Soft-Fermion-Pole Mechanism to Single Spin Asymmetry in Hadronic Pion Production

0.05 MB·English

by  Yuji Koike

Soft-Fermion-Pole Mechanism to Single Spin
Asymmetry in Hadronic Pion Production
Yuji Koike and Tetsuya Tomita
9 DepartmentofPhysics,NiigataUniversity,Ikarashi,Niigata950-2181,Japan
0
0
2 Abstract. Single spin asymmetry (SSA) is a twist-3 observable in the collinear factorization ap-
proach.Wepresentatwist-3single-spin-dependentcrosssectionformulaforthepionproductionin
n pp-collision,p p p X,relevanttoRHICexperiment.Inparticular,wecalculatethesoft-fermion-
a ↑ →
pole (SFP) contributionto the crosssection from the quark-gluoncorrelationfunctions.We show
J
thatitseffectcanbeaslargeasthesoft-gluon-pole(SGP)contributionowingtothelargeSFPpar-
9
tonichardcrosssection,eventhoughthederivativeoftheSFPfunctiondoesnotparticipateinthe
1
crosssection.

] Keywords: Singlespinasymmetry,Twist-3,Soft-fermion-pole
h
PACS: 12.38.Bx,13.88.+e,13.85.Ni
p
-
p
e The observed large single spin asymmetry (SSA) in p p hX (h = p ,K etc) can
↑
h →
not be explained by the usual parton model and perturbative QCD. In the framework
[
of collinear factorization, SSA is a twist-3 observable and can be described in terms of

