Next-to-leading order corrections to Wt production and decay

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

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aCERN-PH-TH/2005-110,DSF-18/2005,hep-ph/0506289Next-to-leadingordercorrectionstoWtproductionanddecayJohnCampbell DepartmentofPhysics,THDivison,CERN,CH-1211Geneva23,SwitzerlandFrancescoTramontano DipartimentodiScienzeFisiche,Universit`adiNapoli,“FedericoII”eINFNsezionediNapoli,ComplessodiMonteS.Angelo,Napoli,Italy(Dated:February2,2008)AbstractWepresenttheresultsofanext-to-leadingordercalculationofWtproduction,includingthedecaysofboththetopquarkandtheWboson.Thee ectsofradiationinthedecayofthetopquarkarealsoincluded.Theseparationofdiagramswhichappearintherealcorrections,intosingly-anddoubly-resonantcontributions,isperformedusingab-jetvetowhichismotivatedbytheuseofthebottomquarkdistributionfunction.We ndthat,forachoiceofscalewhichissuitableforthisapproach,theQCDcorrectionsareverymildandonlychangethecrosssectionbyupto10%attheLHC,dependingontheseverityoftheb-jetveto.Whenfurthercutsareapplied,applicableforaHiggsbosonsearchintheH→WW channel,we ndthattheradiativee ectsgreatlydecreasethenumberofbackgroundeventsexpectedfromthisprocess.Inaddition,theshapesofrelevantdistributionscanbesigni cantlychangedatnext-to-leadingorder.

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

I.INTRODUCTION

AttheLHC,thetopquarkwillbeproducedcopiouslyinmanychannels.Aswellasthe¯pairproductionchannel,thetopquarkmaybeproducedsinglyinassociationwithothertt

particles.Theratesfortheseprocesseswillbesu cienttobothstudythepropertiesofthetopquarkindetailandtoprovideasigni cantsourceofbackgroundeventsforotheranalyses[1].

Inthispaperwewilldiscussthecalculationofthenext-to-leadingorder(NLO)correctionstotheproductionofasingletopquarkinassociationwithaWboson.ThiscalculationhasbeenincludedinthegeneralpurposeNLOprogramMCFM[2,3,4].Thelowestorderprocesswhichweconsideris,

b+g→W +t

||→ν+e++b||→e +ν¯(1)

sothattheleptonicdecaysoftheW andofthetopquarkareincluded.WenotethatbothattheTevatronandtheLHC,therateforthecharge-conjugateprocessinvolvingaW+

¯quarkisidentical[38].Thisprocesshaspreviouslybeenconsideredextensivelyatandat

leadingorder[5,6,7].However,itisonlyatnext-to-leadingorderthatweobtainaccuratepredictionsofeventrateswhicharesensitivetothestructureofjetsinthe nalstate.SuchNLOcalculationshavesofarbeenavailableonlyforthecasewherethedecaysoftheheavyquarkandWbosonarenotincluded[8,9].

WehaveextendedthesepredictionstoincludenotonlythefullspincorrelationsinthedecaysoftheWbosonandthetopquark,butalsotoincludethee ectsofgluonradiationinthetopquarkdecay.Thisisachievedusingthesamemethodthathaspreviouslybeenappliedtoothersingletopproductionchannels[10]andwhichisbrie ydescribed,togetherwithotherdetailsofthecalculation,inSectionII.

Atnext-to-leadingordersomeofthecontributionsrepresentingtheemissionofanaddi-tionalpartonrequirespecialattention.One ndsthatthecorrectionsinvolvingtwogluonsintheinitialstatecontaindiagramsthatwouldnormallybeassignedtothelowestorder

¯productionprocess.Wediscussourtreatmentofthiscalculationofthedoublyresonanttt

complicationinSectionIII.

