\section{INTRODUCTION}
In a series of previous papers \cite{mas0,mas1,mas2,mas3} we analyzed
charmonium hadroproduction in the light of the color-octet model
\cite{braaten}.
We employed a Monte Carlo generator (PYTHIA 5.7) \cite{pythia}
to assess the importance of 
some higher-order QCD effects. As a consequence we concluded that
NRQCD matrix elements \cite{bodwin} determined from Tevatron experimental data
\cite{fermi1}  were considerably overestimated in other analysis
\cite{cho}, since they absorb some
perturbative effects, e.g. initial-state emission of
gluons. The latter gives rise to an effective transverse momentum, 
enhancing the
high-$p_t$ tail of the differential distribution of charmonium.
Once taken into account, the extracted long-distance
parameters have to be lowered significantly. This is especially
apparent for the linear combination of matrix elements (MEs)
 $^1S_0^{(8)}+^3P_0^{(8)}$ 
where, in fact, a large discrepancy w.r.t. HERA results
on $J/\psi$ photoproduction has been pointed out \cite{hera}.\par
However, in our former work we did not consider 
Altarelli-Parisi (AP) evolution of the fragmenting gluon into
charmonium for
the $^3S_1^{(8)}$ contribution at high $p_t$. In this paper we implement
such an effect benefitting from the evolution performed by
PYTHIA itself as will be explained below. We shall show
that although our previous numerical values for the
$^3S_1^{(8)}$ ME were somewhat low, no significant
change occurs for the corresponding $^1S_0^{(8)}+^3P_0^{(8)}$ ME.
We also discuss the sensitivity to the lower cut-off
used to avoid the singular $p_t{\rightarrow}0$ region.
We get started with the latter point in the following section. 

\section{SENSITIVITY TO THE $p_t^{min}$ CUT-OFF}
Production cross sections which are singular at
$p_t=0$ are  automatically regularized in running PYTHIA
by setting a $p_t$ lower cut-off at 1 GeV \cite{pythia}. 
Consistently with our aim of
using by-default options whenever possible in the generation, we
kept this value in all our previous analysis \cite{mas0,mas3}.
\par
In fact, the differential cross section can have a singular
behaviour at low $p_t$ even to next-to-leading order due to 
imperfect cancellation of real and virtual
contributions. However, from a NLO calculation of charmonium 
production by Mangano \cite{mangano}, one may
conclude that our previously used
lower cut-off ($p_t^{min}=1$ GeV) is consistent
with the threshold obtained by Mangano (${\approx}\ 1$ GeV for
the $^1S_0^{(8)}$ channel), such that the averaged cross section is zero 
below this value.

Nevertheless, in order to assess the significance
of the cut-off on the values of the long-distance
parameters we performed new fits to
Tevatron data by changing $p_t^{min}$. We found that, by
varying $p_t^{min}$ in the $[1,2]$ GeV range, the $^3S_1^{(8)}$
ME showed almost no appreciable change, whereas
the $^1S_0^{(8)}+^3P_0^{(8)}$ ME changed by a 
factor two at most \cite{seminar}.\par
Therefore, we keep in our subsequent analysis
the value  $p_t^{min}=1$ GeV as a self-consistent lower
cut-off.


\section{ALTARELLI-PARISI EVOLUTION}
At high transverse momentum, gluon fragmentation via the color
octet mechanism becomes the dominant source of charmonium
production. On the other hand, AP evolution of the 
splitting gluon produces a depletion of its energy which has to be taken
into account. If not so, the long-distance parameter
for the  $^3S_1^{(8)}$ channel would be underestimated.
\par 
In this work the AP evolution of the
fragmenting gluon was carried out from the evolution of
the gluonic partner of the $J/\psi$ in the final state
of the production channel 
\begin{equation}
g\ g\ {\rightarrow}\ g^{\ast}({\rightarrow}J/\psi)\ g
\end{equation}
the technical reason being that the splitting gluon $g^{\ast}$
actually is not generated in our code \cite{mas2}. (Notice
however that the MEs used in the generation include the 
long-distance evolution of the ($c\overline{c}$) bound state 
\cite{mas2}.)

