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%\section{R measurement from energy scan}

In this section we will consider some theoretical and experimental aspects 
of the direct $R$ measurement and related quantities in experiments with energy scan. 
As discussed in the Introduction, the cross section of $e^+e^- $ annihilation 
into hadrons is involved in evaluations of various problems of particle physics and, 
in particular, in the definition of the hadronic contribution to vacuum 
polarisation, which is crucial for the precision tests of the Standard Model 
and searches for new physics.

The ratio of the radiation-corrected hadronic cross sections to the cross
%%%TT
section for muon pair production, calculated in the lowest order, 
is usually denoted as (see Eq.~(\ref{Rhad}))
\begin{equation} \label{r_def}
R \equiv R(s) = \frac{\sigma^0_{\rm had}(s)}
{4\pi \alpha^{2} /(3s)}.
\end{equation}
In the numerator of Eq.~(\ref{r_def})
one has to use the so called {\em undressed} hadronic cross section which does not
include vacuum polarisation corrections. 

The value of  $R$ has been measured in many experiments 
in different energy regions 
from the pion pair production threshold up to the $Z$ mass.
Practically all electron-positron colliders contributed to the global data
set on the hadronic annihilation cross section~\cite{Amsler:2008zzb}.
The value of $R$ extracted from the experimental data is then
widely used for various QCD tests as well as for the calculation
of dispersion integrals.
At high energies and away from resonances, 
the experimentally determined values of $R$ are in good agreement with
predictions of perturbative QCD, confirming, in particular, 
the hypothesis of three colour degrees of freedom for quarks.
On the other hand, for the low energy range the direct $R$ 
measurement~\cite{Amsler:2008zzb,Akhmetshin:2001ig}   
at experiments with energy scan is necessary.\footnote{Lattice QCD computations (see, e.g., Ref.~\cite{Shintani2009xxx}) 
of the hadronic vacuum polarisation 
are in progress, but they are not yet able to provide 
the required precision.} 
Matching between the two regions is performed at energies of a few GeV, where
both approaches for the determination of $R$ are in fair agreement.

For the best possible compilation of $R(s)$, 
data from different channels and different experiments have to be combined. 
For $\sqrt{s} \leq$ 1.4 GeV, the total hadronic cross section 
is a sum of about 25 
exclusive final states. At the present level of precision, 
a careful treatment of the radiative corrections is required. 
As mentioned above, it is mandatory to remove VP effects from the 
observed cross sections, 
but the final state radiation off hadrons should be kept. 
The major contribution to the uncertainty comes from the 
systematic error of the $R(s)$ measurement at low energies
($s<$ 2 GeV$^2$), which, in turn, is dominated by the 
systematic error of the measured cross section $e^+e^-\to\pi^+\pi^-$. 

\subsection{Leading-order annihilation cross sections}

Here we present the lowest-order expressions for the processes of
electron-positron annihilation into pairs of muons, pions and kaons.

For the muon production channel
\begin{eqnarray} \label{eemumu}
e^-(p_-) + e^+(p_+)\to \mu^-(p_-') + \mu^+(p_+')
\end{eqnarray}
within the Standard Model at Born level we have
\begin{eqnarray}
&& \frac{\dd{\sigma}^{\mu\mu}_0}{\dd\Omega_-} 
= \frac{\alpha^2\beta_\mu}{4s}
\left(2-\beta_\mu^2(1-c^2)\right)(1+K_{W}^{\mu\mu}),
\\ \nonumber 
&& s = (p_-+p_+)^2=4\eps^2, \quad c = \cos\theta_-, \quad
\theta_- = \widehat{\vec{p}_-\vec{p}}'_-, 
\end{eqnarray}
where $\beta_\mu=\sqrt{1-m_{\mu}^2/\eps^2}$ is the muon velocity. Small
terms suppressed by the factor $m_e^2/s$ are omitted.
Here $K_W^{\mu\mu}$ represents contributions due to $Z$-boson intermediate states
including $Z-\gamma$ interference, see, e.g., 
Refs.~\cite{Berends:1980yz,Arbuzov:1991pr}:
\begin{eqnarray}
&& K_W^{\mu\mu} =
\frac{s^2(2-\beta_\mu^2(1-c^2))^{-1}}
{(s-M_Z^2)^2+M_Z^2\Gamma_Z^2}\biggl\{(2-\beta_\mu^2(1-c^2))
\nonumber \\ && \quad \times
\left(c_v^2\biggl(3-2\frac{M_Z^2}{s}\biggr)+c_a^2\right) 
- \frac{1-\beta_\mu^2}{2}(c_a^2+c_v^2)
\nonumber \\ && \quad 
+c\beta_\mu\left[4\biggl(1-\frac{M_Z^2}{s}\biggr)c_a^2+8c_a^2c_v^2\right]\biggr\}, 
\nonumber \\ && 
 c_a=-\frac{1}{2\sin 2\theta_W},\quad
c_v=c_a(1-4\sin^2\theta_W),
\end{eqnarray}
where $\theta_W$ is the weak mixing angle.


The contribution of $Z$ boson exchange is suppressed, in the energy range under
consideration, by a factor $s/M_Z^2$ which reaches per mill level only
at $B$ factories. 

In the Born approximation the differential cross section
of the process
\begin{equation} \label{eepipi}
e^+(p_+)\ +\ e^-(p_-)\ \to\ \pi^+(q_+)\ +\ \pi^-(q_-)
\end{equation}
has the form
\begin{eqnarray} \label{seepipi}
&& \frac{\dd\sigma_0^{\pi\pi}}{\dd\Omega}(s)=\frac{\alpha^2\beta_\pi^3}{8s}\sin^2\theta\;
|F_{\pi}(s)|^2, \\ \nonumber 
&& \beta_\pi=\sqrt{1-m_{\pi}^2/\eps^2},\quad 
\theta=\widehat{\vec{p}_-\vec{q}}_- .
\end{eqnarray}
The pion form factor $F_{\pi}(s)$ takes into account
non-pertur\-ba\-tive virtual vertex corrections due to strong 
interactions~\cite{Bramon:1997un,Drago:1997px}.
We would like to emphasise that in the approach under discussion 
the final state QED corrections are not included into $F_{\pi}(s)$. 
The form factor is extracted from the experimental data on the 
same process as discussed below. 

The annihilation process with three pions in the final state was 
considered in Refs.~\cite{Ahmedov:2002tg,Kuraev:1995hc} including 
radiative corrections relevant to the energy region close to the threshold.
A stand-alone Monte Carlo event generator for this channel is 
available~\cite{Ahmedov:2002tg}. The channel was also included in the
MCGPJ generator~\cite{Arbuzov:2005pt} on the same footing as other 
processes under consideration in this report.


In the case of $K_LK_S$ meson pair production the differential
cross section in the Born approximation reads
\begin{equation}
\frac{\dd\sigma_{0}(s)^{K_LK_S}}{\dd\Omega_L}=\frac{\alpha^2\beta_{K}^3}{4s}
\sin^2\theta\; |F_{LS}(s)|^2.
\end{equation}
Here, as well as in the case of pion production, we assume that the
form factor $F_{LS}$ also includes the vacuum polarisation operator
of the virtual photon.
The quantity $\beta_{K}=\sqrt{1-4m_K^2/s}$ is the $K$ meson c.m.s.
velocity, and $\theta$ is the angle between the directions of motion of
the long living kaon and the initial electron.

