%\documentclass[aps,prc,twocolumn,groupedaddress,amssymb]{revtex4}
\documentclass[aps,prc,twocolumn,superscriptaddress,amssymb]{revtex4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%    2002. Dec. 10.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{graphicx}
%%%%%%%%%%%%%%%%%%%%
\def\la{\langle}
\def\ra{\rangle}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\be{\begin{eqnarray}}
\def\ee{\end{eqnarray}}
\def\hs{\hat{s}}
\def\htm{\hat{t}}
\def\hu{\hat{u}}
\def\pt{p_\perp}
\def\at{a_\perp}
\def\bt{b_\perp}
\def\vpt{\vec{p}_\perp}
\def\vat{\vec{a}_\perp}
\def\vbt{\vec{b}_\perp}
\def\kav{\la k_\perp\ra}
\def\k2av{\la k_\perp^2\ra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\f}[2]{\frac{#1}{#2}}
\newcommand{\dd}{ {\textrm d}}

\begin{document}
\title{Intrinsic parton transverse momentum in NLO pion production}
\author{G\'abor Papp}
\email{pg@ludens.elte.hu}
\affiliation{HAS Research Group for Theoretical Physics,
E{\"o}tv{\"o}s University, \\ P{\'a}zm{\'a}ny P. 1/A, Budapest 1117, Hungary}
%\affiliation{CNR, Kent State University, Kent OH 44242, USA}
\author{Gergely G. Barnaf\"oldi}
\email{bgergely@rmki.kfki.hu}
\affiliation{RMKI KFKI, P.O. Box 49, Budapest 1595, Hungary}
%\affiliation{CNR, Kent State University, Kent OH 44242, USA}
\author{P\'eter L\'evai}
\email{plevai@rmki.kfki.hu}
\affiliation{RMKI KFKI, P.O. Box 49, Budapest 1595, Hungary}
\author{George Fai}
\email{fai@cnr4.physics.kent.edu}
\affiliation{CNR, Kent State University, Kent OH 44242, USA}

\date{\today}

 
\begin{abstract}
We study pion production in proton-proton collisions within a
pQCD-improved parton model in next-to-leading order augmented by
intrinsic transverse momentum ($k_\perp$) of the partons. 
We find the introduction of $k_\perp$ necessary
to reproduce the experimental data in the CERN SPS to RHIC energy range.
We study the influence of intrinsic $k_\perp$ on the so-called $K$ factor,
the ratio of the NLO cross section to the Born term.
A strong $p_T$ dependence is seen, especially in the 
$3-6$ GeV transverse momentum region of the outgoing pion, where nuclear
effects (e.g. the Cronin effect) play an important role.
\end{abstract}

\pacs {PACS numbers: 24.85.+p, 13.85.Ni, 13.85.Qk, 25.75.Dw }
\maketitle


\section{Introduction}
Today's collider facilities raise the interest in testing perturbative
QCD (pQCD) at work, and in searching for phenonema beyond its capability.
However -- at least for the experimentally available 
transverse momentum region -- pQCD parton models underestimate the
production of mesons in proton-proton ($pp$) collisions~\cite{LOBp,xnwang},
even at next-to-leading order (NLO)~\cite{aur1}. In order to restore the
consistency with the data, two methods were proposed at leading order
(LO): inclusion of the intrinsic transverse momentum of 
partons~\cite{LOBp,xnwang},
and/or an effective correction factor ($K$ factor), 
accounting for  higher order contributions~\cite{Eskola,Ramona}.
The physical background is to account for both, the
missing higher order perturbative corrections {\em and} initial
multiscattering effects. The latter is not present in
electron-proton processes, however, it becomes important
in $pp$ collisions~\cite{lai}.

