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TransverseSpinAsymmetriesinNeutralStrangeParticleProductionbyThomasBurtonAthesissubmittedtoTheUniversityofBirminghamforthedegreeofDOCTOROFPHILOSOPHYSchoolofPhysicsandAstronomyTheUniversityofBirminghamMay2009.
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AbstractTheoriginofthequantummechanicalspinoftheprotonintermsofitsconstituentsisnotyetfullyunderstood.
Thediscoverythattheintrinsicspinofquarkscontributesonlyasmallfractionofthetotalprotonspinsparkedahugetheoreticalandexperimentalefforttounderstandtheoriginoftheremainder.
Inparticularthetransversespinpropertiesoftheprotonremainpoorlyunderstood.
Signicanttransversespinasymmetriesintheproductionofhadronshavebeenobservedovermanyyears,andarerelatedtoboththetransversepolarisationofquarksinatransverselypolarisedprotonandtothespindependenceoforbitalmotion.
TheseasymmetriesareofinterestbecauseofperturbativeQCDpredictionsthatsuchasymmetriesshouldbesmall.
Measurementsofsuchasymmetriesmayyieldfurtherinsightsintothetransversespinstructureoftheproton.
TheRelativisticHeavyIonCollider(RHIC)istheworld'srstpolarisedprotoncollider,andhasbeentakingprotondatasince2001.
Polarisedprotoncollisionsat√s=200GeVtakenduringthe2006RHICrunhavebeenanalysedandthetransversesingleanddoublespinasym-metriesintheproductionoftheneutralstrangeparticlesK0S,ΛandΛhavebeenmeasuredinthetransversemomentumrange0.
50.
2thevalencequarksdominate.
Notethattheupquarkdistributionhastwicethemagnitudeofthedownquarkdistribution,asexpectedfromthequarkmodeloftheproton.
Thevalencequarkdistributiondecreaseswithdecreasingx,whiletheantiquarkandgluondistributionsallincrease.
Atsmallx,gluonsdomi-natethePDF.
Apictureemergesofthevalencequarksoftheproton,eachcarryingasignicantportionoftheprotonmomentum,plusaseaoflow-momentumgluonsandquark-antiquarkpairs.
1.
3QCDFactorisationCalculatinghighenergyhadroniccrosssectionsreliesuponthefactorisationtheorem.
Considerahighenergycollisionbetweentwohadrons,A+B→C+X,whereCisameasurednalstateparticleandXistheremainingunmeasuredhadronicnalstate.
Atlargemomentumscalesthecollisioncanbeviewedasoccurringbetweentwopartons,aandbfromthehadronsAandBrespectively,producingapartoncthatsubsequentlyfragmentsintotheobservedhadronC.
Thefactorisationtheoremstatesthatthehadroniccrosssectionforthecollision,σA+B→C+Xcanbesplitintothreeseparateparts:thePDFsoftheinitialstatehadronsAandB,thepartoniccrosssectionσa+b→candthefragmentationfunction(FF)ofthescatteredquark:σA+B→C+X=∑a,b,cfa(xA)fb(xB)σa+b→cDc→C(z).
(1.
5)Thesumrunsoverallpartonicspeciesa,b,andcthatcontributetothecrosssectionforA+B→C.
fa(xA)isthePDFofthepartonaasafunctionofitsfraction,xA,ofthemomentumofhadronA.
fb(xB)isdenedcorrespondingly.
Dc→C(z)isthefragmentationfunctiondescribingthefragmentationofpartoncintoahadronCwithafractionzofthemomentumofc.
OnlythehardpartoniccrosssectioncanbecalculatedusingperturbativeQCD,providedthattheQ2of11theinteractionissufcientlyhighthatthestrongcouplingstrengthαSissmall.
ThePDFsandFFarenon-perturbativeandmustbeempiricallydetermined.
BothPDFsandFFareuniversal;theyarethesameforallcollisionprocesses.
Thusifmeasuredinoneprocesstheycanbeusedinpredictingthecrosssectionsofanotherprocess.
Theapplicabilityoffactorisationhasbeendemonstratedforthecrosssectionsofhadroniccollisions(forexamplep+p→jets[20])andlepton-nucleoncollisions(seeforexample[19]),whereonlyonePDFisinvolved.
1.
4NucleonSpinInanaive,non-relativisticquarkmodelinwhichanucleoncontainsthreequarks,allthespinofthenucleoniscarriedbytheintrinsicspinsofthesethreequarks.
Thespinsofthethree(spin-)quarkssumtogivethenucleonspin-.
Modelstakingaccountofrelativisticeffectswithinthenucleonpredictthatsomeofthespinwillbecarriedbytheorbitalangularmomentumofthequarks.
Theamountcarriedbyquarkintrinsicspinisreducedtoabout60%ofthenucleonspin[21].
DISexperimentsshowthatthestructureofthenucleonismorecomplicatedthanathree-quarksystem.
HowdoesthisaffectourunderstandingofthenucleonspinInanalogytotheunpolarisedcase,thespinstructureofthenucleoncanbeprobedusingpolarisedDeepInelasticScattering(pDIS),whereinboththeincidentleptonandnucleontargetarepolarised.
Throughsuchmeasurementsthespin-dependentnucleonstructurefunctiong1,thespin-dependentanalogueoftheF1structurefunction,canbedetermined.
Thefunctiong1isrelatedtothespin-dependentquarkdistributionsviag1(x)=∑qe2q(q↑(x)q↓(x)),(1.
6)wherethesumisoverbothquarkandantiquarkavours,eqisthechargeofthe(anti-)quarkspeciesand↑(↓)indicatesaquarkwithspincomponentparallel(opposite)tothatofitsparentnucleon.
Theintegralofg1overallxgivesthetotal(seaplusvalence)quarkplusanti-quarkintrinsicspincontributiontothenucleon.
PolarisedDISmeasurementsbytheEuropeanMuonCollaboration(EMC)inthelateeight-12ieswerethersttoindicatethattheintrinsicspinsofthequarksinthenucleoncarryasig-nicantlysmallerfractionofthenucleonspinthanhadbeenpredicted[22,23].
EMCresultsfrom++pcollisionsindicatedaquark-plus-antiquarkintrinsicspincontributionintheregionof10to15%.
Theexperimentaluncertaintyonthemeasurementinfactmadeitcompatiblewithzero.
Thiswasmuchsmallerthanthevalueof60%predictedfromrelativisticmodelsofthenucleonspin.
Thisunexpectedlysmallcontributionfromquarkspinhasbeentermedthe'spincrisis'.
Subsequentlymoreprecisemeasurementshavebeenmadeby:SMC(SpinMuonCollaboration,thesuccessortoEMC),SLACE-142,E143,E154andE-155Collaborations,HERMES,J-LabHallAandCOMPASS(seeforexample[24–36]).
Thesehaveindicatedthattheintrinsicspinofquarksandantiquarksaccountsforabout30%oftheprotonspin.
ArecentanalysisofglobalpDISdata[37]givesatotalfractionof0.
27±0.
07.
Thetotalprotonspincomponent,measuredalongaparticulardirection,mustbeonehalf.
Thehelicityspinsumrule12=12Σ+G+Lq+Lg(1.
7)describesallthepossiblecontributionstothenucleonspin:thequarkandanti-quarkintrinsicspin,Σ,thegluonintrinsicspin,G,andthequarkandgluonorbitalangularmomenta,LqandLgrespectively.
AsΣ≈0.
3,theremainderoftheprotonspinmustcomprisegluonintrinsicspinandpartonorbitalangularmomentum.
Untanglingthesecontributionsisamajorobjectiveinspinphysics.
MostDISexperimentsareinclusive,andsoonlyaccessthetotalquark-plus-anti-quarkspincontribution,summedoverallavours.
Semi-inclusivedeepinelasticscattering(SIDIS)canprovideinformationonthecontributionfromdifferentquarkandantiquarkavours.
SIDISdiffersfrominclusiveDISinthatahighenergyhadron,producedfromthefragmentationofthestruckquark,isdetectedincoincidencewiththescatteredlepton.
Thehadronprovidesanindicatoroftheavourofthestruckquark.
Thisisbecauseofthepreferenceforaquarktofragmentintoahadroncontainingavalencequarkofthesameavour.
Forexampleanupquarkismorelikelythanadownquarktofragmentintoaπ+becausetheπ+valencestructureisud.
Differenthadronsprovide'tags'fordifferentquarkandanti-quarkavours,allowing1300.
2xu-0.
20xd-0.
20xu–-0.
20xd–-0.
20xs0.
030.
10.
6xFigure1.
7:Flavour-dependenthelicitydistributionsatQ2=2.
5GeV2fromtheHERMESCol-laboration[34].
Theproductxqisshownforeachlight(anti-)quarkspeciesexcepts.
Thedatawerenotabletoconstrainthes(x)distribution;resultsshownareextractedassumings=0.
thetotalquark-plus-anti-quarkspindistributiontobedecomposedintothecontributionsfromdifferentavours.
TheHERMESCollaborationhaveperformedSIDISmeasurementsusingapolarisede±beamincidentonpolarisedprotonanddeuteriumtargets.
Taggingwithpionsand(forthedeuteriumtargetonly)kaons,thedataprovideinformationabouttheu,u,d,dandshelicitydistributions.
Figure1.
7from[34]showstheextracteddistributionsasafunctionofBjorkenx.
Theupquarkdistributionispositiveforallxandthedownquarkdistributionisnegative,indicatingthesequarksarepolarisedparallelandopposite,respectively,tothenucleonspin.
Theseaquarkdistributions,u(x),d(x)ands(x),wereallfoundtobeconsistentwithzerowithinuncertainties.
141.
5GluonPolarisationThegluonhelicitydistribution,G,cannotbedirectlyaccessedinDISbecausephotonsdonotcouplewithgluons.
Howeverlimitedinformationcanbeinferredaboutgluonspinfromscalingviolations,inthesamewaythatg(x)canbeinferredfromscalingviolationsinF2.
