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Topics in beyond Standard Model Collider Phenomenology

Permanent Link: http://ufdc.ufl.edu/UFE0043989/00001

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Title: Topics in beyond Standard Model Collider Phenomenology
Physical Description: 1 online resource (210 p.)
Language: english
Creator: Sarangi, Gaurab K
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

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Subjects / Keywords: cms -- collider -- diquark -- extra-dimension -- led -- lepto-diquark -- lhc -- monte-carlo -- mssm -- phenomenology -- supersymmetry -- susy -- ued
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is a comprehensive summary of my work as a doctoral student at the University of Florida. This document will eloborate our research in the study of collider phenomenology at the LHC for graviton production inspired by Large Extra-dimension, diquark production as a supermodel (i.e. models with significant signals at early LHC), model independent Minimal Supersymmetric Standard Model (MSSM) and model independent topological analysis of same sign dilepton study by the CMS collaboration.
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 Gaurab K Sarangi.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Matchev, Konstantin T.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-11-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043989:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043989/00001

Material Information

Title: Topics in beyond Standard Model Collider Phenomenology
Physical Description: 1 online resource (210 p.)
Language: english
Creator: Sarangi, Gaurab K
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: cms -- collider -- diquark -- extra-dimension -- led -- lepto-diquark -- lhc -- monte-carlo -- mssm -- phenomenology -- supersymmetry -- susy -- ued
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: This dissertation is a comprehensive summary of my work as a doctoral student at the University of Florida. This document will eloborate our research in the study of collider phenomenology at the LHC for graviton production inspired by Large Extra-dimension, diquark production as a supermodel (i.e. models with significant signals at early LHC), model independent Minimal Supersymmetric Standard Model (MSSM) and model independent topological analysis of same sign dilepton study by the CMS collaboration.
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 Gaurab K Sarangi.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Matchev, Konstantin T.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-11-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043989:00001


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TOPICSINBEYONDSTANDARDMODELCOLLIDERPHENOMENOLOGYByGAURABK.SARANGIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012GaurabK.Sarangi 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmygratitudetoeveryonewhomadethisworkanddissertationpossible,inathreefoldway.Firstandforemost,Iwouldliketothankmyrstteachers,myparentsandmyfamily,whonurturedandencouragedmycuriosityasIwasgrowingupandprovidedconstantmoralandemotionalsupportthroughoutmycareer.Inparticular,IwouldliketothankmybrotherSampad,whowasmorethanfamily-afriendwhostoodbythememyentirelife.Next,Iwouldliketothankmyteachersandmentorswhoshapedmymindthroughvariousstagesofmyeducation.Iwillforeverbeindebtedtomydoctoraladvisor,KonstantinMatchev,aconstantsourceofinspirationandsupportwhoguidedmenotonlyinthewaysofhighenergyphysics,butalsoincriticalthinkingandinexpressingmyideasclearly.IwouldliketoexpressmygratitudetoAseshKDattawhointroducedmetotheworldofFeynmandiagramswhileholdingmyhandsasItookbabystepsintheworldofparticlephysics.IwouldalsoliketoacknowledgemydebttoPrakashSatpathy,myhighschoolmathematicsandphysicsteacherwhointroducedmetoconceptsofabstractionandbeautyinscience.Finally,IwouldliketoacknowledgemygratitudeformyfriendsManoj,Shawn,MJ,Mike,Kathleen,Katie,Joe,JoelandBecca,whomademystayinGainesvillefunandinteresting,amusthaveforahappyandproductiveresearch. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 1.1StandardModel ................................. 12 1.1.1TheSuccessStory ........................... 13 1.1.2So,arewedonethen? ......................... 13 1.2BeyondStandardModelTheories ...................... 15 1.2.1Supersymmetry ............................. 15 1.2.2ExtraDimensions ............................ 16 1.3MotivationforOurWork ............................ 17 1.3.1LargeExtra-Dimension(ADDModel) ................. 17 1.3.2Diquarks ................................. 18 1.3.3Supersymmetry ............................. 19 1.3.4ModelIndependentStudies ...................... 19 2LARGEEXTRA-DIMENSIONSIGNATURETHROUGHZ-PAIRPRODUCTIONATTHELHC ..................................... 20 2.1IntroductiontoExtraDimensions ....................... 20 2.2LargeExtraDimensions:ADDModel ..................... 23 2.3PhenomenologicalSignatures ......................... 25 2.4Limits ...................................... 27 2.5Calculation ................................... 28 2.6Analysis ..................................... 34 2.7Results ..................................... 38 3DIQUARKSINPYTHIA ............................... 39 3.1DiquarkProductionandDecay ........................ 39 3.2ImplementationinPythia ............................ 40 3.2.1ProblemwithImplementation ..................... 40 3.2.2OurWorkaround ............................ 41 3.2.3DescriptionoftheImplementation ................... 41 3.3Results ..................................... 45 5

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4SAME-SIGNDILEPTONS .............................. 47 4.1InterpretationofExperimentalResults .................... 47 4.2CalculatingNsigTheoretically ......................... 47 4.3SimpliedModels ................................ 48 4.4SimpliedModel:AnIllustration ........................ 50 4.5Model-IndependentProcedure ........................ 51 4.5.1Model-IndependentProcedure:FastSimulation ........... 52 4.5.2Model-IndependentProcedure:Emulation .............. 61 4.6Results ..................................... 63 5HOWTOLOOKFORSUPERSYMMETRYUNDERTHELAMPPOSTATTHELHC .......................................... 67 5.1IntroductionandMotivation .......................... 67 5.2TravellingSalesman .............................. 71 5.38-leptonTopologies .............................. 76 5.3.1Case:A ................................. 77 5.3.2Case:BandC ............................. 79 5.4GroupingSignaturesandHierarchies .................... 83 5.5SummaryandOutlook ............................. 85 APPENDIX AFORTRANCODEFOREXTERNALIMPLEMENTATIONOFDIQUARKINPYTHIA ........................................ 88 BPLOTSFROMEMULATION ............................. 117 CHIERARCHYTOSIGNATURES .......................... 122 DSIGNATURETOHIERARCHIES .......................... 168 EW-BTRANSITION .................................. 206 REFERENCES ....................................... 208 BIOGRAPHICALSKETCH ................................ 210 6

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LISTOFTABLES Table page 2-1Compacticationlengthsfordifferentnumbersofextradimensions. ....... 24 3-1Quantumnumbersforvariousparticlesinvolvedindiquarkproductionanddecay. ......................................... 39 4-1Thesevensearchregions. ............................. 53 5-1ThesetofSUSYparticlesconsideredinthisanalysis,shorthandnotationforeachmultiplet,andthecorrespondingsoftSUSYbreakingmassparameter. .. 68 5-2Constructionofasamplehierarchychain ..................... 69 5-3InputsoftSUSYmassparameters(inGeV)forthexxGQWLBEHstudypoints. 78 5-4InputsoftSUSYmassparameters(inGeV)forthexxGUBEWLHstudypoints. 79 5-5InputsoftSUSYmassparameters(inGeV)forthexxGUBEHLWstudypoints. 80 C-1HierarchytoSignatures ............................... 123 D-1SignaturetoHierarchies ............................... 169 7

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LISTOFFIGURES Figure page 2-1Representativediagramsfor(a)realgravitonemissionprocessand(b)virtualgravitonexchangeprocess ............................. 25 2-2FeynmandiagramsfortheleadingorderZpairproductionprocessinSM. ... 29 2-3FeynmandiagramsfortheleadingorderZpairproductionprocessinADDmodel. ......................................... 29 2-4Dependenceoftotalcross-section()forff!ZZ,onEminT,Zata1TeVxedcenter-of-massenergye+e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(collider. ........................ 30 2-5Dependenceoftotalcross-sectionforff!,onEminT,ata1TeVxedcenter-of-massenergye+e)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(collider. ........................ 31 2-6InvariantmassdistributionoftheoutgoingZbosonsforpp!ZZatLHC(14TeV). .......................................... 35 2-7AngulardistributionoftheoutgoingZbosonsforpp!ZZatLHC(14TeV). .. 36 2-8Dependenceofthecross-sectionforpp!ZZonthestringscaleatLHC(14TeV). .......................................... 37 3-1Invariantmassdistributionofthefouroutgoingparticleswiththemassofthelepto-diquarkxedat100GeV. ........................... 45 3-2Invariantmassdistributionofthefouroutgoingparticleswiththemassofthelepto-diquarkxedat300GeV. ........................... 45 4-1Theparameterspace ................................ 49 4-2Diagramaticdescriptionofoursimpliedmodelsetupfortwosamesignedleptonsignal. ..................................... 51 4-3PGS:EfcienciesforthesevensearchregionsforM1=10GeV. ........ 54 4-4PGS:EfcienciesforthesevensearchregionsforM1=100GeV. ....... 55 4-5PGS:EfcienciesforthesevensearchregionsforM1=200GeV. ....... 55 4-6PGS:EfcienciesforthesevensearchregionsforM1=300GeV. ....... 56 4-7PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=10GeV. 57 4-8PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=100GeV. 58 4-9PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=200GeV. 59 4-10PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=300GeV. 60 8

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4-11PGS:SS2Lproductionx-sectionforM1=10,100,200and300GeV. ..... 60 4-12PGS:95%CLlimitonSS2Lproductionx-sectionforM1=10,100and200GeVwithaluminosityof1,10and30fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1. .................... 63 4-13Emulation:95%CLlimitonSS2Lproductionx-sectionforM1=10,100and200GeVwithaluminosityof1,10and30fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1. .................. 64 4-14PGSversustheEmulationprescription:EfciencyofacceptanceofandeasafunctionoftheirrespectivePT). ........................ 65 4-15PGSversustheEmulationprescription:Numberofacceptedjetsandleptons(ande)). ....................................... 65 4-16ComparisionofHTand6ETfromPGSandtheEmulationprescription. ..... 66 5-1Productionrate(inpercentage)of(a)noLCP,(b)onlyoneLCPand(c)bothLCPinaMQ-MUplain ................................ 70 5-2GraphicalrepresentationoftheallowedtransitionsbetweentheSUSYstates. 72 5-3MassspectrumforthehierarchyxxGQWLBEH. .................. 77 5-4Branchingratio,productioncross-sectionandmulti-leptonsignaturesforthecaseofhierarchyxxGQWLBEH ........................... 79 5-5MassspectrumforthehierarchyxxGUBEWLH. .................. 80 5-6Branchingratio,productioncross-sectionandmulti-leptonsignaturesforthecaseofhierarchyxxGUBEWLH ........................... 81 5-7MassspectrumforthehierarchyxxGUBEHLW. .................. 82 5-8Branchingratio,productioncross-sectionandmulti-leptonsignaturesforthecaseofhierarchyxxGUBEHLW ........................... 82 5-9X,Y,Color==#leptons,#groups,#channels. ..................... 84 5-10X,Y,Color==#channels,#groups,#leptons. ..................... 84 5-11X,Y,Color==#hierarchies,#groups,#leptons. .................... 85 5-12X,Y,Color==#hierarchies,#groups,#channels. ................... 86 5-13Thetravellingsalesmandiagramforthecasewhentheelectroweakmultipletsareconsideredseparately. .............................. 87 B-1Emulation:EfcienciesforthesevensearchregionsforM1=10GeV. ..... 117 B-2Emulation:EfcienciesforthesevensearchregionsforM1=100GeV. .... 118 B-3Emulation:EfcienciesforthesevensearchregionsforM1=200GeV. .... 118 9

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B-4Emulation:EfcienciesforthesevensearchregionsforM1=300GeV. .... 119 B-5Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=10GeV. ......................................... 119 B-6Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=100GeV. ......................................... 120 B-7Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=200GeV. ......................................... 120 B-8Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=300GeV. ......................................... 121 E-1Dependenceof~w0decaywidthonM2ine+e)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((blue),Z(red)andh(green)production. ...................................... 207 E-2Dependenceof~w0decaywidthonadditionalparameters ............ 207 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTOPICSINBEYONDSTANDARDMODELCOLLIDERPHENOMENOLOGYByGaurabK.SarangiMay2012Chair:KonstantinT.MatchevMajor:PhysicsThisDissertationisacomprehensivesummaryofmyworkasadoctoralstudentattheUniversityofFlorida.ThisdocumentwilleloborateourresearchinthestudyofcolliderphenomenologyattheLHCforgravitonproductioninspiredbyLargeExtra-dimension,diquarkproductionasasupermodel(i.e.modelswithsignicantsignalsatearlyLHC),modelindependentMinimalSupersymmetricStandardModel(MSSM)andmodelindependenttopologicalanalysisofsamesigndileptonstudybytheCMScollaboration. 11

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CHAPTER1INTRODUCTIONAtthedawnofthe21stcentury,wendourselvesstandingatthebleedingedgeofhighenergyphysicsintermsofbothscienceandtechnology.Beingintheeldofphenomenologyhasgivenmethefrontrowseattoseehowtheycometogethertopropelforwardourknowledgeofreality.Aswetrytoprobeintotheunknownwearefacedwiththefollowingchallenges:a)newcosmologicaldataaskingforanewertheory,b)newertheorieswithpredictionswhichcan'tbevalidatedwithcurrentexperimentalsetups,askingforbetterexperiments/data,andc)newertheorieswithpredictionsthathaveachanceofbeingvalidated(orrejected)incurrentexperimentalsetup.WiththemegabeastcalledtheLargeHadronCollideratCERN,Genevacollidingprotonbeamsataneverseenbefore7TeVcenterofmassenergywenowhaveagreatopportunityinbeingabletodosomeseriousdevelopmentespeciallyinthecase`c'mentionedabove.Buthavingalreadydiscoveredagoodportionofphysicswithasimilarsetup(ppcollider@2TeV)attheFermilab,Batavia,wenowhavetoidentifywhatoncewasconsideredagoodsignaltobemerebackground.Thusoneofthemajorchallengesthatwefacetodayistondthatsignalamidstanoverwhelmingamountofbackground,notveryfarfromtryingtondaneedleinahaystack,exceptforthefactthatwewon'tevenbesureiftheneedlewasproducedatall.Thiscallsforsuperiortechniquesindatahandlingandstatisticalanalysis.InthischapterI'lltrytoshedsomelightonthecurrentstatusofourveriedknowledgeinhighenergyphysics,thetheoreticalideaswhichjustbecameopenforpryingandprodding,andtheanalytictechniquesbeingemployedtodissectthedata. 1.1StandardModelTheStandardModelofparticlephysicsisaQuantumFieldTheorythattriestoexplaintheelectro-magnetic,weakandstronginteractionbetweenvariousparticles.ItincorporatesQuantumChromo-Dynamics(QCD)byPolitzer,WilczekandGross,and 12

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Electro-WeakSymmetryBreaking(EWSB)asprescribedbyGlashowwiththeHiggsmechanismincorporatedintoitbySalamandWeinberg. 1.1.1TheSuccessStoryTheStandardModelsuccessfullyexplainstheinteractionbetweenallobservedparticlesintermsofafewernumberoffundamentalparticlesandinteractions,reducingthenumberoffreeparametersinthetheorytojust19.Itcategorizesallparticlesintomatterparticles(fermions)andforcecarriers(bosons).Furthermore,thematterparticlesaredividedinto3generations.Ineachgenerationwehaveachirallyleftleptondoublet(e.g.lefthandedelectronandelectronnutrino),achirallyrightleptonsinglet,achirallyleftquarkdoubletand2chirallyrightquarksinglets.Theinteractionparticlesare8gluons,3weakbosonsandahyperchargeboson.UponEWSB,theneutralweakandhyperchargegaugebosonsmixtogiverisetoZ0and.TheStandardModelmadepredictionsabouttheWandZbosons,charm,bottomandtopquarks.TheWandZbosonswereobservedattheLargeElectronPositroncolliderandtheirobservedmassesmatchedtheprediction.ThecharmquarkwasdiscoveredatBrookhavenNationalLaboratoryandStanfordLinearAcceleratorCollider(1974)andthebottomandthetopquarkswerediscoveredatFermilab(bottomin1977andtopin1999).TheonlythingleftsofaristheelusiveHiggsparticle.TheHiggsmechanismwasrstpostulatedbyPeterHiggsandwaslaterincorporatedintoSheldonGlashow'sElectro-weaktheorybyStevenWeinbergandAbdusSalam.ThisintroducedthescalarHiggsbosoninthetheory.Ithasbeen45yearssinceitwasproposedin1967andremainsasthelastpiecethatcompletesallStandardModelpredictions. 1.1.2So,arewedonethen?ItlookslikeoncewendHiggswecanhappilysayStandardModelhasexplainedeverythingitsetforthtoexplain.But,thatinnowaymeanstherearen'tanyunansweredquestions.Theseissuescanbegroupedin2categories. 13

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TheoreticalIssues:AlthoughtheStandardModelistheoreticallyverywellformulateditdoesn'texplainawaysomeoftheissuesarisingfromsuchformulations.Atthesametimeitalsodoesn'tattempttoshedlightonotherphysicalfeaturesofreality.a)Hierarchyproblem:ThetoploopcontributiontotheHiggsmassisquadraticallydivergent,whichmeansthiswillleadtoHiggshavingmassontheorderofthecutoffscale,e.g.Planckmass(MPlanck).b)StrongCPProblem:IntheQCDLagrangianthereappeartermswhichinprinciplecanviolateCP(chargeandparity)symmetry.However,wehavenotobservedanysuchcaseofCPviolationintheexperiments.c)19parameters!:Ahugenumberoffreeparametersarerequiredtodescibethetheorycompletely,whichleavesthetheoryabitaestheticallychallenging.d)Whythreegenerations:TheStandardModelorganizesthematterparticles(quarksandleptons)neatlyinthreedifferentfamilies.But,thereisnotheoreticalmotivationbehinditanditdoesn'ttrytoexplainastowhythatshouldbethecase.e)Whataboutgravity?:EventhoughStandardModelexplainstheelectro-magnetic,weakandstronginteractionsverywell,itissilentwhenitcomestogravity.ItisincompatiblewiththeGeneralTheoryofRelativity,themostacceptedandwelltestedtheoryofgravitytodate.ExperimentalIssues:a)Neutrinomasses:TheStandardModeltakesthemassesoftheneutrinosofthethreegenerationstobeexactlyzero.But,wehaveexperimentallyseenthattheneutrinoshavetinymasses.TheyalsooscillatefromoneavortoanotherthusviolatingleptonavornumberwhichispresumedtobeconservedintheStandardModel.b)DarkMatterandDarkEnergy:ThematterparticlesincludedintheStandardModelconstituteallthevisiblematterintheuniverse.Butfromastronomicaldatawendthatthisvisibleportionofmatteronlyconstitutesabout14%ofallmatterand4%ofallmass-energyintheuniverse.Fromtherotationofthegalaxieswenowknowthatthere 14

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issomeotherkindofmatterintheuniversethattakesupabout86%oftheremainingmatterand23%ofallmass-energyintheuniverse.ThiskindofmatterhasbeendubbedasDarkmatter.Therest74%ofallmass-energycontenthasbeennamedDarkEnergywhichaccountsfortheacceleratedexpansionoftheuniverse.c)Matter-AntimatterAsymetry:AccordingtotheStandardModelthereshouldbeequalamountofmatterandantimatterintheuniverse.But,whatweseeisthattheuniverseislledwithmatter. 1.2BeyondStandardModelTheoriesSince,weseethatStandardModelisnotenoughintacklingthechallengesdescribedaboveweneedtoexploretheorieswhichgobeyondthat.Therehavebeenmanysuchtheorieswithvariousmotivations.Here,wementionafew. 1.2.1SupersymmetrySupersymmetryisaBeyondStandardModeltheoryprimarilymotivatedby Gaugeunication,and Thehierarchyproblem, DarkmatterTosolvethehierarchyproblem,itintroducesnewparticles(andthusloopdiagrams)whichcanceloutthequadraticdivergencesencounteredinthehierarchyproblem.Forexample,thetoploopthatcontributestothequadraticdivergencefortheHiggsmassiscanceledoutbyanotherdiagramwithanalmostsimilarparticleintheloopcontributingwithanoppositesign.Thecalculationsrevealthatthisparticlehastobeexactlyliketopquarkexceptforonekeydifference,ithastobeascalar.ThustheSupersymmetriclagrangianhasanewsymmetrycalledSupersymmetrythatrelatesaparticlewithitssuperpartnerwhichhasallofthesamequantumnumbersastheparticlebutwitha 15

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differenceof1 2inthespin.Thus,ifQisasupersymmetrictransformation, QjBoson>=jFermion>andQjFermion>=jBoson>.WhenweincorporateSupersymmetryintotheStandardModelwegettwiceasmanyparticleswithanewsetofsuperpartnersforeveryparticleintheStandardModel.ThesimplestpossiblesupersymmetricmodelconsistentwiththeStandardModeliscalledtheMinimallySupersymmetricStandardModel(MSSM).Apartfromsolvingthehierarchyproblemitalsopredictsthatthegaugecouplingsunifyataveryhighenergyscale(1016GeV).ItalsointroducestheconceptofR-parity,denedasPR=()]TJ /F6 11.955 Tf 9.3 0 Td[(1)2S+3B+L,where,S,BandLaretheparticlespin,baryonnumberandleptonnumberrespectively.IfR-parityisconservedthenasupersymmetricparticlecan'tdecayintoonlyStandardModeldecayproducts.ThisgivesusawayofdescribingDarkmatterparticles.Thisisbecause,ifR-parityisconservedthentheLightestSupersymmetricParticle(LSP)can'tdecayanyfurther.SuchparticlesinprinciplecanconstitutetheobservedDarkmatterintheuniverse,iftheyareelectricallyandcolorneutral. 1.2.2ExtraDimensionsThistheorywasrstpostulatedastheKaluza-Klein(KK)theorybyTheodorKaluzaandOscarKleinin1926seekingtounifygravitationandelectromagnetism.But,sincethenithasdevelopedintoamyriadofdifferentapproachesandinterpretations,andalsoaddressesproblemslikeDarkmatterandhierarchyaswell.Thebasicideaofthetheoryistointroducenmorespatialdimensionstothe3observablespatialdimensions,where,ncanbeanypositiveinteger.Dependingonthenatureoftheextradimensionwehavevarioustheories.LargeExtraDimension:Thismodel(alsocalledADDmodel)wasproposedin1998[ 1 ].Accordingtothismodeltheextradimensioniscompactiedandislarge 16

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comparedtothePlancklength.InthismodeltheStandardModelparticlesandinteractionsareconnedinthe3spatialdimensionsbutgravitycanpropagatethroughallspatialdimensions.ThuswhilethegravitoncanhaveaninnitenumberofKKexcitationmodestheStandardModelparticlesjusthavethezeroexcitation(groundstate)mode.WediscussthismodelinagreaterdetailinChapter 2 .WarpedExtraDimension:Thismodel(alsocalledRandall-Sundrummodel)wasproposedin1999[ 2 3 ]asasolutiontotheHiggshierarchyproblem.Itpostulaesavedimensionalspace-timegeometrywiththefthdimesnionstronglycurved(warped)byalargecosmologicalconstant.ThisvedimensionalmanifoldliesintheantideSitterspace(AdSn)sincethedeSitterspacedescribesawarpedgeometrywithapositivecosmologicalconstant.IntheRSmodel,theStandardModelinteractionsandparticleslieinthe3+1branewhilegravityisspreadoutinthebulkandisthusweakcomparedtotheSMinteractions.UniversalExtraDimension:Inthisframework,themetricofthebulkisatandnotwarped.IncontrasttotheADDmodelandtheRSmodelinUniversalExtra-dimensionnotonlygravitybutalsoalltheStandardModeleldscanpropagateinthebulk[ 4 5 ].Thus,whereasinLargeExtra-dimensionwehaveKKexcitationstatesforgraviton,inUEDwehaveKKexcitationstatesforalltheSMeldsaswell. 1.3MotivationforOurWorkOurworkspannedacrossavarietyofinterestingtopicsinBSMphysics.Here,wewouldbrieydescribethemotivationbehindourworkinthespecictopics. 1.3.1LargeExtra-Dimension(ADDModel)SinceintheADDmodelonlygravitonpropagatesinthebulkandhashigherKKexcitationstates,thecollidersearchesconsidersignaturesfromprocessesinvolvinggravitons.Atthetimeofourwork,suchsearchesprimarilyfocusedon(jets+6ET)and(2)signatures.Since,bothsignaturescomewiththeirshareofoverpowering 17

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backgroundsattheLHC,wewantedtoconsider2-Z0bosonsproducedthroughans-channelgravitonasasignature.Themainbenetsofthissignaturesare,a)VerysmallbackgroundattheLHC,b)Thebackgroundisfromat-channelprocessattreelevel,henceaverydifferentdistribution(the2-Z0'smostlyintheforward/backwarddirection)comparedtothes-channelgravitonprocessandc)LHCwouldbehuntingforthissignalanyway,since,itisoneofthesearchchannelsforaHiggsheavierthanthesumofthemassesofthetwoZ0bosons.ThedetailsofourworkaredescribedinChapter 2 1.3.2DiquarksThemotivationforourworkwithdiquarkscamefromexperimentalneeds.Adiquarkbeingaquark-quark(qq)resonance,itsproductioncross-sectionattheLHCishugecomparedtotheTevatron(R.I.P.).Thiscanbeseenwithoutgoingintomuchdetail,simplybylookingatthepartondistributionfunctions(PDF's)ofthecollidingparticles.AttheTevatronwehaveproton-anti-protoncollidingwitheachother.Toformaqqresonanceoneofthequarkshastobeavalencequarkoftheprotonandtheotherhastobeaseaquarkoftheanti-proton.Incontrast,attheLHC,sinceboththecollidingparticlesareprotons,boththequarksintheqqresonancearevalencequarks.Hence,withtheLHCswitchingonitbecomesimportanttostudythisasasupermodel[ 6 ],sincefortheveryrsttimesuchasignalwillbeavailableevenattheearlyLHC.Thisbringsustothenextstep,i.e.doingtheanalysiswhichinvolveseventgenerationandtheirdetectorsimulation.But,atthetimeofourworktherewasnoexistingeventgeneratortodoso.Moreover,thediquarkunderstudyisacolorsextetandtherewasnoeventgeneratorwheresuchaparticlecouldbeintroduced.So,weimplementedthewholeproductionprocessasanexternalprocessinPythia[ 7 ].WechosePythiabecauseofourfamiliaritywithitaswellasthevastexperienceandexpertiseofthecommunitythatusesit.AdetaildescriptionofourworkispresentedinChapter 3 18

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1.3.3SupersymmetryTherehasbeenmoreliteraturewrittenonsupersymmetrythanonecanreadinalifetime.So,itbecomecrucialforusattheoutsetofthispresentationtoexplainwhyourworkwasimportantintheeld.Asmentioned,beingsohighlyinvestigatedatheory,ithasledtomanymodels.Allofthemconstrainthefullparameterspaceinsomewayoranother,withmSUGRAbeingthemostconstrainedandthemostinvestigated.Withdifferentmodelsdifferentsignaturesbecomeimportantanddifferentsearchstrategiesarerequired.Ourworkwastherstintheeldtotakethemodelindependentsupersymmetricmassspectrumandcategorizethembasedontheirsignatures.Thisisonesteptowardstacklingtheinverseproblem.AdetaildescriptionofourworkispresentedinChapter 4 1.3.4ModelIndependentStudiesWithsomanyBSMtheoriesinthemarket,arigorousanalysiscanonlybedoneinaselectfewofthem.Thus,experimentalcollaborationspickthemodelswhicharethemostreviewedandaremorelikelytogiveadeniteanswer.Ifsuchananalysiscouldberecycledforothermodelsmotivatedbyacompletelydifferenttheorythenitcouldsavealotoftimeandeffortandcouldbearstchecktoseethevalidityofamodel.Chapter 5 ,describesindetailourworkinidentifyingandseparatingthemodelfreeaspectoftheCMSanalysisofsame-signdileptonsearchforsupersymmetryinthemSUGRAm0-m1 2planefromthemodeldependentpart. 19

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CHAPTER2LARGEEXTRA-DIMENSIONSIGNATURETHROUGHZ-PAIRPRODUCTIONATTHELHC 2.1IntroductiontoExtraDimensionsTheideaofextradimensionswasoriginallyproposedbyT.KaluzaandO.Kleinwaybackin1926inanattempttounifyGravitywithElectro-magnetism.InKaluza-Kleintheories,theextradimensionsarespatialdimensionsandtheyaremuchdifferentthanourregularthreedimensions,inthesensethattheyarecompactiedwithsomecompacticationscaleR)]TJ /F7 7.97 Tf 6.59 0 Td[(1c1.Forexample,ifwehaveoneextradimension,itcanbeacircleofradiusRc.Ifwehavemorethanoneextradimensionitcanbeahigherdimensionalsphereoratoruswithourregular(3+1)-dimensionconnedtothe4-dhypersurface.AKaluza-Klein(1+3+d)-dimensionalspace-timewillhavethegeometryofadirectproductM4Xd,M4beingfour-dimensionalMinkowskispace-timeandXdacompactmanifoldoftheextraspatialdimensions.OneofthemostexcitingtheoreticaldevelopmentsofrecentyearshasbeentheideathattheobservableUniversecouldbeconnedtoafour-dimensionalhypersurface,calledbrane,inahigher-dimensional`bulk'spacetime.Thus,onlygravitypropagatesinthebulkandallStandardModeleldsareconnedtothebrane.Westartbyapplyingtheideaofextradimensiontothesimplestofcasesofarealscalareldin5-dimensional(oneextradimension)space-time.TheLagrangianlookslike L=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2@M@M,M=0,1,2,3,4, (2) where,(M)(x,y)(t,~x,y),xbeingthe4-dimensionalco-ordinatewith=0,1,2,3,andyisthecoordinateintheextradimension.Astheextradimensionis 1Forsimplicityofourphenomenologicalanalysis,weassumetorroidalcompacticationwithalldimensionshavingthesamecompacticationradiusRc. 20

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compactiedonacircleofradiusRc, (x,y)=(x,y+2Rc). (2) Thenweexpandthiseldintheharmonicsofacircle (x,y)=+1Xn=n(x)einy=Rc. (2) SubstitutingthisexpansionoftheeldintheexpressionfortheLagrangianweget L=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(1 2+1Xm,n=@n@m)]TJ /F3 11.955 Tf 13.15 8.08 Td[(nm R2cmnei(m+n)y=Rc. (2) Hence,theactiontakestheform S=Zd4xZ2Rc0dyL=Zd4x)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2@0@0)]TJ /F13 11.955 Tf 11.96 16.27 Td[(Zd4x+1Xn=1@n@n+n2 R2cnn. (2) Here,wehaveusedthefactthatn=)]TJ /F4 7.97 Tf 6.58 0 Td[(nandabsorbedthefactorof2Rccomingfromtheintegrationoverybyredeningn=p 2RcnThus,fromthisactionwecanseethatwehave Asinglerealmasslessscalareld,0; Aninnitenumberofmassivecomplexscalarelds(n)withmassesinverselyproportionaltothecompacticationradius,mn=n Rc.ThesestatesarecalledtheKaluza-Kleinmodes,withthemasslessscalareldbeingthezero-mode.Atlowenergyordistancesmuchlargerthanthecompacticationscaleonlythezero-modeisimportant,butathigherenergiesallthemodeshavetobeconsidered.Insteadoftakingascalareldasdoneabove,ifwetakeanAbeliangaugeeld,withtheLagrangian L=)]TJ /F6 11.955 Tf 16.35 8.09 Td[(1 4g2FMNFMN,M,N=0,1,2,3,4 (2) andtreatittheexactsamewayasthescalareld,thenweget AsinglerealmasslessgaugeeldA(0)withgaugecoupling=g2=(2Rc); 21

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Aninnitenumberofmassivegaugebosonswithmasses,mn=n Rc. Amasslessscalareld,A(0)5.Now,usingthesameideaforthegravity,thereasonwhythistheorywasproposedintherstplace,wetakethegravitationalaction S=M3Pl 2Zd4xdyp GR5,. (2) where,MPlisthePlankscalein5-dimension,GMNisthe5-dimensionalmetric,G=GMMandR5istheRicciscalarin5-dimensionalspace-time.Treatingitthesamewayasthescalarandvectorelds,weget Asinglemasslessspin-2graviton,g(0). Aninnitenumberofmassivespin-2gravitonswithmassesinverselyproportionaltothecompacticationradius,mn=n Rc. Amasslessscalareld,g(0)55. Amasslessgaugeeld,g(0)5.Furthermorecomparingthis5-dimensionalactiontothe4-dimensionalactionforgravityweget M2Pl=M3Pl(2Rc). (2) Theexpression(forgeneralizedd-dimension)abovealsocomesfromusingGauss'slawongravitationaleldandcomparingthegravitationalforcebetweentestchargesm1andm2separatedbyadistanceofrin4-dimensionandd-dimension.Gauss'slawfor(3+1)-dimension: I2FdA=1 M2Plm1m2, (2) F=1 M2Plm1m21 r2. (2) 22

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Gauss'slawforD-dimension(D=1+3+dextradimension): I2+dFdA=1 M2+dPlm1m2, (2) F=(4)d=2\(d=2) M2+dPlRdcm1m21 r2, (2) M2Pl(4)d=2\(d=2)=M2+dPlRdc. (2) Where,MPlisthefundamentalscaleofthetheory. 2.2LargeExtraDimensions:ADDModelSuchideasofextradimensionproposedbyKaluzaandKleingaverisetoelegantsolutions[ 1 8 9 ]tothewell-knowngaugehierarchyproblemofhighenergyphysics,whichisjusttheinstabilityagainstquantumcorrections.Whatisevenmoreinteresting,perhaps,isthesuggestion[ 10 11 ]thattherecouldbeobservablesignalsofquantumgravityatcurrentandfutureacceleratorexperiments,andthispossibilityhasspawnedavastandincreasingbodyofworkoverthepastveyears.Thisrelativelynewsetofideas,commonlydubbed`BraneWorldPhenomenology',basesitselfontwomainprinciples:theconceptofhiddencompactdimensionsandthestring-theoreticideaofDp-branes.Thesimplestbrane-worldscenarioistheso-calledArkani-HamedDimopoulosDvali(ADD)model[ 1 8 9 ],inwhichtherearedextraspatialdimensions,compactiedonad-torusofradiusRceachway.TogetherwiththefourcanonicalMinkowskidimensions,thisconstitutesthe`bulk'spacetime.InthisscenariotheradiusRcoftheextradimensionscanbeaslargeas44m[ 12 ].However,theSMeldsareconnedtoafour-dimensionalsliceofspacetime,withthicknessnotmorethan10)]TJ /F7 7.97 Tf 6.59 0 Td[(17cm,whichiscalledthe`brane'.IftheADDmodelisembeddedinastring-theoreticframework,the`brane'is,infactaD3-brane,i.e.a3+1dimensionalhypersurfaceonwhichtheendsofopenstringsareconned.However,itisnotabsolutelyessentialtoembedthemodelinastringtheory,andtheword`brane'or`wall'isthenusedsimplytodenotethehypersurface(orthinslice)wheretheSM 23

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eldsareconned.Acrucialfeatureofthismodelisthatgravity,whichisapropertyofspacetimeitself,isfreetopropagateinthebulk.Asaresulta)Planck'sconstantinthebulkMPlisrelatedtoPlanck'sconstantonthebraneMPl('1.21019GeV)by (MPl)2+d=(4)d=2\(d=2)M2Pl(Rc))]TJ /F4 7.97 Tf 6.59 0 Td[(d. (2) Table2-1.Compacticationlengthsfordifferentnumbersofextradimensions. numberofextradimensionsRc(inm) d=11012d=210)]TJ /F7 7.97 Tf 6.58 0 Td[(3d=310)]TJ /F7 7.97 Tf 6.58 0 Td[(8 b)AtMPl=1TeV,Table 2-1 givesthecompacticationlengthsfordifferentnumberofextradimensions.Thismeans,forRc44m[ 12 ],itispossibletohaveMPlaslowasaTeVford3.(Thenormalizationofreference[ 11 ]hasbeenadoptedhere).Thissolvesthegaugehierarchyproblemsimplybybringingdownthescaleofnewphysics(i.e.stronggravityinthiscase)toaboutaTeVandtherebyprovidinganaturalcut-offtotheSM,sincethestringscaleMPlnowcontrolsgraviton-inducedprocessesonthebrane.c)ThereisahugenumberofmassiveKaluza-Kleinexcitationsofthe(bulk)gravitoneldonthebrane,withmassesmn=n=Rc,andthesecollectivelyproducegravitationalexcitationsofelectroweakstrength,whichmaybeobservableatcurrentexperimentsandthoseplannedinthenearfuture.d)Ashigherenergiesarereached,thedistancesprobedaresmaller,andfordistancescomparabletoorsmallerthanthecompacticationlength,thedependenceofgravitationalforceondistancechangesfrominversesquaretosomethingsteeperdependingonthenumberofextradimensions.ItisonlyfairtomentionthatamajordrawbackoftheADDmodelisthatitcreatesanewhierarchybetweenthe`stringscale'MPl1TeVandthesizeoftheextra 24

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dimensionsR)]TJ /F7 7.97 Tf 6.59 0 Td[(1c1eV.Infact,thehugesizeoftheextradimensions(comparedtothePlancklength)isnotstableunderquantumcorrections,whichtendtoshrinkitdownuntilMPlR)]TJ /F7 7.97 Tf 6.59 0 Td[(1cMPl1019GeV,atwhichstagetheoriginalhierarchyproblemisreinstated.Nevertheless,thereareseveralvariantsoftheADDmodelwhichaddressthisprobleminvariousways,andsomeoftheseideasmaynotbefarfromthetruth.Fromaphenomenologicalpointofview,itis,therefore,reasonabletopostponeaddressingthestabilityissue,andproceedtostudytheminimalADDmodelanditsconsequencesforexperiment. 2.3PhenomenologicalSignaturesThekeyfeaturesinADD-phenomenologyare,a)Massivegravitonscoupletoanythingwithenergyandmomentum.b)Gravitonsarecolor/avorblind.c)Gravitoncouplingincreaseswithenergy.d)Individualgravitons(GnmassivegravitonofnthmodeintheKKtower)escapesdetection.e)Thecollectiveeffectofgravitonsinthewholetowerhasanear-electroweakinteractionstrength.TheexperimentalconsequencesofADDgravityhavebeenmainlystudiedinthecontextof Figure2-1.Representativediagramsfor(a)realgravitonemissionprocessand(b)virtualgravitonexchangeprocess,producingStandardModelparticlesatahadroncollider 25

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RealGravitonEmission:Inthiscase,thenalstategravitonsintheADDmodelare`invisible',escapingthedetectorbecauseoftheirfeebleindividualinteractions(M)]TJ /F7 7.97 Tf 6.59 0 Td[(1Pl)withmatter.Fig. 2-1 (a)isarepresentativediagramofsuchprocess,wheretheinitialstateconsistsofaquarkandananti-quark,annihilatingintoagluonandarealgraviton(thesignaturebeingajetandmissingenergy).Therealgravitoninfactcanbeemittedfromeitherofthetwootherexternallegsorthevertexaswell.Thenalstate,involvingmissingenergyduetogravitons,willbebuiltupbymakinganincoherentsumoverthetowerofgravitonmodes.Thusthecross-sectionwillbe, (q+q!g+Gn)=Xn(mn)=Zp s0dm(m)(m) (2) where,thedensityofstates,(m)isgivenby (m)=2Rncmn)]TJ /F7 7.97 Tf 6.58 0 Td[(1 (4)n=2)]TJ /F3 11.955 Tf 6.77 0 Td[(n=2 (2) andtheintegrationiscutoffatthekinematiclimitofp s.VirtualGravitonExchange:Inthiscasethegravitonisnotproducedasanalstateparticle.Fig. 2-1 (b)isarepresentativediagramofsuchprocess,wheretheinitialstateconsistsofaquarkandananti-quarkandthenalstateconsistsofanelectronandapositron,withans-channelgravitonpropagator.Unlikethecaseofrealgraviton,allnalstatesareexactlysameforvirtualgravitonsfromalltheKKmodes.Thuswedoacoherentsum(atamplitudelevel)tocalculatethecross-section.Whilecalculatingtheamplitudeforeachpropagator(correspondingtodifferentKKmodegraviton),wegetafactorof1 s)]TJ /F4 7.97 Tf 6.59 0 Td[(m2n,where,sisthecenterofmassenergyandmnisthemassofthenthKKmodegraviton.Summingoverallthemodesisdonein[ 10 ]as, 1 M2PlXn1 s)]TJ /F3 11.955 Tf 11.96 0 Td[(m2n=4 4, (2) 26

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whereisthecutoffstringscaleandMPlisthereducedPlanckscale(MPl=MPl 4).Also,forsomeoftheprocesses,wecangetthesamenalstatethroughaStandardModelprocess(forexampleinthecaseofFig. 2-1 (b)thestandardmodelprocesscanbeans-channelprocesswithaphotonorZ-bosonpropagator).Thusthestandardmodelprocessisalsoaddedusingthecoherentsum.Therefore,whenwecalculatethetotalcross-section,alongwithaStandardModelandaBeyondStandardModel(BSM)term,wealsogetaninterferenceterm.Becauseof8suppressioninpureBSMcomparedto4suppressionintheinterferenceterm,theinterferencecontributionisalotmoreimportantthanthepureBSMterm.Ineithercase,itmaybeshown[ 10 11 ]that,aftersumming,thePlanckmassMPlcancelsoutofthecross-section,leavinganinteractionofnear-electroweakstrength.Atahadroncollidermostfrequently,thesignalcomingfromrealgravitonproductionisaccompaniedbyanisolatedjet.Theserealproductionchannelshavestrongdependenceonthenumberofextradimensions.However,onecanexpecttohavebetterreachonthestringscale,as,virtualgravitonexchangeprocesseshaveveryweakornodependenceonthenumberofextradimension.InthispaperwestudytheeffectofgravitonpropagationontheZ-pairproductionattheLHCandcomparetothedominantSMbackground. 2.4LimitsBelowarethelimitsimposedonthesizeandnumberofextra-dimensionfromvariousexperiments.GravitationalInverse-squarelaw:ExperimentswereconductedbyKapneretal.[ 12 ],totestthevalidityofGravitationalInverse-SquareLawbelowDark-EnergyLengthscale.Theyconcludedthatthelawholdsdowntoalengthscaleof56mandthatanextradimensionmusthaveasizesmallerthanthat(Rc56m). 27

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Collidersearch:TheLEPcollaborationshavesearchedforRealGravitonemissionandL3hasthebestlimit.TheirlimitsareMPl>1.5TeV)]TJ /F6 11.955 Tf 12.27 0 Td[(0.5TeVfornumberofextradimensionsn=2to8.CDFputsthelimitonMPl>0.6TeV)]TJ /F6 11.955 Tf 11.96 0 Td[(0.55TeVforn=4to8.CosmologicalData:TherearealsolimitsonLargeExtraDimensionsfromSupernovacooling[ 13 ].Forn=2theyputalimitonMPl>84TeVandRc90m.Forn=3theyputalimitonMPl>7TeVandRc0.19m. 2.5CalculationWeexploretheADDModelforLargeExtraDimensionwithdextra-dimensions,inwhichweconsidertheZ-pairproductionbyproton-protoncollision,mediatedbyaVirtualGraviton.Inourcalculationsweincludefermionanti-fermionaswellasgluon-gluoninitialstates.Weareinterestedinthisparticularprocess,mainlybecause,Z-pairproduction(whichthendecaysintofourleptons)isoneofthemostimportantmodesinthesearchforhiggsandexperimentalistsaredenitelygoingtobelookingforfourleptonsignals.Forourcalculations,weusedtheGiudice,RattaziandWells(GRW)notation[ 10 ].But,incaseofZZwenoticedthatintheGRWnotation,therewasnoFeynmanruleforcouplingofGravitontomassivevectorboson,forwhichweusedtheFeynmanrulesinHan,LykkenandZhang(HLZ)notation[ 11 ](normalizedwiththatoftheGRWnotation).Afterwestartedourcalculations,wesawsimillarworkdonein[ 14 ]and[ 15 ],butourresultdisagreedwiththeirresult.Thenanotherpaper[ 16 ]waspublishedwithcalculationsatnexttoleadingorderandourresultmatchedexactlywiththeirsatleadingorderlevel.In[ 14 ],theytakethecontributionfromqq!ZZtobezeroandtheircross-sectionblowsupformZ!0.Forthetensormanipulationswhiledoingamplitudesquaring,fermiontracecalculationsetc.weusedtheprogramFORM(aSymbolicManipulationSystem).Forcross-sectioncalculationsweusedMathematicaandFortran.Weusedthesub-routine 28

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Figure2-2.FeynmandiagramsfortheleadingorderZpairproductionprocessinSM. Figure2-3.FeynmandiagramsfortheleadingorderZpairproductionprocessinADDmodel. VegasofferedinthepackageCubaforMathematica,fornumericalintegration.WeusedthePartonDistributionFunction,CTEQ5L.Wedene,x=t ^s,z=(MZ p ^s)where,^sisthecenterofmassenergy.For,ff!ZZ 29

