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\begin{document}
\title{Intrinsic Charm Contribution to High-$p_T$ $\psi$ 
Production at the Fermilab Tevatron
\\[2mm]} 
\author{Cong-Feng Qiao\footnote{JSPS Research Fellow. 
E-mail: qiao@theo.phys.sci.hiroshima-u.ac.jp}}
\address{Department of Physics, Faculty of Science,\\
Hiroshima University, Higashi-Hiroshima 739-8526, Japan}
\maketitle
%\date{\today}

\begin{abstract}
We revisit the direct production of $J/\psi(\psi')$ at 
large transverse momentum, $p_T \gg M_{J/\psi}$, at the 
Fermilab Tevatron. It is found that the sea parton  
participating processes dominate in the high-$p_T$ region 
within the framework of color-singlet model, which is 
not widely realized.
We hope this finding may be enlightening for further
investigations of the charmonium production mechanism.

\pacs{PACS numbers: 13.85.Ni, 12.38.Bx.}
\end{abstract}
%\vspace{-1.3cm}
\begin{multicols}{2}

Quarkonium production and decays have long been taken 
as an ideal means to investigate the nature of Quantum 
Chromodynamics(QCD) and other phenomena. Due to the 
approximately non-relativistic nature, the description 
of heavy quark and antiquark system stands as one of the 
simplest applications of QCD. The rich spectrum of its 
radial and orbital excitations provides a suitable play 
ground for testing QCD based models. The heavy, but not 
very heavy, quark mass enables one to get knowledge of 
both perturbative and nonperturbative QCD via investigating 
quarkonium production and decay natures. The clean signals of 
quarkonium leptonic decays make the experiment detection 
with a high precision, and therefore, quarkonia play an 
unique role in investigating other phenomena as well, 
e.g. in detecting the parton distribution, the QGP signal,
and even new physics. However, only with a theory 
which can precisely describe heavy quarkonium production 
and decays, may these advantages come true.

During the past decade, intrigued by the discovery of 
$\jp(\p')$ surplus production at high $p_T$ at the 
Tevatron \cite{cdf1,cdf2,cdf3}, our understanding 
on the natures of quarkonium production and decays has 
experienced dramatic changes. 

Conventionally, the so-called color-singlet model (CSM) 
was widely employed in the study of heavy quarkonium
production and decays \cite{t.a.degrand}. In CSM, 
it is assumed that the $Q\bar{Q}$ pair produced in a high 
energy collision will bind to form a given quarkonium state
only if the $Q\bar{Q}$ pair is created in color-singlet 
state with the same quantum numbers as the produced bound 
states; as well, in the quarkonium decays the annihilating 
$Q\bar{Q}$ pair will be in short distance and singlet with 
the same quantum numbers as its parent bound states.
It is assumed in CSM that the production amplitudes 
can be factorized into short distance and long 
distance parts. The short distance sector is perturbative QCD
applicable, while all the long distance nonperturbative effects 
are attributed to a single parameter, the wave function. 
That is, e.g., 
\ba
d\sigma(\psi_n + {\sc{X}}) = 
d\sigma (c\bar{c_1}(^3S_1) + {\sc{X}}) |R_{\psi_n}(0)|^2\;.
\ea
The wavefunctions can be either determined 
phenomenologically through experiment measurement of 
quarkonium leptonic decay rates, like
\ba
\Gamma(\psi_n \rightarrow l\bar{l})\approx \frac{4
\alpha^2} {9 m_Q^2} |R_{\psi_n}(0)|^2 \;,
\ea
or computed from potential models.

