Comparison of Black Hole Generators for the LHC

DouglasM.Gingrich

DepartmentofPhysics,UniversityofAlberta,Edmonton,ABT6G2G7Canada

TRIUMF,Vancouver,BCV6T2A3Canada

gingrich@ualberta.ca

AbstractWecompareMonteCarloeventgeneratorsdedicatedtosimulatingtheproductionanddecayofextra-dimensionalblackholesattheLargeHadronCollider.

Keywords:blackholes,extradimensions,beyondStandardModel,generators,MonteCarlo

1IntroductionandGeneralComparison

StudiesofblackholeproductionanddecayattheLargeHadronCollider(LHC)havebeenassistedbyMonteCarloeventgenerators.Insomesituationsprivategeneratorshavebeenwritten,buthavenotbeendocumentedinanydetailormadereadilyavailable.SomeexamplesofthesetypesofgeneratorsareTRUENIOR,whichwasusedinthefirststudiesofblackholesattheLHCbyDimopoulosandLandsberg[1,2]andanunnamedgeneratorusedinthefirststudyspecifictoATLASbyTanakaetal.[3].Thesegeneratorsaremadequickandefficientforspecificstudiesbyaveragingoversomedynamicalquantities,ratherthengeneratingthemaccordingtoprobabilisticdistributions.

TwogeneralpurposeMonteCarlogeneratorsforsimulatingblackholesattheLHChavebeensignificantlydocumentedandmadeavailableonpublicwebsites:CHARYBDISandCATFISH.CHARYBDIS1[4,5]isperhapsthemostwidelyusedgeneratorandhasresultedinanumberofstudies(seeforexampleRef.[6]).ArelativelynewgeneratorisCATFISH2[7,8](CollidergrAviTationalFIeldSimulatorforblackHoles).Withtheadventoftwogenerators,itbecomesnecessarytocomparethemandhenceenableaneducatedjudgementtobemadeonwhichonetouseinagivenstudy.

ThispapercomparesCHARYBDISversion1.003(24August2006)withCATFISHver-sion1.1(19October2006).TheinformationonCHARYBDIScomesfromreadingthedoc-umentationandFORTRANcode,whiletheinformationonCATFISHcomesfromreadingthedocumentation.ThesourcecodeforCATFISHisnotyetreadilyavailable.

BothgeneratorsarewritteninFORTRANandinterfacedtothegeneralpurposeMonteCarloprogramPYTHIA[9].CHARYBDIScanalternativelybeinterfacedtoHERWIG[10,11].IntheHERWIGversiontheinitialblackholeisplacedintotheeventrecordandassigned

thePDGcodeIDHEP=40withname’BlacHole’.ThishasnotbeenimplementedinthePYTHIAversion.InbothgeneratorstheinterfaceisdefinedbytheLesHouchesaccord[12].PYTHIAorHERWIGprovidethepartonevolutionandhadronization,aswellas,standardmodelparticledecays.

Theimportanteffectsofangularmomentumintheproductionanddecayofblackholesinextradimensionsarenotaccountedforineithergenerator.

2ComparisonofBlackHoleProduction

MoststudiesofblackholeproductioninhigherdimensionsattheLHChaveusedthesemi-classicalblackdiskapproximationforthepartoncrosssection.Thefactorizationapproxi-mationisthenusedtoconvolutethepartoncrosssectionwiththepartondensityfunctions(PDFs)intheprotontoobtainadifferentialcrosssectiondependingonthemassoftheblackhole.Thiscrosssectionissampledinamassrangetodeterminetherelativeprobabilityforproducingablackholeofaparticularmass.

Bothgeneratorsusetheblackdiskcrosssection,whichdependsonlyonthehorizonradius.TheSchwarzschildradiusischosenforthehorizonradius,anddependsonthenumberofdimensionsandthePlanckscale.TherangeofthetotalnumberofdimensionsallowedbyCHARYBDISis6to11,whilethatallowedbyCATFISHis7to11.ForthePlanckscale,threecommondefinitionsareavailableinCHARYBDIS,whileCATFISHhasonlyonedefinition.ThedefinitionusedinCATFISHisthesameasthedefault(MSSDEF=2)inCHARYBDIS,whichistheDimopoulosandLandsberg[2]definition.ThedefinitionofthePlanckscaleisimportantatLHCenergies,particularlywhencomparingwithexperimentalresults.

