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\newcommand{\be}{\begin{eqnarray}}
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\begin{document}
\draft


\title{
Large-$p_T$ Inclusive $\pi^0$ Production in Heavy-Ion 
Collisions at RHIC and LHC
}
\author{Sangyong Jeon$^{1,2}$, Jamal Jalilian-Marian$^3$ and Ina Sarcevic$^4$}
\address{
$^1$RIKEN-BNL Research Center, Upton, NY 11973-5000\\
$^2$Department of Physics, McGill  University, Montreal, 
QC H3A-2T8, Canada \\
$^3$Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA 
11794\\
$^4$Department of Physics, University of Arizona, Tucson, Arizona
85721, USA\\
}

\wideabs{

\maketitle

\begin{abstract}

\widetext 

We present results for the large-$p_T$ inclusive $\pi^0$ production in 
p-p and A-A collisions  at RHIC and LHC energies.  We include the full 
next-to-leading order radiative corrections, $O(\alpha_s^3)$, and nuclear 
effects such as parton energy loss and nuclear shadowing. We find the 
next-to-leading order corrections and the parton energy loss effect to be 
large and $p_T$ dependent, while the nuclear shadowing effects 
are small. We show how measurements of $\pi^0$ production at RHIC and LHC 
for $p_T>5$ GeV can provide valuable information about the energy loss 
parameters. We calculate the ratio of prompt photons to neutral pions 
produced in heavy ion collisions and show that at RHIC energies this ratio 
increases with $p_T$ approaching one at $p_T \sim 15$ GeV, due to the large 
suppression of $\pi^0$ production. We show that 
at the LHC, this ratio has steep $p_T$ dependence and approaches 
$10\%$ effect at $p_T \sim 40$ GeV.  
We discuss 
 theoretical 
uncertainties inherent in our calculation, such as choice of the 
renormalization, factorization and fragmentation scales and the 
K-factors which signify the size of higher-order corrections.  

\end{abstract}
}

\vskip 0.1true in

\narrowtext
\section{Introduction}

In high energy heavy ion collisions hard scatterings of partons occur 
in the early stages of the reaction, well before a quark gluon plasma 
might have been formed, resulting in production of large 
transverse momentum particles.  Fast partons produced in these 
hard collisions propagate through the hot and dense medium created in the
heavy ion collision and lose their energy. Therefore, a possible 
signature for parton energy loss in a hot, dense medium is the suppression 
of pion production in heavy ion collisions relative to 
hadron-hadron collisions. The Relativistic Heavy-Ion Collider (RHIC) at 
BNL with Au-Au collisions at $\sqrt s=130$ GeV and at $\sqrt s=200$ GeV, 
and the Large Hadron Collider (LHC) at CERN which will collide Pb-Pb at 
$\sqrt s=5.5$TeV provide the best opportunity to study the properties of 
the hot and dense matter and the possible formation of a new phase, the 
quark-gluon plasma. Recent measurements of inclusive $\pi^0$ 
production at RHIC energy of $\sqrt s=130$ GeV \cite{rhic} show a large 
suppression in the low $p_T$ region ($1$GeV$ < p_T < 4$GeV) which has created 
much excitement in the field.  The question of the origin of this suppression 
has inspired many interpretations. Clearly, such a low region of $p_T$ has 
large nonperturbative contributions and thus any perturbative calculation 
is not completely reliable in that region. Here we concentrate on production 
of $\pi^0$ in the large-$p_T$ region where perturbative QCD calculations 
are expected to be more reliable. 

In addition to being of interest for studying nuclear effects, such as 
parton energy loss and nuclear shadowing, large-$p_T$ $\pi^0$ mesons 
form a significant background for the prompt photons.  In principle, 
large-$p_T$ pions could form ``fake prompt photons'' when one photon 
from pion decay escapes detection. Theoretical predictions 
for the ratio of prompt protons to pions at RHIC and LHC energies are 
crucial for studying possible quark-gluon plasma formation via photons. 
In addition, this ratio may reduce some of the theoretical uncertainties, 
such as choice of factorization, renormalization and fragmentation scale, 
or the choice of gluon fragmentation function.  