In this paper we present the first results on pion production
in $pp$ collisions applying a NLO pQCD parton model {\em with}
intrinsic transverse momentum, taken from a
Gaussian distribution with width $\k2av$. In Section II the
intrinsic $k_\perp$ is introduced into the formalism, and the
appropriate NLO expressions are presented.
The necessity of such an extension is demonstrated at $\sqrt{s}=27.4$ GeV
in Section III. 
Next, we display the best fit values
of the width of the intrinsic transverse momentum distribution at NLO level
for the available $pp\to\pi+X$ experiments, in the energy range $20$
GeV $\lesssim\sqrt{s}\lesssim 200$ GeV. Similarly to Ref.~\cite{LOBp},
we study the pion production in the $2$ GeV $< p_T < 6$ GeV transverse
momentum region, where nuclear effects are considered to be important.

Finally, we extract the ratio of the NLO cross section to the Born
term ($K$ factor) at different energies and transverse momenta, and
study its dependence on the amount of the included intrinsic
transverse momentum. This way we provide a numerical foundation to the
correction factors used in LO calculation~\cite{Wong}. 
We demonstrate, that a leading order calculation with a
{\em fitted} $K(s,p_T,Q)$ factor {\em and} intrinsic $k_\perp$
reproduces well the full NLO results at higher energies and momenta,
and can be used as a fast method to get a reasonable estimate of a full 
NLO calculation.


\section{Model}
In order to extend the applicability of the original, infinite momentum 
frame parton model~\cite{field} to smaller transverse momenta, 
we introduce the intrinsic transverse momentum of the partons~\cite{owens}. 
We write the four-momenta of the interacting partons ($a$ and $b$) 
as~\cite{Wong}
\be
 p_a &=& (x_a\frac{\sqrt{s}}2+\frac{k_{\perp,a}^2}{2x_a\sqrt{s}},
        \vec{k}_{\perp,a},x_a\frac{\sqrt{s}}2-
        \frac{k_{\perp,a}^2}{2x_a\sqrt{s}})  \,\, , \\
 p_b &=& (x_b\frac{\sqrt{s}}2+\frac{k_{\perp,b}^2}{2x_b\sqrt{s}},
        \vec{k}_{\perp,b},-x_b\frac{\sqrt{s}}2+\frac{k_{\perp,b}^2}{2x_b\sqrt{s}}) \,\, .
        \nonumber
\ee
In this notation $x$, the momentum fraction carried by the
parton, becomes a parameter. The apparent fraction is $x-k_{\perp}^2/(x
s)$, however, for practical applications at high energy 
($p_T\gtrsim 3$ GeV; $\sqrt{s}\gtrsim 40$ GeV;
$\k2av\lesssim 2$ GeV${}^2$),
the distinction has a negligible ($\lesssim 5\%$) effect.
%At$\sqrt{s}=20$ GeV it causes a 10-15\% drop of the cross section at
%$\k2av=1$ GeV$^2$ and 40\% at $\k2av=2$ GeV$^2$, 10-20\% drop at
%$\k2av=2$ GeV$^2$ and $\sqrt{s}=31$ GeV. -> <k_t^2> drops even
%faster with decreasing energy!
Furthermore, we require that the longitudinal direction of the partons 
does not change due to the transverse momentum, i.e. 
$x>k_\perp/\sqrt{s}$.

%The total cm. energy square of the partonic collision is
%\be
%  \hat{s} = x_a x_b s +\frac{k_{t,1}^2 k_{t,2}^2}{x_ax_b s} -
%       \vec{k}_{t,1} \vec{k}_{t,2} ,
%\ee
%while the other two invariants are
%\be
%  \hat{t} &=& -x_a\sqrt{s}e^{-y}-\frac{k_{t,1}^2}{x_a\sqrt{s}}e^y +
%       \vec{k}_{t,1} \vec{p}_\perp/z ,\\
%  \hat{u} &=& -x_b\sqrt{s}e^{y}-\frac{k_{t,2}^2}{x_b\sqrt{s}}e^{-y} +
%       \vec{k}_{t,2} \vec{p}_\perp/z \,.
%\ee

