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\sectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2}{Above: 2. Theory of Quarkonium Production}%
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\subsection{\usemenu{slac-pub-7173::context::slac-pub-7173-0-0-2-1}{Factorization}}\label{subsection::slac-pub-7173-0-0-2-1}
\label{factorization}
Inclusive quarkonium production involves two distinct scales. First,
a heavy quark pair is produced on a distance scale of order $1/m_Q$.
Then, the quark pair is bound into a quarkonium on a time scale
of order of the inverse binding energy, $\tau\sim 1/(m_Q v^2)$ (in the
quarkonium rest frame), where $v$ is the typical velocity of the
bound quarks. We assume that $v^2$ is small (but do not assume that
the binding force is Coulombic). The creation process can be computed
in PQCD and is insensitive to the details of the bound state. The
binding process can not be computed perturbatively, but long-wavelength
gluons responsible for binding do not resolve the short-distance production
process. The factorization hypothesis for quarkonium production \cite{1}
states that the quarkonium production cross section can be written as a
sum of short-distance coefficients that describe the creation of
a $Q\bar{Q}$ pair in a state $n$ multiplied by a process-independent
matrix element that parameterizes the `hadronization' of the $Q\bar{Q}$
state $n$ into a quarkonium $\psi$ plus light hadrons with energies
of order $m_Q v^2$ in the quarkonium rest frame.
Consequently, in a hadron-hadron collision
$A+B\to \psi+X$, the differential cross section is given by
\be
d\sigma=\sum_{i,j}\int d x_1 d x_2\,f_{i/A}(x_1) f_{j/B}(x_2)\,
\sum_n d\hat{\sigma}_{i+j\to Q\bar{Q}[n]}\,\langle{\cal O}^\psi_n\rangle,
\label{fact}
\ee
where $f_{i/A}$ denotes the parton distribution function. The factorization
formula is diagrammatically represented in Fig.~\docLink{slac-pub-7173-0-0-2.tcx}[fig]{1} for deeply
inelastic scattering. The upper part of the diagram represents the
matrix element $\langle\cal{O}^\psi_n\rangle$. The hard part $H$ is
connected to this matrix element by $Q\bar{Q}$ lines plus additional
lines, if ${\cal O}^\psi_n$ contains more fields. Factorization entails
that soft gluons connecting $S$, $H$ and the remnant jet $J$ cancel up
to `higher twist' effects in $\Lambda$, where $\Lambda$ represents the
QCD low-energy scale. Each element in Eq.~\docLink{slac-pub-7173-0-0-2.tcx}[fact]{1} depends
on a factorization scale. Since gluon emission changes the $Q\bar{Q}$ state
$n$, a change of factorization scale reshuffles contributions
between different terms in the sum over $n$ and only the sum is
physical. The short-distance cross sections $d\hat{\sigma}$ are computed
by familiar matching: One first calculates the cross section for a
perturbative $Q\bar{Q}$ state and then subtracts the matrix elements
computed in this state.
\begin{figure}[t]
\vspace{0cm}
\epsfysize=4.7cm
\epsfxsize=7cm
\centerline{\epsffile{fig.eps}}
\vspace*{0cm}
\caption{\label{fig} Diagrammatic representation of factorization in
$\gamma^*+A\to \psi+X$. For the cross section the diagram has to be
cut.}
\end{figure}
A heuristic argument for factorization can be given starting from the
infrared finiteness of open heavy quark production. The above
short-distance cross sections are obtained by expanding the amplitude
squared for open heavy quark production in the relative three-momentum
of the quarks and by taking projections on color and angular momentum
states. Since soft gluon emission takes one from one state to another,
each projection separately is not infrared safe. However, by construction
the sum in $n$ runs over all states,
so that infrared sensitive contributions can always
be absorbed in some matrix element. In this sense, the sum over all
intermediate $Q\bar{Q}$ states restores the inclusiveness of open heavy
quark production. It is also worth noting that Eq.~\docLink{slac-pub-7173-0-0-2.tcx}[fact]{1}
is valid, up to higher twist effects,
even if the quarkonium is produced predominantly at small transverse momentum
with respect to the beam axis, provided one integrates over
all $p_t$. The transverse momentum distribution is not described by
Eq.~\docLink{slac-pub-7173-0-0-2.tcx}[fact]{1} unless $p_t\gg\Lambda$. A point of concern, however, is,
that higher twist effects might be of order $\Lambda/(m_Q v^2)$,
when a soft gluon from a remnant jet builds the higher Fock state
$Q\bar{Q}g$ together with the quark pair.
If these terms do not cancel, Eq.~\docLink{slac-pub-7173-0-0-2.tcx}[fact]{1}
would be quantitative in hadro-production of quarkonia at low $p_t$ only
for asymptotically large quark masses, since $\Lambda/(m_Q v^2)\sim 1$
both for charmonium and bottomonium.