Polarized Hadroproduction of Open Heavy
Quarks in NLO QCD at JHF and RHIC
Ingo Bojak
CSSM, The University of Adelaide, SA 5005, Australia
Abstract. We present the complete next-to-leading order QCD corrections to the polarized
hadroproduction of heavy flavors. This reaction can be studied experimentally in polarized pp
collisions at the JHF and at the BNL RHIC in order to constrain the polarized gluon density. It
is demonstrated that the dependence on the unphysical renormalization and factorization scales is
strongly reduced beyond the leading order. We also discuss how the high luminosity at the JHF
can be used to control remaining theoretical uncertainties. An effective method for bridging the gap
between theoretical predictions for heavy quarks and experimental measurements of heavy meson
decay products is introduced briefly.
INTRODUCTION
The gluon helicity density ∆g remains weakly constrained [1, 2]. Current data are compatible with ∆g 0 at a low input scale µ 2f 0 04 GeV2 , but even full saturation cannot be excluded [1]. Hence the gluonic contribution to the nucleon spin is unknown:
08 º ∆gn1 5 GeV2 º 17. Polarized DIS data has been the only source of information so far. But the severely restricted kinematical range, in which approximate “scaling”
holds, thwarts attempts to pin down the gluon. Furthermore, the helicity sum rule is useless for constraining ∆g without independent angular momentum measurements. Data
from exclusive processes may ameliorate the polarized parton fits.
Two experiments will be or are collecting data for the hadroproduction of open heavy
quarks: the JHF and the BNL RHIC [3]. We provide here the first corresponding complete calculation in NLO QCD. Note that our calculation is also necessary for obtaining
the resolved photon contributions in NLO QCD for our older NLO photoproduction
analysis [4, 5].
RESULTS FOR THE JHF
In Fig. 1 left we show the ratio of ∆σi j ∆σtot at JHF energies of S 10 GeV, with
the subprocesses i j gg qq̄ gq gq̄ and ∆σtot ∑ ∆σi j . The ratio for the gluongluon fusion is shown in thick lines and the ratio for the quark-antiquark annihilation is
shown by thin lines. The gluon-(anti)quark subprocess contributes little and is omitted.
Note that given the total asymmetry A ∆σ tot σtot , if we require A ∑ Ai j of the subprocesses, then the shown ratio corresponds to the subprocess asymmetry contribution
Ai j ∆σi j ∆σtot .
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338
2
Log x1
-1
-0.75
-0.5
-0.25
GRSV’00 val.
GRSV’00 std.
DS iunpol. GRV’98
1.5
JHF: charm, NLO, Eb=50 GeV
0
0.008
∆σij / ∆σtot
0.006
thick: ij=gg, thin: ij=qq-
0.004
1
0.5
0.002
0
-0.5
-1
0
0.2
0.4
min
0.6
0.8 pT 1
1.2
1.4
1.6
-1
1.8
-0.75
-0.5
-0.25
0
0
Log x2
FIGURE 1. Left: Subprocess importance depending on p T for different helicity densities at the
JHF. Right: Gluon-gluon contribution to ∆σ depending on the x 1 and x2 of the gluons. See text for details.
pmin
T
Acpp(f,c)/Acpp(2.5,1.4) − 1
%
180
120
60
0
-60
-120
-180
-240
0
0
%
180
120
60
0
-60
-120
-180
-240
0
0
− LO
NLO
+30%
1
1
2
f 3
4
1.2
1.3
1.4
c
1.5
1.6
2
f 3
Eb=50 GeV
2
f=µ2
f /mc ,
4
1.2
1.3
1.4
c
1.5
1.6
mc=c GeV, µr=µf
FIGURE 2. Deviation in % of the asymmetry from the marked value as µ 2f µr2 and mc are varied. The
GRSV’00 std. helicity densities [1] are used. LO is multiplied by 1 and 30% contour steps are shown.
Several helicity density sets have been used [1, 2], and σi j σtot with the unpolarized
GRV’98 distributions [6] is shown for comparison. The curves depend on a cut in
transverse momentum pT pmin
T . The polarized GRSV’00 std. and the unpolarized
GRV’98 curves show optimal behavior: the gluon-gluon subprocess dominates and there
is almost no dependence on the pT -cut. For the smaller GRSV’00 val. set ∆g the quark-
339
antiquark subprocess contributes significantly. Also there is now a strong dependence
on the cut (although the “pole” is due to ∆σ tot 0). For the very small DS i- ∆g, quarkantiquark annihilation dominates. Hence at the small JHF energy, one cannot simply
assume gluon-gluon fusion dominance.
2
2
FIGURE 3. The NLO cross section at JHF depending on a cut p T pmin
T as µ f , µr and mc are varied.
64 pb 1 is shown.