The starting point of the parton model is factorization: the hadronic
cross sections up to a power correction may be written as a convolution over
hard partonic (pQCD) processes,
\be
\label{pqcd-def}
        \frac{\dd\sigma}{\dd y\dd^2p_T} 
        \hspace*{-15mm}&\hspace*{15mm}=\hspace*{-15mm}&\hspace*{15mm} 
        \sum_{abc}
        \int\!\dd x_a\dd x_b\dd^2k_{\perp,a}\dd^2k_{\perp,b}\ 
        \frac{\dd z_c}{\pi z_c^2} \\
        & &f_{a/p}(x_a,Q^2;k_{\perp,a}) f_{b/p}(x_b,Q^2;k_{\perp,b}) 
        \frac{\dd\sigma}{\dd\hat{t}} D_{\pi/c}(z_c,Q_f) \,\, ,
        \nonumber 
\ee
where $\dd\sigma/\dd\hat{t}$ is the partonic cross section of
the reaction $a+b\to c+d$ (LO with inserting condition
$\delta(1+(\hat{t}+\hat{u})/\hat{s})$), or $a+b\to c+d+e$ (NLO
with fixed $z_c$) and (at fixed scales) is the function of the 
partonic Mandelstam variables only. In order to avoid 
singularities due to the intrinsic transverse momentum we use regularization
\be
  \hat{s}\to\hat{s}+M^2, \quad \hat{t}\to\hat{t}-M^2/2, \quad
        \hat{u}\to\hat{u}-M^2/2 \,\, ,
\ee
%which conserves energy at Born level. The value of $M=1$ GeV is used in the
%following.
with $M=1$ GeV.
%\be
%  \hat{t} = -(1-v) \hat{s}, \quad \hat{u}=-v w \hat{s}
%\ee
The factorization is done at the {\em factorization} scale $Q$, where 
the content of parton $a$ in the initial hadron $h$ is determined
by the parton distribution
function $f_{a/h}$ (PDF). For simplicity, we assume a Gaussian dependence of
the PDFs on the intrinsic transverse momentum, with a width $\k2av$.
Finally, the hadrons are created collinearly with the outgoing parton
$c$ with momentum fraction $z_c$ at {\em fragmentation} scale $Q_f$.
%The observed meson is coming out at momentum
%\be
%  p_3 = (m_\perp \cosh{y}, \vec{p}_\perp, m_\perp \sinh{y}).
%\ee
%carrying $z$ fraction of collinear(?:eskolanew) parton.
We note that the partonic cross section explicitly depends on three scales,
$Q$, $Q_f$ and the {\it renormalization} scale $Q_r$. 

In principle, the
scales can be determined such that the final result has the
minimum sensitivity to them~\cite{aur1}, and usually are set to be
equal. However, in this paper we use the results of a previous 
study~\cite{BP2002} of $pp$ and $pA$ data, where the scales were fixed
to reproduce the Cronin effect in $pA$ reactions~\cite{Cronin}.

We note that there is some ambiguity in the choice of scales. In the 
literature typical scales are fixed to hadronic 
or partonic variables, $\kappa p_T$ or $\kappa p_T/z_c$, respectively, where
$\kappa$ is an ${\cal O}(1)$ number. 
Other choices are also possible, e.g. an invariant scale,
$Q^2=\kappa^2\,\hat{s}\hat{t}\hat{u}/(\hat{s}^2+\hat{t}^2+\hat{u}^2)$
as proposed in Ref.~\cite{field}.
In our case we found that this choice is equivalent to $\kappa p_T/z_c$.

