NLO QCD corrections to Z b anti-b production with massive bottom quarks at the Fermilab Tev

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

8

2

n

Ju

4

]h

p

-

ep

h

[

1

v

8

8

.

6

8

:0v

i

X

r

aFSU-HEP-2008-0531NLOQCDcorrectionstoZb¯bproductionwithmassivebottomquarksattheFermilabTevatronF.FebresCordero DepartmentofPhysicsandAstronomy,UCLA,LosAngeles,CA90095-1547,USAL.Reina PhysicsDepartment,FloridaStateUniversity,Tallahassee,FL32306-4350,USAD.Wackeroth DepartmentofPhysics,SUNYatBu alo,Bu alo,NY14260-1500,USA(Dated:June4,2008)AbstractWecalculatetheNext-to-LeadingOrder(NLO)QCDcorrectionstoZb¯bproductioninhadroniccollisionsincludingfullbottom-quarkmasse ects.Wepresentresultsforthetotalcrosssectionandtheinvariantmassdistributionofthebottom-quarkjetpairattheFermilabTevatronpp¯collider.Weperformadetailedcomparisonwithacalculationthatconsidersmasslessbottomquarks,asimplementedinMCFM.We ndthatneglectingbottom-quarkmasse ectsoverestimatesthetotalNLOQCDcrosssectionforZb¯bproductionattheTevatronbyabout7%,independentofthechoiceoftherenormalizationandfactorizationscales.Moreover,bottom-quarkmasse ectscanimpacttheshapeofthebottom-quarkpairinvariantmassdistribution,inparticularinthelowinvariantmassregion.

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

I.INTRODUCTION

Oneofthemaingoalsofhigh-energycolliderexperimentsistheelucidationofthemecha-nismofElectroweakSymmetryBreaking(EWSB)aswellastheexplorationofenergyscalesbeyondtheweakscale,wherephysicsbeyondtheStandardModel(BSM)isexpected.Thehadronicproductionofweakgaugebosonsinassociationwithabottom-quarkpairplaysacrucialroleinsomeofthecurrentstudiesofEWSBandBSMphysicsattheFermilabTevatronpp¯collider[1,2,3,4,5,6].Wb¯bandZb¯bproductionprocessesrepresentthemajorirreduciblebackgroundstothemainsearchmodesforalightSM-likeHiggsbosonattheTevatron,i.e.WHandZHwithH→b¯b.Wb¯balsoaccountsforoneofthemostimportantbackgroundstosingle-topproduction,pp¯→t¯b,¯tbwitht(t¯)→Wb(¯b),whichteststhefundamentalstructureoftheWtbvertexattheTevatron[7,8,9,10].Finally,Zb¯bisabackgroundtosearchesforHiggsbosonsinmodelswithenhancedbottom-quarkYukawacouplings,suchastheMinimalSupersymmetricStandardModel(MSSM)withlargetanβ,whereHb¯bwithH→µ+µ ,τ+τ isaninterestingdiscoverychannel[11].

ThehadroniccrosssectionsforW/ZHassociatedproductionhavebeencalculatedinclud-inguptoNext-to-Next-to-LeadingOrder(NNLO)QCDcorrections[12,13,14]andO(α)electroweakcorrections[15].Single-topproductionhasbeencalculatedatNext-to-Leading(NLO)inQCD[16,17,18,19,20,21,22,23,24],andincludingone-loopelectroweak(SMandMSSM)corrections[25],whilethecrosssectionforHb¯bassociatedproductionisknownincludingNLOQCDcorrectionsandfullbottom-quarkmasse ects[26,27,28,29].

TofullyexploittheTevatron’spotentialtodetecttheSMHiggsbosonortoimposelimitsonitsmass,itiscrucialthatthedominantbackgroundprocessesarealsoundergoodtheoreticalcontrol.Inthepresentexperimentalanalyses1,thee ectsofNLOQCDcorrectionsonthetotalcrosssectionandthedijetinvariantmassdistributionoftheW/Zb¯bbackgroundprocesseshavebeentakenintoaccountbyusingtheMCFMpackage[30].InMCFM,theNLOQCDpredictionsofbothtotalanddi erentialcrosssectionsfortheseprocesseshavebeencalculatedinthezerobottom-quarkmass(mb=0)approximation[31,32,33],usingtheanalyticalresultsofRefs.[34,35].FromastudyoftheLeadingOrder(LO)crosssection, nitebottom-quarkmasse ectsareexpectedtoa ectboththetotaland

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

di erentialW/Zb¯bcrosssections,mostlyintheregionofsmallb¯b-pairinvariantmasses[33].Indeed,sincethiskinematicregionofsmallb¯b-pairinvariantmassescontributesconsiderablytoW/Z+njproduction(n=1,2),whereatleastoneofthejetsisab-jet,bottom-quarkmasse ectscannotbeneglectedasdiscussedinRefs.[36,37](forn=2)andinRef.[38](forn=1).GiventhevarietyofexperimentalanalysesinvolvedinthesearchforW/ZHassociatedproduction,single-topandHb¯bproduction,itisimportanttoassesspreciselytheimpactofa nitebottom-quarkmassovertheentirekinematicalreachoftheprocess,includingthecompleteNLOQCDcorrections.

