Hyperon Polarization in Inclusive
Hadronic Production
y
Y. KanazawaandYuji Koike
Department of Physics, Niigata University, Ikarashi, Niigata 9502181, Japan
Abstract.
A QCD formula for the polarization in the large
p
T
hyperon pro
duction in the unpolarized nucleonnucleon collision at large
x
F
is derived. We
focus on the mechanism in which the chiralodd spinindependent twist3 quark
distribution
E
F
(
x; x
) becomes the source of the transversely polarized quarks
fragmenting into the polarized . A simple model estimate for that contribution
shows the possibility that it gives rise to a sizable polarization.
It is a well known experimental fact that the hyperons produced in the
unpolarized nucleonnucleon collisions are polarized transversely to the pro
duction plane [1,2]. In this letter we focus on the polarization of the hyperon
production with large transverse momentum in
pp
collision
N
(
P
)+
N
0
(
P
0
)
!
(
l;
~
S
?
)+
X:
(1)
Ongoing experiment at RHIC is expected to provide more data on the po
larization. The nonzero polarization in this process requires the presence
of particular quarkgluon correlation (higher twist e ect) and/or the e ect
of transverse momentum either in the unpolarized nucleon or the fragmenta
tion function for . According to the generalized QCD factorization theorem,
the polarized cross section for this process consists of two kinds of twist3
contributions:
(A)
E
a
(
x
1
;x
2
)
⊗
q
b
(
x
0
)
⊗
D
c
!
(
z
)
⊗
^
ab
!
c
;
(2)
(B)
q
a
(
x
)
⊗
q
b
(
x
0
)
⊗
D
(3)
c
!
(
z
1
;z
2
)
⊗
^
0
ab
!
c
:
(3)
Here the functions
E
a
(
x
1
;x
2
)and
D
(3)
c
!
(
z
1
;z
2
) are the twist3 quantities rep
resenting, respectively, the unpolarized distribution in the nucleon and the
fragmentation function for the transversely polarized hyperon, and
a
,
b
and
c
stand for the parton’s species. Other functions are twist2;
q
b
(
x
)the
y
)
Proceedings of the talk presented at SPIN2000, Osaka, Oct. 1621, 2000.
unpolarized distribution (quark or gluon) and
D
c
!
(
z
) the transversity frag
mentation function for . The symbol
⊗
denotes convolution. ^
ab
!
c
etc
rep
resents the partonic cross section for the process
a
+
b
!
c
+
anything
which
yields large transverse momentum of the parton
c
. Note that (A) contains
two chiralodd functions
E
a
and
D
c
!
, while (B) contains only chiraleven
functions.
In this report, we derive a QCD formula for the polarized cross section (1)
from the (A) term in the kinematic region
j
x
F
j!
1, using the valence quark
soft gluon approximation proposed by Qiu and Stermann [3]. Employing this
approximation, they reproduced the E704 data for the singletransverse spin
asymmetries in the pion production at
x
F
!
1 reasonably well. The fact
that the perturbative QCD description for the pion production is valid as
low as
l
T
1 GeV encouraged us to apply the method to the polarized
hyperon production (1) for which the data exist only in the relatively small
l
T
region. At large
x
F
>
0, which mainly probes large
x
and small
x
0
region, the
cross section is dominated by the particular terms in (A) which contain the
derivatives of the
valence
twist3 distribution
E
Fa
(
x;x
). The reason for this
observation is the relation
j
@
@x
E
Fa
(
x;x
)
j
E
Fa
(
x;x
) owing to the behavior
of
E
Fa
(
x;x
)
(1
x
)
(
>
0) at
x
!
1. We thus keep only the terms
with the derivative of
E
Fa
for the valence quark (
valence quarksoft gluon
approximation
).
The polarized cross section for (1) is a function of three independent vari
ables,
S
=(
P
+
P
0
)
2
’
2
P
P
0
,
x
F
=2
l
k
=
p
S
(= (
T
U
)
=S
), and
x
T
=2
l
T
=
p
S
.
T
=(
P
l
)
2
’
2
P
l
and
U
=(
P
0
l
)
2
’
2
P
0
l
are
given in terms of these three variables by
T
=
S
q
x
2
F
+
x
2
T
x
F
=
2and
U
=
S
q
x
2
F
+
x
2
T
+
x
F
=
2. In this convention, production of in the for
ward hemisphere in the direction of the incident nucleon (
N
(
P
)) corresponds
to
x
F
>
0. Since
1
<x
F
<
1, 0
<x
T
<
1and
q
x
2
F
+
x
2
T
<
1,
x
F
!
1
corresponds to the region with
U
S
and
T
0.
In the valence quarksoft gluon approximation, the cross section for the (A)
term reads,
E
l
d
3
A
(
S
?
)
dl
3
=
M
2
s
S
X
a;c
Z
1
z
min
dz
z
3
D
c
!
(
z
)
Z
1
x
min
dx
x
1
xS
+
U=z
Z
1
0
dx
0
x
0
x
0
+
xT=z
xS
+
U=z
!
"
lS
?
pn
1
^
u
"
x
@
@x
E
Fa
(
x;x
)
#
"
G
(
x
0
)
b
ag
!
c
+
X
b
q
b
(
x
0
)
b
ab
!
c
#
;
(4)
where
p
and
n
are the two lightlike vectors de ned from the momentum of the
unpolarized nucleon as
P
=
p
+
M
2
n=
2,
p
n
=1 and
"
lS
?
pn
=
"
l
S
?
p
n
sin
with
the azimuthal angle between the spin vector of the hyperon
and the production plane. The invariants in the parton level are de ned as
^
s
=(
p
a
+
p
b
)
2
’
(
xP
+
x
0
P
0
)
2
’
xx
0
S;
^
t
=(
p
a
p
c
)
2
’
(
xP
l=z
)
2
’
xT=z;
^
u
=(
p
b
p
c
)
2
’
(
x
0
P
0
l=z
)
2
’
x
0
U=z
. The lower limits for the integration
variables are
z
min
=
(
T
+
U
)
S
=
q
x
2
F
+
x
2
T
and
x
min
=
U=z
S
+
T=z
.
q
b
(
x
0
)isthe
unpolarized quark distribution, and
G
(
x
0
) is the unpolarized gluon distribu
tion.
b
ag
!
c
and
b
ab
!
c
are partonic cross sections for the quarkgluon and
quarkquark processes, respectively.
E
F
(
x;x
) is the soft gluon component of
the unpolarized twist3 distribution de ned as
E
Fa
(
x;x
)=
i
2
M
Z
d
2
e
i x
h
P
j
a
(0)
6
nγ
?
(
Z
d
2
gF
(
n
)
n
)
a
(
n
)
j
P
i
:
(5)
The summation for the flavor indices of
E
Fa
(
x;x
)istobe over
u
and
d
valence quarks, while that for the twist2 distributions is over
u
,
d
,
u
,
d
,
s
,
s
.
b
ab
!
c
and
b
ag
!
c
can be obtained from the 2
!
2 cut diagrams. The result
reads
^
qq
0
!
q
=
^
s
^
u
^
t
2
!"
2
9
+
1
9
1+
^
u
^
t
!#
;
^
q
q
0
!
q
=
^
s
^
u
^
t
2
!"
7
9
+
1
9
1+
^
u
^
t
!#
;
^
qq
!
q
=
^
s
^
t
!"
10
27
+
1
27
1+
^
u
^
t
!#
;
(6)
for
b
ab
!
c
,and
^
ag
!
c
=
9
8
^
s
^
u
^
t
2
!
+
9
8
^
u
^
t
!
+
1
8
+
"
1
