthechirality tangents.So,wedividedthequestionintotwoparts:Foragi venmassspectrum, Whatarethespins? Whataretheunknownmodelparameters(i.e.theparticle-an tiparticlefractionand thechiralitytangents)? Ourmethodallowsustoaddressthespinquestion independently ofthemodelparameter question.However,the actual answertothespinquestionmaynotbeunique,i.e.two setsofspinsmaytthedata.Inparticularwefoundthatthem odelpairs f FSFS,FSFV g and f FVFS,FVFV g canbeindistinguishable,andtheindistinguishabilityde pendsonthe massspectrum. Sinceweseparatedthespindependencefromthedependenceo ntheparticle-antiparticle fraction,ourmethodisnotlimitedto pp colliderssuchasLHC,andisequallyapplicable totheTevatron.Incontrast,theleptonchargeasymmetrypr oposedin[62]isgreatly aectedbythevalueof f .Forexample,thechargeasymmetryispredictedtobe identicallyzeroattheTevatronandhasnodiscriminatingp owertherewithregardto 60

PAGE 61

spins.Incontrast,ourtechniquenotonlyaddressesthespi nquestioninamodel-independent way,butalsoprovidesameasuredconstrainton f 3.1ClassicationofHelicityCombinations Spincorrelationsininvariantmassdistributionswerealr eadyderived[14];however, weextendedthecalculationbyincluding arbitrarychiral projectionsforbothvertices.We foundthatthespincorrelationdependsononlythe relative chiralprojectionsofthetwo vertices.Ourresultsassumethatallhelicitieshavebeens ummed,forthreereasons: SinceQCDdoesnotdiscriminateleftfromright,weexpectth atthetwopolarizations oftheinitialfermion(orantifermion)areequallylikely. Weassumethatthepolarizationsofthenal-stateparticle swillnotbeobserved. Thenumeratoralgebraforpolarizationsumsprovidessomen icesimplications, especiallywiththeassumptionofmasslessSMfermions. SinceweassumedmasslessSMfermions,thechiralprojectio nscanactuallybeinterpreted asthehelicitiesoftheSMfermions(oroppositethehelicit iesoftheantifermions),sothat, ifonewishestoconsidertheeectsofpolarization,onecan simplymaketheappropriate assignmentstothechiralcouplingcoecients,andthenthe polarizationsumscanstill beused.Infact,theresultsofsuchcalculationsledustoth ehelicitybasisfunctions (Appendix C ). Wefoundthateachtwo-particleinvariantmassdistributio ncanbewritteninthe formofEquation 3{3 dN d ^ m 2p S = 2 X I =1 2 X J =1 K ( p ) IJ ( f;' a ;' b ;' c ) F ( p ) S ; IJ (^ m 2p ; x;y;z )(3{3) Thesub-andsuper-script p denotesoneofthevepossibleSMparticlepairs, j` n j` +n j` f j` +f ,or ` + ` ,and^ m 2p isthesquaredinvariantmassofthispairdividedbyits kinematicmaximum(Equation 3{4 ). ^ m 2p m 2p m 2maxp 0 ^ m 2p 1(3{4) 61

PAGE 62

Weassumedthat m 2maxp wasalreadymeasured(e.g.fromkinematicendpoints).The coecients K IJ dependononlytheparticle-antiparticlefraction( f )andthechirality tangents(tan a ,tan b ,tan c ),andtheirformisindependentofthespinassignment. Thebasisfunctions F ( p ) S ; IJ dependononlythemasses( m A m B m C m D ,and,ofcourse, m p ),buttheirformdependsonthespinassignment(Table 1-2 ).Itisthisdependenceof theformofthebasisfunctionsonthespinassignmentsthata llowsthediscriminationof spin. Oncethespectrumismeasured,thefunctions F ( p ) S ; IJ onlydependon^ m andprovidea uniquebasiswhichcanbettothedataforthemassdistribut ionofeachnal-stateSM two-particlecombination.Sincethesefunctionsdonotdep endonthemodel-dependent parameters f a b and c ,thistcanbedoneinacompletelymodel-independentway, withoutanypriorknowledgeofthecouplings.Thevaluesoft hetcoecients( K IJ ) representconstraints(i.e. measurements )onthecouplingsandmixinganglesoftheheavy partners. Figure 3-1 representsthe8distincthelicitycombinationsofthe3na l-stateSM particles.Table 3-2 furtherdistinguishesthenal-statecombinationsinterm softhe distinctionsbetweenfermionandantifermion,listing32d ierentcoecientscontributing totheprocessofFig. 1-3 A.(Wecount32ratherthan64becauseweassumethat theleptonsare always oppositelycharged,thusremovingonechoiceoffermionvs. antifermion.)The8blue-coloredentriesinTab. 3-2 wereconsideredin[12,14,15].The remaining24red-coloredentriesrepresentourcontributi ontotheanalysis. Weidentiedfourtypesofprocesses(labeledwith IJ )basedonrelativehelicity, wherethevaluesof I;J 2f 1 ; 2 g refertotherelativehelicitiesofthenal-stateSM particles(Fig. 3-1 ). I =1:Thehelicitiesofthejetandnearleptonarethesame.The fourprocessesof\Type 1"in[12,14,15]fallunderthiscategory. I =2:Thehelicitiesofthejetandnearleptonareopposite.Th efourprocessesof\Type 2"in[12,14,15]fallunderthiscategory. 62

PAGE 63

A B j ` n ` f j ` n ` f j ` n ` f j ` n ` f j ` n ` f j ` n ` f j ` n ` f j ` n ` f C D Figure3-1.The8dierenthelicitycombinationsofthenal -stateSMparticles,grouped accordingtorelativehelicities.Forbrevity,weusethete rms\lepton"and \jet"torefertofermionsorantifermionsindiscriminatel y,and\helicity"refers to physical helicity.Thedistinctionbetweenfermionandantifermion ismade explicitinTab. 1-2 .A) I =1 ;J =2 ) allthreehelicitiesarethesame.B) I =2 ;J =2 ) theleptonshavethesamehelicity,buttheyhaveopposite helicitytothejet.C) I =1 ;J =1 ) theleptonshaveoppositehelicity,and thenearleptonhasthesamehelicityasthejet.D) I =2 ;J =1 ) theleptons haveoppositehelicity,andthenearleptonhasoppositehel icitytothejet. J =1:Thehelicitiesofthetwoleptonsareopposite.Theeight processestreatedin [12,14,15]arerestrictedtothistype. J =2:Thehelicitiesofthetwoleptonsarethesame.Processes ofthistypewere completelyneglectedin[12,14,15]. AllcontributionsfromTab. 3-2 foragivenvalueof IJ leadtothesameshapeforagiven invariantmassdistribution.Thenewtypesofprocesses( J =2)giveaqualitativelynew functionaldependenceofthedileptonand j` f invariantmassdistributionsthatwasnot exhibitedin[12,14,15].3.1.1HelicityBasisFunctions F IJ Weformed9invariantmassdistributions,showninEquation s 3{5 through 3{9 ,where weintroducedthefactorof 1 2 ontherighthandsideforconvenience. dN d ^ m 2q` n S = 1 2 2 X I =1 2 X J =1 K ( q` n ) IJ F ( j` n ) S ; IJ (^ m 2q` n )(3{5) 63

PAGE 64

Table3-2.Classicationofmodelparametersaccordingtot heircontributionto K IJ in Equation 3{3 .Thecombinationsshowninbluehavebeenpreviouslyconsid ered in[12,14,15].Thecombinationsshowninredarenewcontrib utionsfromour analysis. I =1 ;J =1 I =1 ;J =2 f q L ` L ` +L gf q L ` +L ` L g f q L ` L ` +R gf q L ` +L ` R g f j c L j 2 j b L j 2 j a L j 2 f j c L j 2 j b L j 2 j a L j 2 f j c L j 2 j b L j 2 j a R j 2 f j c L j 2 j b L j 2 j a R j 2 f q L ` R ` +R gf q L ` +R ` R g f q L ` R ` +L gf q L ` +R ` L g f j c L j 2 j b R j 2 j a R j 2 f j c L j 2 j b R j 2 j a R j 2 f j c L j 2 j b R j 2 j a L j 2 f j c L j 2 j b R j 2 j a L j 2 f q R ` R ` +R gf q R ` +R ` R gf q R ` R ` +L gf q R ` +R ` L g f j c R j 2 j b R j 2 j a R j 2 f j c R j 2 j b R j 2 j a R j 2 f j c R j 2 j b R j 2 j a L j 2 f j c R j 2 j b R j 2 j a L j 2 f q R ` L ` +L gf q R ` +L ` L gf q R ` L ` +R gf q R ;` +L ;` R g f j c R j 2 j b L j 2 j a L j 2 f j c R j 2 j b L j 2 j a L j 2 f j c R j 2 j b L j 2 j a R j 2 f j c R j 2 j b L j 2 j a R j 2 I =2 ;J =1 I =2 ;J =2 f q L ;` L ;` +L gf q L ;` +L ;` L g f q L ;` L ;` +R gf q L ;` +L ;` R g f j c L j 2 j b L j 2 j a L j 2 f j c L j 2 j b L j 2 j a L j 2 f j c L j 2 j b L j 2 j a R j 2 f j c L j 2 j b L j 2 j a R j 2 f q L ;` R ;` +R gf q L ;` +R ;` R g f q L ;` R ;` +L gf q L ;` +R ;` L g f j c L j 2 j b R j 2 j a R j 2 f j c L j 2 j b R j 2 j a R j 2 f j c L j 2 j b R j 2 j a L j 2 f j c L j 2 j b R j 2 j a L j 2 f q R ;` R ;` +R gf q R ;` +R ;` R gf q R ;` R ;` +L gf q R ;` +R ;` L g f j c R j 2 j b R j 2 j a R j 2 f j c R j 2 j b R j 2 j a R j 2 f j c R j 2 j b R j 2 j a L j 2 f j c R j 2 j b R j 2 j a L j 2 f q R ;` L ;` +L gf q R ;` +L ;` L gf q R ;` L ;` +R gf q R ;` +L ;` R g f j c R j 2 j b L j 2 j a L j 2 f j c R j 2 j b L j 2 j a L j 2 f j c R j 2 j b L j 2 j a R j 2 f j c R j 2 j b L j 2 j a R j 2 dN d ^ m 2 q` n S = 1 2 2 X I =1 2 X J =1 K ( q` n ) IJ F ( j` n ) S ; IJ (^ m 2 q` n )(3{6) 0@ dN d ^ m 2q` f 1A S = 1 2 2 X I =1 2 X J =1 K ( q` f ) IJ F ( j` f ) S ; IJ (^ m 2q` f )(3{7) 0@ dN d ^ m 2 q` f 1A S = 1 2 2 X I =1 2 X J =1 K ( q` f ) IJ F ( j` f ) S ; IJ (^ m 2 q` f )(3{8) dN d ^ m 2`` S = 1 2 2 X I =1 2 X J =1 K ( `` ) IJ F ( `` ) S ; IJ (^ m 2`` )(3{9) Thesamesetoffunctions F ( j` n ) S ; IJ enterboththe f q` n g and f q` n g distributions,and similarly,thesamesetoffunctions F ( j` f ) S ; IJ enterboththe f q` f g and f q` f g distributions. F ( q` n ) S ; IJ (^ m 2 )= F ( q` n ) S ; IJ (^ m 2 ) F ( j` n ) S ; IJ (^ m 2 )(3{10) F ( q` f ) S ; IJ (^ m 2 )= F ( q` f ) S ; IJ (^ m 2 ) F ( j` f ) S ; IJ (^ m 2 )(3{11) 64

PAGE 65

Equations 3{5 through 3{9 showthatallinvariantmassdistributionscanbewritten intermsofthreesetsofbasisfunctions: F ( j` n ) S ; IJ (^ m 2 ), F ( j` f ) S ; IJ (^ m 2 )and F ( `` ) S ; IJ (^ m 2 ).Wedene thebasisfunctionstobeunitnormalized. Z 1 0 F ( j` n ) S ; IJ (^ m 2 ) d ^ m 2 =1(3{12) Z 1 0 F ( j` f ) S ; IJ (^ m 2 ) d ^ m 2 =1(3{13) Z 1 0 F ( `` ) S ; IJ (^ m 2 ) d ^ m 2 =1(3{14) Halfoftheprocessesoftype J =1havebeenpreviouslyconsideredin[12,14,15],so thatthefunctions F ( p ) S; 11 and F ( p ) S; 21 inprinciplealreadyappearthere.Wefoundagreement with[12,14,15]forthecaseof F ( p ) S; 11 and F ( p ) S; 21 ,andwesupplementedthoseresultswith theremainingtwotypesoffunctions F ( p ) S; 12 and F ( p ) S; 22 .Notsurprisingly,thebasisfunctions forthe m 2jl n distributionthatdieronlyinthesubscript J areidentical,andsimilarly, thebasisfunctionsforthe m 2ll distributionthatdieronlyinthesubscript I areidentical, since,inbothcasesthedierenceisintherelativehelicit yofaparticlethatisnotusedto formtheinvariantmassdistribution(Tables C-2 and C-1 ). Refs.[12,14,15]missedthefunctions F ( `` ) S; 12 and F ( `` ) S; 22 asaconsequenceoftheir underlyingmodelassumption:theirstudiesassumedverysp ecicxedvaluesofthe chiralitycoecients(namely, c L =1, c R =0, b L =0, b R =1, a L =0, a R =1for thesupersymmetryexampleand c L =1, c R =0, b L =1, b R =0, a L =1, a R =0for theUEDexample)andthereforetheirresults,whilecorrect ,areonlyvalidwithinthis limitedmodel-dependentcontext.Incontrast,the complete setoffunctions F ( `` ) S;IJ forall 4helicitycombinationsallowsustoaddressthespinquesti oninamodel-independent fashion.Similarremarksholdforthe F ( j` f ) S ; IJ functions.Again,thefunctions F ( j` f ) S ;11 and F ( j` f ) S ;21 agree 1 withtheresultsof[14],whilethefunctions F ( j` f ) S ;12 and F ( j` f ) S ;22 arenew. 1 withtheexceptionofatypoin[14]thatwasveriedbytheaut hors 65

PAGE 66

However,whether(andwhattypeof)relationsexistbetween thefourfunctions F ( j` f ) S ; IJ variesfromcasetocase(i.e.thevalueofthespincongurat ionindex S ).Inthethree cases(SFSF,FSFSandFSFV)wherethereisanintermediatehe avyscalarbetweenthe emittedjetandfarlepton,the F ( j` f ) S ; IJ setisagainreducedtoonlytwouniquefunctions; however,symmetryundereither I or J stilldependsonthespecicspinassignments.This isagaineasytounderstand.Forexample,whenthetwolepton sareconnectedbyascalar propagator,thenthedistributionshouldbeinsensitiveto therelativehelicitiesofthetwo leptons(thevalueof J ),whichaccountsfortheequivalenceofthespinbasisfunct ionsin Equations C{1 and C{2 .SimilarremarksholdforEquations C{4 through C{8 withthe replacements ` f j and J I .Itisnotsurprisingthatall4spinbasisfunctions F ( j` f ) S ; IJ areindependentwhenbothpairsofadjacentnal-stateSMpa rticlesareconnectedbya particlewithnontrivialspin(Equations C{10 through C{23 ). 3.1.2HelicityCoecients K IJ UsingthefactorsfromTable 3-2 ,forthe K q` 11 and K q` 11 coecientsweobtain Equations 3{15 through 3{18 K ( q` n ) 11 = K ( q` +f ) 11 = f j c L j 2 j b L j 2 j a L j 2 + f j c R j 2 j b R j 2 j a R j 2 (3{15) K ( q` n ) 11 = K ( q` +f ) 11 = f j c L j 2 j b R j 2 j a R j 2 + f j c R j 2 j b L j 2 j a L j 2 (3{16) K ( q` +n ) 11 = K ( q` f ) 11 = f j c L j 2 j b R j 2 j a R j 2 + f j c R j 2 j b L j 2 j a L j 2 (3{17) K ( q` +n ) 11 = K ( q` f ) 11 = f j c L j 2 j b L j 2 j a L j 2 + f j c R j 2 j b R j 2 j a R j 2 (3{18) The K q` 12 and K q` 12 coecientscanbeobtainedfromEquations 3{15 through 3{18 by substituting a L $ a R K ( q` n ) 12 = K ( q` +f ) 12 = f j c L j 2 j b L j 2 j a R j 2 + f j c R j 2 j b R j 2 j a L j 2 (3{19) K ( q` n ) 12 = K ( q` +f ) 12 = f j c L j 2 j b R j 2 j a L j 2 + f j c R j 2 j b L j 2 j a R j 2 (3{20) K ( q` +n ) 12 = K ( q` f ) 12 = f j c L j 2 j b R j 2 j a L j 2 + f j c R j 2 j b L j 2 j a R j 2 (3{21) K ( q` +n ) 12 = K ( q` f ) 12 = f j c L j 2 j b L j 2 j a R j 2 + f j c R j 2 j b R j 2 j a L j 2 (3{22) 66