At NLO level and no intrinsic transverse momentum Eq.~(\ref{pqcd-def}) is
usually rewritten
with variable change ($x_a$,$x_b$,$z_c$,$k_{\perp,a}$,$k_{\perp,b}$)$\,\to$
($\hat{v}$,$\hat{w}$,$z_c$,$k_{\perp,a}$,$k_{\perp,b}$), where 
$\hat{t}=-(1-\hat{v})\hat{s}$ and $\hat{u}=-\hat{v}\hat{w}\hat{s}$, as
\be
\label{pqcd-vw}
        \frac{\dd\sigma}{\dd y\dd^2p_T} 
        \hspace*{-15mm}&\hspace*{15mm}=\hspace*{-15mm}&\hspace*{15mm} 
        \frac1{\hat{s}}\sum_{abc}
%       \frac1{s}\sum_{abc}
%       \int\!\frac{\dd\hat{v}}{\hat{v}(1-\hat{v})}\frac{\dd\hat{w}}{\hat{w}}
        \int\!\dd\hat{v}\dd\hat{w}
        \frac{\dd z_c}{\pi z_c^2} 
        \ J \\
        & &\dd^2k_{\perp,a}\dd^2k_{\perp,b}\ f_{a/p} f_{b/p} D_{\pi/c}(z_c,Q_f)
        \frac{\dd\sigma^{NLO}}{\dd\hat{v}}  \,\,  ,
        \nonumber 
\ee
with the proper kinematical boundaries, $J$ being the Jacobian of the 
transformation ($1/J=\hat{v}(1-\hat{v})\hat{w}$ for $\k2av$=0), and 
$\dd\sigma^{NLO}/\dd\hat{v}$ is the sum of $2\to2$ and $2\to3$ cross 
sections~\cite{aversa},
\be
 \frac{\dd\sigma^{Born}}{\dd\hat{v}}\delta(1\!-\!\hat{w}) +
 \frac{\alpha_s(Q_r)}{\pi}{\cal K}^{ab\to cd}(\hat{s},\hat{v},\hat{w},Q,Q_r,Q_f)\,\, ,
\ee
with {\em renormalization} scale $Q_r$ (chosen to be equal to the 
factorization scale $Q$). In this paper, however, we use 
Eq.~(\ref{pqcd-def}) directly, since for $\k2av\neq0$ the momentum fraction
$x$ cannot be expressed analytically from $\hat{v}$ and $\hat{w}$, while
the inverse transformation can be done.

Several codes are available for calculating jet cross sections at NLO 
level~\cite{aversa,othernlo}. Here we have chosen to extend the one by 
the Aversa group~\cite{aversa} calculating the partonic cross sections 
at next-to-leading-log level, with the intrinsic transverse momentum 
distribution.
The calculations presented in the following sections were performed
with the MRST-cg PDF~\cite{MRST} and KKP FF~\cite{KKP} parameterization.
 
\section{Results}
\subsection{Comparison to data}
First, we demonstrate the importance of the intrinsic $k_\perp$ to
reproduce experimental data in the transverse momentum region 
$3$ GeV $< \sqrt{s} < 6$ GeV. In this part, we use the scales proposed in 
Ref.~\cite{aur1} to make a direct comparison. 
Fig. 1. shows, that at $\sqrt{s}=27.4$ GeV
the NLO calculations on pion production in $pp$ collision
underpredict the experimental data by a factor of
2 using the scale parameters $Q=Q_r=Q_f=p_T/2$ and neglecting
intrinsic transverse momentum (also done in~\cite{aur1}).
Decreasing the scales (but still keeping them hard for the applicability
of pQCD) the agreement can be improved, however,
reasonably low scales still underestimate data.
%, but with smaller 
%scale we are running out of phase space {\it (ez mi Peter?, MRST-vel 
%$p_t/4$-ig elmehetunk ...)}.

The lower line in Fig.~1. presents the NLO calculation {\em with} a partonic
intrinsic transverse momentum distribution of width $\k2av$ =1 GeV$^2$, and
shows a nice agreement with the data.
\begin{figure}
\centerline{%
\rotatebox{-90}{\includegraphics[height=0.45\textwidth,height=7.5truecm]%
   {Figs/narancs.ps}}
%\framebox[5cm][r]{\parbox[t][5truecm][s]{30mm}{Figure comes here}}
}
\caption{Comparison of experimental data~\cite{Antreasyan} to the NLO
pQCD parton model result in $p+p\to\pi^++X$ reaction at $E_{lab}=400$
GeV, with and without intrinsic transverse momentum.}
\end{figure}
We note, however, that there is a delicate interplay between the choice of the
scales and the intrinsic transverse momentum $\k2av$, needed to reproduce
the data~\cite{usnew}. 
Typically, increasing the scales increases the value of  $\k2av$.


