Hyperon Polarization from Unpolarized
pp and ep Collisions
Yuji Koike
Department of Physics, Niigata University, Ikarashi, Niigata 950–2181, Japan
Abstract. Cross section formulas for the Λ polarization in pp
derived and its characteristic features are discussed.
Λ
X
T
and ep
Λ
X
T
are
In this report we discuss the polarization of Λ hyperon produced in unpolarized
pp and ep collisions relevant for the ongoing RHIC-SPIN, HERMES and COMPASS
experiments. According to the QCD factorization theorem, the polarized cross section
for pp Λ X consists of two twist-3 contributions:
A
Ea x1 x2 B
qa x
q
q
δ q z σ̂
G z z σ̂
b x b x c abc abc c 1 2
c z1 z2 are the twist-3 quantities representing,
where the functions Ea x1 x2 and G
respectively, the unpolarized distribution in the nucleon and the polarized fragmentation
function for Λ . δ qc x is the transversity fragmentation function for Λ . a, b and c
stand for the parton’s species, sum over which is implied. E a and δ qc are chiral-odd.
Corresponding to the above (A) and (B), the polarized cross section for ep Λ X (final
electron is not detected) receives two twist-3 contributions:
δ q z σ̂
z z σ̂
x G
A Ea x1 x2 B qa a a 1 2
eaa eaa The (A) contribution for pp Λ X has been analyzed in [1], where it was shown that
(A) gives rise to growing PΛ at large xF as observed experimentally. Here we extend the
study to the (B) term (see also [2]) at RHIC energy and also for the ep collision.
The unpolarized twist-3 distribution E FD x1 x2 is defined in [1]. Likewise the twist-3
fragmentation function for a polarized Λ (with momentum ) is defined as the lightcone
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
MN
M α 5
α wS / i j ε
GF z1 z2 i N γ5/ i j S
GF z1 z2 2z2
2z2
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
574
(1)
Note that we use the nucleon mass MN to normalize the twist-3 fragmentation function for Λ. There is another twist-3 fragmentation functions which are obtained from
(1) by shifting the gluon-field strength from the left to the right of the cut. The deFR z z and G5FR z z are connected to GF z z by the relation
fined functions G
1 2
1 2
1 2
F z z GFR z z and G5F z z G5FR z z , which follows from hermiticG
1 2
2 1
1 2
2 1
ity and time reversal invariance. Unlike the twist-3 distributions, the twist-3 fragmentation function does not have definite symmetry property. Another class of twist-3 frag5 z1 z2 is also defined from (1) by replacing gF αβ µ ww by
mentation functions G
β
D
Dα µ w ∂ α igAµ w. Note, however, this is not independent from the above (1).
Following the method of [3] we present the analysis of the (C) term. The detailed
F z z appears as soft-gluon-pole contribution (z 1 z2 z), while
analysis shows G
D z z appears as a soft fermion pole (z 0 or z 2). Physically, the latter is
G
1 2
1
2
expected to be suppressed, and we include only the former contribution. This observation also applies to EFD x1 x2 relevant for the (A) term. In the large xF region,
the main contribution comes from large-x and large-z (and small x ) region. Since
F behaves as EF x x 1 xβ and GF z z 1 zβ with β , β 0,
EF and G
F z z
F z z at large x and z. In particud dxEF x x
EF x x , d dzG
G
F dominates in this region. We thus keep only the
lar, the valence component of EF and G
valence quark contribution for the derivative of these soft-gluon pole function (“valencequark soft-gluon approximation”) for the pp collision. For the ep case, we include all
the soft-gluon pole contribution, since the calculation is relatively simple compared to
the pp case.
In general PΛ is a function of S P P 2 2P P , T P 2
2P and
2P where P and P are the momenta of the two nucleons, and U P 2
is the momentum of Λ. In the following we use S, x F S T S U and xT
independent variables. The polarized cross section for the (B) term reads
EΛ
d 3 ∆σ S d 3
δ
2π MN αs2 α wS
∑
ε
S
a
x 2
xT z
xS U z
∑ q xq x a
b
bc
∑ q xq x a
b
bc
a
q xGx z21
1
dz
2
zmin z
1
1
dx
x
xS
U z
xmin
1
0
d x
x
2
T
S
z21
∂ a
G z z
∂ z1 F 1
2p
T
z1 z
α
I
σab
c 2p
U
α
II
σab
c
d a
xpα x pα I
II
z
GF z z
σabc σab
c
dz
xT x U
2
∂ a
G z z
∂ z1 F 1
z1 z
575
2p
T
α
I
σag
a 2p
U
α
II
σag
a
as
a
q xGx d a
xpα x pα I
II
z
GF z z
σaga σag
a
dz
xT x U
2
(2)
where the lower limits for the integration variables are z min T U S x2F x2T
and xmin U zS T z. The partonic hard cross sections are written in terms of
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. They read
I
σab
c 361 ŝ
II
σab
c 7 ŝ2 û2
δac
36 tˆ2
σaIb̄c I
σqg
q II
σqg
q 2 û2
tˆ2
δac 361 ŝ û tˆ δ
2 2
2
ac tˆ2
8
9
1
16
ŝtˆû
2
bc 2
361 ŝ tˆ û δ
1 1 ŝû 2 2
7 ŝ2 tˆ2
1 ŝ2
δ
δ δac bc
36 û2
54 tˆû ab
1 ŝ2
δ δac 54 tˆû ab
7 û2 tˆ2
δab 36 ŝ2
1
288
û
32ŝ
û
ŝ
4ŝû
σaIIb̄c 1 ŝ2 û2
1 û2 tˆ2
δ
δab ac
18 tˆ2
18 ŝ2
ŝ
û
16tˆ 16tˆ
ŝ
û
9û
16tˆ
(3)
I becomes more important at large xF because of
Among these partonic cross sections, σ
the 1T factor in (2).
aF z z Ka qa z
To estimate the above contribution, we introduce a model ansatz as G
with twist-2 unpolarized fragmentation function qa z, noting that the Dirac strucaF z z and qa z is the same [3]. Ka ’s are taken to be Ku K 007
ture of G
d
which are the same values used in the relation GF x x Ka qa x to reproduce AN
F z1 z2 does not have definite
in p p
π X observed at E704 [4]. As noted before, G
symmetry
property unlike the twist-3 distribution E F x1 x2 . Nevertheless we assume
F z1 z
∂ ∂ z1 E
z1 z
F z z.
