
\subsection{Numerical results \label{NUMERICS}}

Before showing the results %and presenting arguments 
which enable us to settle the
technical and theoretical accuracy of the generators, it is worth discussing the impact
of various sources of radiative corrections implemented in the programs used
in the experimental analysis. This allows one to understand which corrections are strictly
necessary to achieve a precision at the per mill level for both the 
calculation of integrated cross sections and the simulation of more exclusive distributions.

\subsubsection{Integrated cross sections \label{xsections}}

The first set of phenomenological results about radiative corrections refer to the Bhabha cross 
section, as obtained by means of the code BabaYaga@NLO, according to different
perturbative and precision levels. %\cite{Balossini:2006wc} 
In Table~\ref{tabnum:1} we show 
the values for the Born cross section $\sigma_0$, the $O (\alpha)$ PS and exact cross section, 
$\sigma_{\alpha}^{\textrm{PS}}$ and $\sigma_{\alpha}^{\textrm{NLO}}$, respectively, as well as the LL PS cross section $\sigma^{\rm PS}$ and the matched cross section $\sigma_{\textrm{matched}}$. 
Furthermore, the cross section in the presence of the vacuum polarisation correction, 
$\sigma_{0}^{\rm VP}$, is also shown. 
The results correspond to the c.m. energies $\sqrt{s} = 1,4,10\ \textrm{GeV}$ and were 
obtained with the selection criteria of Eq.~(\ref{eq:cutsmf}), 
but for an angular acceptance of $55^\circ \leq \theta_{\pm} \leq 125^\circ$ resembling realistic 
data taking at meson factories.
One should keep in mind that the cuts of  Eq.~(\ref{eq:cutsmf}) tend to
single out quasi-elastic Bhabha events and that the energy of final state electron/positron
corresponds to a so-called ``bare'' event selection (i.e. without photon recombination), which 
corresponds to what is done in 
practice at flavour factories. In particular the rather stringent energy and acollinearity 
cuts enhance the impact of soft and collinear radiation with respect to a more inclusive setup.

\begin{table}[t]
\caption{Bhabha cross section (in nb) at meson factories 
according to different precision levels and using the cuts of  
Eq.~(\ref{eq:cutsmf}), 
but with an angular acceptance of 
 $55^\circ \leq \theta_{\pm} \leq 125^\circ$. The numbers in parentheses are 1$\sigma$ 
 MC errors.}
\label{tabnum:1}
\begin{center}
\begin{tabular}{clll}
%\small{      %
%%\begin{tabular}{|p{2.775cm}|c|c|c|c|}
\hline
        $\sqrt{s} (\textrm{GeV})$        &     1.02 &
      4   &     10 \\
      \hline
%      \hline
      $\sigma_{0}   $      &     $529.4631(2)$  &
      $44.9619(1)$   &     $5.5026(2)$    \\
      \hline
      $\sigma_{0}^{\rm VP}$      &     $542.657(6)$  &
      $46.9659(1)$   &     $5.85526(3)$    \\
      \hline
      $\sigma_{\rm NLO}$    &     $451.523\left (6\right )$  &
      $37.1654\left (6\right )$  &     $4.4256\left (2\right )$    \\
      \hline
      $\sigma^{\rm PS}_\alpha$     &     $454.503\left (6\right )$  &
      $37.4186\left (6\right )$  &     $4.4565\left (1\right )$    \\
      \hline
      $\sigma_{\rm matched}$    &     $455.858\left (5\right )$  &
      $37.6731\left (4\right )$   &     $4.5046\left (3\right )$    \\
      \hline
      $\sigma^{\rm PS}$    &     $458.437\left (4\right )$  &
      $37.8862\left (4\right )$   &     $4.5301\left (2\right )$    \\
      \hline
\end{tabular}
%      }
\end{center}
\end{table}

