我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

One-loopQCDcorrectionstothee+e →W+W b¯bprocessatthe

ILC

GuoLei,MaWen-Gan,ZhangRen-You,andWangShao-Ming

DepartmentofModernPhysics,UniversityofScienceandTechnology

ofChina(USTC),Hefei,Anhui230027,P.R.China

Abstract

Westudythefullcontributionsattheleadingorder(LO)andQCDnext-to-leadingor-der(NLO)tothecrosssectionofthee+e →W+W b¯bprocessinthestandardmodel(SM)attheILC.Indealingtheresonanceproblemweadoptedthecomplexmassschemeinbothtree-levelandone-looplevelperturbative√calculations.OurnumericalresultsshowthattheK-factorvariesfrom1.501to0.847whengoesupfrom360GeVto1.5TeV.Weinvestigatethe

+ + ¯dependenceofthe√LOandQCDNLOcorrectedcrosssectionsofprocessee→WWbbon

collidingenergyandHiggs-bosonmass.WealsopresenttheresultsoftheLOandQCDNLOcorrecteddistributionsofthetransversemomentaof nalparticles,andtheinvariantmassesofWb-,b¯b-andWW-pair.

Keywords: ILC，QCD，Top-quark，Higgs-boson

PACS:13.66.Jn,14.65.Ha,14.80.Bn,12.38.Bx

SuppportbyNationalNaturalScienceFoundationofChina:10575094,10575095,10675110andSpecialized

ResearchFundfortheDoctoralProgramofHigherEducation:20050358063

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

I.Introduction

TheHiggsboson,whichgivesmassestotheweakvectorbosonsandfermions,playsanimportantroleinthestandardmodel(SM).Unfortunately,ithasnotbeendirectlydetectedyetinexperiments.SearchingforHiggsbosonwithinthestandardmodel(SM)andstudythephenomenologyconcerningHiggspropertiesaretheimportanttasksatthepresentandupcominghighenergycolliders.LEPIIexperimentshaveprovidedthelowerlimitontheSMHiggsmassas114.4GeVatthe95%con dencelevel,whichisextractedfromtheresultsofsearchesfore+e →Z0H0production[1,2].WhiletheindirectevidencesoftheSMHiggsmassthroughelectroweakprecisionmeasurementsindicatethe95%C.L.upperboundasmH 182GeV,whenthelowerlimitonmHisusedindeterminationofthisupperlimit[2].Ontheotherhand,theheavytop-quarkpracticallyplaysacentralandcrucialroleinprobingtheelectroweaksymmetrybreakingaswellasthe avorprobleminalltheextendedmodelsbeyondtheSMwhichaddressthehierarchyproblem.Recently,anewdatumoftop-quarkmasshasbeenalreadypresentedbytheCDFandD0experimentsatFermilab,andthepreliminaryworldaveragemassofthetop-quarkisknownasmt=172.5±1.3(stat)±1.9(syst)GeV,whichcorrespondstoa20%precisionimprovementrelativetothepreviouscombination[3].

ThefutureInternationalLinearCollider(ILC)isproposedbytheparticlephysicscommunity

√

withtheentirecollidingenergyintherangeof200GeV<<500GeVandanintegrated

√

luminosityofaround500(fb) 1infouryears.Themachineshouldbeupgradeableto～1TeV

withanintegratedluminosityof1(ab) 1inthreeyears[4].MostofthemainphysicstopicswithintheSMoritsbeyondatTeVenergyscalecanbeexploredatsuchamachine.Emphasisisgiventothestudyoftop-quarkphysics,electroweakphysicsintheSM,andthemeasurementsintheextendedSM,suchassupersymmetry.

Comparedwiththehadroncolliders,suchastheTevatronandtheCERNLargeHadronCol-lider(LHC),theILCcanproducetopandHiggsbosonsignaleventsmoreeasilyresolvedfrombackgrounds.Therefore,theILCisanidealfacilitytostudytopandHiggsphysicswithmuchmoreprecisemeasurementfortheirparameters.AttheILCwecanalsocarryoutthestudyofgaugebosoninteractions,andthedelicatecancellationswhicharerelatedtothegaugestructureofthetheoryandessentialtopreserveunitarity.Furthermore,theILCexperimentmightbeableto

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

explorethesignatureofthenewphysics,iftheSMisreallyonlyane ectivetheoryatlowenergy.