TheresultsofourcalculationarepresentedinSectionIV.WediscusstheNLOcorrections

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

attheTevatronandtheLHC,comparingour ndingswiththoseobtainedpreviouslyintheliterature.Wealsoprovideupdatedpredictionsobtainedusingthelatestexperimentalinputsandexaminethee ectofincludinggluonradiationinthedecayofthetopquark.SectionVcontainsastudyofourresultsinthecontextofthesearchforanintermediatemassHiggsbosonattheLHC.InthischanneltheHiggsdecaysviaWW ,withthe nalstatecontainingleptonsandmissingtransversemomentum.SincetheHiggsmasscannotthenbereconstructed,theoreticalinputasaccurateaspossibleisimperative.Tothatend,inthissectionweapplyrealisticacceptanceandsearchcutstoallthe nalstateparticles,thencomparethee ectoftheNLOcorrectionswiththemoreinclusiveresultsalreadypresented.II.CALCULATIONALDETAILS

ToevaluatethematrixelementsfortheproductionanddecayofaW-topsystem,wefollowthesamestrategyasinapreviouscalculationofsingletopproduction[10],whichisbasedontwoapproximations.The rstisthatthetopquarkisproducedanddecaysexactlyon-shell,motivatedbythefactthatdiagramswithoutanonshelltopquarkaresuppressedbyafactorofΓt/mt≈1%.Thisenablesadivisionoftheprocessintoproductionanddecaystages,withthefurtherapproximationthattheinterferencebetweenradiationinthetwostagesisneglected.OngeneralgroundsthecontributionoftheseinterferencetermscanbeshowntobeoftheorderofαsΓt/mtduetothelargedi erencebetweenthecharacteristic

1 1timescalesoftheproduction(m t)anddecay(Γt)stages.Moretechnicaldetailsand

furtherreferencescanbefoundinRef.[10].

ThetreelevelamplitudeisrepresentedinFig.1,wheretheon-shelltoppropagatorisdenotedbytwoshortlines.Duetotheweakverticestheonlytwonon-vanishinghelicityamplitudescorrespondtothetwopolarizationsofthegluon,withallmasslessfermionsleft-handed.Inourexpressionsfortheamplitudeswetakeallmomentaoutgoing,restoringincomingmomentainourimplementationbyperformingtheproperanalyticalcontinuationofthemasslessspinorproducts.Wewritetheamplitudesintermsofthemomentaofthedecayproducts,withlabelsasfollows,

bi+g→W +t

||→νf+ef+bf||→ei+νi

(2)

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.1:ThetwotreeleveldiagramsfortheWtprocess.

andalsousettorepresentthemomentumofthetopquark,sothatt=νf+ef+bf.Bothbottomquarksaretreatedasmasslessparticlesinourapproach.Thetwotree-levelamplitudesarethengivenby,

A =f

[gνi] bi|t|ef ei|t|g m2 be [ge]+2g·t e|t|e iifift gbi

√

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.2:Oneloopdiagramsrepresentingvirtualcorrectionsintheproductionstage.

Thecancellationofthesoftandcollineardivergencesbetweenrealandvirtualcontri-butionshasbeenimplementedthroughthesubtractionmethod[11].Inparticular,forthecontributionfromrealradiationintheproductionstagewehaveadoptedanextensionofthedipolesubtractionscheme[12,13]whichhandlesthecaseofmassivequarksinthe nalstate[14].Wehaveusedageneralizationofthismethod,whereonecanuseatuneableparameterinordertohavebettercontroloverthesizeofthesubtraction.FurtherdetailsandformulaearecontainedinRef.[10]andAppendixA.Forthecaseofrealradiationinthedecaywehaveusedafurtherextensionofthismethod,asinRef.[10],whichensures

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.3:Oneloopdiagramscontributingtothecalculationofvirtualradiationinthedecaystage.

FIG.4:Diagramsrepresentingtheemissionofrealradiationintheproductionstage.Inthecalculation,theadditionalgluonmustalsobecrossedintotheinitialstateanddiagramscontainingtwoquarklines(notshown)arealsoincluded.

thatthetopquarkandtheWbosonremainon-shellwhenthesubtractionisperformed.

Thevirtualcorrectionsinthedecaystageamounttothestudyofthedecayofanon-shelltopquark.Theseamplitudeshavebeenknownforalongtime[15,16]andwedonotreportthemhere.Toevaluatethecontributionofthevirtualradiationintheproductionstage(Fig.2)westartfromtheamplitudeswherethetopquarkisproducedon-shellwithoutdecayingandcalculateamplitudesforthetwopolarizationstatesofthetopquark.This

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.5:Diagramscontributingtorealradiationinthedecayofthetopquark.

isachievedbywritingthespinorsintermsofanauxiliarymasslessfour-vectort0inthefollowingway:

u(t)↑=

u(t)↓=(/t+mt)(/t+mt) gt0

[gt0],.(8)

Thevectort0isconstructedbyformingalinearcombinationoftandg,

tµ0=t µm2t

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

bythethreegluonvertex.WedealwithinfraredandultravioletsingularitiesbyusingdimensionalregularizationinthefourdimensionalhelicityschemeandusethemethodofRef.[17]towritetheamplitudesintermsoftraces.Theappearanceofγ5intheweakvertexisthenhandledbycyclicallyrotatingthesetracessothatγ5appearsatthebeginningofeachtrace,ingthisprescriptionwehavecheckedthattheWardidentityfortheweakcurrentissatis edexactlyandnoadditionalcountertermsarerequired.

