The systematic comparison of Standard Model ({\small SM})
predictions with precise experimental data served in the last 
decades as an invaluable tool to test the theory at the quantum
level. It has also provided stringent constraints on ``new
physics'' scenarios.  The (so far) remarkable agreement between the
measurements of the electroweak observables and their {\small
  SM} predictions is a striking experimental confirmation of the
theory, even if there are a few observables where the agreement is not
so satisfactory.  On the other hand, the Higgs boson has not yet been
observed, and there are clear phenomenological facts (dark matter, 
matter-antimatter asymmetry in the universe) as well as 
strong theoretical arguments hinting at the
presence of physics beyond the {\small SM}. New colliders, like the 
LHC or a future $e^+ e^-$ International Linear Collider
(ILC), will hopefully answer many questions, offering at
the same time great physics potential and a new challenge to provide
even more precise theoretical predictions.
 
Precision tests of the Standard Model require an appropriate inclusion
of higher order effects and the knowledge of very precise input parameters.
One of the basic input parameters is the fine-structure constant 
$\alpha$ , determined from the  anomalous 
 magnetic moment of the electron
%\cite{Hanneke:2008tm}
%which depends logarithmically on the energy scale.
 %It has been determined at zero momentum transfer 
 with an impressive accuracy of 0.37 parts per billion 
(ppb)~\cite{Hanneke:2008tm} 
%from the measurement of the anomalous electron
% magnetic moment \cite{Hanneke:2008tm}, 
relying on the validity of 
 perturbative QED~\cite{Gabrielse:2006gg}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%INCLUDE PASSERA
However, physics at nonzero squared momentum transfer $q^2$
is actually described by an effective electromagnetic
coupling $\alpha(q^2)$ rather than by the low-energy
constant $\alpha$ itself. The shift of the fine-structure
constant from the Thomson limit to high energy involves low
energy non-perturbative hadronic effects which spoil this
precision. In particular, the effective fine-structure
constant at the scale $M_{\scriptscriptstyle{Z}}$,
$\alpha(M_{\scriptscriptstyle{Z}}^2) = \alpha/[1-\Delta
\alpha(M_{\scriptscriptstyle{Z}}^2)]$, plays a crucial role
in basic {\small EW} radiative corrections of the {\small
  SM}. An important example is the {\small EW} mixing
parameter $\sin^2 \!\theta$, related to $\alpha$, the Fermi
coupling constant $G_F$ and $M_{\scriptscriptstyle{Z}}$ via
the Sirlin relation~\cite{Sirlin:1980nh,Sirlin:1989uf,Marciano:1980pb}
%
\begin{equation}
  \sin^2 \!\theta_{\scriptscriptstyle{S}} \cos^2 \!\theta_{\scriptscriptstyle{S}} = 
  \frac{\pi \alpha}
{\sqrt 2 G_F M_{\scriptscriptstyle{Z}}^2 (1-\Delta r_{\scriptscriptstyle{S}})},
\label{eq:sirlin}
\end{equation}
%
where the subscript $S$ identifies the renormalisation
scheme. $\Delta r_{\scriptscriptstyle{S}}$ incorporates the
universal correction $\Delta
\alpha(M_{\scriptscriptstyle{Z}}^2)$, large contributions
that depend quadratically on the top quark
mass $m_t$~\cite{Veltman:1977kh} (which led to its indirect determination
before this quark was discovered), plus all remaining
quantum effects.
%
In the {\small SM}, $\Delta r_{\scriptscriptstyle{S}}$
depends on various physical parameters, including
$M_{\scriptscriptstyle{H}}$, the mass of the Higgs boson. As
this is the only relevant unknown parameter in the {\small
  SM}, important indirect bounds on this missing ingredient
can be set by comparing the calculated quantity in
Eq.~(\ref{eq:sirlin}) with the experimental value of $\sin^2
\!\theta_{\scriptscriptstyle{S}}$ (e.g.\ the effective
{\small EW} mixing angle $\sin^2 \!\theta_{\rm eff}^{\rm
  lept}$ measured at LEP and SLC from the
on-resonance asymmetries) once
$\Delta\alpha(M_{\scriptscriptstyle{Z}}^2)$ and other
experimental inputs like $m_t$ are provided. It is important
to note that an error of $\delta
\Delta\alpha(M_{\scriptscriptstyle{Z}}^2) = 35 \times
10^{-5}$~\cite{Burkhardt:2005se} in the effective
electromagnetic coupling constant dominates the uncertainty
of the theoretical prediction of $\sin^2 \!\theta_{\rm
  eff}^{\rm lept}$, inducing an error $\delta(\sin^2
\!\theta_{\rm eff}^{\rm lept}) \sim 12 \times 10^{-5}$
(which is comparable with the experimental value
$\delta(\sin^2 \!\theta_{\rm eff}^{\rm
  lept})^{\scriptscriptstyle \rm EXP} = 16 \times 10^{-5}$
determined by LEP-I and SLD~\cite{leplsd:2005ema,LEPEWWG}) and
affecting the upper bound for $M_H$~\cite{leplsd:2005ema,LEPEWWG,Passera:2008jk}.  Moreover, as
measurements of the effective {\small EW} mixing angle at a
future linear collider may improve its precision by one
order of magnitude, a much smaller value of
$\delta\Delta\alpha(M_{\scriptscriptstyle{Z}}^2)$ will be
required (see below). It is therefore crucial to assess all
viable options to further reduce this uncertainty.

