ls
\subsection{Theoretical accuracy \label{TH}}

As discussed in Section 
\ref{MOTIVATION}, the total luminosity error crucially depends on the theoretical 
accuracy of the MC programs used by the 
experimentalists.  As emphasised in Section 
\ref{MC}, 
some of these generators like BHAGENF, BabaYaga v3.5 and BKQED miss theoretical ingredients 
which are unavoidable for cross section calculations with a precision at the per mill level.
Therefore, they are inadequate for a highly accurate luminosity determination. 
BabaYaga@NLO, BHWIDE and MCGPJ include, however, both NLO and multiple photon corrections, and 
their accuracy aims at a precision tag of 0.1\%. But also these generators are 
affected by uncertainties which must be carefully considered in the light of the very stringent criteria 
of per mill accuracy. The most important components of the theoretical error of 
BabaYaga@NLO, BHWIDE and MCGPJ are mainly due to approximate or partially included pieces of 
radiative corrections and come from the following sources:
\begin{enumerate}

\item The non-perturbative hadronic contributions to the running of $\alpha$. It can be 
reliably evaluated only using the data of the hadron cross section at low energies. Hence, 
the vacuum polarisation correction receives a data driven error which affects in turn the
prediction of the Bhabha cross section, as emphasised in Section \ref{sec:4}.

\item The complete set of $O (\alpha^2)$ QED corrections. In spite of the impressive progress
in this area, as reviewed in Section \ref{NNLO}, an important piece of NNLO corrections, i.e. 
the exact
NLO SV QED corrections to the single hard bremsstrahlung process $e^+ e^- \to e^+ e^-\gamma$,
 is still missing for the full $s+t$ Bhabha process.\footnote{As already remarked and further discussed in the following, the complete calculation of the NLO corrections to hard photon emission in Bhabha scattering was 
 performed during the completion of this report \cite{Actis:2009uq}.} However, partial results 
 obtained for $t$-channel small-angle Bhabha scattering \cite{Jadach:1995hy,Ward:1998ht}
 and large-angle annihilation processes are available \cite{Glosser:2003ux,Jadach:2006fx}.
 
 \item The $O (\alpha^2)$ contribution due to real and virtual (lepton and hadron) pairs. 
 The virtual contributions originate from the NNLO electron, heavy flavour and hadronic loop corrections discussed in Section \ref{NNLO}, while the real corrections are due to the conversion of an 
 external photon into pairs. The latter, as  discussed in Section \ref{4pfinal}, gives rise to a final state with four 
 particles, two of which to be considered as undetected to contribute to the Bhabha signature.

\end{enumerate}

The uncertainty relative to the first point can be estimated by using the routines available in
the literature for the calculation of the non-perturbative hadronic contribution 
$\Delta\alpha^{(5)}_{\rm hadr}(q^2)$ to the running $\alpha$. 
Actually these routines return, in addition to $\Delta\alpha^{(5)}_{\rm hadr}(q^2)$, 
an error $\delta_{\rm hadr}$ on its value. Therefore an estimate of the 
induced error can be simply obtained by computing the Bhabha cross section with
$\Delta\alpha^{(5)}_{\rm hadr}(q^2)\pm\delta_{\rm hadr}$ and taking the
difference as the theoretical uncertainty due to the hadronic
contribution to vacuum polarisation. In Table~\ref{tabth:1}, the Bhabha cross sections, as obtained 
in the presence of the vacuum polarisation correction according to the parameterisations
of \cite{Jegerlehner:1985gq,Burkhardt:1989ky,Jegerlehner:2006ju} (denoted as J) 
and of \cite{rintpl:2008AA} (denoted as HMNT), respectively, are shown for 
$\mathrm{\phi}$, $\tau$-charm and $B$ factories. The applied angular cuts 
refer to the typically adopted acceptance $55^\circ \leq \theta_{\pm} \leq 125^\circ$.

\begin{table}[h]
\caption{Bhabha scattering cross section in the presence of the vacuum polarisation 
correction, according to \cite{Jegerlehner:1985gq,Burkhardt:1989ky,Jegerlehner:2006ju} (J) and 
\cite{rintpl:2008AA} (HMNT), at meson factories. The 
notation J$_-$/HMNT$_-$, J/HMNT and J$_+$/HMNT$_+$ indicates minimum, central and maximum 
value of the two parametrisations.}
\label{tabth:1}
\begin{center}
%%\begin{tabular}{|c|c|c|c|c|}
\begin{tabular}{clll}
\hline
Parametrisation & $\mathrm{\phi}$ & $\tau$-charm& $B$ \\
\hline
J$_-$ & 542.662(4) & 46.9600(1) & 5.85364(2) \\
\hline
$\!\!\!\!\!$ J & 542.662(4) & 46.9658(1) & 5.85529(2) \\
\hline
J$_+$ & 542.662(4) & 46.9715(1) & 5.85693(2) \\
\hline
HMNT$_-$  & 542.500(5) & 46.9580(1) & 5.85496(1) \\
\hline
$\!\!\!\!$HMNT          & 542.391(5) & 46.9638(1) & 5.85621(1) \\
\hline
HMNT$_+$ & 542.283(5) & 46.9697(1) & 5.85746(2) \\
\hline
\end{tabular}
\end{center}
\end{table}

