Key Distributions for Charmless Semileptonic B Decay

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

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aTheoreticalPhysicsInstituteUniversityofMinnesotaTPI-MINN-97/06-TUMN-TH-1532-97hep-ph/9703437KeyDistributionsforDecayCharmlessSemileptonicBR.DavidDikemanandN.G.Uraltsev TheoreticalPhysicsInstitute,Univ.ofMinnesota,Minneapolis,MN55455AbstractWepresenttheoreticalpredictionsforafewphenomenologicallyinterestingdistributionsinthesemileptonicb→udecayswhicharea ectedbyFermimotion.Theperturbativee ectsareincorporatedattheone-looplevelandappeartobeverymoderate.OurtreatmentofFermimotionisbaseddirectlyonQCD,beingencodedintheuniversaldistributionfunctionF(x).Thedecaydistributionsinthechargedleptonenergy,invariantmassofhadrons,hadronenergy,andq2aregiven.Wenotethattypicallyabout90%ofalldecayeventsareexpectedtohaveMX<MD;thisfeaturecanbeexploited

indeterminationof|Vub|.

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

1Introduction

Themeasurementofheavyquarkdecaydistributionsisimportantinextractingkeyparametersofthestandardmodel.Towardsthisend,theoristsarefacedwiththetoughtaskofseparatingstrongbindinge ects,arisingfromstronginteractionsatlargedistances,fromtherelativelysimplequark-leptonLagrangian,knownatshortdistances.Bothperturbativecorrectionsandlittleunderstoodnonperturbativeef-fectsareverymoderateinbeautydecaysifoneisinterestedinsu cientlyinclusive,integratedcharacteristics[1,2].Inexperimentalstudiesofb→u νdecays,attemptstoseparateb→ctransitionswithoutexplicitlyidentifyingcharmedparticles,oftensuggestsusinganarrowsliceoftheoveralldecaykinematicsre ectingthelowin-variantmassof nal-statehadronicsystemaccessibleonlyinb→udecays.Thestronginteractione ectsaremagni edinthiscase.ThisfactwasrealizedalreadyinearlypaperswheretheQCD-basedtreatmentofinclusivedecayrateswaselabo-rated.Inparticular,itwaspointedoutin[2]thattheQCD-basedOPEitselfleadstotheemergenceofthephenomenonof“Fermimotion”oftheheavyquarkinsideahadron,ane ectintroducedadhocmuchearlier[3]forthedescriptionofheavy avordecaysinphenomenologicalmodels.Forexample,theimpactofFermimotioninsimplequarkmodelsontheinvariantmassofhadrons,MX,wasalsoaddressedin[4].TheconcretemanifestationofFermimotioninQCDhassomepeculiarities

[2,5]makingitsomewhatdi erentfromthesimple-mindedFermimotionofquarkmodels.TheQCDtreatmentwaslaterdescribedindetailinanumberofpubli-cations[5,6,7,8,9].Conceptually,Fermimotionissimilartotheleading-twistnonperturbativee ectsinDIS.Furtherdetailsandmoreextensivereferencescanbefoundin[9].

Inthepresentworkweapplythemethodsof[9]toafewexperimentallyimpor-tantdistributionsintheb→usemileptonicdecays,namely,thedistributionoftheelectronenergydΓ/dE ,theinvarianthadronmassdΓ/dMXandthehadronenergydΓ/dEh;wealsogiveaplotofdΓ/dq2.The rstthreedistributionsaredirectlyaf-fectedbytheFermimotion.Ourtreatmentisincertainaspectssimpli edcomparedto[9];wewillmentiontheseelementsandjusti cationlater.

2GeneralStrategy–IncludingFermiMotionin

HeavyQuarkDecayDistributions

Stronginteractionshavetwofacesinb-decays:one,associatedwiththehardmodesofthegluon elds,istoproduceperturbativecorrections,theother,associatedwiththesoftmodesistoproducenonperturbativecorrections,inparticularFermimotion.ThewaytoseparatethehardandsoftmodesisdictatedbyWilson’sOPE:theintroductionofthehardseparationscaleµ.The‘identity’ofthetworegimeshasnoabsolutesenseanddependsonthechoiceofµ.

