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\title{On the relation between 
 ${\cal A}_{LL}^{\pi^0} / {\cal A}_{LL}^{jet}$ and Gluon Polarization}

\author{NS}

\maketitle 

\abstract{
}


\section{Introduction}
As we know from many experimental results including {\sc Phenix}
and {\sc Star}, the processes 
\begin{eqnarray}
   p+p &\rightarrow &\pi^0 + X  \nonumber \\
   p+p &\rightarrow & jet~ + X  \nonumber 
\end{eqnarray}
are described by the pQCD very well. Unpolarized cross sections are 
well decribed by mixture of $gg$, $gq$ and $qq$ scatterings. 
Therefore the processes are sensitive to the gluon 
contents in the proton and can probe the polarized gluon densities
when done in polarized $pp$ collisions. 

We have estimated the bound on the gluon polarization 
$\Delta g(x)/g(x)$ from the ${\cal A}_{LL}$ for neutral pion production
in $pp$ collisions at $\sqrt{s}=200$~GeV. 

According to the pQCD picture, the process is 
the mixture of the $gg$, $qg$ and $qq$ scatterings. In the lower 
$p_T$ region, the process is dominated by the $gg$ and $gq$ scattering,
and the $A_{LL}$ becomes quadratic in $\Delta g(x)/g(x)$;
\begin{equation}
 A_{LL} \sim
 \alpha \left[ \frac{\Delta g(x)}{g(x)} \right]^2 
 + \beta \left[ \frac{\Delta g(x)}{g(x)}\right] +\gamma,
\end{equation} 
where the coefficients $\alpha, \beta,$ and $\gamma$ reflect
the relative fraction of the $gg$, $gq$, and $qq$ contributions
and the partonic asymmetries $a_{LL}$ for these processes. 

To be more precise, 
\begin{equation} 
{\cal A}_{LL} = 
\frac{ \int_{\Omega} 
\Delta g(x_1) \Delta g(x_2) \frac{d \Delta \sigma}{dt}(gg \rightarrow gg) 
D_{\pi^0/g}(z) + ... d\Omega}{Ed^3 \sigma / dp^3}
\end{equation}
By applying the mean-value theorem for multiple integration, above expression 
can be written as 
\begin{equation}
{\cal A}_{LL} = C \cdot \Delta g(\xi)^2 +... = \alpha \dot 
\left( \frac{\Delta g(\xi)}{g(\xi)} \right)^2 +...
\end{equation} 
Similar expression can be obtained for $\beta$ and $\gamma$, whose main 
contributions are $gq\rightarrow gq$ and $qq\rightarrow qq$, respectively. 

To summarize the approximation to achieve Equation (1) , 
\begin{itemize} 
\item take the limit of $x_1 = x_2$, which is valid in central rapidity 
\item apply mean-value theorem for multiple-integration 
\item assume mean value for the first term ($x_1 \rightarrow \xi_1$) 
  is identical to the one for the second term ($x_2 \rightarrow \xi_2$),
  $\xi_1 = \xi_2 \equiv \xi$. 
\end{itemize} 
Since we measure the asymmetries as a function of $p_T$ the 
coefficients are also functions of $p_T$. To be explicit, 
\begin{equation} 
{\cal A}_{LL}(p_T) = \alpha (p_T) \left[ \frac{\Delta g (\xi)}{g (\xi)} \right]^2
                   + \beta (p_T)  \frac{\Delta g (\xi)}{g (\xi)}
                   + \gamma (p_T) 
\label{E:Asyms}
\end{equation}

