rulesubuntu9.04

PublishedforSISSAbySpringerReceived:April14,2011Revised:June2,2011Accepted:June13,2011Published:June28,2011MadGraph5:goingbeyondJohanAlwall,aMichelHerquet,bFabioMaltoni,cOlivierMattelaercandTimStelzerdaTheoreticalPhysicsDepartment,FermiNationalAcceleratorLaboratory,P.
O.
Box500,Batavia,IL60510,U.
S.
A.
bNikhefTheoryGroup,Kruislaan409,1098SJAmsterdam,TheNetherlandscCentreforCosmology,ParticlePhysicsandPhenomenology(CP3),UniversiteCatholiquedeLouvain,CheminduCyclotron2,B-1348Louvain-la-Neuve,BelgiumdDepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,1110WestGreenStreet,Urbana,IL61801U.
S.
A.
Abstract:MadGraph5isthenewversionoftheMadGraphmatrixelementgen-erator,writteninthePythonprogramminglanguage.
Itimplementsanumberofnew,ecientalgorithmsthatprovideimprovedperformanceandfunctionalityinallaspectsoftheprogram.
Itfeaturesanewuserinterface,severalnewoutputformatsincludingC++processlibrariesforPythia8,andfullcompatibilitywithFeynRulesfornewphysicsmodelsimplementation,allowingforeventgenerationforanymodelthatcanbewrittenintheformofaLagrangian.
MadGraph5buildsonthesamephilosophyasthepreviousversions,anditsdesignallowsittobeusedasacollaborativeplatformwheretheoretical,phenomenologicalandsimulationprojectscanbedevelopedandthendistributedtothehigh-energycommunity.
Wedescribetheideasandthemostimportantdevelopmentsofthecodeandillustrateitscapabilitiesthroughafewsimplephenomenologicalexamples.
Keywords:QCDPhenomenologyOpenAccessdoi:10.
1007/JHEP06(2011)128Contents1Introduction12Overviewandalgorithms32.
1Diagramgeneration42.
2Helicityamplitudecallgenerationandfermionnumberviolation72.
3Colouralgebra82.
4Decaychains113Outputs123.
1MultiprocessgenerationandMadEventeventgeneration123.
2MatrixelementlibrariesforPythia8143.
3Diagramdrawing154Models164.
1InheritingmodelsfromFeynRules:UFOandALOHA164.
2Modelrestrictionles174.
3Consistencychecksforprocessesandmodels185Validationandspeedbenchmarks195.
1Validation195.
2Speedbenchmarks—Processgeneration195.
3Speedbenchmarks—Matrixelementevaluation206BSMexampleapplications216.
1Non-standardcolourstructures:ijkandcoloursextets216.
24-fermionvertices:uu→tt236.
3n-particlevertices:H+4g246.
4Chromo-magneticoperator257Conclusionsandoutlook26AInstallationandonlinewebversion27BCommandlineuserinterface28CProcessgenerationexamples30C.
1Top-quarkpairproduction30C.
2Stoppairproduction31C.
3Sleptonpairproduction31C.
4W+jjproduction32C.
5Graviton-jetproduction32C.
6Gluinodecay32–i–C.
7Top-pairproductionwithoneleptonicdecay33DThetestsuite331IntroductionIdentifyingthefundamentalbuildingblocksofmatteranddescribingtheirinteractionsfromrstprinciplesisthegoalnotonlyofcurrentacceleratorbasedexperiments,suchasthoseoperatingatTevatronandtheLHC,butalsoofmanyotherexperiments,includingavourandneutrinoexperiments,anddarkmatterdetectionexperimentsinundergroundlaboratoriesorsatellites.
Discoveriesfromtheseexperimentsaswellastheirinterpretationwillrelyonourabilitytoperformaccuratesimulationsforboththesignalsandtheirbackgrounds.
AttheLHC,forinstance,extractingphysicsfromthedatawillpresentseveralsignicantchallenges.
First,proton-protoncollisionsatveryhighenergiesproducenalstatesthatinvolvealargenumberofjets,heavy-avourquarks,leptonsandmissingenergy,providinganoverwhelmingbackgroundtomanynewphysicssearches.
Second,eveninthepresenceofaclear"anomaly"withrespecttotheStandardModelprediction,itsinterpretationintermsofanunderlyingphenomenaortheorycouldbeextremelydicult.
ToolsthatareabletomakeprecisepredictionsforwideclassesofBeyondtheStandardModel(BSM)physics,aswellasthosethathelpinbuildingupaneectiveeldtheoryfromthedata,willbeemployed.
Beforeoneorafewcandidatetheoriescanbeselected,accuratemeasurementsofthecorrespondingparameters(masses,couplings,spin,charges)willbeneeded.
Productionratesand/orbranchingratiomeasurements,forexample,willprovideconstraintsonlyifweareabletoconnectthemtothefundamentalparametersofamodelthroughanaccuratecalculation,atleastatnext-to-leadingorderinperturbativeQCD.
Inthiscontext,thereisnodoubtthatMonteCarlosimulationsplayakeyroleateachstageoftheexplorationoftheTeVscale,i.
e.
,fromthediscoveryandidenticationofBSMphysics,tothemeasurementofitsproperties.
TherealizationoftheneedforbettersimulationtoolsfortheLHChasspurredanintenseactivityinrecentyears,thathasresultedinseveralimportantadvancesintheeld.
Generalpurposematrix-elementbasedeventgenerators,suchasCom-pHEP/CalcHEP[1–3],MadGraph/MadEvent[4–6],Sherpa[7]havebeenavailableforseveralyearsnow.
MorerecentlyhighlyeciencientmultipartontechniqueswhichgobeyondusualFeynmandiagramshavebeenintroduced[8–11],andimplementedinpubliclyavailablecodes,suchasWhizard[12,13],Alpgen[14],Helac[15]andComix[16].
Asaresult,theproblemofautomaticallygeneratingtree-levelmatrixelements(andthencrosssectionsandevents)foraverylargeclassofrenormalizablemodelshasbeensolved.
TherecentintroductionofFeynRules[17]hasprovidedanewmethodforimplementingnewphysicsmodelsaswellassettinganewstandardintermsofvalidationandavailability[18,19].
CommunicationbetweenFeynRulesand–1–matrixelementprogramsisbeingstandardizedviathenewUniversalFeynRulesOutputformat,theUFO[20].
AconnectedeortisbeingmadeintheautomationofNLOcomputations.
Thegen-erationoftherealcorrectionswiththeappropriatesubtractionshasbeenachievedinanautomaticway[21–26].
Forvirtualcorrections,severalnewalgorithmsfornumericalcal-culationofloopamplitudeshavebeenproposed(see,e.
g.
,[27]forareview)andsomeofthemsuccessfullyappliedtothecomputationofSMprocessesofphysicalinterest[28–32].
Veryrecently,CutTools[33]hasbeensuccessfullyinterfacedwithMadGraph.
Theresultingtool,MadLoop[34]interfacedtoMadFKS[26],allowsafullyautomaticcal-culationofinfrared-safeobservablesatNLOinQCDforawiderangeofprocessesintheStandardModel.
Lastbutnotleast,anaccuratesimulationofahadroniccollisionrequiresacare-fulintegrationofthematrix-elementhardprocess,withthefullpartonshoweringandhadronizationinfrastructure[35–37].
Hereagain,signicantprogresshasbeenmadeinthedevelopmentofmergingalgorithms,suchasCKKWandMLMmerging[38–44],andintheircomparison[45,46],withapplicationstoSM[41,47,48]andtoBSM[49]processes.
Abreakthroughinmergingxedordercalculationsandpartonshowerswasachievedinrefs.
[50,51],whereitisshownhowtocorrectlyinterfaceanNLOcomputationtoavoiddoublecountinganddeliveredthersteventgeneratoratNLO,MC@NLO.
Morerecently,anothermethodalongthesamelines,dubbedPowheg,hasbeenproposed[52]andappliedtoavarietyofprocessesattheLHCthroughthePowhegBoxgeneralimplementation[53].
ThenewversionofMadGraphhasbeendesignedtosupportandadvancethelinesofdevelopmentmentionedabove,withthreemainobjectives:1.
Lagrangian-basedBSMphysicsviaFeynRulesforanyrenormalizableoreectivetheory.
2.
FullautomationandoptimizationofNLOcomputationsintheSMandbeyond.
3.
Mergingtoshowering/hadronizationcodesforcompleteeventsimulationatLO(viaCKKWandMLMmethods)andatNLO(viaMC@NLOandPowheg),aswellasthecombinationofthetwo("CKKWatNLO").
MadGraph5isopensourcesoftwarewritteninPythonandfeaturesacollaborativedevelopmentstructure.
Itcangeneratematrixelementsatthetree-levelforanyLagrangianbasedmodel(renormalizableoreective)implementedinFeynRulesviatheUFOinter-face,andautomaticgenerationofthecorrespondinghelicityamplitudesubroutinesviatheALOHApackage[54].
WithrespecttoMadGraph4,asignicantimprovementine-ciencyhasbeenattained,andthepossibilitiesfortree-levelmatrixelementgeneration(anddiagramplotting)havebeenextended,includingoptimizationoftheMadEventoutputandreorganizationofmulti-jetnalstatesubprocesses.
ItfeaturesawidesetofexibleoutputformatsinFortran,C++,andPython,anddedicatedmatrixelementoutputforPythia8[55].
ThisworkdocumentsanddescribesthegeneralphilosophyofthenewMadGraphversion,aswellassomeofthemostimportantimprovementsinthecode.
Thepaperis–2–structuredasfollows:Insection2wegiveageneraloverviewonthecodestructureandofthealgorithmsemployed.
Indedicatedsubsectionswedescribethediagramgenera-tionalgorithm,thefermion-owalgorithm,thecolouralgebramodule,andthegenerationofdecaychains.
Wethenpresenttheavailableoutputformats,thenewmultiprocessesoptimizationinMadEventandprocesslibrarygenerationforPythia8aswellasthenewdiagramdrawingalgorithm.
ModelinheritancefromFeynRulesviatheUFOandALOHAispresentedinsection4togetherwithacomprehensivelistofavailablemodelsandthesuiteofmodelteststhatcanbeperformedattheprocesslevel.
Thefollowingsec-tiondescribesthevalidationchecksthathavebeenperformedforSM,MSSM,HEFTandRSprocesses,aswellassomekeyindicatorsfortheperformanceimprovementsinspeedcomparedtopreviousversionsofMadGraph/MadEvent.
Section6providesaselectivesetofexamplesofapplications.
Weleaveourconclusionsandthediscussionoftheoutlooktothelastsection.
Technicalappendicesfollow,wheretheinterestedreadercanndmoredetailsandexamples.
2OverviewandalgorithmsMadGraph[4]isatoolforautomaticallygeneratingmatrixelementsforHighEnergyPhysicsprocesses,suchasdecaysand2→nscatterings.
First,theuserspeciesapro-cessintermsofinitialandnalstateparticles(allowingforanumberofrenedcriteria,includingforcedorforbiddens-channelresonances,excludedinternalparticles,andforceddecaychainsofnalstateparticles).
Multiparticlelabelscanbeusedtospecifyallpossibleprocessesinvolvingarangeofparticles.
Asaresult,MadGraphgeneratesallFeynmandiagramsfortheprocess,andoutputsthecomputercodenecessarytoevaluatethema-trixelementatagivenphasespacepoint.
Thematrixelementevaluationisdoneusingcallstohelicitywavefunctionsandamplitudes,aswererstimplementedintheHELASpackage[56].
Thisimplementationisecientbecauseitnaturallyallowshelicitywavefunc-tionscorrespondingtoidenticalsubdiagramstobereusedacrossdiagrams.
MadGraphalsoproducespictorialoutputoftheFeynmandiagramsfortheprocessinquestion.
ThecomputercodeproducedbyMadGraphcanthenbeusedforcrosssectionordecaywidthcalculationsandeventgeneration,e.
g.
usingtheMadEventpackage[5],whichisincludedwithMadGraph5.
