
\subsection{Tuned comparisons \label{TUNED}}

The typical procedure followed in the literature to establish the technical precision of %a given 
the theoretical tools is to perform tuned comparisons between 
the predictions of independent 
programs using the same
set of input parameters and experimental cuts. This strategy was initiated 
in the 90s during the CERN workshops 
for precision physics at LEP and is still in use when considering processes of interest
for physics at hadron colliders
demanding particularly
accurate theoretical calculations. The tuning procedure is a key step in the validation of generators, 
because it allows to check that the different details entering the complex structure of 
the generators, e.g. the implementation of radiative corrections, event selection routines, MC integration and 
event generation, are under control, and to fix possible mistakes. 

The tuned comparisons discussed in the following were performed switching off the
 vacuum polarisation  correction to the Bhabha scattering cross section. 
Actually, the generators implement the non-perturbative hadronic contribution to the 
running of $\alpha$ according to different parameterisations, which differently affect the 
cross section prediction (see Section \ref{sec:4} for discussion). Hence, this simplification is introduced to avoid possible bias in the interpretation of the results and allows to disentangle the effect of pure QED corrections. 
Also, in order to provide useful results for the experiments, the comparisons take into account {\it realistic} event selection cuts. 

The present Section is a merge of results available in the literature 
\cite{Balossini:2006wc} with those of new studies. The results refer to the Bhabha process at the energies of 
$\mathrm{\phi}$, $\tau$-charm and $B$ factories. No tuned comparisons for the two photon
production process have been carried out.

\subsubsection{$\mathrm{\phi}$ and $\tau$-charm factories \label{phic}}

First we show comparisons between BabaYaga@NLO and BHWIDE according to the 
KLOE selection cuts of Eq.~(\ref{eq:cutsmf}), considering 
also the angular range $20^\circ \leq \vartheta_{\pm} \leq 160^\circ$ 
for cross section results. The predictions of the two codes are reported in 
Table~\ref{tabtun:1} for the two 
acceptance cuts together with their relative deviations. As can be seen the agreement is excellent, 
the relative deviations being well below the 0.1\%. Comparisons between BabaYaga@NLO and 
BHWIDE at the level of differential distributions are
given in Figs.~\ref{figtun:1} and~\ref{figtun:2} where the inset shows the relative deviations
between the predictions of the two codes. As can be seen there is 
very good agreement between the two generators, and the predicted distributions 
appear at a first sight almost indistinguishable. Looking in more detail, there is a 
relative difference of a few per mill for the acollinearity
distribution (Fig.~\ref{figtun:2}) and
of a few per cent for the invariant mass  (Fig.~\ref{figtun:1}), but
only in the very hard tails, where the fluctuations observed are due to limited MC statistics. These
configurations however give a negligible contribution
 to the integrated cross section, a factor $10^{3} \div 10^{4}$ smaller than that around the very dominant peak 
regions. In fact these differences on differential distributions translate into 
agreement on the cross section values well below the one per mill, 
as shown in Table~\ref{tabtun:1}.  

Similar tuned comparisons were performed between the results of 
BabaYaga@NLO, BHWIDE and MCGPJ in the presence of cuts modelling the event selection 
criteria of the CMD-2 experiment at the VEPP-2M collider, for a c.m. energy of $\sqrt{s} = 900$~MeV. 
The cuts used in this case are
\begin{eqnarray}
&& | \theta_- + \theta_+ - \pi | \leq \Delta\theta , \quad 1.1 \leq (\theta_+ - \theta_- + \pi)/2 \leq \pi-1.1, \nonumber\\
&& | |\phi_- + \phi_+|-\pi | \leq 0.15 , \nonumber\\
&& p_- \sin(\theta_-) \geq 90~{\rm MeV} , \qquad \, \qquad p_+ \sin(\theta_+) \geq 90~{\rm MeV},  \nonumber\\
&& (p_- + p_+)/2 \geq 90~{\rm MeV} ,
\label{eq:acmd2}
\end{eqnarray}
where $\theta_-,\theta_+$ are the electron/positron polar angles, respectively, 
$\phi_{\pm}$ their azimuthal angles, and $p_{\pm}$ %= \sqrt{p_{\pm, x}^2+ p_{\pm, y}^2+ p_{\pm, z}^2}$ 
the moduli of their three-momenta. $\Delta\theta$ stands for an acollinearity
cut. 