PAGE 67

Replacing f $ f and q $ q inEquations 3{15 through 3{18 givesthe K q` 21 and K q` 21 coecients. K ( q` n ) 21 = K ( q` +f ) 21 = f j c L j 2 j b L j 2 j a L j 2 + f j c R j 2 j b R j 2 j a R j 2 (3{23) K ( q` n ) 21 = K ( q` +f ) 21 = f j c L j 2 j b R j 2 j a R j 2 + f j c R j 2 j b L j 2 j a L j 2 (3{24) K ( q` +n ) 21 = K ( q` f ) 21 = f j c L j 2 j b R j 2 j a R j 2 + f j c R j 2 j b L j 2 j a L j 2 (3{25) K ( q` +n ) 21 = K ( q` f ) 21 = f j c L j 2 j b L j 2 j a L j 2 + f j c R j 2 j b R j 2 j a R j 2 (3{26) Replacing a L $ a R inEquations 3{23 through 3{26 yieldsthe K q` 22 and K q` 22 coecients. K ( q` n ) 22 = K ( q` +f ) 22 = f j c L j 2 j b L j 2 j a R j 2 + f j c R j 2 j b R j 2 j a L j 2 (3{27) K ( q` n ) 22 = K ( q` +f ) 22 = f j c L j 2 j b R j 2 j a L j 2 + f j c R j 2 j b L j 2 j a R j 2 (3{28) K ( q` +n ) 22 = K ( q` f ) 22 = f j c L j 2 j b R j 2 j a L j 2 + f j c R j 2 j b L j 2 j a R j 2 (3{29) K ( q` +n ) 22 = K ( q` f ) 22 = f j c L j 2 j b L j 2 j a R j 2 + f j c R j 2 j b R j 2 j a L j 2 (3{30) Thecoecients K ( `` ) IJ forthedileptondistributionscanbeexpressedinvariousw ays, forexampleintermsofthecoecientsinvolvingthenearlep ton ` n (Equation 3{31 ), intermsofthecoecientsinvolvingthefarlepton ` f (Equation 3{32 ),intermsofthe coecientsinvolvingthepositivelychargedlepton ` + (Equation 3{33 ),orintermsofthe coecientsinvolvingthenegativelychargedlepton ` (Equation 3{34 ). K ( `` ) IJ = K ( q` n ) IJ + K ( q` n ) IJ + K ( q` +n ) IJ + K ( q` +n ) IJ (3{31) = K ( q` f ) IJ + K ( q` f ) IJ + K ( q` +f ) IJ + K ( q` +f ) IJ (3{32) = K ( q` +n ) IJ + K ( q` +n ) IJ + K ( q` +f ) IJ + K ( q` +f ) IJ (3{33) = K ( q` n ) IJ + K ( q` n ) IJ + K ( q` f ) IJ + K ( q` f ) IJ (3{34) 67

PAGE 68

3.2ObservableDistributions 3.2.1InvariantMassDistributionsintheHelicityBasis fF IJ g Wemadetheconservativeassumption(whichalsohappenstob etrueinmany models)that q isalightravorquark,sothattheexperimentaldistinction betweena q and q cannotbemade.So,theobservabledistributionsmustbethe sumofthedistributions thatdieronlyin q vs. q (Equation 3{35 and 3{36 ). dN d ^ m 2j` n S = dN d ^ m 2q` n S + dN d ^ m 2 q` n S 1 2 2 X I =1 2 X J =1 K ( j` n ) IJ F ( j` n ) S ; IJ (^ m 2j` n )(3{35) 0@ dN d ^ m 2j` f 1A S = 0@ dN d ^ m 2q` f 1A S + 0@ dN d ^ m 2 q` f 1A S 1 2 2 X I =1 2 X J =1 K ( j` f ) IJ F ( j` f ) S ; IJ (^ m 2j` f )(3{36) Sincethespinbasisfunctions F ( p ) S ; IJ donotdependonthe q q ambiguity,thenewsetof coecients K ( j` n ) IJ and K ( j` f ) IJ aresimplyrelatedtotheoldonesbyEquation 3{37 and 3{38 K ( j` n ) IJ = K ( q` n ) IJ + K ( q` n ) IJ (3{37) K ( j` f ) IJ = K ( q` f ) IJ + K ( q` f ) IJ (3{38) SubstitutingEquations 3{15 through 3{30 intoEquations 3{37 and 3{38 givesEquations 3{39 through 3{42 K ( j` n ) 11 =( f j c L j 2 + f j c R j 2 ) j b L j 2 j a L j 2 +( f j c L j 2 + f j c R j 2 ) j b R j 2 j a R j 2 (3{39) K ( j` n ) 12 =( f j c L j 2 + f j c R j 2 ) j b L j 2 j a R j 2 +( f j c L j 2 + f j c R j 2 ) j b R j 2 j a L j 2 (3{40) K ( j` n ) 21 =( f j c L j 2 + f j c R j 2 ) j b L j 2 j a L j 2 +( f j c L j 2 + f j c R j 2 ) j b R j 2 j a R j 2 (3{41) K ( j` n ) 22 =( f j c L j 2 + f j c R j 2 ) j b L j 2 j a R j 2 +( f j c L j 2 + f j c R j 2 ) j b R j 2 j a L j 2 (3{42) Theremaining K ( j` ) IJ coecientsarerelatedtothesebyEquations 3{43 through 3{46 K ( j` n ) 11 = K ( j` +f ) 11 = K ( j` +n ) 21 = K ( j` f ) 21 (3{43) K ( j` n ) 12 = K ( j` +f ) 12 = K ( j` +n ) 22 = K ( j` f ) 22 (3{44) 68

PAGE 69

Figure3-2.Acontourplotofcos~ c asafunctionofcos c and f K ( j` n ) 21 = K ( j` +f ) 21 = K ( j` +n ) 11 = K ( j` f ) 11 (3{45) K ( j` n ) 22 = K ( j` +f ) 22 = K ( j` +n ) 12 = K ( j` f ) 12 (3{46) Thedependenceofthesecoecientson f and c alwaysappearsthroughthecombinations f j c L j 2 + f j c R j 2 = f cos 2 c + f sin 2 c and f j c L j 2 + f j c R j 2 = f cos 2 c + f sin 2 c .Wetherefore introducedanalternativemodelparameter~ c denedbyEquation 3{47 3{48 ,or 3{49 cos 2 ~ c = f cos 2 c + f sin 2 c (3{47) sin 2 ~ c = f cos 2 c + f sin 2 c (3{48) cos2~ c =( f f )cos2 c (3{49) Fig. 3-2 showstherelationshipbetweentheoldparameters, f f;' c g andthenewly introducedparameter~ c The K ( j` ) IJ coecientsareunit-normalized(Equations 3{50 and 3{51 ). 2 X I =1 2 X J =1 K ( j` n ) IJ =1(3{50) 69

PAGE 70

2 X I =1 2 X J =1 K ( j` f ) IJ =1(3{51) GivenEquations 3{12 through 3{14 ofourbasisfunctions F ( p ) S ; IJ ,Equations 3{50 and 3{51 implythatthe f j` n g and f j` f g distributions(Equations 3{35 and 3{36 )arehalf-unit normalized(Equations 3{52 and 3{53 ). Z 1 0 dN d ^ m 2j` n S d ^ m 2j` n = 1 2 ; (3{52) Z 1 0 0@ dN d ^ m 2j` f 1A S d ^ m 2j` f = 1 2 : (3{53) Thelast(andbyfarmostcomplicated)stepinderivingthe observable distributionsis toformthejet-leptondistributionswhicharebasedonden iteleptoncharge.Thisisdone byaddingtogetherthecontributionsfromthe m 2jl n and m 2jl f foreachchargeaccordingto Equation 3{54 dN dm 2j` S dN dm 2j` n S + 0@ dN dm 2j` f 1A S (3{54) NoticethatEquation 3{54 isexpressedintermsofadimensionfulinvariantmass, m 2jl ratherthanaunit-normalizedmassparameter,^ m 2jl (e.g.Equations 3{52 and 3{53 ). However,wepreferredourdistributionstobegiveninterms oftheunit-normalized massparameter^ m .Tothisend,wenormalizedthegenericjet-leptonmasstoth e m 2maxjl ( hi ) endpoint(Equation 2{2 )accordingtoEquation 3{55 ^ m 2jl m 2jl m 2maxjl ( hi ) (3{55) WealsousedtheratiosinEquations 3{56 and 3{57 r n m maxj` m maxj` n (3{56) r f m maxj` m maxj` f (3{57) 70

PAGE 71

Equation 3{58 showsthecombinedjet-leptondistributionsforeachlepto nchargeinterms oftheunit-normalizedsquaredmassvariable. dN d ^ m 2j` S = 1 2 2 X I =1 2 X J =1 K ( j` n ) IJ r 2 n F ( j` n ) S ; IJ ( r 2 n ^ m 2j` )+ K ( j` f ) IJ r 2 f F ( j` f ) S ; IJ ( r 2 f ^ m 2j` ) (3{58) Wheneverthetwoendpoints m maxj` n and m maxj` f aredierent,oneofthetworatios r n and r f isguaranteedtoexceed1,sothatthereisarangeofmassesfo rwhichthecorresponding argument( r 2 n ^ m 2j` or r 2 f ^ m 2j` )inthecorrespondingspinbasisfunction( F ( j` n ) S ; IJ or F ( j` f ) S ; IJ ) exceeds1.Inthismassrange,onlytheotherspinbasisfunct ion( F ( j` f ) S ; IJ or F ( j` n ) S ; IJ ) contributes.Bothoftheobservabledistributions(Equati on 3{58 )areunitnormalized. Z 1 0 dN d ^ m 2j` S d ^ m 2j` =1(3{59) Weobtainedanalyticalexpressionsforthethree observable massdistributions,the dileptondistribution(Equation 3{9 )andthe2jet-leptondistributions(Equation 3{58 ), thatexhibitspincorrelations.Allthreeofourformulasar eunitnormalizedandcanbe readilyrescaledfortheactualobservednumberofevents(w hichisthesameforeachof thethreedistributions).Ourformulasarewritteninterms ofasetofknownfunctions F ( p ) S ; IJ ,whichareexplicitlydenedinAppendix C .Thecoecients K ( p ) IJ appearinginour formulasdependonthreemodel-dependentparameters: a b and~ c (Equations 3{2 and 3{47 through 3{49 ),buttheydo not dependonthespinassignment, S .Theseresults couldinprinciplebeusedforspindetermination,butwecho seanalternativeapproach. 3.2.2InvariantMassDistributionsintheObservableBasis fF ; F ; F r ; F g Weactuallyfoundamuchmoreconvenientsetofspinbasisfun ctionsthataresimple linearcombinationsoftheoriginalhelicitybasisfunctio ns F ( p ) S ; IJ givenbyEquations 3{60 through 3{63 F ( p ) S ; = 1 4 n F ( p ) S ;11 F ( p ) S ;12 + F ( p ) S ;21 F ( p ) S ;22 o (3{60) F ( p ) S ; = 1 4 n F ( p ) S ;11 + F ( p ) S ;12 F ( p ) S ;21 F ( p ) S ;22 o (3{61) 71

PAGE 72

F ( p ) S ; r = 1 4 n F ( p ) S ;11 F ( p ) S ;12 F ( p ) S ;21 + F ( p ) S ;22 o (3{62) F ( p ) S ; = 1 4 n F ( p ) S ;11 + F ( p ) S ;12 + F ( p ) S ;21 + F ( p ) S ;22 o (3{63) Whereastheoriginalhelicitybasisfunctionsaredistingu isheddirectlyintermsofthe 4dierentrelativehelicitycombinations IJ (whichweassumeareunobservable),the newbasisfunctions(Equations 3{60 through 3{63 )aredistinguishedintermsofthe model-dependenceoftheobservabledistributions.Thus,t henewbasisemphasizeswhat partofthespincorrelationismodel-dependent,andinwhat way.Furthermore,this newbasis(Equations 3{60 through 3{63 )revealsthatthere can indeedbeanontrivial model-independentspincorrelation .Theexplicitformofthesebasisfunctionsisgivenin Appendix D Theadvantageofthenewsetofbasisfunctionsbecomesappar entifwerewriteour resultsforthedierentinvariantmassdistributionsacco rdingtoEquations 3{64 through 3{66 dN d ^ m 2`` S L + S = F ( `` ) S ; (^ m 2`` )+ F ( `` ) S ; (^ m 2`` )(3{64) dN d ^ m 2j` n S = 1 2 F ( j` n ) S ; (^ m 2j` n ) F ( j` n ) S ; (^ m 2j` n ) (3{65) 0@ dN d ^ m 2j` f 1A S = 1 2 F ( j` f ) S ; (^ m 2j` f )+ F ( j` f ) S ; (^ m 2j` f ) F ( j` f ) S ; (^ m 2j` f ) r F ( j` f ) S ; r (^ m 2j` f ) (3{66) Thethreenewparameters f ;;r g are observable modelparametersthatplayan analogousroletothecoecients K IJ oftheoldbasis.Theyarerelatedtothechiral couplingsaccordingtoEquations 3{67 through 3{69 j a L j 2 j a R j 2 j b L j 2 j b R j 2 (3{67) (2 f 1) j b L j 2 j b R j 2 j c L j 2 j c R j 2 (3{68) 72

PAGE 73

r (2 f 1) j a L j 2 j a R j 2 j c L j 2 j c R j 2 (3{69) Eachoneofthe and r parameterscantakevaluesintheinterval[ 1 ; 1].However, and r arenotcompletelyunrelated.Giventheirdenitions,they mustsatisfycertain relationsamongthemselves,andthosearelistedinAppendi x E The F ( p ) S ; termappearswithoutanymodelcoecient.Formanyspinassi gnments andnal-stateparticlecombinations, F ( p ) S ; simplygivestheinvariantmassdistributionas predictedbypurephasespace,i.e.withoutspincorrelatio ns.Thisistruewheneverthe intermediateheavypartnersarescalarsorfermions.Howev er,ifaheavy vector appears amongtheintermediateheavypartners,thenthe F ( p ) S ; function always deviatesfrom purephasespace.Furthermore,thisdeviationleadstoadev iationoftheentireinvariant massdistributionthatcannotbecanceledbyanychoiceof and r .Therefore,one ofourgeneralconclusionsisthatanintermediateheavyvec torboson always leadsto deviationsfrompurephasespaceandconversely,whenevera purephasespacedistribution isobserved,aheavyvectorbosonisruledout. Againthe f j` n g and f j` f g distributions(Equations 3{65 and 3{66 )arenotseparately observable,andinsteadmustbecombinedinordertoformthe observable f j` + g and f j` g distributions.The f j` f g distributiondependsonallthreeparameters and r CombiningEquations 3{65 and 3{66 givestheobservablejet-leptondistributionsthatare distinguishedbythechargeofthelepton(Equation 3{70 ).Bothofthesedistributions dependonallthreemodelparameters. dN d ^ m 2j` S = 1 2 r 2 n F ( j` n ) S ; ( r 2 n ^ m 2j` )+ r 2 f F ( j` f ) S ; ( r 2 f ^ m 2j` )+ r 2 f F ( j` f ) S ; ( r 2 f ^ m 2j` )(3{70) rr 2 f F ( j` f ) S ; r ( r 2 f ^ m 2j` ) r 2 f F ( j` f ) S ; ( r 2 f ^ m 2j` ) r 2 n F ( j` n ) S ; ( r 2 n ^ m 2j` ) However,thesame and r termsinEquation 3{70 appearwithoppositesignsinthe f j` + g andthe f j` g distribution.Thissuggeststhat,insteadofthetwoindivi dual distributions(Equation 3{70 )weshouldconsidertheirsum(Equation 3{71 )anddierence 73

