Hard Exclusive Electroproduction of Pions

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

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aTUM/T39-98-33HardExclusiveElectroproductionofPions L.Mankiewiczab,G.PilleraandA.Radyushkincd aPhysikDepartment,TechnischeUniversit¨atM¨unchen,D-85747Garching,GermanybN.CopernicusAstronomicalCenter,ul.Bartycka18,PL–00-716Warsaw,PolandcPhysicsDepartment,OldDominionUniversity,Norfolk,VA23529,USAdJe ersonLab,NewportNews,VA23606,USAAbstractWeinvestigatetheexclusiveelectroproductionofπ+mesonsfromnucleons.Tolead-ingtwist,leadingorderαsaccuracythecorrespondingproductionamplitudecanbedecomposedintopseudovectorandpseudoscalarparts.Bothcanbeexpressedintermsofquarkdoubledistributionfunctionsofthenucleon.Whilethepseudovectorcontributionisconnectedtoordinarypolarizedquarkdistributions,thepseudoscalar

partcanberelatedtoonepiont-channelexchange.Weobservethatthepseudovec-torpartoftheproductionamplitudeisimportantatxBj<0.1.Ontheotherhand,for0.1<xBj<0.4contributionsfromonepionexchangedominate.

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

1Introduction

Hardexclusiveelectroproductionofmesonsfromnucleonshasbecomea eldofgrowinginterest.TherecentprogressintheQCDanalysisofsuchprocessesisbasedonafactoriza-tiontheorem[1](seealso[2,3,4,5])validinthecaseoflongitudinallypolarizedphotons:atlargephotonvirtualities,Q2 Λ2QCD,theunderlyingphoton-partonsub-processesaredominatedbyshortdistancesand,hence,canbecalculatedperturbatively.Ontheotherhand,allinformationonthelongdistancedynamicsofquarksandgluonscanbecollectedinthedistributionamplitudeoftheproducedmesonandingeneralized(skewed[2,6]ordouble[2])partondistributionfunctionsofthenucleon.Asaconsequence,experimentalinvestigationsofmesonproductionprocessesallowtoprobedetailsofthequark-gluondynamicsinnucleonsbeyondourcurrentknowledgeobtainedmainlyfrominclusivehigh-energyprocesses.OngoingandplannedmeasurementsatDESY(HERA[7],HERMES

[8]),CERN(COMPASS[9]),andJe ersonLab[10]arethereforeofgreatinterest.Inourpreviouswork[4,11]wehavederivedtheleadingorderQCDproductionam-plitudesforneutralpseudoscalarmesonsandvectormesons.Thecorrespondingcrosssectionshavebeendiscussedwithinaspeci cmodel[12]fordoublepartondistributions.Inthispaperweconsiderhardelectroproductionofchargedpions.Interestinthispro-cessarisesinparticularfromthefactthatitreceivessigni cantcontributionsfromtwoproductionmechanisms:either(i)thephotoninteractswithaquarkfromthenucleonwhich,afterhardgluonexchange,combineswithasecondquarkofthetargettothe nalpion,or(ii)thepionisproducedfromthemesoncloudofthenucleon.Thelattermech-anismisoftenreferredtoasthepionpolecontribution.Inmeson-cloudmodelsofthenucleonthesecondmechanismdirectlyinvolvestheelectromagneticformfactorFπ(Q2)ofthepion(seee.g.[13]andreferencestherein).PionelectroproductionhasthereforeoftenbeenconsideredasagenericprocesstodetermineFπ(Q2)(seee.g.[14]).However,asdiscussedinref.[15],theremaybesigni cantuncertaintiesduetotype(i)contribu-tions.Thelatterhavebeenmodeledinref.[15]intermsofhardgluonexchangediagramswithnonperturbativefactorsestimatedbyanoverlapoflight-conewavefunctionswithChernyak-Zhitnitskytypedistributionamplitudes[17].