In perturbative QCD, the inclusive cross section
for pion production in a hadronic collision is given by: 

\be
E_\pi \frac{d^3\sigma}{d^3p_\pi}(\sqrt s,p_\pi)
&=&
\int dx_{a}\int dx_{b} \int dz \sum_{i,j}F_{i}(x_{a},Q^{2}) \nonumber \\
&&
F_{j}(x_{b},Q^{2}) D_{c/\pi}(z,Q^2_f) E_\pi 
{d^3\hat{\sigma}_{ij\rightarrow c X}\over d^3p_\pi} 
\label{eq:factcs}
\ee

\noindent
where $F_{i}(x,Q^{2})$ is the i-th parton distribution in a nucleon,
$x_a$ and $x_b$ are the fractional momenta of incoming partons,
$D_{c/\pi}(z,Q^2_f)$ is the pion fragmentation function, $z$ is the fraction 
of parton energy carried by the pion and 
${d^3\hat{\sigma}_{ij\rightarrow c X}\over d^3p_c}$ are parton-parton cross 
sections which include leading-order, $O(\alpha_s^2)$, subprocesses such as: 
\be
q + q &\rightarrow &  q + q  \nonumber\\
q + \bar q &\rightarrow & \bar q + q  \nonumber\\
q +  g &\rightarrow & g   + q \nonumber\\
g + g &\rightarrow & g  + g   
\ee

\noindent 
and the next-to-leading order, $O(\alpha_s^3)$, subprocesses such as:
\be
q + q &\rightarrow &  q + q + g  \nonumber\\
q + \bar q &\rightarrow & q + \bar q + g  \nonumber\\
q + q' &\rightarrow & q + q' + g  \nonumber \\
q + \bar q &\rightarrow & q' + \bar q' +  g  \nonumber\\
g + g &\rightarrow & g + g + g 
\ee

The running coupling  constant $\alpha_{s}(\mu^{2})$, calculated to
next-to-leading order, is given by

{
\small
\begin{eqnarray*}
\alpha_s (\mu^2)={12\pi\over (33-2N_f)\ln \mu^2/\Lambda^2}
\bigg[1-{6 (153-19N_f)\ln\ln \mu^2/\Lambda^2 \over
(33-2N_f)^2\ln \mu^2/\Lambda^2}\bigg] \nonumber\\
\end{eqnarray*}
}

\noindent
where $\mu$ is the renormalization scale, $N_F$ is the
number of flavors and $\Lambda$ is the $\Lambda_{QCD}$ scale.

The parton distribution functions $F_{i}(x,Q^{2})$ are measured in Deep
Inelastic Scattering experiments such as those at HERA \cite{hera} while
fragmentation functions, $D_{c/\pi}(z,Q^2_f)$, that describe the transition 
of the partons into the final-state pions are extracted from $e^+e^-$ 
annihilation data from PETRA, PEP and LEP \cite{frag}.  
The gluon fragmentation function, which gives the dominant contribution 
to $\pi^0$ production at the LHC, is not well determined by $e^+e^-$ data, 
since it appears only at NLO.
Nevertheless, it is possible to get some 
constraint on the gluon fragmentation function from measurements of large 
$p_T$ pion production in hadronic collisions at high energies. Such a study 
has been done using UA1 data in the range $5$GeV$ <p_T<20 $GeV \cite{UA1}.  
The gluon fragmentation function of 
Binnewies, Kniehl and Kramer (BKK) \cite{frag} 
 is found to be consistent with 
the UA1 data.  
The fragmentation scale, $Q_f$, when taken to be too small, i.e. 
$Q_f=p_T/3$, probes the region currently not tested by the data. In 
addition, theoretical improvement is needed in resumming large $ln(1-z)$ 
terms present in the higher-order corrections.  

We remark on 
divergencies which are present 
in the calculation of parton cross section,
$d\hat{\sigma}_{ij}$, require careful
treatment of collinear and ``soft'' singularities in the
matrix elements.
Divergences cancel between real and virtual graphs when the
physical $\pi^0$ cross section is calculated.
Infrared or ``soft'' divergences
($k\rightarrow 0$) cancel between virtual amplitudes
and real amplitudes.
Collinear divergences ($\theta \rightarrow 0$ between initial quark and
radiated gluon, for example) are reabsorbed
in structure function renormalization (``mass'' singularities).