Indeed, on the average the fragmenting gluon should evolve in the 
transverse plane similarly to the other final-state gluon in (1), 
once corrected its virtuality  to become of the order of the 
charmonium mass.

Thereby we get a correcting factor to be applied event by event
to the transverse momentum of the generated
$J/\psi$ (for the $^3S_1^{(8)}$ channel only):
\begin{equation}
x_p\ =\ \frac{\sqrt{P_t^{{\ast}2}+m_{J/\psi}^2}}
{\sqrt{P_t^{2}+m_{J/\psi}^2}} 
\end{equation}
where $P_t$ ($P_t^{\ast}$) is the transverse momentum of 
the final-state gluon without (with) AP evolution.

At high $p_t$,
\begin{equation}
 p_t^{AP}\ =\ x_p\ {\times}\ p_t
\end{equation}
where $p_t$ is the transverse momentum of the $J/\psi$
as generated by PYTHIA (i.e. without AP evolution).\par
Although the above way to implement AP evolution may be
somewhat indirect, it remains in the spirit of our whole
analysis, i.e. using PYTHIA algorithms whenever possible. In fact it
provides an energy depletion of the fragmenting gluon
in accordance with Cho and Leibovich's work \cite{cho}.

\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{file=jpsi18_cte_ap.eps,height=6.5cm,width=7.5cm}}\end{center}
\caption{Curves obtained from PYTHIA for $J/\psi$ production
at the Tevatron, including AP evolution and
initial-state radiation on for the CTEQ 2L parton distribution function.
The charm mass was set equal to $1.48$ GeV.
(i) dotted line: Color-singlet model, (ii) dashed line: 
$^1S_0^{(8)}+^3P_0^{(8)}$, (iii) dot-dashed line: $^3S_1^{(8)}$, 
(iv) solid line: all contributions.}
\end{figure}


\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{file=jpsi14_cte_nap.eps,height=6.5cm,width=7.5cm}}
\end{center}
\caption{Curves obtained from PYTHIA for $J/\psi$ production at
the LHC, without AP evolution and
initial-state radiation on for the CTEQ 2L parton distribution function.
(i) dotted line: Color-singlet model, (ii) dashed line: 
$^1S_0^{(8)}+^3P_0^{(8)}$, (iii) dot-dashed line: $^3S_1^{(8)}$, 
(iv) solid line: all contributions.}
\end{figure}



\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{file=jpsi14_cte_ap.eps,height=6.5cm,width=7.5cm}}
\end{center}
\caption{Curves obtained from PYTHIA for $J/\psi$ production
at the LHC, including AP evolution and
initial-state radiation on for the CTEQ 2L parton distribution function.
(i) dotted line: Color-singlet model, (ii) dashed line: 
$^1S_0^{(8)}+^3P_0^{(8)}$, (iii) dot-dashed line: $^3S_1^{(8)}$, 
(iv) solid line: all contributions.}
\end{figure}


\section{NEW EXTRACTION OF COLOR-OCTET MEs}
We have performed new fits to the Tevatron data \cite{fermi1}
using three PDFs (as in \cite{mas2}) but now 
with AP evolution implemented in the generation as described in the
preceding section. In table 1 we show the numerical values for
 $<O_8(^3S_1)>$ and the linear combination 
$M_3=3{\times}(\frac{<O_8(^3P_0)>}{m_c^2}+
\frac{<O_8(^1S_0)>}{3})$ obtained in our analysis.
The new values for the  $<O_8(^3S_1)>$
matrix elements have been increased by a factor about 3 w.r.t.
AP off \cite{mas2}. Conversely
the $M_3$ value is slightly decreased, moreover remaining quite
smaller than in other analysis \cite{cho}. Therefore we conclude that
the $M_3$ value obtained in our work is reliable and in a
better agreement with HERA results on $J/\psi$ photoproduction \cite{kniehl}. 



