\subsection{Comparison of PYTHIA and DIPHOX for $\gaga$ and $\pi^{0} \gamma$ 
productions \protect\footnote{Contributing authors: T.~Binoth, J.-Ph.~Guillet, 
K.~Lassila-Perini and E.~Tournefier}}\label{intro}

Photon pair production plays a prominent role in the search  for a neutral Higgs
boson at the LHC. In this respect it is relevant to directly compare the
extensively used code PYTHIA~\cite{Sjostrand:1994yb} with a recent
next--to--leading order (NLO) code DIPHOX~\cite{Binoth:1999qq}. DIPHOX is a
computer code of partonic event generator type, describing the production of pairs
of particles in hadronic collisions at full NLO accuracy. PYTHIA is a computer
code of hadronic event generator type which fully describes an hadron-hadron
collision at leading--order (LO) accuracy. The comparisons performed for
the production of $\gamma \gamma$ and $\pi^0 \gamma$ are presented in subsects. 
\ref{sec:gamgam,gamjet,qcdsm} and \ref{sec:pigam,gamjet,qcdsm} respectivelly. 

\subsubsection{Comparison of PYTHIA and DIPHOX for $\gaga$ 
production}\label{sec:gamgam,gamjet,qcdsm}

Here, PYTHIA version 6.152~\cite{Sjostrand:1994yb} is compared to
DIPHOX for the $\gaga$ production and  K-factors are extracted. The scales used
for PYTHIA are the default ones, whereas for DIPHOX all the scales have been
chosen as $\sqrt{\ptgu + \ptgd}$. Note that a K-factor depends, among other
things, on the factorization and renormalization scale chosen and must be used
with care. In order to make easier the comparison, the direct contribution of
the $\gaga$ production (c.f. \cite{Binoth:1999qq}) has been split into two
parts.
\begin{itemize}
\item[a)] A first part, called``initial direct", groups together terms which
have no final state collinear singularities. It contains the Born term  $q +
\bar{q}\rightarrow \gamma + \gamma$, related higher order term $q +
\bar{q}\rightarrow \gamma + \gamma + g$ and the box term $g + g\rightarrow
\gamma + \gamma$.
\item[b)] A second part, called  ``Bremsstrahlung'', contains the left-over term of
the direct contribution, i.e. $q + g\rightarrow \gamma + \gamma + q$ and the LO
terms of the one fragmentation contribution: $q + g\rightarrow \gamma + q$, $q
+ \bar{q}\rightarrow \gamma + g$ where the $q/g$ fragments into a photon.
\end{itemize}

\vspace{0.2cm}

\noindent
{\it a) The initial direct contribution}

\vspace{0.2cm}

\indent
{\it a1) Comparison of PYTHIA without ISR/FSR and hadronisation with
DIPHOX at LO}\\
For this comparison, the CTEQ5L partonic densities and $\asgj$ at 1 loop are
used. To start with, a symmetric cut is applied on the transverse momentum of
the two photons $\ptg>20$~GeV. Then more realistic cuts (the $\Hgg$ selection
cuts) have been applied: $|\etag|<2.5$, $\ptgu>40$~GeV, $\ptgd>25$~GeV. As
shown in Table~\ref{LO} PYTHIA and DIPHOX agree at the few percent level. The
distributions of the variables $\etag$, $\ptg$ and $\mgg$  also agree well.
\begin{table}[h]
\begin{center}
  \begin{tabular}{|l|r|r||r|r|}
    \hline
    & \multicolumn{2}{|c|}{$\ptg>20$ GeV} & \multicolumn{2}{|c|}{$\Hgg$ selection cuts }  \\
    \hline
          & PYTHIA & DIPHOX & PYTHIA & DIPHOX\\
    Born  & 82.3 & 83.2 & 10.2 & 9.8\\
    \hline
    Box   & 82.3 & 83.2 & 5.7 & 5.6\\
    \hline
    Total & 164.6 & 166.4  & 15.9 & 15.5\\
    \hline
\end{tabular}
\end{center}
\caption{Comparison of DIPHOX and PYTHIA cross sections (in pb) 
for the Born and the 
Box terms with minimal cuts ($\ptg>20$ GeV) and after $\Hgg$ selection cuts.
\label{LO}}
\end{table}

