Factorization of heavy-to-light form factors in soft-collinear effective theory

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

PITHA03/11

CERN-TH/2003-286

hep-ph/0311335

November25,2003

arXiv:hep-ph/0311335v1 26 Nov 2003Factorizationofheavy-to-lightformfactorsinsoft-collineare ectivetheoryM.BenekeaandTh.FeldmannbaInstitutf¨urTheoretischePhysikE,RWTHAachenD–52056Aachen,GermanybCERN,TheoryDivision,CH-1211Geneva,SwitzerlandAbstractHeavy-to-lighttransitionformfactorsatlargerecoilenergyofthelightmesonhavebeenconjecturedtoobeyafactorizationformula,wherethesetofformfactorsisreducedtoasmallernumberofuniversalformfactorsuptohard-scattering

corrections.Inthispaperweextendourpreviousinvestigationofheavy-to-lightcurrentsinsoft-collineare ectivetheoryto nalstateswithinvariantmassΛ2asisappropriatetoexclusiveBmesondecays.Thee ectivetheorycontainssoftmodesandtwocollinearmodeswithvirtualitiesofordermbΛ(‘hard-collinear’)andΛ2.Integratingoutthehard-collinearmodesresultsinthehardspectator-scatteringcontributionstoexclusiveBdecays.Wediscusstherepresentationofheavy-to-lightcurrentsinthee ectivetheoryafterintegratingoutthehard-collinearscale,andshowthatthepreviouslyconjecturedfactorizationformulaisvalidtoallordersinperturbationtheory.Thenaivefactorizationofmatrixelementsinthee ectivetheoryintocollinearandsoftmatrixelementsmaybeinvalidatedbydivergencesinconvolutionintegrals.Inthefactorizationproofwecircumventtheexplicitregularizationofendpointdivergencesbyade nitionoftheuniversalformfactorsthatincludeshard-collinear,collinearandsofte ects.

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

1Introduction

ThepurposeofthispaperistodevelopfurtherthetheoryofexclusiveBdecaystolightenergeticmesons.Wearespeci callyinterestedin“hard-spectatorscattering”,i.e.scatter-¯meson.ThesoftexternalquarkingmechanismsthatinvolvethelightantiquarkintheB

lineisoneofthecrucialdi erencestothestandardsituationofhardexclusiveprocesses

[1]involvingonlylighthadrons,whereallexternallinescarrylargemomentum.Aconse-quenceofthisdi erenceisthatexclusiveBmesondecaysinvolvetwohardscales,m2bandmbΛ,whereΛisoforderofthestronginteractionscale.

Hard-spectatorscatteringisanimportantingredientinQCDfactorizationfornon-leptonicBdecaystocharmless nalstates[2,3],andisevenmoreimportantintheso-calledPQCDapproach[4].Abetterunderstandingofspectatorinteractionsisneededtojustifythefactorizationhypothesesofthetwoapproachestoallordersinperturbationtheoryandtoleadingorderintheheavy-quarklimit.However,evenapparentlysimplerprocessessuchasthesemi-leptonicdecayB→πlνatlargemomentumtransfertothepionarecurrentlynotcompletelyunderstood.TheoneexceptionisB→γlν,whichhasreceivedmuchattentionrecently[5,6,7,8].AfactorizationformulaoftheschematicformF=T φB,wherethestar-productdenotesconvolutionofahard-scatteringkernelwiththeBmesonlight-conedistributionamplitude,hasnowbeenshowntobevalidtoallordersinperturbationtheory[7,8].InthispaperweconsiderB→πlν(heavy-to-lightformfactors)atlargepionenergy.Asummaryoftheresultshasalreadybeengivenin[9].AstraightforwardextensionoftheresultsforB→γlνdecaytotheB→πformfactorsrelevanttosemi-leptonicdecaysfails.Ifonewritestheformfactorsasφπ T φBinanalogytotheπ→πtransitionformfactoratlargemomentumtransfer,one ndsthattheconvolutionintegralsdonotconvergeattheirendpoints.Inotherwords,theformfactorsreceiveleadingcontributionsfrommomentumcon gurationswheresomepartonsinthepionappeartohavesmallmomentum[10,11].In[12]thefactorizationformula

Fi=Ciξπ+φB Ti φπ(1)

hasbeenconjecturedforthethreeLorentzinvariantB→πformfactorsandshowntobevalidatorderαs.TheadditionaltermCiξπinvolvesashort-distancecoe cientandasingle“softformfactor”ξπ,whichobeysthelarge-recoilsymmetries[13].FactorizationfortheB→πformfactorsismorecomplicatedthanforboth,theπ→πandB→Dformfactors.Atlargemomentumtransfersoftinteractionscancelintheπ→πtransitionatleadingpower.Theremaininghardandcollineare ectsarefactoredintoconvolutionsasinthesecondtermontheright-handsideof(1).Whenbothmesonsareheavy,suchasinB→D,collineare ectsareirrelevant.Theremaininghardandsoftinteractionsfactorintoashort-distancecoe cientandtheIsgur-Wiseformfactor[14]similartothe rsttermontheright-handsideof(1).TheB→πformfactor,however,involveshard,collinearandsofte ects.Furthermore,duetothepresenceofseveralscalesonemustdistinguishshort-andlong-distancecollineare ects,aswediscussinmoredetailbelow.

Aseparationofallthesee ectsandanoperatorde nitionofthevariousshort-andlong-distancequantitiesin(1)toallordersinαsandtoleadingorderin1/mbhasnot

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Table1:TerminologyforthevariousmomentummodesrelevanttoexclusiveBde-

cays.Themomentumcomponentsaregivenas(n+p,p⊥,n p),butmassdimension

hastoberestoredbymultiplyingappropriatefactorsofmb.Twodi erenttermi-

nologiesforthesamemomentummodesareusedintheliterature.Inphysicalunits

λisoforder(Λ/mb)1/2,whereΛisthestrong-interactionscale.