\subsection{1st Approach: Assume Relevant $x$} 
If we can know the values of relevant $x$, then we can determine the coefficients 
by using the asymmetries with three or mode models on the gluon polarization. 
Since we have $\Delta g(x) = +g(x)$, standard, and $\Delta g(x) = +g(x)$ at least, 
we can write asymmetries in a partiular $p_T$ bin as 
\begin{eqnarray}
\left(
\begin{array}{l} 
{\cal A}_{LL}^{\rm max}(p_T) \\
{\cal A}_{LL}^{\rm std}(p_T) \\
{\cal A}_{LL}^{\rm min}(p_T) \\
\end{array}
\right)
=
\left(
\begin{array}{llc} 
\left[\frac{\Delta g(\xi)}{g(\xi)}^{\rm max}\right]^2 &
\frac{\Delta g(\xi)}{g(\xi)}^{\rm max} & 1\\
\left[\frac{\Delta g(\xi)}{g(\xi)}^{\rm std}\right]^2 &
\frac{\Delta g(\xi)}{g(\xi)}^{\rm std} & 1\\
\left[\frac{\Delta g(\xi)}{g(\xi)}^{\rm min}\right]^2 &
\frac{\Delta g(\xi)}{g(\xi)}^{\rm min} & 1\\
\end{array}
\right)
\left(
\begin{array}{l} 
\alpha(p_T)\\
\beta(p_T)\\
\gamma(p_T)\\
\end{array}
\right)
\end{eqnarray}
Here we know the values of $\xi$ thus $\Delta g(\xi)/g(\xi)$, and 
asymmetries ${\cal A}_{LL}(p_T)$ from NLO calculations.  
The coefficnets $\alpha(p_T)$, $\beta(p_T)$ and $\gamma(p_T)$, an be obtained 
by solving this simultaneous equations as follows: 
\begin{eqnarray}
\left(
\begin{array}{l} 
\alpha(p_T)\\
\beta(p_T)\\
\gamma(p_T)\\
\end{array}
\right)
=
\left(
\begin{array}{llc} 
\left[\frac{\Delta g(\xi)}{g(\xi)}^{\rm max}\right]^2 &
\frac{\Delta g(\xi)}{g(\xi)}^{\rm max} & 1\\
\left[\frac{\Delta g(\xi)}{g(\xi)}^{\rm std}\right]^2 &
\frac{\Delta g(\xi)}{g(\xi)}^{\rm std} & 1\\
\left[\frac{\Delta g(\xi)}{g(\xi)}^{\rm min}\right]^2 &
\frac{\Delta g(\xi)}{g(\xi)}^{\rm min} & 1\\
\end{array}
\right)^{-1}
\left(
\begin{array}{l} 
{\cal A}_{LL}^{\rm max}(p_T) \\
{\cal A}_{LL}^{\rm std}(p_T) \\
{\cal A}_{LL}^{\rm min}(p_T) \\
\end{array}
\right)
\end{eqnarray}
Once $(\alpha(p_T),\beta(p_T),\gamma(p_T))$ is obtained, 
we can apply Equation~\ref{E:Asyms} to determine the 
gluon polarization $\Delta g(\xi)/g(\xi)$ and its error directly from the 
asymmetry ${\cal A}_{LL}$ and its error. 
Because of the quadratic nature of Equation~\ref{E:Asyms}, 
experimental results on ${\cal A}_{LL}$ could provide two possible ranges of 
gluon polarization. This situation is unchanged for 
jet production, since it is also a mixture of $gg$, $gq$ 
and $qq$ scatterings. 

Figure~\ref{F:Run-5} shows Run-5 projection on the gluon polarization 
measurements. The evaluation of the Run-5 sensitivity is performed 
assuming GRSV-std. 
One can see that data center of
Run-5 projetions are deviated from the GRSV-std. Actually
we used the GRSV-std ${\cal A}_{LL}^{\pi^0}$ as an input, but the 
output $\Delta g(x)/g(x)$ was
systematically higher than GRSV-std. 

Possible origins of this deviation are 
\begin{itemize} 
  \item $x_1$=$x_2$ assumption is not so good, 
  \item choice of $\xi$ are not good in followings 
    \begin{itemize} 
       \item currently $\xi$ is chosen simply as 
             $x_T$ for jet prodution and $x_T/0.4$ simply 
             reflecting average value of energy fration 
             carried by $\pi^0$ 
       \item mean-value for $x$, $\xi$ for the first term 
             and $\xi$ for the seond term can be different, 
    \end{itemize} 
  \item quadratic formula is too simplistic and completely wrong. 
\end{itemize} 

\subsection{2nd Approach: No Assumptions on $\xi$} 
In general, we do not know the value of $\xi$, thus the
value of $\Delta g(\xi)/g(\xi)$, {\it apriori}. 
One natural choice of $\xi$ is the average value of $x$
\begin{equation}
\langle x \rangle = \frac{\int_{\Omega(p_T-{\rm bin})} x \cdot 
 \sigma(x,s,t,u) d\Omega}{\int_{\Omega(p_T-{\rm bin})}  
 \sigma(x,s,t,u) d\Omega} 
\end{equation}
However, one can easily imagine that $\langle x \rangle$ 
would depnd on the $x$-dependence of $\sigma(x,...)$ originated 
in parton distribution functions. For example, when we 
calculate the average $x$ for polarized cross section, which 
can be obtainbed by replacing the unpolarzed PDFs and cross sections
to the polarized ones, $\langle x \rangle$ can be different. 