The GRV’98 distributions [6] are used. The statistical error after one day using
Given a large enough ∆g, which regions of x does gluon-gluon fusion probe? From
kinematics we have x1 x2 4m2 S for the momentum fractions of the gluons. In
Fig. 1 right we show
x1 x2
∆gx1 µ 2f ∆gx2 µ 2f ∆σ̂gg x1 x2 (1)
αs2 m2
with µ 2f 25 14 GeV2 , ∆g of the GRSV’00 std. set, and the partonic cross section
∆σ̂gg . The x1 x2 is multiplied to give the appropriate volume impression with logarithmic
axes and a general αs2 m2 dependence has been divided out. Apart from that the integrals
over x1 and x2 of (1) gives the hadronic ∆σgg . The main contribution comes from the
kinematic edge x1 x2 4m2 S. Furthermore it is peaked at x1 x2 . Hence x̂ 4m2 S
is mainly
probed. At the JHF for charm x̂c 03 and at the BNL RHIC x̂c 001 0006
for S 200 500 GeV. For photoproduction with x 2 1 one can similarly show
x̂ 4m2 S. Hence the COMPASS experiment [7], at the same center-of-mass energy
S 10 GeV as the JHF, further complements the probed range with x̂c 008.
In Fig. 2 we show the deviation in % from a central prediction of the asymmetry
(µr2 µ 2f 25 m2c and mc 14 GeV) as renormalization and factorization scales,
which are set equal, and the charm mass are varied. We see a massive reduction in the
the theoretical uncertainty in NLO as compared to LO, basically only the dependence on
mc of 30% is left. However, Fig. 3 shows that this amazing improvement is partly due
to strong cancellations in ∆σ σ . The underlying uncertainty of σ is massive. However,
340
c
App
’00: GRSV val.
GRSV std.
AAC 1
AAC 2
DS iDS iii+
0.2
0.15
0.1
0.05
0
-0.05
’96: GRSV val.
GRSV std.
GS A
GS C
δAcpp with εc=0.001
-0.1
-0.15
0
0.2
0.4
JHF: Eb=50 GeV
-1
120 days (7.66 fb )
δApp(COMPASS) scaled
c
min
0.6
0.8 pT
1
1.2
1.4
1.6
FIGURE 4. The NLO charm asymmetry A at S 10 GeV for JHF depending on a cut p T
estimate for the statistical error after 120 days using
766 fb 1 is shown.
1.8
pmin
T . An
the error bars of the shown statistical accuracy of one day of measurements at the JHF
are too small to be seen. Hence there is hope that the scales can be pinned down first in
precise unpolarized measurements.
Finally, we present in Fig. 4 predictions for the JHF with a range of older and newer
helicity distributions [1, 2]. When compared to the expected statistical uncertainty at the
JHF with a detection efficiency of εc 0001, we see that one will be able to see any but
the smallest gluons and that one can even clearly distinguish between several groups of
sets. The situation is similar at the BNL RHIC, as we will see below.
RESULTS FOR THE BNL RHIC
Obtaining similar predictions for the PHENIX experiment at the BNL RHIC is more involved, because its detector cannot reasonably be approximated by an uniform detection
efficiency εc . We use the software employed by PHENIX to generate a εeff depending
on pT and the pseudo-rapidity η , which takes into account hadronization, decay product cuts, and the detector acceptance. For Fig. 5, see [8] for more details, εeff is then
convoluted with our double differential partonic results
σ̃eff peT
1 GeV pmax
T
0
d pT
η max
η max
d η εeff pT η ; peT
341
1 GeV
d 2 σ̃
d pT d η
(2)
0
PHENIX, peT > 1 GeV, charm, NLO
-0.1
-0.2
-0.3
A / xmin
T
-0.4
-0.5
GRSV’00 val.
GRSV’00 std.
AAC 1
AAC 2
DS iDS iii+
-0.6
-0.7
-0.8
-0.9
0.01
GRSV’96 val.
GRSV’96 std.
0.7 × GS A
GS C
−
√S=200 GeV
-1
L=320 pb
min max
xmin
T =pT /pT
0.1
FIGURE 5. The NLO charm asymmetry A at S 200 GeV for PHENIX depending on x min
T
1
max
min
pmin
p
.
A
is
rescaled
by
1
x
.
An
estimate
for
the
statistical
error
using
320
pb
is
shown.
T
T
T
CONCLUSIONS
We have shown that the NLO predictions for the JHF and the BNL RHIC show great
promise for pinning down ∆g at x values of about 0.3 and 0.01 (200 GeV), 0.006 (500
GeV), respectively. The theoretical uncertainties for the asymmetry are much improved
in NLO, but the large uncertainty of the unpolarized cross section should be reduced
first by employing the fantastic luminosity of the JHF. We have also shown a method of
taking into account the complicated experimental setup of PHENIX at the BNL RHIC.
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