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\begin{document}
\title{Detect $\Delta G$ at BNL-RHIC via Double Quarkonium 
Production\\[2mm]} 
\author{Jiro Kodaira
\footnote{E-mail: kodaira@theo.phys.sci.hiroshima-u.ac.jp}
\, and \, Cong-Feng Qiao\footnote{
E-mail: qiao@theo.phys.sci.hiroshima-u.ac.jp}}
\address{Department of Physics, Hiroshima University\\
Higashi-Hiroshima 739-8526, Japan}
\maketitle

\begin{abstract}
The double spin asymmetry for the $J/\psi$ pair productions
in the polarized $p-p$ collisions at RHIC energy is investigated.
Our study shows that the asymmetry measurement at RHIC-SPIN
experiments is pretty realistic to extract the
polarized gluon distribution function $\Delta G (x)$.
 
\pacs{PACS numbers: 13.88.+e, 21.10.Hw, 13.60.Le}

\end{abstract}

\begin{multicols}{2}

Intrigued by the measurement of the European Muon Collaboration
in polarized-target experiment \cite{emc}, an enormous amount of
researches have been done on the nucleon spin structure
both experimentally and theoretically. 
The unpolarized deep inelastic scattering (DIS) experiments
show that the gluon shares a large portion of the parent proton's momentum.
However, how the gluons share the spin of proton is still an open question.
In the conventional DIS process,
the gluon contributions are the QCD (Quantum Chromodynamics) higher
order effects and we can not avoid some ambiguities coming from {\it e.g.}
the factorization scheme dependence, to determine the polarized
gluon distributions from the DIS experimental data.
Although there are some efforts \cite{GS,GRVO} to parameterize them,
it seems difficult to obtain clear understanding of the role of gluons. 

Therefore it is desirable to study other processes than DIS
to expose the behavior of gluon in the polarized hadrons.
It is now expected that the polarized proton-proton collisions
(RHIC-Spin) at BNL relativistic heavy-ion collider RHIC 
will provide copious experimental data to unveil the 
polarized parton distributions. 
Since the major emphasis and strength of RHIC-Spin is to measure the gluon 
polarization, it is important and interesting to investigate various 
processes which are attainable experimentally to this aim. 

Up to now, three main kinds of means in measuring the gluon polarization 
are proposed, which are thought to be feasible at RHIC technically. 
Those are
%
\begin{itemize}
\item{High-$p_T$ Prompt Photon Production}
\item{Jet production}
\item{Heavy Flavor Production}
\end{itemize}
%
There are some advantages and drawbacks in each of these schemes.
For detailed discussions, see recent review paper in Ref. \cite{bssv}. 
The prompt photon production with polarized beams at RHIC is a 
promising method to measure gluon polarization, but we must cooperate 
with the polarized DIS results on quarks.
For the jet and heavy flavor production, the tremendous effects of 
hadronization and higher order corrections are not yet well controlled 
theoretically. 
Therefore, to develop more practical ways for running RHIC spin project in 
measuring the gluon polarization is one of the urgent
theoretical tasks today.

Quarkonium production and decays have long been taken as an ideal 
means to investigate the nature of 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 very clean signals of quarkonium leptonic decays make 
the experimental detection with a high precision, and therefore 
quarknoium plays a crucial role in investigating other phenomena as well,
{\it e.g.} detecting the parton distribution, the QGP signal, 
and even new physics. 

Nevertheless, it is necessary to point out that although
the heavy quarkonium physics has been investigated for
more than twenty years, theoretical description for 
quarkonium production is still not mature. 
In explaining the high-$p_T$ $J/\psi$ surplus production discovered by 
CDF group \cite{cdf1,cdf2,cdf3} at the Fermilab Tevatron, 
the color-octet scenario \cite{fleming} was proposed based on 
a novel effective theory, the non-relativistic QCD(NRQCD) \cite{nrqcd}. 
The appearing of the new quarkonium production mechanism
brought deep effects on the quarkonium physics.
Having achieved the first-step success in explaining the CDF data, 
the color-octet mechanism also encounters difficulties in confronting 
other phenomena \cite{rothstein}. 
Due to the recent discovery \cite{qcf1}, the
extent of importance of color-octet contributions in
Charmonium production remains to be unfixed.
Because of this and as well the large QCD and relativistic corrections
especially in Charmonium system, 
to obtain new insights through the quarknoium production, 
one must be careful on these respects.