Inthisworkweaddressquestionssimilartothoseraisedin[15].ThedistinctivefeatureofourapproachisasystematicuseofperturbativeQCD(PQCD)factorization

[1,2].WithinaconsistentPQCDanalysisallnecessaryinformationaboutthenucleonstructureiscontainedinquarkdoubledistributions.Thelatterareconstrainedthroughthebehaviorofordinaryquarkdistributionfunctions,aswellasthroughsumrulesandsymmetryproperties(foradiscussionsee[4,16]).Inparticularweshowthatactuallybothproductionmechanisms,(i)and(ii),canbeformulatedintermsofquarkdoubledistributionfunctionsofthenucleon,i.e.theexistenceofthepionpolecontribution tsintothePQCDfactorizationframework.

Aftermodelingtheinvolveddoubledistributionfunctionswecalculatethedi erentialπ+productioncrosssectionintheregionofsmallmomentumtransfers, t<0.5GeV2.We ndthatbothmechanisms,(i)and(ii),contributesigni cantly.Whileonepion

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

exchangedominatesatintermediatevaluesofBjorken-x,perturbativequarkexchangeisrelevantmainlyatsmallxBj.

Thepaperisorganizedasfollows:inSec.2wepresenttheamplitudesforchargedpionelectroproduction.Pseudovectorandpseudoscalarpartsoftherelevantdoubledis-tributionfunctionsareassociatedinSec.3and4withtheperturbativeandpionpolecontribution,respectively.ResultsarepresentedinSec.5.Finallywesummarize.2Amplitudes

ThecalculationofmesonelectroproductionamplitudesisbasedonPQCDfactorization[1].Itappliestoincidentlongitudinallypolarizedphotonswithlargespace-likemomenta,<2Q2= q2 Λ2QCD,andmoderatemomentumtransfertothenucleontarget,|t|～Λingthetechniquesoutlinedinrefs.[1,2,4,11]givesthefollowingresultfortheπ+virtualphotoproductionamplitudeinleadingtwist,leadingorderαsaccuracy:Aπ+=i

g2CF

Q¯(P′,S′)γ5nN N(P,S)

ττ¯

ω¯

d[x,y]x+2y xω¯ i ¯du)×(ed Fdu eu F

ig2CF

Q¯(P′,S′)γ5N(P,S)N

ττ¯

1d[x,y]

x+2y xω¯ i ¯du)×(ed Kdu eu K

.(1)

¯Herethenotationd[x,y]...=01dx0xdy...,withx¯=1 xisused.N(P,S)and¯(P′,S′)aretheDiracspinorsoftheinitialandscatterednucleon,respectively,withtheN

correspondingfour-momentaP,P′andspinsS,S′.Mstandsforthenucleonmass.The¯=(P+P′)/2,andthemomentumtransferaveragenucleonmomentumisdenotedbyP

isr=P P′witht=r2.Theproducedmesoncarriesthefour-momentumq′and¯/( qq¯=(q+q′)/2.Furthermore,wehaveintroducedthevariableω¯=2¯q·P¯2).Finally,nisalight-likevectorwithn·a=a+=a0+a3foranyvectora,andn =γµnµ.

Thelongdistancedynamicsofthepioninthe nalstateiscontainedinthedecayconstantfπ=133MeVandthedistributionamplitudeΦπ(τ)[18,19]:

¯u(x)γ5n π(q)|ψ ψd(y)|0 = ifπ(q′·n)+′ 1

0τy)Φπ(τ)eiq·(τx+¯dτ,′(2)

wherequark eldsψwithproper avorquantumnumbersenter.

Thenucleonpartoftheamplitude(1)isdeterminedbydoubledistributionfunctions Fduand Kduwhicharenondiagonalin avor.Theyarede nedasmatrixelementsof

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

anonlocalquarkoperatorsandwichedbetweenprotonandneutronstates:

¯d(0)γ5n n(P′,S′)|ψ [0,z]ψu(z)|p(P,S) z2=0=

¯(P,S)γ5nN N(P,S)′′

¯(P′,S′)γ5N(P,S)r·n N d[x,y]e ix(P·z) iy(r·z) Fdu+eix(P·z) iy¯(r·z)¯du F

fπ

8Ncπ(τ)P¯·n 10dτΦ

x+2y+xω¯ i +ω¯

ω¯x+2y+xω¯ i

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

prefactorstheρ+amplitudecanbeobtainedfrom(6)byreplacingpolarizedquarkdoubledistributionfunctionsbyunpolarizedones,withproperaccountoftheiroppositechargeconjugationproperties.