From these cross section values, it is possible to calculate the relative effect of various 
corrections, namely the contribution of vacuum polarisation and 
 exact $O (\alpha)$ QED corrections, 
of non-logarithmic (NLL) terms entering the $O (\alpha)$ cross section, 
of HO corrections in the $O (\alpha)$ matched PS scheme, and finally of 
NNL effects beyond order $\alpha$ largely dominated by $O (\alpha^2 L)$ 
contributions.
%, of exponentiation with respect to the exact $\Obig{\alpha}$ cross section and, finally, of %and present in the improved PS algorithm. 
The above corrections are shown in Table~\ref{tabnum:2} in per cent and 
can be derived from the cross section results of Table~\ref{tabnum:1} with the following definitions:

\begin{eqnarray*}
   \delta_{\rm VP}       &\equiv& \frac{\sigma_{0}^{\rm VP}-\sigma_{0}}{\sigma_{0}}, ~~~~~~~~~~~~~~
   \delta_{\alpha}    \equiv  \frac{\sigma_{\rm NLO}-\sigma_0}{\sigma_0} , \nonumber\\
\delta_{\alpha}^{\rm NLL} &\equiv&
                              \frac{\sigma_{\rm NLO}-\sigma^{\rm PS}_\alpha}{\sigma_{\rm NLO}} ,~~~~~~~~
   \delta_{\rm HO}       \equiv \frac{\sigma_{\rm matched}-\sigma_{\rm NLO}}{\sigma_{\rm NLO}} ,
   \nonumber\\
   \delta_{\alpha^2L}&\equiv& \frac{\sigma_{\rm matched}-\sigma_{\rm NLO} -\sigma^{\rm PS}+\sigma^{\rm PS}_\alpha}{\sigma_{\rm NLO}} . \nonumber
\label{eq:deltas}                                   
\end{eqnarray*}

From Table~\ref{tabnum:2}  it can be seen that $O (\alpha)$ corrections decrease the Bhabha cross section
by about 15$\div$17\% at the $\mathrm{\phi}$ and $\tau$-charm factories, and 
by about 20\% at the $B$ factories. Within the full set
of $O (\alpha)$ corrections, non-logarithmic terms are of the order of 0.5\%, as expected almost independent of the 
c.m. energy, and with a mild dependence 
on the angular acceptance cuts 
due to box and interference contributions. The effect of HO corrections due to multiple photon
emission is about 1\% at the $\mathrm{\phi}$ and $\tau$-charm factories and reaches about 2\% at the $B$ factories. The 
contribution of (approximate) $O (\alpha^2 L)$ corrections is at the 0.1\% level, 
while vacuum polarisation increases the cross section by about 2\% around 1 GeV, and by 
about 5\% and 6\% at 4~GeV and 10~GeV, respectively. Concerning the latter correction the 
non-perturbative hadronic contribution to the running of $\alpha$ was parameterised in terms of the 
 HADR5N routine \cite{Jegerlehner:1985gq,Burkhardt:1989ky,Jegerlehner:2006ju} included in BabaYaga@NLO both in the 
 LO and NLO diagrams. We have checked that 
the results obtained for the vacuum polarisation correction in terms of the 
parametrisation \cite{rintpl:2008AA} agree at the $10^{-4}$ level with those 
obtained with HADR5N, as shown in detail in Section \ref{TH}. Those routines return a
data driven error, thus affecting the theoretical precision of the calculation of the Bhabha cross 
cross section as will be discussed in Section \ref{CONCLUSIONS}.

Analogous results for the size of radiative 
corrections to the process $e^+ e^- \to \gamma\gamma$ are given in Table~\ref{tabnum:3}
\cite{Balossini:2008xr}. 
They were obtained using BabaYaga@NLO, according 
to the experimental cuts of Eq.~(\ref{eq:cutsmfg}) for the c.m. energies 
$\sqrt{s} = 1, 3, 10$~GeV.