PAGE 74

(Equation 3{72 ). S + S dN d ^ m 2j` + S + dN d ^ m 2j` S = r 2 n F ( j` n ) S ; ( r 2 n ^ m 2j` )+ r 2 f F ( j` f ) S ; ( r 2 f ^ m 2j` )+ r 2 f F ( j` f ) S ; ( r 2 f ^ m 2j` )(3{71) D + S dN d ^ m 2j` + S dN d ^ m 2j` S = rr 2 f F ( j` f ) S ; r ( r 2 f ^ m 2j` )+ r 2 f F ( j` f ) S ; ( r 2 f ^ m 2j` ) r 2 n F ( j` n ) S ; ( r 2 n ^ m 2j` )(3{72) Thenormalizationconditionsforthenewlydenedquantiti es S + S and D + S aregivenin Equations 3{73 and 3{74 Z 1 0 S + S d ^ m 2j` =2(3{73) Z 1 0 D + S d ^ m 2j` =0(3{74) Equations 3{71 and 3{72 revealoneofourmostimportantresults{thatthesum ofthetwojet-leptondistributionsdependsonasinglemode l-dependentparameter,and thisisthesameparameter thatdeterminesthedileptondistribution.Therefore,onc e ismeasuredfromtherelativelycleandileptondata,theobs ervable S + S distributionis completelyspecied.The D + S distributioncanprovideameasurementoftheothertwo model-dependentparameters and r .However,the r parametercannotbedetermined if S =1 ; 2 ; 3,sincethebasisfunction F ( j` ) S ; r vanishesidenticallysothat D + S becomes r -independent(Appendix D ).Similarly,theparameter cannotbedeterminedif S =2 ; 3. Webelievethatourbasisfunctionsprovideasuperioralter nativetotheleptoncharge asymmetry(Equation 3{75 )thathasbeensuggestedinpreviousstudies[62]. A + S D + S S + S (3{75) Ingeneral, A + S ismoremodel-dependentthaneither S + S or D + S .While S + S depends onthesinglemodelparameter ,and D + S dependsonthetwoothermodelparameters 74

PAGE 75

and r A + S dependson allthree ofthesemodelparameters and r .Moreimportantly, A + S isasingledistribution,derivedfrom S + S and D + S ,thereforeitisboundtocontain lessinformationthanthetwoseparatedistributions S + S and D + S .Theinformationgain fromusingtheseparate S + S and D + S ismoststrikingforthecaseofa p p colliderlikethe Tevatron(Appendix F.6 ). 3.3DeterminationofModelParameters f ;;r g Ourspindeterminationassumesthegeneralformoftheobser vableinvariantmass distributionsgiveninEquations 3{64 3{71 ,and 3{72 .Thefunctions F ( p ) S ; F ( p ) S ; F ( p ) S ; r and F ( p ) S ; aregiveninAppendix D ,whilethemodelparameters and r arerelated tothechiralcouplingsandparticle-antiparticlefractio n(Equations 3{67 through 3{69 ).Giventhedataforthethreeobservabledistributions(Eq uations 3{64 3{71 and 3{72 ),ourtechniqueistotfortheunknownmodelparameters, and r (Appendix E )consideringeachofthe6dierentspinassignments(Tab. 1-2 )oneata time.Ourtechniqueresultsin6dierentsetsof\bestt"va luesforthemodelparameters (Appendix F ),andanaccompanyingmeasureforthegoodnessoftineachc ase.The goodnessoftforeachspinassignmentisanindicationofth econsistencyofthedata withthatspinassignment,andonecanalsoassigncondence levelprobabilitiestothose statements.Thisprocedureismodel-independent,andinfa ctprovides independent measurementsofthemodelparameters(Table 3-3 ).Forexample,whenallthreemodel parameters and r aremeasuredandfoundtobenon-zero,therelativechiralit iesof thethreevertices(Fig. 1-3 A)aredeterminedaccordingtoEquations 3{76 through 3{78 j a L j 2 j a R j 2 2 = r (3{76) j b L j 2 j b R j 2 2 = r (3{77) j c L j 2 j c R j 2 2 = r (2 f 1) 2 (3{78) 75

PAGE 76

Sincethespincorrelationdependsononly relative chiralities,wecannotdeterminethe actualhandednessofthevertices(asindicatedbythesquar inginEquations 3{76 through 3{78 ),onlyhowstronglychiraltheyare.Whileingeneral and r canbeeitherpositiveornegative,Equations 3{67 through 3{69 implythat r r ,and r arealwaysnon-negative.Furthermore,itfollows that j jj r j j r jj j and j r jj j .Weseethatforanygivenmeasurementof and r ,thechiralityoftheleptonverticescanbeuniquelydeterm ined,uptothe ambiguitybetweenleft-handedandright-handed.Inotherw ords,theparticle-antiparticle ambiguityonlyaectsthedeterminationofthechiralityof thejetvertex,whichisnot atallsurprising.Thechiralityofthejetvertexisnotuniq uelydetermined,andinstead isparametrizedasafunctionof f (Equation 3{78 ).Although f cannotbedetermined, consistency(Equations 3{67 through 3{69 and 3{76 through 3{78 )restrictstheallowed valuesof f tobeintherangegiveninEquation 3{79 0 f 1 2 1 r r or 1 2 1+ r r f 1(3{79) Ata pp colliderliketheLHC,ingeneralweexpect f> 1 2 ,sowewouldselectthehigher f range,whilethelower f rangewouldberelevantforahypothetical p p collider.While Equation 3{79 isonlyarestrictiononthevalueof f attheLHC,ifthemeasuredvalues of and r happentobesuchthat j r jj j ,then f isrestrictedtobeverycloseto unity,andthejetvertexmustbestronglychiral. 3.4TwinSpinScenariosFSFS/FSFVandFVFS/FVFV ConsultingAppendix D ,onecanseethatthespinbasisfunctionsforFSFS( S =2) andFSFV( S =3)generallysatisfytherelationshipsinEquations 3{80 through 3{83 F ( p ) 3; = F ( p ) 2; 1 2 z 1+2 z (3{80) F ( p ) 3; = F ( p ) 2; =0(3{81) F ( p ) 3; r = F ( p ) 2; r =0(3{82) 76

PAGE 77

Table3-3.Availablemeasurementsofthemodelparameters and r foreachofthe6 dierentspinassignments.ForSFSF,only canbedetermined.ForFSFSand FSFV,only canbedetermined.Fortheotherthreespinassignments,all threemodelparameterscanbedetermined. SpinParametersmeasuredfromdistribution:chain L + S + D + L + S + D + SFSF FSFS FSFV FVFS ;r;;r FVFV ;r;;r SFVF ;r;;r F ( p ) 3; = F ( p ) 2; (3{83) ThereforeEquation 3{84 issucienttoguaranteethat allthree invariantmassdistributions (Equations 3{85 through 3{87 ),inthecaseofFSFS( S =2)andFSFV( S =3),are exactlythesame. 2 = 3 1 2 z 1+2 z (3{84) L + 2 ^ m 2`` ; x;y;z; 3 1 2 z 1+2 z = L + 3 ^ m 2`` ; x;y;z; 3 (3{85) S + 2 ^ m 2j` ; x;y;z; 3 1 2 z 1+2 z = S + 3 ^ m 2j` ; x;y;z; 3 (3{86) D + 2 ^ m 2j` ; x;y;z; 2 ;r 2 = D + 3 ^ m 2j` ; x;y;z; 3 ;r 3 : (3{87) Equations 3{85 through 3{87 holdidenticallyfor any valuesoftheveparameters 3 3 r 3 2 and r 2 .Theabilitytodiscriminatedependsonthevaluesof 2 and z (Equation 3{84 ). isdenedintherange[ 1 ; 1],while z isdenedin(0 ; 1).Then,foranygiven valueof 3 2 [ 1 ; 1], 2 fallsintoitsallowedrange,andanexactduplicationisine vitable. However,noteveryvalueof 2 leadstoavalidsolutionfor 3 ,sinceforlargeenough valuesof j 2 j ,thevalueof j 3 j wouldexceed1,whichisnotallowed.Thetwomodelswill alwaysbeconfusedwitheachotherifnaturehappenstochoos eFSFV( S =3),whereas ifnaturechoosesFSFS( S =2),thentheconfusionarisesonlyifnaturealsosatises 77

PAGE 78

Equation 3{88 j 2 j 1 2 z 1+2 z (3{88) AsimilarambiguityalsoexistsbetweentheFVFS( S =4)andFVFV( S =5)spin assignments.Thespinbasisfunctionsexhibittherelation shipsshowninEquations 3{89 through 3{92 F ( p ) 5; = F ( p ) 4; 1 2 z 1+2 z (3{89) F ( p ) 5; = F ( p ) 4; (3{90) F ( p ) 5; r = F ( p ) 4; r 1 2 z 1+2 z (3{91) F ( p ) 5; = F ( p ) 4; (3{92) So,ifEquations 3{93 through 3{95 canbesatisedbythemodelparameters,thenthese twospinassignmentscannotbedistinguished. 4 = 5 1 2 z 1+2 z (3{93) 4 = 5 (3{94) r 4 = r 5 1 2 z 1+2 z (3{95) IfnaturehappenstochooseFVFV,thenthemodelwillalwaysb econfusedwithFVFS. However,ifnaturechoosesFVFS,theconfusionarisesonlyi fnaturealsosatises Equations 3{96 and 3{97 j 4 j 1 2 z 1+2 z (3{96) j r 4 j 1 2 z 1+2 z (3{97) However,theambiguityisresolvedunlessthevaluesof 4 4 and r 4 alsosatisfythe domainconstraints(Equations E{2 through E{5 ). 78

PAGE 79

CHAPTER4 CONCLUSIONS 4.1KinematicBoundaries Onceweignoredtheendpoint m maxjll weonlyneededtoconsider4dierentcases, R i i =1 ; 2 ; 3 ; 4,asillustratedwiththecolor-codedregionsinFig. 1-4 .Incontrast, previousstudiesthatusedthe m maxjll endpoint[11,31,32]wereforcedtoconsiderall 11dierentpossibilities( N jll ;N jl )showninFig. 1-4 .Weprovideanalyticalinversion formulas(Equations 2{13 through 2{16 with 2{17 through 2{21 )thatallowtheimmediate calculationofthemassspectrum m A m B m C and m D intermsofasetoffourmeasured invariantmassendpoints f a;c;d;e g (Equations 2{7 through 2{11 ),eliminatingtheneed fornumericalscanningofsolutionspace.Ourformulasarev alidin all parameterspace regions,sincewedonotusetheendpoint b = m maxjll ,whichisproblematicinregions(3,1), (3,2)and(2,3)(Equation 2{4 ). Weinvestigatedthepossibilityofndingmultiplesolutio nsforthemassspectrum, inthecaseofaperfectexperimentthatmeasuresthevalueso f allve invariantmass endpoints f a;b;c;d;e g (Equations 2{7 through 2{11 )withzeroerrorbars.Wefound thatthereisacertainportionofparameterspace(Fig. 2-4 )whereexactduplication occurs,i.e.twoverydierentmassspectrayieldidentical valuesforallvemeasurements f a;b;c;d;e g .Thesituationonlyworsensiftheinevitableexperimental errorsonthe endpointmeasurementsareincludedintheanalysis. Weadvertiseanewapproachtothestudyoftheusualsingle-v ariableinvariantmass distributions.Inparticular,wepointoutthatthemultiva riateinvariantmassdistributions containmoreusefulinformationthantheindividualsingle -variablehistogramsthat areusuallyconsidered.Astwoillustrativeexamples,weco nsideredthetwo-variable ( m 2jl ( lo ) ;m 2jl ( hi ) )and( m 2ll ;m 2jll )distributions(Figs. 2-7 and 2-9 ).Thesetwochoices areactuallyquitenatural,since m 2jl ( lo ) and m 2jl ( hi ) arealreadyrelatedtoeachother throughtheirdenitionsintermsof m 2jl n and m 2jl f (Equations 2{1 and 2{2 ),andsince 79

PAGE 80

the( m 2ll ;m 2jll )distributionprovidesaconvenientwaytoseethethreshol d e (Fig. 2-9 ). The( m 2jl ( lo ) ;m 2jl ( hi ) )distributionisalwaysboundedbystraightlines(Fig. 2-7 ),whilethe ( m 2ll ;m 2jll )distributionisboundedbyahyperbola(Equations 2{58 and 2{59 ),and(inthe on-shellcase)bythestraightline UV (Fig. 2-9 ).Onecouldalsoconsiderotherchoicesof two-dimensionaldistributions,forexample f m 2ll ;m 2jl g f m 2jl ;m 2jll g [70],or f m 2ll ;m 2jl ( lo ) g f m 2ll ;m 2jl ( hi ) g [71].Thosedistributionsalsoprovidediscriminationbet weenthe\near"and \far"leptonendpoints,andwillcontributeevenmoredatap oints. Theboundarylinesexhibittwousefulfeatures.First,thes hapesofthekinematic boundariesarecharacteristicofthecorrespondingparame terspaceregion R i (Fig. 2-7 ). Thisobservationcanbeusedtoidentifytherelevantparame terspaceregion,andresolve potentialambiguitiesintheextractionofthemassspectru m.Second,theboundary linesexhibitanumberofspecialpoints,whosecoordinates caninprinciplebemeasured, providingadditionalexperimentalinformationaboutthem assspectrum.Ofcourse,the locationsofthespecialpointsarenotindependentfromeac hother,sincetheyareall givenintermsofonly4inputmasses.Nevertheless,itiscer tainlypreferabletohaveas manymeasurementsaspossible.Theinversionformulasmays implifyconsiderablyifwe replace e ,whoseanalyticalexpression(Equation 2{11 )israthercomplicated,withsomeof theothermeasurementsthatcanbemadefromthetwo-variabl eboundarylines.Onesuch exampleisshowninAppendix B Animportantadvantageofthetwo-variableapproachisthat onecanreadilyresolve theambiguitybetweentheendpoints m 2maxjl f and m 2maxjl n .Indeed,theunobservable m 2maxjl n and m 2maxjl f endpointsareRegion-independent(amongthethreeon-shel lRegions),and canbedirectlyobservedfromtheboundarylines.Anotherad vantageisthatonecan performattotheboundarylinesofthescatterplotinstead ofattotheendpointsin theone-dimensionaldistributions.Thisimprovestheprec isionofthemassdetermination, asdemonstratedin[71]usingtheSPS1aSUSYbenchmarkexamp le. 80

PAGE 81

Perhapsthemostpressingconsiderationishowwellthemeth odproposedhere willsurvivetheexperimentalcomplicationsofafull-blow nanalysis,includingdetector simulation,backgrounds(bothSMandBSM),nitewidthsoft heparticles B C and D varyingpopulationdensityofthescatterplots,etc..TheC MSSUSYworkinggroupis currentlyinvestigatingtheseissues,andafuturepublica tionisplannedtodisplaythe results. 4.2SpinCorrelations Wefoundanalyticalexpressionsfortheobservabletwo-par ticleinvariantmass distributionsin m 2jl + m 2jl and m 2ll exhibitingspincorrelationsbetweenthetwoparticles. Wepresentedourexpressionsinthemodel-independentcont extofarbitrarycouplingsand arbitraryparticle-antiparticlefraction,for6possible spinassignments(Table 1-2 ).Our resultsgeneralizethoseofRefs.[11,12,14,15].Wealsode rivedtheexactcombinationsof couplingsthatcanbedeterminedasabyproductofthespinme asurement. WedemonstratedourmethodusingtheSPS1amassspectrum,co uplings,and particle-antiparticlefractiontogeneratethesimulated invariantmassdistributions.By assumingthatthedistributionscomefromeachoneofour6sp inscenarios(Table 1-2 ) inturn,wethendeterminediftheother5spinassignmentsco uldleadtoidentical invariantmassdistributionsforsome(possiblydierent) choiceofcouplingsand particle-antiparticlefraction.Wealsoprovedthe general existenceofambiguityinthe determinationofthe f FSFS,FSFV g and f FVFS,FVFV g modelpairs(Table 4-1 )by derivingtherelationbetweenthecouplingsandmixingangl eswithineachpairofmodels thatwouldresultin identical observablemassdistributionsforthosemodelpairs.This ambiguityisexhibitedinourcasestudiesfortheFSFVandFV FVspinassignments (Appendices F.3 and F.4 ). Weconsideredtheexampleofaquarkjetfollowedbytwolepto ns,whichiscommonly encounteredinmodelsofsupersymmetryorextradimensions .Thethreeobservable invariantmassdistributionsforeachpairofwell-denedo bjects(inourcase f ` + ` g 81