Another way to estimate the relevant $x$ is to use the 
relation between asymmetries and gluon polarization. 
In order to relate the cross section asymmetries with
the gluon polarzation, we calculated the asymmetries with 
following gluon polarizations: 
\begin{equation} 
\Delta g_i(x) = \Delta g_{\rm std}(x) + 0.1 i \delta (\Delta g(x)). 
\end{equation} 
Resulted asymmetries are described by quadratic function in $i$
very well, which is natural since the pion production is 
mixture of $gg$, $gq$ and $qq$ scattering. 
\begin{equation}
{\cal A}_{LL}_{i}^{\pi^0}(p_T) 
   = a(p_T) \cdot i^2 + b(p_T) \cdot i + c(p_T)
\end{equation}


If we know the value of $\xi$, we should be able to relate the
coeeficients $a(p_T),b(p_T),$ and $c(p_T)$ to the $\alpha(p_T)$,
$\beta(p_T)$, and $\gamma(p_T)$
\begin{equation} 
\alpha(p_T)\left[ \frac{\Delta g(\xi)}{g(\xi)} +0.1 i \cdot 
                  \delta \frac{\Delta g (\xi)}{g(\xi)} \right]^2 
+\beta(p_T) \left[ \frac{\Delta g(\xi)}{g(\xi)} +0.1 i \cdot 
                  \delta \frac{\Delta g (\xi)}{g(\xi)} \right]
+\gamma(p_T)
\end{equation}
\begin{equation} 
a(p_T)=\alpha(p_T)\left[0.1 \cdot 
                  \delta \frac{\Delta g (\xi)}{g(\xi)} \right]^2
\end{equation}  
\begin{equation}
b(p_T)=0.2 \alpha(p_T) \frac{\Delta g(\xi)}{g(\xi)} \delta 
                    \frac{\Delta g(\xi)}{g(\xi)}
  + 0.1 \beta(p_T) \delta \frac{\Delta g(\xi)}{g(\xi)}
\end{equation} 
\begin{equation} 
c(p_T)= \alpha(p_T)\left[\frac{\Delta g(\xi)}{g(\xi)}\right]^2 
  + \beta(p_T) \left[\frac{\Delta g(\xi)}{g(\xi)}\right]
  +\gamma(p_T)
\end{equation} 
Now if we have one more set of asymmetries with 
\begin{equation}
\Delta g_{i} (x) = 0.1 \cdot i \cdot g(x) , 
\end{equation} 
we will be able to extract
$\alpha(p_T)$, $\beta(p_T)$, $\gamma(p_T)$, 
$\Delta g(\xi)/g(\xi)$ and $\delta \Delta g(\xi)/g(\xi)$.  
Then the asymmetriesw will become 
\begin{equation} 
{\cal A}_{LL}_{i}^{\pi^0}(p_T) = a'(p_T)\cdot i^2 + b'(p_T)\cdot i +c'(p_T) 
\end{equation} 
If we could assume $\alpha(p_T)$, $\beta(p_T)$, and $\gamma(p_T)$
are unchanged, then the new asymmetries will be 
\begin{equation} 
a'(p_T)=\alpha(p_T) (0.1)^2 
\end{equation} 
\begin{equation} 
b'(p_T)=\beta(p_T) 0.1 
\end{equation} 
\begin{equation} 
c'(p_T)=\gamma(p_T)
\end{equation} 
A part of the asymmetries and the derived parameters are listed in Table~\ref{T:Asyms}. 
In the lowest $p_T$ bin the relavant $x$ is smaller than the average 
value of $x$, $\langle x \rangle$. In the higher $p_T$ region beyond Table~\ref{T:Asyms},
relevant-$x$ reahes 20\% higher values than $\langle x \rangle$. 