\begin{table}[phtb]
\caption{Relative size of different sources of corrections (in per cent) to the large-angle Bhabha 
cross section for typical selection cuts at $\mathrm{\phi}$, $\tau$-charm and $B$ factories.}
\label{tabnum:2}
\begin{center}
%\small{      %
\begin{tabular}{clll}
%\begin{tabular}{|l|c|c|c|}
      \hline
                                      $\sqrt{s} \mathrm{(GeV)}$ &     1.02  &
      4.   &      10.   \\
%      \hline
      \hline
      $\delta_{\alpha}^{}$                          &
      $-14.73$       &     $ -17.32$       &  $-19.57$       \\
      \hline
      $\delta_{\alpha}^{\rm NLL}$                  &
      $-0.66$       &     $-0.68$       &  $-0.70$       \\
      \hline
      $\delta_{\mathrm{HO}}$                      &
      $0.97$        &     $1.35$        &  $1.79$        \\
      \hline
      $\delta_{\alpha^{2}L}$                    &
      $0.09$        &     $0.09$        &  $0.11$        \\
      \hline
      $\delta_{\mathrm{VP}}$                      &
      $2.43$       &     $4.46$        &  $6.03$        \\
      \hline
      \end{tabular}
%      }
\end{center}
\end{table}

\begin{table}[!htbp]
\caption{Photon pair production cross sections (in nb) at different 
accuracy levels and relative corrections (in per cent) for the setup 
of Eq.~(\ref{eq:cutsmfg}) and the c.m. energies $\sqrt{s} = 1, 3, 10$~GeV.}
%specified in the text.} 
\label{tabnum:3}
\begin{center}
%\begin{tabular}{|l|c|c|c|}
\begin{tabular}{clll}
\hline
$\sqrt{s}\ \left(\textrm{GeV}\right)$ & $1$ & $3$ & $10$ \\
\hline
%\hline
$\sigma_0$ & $137.53$ & $15.281$ & $1.3753$ \\
\hline
$\sigma_{\rm NLO}$ & $129.45$ & $14.211$ & $1.2620$ \\
\hline
$\sigma_{\alpha}^{\rm PS}$ & $128.55$ & $14.111$ & $1.2529$ \\
\hline
$\sigma_{\rm matched}$ & $129.77$ & $14.263$ & $1.2685$ \\
\hline
$\sigma^{\rm PS}$ & $128.92$ & $14.169$ & $1.2597$ \\
\hline
%\hline
$\delta_{\alpha}$ & $-5.87$ & $-7.00$ & $-8.24$ \\
%\hline
%$\delta_{\infty}$ & $-5.65$ & $-6.66$ & $-7.77$ \\
\hline
$\delta_{\alpha}^{\rm NLL}$ & $0.70$ & $0.71$ & $0.73$ \\
\hline
$\delta_{\rm HO}$ & $0.24$ & $0.37$ & $0.51$ \\
%\hline
%$\delta_{\infty}^{\textrm{NLL}}$ & $0.66$ & $0.66$ & $0.69$ \\
\hline
\end{tabular}
\end{center}
\end{table}

The numerical errors coming from the MC integration are not shown in Table~\ref{tabnum:2} because
they are beyond the quoted digits. From Table~\ref{tabnum:2} it can be seen that the exact $O(\alpha)$ corrections %, measured by the relative contribution $\delta_{\alpha}$, 
lower the Born cross section by about $5.9\%$ (at the $\mathrm{\phi}$ resonance), $7.0\%$ 
(at $\sqrt{s} = 3$~GeV) and $8.2\%$ (at the $\Upsilon$ resonance).  
The effect due to $O (\alpha^{n}L^{n})$ 
(with $n \geq 2$) terms is quantified by the contribution $\delta_{\rm HO}$, which is 
a positive correction of about $0.2\%$ (at the $\mathrm{\phi}$ resonance), $0.4\%$ ($\tau$-charm factories)
%($J/\psi$ resonance) 
and $0.5\%$ (at the $\Upsilon$ resonance), and therefore important in the light of the per mill accuracy aimed at. 
On the other hand, also next-to-leading $O (\alpha)$ corrections, quantified
by the  contribution $\delta_{\alpha}^{\rm NLL}$, are necessary at the precision level 
of 0.1\%, since their contribution is of about $0.7\%$ almost independent of the c.m. energy. 
To further corroborate
the precision reached in the cross section calculation of 
$e^+ e^- \to \gamma\gamma$, we also evaluated the effect due to the 
most important sub-leading $O (\alpha^2)$ photonic corrections given by 
order $\alpha^2 L$ 
contributions. It turns out that the effect due to  $O (\alpha^2 L)$ corrections 
does not exceed the 0.05\% level. Obviously, the contribution of vacuum polarisation is absent in $\gamma\gamma$ 
production. This is an advantage for particularly precise predictions, %because 
as the uncertainty associated with the
hadronic part of vacuum polarisation does not affect the cross section calculation.


