Effect of polarized gluon distribution on spin asymmetries for neutral and charged pion production

0.21 MB·English

by  Masanori Hirai

Effect of polarized gluon distribution on spin asymmetries
for neutral and charged pion production
M. Hirai1,∗ and K. Sudoh2,†
1Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization,
1-1, Ooho, Tsukuba, Ibaraki, 305-0801, JAPAN
2Radiation Laboratory, RIKEN (The Institute of Physical and Chemical Research),
Wako, Saitama 351-0198, JAPAN
(Dated: February 1, 2008)
5 A longitudinal double spin asymmetry for π0 production has been measured by the PHENIX
0
collaboration. Theasymmetryissensitivetothepolarized gluondistributionandisindicatedtobe
0
positivebytheoreticalpredictions. Westudyacorrelation betweenbehavioroftheasymmetryand
2
polarized gluon distribution in neutral and charged pion production at RHIC.

n
a PACSnumbers: 13.85.Ni,13.88.+e
J

3
1 I. INTRODUCTION pT = 1 3 GeV. The lower bound of the π0 asym-
∼
metry at low p has been considered, and a slight nega-
T
2 tiveasymmetrybymodifying∆g(x,Q2)hasbeendemon-
v Determination of the polarized parton distribution strated in Ref. [9]. However, there is no theoretical pre-
2 functions (PDFs) is crucial for understanding the spin
dictions indicating large negative asymmetry.
0 structureofthenucleon[1]. Asisknownwell,theproton
In this paper, we study the behavior of the π0 dou-
1 spiniscomposedofthe spinandorbitalangularmomen-
3 ble spin asymmetry correlated with ∆g(x,Q2) in Sec II.
tum of quarks and gluons. Several parametrizations of
0 By using three types of ∆g(x,Q2), we suggest that the
the polarized PDFs have been proposed, and have suc-
4 asymmetry in large p region is more sensitive to the
cessfully reproduced experimental data [2, 3, 4, 5, 6]. T
0 functionalformof∆g(x,Q2). Animpactofthenewdata
/ In particular, the amount of the proton spin carried by
h on determination of ∆g(x) is discussed in terms of un-
quarksisdeterminedwellbyglobalanalyseswiththepo-
p certainty of the asymmetry coming from the polarized
larizeddeepinelastic scattering(DIS) data. The value is
- PDFs. Furthermore, we discuss a spin asymmetry for
p about ∆Σ = 0.1 0.3, whereas the prediction from the
∼ chargedpionproductionin Sec. III. An asymmetry tak-
e naivequarkmodelis∆Σ=1. Thissurprisingresultleads
h ing the difference of cross sections for π+ and π− pro-
to extensive study on the gluon polarization. The cur-
: duction is proposed, and it is useful to discuss the sign
v rent parametrizations suggest a large positive polariza-
of ∆g(x) in the whole p region. The Summary is given
i tion of gluon. However, our knowledge about the polar- T
X in Sec. IV.
ized gluon distribution ∆g(x,Q2) is still poor, since the-
r
a oretical and experimental uncertainties are rather large.
The determination of ∆g(x,Q2) gives us a clue to the
proton spin puzzle. II. SPIN ASYMMETRY FOR NEUTRAL PION
PRODUCTION
The RHIC is the first high energy polarized proton-
proton collider to measure ∆g(x,Q2) [7]. We can ex-
tract information about ∆g(x,Q2) through various pro- A. Ambiguity of the polarized cross section
cesses, e.g., prompt photon production, jet production,
and heavy flavor production. These processes are quite First, we describe the longitudinal double spin asym-
sensitive to ∆g(x,Q2), since gluons in the initial state metry for π0 production. It is defined by
associate with the cross section in leading order (LO).
Recently, the PHENIX collaboration has reported re- Aπ0 [dσ++−dσ+−]/dpT = d∆σ/dpT, (1)
sults for inclusive π0 production pp π0X [8] which LL ≡ [dσ+++dσ+−]/dpT dσ/dpT
is also likely to be sensitive to ∆g(x,→Q2). The double
where p is the transversemomentum of produced pion.
spin asymmetry was measured in longitudinally polar- T
ized proton-protoncollisionsat RHIC in the kinematical dσhh′ denotes the spin-dependent cross section with def-
inite helicity h and h′ for incident protons.
