Transversity Distribution and Polarized Fragmentation Function from Semi-inclusive Pion Electroproduction

5 Pages·0.14 MB·English

by  V. A. Korotkov

EPJmanuscriptNo.
(willbeinsertedbytheeditor)
Transversity Distribution and Polarized Fragmentation
Function from Semi-inclusive Pion Electroproduction
V.A.Korotkov1,2,W.–D.Nowak2,andK.A.Oganessyan2,3,4
1
0
1 IHEP,142284Protvino,Russia
0
2 DESYZeuthen,15738Zeuthen,Germany
2
3 INFN,LaboratoriNazionalidiFrascati,00044Frascati,Italy

n 4 YerevanPhysicsInstitute,375036Yerevan,Armenia
a
J
Received:date/Revisedversion:date
2
2
Abstract. A method is discussed to determine the hitherto unknown u-quark transversity distribution δu(x) from
2 a planned HERMES measurement of a single target-spin asymmetry in semi-inclusive pion electroproduction off a
v transverselypolarizedtarget.Assumingu-quarkdominance,themeasurementyieldstheshapesofthetransversitydis-
8 tribution δu(x) and of the ratio H1⊥(1)u(z)/D1u(z), of polarized and unpolarized u-quark fragmentation functions.
6 Theunknownrelativenormalizationcanbeobtainedbyidentifyingthetransversitydistributionwiththewell-known
2 helicitydistributionatlargexandsmallQ2.Thesystematicuncertaintyofthemethodisdominatedbytheassumption
2 ofu-quarkdominance.
0
0
0 1 Introduction at some scale, they will be differentat Q2 values higher than
/
h that.
p Deepinelasticchargedleptonscatteringoffatransverselypo-
The chiral-odd nature of transversity distributions makes
- larizednucleontargetisanimportanttooltofurtherstudythe
p theirexperimentaldeterminationdifficult;uptonownoexper-
internal spin structure of the nucleon. While a lot of experi-
e imentalinformationonδq(x,Q2)isavailable.Itcannotbeac-
mental data on the longitudinal spin structure of the nucleon
h
cessed in inclusive deep inelastic scattering (DIS) due to chi-
: hasbeencollectedoverthelast10years,thestudyofitstrans-
v rality conservation; it decouples from all hard processes that
versespinstructureisjustabouttobegin.Onlyaverylimited
i involveonlyonequarkdistribution(orfragmentation)function
X number of preliminary experimentalresults is available up to
(seee.g.[8]).Thisisin contrastto thecase ofthe chiral-even
now:
r
a number density and helicity distribution functions, which are
(1) measurements of the nucleon structure function g2(x) at directlyaccessibleininclusiveleptonDIS.
CERN[1]andSLAC[2]-[4],
(2) a first measurementof a single target-spin asymmetry for Inprinciple,transversitydistributionscanbeextractedfrom
pions produced in lepton scattering off longitudinally po- cross section asymmetries in polarized processes involving a
larizedprotonsatHERMES[5], transverselypolarizednucleon.Thecorrespondingasymmetry
(3) afirststudyofhadronazimuthaldistributionsinDISoflep- can be expressed through a flavor sum involving products of
tonsoffatransverselypolarizedtargetatSMC[6]. twochiral-oddtransversitydistributionsinthecaseofhadron-
hadronscattering,whileinthecaseofsemi-inclusiveDIS(SIDIS)
A quarkofa givenflavouris characterizedbythreetwist-
achiral-oddquarkdistributionfunctionalwaysappearsincom-
2 parton distributions. The quark number density distribution
binationwithachiral-oddquarkfragmentationfunction.These
q(x,Q2)hasbeenstudiednowfordecadesandiswellknown
fragmentationfunctionscaninprinciplebemeasuredine+e−
for all flavors. The helicity distribution ∆q(x,Q2) was only
annihilation.
