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\section{\usemenu{slacpub7129::context::slacpub7129001}{Introduction}}\label{section::slacpub7129001}
Quarkonium production has traditionally been calculated in the
color singlet model (CSM) \cite{1}. Although the model successfully
describes the production rates for some quarkonium states,
it has become clear that it fails to provide a theoretically
and phenomenologically consistent picture of all production
processes. In hadroproduction of charmonia at fixed target
energies ($\sqrt{s} < 50\,$ GeV), the ratio of the number of $J/\psi$
produced directly
to those arising from decays of higher
charmonium states is underpredicted by at least a factor five
\cite{2}. The $\chi_{c1}$ to $\chi_{c2}$ production ratio
is far too low, and the observation of essentially unpolarized
$J/\psi$ and $\psi'$ can not be reproduced. At Tevatron
collider energies, when fragmentation production dominates,
the deficit of direct $J/\psi$ and $\psi'$ in the color singlet
model is even larger. This deficit has been referred to as the
`$\psi'$anomaly'
\cite{3,4}.
These discrepancies suggest that the color singlet model is
too restrictive and that other production mechanisms
are necessitated. Indeed, the CSM requires that the quarkantiquark
pair that binds into a quarkonium state be produced
on the time scale $\tau\simeq 1/m_Q$ with the same
color and angular momentum quantum numbers as the eventually formed
quarkonium.
Consequently, a hard
gluon has to be emitted to produce a ${}^3 S_1$ state in the
CSM and costs one power of $\alpha_s/\pi$.
Since the time scale for quarkonium formation is of order
$1/(m_Q v^2)$, where $v$ is the relative quarkantiquark
velocity in the quarkonium bound state, this suppression
can be overcome if one allows for the possibility that the
quarkantiquark pair is in any angular momentum or color
state when produced on time scales $\tau\simeq 1/m_Q$.
Subsequent evolution into the physical
quarkonium state is mediated by emission of soft gluons with momenta
of order $m_Q v^2$. Since
the quarkantiquark pair is small in size, the emission of these
gluons can be analyzed within a multipole expansion.
A rigorous formulation \cite{5} of this picture can be given in
terms of nonrelativistic QCD (NRQCD). Accordingly, the production
cross section for a quarkonium state $H$ in the process
\begin{equation}
\label{proc}
A + B \longrightarrow H + X,
\end{equation}
\noindent can be written as
\begin{equation}
\label{fact}
\sigma_H = \sum_{i,j}\int\limits_0^1 d x_1 d x_2\,
f_{i/A}(x_1) f_{j/B}(x_2)\,\hat{\sigma}(ij\rightarrow H)\,,
\end{equation}
\begin{equation}
\label{factformula}
\hat{\sigma}(ij\rightarrow H) = \sum_n C^{ij}_{\bar{Q} Q[n]}
\langle {\cal O}^H_n\rangle\,.
\end{equation}
\noindent Here the first sum extends over all partons in the colliding
hadrons and $f_{i/A}$ etc. denote the corresponding distribution
functions. The shortdistance ($x\sim 1/m_Q \gg 1/(m_Q v)$)
coefficients $C^{ij}_{\bar{Q} Q[n]}$
describe the production of a quarkantiquark pair
in a state $n$ and have expansions in $\alpha_s(2 m_Q)$. The
parameters\footnote{Their precise definition is given in
Sect.~VI of \cite{5}.}
$\langle {\cal O}^H_n\rangle$ describe the subsequent
hadronization of the $Q\bar{Q}$ pair into a jet containing
the quarkonium $H$ and light hadrons. These matrix elements
can not be computed
perturbatively, but their relative importance in powers
of $v$ can be estimated from the selection rules for multipole
transitions.
The color octet picture has led to the most plausible explanation of
the `$\psi'$anomaly' and the direct $J/\psi$ production deficit.