Finally,-ingPassarino-Veltmann-dimensionaldecomposition,weobtainaresultintermsofscalarintegrals.Duetothenatureofourapproach,otherspuriousdivergencesarestillpresentatthisstage.Individualtermsintheresultappeartobedivergentasfactorsinthede-nominatorapproachzero.However,inthislimit,acombinationofsuchapparently-singulartermsis nite.Bycollectingallsuchtermsoveracommondenominator,onecanidentifynewfunctionsthatarewell-behavedintheselimits.Thesearecombinationsofrationalandlogarithmicfunctions,asinRef.[18].Followingthisprocedure,weareabletore neour rstresultconsiderably.Toshowthelevelofsimpli cationthatwehavereached,oneoftheamplitudesisreportedinAppendixB.Theothersareslightlylargerinsizeandwedonotreproducethemhere.Theywillbeavailable,togetherwiththerestofourcalculation,aspartofthenextreleaseoftheMonteCarloprogramMCFM.

III.¯DIAGRAMSSEPARATIONOFWtANDtt

Whencalculatingtherealradiationcorrections,allappropriatecrossingsofthediagramsshowninFig.4shouldbeincluded.Someofthecrossings,inwhichtheadditionalpartonisa¯bquarkinthe nalstate,areparticularlyproblematic.ThesediagramsareshowninFig.6andinvolvegluon-gluonandsame avourquark-antiquarkinitialstates,whichareimportantattheLHCandTevatronrespectively.Allthesediagramsproducea nalstateconsistingofaW ,anon-shelltopquarkanda¯bquark.However,thediagramsin

¯propagatorandrepresenttheproductionofatt¯pairwiththepanel(b)containaresonantt

¯intotheW and¯subsequentdecayofthetbquark.Assuch,thecontributionfromthese

diagramswhenintegratedoverthetotalavailablephasespacecanbemuchlargerthanthelowestorderWtcrosssection(anorderofmagnitudeattheLHC).Inordertodisentangle

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.6:RealcorrectionstoW tproductionwhichinvolveanadditional¯bquark.Thedoublebarsindicatetheon-shelltopquarkwhichsubsequentlydecaysintoW+b.Diagramsobtainedbyinterchangingtwogluonsarenotshown.The3representativediagramsintherightpanel(b)

¯propagator,whilethoseontheleft(a)donot.containaresonantt

thesetwoprocesses,twomethodshavebeenoutlinedintheliterature.

The rstinvolvesmakingacutontheinvariantmassoftheW ¯bsystemtoprevent¯propagatorfrombecomingresonant[5].Thesecondmethodinsteadsubtractsthethet

contributionfromtheresonantdiagramssothatnoon-shellpieceremains[6].Acomparisonofthesetwoapproaches[7]showsthatthemethodsyieldthesametotalcrosssectionwhenmethodsdonotlendthemselvestoaMonteCarloimplementationwhereonewishestostudydistributionsof nalstateparticlesaswellastotalcrosssections.Thereforeweshallamasswindowof15Γt≈25GeVeithersideofthetopmassischosen.However,these

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

adoptneitheroftheseprescriptionsbutinsteadfollowaproceduremotivatedbyouruseofthebottomquarkPDF.

Intheb-PDFapproach,thebquarkdistributionfunctionisderivedperturbativelyfromacollinearg→b¯bsplittingthatoccursintheinitialstate.Itimplicitlyincludesallsplit-tingsuptoapTofthe¯b-quarkequaltothefactorizationscale,µF.Thismeansthatthecontributionfromthecorrespondinggg→Wtbdiagrams(containedinpanel(a)ofFig.6)hasalreadybeenincludedinthelowestordercalculation.Thereforethenetcontributionfromthesediagrams,includingappropriatecounter-termsandintegratingoverall¯bquarktransversemomentauptoµF,shouldbeapproximatelyzero.ForasuitablechoiceofµFwehavecheckedthatthisisindeedthecase.ThechoiceofµFismadesuchthatthecollinearapproximationusedinderivingtheb-PDFisaccurate,whichforthisprocessimpliesthatµF (mW+mt)/4≈65GeV[39].