ranges: center-of-mass(c.m.) energy √s=200GeV and
The cross sections can be separated short distance
central rapidity η 0.38. The data imply that the
| | ≤ parts from long distance parts by the QCD factorization
asymmetry might be negative at transverse momentum
theorem. The short distance parts represent interaction
amplitudes of hard partons, and are calculable in the
framework of perturbative QCD (pQCD). On the other
∗E-mail: [email protected] hand,thelongdistancepartssuchasPDFsandfragmen-
†E-mail: [email protected] tation functions should be determined by experimental
2
data. The polarized cross section ∆σ is written by on the fragmentation functions. Significant modification
of them would not be expected. Therefore, the fragmen-
d∆σpp→π0X ηmax 1 1 tationfunctions arenotthe sourceofthe negativeasym-
= dη dx dx
a b metry even if they have uncertainty to some extent.
dp
T aX,b,cZηmin Zxmain Zxmbin In the polarized reaction, kinematical ranges and the
∆fa(xa,Q2)∆fb(xb,Q2) fragmentationfunctions are the same as the unpolarized
×
∂(tˆ,z) ∆σˆab→cX(sˆ,tˆ) caseexceptthe polarizedPDFs. For the polarizedquark
distributions ∆q(x) and ∆q¯(x), the antiquark distribu-
×J (cid:18)∂(pT,η)(cid:19) dtˆ tions and their flavor structure are not well known. For
Dπ0(z,Q2), (2) π0production,subprocessesare(lightquark)flavorblind
× c
reaction,andthepredominantqg processdependsonthe
where ∆f (x,Q2) is the polarizedPDFs, andDπ0(z,Q2) sum ∆q(x)+∆q¯(x) which is relatively determined well
i c
is the spin-independent fragmentationfunction decaying by the polarized DIS data [9], and so ambiguities of the
into pion c π0 with a momentum fractionz. The sum polarized quark distributions can be neglected. In con-
→
is over the partonic processes a+b c+X associated sequence, the undetermined polarized gluon distribution
with π0 production. is the Jacobia→nwhich transforms ∆g(x) remains as the source of the uncertainty of the
kinematical variablesJfrom tˆand z into p and η of the asymmetry.
T
produced π0. ∆σˆ describes the polarized cross section
of subprocesses. The partonic Mandelstam variables sˆ
and tˆare defined by sˆ = (p +p )2 and tˆ= (p p )2
a b a − c B. Correlation between the spin asymmetry and
with partonic momentum p , respectively. The squared
i the polarized gluon distribution ∆g(x)
c.m. energy s is related to sˆthrough sˆ= x x s and set
a b
as √s = 200 GeV. The pseudo-rapidity η is limited as
To investigate a role of ∆g(x) for the behavior of the
η 0.38 in the PHENIX acceptance.
| |≤ asymmetry, we prepare three functional forms as shown
In this analysis, the cross sections and the spin asym-
in Fig 1. Solid curve shows ∆g(x) by the global analysis
metry are calculated in LO level. Rigorous analysis of
with the polarized DIS data [2]. Dashed and dot-dashed
(α3) next-to-leading order (NLO) calculationhas been
O s curves show two artificial modified ∆g(x), respectively.
established in Ref. [10]. We believe that the qualitative
The sample-1 distribution has a node. The gluon dis-
behavioroftheasymmetrydoesnotchange,evenifNLO
tribution with a node has been indicated in the paper
correctionsareincludedinourstudy. Innumericalcalcu-
by J¨ager et al., [9]. Our distribution is negative in the
lations, we adopt the AAC set [2] as the polarizedPDFs
small x region, and positive in the large x region. It
and the Kretzer set [11] as the fragmentation functions.
has opposite signs of ∆g(x) shown in Fig. 2 of their
We choose the scale Q2 =p2.
T paper. The sample-2 distribution is small negative in
The partonic subprocesses in LO are composed of
the whole x region. Their distribution is similar to the
(α2) 2 2 tree-level channels listed as gg q(g)X,
qOg s q(g→)X, qq qX, qq¯ q(g,q′)X, qq′ →qX, and sample-2 rather than the sample-1. It shows barely pos-
qq¯′→ qX includi→ng channel→s of the permuta→tion q q¯. itive at small x, while the sample-2 is negative. Since
→ ↔ the sample-1 and 2 are within the ∆g(x) uncertainty by
Main contribution to the polarized cross section comes
the AAC analysis, these distributions can be adopted as
from gg q(g)X and qg q(g)X channels with con-
→ →
ventional PDFs and fragmentation functions. The gg
contribution dominates in low pT region and steeply de- 0.4
creaseswith pT increases. Then, the qg process becomes AAC
dominantinlargerpT region. Thecrossingpointofthese 0.3 sample-1
contributionshoweverdependsonparametrizationofthe
polarized PDFs. In both cases, the spin asymmetry for 0.2 sample-2
π0 production is sensitive to the gluon polarization.