The shift $\Delta \alpha(M_{\scriptscriptstyle{Z}}^2)$ can
be split in two parts:
$\Delta\alpha(M_{\scriptscriptstyle{Z}}^2) =
\Delta\alpha_{\rm lep}(M_{\scriptscriptstyle{Z}}^2) + \Delta
\alpha_{\rm had}^{(5)}(M_{\scriptscriptstyle{Z}}^2)$. The
leptonic contribution is calculable in perturbation theory
and known up to three-loop accuracy: $\Delta\alpha_{\rm
  lep}(M_{\scriptscriptstyle{Z}}^2) = 3149.7686\times
10^{-5}$~\cite{Steinhauser:1998rq}. The hadronic contribution $\Delta
\alpha_{\rm had}^{(5)}(M_{\scriptscriptstyle{Z}}^2)$ of the
five light quarks ($u$, $d$, $s$, $c$, and $b$) can be
computed from hadronic $e^+ e^-$ annihilation data via the
dispersion relation~\cite{Cabibbo:1961sz}
%
\begin{equation} 
  \Delta \alpha_{\rm had}^{(5)}(M_{\scriptscriptstyle{Z}}^2) = 
-\left(\frac{\alpha M_{\scriptscriptstyle{Z}}^2}{3\pi}
  \right) \mbox{Re}\int_{m_\pi^2}^{\infty} {\rm d}s 
\frac{R(s)}{s(s- M_{\scriptscriptstyle{Z}}^2
  -i\epsilon)},
\label{eq:delta_alpha_had}
\end{equation}
%
where $R(s) = \sigma^{0}_{\rm had}(s)/(4\pi\alpha^2\!/3s)$ and
$\sigma^{0}_{\rm had}\!(s)$ is the total cross section for $e^+ e^-$
annihilation into any hadronic states, with 
%extraneous {\small QED} and vacuum polarisation 
vacuum polarisation and initial state {\small QED}
corrections subtracted off. The current
accuracy of this dispersion integral is of the order
of 1\%, dominated by the error of the hadronic cross section
measurements in the energy region below a few GeV~\cite{Eidelman:1995ny,Davier:1997kw,Burkhardt:2001xp,Burkhardt:2005se,Jegerlehner:2001wq,Jegerlehner:2003rx,Jegerlehner:2006ju,Jegerlehner:2008rs,Jegerlehner:2003qp,Jegerlehner:2003ip,Hagiwara:2003da,Hagiwara:2006jt}.
%%%END PASSERA%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
%At the $M_{\scriptscriptstyle{Z}}$ scale the shift of the fine-structure constant 
%involves low energy non-perturbative hadronic effects, which  
%cause a dramatic reduction of accuracy, by several orders of magnitude~\cite{Jegerlehner:2001wq,Jegerlehner:2003rx,Jegerlehner:2006ju,Jegerlehner:2008rs}.
%These low energy strong-interaction effects can be computed from hadronic 
%$e^+e^-$ annihilation data via the dispersion relation~\cite{Cabibbo:1961sz}:
%
%\begin{equation} 
%  \Delta \alpha_{\rm had}^{(5)}(M_{\scriptscriptstyle{Z}}^2) = 
%-\left(\frac{\alpha M_{\scriptscriptstyle{Z}}^2}{3\pi}
%  \right) \mbox{Re}\int_{m_\pi^2}^{\infty} ds 
%\frac{R(s)}{s(s- M_{\scriptscriptstyle{Z}}^2
%  -i\epsilon)},
%\label{eq:delta_alpha_had}
%\end{equation}
%%
%where $R(s) = \sigma^{0}(s)/(4\pi\alpha^2\!/3s)$ and
%$\sigma^{0}\!(s)$ is the total cross section for $e^+ e^-$
%annihilation into hadronic state.

