πX
Single Transverse-Spin Asymmetry in pp
and ep π X
Yuji Koike
Department of Physics, Niigata University, Ikarashi, Niigata 950–2181, Japan
Abstract. Cross section formulas for the single spin asymmetry in p p
π T X are derived and its characteristic features are discussed.
π T X
and ep
In this report we discuss the single transverse-spin asymmetry for the pion production with large transverse momentum A N in pp
π X and p e
π X relevant
for RHIC-SPIN, HERMES and COMPASS experiments. According to the QCD factorization theorem, the polarized cross section for pp π X consists of three twist-3
contributions:
D z σ̂
x E x x D z σ̂
x q x E z z σ̂
q
b x A
Ga x1 x2 B
δ qa b 1
C δ qa b abc c 2
abc
2
abc c 1
c where the functions Ga x1 x2 , Eb x1 x2 and Ec z1 z2 are the twist-3 quantities representing, respectively, the transversely polarized distribution, the unpolarized distribution,
and the fragmentation function for the pion. δ q a x is the transversity distribution in p .
a, b and c stand for the parton’s species, sum over which is implied. δ q a , Eb and Ec
are chiral-odd. Corresponding to the above (A) and (C), the polarized cross section for
ep π X (final electron is not detected) receives two twist-3 contributions:
D z σ̂ C
δ q x E z z σ̂
The (A) and (B) contributions for pp π X have been analyzed in [1] and [2], respecA Ga x1 x2 a a a 1
2
eaa
eaa
tively, and it has been shown that (A) gives rise to large A N at large xF as observed in
E704, and (B) is negligible in all kinematic region. Here we extend the study to the (C)
term (see also [3]) at RHIC energy and also the asymmetry in ep collision.
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
449
The transversely polarized twist-3 distributions G F x1 x2 relevant to the (A) term is
given in [2]. Likewise the twist-3 fragmentation function for the pion (with momentum
) is defined as thelightcone correlation function as (w2 0, w 1)
d λ d µ i zλ iµ z1 z1 1
2
1 0 ψ 0 π X
e 1e
π X gF αβ µ wwβ ψ̄ j λ w 0
i
Nc ∑
2
π
2
π
X
ναβ λ
MN γ / γ
ε
wβ λ EF z1 z2 2z2 5 ν i j
(1)
Note that we use the nucleon mass MN to normalize the twist-3 pion fragmentation
function. There is another twist-3 fragmentation function which is obtained from (1) by
shifting the gluon-field strength from the left to the right of the cut. The defined function
EFR z1 z2 is connected to EF z1 z2 by the relation EF z1 z2 EFR z2 z1, which
follows from hermiticity and time reversal invariance. Unlike the twist-3 distributions,
the twist-3 fragmentation function does not have definite symmetry property.
Following [1]-[3], we analyze the asymmetries focussing on the soft-gluon pole contributions with GF x x and EF z z. In the large xF region, i.e. production of pion in
the forward direction of the polarized nucleon, the main contribution comes from the
large-x and large-z region of distribution and fragmentation functions, respectively. Since
GF and EF behaves as GF x x 1 xβ and EF z z 1 zβ with β , β 0,
d dxGF x x
GF x x , d dzEF z z
EF z z at large x and z. In par
ticular, the valence component of GF and EF dominate in this region. We thus keep
only the valence quark contribution for the derivative of these soft-gluon pole functions
(“valence-quark soft-gluon approximation”) for the pp collision. For the ep case, we
include all the soft-gluon pole contribution.