\begin{table}
\caption{Cross section predictions [nb] of BabaYaga@NLO and BHWIDE for the Bhabha cross section 
corresponding to two different angular acceptances, for 
the KLOE experiment at DA$\mathrm{\Phi}$NE, and their relative differences (in per cent).}
\label{tabtun:1}
\begin{center}
\begin{tabular}{clll}
%  \begin{array}{|c|c|c|c|}
    \hline
    angular acceptance & BabaYaga@NLO & BHWIDE & $\delta(\%)$ \\ %%[1mm]
    \hline
%    \hline
    $20^\circ \div 160^\circ$ & 6086.6(1) & 6086.3(2)& 0.005\\ %[1mm]
     \hline 
    $55^\circ \div 125^\circ$ & 455.85(1) & 455.73(1)& 0.030\\ %[1mm]
    \hline
 \end{tabular}
 \end{center}
\end{table}

Figure~\ref{figtun:3} shows the relative differences between the results of BHWIDE and MCGPJ according
to the criteria of Eq.~(\ref{eq:acmd2}), as a function of the acollinearity cut $\Delta\theta$. The
relative deviations between the results of BabaYaga@NLO and MCGPJ for the same cuts
are given in Fig.~\ref{figtun:4}. It can be seen that the predictions of the three generators lie 
within a $0.2\%$ band with differences of $\sim 0.3\%$ for extreme values of the acollinearity cut. 
This agreement can be considered satisfactory since for the acollinearity cut of 
real experimental interest ($\Delta\theta \approx 0.2$~rad) the generators agree 
within one per mill.

A number of comparisons were also performed for a c.m. energy of 3.5~GeV relevant to the experiments 
at $\tau$-charm factories. An example is given in Table~\ref{tabtun:2}
 where the predictions of BabaYaga@NLO and
MCGPJ are compared, using cuts similar to those of Eq.~(\ref{eq:acmd2}) 
and for an acollinearity cut 
of $\Delta\theta = 0.25$~rad. The agreement between the two codes is below
 one per mill. 
Comparisons between the two codes were also done at the level of differential cross 
sections, showing satisfactory agreement in the statistically relevant phase space regions.
Preliminary results \cite{Ping:2009xxxx} for a c.m. energy on top of the $J/\Psi$ resonance show good agreement between BabaYaga@NLO and BHWIDE predictions too.

\begin{table}
\caption{Cross section predictions [nb] of BabaYaga@NLO and MCGPJ for the Bhabha cross section 
at $\tau$-charm factories ($\sqrt{s} = 3.5$~GeV) and their relative difference (in per cent).}
\label{tabtun:2}
\begin{center}
\begin{tabular}{lll}
%  \begin{array}{|c|c|c|c|}
    \hline
    BabaYaga@NLO & MCGPJ & $\delta(\%)$ \\ %%[1mm]
    \hline
%    \hline
    35.20(2) & 35.181(5)& 0.06 \\ %[1mm]
     \hline 
 \end{tabular}
 \end{center}
\end{table}

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{invmass-bhwide.eps}
}
\caption{Invariant mass distribution of the Bhabha process 
according to BHWIDE and BabaYaga@NLO,  for the KLOE experiment at 
DA$\mathrm{\Phi}$NE, and relative differences of the program predictions (inset). From \cite{Balossini:2006wc}.}
\label{figtun:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{acoll-bhwide.eps}
}
\caption{Acollinearity distribution of the Bhabha process 
according to BHWIDE and BabaYaga@NLO,  for the KLOE experiment at 
DA$\mathrm{\Phi}$NE, and relative differences of the program predictions (inset). From \cite{Balossini:2006wc}.}
\label{figtun:2}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{bhwide_mcgpj-alexei.eps}
}
\caption{Relative differences between BHWIDE and MCGPJ Bhabha cross sections as a function 
of the acollinearity cut, for the CMD-2 experiment at VEPP-2M.}
\label{figtun:3}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{BabaYagaatNLO-al.eps}
}
\caption{ Relative differences between BabaYaga@NLO and MCGPJ Bhabha cross sections as a function 
of the acollinearity cut, for the CMD-2 experiment at VEPP-2M.}
\label{figtun:4}
\end{center}
\end{figure}

\subsubsection{$B$ factories \label{bfac}}

Concerning the $B$ factories, a considerable effort was done to establish the level of 
agreement between the generators BabaYaga@NLO and BHWIDE in comparison with BabaYaga v3.5 too.
This study made use of the realistic luminosity cuts 
quoted in Section \ref{4pfinal} for the BaBar experiment. The
cross sections predicted by BabaYaga@NLO and BHWIDE are shown in 
Table~\ref{tabtun:3}, together with
the corresponding relative differences as a function of the considered angular range.  
The latter are also shown in Fig.~\ref{figtun:5},  where the 1$\sigma$ numerical error due to MC statistics 
is also quoted. As can be seen, the two codes agree nicely, the predictions for the 
central value being in general in agreement at the 0.1\% level or statistically compatible whenever a 
two to three per mill difference is present.