WhilepreviousversionsofMadGraphwerewritteninFortran77,MadGraph5iswritteninPython.
Thisobjectorientedcomputerlanguageallowsforcompletelynewalgo-rithms,andremovesmanyoftherestrictionsthatwereinherentintheFortranversions.
Asaresult,bothMadGraph5andthecodeitproducesrunsignicantlyfasterthanpreviousversions.
Evenmoreimportanthowever,isthatthestructureofthenewimplementationgreatlyfacilitatesselectiveuseofmodulesandadditionsofnewfeatures.
Inthispaperwewilldiscussafewsuchadditions,suchasimplementationofnewcolourrepresentations(coloursextetsandijk),implementationofmulti-fermionverticesandadditionofnewoutputformats,includingoutputinC++andPython.
–3–2.
1DiagramgenerationThediagramgenerationalgorithmusedinMadGraph5isfaster,moreecientandproducesbetteroptimizedcodethanearlierMadGraphversions.
InMadGraph4diagramsgenerationisbasedonthefollowingalgorithm:1.
Generatealltopologieswithappropriatenumberofexternallegs.
2.
Assignparticlestoexternallegs.
3.
Identifyverticeswithatmostoneunassignedline.
4.
Checktoseeifthereisanyinteractioninthemodelthatwillaccomodatetheas-signedlines.
5.
Ifyes,(a)ifavertexhadanunassignedline,assigntoittheappropriateparticleIDfromtheinteraction,thenchecknextvertex.
(b)ifalllinesinthevertexareknown,diagramiscomplete.
Writeittole.
6.
Ifno,diagramfails,trynexttopology.
Thisalgorithmisstraightforwardtoimplement,butthetimerequirementgrowsquicklywiththenumberofexternalparticlessinceeverytopologymustbeexplicitlychecked,evenifonlyasmallfractioncontributetoviablediagrams.
ThealgorithminMadGraph5eliminatesthisineciencybymakinguseofthemodelinformationtoeectivelyonlyconstructtopologiesthatwillyieldvaliddiagrams.
Fur-thermore,itrecursivelygeneratesallofthediagramsinparallel.
Thisensuresthatanycombinationofexternallegs(a,b),(a,b,c)etc.
thatiscommontomultiplediagramswillberecognizedassuch,allowingforoptimalrecyclingofalreadycalculatedsubdiagramsintheresultinghelicityamplitudecode.
Italsoremovesrestrictionsonthenumberofparticlesinaninteractionvertex,pavingthewayforimplementationofhigher-dimensionaleectiveinteractionswith5ormoreelds.
Thealgorithm,presentedbelow,isbasedonrecursivelycreatingsub-diagramsfromthediagramsbymerginglegs,withthecrucialadditionofaag,fromgroup,whichisusedtoindicatewhetheragivenparticleresultsfromamergingofparticles(i.
e.
,itisconnectedtoagivensetofparticles)inthepreviousstep(True),orifitissimplycopiedfromthepreviousstep(False).
Thisaghelpsensurethatnodiagramsaredouble-countedbythealgorithm.
1.
Giventhemodel,generatetwohashmaps(calleddictionariesinPython),contain-inginformationabouttheinteractionsinthemodel.
Therstdictionary(calledVertices)mapsallcombinationsofnparticlestoalln-pointinteractionscombiningtheseparticles,andmapsallpairsparticle-antiparticleto"0".
Theseconddictionary(calledCurrents)maps,foralln-pointinteractions,n1particlestoallcombinationsofresultingparticlesfortheinteractions.
–4–1stiterationGroupingsAfterreplacementsResulte,e+,u,u,g(e,e+),u,u,g(γ),u,u,gFailed(only1FG=True)(Z),u,u,gFailed(only1FG=True)e,e+,(u,u),ge,e+,(γ),gFailed(only1FG=True)e,e+,(Z),gFailed(only1FG=True)e,e+,(g),gFailed(only1FG=True)e,e+,(u,g),ue,e+,(u),uFailed(only1FG=True)e,e+,u,(u,g)e,e+,u,(u)Failed(only1FG=True)(e,e+),(u,u),g(γ),(γ),gFailed(novertex)(γ),(Z),gFailed(novertex)(γ),(g),gFailed(novertex)(Z),(γ),gFailed(novertex)(Z),(Z),gFailed(novertex)(Z),(g),gFailed(novertex)(e,e+),(u,g),u(γ),(u),uDiagram1(Z),(u),uDiagram2(e,e+),u,(u,g)(γ),u,(u)Diagram3(Z),u,(u)Diagram4Table1.
Tabeltoillustratethestepsofthediagramgenerationalgorithm.
Seetextforexplana-tions.
fromgrouphasbeenabbreviatedasFG.
2.
Flipparticle/antiparticlestatusforincomingparticlesintheprocess(i.
e.
,consideralltheparticlesoutgoing).
Settheagfromgroup=Trueforallexternalparticles.
3.
IfthereisanentryintheVerticesdictionarycombiningallexternalparticles,createthecombination[(1,2,3,4,ifatleasttwoparticleshavefromgroup=True.
4.
Createallallowedgroupingsofparticleswithatleastonefromgroup=TruepresentintheCurrentsdictionary.
5.
Setfromgroup=Trueforthenewlycombinedparticles,andFalseforanyparticlethathasnotbeencombinedinthisiteration.
Repeatfrom3forthereducedsetofexternalparticles.
6.
Stopalgorithmwhenatmost2externalparticlesremain.
Asasimple,yetcompleteexample,letusconsidertheprocesse+e→uuginthestandardmodel.
Theprocedureisillustratedintable.
1,anddescribedindetailbelow.
Therelevantinteractionsare(e+eγ),(e+eZ),(uuγ),(uuZ),and(uug).
Firstiteration:afterippingtheparticle/antiparticleidentitiesfortheinitialstate,wehavetheexternalparticlese,e+,u,u,g.
1.
Nogrouping(e,e+,u,u,g)ispossible.
–5–e+1e-2au3g5uu~4diagram1e+1e-2zu3g5uu~4diagram2e+1e-2au~4g5u~u3diagram3e+1e-2zu~4g5u~u3diagram4Figure1.
Diagramsfortheprocesse+e→uug.
Notethatuisdenotedby"u"andγby"a"inthediagrams.
2.
Createallpossibleparticlegroupings(seetable1):[(e,e+),u,u,g],[e,e+,(u,g),u],[e,e+,u,(u,g)],[(e,e+),(u,u),g],[(e,e+),(u,g),u],[(e,e+),u,(u,g)]andreplacethegroupedparticleswiththeresultingparticlesfromtheinteractions:[(γ),u,u,g],[(Z),u,u,g],[e,e+,(γ),g],[e,e+,(Z),g],[e,e+,(g),g],[e,e+,(u),u],[e,e+,u,(u)],[(γ),(γ),g],[(Z),(γ),g],[(γ),(Z),g],[(Z),(Z),g],[(γ),(g),g],[(Z),(g),g],[(γ),(u),u],[(Z),(u),u],[(γ),u,(u)],[(Z),u,(u)].
Notethatonlytheparticlesinparenthesesnowhavefromgroup=True.
Seconditeration:theresultingreducedsetswithfourparticlesallhaveonlyonefromgroup=True,andcanthereforenotgivevaliddiagrams.
Wethereforeignoretheseandfocusonthereducedsetswiththreeparticles.
3.
Thecombinationsallowedbytheinteractionsare:((γ),(u),u),((Z),(u),u),((γ),u,(u))and((Z),u,(u)).
4.
AnyfurthercombinationintheCurrentsdictionarywillresultinanexternalstatewithonlyonefromgroup=True,whichcannotgiveanydiagrams.
5.
Theiterationstops,sincenoexternalparticlesareleft.
Theresultingdiagramsarefoundingure1.
–6–2.
2HelicityamplitudecallgenerationandfermionnumberviolationThecodeformatrixelementevaluationgeneratedbyMadGraph(previousversionsaswellasMadGraph5)iswrittenintermsofsuccessivecallstoahelicityamplitudefunc-tionlibrary(originallytheHELASlibrary,inMadGraph5eitherHELASorhelicityamplitudefunctionsautomaticallygeneratedbyALOHA[54]).
Ahelicitywavefunctionisgeneratedforeachexternalleginadiagram,andthesewavefunctionsarecombinedintonewwavefunctionscorrespondingtothepropagatorsinthediagrambysuccessivehe-licitywavefunctioncalls.
Thenalvertexcorrespondstoahelicityamplitudecallwhichreturnsthevalueoftheamplitudecorrespondingtothisdiagram.
Inthisprocedure,wave-functionscorrespondingtoidenticalsubdiagramscontributingtodierentdiagramscanbereusedbetweenthediagrams,leadingtoaconsiderableoptimization,eectivelygivinguptoafactorhundredfewerwavefunctioncallsthannaivelyexpectedbythenumberofFeynmandiagrams.
InthepresenceofMajoranaparticlesorfermionnumberviolatingvertices,specialcareisneededtoallowforallpossiblecontractionsoffermions.
MadGraph5,likeitsprede-cessors[6,57],usesthe4-spinorFeynmanrulesforfermionnumberviolationdevelopedin[58].
Inthisformulation,afermionowisdenedforeachfermionlineinadiagram,andfermionnumberviolation(duetofermionowclashes)istakenintoaccountusingspecialchargeconjugateversionsoftheΓmatrices.
Withthismodication,thefermionlinesmeetingatthevertexshouldbetreatedasiftheyhadasinglefermionow.
Theowthereforeneedstobeinvertedalongoneofthelines,resultinginacontinuousfermionow.
InearlierversionsofMadGraph,thisfermionowwasimplementedbycreatingacharge-conjugateparticleforeachfermioninthemodel.
Inordertocheckfordiagramswithclashingarrowsitwasnecessarytocheckeachtopologywithboththeregularfermion,anditschargeconjugate.
Thiswouldincreasethetimeforgeneratingcodebyafactorof2Nfermions1.
InMadGraph5,thediagramsresultingfromthediagramgenerationareindependentofthefermionow,andthedenitionoftheowispostponedtothetimeofhelicityamplitudecallgeneration.
FermionswithpositivePDGcode("particles")aretentativelyassignedtobeincoming(outgoing)iftheyareintheinitial(nal)state,andviceversaforfermionswithnegativePDGcode("antiparticles").
Ifafermionowclashisdetectedatthemeetingoftwofermionlines(i.
e.
,thetwofermionshavethesame"incoming/outgoing"status),thefermionowdirectionofoneofthelineshastobeinverted,andchargeconjugateverticeshavetobeintroducedstartingfromthepositionofthevertexorMajoranaparticlelineresponsibleforthefermionowclash.
Theprocedureisthefollowing:Fermionlinesinvolvedintheclasharetraversed,lookingforMajoranaparticles.
IfaMajoranaparticleisfoundalongoneofthelines,the"incoming/outgoing"statusandparticle/antiparticleidisreversedforallparticlesupto(andincluding)thelastMajoranaparticlealongtheline.
ForparticlesbeyondtheMajoranaparticlealongthesamefermionowline,aagfermionflowissetto-1.
Theotherlineisleftunchanged.
IfnoMajoranaparticleisfoundalongeitherofthelines–7–u1~u3~gd2~d4Figure2.
Diagramforthefermionowviolatingprocessud→ud.
(whichisthecasewhentheclashisduetoafermionowviolatingvertex),allfermionsalongtherstleghavetheirfermionflowagissetto-1.
Ahelicityamplitudeorwavefunctionwithanyfermionwithnegativefermionflowagisrequiredtousethechargeconjugateversionoftheamplitudeofwavefunction,inaccordancetotheFeynmanrulesin[58].
Foranexternalleg,theincoming/outgoingstatusisreversed.
Formultifermionvertices,thefermionsaregroupedinpairs,eachconstitutingafermionline(seesection6.
2).