PAGE 82

Table4-1.Expectedoutcomesfromourspindiscriminationa nalysis,barringnumerical accidentsduetoveryspecialmassspectra.The4problement riesarecolored redforemphasis.Thetwocaseslabeled\maybe"correspondt othepotential confusionofanFSFS(FVFS)chainwithanFSFV(FVFV)chainth atis certaintooccurforonlyarestrictedrangeofthemodelpara meters(Equation 3{88 orEquations 3{96 and 3{97 ). DataIsthismodelcertaintotthedata?fromSFSFFSFSFSFVFVFSFVFVSFVF SFSFofcoursenononononoFSFSnoofcourse maybe nonono FSFVno yes ofcoursenonono FVFSnononoofcourse maybe no FVFVnonono yes ofcourseno SFVFnononononoofcourse f j` + g and f j` g )werederived.Inordertoremovethecombinatorialambigui ties,onecan performravorsubtractionontheleptonsandmixed-eventsu btractiononthejet.One mayalsoapplycutsinordertosuppressanySMandnewphysics backgrounds.Theend productfromthisstepisthe3distributions L + S + and D + (Equations 3{64 3{71 and 3{72 ). Nosinglemethodisuniversallyapplicable,thereforethea vailabilityofdierentand complementarytechniquesisimportant.Thesuccessofanyg ivenmethoddependsonthe specicnewphysicsscenario.Weidentifysomefeaturesofo urmethodthatarelikelyto makeitrelevantandsuccessful,ifamissingenergysignalo fnewphysicsisseenatthe LHCand/ortheTevatron. Manyoftheexistingtechniquesforspindeterminations(se e,forexample,[73,74,84, 85])havebeenoriginallydevelopedinthecontextoflepton colliders,wherethetotal centerofmassenergyineacheventisknown.Consequently,a thadroncolliders, thosemethodsareapplicableonlyiftheeventscanbefullyr econstructed.Innew physicsscenarioswithdarkmatterWIMPs,thisappearstobe ratherchallenging, sincethereare two invisibleWIMPparticlesescapingthedetector.Insomespe cial circumstances,wheretwosucientlylongdecaychainscanb eidentiedinthe event,fullreconstructionmightbepossible[39,46,51],b utinanycase,thisappears torequireverylargedatasamples.Incontrast,ourmethodr eliesoninvariantmass distributions,whichareframe-independent,andwedonotr equirefullreconstruction oftheevents.Furthermore,ourmethoddoesnotrequirethep resenceoftwo 82

PAGE 83

separatedecaychainsintheevent,andcanbeinprincipleal soappliedtothe associatedproductionofaWIMPwithonlyoneotherheavypar tner. Onemajoradvantageofourmethodincomparisontovariousev entcounting techniques[35,76,83,86]isthatwedonotneedtoknowanyth ingaboutthe productioncross-sectionsforthedierentparton-leveli nitialstates,thebranching fractions,theexperimentaleciencies,etc..Ourmethodr eliesononlytheshapesof thedistributionsfromthesequentialdecay(Fig. 1-3 A),andsoitisinsensitivetothe model-dependentcross-sectionandbranchingratio. Incomparisontostudiesthathavealsousedinvariantmassd istributions[12,14, 15,17,18,62,75,77{82],themainadvantageofourapproach isthatwemakeno assumptionsaboutthetypeofcouplingsineachvertexofFig 1-3 A,oraboutthe particle-antiparticlefraction f .Asaresult,weevendevised measurements ofcertain combinationsofthecouplingsandthe f parameter. 4.3GeneralRemarks Ouranalysiscontainstwogeneralthemes:model-independe nceandambiguities. Ourbasicconclusionregardingthesethemesisthat,forax edsetofobservables, determinationofnewphysicsbecomesmoreambiguousasthem ethodusedtomake thedeterminationbecomesmoremodel-independent.Howeve r,werealizedafortunate side-eectofmodel-independence.Byforcingourselvesto approachthemassandspin determinationinamodel-independentway,weencountereda lternativemeasurements thatcanbeusedtoresolvesomeambiguityinthedeterminati on.Furthermore,whereas previoustechniquesrequirearticialmodelassumptions, ourtechnique automatically restrictsthepossiblemodelsofnewphysics(e.g.discrimi nationbetweenon-shelland o-shell,anddeterminationof ,and r ). 83

PAGE 84

APPENDIXA ABSMEXAMPLE:SUSY SUSYisasymmetryoftheniterepresentationsoftheLorent zGroupthatmay beimposedontheparticlephysicsLagrangian.Itrequirest heactionfunctionalof particlephysicstobeinvariantundercontinuousexchange sofniteLorentzGroup representationsthatdierbyhalf-integerspin[1].Howev er,thegeneratorsofSUSY arealsorequiredtocommutewiththegeneratorsoftheinter nalsymmetrygroupof theSM[1], SU (3) C SU (2) W U (1) Y 1 So,leptonandquarkeldsareeachin distinctrepresentationsofSUSY,theHiggseldisinadist inctrepresentationofSUSY, andthegaugeeldisinadistinctrepresentationofSUSY.(I nfact,left-handedand right-handedeldsareeachindistinctrepresentationsof SUSY,becausetheyaredistinct representationsof SU (2) W .)Thus,therearenoeldsintheSMthatcanbeexchanged witheachotherunderSUSY,whichimpliesthatthenumberof eldsrequiredforSUSY mustbe atleast twicethenumberofeldsrequiredfortheSM[90].Inotherwo rds,SUSY requireseverySMeldthatisuniqueundertheinternalgrou poftheSMtohaveaunique \partner"eldthatdoesnotexistintheSM,commonlyreferr edtoasa\superpartner". Aquantumofasuperpartneriscommonlyreferredtoasa\supe rparticle"or\sparticle", althoughtheterm\superpartner"isalsocommonlyreusedfo rthispurposeaswell.A \supermultiplet"(orsometimes\supereld")isanirreduc iblerepresentationofSUSY, whichissomelinearcombinationofSMeldsandsuperpartne rs.[1] OnemaywonderwhatistheexperimentalevidenceforSUSY.Th eansweris\none" {sofar[91].Apparently,SUSYdoes not representnature{notexactly.Onewayto maintainconsistencybetweenSUSYandtheexperimentalevi denceagainstitisto 1 Iuseasubscript\W"(for\weak")ratherthanasubscript\L" (for\left")todenote thegaugegroupofweakisospin.Ibelievethattheuseofthes ubscript\L"isconfusing for2reasons:1)thereisanother SU (2) L intheSMfortheleft-chiralblockofthespinor representationoftheLorentzgroup,and2)actually,right -handedratherthanleft-handed (helicity)antifermionsareinthefundamentalrepresenta tionof SU (2) W [89]. 84

PAGE 85

assumethatSUSYisonly approximate (i.ebroken).Oneimportantphenomenological manifestationofSUSYviolationisthatasuperparticlecan haveadierentmassfromits SMcounterpart[1]. Another,moresubtleissueisthestabilityoftheproton.SU SYallowsfortermsin theLagrangianthatcanleadtopromptdecayoftheproton[1] .Assumingthatprotons are,astheyseemtobe,stablewithahalf-lifegreaterthant heageoftheuniverse[9], thispresentsapotentialproblemforSUSY.Apopularresolu tiontothisproblemisto introduceyetanothersymmetry:R-parityconservation.Th eeldsofthemodelrepresent R-parity,similarlytospatialparity,aseitheratrivialo rafundamentalrepresentationof some Z 2 .AllSMeldsareassignedevenR-parity(i.e.theyaretrivi alrepresentations ofthis Z 2 ).So,R-parityhasnoinruenceonphenomenathatarerestric tedtotheSM. AllsuperpartnersareassignedoddR-parity(i.e.theyaref undamentalrepresentations ofthis Z 2 ).So,animportantphenomenologicalconsequenceofR-pari tyisthatthe numberofsuperparticlesmustchangebyamultipleoftwo.Th isimpliesthatthelightest superparticle(LSP)isstable[1].AneutralinoLSPmakesap opularDMcandidate[92]. Thecascade~ q L ~ 02 ~ ` R ~ 01 ofminimalSupergravity(mSUGRA),depictedin Fig. A-1 ,providesapopularexampleofthedecaytopologyandnal-s tatesignaturethat weexamined,withtheidenticationsgiveninExpression A{1 ~ q L D ~ 02 C ~ ` R B ~ 01 Aq j (A{1) Infact,sincemSUGRAisalreadycodedintoPYTHIA[93],apop ularhigh-energy collisionsimulator,thiscascadedecaymadeaconvenients tartingpointforouranalysis. Weassumedthatthesecondlightestneutralinohasalargewi nocontentandthatthe lightestneutralino(andLSP)hasalargebinocontent.Wein itiatedthecascadewitha left-squarkinsteadofaright-squarkbecausethedecayoft heright-squarktothewinois prohibited(justasthe W bosondoesnotcoupletoright-handedfermions),which,byo ur assumption,greatlysuppressesthedecay. 85

PAGE 86

~ q L ~ 02 ~ ` R ~ 01 q ` ` FigureA-1.AnexampleofFig. 1-3 AfrommSUGRAinwhichasquarkdecaystoa neutralino,whichthendecaystoaslepton,whichthennall ydecaysto anotherneutralino(DMcandidate).Thisspeciccascadede cayisactually quitepopular[11{14,26,31,36,46]. WeshowcasestudiesforapopularSUSYmodelpoint(SPS1a)in Appendix F Inparticular,thedecaychaininFig. A-1 isanexampleoftheSFSFspinassignment. However,theothervecasestudiesare not examplesofSUSY. 86

PAGE 87

APPENDIXB EXAMPLEINVERSIONFORMULASFOR N JL =1 ; 2 ; 3 Wediscoveredthattheshapeofthekinematicboundariesoft he( m 2jl ( lo ) ;m 2jl ( hi ) ) distributionrevealssomenewmeasurements.Weprovideane xampleofsimpleinversion formulasintermsofthesenewmeasurements(Equations B{1 through B{3 ),assuming thattheshapealsoindicatesoneofthethreeon-shellRegio nsofmassparameterspace. f ( m maxjl f ) 2 = m 2D (1 R CD )(1 R AB )(B{1) p m ( p ) jl f 2 = m 2D R BC (1 R CD )(1 R AB )= fR BC (B{2) n ( m maxjl n ) 2 = m 2D (1 R CD )(1 R BC )(B{3) Inordertodeterminethe4unknowns( m A m B m C and m D ),weneedonemore independentmeasurement. Fortunately,thedileptonmassedgemeasurementissimple, robust,andindependent oftheon-shellmassRegion(Equation B{4 ),sowechosetouseitasourfourthindependent measurement. a ( m maxll ) 2 = m 2D R CD (1 R BC )(1 R AB )(B{4) Equations B{5 through B{8 showtheinversionforthesquaredmassratios,andEquation s B{9 through B{12 showtheinversionforthesquaredmassesthemselves. R AB =1 f p n (B{5) R BC = p f (B{6) R CD = 1+ f p a 1 (B{7) m 2D = afn ( f p ) 2 1+ f p a (B{8) m 2A = anp ( f p ) 2 1 f p n (B{9) m 2B = anp ( f p ) 2 (B{10) 87

PAGE 88

m 2C = anf ( f p ) 2 (B{11) m 2D = anf ( f p ) 2 1+ f p a (B{12) TheseinversionformulasaremuchsimplerthanEquations 2{13 through 2{16 with 2{17 through 2{21 .Thesimplicationisduetothefactthatwearenotusingthe threshold measurement(Equation 2{11 )whoseanalyticalexpressionissomewhatcomplicated, and thatwe(implicitly)usedtheshapeofthe( m 2jl ( lo ) m 2jl ( hi ) )boundarylinestodeterminethe on-shellscenario(Fig. 2-7 ). Equations B{9 through B{12 areonlyoneexampleofthevirtuallylimitless possibilitiesprovidedbythevirtuallylimitlesschoices forobservablepointsonthe kinematicboundariesofthetwo-variabledistributions.F urthermore,anypointona kinematicsboundarycannowbeunambiguouslyexpressedint ermsofthemassesofthe heavypartners,thankstothecharacteristicshapeofthebo undarylines.Asawordof caution,though,onemusttakecaretoselect independent observables.Ultimately,the bestprocedureisprobablytottheentirekinematicsbound aryatmanydierentpoints, thusover-constrainingthemassdeterminationandincorpo ratingredundancytoimprove thecondenceofthemeasurement.However,weemphasizetha t,inprinciple,wehave providedtwoalternativesetsofobservablesthatcanbeimm ediatelyinvertedforthe massesoftheheavypartnersinthedecaychain,withoutanyn eedfornumericaltting, butattheexpenseofsomeamountofadditionaluncertaintyc omparedtothenumerical t. 88

PAGE 89

APPENDIXC HELICITYBASISFUNCTIONS fF IJ g Thedileptonhelicitybasisfunctions F ( `` ) S ; IJ (^ m 2 ; x;y;z )aregiveninTable C-1 .The j` n helicitybasisfunctions F ( j` n ) S ; IJ (^ m 2 ; x;y;z )aregiveninTable C-2 The j` f helicitybasis functions F ( j` f ) S ; IJ (^ m 2 ; x;y;z )aregiveninEquations C{1 through C{23 Theparameters x y ,and z arethesquaredmassratiosdenedbyEquation 1{3 .In ordertosimplifytheexpressions,wedenednormalization constants, N IJ S ,foreachspin assignment, S SFSF ( S =1) F ( j` f ) 1;11 (^ m 2 )= F ( j` f ) 1;12 (^ m 2 )= N IJ 1 8>>>><>>>>: (1 y ) log y if^ m 2 y 1+^ m 2 log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{1) F ( j` f ) 1;21 (^ m 2 )= F ( j` f ) 1;22 (^ m 2 )= N IJ 1 8>>>><>>>>: (1 y )+ y log y if^ m 2 y 1 ^ m 2 + y log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{2) N IJ 1 = 2 (1 y ) 2 (C{3) FSFS ( S =2) F ( j` f ) 2;11 (^ m 2 )= F ( j` f ) 2;21 (^ m 2 )= N IJ 2 8>>>><>>>>: (1 y ) log y if^ m 2 y 1+^ m 2 log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{4) F ( j` f ) 2;12 (^ m 2 )= F ( j` f ) 2;22 (^ m 2 )= N IJ 2 8>>>><>>>>: (1 y )+ y log y if^ m 2 y 1 ^ m 2 + y log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{5) N IJ 2 = N IJ 1 = 2 (1 y ) 2 (C{6) 89

PAGE 90

FSFV ( S =3) F ( j` f ) 3;11 (^ m 2 )= F ( j` f ) 3;21 (^ m 2 )= N IJ 3 8>>>><>>>>: (1 y )(1 2 z ) (1 2 yz )log y if^ m 2 y (1 ^ m 2 )(1 2 z ) (1 2 yz )log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{7) F ( j` f ) 3;12 (^ m 2 )= F ( j` f ) 3;22 (^ m 2 )= N IJ 3 8>>>><>>>>: (1 y )(1 2 z )+( y 2 z )log y if^ m 2 y (1 ^ m 2 )(1 2 z )+( y 2 z )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{8) N IJ 3 = N IJ 2 1+2 z = 2 (1 y ) 2 (1+2 z ) (C{9) FVFS ( S =4) F ( j` f ) 4;11 (^ m 2 )= N IJ 4 8>>>>>>>>>><>>>>>>>>>>: (1 y )[4 x y 4^ m 2 (2 3 x )] +[( 1+4 x ) y +4^ m 2 f 1 (2+ y )(1 x ) g ]log y if^ m 2 y (1 ^ m 2 )[4 x (2 y +1) 5 y 4^ m 2 (1 x )] +[( 1+4 x ) y +4^ m 2 f 1 (2+ y )(1 x ) g ]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{10) F ( j` f ) 4;12 (^ m 2 )= N IJ 4 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: (1 y )[2+3 y 2 x (5+ y )+4^ m 2 (2 3 x )] +[ y (4+ y ) 4 x (1+2 y ) 4^ m 2 f 1 (2+ y )(1 x ) g ]log y if^ m 2 y (1 ^ m 2 )[2+9 y 2 x (5+6 y )+2^ m 2 (1 x )] +[ y (4+ y ) 4 x (1+2 y ) 4^ m 2 f 1 (2+ y )(1 x ) g ]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{11) 90