recently measured more accurately for u and d quarks [7]
− −
andisstillessentiallyunknownfors quarks.Thethirdparton ThetransversitydistributionwasfirstdiscussedbyRalston
−
distribution,knowngenerallyas‘transversitydistribution’and andSoper[9]indoublytransversepolarizedDrell-Yanscatter-
denotedδq(x,Q2)characterizesthedistributionofthequark’s ing.Itsmeasurementisoneofthemaingoalsofthespinpro-
transversespininatransverselypolarizednucleon. gramatRHIC[10].Anevaluationofthecorrespondingasym-
Fornon-relativisticquarks,whereboostsandrotationscom- metry A was carried out [11] by assuming the saturation
TT
mute,δq(x) = ∆q(x).Sincequarksinthenucleonareknown ofSoffer’sinequality[12]forthetransversitydistribution.The
toberelativistic,thedifferencebetweenbothdistributionswill maximumpossibleasymmetryatRHICenergieswasestimated
providefurtherinformationontheirrelativisticnature.Thetrans- to be A = 1 2%. At smaller energies, e.g. for a possi-
TT
versitydistributiondoesnotmixwithgluonsunderQCDevolu- blefixed-targetha÷dron-hadronspinexperimentHERA-N[13]
tion,i.e.eveniftransversityandhelicitydistributionscoincide (√s 40GeV),theasymmetryisexpectedtobehigher.
≃
2 V.A.Korotkovetal.:TransversityDistributionandPolarizedFragmentationFunction
Insemi-inclusivedeepinelasticleptonscatteringofftrans- wherethesuperscript(1)indicatesthatanoriginallyk depen-
T
versely polarized nucleons there exist several methods to ac- dent function was integrated over k with the weight kT2 .
cess transversity distributions; all of them can in principle be T 2Mh2
Here,M isthemassoftheproducedhadronh.
realized at HERMES. One of them, namely twist-3 pion pro- h
To facilitate access to transversityand polarizedfragmen-
duction[14], useslongitudinallypolarizedleptonsand a dou-
tation functionsfrom SIDIS, single-spin asymmetriesmay be
blespinasymmetryismeasured.Theothermethodsdonotre-
formedthroughintegrationofthepolarizedcrosssectionover
quire a polarized beam; they rely on polarimetry of the scat-
P , the transverse momentum of the final hadron, with ap-
teredtransverselypolarizedquark: h⊥
propriateweights.Intheparticularcaseofanunpolarizedbeam
(1) measurement of the transverse polarization of Λ’s in the andatransverselypolarizedtargetthefollowingweightedasym-
currentfragmentationregion[15]-[17], metryprovidesaccesstothequarktransversitydistributionvia
(2) observation of a correlation between the transverse spin theCollinseffect[25]:
vectorofthetargetnucleonandthenormaltothetwome-
A (x,y,z)
sonplane[8,18], T
≡
(3) observation of the Collins effect in quark fragmentation dφℓ d2P |Ph⊥|sin(φℓ +φℓ) dσ↑ dσ↓
throughthe measurementof pionsingle target-spinasym- h⊥ zMh s h − . (3)
R R dφℓ d2P (dσ↑+dσ(cid:0)↓) (cid:1)
metries[19]-[23]. h⊥
R R
Here ( ) denotes target up (down) transverse polarization.
TheHERMESexperiment[24]hasexcellentcapabilitiestoin-
↑ ↓
Theazimuthalanglesaredefinedinthetransversespacegiving
vestigate semi-inclusive particle production. Taking the mea-
the orientation of the lepton plane (φℓ) and the orientation of
surementoftheCollinseffectasanexample,itwillbeshown
the hadron plane (φℓ = φ φℓ) or spin vector (φℓ = φ
inthefollowingthatHERMESwillbecapabletoextractboth h h − s s −
transversityandchiral-oddfragmentationfunctionatthesame φℓ) withrespecttothe leptonplane.Theanglesare measured
timeandwithgoodstatisticalprecision. aroundthez-axiswhichisdefinedbythemomentaqandP of
thevirtualphotonandthetargetnucleon,respectively.Theraw
asymmetry(3)canbeestimated[25]using
2 Single Target-Spin Asymmetry in Pion
e2δq(x)H⊥(1)q(z)
Electroproduction A (x,y,z)=P D q q 1 , (4)
T T · nn· P e2q(x)Dq(z)
q q 1
P
A complete analysis of polarized SIDIS with non-zero trans-
whereP isthetargetpolarizationandD =(1 y)/(1 y+
T nn
versemomentumeffectsinboththequarkdistributionandfrag- y2/2)isthetransversespintransfercoefficient.Th−emagn−itude
mentation functions was performed in the framework of the
oftheasymmetrydependsontheunknownfunctionsδq(x)and
quark-partonmodelin [21] and in the field theoreticalframe- H⊥(1)q(z).