Inadditiontotheblackdiskcrosssection,CATFISHcanproducedblackholesinelas-ticallyusingtwogravitationalmodels.Thegravitationalmodelsusethetrapped-surfaceapproach,whichsetslimitsontheminimumenergythatgetstrappedbehindtheeventhorizon.Thesemodelsalsopredictvaluesfortheformfactor.ThemodelbyYoshinoandNambu[13]determinestheapparenthorizonattheinstanceofcollision,whilethemodelbyYoshinoandRychkov[14]determinestheapparenthorizonusingthe“optimalslice”.CATFISHassumestheremainingenergynotformingtheblackholeisradiatedasgravitons.ThisistreadedbygivingtheprotonbeamremnantsallthelostenergyandnotpassingthemontoPYTHIA.

CHARYBDIScanbeusedforeitherproton-protonorproton-antiprotoncollisions,andcanbeeasilyinterfacedtoanyofthecommonpartondensityfunctionsusingtheLesHouchesaccordorthePDFLIBlibrary.CATFISHsimulatesproton-protoncollisionsandonlyusesthe(stable)CTEQ5partondensityfunctionsfortheproton.BothgeneratorsallowthemomentumscaleofthepartondensityfunctionstobeeitherthemassoftheblackholeortheinverseSchwarzchildradius.

Whenusingthegenerators,onespecifiesarangeofbackholemassestogenerate.InbothgeneratorstheminimummassshouldnotbetooclosetothePlanckscale.CATFISHallowstheminimummasstobespecifieddirectlyorbyspecifyingaminimumspacetimelength.

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3ComparisonofBlackHoleDecay

Thedecayofablackholecanbeviewasafour-phaseprocesses:baldingphase,spin-downphase,Hawkingevaporationphase,andPlanckphase.Thebaldingandspin-downphasesareneglectedinbothgenerators.Thetrapped-surfacemodelsinCATFISHperhapscompensatefortheseneglectedphasesbyallowingforlostenergyinblackholeformation.

TheHawkingevaporation,orSchwarzschild,phaseisthemostwellstudied.Inbothgen-erators,theparticlesaretreatedasmassless,includingthegaugebosonsandheavyquarks.Baryonnumber,colour,andelectricchargeareconservedintheblackholeproductionanddecayinCHARYBDIS.CATFISHconservescolourbutnotbaryonnumber.ElectricchargemayormaynotbeconservedinCATFISHdependingontheoptionchosen.Theblackholeisallowedtodecaytoallstandardmodelparticles,includingtheHiggsboson.AfewoptionsexistinCHARYBDISforexcludingtheHiggs,ortheHiggs,gaugebosons,andtopquark.InCHARYBDISthegravitonisignored,whileinCATFISHitisincluded[15,16].

Forthegaugebosons,theirlongitudinaldegreesoffreedomcomingformtheHiggsmech-anismaretakenintoaccountslightlydifferently.InCHARYBDISthelongitudinaldegreesoffreedomareconsideredasscalars,whileinCATFISHtheyareincludedinthecountingofthevectorbosons.

WecanthinkoftheHawkingevaporationphaseasconsistingoftwoparts:determinationoftheparticlespeciesandassigningenergytothedecayproducts.Aparticlespicesisselectedrandomlywithaprobabilitydeterminedbyitsnumberofdegreesoffreedomandtheratiosofemissivities.Thedegreesoffreedomtakeintoaccountpolarization,charge,andcolour.InCHARYBDIStheemittedchargeischosensuchthatthemagnitudeoftheblackholechargedecreases.Thisreproducessomeofthefeaturesofthecharge-dependentemissionspectrawhilstatthesametimemakingiteasierfortheeventgeneratortoensurethatchargeisconservedforthefulldecay.

TheenergyassignmenttothedecayparticlesintheHawkingevaporationphasehasbeenimplementedineachgeneratordifferently.InCHARYBDIStheparticlespicesselectedbythemethoddescribedaboveisgivenanenergyrandomlyaccordingtoitsextra-dimensiondecayspectrum.Adifferentdecayspectrumisusedforfermionsandvectorbosons,i.e.thespinstatisticsfactoristakenintoaccount.AGrey-bodyorapureblack-bodyspectrumcanbeused.Grey-bodyeffectsareincludedwithoutapproximations[17].Thegrey-bodyfactorsarespin-dependentanddependonthenumberofdimensions.Thechoiceofenergyismadeintherestframeoftheblackholebeforeemission.TheHawkingtemperatureofthespectrumiseitherfixedatthebeginningofthedecayorupdatedaftereachdecay.IftheHawkingtemperatureisallowedtovary,itisassumedthedecayisquasi-stationaryinthesensethattheblackholehastimetocomeintoequilibriumateachnewtemperaturebeforethenextparticleisemitted.IftheHawkingtemperatureisfixed,itisassumedthedecayissuddeninthesensethatthebackholespendsmostofitstimenearitsoriginalmassandtemperaturebecausethatiswhenitevolvestheslowest.Theenergyoftheparticlegivenbythespectrummustbeconstrainttoconserveenergyandmomentum.Ifthedecayisnotkinematicallypossible,twooptionsexist:1)tryagainor2)godirectlytothefinalstage(Planckphase).Heavyparticleproductionspectramaybeunreliableforchoicesof