Inordertoobtainanumericalestimateforthepseudovectorcontribution(6)toπ+productionmodelsfortheinvolveddoubledistributions Fhavetobeconstructed.Weareguidedherebytheappropriateforwardlimitofdoubledistributions,theirsymmetryproperties,andsumruleswhichrelatethemtonucleonformfactors.Followingrefs.[20]and[16]weuse:

A: F(x,y;t)=h(x,y) q(x)f(t),

B: F(x,y;t)=h(x,y) q(x)exp t

x(1 x) .(7)

Hereh(x,y)=6y(1 x y)/(1 x)3,and q(x)denotesthecorrespondingordinarypolarizedquarkdistribution.InnumericalcalculationspresentedinthispaperwehaveusedGehrman-StirlingLOset-Aparametrizationsofpolarizedquarkdistributions[21].Furthermore,wehaveusedtheoneloopexpressionfortherunningcouplingconstantwithΛQCD=200MeV.Notethatinadditiontotheabovementionedconstraints,themodels

(7)areconsistentwithasymptoticsolutionsofevolutionequationsfordoubledistributionsandsatisfythesymmetriesofthelatter[4].Thet-dependenceofdoubledistributionsinmodelAisgovernedbytheformfactorf(t)which,likein[4],istakentobeequaltothenucleonpseudovectorformfactor[22]:

f(t)=1

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

moredi cult.Nevertheless,someintuitionabouttheirmagnitudehasbeenobtainedfrommodelcalculations[23].Still,contributions

oftheso-calledK-termstohardexclu-sivemesonproductionhavebeenneglectedsofar.Inthepresentcaseofπ+productionthesituationisdi erent.Aswewillshow,thepseudoscalarpieceoftheamplitudeAπ+canbeassociatedwiththevirtualphotoproductionofpionsfromthenucleonpioncloud.Duetothesmallpionmassthismechanismisexpectedtobeimportant,especiallyatsmallmomentumtransferst.

Weareinterestedinevaluatingthecontributiontothenucleonmatrixelement(3)arisingfromasituationwhenaquark-antiquarkpairwithlowinvariantmasspropagatesinthet-channel,suchthatitbecomesclosetoaπ+boundstate.Toconstructthecorre-spondingdoubledistributionfunctionweintroduceane ectiveLagrangiandescribingapseudovectorpion-nucleoninteraction[22]:

LπNN=gπNN

× ¯(P′,S′)γ5N(P,S)2gπNNN

¯d(0)γ5nd4xe ir·x 0|T[ψ [0,z]ψu(z)φ+(x)]|0 z2=0,(10)

where +=1

2( 1+i 2)isthepion eldassociatedwiththeπ+meson.Theleading-twist

contributiontothematrixelement(10)arisesfromtheregionofphasespacewherethequark eldsarelocalizednearthelight-cone,z+～0.Insertingafullsetofintermediatepionstatesbetweenthepion eldandthenon-localquarkoperatorgives:

¯d(0)γ5n n(P′,S′)|ψ [0,z]ψu(z)|p(P,S) z2=0√ i+¯d(0)γ5n 0|ψ [0,z]ψ(z)|π(r) z2=0.=N(P′,S′)γ5N(P,S)um2 tπ(11)

Usingthepiondistributionamplitude(2)andcomparingwiththede nitionofthepseu-doscalardoubledistributionfunctionsineq.(3)yields:

√ 2¯du(x,y,t≈0)= δ(x)Φπ(y).(12) Kdu+ K2mπ t

Asthiscontributioncanbeexpressedentirelyintermsof Kthereisnodangerofdoublecountingifweaddthistermtothecontributionarisingfromamodelfor F,asdiscussedintheprevioussection.