\begin{table*}[hbt]
\setlength{\tabcolsep}{1.5pc}
\caption{Color-octet matrix elements (in units of $10^{-3}$ GeV$^3$) from 
the best fit to Tevatron data on direct $J/\psi$ production
for different PDFs. Initial-state radiation of gluons and AP evolution as
explained in the text were
swichted on. For 
comparison we quote the values given by Cho and Leibovich: 
$(6.6{\pm}2.1)$ and $(66{\pm}15)$  
respectively.}
\label{FACTORES}

\begin{center}
\begin{tabular}{lcc}    \hline
ME:  & $<O_8(^3S_1)>$ & 
$M_3=3{\times}\biggl(\frac{<O_8(^3P_0)>}{m_c^2}+
\frac{<O_8(^1S_0)>}{3}\biggr)$ \\
\hline

CTEQ2L & $9.6{\pm}1.5$ & $13.2{\pm}2.1$ \\
MRSD0 & $6.8{\pm}1.6$ & $13.2{\pm}2.1$ \\
GRVHO& $9.2{\pm}1.1$ & $4.5{\pm}0.9$ \\
\hline
\end{tabular}
\end{center}
\end{table*}


\section{$J/\psi$ DIRECT PRODUCTION RATE AT THE LHC}
In order to get an estimate of the expected charmonium
production rate at the LHC, we ran PYTHIA with the color-octet
model implemented in. In figure 3 we show the predicted
differential cross section times the muonic branching fraction
of $J/\psi$ with AP evolution included. In figure 2 we show
the same curve with AP off. By comparison, one may conclude that, 
once adjusted
the long-distance parameters to fit Tevatron data, there is no
significant difference between both generations.\par
Let us stress that because of theoretical uncertainties 
associated to different choices for the charm mass, energy scales,
parton distribution function ...,  
the theoretical curves of figures 2 and 3
 have to be considered as order-of-magnitude
predictions. Nevertheless the relatively high production rate
at high-$p_t$ (of the order of the pb at $p_t=50$ GeV) makes especially
interesting the analysis of charmonia production at the LHC. A
preliminary study on $\Upsilon(1S)$ prompt production 
leads to a similar conclusion for the bottomonia family.


\section{CONCLUSIONS}
After checking our analysis by varying the lower $p_t$ cut-off
set in the Monte Carlo generation to avoid the
problematic $p_t{\rightarrow}0$ region, we conclude that our
former choice $p_t^{min}=1$ GeV was justified and our
study of charmonium hadroproduction trustworthy. 
On the other hand, once implemented Altarelli-Parisi
evolution in our framework, the  new extracted values of
the color-octet ME  $<O_8(^3S_1)>$ increase (table 1), recovering
a similar result as in previous extractions \cite{cho}. On the
other hand, the numerical values obtained for 
$M_3$ even decrease slightly, reinforcing the conclusions reported in our
former work \cite{mas0,mas1,mas2,mas3}. 

With respect to the extrapolation of the
$J/\psi$ production rate up to LHC energy, the theoretical
prediction incorporating AP evolution (figure 3) does not differ 
significantly from the
curve obtained without AP evolution (figure 2). This means that, from 
a practical point of view, there is no need to modify the code
for a fast generation as shown in \cite{mas2} if the corresponding
color-octet matrix elements are duly employed. 
Finally we conclude that the analysis of charmonia (and bottomonia) 
prompt production at the LHC deserves special attention in its own right.