In the case of $K^+K^-$ meson production near threshold, the Sakharov-Sommerfeld
factor for the Coulomb final state interaction 
should additionally be taken into account:
\begin{eqnarray} \label{bornk}
&& \frac{\dd\sigma_{0}(s)^{K^+K^-}}{\dd\Omega_-}=\frac{\alpha^2\beta_{K}^3}{4s}
\sin^2\theta |F_{K}(s)|^2\frac{Z}{1-\exp(-Z)}, 
\nonumber \\ 
&& Z=\frac{2\pi\alpha}{v}, \ \ 
v = 2\sqrt{\frac{s-4m_K^2}{s}}\left(1+\frac{s-4m_K^2}{s}\right)^{-1},
\end{eqnarray}
where $v$ is the relative velocity of the kaons~\cite{Arbuzov:1993qc} which has
the proper non-relativistic and ultra-relativistic limits. 
When $s=m_{\phi}^2$, we have $v\approx 0.5$ and the final state interaction
correction gives about 5\%  enhancement in the cross section. 


\subsection{QED radiative corrections}

One-loop radiative corrections (RC) for the 
processes~(\ref{eemumu},\ref{eepipi})  
can be separated into two natural parts according to the parity 
with respect to the substitution $c\to -c$.

The $c$-even part of the one-loop soft and virtual contribution to the muon 
%%%TT
pair creation channel can be combined from the well known Dirac and Pauli
form factors and the soft photon contributions. It reads
\begin{eqnarray} \label{beven_mumu}
&& \frac{\dd\sigma^{B+S+V}_{\mu\mu-\mathrm{even}}}{\dd\Omega} =
\frac{\dd\sigma_0^{\mu\mu}}{\dd\Omega}\,\frac{1}{|1-\Pi(s)|^2}\Biggl\{1
+ \frac{2\alpha}{\pi}\Biggl[\biggl[L - 2
\nonumber \\ && \quad 
+ \frac{1+\beta_\mu^2}{2\beta_\mu}l_\beta \biggr]
\ln\frac{\Delta\eps}{\eps} 
+ \frac{3}{4}(L-1)
+ K^{\mu\mu}_{\mathrm{even}} \Biggr]\Biggr\}, \\ \nonumber\label{js}
&& K_{\mathrm{even}}^{\mu\mu} = \frac{\pi^2}{6} - \frac{5}{4}
+ \rho\biggl(\frac{1+\beta_\mu^2}{2\beta_\mu} - \frac{1}{2}
+ \frac{1}{4\beta_\mu}\biggr) 
\nonumber \\ && \quad 
+ \ln\frac{1+\beta_\mu}{2}\left(\frac{1}{2\beta_\mu}
+ \frac{1+\beta_\mu^2}{\beta_\mu}\right) 
\\ \nonumber && \quad
- \frac{1-\beta_\mu^2}{2\beta_\mu}\frac{l_\beta}{2-\beta_\mu^2(1-c^2)}
\\ \nonumber && \quad
+ \frac{1+\beta_\mu^2}{2\beta_\mu}\biggl[ \frac{\pi^2}{6}
+ 2{\mathrm{Li}}_2\left(\frac{1-\beta_\mu}{1+\beta_\mu}\right)
\\ \nonumber && \quad
+ \rho\ln\frac{1+\beta_\mu}{2\beta_\mu^2} 
+ 2\ln\frac{1+\beta_\mu}{2}\ln\frac{1+\beta_\mu}{2\beta_\mu^2} \biggr], \\ 
\nonumber
l_{\beta} &=& \ln\frac{1+\beta_\mu}{1-\beta_\mu}\, , \quad
\rho=\ln\frac{s}{m_{\mu}^2}\quad L=\ln\frac{s}{m_{e}^2},
\end{eqnarray}
where ${\rm Li}_2(z)=-\int_0^z dt \ln(1-t)/t$ is the dilogarithm and 
$\Delta\eps\ll\eps$ is the maximum energy of soft photons
in the centre--of--mass (c.m.) system.
$\Pi(s)$ is the vacuum polarisation operator. Here we again see
that the terms with the large logarithm $L$ dominate numerically.


The $c$-odd part of the one--loop correction comes from the interference
of Born and box Feynman diagrams and from the interference part
of the soft photon emission contribution.
It causes the charge asymmetry of the process:
\begin{equation}
\eta={\dd\sigma(c)-\dd\sigma(-c)\over\dd\sigma(c)+\dd\sigma(-c)}\neq 0.
\end{equation}
The $c$-odd part of the differential cross section has the following 
form~\cite{Arbuzov:1997pj}:
\begin{eqnarray} \label{eqll}
\frac{\dd\sigma^{S+V}_{\mathrm{odd}}}{\dd\Omega} &=&
\frac{\dd\sigma_0^{\mu\mu}}{\dd\Omega}\,\frac{2\alpha}{\pi}\Biggl[
2\ln\frac{\Delta\eps}{\eps}\ln\frac{1-\beta c}{1+\beta c}
+ K_{\mathrm{odd}}^{\mu\mu} \biggr]. 
\end{eqnarray}
The expression for the $c$-odd form factor can be found 
in Ref.~\cite{Arbuzov:1997pj}.
Note that in most cases the experiments have a symmetric angular 
acceptance, so that the odd part of the cross section does not 
contribute to the measured quantities.