Eachfermionlinethenhasitsownchargeconjugate,whichcanleadtomultiplechargeconjugateagsforasinglehelicityamplitudeorwavefunction.
Asanexplicitexampleofbothhelicityamplitudecallgenerationandthetreatmentoffermionowviolation,wetakethefermionnumberviolatingprocessud→udwitht-channelexchangeofaMajoranagluino(seegure2).
Thediagramisrepresentedinternally(withtentativeincoming/outgoingfermionstategiveninparentheses)as(u(in),u→g(in)),(g(in),d(in),d)Theuu→gvertexwouldresultinanincomingspin12helicitywavefunctionwithincomingow,andthegddvertexcorrespondtoahelicityamplitudecall.
However,afermionowclashisdetectedinthelastvertex,sincetherearetwoincomingfermions.
Thetwofermionlinescomingfromthisvertex(initiatedbytheganddrespectively)arenowtraversed,startingwiththegluino.
SincethegluinoisaMajoranaparticle(andtherearenootherMajoranaparticlesalongtheline),itsowisreversedto"outgoing".
Theu,whichisbeyondthegluinoalongtheline,getstheagfermionflowsetto-1.
Thisindicatesthattheuwavefunctionshouldbethatofanoutgoingexternalspin12particle,andtheuu→gwavefunctionshouldbeconjugated(sinceoneoftheparticlesinthiswavefunctionhasfermionflowag-1).
Thegddhelicityamplitudedoesnotusetheconjugatedversion,sincenoneoftheinvolvedwavefunctionshavenegativefermionflowag.
2.
3ColouralgebraInthissectionwediscusshowthecomputationofcolourcoecientsinscatteringam-plitudesofstateswhicharechargedunderSU(3)Cisperformed.
Therearetwomainmotivationstodedicatespecialattentiontothisaspect.
–8–First,multipartonamplitudes,i.
e.
,amplitudeswithmanyexternalquarksandgluons,arephenomenologicallyveryimportantastheycorrespondtotheleadingorderapprox-imationofmulti-jetproduction(bythemselvesorinassociationwithotherparticles)inhighenergycollisionsandespeciallyathadroncolliders.
Theseprocessesarethemajorbackgroundstomanynew-physicssignals,soanaccuratedescriptionofthesenalstatesisessential.
Inthiscase,notonlythecomplexityofcolourcomputationgrowsfactoriallybutalsotheamountofinformationtobestoredgrowsatthesamerate.
Eciencyandimprovedalgorithmsarethereforenecessary.
Second,new-physicsmodelspossiblyfeaturestatesinexoticcolourrepresentationsornon-standardcolourstructuresininteractionvertices,whichneedtobetakenintoaccount.
AgenericinteractionvertexmayalsohaveacomplicatedstructurewithseveralcolourfactorsinfrontofdierentLorentzstructures.
Theneedsaboverequirebothexibilityandeciencyofthecolouralgebramodule,achallengethatisnoteasytomeet.
PriorversionsofMadGraphhard-codedthreecolourstructuresδij,Taijandfabcaswellasidentitiesforsummingoverthecolourindices.
Thenumberofcolourswasexplic-itlysetto3intheseidentities.
Numericvaluesforthecolourmatrixwherecomputedbysummingoverthecolourindicesusingintegeroperationsonthenumeratorandde-nominatorseparately.
Theresultingcolourmatrixwasexact,butlackedexibility.
Thecolourstructureoftheinteractionwasinferredbasedonthecolouroftheparticipatingparticles.
Modelsthathadnewcolourinteractionsrequiredtheusertoexplicitlycodenewcolourstructures,whichrequireddetailedknowledgeoftheMadGraphcode,signicantlyrestrictingthetypesofnewinteractionsthatcouldbeimplemented.
ThesolutionadoptedinMadGraph5istohaveallthecolourobjectsandtheiralgebracodedinasymbolicway.
Thisapproachhasmanyadvantages,themostimportantonebeingcompleteexibility.
Forexample,oneimportantaspectinhavingecientcolourcomputationsisthatofthechoiceofthecolourbasis.
Severalpossibilitieshavebeenproposedintheliterature[59–61]withtheaimtomakethecomputationofcoloursymbolicallyand/ornumericallyecientattheamplitudelevel.
Allthesechoicescanbeeasilyadoptedinourimplementationasthecolouralgebraisdealtwithsymbolicallyandseveraldierentbasisandrepresentationscanbepresentatthesametimeandusedasneeded.
Thecolouralgebraisrealizedthroughobjects,suchastheGell-Mannmatricesorstructurefunctions,whoseproductsandcombinationscanbeeasilysimpliedbyalgebraicreductionrules,amongwhich(Ta)ij(Ta)kl=12δilδkj1Ncδijδkk(2.
1)playsacentralrole.
Otherobjects,suchas,δab,fabcanddabcaredenedintermsoflinearcombinationsoftracesofGell-Mannmatrices,andanycolourfactorcanbeeasilysimpliedinarecursiveway.
Ourdefaultalgorithmgoesasfollows.
Thecolourfactorcorrespondingtoeachdiagramislinearlydecomposedoveracompleteandorthogonal(inthelargeNclimit)basiswhichisconstructedatthesametime.
Thisallowstheautomaticorganizationofthefullamplitude–9–intogaugeinvariantsubamplitudesAi(normallycalleddualorcolour-orderedamplitudes),eachdefactocorrespondingtoagivencolourowi,sothatM=iCiAi.
(2.
2)Theformaboveisthebasisforfurthermanipulations,asitcanbenowusedindierentwaysdependingonthecomplexityofthecalculationitself.
Inthecaseofalimitednumberofpartonsintheamplitude,itissquaredbyanalyticallycomputingthecolourmatrixCij=coloursCiCj,whichisthenstoredinmemoryandwrittenintheoutputle.
ThisisthedefaultapproachfollowedinthisversionofMadGraph5wherealldiagramsarecomputedandthecolourorderedamplitudesobtained.
Atthispointwenotethatinthepresenceofidenticalexternalparticles,typicallygluons,manyoftheentriesofthecolourmatrixareequalduetosimplesymmetryproperties.
SuchsymmetriesareecientlyexploitedtoreducetheeectivenumberofcomputationsneededtodeterminethefullCij.
ThisallowsMadGraphtocalculateamplitudeswithupto7gluons.
Beyond7gluons,explicitlycalculatingtheamplitudeassociatedwitheachFeynmandiagramisnotviableandrecursiverelationsneedtobeemployed[62,63].
TheseallowonetocalculatetheAidirectlyandprovethattheircomplexityisonlypolynomial.
Theproblemofthefactorialgrowththereforeremainsonlyforthecoloursum.
Severaltechniqueshavebeenproposed,themostcommonlyemployedistheideaofrecursivelycomputingMatxedcolourfortheexternalstatesandthenrandomlychoosecolourcongurations[10,14,15].
Anotherpossibility,whichisbornoutofeq.
2.
2,istoorganizethecalculationasanexpansionin1/Nc.
ThisistheapproachthatiscurrentlyunderinvestigationtocalculatemultipartonamplitudesatthetreeandlooplevelandalsotocombinerealandvirtualcorrectionsinNLOcomputations,followingtheapproachofref.
[64].
Theothermainadvantageoftreatingthecolouralgebrasymbolicallyisastraightfor-wardimplementationofnewcolourstructures.
Asanexample,weheregiveadetaileddescriptionofthenecessaryalgebraforcoloursextetsandcolourtriplettensors.
Higherdimensionalitycolourrepresentationscanbeimplementedinasimilarway.
Theijk(ijk)objectisthetotallyantisymmetrictensorofthreecolourtriplet(an-titriplet)indices.
Thealgebraicrelationsneededforandare:ijkilm=δjlδkmδjmδklwhere(ijk)=ijk.
Forcoloursextets,weneedthreenewcolourobjects(K6)Aij,(K6)ijAandthesextetrepresentation(Ta6)AB.
K6(K6)isthesymmetrictensorcontractingacoloursextet(antisextet)indexandtwocolourantitriplet(triplet)indices(i.
e.
,theClebsch-Gordancoecient),whiletheT6describestheinteractionofagluonwithasextet.
Asinthecolourtripletimplementationdescribedabove,thesextetdeltafunctionisdenotedasatwo-indexδmn.
Thecompletesetofneededalgebraicrelationsfortheseobjectscanbefoundintheappendixofref.
[65].
Theycanbeeasilyreproducedbystartingfrom(K6)Aij(K6)klA=12(δliδkj+δkiδlj).
–10–Togetherwithfundamentalanti-commutationrelation,thecolourmatrixandcolourowsofanydiagraminvolvingcoloursextetparticlescanbecalculated.
Asanalremark,wenotethatasmalltechnicalcomplicationarisesifthepartonleveleventsarepassedtoapartonshower.
InthiscasethewritingoftheeventintheLesHouchesAccord(LHA)atleading-NCcolourstringsisneeded.
Thetensorcanbehandledbyinsertingnewcolourlabelsintheeventandeventhoughanapparentviolationofthecolourowarises,thiscanbeinterpretedcorrectlybytheparton-shower(seeref.
[66]).
Forcoloursextetsnoconventionhasbeenestablished.
Asimplesolutionistonotethatintheplanar"doubleline"colourownotation,acoloursextetiscontractedwithaK6,togivetheequivalentpairoftripletlines.
ThiscanthenbewrittenintotheLHAevent,byusinganegativeantitripletlabelforthesecondtriplet,andviceversaforasecondantitriplet.
Apartonshowerprogramreadingtheeventmustthentreatthecontinuedcolourow,keepingtrackofcoloursextetsandantisextetsappropriately.
2.
4DecaychainsWhereasdecaychainsinMadGraph4weregeneratedinthesamewayasregularpro-cessgeneration(bydressingtopologieswithparticleandinteractioninformationtocreatediagrams),decaychainsinMadGraph5aredenedusingsuccessivechainsofprocesses.
Thecoreprocesscanhaveanumberofdecayprocessesdened,andeachdecayprocesscanagainhaveanumberofdecayprocessesdened,andsoon.
Thistreatmentallowsforquickandecientgenerationofdecaychainsofvirtuallyunlimitedlength.
Thehelicityamplitudecallsfortheindividualprocesses(coreprocessandeachde-cayprocess)aregeneratedseparately,whichallowsforveryecienttreatmentofmul-tiprocesses,wheremultipleprocesseshavethesamedecayingnalstate.
Asasimpleexampleconsiderpp→W+withW+→l+νl,whichgivesthecoreprocessesud→W+,du→W+,cs→W+andsc→W+,andthedecayprocessesW+→e+νe,W+→+νandW+→τ+ντ.
Thetotalnumberofsubprocessesisobtainedbycombiningthefourcoreprocesseswiththethreedecayprocesses.
Withthehelicityamplitudesforeachofthecoreprocessesandeachofthedecayprocessesalreadygenerated,creatingthesubprocessesonlyamountstoreplacingthenalstatewavefunctioncorrespondingtotheW+bythewavefunctionscorrespondingtothedecaysintoe+νe,+νandτ+ντrespectively.
Theproceduregetsslightlymorecomplicatedwhendecayprocesseshavemultiplediagrams-inthiscase,thediagramsforthecoreprocessneedtobemultipliedtogetherwiththediagramsforthedecay.
TheresultingmatrixelementscontainfullspincorrelationsandBreit-Wignereects,butarenotvalidfarfromthemasspeak,wherenon-resonantdiagramsmightgivesig-nicantinterferenceeects.
ThetailsoftheBreit-WignerdistributionsforthespecieddecayingparticlesarethereforecutoinMadEvent,usingtherunparameterbwcutoff,atM±Γbwcutoff(bydefaultsetto15).
Processgeneration,aswellaseventgenerationwithMadEvent,hasbeensuccessfullytestedupto14nalstateparticlesatthetimeofwritingthispaper.
–11–3OutputsPreviousversionsofMadGraphcouldonlyoutputmatrixelementsinFortran77.