Detailed study of large-$p_T$ inclusive $\pi^0$ production in 
hadronic collisions show very good agreement between theory and experiments, 
apart from an overall normalization \cite{se}.  

\section{Large-$p_T$ Inclusive $\pi^0$ Production in Heavy Ion Collisions} 

To calculate the inclusive cross section for pion production in heavy
ion collisions, we will use (\ref{eq:factcs}) with the distribution
and fragmentation functions appropriately modified to include nuclear
effects such as shadowing and energy loss.

It is a well known experimental fact that the distribution of quarks 
and gluons inside nuclei is modified compared to that in a free nucleon. 
This modification is known as nuclear shadowing (for a review of nuclear 
shadowing, we refer the reader to \cite{arneodo}). The parton 
distribution in a nucleus $F_{a/A}(x,Q^2,b_t)$, can be written as 
\begin{eqnarray*}
F_{a/A}(x,Q^2,b_t)=T_A(b_t)\,S_{a/A}(x,Q^2)\,F_{a/N}(x,Q^2)
\end{eqnarray*}
\noindent
where $T_A(b_t)$ is the nuclear thickness function, $F_{a/N}(x,Q^2)$ is the
parton distribution function in a nucleon and $S_{a/A}(x,Q^2)$ is the
parton shadowing function, 
$S_{a/A}(x,Q^2) = F_{a/A}(x,Q^2)/A F_{a/N}(x,Q^2)$.  
  It should be emphasized that the $Q^2$ dependence
of nuclear shadowing is poorly known, specially in the small $x_{bj}$ 
region. Also, gluon shadowing is measured only indirectly through scaling
violation of $F_2$ structure function which leads to large uncertainties
in the gluon distribution function in nuclei. In this work, we use two
rather different parametrizations of nuclear shadowing in order to
investigate the sensitivity of our results to uncertainties in nuclear 
distribution functions of quarks and gluons. One parametrization is due
to Benesh, Qiu and Vary (BQV) \cite{bqv} which 
is $Q^2$ independent and treats
shadowing of quarks and gluons on the same footing. In this parametrization,
the shadowing function  $S_{a/A}(x,Q^2)$ is given by

\begin{eqnarray}
S_{a/A}(x)
=
\left\{\begin{array}{ll}
\alpha_3 -\alpha_4 x & x_0 <x\leq 0.6 \\
(\alpha_3 -\alpha_4 x_0)\frac{1+k_q \alpha_2 ({1/x}-1/x_{0})}
{1+k_q A^{\alpha_1}({1/x}-1/x_{0})}
&x\leq x_{0}. \nonumber
\end{array}
\right.
\end{eqnarray}

\noindent
where the parameters $\alpha_i$, $k_q$ and $x_0$ are determined by 
fitting the EMC, NMC, and E665 data and given in \cite{bqv}. 

A more recent parametrization is due to Eskola, Kolhinen and Salgado (EKS), 
which is $Q^2$ dependent and distinguishes between quarks and gluons 
\cite{eks98} was shown to be in very good agreement with the NMC data 
on $Q^2$ dependence of $F_2^{Sn}/F_2^C$ \cite{NMC}, while some 
other parametrizations which have large gluon suppression due to 
nuclear shadowing are ruled out \cite{ehks}. It is also shown that 
modifications to the DGLAP evolution due to gluon fusion are small for 
the kinematic region of relevance to RHIC and that strong gluon 
shadowing is disfavored.  In Fig. (\ref{fig:eksbqv}) we show EKS98 
nuclear shadowing function at $Q^2=2.25$GeV$^2$ and 
the BQV nuclear shadowing functions.  

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{eksbqv.eps}}
\caption{The nuclear shadowing ratio as parametrized by BQV 
and EKS98.}
\label{fig:eksbqv}
\end{figure}

In Fig. (\ref{fig:shadQ2}) we show the $Q^2$ dependence of the
shadowing function for gluons in EKS parametrization
\cite{ekrs}.  Clearly, there is a strong $Q^2$ dependence
in the EKS parametrization of nuclear shadowing, especially
for gluons, as well as large anti-shadowing effect 
in the large $x_{bj}$
region.