0.1
As mentioned above, the partonic cross section ∆σˆ is
well-defined in the pQCD framework. Hence, as a cause
0
ofinconsistencywith the PHENIXdata,weconsiderthe
ambiguity of long distance parts: fragmentation func-
-0.1
tions and PDFs.
The fragmentation into π0 includes all channels -0.2
q,q¯,g π0. Eachcomponentofthe fragmentationfunc- 0.001 0.01 0.1 1
tions D→π0 can be determined by global analyses with x
c
several experiments [11, 12]. The unpolarized cross sec-
tion measured by the PHENIX [13] are consistent with FIG. 1: Polarized gluon distributions ∆g(x) at pT = 2.5
NLO pQCD calculations within model dependence of GeV.Solid,dashed,anddot-dashedcurvesindicatetheAAC,
Dπ0. These precise measurements give strong constraint sample-1, and 2 distributions, respectively.
c
3
0.05 Thisisbecausethatthenoderapidlyshiftstowardlow-x
AAC
direction due to Q2 evolution with p . Therefore, the
T
0.04 sample-1 positive polarization for ∆g(x) at medium x contributes
sample-2 predominantlytothepositiveasymmetryviathegg pro-
0.03
cess. Furthermore,theasymmetryatlargep issensitive
T
0.02 to the behavior of ∆g(x) at medium x.
As another possibility of the negative asymmetry, we
0.01 choose slight negative polarization for ∆g(x). In this
case, the gg contribution is positive while the qg con-
0 tribution is negative. The asymmetry is determined by
the difference between two contributions. The gg and
-0.01
qg contributions are proportional to (∆g)2 and ∆g, re-
0 2 4 6 8 10 12 14
spectively. The gg contribution is more sensitive to the
p (GeV)
T behavior of ∆g(x). In particular, the behavior at low x
significantly affects on the contribution at low p since
T
FIG. 2: Spin asymmetries for π0 production by using three the value of xmin in Eq. (2) is rather small. In order
different ∆g(x)in Fig. 1. to make the positive gg contribution smaller, the ∆g(x)
for the sample-2 is taken small polarization at low x as
shown in Fig. 1.
a model of ∆g(x). These are taken account of the Q2 InFig. 2,asfarasthesample-2isconcerned,theasym-
dependence by the Dokshitzer-Gribov-Lipatov-Altarelli- metry indeed becomes negative in the whole p region.
T
Parisi (DGLAP) equation with the polarized quark and In the region p < 3 GeV, the small negative polar-
T
antiquark distributions. ization for ∆g(x) generates slight positive contribution
We discuss the behavior of the spin asymmetry asso- of the gg process. In this case, the gg contribution is
ciated with the functional form of ∆g(x). The obtained the same order ofmagnitude as the qg contribution,and
asymmetries with these gluon distributions are shownin almost cancel out the negative contribution. The asym-
Fig. 2. We find that the asymmetry for the AAC ∆g(x) metry is therefore determined by other processes. The
is positive in the whole pT region. The asymmetries for totalcontributionofthe processesexceptabovetwopro-
thesample-1and2becomenegativeatlowpT. Inpartic- cesses becomes slight negative. Above the region,the gg
ular, we obtained the negative asymmetry in the whole contributionrapidlydecreaseswithp increases. The qg
T
pT region by using the sample-2 ∆g(x). Furthermore, process becomes dominant contribution, which provides
one cansee variationsof these asymmetriesatlargepT. thenegativeasymmetry[9]. Thus,thenegativeasymme-
The asymmetry for the AAC is positive and increases try can be obtained in the whole p region by using the
T
with p . The positive polarization for ∆g(x) generates negative ∆g(x) which makes the qg contribution larger
T
positivecontributionsofgg andqg processeswhichdom- than the gg contribution.