¯isagoodwaytostudyAttheILC,detectingthetop-quarkpairproductionprocesse+e →tt

thetop-quarkproperties,andtheassociatedHiggsproductionwithZ0bosone+e →H0Z0isoneofthecleanestsignatureindiscoveringHiggsbosonifthetheb-quarktriggersystemhashighperformancesexceptvertexdetectors[5].Theformerprocesswillbefollowedbythesubsequential¯→W+W b¯decaythroughttb[6],whilethelaterprocessgoesviaH0Z0→W+W b¯bthroughdecaysH0→W+W andZ0→b¯biftheHiggsbosonmassislargerthan2mW[7].Therefore,thesignatureofe+e →W+W b¯battheILCservesasnon-resonantbackgroundtobothtop-quarkpairproductionandassociatedproductionofHiggsbosonwithZ0boson.WecanseethatitiscrucialtoseparatethetopandHiggssignaturesfromtheotherW+W b¯bproductionbackgroundsinILCexperimentaldataanalyzing.Intheprecisemeasurementsofthesignalsof¯pairandH0Z0associatedproductionprocesses,therelevantirreduciblebackgroundboththett

frome+e →W+W b¯bshouldbecarefullyinvestigated.

¯InRefs.[6,8,9,10,11,12]theNLOelectroweakandQCDcorrectionstotheprocesse+e →ttanddecayt→W+bhavebeenalreadyextensivelystudied.Andthenon-relativistice ectnearthe¯productionisalsostudiedcarefullyinRef.[13],whichcannotbereliablydescribedthresholdoftt

with xedQCDordersinperturbativetheory.TheHiggs-strahlungBjorkenprocesse+e →H0Z0¯→W+W b¯wasinvestigatedinRef.[14],andtheprocesse+e →ttb→6fwithsixfermion nalstatesafterWpairdecayshasbeenalsocalculatedatthelowestorderinRef.[15].Theevaluationofthee+e →W+W b¯bprocesswith nitewidthmethodatthetree-levelisalsopresentedinRef.[7,16].Allthosestudiesindicatethatthepreciseinvestigationsofthecharacteristicsoftop-quarkandtheHiggs-bosonaresigni cantforthefuturee+e ILCexperiments.

Inthispaperwepresentthecalculationsofthecrosssectionoftheprocesse+e →W+W b¯battheleadingorder(LO)anditsQCDnext-to-leadingorder(NLO)(O(αs))corrections.Thepaperisorganizedasfollows:Inthefollowingsectionwepresenttheanalyticalcalculationsforprocesse+e →W+W b¯battheLOandQCDNLO.Theveri cationsofthecorrectnessofourcalculationsaredeclaredinsectionIII.ThenumericalresultsanddiscussionsaregiveninsectionIV.Inthelastsectionwegiveashortsummary.

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

II.Calculations

Thecalculationsfortheprocesse+e →W+W b¯barecarriedoutin’tHooft-Feynmangauge.IntheQCDNLOcalculations,weusethedimensionalregularization(DR)methodtoisolatetheultraviolet(UV)andinfrared(IR)singularities.Inordertopreservegaugeinvariance,weadopttheapproachofthecomplexmassschemetodealwiththeunstableparticlesinthecalculationsforthetree-levelcrosssectionandQCDNLOradiativecorrection[17,18].Theon-mass-shell(OS)schemeisusedtorenormalizethemassesand eldsofrelatedbosonsandfermions.TheFeynArts3.2package[19]isadoptedtogenerateFeynmandiagramsandconvertthemtocorrespondingampli-tudes.TheamplitudecalculationsaremainlyimplementedbyapplyingFormCalc4.1programs[20].TheformulaforcalculatingtheIRdivergentintegralswithcomplexinternalmassesinDRschemeareobtainedbyanalyticallyextendingtheexpressionsinRef.[21]tothecomplexplane.Thenumer-icalevaluationsofIRsafeone-point,two-point,three-pointandfour-pointintegralswithinternalcomplexmasses,areimplementedbyusingtheexpressionsanalyticallycontinuedtocomplexplanefromthosepresentedinRefs.[22,23].Andthe5-pointscalarintegralcanbeexpressedintermsofmultiplescalarfour-pointintegrals[24].Thesubroutinesforone-loopintegralswithcomplexmassesarecodedbasedontheLoopTools2.1[20]packagewhichcomesfromFFlibrary[25].The2→4phase-spaceintegrationroutine[27]iscreatedbasedonthe2to3.FprograminFormCalc4.1package.The ve-bodyphase-spaceintegrationforhardgluonradiationprocesse+e →W+W b¯bgisaccomplishedbyusingCompHEP-4.4p3program[26].