\vspace{0.1cm}

\indent
{\it a2) NLO corrections to Born}\\
The K-factor K$_{\rm{NLO/LO}}$ is defined as the ratio of the NLO to the LO 
cross sections obtained with DIPHOX. The CTEQ5L pdf and $\asgj$ at  1 loop have
been used to compute the LO cross section whereas the CTEQ5M pdf and $\asgj$ at 2
loops have been used to compute the NLO cross section. The Table~\ref{DiNLO}
gives the value of the cross sections and of K$_{\rm{NLO/LO}}$ for different
stages of the $\Hgg$ selection.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|r|r|c|}
\hline
Cut                           & LO   & NLO  & K$_{\rm{NLO/LO}}$ \\
\hline
$\ptg>20$ GeV                 & 83.2 & 94.9 & 1.14 \\
\hline
+ $|\etag|<2.5$               & 32.5 & 41.0 & 1.26 \\ 
\hline
+ $\ptg>25$ GeV               & 18.8 & 23.6 & 1.26 \\
\hline
+ Max($\ptgu$,$\ptgd$)$>$40 GeV & 5.7 &10.4 & 1.83 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of LO and NLO cross section (in pb) for the Born term
at different stage of the $\Hgg$ selection.\label{DiNLO} }
\end{table}
The asymmetric cut results in an enhancement of the K-factor. The reason is the
following: at LO the two photons have the same $p_T$ whereas this is not the
case at NLO because the $p_T$  balance is distorted. Therefore, at LO, the cut
Max($\ptgu$,$\ptgd$)$>$40 GeV  is equivalent to a symmetric cut of 40 GeV on
both photons.  This cut acts more efficiently on LO than on NLO contributions.
After all selection cuts, and if one restricts  in addition to $\mgg>90$~GeV,
K$_{\rm{NLO/LO}}$ is  independent of $\mgg$  and is equal to 1.57.\\

ATLAS and CMS have based their prediction on PYTHIA with ISR. In order to
obtain the correction which has to be applied to their numbers
(K$_{\rm{NLO/ISR}}$) we  have compared PYTHIA with ISR to DIPHOX at NLO. The
cross sections are given in Table~\ref{PytDi}. PYTHIA  with ISR includes part
of the NLO corrections; therefore the effect of the  asymmetric cut on $\ptg$
is reduced and  K$_{\rm{NLO/ISR}}$ is smaller than K$_{\rm{NLO/LO}}$. Since
there is no hard radiation in PYTHIA, the cross section in the high $q_T$
region  is underestimated as is shown in Fig.~\ref{Qtborn}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|r|l|l|c|}
\hline
Cut                           & PYTHIA with ISR   & DIPHOX at NLO  & K$_{\rm{NLO/ISR}}$ \\
\hline
$\ptg>20$ GeV                 & 68.8 & 94.9 & 1.38 \\
\hline
+ $|\etag|<2.5$               & 27.6 & 41.0 & 1.49 \\ 
\hline
+ $\ptg>25$ GeV               & 16.0 & 23.6 & 1.48 \\
\hline
%+ Max($\ptgu$,$\ptgd$)$>$40 GeV & 6.42 & 9.53 & 1.48 \\ avec Mgg>80
+ Max($\ptgu$,$\ptgd$)$>$40 GeV & 6.9 &10.4 & 1.51 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of PYTHIA (CTEQ5L+ $\asgj$ at 1 loop) with ISR and DIPHOX 
NLO cross section (in pb) for the Born term
at different stage of the $\Hgg$ selection.\label{PytDi} }
\end{table}