During the past years, a series of efforts
have been made on detecting polarized parton distributions
through the quarkonium production process \cite{bs,cp,dr,mty,jk}. 
Unfortunately, most of the previous investigations are not directly
applied to the RHIC physics and are spoiled by
the uncertainties aforementioned.
In this letter, we want to show that the double heavy quarkonium 
production in the polarized proton-proton collision would provide an 
ideal means of detecting the polarized gluon distributions, and which 
may at least play a supplemental role to the presently proposed 
program at RHIC to this end. 
It should be stressed that the double quarkonium production has
several advantages in reducing the theoretical uncertainties
mentioned above.
(1) By considering double production, the relativistic
corrections and color-octet uncertainties are reasonable highly
suppressed, especially without or with a lower  
transverse momentum cut \cite{qcf2}. 
(2) The total contribution from the higher excited states 
are also doubly suppressed. 
(3)The higher order QCD correction can be
well controlled by applying a suitable $p_T$ cut for the 
Charmonium system. 
(4) Since the prevailing partonic process is the gluon-gluon 
fusion into double quarkoniums, it stands as very sensitive method 
in measuring the gluon polarization. 
In the RHIC energy region, and for the $p-p$ collision, 
the $q \bar{q}$ initiated process is negligible.
Therefore, we will restrict ourselves to the gluon-gluon channel. 

In fact, we found that there were discussions of detecting the gluon 
polarization by means of double $J/\psi$ production in literatures \cite{bj,gm}. 
However, their main emphasis were not on RHIC physics 
and the analytical expressions for the polarized process were not given,
since their expressions are too long to be presented.
So, this work should be meaningful both phenomenologically and theoretically.

\vskip 2mm
\begin{figure}%[tbh]
\begin{center}
\psfig{file=feynman.ps, bbllx=70pt,bblly=280pt,bburx=500pt,
bbury=672pt,width=8cm,height=6cm,clip=0}
\end{center}
\caption[]{Typical Feynman diagrams for $g\, + \,g \rightarrow
J/\psi + J/\psi$. The topological groups (A) and (B)  
show up in the photon-photon to double charmonium production
process. The group (C) is necessary for guaranteeing 
the gauge invariance.}
\end{figure}

At the partonic level, the concerned process is similar 
to the photon-photon scattering to double quarkonium production \cite{qcf3}. 
Because of the non-abelian nature of QCD, 
here we should consider more diagrams as show in Figure 1. 
When calculating the helicity-dependent matrix elements, one
must project onto definite helicity states for the initial gluons.
In our calculation, this is achieved by taking the ghost free 
expression for the gluon state with helicity $\lambda = \pm$ ,
%
\bea
\epsilon_{\mu}(k_1,\,\lambda)\;\epsilon^\star_{\nu}(k_1,\,\lambda)
&=&\frac{1}{2}\left[-g_{\mu\nu} + \frac{k_{1\mu} k_{2\nu} + k_{1\nu} k_{2\mu}}
{k_1 \cdot k_2}\right. \nonumber \\
&+& \left. i \,\lambda\;
\epsilon_{\mu\nu\alpha\beta}\frac{k_{1\alpha} k_{2\beta}}
{k_1 \cdot k_2}\right]
\; .
\label{eq0}
\eea
%