InRef.[39]wehaveperformedastudyofNLOQCDcrosssectionsandinvariantmassdistributionsofthebottomquarkpairinWb¯bproductionattheTevatron,includingfullbottom-quarkmasse ects.Wefoundthatbottom-quarkmasse ectsamounttoabout8%ofthetotalNLOcrosssectionattheTevatronandaremostlyvisibleintheregionoflowb¯b-pairinvariantmass.

Inthispaper,wecomputetheNLOQCDcorrectionstoZb¯bhadronicproduction,includ-ingthefullbottom-quarkmasse ects.WeconsiderallpartonicprocessesthatcontributeatO(αα3),i.e.NLOQCDcorrectionstoqq¯→Zb¯bandgg→Zb¯bandthetree-levelprocesss

q(¯q)g→Zb¯bq(¯q).Wepresentnumericalresultsforthetotalcrosssectionandtheinvariantmassdistributionoftheb¯bpairfortheTevatronpp¯collider,includingkinematiccutsandajet- ndingalgorithm.Inparticular,weapplythekTjetalgorithmandrequiretwotaggedb-jetsinthe ingtheMCFMpackage[30],wecompareourresultswiththecorrespondingresultsobtainedinthemb=0limit.NumericalresultsforbothZb¯bandWb¯bproductionattheLargeHadronColliderwillbepresentedinaseparatepublication[40].Theoutlineofthepaperisasfollows:inSectionIIwebrie ydiscussthetechnicaldetailsofourcalculation,whilewepresentnumericalresultsandadiscussionofthebottom-quarkmasse ectsinSectionIII.SectionIVcontainsourconclusions.

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

qb

FIG.1:Tree-levelFeynmandiagramsforqq¯→Zb¯b.Thecircledcrossesindicateallpossibleinsertionsofthe nal-stateZbosonleg,eachinsertioncorrespondingtoadi erent

diagram.FIG.2:Tree-levelFeynmandiagramsforgg→Zb¯b.Thecircledcrossesindicateallpossibleinsertionsofthe nal-stateZbosonleg,eachinsertioncorrespondingtoadi erentdiagram.II.CALCULATION

A.Basics

3Thetotalcrosssectionforpp¯(pp)→Zb¯batO(ααs)canbewrittenasfollows:

σNLO(pp¯(pp)→Zb¯b)= ij1

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

s,isgivenintermsofthehadroniccenterofmassenergysquared,sH,bys=x1x2sH.TheNLOQCDpartoniccrosssectionreads:

σ ijijij

NLO(x1,x2,µ)=σ LO(x1,x2,µ)+δσ NLO(x1,x2,µ),(2)

whereσ ij

LO(x1,x2,µ)denotestheO(αα2s)LOpartoniccrosssectionandδσ ij

NLO(x1,x2,µ)

describestheO(αs)correctionstoσ ij

LO(x1,x2,µ).TheLOpp¯(pp)→Zb¯bprocessreceives

contributionsfromqq¯andgginitiatedprocesses,asshowninFig.1andFig.2,respectively.TheNLOQCDcorrections,δσ ij

NLO,receivecontributionsfromqq¯,gg,qgandq¯ginitiated

processesandcanbedecomposedinthefollowingway:

δσ ij

NLO= d(PS3)|Areal(ij→Zb¯b+l)|2

≡σ ij+σ ij

virtreal, (3)

wherethetermintegratedoverthephasespacemeasured(PS3)correspondstothevirtualone-loopcorrectionswiththreeparticlesinthe nalstate,whiletheoneintegratedoverthephasespacemeasured(PS4)correspondstotherealtree-levelcorrectionswithoneadditionalemittedparton.Thesum