Thereisalsoacontributionfromthediagramsinpanel(b)ofFig.6whenthepTofthe

¯¯bquarkissmall,pbT<µF.However,thesediagramssimplyrepresentthedoubly-resonant

¯productionprocesswhichiscurrentlyknownuptoNLO[40].Thereforeitispreferablett

toseparatethiscontributionfromthe‘genuine’NLOcorrectionstotheWtprocess.As

¯diagramsinthisregionofphaseweshallshowlater,althoughthecontributionfromthett

spaceisrathersmallinrelationtothetotalcrosssection,itisstillcompetitivewiththeWtresult.Thissuppressionmeansthattheinterferencee ectsbetweenthetwosetsofdiagramsisverysmallwhenusingab-jetveto,incontrasttothecasewhenpbTisunconstrained[7].Whena¯bquarkisobservedwithapTaboveµFthenourdescriptionofthe nalstateisalowestorderone.Thecontributionfromthedoubly-resonantdiagramsdominatesand,as

¯process.Alternatively,onecouldabove,abetterpredictionwouldbeobtainedbyusingthett

useacalculationincludingallthediagramsforgg→tWb,includingthet→bW→b νdecay(retainingthebquarkmass)andalso nitewidthe ects[21].However,currentlythisstudywouldbelimitedtoleadingorderinαsonly.

Tosummarize,weshallperformourcalculationoftheWtprocessbyapplyingavetoonthepToftheadditionalbquarkthatappearsatnext-to-leadingorder.Thisaidsthe

¯production.Whenapplyingthisveto,separationofthisprocessfromdoubly-resonanttt

oneshouldchoosethefactorizationscaleequalto(oratleastofthesameorderas)the

bmaximumpTofthebquarkthatisallowed,pT¯veto.Thischoicerespectstheapproximations

¯vetobthatwereoriginallyusedtode nethebquarkPDF.ForµF=pTandforlessinclusive

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

quantities,thecontributionfromthegg→Wtbdiagramsiscalculatedbysimplyomittingthedoubly-resonantdiagrams[41].Theresultforthispieceremainsatthelevelofafewpercentofthelowestordercrosssection.

IV.RESULTS

Beforediscussingthee ectofincludingradiationinthedecayofthetopquark,wewill rstconsiderjusttheWttotalcrosssectioninordertodiscusssomefeaturesofourapproachandtocompareourresultswiththoseavailableintheliterature.

parisonwithnotopquarkdecay

TheNLOcorrectionstothetotalWtcrosssection,wherenodecayofthetopquarkisincluded,werepreviouslypresentedinRef.[9].Forthesakeofcomparison,inthissectionwewilladopttheparametersthereinascloselyaspossible.Inparticular,wechoosemt=175GeVandtheCTEQ5setofpartondistributionfunctions[42].Theotherelectroweakparametersthatenterourcalculationarechosentobe,

MW=80.419GeV,2gW=0.4267.(11)

WeperformourcomparisonattheLHCandexaminethedependenceofourresultsonthecommonrenormalizationandfactorizationscaleµ.Aswehavealreadyargued,theb-PDFapproachismostwell-motivatedwhenchoosingavalueofµlessthanabout65GeV.However,suchavalueismuchsmallerthanthemoretypicalchoice,µ=mt+mWwhichisthecentralvaluechoseninRef.[9].Therefore,forthesakeofillustration,wechoosetostudythescaledependenceoveralargerangefromµ=25GeVuptoµ=mt+mW=255GeV.Aswehavediscussedabove,welimitthepTofthebquarkthatappearsatnext-to-leading

bordertohaveamaximumvaluepT¯veto,whichwechooseheretobe50GeV.

OurresultsatLOandNLOareshowninFig.7.Weseethatthedependenceofthelowestordercurveonacommonscalechoiceisalreadyremarkablysmall.Atnext-to-leadingorderweseethatthisisimprovedstillfurther,withthecrosssectionvaryingbyabout3%overtherangeofscalesshowninthe gure.

