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%...spectral functions
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\def\Sfs{Spectral functions}
\def\SF{Spectral Function}
\def\SFs{Spectral Functions}
%
%...physical variables
\def\as{$\alpha_s$}
\def\asm{$\alpha_s(M_\tau^2)$}
\def\asZ{$\alpha_s(M_{\rm Z}^2)$}
\def\ass{$\alpha_s(s)$}
\def\assz{$\alpha_s(s_0)$}
\def\eee{$e^+e^-$}
\def\eeto{$e^+e^-\!\rightarrow\,$}
\def\eetohad{$e^+e^-\!\rightarrow{\rm hadrons}$}
\def\MZ{$M_{\rm Z}$}
\def\pms{$\,\pm\,$}
\def\aqed{$\alpha(s)$}
\def\aqedZ{$\alpha(M_{\rm Z}^2)$}
\def\daqed{$\Delta\alpha(s)$}
\def\daqedZ{$\Delta\alpha(M_{\rm Z}^2)$}
\def\daqedh{$\Delta\alpha_{\rm had}(s)$}
\def\daqedhZ{$\Delta\alpha_{\rm had}(M_{\rm Z}^2)$}
\def\amuhad{$a_\mu^{\rm had}$}
\def\aehad{$a_e^{\rm had}$}
%
%...abbreviations
\def\br{branching ratio}
\def\brs{branching ratios}
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\begin{document}



\title{EVALUATION OF \aqedZ\ AND $(g-2)_\mu$}

\author{A.~H\"ocker}

\address{Laboratoire de l'Acc\'el\'erateur Lin\'eaire,\\
	 B\^at. 200, B.P. 34, F-91898 Orsay CEDEX, France \\
         E-mail: hoecker@lal.in2p3.fr}   

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\twocolumn[\maketitle\abstracts{ This talk summarizes the recent 
development in the evaluation of the leading order hadronic contributions 
to the running of the QED f\/ine structure constant \aqed, at $s=M_{\rm Z}^2$,
and to the anomalous magnetic moments of the muon $(g-2)_\mu$. The 
accuracy of the theoretical prediction of these observables
is limited by the uncertainties on the hadronic contributions. 
Signif\/icant improvement has been achieved in a series of new analyses
which is presented historically in three steps: (I), use of $\tau$
spectral functions in addition to \eee\ cross sections, (II), extended
use of perturbative QCD and (III), application of QCD sum rule techniques.
The most precise values obtained are: \daqedhZ\,$=(276.3\pm1.6)\times10^{-4}$,
yielding $\alpha^{-1}(M_{\rm Z}^2)=128.933\pm0.021$, and
$a_\mu^{\rm had}=(692.4\pm6.2)\times 10^{-10}$ with which one
f\/inds for the complete Standard Model prediction
$a_\mu^{\rm SM}=(11\,659\,159.6\pm6.7)\times10^{-10}$.
For the electron $(g-2)_e$, the hadronic contribution is
$a_e^{\rm had}=(187.5\pm1.8)\times 10^{-14}$.
}]

\section{ Introduction }

The running of the QED f\/ine structure constant $\alpha(s)$
and the anomalous magnetic moment of the muon are famous observables for
which the theoretical precision is limited by second order loop 
ef\/fects from hadronic vacuum polarization. Both quantities are related \via\
dispersion relations to the hadronic production rate in \eee\
annihilation,
\beq
\label{eq_rsigma}
      R(s) = \frac{\sigma_{\rm tot}(e^+e^-\!\rightarrow {\rm hadrons})}
                  {\sigma_0(e^+e^-\!\rightarrow \mu^+\mu^-)}~,
\eeq
with $\sigma_0(e^+e^-\!\rightarrow \mu^+\mu^-) = 4\pi\alpha^2/(3s)$.
While far from quark thresholds and at suf\/f\/iciently high energy
$\sqrt{s}$, $R(s)$ can be predicted by perturbative QCD, theory  
fails when resonances occur, \ie, local quark-hadron duality is broken. 
Fortunately, one can circumvent this drawback by using \eee\ 
annihilation data of $R(s)$ and, as proposed in 
Ref.~\cite{g_2pap}, hadronic $\tau$ decays benef\/itting from the 
largely conserved vector current (CVC), to replace theory in the 
critical energy regions.

There is a strong interest in the electroweak phenomenology to
reduce the uncertainty on \aqedZ\ which used to be a serious limit
to progress in the determination of the Higgs mass from 
radiative corrections in the Standard Model. Table~\ref{sin2} gives
the uncertainties of the dif\/ferent Standard Model input expressed
as errors on ${\rm sin}^2\theta_{\rm W}$~\cite{blondel}. Using the
former value~\cite{eidelman} for \aqedZ, the dominant uncertainties
stem from the experimental ${\rm sin}^2\theta_{\rm W}$ determination
and from the running f\/ine structure constant. Thus, any useful 
experimental amelioration on ${\rm sin}^2\theta_{\rm W}$ requires a
better precision of \aqedZ, \ie, an improved determination of
its hadronic contribution.