Consider now the process of hard photon emission
\begin{eqnarray}
e^+(p_+) + e^-(p_-) \rightarrow \mu^+(q_+) + \mu^-(q_-) + \gamma(k).
\end{eqnarray}
It was studied in detail in Refs.~\cite{Arbuzov:1997pj,Eidelman:1979gc}.
The photon energy is assumed to be larger than $\Delta\eps$.
The differential cross section has the form
\begin{eqnarray} \label{rmu}
&& \dd\sigma^{\mu\mu\gamma}=\frac{\alpha^3}{2\pi^2 s^2}R \dd\Gamma, 
\\ \nonumber 
&& \dd\Gamma = \frac{\dd^3q_-\dd^3q_+\dd^3k}
{q_-^0q_+^0k^0}\delta^{(4)}(p_++p_--q_--q_+-k), 
\\ \nonumber
&& R=\frac{s}{16(4\pi\alpha)^3}\sum\limits_{spins}^{}
|M|^2=R_e + R_{\mu} + R_{e\mu}.
\end{eqnarray}
The quantities $R_i$ are found directly from the matrix elements 
and read
\begin{eqnarray*}
&& R_e = \frac{s}{\chi_-\chi_+}B
- \frac{m_e^2}{2\chi_-^2}\;\frac{(t_1^2+u_1^2+2m_{\mu}^2s_1)}{s_1^2}
\\ && \quad
- \frac{m_e^2}{2\chi_+^2}\;\frac{(t^2+u^2+2m_{\mu}^2s_1)}{s_1^2}
+ \frac{m_{\mu}^2}{s_1^2} \Delta_{s_1s_1},
\nonumber \\
&& R_{e\mu} = B\biggl( \frac{u}{\chi_-\chi_+'} + \frac{u_1}{\chi_+\chi_-'}
- \frac{t}{\chi_-\chi_-'} 
\\ && \quad
- \frac{t_1}{\chi_+\chi_+'}\biggr)
+ \frac{m_{\mu}^2}{ss_1} \Delta_{ss_1},
\\ \nonumber
&& R_{\mu} = \frac{s_1}{\chi_-'\chi_+'}B + \frac{m_{\mu}^2}{s^2} \Delta_{ss}\, ,
\\ \nonumber
&& B = \frac{u^2+u_1^2+t^2+t_1^2}{4ss_1}\, ,
\\ \nonumber
&& \Delta_{s_1s_1} = - \frac{(t+u)^2+(t_1+u_1)^2}{2\chi_-\chi_+}\, ,
\\ \nonumber
&& \Delta_{ss} = - \frac{u^2+t_1^2+2sm_{\mu}^2}{2(\chi_-')^2}
- \frac{u_1^2+t^2+2sm_{\mu}^2}{2(\chi_+')^2}
\\ && \quad
+ \frac{1}{\chi_-'\chi_+'}\bigl( ss_1 - s^2 + tu +t_1u_1 - 2sm_{\mu}^2\bigr),
\\ \nonumber
&& \Delta_{ss_1} = \frac{s+s_1}{2}\biggl( \frac{u}{\chi_-\chi_+'}
+ \frac{u_1}{\chi_+\chi_-'} - \frac{t}{\chi_-\chi_-'}
\\ && \quad
- \frac{t_1}{\chi_+\chi_+'} \biggr)
+ \frac{2(u-t_1)}{\chi_-'} + \frac{2(u_1-t)}{\chi_+'},
\end{eqnarray*}
where
\begin{eqnarray*}
&& s_1=(q_++q_-)^2, \ \ 
t=-2p_-q_-, \ \ 
t_1=-2p_+q_+,
\\ 
&& u=-2p_-q_+,\ \ 
u_1=-2p_+q_-, \ \
\chi_{\pm}=p_{\pm}k, \ \ 
\chi_{\pm}'=q_{\pm}k.
\end{eqnarray*}

The bulk of the hard photon radiation comes from ISR in collinear regions.
If we consider photon emission inside two narrow cones along the beam axis 
with restrictions
\begin{eqnarray}
\widehat{\vec{p}_{\pm}\vec{k}}=\theta\leq\theta_0\ll 1,\quad
\theta_0\gg\frac{m_e}{\eps},
\end{eqnarray}
we see that the corresponding contribution takes the factorised form
\begin{eqnarray} \label{eqcd}
&& \left(\frac{\dd\sigma^{\mu\mu}}{\dd\Omega_-}\right)_{\mathrm{coll}}=
C_e + D_e, 
\\ \nonumber 
&& C_e=\frac{\alpha}{2\pi}\biggl(\ln\frac{s}{m_e^2}-1\biggr)
\int\limits_{\Delta}^{1}\!\!\dd x \frac{1+(1-x)^2}{x} A_0,
\\ \nonumber
&& D_e=\frac{\alpha}{2\pi}\int\limits_{\Delta}^{1}\!\!\dd x \biggl\{x
+ \frac{1+(1-x)^2}{x}\ln\frac{\theta_0^2}{4}\biggr\} A_0, 
\\ \nonumber
&& A_0 = \biggl[\frac{\dd\tilde{\sigma}_0(1-x,1)}{\dd\Omega_-}
+ \frac{\dd\tilde{\sigma}_0(1,1-x)}{\dd\Omega_-}\biggr],
\end{eqnarray}
where the {\em shifted\/} 
Born differential cross section describes the process
$e^+(p_+(1-x_2)) + e^-(p_-(1-x_1)) \to \mu^+(q_+) + \mu^-(q_-)$,
\begin{eqnarray}
&& \frac{\dd\tilde{\sigma}_0^{\mu\mu}(z_1,z_2)}{\dd\Omega_-}
= \frac{\alpha^2}{4s}
\nonumber \\ && \quad \times
\frac{y_1[z_1^2(Y_1-y_1c)^2
+ z_2^2(Y_1+y_1c)^2+8z_1z_2m_{\mu}^2/s]}
{z_1^3z_2^3[z_1+z_2-(z_1-z_2)cY_1/y_1]}, 
\nonumber \\
&& y_{1,2}^2=Y_{1,2}^2-\frac{4m_{\mu}^2}{s},\quad
Y_{1,2}=\frac{q_{-,+}^0}{\eps}, \quad z_{1,2}=1-x_{1,2},
\nonumber \\
&& Y_1= \frac{4m_{\mu}^2}{s}(z_2-z_1)c \biggl[
2z_1z_2
\nonumber \\ && \quad 
+\sqrt{4z_1^2z_2^2-4(m_{\mu}^2/s)((z_1+z_2)^2
-(z_1-z_2)^2c^2)}\biggr]^{-1} 
\nonumber \\ && \quad
+ \frac{2z_1z_2}{z_1+z_2-c(z_1-z_2)}.
\end{eqnarray}
One can recognise that the large logarithms related to the collinear
photon emission appear in $C_e$ in agreement with the structure 
function approach
discussed in the Luminosity Section. In analogy to the definition of
the QCD structure functions, one can move the factorised logarithmic 
corrections $C_e$
into the QED electron structure function. 
Adding the higher-order radiative corrections in the leading logarithmic
approximation to the complete one-loop result,
the resulting cross section can be written as
\begin{eqnarray} \label{eemmg}
&& \frac{\dd\sigma^{e^+e^-\to\mu^+\mu^-(\gamma)}}{\dd \Omega_-}
=\int\limits_{z_{\mathrm{min}}}^{1}
\int\limits_{z_{\mathrm{min}}}^{1}\dd z_1\dd z_2
\frac{{\cal D}(z_1,s){\cal D}(z_2,s)
}{|1-\Pi(sz_1z_2)|^2} 
\nonumber \\ && \quad \times
\frac{\dd\tilde{\sigma}_0^{\mu\mu}(z_1,z_2)}{\dd \Omega_-}
\biggl(1+\frac{\alpha}{\pi}K_{\mathrm{odd}}^{\mu\mu}
+\frac{\alpha}{\pi}K_{\mathrm{even}}^{\mu\mu}\biggr)
\nonumber \\ \nonumber && \quad
+ \biggl\{
\frac{\alpha^3}{2\pi^2s^2} \int\limits_{\stackrel{k^0>\Delta\eps}
{\widehat{k p}_{\pm}>\theta_0}}
\frac{R_e|_{m_e=0}}{|1-\Pi(s_1)|^2}\frac{\dd\Gamma}{\dd \Omega_-}
 + \frac{D_e}{|1-\Pi(s_1)|^2}\biggr\}
\\  && \quad
+ \biggl\{\frac{\alpha^3}{2\pi^2s^2}\int\limits_{k^0>\Delta\eps}\!\!\!
\biggl(\mbox{Re}\, \frac{R_{e\mu}}{(1-\Pi(s_1))(1-\Pi(s))^{*}}
\nonumber \\ && \quad 
+ \frac{R_{\mu}}{|1-\Pi(s)|^2}\biggr)\frac{\dd\Gamma}{\dd \Omega_-} 
+ \mbox{Re}\,\frac{C_{e\mu}}{(1-\Pi(s_1))(1-\Pi(s))^{*}}
\nonumber \\  && \quad
+ \frac{C_{\mu}}{|1-\Pi(s)|^2} \biggr\},                     
\\ && \nonumber
C_{\mu} = \frac{2\alpha}{\pi}\frac{\dd\sigma_0^{\mu\mu}}{\dd\Omega_-}
\ln\frac{\Delta\eps}{\eps}
\biggl(\frac{1+\beta^2}{2\beta}\ln\frac{1+\beta}{1-\beta}-1\biggr),
\\ \nonumber &&
C_{e\mu} = \frac{4\alpha}{\pi}\frac{\dd\sigma_0^{\mu\mu}}{\dd\Omega_-}
\ln\frac{\Delta\eps}{\eps}\ln\frac{1-\beta c}{1+\beta c},
\end{eqnarray}
where $D_e$, $C_{e\mu}$ and $C_{\mu}$ are compensating terms,
which provide the cancellation of the auxiliary parameters $\Delta$ and $\theta_0$
inside the curly brackets. In the first term, containing ${\cal D}$ functions,
we collect all the leading logarithmic terms. 
A part of non-leading terms proportional
to the Born cross section is written as the $K$-factor. The rest of the 
non-leading contributions are written as two additional terms. The compensating
term $D_e$ (see Eq.~(\ref{eqcd})) comes from the integration in the collinear
region of hard photon emission. The quantities $C_{\mu}$ and $C_{e\mu}$
come from the even and odd parts of the differential cross section
(arising from soft and virtual corrections), respectively.
Here we consider the phase space of two
$(\dd\Omega_-)$ and three $(\dd\Gamma)$ final particles as
those that already include all required experimental cuts.
Using specific experimental conditions one can determine the lower
limits of the integration over $z_1$ and $z_2$ instead of the kinematical
limit $z_{\mathrm{min}} = 2m_{\mu}/(2\eps-m_{\mu})$.