Nowwepresenttheanalyticallycalculationsofthetree-levelcrosssectionfore+e →W+W b¯banditsQCDNLOradiativecorrections.Thenotationsfortheprocessarede nedas

e+(p1)+e (p2)→W+(p3)+W (p4)+b(p5)+¯b(p6),

(2.1)

wherepi(i=1 6)labelthefour-momentaofincominge+,e andoutgoing nalparticles,respectively.Thereare64generictree-leveldiagramsfortheprocesse+e →W+W b¯bpresentedinFig.1,whereinternalwavy-linerepresentsγ,Z0,orW±andinternaldash-linerepresentsaHiggs-bosonH0oraGoldstoneG0(G±).Wecaneasily ndthatinFig.1thereincludesthetree-¯ →W+W b¯leveldiagramsfortheprocessese+e →t tbande+e →H0 Z0 →W+W b¯b.

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

Figure1:Thegenerictree-levelFeynmandiagramsforthee+e →W+W b¯bprocess.Internalwavy-linerepresentsγ-,Z0-,orW±-propagator.Internaldash-linerepresentsaHiggsbosonH0oraGoldstoneG0(G±).

Thedi erentialcrosssectionfortheprocesse+e →W+W b¯batthetree-levelisobtainedbythetree-levelisobtainedby

dσtree=412e

(2π)4

|Mtree|2dΦ4,

(2.2)

wheredΦ4isthefour-bodyphasespaceelementgivenby 6

6

(4)

dΦ4=δp1+p2 pi

i=3

i=3

d3pi

.

(2π)2Ei

(2.3)

ThesummationinEq.(2.2)istakenoverthespinsandcolorsof nalstates,andthebaroverthesummationrecallsaveragingoverinitialspinstates.Inthecalculation,theinternalZ0and

√

Higgsbosoncanbereal,andthetop-quarkpropagatorcanalsoberesonancewhen>2mt.

Todealwiththeseresonantsingularities,weusetheso-calledcomplexmassscheme(CMS)inourperturbativecalculations[17,18].ThecomplexmassesofW-,Z-,H-bosonandtop-quarkarede ned

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

as

2

µ2X=mX imXΓX,

(X=W,Z,H,t).(2.4)

IntheCMSapproachthecomplexmassesforallrelatedunstableparticlesshouldbetakenev-erywhereinbothtree-levelandone-looplevelcalculations.Thenthegaugeinvariancecanbeconservedandsingularitypolesofpropagatorsareavoided.

IncalculatingthecompleteQCDNLOcorrections,weshouldconsiderthecontributionsof30self-energydiagrams,94trianglediagrams,17boxdiagramsand6pentagondiagrams.Asarepresentativeselection,wepresentthepentagonFeynmandiagramsofthee+e →W+W b¯bprocessinFig.2.WeadopttheEqs.(4.26)and(4.27)inRef.[18]fortherenormalizedQCDself-2)byusingenergyandcounter-termsoftop-quarkwithcomplexmassneglectingtermsofO(αs

OS-scheme.

Figure2:ThepentagonFeynmandiagramsforthee+e →W+W b¯bprocess.

