ORNL-CTP-96-02/UCLA-NT-9602 and hep-ph/9604282
Color-Octet Fraction in J/ψ Production
Cheuk-Yin Wong
arXiv:hep-ph/9604282v1 10 Apr 1996
Oak Ridge National Laboratory, Oak Ridge, TN 37831
Chun Wa Wong
University of California, Los Angeles, CA 90095-1547
(April 3, 2008)
Abstract
The cross section between a cc̄ object and a nucleon is small if the object is
in a color-singlet state, but very large and insensitive to its size if it is in a
color-octet state. We use this property to estimate the fraction of cc̄ coloroctet states formed after the initial hard-scattering processes by studying the
effective J/ψ absorption in a nucleus. The experimental NA3 and E772 data
of pA → ψX reveal the presence of anomalously large effective absorption of
J/ψ at large xF , which indicates a large color-octet fraction in J/ψ production
at large xF .
PACS number(s): 13.85-t, 25.75.-q, 13.90.+i
Typeset using REVTEX
1
Recently, there has been much interest in the mechanism of heavy quarkonium production. Bodwin, Braaten, and Lepage have developed a factorization formalism that allows a
systematic calculation of inclusive J/ψ cross sections [1]. The formalism takes into account
the production of color-singlet and color-octet states before the formation of the quarkonium. The model has been applied to study many heavy quarkonium production processes
[2–6].
It has been pointed out that the production of color-octet states in the e+ e− annihilation may dominate near the upper endpoint of the J/ψ energy spectrum, and that the
production of these color-octet objects can be identified by a drastic change in the J/ψ
angular distribution near the upper endpoint [7]. We would like to suggest here that the
occurrence of the cc̄ color-octet states can also be deduced from their anomalously large
absorption when propagating in nuclear matter. This possibility arises in the following way:
High-energy hadron-hadron cross sections are known to be dominated by Pomeron exchange
[8]. In the Two-Gluon Model of the Pomeron (TGMP) [9,10], the (color-octet cc̄)-nucleon
[(cc̄)8 -N] cross sections are known to be significantly larger than the color-singlet (cc̄)1 -N
cross sections (by an order of magnitude) [11,12]. This large difference allows us to deduce the color-octet fraction of the produced cc̄ pair from the effective J/ψ absorption cross
section in nuclei. Analysis of the NA3 [13] and E772 [14] data on the xF dependence of
pA → ψX reveals the presence of anomalously large effective absorption cross sections for
J/ψ production at large xF , which can be attributed to a large (cc̄)8 fraction near the upper
endpoint of the J/ψ spectrum, as in the analogous J/ψ production in the e+ e− annihilation
predicted by [7].
The propagation of a cc̄ in nuclear matter or other exotic matter has been the subject
of intense study from another viewpoint. It has been suggested that in a quark-gluon
plasma, the cc̄ pair becomes unbound when the temperature exceeds a critical value and
the production of cc̄ bound states will then be suppressed [15]. Although J/ψ suppression
has indeed been observed experimentally [16], the characteristics of the suppression can also
be explained by the breakup of the cc̄ pair as it interacts with the nucleons and/or other
2
produced soft particles [17–23]. In the following, we are interested in the difference in the
effective absorption of cc̄ in nuclear matter when the cc̄ pair is in different color states.
In the quarkonium production model of Bodwin et al [1], the factorization property of the
production process implies that a cc̄ pair is produced in both color-singlet and octet states
in a hard-scattering process on a short time scale. An octet pair produced will eventually
hadronize into a color-singlet state by color neutralization through the absorption or emission
of soft gluons. This takes place over a much longer time scale of about 1 fm/c, typical of
nonperturbative processes. In a pA reaction, the dynamics of the produced cc̄ before the
neutralization of its color is simplest to describe in the cc̄ rest frame. In this frame, the
longitudinal spacing between target nucleons is d/γ where d = 2 fm is the internucleon
spacing in a nucleus at rest and γ is the relativistic energy/mass ratio of the moving target
nucleons in this cc̄ rest frame. Since the value of γ for target nucleons is large in high-energy
pA collisions, d/γ << 1 fm/c. Therefore, collisions between the produced (cc̄)8 and target
nucleons, when they occur, will take place before its color is neutralized.
The high-energy (cc̄)8 -N and (cc̄)1 -N cross sections can be estimated by using the TGMP
studied by Low, Nussinov and others [9–11,24–27]. In this model, the total cross section
for color-singlet states can be expressed as S1 − S2 , where Sn is the contribution in which
the two exchanged gluons interact with n particles (here quarks) in the projectile. The
contribution S1 is independent of the meson size. It is larger than S2 , which decreases with
increasing meson size, except in the limit of a point meson where the distinction between
these contributions disappears. Then S2 = S1 , and the singlet cross section vanishes. For
small mesons, the cross section can be shown to be proportional to the mean-square radius
of the meson [24]. For light mesons of large size, on the other hand, S2 is much smaller than
S1 . This feature allows the model to reproduce the quark additivity property of high-energy
cross sections expressed so appealingly by the Additive Quark Model [28].