\subsection{Intrinsic transverse momentum}

In this section we summarize the results on the intrinsic
transverse momentum width of the partons in the nucleon
fitting the NLO pQCD calculations to the data, restoring the scale choice
$Q=Q_r=p_T/z_c$, $Q_f=p_T/2$~\cite{BP2002}. Our previous 
study~\cite{BP2002} also showed, that the reproduction of the Cronin peak
in $pA$ collision requires the width of the intrinsic transverse momentum
distribution to be on the order of $\k2av\approx$ 2 GeV$^2$ at energies 
$\sqrt{s}\sim$ 30 GeV, which is achieved by the above choice of scales. A
similar
value of $\k2av$ was extracted from the experimental analysis of jet-angle
distribution~\cite{EXPBROAD}.

% This corresponds to R=0.7 jet cone angle cone:
% R \approx \sqrt{2} \atan{Q_t/p_q}; R=1 corresponds to Q_f=p_T/z
Analyzing the $pp\to\pi+X$ experimental data~\cite{Antreasyan,Exp},
we deduced the best fit value $\k2av$ for each experiment, similarly to
what is shown in Fig.~1, minimizing the $\chi^2(D/T-1)$ (data over theory) 
ratio 
in the range $3-6$ GeV. The result is presented in Fig.~\ref{fig-kt2},
separately indicating the $\pi^\pm$ and $\pi^0$ values. Despite some
uncertainties, we conclude that the width does not depend on the charge
of the pion, and we need a considerable amount of partonic transverse
momentum in order to reproduce experimental data.

%We note that changing the fragmentation scale to $Q_f=b p_T$, 
%where $b < 0.5$\marginpar{\it is this OK?} supresses
%the $\k2av(\sqrt{s})$ curve at high energies ($\sqrt{s}\gtrsim100$ GeV), and
%with such a choice of scales the role of the intrinsic transverse momentum
%is negligble already at RHIC energies. Keeping the fragmentation scale fixed
%to one-half of the jet transverse momentum, the corresponding transition  
%happens at around 500 GeV.

%The factorization (and renormalization) scales are even less sensitive to
%the $p_T/z_c\top_T$ change, their effect in the energy range
%presented is negligible.
\begin{figure}
\centerline{%
{\includegraphics[width=0.49\textwidth,height=8.1truecm]{Figs/ffpgskt3.eps}}
}
\caption{Best fits of $\k2av$ to experimental data in $pp\to\pi+X$
reactions from 19.5 to 200 GeV. Full dots: $\pi^+$, open dots:
$\pi^-$ in CP, E605, ISR, open crosses: $\pi^0$ production in 
R806, CCRS and RHIC experiments~\cite{Antreasyan,Exp}.}
\label{fig-kt2}
\end{figure}
Comparing NLO result to the previous LO ones~\cite{LOBp} one notices
that keeping the scales yields a reduction of $\k2av$ at NLO level,
indicating that part of the transverse momentum is generated by higher 
order processes. To keep $\k2av$ at the same level as in LO, we had to
choose to a larger scale~\cite{BP2002}.
%{\it Peter: erre azt mondtad, hogy nem igaz, akkor nem kene kivenni?}
%The discrepancy between experiments and pQCD calculations in the pion 
%fraction 
%of the hadron yield may originate in the uncertainty of the fragmentation
%functions at high momentum fractions ($z> 0.5-0.7$),
%especially for kaons and protons~\cite{FFwrong}.