MomentumscalingTerminologyII

([20],thiswork)

(1,1,1)

soft

collinear

(λ2,λ2,λ2)

(1,λ2,λ4)hardsoftcollinear

yetbeenachieved.Acomplicationwaspointedoutin[15],whereitwasshownthatsuper ciallysub-leadinginteractionsin1/mbcontributetotheB→πformfactorsatleadingpower.Someofthepreviousarguments[16,17]tojustifyorextendsomeaspectsof(1)mustthereforeberevised.Theformfactorshavebeenreconsideredin[15]intheframeworkofsoft-collineare ective eldtheory(SCET)[16,18]andthestructureoftheformula(1)wasseentoemerge.However,in[15]atechnicalde nitionof“factorizable”and“non-factorizable”termshasbeenadoptedthatdoesnotcorrespondtotheusualnotions,sothattheissuesofendpointsingularitiesandconvergenceofconvolutionintegralscouldnotbeclari ed.BelowweextendSCETintheposition-spaceformulation[17,19]tocoverthecaseofexclusivedecays,wheretheexternalcollinearlineshaveinvariantmassoforderΛ2asappropriateforexclusivedecays.Toobtainthefactorizationformula(1)wematchSCETtoane ectivetheoryfromwhichshort-distancecollinearmodeswithvirtualitymbΛareremoved.Thispointwas rstaddressedin[20]andtheformalismhasbeenworkedouttotheextentthatafactorizationtheoremforB→γlνwasestablished[8].¯(p) ,Themomentummodesrelevanttothefactorizationofformfactors M(p′)|q¯Γb|BwhereMisalightmesonoraphotonwithmomentumofordermb,andΓisaDiracmatrix,aresummarizedinTable1.Ingeneralwedecomposeamomentumas

p=(n+p)µnµ

2,(2)

22wherenµ±aretwolight-likevectors,n+=n =0withn+n =2.Thereferencedirections

n±arechosensuchthattheenergeticmasslessexternallineshaven+pofordermb.AsindicatedinTable1twodi erentterminologieshavebeenusedintheliteraturewhichhas

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

beenthecauseofsomeconfusion.Sinceinthispaperwewillconstructane ectivetheoryformodeswithvirtualityΛ2only,wewillusethesecondterminology.Thee ectivetheorythencontainssoftandcollinearmodesinagreementwiththestandardQCDterminology.Forpowercountingwede nethescalingparameterλtobeoforder(Λ/mb)1/2.Thisdi ersfromtheconventionin[20]whereλisoforderΛ/mb.

TheexistenceofthevariousmodesfollowsfromtheassumptionthattheexternalmomentaofscatteringamplitudesforexclusiveBdecaysatlargemomentumtransfer

2aresoftorcollinear.1One ndsthethreecharacteristicvirtualitiesm2b,mbΛandΛby

combiningexternalmomenta.Forinstance,m2bisobtainedbyaddingandsquaringaheavyquarkandacollinearmomentum,orbysquaringtheheavyquarkmomentum.Theintermediatevirtualityistypicalforinteractionsofcollineargluonsorlightquarkswithsoftgluonsorlightquarks,whileΛ2arisesintheself-interactionsofcollinearorsoftmodes.Soft-collineare ectivetheoryasde nedin[16,17]isthee ectivetheoryobtainedafterintegratingouthardmodesofvirtualitym2b.Thistheorystillcontainstwotypesofsoftmodes,called“semi-hard”(virtualityofordermbΛ)and“soft”.Thesemi-hardmodescanbeintegratedoutperturbatively,butitappearsthatsemi-hardloopintegralsalwaysvanishindimensionalregularization[17],sosemi-hardmodescanbeignoredinpractice.Thetheoryalsocontainstwotypesofcollinearmodes,called“hard-collinear”and“collinear”accordingtotheirvirtuality.AlthougheachoneofthesetwohasbeendiscussedinpreviousapplicationsofSCET,thesimultaneouspresenceoftwodistinctcollinearmodeshasnotbeenconsideredinmuchdetailuptonow.ThereasonforthisisthatpreviousapplicationsofSCETtosemi-inclusiveBdecay,suchasB→Xsγnearmaximalphotonenergy[18],andtoB→γlν(atleadingorderin1/mb)aresensitiveonlytohard-collinearmodes[7,8].OnecanthereforematchSCETdirectlytothestandardheavyquarke ectivetheory,whichcontainsonlysoftmodes.Ontheotherhand,intheexclusivedecayB→Dπ[3,21]orhardexclusivescatteringoflighthadrons[22]thee ectsofhard-collinearandsoftmodescancelalmosttriviallyatleadingpowerinthepowerexpansion.Thee ectivetheoryatleadingpowercanthenbeformulatedentirelyintermsofcollinearmodes.

Theoutlineofthispaperisasfollows:inSection2westudyascalarintegral,whichwouldberelevanttoB→γlνdecayatsub-leadingorderin1/mb,usingthemethodofexpandingbyregions[23].Wedemonstratewiththisexamplethatseparatecontributionsfromhard-collinearandcollinearloopmomentummustbeincludedtoreproducethein-tegralinfull“QCD”.Weshallalso ndthatthecollinearandsoftcontributionsarenotwell-de nedindividuallyindimensionalregularization.Theinterpretationofthesead-ditionaldivergencesprovidesanimportantcluetotheproblemofendpointdivergences.Inthecontextofe ective eldtheorytheadditionaldivergencesshowthatthematrixelementsinthee ectivetheoryofsoftandcollinear eldsdonotfactorize(naively)intoaproductofsoftandcollinearmatrixelementsasonemighthaveconcludedfromthedecouplingofsoftandcollinear eldsintheLagrangian.

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

InSection3weturnourattentiontotherepresentationoftheheavy-to-lightcurrentinthee ectivetheorywiththehard-collinearscaleremoved.Weintegrateoutthehard-collinearmodesintreegraphsbysolvingtheclassical eldequationsforthehard-collinearquarkandgluon elds.Despitethecomplexbranchingsoftherelevanttrees,thesolu-tioncanbefoundbychoosingaspecialgaugeforthecalculationandreconstructingthe¯ completeresultthroughgaugeinvariance.The rsttermwithanon-vanishing π|...|B

3/2matrixelementintheexpansionofthecurrentisλ3suppressed,whichexplainsthe1/mbsuppressionofheavy-to-lightformfactorsatlargerecoil.Thecalculationalsoshowsthat

thee ectiveoperatorishighlynon-local,implyingconvolutionsintwolight-likedirections.Theconvolutionsaredivergent,asexpected,butwealso ndthatquark-antiquark-gluonamplitudesintheBmesonandinthelightmesoncontributeatleadingpower,whichisanewfeatureofheavy-to-lighttransitions.Attheendofthissectionwebrie ydis-cusshard-collinearquantumcorrectionstodeterminethegeneralformofoperatorsandshort-distancekernelsinthee ectivetheoryofsoftandcollinearmodes.

