% VP contribution to MC-Sighad proceedings from Daisuke Nomura and Thomas Teubner

\def\lapproxeq{\lower .7ex\hbox{$\;\stackrel{\textstyle <}{\sim}\;$}}
%\def\gapproxeq{\lower .7ex\hbox{$\;\stackrel{\textstyle >}{\sim}\;$}}
\def\Re{\mathop{\rm Re}\nolimits} \def\Im{\mathop{\rm Im}\nolimits}

%
%\section{Vacuum Polarisation}

\subsection{Introduction}
%
The vacuum polarisation (VP) of the photon is a quantum effect which
leads, through renormalisation, to the scale dependence (`running')
of the electromagnetic coupling, $\alpha(q^2)$. It therefore plays an
important role in many physical processes and its knowledge is crucial
for many precision analyses. A prominent example is the precision
fits of the Standard Model as performed by the electroweak working
group, where the QED coupling $\alpha(q^2 = M_Z^2)$ is the least well
known of the set of fundamental parameters at the $Z$ scale, $\{G_{\mu}, M_Z,
\alpha(M_Z^2)\}$. Here we are more concerned about the VP at lower
scales as it enters all photon-mediated hadronic cross sections. 
These are used, e.g., in the determination of the strong coupling
$\alpha_s$, the charm and bottom quark masses from $R_{\rm had}$ as
well as in the evaluation of the hadronic contributions to the muon
$g-2$ and $\alpha(q^2)$ itself.
It also appears in Bhabha scattering in higher orders of perturbation
theory needed for a precise determination of the luminosity. 
It is hence clear that VP also has to be included in the corresponding
Monte Carlo programs.

In the following we shall first define the relevant notations, then
briefly discuss the calculation of the leptonic and hadronic VP
contributions, before comparing available VP parametrisations. 

\begin{figure}[h]
\begin{center}
\includegraphics[width=5cm]{vp.ps}
\end{center}
\vspace{-0.2cm}
\caption{\label{fig:VP} Photon vacuum polarisation $\Pi(q^2)$.} 
\end{figure}
Conventionally the vacuum polarisation function is denoted by $\Pi(q^2)$
where $q$ is a space- or time-like momentum. The shaded blob in
Fig.~\ref{fig:VP} stands for all possible one-particle irreducible leptonic or hadronic
contributions. The full photon propagator is then the sum of the bare
photon propagator and arbitrarily many iterations of VP insertions,
\begin{eqnarray}
\mbox{full photon propagator} \ \sim\ \frac{-i}{q^2}\,\cdot
\qquad\qquad\qquad\qquad & &\nonumber\\
\quad\left( 1\,+\,\Pi\,+\,\Pi\cdot\Pi\,+\,\Pi\cdot\Pi\cdot\Pi\,+\,\ldots\right)\,.&&
\label{eq:photonprop}
\end{eqnarray}
The Dyson summation of the real part of the one-particle irreducible
blobs then defines the effective QED coupling
\begin{equation}
\qquad\alpha(q^2)\ =\ \frac{\alpha}{1-\Delta\alpha(q^2)}\ =
\ \frac{\alpha}{1-{\rm Re} \Pi(q^2)}\,,
\label{eq:defalphaqed}
\end{equation}
where $\alpha \equiv \alpha(0)$ is the usual fine structure constant,
$\alpha \sim 1/137$.
% PDG: $\alpha = 1/137.035999679(94)$ 
It is determined most precisely through the anomalous magnetic moment
of the electron, $a_e$, as measured by the Harvard group to an amazing $0.24$
ppb~\cite{Hanneke:2008tm}, in agreement with less precise
determinations from caesium and rubidium atom experiments. The most
precise value for $\alpha$, which includes the updated calculations
of $O (\alpha^4)$ contributions to $a_e$~\cite{Aoyama:2007mn},
is given by $1/\alpha = 137.035\, 999\, 084\, (51)$. 

By using Eq.~(\ref{eq:defalphaqed}) we have defined $\Pi$ to include
the electric charge squared, $e^2$ for leptons, but note that
different conventions are used in the literature, and sometimes $\Pi$
is also defined with a different overall sign. 