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

¯paperscitedaboveandtoRef.[50]formorecalculationandrefertotheWb¯bandHtt

details.TheO(αs)correctionstoqq¯→Zb¯bcanbederivedfromtheO(αs)corrections

¯[48](witht bandtoqq¯′→Wb¯bproduction[39](withW Z)andtoqq¯→Htt

H Z),whiletheO(αs)correctionstothegginitiatedZb¯bproductionprocesscanbe

¯[49](witht bandH Z).TheobtainedfromtheO(αs)correctionstogg→Htt

¯NLOQCDcalculationswithW Zqg,q¯ginitiatedprocessesappearinbothWb¯bandHtt

¯andH Z,t b,respectively.Notethat,whenapplyingtheresultsofWb¯bandHtt

productionofRef.[39]andRefs.[48,49],onealsoneedstoreplacerespectivelytheWff′andfermionYukawacouplingsbytheV AcouplingoffermionstotheZ

boson:

=

2MZ,(6)

whereMZisthemassoftheZboson.

ijThevirtualcrosssectionσ virtB.

TheO(αs)virtualcorrectionstothepartonictree-levelqq¯→Zb¯bandgg→Zb¯bproduc-tionprocessesconsistofself-energy,vertex,boxandpentagondiagrams,asshown,forthe¯-likepart,inFigs.2-4ofRef.[48]andFigs.2-5ofRef.[49],respectively(forafullsetHtt

virtofdiagramsseealsoRef.[50]).Thecontributionstoσ ijinEq.(3)canbewrittenas:

A0AD+A0AD=

D

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

whereA0isthetreelevelamplitudeandADdenotestheamplitudefortheone-loopdiagramD,withDrunningoverallself-energy,vertex,boxandpentagondiagramscorrespondingtotheij-initiatedsubprocess.

Thecalculationofeachvirtualdiagram(AD)isperformedinthesamewayasoutlinedinRefs.[48,49],i.e.ADiscalculatedasalinearcombinationofDiracstructureswithcoe cientsthatdependonbothscalarandtensorone-loopFeynmanintegralswithupto vedenominators.Wesolvetheone-loopintegralsinthecoe cientseitherattheleveloftheamplitudeorattheleveloftheamplitudesquared(seeEq.(7)).Thesetwoin-dependentapproachesallowustothoroughlycross-checkthecalculationofeachindividualdiagram.Indeed,thetensorstructurespresentintheone-loopintegralsoftheamplitudearetypicallydi erentfromtheonespresentintheamplitudesquared,asonecanperformnon-trivialreductionsofthelatterbycancelingdot-productsoftheintegrationmomentuminthenumeratorwithdenominatorsintheFeynmanintegrals.Inthisway,the nalanalyticalexpressionofagivendiagramendsupbeingrepresentedintermsofdi erentbuildingblocks.Apossibleincorrectrelationbetweenthebuildingblockswouldthennaturallyproduceadiscrepancybetweenthetwoapproaches.

ingthePassarino-Veltman(PV)method[51,52],thetensorintegralsareexpressedasalinearcombinationoftensorstructuresandcoe cients,wherethetensorstructuresdependontheexternalmomentaandthemetrictensor,whilethecoe cientsdependonscalarintegrals,kinematicsinvariantsandthedimensionoftheintegral.Numericalstabilityissuesmayariseatthislevelasaconsequenceoftheproportionalityofthetensorintegralcoe cientstopowersofinverseGramDeterminants(GDs),asdiscussedindetailinRef.[49],althoughforZb¯bproductiontheproblemisconsiderablymoreserious.

Thesenumericalinstabilitiescanbeconsideredas“spurious”or“unphysical”divergences,sinceitiswellknownthatonlytwo-particleinvariantscangiverisetoaphysicalsingularity.Indeed,thesespuriousdivergencescancelwhenlargesetsofdiagramsarecombined[35],suchas,forexample,whenonecombinesgaugeinvariantsetsofcoloramplitudes(i.e.amplitudeswithacommoncolorfactor).Aswehaveexpressedourcalculationintermsofinvariants,andweemployastandardbasisofscalarintegrals,thefullcancellationonlyoccursbetweennumeratoranddenominatoratthenumericallevel,oftenbetweenfairlylargeexpressions.Forthisreasonwehavechosentoorganizethediagrams,atcertainstages,intogaugein-

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

variantcoloramplitudes,thatis,intocoe cientsofthesamecolorstructure(seeRef.[50]).Thisallowsabetterhandlingofthespurioussingularitiesandanaturalwaytomakein-ternalcross-checksandcross-checkswithnewtechniques.Whenweconsiderthesegaugeinvariantsetsofcoloramplitudesandfullanalyticalreductionsofalltensorintegrals,we ndcancellationofsomepowersofGDs,whichimprovesthenumericalstabilityofourcode,sothatwhenintegratingovertheZb¯bphasespace,usingMCtechniques,weobtainstatisticalerrorsbelow0.1%fortotalcrosssections.