Theexistenceofdivergentconvolutionssignalsthatheavy-to-lightformfactorsdonotfactorizestraightforwardly.InSection4wereturntothefactorizationformula(1),andshowthatitisindeedvalidtoallordersinthestrongcouplingandtoleadingpowerin1/mb.Weshallde netheuniversalformfactorξπasamatrixelementinSCETbeforeintegratingouthard-collineare ects.Wethenshowthatthetermsnotcontainedinthisde nitionfactorizeintoconvolutionsoflight-conedistributionamplitudeswithconvergentintegralsafterintegratingouthard-collinearmodes.Theproofofconvergencereliesonpowercounting,boostinvariance,andthecorrespondenceofcollinearandsoftendpointdivergencesthroughsoft-collinearfactorization.WeconcludeinSection5.

2Thescalar“photon”vertex

Thepurposeofthissectionistodemonstratebytheexampleofaspeci cFeynmaninte-gralthatthedistinctionofhard-collinearandcollinearmodeshasatechnicalmeaning.Weshallalsoseehowthefactorizationofcollinearandsoftmodesintroduces“endpointsingu-larities”inthelongitudinalintegrations,andhowthesingularitiesarerelatedtocancelinthesumofallterms.Finally,wesketchhowthediagrammaticresultwouldbeinterpretedinthecontextofe ective eldtheory.

Weconsiderthescalarintegral

I= [dk]1

i µ2eγE(2π)d=µe2 γEddk

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

¯→γlν.Γdenotestheweakb→udecayFigure1:PhotonvertexcorrectiontoB

vertex.Weconsiderthecorrespondingvertexintegralwithalllinessimpli edto

scalarpropagatorsandallvertexfactorssetto1.

TheintegralIisultravioletandinfrared nite,butdimensionalregularizationwillbeneededtoconstructtheexpansion.The+i prescriptionforthepropagatorsisunderstood.

Theexternalmomentaofthevertexsubgraphare:acollinear“photon”momentum′p=(n+p′,p′⊥,n p′)～(1,0,0)withp′2=0;asoft“lightquark”momentuml～(λ2,λ2,λ2)withl2=m2;ahard-collinear“lightquark”momentump′ l～(1,λ2,λ2)withvirtualityλ2.Thetwoinvariantsare2p′·l～λ2andm2～λ4,soImustbeafunctionofthesmalldimensionlessratiom2/(2p′·l).Astraightforwardcalculationgives

I=1

m2 π2

2p′·l 1

2p′·l+π2

whichvanishindimensionalregularization,sincetheonlypossibleinvariantp′2=0.Thisisnotsurprising,becausethereisnoexternalinvariantoforder1.Proceedingfordi erent

[k2]a[ 2p′·k]b×polynomial,(6)

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

n,theresultisthatonlythesoftmomentumregion,n=2,contributes.Thecorrespondingintegralwillbecalculatedbelow.

Sincethereareexternallineswithlargemomentumandsmallvirtuality,weshouldalsoconsiderloopmomentumcon gurations,wheren+kisthelargestcomponent.Thatis,wetaken+k～λnandk2～λ2mwithm<n,expandtheintegrand,anddeterminetheintegralsthatdonotvanish.2Theresultistwonon-vanishingcontributions,onefromn+k～1andk2～λ2,whichweidentifyashard-collinear,andtheotherfromn+k～1andk2～λ4,whichwecallcollinear,seeTable1.Regionswithk2<λ4donotappearduetotheinternalmassesm2～λ4.Wewillnowverify(atleadingorder)thatthesumofthethreeregionsconstructstheexpansionofourintegralI.

Thehard-collinearregion.Expandingthepropagatorssystematicallytheleadinghard-collinearintegralis

Ihc= [dk]

1

1

2p·l′2p′·lln2ln22p·l′

12 .(7)

Theexpansionhasrenderedtheintegralinfrareddivergent.Ifweperformthen kinte-grationbycontourmethods,thek⊥integralisdivergentfork⊥→0(physically,k⊥ λ)foranyn+k,butthen+kintegralconvergesat xedk⊥.Thedoublepoleoriginatesfromn±k→0,k⊥→0simultaneously.

Thecollinearregion.Inthisregionthe“lightquark”propagatorswithmomentap′ kand karecollinearandhavevirtualityoforderλ4.The“gluon”propagatorishard-collinearwithvirtualityλ2.One ndsthatthecollinearandsoftintegralsarenotwell-de nedseparatelyindimensionalregularization.Thisalsooccurredinpreviousapplicationsofthemethodofexpansionbyregionstocollinearintegrals[24],andisrelatedtothefactthedimensionalregulatorisattachedtothetransversemomentumcomponents.Ifadditionaldivergencesarisefromthen+korn kintegrations,theymaynotberegularized.Asin

[24]weintroduceanadditional“analytic”regularizationbysubstituting

1

[(k l)2]1+δ,(8)

whereνisaparameterwithmassdimensionone.Theleadingcollinearintegralis

Ic=

2 [dk][ ν2]δSomeintegralsvanishindependentofanyregularization,becauseallpoleslieinoneofthecomplexhalf-planes.Otherintegralsvanishonly,becauseweassumearegularizationthatdoesnotintroduceanadditionalscaleintotheintegral.