It has been pointed out by Dolejší and Hüfner [11] that the cross section is very different
for (q q̄)8 -N scattering because the contributions add together in the form of S1 + S2/8. The
result is then insensitive to the (q q̄)8 size. It is also very large, typically of the order of
3
30-60 mb when a perturbative propagator is used for gluons with a nonzero effective mass.
Recently, this TGMP for both singlet and octet (q q̄)8 -N scattering has been studied in
detail by one of us [12] with a number of important refinements: (1) A nonperturbative gluon
propagator (the Cornwall propagator) is used [29,30] with the gluon mass obtained by fitting
the πN and KN total cross sections. (2) Complete meson form factors are used without
making the small meson approximation. (3) For singlet meson-N scattering, a coupledchannel problem [22] is solved using many scattering channels containing radially excited
mesons. By fitting NN cross sections and the ratios of (meson-N)/NN cross sections,
the extrapolated octet (cc̄)8 − N cross section turns out to be 48 mb, which falls within
the range of 30 − 60 mb (from Dolejší and Hüfner [11]). The color-octet cross section is
relatively insensitive to the meson size and flavor content. An intuitive understanding of the
very large size of the cross section, by analogy with electrodynamics, has been presented in
Ref. [11].
Experimental J/ψ photoproduction data [31] and the Vector Dominance Model give a
√
total ψ − N cross section of about 1.75 mb at s = 20 GeV. The photon is in a colorsinglet state, and the vector meson produced from the photon is also in a color-singlet state.
At lowest order, inclusive photoproduction should involve predominantly color-singlet cc̄
states. Thus, the total color-singlet cross section for the collision of (cc̄)1 with a nucleon
√
is approximately about 1.75 mb at s = 20 GeV. Given the inherent uncertainty of the
Vector-Dominance Model, which underestimates the ρ − N cross section by about 15% and
φ − N by about 50% [32], we estimate that the color-singlet ψ − N cross section is 2 to
3.5 mb. There is thus an order of magnitude difference in the (cc̄)8 -N and (cc̄)1 -N cross
sections, which makes it possible to extract a signal for (cc̄)8 from its stronger absorption in
nuclei.
We must first determine the effective (cc̄)-N absorption cross section in nuclei. This is
done by using the Glauber multiple-scattering model for the absorption of the produced cc̄
pair as it traverses the nucleus [20,21]. For completeness, we have used a complex (cc̄)-N
scattering amplitude with α = Re/Im. The meson production cross section in a nuclear
4
target is then
pA
dσJ/ψ
/dxF
NN
AdσJ/ψ
/dxF
=
Z
dbA
Z
Z
dzA ρ(bA , za )1 −
∞
zA
σtot (cc̄
dzA′ ρ(bA , zA′ )
2
− N)
2(A−1)
(1 − iα)
,
(1)
which is a generalization of the absorption model used in Ref. [20]. Here, ρ(bA , zA ) is
normalized to ρ(bA , zA )dbA dzA = 1. For numerical calculations, we use a real-to-imaginary
R
ratio α = 0.19, the value in Eq. (4.5a) of Ref. [32] for p = 10 GeV. Note, however, that when
α is so small, the extracted total cross section σtot (cc̄ − N) is quite insensitive to it. The
effective (cc̄) − N absorption cross section σabs (cc̄ − N), which we take to be the reaction
cross section, can be deduced from σtot (cc̄ − N ) and α.
The results for σabs (cc̄ − N) from the analysis of the NA3 and E772 data are shown
in Fig. 1. One finds that the NA3 and E772 data approximately scale with each other,
suggesting that the phenomenon has a common origin. The cross sections are about 4-8 mb
at small xF , but increase dramatically beyond xF ≃ 0.4 to about 20 mb as xF reaches 0.6.
To account for these large cross sections, Badier et al. [13] have to postulate the existence of
a new mechanism of J/ψ production. We instead attribute these large cross sections to the
production of (cc̄)8 states and to the strong absorption of the (cc̄)8 objects as they propagate
in nuclear matter.
If we assume that the color-octet and color-singlet fraction of cc̄ pairs produced after the
hard-scattering processes are f8 and (1 − f8 ) respectively, then we can extract f8 from the
effective absorption cross section σabs (cc̄ − N ) inferred from the absorption of the produced
cc̄ through nuclear matter. Because the cc̄-N collision takes place at high energies, a cc̄-N
collision will cause the singlet or octet cc̄ pair to be broken up, or absorbed. From [31] and
[32], we use the color-singlet absorption cross section σabs ((cc̄)1 − N) = 2.75 mb. From [11]
and [12], we take the color-octet absorption cross section σabs ((cc̄)8 − N) = 45 mb. The
color-octet fraction f8 is then
f8 =
σabs (cc̄ − N ) − σabs ((cc̄)1 − N)
.
