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,