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Theintegralcanbedonewithstandardmethods,butitwillbeusefultoobtainanintermediateresult,whereonlythen kintegrationisperformed.Thevariablen+p′isrelatedtotheenergyofthe“photon”,son+p′>0.Wethenclosethecontourinthelower

2halfplaneandpickupthepoleat( k⊥+m2 i )/n+kfor0<n+k<n+p′.(Inour

2conventionk⊥isnegative.)Thisgives

Ic=

= 12p′1

δ·l δ n+p′0dn+kn+k ln δ µe2γE dd 2k⊥2k⊥ m2+ln2p′·lm2

Hereweperformthen+kintegral rst.Assumingn l>0,weclosethecontourinthe

2lowerhalfplaneandpickupthepoleat( k⊥+m2 i )/n kfor0<n k<n l.This

gives

Is= 1

n k

1

m2 [(k l)2]1+δ[k2 m2][ n+p′n k].(11)n lπd/2 1 ν2δn l= Γ( )m2

δΓ(δ+ )Γ(1 2δ 2 ) 221+δ k⊥)

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Thereisasingularityfork⊥→∞foranyn k.Thepoleatδ=0isanendpointdivergencefromn k→0foranyk⊥.Thisimpliesthatn+kbecomeslargefor xedk⊥～λ2,andhencethe“quark”withmomentumkbecomescollinear.Inthesoftregionthetransversemomentumandlongitudinalmomentumintegralsdonotfactorize,andthereisalsoadivergencewhenk⊥→∞andn k→0simultaneously,whichcorrespondstothedoublepoleinthehard-collinearintegral.

Sincewedidnotregularizethehard-collinearcontributionanalytically,thecorrectprocedureistoexpand rstinδandthenin .Infactthepoleinδcancelswiththecollinearcontributionbeforeexpandingin .However,performingbothexpansionstocomparewith(10)weobtain

Is= 1

δ lnm2

lnm2

2+1

µ2 1

µ2+5π2

2p′·l 1

ln2p′·lµ2lnm2

2ln2m2

12 .(14)

Finallyaddingtothisthehard-collinearcontribution(7),thesingularityin alsocancels,andweobtain ′2p·l1,(15)ln2Ic+Is+Ihc= 23

inagreementwiththeexpansion(5)ofthefullintegral.Weconcludethatingeneralhard-collinear,collinearandsoftmomentumregionsmustbeconsidered.Inthescalarintegral

(3)allthreeregionscontributealreadytotheleadingtermintheexpansion.Semi-hardmodeswithscaling(λ,λ,λ)arenotneededinthiscalculation,sincethecorrespondingintegralsarescaleless.

InQCDthephoton-vertexintegralcontainsanumeratorproportionalton kwhichsuppressesthecollinearregionbyafactorofλ2relativetothehard-collinearandsoftregion.Forthisreasonitissu cienttoconsideronlyhard-collinearandsoftcon gurationsinthefactorizationtheoremforB→γlνatleadingpowerin1/mb,ashasbeendonein

[6,7,8].Hard-collinearmodesareperturbativeandcanbeintegratedout,resultinginhard-scatteringkernels.Softandcollinearmodeshavevirtualityλ4～Λ2,andcannotbetreatedinperturbationtheory.The1/mbsuppressionofthecollinearcontributioninQCDimpliesthatthehadronicstructureofthephotonisasub-leadinge ectinB→γlνdecay.

2.2O -shellregularization

Thescalarintegral(10)hasrecentlybeendiscussedin[25],howeverwithm=0andtheexternalcollinearandsoftlineso -shell,l2≡ L2=n+ln l～λ4,and(p′)2≡ (P′)2=n+p′n p′～λ4.Itisinstructivetodiscussthedi erencetothecaseabove.

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Thehard-collinearintegralsareidenticalastheyshouldbe,becausethetwointegralsdi eronlyatthesmallscaleλ4.Thecollinearcontributionisnowgivenby

′Ic= 1

2p′·l δ 1

0duπd/2 1 1

Itisverylikelythatwithmoreloops,moremodesofsuccessivelysmallervirtualitymustbeintroduced,withnolowerlimitonthevirtualityasthenumberofloopsincreases.3

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Figure2:Divergencestructureoftheintegral(3)anditso -shell,masslessver-

sionwhenexpandedbyregions.Thearrowsindicatethedivergencesindi erent

regionsthatarerelatedandcanceleachother.Thedashedlinesmarkthevarious

factorizationsteps.

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

occursatweakcoupling.Ifitdoesnot,factorizationofsoftandcollinearmodescannotbeimplementedperturbativelyinQCD.Ifendpointdivergencescanbeshowntobeabsenttoallordersinperturbationtheory,sothatnoregularizationisrequired,thisindicatesthatnaivesoft-collinearfactorizationisvalid.

2.3Interpretationoftheresult

Hard-collinearmodesinSCET.Thetoyintegralclari esthatSCET,de nedasthee ec-tivetheoryafterintegratingouthardmodes,containstwocollinearmodeswithdi erentvirtuality.IfSCETisformulatedwithasinglecollinearquarkandgluon eld,thecollinear eldscannotbeassignedanunambiguousscalinglaw,andpowercountingisnolongermanifestintheverticesofthee ectivetheory,4unlessoneofthetwocollinearmodesisirrelevantforaspeci cprocess.Analternativeistointroduceseparatehard-collinearandcollinear eldsinSCET.ThecorrespondingLagrangiancanbetakenasastartingpointforthesecondmatchingstep,inwhichhard-collinearmodesareintegratedoutandSCETisreducedtoane ectiveLagrangianforsoftandcollinearmodesonly.Thisformulationwillbeusedinsomeofthetechnicalstepsinthefollowingtwosections.

Acommentisnecessaryonthetransversemomentumscalingofhard-collinearmodes.Whenasoftandcollinearmomentumcombinetoahard-collinear uctuationofvirtualityλ2,thehard-collineartransversemomentummustbeoforderλ2bymomentumconser-vation.Thisisthecaseforthehard-collinearpropagatorsinthesoftandcollinearcon-tributionstothetoyintegral(seetheupperrowofFigure3).However,inhard-collinearloopsthetransversemomentumisoforderλ,asonecaneasilyverifyfromthelocationofpolesofthehard-collinearintegrand.Assumingk⊥～λ2wouldmakeallhard-collinear

2loopintegralsvanish,sincetheintegralswouldhavetobeexpandedink⊥.Thiswould

obviouslyfailtoreproducetheexpansionoftheexactintegral.Wethereforeassignthegenericscaling(1,λ,λ2)tohard-collinearmodes,asgiveninTable1.Non-genericscalingintreesubgraphsisnotparticulartothepresentcaseofhard-collinearmodes.Whenoneintegratesouthardheavyquark uctuationsgeneratedbytheinteractionofnearon-shellheavyquarkswithhard-collinearorcollineargluons[16],theo -shellmodeshavemomentum(1,λ,1)or(1,λ2,1),unlikethegenerichardmomentum(1,1,1).