σabs ((cc̄)8 − N) − σabs ((cc̄)1 − N)
(2)
The extracted f8 as a function of xF is shown in Fig. 2. The actual numbers will change a
5
little for a different choice of σabs ((cc̄)8 − N) and σabs ((cc̄)1 − N) [12], but the qualitative
behavior shown in this figure should remain the same.
We see that the octet fraction f8 is small for small values of xF and becomes large for
large xF , reaching about 0.4 at xF ∼ 0.6. This indicates that the color-singlet fraction still
dominates the production process in the region of low xF , which is the region of greatest
probability. However, the effective xF -averaged absorption cross section is much enhanced
over the color-singlet value, even with a small octet fraction. This provides a simple explanation for the large effective xF -averaged absorption cross section of 5-7 mb which one finds
in the conventional absorption models [17–21].
The variation of the octet fraction as a function of xF extracted here will be useful to
provide a constraint on the construction of J/ψ production models. To produce a J/ψ with a
large xF , the projectile parton needs to have a large light-cone variable x. From the structure
functions, the valence quark distribution decreases less rapidly than the gluon distribution
as x approaches unity. Thus, in the region of large xF , the dominant mode of cc̄ (or J/ψ)
production in pp collisions is from the q q̄ annihilation (see e.g. [33]). The cc̄ pair formed by
the annihilation of a light q q̄ pair can only be in the color-octet state in the lowest-order
process. Higher-order processes such as those involving the emission of hard gluons could
change the color state, but they will also reduce its xF , thus preserving the octet dominance
at large xF . On the other hand, at the central rapidity y = 0, which corresponds to xF = 0,
the differential cross section dσ/dy can be described by a color-singlet model including the
effect of bound-state wave functions in the hard-scattering processes [34]. Recent theoretical
investigations of pp → ψX [4,5] indicate that the singlet hard-scattering cross section for
J/ψ production from gg collisions go as αs3 v 0 , while the octet cross section from gg and
q q̄ collisions go as αs2 v 4 , where αs is the strong coupling constant and v is the c-c̄ relative
velocity. Hence octet production is suppressed by a relative factor of v 4 /αs relative to singlet
production. One also needs to fold these cross sections with the parton structure functions.
One expects the singlet contributions to be dominant over most of the phase space region,
except in some special kinematic regions such as the region of large xF mentioned above.
6
Having described one color-octet scenario, it is useful to discuss a variation of this coloroctet model. Kharzeev and Satz [23] have recently suggested that the difference between
the singlet cross section σtot ((cc̄)1 − N) of about 3 mb, and the effective total cross section
of 5-7 mb for J/ψ absorption in nuclei can be understood if a color-singlet (cc̄)8 g hybrid
meson is formed after the initial hard scattering and if this hybrid then traverses the nucleus unchanged before its (cc̄)8 constituent eventually hadronizes with the gluon partner.
The total cross section σtot ((cc̄)8 g − N) is approximately (9/4)σtot ((cc̄)1 − N) if the (cc̄)8
constituent is treated as a point particle, and if the average (cc̄)8 -g separation is the same
as that between the quark and the antiquark in (cc̄)1 . In this model, there are no singlet
(cc̄)1 contributions from the hard-scattering processes to J/ψ production, but the theoretical
results of [34] suggests that singlet contributions dominate experimental data at xF = 0.
Moreover, to describe the large absorption at large xF using this hybrid model, the rms
separation between the (cc̄)8 and g needs to be about 0.5±0.15 fm [12]. Because the hybrid
collides with target nucleons at a time t much shorter than 0.5 fm/c (in its rest frame),
there is the question whether there is sufficient time for the cc̄ to form such a large hybrid
state. Further studies will be needed to clarify the relative importance of singlet and octet
contributions and to discriminate between different models of J/ψ absorption.
In conclusion, the production of (cc̄)8 pairs should be accompanied by an anomalously
large effective J/ψ absorption in nuclear matter due to the large (cc̄)8 -N total cross section.
The NA3 and E772 data seem to support the idea that the color-octet production dominates
near the upper endpoint of J/ψ production and that the color-singlet production is still
predominant for small values of xF .
ACKNOWLEDGMENTS
We would like to thank Dr. Yu-Qi Chen for helpful discussions. This research was
supported by the the Division of Nuclear Physics, U.S.D.O.E. under Contract DE-AC0596OR22464 managed by Lockheed Martin Energy Research Corp.
