Global analysis of AAC for determining polarized parton distribution functions

0.16 MB·English

by  M. Hirai

Global analysis of AAC for determining
polarized parton distribution functions
M. Hirai∗, S. Kumano∗† and N. Saito∗∗
6 ∗InstituteofParticleandNuclearStudies,HighEnergyAcceleratorResearchOrganization(KEK)
0 †DepartmentofParticleandNuclearStudies,TheGraduateUniversityforAdvancedStudies
0 1-1,Ooho,Tsukuba,Ibaraki,305-0801,Japan
2 ∗∗DepartmentofPhysics,KyotoUniversity,Kyoto,606-8502,Japan

n
a
Abstract. We report global analysis results for polarized parton distribution functions in the nu-
J
cleon. The optimumdistributionsare determinedby using spin asymmetrydata on polarizedlep-
1
ton scattering on proton, neutron, and deuteron. Their uncertainties are estimated by the Hessian
1
method.Asaresult,polarizedquarkdistributionsarerelativelywelldetermined,whereasthepolar-
1 izedgluondistributionhasalargeuncertaintyband.Wefindthattheobtainedgluondistributionis
v compatiblewithrecentD g/gmeasurementsinhigh-pT hadronproductions.
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8
0
INTRODUCTION
1
0
6
The nucleon spin is one of the fundamental quantities in physics, yet we do not under-
0
/ stand its origin. Internal structure of the nucleon spin has been investigated mainly by
h
polarizedlepton-nucleonscatteringexperiments.Polarizationsofquarksand gluonsare
p
- expressed by polarized parton distribution functions (PDFs), and we extract them from
p
e theexperimentalmeasurements.
h The standard method is to obtain the polarized PDFs, which are expressed by a
v: number of free parameters, by a c 2 analysis of available polarization-asymmetry data.
i
X Among such analyses, we report the results by the Asymmetry Analysis Collaboration
(AAC) [1, 2]. In the recent version [2], the polarized PDFs and their uncertainties are
r
a
obtained by using inclusive deep inelastic scattering (DIS) data. However, these data
are not enough to impose considerable constraints on each polarized PDF, especially
the gluon distribution. Fortunately, the situation is changing because the RHIC-Spin
started producing data and there are reports on the gluon polarization by using semi-
inclusive lepton scattering data. In the following, we explain the AAC results [2] and
alsoacomparisonwithrecent D g/gmeasurements.
GLOBAL ANALYSIS OF SPIN ASYMMETRY DATA
Forprobingtheinternalstructureofthenucleon,crosssectionsforpolarizeddeepinelas-
ticlepton-nucleonscatteringhavebeenmeasured.Theirexperimentaldataareshownby
the spin asymmetry A . It is expressed by the ratio of the polarized structure function
1
g to the unpolarized one F , which is usually expressed by F and the longitudinal-
1 1 2
transverseratioR:
g (x,Q2)
A (x,Q2)= 1 2x[1+R(x,Q2)]. (1)
1 F (x,Q2)
2
The unpolarized PDFs of the GRV98 are used for calculating F , and the SLAC
2
parametrization is used for R. The polarized structure function is expressed in terms
ofpolarized PDFs:
n
g (x,Q2)= 1 (cid:229) f e2 D C (x,a )⊗[D q(x,Q2)+D q¯ (x,Q2)]+D C (x,a )⊗D g(x,Q2) ,
1 2 i(cid:26) q s i i g s (cid:27)
i=1
(2)
whereD C and D C are coefficient functions.
q g
A polarized PDF, D f (= D q , D q¯, or D g), is defined by a number of parameters
i i
at the initial scale Q2=1 GeV2: D f(x) = [d xn −k (xn −xm )]f(x), where f(x) is the
corresponding unpolarized distribution. This functional form is convenient especially
for imposing the positivity condition, |D f(x)| ≤ f(x), in the global analysis. Flavor
symmetric antiquark distributions are assumed at the initial scale because of the lack
ofexperimentalinformation.ThedistributionsareevolvedtoexperimentalQ2 pointsof
the spin asymmetry by the DGLAP evolution equations in order to calculate c 2 values
at the same Q2 points. All the calculations are done in the next-to-leading order (NLO)
of a and theMS scheme is used. The parameters are determined so as to minimizethe
s
totalc 2.Then,theuncertaintiesofobtainedPDFsareestimatedbytheHessianmethod.