Matching of the complete $O (\alpha)$ RC with higher-order leading
logarithmic corrections can be performed in different schemes. The above approach
is implemented in the MCGPJ event generator~\cite{Arbuzov:2005pt}. The solution of the
QED evolution equations in the form of parton showers (see the Luminosity Section), 
matched again with the first order corrections, is implemented in the BabaYaga@NLO 
generator~\cite{CarloniCalame:2003yt}. Another possibility is realised in the 
KKMC code~\cite{Jadach:1999vf,Jadach:2000ir} with the Yennie-Frautschi-Suura exponentiated 
representation of the photonic higher-order corrections.
A good agreement was obtained in~\cite{Arbuzov:2005pt} for various differential 
distributions for the $\mu^+\mu^-$ channel between MCGPJ, BabaYaga@NLO and KKMC,
see Fig.~\ref{mumu_kkmc} for an example.

\begin{figure}[htb]
\centering\includegraphics[width=0.49\textwidth]{kkmc_diff_fsr.eps}
\caption{Comparison of the $e^+e^-\to\mu^+\mu^-$ total cross sections
computed by the MCGPJ and KKMC generators versus the c.m. energy.}
\label{mumu_kkmc} 
\end{figure}

Since the radiative corrections to the initial $e^+e^-$ state 
are the same for annihilation into hadrons and muons
as well as that into pions, they cancel out in part in the ratio~(\ref{piform}). 
However, the experimental conditions and systematic are different 
for the muon and hadron channels.
Therefore, a separate treatment of the processes has to be performed and the 
corrections to the initial states have to be included in the analysis.

For the $\pi^+\pi^-$ channel the complete set of $O (\alpha)$ 
corrections matched with the leading logarithmic electron structure functions
can be found in Ref.~\cite{Arbuzov:1997je}. There the RC calculation 
was performed within scalar QED. 

Taking into account only final state corrections calculated within scalar QED,
it is convenient to introduce the {\em bare} 
$e^+e^-\to\pi^+\pi^-(\gamma)$ cross
section as 
\begin{equation}
\label{bare}
\sigma^0_{\pi\pi(\gamma)} = 
\frac{\pi\alpha^2}{3s}\beta^3_\pi \left| F_\pi(s)\right|^2 |1-\Pi(s)|^2 
\left( 1+\frac{\alpha}{\pi}\Lambda(s) \right),
\end{equation}
where the  factor $|1-\Pi(s)|^2$ with the polarisation operator 
$\Pi(s)$ gives the effect of leptonic and hadronic vacuum 
polarisation. The final state radiation (FSR) correction is
denoted by $\Lambda(s)$. For an inclusive measurement without cuts
it reads~\cite{Schwinger:1989ix,Drees:1990te,Melnikov:2001uw,Hoefer:2001mx}
\begin{eqnarray}
&& \Lambda(s)  =  \frac {1+\beta_{\pi}^2} {\beta_{\pi}} 
\biggl\{ 4\mathrm{Li}_2 (\frac {1-\beta_{\pi}} {1+\beta_{\pi}})+
2\mathrm{Li}_2 (-\frac {1-\beta_{\pi}} {1+\beta_{\pi}})
\nonumber \\ && \quad
- \biggl[3\ln(\frac{2}{1+\beta_{\pi}})
+ 2\ln\beta_{\pi} \biggr]\ln\frac{1+\beta_{\pi}}{1-\beta_{\pi}}\biggr\} 
-  3 \ln  (\frac {4}{1-\beta_{\pi}^2})
\nonumber \\ && \quad
-4\ln\beta_{\pi} 
+  \frac {1} {\beta_{\pi}^3}
 \left[\frac{5}{4}(1+\beta_{\pi}^2)^2-2\right]\ln\frac{1+\beta_{\pi}}{1-\beta_{\pi}}
\nonumber \\ && \quad
 + \frac {3} {2} \frac{1+\beta_{\pi}^2}{\beta_{\pi}^2}.
\end{eqnarray}