The anomalous magnetic moment $a_\mu=(g-2)/2$ of the muon is experimentally
and theoretically known to very high accuracy. In addition, the contribution 
of heavier objects to $a_\mu$ relative to the anomalous moment of the electron 
scales as $(m_\mu/m_e)^2\sim4\times10^{4}$. These properties allow an 
extremely sensitive test of the validity of electroweak theory. The present 
value from the combined $\mu^+$ and $\mu^-$ measurements~\cite{bailey},
\beq
    a_\mu \:=\: (11\,659\,230 \pm 85)\times10^{-10}~,
\eeq
is expected to be improved to a precision of at least
$4\times10^{-10}$ by the E821 experiment at Brookhaven~\cite{bnl}
(see also the contribution to this conference~\cite{bnlnew}). Again, the
precision of the theoretical prediction of $a_\mu$ is limited
by the contribution from hadronic vacuum polarization determined
analogously to \aqedZ\ by evaluating a dispersion integral using
\eee\ cross sections and perturbative QCD.
\begin{table}[t]
\caption[.]{ \label{sin2} 
	      Uncertainties of the Standard Model constraints
              expressed in terms of $\Delta{\rm sin}^2\theta_{\rm W}$.
              Downward arrows indicate future experimental improvement. }
\setlength{\tabcolsep}{0.2pc}
\vspace{0.2cm}
\begin{center}
\begin{tabular}{lcc} \\ \hline
 Input               & $\Delta{\rm sin}^2\theta_{\rm W}\times10^{-5}$   
	                    & Uncertainty/Source \\ \hline
 Experiment          &  19  & (LEP+SLD)~\cite{sinth} $\downarrow$ \\
 $\alpha(M_{\rm Z})$ &  23  & $\Delta\alpha(M_{\rm Z}^2) = 0.09$~\cite{eidelman} \\
 $m_t$               &  15  & (CDF+D0)~\cite{topmass} $\downarrow$ \\
 Theory              & 5-10 & 2-loop EW prediction~\cite{degrassi} \\
 $M_{\rm Higgs}$     & 150  & 65 -- 1000 GeV \\ \hline
\end{tabular}
\end{center}
\end{table}

%
% ------------------ Running of alpha  ----------------------------------
%

\section{Running of the QED Fine Structure Constant}

The running of the electromagnetic f\/ine structure constant \aqed\
is governed by the renormalized vacuum polarization function,
$\Pi_\gamma(s)$. For the spin 1 photon, $\Pi_\gamma(s)$ is given 
by the Fourier transform of the time-ordered product of the 
electromagnetic currents $j_{\rm em}^\mu(s)$ in the vacuum 
$(q^\mu q^\nu-q^2g^{\mu\nu})\,\Pi_\gamma(q^2)=
i\int d^4x\,e^{iqx}\langle 0|T(j_{\rm em}^\mu(x)j_{\rm em}^\nu(0))|0\rangle$.\linebreak  
With $\Delta\alpha(s)=-4\pi\alpha\,{\rm Re}
\left[\Pi_\gamma(s)-\Pi_\gamma(0)\right]$ and 
$\Delta\alpha(s)=\Delta\alpha_{\rm lep}(s)+\Delta\alpha_{\rm had}(s)$,
which subdivides the running contributions into a leptonic and a 
hadronic part, one has
\beq
    \alpha(s) \:=\: \frac{\alpha(0)}
                         {1 - \Delta\alpha_{\rm lep}(s)
                            - \Delta\alpha_{\rm had}(s)}~,
\eeq
where $4\pi\alpha(0)$ is the square of the electron charge in the 
long-wavelength Thomson limit. 

For the case of interest, $s=M_{\rm Z}^2$, the leptonic contribution 
at three-loop order has been calculated to be~\cite{steinh}
\beq
   \Delta\alpha_{\rm lep}(M_{\rm Z}^2) = 314.97686\times10^{-4}~.
\eeq
Using analyticity and unitarity, the dispersion integral for the 
contribution from hadronic vacuum polarization 
reads~\cite{cabibbo}
\beq\label{eq_int_alpha}
    \Delta\alpha_{\rm had}(M_{\rm Z}^2) \:=\:
        -\frac{\alpha(0) M_{\rm Z}^2}{3\pi}\,
         {\rm Re}\!\!\!\intl_{4m_\pi^2}^{\infty}\!\!ds\,
            \frac{R(s)}{s(s-M_{\rm Z}^2)-i\epsilon}~,
\eeq
and, employing the identity $1/(x^\prime-x-i\epsilon)_{\epsilon\rightarrow0}
={\rm P}\{1/(x^\prime-x)\}+i\pi\delta(x^\prime-x)$, the above
integral is evaluated using the principle value integration 
technique.

%
% ------------------- Muon Magnetic Anomaly ------------------------------------
%
\section{Muon Magnetic Anomaly}

It is convenient to separate the Standard Model prediction for the
anomalous magnetic moment of the muon,
$a_\mu\equiv(g-2)_\mu/2$, into its dif\/ferent contributions,
\beq
    a_\mu^{\rm SM} \:=\: a_\mu^{\rm QED} + a_\mu^{\rm had} +
                             a_\mu^{\rm weak}~,
\eeq
where $a_\mu^{\rm QED}=(11\,658\,470.6\,\pm\,0.2)\times10^{-10}$ is 
the pure electromagnetic contribution (see~\cite{krause1} and references 
therein), \amuhad\ is the contribution from hadronic vacuum polarization,
and $a_\mu^{\rm weak}=(15.1\,\pm\,0.4)\times10^{-10}
$~\cite{krause1,peris,weinberg} accounts for corrections due to
exchange of the weak interacting bosons up to two loops.