ThemodularizeddesignofMadGraph5allowseasyimplementationofoutputsinanylanguageoruserdesiredformat.
TheUFOoutputandALOHAimplementationsallowforthesameexibilityintheoutputofmodelsandhelicityamplituderoutines.
Atpresent,matrixelementoutputisavailableinFortran77,C++andPython.
TheFortran77outputisintheformofMadEventdirectoryoutput(seesection3.
1),orstan-dalonematrixelementevaluationintheformofMadGraphstandalonedirectoryoutput.
ForC++,presentlyavailableoutputformatsarestandalonematrixelementevaluationoutput,anddedicatedoutputforPythia8(seesection3.
2).
ThePythonmatrixelementcodeoutputiscurrentlyusedinternallyinMadGraph5toperformmodelconsistencychecksduringprocessgenerationasdescribedinsection4.
3below.
Implementationofotheroutputformats,eitherinanyofthecurrentlysupportedlanguagesordierentones,canbeeasilydonebasedontheexistingoutputformatimplementations.
Finally,oneofmostusefuloutputsofMadGraphhasalwaysbeenthedrawingsoftheFeynmandiagrams.
Currentnewfeatures(e.
g.
,generationofprocesseswithlongdecaychains)aswellasplannedones(e.
g.
,generationofloopdiagrams)requiredanewalgorithmtobeimplemented.
Thisisdescribedinthelastsubsection.
3.
1MultiprocessgenerationandMadEventeventgenerationSimultaneousgenerationofmultipleprocesses(e.
g.
pp→jj,usingmultiparticlelabelssuchasp/j=g/d/u/s/c/d/u/s/c)hasbeenconsiderablyoptimizedinMadGraph5,usingimprovedalgorithmsbothforprocessgenerationandforeventgeneration.
Ontheprocessgenerationside,thetimespentattemptingtogenerateprocesseswhichhavenodiagrams(suchasud→gg)isminimizedononehandbycheckinganyconservedquantumnumbers,suchaselectricalcharge,providedbythemodel,andontheotherhandinamodel-independentmannerbykeepingamemoryoffailedprocesses,andignoringanyprocesswhichcorrespondstoacrossingofsuchafailedprocess.
Thisisdonebeforeapplyingcrossingsymmetrybreakingconditionssuchasrequiredorforbiddens-channelpropagators.
Furthermore,processeswithmirroredinitialstate(suchasgu→γuandug→γu)areautomaticallyrecognizedandcombinedintoasingleprocess.
Tofurtherspeedupthegeneration,diagramsfromcrossedprocessesarereusedinthediagramgeneration(sothatthediagramsfor,e.
g.
,gg→uu,ug→uganduu→ggaregeneratedonlyonce,andthenreusedwithlegnumbersreplacedasneeded).
TheorganizationofsubprocessdirectoriesformultiprocesseventgenerationinMadE-venthasbeenrevamped.
WhileMadEvent4wasalreadycombiningprocesseswithidenticalmatrixelements(suchasgg→uuandgg→dd),MadGraph5combinesallprocesseswiththesamespin,colourandmassofexternalparticlesintoasinglesubpro-cessdirectory.
Thediagramsofthesesubprocessesarematchedtocombinedintegrationchannels,sothatasingleintegrationchannelwillperformsinglediagramenhancedphasespaceintegration[5]forallsubprocesseswiththecorrespondingdiagram.
Diagramswhosepolestructuredieronlybypermutationsofthenalstatemomentaarealsocombined,–12–ProcessSubproc.
dirs.
ChannelsDirectorysizeEventgen.
timeMG4MG5MG4MG5MG4MG5MG4MG5pp→W+j6212479MB35MB3:15min1:55minpp→W+jj41413824438MB64MB9:15min4:19minpp→W+jjj7351164120842MB110MB21:41min*8:14min*pp→W+jjjj2967150296093.
8GB352MB2:54h*46:50min*pp→W+jjjjj-8-2976-1.
5GB-11:39h*pp→l+lj122488149MB44MB21:46min3:00minpp→l+ljj54458648612MB83MB2:40h11:52minpp→l+ljjj86554082401.
2GB151MB49:18min*16:38min*pp→l+ljjjj23576547212185.
3GB662MB7:16h*2:45h*pp→tt325349MB39MB2:39min1:55minpp→ttj73451797MB56MB10:24min3:52minpp→ttjj225417103274MB98MB1:50h32:37minpp→ttjjj3463816545620MB209MB2:45h*23:15min*Table2.
Numberofsubprocessdirectories,numberofintegrationchannelsfortheinitialrun("survey")oftheeventgeneration,sizeofthedirectoryafteronerungenerating10,000events,andruntimesforgenerating10,000events,comparingMadGraph/MadEvent4output("MG4")withgroupedsubprocessoutput("MG5").
Forallprocesses,p=j=g/u/u/c/c/d/d/s/s,l±=e±/±.
Theruntimesfor0-,1-and2-jetprocessesareforaSonyVAIOTZlaptopwith1.
06GHzIntelCoreDuoCPUrunningUbuntu9.
04,gFortran4.
3andPython2.
6,whilethe3-,4-and5-jetruntimes(markedby*)arefora128-corecomputerclusterwithIntelXeon2.
50GHzCPUs.
pp→W++5jisnotpossibletorunwithMadGraph/MadEvent4.
tofurtherminimizethenumberofintegrationchannels.
Thismeansthatforaprocesslikepp→l+l+3j,therewillbeonly5subprocessdirectories,correspondingtothesubprocessgroupsgg→l+lgqq,gq→l+lggq,gq→l+lqqq,qq→l+lggg,andqq→l+lgqq,ascomparedto86directoriesinMadGraph4(seetable2).
Takentogetherwiththeidenticationofinitialstatemirrorprocesses,thenumberofintegrationchannelsfortheinitialcrosssectiondeterminationrun("survey")issignicantlyreduced,seetable2.
Alsoforthesubsequenteventgenerationrun("rene"),thenumberofintegrationchannelsisusuallyreduced.
Therequireddiskspaceisreducedbyafactorcorrespondingtothereductioninnumberofintegrationchannels.
Tofurtherimproveparallelrunningoftheresultingcongurations,wehaveimple-mentedtheabilitytosplitupchannelswithlargecontributiontothecrosssectionintomultiplesub-channels,eachgeneratingafractionoftheeventsforthechannelinquestion.
Thisresultsinshortergenerationtimesandmoreequalworkloadforjobssubmittedtoacluster,aswellasconsiderablymorestableunweightingeciencyfortheintegration.
Manyfurthermeasurestospeedupandimprovethestabilityofthegenerationhavealsobeentaken,includinganewinitialguessfortheshapesoftheintegrationvariables,leadingtoconsiderablybetterstabilityoftheresults.
Theresultingreductioninruntimesforafewsampleprocessesarealsogivenintable2.
–13–3.
2MatrixelementlibrariesforPythia8Pythiaisoneofthemostwidelyusedmultipurposeeventgenerators,whichincludesma-trixelementevaluation,partonshowering,hadronization,particledecaysandunderlyingeventsinasingleframework.
MatrixelementsforPythiahavehistoricallybeenimple-mentedbyhand.
ThemostrecentimplementationofPythia,theC++versionPythia8,allowsmatrixelementsfor2→1,2→2and2→3processestobeprovidedbyexternalprograms.
TheexibilityinoutputformatsinMadGraph5hasallowedustoimplementdedicatedmatrixelementoutputforPythia8,therebyeectivelyremovingtheneedforimplementationofanymatrixelementsforPythiabyhand.
Letusnowdescribethemainfeaturesofthisnewimplementationforthereadersinterestedinmoretechnicaldetails.
ThenewmatrixelementsaregivenintheformofclassesinheritingfromtheinternalbaseclassSigmaProcess.
Suchprocessclassesneedtoimplementanumberofmemberfunctions,providingPythiawithinformationabouttheprocess(initialstates,externalparticlemasses,s-channelresonances,etc.
),aswellasfunctionstoevaluatethematrixelementsforallincludedsubprocessesandselectnal-stateparticleid'sandcolourowforeachevent.
Duringeventgeneration,Pythia8callsthematrixelementclasseswithgivenmomentafortheexternalparticles.
Startingfromv.
8.
150,Pythiaalsomakesmodelparametersfornewmodels(readinusingtheLesHouchesinterfaceforBSMmodelpa-rameters[67])availabletotheresultingprocessclassthroughaninstanceoftheclassSusyLesHouches,allowingfortheimplementationandsimulationofprocessesinanynewphysicsmodel.
TheresultingmatrixelementsaretreatedbyPythiainexactlythesamewayastheprocessesavailableinthePythiacorecode.
MadGraph5alsoprovidesclassesforevaluationofallmodelparametersneededforthematrixelementevaluationaswellasthehelicityamplitudesusedbythematrixelements.
StandardmodelparametersareextractedfrominternalPythiaparameters,whilenewphysicsparametersarereadinfromBSM-LHAles.
JustasintheregularMadEventoutput,anymodelparametersbasedonthestrongcouplingconstantarerecalculatedeventbyeventtousetherunningvalueofαs(asprovidedbyPythia),whilexedmodelparametersareinitializedatthetimeofprocessinitialization.
ThePythia8outputofMadGraph5isalibrarycalledProcessesmodelname,whichisautomaticallycreatedinanewdirectoryunderthePythia8basedirectory.
Thislibrarycontainssourcecodelesforallgeneratedprocesses,modelparameters,andhelicityamplitudes,aswellasamakeletocompilethelibraryandplaceitinthelibdirectoryofPythia.
Asanextrahelptotheinexperienceduser,anexamplemainprogramleisalsocreatedintheexamplesdirectorytogetherwithadedicatedmakele,whichshowshowtogenerateeventsfromtheimplementedprocesses.
Thesemainprogramlescanbeedited,compiled,andrundirectlyfromtheMadGraph5commandlinebyrunningthelaunchcommand,orcompiledandrunexternally.
IfmultipleprocessesaregeneratedinMadGraph5,thoseprocesseswillautomaticallybearrangedindierentprocessclassesaccordingtotheinitialandnalstates.
This–14–orderingusesthesamemachineryastheneworganizationofMadEventsubdirectoriesdescribedinsection3.
1above.
InorderforPythiatoperformeventgeneration,itneedsallsubprocessesinsideagivenprocesstohavethesamespinandmassofexternalparticles.
Thiscombinationofsubprocessesintosingleclassesallowsforfurtheroptimizationofthematrixelementcalculation,inthatthehelicitywavefunctionsandamplitudescanbecalculatedonceandforall,andthenbereusedbetweenallthedierentsubprocessesinaprocessclass.
Asacrosscheck,wehavecomparedthecrosssectionresultsfortheautomaticallygeneratedprocesseswithinternalPythiaprocessesforalargevarietyofprocessesintheStandardModelandtheMSSM,withperfectagreement.
3.
3DiagramdrawingInMadGraph4theamplitudegenerationanddiagramdrawingislimitedtohandlingthreeandfourpointvertices.
Inaddition,thediagramdrawingisbasedonalengthmin-imizationprocedure,whoseconvergenceisnotaprioriguaranteedandsometimescreateslineswithzerolength.
Toovercometheselimitations,acompletelynewalgorithmhasbeenimplementedinMadGraph5.
Thebasicideaistoassociatetoeachvertexalevel(i.
e.
,toorganisethevertecesinclasseseachonecharacterizedbya"distance"fromtheleftendofthediagram)denedbythefollowingsimplerules:Theverticesassociatetoinitialparticlesarealwaysatlevelone.
Allverticesattachedtoat-channelpropagatoraresetatlevelone.
Thedierenceoflevelbetweentheendpointsofans-channelpropagatorequalsone.
Asaboundarycondition,allexternallinesareassociatedtoalevelatthestartoftheprocedure:initialstateparticlesaresettolevelzero,whileallnalstateparticle'slevelsaresettothemaximalpossibleoneplusunity.