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Figure2-4.Dependenceoftotalcross-section()forff!ZZ,onEminT,Zata1TeVxedcenter-of-massenergye+e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(collider.EminT,ZistheminimumvalueofET(TransverseEnergy)ofoutgoingZ-bosons.IntheplotsthesymbolsrepresentresultsfromtheeventgeneratorSherpaandthelinesrepresentouranalyticalresultusing( 2 ).Thelowerlinerepresentsthecross-sectioncomingfromonlyStandardModelandtheotherthreerepresentcross-sectionsincludingthecontributionfromextradimensionandcorrespondtothreedifferentvaluesofthestringscalenamely2.3TeV,3.0TeVand3.8TeV. dff!ZZ dx=)]TJ /F5 11.955 Tf 20.84 8.08 Td[(^s3 8812z8)]TJ /F6 11.955 Tf 11.95 0 Td[(12z6(4x+1)+z4(72x2+48x+5))]TJ /F6 11.955 Tf 9.3 0 Td[(2z2(24x3+30x2+11x+2)+x(12x3+24x2+17x+5)+2^s 42z8)]TJ /F3 11.955 Tf 11.96 0 Td[(z6()]TJ /F6 11.955 Tf 9.3 0 Td[(8x+1)+z4x(12x2+2x)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(z2x(8x2+7x+1)+x(2x3+4x2+3x+1)C1)]TJ /F6 11.955 Tf 20.46 8.09 Td[(22 ^s4z8)]TJ /F6 11.955 Tf 11.95 0 Td[(4z6(3x+1)+z4(14x2+6x+1))]TJ /F6 11.955 Tf 11.95 0 Td[(2z2x(2x+1)2+x(2x3+4x2+3x+1)C2, (2) 30

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Figure2-5.Dependenceoftotalcross-sectionforff!,onEminT,ata1TeVxedcenter-of-massenergye+e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(collider.EminT,istheminimumvalueofET(TransverseEnergy)ofoutgoingphotons.IntheplotsthesymbolsrepresentresultsfromtheeventgeneratorSherpaandthelinesrepresentouranalyticalresultusing( 2 ).Thelowerlinerepresentsthecross-sectioncomingfromonlyStandardModelandtheotherthreerepresentcross-sectionsincludingthecontributionfromextradimensionandcorrespondtothreedifferentvaluesofthestringscalenamely2.3TeV,3.0TeVand3.8TeV. where C1=(g2Af+g2Vf) xw(xw)]TJ /F6 11.955 Tf 11.96 0 Td[(1)x()]TJ /F6 11.955 Tf 9.3 0 Td[(2z2+x+1),C2=(g4Af+6g2Afg2Vf+g4Vf) x2w(xw)]TJ /F6 11.955 Tf 11.96 0 Td[(1)2x2()]TJ /F6 11.955 Tf 9.3 0 Td[(2z2+x+1)2,andgAfandgVfaretheaxialvectorandvectorcouplingoftheincomingquarktoZboson.For,ff! 31

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dff! dx=)]TJ /F5 11.955 Tf 20.84 8.09 Td[(^s3 884x(x+1)(2x2+2x+1)+2^s 42x2+2x+1)]TJ /F6 11.955 Tf 20.45 8.09 Td[(22 ^s(2x2+2x+1) x(x+1), (2) whichissameasthatofGRW[ 10 ].Forgg!ZZ, dgg!ZZ dx=)]TJ /F5 11.955 Tf 23.99 8.09 Td[(^s3 24812z8)]TJ /F6 11.955 Tf 11.95 0 Td[(12z6(4x+1)+z4(72x2+48x+5))]TJ /F6 11.955 Tf 19.27 0 Td[(2z2(24x3+30x2+11x+2)+(12x3+24x2+17x+5), (2) andfor,gg! dgg! dx=)]TJ /F5 11.955 Tf 23.98 8.09 Td[(^s3 168(2x4+4x3+6x2+4x+1), (2) ForcomparisonwithnumericalresultsfromeventgeneratorSherpa,wemadeplotswiththecross-sectionvsthePTcutoftheoutgoingZboson(and).WewantedtomatchouranalyticalresultforthecaseofhadroncolliderwiththenumericalresultfromtheeventgeneratorSherpa.Butwecouldn'tgetSherpatocalculatethecross-sectionfortheprocessgg!ZZandgg!.WefoundoutthatSherpacouldn'tcalculatethecross-sectionofaprocessthatdoesn'thaveaStandardModelcounterpart.So,tocross-checkouranalyticalresultwiththenumericalonefromSherpawechosee)]TJ /F6 11.955 Tf 10.11 -4.34 Td[(+e+!Z+Zande)]TJ /F6 11.955 Tf 10.11 -4.34 Td[(+e+!+,wherewexedthecenterofmassenergytobes=1TeV.InFigures( 2-5 )and( 2-4 )thedottedlinesrepresentresultfromSherpaandthecontinuouslinesrepresentouranalyticalresult.Thex-axiscorrespondstotheEminT(transverseenergycutoftheoutgoingZor)inTeV.They-axiscorrespondstothecross-sectioninpb.The4setsoflinesrepresentSMand3ADDresultsbasedon3valuesofnamely2.3TeV,3.0TeVand3.8TeV.Oneinterestingthingtonoteisthatthe 32

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percentagechangeinStandardModelcross-sectionwithadditionofADDcontributionincreaseswithincreasingEminT.Moreoveraswedecreasethecutoffstringscalethecontributionfromtheextradimensionprocessincreases.AlsoaswegofromtoZZthisdifferenceisevenmoremagnied.Tocomparewithexistingresults,wealsoconsideredseveralotherprocessesnamelypairproductionfromgluon-gluonaswellasfermionanti-fermioninitialstate,suchas,anelectronandapositronannihilatingintoamasslessfermion-anti-fermionpair.Forthesakeofcompletenesswealsoincludedanelectronandapositronannihilatingintoapairoftopquarkandanti-topquark.For,ee!ff, dee!ff dx=d(ee!ff)SM dx+d(ee!ff)INT dx+d(ee!ff)ADD dx=d(ee!ff)SM dx+Nf^s3 32832x4+64x3+42x2+10x+1)]TJ /F3 11.955 Tf 20.46 8.08 Td[(Nf^s 24QeQf(8x3+12x2+6x+1)+1 xw(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xw)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z2))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(gVegVf(8x3+12x2+6x+1)+gAegAf(6x2+6x+1)+ef)]TJ /F5 11.955 Tf 10.49 8.09 Td[(^s 24Q2e(x3+11x2+24x+22+9 x)+g2Ve+g2Ae xw(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xw)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z2)(x3+6x2+9x+4)+1 xw(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xw)(x)]TJ /F3 11.955 Tf 11.96 0 Td[(z2))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(g2Ve(5x3+15x2+18x+9)+g2Ae(5x3+15x2+10x+1)+^s3 328(9x4+60x3+126x2+114x+40). (2) 33

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For,ff!tt dee!tt dx=d(ee!tt)SM dx+d(ee!tt)INT dx+d(ee!tt)ADD dx=d(ee!tt)SM dx+3 32^s840m8t)]TJ /F6 11.955 Tf 11.95 0 Td[(8m6t^s(18x+5)+4m4t^s2(50x2+34x+5))]TJ /F6 11.955 Tf 20.45 0 Td[(2m2t^s3(64x3+80x2+26x+1)+^s4(32x4+64x3+42x2+10x+1))]TJ /F5 11.955 Tf 24.68 8.08 Td[( ^s24)]TJ /F6 11.955 Tf 9.3 0 Td[(12m6t+2m4t^s(14x+5))]TJ /F6 11.955 Tf 20.45 0 Td[(4m2t^s2(6x2+5x+1)+(8x3+12x2+6x+1)+3 2^s24xw(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xw)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z2)2gVegVt)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 11.955 Tf 9.3 0 Td[(12m6t+2m4t^s(14x+5))]TJ /F6 11.955 Tf 11.95 0 Td[(4m2t^s2(6x2+5x+1)+(8x3+12x2+6x+1)+gAegAt)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(2m2t+^s)(6m4t)]TJ /F6 11.955 Tf 11.95 0 Td[(4m2t^s(3x+1)+^s2(6x2+6x+1). 2.6AnalysisWehaveallthenecessaryexpressionsforthedifferentialcross-sectionforZ-pairproductionasweseeinEquations( 2 )and( 2 ).SoweareinapositiontodoananalysistoseehowtheZ-pairsignalstandsoutinthepresenceoftheStandardModelbackground.InFigure( 2-6 )wehavethedifferentialcross-sectionforpp!ZZatLHC(14TeV).Onthex-axiswehavetheinvariantmassofthetwooutgoingZ-bosons(mZZ)inGeVandonthey-axiswehavethedifferentialcross-sectioninpb/GeV.TheareasunderthedottedlinesrepresentsADDcontributionsfromuu,ddandggseparately.ThecontributionfromStandardModelfallssharplywithincreasingmZZ.Thisbehaviourcanbeattributedtothepartondistributionfunctionofthequarksandgluons(whichbehaveslikee)]TJ /F7 7.97 Tf 6.4 0 Td[(^s).ButthecontributionfromextradimensionincreaseswithincreasingmZZ.Thisisbecauseinthedifferentialcross-sectiontermscomingfromextradimensionwehavepositivepowersof^s(^s3inthepureextradimensiontermand^sintheinterferenceterm). 34

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Figure2-6.Differentialcross-sectionforpp!ZZatLHC(14TeV).Onthex-axiswehavetheinvariantmassofthetwooutgoingZ-bosons(mZZ)inGeVandonthey-axiswehavethedifferentialcross-sectioninpb/GeV.TheareasunderthedottedlinesrepresentsADDcontributionsfromuu,ddandggseparately.ThecontributionfromStandardModelfallssharplywithincreasingmZZ.Thisbehaviourcanbeattributedtothepartondistributionfunctionofthequarksandgluons(whichbehaveslikee)]TJ /F7 7.97 Tf 6.4 0 Td[(^s).ButthecontributionfromextradimensionincreaseswithincreasingmZZ.Thisisbecauseinthedifferentialcross-sectiontermscomingfromextradimensionwehavepositivepowersof^s(^s3inthepureextradimensiontermand^sintheinterferenceterm). InFigure( 2-7 )wehavethenormalizeddifferentialcross-sectionforpp!ZZatLHC(14TeV).Thex-axiscorrespondstoCosineoftheangle()betweenthebeamdirectionandtheoutgoingZ-bosonandthey-axishasthedifferentialcross-section.Forgure(a)ismeasuredincenterofmassframeandingure(b)ismeasuredinthelabframe.Inboththecasesthered(dashed)linecorrespondstotheresultfromonlyStandardModelandtheblue(solid)linecorrespondstotheresultincludingcontributionsextradimension.Thestringcutoffscaleissetat1.5TeV.InFigure( 2-8 )wehavethedependenceofcross-sectiononthestringscaleforpp!ZZatLHC(14TeV).The 35

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Figure2-7.AngulardistributionoftheoutgoingZbosonsforpp!ZZatLHC(14TeV).Thex-axiscorrespondstoCosineoftheangle()betweenthebeamdirectionandtheoutgoingZ-bosonandthey-axishasthedifferentialcross-section.Forgure(a)ismeasuredincenterofmassframeandingure(b)ismeasuredinthelabframe.Inbothcasesthered(dashed)linecorrespondstotheresultfromonlyStandardModelandtheblue(solid)linecorrespondstotheresultincludingcontributionsextradimension.Thestringcutoffscaleissetat1.5TeV. dottedlinesrepresentADDcontributionsfromuu,ddandggseparately.Thex-axisrepresentsthestringcutoffscaleandthey-axisrepresentsthecross-section.Asthe 36

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Figure2-8.Dependenceofthecross-sectionforpp!ZZonthestringscaleatLHC(14TeV).Thedottedlinesrepresentsaddcontributionsfromuu,ddandggseparately.Thex-axisrepresentsthestringcutoffscaleandthey-axisrepresentsthecross-section.Asthestringcutoffscaleincreases,thecontributionfromextradimensiondecreasesrapidly.Thisisduetothefactthatthepureextradimensioncontributiontermhasafactorof)]TJ /F7 7.97 Tf 6.58 0 Td[(8andtheinterferencetermhasafactorof)]TJ /F7 7.97 Tf 6.59 0 Td[(4.Itcanbeseenfromtheplotthattheextradimensioncontributionfromuuistwicethatofddwhichcanbeattributedtothepartondistributionfunction,apartfromwhichtheamplitudeandphasespaceareexactlysameforbothprocesses. stringcutoffscaleincreasesthecontributionfromextradimensiondecreasesrapidly.Thisisduetothefactthatthepureextradimensioncontributiontermhasafactorof)]TJ /F7 7.97 Tf 6.59 0 Td[(8andtheinterferencetermhasafactorof)]TJ /F7 7.97 Tf 6.59 0 Td[(4.Itcanbeseenfromtheplotthattheextradimensioncontributionfromuuistwicethatofddwhichcanbeattributedtothepartondistributionfunction,apartfromwhichtheamplitudeandphasespaceareexactlysameforboththeprocesses. 37

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2.7ResultsInscenarioswithquantumgravitypropagatinginlargeextradimensions,wecomputedtheeffectsofthevirtualKKgraviton-exchangeamplitude.Wenoticedthatnotonlytheinterferenceterm(ADDandSM)contributestothesignalbutthepureADDamplitudeisalsoquitesignicant.Asmentionedearlier,wewereunabletonumericallycomputesuchpureADDcontributionusingtheeventgeneratorSherpaforprocesseswithoutaStandardModelcounterpart.Inrecentliteraturethesecalculationshavebeendoneforpp!ZZ.Inthisworkwewereabletocross-checktheirresultsandidentifythecorrectonesfromthewrongones. 38

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CHAPTER3DIQUARKSINPYTHIA 3.1DiquarkProductionandDecayAdiquarkisqqresonancewithspinzeroorone,baryonnumber2 3,andelectriccharge)]TJ /F7 7.97 Tf 10.49 4.71 Td[(4 3,1 3or)]TJ /F7 7.97 Tf 10.49 4.71 Td[(2 3.Itcantransformeitherasa6or3underSU(3)[ 6 ].Forconcreteness,thediquarkisconsideredtobeaspin-0sextet(transformsunderSU(3)asa6). Table3-1.Quantumnumbersforvariousparticlesinvolvedindiquarkproductionanddecay. FieldSpinSU(3)CSU(2)LU(1)Y ec1=2)]TJ 50.38 0 Td[()]TJ /F6 11.955 Tf 61.97 0 Td[(1uc1=23)]TJ /F6 11.955 Tf 49.84 0 Td[(2=3D06)]TJ /F6 11.955 Tf 49.84 0 Td[(4=3L1=26)]TJ /F6 11.955 Tf 49.84 0 Td[(7=3Lc1=2 6)-2613()]TJ /F6 11.955 Tf 49.84 0 Td[(7=3 IthascouplingstoSU(2)singletandSU(3)tripletup-typequarks(andissymmetricinavorindices).So,thisdiquark(D)canthusbeproducedasaresultofcollisionbetweentwoup-typequarks(uc)intermsoftheproductionoperatorasshownbelow. OD=D 2Ducuc,(3)where,DisthecouplingconstantwhosenormalizationdependsonthenormalizationofthecolormatricesRawithwhichDisexpandedsuchthatD=DaRa.WithTr(RaRb)=ab,thepartoniccross-sectionfortheproductionofthediquarkis, (uu!D)= 62D(^s)]TJ /F3 11.955 Tf 11.96 0 Td[(m2D).(3)Since,itdoesn'thaveanyothercouplingavailableitwoulddecaybackinto2upquarkswithadecaywidthgivenby \(D!uu)=1 162DmD.(3)AnotherresonancecalledLepto)]TJ /F3 11.955 Tf 12.33 0 Td[(diquarkhasbeenintroducedintothispicture,whichisavector-likefermionthattransformsunderSU(3)thesamewayasthediquarkand 39

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hasthesamebaryonnumber,butdiffersintermsofotherquantumnumbers.Italsohastobelighterthanthediquarksuchthatthedecayofdiquarkintolepto-diquarkcanbeallowedbyphasespace.Thedecayoperatorwillbe, DDLcec,(3)andtheparticaldecaywidthwillbe, \(D!Le)=1 16D2mD.(3)Thislepto-diquarkwillfurtherdecayinto2quarksandaleptonviathetwooperatorsaboveinthissection, Lc!ecucuc(3)throughanoff-shelldiquark.Thus,thenalstatehas2jetsandapairofoppositesigndileptons. 3.2ImplementationinPythia 3.2.1ProblemwithImplementationTobeabletoanalysethismodelweneedtobeabletogeneratesucheventsinoneofthemanywidelyavailablemonte-carloeventgenerators.Since,thismodelwasnotyetimplementedinanyoftheavailableeventgeneratorsthisneededtobedonetodoanypossibleanalysis.ForthispurposewechosePythia[ 7 ],since,ithasaveryneatfeaturewithinstructionforincorporatingexternalprocesses.Also,Pythiacomeswiththeexibilityfordeninganewparticlethatcanbeincorporatedintotheexternalprocess.However,wefacedauniqueproblemindoingso.AparticleinPythiacanhaveuptoonecolorlabelandoneanti-colorlabel.Thiswouldposenoissueifourdiquarkwasananti-triplet(wecoulddeneittohavenocolorlabelandananti-colorlabel).But,sincethediquarkwechoseisasextetitmeansweneedtwocolorlabelsforitwhichisnot-triviallypossibletodoinPythia. 40

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3.2.2OurWorkaroundWedenedneitherthediquarknorthelepto-diquarkasparticlesinPythia'sdecaytable.Weimplementedtheexternalprocessonlyintermsofthetwoincomingupquarksandoutgoingsamesigndileptonsandthetwooutgoingup-quarks,thuseliminatingtheneedtohaveparticleswithtwocolorlabels.Colorstretchingwasdoneonlywithrespecttotheincomingandoutgoingupquarkssuchthatitwasconservedintheprocess. 3.2.3DescriptionoftheImplementationThePythia6.4manualhasspeciedaprescriptioninsection9.9toincludeanexternalprocessinPythiawhichwefollowedforourimplementation.Accordingtotheprescription,weneedtowriteourownsubroutines. UPINIT:Thissubroutineinitializesthecenterofmassenergy,theincomingbeamsandtheexternalprocesses. UPEVNT:Thissubroutinesamplesthephasespace,evaluatestheprocesscross-sectionandsetsthecolortopologyandpartonshowerscales.ThesesubroutinesbeingalreadypresentinPythiaasdummysubroutinesweneedtoremovethemfromthePythiasourcecodeandcompileittomakeitcompatibleforusewithourexternalprocesscode.SincetheactualphysicsisimplementedinthesubroutineUPEVNTwe'llelaborateitindetailedsteps.SamplingofthePhaseSpace:Theveryrstthingtodois,decidehowthephasespaceshouldbesampled.Thetotalcrosssectionfora2!1processcanbewrittenas =Zdx1Zdx2f1(x1,Q2)f2(x2,Q2)^(^s),(3)or,intermsofx1x2andy1 2lnx1 x2,as =Zd Zdyx1f1(x1,Q2)x2f2(x2,Q2)^(^s).(3)So,asimplerecipewouldbetopickandyuniformly.However,theresonantcross-sectionandpartondistributionsarepeaked,sothiswouldbeinefcient.In 41

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particlar,thecross-sectionispeakedat^s.So,followingtheinstructionsfromsection7.4inthePythiausermanual,theintegralisrewrittenas = SZh()dZhy(y)dyx1f1(x1,Q2)x2f2(x2,Q2) 2h()hy(y)^s ^(^s), (3) where,andyaregeneratedaccordingtothedistributionsh()andhy(y)givenas, h()=e2cmmD)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(D B)]TJ /F5 11.955 Tf 11.95 0 Td[(A1 (^s)]TJ /F3 11.955 Tf 11.96 0 Td[(mD)2+m2D)]TJ /F7 7.97 Tf 6.77 4.12 Td[(2Dwhere,A=arctan^smin)]TJ /F3 11.955 Tf 11.96 0 Td[(m2D mD)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(DB=arctan^smax)]TJ /F3 11.955 Tf 11.95 0 Td[(m2D mD)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(Dandhy(y)=1 ymax)]TJ /F3 11.955 Tf 11.96 0 Td[(yminwhere,ymax=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2ln()ymin=+1 2ln()h()isdenedso,becausetheprocessbeingonlyinthes-channel=^s SbehaveslikeaBreit-Wigner.Thedependenceonybeinguniform,hy(y)isdenedasaconstantfunctionappropriatelynormalized.Montecarlo:NextstepistogeneraterandompointsfortheMontecarlointegration.Here,rstofallwecalculatevariouslimitsbasedonthemassofthelepto-diquark(L)randomlygeneratedintheeventaccordingtotheBreit-Wignerdistributionwiththe 42

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lepto-diquarkwidth. ^smin=pTmin+q m2L+p2Tmin2min=^smin E2cm^smax=E2cm)]TJ /F6 11.955 Tf 11.71 0 Td[(^smaxmax=^smax E2cmThenwerandomlygeneratetherequiredvariables(andy)whileobeyingthelimitsabove. =1 E2cmm2D+mD)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(Dtan(A+(B)]TJ /F5 11.955 Tf 11.96 0 Td[(A)r)y=ymin+(ymax)]TJ /F3 11.955 Tf 11.95 0 Td[(ymin)r,where,`r'isarandomrealnumbersampleduniformlyfrom0to1.Werejectanyeventfallingoutsidethephasespaceregion.Foreachsuchpointinphasespacewecalculatethecross-section, =d^PSvolPDFCONV, (3) where,fromthematrixelementwehave, d^=2D 6mD)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(D (^s)]TJ /F3 11.955 Tf 11.96 0 Td[(m2D)2+(mD)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(D)2,thePDFissuppliedbyPythia(withscaleQ2=m2D)as PDF=x1f1(x1,Q2)x2f2(x2,Q2),PSvolisthevolumeoftheselectedphasespace, PSvol=^s E2cm2h()hy(y)andCONVisconversionfactorfromGeV)]TJ /F7 7.97 Tf 6.59 0 Td[(2topb. 43

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Settinguptheparticlesinvolvedinthepartonicprocess:Here,wesetupthenumberofincomingandoutgoingparticles,theidentityoftheparticles,theirstatuscodes,theirmotherparticlecodes(0forincomingparticles)andtheircolortopology.Wealsoinitializethemomentaoftheoutgoingparticlestozeroandthatoftheincomingparticlesistakenfromthevariablesfrommontecarlointheprevioussection(i.e.x1andx2fromthePDFs').EventGeneration:Atthispoint,wehavecreatedanobject(asa2!1processinthemontecarlosection)thathasthe4-momentumcongurationofthediquarkandotherpartonicobjectswithcorrectquantumnumbersandcolorconnections.Now,weneedtogeneratethe4-momentumcongurationoftheoutgoingparticles.Thiswedobydecayingthediquarkobjectintothe4outgoingparticles(2up-quarksandapairofoppositesigndileptons).Thiswedointhreesteps.Firstwedecayitintoaleptonandalepto-diquarkobjectandthenthelepto-diquarkobjectintoaleptonandcompositeobjectwhichnallydecaysinto2up-quarks.Intherststepweboosttotherestframeofthediquarkandthenassignthemomentaofthetwodecayproductswithonerandomnumberdeterminingtheazimuthalangle,whichhasauniformatdistributioninf0,2g.Then,weboostbacktothelabframeandgetthemomentafortherstoutgoinglepton.Then,weboosttotherestframeofthelepto-diquarkobjectandassignthemomentaofthetwodecayproductswithtworandomnumbers(oneasbeforeandonefortheinvariantmassofthecompositeobjectwithtwoupquarksrandomlysampledfrom0tomL).Then,weboostbacktothelabframetogetthemomentaforthesecondoutgoinglepton.Finally,weboosttotherestframeofthecompisiteobjectwithtwoupquarksandwithonerandomnumberforweassignthemomentaforthetwooutgoingupquarks.Again,weboostbacktothelabframeandgetthemomentaofthetwoupquarks.Thus,attheendwehavethemomentacongurationforthefouroutgoingparticlesinthelabframe. 44

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3.3ResultsTotestthecode,wegeneratedplotsfortheinvariantmassdistributionofthefouroutgoingparticlesforvariousdiquarkandlepto-diquarkmasses,whichshowsaBreit-Wignershapewiththecorrectwidth.TheseplotsareshowninFigure 3-1 and 3-2 .Figure 3-1 corrspondtoalepto-diquarkofmass100GeVandgure 3-2 correspondstoalepto-diquarkofmass300GeV. Figure3-1.Invariantmassdistributionofthefouroutgoingparticleswiththemassofthelepto-diquarkxedat100GeV. Figure3-2.Invariantmassdistributionofthefouroutgoingparticleswiththemassofthelepto-diquarkxedat300GeV. 45

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WiththedistributionforourMonte-Carlosamplingoptimizedforthegivenprocess,theefciencyturnedouttobe0.3,whichmeanstogenerate1000eventsthesubroutinehastotryaroundmerely3400times,whichcansavealotofcomputingtime. 46

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CHAPTER4SAME-SIGNDILEPTONS 4.1InterpretationofExperimentalResultsWhenexperimentalcollaborationsdoananalysis,theypublishtheresultsoftheirsearchi.e.whetherthetheoreticalmodeltheyconsiderfortheiranalysiscanberuledoutornot.Tomakesuchaclaimtheyneedtoestimatethenumberofbackgroundeventsthatispredictedbyexistingveriedtheoriesandcompareittothepredictednumberofsignaleventsfromthetheoreticalmodelthatpasstheiranalysiscuts.Thislimitonthemaximumnumberofsignalevents(Nsig)canbeusedforothermodelshavingsimilarsignatureastheoneconsideredbythecollaboration.Atheoristcancomputethenumberofsignaleventsfromhis/hertheoreticalmodelpassingthesamecutsasintheexperimentandifthenumberofsucheventsislargerthanthelimitNsiggivenbytheexperimentthenthemodelcanberuledout. 4.2CalculatingNsigTheoreticallyThenumberofsignaleventsdependonthecross-sectionoftheparticlesproducedatpartonlevelcollision,theirbranchingratiosintothenalproducts,theintegratedluminosityandtheefciencywithwhichtheeventscanpassthecutsisgiveninthefollowingrelationship. Nsig=(Pi)BR(Pi)(Pi)L, (4) where,(Pi)isthepartonlevelproductioncross-section,BR(Pi)isthebranchingratiooftheparticlesproducedatpartonlevelintothenaldecayproducts,(Pi)istheefciencyfortheeventtopassthecuts,ListheintegratedluminosityandPi'sarethevariousmodelparameters(e.g.m0,m1 2,tan()etc.formsugra).Onceamodelischosenwecancalculatethemassesofalltheparticlesinvolvedintheprocess(Mi's)andexpressNsigintermsofthoserelevantmasses.Listhesameluminositythattheexperimentalistsusedfortheiranalysis.(Mi)andBR(Mi)canbecalculated 47

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eitheranalyticallyfromthelagrangianorusinganeventgeneratorlikeCalcHEPorPythia.(Mi)ishardertocomputeanalytically,since,1)thereisnoanalyticalformulaforMonte-Carlointegrationrequiredtogenerateeventsand2)detectorsimulationofthenalparticlesintoreconstuctedobjectsinvolvesconvolutedintegrations.WhenwegenerateeventsusingaMonte-Carloeventgenerator,wegettheobjectswiththeirtruemomentum(ptrueandtrueforward/backwardangletrue).Butweareinterestedinndingtheirobservedvaluesatthedetector(pobsandobs).ThuswearelookingfortransferfunctionsP(pobsjptrue)andP(obsjtrue)suchthatforaparticlewithtruemomentumptrue,P(pobsjptrue)istheprobabilitydistributionofobservingpobsinthedetector,suchthat, Z10dptrueP(pobsjptrue)=1 (4) Z10dtrueP(obsjtrue)=1. (4) Hence,theobservedmatrixelementfortheprocessrelatestothetruematrixelement,as, jM(pobs,obs)j2=Z10dptrueZ10dtruejM(ptrue,true)j2P(pobsjptrue)P(obsjtrue). (4) Thus,withtherelevantcutsonandpT,we'llhavetosolvethefollowingconvolutedintegral. Nsig=LZjcutjjcutjdpobsZ1pT p 1)]TJ /F14 5.978 Tf 5.75 0 Td[(cos2dobsjM(pobs,obs)j2. (4) As,wedon'thaveanalyticalformulaeforthetransferfunctions,theaboveintegralhastobecomputednumericallyusingMonte-CarlowhichconsumesalotofCPUtime. 4.3SimpliedModelsAsimpliedmodelisdenedbyaneffectiveLagrangiandescribingtheinteractionsofasmallnumberofnewparticles[ 17 ].Weuseamethodsuchthatwecandivideouranalysisintomodeldependentandmodelindependentcomponents.Themodeldependentpartgovernstheproductioncross-sectionoftheBeyondStandardModel 48

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Figure4-1.Theparameterspace particlesandthebranchingfractionsfortheirdecayintostableparticles.Themodelindependentpartdependsonthetopologyandgovernstheefciencyofsuchaneventpassingtheanalysiscuts.Westartbydescribingthestepsofouranalysis,beginingwiththemodeldependentpart.Forouranalysiswechose(detailsinnextsub-section): searchchannel:SS2LanalysisbyCMS productionchannel:pp!~g~g!~+1~+1(~)]TJ /F7 7.97 Tf 0 -7.97 Td[(1~)]TJ /F7 7.97 Tf 0 -7.97 Td[(1)4j!l+l+(l)]TJ /F3 11.955 Tf 7.08 -4.33 Td[(l)]TJ /F6 11.955 Tf 7.09 -4.33 Td[()4j22~01Becauseofsuchchoices,ourboundsareconservative,sincethereusuallyare,extraproductionsubprocessesandextradecaychannels,bothofwhichwillresultinanincreaseinsignal.Next,weacquirethecutsandthenumberofbackgroundeventsforthesearchchannelfromtheexperimentalpaper,whichinourcaseis[ 18 ].Thenweruntheeventgeneratoranddetectorsimulationinthechosenparameterspace.Wechoseourparameterspacetobe(M1,M2,M3) 4-1 ,whereM1,M2andM3arethebino,winoandgluinomassparameters.ForM1=10GeV,100GeV,200GeV,300GeVand400GeV,wevariedM2andM3suchthatM1
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(=Ncut Ntot).Finallywecalculateandplotthereachusingthefollowingexpression (.BR)max(Mi)=95%CLULyield L.(Mi), (4) asafunctionofthegauginomassparametersMi(i=1,2,3),where,95%CLULyieldrepresentsobservedupperlimitoneventyieldsfromnewphysics(fromtheexperimentalanalysispaper)andListheintegratedluminosity.Inourcase,Lwastakentobe1fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1and95%CLULyieldwastakenfromtheCMSanalysisnotes[ 18 ]. 4.4SimpliedModel:AnIllustrationForourillustration,wechoosetostudytwosame-signisolatedleptons(2SSL)search.Withinasimpliedmodelapproach,wechoosethreerelevantparticles,~G,~W,and~N0.~GisaSU(3)gaugepartner.Forexample,~GcanbeagluinointheMSSM,orexcitedKK-gluoninMUED.~WisthepartnerofSU(2)and~N0isofU(1).Thisisaminimalsetuppreservingthegaugestructureofthestandardmodel.Thus~GisanewcoloredparticlesofSU(3)octet,hasatransitionto~Wwithtwojets.ComparedtoSU(3)tripletpairproduction,thisoctetpairproductionisusuallypreferableduetothegluionfusionscomparedtosinglediagramfromavorpreservinginteractionoftripletpairproduction.Atransitionof~G!~Wwillbeathreebodydecayprocess.Transitionbetween~Wand~N0willgiveachargedleptonandneutrinoifnoneof~Wand~N0willcarryaleptonnumberwhichiscommoninmostBSMs.Thistransitionwillhavetwomodesdependingontheavaiablephasespace.Forexample,ifthemassdifferencebetween~Wand~N0issmallerthanthemassofthestandardmodelWgaugeboson,thenchargedleptonswillbeproducedthroughthethree-bodydecay.Ontheotherhand,whenthephasespaceisavaiableforanonshellWboson,thenthe~WwilldecayintoWand~N0,followedbyasubsequentdecayoftheWboson.AsforthebranchingratioofWintooneleptonandneutrinofromthesetwoprocesses,itwillbemodeldependent.Inshort,oursetupisfollowing: 50

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Figure4-2.Diagramaticdescriptionofoursimpliedmodelsetupfortwosamesignedleptonsignal. ~Gtwojets)338()222()222()339(!~W:Threebody, (4) ~Wl,)287(!~N0:8>><>>:ThreebodyifMMW, (4) where, M=M~W)]TJ /F3 11.955 Tf 11.96 0 Td[(M~N0.(4)ThisprocedureisdiagrammaticallyexpressedinFIG 4-2 .Duetothenatureofsimpliedmodelofthisapproach,itwouldbeimplementedeasilyinvariousnewphysicsmodels.Inthenextsub-section,webreiyexplainhowthissetupcanbederivedfromtheMSSM. 4.5Model-IndependentProcedureWehaveappliedCMSsame-signdileptonanalysis(2SSL)[ 18 ]tothesimpliedmodel,whichinturncanbeapplicabletoanynewphysicswhichhas2SSLsignaturesattheLHC.Fromthevariouscutsexperimetalistsapplyontheiranalysis,wecangettheefcienciesofnewphysicswithrespecttotherelevantparameters.Thisprocedurewilldependontheeventtopologywhichresemblesoffeynmandiagram.CMSanalyisisfor2SSL. 51

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ThemuonandelectroncandidatesmusthavepT>5and10GeVrespectively,andjj<2.4.Taucandidatesareexcluded.Eventswith2ormorejetswithpT>40GeVareselected.Thepreliminarycutsforobjectacceptanceweredenedasfollows, jj=e=j<2.4, pT>5GeV, pTe>10GeV, pTj>40GeV.Thechoiceoftheselectioncutsneedthefollowingconsiderations. Theefciencyofeventselectiondependsonthecutsemployed. Theamountofbackgroundsupressionalsodependsonthecuts. Asetofcutsmightbebetterinoneregionintheparameterspaceandanothersetofcutsmightbebetterelsewhere.Thus,it'simportanttodoananalysiswithavarietyofcutsandpickdifferentcutsforexploringdifferentregionsintheparameterspace.Sevenselectioncriteria(calledthesevensearchregions)areemployedbytheCMScollaboration.Sincewewerefollowingtheiranalysis(withtheirnumbersforbackgroundestimation)wehadtosticktotheirsearchregions.Thetwobaselineselectionsusedare: inclusivedileptons:eventswithapairofsamesigndileptoncandidateandHT>200GeV; high)]TJ /F3 11.955 Tf 12.36 0 Td[(pTdileptons:eventswithapairofsamesigndileptoncandidatewithboththeleptonshavingpT>10GeV,andatleastoneleptonhavingpT>20GeV.FurtherdivisionintosevensearchregionsandthecorrespondingcutsarelistedinTable 4-1 4.5.1Model-IndependentProcedure:FastSimulationWeusedPythia[ 7 ]foreventgenerationandPGS[ 19 ]fordetectorsimulation.Since,PGSsimulatesonlyaportionofthedetectorbehavioritisnotafullsimulation.Atthesametime,asitneglectssomeoftheotherrealdetectoreffects(likethemagneticelds) 52

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Table4-1.Thesevensearchregions. BaselineRegionOurnotationmin.HTmin.EmissT95%CLULHTEmissT(GeV)(GeV)(events) highhighI/400/1204001203.7InclusivehighlowI/400/50400508.9mediumhighI/200/1202001207.3highhighH/400/1204001203.0highpThighlowH/400/50400507.5mediumhighH/200/1202001205.2lowlowH/80/120801206.0 thatdemandalotofcomputingpower,itismuchfasterthanafulldetectorsimulation.ForthisreasonitiscalledFastSim.ThevariablemodelparametersofinterestwereM1,M2andM3.For5separatevaluesofM1,namely,10,100,200,300and400GeV,weevenlysampledM2andM3inincrementsof10GeVwithM1>M2>M3.Foreachsuchpointwegeneratedandsimulated10000events.Aswewereonlyinterestedindecaysfromgluinos,onlysub-processes243and244(gluinoproduction)wereturnedon.Thexedmodelparameterswere.=700GeV,tan=10,msleptons,msquarks=700GeVandtrilinearcouplingparametersRMSS(15)=800,RMSS(16)=800andRMSS(17)=0.IMSS(3)wassetto1,tomakegluinopolemasstobesameasM3.ISR,FSR,multipleinteractions,andfragmentationanddecaywereturnedon(defaultconguration).Wewereinterestedinthedecayfromgluinotochargino.So,onlydecaysofgluinosintopositivelychargedcharginoswereallowed.EachofthecharginowasthenallowedtodecayeitherintoaWbosonandtheneutralino-LSP(2body)oralepton(e,),aneutrinoandtheneutralino-LSP(3-body).IfitdecaystogiveaW-boson,weallowtheWbosontodecayintoalepton(e,)andaneutrino.Thenthebranchingratiofor2gluinostodecayinto2same-signdileptons(e,), BR(~g~g!ss2l)=2"BR(~g!~+1+2j) BR(~+1!~01+e+=++e=)+BR(~+1!~01+W+)BR(W+!e+=++e=)!#2. (4) 53

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Intheexpressionabove,theprefactor2comesfromthefactthatweneedtoaccountforthenegativelychargedcharginosthatwehadneglected.Thesquaringisdonetotakeintoaccountthedecaysfromboththegluinosproducedatpartonlevel. Figure4-3.PGS:EfcienciesforthesevensearchregionsforM1=10GeV. InFigures( 4-3 4-4 4-5 4-6 B-1 B-2 B-3 B-4 ),thesevenplotscorrespondtothesevensearchregions.Thex-axisrepresentsthecharginomass,they-axisrepresentsthegluinomassandthez-axis(colorcoded)representstheefciencyofaneventpassingthecutsforagivenxandyvalue.Ahighervalueforefciencyisdesirable.Ineachoftheplots,wecancategorizetheparameterspaceintoregionsintwoways,namely,Lowvshighgluinomass(M3)foraxedcharginomass(M2):Asgluinomassincreases,thesplittingbetweenthegluinoandcharginomassincreases.Since,inourprocessofinterest,thegluinodecaysintoacharginoandaquark.IfthemasssplittingisbigthenthequarksaremorelikelytohaveahigherpT,andthuspassthecutsonjetpTandHT.ThiswecanseeintheplotsbylookingataxedvalueofM2onthex-axis 54

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Figure4-4.PGS:EfcienciesforthesevensearchregionsforM1=100GeV. Figure4-5.PGS:EfcienciesforthesevensearchregionsforM1=200GeV. andlookingattheefciencyaswegoverticallyalongthey-axis.Wecanclearlyseeinalltheplotsthatefciencygoesupasitshould. 55

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Figure4-6.PGS:EfcienciesforthesevensearchregionsforM1=300GeV. Lowvshighcharginomass(M2)foraxedgluinomass(M3):ThecharginomassisfreetovarywithintherangeofM1,M3.a)WhenM2isclosertoM1:AsM2increasesthemasssplittingbetweenthecharginoandneutralinoincreases.ThustheleptonsproducedasthedecayproductofthecharginohavehigherpTandaremorelikelytopasstheleptonpTcuts.b)WhenM2isclosertoM3:AsM2increasesfurther,themasssplittingbetweenthecharginoandgluinodecreases.ThusthejetsproducedasthedecayproductofthegluinohavelowerpTandarelesslikelytopassthejetpTandHTcuts.Intheplots,wecaneasilyseethisbehaviour,aswegoalongahorizontallineforaxedvalueofM3.Theefciencyseemstoincreaseintitiallyasthemasssplittingbetweenthecharginoandtheneutralinoincreasesandthenafterreachingamaximumvalueitstartstogodownasthemasssplittingbetweenthecharginoandthegluinostartstodecrease. 56

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It'salsoimportanttonoticethedifferencesamongthesevenplotsineachgure(i.e.foragivenvalueofM1)a)lowpTleptonsvshighpTleptons:withothercutsthesamewenoticethattheefciencyforLcasesarehigher(noticeablemoreforlowervaluesofM2)astheyshouldbebecauseofmoreleptonscanpassthecuts.b)lowHTcutvshighHTcut:againtheefciencyforlowercutsincreases,andnoticeableepeciallyforhigerM3andhighermasssplittingbetweenM2andM3.c)low6ETvshigh6ETcut:efciencyincreasewithlowercuts. Figure4-7.PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=10GeV. InFigures( 4-7 4-8 4-9 4-10 B-5 B-6 B-7 B-8 ),thesevenplotscorrespondtothesevensearchregions.Weusethefollowingequationtocomputethelimitonthecross-section. Limit=Nbkg L 57

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Figure4-8.PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=100GeV. Where,Nbkgistheestimatednumberofbackgroundevents(fromtheCMSanalysisnotes)forthegivensearchregion,Listheintegratedluminosityandistheefciencyofeventselection.ThislimitatagivenregionintheparameterspaceisthencomparedtotheBR(productioncross-sectionoftwosamesigndileptonthroughthemodelunderinvestigation)forthesamevaluesoftheparameters.Ifthelimitislower,thenthatregionoftheparameterspacegetsexcludedforthemodel.Thus,alowervalueonthelimitisdesirable.Intheplots,thex-axisrepresentsthecharginomass,they-axisrepresentsthegluinomassandthez-axis(colorcoded)representsthelimitontheforagivenxandyvalue.Ineachoftheplots,wecancategorizetheparameterspaceintoregionsintwoways,inthesamewayastheefciencyplots,since,theseplotsmoreorlessplotthereciprocalsoftheefciency.But,sincethenumeratorhasachangingvaluefortheestimatednumberofbackgroundevents,anefciencyplotwhichisnotparticularlyverygood(lowervalues)canhaveagoodlimitplotandviceversa,e.g.L/400/120hasavisiblyworseefciencyplotthanL/400/50 4-3 ,butintheirlimitplots 4-7 L/400/120fares 58

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muchbetter,sinceitisveryefcientinreducingthenumberofuncerlyingbackgroundevents(3.7comparedto8.9for95%CLatL=1fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1). Figure4-9.PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=200GeV. InFigure( 4-11 ),thefourplotscorrespondtofourchoicesforthevalueoftheparameterM1.Thex-axisrepresentsthecharginomass,they-axisrepresentsthegluinomassandthez-axis(colorcoded)representstheproductioncross-sectionoftwosamesignleptonscomputedbymultiplyingtheproductioncross-sectionofgluinos~g~gandsquareofthebranchingratioofagluinodecayingintoabinoandaleptonviathechargino.Ineachoftheplots,wecancategorizetheparameterspaceintoregionsintwoways,namely,Lowvshighgluinomass(M3)foraxedcharginomass(M2):Asgluinomassincreases,thesplittingbetweenthegluinoandcharginomassincreasesandhencethebranchingratioofthegluinodecayingintoacharginoandaquarkincreases.Ontheotherhandwithaincreaseinthegluinomassitsproductioncross-sectionattheLHCdecereases.Since,theproductioncross-sectiondecreasesfasterthantheincrease 59

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Figure4-10.PGS:Limitsonthex-sectionsforthesevensearchregionsforM1=300GeV. Figure4-11.PGS:SS2Lproductionx-sectionforM1=10,100,200and300GeV. inthebranchingratiolowervaluesofM3giveabettersamesigndileptonproductioncross-section. 60

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Lowvshighcharginomass(M2)foraxedgluinomass(M3):ThecharginomassisfreetovarywithintherangeofM1,M3.Since,M2isalwaysgoingtobebiggerthanM1withnootherparticlehavingintermediatemass,thebranchingfractionofthecharginodecayingintoaleptonandabinodoesn'tdependonthemassofthechargino.Also,sinceM3isxedtheproductioncross-sectionfortheproductionofthetwogluinosdoesn'tchangewithM2.But,asM2getsclosertoM3thebanchingratioofthegluinodecayintocharginogoesdownandhencetheoverallsamesigndileptonproductiongoesdown. 4.5.2Model-IndependentProcedure:EmulationTheCMScollaborationcameupwithanemulationprescriptionoptimizedfortheLM0studypoint[ 20 ].Theideaistorunaneventgenerator(e.g.Pythia)andgetthepartonlevelinformationaswellasthenumberoftrackscreatedbythestableparticles.BelowisadescriptionofthepresciptionusingPythiaastheeventgenerator.a)Setuptheeventgeneratorforthespecicphysicsmodelofinterestandgenerateevents.b)Storethe3-momentumandenergyinformationofthepartonlevel(isthep=3)leptons,quarks(u,d,s,candb)and/orgluonsthatpassthecorrespondingandPTcuts.c)Storethe3-momentumandenergyinformationofallthenon-ineractingparticles(neutralinosandneutrinos).d)Countthenumberoftracksofthenaloutgoingparticles(i.e.numberofcoloredorelectricallychargedparticlesinisthep=1).e)Additionaly,accepttheleptonsfromstep2abovewithanefciency, 61