Equivalently to \daqedhZ, by virtue of the analyticity of the 
vacuum polarization correlator, the contribution of the hadronic 
vacuum polarization to $a_\mu$ can be calculated \via\ the dispersion 
integral~\cite{rafael}
\beq\label{eq_int_amu}
    a_\mu^{\rm had} \:=\: 
           \frac{\alpha^2(0)}{3\pi^2}
           \intl_{4m_\pi^2}^\infty\!\!ds\,\frac{K(s)}{s}R(s)~,
\eeq
where $K(s)$ denotes the QED kernel~\cite{rafael2}~,
\beqn
    \lefteqn{K(s) \:=\: x^2\left(1-\frac{x^2}{2}\right) \,+\,
                 (1+x)^2\left(1+\frac{1}{x^2}\right)} \nonumber\\
    & &\hspace{0.5cm}\times\left({\rm ln}(1+x)-x+\frac{x^2}{2}\right) \,+\,
                 \frac{(1+x)}{(1-x)}x^2\,{\rm ln}x~,
\eeqn
with $x=(1-\beta_\mu)/(1+\beta_\mu)$ and $\beta_\mu=(1-4m_\mu^2/s)^{1/2}$.
The function $K(s)$ decreases monotonically with increasing $s$. It gives
a strong weight to the low energy part of the integral~(\ref{eq_int_amu}).
About 92\pc\ of the total contribution to \amuhad\ is accumulated at c.m. 
energies $\sqrt{s}$ below 1.8~GeV and 72\pc\ of \amuhad\ is covered by 
the two-pion f\/inal state which is dominated by the $\rho(770)$ 
resonance. Data from vector hadronic $\tau$ decays published by the 
ALEPH Collaboration~\cite{aleph_vsf} provide a precise spectrum for
the two-pion f\/inal state as well as new input for the lesser known 
four-pion f\/inal states. This new information improves signif\/icantly
the precision of the \amuhad\ determination~\cite{g_2pap}.

\section{Improvement in 3 Steps}
\label{sec_improve}

A very detailed and rigorous evaluation of both \aqedZ\ and $(g-2)_\mu$
was performed by S.~Eidelman and F.~Jegerlehner in 1995~\cite{eidelman}
which since then is frequently used as standard reference. In their
numerical calculation of the integrals~(\ref{eq_int_alpha}) and 
(\ref{eq_int_amu}), the authors use exclusive \eetohad\ cross section 
measurements below 2~GeV c.m. energy, inclusive $R$ measurements 
up to 40~GeV and f\/inally perturbative QCD for above 40~GeV.
Their results to which I later will refer are
\beqn
\label{eq_eidel}
     \Delta\alpha_{\rm had}(M_{\rm Z}^2)
         &=& (279.7 \pm 6.5)\times10^{-4}~, \nonumber\\
     a_\mu^{\rm had}
      &=& (702.4 \pm 15.3)\times10^{-10}~.
\eeqn

Due to improvements on the experimental sides these theoretical 
evaluations are insuf\/f\/icient for present needs. Fortunately, new 
data and a better understanding of the underlying QCD phenomena led 
to new and signif\/icantly more accurate determinations of the hadronic
contributions to both observables.

\subsection*{(I) Addition of Precise $\tau$ Data}
\label{sec_i}

Using the conserved vector current (CVC) it was shown in 
Ref.~\cite{g_2pap} that the addition of precise $\tau$
\sfs, in particular of the $\tau^-\rightarrow\pi^-\pi^0\,\nu_\tau$ 
channel, to the \eee\ annihilation cross section measurements
improves the low-energy evaluation of the 
integrals~(\ref{eq_int_alpha}) and (\ref{eq_int_amu}). Hadronic
$\tau$ decays into $\bar{u}d^\prime$ isovector f\/inal states occur 
via exchange of a virtual $W^-$ boson and have therefore contributions 
from vector and axial-vector currents. On the contrary, f\/inal states
produced via photon exchange in \eee\ annihilation are always vector
but have isovector and isoscalar parts. The CVC
relation between the vector two-pion $\tau$ spectral function 
$v_{J=1}(\tau\rightarrow\pi\pi^0\,\nu_\tau)$ and the 
corresponding isovector \eee\ cross section at energy-squared $s$ reads
\beqns
   \sigma^{I=1}(e^+e^-\longrightarrow \pi^+\pi^-) = 
         \frac{4\pi\alpha^2(0)}{s}v_{J=1}(\tau\rightarrow\pi\pi^0\,\nu_\tau)~,
\eeqns
where $v_{J=1}(\tau\rightarrow\pi\pi^0\,\nu_\tau)$ is essentially the
hadronic invariant mass spectrum normalized to the two-pion
branching ratio and corrected by a kinematic factor appropriate to
$\tau$ decays with spin $J=1$ ~\cite{aleph_vsf}. The two-pion
cross sections (incl. the $\tau$ contribution) in dif\/ferent 
energy regions are depicted in F\/ig.~\ref{fig_2pi}. Excellent
agreement between $\tau$ and \eee\ data is observed.
For the four pion f\/inal states, isospin rotations must be performed 
to relate the respective $\tau$ charges to the corresponding \eee\ 
topologies~\cite{aleph_vsf}. 

\subsubsection*{\it Effects from $SU(2)$ violation}

Hadronic \sfs\ from $\tau$ decays are directly related to the 
isovector vacuum polarization currents when isospin invariance (CVC) and 
unitarity hold. For this purpose one has to worry whether the breakdown 
of CVC due to quark mass ef\/fects ($m_u \neq m_d$ generating 
$\partial_{\mu}J^\mu\sim(m_u - m_d)$ for a charge-changing hadronic 
current $J^\mu$ between $u$ and $d$ quarks) or unknown isospin-violating 
electromagnetic decays have non-negligible contributions within the 
present accuracy. Expected deviations from CVC due to so-called {\it second 
class currents} as, \eg, the decay $\tau^-\rightarrow\pi^-\eta\,\nu_\tau$ 
where the corresponding \eee\ f\/inal state
$\pi^0\eta$ (C=+1) is strictly forbidden, have estimated branching 
fractions of the order of $(m_u-m_d)^2\simeq10^{-5}$~\cite{etapi}, 
while the experimental upper limit amounts to 
B($\tau\rightarrow\pi^-\eta\,\nu_\tau$)\,$\,<1.4\times10^{-4}$~\cite{pdg} with 
95\pc\ CL. $SU(2)$ symmetry breaking caused by electromagnetic interactions
can occur in the $\rho^\pm$--$\rho^0$ masses and widths. Hadronic 
contributions to the $\rho^\pm$--$\rho^0$ width dif\/ference are expected 
to be much smaller since they are proportional to $(m_u-m_d)^2$. 
The total expected $SU(2)$ violation in the $\rho$
width is estimated in Ref.~\cite{g_2pap} to be
$(\Gamma_{\rho^\pm}-\Gamma_{\rho^0})/\Gamma_{\rho}=(2.8 \pm 3.9)\times 10^{-3}$,
yielding the corrections
\beqn
\label{eq_adif}
   \delta a_\mu^{\rm had} 
       &=& 
         -(1.3 \pm 2.0)\times10^{-10}~, \nonumber\\
   \delta \Delta\alpha_{\rm had}^{(5)}(M_{\rm Z}^2)
       &=&
         -(0.09 \pm 0.12)\times10^{-4}~,
\eeqn
to the respective dispersion integrals when using the 
$\tau^-\!\rightarrow\pi^-\pi^0\nu_\tau$ \sf\ in addition to \eee\ data.