1Thepositionoftheverticesarethencomputedsuchthatallverticesatagivenlevellieequallyspacedonthesameverticalline.
Themainchallengeofthismethodconsistsinavoidinglinecrossingbetweenpropa-gators/nalstateparticles.
Inmostsituations,asimplere-orderingoftheverticesateachlevelisenoughtoavoidlinecrossings.
Theordershouldbesuchasverticesconnectedtotherstvertexofthepreviouslevelcomesrst,thentheoneslinkedtothesecondoneandsoon.
Ifalineconnectstwonon-adjacentlevels,whichhappensfornalstateparticles,thismethodsneedsanadditionaltrick.
Then,theintermediatelevelneedsaspecialcon-guration—i.
e.
,notequallyspaced—inordertoavoidanypossiblelinecrossing.
Thisisdealtwithbyaddingafakevertex(whichisnotdrawn)totheintermediatelevel.
Sincewekeepthe"equallyspaced"rulewiththefakevertex,thiscreatestheappropriategap.
Theresultingalgorithmisveryfastandecientlygeneratescleandiagramswithanynumberofexternalparticles.
1Optionsareavailablethatallowtomodifytheleveloftheexternalpartonsedges,resultingindierent-lookinggraphs.
–15–ModelClassModelnameDescription/CommentsInMG5StandardModelsmSeveralrestrictions/simplicationsavailableheftTopandWloopsfortheHiggsthroughdim-5operatorsSMextensions4GenFourthGenerationmodelwithfullCKM4SMScalarsExtraO(n)scalarsectorHiddenHiddenAbelianHiggsModelHillHillModel2HDMThegeneralTwoHiggs-DoubletModelTripletDiquarksSMplustripletdiquarksSextetDiquarksSMplussextetdiquarksSUSYmodelsmssmMinimalSupersymmetricextensionoftheSMnmssmNext-to-MinimalMSSMrmssmR-symmetricMSSMrpvmssmRparityviolatingMSSMExtra-Dimmodels3-siteMinimalHigglessModel(3-siteModel)MUEDMinimalUEDLEDLargeExtraDimensionsRSRandall-SundrumHEIDICompactHEIDIEFT'sChiPTChiralperturbationtheorySILHStronglyInteractingLightHiggsMWTTechnicolorTable3.
SelectionofmodelsthatarecurrentlyavailableinFeynRulesandcanbeusedinMadGraph5.
ThelastcolumnindicatemodelwhicharepresentbydefaultinthecurrentreleaseofMadGraph5.
Anasterisk()indicatesthatthemodelintheMadGraph5libraryisasimpliedversionofthecompletemodel.
4ModelsTheMadGraph5libraryofmodelsisbuiltuponthatofFeynRules[17]andwrittenintheUFOformat[20].
BackwardcompatibilitywiththecurrentmodelsofMadGraph4issupportedaslongasnoparticleswithspin3/2orhigherarepresentinthemodel.
ThesetofcurrentlypubliclyavailablemodelsfromtheFeynRuleswikipageisshowninthetable3.
SeveralmodelsarecurrentlyavailableintheMadGraph5release-thesearemarkedwitha""inthetable.
Besidesthemodelsinthetable,alsosomeadditionalmodelsusedforexamplesinthispaperareincludedintherelease,e.
g.
,thefour-fermioninteractionmodelsusedinsection6.
2.
4.
1InheritingmodelsfromFeynRules:UFOandALOHAAnylocalquantumeldtheorycanbeidentiedby:Asetofparticlesandtheirquantumnumbers(spin,charges,etc.
).
Asetofparameters(masses,couplingconstants,etc.
).
Asetofinteractionsamongthedierentparticles.
Themostecient,reliableandcompactwaytoencodethatinformationisbydirectlywritingaLagrangianwith(matterandinteraction)eldscarryingthedesiredquantum–16–numbersandbyusingwell-knowntext-bookrulestoextracttheFeynmanvertices.
ThisiswhatFeynRulesdoesinafullyautomaticway.
Oncetheverticesareobtained,theissueofpassingthisinformationtoamatrixelementgeneratorlikeMadGrapharises.
Forexample,forMadGraph4,FeynRuleswritestheoutputlesexactlyintheformatneededbythecode.
Thisprocedure,however,inadditiontobeingveryheavytomain-tainforFeynRulesdevelopers,hasthedrawbackthatitsuersfromthesameintrinsiclimitationsoftheMadGraph4modelformatitself.
ThepurposeoftheUniversalFeynRulesOutput(UFO)[20]istoovercomepossiblelimitationsduetospecicmatrixelementgeneratorsandtranslatealltheinformationaboutagivenparticlephysicsmodelintoaPythonmodulethatcaneasilybelinkedtoanyexistingcode.
Thisoutputiscompleteandindependentofthematrixelementgenerator,allowingfullexibilityandimprovements.
Itsavesthemodelinformationinanabstract(generator-independent)wayintermsofPythonobjectsandclasses,whichincaseofMadGraph5canbedirectlylinkedtothecode.
OncetheinformationofamodelisavailableintheUFO,itcanbeusedbyALOHAtoautomaticallywritetheHELASlibrarycorrespondingtothecorrespondingFeynmanrules.
ALOHA,whichiswritteninPython,producesthecompletesetofroutines(wave-functionsandamplitudes)thatareneededforthecomputationofFeynmandiagramsatleadingaswellasathigherorders.
TherepresentationislanguageindependentandoutputsinFortran,C++,Pythonarecurrentlyavailable.
InsodoingalltheintrinsiclimitationsregardingthepossibilityofgeneratingarbitrarynewphysicsmodelprocessofMadGraph4areovercome.
Asalreadyexplainedabove,MadGraph4isbasedontheHELASlibrarythatencodesalimitedsetofLorentzstruc-tures.
Whileextensionsarepossibleandhavebeendoneforseveralimportantcases(suchasspin-2[68]andspin-3/2[69]particles),theyentailatediousworkofwritingandtestingallnewroutinesbyhand.
ALOHAviatheUFOfullyautomatesthisprocedure.
Inaddi-tion,complicatedinteractionsthatfeaturenon-factorizablecolourandLorentzstructures,suchasthoseshowingupinthecountertermsandR2verticesatNLO[70],whichcannotbehandledbyMadGraph4,arenowfullysupported.
4.
2ModelrestrictionlesInphenomenologyapplications,itisoftenconvenienttoxsomeparametersatsomegivenvalues(e.
g.
,masses/CKMparameters/couplings,etc.
).
Suchrestrictionsmightallowimportantgainsbothintermsofspeedandalsoinsizeofthegeneratedmatrixelementcode.
MadGraph5allowstorestrictagivenUFOmodelbasedonanumericalevaluationofallcouplingswithparametersfromaBSM-LHAle[67](notethatanalyticalmodelrestrictionscanalsobeperformeddirectlyinsideFeynRules).
FromagivenBSM-LHAle(thatwecallarestrictionle),MadGraphcanevaluatethevalueoftheinternalparametersandofthecouplings.
Themodelisthenmodiedaccordingtothefollowingrules:Allvanishingparametersareremovedfromthemodel,andallparameterswithvalueequaltounityarexedtothisvalue.
–17–AllparametersbelongingtothesameLHAblockwithidenticalvaluesarerepresentedbyasingleparameter.
Allcouplingswithidenticalvaluesarerepresentedbyasinglecoupling.
Allinteractionslinkedtoavanishingcouplingareremovedfromthelistofinteractions.
ThesestepsallowMadGraph5tooptimisethematrixelementoutput,andtheoutputofmulti-processgeneration(seesection3.
1).
Inordertoavoidtheusersettingtheparametersofthemodelinawaywhichisinconsistentwiththerestrictedmodel,wealsomodifytheparamcardassociatedwiththemodel,removingallparametersthathavebeenxedbytherestriction.
Thedefaultvalueforallremainingparametersissettothatgivenintherestrictionle.
Notethatcareisneededwhentheuserdesignstherestrictionle,toensurethattheresultcorrespondstowhatitisexpected,andthattheresultingparamcardincludesalldesideredparameters.
ModelrestrictionsareusedforseveralmodelspresentinMadGraph5,includingSMandMSSM.
Bydefaultwhenamodelisloaded,MadGraph5appliestherestrictiondenedinthelerestrictdefault.
txtinthecorrespondingmodeldirectory.
FortheStandardModel,thedefaultrestrictionsetstheCKMmatrixtobediagonalandsetsthemassoftherstandsecondgenerationfermionstozero.
Theusercanbypassthedefaultrestrictionbyadding"-full"tothemodelnamewhenimportingthemodel,orapplyadierentrestrictionlebyadding"-restriction"tothemodelname,correspondingtoarestrictionlerestrictrestriction.
dat.
4.
3ConsistencychecksforprocessesandmodelsFornewmodelimplementations,eitherfromFeynRules[17]orbydirectlyprovidingtheUFOmodelformatusedinMadGraph5,itiscrucialtobeabletochecktheconsistencyofthenewmodels.
Tothisend,MadGraph5featuresaseriesofconsistencychecksforprocesses:1.
Helicityamplitudecallsandthehelicityamplitudeimplementationsarecheckedbycalculatingthespeciedprocesseswithmultiplepermutationsoftheexternalparticlesinthediagramgenerationinagivenphasespacepoint,checkingthatthevalueofthematrixelementisidenticalforthedierentpermutations.
Thisecientlychecksseveralaspectsofthemodelimplementation:therelationbetweentheLorentzandcolourstructures,theimplementationoftherelatedhelicityamplitudeswithdierentwavefunctiondecomposition,andtheeectsoffermionowviolationandchargeconjugation.
2.
Gaugeinvarianceischeckedbycalculatingthematrixelementinarandomphasespacepointwiththewavefunctionofanexternalmasslessvectorbosonreplacedbyitsmomentum(forprocesseswithexternalmasslessvectorbosons).
Ifgaugeinvarianceissatised,theresultingmatrixelementiszero(withinnumericalprecision).
–18–3.
InvarianceofthematrixelementbyLorentztransformationsischeckedbycomparingthematrixelementvaluebeforeandafteraseriesofLorentzboosts.
Wehavefoundthataltogether,thesechecksprovideapowerfulwaytovalidatemodelimplementationsinMadGraph5.
5Validationandspeedbenchmarks5.
1ValidationOncethechecksbasedonsymmetries,gaugeinvariance,andLorentzinvariancedescribedabovearesatised,onecanperformthe"physics"validationbycomparingresultsfortheevaluationofthematrixelementingivenpointsofthephasespaceand/orintegratedcrosssectionswithothergeneratorsoranalyticalcalculations.
TovalidatebothMadGraph5andthemodelsprovidedwithit,wehaveextensivelycomparedsquaredmatrixelementscomputedpoint-by-pointinthephase-spacewiththoseobtainedinMadGraph4.
SinceMadGraph5supportsdierentinputmodelformats(theMadGraph4formatmodelsandthenewUFOmodels)andisabletocreateoutputindierentlanguages(Fortran77,C++),wehavecomparedtheMadGraph4resultswithMadGraph5inthefollowingthreecongurations:UsingtheUFOmodelasinputandchoosingtoexportthematrixelementinFortran77.
(InthiscontextthehelicityamplituderoutinesarecreatedbyALOHA).
UsingtheUFOmodelasinputandchoosingtoexportthematrixelementinC++.
(InthiscontextthehelicityamplituderoutinesarecreatedbyALOHA).
UsingtheMadGraph4modelasinput(thereforetheonlyoutputavailableisFortran77andweusetheHELASpackage).
Asummaryofthechecksperformedispresentedintable4.
Thoseprocesses(morethanthreethousandintotal)areallinperfectagreementbetweenallthreecongurationsandtheMadGraph4value.
Inadditiontothesquaredmatrixelementtests,wehavealsoperformedasystematiccomparisonatthecross-sectionlevelfor2→2and2→3bothintheSMandintheMSSM.