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lepton(PT)=max,lepton+Aleptonerf(PT)]TJ /F3 11.955 Tf 11.95 0 Td[(PTcut Blepton)]TJ /F6 11.955 Tf 11.95 0 Td[(1)+0.125)]TJ /F3 11.955 Tf 11.95 0 Td[(Ntrack 15,where,max=0.75,maxe=0.66,A=0.32,Ae=0.4,B,e=18,PT,=5GeVandPTe=10GeV.f)CalculateHTasascalarsumofPTofthe4partonlevelquarksfrom~gdecays.g)Calculate6ETasthemagnitudeofthevectorsumofthePT0softhenon-ineractingparticles.h)TheratioofthemeandetectorresponsesforHTand6ETabovetotheirtruevaluesare0.940.05and0.950.05.So,HTand6ETcomputedabovearesmearedwithaGaussianwiththesemeanvaluesandwidths.i)Additionaly,the6ETresolutiondependsonthehadronicactivityintheevent.Itrangesfrom7to25GeVforeventswithHTintherangeof60to350GeV.So,the6ETissmearedagainwithaGaussianofmean1andaesulutionbasedonHT.j)TheHTresolutiondecreasesfromabout26%at200GeVto19%at300GeVandto18%for350GeV.So,HTissmearedagainwithaGaussianofmean1andresolutionthuscalculatedfromtheHT.k)Finally,wehavetheobjects,jets(partonlevelquarks),leptons(.e),HTand6ETwhichareanalogoustosuchobjectsreconstructedusingadetectorsimulationlikePGS. 62

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Finally,usingemulation,wereproducedthepreviousplotsgeneratedusingPGS.TheycanbefoundinAppendix B .TheexplanationfortheplotsaresameasthosegeneratedusingPGS. Figure4-12.PGS:95%CLlimitonSS2Lproductionx-sectionforM1=10,100and200GeVwithaluminosityof1,10and30fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1. 4.6ResultsInFigure 4-12 ( 4-13 ),wehaveplottedthecontouroftheratiobetweenthesignalcross-sectionandthelimitfromPGS(emulation).Thecontourslinesaredrawnwheretheratioisone.Thismeanstheregionresidingwithinthecontourlineshavearatio 63

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greaterthanone,i.e.sucharegionisexcludedbymeansoftheselectioncutsspeciesbythecolorofthecontourline. Figure4-13.Emulation:95%CLlimitonSS2Lproductionx-sectionforM1=10,100and200GeVwithaluminosityof1,10and30fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1. Lowvshighbinomass(M1):Theseplotsalsohighlightthetheeffectofdifferentbinomasses.As,weincreasethemassofthebinofrom10GeVto100GeV,wecanseeadecreaseinthenumberofcontourlines.Byincreasingthebinomasswedecreasetherangeofmassesthatthecharginocanhave.Forlowervaluesforthecharginomass,itdecaysintotheLSPwithsofterleptonsthatareunabletopassthe 64

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selectioncuts.Atthesametimeforhighervalues(valuesclosertothegluinomass)thegluinodecaysintothecharginowithsofterjetsunabletopasstheselectioncuts.Wealsonotethatasweincreasetheluminosityforanyxedbinomass,wesethesizeoftheexclusionzoneincrease.Thisissimplybecauseoftheincreasedamountofdataopenforanalysis.InFigure( 4-14 ),wecanseethecomparisonbetweenthe Figure4-14.PGSversustheEmulationprescription:EfciencyofacceptanceofandeasafunctionoftheirrespectivePT). transversemomentumofthemuonsandtheelectronsselectedbytheeventselectioncutsemployingPGSaswellastheCMSemulationprescription.ThiscomparisionisdonefortheLM0pointsincetheemulationisbasedonthatanalysis.InFigure( 4-15 ), Figure4-15.PGSversustheEmulationprescription:Numberofacceptedjetsandleptons(ande)). 65

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wecomparethenumberofleptonsandjetsacceptedbyemulationandthenumberofleptonandjetobjectsreconstructedbyPGS.Wecanseethatthenumberofjetsforemulationdon'texceedingfour.Thisisalreadyexpectedsinceeachgluinodecaysintotwoquarksandacharginoandaccordingtotheemulationprescriptionweselectthosefourpartonlevelquarks.OntheotherhandthenumberofjetsfromPGSismuchmore(upto12innumber).Thisisaresultoftheinitialstateradiation(ISR)andnalstateradiation(FSR)intheevent.InFigure( 4-16 ),wecomparetheHTand6ETforeach Figure4-16.ComparisionofHTand6ETfromPGSandtheEmulationprescription. eventasreconstructedbyPGSandemulation.AswecanseeforagiveneventtheHTreconstructedbyPGSislikelytobemuchhighercomparedtothevaluefromemulation.ThiscanbeexplainedusingFigure( 4-15 ).Wecanseethatthenumberofjetsishigher(fromISRandFSR)forPGS.Hence,HT,whichisthescalarsumoftheirpT'sisalsohigher. 66

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CHAPTER5HOWTOLOOKFORSUPERSYMMETRYUNDERTHELAMPPOSTATTHELHC 5.1IntroductionandMotivationLargeHadronCollider(LHC)atCERNstartedit'slongwaitedhistoricjourneywithmuchexpectationofndingsomeformsignatureforphysicsbeyondthestandardelectroweaktheory.Nodoubt,observationofanyhallmarkfrombeyondstandardmodelrequiresmuchoftheeffortdirectingthepreciouspredictionofSMbackgroundandreducingthem.EspeciallyinahadroncolliderlikeLHCthejettysignalscanbeeasilyburiedintooverwhelmingSM-QCDbackground.Inacomplexsignatureproleofhadroncollider,leptons(electronandmuon)effectivelyholdthekeytountanglethisproblem.Generalperceptiontellsusthatthemultileptonsignaturesareclean,however,rarewithincreasingnumberofleptonsandonlyleptonicsignalscangiveacleanguidelinetocatchasignaloverthebackgrounds.Naturally,thequestiononecanaskiswhatcanbetheeasilydiscoverableinterestingsignaturesbearingthetracesofnewphysics.Amongthenumerousmodelsofnewphysics,lowenergysupersymmetry(SUSY)constitutesavariedwellmotivatedclassofmodelsandexpectationishightonditattheLHCexperimentatoneformorother.Byitself,SUSYisverypredictive,asitxesthespinsandcouplingsofthenewparticles(thesuperpartners).Unfortunately,thisisnotsufcienttopindownitsprecisecolliderdiscoverysignatures.SUSYmassspectrum,whichisinturndeterminedbythemechanismofsupersymmetrybreakingdeterminesthewaysupersymmetrywouldmanifestitselfintheexperiment.However,ourignoranceaboutthedynamicalmechanismhowSUSYisbroken,easilyleavesusguessingthepossiblemassspectrumwithasignicantlylargenumberofparameters.Inthischapterweadoptaconservative,agnosticapproach,wherethemassesofallsuperpartnersaretreatedasfreeinputsattheweakscale.Weshallthenconsiderallpossiblehierarchicalpatternsamongthem,andidentifythesetofdominant(inthesensedenedbelow)collidersignaturesineachcase.Weprovidethecompletelist 67

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Table5-1.ThesetofSUSYparticlesconsideredinthisanalysis,shorthandnotationforeachmultiplet,andthecorrespondingsoftSUSYbreakingmassparameter.Thethirdrowisthenumericalcodeassignedforeachparticlewhichweuseintheappendix.NoteourunconventionalnotationfortheHiggsinoparameterMHandthethreegauginomassesM1MB,M2MWandM3MG. ~uL,~dL~uR~dR~eL,~L~eR~h,~h0u,~h0d~b0~w,~w0~gQUDLEHBWG 23458761MQMUMDMLMEMHM1MBM2MWM3MG ofminimallysuppressedsignaturesforeachofthehierarchicalpatterninAppendix C-1 summaryofwhichwereproducedin[ 21 ].InAppendix D-1 wealsoreorganiseforcategoricaldisplayofallhierarchycorrespondtoeachclassofsignatures.Ourexhaustivesearchuncoveredseveraldramaticyetunexploredmultileptonsignatures.OneofourmainresultsherewillbetheidenticationofanumberofnewSUSYmasspatternswhosedominantsignatureshaveuptoeightleptonsinthenalstate.InSec. 5.3 weclassifyallsuchmasspatternwithexclusiveanalysingthreescenarios.Westartwithwidelyestablishedmodelofminimalsupersymmetricstandardmodel(MSSM).InTable 5-1 wesummarizethesupersymmetricparticles,softmassesandournotationforthem.Forsimplicity,weconsiderjusttwodegeneratelightgenerationsofsfermions.Thirdgenerationeffectscanbetriviallyincorporatedinthediscussion[ 22 ],andonlycomplicatethebookkeeping.ThemasssplittingswithintheQ,L,HandWmultipletsariseaselectroweaksymmetrybreakingeffectsandcanbesafelyignoredforourpurposes.Giventhe9inputmassparametersinTable 5-1 ,ingeneralthereare9!=362,880possibleorderingsamongthem,eachleadingtoadistinctpattern(hierarchy)ofsparticlemasses.WeshallusetheshorthandnotationfromTable 5-1 tolabeleachhierarchy:forexample,GQUDHLWEBisamodelwithMG>MQ>MU>MD>MH>ML>MW>ME>MB.Ourrstgoalistoidentifythemaincollidersignaturesforeachhierarchy.Indeterminingso,natureoftheLightestSupersymmetricParticle(LSP)carryoneofthe 68

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Table5-2.Constructionofasamplehierarchyaccordingtosuperparticlemassesandrepresentationofthehierarchyfollowingournotation.WeusedbothofthesealphabeticaswellasnumericnotationsafterwardstoclassifyallpossiblehierarchyandsignaturesatAppendix. Samplehierarchymodel:ConsequenceGQU| {z }xxxD|{z}CHLWE| {z }yyyyB|{z}LMG>MQ>MU>MD>MH>ML>MW>ME>MB OurnotationfromTab. 5-1 :ConsequencexxxDHLWEB000384657MD>MH>ML>MW>ME>MB mostcrucialrole.WeassumethatR-parityisconserved(orveryweaklybroken),sothatLSP,whichwegenericallydenotebyLisstableonthescaleofthedetector.Then,theoriginal9!modelhierarchiescanbeclassiedintothefollowingthreecategories:I.CHAMPs.Inthe8!=40,320caseswithL=E,theLSPisanelectricallycharged,color-neutralparticle(theright-handedslepton~eR).Thecorrespondinggenericcollidersignatureisalong-livedchargedmassiveparticle(CHAMP)[ 23 ],regardlessoftheparticularorderingoftheheaviersparticles.II.R-hadrons.In48!=161,280oftheremaininghierarchiesL2fQ,U,D,Gg,theLSPiscolored,andthegenericsearchesforstableR-hadronsapply[ 24 ].Againtheorderingoftheheavierparticlesisunimportant.III.Missingtransverseenergy.Intheremaining48!=161,280casesL2fL,B,W,HgandtheLSPisaweakly-interacting,electricallyneutralparticle.Itsproductionwillleadtomissingtransverseenergy(6ET)inthedetector.Now,however,thesignaturescruciallydependontheorderingoftheheavierparticles,sinceitisnotfeasibletolookfor6ETinclusively.Bothofthecurrentlyoperatinghighenergycolliders(theTevatronatFermilabandtheLHCatCERN)arehadronmachines,atwhichthetotalproductionisexpectedtobedominatedbythestrongproductionofcoloredsuperpartners.Natureofthelightestcoloredsuperpartner(LCP)playsanotherimportantroleindeterminingthedecaychainwithsignicantproduction.Here,wemakeanassumptionthatheaviercolorproductions 69

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areprobablysmallortheysubsequentlydecaytoproduceLCP.EffectivelyanyhierarchychainalwaysstartswithLCP,denotedbyC.Thevalidityofthisassumptioncanbeseen Figure5-1.Productionrate(inpercentage)of(a)noLCP,(b)onlyoneLCPand(c)bothLCPinaMQ-MUplainforachoiceofbinomassMB=100GeV,winomassMW=200GeVandgluinomassMG=400GeV,. inFigure 5-1 .Inthegurewehaveplotted(colorcodedonthez-axis)theproductionrate(inpercent)ofa)noLCP,b)oneLCPandc)2LCP's.Inallthethreeplotswexedthemassofgluinoat400GeV,binoat100GeVandwinoat200GeVwhilevaryingthemassesoftheleft-handedandright-handedsquarksonthexandyaxisrespectively.ThewhitelinesinthegurescorrespondtodifferentmassdegeneraciesseparatingregionswhereonlyoneofthecoloredSUSYparticlesistheLCP.Alongthewhitediagonalline,themassesoftheright-handedandleft-handedsquarksaredegenerate.Alongtheverticallinethemassesoftheleft-handedsquarkandthegluinoaredegenerateandalongthehirizontallinethemassesoftheright-handedsquarkandgluinoaredegenerate.InthetoprightregiongluinoistheLCP,inthebottomrightregion(belowthewhitediagonal)theright-handedsquarkistheLCPandinthetopleftcornertheleft-handedsquarkistheLCP.Fromsub-gure`c',itcanclearlybeinferredthataswemoveawayfromthewhitelinestheproductionrateoftheLCP'sincreaseallthewayuptoa100%.OncebothLSPLandLCPCareidentied,remainingentriesofthehierarchychainaredividedintotwogroupsaccordingtotheirroleinconstructingsignatures.Heavier 70

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entries(thanC)carrylittlesignicanceindeterminingthephenomenologicalsignaturebasedonthatmodel.Whereaslighterentrieshavesignicantroleinit.Ourgoalhereistofullyclassifyall161,280missingtransverseenergymodelsaccordingtotheircolliderphenomenology.Unlikepreviousgeneralapproaches[ 25 ],whichemployedscansofthemulti-dimensionalSUSYparameterspace,herewewouldliketoavoidscanning,keepingthediscussionsimpleandqualitative.Here,wesummarisethesetup:i.AnydecaychainstartswithlightestColorSupersymmetricParticle(C2fG,Q,U,Dg)whichdeterminethesignicantproductionmechanisminsomemodel.ii.DecaychainnallyproduceschargeandcolorsingletLSP(L2fL,H,B,Wg).So,eachofthehierarchycanberepresentedbyaparticularordering x...xCy...yL.(5)Thedominantcollidersignatureforeachmodelhierarchy( 5 )willbedeterminedbytheinclusivepairproductionofCanditsdominantsubsequentdecaysthroughy2fL,B,W,H,Eg.Here'x'sareanythingheavierthanCandmostlyinconsequentialentriesforourdecaychain.iii.Weignoretheroleof3rdgenerationsleptonorsquarksinthispresentanalysis.Wearealsorestrictingourselfwiththewellmotivatedweakly-interacting,electricallyneutralLSP. 5.2TravellingSalesmanOurkeyideahereisthatonceagivenhierarchy( 5 )isassumed,thedominantdecaymodesofCareuniquelydetermined,sincesupersymmetrypredictsallsuperpartnercouplings.Inouranalysis,weshallassumethattherearenoaccidentalphasespacesuppressionsduetoanytwomassparametersfromTable 5-1 beingveryclose.Thisassumptionalsoguaranteesthatthecharginoandneutralinomixinganglesaresmallandthemasseigenstatesareroughlyalignedwiththeinteractioneigenstates.Onecan 71

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Nosuppression Mildsuppression Strongsuppression Summary Figure5-2.GraphicalrepresentationoftheallowedtransitionsbetweentheSUSYstatesfromTable 5-1 .Differentmodesoftransitionsseparatedaccordingtotheirrelativesuppressionindicatedbythelinetype.(a)solidlinetyperepresentunsuppressedonewith2-bodydecayanddonotsufferfromany(charginoorneutralino)mixinganglesuppression(MAS).Dashedlineusedeitherfortransitions(b)MASaffected2-bodydecayor(c)3-bodybutunsuppressedbymixingangle.Similarlydottedlinefor(d)MASaffected3-bodydecayor(e)MASunaffected4-bodydecay.Figure(f)showsthecompletepictureincludingallsuchtransitions.One(two,three)parallellinesrepresenttwo-(three-,four-)bodydecays.TheidentityoftheresultingSMdecayproductsisdenotedbythelinecolor:redforajetj,blueforaleptonlandgreenforamassivebosonvW,Z,h(whichmaybeeitheron-shelloroff-shell).LogicchartdepictingthekindofdecaytransitionsbetweendifferentSUSYparticles. thenusethesimplechartinFigure 5-2 toidentifythedominant(i.e.leastsuppressed)decaymodesofCandintermediatemodesofy,whichwelabelbythenumberofleptonsn`(bluelines),numberofjetsnj(redlines)andnumberofmassivebosonsnv(greenlines)encounteredalongtheway.Figure 5-2 categorizedallsignicanttransitions 72

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intothreeparts.SolidlinesintheFigure 5-2 (a)correspondto2-bodydecayswhichdonotsufferfromany(charginoorneutralino)mixinganglesuppression(MAS).DashedlinesinFigure 5-2 (b,c)indicateeither2-bodydecayswithMASor3-bodydecayswithnoMASrespectively.Andnally,dottedlinesinFigure 5-2 (d,e)standforeither3-bodydecayswithMASor4-bodydecayswithnoMAS.Wesummarisedallthesetransitionsasacombinedchartattheend.Tounderstandthesuppressionlevelofeachtransitionalongthehierarchy( 5 )betweenanypairofsupersymmetricstatesC!y,y1!y2andy!L,wediscussinfollowingcategory.(i)Gaugino-Scalartransition:Onepartofthiskindoftransitionconsistofinteractionbetweencolorstatesofgluino(G)andsquarks(U/D)whichisdeterminedpurelybygaugeinteractionofstrongcoupling.ProducescorrespondingquarkrepresentedbysingleredlineinFigure 5-2 (a).ItshouldbenotedthatthistransitiondoesnotoccurwithintheeffectivedecaychainstartingatLCP(C)production;stillitisquiteimportanteffectivelyincreasingLCPproductionattherstplace.Similarly,electro-weaktransitionbetweencoloredsquarks(Q,U/D)orsleptons(L,E)andgauginos(W,B)arealsospeciedbyfullstrengthgaugecouplingsasidentiedbythesolidlinesinFigure 5-2 (a).(ii)Higgsino-Scalartransition:Sincehiggsino(H)coupleswithscalars(Q,U/D,L,E)inYukawaterm,inthemasslesslimitofcorrespondingfermions,Hcompletelydecouplesfromscalarsector.Sotheirtransitionwithscalarsaresuppressedthroughsmallfermionmassormixingatthechargino/neutralinosector.DashedlinesofthesekindoftransitionsareshowninFigure 5-2 (b).NotethatYukawainteractioncouplesleft(right)fermionstotheright(left)componentofsfermionsunlikethepreviouslydiscussedgaugeinteraction.(iii)Gaugino-Higgsinotransition:Couplingsforgauginos(W,B)withHiggs-Higgsino(H)pairdeterminedfullybygaugeinteractionindicatedinsolidlineinFigure 5-2 (a). 73

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(iv)Gaugino-Gauginotransition:ThereisnodirectW)]TJ /F3 11.955 Tf 12.52 0 Td[(Btransition.Sothistransitionissuppressedthroughthemixinginchargino/neutralinosector.Gaugeinteractionthroughgaugino-higgsino-higgsandSU(2)Lgaugino-gaugino-vectorbosonplayaroleincoupling.DashedlinesofthesekindoftransitionsareshowninFigure 5-2 (b).Kinematicinaccessibilityoftwobodydecaymaydrivetowards3-bodydecaythroughoff-shellsfermion.Thoughhavingfullgaugestrengththistransitionsaresuppressedfrom3-bodyphase-spaceandindicatedbydashedlineinFigure 5-2 (c).WediscussedbothofthesetransitionsinAppendix E indetails.Wefurnishedapproximateanalyticexpressionsofdecaywidthbothfor2and3-bodydecayandpresenteddependenceinplots.(v)Scalar-Scalartransition:Gaugecouplingstrengthbutsuppressedfrom3-bodydecaythroughgauginospresentedinFigure 5-2 (c).Wethencountthenumberofmasshierarchies( 5 )whichexhibitadominantdecaychannelforCwithagivensetof(n`,nv,nj),andclassiedallresultsinAppendix C-1 andAppendix D-1 .Howeverbeforediscussingthatitisworthwhiletogothroughfewexamplestoelaboratetheprocedureandestablishthereasoning.Inthevariousotherexampleslistedbelow,weusethefollowingconvention:jstandsforajet,`foralepton(electronormuon)andvforavectorboson.a)xxxxxxxGjj B:InthisshortesthierarchychaingluinoNLSP[ 26 ]islightestcolorparticleandreadilyproduceLSPwithtwojet.Signicantchannelisgluinopairproduction.Allothergauginos,sleptonandsquarks(denotedbyx)areheavyandtheydon'thaveanymajorrole.Sodominantsignaturewecanexpectisdi-jetwithmissingtransverseenergy.b)xxxxGjj'' EHLB:ThisonehavingsameCandLaswehaveseeninourlastexampleaddedwiththreeintermediatestates.Thismodelisinterestingsinceaftersignicantproductionofgluinopair,eachgluinoreadilyproducesbinoLSPinthesame 74

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wayasbefore.Sowearehavingexactlysamedominantsignaturedi-jetwithmissingtransverseenergy.c)xxxxxQj j88 W` L` B:ToillustratetheprocedureofselectingmostinterestingsignatureinthispresentexamplewedemonstratetwowaysQcandecay.Clearly,singlejetwithoppositesigndi-leptonfromcascadedecayisbettersignalcomparedtotheotheronedecayingdirectlytoLSP.d)xxxxQj j:: W`"" B`DD L` `DD H:Toillustratethesituationwheninterestingmultileptonsignaturesnotonlycomingfromoneuniquepathalsofromtwoindependentcombinations.Noticethatonecanget(n`,nv,nj)=(0,1,1)intwoways:Q!W!HorQ!B!H.Itisalsopossibletohave(n`,nv,nj)=(2,0,1)intwodifferentways:Q!W!L!HorQ!B!L!H.Therefore,xxxxQWBLHcontributestoeachofthe(n`,nv,nj)=(0,1,1)and(n`,nv,nj)=(2,0,1)inAppendix C-1 and D-1 .e)xxxxQj jDD B` `DD E` ``<< H` L:Toachievesignicantcontributioninmultileptonchanneloneneedtotakecareoftwokindofsuppressionineachdecaychannel.Inthisexampletwostepdecaychannelissuppressedbythemixingwhereasdirect3-bodydecayofEissuppressedbythephasespace(dottedlinetorepresentsuppression).Exactquantitativevaluesofthesesuppressionscanbecalculatedonceweknowallmassesandmixing.Howevertokeepouranalysisindependentofexactmasses,weconsiderthesetwotypeofsuppressioninsamefooting.Inaboveanalysiswetakeintoaccountvectorbosonsassignals,becausetheycontributeleptonicsignalstherebyeventuallyincreasingtheleptoniccount.Butwhenwehavetochoosebetweenavectorbosonoraleptonwegofortheleptonforthathierarchy.Becauserstandforemost,indecaysproducingvectorbosons,sometimesHiggsbosonisalsoproducedreducingthebranchingratiosforvectorbosonproduction.Also,ifthevectorbosonisa(i)Wboson,it'sbranchingratiotoamuonorelectronis 75

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20%whichreducestheoverallleptonicproductionbranchingratiobyafactorof52.Ontheotherhandifit'sa(ii)Zboson,granteditwillgive2leptonsinsteadofone,thebranchingratioofit'sdecayinginto2electronsor2muonsis6%whichreducestheoverallleptonicproductionbranchingratiobyafactorof172.Forexample,inthecaseofthehierarchy,xxxGLBHEW,wehave2choices(i)G-B-E-Wgiving2jetsand2leptonsand(ii)G-B-H-Wgiving2vectorbosons.Inthiscasewechoose(i)forthereasonexplainedabove.NowweareinapositiontodescribetheresultpresentedinAppendix.Thecompletelistofall161,280hierarchiesanalyzedinthispaperandtheirdominantsignaturesarepresentedAppendix C-1 .Column1listsnumericalcodesforeachhierarchygiveninalphabeticnotationincolumn2.Column3showsthenumberofminimallysuppressedsignaturespossibleandcolumn4listscorrespondingdecaychainswithequalsuppression(withoutanydegeneracyofdecaychainsi.e.foragivenhierarchyiftwodecaychainshavethesamesignaturethenwewriteonlyoneofthemincolumn4).Column5showsthecorrespondingsignatureinasetof(n`,nv,nj)andnally,column6showsthemultiplicity.IfmultiplicityisNxM!,itmeansthereareNLCPsweconsidereddegenerateandMnumberofx's(numberofsoftmassesheavierthantheLCP).MoreinstructivedescriptioncanbeavailableoncewereformulatethelistinAppendix C-1 toassignthepossiblehierarchiescorrespondtoanyparticularsignatureinasetof(n`,nv,nj).Foreachsignatureprolealphabetic(numerical)hierarchyisshownincolumn3(2)withmultiplicity.Otherpossibledominantsignaturesarealsolistedforsuchhierarchy. 5.38-leptonTopologiesInthissectionwestudyindetailseveralexamples(includingoneindicatedin[ 21 ])ofamaximallyleptonicdecaychainwith4leptons,wherethecorrespondingcollidersignalis8isolatedleptonsplusjetsplusmissingenergy. 76

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5.3.1Case:AForconcreteness,rstweconsiderthehierarchyxxGQWLBEHatseveraldifferentstudypoints,denedinTable 5-3 andchosenunderthelamppost,i.e.tomaximizethe8-leptonsignalrateforagivenvalueofMQ.Forthishierarchy,8leptoneventsarisefromtheinclusivepairproductionofleft-handedsquarks~uL,~dL,followedby ~uL,~dLj!~w0`!~`L`!~b0`!~`R`!~h0u,~h0d.(5) Figure5-3.MassspectrumforthehierarchyxxGQWLBEH. Ignoringphasespacesuppressionfactors,thirdgenerationeffectsandthemassesoftheW,Zandh,itiseasytoestimatetheindividualbranchingratiosforthischainasfollows[ 22 ]:BR(~g!~qL+j)=1;BR(~qL!~w0+j)'1 3;BR(~w0!~`L+`)'1 3;BR(~`L!~b0+`)=1;BR(~b0!~`R+`)'4 5;BR(~`R!~h0u=d+`)=1. 77

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Multiplyingtheseresultsandsquaring,weexpectthecumulativebranchingfractionfor8leptoneventstobearound0.7%.ThisisconrmedbythedashedlineinFigure 5-4 ,whichapproachesthisasymptoticvalueforlargemasssplittings(largeMQ).Weseethattomaximizetheoverallbranchingfraction,wemustscaletheSUSYmassspectrumup.Yettomaximizetheproductionrate,wemustscalethespectrumdown.AnoptimumcompromiseisthereforefoundforintermediatevaluesoftheSUSYmasses,asshowninFigure 5-4 Table5-3.InputsoftSUSYmassparameters(inGeV)forthexxGQWLBEHstudypoints. MGMQMWMLMBMEMH 4003002201901301301304503502801901201201205004002801901201201205504503102001201201206005003502101301201207006004202301501301208007004802501601301209008005002501701301201000900510250170130120 ThestudypointsinTable 5-3 werepickedbyvaryingMQ,xingMG=MQ+100GeV,andchoosingtherestofthespectrumfromacoarsescantomaximizethe8leptonrateshownbythesolidlinesinFigure 5-4 .ThesestudypointswerethenprocessedwiththePYTHIA[ 7 ]eventgeneratorandthePGS[ 19 ]detectorsimulatorforthecaseofanLHCat7TeVcenter-of-massenergyandjust1fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1ofdata.Becausetheseeventsareessentiallybackgroundfree,wesimplycountPGS-reconstructedisolatedleptonswithdefaultpTcutsof3GeVformuonsand10GeVforelectrons,anddisplaytheresultinFigure 5-4 .Duetotheimperfectdetectoracceptance,wegotonlyone8leptoneventin1fb)]TJ /F7 7.97 Tf 6.59 0 Td[(1.Nevertheless,forMQ<500GeVtypicallythereareahandfulof7leptoneventsandhundredsof4leptonevents,whicharealreadyveryclean.Sinceonly5cleaneventsaresufcientfordiscovery,theregionMQ<500GeVcanbeprobedwithaslittleas10pb)]TJ /F7 7.97 Tf 6.58 0 Td[(1ofdata. 78

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Figure5-4.Branchingratio,productioncross-sectionandmulti-leptonsignaturesforhierarchyxxGQWLBEH.Figure(a)showsthebranchingratioandcross-sectionforthedifferentmassspectraincaseAcorrespondingtodifferentLCPmasses.Themassspectraareoptimizedtoyieldmaximal8-leptonbranchingratioandareshowninTable 5-3 .Figure(b)showsnumberofmulti-leptonsignaturesforsuchdifferentmassspectrawiththehelpofPGS. 5.3.2Case:BandCInthenexttwoexamplesBandC,weconsiderthehierarchyxxGUBEWLHandxxGUBEHLWrespectively.WerepeatouranalysisofthesetwohierarchiesintheexactsamefashionasinthecaseofhierarchyA.Westudythehierarchiesatseveraldifferentstudypoints,denedinTable 5-4 and 5-5 andchosenunderthelamppost,i.e.tomaximizethe8-leptonsignalrateforagivenvalueofMU. Table5-4.InputsoftSUSYmassparameters(inGeV)forthexxGUBEWLHstudypoints. MGMUMBMEMWMLMH 4003002602401601601604503502802401601601605004003202601601601605504503202601601601606005003802801601601607006005003201601601608007005603401601601609008006203601601601601000900640360160160160 79

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Figure5-5.MassspectrumforthehierarchyxxGUBEWLH. Table5-5.InputsoftSUSYmassparameters(inGeV)forthexxGUBEHLWstudypoints. MGMUMBMEMHMLMW 4003002602401601601604503503002601801601605004003402802001601605504503402802001601606005003803002201601607006004803202201601608007005603402201601609008006203602201601601000900620360220160160 Forbothofthesehierarchies,8leptoneventsarisefromtheinclusivepairproductionofright-handedsquarks~uR,~dR,followedby ~uR,~dRj!~b0`!~`R`!~w0`!~`L`!~h0u,~h0d.(5)forhierarchyBand ~uR,~dRj!~b0`!~`R`!~h0u,~h0d`!~`L`!~w0.(5) 80

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Figure5-6.Branchingratio,productioncross-sectionandmulti-leptonsignaturesforhierarchyxxGUBEWLH.Figure(a)showsthebranchingratioandcross-sectionforthedifferentmassspectraincaseBcorrespondingtodifferentLCPmasses.Themassspectraareoptimizedtoyieldmaximal8-leptonbranchingratioandareshowninTable 5-4 .Figure(b)showsnumberofmulti-leptonsignaturesforsuchdifferentmassspectrawiththehelpofPGS. forhierarchyC.UnlikecaseA,wecannotanalyticallycomputetheasymptoticlimitonthebranchingratio.Thisisbecause,incaseA,allthedecaysinthecascadedecayoftheLCP,aresupressedbyneitherelectroweak-inomixingnormultibodydecay(morethan2decayproducts)andcanbeanalyticallyapproximatedifwepicklargemasssplittingbetweenvariouselectroweak-inos.Ontheotherhand,forBandC,thecascadedecaysinvolvedecayssupressedbyelectroweak-inomixing.Thebranchingratiosofsuchdecaysgetsmallera)asthemasssplittinggetssmaller(becausethephasespaceisreduced)aswellasb)asthemasssplittinggetslarger(becausethemixingisreduced).Hence,thebranchingratioismaximumforsomeintermediatemasssplittingwhichcannotbecomputedanalyticallybytakingsomeasymptoticlimit.Inadditiontoabovehierarchies,weclassiedsomemorecomplexhierarchieswhichcanproducesimilarinterestingmultileptonsignature.Howevertheyinvolve3bodydecaywhichisabsentinPYTHIA.: 81

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Figure5-7.MassspectrumforthehierarchyxxGUBEHLW. Figure5-8.Branchingratio,productioncross-sectionandmulti-leptonsignaturesforhierarchyxxGUBEHLW.Figure(a)showsthebranchingratioandcross-sectionforthedifferentmassspectraincaseAcorrespondingtodifferentLCPmasses.Themassspectraareoptimizedtoyieldmaximal8-leptonbranchingratioandareshowninTable 5-5 .Figure(b)showsnumberofmulti-leptonsignaturesforsuchdifferentmassspectrawiththehelpofPGS. 82

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(a)xxxCBELHW:(4`,0v)(b)xxxCBELWH:(4`,0v)(c)xxxCBWELH:(4`,0v)(d)xxxCBWLEH:(4`,0v)(e)xxxCWBELH:(4`,0v)(f)xxxCWBLEH:(4`,0v)(g)xxxCWHBEL:(3`,2v)-withSneutrinoLSP. 5.4GroupingSignaturesandHierarchiesForagivenhierarchy,asshowninAppendix C-1 ,wecandeterminethevariousmodesofequallysupresseddecaychannelsandthesignaturestheycorrespondto.ThisinthelightoftheLHCInverseProblem[ 28 ]raisesaverythequestion:Canwesaysomethingaboutthehierarchyifweknowsomethingaboutthesignatures?Wetriedtoapproachthisproblembythinkingnotaboutthenatureofindividualsignaturesandhierarchies,butratherbygroups.Since,ahierarchycanhavemorethanonekindofequallysupressedsignatures,wegroupallsuchhierarchieswithsamesetofsuchsignatures.InAppendix D-1 forvarioussignatureswelistallofthecorrespondinghierarchiesandotherpossiblesignaturesandputthemtogetherinagroupasdenedabove.InFigure 5-9 ,wemakea3dimensionalplotthatshowsforagivengroup,a)thenumberofleptonspresentinthemaximallyleptonicsignature(x-axis),b)thetotalnumberofsignaturesinthegroup(z-axis,colorcoded)andc)thenumberofsuchgroups(y-axis).Forexample,theredblockforx=0,hasaheightof5.Thismeans,thereare5groupswhichhave0leptonsintheirmaximallyleptonicsignatureandallofthemhavejustonedominantsignature.Similarly,theaquablockforx=8withaheightof4,meansthereare4suchgroupswith8leptonsintheirmaximallyleptonicsignature.Theaquablockappearsonlyinoneplaceintheplot,whichmeansallgroupswith5typesofdominantsignatureshavemaximallyleptonicsignaturewith8leptons.Suchan 83

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Figure5-9.X,Y,Color==#leptons,#groups,#channels. approachreducesthedegeneracybyagreatdeal.Figure 5-10 ,isbasicallysameas Figure5-10.X,Y,Color==#channels,#groups,#leptons. Figure 5-9 ,exceptforthefactthatthexandzcoordinatesareswitched.Eventhough 84

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theybothcontainthesameamountofinformation,ithelpstovisualizethedatainadifferentway.InFigure 5-11 ,wemakea3dimensionalplotthatshowsforagiven Figure5-11.X,Y,Color==#hierarchies,#groups,#leptons. group,a)thetotalnumberofduplicatehierarchiesinthegroup(x-axis),b)thenumberofleptonspresentinthemaximallyleptonicsignature(z-axis,colorcoded)andc)thenumberofsuchgroups(y-axis).Wecanclearlyseeherethat,foragroupwith8leptonsinitsmaximallyleptonicchannel(aquacoloredbars)therearefewernumberofduplicatehierarchies.Ontheotherhand,foragroupwith0leptonsinitsmaximallyleptonicchannel,thenumberofduplicatehierarchiescangetaslargeas100.Wealsonotethatasthenumberofleptonsinthemaximallyleptonicchanneldecreasesnumberofsuchgroupsalsodecreases.Figure 5-12 ,issameasFigure 5-11 exceptforthez-axis,whichshowsthenumberofdominantsignaturesthatthegrouphas.weseethat,foragroupwith5dominantsignatures,therearefewernumberofduplicatehierarchies.Ontheotherhand,foragroupwithjustonedominantsignature,thenumberofduplicatehierarchiescangetaslargeas100.Wealsonotethatasthenumberofdominantsignaturesforagroupdecreasesnumberofsuchgroupsalsodecreases. 5.5SummaryandOutlookInconclusion,wehaveshownthatalreadywithitsrst10pb)]TJ /F7 7.97 Tf 6.59 0 Td[(1ofdata,theLHCcanstartprobingSUSY,providedonelooksintherightplace.Tothisend,it'simperative 85

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Figure5-12.X,Y,Color==#hierarchies,#groups,#channels. tosearchforthemodelhierarchiesthatarethemostlikelytobediscoveredrst.Ourworkshowninthecurrentchapteroutlinesthemostgeneralstrategytodojustthat.Wehavealsoshownthatbyidentifyingsets(groups)ofsignatureswecanstartinvestigatingtheLHCinverseproblem.Thenextfewstepsweshouldtakeareveryclear.Wecandividethesignaturespaceevenmorebydistinguishingdecaysofandintotheparticlesinvariouselectroweakmultipletsseparately.ItwillneedachangeinthetravellingsalesmandiagramasshowninFigure 5-13 86

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Figure5-13.Thetravellingsalesmandiagramforthecasewhentheelectroweakmultipletsareconsideredseparately.WehavenotshownthecoloredSUSYparticlesastheydon'tchangetheirsignatureswithsuchsplittingofthemultiplets. 87

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APPENDIXAFORTRANCODEFOREXTERNALIMPLEMENTATIONOFDIQUARKINPYTHIA C...Thisisauserdefinedexternalprocess C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -104.1 Td[(C...Namesofparticles: C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -151.92 Td[(C........u:u)]TJ /F1 11.955 Tf 9.23 0 Td[(quark C........uu:Intermediateparticlecomposedof2u's. C........l)]TJ /F1 11.955 Tf 9.87 0 Td[(:Lepton C........l+:Anti)]TJ /F1 11.955 Tf 9.73 0 Td[(lepton C......Diq:Diquark C......LDQ:Leptodiquark C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -319.27 Td[(C...Process:u,u)]TJ /F5 11.955 Tf 7.27 0 Td[(>u,u,l)]TJ /F1 11.955 Tf 10.16 0 Td[(,l+,in2steps C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -367.09 Td[(C......Step1:u,u)]TJ /F5 11.955 Tf 7.27 0 Td[(>Diq,resonantproduction C......Step2:Diq)]TJ /F5 11.955 Tf 7.27 0 Td[(>LDQ,l)]TJ ET BT /F1 11.955 Tf -.73 -414.9 Td[(C......Step3:LDQ)]TJ /F5 11.955 Tf 7.27 0 Td[(>uu,l+ C......Step4:uu)]TJ /F5 11.955 Tf 7.27 0 Td[(>u,u C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -486.63 Td[(C...Theinputparametersinthecodeare C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -534.44 Td[(C...NEV:Numberofevents C...ECM:Centerofmassenergy C...DefineapTmincutofffortheprocess. C...MDiq:Diquarkmass C...MLD:Lepto)]TJ /F1 11.955 Tf 9.62 0 Td[(diquarkmass 88

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C...KD1:CouplingbetweenD,u,u C...KD2:CouplingbetweenD,L,l)]TJ ET BT /F1 11.955 Tf -.49 -64.55 Td[(C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -88.46 Td[(C...OPTIONALInputs C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -136.27 Td[(C.....WDiq:Diquarkwidth C.....WLDQ:Lepto)]TJ /F1 11.955 Tf 9.61 0 Td[(diquarkwidth C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -208 Td[(C...OPTIONALInputsabovechosenwithSWITCHES C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -255.81 Td[(C.....WDSWITCH:0tocalculateWDiqinternally C.............:1tobespecifiedbytheuser C.....WLSWITCH:0tocalculateWLDQinternally C.............:1tobespecifiedbytheuser C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -375.35 Td[(C...Outputofthecode C....Totalcross)]TJ /F1 11.955 Tf 9.78 0 Td[(section C....Eventsgenerated C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -470.98 Td[(C...Codewrittenby:TheGATORTeam C...Date:09/30/2010 C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -566.62 Td[(C...Preamble:declarations. C...Allrealarithmeticindoubleprecision. IMPLICITDOUBLEPRECISION(A)]TJ /F1 11.955 Tf 7.75 0 Td[(H,O)]TJ /F1 11.955 Tf 8.17 0 Td[(Z) 89

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C...ThreePythiafunctionsreturnintegers,soneeddeclaring. INTEGERPYK,PYCHGE,PYCOMP C...EXTERNALstatementlinksPYDATAonmostmachines. EXTERNALPYDATA C...Commonblocks. C...Theeventrecord. COMMON/PYJETS/N,NPAD,K(4000,5),P(4000,5),V(4000,5) C...Parameters. COMMON/PYDAT1/MSTU(200),PARU(200),MSTJ(200),PARJ(200) C...Parameters. COMMON/PYPARS/MSTP(200),PARP(200),MSTI(200),PARI(200) C...Theuser'sowntransferofinformation. DoublePrecisionECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD,WDiq1,WDiq2 COMMON/MYCOMM/ECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD,WDiq1,WDiq2 SAVE/MYCOMM/ C...HEPEVTcommonblock(added07/24/2005KM) PARAMETER(NMXHEP=4000) COMMON/HEPEVT/NEVHEP,NHEP,ISTHEP(NMXHEP),IDHEP(NMXHEP) >,JMOHEP(2,NMXHEP),JDAHEP(2,NMXHEP),PHEP(5,NMXHEP) >,VHEP(4,NMXHEP) DOUBLEPRECISIONPHEP,VHEP 90

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integermint real8vint COMMON/PYINT1/MINT(400),VINT(400) C...Counterfornumberofgeneratedeventsofeachtype. DIMENSIONNCOUNT(2) DATANCOUNT/20/ DATAPI/3.141592653589793D0/ CdoubleprecisionLSWITCH,DSWITCH INTEGERWDSWITCH,WLSWITCH C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.49 -303.63 Td[(C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -327.54 Td[(C...Firstsection:USERINPUT C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -375.35 Td[(C...NEV:Numberofevents NEV=1000 C...ECM:Centerofmassenergy ECM=7000D0 C...DefineapTmincutofffortheprocess. PTMIN=10D0 C...MDiq:Diquarkmass MDiq=600D0 C...MLD:Lepto)]TJ /F1 11.955 Tf 9.62 0 Td[(diquarkmass MLD=400D0 91

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C...KD1:CouplingbetweenD,u,u KD1=1D0 C...KD2:CouplingbetweenD,L,l)]TJ ET BT /F1 11.955 Tf 42.61 -88.46 Td[(KD2=1.5D0 C...WDiqI:Diquarkwidth C...WDSWITCH=0(Widthcalculatedintenally) C...WDSWITCH=1(Userspecifiedwidth) WDSWITCH=0 WDiq=300D0 C...Calculatewidthinternallyandcomparewithuser C...inputifswitchison. C...Ifyourinputislessthanthecalculatedwidth C...thentheprogramwillgiveerrorandstop. CALLDWIDTH(WDSWITCH,WDiq,WDiq1,WDiq2) C...WLDQ:Lepto)]TJ /F1 11.955 Tf 9.62 0 Td[(diquarkwidth C...WLSWITCH=0(Widthcalculatedintenally) C...WLSWITCH=1(Userspecifiedwidth) WLSWITCH=0 WLDQ=5D0 C...Calculatewidthinternallyandcomparewithuser C...inputifswitchison. C...Ifyourinputislessthanthecalculatedwidth C...thentheprogramwillgiveerrorandstop. 92

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CALLLWIDTH(WLSWITCH,WLDQ) C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -88.46 Td[(C...Switchoffunnecessaryaspects:initialandfinalstate C...showers,multipleinteractions,hadronization. C...(Optionalforfastersimulationoftheparton)]TJ /F1 11.955 Tf 10.12 0 Td[(level C...processesonly.) MSTP(61)=0 MSTP(71)=0 MSTP(81)=0 MSTP(111)=0 C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -303.63 Td[(C...Userinputendshere C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -399.26 Td[(C...InitializePythia. CALLPYINIT('USER','','',0D0) C...Secondsection:eventloop. C...Generateeventsandlookatfirstfew. DO200IEV=1,NEV IF(MOD(IEV,500).EQ.0)WRITE(6,) &'Nowateventnumber',IEV CALLPYEVNT CALLPYHEPC(1) if(IEV.lt.5)callpylist(1) 93

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C...Endofloopoverevents. 200CONTINUE C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -136.27 Td[(C...Thirdsection:produceoutputandend. C...Crosssectiontable. CALLPYLIST(2) CALLPYSTAT(1) END C SUBROUTINEUPINIT C...Doubleprecisionandintegerdeclarations. IMPLICITDOUBLEPRECISION(A)]TJ /F1 11.955 Tf 7.75 0 Td[(H,O)]TJ /F1 11.955 Tf 8.17 0 Td[(Z) IMPLICITINTEGER(I)]TJ /F1 11.955 Tf 7.51 0 Td[(N) C...Selectionofhardscatteringsubprocesses. COMMON/PYSUBS/MSEL,MSELPD,MSUB(500),KFIN(2,)]TJ /F1 11.955 Tf 10.42 0 Td[(40:40) &,CKIN(200) C...Userprocessinitializationcommonblock. INTEGERMAXPUP PARAMETER(MAXPUP=100) INTEGERIDBMUP,PDFGUP,PDFSUP,IDWTUP,NPRUP,LPRUP 94