Thehadronicpartofthesemileptonicdecaysisdescribedby vestructurefunc-

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

tionswi(q0,q2),onlythreeofwhicharerelevantfor =eorµ(forde nitions,see[10]).Thee ect

of

Fermi

motionisencodedintheheavyquarkdistributionfunctionF(x);notethattheprimordialmomentumisnormalizedto

Λ2 q2,q2q0

(1)

Herewpertisapartonstructurefunctiondressedwithshort-distance(perturbative)corrections.ThisexpressioncoincideswiththoseofRef.[5]throughleading-twistterms,summationofwhichyieldsthee ectoftheprimordialmomentumdistributionoftheheavyquark.TheformofEq.(1)isconvenientsinceitcloselyfollowsthesimplepictureofthedecayoftheheavyquarkboostedtothevelocityxF(x) 1 xMBq02q0 q2 dx.

3

13

2MB )2b|B , B|¯b(iDρ3D=

q2<～MXwhereMXistheinvariantmassofthe nal

hadronicsystem.Forexample,relations(2)aremodi ed.InthepresenceofgluonbremsstrahlungMXcanbecomelarge, ΛQCD,evenforb→utransitions.Certain

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

relationsbetweenthemomentsofFatdi erentq2stillhold,1andweexpectthatthee ectsofchangingF(x)areinsigni cantinwhatconcernssingle-di erentialdistributions.

AnothercomplicationoftheQCDdescriptionisthatF(x)togetherwith

Λ(µ)[9].

3FermimotionindΓ/dEl,dΓ/dMX,anddΓ/dEh

Withthestructurefunctions(1)wearriveatthefollowingleptonspectruminb→

u νdecays:

dΓ

dEE ¯E ΛΛ

Λ+

Λ.

Fromthisweget

dΓ

2MB (7)dq2

dq0dq2

xdxF(x) 1 MBΛ 1 24q2MB222 MB+q MX22MB+q2 MXdEh=

dΓdq2 x 22 1/2dxF(x)1 1 q/(MB Eh)MB ×(9)2pert

Λ 1 x MB ,q2 .

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

4Perturbativecorrections

Perturbativecorrectionsareparametricallyenhancedinthekinematicsclosetothefree-quarkdecay.Inparticular,inb→u(b→s)transitionsdoubleloge ectsap-pear.Theseenhancedcorrectionsmustbesummedupinthedecayb→s+γwhereq2=0,whichwasdonein[9],includingthee ectofrunningαs.Insemileptonictransitionsthebremsstrahlunge ectsaremuchsoftersincethetypicalcon gura-tioncorrespondstoasigni cantinvariantmassoftheleptonpair.Therefore,weuseinsteadtheexactone-loopradiativecorrectionscalculatedinRef.[11];therunningofαsisthusdiscardedaswell.Itseemsjusti edsinceeveninthecaseofb→s+γthesee ects–althoughveryimportantinpurelyperturbativecalculationsincludingintegrationoverlow-momentumgluons,–aremarginallyseenwhenoneproceedswiththeWilson’sOPEwheretheevolutionofthecoe cientfunctionsande ectivelow-energyparametersisdoneonthesamefooting.Wewilldemonstratealsothattheresultsarefairlyinsensitivetovariationofαs,whichisanotherjusti cation.Ontheotherhand,limitingourselvestotheone-loop xed-αscalculations,andkeepinginmindtheextremelyweaknumericalµ-dependenceoftheobtainedphysicalspectrum,2weformallyputµ=0whichallowsusingtheexpressionsofRef.[11]literally.This,however,necessitatesusingtheperturbativeone-loopvalueoftheb-quarkpolemass

m m16b b(µ)+πµ,(10)

likewiseasimilarsubtractedvalueofµ2π

µ 2π µ2π(µ) 4

πµ2,(11)

etc.(formoredetails,seereview[12]).Weuseαs/π 0.1intheanalysis;atµ≈1GeVitcorrespondstom b