\subsubsection*{\it Evaluation of the dispersion integrals~(\ref{eq_int_alpha}) and
                (\ref{eq_int_amu})}

Details about the non-trivial task of evaluating in a coherent way 
numerical integrals over data points which have statistical and 
correlated systematic errors between measurements {\it and}
experiments are given in Ref.~\cite{g_2pap}. The procedure is based
on an analytical $\chi^2$ minimization, taking into account all initial
correlations, and it provides the averages and the covariances of the 
cross sections from dif\/ferent experiments contributing to a 
certain f\/inal state in a given range of c.m. energies. One then 
applies the trapezoidal rule for the numerical integration of the 
dispersion integrals~(\ref{eq_int_alpha}) and (\ref{eq_int_amu}),
\ie, the integration range is subdivided into
suf\/f\/iciently small energy steps and for each of these steps 
the corresponding covariances (where additional correlations induced by 
the trapezoidal rule have to be taken into account) are calculated. This 
procedure yields error envelopes between adjacent measurements as 
depicted by the shaded bands in Fig.~\ref{fig_2pi}.
\begin{figure}
\psfig{figure=fig2.eps,height=3.5in}
\caption[.]{ \label{fig_2pi}
	     Two-pion cross section as a function of the c.m.
             energy-squared. The band represents the result of the averaging 
             procedure described in the text within the diagonal errors. 
             The lower left hand plot shows the chiral expansion of the 
             two-pion cross section used (see Ref.~\cite{g_2pap}).}
\end{figure}

\subsubsection*{\it Results}

With the inclusion of the $\tau$ vector \sfs, the hadronic contributions 
to the running \aqedZ\ and to $a_\mu$ are found to be~\cite{g_2pap}
\beqn
\label{eq_res1}
     \Delta\alpha_{\rm had}(M_{\rm Z}^2)
             & = &  (281.0 \pm 6.2)\times 10^{-4}~,\nonumber \\
     a_\mu^{\rm had} & = & (701.1\,\pm\, 9.4)\times 10^{-10}~, 
\eeqn
with an improvement for $a_\mu^{\rm had}$ of about $40\%$ compared to the
previous evaluation~(\ref{eq_eidel}), while there is only a marginal
improvement of \daqedZ\ for which the dominant uncertainties stem from 
higher energies.

\subsection*{(II) Extended Theoretical Approach}
\label{sec_ii}

The above analysis shows that in order to improve the precision on 
\daqedZ, a more accurate determination of the hadronic cross section between 
2~GeV and 10~GeV is needed. On the experimental side there are
ongoing $R$ measurements performed by the BES Collaboration~\cite{ichep_bes}.
On the other hand, QCD analyses using hadronic $\tau$ decays performed by 
ALEPH~\cite{aleph_asf} and recently by OPAL~\cite{opal_asf,ichep_opal} 
revealed excellent applicability of the Wilson {\it Operator Product 
Expansion} (OPE)~\cite{wilson} (also called SVZ 
approach~{\cite{svz}), organizing perturbative and nonperturbative contributions
to a physical observable using the concept of global quark-hadron duality, at
the scale of the $\tau$ mass, $M_\tau\simeq1.8~{\rm GeV}$. Using moments of 
spectral functions, dimensional nonperturbative operators contributing  to the 
$\tau$ hadronic width have been f\/itted simultaneously and turned out to be
small. This encouraged the authors of Ref.~\cite{alphapap} to apply 
a similar approach based on spectral moments to determine the size of
the nonperturbative contributions to integrals over total cross sections in 
\eee\ annihilation, and to f\/igure out whether or not the OPE, \ie, global 
duality is a valid approach at relatively low energies.