From Table~\ref{tabth:1} it can be seen that the two treatments of $\Delta\alpha^{(5)}_{\rm hadr}(q^2)$ 
induce effects on the Bhabha cross section in very good agreement, the relative differences 
between the central values being 0.05\% ($\mathrm{\phi}$ factories), 0.005\% ($\tau$-charm 
factories) and 
0.02\% ($B$ factories). This can be understood in terms of the dominance of $t$-channel 
exchange for large-angle Bhabha scattering at meson factories. Indeed, the two routines provide 
results in excellent agreement for space-like momenta, as we explicitly checked, whereas 
differences in the predictions show up for time-like momenta which, however, contribute only marginally to the
Bhabha cross section. Also the spread between the minimum/maximum values and
the central one as returned by the two routines agrees rather well, 
also a consequence of the dominance of 
$t$-channel exchange. This spread amounts to a few units in $10^{-4}$ and is presented in detail
in Table~\ref{tabcon:1} in the next Section.

Concerning the second point a general strategy to evaluate the size of missing 
NNLO corrections consists in deriving a cross section expansion up to 
 $O (\alpha^2)$ from the theoretical
formulation implemented in the generator of interest. It can be cast in
general into the following form
\begin{eqnarray}
\sigma^{\alpha^2} \, = \, \sigma^{\alpha^2}_{\rm SV}
+ \sigma^{\alpha^2}_{\rm SV,H} + \sigma^{\alpha^2}_{\rm HH} ,
\label{eq:a2}
\end{eqnarray}
where in principle each of the $O (\alpha^2)$ contributions is affected by an uncertainty 
to be properly estimated. In Eq.~(\ref{eq:a2}) the first contribution is the cross section 
including $O (\alpha^2)$ SV corrections, whose uncertainty can be evaluated 
through a comparison with some of the available NNLO calculations 
reviewed in Section \ref{NNLO}. In particular, in 
\cite{Balossini:2006wc} the $\sigma^{\alpha^2}_{\rm SV}$ of 
the BabaYaga@NLO generator was compared with
the calculation of photonic corrections by Penin \cite{Penin:2005kf,Penin:2005eh} 
and the calculations by Bonciani 
{\it et al.}~\cite{Bonciani:2005im,Bonciani:2003te,Bonciani:2003cj0,Bonciani:2004gi,Bonciani:2004qt} 
who
computed two-loop fermionic corrections (in the one-family approximation $N_F = 1$) with finite mass
terms and the addition of soft bremsstrahlung and real pair contributions.\footnote{To provide 
meaningful results, the contribution of the vacuum polarisation was switched off in BabaYaga@NLO
to compare with the calculation by Penin consistently. For the same reason the real soft and 
some pieces of virtual electron pair corrections
%partially implemented in BabaYaga@NLO 
were neglected in the 
comparison with the calculation by Bonciani {\it et al.}} The 
results of such comparisons are shown in Figs.~\ref{figth:1} 
and \ref{figth:2} for realistic cuts at the $\mathrm{\phi}$ factories.
In  Fig.~\ref{figth:1} $\delta\sigma$ is the difference between $\sigma^{\alpha^2}_{\rm SV}$ of BabaYaga@NLO and the cross sections of the two $O (\alpha^2)$ calculations, denoted
as photonic (Penin) and $N_F = 1$ (Bonciani {\it et al.}), as a function of the logarithm of the
infrared regulator $\epsilon$. It can be seen that the differences are given by flat functions,
demonstrating that such differences are infrared-safe, as expected, a consequence of
the universality and factorisation properties of the infrared divergences. In Fig.~\ref{figth:2},  
$\delta\sigma$ is shown as a function of the logarithm of a fictitious electron mass and for a
fixed value of $\epsilon = 10^{-5}$. Since the difference with the calculation by Penin is given by
a straight line, this indicates that the soft plus virtual two-loop photonic corrections missing in BabaYaga@NLO are $O (\alpha^2 L)$ contributions, as 
already remarked. On the other hand, the difference with the
calculation by Bonciani {\it et al.} is fitted by a quadratic function, showing that the 
electron two-loop
effects missing in BabaYaga@NLO are of the order of $\alpha^2 L^2$. 
However, it is important to emphasise
that, as shown in detail in~\cite{Balossini:2006wc}, the sum of the 
relative differences with the two $O (\alpha^2)$ calculations 
does not exceed the $2 \times 10^{-4}$ level for experiments at 
$\mathrm{\phi}$ and $B$ factories. 
%and is, therefore, an order of magnitude smaller than the required theoretical accuracy. 