1
thetwist-3quark-gluon-correlationcorrelationfunctionsin thenucleonand/orpion[1]-
v
6 [9]. Among various components of the twist-3 cross sections, a large contribution is
5 expected to come from the twist-3 distribution functions in the transversely polarized
7
nucleon.(See[4],however.)Therearetwoindependenttwist-3distributionsGa(x ,x )
2 F 1 2
1. andGaF(x1,x2)whicharedefinedfromalight-conecorrelationfunctioninthenucleon
0 y¯aFa +y a wherey a isthequarkfieldofflavoraandFa + isthegluon’sfieldstrengt∼h.
9 h i
Forteheexplicitdefinitionof Ga(x ,x ),Ga(x ,x ) and therelation between different
0 F 1 2 F 1 2
{ }
: expressions for the twist-3 distribution functions, see [5, 7]. Due to the naively T-odd
v
i nature of SSA, the relevant cross sectioneoccurs as a pole contribution of an internal
X propagator in the partonic hard part. For p p p X, two kinds of poles contribute:
↑
r →
a soft-gluon-pole(SGP) leading to x1 =x2 and soft-fermion-pole(SFP) leading to xi =0
(i = 1 or 2). Thus the contribution to the cross section from these functions can be
schematicallywrittenas
dG (x,x)
D s tw3 = G (x,x) x F f(x ) D(z) sˆ
F ′ SGP
− dx ⊗ ⊗ ⊗
(cid:18) (cid:19)
+ G (0,x) f(x) D(z) sˆ +G (0,x) f(x) D(z) sˆ , (1)
F ′ SFP F ′ S′FP
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
where we have used the fact that the SGP contribution appears in the combination
e
of G (x,x) xdGF(x,x) [6, 9]. Note only G (x,x) contributes through the SGP, since
F − dx F
G (x,x)=0 due to the anti-symmetric nature of G (x ,x ) under x x . The origin
F F 1 2 1 2
ofthiscombinationandtheconnectionofsˆ tothetwist-2crosssec↔tionwasclarified
SGP
ien [9, 8]. Previous analyses[2, 6] focussed on theeSGP contribution (first term of (1)),
assumingthatthisoneisadominantcontribution.However,thereisnocluethattheSFP
functionsG (0,x)andG (0,x)themselvesaresmall,anditsimportancedependsonthe
F F
behaviorofthepartonichard cross sectionssˆ and sˆ .
SFP S′FP
The purpose of this reeport is to present the SFP cross section for p p p X and to
↑
→
study its impact compared with the SGP contribution, assuming that the SFP functions
G (0,x)and G (0,x)havethesameorderofmagnitudeas theSGPfunctionG (x,x).
F F F
For the calculation of the SFP contribution to p (p,S )+ p(p ) p (P )+X, we
↑ ′ h
followed the eformalism in [7]. With the Mandelstam va⊥riables for→this process, S =
(p+p )2,T =(p P )2 andU =(p P )2,theSFPcontributioncanbewrittenas[10]:
′ h ′ h
− −
E d3D s SFP = a s2MNp e +PhS (cid:229) 1 dz 1 dx′ dx 1
h dPh3 S 2 − ⊥a,b,cZzmin z3 Zx′min x′ Z x x′S+T/z
d x −x′U/z (cid:229) Ga(0,x)+Ga(0,x)
× −xS+T/z F F
(cid:18) ′ (cid:19)"a,b,c
(cid:16) (cid:17)
e
qb(x) Dc(z)sˆ +Dc¯(z)sˆ +qb¯(x) Dc(z)sˆ +Dc¯(z)sˆ
′ ab c ab c¯ ′ ab¯ c ab¯ c¯
× → → → →
n o
+(cid:229) (cid:0)Ga(0,x)+Ga(0,x) qb((cid:1)x)Dg(z)sˆ(cid:0) +qb¯(x)Dg(z)sˆ (cid:1)
F F ′ ab g ′ ab¯ g
a,b → →
(cid:16) (cid:17)(cid:16) (cid:17)
+(cid:229) Ga(0,x)+Gea(0,x) G(x) Dc(z)sˆ +Dc¯(z)sˆ
F F ′ ag c ag c¯
→ →
a,c
(cid:16) (cid:17)
(cid:0) (cid:1)
+(cid:229) Ga(0,x)+Gea(0,x) G(x)Dg(z)sˆ , (2)
F F ′ ag g
→
a (cid:21)
(cid:16) (cid:17)
where the summation (cid:229) impelies that the sum of a is over all quark and anti-quark
a,b,c
flavors (a = u,d,s,u¯,d¯,s¯, ) and the sum of b and c is restricted onto quark flavors
···
when a is a quark and onto anti-quark flavors when a is an anti-quark. (For an anti-
quark b, b¯ denotes quark flavor.) The integration region in the convolution formula is
T/z (T+U)
specified by x′min = S−+U/z and zmin = − S . G(x′) represents the gluon distribution
andDg(z)isthegluonfragmentationfunctionforthepion.sˆ etcrepresentspartonic
ab c
hard cross sections where c is the flavor of the parton fragme→nting into the pion. They
are functions of the Mandelstam variables in the parton level; sˆ=(xp+x p )2 =xx S,
′ ′ ′
tˆ=(xp P /z)2 = xT and uˆ=(x p P /z)2 = x′U and are givenas follows(N =3 is
− h z ′ ′− h z
thenumberofcolor):
(N2sˆ+2tˆ)(sˆ2+uˆ2) (N2tˆ+uˆ sˆ)sˆ
sˆ = − d +− − d d ,
ab→c N2tˆ3uˆ ac N3tˆuˆ2 ab ac
sˆ = 0,
ab c¯
→
(N2uˆ+2tˆ)(sˆ2+uˆ2) (N2uˆ+2sˆ)(tˆ2+uˆ2) (N2 1)uˆ2
sˆab¯ c = N2tˆ3uˆ d ac+ N2sˆ2tˆuˆ d ab− N−3sˆtˆ2 d abd ac,
→
(N2tˆ+2sˆ)(tˆ2+uˆ2) N2sˆ+tˆ uˆ
sˆab¯ c¯ = − N2sˆ2tˆuˆ d ab+− N3uˆ2− d abd ac, (3)
→
(N2sˆ+2tˆ)(sˆ2+uˆ2) 1
sˆ = + − N2(sˆ3+3sˆ2uˆ 2uˆ3)+sˆ3 sˆ2uˆ d ,