CSM provides a prescription for 
calculating not only the inclusive production rate 
of quarkonium states, but also their inclusive decay 
rates into light hadrons, leptons, and photons. 
Based on it many investigations had been carried
out in past more than two decades, and at
least qualitative descriptions of quarkonium decay
and production natures were achieved.
Nevertheless, color-singlet factorization is only an
ad hoc hypothesis. There are no general arguments to 
guarantee such a naive assumption to be held up 
to higher order radiative corrections. Even
the straightforward attempt to incorporate
relativistic corrections also met difficulties 
\cite{barbieri}. Although the difficulties 
color-singlet model met are well-known for 
many years and disagreements of theoretical predictions
with experiment data loomed up sometimes, 
what people did was amending it until the CDF
group released \cite{cdf1} the data on large-$p_T$ $J/\psi$
production collected in the 1992-1993 run.
The new data \cite{cdf2}, benefited from the 
advanced technology of vertex detector, were free of 
the large background from $B$ decays and put 
CSM in an awkward situation, as the 
data differ very much from the leading order(LO)   
CSM predictions in both normalization and $p_T$ 
scaling. 

In 1993, Braaten and Yuan \cite{braaten1} noticed 
that at sufficiently high $p_T$ the dominant
charmonium production mechanism is the production
of a parton with large transverse momentum followed by
its fragmenting into a charmonium state. After including
the fragmentation contribution, the prompt $J/\psi$
data can be explained within a small amount of error
\cite{cacciari,braaten2}, where the charm quark splitting 
into $\chi$ states and then feeding down to $J/\psi$ 
contributes overwhelmingly. Nevertheless, one can merely get 
a similar $p_T$ asymptotic behavior for $\psi'$ production
with the same scenario, and large discrepancy in normalization 
remained. This phenomenon was referred as the so-called 
"$\psi'$-surplus" production or "-anomaly". 

In 1997, a measurement of direct $J/\psi$ production 
exposed \cite{cdf3}, in which the higher excited states 
feeddown were stripped off. To one's surprise
the new experiment result excesses the CSM 
prediction by a factor of $\sim$ 30, the same as the 
situation in $\psi'$ production. Nowadays, the former 
"$\psi'$-surplus" problem turns to be the generic 
"$\psi$-surplus" problem terminologically.

A general factorization formalism \cite{nrqcd} developed
from the non-relativistic QCD(NRQCD) \cite{caswell}, 
which describes the inclusive heavy quarkonium production 
and decays, were established by Bodwin, Braaten and Lepage(BBL). 
NRQCD is formulated from first principles and the BBL approach 
allows relativistic and radiative corrections to be performed 
safely to any desired order. One of the striking 
advancements of the new development from CSM is that within 
the BBL framework the intermediate $Q\bar{Q}$ state, which 
subsequently evolves into quarkonium states nonperturbatively, 
can be in both color-singlet and -octet configurations. At 
first order in $v$, the relative velocity of heavy quark, 
BBL and CSM are coincident in describing the S-wave quarkonium 
production.

Based on the BBL formalism, Braaten and Fleming suggested to 
solve the $\psi'$ surplus production puzzle via color-octet 
mechanism(COM) \cite{fleming}. They proposed that the 
dominant $\psi'$ production at high $p_T$ is through the 
fragmentation of a gluon into a $c\bar{c}$ pair in 
color-octet configuration, which will evolve into 
$\psi'$ non-perturbatively. Indeed they gave a well-fitted 
curve to the data and from which the non-perturbative matrix 
element $<{\cal{O}}_8^{\psi'}(^3S_1)>$ was extracted with
a magnitude being consistent with the estimation from
NRQCD "velocity scaling rules". After their pioneer work, 
hundreds of investigations have been performed in order 
to find either the signatures of color-octet states or 
any implication of the new proposal to other phenomena 
\cite{rothstein}.

The present situation is that on one hand the COM 
stands as the most plausible approach, up to now,
in explaining the $\jp(\psi')$ production "anomaly"; 
on the other hand, this scenario encounters 
some difficulties in confronting with other phenomena 
\cite{rothstein}. The most striking crisis is the absence
of high-$p_T$ transversely polarized $J/\psi$ and $\psi'$  
at the Tevatron in the first measurement
from CDF \cite{cdf4}. According to NRQCD spin-symmetry,
and the prescription that the dominant charmonium production
mechanism at high $p_T$ is of a gluon splitting into a 
color-octet $^3S_1$ charm quark pair, such polarized states 
should appear \cite{wise}. Therefore, to what degree the COM 
plays the role in quarkonium production is still an open 
question to my understanding. To find 
distinctive signature of color-octet and eliminate the 
large errors remaining in the different fits for corresponding
matrix elements are currently urgent tasks in this research realm,
both theoretically and experimentally.