\subsubsection*{Acknowledgments}
I thank the LHC Workshop b-production subgroup for interesting
discussions and suggestions. 
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(Proc. Suppl.) 55A (1997) 277.
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(1997) 232.
\bibitem{mas2} B. Cano-Coloma and M.A. Sanchis-Lozano, Nucl. Phys.  B 508
 (1997) 753.
\bibitem{mas3} M.A. Sanchis-Lozano, Nucl. Phys. B (Proc. Suppl.) 75B (1999)
191.
\bibitem{braaten} E. Braaten and S. Fleming, Phys. Rev. Lett. 74 
(1995) 3327.
\bibitem{pythia} T. Sj\"{o}strand, Comp. Phys. Comm. 82 (1994) 74.
\bibitem{bodwin} G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 51
(1995) 1125. 
\bibitem{fermi1} CDF Collaboration, F. Abe at al., Phys. Rev. Lett. 
69 (1992) 3704; 71 (1993) 2537; 75 (1995) 1451 ; 79 (1997) 578.
\bibitem{cho} P. Cho and A.K. Leibovich, Phys. Rev. D 53 (1996) 6203.
\bibitem{hera} M. Cacciari and M. Kr\"{a}mer, Phys. Rev. Lett. 76
(1996) 4128.
\bibitem{mangano} M. Mangano, private communication.
\bibitem{seminar} M.A. Sanchis-Lozano, 2nd meeting of the b-production
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\end{document}






\begin{table*}[htb]
\setlength{\tabcolsep}{1.5pc}
\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
\caption{Color-octet matrix elements (in units of GeV$^3$) from 
best fits to old and new 
Tevatron data on prompt $J/\psi$ production. 
($M_r^{J/\psi}=<0{\mid}O_8^{J/\psi}(^1S_0){\mid}0>\ +\ 
r{\cdot}<0{\mid}O_8^{J/\psi}(^3P_0){\mid}0>/m_c^2$.) 
\hrule
}
\label{FACTORES}

\begin{center}
\begin{tabular}{lcc}    \hline
NRQCD matrix element:  & $<0{\mid}O_8^{J/\psi}(^3S_1){\mid}0>$ & 
$M_r^{J/\psi}$ \\
\hline

Ref. \cite{mas} (GRV, $r=3$): &$(3.4{\pm}0.4){\times}10^{-3}$& 
$(6.0{\pm}1.2){\times}10^{-3}$ \\
Ref. \cite{kniehl} ($r=3.5$): & $(2.73{\pm}0.45){\times}10^{-3}$ & 
$(5.72{\pm}1.84){\times}10^{-3}$ \\
\hline
\end{tabular}
\end{center}
\end{table*}



\section{THEORETICAL UNCERTAINTIES}
	There is no space to review all the theoretical uncertainties
related to the extraction of matrix elements from charmonium
production which on the other hand have extensively been reviewed in
the literature (see for example \cite{schuler}). Let us only
mention:
\begin{itemize}
\item Ambiguities in the choice of energy scales, charm mass, PDF, ...
\item Partial breaking of the heavy-quark spin symmetry in charmonia 
systems, ...
\item Subleading $O(>v^7)$ contributions, higher-order ${\alpha}_s$
effects, ...
\item Possible kinematic effects near boundaries of phase space
\item Possible effects due to the surrounding medium breaking universality 
\end{itemize}
\par
Below we comment in some detail on the possible effects due to
the sensitivity to the kinematics of soft
gluon radiation in the long-distance evolution of the
$c\overline{c}$ pair into final charmonium \cite{beneke}.




\subsection{Finite width effects}
	Finite width effects can have consequences
as Ernstr\"{o}m, L\"{o}nnblad and V\"{a}nttinen discussed some time ago 
\cite{vanti}. Indeed,  
even a small energy difference (of order $Mv^2$) between the mass of final 
charmonium and the invariant mass of the created $c\overline{c}$ pair 
should have an effect on the (steep) momentum distribution. Nevertheless, 
the point is that the formation of an intermediate colored $c\overline{c}$ 
state does not preclude the possibility of an energy deficit and not 
necessarily an excess w.r.t. charmonium mass, the 
latter implying the emission of soft gluons as usually considered. In other 
words, the $c\overline{c}$ pair could get energy from the 
surrounding medium. 
\par
This viewpoint is in fact not surprising since in the color 
evaporation model \cite{frit} based on duality arguments, the 
cross section for charmonium production is written as an integral over the
$\hat{s}$ range $4m_c^2$ and $4m_D^2$ (i.e. the upper limit equal to open
charm production). However, charmonium production takes place in a $\lq\lq$hot" 
medium as Hoyer pointed out in his talk (for more
details see Ref. \cite{hoyer}), so sub-thresholds effects likely should
have an influence. In the improved color evaporation model this means
that the heavy quark pair can get energy from the surrounding color field
to give open charm production beneath $\hat{s}=4m_D^2$ \cite{vogt}. 
\par
Analogously in the COM, those dipole transitions occurring 
during the evolution into final particles should be
associated to absorption of soft gluons from the neighborhood as well.
For example a $c\overline{c}[^3S_1^{(8)}]$ pair may absorb two 
non-perturbative gluons
becoming a $O(v^2)$ Fock component of the $J/\psi$ state or may radiate
two soft gluons thus with nearly unity overlap with the $J/\psi$ wave function.
The conclusion is that on the average one should consider both 
possibilities, likely decreasing the net impact of the above-mentioned
effect on the $p_t$ distribution.