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DOUBLEPRECISIONEBMUP,XSECUP,XERRUP,XMAXUP COMMON/HEPRUP/IDBMUP(2),EBMUP(2),PDFGUP(2),PDFSUP(2), &IDWTUP,NPRUP,XSECUP(MAXPUP),XERRUP(MAXPUP) &,XMAXUP(MAXPUP),LPRUP(MAXPUP) SAVE/HEPRUP/ C...Userprocesseventcommonblock. INTEGERMAXNUP PARAMETER(MAXNUP=500) INTEGERNUP,IDPRUP,IDUP,ISTUP,MOTHUP,ICOLUP DOUBLEPRECISIONXWGTUP,SCALUP,AQEDUP,AQCDUP,PUP,VTIMUP >,SPINUP COMMON/HEPEUP/NUP,IDPRUP,XWGTUP,SCALUP,AQEDUP,AQCDUP >,ISTUP(MAXNUP),MOTHUP(2,MAXNUP),ICOLUP(2,MAXNUP) >,VTIMUP(MAXNUP),SPINUP(MAXNUP),IDUP(MAXNUP),PUP(5,MAXNUP) SAVE/HEPEUP/ C...Theuser'sowntransferofinformation. DoublePrecisionECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq,MLDQ >,WLDQ,MLD,WDiq1,WDiq2 COMMON/MYCOMM/ECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq,MLDQ >,WLDQ,MLD,WDiq1,WDiq2 SAVE/MYCOMM/ C...Setupincomingbeams. IDBMUP(1)=2212 IDBMUP(2)=2212 95

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EBMUP(1)=0.5D0ECM EBMUP(2)=0.5D0ECM C...Setuptheexternalprocess. IDWTUP=1 NPRUP=1 LPRUP(1)=1001 C...Findsomereasonablemaximumforexternalprocess. C...(ThisisdonebyacheatingcalltoUPEVNTwithspecial C...switch.) INIT=1 CALLUPEVNT XMAXUP(1)=XWGTUP INIT=0 RETURN END C C...Thisroutinesamplesthephasespace,evaluatestheprocess C...crossandsetsthecolourtopologyandpartonshowerscales. SUBROUTINEUPEVNT C...Allrealarithmeticindoubleprecision. IMPLICITDOUBLEPRECISION(A)]TJ /F1 11.955 Tf 7.75 0 Td[(H,O)]TJ /F1 11.955 Tf 8.17 0 Td[(Z) 96

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C...Userprocesseventcommonblock. INTEGERMAXNUP PARAMETER(MAXNUP=500) INTEGERNUP,IDPRUP,IDUP,ISTUP,MOTHUP,ICOLUP DOUBLEPRECISIONXWGTUP,SCALUP,AQEDUP,AQCDUP,PUP >,VTIMUP,SPINUP COMMON/HEPEUP/NUP,IDPRUP,XWGTUP,SCALUP,AQEDUP &,AQCDUP,IDUP(MAXNUP),ISTUP(MAXNUP),MOTHUP(2,MAXNUP) &,ICOLUP(2,MAXNUP),PUP(5,MAXNUP),VTIMUP(MAXNUP) &,SPINUP(MAXNUP) SAVE/HEPEUP/ C...Theuser'sowntransferofinformation. DoublePrecisionECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD,WDiq1,WDiq2 COMMON/MYCOMM/ECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD,WDiq1,WDiq2 SAVE/MYCOMM/ C...Localarraysandparameters. DIMENSIONXPP1()]TJ /F1 11.955 Tf 10.47 0 Td[(25:25),XPP2()]TJ /F1 11.955 Tf 10.47 0 Td[(25:25),TERM(5,5) DATAPI/3.141592653589793D0/ DATACONV/0.3894D9/ INTEGERKFLAV1,KFLAV2 DoublePrecisionPUU(5),PLDQ(5),Pl1(5),PU1(5),PU2(5),X1buf DoublePrecisionbetaLDQ(3),betaUU(3),P4,PlMAX,Plabs,UHATP 97

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DoublePrecisionCTH,CTHMIN,CTHMAX,MDiqX DoublePrecisionSHMIN,SHMAX,TAUMIN,TAUMAX,TAUA,TAUB,TI4 &,TAU,YMAX,YMIN,YCUR,ZMAX,ZMIN,ZCUR,HTAU,HY,HZ C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -160.18 Td[(C...Defaultreturnvalue. XWGTUP=0D0 C...Defaultflagforeventrejection. IREJ=0 C...CalculatetheMassofLepto)]TJ /F1 11.955 Tf 9.36 0 Td[(DiquarkaccordingtoBreit)]TJ /F1 11.955 Tf 8.98 0 Td[(Wigner C...distribution.Thefollowingconditionsareforputtingthe C...maximumandminimumcutsonthemass IF(MDiq)]TJ /F1 11.955 Tf 7.07 0 Td[(MLDQ.gt.MLD)]TJ /F1 11.955 Tf 8.76 0 Td[(0D0)then MLDQ=MLD+0.5D0WLDQTAN((2D0PYR(0))]TJ /F1 11.955 Tf 10.06 0 Td[(1D0) &ATAN(2D0(MLD)]TJ /F1 11.955 Tf 8.77 0 Td[(3D0)/WLDQ)) ELSE MLDQ=MLD+0.5D0WLDQTAN((2D0PYR(0))]TJ /F1 11.955 Tf 10.06 0 Td[(1D0) &ATAN(2D0(MDiq)]TJ /F1 11.955 Tf 7.3 0 Td[(MLD)/WLDQ)) ENDIF C................................................................. C...Step1:ResonanceproductionofDiquark(uu)]TJ /F5 11.955 Tf 7.28 0 Td[(>D) C)177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()178()177()178()178()178()177()178()178()178()177()178()178()]TJ ET BT /F1 11.955 Tf -.73 -614.43 Td[(C...Thefirstthingtodois,decidehowthephasespaceshould C...besampled. 98

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C...Thetotalcrosssectionfora2)]TJ /F5 11.955 Tf 7.68 0 Td[(>1processcanbewritten C...sigma=Integrald(x 1)Integrald(x 2) C...f 1(x 1,Q2)f 2(x 2,Q2)(sigma)]TJ /F1 11.955 Tf 9.46 0 Td[(hat(s)]TJ /F1 11.955 Tf 9.46 0 Td[(hat)) C...Or,intermsoftauandy, C...sigma=Integral(d(tau)/tau)Integrald(y) C...x 1f 1(x 1,Q2)x 2f 2(x 2,Q2)(sigma)]TJ /F1 11.955 Tf 9.46 0 Td[(hat(s)]TJ /F1 11.955 Tf 9.46 0 Td[(hat)) C...SeeSection7.3PythiaManual C...Soasimplerecipewouldbetopicktauandyuniformly. C...However,theresonantcross)]TJ /F1 11.955 Tf 9.77 0 Td[(sectionand C...partondistributionsarepeaked,sothiswouldbeinefficient. C...Inparticlar,thecross)]TJ /F1 11.955 Tf 9.78 0 Td[(sectionispeakedats)]TJ /F1 11.955 Tf 9.21 0 Td[(hat, C...so,theintegralisrewrittenas C...sigma=(PI/S)h)]TJ /F1 11.955 Tf 9.46 0 Td[(tau(tau)Integrald(tau)h)]TJ /F1 11.955 Tf 8.83 0 Td[(y(y) C...Integrald(y)x 1f 1(x 1,Q2)x 2f 2(x 2,Q2) C...(s)]TJ /F1 11.955 Tf 9.46 0 Td[(hat/PI)sigma)]TJ /F1 11.955 Tf 9.46 0 Td[(hat(s)]TJ /F1 11.955 Tf 9.47 0 Td[(hat) C.../(tau2h)]TJ /F1 11.955 Tf 9.46 0 Td[(tau(tau)h)]TJ /F1 11.955 Tf 8.83 0 Td[(y(y)) C...See,Section7.4.1and7.4.2(PythiaManual) C...Where,tauandyaregeneratedaccordingtothedistributions C...h)]TJ /F1 11.955 Tf 8.84 0 Td[(y(y),h)]TJ /F1 11.955 Tf 9.46 0 Td[(tau(tau) C...Inourcaseh)]TJ /F1 11.955 Tf 9.46 0 Td[(tauisaBreit)]TJ /F1 11.955 Tf 9.35 0 Td[(wignerpeakedattheDiquarkmass& C...h)]TJ /F1 11.955 Tf 8.84 0 Td[(yisflat(noforward/backwardassymetry). C...The1stpartofthecodeisgenerating2)]TJ /F5 11.955 Tf 8.44 0 Td[(>1resonant C...cross)]TJ /F1 11.955 Tf 9.77 0 Td[(sectiondoesn'thaveanyscatteringangle. C...Butsincewedecaytheresonantparticle,andtheoutgoing C...particleshavePTcuts,wecalculatethelimitsonthe C...scatteringanglealongwithy(inthecodeit'snamedYCUR). 99

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C...Thecosineofthescatteringangle(z) C...(z,inthecodeit'scalledZCUR)isgenerated C...asaflatdistribution. C...ThepTmincuttranslatesintocutsonx 1,x 2,andhenceinto C...cutsoftauandy. C...TAUlimitsarecalculatedhere.Limitsonydependsontauand C...willbecalculatedoncewesetthevalueoftau. SHMIN=MLDQ2+2D0(PTMINdsqrt(MLDQ2+PTMIN2)+PTMIN2) TAUMIN=SHMIN/ECM2 SHMAX=ECM2)]TJ /F1 11.955 Tf 8.99 0 Td[(SHMIN TAUMAX=SHMAX/ECM2 TAUA=DATAN((SHMIN)]TJ /F1 11.955 Tf 8.4 0 Td[(MDiq2)/(MDiqWDiq)) TAUB=DATAN((SHMAX)]TJ /F1 11.955 Tf 8.4 0 Td[(MDiq2)/(MDiqWDiq)) TI4=(1D0/(ECM2MDiqWDiq))(TAUB)]TJ /F1 11.955 Tf 7.6 0 Td[(TAUA) C...INIT=1or0correspondstoinitializationorsubsequent C...eventgeneration. IF(INIT.EQ.0)THEN C...SelectTAUhere.ThenselectYMIN,YMAX,ZMINandZMAX C...ThenevaluatethevaluesofX1,X2andUHAT TAU=(1D0/ECM2) &(MDiq2+MDiqWDiqDTAN(TAUA+(TAUB)]TJ /F1 11.955 Tf 7.6 0 Td[(TAUA)PYR(0))) SHAT=TAUECM2 RTSHAT=DSQRT(SHAT) 100

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YMAX=)]TJ /F1 11.955 Tf 9.72 0 Td[(0.5D0DLOG(TAU) YMIN=)]TJ /F1 11.955 Tf 7.78 0 Td[(YMAX YCUR=YMIN+(YMAX)]TJ /F1 11.955 Tf 8 0 Td[(YMIN)PYR(0) ZMAX=DSQRT(1D0)]TJ /F1 11.955 Tf 8.76 0 Td[(4D0PTMIN2/SHAT) ZMIN=)]TJ /F1 11.955 Tf 7.91 0 Td[(ZMAX ZCUR=ZMIN+(ZMAX)]TJ /F1 11.955 Tf 8.13 0 Td[(ZMIN)PYR(0) ELSE C...Forspecialinitializationcallpicklowestpointas C...likelytorepresentmaximumcrosssection. TAU=DSQRT(MDiq2/ECM2) X1=DSQRT(TAU) X2=X1 SHAT=TAUECM2 RTSHAT=DSQRT(SHAT) YMAX=)]TJ /F1 11.955 Tf 9.72 0 Td[(0.5D0DLOG(TAU) YMIN=)]TJ /F1 11.955 Tf 7.78 0 Td[(YMAX YCUR=YMIN+(YMAX)]TJ /F1 11.955 Tf 8 0 Td[(YMIN)0.5D0 ZMAX=DSQRT(1D0)]TJ /F1 11.955 Tf 8.76 0 Td[(4D0PTMIN2/SHAT) ZMIN=)]TJ /F1 11.955 Tf 7.91 0 Td[(ZMAX ZCUR=ZMIN+(ZMAX)]TJ /F1 11.955 Tf 8.13 0 Td[(ZMIN)0.5D0 ENDIF 101

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C...Deriveotherkinematicalquantities. X1=DSQRT(TAU)DEXP(YCUR) X2=DSQRT(TAU)DEXP()]TJ /F1 11.955 Tf 8.53 0 Td[(YCUR) UHAT=)]TJ /F1 11.955 Tf 9.72 0 Td[(0.5D0(SHAT)]TJ /F1 11.955 Tf 7.07 0 Td[(MLDQ2)(1D0+ZCUR) THAT=)]TJ /F1 11.955 Tf 8.29 0 Td[(SHAT)]TJ /F1 11.955 Tf 7.46 0 Td[(UHAT+MLDQ2 PT2=THATUHAT/SHAT C...Bydecouplingthex 1,x 2andu)]TJ /F1 11.955 Tf 9.46 0 Td[(hatvariablesweoverestimate C...thephasespaceregion.Wethereforehavetorejectevents C...outsideallowedphasespacebyputtingthecrosssection=0. C...However,tokeepbookkeepingcorrect,onestillneedstoset C...variablesasifthereisanevent,e.g.pickingthesame C...pointasfortheinitializationcall. IF(INIT.EQ.0.AND.PT2.LT.PTMIN2)THEN IREJ=1 TAU=DSQRT(MDiq2/ECM2) X1=DSQRT(TAU) X2=X1 SHAT=TAUECM2 RTSHAT=DSQRT(SHAT) YMAX=)]TJ /F1 11.955 Tf 9.72 0 Td[(0.5D0DLOG(TAU) YMIN=)]TJ /F1 11.955 Tf 7.78 0 Td[(YMAX YCUR=YMIN+(YMAX)]TJ /F1 11.955 Tf 8 0 Td[(YMIN)0.5D0 102

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ZMAX=DSQRT(1D0)]TJ /F1 11.955 Tf 8.76 0 Td[(4D0PTMIN2/SHAT) ZMIN=)]TJ /F1 11.955 Tf 7.91 0 Td[(ZMAX ZCUR=ZMIN+(ZMAX)]TJ /F1 11.955 Tf 8.13 0 Td[(ZMIN)0.5D0 UHAT=)]TJ /F1 11.955 Tf 9.72 0 Td[(0.5D0(SHAT)]TJ /F1 11.955 Tf 7.06 0 Td[(MLDQ2)(1D0+ZCUR) THAT=)]TJ /F1 11.955 Tf 8.28 0 Td[(SHAT)]TJ /F1 11.955 Tf 7.47 0 Td[(UHAT+MLDQ2 PT2=THATUHAT/SHAT ENDIF C...Herewecalculatethephasespacevolume.Todoso, C...weneedtocalculatetheJacobiantocompensateforthe C...changeofvariablesastheyaregeneratedaccordingto C...specificdistributions. C...ThephasespaceiscalculatedincludingthecorrectJacobians. C...JacobianfromPythiaManual(Section)]TJ /F1 11.955 Tf 10.3 0 Td[(7.2) HTAU=(1D0/TI4)(1D0/((SHAT)]TJ /F1 11.955 Tf 8.4 0 Td[(MDiq2)2+MDiq2WDiq2)) HY=1/(YMAX)]TJ /F1 11.955 Tf 7.99 0 Td[(YMIN) HZ=1/(ZMAX)]TJ /F1 11.955 Tf 8.13 0 Td[(ZMIN) PHSPV=SHAT/((ECM2)(TAU2HTAUHY)) C................................................................. C...Thesecondpartistoevaluatethecrosssectioninthe C...phasespacepointselectedabove. C...PickQ2scale(whichinvolvessomearbitrariness) Q2=MDiq2 103

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C...Theresonantcross)]TJ /F1 11.955 Tf 9.78 0 Td[(sectionDSIGafterreplacingthe C...Delta)]TJ /F1 11.955 Tf 9.96 0 Td[(functionwithbytheappropriateBrite)]TJ /F1 11.955 Tf 8.98 0 Td[(Wigner C...shape(accordingtotheprescriptioninSection7.3 C...ofPythiaManual)isgivenhere. cDSIG=(KD12KD22/(96D0PI)) c&(SHAT) c&/((SHAT)]TJ /F1 11.955 Tf 8.4 0 Td[(MDiq2)2+(MDiqWDiq)2) DSIG=(KD12/(6D0))MDiqWDiq2 &/((SHAT)]TJ /F1 11.955 Tf 8.4 0 Td[(MDiq2)2+(MDiqWDiq)2) C...Partondistributions(multipliedbyx)ofthe2incoming C...protons. CALLPYPDFU(2212,X1,Q2,XPP1) CALLPYPDFU(2212,X2,Q2,XPP2) C...PDFevaluatedforX1andX2fromstepabove SUM=XPP1(2)XPP2(2) C...NowpossibletodefinetheSIGEVreturnvalue. C...TheCONVfactorconvertsfromGeV()]TJ /F1 11.955 Tf 10.19 0 Td[(2)topb. IF(IREJ.EQ.0)XWGTUP=CONVPHSPVDSIGSUM C...Donewheninitializingformaximumcrosssection. IF(INIT.EQ.1)RETURN C................................................................. C...Thethirdstepistosetupthepartonicprocessthatis C...selected. 104

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C...Definenumberofpartons)]TJ /F1 11.955 Tf 16.49 0 Td[(twoincoming(uu) C...andfouroutgoing(uull) NUP=6 C...KFcodesoftheparticles IDUP(1)=2 IDUP(2)=2 IDUP(3)=2 IDUP(4)=2 IDUP(5)=11 IDUP(6)=)]TJ /F1 11.955 Tf 9.97 0 Td[(11 IF(3D0PYR(0).lt.2D0)then IDUP(5)=13 IDUP(6)=)]TJ /F1 11.955 Tf 9.97 0 Td[(13 IF(2D0PYR(0).lt.1D0)then IDUP(5)=15 IDUP(6)=)]TJ /F1 11.955 Tf 9.97 0 Td[(15 ENDIF ENDIF C...Statuscodes.()]TJ /F1 11.955 Tf 9.62 0 Td[(1forincoming,+1foroutgoing) ISTUP(1)=)]TJ /F1 11.955 Tf 9.99 0 Td[(1 ISTUP(2)=)]TJ /F1 11.955 Tf 9.99 0 Td[(1 ISTUP(3)=1 ISTUP(4)=1 ISTUP(5)=1 105

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ISTUP(6)=1 C...Mothercodes.(Forincomingparticles,it's0) C...Foroutgoingparticlesthetwomothersforeachofthem C...arethetwoincomingparticles MOTHUP(1,1)=0 MOTHUP(2,1)=0 MOTHUP(1,2)=0 MOTHUP(2,2)=0 MOTHUP(1,3)=1 MOTHUP(2,3)=2 MOTHUP(1,4)=1 MOTHUP(2,4)=2 MOTHUP(1,5)=1 MOTHUP(2,5)=2 MOTHUP(1,6)=1 MOTHUP(2,6)=2 C...Colourstretchedfrominitialu)]TJ /F1 11.955 Tf 9.26 0 Td[(quarkstofinalu)]TJ /F1 11.955 Tf 9.25 0 Td[(quarks. ICOLUP(1,1)=501 ICOLUP(2,1)=0 ICOLUP(1,2)=502 ICOLUP(2,2)=0 ICOLUP(1,3)=502 ICOLUP(2,3)=0 ICOLUP(1,4)=501 ICOLUP(2,4)=0 106

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ICOLUP(1,5)=0 ICOLUP(2,5)=0 ICOLUP(1,6)=0 ICOLUP(2,6)=0 C...Resetmomentatozero. DO130I=1,6 DO120J=1,5 PUP(J,I)=0D0 120CONTINUE 130CONTINUE PLDQ(1)=0D0 PLDQ(2)=0D0 PLDQ(3)=0D0 PLDQ(4)=0D0 PLDQ(5)=MLDQ C...Massesoffinalstateentries;initialassumedmassless. PUP(5,3)=PYMASS(IDUP(3)) PUP(5,4)=PYMASS(IDUP(4)) PUP(5,5)=PYMASS(IDUP(5)) PUP(5,6)=PYMASS(IDUP(6)) C...Four)]TJ /F1 11.955 Tf 8.29 0 Td[(momentaoftheincomingpartonssimple. PUP(4,1)=0.5D0X1ECM PUP(3,1)=PUP(4,1) PUP(4,2)=0.5D0X2ECM 107

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PUP(3,2)=)]TJ /F1 11.955 Tf 9.47 0 Td[(PUP(4,2) C................................................................. C...STEP2:DecayoftheDiquarkintolepto)]TJ /F1 11.955 Tf 9.62 0 Td[(diquarkand1stlepton C................................................................. C...Energiesandabsolutemomentumlepto)]TJ /F1 11.955 Tf 9.62 0 Td[(diquarkand1stlepton C...thesubsystemframe. RTSHAT=DSQRT(X1X2)ECM PABS=0.5D0DSQRT((RTSHAT2)]TJ /F1 11.955 Tf 8.66 0 Td[(PUP(5,6)2)]TJ ET BT /F1 11.955 Tf 35.47 -231.91 Td[(&PLDQ(5)2)2)]TJ /F1 11.955 Tf 10.32 0 Td[(4D0PUP(5,6)2PLDQ(5)2)/RTSHAT PE3=0.5D0(RTSHAT2+PUP(5,6)2)]TJ /F1 11.955 Tf 9.53 0 Td[(PLDQ(5)2)/RTSHAT PE4=RTSHAT)]TJ /F1 11.955 Tf 7.97 0 Td[(PE3 C...Subsystemscatteringangledefinedneglectingquarkmass. COSTHE=ZCUR SINTHE=DSQRT(1D0)]TJ /F1 11.955 Tf 7.26 0 Td[(COSTHE2) C...Azimuthalangleatrandom. PHI=2D0PIPYR(0) C...MomentaofoutgoingpartonsintheCoMframe. PUP(1,6)=PABSSINTHEDCOS(PHI) PUP(2,6)=PABSSINTHEDSIN(PHI) PZ3=PABSCOSTHE PLDQ(1)=)]TJ /F1 11.955 Tf 9.24 0 Td[(PUP(1,6) PLDQ(2)=)]TJ /F1 11.955 Tf 9.24 0 Td[(PUP(2,6) PZ4=)]TJ /F1 11.955 Tf 8.71 0 Td[(PZ3 108

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C...Longitudinalboostofoutgoingpartonstocmframe. BETA=(X1)]TJ /F1 11.955 Tf 8.15 0 Td[(X2)/(X1+X2) GAMMA=0.5D0(X1+X2)/DSQRT(X1X2) PUP(3,6)=GAMMA(PZ3+BETAPE3) PUP(4,6)=GAMMA(PE3+BETAPZ3) PLDQ(3)=GAMMA(PZ4+BETAPE4) PLDQ(4)=GAMMA(PE4+BETAPZ4) C................................................................. C...Step3:DecayofLDQ)]TJ /F5 11.955 Tf 7.27 0 Td[(>UU(PUU)+l1(PUP(5,5)) C................................................................. C...CalculatetheboostfactoroftheLepto)]TJ /F1 11.955 Tf 9.62 0 Td[(diquarksubsystem betaLDQ(1)=PLDQ(1)/PLDQ(4) betaLDQ(2)=PLDQ(2)/PLDQ(4) betaLDQ(3)=PLDQ(3)/PLDQ(4) C...Initializethe4)]TJ /F1 11.955 Tf 8.94 0 Td[(momentaofUUsub)]TJ /F1 11.955 Tf 8.97 0 Td[(systemandthe2nd C...outgoinglepton Pl1(1)=0D0 Pl1(2)=0D0 Pl1(3)=0D0 Pl1(4)=0D0 Pl1(5)=PUP(5,5) PUU(1)=0D0 PUU(2)=0D0 PUU(3)=0D0 PUU(4)=0D0 PU1(1)=0D0 109

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PU1(2)=0D0 PU1(3)=0D0 PU1(4)=0D0 PU1(5)=PUP(5,3) PU2(1)=0D0 PU2(2)=0D0 PU2(3)=0D0 PU2(4)=0D0 PU2(5)=PUP(5,4) C...Calculatethemaximumvalueofthe3)]TJ /F1 11.955 Tf 8.63 0 Td[(momentumsquare C...oftheoutgoingleptonintheleptodiquarkrestframe PlMAX=(PLDQ(5)4+Pl1(5)4+(PU1(5)+PU2(5))4 &)]TJ /F1 11.955 Tf 16.33 0 Td[(2PLDQ(5)2Pl1(5)2)]TJ /F1 11.955 Tf 16.34 0 Td[(2PLDQ(5)2(PU1(5)+PU2(5))2 &)]TJ /F1 11.955 Tf 16.99 0 Td[(2(PU1(5)+PU2(5))2Pl1(5)2)/(4PLDQ(5)2) C...Randomlyselectthemagnitudeofthe3)]TJ /F1 11.955 Tf 8.63 0 Td[(momentumofthe C...outgoinglepton(whichisalsosamefortheUUsubsystem) C...intheleptodiquarkrestframe Plabs=DSQRT(PlMAXPYR(0)) PUU(5)=DSQRT(PLDQ(5)2+Pl1(5)2 &)]TJ /F1 11.955 Tf 16.33 0 Td[(2PLDQ(5)DSQRT(Plabs2+Pl1(5)2)) C...CallDK2subroutineforthe2)]TJ /F1 11.955 Tf 9.45 0 Td[(bodydecayoftheLepto)]TJ /F1 11.955 Tf 9.61 0 Td[(diquark CALLDK2(PLDQ(5),PUU(5),Pl1(5),PUU,Pl1) 110

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C...BoostthemomentumoftheleptonandtheUUsubsystem CALLBOOST(Pl1,betaLDQ(1),betaLDQ(2),betaLDQ(3)) CALLBOOST(PUU,betaLDQ(1),betaLDQ(2),betaLDQ(3)) PUP(1,5)=Pl1(1) PUP(2,5)=Pl1(2) PUP(3,5)=Pl1(3) PUP(4,5)=Pl1(4) C................................................................. C...Step4:DecayofUU(PUU))]TJ /F5 11.955 Tf 7.68 0 Td[(>u+u C................................................................. C...CalculatetheboostfactoroftheUUsubsystem betaUU(1)=PUU(1)/PUU(4) betaUU(2)=PUU(2)/PUU(4) betaUU(3)=PUU(3)/PUU(4) C...CallDK2subroutineforthe2)]TJ /F1 11.955 Tf 9.45 0 Td[(bodydecayoftheUUsubsystem CALLDK2(PUU(5),PUP(5,3),PUP(5,4),PU1,PU2) C...Boostthemomentumofthe2outgoingu)]TJ /F1 11.955 Tf 9.26 0 Td[(quarks CALLBOOST(PU1,betaUU(1),betaUU(2),betaUU(3)) CALLBOOST(PU2,betaUU(1),betaUU(2),betaUU(3)) PUP(1,3)=PU1(1) PUP(2,3)=PU1(2) PUP(3,3)=PU1(3) PUP(4,3)=PU1(4) PUP(1,4)=PU2(1) 111

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PUP(2,4)=PU2(2) PUP(3,4)=PU2(3) PUP(4,4)=PU2(4) C...Done. RETURN END C................................................................. C.....SubroutinetodecayparticlewithmassXtoparticlesaandb C.....withmassesmaandmbrespectively.Thissubroutinereturns C.....the4)]TJ /F1 11.955 Tf 8.63 0 Td[(momentumofparticleaandb. SUBROUTINEDK2(X,ma,mb,pa,pb) DoublePrecisionX,ma,mb,rand1,rand2,p1,pa(4),pb(4),phai,thet DATAPI/3.141592653589793D0/ p1=0.5D0DSQRT((X2)]TJ /F1 11.955 Tf 8.67 0 Td[(ma2)]TJ /F1 11.955 Tf 8.68 0 Td[(mb2)2)]TJ /F1 11.955 Tf 9.98 0 Td[(4D0ma2mb2)/X thet=DACOS(2D0PYR(0))]TJ /F1 11.955 Tf 10.06 0 Td[(1D0)!PIPYR(0) phai=2D0PIPYR(0) pa(1)=p1DSIN(thet)DCOS(phai) pa(2)=p1DSIN(thet)DSIN(phai) pa(3)=p1DCOS(thet) pa(4)=DSQRT(ma2+pa(1)2+pa(2)2+pa(3)2) pb(1)=)]TJ /F1 11.955 Tf 10.36 0 Td[(pa(1) pb(2)=)]TJ /F1 11.955 Tf 10.36 0 Td[(pa(2) pb(3)=)]TJ /F1 11.955 Tf 10.36 0 Td[(pa(3) pb(4)=DSQRT(mb2+pb(1)2+pb(2)2+pb(3)2) RETURN END 112

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C................................................................. C.....Subroutinetoboostmomentumpxwithgivenbeta. C.....Itreturnsthemomentumpxintheboostedframe SUBROUTINEBOOST(px,betx,bety,betz) DoublePrecisionpx(5),betx,bety,betz,gam,betp,gamp gam=1D0/DSQRT(1D0)]TJ /F1 11.955 Tf 9.46 0 Td[(betx2)]TJ /F1 11.955 Tf 10.65 0 Td[(bety2)]TJ /F1 11.955 Tf 10.65 0 Td[(betz2) betAP=betxpx(1)+betypx(2)+betzpx(3) gamp=gam(gambetAP/(1D0+gam)+px(4)) px(1)=px(1)+gampbetx px(2)=px(2)+gampbety px(3)=px(3)+gampbetz px(4)=gam(px(4)+betAP) RETURN END C................................................................. C.....SubroutinetocalculatethewidthofDiquark C.....andcompareittouserspecifiedwidth SUBROUTINEDWIDTH(SWITCH,WDiqI,WDiqC1,WDiqC2) DoublePrecisionECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD,WDiqC1,WDiqC2 COMMON/MYCOMM/ECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD SAVE/MYCOMM/ 113

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DATAPI/3.141592653589793D0/ DOUBLEPRECISIONWDiqI INTEGERSWITCH WDiqC1=MDiqKD12/(16D0PI) WDiqC2=MDiqKD22(1)]TJ /F1 11.955 Tf 10.19 0 Td[((MLD/MDiq)2)/(16D0PI) WDiqC=WDiqC1+WDiqC2 C...Comparetheuserspecifieddecaywidthswiththecalculated C...widths IF(SWITCH.eq.0)then WDiqI=WDiqC ELSEIF(SWITCH.eq.1)then IF(WDiqI.lt.WDiqC)then WRITE(,)WARNING:UserspecifieddecaywidthofDiquark WRITE(,)islessthanthecalculateddecaywidth. WRITE(,)Calculated=,WDiqC WRITE(,)UserInput=,WDiqI STOP6 ENDIF WDiqC1=WDiqC1WDiqI/WDiqC WDiqC2=WDiqC2WDiqI/WDiqC ELSE WRITE(,)SelecttheDSWITCHtobe0or1 STOP6 ENDIF RETURN END 114

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C................................................................. C.....SubroutinetocalculatethewidthofLepto)]TJ /F1 11.955 Tf 9.61 0 Td[(diquark C.....andcompareittouserspecifiedwidth SUBROUTINELWIDTH(SWITCH,WLDQI) DoublePrecisionECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD COMMON/MYCOMM/ECM,PTMIN,INIT,KD1,KD2,MDiq,WDiq >,MLDQ,WLDQ,MLD SAVE/MYCOMM/ DATAPI/3.141592653589793D0/ DOUBLEPRECISIONWLDQI INTEGERSWITCH WLDQC=KD12KD22MLD5/(1536D0PI3MDiq4) write(,)MLD,PI,MDiq C...Comparetheuserspecifieddecaywidthswiththecalculated C...widths IF(SWITCH.eq.0)then WLDQI=WLDQC ELSEIF(SWITCH.eq.1)then IF(WLDQI.lt.WLDQC)then WRITE(,)WARNING:Userspecifieddecaywidthof WRITE(,)LeptoDiquarkislessthanthe WRITE(,)calculateddecaywidth. 115

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WRITE(,)Calculated=,WLDQC WRITE(,)UserInput=,WLDQI STOP6 ENDIF ELSE WRITE(,)SelecttheLSWITCHtobe0or1 STOP6 ENDIF RETURN END 116

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APPENDIXBPLOTSFROMEMULATIONInthisappendixwelisttheplotsgeneratedfortheModelIndependentAnalysisoftheCMSsamesigndileptonsearch,usingtheCMSemulationprescription.Theseplotsverycloselyresembletotheplotscorrespondingtoouranalysisusingfastsim(PGS)andsharethesameexplanation. FigureB-1.Emulation:EfcienciesforthesevensearchregionsforM1=10GeV. 117

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FigureB-2.Emulation:EfcienciesforthesevensearchregionsforM1=100GeV. FigureB-3.Emulation:EfcienciesforthesevensearchregionsforM1=200GeV. 118

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FigureB-4.Emulation:EfcienciesforthesevensearchregionsforM1=300GeV. FigureB-5.Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=10GeV. 119

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FigureB-6.Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=100GeV. FigureB-7.Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=200GeV. 120

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FigureB-8.Emulation:Limitsonthex-sectionsforthesevensearchregionsforM1=300GeV. 121

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APPENDIXCHIERARCHYTOSIGNATURESInthisappendix,welistallthehierarchiesalongwiththeirvariousdecaychannelswithdifferentsignatures.Thetablehassixcolumns.Therstcolumnliststhehierarchyrepresentedintermsofit'snumericalcode,whereeachdigitcorrespondsofoneofthe9particlesinTable 5-1 ,suchthataswereadthedigitsfromlefttorightthemassesofthecorrespondingparticlestheyrepresentdecreases.Thesecondcolumnisthesamehierarchyrepresentedintermsofit'slettercode.Ineithercasesinceweonlycareaboutthemassspectrumofparticleslighterthanthelightestcoloredsupersymmetricparticle(LCP)theX's(andforthenumericalcodethe0's)representtheparticlesheavierthantheLCP.Sinceahierarchycanhavemorethanoneequallysupresseddecaychannelwithdifferentsignatures,welistthemallincolumnfourandthecorrespondingsignatureincolumnve.Thethirdcolumncontainsthedecaychannelnumberforthedecaychain.Forexample,ifahierarchyhas3decaychains,welistallofthemin3separaterowsincolumnfourandincolumnthreewelisttheirmodenumberas1,2and3.Inthesixthcolumnwelistthehierarchymultiplicity,whichisthenumberofarrangementofthemassspectrumforwhichwehavethecorrespondinghierarchy. 122

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TableC-1.HierarchytoSignatures Hier.#Hier.Mode##ChainsSign.Mul. 000000014xxxxxxxGL1Gjjl!L(1,0,2)1x7!000000016xxxxxxxGW1Gjj!W(0,0,2)1x7!000000017xxxxxxxGB1Gjj!B(0,0,2)1x7!000000018xxxxxxxGH1Gjj!H(0,0,2)1x7!000000024xxxxxxxQL1Qjl!L(1,0,1)1x7!000000026xxxxxxxQW1Qj!W(0,0,1)1x7!000000027xxxxxxxQB1Qj!B(0,0,1)1x7!000000028xxxxxxxQH1Qj!H(0,0,1)1x7!000000034xxxxxxxUL1Ujl!L(1,0,1)2x7!000000036xxxxxxxUW1Uj!W(0,0,1)2x7!000000037xxxxxxxUB1Uj!B(0,0,1)2x7!000000038xxxxxxxUH1Uj!H(0,0,1)2x7!000000146xxxxxxGLW1Gjj!W(0,0,2)1x6!000000147xxxxxxGLB1Gjj!B(0,0,2)1x6!000000148xxxxxxGLH1Gjjl!Ll!H(2,0,2)1x6!2Gjj!H(0,0,2)1x6!000000154xxxxxxGEL1Gjjl!Ell!L(3,0,2)1x6!2Gjjl!L(1,0,2)1x6!000000156xxxxxxGEW1Gjj!W(0,0,2)1x6!000000157xxxxxxGEB1Gjj!B(0,0,2)1x6!000000158xxxxxxGEH1Gjjl!El!H(2,0,2)1x6!2Gjj!H(0,0,2)1x6!000000164xxxxxxGWL1Gjj!Wl!L(1,0,2)1x6!000000167xxxxxxGWB1Gjj!WV!B(2,0,2)1x6!2Gjj!B(0,0,2)1x6!000000168xxxxxxGWH1Gjj!WV!H(0,1,2)1x6!000000174xxxxxxGBL1Gjj!Bl!L(1,0,2)1x6!000000176xxxxxxGBW1Gjj!BV!W(2,0,2)1x6!2Gjj!W(0,0,2)1x6!000000178xxxxxxGBH1Gjj!BV!H(0,1,2)1x6!000000184xxxxxxGHL1Gjjl!L(1,0,2)1x6!000000186xxxxxxGHW1Gjj!W(0,0,2)1x6!000000187xxxxxxGHB1Gjj!B(0,0,2)1x6! 123

PAGE 124

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000000246xxxxxxQLW1Qj!W(0,0,1)1x6!000000247xxxxxxQLB1Qj!B(0,0,1)1x6!000000248xxxxxxQLH1Qjl!Ll!H(2,0,1)1x6!2Qj!H(0,0,1)1x6!000000254xxxxxxQEL1Qjl!Ell!L(3,0,1)1x6!2Qjl!L(1,0,1)1x6!000000256xxxxxxQEW1Qj!W(0,0,1)1x6!000000257xxxxxxQEB1Qj!B(0,0,1)1x6!000000258xxxxxxQEH1Qjl!El!H(2,0,1)1x6!2Qj!H(0,0,1)1x6!000000264xxxxxxQWL1Qj!Wl!L(1,0,1)1x6!000000267xxxxxxQWB1Qj!WV!B(2,0,1)1x6!2Qj!B(0,0,1)1x6!000000268xxxxxxQWH1Qj!WV!H(0,1,1)1x6!000000274xxxxxxQBL1Qj!Bl!L(1,0,1)1x6!000000276xxxxxxQBW1Qj!BV!W(2,0,1)1x6!2Qj!W(0,0,1)1x6!000000278xxxxxxQBH1Qj!BV!H(0,1,1)1x6!000000284xxxxxxQHL1Qjl!L(1,0,1)1x6!000000286xxxxxxQHW1Qj!W(0,0,1)1x6!000000287xxxxxxQHB1Qj!B(0,0,1)1x6!000000346xxxxxxULW1Ujl!Ll!W(2,0,1)2x6!2Uj!W(0,0,1)2x6!000000347xxxxxxULB1Uj!B(0,0,1)2x6!000000348xxxxxxULH1Ujl!Ll!H(2,0,1)2x6!2Uj!H(0,0,1)2x6!000000354xxxxxxUEL1Ujl!Ell!L(3,0,1)2x6!2Ujl!L(1,0,1)2x6!000000356xxxxxxUEW1Ujl!El!W(2,0,1)2x6!2Uj!W(0,0,1)2x6!000000357xxxxxxUEB1Uj!B(0,0,1)2x6!000000358xxxxxxUEH1Ujl!El!H(2,0,1)2x6!2Uj!H(0,0,1)2x6! 124

PAGE 125

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000000364xxxxxxUWL1Ujl!L(1,0,1)2x6!000000367xxxxxxUWB1Uj!B(0,0,1)2x6!000000368xxxxxxUWH1Uj!WV!H(0,1,1)2x6!2Uj!H(0,0,1)2x6!000000374xxxxxxUBL1Uj!Bl!L(1,0,1)2x6!000000376xxxxxxUBW1Uj!BV!W(2,0,1)2x6!000000378xxxxxxUBH1Uj!BV!H(0,1,1)2x6!000000384xxxxxxUHL1Ujl!L(1,0,1)2x6!000000386xxxxxxUHW1Uj!HV!W(0,1,1)2x6!2Uj!W(0,0,1)2x6!000000387xxxxxxUHB1Uj!B(0,0,1)2x6!000001456xxxxxGLEW1Gjj!W(0,0,2)1x5!000001457xxxxxGLEB1Gjj!B(0,0,2)1x5!000001458xxxxxGLEH1Gjjl!Lll!El!H(4,0,2)1x5!2Gjjl!El!H(2,0,2)1x5!3Gjj!H(0,0,2)1x5!000001467xxxxxGLWB1Gjj!WV!B(2,0,2)1x5!2Gjj!B(0,0,2)1x5!000001468xxxxxGLWH1Gjj!WV!H(0,1,2)1x5!000001476xxxxxGLBW1Gjj!BV!W(2,0,2)1x5!2Gjj!W(0,0,2)1x5!000001478xxxxxGLBH1Gjj!BV!H(0,1,2)1x5!000001486xxxxxGLHW1Gjj!W(0,0,2)1x5!000001487xxxxxGLHB1Gjj!B(0,0,2)1x5!000001546xxxxxGELW1Gjj!W(0,0,2)1x5!000001547xxxxxGELB1Gjj!B(0,0,2)1x5!000001548xxxxxGELH1Gjjl!Ell!Ll!H(4,0,2)1x5!2Gjjl!Ll!H(2,0,2)1x5!3Gjj!H(0,0,2)1x5!000001564xxxxxGEWL1Gjj!Wl!L(1,0,2)1x5!000001567xxxxxGEWB1Gjj!WV!B(2,0,2)1x5!2Gjj!B(0,0,2)1x5!000001568xxxxxGEWH1Gjj!WV!H(0,1,2)1x5! 125

PAGE 126

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000001574xxxxxGEBL1Gjj!Bl!L(1,0,2)1x5!000001576xxxxxGEBW1Gjj!BV!W(2,0,2)1x5!2Gjj!W(0,0,2)1x5!000001578xxxxxGEBH1Gjj!BV!H(0,1,2)1x5!000001584xxxxxGEHL1Gjjl!Ell!L(3,0,2)1x5!2Gjj!Hl!L(1,0,2)1x5!000001586xxxxxGEHW1Gjj!W(0,0,2)1x5!000001587xxxxxGEHB1Gjj!B(0,0,2)1x5!000001647xxxxxGWLB1Gjj!Wl!Ll!B(2,0,2)1x5!2Gjj!B(0,0,2)1x5!000001648xxxxxGWLH1Gjj!Wl!Ll!H(2,0,2)1x5!2Gjj!WV!H(0,1,2)1x5!000001654xxxxxGWEL1Gjj!Wl!L(1,0,2)1x5!000001657xxxxxGWEB1Gjj!WV!B(2,0,2)1x5!2Gjj!B(0,0,2)1x5!000001658xxxxxGWEH1Gjj!WV!H(0,1,2)1x5!000001674xxxxxGWBL1Gjj!Bl!L(1,0,2)1x5!000001678xxxxxGWBH1Gjj!BV!H(0,1,2)1x5!000001684xxxxxGWHL1Gjj!WV!Hl!L(1,1,2)1x5!2Gjj!Wl!L(1,0,2)1x5!000001687xxxxxGWHB1Gjj!WV!HV!B(0,2,2)1x5!2Gjj!B(0,0,2)1x5!000001746xxxxxGBLW1Gjj!Bl!Ll!W(2,0,2)1x5!2Gjj!W(0,0,2)1x5!000001748xxxxxGBLH1Gjj!Bl!Ll!H(2,0,2)1x5!2Gjj!BV!H(0,1,2)1x5!000001754xxxxxGBEL1Gjj!Bl!Ell!L(3,0,2)1x5!2Gjj!Bl!L(1,0,2)1x5!000001756xxxxxGBEW1Gjj!Bl!El!W(2,0,2)1x5!2Gjj!W(0,0,2)1x5!000001758xxxxxGBEH1Gjj!Bl!El!H(2,0,2)1x5!2Gjj!BV!H(0,1,2)1x5!000001764xxxxxGBWL1Gjj!Wl!L(1,0,2)1x5! 126

PAGE 127

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000001768xxxxxGBWH1Gjj!WV!H(0,1,2)1x5!000001784xxxxxGBHL1Gjj!BV!Hl!L(1,1,2)1x5!2Gjj!Bl!L(1,0,2)1x5!000001786xxxxxGBHW1Gjj!BV!HV!W(0,2,2)1x5!2Gjj!W(0,0,2)1x5!000001846xxxxxGHLW1Gjj!W(0,0,2)1x5!000001847xxxxxGHLB1Gjj!B(0,0,2)1x5!000001854xxxxxGHEL1Gjjl!Ell!L(3,0,2)1x5!2Gjjl!L(1,0,2)1x5!000001856xxxxxGHEW1Gjj!W(0,0,2)1x5!000001857xxxxxGHEB1Gjj!B(0,0,2)1x5!000001864xxxxxGHWL1Gjj!Wl!L(1,0,2)1x5!000001867xxxxxGHWB1Gjj!WV!B(2,0,2)1x5!2Gjj!B(0,0,2)1x5!000001874xxxxxGHBL1Gjj!Bl!L(1,0,2)1x5!000001876xxxxxGHBW1Gjj!BV!W(2,0,2)1x5!2Gjj!W(0,0,2)1x5!000002456xxxxxQLEW1Qj!W(0,0,1)1x5!000002457xxxxxQLEB1Qj!B(0,0,1)1x5!000002458xxxxxQLEH1Qjl!Lll!El!H(4,0,1)1x5!2Qjl!El!H(2,0,1)1x5!3Qj!H(0,0,1)1x5!000002467xxxxxQLWB1Qj!WV!B(2,0,1)1x5!2Qj!B(0,0,1)1x5!000002468xxxxxQLWH1Qj!WV!H(0,1,1)1x5!000002476xxxxxQLBW1Qj!BV!W(2,0,1)1x5!2Qj!W(0,0,1)1x5!000002478xxxxxQLBH1Qj!BV!H(0,1,1)1x5!000002486xxxxxQLHW1Qj!W(0,0,1)1x5!000002487xxxxxQLHB1Qj!B(0,0,1)1x5!000002546xxxxxQELW1Qj!W(0,0,1)1x5!000002547xxxxxQELB1Qj!B(0,0,1)1x5!000002548xxxxxQELH1Qjl!Ell!Ll!H(4,0,1)1x5! 127