\begin{figure}
  \begin{minipage}{.48\linewidth}
    \bc 
    \mbox{\hspace{-.8cm}\includegraphics[height=90mm]{qt_born.eps}}
%     \mbox{\hspace{-.8cm}\epsfig{file=qt_born.eps,height=90mm}}
    \ec
    \caption{$Q_t$ distribution for PYTHIA with ISR and DIPHOX NLO after all selection cuts
      for the Born contribution.\label{Qtborn}}
    \end{minipage} \hspace{.3cm}
    \begin{minipage}{.48\linewidth}
    \bc \vspace{.7cm}
%     \mbox{\epsfig{file=mgg_born+box.eps,height=81mm}}
    \mbox{\includegraphics[height=81mm]{mgg_born+box.eps}}
    \ec
    \caption{$\mgg$ distribution for PYTHIA with ISR and DIPHOX NLO after all selection cuts
      for the Born plus Box contribution.\label{mggborn+box}}
    \end{minipage} 
\end{figure}
%
% After all cuts and for $\mgg >80$~GeV, the K-factor is independent of $\mgg$ 
% and is equal to 1.48 GeV.
\indent
{\it a3) The Box contribution}\\
Since there is no higher order calculation for the Box contribution, we have
studied the effect of switching on ISR and of including  the running of $\asgj$
at 2 loops. PYTHIA with ISR results in a cross section larger by 40$\%$ after
all selection cuts.  This increase is again an effect of the asymmetric cuts.
For $\mgg>80$~GeV the ratio is equal to 1.19. The inclusion of $\asgj$ at 2 loops
decreases the cross section by $23\%$. Therefore, the effect of including ISR
in PYTHIA almost cancels the effect of the  running of $\asgj$. Thus we keep the
LO calculation with $\asgj$ at one loop for the DIPHOX 'NLO' calculation in the
following.

\vspace{0.2cm}

\indent
{\it a4) K-factor for the direct $\gaga$ production}\\
In this section, we compare the cross sections obtained by PYTHIA with ISR
(which correspond to the experiment's Monte--Carlo) to the NLO prediction from
DIPHOX for the direct $\gaga$ production (i.e. Born plus Box). PYTHIA is used
with CTEQ5L and $\asgj$ at 1 loop. The DIPHOX 'NLO' contribution includes Born at
NLO with CTEQ5M and $\asgj$ at 2 loops (as in {\it a2)}) and Box at LO
with CTEQ5L and $\asgj$ at 1 loop (as in {\it a3)}). 
% As has been shown in \ref{box} some of the 
% higher order effects (ISR) almost cancel the inclusion of $\asgj$ at 2 loops, 
% therefore it was decided to keep $\asgj$ at 1 loop for the estimation of the Box
% contribution in DIPHOX.
The cross sections  are given in Table~\ref{sum} and Fig.~\ref{mggborn+box}
shows the $\mgg$ distribution after all selection cuts for PYTHIA and DIPHOX.

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|r|r|r||r|r|r|c|}
\hline
Cut      & \multicolumn{3}{|c|}{PYTHIA with ISR}   & \multicolumn{3}{|c|}{DIPHOX 'NLO'} & K$_{\rm{NLO/ISR}}$ \\
\hline
                       & Box & Born &Total& Box & Born &Total& \\
\hline
$\ptg>20$ GeV                 & 60.3 & 68.8 &129.1 & 83.2 & 94.9 & 178.1&1.38 \\
\hline
+ $|\etag|<2.5$               & 41.4 & 27.6 & 69.0 &56.3 & 41.0 & 97.3 &1.41 \\ 
\hline
+ $\ptg>25$ GeV               & 18.9 & 16.0 & 34.9 & 25.7 & 23.6 & 49.3& 1.41 \\
\hline
+ Max($\ptgu$,$\ptgd$)$>$40 GeV & 6.6 & 6.9 & 13.5 &4.7 & 10.4 & 15.1& 1.12 \\
\hline
+ $\mgg>80$GeV                & 5.6 & 6.4 & 12.0&4.7 & 9.5 & 14.2&1.18 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of PYTHIA (CTEQ5L+ $\asgj$ at 1 loop) with ISR and DIPHOX 
'NLO' cross section (in pb) at different stage of the $\Hgg$ selection for Born plus
Box contributions.
The K factor is the ratio of the total cross sections for PYTHIA with 
ISR and DIPHOX 'NLO'.
\label{sum} }
\end{table}