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{eksQevol.eps}}
\caption{$Q^2$ dependence of gluon shadowing function 
in a nucleus ($A=208$) in EKS parametrization.}
\label{fig:shadQ2}
\end{figure}
 
Another nuclear effect which we include is the medium induced 
energy loss. It is expected that fast partons propagating through
the hot and dense medium created after a high energy heavy ion
collision will scatter from the partons in the medium, lose
part of their energy and then fragment into hadrons with a reduced 
energy. As a result, the spectrum of the final state hadrons observed
in a heavy ion collision is expected to be suppressed compared to
hadronic collisions. This has been extensively studied and we refer
the reader to \cite{eloss} for some recent work. The 
Landau-Pomeranchuk-Migdal (LPM) effect occurs when an ultrarelativistic 
particle emits low energy bremsstrahlung photons (gluons) as the particle 
passes through dense matter; fewer photons (gluons) are emitted than 
predicted by bremsstrahlung theory for isolated atoms.  

While a dynamical study of the parton propagation in a hot and dense medium
created in a realistic heavy ion collision and the modification of the 
hadronization is more desirable, we will use a phenomenological model 
\cite{hsw} here to demonstrate how sensitive pion production is to the 
average energy loss suffered by a parton in a hot and dense medium.
We restrict ourselves to the central rapidity region so that a parton will 
only propagate in the transverse direction in a cylindrical system. Given the 
inelastic scattering mean-free-path, $\lambda_a$, the probability for a 
parton to scatter $n$ times within a distance $\Delta L$ before it escapes 
the system is assumed to be given by the Poisson distribution

\be
P_a(n) = \frac{(\Delta L/\lambda_a)^n}{n!} e^{-\Delta L/\lambda_a}.
\nonumber 
\end{eqnarray}

In the Huang-Sarcevic-Wang model of energy loss, the hadronic 
fragmentation function $D_{c/\pi}(z,Q^2)$ is modified in order to include 
multiple scattering of a parton in a nuclear medium. If the average energy 
loss per scattering suffered by the parton $a$ is $\epsilon_a$, the nuclear
fragmentation functions can be modeled as \cite{hsw}, 

\be
z D_{c/\pi}(z,\Delta L,Q^2)& =&
 \sum_{n=0}^NP_a(n) z^a_n D^0_{c/\pi}(z^a_n,Q^2)
  \nonumber \\
&+&\langle n_a\rangle z'_aD^0_{g/\pi}(z'_a,Q_0^2), 
\label{eq:mfrag}
\ee

where $z^a_n=z/(1-n\epsilon_a/E_T)$, $z'_a=zE_T/\epsilon_a$, $N$ is 
the maximum number of collisions for which $z_n^a \le 1$ and 
 $D^0_{c/\pi}$ is the hadronic 
fragmentation function which gives the probability that quark or a gluon 
would fragment into a pion. 
The first term corresponds to the fragmentation of the leading partons 
with reduced energy $E_T-n\epsilon_a$ and the second term comes from the 
emitted gluons each having energy $\epsilon_a$ on the average. 
The average number of scatterings within a distance $\Delta L$
is $\langle n_a\rangle =  \Delta L/\lambda_a$.  We take 
$\lambda_a=1fm$ and $\Delta L= R_A$.  

In this work, we use MRS99 parametrization of nucleon structure functions 
\cite{mrs99}, BKK pion fragmentation functions \cite{frag}, 
EKS98 
shadowing functions to include nuclear shadowing and we modify the BKK 
fragmentation functions according to the model of Huang-Sarcevic-Wang 
\cite{hsw} in order to take into account medium induced parton 
energy loss effects. 
We calculate the invariant cross section for $\pi^0$ production in heavy-ion 
collision normalized 
to the number of binary nucleon-nucleon collisions, $N_{coll}$, 
where $N_{coll}$ can theoretically be 
determined from nuclear 
overlapping function, i.e.  
$N_{coll}= T_{AA}(b)/\sigma_{inel}^{NN}$ and $T_{AA}(b)= 
 \int d^{2}b_{1}
T_{A}(\mid\vec{b}_{1}\mid)
T_{A}(\mid\vec{b} - \vec{b}_{1}\mid)$.  The number of N-N collisions 
depends on the centrality that experiment triggers on.  
In order to investigate the sensitivity of our results to the choice of
energy loss parameters in the model of \cite{hsw}, we vary the average 
parton energy loss parameter, $\epsilon$, which we take to be independent 
of the initial parton energy, from $0.3$ GeV to $2$ GeV. We set all the
scales appearing in the parton distributions and fragmentation functions, 
 as well 
as the renormalization scale in the running coupling constant, 
$Q = Q_F = \mu = \kappa p_T$ where we take 
$\kappa=1$ and $\kappa=2$.    
We discuss 
sensitivity of inclusive $\pi^0$ 
production to the choice of scale.  
 