workofQCDin[22].Animportantingredientofthisanalysis 1
isthefactorizationpropertythatwasprovenfork integrated
T
functionsandthatcanreasonablybeassumedfork depending
T 3 Transversity Distribution and Polarized
functions[22]. In the situation thatthe final state polarization
Fragmentation Function
is not considered, two quark fragmentation functions are in-
volved:Dq(z,z2k2)andH⊥q(z,z2k2).Herek istheintrin-
1 T 1 T T No experimentaldata are available on any of the transversity
sic quarktransversemomentumandz isthefractionofquark
distributions δq(x), while their behaviour under QCD-evolu-
momentumtransferedto the hadronin the fragmentationpro-
tionis theoreticallywellestablished [16]. An exampleforthe
cess. The ‘polarized’ fragmentation function H1⊥q allows for leadingorderevolutionoftheprotonstructurefunctionsg (x,Q2)
a correlation between the transverse polarization of the frag- = 1 e2∆q (x,Q2) and h (x,Q2) = 1 e2δq (x,Q12) is
mentingquarkandthe transversemomentumofthe produced sho2wPniiniFigi. 1. It was ass1umed that h2p(Px)i cioinicides with
hadron. It may be non-zero because time reversal invariance gp(x) at the scale Q2 = 0.4 GeV2 and1both functions were
is notapplicablein a decayprocess,as was first discussed by 1 0
evolvedto the scale Q2 = 10 GeV2. The evolutionwas per-
Collins[19].
formedusingtheprogramsfrom[27,28]forg andh ,respec-
Since quark transverse momenta cannot be measured di- 1 1
tively.Theimportantconclusion,whichwasalreadydiscussed
rectly,integralsoverk (withsuitable weights)aredefinedto
T earlier(seee.g.[29]),followsthatwithincreasingQ2 thetwo
arriveatexperimentallyaccessiblefragmentationfunctions:
functionsarebecomingmoreandmoredifferentfordecreasing
xwhileatlargexthedifferenceremainsquitesmall.