For the neutral kaon channel the corrected cross section has the form
\begin{eqnarray} \nonumber
\frac{\dd\sigma^{e^+e^-\to K_LK_S}(s)}{\dd\Omega_L}=
\int\limits_{0}^{\Delta}\dd x\;
\frac{\dd\sigma_0^{e^+e^-\to K_LK_S}(s(1-x))}{\dd\Omega_L}
F(x,s).
\end{eqnarray}
The radiation factor $F$ takes into account radiative
corrections to the initial state within the leading logarithmic 
approximation with exponentiation of the numerically important
contribution of soft photon radiation, see Ref.~\cite{Kuraev:1985hb}:
\begin{eqnarray}
&& F(x,s) = bx^{b-1}\biggl[ 1 + \frac{3}{4}b
+ \frac{\alpha}{\pi}\biggl(\frac{\pi^2}{3}-\frac{1}{2}\biggr)
- \frac{b^2}{24}\biggl(\frac{1}{3}L  - 2\pi^2 
\nonumber \\ \nonumber && \quad 
- \frac{37}{4}\biggr)\biggr] 
- b\biggl(1-\frac{x}{2}\biggr)
+ \frac{1}{8}b^2\biggl[4(2-x)\ln\frac{1}{x}
\\ \nonumber && \quad
+ \frac{1}{x}(1+3(1-x)^2)\ln\frac{1}{1-x} - 6 + x\biggr] 
\\ \nonumber && \quad
+ \biggl(\frac{\alpha}{\pi}\biggr)^2\biggl\{\frac{1}{6x}
\biggl(x-\frac{2m_e}{\eps}\biggr)^{b}\biggl[ (2-2x+x^2)
\biggl(\ln\frac{sx^2}{m_e^2}-\frac{5}{3}\biggr)^2
\\ \nonumber && \quad
+ \frac{b}{3}\biggl(\ln\frac{sx^2}{m_e^2}-\frac{5}{3}\biggr)^3\biggr]
+ \frac{1}{2}L^2\biggl[\frac{2}{3}\;\frac{1-(1-x)^3}{1-x}
\\ \nonumber && \quad
+ (2-x)\ln(1-x) + \frac{x}{2}\biggr]\biggr\} \Theta(x-\frac{2m_e}{\eps}).
\end{eqnarray}

Radiative corrections to the $K^+K^-$ channel in the point-like particle
approximation are the same as for the case of charged pion pair 
(with the substitution $m_\pi\to m_K$).
Usually, for the kaon channel we deal with the energy range
close to $\phi$ mass. There one may choose the maximal energy of a radiated
photon as
\begin{equation}
\omega\leq\Delta E=m_{\phi}-2m_K\ll m_K,\quad
\Delta\equiv\frac{\Delta E}{m_K}\approx\frac{1}{25}.
\end{equation}
For these photons one can use the soft photon approximation. 

\subsection{Experimental treatment of hadronic cross sections and $R$}

For older low energy data sets obtained at various $e^+e^-$ colliders, 
the correct treatment of radiative corrections is difficult and
sometimes ambiguous. 
So, to avoid uncontrolled possible
systematic errors, it may be reasonable not to include all previous 
results except the recent data from CMD-2 and SND. 
Both experiments at the VEPP-2M collider in
Novosibirsk have delivered independent new measurements. 
The covered energy range is crucial for
($g_{\mu}$-2)/2 of muon and for running $\alpha$. 
As for the two-pion channel $\pi^+\pi^-$, which gives more than
70\% of the total hadronic contribution, both experiments have very good
agreement over the whole energy range. The relative deviation ``SND -
CMD-2'' is (-0.3 $\pm$ 1.6)\% only, well within the quoted errors.       

The CMD-2 and SND detectors were located in the opposite straight sections of
VEPP-2M and were taking data in parallel until the year 2000 when the   
collider was shut down to prepare for the construction of the new collider 
VEPP-2000. Some important features of the CMD-2
detector allowed one to select a sample of the ``clean'' 
collinear back-to-back 
events. The drift chamber (DC) was used to 
separate $e^+e^-$, $\mu^+\mu^-$, $\pi^+\pi^-$ and $K^+K^-$ events from other
particles. The Z-chamber allowed one to significantly improve 
the determination 
of the polar angle of charged particle tracks in the DC that, in turn, 
provided the detector acceptance with 0.2\% precision. 
The barrel electromagnetic calorimeter based on CsI crystals 
helped to separate the Bhabha from other collinear events. 

The SND detector consisted of three spherical layers of the
electromagnetic calorimeter with 1620 crystals (NaI) and a total
weight of 3.6 tons.  The solid angle of the calorimeter is about 90\% of
$4\pi$ steradians, which makes the detector practically hermetic for photons
coming from the interaction point. The angular and energy resolution 
for photons was found to be $1.5^{\circ}$ and 
$\sigma(E)/E = 4.2\%/E$(GeV)$^{1/4}$, respectively. 
More detail about CMD-2 and SND can be found 
elsewhere~\cite{Anashkin1988:xxx,Achasov:1999ju}.\\
 

\subsubsection{Data taking and analysis of the $\pi^+\pi^-$ channel}
The detailed data on the pion form factor are crucial for a number of problems
in hadronic physics and they are used to extract $\rho(770)$ meson
parameters and its radial excitations. Besides, the detailed data allow to 
extrapolate the pion form factor to the point $s=0$ and determine 
the value of the pion electromagnetic radius.

From the experimental point of view the form factor
can be defined as~\cite{Akhmetshin:2001ig}
\begin{eqnarray} \label{piform}
|F_\pi|^2 &=& \frac{N_{\pi\pi}}{N_{ee}+N_{\mu\mu}}
\frac{
\sigma_{ee}(1+\delta_{ee})\varepsilon_{ee} +
\sigma_{\mu\mu}(1+\delta_{\mu\mu})\varepsilon_{\mu\mu}
}{\sigma_{\pi\pi}(1+\delta_{\pi\pi})
(1+\Delta_N)(1+\Delta_D)\varepsilon_{\pi\pi}}
\nonumber \\
&-& \Delta_{3\pi},
\end{eqnarray}
where the ratio $N_{\pi\pi}/(N_{ee}+N_{\mu\mu})$ is derived from the
observed numbers of events,
$\sigma$ are the corresponding Born cross sections,
$\delta$ are the radiative corrections (see below),
$\epsilon$ are the detection efficiencies,
$\Delta_D$ and $\Delta_N$ are the corrections for the pion losses
caused by decays in flight and nuclear interactions respectively,
and $\Delta_{3\pi}$  is the correction for misidentification of
$\omega\to\pi^+\pi^-\pi^0$ events as $e^+e^-\to\pi^+\pi^-$. In the case of the
latter process, $\sigma_{\pi\pi}$ corresponds to point-like pions. \\

The data were collected in the whole energy range of VEPP-2M and 
the integrated luminosity of about 60~pb$^{-1}$ was recorded by both
detectors. The beam energy was controlled and measured with a relative 
accuracy not worse than $\sim 10^{-4}$ by using the method of resonance depolarisation. A sample of the $e^+e^-$, $\mu^+\mu^-$ and $\pi^+\pi^-$  
events was selected for analysis. 
As for CMD-2, the procedure of the
$e/\mu/\pi$ separation for energies 2E $\leq$ 600 MeV was based on the momentum
measurement in the DC. For these energies the average difference 
between the momenta of $e/\mu/\pi$ is large enough with respect to the momentum 
resolution (Fig.~\ref{fig:p1-vs-p2}). On the contrary, for energies 
2E $\geq$ 600~MeV,
the energy deposition of the particles in the calorimeter is quite different 
and allows one to separate electrons from muons and pions 
(Fig.~\ref{fig:e1-vs-e2}).
At the same time, muons and pions cannot be separated by their energy
depositions in the calorimeter. So, the ratio 
$N(\mu^+ \mu^-)/N(e^+e^-)$ was fixed
according to QED calculations taking into account the detector acceptance and
the radiative corrections. Since the selection criteria were the same for  
all collinear events, many effects of the detector
imperfections were partly cancelled out. It allowed one to measure the cross
section of the process $e^+e^- \to \pi^+\pi^-$ with better precision than
that of the luminosity.  