Equation~(\ref{eq:defalphaqed}) is the usual definition of the running
effective QED coupling and has the advantage that one obtains a real
coupling. However, the imaginary part of the VP function $\Pi$ is
completely neglected, which is normally a good approximation as the
contributions from the imaginary part are formally suppressed. This
can be seen, e.g., in the case of the `undressing' of the experimentally
measured hadronic cross section $\sigma_{\rm had}(s)$. 
The measured cross section $e^+e^- \to \gamma^* \to
hadrons$ contains $|\mbox{full photon propagator}|^2$, i.e. the
modulus squared of the infinite sum (\ref{eq:photonprop}). Writing
$\Pi = e^2 (P+iA)$ one easily sees that 
\begin{eqnarray}
& & |1 + e^2 (P+iA) + e^4 (P+iA)^2 + \ldots|^2 \ =\ \nonumber\\
& & \ 1\, +\, e^2\, 2 P\, +\, e^4\, ( 3 P^2 - A^2)\, +\, e^6\, 4P(P^2
- A^2)\, +\, \ldots \nonumber
\end{eqnarray}
and that the imaginary part $A$ enters only at order $O (e^4)$
compared to $O (e^2)$ for the leading contribution from the real
part $P$. To account for the imaginary part of $\Pi$ one may therefore
apply the summed form of the `(un)dressing' factor with the relation
\begin{equation}
\sigma_{\rm had}(s) = \frac{\sigma^0_{\rm had}(s)}{|1-\Pi|^2}
\label{eq:defsigma0}
\end{equation}
instead of the traditionally used relation with the real effective coupling,
\begin{equation}
\sigma_{\rm had}(s) = \sigma^0_{\rm had}(s) \left(\frac{\alpha(s)}{\alpha}\right)^2\,.
\label{eq:defsigma0re}
\end{equation}
We shall return to a comparison of the different approaches below for
the case of the hadronic VP.

It should be noted that the summation breaks down and hence can not be
used if $|\Pi(s)| \sim 1$. This is the case if $\sqrt{s}$ is very
close to or even at narrow resonance energies. In this case one can
not include the narrow resonance in the definition of the effective
coupling but has to rely on another formulation, e.g. through a
Breit-Wigner propagator (or a narrow width approximation with a
delta-function). For a discussion of this issue see~\cite{hmnt}.
Also note that the VP summation covers only the class of one-particle
irreducible diagrams of factorisable bubbles depicted in
Fig.~\ref{fig:VP}. This includes photon radiation within and between
single bubbles, but clearly does not take into account
higher-order corrections from initial state radiation or
initial-final state interference effects in $e^+e^-\to hadrons$.

As will be discussed in the following, leptonic and hadronic
contributions to $\Delta\alpha$ are normally calculated separately and
then added, $\Delta\alpha(q^2) = \Delta\alpha_{\rm lep}(q^2) +
\Delta\alpha_{\rm had}(q^2)$. While the leptonic contributions can be
predicted within perturbation theory, the precise determination of the
ha\-dronic contributions relies on a dispersion relation using
experimental data as input. 


\subsection{Leptonic contributions}
%
The leptonic contributions $\Delta\alpha_{\rm lep}$ have been
calculated to sufficiently high precision. The leading order (LO) and
next-to-leading order (NLO) contributions are known as analytic
expressions including the full mass dependence~\cite{Kallen:1955fb},
where LO and NLO refer to the expansion in terms of $\alpha$. The
next-to-next-to-leading order (NNLO) contribution is available as
an expansion in terms of $m_{\ell}^2/q^2$~\cite{Steinhauser:1998rq},
where $m_{\ell}$ is the lepton mass. To evaluate $\Delta \alpha_{\rm
  lep}(q^2)$ for $|q^2| \lapproxeq m_\tau^2$, this expansion is not
appropriate, but this is exactly the region where the hadronic
uncertainties are dominant. Also from the smallness of the NNLO
contribution, we conclude that we do not need to further improve the
leptonic contributions beyond this approximation. 