Operatorinterpretationofthetoyintegral.Weproceedtodiscussthethreecontributionstothetoyintegralintermsofoperatorsandmatrixelementsofane ectivetheoryforsoftandcollinearmodes.Thisdiscussionwillbeheuristic,sinceweabstractfromthescalarintegralanduseQCDterminology,butwithoutmakingthenotationcompletelyexplicit.WeimaginethatFigure1representsacorrectiontothematrixelement γ|J|q¯b oftheb→utransitioncurrentbetweenaq¯bstatewith xedlightquarkmomentumn landaphotonwithlargeenergyE=n+p′/2.Thecorrespondingtreediagramhasonehard-collinearlinejoiningtheweakvertextothephotonvertex.Inthee ectivetheory(of

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Figure3:Diagrammaticandoperator/matrixelementrepresentationofthehard-

collinear(leftcolumn),soft(middlecolumn)andcollinearcontributiontothedia-

gramofFigure1.Eachcolumnshows:theoriginaldiagramwiththehard-collinear

subgraphmarkedbybold-facelines(upperrow),andwiththedashedlineindicating

wherethegraphfactorizesintoashort-distanceandlong-distancesubgraph;theop-

eratorvertexinthee ectivetheorycorrespondingtothecontractedhard-collinear

subgraph(middlerow);thecontributiontotheoperatormatrixelement γ|Oi|q¯b correspondingtotheoriginaldiagram(lowerrow).

softandcollinearmodes)thisiswrittenas

C0(E,n l)FT γ|[Aγ(sn+)]c[¯q(tn )hv(0)]s|q¯b (E,n l).(17)

ThesymbolFT ... meansthataFouriertransformofthematrixelementwithrespecttothepositionargumentsofthe eldsistaken,with(E,n l)thevariablesconjugateto(sn+,tn ).Theindexonproductsof eldsindicateswhethertheyaresoftorcollinear,andthenon-localityoftheoperatorisrelatedtothenon-polynomialdependenceofthehard-collinearpropagatoronthemomentumcomponentn lofthelightexternalquark,andthemomentumcomponentn+p′oftheexternalphoton.Thematrixelementfactorizestriviallyinto

q(tn )hv(0)]s|q¯b (n l).(18)FT γ|[Aγ(sn+)]c|0 (E)FT 0|[¯¯meson,Thephotonmatrixelementcanbecalculated.Whentheq¯bstateisreplacedbyaB

thesoftmatrixelementgivestheBmesonlight-conedistributionamplitude.Hence(17)assumestheformofaconvolutionofahard-collinearcoe cientfunctionwiththeBmesonlight-conedistributionfunction,whichreproducesthefactorizationpropertyoftheB→γtransitionatleadingorderin1/mb,andatleadingorderinαs[5].

Thehard-collinearcontributiontothetoyintegralanditsoperatorinterpretationisshownintheleftcolumnofFigure3.Whenthehard-collinearsubgraphiscontractedtoa“point”,thecorrespondingoperatorhasthesame eldcontentasin(17),butwitha

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

di erentcoe cientfunctionC1(E,n l)asaresultoftheloopintegration.WeidentifyC1asa1-loopcorrectiontothehard-scatteringkernel.Theexplicitcalculationshowsthatthereisadoublepolein1/ ,leavingadouble-logarithmicdependenceonthefactorizationscaleµ.

Considernowthesoftcontribution(middlecolumnintheFigure).Thehard-collinearsubgraphhasanadditionalexternalsoftgluonline,sotheoperatorinthee ectivetheoryhasthestructure[¯qAhv]s[Aγ]c(secondlineintheFigure).ThematrixelementinthethirdlineofFigure3takestheform

FT γ|[Aγ(sn+)]c|0 (E) dωC2(E,n l,ω)FT 0|[¯q(t2n )A(t1n )hv(0)]s|q¯b (n l,ω).(19)Thesoftmatrixelementcanbeidenti edwithathree-particlelight-conedistributionamplitudeφq¯paringthisexpressionto(12),weseethatthen k¯bgoftheq

integralin(12)correspondstotheintegrationoverω,whilethetransversemomentumintegralisincludedinthede nitionofthelight-conedistributionamplitude.

Intheconventionalhard-scatteringformalismthescaledependenceofthedistributionamplitudewouldcancelagainstthescaledependenceofahard-scatteringkernel(suchasC1).Thiscannotbecompletelycorrecthere,sincetheω-integralhasanendpointdivergenceasω→0,whichcorrespondstothe1/δsingularityin(12).Theassociatedν-dependenceisnotcancelledbyahard-scatteringkernel,butbythecollinearcontributionasseenfromthetoyexample.Theexistenceofanendpointdivergenceimpliesthatexpression

(19)initsentiretyhasascale-dependencedi erentfromthetwomatrixelementsinthefactorizedexpression.Thispossibilityisnotconsideredintheconventionalhard-scatteringformalism.5

TheoperatorinterpretationofthecollinearintegralisillustratedinthethirdcolumnofFigure3.Thephotonlineisnotdirectlyconnectedtothehard-collinearsubgraphinthiscase.Rathertheoperatorthatresultsaftercontractingthehard-collinearsubgraphhas eldcontent[¯qhv]s[¯qq]c(secondlineintheFigure).Thematrixelement(thirdline)canbewrittenas

q(tn )hv(0)]s|q¯b (n l)FT 0|[¯ 1

0duC3(E,u,n l)FT γ|[¯q(s1n+)q(s2n+)]c|0 (E,u).(20)

Thisseemstorepresenttheconvolutionofahard-scatteringkernelC3withthetwo-particlelight-conedistributionamplitudeoftheq¯bstateφq¯light-conedistribution¯bandtheqq

amplitudeofthephotonφγqq¯.Thisisonlycorrectwiththeunderstandingthattheu-integralisdivergentandmustberegularizedinawaythatisconsistentwiththeregularizationoftheω-integralinthesoftcontribution.Theadditionaldivergence,whichisnotrelatedto

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

therenormalizationoftheconventionallight-conedistributionamplitudes,istheendpointdivergenceofthen+kintegralin(10).Theassociatedν-dependencecancelsagainsttheν-dependenceofthesoftcontribution.Ingeneral,thedistributionamplitudesmaythemselvesdependontheadditionalregularization,andhencedi erfromthedistributionamplitudesthatappearinthehard-scatteringformalism.