z2 d2k Dq(z,z2k2) Dq(z) (1) Results fromtwo independentmeasurementsindicate that
Z T 1 T ≡ 1
the polarized fragmentation function H⊥(1)q(z) may be non-
1
isthefamiliarunpolarizedfragmentationfunction,normalized zero:i)azimuthalcorrelationsmeasuredbetweenparticlespro-
by the momentumsum rule dzzDq→h(z) = 1. Corre- ducedfromoppositejetsinZdecayatDELPHI[30]andii)the
h 1
spondingly,thepolarizedfragPmenRtationfunctionisobtainedas single target-spinasymmetry measured for pions producedin
SIDISofleptonsoffalongitudinallypolarizedtargetatHER-
k2 MES[5].Theapproachof[25]isadoptedtoestimatethepos-
z2Z d2kT (cid:18)2MTh2(cid:19) H1⊥q(z,z2kT2)≡ H1⊥(1)q(z), (2) siblevalueofH1⊥(1)q(z).Collins[19]suggestedthefollowing
V.A.Korotkovetal.:TransversityDistributionandPolarizedFragmentationFunction 3
4 Projected Statistical Accuracy and
22px,Q), g(x,Q)10.715 ghgppp111,,, hQQp122,== Q11002= GG0.ee4VV G22eV2 SAyfusllteanmalaystiisctsoextracttransversityandpolarizedfragmenta-
ph(1 tionfunctionsthrough(4)requiresonetotakeintoaccountall
0.5 quark flavours contributing to the measured asymmetry. Ac-
cording to calculations with the HERMES Monte Carlo pro-
gramHMC,thefractionofpositivepionsoriginatingfromthe
0.25 fragmentation of a struck u-quark ranges, depending on the
valueofx,between70and90%foraprotontargetandisonly
slightlysmallerforadeuterontarget.Therefore,inafirstanal-
0 -2 -1 ysis,theassumptionofu-quarkdominanceintheπ+ produc-
10 10 1
x tion cross-section appears to be reasonable.This is supported
Fig. 1. The transversity distribution hp(x,Q2) (continuous line) bythesumruleforT-oddfragmentationfunctionsrecentlyde-
1 0
whichcoincideswiththehelicitydistributiongp(x,Q2)at thescale rivedin[33].Theseauthorsconcludedthatcontributionsfrom
1 0
Q2 = 0.4 GeV2 (asgiven by theGRSV LOparameterization [26] non-leading parton fragmentation, like d π+, is severely
in0the’standard’scenario).TheirevolvedLOdistributionshp(x,Q2) suppressedforallT-oddfragmentationfunc→tions.
1
(dotted)andg1p(x,Q2)(dot-dashed)areshownatQ2 =10GeV2. Consequently, the assumption of u-quark dominance was
usedtocalculateprojectionsforthestatisticalaccuracyinmea-
suringtheasymmetryAπ+(x).Theexpectedstatisticsforscat-
parameterizationfortheanalyzingpowerintransverselypolar- T
teringatHERMESunpolarizedleptonsoffatransverselypolar-
izedquarkfragmentation:
izedtarget(protonordeuteronoptionsareunderconsideration)
k H⊥q(z,z2k2) M k willconsistofaboutsevenmillionsreconstructedDISevents.
AC(z,kT)≡ |MT|D1q(z,z2k2T) = M2C+| Tk2| , (5) ThestandarddefinitionofaDISeventatHERMESisgivenby
h 1 T C | T| thefollowingsetofkinematiccuts1:
with M 0.3 1.0 GeV being a typical hadronic mass.
C
Choosinga≃Gaussi÷anparameterizationforthequarktransverse Q2 >1GeV2,W >2GeV,0.02 < x < 0.7,y < 0.85.
momentumdependenceintheunpolarizedfragmentationfunc-
tion AnadditionalcutW2 >10GeV2wasintroducedintheanaly-
R2 sistoimprovetheseparationofthestruckquarkfragmentation
D1q(z,z2kT2)=D1q(z)πz2 exp(−R2kT2), (6) region.AnaveragetargetpolarizationofPT =75%isusedfor
theanalysis.
leadsto
Considering only u-quarks the expression for the asymmetry
H⊥(1)q(z)= (4)reducestothesimpleform
1
M ∞ exp( x)
D1q(z)2MCh (cid:18)1−MC2R2Z0 dxx+M−C2R2(cid:19). (7) AT(x,y,z)=PT ·Dnn· δuu((xx)) · H1D⊥(u1()uz()z) (10)
1
Here R2 = z2/b2, and b2 is the mean-squaremomentumthe
hadronacquiresinthequarkfragmentationprocess.Inthefol- foraprotontarget,and
lowingtheparametersettingsM = 0.7GeV andb2 = 0.25
C
GeV2 are used because they are consistent [31] with the sin- A (x,y,z)=
T
gletarget-spinasymmetrymeasuredatHERMES[5].Theyare
also compatible with the analysis of [30], as can be seen by (1 3ω ) P D δu(x)+δd(x) H1⊥(1)u(z) (11)
evaluatingtheratio − 2 D · T · nn· u(x)+d(x) · Du(z)
1
1 dzH⊥(z) foradeuterontarget.Hereω =0.05 0.01istheprobability
R(z )= zmin 1 , (8) D ±
min R 1 ofthedeuterontobeintheD-state.