The evaluation of the LO contribution is rather simple, and we briefly
summarise the results below. Hereafter, it is understood that we
impose the renormalisation condition $\Pi(0)=0$ on $\Pi(q^2)$. For
$q^2 < 0$, the VP function reads 
%
\begin{eqnarray}
 \Pi(q^2)
& = &
 -\frac{e^2}{36\pi^2} 
 \Big( 5-12\eta\\
& & + 3 ( -1 + 2 \eta ) \sqrt{1+4 \eta} 
      \,\ln \frac{\sqrt{1+4 \eta}+1}{\sqrt{1+4 \eta}-1}
 \Big)\,,\nonumber
\label{eq:LOleptonic} 
\end{eqnarray} 
%
where $\eta \equiv m_{\ell}^2/(-q^2)$. For $0\le q^2 \le 4 m_{\ell}^2$
one obtains
%
\begin{eqnarray}
 \Pi(q^2)
& = &
 -\frac{e^2}{36\pi^2} 
 \Big( 
     5-12\eta\\
& & + 3 ( -1 + 2 \eta ) \sqrt{- 1 - 4 \eta}
      \,\arctan \frac{\sqrt{-1-4 \eta}}{-1-2 \eta}
 \Big)\,,\nonumber
\end{eqnarray}
%
and for $q^2 \ge 4 m_{\ell}^2$
%
\begin{eqnarray}
 \Pi(q^2)
& = &
 -\frac{e^2}{36\pi^2} 
 \Big( 
     5-12\eta + 3 ( -1 + 2 \eta ) \sqrt{1+4 \eta}\\
 & & 
      \cdot\ln \frac{1+\sqrt{1+4 \eta}}{1-\sqrt{1+4 \eta}}
 \Big) 
- \frac{i\,e^2}{12\pi} ( 1 - 2 \eta ) \sqrt{1+4 \eta}\,.\nonumber
\end{eqnarray}
%
An easily accessible reference which gives the NLO contributions is,
for instance, Ref.~\cite{Djouadi:1993ss,PhysRevD.53.4111}. As
mentioned above, the NNLO contribution is given in
Ref.~\cite{Steinhauser:1998rq}. For all foreseeable applications the
available formulae can be easily implemented and provide a sufficient
accuracy. While the uncertainty from $\alpha$ is of course completely
negligible, the uncertainty stemming from the lepton masses is only
tiny. Therefore the leptonic VP poses no problem. 


\subsection{Hadronic contributions}
%
In contrast to the leptonic case, the hadronic VP $\Pi_{\rm had}(q^2)$
can not be reliably calculated using perturbation theory. This is
clear for time-like momentum transfer $q^2 > 0$, where, via the
optical theorem $\Im\Pi_{\rm had}(q^2) \sim \sigma(e^+e^- \to
hadrons)$ goes through all the resonances in the low energy
region. However, it is possible to use a dispersion relation to
obtain the real part of $\Pi$ from the imaginary part. The dispersion
integral is given by
\begin{equation}
\Delta\alpha_{\rm had}^{(5)}(q^2)\ =\ 
  -\frac{q^2}{4\pi^2\alpha}\, {\rm P}\int_{m_{\pi}^2}^\infty
\frac{\sigma_{\rm had}^0(s)\,{\rm d}s}{s-q^2}\,,
% ,\quad \sigma_{\rm had}(s) = \frac{\sigma^0_{\rm had}(s)}{|1-\Pi|^2}
\label{eq:defdelalhad5}
\end{equation}
where $\sigma_{\rm had}^0(s)$ is the (undressed) hadronic cross
section which is determined from experimental data. Only away from
hadronic resonances and (heavy) quark thresholds one can apply
perturbative QCD to calculate $\sigma_{\rm had}^0(s)$. In this region
the parametric uncertainties due to the values of the quark masses and
$\alpha_s$, and due to the choice of the renormalisation scale, are
small. Therefore the uncertainty of the hadronic VP is dominated by
the statistical and systematic uncertainties of the experimental data
for $\sigma_{\rm had}^0(s)$ used as input in (\ref{eq:defdelalhad5}). 

