Single Transverse Spin Asymmetry in $p^uparrow p to pi X$ and $ep^uparrow to pi X$

Single transverse spin asymmetry for the pion production in pp and ep collisions is studied in the framework of collinear factorization. The soft-gluon pole contributions with twist-3 distribution and fragmentation functions are identified and its characte

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arXiv:hep-ph/0211400v1 26 Nov 2002

SingleTransverseSpinAsymmetryinp↑p→πXandep↑→πX

a

YujiKoikeaMCSD]DepartmentofPhysics,NiigataUniversity,Niigata950-2181,Japan[

Singletransversespinasymmetryforthepionproductioninppandepcollisionsisstudiedintheframeworkofcollinearfactorization.Thesoft-gluonpolecontributionswithtwist-3distributionandfragmentationfunctionsareidenti edanditscharacteristicfeaturesarediscussed.

Inthisreportwediscussthesingletransverse-spinasymmetryANforthepionpro-ductionwithlargetransversemomentuminpp↑→π( )Xandp↑e→π( )XrelevantforRHIC-SPIN,HERMESandCOMPASSexperimentsetc.AccordingtotheQCDfac-torizationtheorem,thepolarizedcrosssectionforpp↑→πXconsistsofthreetwist-3contributions:

(A)(B)(C)

Ga(x1,x2) qb(x′) Dc(z) σ ab→c,

′

δqa(x) Eb(x′1,x′2) Dc(z) σ ab→c,

′′ (z,z) σδqa(x) qb(x′) E abc12→c,

(z,z)arethetwist-3quantitiesrepre-wherethefunctionsGa(x1,x2),Eb(x′1,x′2)andEc12

senting,respectively,thetransverselypolarizeddistribution,theunpolarizeddistribution,andthefragmentationfunctionforthepion.Otherfunctionsaretwist-2:qb(x′)istheunpolarizedquarkorgluondistribution,δqa(x)isthetransversitydistributioninp↑,andDc(z)isthefragmentationfunctionforthepion.a,bandcstandfortheparton’sspecies,

arechiral-odd.σsumoverwhichisimplied.δqa,EbandE ab→cetcarethepartonicc

hardcrosssectionproducinglargetransversemomentumofthepion.Correspondingtotheabove(A)and(C),thepolarizedcrosssectionforep↑→πX( nalelectronisnotdetected)receivestwotwist-3contributions:

(A′)(C′)

The(A)and(B)contributionsforpp↑→πXhavebeenanalyzedin[1]and[2],respec-tively,andithasbeenshownthat(A)givesrisetolargeANatlargexFasobservedinE704,and(B)isnegligibleinallkinematicregion.Hereweextendthestudytothe(C)termatRHICenergyandalsotheasymmetryinepcollision.Thetransverselypolarizedtwist-3distributionGaF(x1,x2)relevanttoGainthe(A)

c(z,z)forE and(A’)termsisgivenin[2],andthetwist-3fragmentationfunctionEcF12

ca contributein(C)and(C’)isde nedin[4].Accordingtoadetailedanalysis,GFandEF

tothecrosssectionatx1=x2=xandz1=z2=z,respectively(soft-gluon-pole

Ga(x1,x2) Da(z) σ ea→a,

′ (z,z) σδqa(x) E eaa12→a.

Single transverse spin asymmetry for the pion production in pp and ep collisions is studied in the framework of collinear factorization. The soft-gluon pole contributions with twist-3 distribution and fragmentation functions are identified and its characte

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ccontribution).Inparticular,thederivarives(d/dx)GaF(x,x)and(d/dz)EF(z,z)appear

inthecrosssectionformula.InthelargexFregion,i.e.fortheproductionofpionintheforwarddirectionofthepolarizednucleon,themaincontributioncomesfromthelarge-xandlarge-zregionofdistributionandfragmentationfunctions,respectively.

c(z,z)～(1 z)β′withβ, cbehavesasGa(x,x)～(1 x)βandEandESinceGaFFFF

ca′a c(z,z)|atlargexandβ>0,|(d/dx)GF(x,x)| |GF(x,x)|,|(d/dz)EF(z,z)| |EFac z.Inparticular,thevalencecomponentofGFandEFdominateinthisregion.We

thuskeeponlythevalencequarkcontributionforthederivativeofthesesoft-gluonpolefunctions(“valence-quarksoft-gluonapproximation”)fortheppcollision.Fortheepcase,weincludeallthesoft-gluonpolecontribution.