ThiscomparisonwasdonewiththeFeynruleswebvalidationinterface[71]withrespecttoMadGraph4,CompHEP/CalcHEP,andWhizard.
5.
2Speedbenchmarks—ProcessgenerationItcanbeinterestingtocomparethetimerequiredforthegenerationofcompleteMadE-ventdirectoriesfordierentprocessesinMadGraph4andMadGraph5.
Notethatthisisthetimeforthegenerationofthematrixelementoutputcodeandadditionallesneededforphasespaceintegrationandeventgeneration,notthetimeforevaluatingthematrixelementvaluesusingthegeneratedcode.
Atimecomparisonforevaluationthegeneratedmatrixelementisgiveninthenextsection.
–19–modelprocessclassinformationnumberofprocessesSMAA→AAOnlyrstgenerationforfermions249SMAA→AANorstgenerationoffermions589SMBB→BBB86SMBB→BF1,2F1,246SMF1,2F1,2→BF1,2F1,240SMF1,2F1,2→BF1,2F1,2216SMBB→BBBB55MSSMPP→χ0χ050MSSMPP→χ+χ/gg26MSSMPP→LL55MSSMPP→Q1,2Q1,2188MSSMPP→Q3Q371MSSMQ1,2Q1,2→LL208MSSMQ1,2Q1,2→Q1,2Q1,2285MSSMPP→Q1,2Q1,2VonlySMvectorbosonsincluded564MSSMVV→Vχ+χ71MSSMPP→L+Lχ0χ0200MSSMVV→VVχ+χ177HEFTBB→BBincludingtheCP-oddHiggs62HEFTgg→H+ngn=1,2,3,44RSAA→AAOnlyrstgenerationforfermions362RSF3F3→AA248RSF1,3F1,3→F1,3F1,3B452Table4.
ClassesofprocessesusedtocomparetheoutputofMadGraph5withMadGraph4.
Thedierentlettersdesignateclassesofparticles:Acontainsalltheparticlesofthemodel;Ficontainstheithgenerationoffermionsofthemodel(Noindicesmeansallgenerationsallowed);L±containsalltheleptonsofthemodel;Vcontainsallthevectorbosonsofthemodel;Bcontainsallthebosonsofthemodel(i.
e.
,thevectorandthescalarparticles);χ0containsalltheneutralinos;χ±containsallthecharginos;Qicontainstheithgenerationofsquarks;Lcontainsthefullsetofsleptons.
Table5showsthetimeneededforanumberofexampleprocesses,includingmulti-processes,high-multiplicitynalstateprocesses,anddecaychainprocesses.
Ascanbeseenfromthetable,themaingainsinspeedareincomplicatedprocesses:processeswithalargenumberofsubprocessesduetolargemultiplicitiesofmultiparticlelabels(suchaspp→jjje+e),processeswithmanyexternalparticles(suchasgg→5gore+e→6e),andindecaychainprocesses,wherethespeedupcanbeseveralordersofmagnitudewithrespecttopreviousversionsofMadGraph.
5.
3Speedbenchmarks—MatrixelementevaluationMadGraph5isnotonlyfasterthanitspredecessorsingeneratingcodeforcomplicatedprocesses,theproducedmatrixelementcodeisalsofasterandmorecompact.
Thisis–20–ProcessMadGraph4MadGraph5SubprocessesDiagramspp→jjj29.
0s25.
8s34307pp→jjl+l341s103s1081216pp→jjje+e1150s134s1419012uu→e+ee+ee+e772s242s13474gg→ggggg2788s1050s17245pp→jj(W+→l+νl)146s25.
7s82304pp→tt+fulldecays5640s15.
7s2745pp→q/gq/g222s107s3134757particledecaychain383s13.
9s16gg→(g→uuχ01)(g→uuχ01)70s13.
9s148pp→(g→jjχ01)(g→jjχ01)—251s14411008Table5.
TimeforgenerationofcompleteMadEventdirectories(withtheexceptionofgg→5g,forwhichaFortranstandalonedirectorywasgenerated)foraselectionofprocesses,forMad-Graph4andMadGraph5.
Allprocesseshavep=j=g/u/u/c/c/d/d/s/s,l±=e±/±/τ±,νl=νe/ν/ντandνl=νe/ν/ντ.
q/ginthetablecorrespondstod()l/r/u()l/r/s()l/r/c()l/r/g.
Fortt+fulldecays(meaningpp→(t→bq/l+q/νl)(t→bq/lq/νl)),theMadGraph4processgenerationwassplitupin12dierentprocessdenitionstoreducethenumberoffailedprocessattempts.
The"sevenparticledecaychain"wasgg→(g→u(ul→u(χ02→Zχ01)))(g→udχ1).
Thenumberofsubprocessesanddiagramsarequotedaftercombinationofsubprocesseswithidenticalmatrixelements.
AllprocessesaregeneratedwithmaximalnumberofQCDvertices.
AllnumbersareforaSonyVAIOTZlaptopwith1.
06GHzIntelCoreDuoCPUrunningUbuntu9.
04,gFortran4.
3andPython2.
6.
thankstothenewdiagramgenerationalgorithm,whichallowsforimprovedrecyclingofsubdiagramwavefunctionsbetweendierentdiagrams,reducingthenumberofhelicitywavefunctioncalls(asdiscussedinsections2.
1and2.
2).
Table6showsacomparisonofthenumberoffunctioncallsandruntimeformatrixelementevaluationusingHELASandALOHAroutines,relativetoMadGraph4.
Fromthetable,weseehowtheimprovedwavefunctioncalloptimisationtranslatestoconsiderablyimprovedruntimes,especiallyforcomplicatedprocesses.
6BSMexampleapplications6.
1Non-standardcolourstructures:ijkandcoloursextetsThephenomenologyofdiquarkresonanceshasrecentlybecomepopular,sincetheseparti-clescouldbeamongtherstnewphysicsparticlestobeobservedattheprotononprotoncolliderLHC[65].
Suchdiquarkshavethequantumnumbersoftwovalencequarks,andmustthereforebeeithercoloursextetsorcolouranti-triplets.
Inthelattercase,thecou-plingtoquarksisthroughacompletelyantisymmetriccolourtriplettensor,whichistheonlywaytocontractthreecolourtripletindices.
ThetriplettensorisalsoimportantintheformulationofR-parityviolationsupersymmetricmodels,wheree.
g.
ascalarquarkcandecayintoapairofStandardModelantiquarks.
Asdiscussedinsection2.
3,boththesextetcolouralgebraandthetensorhavebeenimplementedinMadGraph5.
–21–ProcessFunctioncallsRuntimerelativetoMG4MG4MG5MG5+HELASMG5+ALOHAuu→e+e881.
01.
1uu→e+ee+e110800.
521.
4uu→e+ee+ee+e666837750.
330.
57gg→gg13130.
900.
81gg→ggg86780.
940.
94gg→gggg8116210.
990.
66uu→dd661.
01.
0uu→ddg16161.
01.
2uu→ddgg85670.
740.
86uu→ddggg7485150.
680.
52uu→uugg1601160.
670.
70uu→uuggg14689600.
480.
36uu→dddd42330.
991.
2uu→ddddg3101970.
610.
74uu→ddddgg337218760.
240.
19uu→dddddd13707530.
180.
19Table6.
NumberofhelicityfunctioncallsandruntimeratiotoMadGraph4formatrixelementevaluationofmatrixelementcodeproducedbyMadGraph4andMadGraph5usingHELAS,andMadGraph5usingALOHAroutines.
ForMadGraph4,theHELASlibraryhasbeenused.
Notethatintheuu→qq+Xprocessgenerations,onlyQCDinteractionshavebeenallowed(QED=0).
ThenumberoffunctioncallsforMadGraph5doesnotdependonwhetherHELASorALOHAisused.
Asanexampleofphenomenologyusingtheseimplementations,weshowingure3athecrosssectionsfordierentspeciesofcoloursextetandantitripletscalardiquarksDatLHCwith7TeVc.
m.
energy.
Wehaveincludedcoloursextetdiquarkscouplingtouu/cc/tt,dd/ss/bbandud/cs/tb,andcolourantitripletdiquarkscouplingtoud/cs/tb.
Notethatduetotheantisymmetryoftheijkcolourcouplingofcolourtripletdiquarkstoquarks,colourtripletdiquarkscanonlycoupletoo-diagonalavourquarkcombinations.
TheDqq(′)Yukawacouplingconstantshavebeensetto102inthegure.
Notethefactor2betweenthepp→Dproductioncrosssectionsforsextetandtripletdiquarks(foridenticalYukawacouplings),duetothedierentcolourfactors.
Ingure3b,weshowtheeectofjetmatchingbetweenmatrixelementsandpartonshowersforcharge+43sextetdiquarkproductionat7TeVLHC.
pTdistributionsfortheradiatedjetsarecomparedbetweenmatchedproductionwithMadGraph(usingthekT-MLMmatchingschemethatisdefaultinMadGraph,withmatrixelementsforpp→D+0,1,2jets)andPythia6.
4withpT-orderedshowers,andjustusingtheleadingorderprocesspp→DwithpT-orderedPythiadefaultsettings.
Themassofthediquarkis500GeV.
Itisclearthatmatchingisnecessaryforaprecisedescriptionofhigh-pTjetradiationinassociationwithdiquarkproduction.
–22–10100100010000100000400600800100012001400160018002000Crosssection(pb)at7TeVLHCDiquarkmass(GeV)Sextet(uu/cc/tt)Sextet(dd/ss/bb)Sextet(ud/cs/tb)Triplet(ud/cs/tb)ofjetTP050100150200250300350400pb/bin-310-210-1101pp->sextet,matchedpp->sextet,unmatched1st2nd3rdFigure3.
Upper:Crosssectionsfordierenttypesofdiquarkresonancesat7TeVLHC.
Seetextfordetails.
Lower:ComparisonofpTforradiatedjetsbetweensinglecoloursextetdiquarkproductionwithjetmatching(usingMadGraphandPythia)andwithoutjetmatching(usingPythiapartonshowersfortheleadingorderprocesspp→Donly).
Seetextfordetails.
6.
24-fermionvertices:uu→ttWiththepossibilityofspecifyingverticeswitharbitrarynumberofparticles,aparticulardicultyariseswhenavertexhasmorethantwofermions,inwhichcaseitisnecessarytodenethefermionowinanunambiguousway.
TheconventionchosenbytheMad-Graph5andFeynRulesauthorsisthatthefermionowisdenedbythepositionoftheparticleintheinteraction,withtheorderbeingIOIO.
.
.
whereIstandsfor"incom-ing"andOstandsfor"outgoing"fermion.
Anynumberofbosonscanbeaddedafterthefermions.
Thismeansthat,fromtheMadGraph5pointofview,theinteractionsututanduuttaretreatedindierentways-intheformercase,thefermionowsgobetweenuandt,whileinthelattercasewehavefermionnumberviolatingowsu→uandt→t.
Suchfermionnumberviolatingmulti-fermionverticesarereadilytreatedbythealgorithmdescribedinsection2.
2,bytheuseofconjugateΓmatricesforeachfermionnumberviolatingfermionline.
Ofparticularinterestarefour-fermionvertices,whichareacommonfeatureofeectivetheoryformulationsforphysicsbeyondtheStandardModel,andhaverecentlybeenstudied–23–1e-101e-091e-081e-071e-061e-0530040010002000300010000Crosssection(pb)C.
M.
energy(GeV)T-channelpropagator4-fermionvertex1e-101e-091e-081e-071e-061e-050.
00010.
0010.
010.
1130040010002000300010000Crosssection(pb)C.
M.
energy(GeV)S-channelresonance4-fermionvertexFigure4.
Crosssectionforuu→ttasafunctionof(xed)beamenergy,comparing4-fermionimplementationsandthecorrespondingimplementationswithexplicitpropagators.
Left:t-channelscalarexchange.
Right:s-channelscalarexchange.