Note that the dispersion integral (\ref{eq:defdelalhad5}) leads to a
smooth function for space-like momenta $q^2 < 0$, whereas in the
time-like region it has to be evaluated using the principal value description
and shows strong variations at resonance energies, as demonstrated
e.g. in Fig.~\ref{fig:dalphaFJ03}.
In Eq.~(\ref{eq:defdelalhad5}) $\Delta\alpha_{\rm had}^{(5)}$ denotes the
five-flavour hadronic contribution. At energies we are interested in,
i.e. far below the $t\bar t$ threshold, the contribution from the top
quark is small and usually added separately. The analytic expressions
for $\Delta\alpha^{\rm top}(q^2)$ obtained in perturbative QCD are
the same as for the leptonic contributions given above, up to
multiplicative factors taking into account the top quark charge and the
corresponding SU(3) colour factors, which read $Q_t^2 N_c$ at LO and
$Q_t^2 \frac{N_c^2-1}{2N_c}$ at NLO. 

Contributions from narrow resonances can easily be treated using the
narrow width approximation or a Breit-Wigner form. For the latter one
obtains
\begin{equation}
\Delta\alpha^{\rm Breit-Wigner}(s) =  \frac{3\Gamma_{ee}}{\alpha
  M}\,\frac{s(s-M^2-\Gamma^2)}{(s-M^2)^2+M^2\Gamma^2}\,, 
\label{eq:BW}
\end{equation}
with $M$, $\Gamma$ and $\Gamma_{ee}$ the mass, total and electronic
width of the resonance. For a discussion of the undressing of
$\Gamma_{ee}$ see~\cite{hmnt}. 

Although the determination of $\Delta\alpha_{\rm had}^{(5)}(q^2)$ via the
dispersion integral~(\ref{eq:defdelalhad5}) may appear straightforward,
in practice the data combination for $\sigma_{\rm had}^0(s)$ is far from
trivial. In the low energy region up to about $1.4 - 2$ GeV many data sets
from the different hadronic exclusive final states (channels) from
various experiments have to be combined, before the different channels
which contribute incoherently to $\sigma_{\rm had}^0(s)$ can be
summed. For higher energies the data for the fast growing number of possible
multi-hadronic final states are far from complete, and instead
inclusive (hadronic) measurements are used. For the details of the
data input, the treatment of the data w.r.t. radiative corrections,
the estimate of missing threshold contributions and unknown
subleading channels (often via isospin correlations) and the
combination procedures we refer to the publications of the different
groups cited below.

In the following we shall briefly describe and then compare the
evaluations of the (hadronic) VP available as para\-me\-tri\-sations or
tabulations from different groups. 


\subsection{Currently available VP parametrisations}
%
For many years Helmut Burkhardt and Bolek Pietrzyk have been providing
the Fortran function named REPI for the leptonic and hadronic
VP \cite{Burkhardt:1981jk,Burkhardt:1982kr,Burkhardt:1989ky,Burkhardt:1995tt,Burkhardt:2001xp}. While the leptonic VP is coded in analytical form
with one-loop accuracy, the hadronic VP is given as a very compact
parametrisation in the space-like region, but does not cover the
time-like region. For their latest update
see~\cite{Burkhardt:2005se}. The code can be obtained from Burkhardt's
web-pages which contain also a short introduction and a list of older
references, see \newline{\tt http://hbu.web.cern.ch/hbu/aqed/aqed.html}.

Similarly, Fred Jegerlehner has been providing a package of
Fortran routines for the running of the effective QED
coupling \cite{Jegerlehner:1985gq,Eidelman:1995ny,Jegerlehner:2003ip,Jegerlehner:2003qp,Jegerlehner:2006ju,Jegerlehner:2008rs}.
It provides leptonic and hadronic
VP both in the space- and time-like region. For the leptonic VP the
complete one- and two-loop results and the known high energy
approximation for the three-loop corrections are included. The
hadronic contributions are given in tabulated form in the subroutine
HADR5N. The full set of routines can be downloaded from
Jegerlehner's web-page \newline{\tt
  http://www-com.physik.hu-berlin.de/$\sim$fjeger/}. The version available
from there is the one we use in the comparisons below and was last
modified in November 2003. It will be referred to as J03 in the
following. An update is in progress and other versions
may be available from the author upon request. Note that for quite
some time his routine has been the only available code for the time-like
hadronic VP. Fig.~\ref{fig:dalphaFJ03} shows the leptonic and hadronic
contributions together with their sum as given by Jegerleh\-ner's routine.
%
\begin{figure} \begin{center}
\includegraphics[bb=0 0 382 245,width=8.9cm]{dalpslow.ps}
\end{center}
\vspace{-0.1cm}
\caption{Different contributions to $\Delta\alpha(s)$ in the time-like
  region as given by the routine from Fred Jegerlehner. Figure
  provided with the package {\tt alphaQED.uu} from his homepage.} 
\label{fig:dalphaFJ03}
\end{figure}
%