In order to overcome the difficulties COM met, 
people tried to attribute large amount of high-$p_T$ 
events to intrinsic transverse momentum of the 
interacting partons, suppose that the large uncertainties 
existed in the $k_t$-factorization are manageable and the 
$k_t$ would still manifest itself in not very small-$x$ 
\cite{hagler}. To be noted that in the $k_t$-factorization
formalism, the analyses suggest that the direct $J/\psi$
production is still dominated by color-octet contributions, 
but from $^1S_0^{(8)}$ and $^3P_J^{(8)}$, up to large 
transverse momenta of the order $p_T \le 20$ GeV.

Now that the difficulties for direct $J/\psi$ 
and its radial excitation $\psi'$ production at the 
Tevatron are the same within a small amount of error, as 
aforementioned, one may reasonably infer that the origins 
accounting for the large discrepancies between experimental
data and the color-singlet description for both states 
would be the same. On this premise our investigation 
in this work will be restricted to $J/\psi$ for simplicity. 
The results and conclusions are applicable to $\psi'$.

We notice  that within the framework of collinear factorization, 
processes initiated by the sea quark participating
interactions have been paid less attention in previous
calculations with CSM prescription. Ref. \cite{kniehl}
first claimed that the sea initiated processes were considered 
for the large-$p_T$ $J/\psi(\psi')$ production, where
the relative importance of constituent and sea parton
interacting processes was not distinguished.
The LO intrinsic(sea) charm interacting process  
was investigated in ref. \cite{saleev}, and found that
it contributes negligiblely to the large-$p_T$ $\psi$
production as expected, since CDF 
data indicates that the $d\hat{\sigma}/dp_T^2$ scaling 
favors fragmentation process behaving like $1/p_T^4$. 
Therefore, to see the sea quark contributions 
we need to consider possible processes beyond LO. 
Superficially, the intrinsic quark interacting
processes are negligible since the sea distribution
probabilities are pretty small comparing to those of
constituent partons. However, due to being
in high energy and at high, but not very high, $p_T$ region, 
the $\hat{s}/\hat{t}$, the ratio of Mandelstam variables 
of the squares of the total center-of-mass energy in 
$s$ and $t$-channels respectively, will suppress the 
constituent quark and gluon initiated processes
relative to the sea interacting ones. And in the meantime 
high energy and large $p_T$ enhance the sea quark densities 
inside the incident hadrons. To see this picture more clearly, 
let us have a close look at the fragmentation prescription 
for quarkonium production. Generally, quarkonium fragmentation 
production, $A\; +\; B \rightarrow \psi\; + \; {\sc X}$, can be 
expressed as
\bea
d\sigma (A\; &+&\; B \rightarrow H(p_T)\; +\; {\sc X}) =
\sum_{a,b,c} \int_0^1 dx_a f_{a/A}(x_a) \nonumber \\ 
&\times& \int_0^1 dx_b f_{b/B}(x_b) \int_0^1 dz
d\hat{\sigma}(a + b \rightarrow c(p_T/z) + {\sc X})
\nonumber\\
&\times& D_{c \rightarrow H}(z,\; \mu)\;,
\label{eq0}
\eea
where $c$ is the fragmenting parton, either a gluon or
a (anit)charm quark, and the sum runs over all possible
partons. $D(z, \mu)$ is the fragmentation
function and $z$ is the momentum fraction of the
the fragmenting parton carried by quarkonium state.
The evolution of the fragmentation function 
$D_{c \rightarrow H}(z,\; \mu)$ with scale $\mu$ 
in Eq.(\ref{eq0}) is accomplished by the utilization of 
Alterelli-Parisi(AP) equations
\bea
\mu \frac{\partial}{\partial \mu} D_{i\rightarrow \psi}
(z,\; \mu) = \sum_j \int_z^1 \frac{dy}{y} P_{ij}(z/y,\mu)
D_{j\rightarrow \psi}(y,\; \mu)\;,
\label{ev}
\eea
where the $P_{ij}$ are the splitting functions of a parton
$j$ into a parton $i$.