PAGE 128

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qjl!Ll!H(2,0,1)1x5!3Qj!H(0,0,1)1x5!000002564xxxxxQEWL1Qj!Wl!L(1,0,1)1x5!000002567xxxxxQEWB1Qj!WV!B(2,0,1)1x5!2Qj!B(0,0,1)1x5!000002568xxxxxQEWH1Qj!WV!H(0,1,1)1x5!000002574xxxxxQEBL1Qj!Bl!L(1,0,1)1x5!000002576xxxxxQEBW1Qj!BV!W(2,0,1)1x5!2Qj!W(0,0,1)1x5!000002578xxxxxQEBH1Qj!BV!H(0,1,1)1x5!000002584xxxxxQEHL1Qjl!Ell!L(3,0,1)1x5!2Qj!Hl!L(1,0,1)1x5!000002586xxxxxQEHW1Qj!W(0,0,1)1x5!000002587xxxxxQEHB1Qj!B(0,0,1)1x5!000002647xxxxxQWLB1Qj!Wl!Ll!B(2,0,1)1x5!2Qj!B(0,0,1)1x5!000002648xxxxxQWLH1Qj!Wl!Ll!H(2,0,1)1x5!2Qj!WV!H(0,1,1)1x5!000002654xxxxxQWEL1Qj!Wl!L(1,0,1)1x5!000002657xxxxxQWEB1Qj!WV!B(2,0,1)1x5!2Qj!B(0,0,1)1x5!000002658xxxxxQWEH1Qj!WV!H(0,1,1)1x5!000002674xxxxxQWBL1Qj!Bl!L(1,0,1)1x5!000002678xxxxxQWBH1Qj!BV!H(0,1,1)1x5!000002684xxxxxQWHL1Qj!WV!Hl!L(1,1,1)1x5!2Qj!Wl!L(1,0,1)1x5!000002687xxxxxQWHB1Qj!WV!HV!B(0,2,1)1x5!2Qj!B(0,0,1)1x5!000002746xxxxxQBLW1Qj!Bl!Ll!W(2,0,1)1x5!2Qj!W(0,0,1)1x5!000002748xxxxxQBLH1Qj!Bl!Ll!H(2,0,1)1x5!2Qj!BV!H(0,1,1)1x5!000002754xxxxxQBEL1Qj!Bl!Ell!L(3,0,1)1x5! 128

PAGE 129

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qj!Bl!L(1,0,1)1x5!000002756xxxxxQBEW1Qj!Bl!El!W(2,0,1)1x5!2Qj!W(0,0,1)1x5!000002758xxxxxQBEH1Qj!Bl!El!H(2,0,1)1x5!2Qj!BV!H(0,1,1)1x5!000002764xxxxxQBWL1Qj!Wl!L(1,0,1)1x5!000002768xxxxxQBWH1Qj!WV!H(0,1,1)1x5!000002784xxxxxQBHL1Qj!BV!Hl!L(1,1,1)1x5!2Qj!Bl!L(1,0,1)1x5!000002786xxxxxQBHW1Qj!BV!HV!W(0,2,1)1x5!2Qj!W(0,0,1)1x5!000002846xxxxxQHLW1Qj!W(0,0,1)1x5!000002847xxxxxQHLB1Qj!B(0,0,1)1x5!000002854xxxxxQHEL1Qjl!Ell!L(3,0,1)1x5!2Qjl!L(1,0,1)1x5!000002856xxxxxQHEW1Qj!W(0,0,1)1x5!000002857xxxxxQHEB1Qj!B(0,0,1)1x5!000002864xxxxxQHWL1Qj!Wl!L(1,0,1)1x5!000002867xxxxxQHWB1Qj!WV!B(2,0,1)1x5!2Qj!B(0,0,1)1x5!000002874xxxxxQHBL1Qj!Bl!L(1,0,1)1x5!000002876xxxxxQHBW1Qj!BV!W(2,0,1)1x5!2Qj!W(0,0,1)1x5!000003456xxxxxULEW1Ujl!El!W(2,0,1)2x5!2Uj!W(0,0,1)2x5!000003457xxxxxULEB1Uj!B(0,0,1)2x5!000003458xxxxxULEH1Ujl!Lll!El!H(4,0,1)2x5!2Ujl!El!H(2,0,1)2x5!3Uj!H(0,0,1)2x5!000003467xxxxxULWB1Uj!B(0,0,1)2x5!000003468xxxxxULWH1Ujl!Ll!WV!H(2,1,1)2x5!2Uj!WV!H(0,1,1)2x5!3Uj!H(0,0,1)2x5! 129

PAGE 130

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000003476xxxxxULBW1Uj!BV!W(2,0,1)2x5!000003478xxxxxULBH1Uj!BV!H(0,1,1)2x5!000003486xxxxxULHW1Ujl!Ll!W(2,0,1)2x5!2Uj!HV!W(0,1,1)2x5!3Uj!W(0,0,1)2x5!000003487xxxxxULHB1Uj!B(0,0,1)2x5!000003546xxxxxUELW1Ujl!Ell!Ll!W(4,0,1)2x5!2Ujl!Ll!W(2,0,1)2x5!3Uj!W(0,0,1)2x5!000003547xxxxxUELB1Uj!B(0,0,1)2x5!000003548xxxxxUELH1Ujl!Ell!Ll!H(4,0,1)2x5!2Ujl!Ll!H(2,0,1)2x5!3Uj!H(0,0,1)2x5!000003564xxxxxUEWL1Ujl!Ell!L(3,0,1)2x5!2Uj!Wl!L(1,0,1)2x5!000003567xxxxxUEWB1Uj!B(0,0,1)2x5!000003568xxxxxUEWH1Ujl!El!WV!H(2,1,1)2x5!2Ujl!El!H(2,0,1)2x5!3Uj!WV!H(0,1,1)2x5!4Uj!H(0,0,1)2x5!000003574xxxxxUEBL1Uj!Bl!L(1,0,1)2x5!000003576xxxxxUEBW1Uj!BV!W(2,0,1)2x5!000003578xxxxxUEBH1Uj!BV!H(0,1,1)2x5!000003584xxxxxUEHL1Ujl!Ell!L(3,0,1)2x5!2Uj!Hl!L(1,0,1)2x5!000003586xxxxxUEHW1Ujl!El!HV!W(2,1,1)2x5!2Ujl!El!W(2,0,1)2x5!3Uj!HV!W(0,1,1)2x5!4Uj!W(0,0,1)2x5!000003587xxxxxUEHB1Uj!B(0,0,1)2x5!000003647xxxxxUWLB1Uj!B(0,0,1)2x5!000003648xxxxxUWLH1Ujl!Ll!H(2,0,1)2x5!2Uj!WV!H(0,1,1)2x5! 130

PAGE 131

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 3Uj!H(0,0,1)2x5!000003654xxxxxUWEL1Ujl!Ell!L(3,0,1)2x5!2Uj!Wl!L(1,0,1)2x5!000003657xxxxxUWEB1Uj!B(0,0,1)2x5!000003658xxxxxUWEH1Ujl!El!H(2,0,1)2x5!2Uj!WV!H(0,1,1)2x5!3Uj!H(0,0,1)2x5!000003674xxxxxUWBL1Uj!Bl!L(1,0,1)2x5!000003678xxxxxUWBH1Uj!BV!H(0,1,1)2x5!000003684xxxxxUWHL1Uj!WV!Hl!L(1,1,1)2x5!2Uj!Hl!L(1,0,1)2x5!000003687xxxxxUWHB1Uj!B(0,0,1)2x5!000003746xxxxxUBLW1Uj!Bl!Ll!W(2,0,1)2x5!000003748xxxxxUBLH1Uj!Bl!Ll!H(2,0,1)2x5!2Uj!BV!H(0,1,1)2x5!000003754xxxxxUBEL1Uj!Bl!Ell!L(3,0,1)2x5!2Uj!Bl!L(1,0,1)2x5!000003756xxxxxUBEW1Uj!Bl!El!W(2,0,1)2x5!000003758xxxxxUBEH1Uj!Bl!El!H(2,0,1)2x5!2Uj!BV!H(0,1,1)2x5!000003764xxxxxUBWL1Uj!Bl!L(1,0,1)2x5!000003768xxxxxUBWH1Uj!BV!H(0,1,1)2x5!000003784xxxxxUBHL1Uj!BV!Hl!L(1,1,1)2x5!2Uj!Bl!L(1,0,1)2x5!000003786xxxxxUBHW1Uj!BV!HV!W(0,2,1)2x5!000003846xxxxxUHLW1Ujl!Ll!W(2,0,1)2x5!2Uj!HV!W(0,1,1)2x5!3Uj!W(0,0,1)2x5!000003847xxxxxUHLB1Uj!B(0,0,1)2x5!000003854xxxxxUHEL1Ujl!Ell!L(3,0,1)2x5!2Ujl!L(1,0,1)2x5!000003856xxxxxUHEW1Ujl!El!W(2,0,1)2x5!2Uj!HV!W(0,1,1)2x5! 131

PAGE 132

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 3Uj!W(0,0,1)2x5!000003857xxxxxUHEB1Uj!B(0,0,1)2x5!000003864xxxxxUHWL1Uj!HV!Wl!L(1,1,1)2x5!2Uj!Wl!L(1,0,1)2x5!000003867xxxxxUHWB1Uj!B(0,0,1)2x5!000003874xxxxxUHBL1Uj!Bl!L(1,0,1)2x5!000003876xxxxxUHBW1Uj!BV!W(2,0,1)2x5!000014567xxxxGLEWB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000014568xxxxGLEWH1Gjj!WV!H(0,1,2)1x4!000014576xxxxGLEBW1Gjj!BV!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000014578xxxxGLEBH1Gjj!BV!H(0,1,2)1x4!000014586xxxxGLEHW1Gjj!W(0,0,2)1x4!000014587xxxxGLEHB1Gjj!B(0,0,2)1x4!000014657xxxxGLWEB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000014658xxxxGLWEH1Gjj!WV!H(0,1,2)1x4!000014678xxxxGLWBH1Gjj!WV!H(0,1,2)1x4!000014687xxxxGLWHB1Gjj!WV!HV!B(0,2,2)1x4!2Gjj!B(0,0,2)1x4!000014756xxxxGLBEW1Gjj!Bl!El!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000014758xxxxGLBEH1Gjj!Bl!El!H(2,0,2)1x4!2Gjj!BV!H(0,1,2)1x4!000014768xxxxGLBWH1Gjj!BV!H(0,1,2)1x4!000014786xxxxGLBHW1Gjj!BV!HV!W(0,2,2)1x4!2Gjj!W(0,0,2)1x4!000014856xxxxGLHEW1Gjj!W(0,0,2)1x4!000014857xxxxGLHEB1Gjj!B(0,0,2)1x4!000014867xxxxGLHWB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000014876xxxxGLHBW1Gjj!BV!W(2,0,2)1x4! 132

PAGE 133

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Gjj!W(0,0,2)1x4!000015467xxxxGELWB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000015468xxxxGELWH1Gjj!WV!H(0,1,2)1x4!000015476xxxxGELBW1Gjj!BV!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000015478xxxxGELBH1Gjj!BV!H(0,1,2)1x4!000015486xxxxGELHW1Gjj!W(0,0,2)1x4!000015487xxxxGELHB1Gjj!B(0,0,2)1x4!000015647xxxxGEWLB1Gjj!Wl!Ll!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000015648xxxxGEWLH1Gjj!Wl!Ll!H(2,0,2)1x4!2Gjj!WV!H(0,1,2)1x4!000015674xxxxGEWBL1Gjj!Wl!L(1,0,2)1x4!000015678xxxxGEWBH1Gjj!WV!H(0,1,2)1x4!000015684xxxxGEWHL1Gjj!WV!Hl!L(1,1,2)1x4!2Gjj!Wl!L(1,0,2)1x4!000015687xxxxGEWHB1Gjj!WV!HV!B(0,2,2)1x4!2Gjj!B(0,0,2)1x4!000015746xxxxGEBLW1Gjj!Bl!Ll!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000015748xxxxGEBLH1Gjj!Bl!Ll!H(2,0,2)1x4!2Gjj!BV!H(0,1,2)1x4!000015764xxxxGEBWL1Gjj!Bl!L(1,0,2)1x4!000015768xxxxGEBWH1Gjj!BV!H(0,1,2)1x4!000015784xxxxGEBHL1Gjj!BV!Hl!L(1,1,2)1x4!2Gjj!Bl!L(1,0,2)1x4!000015786xxxxGEBHW1Gjj!BV!HV!W(0,2,2)1x4!2Gjj!W(0,0,2)1x4!000015846xxxxGEHLW1Gjj!W(0,0,2)1x4!000015847xxxxGEHLB1Gjj!B(0,0,2)1x4!000015864xxxxGEHWL1Gjj!Wl!L(1,0,2)1x4!000015867xxxxGEHWB1Gjj!WV!B(2,0,2)1x4! 133

PAGE 134

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Gjj!B(0,0,2)1x4!000015874xxxxGEHBL1Gjj!Bl!L(1,0,2)1x4!000015876xxxxGEHBW1Gjj!BV!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000016457xxxxGWLEB1Gjj!Wl!Ll!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000016458xxxxGWLEH1Gjj!Wl!Lll!El!H(4,0,2)1x4!2Gjj!Wl!Ll!H(2,0,2)1x4!3Gjj!WV!H(0,1,2)1x4!000016478xxxxGWLBH1Gjj!Wl!Ll!BV!H(2,1,2)1x4!2Gjj!WV!H(0,1,2)1x4!000016487xxxxGWLHB1Gjj!Wl!Ll!B(2,0,2)1x4!2Gjj!WV!HV!B(0,2,2)1x4!3Gjj!B(0,0,2)1x4!000016547xxxxGWELB1Gjj!Wl!Ll!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000016548xxxxGWELH1Gjj!Wl!Ll!H(2,0,2)1x4!2Gjj!WV!H(0,1,2)1x4!000016574xxxxGWEBL1Gjj!Wl!L(1,0,2)1x4!000016578xxxxGWEBH1Gjj!WV!H(0,1,2)1x4!000016584xxxxGWEHL1Gjj!WV!Hl!L(1,1,2)1x4!2Gjj!Wl!L(1,0,2)1x4!000016587xxxxGWEHB1Gjj!WV!HV!B(0,2,2)1x4!2Gjj!B(0,0,2)1x4!000016748xxxxGWBLH1Gjj!Wl!Ll!H(2,0,2)1x4!2Gjj!WV!H(0,1,2)1x4!000016754xxxxGWBEL1Gjj!Bl!Ell!L(3,0,2)1x4!2Gjj!Wl!L(1,0,2)1x4!000016758xxxxGWBEH1Gjj!Bl!El!H(2,0,2)1x4!2Gjj!WV!H(0,1,2)1x4!000016784xxxxGWBHL1Gjj!WV!Hl!L(1,1,2)1x4!2Gjj!Wl!L(1,0,2)1x4!000016847xxxxGWHLB1Gjj!Wl!Ll!B(2,0,2)1x4! 134

PAGE 135

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Gjj!WV!HV!B(0,2,2)1x4!3Gjj!B(0,0,2)1x4!000016854xxxxGWHEL1Gjj!WV!Hl!Ell!L(3,1,2)1x4!2Gjj!WV!Hl!L(1,1,2)1x4!3Gjj!Wl!L(1,0,2)1x4!000016857xxxxGWHEB1Gjj!WV!HV!B(0,2,2)1x4!2Gjj!B(0,0,2)1x4!000016874xxxxGWHBL1Gjj!WV!HV!Bl!L(1,2,2)1x4!2Gjj!Wl!L(1,0,2)1x4!000017456xxxxGBLEW1Gjj!Bl!El!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000017458xxxxGBLEH1Gjj!Bl!Lll!El!H(4,0,2)1x4!2Gjj!Bl!El!H(2,0,2)1x4!3Gjj!BV!H(0,1,2)1x4!000017468xxxxGBLWH1Gjj!Bl!Ll!WV!H(2,1,2)1x4!2Gjj!BV!H(0,1,2)1x4!000017486xxxxGBLHW1Gjj!Bl!Ll!W(2,0,2)1x4!2Gjj!BV!HV!W(0,2,2)1x4!3Gjj!W(0,0,2)1x4!000017546xxxxGBELW1Gjj!Bl!Ell!Ll!W(4,0,2)1x4!2Gjj!Bl!Ll!W(2,0,2)1x4!3Gjj!W(0,0,2)1x4!000017548xxxxGBELH1Gjj!Bl!Ell!Ll!H(4,0,2)1x4!2Gjj!Bl!Ll!H(2,0,2)1x4!3Gjj!BV!H(0,1,2)1x4!000017564xxxxGBEWL1Gjj!Bl!Ell!L(3,0,2)1x4!2Gjj!Bl!L(1,0,2)1x4!000017568xxxxGBEWH1Gjj!Bl!El!WV!H(2,1,2)1x4!2Gjj!Bl!El!H(2,0,2)1x4!3Gjj!BV!H(0,1,2)1x4!000017584xxxxGBEHL1Gjj!Bl!Ell!L(3,0,2)1x4!2Gjj!BV!Hl!L(1,1,2)1x4!3Gjj!Bl!L(1,0,2)1x4! 135

PAGE 136

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000017586xxxxGBEHW1Gjj!Bl!El!HV!W(2,1,2)1x4!2Gjj!Bl!El!W(2,0,2)1x4!3Gjj!BV!HV!W(0,2,2)1x4!4Gjj!W(0,0,2)1x4!000017648xxxxGBWLH1Gjj!Bl!Ll!H(2,0,2)1x4!2Gjj!BV!H(0,1,2)1x4!000017654xxxxGBWEL1Gjj!Bl!Ell!L(3,0,2)1x4!2Gjj!Wl!L(1,0,2)1x4!000017658xxxxGBWEH1Gjj!Bl!El!H(2,0,2)1x4!2Gjj!WV!H(0,1,2)1x4!000017684xxxxGBWHL1Gjj!BV!Hl!L(1,1,2)1x4!2Gjj!Bl!L(1,0,2)1x4!000017846xxxxGBHLW1Gjj!Bl!Ll!W(2,0,2)1x4!2Gjj!BV!HV!W(0,2,2)1x4!3Gjj!W(0,0,2)1x4!000017854xxxxGBHEL1Gjj!BV!Hl!Ell!L(3,1,2)1x4!2Gjj!Bl!Ell!L(3,0,2)1x4!3Gjj!BV!Hl!L(1,1,2)1x4!4Gjj!Bl!L(1,0,2)1x4!000017856xxxxGBHEW1Gjj!Bl!El!W(2,0,2)1x4!2Gjj!BV!HV!W(0,2,2)1x4!3Gjj!W(0,0,2)1x4!000017864xxxxGBHWL1Gjj!BV!HV!Wl!L(1,2,2)1x4!2Gjj!Bl!L(1,0,2)1x4!000018456xxxxGHLEW1Gjj!W(0,0,2)1x4!000018457xxxxGHLEB1Gjj!B(0,0,2)1x4!000018467xxxxGHLWB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000018476xxxxGHLBW1Gjj!BV!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000018546xxxxGHELW1Gjj!W(0,0,2)1x4!000018547xxxxGHELB1Gjj!B(0,0,2)1x4!000018564xxxxGHEWL1Gjj!Wl!L(1,0,2)1x4! 136

PAGE 137

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000018567xxxxGHEWB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000018574xxxxGHEBL1Gjj!Bl!L(1,0,2)1x4!000018576xxxxGHEBW1Gjj!BV!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000018647xxxxGHWLB1Gjj!Wl!Ll!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000018654xxxxGHWEL1Gjj!Wl!L(1,0,2)1x4!000018657xxxxGHWEB1Gjj!WV!B(2,0,2)1x4!2Gjj!B(0,0,2)1x4!000018674xxxxGHWBL1Gjj!Wl!L(1,0,2)1x4!000018746xxxxGHBLW1Gjj!Bl!Ll!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000018754xxxxGHBEL1Gjj!Bl!Ell!L(3,0,2)1x4!2Gjj!Bl!L(1,0,2)1x4!000018756xxxxGHBEW1Gjj!Bl!El!W(2,0,2)1x4!2Gjj!W(0,0,2)1x4!000018764xxxxGHBWL1Gjj!Bl!L(1,0,2)1x4!000024567xxxxQLEWB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000024568xxxxQLEWH1Qj!WV!H(0,1,1)1x4!000024576xxxxQLEBW1Qj!BV!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000024578xxxxQLEBH1Qj!BV!H(0,1,1)1x4!000024586xxxxQLEHW1Qj!W(0,0,1)1x4!000024587xxxxQLEHB1Qj!B(0,0,1)1x4!000024657xxxxQLWEB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000024658xxxxQLWEH1Qj!WV!H(0,1,1)1x4!000024678xxxxQLWBH1Qj!WV!H(0,1,1)1x4!000024687xxxxQLWHB1Qj!WV!HV!B(0,2,1)1x4!2Qj!B(0,0,1)1x4!000024756xxxxQLBEW1Qj!Bl!El!W(2,0,1)1x4! 137

PAGE 138

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qj!W(0,0,1)1x4!000024758xxxxQLBEH1Qj!Bl!El!H(2,0,1)1x4!2Qj!BV!H(0,1,1)1x4!000024768xxxxQLBWH1Qj!BV!H(0,1,1)1x4!000024786xxxxQLBHW1Qj!BV!HV!W(0,2,1)1x4!2Qj!W(0,0,1)1x4!000024856xxxxQLHEW1Qj!W(0,0,1)1x4!000024857xxxxQLHEB1Qj!B(0,0,1)1x4!000024867xxxxQLHWB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000024876xxxxQLHBW1Qj!BV!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000025467xxxxQELWB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000025468xxxxQELWH1Qj!WV!H(0,1,1)1x4!000025476xxxxQELBW1Qj!BV!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000025478xxxxQELBH1Qj!BV!H(0,1,1)1x4!000025486xxxxQELHW1Qj!W(0,0,1)1x4!000025487xxxxQELHB1Qj!B(0,0,1)1x4!000025647xxxxQEWLB1Qj!Wl!Ll!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000025648xxxxQEWLH1Qj!Wl!Ll!H(2,0,1)1x4!2Qj!WV!H(0,1,1)1x4!000025674xxxxQEWBL1Qj!Wl!L(1,0,1)1x4!000025678xxxxQEWBH1Qj!WV!H(0,1,1)1x4!000025684xxxxQEWHL1Qj!WV!Hl!L(1,1,1)1x4!2Qj!Wl!L(1,0,1)1x4!000025687xxxxQEWHB1Qj!WV!HV!B(0,2,1)1x4!2Qj!B(0,0,1)1x4!000025746xxxxQEBLW1Qj!Bl!Ll!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000025748xxxxQEBLH1Qj!Bl!Ll!H(2,0,1)1x4! 138

PAGE 139

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qj!BV!H(0,1,1)1x4!000025764xxxxQEBWL1Qj!Bl!L(1,0,1)1x4!000025768xxxxQEBWH1Qj!BV!H(0,1,1)1x4!000025784xxxxQEBHL1Qj!BV!Hl!L(1,1,1)1x4!2Qj!Bl!L(1,0,1)1x4!000025786xxxxQEBHW1Qj!BV!HV!W(0,2,1)1x4!2Qj!W(0,0,1)1x4!000025846xxxxQEHLW1Qj!W(0,0,1)1x4!000025847xxxxQEHLB1Qj!B(0,0,1)1x4!000025864xxxxQEHWL1Qj!Wl!L(1,0,1)1x4!000025867xxxxQEHWB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000025874xxxxQEHBL1Qj!Bl!L(1,0,1)1x4!000025876xxxxQEHBW1Qj!BV!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000026457xxxxQWLEB1Qj!Wl!Ll!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000026458xxxxQWLEH1Qj!Wl!Lll!El!H(4,0,1)1x4!2Qj!Wl!Ll!H(2,0,1)1x4!3Qj!WV!H(0,1,1)1x4!000026478xxxxQWLBH1Qj!Wl!Ll!BV!H(2,1,1)1x4!2Qj!WV!H(0,1,1)1x4!000026487xxxxQWLHB1Qj!Wl!Ll!B(2,0,1)1x4!2Qj!WV!HV!B(0,2,1)1x4!3Qj!B(0,0,1)1x4!000026547xxxxQWELB1Qj!Wl!Ll!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000026548xxxxQWELH1Qj!Wl!Ll!H(2,0,1)1x4!2Qj!WV!H(0,1,1)1x4!000026574xxxxQWEBL1Qj!Wl!L(1,0,1)1x4!000026578xxxxQWEBH1Qj!WV!H(0,1,1)1x4!000026584xxxxQWEHL1Qj!WV!Hl!L(1,1,1)1x4!2Qj!Wl!L(1,0,1)1x4! 139

PAGE 140

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000026587xxxxQWEHB1Qj!WV!HV!B(0,2,1)1x4!2Qj!B(0,0,1)1x4!000026748xxxxQWBLH1Qj!Wl!Ll!H(2,0,1)1x4!2Qj!WV!H(0,1,1)1x4!000026754xxxxQWBEL1Qj!Bl!Ell!L(3,0,1)1x4!2Qj!Wl!L(1,0,1)1x4!000026758xxxxQWBEH1Qj!Bl!El!H(2,0,1)1x4!2Qj!WV!H(0,1,1)1x4!000026784xxxxQWBHL1Qj!WV!Hl!L(1,1,1)1x4!2Qj!Wl!L(1,0,1)1x4!000026847xxxxQWHLB1Qj!Wl!Ll!B(2,0,1)1x4!2Qj!WV!HV!B(0,2,1)1x4!3Qj!B(0,0,1)1x4!000026854xxxxQWHEL1Qj!WV!Hl!Ell!L(3,1,1)1x4!2Qj!WV!Hl!L(1,1,1)1x4!3Qj!Wl!L(1,0,1)1x4!000026857xxxxQWHEB1Qj!WV!HV!B(0,2,1)1x4!2Qj!B(0,0,1)1x4!000026874xxxxQWHBL1Qj!WV!HV!Bl!L(1,2,1)1x4!2Qj!Wl!L(1,0,1)1x4!000027456xxxxQBLEW1Qj!Bl!El!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000027458xxxxQBLEH1Qj!Bl!Lll!El!H(4,0,1)1x4!2Qj!Bl!El!H(2,0,1)1x4!3Qj!BV!H(0,1,1)1x4!000027468xxxxQBLWH1Qj!Bl!Ll!WV!H(2,1,1)1x4!2Qj!BV!H(0,1,1)1x4!000027486xxxxQBLHW1Qj!Bl!Ll!W(2,0,1)1x4!2Qj!BV!HV!W(0,2,1)1x4!3Qj!W(0,0,1)1x4!000027546xxxxQBELW1Qj!Bl!Ell!Ll!W(4,0,1)1x4!2Qj!Bl!Ll!W(2,0,1)1x4!3Qj!W(0,0,1)1x4! 140

PAGE 141

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000027548xxxxQBELH1Qj!Bl!Ell!Ll!H(4,0,1)1x4!2Qj!Bl!Ll!H(2,0,1)1x4!3Qj!BV!H(0,1,1)1x4!000027564xxxxQBEWL1Qj!Bl!Ell!L(3,0,1)1x4!2Qj!Bl!L(1,0,1)1x4!000027568xxxxQBEWH1Qj!Bl!El!WV!H(2,1,1)1x4!2Qj!Bl!El!H(2,0,1)1x4!3Qj!BV!H(0,1,1)1x4!000027584xxxxQBEHL1Qj!Bl!Ell!L(3,0,1)1x4!2Qj!BV!Hl!L(1,1,1)1x4!3Qj!Bl!L(1,0,1)1x4!000027586xxxxQBEHW1Qj!Bl!El!HV!W(2,1,1)1x4!2Qj!Bl!El!W(2,0,1)1x4!3Qj!BV!HV!W(0,2,1)1x4!4Qj!W(0,0,1)1x4!000027648xxxxQBWLH1Qj!Bl!Ll!H(2,0,1)1x4!2Qj!BV!H(0,1,1)1x4!000027654xxxxQBWEL1Qj!Bl!Ell!L(3,0,1)1x4!2Qj!Wl!L(1,0,1)1x4!000027658xxxxQBWEH1Qj!Bl!El!H(2,0,1)1x4!2Qj!WV!H(0,1,1)1x4!000027684xxxxQBWHL1Qj!BV!Hl!L(1,1,1)1x4!2Qj!Bl!L(1,0,1)1x4!000027846xxxxQBHLW1Qj!Bl!Ll!W(2,0,1)1x4!2Qj!BV!HV!W(0,2,1)1x4!3Qj!W(0,0,1)1x4!000027854xxxxQBHEL1Qj!BV!Hl!Ell!L(3,1,1)1x4!2Qj!Bl!Ell!L(3,0,1)1x4!3Qj!BV!Hl!L(1,1,1)1x4!4Qj!Bl!L(1,0,1)1x4!000027856xxxxQBHEW1Qj!Bl!El!W(2,0,1)1x4!2Qj!BV!HV!W(0,2,1)1x4!3Qj!W(0,0,1)1x4! 141

PAGE 142

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000027864xxxxQBHWL1Qj!BV!HV!Wl!L(1,2,1)1x4!2Qj!Bl!L(1,0,1)1x4!000028456xxxxQHLEW1Qj!W(0,0,1)1x4!000028457xxxxQHLEB1Qj!B(0,0,1)1x4!000028467xxxxQHLWB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000028476xxxxQHLBW1Qj!BV!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000028546xxxxQHELW1Qj!W(0,0,1)1x4!000028547xxxxQHELB1Qj!B(0,0,1)1x4!000028564xxxxQHEWL1Qj!Wl!L(1,0,1)1x4!000028567xxxxQHEWB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000028574xxxxQHEBL1Qj!Bl!L(1,0,1)1x4!000028576xxxxQHEBW1Qj!BV!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000028647xxxxQHWLB1Qj!Wl!Ll!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000028654xxxxQHWEL1Qj!Wl!L(1,0,1)1x4!000028657xxxxQHWEB1Qj!WV!B(2,0,1)1x4!2Qj!B(0,0,1)1x4!000028674xxxxQHWBL1Qj!Wl!L(1,0,1)1x4!000028746xxxxQHBLW1Qj!Bl!Ll!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000028754xxxxQHBEL1Qj!Bl!Ell!L(3,0,1)1x4!2Qj!Bl!L(1,0,1)1x4!000028756xxxxQHBEW1Qj!Bl!El!W(2,0,1)1x4!2Qj!W(0,0,1)1x4!000028764xxxxQHBWL1Qj!Bl!L(1,0,1)1x4!000034567xxxxULEWB1Uj!B(0,0,1)2x4!000034568xxxxULEWH1Ujl!El!WV!H(2,1,1)2x4!2Ujl!El!H(2,0,1)2x4!3Uj!WV!H(0,1,1)2x4! 142

PAGE 143

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 4Uj!H(0,0,1)2x4!000034576xxxxULEBW1Uj!BV!W(2,0,1)2x4!000034578xxxxULEBH1Uj!BV!H(0,1,1)2x4!000034586xxxxULEHW1Ujl!El!HV!W(2,1,1)2x4!2Ujl!Ll!W(2,0,1)2x4!3Uj!HV!W(0,1,1)2x4!4Uj!W(0,0,1)2x4!000034587xxxxULEHB1Uj!B(0,0,1)2x4!000034657xxxxULWEB1Uj!B(0,0,1)2x4!000034658xxxxULWEH1Ujl!Ll!WV!H(2,1,1)2x4!2Ujl!El!H(2,0,1)2x4!3Uj!WV!H(0,1,1)2x4!4Uj!H(0,0,1)2x4!000034678xxxxULWBH1Uj!BV!H(0,1,1)2x4!000034687xxxxULWHB1Uj!B(0,0,1)2x4!000034756xxxxULBEW1Uj!Bl!El!W(2,0,1)2x4!000034758xxxxULBEH1Uj!Bl!El!H(2,0,1)2x4!2Uj!BV!H(0,1,1)2x4!000034768xxxxULBWH1Uj!BV!H(0,1,1)2x4!000034786xxxxULBHW1Uj!BV!HV!W(0,2,1)2x4!000034856xxxxULHEW1Ujl!El!W(2,0,1)2x4!2Uj!HV!W(0,1,1)2x4!3Uj!W(0,0,1)2x4!000034857xxxxULHEB1Uj!B(0,0,1)2x4!000034867xxxxULHWB1Uj!B(0,0,1)2x4!000034876xxxxULHBW1Uj!BV!W(2,0,1)2x4!000035467xxxxUELWB1Uj!B(0,0,1)2x4!000035468xxxxUELWH1Ujl!Ell!Ll!WV!H(4,1,1)2x4!2Ujl!Ll!WV!H(2,1,1)2x4!3Ujl!El!H(2,0,1)2x4!4Uj!WV!H(0,1,1)2x4!5Uj!H(0,0,1)2x4!000035476xxxxUELBW1Uj!BV!W(2,0,1)2x4! 143

PAGE 144

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000035478xxxxUELBH1Uj!BV!H(0,1,1)2x4!000035486xxxxUELHW1Ujl!Ell!Ll!W(4,0,1)2x4!2Ujl!El!HV!W(2,1,1)2x4!3Ujl!Ll!W(2,0,1)2x4!4Uj!HV!W(0,1,1)2x4!5Uj!W(0,0,1)2x4!000035487xxxxUELHB1Uj!B(0,0,1)2x4!000035647xxxxUEWLB1Uj!B(0,0,1)2x4!000035648xxxxUEWLH1Ujl!Ell!Ll!H(4,0,1)2x4!2Ujl!El!WV!H(2,1,1)2x4!3Uj!Wl!Ll!H(2,0,1)2x4!4Uj!WV!H(0,1,1)2x4!5Uj!H(0,0,1)2x4!000035674xxxxUEWBL1Uj!Bl!L(1,0,1)2x4!000035678xxxxUEWBH1Uj!BV!H(0,1,1)2x4!000035684xxxxUEWHL1Ujl!El!WV!Hl!L(3,1,1)2x4!2Ujl!El!Hl!L(3,0,1)2x4!3Uj!WV!Hl!L(1,1,1)2x4!4Uj!Hl!L(1,0,1)2x4!000035687xxxxUEWHB1Uj!B(0,0,1)2x4!000035746xxxxUEBLW1Uj!Bl!Ll!W(2,0,1)2x4!000035748xxxxUEBLH1Uj!Bl!Ll!H(2,0,1)2x4!2Uj!BV!H(0,1,1)2x4!000035764xxxxUEBWL1Uj!Bl!L(1,0,1)2x4!000035768xxxxUEBWH1Uj!BV!H(0,1,1)2x4!000035784xxxxUEBHL1Uj!BV!Hl!L(1,1,1)2x4!2Uj!Bl!L(1,0,1)2x4!000035786xxxxUEBHW1Uj!BV!HV!W(0,2,1)2x4!000035846xxxxUEHLW1Ujl!Ell!Ll!W(4,0,1)2x4!2Ujl!El!HV!W(2,1,1)2x4!3Ujl!Ll!W(2,0,1)2x4!4Uj!HV!W(0,1,1)2x4!5Uj!W(0,0,1)2x4! 144

PAGE 145

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000035847xxxxUEHLB1Uj!B(0,0,1)2x4!000035864xxxxUEHWL1Ujl!El!HV!Wl!L(3,1,1)2x4!2Ujl!El!Wl!L(3,0,1)2x4!3Uj!HV!Wl!L(1,1,1)2x4!4Uj!Wl!L(1,0,1)2x4!000035867xxxxUEHWB1Uj!B(0,0,1)2x4!000035874xxxxUEHBL1Uj!Bl!L(1,0,1)2x4!000035876xxxxUEHBW1Uj!BV!W(2,0,1)2x4!000036457xxxxUWLEB1Uj!B(0,0,1)2x4!000036458xxxxUWLEH1Uj!Wl!Lll!El!H(4,0,1)2x4!2Ujl!El!H(2,0,1)2x4!3Uj!WV!H(0,1,1)2x4!4Uj!H(0,0,1)2x4!000036478xxxxUWLBH1Uj!BV!H(0,1,1)2x4!000036487xxxxUWLHB1Uj!B(0,0,1)2x4!000036547xxxxUWELB1Uj!B(0,0,1)2x4!000036548xxxxUWELH1Ujl!Ell!Ll!H(4,0,1)2x4!2Ujl!Ll!H(2,0,1)2x4!3Uj!WV!H(0,1,1)2x4!4Uj!H(0,0,1)2x4!000036574xxxxUWEBL1Uj!Bl!L(1,0,1)2x4!000036578xxxxUWEBH1Uj!BV!H(0,1,1)2x4!000036584xxxxUWEHL1Ujl!Ell!L(3,0,1)2x4!2Uj!WV!Hl!L(1,1,1)2x4!3Uj!Wl!L(1,0,1)2x4!000036587xxxxUWEHB1Uj!B(0,0,1)2x4!000036748xxxxUWBLH1Uj!Bl!Ll!H(2,0,1)2x4!2Uj!BV!H(0,1,1)2x4!000036754xxxxUWBEL1Uj!Bl!Ell!L(3,0,1)2x4!2Uj!Bl!L(1,0,1)2x4!000036758xxxxUWBEH1Uj!Bl!El!H(2,0,1)2x4!2Uj!BV!H(0,1,1)2x4!000036784xxxxUWBHL1Uj!BV!Hl!L(1,1,1)2x4! 145

PAGE 146

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Uj!Bl!L(1,0,1)2x4!000036847xxxxUWHLB1Uj!B(0,0,1)2x4!000036854xxxxUWHEL1Uj!WV!Hl!Ell!L(3,1,1)2x4!2Ujl!Ell!L(3,0,1)2x4!3Uj!WV!Hl!L(1,1,1)2x4!4Uj!Wl!L(1,0,1)2x4!000036857xxxxUWHEB1Uj!B(0,0,1)2x4!000036874xxxxUWHBL1Uj!Bl!L(1,0,1)2x4!000037456xxxxUBLEW1Uj!Bl!El!W(2,0,1)2x4!000037458xxxxUBLEH1Uj!Bl!Lll!El!H(4,0,1)2x4!2Uj!Bl!El!H(2,0,1)2x4!3Uj!BV!H(0,1,1)2x4!000037468xxxxUBLWH1Uj!Bl!Ll!WV!H(2,1,1)2x4!2Uj!BV!H(0,1,1)2x4!000037486xxxxUBLHW1Uj!Bl!Ll!W(2,0,1)2x4!2Uj!BV!HV!W(0,2,1)2x4!000037546xxxxUBELW1Uj!Bl!Ell!Ll!W(4,0,1)2x4!2Uj!Bl!Ll!W(2,0,1)2x4!000037548xxxxUBELH1Uj!Bl!Ell!Ll!H(4,0,1)2x4!2Uj!Bl!Ll!H(2,0,1)2x4!3Uj!BV!H(0,1,1)2x4!000037564xxxxUBEWL1Uj!Bl!Ell!L(3,0,1)2x4!2Uj!Bl!L(1,0,1)2x4!000037568xxxxUBEWH1Uj!Bl!El!WV!H(2,1,1)2x4!2Uj!Bl!El!H(2,0,1)2x4!3Uj!BV!H(0,1,1)2x4!000037584xxxxUBEHL1Uj!Bl!Ell!L(3,0,1)2x4!2Uj!BV!Hl!L(1,1,1)2x4!3Uj!Bl!L(1,0,1)2x4!000037586xxxxUBEHW1Uj!Bl!El!HV!W(2,1,1)2x4!2Uj!Bl!El!W(2,0,1)2x4!3Uj!BV!HV!W(0,2,1)2x4!000037648xxxxUBWLH1Uj!Bl!Ll!H(2,0,1)2x4! 146

PAGE 147

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Uj!BV!H(0,1,1)2x4!000037654xxxxUBWEL1Uj!Bl!Ell!L(3,0,1)2x4!2Uj!Bl!L(1,0,1)2x4!000037658xxxxUBWEH1Uj!Bl!El!H(2,0,1)2x4!2Uj!BV!H(0,1,1)2x4!000037684xxxxUBWHL1Uj!BV!Hl!L(1,1,1)2x4!2Uj!Bl!L(1,0,1)2x4!000037846xxxxUBHLW1Uj!Bl!Ll!W(2,0,1)2x4!2Uj!BV!HV!W(0,2,1)2x4!000037854xxxxUBHEL1Uj!BV!Hl!Ell!L(3,1,1)2x4!2Uj!Bl!Ell!L(3,0,1)2x4!3Uj!BV!Hl!L(1,1,1)2x4!4Uj!Bl!L(1,0,1)2x4!000037856xxxxUBHEW1Uj!Bl!El!W(2,0,1)2x4!2Uj!BV!HV!W(0,2,1)2x4!000037864xxxxUBHWL1Uj!BV!HV!Wl!L(1,2,1)2x4!2Uj!Bl!L(1,0,1)2x4!000038456xxxxUHLEW1Ujl!El!W(2,0,1)2x4!2Uj!HV!W(0,1,1)2x4!3Uj!W(0,0,1)2x4!000038457xxxxUHLEB1Uj!B(0,0,1)2x4!000038467xxxxUHLWB1Uj!B(0,0,1)2x4!000038476xxxxUHLBW1Uj!BV!W(2,0,1)2x4!000038546xxxxUHELW1Ujl!Ell!Ll!W(4,0,1)2x4!2Ujl!Ll!W(2,0,1)2x4!3Uj!HV!W(0,1,1)2x4!4Uj!W(0,0,1)2x4!000038547xxxxUHELB1Uj!B(0,0,1)2x4!000038564xxxxUHEWL1Ujl!Ell!L(3,0,1)2x4!2Uj!HV!Wl!L(1,1,1)2x4!3Uj!Wl!L(1,0,1)2x4!000038567xxxxUHEWB1Uj!B(0,0,1)2x4!000038574xxxxUHEBL1Uj!Bl!L(1,0,1)2x4! 147

PAGE 148

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000038576xxxxUHEBW1Uj!BV!W(2,0,1)2x4!000038647xxxxUHWLB1Uj!B(0,0,1)2x4!000038654xxxxUHWEL1Ujl!Ell!L(3,0,1)2x4!2Uj!HV!Wl!L(1,1,1)2x4!3Uj!Wl!L(1,0,1)2x4!000038657xxxxUHWEB1Uj!B(0,0,1)2x4!000038674xxxxUHWBL1Uj!Bl!L(1,0,1)2x4!000038746xxxxUHBLW1Uj!Bl!Ll!W(2,0,1)2x4!000038754xxxxUHBEL1Uj!Bl!Ell!L(3,0,1)2x4!2Uj!Bl!L(1,0,1)2x4!000038756xxxxUHBEW1Uj!Bl!El!W(2,0,1)2x4!000038764xxxxUHBWL1Uj!Bl!L(1,0,1)2x4!000145678xxxGLEWBH1Gjj!WV!H(0,1,2)1x3!000145687xxxGLEWHB1Gjj!WV!HV!B(0,2,2)1x3!2Gjj!B(0,0,2)1x3!000145768xxxGLEBWH1Gjj!BV!H(0,1,2)1x3!000145786xxxGLEBHW1Gjj!BV!HV!W(0,2,2)1x3!2Gjj!W(0,0,2)1x3!000145867xxxGLEHWB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000145876xxxGLEHBW1Gjj!BV!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000146578xxxGLWEBH1Gjj!WV!H(0,1,2)1x3!000146587xxxGLWEHB1Gjj!WV!HV!B(0,2,2)1x3!2Gjj!B(0,0,2)1x3!000146758xxxGLWBEH1Gjj!Bl!El!H(2,0,2)1x3!2Gjj!WV!H(0,1,2)1x3!000146857xxxGLWHEB1Gjj!WV!HV!B(0,2,2)1x3!2Gjj!B(0,0,2)1x3!000147568xxxGLBEWH1Gjj!Bl!El!WV!H(2,1,2)1x3!2Gjj!Bl!El!H(2,0,2)1x3!3Gjj!BV!H(0,1,2)1x3!000147586xxxGLBEHW1Gjj!Bl!El!HV!W(2,1,2)1x3! 148

PAGE 149

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Gjj!Bl!El!W(2,0,2)1x3!3Gjj!BV!HV!W(0,2,2)1x3!4Gjj!W(0,0,2)1x3!000147658xxxGLBWEH1Gjj!Bl!El!H(2,0,2)1x3!2Gjj!BV!H(0,1,2)1x3!000147856xxxGLBHEW1Gjj!Bl!El!W(2,0,2)1x3!2Gjj!BV!HV!W(0,2,2)1x3!3Gjj!W(0,0,2)1x3!000148567xxxGLHEWB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000148576xxxGLHEBW1Gjj!BV!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000148657xxxGLHWEB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000148756xxxGLHBEW1Gjj!Bl!El!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000154678xxxGELWBH1Gjj!WV!H(0,1,2)1x3!000154687xxxGELWHB1Gjj!WV!HV!B(0,2,2)1x3!2Gjj!B(0,0,2)1x3!000154768xxxGELBWH1Gjj!BV!H(0,1,2)1x3!000154786xxxGELBHW1Gjj!BV!HV!W(0,2,2)1x3!2Gjj!W(0,0,2)1x3!000154867xxxGELHWB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000154876xxxGELHBW1Gjj!BV!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000156478xxxGEWLBH1Gjj!Wl!Ll!BV!H(2,1,2)1x3!2Gjj!WV!H(0,1,2)1x3!000156487xxxGEWLHB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000156748xxxGEWBLH1Gjj!Wl!Ll!H(2,0,2)1x3!2Gjj!WV!H(0,1,2)1x3! 149