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parametersforwhichtheinitialHawkingtemperatureisbelowtherestmassoftheparticlebeingconsidered.

InCATFISHtheinitialenergyoftheblackholeisdistributeddemocraticallyamongallHawkingquantawitharandomsmearingof±10%.Iftheminimumspacetimelengthiszero,thedecayproceedsaccordingtotheHawkingevaporationtheory.Iftheminimumspacetimelengthisnon-zero,thedecayproceedsaccordingtomodifiedthermodynamicsbasedonminimallength.Themodifiedthermodynamicmodelisaresultofassumingageneralizeduncertaintyprinciple[18,19],whichcanbemotivatedbystringtheoryandnon-commutativequantummechanics.

HowtheHawkingevaporationphaseends,andthesubsequentfateoftheblackholeisnotknow.Thegeneratorshandletheterminationofthedecayprocessslightlydifferently.InCHARYBDIStheevaporationphaseendswhentheblackholemassand/ortheHawk-ingtemperaturereachesthePlanckscale.AnoptionexistsinCHARYBDIStocausetheevaporationphasetoendwhenaparticleemissionisselectedwithanenergygreaterthantheblackholemass.InCATFISHtheblackholemasssignallingtheendoftheevaporationphaseisaparameter,ortheminimumlengthisusedtosetthismass.TheminimumenergyforblackholeproductionisnotthesameasthemassatwhichtheHawkingradiationstops.AftertheHawkingevaporationphase,anisotropicN-bodyphase-spacedecayisper-formed,whereNisaparameter.Theprobabilityofaparticularspicesofparticleisagaingivenbyitsnumberofdegreesoffreedom.InCHARYBDISNcanbebetween2and5,whileinCATFISHitcanbebetween2and18.BysettingthisparametertozeroinCAT-FISHablackholeremnantisformed.AnadditionalmechanisminCATFISHexisttomaketheremnantchargedorneutral.Theblackholeremnantisnotimplementedasanactualparticlethatcanbetreatedbythepartonevolutioncode,butisratheranamountofenergy-momentum,andpossiblycharge,whichissimplylost.InCHARYBDISanoptionexisttousethe“boilingremnant”model,inwhichtheblackholedecaycontinuesbelowthePlanckscale.Inthiscase,theminimumremnantmassneedstobeset(belowthePlanckmass)andthetemperaturemaybelimitedtoamaximumvalue,bothspecifiedbyparameters.

4DiscussionandRecommendations

Table1summarisesthedifferencesbetweentheCHARYBDISandCATFISHgenerators.Ifthetwogeneratorstreatanaspectsoftheproductionordecayinasimilarway,itisnotmentionedinthetable.

Althoughtheminimumnumberofextradimensionsisconstrainedbyexperimentalbounds,Iseenotechnicalreasonwhyeithergeneratorshoulddisallowlowvaluesforthenumberofdimension.Somecautionwouldbeneededinusingthegeneratorswith4dimen-sions.TheonlyreasonforrestrictinghighvaluesforthenumberofdimensionsistoallowasimpleandefficientcodingoftheEuler-Gammafunction.

WhencomparingresultsforthegeneratorswithexperimentsitisimportanttousethesamedefinitionofthePlanckscale.Itisconvenienttousetheproperdefinitioninthegeneratortoavoidhavingtoscalethehorizonradiusandthenpropagatethisscaletothe

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Table1:SummaryofdifferencesbetweenCHARYBDISandCATFISH.Feature

CATFISH

yes

LesHouchesorPDFLIBporp¯6–11

3definitionsno

optionalno

variableorfixedblack-body2–5no

Acknowledgments

Iwouldliketothanktheauthorsofthegenerators,BryanWebberandMarcoCavagli`a,forusefulcommentsonthefirstdraftofthispaper.

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