To show the importance of the sea quark interacting
processes in the fragmentation approach (\ref{eq0}), we do
a simple comparison, for example, of the hard-scattering
processes  
\bea
g + g \rightarrow C + \bar{C}
\label{eq1}
\eea
and
\bea
g + C(\bar{C}) \rightarrow g + C (\bar{C})\;,
\label{eq2}
\eea
where the $C$ and $\bar{C}$ stand for charm
and anticharm quarks, which are produced slightly 
off-shell and in high energy with large $p_T$. In LO
and massless limit, the differential cross 
sections for processes (\ref{eq1}) and (\ref{eq2}) 
are:
\bea
\frac{d\sigma_5}{d\hat{t}} (\hat{s},\hat{t}) =
\frac{\pi \alpha_s^2}{\hat{s}^2} \left\{
\frac{1}{6}\left(\frac{\hat{u}^2 + \hat{t}^2}
{\hat{u}\hat{t}}\right) - \frac{3}{8}\left(
\frac{\hat{u}^2 + \hat{t}^2}{\hat{s}^2}\right)\right\}
\label{eq3}
\eea
and
\bea
\frac{d\sigma_6}{d\hat{t}} (\hat{s},\hat{t}) =
\frac{\pi \alpha_s^2}{\hat{s}^2} \left\{
\left(\frac{\hat{u}^2 + \hat{s}^2}
{\hat{t}^2}\right) - \frac{4}{9}\left(
\frac{\hat{u}^2 + \hat{s}^2}{\hat{s}\hat{u}}\right)
\right\}\;,
\label{eq4}
\eea
respectively. Since we are interested in large-$p_T$ 
$J/\psi(\psi')$ hadroproduction at high energy, obviously
in a certain scope of phase space, the process (\ref{eq1})
is suppressed relative to process (\ref{eq2}) by the 
factor of $\hat{s}/\hat{t}$. However, this is just a 
schematic argument; strictly, we need to 
convolute the hard scattering cross section with 
parton distributions and fragmentation functions. 
Without losing qualitative correctness, for simplicity 
we do the comparison of the subsets 
$g + g \rightarrow C+\bar{C}$
and 
$g + C(\bar{C}) \rightarrow g + \;C(\bar{C})$ 
induced processes in $p\bar{p}$ collision by convoluting
the hard part with only the parton densities, or in other words,
integrating out the common fragmentation probability 
$\int D_C^{\psi}$.
Direct numerical calculation shows that the cross section
induced by hard process (\ref{eq2}) overtakes
what induced by (\ref{eq1}) by a factor of
7 at $p_T = 15$ GeV.

For completeness we consider processes
\bea
q_i(\bar{q_i}) + C(\bar{C}) \rightarrow q _i(\bar{q_i})
+\; C(\bar{C})\;,
\label{eq5}
\eea
in our numerical calculation as well. Here, the $q_i(\bar{q_i})$ 
represents partons of both constituent and sea 
quarks(antiquarks) of the colliding nucleons. 
Practical exercise, similar as performed in 
preceding paragraph, shows that this kind of hard interacting 
processes also contributes more to $J/\psi$ 
large-$p_T$ production than via subprocess 
$g + g \rightarrow C\;+ \bar{C}$. And, to be noted that 
the latter was taken to be the dominant process in many of 
previous analyses within CSM. 

In the numerical calculation of the differential cross section, 
we need to choose a set of parton distributions. 
In this work we take CTEQ5M \cite{cteq} 
parameterization as our input. We have also tried another 
set of parton distributions, the MRST99 \cite{mrst}, and found 
that different parton distribution functions give similar 
results within tens of percent. That means the conclusions given
in this paper will not be spoiled by taking a different set of 
parton distributions for convolution.