Thereexistbothultraviolet(UV)divergencyandinfrared(IR)softsingularityinthecontribu-tionsoftheQCDone-loopdiagramsfore+e →W+W b¯bprocess,butnocollinearIRsingularityduetothemassivetop-andbottom-quark.Afterdoingtherenormalizationprocedure,theUVsingularityisvanished.

TocanceltheIRsoftdivergencyappearedinthevirtualcorrection,weshouldconsiderthecontributionoftherealgluonemissionprocesse+e →W+W b¯bg.Wedenotetherealgluonemissionprocessas

e+(p1)+e (p2)→W+(p3)+W (p4)+b(p5)+¯b(p6)+g(p7).

(2.5)

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

Figure1:Thegenerictree-levelFeynmandiagramsforthee+e →W+W b¯bprocess.Internalwavy-linerepresentsγ-,Z0-,orW±-propagator.Internaldash-linerepresentsaHiggsbosonH0oraGoldstoneG0(G±).

Thedi erentialcrosssectionfortheprocesse+e →W+W b¯batthetree-levelisobtainedbythetree-levelisobtainedby

dσtree=412e

(2π)4

|Mtree|2dΦ4,

(2.2)

wheredΦ4isthefour-bodyphasespaceelementgivenby 6

6

(4)

dΦ4=δp1+p2 pi

i=3

i=3

d3pi

.

(2π)2Ei

(2.3)

ThesummationinEq.(2.2)istakenoverthespinsandcolorsof nalstates,andthebaroverthesummationrecallsaveragingoverinitialspinstates.Inthecalculation,theinternalZ0and

√

Higgsbosoncanbereal,andthetop-quarkpropagatorcanalsoberesonancewhen>2mt.

Todealwiththeseresonantsingularities,weusetheso-calledcomplexmassscheme(CMS)inourperturbativecalculations[17,18].ThecomplexmassesofW-,Z-,H-bosonandtop-quarkarede ned

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

Table1:ThecomparisonofthenumericalresultsofLOcrosssectionneglectingthediagramswithHiggs-bosoninterchangingbyusingCompHEP-4.4p3system,ourin-house2→4phase-spaceintegrationroutinewiththecorrespondingselectedresultspresented

√

inRef.[16]when=500GeV.

t180200LO576.26497.63LO576.19(1±0.04%)497.58(1±0.04%)LO576.52(1±0.04%)497.68(1±0.04%)

TheLOcrosssectionfortheprocesse+e →W+W b¯biscalculatedintheconditionsof

√

taking=500GeVandneglectingthecontributionofthediagramswithinternalHiggs-

bosonexchangewhicharetakeninRef.[16].ThenumericalresultsoftheLOcrosssectionfortheprocesse+e →W+W b¯barelistedinTable1.ThereourresultsareobtainedbyusingbothCompHEP-4.4p3programandourcreated2→4phase-spaceintegrationroutine,andcomparedwiththecorrespondingonespresentedinRef.[16].WecanseethereisagoodagreementbetweenoursandthosepresentedinRef.[16].Thein-house2→4phase-spaceintegrationroutinewasalsoonceveri edinourpreviouswork[27].

Weuseourcreatedcodesfornumericalevaluationoftheone-loopintegralswithcomplexinternalmasses.ThecomparisonsaremadebetweentheresultsandthoseobtainedbydoingdirectlytheintegrationofFeynman-parameter.Thereexistsagoodagreement.Theresultsfrombothcalculationsforscalartwo-,three-,four-pointintegralsarecoincidentwitheachotheratleastuptosixdigits,respectively.

TheexactcancellationsofUV-andIR-divergenciesareveri edbothanalyticallyandnumer-icallyinourcalculation.

TheindependenceofthetotalcrosssectionincludingtheNLOQCDcorrectionsonthesoft

√

cuto δs(=2 E/iscon rmednumerically.Ourcalculationshowstheerrorsofthe

independencearelessthan0.6%intheδsregionof[10 4,5×10 2].Infurthernumericalcalculationwe xδs=10 3.

Inthefollowingsection,weshallclarifyotherveri cations.