\subsubsection{Distributions \label{distributions}}

Besides the integrated cross section, various differential 
cross sections are used by the
experimentalists to monitor the collider luminosity. 
In Figs.~\ref{fignum:1} and \ref{fignum:2} we show two
distributions which are particularly sensitive to the details of
photon radiation, i.e.  the $e^+ e^-$ invariant mass and acollinearity
distribution, in order to quantify the size of NLO and HO corrections. The distributions
are obtained according to the exact $O (\alpha)$ calculation and 
with the two BabaYaga
versions, BabaYaga v3.5 and BabaYaga@NLO. From Figs.~\ref{fignum:1} and \ref{fignum:2} it can be clearly
seen that multiple photon
corrections introduce significant deviations with respect to 
an $O (\alpha)$ simulation, especially in the hard tails of
the distributions where they amount to 
several per cent. To make the %improvements 
contribution of exact  $O (\alpha)$ non-logarithmic terms clearly visible, the inset shows the relative 
differences between the predictions of BabaYaga v3.5 (denoted as OLD) and BabaYaga@NLO 
(denoted as NEW). Actually, as discussed in Section \ref{NLO-HO}, these differences mainly come from 
non-logarithmic NLO contributions and to a smaller extent
 from $O (\alpha^2 L)$ terms. Their effect is flat and at 
the level of 0.5\% for the acollinearity distribution, while they reach the several per cent level 
in the hard tail of the invariant mass distribution.

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{invmass-old-new.eps}
}
\caption{Invariant mass distribution of the Bhabha process at KLOE, according to BabaYaga v3.5 
(OLD), BabaYaga@NLO (NEW) 
and an exact NLO calculation. The inset shows the relative effect of NLO corrections, given by the difference of BabaYaga v3.5 and BabaYaga@NLO predictions. From 
 \cite{Balossini:2006wc}.} 
\label{fignum:1}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{acoll-old-new.eps}
}
\caption{Acollinearity distribution of the Bhabha process at KLOE, according to BabaYaga v3.5 
(OLD) and BabaYaga@NLO (NEW). The inset shows the relative effect of NLO corrections, given by the difference of BabaYaga v3.5 and BabaYaga@NLO predictions. From  \cite{Balossini:2006wc}.}
\label{fignum:2}
\end{center}
\end{figure}

It is also worth noticing that LL radiative corrections
beyond $\alpha^2$ can be quite important for accurate 
simulations, at least when considering
differential distributions. This means that even
with a complete NNLO calculation at hand it would be
desirable to match such corrections with the resummation of all the remaining
LL effects. In Fig.~\ref{fignum:3},
%and \ref{fig:invmassa3}, 
the relative effect of  HO corrections beyond $\alpha^2$ dominated by the $\alpha^3$ 
contributions (dashed line) is shown in comparison with that of the $\alpha^2$ 
corrections (solid line) on the acollinearity
%and the final state
%$e^+e^-$ invariant mass distributions,
distribution for the Bhabha process at DA$\mathrm{\Phi}$NE. As can be seen, 
the $\alpha^3$ effect can be as large as
% $20\%$
$10\%$ in the phase space region of soft photon emission, corresponding to
small acollinearity angles  with almost back-to-back final state fermions.