\noindent 
{\it b) The Bremsstrahlung contribution}\\
In PYTHIA the Bremsstrahlung contribution is obtained with the processes
$qg\rightarrow q\gamma$ and $q\bar{q}\rightarrow g \gamma$ with QCD ISR and
FSR. The second photon is produced mainly by QED FSR. The cross sections are
summarized in Table~\ref{bremtab}. Note that the isolation cut reduces the
cross section by less than a factor 2. All the cross sections are obtained with
CTEQ5L and $\asgj$ at 1 loop. ISR/FSR in PYTHIA does not produce as high $\ptg$
as DIPHOX NLO, therefore the cross section is smaller in PYTHIA. Moreover the
asymmetric cut on $\ptg$ enhances this difference: since PYTHIA is a quasi
$2\rightarrow 2$ generator it does not produce high $p_T$ difference between
the photons.\\  An isolation cut is defined by imposing that $\Delta R > 0.4$
{\bf or} $\ptq<10$~GeV.  For DIPHOX, $\ptq$ is either the $p_T$ of the parton
$p$ produced with the photon pair or the transverse energy  of the residue of
the fragmentation flowing along the photon direction.  $\Delta R$ is  the
smallest distance between the parton $p$ produced with the photon pair and the
photons \footnote{In the case of the two to two kinematics, $\Delta R = 0$},
$\Delta R = \rm{Min} ((\eta_p-\eta_{\gamma_{1,2}})^2 + (\phi_p -
\phi_{\gamma_{1,2}})^2)$. For PYTHIA, $\ptq = || \sum_i \vec{p}_{T \, i} ||$
where the sum runs on all the partons produced in the shower going with the
photon and $\Delta R$ is  the smallest distance in rapidity-azimuthal angle
plane between the parton produced in the $2 \rightarrow 2$ hard scattering and
the photons. The isolation cut also slightly enhances the K-factor. It might be
due to the fact that  the quark tends to be more collinear to the radiated
photon in  PYTHIA. Figure~\ref{mggbrem} shows the $\mgg$ distribution after all
selection cuts and Figure~\ref{qtbrem} shows the $q_T$ distribution.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|r|r|c|}
\hline
Cut      &              PYTHIA (ISR/FSR)   & DIPHOX  & K$^{\rm{brem}}_{\rm{DIPHOX/PYTHIA}}$ \\
\hline
$|\etag|<2.5$,  $\ptg>25$ GeV & 22.2 & 27.5 &  1.24\\ 
\hline
+ Max($\ptgu$,$\ptgd$)$>$40 GeV & 14.9 & 24.7  & 1.66 \\
\hline
+ $\mgg>80$GeV                & 11.9 & 19.7  & 1.66 \\
\hline
$\Delta r > 0.4$ or $\ptq<10$~GeV & 7.4 & 12.8 & 1.72 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of PYTHIA (CTEQ5L+ $\asgj$ at 1 loop) with ISR/FSR and 
DIPHOX  cross section (in pb) at different stage of the $\Hgg$ selection
for the Bremsstrahlung contribution.
\label{bremtab} }
\end{table} 
%
\begin{figure}
  \bmini
  \bc
%   \mbox{\hspace{-.8cm}\epsfig{file=,height=90mm}}
  \mbox{\hspace{-.8cm}\includegraphics[height=90mm]{mgg_brem.eps}}
  \ec
  \caption{$\mgg$ distribution for PYTHIA and DIPHOX after all selection cuts
    for the Bremsstrahlung contribution.\label{mggbrem}}
  \emini \hspace{.3cm}
                                %\end{figure}
                                %\begin{figure}
  \bmini
  \bc
%   \mbox{\epsfig{file=qt_brem1.eps,height=90mm}}
  \mbox{\includegraphics[height=90mm]{qt_brem1.eps}}
  \ec
  \caption{$Q_t$ distribution for PYTHIA and DIPHOX after all selection cuts
    for the Bremsstrahlung contribution.\label{qtbrem}}
  \emini
\end{figure}
% \begin{figure}
%   \bc
%   \mbox{\epsfig{file=dr_brem.eps,height=90mm}}
%   \ec
%   \caption{$\Delta r$ distribution for PYTHIA and DIPHOX before the isolation cut
%     for the Bremsstrahlung contribution. Only events with $\ptq>10$~GeV are shown.
%     \label{deltar}}
% \end{figure}
%

\noindent
{\it Conclusion}\\
The contribution of the two photon background for the $H\rightarrow \gaga$
search  at the LHC has been estimated at NLO with DIPHOX and compared to PYTHIA
Monte--Carlo. The K-factor has to be used with care since it depends on the
selection cuts notably on the isolation criteria.  Note that the asymmetric cut
on $\ptg$ is responsible  for an enhancement of the K-factor. 