In general AN is a function of S P P 2 2P P , T P 2
2P and
2P where P, P and are the momenta of p , unpolarized p (or e),
U P 2
2
and the pion respectively. In the following we use S, x F S T SU and xT
independent variables. The polarized cross section for the (C) term reads
d 3 ∆σ S Eπ
d 3
δ
x 2π MN αs2 α wS
ε
S
xT z
xS U z
∑ δ q xq x a
b
∑ δ q xq x a
b
b
a
1
dz
2
zmin z
1
1
dx
x
xS
U z
xmin
1
0
d x
x
b
∑
2
T
S
z21
∂ a
E z z
∂ z1 F 1
z1 z
2p
T
α
I
σab
a 2p
α
U
II
σab
a
d a
xpα x pα I
II
z
EF z z
σaba σab
a
dz
xT xU
2
450
as
δ q xGx a
z21
δ q xGx a
∂ a
E z z
∂ z1 F 1
2p
α
T
z1 z
I
σag
a U
α
II
σag
a
d a
xpα x pα I
II
z
EF z z
σaga σag
a
dz
xT x U
2
where the lower limits for the integration variables are z min 2p
T
(2)
U S x2F x2T
and xmin U zS T z. Using the invariants in the parton level, ŝ xp x p 2 xx S, tˆ xp z2 xT z and û x p z2 xU z, the partonic hard cross
sections read
54ŝ tˆδ
I
σqq
q ŝû
18tˆ2
σq̄Iq̄ q̄ 18ŝûtˆ
II
σq̄q
q̄ ŝû
9tˆ2
I
σqg
q ŝû ŝ û
5
8tˆ2
16tˆ
72
2
σqIq̄ q qq ŝ
δ 54tˆ qq
σq̄IIq̄ q̄ T ), σ
At large xF ( U
I
ŝû
18tˆ2
II
σqq
q 18ŝûtˆ 2
σqIIq̄ q qq 2
9ŝtˆû 2
7ŝû
ŝ
δ 18tˆ2 54tˆ qq
II
σqg
q 169ŝtˆû 169ûtˆ 161 (3)
2
becomes more important because of 1T factor in (2).
0.4
Chiral-odd
Chiral-even
0.2
0.2
0.0
AN
AN
187ŝtˆû 54ŝ tˆδ
0.4
π
-0.2
π
π
-0.4
0.0
I
σq̄q
q̄ 0.2
0.0
+
−
π
-0.2
+
−
π
0
0
0.4
xF
0.6
FIGURE 1. ANpp at
-0.4
0.0
0.8
S
20
π
0.2
GeV and T
0.4
xF
15
0.6
0.8
GeV.
To estimate the above contribution, we introduce a simple model ansatz as EFa z z Ka Da z with a flavor dependent factor Ka . Ka ’s are determined to be Ku 011
and Kd 019 so that (2) approximately gives rise to A Npp observed in E704 data
at S 20 GeV and T 15 GeV. As noted before, EF z1 z2 does not have definite symmetry
property unlike the twist-3 distribution G F x1 x2 . Nevertheless we as
sume
F z1 z
∂ ∂ z1 E
z1 z
F z z.
12d dzE
We refer the readers to [2] for the
adopted distribution and fragmentation functions. The result for A Npp from the (C) (chiralodd) term is shown in Fig. 1 in comparison with the (A) (chiral-even) contribution. (See
451
[2] for the detail.) Both effects give rise to ANpp similar to the E704 data. The origin of
the growing AN at large xF is (i) large partonic cross sections in (3) ( 1tˆ2 term) and
(ii) the derivative of the soft-gluon pole functions. With the parameters K a fixed, ANpp at
RHIC energy ( S 200 GeV) is shown in Fig.2 at l T 15 GeV. Both (A) and (C) contributions give slightly smaller A Npp than the STAR data reported at this conference [4].
Fig.3 shows the ANpp as a function of T at S 200 GeV and xF 06, indicating quite
large lT dependence in both (A) and (C) contributions at 1 lT 4 GeV region.
0.4
0.4
Chiral-even
2
S=200 GeV
lT=1.5 GeV
Chiral-odd
2
2
0.2
0.0
AN
AN
0.2
0.0
+
-0.2
-0.4
0.0
π
π
π
0.2
2
S=200 GeV
lT=1.5 GeV
+
−
π
-0.2
π
0
0.4
xF
0.6
0.8
FIGURE 2. ANpp at
S
π
-0.4
0.0
200
p e
−
0
0.2
0.4
xF
GeV and T
15
0.6
0.8
GeV.