PAGE 150

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000156784xxxGEWBHL1Gjj!WV!Hl!L(1,1,2)1x3!2Gjj!Wl!L(1,0,2)1x3!000156847xxxGEWHLB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000156874xxxGEWHBL1Gjj!WV!HV!Bl!L(1,2,2)1x3!2Gjj!Wl!L(1,0,2)1x3!000157468xxxGEBLWH1Gjj!Bl!Ll!WV!H(2,1,2)1x3!2Gjj!BV!H(0,1,2)1x3!000157486xxxGEBLHW1Gjj!Bl!Ll!W(2,0,2)1x3!2Gjj!BV!HV!W(0,2,2)1x3!3Gjj!W(0,0,2)1x3!000157648xxxGEBWLH1Gjj!Bl!Ll!H(2,0,2)1x3!2Gjj!BV!H(0,1,2)1x3!000157684xxxGEBWHL1Gjj!BV!Hl!L(1,1,2)1x3!2Gjj!Bl!L(1,0,2)1x3!000157846xxxGEBHLW1Gjj!Bl!Ll!W(2,0,2)1x3!2Gjj!BV!HV!W(0,2,2)1x3!3Gjj!W(0,0,2)1x3!000157864xxxGEBHWL1Gjj!BV!HV!Wl!L(1,2,2)1x3!2Gjj!Bl!L(1,0,2)1x3!000158467xxxGEHLWB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000158476xxxGEHLBW1Gjj!BV!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000158647xxxGEHWLB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000158674xxxGEHWBL1Gjj!Wl!L(1,0,2)1x3!000158746xxxGEHBLW1Gjj!Bl!Ll!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000158764xxxGEHBWL1Gjj!Bl!L(1,0,2)1x3!000164578xxxGWLEBH1Gjj!Wl!Ll!BV!H(2,1,2)1x3!2Gjj!WV!H(0,1,2)1x3! 150

PAGE 151

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000164587xxxGWLEHB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000164758xxxGWLBEH1Gjj!Wl!Ll!Bl!El!H(4,0,2)1x3!2Gjj!Wl!Ll!BV!H(2,1,2)1x3!3Gjj!Bl!El!H(2,0,2)1x3!4Gjj!WV!H(0,1,2)1x3!000164857xxxGWLHEB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000165478xxxGWELBH1Gjj!Wl!Ll!BV!H(2,1,2)1x3!2Gjj!WV!H(0,1,2)1x3!000165487xxxGWELHB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000165748xxxGWEBLH1Gjj!Wl!Ll!H(2,0,2)1x3!2Gjj!WV!H(0,1,2)1x3!000165784xxxGWEBHL1Gjj!WV!Hl!L(1,1,2)1x3!2Gjj!Wl!L(1,0,2)1x3!000165847xxxGWEHLB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000165874xxxGWEHBL1Gjj!WV!HV!Bl!L(1,2,2)1x3!2Gjj!Wl!L(1,0,2)1x3!000167458xxxGWBLEH1Gjj!Wl!Lll!El!H(4,0,2)1x3!2Gjj!Bl!El!H(2,0,2)1x3!3Gjj!WV!H(0,1,2)1x3!000167548xxxGWBELH1Gjj!Bl!Ell!Ll!H(4,0,2)1x3!2Gjj!Wl!Ll!H(2,0,2)1x3!3Gjj!WV!H(0,1,2)1x3!000167584xxxGWBEHL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!WV!Hl!L(1,1,2)1x3!3Gjj!Wl!L(1,0,2)1x3! 151

PAGE 152

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000167854xxxGWBHEL1Gjj!WV!Hl!Ell!L(3,1,2)1x3!2Gjj!Bl!Ell!L(3,0,2)1x3!3Gjj!BV!Hl!L(1,1,2)1x3!4Gjj!Wl!L(1,0,2)1x3!000168457xxxGWHLEB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000168547xxxGWHELB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!WV!HV!B(0,2,2)1x3!3Gjj!B(0,0,2)1x3!000168574xxxGWHEBL1Gjj!WV!HV!Bl!L(1,2,2)1x3!2Gjj!Wl!L(1,0,2)1x3!000168754xxxGWHBEL1Gjj!WV!HV!Bl!Ell!L(3,2,2)1x3!2Gjj!Bl!Ell!L(3,0,2)1x3!3Gjj!WV!HV!Bl!L(1,2,2)1x3!4Gjj!Wl!L(1,0,2)1x3!000174568xxxGBLEWH1Gjj!Bl!El!WV!H(2,1,2)1x3!2Gjj!Bl!El!H(2,0,2)1x3!3Gjj!BV!H(0,1,2)1x3!000174586xxxGBLEHW1Gjj!Bl!El!HV!W(2,1,2)1x3!2Gjj!Bl!Ll!W(2,0,2)1x3!3Gjj!BV!HV!W(0,2,2)1x3!4Gjj!W(0,0,2)1x3!000174658xxxGBLWEH1Gjj!Bl!Ll!WV!H(2,1,2)1x3!2Gjj!Bl!El!H(2,0,2)1x3!3Gjj!BV!H(0,1,2)1x3!000174856xxxGBLHEW1Gjj!Bl!El!W(2,0,2)1x3!2Gjj!BV!HV!W(0,2,2)1x3!3Gjj!W(0,0,2)1x3!000175468xxxGBELWH1Gjj!Bl!Ell!Ll!WV!H(4,1,2)1x3!2Gjj!Bl!Ll!WV!H(2,1,2)1x3!3Gjj!Bl!El!H(2,0,2)1x3!4Gjj!BV!H(0,1,2)1x3! 152

PAGE 153

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000175486xxxGBELHW1Gjj!Bl!Ell!Ll!W(4,0,2)1x3!2Gjj!Bl!El!HV!W(2,1,2)1x3!3Gjj!Bl!Ll!W(2,0,2)1x3!4Gjj!BV!HV!W(0,2,2)1x3!5Gjj!W(0,0,2)1x3!000175648xxxGBEWLH1Gjj!Bl!Ell!Ll!H(4,0,2)1x3!2Gjj!Bl!El!WV!H(2,1,2)1x3!3Gjj!Bl!Ll!H(2,0,2)1x3!4Gjj!BV!H(0,1,2)1x3!000175684xxxGBEWHL1Gjj!Bl!El!WV!Hl!L(3,1,2)1x3!2Gjj!Bl!El!Hl!L(3,0,2)1x3!3Gjj!BV!Hl!L(1,1,2)1x3!4Gjj!Bl!L(1,0,2)1x3!000175846xxxGBEHLW1Gjj!Bl!Ell!Ll!W(4,0,2)1x3!2Gjj!Bl!El!HV!W(2,1,2)1x3!3Gjj!Bl!Ll!W(2,0,2)1x3!4Gjj!BV!HV!W(0,2,2)1x3!5Gjj!W(0,0,2)1x3!000175864xxxGBEHWL1Gjj!Bl!El!HV!Wl!L(3,1,2)1x3!2Gjj!Bl!El!Wl!L(3,0,2)1x3!3Gjj!BV!HV!Wl!L(1,2,2)1x3!4Gjj!Bl!L(1,0,2)1x3!000176458xxxGBWLEH1Gjj!Wl!Lll!El!H(4,0,2)1x3!2Gjj!Bl!El!H(2,0,2)1x3!3Gjj!WV!H(0,1,2)1x3!000176548xxxGBWELH1Gjj!Bl!Ell!Ll!H(4,0,2)1x3!2Gjj!Bl!Ll!H(2,0,2)1x3!3Gjj!WV!H(0,1,2)1x3!000176584xxxGBWEHL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!BV!Hl!L(1,1,2)1x3!3Gjj!Wl!L(1,0,2)1x3!000176854xxxGBWHEL1Gjj!WV!Hl!Ell!L(3,1,2)1x3!2Gjj!Bl!Ell!L(3,0,2)1x3! 153

PAGE 154

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 3Gjj!BV!Hl!L(1,1,2)1x3!4Gjj!Wl!L(1,0,2)1x3!000178456xxxGBHLEW1Gjj!Bl!El!W(2,0,2)1x3!2Gjj!BV!HV!W(0,2,2)1x3!3Gjj!W(0,0,2)1x3!000178546xxxGBHELW1Gjj!Bl!Ell!Ll!W(4,0,2)1x3!2Gjj!Bl!Ll!W(2,0,2)1x3!3Gjj!BV!HV!W(0,2,2)1x3!4Gjj!W(0,0,2)1x3!000178564xxxGBHEWL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!BV!HV!Wl!L(1,2,2)1x3!3Gjj!Bl!L(1,0,2)1x3!000178654xxxGBHWEL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!BV!HV!Wl!L(1,2,2)1x3!3Gjj!Bl!L(1,0,2)1x3!000184567xxxGHLEWB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000184576xxxGHLEBW1Gjj!BV!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000184657xxxGHLWEB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000184756xxxGHLBEW1Gjj!Bl!El!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000185467xxxGHELWB1Gjj!WV!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000185476xxxGHELBW1Gjj!BV!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000185647xxxGHEWLB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000185674xxxGHEWBL1Gjj!Wl!L(1,0,2)1x3!000185746xxxGHEBLW1Gjj!Bl!Ll!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000185764xxxGHEBWL1Gjj!Bl!L(1,0,2)1x3! 154

PAGE 155

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000186457xxxGHWLEB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000186547xxxGHWELB1Gjj!Wl!Ll!B(2,0,2)1x3!2Gjj!B(0,0,2)1x3!000186574xxxGHWEBL1Gjj!Wl!L(1,0,2)1x3!000186754xxxGHWBEL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!Wl!L(1,0,2)1x3!000187456xxxGHBLEW1Gjj!Bl!El!W(2,0,2)1x3!2Gjj!W(0,0,2)1x3!000187546xxxGHBELW1Gjj!Bl!Ell!Ll!W(4,0,2)1x3!2Gjj!Bl!Ll!W(2,0,2)1x3!3Gjj!W(0,0,2)1x3!000187564xxxGHBEWL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!Bl!L(1,0,2)1x3!000187654xxxGHBWEL1Gjj!Bl!Ell!L(3,0,2)1x3!2Gjj!Bl!L(1,0,2)1x3!000245678xxxQLEWBH1Qj!WV!H(0,1,1)1x3!000245687xxxQLEWHB1Qj!WV!HV!B(0,2,1)1x3!2Qj!B(0,0,1)1x3!000245768xxxQLEBWH1Qj!BV!H(0,1,1)1x3!000245786xxxQLEBHW1Qj!BV!HV!W(0,2,1)1x3!2Qj!W(0,0,1)1x3!000245867xxxQLEHWB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000245876xxxQLEHBW1Qj!BV!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000246578xxxQLWEBH1Qj!WV!H(0,1,1)1x3!000246587xxxQLWEHB1Qj!WV!HV!B(0,2,1)1x3!2Qj!B(0,0,1)1x3!000246758xxxQLWBEH1Qj!Bl!El!H(2,0,1)1x3!2Qj!WV!H(0,1,1)1x3!000246857xxxQLWHEB1Qj!WV!HV!B(0,2,1)1x3!2Qj!B(0,0,1)1x3! 155

PAGE 156

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000247568xxxQLBEWH1Qj!Bl!El!WV!H(2,1,1)1x3!2Qj!Bl!El!H(2,0,1)1x3!3Qj!BV!H(0,1,1)1x3!000247586xxxQLBEHW1Qj!Bl!El!HV!W(2,1,1)1x3!2Qj!Bl!El!W(2,0,1)1x3!3Qj!BV!HV!W(0,2,1)1x3!4Qj!W(0,0,1)1x3!000247658xxxQLBWEH1Qj!Bl!El!H(2,0,1)1x3!2Qj!BV!H(0,1,1)1x3!000247856xxxQLBHEW1Qj!Bl!El!W(2,0,1)1x3!2Qj!BV!HV!W(0,2,1)1x3!3Qj!W(0,0,1)1x3!000248567xxxQLHEWB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000248576xxxQLHEBW1Qj!BV!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000248657xxxQLHWEB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000248756xxxQLHBEW1Qj!Bl!El!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000254678xxxQELWBH1Qj!WV!H(0,1,1)1x3!000254687xxxQELWHB1Qj!WV!HV!B(0,2,1)1x3!2Qj!B(0,0,1)1x3!000254768xxxQELBWH1Qj!BV!H(0,1,1)1x3!000254786xxxQELBHW1Qj!BV!HV!W(0,2,1)1x3!2Qj!W(0,0,1)1x3!000254867xxxQELHWB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000254876xxxQELHBW1Qj!BV!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000256478xxxQEWLBH1Qj!Wl!Ll!BV!H(2,1,1)1x3!2Qj!WV!H(0,1,1)1x3!000256487xxxQEWLHB1Qj!Wl!Ll!B(2,0,1)1x3! 156

PAGE 157

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000256748xxxQEWBLH1Qj!Wl!Ll!H(2,0,1)1x3!2Qj!WV!H(0,1,1)1x3!000256784xxxQEWBHL1Qj!WV!Hl!L(1,1,1)1x3!2Qj!Wl!L(1,0,1)1x3!000256847xxxQEWHLB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000256874xxxQEWHBL1Qj!WV!HV!Bl!L(1,2,1)1x3!2Qj!Wl!L(1,0,1)1x3!000257468xxxQEBLWH1Qj!Bl!Ll!WV!H(2,1,1)1x3!2Qj!BV!H(0,1,1)1x3!000257486xxxQEBLHW1Qj!Bl!Ll!W(2,0,1)1x3!2Qj!BV!HV!W(0,2,1)1x3!3Qj!W(0,0,1)1x3!000257648xxxQEBWLH1Qj!Bl!Ll!H(2,0,1)1x3!2Qj!BV!H(0,1,1)1x3!000257684xxxQEBWHL1Qj!BV!Hl!L(1,1,1)1x3!2Qj!Bl!L(1,0,1)1x3!000257846xxxQEBHLW1Qj!Bl!Ll!W(2,0,1)1x3!2Qj!BV!HV!W(0,2,1)1x3!3Qj!W(0,0,1)1x3!000257864xxxQEBHWL1Qj!BV!HV!Wl!L(1,2,1)1x3!2Qj!Bl!L(1,0,1)1x3!000258467xxxQEHLWB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000258476xxxQEHLBW1Qj!BV!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000258647xxxQEHWLB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000258674xxxQEHWBL1Qj!Wl!L(1,0,1)1x3!000258746xxxQEHBLW1Qj!Bl!Ll!W(2,0,1)1x3! 157

PAGE 158

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qj!W(0,0,1)1x3!000258764xxxQEHBWL1Qj!Bl!L(1,0,1)1x3!000264578xxxQWLEBH1Qj!Wl!Ll!BV!H(2,1,1)1x3!2Qj!WV!H(0,1,1)1x3!000264587xxxQWLEHB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000264758xxxQWLBEH1Qj!Wl!Ll!Bl!El!H(4,0,1)1x3!2Qj!Wl!Ll!BV!H(2,1,1)1x3!3Qj!Bl!El!H(2,0,1)1x3!4Qj!WV!H(0,1,1)1x3!000264857xxxQWLHEB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000265478xxxQWELBH1Qj!Wl!Ll!BV!H(2,1,1)1x3!2Qj!WV!H(0,1,1)1x3!000265487xxxQWELHB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000265748xxxQWEBLH1Qj!Wl!Ll!H(2,0,1)1x3!2Qj!WV!H(0,1,1)1x3!000265784xxxQWEBHL1Qj!WV!Hl!L(1,1,1)1x3!2Qj!Wl!L(1,0,1)1x3!000265847xxxQWEHLB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000265874xxxQWEHBL1Qj!WV!HV!Bl!L(1,2,1)1x3!2Qj!Wl!L(1,0,1)1x3!000267458xxxQWBLEH1Qj!Wl!Lll!El!H(4,0,1)1x3!2Qj!Bl!El!H(2,0,1)1x3!3Qj!WV!H(0,1,1)1x3!000267548xxxQWBELH1Qj!Bl!Ell!Ll!H(4,0,1)1x3!2Qj!Wl!Ll!H(2,0,1)1x3! 158

PAGE 159

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 3Qj!WV!H(0,1,1)1x3!000267584xxxQWBEHL1Qj!Bl!Ell!L(3,0,1)1x3!2Qj!WV!Hl!L(1,1,1)1x3!3Qj!Wl!L(1,0,1)1x3!000267854xxxQWBHEL1Qj!WV!Hl!Ell!L(3,1,1)1x3!2Qj!Bl!Ell!L(3,0,1)1x3!3Qj!BV!Hl!L(1,1,1)1x3!4Qj!Wl!L(1,0,1)1x3!000268457xxxQWHLEB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000268547xxxQWHELB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!WV!HV!B(0,2,1)1x3!3Qj!B(0,0,1)1x3!000268574xxxQWHEBL1Qj!WV!HV!Bl!L(1,2,1)1x3!2Qj!Wl!L(1,0,1)1x3!000268754xxxQWHBEL1Qj!WV!HV!Bl!Ell!L(3,2,1)1x3!2Qj!Bl!Ell!L(3,0,1)1x3!3Qj!WV!HV!Bl!L(1,2,1)1x3!4Qj!Wl!L(1,0,1)1x3!000274568xxxQBLEWH1Qj!Bl!El!WV!H(2,1,1)1x3!2Qj!Bl!El!H(2,0,1)1x3!3Qj!BV!H(0,1,1)1x3!000274586xxxQBLEHW1Qj!Bl!El!HV!W(2,1,1)1x3!2Qj!Bl!Ll!W(2,0,1)1x3!3Qj!BV!HV!W(0,2,1)1x3!4Qj!W(0,0,1)1x3!000274658xxxQBLWEH1Qj!Bl!Ll!WV!H(2,1,1)1x3!2Qj!Bl!El!H(2,0,1)1x3!3Qj!BV!H(0,1,1)1x3!000274856xxxQBLHEW1Qj!Bl!El!W(2,0,1)1x3!2Qj!BV!HV!W(0,2,1)1x3!3Qj!W(0,0,1)1x3! 159

PAGE 160

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000275468xxxQBELWH1Qj!Bl!Ell!Ll!WV!H(4,1,1)1x3!2Qj!Bl!Ll!WV!H(2,1,1)1x3!3Qj!Bl!El!H(2,0,1)1x3!4Qj!BV!H(0,1,1)1x3!000275486xxxQBELHW1Qj!Bl!Ell!Ll!W(4,0,1)1x3!2Qj!Bl!El!HV!W(2,1,1)1x3!3Qj!Bl!Ll!W(2,0,1)1x3!4Qj!BV!HV!W(0,2,1)1x3!5Qj!W(0,0,1)1x3!000275648xxxQBEWLH1Qj!Bl!Ell!Ll!H(4,0,1)1x3!2Qj!Bl!El!WV!H(2,1,1)1x3!3Qj!Bl!Ll!H(2,0,1)1x3!4Qj!BV!H(0,1,1)1x3!000275684xxxQBEWHL1Qj!Bl!El!WV!Hl!L(3,1,1)1x3!2Qj!Bl!El!Hl!L(3,0,1)1x3!3Qj!BV!Hl!L(1,1,1)1x3!4Qj!Bl!L(1,0,1)1x3!000275846xxxQBEHLW1Qj!Bl!Ell!Ll!W(4,0,1)1x3!2Qj!Bl!El!HV!W(2,1,1)1x3!3Qj!Bl!Ll!W(2,0,1)1x3!4Qj!BV!HV!W(0,2,1)1x3!5Qj!W(0,0,1)1x3!000275864xxxQBEHWL1Qj!Bl!El!HV!Wl!L(3,1,1)1x3!2Qj!Bl!El!Wl!L(3,0,1)1x3!3Qj!BV!HV!Wl!L(1,2,1)1x3!4Qj!Bl!L(1,0,1)1x3!000276458xxxQBWLEH1Qj!Wl!Lll!El!H(4,0,1)1x3!2Qj!Bl!El!H(2,0,1)1x3!3Qj!WV!H(0,1,1)1x3!000276548xxxQBWELH1Qj!Bl!Ell!Ll!H(4,0,1)1x3!2Qj!Bl!Ll!H(2,0,1)1x3!3Qj!WV!H(0,1,1)1x3!000276584xxxQBWEHL1Qj!Bl!Ell!L(3,0,1)1x3! 160

PAGE 161

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Qj!BV!Hl!L(1,1,1)1x3!3Qj!Wl!L(1,0,1)1x3!000276854xxxQBWHEL1Qj!WV!Hl!Ell!L(3,1,1)1x3!2Qj!Bl!Ell!L(3,0,1)1x3!3Qj!BV!Hl!L(1,1,1)1x3!4Qj!Wl!L(1,0,1)1x3!000278456xxxQBHLEW1Qj!Bl!El!W(2,0,1)1x3!2Qj!BV!HV!W(0,2,1)1x3!3Qj!W(0,0,1)1x3!000278546xxxQBHELW1Qj!Bl!Ell!Ll!W(4,0,1)1x3!2Qj!Bl!Ll!W(2,0,1)1x3!3Qj!BV!HV!W(0,2,1)1x3!4Qj!W(0,0,1)1x3!000278564xxxQBHEWL1Qj!Bl!Ell!L(3,0,1)1x3!2Qj!BV!HV!Wl!L(1,2,1)1x3!3Qj!Bl!L(1,0,1)1x3!000278654xxxQBHWEL1Qj!Bl!Ell!L(3,0,1)1x3!2Qj!BV!HV!Wl!L(1,2,1)1x3!3Qj!Bl!L(1,0,1)1x3!000284567xxxQHLEWB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000284576xxxQHLEBW1Qj!BV!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000284657xxxQHLWEB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000284756xxxQHLBEW1Qj!Bl!El!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000285467xxxQHELWB1Qj!WV!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000285476xxxQHELBW1Qj!BV!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000285647xxxQHEWLB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!B(0,0,1)1x3! 161

PAGE 162

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000285674xxxQHEWBL1Qj!Wl!L(1,0,1)1x3!000285746xxxQHEBLW1Qj!Bl!Ll!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000285764xxxQHEBWL1Qj!Bl!L(1,0,1)1x3!000286457xxxQHWLEB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000286547xxxQHWELB1Qj!Wl!Ll!B(2,0,1)1x3!2Qj!B(0,0,1)1x3!000286574xxxQHWEBL1Qj!Wl!L(1,0,1)1x3!000286754xxxQHWBEL1Qj!Bl!Ell!L(3,0,1)1x3!2Qj!Wl!L(1,0,1)1x3!000287456xxxQHBLEW1Qj!Bl!El!W(2,0,1)1x3!2Qj!W(0,0,1)1x3!000287546xxxQHBELW1Qj!Bl!Ell!Ll!W(4,0,1)1x3!2Qj!Bl!Ll!W(2,0,1)1x3!3Qj!W(0,0,1)1x3!000287564xxxQHBEWL1Qj!Bl!Ell!L(3,0,1)1x3!2Qj!Bl!L(1,0,1)1x3!000287654xxxQHBWEL1Qj!Bl!Ell!L(3,0,1)1x3!2Qj!Bl!L(1,0,1)1x3!000345678xxxULEWBH1Uj!BV!H(0,1,1)2x3!000345687xxxULEWHB1Uj!B(0,0,1)2x3!000345768xxxULEBWH1Uj!BV!H(0,1,1)2x3!000345786xxxULEBHW1Uj!BV!HV!W(0,2,1)2x3!000345867xxxULEHWB1Uj!B(0,0,1)2x3!000345876xxxULEHBW1Uj!BV!W(2,0,1)2x3!000346578xxxULWEBH1Uj!BV!H(0,1,1)2x3!000346587xxxULWEHB1Uj!B(0,0,1)2x3!000346758xxxULWBEH1Uj!Bl!El!H(2,0,1)2x3!2Uj!BV!H(0,1,1)2x3!000346857xxxULWHEB1Uj!B(0,0,1)2x3!000347568xxxULBEWH1Uj!Bl!El!WV!H(2,1,1)2x3!2Uj!Bl!El!H(2,0,1)2x3! 162

PAGE 163

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 3Uj!BV!H(0,1,1)2x3!000347586xxxULBEHW1Uj!Bl!El!HV!W(2,1,1)2x3!2Uj!Bl!El!W(2,0,1)2x3!3Uj!BV!HV!W(0,2,1)2x3!000347658xxxULBWEH1Uj!Bl!El!H(2,0,1)2x3!2Uj!BV!H(0,1,1)2x3!000347856xxxULBHEW1Uj!Bl!El!W(2,0,1)2x3!2Uj!BV!HV!W(0,2,1)2x3!000348567xxxULHEWB1Uj!B(0,0,1)2x3!000348576xxxULHEBW1Uj!BV!W(2,0,1)2x3!000348657xxxULHWEB1Uj!B(0,0,1)2x3!000348756xxxULHBEW1Uj!Bl!El!W(2,0,1)2x3!000354678xxxUELWBH1Uj!BV!H(0,1,1)2x3!000354687xxxUELWHB1Uj!B(0,0,1)2x3!000354768xxxUELBWH1Uj!BV!H(0,1,1)2x3!000354786xxxUELBHW1Uj!BV!HV!W(0,2,1)2x3!000354867xxxUELHWB1Uj!B(0,0,1)2x3!000354876xxxUELHBW1Uj!BV!W(2,0,1)2x3!000356478xxxUEWLBH1Uj!BV!H(0,1,1)2x3!000356487xxxUEWLHB1Uj!B(0,0,1)2x3!000356748xxxUEWBLH1Uj!Bl!Ll!H(2,0,1)2x3!2Uj!BV!H(0,1,1)2x3!000356784xxxUEWBHL1Uj!BV!Hl!L(1,1,1)2x3!2Uj!Bl!L(1,0,1)2x3!000356847xxxUEWHLB1Uj!B(0,0,1)2x3!000356874xxxUEWHBL1Uj!Bl!L(1,0,1)2x3!000357468xxxUEBLWH1Uj!Bl!Ll!WV!H(2,1,1)2x3!2Uj!BV!H(0,1,1)2x3!000357486xxxUEBLHW1Uj!Bl!Ll!W(2,0,1)2x3!2Uj!BV!HV!W(0,2,1)2x3!000357648xxxUEBWLH1Uj!Bl!Ll!H(2,0,1)2x3!2Uj!BV!H(0,1,1)2x3!000357684xxxUEBWHL1Uj!BV!Hl!L(1,1,1)2x3! 163

PAGE 164

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Uj!Bl!L(1,0,1)2x3!000357846xxxUEBHLW1Uj!Bl!Ll!W(2,0,1)2x3!2Uj!BV!HV!W(0,2,1)2x3!000357864xxxUEBHWL1Uj!BV!HV!Wl!L(1,2,1)2x3!2Uj!Bl!L(1,0,1)2x3!000358467xxxUEHLWB1Uj!B(0,0,1)2x3!000358476xxxUEHLBW1Uj!BV!W(2,0,1)2x3!000358647xxxUEHWLB1Uj!B(0,0,1)2x3!000358674xxxUEHWBL1Uj!Bl!L(1,0,1)2x3!000358746xxxUEHBLW1Uj!Bl!Ll!W(2,0,1)2x3!000358764xxxUEHBWL1Uj!Bl!L(1,0,1)2x3!000364578xxxUWLEBH1Uj!BV!H(0,1,1)2x3!000364587xxxUWLEHB1Uj!B(0,0,1)2x3!000364758xxxUWLBEH1Uj!Bl!El!H(2,0,1)2x3!2Uj!BV!H(0,1,1)2x3!000364857xxxUWLHEB1Uj!B(0,0,1)2x3!000365478xxxUWELBH1Uj!BV!H(0,1,1)2x3!000365487xxxUWELHB1Uj!B(0,0,1)2x3!000365748xxxUWEBLH1Uj!Bl!Ll!H(2,0,1)2x3!2Uj!BV!H(0,1,1)2x3!000365784xxxUWEBHL1Uj!BV!Hl!L(1,1,1)2x3!2Uj!Bl!L(1,0,1)2x3!000365847xxxUWEHLB1Uj!B(0,0,1)2x3!000365874xxxUWEHBL1Uj!Bl!L(1,0,1)2x3!000367458xxxUWBLEH1Uj!Bl!Lll!El!H(4,0,1)2x3!2Uj!Bl!El!H(2,0,1)2x3!3Uj!BV!H(0,1,1)2x3!000367548xxxUWBELH1Uj!Bl!Ell!Ll!H(4,0,1)2x3!2Uj!Bl!Ll!H(2,0,1)2x3!3Uj!BV!H(0,1,1)2x3!000367584xxxUWBEHL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!BV!Hl!L(1,1,1)2x3!3Uj!Bl!L(1,0,1)2x3! 164

PAGE 165

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 000367854xxxUWBHEL1Uj!BV!Hl!Ell!L(3,1,1)2x3!2Uj!Bl!Ell!L(3,0,1)2x3!3Uj!BV!Hl!L(1,1,1)2x3!4Uj!Bl!L(1,0,1)2x3!000368457xxxUWHLEB1Uj!B(0,0,1)2x3!000368547xxxUWHELB1Uj!B(0,0,1)2x3!000368574xxxUWHEBL1Uj!Bl!L(1,0,1)2x3!000368754xxxUWHBEL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!Bl!L(1,0,1)2x3!000374568xxxUBLEWH1Uj!Bl!El!WV!H(2,1,1)2x3!2Uj!Bl!El!H(2,0,1)2x3!3Uj!BV!H(0,1,1)2x3!000374586xxxUBLEHW1Uj!Bl!El!HV!W(2,1,1)2x3!2Uj!Bl!Ll!W(2,0,1)2x3!3Uj!BV!HV!W(0,2,1)2x3!000374658xxxUBLWEH1Uj!Bl!Ll!WV!H(2,1,1)2x3!2Uj!Bl!El!H(2,0,1)2x3!3Uj!BV!H(0,1,1)2x3!000374856xxxUBLHEW1Uj!Bl!El!W(2,0,1)2x3!2Uj!BV!HV!W(0,2,1)2x3!000375468xxxUBELWH1Uj!Bl!Ell!Ll!WV!H(4,1,1)2x3!2Uj!Bl!Ll!WV!H(2,1,1)2x3!3Uj!Bl!El!H(2,0,1)2x3!4Uj!BV!H(0,1,1)2x3!000375486xxxUBELHW1Uj!Bl!Ell!Ll!W(4,0,1)2x3!2Uj!Bl!El!HV!W(2,1,1)2x3!3Uj!Bl!Ll!W(2,0,1)2x3!4Uj!BV!HV!W(0,2,1)2x3!000375648xxxUBEWLH1Uj!Bl!Ell!Ll!H(4,0,1)2x3!2Uj!Bl!El!WV!H(2,1,1)2x3!3Uj!Bl!Ll!H(2,0,1)2x3!4Uj!BV!H(0,1,1)2x3!000375684xxxUBEWHL1Uj!Bl!El!WV!Hl!L(3,1,1)2x3! 165

PAGE 166

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Uj!Bl!El!Hl!L(3,0,1)2x3!3Uj!BV!Hl!L(1,1,1)2x3!4Uj!Bl!L(1,0,1)2x3!000375846xxxUBEHLW1Uj!Bl!Ell!Ll!W(4,0,1)2x3!2Uj!Bl!El!HV!W(2,1,1)2x3!3Uj!Bl!Ll!W(2,0,1)2x3!4Uj!BV!HV!W(0,2,1)2x3!000375864xxxUBEHWL1Uj!Bl!El!HV!Wl!L(3,1,1)2x3!2Uj!Bl!El!Wl!L(3,0,1)2x3!3Uj!BV!HV!Wl!L(1,2,1)2x3!4Uj!Bl!L(1,0,1)2x3!000376458xxxUBWLEH1Uj!Bl!Lll!El!H(4,0,1)2x3!2Uj!Bl!El!H(2,0,1)2x3!3Uj!BV!H(0,1,1)2x3!000376548xxxUBWELH1Uj!Bl!Ell!Ll!H(4,0,1)2x3!2Uj!Bl!Ll!H(2,0,1)2x3!3Uj!BV!H(0,1,1)2x3!000376584xxxUBWEHL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!BV!Hl!L(1,1,1)2x3!3Uj!Bl!L(1,0,1)2x3!000376854xxxUBWHEL1Uj!BV!Hl!Ell!L(3,1,1)2x3!2Uj!Bl!Ell!L(3,0,1)2x3!3Uj!BV!Hl!L(1,1,1)2x3!4Uj!Bl!L(1,0,1)2x3!000378456xxxUBHLEW1Uj!Bl!El!W(2,0,1)2x3!2Uj!BV!HV!W(0,2,1)2x3!000378546xxxUBHELW1Uj!Bl!Ell!Ll!W(4,0,1)2x3!2Uj!Bl!Ll!W(2,0,1)2x3!3Uj!BV!HV!W(0,2,1)2x3!000378564xxxUBHEWL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!BV!HV!Wl!L(1,2,1)2x3!3Uj!Bl!L(1,0,1)2x3!000378654xxxUBHWEL1Uj!Bl!Ell!L(3,0,1)2x3! 166

PAGE 167

Table C-1 continued Hier.#Hier.Mode##ChainsSign.Mul. 2Uj!BV!HV!Wl!L(1,2,1)2x3!3Uj!Bl!L(1,0,1)2x3!000384567xxxUHLEWB1Uj!B(0,0,1)2x3!000384576xxxUHLEBW1Uj!BV!W(2,0,1)2x3!000384657xxxUHLWEB1Uj!B(0,0,1)2x3!000384756xxxUHLBEW1Uj!Bl!El!W(2,0,1)2x3!000385467xxxUHELWB1Uj!B(0,0,1)2x3!000385476xxxUHELBW1Uj!BV!W(2,0,1)2x3!000385647xxxUHEWLB1Uj!B(0,0,1)2x3!000385674xxxUHEWBL1Uj!Bl!L(1,0,1)2x3!000385746xxxUHEBLW1Uj!Bl!Ll!W(2,0,1)2x3!000385764xxxUHEBWL1Uj!Bl!L(1,0,1)2x3!000386457xxxUHWLEB1Uj!B(0,0,1)2x3!000386547xxxUHWELB1Uj!B(0,0,1)2x3!000386574xxxUHWEBL1Uj!Bl!L(1,0,1)2x3!000386754xxxUHWBEL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!Bl!L(1,0,1)2x3!000387456xxxUHBLEW1Uj!Bl!El!W(2,0,1)2x3!000387546xxxUHBELW1Uj!Bl!Ell!Ll!W(4,0,1)2x3!2Uj!Bl!Ll!W(2,0,1)2x3!000387564xxxUHBEWL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!Bl!L(1,0,1)2x3!000387654xxxUHBWEL1Uj!Bl!Ell!L(3,0,1)2x3!2Uj!Bl!L(1,0,1)2x3! 167

PAGE 168

APPENDIXDSIGNATURETOHIERARCHIESThisappendixistheinverseofAppendix C-1 .Inthisappendixwelistthesignaturesaliongwithallthehierarchiesthathaveadecaychannelcorrespondingtosuchasignature.Therstcolumncontainsthesignatureinthe(nl,nv,nj)format,i.e.thenumberofleptons,thenumberofbosons(W,ZandH)andthenumberofjets.Thesecondandthirdcolumnscontainsthecorrespondinghierarchiesintheirnumericalandlettercoderespectively.Incolumnfourwelisttemultiplicityforthehierarchy.Incolumnsvethrougheightwelisttherestoftheequallysupressedsignaturesforthehierarchy.Foreverysignature,therowsaregroupedtogetherinsuchawaythateverygroupcontainsthehierarchieswiththeexactsamesetofsignatures.Everyhierarchyfallingunderthesamegroupisthuslistedinconsecutiverows.Anytwogroupsunderagivensignatureareseparatedbyahorizontallinerunningfromcolumntwothrougheight.Alinerunningfromcolumnonethrougheightdividesthetableinchunkswithdifferentsignaturesincolumnone. 168

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TableD-1.SignaturetoHierarchies Signaturetohierarchy SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) (4,1,2)000175468xxxGBELWH1x3!(2,1,2)(2,0,2)(0,1,2)(4,1,1)000035468xxxxUELWH2x4!(2,1,1)(2,0,1)(0,1,1)(0,0,1)000375468xxxUBELWH2x3!(2,1,1)(2,0,1)(0,1,1)000275468xxxQBELWH1x3!(2,1,1)(2,0,1)(0,1,1)(4,0,2)000175846xxxGBEHLW1x3!(2,1,2)(2,0,2)(0,2,2)(0,0,2)000175486xxxGBELHW1x3!(2,1,2)(2,0,2)(0,2,2)(0,0,2)000175648xxxGBEWLH1x3!(2,1,2)(2,0,2)(0,1,2)000164758xxxGWLBEH1x3!(2,1,2)(2,0,2)(0,1,2)000178546xxxGBHELW1x3!(2,0,2)(0,2,2)(0,0,2)000176548xxxGBWELH1x3!(2,0,2)(0,1,2)000176458xxxGBWLEH1x3!(2,0,2)(0,1,2)000167548xxxGWBELH1x3!(2,0,2)(0,1,2)000167458xxxGWBLEH1x3!(2,0,2)(0,1,2)000017548xxxxGBELH1x4!(2,0,2)(0,1,2)000017458xxxxGBLEH1x4!(2,0,2)(0,1,2)000016458xxxxGWLEH1x4!(2,0,2)(0,1,2)000187546xxxGHBELW1x3!(2,0,2)(0,0,2)000017546xxxxGBELW1x4!(2,0,2)(0,0,2)000001548xxxxxGELH1x5!(2,0,2)(0,0,2)000001458xxxxxGLEH1x5!(2,0,2)(0,0,2)(4,0,1)000275846xxxQBEHLW1x3!(2,1,1)(2,0,1)(0,2,1)(0,0,1)000275486xxxQBELHW1x3!(2,1,1)(2,0,1)(0,2,1)(0,0,1)000035846xxxxUEHLW2x4!(2,1,1)(2,0,1)(0,1,1)(0,0,1)000035648xxxxUEWLH2x4!(2,1,1)(2,0,1)(0,1,1)(0,0,1)000035486xxxxUELHW2x4!(2,1,1)(2,0,1)(0,1,1)(0,0,1)000375846xxxUBEHLW2x3!(2,1,1)(2,0,1)(0,2,1)000375486xxxUBELHW2x3!(2,1,1)(2,0,1)(0,2,1)000375648xxxUBEWLH2x3!(2,1,1)(2,0,1)(0,1,1)000275648xxxQBEWLH1x3!(2,1,1)(2,0,1)(0,1,1)000264758xxxQWLBEH1x3!(2,1,1)(2,0,1)(0,1,1)000278546xxxQBHELW1x3!(2,0,1)(0,2,1)(0,0,1)000038546xxxxUHELW2x4!(2,0,1)(0,1,1)(0,0,1)000036548xxxxUWELH2x4!(2,0,1)(0,1,1)(0,0,1)000036458xxxxUWLEH2x4!(2,0,1)(0,1,1)(0,0,1)000378546xxxUBHELW2x3!(2,0,1)(0,2,1)000376548xxxUBWELH2x3!(2,0,1)(0,1,1)000376458xxxUBWLEH2x3!(2,0,1)(0,1,1)000367548xxxUWBELH2x3!(2,0,1)(0,1,1)000367458xxxUWBLEH2x3!(2,0,1)(0,1,1)000276548xxxQBWELH1x3!(2,0,1)(0,1,1) 169

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000276458xxxQBWLEH1x3!(2,0,1)(0,1,1)000267548xxxQWBELH1x3!(2,0,1)(0,1,1)000267458xxxQWBLEH1x3!(2,0,1)(0,1,1)000037548xxxxUBELH2x4!(2,0,1)(0,1,1)000037458xxxxUBLEH2x4!(2,0,1)(0,1,1)000027548xxxxQBELH1x4!(2,0,1)(0,1,1)000027458xxxxQBLEH1x4!(2,0,1)(0,1,1)000026458xxxxQWLEH1x4!(2,0,1)(0,1,1)000287546xxxQHBELW1x3!(2,0,1)(0,0,1)000027546xxxxQBELW1x4!(2,0,1)(0,0,1)000003548xxxxxUELH2x5!(2,0,1)(0,0,1)000003546xxxxxUELW2x5!(2,0,1)(0,0,1)000003458xxxxxULEH2x5!(2,0,1)(0,0,1)000002548xxxxxQELH1x5!(2,0,1)(0,0,1)000002458xxxxxQLEH1x5!(2,0,1)(0,0,1)000387546xxxUHBELW2x3!(2,0,1)000037546xxxxUBELW2x4!(2,0,1)(3,2,2)000168754xxxGWHBEL1x3!(3,0,2)(1,2,2)(1,0,2)(3,2,1)000268754xxxQWHBEL1x3!(3,0,1)(1,2,1)(1,0,1)(3,1,2)000175864xxxGBEHWL1x3!(3,0,2)(1,2,2)(1,0,2)000176854xxxGBWHEL1x3!(3,0,2)(1,1,2)(1,0,2)000175684xxxGBEWHL1x3!(3,0,2)(1,1,2)(1,0,2)000167854xxxGWBHEL1x3!(3,0,2)(1,1,2)(1,0,2)000017854xxxxGBHEL1x4!(3,0,2)(1,1,2)(1,0,2)000016854xxxxGWHEL1x4!(1,1,2)(1,0,2)(3,1,1)000375864xxxUBEHWL2x3!(3,0,1)(1,2,1)(1,0,1)000275864xxxQBEHWL1x3!(3,0,1)(1,2,1)(1,0,1)000376854xxxUBWHEL2x3!(3,0,1)(1,1,1)(1,0,1)000375684xxxUBEWHL2x3!(3,0,1)(1,1,1)(1,0,1)000367854xxxUWBHEL2x3!(3,0,1)(1,1,1)(1,0,1)000276854xxxQBWHEL1x3!(3,0,1)(1,1,1)(1,0,1)000275684xxxQBEWHL1x3!(3,0,1)(1,1,1)(1,0,1)000267854xxxQWBHEL1x3!(3,0,1)(1,1,1)(1,0,1)000037854xxxxUBHEL2x4!(3,0,1)(1,1,1)(1,0,1)000036854xxxxUWHEL2x4!(3,0,1)(1,1,1)(1,0,1)000035864xxxxUEHWL2x4!(3,0,1)(1,1,1)(1,0,1)000035684xxxxUEWHL2x4!(3,0,1)(1,1,1)(1,0,1)000027854xxxxQBHEL1x4!(3,0,1)(1,1,1)(1,0,1)000026854xxxxQWHEL1x4!(1,1,1)(1,0,1)(3,0,2)000168754xxxGWHBEL1x3!(3,2,2)(1,2,2)(1,0,2) 170

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n)000175864xxxGBEHWL1x3!(3,1,2)(1,2,2)(1,0,2)000176854xxxGBWHEL1x3!(3,1,2)(1,1,2)(1,0,2)000175684xxxGBEWHL1x3!(3,1,2)(1,1,2)(1,0,2)000167854xxxGWBHEL1x3!(3,1,2)(1,1,2)(1,0,2)000017854xxxxGBHEL1x4!(3,1,2)(1,1,2)(1,0,2)000178654xxxGBHWEL1x3!(1,2,2)(1,0,2)000178564xxxGBHEWL1x3!(1,2,2)(1,0,2)000176584xxxGBWEHL1x3!(1,1,2)(1,0,2)000167584xxxGWBEHL1x3!(1,1,2)(1,0,2)000017584xxxxGBEHL1x4!(1,1,2)(1,0,2)000187654xxxGHBWEL1x3!(1,0,2)000187564xxxGHBEWL1x3!(1,0,2)000186754xxxGHWBEL1x3!(1,0,2)000018754xxxxGHBEL1x4!(1,0,2)000017654xxxxGBWEL1x4!(1,0,2)000017564xxxxGBEWL1x4!(1,0,2)000016754xxxxGWBEL1x4!(1,0,2)000001854xxxxxGHEL1x5!(1,0,2)000001754xxxxxGBEL1x5!(1,0,2)000001584xxxxxGEHL1x5!(1,0,2)000000154xxxxxxGEL1x6!(1,0,2)(3,0,1)000268754xxxQWHBEL1x3!(3,2,1)(1,2,1)(1,0,1)000375864xxxUBEHWL2x3!(3,1,1)(1,2,1)(1,0,1)000275864xxxQBEHWL1x3!(3,1,1)(1,2,1)(1,0,1)000376854xxxUBWHEL2x3!(3,1,1)(1,1,1)(1,0,1)000375684xxxUBEWHL2x3!(3,1,1)(1,1,1)(1,0,1)000367854xxxUWBHEL2x3!(3,1,1)(1,1,1)(1,0,1)000276854xxxQBWHEL1x3!(3,1,1)(1,1,1)(1,0,1)000275684xxxQBEWHL1x3!(3,1,1)(1,1,1)(1,0,1)000267854xxxQWBHEL1x3!(3,1,1)(1,1,1)(1,0,1)000037854xxxxUBHEL2x4!(3,1,1)(1,1,1)(1,0,1)000036854xxxxUWHEL2x4!(3,1,1)(1,1,1)(1,0,1)000035864xxxxUEHWL2x4!(3,1,1)(1,1,1)(1,0,1)000035684xxxxUEWHL2x4!(3,1,1)(1,1,1)(1,0,1)000027854xxxxQBHEL1x4!(3,1,1)(1,1,1)(1,0,1)000378654xxxUBHWEL2x3!(1,2,1)(1,0,1)000378564xxxUBHEWL2x3!(1,2,1)(1,0,1)000278654xxxQBHWEL1x3!(1,2,1)(1,0,1)000278564xxxQBHEWL1x3!(1,2,1)(1,0,1)000376584xxxUBWEHL2x3!(1,1,1)(1,0,1) 171