ComparingwithZhu[9],we ndanumberofdi erences.Evenatlowestorderourresultforµ=mt+mWishigherandfurthermore,thedependenceoftheresultonthisscale

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.7:ScaledependenceofthecrosssectionsforW tproductionattheLHCformt=175GeV.ThebranchingratiosforthedecayofthetopquarkandtheW’sarenotincluded.CrosssectionsareevaluatedwithCTEQ5L1(αs(MZ)=0.127)andCTEQ5M1(αs(MZ)=0.118)PDFs[22].Wechooseasinglecommonrenormalizationandfactorizationscale,µ.Thelowestordercrosssection

bisthedashedcurve,whilsttheNLOone–calculatedwithpT¯veto=50GeV–issolid.

appearslessmild(c.f.Fig.2ofRef.[9]).However,wehavecheckedthelowestorderresultsofourprogramagainstthoseobtainedwithMadEvent[23]andfoundgoodagreement.Atnext-to-leadingorderwealso ndaslightlydi erentresult,lowerandwithastrongerdependenceonthescale.Inthiscase,weexpectsomediscrepancyduetoourdi erentmethodofhandlingthegg→Wtbcontribution.Wenotethatthecombinationofthe¯b-jetvetoandourpreferredchoiceofamuchsmallerscale,leadstoanext-to-leadingordercrosssectionthatisabout15%smallerthanthatfoundbyZhu.Inaddition,theK-factor,de nedastheratioofNLOandLOcrosssections,ismuchsmallerandintherange1.2–1.3dependingonthescalechoice.

bFinally,weconsiderthedependenceofourresultonthechoiceforpT¯veto.InFig.8

weshowthescaledependenceforthreedi erentchoicesofthevetothreshold.Wehaveconcentratedontheregionofsmallerscalesandvariedthecommonscalesbyafactoroftwo

baboutthecentralvalueµ0=pT¯veto.Oneseesthat,withinthiswindow,thescaledependence

ofthenext-to-leadingordercalculationisagainsomewhatsmallerthanthatfoundatLO

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.8:ScaledependenceofthecrosssectionsforW tproductionattheLHC,forthreedi erent

bchoicesofpT¯veto=µ0.Frombottomtotop,thesolidcurvesrepresentvaluesof30,40and50GeV.

Thelowestorderdashedcurve(calculatedwithµ0=50GeV)isalsoshownforreference.

paredtoathresholdof50GeV,thecrosssectiondecreasesbyabout15%whenloweringthevetoto30GeV.Thissubstantiallyreduesthee ectofthenext-to-leadingordercorrectionsonthecrosssection,leavingaK-factorclosetounityforourcentralscalechoice.

B.Updatedresultsandradiationinthedecay

First,werepeatthecalculationofthetotalWtcrosssection,butusingthemostrecentdeterminationofthetopquarkmass[24],whichyieldsmt=178GeV.WealsouserecentPDFsetsfromtheMRSTandCTEQgroups.

OurpredictionsfortheTevatronandtheLHCareshowninTableI,wherewehaveusedbpT¯veto=50GeVandfactorizationandrenormalizationscalesalsoequaltothisvalue.ItisclearfromthequotedcrosssectionsthatthisprocessisoflittlephenomenologicalrelevanceattheFermilabcollider,althoughweincludetheresulthereforcompleteness.TheNLOcorrectionsattheTevatronincreasethecrosssectionbyafactorof1.35whenusingtheCTEQPDFset,butdecreaseitbyalittleunder10%fortheMRSTparametrization.The

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

TABLEI:LOandNLOcrosssectionsforsingletop-quarkproductioninassociationwithaW attheTevatronandLHC,formt=178GeV.ThebranchingratiosforthedecaysofthetopquarkandtheW arenotincluded.Crosssections

are

evaluated

withCTEQ6L1(αs(MZ)=0.130)andCTEQ6M(αs(MZ)=0.118)PDFs[25]andalsowiththeMRST2002NLOPDFset(αs(MZ)=0.1197)[26].Therenormalizationandfactorizationscalesaresettothe¯bjetvetothresholdof50GeV.TheerrorsrepresentMonteCarlostatisticsonly.