PAGE 91

F ( j` f ) 4;21 (^ m 2 )= N IJ 4 8>>>>>>>>>><>>>>>>>>>>: (1 y )[ y 4^ m 2 (2 x )] [ y +4^ m 2 f 1+ y (1 x ) g ]log y if^ m 2 y (1 ^ m 2 )[ 5 y 4^ m 2 (1 x )] [ y +4^ m 2 f 1+ y (1 x ) g ]log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{12) F ( j` f ) 4;22 (^ m 2 )= N IJ 4 8>>>>>>>>>><>>>>>>>>>>: (1 y )[2+3 y +2 x (1 y )+4^ m 2 (2 x )] +[ y (4+ y )+4^ m 2 f 1+ y (1 x ) g ]log y if^ m 2 y (1 ^ m 2 )[2+9 y +2 x (1 2 y )+2^ m 2 (1 x )] +[ y (4+ y )+4^ m 2 f 1+ y (1 x ) g ]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{13) N IJ 4 = 3 N IJ 1 (1+2 x )(2+ y ) = 6 (1+2 x )(2+ y )(1 y ) 2 (C{14) FVFV ( S =5) F ( j` f ) 5;11 (^ m 2 )= N IJ 5 8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>: [4 x y +2 z f 2+3 y 2 x (5+ y ) g 4^ m 2 (2 3 x )(1 2 z )](1 y ) [ y 2 yz (4+ y )+4 x f 2 z y (1 4 z ) g +4^ m 2 f 1+ y x (2+ y ) g (1 2 z )]log y if^ m 2 y [4 x f 1 5 z +2 y (1 3 z ) g 5 y +2 z (2+9 y ) 4^ m 2 (1 x )(1 z )](1 ^ m 2 ) [ y 2 yz (4+ y )+4 x f 2 z y (1 4 z ) g +4^ m 2 f 1+ y x (2+ y ) g (1 2 z )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{15) 91

PAGE 92

F ( j` f ) 5;12 (^ m 2 )= N IJ 5 8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>: [2+3 y 2 x (5+ y )+2(4 x y ) z +4^ m 2 (2 3 x )(1 2 z )](1 y ) [4 x f 1+2 y (1 z ) g y (4+ y 2 z ) 4^ m 2 f 1+ y x (2+ y ) g (1 2 z )]log y if^ m 2 y [2 2 x f 5 4 z +2 y (3 4 z ) g + y (9 10 z ) +2^ m 2 (1 x )(1 4 z )](1 ^ m 2 ) [4 x f 1+2 y (1 z ) g y (4+ y 2 z ) 4^ m 2 f 1+ y x (2+ y ) g (1 2 z )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{16) F ( j` f ) 5;21 (^ m 2 )= N IJ 5 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: [ y +2 f 2+2 x (1 y )+3 y g z 4^ m 2 (2 x )(1 2 z )](1 y ) [ y f 1 2(4+ y ) z g +4^ m 2 (1+ y xy )(1 2 z )]log y if^ m 2 y [4(1+ x ) z y f 5 2(9 4 x ) z g 4^ m 2 (1 x )(1 z )](1 ^ m 2 ) [ y f 1 2(4+ y ) z g +4^ m 2 (1+ y xy )(1 2 z )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{17) F ( j` f ) 5;22 (^ m 2 )= N IJ 5 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: [2+2 x (1 y )+ y (3 2 z ) +4^ m 2 (2 x )(1 2 z )](1 y ) +[ y (4+ y 2 z )+4^ m 2 (1+ y xy )(1 2 z )]log y if^ m 2 y [2+2 x (1 2 y )+ y (9 10 z ) +2^ m 2 (1 x )(1 4 z )](1 ^ m 2 ) +[ y (4+ y 2 z )+4^ m 2 (1+ y xy )(1 2 z )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{18) N IJ 5 = N IJ 4 1+2 z = 6 (1+2 x )(2+ y )(1 y ) 2 (1+2 z ) (C{19) 92

PAGE 93

SFVF ( S =6) F ( j` f ) 6;11 (^ m 2 )= N IJ 6 8>>>>>>>>>><>>>>>>>>>>: (1 y )[2 3 z 2 y (1+ z )+4^ m 2 (1 2 z )] [ z (1+4 y ) 4^ m 2 (1 z yz )]log y if^ m 2 y (1 ^ m 2 )[2 3 z 8 yz +2^ m 2 (1 z )] [ z (1+4 y ) 4^ m 2 (1 z yz )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{20) F ( j` f ) 6;12 (^ m 2 )= N IJ 6 8>>>>>>>>>><>>>>>>>>>>: (1 y )[2 3 z +2 y (5 z )+4^ m 2 (3 2 z )] [ z (1+4 y ) 4 y (2+ y ) 4^ m 2 (1+2 y z yz )]log y if^ m 2 y (1 ^ m 2 )[2 3 z +4 y (5 2 z )+2^ m 2 (1 z )] [ z (1+4 y ) 4 y (2+ y ) 4^ m 2 (1+2 y z yz )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{21) F ( j` f ) 6;21 (^ m 2 )= N IJ 6 8>>>>>>>>>><>>>>>>>>>>: (1 y )[ z 4^ m 2 (1 2 z )] +[ yz 4^ m 2 (1 z yz )]log y if^ m 2 y (1 ^ m 2 )[ z (1+4 y ) 4^ m 2 (1 z )] +[ yz 4^ m 2 (1 z yz )]log^ m 2 if y ^ m 2 1 0if^ m 2 1 (C{22) F ( j` f ) 6;22 (^ m 2 )= N IJ 6 8>>>>>>>>>><>>>>>>>>>>: (1 y )[ z 4 y 4^ m 2 (3 2 z )] [ y (4 z )+4^ m 2 (1+2 y z yz )]log y if^ m 2 y (1 ^ m 2 )[ z 4 y (3 z ) 4^ m 2 (1 z )] [ y (4 z )+4^ m 2 (1+2 y z yz )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (C{23) N IJ 6 = 3 N IJ 1 (1+2 y )(2+ z ) = 6 (1 y ) 2 (1+2 y )(2+ z ) (C{24) 93

PAGE 94

TableC-1.Helicitybasisfunctionsforthedileptoninvari antmassdistribution.Thenormalizationconstants, N IJ S ,aregiven inEquations C{3 C{6 C{9 ,and C{24 S Spins F ( `` ) S ;11 (^ m 2 ; x;y;z )= F ( `` ) S ;21 (^ m 2 ; x;y;z ) F ( `` ) S ;12 (^ m 2 ; x;y;z )= F ( `` ) S ;22 (^ m 2 ; x;y;z ) 1 SFSF 1 1 2 FSFS 2(1 ^ m 2 )2^ m 2 3 FSFV 2 N IJ 3 N IJ 2 f 1 (1 2 z )^ m 2 g 2 N IJ 3 N IJ 2 f 2 z +(1 2 z )^ m 2 g 4 FVFS 2 2+ y f y +(2 y )^ m 2 g 2 2+ y f 2 (2 y )^ m 2 g 5 FVFV 2 N IJ 3 N IJ 2 (2+ y ) f y +4 z +(2 y )(1 2 z )^ m 2 g 2 N IJ 3 N IJ 2 (2+ y ) f 2+2 yz (2 y )(1 2 z )^ m 2 g 6 SFVF N IJ 6 N IJ 1 4 y + z +4(1 2 y z + yz )^ m 2 4(1 y )(1 z )^ m 4 N IJ 6 N IJ 1 z +4(1 z + yz )^ m 2 4(1 y )(1 z )^ m 4 TableC-2.Helicitybasisfunctionsforthe j` n invariantmassdistribution.Thenormalizationconstants N IJ S ,aregivenin Equations C{3 and C{14 S Spins F ( j` n ) S ;11 (^ m 2 ; x;y;z )= F ( j` n ) S ;12 (^ m 2 ; x;y;z ) F ( j` n ) S ;21 (^ m 2 ; x;y;z )= F ( j` n ) S ;22 (^ m 2 ; x;y;z ) 1 SFSF 2^ m 2 2(1 ^ m 2 ) 2 FSFS 1 1 3 FSFV 1 1 4 FVFS N IJ 4 N IJ 1 y +4(1 y + xy )^ m 2 4(1 x )(1 y )^ m 4 N IJ 4 N IJ 1 4 x + y +4(1 2 x y + xy )^ m 2 4(1 x )(1 y )^ m 4 5 FVFV N IJ 4 N IJ 1 y +4(1 y + xy )^ m 2 4(1 x )(1 y )^ m 4 N IJ 4 N IJ 1 4 x + y +4(1 2 x y + xy )^ m 2 4(1 x )(1 y )^ m 4 6 SFVF 2 1+2 y f 2 y +(1 2 y )^ m 2 g 2 1+2 y f 1 (1 2 y )^ m 2 g 94

PAGE 95

APPENDIXD OBSERVABLESPINBASISFUNCTIONS fF ; F ; F r ; F g Thedileptonobservablebasisfunctions F ( `` ) S ; F ( `` ) S ; F ( `` ) S ; r and F ( `` ) S ; aregivenin Table D-1 .The j` n observablebasisfunctions F ( j` n ) S ; F ( j` n ) S ; F ( j` n ) S ; r and F ( j` n ) S ; aregivenin Table D-2 .The j` f observablebasisfunctions F ( j` f ) S ; F ( j` f ) S ; F ( j` f ) S ; r and F ( j` f ) S ; aregivenin Equations D{1 through D{26 Theparameters x y ,and z arethesquaredmassratiosdenedbyEquation 1{3 .In ordertosimplifytheexpressions,wedenednormalization constants, N S ,foreachspin assignment, S SFSF ( S =1) F ( j` f ) 1; (^ m 2 )= F ( j` f ) 1; r (^ m 2 )=0(D{1) F ( j` f ) 1; (^ m 2 )= N 1 8>>>><>>>>: 2(1 y ) (1+ y )log y if^ m 2 y 2(1 ^ m 2 ) (1+ y )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{2) F ( j` f ) 1; (^ m 2 )= 8>>>><>>>>: log y 1 y if^ m 2 y log^ m 2 1 y if y ^ m 2 1 0if^ m 2 1 (D{3) N 1 = 1 (1 y ) 2 (D{4) FSFS ( S =2) F ( j` f ) 2; (^ m 2 )= F ( j` f ) 1; (^ m 2 )= N 2 8>>>><>>>>: 2(1 y ) (1+ y )log y if^ m 2 y 2(1 ^ m 2 ) (1+ y )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{5) F ( j` f ) 2; (^ m 2 )= F ( j` f ) 2; r (^ m 2 )=0(D{6) 95

PAGE 96

F ( j` f ) 2; (^ m 2 )= F ( j` f ) 1; (^ m 2 )= 8>>>><>>>>: log y 1 y if^ m 2 y log^ m 2 1 y if y ^ m 2 1 0if^ m 2 1 (D{7) N 2 = N 1 = 1 (1 y ) 2 (D{8) FSFV ( S =3) F ( j` f ) 3; (^ m 2 )= N 3 8>>>><>>>>: 2(1 y ) (1+ y )log y if^ m 2 y 2(1 ^ m 2 ) (1+ y )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{9) F ( j` f ) 3; (^ m 2 )= F ( j` f ) 3; r (^ m 2 )=0(D{10) F ( j` f ) 3; (^ m 2 )= F ( j` f ) 2; (^ m 2 )= F ( j` f ) 1; (^ m 2 )= 8>>>><>>>>: log y 1 y if^ m 2 y log^ m 2 1 y if y ^ m 2 1 0if^ m 2 1 (D{11) N 3 = N 2 1 2 z 1+2 z = 1 2 z (1 y ) 2 (1+2 z ) (D{12) FVFS ( S =4) F ( j` f ) 4; (^ m 2 )= N 4 8>>>>>>>>>><>>>>>>>>>>: (1 y )[ 2(1+2 y )+2 x (3+ y ) 16^ m 2 (1 x )] [ y (5+ y ) 2 x (1+3 y )+8^ m 2 (1 x )(1+ y )]log y if^ m 2 y (1 ^ m 2 )[ 2(1+7 y )+6 x (1+2 y ) 6^ m 2 (1 x )] [ y (5+ y ) 2 x (1+3 y )+8^ m 2 (1 x )(1+ y )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (D{13) F ( j` f ) 4; (^ m 2 )=2 xN 4 8>>>><>>>>: 2(1 y ) (1+ y )log y if^ m 2 y 2(1 ^ m 2 ) (1+ y )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{14) 96

PAGE 97

F ( j` f ) 4; r (^ m 2 )=2 xN 4 8>>>><>>>>: 4(1 y )(1+^ m 2 )+[(1+3 y )+4^ m 2 ]log y if^ m 2 y 4(1 ^ m 2 )(1+ y )+[(1+3 y )+4^ m 2 ]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (D{15) F ( j` f ) 4; (^ m 2 )= N 4 8>>>>>>>>>><>>>>>>>>>>: 2(1 y )(1+ y )(1 x ) +[ 2 x (1+ y )+ y (3+ y )]log y if^ m 2 y 2(1 ^ m 2 )(1 x ) f (1+2 y ) ^ m 2 g +[ 2 x (1+ y )+ y (3+ y )]log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{16) N 4 = 3 N 1 (1+2 x )(2+ y ) = 3 (1+2 x )(2+ y )(1 y ) 2 (D{17) FVFV ( S =5) F ( j` f ) 5; (^ m 2 )= N 5 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: (1 y )[ 2(1+2 y )+2 x (3+ y ) 16^ m 2 (1 x )] [ y (5+ y ) 2 x (1+3 y ) +8^ m 2 (1 x )(1+ y )]log y if^ m 2 y (1 ^ m 2 )[ 2(1+7 y )+6 x (1+2 y ) 6^ m 2 (1 x )] [ y (5+ y ) 2 x (1+3 y ) +8^ m 2 (1 x )(1+ y )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (D{18) F ( j` f ) 5; (^ m 2 )=2 xN 4 8>>>><>>>>: 2(1 y ) (1+ y )log y if^ m 2 y 2(1 ^ m 2 ) (1+ y )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{19) F ( j` f ) 5; r (^ m 2 )=2 xN 5 8>>>>>>>>>><>>>>>>>>>>: 4(1 y )(1+^ m 2 ) +[(1+3 y )+4^ m 2 ]log y if^ m 2 y 4(1 ^ m 2 )(1+ y ) +[(1+3 y )+4^ m 2 ]log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{20) 97

PAGE 98

F ( j` f ) 5; (^ m 2 )= N 4 8>>>>>>>>>><>>>>>>>>>>: 2(1 y )(1+ y )(1 x ) +[ 2 x (1+ y )+ y (3+ y )]log y if^ m 2 y 2(1 ^ m 2 )(1 x ) f (1+2 y ) ^ m 2 g +[ 2 x (1+ y )+ y (3+ y )]log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{21) N 5 = N 4 1 2 z 1+2 z = 3(1 2 z ) (1+2 x )(2+ y )(1 y ) 2 (1+2 z ) (D{22) SFVF ( S =6) F ( j` f ) 6; (^ m 2 )=2 yN 6 8>>>><>>>>: 2(1 y ) (1+ y )log y if^ m 2 y 2(1 ^ m 2 ) (1+ y )log^ m 2 if y ^ m 2 1 0if^ m 2 1 (D{23) F ( j` f ) 6; (^ m 2 )= N 6 8>>>>>>>>>><>>>>>>>>>>: (1 y )[2(1+3 y ) 2(2+ y ) z +16^ m 2 (1 z )] +[2 y (3+ y ) (1+5 y ) z +8^ m 2 (1+ y )(1 z )]log y if^ m 2 y (1 ^ m 2 )[2(1+8 y ) 4(1+3 y ) z +6^ m 2 (1 z )] +[2 y (3+ y ) (1+5 y ) z +8^ m 2 (1+ y )(1 z )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (D{24) F ( j` f ) 6; r (^ m 2 )=2 N 6 8>>>><>>>>: 4(1 y )( y +^ m 2 ) [ y (3+ y )+4 y ^ m 2 ]log y if^ m 2 y 8 y (1 ^ m 2 ) [ y (3+ y )+4 y ^ m 2 ]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (D{25) F ( j` f ) 6; (^ m 2 )= N 6 8>>>>>>>>>><>>>>>>>>>>: 2(1 y )(1+ y )(1 z ) +[ (1 y )(1+2 y )+(1+3 y )(1 z )]log y if^ m 2 y 2(1 ^ m 2 )(1 z ) f (1+2 y ) ^ m 2 g +[ (1 y )(1+2 y )+(1+3 y )(1 z )]log^ m 2 if y ^ m 2 1 0 if^ m 2 1 (D{26) N 6 = 3 N 1 (1+2 y )(2+ z ) = 3 (1 y ) 2 (1+2 y )(2+ z ) (D{27) 98