\subsection{$K$ factor}
\begin{figure}
\centerline{%
\rotatebox{-90}{\includegraphics[height=0.45\textwidth]{Figs/kfac0.ps}}
}
\caption{$K$ factor at $\k2av=0$ and energies from $\sqrt{s}=20$ GeV 
to 200 GeV.}
\label{fig-kt0}
\end{figure}

Since most of the calculation (especially, for nucleus-nucleus collisions)
are still based on LO, it is useful to provide a well-founded $K$ factor
for these faster calculations. Fig.~\ref{fig-kt0} shows the ratio of the
full NLO calculation to the Born term with no intrinsic transverse momentum.
Indeed, at high energy and transverse momenta $K_\pi$ approaches the
well known value of 2, however, in the low transverse momentum region
it has a strong $p_T$ dependence. As a first approximation, for
$\sqrt{s}\gtrsim$ 60 GeV, the pionic $K$ factor can be taken energy
independent~\cite{JPG}.

Since the intrinsic transverse momentum may have a different effect on the
Born term than on the higher order (HO) processes, the latter contributing the
dominant part, it is worth to study the dependence of the $K$ factor on the
width $\k2av$. We present this behavior in Fig.~\ref{fig-kt}, indicating
the ratio $K(\k2av)/K(0)$ at $\k2av=$ 1 and 2 GeV$^2$ (with the usual
$Q=Q_r=p_T/z_c$, $Q_f=p_T/2$ scale fixing) and at different energies
from $\sqrt{s}=$ 20 to 60 GeV. While at low energies the $K$ factor shows
a large decrease with increasing $\k2av$ (mainly due to the efficiency of the
intrinsic transverse momentum at low energies in enhancing the Born term faster
than the HO corrections), from $\sqrt{s}\gtrsim$ 60 GeV there is less than
10\% correction due to $\k2av$ for $p_T\gtrsim$ 4 GeV, with vanishing
contribution at high transverse momenta, validating LO calculations
in this range with $\k2av$ independent $K$ factors~\cite{Wong,BP2002}.
\begin{figure*}
\centerline{%
\rotatebox{-90}{\includegraphics[height=0.8\textwidth]{Figs/kfac-xf.ps}}
}
\caption{Ratio of the pionic $K$ factor at $\k2av=1$ GeV${}^2$ (left) and 
$2$ GeV${}^2$ (right) to $K$ factor at $\k2av=0$ at energies from 
$\sqrt{s}=20$ GeV to 200 GeV.}
\label{fig-kt}
\end{figure*}



\section{Conclusions}

We introduced initial transverse momentum distributions into the
parton based description of the hadron production in NLO level
pQCD calculations. Such an extension is necessary for reproducing the
experimental data with high precision. We investigated in detail
the modifications of the NLO contributions corresponding to the presence
of an intrinsic $k_\perp$.

We have shown that the dependence of the pionic $K$ factor (full NLO
cross section to the Born term) on the intrinsic transverse momentum is not
too strong for reasonable values of $\k2av$, and hence $\k2av$ independent
$K$ factor calculations~\cite{Wong,BP2002} are good approximations
for $p_T\gtrsim$ 4 GeV. However, at smaller transverse momenta
more care should be taken in connection with the calculation of the
Cronin peak.

The fitting
of experimental data in the 3 GeV $\lesssim p_T \lesssim$ 6~GeV range 
indicates that at least in the energy range
$20$ GeV $\gtrsim \sqrt{s}\gtrsim $ 60 GeV 
we need an intrinsic transverse momentum  with a reasonable large
value $\k2av\approx\ 2$ GeV$^2$.

\section*{Acknowledgements}

We thank G. David for many useful discussions.
This work was supported in part by  U.S. DOE grant DE-FG02-86ER40251, NSF grant
 and Hungarian grants  FKFP220/2000, OTKA-T032796 and OTKA-T034842.
The support of the Bergen Computational Physics Laboratory in the
framework of the European Community - Access to Research Infrastructure 
action of the Improving Human Potential Programme is gratefully acknowledged.