Collider,√

CTEQ6

MRST20020.047960.080830.06458±0.00010.07414±0.0001pp¯,1.96TeVpp¯,1.96TeV

ΓNLOt

1(σBt→bνe σ0Bt→bνe). (13)

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

TABLEII:ComparisonofLOandNLOcrosssectionsforW tproductionattheTevatronandLHC,withleptonicdecaysofboththeW andthetopquark.TheNLOcalculationisper-formedbothwithoutincludingQCDe ectsinthedecay(σBt→bνe)andalsowhenitisincluded(σBt→bνe+X).Thetopquarkmassismt=178GeVandcrosssectionsareevaluatedusingtheCTEQ6MPDFsetwithallscalesequalto50GeV.TheerrorsrepresentMonteCarlostatisticsonly.NotethatthevaluesofΓtatLOandNLOare1.651GeVand1.480GeVrespectivelyandthebranchingratiooftheWintoleptonsisBr(W→eν)=0.1105.

Collider,√

0.8564±0.0006

356.9±0.20.7887±0.0005391.7±0.30.7806±0.0005395.7±0.3pp¯,1.96TeVpp,14TeV

Theresultsshowninthetableagreewiththisexpectation.Since,inourapproach,thee ectoftheNLOcorrectionsintheproductionisfairlysmall,thenumericaldi erenceisonly1%atbothcolliders.

WeconcludethissectionwithamoredetailedpresentationofthescaledependenceofourcalculationwhenusingtheupdatedparametersandPDFset.Anticipatingthestudyofthefollowingsection,wealsochoosealowervalueforthe¯bjetvetointhenext-to-leadingorder

bcalculation,pT¯veto=30GeV.ThisresultsinthescaledependenceshowninFig.9,where

thecrosssectionsateachchoiceofscaleareexpressedasaratiowiththecentralresultat

bµ0=pT¯vetoandwevarythescalesbyafactoroftwoaboutµ0.Wealsoshowthecurvesobtainedwhenvaryingtherenormalizationandfactorizationscalesseparately.Oneseesthattherelativelysmallscaledependenceatlowestorderistheresultofalargecancellationbetweenthedependenceonthefactorizationandrenormalizationscalesindividually.Incontrast,thedependenceonthesescalesatnext-to-leadingorder,eitherontheirownorwhenvariedtogether,issmall–lessthan10%overthisrange.Wealsonotethat,forthischoiceofparametersandveto,thenext-to-leadingordercorrectionsdonotalterthetreelevelcrosssection,whichremainsat346fb.

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.9:ScaledependenceofthecrosssectionsforW (→e ν¯)t(→νe+b)productionattheLHC,atLO(left)andNLO(right).TheNLOcalculationincludesthee ectofradiationinthedecayofthe

btopquark.Thescaleµisexpressedastheratiowiththecentralscaleµ0=pT¯veto=30GeVand

thecrosssectionsarescaledtothecentralresultatµ=µ0.Thesolidlinesrepresentthevariationofbothrenormalizationandfactorizationscalestogether(µR=µF=µ),thedashedonestheresultwhenonlyµFisvaried(µR=µ0)andthedot-dashedcurvesrepresentthedependenceontherenormalizationscalealone(µF=µ0).

V.THEWtBACKGROUNDTOH→WW

Asanexampleoftheutilityofourcalculation,inthissectionweconsiderthee ectofNLOcorrectionsinthecontextofasearchfortheHiggsbosonattheLHC.Intheintermediatemassrange,155<mH<180GeV,oneofthesearchstrategiesinvolvesHiggsproductionviagluonfusion,withthesubsequentdecayoftheHiggsbosonintoo -shellWpairswhichthendecayleptonically[27],

g+g→H→W +W+

||→ν+e+||→e +ν¯(14)

ThelargestbackgroundinthischannelisfromthecontinuumproductionofWpairs,bothfromdibosonproductionviaquark-antiquarkscatteringandfromloop-inducedgluon-gluonfusion[28,29].Afurthersigni cantsourceofbackgroundeventscomesfromprocessesproducingtop-quarksthatdecayleptonically.SincethepresenceofneutrinosinthesignalpreventsafullreconstructionoftheHiggsmasspeak,anaccuratepredictionofallthebackgroundsisnecessary.

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

Inpreviousstudies[30,31]twotopbackgroundshavebeenconsideredusingPythia[32].