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

IV.Numericalresultsanddiscussion

Inournumericalcalculationwetakethefollowinginputparameters[29,30]:

α(mZ) 1=127.918,αs(m2Z)=0.1176,mZ=91.1876GeV,mW=80.403GeV,ΓZ=2.495GeV,ΓW=2.141GeV,mt=172.5GeV,mb=4.7GeV,me=0.5109991MeV.

(4.1)

DuetotheapplicationoftheCMSapproach,weusethecomplexweakmixinganglede nedas

c2w

=1

s2w

µ2

.=WµZ

(4.2)

InourLOandNLOnumericalcalculationswesettheQCDrenormalizationscaleµasµ=mW+mb,andtakethestrongcouplingαs(µ2)=0.11885,whichisobtainedbyusingtheformulaatthree-looplevel(scheme)withthe veactive avors[29].

Sincethewidthsoftop-quarkandHiggsbosonhaven’tbeenwellprovidedormeasuredexper-imentallybynow,weusetheirtheoreticalresultsfromperturbativecalculations.Consideringthefactthattop-quarkmassisabovemW+mb,andVtb～1,thedecayoftop-quarkisdominatedbyundergoingtwo-bodydecayt→W+b,andthetotaldecaywidthoftop-quarkisapproximately

2222equaltothedecaywidthoft→W+b.Neglectingtermsoforderm2b/mt,αsand(αs/π)MW/mt,

thewidthpredictedintheSMis[31]:

αm3t

Γt=16mW(1 mW/mZ)

m2

1 mt

2 2

m252π2αs

. 1 1+2233πmt

(4.3)

ThereasonablephysicaldecaywidthofHiggsbosonisobtainedbyemployingtheprogramHdecay[32],

3)QCDradiativecor-wherethepartialdecaywidthΓ(H0→qq¯)iscalculatedincludingO(αs

rections.ThenweobtainΓt=1.3745GeV,ΓH(mH=120GeV)=0.3692×10 2GeVandΓH(mH=180GeV)=0.6286GeV.

ThenumericalresultsoftheLO,QCDNLOcorrectedcrosssectionsandthecorrespondingK-factor(K≡

σNLO

LO)

fortheprocesse+e →W+W b¯bareplottedinFigs.3(a)and(b)respectively,

whenmH=120GeV.AsindicatedinFig.3(a),bothcurvesforthecrosssectionsattheLOand

√√

NLOincreasequicklyintheregionof[350GeV,400GeV]anddecreasewhen>430GeV.

√

Fig.3(b)showsthatthecorrespondingK-factordecreasesslowlyfrom1.501to0.847asrunning

¯thresholdinFig.3(b)isduetoafrom360GeVto1.5TeV.Thelargepositivepeaknearthett

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

(fb)

K

s(GeV)Figure3:(a)TheLOandNLOQCDcorrectedcrosssectionsfortheprocesse+e →W+W b¯bas

√

thefunctionsofc.m.s.collidingenergy(withmH=120GeV,(b)thecorrespondingK-factor

√

versusTable2:TheLOandNLOQCDcorrectedcrosssections,K-factorsfore+e →W+W b¯b

√

processwithmH=120GeVand=500GeV,1000GeV,1500GeV,respectively.

(GeV)10001500tree182.24(1±0.04%)82.73(1±0.04%)NLO156.7(1±0.38%)70.1(1±0.37%)0.860(4)0.847(4)

Coulombsingularitye ectcomingfromtheinstantaneousgluonexchangebetweenheavyquarkswhichhasasmallspatialmomentum.InTable2welistthevaluesofσtree,σNLOandK-factor

√

atsometypicalpoints,whicharereadoutfromFigs.3(a-b).SincetheQCDcorrectiontothe

e+e →W+W b¯bprocesswithhighcollidingenergycanbeapproximatelydecomposedintothe¯productionplusthecorrectionstothet(t¯)→W+b(W ¯QCDcorrectiontothettb)decayswhenmH<2mW,wemakefollowingveri cationtocheckourresults.WeevaluatetheQCDcorrection¯productionandtoe+e →W+W b¯bprocessbycombiningtheQCDcorrectionstoe+e →tt¯)→W+b(W ¯b)decaystogether,andgettheK-factorstoprocesse+e →W+W b¯bas0.8562(1)t(t

√√

for=1TeVand0.8433(1)for=1.5TeV,whicharecoincidentwiththecorresponding

onesinTable2inerrorranges.