dzD (z)
zmin 1 TosimulateameasurementofAT theapproximationδq(x) =
R ∆q(x) could be used in view of the relatively low Q2-values
where H⊥(z), in contrast to Eq.(2), is the unweighted polar-
1 at HERMES, in accordance with the above discussion. The
izedfragmentationfunctionusedin[30]
Gehrmann-Stirlingparameterizationinleadingorder[34]was
takenfor∆q(x) and the GRV94LO parameterization[35] for
z2Z d2kT H1⊥q(z,z2kT2) ≡ H1⊥q(z). (9) q(x).TheQ2evolutionofthequarkdistributionswasneglected
and Q2 = 2.5 GeV2 was taken as an average value for the
The BKK parameterization [32] was used to estimate the in-
tegralovertheunpolarizedfragmentationfunctionD1(z).The 1 Q2 andν arethephoton’svirtualityandlaboratoryenergy,x =
values obtained for the ratio, R(0.1) = 0.048 and R(0.2) = Q2/2Mν istheBjorkenscalingvariable,y = ν/E isthefractional
0.070,aretobecomparedtotheexperimentalresult[30]:0.063 photonenergyandW isthec.m.energyofthephoton-nucleon sys-
±
0.017. tem;E =27.5GeVatHERMES.
4 V.A.Korotkovetal.:TransversityDistributionandPolarizedFragmentationFunction
HERMESkinematicalregion.TheHERMESMonteCarlopro- thefunctionK(x ,z ),asopposedto9unknownfunctionval-
i j
gram HMC was used to account for the spectrometer accep- ues:4valuesforδu(x )and5valuesforH⊥(1)u(z )/Du(z ),
tance. The following cuts were applied to the kinematic vari- wheretheindicesiandi j enumeratetheexp1erimentjalint1ervajls
ablesofthepion2: inxandinz,respectively.Thestandardprocedureofχ2 min-
x >0,z >0.1,P >0.05GeV. imizationwasappliedtoreconstructthevaluesforbothδu(x)
F h⊥
andH⊥(1)u(z)/Du(z)andtoevaluatetheirprojectedstatisti-
Thesimulateddataweredividedinto5x5binsin(x,z).The 1 1
expectationsfortheasymmetryAπ+(x)aswouldbemeasured cal accuracies expected for a real measurementat HERMES.
T TheresultsareshowninFigs.3a,b,respectively.
byHERMESusingaprotontarget,arepresentedinFig.2ain
Inananalogouswaytheconsiderationofthedeuteronasym-
differentintervalsofthepionvariablez.Theprojectedaccura-
metry(11)allowstheevaluationoftheprojectedstatisticalac-
ciesfortheasymmetrywereestimatedaccordingto
curaciesforameasurementofthefunctionsδu(x)+δd(x)and
H⊥(1)u(z)/Du(z)(seeFigs.4a,b,respectively).Theprojected
1 1
statisticalaccuracyisconsiderablyworsethanthatforthepro-
π+AT0.18 00..12e<<pzz<< →00..23 eπ+X K(x, z) 56 ep → eπ+X toofnthtaeragseyt;mthmisetirsyc(a1u1s)e,dwmhiacihnliynbtuyrtnheisedxupeectotetdhesmloawlleerrvvaalluuee
0.3<z<0.5 4 of(δu(x)+δd(x))/(u(x)+d(x))comparedtoδu(x)/u(x).
0.6 0.5<z<0.7
0.7<z<1.0 3
0.4
2
0.2 x) 3 z)2.5
0 -1 a) 01 b) -1 δu(2.4 ep → eπ+Xa) uz)/D(1 2 ep → eπ+Xb)
10 x 10 x 1.8 u(1.5
1)
Fig.2.Protontarget.a)TheweightedasymmetryATπ+(x)indifferent 1.2 ⊥(H1 1
intervalsofz;b)thefunctionK(x,z).