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000367584xxxUWBEHL2x3!(1,1,1)(1,0,1)000276584xxxQBWEHL1x3!(1,1,1)(1,0,1)000267584xxxQWBEHL1x3!(1,1,1)(1,0,1)000038654xxxxUHWEL2x4!(1,1,1)(1,0,1)000038564xxxxUHEWL2x4!(1,1,1)(1,0,1)000037584xxxxUBEHL2x4!(1,1,1)(1,0,1)000036584xxxxUWEHL2x4!(1,1,1)(1,0,1)000027584xxxxQBEHL1x4!(1,1,1)(1,0,1)000387654xxxUHBWEL2x3!(1,0,1)000387564xxxUHBEWL2x3!(1,0,1)000386754xxxUHWBEL2x3!(1,0,1)000368754xxxUWHBEL2x3!(1,0,1)000287654xxxQHBWEL1x3!(1,0,1)000287564xxxQHBEWL1x3!(1,0,1)000286754xxxQHWBEL1x3!(1,0,1)000038754xxxxUHBEL2x4!(1,0,1)000037654xxxxUBWEL2x4!(1,0,1)000037564xxxxUBEWL2x4!(1,0,1)000036754xxxxUWBEL2x4!(1,0,1)000028754xxxxQHBEL1x4!(1,0,1)000027654xxxxQBWEL1x4!(1,0,1)000027564xxxxQBEWL1x4!(1,0,1)000026754xxxxQWBEL1x4!(1,0,1)000003854xxxxxUHEL2x5!(1,0,1)000003754xxxxxUBEL2x5!(1,0,1)000003654xxxxxUWEL2x5!(1,0,1)000003584xxxxxUEHL2x5!(1,0,1)000003564xxxxxUEWL2x5!(1,0,1)000002854xxxxxQHEL1x5!(1,0,1)000002754xxxxxQBEL1x5!(1,0,1)000002584xxxxxQEHL1x5!(1,0,1)000000354xxxxxxUEL2x6!(1,0,1)000000254xxxxxxQEL1x6!(1,0,1)(2,1,2)000175846xxxGBEHLW1x3!(4,0,2)(2,0,2)(0,2,2)(0,0,2)000175486xxxGBELHW1x3!(4,0,2)(2,0,2)(0,2,2)(0,0,2)000175468xxxGBELWH1x3!(4,1,2)(2,0,2)(0,1,2)000175648xxxGBEWLH1x3!(4,0,2)(2,0,2)(0,1,2)000164758xxxGWLBEH1x3!(4,0,2)(2,0,2)(0,1,2)000174586xxxGBLEHW1x3!(2,0,2)(0,2,2)(0,0,2)000147586xxxGLBEHW1x3!(2,0,2)(0,2,2)(0,0,2) 172

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000017586xxxxGBEHW1x4!(2,0,2)(0,2,2)(0,0,2)000174658xxxGBLWEH1x3!(2,0,2)(0,1,2)000174568xxxGBLEWH1x3!(2,0,2)(0,1,2)000147568xxxGLBEWH1x3!(2,0,2)(0,1,2)000017568xxxxGBEWH1x4!(2,0,2)(0,1,2)000165478xxxGWELBH1x3!(0,1,2)000164578xxxGWLEBH1x3!(0,1,2)000157468xxxGEBLWH1x3!(0,1,2)000156478xxxGEWLBH1x3!(0,1,2)000017468xxxxGBLWH1x4!(0,1,2)000016478xxxxGWLBH1x4!(0,1,2)(2,1,1)000035468xxxxUELWH2x4!(4,1,1)(2,0,1)(0,1,1)(0,0,1)000275846xxxQBEHLW1x3!(4,0,1)(2,0,1)(0,2,1)(0,0,1)000275486xxxQBELHW1x3!(4,0,1)(2,0,1)(0,2,1)(0,0,1)000035846xxxxUEHLW2x4!(4,0,1)(2,0,1)(0,1,1)(0,0,1)000035648xxxxUEWLH2x4!(4,0,1)(2,0,1)(0,1,1)(0,0,1)000035486xxxxUELHW2x4!(4,0,1)(2,0,1)(0,1,1)(0,0,1)000375468xxxUBELWH2x3!(4,1,1)(2,0,1)(0,1,1)000275468xxxQBELWH1x3!(4,1,1)(2,0,1)(0,1,1)000375846xxxUBEHLW2x3!(4,0,1)(2,0,1)(0,2,1)000375486xxxUBELHW2x3!(4,0,1)(2,0,1)(0,2,1)000375648xxxUBEWLH2x3!(4,0,1)(2,0,1)(0,1,1)000275648xxxQBEWLH1x3!(4,0,1)(2,0,1)(0,1,1)000264758xxxQWLBEH1x3!(4,0,1)(2,0,1)(0,1,1)000274586xxxQBLEHW1x3!(2,0,1)(0,2,1)(0,0,1)000247586xxxQLBEHW1x3!(2,0,1)(0,2,1)(0,0,1)000027586xxxxQBEHW1x4!(2,0,1)(0,2,1)(0,0,1)000034658xxxxULWEH2x4!(2,0,1)(0,1,1)(0,0,1)000034586xxxxULEHW2x4!(2,0,1)(0,1,1)(0,0,1)000034568xxxxULEWH2x4!(2,0,1)(0,1,1)(0,0,1)000003586xxxxxUEHW2x5!(2,0,1)(0,1,1)(0,0,1)000003568xxxxxUEWH2x5!(2,0,1)(0,1,1)(0,0,1)000374586xxxUBLEHW2x3!(2,0,1)(0,2,1)000347586xxxULBEHW2x3!(2,0,1)(0,2,1)000037586xxxxUBEHW2x4!(2,0,1)(0,2,1)000374658xxxUBLWEH2x3!(2,0,1)(0,1,1)000374568xxxUBLEWH2x3!(2,0,1)(0,1,1)000347568xxxULBEWH2x3!(2,0,1)(0,1,1)000274658xxxQBLWEH1x3!(2,0,1)(0,1,1)000274568xxxQBLEWH1x3!(2,0,1)(0,1,1) 173

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000247568xxxQLBEWH1x3!(2,0,1)(0,1,1)000037568xxxxUBEWH2x4!(2,0,1)(0,1,1)000027568xxxxQBEWH1x4!(2,0,1)(0,1,1)000003468xxxxxULWH2x5!(0,1,1)(0,0,1)000357468xxxUEBLWH2x3!(0,1,1)000265478xxxQWELBH1x3!(0,1,1)000264578xxxQWLEBH1x3!(0,1,1)000257468xxxQEBLWH1x3!(0,1,1)000256478xxxQEWLBH1x3!(0,1,1)000037468xxxxUBLWH2x4!(0,1,1)000027468xxxxQBLWH1x4!(0,1,1)000026478xxxxQWLBH1x4!(0,1,1)(2,0,2)000175846xxxGBEHLW1x3!(4,0,2)(2,1,2)(0,2,2)(0,0,2)000175486xxxGBELHW1x3!(4,0,2)(2,1,2)(0,2,2)(0,0,2)000175468xxxGBELWH1x3!(4,1,2)(2,1,2)(0,1,2)000175648xxxGBEWLH1x3!(4,0,2)(2,1,2)(0,1,2)000164758xxxGWLBEH1x3!(4,0,2)(2,1,2)(0,1,2)000178546xxxGBHELW1x3!(4,0,2)(0,2,2)(0,0,2)000174586xxxGBLEHW1x3!(2,1,2)(0,2,2)(0,0,2)000147586xxxGLBEHW1x3!(2,1,2)(0,2,2)(0,0,2)000017586xxxxGBEHW1x4!(2,1,2)(0,2,2)(0,0,2)000176548xxxGBWELH1x3!(4,0,2)(0,1,2)000176458xxxGBWLEH1x3!(4,0,2)(0,1,2)000167548xxxGWBELH1x3!(4,0,2)(0,1,2)000167458xxxGWBLEH1x3!(4,0,2)(0,1,2)000017548xxxxGBELH1x4!(4,0,2)(0,1,2)000017458xxxxGBLEH1x4!(4,0,2)(0,1,2)000016458xxxxGWLEH1x4!(4,0,2)(0,1,2)000187546xxxGHBELW1x3!(4,0,2)(0,0,2)000017546xxxxGBELW1x4!(4,0,2)(0,0,2)000001548xxxxxGELH1x5!(4,0,2)(0,0,2)000001458xxxxxGLEH1x5!(4,0,2)(0,0,2)000174658xxxGBLWEH1x3!(2,1,2)(0,1,2)000174568xxxGBLEWH1x3!(2,1,2)(0,1,2)000147568xxxGLBEWH1x3!(2,1,2)(0,1,2)000017568xxxxGBEWH1x4!(2,1,2)(0,1,2)000178456xxxGBHLEW1x3!(0,2,2)(0,0,2)000174856xxxGBLHEW1x3!(0,2,2)(0,0,2)000168547xxxGWHELB1x3!(0,2,2)(0,0,2)000168457xxxGWHLEB1x3!(0,2,2)(0,0,2) 174

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000165847xxxGWEHLB1x3!(0,2,2)(0,0,2)000165487xxxGWELHB1x3!(0,2,2)(0,0,2)000164857xxxGWLHEB1x3!(0,2,2)(0,0,2)000164587xxxGWLEHB1x3!(0,2,2)(0,0,2)000157846xxxGEBHLW1x3!(0,2,2)(0,0,2)000157486xxxGEBLHW1x3!(0,2,2)(0,0,2)000156847xxxGEWHLB1x3!(0,2,2)(0,0,2)000156487xxxGEWLHB1x3!(0,2,2)(0,0,2)000147856xxxGLBHEW1x3!(0,2,2)(0,0,2)000017856xxxxGBHEW1x4!(0,2,2)(0,0,2)000017846xxxxGBHLW1x4!(0,2,2)(0,0,2)000017486xxxxGBLHW1x4!(0,2,2)(0,0,2)000016847xxxxGWHLB1x4!(0,2,2)(0,0,2)000016487xxxxGWLHB1x4!(0,2,2)(0,0,2)000165748xxxGWEBLH1x3!(0,1,2)000157648xxxGEBWLH1x3!(0,1,2)000156748xxxGEWBLH1x3!(0,1,2)000147658xxxGLBWEH1x3!(0,1,2)000146758xxxGLWBEH1x3!(0,1,2)000017658xxxxGBWEH1x4!(0,1,2)000017648xxxxGBWLH1x4!(0,1,2)000016758xxxxGWBEH1x4!(0,1,2)000016748xxxxGWBLH1x4!(0,1,2)000016548xxxxGWELH1x4!(0,1,2)000015748xxxxGEBLH1x4!(0,1,2)000015648xxxxGEWLH1x4!(0,1,2)000014758xxxxGLBEH1x4!(0,1,2)000001758xxxxxGBEH1x5!(0,1,2)000001748xxxxxGBLH1x5!(0,1,2)000001648xxxxxGWLH1x5!(0,1,2)000187456xxxGHBLEW1x3!(0,0,2)000186547xxxGHWELB1x3!(0,0,2)000186457xxxGHWLEB1x3!(0,0,2)000185746xxxGHEBLW1x3!(0,0,2)000185647xxxGHEWLB1x3!(0,0,2)000185476xxxGHELBW1x3!(0,0,2)000185467xxxGHELWB1x3!(0,0,2)000184756xxxGHLBEW1x3!(0,0,2)000184657xxxGHLWEB1x3!(0,0,2)000184576xxxGHLEBW1x3!(0,0,2) 175

PAGE 176

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000184567xxxGHLEWB1x3!(0,0,2)000158746xxxGEHBLW1x3!(0,0,2)000158647xxxGEHWLB1x3!(0,0,2)000158476xxxGEHLBW1x3!(0,0,2)000158467xxxGEHLWB1x3!(0,0,2)000154876xxxGELHBW1x3!(0,0,2)000154867xxxGELHWB1x3!(0,0,2)000148756xxxGLHBEW1x3!(0,0,2)000148657xxxGLHWEB1x3!(0,0,2)000148576xxxGLHEBW1x3!(0,0,2)000148567xxxGLHEWB1x3!(0,0,2)000145876xxxGLEHBW1x3!(0,0,2)000145867xxxGLEHWB1x3!(0,0,2)000018756xxxxGHBEW1x4!(0,0,2)000018746xxxxGHBLW1x4!(0,0,2)000018657xxxxGHWEB1x4!(0,0,2)000018647xxxxGHWLB1x4!(0,0,2)000018576xxxxGHEBW1x4!(0,0,2)000018567xxxxGHEWB1x4!(0,0,2)000018476xxxxGHLBW1x4!(0,0,2)000018467xxxxGHLWB1x4!(0,0,2)000017456xxxxGBLEW1x4!(0,0,2)000016547xxxxGWELB1x4!(0,0,2)000016457xxxxGWLEB1x4!(0,0,2)000015876xxxxGEHBW1x4!(0,0,2)000015867xxxxGEHWB1x4!(0,0,2)000015746xxxxGEBLW1x4!(0,0,2)000015647xxxxGEWLB1x4!(0,0,2)000015476xxxxGELBW1x4!(0,0,2)000015467xxxxGELWB1x4!(0,0,2)000014876xxxxGLHBW1x4!(0,0,2)000014867xxxxGLHWB1x4!(0,0,2)000014756xxxxGLBEW1x4!(0,0,2)000014657xxxxGLWEB1x4!(0,0,2)000014576xxxxGLEBW1x4!(0,0,2)000014567xxxxGLEWB1x4!(0,0,2)000001876xxxxxGHBW1x5!(0,0,2)000001867xxxxxGHWB1x5!(0,0,2)000001756xxxxxGBEW1x5!(0,0,2)000001746xxxxxGBLW1x5!(0,0,2) 176

PAGE 177

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000001657xxxxxGWEB1x5!(0,0,2)000001647xxxxxGWLB1x5!(0,0,2)000001576xxxxxGEBW1x5!(0,0,2)000001567xxxxxGEWB1x5!(0,0,2)000001476xxxxxGLBW1x5!(0,0,2)000001467xxxxxGLWB1x5!(0,0,2)000000176xxxxxxGBW1x6!(0,0,2)000000167xxxxxxGWB1x6!(0,0,2)000000158xxxxxxGEH1x6!(0,0,2)000000148xxxxxxGLH1x6!(0,0,2)(2,0,1)000035468xxxxUELWH2x4!(4,1,1)(2,1,1)(0,1,1)(0,0,1)000275846xxxQBEHLW1x3!(4,0,1)(2,1,1)(0,2,1)(0,0,1)000275486xxxQBELHW1x3!(4,0,1)(2,1,1)(0,2,1)(0,0,1)000035846xxxxUEHLW2x4!(4,0,1)(2,1,1)(0,1,1)(0,0,1)000035648xxxxUEWLH2x4!(4,0,1)(2,1,1)(0,1,1)(0,0,1)000035486xxxxUELHW2x4!(4,0,1)(2,1,1)(0,1,1)(0,0,1)000375468xxxUBELWH2x3!(4,1,1)(2,1,1)(0,1,1)000275468xxxQBELWH1x3!(4,1,1)(2,1,1)(0,1,1)000375846xxxUBEHLW2x3!(4,0,1)(2,1,1)(0,2,1)000375486xxxUBELHW2x3!(4,0,1)(2,1,1)(0,2,1)000375648xxxUBEWLH2x3!(4,0,1)(2,1,1)(0,1,1)000275648xxxQBEWLH1x3!(4,0,1)(2,1,1)(0,1,1)000264758xxxQWLBEH1x3!(4,0,1)(2,1,1)(0,1,1)000278546xxxQBHELW1x3!(4,0,1)(0,2,1)(0,0,1)000038546xxxxUHELW2x4!(4,0,1)(0,1,1)(0,0,1)000036548xxxxUWELH2x4!(4,0,1)(0,1,1)(0,0,1)000036458xxxxUWLEH2x4!(4,0,1)(0,1,1)(0,0,1)000274586xxxQBLEHW1x3!(2,1,1)(0,2,1)(0,0,1)000247586xxxQLBEHW1x3!(2,1,1)(0,2,1)(0,0,1)000027586xxxxQBEHW1x4!(2,1,1)(0,2,1)(0,0,1)000034658xxxxULWEH2x4!(2,1,1)(0,1,1)(0,0,1)000034586xxxxULEHW2x4!(2,1,1)(0,1,1)(0,0,1)000034568xxxxULEWH2x4!(2,1,1)(0,1,1)(0,0,1)000003586xxxxxUEHW2x5!(2,1,1)(0,1,1)(0,0,1)000003568xxxxxUEWH2x5!(2,1,1)(0,1,1)(0,0,1)000378546xxxUBHELW2x3!(4,0,1)(0,2,1)000376548xxxUBWELH2x3!(4,0,1)(0,1,1)000376458xxxUBWLEH2x3!(4,0,1)(0,1,1)000367548xxxUWBELH2x3!(4,0,1)(0,1,1)000367458xxxUWBLEH2x3!(4,0,1)(0,1,1) 177

PAGE 178

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000276548xxxQBWELH1x3!(4,0,1)(0,1,1)000276458xxxQBWLEH1x3!(4,0,1)(0,1,1)000267548xxxQWBELH1x3!(4,0,1)(0,1,1)000267458xxxQWBLEH1x3!(4,0,1)(0,1,1)000037548xxxxUBELH2x4!(4,0,1)(0,1,1)000037458xxxxUBLEH2x4!(4,0,1)(0,1,1)000027548xxxxQBELH1x4!(4,0,1)(0,1,1)000027458xxxxQBLEH1x4!(4,0,1)(0,1,1)000026458xxxxQWLEH1x4!(4,0,1)(0,1,1)000287546xxxQHBELW1x3!(4,0,1)(0,0,1)000027546xxxxQBELW1x4!(4,0,1)(0,0,1)000003548xxxxxUELH2x5!(4,0,1)(0,0,1)000003546xxxxxUELW2x5!(4,0,1)(0,0,1)000003458xxxxxULEH2x5!(4,0,1)(0,0,1)000002548xxxxxQELH1x5!(4,0,1)(0,0,1)000002458xxxxxQLEH1x5!(4,0,1)(0,0,1)000374586xxxUBLEHW2x3!(2,1,1)(0,2,1)000347586xxxULBEHW2x3!(2,1,1)(0,2,1)000037586xxxxUBEHW2x4!(2,1,1)(0,2,1)000374658xxxUBLWEH2x3!(2,1,1)(0,1,1)000374568xxxUBLEWH2x3!(2,1,1)(0,1,1)000347568xxxULBEWH2x3!(2,1,1)(0,1,1)000274658xxxQBLWEH1x3!(2,1,1)(0,1,1)000274568xxxQBLEWH1x3!(2,1,1)(0,1,1)000247568xxxQLBEWH1x3!(2,1,1)(0,1,1)000037568xxxxUBEWH2x4!(2,1,1)(0,1,1)000027568xxxxQBEWH1x4!(2,1,1)(0,1,1)000278456xxxQBHLEW1x3!(0,2,1)(0,0,1)000274856xxxQBLHEW1x3!(0,2,1)(0,0,1)000268547xxxQWHELB1x3!(0,2,1)(0,0,1)000268457xxxQWHLEB1x3!(0,2,1)(0,0,1)000265847xxxQWEHLB1x3!(0,2,1)(0,0,1)000265487xxxQWELHB1x3!(0,2,1)(0,0,1)000264857xxxQWLHEB1x3!(0,2,1)(0,0,1)000264587xxxQWLEHB1x3!(0,2,1)(0,0,1)000257846xxxQEBHLW1x3!(0,2,1)(0,0,1)000257486xxxQEBLHW1x3!(0,2,1)(0,0,1)000256847xxxQEWHLB1x3!(0,2,1)(0,0,1)000256487xxxQEWLHB1x3!(0,2,1)(0,0,1)000247856xxxQLBHEW1x3!(0,2,1)(0,0,1) 178

PAGE 179

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000027856xxxxQBHEW1x4!(0,2,1)(0,0,1)000027846xxxxQBHLW1x4!(0,2,1)(0,0,1)000027486xxxxQBLHW1x4!(0,2,1)(0,0,1)000026847xxxxQWHLB1x4!(0,2,1)(0,0,1)000026487xxxxQWLHB1x4!(0,2,1)(0,0,1)000038456xxxxUHLEW2x4!(0,1,1)(0,0,1)000034856xxxxULHEW2x4!(0,1,1)(0,0,1)000003856xxxxxUHEW2x5!(0,1,1)(0,0,1)000003846xxxxxUHLW2x5!(0,1,1)(0,0,1)000003658xxxxxUWEH2x5!(0,1,1)(0,0,1)000003648xxxxxUWLH2x5!(0,1,1)(0,0,1)000003486xxxxxULHW2x5!(0,1,1)(0,0,1)000387546xxxUHBELW2x3!(4,0,1)000037546xxxxUBELW2x4!(4,0,1)000378456xxxUBHLEW2x3!(0,2,1)000374856xxxUBLHEW2x3!(0,2,1)000357846xxxUEBHLW2x3!(0,2,1)000357486xxxUEBLHW2x3!(0,2,1)000347856xxxULBHEW2x3!(0,2,1)000037856xxxxUBHEW2x4!(0,2,1)000037846xxxxUBHLW2x4!(0,2,1)000037486xxxxUBLHW2x4!(0,2,1)000365748xxxUWEBLH2x3!(0,1,1)000364758xxxUWLBEH2x3!(0,1,1)000357648xxxUEBWLH2x3!(0,1,1)000356748xxxUEWBLH2x3!(0,1,1)000347658xxxULBWEH2x3!(0,1,1)000346758xxxULWBEH2x3!(0,1,1)000265748xxxQWEBLH1x3!(0,1,1)000257648xxxQEBWLH1x3!(0,1,1)000256748xxxQEWBLH1x3!(0,1,1)000247658xxxQLBWEH1x3!(0,1,1)000246758xxxQLWBEH1x3!(0,1,1)000037658xxxxUBWEH2x4!(0,1,1)000037648xxxxUBWLH2x4!(0,1,1)000036758xxxxUWBEH2x4!(0,1,1)000036748xxxxUWBLH2x4!(0,1,1)000035748xxxxUEBLH2x4!(0,1,1)000034758xxxxULBEH2x4!(0,1,1)000027658xxxxQBWEH1x4!(0,1,1) 179

PAGE 180

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000027648xxxxQBWLH1x4!(0,1,1)000026758xxxxQWBEH1x4!(0,1,1)000026748xxxxQWBLH1x4!(0,1,1)000026548xxxxQWELH1x4!(0,1,1)000025748xxxxQEBLH1x4!(0,1,1)000025648xxxxQEWLH1x4!(0,1,1)000024758xxxxQLBEH1x4!(0,1,1)000003758xxxxxUBEH2x5!(0,1,1)000003748xxxxxUBLH2x5!(0,1,1)000002758xxxxxQBEH1x5!(0,1,1)000002748xxxxxQBLH1x5!(0,1,1)000002648xxxxxQWLH1x5!(0,1,1)000287456xxxQHBLEW1x3!(0,0,1)000286547xxxQHWELB1x3!(0,0,1)000286457xxxQHWLEB1x3!(0,0,1)000285746xxxQHEBLW1x3!(0,0,1)000285647xxxQHEWLB1x3!(0,0,1)000285476xxxQHELBW1x3!(0,0,1)000285467xxxQHELWB1x3!(0,0,1)000284756xxxQHLBEW1x3!(0,0,1)000284657xxxQHLWEB1x3!(0,0,1)000284576xxxQHLEBW1x3!(0,0,1)000284567xxxQHLEWB1x3!(0,0,1)000258746xxxQEHBLW1x3!(0,0,1)000258647xxxQEHWLB1x3!(0,0,1)000258476xxxQEHLBW1x3!(0,0,1)000258467xxxQEHLWB1x3!(0,0,1)000254876xxxQELHBW1x3!(0,0,1)000254867xxxQELHWB1x3!(0,0,1)000248756xxxQLHBEW1x3!(0,0,1)000248657xxxQLHWEB1x3!(0,0,1)000248576xxxQLHEBW1x3!(0,0,1)000248567xxxQLHEWB1x3!(0,0,1)000245876xxxQLEHBW1x3!(0,0,1)000245867xxxQLEHWB1x3!(0,0,1)000028756xxxxQHBEW1x4!(0,0,1)000028746xxxxQHBLW1x4!(0,0,1)000028657xxxxQHWEB1x4!(0,0,1)000028647xxxxQHWLB1x4!(0,0,1)000028576xxxxQHEBW1x4!(0,0,1) 180

PAGE 181

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000028567xxxxQHEWB1x4!(0,0,1)000028476xxxxQHLBW1x4!(0,0,1)000028467xxxxQHLWB1x4!(0,0,1)000027456xxxxQBLEW1x4!(0,0,1)000026547xxxxQWELB1x4!(0,0,1)000026457xxxxQWLEB1x4!(0,0,1)000025876xxxxQEHBW1x4!(0,0,1)000025867xxxxQEHWB1x4!(0,0,1)000025746xxxxQEBLW1x4!(0,0,1)000025647xxxxQEWLB1x4!(0,0,1)000025476xxxxQELBW1x4!(0,0,1)000025467xxxxQELWB1x4!(0,0,1)000024876xxxxQLHBW1x4!(0,0,1)000024867xxxxQLHWB1x4!(0,0,1)000024756xxxxQLBEW1x4!(0,0,1)000024657xxxxQLWEB1x4!(0,0,1)000024576xxxxQLEBW1x4!(0,0,1)000024567xxxxQLEWB1x4!(0,0,1)000003456xxxxxULEW2x5!(0,0,1)000002876xxxxxQHBW1x5!(0,0,1)000002867xxxxxQHWB1x5!(0,0,1)000002756xxxxxQBEW1x5!(0,0,1)000002746xxxxxQBLW1x5!(0,0,1)000002657xxxxxQWEB1x5!(0,0,1)000002647xxxxxQWLB1x5!(0,0,1)000002576xxxxxQEBW1x5!(0,0,1)000002567xxxxxQEWB1x5!(0,0,1)000002476xxxxxQLBW1x5!(0,0,1)000002467xxxxxQLWB1x5!(0,0,1)000000358xxxxxxUEH2x6!(0,0,1)000000356xxxxxxUEW2x6!(0,0,1)000000348xxxxxxULH2x6!(0,0,1)000000346xxxxxxULW2x6!(0,0,1)000000276xxxxxxQBW1x6!(0,0,1)000000267xxxxxxQWB1x6!(0,0,1)000000258xxxxxxQEH1x6!(0,0,1)000000248xxxxxxQLH1x6!(0,0,1)000387456xxxUHBLEW2x3!000385746xxxUHEBLW2x3!000385476xxxUHELBW2x3! 181

PAGE 182

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000384756xxxUHLBEW2x3!000384576xxxUHLEBW2x3!000358746xxxUEHBLW2x3!000358476xxxUEHLBW2x3!000354876xxxUELHBW2x3!000348756xxxULHBEW2x3!000348576xxxULHEBW2x3!000345876xxxULEHBW2x3!000038756xxxxUHBEW2x4!000038746xxxxUHBLW2x4!000038576xxxxUHEBW2x4!000038476xxxxUHLBW2x4!000037456xxxxUBLEW2x4!000035876xxxxUEHBW2x4!000035746xxxxUEBLW2x4!000035476xxxxUELBW2x4!000034876xxxxULHBW2x4!000034756xxxxULBEW2x4!000034576xxxxULEBW2x4!000003876xxxxxUHBW2x5!000003756xxxxxUBEW2x5!000003746xxxxxUBLW2x5!000003576xxxxxUEBW2x5!000003476xxxxxULBW2x5!000000376xxxxxxUBW2x6!(1,2,2)000168754xxxGWHBEL1x3!(3,2,2)(3,0,2)(1,0,2)000175864xxxGBEHWL1x3!(3,1,2)(3,0,2)(1,0,2)000178654xxxGBHWEL1x3!(3,0,2)(1,0,2)000178564xxxGBHEWL1x3!(3,0,2)(1,0,2)000168574xxxGWHEBL1x3!(1,0,2)000165874xxxGWEHBL1x3!(1,0,2)000157864xxxGEBHWL1x3!(1,0,2)000156874xxxGEWHBL1x3!(1,0,2)000017864xxxxGBHWL1x4!(1,0,2)000016874xxxxGWHBL1x4!(1,0,2)(1,2,1)000268754xxxQWHBEL1x3!(3,2,1)(3,0,1)(1,0,1)000375864xxxUBEHWL2x3!(3,1,1)(3,0,1)(1,0,1)000275864xxxQBEHWL1x3!(3,1,1)(3,0,1)(1,0,1)000378654xxxUBHWEL2x3!(3,0,1)(1,0,1)000378564xxxUBHEWL2x3!(3,0,1)(1,0,1) 182

PAGE 183

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000278654xxxQBHWEL1x3!(3,0,1)(1,0,1)000278564xxxQBHEWL1x3!(3,0,1)(1,0,1)000357864xxxUEBHWL2x3!(1,0,1)000268574xxxQWHEBL1x3!(1,0,1)000265874xxxQWEHBL1x3!(1,0,1)000257864xxxQEBHWL1x3!(1,0,1)000256874xxxQEWHBL1x3!(1,0,1)000037864xxxxUBHWL2x4!(1,0,1)000027864xxxxQBHWL1x4!(1,0,1)000026874xxxxQWHBL1x4!(1,0,1)(1,1,2)000176854xxxGBWHEL1x3!(3,1,2)(3,0,2)(1,0,2)000175684xxxGBEWHL1x3!(3,1,2)(3,0,2)(1,0,2)000167854xxxGWBHEL1x3!(3,1,2)(3,0,2)(1,0,2)000017854xxxxGBHEL1x4!(3,1,2)(3,0,2)(1,0,2)000016854xxxxGWHEL1x4!(3,1,2)(1,0,2)000176584xxxGBWEHL1x3!(3,0,2)(1,0,2)000167584xxxGWBEHL1x3!(3,0,2)(1,0,2)000017584xxxxGBEHL1x4!(3,0,2)(1,0,2)000165784xxxGWEBHL1x3!(1,0,2)000157684xxxGEBWHL1x3!(1,0,2)000156784xxxGEWBHL1x3!(1,0,2)000017684xxxxGBWHL1x4!(1,0,2)000016784xxxxGWBHL1x4!(1,0,2)000016584xxxxGWEHL1x4!(1,0,2)000015784xxxxGEBHL1x4!(1,0,2)000015684xxxxGEWHL1x4!(1,0,2)000001784xxxxxGBHL1x5!(1,0,2)000001684xxxxxGWHL1x5!(1,0,2)(1,1,1)000376854xxxUBWHEL2x3!(3,1,1)(3,0,1)(1,0,1)000375684xxxUBEWHL2x3!(3,1,1)(3,0,1)(1,0,1)000367854xxxUWBHEL2x3!(3,1,1)(3,0,1)(1,0,1)000276854xxxQBWHEL1x3!(3,1,1)(3,0,1)(1,0,1)000275684xxxQBEWHL1x3!(3,1,1)(3,0,1)(1,0,1)000267854xxxQWBHEL1x3!(3,1,1)(3,0,1)(1,0,1)000037854xxxxUBHEL2x4!(3,1,1)(3,0,1)(1,0,1)000036854xxxxUWHEL2x4!(3,1,1)(3,0,1)(1,0,1)000035864xxxxUEHWL2x4!(3,1,1)(3,0,1)(1,0,1)000035684xxxxUEWHL2x4!(3,1,1)(3,0,1)(1,0,1)000027854xxxxQBHEL1x4!(3,1,1)(3,0,1)(1,0,1)000026854xxxxQWHEL1x4!(3,1,1)(1,0,1) 183

PAGE 184

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000376584xxxUBWEHL2x3!(3,0,1)(1,0,1)000367584xxxUWBEHL2x3!(3,0,1)(1,0,1)000276584xxxQBWEHL1x3!(3,0,1)(1,0,1)000267584xxxQWBEHL1x3!(3,0,1)(1,0,1)000038654xxxxUHWEL2x4!(3,0,1)(1,0,1)000038564xxxxUHEWL2x4!(3,0,1)(1,0,1)000037584xxxxUBEHL2x4!(3,0,1)(1,0,1)000036584xxxxUWEHL2x4!(3,0,1)(1,0,1)000027584xxxxQBEHL1x4!(3,0,1)(1,0,1)000365784xxxUWEBHL2x3!(1,0,1)000357684xxxUEBWHL2x3!(1,0,1)000356784xxxUEWBHL2x3!(1,0,1)000265784xxxQWEBHL1x3!(1,0,1)000257684xxxQEBWHL1x3!(1,0,1)000256784xxxQEWBHL1x3!(1,0,1)000037684xxxxUBWHL2x4!(1,0,1)000036784xxxxUWBHL2x4!(1,0,1)000035784xxxxUEBHL2x4!(1,0,1)000027684xxxxQBWHL1x4!(1,0,1)000026784xxxxQWBHL1x4!(1,0,1)000026584xxxxQWEHL1x4!(1,0,1)000025784xxxxQEBHL1x4!(1,0,1)000025684xxxxQEWHL1x4!(1,0,1)000003864xxxxxUHWL2x5!(1,0,1)000003784xxxxxUBHL2x5!(1,0,1)000003684xxxxxUWHL2x5!(1,0,1)000002784xxxxxQBHL1x5!(1,0,1)000002684xxxxxQWHL1x5!(1,0,1)(1,0,2)000168754xxxGWHBEL1x3!(3,2,2)(3,0,2)(1,2,2)000175864xxxGBEHWL1x3!(3,1,2)(3,0,2)(1,2,2)000176854xxxGBWHEL1x3!(3,1,2)(3,0,2)(1,1,2)000175684xxxGBEWHL1x3!(3,1,2)(3,0,2)(1,1,2)000167854xxxGWBHEL1x3!(3,1,2)(3,0,2)(1,1,2)000017854xxxxGBHEL1x4!(3,1,2)(3,0,2)(1,1,2)000016854xxxxGWHEL1x4!(3,1,2)(1,1,2)000178654xxxGBHWEL1x3!(3,0,2)(1,2,2)000178564xxxGBHEWL1x3!(3,0,2)(1,2,2)000176584xxxGBWEHL1x3!(3,0,2)(1,1,2)000167584xxxGWBEHL1x3!(3,0,2)(1,1,2)000017584xxxxGBEHL1x4!(3,0,2)(1,1,2) 184

PAGE 185

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000187654xxxGHBWEL1x3!(3,0,2)000187564xxxGHBEWL1x3!(3,0,2)000186754xxxGHWBEL1x3!(3,0,2)000018754xxxxGHBEL1x4!(3,0,2)000017654xxxxGBWEL1x4!(3,0,2)000017564xxxxGBEWL1x4!(3,0,2)000016754xxxxGWBEL1x4!(3,0,2)000001854xxxxxGHEL1x5!(3,0,2)000001754xxxxxGBEL1x5!(3,0,2)000001584xxxxxGEHL1x5!(3,0,2)000000154xxxxxxGEL1x6!(3,0,2)000168574xxxGWHEBL1x3!(1,2,2)000165874xxxGWEHBL1x3!(1,2,2)000157864xxxGEBHWL1x3!(1,2,2)000156874xxxGEWHBL1x3!(1,2,2)000017864xxxxGBHWL1x4!(1,2,2)000016874xxxxGWHBL1x4!(1,2,2)000165784xxxGWEBHL1x3!(1,1,2)000157684xxxGEBWHL1x3!(1,1,2)000156784xxxGEWBHL1x3!(1,1,2)000017684xxxxGBWHL1x4!(1,1,2)000016784xxxxGWBHL1x4!(1,1,2)000016584xxxxGWEHL1x4!(1,1,2)000015784xxxxGEBHL1x4!(1,1,2)000015684xxxxGEWHL1x4!(1,1,2)000001784xxxxxGBHL1x5!(1,1,2)000001684xxxxxGWHL1x5!(1,1,2)000186574xxxGHWEBL1x3!000185764xxxGHEBWL1x3!000185674xxxGHEWBL1x3!000158764xxxGEHBWL1x3!000158674xxxGEHWBL1x3!000018764xxxxGHBWL1x4!000018674xxxxGHWBL1x4!000018654xxxxGHWEL1x4!000018574xxxxGHEBL1x4!000018564xxxxGHEWL1x4!000016574xxxxGWEBL1x4!000015874xxxxGEHBL1x4!000015864xxxxGEHWL1x4! 185

PAGE 186

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000015764xxxxGEBWL1x4!000015674xxxxGEWBL1x4!000001874xxxxxGHBL1x5!000001864xxxxxGHWL1x5!000001764xxxxxGBWL1x5!000001674xxxxxGWBL1x5!000001654xxxxxGWEL1x5!000001574xxxxxGEBL1x5!000001564xxxxxGEWL1x5!000000184xxxxxxGHL1x6!000000174xxxxxxGBL1x6!000000164xxxxxxGWL1x6!000000014xxxxxxxGL1x7!(1,0,1)000268754xxxQWHBEL1x3!(3,2,1)(3,0,1)(1,2,1)000375864xxxUBEHWL2x3!(3,1,1)(3,0,1)(1,2,1)000275864xxxQBEHWL1x3!(3,1,1)(3,0,1)(1,2,1)000376854xxxUBWHEL2x3!(3,1,1)(3,0,1)(1,1,1)000375684xxxUBEWHL2x3!(3,1,1)(3,0,1)(1,1,1)000367854xxxUWBHEL2x3!(3,1,1)(3,0,1)(1,1,1)000276854xxxQBWHEL1x3!(3,1,1)(3,0,1)(1,1,1)000275684xxxQBEWHL1x3!(3,1,1)(3,0,1)(1,1,1)000267854xxxQWBHEL1x3!(3,1,1)(3,0,1)(1,1,1)000037854xxxxUBHEL2x4!(3,1,1)(3,0,1)(1,1,1)000036854xxxxUWHEL2x4!(3,1,1)(3,0,1)(1,1,1)000035864xxxxUEHWL2x4!(3,1,1)(3,0,1)(1,1,1)000035684xxxxUEWHL2x4!(3,1,1)(3,0,1)(1,1,1)000027854xxxxQBHEL1x4!(3,1,1)(3,0,1)(1,1,1)000026854xxxxQWHEL1x4!(3,1,1)(1,1,1)000378654xxxUBHWEL2x3!(3,0,1)(1,2,1)000378564xxxUBHEWL2x3!(3,0,1)(1,2,1)000278654xxxQBHWEL1x3!(3,0,1)(1,2,1)000278564xxxQBHEWL1x3!(3,0,1)(1,2,1)000376584xxxUBWEHL2x3!(3,0,1)(1,1,1)000367584xxxUWBEHL2x3!(3,0,1)(1,1,1)000276584xxxQBWEHL1x3!(3,0,1)(1,1,1)000267584xxxQWBEHL1x3!(3,0,1)(1,1,1)000038654xxxxUHWEL2x4!(3,0,1)(1,1,1)000038564xxxxUHEWL2x4!(3,0,1)(1,1,1)000037584xxxxUBEHL2x4!(3,0,1)(1,1,1)000036584xxxxUWEHL2x4!(3,0,1)(1,1,1) 186

PAGE 187

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000027584xxxxQBEHL1x4!(3,0,1)(1,1,1)000387654xxxUHBWEL2x3!(3,0,1)000387564xxxUHBEWL2x3!(3,0,1)000386754xxxUHWBEL2x3!(3,0,1)000368754xxxUWHBEL2x3!(3,0,1)000287654xxxQHBWEL1x3!(3,0,1)000287564xxxQHBEWL1x3!(3,0,1)000286754xxxQHWBEL1x3!(3,0,1)000038754xxxxUHBEL2x4!(3,0,1)000037654xxxxUBWEL2x4!(3,0,1)000037564xxxxUBEWL2x4!(3,0,1)000036754xxxxUWBEL2x4!(3,0,1)000028754xxxxQHBEL1x4!(3,0,1)000027654xxxxQBWEL1x4!(3,0,1)000027564xxxxQBEWL1x4!(3,0,1)000026754xxxxQWBEL1x4!(3,0,1)000003854xxxxxUHEL2x5!(3,0,1)000003754xxxxxUBEL2x5!(3,0,1)000003654xxxxxUWEL2x5!(3,0,1)000003584xxxxxUEHL2x5!(3,0,1)000003564xxxxxUEWL2x5!(3,0,1)000002854xxxxxQHEL1x5!(3,0,1)000002754xxxxxQBEL1x5!(3,0,1)000002584xxxxxQEHL1x5!(3,0,1)000000354xxxxxxUEL2x6!(3,0,1)000000254xxxxxxQEL1x6!(3,0,1)000357864xxxUEBHWL2x3!(1,2,1)000268574xxxQWHEBL1x3!(1,2,1)000265874xxxQWEHBL1x3!(1,2,1)000257864xxxQEBHWL1x3!(1,2,1)000256874xxxQEWHBL1x3!(1,2,1)000037864xxxxUBHWL2x4!(1,2,1)000027864xxxxQBHWL1x4!(1,2,1)000026874xxxxQWHBL1x4!(1,2,1)000365784xxxUWEBHL2x3!(1,1,1)000357684xxxUEBWHL2x3!(1,1,1)000356784xxxUEWBHL2x3!(1,1,1)000265784xxxQWEBHL1x3!(1,1,1)000257684xxxQEBWHL1x3!(1,1,1)000256784xxxQEWBHL1x3!(1,1,1) 187

PAGE 188

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000037684xxxxUBWHL2x4!(1,1,1)000036784xxxxUWBHL2x4!(1,1,1)000035784xxxxUEBHL2x4!(1,1,1)000027684xxxxQBWHL1x4!(1,1,1)000026784xxxxQWBHL1x4!(1,1,1)000026584xxxxQWEHL1x4!(1,1,1)000025784xxxxQEBHL1x4!(1,1,1)000025684xxxxQEWHL1x4!(1,1,1)000003864xxxxxUHWL2x5!(1,1,1)000003784xxxxxUBHL2x5!(1,1,1)000003684xxxxxUWHL2x5!(1,1,1)000002784xxxxxQBHL1x5!(1,1,1)000002684xxxxxQWHL1x5!(1,1,1)000386574xxxUHWEBL2x3!000385764xxxUHEBWL2x3!000385674xxxUHEWBL2x3!000368574xxxUWHEBL2x3!000365874xxxUWEHBL2x3!000358764xxxUEHBWL2x3!000358674xxxUEHWBL2x3!000356874xxxUEWHBL2x3!000286574xxxQHWEBL1x3!000285764xxxQHEBWL1x3!000285674xxxQHEWBL1x3!000258764xxxQEHBWL1x3!000258674xxxQEHWBL1x3!000038764xxxxUHBWL2x4!000038674xxxxUHWBL2x4!000038574xxxxUHEBL2x4!000036874xxxxUWHBL2x4!000036574xxxxUWEBL2x4!000035874xxxxUEHBL2x4!000035764xxxxUEBWL2x4!000035674xxxxUEWBL2x4!000028764xxxxQHBWL1x4!000028674xxxxQHWBL1x4!000028654xxxxQHWEL1x4!000028574xxxxQHEBL1x4!000028564xxxxQHEWL1x4!000026574xxxxQWEBL1x4! 188

PAGE 189

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000025874xxxxQEHBL1x4!000025864xxxxQEHWL1x4!000025764xxxxQEBWL1x4!000025674xxxxQEWBL1x4!000003874xxxxxUHBL2x5!000003764xxxxxUBWL2x5!000003674xxxxxUWBL2x5!000003574xxxxxUEBL2x5!000002874xxxxxQHBL1x5!000002864xxxxxQHWL1x5!000002764xxxxxQBWL1x5!000002674xxxxxQWBL1x5!000002654xxxxxQWEL1x5!000002574xxxxxQEBL1x5!000002564xxxxxQEWL1x5!000000384xxxxxxUHL2x6!000000374xxxxxxUBL2x6!000000364xxxxxxUWL2x6!000000284xxxxxxQHL1x6!000000274xxxxxxQBL1x6!000000264xxxxxxQWL1x6!000000034xxxxxxxUL2x7!000000024xxxxxxxQL1x7!(0,2,2)000175846xxxGBEHLW1x3!(4,0,2)(2,1,2)(2,0,2)(0,0,2)000175486xxxGBELHW1x3!(4,0,2)(2,1,2)(2,0,2)(0,0,2)000178546xxxGBHELW1x3!(4,0,2)(2,0,2)(0,0,2)000174586xxxGBLEHW1x3!(2,1,2)(2,0,2)(0,0,2)000147586xxxGLBEHW1x3!(2,1,2)(2,0,2)(0,0,2)000017586xxxxGBEHW1x4!(2,1,2)(2,0,2)(0,0,2)000178456xxxGBHLEW1x3!(2,0,2)(0,0,2)000174856xxxGBLHEW1x3!(2,0,2)(0,0,2)000168547xxxGWHELB1x3!(2,0,2)(0,0,2)000168457xxxGWHLEB1x3!(2,0,2)(0,0,2)000165847xxxGWEHLB1x3!(2,0,2)(0,0,2)000165487xxxGWELHB1x3!(2,0,2)(0,0,2)000164857xxxGWLHEB1x3!(2,0,2)(0,0,2)000164587xxxGWLEHB1x3!(2,0,2)(0,0,2)000157846xxxGEBHLW1x3!(2,0,2)(0,0,2)000157486xxxGEBLHW1x3!(2,0,2)(0,0,2)000156847xxxGEWHLB1x3!(2,0,2)(0,0,2) 189