normalizedatthescale0 .5–14.82GeV.GeV,andµ 2πbeingabout0.1GeV2smallerthanµ2

Althoughπtheperturbativedistributionsareknowntoonelooponly,onecaneasilywritetheminexponentiatedform(see,e.g.[9]).Thisistechnicallyconvenientsincetheelasticpeaka ectedbyvirtualcorrectionsdisappearsandonecanthendealwithsmoothperturbativedistributions.OnerelativelysimpleformoftheexponentiatedcorrectionsforthedoubledistributionwassuggestedinEq.(17)ofRef.[13].TosimplifythecalculationsweusedthisformofradiativecorrectionsfordΓ/dMXanddΓ/dEhwheredouble-di erentialdistributionsmustbeused.Thenumericaldi erenceofexponentiationisnegligibleafterincorporatingthee ectoftheprimordialFermimotion.

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

5ResultsandDiscussion

TheresultsofcalculationofthedistributionsareshowninFigs.1–3(dΓ/dE ),Fig.4(dΓ/dMX)andFig.5(dΓ/dEh).Ourbasicsetofparametersism b=4.82GeV,2µ 2π=0.2,0.4and0.6GeVwhichcorrespondto

[α,a,b,c]=[1.5,3,16, 3.3],[0.5,2,0.76, 1.75]and[0.1,1.2,0, 1.1],(12)respectively,intheansatzforF(x)(weusek=1);thevalueofαsissetto0.3.ToillustratedependenceonmbweshowalsotheplotfordΓ/dE form b=

4.73GeV(Fig.2).Figure3showsthedependenceoftheleptonspectrumonαs;itisclearlyratherweak.

Asinthecaseofb→s+γ,thee ectoftheFermimotion–wherepresent–ismorepronouncedthanperturbativemodi cations.Asexpected,itshowsupinamuchsofterwayindΓ/dE comparedtodΓ/dEγinb→s+γ.Thesituationwithb→s+γisincloseranalogywithdΓ/dMX,butevenherethee ectissuppressedduetosigni cantaverageinvariantmassoftheleptonpair,i.e.e ectivelysmallerenergyreleaseandrecoilmomentum.

ApartfromsmearingbyFermimotionofnon-smoothpartsofthepartonicdis-tributions,themostsigni cantnonperturbativee ectindΓ/dMXanddΓ/dEhisassociatedwiththedi erence

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

2ratherthanoverMX–distributionsmodi edinsuchawayareformallyequivalent

totheorderin1/mbexpansionwework.

ThedecaydistributiondΓ/dq2isnotessentiallya ectedbystronginteractions.Atmaximalq2particularlyclosetom2b,nevertheless,thesee ectsblowup.Itisimportanttorealizethatitisnotliterallythee ectofFermimotionthatshapesdΓ/dq2.Forexample,theintegraloverthelarge-q2domainisgovernedbythe avor-dependentexpectationvalueofthesix-quarkoperatorandthuscanbecompletelydi erentindecaysofB±andB0.Thee ectsoriginatingatq2→m2bweredis-cussedindetailin[14].Hereweremindthatstudyingthedi erenceofthedecaydistributionsforB±andB0nearmaximalq2,atsmallMXorEh–ormerelyintheend-pointelectronspectrum–isawaytodirectlymeasurethefour-fermionexpectationvalueswhichareimportantforBphysics.

ThedecaydistributiondΓ/dq2isshowninFig.6;exceptnearthekinematicboundary,thedeviationfromthetree-levelpartonshapeisverysmall.Weob-tainedthisdistributionmerelyaddingknownnonperturbative1/m2e ects[10]tothepartonexpressionscalculatedthroughorderαs[11].