%The current accuracy of this dispersion integral is of the order of 1\%, 
%dominated by the error of the hadronic cross section measurement in the energy region below a few GeV~\cite{Eidelman:1995ny,Davier:1997kw,Burkhardt:2001xp,Burkhardt:2005se,Jegerlehner:2001wq,Jegerlehner:2003rx,Jegerlehner:2006ju,Jegerlehner:2008rs,Jegerlehner:2003qp,Jegerlehner:2003ip,Hagiwara:2003da,Hagiwara:2006jt}.
% This error represents the largest source of the uncertainty of the effective electromagnetic coupling constant,  (recent evaluations give $\Delta\alpha(M^2_Z) = 22\times 10^{-5}$~\cite{Jegerlehner:2008rs,Hagiwara:2006jt}), becoming a limiting factor for the effective electroweak mixing parameter $\sin^2{\theta_f}$, and in turn for the prediction of the Higgs mass~\cite{Collaboration:2008ub}.
Table~\ref{tab:future} (from Ref.~\cite{Jegerlehner:2001wq}) shows that an
uncertainty $\delta \Delta\alpha_{\rm had}^{(5)} \sim 5 \times
10^{-5}$, needed for precision physics at a future linear collider,
requires the measurement of the hadronic cross section with a
precision of $O(1\%)$ from threshold up to the $\Upsilon$ peak.
%
\begin{table}[h]
\begin{center}
 \renewcommand{\arraystretch}{1.4}
 \setlength{\tabcolsep}{1.6mm}
{\footnotesize
\begin{tabular}{|c|c|c|c|}
\hline
$\delta \Delta\alpha_{\rm had}^{(5)} \!\times \! 10^{5} $ 
& $\delta(\sin^2 \!\theta_{\rm eff}^{\rm lept}) \! \times \! 10^{5}$  
& Request on $R$\\
\hline \hline
22   &  7.9 & Present \\
\hline
7   &   2.5 & $\!\delta R/R \sim  1\%$ up to ${J/\psi}$\\
\hline   
5   &   1.8 & $\delta R/R \sim  1\%$ up to $\Upsilon$\\
\hline   
\end{tabular}
}
\caption{\label{tab:future} 
Values of the uncertainties $\delta
\Delta\alpha_{\rm had}^{(5)}$ (first column) and the errors induced by these
uncertainties on the theoretical {\small SM} prediction for $\sin^2
\!\theta_{\rm eff}^{\rm lept}$ (second column). The third column indicates
the corresponding requirements for the $R$ measurement. From Ref.~\cite{Jegerlehner:2001wq}.}
\end{center}
\end{table}

Like the effective fine-structure constant at the scale
$M_{\scriptscriptstyle{Z}}$, the {\small SM} determination of the
anomalous magnetic moment of the muon $a_\mu$ is presently limited by the
evaluation of the hadronic vacuum polarisation effects, which cannot be 
computed perturbatively at low energies. 
However, using analyticity and unitarity, it was shown
long ago that this term can be computed from hadronic $e^+ e^-$
annihilation data via the dispersion integral~\cite{Gourdin:1969dm}:
%
\begin{eqnarray}
      a_{\mu}^{\mbox{$\scriptscriptstyle{\rm HLO}$}} &=& 
      \frac{1}{4\pi^3}
      \int^{\infty}_{m_{\pi}^2} {\rm d}s \, K(s) \sigma^{0}\!(s) \nonumber \\
                                                                  &=&
      \frac{\alpha^2}{3\pi^2}
      \int^{\infty}_{m_{\pi}^2} {\rm d}s \, K(s) R(s)/s \, .
\label{eq:amu_had}
\end{eqnarray}
%
The kernel function $K(s)$ decreases monotonically with increasing~$s$. This
integral is similar to the one entering the evaluation of the hadronic
contribution $\Delta \alpha_{\rm had}^{(5)}(M_{\scriptscriptstyle{Z}}^2)$ in
Eq.~(\ref{eq:delta_alpha_had}). Here, however, the weight function in the
integrand gives a stronger weight to low-energy data.
A recent compilation of $e^+e^-$ data gives~\cite{Davier:2009zi}:
\begin{equation}
a_\mu^{\scriptscriptstyle{\rm HLO}} = (695.5\pm4.1)\times 10^{-10} \, .
\end{equation}
Similar values are obtained by other
 groups~\cite{Hagiwara:2006jt,Jegerlehner:2008zz,Jegerlehner:2009ry,Eidelman:2009ft}. 