0.6 0.5
0 0
1 10-1 x 0 0.2 0.4 0.6 0.8 z1
δA = Ph⊥ sin(φℓ +φℓ) 2 2 1 , (12)
T (cid:28)(cid:16)zmπ s h (cid:17) (cid:29) · √Nπ Fig.3.a)Thetransversitydistributionδu(x),andb)theratioofthe
fragmentationfunctionsH⊥(1)u(z)andDu(z)aswouldbemeasured
whereN isthetotalnumberofmeasuredpositivepionsafter 1 1
π byHERMESwithaprotontarget.Theasteriskina)showsthenormal-
kinematiccuts,and< >meansaveragingoverallaccepted izationpoint.
···
events.In thisway the productofthe transversitydistribution
andtheratioofthefragmentationfunctions,
x) 3 z)2.5
K(x,z)=δu(x)· H1D⊥(1u1()uz()z), (13) δx) + d(2.25 ed → eπ+Xa) uu(z)/D(11.25 ed → eπ+Xb)
aswellastheprojectedstatisticalaccuracyforameasurement δu(1.5 ⊥(1)H1
1
ofthisfunctionwerecalculatedandareshowninFig.2b,again 1
forthecaseofaprotontarget. 0.5
0.5
Thefactorizedformofexpression(10)with respectto the
svhaaripaebfleosrtxheantwdozuanllkonwoswthnefusnimctuioltnasnδeouu(xs)reacnodnHstr⊥u(c1t)iuo(nzo)f/tDheu(z), 0 10-1 x 00 0.2 0.4 0.6 0.8 z1
1 1
whiletherelativenormalizationcannotbefixedwithoutafur- Fig. 4. a) The transversity distribution δu(x) + δd(x), and b) the
therassumption.Aswasdiscussedabove,thetransversitydis- ratioofthefragmentationfunctionsH⊥(1)u(z)andDu(z)aswould
tributionδq(x)conceivablycoincideswiththehelicitydistribu- bemeasured by HERMESwithadeut1eron target. The1asteriskina)
tion∆q(x)atsmallvaluesofQ2 wheretherelativisticeffects showsthenormalizationpoint.
are expected to be small. According to Fig. 1 the differences
aresmallestintheregionofintermediateandlargevaluesofx.
Hencetheassumption Two sources of systematic uncertainties arising from ap-
proximationsusedintheanalysiswereinvestigated.Toevalu-
δq(x0)=∆q(x0) (14) ate the contributionof the normalizationassumption(14),the
relativedifferencebetweentransversitydistributionδu(x,Q2)
atx0 =0.25wasmadetoresolvethenormalizationambiguity. andhelicity distribution∆u(x,Q2) was studied as a function
The experimental data then consist of 25 measured values of ofxandQ2intheHERMESkinematics.Startingfromδu(x)=
2 xF = 2pL/W, where pL is thelongitudinal momentum of the ∆u(x)atthescaleQ20 = 0.4GeV2bothfunctionswereevolved
hadronwithrespecttothevirtualphotoninthephoton-nucleonc.m.s., to higher values of Q2. The results are shown in Fig. 5 and
andz=Eh/ν,whereEhistheenergyoftheproducedhadron. allow the conclusion that the relative difference is small for
V.A.Korotkovetal.:TransversityDistributionandPolarizedFragmentationFunction 5
x above 0.2 0.3; the corresponding systematic uncertainty Acknowledgements
÷
is on the level of 2 5%. The same conclusion is valid for
÷ WethankKlausRithfordiscussionswhichledtothedevelop-
the evolution of δu(x) + δd(x). A larger contribution to the
mentofthemethoddescribedinthispaperandtheHERMES
CollaborationforthekindpermissiontousetheirMonteCarlo
Program.WeareindebtedtoRalfKaiserforthecarefulreading
2Q)0.1 ofthemanuscript.
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