Thefullyreducednumericalcodesareoftenmoredemandingcomputationally,andbe-causeofthatwehavebuiltmastercodesthatusethemonlywhenclosetoregionsofphasespacewherecertainproblematicGDsbecomesmall.AllthisisfoundparticularlyusefulwhenconsideringhigherrankD-PVfunctions(wehaveuptoD4-PVfunctionsinourcalcu-lation)aswellasE-PVfunctions.Probably,thistechniquewouldbreakdown,ifoneweretoextendittoprocesseswithevenmorelegs,andtheuseofhelicityamplitudeswouldinthiscasebepreferable.

Inthecaseofpentagondiagrams,apowerfulandconvenientcheckconsistsofreducingconsistentlyallE-PVfunctionsbycancelingsystematically,attheleveloftheamplitudesquaredinEq.(7),allpossiblevectorproductscontainingtheloopmomentuminthenumer-atorwithsomedenominators.Thisispossibleas,inthepentagontopologyofourprocess,eachleghasanoutgoingmomentumwhichison-shell,correspondingbasicallytooneoftheexternalinitialor nalparticlesofthesubprocess.Onethenendswithexpressionsforeachpentagondiagramcontainingpurelyscalarpentagonintegrals,ortensorintegralswithfewerthan vedenominators,improvingconsiderablythenumericalstability.Wecomparedanalyticallythesereductionstothenon-reducedexpressionsbyusingthefullreductionofalltensorintegralstoscalarintegrals,andfoundagreement.

Wealsocheckedpartsofourresultbyusingunitaritytechniques[35],speci callythequadruple-cuttechnique[53].AsshownbyBritto,CachazoandFeng(BCF),fromanysetofFeynmandiagrams(ormoregenerallyfromanytensorintegral[54])onecanextractthecoe cientofagivenscalarboxintegralbycuttingthefourcorrespondingpropagators(seemomentumpandmassm.Thise ectivelyfreezesthemomentumintegration,andreplacesitbyasetofalgebraicequationswhichdeterminetheloopmomentumentirely.WesolvedthissetofequationsbyusingaBCFansatz[53],andthencomparedtheresulttothe

Fig.3),i.e.byreplacingi/(p2 m2+i )→2πδ(+)(p2 m2)foreachcuttedpropagatorof

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

FIG. 3: Quadruple cut[53] check of the calculation of a box diagram involving a top-quark loop. It corresponds to two Feynman diagrams given by the two possible orientations of the fermion line.

correspondi

ng box coe cient extracted from our analytic expression, and found agreement (for more details and speci c solutions for the topology in Fig. 3 see Ref.[50]). This is a rather non-trivial check for the set of E-PV and D-PV functions we have employed at di erent stages, since they all contribute to the coe cients of the scalar D-functions occurring in the 1-loop Zb¯ amplitude. For instance, it has been particularly useful in the case of box b diagrams like the one shown in Fig. 3, since this diagram and related ones contain up to D4-PV functions that cannot be reduced even at the level of the amplitude squared. Since they involve up to four powers of inverse GDs, they are particularly subject to numerical instabilities and it is important to have their analytic expressions as compact as possible. After the tensor integral reduction is performed, the fundamental building blocks are one-loop scalar integrals with up to ve denominators. They may be nite or contain both ultraviolet (UV) and infrared (IR) divergences. The nite scalar integrals are evaluated using the method described in Ref.[52] and cross-checked with the numerical package FF[44]. The UV and IR singular scalar integrals are calculated analytically by using dimensional regularization in d= 4 2 dimensions. The most di cult integrals arise from IR divergent pentagon diagrams with several external and internal massive particles. We calculate them as linear combinations of box integrals using the method of Refs.[55, 56] and of Ref.[52]. Details of the box scalar integrals (see also Ref.[57]) and the pentagon reduction, as well as the set of IR-divergent three and two-point functions used in this calculation, are given in Ref.[50]. The UV singularities of the virtual cross section are removed by introducing a suitable set of counterterms (see Refs.[48, 49, 50] for details), while the residual renormalization 9

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

scaledependenceischeckedfrom rstprinciplesusingrenormalizationgroupargumentsasinEq.(4)ofRef.[49].Notethatweusetheon-shellsubtractionschemewhen xingthewavefunctionrenormalizationconstantoftheexternalbottomquark eld(δZ2)andthemassrenormalizationconstant(δmb).TheIRsingularitiesofthevirtualcrosssectionare

3canceledbyanalogoussingularitiesintheO(ααs)realcrosssection.(b)