MeasurementoftheQ2-dependenceoftheg1structurefunctionallowslimitstobeplacedonthegluonpolarisation.
Analysesofglobalg1data(forexample[38,39])provideameasureofG,buttheuncertaintiesareverylarge;forexample[38]reportsatotalgluonspincontributionof0.
499±1.
266.
Thoughapositivegluoncontributionisfavouredbythetstothedata,anegativegluondistribution,oronewhichchangessignasafunctionofx,cannotbedismissed,asdiscussedin[39].
DISdataalonedonotthereforestronglyconstrainG(gure1.
8).
OtherconstraintsonGusinglepton-nucleoncollisionscomefrommeasurementsofjets,charmmesonsandhadronsproducedatlargemomentumtransversetothebeam(pT).
Theproductionofallofthesearesensitivetoprocessesinvolvinggluons.
MeasurementshavebeencarriedoutofjetsandhighpThadronsbyHERMES,SMC,andCOMPASS[41–43]andofcharmmesonsbyCOMPASS[44].
Anotherpromisingavenueispolarisedproton-protoncollisionsatRHIC.
Jets,highpThadrons,heavyavourproductionandpromptphotonsaresensitivetothegluonpolarisa-tion,andmeasurementsbytheSTARandPHENIXcollaborationsareexpectedtoputmuchstrongerconstraintsonthegluonpolarisation;indeed,earlymeasurementsfrombothPHENIXandSTARhavealreadydoneso[20,45–48].
BothexperimentshaveperformedmeasurementsoflongitudinaldoublespinasymmetriesoftheformALL=σ++σ+σ+++σ+,(1.
8)whereσ++(+-)isthecrosssectionforprotonswiththesame(opposite)helicities.
STARmea-surementsofALLininclusivejetproduction,p+p→jet+X,disfavouralargepositivegluonpolarisation[20],suggestingamaximumvalueof65%oftheprotonspinata90%condencelevel(gure1.
9).
PHENIXmeasurementsofALLforp+p→π0+X[45]havebeenincorpo-15x10-210-1x-0.
0100.
010.
020.
03COMPASSSMCg1d(a)Measurementsoftheg1structurefunctionofthedeuteronbySMCandCOM-PASS[36].
xxgQ2=5GeV2GRSVAACBB-0.
500.
5110-310-210-11(b)ComparisonofanumberofNLOanalysesofpDISdata[40].
Figure1.
8:Measurementsoftheg1structurefunction,suchasthoseshownin(a),areusedinQCDanalysestoextractthepolarisedgluondistribution.
Theconstraintsobtainedbyanumberofanalysesarecomparedin(b).
160.
20.
40.
60.
82/c2=100GeV2Q-310-210-11010Xgluonp=28GeV/cTp=5.
6GeVT/ca)1Gx1050.
250.
50.
751.
00fracdN/d(logx)-410-310-21010CL/c)2=0.
4GeV2G(Q20-1-0.
500.
51-11STARjet+X→ppg=-gGRSVg=0GRSVGRSV-stdg=gGRSVb)Pol.
uncertaintyFigure1.
9:ConstraintsonGfromjetmeasurementsbytheSTARCollaboration.
CListhecondencelevelforvariousgluondistributions.
Themaximallypositive(G=g)distributionisstronglydisfavoured.
ratedintoglobalanalyseswithpDISdatatosignicantlyreducetheuncertaintyonG[37].
Thoughtheuncertaintyremainslargecomparedtothatofthequarkspincontribution,theanal-ysisstronglyfavoursapositivegluonhelicitydistribution.
17Chapter2TransverseSpinPhysics2.
1TheTransversityDistributionTofullydescribethenucleon,athirdcategoryofpartondistributionfunctionsisrequired,inadditiontotheunpolarisedpartondistributionsq(x)andthehelicitydistributionsq(x).
Thesearethetransversitydistributions,δq(x).
TransversityisalsofrequentlydenotedTq(x)orhq1(x)intheliterature.
Transversitycanbeconsideredasthetransverse-spinanalogueofthehelicitydistribution.
Itdescribesthedifferencebetweenthedistributionsofquarkswithspinparallelandoppositetothatoftheirtransverselypolarisedparentnucleon,δq(x)=q↑(x)q↓(x).
(2.
1)indicatesthenucleonspindirectionand↑(↓)thequarkspindirection.
q↑(↓)(x)isthedistributionofquarksofavourqwithinthenucleonwithspinparallel(opposite)tothatoftheparentnucleon.
Thustransversitydescribesthedegreetowhichthetransversequarkspiniscorrelatedwiththetransversenucleonspin.
Fromthedenitioninequation(2.
1)itfollowsthatthetransversitydistributionsmustobeytheboundδq(x)≤q(x)(2.
2)18inordertoalwaysbepositive.
TheSofferinequality[49]providesanothermodel-independentconstraintonthetransversitydistributionandrelatesittotheunpolarisedandhelicitydistribu-tionsatleadingorderinQCD:2|δq(x)|≤q(x)+q(x).
(2.
3)Thefactthatthehelicityandtransversitydistributionsdifferreectstherelativisticnatureofthenucleon'sconstituents.
Inanon-relativisticcaseaseriesoftransformationsandrotationscanbeusedtochangefromalongitudinallypolarisedtoatransverselypolarisedproton.
IntherelativisticcaseLorentzboostsandrotationsdonotcommute.
Asaresultthetransversityandhelicitydistributionsneednotbethesame.
Bytheopticaltheorem,thetransversitydistributionisrelatedtoscatteringamplitudesthatinvolveaipofthequarkandnucleonhelicities[50].
Transversitydistributionsarethusde-scribedas'chiral-odd'becausetheyareinvolvedwithahelicityip.
Thiscontrastswiththeunpolarisedandhelicitydistributionfunctionswhichare'chiral-even':theyinvolvenohelicityip.
Thereisnoleading-twist1gluontransversitydistributionforapolarisedspin-targetbe-causeofhelicityconservation.
Gluonsarespin-1bosons,sohavehelicity±1.
Agluonhelicityipthereforeinvolvesahelicitychangeof±2,whichaspin-nucleoncannotaccomodate.
Ininclusivedeepinelasticscattering,whichprovidesthemajorityofourunderstandingofpartondistributions,chiral-oddprocessesarenotobserved,becausehelicityisconservedinper-turbativeQCDinteractions.
Forthisreasontransversitydistributionsaremuchlesswellunder-stoodthanhelicitydistributions.
Inorderforaprocessrelatedtotransversitytobeobservable,asecondchiral-oddfunctionmustbeinvolved.
Thecombinationoftwochiral-oddfunctionsthenconserveshelicityoverall.
Inhadron-hadroncollisions,thetwochiral-oddfunctionscanbeprovidedbythetransversitydistributionsofthetwonucleons.
Transversitymaythenbestudiedviatransversedoublespinasymmetriesinparticleproductionresultingfromcollisionsbetweentwotransverselypolarisedhadrons.
Anotherpossibilityisachiral-odd,spin-dependent1twistdescribestheorderin1/Qatwhichaneffectisseeninexperiment.
Aneffectoftwistt,wheret>1,issuppressedbyafactorQ(2t).
Aneffectatthelowesttwist,t=2,isreferredtoas"leadingtwist",andisnotsuppressedbyafactorofQ,whilesuppressedeffects,associatedwithlargervaluesoft,are"highertwist".
19fragmentationprocess.
This,combinedwithtransversity,cangiverisetotransversespinasym-metries.
Transversityisthereforerelatedtotheobservationoftransversespinasymmetriesinhadroniccollisions.
Bymeasuringthese,itmaybepossibletoinferinformationaboutthetransver-sitydistributions.
Transversespinasymmetriesarealsorelatedtothestudyoftransverse-momentum-dependentpartondistributionsandpartonorbitalmotion.
Iwillrstgiveasummaryofexperimentalmeasurementsofsuchtransversespinasymmetries,whichhaveahistoryspan-ninganumberofdecades.
Afterdescribingthemeasurements,Iwillhighlightthemechanismsproposedtoexplaintheirexistence.
Theserelatetotransversity,spin-dependentfragmentationandtransverse-momentum-dependentpartondistributions.
Finally,Ishalldiscussrecentexper-imentalworkthatisbeginningtoprovidetherstinformationonthetransversitydistribution.
2.
2TransverseSpinAsymmetriesSincethe1970's,signicanttransversespineffectshavebeenobservedinhadroniccollisions.
TherstobservationwasthatΛhyperonsproducedininelasticproton-berylliumcollisions,p+Be→Λ+X,arestronglyspin-polarisedtransversetotheirproductionplane[51].
Laterexperimentsfoundunexpectedlylargetransverseproductionasymmetriesformanyspeciesininclusiveproton-protoncollisions.
Considercollisionsbetweenatransverselypo-larisedandanunpolarisedproton:p+p→d+X,wherepdenotesthetransverselyspin-polarisedproton.
Theparticled,ofaspeciesofinterest,isdetected,whileXindicatestheremainingunmeasuredhadronicnalstate.
Thesinglespinasymmetry(SSA)oranalysingpower(AN)intheproductionofdcanbedenedasAN=1PNleftNleftNleft+Nleft,(2.
4)wherePistheaveragetransversespin-polarisationofthepolarisedprotonbeamortarget.
N()leftisthenumberofparticlesproducedtotheleftofthebeamwhenthetransverselypolarisedbeamdirectionisup(down).
Nleft+Nleftissimplythetotalnumberofparticlesproducedtotheleftof20thebeam.
Theasymmetrythusmeasuresthedifferenceinparticleproductiontobeam-leftuponippingofthebeampolarisation.
Rotationalinvariancerequiresproductiontotheleftwhenpolarisationisuptoequalproductiontotherightwhenpolarisationisdown:Nleft=Nright.