\begin{figure}[tbh]
\begin{center}
%\vskip -4cm
\psfig{file=Graph1.EPS,width=8.5cm,height=5cm,clip=0}
\end{center}
%\vskip 6cm
\caption[bt]{Fragmentation Contributions to the 
differential cross section for direct $J/\psi$ production 
at the Fermilab Tevatron, compared with the CDF experiment 
data read from \cite{cdf3}. The upper curve corresponds to 
$\mu_R = \mu_F = \mu_{\rm frag} = p_T/2$, 
while the lower curve to 
$\mu_R = \mu_F= \mu_{\rm frag}  = 2 p_T$.}
\label{graph2}
\end{figure}    

In figure 1, various fragmentation contributions 
to direct $\jp$ production are shown, and the sum of
them is confronted to the CDF data. Since we merely 
want to sketch the importance of sea parton contributions 
to the large-$p_T$ charmonium production, the lowest order 
hard scattering cross sections and fragmentation function 
are employed. We also neglect the effect from non-diagonal 
splitting function, $P_{cg}$, in Eq. (\ref{ev}), which, 
as pointed out in Ref. \cite{cacciari,braaten2}, might be as 
large as a factor of 1.5 to the charm fragmentation process. 
We estimate the uncertainties of higher order
corrections by varying the scale, 
given that a lower value of scale will account for, 
in a certain degree, the contributions from higher order 
corrections and non-diagonal splitting functions. 
The upper and lower lines in the figure are 
obtained by varying the scales of factorization, 
renormalization and fragmentation. The upper 
curve corresponds to $\mu_R = \mu_F = \mu_{\rm frag} = 
p_T/2$, while the lower curve to $\mu_R = \mu_F
= \mu_{\rm frag}  = 2 p_T$. In drawing the diagram we 
use the fragmentation function given in Ref.\cite{braaten1}
and values quoted thereof ($R_0^2 = 0.8 {\rm GeV}^3$, 
$\alpha_s = 0.26$, $m_c = 1.5$ GeV). The symmetry 
of a sea quark and its antiquark in hard scattering
and fragmentation precesses is invoked. We limit 
our calculation in the $p_T \ge 10$ GeV region, where 
contributions from lowest order parton fusion processes
can be safely neglected,

From figure 1 we see that the discrepancy between direct
$\jp$ production data and the CSM prediction, the sum of
different processes, is less than an order in optimal case
(the upper solid line). The finding that the intrinsic charm 
contributes to high $p_T$ charmonium production dominantly, 
shown in the figure, within CSM looks surprise, whereas, 
in fact it is not really an unthinkable thing. Similar cases 
exist in some other quarkonium high energy production processes 
as well. For example, in photon-photon collision, 
quarkonium production via resolved processes is not always minor 
to the direct one. In addition, it should be noted that although 
the charm sea originates from the high order constituent quark 
and gluon interactions, the naive NLO QCD result for 
$J/\psi(\psi')$ hadroproduction can not simply substitute or cover 
the one from sea parton interacting processes. Because in the latter 
case we are considering 
the production via fragmentation mechanism, which is already
in high order in $\alpha_s$.

To conclude, in this work we study the relative 
importance of the sea quark initiated processes relative
to the constituent parton initiated ones for large-$p_T$ 
$\jp (\psi')$ production within the CSM. 
It is found that the former may contribute more than
the latter by a factor of six. We notice that to many people
within the community the constituent parton initiated processes
are still taken to be the dominant ones for the high-$p_T$ 
$\jp (\psi')$ production from CSM calculations; we 
hope this work may elucidate it somehow.
With including the new production scheme, the total cross section
from CSM prediction fall off the experiment data by less 
than an order at most. Therefore, to 
explain the CDF data, COM is still dominant and indispensable. 
Nevertheless, the increase of the contribution from CSM 
means a shrinkage of the contribution from COM, which might be 
as large as twenty percent. Finally, we would like to
point out that since the intrinsic charm is taken to be massless 
in our calculation, some uncertainties from this measure, 
as large as a factor of two, exist in our analysis. 

This work was supported by the Grant-in-Aid of JSPS 
committee. The author would like to thank  
E. Braaten, S. Brodsky, M. Cacciari, B. Kniehl and
P. Hoyer for communications and K. Hagiwara for helpful 
discussion.

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\end{multicols}
\end{document}