Inourcalculationwetreatγ5accordingtothenaivedimensionalregularizationapproach,i.e.weenforcethefactthatγ5anticommuteswithallotherγmatricesind=4 2 dimensions.Thisisknowntogiverisetoinconsistencieswhen,atthesametime,thed-dimensionaltraceoffourγmatricesandoneγ5isforcedtobenon-zero(asind=4,whereTr(γµγνγργσγ5)=4i µνρσ)[58].Inourcalculation,bothUVandIRdivergencesarehandledinsuchawaythatweneverhavetoenforcesimultaneouslythesetwopropertiesoftheDiracalgebrainddimensions.Forinstance,theUVdivergencesareextractedandcanceledattheamplitudelevel,afterwhichthed→4limitistakenandtherenormalizedamplitudeissquaredusingd=4.Thus,allfermiontracesappearingatthispointarecomputedinfourdimensionsandthereforehavenoambiguities.

Wenotethatthetree-levelamplitudeA0inEq.(7)hasgenericallytobeconsideredasad-dimensionaltree-levelamplitude.ThismatterswhentheADamplitudesinEq.(7)areUVorIRdivergent.Actually,asithasbeenshowninRefs.[48,49],bothUVandIRdivergencesarealwaysproportionaltothetreelevelamplitudesandtheycanbeformallycanceledwithouthavingtoexplicitlyspecifythedimensionalityofthetreelevelamplitudeitself.AfterUVandIRsingularitieshavebeencanceled,theremainingphasespaceintegrationiscomputedind=4dimensionsusingstandardMCtechniques.

ijTherealcrosssectionσ realC.

ijTheNLOQCDrealcrosssectionσ realinEq.(3)correspondstotheO(αs)correctionsto

ij→Zb¯bduetotheemissionofanadditionalrealparton,i.e.totheprocessij→Zb¯b+g,andthetree-levelprocessq(¯q)g→Zb¯b+q(¯q).σ ijcontainsIRsingularitieswhichcancelthereal

analogoussingularitiespresentintheO(αs)virtualcorrectionsandintheNLOPDFs(seeRefs.[48,49,50]fordetails).Thesesingularitiescanbeeithersoft,whentheemittedextrapartonisagluonanditsenergybecomesverysmall,orcollinear,whenthe nal-statepartonisemittedcollineartooneofthepartonsintheinitialstate.Thereisnocollinearsingularity

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

arisingfromtheradiationo the nal-statebottomquarks,sincetheyareconsideredtobemassive.

Wehavecalculatedthecrosssectionsfortheprocesses

i(q1)+j(q2)→b(pb)+¯b(p¯b)+Z(pZ)+g(k)

and

¯)(k),(q,q¯)(q1)+g(q2)→b(pb)+¯b(p¯b)+Z(pZ)+(q,q

withq1+q2=pb+p¯b+pZ+k,usingthetwo-cuto PhaseSpaceSlicing(PSS)method.

ThisimplementationofthePSSmethodwasoriginallydevelopedtostudyQCDcorrectionstodihadronproduction[59]andhassincethenbeenappliedtoavarietyofprocesses(anicereviewcanbefoundinRef.[60]).WefollowcloselytheapplicationofthePSSmethodto¯productionaspresentedinRefs.[48,49]towhichwereferformoreextensivereferencesHtt

andfulldetails.AlthoughweareconsideringZb¯bproduction,thekinematicsareequivalent,andthecolorstructureandIRbehaviorarethesame,sonecessarilytheirsoftandcollinearkernelsarethesame.Inthefollowingwebrie ysummarizeourimplementationofthetwo-cuto PSSmethod.

UsingthePSSmethod,theIRsingularitiescanbeconvenientlyisolatedbyslicingthephasespaceofthe nal-stateparticlesintodi erentregionsde nedbysuitablecuto s.Toisolatethesoftandcollinearsingularitiesweimposesoft(δs)andcollinear(δc)cuto sonthephasespaceoftheemittedpartonasfollows.Byintroducinganarbitrarysmallsoftcuto δs,weseparatetheoverallintegrationoftheqq¯,gg→b¯bZ+gphasespaceintotworegions√0accordingtowhethertheenergyofthe nalstategluon(k=Eg)issoft,i.e.Eg≤δs

s/2.Inordertoisolatethecollinearsingularitieswefurtherdivide

thehardregionoftheqq¯,gg→b¯bZ+gphasespaceintoahard/collinearandahard/non-collinearregion,byintroducingasecondsmallcollinearcuto δc.Thehard/non-collinearregionisde nedbytheconditionthatboth