The second term in Eq.~(\ref{eq:a2}) is the cross section containing the one-loop corrections to single
hard photon emission, and its uncertainty can be estimated by relying on partial results 
existing in the literature. Actually the exact perturbative expression of $\sigma^{\alpha^2}_{\rm SV,H}$
is not yet available for full $s+t$ Bhabha scattering, but using the results valid for 
small-angle Bhabha scattering~\cite{Jadach:1995hy,Ward:1998ht} and large-angle 
annihilation processes \cite{Glosser:2003ux,Jadach:2006fx} the
relative uncertainty of the theoretical tools in the calculation of $\sigma^{\alpha^2}_{\rm SV,H}$ can be 
conservatively estimated to be at the level of 0.05\%. Indeed the papers 
~\cite{Jadach:1995hy,Ward:1998ht,Glosser:2003ux,Jadach:2006fx} show that a YFS 
matching of NLO and HO corrections gives SV one-loop results for the $t$-channel process $e^+ e^- 
\to e^+ e^- \gamma$ and $s$-channel annihilation $e^+ e^- \to f \bar{f} \gamma$ ($f$ = fermion) 
differing from the exact perturbative calculations by a few units in $10^{-4}$ at most. This
conclusion also holds when photon energy cuts are varied. It is worth noting that 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 appeared in the literature \cite{Actis:2009uq}, along the lines described in Section \ref{Penta}. Explicit comparisons between the results of such an exact calculation with the predictions of the most accurate MC tools according to the typical luminosity cuts used at meson factories would be worthwhile to 
make the present error estimate related to the calculation of $\sigma^{\alpha^2}_{\rm SV,H}$ 
more robust.

The third contribution in Eq.~(\ref{eq:a2}) is the double hard 
bremsstrahlung cross section whose uncertainty can be directly evaluated by 
explicit comparison with the 
exact $e^+ e^- \to e^+ e^- \gamma\gamma$ cross section. It was shown in~\cite{Balossini:2006wc} that the differences between $\sigma^{\alpha^2}_{\rm HH}$ as in BabaYaga@NLO 
and the matrix element calculation, which exactly describes the 
contribution of two hard photons, are really negligible, being at the $10^{-5}$ level.

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{epsscan.eps}
}
\caption{Absolute differences (in nb) between the $\sigma^{\alpha^2}_{\rm SV}$ 
prediction of BabaYaga@NLO and 
the NNLO calculations of the photonic corrections
\cite{Penin:2005kf,Penin:2005eh} (photonic) and of the electron loop corrections
\cite{Bonciani:2005im,Bonciani:2003te,Bonciani:2003cj0,Bonciani:2004gi,Bonciani:2004qt} ($N_F = 1$)
 as a function of the infrared regulator $\epsilon$ for typical KLOE cuts. From 
 \cite{Balossini:2006wc}.}
\label{figth:1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{mescan.eps}
}
\caption{Absolute differences (in nb) between the $\sigma^{\alpha^2}_{\rm SV}$ 
prediction of BabaYaga@NLO and 
the NNLO calculations of the photonic corrections 
\cite{Penin:2005kf,Penin:2005eh} (photonic) and of the electron loop corrections 
\cite{Bonciani:2005im,Bonciani:2003te,Bonciani:2003cj0,Bonciani:2004gi,Bonciani:2004qt} ($N_F = 1$) as a function of a fictitious electron mass for typical KLOE cuts. From \cite{Balossini:2006wc}.}
\label{figth:2}
\end{center}
\end{figure}

The relative effect due to lepton ($e,\mu,\tau$) and hadron ($\pi$) pairs has been numerically analysed in 
Section \ref{4pfinal}, in the presence of realistic selection cuts. %for KLOE and Babar experiments. 
This evaluation makes use of the complete NNLO virtual corrections combined with an 
exact matrix element calculation of the four-particle production processes. It 
supersedes previous approximate estimates which
underestimated the impact of those corrections. According to this new evaluation, the pair contribution, 
dominated by the electron pair correction, amounts to about 0.3\% for KLOE and 0.1\% 
for BaBar. These contributions are partially included in the BabaYaga@NLO code, as well as in other generators, through the insertion of the vacuum polarisation correction in the NLO diagrams, and detailed
comparisons between the exact calculation and the BabaYaga@NLO predictions are in progress 
\cite{pairs}.