¯productionandWtproduction,theprocessthatweconsiderhere.Theseareresonanttt

Aswehavediscussedpreviously,thesetwoprocessesbecomeentangledatNLO.WewillseparatethemaccordingtotheprocedurethatweoutlinedinSectionIII.Therefore,inadditiontoourNLOcalculationoftheWtprocess,wewillalsoconsiderthecontribution

¯production,withtheappropriatetopquarkdecays.fromresonanttt

Sincethesignalprocesscontainsnojetsatlowestorder,itise cienttoimposeavetoonalljetactivitytoreducethesizeofthesebackgrounds.Ourapplicationofacutonthetransversemomentumofthe¯bjetintheWtprocess tsnaturallyintothisprocedure.Wesimplyextendourvetotodisallowallcontributionswithanyjetobservedabovethevetothreshold.Wenotethatthereisaslightmismatchduetothefactthatourtheoretically-motivatedvetoappliesatallrapidityvalueswhilsttheexperimentalapproachonlyvetoesajetuptoafewunitsofrapidity.However,wedonotexpectthistogreatlya ectourresults.Forourparton-levelstudyweadoptaminimalsetofcutsandexaminethee ectoftheNLOcorrectionsonaselectionofobservablesthataretypicallyusedinmoredetailedexperimentalstudies[27,29,30,31].Ourbasiccutsrepresentthe niteacceptancelimitsofthedetectorsattheLHC,

pT(lepton)>20GeV,|η(lepton)|<2.5,(15)

appliedtobothoftheleptonsproducedintheWdecays,withthemissingtransversemomentumalsoconstrainedby,

pT(missing)>30GeV.(16)

The nalcutthatweapplyisthejetveto,afterpotentialjetshavebeenclusteredaccordingtothekTalgorithmwithajetseparationparameter R=1.0.Eventsarenotincludedifanyjetisobservedwith,

pT(jet)>30GeV,|η(jet)|<3.(17)

Wenotethat,inadditiontoexcludingadditionalradiationatnext-to-leadingorder,thisvetoalsoappliestothebjetthatisproducedinthetopquarkdecay.Thisresultsinasubstantialdecreaseincrosssectioncomparedtothetotallyinclusivecase.

Toexploitthespincorrelationbetweentheleptonsinthesignalevents[33],onecanmakequitestringentcutsontheopeninganglebetweentheleptonsinthetransverseplane, φ .

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.10:Thedistributionoftheopeninganglebetweentheleptonsinthetransverseplane,forthesignalandthetwobackgroundprocessesconsideredhere.Allcurvesarelowestorderpredictionsandarenormalizedtounity.ThesignalcalculationusesaHiggsmassof155GeVand

¯(dashed).thebackgroundsareW t(solid)andtt

ThisisillustratedinFig.10,whereweshowtheshapesofthelowestorderpredictionsfor

¯.SignaleventspredominantlytheHiggssignalandthetwotopbackgrounds,Wtandtt

containleptonswithasmallopeninganglebetweenthem,whereasbothbackgroundstendtoproduceleptonsthatarealmostback-to-backinthetransverseplane.

WenowexaminetheextenttowhichthisischangedinourNLOcalculationoftheWtbackground,withourresultsshowninFig.11.Thee ectoftheNLOcorrectionsistochangetheshapeofthedistributionconsiderablywhenthesecutsareapplied.Thepeakatlarge φ isshiftedtoasmallervalueandbecomesmuchlesspronounced.Thiscouldhavequitealargeimpactonastrategyinwhichthisbackgroundismeasuredusingeventsatlarge φ andthenextrapolatedviathetheoreticalshapetotheHiggssignalregion.Onealsoseesthattheshapeischangedagainwhenincludingthee ectsofradiationinthetopquarkdecay,withthepeakbeingsharpenedoncemore,althoughthee ectisfairlyminor.OnecanalsoimagineconstructingthetransversemassoftheputativeHiggsboson(mT)fromthetransversemomentaofthedileptonsystemandthemissingpT,

mT=

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.11:ThedistributionoftheopeninganglebetweentheleptonsinthetransverseplaneforW teventsattheLHC,calculatedatLO(dashed)andNLO(solid)fortheLHC.TheNLOcurvesarelabelledaccordingtowhetherornottheyincludethee ectofradiationinthedecay.Alltheratesarenormalizedtounity.

where φistheanglebetweenthetwovectorsinthetransverseplane.Cuttinginasuitablemasswindowcanfurtherhelptoreducethebackgroundsforonlyasmalllossinsignal.Theimpactofournext-to-leadingordercalculationonthisdistributionisillustratedinFig.12.Inthis gurewehaveonlyshowntheresultwhenincludingradiationinthedecaybutnotethatthisdistributionchangeslittlewhenitisexcluded.OnecanseethattheshapeofthisdistributionisrelativelyunchangedatNLO,althoughmoreeventsareproducedathighvaluesofMT,beyondthepeakofthedistribution,thanatLO.