Thesinglespinasymmetryisthusequivalenttothedifferencebetweenparticleproductiontotheleftandrightofthebeamforaxedpolarisation.
Hencethesinglespinasymmetryisoftenreferredtoastheleft-rightasymmetry.
Equation(2.
4)isdenedsuchthatthattheasymmetryispositiveifparticleproductiontotheleftofthebeammomentum-polarisationplaneexceedsthattotherightwhenthebeampolarisationdirectionisup.
InitialexpectationsfromperturbativeQCDargumentswerethatsuchasymmetriesshouldbesmallathighenergies[52].
AtleadingorderinQCD,ANispredictedtobeapproximatelyzero,beingsuppressedbyafactorofmquark/√s,where√sisthecentreofmassenergyofthecollision.
Howeversuchasymmetrieshavebeenobservedandareinmanycasesverylarge.
ResultstakenattheArgonneZeroGradientSynchrotron(ZGS)inthe1970sfoundlargeasym-metries,inexcessof10%,intheproductionofchargedpionsandkaonsinp+pandp+2Hcollisionsat6and11.
8GeV/c[53,54].
AsymmetrieswerefoundtobesmallatsmallFeynmanx(xF=2pL/√s,wherepListheparticle'slongitudinalmomentum)andlargeatlargexF.
Asignicantasymmetryinπ0productionnearxF=0wasfoundatCERNin24GeV/cp+pcol-lisions,whichincreasedwiththepTofthepion[55].
AnumberofmeasurementswerecarriedoutattheBrookhavenNationalLaboratory(BNL)AlternatingGradientSynchrotron(AGS)withbeammomentaof13.
3and18.
5GeV/c[56–59].
Thespeciesmeasuredwereπ±,p,K0SandΛ.
π+showedaclearpositiveasymmetryatforwardangles,xF>0.
2,increasingwithpTtoaround25%atpTof2GeV/c.
AtsmallervaluesofxFtheasymmetrywasconsistentwithzero.
K0Sshowedasignicantnegativeasymmetryof-10%forxF0.
6andmoderatetransversemomentum,0.
60.
2,andmoder-atepT1GeV/cfrompolarisedprotoncollisionsat√s=62.
4GeV[72].
Atbackwardangles22Figure2.
1:SinglespinasymmetriesforpionsmeasuredbytheE704Collaborationforp+p→π+Xwitha200GeV/cpolarisedbeam.
(GeV/c)Tp0.
511.
522.
533.
544.
55NA-0.
2-0.
15-0.
1-0.
05-00.
050.
10.
150.
2h+h-035%notincludedscaleuncertaintyofNAFigure2.
2:Singlespinasymmetryinmid-rapidityπ0andchargedhadron(h±)productionfrompolarisedproton-protoncollisionsat√s=200GeV,measuredbythePHENIXCollaboration[69].
23Figure2.
3:Singlespinasymmetryintheproductionofπ0mesonsfrompolarisedproton-protoncollisionsat√s=200GeV,measuredbytheSTARCollaboration[71].
Largeasymmetriesareseentopersisttolargetransversemomentaatforwardangles(xF>0.
4).
asymmetriesareallconsistentwithzero,butpionsandkaonsshowsignicantasymmetriesatforwardangles,increasingwithxFtoAN≈0.
2atxF=0.
6(gure2.
4).
Protonsshowzeroasymmetryatforwardangles,unlikethemesonspecies.
Experimentalresultstodatehaveshownsignicantnon-zeroasymmetriesinavarietyofspecies.
Themagnitudesandsignsoftheasymmetriesarehighlydependentontheavouroftheproducedparticle.
Itisthereforeusefultomeasuretheasymmetriesforawidevarietyofidentiedspecies,asinformationontheavourdependenceoftheasymmetriesmayprovideinsightsintotheirphysicalorigin.
Additionally,thedependenciesoftheasymmetriesonpTandxFarenotyetunderstood.
Measurementsoverdifferentkinematicrangesarethereforealsousefulinconstrainingmodelsoftheasymmetry.
Understandingtheoriginoftransversespinasymmetrieswillaideinelucidatingthetransversespinstructureofthenucleon.
ResultsbeforetheRHICerawereatlowtransversemomenta,typicallymeasuringtheasym-metriesinparticlesproducedwithpT≈1GeV/candsmaller.
ThisistoolowforpQCDtobeapplicableintheanalysisofthedata.
HigherenergyexperimentsatRHIChavedoneandcon-tinuetomeasureasymmetriesatmuchlargertransversemomenta,facilitatingtheapplicabilityofpQCDtotheanalysisofthedata.
ItsexcellentparticletrackingcapabilitymeansthattheSTARexperimentinparticulariswellsuitedtotheidenticationofavarietyofspeciesatlarge24Figure2.
4:SinglespinasymmetryintheproductionofK±mesonsfrompolarisedproton-protoncollisionsat√s=62.
4GeV,measuredbytheBRAHMSCollaboration[72].
momenta,especiallynearmid-rapidity.
2.
3CollinsFragmentationFunctionsAsdiscussedabove,thechiral-oddnatureoftransversitymeansthatitcanonlybestudiedincombinationwithanotherchiral-oddfunction.
Collins[73]proposedachiral-odd,transverse-momentum-dependentfragmentationfunctioninwhichtheazimuthaldistributionofhadronsproducedbyafragmentingquarkiscorrelatedwiththequark'stransversespindirection.
Inacollisionsuchasp+p→π+Xthiscorrelation,combinedwiththetransversitydistribution,cangiverisetoaspin-dependenttransverseasymmetryintheproductionofthepion.
TheCollinsfragmentationfunctionactsasananalyserofthetransversequarkpolarisationinatransverselypolarisedhadron.
Thefragmentationofatransverselypolarisedquark,q↑,intoanunpolarisedhadron,h,canbeexpressedas[74]Dh/q↑(z,Ph⊥)=Dq1z,P2h⊥+H⊥q1z,P2h⊥k*Ph⊥·SqzMh.
(2.
5)kisthemomentumdirectionofthequarkandSqisitstransversespin.
TheproducedhadronhasmassMhandcarriesafractionzofthemomentumofthequark.
Ph⊥isthetransversemomentum25ofthehadronwithrespecttotheoriginalquarkdirection.
Therstterminequation(2.
5)containsthespin-independentpartofthefragmentationprocess.
Thesecondtermdescribesthetransverse-spin-dependentpartoffragmentation.
ThefunctionH⊥q1iscalledtheCollinsfunctionanddescribesthemomentumdependenceofthespin-dependentpart.
Theterm(k*Ph⊥)·Sqchangessignunderaipofspin,andgeneratesaspin-dependentazimuthalvariationinhadronproduction.
ExperimenthasbeguntoprovidetherstinformationabouttheCollinseffectandindicatesthatitisnon-vanishing.
TheHERMESCollaborationhavereportedresultsfromsemi-inclusiveDISofpositronsincidentonatransverselypolarisedprotontarget[75].
Anon-zeroasymme-tryinchargedpionproduction,arisingfromthecombinationoftransversityandtheCollinsfunctions,isobserved.
ThisindicatesthatboththetransversitydistributionandtheCollinsfunctionarenon-vanishing.
Theobservedasymmetryhasoppositesignandcomparablemag-nitudeforpositiveandnegativepions.
ThelargemagnitudeandoppositesignofthenegativepionasymmetrycanbeexplainedifthedisfavouredCollinsfunctionhasasigncantmagnitudeandoppositesigncomparedtothefavoured2Collinsfunction.
TheCOMPASSCollaborationhavemeasuredchargedhadronproductionincollisionsbe-tweenmuonsandpolariseddeuteronsandfoundallasymmetriestobesmallandcompatiblewithzero[76].
Whentakenwiththenon-zeroresultswithaprotontargetfromHERMES,theCOMPASSresultssuggestthecancellationofasymmetriesfromtheprotonandtheneutroninthetarget.
TheBelleCollaborationhaveobservedazimuthalasymmetriesofafewpercentindihadronproductionine+ecollisionsat√s=10.
58GeV[77,78].
Becausetheseareleptoniccolli-sions,transversityisnotinvolvedandtheBelleresultsprovideadirectindicationoftheCollinsfunctions.
TheseresultsconrmtheHERMESobservationthatthefavouredanddisfavouredCollinsfunctionshaveoppositesign.
2Afragmentationfunctiondescribingaquarkfragmentingintoahadronissaidtobefavouredifthehadroncontainsavalencequarkofthesameavourasthefragmentingquark,forexampleauquarkfragmentingintoaπ+.
Iftheproducedhadrondoesnotcontainaquarkofthesameavour,thefragmentationfunctionisdisfavoured.
262.
4TheSiversMechanismAnothermechanismforgeneratingtransversesinglespinasymmetrieswasproposedbySivers,involvingtheintrinsictransversemomentumofthenucleonconstituents,kT[79].
Siversas-sumedthattherecouldbeacorrelationbetweenthespinofaprotonandtheorbitalmotionofthe(unpolarised)partonconstituents.
ThisgivesthepossibilityofanasymmetryinthepartonickTdistributioninthedirectionnormaltotheplanedenedbytheprotonmomentumandspindirections.
IftheintrinsickTsurvivesthefragmentation/hadronisationprocessfollowingscat-tering,theimbalanceintheintrinsicmomentumcanbeobservedasaleft-rightimbalanceinthepTdistributionoftheproducedhadrons.
TheSiversmechanismisthereforerelatedtopartonicmotionwithinthenucleon.
TheSiversmechanismisnotrelatedtotransversity;itisaseparatemechanisminvolvedinSSAs.
Thepartondistributionfunctioncanbeexpressedasa(conven-tional)spin-independentterm,plusaspin-dependenttermmultipliedbyaspecialpartondensityfunction.
ThisspecialpartondensityiscommonlyreferredtoastheSiversdistributionfunc-tionandisdenotedf⊥1T(x,kT).
Notethatitisatransverse-momentum-dependentdistribution,incontrasttousual,transverse-momentum-integratedPDFs,q(x).