The measurable double spin asymmetry $A$ for $J/\psi$ pair
production is defined as,
%
\bea
A &=& \frac{d \sigma (p_+ p_+ 
\rightarrow J/\psi J/\psi) - d \sigma (p_+ p_- 
\rightarrow J/\psi J/\psi)}{d \sigma (p_+ p_+ 
\rightarrow J/\psi J/\psi) + d \sigma (p_+ p_- 
\rightarrow J/\psi J/\psi)}\nonumber \\
&=& \frac{E d \Delta\sigma/d^3p}{E d \sigma/d^3p}\; ,
\label{eq1}
\eea
%
where $p_+$ and $p_-$ denote the helicity projection of
the incident protons being positive and negative, respectively. 
Here and in the following, the parity conservation is taken for granted.
In terms of the gluon densities and the partonic cross sections,
this asymmetry reads,
%
\be A 
     =  \frac{ \int dx_1 dx_2  d \Delta \hat{\sigma}
             \Delta G(x_1, Q^2 ) \Delta G(x_2, Q^2 )}
         {\int dx_1 dx_2  d\hat{\sigma} G(x_1, Q^2 ) G(x_2, Q^2)} \; ,
\ee
%
where $\Delta G(x, Q^2) = G_+ (x, Q^2 ) - G_- (x, Q^2)$
and $G(x, Q^2) = G_+ (x, Q^2) + G_- (x, Q^2)$
are the polarized and unpolarized gluon distributions defined at the
scale $Q^2$. The unpolarized (polarized) partonic cross section
$\hat{\sigma}$ ($\Delta \hat{\sigma}$) is defined as,
%
\be 
\hat{\sigma} = \frac{1}{4}\ \sum_{\lambda,\lambda'}\ \hat{\sigma}\
(\lambda,\;\lambda'),\;
\Delta\hat{\sigma} = \frac{1}{4}\ \sum_{\lambda,\lambda'}\ 
\lambda\lambda' \ \hat{\sigma} \ (\lambda,\;\lambda').
\ee
%
The polarized partonic differential cross section is given by,
%
\bea
&&\frac{d\Delta \hat{\sigma}}{dt}=\frac{-16 \alpha_s^4 \pi |R(0)|^4}
{81 s^8 (m^2 - t)^4 (m^2 - u)^4}
\times \Bigl[ \ 2744 m^{24}\nonumber\\  
&-& 15240 m^{22} (t + u)\nonumber\\
&+& m^{20} (32110 t^2 + 90076 t u + 32110 u^2) \nonumber\\ 
&-& 16 m^{18} (2025 t^3 + 12673 t^2 u + 12673 t u^2 + 2025 u^3) \nonumber\\ 
&+& 2 t^4 u^4 (349 t^4 - 908 t^3 u + 1374 t^2 u^2 - 908 t u^3 + 349 u^4) 
\nonumber\\
&+& 4 m^{16} (3903 t^4 + 57292 t^3 u + 117766 t^2 u^2 + 57292 t u^3 \nonumber\\ 
&+& 3903 u^4) - 4 m^{14} (510 t^5 + 36713 t^4 u + 135685 t^3 u^2 \nonumber\\
&+& 135685 t^2 u^3 + 36713 t u^4 + 510 u^5)\nonumber\\
&+& m^{12} (-1461 t^6 + 58600 t^5 u + 364313 t^4 u^2 \nonumber\\
&+& 594840 t^3 u^3 + 364313 t^2 u^4 + 58600 t u^5 - 1461 u^6) \nonumber\\
&+& 4 m^2 t^2 u^2 (9 t^7 - 505 t^6 u + 44 t^5 u^2 - 556 t^4 u^3\nonumber\\ 
&-& 556 t^3 u^4 + 44 t^2 u^5 - 505 t u^6 + 9 u^7) \nonumber\\
&+& 2 m^{10} (381 t^7 - 7111 t^6 u - 83783 t^5 u^2 - 180639 t^4 u^3\nonumber\\ 
&-& 180639 t^3 u^4 - 83783 t^2 u^5 - 7111 t u^6 + 381 u^7)
\label{eq2} \\
&+& m^8 (-79 t^8 + 1272 t^7 u + 54526 t^6 u^2 + 156224 t^5 u^3 \nonumber\\ 
&+& 163850 t^4 u^4 + 156224 t^3 u^5 \nonumber\\
&+& 54526 t^2 u^6 + 1272 t u^7  - 79 u^8) \nonumber\\
&+& m^4 t u (-36 t^8 + 1471 t^7 u + 9764 t^6 u^2 + 12863 t^5 u^3 \nonumber\\ 
&+& 7196 t^4 u^4 + 12863 t^3 u^5 + 9764 t^2 u^6 \nonumber\\
&+& 1471 t u^7 - 36 u^8) \nonumber\\
&-& 2 m^6 (2 t^9 + 17 t^8 u + 5151 t^7 u^2 + 25947 t^6 u^3\nonumber\\
&+& 24439 t^5 u^4 + 24439 t^4 u^5 + 25947 t^3 u^6\nonumber\\
&+& 5151 t^2 u^7 + 17 t u^8 + 2 u^9) \ \Bigr] \; ,\nonumber
\eea
%
where $R(0)$ is the wave function of $J/\psi$ at the origin
and $m$ is the rest mass of $J/\psi$;\  $s,\,t$ and $u$ are the 
Mandelstam variables for the partonic system.
The expression (\ref{eq2}) is the first analytical result 
obtained in this work for the polarized process. 
Our calculation for the unpolarized partonic differential cross section
$\hat{\sigma}$ in (4) reproduces the results
in Refs. \cite{qcf2,hm}.