Onecanexpressthet-channelonepionexchangecontributionalsointheform:

Aπ+√= m2π t

εL·(q′+r)Fπ+(Q2),(13)

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

where

εL=i

2fπ

2Ncu 2(15)

istheleadingQCDcontributiontotheπ+electromagneticformfactoratlargeQ2[18,19].2Totwist-2accuracy,i.e.neglectingtermsofordert/Q2andm2π/Q,theamplitude

(13)isexplicitlygaugeinvariant.ThisisguaranteedbythefactorεL·(q′+r)whicharisesfromthehardphoton-quarkinteraction.Intheonepionexchangeapproximation¯dudecouplesfromtheconsideredproductionprocess.Finally,thedi erence Kdu K

notethatinthecaseofπ0productionat-channelexchangecontributionsimilarto(12)appearsintheproductionamplitudewithazerocoe cientfromthedi erenceoftwoequalquarkcharges.

Tothisendsomecommentsregardingonepiont-channelexchange,asusedabove,areinorder.Ingeneraloneexpectsthatatlargecenter-of-massenergiesanyt-channelexchangeofanhadronicstateshouldbereplacedbythecorrespondingReggetrajectory.Therefore,atsmallvaluesofBjorken-xadditionalcontributionstothepseudoscalarpartoftheproductionamplitudeAπ+couldoccur.Oncethepionpolecontributionisreplacedbyreggeizedpion(andrho)exchange[24]therelationtothepionelectromagneticformfactorseemstobelost(see,however,ref.[25]foranattempttointerpretpionelectroproductiondataintheframeworkofaReggeexchangemodel).Werestrictourselvesinthisworktothesimplestmodelforthepseudoscalarpartoftheproductionamplitudeinthehopethatitcontainsthedominantcontributionatsmallmomentumtransfers,i.e.closetothepionpole.

5Results

InFigs.1and2wepresentresultsforthecrosssectionofexclusiveπ+productioninthescatteringoflongitudinallypolarizedphotonsfromnucleonsatQ2=10GeV2.Uptologarithmiccorrectionsthecrosssectionisproportionalto1/Q6asonecaninferfromeq.(1).

InFig.1weshowthedi erentialproductioncrosssectionfortheminimalkinematically

2allowedvalueoft=tmin= x2BjM/(1 xBj).We ndthatthepseudovectorcontribution

(6)dominatesatsmallvaluesoftheBjorkenvariablexBj.ItsxBj-dependencere ectsthebehavioroftherelevantforwardpartondistributions(7).Atx～0.3thecrosssectionisgovernedbythepseudoscalarpioncloudcontribution(11).

WhilemodelsAandBgivequalitativelysimilarcrosssectionsatsmallmomentumtransfers,theystarttodi erforincreasing|t|.ThisisillustratedinFig.2wherewepresentresultsforthedi erentialproductioncrosssectiontakenatt= 0.4GeV2.The

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

maximumvalueofBjorken-xisdeterminedbythecondition|t|≥|tmin|.FormodelAthepseudovectorcontributiondecreasesslowlywithdecreasingxBjanddominatesthecrosssectionatxBj<～0.1.AtxBj～xmaxthetotalpionproductioncrosssectionisabouthalfaslargeasobtainedfromthepioncloudcontributionaloneduetostronginterferencee ects.Ontheotherhand,inmodelBthexBj-dependenceofthepseudovectorcontributionissimilartotheonefromthepioncloud.Also,contrarytothepreviouscase,atxBj～xmaxtheproductioncrosssectionisabout20%largerthanobtainedfromthepioncloudalone.FinallywecompareinFig.3totheleadingtwistproductioncrosssectionforπ0mesonswhichhasbeeninvestigatedin[4].Att=tminthepseudovectorcontributiontoπ+productionisaboutafactor5–10largerascomparedtotheπ0case.AtxBj～0.3largecontributionsfromthepioncloud,whichareabsentinπ0production,addfurthertothecrosssectionratio.