\begin{table}[hbt]
{\begin{tabular}{r|ccccc}
\hline
\hline
  $i$ & 1.25 GeV/$c$&1.75 GeV/$c$      &       2.25 GeV/$c$    &       2.75 GeV/$c$    &       3.25 GeV/$c$ \\
\hline
-10     &       0.09588 &       0.1305  &       0.1435  &       0.1468  &       0.144 \\
-9      &       0.0777  &       0.1053  &       0.1163  &       0.1182  &       0.1169\\
-8      &       0.06107 &       0.08351 &       0.09158 &       0.09308 &       0.09147\\
-7      &       0.04675 &       0.06335 &       0.06951 &       0.07094 &       0.06927\\
-6      &       0.03439 &       0.04658 &       0.05072 &       0.05167 &       0.05028\\
-5      &       0.02377 &       0.03215 &       0.03515 &       0.03529 &       0.03431\\
-4      &       0.01506 &       0.02048 &       0.02219 &       0.02224 &       0.02141\\
-3      &       0.00842 &       0.01138 &       0.0122  &       0.01211 &       0.01139 \\
-2      &       0.00366 &       0.00490 &       0.00517 &       0.00499 &       0.00455 \\
-1      &       0.00086 &       0.00111 &       0.00111 &       0.00094 &       0.00071 \\
0       &      -0.00002 &      -0.00002 &      -0.00004 &      -0.00006 &      -0.00009 \\ 
1       &       0.00105 &       0.00149 &       0.00176 &       0.00198 &       0.00215 \\
2       &       0.00407 &       0.00565 &       0.00649 &       0.00705 &       0.00742 \\
3       &       0.00902 &       0.01246 &       0.01422 &       0.01513 &       0.01574\\
4       &       0.01585 &       0.02199 &       0.02487 &       0.02636 &       0.02717\\
5       &       0.02475 &       0.03412 &       0.03844 &       0.04051 &       0.04141\\
6       &       0.03554 &       0.04864 &       0.05499 &       0.05782 &       0.05896\\
7       &       0.04813 &       0.0662  &       0.07427 &       0.07806 &       0.07928\\
8       &       0.06275 &       0.08619 &       0.09702 &       0.1014  &       0.103\\
9       &       0.07919 &       0.1088  &       0.1222  &       0.128   &       0.1294\\
10      &       0.09755 &       0.1341  &       0.1501  &       0.1569  &       0.1593\\
\hline 
\hline 
$c'(p_T)$       &1.79E-05       &1.28E-05       &-1.65E-05      &-3.70E-05      &       -7.56E-05 \\
$b'(p_T)$       &9.22E-05       &1.85E-04       &3.35E-04       &5.19E-04       &       7.26E-04  \\ 
$a'(p_T)$       &9.68E-04       &1.32E-03       &1.47E-03       &1.52E-03       &       1.52E-03\\
\hline 
$c(p_T)$        &3.24E-04       &9.68E-04       &1.79E-03       &2.90E-03       &4.29E-03 \\
$b(p_T)$        &1.63E-04       &4.48E-04       &7.98E-04       &1.26E-03       &1.84E-03\\
$a(p_T)$        &1.98E-05       &5.00E-05       &8.61E-05       &1.33E-04       &1.90E-04 \\
\hline  
${\cal D}$     &0.143000       &0.194430        &0.241960       &0.295325       &0.353527 \\ 
$\Delta g(\xi)/g(\xi)$  &0.054213&0.080057      &0.100821       &0.123376       &0.147139\\
$\alpha(p_T)$  &0.0968         &0.1323          &0.1470         &0.1520         &0.1519\\
$\beta(p_T)$    &9.22E-04       &1.85E-03       &3.35E-03       &5.19E-03       &7.26E-03 \\
$\gamma(p_T)$   &1.79E-05       &1.28E-05       &-1.65E-05      &-3.70E-05      &-7.56E-05 \\
check   &       -2.85E-05       &-4.01E-05      &-3.02E-05      &-1.69E-05      &1.24E-05 \\
\hline
$\langle x \rangle$&0.097       &0.092  &0.098  &0.105  &       0.112 \\
relevant-$x$            &0.068       &0.086  &0.098  &0.1105 &       0.123\\
$\Delta g^{\rm AAC}(\langle x \rangle)/g(\langle x \rangle)$&0.0977&0.0909&0.1007       &0.1128 &0.1271 \\
\hline 
\hline 
\end{tabular}}
\caption{Asymmetries with constant gluon polarization and parameters derived 
  with the quadratic form assumption.} 
\label{T:Asyms}
\end{table}

In conclusion, the values of $\xi$ are close to the normal 
average values of $x$, but differ by order 20\%. 


\thebibliography{99}
\bibitem{test}nothing so far... 
\end{document}

%%\begin{eqnarray}
%\begin{equation} 
%E\frac{d^3 \sigma}{dp^3} = 
%\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!   
%( g(x_1) g(x_2) \frac{d \sigma}{dt} (gg\rightarrow gg) D_{\pi^0/g}(z) 
%\end{equation}
%\begin{equation} 
%+q(x_1) g(x_2) \frac{d \sigma}{dt} (qg\rightarrow qg) D_{\pi^0/q}(z) 
%\end{equation}
%\begin{equation} 
%+g(x_1) q(x_2) \frac{d \sigma}{dt} (gq\rightarrow gq) D_{\pi^0/g}(z) 
%\end{equation}
%\begin{equation} 
%+ q(x_1) q(x_2) \frac{d \sigma}{dt} (qq\rightarrow qq) D_{\pi^0/q}(z) )
%dx_1 dx_2 dt dz 
%\end{equation}
%%\end{eqnarray}