Forbothcases,themassofthepropagatorissetto10TeV.
inthecontextoftop-quarkLHCphenomenology[72–74]Weareherepresentingtwoexam-plesoffour-fermionverticesleadingtotheprocessuu→tt[75],onewhichcorrespondstotheexchangeofaheavys-channelpropagator(inthiscaseascalarcoloursextetdiquark)andonewhichcorrespondstoaheavyt-channelpropagator(aneutralcoloursingletavourchangingscalar).
Inbothcases,weuseamassof10TeVforthepropagators,andrepresentthefour-fermionverticesbyintegratingouttheappropriatescalarpropagator.
Ingure4wecomparethefulltheories,includingtheexplicitpropagators,withthe4-fermionvertexversionsofthetheories.
Thegureshowsuu→ttcrosssectionsforthetwoscenariosasafunctionofxedcenterofmassenergy,forsomeparticularcouplingvalues;tocomparewithaparticularcollider,thecrosssectionsneedtobeconvolutedwiththeuupartonluminosity.
Thesuddenturn-onofthecrosssectionisduetothekinematicalsuppressionfromthenal-statetopquarkmass.
Asexpected,thefourfermionvertexformulationagreeswiththeexplicitpropagatorformulationuptoac.
m.
energyofabout1/10ofthepropagatormass.
Theexplicitt-channelpropagatormakesthecrosssectionleveloastheenergygetsclosetothemassandtheexchangemomentumterminthedenominatorofthepropagatorstartsdominatingoverthemassterm,whiletheexplicits-channelpropagatordisplaystheusualBreit-Wignerpeakastheenergygetsclosetothemass.
Inthiscase,thewidthofs-channelpropagatorisΓS=200GeV.
6.
3n-particlevertices:H+4gMadGraph5allowsverticeswithanynumberofexternalparticles.
Suchverticesfre-quentlyappearineectiveeldtheories,wherenon-renormalizableoperatorsareincludedwithsomeappropriatescalesuppression.
Oneofthemostphenomenologicallyimportanteectiveeldtheoriesforhigh-energyphysicsistheadditiontotheStandardModelofeectivecouplingsbetweentheHiggsbosonandgluonsthroughatopquarkloop,withthetopmasstakenmuchlargerthantheHiggsbosonmass.
–24–g1g2h3g4g5Figure5.
5-particlediagrambyMadGraph5.
Thisisoneof38diagramsfromtheprocessgg→HggintheStandardModelwitheectivecouplingsoftheHiggsbosontogluonsthroughatoploop.
WhilethesimplestvertexwecanwritedowninthistheoryisHgg,thenon-AbeliannatureofQCDrequireustoincludetwoadditionaloperatorsintheLagrangian,HgggandHgggg,couplingtheHiggsdirectlytothreeandfourgluonsrespectively.
WhilepreviousversionsofMadGraphhadtosplituptheve-particleHggggvertexusinganauxiliarynon-propagatingtensorparticle,MadGraph5,inconjunctionwithALOHA,candirectlyhandlethisvertexinexactlythesamewaythatithandlesverticeswithlowermultiplicity.
Thecorrespondingdiagram,fromtheprocessgg→Hgg,isfoundingure5.
Notethatforconsistency,onlyoneeectiveHiggs-gluoncouplingvertexcanbepresentinagivendiagram.
Thisismadepossiblebyspecifyingaseparatecouplingorder,HIG,forthesevertices.
AnyprocessgenerationinthismodelshouldthereforebespeciedwithamaximumorderHIG=1.
ThenewformulationofHiggsbosoncouplingstogluonshasbeenthoroughlycheckedagainsttheimplementationinMadGraph4,andisoneofthemodelsincludedintheMadGraph5package.
Thesameissuehappensforagravitoninteractingwithfourgaugebosons.
AfullvalidationoftheMadGraph5implementationtheFeynRulesRSmodelagainsttheMadGraph4[68]implementationhasbeenperformed.
6.
4Chromo-magneticoperatorRecentmeasurementsintopquarkpairproductionperformedbothatTevatronandLHCoeranidealgroundtosearchfornewphysicseects[74].
Ifsuchnewphysicsisatscaleshigherthanthoseexploredatthecurrentcollidersitcanbeecientlymodeledbyaneectivetheorywithanon-renormalizableoperatorsuppressedbyaenergy-scaleΛ.
Inthecaseofthetopquark,onlyoneoperatorofdimension6existsthatisnota4-fermionoperator,thesocalledchromo-magneticoperator:L=(HQ)σνTAtGAνΛ2+h.
c.
,whereΛrepresentsthecutooftheeectivetheory.
AddingsuchatermtotheLagrangianofthestandardmodelleadstoadditionalinteractions,seegure6.
Forconsistency,any–25–g1g2t3t~4diagram1NP=2,QCD=1t2t~3g1diagram2NP=2g1g2t3t~4h5diagram3NP=2,QED=1,QCD=1t2t~3h4g1diagram4NP=2,QED=1Figure6.
Interactionsinducedbythechromomagneticoperator.
matrixelementshouldbecomputeduptoorderΛ2andthereforerequiringoneandonlyoneeectivecouplingtoenteratthesquaredmatrixelementlevel.
Thiscanbeobtainedbydeningathirdtypeofcoupling(inadditiontoQEDandQCD)associatedwiththenewinteractions.
AgaintheautomationoftheHELASroutinesandthepossibilityforMadGraph5todealwithverticeswitharbitrarynumberoflegs,allowsastraightforwardtreatmentofsuchoperators.
ThevalidationofthismodelhasbeenperformedbycomparingtheMad-Graph5resultstothoseobtainedinaprivate(andnotfullyautomatic)implementationofthismodelinsidetheMadGraph4framework.
Inadditiontoautomaticallyprovidingthematrixelementforanyprocessinvolvingthenewoperators,theevaluationofthecrosssectionsareapproximativelyfourtimesfasterthanthepreviousversion.
7ConclusionsandoutlookThecompleterewritingoftheMadGraphmatrix-elementgeneratorinPythonhasallowedustobuildonthetwentyyearsofexperiencegainedwiththeFortranversionandpushthecodeaswellasitsfunctionalitiestoanewlevel.
Theresultingcode,modularinstructureandwithembeddedrobustnessandsanitychecks,isnaturallyorganisedasacollaborativeplatform.
Anysucientlyskilledusercanexploit,modifyandextendthefunctionalitiesofthecurrentversion.
Thecodeisavailableviaamajoropen-sourceprojectdevelopmenthostingserviceusingtheBazaarversioncontrolsystem.
–26–Thenewcodestructureandnewfunctionalitiesopenthewaytodevelopmentsinthreemaindirections:BSM,NLO,andmergingwithshower/hadronizationcodes.
ThedirectlinktotheFeynRulesmodeldatabasewillallowquickandrobustimplementationnotonlyofnewphysicsmodelsbutalsoofanytypeof1-loopcounterterms,essentialingredientstoachieveNLOautomaticcomputationsintheSMandbeyond.
Workinallthesedirectionsisinprogress.
AcknowledgementsItisagreatpleasureforustothankallthepeoplewho,directlyorindirectly,helpandsupportoureortsandservicestothehigh-energycommunityandallourusersfortheircontinuousandpatientfeedback.
Inparticular,forthehelpintheextensivetestingofthenewversion,wethankAlexisKalogeropoulos;forthevalidationofnewphysicsmodels(andmuchmore)wethanktheFeynRulescoreauthors(NeilChristensen,ClaudeDuhr,BenjaminFuks)andassociates(PrisciladeAquino,CelineDegrande);forourclustermanagementwearegratefultoVincentBoucher,JeromedeFavereau,PavelDemin,andLarryNelson;forthegreatphysicsworkandthefun,weinparticularthankourcolleaguesandcollaborators:PierreArtoisenet,SimondeVisscher,RikkertFrederix,StefanoFrixione,NicolasGreiner,KaoruHagiwara,JunichiKanzaki,ValentinHirschi,QiangLi,KentarouMawatari,RobertoPittau,TilmanPlehn,MarcoZaro.
ThisworkispartiallysupportedbytheBelgianFederalOceforScientic,TechnicalandCulturalAairsthroughthe'InteruniversityAttractionPoleProgram-BelgiumSciencePolicy'P6/11-PandbytheIISN"MadGraph"convention4.
4511.
10.
AInstallationandonlinewebversionMadGraph5canbeeitheruseddirectlyonlineontheweborlocally.
Thecodecanbedownloadedfromthewebpagehttps://launchpad.
net/madgraph5.
InordertorunMadGraph5locally,Python2.
6orhigher(butnot3.
x)mustbeinstalled.
Thepackagedoesnotrequireanycompilationorconguration;afterunpacking,simplylaunchthemainscript:tar-xzpvfMadGraph5_v1.
x.
x.
tar.
gzcdMadGraph5_v1_x_x.
/bin/mg5TolearnhowtouseMadGraph5,entertutorialinthecommandlineinterface.
ThefulllistofpresentlyavailablecommandsaredescribedinappendixB.
MadGraph5canalsobeusedontheweb(afterfreeregistration).
Wecurrentlyhavethreepublicclusters:http://madgraph.
phys.
ucl.
ac.
be/http://madgraph.
hep.
uiuc.
edu/–27–http://madgraph.
roma2.
infn.
it/Atthisstage,MadEvent(Fortran)outputcanbegeneratedonline,anddownloadedasastandaloneprocessdirectory.
Inthenearfuture,alsootheroutputformatwillbeavailableonline.
Basedonauserrequest,wealsograntaccesstoruneventgenerationdirectlyononeoftheclusters.
Inthatcase,theusercanalsochoosetodirectlypasstheeventsthroughPythia6forhadronization,anduseafastdetectorsimulation,eitherDelphes[76]orPGS[77].
BCommandlineuserinterfaceThecommandlineinterfaceforMadGraph5isbuiltonthePythonmodulecmd.
Thismoduleallowsforaexibletreatmentofuserinputandsupportoffeaturessuchastabcompletion,commandhistory(accessedbytheupkey),helptexts,andaccesstoshellcommandsfrominsidethecommandlineinterface.
Usingthecommandlineinterface,theusercanconvenientlyaccessthefullfunctionalityofMadGraph5,includingimportingmodels,generatingprocesses,drawingFeynmandiagrams,generatingoutputinallavail-ableoutputformats(atpresent,MadEvent,Pythia8andstandaloneprocessandmodeloutputinFortranorC++),performingmodelchecks,andlaunchingeventgenerationinpreviouslycreatedprocessdirectories.
Thecommandlineinterfaceisstraightforwardtoextend,andmorefunctionalityiscontinuouslyaddedbasedonuserrequestsandcodedevelopment.
ThesyntaxforprocessgenerationinMadGraph5isverysimilartothatforMad-Graph4,withafewexceptionsreectingtheextendedfunctionalityofMadGraph5—mostnotably,spacesareneededbetweenparticlenames,sincethereisnolongeranylimitonparticlenamelength,andthesyntaxforgeneratingdecaychainsismodiedtoaccommodatethegreaterexibilityindecaychaingeneration.
Somesyntaxexamplesaregivenintable7.
However,theinterfacealsohastheabilitytoreadprocesscardswrittenforMadGraph4.
Theusercanstartthecommandlineinterfacebyrunningbin/mg5fromtheMad-Graph5directory.
IfthenameofalecontainingMadGraph5commandsisgivenasargument,thenthecommandsintheleareperformed.
Suchalecanbegeneratedfromtheseriesofcommandsusedinasessionbythehistorycommand,andisautomaticallyplacedintheCards/directorywhenaMadEventdirectoryiscreated.
ProcessgenerationcanalsobedoneasinMadGraph4,byrunningbin/newprocessmg5inacopyoftheTemplatedirectorywithanappropriateproccard.
datorproccardmg5.
datplacedintheCardsdirectory.
Welistherethepresentlyavailablecommandsinthecommandinterface(inalphabet-icalorder).
addprocess:Addandgeneratediagramsforaprocess,keepingpreviouslygeneratedprocesses.
