
\subsection{Conclusions and open issues \label{CONCLUSIONS}}

During the last few years a remarkable progress occurred in reducing the error 
of the luminosity measurements at flavour factories. 
%These 
%advances were presented during the workshop, together with 
%original work towards more and more reliable theoretical predictions and 
%a better control of the technical precision of the analysis tools.

Dedicated event generators like
BabaYaga@NLO and MCGPJ were developed in 2006 to provide predictions for the cross section 
of the large-angle Bhabha process, as well as for other QED reactions of interest, with a
theoretical accuracy at the level of  0.1\%. In parallel codes well-known since 
the time of the LEP/SLC operation such as BHWIDE were extensively used by the experimentalists in
data analyses. All these MC programs include, albeit according to different formulations, 
exact $O (\alpha)$ QED corrections matched with LL contributions describing
multiple photon emission. Such ingredients, together with the vacuum polarisation 
correction, are strictly necessary to achieve a physical 
precision down to the per mill level. Indeed, when considering typical selection, cuts
the NLO photonic corrections amount to about 15$\div$20\%, vacuum polarisation contributes 
at the several per cent level and HO effects lie between 1$\div$2\%.

The generators mentioned are, however, affected by 
an uncertainty due to HO effects neglected in their formulation, such as light pair
corrections or exact perturbative contributions present in NNLO calculations. 
From this point of view  the great progress in the calculation of two-loop corrections to the Bhabha scattering cross section was essential to establish the 
theoretical accuracy of the existing generators and 
will be crucial if an improvement 
of the precision below the one per mill level will be required.

A particular effort was done to compare the predictions of the generators 
consistently, in order to assess the technical precision obtained by the implementation of radiative corrections 
and related computational details. These comparisons were performed in the presence of realistic 
event selection criteria and at different c.m. energies. For the KLOE and CMD-2 experiments around the
$\mathrm{\phi}$-resonance, where the statistics of Bhabha events is the highest and the experimental luminosity
error at a few per mill level, the cross section results of BabaYaga@NLO, BHWIDE and MCGPJ agree 
within $\sim 0.1$\%. If (slightly) larger discrepancies are observed, they show up only for particularly
tight cuts or exclusive distributions in specific phase space regions which do not influence the 
luminosity determination. Very similar results were obtained for $\tau$-charm and $B$ factories.
The main conclusion of the work on tuned comparisons is that the technical precision of MC programs
is well under control, the discrepancies being 
%definitely smaller than the %respective experimental error. 
due to different details in the treatment of the same sources of radiative corrections and their
technical implementation. For example, BabaYaga@NLO and 
BHWIDE adopt a fully factorised prescription for the matching of NLO and HO corrections, 
whereas MCGPJ implement some radiative corrections pieces in additive form. This can give
rise to some discrepancies between their predictions, especially in the presence of
tight cuts enhancing the effect of soft radiation. Furthermore, different choices are adopted in the
generators for the energy scale in the treatment of 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 for the specific effect due to the 
 exponentiation of soft and collinear logarithms. This would certainly shed light on the origin
of the (minor) discrepancies still registered at present.

On the theoretical side, a new exact evaluation of lepton and hadron pair corrections to the Bhabha scattering cross section was carried out, taking into account realistic cuts. This calculation provides results in 
substantial agreement with estimates based on singlet SF but supersedes previous evaluations in the soft-photon approximation. The results of the new exact calculation were preliminarily compared with the predictions of BabaYaga@NLO, which includes the bulk of such corrections (due to reducible contributions) through the insertion of the vacuum polarisation correction in the NLO diagrams, but neglects the effect of real pair radiation and two-loop form factors. It turns out that the error induced by the approximate treatment of pair corrections amounts to a few units in $10^{-4}$, both at KLOE and BaBar. Further work is in progress to arrive at a 
more solid and 
quantitative error estimate for these corrections when considering other 
selection criteria and c.m. energies too \cite{pairs}.
Also, the contribution induced by the uncertainty 
related to the non-perturbative contribution to the running of $\alpha$ was revisited,
making use of and comparing the two independent parameterisations 
derived in \cite{Jegerlehner:1985gq,Burkhardt:1989ky,Jegerlehner:2006ju} and \cite{rintpl:2008AA}.


A summary of the different sources of theoretical error and their relative impact on the Bhabha 
cross section is given in Table~\ref{tabcon:1}.  In Table~\ref{tabcon:1}, $|\delta^{\rm err}_{\rm VP}|$ 
is the error induced by the hadronic component of the vacuum polarisation, 
$|\delta^{\rm err}_{\rm pairs}|$ the error due to missing pair corrections, 
$|\delta^{\rm err}_{\rm SV}|$ the uncertainty coming from SV NNLO corrections, 
 $|\delta^{\rm err}_{\rm HH}|$ the uncertainty in the calculation of the double 
 hard bremsstrahlung 
 process %beyond the collinear approximation 
 and $|\delta^{\rm err}_{\rm SV,H}|$ the error 
 estimate for one-loop corrections to single hard bremsstrahlung. As can be seen,
 pair corrections and exact NLO corrections to $e^+ e^- \to e^+ e^-\gamma$ are the
 dominant sources of error. 

