QCD-corrected spin analysing power of jets in decays of pola

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

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aPITHA02/07April2002DESY02-055TTP02-04QCD-correcteddecaysofspinpolarizedanalysingtoppowerquarksofjetsinA.Brandenburga, ,Z.G.Sib, andP.UwercabDESY-Theorie,22603Hamburg,GermanyInstitutf.TheoretischePhysik,RWTHAachen,52056Aachen,GermanycInstitutf.TheoretischeTeilchenphysik,UniversitätKarlsruhe,76128Karlsruhe,GermanyAbstract:Wepresentresultsforthedifferentialdistributionsofjetsfromnon-leptonicdecaysofpolarizedtopquarkswithintheStandardModel,includingQCDradiativecorrections.Ourworkextendsexistingresultswhichareonlyavailableforsemileptonictopquarkdecaysatthepartonlevel.Fort(↑)→b-jet+2lightjetswecomputeinparticulartheQCD-correctedtop-spinanalysingpoweroftheb-quarkjetandtheleastenergeticlight

jet.Thedependenceoftheresultsonthechoiceofthejetrecombinationschemeisfoundtobesmall.Inadditionwecomputethespinanalysingpowerofthethrustaxis.Ourresultsconstituteasofarmissingingredienttoanalysetopquarkproductionandsubse-quentnon-leptonicdecayatnext-to-leadingorderinαs,keepingthefullinformationonthetopquarkpolarization.

PACSnumber(s):12.38.Bx,13.88.+e,14.65.Ha

Keywords:topquarks,polarization,QCDcorrections,jets

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

1Introduction

Thedetailedanalysisofthedynamicsoftopquarkproductionanddecayisamajorobjec-tiveofexperimentsattheTevatron,theLHC,andapossiblefuturelineare+e collider.Aspecialfeatureofthetopquarkthatmakessuchstudiesveryattractiveisitslargedecaywidth:IncontrasttothelightquarksthelargetopdecaywidthΓt≈1.5GeVservesasacut-offfornon-perturbativeeffectsintopquarkdecays.Asaconsequenceprecisetheo-reticalpredictionsofcrosssectionsanddifferentialdistributionsinvolvingtopquarksarepossiblewithintheStandardModelanditsextensions.Aconfrontationofsuchpredic-tionswithforthcominghigh-precisiondatawillleadtoaccuratedeterminationsofStan-dardModelparametersandmaybehintstonewphenomena.Inparticular,observablesrelatedtothespinofthetopquarkcanbestudiedandutilizedtosearchfornewinterac-tionsofthetopquark[1].Ine+e collisions,topquarksareproducedhighlypolarized,especiallyifonetunesthepolarizationoftheincomingbeams,aspossiblee.g.attheTESLAcollider[2].Furthermore,evenforpurelyQCD-inducedproductionoftopquark

¯areingeneralhighlycorrelated[3].pairs,thespinsoftandt

Thepolarizationandspincorrelationsoftopquarksmustbetracedinthedifferentialdistributionsofthedecayproducts.Thisispossiblesincetheinformationonthetoppolarizationistransferredtotheangulardistributionofthedecayproductsthroughitsweak,parityviolatingdecays.Tobemoreprecise:ConsiderapolarizedensembleoftopquarksatrestwithpolarizationvectorP,0≤|P|≤1.Thedifferentialdecaydistribution

ofagivendecayproductreads:withrespecttotheangle betweenPandthedirectionp

1

dcos =1

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

non-factorizablecontributions,seeref.[7].)QCDcorrectionstotheproductionoftopquarkpairs,includingthefullinformationabouttheirspins,canbefoundinref.[8]fore+e collisionsandinref.[9]forhadron-hadroncollisons.Inthecaseofe+e collisions,alsofullyanalyticresultsforthetopquarkpolarization[10]andaspeci cspincorrelation

¯Xproduction[11]toorderαsareavailable.Inref.[12],helicityamplitudesfore+e →tt

andsemileptonicdecaysarecomputedtoorderαsandareusedtoconstructanevent¯istreatedinref.[13],includingboth¯→W+bW bgenerator.Thereactione+e →tt

factorizableandnon-factorizableQCDcorrections.Thetheoreticalstatusofpolarizedtopquarkdecayisasfollows:Acompletecalculationoftheangulardecaydistributionfort(↑)→W+btoorderαscanbefoundinref.[14].TheQCDcorrectionsforsemileptonicpolarizedtopquarkdecayshavebeencomputedinref.[15].Wewillcompareourresults,fromwhichthesemileptoniccasecanbeeasilyderived,tothoseofref.[15]insection3.Theoutlineofthispaperisasfollows:Inthenextsectionweshortlyreviewthetreelevelresultsforthedecayofpolarizedtopquarks.Insection3wediscussthecalculationoftheQCDcorrections.Section4containsournumericalresults,whichwediscussinsection5.