The experiments CMD-2 and SND at Novosibirsk are using their own VP
compilation to undress hadronic cross sections, and the values used
%for $1/|1-\Pi|^2$ 
are given in tables in some of their
publications. Recently CMD-2 has made their compilation publicly
available, see Fedor Ignatov's web-page {\tt
  http://cmd.inp.nsk.su/$\sim$ignatov/vpl/}.\ \ There links are given to a
corresponding talk at the `4th meeting of the Working Group on
Radiative Corrections and Monte Carlo Generators for Low Energies'
(Beijing 2008), to the thesis of Ignatov (in Russian) and to a file
containing the tabulation, which can be used together with a
downloadable package. The tabulation is given for the real and
imaginary parts of the sum of leptonic and hadronic VP, for both
space- and time-like momenta, and for the corresponding
errors. Fig.~\ref{fig:cmd2vp}, also displayed on their web-page,
shows the results from CMD-2 for $|1+\Pi|^2$ both for the space- and
time-like momenta in the range $-(15\ {\rm GeV})^2 < q^2 < (15\ {\rm
  GeV})^2$ (upper panel) and for the important low energy region $-(2\
{\rm GeV})^2 < q^2 < (2\ {\rm GeV})^2$. The solid (black) lines are
the sum of leptonic and hadronic contributions, while the dotted (red)
lines are for the leptonic contributions only. 
%
\begin{figure} \begin{center}
\includegraphics[bb=0 0 530 380,width=8.8cm]{polfinal.eps}\\
\includegraphics[bb=0 0 530 380,width=8.8cm]{polfinal_2GeV.eps}
\end{center}
\vspace{-0.1cm}
\caption{$|1+\Pi|^2$ from CMD-2's compilation for space- and
  time-like momenta (labelled $\sqrt{s}$); solid (black) lines: leptonic plus hadronic
  contributions, dotted (red) lines: only leptonic
  contributions. Upper panel: $-(15\ {\rm GeV})^2 < q^2 < (15\ {\rm GeV})^2$. Lower panel:
  $-(2\ {\rm GeV})^2 < q^2 < (2\ {\rm GeV})^2$. Figures provided by Fedor Ignatov.} 
\label{fig:cmd2vp}
\end{figure}
%


Another independent compilation of the hadronic VP is available from
the group of Hagiwara et al.~\cite{hmnt} (HMNT), at present upon
request from the authors. They provide tabulations (with a simple
interpolation routine in Fortran) of $\Delta\alpha^{(5)}_{\rm had}(q^2)$ both in
the space- and time-like region, and also a compilation of
$R_{\rm had}(s)$. Currently available routines are based on the
analysis~\cite{Hagiwara:2003da,Hagiwara:2006jt}. Two
different versions are provided, one including the narrow resonances
$J/\psi, \psi^{\prime}$ and the Upsilon family,
$\Upsilon(1S)-\Upsilon(3S)$, in Breit-Wigner form, one excluding them.
However, for applications of $\Delta\alpha$ it should be remembered
that close to narrow resonances the resummation of such large
contributions in the effective coupling breaks down. 
In this context, note that the compilation from Novosibirsk contains
these narrow resonances, whereas the routine from Jegerlehner does contain
$J/\psi$ and $\psi^{\prime}$, but seems to exclude (or smear over) the
Upsilon resonances. When called in the charm or bottom resonance
region Jegerlehner's routine gives a warning that the ``results may
not be reliable close to J/Psi and Upsilon resonances''. 

In the following we shall compare the parametrisations from the
different groups. 