2q1·k>δcsand2q2·k>δcs(8)

aretrue.Weapplythesamecollinearcuto tothetree-levelprocessq(¯q)g→Zb¯b+q(¯q).

qq¯,gg,qgThehardnon-collinearpartsoftherealcrosssections,σ hard/non coll,are niteandcanbe

computednumerically.Thepartonicrealcrosssectionscanthenbewrittenasfollows:

qq¯,gg,qgqq¯,ggqq¯,gg,qgqq¯,gg,qgσ real=σ soft+σ hard/coll+σ hard/non coll,(9)

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

qq¯,ggqq¯,gg,qgwhereσ softandσ hard/collisobtainedbyintegratinganalyticallyoverthesoftandcollinear

regionsofthephasespaceoftheemittedparton,respectively,andcontainsalltheIR

qq¯,gg,qgdivergencesofσ real.Thedependenceonthesearbitrarycuto s,δs,δc,isnotphysical,

ijandcancelsattheleveloftherealcrosssection,i.e.inσ real.Thiscancellationconstitutes

animportantcheckofthecalculation.

WeconcludethissectionbyshowingexplicitlythatthetotalhadroniccrosssectionatNLOQCDdoesnotdependonthearbitrarycuto sintroducedbythePSSmethod,i.e.on

ijδsandδc.ThecancellationofthePSScuto dependenceisrealizedinσ realbymatching

ijijcontributionsthatarecalculatedeitheranalytically( σsoftandσ hard/coll),intheIR-unsafere-

ijgionbelowthecuto s,ornumerically,intheIR-saferegionabovethecuto s( σhard/non coll).

WhiletheanalyticalcalculationintheIR-unsaferegionreproducestheformofthecrosssectioninthesoftorcollinearlimitsandisthereforeonlyaccurateforsmallvaluesofthecuto s,thenumericalintegrationintheIR-saferegionbecomesunstableforverysmallval-uesofthecuto s.Therefore,obtainingaconvincingcuto independenceinvolvesadelicatebalancebetweenthepreviousantagonisticrequirementsandultimatelydictatesthechoiceofvaluesthatareneithertoolargenortoosmallforthecuto s.InFigs.4and5wedemon-stratetheindependenceofσNLO(pp¯→Zb¯b)onδsandδcseparately,byvaryingonlyoneofthetwocuto soveranextendedrange,whiletheotheriskept xed.InFig.4,δsisvariedbetween10 5and10 2withδc=10 5,whileinFig.5,δcisvariedbetween10 7and10 4withδs=10 3.Inbothplots,weshowintheupperwindowtheoverallcuto dependence ijijijcancellationbetweenthehadroniccrosssectionsij(σsoft+σhard/coll)andijσhard/non coll ijin¯andqg.Notethatwealsotakeintoaccountijσreal,includingallchannels,gg,qq

contributionsfromtheLOandthevirtualcrosssectionswhicharecuto independent.InthelowerwindowofthesameplotsweshowthefullσNLO,includingallchannels,onascale

thatmagni esthedetailsofthecuto -dependencecancellation.ThestatisticalerrorsfromtheMCphasespaceintegrationarealsoshown.BothFigs.4and5showaclearplateauoverawiderangeofδsandδcandtheNLOcrosssectionisproventobecuto independent.ThenumericalresultspresentedinSectionIIIhavebeenobtainedbyusingthetwo-cuto PSSmethodwithδs=10 3andδc=10 5.

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

FIG.4:NLOµ=mb+MZ/2,andδc=10 5.Theupperplotshowsthecancellationoftheδs-dependencebetweenσsoft+σhard/collandσhard/non coll.Thelowerplotshows,onanenlargedscale,thedependence

ggqq¯qgofthefullσNLO=σNLO+σNLO+σNLOonδswiththecorrespondingstatisticalerrorsoftheMC

integration.

III.NUMERICALRESULTS

TheresultsforZb¯bobservablespresentedinthispaperareobtainedfortheTevatronpp¯collideratsH=1.96TeV.Ifnotstatedotherwise,weassumeanon-zerobottom-quarkmass, xedatmb=4.62GeV.Themassofthetopquark,enteringinthevirtualcorrections,issettomt=170.9GeV.TheZbosonmassistakentobeMZ=91.1876GeV[61]andtheWbosonmassiscalculatedfromMW=MZcosθwwithsin2θw=0.223.WeworkintheelectroweakGµinputschemeandreplacethe nestructureconstantα(0)=e2/(4π)byα(Gµ)=√

