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%% subsection 5.1 Quarkonium production at large $p_t$ [slac-pub-7129-0-0-5-1 in slac-pub-7129-0-0-5: ^slac-pub-7129-0-0-5 >slac-pub-7129-0-0-5-2]
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\subsection{\usemenu{slac-pub-7129::context::slac-pub-7129-0-0-5-1}{Quarkonium production at large $p_t$}}\label{subsection::slac-pub-7129-0-0-5-1}
An extensive analysis of charmonium production data at $p_t>5\,$GeV
has been carried out by Cho and Leibovich \cite{8,9}, who
relaxed the fragmentation approximation employed earlier
\cite{6,7}. At the lower $p_t$ boundary, the theoretical
prediction is dominated by the ${}^1 S_0^{(8)}$ and ${}^3 P_J^{(8)}$
subprocesses and the fit yields
\begin{eqnarray}
\label{tevme}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{3}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 6.6\cdot 10^{-2}\nonumber\\
\langle {\cal O}^{\psi'}_8 ({}^1 S_0) \rangle +
\frac{3}{m_c^2}\langle {\cal O}^{\psi'}_8 ({}^3 P_0) \rangle\,
= 1.8\cdot 10^{-2}\,,
\end{eqnarray}
\noindent to be compared with the fixed target values\footnote{
Since there is a strong correlation between the charm quark mass
and the extracted NRQCD matrix elements, we emphasize that both
(\docLink{slac-pub-7129-0-0-5.tcx}[tevme]{36}) and (\docLink{slac-pub-7129-0-0-5.tcx}[fixme]{37}) as well as (\docLink{slac-pub-7129-0-0-5.tcx}[photome]{39}) below
have been obtained with the same $m_c=1.5\,$GeV (or $m_c=1.48\,$ GeV,
to be precise). On the other hand, the apparent agreement of
predictions for fixed target experiments with data claimed in
\cite{14} is obtained from (\docLink{slac-pub-7129-0-0-5.tcx}[photome]{39}) in conjunction with
$m_c=1.7\,$GeV.}
\begin{eqnarray}\label{fixme}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 3.0\cdot 10^{-2}\nonumber\\
\langle {\cal O}^{\psi'}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{\psi'}_8 ({}^3 P_0) \rangle\,
= 0.5\cdot 10^{-2}\,.
\end{eqnarray}
\noindent If we assume $\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle =
\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle/m_c^2$, the
fixed target values are a factor seven (four) smaller than the
Tevatron values for $J/\psi$ ($\psi'$). The discrepancy would be lower
for the radical choice $\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle=0$.
While this comparison looks like a flagrant violation of the
supposed process-independence of NRQCD production matrix
elements, there are at least two possibilities that could lead to
systematic differences:
(i) The $2\to 2$ color octet parton processes are
schematically of the form
\begin{equation}
\frac{\langle {\cal O}\rangle}{2 m_c}\,\frac{1}{M_f^2}\,
\delta(x_1 x_2 s-M_f^2)\,,
\end{equation}
\noindent where $M_f$ denotes the final state invariant mass. To leading
order in $v^2$, we have $M_f=2 m_c$. Note, however, that this is
physically unrealistic. Since color must be emitted from the quark pair
in the octet state and neutralized by final-state interactions, the
final state is a quarkonium accompanied by light hadrons with
invariant mass squared of order $M_f^2\approx (M_H+M_H v^2)^2$
since the soft gluon emission carries an energy of order $M_H v^2$,
where $M_H$ is the quarkonium mass. The kinematic effect of this
difference in invariant mass is very large since the gluon
distribution rises steeply at small $x$ and reduces the
cross section by at least a factor two.
The `true' matrix elements would therefore be larger
than those extracted from fixed target experiments at leading order
in NRQCD. Since the $\psi'$ is heavier than the $J/\psi$, the effect
is more pronounced for $\psi'$, consistent with the larger
disagreement with the Tevatron extraction for $\psi'$. Note that
the effect is absent for large-$p_t$ production, since in this
case, $x_1 x_2 s > 4 p_t^2 \gg M_f^2$. If we write $M_f=2 m_c+{\cal O}
(v^2)$, then the difference between fixed target and large-$p_t$ production
stems from different behaviors of the velocity expansion in the two cases.
(ii) It is known that small-$x$ effects increase the
open bottom production cross section at the Tevatron as compared
to collisions at lower $\sqrt{s}$. Since even at large $p_t$,
the typical $x$ is smaller
at the Tevatron than in fixed target experiments, this effect would
enhance the Tevatron prediction more than the fixed target prediction.
The `true' matrix elements would therefore be smaller than
those extracted from the Tevatron in \cite{9}.
While a combination of both effects could well account for the
apparently different NRQCD matrix elements, one must keep in mind
that we have reason to suspect important higher twist effects
for charmonium production at fixed target energies. Theoretical
predictions for fixed target production are intrinsically less
accurate than at large $p_t$, where higher-twist contributions
due to the initial hadrons are expected to be suppressed by
$\Lambda_{QCD}/p_t$ (if not $\Lambda_{QCD}^2/p_t^2$)
rather than $\Lambda_{QCD}/m_c$.