PAGE 99

TableD-1.Observablebasisfunctionsforthedileptoninva riantmassdistribution.Thenormalizationconstants, N S ,are giveninEquations D{4 D{8 D{12 D{17 D{22 ,and D{27 S Spins F ( `` ) S ; (^ m 2 ; x;y;z ) F ( `` ) S ; (^ m 2 ; x;y;z ) F ( `` ) S ; (^ m 2 ; x;y;z ) F ( `` ) S ; r (^ m 2 ; x;y;z ) 1 SFSF 1000 2 FSFS 11 2^ m 2 00 3 FSFV 1 N 3 N 2 f 1 2^ m 2 g 00 4 FVFS 1 2 y 2+ y f 1 2^ m 2 g 00 5 FVFV 1 (2 y ) N 5 (2+ y ) N 4 f 1 2^ m 2 g 00 6 SFVF N 6 N 1 2 y + z +4(1 y )(1 z )^ m 2 4(1 y )(1 z )^ m 4 2 yN 6 N 1 f 1 2^ m 2 g 00 TableD-2.Observablebasisfunctionsforthe j` n invariantmassdistribution.Thenormalizationconstants N S ,aregivenin Equations D{4 and D{17 S Spins F ( j` n ) S ; (^ m 2 ; x;y;z ) F ( j` n ) S ; (^ m 2 ; x;y;z ) F ( j` n ) S ; (^ m 2 ; x;y;z ) F ( j` n ) S ; r (^ m 2 ; x;y;z ) 1 SFSF 10 f 1 2^ m 2 g 0 2 FSFS 1000 3 FSFV 1000 4 FVFS N 4 N 1 2 x + y +4(1 x )(1 y )(^ m 2 ^ m 4 0 2 xN 4 N 1 f 1 2^ m 2 g 0 5 FVFV N 4 N 1 2 x + y +4(1 x )(1 y )(^ m 2 ^ m 4 0 2 xN 4 N 1 f 1 2^ m 2 g 0 6 SFVF 10 1 2 y 1+2 y f 1 2^ m 2 g 0 99

PAGE 100

APPENDIXE SIMPLESPINFITTINGPROCEDURE Withvanishingerrorbars,weusetherathernaivematchingc riteriongivenby Equation E{1 2 ( ;;r ) Z 1 0 f 0 (^ m 2 ; 0 ; 0 ;r 0 ) f (^ m 2 ;;;r ) 2 d ^ m 2 (E{1) Thefunction f 0 (^ m 2 ; 0 ; 0 ;r 0 )representstheobserveddistributiontobetted,and f (^ m 2 ;;;r )isthetheoreticaltestdistribution.Weminimizethe 2 ( ;;r )functionfor and/or r ,asappropriate. 0 0 and r 0 arexedconstantvaluesthatweselectedfor thegivenstudypoint(Equations F{4 and F{9 ).Wesimplyinterpretedanynonzerovalue of 2 asamismatch.Onethingthatwefoundinteresting,then,was thatwestillwere abletotsome\wrong"spinassignments(Appendix F ). Thettothe L + or S + distributionisasimpleone-parametertfor ,whilethe ttothe D + distributionisatwo-parametertfor and r .Fig. E-1 showssample resultsfromour D + tsfor and r (Appendix F ).IneachplotinFig. E-1 ,the f 0 (^ m 2 ; 0 ; 0 ;r 0 )distributioncomesfromtherstspinchain(red)writtena tthetop ofeachplot,whichisthenttothe f (^ m 2 ;;;r )distributionpredictedbythesecond spinchain(blue)writtenatthetopofeachplot.Thecontour linesrepresentconstant valuesof 2 ( ;;r ),where hasalreadybeenxedbyttingto L + .Thebluedot correspondstotheabsoluteminimumof 2 ,ignoringanyrestrictionson and r Theparameters and r arenotcompletelyindependentfromeachother.Forany given ,thephysicallyallowedregioninthe( ;r )parameterspaceisdescribedbyan envelope(Equations E{2 through E{5 ). r;r ;r if > 0 ;> 0and r> 0(E{2) r;r ;r if > 0 ;< 0and r< 0(E{3) r;r ;r if < 0 ;> 0and r< 0(E{4) r;r ;r if < 0 ;< 0and r> 0(E{5) 100

PAGE 101

FigureE-1.Contourplotsof 2 ( ;;r )asafunctionof and r ,with alreadyxedby thettothe L + data.Theregionsatisfyingtheconstraints(Equations E{2 through E{5 )isshadedinwhite.Thebluedotdenotesthe global 2 minimum,whichdoesnotnecessarilyrepresentaconsistent setofmodel parameters.Thegreentriangledenotesthelocationofthe 2 minimum withintheconsistentregion.The f 0 (^ m 2 ; 0 ; 0 ;r 0 )distributioncomesfrom therstspinassignment(showninred)atthetopofeachplot ,whichisthen ttedtothe f (^ m 2 ;;;r )distribution(showninblue). 101

PAGE 102

ThisenvelopeisrepresentedbythewhiteregioninFig. E-1 .Thegreentriangle correspondstotheminimumofthe 2 functionwithinthisrestrictedparameterspace. Thegreentrianglesolutionfor and r wasthenusedforourplotsinFig. F-3 .For thetwoFVFV(S=5)cases,theglobalminimumhappenstoliewi thintheregionof consistency(white),andsothebluedotandthegreentriang lecoincide.Fortheextreme valuesof ,theregionofconsistency(white)collapsestooneortwoli nes,asindicatedby Equations E{6 and E{7 =0or r =0if =0(E{6) r = if = 1(E{7) 102

PAGE 103

APPENDIXF SPINDETERMINATIONEXAMPLES F.1TheSPS1aStudyPoint Weprovideanexplicitdemonstrationofourmethod.Wersts imulatedobservable distributionsforeachofthe6spincongurationsthatarel istedinTable 1-2 .Then,for eachcaseweusedthesimplettingprocedureinAppendix E todeterminewhetherthe observabledistributionswereconsistentwitheachofther emaining5alternatives. Sincewedonotyethaverealdataavailable,wearticiallyc hoseinputparameter values(butonlyforthesakeofdemonstration).Inordertof acilitatecomparisonsto previousstudies,ourinputparameters(Equations F{1 through F{9 )correspondtothe SPS1astudypointinsupersymmetry.However,thespinassig nmentsonlycorrespondto thisstudypointinthecaseofSFSF.WeusedthevaluesfromRe fs.[12,14]. m A =96GeV m B =143GeV m C =177GeV m D =537GeV(F{1) x =0 : 109 y =0 : 653 z =0 : 451(F{2) a L =0 a R =1 b L =0 b R =1 c L =1 c R =0(F{3) f =0 : 7 f =0 : 3(F{4) m max`` = m D p x (1 y )(1 z )=77 : 31GeV(F{5) m maxj` n = m D p (1 x )(1 y )=298 : 77GeV(F{6) m maxj` f = m D p (1 x )(1 z )=375 : 76GeV(F{7) m maxj`` = m D p (1 x )(1 yz )=425 : 94GeV(F{8) =1 = 0 : 4 r = 0 : 4(F{9) Notethat =1necessarilyimpliesthat = r (Equations 3{67 through 3{69 ).Equation F{9 denestheinputvaluesofthemodelparametersusedinourca sestudy. 103

PAGE 104

FigureF-1.Dileptoninvariantmassdistributions, L + S .Thesolid(magenta)lineineach plotrepresentsthesimulatedobserveddileptondistribut ionforeachofthe6 spinassignments:A)SFSF;B)FSFS;C)FSFV;D)FVFS;E)FVFV; F) SFVF.Theother(dottedordashed)linesrepresentourbest tstothisinput distribution,foreachoftheremaining5spinconguration s.Adashed(green) linerepresentsanexactt,andadottedlinerepresentsani nexactt(colored otherthangreenormagenta).Thebesttvalueof foreachcaseisalso shown,exceptforcaseswhereitisleftundetermined(NA). 104

PAGE 105

FigureF-2.Sumof j` + and j` invariantmassdistributions, S + .Thedierentline stylesbasicallyrepresentthesameconditionsasinFig. F-1 ,exceptthatnot wasperformed,andinsteadthedistributionswereplottedw iththespecic valueof thatwasobtainedfromthetto L + 105

PAGE 106

FigureF-3.ThesameasinFig. F-1 butfor D + insteadof L + ,andatto and r insteadof 106

PAGE 107

Althoughthetcanbedonesimultaneouslyforallthreepara meters and r weperformeditsequentially,usingthefactthatthe L + S and S + S distributionsdepend onlyon andnoton and r .Therefore,webeganwiththe L + S distributionandrst determinedthevalueof ,whichwethenusedtocomparethe S + S testdistributionto the S + S observabledistribution.Alreadyatthisstageitmaybepos sibletoruleout allbutthecorrectspinconguration,andweencounteredsu chexamples.Sometimes, however,theremaystillbeseveralalternativesleft,inwh ichcasethe D + distribution mustalsobeconsidered,wherewetforthevaluesofthecoe cients and r .Atany rate,thereisnoharminusingallthreedistributions, L + S S + S ,and D + S OurresultsaresummarizedinFigures F-1 through F-3 ,whichshowthe L + S S + S and D + S distributionsforeachofthe6inputspinassignments,aswe llasthe5 testdistributionsforeachcase.Intheseguresthesolid( magenta)linesrepresentthe observeddistributions,andtheother(dottedordashed)li nesrepresentourbestts ofthetestdistributionstotheobserveddistributions,fo reachoftheremaining5spin congurations(Table 1-2 ).Adashed(green)linerepresentsanexactttotheobserve d distribution.The(color-coded)dottedlinesrepresentdi stributionsthatcouldnotbet exactlytotheinputdistribution.Thebesttvaluesof and r foreachcasearealso shown,exceptforthosecases(labeledby\NA")wheretheyar eleftundeterminedbythe t. F.2InputfromSFSF( S =1 ) WebegantheSFSFcasestudywiththeobservabledistributio nsinEquations F{10 through F{12 .Fig. A-1 givesanexampleofapossibledecaydiagramforthisspin assignment. L + 1 =1 (F{10) S + 1 = 8>>>><>>>>: 2 : 810 ^ m 2j` 0 : 632 1 : 2280 : 632 ^ m 2j` 0 : 653 2 : 880log^ m 2j` 0 : 653 ^ m 2j` (F{11) 107

PAGE 108

D + 1 = 8>>>><>>>>: 0 : 668+2 : 002^ m 2j` ^ m 2j` 0 : 632 0 : 0350 : 632 ^ m 2j` 0 : 653 6 : 633 6 : 633^ m 2j` +5 : 481log^ m 2j` 0 : 653 ^ m 2j` (F{12) Thesedistributionsareshownwithsolid(magenta)linesin Figs. F-1 A, F-2 Aand F-3 A,respectively.WersttthedileptondatainFig. F-1 A.Duetothepresenceof anintermediatescalarparticleB,the L + distributionfortheSFSFchain(S=1),is completelyrat.However,thatdoes not implythatthespinofparticleBisdetermined tobezero.Infact,asseenfromFig. F-1 A,allotherspincongurationsexceptSFVF canalsotthisratdistribution,simplybychoosingavanis hing parameter.Eventhe caseof S =6(SFVF),whose\bestt"predictionisdierentfromthein putdata,may stillbediculttodiscriminateinpractice,onceexperime ntalerrorsareintroduced.The badnews,therefore,isthatwecannotimmediatelydetermin ethespinsfromthe L + distributionalone,butthegoodnewsisthat,asanticipate d,weobtainedameasurement ofthe parameter(i.e. =0),whichrepresentssomecombinationofheavyparticle couplingsandmixingangles(Equation 3{67 ). WecompareourFig. F-1 AtoFig.2ainRef.[14],whereaverysimilarexercisewas performed.Thetworesultsarequitedierent.Forexamplew efoundthat4outofthe5 \wrong"modelscanperfectlytthedileptondistribution, whileinRef.[14]all6models givedistinctdileptonshapes.Thedierencearisesduetoo urdierentphilosophy.In Ref.[14]theparameterscorrespondingto and r inournotationwereallkeptxedto theSPS1avalues(Equation F{9 ),whereasourtechniqueallowsthemtoroat,sincethey wouldnothavebeenmeasuredinadvance.Asaresult,oncewef actorintheexperimental realism,wedeterminedthattheactualspinmeasurementsar emorechallengingthan previouslyanticipated. S + dependsonexactlythesamemodelparameter asthedileptondistribution L + .Sincewemeasured byttingtothe L + distribution,therewerenofree parametersleftinthe S + distribution,anditwasuniquelypredictedforeachspin 108

PAGE 109

assignment.Fig. F-2 Ashowstheresultingpredictionsforthe6spinmodels,usin gineach casethecorrespondingvalueof fromthe L + t.Inthiscase,the S + distribution can dierentiatesomeofthespinassignments,e.g.itcanruleo ut(inprinciple)theFVFSand FVFVspinassignments.Interestingly,SFVFgivesaperfect match,butfortunately,itwas alreadyeliminatedfromconsiderationbythe L + t.Unfortunately,the\wrong"spin scenariosFSFSandFSFVgiveaperfectmatchforboth L + and S + ,leaving3distinct possibilitiesforthespinsoftheheavypartners. Wewerethereforeforcedtoconsiderourthirdpieceofdata, the D + distribution (Equation 3{72 ).Thisdistributiondoesnotdependonthepreviouslytted parameter ,andinsteadmustbettedwiththeothertwomodelparameter s, and r .Even though D + itselfdoesnotexplicitlydependon ,thetisneverthelessimpactedby themeasuredvalueof ,as restrictstheallowedrangeofvaluesfor and r (Appendix E ).Theresultsfromourttingexercisefor D + areshowninFig. F-3 A.Weseethat D + cannoweliminatetheremainingtwo\wrong"spinscenariosF SFSandFSFV, and,asaresultofallthreetypesofts,wewereabletodeter mineuniquelythespin assignmentasSFSF(fromtheidealdistributions).Inaddit ion,wewerealsoableto obtainameasurementoftheparameter ,whichcarriesinformationaboutthecouplings andmixinganglesoftheheavypartners D C and B .Unfortunately,theparameters and r arenotexperimentallyaccessibleforSFSF,sincetheircor respondingbasisfunctions F ( p ) 1; and F ( p ) 1; r areidenticallyzero(Appendix D ). F.3InputfromFSFS( S =2 )andFSFV( S =3 ) WebegantheFSFScasestudywiththeobservabledistributio nsinEquations F{13 through F{15 ,andwebegantheFSFVcasestudywiththeobservabledistrib utionsin Equations F{16 through F{18 .Fig. F-4 showsthediagramsforthesespinassignments. L + 2 =2 2^ m 2`` (F{13) 109

PAGE 110

DCBA q ` ` DCBA q ` ` AB FigureF-4.DecaydiagramsforA)FSFSandB)FSFV.Theonlydi erencebetween thesespinassignmentsisthespinofthenal-stateparticl e A ,whichcanbe eitherascalaroravector. S + 2 = 8>>>><>>>>: 2 : 898 ^ m 2j` 0 : 632 1 : 316 0 : 632 ^ m 2j` 0 : 653 16 : 583+16 : 583^ m 2j` 16 : 583log^ m 2j` 0 : 653 ^ m 2j` (F{14) D + 2 =0 (F{15) L + 3 =1 : 052 0 : 104^ m 2`` (F{16) S + 3 = 8>>>><>>>>: 2 : 815 ^ m 2j` 0 : 632 1 : 2330 : 632 ^ m 2j` 0 : 653 0 : 860+0 : 860^ m 2j` 3 : 590log^ m 2j` 0 : 653 ^ m 2j` (F{17) D + 3 =0 (F{18) Perhapsthemoststrikingfeatureisthatthe D + distribution,andconsequentlythe leptonchargeasymmetry A + ,arebothidenticallyzero.Therefore,theydonotconvey anyinformationaboutthespins,sinceanyspinconguratio ncantthosedistributions withtheproperchoiceofparametersasshowninFigs. F-3 Band F-3 C.Thisbeingthe case,weconcentratedonthe L + and S + distributions. WerstconsideredtheFSFSspinassignment.Again,webegan ouranalysiswith L + ,whichinthiscaseshowsverygooddiscrimination,andcana lreadyruleout all of the\wrong"spincombinations.And,whileFSFSdistributio nscansometimesbefaked byFSFVdistributions,thiscouldonlyhappeniftheinput parametersatisesEquation 110