2Kinematicsandtreelevelresults

ConsideraninitialstateconsistingoftopquarksatrestwithpolarizationP.Forthenon-leptonicdecay

t(pt)→b(pb)+u(pu)+d¯(pd¯),(2)

thephasespaceR3ofthe nalstatemaybeparametrizedbytwoscaledenergiesandtwoangles:

dR3=mt2

2mt|M(pb,pu,pd¯)|2dR3,(4)

where|M(pb,pu,pd¯)|2standsforthesquaredmatrixelementaveragedoverthecolouroftheinitialstateandsummedovercolourandspinsofthe nalstate.Thefullydifferentialdistributionforreaction(2)readsattreelevel:

dΓ0

(1 xb+zb ξ)2+η2ξ2

(1+|P|cosθ),(5)

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

where

c=NC|Vud|2e4|Vtb|2mt

4,(6)

with4(2π)

m2

ξ=W

mW,zb=m2b

Γ dΓcos =κp|P|

(k2 m2)2+m2Γ2→π

WWW

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

Table1:Bornresultsforspinanalysingpowerofd¯,b,u,leastenergeticlightjetandthrustaxis.

mb=0,ΓWkept

1

0.40622

0.31817

0.50774

0.317121 0.40553 0.319640.50708 0.31597mb=5GeV,ΓWkept

dxd¯dxbdχdcosθ=c

4πΓ(1 ε)

×[f1(xd¯,xb,zb)(1+|P|cosθ)+f2(xd¯,xb,zb)|P|sinθcosχ],(11)

4πµ2

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

with

f1(xd¯,xb,zb)= ln(ω)+ln2(ω)+4ln(ω)ln1 xb(1 ω)whereε1+ω x 4Libω2 1+ω xb 8+ln(zb)+gWud¯¯ zb) ε xd¯(1 xd+ln(ω)x(2zbβb xb+2xd¯ xd¯xb)+O(ε),f2(xβsinθdb¯xd¯(1 xd¯,xb,zb)=d¯ zb) ln(zb)+2zb 1 xb+zbβ β +O(ε),

β=

1+β,

gWud¯=2 mt23

ε2

4πµ2

ε Γ(1 ε)→ln λ2(12)(13)(15)

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

Thiscontributioncanalsobesplitintotwoseparateparts,whichdonotinterferewitheachother:thecasewherethegluonisemittedfromaheavyquark(torb),andthecontributionwherethegluonisemittedfromalightquark.Thecalculationofthecor-respondingamplitudesisstraightforward.Asalwaysinthecaseofajet-calculationwehavetoaddressthequestionhowtocanceltheinfraredandcollinearsingularities.Forthecontributionwherethegluonisemittedfromthesecondaryfermionlineweusethedipoleformalism[16].Droppingoverall

factorsthesubtractiontermforthisprocessisgivenbyVug,d¯2 |M(pb,pu,p dg(19)¯)|2(pd¯·pg)

withM(pb,pu,pd¯)beingthematrixelementatleadingorderandp ,Varede nedinEq.(5.3)andEq.(5.7)ofref.[16].Thesubtractiontermintegratedoverthesingularregionisalsogiveninref.[16]andreads:

αsCF

k2 ε1ε+23

2dRd4|M(pb,pu,pd¯,pg)|Θ(xminmbmt 2(pt·pg))2mt ε 1αsCF1=ln(ω)22mβx2mtmint

12 ln(ω))|M(pb,pu,pd¯)|2dRd3.2 (22)

Inordertobeconsistentwithourde nitionofthephasespacemeasuredR3forthevirtuald=http://paringwithEq.(12)itcorrections,wehavetoevaluatetheaboveformulafordR3isstraightforwardtoseethattheinfraredsingularitiescancel.Thenumericalimplemen-tationofthesubtractiontermgiveninEq.(19)andtheslicinggiveninEq.(21)doesnotimposeanyproblem.Oneshouldkeepinmindthatthemethodpresentedaboveallowsthecalculationofarbitraryinfraredsaveobservables.

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

4Numericalresults

Whileforthethreeparton nalstateonecansimplyidentifythepartonswiththejets,inthecaseofrealgluonemissionade nitionofjetsintermsofpartonsisneededwhichful llstheconditionofinfrared-safeness.Therearemanyvariationsofsuchde nitions,andingeneraltheNLOresultforκpwilldependonthechosende nition.Inordertostudythisdependence,wechoosetwodifferentjetclusteringschemes:TheE-algorithmandtheDurham-algorithm.The rststepistocomputeforallpairs(i,j)ofthemomentaofthe nalstatepartonsthejetmeasureyij,whichreads

yij=(pi+pj)2

(1 cosθij)Durham-algorithm,(24)mt2

whereθijistheanglebetweenpartoniandpartonjinthetopquarkrestframe.Thesecondstepistorecombinethetwopartonswiththesmallestyijintoapseudoparticlewithmomentumpk=pi+pj.Thenthedirectionoftheb-jetandthelightquarkjetwiththesmallestenergycanbereadilyobtainedfrompkandtheremainingtwomomenta.Forbscg¯eventsalsothedirectionofthecharmjetcanbede nedifthecharmistagged.Theconstructionofthe“s¯-jet”frombscg¯eventsisnotstraightforward:Onlyeventswherebothab-jetandac-jetaretaggedcanbeused(inparticular,the(rare)eventswherebandcareclusteredintoasinglejethavetobediscarded),andforthoseeventstheremainingthirdjetisde nedtobethe“s¯-jet”.Thisde nitionincludescaseswherethes¯isrecombinedwiththeb-orc-quarkandtheremainingthirdjetconsistsofahardgluonratherthanans¯-quark.