In Fig. (\ref{fig:rhic130dsigma}) we show the invariant cross section 
${E\, d^3\sigma \over d^3\,p}$ for inclusive $\pi^0$ production 
in proton-proton and nucleus-nucleus ($A=200$) collisions at 
$\sqrt s=130$ GeV. There is a clear suppression of the nuclear cross
section compared to the hadronic one. The value of the average energy
loss parameter $\epsilon$ is $0.3$ GeV and EKS98 shadowing is used.
We have chosen this value of $\epsilon$ in order to make our cross
sections in qualitative agreement with the 
experimental results on $\pi^0$ spectrum
at RHIC at $\sqrt{s} = 130$ GeV \cite{rhic} even though we have not made any 
attempt to fit the data.  At higher energies, we use larger values 
of $\epsilon$ because we expect that the parton energy loss  would 
increase with energy.  

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic130dsigma.eps}}
\caption{Inclusive $\pi^0$ distribution at $\sqrt s=130$ GeV.}
\label{fig:rhic130dsigma}
\end{figure}

From Fig. 
 (\ref{fig:rhic130dsigma}) we see that 
the $\pi^0$ production cross section in heavy
ion collisions is suppressed compared to hadronic collisions even at very
large $p_T$ ($p_T>10$ GeV). This is due to the fact that high $p_T$ partons 
lose their energy as they propagate through the dense medium and contribute 
to the lower $p_T$ pions. 

The next-to-leading order contributions, $O(\alpha_s^3)$, which are included 
in our calculations, are large and $p_T$ dependent. For example, the nuclear 
K-factor defined as a ratio of full next-to-leading order 
 calculation to the leading order 
 is 
varying between 1.6 and 2.2 for $p_T$ between 4GeV and 20GeV. This is
shown in Fig. (\ref{fig:rhic130k}).

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic130k.eps}}
\caption{K-factor, defined as the ratio of full NLO and LO 
$\pi^0$ inclusive cross section at $\sqrt s=130$ GeV.}
\label{fig:rhic130k}
\end{figure}
  
In Fig. (\ref{fig:rhic130ratio}) we show the ratio of inclusive $\pi^0$ 
cross section in nucleus-nucleus collisions to the one in p-p collisions.  
We note that nuclear shadowing effects are small and that most of the
observed suppression is due to energy loss effects. For $\epsilon=0.3$ GeV, 
the ratio is increasing from $30\%$ at $p_T =4$ GeV to $40\%$ at $p_T=20$ 
GeV.  Suppression at low $p_T$ is in qualitative agreement with the PHENIX 
data \cite{rhic}.  Nuclear shadowing effect varies from $2\%$ at low $p_T$ 
to about $10\%$ at large $p_T$.  
  At larger $Q^2$ 
($Q > 10$ GeV), EKS distributions exhibit small shadowing effect.  
  Therefore, the effect that nuclear 
shadowing has at low $p_T$ is slight increase of the ratio shown in Fig. 
 (\ref{fig:rhic130ratio}), 
 while at larger $p_T$, it gives about $10\%$ suppression due to 
the shadowing effect and 
 the $Q^2$ evolution of the EKS shadowing function.
In order to study the sensitivity of the ratio plotted in Fig. 
 (\ref{fig:rhic130ratio}) 
to the choice of the 
nuclear shadowing function, we also present our results using BQV 
shadowing function.  We find that in this case, there is small suppression 
for all values of $p_T$, of a few percent.  This is due to the fact that 
BQV shadowing function plotted in Fig. (\ref{fig:eksbqv}) 
 has shadowing effect for 
all values of $x$ and is $Q^2$ independent.  
  To illustrate sensitivity of the ratio of 
inclusive cross sections to energy loss effects, we show this ratio for
different choices of $\epsilon$. We find that the ratio is about 
$15-25\%$ 
for $\epsilon=0.5$ GeV and about $5-10\%$ for $\epsilon=1$ GeV.  
Nuclear shadowing effect is less than $2\%$ in the later case for 
$p_T$ between $4$ GeV and $20$ GeV.  