\begin{thebibliography}{99}
\bibitem{LOBp}
        Y. Zhang, G. Fai, G. Papp, G.G. Barnaf\"oldi, and P. L\'evai, 
        Phys. Rev. {\bf C65}, 34903 (2002).
\bibitem{xnwang}
  X.N. Wang, %Phys. Rep. {\bf 280} 287 (1997); 
  Phys. Rev. {\bf C61}, 64910, (2000).
\bibitem{aur1} 
  P. Aurenche, M. Fontannaz, J.-Ph. Guillet, et al.,
  Eur. Phys. J. {\bf C13}, 347 (2000).
\bibitem{Eskola}
        K.J. Eskola, H. Honkanen, 
		Nucl. Phys. {\bf A713}, 67 (2003).
%  .
\bibitem{Ramona}
        R. Vogt, .
%  Heavy Ion Physics ????
\bibitem{lai}
  H.L. Lai, Phys. Rev. {\bf D58}, 114020 (1998).
\bibitem{Wong}
  C.Y. Wong, H. Wang, Phys. Rev. {\bf C58}, 376 (1998).
\bibitem{field} R.D. Field, {\it Applications of Perturbative QCD}, 
        Addison-Wesley, 1989, USA.
\bibitem{owens} 
  J.F. Owens, Rev. Mod. Phys. {\bf 59}, 465 (1987).
\bibitem{BP2002} 
  G.G. Barnaf\"oldi, P. L\'evai, G. Papp, G. Fai, Y. Zhang,
        .
\bibitem{Cronin}
  J.W. Cronin {\it et al.}
   Phys. Rev. D {\bf 11}, 3105 (1975).
\bibitem{aversa}
  F. Aversa, P. Chiappetta, M. Greco and J. Ph. Guillet,
        Nucl. Phys. {\bf B327}, 105 (1989); 
{\tt http://web13.cern.ch/monicaw/readme\_inc.html}.
%http://monicaw.home.cern.ch/monicaw/phonll.html}.
\bibitem{othernlo} 
%       R.K. Ellis, J.C. Sexton, Nucl. Phys. {\bf B269}, 445 (1986);
        S.D. Ellis, Z. Kunszt, D.E. Soper, Phys. Rev. Lett
        {\bf 69}, 1496 (1992);
        Z. Nagy, Phys. Rev. Lett.  {\bf 88}, 122003 (2002);
        {\tt http://www.cpt.dur.ac.uk/\~{}nagyz/nlo++/}.
\bibitem{MRST}
  A.D. Martin, R.G. Roberts, W.J. Stirling, R.S. Thorne
    Eur. Phys. Journal {\bf C23}, 73 (2002).
\bibitem{KKP}
  B.A. Kniehl, G. Kramer, and B. P{\"o}tter,
  Nucl. Phys. B {\bf 597}, 337 (2001).
\bibitem{usnew}
   G.G Barnaf\"oldi, P. L\'evai, G. Papp and G. Fai,
   to be published.
\bibitem{Antreasyan}
%CP 
  D. Antreasyan {\it et al.}, Phys. Rev. D {\bf 19}, 764 (1979).
\bibitem{EXPBROAD}
  M.D. Corcoran {\it et al.}, Phys. Lett. {\bf B259}, 209 
(1991).
\bibitem{Exp} 
%R806
  C. Kourkoumelis {\it et al.}, Z. Phys. C {\bf 5}, 95 (1980);
%CCRS
  F.W. B{\"u}sser {\it et al.}, Nucl. Phys. B {\bf 106}, 1 (1976);
%E605
  D.E. Jaffe {\it et al.}, Phys. Rev. D {\bf 40}, 2777 (1989);
%P.B. Straub {\it et al.}, Phys. Rev. Lett. {\bf 68}, 452 (1992);
%ISR
  B. Alper {\it et al.}, Nucl. Phys. B {\bf 100}, 237 (1975);
%L. Apanasevich, et al., .
%PHENIX
   S. Mioduszewski {\it et al.},;
%% NPA, QM2002 proceedings
%\bibitem{GABOR} G. David, private communication.
%\bibitem{FFwrong} X. Zhang, G. Fai, P. L\'evai, Phys. Rev. Lett, in press;
%.
\bibitem{JPG}
G.G. Barnaf\"oldi, G. Fai, P. L\'evai, G. Papp, Y. Zhang
	 J.Phys. {\bf G27}, 1767 (2001).
\end{thebibliography}

\end{document}