ab→g N2tˆ3uˆ N3sˆtˆuˆ2 − − ab
(N2uˆ+2tˆ)(sˆ2+uˆ2) (cid:0) (cid:1)
sˆab¯ g = − N2tˆ3uˆ
→
1 uˆ 1 1 sˆ2+sˆtˆ+tˆ2 uˆ N(uˆ3 tˆ3)(tˆ2+uˆ2)
+ + + + − d , (4)
N3 sˆtˆ uˆ N sˆuˆ2 −tˆ2 sˆ2tˆ2uˆ2 ab
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
N2(sˆ3 uˆ3)(sˆ2+uˆ2)
sˆ = −
ag→c (N2 1)sˆtˆ3uˆ2
(cid:26) −
sˆuˆ(sˆ2+sˆuˆ uˆ2) N2(sˆ4+sˆ3uˆ+sˆ2uˆ2+sˆuˆ3+uˆ4) (N2uˆ+2sˆ)(tˆ2+uˆ2)
+ − − d + ,
N2(N2 1)sˆtˆ2uˆ2 ac N(N2 1)sˆ2tˆuˆ
− (cid:27) −
sˆ+2tˆ N2sˆ (N2tˆ+2sˆ)(tˆ2+uˆ2)
sˆ = − d +− , (5)
ag→c¯ N2(N2 1)uˆ2 ac N(N2 1)sˆ2tˆuˆ
− −
N2
sˆ = − 4sˆ6+11sˆ5tˆ+19sˆ4tˆ2+22sˆ3tˆ3+19sˆ2tˆ4+11sˆtˆ5+4tˆ6
ag→g (N2 1)sˆ2tˆ3uˆ2
− (cid:16) (cid:17)
1
+ sˆtˆuˆ2+N2(sˆ4+sˆ3tˆ+2sˆ2tˆ2+sˆtˆ3+tˆ4) . (6)
N2(N2 1)sˆtˆ2uˆ2 −
−
(cid:8) (cid:9)
A remarkable feature of (2) is that the partonic hard cross sections for G (0,x) and
F
G (0,x) are the same, even though each diagram gives different contributions for the
F
two functions[10]. Accordingly, they appear in the combination of Ga(0,x)+Ga(0,x)
F F
ien (2). We also note that some terms in the hard cross section (the first terms in sˆ ,
ab c
sˆab g, sˆag c, sˆag g etc) accompany the large color factors compared to tehe S→GP
cros→s sectio→n derive→d in [6], and show the steep rising behavior in the forward direction
(i.e. large S/T). This suggests that the SFP contribution gives rise to the nonnegligible
contributiontotheasymmetry.
ToseetheimpactoftheSFPcontribution,wehaveperformedanumericalcalculation
ofA for p p p X,assumingthat theSFP functionisofthesameorderofmagnitude
N ↑
as the SGP fun→ction. Kouvaris et al. [6] have parametrized the SGP function Ga(x,x)
F
so that the SGP contribution reproduces the A data obtained from RHIC and FNAL
N
data. Their analysis shows that both data are reasonably well reproduced. In particular,
they found that the derivativeterm in (1) brings dominant contributioncompared to the
nonderivativecontribution.HereweadoptFit(I)of[6]andassumeGa(0,x)+Ga(0,x)=
F F
Ga(x,x) (a=u,d). Fig. 1 shows A for the pion at √S=200 GeV and P =1.5 GeV
F N hT
withand withoutSFP contribution.As is seen from thefigure that theSFP coentribution
brings large effect in the positivex region, while its effect is negligiblein the negative
F
x region.FromthisfigureitisclearthattheSFPcontributioncanaffectA significantly
F N
eventhoughit doesnot receiveenhancement by thederivativeunlikeSGPcontribution,
unlesstheSFP functionitselfis small.
To summarize, we have calculated the SFP contribution to the cross section for
p p hX associatedwiththetwist-3quark-gluoncorrelationfunctionsinthepolarized
↑
→
A
0.4 N
s = (200 GeV)2
P =1.5 GeV
0.2 T
0.0
+
p (SGP)
+
p (SGP+SFP)
-
-0.2 p (SGP)
-
p (SGP+SFP)
0
p (SGP)
-0.4 0
p (SGP+SFP)
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
x
F
FIGURE1. A for p p p X at√S=200GeVandP =1.5GeV.Solid,long-dashed,anddash-dot
N ↑ hT
lines are, respectively, A →for p +, p and p 0 obtained with only the SGP contribution. Dotted, short-
N −
dashed,anddash-double-dotlinesare,respectively,A forp +, p andp 0 obtainedwith bothSGP and
N −
SFPcontributions.
nucleon, and have shown that its effect is significant and should be included in the
analysis of A . For a more complete analysis, one needs to include the contribution
N
from the triple-gluontwist-3distributionfunction as well. Recent study[11] showsthat
A for the open charm production has a potential to determine the function. We hope
N
that the global analysis of all those data including all the QCD effects will clarify the
originofobservedSSA.
ACKNOWLEDGMENTS
TheworkofY.K.issupportedinpartbyUchidaEnergySciencePromotionFoundation.
REFERENCES
1. J.QiuandG.Sterman,Nucl.Phys.B378(1992)52.
2. J.QiuandG.Sterman,Phys.Rev.D59(1999)014004.
3. Y.KanazawaandY.Koike,Phys.Lett.B478(2000)121;Phys.Lett.B490(2000)99.
4. Y.Koike,AIPConf.Proc.675(2003)449[hep-ph/0210396];Nucl.Phys.A721(2003)364.
5. H.Eguchi,Y.KoikeandK.Tanaka,Nucl.Phys.B752(2006)1.
6. C.Kouvaris,J.W.Qiu,W.VogelsangandF.Yuan,Phys.Rev.D74,114013(2006).
7. H.Eguchi,Y.KoikeandK.Tanaka,Nucl.Phys.B763193(2007).
8. Y.KoikeandK.Tanaka,Phys.Lett.B646,232(2007)[Erratum-ibid.B668,458(2008)]
9. Y.KoikeandK.Tanaka,Phys.Rev.D76,011502(2007)
10. Y.KoikeandT.Tomita,inpreparation.
11. Z.B.Kang,J.W.Qiu,W.VogelsangandF.Yuan,Phys.Rev.D78,114013(2008)