\subsection{Comparison of the results from different groups}
%
In Fig.~\ref{fig:delalfcomp}, we compare the parametrisations from
Burk\-hardt and Pietrzyk (BP05), Jegerlehner
(J03) and Hagiwara et al. (HMNT) in
the space-like (upper) and time-like region (lower panel). For the
space-like region the differences among the three parametrisations are
roughly within one standard deviation in the whole energy range
shown. However, for the time-like region, there is disagreement
between HMNT and J03 at several energy regions, most notably at $1\
{\rm GeV} \lapproxeq \sqrt{s} \lapproxeq 1.6$ GeV, and at $0.8$ GeV
$\lapproxeq \sqrt{s}$ $\lapproxeq 0.95$ GeV. As for the
discrepancy at $1$ GeV $\lapproxeq \sqrt{s} \lapproxeq 1.6$ GeV,
checking the routine from Jeger\-leh\-ner, one finds that a too sparsely
spaced energy grid in this region seems to be the reason. The
discrepancy at $0.8$ GeV $\lapproxeq \sqrt{s} \lapproxeq$ $0.95$
GeV is further scrutinised in Fig.~\ref{fig:compzoom}, where in
addition to the two parametrisations HMNT (solid (red) line) and J03
(dotted (blue) line), the result for $\Delta\alpha_{\rm
  had}^{(5)}(s)/\alpha$ obtained by integrating over the $R$-data as
compiled by the PDG~\cite{Amsler:2008zzb}\footnote{The actual
  compilation of the data is available in electronic form from {\tt
    http://pdg.lbl.gov/2008/hadronic-xsections} {\tt
    /hadronicrpp\_page1001.dat}\,.} is shown as the dashed (green)
line. While the results from HMNT and the one based on the PDG $R$-data
agree rather well, their disagreement with the J03 compilation in the
region $0.8$ GeV $\lapproxeq \sqrt{s} \lapproxeq 0.95$ GeV is
uncomfortably large compared to the error but may be due to a
different data input of the J03 parametrisation.
%
\begin{figure}
\begin{center}
\includegraphics[bb=80 280 480 540,width=8.8cm]{dahadvglspacel_mcsh.ps}\\
\includegraphics[bb=80 280 480 540,width=8.8cm]{dahadvgltimel_mcsh.ps}
\end{center}
\vspace{-0.3cm}
\caption{Comparison of the results from Hagiwara et
  al. (HMNT~\cite{hmnt}) for $\Delta\alpha_{\rm had}^{(5)}(q^2)$ in
  units of $\alpha$ with parametrisations from Burkhardt and Pietrzyk
  (BP05~\cite{Burkhardt:2005se}) and Jegerlehner (J03). Upper panel:
  $\Delta\alpha_{\rm had}^{(5)}(Q^2)/\alpha$ for space-like momentum transfer
  ($Q^2<0$), where the three parametrisations are indistinguishable.
  The differences (normalised and multiplied by 100) are highlighted
  by the dashed and dotted curves; the wide light (blue) band is
  obtained by using the error band of HMNT in the normalised
  difference to J03, labelled `(J03-HMNT)/HMNT ($\times 100$)'. Lower
  panel: $\Delta\alpha_{\rm had}^{(5)}(s)/\alpha$ from J03 and HMNT (as
  labelled) for time-like momenta ($q^2=s$). For readability, only the
  error band of HMNT is displayed.} 
\label{fig:delalfcomp}
\end{figure}
%
\begin{figure} \begin{center}
\includegraphics[width=8.9cm]{comparis_J.eps}
\end{center}
\vspace{-0.1cm}
\caption{\label{fig:compzoom}
  Comparison of the results from Hagiwara et al. (HMNT, solid (red)
  line) for $\Delta\alpha_{\rm had}^{(5)}(s)/\alpha$ with the
  parametrisation from Jegerlehner (J03, dotted
  (blue) line) in the time-like region in the range $\sqrt{s} = 0.7 -
  1$ GeV. The dashed (green) line shows the result if the data
  compilation from the PDG~\cite{Amsler:2008zzb} is used.} 
\end{figure}
%