\subsubsection*{\it Theoretical prediction of $R(s)$}

The optical theorem relates the total hadronic cross section in \eee\ 
annihilation, $R(s_0)$, at a given energy-squared, $s_0$, to the absorptive 
part of the photon vacuum polarization correlator
\beq
\label{eq_rimpi}
     R(s_0) = 12\pi{\rm Im}\Pi(s_0+i\epsilon)~.
\eeq
Perturbative QCD predictions up to next-to-next-to leading order
$\alpha_s^3$ as well as second order quark mass corrections far from
the production threshold and
the f\/irst order dimension $D=4$ nonperturbative term are available 
for the Adler $D$-function~\cite{adler},
which is the logarithmic derivative of the correlator $\Pi$,
carrying all physical information:
\beq
\label{eq_adler}
     D(s) = - 12 \pi^2 s\frac{d\Pi(s)}{ds}~.
\eeq
This yields the relation
\beq
\label{eq_radler}
     R(s_0) = \frac{1}{2\pi i}
              \hm\ointl_{|s|=s_0}\hm\hm\frac{ds}{s} D(s)~,
\eeq
where the contour integral runs counter-clockwise around the 
circle from $s=s_0+i\e$ to $s=s_0-i\e$.
The Adler function is given by~\cite{3loop,kataev,bnp}
\beqn
\label{eq_d}
   \lefteqn{D_{f_i}(-s) = N_C\sum_f Q_f^2
            \Bigg\{} \nonumber \\
         & & \hspace{0.0cm} 
            1 + d_0\frac{\alpha_s(s)}{\pi}
                     + d_1\left(\frac{\alpha_s(s)}{\pi}\right)^{\!\!2}
                     + \tilde{d}_2\left(\frac{\alpha_s(s)}{\pi}\right)^{\!\!3} 
                     \nonumber \\
         & & \hspace{0.0cm} 
           -\; \frac{m_f^2(s)}{s}
                \left( 6 + 28\,\frac{\alpha_s(s)}{\pi}
                        + (295.1-12.3\,n_f)
                          \left(\frac{\alpha_s(s)}{\pi}\right)^{\!\!2}
                 \right)  \nonumber \\
         & & \hspace{0.0cm} 
           +\; \frac{2\pi^2}{3}\left(1 - \frac{11}{18}\frac{\alpha_s(s)}{\pi}
                         \right)\frac{\left\langle\frac{\alpha_s}{\pi} 
                                           GG\right\rangle}{s^2} 
      \nonumber \\
         & & \hspace{0.0cm} 
           + \;8\pi^2\left(1 - \frac{\alpha_s(s)}{\pi}
                  \right)\frac{\langle m_f\bar{q_f}q_f\rangle}{s^2}
      \nonumber \\
         & & \hspace{0.0cm} 
           + \;\frac{32\pi^2}{27}\frac{\alpha_s(s)}{\pi}
              \sum_k\frac{\langle m_k\bar{q_k}q_k\rangle}{s^2}
      \nonumber \\
         & & \hspace{0.0cm} 
           + \;12\pi^2\frac{\langle{\cal O}_6\rangle}{s^3}
            \;+\; 16\pi^2\frac{\langle{\cal O}_8\rangle}{s^4}
      \Bigg\}~, 
\eeqn
where additional logarithms occur when $\mu^2\neq s$ and $\mu$ 
being the renormalization scale\footnote
{
   The negative energy-squared in $D(-s)$ of Eq.~(\ref{eq_d})
   is introduced when continuing the Adler function from the spacelike
   Euclidean space, where it is originally def\/ined, to the timelike
   Minkowski space by virtue of its analyticity property.
}.
The coef\/f\/icients of the massless perturbative part are 
$d_0=1$, $d_1=1.9857 - 0.1153\,n_f$,
$\tilde{d}_2=d_2+\beta_0^2\pi^2/48$ with $\beta_0=11-2n_f/3$,
$d_2=-6.6368-1.2001\,n_f-0.0052\,n_f^2-1.2395\,(\sum_f Q_f)^2/N_C\sum_f Q_f^2$
and $n_f$ being the number of involved quark f\/lavours.
The nonperturbative operators in Eq.~(\ref{eq_d}) are the gluon condensate, 
$\langle(\alpha_s/\pi) GG\rangle$, and the quark condensates,
$\langle m_f\bar{q_f}q_f\rangle$. The latter obey approximately the PCAC 
relations
\beqn
\label{eq_pcac}
     (m_u + m_d)\langle\bar{u}u + \bar{d}d\rangle
       & \simeq & - 2 f_\pi^2 m_\pi^2~, \nonumber \\
     m_s\langle\bar{s}s\rangle 
       & \simeq & - f_\pi^2 (m_K^2-m_\pi^2)~,
\eeqn
with the pion decay constant $f_\pi=(92.4\pm0.26)$\MeVE~\cite{pdg}.
In the chiral limit the equations $f_\pi=f_K$ and
$\langle\bar{u}u\rangle=\langle\bar{d}d\rangle=\langle\bar{s}s\rangle$
hold. The complete dimension $D=6$ and $D=8$ operators are parametrized 
phenomenologically in Eq.~(\ref{eq_d}) using the saturated vacuum 
expectation values $\langle{\cal O}_6\rangle$ and 
$\langle{\cal O}_8\rangle$, respectively.

Although the theoretical prediction of $R$ using Eqs.~(\ref{eq_radler})
and (\ref{eq_d}) assumes local duality and therefore suf\/fers from 
unpredicted low-energy resonance oscillations, the following 
integration, Eqs.~(\ref{eq_int_alpha})/(\ref{eq_int_amu}), turns duality 
globally, \ie, the nonperturbative oscillations are averaged over the energy 
spectrum. However, a systematic uncertainty is introduced through the cut
at explicitly 1.8~GeV so that non-vanishing oscillations could give rise to 
a bias after integration. The associated systematic error is estimated
in Ref.~\cite{apap} by means of f\/itting dif\/ferent oscillating curves 
to the data around the cut region, yielding the error estimates
$\Delta(\Delta \alpha_{\rm had}(M_{\rm Z}^2))=0.15\times10^{-4}$ and
$\Delta a_\mu^{\rm had} = 0.24\times10^{-10}$,
from the comparison of the integral over the oscillating simulated
data to the OPE prediction. These numbers are added as systematic
uncertainties to the corresponding low-energy integrals.

In asymptotic energy regions we use the formulae of Ref.~\cite{kuhn1}
which include complete quark mass corrections up to order $\alpha_s^2$ to 
evaluate the perturbative prediction of $R(s)$ entering into the
integrals~(\ref{eq_int_alpha}) and (\ref{eq_int_amu}).