ForalongtimetheSiversdistributionwasbelievedtoberequiredtobezeroduetoargu-mentsrelatedtotime-reversalsymmetryinQCD[73].
Morerecentlyhowever[80–83],workhasshownthatsuchanasymmetryisallowed,byaccountingfornalstateinteractionsbetweentheoutgoing,scatteredquarkandthespectatorhadronicremnant.
'Finalstate'referstothefactthattheinteractionsoccurafterthescatteringofthequark.
However,thisinteractionisnot'-nalstate'inthesenseofbeingrelatedtofragmentationorhadronisationofthequark;thegluonexchangenalstateinteractionsoccurbeforethis.
IthasalsobeenshownthattheSiversdistributionisnon-universal[81,82];thatis,themeasuredfunctiondependsontheprocessstudied.
Thisisincontrasttotheconventional(transverse-momentum-integrated)PDFs,whichareuniversal(thesameineveryscatteringpro-cess).
Forexampleapredictiongivenin[81]isthattheSiversdistributionforDrell-Yanpro-ductionisequalinmagnitudebutdiffersinsigntothatindeepinelasticscattering.
Aqualitativeunderstandingisprovided[40]byrecallingthatthequarkmustundergoadditionalinteractions27inorderfortheSiverseffecttobenon-vanishing.
Thisinteractioncanbethoughtofasthequarkscatteringinthecoloureldofthespectatorremnant.
Differentcollisionprocesseswillresultindifferentforcesactingonthequark,givingrisetodifferentSiversfunctions.
TheHERMESCollaborationhavemadetherstreportofanon-zeroSiversfunction[75].
SIDISproductionofπ+withtheHERA27.
5GeVpositronbeamshowedanasymmetrycorre-spondingtoanegative,non-zeroSiversfunction.
TheCOMPASSCollaborationhavereportedmeasurementsofSiversasymmetriesforinclusivepositivelyandnegativelychargedhadrons[76],andforpionsandkaons[84]inSIDISwitha160GeV/cmuonbeamanddeuterontarget.
Allasymmetrieswerefoundtobesmallandconsistentwithzero,suggestingcancellationofupanddownquarkcontributionsfromthedeuterontarget.
TheSTARexperimenthaspresentedresultsfordi-jetproductioninp+pcollisionsat√s=200GeV[85].
Measurementsweremadeoftheopeninganglebetweenthejets.
ThekTasymmetryproducedbytheSiversmech-anismmaymanifestasanopeningangleotherthan180degrees(back-to-backjets).
Observedasymmetrieswerefoundtobesmallandconsistentwithzero,andsmallerthanSIDISresultsfromHERMES.
pQCDcalculationssuggestthedifferenceisduetocancellationbetweenupanddownquarkcontributions[86],andbetweennal-andinitial-stateinteractions,bothofwhichcontributeinthejetproductionmechanism[87].
2.
5MeasurementsofTransversityRecentlythetransversitydistributionsofuanddquarksintheprotonhavebeenextractedforthersttime[88].
SIDISdatafromtheHERMESandCOMPASSCollaborations,measuring+p→+π+X,ande++e→h+h+XdatafromtheBelleCollaborationwerestudied.
TheHERMEStransverseasymmetriesinvolvethecombinationoftransversitywiththeCollinsmechanism.
TheBelledataprovideadirectmeasureoftheCollinsfunctions,andsoallowthetransversitydistributionstobedeterminedfromtheHERMESdata.
Thetransversitydistribu-tionswereparameterisedandthebest-tparametersdeterminedfromaglobalttothedata.
Theextractedtransversitydistributionsareshowningure2.
5.
Theextractedupquarkdistribu-28tionispositiveforallxwhilethedownquarkdistributionisnegative.
Theupquarkdistributionisgreaterinmagnitudethanthedownquarkdistribution,|δu(x)|>|δd(x)|andbotharesmallerinmagnitudethantheSofferboundgiveninequation(2.
3).
Therststepsarenowbeingmadetowardunderstandingthequarktransversitydistribu-tions,thoughtheyremainmuchlesswellunderstoodthantheunpolarisedandhelicitydistribu-tions.
2.
6AimsofThisThesisTheworktobepresentedherewasperformedusingdatatakenbytheSTARexperimentattheRelativisticHeavyIonCollider(RHIC).
TheRHICphysicsprogrammeencompassesstudiesofheavyioncollisionsandpolarisedprotoncollisions,ofwhichthepolarisedprotonprogrammeisofinteresthere.
RHICiscapableofprovidingbothlongitudinallyandtransverselypolarisedprotons,andanextensivespinprogrammehasbeeninoperationsince2002.
Studiesusinglon-gitudinallypolarisedprotonshaveyieldedconstraintsonthepolarisedgluoncontribution,G,tothespinoftheprotonbymeasuringdoublehelicityasymmetries,ALL,intheproductionofjetsandpions.
Withtransverselypolarisedprotons,investigationshavebeenperformedintothetransversespinstructureoftheprotonviameasurementsoftransversesinglespinasymmetriesinhadronandjetproduction.
Thisthesispresentsastudyofsingleanddoubletransversespinasymmetriesinthepro-ductionoftheneutralstrangeparticlesK0S,ΛandΛ,usingtransverselypolarisedprotondataacquiredduringthe2006RHICrun.
TheseparticlesarewellsuitedtostudybySTAR,astheycanbeidentiedoveralargemomentumrangeusingtopologicalreconstructionoftheirdecayproducts,whilealsohavingareasonableproductioncrosssection.
Bycontrast,chargedspeciessuchaspionsandprotonscanonlybemeasuredoveralimitedmomentumrange.
TransversesinglespinasymmetriesintheproductionoftheseparticleshavebeenmeasuredbeforeatAGSandbytheE704Collaboration,asmentionedinsection2.
2.
STARprovidestheopportunitytoextendthesestudiestosignicantlyhighercollisionenergyandparticlemomentumthanhas29Figure2.
5:Transversitydistributionsofuanddquarksintheprotonextractedfromaglobalttodata,takenfrom[88].
Theshadedregionshowsaone-sigmauncertaintyaroundthebest-tdistribution.
TheboldlinesoutsidetheshadedregionindicatetheSofferbound.
30previouslybeenattained.
Theremainderofthethesisisorganisedasfollows.
Inchapter3theRHICcomplex,itsoperationasapolarisedprotoncolliderandtheSTARexperimentaredescribed.
ThetechniquesusedtoidentifyK0S,ΛandΛparticlesarethendetailed.
Inchapter4thedatasetusedintheanalysisispresentedandtheextractionofK0S,ΛandΛyields,usingthetechniquesoutlinedinchapter3,isdescribed.
Chapter5presentsthemethodsusedtocalculatethetransversesinglespinasymmetriesintheproductionofeachparticlespecies,anddescribesanumberofsystematicchecksperformedontheresults.
Chapter6presentstheanalysisoftransversedoublespinasymmetries.
Finally,chapter7summarisestheresultsandprovidesanoverviewoffuturetransversespinphysicsexperimentsplannedatRHICandelsewhere.
31Chapter3TheExperiment3.
1TheRelativisticHeavyIonColliderTheRelativisticHeavyIonCollider(RHIC)[89,90]islocatedatBrookhavenNationalLabo-ratory,NewYork.
Itisanintersectingstorageringthatacceleratestwoindependentbeamsofionswithmassnumbersfromonetoaround200usingsuperconductingmagnets.
Themaxi-mumenergypernucleondecreaseswithmassnumberfrom250GeVforprotonbeamsto100GeVfortheheaviestionssuchasgold,A=197u.
Becausethebeamsareindependentofoneanothereachneednotbeofthesamespeciesandasymmetriccollisionscanbeandhavebeenperformed(forexamplebetweendeuteronsandgoldions).
Figure3.
1showsaschematicviewofalltheelementsoftheRHICcomplex.
TheRHICringhasa3.
8kmcircumferenceandisapproximatelycircular,withsixarcsectionsandsixstraightregions.
Inthestraightsectionsthebeamsaresteeredtointersectsothatcollisionscanoccur.
Collisionpointsareatthetwo,four,six,eight,tenandtwelveo'clockpositions,inthemiddleofthestraightsections.
Experimentalhallsaresituatedatthecollisionspoints:BRAHMSattwoo'clock,STARatsix,PHENIXateightandPHOBOSatten.
TheboosteracceleratorandAGSareusedtoaccelerateionstoRHICinjectionenergy.
TheLINACaccelerateshydrogenionsforuseinprotoncollisions,whiletheTandemvanderGraaffgeneratorisusedtoaccelerateheavierions.
Onlyoperationsrelatingtoproton-protoncollisionswillbediscussedhere.
32Figure3.
1:TheRHICcomplex.
ProtonsfollowapaththroughtheLinac,Booster,AGS,ATR(AGStoRHICline)andintotheRHICring.
TheTandemVandeGraaffgeneratorisusedintheaccelerationofheavyions.
Polarisedprotonsareproducedusinganoptically-pumpedpolarisedionsource[91,92],whichtypicallygenerates0.
5mA,300spulsesofions,correspondingto9x1011ionsperpulse,withupto90%polarisation.
TobetterachievehighluminositytheRHICbeamsarenotcontinuousbutareinsteadcompressedinto120'bunches'ofparticles,eachlessthan30cminlength.
BecauseoflossesduringaccelerationandtransfertoRHIC,pulsesof9x1011ionsareneededfromthesourcetoprovidetherequiredRHICluminosityof1.
4x1031cm-2s-1,whichcorrespondstobunchesof2x1011protons.
Protonsarepassedthrougharubidiumvapourpumpedwithcircularlypolarisedlaserlightinastrongmagneticeld,wherebytherubidiumelectronsare95-100%polarised[93].
Apolarisedelectronistransferredtotheprotonsthroughcollisions,andmagneticeldsareusedtotransfertheelectronpolarisationtothehydrogennucleus.