PAGE 111

3{88 .SinceforSPS1a =1(Equation F{9 ),thisconditionwasnotsatisedinour speciccasestudy,andtheFSFVmodeldidnotfaketheFSFSdi stribution.Thisis conrmedbyourresultinFig. F-1 B. Sincethe L + distributionalonealreadysinglesoutthecorrectspincon guration, wedidnotevenneedtoconsiderthe S + distribution.Itisworthpointingout,however, that S + inthisidealcasealsocanruleoutall\wrong"spinmodels(a lthoughthe dierencesarenotsopronouncedasfor L + evenusingtheseidealdistributions).Wealso obtainedameasurementoftheparameter .However,theparameters and r remain undetermined,sincetheircorrespondingbasisfunctions F ( p ) 2; and F ( p ) 2; r areidenticallyzero forany p 2f ``;j` n ;j` f g (Appendix D ). WenextconsideredtheFSFVspinassignment,whichprovides anexampleofan inconclusivespindetermination.Ofcourse,thisresultwa salreadyanticipated,sincewe alreadydeterminedthatFSFScan always provideaperfectttotheFSFVdistributions. Thevalueof 2 thatismeasuredforthefakeFSFSmodelispredictedbyEquat ion F{19 2 = 3 1 2 z 1+2 z 0 : 05(F{19) Ournumericalstudyexplicitlyconrmsthisgeneralexpect ationasshowninFigs. F-1 C, F-2 Cand F-3 C.Inaddition,weveriedthatthe m j`` distributionsforthesetwo\twin" spinmodelsarealsoidentical. F.4InputfromFVFS( S =4 )andFVFV( S =5 ) WebegantheFVFScasestudywiththeobservabledistributio nsinEquations F{20 through F{22 ,andwebegantheFVFVcasestudywiththeobservabledistrib utionsin Equations F{23 through F{25 .Fig. F-5 showsthediagramsforthesespinassignments. L + 4 =0 : 492+1 : 016^ m 2`` (F{20) 111

PAGE 112

DCBA q ` ` DCBA q ` ` AB FigureF-5.DecaydiagramsforA)FVFSandB)FVFV.Theonlydi erencebetween thesespinassignmentsisthespinofthenal-stateparticl e A ,whichcanbe eitherascalaroravector.Thesediagramsarerelatedtotho seinFig. F-4 by changingparticle C fromascalartoavector. S + 4 = 8>>>>>>><>>>>>>>: 2 : 307+3 : 455^ m 2j` 4 : 553^ m 4j` ^ m 2j` 0 : 632 1 : 028+0 : 577^ m 2j` 0 : 632 ^ m 2j` 0 : 653 42 : 563 12 : 368^ m 2j` +54 : 931^ m 4j` 7 : 871+90 : 785^ m 2j` log^ m 2j` 0 : 653 ^ m 2j` (F{21) D + 4 = 8>>>>>>><>>>>>>>: 0 : 22+0 : 616^ m 2j` ^ m 2j` 0 : 632 0 : 092+0 : 212^ m 2j` 0 : 632 ^ m 2j` 0 : 653 3 : 087+3 : 087^ m 2j` 0 : 874+2 : 678^ m 2j` log^ m 2j` 0 : 653 ^ m 2j` (F{22) L + 5 =0 : 974+0 : 053^ m 2`` (F{23) S + 5 = 8>>>>>>><>>>>>>>: 2 : 496+2 : 908^ m 2j` 4 : 553^ m 4j` ^ m 2j` 0 : 632 1 : 217+0 : 030^ m 2j` 0 : 632 ^ m 2j` 0 : 653 27 : 809 43 : 679^ m 2j` +15 : 870^ m 4j` + 14 : 382 4 : 710^ m 2j` log^ m 2j` 0 : 653 ^ m 2j` (F{24) D + 5 = 8>>>>>>><>>>>>>>: 0 : 139+0 : 415^ m 2j` ^ m 2j` 0 : 632 0 : 011+0 : 011^ m 2j` 0 : 632 ^ m 2j` 0 : 653 1 : 109 1 : 109^ m 2j` + 1 : 004 0 : 139^ m 2j` log^ m 2j` 0 : 653 ^ m 2j` ; (F{25) 112

PAGE 113

DCBA q ` ` FigureF-6.DecaydiagramforSFVF. Theendresultofthesetwocasestudiesissimilartowhatweo btainedfortheother \twin"modelpairFSFSandFSFV,whichwasalsoalreadyantic ipated,sincewealready determinedthatFVFScan always provideaperfectttotheFVFVdistributions.When beginningwithFVFSandttingtheother5spinassignments, wedidnotencounterany spinambiguities.AswealreadydeterminedforFSFS,thisis anumericalaccidentdueto theparticularchoiceoftheSPS1astudypoint(namely =1),whichdoesnotsatisfy thenecessarycondition(Equation 3{96 )fortheFVFVspinassignmenttofaketheFVFS distributions.Asaresult,thetwo L + and S + distributionsweresucienttodetermine theFVFSscenario,andthe D + distributioncouldthenbeusedasacross-check,andto measurethe and r parameters. However,whenstartingwithFVFVandttingtheother5spina ssignments,we did encounteraspinambiguity.Again,thereasonforthiswasal readyanticipated.In agreementwithouranalyticalresults,Figs. F-1 E, F-2 Eand F-3 Eshowthatall3ofthe FVFSdistributionsprovideidenticalmatchestothecorres pondingFVFVdistributions. However,whileweencounteredatwo-foldambiguitywithres pecttothespinofparticle A foreachspinassignmenttheparameters and r wereallmeasured. F.5InputfromSFVF( S =6 ) WebegantheSFVFcasestudywiththeobservabledistributio nsinEquations F{26 through F{28 .Fig. F-6 showsthedecaydiagramforthisspinassignments. L + 6 =1 : 626 0 : 981^ m 2`` 0 : 405^ m 4`` (F{26) 113

PAGE 114

S + 6 = 8>>>>>>><>>>>>>>: 2 : 87 ^ m 2j` 0 : 632 1 : 2880 : 632 ^ m 2j` 0 : 653 0 : 344 4 : 493^ m 2j` +4 : 837^ m 4j` 5 : 870log^ m 2j` 0 : 653 ^ m 2j` (F{27) D + 6 = 8>>>>>>><>>>>>>>: 0 : 322+0 : 786^ m 2j` ^ m 2j` 0 : 632 0 : 406+1 : 051^ m 2j` 0 : 632 ^ m 2j` 0 : 653 5 : 870 11 : 674^ m 2j` +5 : 804^ m 4j` + 3 : 384 3 : 595^ m 2j` log^ m 2j` 0 : 653 ^ m 2j` (F{28) TheSFVFspinassignmentisspecial,sinceinthiscasethedi leptoninvariantmass distributionexhibitsacharacteristic^ m 4 termwhichisnotpresentforanyoftheother5 spincongurations.Notethattheexistenceofan^ m 4 terminthedileptondistribution doesnotrequireanyspecialvaluesofthemodelparameters. Actually,the^ m 4 termcannot evenberemovedbyanychoiceofmodelparametervalues.More generally,thedilepton invariantmassdistributionisingeneralgivenbysomepoly nomialintermsof^ m 2 ,whose orderisequaltotwicethespinoftheintermediateparticle B .OnlyintheSFVFcase isthereaspin1intermediateparticlewhichintroducesthe ^ m 4 termin L + .Therefore, observationofthe^ m 4 dependenceinthedileptondistributionunambiguouslysel ects SFVFoutofour6possiblespinassignments.Thesizeoftheco ecientofthe^ m 4 term dependsonthemassspectruminthemodel,butitcannotbevan ishinglysmall{this wouldrequireeither z =1or y =1,whichwouldcorrespondinglycloseothe B A` or the C B` decay,andthenthewholedecaychainwouldbeunobservable. Ournumerical resultsinFig. F-1 Fconrmthisconclusion{wesawthatnoneoftheothervemod els couldreproducetheSFVFdileptondistribution,duetothep resenceofthe^ m 4 term. F.6RemarksonSpinDeterminationattheTevatron Ata p p collidersuchastheTevatronweexpecttheparticle-antipa rticlefractionto be f = 1 2 .Onthesurface,itmayappearthatthisconstrainteliminat esonlyoneout 114

PAGE 115

ofthefourmodel-dependentdegreesoffreedom( f a b and c )thatweoriginally startedwith.However,ascanbededucedfromEquations 3{47 and 3{48 ,thisassumption completelyxesthe~ c parameter(Equation F{29 ),andasaresultboth and r vanish identically. ~ c = 4 (F{29) Inthiscase,bothour D + S distributionandtheleptonchargeasymmetrydistribution ( A + )alsovanishidentically,sothatthesedistributionsareu selessforspindetermination ata p p collidersuchastheTevatron. However,ourresultsfor L + S and S + S stillhold,andcontainnon-trivialspin information,sothatthespinanalysisfollowingourmethod canstillbeperformed. Infact,ourmethodcanalreadybetestedinthetopquarksemi leptonicanddilepton samplesattheTevatronbylookingattheinvariantmassdist ributionofthe b -jetandthe lepton[63,69].Indeed,ourdecaychainfromFig. 1-3 canbeappliedtotopquarkdecays, forexamplebyidentifying C = t B = W + and A = ` ,andreinterpreting ` n asthe b -jetand ` f astheleptoncomingfrom W decay.Inthatcase,the m b` distributionshould bedescribedbyEquation 3{64 for L + 6 .Alternatively,onecanidentifytheparticlesin Fig. 1-3 as D = t C = W + B = ` q = b and ` n = ` .Inthiscase,the m b` distribution shouldbedescribedbyEquation 3{65 for S =4or S =5.Ineithercase,thecharacteristic ^ m 4 termsignalsthatthe W isspin1andthetopquarkandtheneutrinoarebothspin 1/2(theFVFpartofthespinassignment). 115

PAGE 116

REFERENCES [1]StephenP.Martin, ASupersymmetryPrimer ,Dept.ofPhys.,N.ILUniv.,DeKalb, IL60115andFNALP.O.Box500,Batavia,IL60510[arXiv:hepph/9709356v4]. [2]MichaelE.Peskin, Supersymmetry:Theory ,SLAC-PUB-7125,March,1996. [3]P.Igo-Kemenes, SearchesforHiggsBosons ,J.Phys.G,NuclearandParticle Physics,Volume33July2006,pp.388-397(i.e.thePDGRevie wofParticlePhysics). [4]TevatronNewPhenomena,Higgsworkinggroup,fortheCDF collaboration, DZerocollaboration, CombinedCDFandDZeroUpperLimitsonStandardModel Higgs-BosonProductionwithupto4.2fb-1ofData ,FERMILAB-PUB-09-060-E [arXiv:0903.4001v1[hep-ex]]. [5]G.Abendi,et.al.,(theALEPHCollaboration,theDELPHI Collaboration,the L3CollaborationandtheOPALCollaboration,TheLEPWorkin gGroupfor HiggsBosonSearches), SearchfortheStandardModelHiggsBosonatLEP Phys.Lett.B565:61-75,2003[arXiv:hep-ex/0306033v1]. [6]SallyDawson,MarkOreglia, PhysicsOpportunitieswithaTeVLinearCollider Ann.Rev.Nucl.Part.Sci.54(2004)269-314,p.2[arXiv:hep -ph/0403015v1]. [7]TaoHan, ColliderPhenomenologyBasicKnowledgeandTechniques MADPH-05-1434[arXiv:hep-ph/0508097v1]. [8]D.Chakraborty,T.Sumiyoshi,K.F.Johnson,C.L.Woody, R.-Y.Zhu, B.N.Ratcli,D.Casper,D.Froidevaux,A.Cattai,G.Roland i,M.TRonan, H.R.Band,J.Zhang,M.TRonan,H.Spieler,R.-Y.Zhu ParticleDetectors J.Phys.G,NuclearandParticlePhysics,Volume33July2006 ,pp.271-285,286-288 (i.e.thePDGReviewofParticlePhysics). [9] SummaryTablesofParticlePhysics ,J.Phys.G,NuclearandParticlePhysics, Volume33July2006,pp.31-82(i.e.thePDGReviewofParticl ePhysics),Extracted fromtheParticleListingsoftheReviewofParticlePhysics ,W.-M.Yaoet.al., J.Phys.G 33 ,1(2006) [10]E.Accomandoet.al.,Phys.Reports 299 ,pp.1-78(1998). [11]D.J.Miller,P.OslandandA.R.Raklev, Invariantmassdistributionsincascade decays ,JHEP 0603 ,034(2006)[arXiv:hep-ph/0510356]. [12]J.M.SmillieandB.R.Webber, Distinguishingspinsinsupersymmetricand universalextradimensionmodelsattheLargeHadronCollid er ,JHEP 0510 ,069 (2005)[arXiv:hep-ph/0507170]. [13]C.G.Lester,M.A.ParkerandM.J..White, Threebodykinematicendpoints inSUSYmodelswithnon-universalHiggsmasses ,JHEP 0710 (2007)051 [arXiv:hep-ph/0609298]. 116

PAGE 117

[14]C.Athanasiou,C.G.Lester,J.M.SmillieandB.R.Webbe r, Distinguishing spinsindecaychainsattheLargeHadronCollider ,JHEP 0608 ,055(2006) [arXiv:hep-ph/0605286]. [15]C.Athanasiou,C.G.Lester,J.M.SmillieandB.R.Webbe r, Addendum to'DistinguishingspinsindecaychainsattheLargeHadron Collider' arXiv:hep-ph/0606212. [16]A.J.Barr, MeasuringsleptonspinattheLHC ,JHEP 0602 ,042(2006) [arXiv:hep-ph/0511115]. [17]L.T.WangandI.Yavin, SpinMeasurementsinCascadeDecaysattheLHC ,JHEP 0704 ,032(2007)[arXiv:hep-ph/0605296]. [18]A.Alves,O.EboliandT.Plehn, It'sagluino ,Phys.Rev.D 74 ,095010(2006) [arXiv:hep-ph/0605067]. [19]I.Hinchlie,F.E.Paige,M.D.Shapiro,J.Soderqvista ndW.Yao, PrecisionSUSY measurementsatLHC ,Phys.Rev.D 55 ,5520(1997)[arXiv:hep-ph/9610544]. [20]N.Ozturk[ATLASCollaboration], SUSYParametersDeterminationwithATLAS arXiv:0710.4546[hep-ph]. [21]C.G.LesterandD.J.Summers, Measuringmassesofsemi-invisiblydecayingparticlespairproducedathadroncolliders ,Phys.Lett.B 463 ,99(1999) [arXiv:hep-ph/9906349]. [22]H.Bachacou,I.HinchlieandF.E.Paige, MeasurementsofmassesinSUGRA modelsatLHC ,Phys.Rev.D 62 ,015009(2000)[arXiv:hep-ph/9907518]. [23]I.HinchlieandF.E.Paige, MeasurementsinSUGRAmodelswithlargetan(beta) atLHC ,Phys.Rev.D 61 ,095011(2000)[arXiv:hep-ph/9907519]. [24]ATLAScollaboration, ATLASdetectorandphysicsperformance ,ATLASTDR15, CERN/LHCC99-15 [25]M.M.Nojiri,D.ToyaandT.Kobayashi, LeptonEnergyAsymmetryand PrecisionSUSYstudyatHadronColliders ,Phys.Rev.D 62 ,075009(2000) [arXiv:hep-ph/0001267]. [26]B.C.Allanach,C.G.Lester,M.A.ParkerandB.R.Webber Measuringsparticle massesinnon-universalstringinspiredmodelsattheLHC ,JHEP 0009 ,004(2000) [arXiv:hep-ph/0007009]. [27]A.Barr,C.LesterandP.Stephens, m(T2):Thetruthbehindtheglamour ,J.Phys. G 29 ,2343(2003)[arXiv:hep-ph/0304226]. [28]M.M.Nojiri,G.PoleselloandD.R.Tovey, Proposalforanewreconstruction techniqueforSUSYprocessesattheLHC ,arXiv:hep-ph/0312317. 117

PAGE 118

[29]ChristopherGorhamLester, Modelindependentsparticlemassmeasurementsat ATLAS ,AdissertationsubmittedtotheUniversityofCambridgefo rthedegreeof DoctorofPhilosophyDecember2001. [30]K.Kawagoe,M.M.NojiriandG.Polesello, AnewSUSYmassreconstructionmethodattheCERNLHC ,Phys.Rev.D 71 ,035008(2005) [arXiv:hep-ph/0410160]. [31]B.K.Gjelsten,D.J.MillerandP.Osland, MeasurementofSUSYmassesvia cascadedecaysforSPS1a ,JHEP 0412 ,003(2004)[arXiv:hep-ph/0410303]. [32]B.K.Gjelsten,D.J.MillerandP.Osland, Measurementofthegluinomassvia cascadedecaysforSPS1a ,JHEP 0506 ,015(2005)[arXiv:hep-ph/0501033]. [33]A.Birkedal,R.C.GroupandK.Matchev, SleptonmassmeasurementsattheLHC IntheProceedingsof2005InternationalLinearColliderWo rkshop(LCWS2005), Stanford,California,18-22Mar2005,pp0210 [arXiv:hep-ph/0507002]. [34]C.G.Lester,M.A.ParkerandM.J..White, DeterminingSUSYmodelparameters andmassesattheLHCusingcross-sections,kinematicedges andotherobservables JHEP 0601 ,080(2006)[arXiv:hep-ph/0508143]. [35]P.MeadeandM.Reece, ToppartnersattheLHC:Spinandmassmeasurement Phys.Rev.D 74 ,015010(2006)[arXiv:hep-ph/0601124]. [36]C.G.Lester, Constrainedinvariantmassdistributionsincascadedecay s:Theshape ofthe'm(qll)-threshold'andsimilardistributions ,Phys.Lett.B 655 ,39(2007) [arXiv:hep-ph/0603171]. [37]B.K.Gjelsten,D.J.Miller,P.OslandandA.R.Raklev, Massdeterminationincascadedecaysusingshapeformulas ,AIPConf.Proc. 903 ,257(2007) [arXiv:hep-ph/0611259]. [38]S.Matsumoto,M.M.NojiriandD.Nomura, Huntingforthetoppartnerinthe littlestHiggsmodelwithT-parityattheLHC ,Phys.Rev.D 75 ,055006(2007) [arXiv:hep-ph/0612249]. [39]H.C.Cheng,J.F.Gunion,Z.Han,G.MarandellaandB.McE lrath, MassDeterminationinSUSY-likeEventswithMissingEnergy ,JHEP 0712 ,076(2007) [arXiv:0707.0030[hep-ph]]. [40]C.LesterandA.Barr, MTGEN:Massscalemeasurementsinpair-productionat colliders ,JHEP 0712 ,102(2007)[arXiv:0708.1028[hep-ph]]. [41]W.S.Cho,K.Choi,Y.G.KimandC.B.Park, GluinoStransverseMass ,Phys. Rev.Lett. 100 ,171801(2008)[arXiv:0709.0288[hep-ph]]. [42]B.Gripaios, TransverseObservablesandMassDeterminationatHadronCo lliders JHEP 0802 ,053(2008)[arXiv:0709.2740[hep-ph]]. 118