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic130ratio.eps}}
\caption{Ratio of inclusive $\pi^0$ distributions in 
heavy ion and p-p collisions at $\sqrt s=130$ GeV.}
\label{fig:rhic130ratio}
\end{figure}

In Fig. (\ref{fig:rhic130gratio_pratio}) we show the inclusive cross 
section for prompt photon production in heavy ion collisions at 
$\sqrt s=130$ GeV, normalized to proton-proton collisions, together with 
the same ratio for inclusive $\pi^0$ production. We note that 
for the same energy loss parameter $\epsilon =0.3$ GeV, $\pi^0$ 
production is suppressed more than prompt photons. This is due to the
fact that about $75\%$ of the photon contributions comes from "direct" 
processes which are not affected by parton energy loss, while in case 
of pions, all subprocesses are convoluted with the modified 
fragmentation function.  Nuclear shadowing effect is small both in 
photon and in pion production.  
 
\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic130gratio_pratio.eps}}
\caption{Prompt photon and inclusive $\pi^0$ cross sections in heavy ion 
collisions normalized to p-p at $\sqrt s=130$ GeV.}
\label{fig:rhic130gratio_pratio}
\end{figure}
 
We show our results for inclusive $\pi^0$ and prompt photon production at 
RHIC at $\sqrt{s}=200$ in Figs. (\ref{fig:rhic200dsigma}, 
\ref{fig:rhic200ratio}, \ref{fig:rhic200gratio_pratio}) for $\epsilon =0.5$.  
We choose average energy loss parameter
$\epsilon$ slightly larger than at lower energies of $\sqrt s= 130$ GeV, 
in order to take into account that at higher energies there will be an 
 increase in 
temperature and particle density.  
Again, we find that nuclear shadowing effects are small (less than $5\%$)
and the suppression is mostly due to energy loss effects. We find that the 
nuclear K-factor for pions varies between 1.3 and 1.9 for parton 
energy loss of 
$\epsilon = 0.5$ GeV. 

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic200dsigma.eps}}
\caption{Inclusive $\pi^0$ spectrum at $\sqrt s=200$ GeV.}
\label{fig:rhic200dsigma}
\end{figure}
 
\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic200ratio.eps}}
\caption{Ratio of inclusive $\pi^0$ cross sections in heavy ion and  
p-p collisions at $\sqrt s=200$ GeV.}
\label{fig:rhic200ratio}
\end{figure}

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic200gratio_pratio.eps}}
\caption{Prompt photon and inclusive $\pi^0$ cross sections in heavy ion 
collision normalized to p-p at $\sqrt s=200$ GeV.}
\label{fig:rhic200gratio_pratio}
\end{figure}

Clearly measurements of inclusive pion production at RHIC energies for 
$p_T>4$ GeV could provide valuable information about the medium induced 
parton energy loss since nuclear shadowing effects are very small 
(few $\%$) and most of the observed suppression of hadronic spectra
in heavy ion collisions is due to energy loss. 

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{uncerscale.eps}}
\caption{Theoretical uncertainty in inclusive $\pi^0$ cross section 
 due to choice of scale for $Q=p_T$ and 
$Q=2 p_T$. We assume that $Q=Q_F=\mu$.}  
\label{fig:uncerscale}
\end{figure} 