Theneutral,nuclear-polarisedhydrogenatomsarenallyionisedtoH-bycollisionswithasodiumvapour.
Accelerationoccursinfourstages:LINAC,booster,AGSandRHIC.
FirsttheH-ionsare33acceleratedtoakineticenergyof200MeVintheLINAC,withanefciencyofaround50%,andthenstrippedoftheirelectronsandinjectedasasingle4x1011ionbunchintotheboosterring.
Theboosteracceleratestheprotonstoakineticenergyof1.
5GeVanddeliversthemtotheAlternatingGradientSyncrotron(AGS),whichacceleratestheprotonsto25GeVforinjectionintoRHIC.
RHICthenaccelerateseachbeamtothedesiredcollisionenergy;forprotonsthedesignminimumis30GeVandthemaximum250GeV.
TheRHICcomplexisdescribedinmoredetailinreferences[89,90,94].
ThestableprotonspindirectionintheRHICringisverticali.
e.
transverselypolarisedbeams.
TheprotonspinsprecessaroundtheverticalmagneticeldintheRHICringatarateofGγprecessionsperorbit,whereG=1.
7928,theanomalousmagneticmomentoftheproton,andγistherelativisticfactor.
Imperfectionsinelectricandmagneticeldscanperturbthespindirection.
Theseperturbationstypicallycancelovermanyorbitsbecausethespinisatadifferentpointinitsprecessionineachorbit.
However,whentheprotonspinisatthesamepointinitsprecessiononeachorbit,correspondingtoGγ=integer,the'kicks'tothespindirectionsadduponconsecutiveorbitsandleadtodepolarisationofthebeam.
Spinresonancesofthissortoccurevery523MeVforprotons,somanyareencounteredduringbeamacceleration.
Topreventlossofbeampolarisationduetothesespinresonances,twohelicalSiberianSnakemagents[95]areinstalledineachbeam,oneatthethreeo'clockandoneatthenineo'clockposition.
ASiberianSnakerotatesthestablespindirectionby180degreesaroundahorizontaldirection,whicheliminatespolarisationlossesfromspinresonances.
ThedesignmaximumpolarisationforRHICis70%.
The80-90%polarisationatthepolarisedsourceisreducedbylossesduringthevariousaccelerationandtransportstages.
SpinrotatorsplacedeithersideofbothPHENIXandSTARallowrotationofthestabledirectionattheseexperimentsifsodesired,inordertostudylongitudinallypolarisedprotons.
AlldatausedinthisanalysiswereacquiredwithtransversebeampolarisationsatSTAR.
Eachbeam'spolarisationismonitoredusingitsowncarbon-targetCoulomb-NuclearInter-ference(CNI)polarimeter.
Thesemeasuretheasymmetryinrecoilcarbonatomsfromelasticp+CscatteringbetweenthepolarisedRHICprotonbeamandthecarbontargetofthepo-34larimeter.
Atthesmallmomentumtransfers(0.
002to0.
01GeV2)atwhichthescatteringsoccur,thep+Cscatteringprocesshasasignicantanalysingpowerofaround4%.
Thescat-teringcross-sectionislargeandonlyweaklydependentonbeamenergyovertheRHICrange,allowingquickmeasurementsofthebeampolarisation.
Measurementsof107carbonatoms,sufcientforastatisticalprecisionofafewpercentonthepolarisationmeasurement,canbeacquiredin30seconds.
Thecarbontargetsusedareverythin(≈5mwideand150thickforatarget2.
5cminlength),allowingtherecoilcarbonatomstoescapethetargetandbedetected.
Theuseofthintargetsalsokeepsthelossofbeamluminosityduetoscatteringtoanacceptablelevel.
Siliconstripdetectorssurroundingthecarbontargetallowmeasurementofboththeverticalandradialtransversecomponentsofthebeampolarisation.
Becauseoftheoreticaluncertain-tiesinthep+Canalysingpower,thepCpolarimetersarecalibratedusingasinglepolarisedhydrogenjettargetsharedbetweenthebeams.
AmeasurementofonetotwodaysisrequiredtocalibratethepCpolarimeterstowithin5%.
ThecalibratedpCpolarimetersarethenusedtomeasurerelativevariationsinthebeampolarisation.
EachRHICbeamstoretypicallylastseighttotenhours.
Thepolarisationofeachbeamismeasuredatthestartofthestoreandagainatapproximatelytwo-tothree-hourintervalsduringthestoreusingthepCpolarimeters.
Thisintervalisacompromisebetweenthedesirestomonitorthepolarisationfrequentlyandtominimisebeamlossesincurredduringpolarisationmeasurement(0,forbeam-right,(5.
3)wherepV0istheV0momentumvectorandpbeamandPbeamarerespectivelythebeammomen-tumandpolarisationvectors.
BeamleftcorrespondstonegativexintheSTARcoordinatesys-68temfortheclockwisebeam,andpositivexfortheanticlockwisebeam.
Becausetheasymmetryinvolvestakingratiosofparticleyields,theacceptanceofthedetectorandtheefcienciesoftheSTARtrigger,V0reconstructionandanalysiscutscancelout,greatlysimplifyingtheanalysis.
Thefactorofcosφinequation(5.
2)andhowthisisaccountedforintheanalysisarediscussedfurtherinsection5.
2.
Singlespinasymmetrymeasurementsrequireonlyonebeamtobepolarised.
BecausethetwoRHICbeamsareindependentlypolarised,twomeasurementsofthesinglespinasymmetrycanbemadeusingthesamedata.
Ineachcase,onebeamistreatedasthepolarisedbeam,whiletheotheristreatedasunpolarisedbysummingparticleproductionoverboththepolarisationstatesofthatbeam.
Twomethodswereusedtocalculatethesinglespinasymmetriesandtheresultscomparedtocheckconsistency.
Onemethodinvolvedusingtherelativeluminositiesofthebeamstocorrectthecountsforeachpolarisationpermutation.
Theotherusesacombinationofparticleproductiontobothsidesofthebeamtocanceltheeffectsofdifferingbeamluminosities.
Thesetwomethodswillbediscussedindetailnow.
5.
1.
1RelativeluminosityMethodThebunchesineachRHICbeamarenotidenticalintheirspatialprole,andsoeachprovidesaslightlydifferentluminosity.
Themoretightlytheprotonsarebunched,thehigherthelumi-nositywillbe.
Thismeansthatthereiseffectivelyadifferentluminosityforeachpermutationofthebeampolarisationsand)inanygivenbeamstore.
Toaccountforthis,whencalculatingtheasymmetrytheyieldforeachpolarisationstatemustbescaledbythecorrespondingluminosity.
Accountingforthis,equation(5.
2)thenbecomes:ANP=1cosφNLNLNL+NL.
(5.
4)whereLandLindicatethebeamluminosityforbunchesofupanddownpolarisationrespectively.
Totreatonebeamaseffectivelyunpolarised,yieldsfromthetwopolarisation69statesofthatbeamaresummed,N=N+N,N=N+N,(5.
5)wheretherstsuperscriptarrowineachtermontherightofequation(5.
5)indicatesthepolari-sationdirectionofthe'polarisedbeam'andthesecondarrowindicatesthepolarisationdirectionofthe'unpolarisedbeam'.
Astheluminositiesforthesestatesarenotnecessarilythesame,theymustbescaledbytheappropriateluminosityindividually:NL→NL+NL,NL→NL+NL.
(5.
6)Becausetheasymmetrymeasurementinvolvestakingaratio,itisadequatetouseonlytherel-ativeluminositiesbetweenbeams,meaningthattheabsolutenormalisationoftheluminositiesdidnotneedtobeknown.
Buncheswithbothbeamspolariseddown(bunches)wereusedasthereferenceforcalculatingrelativeluminosities.
Thefourrelativeluminositieswerethusdened:R=LL,R=LL,R=LL,R≡1.
(5.
7)TherelativeluminositieswerecalculatedbytakingtheratiosofBBCcountratesforcollisions70witheachpermutationofbeampolarisations.
Particleyieldsforeachpermutationofbeampolarisationswerescaledbytherelativelumi-nosityforthatpermutation.
Thefullexpressionfortheasymmetrywasthen:ANP=1cosφN/R+N/RN/R+NN/R+N/R+N/R+N.
(5.
8)Theasymmetrycanbecalculatedusingcountstobeamleftorcountstobeamright.
Byro-tationalinvariance,theasymmetryusingparticleproductiontotheleftisequivalenttothenegativeofthatusingproductiontotheright.
Foreachbeam,boththeleft-asymmetryandtheright-asymmetrywerecalculated,andtheleft-asymmetryaveragedwiththenegativeoftherightasymmetry.
Asymmetrieswerecalculatedseparatelyforforwardandbackwardanglesrelativetothebeamdirection.
Theresultsforeachbeamwerethensummedtogiveanaveragevaluefortheasymmetryatforwardanglesandatbackwardangles.
5.
1.
2CrossRatioMethodForthe'cross-ratio'method,NandNinequation(5.
2)weredenedasN=LR,N=LR,(5.
9)whereLandRindicateparticleyieldtobeam-leftandbeam-rightrespectively.
Thiscombinedtheparticleproductiontoeachsideofthebeamforoppositebeampolarisationsatthestart,ratherthancombiningbyaveraging'left'and'right'asymmetriesattheendasintheafore-mentionedmethod.
Becauseofrotationalinvariance,particleproductiontothebeam-leftforonepolarisationmustbeequivalenttothattobeam-rightfortheoppositepolarisation.
Equation(5.
9)thusdenes'effective'yieldstobeamleft.
Tomeasurethesinglespinasymmetry,onebeamisagaintreatedasunpolarisedbysummingoveritspolarisationstates.
Thetermsinequation(5.
9)arethendenedas71L=L+L,L=L+L,R=R+R,R=R+R.
(5.
