Charmed Hadron Production in Polarized pp
Collisions
Toshiyuki Morii and Kazumasa Ohkuma†
Division of Sciences for Natural Environment, Faculty of Human Development,
Kobe University, Nada, Kobe 657-8501, JAPAN
Electronic address:[email protected]
†
Department of Physics, Faculty of Engineering, Yokohama National University,
Hodogaya, Yokohama 240-8501, JAPAN
Electronic address:[email protected]
Abstract. To extract information about polarized gluons in the proton, production of charmed
hadrons, in particular, Λ c baryon in pp collisions was studied. We calculated the transverse momentum distribution and the pseudo-rapidity distribution of the spin correlation asymmetry A LL
between the initial proton and the produced Λ c . Those statistical sensitivities were also calculated
under the condition of RHIC experiment. We found that the pseudo-rapidity distribution of A LL
is promising for testing the model of polarized gluons in the proton and also the spin-dependent
fragmentation model of a charm quark decaying into Λ c baryon.
INTRODUCTION
The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory has
just started to explore the internal structure of proton. One of the important purposes of
those RHIC experiments is to study the behavior of polarized gluons in the proton. As
is well known, the proton spin is given by the sum of the spin carried by quarks ∆Σ and
gluons ∆G, and their orbital angular momenta L Z . In these years, a great deal of efforts
have been made for extracting those components from polarized structure functions
of nucleons[1]. Based on the next–to–leading order QCD analyses on the polarized
structure functions g1 x, the contribution of quarks to the proton spin is well known.
However, knowledge on polarized gluons in the proton is still poor. To understand the
origin of the nucleon spin, it is very important to know how gluons polarize in the
nucleon. So far, several interesting processes have been proposed for extracting ∆G. Here
we also propose a different process to obtain more detailed information on polarized
gluons, expecting the forthcoming RHIC experiment. The processes which we propose
here are the polarized charmed hadron production, i.e. pp Λ
p D X , in
c X and p
the polarized proton–unpolarized proton collision, 1 which will be observed at the RHIC
experiment. We study which observables are useful for extracting information about
polarized gluons in the proton and also discuss its sensitivity.
Though we have calculated even for D production, we focus only on the Λ c production in this report,
because the main point of the result remain unchanged.
1
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
333
ΛC BARYON PRODUCTION IN PROTON–PROTON COLLISION
In the process on which we focus here, the Λ
c baryon is expected to have some
advantageous properties for probing behavior of polarized gluons in the proton. Those
properties are as follows;
1. A charm quark which is one of constituents of the Λ c baryon is dominantly
produced via gluon-gluon fusion in pp reaction, because charm quarks are tiny
contents in the proton. Thus, the cross section of this process is directly proportional
to the gluon distribution in the proton.
2. Since the Λ
c baryon is composed of a charm quark and antisymmetrically combined light up and down quarks, the spin of Λ c baryon is expected to be almost
equal to the spin of its constituent charm quark.
3. Since a charm quark is heavy, it must be very rare for the charm quark to change
its spin arrangement during its fragmentation into a Λ c baryon. In other words, the
spin direction of the charm quark produced in the subprocess is expected to be kept
in the Λ
c baryon produced in the final state.
After all, the spin of the Λ
c is in strong correlation to the polarization of gluons in the
proton. Therefore, by observing the spin correlation between the polarized proton in the
initial state and the polarized Λ
c in the final state, we can get, rather clearly, information
on the polarized gluon in the proton.
SPIN CORRELATION ASYMMETRY AND ITS STATISTICAL
SENSITIVITY
To study the polarized gluon distribution in the proton, we introduced the spin correlation asymmetry of the target polarized-proton and produced Λ c baryon [2];
ALL
d σ d σ d σ d σ
d σ d σ d σ d σ
d∆σ dX
X pT or η d σ dX
(1)
where d σ , for example, denotes the spin-dependent differential cross section with
the positive helicity of the target proton and the negative helicity of the produced Λ c
baryon. pT and η , which are represented by X in Eq.(1), are transverse momentum and
pseudo-rapidity of produced Λ
c , respectively.
According to the quark-parton model, d∆σ dX can be expressed as
d∆σ
dX
Y max 1
min
xa
Y min
1
min
x
b
Gp
A
2
2
ga xa Q ∆Gp g xb Q ∆DcΛ z
B
d∆σ̂
Jdxa dxb dY d tˆ
334
X Y
η
c
b
or pT X
Y (2)
with
J
where G p
A
2
ga xa Q ,
2sβ p2T coshη
zŝ m2c p2T cosh2 η
∆Gp
B
2
gb xb Q β
and ∆D
1
z
cΛ
c
4m2p
s
represent the unpolarized
gluon distribution function, the polarized gluon distribution function and the spindependent fragmentation function of the outgoing charm quark decaying into a polarized
Λ
c , respectively. d∆σ̂ d tˆ is the spin-dependent differential cross section of the subprocess and J is the Jacobian which transforms the variables z and tˆ into pT and η . In the
expression of Eq.(2), pT and η are described as X or Y .
Statistical sensitivities of A LL for the pT and η distribution are estimated by using the
following formula;
1
1
δ ALL (3)
P b εLT σ
Λc
To numerically estimate the value of δ ALL , here we use following parameters: operating
time; T 100-day, the
beam polarization; P 70%, a luminosity; L 8 10 31 2 32
2
1
10 ) cm sec for s 200 500 GeV, the trigger efficiency; ε 10% for detecting
produced Λ
BrΛ
c events and a branching ratio; bΛ
c pK π 5% [3]. The
c
purely charged decay mode is needed to measure the polarization of produced Λ
c. σ
denotes the unpolarized cross section integrated over suitable p T or η region.