Now let us investigate the asymmetry at the RHIC-Spin experiment
at $\sqrt{s} = 500$ GeV which is the projected highest RHIC energy. 
In our numerical calculations,
the scale $Q^2$ of the parton distribution function and
the strong coupling constant is taken
to be the transverse momentum of $J/\psi$ for the $p_T$ distributions.
Whereas, in the calculation of the angular 
distribution of the spin asymmetries and the integrated cross sections,
the scale is taken to be $Q^2 = m^2$.
The nonrelativistic relation $m = 2\ m_c$, with
$m_c = 1.5$ GeV, is used and $|R(0)|^2\ = \ 0.8\ \rm{GeV}^3$.

\begin{figure}[tbh]
\begin{center}
\epsfig{file=Graph1.eps,bbllx=30pt,bblly=710pt,bburx=330pt, 
bbury=912pt,width=8cm,height=6cm,clip=0}
\end{center}
\caption[]{Spin asymmetry versus transverse momentum for different 
set of GS polarized parton distributions at colliding energy 
$\sqrt{s} = 500$ GeV.}
\label{graph1}
\end{figure}

We plot in Figure 2 the double spin asymmetry versus the
transverse momentum of $J/\psi$ with respect to the proton 
beams for the different set of 
the Gehrmann and Stirling parameterizations \cite{GS}.
Figure 3 shows the angular distribution of the asymmetry
in the parton center-of-mass frame.
For the $J/\psi$ pair productions, it may be easy to reconstruct
the parton center-of-mass system.
For consistency with the polarized distributions used
in this work, GS parameterization \cite{GS},
the unpolarized gluon distribution used in our prediction for 
the asymmetry is the MRST parameterization \cite{mrst}.
From these results, one can see
that the asymmetries obtained with these parameterization
are rather different from each other. 
By combining the $p_T$ distribution and the angular distribution,
we will be able to discriminate between various parameterizations
for the polarized gluon.

\begin{figure}[tbh]
\begin{center}
\epsfig{file=Graph2.eps,bbllx=20pt,bblly=1024pt,bburx=325pt,bbury=1230pt,
width=8cm,height=6cm,clip=0}
\end{center}
\caption[]{Angular differential asymmetry distribution of the 
$J/\psi$ pair in the parton center-of mass frame at RHIC with
colliding energy $\sqrt{s}$ = 500 GeV.}
\label{graph2}
\end{figure}    