In Fig. (\ref{fig:uncerscale}) we show theoretical uncertainty in 
predicting inclusive
$\pi^0$ cross section 
by varying scales from $Q=p_T$ to 
$Q=2 p_T$, with the assumption that 
$Q=Q_F=\mu$.  We note that uncertainty is about $40\%$ and that 
the shape of the $p_T$ spectrum is insensitive to the choice of 
scale.   Similar conclusion was found in case of $\pi^0$ production 
in hadronic collisions \cite{se}.  In hadronic collisions 
Aurenche et al. \cite{se} 
have found that NLO predictions give very 
good agreement with the data, but even with the 
 choice of scale of $p_T/2$, the overall normalization was found to 
be low.  Thus, we expect that our results for inclusive $\pi^0$ 
production would also require overall normalization factor in 
order to give quantitative description of the data.  However, 
we find that the ratio of $\pi^0$ production in heavy-ion collisions 
and in hadronic collisions, has very weak dependence on the 
choice of scales, thus providing a better way to probe perturbative 
predictions.  
  


In Fig. (\ref{fig:lhcdsigma}) we show the inclusive $\pi^0$ cross sections
in p-p and heavy ion collisions at LHC energy of $\sqrt s=5.5$ TeV.
In case of heavy ion collisions, we choose energy loss parameter $\epsilon=1$
GeV and we include EKS98 parametrization of nuclear shadowing. In 
Fig. (\ref{fig:lhcratio}), we present the ratio of $\pi^0$ inclusive
cross section in heavy ion collisions to p-p and find that enery loss 
effects reduce the nuclear cross section by $60\%-90\%$ at low $p_T$, 
and by about $40\%-10\%$ at $p_T=40$ GeV, depending on the size of the 
parton energy loss. Nulear shadowing effects are still very small, a few 
percent at all $p_T$. This is mostly due to strong $Q^2$ dependence of 
EKS nuclear shadowing function and that the scale in the distribution
function is $2 p_T$. Higher-order corrections are large, ranging between 
$1.6$ and $2$ in p-p and from $1.5$ to $1.6$ in the nuclear case when 
$\epsilon=2$ GeV. In Fig. (\ref{fig:lhcgratio_pratio}) we show cross 
sections for prompt 
photon and inclusive $\pi^0$ production normalized to p-p with the same 
nuclear parameters.  At this energy we note that suppression of 
prompt photons is similar to the $\pi^0$ case, because at LHC 
energy prompt photon production is dominated by bremsstrahlung processes 
(about $60\%$ contribution) 
which are modified due to the energy loss in a similar way to the 
$\pi^0$ case. We find that $\pi^0$ production is sensitive to the energy 
loss parameters. Varying $\epsilon$ from $0.5$ GeV to 
$2$ GeV results in reducing 
the cross section by about $30\%$.


\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{lhcdsigma.eps}}
\caption{Inclusive $\pi^0$ cross sections at LHC energy, 
$\sqrt s=5.5$ TeV.} 
\label{fig:lhcdsigma}
\end{figure}

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{lhcratio.eps}}
\caption{Ratio of inclusive $\pi^0$ cross sections in 
heavy ion and p-p collisions at LHC energy, $\sqrt s=5.5$TeV.}
\label{fig:lhcratio}
\end{figure}

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{lhcgratio_pratio.eps}}
\caption{Prompt photon and inclusive $\pi^0$ cross sections in heavy
ion collisions normalized to p-p at $\sqrt s=5.5$ TeV.}
\label{fig:lhcgratio_pratio}
\end{figure}

Finally, in Figs. (\ref{fig:rhic130_goverp}, \ref{fig:rhic200_goverp}, 
\ref{fig:lhc_goverp})  we show the ratio of prompt photons to $\pi^0$ 
as a function of transverse momentum for energies $\sqrt s=130$ GeV, 
$\sqrt s=200$ GeV and $\sqrt s=5.5$ TeV, including nuclear shadowing and 
energy loses of $0.5$ GeV, $1$ GeV and $2$ GeV respectively.  We also show 
this ratio for proton-proton collisions at the same energy.  Because of the 
large $\pi^0$ suppression relative to prompt photons at RHIC energies, this 
ratio increases with $p_T$ approaching $1$ at $p_T~14$ GeV. However, at LHC 
energies this ratio increases from 0.03 at low $p_T$ to about 0.13 at large 
$p_T$ for $\epsilon=2$ GeV.  