Tosummarizethisdiscussion,wedistinguishtwostepsoffactorization.Inthe rststep,weintegrateoutthelarge-virtualityhard-collinearmodes,andrepresenttheresultintermsofoperatorsofsoftandcollinear elds.Theseoperatorswillbenon-local,re ectingthefactthatthehard-scatteringkernelsappearinconvolutionsratherthanasmultiplicativefactors.Thesecondfactorizationstepreferstotheseparationofsoftandcollinearmodeswithinthee ectivetheoryofsoftandcollinearmodes.Inourexample,thephotoncouplesonlytocollinearlines,andtheq¯bstatecouplesonlytosoftlines,sowewouldexpectthee ectivetheorymatrixelementstofactorizeintoamatrixelementofcollinear eldsbetweenthephotonandthevacuum,andamatrixelementofsoft eldsbetweenthevacuumandtheq¯bstate.Ifthiswerethecase,theprocesswouldfactorizeintoS T Φ.ThefactorizationscaledependenceofthesoftfactorSandofthecollinearfactorΦwouldcancelseparatelywiththatofthehard-scatteringkernelT,butthesoftandcollinearfactorswouldbeunrelated.Theendpointdivergencespreventsuchacompletefactorization.Forourtoyexamplewe ndinsteadafactorizationformulathattakestheschematicform

γ|J|q¯b =(C0+C1) φq¯b+C2 φq¯bg ν+

The rsttermontheright-handsiderepresentsadirectphotoncontribution;inthethirdtermthepartonicstructureofthephotonisresolved.Thesquarebracketsindicatetheadditionalscale-dependenceintroducedbytheendpointdivergences,whichconnectthesecondwiththethirdterm.Ifthescaleνischosensuchthatthethirdtermcontainsnolargelogarithmrelatedtotheendpointdivergence,wecaninterpretitasaendpoint-subtractedhard-scatteringcontributiontotheq¯b→γtransition.Forourtoyintegral,

2(10,13)showthatthiscorrespondstotakingνoforder2p′·l.Thecorrespondingendpointlogarithmthenresidesinthesecondterm,whichwemaycallthe“softoverlap”contribution

(sinceasoftlineconnectstheinitialstatewiththephotonasseenfromthemiddlecolumnofFigure3).Thetwotermsarerelatedviatheirν-dependence,suchthatthesumisindependentoftheimplementationofsoft-collinearor“endpoint”factorization.AsimilarstructureisexpectedfortheB→πformfactor[12]. φγqq¯ C3 ν φq¯b.(21)

3Heavy-to-lighttransitionsinSCET(c,s)

¯ΓQisob-Thee ectivetheoryrepresentationoftheheavy-to-lighttransitioncurrentsψ

tainedintwosteps: rstthehardmodesareintegratedout,andthecurrentisdescribedinsoft-collineare ectivetheoryincludinghard-collinearmodes.Weshalldenotethisthe-orybySCET(hc,c,s)(alsocalledSCETIintheliterature[15]).Thisstep,inwhichitisnotnecessarytodistinguishhard-collinearandcollinear,hasalreadybeendiscussedin

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Figure 4: Kinematics of an exclusive heavy-to-light transition in SCET(c,s). The heavy quark and the soft pa

rtons in the B meson must be converted into a cluster of collinear partons.[16, 17]. We will be mainly concerned with the second matching step, in which the hardcollinear modes are integrated out and the transition current is nally represented in terms of operators constructed only from soft and collinear elds. We refer to the theory of soft and collinear elds as SCET(c,s) (also called SCETII ). The kinematics of a heavy-to-light transition is illustrated in Figure 4. In contrast to[17] the invariant mass of the nal state is restricted to orderλ4～Λ2 as appropriate to an exclusive decay. This implies that the nal state must now consist only of collinear lines, since the addition of a soft line would increase the virtuality toλ2 . The initial state is described by a heavy quark and soft lines with total invariant mass near m2 . The SCET(c,s) transition current has to turn a cluster b¯ of soft modes with the quantum numbers of the B meson into a cluster of collinear lines with the quantum numbers of the nal-state meson. We begin with a brief description of the SCET(c,s) elds, gauge symmetries and Lagrangian. This theory is in many ways simpler than SCET(hc,c,s), because the dynamics of the soft-collinear transition resides only in the e ective current. We then discuss in detail the representation of the heavy-to-light current for an exclusive decay. In this section we restrict ourselves to tree-level matching. The general case will be considered in Section 4 to the extent that is necessary to prove the factorization of form factors. However, we brie y sketch the structure of transition operators and their coe cient functions beyond tree level at the end of this section.

3.1

Elements of SCET(c,s)

Fields. The SCET(c,s) Lagrangian and operators are built from a collinear light quark eldξc, a collinear gluon eld Ac, and soft light quark, heavy quark and gluon elds, denoted by qs, hv, As, respectively. As in[17] we assume that collinear elds describe particles with large momentum in the direction of the light-like vector n . n+ is another light-like vector, satisfying n n+= 2, and v will be the velocity vector labelling soft heavy quark elds. We will present our results in a general frame subject to the conditions n+ v～ 1, n v～ 1, v⊥～λ2 . The scaling of quark and gluon elds can be read o from the corresponding propagators

15

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

inmomentumspace.Forthequark eldsone nds

(22)2

The“opposite”projectionsofthefullcollinearquark eldψcandthefullsoftheavyquark eldsQv(de nedastheheavyquark eldwiththerapidvariationse imbvxremoved)areλ2suppressed,andintegratedout.Gluon eldsscaleasthecorrespondingderivatives,ξc=

n+Ac～1,A⊥c～λ2,n Ac～λ4,As～λ2.(23)n/ n/+Qv～λ3,qs～λ3.