\begin{figure}[htb]
\centering\includegraphics[width=0.49\textwidth]{p1-vs-p2.eps}
\caption{Two-dimensional plot of the $e/\mu/\pi$ events. 
Cosmic events are distributed predominantly along a corridor which
extends from the right upper to the left bottom corner. 
Points in this plot correspond to the momenta of particles 
for the beam energy of 195~MeV. }
\label{fig:p1-vs-p2} 
\end{figure}


\begin{figure}[htb]
\centering\includegraphics[width=0.49\textwidth]{e1-vs-e2.eps}
\caption{ Energy deposition of collinear events 
for the beam energy of 460~MeV.}
\label{fig:e1-vs-e2}
\end{figure}


Separation of $e^+e^-$, $\mu^+\mu^-$ and $\pi^+\pi^-$
events was based on the minimisation of the unbinned likelihood
function. This method is described in detail 
elsewhere~\cite{Akhmetshin:1999uj}. 
To simplify the error calculation of the pion form
factor, the likelihood function had the global fit parameters $(N_{ee}
+ N_{\mu\mu})$ and $N_{\pi\pi}/(N_{ee} + N_{\mu\mu})$, 
through $|F_{\pi}(s)|^2$ 
given by Eq.~(\ref{piform}). 
The pion form factor measured by CMD-2 has a systematic 
error of about 0.6-0.8\% for $\sqrt{s} \leq $ 1 GeV. 
For energies above 1 GeV it varies from 1.2\% to 4.2\%. 

Since at low energies all three final states could be separated independently, 
the cross section of the process $e^+e^- \to \mu^+\mu^-$ was also measured,
providing an additional consistency test. The experimental value
$\sigma^{\rm exp}_{\mu\mu} / \sigma^{\rm QED}_{\mu\mu}$ = (0.980 $\pm$ 0.013 $\pm$
0.007) is in good agreement with the expected value of 1 within 1.4
statistical deviations.  

Another method  to discriminate electrons and pions from other
particles was used in SND. The event separation was 
based on the difference in longitudinal energy deposition profiles 
(energy deposition in three calorimeter layers) 
for these particles. To use the correlations between energy 
depositions in calorimeter layers in the most complete way, 
the separation parameter was
based on the neural network approach~\cite{Achasov:2005rg,Achasov:2006vp}. 
The network had an input layer consisting of 7 neurons, two hidden layers with
20 neurons each, and the output layer with one neuron. As input data, the
network used the energy depositions of the particles in calorimeter layers
and the polar angle of one of the particles. The output signal $R_{e/\pi}$ is
a discrimination parameter between different particles. The network was
tuned by using simulated events and was checked with experimental 3$\pi$ and
$e^+e^-$ events. The misidentification ratio between electrons and pions was
found to be 0.5 - 1\%. SND measured the
$e^+e^- \to \pi^+\pi^-$ cross section in the energy range 0.36 - 0.87
GeV with a systematic error of 1.3\%.   

The  Gounaris-Sakurai (GS) parametrisation was used to fit the pion form factor.    
Results of the fit are shown in Fig.~\ref{fig:fpicmd2}.
The 
%minimal value of 
$\chi^2$
 was found to be $\chi^2_{\rm min}/\mathrm{n.d.f.} = 122.9/111$ that corresponds to the
 probability P$(\chi^2_{\rm min}/\mathrm{n.d.f.})$ = 0.21.
 The average deviation between SND \cite{Achasov:2005rg,Achasov:2006vp} 
and CMD-2 \cite{Akhmetshin:2003zn} data
 is: $\Delta$(SND -- CMD-2) $\sim (1.3 \pm 3.6)\%$ for the energy range
 $\sqrt{s}$ $\leq$ 0.55 GeV and $\Delta$(SND -- CMD-2)
 $\sim (-0.53 \pm 0.34)\%$ for the energy range $\sqrt{s}$ $\geq$ 0.55 GeV.
The obtained $\rho$ meson parameters are:\\ 
CMD-2 -- $M_{\rho} = 775.97 \pm 0.46 \pm 0.70 $ MeV,\\
$\Gamma_{\rho} = 145.98 \pm 0.75 \pm 0.50 $ MeV,\\
$\Gamma_{ee} = 7.048 \pm 0.057 \pm 0.050 $ keV,\\
$\mathrm{Br}(\omega \to \pi^+\pi^-) = (1.46 \pm 0.12 \pm 0.02) \%$;\\
SND -- $M_{\rho} = 774.6 \pm 0.4 \pm 0.5$ MeV,\\
$\Gamma_{\rho} = 146.1 \pm 0.8 \pm 1.5  $ MeV,\\
$\Gamma_{ee} = 7.12 \pm 0.02 \pm 0.11 $ keV,\\
$\mathrm{Br}(\omega \to \pi^+\pi^-) = (1.72 \pm 0.10 \pm 0.07) \%$.\\ 
The systematic errors were carefully studied and are listed in 
Table~\ref{tsyst-err}.   

\begin{table*}
\begin{center}
\begin{tabular}{|l|c|c|c|}
    \hline
    Sources of errors & CMD-2             & SND & CMD-2 \\
    & $\sqrt {s} < 1 $ GeV &     & $1.4 > \sqrt {s} > 1$ GeV \\
    \hline
    Event separation method & 0.2 - 0.4\% & 0.5\% & $0.2 - 1.5\%$ \\
    Fiducial volume &  0.2\%  & 0.8\% & $0.2 - 0.5\%$ \\
    Detection efficiency & 0.2 - 0.5\% & 0.6\% & $0.5 - 2\%$ \\
    Corrections for pion losses & 0.2\% & 0.2\% & 0.2\% \\
    Radiative corrections & 0.3 - 0.4\% & 0.2\% & $0.5 - 2\%$ \\
    Beam energy determination & 0.1 - 0.3\% & 0.3\% & $0.7 - 1.1\%$  \\
    Other corrections & 0.2\% & 0.5\% & $0.6 - 2.2\%$ \\
    \hline
    The total systematic error & 0.6 - 0.8\% & 1.3 \% & $1.2 - 4.2\%$ \\
    \hline
\end{tabular}
\caption{ The main sources of the systematic errors for different energy regions.}
\label{tsyst-err}
\end{center}
\end{table*}

\begin{figure}[htb]
  \centering\includegraphics[width=0.49\textwidth]{fpicmd2.eps}
  \caption{ Pion form factor data from CMD-2 and GS
    fit. The energy range around the $\omega$ meson is scaled up and
    presented in the inset.}
  \label{fig:fpicmd2}
\end{figure}
%%%TT
The comparison of the $\rho$ meson parameters determined by CMD-2 and
SND with the values from the PDG is presented in
Fig.~\ref{fig:rho-par}. Good agreement is observed for all parameters, 
except for the branching fraction of $\omega$ decaying to $\pi^+\pi^-$,
where a difference $\sim$ 1.6 standard deviations is observed. 