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic130_goverp.eps}}
\caption{Ratio of prompt photon to $\pi^0$ cross sections at 
$\sqrt s=130$ GeV.}
\label{fig:rhic130_goverp}
\end{figure}

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{rhic200_goverp.eps}}
\caption{Ratio of prompt photon to $\pi^0$ cross sections at 
$\sqrt s=200$ GeV.}
\label{fig:rhic200_goverp}
\end{figure}

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=7.5cm}
\centerline{\epsffile{lhc_goverp.eps}}
\caption{Ratio of prompt photon to $\pi^0$ cross sections at 
$\sqrt s=5.5$ TeV.}
\label{fig:lhc_goverp}
\end{figure}

\section{Summary and Conclusions}

We have calculated inclusive pion and prompt photon production
cross sections in heavy ion and hadronic collisions at RHIC
and LHC energies. We have incorrporated next-to-leading order 
contributions,  
initial state parton distribution functions in a nucleus and 
 medium induced parton energy loss by modifying the 
final state pion fragmentation function.  

We have shown that the nuclear K-factor, the ratio of NLO to LO
pion cross sections, is large and $p_T$ dependent. This $p_T$ dependence
is specially strong at lower energies, i.e. RHIC at $\sqrt{s} = 130$ GeV.
We find that nuclear shadowing effects are quite small, specially at
RHIC at all $p_T$'s considered. The EKS98 parametrization of nuclear
shadowing has strong $p_T^2$ dependence which affects the cross sections 
through dependence of the factorization scale on $p_T$. If one is able to
measure the $\pi^0$ spectra at very large $p_T$ at RHIC to high accuracy, 
one could in principle rule out some models of nuclear shadowing which are 
$Q^2$ independent.

The medium induced energy loss effects in pion production are large even 
at $\sqrt{s} =130$ GeV. We use the existing low $p_T$ data at 
RHIC at $\sqrt{s} =130$ GeV
to determine a value for the average energy
loss per collision, $\epsilon$.  Then we predict inclusive $\pi^0$ production 
in Au-Au collisions normalized to the p-p case at high $p_T$ ($4$ 
GeV$ <p_T<20$ GeV).  
We also present the inclusive $\pi^0$ production cross section
in heavy ion collisions normalized to p-p at $\sqrt{s} =200$ GeV at RHIC 
and $\sqrt{s} =5.5$ TeV at LHC for energy loss of $0.5$ GeV, $1$ GeV and 
$2$ GeV per collision.  We demonstrate that perturbative calculation predicts 
 this ratio with very small ($<10\%$)  
 theoretical uncertainty 
due to the choice of the scales.  

Finally, we show results for the ratio of prompt photon to $\pi^0$ cross sections
at RHIC and LHC, of relevance to separating 
 different sources of photon production. We show that contribution
of direct photons is large at RHIC (about $75\%$) 
but decreases as the energy is increasing, becoming 
$40\%$ contribution (with $60\%$ contribution coming from bremsstrahlung 
processes) 
at LHC energies.


\vskip 0.1true in

\leftline{\bf Acknowledgments}

We would like to thank P. Aurenche and J. P. Guillet for providing us with
the fortran routines for calculating double differential distributions for 
$\pi^0$ and photon 
production in hadronic collisions and for many useful discussions.  
We would like to thank D. d'Enterria and M. Tannenbaum for many helpful 
discussions and suggestions. We would also like to thank ITP, Santa Barbara 
where part of this work was done. J. J-M. is grateful to LBNL nuclear theory 
group for the use of their computing resources. This work was supported in 
part through U.S. Department of Energy Grants Nos. DE-FG03-93ER40792 and 
DE-FG02-95ER40906. S.J. is supported in part by the Natural Sciences and
 Engineering Research Council of Canada and by le Fonds pour la Formation
 de Chercheurs et l'Aide \`a la Recherche du Qu\'ebec.  
J.J-M. is supported in part by a PDF from BSA and by 
U.S. Department of Energy under Contract No. DE-AC02-98CH10886.


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n:Jalilian-Marian;Jamal
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note;quoted-printable:Jamal Jalilian-Marian=0D=0APhysics Department, Bldg 510A=0D=0ABrookhaven National Laboratory=0D=0AUpton, NY 11973-5000=0D=0A(631)344-5805 office=0D=0A(631)344-7651 fax
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fn:Jamal Jalilian-Marian
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