–28–SyntaxDescriptionpp>l+l-Generatethe(valid)processesuu→e+e,dd→+etc.
pp>jjj(Nospecicationoforders.
)OnlyallowdiagramwithmaximalQCD/minimalQEDorder.
pp>jjjQED=2Allowupto2QEDvertices(andunlimitedQCDver-tices)indiagrams.
pp>z>l+l-jIncludeonlydiagramswithans-channelZ.
l+l->z|a>l+l-Includeonlydiagramswithans-channelZorans-channelγ.
bb~>tt~/gExcludeanydiagramswithagasinternalpropagator.
pp>bw+t~$tExcludeanydiagramswithans-channeltpropagator.
pp>tt~,t>bw+,\(t~>b~w-,w->l-vl~)Generateadecaychainwitht,tandWrequiredtobenear-onshell.
Notetheuseofparenthesestospecifydecayswithinadecaychain.
Table7.
SomeexamplesofprocessgenerationsyntaxintheMadGraph5commandlineinterface.
Themaindierencesw.
r.
t.
MadGraph4arethatspacesareneededbetweenparticlenames,thatbydefaultminimalQEDcouplingorderisassumed(ifthereareonlyQEDandQCDcouplingsinthemodel),andfurthermorethedecaychainsyntaxwhichnowallowsfullspecicationofalldecayprocessesincludingcouplingorders,requiredandexcludedparticles,etc.
check:Runmodelconsistencychecksforspeciedprocesses.
Availablechecksare:processpermutationchecks,gaugeinvariancecheck,andLorentzinvariancecheck(seesection4.
3).
define:Deneamultiparticlelabelusedforimplicitsummingoverprocesses.
Somecommonlyusedmultiparticlelabels(p,j,l+,l-,vl,vl~)areautomaticallydenedwhenamodelisimported.
display:Displayparticles,interactions,denedmultiparticlelabels,generatedpro-cesses,generateddiagrams,ortheresultsofprocesschecks.
generate:Generatediagramsforaprocess,replacinganypreviouslygeneratedpro-cesses.
history:Listthehistoryofpreviouscommandstothescreenortoale.
Theresult-inglecanbeusedtorepeatasequenceofcommandsusingtheimportcommand.
import:Importamodel(eitherintheUFOformatorMadGraph4format)oraprocesscard(inMadGraph5orMadGraph4format).
launch:Launcheventgenerationormatrixelementevaluationforcreatedprocessdirectories.
load:Loadaprocessormodelpreviouslysavedtoleusingthesavecommand.
–29–output:Outputprocesslesformatrixelementintegration.
Presentlyavailableout-putformatsare:MadEventformat(default),stand-aloneFortranformat,Pythia8format,andstand-aloneC++format.
save:Saveaprocessormodeltole,usingthePython"pickle"format.
set:Modifysettings,includingturningon/osubprocessgrouping(seesection3.
1).
tutorial:Startashorttutorialshowinghowtousethemostcommoncommands.
Notethatthehelpcommandgivesthefulllistofavailablecommands.
Typinghelpcommandgivesinformationabouteachcommand.
Thebuilt-incommandsquitandexit(orpressingcontrol-D)quitsMadGraph5,andshell(orstartingthelinewitha"!
")allowstoaccessanyshellcommand.
Finally,theinteractiveinterfacehasatutorialmodewhichallowstoquicklylearnthemostcommoncommandsusedintheinterface.
CProcessgenerationexamplesThisappendixpresentsexamplesoftheseriesofcommandsrequiredtogeneratethecodecorrespondingtothesquarematrixelementofvariousprocess.
Thosecommandcanbeeithercopied-pasteddirectlyintheinteractiveinterface(accessedbyrunning.
/bin/mg5)orwritteninatextleexecutedbyMadGraph5(executedby.
/bin/mg5commandfile).
C.
1Top-quarkpairproductionTherstexampleshowshowtoevaluateacross-section(andhowtogeneratepartonicevents)fortop-quarkpairproductionintheStandardModel:generatepp>tt~QED=2QCD=2outputMyOutputDirlaunchBydefault,MadGraph5importsthestandardmodel.
Therefore,nospeciccommandisneededforthat.
Asstatedabove,theprocessdenitionsyntaxisslightlydierentfromtheMadGraph4one.
InMadGraph5,spacesbetweenparticlesnamesaremandatory.
AsinMadGraph4,itispossibletospecifycouplingorders.
Ifthecouplingordersarenotspecied,thenMadGraph5guesseswhichinteractionstoallowbasedonthefollowingrules:IftheordersdenedinthemodelareQEDandQCDonly:Thestrongcouplingisassumedtobedominantovertheelectroweakcouplings,andtheQEDorderisthereforesettoitsminimalpossiblevalue,whileputtingQCDtoitsmaximalpossiblevalue.
Thisprovidesthedominantcontributiontothecrosssection,withoutthe(negligible)sub-leadingdiagramswithadditionalQEDcouplings.
–30–IfadditionalordersarepresentbesidesQEDandQCD,thenweallowanycouplingorderforanycoupling,(ifgivencouplingsarepreferred,theuserneedstosupplymaximumordersintheprocessdenition).
Thecomputationofthecross-sectionandthegenerationofpartoniceventsisdonewiththecommandlaunch.
DierentoptionsforcrosssectioncalculationandeventgenerationormatrixelementevaluationcanbefoundeitherwiththehelplaunchcommandorintheleMyOutputDir/README.
C.
2StoppairproductionThisexampleshowshowtoevaluatethevalueofthesquarematrixelementforagivenpointinphase-space.
Thisisoftenusedfortestingthecodeand/ortointerfaceMadGraphwithanexternalprogram.
importmodelmssmgeneratepp>t1t1~addprocesspp>t2t2~outputstandalonelaunchFirst,themssmmodelisimported.
Then,theexampleshowshowtocreatemultiplesubprocesses.
Thegeneratecommandclearsallpreviouslygeneratedprocesses,whiletheaddprocesskeepsallpreviousprocessesandaddsthenewprocesstotheset.
Ifnooutputdirectoryisgiventotheoutputcommand,theoutputwillbeplacedinanautomaticallynameddirectoryPROCmssm0.
C.
3SleptonpairproductionInthisexample,weshowhowtogenerateallsleptonpairproductionmatrixelementsforuseinPythia8(Seesection3.
2).
Togeneratetheneededsubprocesses,wecouldusetheaddprocesscommand.
However,giventhatthenumberofsub-processesisquitelarge,thisisnotveryconvenient.
Amuchmorehandysolutionistouseamulti-particlelabel,similartop/jforproton/jet:importmodelmssmdefinesl-=el-mul-ta1-er-mur-ta2-definesl+=el+mul+ta1+er+mur+ta2+generatepp>sl+sl-outputpythia8launchInordertoruneventgenerationfromtheprocess,youwillneedtohavePythia8installedonyourcomputer.
IfthepathtothePythia8maindirectoryisnotgivenintheoutputcommand,MadGraph5willlookforitinthedefaultlocation(.
/pythia8).
Thedefaultlocationcanbemodiedbyeditingthele.
/input/mg5configuration.
txt.
–31–C.
4W+jjproductionInthissimpleexampleweshowhowtocreateepslescontainingtheFeynmandiagramsforasetofprocesses:generatepp>W+jjdisplaydiagrams.
/The'.
eps'leswillbewrittenintheoutputdirectory.
Notethatunlessacouplingorderisspecied,MadGraph5willminimizethenumberofallowedQEDorders,asexplainedinsectionC.
1above;hereQED=1.
Ifyouwantthefullexpansioninαem,youwillneedtospecifywheretostoptheexpansion,i.
e.
,generatepp>W+jjQED=3displaydiagrams.
/C.
5Graviton-jetproductionInthisexample,weshowhowtoevaluatethesquaredmatrixelementinC++foramoreexoticmodel.
WewillusethespeciccaseofthegravitonproductionwithoneadditionaljetintheRandall-Sundrummodel[78,79].
Thelistofcommandisthefollowing:importmodelRSgeneratepp>yjoutputstandalone_cpplaunchInthemodelimplementation,thenameforthegravitonisy.
Inordertoknowthenameofallparticlespresentinthemodel,youcanusethecommanddisplayparticlesandinordertogetmoreinformationaboutaspecicparticleyoucanenterdisplayparticlesy.
InthisexamplestheyarethreedierentcouplingorderlabelsQED/QCD/QTD;thelatterislinkedtothegravitonsectorofthetheory.
Sincetheyarethreecouplingorders,MadGraph5isnotabletoguessasuitablehierarchybetweenthoseorderandsetbydefaultallorderstotheirmaximalpossiblevalue.
C.
6GluinodecayInthisexampleweevaluatethepartialdecaywidthforgluinodecayintouuχ01throughaleft-handedsquark:generatego>ul>u~un1outputmadeventlaunchTheparticle(s)betweenthetwo>arerequestedtobepresentinalldiagramsass-channelpropagators.
Notethatthisconditiondoesnotimplythattheparticleisstrictlyon-shell.
AlsonotethattheresultofthedecaywidthcalculationisgiveninGeV.
–32–C.
7Top-pairproductionwithoneleptonicdecayInMadGraph5,itispossibletospecifythedecayofa(nearly)on-shellparticle(seesection2.
4).
Thesyntaxfollowsthelogicofrststatingthecoreprocessandthenindicatingthedecay(s).
Ifsub-decaysarerequested,theyshouldbeenclosedbetweenparenthesis:generatepp>tt~QED=0,\(t>W+b,W+>jj),\(t~>w-b\,,W->l-vl~)outputmadeventlaunchThefullprocesscanbewrittenononeline,orusingthelinecontinuationsymbol\todividelines.
DThetestsuiteMadGraph5featuresatestsuitethatallowstotesttheinstallationaswellasanyfurtherdevelopmentofthecode.
Duringcodedevelopment,thistestsuiteisextremelyimportantinordertoavoidbugs,ensurethestabilityofthepackageandallowarobustmulti-developerapproach.
Inthisrespect,itisconsideredmandatorythattheimplementationofanynewfunctionalityisaccompaniedbyrelatedtests.
Generaladviceisthatthetestsuiteshouldbeasimportantanddevelopedasthecodeitselfandshouldbesplitintothreedierentlevels:Unittests:Eachpartofthecode(classorfunction)istestedbyaseriesofdedicatedtests.
Thepurposeistofullycheckthebehaviouroftheroutine/class,i.
e.
,checktheoutputinsomespeciccase,checkthebehaviorofthecodeincaseofwronginput,etc.
CurrentlyMadGraph5includesmorethan400independentunittests.
Acceptancetests:Thispartsimulatestheinstructionsenteredbytheuserandchecksthatallmodulesarecorrectlyinterfaced.
Thiscorrespondstocheckthataseriesofexamplesarecorrectlyrunningandprovidetheexpectedresults.
Paralleltests:ThesetestscheckthattheoutputofMadGraph5arethesameofthoseobtainedbyMadGraph4.
Thesechecksaremadefordierentmodels(SM/MSSM/HEFT)andusingbothUFOandversion4models.
Inordertoecientlyrunthesetests,wehaveimplementascriptwhichautomaticallydetectsallthetestsavailable.
Thetestscanbelteredbynameorle,inordertorunonlyasubsetoftests.
AsimilartestmoduleisnowpresentbydefaultinPython2.
7.
However,sinceMadGraph5isbydesignalsocompatiblewithPython2.
6,adedicatedtestcodeisincludedinthedistribution.
Inordertorunthedierenttestsuites,theusercantype(respectivelyforunittest/acceptancetests/paralleltests):–33–.
/tests/tests_manager.
py.
/tests/tests_manager.
py-pA.
/tests/tests_manager.
py-pPOpenAccess.
ThisarticleisdistributedunderthetermsoftheCreativeCommonsAttributionNoncommercialLicensewhichpermitsanynoncommercialuse,distribution,andreproductioninanymedium,providedtheoriginalauthor(s)andsourcearecredited.
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