Aninterestingobservationweinferfromtheplotsisthatasigni cantfraction,≈90%,ofthedecayeventsisexpectedtohaveMX<1.87GeV,i.e.liebelowthecharmedstates.Itisincontrastwiththecaseofsemileptonicspectrumwhereonlyasmallfractionofthedecaysproceedtothedomainabovethekinematicalboundforb→ctransitions,andevensmallmeasurementerrorsleadtosizeablebias.Forthedistributionininvariantmass,withalowercuto onMXbetween1.5and1.6GeV(toallowforapossibleleakingofhigher-massstatesduetoexperimentaluncertain-ties),themajorityofdecaysappearsinthelow-MXregion(80%forMX<1.6GeVandabout75%forMX<1.5GeV),andcanbereliablycalculatedtheoretically.Ontheotherhand,thiscanpossiblybedeterminedinexperiment.Thiswouldsuggestawayforatrustworthydeterminationof|Vub|witharelativelygoodaccuracy.2IfameasurementofthedistributionoverMXispossible,ityieldsthepossibilitytodetermineindependentlythehadronicparameters.Inanalogytotheb→s+γdecays,thecenterofgravityofthedistributionissensitivetothebquarkmassanditswidthdeterminesµ2π.Figure4showsareasonablesensitivityofthedistribution

2overMXtoµ2π.

6OnaPossibleImprovementofTheoreticalPre-

dictions

Thepresentedanalysisofthedecaydistributionsisinmanyaspectssimpli ed.Whilethebquarkmassislargeenoughforfullyintegratedcharacteristics,itisnotthecasewhenthedetaileddi erentialpredictionsareattempted.Itisnotthereforeclearthatfurtherre nementsbasedonmoreadvancedtreatmentofperturbativecorrectionsandnonperturbativee ectscanyieldanessentialandtrustworthyim-provement.The rstcandidateforthere nementis,obviously,inclusionofknown

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

1/m2correctionswhicharenotincorporatedintheleading-twiste ectssummedupbytheFermimotion.Theanticipatedscaleofthesee ectswasmentionedabove.Anotherdirectionforre nementsisimprovingtheperturbativedescription.TheimpactoftheperturbativecorrectionsonthedistributionseveninBdecaysappearstobesmallerthanthoseofnonperturbativee ects.Inparticular,weexpectasmalle ectfrominclusionofhigher-orderαs-corrections.Thisapplies,however,onlywhenapropertreatmentoftheinfrareddomainisdone–otherwisetherunningofαs,typicallythedominante ectamongthesecond-ordercorrections,apparentlygeneratese ectswhichblowup.ThisdoesnothappenwhenoneliterallyfollowsWilson’sprocedureofconstructingtheOPE.Experienceshowsthatthisprocedureisnecessaryalreadywhenthesecond-ordercorrectionsinbdecaysareaddressed-theprocedureenforcesthesafeguardfrominconvenienceofthemiracleouscompensationofsigni cante ectscomingfromdi erentsources[15,16].

WhileintroductionoftheIRcuto incalculatingtheperturbativecoe cientfunctionsismoreorlessstraightforwardinusualperturbativecalculations,thisisnotsosimplewhenexponentiationofsoft/collineare ectsisemployed.Amethodap-plicabletotheseproblemswasdescribedin[9];itcombinesafewessentialelements:reproducingexactone-loopand(all-order)BLM-improvedanswerand,simultane-ously,properexponentiationofsingulardouble-logcorrections.Thisfeasiblewayisexpectedtoyieldmorethanenoughaccuracyinevaluatingperturbativecorrections.Whilesuchatreatmentisnecessarywhenoneintendstodetermineunderly-ingparametersoftheheavyquarkexpansionlikembandµ2πwiththeaccuracywhentheirscale-dependenceisessential,weanticipateasmalloverallimpactofthehigher-orderperturbativecorrectionsonvarioushadroniccharacteristicsinvolved,forexample,inmeasuring|Vub|,wellbelowthee ectoftheleadingnonperturbativephenomena.

7Outlook

Oneofthemostimportantpracticalapplicationsoftheanalysisoftheb→u νdistributionsistheextractionof|Vub|.Aswasrealizedlongago[17],thetheoreticallycleanestwaytodetermineitisfromthetotalb→u νwidth.Inparticular[15],

|Vub| 0.00415 BR(B→Xu ν)2 1.55ps2(13)

wheretheoreticaluncertaintiesliewellbelowthelevelwhichisexperimentallyrele-vantnowandinthenearfuture.However,acompletelymodel-independentexper-imentalmeasurementofsuchawidthisembryonicatpresent.