Inderivingthis,weusedthattheintegrationmeasured4x～1/λ8,whentheintegralisoverproductsofonlycollinear eldsorproductsofonlysoft elds.Thisfollowsfromthefactthatanx-componentscalesinverselytothecorrespondingmomentumcomponent.Multipoleexpansion.Sincesoftandcollinear eldshavesigni cantvariationsoverdif-ferentlengthscalesinthen andn+directions,theyhavetobemultipole-expandedinproductsofsoftandcollinear elds.ThemultipoleexpansioninSCET(c,s)isdi erentfromthemultipoleexpansionde nedin[17],whichappliestoatheorywithhard-collinearandsoft elds,andnocollinear elds.Hereweneed

n xφs(x)=φs(x )+

2

where

x ≡n+xn

2+x⊥.(25)[n φc](x+)+...,(24)

Thecorrectiontermsinthetwoexpansionsof(24)arebothλ2suppressedrelativetotheleadingterms.

Gaugesymmetry.Thee ectivetheoryshouldbeinvariantundercollinearandsoftgaugetransformations,de nedastherestrictionofgaugefunctionsU(x)tothecorrespondingspatialvariations.Theimplementationofgaugetransformationsinthee ectivetheoryisnotunique,since eldrede nitionsorapplicationsofthe eldequationscanbeusedtoalterthegauge-transformationproperties[17,19,27].

InSCET(c,s)collinearandsoft eldsdecoupleatleadingpowerintheλexpansionaswillbeseenbelow.Furthermore,theproductofacollinearandasoft eldhashard-collinearmomentummodes,thereforegeneralsoftgaugetransformationsactingoncollinear elds(andviceversa)arenotallowed.Anaturalchoiceisthentode ne[20]

ξc→Ucξc, Ac→UcAcUc+i

Us ,Us g

.(26)

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Lagrangian.TheSCET(c,s)Lagrangiandescribesinteractionsofsoftandcollinear elds.Ingeneral,therecanbescatteringprocessesofthetypes+c→s+c.Ithasbeenshownin[20]forquarkscatteringthattheseinteractionsarepower-suppressed.Belowweextendthistogluonsandderivetheexplicitformoftheleadingpower-suppressedinteractions.Becauseofthedecouplingofsoftandcollinearmodesatleadingpower,theLagrangianissimply

L(0)=Ls+Lc(27)

atleadingpower,with

Lc= 1

in+Dc

2tr(FµνsFsµν)+q¯s(iD/s m)qs+LHQET (iD/⊥c+m)n/+(28)

andm～λ2thelightquarkmass.Theheavyquarkinteractionswithsoft eldsaredescribedbythestandardheavyquarke ectivetheory(HQET)LagrangianLHQET=¯viv·Dshv+...h

ForthekinematicsituationshowninFigure4scatteringprocessess+c→s+ccannotoccur.Ontheotherhand,thecrossedprocesss+s→c+cisnotpossiblebymomentumconservationinthen directions,sincen+p>0foracollinearmomentumandn l>0forasoftmomentum.Itfollowsthatinsertionsofthesoft-collinearinteractiontermsfrom¯ matrixelements,sothatwecansimplythesub-leadingLagrangianhavezero π|...|B

workwiththeLagrangianwithoutsoft-collinearinteractionstoanyaccuracy.(See[25]forarelateddiscussion.)BeyondleadingpowerthereareadditionaltermsinthecollinearLagrangianLc,whichariseuponintegratingoutheavy-quarkloopswithexternalcollinearlines.Heavy-quarkloopsalsogenerateadditionalsoftinteractionterms,whichcorrectLs,whichalsoincludesthe1/mbsuppressedtermsfromtheHQETLagrangian.However,thesoft-collinearinteractionsallresideinthee ectivecurrent.

Statesandhadronicmatrixelements.TheBmesonstatesinthee ectivetheorymaybede nedastheeigenstatesoftheleading-ordersoftHamiltonian.ThesestatesareidenticaltothoseusedinHQET.Alternatively,ifweregardthestatesastheeigenstatesoftheexactsoftHamiltonian,thesestatescoincidewiththeBmesonstatesoffullQCD,sincethesoftLagrangiantoallordersisequivalenttothefullQCDLagrangian.Similarly,thelightmesonstateinSCET(c,s)maybede nedastheeigenstateoftheleading-ordercollinearHamiltonian.ThisHamiltonianisequivalenttoQCDwithoutheavyquarks,sothepionstateinthee ectivetheoryisthesameasinQCD(withoutheavyquarks).De ningthepionwithrespecttotheexactcollinearHamiltonianimpliesthatthepionstateisthesameasinfullQCDwithheavyquarks.Inthefollowingweadopttheconventionthatthestatesarede nedwithrespecttotheexactsoftorcollinearHamiltonians,sothatwedonotdistinguishthee ectivetheorystatesfromthoseinQCD.Itwouldbeasimplemattertomaketheλdependenceofthestatesexplicit.Animplicitassumptionhereis

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

thattheseparationofcollinearandsoftmodesisdonewithoutanexplicitcut-o (dimen-sionaloranalyticregularization).WitharegularizationthatbreaksboostinvariancethecollinearLagrangianisnotequivalenttothefullQCDLagrangian.Fromtheconventionalnormalizationofhadronicstatesitfollowsthat

|B(p) ～λ 3,|π(p′) ～λ 2,(29)

wherethelightmesonisassumedtobeenergetic.

ThatapionismadeonlyofcollinearpartonscanbeunderstoodbynotingthatacollinearpionstatecanbeobtainedfromapionatrestbyalargeLorentzboost.InapionatrestallpartonshavemomentaoforderΛ,sotheboostedsystemcontainsonlycollinearmodes.Addingasoftpartontothecollinearmodesproducesacon gurationofinvariantmassmbΛ,whichcannotcontributetothepionpole.Ontheotherhand,withthesamelineofreasoning,aBmesonconsistsonlyofsoftpartons(andaheavyquark),sinceaddingacollinearmodeproducesacon gurationfarawayfromtheBmesonpole.Anapparentconsequenceoftheabsenceofsoft-collinearinteractionsandthenatureofthestatesisthatanexpression

¯ C π|f(φc)g(φs)|B(30)

wheref(φc)(g(φs))isanon-localproductofcollinear(soft) eldsandthestardenotesconvolutions,factorizesinto

¯ . π|f(φc)|0 C 0|g(φs)|B(31)

AsshowninSection2thisshouldbeconsideredasformal,sincethecollinearandsoftconvolutionintegralscanbedivergent.