\begin{table}[thb]
\caption{Summary of different
sources of theoretical uncertainty for the most precise generators used
for luminosity measurements and the corresponding total
theoretical errors for the calculation of the large-angle Bhabha cross section 
at meson factories.}
\label{tabcon:1}
\begin{center}
%%\begin{tabular}{|c|c|c|c|c|}
\begin{tabular}{llll}
\hline
Source of error (\%) & $\mathrm{\phi}$  & $\tau$-charm & $B$ \\
\hline
 $|\delta^{\rm err}_{\rm VP}|$~\cite{Jegerlehner:1985gq,Burkhardt:1989ky,Jegerlehner:2006ju}        & 0.00 & 0.01 & 0.03 \\
\hline
 $|\delta^{\rm err}_{\rm VP}|$~\cite{rintpl:2008AA}      & 0.02 & 0.01 & 0.02 \\ 
  \hline
 $|\delta^{\rm err}_{\rm SV}|$ & 0.02 & 0.02 & 0.02\\
 \hline
 $|\delta^{\rm err}_{\rm HH}|$       & 0.00 & 0.00 & 0.00\\
 \hline
 $|\delta^{\rm err}_{\rm SV,H}|$      & 0.05 & 0.05 & 0.05 \\
 \hline
 $|\delta^{\rm err}_{\rm pairs}|$     & 0.05 & 0.1 %\footnote{This is a guess, waiting for  the final numbers} 
 & 0.02 \\
\hline
 $|\delta^{\rm err}_{\rm total}|$     & 0.12$\div$0.14 & 0.18 &  0.11$\div$0.12\\
\hline
\end{tabular}
\end{center}
\end{table}


The total theoretical uncertainty as
obtained by summing the different contributions linearly is 0.12$\div$0.14\% 
at the $\mathrm{\phi}$ factories, 
0.18\% at the $\tau$-charm factories and $0.11\div 0.12$\% at the $B$ factories.  As can be seen from Table~\ref{tabcon:1}, the slightly larger uncertainty at the $\tau$-charm factories is mainly due to the pair contribution error, which is presently based on a very preliminary evaluation and for which a deeper analysis is ongoing \cite{pairs}. The
total uncertainty is slightly affected by the particular choice of the 
routine for the calculation of  $\Delta\alpha^{(5)}_{\rm hadr}(q^2)$, 
 since the two parameterisations considered here give rise to 
 similar errors, with the exception of the $\mathrm{\phi}$ factories 
 for which the two recipes return uncertainties differing by $2 \times 10^{-4}$. However 
 the ``parametric'' error induced by the hadronic contribution to the vacuum polarisation may become 
 a relevant source of uncertainty  when considering predictions for a c.m. energy on top of and closely 
 around very narrow resonances. For such a specific situation of interest, for instance for the BES experiment, 
%%%TT
 the appropriate treatment of the running $\alpha$ in the calculation of the Bhabha cross section should be scrutinised deeper because of the differences observed between the predictions for $\Delta\alpha^{(5)}_{\rm hadr}(q^2)$ obtained by means of the different parametrisation routines available (see Section \ref{sec:4} for a more detailed discussion).
 
Although the theoretical uncertainty quoted in Table~\ref{tabcon:1} could be put on firmer 
ground thanks to further studies in progress, it appears to be quite robust and sufficient for 
present and planned precision luminosity measurements at meson factories, where the experimental 
error currently is about a factor of two or three larger. 
Adopting the strategy followed during the LEP/SLC operation
one could arrive at a more aggressive error estimate by summing the relative
contributions in quadrature. However, for the time
being, this does not seem to be necessary in the light of the current experimental errors.

%It is worth noticing, in conclusion, that 
In conclusion, the precision presently reached by large-angle Bhabha programs used in the luminosity measurement at meson factories is comparable 
with that achieved about ten years ago for luminosity monitoring 
through small-angle Bhabha scattering at LEP/SLC. 

%The work done during the workshop left open some issues. 
Some issues are still left open.
In the context of tuned comparisons, no 
effort was done to compare the available codes for the process of photon pair production. Since it contributes relevantly to the luminosity determination and as precise predictions for its 
cross section can be obtained by means of the codes BabaYaga@NLO and MCGPJ, this work should 
be definitely carried out. This would lead to a 
better understanding of the luminosity on the experimental side. In the framework of new theoretical advances, 
an evaluation of %the light 
%pair contribution 
NNLO contributions to the process $e^+ e^- \to \gamma\gamma$ would be 
worthwhile to better assess the
precision of the generators which, for the time being, do not include such corrections 
exactly. 
More importantly, the exact one-loop corrections to the radiative process $e^+ e^- \to e^+ e^- \gamma$ should be calculated going beyond the partial results scattered in the literature 
(and referring to selection criteria valid for high-energy $e^+ e^-$ colliders)  
or limited to the soft-photon approximation.\footnote{As already remarked in Section 
\ref{TH}, during the completion of the present work a complete calculation of the NLO QED corrections to hard bremsstrahlung emission in full $s+t$ Bhabha scattering was performed in \cite{Actis:2009uq}. However, explicit comparisons between the predictions of this new calculation and the corresponding results of the most precise luminosity tools  are still missing and would be needed to better assess the theoretical error induced by such 
contributions in the calculation of the luminosity cross section.} Furthermore, to get a better control of the theoretical uncertainty in the sector of NNLO corrections to Bhabha scattering, the radiative Bhabha process at one-loop should be evaluated taking into account the typical experimental cuts used at meson factories. Incidentally this calculation would be also
of interest for other studies at $e^+ e^-$ 
colliders of moderately high energy, such as the search for new physics 
phenomena (e.g. dark matter candidates), for which radiative Bhabha scattering is a very important background. \\
 

% Useful discussions with S. Eidelman, F. Jegerlehner, G. Venanzoni {\em ... ? others ... to be included by all of who, if any} are gratefully acknowledged. 