inate in the π0 production. In this case, the asymmetry In the sample-2, we should note that the magnitude
cannot become negative. of ∆g(x) at the minimum point cannot be large. This
The positive ∆g(x) is suggested by the recent global is because that the shape of ∆g(x) is rapidly varied by
analyses with the polarized DIS data [2, 3, 4, 5, 6]. Al- the Q2 evolution, the minimum point of ∆g(x) shifts to-
though these analyses obtain good agreement with the ward low-x and the width broadens. At moderate p ,
T
experimentaldata, the ∆g(x) cannot be determined and the gg process is more sensitive to the low-x behavior of
it has large uncertainty. Therefore, we cannot rule out the evolved ∆g(x) than the qg process. If the ∆g(x) is
thenegativepolarizationfor∆g(x). Thereisapossibility taken large negative polarization at the minimum point,
of the negative asymmetry with the modified ∆g(x). the magnitude of the gg contribution becomes rapidly
For the sample-1 in Fig. 2, the asymmetry is slight large compared with the qg contribution, and then the
negative at low pT and changes into positive at pT = 3 asymmetry becomes positive at moderate pT. The small
GeV.As is mentionedinRef. [9],the ∆g(x)with anode negative ∆g(x) is therefore required to obtain the nega-
has the possibility of making the small negative asym- tive asymmetry in the whole pT region.
metry at low p . In the region p < 3 GeV, we find In above two cases at low p , we cannot also obtain
T T T
that contributions of gg and qg processes are negative, negative value exceeded the lower limit 0.1% that is
−
respectively. To make negative gg contribution would suggested in Ref. [9]. Furthermore, even if the asym-
be needed opposite polarizations of ∆g(x) at x and x . metry is positive, the magnitude is below 1%. As we
a b
Computedbyusingseveralshapesof∆g(x)withanode, discussed, the functional form of ∆g(x) needs some re-
the gg contribution is not always negative. The contri- straints to make the asymmetry negative. It is difficult
bution basically depends on the shape of ∆g(x) even if to obtain sizable value in comparison with the positive
it has a node. case.
Intheregionp >3GeV,thegg contributionchanges Atlargep ,thedifferenceoftheobtainedasymmetries
T T
into positive, and dominates in the regionp <10 GeV. remarkablyreflectsthemedium-xbehaviorof∆g(x). Ex-
T
4
perimental data in the region is useful to determine the 0.1
∆g(x). Forinstance,theasymmetryforthesample-2be-
comes rather larger to negative direction. If future pre-
cise data indicate the negative asymmetry in the region,
the ∆g(x) requires significant modification of its func-
0
tional form. It has the potential of the negative gluon
contributionto the nucleonspin. In orderto understand
the behavior of ∆g(x) in detail, we require experimental
data covering a wide p region.
T
-0.1
1 2 3 4 5 6
C. Uncertainty of the spin asymmetry
p (GeV)
T
Next,weconsidertheeffectoftheπ0dataonthe∆g(x)
determination in terms of the uncertainty estimation for FIG. 3: Comparison of the asymmetry uncertainty δAπL0L
the spin asymmetry. The large uncertainty of ∆g(x) im- with thestatistical errors for √s=200 GeV.
plies the difficulty of extracting the gluon contribution
fromthepolarizedDISdata. Wearethereforeinterested
in constraint power of the new data on ∆g(x). If the In Fig. 3, the asymmetry uncertainty is compared
experimental data are included in a global analysis, the to the statistical errors of the experimental data by the
asymmetry uncertainty will be bounded within statisti- PHENIX [8]. In this comparison, we exclude the data
cal error range. See, for example, Fig. 2 of Ref. [2]. As at pT = 1.5 GeV. This is because that the data might
far as evaluation of the constraint is concerned, the un- have contribution from soft physics, and it might not be
certainty can be compared with statistical errors of the explained as physics of a hard process. We have no idea
data, although it is rough evaluation. whether such data can be included in the global analy-
The asymmetry uncertainty coming from the polar- sis. Fromthisfigure,wefindthattheuncertaintyalmost
ized PDFs is defined by a polarized cross section uncer- corresponds to the experimental errors, and is mainly
tainty divided by a unpolarized cross section: δAπ0 = composed of the uncertainty of ∆g(x). This fact indi-
δ∆σπ0/σπ0. ThecrosssectionuncertaintyisobtainLedLby cates that the present π0 data have the same constraint
on ∆g(x) as the polarized DIS data. At this stage, one
taking a root sum square of uncertainties of all subpro-
cannot expect to reduce the ∆g(x) uncertainty even if
cesses. These uncertainties are estimated by the Hessian
these data are included into the global analysis. How-
method, and are given by
ever, the asymmetry uncertainty is very sensitive to the
δ∆σkπ0 2 =∆χ2 ∂∆σ∂kπa0(pT) Hi−j1 ∂∆σ∂kπa0(pT) , t∆ogb(exc)oumneceargtaoiondtyp.rTobheefπo0r∆prgo(dxu)cbtyiofnuthuarsetphreecpisoetednattiaal.