π2GµMWsin2θwwiththeFermiconstantGµ=1.16639·10 5GeV 2.TheLO

LOresultsusethe1-loopevolutionofαsandtheCTEQ6L1setofPDFs[62],withαs(MZ)=

0.130,whiletheNLOresultsusethe2-loopevolutionofαsandtheCTEQ6MsetofPDFs,

NLOwithαs(MZ)=0.118.WeimplementthekTjetalgorithm[63,64,65,66]witha

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

FIG.5:NLOµ=mb+MZ/2,andδs=10 3.Theupperplotshowsthecancellationoftheδs-dependencebetweenσsoft+σhard/coll,andσhard/non coll.Thelowerplotshows,onanenlargedscale,thedependence

ggqq¯qgofthefullσNLO=σNLO+σNLO+σNLOonδcwiththecorrespondingstatisticalerrorsoftheMC

integration.

pseudo-conesizeR=0.7andwerecombinethepartonmomentawithinajetusingthesocalledcovariantE-scheme[64].WecheckedthatourimplementationofthekTjetalgorithmcoincideswiththeoneinMCFM.Werequirealleventstohaveab¯bjetpairinthe nalstate,

b,bwithatransversemomentumlargerthan15GeV(pT>15GeV)andapseudorapiditythat¯

satis es|ηb,b|<2.WeimposethesamepTand|η|cutsalsoontheextrajetthatmayariseduetohardnon-collinearrealemissionofaparton,i.e.intheprocessesZb¯b+gorZb¯b+q(¯q).Thishardnon-collinearextrapartonistreatedeitherinclusivelyorexclusively.Intheinclusivecaseweincludebothtwo-andthree-jetevents,whileintheexclusivecasewerequireexactlytwojetsintheevent.Two-jeteventsconsistofabottom-quarkjetpairthatmayalsoincludea nal-statelightparton(gluonorquark)duetotheappliedrecombinationprocedure.Resultsinthemasslessbottom-quarkapproximationhavebeenobtainedusingtheMCFMcode[30].

¯

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

InTableIwepresentresultsforthetotalLOandNLOQCDpp¯→Zb¯bcrosssections,obtainedwiththescaleµr=µf=MZ+2mb,forbothourfullymassivecalculationandinthemasslessapproximation.Ascanbeseen,theNLOQCDcorrectionsincreaseconsiderablyTABLEI:LOandNLOtotalZb¯bcrosssectionsattheTevatronformassiveandmasslessbottomquarks,usingµr=µf=MZ+2mb.ThenumbersinsquarebracketsaretheratiosoftheNLOandLOcrosssections,thesocalledK-factors.StatisticalerrorsoftheMCintegrationamounttoabout0.1%.

mb=0(pb)[ratio]mb=0(pb)[ratio]

2.21[ ]2.37[ ]

σNLOinclusive

2.80[1.27]3.01[1.27]

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

FIG.6:DependenceoftheLO(blacksolidband),NLOinclusive(bluedashedband),andNLOexclusive(reddottedband)Zb¯btotalcrosssectionsontherenormalization/factorizationscales,includingfullbottom-quarkmasse ects.Thebandsareobtainedbyindependentlyvaryingbothµrandµfbetweenµ0/2and4µ0(withµ0=mb+MZ/2).

and7(b)wealsocomparethescaledependenceofourresultstothescaledependenceofthecorrespondingresultsobtainedwithmb=0(usingMCFM),inganon-zerovalueofmbisexpectedtohaveasmallimpactonthescaledependenceoftheresults2,sincetheonlymodi cationtotherenormalizationscaledependenceoriginatesfromthebottom-quarkmassand eldrenormalization,asdiscussedinSectionIIBofRef.[27],wherewecomparetheminimalandon-shellsubtractionschemes.Indeed,ascanbeseeninFigs.7(a),7(b)thescaledependenceoftheLOandNLOcurvesisverysimilarforboththecaseofamassiveandmasslessbottomquark.WhiletheLOcrosssectionstillhasa45%uncertaintyduetoscaledependence,thisuncertaintyisreducedatNLOtoabout20%fortheinclusiveandtoabout11%fortheexclusivecrosssections.Theuncertaintieshavebeenestimatedasthepositive/negativedeviationwithrespecttothemid-pointofthebandsplottedinFig.6,whereeachbandrangeisde nedbytheminimumandmaximumvalueintheband.Wenoticeincidentallythatthedi erenceinthetotalcrosssectiondueto nitebottom-quarkmasse ectsislesssigni cantthanthetheoreticaluncertaintydueto

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

5 4.5σtotal (pb) 4 3.5 3 2.5 2 1.5 0.5cuts: pt> 15 GeV|η|< 2 R= 0.7

NLO massless NLO massive LO massless LO massiveσtotal (pb)