Sofar,onlytheinclusiveDISdataareusedfortheanalysisintheAACanalysis.Because
there are measurements after the publication [2], an analysis is in progress with JLab,
HERMES, and COMPASS data. In future, we need to include other data such as the
onesfrom semi-inclusiveDISand RHIC-Spin.
RESULTS
We analyzed the data of the spin
0.5 2
asymmetries A in Eq. (1) for the Q2 = 1 GeV2 AAC03
1
0.4 GRSV
proton, neutron, and deuteron in po- 1 BB
0.3 LSS
larized lepton scattering. The polar-
ized PDFs obtained by the analysis 0.2 x∆∆∆∆uv(x) 0
at Q2=1 GeV2 are shown in Fig. 1, 0.1 x∆∆∆∆g(x)
0 -1
0.001 0.01 0.1 1 0.001 0.01 0.1 1
where the solid curves indicate the
0 0.01
AAC03 distributions and their un-
0
certainties are shown by the bands. x∆∆∆∆d (x)
v
-0.01
In comparison, other analysis results -0.1
AAC03 -0.02
(BB, GRSV, LSS) are also shown in GRSV
the figure. The valence-quark distri- BLBSS -0.03 x∆∆∆∆q(x) Q2 = 1 GeV2
-0.2 -0.04
0.001 0.01 0.1 1 0.001 0.01 0.1 1
butions are well determined; how- x x
ever, the antiquark and gluon distri-
FIGURE1.PolarizedPDFswithuncertainties[2].
butions have large uncertainties. In
particular,thepolarizedgluondistributionhasthehugeuncertaintywhichismuchlarger
thanthedistributionitself.Iftheuncertaintyis takenintoaccount,evenD g(x)=0isal-
lowed at this stage. Although there are variations among the four parametrizations in
Fig. 1, they are consistent with each other within the PDF uncertainties. From the PDF
determination,we find that thequark spincontent is DS =0.21±0.14and the first mo-
mentofD g(x)isD g=0.50±1.27at Q2=1 GeV2.
The results indicate that it is impossible to
determine a reliable gluon polarization at this 1.2
HERMES
stage. There are two types of contributions to 0.8 SMC
g from the polarized gluon distribution. One COMPASS Q2<1 GeV2
is1through the NLO coefficient function in Eq. xg() 0.4
(2), and the other is through the DGLAP Q2 x)/ 0
g(
evolution. Current inclusive DIS experimental -0.4
D AAC03
data are not accurate enough to specify these Uncertainty
-0.8 of AAC03
gluonic effects, which results in the huge glu-
-1.2
onic uncertainty. On the other hand, there are 0.001 0.01 0.1 1
x
reports on the gluon polarization by observing
high-pT hadrons in DIS. The HERMES, SMC, FIGURE2.AACD g/gincomparison
and COMPASS results are shown in Fig. 2 in withhigh-p hadron-productiondata.
T
comparison with the AAC03 gluon distribution
withtheuncertaintyatQ2=1GeV2.TherearealsopreliminaryresultsonD g/gfromthe
COMPASS collaboration by using large-Q2 and charmed-meson data. We find that the
AAC03curveagrees withthree datapointsinFig. 2.
After the AAC03 analysis, measurements were reported by JLab, HERMES, and
COMPASScollaborationsonthespinasymmetryA . InadditiontotheseDISmeasure-
1
ments, high precision data of A (p 0) from PHENIX and A (jet) from STAR experi-
LL LL
ment measured in polarized pp collisions at RHIC are reported at this conference. The
AAC03 distributions are compatible with the reported A (p 0). These new data should
LL
improve the situation on the determination of polarized PDFs. In particular, the RHIC
data as well as the high-p DIS data can impose considerable constraints on the gluon
T
polarization.
The AAC polarized PDF code is available at the web site [3], and it can be used for
calculating the PDFs in a given kinematical point of x and Q2. It is noteworthy that a
codeisprovidedalso forcalculatingthePDFuncertainties.
ACKNOWLEDGMENTS
S.K. was supported by the Grant-in-Aid for Scientific Research from the Japanese
MinistryofEducation,Culture,Sports,Science,andTechnology.Thisworkissupported
bytheJapan-U.S. CooperativeScienceProgram.
REFERENCES
1. Y.Goto,N.Hayashi,M.Hirai,H.Horikawa,S.Kumano,M.Miyama,T.Morii,N.Saito,T.-A.Shibata,
E.Taniguchi,T.Yamanishi(AsymmetryAnalysisCollaboration),Phys.Rev.D62,034017(2000).
2. M.Hirai,S.Kumano,N.Saito(AAC),Phys.Rev.D69,054021(2004).
3. TheAACcodeforpolarizedPDFsisavailableathttp://spin.riken.bnl.gov/aac/.