In this picture
gluons fragment into quarkantiquark pairs in a coloroctet
${}^3 S_1^{(8)}$ state which then hadronizes into a $\psi'$ (or $J/\psi$)
\cite{6,7,8}. Aside from this striking prediction, the
color octet mechanism remains largely untested. Its verification
now requires considering quarkonium production in other processes
in order to demonstrate the processindependence (universality) of
the production matrix elements $\langle {\cal O}_n^H\rangle$, which is
an essential prediction of the factorization formula (\docLink{slacpub7129001.tcx}[factformula]{3}).
Direct $J/\psi$ and $\psi'$ production at large $p_t\gg 2 m_Q$ (where
$m_Q$ denotes the heavy {\em quark} mass) is rather unique in that a
single term, proportional to $\langle {\cal O}_8^H ({}^3 S_1)\rangle$,
overwhelmingly dominates the sum (\docLink{slacpub7129001.tcx}[factformula]{3}). On the other hand,
in quarkonium
formation at moderate $p_t\sim 2 m_Q$ at colliders and in photoproduction
or fixed target experiments ($p_t\sim 1\,$GeV),
the signatures of color octet production
are less dramatic, because they are not as enhanced
by powers of $\pi/\alpha_s$
or $p_t^2/m_Q^2$ over the singlet mechanisms. Furthermore,
theoretical predictions are parameterized by
more unknown octet matrix elements and are
afflicted by larger uncertainties. In particular, there are large
uncertainties
due to the increased
sensitivity to the heavy quark mass close to threshold. (The production
of a quarkantiquark pair close to threshold is favored by the rise of
parton densities at small $x$.)
These facts complicate establishing color
octet mechanisms precisely in those processes where experimental data is
most abundant.
Cho and Leibovich \cite{9} studied direct quarkonium
production at moderate $p_t$ at the Tevatron collider and were able
to extract a value for a certain combination of
unknown parameters $\langle {\cal O}_8^H({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^H({}^3 P_0)\rangle$ ($H=J/\psi,\psi',\Upsilon(1S),
\Upsilon(2S)$). A first test of universality comes from
photoproduction \cite{10,11,12}, where a different combination
of these two matrix elements becomes important near the elastic
peak at $z\approx 1$, where $z=p\cdot k_\psi / p \cdot k_\gamma$, and
$p$ is the proton momentum. A fit to photoproduction data requires
much smaller matrix elements than those found in \cite{9}.
Taken at face value, this comparison would imply failure of the
universality assumption underlying the nonrelativistic QCD approach.
However, the extraction from photoproduction should be regarded
with caution since the NRQCD formalism describes
inclusive quarkonium production only after sufficient smearing in $z$
and is not applicable in the exclusive region close to $z=1$, where
diffractive quarkonium production is important.
In this paper we investigate the universality of the color octet
quarkonium production matrix elements
in fixed target hadron collisions and
reevaluate the failures of the CSM in fixed target production
\cite{2} after inclusion of color octet
mechanisms. Some of the issues involved have already been
addressed by Tang and V\"anttinen \cite{13} and by Gupta and
Sridhar \cite{14}, but a complete survey is still missing. We
also differ from \cite{13} in the treatment of polarized
quarkonium production and the assessment of the importance of color
octet contributions and from \cite{14} in the color
octet shortdistance coefficients.
The paper is organized as follows: In Sect.~2 we compile the leading
order color singlet and color octet contributions to the production
rates for $\psi^\prime,~\chi_J,~J/\psi$ as well as bottomonium.
In Sect.~3 we present our numerical results for proton and
pion induced collisions. Sect.~4 is
devoted to the treatment of polarized quarkonium production. As
polarization remains one of the cleanest tests of octet quarkonium
production at large $p_t$ \cite{15,16}, we clarify in detail
the conflicting treatments of polarized production in \cite{16} and
\cite{9}. Sect.~5 is dedicated to a comparison of the extracted
coloroctet matrix elements from fixed target experiments with those
from photoproduction and the Tevatron. We argue that
kinematical effects and small$x$ effects can bias the extraction
of NRQCD matrix elements so that a fit to Tevatron data at large $p_t$
requires larger matrix elements than the fit to fixed
target and photoproduction data. The
final section summarizes our conclusions.
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