5 4 3 2 1 0 0.5

NLO massive _ qq initiated gg initiated qg initiated

Inclusive caseµ0= MZ/2+ mb

1

µ/µ0

2

4

1

µ/µ0

2

4

(a)Inclusive case

4.5 4σtotal (pb) 3.5 3 2.5 2 1.5 0.5cuts: pt> 15 GeV|η|< 2 R= 0.7

NLO massless NLO mass

ive LO massless LO massiveσtotal (pb)

4 3 2

NLO massive qq initiated gg initiated qg initiated

Exclusive case

1µ0= MZ/2+ mb

0 1µ/µ0 2 4 0.5 1µ/µ0 2 4

(b)Exclusive case

FIG. 7: Dependence of the LO and NLO inclusive and exclusive Zb¯ total cross section on the b renormalization/factorization scale, whenµr=µf=µ. The LHS plots compare both LO and NLO total cross sections for the case in which the bottom quark is treated as massless (MCFM) or massive (our calculation). The RHS plots show separately, for the massive case only, the scale dependence of the q q, gg and qg+ q g contributions, as well as their sum.¯¯

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

FIG.8:ourNLOtheinclusiveandexclusivecases(withµr=µf).TheerrorbarsindicatethestatisticaluncertaintyoftheMCintegration.

theresidualscaledependenceintheinclusivecase,butiscomparableinsizeintheexclusivecase.Indeed,the nitebottom-quarkmasse ectsamounttoareductionofthetotalcrosssectionsbyabout7%comparedtothemasslesscaseatbothLOandNLOQCD.

InFig.8,weshowtherescaleddi erencebetweentheNLOtotalcrosssectionsobtainedfromourcalculation(withmb=0)andwithMCFM(withmb=0)de nedasfollows:

σ=σNLO(mb=0) σNLO(mb=0)σLO(mb=0)

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

MCFM.Asexpected,mostofthedi erencebetweenthemasslessandmassivebottom-

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

100µr=µf= MZ+ 2mb_ dσ/dmbb (fb/GeV)

1.5 dσ(massive)/ dσ(massless)

Inclusive case

1cuts: pt> 15 GeV

10

0.5

|η|< 2 R= 0.7

1 30

NLO masslessσtotal= 3.64 pb NLO massiveσtotal= 3.40 pb 60 90 120 150 _ mbb (GeV) 180

0 30

NLO ratio 60 90 120 150 180 _ mbb (GeV)

(a)Inclusive case

100µr=µf= MZ+ 2mb_ dσ/dmbb (fb/GeV)

1.5 dσ(massive)/ dσ(massless)

Exclusive case

1cuts: pt> 15 GeV

10

0.5

|η|< 2 R= 0.7

1 30

NLO masslessσtotal= 3.01 pb NLO massiveσtotal= 2.80 pb 60 90 120 150 _ mbb (GeV) 180

0 30

NLO ratio 60 90 120 150 180 _ mbb (GeV)

(b)Exclusive case

FIG. 10: The inclusive and exclusive NLO QCD distributions dσ/dmb¯ derived from our calculation b (with mb= 0) and from MCFM (with mb= 0). The RHS plots show the ratio of the two distributions, dσ(mb= 0)/dσ(mb= 0).

We calculate the Next-to-Leading Order (NLO) QCD corrections to Z b anti-b production in hadronic collisions including full bottom-quark mass effects. We present results for the total cross section and the invariant mass distribution of the bottom-quark je

quarkcrosssectionsiscomingfromtheregionoflowmb¯binvariantmass,bothforthetheoneswithmb=0.ThisisemphasizedintheRHSplots,whereweshowtheratioofthetwodistributions,dσ(mb=0)/dσ(mb=0).Forcompleteness,wealsoshowinFig.

11QCD.TheLOmb¯bdistributionformassivebottom-quarkshasbeenobtainedbothfromourcalculationandfromMCFM,whichimplementsthemb=0optionattreelevel,andbothresultsagreeperfectly.Ingeneral,masse ectsaresimilaratLOandNLO.ToillustratethisinmoredetailweshowinFig.12therescaleddi erencebetweenthemb¯bdistributionsasfollows:obtainedwithourNLOcalculation(withmb=0)andwithMCFM(withmb=0)de ned

dσdσNLOthecomparisonbetweenmassive(mb=0)andmassless(mb=0)calculationsatLOininclusiveandexclusivecase,wherethecrosssectionsformb=0areconsistentlybelow

fullbottom-quarkmasse ects.Wehavepresentednumericalresultsforthetotalcross

本文来源：https://www.bwwdw.com/article/5vym.html