\subsubsection*{\it Theoretical uncertainties}

Details about the parameter errors used to estimate the uncertainties 
accompanying the theoretical analysis are given in Refs.~\cite{alphapap,apap}.
Theoretical uncertainties arise from essentially three sources
\begin{itemize}
  \item[(\it i)] {\it The perturbative prediction.} The estimation
                of theoretical errors of the perturbative series is
                strongly linked to its truncation at f\/inite order
                in $\alpha_s$. This introduces a non-vanishing dependence 
                on the choice of the renormalization scheme and the 
                renormalization scale. Furthermore, one has
                to worry whether the missing four-loop order 
                contribution $d_3(\alpha_s/\pi)^4$ gives rise to
                large corrections to the perturbative series.
	        An additional uncertainty stems from the ambiguity
	        between the results on $R$ obtained using contour-improved 
                f\/ixed-order perturbation theory (\FOPTCI) and FOPT only 
                (see Ref.~\cite{aleph_asf}).
  \item[(\it ii)] {\it The quark mass correction.} Since a theoretical 
                evaluation of the integrals~(\ref{eq_int_alpha}) and 
                (\ref{eq_int_amu}) is only applied far from quark production 
                thresholds, quark mass corrections and the corresponding 
                errors are small.
  \item[(\it iii)] {\it The nonperturbative contribution}. In order to
                detach the measurement from theoretical constraints
                on the nonperturbative parameters of the OPE, the dominant
                dimension $D=4,6,8$ terms are determined experimentally
                by means of a simultaneous f\/it of weighted integrals over 
                the inclusive low energy \eee\ cross section, so-called spectral 
	        moments, to the theoretical prediction obtained from 
	        Eq.~(\ref{eq_d}). Small uncertainties are introduced from 
                possible deviations from the PCAC relations~(\ref{eq_pcac}).
\end{itemize}

The spectral moment f\/it of the nonperturbative operators results in 
a very small contribution from the OPE power terms to the lowest moment 
at the scale of $1.8~{\rm GeV}$ (repeated and conf\/irmed at $2.1~{\rm GeV}$), 
as expected from the $\tau$ analyses~\cite{aleph_asf,opal_asf}. The value of
$\langle\frac{\alpha_s}{\pi} GG\rangle=(0.037\pm0.019)~{\rm GeV}^4$
found for the Gluon condensate is compatible with other 
evaluations~\cite{reinders,bertlmann}. The analysis proved that 
global duality holds at $1.8~{\rm GeV}$ and nonperturbative ef\/fects
contribute only negligibly, so that above this energy perturbative QCD 
can replace the rather imprecise data in the dispersion 
integrals~(\ref{eq_int_alpha}) and (\ref{eq_int_amu}).

\subsubsection*{\it Results}

The $R(s)$ measurements and the corresponding theoretical 
prediction are shown in F\/ig.~\ref{fig_r}. The shaded bands indicate the 
regions where data are used instead of theory to evaluate the dispersion 
integrals. This is below 1.8~GeV and at $c\bar{c}$ threshold
energies. Good agreement between data and QCD is found above 8~GeV, 
while at lower energies systematic deviations are observed. The $R$ 
measurements in this region are essentially provided by the 
$\gamma\gamma2$~\cite{E_78} and MARK~I~\cite{E_96} collaborations. 
MARK~I data above 5~GeV lie systematically above the measurements of 
the Crystal Ball~\cite{CB} and MD1~\cite{MD1} Collaborations as well 
as above the QCD prediction.
\begin{figure}
\psfig{figure=data.eps,height=3.55in}
\caption[.]{\label{fig_r}
            Inclusive hadronic cross section ratio in \eee\
            annihilation versus the c.m. energy $\sqrt{s}$. 
            Additionally shown is the QCD prediction of the continuum 
            contribution as explained in the text. The shaded areas 
            indicate regions were experimental data are used for the 
            evaluation of \daqedhZ\ and \amuhad\ in addition to the 
            measured narrow resonance parameters. The exclusive 
            \eee\ cross section measurements at low c.m. energies
            are taken from DM1,DM2,M2N,M3N,OLYA,CMD,ND and
            $\tau$ data from ALEPH (see Ref.~\cite{g_2pap}
            for detailed information).}
\end{figure}

The combination of the theoretical and experimental evaluations of the 
integrals yields the results~\cite{alphapap}
\beqn
\label{eq_res2}
     \Delta\alpha_{\rm had}(M_{\rm Z}^2)
         &=& (277.8 \pm 2.2_{\rm exp} \pm 1.4_{\rm theo})\times10^{-4}~,~~~~ \nonumber\\
     a_\mu^{\rm had}
      &=& (695.1 \pm 7.5_{\rm exp} \pm 0.7_{\rm theo})\times10^{-10}~,~~~~
\eeqn
with a signif\/icant improvement by more than a factor of two for \daqedZ,
and a $20\%$ better accuracy on $a_\mu^{\rm had}$ compared to the 
numbers~(\ref{eq_res1}).

The authors of Ref.~\cite{kuhnstein} improved the above 
analysis in the charm region by normalizing experimental results in the
theoretically not accessible region (at least locally) so that they match 
perturbative QCD at safe energies below and above the occurrence of 
resonances. The so-renormalized data show excellent agreement among 
dif\/ferent experiments which supports the hypothesis made that experimental 
systematic errors are completely correlated over the whole involved energy 
regime. The result (after correcting for the small top quark contribution)
reads~\cite{kuhnstein}
\beq
\label{eq_kuhnstein}
     \Delta\alpha_{\rm had}(M_{\rm Z}^2)
         = (276.7 \pm 1.7)\times10^{-4}~.
\eeq

Another, very elegant method based on an analytical calculation of the 
unsubtracted dispersion relation, corresponding to the subtracted
integral~(\ref{eq_int_alpha}), was presented in Ref.~\cite{erler}.
Only the low-energy pole contribution is taken from data,
while the contribution from higher energies is calculated analytically
using the two-point correlation function given in Ref.~\cite{kuhn1},
and the renormalization group equations for the running quantities.
This leads to the precise result~\cite{erler}
\beq
\label{eq_erler}
     \Delta\alpha_{\rm had}(M_{\rm Z}^2)
         = (277.2 \pm 1.9)\times10^{-4}~.
\eeq

Both numbers~(\ref{eq_kuhnstein}) and (\ref{eq_erler}) are in agreement
with \daqedZ\ from Eq.~(\ref{eq_res2}).