IngeneralANisafunctionofS=(P+P′)2 2P·P′,T=(P )2 2P· andU=(P′ )2 2P′· whereP,P′and arethemomentaofp↑,unpolarizedp(ore),

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√andthepionrespectively.InthefollowingweuseS,xF=S=T U

S=20GeVand T=1.5GeVtogetherwithE704data.

a(z,z)=Toestimatethe(C)contribution,weintroduceasimplemodelansatzasEF

KaDa(z)witha avordependentfactorKa.Assumingthatthe(C)isthesoleoriginofAN,Ka’saredeterminedtobeKu= 0.11√andKd= 0.19sothatitcanapproximatelyreproduceAppNobservedinE704dataat

S=200GeV)isshowninFig.2atlT=1.5GeV.Both(A)and(C)contributionsgiveslightlysmallerAppN√thantheppSTARdatareportedatSPIN2002.Fig.3showsthe T-dependenceofANat

Single transverse spin asymmetry for the pion production in pp and ep collisions is studied in the framework of collinear factorization. The soft-gluon pole contributions with twist-3 distribution and fragmentation functions are identified and its characte

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stableatlarger T.

Figure2.

AppN

at

S=200GeVand

xF=0.6.

↑

WenextdiscusstheasymmetryAepNforpe→π( )Xwherethe nalelectronisnot

observed.InourO(αs)calculation,theexchangedphotonremainshighlyvirtualasfarastheobservedπhasalargetransversemomentum Twithrespecttotheepaxis.Thereforeexperimentallyoneneedstointegratesonlyoverthosevirtualphotonevents (Q2>(S/2)[1 xF/

S=20GeVand T=1.5GeV.

Usingthetwist-3distributionandfragmentationfunctionsusedtodescribeppdata,weshowinFig.4AepNcorrespondingto(A’)(chiral-even)and(C’)(chiral-odd)contributions.RemarkablefeatureofFig.3isthatinbothchiral-evenandchiral-oddcontributions

ppep

(i)thesignofAepNisoppositetothesignofANand(ii)themagnitudeofANismuchlargerthanthatofAppN,inparticular,atlargexF,anditevenovershootsone.(Inourconvention,xF>0correspondstotheproductionofπintheforwardhemisphereoftheinitialpolarizedprotonbothinp↑pandp↑ecase.)Theoriginofthesefeaturescanbetracedbacktothecolorfactorinthedominantdiagramsforthetwist-3polarizedcrosssectionsinepandppcollisions.Ofcourse,theasymmetrycannotexceedsone,andthusourmodelestimateneedstobemodi ed.First,theappliedkinematicrangeofourformulashouldbereconsidered:ANat TsmallerthanafewGeVmaynotbeascribed

aa atothetwist-3e ect.Second,ourmodelansatzofGaF(x,x)～q(x)andEF(z,z)～D(z)

Single transverse spin asymmetry for the pion production in pp and ep collisions is studied in the framework of collinear factorization. The soft-gluon pole contributions with twist-3 distribution and fragmentation functions are identified and its characte

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isnotappropriateatx→1andz→1,respectively.Thederivativeofthesefunctions,

pp

whichisimportantforthegrowingAppNatlargexF,eventuallyleadstodivergenceofANatxF→1as～1/(1 xF).

Asapossibleremedytothispathologywetriedthefollowing.Forthe(A)and(A’)(chiral-even)contribution,asanexample,wehaveamodelGaF(x,x)～qa(x)～x→1(1 x)βawhereβu=3.027andβd=3.774intheGRVdistributionweadopted.Tentativelyweshiftedβu,dasβa→βa(x)=βa+x3,whichsuppressesthedivergenceofANatxF→1butstillcausesrisingbehaviorofANatlargexF.TheresultforANwiththismodi cation

pppp

ofGF(x,x)isshowninFig.5(AepN),Fig.6(ANatE704energy)andFig.7(ANatppRHICenergy).ThisavoidsovershootingofoneinAepNbutreducesAN,typicallybyfactor2,whichmakesthedeviationfromE704andSTARdatalarger.Asimilarmodi cation (z,z)inthe(C)and(C’)(chiral-odd)contributionalsoleadstothesamechangeinofEFAN.

Figure5.AepNwithamodi- edGF.pp

Figure6.ANwithamodi-√ edGFat

S=200GeV.

TosummarizewehavestudiedtheANforpionproductioninppandepcollisions.Wehavepointedoutthatboth(A)(chiral-even)and(C)(chiral-odd)((A’)and(C’)forepcollisions)contributionscanbeequallyimportantsourcesfortheasymmetry.Althoughourapproachprovidesasystematicframeworkforthelarge Tproduction,applicabilityoftheformulatothecurrentlyavailablelow Tdatastillneedstobetested,inpartic-ular,detailedcomparisonof T-dependenceandtheglobalanalysisofppandepdataisnecessary.

Acknowledgement:ThisworkissupportedinpartbytheGrant-in-AidforScienti cResearchofMonbu-Kagaku-sho.REFERENCES1.2.3.4.

J.QiuandG.Sterman,Phys.Rev.D59(1999)014004.

Y.KanazawaandY.Koike,Phys.Lett.B478(2000)121;B490(2000)99.

Y.Koike,hep-ph/0106260(ProceedingsofDIS2001,Bologna,Italy,April,2001.)Y.Koike,hep-ph/0210396(ProceedingsofSPIN2002,LongIsland,USA,Sep.2002)

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