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% \subsection{Pion photon production at the LHC: PYTHIA vs. DIPHOX}
\subsubsection{Comparison of PYTHIA and DIPHOX for $\pi^0 \, \gamma$ production}
\label{sec:pigam,gamjet,qcdsm}
Because of the huge jet rates at the LHC, any photonic observable is heavily
contaminated by neutral pions which appear as fake photons inside the
electro-magnetic calorimeter. Thus, concerning the search for a light neutral
Higgs boson in the mass window between 80 to 140 GeV, a detailed understanding
of not only $\gamma\gamma$ but also $\gamma\pi^0$ and $\pi^0\pi^0$ rates is
mandatory
\cite{Armstrong:1994it,ATLAS:1999tdr,CMS:1994tp,CMS:1997tdr,Branson:2001pj,Catani:2000jh}. 
The experimental studies for LHC depend heavily on Monte--Carlo event
generators like  PYTHIA which treat QCD observables on the leading order level
along with the modelling of hadronization and radiation effects. A comparison
with theoretical results which  include up-to-date knowledge is a way to test
the reliability of these programs. An adequate tool to do such a comparison is
the DIPHOX code which is a partonic event generator designed for the pair
production of hadrons and/or photons in hadronic collisions at full
next--to--leading order \cite{Binoth:1999qq,Binoth:2001jd}.

Here we present a comparison of $\pi^0\gamma$ observables  relevant for Higgs
search, namely the invariant mass, $M_{\gamma\pi}$  and the transverse momentum
distribution of the pair, $q_T$, with a special emphasis on isolation
criteria.  Both distributions were calculated with PYTHIA
6.152\cite{Sjostrand:1994yb} and DIPHOX. In the former, in order to allow a
comparison, the multiple interactions within the p-p collision were switched
off and the pile-up effects from the collisions within the same bunch crossing
were not taken into account. The comparisons were made with the initial and
final state radiation allowed.

Two isolation criteria were implemented: the transverse energy 
flow isolation and the charged track isolation. In the former,
a threshold is set to the sum of $E_T$ of all particles in the
isolation cone $\Delta R = \sqrt{(\Delta \eta)^2 + (\Delta \phi)^2}$.
In the latter, a threshold is set to the maximum $p_T$ of any
charged particle in the isolation cone. The correlation 
between the two isolation criteria is plotted in Fig.~\ref{Fig:isol_crit}.
\begin{figure}[th]
\begin{center}
\includegraphics[width=14cm]{isolation_comp.eps}
\end{center}
\caption{\label{Fig:isol_crit}{\em Comparison between the 
cone isolation criterion vs. charged track isolation.}}
\end{figure}
As DIPHOX generates only partonic events with a subsequent  collinear
fragmentation of partons into pions, only  the transverse energy  flow
isolation can be implemented. From Fig.~\ref{Fig:isol_crit} it can be inferred
that the maximal hadronic transverse energy inside a cone   lies between 5 and
15 GeV in an experimentally realistic situation. As a reminder, this is the
transverse energy stemming from the partonic reaction only and not from
multiple interactions and/or pile-up.  In this window, the partonic reaction, 
$q g\rightarrow q\gamma$, is by far dominant in the cross section. This 
motivates us to focus on this case only. For the following plots we fixed the
isolation cone size to be $\Delta R=0.4$.

With PYTHIA the process $q + g \rightarrow q + \gamma$ was generated and a
requirement was made that the photon candidates (real photon or a hard $\pi^0$
from the quark jet) have $p_T >$ 25, 40 GeV, and that they are within the
pseudo-rapidity range of $|\eta| < 2.5$. A invariant mass cut of $80 \,
\mbox{GeV} < M_{\pi\gamma} <140\, \mbox{GeV}$ was imposed.  Furthermore, it was
required that one of the photon candidates is a  $\pi^0$, thus rejecting events
were the second photon comes from the jet fragmentation.  With DIPHOX the same
reaction, including next-to-leading corrections, was calculated. The
next-to-leading order parton distribution functions  GRV94\cite{Gluck:1995uf}
were used to compare with the PYTHIA default value.  Only direct photons  were
considered in this study, since in the case of severe isolation, the
contribution from fragmentation is suppressed below the 10 per cent level. For
the fragmentation scale we used the value $M_{\gamma\pi}/8$. This non-canonical
choice is dictated by the fact that in the case  of isolation cuts the typical
fragmentation scale should be  governed by values $\sim E_{T max}$
\cite{Binoth:2002wa}.  