7
REFERENCES
[1] G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D51, 1125 (1995).
[2] For a review, see E. Braaten, Northwestern University Report NUHEP 95-11, hepph/9509210, and E. Braaten, S. Fleming, Tzu Chiang Yuan, OHSTPY-HEP-T-96-001,
(hep-ph/9602374).
[3] Wai-Keung Tang and M. Vänttinen, hep-ph/9603266.
[4] P. Cho and A. Leibovich, Phys. Rev. D53, 150 (1996); P. Cho and A. Leibovich, CALT68-2026 (hep-ph/9511315).
[5] S. Fleming and I. Maksymyk, hep/ph9512320.
[6] M. Cacciari and M. Krämer, DESY-96-005 (hep-ph9601276); P. Ko, J. Lee, and H. S.
Song, hep-ph/9602223.
[7] E. Braaten and Yu-Qi Chen, Phys. Rev. Lett. 76, 730 (1996).
[8] H. J. Lipkin, Phys. Rev. D11, 1827 (1975); A. Donnachie and P.V. Landshoff, Phys.
Lett. B 296, 227 (1992).
[9] F. E. Low, Phys. Rev. D12, 163 (1975).
[10] S. Nussinov, Phys. Rev. Lett. 34, 1286 (1975).
[11] J. Dolejší and J. Hüfner, Z. Phys. C54, 489 (1992).
[12] C. W. Wong, Coupled-channel effects in high-energy hadron collisions, UCLA-NT-9603
(hep-ph/9604277).
[13] J. Badier et al., NA3 Collaboration, Z. Phys. C20, 101 (1983).
[14] D. M. Alde et al, E772 Collaboration, Phys. Rev. Lett. 66, 133 (1991).
[15] T. Matsui and H. Satz, Phys. Lett. B178, 416 (1986).
8
[16] C. Baglin et al., NA38 Collaboration, Phys. Lett. B220, 471 (1989); M. C. Abreu et al.,
NA38 Collaboration, Nucl. Phys. A544, 209c (1992).
[17] R. L. Anderson et al., Phys. Rev. Lett. 38, 263 (1977).
[18] C. Gerschel and J. Hüfner, Phys. Lett. B207, 253 (1988); C. Gerschel and J. Hüfner,
Nucl. Phys. A544, 513c (1992).
[19] S. Gavin and R. Vogt, Nucl. Phys. B345, 104 (1990); S. Gavin, Nucl. Phys. A566, 383c
(1994).
[20] C. Y. Wong, Introduction to High-Energy Heavy-Ion Collisions, World Scientific Publishing Company, 1994.
[21] C. Y. Wong, Phy. Rev. Lett. 76, 196 (1996).
[22] J. Hüfner and B. Kopeliovich, Phys. Rev. Lett. 76, 192 (1996).
[23] D. Kharzeev and H. Satz, Phys. Lett. B366, 316 (1996).
[24] J. F. Gunion and D. E. Soper, Phys. Rev. D15, 2617 (1977).
[25] E.M. Levin and M.G. Ryskin, Sov. J. Nucl. Phys. 34, 619 (1981).
[26] P. V. Landshoff and O. Nachtmann, Z. Phys. C35, 405 (1987).
[27] M. B. G. Ducati, F. Halzen, and A. A. Natale, Phys. Rev. D48, 2324 (1993).
[28] E. M. Levin and L. L. Frankfurt, JETP Lett. 2, 65 (1965); H. J. Lipkin and F. Scheck,
Phy. Rev. Lett. 16, 71 (1966).
[29] J. M. Cornwall, Phys. Rev. D26, 1453 (1982).
[30] F. Halzen, G. Krein, and A. A. Natale, Phys. Rev. D47, 295 (1993).
[31] S. D. Holmes, W. Lee, and J. E. Wiss, Ann. Rev. Nucl. Part. Sci. 35, 397 (1985).
[32] T. H. Bauer, R. D. Spital, D. R. Yennie, and F. M. Pipkin, Rev. Mod. Phys. 50, 261
9
(1978).
[33] M. S. Kowitt et al., Phys. Rev. Lett. 72, 1318 (1994).
[34] G. A. Schuler, CERN-TH.7170/94 (hep-ph 9403387), to be published in Physics Reports.
10
FIGURES
FIG. 1. The effective (cc̄)-N absorption cross section as a function of xF .
FIG. 2. The color-octet fraction as a function of xF .
11
50.0
NA3 (200 GeV)
E772 (800 GeV)
45.0
40.0
35.0
σabs (mb)
30.0
25.0
20.0
15.0
10.0
5.0
0.0
0.0
0.2
0.4
0.6
xF
0.8
1.0
Fig. 1
NA3 (200 GeV)
E772 (800 GeV)
Color−Octet Fraction f8
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
xF
0.8
1.0
Fig. 2