In the following we shall compare the parametrisation from HMNT with
the one from the CMD-2 collaboration which has become available very
recently. Note that for undressing their experimentally measured hadronic
cross sections, CMD-2 includes the imaginary part of the VP function
$\Pi(q^2)$ in addition to the real part. Before coming to the
comparison with CMD-2, let us discuss some generalities
about $\Im\Pi(q^2)$. If we are to include the imaginary part, then the
VP correction factor $\alpha(q^2)^2$ should be replaced as
%
\begin{eqnarray}
& &\left(\frac{\alpha}{1 - \Delta \alpha(q^2)}\right)^2
= \left(\frac{\alpha}{1 - \Re \Pi(q^2)}\right)^2
\to\\ 
& &\left| \frac{\alpha}{1 - \Pi(q^2)} \right|^2
= \frac{\alpha^2}{(1 - \Re\Pi(q^2))^2 + (\Im \Pi(q^2))^2}\,.\nonumber
\end{eqnarray}
%
Note that, as mentioned already
in the introduction, the contribution from the real part appears at
$O (e^2)$ in the denominator, while that from the imaginary part
starts only at $O (e^4)$. Because of this suppression we expect
the effects from the imaginary part to be small. Nevertheless we would
like to stress two points. First, field-theoretically, it is more
accurate to include the imaginary part which exists above
threshold. Including only $\Re\Pi(q^2)$ in the VP correction is an
approximation which may be sufficient in most cases. Second, it is
expected that the contribution from the imaginary part is of the order
of a few per mill of the total VP corrections. While this seems
small, it can be non-negligible at the $\rho$ meson region where the
accuracy of the cross section measurements reaches the order of (or even
less than) 1\%. Similarly, in the region of the narrow $\phi$
resonance, the contributions from the imaginary part become
non-negligible and should be taken into account.

%
\begin{figure}[htb]
\begin{center}
\includegraphics[bb=60 55 405 300, width=8.9cm]{vpcorr.eps}\\
\includegraphics[bb=60 55 405 300, width=8.9cm]{vpcorr_diff.eps}
\end{center}
\vspace{-0.2cm}
\caption{Upper panel: Correction factor $\left | 1 - \Pi(s) \right
  |^2$ as used for `undressing' by the CMD-2 collaboration in~\cite{Akhmetshin:2006bx}
  (dashed line) compared to the same quantity using the HMNT
  compilation for the $e^+e^- \to hadrons$ data (solid line). Also
  shown is the correction factor $(1 - \Re \Pi)^2 =
  (\alpha/\alpha(s))^2$, based on $\alpha(s)$ in the time-like region
  from HMNT (dotted line). Lower panel: Differences of the quantities
  as indicated on the plot.} 
\label{fig:cmd2comp}
\end{figure}
%
In Fig.~\ref{fig:cmd2comp} the VP correction factor, based on the
compilation from HMNT, with and without $\Im\Pi(q^2)$ is compared to
$|1-\Pi(s)|^2$ as used by the CMD-2 collaboration in their recent
analysis of the hadronic cross section in the $2\pi$ channel in the
$\rho$ central region~\cite{Akhmetshin:2006bx}.\footnote{We thank
  Gennadiy Fedotovich for providing us with a table including the VP
  correction factors not included in~\cite{Akhmetshin:2006bx}.} In the
%
\begin{figure}[htb]
\begin{center}
\includegraphics[bb=75 280 480 535, width=8.9cm]{cmd2timel1.ps}\\
\includegraphics[bb=75 280 480 535, width=8.9cm]{cmd2timel2.ps}
\end{center}
\vspace{-0.3cm}
\caption{$\Delta\alpha(s)$ in the time-like region as given by the
  parametrisation from CMD-2 (solid (blue) band) compared to the same quantity
  from HMNT (dotted (red) line). Upper panel: $0 < \sqrt{s} < 2$ GeV, lower
  panel: $2$ GeV $< \sqrt{s} < 10$ GeV.}
\label{fig:cmd2mnhtda}
\end{figure}
%
upper panel the VP correction factors are given, whereas in the lower
panel the differences are shown. As expected, the differences between
the three are visible, and are about a few per mill at most. The
difference between the CMD-2 results and the one from HMNT including
$\Im\Pi(q^2)$ (solid (red) curve in the lower panel of
Fig.~\ref{fig:cmd2comp} shows a marked dip followed by a peak in the
$\rho - \omega$ interference region where the $\pi^+\pi^-$ cross
section falls sharply. This is most probably a direct consequence of
the different data input used. However, in most applications such a
difference will be partially cancelled when integrated over an energy
region including the $\rho$ peak. 