InFig.4(a)wepresenttheplotoftheLOandQCDNLOcorrectedcrosssectionsasthefunctions

√

ofHiggs-bosonmass,with=500GeVandmHrunningform60GeVto200GeV.We ndfrom

Fig.4(a)thattheLOandQCDNLOcorrectedcrosssectionsarenon-sensitivetotheHiggs-boson

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

(fb)

K

mm

Figure4:(a)TheLOandQCDNLOcorrectedcrosssectionsfortheprocesse+e →W+W b¯bas

√

thefunctionsofHiggsmass(mH)with=500GeV.(b)ThecorrespondingrelativeQCDNLOcorrectionsversusmH.

massexceptinthevicinityofmH～2mW,fromtheretheHiggsmassbecomeslargerthan2mW,andH0-,Z0-bosonaresimultaneouslyresonances.Wecanseealsofromthe guresthatthecontribution¯ →W+W b¯viae+e →t tbchannelismuchlargerthanthatfrome+e →H0 Z0 →W+W b¯basconcludedinRef.[16].Fig.4(b)showsthecorrespondingK-factorhasthevaluesaround0.956intherangeofmH∈[60GeV,200GeV].

DuetotheCP-conservation,thedistributionsoftransversemomentaofW -bosonand¯bquarkshouldbethesameasthoseofpWTofthepWT

+

andpbTwithmH

andpbT,respectively.Weonlypresentthedistributions

√

=120GeVand=500GeVinFigs.5(a)and(b).Inthese

+

two gureswecanseethattheQCDNLOcorrectionssuppresstheLOdi erentialcrosssectionsdσLO/dpWT

+

WanddσLO/dpbT.Theyalsoshowthatthedi erentialcrosssectionsofdσLO,NLO/dpT

+

+

WanddσLO,NLO/dpbThavetheirmaximalvaluesataboutpT

～70GeVandpbT～30GeVrespec-

tively.Weseethatthelineshapesofthedi erentialcrosssectionsinthesetwo guresaremainly¯ →W+W b¯determinedbythecontributionsoftheofe+e →t tbproduction.

Weplottheinvariantmassdistributionsof(W+b)-pair,denotedasM(W+b),attheLOand

√

QCDNLOinFig.6withmH=120GeVand=500GeV.ThedistributionofM(W ¯b)should

bethesameasthatof(W+b)-pairduetotheCP-conservation.Wecanseefromthe gurethatmostoftheeventsareconcentratedaroundapeaklocatedatthepositionofM(W+b)～mt.Thatdemonstratesagainthemaincontributiontothecrosssectionoftheprocesse+e →W+W b¯b

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

(fb/GeV)

(fb/GeV)

W

b

dp

dp

d

d

W

p

(GeV)

b

p

(GeV)

Figure5:ThedistributionsofthetransversemomentaofW+andbottom-quarkforthee+e →

√

W+W b¯bprocessattheLOandQCDNLOwith=500GeVandmH=120GeV.(a)forW+,(b)forbottom-quark.

(fb/GeV)

dM

Figure6:Thedistributionsofthe(or(W ¯b)-pair)attheLOand

QCDNLOwithmH=120GeV¯andfollowedbythewithhighcollidingenergy,isfromtop-pairproductionchannele+e →tt

¯)→W+b(W ¯decayoft(tb).HerewecanseethattheQCDNLOcorrectionslightlysuppressestheLOdi erentialcrosssectiondσLO/dM(W+b).