12d dzE
The result for the Λ polarization PΛpp
at S 62 GeV is shown in Fig. 1(a) together with the R608 data.There (A) (chiralodd) contribution studied in [1] is also shown for comparison. (For the adopted distribution and fragmentation functions, see [1].) One sees that the tendency of PΛpp from the
(B)(chiral-even) contribution is quite similar to the R608 data. Rising behavior of PΛpp
at large xF comes from (i) the large partonic cross sections in (3) ( 1tˆ2 term) and (ii)
the derivative of the soft-gluon pole functions. With these parameters Ka , PΛpp at RHIC
energy ( S 200 GeV) is shown in Fig. 1(b) at lT 15 GeV.Fig. 2(a) shows the lT
dependence of PΛpp of the (B) term,indicating large T dependence at 1 T 3 GeV.
Experimentally, PΛpp grows up as T increases up to T 1 GeV and stays constant at
1 T 3 GeV. So the PΛpp observed at R608 can not be wholly ascribed to the twist-3
effect studied here which is designed to describe large T polarization.
576
0.2
0.2
Odd (sc. 3)
Odd (sc.3)
0.0
PΛ
0.0
PΛ
Odd (sc. 2)
Odd (sc.2)
Even
-0.2
-0.2
Even
2
R608 data
-0.4
2
S=62 GeV
lT=1.1 GeV
2
-0.6
0.0
-0.6
0.2
S=200 GeV
lT= 1.5 GeV
-0.4
0.4
0.6
0.8
xF
FIGURE 1. (a) PΛpp at
0.2
2
0.4
0.6
0.8
xF
S
62
GeV. (b) PΛpp at
S
200
GeV.
We next discuss the polarization PΛep in pe
Λ X where the final electron is not
observed. In our Oαs0 calculation, the exchanged photon remains highly virtual as far
as the observed Λ has a large transverse 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.
0.00
1.0
0.8
-0.05
xF=0.5
0.6
ep
0.4
PΛ
PΛ
-0.15
Chiral-even
-0.20
2
2
chiral-odd(sc.2)
xF=0.7
-0.10
S= 400 GeV
lT=2.0 GeV
chiral-even
0.2
0.0
-0.2
2
S=62 GeV
chiral-odd(sc.3)
-0.4
-0.25
1
2
3
4
5
6
7
8
-0.8
lT (GeV)
FIGURE 2. (a) T dependence of PΛpp . (b)PΛep at
-0.4
S
0.0
xF
20
0.4
0.8
GeV.
Using the twist-3 distribution and fragmentation functions used to describe PΛpp , we
show in Fig. 2(b) the obtainedPΛep corresponding to (A’)(chiral-odd) and (B’)(chiraleven) contributions. Remarkable feature of Fig. 2(b) is that in both chiral-even and
chiral-odd contributions (i) the sign of PΛep is opposite to the sign of PΛpp and (ii) the
magnitude of PΛep is much larger than that of PΛpp , in particular, at large xF , and it even
overshoots one. (In our convention, xF 0 corresponds to the production of Λ in the
forward hemisphere of the initial proton in the ep case.) The origin of these features can
577
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 PΛ 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 simple model ansatz of E Fa x x δ qa x (in (A) term) and
aF z z qa z should be modified at x 1 and z 1, respectively. The derivative of
G
these functions, which is important for the growing PΛpp at large xF , eventually leads to
divergence of PΛ at xF
1 as 11 xF .
As a possible remedy for this pathology we tried the following: As an example for
aF z z qa z
the (B) (chiral-even) contribution we have a model G
zβ
z1 1
where β 183 in the fragmentation function we adopted. Tentatively we shifted β
as β
β z β z8 , which suppresses the divergence of PΛ at xF
1 but still keeps
rising behavior of PΛ at large xF . This avoids overshooting of one in PΛep but reduces PΛpp
seriously. The result obtained by this modification is shown in Figs. 3(a) and (b)
To summarize we have studied the Λ polarization in pp and ep collisions in the
framework of collinear factorization. Our approach includes all effects for the large T
production. One needs to be causious in interpreting the available pp data at relatively
low T in terms of the derived formula. Determination of the participating twist-3
functions requires global analysis of future pp and ep data.
0.8
lT=2 GeV
PΛ
lT=1.1GeV
2
S= 400 GeV
lT=2.0 GeV
0.6
ep
0.0
PΛ
1.0
0.2
-0.2
2
0.4
0.2
0.0
2
S=62 GeV
-0.4
-0.2
R608 (lT=1.1GeV)
-0.4
-0.6
0.2
0.4
0.6
0.8
-0.8
-0.4
xF
0.0
xF
0.4
0.8
F . (b) Pep with modified GF .
FIGURE 3. (a) PΛpp with modified G
Λ
ACKNOWLEDGMENTS
This work is supported in part by the Grant-in-Aid for Scientific Research of Monbusho.
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578