%The fractional accuracy of 0.6\% represents 
%a reduction of more than a factor two of the error respect to 
%8 years ago. 
By adding this contribution to the rest of the
{\small SM} contributions, a recent update of the
 SM prediction of $a_{\mu}$, which uses the hadronic light-by-light
result from~\cite{Prades:2009tw} gives \cite{Davier:2009zi,Prades:2009qp}:
%
\noindent $a_{\mu}^{\mbox{$\scriptscriptstyle{\rm SM}$}}= 116591834 (49)  
\times 10^{-11}$.
%
The difference between the experimental average~\cite{Bennett:2006fi},
$     a_{\mu}^{\mbox{$\scriptscriptstyle{\rm exp}$}}= 11659 2080 (63)  
\times 10^{-11}$
%
and the SM prediction is then
%
$\Delta a_{\mu} = a_{\mu}^{\mbox{$\scriptscriptstyle{\rm exp}$}}-
a_{\mu}^{\mbox{$\scriptscriptstyle{\rm SM}$}} = +246 (80) \times 10^{-11}$,
%
i.e. 3.1 standard deviations (adding all errors in quadrature).
Slightly higher discrepancies are obtained 
%employing the values of the
%leading-order hadronic contribution reported 
in 
Refs.~\cite{Hagiwara:2006jt,Jegerlehner:2009ry,Eidelman:2009ft}.
%
%Recent reviews of the muon $g$$-$$2$ can be found in
%Refs.~\cite{Passera:2004bj, Jegerlehner:2008zz, Jegerlehner:2009ry, Reviews}.
As in the case of $\alpha(M^2_Z)$, the uncertainty 
of the theoretical evaluation of
 $a_{\mu}^{\mbox{$\scriptscriptstyle{\rm SM}$}}$
 is still dominated by the  hadronic contribution at low energies, and
a reduction of the uncertainty is necessary in order to match 
the increased precision of  the proposed muon g-2 experiments 
at FNAL~\cite{Carey:2009zz} and J-PARC~\cite{Imazato:2004fy}.
%to measure the muon $g$$-$$2$ to a precision of 0.14 ppm at FNAL~\cite{Carey:2009zz}. 


%The analysis of this section shows that while the {\small QED} and
%{\small EW} contributions to the anomalous magnetic moment of the muon
%appear to be ready to rival the forecasted precisions of future
%experiments, much effort will be needed to reduce the hadronic
%uncertainty.
%$e^+ e^-$ 
%experimental uncertainty 8 years ago.
%Such an achievement was possible thanks to 


The precise determination of the hadronic cross sections (accuracy $\lesssim 1\%$) 
requires an excellent control of higher order effects like Radiative
 Corrections (RC) and the 
non-perturbative hadronic contribution to the running of $\alpha$ 
(i.e. the vacuum polarisation, VP)
in Monte Carlo (MC) programs used for the analysis of the data.
%of the analysis tools, 
%like Monte Carlo (MC) event generators and reliable calculations
%of high order effects, like Radiative Corrections (RC).
Particularly in the last years, the increasing precision reached 
on the experimental side at the $e^+e^-$ colliders (VEPP-2M, DA$\mathrm{\Phi}$NE, BEPC, PEP-II and KEKB)
%(detector \emph{BES}), at the $J/\psi$ and ${\psi}(2S)$-resonances in Beijing,
%\emph{PEP-II} (detector \emph{BABAR}) at Stanford and \emph{KEKB} (detector
%\emph{Belle}) at ... at the $\Upsilon(4S)$-resonance,
led to the development of dedicated high precision theoretical tools:
BabaYaga (and its successor BabaYaga@NLO) 
for the measurement of the luminosity, 
MCGPJ for the simulation of the exclusive 
QED channels, and 
PHOKHARA for the simulation of the process  with Initial State Radiation (ISR) $e^+e^-\to hadrons+\gamma$,
are examples of MC generators which include NLO corrections with per mill accuracy.
In parallel to these efforts, well-tested codes such as 
BHWIDE (developed for LEP/SLC colliders) were adopted.

