Higher Twist Distribution Amplitudes of Vector Mesons in QCD Twist-4 Distributions and Meso

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arXiv:hep-ph/9810475v1 26 Oct 1998CERN–TH/98–333

NORDITA–98–62–HE

hep-ph/9810475

HigherTwistDistributionAmplitudesofVector

MesonsinQCD:Twist-4Distributionsand

MesonMassCorrections

PatriciaBall1,∗andV.M.Braun2,†

1CERN–TH,CH–1211Geneva23,Switzerland

2NORDITA,Blegdamsvej17,DK-2100Copenhagen,Denmark

Abstract

Wepresentasystematicstudyoftwist-4light-conedistributionamplitudesofvectormesons

inQCD,whichisbasedonconformalexpansion.Thestructureofmesonmasscorrections

isstudiedindetail.Acompletesetofdistributionamplitudesisconstructed,whichsatisfies

all(exact)equationsofmotionandconstraintsfromconformalexpansion.Nonperturbative

inputparametersareestimatedfromQCDsumrules.Ourstudysuggeststhatmesonmass

correctionsmaypresentadominantsourceofhighertwisteffectsinexclusiveprocesses.

SubmittedtoNuclearPhysicsB1Introduction

Thenotionofdistributionamplitudesreferstomomentumfractiondistributionsofpartons

inameson,inaparticularFockstate,withafixednumberofconstituents.Fortheminimal

numberofconstituents,thedistributionamplitudeφisrelatedtothemeson’sBethe-Salpeter

wavefunctionφ

BSby

φ(x)∼󰀇

|k⊥|

d2k

⊥φ

BS(x,k

⊥).(1.1)

Thestandardapproachtodistributionamplitudes,whichisduetoBrodskyandLepage[1],

considersthehadron’spartondecompositionintheinfinitemomentumframe.Aconceptually

different,butmathematicallyequivalentformalismisthelight-conequantization[2].Either

way,powersuppressedcontributionstoexclusiveprocessesinQCD,whicharecommonly

referredtoashighertwistcorrections,arethoughttooriginatefromthreedifferentsources:

•contributionsof“bad”componentsinthewavefunctionandinparticularofcomponents

with“wrong”spinprojection;

•contributionsoftransversemotionofquarks(antiquarks)intheleadingtwistcompo-

nents,givenforinstancebyintegralsasabovewithadditionalfactorsofk2

⊥;

•contributionsofhigherFockstateswithadditionalgluonsand/orquark-antiquarkpairs.

Inthispaperwecontinuethesystematicstudyofhighertwistlight-conedistribution

amplitudesstartedinRef.[3].Inparticular,weextendtheanalysisof[3]toincludetwist-

4distributionamplitudesand,mostsignificantly,mesonmasscorrections.Apreliminary

accountofsomeofourresultshasbeenreportedin[4].

Following[3],wedefinelight-conedistributionamplitudesasmeson-to-vacuumtransition

matrixelementsofnonlocalgaugeinvariantlight-coneoperators.Thisformalismisperhaps

lessintuitivethantheinfinitemomentumframeformulation,butitismoreconvenientforthe

studyofhighertwistdistributionsasitisLorentzandgaugeinvariant.Itallowsallequations

ofmotiontobesolvedexplicitly,relatingdifferenthighertwistdistributionstooneanother.

Wewillfindthat,muchlikeinthetwist-3case[3],alldynamicaldegreesoffreedomarethose

describingcontributionsofhigherFockstates,whileallotherhighertwisteffectsaregivenin

termsofthelatterwithoutanyfreeparameters.

Asystematicstudyofmesonmasscorrectionspresentstheprincipalnewcontributionof

thiswork.Bycountingdimensions,foranyexclusiveobservableinvolvingalargemomentum

transferQ,powersuppressedhighertwistcorrectionshavethegenericstructure1Thestructureofsuchkinematiccorrectionsiswellknownfordeep-inelasticlepton-hadron

scattering,inwhichcasetheycanbeabsorbedintoaredefinitionofthescalingvariable[5].Thecrucialobservationleadingtothis“Nachtmannscaling”isthathadronmasscorrections

(“targetmasscorrections”inthiscontext)ariseexclusivelyfromthedefinitionoftherelevant

leadingtwistmatrixelementsanddonotinvolvenew(highertwist)operators.Thissim-

plificationdoesnotholdinexclusiveprocessesbecausethereareadditionalcontributionsof

operatorscontainingtotalderivatives.Specifically,totwist-4accuracy,inadditiontoNacht-

mann’scorrections,therearealsocontributionsofoperatorsoftype

∂2O(2)

µ1µ2...µn

and

µ1O(2)

µ1µ2...µn,

whereO(2)isaleadingtwistoperator.Wefindthatcontributionsofthefirsttypecanbe

takenintoaccountconsistentlyforallmoments,whilecontributionsofthesecondtypeare

morecomplicatedandcanbeunravelledonlyorderbyorderintheconformalexpansion.

Theoutlineofthispaperisasfollows:definitionsofandnotationsfordistributionampli-

tudesarepresentedinSec.2togetherwithgeneralremarksaboutspecificfeaturesoftheoper-

atorproductexpansion(OPE)forexclusiveprocessesandaboutconformalexpansion.Section

3givesageneraldiscussionofmesonmasscorrectionsforasimpleexample.Thesubsequent

Secs.4and5containadetailedderivationofchiral-evenandchiral-odddistributionampli-

tudes,respectively.Wetakeintoaccountcontributionsoftheleadingandnext-to-leading

conformalspinandderiveaself-consistentapproximationforthedistributionamplitudes,

whichrespectstheexactQCDequationsofmotion.Thechiral-evenandchiral-oddasymp-

toticdistributionamplitudesinvolvethreenonperturbativeparameters,andfouradditional

parametersarerequiredforthedescriptionoftheleadingcorrections.Thecorrespondingesti-

matesareworkedoutusingtheQCDsumruleapproach[6].Onthebasisoftheseestimates,