i ! j !
h i Xi,j Itshouldbenotedthatsymmetricuncertaintyisshown
(3)
in order to compare with the statistical errors in Fig. 3.
where k is the index of subprocesses. a is a optimized
i The lower bound is however incorrect because a lower
parameter in the polarized PDFs. H is the Hessian
ij limit of the asymmetry is not taken into account. As
matrix which has the information of the parameter er-
mentioned in previous subsection, the asymmetry can-
rors and the correlation between these parameters. The
notexceed 0.1%atlowp where the ggprocessdomi-
∆χ2determinesaconfidenceleveloftheuncertainty,and − T
nates. Although asymmetric uncertainty should be esti-
isestimatedsothatthe levelcorrespondstothe 1σ stan-
mated, such uncertainty cannot be obtained by the Hes-
dard error. We choose the same value as the AAC anal-
sian method. We therefore need further investigation of
ysis [2]. Further,the gradientterms for the subprocesses
the lower bound for the asymmetry uncertainty.
∂∆σπ0(p )/∂a are obtained by
k T i
d∆σπ0 ηmax 1 1
k = dη dxa dxb III. SPIN ASYMMETRY FOR CHARGED PION
dp
T a,b,cZηmin Zxmain Zxmbin PRODUCTION
X
∂∆f (x ) ∂∆f (x )
a a b b
∆f (x )+∆f (x )
× ∂a b b a a ∂a We discuss the spin asymmetry for charged pion pro-
(cid:20) i i (cid:21) duction, π+ and π−. Unpolarized and polarized cross
∂(tˆ,z) ∆σˆab→cX(sˆ,tˆ)Dπ0(z), (4) sections can be similarly calculated by using the frag-
×J (cid:18)∂(pT,η)(cid:19) dtˆ c mentation functions decaying into charged pion Dπ± in
ThegradienttermsforthepolarizedPDF∂∆f (x )/∂a Eq. (2). We show asymmetries with the AAC ∆g(x)
a a i
are analytically obtained at initial scale Q2, and are nu- and sample-2 ∆g(x) in Fig. 4. In the asymmetries for
0
merically evolved to arbitrary scale Q2 by the DGLAP the AAC ∆g(x), one can see differences among them in
equation. large p region where the qg process is dominant. The
T
5
Thebehaviorofthe asymmetryissensitivetothesignof
0.06 p 0
∆g(x) because the contribution of the gg processes are
p + eliminated and one of the qg process becomes dominant
0.04 in the whole pT region. The polarized cross section for
p - gg gg process is given by
AAC03 →
0.02 ∆σgπg± =∆g⊗∆g⊗Dgπ±⊗∆σˆgg→gg. (9)
0 This contribution is cancelled out due to Dπ+ = Dπ−.
g g
sample-2 For the same reason, gg qq¯process does not also con-
→
-0.02 tribute by summing fragmentation functions for flavors:
0 2 4 6 8 10 12 14 Dπ+ = Dπ−. As the similar case, the contribu-
i i i i
p (GeV) tions of qq¯ gg, qq¯ q′q¯′ processes are also vanished.
T
P →P →
Theunpolarizedcrosssectioncanbesimilarlycalculated
FIG. 4: Asymmetries for neutral and charged pion produc- with unpolarized PDF’s and partonic cross sections.
tions with theAACand sample-2 ∆g(x)sets. The asymmetry can be obtained by the difference of
qg process. The second term of Eq. (5) is cancelled out
for the same reason of gg gg process. And then, the
polarizedcrosssections ofqg processforπ+ andπ− pro- →
asymmetry is consequently given by
duction are written by
∆σqπg± = ∆g⊗ i ∆fi⊗Diπ±!⊗∆σˆqg→qg AπL+L−π− ≃ ∆g⊗g(⊗∆(uuvv−−d∆vd)v⊗)(⊗D(1Dπ−1π−D2Dπ)2π⊗)σ⊗ˆq∆g→σˆqqgg→q(g10,)
X where∆f (=∆f ∆f¯)is apolarizedvalence quarkdis-
v
+ ∆g⊗ ∆fi ⊗Dgπ±⊗∆σˆqg→qg . (5) tribution. The fol−lowing relations among the fragmenta-
i ! tion functions are assumed by the isospin symmetry,
X
where the symbol ⊗ denotes convolution integral in Eq.