\subsection*{(III) Constraints from QCD Sum Rules}
\label{sec_iii}

It was shown in Ref.~\cite{apap} that the the previous determinations 
can be further improved by using f\/inite-energy QCD sum rule techniques 
in order to access theoretically energy regions where locally perturbative 
QCD fails. This idea was f\/irst presented in Ref.~\cite{schilcher}. In 
principle, the method uses no additional assumptions beyond those applied in 
Section~\ref{sec_ii}. However, parts of the dispersion integrals evaluated at 
low-energy and the $c\bar{c}$ threshold are obtained from values of the Adler
$D$-function itself, for which local quark-hadron duality is assumed to hold.
One therefore must perform an evaluation at rather high energies (3~GeV for 
$u,d,s$ quarks and 15~GeV for the $c$ quark contribution have been chosen in 
Ref.~\cite{apap}) to suppress deviations from local duality due to 
nonperturbative phenomena.

The idea of the approach is to reduce the data contribution to the
dispersion integrals by subtracting analytical functions from the singular
integration kernels in Eqs.~(\ref{eq_int_alpha}) and (\ref{eq_int_amu}),
and adding the subtracted part subsequently by using theory only.
Two approaches have been applied in Ref.~\cite{apap}: f\/irst, a method
based on spectral moments is def\/ined by the identity
\beqn
\label{eq_f}
     \lefteqn{F = 
          \intl_{4m_\pi^2}^{s_0}\!\!ds\,R(s)\left[f(s) - p_{n}(s)\right]}
          \nonumber \\
     & & \hspace{0.6cm}    
             +\; \frac{1}{2\pi i}\!\!\ointl_{|s|=s_0}\!\!\!\frac{ds}{s}
             \left[P_{n}(s_0) - P_{n}(s)\right] D_{uds}(s)~,
\eeqn
with $P_{n}(s)=\int_0^sdt\,p_{n}(t)$ and 
$f(s)=\alpha(0)^2K(s)/(3\pi^2s)$ for 
$F\equiv a_{\mu,\,[2m_\pi,\;\sqrt{s_0}]}^{\rm had}$,
as well as $f(s)=\alpha(0)M_{\rm Z}^2/(3\pi s(s-M_{\rm Z}^2))$ for 
$F\equiv \Delta\alpha_{\rm had}(M_{\rm Z}^2)_{[2m_\pi,\;\sqrt{s_0}]}$.
The regular functions $p_{n}(s)$ approximate the kernel $f(s)$ in 
order to reduce the contribution of the f\/irst integral in 
Eq.~(\ref{eq_f}) which has a singularity at $s=0$ and 
is thus evaluated using experimental data. The second integral in 
Eq.~(\ref{eq_f}) can be calculated theoretically in the framework of 
the OPE. The functions $p_{n}(s)$ are 
chosen in order to reduce the {\it uncertainty} of the data integral.
This approximately coincides with a low residual value of the 
integral, \ie, a good approximation of the integration kernel $f(s)$
by the $p_{n}(s)$ def\/ined as~\cite{apap}
\beq
\label{eq_pol}
    p_{n}(s) \equiv \sum_{i=1}^n c_i
                    \left(1 - \left(\frac{s}{s_0}\right)^{\!\!i}\right)~,
\eeq
with the form $(1-s/s_0)$ in order to ensure a vanishing integrand at
the crossing of the positive real axis where the validity of the OPE 
is questioned~\cite{svz}. Polynomials of order $s^n$ involve leading order
nonperturbative contributions of dimension $D=2(n+1)$. The analysis is 
therefore restricted to the linear $n=1$ case only.

A second approach uses the dispersion relation of the Adler $D$-function
\beq
\label{eq_dispd}
    D_f(Q^2) = Q^2\!\!\intl_{4m_f^2}^\infty \!\!ds\,\frac{R_f(s)}{(s+Q^2)^2}~,
\eeq
for space-like $Q^2=-q^2$ and the quark f\/lavour $f$. The 
above integrand approximate the integration kernels in 
Eqs.~(\ref{eq_int_alpha}) and (\ref{eq_int_amu}), so that the 
modif\/ied Eq.~(\ref{eq_f}) reads
\beqn
\label{eq_disp}
  \lefteqn{F = 
        \intl_{4m_\pi^2}^{s_0}\!\!ds\,R^{\rm Data}(s)
               \left[f(s) - \frac{A_FQ^2}{(s+Q^2)^2}\right]} \nonumber \\
    & & \hspace{0.5cm}
         +\; A_F\left(D_{uds}(Q^2) - Q^2\!\intl_{s_0}^\infty\!ds\,
           \frac{R_{uds}^{\rm QCD}(s)}{(s+Q^2)^2}\right)~,
\eeqn
with a normalization constant $A_F$ to be optimized for both \daqedZ\ and 
\amuhad.