NUMERICAL CALCULATIONS
To carry out the numerical calculation of ALL , we used, as input parameters, mc 120
GeV, m p 0938 GeV and mΛ 228 GeV[3]. We limited the integration region
c
of η and pT of produced Λ
as 13 η 13 and 3 GeV pT 15(40) GeV,
c
respectively, for s 200500 GeV. The range of η and the lower limit of p T were
selected in order to get rid of the contribution from the diffractive Λ c production. As
for the upper limit of pT , we took it as described above, for simplicity, though the
kinematical maximum of p T of produced Λ
c is slightly larger than 15 GeV and 40
Gev for s 200 GeV and 500 GeV, respectively. In addition, we took the AAC[4]
and GRSV01 [5] parameterization models for the polarized gluon distribution function
and the GRV98 [6] model for the unpolarized one. Though both of AAC and GRSV01
models excellently reproduce the experimental data on the polarized structure function
of nucleons g1 x, the polarized gluon distributions for those models are quite different.
In other words, the data on polarized structure function of nucleons g 1 x alone are
not enough to distinguish the model of gluon distributions. Since the process is semiinclusive, the fragmentation function of a charm quark to Λ c is necessary to carry out
numerical calculations. For the unpolarized fragmentation function, we used Peterson
fragmentation function, DcΛ z [3, 7]. However, since we have no data, at present,
c
about polarized fragmentation functions for the polarized Λ c production, we took the
335
0
AAC ; CΛ=1
AAC ; CΛ=z
GRSV01 ; CΛ=1
GRSV01 ; CΛ=z
0.2
AAC
GRSV01
0.1
AAC ; CΛ=1
AAC ; CΛ=z
GRSV01 ; CΛ=1
GRSV01 ; CΛ=z
ALL
-0.01
-0.015
ALL
-0.005
0
GRSV01
-0.02
-0.025
2
Q =(PT)
-0.03
3
4
6
8
pT
10
12
3GeV ≤pT≤ 15GeV
-0.1
-1.3≤η≤1.3
AAC
2
2
Q =(PT)
14 15
[GeV]
-0.2
-1.3
-1.0
2
-0.5
FIGURE 1. ALL as a function of p T (left panel) and η (right panel) at
0
η
0.5
1.0
s 200 GeV
following ansatz for the polarized fragmentation function ∆D
x,
cΛ
c
∆D
cΛ
c
z CcΛ DcΛ c
c
(4)
where CcΛ is a scale-independent spin transfer coefficient. In this analysis, we studc
ied two cases: (A) CcΛ 1 (non-relativistic quark model) and (B) CcΛ z (Jet
c
c
fragmentation model [8]). As we discussed before, if the spin of Λ c is same as the spin
of charm quark produced in subprocess, the model (A) might be a reasonable scenario.
Numerical results of ALL are shown in Fig. 1 and Fig. 2. In those figures, statistical
sensitivities, δ ALL , are also attached to the solid line of ALL which is calculated for the
case of the GRSV01 parametrization model of polarized gluon and the non-relativistic
fragmentation model. 2 From these results, we see that the
η distributions of A LL are
more effective observables than the pT distributions at s 200 GeV and 500 GeV.
As shown in the right panel of Fig. 2 given at s 500 GeV, we could distinguish the
parametrization models of polarized gluon as well as the models of the spin-dependent
fragmentation function though the magnitude of A LL is rather small. At s 200 GeV,
the magnitude of ALL for η distribution becomeslarger, though statistical sensitivities
are not so small. If the integrated luminosity at s 200 GeV could be large and the
detection efficiency, ε , is improved, this observable could be promising to distinguish
not only the models of ∆Gx but also the models of ∆Dz. For the p T distribution of
ALL , δ ALL become rapidly larger with increasing pT and we cannot say anything from
those region. However, if we confine the kinematical region in rather small p T range like
pT 3 5(10) GeV at s 200500 GeV, it might be still effective.
Note that as shown from Eq.(4), δ A LL does not depend on both of the model of polarized gluons and the
model of fragmentation functions.
2
336
1.3
AAC ; CΛ=1
AAC ; CΛ=z
GRSV01 ; CΛ=1
GRSV01 ; CΛ=z
0
3GeV≤ pT≤40GeV
0.02
-0.01 GRSV01
2
AAC ; CΛ=1
AAC ; CΛ=z
GRSV01 ; CΛ=1
GRSV01 ; CΛ=z
GRSV01
ALL
ALL
-0.02
-0.03
-0.02
-1.3≤η≤1.3
AAC
-0.04
-0.05
3 5
2
Q =(PT)
0
2
-0.04
2
Q =(PT)
10
15
20
pT
25
30
35
40
[GeV]
-0.06
-1.3
AAC
-1.0
FIGURE 2. The same as in Fig. 1, but for
-0.5
s 500 GeV
0
η
0.5
1.0
CONCLUDING REMARK
To extract information on the polarized gluon distribution in the proton, the charmed
hadron production processes at RHIC experiments have been proposed. (Actually, only
Λ
c process was discussed in this report.) The spin correlation asymmetry A LL between
the initial proton and the produced Λ
c was calculated for pT and η distributions with
statistical sensitivities which were estimated using RHIC parameters. We found that A LL
is rather sensitive to the model of ∆Gx and ∆Dz. The η distribution of A LL could be
promising for distinguishing the parametrization model of polarized gluons as well as
the model of spin-dependent fragmentation of a charm quark into Λ c.
ACKNOWLEDGMENTS
One of the authors, (K.O), would like to thank the organizers and fellowship committee
of SPIN2002 for giving a chance to present this contribution at the symposium.
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