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{acolla3.eps}
}
\caption{Relative effect of HO corrections $\alpha^2 L^2$ and
$\alpha^n L^n$ ($n \geq 3$) to the acollinearity distribution
of the Bhabha process at KLOE. From  \cite{Balossini:2006wc}.}
\label{fignum:3}
\end{center}
\end{figure}

Concerning the process $e^+ e^- \to \gamma\gamma$ 
 we show in  Fig.~\ref{fignum:4}  the energy distribution 
of the most energetic photon,  while the acollinearity distribution of 
the two most energetic photons is represented in Fig.~\ref{fignum:5}. The distributions 
%correspond to the experimental set up of Eq. (\ref{eq:KLOE}) and 
refer to exact  $O (\alpha)$ corrections matched with 
the PS algorithm (solid line), to the exact NLO calculation %as in Eq.~(\ref{eq:3})  
 (dashed line) and to all-order pure PS predictions of BabaYaga v3.5
 (dashed-dotted line). In the inset of each plot, the relative effect due to multiple photon contributions
 ($\delta_{\rm HO}$) and non-logarithmic terms entering the improved PS algorithm
($\delta_{\alpha}^{\rm NLL}$) is also shown, according to the definitions given in 
Eq.~(\ref{eq:deltas}). 

For the energy distribution of the most energetic photon 
particularly pronounced effects due to exponentiation
are present. In the statistically 
dominant region, %around 0.5~GeV, 
HO corrections reduce the $O (\alpha)$ 
distribution by about 20\%, while they give rise to a significant hard tail close
to the energy threshold of 0.3$\sqrt{s}$ as a consequence of the higher photon multiplicity 
of the resummed calculation with respect to the fixed-order NLO prediction. 
Needless to say, the relative effect of multiple photon corrections below about 0.46 GeV not shown in the inset is finite but huge. This representation with the inset was chosen to make also the contribution of $O (\alpha)$ non-logarithmic terms visible, which otherwise
would be hardly seen in comparison with the multiple photon corrections. Concerning
the acollinearity distribution, the contribution of higher-order corrections is positive and of about
10\% for quasi-back-to-back photon events, whereas it is negative and decreasing
from $\sim -30$\% to $\sim -10$\% for increasing acollinearity values. As far as the contributions of non-logarithmic effects
dominated by next-to-leading  $O (\alpha)$
corrections are concerned, they contribute at the level of several per mill for the acollinearity distribution, while they lie in 
the range of several per cent for the energy distribution.

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{Fig4_gg.eps}
}
\caption{Energy distribution of the most energetic photon in the process 
$e^+ e^- \to \gamma\gamma$, according to the PS matched 
with $O (\alpha)$ corrections denoted as exp (solid line), the exact $O (\alpha)$ 
calculation (dashed line) and the pure all-order PS as in BabaYaga v3.5 
(dashed-dotted line). lnset: relative effect (in per cent) of multiple photon corrections (solid line) 
and of non-logarithmic contributions of the matched PS algorithm (dashed line). 
From \cite{Balossini:2008xr}.}
\label{fignum:4}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{0.475\textwidth}{!}{%
\includegraphics{Fig5_gg.eps}
}
\caption{Acollinearity distribution for the process 
$e^+ e^- \to \gamma\gamma$, according to the PS matched 
with $O (\alpha)$ corrections denoted as exp (solid line), the exact $O (\alpha)$ 
calculation (dashed line) and the pure all-order PS as in BabaYaga v3.5 
(dashed-dotted line). lnset: relative effect (in per cent) of multiple photon corrections (solid line) 
and of non-logarithmic contributions of the matched PS algorithm (dashed line). From \cite{Balossini:2008xr}.}
\label{fignum:5}
\end{center}
\end{figure}

As a whole, the results of the present Section emphasise that, for a 0.1\% theoretical precision in the calculation of both the cross sections and distributions, both exact $O (\alpha)$ and HO photonic corrections are necessary, as well as the running of $\alpha$.