Finally,togivesomeideaofthee ectoftheNLOcorrectionsonthenumberofeventsthatshouldbeobservedinthischannel,inTableIIIweshowthecrosssectionsthatwe nd

¯processes.TheW’sdecayintoelectronsonlyandfortheWtprocess,bothfortheWtandtt

¯areincluded.ResultsareshownatLO,NLOandatNLOwhenincludingW tandW+t

¯process,thelowestorderdiagramsofFig.6(b)radiationinthetopquarkdecay.Forthett

arecalculated.AsanapproximationtotheNLOresult,aK-factorisappliedfromtheNLOcalculationinvolvingnotopquarkdecay.Inordertomatchthestudymoreclosely,wehaveusedaK-factorobtainedwhenapplyingthejetvetoofEq.(17).Usingthecommonscales

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

FIG.12:ThedistributionofthetransversemassforW teventsattheLHC,calculatedatLO(dashed)andNLO(solid)fortheLHC.TheNLOresultincludesthee ectofradiationinthedecay.Theratesarenormalizedtounity.

µR=µF=mt,we ndthatthisfactorisK=0.7.

ThecrosssectionsshowninTableIIIareevaluatedusingthreedi erentsetsofcuts.The rstcolumnusesjustthebasicsetofcuts(Eqs.(15)–(17)),thentheothertwocolumnsrepresentextensionsofthesecutsthatmightbeusedinthesearchforaHiggsbosonofgivenmass.Boththefurthersetsalsoimpose,

φ <π/4,

m <35GeV,(19)

toselecttheHiggssignalregion.InadditionwehaveusedacutonthereconstructedtransversemassaroundtheJacobianpeakoftheputativeHiggsmass.Inthe rstcase,thecutisconstructedforaHiggsmassof155GeV,byconstraining125<mT<155GeV.Thesecondsetrequiresthat140<mT<180GeVandisaimedatasearchfora180GeVHiggs.Oneseesthatthee ectoftheNLOcorrectionsinthisregionofphasespaceissigni cantlydi erentfromtheinclusivecase.Thecrosssectionisdecreasedsubstantially,withaK-factor(forthecalculationincludingradiationinthedecay)ofapproximately0.6whenapplyingtheHiggssearchcuts.Wealsoseethatthecontributionsfromthetwotopquarkprocesses

We present the results of a next-to-leading order calculation of Wt production, including the decays of both the top quark and the W boson. The effects of radiation in the decay of the top quark are also included. The separation of diagrams which appear in

TABLEIII:ComparisonofLOandNLOcrosssectionsfortopquarkbackgroundsinaninter-

¯)andmediatemassHiggssearchattheLHC.Resultsareshowninfemtobarnsfor(W t+W+t

¯production,withleptonicdecaysofboththeWandthetopquark.Threesetsofcutsarett

considered,whicharedescribedindetailinthetext.TheNLOWtcalculationisperformedbothwithoutincludingQCDe ectsinthedecay(σBt→bνe)andalsowhenitisincluded(σBt→bνe+X).

basicmH=155GeVmH=180GeV

¯processtt

σLO×K[fb]30.520.420.43arecomparablewhencalculatedinthisway.

VI.CONCLUSIONS

AttheLHC,thetopquarkwillbereadilyproducedinassociationwithaWboson.Inthispaperwehaveperformedanext-to-leadingordercalculationofthisprocess,includingboththesubsequentleptonicdecaysW →e ν¯andt→νe+b,aswellastheemissionofrealradiationinthetopdecay.

Fortotalinclusivecrosssections,wherethetopquarkandW bosondonotdecay,comparisonwithpreviouscalculationsispossibleandwe ndresultswhicharebroadly

¯diagramsatNLO–whichsimilar.However,duetothepresenceofdoubly-resonanttt

arehandleddi erentlyinourcalculation–wedonot ndexactagreement.Ourmethodmaintainstheconsistencyoftheb-PDFapproachandrequiresthesimultaneoususeofarelativelylowfactorizationscaleµF～(mt+mW)/4andavetoon¯bquarkswithatransversemomentumlargerthanthisvalueofµF.

Withthisapproach,we ndthattheNLOcorrectionstotheinclusiverateattheTevatroncanbelarge,buttheprocessremainsphenomenologicallyirrelevantthere.AttheLHC

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