\begin{figure}[htb]
  \centering\includegraphics[width=0.49\textwidth]{rho-par.eps}
%%%TT
  \caption{Comparison of $\rho$ meson parameters from CMD-2 and SND
    with corresponding PDG values. The panels (top-left to
    bottom-right) refer to the mass (MeV), width (MeV), leptonic width
    (keV) and the branching fraction of the decay
    $\omega \to \pi^+\pi^-$ (\%).} 
  \label{fig:rho-par} 
\end{figure}

\subsubsection{Cross section of the process $e^+e^- \to \pi^+\pi^-\pi^{0}$}

This channel was studied
by SND in the energy range $\sqrt{s}$ from 0.6 to 1.4 
GeV~\cite{Achasov:2002ud,Achasov:2003ir}, 
while CMD-2 has reported results of the measurements in
vicinity of the $\omega$~\cite{Akhmetshin:2003zn} and 
$\phi$ meson peaks~\cite{Akhmetshin:2006sc}.   
For both the $\omega$ and $\phi$ resonances CMD-2 and SND obtain consistent 
results for the product of the resonance branching fractions into
$e^+e^-$ and $\pi^+\pi^-\pi^0$, for which they have the world's best accuracy
(SND for the $\omega$ and CMD-2 for the $\phi$ resonance).

CMD-2 has also performed a detailed Dalitz plot analysis of the 
dynamics of $\phi$ decaying to $\pi^+\pi^-\pi^{0}$. 
Two models of $3\pi$ production were used: a $\rho\pi$ mechanism 
and a contact amplitude.  
The result obtained for the ratio of the contact and $\rho\pi$
amplitudes is in good agreement with that of KLOE~\cite{Aloisio:2003ur}. 
 
The systematic accuracy of the measurements is about 1.3\% around  the $\omega$
meson energy region, 2.5\% in the $\phi$ region, and about 5.6\%  for  
higher energies. The results of different experiments are collected in
Fig.~\ref{fig:3pi-crsect}. The curve is the fit which takes into account
the $\rho,\ \omega,\ \phi,\ \omega' $ and $\omega''$ mesons.  

\begin{figure}[htb]

\centering\includegraphics[width=0.49\textwidth]{3pi-crsect.eps}
  \caption{Cross section of the process 
  $e^+e^- \to \pi^+\pi^-\pi^{0}$.}
  \label{fig:3pi-crsect} 
\end{figure}

\subsubsection{Cross section of the process $e^+e^- \to 4\pi$}

This cross section becomes 
important for energies above the $\phi$ meson region. CMD-2  
showed that the $a_1(1260)\pi$ mechanism
is dominant for the process  $e^+e^- \to \pi^+\pi^-\pi^+\pi^-$, whereas for
%%%TT
the channel $e^+e^- \to \pi^+\pi^-\pi^{0}\pi^{0}$ in addition the
intermediate state $\omega\pi$ is required to describe the energy
dependence of the cross section~\cite{Akhmetshin:1998df}.
The SND analysis confirmed these conclusions~\cite{Achasov:2003bv}. 
The knowledge of the dynamics of $4\pi$ production allowed to determine
%%%TT
the detector acceptance and efficiencies with better precision
compared to the previous measurements. 

The cross section of the process  $e^+e^- \to \pi^+\pi^-\pi^+\pi^-$ was
measured with a total systematic error of 15\% for CMD-2 and
7\% for SND. For the channel $e^+e^- \to \pi^+\pi^-\pi^{0}\pi^{0}$
the systematic uncertainty was 15 and 8\%, respectively. 
The CMD-2 reanalysis of the process
$e^+e^- \to \pi^+\pi^-\pi^+\pi^-$, 
with a better procedure for the efficiency determination, reduced the
systematic error to (5-7)\%~\cite{Akhmetshin:2004dy}, and these new
results are now in remarkable agreement with other experiments. \\ 

\subsubsection{Other modes}

CMD-2 and SND have also measured the cross sections of the processes 
$e^+e^- \to K_{S} K_{L}$ and $e^+e^- \to K^+K^-$ from threshold and 
up to 1.38 GeV with much better accuracy than before~\cite
{Akhmetshin:2002vj,Achasov:2006bv,Achasov:2007kg}. 
These cross sections were studied 
thoroughly in the vicinity of the
$\phi$ meson, and 
their systematic errors were determined 
with a precision of about 1.7\% (SND) and 4\% (CMD-2), 
respectively. The analyses were based
on two decay modes of the $K_{S}$: 
$K_{S} \to \pi^{0}\pi^{0}$ and $\pi^+\pi^-$.
As for the process $e^+e^- \to K^+ K^-$, 
the systematic uncertainty was studied in detail 
and found to be 2.2\% (CMD-2) and 7\% (SND). 

At energies $\sqrt{s}$ above 1.04 GeV the cross 
sections of the processes $e^+e^- \to K_{S} K_{L}, 
K^+ K^-$ were measured with a statistical 
accuracy of about 4\% and systematic errors of about 
4-6\% and 3\%, respectively, and  are in good agreement
with other experiments. 

To summarise, the experiments performed in 1995--2000 with 
the CMD-2 and SND detectors at VEPP-2M allowed one to measure
the exclusive cross sections of $e^+e^-$ annihilation into
hadrons in the energy range $\sqrt{s}$ = 0.36 - 1.38 GeV with
larger statistics and smaller systematic errors compared to the previous
experiments. 
%%%TT
Figure~\ref{fig:crscmd2snd-all} summarises the cross
section measurements from CMD-2 ans SND.
%
\begin{figure}[htb]
\centering\includegraphics[width=0.49\textwidth]{crscmd2snd-all.eps}
  \caption{Hadronic cross sections measured by
  CMD-2 and SND in the whole energy range of VEPP-2M. The curve represents
  a smooth spline of the sum of all data.}
  \label{fig:crscmd2snd-all} 
  \end{figure}
%
The results of these experiments determine the current accuracy of the
  calculation of the muon anomaly, and they are one of the main
  sources of information about physics of vector mesons at low energies. 

\subsubsection{$R$ measurement at CLEO}

Two important measurements of the $R$ ratio have been recently
reported by the CLEO Collaboration~\cite{:2007qwa,CroninHennessy:2008yi}.