\begin{table}
\caption{Cross section predictions [nb] of BabaYaga@NLO and BHWIDE for the Bhabha cross section 
as  a function of the angular selection cuts for the BaBar experiment at PEP-II and 
absolute value of their relative differences (in per cent).}
\label{tabtun:3}
\begin{center}
\begin{tabular}{clll}
%%  \begin{array}{|c|c|c|c|}
    \hline
    angular range (c.m.s.) & BabaYaga@NLO & BHWIDE &$|\delta (\%)|$ \\ %[1mm]
%    \hline
    \hline
    $15^{\circ}\div 165^{\circ}$ & 119.5(1) & 119.53(8)&0.025\\ %[1mm]
     \hline 
    $30^{\circ}\div 150^{\circ}$ & 24.17(2) & 24.22(2)&0.207\\
        \hline
    $40^{\circ}\div 140^{\circ}$ & 11.67(3) & 11.660(8)&0.086\\
        \hline
    $50^{\circ}\div 130^{\circ}$ &  6.31(3) & 6.289(4)&0.332\\
        \hline
    $60^{\circ}\div 120^{\circ}$ & 1.928(2) & 1.931(3)&0.141\\
        \hline
    $70^{\circ}\div 110^{\circ}$ & 3.554(6) & 3.549(3)&0.155\\
        \hline
    $80^{\circ}\div 100^{\circ}$ & 0.824(2) & 0.822(1)&0.243\\
    \hline
 \end{tabular}
 \end{center}
\end{table}

To further investigate how the two generators compare with each other
 a number of differential cross sections were
studied. The results of this study are shown in Figs.~\ref{figtun:6} and \ref{figtun:7} 
for the distribution of the electron energy and the polar
angle, respectively, and in Fig.~\ref{figtun:8} for the acollinearity. For both the energy and
scattering angle distribution, the two programs agree within the statistical errors showing deviations 
below 0.5\%. For the acollinearity dependence of the cross section, BabaYaga@NLO
and BHWIDE agree within $\sim 1\%$. Therefore, the level of the agreement between the two codes 
around 10~GeV is the same as that observed at 
the $\mathrm{\phi}$ factories.

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{sigmaBHvsBNLO-Babar.eps}
}
\caption{Relative differences between BabaYaga@NLO and BHWIDE Bhabha cross sections as a function of the angular acceptance cut for the BaBar experiment at PEP-II. From \cite{andreas:2009xxxx}.}
\label{figtun:5}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{ele-andreas.eps}
}
\caption{Electron energy distributions according to BHWIDE, BabaYaga@NLO and BabaYaga v3.5
for the BaBar experiment at PEP-II and relative differences of the 
predictions of the programs. From \cite{andreas:2009xxxx}.}
\label{figtun:6}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{elangle-andreas.ps}
}
\caption{Electron polar angle distributions according to BHWIDE, BabaYaga@NLO and BabaYaga v3.5
for the BaBar experiment at PEP-II and relative differences of the predictions of the programs. 
From \cite{andreas:2009xxxx}.}
\label{figtun:7}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{acoll-andreas.eps}
}
\caption{Acollinearity distributions according to BHWIDE, BabaYaga@NLO and BabaYaga v3.5
for the BaBar experiment at PEP-II and relative differences of the predictions of the programs. 
From \cite{andreas:2009xxxx}.}
\label{figtun:8}
\end{center}
\end{figure}

The main conclusions emerging from the tuned comparisons 
discussed in the present Section can be summarised as follows:

\begin{itemize}

\item The predictions for the Bhabha  cross section of the most precise tools, i.e. BabaYaga@NLO, BHWIDE and MCGPJ, generally 
agree within 0.1\%. If (slightly) lar\-ger differences are present they show up for particularly 
tight cuts
or are due to limited MC statistics. When statistically meaningful discrepancies are observed they 
can be ascribed to the different theoretical recipes for the treatment of radiative corrections and their
technical implementation. For example, as already emphasised, BabaYaga@NLO and 
BHWIDE adopt a fully factorised prescription for the matching of NLO and HO corrections, 
whereas MCGPJ implement some pie\-ces of the 
radiative corrections in additive form. This can give
rise to discrepancies between the programs' predictions, especially in the presence of
tight cuts enhancing the effect of soft radiation. Furthermore, different choices are adopted in the
generators for the scale entering the collinear logarithms in HO corrections 
beyond $O (\alpha)$, 
which are
another possible source of the observed differences. To go beyond the present situation, a 
further nontrivial effort should be done by comparing, for instance, the programs in the 
presence of NLO corrections only (technical test) and by analysing their different 
treatment of the
exponentiation of soft and collinear logarithms. This would certainly shed light on the origin
of the (small) discrepancies still registered at present.

\item Also the distributions predicted by the generators agree well, with relative differences 
below the 1\% level. Slight\-ly larger discrepancies are
only seen in sparsely populated phase space regions corresponding to very hard photon emission 
which do not influence the luminosity measurement noticeably.


\end{itemize}