However, it should be noted that to observe the spin asymmetry both in 
$p_T$ and angular distributions, a very high luminosity is required.
To estimate the event rate for the $J/\psi$ pair productions, 
we calculate the total cross section with
different parton parameterizations and results are summarized in table I. 
The discrepancies among  

\vskip 6mm
{\small{TABLE I. Total cross sections for $J/\psi$ pair production
at RHIC with $\sqrt{s} = 500$ GeV, evaluated with different
parton distributions.}}
\begin{center}
\begin{tabular}{ccc}\hline\hline
\rule[-1.2ex]{0mm}{4ex}&\hspace{-0.6cm} 
$\sigma^{tot}_{\mu^+\mu^-}$ \hspace{0.6cm}&
$\sigma_{\mu^+\mu^-}(|p_T|>1 \;\mbox{GeV})$ \\ \hline
\rule[-1.2ex]{0mm}{4ex} CTEQ5L\cite{cteq} 
\hspace{0.6cm} & $11.8 \; \mbox{pb}$ \hspace{0.6cm}
& $7.3\; \mbox{pb}$ \\ \hline
\rule[-1.2ex]{0mm}{4ex} MRST \cite{mrst}\hspace{0.6cm} & $6.5 
\; \mbox{pb}$\hspace{0.6cm} 
& $4.3\; \mbox{pb}$ \\ \hline
\rule[-1.2ex]{0mm}{4ex} GRV \cite{grv}
\hspace{0.6cm} & $7.4 \; \mbox{pb}$ \hspace{0.6cm}
& $4.7\; \mbox{pb}$ \\ \hline\hline
\end{tabular}
\end{center}
\vskip 5mm

\noindent
these predictions are not very large as expected. 
Here, the notation $\sigma_{\mu^+\mu^-}$ means that the branching ratio of 
$B(\psi \rightarrow {\mu^+\mu^-}) =  0.0588$ as the practical 
measuring mode to reconstruct the charmonium state, is included.
From the predicted cross sections, we see
that with the integrated luminosity of $800\;\rm{pb}^{-1}$ 
in the next run of RHIC, there will be thousands of $J/\psi$
pair events detected, which can certainly give
us some information on the gluon polarization within the hadron.

To conclude, in this work we have shown that the the $J/\psi$ 
pair production at RHIC may stand as independent means in
observing the gluon spin distributions within the hadron.
The large mass of heavy quark guarantees that perturbative
calculation is applicable to this process; the asymmetry, 
rather than cross sections, eliminates large amount of
uncertainties which come from the non-perturbative hadronization.
Since we are interested in the contribution from low $p_T$ region,
which dominates the process, our treatment is reliable
by neglecting the relative color-octet process.
In the high $p_T$ region, there might be a possibility that
the color-octet mechanism gives some contributions.
However, this issue is beyond the scope of this paper and
a detailed study will be presented elsewhere.
The effects from higher excited states and 
higher order QCD corrections will be suppressed by the
dual production of $J/\psi$ and measuring the asymmetry. 
We have also discussed the feasibility of the observation of
the double spin asymmetry for the $J/\psi$ pair productions:
for the time being, the accumulated data at RHIC detectors are far
from enough to analyze this process
for the purpose of measuring the gluon helicity distributions:
in the next run, that is to say with colliding energy of 500 GeV
and accumulated luminosity 800 $\rm{pb}^{-1}$, 
the $J/\psi$ pair events could be detected
and the gluon polarization could be measured. 
With the expected upgrade of RHIC in the near future, the concerned 
process would show up in chasing the goal
of uncovering the nucleon spin structures. 

The work of J.K. was supported in part by the 
Monbu-kagaku-sho Grant-in-Aid for Scientific Research 
No.9. The work of C-F.Q. was supported by the 
Grant-in-Aid of JSPS committee. Authors would like thank
M.G. Perdekamp and K. Hagiwara for helpful comments.

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\end{multicols}
\end{document}



