%% slac-pub-7173: page file slac-pub-7173-0-0-2-3.tcx.
%% subsection 2.3 Fragmentation [slac-pub-7173-0-0-2-3 in slac-pub-7173-0-0-2: slac-pub-7173-0-0-2-4]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\newmenu{slac-pub-7173::context::slac-pub-7173-0-0-2-3}{
\docLink{slac-pub-7173.tcx}[::Top]{Top}%
\sectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2}{Above: 2. Theory of Quarkonium Production}%
\subsectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2-2}{Previous: 2.2. NRQCD and Velocity Scaling}%
\subsectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2-4}{Next: 2.4. Quarkonium Polarization}%
}
%%%% end of latex2techexplorer block.
\subsection{\usemenu{slac-pub-7173::context::slac-pub-7173-0-0-2-3}{Fragmentation}}\label{subsection::slac-pub-7173-0-0-2-3}
We now consider the transverse momentum distribution $d\sigma/d p_t^2$.
The leading order contributions (in $\alpha_s$ and $v^2$) decrease as
$1/p_t^6$ (P-waves) or $1/p_t^8$ (S-waves), when $p_t$ is large compared
to $2 m_Q$. This steep decrease is a penalty for preforming the quarkonium
state at very small distances $1/p_t$ rather than $1/m_Q$ and is not what
would be expected from a high-energy cross section in QCD. For
$A+B\to H+X$, where $H$ can be any hadron, we expect scaling: When the
cms energy and $p_t$ are large compared to all hadron masses, the
short-distance cross section $d\hat{\sigma}/dp_t^2$ scales as
$1/p_t^4$ on dimensional grounds, modulo logarithms of $p_t$ and higher
twist corrections of order $m_H/p_t$ and $\Lambda/p_t$. Moreover, the
leading twist cross section can be written as a convolution of distribution
functions, a short-distance cross section and a fragmentation function.
From their $p_t$-behaviour we deduce that the leading order (in $\alpha_s$
and $v^2$) $J/\psi$ production mechanisms are higher twist at large $p_t$
(but calculable, since $m_H$ is large compared to $\Lambda$).
In general, fragmentation functions remain uncalculable. If $H$ is a
quarkonium, however, the dependence on the energy fraction
$z$ can be calculated \cite{2,8}, because the quark mass
$m_Q\gg\Lambda$ provides another large mass scale. Thus, a parton fragments
first into a $Q\bar{Q}$ pair, which subsequently hadronizes.
Since the hadronization
of the $Q\bar{Q}$ pair takes place by emission of gluons with momenta
of order $m_Q v^2$ in the quarkonium rest frame,
the energy fraction of the quarkonium relative to
the fragmenting parton, differs from that of the $Q\bar{Q}$ pair
only by an
amount $\delta z\sim v^2\ll 1$. As a result, the fragmentation functions
are expressed as a sum over perturbatively calculable,
$z$-dependent coefficient functions that
describe the fragmentation process $i\to Q\bar{Q}[n]$ multiplied by the
same $z$-independent `hadronization matrix elements' encountered
earlier.
\renewcommand{\arraystretch}{1.5}
\begin{table}[t]
\caption{Parametric dependence of various $J/\psi$ production mechanisms
in hadron-hadron collisions.
\label{tab1}}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& color singlet & color octet \\ \hline
$p_t\sim 0$ & $\alpha_s^3$ & $\alpha_s^2 v^4$ \\ \hline
$p_t \gg 2 m_Q$ & $\alpha_s^5 \left(p_t^2/(4 m_Q^2)\right)^2$ &
$\alpha_s^3 v^4 \left(p_t^2/(4 m_Q^2)\right)^2$\\
\hline
\end{tabular}
\end{center}
\end{table}
At large $p_t$ quarkonium production depends on three small parameters:
$\alpha_s/\pi\sim 0.1$, $v^2\sim 0.25-0.3$, $4 m_Q^2/p_t^2$ (numbers for
charmonium). The leading contributions for octet vs. singlet and
fragmentation vs. non-fragmentation production of $J/\psi$ are shown
in Tab.~\docLink{slac-pub-7173-0-0-2.tcx}[tab1]{1}. The scaling with $v^2$ and $4 m_c^2/p_t^2$ is
measured relative to the color-singlet non-fragmentation term. We see
that at $p_t\sim 10\,$GeV gluon fragmentation into a color-octet
quark pair dominates \cite{4} all other mechanisms by at least a
factor ten.