\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{file=tres_jpsi_14.eps,height=6.5cm,width=8.cm}}\end{center}
\caption{Our predictions for $J/\psi$ direct production at the LHC
according to the color-octet model implemented in PYTHIA for the CTEQ
PDF [10]. Dotted line: CSM; dashed line:
$^1S_0^{(8)}+^3P_J^{(8)}$ contributions; dot-dash line: $^3S_1^{(8)}$
contribution; solid line: all contributions. Notice that asymptotically
the $^3S_1^{(8)}$ contribution becomes dominant, likely
implying a transverse polarization of charmonium at high $p_t$.}
\end{figure}



\section{LHC PREDICTIONS}
As outlined in the previous section, the extraction of NRQCD matrix 
elements from charmonium
production is not yet fully satisfactory
due to several theoretical uncertainties. Still more analysis
has to be done to check the universality hypothesis allowing
comparisons between different production reactions. Polarization
of charmonium from gluon fragmentation at high $p_t$ might be decisive
to verify the validity of the COM \cite{beneke2}.\par 
Nevertheless, under the assumption of the validity of the COM
to explain charmonia production, long-distance matrix elements obtained 
from Tevatron data should parameterize those uncertainties to a large
extent. Therefore, in extrapolating to
LHC energies, predictions should be reliable if correctly
performed. In our case, this actually means running PYTHIA with the
same parameters and options (e.g. radiation on) as previously employed
in the fit to Tevatron experimental points.\par 
In Figure 2 we show
our predictions for prompt $J/\psi$ direct production at the LHC
using the CTEQ PDF. A caveat is in order, however. 
In changing the choice for the PDF there are noticeable changes on the 
production rate by about $50\%$ in certain regions of $p_t$ 
(see Ref.\cite{mas}). 

\par

\subsection{Muon cross sections from prompt $J/\psi$ production at
the LHC}
We have estimated the cross sections for events with
muons from prompt $J/\psi$'s, passing typical pseudorapidity
and transverse momentum cuts, foreseen in the first and
second level triggers for $B$ physics at the two LHC experiments
ATLAS and CMS. We shall denote as ${\mu}3$, ${\mu}6$ 
(${\mu}3{\mu}3$, ${\mu}6{\mu}3$) those events with at least one (two)
muon(s) whose $p_t$ is (are) larger than 3 GeV or 6 GeV, respectively. 
\par
From our modified version of PYTHIA with the COM implemented in, we
obtain 
\begin{eqnarray}
\sigma({\mu}3)\ =\  622 & nb & \nonumber \\
\sigma({\mu}6)\ =\ 52 & nb & \nonumber \\
\sigma({\mu}3{\mu3})\ =\  30 & nb & \nonumber \\ 
\sigma({\mu}6{\mu3})\ =\ 9 & nb & \nonumber 
\end{eqnarray}
\par
In all cases an additional pseudorapidity cut ${\mid}{\eta}{\mid}<2.5$
for muons was required on generated events.\par
Those numbers should be useful to evaluate an order-of-magnitude
estimate of background sources
for the $B$ physics programme at LHC experiments, especially regarding
the search for CP violation via the golden decay mode 
$B_d^0{\rightarrow}{J/\psi}K_s$.

\end{document}
 