(u2,)d.,si,u¯i,ndd¯i,caatneds st¯h.eAqcutaurakl flcaalvcourl,atainond iinsctluadkeens paesrmi u=- (DDuuππ−+ ==DDddππ¯¯+− ==DDuπu¯π¯−+ ==DDddππ−+ ≡≡DD21ππ (11)
tated terms of x and x . There are following relations
a b
among the fragmentation functions for charged pion: This relation is used in parametrization of the fragmen-
tation functions [11].
Dπ+ >Dπ−, Dπ+ <Dπ−,
u u d d (6) In the asymmetry in Eq. (10), ambiguity of fragmen-
Dqπ+ =Dqπ¯−, Dgπ+ =Dgπ−, tationfunctionDπ± isremovedbythecancellationofthe
g
and the fragmentation functions for neutral pion are de- convolution part. Another ambiguity from the fragmen-
fined by tationfunctionscanbealsocancelledbetweennumerator
anddenominator. Inaddition, ∆u ∆d is determined
Dπ0 =(Dπ+ +Dπ−)/2 . (7) well,sinceitsfirstmomentisconstvra−inedvbyneutronand
i i i
hyperon beta decay constants [2, 3, 4, 5]. Of course, un-
For π+ production, the contribution associated with
polarized PDF’s are precisely determined in comparison
∆u distribution is enhanced by the fragmentation func-
with the polarized PDF’s. This asymmetry can be de-
tion Dπ+. Increasing asymmetry for π+ production is
u fined by well knowndistributions without ∆g; therefore,
caused by positive contribution from ∆u distribution,
we can effectively extract information about ∆g(x) in-
whereas decreasing asymmetry for π− production comes
cluding its sign.
from negative ∆d distribution.
Figure. 5 shows the asymmetry defined by Eq. (8).
On the other hand, the asymmetries for the sample-2
Solid and Dotted curves are asymmetries with AAC
∆g(x)arealmostthe same. Thedifferencesamongthem
∆g(x) and ∆g(x). We find large asymmetries in both
dependsonthemagnitudeof∆g(x),sincetheasymmetry −
cases. Inparticular,the asymmetrywith ∆g(x)isneg-
is proportional to ∆g(x) as written in Eq. (5). If the −
ative and the absolute value is large in comparison with
absolutevalueof∆g(x)issmall,therearenotsignificant
single pion production. Since the asymmetry is domi-
differences among the asymmetries for π0, π+, and π−
nated by qg process in the whole p region, the differ-
productions. T
ence of the sign of ∆g(x) is markedly reflected in the
In order to determine ∆g(x) with its sign by using
asymmetry.
charged pion production, let us propose an interesting
We mention the contribution of qq process to the
observable which is defined by
asymmetry. In the region 8 < p < 13 GeV, the qq
T
AπL+L−π− = ∆σσππ++−−ππ−− ≡ ∆σσππ++ −σ∆πσ−π− . (8) c(∆onσtπri+butio∆nσaπ−cc)o,uanntsdf2o7r-5160%-15o%f othfethuenppoollaarriizzeedd ppaarrtt
−
−
6
0.25 constraint power of experimental data are weaker than
S =200GeV that of the π0 asymmetry below the region. However,
h £ 0.35 D DDD g the value of Rasym becomes larger than Rsta above the
AAC03 region. The asymmetry would have the same impact on
∆g(x) as the π0 production. Although more luminosity
is needed in comparison with π0 production, it is useful
0
TABLE I:The value of parameters α, Rsta, and Rasym.
pT (GeV) 9 10 11 12 13
Ap +-p - -D ---DDD g α 0.82 0.80 0.78 0.76 0.74
LL AAC03 Rsta 7.2 6.4 5.7 5.2 4.7
-0.250 2 4 6 8 10 12 14 Rasym 5.0 5.1 5.3 5.5 5.6
p (GeV)
T
to determine effectively the behavior of ∆g(x) with the
FIG. 5: Asymmetries for the difference of charged pion pro-
sign.
duction with ∆g(x)and ∆g(x).