%
\begin{figure}[htb]
\begin{center}
\includegraphics[bb=75 280 480 535, width=8.9cm]{cmd2timel1e.ps}\\
\includegraphics[bb=75 280 480 535, width=8.9cm]{cmd2timel2e.ps}
\end{center}
\vspace{-0.3cm}
\caption{Solid (black) lines: Normalised difference $(\Delta\alpha^{\rm
    CMD-2}(s)-\Delta\alpha^{\rm HMNT}(s))/\Delta\alpha^{\rm HMNT}(s)$ in the
  time-like region. The dashed (blue) and dotted (red) lines indicate
  the relative error for the CMD-2 and HMNT parametrisations. Upper
  panel: $0 < \sqrt{s} < 2$ GeV, lower panel: $2$ GeV $< \sqrt{s} < 10$ GeV.}
\label{fig:cmd2mnhtdae}
\end{figure}
%
In Figs.~\ref{fig:cmd2mnhtda} and~\ref{fig:cmd2mnhtdae} we compare
$\Delta\alpha(s)$ in the time-like region as given by the
parametrisation from CMD-2 with the one from HMNT, where for HMNT we
have calculated the leptonic contributions (up to including the NNLO
corrections) as described above. The two panels in
Fig.~\ref{fig:cmd2mnhtda} (upper panel: $0 < \sqrt{s} < 2$ GeV, lower
panel: $2$ GeV $< \sqrt{s} < 10$ GeV) show $\Delta\alpha(s)$ with the 
$1\sigma$ error band from CMD-2 as a solid (blue) band, whereas for
HMNT the mean value for $\Delta\alpha(s)$ is given by the dotted (red)
line, which can hardly be distinguished. To highlight the differences
between the two parametrisations, Fig.~\ref{fig:cmd2mnhtdae} displays
the normalised difference $(\Delta\alpha^{\rm CMD-2}(s) -
\Delta\alpha^{\rm HMNT}(s))/\Delta\alpha^{\rm HMNT}(s)$ as a so\-lid
(black) line, and also shows the relative errors of CMD-2 and HMNT as
dashed (blue) and red (dotted) lines, respectively. As visible in
Fig.~\ref{fig:cmd2mnhtdae}, the error as given by the CMD-2
parametrisation is somewhat smaller than the one from HMNT. Both
parametrisations agree fairly well, and for most energies the
differences between the parametrisations are about as large or smaller
than the error bands. Close to narrow resonances the estimated
uncertainties are large, but as discussed above, there the
approximation of the effective coupling $\alpha(s)$ breaks down and
resonance contributions should be treated differently. 


\subsection{Summary}
%
Vacuum polarisation of the photon plays an important role in many
physical processes. It has to be taken into account, e.g., in Monte Carlo
generators for hadronic cross sections or Bhabha scattering. When
low energy data are used in dispersion integrals to predict the
hadronic contributions to muon $g-2$ or $\Delta\alpha(q^2)$, undressed data
have to be used, so VP has to be subtracted from measured cross
sections. The different VP contributions have been discussed,
and available VP compilations have been briefly described and
compared. Until recently only one parametrisation has been available
in the time-like region, now three routines in the space- and time-like
regions exist,
from Jegerlehner, CMD-2 and HMNT, and a fourth from Burkhardt and
Pietrzyk in the space-like region. While the accuracy of the hadronic cross
section data themselves is the limiting factor in the precise
determination of $g-2$ and $\Delta\alpha(M_Z^2)$, the error of the VP
(or $\Delta\alpha(q^2)$) is not the limiting factor in its current
applications. With the ongoing efforts to measure $\sigma_{\rm
  had}(s)$ with even better accuracy in the whole low energy region,
further improvements of the various VP parametrisations are foreseen. 