In Fig.~\ref{Fig:mpg} the comparison between PYTHIA and DIPHOX is shown for the
invariant mass distribution of the pion photon pair. Good agreement is found
for the case of very loose isolation,  $E_{T max}=100$ GeV and the more
relevant value of  $E_{T max}=15$ GeV.     Note that the NLO prediction is
plagued by large scale uncertainties around $\pm 30$ to $40$ per cent, as the
hard experimental cuts spoil compensations of higher order terms with opposite
signs. In the case of very strict isolation, $E_{T max}=5$ GeV, PYTHIA produces
somewhat higher rates. The discrepancy has presumably three sources which have
to do with the small value of $E_{T max}$. Small $E_{T max}$ means that the
fragmentation variable $z$, the ratio of transverse pion momentum to parent
parton momentum, is pressed towards one, $z>z_{min}=p_{T min}/(p_{T min}+E_{T
max})\sim 0.8$.  In this regime, first of all, the available fragmentation
functions are not well constrained by  experimental data \cite{Kniehl:2000fe}
and there may be differences in the  fragmentation/hadronization models for
large $z$  implemented in PYTHIA and DIPHOX.  Second, large logarithms 
($\sim\log(1-z)^2$) are present  in the next-to-leading order calculation
which    may render the perturbative contributions unreliable.  Third, the
above mentioned fact that the fragmentation scale choice should be related to 
$E_{T max}$, indeed, indicates that the fragmentation scale should be chosen
smaller here, which  increases the DIPHOX prediction in the right way. With the
value $M_f$=$M_{\gamma\pi}/16$ the DIPHOX curve shifts upwards by about 40 per
cent leading to a better agreement. Altogether one can conclude that PYTHIA and
DIPHOX show a  reasonable agreement in the invariant mass distribution apart
from the regime of very hard isolation which deserves further investigation
from the theoretical side \cite{Binoth:2002wa}.  
% \begin{figure}[th]
% \begin{center}
% \includegraphics[width=10cm]{m_piga_diphox_vs_pythia.eps}
% \end{center}
% \caption{\label{Fig:mpg}{\em Comparison PYTHIA vs. DIPHOX:
% The invariant $\gamma\pi$-mass spectrum 
% for different cone isolation criteria and standard LHC cuts.}}
% \end{figure}
In Fig.~\ref{Fig:qt}, the $q_T$ spectrum is plotted for the same  isolation
criteria. For $E_{T max}=15, 100$ GeV good agreement  is found for the used
scales for the first bins.  The shape for these contributions is not quite the
same, with the PYTHIA prediction being slightly steeper. This is to be
expected, as the NLO calculation  encoded in DIPHOX contains $2\rightarrow 3$
matrix elements, enhancing the tail of the distribution. As a reminder, the
tail in  PYTHIA is filled by a parton showering model which is not as reliable
as explicit higher order matrix elements. Still, the disagreement  in shape for
the relevant $q_T$ domain seems not too alarming in the plotted range.  Note
that in the case of hard isolation the PYTHIA prediction of the last 4 bins is
plagued by low  statistics. The apparent overall discrepancy in the case $E_{T
max}=5$ GeV can be understood by the same reasoning as given above for the 
$M_{\gamma\pi}$ spectrum. 
% \begin{figure}[th]
% \begin{center}
% \includegraphics[width=10cm]{qt_diphox_vs_pythia.eps}
% \end{center}
% \caption{\label{Fig:qt}{\em Comparison PYTHIA vs. DIPHOX:
% The transverse momentum spectrum of $\gamma\pi$ pairs 
% for different cone isolation criteria and standard LHC cuts.}}
% \end{figure}
\begin{figure}[th]
  \bmini
  \bc
  \mbox{\hspace{-.8cm}\includegraphics[height=90mm]{m_piga_diphox_vs_pythia.eps}}
  \ec
  \caption{\label{Fig:mpg}{\em Comparison PYTHIA vs. DIPHOX:
   The invariant $\gamma\pi$-mass spectrum 
   for different cone isolation criteria and standard LHC cuts.}}
  \emini \hspace{.3cm}
                                %\end{figure}
                                %\begin{figure}
  \bmini
  \bc
  \mbox{\includegraphics[height=90mm]{qt_diphox_vs_pythia.eps}}
  \ec
\caption{\label{Fig:qt}{\em Comparison PYTHIA vs. DIPHOX:
The transverse momentum spectrum of $\gamma\pi$ pairs 
for different cone isolation criteria and standard LHC cuts.}}
  \emini
\end{figure}

In conclusion, the comparison  between PYTHIA and the NLO DIPHOX code shows a
reasonable agreement. Whereas infrared insensitive spectra, such as the
invariant mass distribution,  differ only by an overall normalisation which can
be accounted for by an adequate scale choice, infrared sensitive observables
like the transverse momentum distribution of the pair   are typically  steeper
in PYTHIA due to missing NLO matrix elements. In the case of very hard
isolation an apparent difference  between the PYTHIA and DIPHOX predictions can
be traced back to a different modelling of the production of high $p_T$ pions,
a too high fragmentation scale choice and/or subtleties in the interplay
between higher order corrections and severe experimental cuts. This will be
extensively  discussed elsewhere \cite{Binoth:2002wa}. 