We next discuss the asymmetry
for
π X where the final electron is not
0
observed. In our Oαs calculation, the exchanged photon remains highly virtual as
far as the observed π has a large trasverse momentum T with respect to the ep axis.
Therefore experimentally one needs to integrates only over those virtual photon events
to compare with our formula.
Aep
N
0.14
0
π
2
2
S=200 GeV
xF=0.6
0.12
AN
0.10
0.08
Chiral-odd
0.06
0.04
Chiral-even
0.02
0.00
1
2
3
4
5
6
7
8
lT (GeV)
FIGURE 3. T dependence of A Npp at
S
200 GeV
and x F
06.
Using the twist-3 distribution and fragmentation functions used to describe pp data,
corresponding to (A’)(chiral-even) and (C’)(chiral-odd) contribuwe show in Fig. 4 Aep
N
tions. Remarkable feature of Fig. 4 is that in both chiral-even and chiral-odd contributions (i) the sign of Aep
is opposite to the sign of A Npp and (ii) the magnitude of Aep
is
N
N
much larger than that of ANpp , in particular, at large xF , and it even overshoots one. (In
our convention, xF 0 corresponds to the production of π in the forward hemisphere of
452
the initial polarized proton both in p p and p e case.) The origin of these features can be
traced back to the color factor in the dominant diagrams for the twist-3 polarized cross
sections in ep and pp collisions. Of course, the asymmetry can not exceeds one, and
thus our model estimate needs to be modified. First, the applied kinematic range of our
formula should be reconsidered: Application of the twist-3 cross section at such small T
may not be justified. Second, our model ansatz of G aF x x qa x and EFa z z Da z
is not appropriate at x
1 and z
1, respectively. The derivative of these functions,
which is important for the growing A Npp at large xF , eventually leads to divergence of
1 as 11 xF .
ANpp at xF
1.0
1.0
Chiral-even
Chiral-odd
2
0.5
ep
S=400 GeV
lT=1.5 GeV
0.0
AN
AN
ep
0.5
+
S=400 GeV
lT=1.5 GeV
0.0
π
π
-0.5
π
π
-1.0
-0.8
−
+
−
π
-0.5
0
0
-0.4
2
π
0.0
xF
0.4
0.8
-1.0
-0.8
-0.4
FIGURE 4. Aep
at
N
S
0.0
xF
0.4
0.8
20 GeV.
As a possible remedy to this pathology we tried the following. For the (A) (chiraleven) contribution we have a model GaF x x qa x x1 1 xβa where βu 3027
and βd 3774 in the GRV distribution we adopted. Tentatively we shifted β ud as
βa βa x βa x3 , which suppresses the divergence of AN at xF
1 but still causes
rising behavior of AN at large xF . This avoids overshooting of one in A ep
but reduces
N
pp
pp
AN , typically by factor 2. So the twist-3 contribution to A N shown in Fig. 1 and 2 is
further reduced, making the deviation from E704 and STAR data bigger.
To summarize we have studied the AN for pion production in pp and ep collisions.
Although our approach provides a systematic framework for the large T production,
applicability of the formula to the currently available low T data still needs to be tested,
in particular, comparison of T dependence with data is needed.
ACKNOWLEDGMENTS
I would like to thank Daniel Boer, Jianwei Qiu and George Sterman for useful discussions. This work is supported in part by the Grant-in-Aid for Scientific Research of
Monbusho.
REFERENCES
1.
2.
3.
4.
J. Qiu and G. Sterman, Phys. Rev. D59 (1999) 014004.
Y. Kanazawa and Y. Koike, Phys. Lett. B478 (2000) 121; B490 (2000) 99.
Y. Koike, hep-ph/0106260 (Proceedings of DIS2001, Bologna, Italy, April, 2001.)
G. Rakness, these proceedings.
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