PAGE 119

[43]A.J.Barr,B.GripaiosandC.G.Lester, WeighingWimpswithKinksatColliders: InvisibleParticleMassMeasurementsfromEndpoints ,JHEP 0802 ,014(2008) [arXiv:0711.4008[hep-ph]]. [44]W.S.Cho,K.Choi,Y.G.KimandC.B.Park, Measuringsuperparticlemasses athadroncolliderusingthetransversemasskink ,JHEP 0802 ,035(2008) [arXiv:0711.4526[hep-ph]]. [45]G.G.RossandM.Serna, MassDeterminationofNewStatesatHadronColliders Phys.Lett.B 665 ,212(2008)[arXiv:0712.0943[hep-ph]]. [46]M.M.Nojiri,G.PoleselloandD.R.Tovey, AhybridmethodfordeterminingSUSY particlemassesattheLHCwithfullyidentiedcascadedeca ys ,JHEP 0805 ,014 (2008)[arXiv:0712.2718[hep-ph]]. [47]P.Huang,N.KerstingandH.H.Yang, HiddenThresholds:ATechniquefor ReconstructingNewPhysicsMassesatHadronColliders ,arXiv:0802.0022[hep-ph]. [48]M.M.Nojiri,Y.Shimizu,S.OkadaandK.Kawagoe, Inclusivetransversemass analysisforsquarkandgluinomassdetermination ,JHEP 0806 ,035(2008) [arXiv:0802.2412[hep-ph]]. [49]D.R.Tovey, Onmeasuringthemassesofpair-producedsemi-invisiblyde caying particlesathadroncolliders ,JHEP 0804 ,034(2008)[arXiv:0802.2879[hep-ph]]. [50]M.M.NojiriandM.Takeuchi, Studyofthetopreconstructionintop-partnerevents attheLHC ,arXiv:0802.4142[hep-ph]. [51]H.C.Cheng,D.Engelhardt,J.F.Gunion,Z.HanandB.McE lrath, Accurate MassDeterminationsinDecayChainswithMissingEnergy ,Phys.Rev.Lett. 100 252001(2008)[arXiv:0802.4290[hep-ph]]. [52]W.S.Cho,K.Choi,Y.G.KimandC.B.Park, Measuringthetopquarkmasswith m T 2 attheLHC ,Phys.Rev.D 78 ,034019(2008)[arXiv:0804.2185[hep-ph]]. [53]M.Serna, Ashortcomparisonbetween m T 2 and m CT ,JHEP 0806 ,004(2008) [arXiv:0804.3344[hep-ph]]. [54]M.Bisset,R.LuandN.Kersting, ImprovingSUSYSpectrumDeterminationsatthe LHCwithWedgeboxandHiddenThresholdTechniques ,arXiv:0806.2492[hep-ph]. [55]A.J.Barr,G.G.RossandM.Serna, ThePrecisionDeterminationofInvisibleParticleMassesattheLHC ,arXiv:0806.3224[hep-ph]. [56]N.Kersting, OnMeasuringSplit-SUSYGauginoMassesattheLHC arXiv:0806.4238[hep-ph]. [57]M.M.Nojiri,K.Sakurai,Y.ShimizuandM.Takeuchi, Handlingjets+missing E T channelusinginclusivemT2 ,arXiv:0808.1094[hep-ph]. 119

PAGE 120

[58]H.C.ChengandZ.Han, MinimalKinematicConstraintsand M T 2 ,arXiv:0810.5178 [hep-ph]. [59]M.Burns,K.Kong,K.T.MatchevandM.Park, UsingSubsystem M T 2 forCompleteMassDeterminationsinDecayChainswithMissingEner gyatHadronColliders ,JHEP 0903 ,143(2009)[arXiv:0810.5576[hep-ph]]. [60]A.J.Barr,A.PinderandM.Serna, PrecisionDeterminationofInvisible-Particle MassesattheCERNLHC:II ,arXiv:0811.2138[hep-ph]. [61]P.Konar,K.KongandK.T.Matchev, p s min :aglobalinclusivevariablefor determiningthemassscaleofnewphysicsineventswithmiss ingenergyathadron colliders ,arXiv:0812.1042[hep-ph]. [62]A.J.Barr, Usingleptonchargeasymmetrytoinvestigatethespinofsup ersymmetric particlesattheLHC ,Phys.Lett.B 596 ,205(2004)[arXiv:hep-ph/0405052]. [63]M.Burns,K.Kong,K.T.MatchevandM.Park, AGeneralMethodforModelIndependentMeasurementsofParticleSpins,Couplingsand MixingAnglesin CascadeDecayswithMissingEnergyatHadronColliders ,JHEP 0810 ,081(2008) [arXiv:0808.2472[hep-ph]]. [64]N.Arkani-Hamed,G.L.Kane,J.ThalerandL.T.Wang, Supersymmetryandthe LHCinverseproblem ,JHEP 0608 ,070(2006)[arXiv:hep-ph/0512190]. [65]B.K.Gjelsten,D.J.MillerandP.Osland, Resolvingambiguitiesinmassdeterminationsatfuturecolliders IntheProceedingsof2005InternationalLinear ColliderWorkshop(LCWS2005),Stanford,California,18-2 2Mar2005,pp0211 [arXiv:hep-ph/0507232]. [66]B.K.Gjelsten,D.J.Miller,P.OslandandA.R.Raklev, Massambiguitiesin cascadedecays ,arXiv:hep-ph/0611080. [67]M.Burns, GeneralizingtheMethodofKinematicalEndpoints ,talkgivenatthe Pheno2008Symposium\LHCTurnOn",MadisonWI,April28,200 8 [68]M.Park, AmbiguitiesinSUSYmassdeterminationfromkinematicendp ointsat LHC ,talkgivenatthePheno2008Symposium\LHCTurnOn",Madiso nWI,April 28,2008. [69]K.Matchev, NewPhysicsSignaturesandPrecisionMeasurementsattheLH C ,talk givenattheKITPConference:\AnticipatingPhysicsattheL HCCollider",UC SantaBarbara,June5,2008. [70]GeorgiaKarapostoli, FeasibilityofSUSYparticlemassmeasurementsfromendpointsindi-leptonevents ,talkgivenattheCMSSUSYMeeting,December16 2008,CERN.SeealsoLucPape,\Reconstructionofsparticle massesfromendpoints (andothers)atLHC",CMSInternalNoteCMSIN-2006/12. 120

PAGE 121

[71]D.CostanzoandD.R.Tovey, Supersymmetricparticlemassmeasurementwith invariantmasscorrelations ,arXiv:0902.2331[hep-ph]. [72]M.Burns,K.T.Matchev,M.Park, Usingkinematicboundarylinesforparticle massmeasurementsanddisambiguationinSUSY-likeeventsw ithmissingenergy JHEP 0905 ,094(2009)[arXiv:0903.4371[hep-ph]]. [73]M.Battaglia,A.Datta,A.DeRoeck,K.KongandK.T.Matc hev, Contrasting supersymmetryanduniversalextradimensionsattheCLICmu lti-TeVe+e-collider ,JHEP 0507 ,033(2005)[arXiv:hep-ph/0502041]. [74]M.Battaglia,A.K.Datta,A.DeRoeck,K.KongandK.T.Ma tchev, Contrasting supersymmetryanduniversalextradimensionsatcolliders IntheProceedingsof 2005InternationalLinearColliderWorkshop(LCWS2005),S tanford,California, 18-22Mar2005,pp0302 [arXiv:hep-ph/0507284]. [75]A.Datta,K.KongandK.T.Matchev, Discriminationofsupersymmetryand universalextradimensionsathadroncolliders ,Phys.Rev.D 72 ,096006(2005) [Erratum-ibid.D 72 ,119901(2005)][arXiv:hep-ph/0509246]. [76]A.Datta,G.L.KaneandM.Toharia, IsitSUSY? ,arXiv:hep-ph/0510204. [77]S.Abdullin etal. [TeV4LHCWorkingGroup], Tevatron-for-LHCreport:Preparationsfordiscoveries ,arXiv:hep-ph/0608322. [78]J.M.Smillie, SpinCorrelationsinDecayChainsInvolvingWBosons ,Eur.Phys.J. C 51 ,933(2007)[arXiv:hep-ph/0609296]. [79]K.KongandK.T.Matchev, Phenomenologyofuniversalextradimensions ,AIP Conf.Proc. 903 ,451(2007)[arXiv:hep-ph/0610057]. [80]C.Kilic,L.T.WangandI.Yavin, OntheExistenceofAngularCorrelationsinDecayswithHeavyMatterPartners ,JHEP 0705 ,052(2007)[arXiv:hep-ph/0703085]. [81]A.AlvesandO.Eboli, UnravellingthesbottomspinattheCERNLHC ,Phys.Rev. D 75 ,115013(2007)[arXiv:0704.0254[hep-ph]]. [82]C.Csaki,J.HeinonenandM.Perelstein, TestingGluinoSpinwithThree-Body Decays ,JHEP 0710 ,107(2007)[arXiv:0707.0014[hep-ph]]. [83]A.Datta,P.Dey,S.K.Gupta,B.MukhopadhyayaandA.Ny eler, Distinguishing theLittlestHiggsmodelwithT-parityfromsupersymmetrya ttheLHCusing trileptons ,Phys.Lett.B 659 ,308(2008)[arXiv:0708.1912[hep-ph]]. [84]M.R.Buckley,H.Murayama,W.KlemmandV.Rentala, Discriminatingspin throughquantuminterference ,arXiv:0711.0364[hep-ph]. 121

PAGE 122

[85]M.R.Buckley,B.Heinemann,W.KlemmandH.Murayama, QuantumInterference EectsAmongHelicitiesatLEP-IIandTevatron ,Phys.Rev.D 77 ,113017(2008) [arXiv:0804.0476[hep-ph]]. [86]G.L.Kane,A.A.Petrov,J.ShaoandL.T.Wang, Initialdeterminationofthe spinsofthegluinoandsquarksatLHC ,arXiv:0805.1397[hep-ph]. [87]T.Appelquist,H.C.ChengandB.A.Dobrescu, Boundsonuniversalextradimensions ,Phys.Rev.D 64 ,035002(2001)[arXiv:hep-ph/0012100]. [88]B.A.Dobrescu,K.KongandR.Mahbubani, Leptonsandphotonsat theLHC:Cascadesthroughspinlessadjoints ,JHEP 0707 ,006(2007) [arXiv:hep-ph/0703231]. [89]See,forexample,ChrisQuigg, GaugeTheoriesoftheStrong,Weak,andElectromagneticInteractions ,FNAL,Batavia,IL,WestviewPress(1983,1997),p.92. [90]HowardE.Haber, SupersymmetricParticleSearches ,J.Phys.G,Nuclearand ParticlePhysics,Volume33July2006,p.1106(i.e.thePDGR eviewofParticle Physics). [91]M.Schmitt, Supersymmetry,PartII(Experiment) ,J.Phys.G,Nuclearand ParticlePhysics,Volume33July2006,p.1120(i.e.thePDGR eviewofParticle Physics). [92]M.Drees,G.Gerbier, DarkMatter ,J.Phys.G,NuclearandParticlePhysics, Volume33July2006,p.234(i.e.thePDGReviewofParticlePh ysics). [93]TorbjornSjostrand,StephenMrenna,PeterSkands, PYTHIA6.4Physicsand Manual ,LUTP06-13,FERMILAB-PUB-06-052-CD-T,March2006,pp.16 5-175 [arXiv:hep-ph/0603175]. [94]MichaelE.PeskinandDanielV.Schroeder, AnIntroductiontoQuantumField Theory ,WestviewPress,1995. 122

PAGE 123

BIOGRAPHICALSKETCH MichaelBurnswasborninLubbock,Texason1978October8.Hi sfamilymoved himtoLakeJackson,Texaswhenhewasstillaninfant.Heresi dedhisentirechildhood inthesmallcoastaltownsouthofHouston.Heenjoyedmanyou tdooractivitieswiththe menfolkofhisfamily,andhespentmuchofhisfreetimehunti ng,shing,camping,hiking, andswimmingintheTexasandColoradowilderness. MichaelattendedBrazosportChristianSchool,asmallpriv atereligiousschool,from4 yearsofageuntilhewas14yearsofage.Heattendedtheonlyn earbypublichighschool, BrazoswoodHighschool(\B-wood"),intheadjacenttownofC lute,Texas.Duringhis juniorandsenioryearsofhighschool,hetookseveralcours esatthelocalcommunity college,BrazosportCollege,andalsoenrolledinanautomo tivetechnologyprogramata dierenthighschool,BrazosportHighschool. Duetoareligiousdisagreementwithhisfather,Michaelmov edoutofhisfather's housetosupporthimselfbeforetheendofhissenioryearinh ighschool.Hedidsorashly, andwasnotpreparedforthenancialresponsibility.After severalmonthshereconciled withhisfather,whoagreedtopayhistuitionandhousingcos tsatapopularstateschool, TexasA&MUniversity(TAMU),fromwherehisfatherhadalsor eceivedadegree. There,Michaelchosetomajorinelectricalengineeringdue toafascinationwithcomplex numbers.Feelingsomewhattrappedbythearrangementwithh isfather,Michaeltook heavycourseloadsandsummerschool,andmanagedtograduat ewithin3yearsin2000 May. Afterhisgraduation,Michaelwashiredbyaglobaltelecomm unicationsrm,Marconi Communications(Marconi),InIrving,Texas.Hesoonrealiz edthathedidnotenjoy engineering,andrequesteddismissalfromthecompany.The directorofhisdepartment expressedtoMichaelangerattherequest,andsuggestedtha tMichaelshouldatleast giveanotheryearofservicetoMarconi.However,withinmon ths,thetelecommunications companyhitaneconomicdownturn,andMichaelwaslaid-o(p erhapstofulllhis 123

PAGE 124

request),alongwithhundredsofotheremployeesfromMarco ni,andthousandsof employeesintheDallasarea. Facingtheendofhisunemploymentbenets,andstretchingh issavingsthin,Michael decidedtopursuehispassionforphysicsbyenrollinginama ster'sprogramatthenearby UniversityofNorthTexas(UNT),whooeredhimateachingas sistantship.Henally foundenjoymentinhisworkatUNT,especiallyteaching,whe rehisresponsibilities includedtutoring,labmanaging,andevenlecturingphysic scoursesduringhislast summeratUNT.HisacademicfocusatUNTwasGeneralRelativi ty.Hereceiveda master'sinphysicsfromtheUniversityofNorthTexasin200 4May.Eventually,he decidedthathewantedtolearnabouthighenergyphysicsand thestandardmodelof particlephysics,buthecouldnotndanappropriateadviso rforthisatUNT.So,aftera yearofcontinuinghisassistantshipworkingonalaserexpe rimentatUNT,hetransferred totheUniversityofFlorida(UF)in2005August,whereherec eiveda4-yearfellowship. Michael'sfocusatUFwasHighEnergyPhenomenology.Hegrad uatedwitha doctorateinphysicsin2009August,after4yearsatUF.Whil eatUF,hecoauthored threepaperswithhisadvisorandothercollaborators. 124