10)Therstsuperscriptarrowsinthetermsontherightindicatethepolarisationstateofthepo-larisedbeamandthesecondthestateofthe'unpolarised',beam.
Thecompleteexpressionfortheasymmetryusingthecross-ratiomethodisANP=1cosφL+LR+RL+LR+RL+LR+R+L+LR+R.
(5.
11)Thecross-ratiomethodofcalculatingtheasymmetryhastwoadvantages.
First,itcancelsouttheeffectsofdetectoracceptance,aswiththeabovemethod.
Second,becauseeachterminthenumeratoranddenominatorinequation(5.
2)containsacontributionfrombuncheswithbothupanddownpolarisationviaequation(5.
9),polarisation-dependentluminositydifferencesbetweenbunchesalsocancelout.
Thismeansthattheasymmetrycanbecalculatedwithoutknowledgeofthebeamluminosities.
Itisthereforeunnecessarytoscaleparticleyieldsbytheluminosityforthecorrespondingpolarisationstateofthecollisionscreatingthem.
Thissimpliesthecalculationoftheasymmetryandallowsmorerunstobeusedintheanalysis,asrelativeluminositymeasurementswerenotavailableforallruns.
Italsonegatestheeffectofanysystematicerrorsintheluminositymonitoring.
Todemonstratethecancellationoftheluminosity,theasymmetrywascalculatedforanum-berofrunsusingequation(5.
11).
Theasymmetrieswerethenrecalculated,usingyieldsscaledbytherelativeluminosities.
Theresultsofthesecalculationsarecomparedingure5.
2.
Thesinglespinasymmetrycalculatedwhentherelativeluminosityeffectsareexplicitlyaccounted72STARrun71290017129002712900371290187129020712902371290317129032712903571290367129041NAnalysingpower,A-1-0.
8-0.
6-0.
4-0.
200.
20.
40.
60.
81Figure5.
2:Thecancellationofbeamluminosityusingthecross-ratiomethodofasymmetrycalculation(equation(5.
11)).
TheshownasymmetriesarecalculatedusingyieldsofΛhyperonsatforwardangles,treatingtheclockwisebeamasthepolarisedbeamandtheanticlockwisebeamasunpolarised.
Solidpointsshowtheasymmetrycalculatedwithnoaccounttakenofbeamluminosity.
Theempty,offsetpointsshowtheeffectofexplicitlyincludingluminosity.
Thedifferenceisnegligible.
forbarelydifferfromthosewithouttheluminosityconsidered.
Deviationsarenegligiblecom-paredtothestatisticaluncertaintiesinallcases,showingthatthecross-ratiomethodsuccessfullyaccountsfortheeffectofbeamluminosity.
Forthisreasonthecross-ratiomethodisthepreferredmethodforcalculatingtheasymme-try.
Therelativeluminositymethodwasusedasacheckonthecross-ratioresults,tolookforsystematicerrors.
5.
2AzimuthalWeightingCaremustalsobetakentoaccountfortheazimuthalparticledistributionwhencalculatingtheasymmetry.
Theasymmetryisstrongestinthedirectionnormaltothebeam-polarisationplaneandgoestozeroalongthepolarisationdirection.
Thefactorof1/cosφinequation(5.
2)accountsforthis.
Inpreviousxed-targetexperimentssuchasthoseattheAGSthedetectoracceptedonlyparticlesproducedinasmallrangeinazimuth,wherebyasinglevalueforcosφ73couldbeused[56].
TheSTARdetectorcoversinsteadafull2πradiansinazimuth.
Theφdistributionofparticles,producedfromabeamwithpolarisationPandwithanalysingpowerANisgivenbyequation(5.
1),N(φ)∝1+ANPcosφ.
Theanalysingpowercaninprinciplebeextractedbybinningtheparticleyieldsintorangesincosφandttingastraightlinetotheresults.
Theanalysingpowerwouldthenbefoundfromthegradientofthelineusingtheknownbeampolarisation.
Howeveritwasfoundthatthestatisticsinthedataavailablewereinsufcienttoallowbinningintoasufcientnumberofcosφrangesforameaningfult.
Thereforeyieldswereintegratedoverallanglesineachhemisphere:beam-left(|φ|π/2).
Theseyieldswereenteredinequation(5.
11)tocalculatetheasymmetry.
Usingthefullazimuthalacceptance'waters-down'themeasuredasymmetryduetoproductionatsmallvaluesofcosφ,resultingintheanalysingpowermeasuredbyequation(5.
11)beingsmallerbyafactorof2/πthanthephysicalanalysingpowerinequation(5.
1).
Toaccountfortheazimuthaldistributionwithoutdividingthedataintocosφranges,theparticlecountsusedinthedenominatorwereweightedby|cosφ|.
Doingthisequation(5.
2)becomesANP=1cosφNNN+N→NN∑Ni=1|cosφ|+∑Ni=1|cosφ|(5.
12)Equations(5.
8)and(5.
11),derivedfromequation(5.
2),weresimilarlymodied.
Thisaccountsfortheazimuthaldependenceoftheasymmetryandmeansthatthesinglespinasymmetrydeterminedusingequations(5.
8)or(5.
11)correspondstothephysicalanalysingpowerANinequation(5.
1).
Tochecktherelationshipbetweenaphysicalasymmetryandthatextractedbymeasurement,K0Scandidateswererandomlyassignedazimuthalanglesfromthedistributioninequation(5.
1)withdifferentchosenvaluesofANP.
TheasymmetryANinthemodelledparticleswascalcu-latedusingequation(5.
11),bothwithandwithoutthecosφweightingprocedure.
Theresults74Recovered(asymmetryxpolarisation)0.
50.
60.
70.
80.
911.
11.
200.
050.
10.
150.
20.
250.
30.
350.
4WeightedUnweightedFigure5.
3:Effectof|cosφ|weightingindenominatortermsforasymmetrycalculation.
10,000K0Swererandomlyassignedaφangle,usingapolarisationof100%andananalysingpowerof1.
0,andtheasymmetrycalculatedusingthecross-ratiomethod.
Theimageshowstheresultsfrom100repetitionsofthiscalculation.
Thedashedpeakshowstheasymmetrycalculatedbyequation(5.
11)withnoweighting.
Thedottedlineindicatesthevalueof2/πexpected.
Thesolidlineshowsthesameasymmetrycalculatedwithweighting,recoveringtheinputvalueof1.
0.
areshowningure5.
3.
5.
3ResultsThecross-ratioasymmetrywascalculatedforeachRHICstoreindividually,summingparticleyieldsfromeverySTARrunthatoccurredinagivenstore.
Asymmetrieswerecalculatedforeachbeam,separatelyforforwardandbackwardangles.
Poisson(√N)errorswerecalculatedforeachparticleyieldinequation(5.
11)andpropagatedtocalculatethestatisticaluncertaintyontheasymmetryforeachstore.
Theasymmetryforeachstorewasthenscaledusingthemeanpolarisationmeasuredforthecorrespondingstore.
Theuncertaintiesontheasymmetrieswere75/ndf2χ32.
99/32Mean0.
019262±0.
003152RHICstore771877227724772577297739774077817785778677887790779177927794779577967797780078037804781078117815781778207823782478257826782778307831NAnalysingpower,A-1-0.
8-0.
6-0.
4-0.
200.
20.
40.
60.
81/ndf2χ23/32Mean0.
02017±0.
05217Figure5.
4:ANcalculatedusingthecross-ratiomethodforK0S,1.
00fromboththeclockwiseandanticlockwisebeamsforasingletransversemomentumbin.
Notethattheresultsarenotcorrectedfortheglobaluncertaintyinbeampolar-isation(4.
7%clockwisebeam,4.
8%anticlockwisebeam).
Consistencybetweenthetwobeamsisgoodonastore-by-storebasis.
Foreachbeamanddirectionrelativetothebeam(forwardandbackwardangles)abest-tlinewasappliedtothestore-by-storeresultstogiveaweightedmeanasymmetry.
Ineachcasethebest-tlineshaveaχ2perdegreeoffreedomclosetoone,indicatingagoodqualityoftandshowingstore-to-storesystematicdifferencesaresmallcomparedtothestatisticaluncertainties.
Theresultsfromeachbeamprovideindependentmeasurementsoftheasymmetry,sothemeanasymmetriescalculatedfromeachshouldshowconsistencywithinstatisticaluncertain-ties.
Figure5.
5showscomparisonsoftheasymmetryinK0Sproduction,asafunctionofK0Stransversemomentum,calculatedforbothbeams.
Thedifferencesbetweenthetwobeamsaretypicallyonestandarddeviationorsmaller.
Largerdifferencesareseenatforwardanglesinthe76pTrange1.
0to2.
0GeV/c,butthedeviationsarenotfoundtobeinconsistentwithstatisticaluctuations.
TheresultsforΛandΛatbothforwardandbackwardproductionangles(notshown),showedgoodagreementbetweentheresultsforthetwobeams.
Inallcasesaatbest-tlinettedtotheresultsfromeachbeamshowedresultsconsistentwithazeroasymmetryatallvaluesoftransversemomentum.
Theresultsfromeachbeamatthesamerelativeproductionangleswerethenaveragedtogiveameanasymmetryatforwardanglesandameanasymmetryatbackwardangles.
Theseaveragedasymmetries,asafunctionofparticletransversemomentum,areshowningures5.
6(K0S),5.
7(Λ)and5.
8(Λ)andarelistedintable5.
1.
Alluncertaintiesshownarestatisticalonly,exceptforthesmallcontributionfromthepolar-isationsystematicuncertaintyforeachstore.
FortheΛandΛ,theasymmetriesarefoundtobeconsistentwithzerowithinstatisticaluncertaintiesatallmomentastudied,forbothforwardandbackwardangles.
ThesameisobservedforK0Satbackwardangles.
Smallnon-zeroasymme-triesareseenfortheK0Satforwardangles.
Apositiveasymmetryisobservedattheleveloftwostandarddeviationsintherange1.
00intheSTARco-ordinatesystem.