PAGE 190

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000156487xxxGEWLHB1x3!(2,0,2)(0,0,2)000147856xxxGLBHEW1x3!(2,0,2)(0,0,2)000017856xxxxGBHEW1x4!(2,0,2)(0,0,2)000017846xxxxGBHLW1x4!(2,0,2)(0,0,2)000017486xxxxGBLHW1x4!(2,0,2)(0,0,2)000016847xxxxGWHLB1x4!(2,0,2)(0,0,2)000016487xxxxGWLHB1x4!(2,0,2)(0,0,2)000154786xxxGELBHW1x3!(0,0,2)000154687xxxGELWHB1x3!(0,0,2)000146857xxxGLWHEB1x3!(0,0,2)000146587xxxGLWEHB1x3!(0,0,2)000145786xxxGLEBHW1x3!(0,0,2)000145687xxxGLEWHB1x3!(0,0,2)000016857xxxxGWHEB1x4!(0,0,2)000016587xxxxGWEHB1x4!(0,0,2)000015786xxxxGEBHW1x4!(0,0,2)000015687xxxxGEWHB1x4!(0,0,2)000014786xxxxGLBHW1x4!(0,0,2)000014687xxxxGLWHB1x4!(0,0,2)000001786xxxxxGBHW1x5!(0,0,2)000001687xxxxxGWHB1x5!(0,0,2)(0,2,1)000275846xxxQBEHLW1x3!(4,0,1)(2,1,1)(2,0,1)(0,0,1)000275486xxxQBELHW1x3!(4,0,1)(2,1,1)(2,0,1)(0,0,1)000375846xxxUBEHLW2x3!(4,0,1)(2,1,1)(2,0,1)000375486xxxUBELHW2x3!(4,0,1)(2,1,1)(2,0,1)000278546xxxQBHELW1x3!(4,0,1)(2,0,1)(0,0,1)000274586xxxQBLEHW1x3!(2,1,1)(2,0,1)(0,0,1)000247586xxxQLBEHW1x3!(2,1,1)(2,0,1)(0,0,1)000027586xxxxQBEHW1x4!(2,1,1)(2,0,1)(0,0,1)000378546xxxUBHELW2x3!(4,0,1)(2,0,1)000374586xxxUBLEHW2x3!(2,1,1)(2,0,1)000347586xxxULBEHW2x3!(2,1,1)(2,0,1)000037586xxxxUBEHW2x4!(2,1,1)(2,0,1)000278456xxxQBHLEW1x3!(2,0,1)(0,0,1)000274856xxxQBLHEW1x3!(2,0,1)(0,0,1)000268547xxxQWHELB1x3!(2,0,1)(0,0,1)000268457xxxQWHLEB1x3!(2,0,1)(0,0,1)000265847xxxQWEHLB1x3!(2,0,1)(0,0,1)000265487xxxQWELHB1x3!(2,0,1)(0,0,1)000264857xxxQWLHEB1x3!(2,0,1)(0,0,1) 190

PAGE 191

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000264587xxxQWLEHB1x3!(2,0,1)(0,0,1)000257846xxxQEBHLW1x3!(2,0,1)(0,0,1)000257486xxxQEBLHW1x3!(2,0,1)(0,0,1)000256847xxxQEWHLB1x3!(2,0,1)(0,0,1)000256487xxxQEWLHB1x3!(2,0,1)(0,0,1)000247856xxxQLBHEW1x3!(2,0,1)(0,0,1)000027856xxxxQBHEW1x4!(2,0,1)(0,0,1)000027846xxxxQBHLW1x4!(2,0,1)(0,0,1)000027486xxxxQBLHW1x4!(2,0,1)(0,0,1)000026847xxxxQWHLB1x4!(2,0,1)(0,0,1)000026487xxxxQWLHB1x4!(2,0,1)(0,0,1)000378456xxxUBHLEW2x3!(2,0,1)000374856xxxUBLHEW2x3!(2,0,1)000357846xxxUEBHLW2x3!(2,0,1)000357486xxxUEBLHW2x3!(2,0,1)000347856xxxULBHEW2x3!(2,0,1)000037856xxxxUBHEW2x4!(2,0,1)000037846xxxxUBHLW2x4!(2,0,1)000037486xxxxUBLHW2x4!(2,0,1)000254786xxxQELBHW1x3!(0,0,1)000254687xxxQELWHB1x3!(0,0,1)000246857xxxQLWHEB1x3!(0,0,1)000246587xxxQLWEHB1x3!(0,0,1)000245786xxxQLEBHW1x3!(0,0,1)000245687xxxQLEWHB1x3!(0,0,1)000026857xxxxQWHEB1x4!(0,0,1)000026587xxxxQWEHB1x4!(0,0,1)000025786xxxxQEBHW1x4!(0,0,1)000025687xxxxQEWHB1x4!(0,0,1)000024786xxxxQLBHW1x4!(0,0,1)000024687xxxxQLWHB1x4!(0,0,1)000002786xxxxxQBHW1x5!(0,0,1)000002687xxxxxQWHB1x5!(0,0,1)000354786xxxUELBHW2x3!000345786xxxULEBHW2x3!000035786xxxxUEBHW2x4!000034786xxxxULBHW2x4!000003786xxxxxUBHW2x5!(0,1,2)000175468xxxGBELWH1x3!(4,1,2)(2,1,2)(2,0,2)000175648xxxGBEWLH1x3!(4,0,2)(2,1,2)(2,0,2) 191

PAGE 192

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000164758xxxGWLBEH1x3!(4,0,2)(2,1,2)(2,0,2)000176548xxxGBWELH1x3!(4,0,2)(2,0,2)000176458xxxGBWLEH1x3!(4,0,2)(2,0,2)000167548xxxGWBELH1x3!(4,0,2)(2,0,2)000167458xxxGWBLEH1x3!(4,0,2)(2,0,2)000017548xxxxGBELH1x4!(4,0,2)(2,0,2)000017458xxxxGBLEH1x4!(4,0,2)(2,0,2)000016458xxxxGWLEH1x4!(4,0,2)(2,0,2)000174658xxxGBLWEH1x3!(2,1,2)(2,0,2)000174568xxxGBLEWH1x3!(2,1,2)(2,0,2)000147568xxxGLBEWH1x3!(2,1,2)(2,0,2)000017568xxxxGBEWH1x4!(2,1,2)(2,0,2)000165478xxxGWELBH1x3!(2,1,2)000164578xxxGWLEBH1x3!(2,1,2)000157468xxxGEBLWH1x3!(2,1,2)000156478xxxGEWLBH1x3!(2,1,2)000017468xxxxGBLWH1x4!(2,1,2)000016478xxxxGWLBH1x4!(2,1,2)000165748xxxGWEBLH1x3!(2,0,2)000157648xxxGEBWLH1x3!(2,0,2)000156748xxxGEWBLH1x3!(2,0,2)000147658xxxGLBWEH1x3!(2,0,2)000146758xxxGLWBEH1x3!(2,0,2)000017658xxxxGBWEH1x4!(2,0,2)000017648xxxxGBWLH1x4!(2,0,2)000016758xxxxGWBEH1x4!(2,0,2)000016748xxxxGWBLH1x4!(2,0,2)000016548xxxxGWELH1x4!(2,0,2)000015748xxxxGEBLH1x4!(2,0,2)000015648xxxxGEWLH1x4!(2,0,2)000014758xxxxGLBEH1x4!(2,0,2)000001758xxxxxGBEH1x5!(2,0,2)000001748xxxxxGBLH1x5!(2,0,2)000001648xxxxxGWLH1x5!(2,0,2)000154768xxxGELBWH1x3!000154678xxxGELWBH1x3!000146578xxxGLWEBH1x3!000145768xxxGLEBWH1x3!000145678xxxGLEWBH1x3!000016578xxxxGWEBH1x4! 192

PAGE 193

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000015768xxxxGEBWH1x4!000015678xxxxGEWBH1x4!000015478xxxxGELBH1x4!000015468xxxxGELWH1x4!000014768xxxxGLBWH1x4!000014678xxxxGLWBH1x4!000014658xxxxGLWEH1x4!000014578xxxxGLEBH1x4!000014568xxxxGLEWH1x4!000001768xxxxxGBWH1x5!000001678xxxxxGWBH1x5!000001658xxxxxGWEH1x5!000001578xxxxxGEBH1x5!000001568xxxxxGEWH1x5!000001478xxxxxGLBH1x5!000001468xxxxxGLWH1x5!000000178xxxxxxGBH1x6!000000168xxxxxxGWH1x6!(0,1,1)000035468xxxxUELWH2x4!(4,1,1)(2,1,1)(2,0,1)(0,0,1)000035846xxxxUEHLW2x4!(4,0,1)(2,1,1)(2,0,1)(0,0,1)000035648xxxxUEWLH2x4!(4,0,1)(2,1,1)(2,0,1)(0,0,1)000035486xxxxUELHW2x4!(4,0,1)(2,1,1)(2,0,1)(0,0,1)000375468xxxUBELWH2x3!(4,1,1)(2,1,1)(2,0,1)000275468xxxQBELWH1x3!(4,1,1)(2,1,1)(2,0,1)000375648xxxUBEWLH2x3!(4,0,1)(2,1,1)(2,0,1)000275648xxxQBEWLH1x3!(4,0,1)(2,1,1)(2,0,1)000264758xxxQWLBEH1x3!(4,0,1)(2,1,1)(2,0,1)000038546xxxxUHELW2x4!(4,0,1)(2,0,1)(0,0,1)000036548xxxxUWELH2x4!(4,0,1)(2,0,1)(0,0,1)000036458xxxxUWLEH2x4!(4,0,1)(2,0,1)(0,0,1)000034658xxxxULWEH2x4!(2,1,1)(2,0,1)(0,0,1)000034586xxxxULEHW2x4!(2,1,1)(2,0,1)(0,0,1)000034568xxxxULEWH2x4!(2,1,1)(2,0,1)(0,0,1)000003586xxxxxUEHW2x5!(2,1,1)(2,0,1)(0,0,1)000003568xxxxxUEWH2x5!(2,1,1)(2,0,1)(0,0,1)000376548xxxUBWELH2x3!(4,0,1)(2,0,1)000376458xxxUBWLEH2x3!(4,0,1)(2,0,1)000367548xxxUWBELH2x3!(4,0,1)(2,0,1)000367458xxxUWBLEH2x3!(4,0,1)(2,0,1)000276548xxxQBWELH1x3!(4,0,1)(2,0,1) 193

PAGE 194

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000276458xxxQBWLEH1x3!(4,0,1)(2,0,1)000267548xxxQWBELH1x3!(4,0,1)(2,0,1)000267458xxxQWBLEH1x3!(4,0,1)(2,0,1)000037548xxxxUBELH2x4!(4,0,1)(2,0,1)000037458xxxxUBLEH2x4!(4,0,1)(2,0,1)000027548xxxxQBELH1x4!(4,0,1)(2,0,1)000027458xxxxQBLEH1x4!(4,0,1)(2,0,1)000026458xxxxQWLEH1x4!(4,0,1)(2,0,1)000374658xxxUBLWEH2x3!(2,1,1)(2,0,1)000374568xxxUBLEWH2x3!(2,1,1)(2,0,1)000347568xxxULBEWH2x3!(2,1,1)(2,0,1)000274658xxxQBLWEH1x3!(2,1,1)(2,0,1)000274568xxxQBLEWH1x3!(2,1,1)(2,0,1)000247568xxxQLBEWH1x3!(2,1,1)(2,0,1)000037568xxxxUBEWH2x4!(2,1,1)(2,0,1)000027568xxxxQBEWH1x4!(2,1,1)(2,0,1)000003468xxxxxULWH2x5!(2,1,1)(0,0,1)000038456xxxxUHLEW2x4!(2,0,1)(0,0,1)000034856xxxxULHEW2x4!(2,0,1)(0,0,1)000003856xxxxxUHEW2x5!(2,0,1)(0,0,1)000003846xxxxxUHLW2x5!(2,0,1)(0,0,1)000003658xxxxxUWEH2x5!(2,0,1)(0,0,1)000003648xxxxxUWLH2x5!(2,0,1)(0,0,1)000003486xxxxxULHW2x5!(2,0,1)(0,0,1)000357468xxxUEBLWH2x3!(2,1,1)000265478xxxQWELBH1x3!(2,1,1)000264578xxxQWLEBH1x3!(2,1,1)000257468xxxQEBLWH1x3!(2,1,1)000256478xxxQEWLBH1x3!(2,1,1)000037468xxxxUBLWH2x4!(2,1,1)000027468xxxxQBLWH1x4!(2,1,1)000026478xxxxQWLBH1x4!(2,1,1)000365748xxxUWEBLH2x3!(2,0,1)000364758xxxUWLBEH2x3!(2,0,1)000357648xxxUEBWLH2x3!(2,0,1)000356748xxxUEWBLH2x3!(2,0,1)000347658xxxULBWEH2x3!(2,0,1)000346758xxxULWBEH2x3!(2,0,1)000265748xxxQWEBLH1x3!(2,0,1)000257648xxxQEBWLH1x3!(2,0,1) 194

PAGE 195

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000256748xxxQEWBLH1x3!(2,0,1)000247658xxxQLBWEH1x3!(2,0,1)000246758xxxQLWBEH1x3!(2,0,1)000037658xxxxUBWEH2x4!(2,0,1)000037648xxxxUBWLH2x4!(2,0,1)000036758xxxxUWBEH2x4!(2,0,1)000036748xxxxUWBLH2x4!(2,0,1)000035748xxxxUEBLH2x4!(2,0,1)000034758xxxxULBEH2x4!(2,0,1)000027658xxxxQBWEH1x4!(2,0,1)000027648xxxxQBWLH1x4!(2,0,1)000026758xxxxQWBEH1x4!(2,0,1)000026748xxxxQWBLH1x4!(2,0,1)000026548xxxxQWELH1x4!(2,0,1)000025748xxxxQEBLH1x4!(2,0,1)000025648xxxxQEWLH1x4!(2,0,1)000024758xxxxQLBEH1x4!(2,0,1)000003758xxxxxUBEH2x5!(2,0,1)000003748xxxxxUBLH2x5!(2,0,1)000002758xxxxxQBEH1x5!(2,0,1)000002748xxxxxQBLH1x5!(2,0,1)000002648xxxxxQWLH1x5!(2,0,1)000000386xxxxxxUHW2x6!(0,0,1)000000368xxxxxxUWH2x6!(0,0,1)000365478xxxUWELBH2x3!000364578xxxUWLEBH2x3!000356478xxxUEWLBH2x3!000354768xxxUELBWH2x3!000354678xxxUELWBH2x3!000346578xxxULWEBH2x3!000345768xxxULEBWH2x3!000345678xxxULEWBH2x3!000254768xxxQELBWH1x3!000254678xxxQELWBH1x3!000246578xxxQLWEBH1x3!000245768xxxQLEBWH1x3!000245678xxxQLEWBH1x3!000036578xxxxUWEBH2x4!000036478xxxxUWLBH2x4!000035768xxxxUEBWH2x4! 195

PAGE 196

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000035678xxxxUEWBH2x4!000035478xxxxUELBH2x4!000034768xxxxULBWH2x4!000034678xxxxULWBH2x4!000034578xxxxULEBH2x4!000026578xxxxQWEBH1x4!000025768xxxxQEBWH1x4!000025678xxxxQEWBH1x4!000025478xxxxQELBH1x4!000025468xxxxQELWH1x4!000024768xxxxQLBWH1x4!000024678xxxxQLWBH1x4!000024658xxxxQLWEH1x4!000024578xxxxQLEBH1x4!000024568xxxxQLEWH1x4!000003768xxxxxUBWH2x5!000003678xxxxxUWBH2x5!000003578xxxxxUEBH2x5!000003478xxxxxULBH2x5!000002768xxxxxQBWH1x5!000002678xxxxxQWBH1x5!000002658xxxxxQWEH1x5!000002578xxxxxQEBH1x5!000002568xxxxxQEWH1x5!000002478xxxxxQLBH1x5!000002468xxxxxQLWH1x5!000000378xxxxxxUBH2x6!000000278xxxxxxQBH1x6!000000268xxxxxxQWH1x6!(0,0,2)000175846xxxGBEHLW1x3!(4,0,2)(2,1,2)(2,0,2)(0,2,2)000175486xxxGBELHW1x3!(4,0,2)(2,1,2)(2,0,2)(0,2,2)000178546xxxGBHELW1x3!(4,0,2)(2,0,2)(0,2,2)000174586xxxGBLEHW1x3!(2,1,2)(2,0,2)(0,2,2)000147586xxxGLBEHW1x3!(2,1,2)(2,0,2)(0,2,2)000017586xxxxGBEHW1x4!(2,1,2)(2,0,2)(0,2,2)000187546xxxGHBELW1x3!(4,0,2)(2,0,2)000017546xxxxGBELW1x4!(4,0,2)(2,0,2)000001548xxxxxGELH1x5!(4,0,2)(2,0,2)000001458xxxxxGLEH1x5!(4,0,2)(2,0,2)000178456xxxGBHLEW1x3!(2,0,2)(0,2,2) 196

PAGE 197

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000174856xxxGBLHEW1x3!(2,0,2)(0,2,2)000168547xxxGWHELB1x3!(2,0,2)(0,2,2)000168457xxxGWHLEB1x3!(2,0,2)(0,2,2)000165847xxxGWEHLB1x3!(2,0,2)(0,2,2)000165487xxxGWELHB1x3!(2,0,2)(0,2,2)000164857xxxGWLHEB1x3!(2,0,2)(0,2,2)000164587xxxGWLEHB1x3!(2,0,2)(0,2,2)000157846xxxGEBHLW1x3!(2,0,2)(0,2,2)000157486xxxGEBLHW1x3!(2,0,2)(0,2,2)000156847xxxGEWHLB1x3!(2,0,2)(0,2,2)000156487xxxGEWLHB1x3!(2,0,2)(0,2,2)000147856xxxGLBHEW1x3!(2,0,2)(0,2,2)000017856xxxxGBHEW1x4!(2,0,2)(0,2,2)000017846xxxxGBHLW1x4!(2,0,2)(0,2,2)000017486xxxxGBLHW1x4!(2,0,2)(0,2,2)000016847xxxxGWHLB1x4!(2,0,2)(0,2,2)000016487xxxxGWLHB1x4!(2,0,2)(0,2,2)000187456xxxGHBLEW1x3!(2,0,2)000186547xxxGHWELB1x3!(2,0,2)000186457xxxGHWLEB1x3!(2,0,2)000185746xxxGHEBLW1x3!(2,0,2)000185647xxxGHEWLB1x3!(2,0,2)000185476xxxGHELBW1x3!(2,0,2)000185467xxxGHELWB1x3!(2,0,2)000184756xxxGHLBEW1x3!(2,0,2)000184657xxxGHLWEB1x3!(2,0,2)000184576xxxGHLEBW1x3!(2,0,2)000184567xxxGHLEWB1x3!(2,0,2)000158746xxxGEHBLW1x3!(2,0,2)000158647xxxGEHWLB1x3!(2,0,2)000158476xxxGEHLBW1x3!(2,0,2)000158467xxxGEHLWB1x3!(2,0,2)000154876xxxGELHBW1x3!(2,0,2)000154867xxxGELHWB1x3!(2,0,2)000148756xxxGLHBEW1x3!(2,0,2)000148657xxxGLHWEB1x3!(2,0,2)000148576xxxGLHEBW1x3!(2,0,2)000148567xxxGLHEWB1x3!(2,0,2)000145876xxxGLEHBW1x3!(2,0,2)000145867xxxGLEHWB1x3!(2,0,2) 197

PAGE 198

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000018756xxxxGHBEW1x4!(2,0,2)000018746xxxxGHBLW1x4!(2,0,2)000018657xxxxGHWEB1x4!(2,0,2)000018647xxxxGHWLB1x4!(2,0,2)000018576xxxxGHEBW1x4!(2,0,2)000018567xxxxGHEWB1x4!(2,0,2)000018476xxxxGHLBW1x4!(2,0,2)000018467xxxxGHLWB1x4!(2,0,2)000017456xxxxGBLEW1x4!(2,0,2)000016547xxxxGWELB1x4!(2,0,2)000016457xxxxGWLEB1x4!(2,0,2)000015876xxxxGEHBW1x4!(2,0,2)000015867xxxxGEHWB1x4!(2,0,2)000015746xxxxGEBLW1x4!(2,0,2)000015647xxxxGEWLB1x4!(2,0,2)000015476xxxxGELBW1x4!(2,0,2)000015467xxxxGELWB1x4!(2,0,2)000014876xxxxGLHBW1x4!(2,0,2)000014867xxxxGLHWB1x4!(2,0,2)000014756xxxxGLBEW1x4!(2,0,2)000014657xxxxGLWEB1x4!(2,0,2)000014576xxxxGLEBW1x4!(2,0,2)000014567xxxxGLEWB1x4!(2,0,2)000001876xxxxxGHBW1x5!(2,0,2)000001867xxxxxGHWB1x5!(2,0,2)000001756xxxxxGBEW1x5!(2,0,2)000001746xxxxxGBLW1x5!(2,0,2)000001657xxxxxGWEB1x5!(2,0,2)000001647xxxxxGWLB1x5!(2,0,2)000001576xxxxxGEBW1x5!(2,0,2)000001567xxxxxGEWB1x5!(2,0,2)000001476xxxxxGLBW1x5!(2,0,2)000001467xxxxxGLWB1x5!(2,0,2)000000176xxxxxxGBW1x6!(2,0,2)000000167xxxxxxGWB1x6!(2,0,2)000000158xxxxxxGEH1x6!(2,0,2)000000148xxxxxxGLH1x6!(2,0,2)000154786xxxGELBHW1x3!(0,2,2)000154687xxxGELWHB1x3!(0,2,2)000146857xxxGLWHEB1x3!(0,2,2) 198

PAGE 199

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000146587xxxGLWEHB1x3!(0,2,2)000145786xxxGLEBHW1x3!(0,2,2)000145687xxxGLEWHB1x3!(0,2,2)000016857xxxxGWHEB1x4!(0,2,2)000016587xxxxGWEHB1x4!(0,2,2)000015786xxxxGEBHW1x4!(0,2,2)000015687xxxxGEWHB1x4!(0,2,2)000014786xxxxGLBHW1x4!(0,2,2)000014687xxxxGLWHB1x4!(0,2,2)000001786xxxxxGBHW1x5!(0,2,2)000001687xxxxxGWHB1x5!(0,2,2)000018547xxxxGHELB1x4!000018546xxxxGHELW1x4!000018457xxxxGHLEB1x4!000018456xxxxGHLEW1x4!000015847xxxxGEHLB1x4!000015846xxxxGEHLW1x4!000015487xxxxGELHB1x4!000015486xxxxGELHW1x4!000014857xxxxGLHEB1x4!000014856xxxxGLHEW1x4!000014587xxxxGLEHB1x4!000014586xxxxGLEHW1x4!000001857xxxxxGHEB1x5!000001856xxxxxGHEW1x5!000001847xxxxxGHLB1x5!000001846xxxxxGHLW1x5!000001587xxxxxGEHB1x5!000001586xxxxxGEHW1x5!000001547xxxxxGELB1x5!000001546xxxxxGELW1x5!000001487xxxxxGLHB1x5!000001486xxxxxGLHW1x5!000001457xxxxxGLEB1x5!000001456xxxxxGLEW1x5!000000187xxxxxxGHB1x6!000000186xxxxxxGHW1x6!000000157xxxxxxGEB1x6!000000156xxxxxxGEW1x6!000000147xxxxxxGLB1x6! 199

PAGE 200

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000000146xxxxxxGLW1x6!000000018xxxxxxxGH1x7!000000017xxxxxxxGB1x7!000000016xxxxxxxGW1x7!(0,0,1)000035468xxxxUELWH2x4!(4,1,1)(2,1,1)(2,0,1)(0,1,1)000275846xxxQBEHLW1x3!(4,0,1)(2,1,1)(2,0,1)(0,2,1)000275486xxxQBELHW1x3!(4,0,1)(2,1,1)(2,0,1)(0,2,1)000035846xxxxUEHLW2x4!(4,0,1)(2,1,1)(2,0,1)(0,1,1)000035648xxxxUEWLH2x4!(4,0,1)(2,1,1)(2,0,1)(0,1,1)000035486xxxxUELHW2x4!(4,0,1)(2,1,1)(2,0,1)(0,1,1)000278546xxxQBHELW1x3!(4,0,1)(2,0,1)(0,2,1)000038546xxxxUHELW2x4!(4,0,1)(2,0,1)(0,1,1)000036548xxxxUWELH2x4!(4,0,1)(2,0,1)(0,1,1)000036458xxxxUWLEH2x4!(4,0,1)(2,0,1)(0,1,1)000274586xxxQBLEHW1x3!(2,1,1)(2,0,1)(0,2,1)000247586xxxQLBEHW1x3!(2,1,1)(2,0,1)(0,2,1)000027586xxxxQBEHW1x4!(2,1,1)(2,0,1)(0,2,1)000034658xxxxULWEH2x4!(2,1,1)(2,0,1)(0,1,1)000034586xxxxULEHW2x4!(2,1,1)(2,0,1)(0,1,1)000034568xxxxULEWH2x4!(2,1,1)(2,0,1)(0,1,1)000003586xxxxxUEHW2x5!(2,1,1)(2,0,1)(0,1,1)000003568xxxxxUEWH2x5!(2,1,1)(2,0,1)(0,1,1)000287546xxxQHBELW1x3!(4,0,1)(2,0,1)000027546xxxxQBELW1x4!(4,0,1)(2,0,1)000003548xxxxxUELH2x5!(4,0,1)(2,0,1)000003546xxxxxUELW2x5!(4,0,1)(2,0,1)000003458xxxxxULEH2x5!(4,0,1)(2,0,1)000002548xxxxxQELH1x5!(4,0,1)(2,0,1)000002458xxxxxQLEH1x5!(4,0,1)(2,0,1)000003468xxxxxULWH2x5!(2,1,1)(0,1,1)000278456xxxQBHLEW1x3!(2,0,1)(0,2,1)000274856xxxQBLHEW1x3!(2,0,1)(0,2,1)000268547xxxQWHELB1x3!(2,0,1)(0,2,1)000268457xxxQWHLEB1x3!(2,0,1)(0,2,1)000265847xxxQWEHLB1x3!(2,0,1)(0,2,1)000265487xxxQWELHB1x3!(2,0,1)(0,2,1)000264857xxxQWLHEB1x3!(2,0,1)(0,2,1)000264587xxxQWLEHB1x3!(2,0,1)(0,2,1)000257846xxxQEBHLW1x3!(2,0,1)(0,2,1)000257486xxxQEBLHW1x3!(2,0,1)(0,2,1) 200

PAGE 201

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000256847xxxQEWHLB1x3!(2,0,1)(0,2,1)000256487xxxQEWLHB1x3!(2,0,1)(0,2,1)000247856xxxQLBHEW1x3!(2,0,1)(0,2,1)000027856xxxxQBHEW1x4!(2,0,1)(0,2,1)000027846xxxxQBHLW1x4!(2,0,1)(0,2,1)000027486xxxxQBLHW1x4!(2,0,1)(0,2,1)000026847xxxxQWHLB1x4!(2,0,1)(0,2,1)000026487xxxxQWLHB1x4!(2,0,1)(0,2,1)000038456xxxxUHLEW2x4!(2,0,1)(0,1,1)000034856xxxxULHEW2x4!(2,0,1)(0,1,1)000003856xxxxxUHEW2x5!(2,0,1)(0,1,1)000003846xxxxxUHLW2x5!(2,0,1)(0,1,1)000003658xxxxxUWEH2x5!(2,0,1)(0,1,1)000003648xxxxxUWLH2x5!(2,0,1)(0,1,1)000003486xxxxxULHW2x5!(2,0,1)(0,1,1)000287456xxxQHBLEW1x3!(2,0,1)000286547xxxQHWELB1x3!(2,0,1)000286457xxxQHWLEB1x3!(2,0,1)000285746xxxQHEBLW1x3!(2,0,1)000285647xxxQHEWLB1x3!(2,0,1)000285476xxxQHELBW1x3!(2,0,1)000285467xxxQHELWB1x3!(2,0,1)000284756xxxQHLBEW1x3!(2,0,1)000284657xxxQHLWEB1x3!(2,0,1)000284576xxxQHLEBW1x3!(2,0,1)000284567xxxQHLEWB1x3!(2,0,1)000258746xxxQEHBLW1x3!(2,0,1)000258647xxxQEHWLB1x3!(2,0,1)000258476xxxQEHLBW1x3!(2,0,1)000258467xxxQEHLWB1x3!(2,0,1)000254876xxxQELHBW1x3!(2,0,1)000254867xxxQELHWB1x3!(2,0,1)000248756xxxQLHBEW1x3!(2,0,1)000248657xxxQLHWEB1x3!(2,0,1)000248576xxxQLHEBW1x3!(2,0,1)000248567xxxQLHEWB1x3!(2,0,1)000245876xxxQLEHBW1x3!(2,0,1)000245867xxxQLEHWB1x3!(2,0,1)000028756xxxxQHBEW1x4!(2,0,1)000028746xxxxQHBLW1x4!(2,0,1) 201

PAGE 202

Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000028657xxxxQHWEB1x4!(2,0,1)000028647xxxxQHWLB1x4!(2,0,1)000028576xxxxQHEBW1x4!(2,0,1)000028567xxxxQHEWB1x4!(2,0,1)000028476xxxxQHLBW1x4!(2,0,1)000028467xxxxQHLWB1x4!(2,0,1)000027456xxxxQBLEW1x4!(2,0,1)000026547xxxxQWELB1x4!(2,0,1)000026457xxxxQWLEB1x4!(2,0,1)000025876xxxxQEHBW1x4!(2,0,1)000025867xxxxQEHWB1x4!(2,0,1)000025746xxxxQEBLW1x4!(2,0,1)000025647xxxxQEWLB1x4!(2,0,1)000025476xxxxQELBW1x4!(2,0,1)000025467xxxxQELWB1x4!(2,0,1)000024876xxxxQLHBW1x4!(2,0,1)000024867xxxxQLHWB1x4!(2,0,1)000024756xxxxQLBEW1x4!(2,0,1)000024657xxxxQLWEB1x4!(2,0,1)000024576xxxxQLEBW1x4!(2,0,1)000024567xxxxQLEWB1x4!(2,0,1)000003456xxxxxULEW2x5!(2,0,1)000002876xxxxxQHBW1x5!(2,0,1)000002867xxxxxQHWB1x5!(2,0,1)000002756xxxxxQBEW1x5!(2,0,1)000002746xxxxxQBLW1x5!(2,0,1)000002657xxxxxQWEB1x5!(2,0,1)000002647xxxxxQWLB1x5!(2,0,1)000002576xxxxxQEBW1x5!(2,0,1)000002567xxxxxQEWB1x5!(2,0,1)000002476xxxxxQLBW1x5!(2,0,1)000002467xxxxxQLWB1x5!(2,0,1)000000358xxxxxxUEH2x6!(2,0,1)000000356xxxxxxUEW2x6!(2,0,1)000000348xxxxxxULH2x6!(2,0,1)000000346xxxxxxULW2x6!(2,0,1)000000276xxxxxxQBW1x6!(2,0,1)000000267xxxxxxQWB1x6!(2,0,1)000000258xxxxxxQEH1x6!(2,0,1)000000248xxxxxxQLH1x6!(2,0,1) 202

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000254786xxxQELBHW1x3!(0,2,1)000254687xxxQELWHB1x3!(0,2,1)000246857xxxQLWHEB1x3!(0,2,1)000246587xxxQLWEHB1x3!(0,2,1)000245786xxxQLEBHW1x3!(0,2,1)000245687xxxQLEWHB1x3!(0,2,1)000026857xxxxQWHEB1x4!(0,2,1)000026587xxxxQWEHB1x4!(0,2,1)000025786xxxxQEBHW1x4!(0,2,1)000025687xxxxQEWHB1x4!(0,2,1)000024786xxxxQLBHW1x4!(0,2,1)000024687xxxxQLWHB1x4!(0,2,1)000002786xxxxxQBHW1x5!(0,2,1)000002687xxxxxQWHB1x5!(0,2,1)000000386xxxxxxUHW2x6!(0,1,1)000000368xxxxxxUWH2x6!(0,1,1)000386547xxxUHWELB2x3!000386457xxxUHWLEB2x3!000385647xxxUHEWLB2x3!000385467xxxUHELWB2x3!000384657xxxUHLWEB2x3!000384567xxxUHLEWB2x3!000368547xxxUWHELB2x3!000368457xxxUWHLEB2x3!000365847xxxUWEHLB2x3!000365487xxxUWELHB2x3!000364857xxxUWLHEB2x3!000364587xxxUWLEHB2x3!000358647xxxUEHWLB2x3!000358467xxxUEHLWB2x3!000356847xxxUEWHLB2x3!000356487xxxUEWLHB2x3!000354867xxxUELHWB2x3!000354687xxxUELWHB2x3!000348657xxxULHWEB2x3!000348567xxxULHEWB2x3!000346857xxxULWHEB2x3!000346587xxxULWEHB2x3!000345867xxxULEHWB2x3!000345687xxxULEWHB2x3! 203

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000038657xxxxUHWEB2x4!000038647xxxxUHWLB2x4!000038567xxxxUHEWB2x4!000038547xxxxUHELB2x4!000038467xxxxUHLWB2x4!000038457xxxxUHLEB2x4!000036857xxxxUWHEB2x4!000036847xxxxUWHLB2x4!000036587xxxxUWEHB2x4!000036547xxxxUWELB2x4!000036487xxxxUWLHB2x4!000036457xxxxUWLEB2x4!000035867xxxxUEHWB2x4!000035847xxxxUEHLB2x4!000035687xxxxUEWHB2x4!000035647xxxxUEWLB2x4!000035487xxxxUELHB2x4!000035467xxxxUELWB2x4!000034867xxxxULHWB2x4!000034857xxxxULHEB2x4!000034687xxxxULWHB2x4!000034657xxxxULWEB2x4!000034587xxxxULEHB2x4!000034567xxxxULEWB2x4!000028547xxxxQHELB1x4!000028546xxxxQHELW1x4!000028457xxxxQHLEB1x4!000028456xxxxQHLEW1x4!000025847xxxxQEHLB1x4!000025846xxxxQEHLW1x4!000025487xxxxQELHB1x4!000025486xxxxQELHW1x4!000024857xxxxQLHEB1x4!000024856xxxxQLHEW1x4!000024587xxxxQLEHB1x4!000024586xxxxQLEHW1x4!000003867xxxxxUHWB2x5!000003857xxxxxUHEB2x5!000003847xxxxxUHLB2x5!000003687xxxxxUWHB2x5! 204

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Table D-1 continued SignatureHierarchyOtherSignatures(n`,nv,n)Code#x...xCy...yLMul(n`,nv,n) 000003657xxxxxUWEB2x5!000003647xxxxxUWLB2x5!000003587xxxxxUEHB2x5!000003567xxxxxUEWB2x5!000003547xxxxxUELB2x5!000003487xxxxxULHB2x5!000003467xxxxxULWB2x5!000003457xxxxxULEB2x5!000002857xxxxxQHEB1x5!000002856xxxxxQHEW1x5!000002847xxxxxQHLB1x5!000002846xxxxxQHLW1x5!000002587xxxxxQEHB1x5!000002586xxxxxQEHW1x5!000002547xxxxxQELB1x5!000002546xxxxxQELW1x5!000002487xxxxxQLHB1x5!000002486xxxxxQLHW1x5!000002457xxxxxQLEB1x5!000002456xxxxxQLEW1x5!000000387xxxxxxUHB2x6!000000367xxxxxxUWB2x6!000000357xxxxxxUEB2x6!000000347xxxxxxULB2x6!000000287xxxxxxQHB1x6!000000286xxxxxxQHW1x6!000000257xxxxxxQEB1x6!000000256xxxxxxQEW1x6!000000247xxxxxxQLB1x6!000000246xxxxxxQLW1x6!000000038xxxxxxxUH2x7!000000037xxxxxxxUB2x7!000000036xxxxxxxUW2x7!000000028xxxxxxxQH1x7!000000027xxxxxxxQB1x7!000000026xxxxxxxQW1x7! 205

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APPENDIXEW-BTRANSITIONW-Btransitionscangiveeither2leptons(3bodydecay)oraboson(ZorHiggsvia2bodydecay)dependingonthemasssplittingbetweenthem.Herewegivetheanalyticformulaeforsuchdecaywidths.Wealsomakeaplotshowinghow/wherethe3bodydecaydominatesoverthe2bodydecay. \(~w0!~b0ee)=g4M52s2W 641203c2WM4L 1)]TJ /F6 11.955 Tf 11.96 0 Td[(8M21 M22+8M1 M26)]TJ /F13 11.955 Tf 11.95 16.85 Td[(M1 M28+12M1 M24lnM22 M21!(E) \(~w0!~b0Z)=g2M2Zs2Wc22 64M32(2)]TJ /F3 11.955 Tf 11.95 0 Td[(M21)2(2)]TJ /F3 11.955 Tf 11.95 0 Td[(M22)2)]TJ /F6 11.955 Tf 5.48 -9.68 Td[((M1+M2)2)]TJ /F3 11.955 Tf 11.95 0 Td[(M2Z)]TJ /F6 11.955 Tf 12.95 -9.68 Td[((M1)]TJ /F3 11.955 Tf 11.96 0 Td[(M2)2+2M2Zq (M22)]TJ /F6 11.955 Tf 11.96 0 Td[((M1)]TJ /F3 11.955 Tf 11.95 0 Td[(MZ)2)(M22)]TJ /F6 11.955 Tf 11.96 0 Td[((M1+MZ)2)(2)]TJ /F3 11.955 Tf 11.95 0 Td[(M1M2)2 (E) \(~w0!~b0h)=g2M2Zs2W 128M32q (M22)]TJ /F6 11.955 Tf 11.95 0 Td[((M1)]TJ /F3 11.955 Tf 11.96 0 Td[(Mh)2)(M22)]TJ /F6 11.955 Tf 11.96 0 Td[((M1+Mh)2)M21+M1M2 2+M22)]TJ /F3 11.955 Tf 11.95 0 Td[(M2hc2WM2Z(M1+s2) (M1)]TJ /F3 11.955 Tf 11.95 0 Td[(M2)(M21)]TJ /F5 11.955 Tf 11.96 0 Td[(2)(M22)]TJ /F5 11.955 Tf 11.95 0 Td[(2)+(M1+s2) M21)]TJ /F5 11.955 Tf 11.95 0 Td[(2 1)]TJ /F3 11.955 Tf 13.15 8.85 Td[(M2Zc2W)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(M22+2+2M2s2 2(M22)]TJ /F5 11.955 Tf 11.96 0 Td[(2)2!+(M2+s2) M22)]TJ /F5 11.955 Tf 11.95 0 Td[(2 1)]TJ /F3 11.955 Tf 13.15 8.85 Td[(M2Zs2W)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M21+2+2M1s2 2(M21)]TJ /F5 11.955 Tf 11.96 0 Td[(2)2!)2 (E) 206

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FigureE-1.Dependenceof~w0decaywidthonM2ine+e)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((blue),Z(red)andh(green)production.ThedashedlinesrepresentnumericresultfromCalcHEP,matchingquitewellwithourapproximateanalyticformula(indottedline)asprovidedinEqns. E E and E .Usedparametersareindicatedatthegure. FigureE-2.ThesameasFig. E-1 ,butdependenceof~w0decaywidthonadditionalparameters(a)MLinGeVinleptonpairproductionand(b)intheZ(red)andh(green)production. 207

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[17] D.Alvesetal.,SimpliedModelsforLHCNewPhysicsSearches,arXiv:1105.2838[hep-ph]. [18] CMSCollaboration,Searchfornewphysicswithsame-signisolateddileptoneventswithjetsandmissingenergy,CMS-PAS-SUS-11-010(2011). [19] http://www.physics.ucdavis.edu/conway/research/software/pgs/pgs4-general.htm [20] S.Chatrchyanetal.[CMSCollaboration],Searchfornewphysicswithsame-signisolateddileptoneventswithjetsandmissingtransverseenergyattheLHC,JHEP1106,077(2011).[arXiv:1104.3168[hep-ex]]. [21] P.Konar,K.T.Matchev,M.ParkandG.K.Sarangi,HowtolookforsupersymmetryunderthelamppostattheLHC,Phys.Rev.Lett.105,221801(2010)[arXiv:1008.2483[hep-ph]]. [22] P.Konar,K.Matchev,M.ParkandG.Sarangi,toappear. [23] T.Aaltonenetal.[CDFCollaboration],SearchforLong-LivedMassiveChargedParticlesin1.96-TeVpanti-pCollisions,Phys.Rev.Lett.103,021802(2009);V.M.Abazovetal.[D0Collaboration],SearchforLong-LivedChargedMassiveParticleswiththeD0Detector,Phys.Rev.Lett.102,161802(2009). [24] A.C.Kraan,J.B.Hansen,P.Nevski,DiscoverypotentialofR-hadronswiththeATLASdetector,Eur.Phys.J.C49,623-640(2007). [25] See,e.g.D.Feldman,Z.Liu,P.Nath,TheLandscapeofSparticleMassHierarchiesandTheirSignatureSpaceattheLHC,Phys.Rev.Lett.99,251802(2007);C.F.Bergeretal.,SupersymmetryWithoutPrejudice,JHEP0902,023(2009);C.Horn,J.Phys.G36,105005(2009). [26] D.Feldman,Z.LiuandP.Nath,GluinoNLSP,DarkMatterviaGluinoCoannihilation,andLHCSignatures,Phys.Rev.D80,015007(2009)[arXiv:0905.1148[hep-ph]]. [27] D.J.H.Chungetal.,D.J.H.Chung,L.L.Everett,G.L.Kane,S.F.King,J.D.LykkenandL.T.Wang,Thesoftsupersymmetry-breakingLagrangian:Theoryandapplications,Phys.Rept.407(2005)1. [28] N.Arkani-Hamed,G.L.Kane,J.ThalerandL.-T.Wang,JHEP0608,070(2006)[hep-ph/0512190]. [29] M.Battagliaetal.,Proposedpost-LEPbenchmarksforsupersymmetry,Eur.Phys.J.C22,535-561(2001). [30] J.L.Feng,K.T.Matchev,T.Moroi,Focuspointsandnaturalnessinsupersymmetry,Phys.Rev.D61,075005(2000). 209

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BIOGRAPHICALSKETCH GaurabK.SarangiwasborninSambalpur,India.Hewasbroughtupinalargefamilywithhisparents,grandparents,uncle,auntsandayoungerbrother.Fromhischildhoodhewasveryinquisitiveaboutallthingsnaturalaswellasmanmade.Ashegrewup,thisnatureofhisdevelopedintoadeeppassionforscienceandphilosophy.AfternishinghishighersecondaryeducationinSambalpur,heearnedhisbachelorsdegreeinElectricalEngineeringfromNationalInstituteofTechnology,Rourkela,India.Upongraduatingin2003,heworkedforPRADANbasedinWestSinghboomforayearandhalf,wherehehelpedtheHoandMundaritribalcommunitiesthroughvariousgrassrootlevellivelihoodinterventions(inagriculture,sericultureandmicronance).ForayearafterthatheworkedasamicronanceconsultantinMFTech,Hyderabad.Duringthisperiodhespentmuchofhissparetimetryingtolearnphysicsfrompopularsciencebooks.Thisreinvigoratedhischildhoodpassionforunderstandingthewaysofnature,especiallyinphysics,andhedecidedtotakeitupacademically.Helefthisjobin2006toworkatHarish-ChandraResearchInstituteinAllahabadasaresearchassistantinhighenergyphysics.In2007hejoinedthegraduateprogramattheUniversityofFloridatopursuefurthereducationinhighenergyphysics.Whileinthegraduateprogram,hegotinterestedinavarietyofothertopicslikecosmology,evolutionarybiology,linguisticsandneuroscienceaswellasvariouswaysinwhichstatisticaltoolsfromphysicscanbeappliedinotherelds.HereceivedhisPh.D.inphysicsin2012andjoinedMaxPlanckInstituteforEvolutionaryAnthropology,LeipzigforpostdoctoralresearchintheeldofPopulationGenetics,workingonGenomicanalysisofrecombinationandadaptationinbacteria. 210