Asweknow,ifHiggsbosonhasamasslargerthan2mW,thee+e →H0 Z0 →W+W b¯bchannelwillcertainlyslightlyincreaseboththeLOandQCDNLOcorrectedcrosssectionsforW+W b¯bproductionduetotheHiggs-bosonresonante ectasshowninFig.4(a).Itwillbringaspikeonthedistributionofthe(W+W )-pairinvariantmassatthepositionofM(WW)=mH.Analogously,theassociatedrealZ0-bosonproducedviae+e →H0Z0willinduceaspikeonthedistributionoftheinvariantmassM(b¯b)atthepositionofM(b¯b)=mZ.InFigs.7(a)and(b)we

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

(fb/GeV)

(fb/GeV)

dM

dM

d

MM

Figure7:Thedistributionsoftheinvariantmassesof(b¯b)-pairand(WW)-pairwithmH=180GeV

√

b)-pair.and=500GeV.(a)isfor(WW)-pair,(b)isfor(b¯

√

showthethedistributionsoftheWW-andb¯b-pairinvariantmasseswith=500GeVand

mH=180GeV,respectively.WecanseespikesaroundthevicinitiesofM(WW)～mH～180GeVandM(b¯b)～mZ～90GeVinFig.7(a)andFig.7(b)respectively,whichmaybeusedtodistinguishthee+e →H0Z0→W+W b¯bsignaturefromthecorrespondingirreduciblebackgrounde+e →W+W b¯b.ItshowsalsotheQCDNLOcorrectionobviouslymodi estheLOdi erentialcrosssectionsofdσLO/dM(WW)anddσLO/dM(b¯b).

Theoretically,theNLOQCDcorrectiontotheprocesse+e →Z0H0→W+W b¯bwithrealZ0-andHiggs-bosonasintermediateparticles,shouldbedeterminedonlybytheNLOQCDcorrectionstotheZ0→b¯bdecay.Asachecktoverifyourcalculations,wealsocalculatethecorrectionto

√

thedecayZ0→b¯bwith=500GeVandmH=180GeV>2mWandgettheK-factors

fore+e →Z0 H0 →W+W b¯bprocessbeing1.0466(1),whichiscoincidentwiththeresultbycalculatingthee+e →Z0 H0 →W+W b¯bprocesswithfullQCDNLOdiagrams,wheretheK-factorsare1.046(2).

V.Summary

Inthispaperwecalculatethecompleteone-loopQCDcorrectionsintheSMtotheprocesse+e →W+W b¯battheILC.WestudythedependenceoftheLOandQCDNLOcorrectedcrosssections

√

ofprocesse+e →W+W b¯boncollidingenergyandHiggs-bosonmass.Weinvestigatethe

LOandQCDNLOcorrecteddistributionsofthetransversemomentaof nalparticlesandthe

我们研究了标准模型下在ILC上$e^+e^- \to W^+ W^- b\bar b$过程在领头阶(LO)和QCD修正的次领头(NLO)的所有贡献。为了处理共振问题,我们在树图和一圈图的微扰计算中采用了复质量方案。我们得到的数值结果表明在$\sqrt{s}$从$360~GeV$到$1.5~TeV$时,K因子从$1.501$变化到$

http://

LOandQCDNLOcorrecteddi erentialcrosssectionsofinvariantmassesofWb-,b¯b-andWW-pair.ItshowsthatNLOQCDcorrectionobviouslymodi estheLOcrosssectionoftheprocess

√

e+e →W+W b¯b,andwhenthecollidingenergygoesupfrom360GeVto1.5TeV,theK-

factorvariesfrom1.501to0.847.ThenumericalresultsshowthatifmH>2mW,theresonante ectofH0-bosonappearinginthee+e →H0Z0→W+W b¯bchannelwillinducealittleenhancementtotheLOandQCDNLOcorrectedcrosssectionsfore+e →W+W b¯bprocess.We ndthatitmaybepossibletoselectthee+e →H0Z0→W+W b¯beventsfromthecorrespondingirreducible¯→W+W b¯backgrounde+e →W+W b¯bwhichisdominantlyproducedbythee+e →ttbchannelbyanalyzingtheinvariantmassesof nalWW-andb¯b-pair.

Acknowledgments:ThisworkwassupportedinpartbytheNationalNaturalScienceFounda-tionofChina,SpecializedResearchFundfortheDoctoralProgramofHigherEducation(SRFDP)andaspecialfundsponsoredbyChineseAcademyofSciences.

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