AnotherconsequenceisthattheQCDcurrentmatrixelement π(p′)|u¯Γb|B(p) simplymatchesto π(p′)|Je |B(p) ,sotheproblemreducestoobtainingthee ectivecurrent.Alreadyatthispointwemaynotethattheapparentlyleadingtermvanishes,

¯cΓhv|B¯(p) =0, π(p′)|ξ(32)

¯c)dobecausethequantumnumbersoftheproductofcollinear elds(herethesingle eldξ

notmatchthoseofapion,andthequantumnumbersofthesoft elds(hereonlyhv)donot¯meson.Thiscanbeformalizedbysayingthatthee ectiveLagrangianmatchthoseoftheB

isinvariantunderseparatephasetransformationsofthecollinear,softandheavyquark elds,sowecanassign“collinearquarknumber”toproductsofoperators,withξ elds¯havingcharge 1andallotherfundamental eldscarryingcollinearquarkcharge+1,ξ

charge0.Weshallseelaterthatthe rstnon-zeromatrixelementissuppressedbythree3/2powersofλ.Thisistheoriginofthewell-known1/mbsuppressionofheavy-to-lightformfactorsatlargerecoil[11].

Thisleadstotheimportantobservation[15]thatpower-suppressedcurrentsinthee ectivetheorybecomerelevanttotheB→πformfactoratleadingpower.ThederivationofthesecurrentsinSCET(c,s)willbeworkedoutattreelevelbelow,andinmoregenerality

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

butlessexplicitlyinSection4.NotethatintheintermediateSCET(hc,c,s),whereonlyhardmodesareintegratedout,thepower-countingforhadronicmatrixelementsisnotexplicit.Inparticularin[17]thepionstateincludedsoft-collinearinteractionsandthe¯hcΓhv(withξhcdenotingthehard-collinearquarksuppressionofthematrixelementofξ

eld)withthesepionstateswasnotdetermined(seethediscussionattheendofSection

5.3of[17]andin[20]).

Light-conegaugeandWilsonlines.Itwillsometimesbeconvenient–especiallyforthefollowingtree-levelmatchingofSCET(hc,c,s)toSCET(c,s)–tochoosethegauge

n+Ahc=n+Ac=n As=0.(33)

Theusefulnessofn+Ahc=n+Ac=0gaugefollowsfromthefactthatSCET(hc,c,s)isnon-localonlyduetothepresenceofWilsonlinesinthedirectionofn+,andduetotheappearanceof(in+ ) 1.Withn+Ahc=n+Ac=0allWilsonlinesreduceto1,andthereareno eldsoforder1.Theusefulnessofn As=0gaugeisrelatedtothefactthatinSCET(hc,c,s)softgluonsdecouplefromcollinearandhard-collineargluonsatleadingorderinλinthisgauge.

Onceaparticularresulthasbeenderivedinthisgauge,thecollinearandsoftgaugeinvarianceisrecoveredbytransformingthe eldsbacktoageneralgaugeusingthegaugetransformations(26).Asexplainedin[19],thetransformationmatricesUcandUsthataccomplishthisarethelight-likeWilsonlines

Uc(x)=Wc(x)=Pexpig

Us(x)=Ys (x)=Pexpig 0

∞

∞dsn+Ac(x+sn+),

0dtn As(x+tn ),(34)

andthecorrespondinggaugetransformationofthe eldscanbewrittenas

ξc→Wc ξc,

hv→Ys hv,gAc→Wc [iDcWc]≡Ac,qs→Ys qs,gAs→Ys [iDsYs]≡As.(35)

(Hereandinthefollowingderivativesinsquarebracketsactonlyontheexpressiontotheirrightinsidethebracket.)BecausetheWilsonlinestransformas

cYs,Ys→UsUsYs,Ys→UcUcWc,Wc→UsWc,Wc→U(36)

theexpressionsontheright-handsideof(35)aregauge-singlets.The eldsAc,Ashavebeenintroducedin[20]asbuildingblocksformanifestlygauge-invariantoperators.

InageneralgaugetheWilsonlinesemergeautomaticallyfrommatchinganin nitesetofunsuppressedtree-levelFeynmandiagramswithattachmentsofn+Actosoft elds,andn Asgluonstocollinear eldsassketchedinFigure5.Indeed,atleadingpowerthese

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Figure5:In nitesetsofFeynmangraphswithattachmentsofsoftgluonsto

collinearquarksandvice-versa.Integratingouttheintermediatehard-collinear

propagatorsleadstotheWilsonlinesYs andWc,respectively,see(37).

diagramsresultin6

¯c1 gAψ/s

1← in Ds

in+Dc ¯cY , ξs(37)1 1gn+Acqs Wcqs.

Inpracticeusingthe xedgauge(33)inintermediatestepsandrestoringthegaugesym-metryvia(35)ismoree cient,inparticularwhenkeepingtrackofWilsonlinesarisingfrommulti-gluonverticesinthenon-Abeliantheory.

3.2E ectivecurrentattreelevel

¯ΓQ,whereWenowconsidertherepresentationofheavy-to-lighttransitioncurrentsJ=ψ

ΓisaDiracmatrix,inSCET(c,s).Weareonlyconcernedwithtreediagramsinthissubsection.

Inthefollowingweintroducedi erent eldsforhard-collinearandcollinearmodes.Weworkwiththegaugen+Ac=n+Ahc=n As=0andpresentthegauge-invariantresultonlyattheend.Integratingouthardintermediateheavy-quarkpropagatorsintreediagrams,weobtainthecurrentinSCET(hc,c,s)intheform

J(x)=e

where

ψ=ξc+ηc+ξhc+ηhc+qs

1=ξc+ξhc+qs

Q=

6 imbvx ¯ΓQ(x),ψ (38)2((iD/⊥+m)(ξc+ξhc)+(gA/⊥c+gA/⊥hc)qs), 1+iD/sn vn/ Wedonotwriteoutthe+i prescriptiononthepropagators,whichreads+i for1/(in+ )and i for1/(in ).Thisfollows,becausetheinternalhard-collinearpropagatorsarealwaysspace-like,with(p′ l)2 n ln+p′<0,wheren+p′>0describesanoutgoingcollinearmomentum,andn l>0anincomingsoftmomentum.Henceinpositionspace1/(in+ in +i )is1/(in+ +i )1/(in i ).

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