−
(σπ+ σπ−) of the asymmetry in Eq. (8). These contri- IV. SUMMARY
−
butions are not negligible. In particular, effect of the qq
In summary, we have investigated the correlation be-
contribution in the polarized part appears as the differ-
tween the behavior of the spin asymmetry for pion pro-
encebetweentheabsolutevaluesofasymmetries. Contri-
duction and the functional form of ∆g(x). The experi-
butions of all sub-processes are taken into account,how-
ever the qq¯(′) qq¯(′) and qq′ qq′ processes are negli- mentaldatabythePHENIXindicatesthenegativeasym-
→ → metry at low p , and motivate us to modify the func-
gible. The difference is due to the positive contribution T
tional form of ∆g(x) drastically. In order to obtain neg-
of qq process. The asymmetry with ∆g(x) is therefore
− ative asymmetry, the functional form of ∆g(x) requires
suppressed.
some restraints. By modifying ∆g(x), the slight nega-
Next, we evaluate the experimental sensitivity of this
tive asymmetry can be obtained at low p . Moreover,
spin asymmetry. We compare the statistical error of the T
asymmetry δAπ+−π− with one of π0 production δAπ0 . we have indicated the existence of the negative polariza-
LL LL tionof∆g(x) whichkeepsthe asymmetryto be negative
An expected statistical error δALπ+L−π− is given by in the whole pT regions. The large negative asymmetry
is inconsistent with the theoretical predictions by using
δAπ+−π− = 1 1 √1+α, (12) ∆g(x) from polarized DIS data. However, experimental
LL P2√Nπ+ 1 α uncertaintiesarelargeatpresent. Itisprematuretocon-
−
clude that the pQCD framework is not applicable to π0
where P is the beam polarization. α is the ratio of the
production in polarized pp collisions.
number of event for π− and π+: α = Nπ−/Nπ+. Nπ± Uncertaintyoftheπ0 asymmetrycomingfromthe po-
are obtained by the integratedluminosity and the un- larizedPDF’swithDISdataiscorrespondtothecurrent
polarized total cross section σπ±: Nπ± =L σπ±. The statistical errors by the PHENIX. These data have the
L
ratio of these statistical errors can be obtained by same constraint power on ∆g(x) as present DIS data.
Thefuturemeasurementswillprovideusefulinformation
δAπ+−π− 1 1+α for clarifying the gluon spin content.
R LL = . (13)
sta ≡ δAπ0 √21 α Furthermore, we have proposed the spin asymmetry
LL − defined by the difference of cross sections for π+ and π−
The parameter α has energy dependence, and decreases production. We have discussed an impact of the asym-
with p increases. metry on determination of ∆g(x). In the asymmetry
T
Table. I represents the value of these parameters. Aπ+−π−, the gg processes are cancelled out, and qg pro-
R Aπ+−π−/Aπ0 is the ratio of asymmetries. The ceLssLbecomes dominant. Ambiguity ofthe fragmentation
π+asymπ≡− asLyLmmetryLiLs about 5 times larger than the π0 functions can be reduced. The behavior of the asymme-
−
asymmetry. Intheregionp <11GeV,thestatisticaler- tryissensitivetothe signof∆g(x). Onecanobtainnew
T
ror becomes larger than the rate of the asymmetry. The probe for ∆g(x) in pion production at RHIC.
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332, 1 (2000). mano, and N. Saito, Phys. Rev.D 69, 054021 (2004).
7
[3] J. Blu¨mlein and H. B¨ottcher, Nucl. Phys. B 636, 225 [9] B. J¨ager, M. Stratmann, S. Kretzer, and W. Vogelsang,
(2002). Phys. Rev.Lett. 92, 121803 (2004).
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Phys.J. C 23, 479 (2002). Phys. Rev. D 67, 054005 (2003); D. de Florian, Phys.
[5] M. Glu¨ck, E. Reya, M. Stratmann, and W. Vogelsang, Rev. D 67, 054004 (2003).
Phys.Rev.D 63, 094005 (2001). [11] S. Kretzer, Phys. Rev. D 62, 054001 (2000), and refer-
[6] D. de Florian and R. Sassot, Phys. Rev. D 62, 094025 ences therein.
(2000). [12] B. A. Kniehl, G. Kramer, and B. P¨otter, Nucl. Phys. B
[7] For example, G. Bunce, N. Saito, J. Soffer, and W. Vo- 582, 514 (2000).
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