Theazimuthalangleφwasdenedtocovertherangeπto+π.
Withthisdenition,thefourquadrantsspannedtheφrangesgivenintable6.
1.
CalculatingATTinthiswayutilisestheparticleproductionthroughoutthewholedetec-tor,minimisingstatisticaluncertainties.
Becausebothbeamsenterequation(6.
2)equivalently,anyasymmetrymustbesymmetricaboutη=0.
Yieldsatbothforwardandbackwardanglesarethereforesummed.
Inanalogywiththesinglespincase,integratingoveranangularrange89QuadrantφminφmaxTopπ/43π/4Left-π/4π/4Bottom-3π/4-π/4Right3π/4-3π/4Table6.
1:φanglerangesdeningthefourquadrantsusedforcalculatingATT.
dilutestheasymmetrybecauseofitsangulardependence.
Thisresultsinthemeasuredasym-metrybeingsmallerthanthephysicalasymmetry.
Thecos2φdependenceisaccountedforbyweightingeachcountinthedenominatorby|cos2φ|oftheparticle.
Yieldsforeachspeciesareextracted,asdiscussedpreviously,foreachSTARrun.
Thesearescaledbytheappropriaterelativeluminosity,R.
Thesinglespinasymmetryresultscal-culatedusingtheluminosity-dependentmethodandthecross-ratiomethodcorrespondedwell.
Thisindicatesthattherelativeluminosityscalingprocedureusedisreliable,andsystematicuncertaintiesduetotheluminositymeasurementsarenotlarge.
ScaledyieldsfromeachruninagivenRHICstoreareaddedtogivetotalstore-by-storeyields.
TheasymmetryforeachRHICstoreisthencalculatedusingequations(6.
3)and(6.
4).
Therawasymmetriesarecorrectedfortheproductofthetwobeampolarisations.
Abesttlinetothestore-by-storeresultsisusedtoobtainaweightedmeanvaluefortheasymmetry.
Thecalculatedasymmetryisshownforeachbeamstoreingure6.
1.
Resultsareintegratedoverallparticletransversemomentatominimisestatisticaluncertainties.
Theglobaluncertaintyintheproductofthebeampolarisations,δ(PAPC)/(PAPC)isnotincorporated.
Nosignicantstore-to-storeuctuationsareobserved.
Asymmetriesaresmall,inagreementwiththeoreti-calpredictions.
Thebest-tmeanasymmetriesareconsistentwithzerowithinthestatisticaluncertainties.
ThemeanvaluesofATTextractedasafunctionoftransversemomentumaresummarisedingure6.
2andarelistedintable6.
2.
TheresultsareatandconsistentwithzeroateachpTbinstudied.
90RHICstore77227724772577297739774077817785778677887790779177927795779677977815781778207823782478257826782778307831TTDoubleasymmetry,A-1-0.
8-0.
6-0.
4-0.
200.
20.
40.
60.
81/ndf2χ12.
12/25Mean0.
01685±-0.
01047(a)K0SRHICstore77227724772577297739774077817785778677887790779177927795779677977815781778207823782478257826782778307831TTDoubleasymmetry,A-1-0.
8-0.
6-0.
4-0.
200.
20.
40.
60.
81/ndf2χ30.
5/25Mean0.
0273±-0.
0322(b)ΛRHICstore77227724772577297739774077817785778677887790779177927795779677977815781778207823782478257826782778307831TTDoubleasymmetry,A-1-0.
8-0.
6-0.
4-0.
200.
20.
40.
60.
81/ndf2χ18.
98/25Mean0.
028641±0.
005922(c)ΛFigure6.
1:ATTforeachRHICstoreintegratedoverallparticletransversemomenta.
Thehorizontalstraightlinesshowbesttstothedataoverallstores.
91Transversemomentum(GeV/c)0.
511.
522.
533.
54TTMeanasymmetry,A-0.
2-0.
15-0.
1-0.
05-00.
050.
10.
150.
2(a)K0STransversemomentum(GeV/c)0.
511.
522.
533.
54TTMeanasymmetry,A-0.
2-0.
15-0.
1-0.
05-00.
050.
10.
150.
2(b)ΛTransversemomentum(GeV/c)0.
511.
522.
533.
54TTMeanasymmetry,A-0.
2-0.
15-0.
1-0.
05-00.
050.
10.
150.
2(c)ΛFigure6.
2:ATTofeachV0speciesasafunctionofparticletransversemomentum.
92pTinterval(GeV/c)DoublespinasymmetryUncertaintyK0S0.
5to1.
0-0.
00970.
02931.
0to1.
5-0.
01430.
02821.
5to2.
0-0.
06060.
03402.
0to3.
0-0.
02020.
02983.
0to4.
00.
02690.
0419Λ0.
5to1.
0-0.
02770.
07171.
0to1.
5-0.
04520.
05931.
5to2.
0-0.
03520.
05172.
0to3.
0-0.
04620.
04833.
0to4.
00.
05510.
0833Λ0.
5to1.
00.
02000.
08471.
0to1.
50.
04170.
06581.
5to2.
0-0.
07540.
05692.
0to3.
00.
03050.
04873.
0to4.
00.
01500.
0693Table6.
2:DoublespinasymmetriesandassociatedstatisticaluncertaintiesasafunctionofparticlepT.
936.
1SummaryTransverselypolarisedproton-protoncollisionshavebeenanalysedforacos2φdoublespinasymmetry,ATT,intheproductionoftheneutralstrangespeciesK0S,ΛandΛ.
Theasymme-tryisfoundtobesmallandconsistentwithzerowithinstatisticaluncertaintiesof0.
017fortheK0Sand0.
028forthe(anti-)Λwhenthedataareintegratedoveralltransversemomentum.
Nonon-zeroresultsareseenforparticleproductionintherange0.
50GeV/cwhentheresultsareplottedasafunctionofpT.
TheasymmetriesaremeasuredatxF≈0,wherecollisionsinvolvinggluonsdominate.
Whilequarkscanpossessanon-zerotransversitydistri-butionatleadingtwist,gluonscannot.
Thereforethedoublespinasymmetry,whichinvolvesthetransversitydistributionsofboththepartonsinvolvedinacollision,ispredictedtobeverysmallatsmallxF.
Thereresultspresentedhereareconsistentwiththisprediction.
94Chapter7OverviewandOutlookIshallrstgiveanoverviewoftheworkpresentedanditssignicanceintheanalysisoftrans-versespinasymmetries.
IwillthengiveabriefoverviewofsomeoftheworkbeingcarriedoutnowandproposedforthefutureatRHICandotherfacilitiesthatwillincreaseourunderstandingofthetransversespinofthenucleon.
7.
1AnOverviewoftheWorkPresentedTheresultspresentedinthisthesisshowmeasurementsofthetransversesinglespinasymmetryANandtransversedoublespinasymmetryATToftheneutralstrangeparticlesK0S,ΛandΛatmid-rapidity(|xF|<0.
05)andtransversemomentumintherange0.
5ForeachspeciesthemeasurementsofthesinglespinasymmetryareallsmallandconsistentwithzeroacrossthewholepTrangestudied,withinstatisticaluncertaintiesofafewpercent.
Noevidencehasbeenfoundofsystematiceffectsintheresultsofasizesignicantatthecurrentlevelofstatisticalprecision.
TheresultsfortheΛhyperonareconsistentwiththoseobtainedbytheAGSexperimentat√s=13.
3and18.
5GeVintheregionofkinematicaloverlap[57],andtwiththelow-xFtrendseeninE704dataat√s=20GeV[65].
TheK0SasymmetryisnotinagreementwiththesignicantnegativeasymmetryAN(K0S)≈-0.
10obtainedattheAGSat√s=18.
5GeV[57].
Theresultisconsistentwiththatobservedforneutralpions,anotherme-95sonspeciesmeasuredatthesamecentre-of-massenergyandkinematicrangebythePHENIXCollaboration[69].
Moreworkwouldberequiredtounderstandtheproductionmechanismleadingtothenon-zeroK0Sasymmetryatlowbeamenergies.
TherearetwomeasurementsthatcouldbemadeoftheK0Sasymmetrythatwouldbeinformative.
First,itwouldbeinterestingtomeasuretheK0SasymmetryatlargexFtoseeiftheasymmetrythereremainsathighenergies,asisthecaseforpions,orifittoovanishes.
Secondly,itwouldbeinterestingtomeasuretheK0SasymmetryatenergiesintermediatebetweenthoseoftheAGSmeasurementandthismea-surement,toobservetheevolutionoftheasymmetrywithbeamenergy.
ThoughRHIChasrunat√s=62.
4GeV/c,thedataacquiredbySTARtodateatthisenergyareinsufcienttomakeameasurementoftheK0Sasymmetryattherequiredprecision.
AmeasurementoftheΛasymmetryattheAGSat√s=18.
5GeVandsmallpTgaveAN(Λ)=0.
03±0.
10.
Theresultspresentedhereareconsistentwiththisvalue,butprovideimprovedstatisticalprecisionandareoveramuchlargerrangeofpT.
Themeasurementshereextendthetransversemomentumrangeatwhichtheasymmetriesaremeasuredsignicantlybeyondthatpreviouslyachieved.
ThemeasurementsareatsufcientmomentathatpQCDwillbeapplicableintheiranalysis,unlikepreviousmeasurementsofthesamespecies[111].
CombinedwithchargedkaonresultsfromtheBRAHMSCollaboration[72],RHIChasnowproducedhigh-energymeasurementsofsinglespinasymmetriesintheproductionofseveralstrangespecies,providingconstraintsonstrangequarkcontributionstonucleonspin.
Theresultspresentedhere,thoughinadifferentxFrangefromthereportedBRAHMSK±results,tthetrendseenthere.
TransversedoublespinasymmetriesatxF≈0involvetheseaquarkandantiquarktransver-sitydistributions.