Itshouldbeemphasizedthatthedominanceofthehardgluonexchangeprocessesconsideredinthispapercanonlybeensuredforsu cientlylargevaluesofQ2.Justlikeinthecaseofthepionformfactor,competingcontributionsareprovidedbysoftmechanismswhich,inthiscase,correspondtothenonperturbativeoverlapofthreehadronicwavefunctions.TheinterplayofsoftandhardinteractionmechanismshasbeenstudiedrecentlyforthepionformfactorinaQCDsumrulerelatedmodel[26].GuidedbytheresultsofthisinvestigationweexpectthedominanceofhardgluonexchangeforvaluesofQ2above10GeV2.

6Summary

Wehavediscussedthehardexclusiveelectroproductionofπ+mesonsfromnucleons.Forlongitudinallypolarizedphotonsonecanexpressthecorrespondingphotoproductionam-plitudetoleadingtwist,leadingorderαsaccuracyintermsofquarkdoubledistributionfunctionsofthenucleon.We ndthattheamplitudereceivescontributionsfrompseu-dovectorandpseudoscalarpieces.Thepseudovectorcontributionisdeterminedbyquarkdoubledistributionswithwellknownpropertiesintheforwardlimit,asgivenbyordinarypolarizedquarkdistributionfunctions.However,forthepseudoscalarpartsuchinforma-tionisnotavailable.Inthisworkithasbeenmodeledintermsofaninteractionofthevirtualphotonwiththenucleonpioncloud.Asaconsequence,therelateddoubledistri-butionisdeterminedbythepiondistributionamplitude.Thecorrespondingcontributiontothepionproductionamplitudeturnsouttobeproportionaltothepionformfactor.Wehavefoundthatboth,pseudovectorandpseudoscalartermsshouldbeincludedifoneconsidersthewhole0<xBj<1range.Inparticular,wehaveobservedthatthepseudovectorpartdominatesatsmallxBj<0.1.Ontheotherhand,thepioncloudcontributioncontrolstheπ+productionatsmallt～tminand0.1<xBj<0.4.Atlargervaluesof|t|～0.4GeV2therelativeweightofpseudoscalarandpseudovectorcontributionshasbeenfoundtobesensitivetodetailsoftherelevantdoubledistributionfunctions.Furtherinvestigationsofdoubledistributionfunctions,aswellasexperimentaldataonexclusivemesonproductionarecertainlyneeded.

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

Acknowledgments:

G.P.wouldliketothanktheSLACtheorygroupfortheirhospitality.Furthermore,wewouldliketoacknowledgediscussionswithV.Burkert,S.J.Brodsky,L.Frankfurt,M.V.PolyakovandP.Stoler.TheworkofA.R.wassupportedbytheUSDepartmentofEnergyundercontractDE-AC05-84ER40150.

Afterthispaperhasbeencompleted,welearnedaboutanindependentcalculationofπ+electroproductionwhichusesskewedpartondistributionscomputedinachiralquarksolitonmodel[27].Thecorrespondingresultsturnouttobequitesimilartoours.Wethanktheauthorsof[27]fordiscussionsandanexchangeofresults.

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We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

Figure1:Di erentialproductioncrosssectionforexclusiveπ+productionthroughthescatteringoflongitudinallypolarizedphotonsfromprotonsatt=tminandQ2=10GeV2.Thedottedanddot-dashedcurvesshowthepseudovectorandpseudoscalarcontributions,respectively.Theupper gurecorrespondstomodelA,theloweronetomodelB.

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

Figure2:Di erentialproductioncrosssectionforexclusiveπ+productionthroughthescatteringoflongitudinallypolarizedphotonsfromprotonsatt= 0.4GeV2andQ2=10GeV2.Thedottedanddot-dashedcurvesshowthepseudovectorandpseudoscalarcontributions,respectively.Theupper gurecorrespondstomodelA,theloweronetomodelB.

We investigate the exclusive electroproduction of pi^+ mesons from nucleons. To leading twist, leading order alpha_s accuracy the corresponding production amplitude can be decomposed into pseudovector and pseudoscalar parts. Both can be expressed in terms

ππ+o

Figure3:Ratioofthedi erentialproductioncrosssectionsforexclusiveπ+andπ0productionthroughthescatteringoflongitudinallypolarizedphotonsfromprotonsatt=tminandQ2=10GeV2formodelA.Thedottedcurvecorrespondstothepseudovectorcontributionalone.

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