Theoretical accuracies of these generators were estimated, whenever possible, 
 by evaluating  missing higher order contributions. From this point of view, the great pro\-gress in the calculation of two-loop corrections to the Bha\-bha scattering cross section was essential to 
establish the high
theoretical accuracy of the existing generators for the luminosity measurement.
% and is crucial if an improvement of the precision below the one per mill will be required.
However, usually only analytical or semi-analytical estimates
 of missing terms exist which don't take into account 
realistic experimental cuts.
In addition, MC event generators include different parametrisations for the VP
 which affect the prediction (and the precision) 
of the cross sections and also the RC are usually implemented differently. 

These arguments evidently imply the importance of 
comparisons of MC generators with a common set of input parameters 
and experimental cuts. 
Such  {\it tuned} comparisons, which started in the LEP era, are a key step for the validation of the generators, since they allow
 to check that the details entering the complex structure of the generators are under control and free of possible bugs.
This
% tuned comparison of different MC programs 
was the main motivation for the 
{\it ``Working Group on Radiative Corrections  and  Monte Carlo Generators for Low Energies'' (Radio MontecarLow)},
which was formed a few years ago bringing together experts 
(theorists and experimentalists) working in the field of low energy $e^+e^-$ physics and partly also the $\tau$ community.

In addition to tuned comparisons, technical details of the MC
 generators, recent progress (like new calculations) and remaining
 open issues were also discussed in regular meetings.

This report is a summary of all these efforts:
it provides a self-contained and up-to-date description of the progress which 
occurred in the last years towards precision hadronic 
 phy\-sics at low energies, together with new results like comparisons and estimates of high order effects (e.g. of the 
pion pair correction to the Bhabha process) in the 
presence of realistic experimental cuts. 

The report is  divided into five sections: 
Sections~\ref{sec:1},~\ref{sec:2} and~\ref{sec:3} are devoted to the status of the MC tools for Luminosity,
the $R$-scan and Initial State Radiation (ISR).

%MC issues for  Tau decays and the theoretical treatment of the 
%Vacuum Polarisation are the subjects of the last two sections.

Tau spectral functions of hadronic decays are also used to estimate  
$a_{\mu}^{\mbox{$\scriptscriptstyle{\rm HLO}$}}$, since they can be related to
 $e^+e^-$ annihilation cross section via isospin
symmetry~\cite{Alemany:1997tn,Davier:2002dy,Davier:2003pw,Davier:2009ag}.
The substantial difference
between the $e^+e^-$- and $\tau$-based determinations of
$a_{\mu}^{\mbox{$\scriptscriptstyle{\rm HLO}$}}$, even if isospin
violation corrections are taken into account, shows that further
common theoretical and experimental efforts are necessary
to understand this phenomenon. 
% The
%$\tau$-based value is significantly higher, leading to a small ($\sim
%1 \sigma$) $\Delta a_{\mu}$ difference. 
In
Section~\ref{sec:5}  the experimental status  and 
MC tools for tau decays are discussed. The recent improvements
of the generators  TAUOLA and PHOTOS are discussed and prospects
  for further developments are sketched.

 

%T%he first three sections are directly connected with the measurement of hadronic cross sections and describes the status and the open issues in these fields.
%The four section describes the experimental status of the measurement and 
%the MC generators for $\tau$. 

Section~\ref{sec:4}  discusses vacuum polarisation at low energies, which is a
key ingredient for 
%focuses on the theoretical treatment of non-perturbative 
%contributions of light quarks to the Vacuum Polarisation, which is a key ingredient for 
the high precision determination of the hadronic cross section, focusing on
the description and comparison of available parametrisations. 
Finally, Section~\ref{sec:6} contains a brief summary of the report.