Anotherextremeway,suggestedlongago,wastoconsiderdecayeventswith

22E >～(MB MD)/2MB=2.31GeV,wherecharmdecayscannotcontribute.Thedecayrateinthistoonarrowsliceofkinematics,ontheotherhand,dependsonpoorlyknowndetailsofstrongdynamics.Withinourcalculationthisfractionofall

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

decayeventsdoesnotvarytoosigni cantly;nevertheless,moreworkisneededtoputthisobservationonamore rmlygroundedquantitativebasis.

Therearemanyintermediateoptionswhoseadvantagesmainlydependonexper-iment.Thelargerthekinematicdomain,themoreitencompassestheparton-levelkinematics,thebetter,ingeneral,isthetheoreticaldescriptionofsuchaninclusivewidth.WefoundthatthedistributionoverMXstudiedinexperimentisratherpromisinginthisrespect.Asimilarsuggestionwasmadetolookatthedistri-butionoverEh[13].Itisclearthatanaccuratedeterminationof|Vub|requiresadetailedanalysisofthekinematicsofthedecayevents,whichwouldultimatelyleadtodeterminationofjustthedouble-di erentialdistributionsoverq0andq2.Whichparticularintegralappearstobemostsuitableforextracting|Vub|willeventuallydependontheavailableexperimentaltechnique.

ACKNOWLEDGMENTS:WearegratefultoP.HenrardandhisALEPHcollaboratorsforsuggestingtoustheextensionofourb→s+γconsiderationstothecalculationofthesemileptonicb→udistributions,usefuldiscussions,criticalcommentsontheresultsandincorporatingthemintheanalysisoftheALEPHdata

[18].WearethankfultoI.BigiandM.Shifmanforstimulatingcommentsandtheirencouraginginterest.N.U.thankstheCERNTheoryDivisionforkindhospitalityduringtheperiodwhenthemaincalculationsofthisworkweredone.ThisworkwassupportedinpartbyDOEunderthegrantnumberDE-FG02-94ER40823andbyNSFunderthegrantnumberPHY92-13313.

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We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

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We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

0.8

0.6

0.4

0.2

Figure1:TheelectronenergydistributiondΓsl(b→u)/dEl,arbitraryunits.Long-2dashed,solid,andshort-dashedlinescorrespondtoµ 2π=0.2,0.4and0.6GeV.Theb-quarkmassm b=4.82GeV,αs=0.3.0.8

0.6

0.4

0.2

0.52.5E_l, GeV

Figure2:Dependenceof1/Γsl(b→u)dΓ/dElonmb.Thesolidlinecorresponds2tomb=4.82GeV,andthedashedlineisformb=4.72GeV,whileµ 2π=0.4GeViskept xed.

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

0.6

0.4

0.2

Figure3:E ectofradiativecorrectionsondΓ/dEl:solidlineshowsαs=0.3and2dashedlineisαs=0.Herem b=4.82GeVandµ 2π=0.4GeV.0.8

0.6

0.4

0.2

0.52.53

Figure4:TheinvariantmassdistributiondΓ/dMx;long-dashed,solid,andshort-2dashedlinescorrespondtoµ 2 b=π=0.2,0.4and0.6GeV.Theb-quarkmassm

4.82GeV,αs=0.3.Alldistributionsarenormalizedtothesametotalwidth.

We present theoretical predictions for a few phenomenologically interesting distributions in the semileptonic b->u decays which are affected by Fermi motion. The perturbative effects are incorporated at the one-loop level and appear to be very moderate. Ou

0.7

0.60.5

0.4

0.3

0.2

Figure5:ThehadronicenergydistributiondΓ/dEhinthesamesettingasinFigs.1and4.

2

1.5

1

0.5

Figure6:Theq2-distributiondΓ/dq2ismadewiththesameconventionsasforFigs.1and4.

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