Notethatnojetresolutionparameterentersintheabovede nitions;thisisnotnecessary,sincetheleadingorderprocessisfreefromsoftandcollinearsingularities.

Inanexperimentwhichproducestopquarksthatdecayintojets,the rststepintheanalysisistoidentifythesignalbyusingajet ndingalgorithmandbyapplyinganumberofcuts.Forexample,attheTevatronaconejetalgorithmisusedtoclassifytheeventsaccordingtothenumberofjets.Tobemorespeci c,considertopquarkpairswherethe

¯decaysintojets.Thentheeventcontainsatleasttwotdecayssemileptonicallyandthet

¯,ourb-jetsandtwolightjets.Forthoseeventswith4ormorejetsoriginatingfromthet

algorithmshouldbeusedinadditiontotheproduction-speci cjetalgorithm,thusleaving

¯decay.Inprinciplethevalueforκpcandependonlyeventswithexactly3jetsfromthet

onthedetailsofthe“pre-clustering”.ThisshouldbestudiedinMonte-Carlosimulations.Wecanalsocomputeκpforbareb,d¯(s¯)andu(c)quarks,sincethedirectionsofaquarkandaquarkplusacollinearorsoftgluonareidenticalandthustheconditionofinfrared/collinearsafenessisful lled.Thisservesasabenchmarkfortherealisticcaseofκpforjets.Notehoweverthatrecombinationisneededtode nethedirectionoftheleastenergeticlightjet.Inthecaseofthethrustaxistheconceptofajetisnotneeded.

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

Wewriteourresultstoorderαsinthefollowingform

κp=11Γ0κ0p+αsΓκp 1 020QCD2κ κ+O(α)≡κ[1+δ]+O(αppspps),(25)Γ0

QCD,wherewewhereκ0denotestheBornresult.Table2givesourresultsforκandδpppusetheexpandedformofEq.(25).Thestrongcouplingconstantissettoαs(mt)=0.108.

Table2:QCD-correctedresultsforspinanalysingpowers.

κd¯

[%]δQCD

bκu 3.36±0.07 0.3925(6) 4.24±0.15 0.3032(6)

+1.39±0.19

7.18±0.13

2.65±0.19 2.22±0.190.4734(7)jets,E-alg.0.9379(8) 6.73±0.08 0.3910(6)δQCD[%]jκT

5Discussionandconclusions

OurresultslistedinTable2showthatthetop-spinanalysingpowersofthe nalstatesinnon-leptonictopquarkdecaysreceiveQCDcorrectionsintherange+1.4%to 7.2%.Thishastobecontrastedwiththespinanalysingpowerofthechargedleptonindecayst(↑)→bl+νl:theQCDcorrectedresult(formb=0)reads[15]κl=1 0.015αs,i.e.thecorrectionisatthepermilllevel.TheQCDcorrectionstothespinanalysingpowerofagivenquarkaremuchlargerduetohardgluonemissionfromthatquark.Thespinanalysingpowerofjetsissmallerthanthatofthecorrespondingbarequarks.Thiseffectislargestforthe“s¯-jet”.We ndonlyasmall(atmost0.7%)dependenceoftheresultsonthejetalgorithm.Inpracticethemostimportantspinanalysersare,asfarasnon-leptonictopdecaysareconcerned,theb-quarkjetandtheleastenergeticlight(non-b-quark)jet.TheQCDcorrectedresultsareκb≈ 0.39andκj≈0.47.Fortheb-jetthedifferencebetweenthepartonlevelresultandthejetresultissmall.Theorientedthrustaxis,forwhichκT≈ 0.31,mayalsoserveasagoodspinanalyser,sinceitiseasilymeasurable.

We present results for the differential distributions of jets from non-leptonic decays of polarized top quarks within the Standard Model, including QCD radiative corrections. Our work extends existing results which are only available for semileptonic top q

Insummary,wehavecomputedtheQCDcorrectionstothetop-spinanalysingpowerofjetsandthethrustaxisinnon-leptonicpolarizedtopquarkdecays.OurresultscanbeusedinconjunctionwiththeknownNLOQCDresultsfortheproductionofpolarizedtopquarksbothatleptonandhadroncolliders.

Acknowledgements

WewouldliketothankW.BernreutherforsuggestingthisworkandhimandS.Dittmaierforusefuldiscussions.

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