In the energy range just above the open charm threshold,
they collected statistics at thirteen c.m. energy points
from 3.97 to 4.26~GeV~\cite{CroninHennessy:2008yi}.
Hadronic cross sections 
in this region exhibit a
rich structure, reflecting the production of $c\bar{c}$ resonances.
Two independent measurements 
have been performed. In one of them they determined a sum of the 
exclusive cross sections for final
states consisting of two charm mesons ($D\bar{D}$, $D^*\bar{D}$,
 $D^*\bar{D}^*$, $D^+_sD^-_s$,  $D^{*+}_sD^-_s$, and
$D^{*+}_sD^{*-}_s$)
and of processes in which the charm-meson pair is accompanied by a
pion. In the second one they measured the inclusive cross section
with a systematic uncertainty between 5.2 and 6.1\%.
The results of  both measurements are in excellent agreement, which
leads to the 
important conclusion that in this energy range the sum of the two- and 
three-body cross sections
saturates the total cross section for charm production. In 
Fig.~\ref{fig:low} the inclusive cross section measured by CLEO 
is compared with the previous measurements by
Crystal Ball~\cite{Osterheld:1986hw} and BES~\cite{Bai:2001ct}. 
Good agreement is observed between the data.

\begin{figure}
\centering\includegraphics[width=0.49\textwidth]{cleorl.eps}
\caption{Comparison of the $R$ values from CLEO
(the inclusive determination) with those from Crystal Ball and BES.}
\label{fig:low}
\end{figure}

CLEO has also performed a new measurement of $R$ at higher energy. 
They collected statistics at seven c.m.
energy points from 6.964 to 10.538~GeV~\cite{:2007qwa} and reached
a very small systematic uncertainty of 2\% only. Results of their 
scan are presented in Fig.~\ref{fig:high} and are in good 
agreement with those of Crystal Ball~\cite{Osterheld:1986hw}, 
MD-1~\cite{Blinov:1993fw} and the previous 
measurement of CLEO~\cite{Ammar:1997sk}. However, they are obviously  
inconsistent with those of the old MARK I 
measurement~\cite{Siegrist:1981zp}. 
    
\begin{figure}
\centering\includegraphics[width=0.49\textwidth]{cleorh.eps}
\caption{Top plot: comparison of the $R$ values from CLEO
with those from MARK I, Crystal Ball and MD-1; bottom plot: comparison
of the new CLEO results with the QCD prediction at $\Lambda=$0.31~GeV.}
\label{fig:high}
\end{figure}

\subsubsection{$R$ measurement at BES}

Above 2 GeV the number of final states becomes too large for completely
exclusive measurements, so that  the values of $R$ are measured 
inclusively.

In 1998, as a feasibility test of $R$ measurements, 
BES took data at six c.m. energy
points between 2.6 and 5.0 GeV~\cite{Bai:1999pk}. The integrated
luminosity collected at each energy point changed from 85 to 292~nb$^{-1}$.
The statistical error was  around 3\% per point 
and the systematic error ranged from 7 to 10\%.

Later, in 1999, BES performed a systematic fine scan over the
c.m. energy range from 2 to 4.8~GeV~\cite{Bai:2001ct}. 
Data were taken at 85 energy
points, with an integrated luminosity varying from 
9.2 to 135~nb$^{-1}$ per point. In this experiment, besides 
the continuum region below the 
charmonium threshold, the high charmonium states from 3.77 to
4.50~GeV were studied~\cite{Ablikim:2007gd} in detail. The statistical error 
was between 2 to 3\%,
while the systematic error ranged from 
5 to 8\%, due to improvement 
on hadronic event selection and Monte Carlo simulation of hadronisation
processes. The uncertainty due to the luminosity
determination varied from 2 to 5.8\%.

More recently, in 2003 and 2004, before BES-II
was shut down for the upgrade to BES-III, a high-statistics 
data sample was taken at 2.6, 3.07 and 3.65~GeV, with an integrated luminosity
of 1222, 2291 and 6485~nb$^{-1}$, respectively~\cite{:2009js}. The systematic
error, which exceeded the statistical error, was reduced to
3.5\%  due to further refinement on
hadronic event selection and Monte Carlo simulation.

For BES-III, the main goal of the $R$ measurement is to perform a fine scan
over the whole energy region which BEPC-II can cover. For a continuum region
(below 3.73~GeV), the step size should not exceed 100~MeV, and for
the resonance region (above 3.73~GeV), the step size should be 10 to
20~MeV. Since the luminosity of BEPC-II is two orders of magnitude
higher than at BEPC, the scan of the resonance region will provide
precise information on the $1^{--}$  charmonium states up to
4.6~GeV.


\subsection{Estimate of the theoretical accuracy}

Let us discuss the accuracy of the description of the
processes under consideration. This accuracy can be subdivided into
two major parts: theoretical and technical one.
The first one is related to the precision in the actual computer codes.  
It usually does not take into account all known contributions
in the {\em best} approximation. The technical precision can be verified
by special tests within a given code (e.g., by looking at the
numerical cancellation of the dependence on auxiliary parameters)
and tuned comparisons of different codes. 

The pure theoretical precision consists of unknown high\-er-order 
corrections, of uncertainties in the treatment of photon radiation off
hadrons, and of errors in the phenomenological definition of such
quantities as the hadronic vacuum polarisation and the pion form factor.

Many of the codes used at meson factories do not include contributions
from weak interactions even at Born level. As discussed above, these
contributions are suppressed at least by a factor  of $s/M_Z^2$ and do not
spoil the precision up to the energies of $B$ factories.

Matching the complete one-loop QED corrections with the higher-order 
corrections in the leading logarithmic approximation, 
certain parts of the second-order next-to-leading corrections 
are taken into account~\cite{Balossini:2006wc}. For the case of Bhabha
scattering, where, e.g., soft and virtual photonic corrections 
in $O (\alpha^2L)$ are known analytically,
one can see that their contribution in the relevant kinematic 
region does not exceed 0.1\%.\footnote{The proper choice of the factorisation 
scale~\cite{Arbuzov:2006mu} is important here.}

The uncertainty coming from the 
the hadronic vacuum polarisation has been estimated~\cite{Eidelman:1995ny}
to be of order 0.04\%. For measurements performed with the c.m.
energy at a narrow resonance (like at the $\phi$-meson factories), 
a systematic error in the determination of the resonance contribution to vacuum
polarisation is to be added.

The next point concerns non-leading terms of order $(\alpha/\pi)^2L$.
There are several sources of them. One is the emission of two extra
hard photons, one in the collinear region and one
at large angles. Others are related to virtual and soft-photon
radiative corrections to single hard photon emission and Born processes.
Most of these contributions were not considered up to now.
Nevertheless we can estimate the coefficient in front of the quantity
$(\alpha/\pi)^2L\approx 1\cdot 10^{-4}$ to be of order one.
This was indirectly confirmed by our complete calculations of
these terms for the case of small--angle Bhabha scattering.

Considering all sources of uncertainties mentioned above as
independent, we conclude that the systematic error of our formulae
is about 0.2\% or better, both for muons and pions.
For the former it is a rather safe estimate.
Comparisons between different codes which treat higher-order QED
corrections in different ways typically show agreement at the 
0.1\% level. Such comparisons test the technical and partially  
the theoretical uncertainties.
As for the $\pi^+\pi^-$ and two kaon channels, the uncertainty is 
enhanced due to the presence of form factors and due to the application
of the point-like approximation for the final state hadrons. 

%\subsection*{Acknowledgements}

%This work was supported in part by the INTAS grant 05-1000008-8328 
%and RFBR grants 
%03-02-16477, 04-02-16217, 04-02-1623, 04-02-16443 and 04-02-16181-a,
%04-02-16184-a, 05-02-16250-a, 06-02-16192-a, 07-02-00816-a,
%08-02-13516 and 08-02-91969.