ThemeasurementsofATTareconsistentwithzeroforeachspeciesasafunc-tionoftransversemomentum.
Thisisthersttimesuchameasurementhasbeenattemptedforanyofthesespecies.
Calculationspredictthatdoublespinasymmetriesinthismomentumrangeinhadroniccollisionsshouldbevanishinglysmall,duetosmallasymmetriesandlargebackgroundsfromgluoniccollisions.
Theresultsagreewiththesepredictions,andruleoutunexpectedlylargevaluesforthetransversedoublespinasymmetry.
967.
1.
1GluonicSiversEffectAtlargexF,particleproductionprocessesaredominatedbyvalencequark-quark(qq)collisions.
AtsmallxFgluon-quark(gq)collisions(involvingseaquarks)andgluon-gluon(gg)collisionsaredominant.
ThismeansthatmeasuringasymmetriesindifferentrangesinxFgivesaccesstothespindistributionsanddynamicsofdifferentconstituentsofthenucleon.
ThetrendintransversesinglespinasymmetriestoincreasewithlargexFindicatesthattheasymmetryisassociatedwithmechanismsinvolvingthevalencequarksofthepolarisednucleon.
ConverselysmallasymmetriesatxF≈0indicatethateffectsduetoseaquarksandgluonsaresmall.
TheresultsatsmallxFpresentedherearethereforerelatedtotheseaquarkandgluoncon-tentofthenucleon.
ThedominanceofgluonicscatteringinthisxFrangemeansthattheCollinsmechanismisunlikelytobesignicantintheanalysis.
HowevertheSiversmechanism(spin-dependenttransverse-momentumdistribution)willbeinvolved.
ThoughmostworktodatehasconcentratedonthequarkSiverseffect-thatis,thecorrelationbetweenthetransversemomen-tumofthequarksandthenucleonspin-therecanalsobeacorrelationwiththemotionofthegluons.
ThesmallresultsforANpresentedhereindicatethatthegluonicandsea-quarkSiversfunctionsaresmall.
AnanalysisusingPHENIXmeasurementsofp+p→π0+X,whichcoveredacomparablekinematicrangetothiswork,hasbeenperformedtoprovideaconstraintonthegluonicSiversfunction(GSF)[112].
ItfoundthatthePHENIXdata,combinedwithotherdataonpionproductioninhadroniccollisionsandSIDIS,areconsistentwithanon-zerovalence-likequarkSiversdistributionandavanishingsea-quarkandgluoniccontribution.
ThePHENIXresultswereusedtoprovideanupperlimitonthesizeofthegluonicSiversfunction,showningure7.
1.
TheresultspresentedherecanbeusedtoprovidefurtherconstraintsonthegluonicSiverseffect.
7.
2TheFutureWhatdoesthefutureholdfortransversespinphysicsTherehavebeengreatstridesinthepastfewyearsinunderstandingtransversespineffects:therstmeasurementshavebeenmadeof97Figure7.
1:UpperlimitonthegluonicSiversfunction,normalisedtothepositivitylimit,ob-tainedusingattoPHENIXp+p→π0+Xdata[112].
Thedataconstrainthefunctionwellatsmallxwheregluonsdominate.
Thesolidlineshowstheresultsassumingavanishingseaquarkcontribution.
Thedashedlineassumesamaximalseaquarkdistributionthatbalancesthegluoncontribution.
98CollinsandSiversfunctions,andtherstextractionofuanddquarktransversitydistributionshavebeenperformed.
Howeveruncertaintiesarestilllargecomparedtounpolarisedandhelicitydistributions,somuchmoreworkisneededbeforetransversespinphenomenaareunderstoodtothesamedegree.
Transversespinprogrammesarecurrentlybeingperformedorplannedatarangeofexper-iments.
Ishallgivehereabriefoverviewofsometheadvancesandmeasurementsanticipatedoverthenextfewyears,andhowtheywillimpactonourunderstandingoftransversespinphenomena.
7.
2.
1AtRHICRHICissoontobeginacceleratingprotonbeamsto250GeV,providingacentreofmassenergy√s=500GeV,animportantmilestoneinitsoperation.
Thisisasignicantincreaseover√s=200GeVatwhichdatahasbeenacquiredtodate,andwillopenuptheabilitytostudyprocessesinvolvingWbosondecays.
MeasurementsofWasymmetrieswillprovideinformationontheavourdependenceofspincontributionsfrombothseaandvalencequarksandantiquarksintheproton.
InthelongertermthereareplansforaRHICluminosityupgrade(RHICII),andthereareproposalstoupgradeRHICtoapolarisedelectron-protoncollider(eRHIC).
BothongoingRHICexperiments,PHENIXandSTAR,maintainstrongspinprogrammes.
ImprovementstotheSTARdataacquisitionsystemarenowallowingittotakemuchlargerdatasetsthanpreviously.
Thiswillcontributegreatlytoimprovingtheprecisionofitsspinmeasurements.
STARisintheprocessofdevelopingaforwardmesonspectrometer(FMS),anadvancedversionoftheforwardpiondetectorthathasbeenusedinmeasurementsofπ0singlespinasymmetries.
Non-zerovaluesforbothSiversdistributionfunctionsandCollinsfragmentationfunctionshaverecentlybeenshownbyHERMESandBELLE[75,77,78],andbothmechanismsareexpectedtocontributetotransversesinglespinasymmetriesinhadron-hadroncollisions.
TheSTARFMSwillallowdisentanglingoftherelativecontributionsoftheSiversdistributionfunctionandCollinsfragmentationfunctiontotheneutralpionasymmetry.
ThePHENIXexperimentcandetectmuonsfromthedecaysu+d→W+→++νand99d+u→W→+ν.
StudyingthesinglespinasymmetryinW±decayswillseparatetheu,d,uandd(anti)quarkSiversdistributions.
PHENIXisreceivinganupgradetoitsmuontriggeringsystemsinpreparationfortheincreaseinbeamenergy.
7.
2.
2SIDISMeasurementsThoughHERMEShasnowceasedtakingdata,theCOMPASSexperimentcontinuesapro-grammeoftransversespinstudiesusingSIDISmeasurementswithdeuteronandprotontargets,sheddingfurtherlightonthequarkavourdependenceoftransversityandtheCollinseffect.
ThespinprogrammeatJeffersonLaboratoryplanscontinuedtransversespinmeasurements[113].
TheJeffersonLaboratoryHallAexperimentispreparingameasurementoftheneutrontransversityusingapolarisedhelium-3target[114].
Measurementswillbemadeofthesinglespinasymmetryinsemi-inclusivechargedpionproduction,e+n→e+π±+X,witha6GeVelectronbeam.
ThiswillcomplementSIDISmeasurementsmadebyHERMESusingaprotontargetandcontinuingmeasurementsbyCOMPASSusingadeuterontarget,andwillaidinconstrainingtheuanddvalencequarktransversitydistributions.
JLabexperimentswillalsohaveafuture12GeVbeamupgrade.
Oncethisisimplemented,datawillgiveaccesstoboththeSiversandCollinsfunctionsatlargemomentumfractions[115].
7.
2.
3PolarisedAntiprotonsThePAX(PolarisedAntiprotoneXperiments)Collaborationhaveproposedapolarisedproton-antiprotoncollider.
MeasurementsofDrell-YanproductioncantestthepredictionthattheSiversfunctionsinpDISandDrell-Yanproductionareequalinmagnitudebutoppositeinsign,aresultofthenon-universalityoftheSiversfunctions.
PAXcouldalsostudytransversitybymeasure-mentsofATTinpolarisedDrell-Yanproduction.
Inthisprocesstheasymmetryresultssolelyfromthetransversitydistributionsofthepolarisedprotonandantiproton.
TheCollinsfrag-mentationfunctionsarenotinvolvedintheasymmetry,unlikethecase,forexample,ofHER-MESandCOMPASSSIDISdata.
InthiswayPAXcoulddirectlyaccesstransversity.
OthermeasurementsmaybeabletodisentanglethecontributionsoftheSiversandCollinsfunc-100tionstosinglespinasymmetries,viameasurementsofcharmmesonproduction,forexampleinp+p↑→D+X.
7.
2.
4GeneralisedPartonDistributionsTheSiverseffectisrelatedtotheorbitalmotionofpartonswithinthenucleon.
However,moredirectaccesstoorbitalinformationmaybeprovidedbystudyingGeneralisedPartonDistribu-tions(GPDs)[116].
Ordinarypartondistributionscontaininformationaboutthelongitudinalmomentumfractionofthepartons(x),butdonotcontainanyinformationabouttransversemotion.
GPDshowevercontaininformationaboutboththetransverseandlongitudinalpartonmomenta.
Theyarecharacterisedbythreekinematicvariablesinsteadofone;xandξ,whichcharacterisethelongitudinalpartonmomentum,andt,thesquareofthefour-momentumtrans-fertothetarget,whichinvolvestransversemomentum.
GPDsthereforeprovideameanstogivea'multi-dimensional'descriptionofpartonsinthenucleon.
Theyareofinterestinrelationtospinbecausetheycanyieldinformationonthetotalpartonangularmomentum:boththeintrin-sicandorbitalcontributions.
TwoGPDs,denotedEandH,canberelatedtothetotalangularmomentumofapartonspeciesbytheintegralJ=limt→010x(H+E)dx(7.
1)ThereforeiftheGPDscanbesufcientlywelldetermined,thetotalangularmomentumcontri-butionsofthepartonswillbecomeaccessible.
Combinedwithknowledgeoftheintrinsicspincontributionsfromothermeasurements,theorbitalcontributionscanbedetermined.
DeeplyVirtualComptonScattering(DVCS)(e+p→e′+p+γ)hasbeenusedasameansaccesstheGPDs.
InvestigationofGPDsviaDVCShasbeenperformedbytheexperimentsH1,ZEUS,HERMESandJLabHallAandCLAS[117–121].
ThereisalsoaproposedprogrammeofstudyattheCOMPASSexperiment.
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