The theoretical errors are evaluated correspondingly to the analysis 
presented in Section~\ref{sec_ii}.
\begin{table*}[t]
\caption[.]{\label{tab_alphares}
            Contributions to \daqedhZ, \amuhad\ and to \aehad\ from the 
            dif\/ferent energy regions. The subscripts in the f\/irst column
            give the quark flavours involved in the calculation.}
\setlength{\tabcolsep}{0.9pc}
\vspace{0.2cm}
\begin{center}
{\small
\begin{tabular}{lccc} \hline \\[-0.33cm]
Energy~(GeV)
                         & $\Delta \alpha_{\rm had}(M_{\rm Z}^2)\times10^{4}$
                              & $a_\mu^{\rm had}\times10^{10}$ 
                                   & $a_e^{\rm had}\times10^{14}$ 
\\[0.07cm]
\hline \\[-0.33cm]
$(2m_\pi$ -- $1.8)_{uds}$
                         & $56.36\pm0.70_{\rm exp}\pm0.18_{\rm theo}$
                              & $634.3\pm5.6_{\rm exp}\pm2.1_{\rm theo}$ 
                                   & $173.67\pm1.7_{\rm exp}\pm0.6_{\rm theo}$ 
\\[0.07cm]
$(1.8$ -- $3.700)_{uds}$ & $24.53\pm0.28_{\rm theo}$  
                              & $33.87\pm0.46_{\rm theo}$     
                                   & $8.13\pm0.11_{\rm theo}$
\\[0.07cm]
$\psi(1S,2S,3770)_c$ 
$+~(3.7$ -- $5)_{udsc}$
                         &$24.75\pm0.84_{\rm exp}\pm0.50_{\rm theo}$
                              & $14.31\pm0.50_{\rm exp}\pm0.21_{\rm theo}$
                                   & $3.41\pm0.12_{\rm exp}\pm0.05_{\rm theo}$
\\[0.07cm]
$(5$ -- $9.3)_{udsc}$    & $34.95\pm0.29_{\rm theo}$   
                              & $6.87\pm0.11_{\rm theo}$ 
                                   & $1.62\pm0.03_{\rm theo}$ 
\\[0.07cm]
$(9.3$ -- $12)_{udscb}$
                         &$15.70\pm0.28_{\rm theo}$
                              & $1.21\pm0.05_{\rm theo}$ 
                                   & $0.28\pm0.02_{\rm theo}$ 
\\[0.07cm]
$(12$ -- $\infty)_{udscb}$
                         & $120.68\pm0.25_{\rm theo}$   
                              & $1.80\pm0.01_{\rm theo}$ 
                                   & $0.42\pm0.01_{\rm theo}$ 
\\[0.07cm]
$(2m_t$ -- $\infty)_t$
                         &$-0.69\pm0.06_{\rm theo}$  
                              & $\approx0 $ 
                                   & $\approx0 $ 
\\[0.07cm]
\hline\\[-0.33cm]
$(2m_\pi$ -- $\infty)_{udscbt}$
                         & $276.3\pm1.1_{\rm exp}\pm1.1_{\rm theo}$
                              & $692.4\pm5.6_{\rm exp}\pm2.6_{\rm theo}$ 
                                   & $187.5\pm1.7_{\rm exp}\pm0.7_{\rm theo}$ 
\\[0.07cm]
\hline
\end{tabular}
}
\end{center}
\end{table*}

\subsubsection*{\it Results}

A $\chi^2$ f\/it taking into account the experimental and theoretical 
correlations between the polynomial moments yields for the f\/irst (spectral
moment) approach (hadronic contribution from 
$2m_\pi$ to 1.8~GeV)~\cite{apap}{\small
\beqns
\label{eq_fitres}
   \Delta\alpha_{\rm had}(M_{\rm Z}^2)_{[1.8~{\rm GeV}]}
      &=& (56.53 \pm 0.73_{\rm exp} \pm 0.39_{\rm theo})\times 10^{-4}~,\nonumber\\
   a_{\mu,\,[1.8~{\rm GeV}]}^{\rm had}
      &=& (634.3 \pm 5.6_{\rm exp} \pm 2.1_{\rm theo})\times 10^{-10}~, 
\eeqns
}while the dispersion relation approach gives 
($\sqrt{Q^2}=3~{\rm GeV}$)~\cite{apap}{\small
\beqns
   \Delta\alpha_{\rm had}(M_{\rm Z}^2)_{[1.8~{\rm GeV}]}
      &=& (56.36 \pm 0.70_{\rm exp} \pm 0.18_{\rm theo})\times 10^{-4}~,\nonumber\\
   a_{\mu,\,[1.8~{\rm GeV}]}^{\rm had}
      &=& (632.5 \pm 6.2_{\rm exp} \pm 1.6_{\rm theo})\times 10^{-10}~.
\eeqns
}Only the most precise of the above numbers are used for the f\/inal results.
The above theory-improved results can be compared to the corresponding
pure experimental values,
$\Delta\alpha_{\rm had}(M_{\rm Z}^2)_{[1.8~{\rm GeV}]}=(56.77\pm1.06)\times 10^{-4}$
and $a_{\mu,\,[1.8~{\rm GeV}]}^{\rm had}=(635.1 \pm 7.4)\times 10^{-10}$,
showing clear improvement.

For the charm threshold region only the approach~(\ref{eq_disp}) is 
used giving for the hadronic contributions from 3.7~GeV to 5~GeV
($\sqrt{Q^2}=15~{\rm GeV}$)~\cite{apap}{\small
\beqns
   \Delta\alpha_{\rm had}(M_{\rm Z}^2)_{[3.7,\;5~{\rm GeV}]}
      &=& (24.75 \pm 0.84_{\rm exp} \pm 0.50_{\rm theo})\times 10^{-4}~,\nonumber\\
   a_{\mu,\,[3.7,\;5~{\rm GeV}]}^{\rm had}
      &=& (14.31 \pm 0.50_{\rm exp} \pm 0.21_{\rm theo})\times 10^{-10}~, 
\eeqns
}for which compared with the pure data results,
$\Delta\alpha_{\rm had}(M_{\rm Z}^2)_{[3.7,\;5~{\rm GeV}]}=(25.04\pm1.21)\times 10^{-4}$
and $a_{\mu,\,[3.7,\;5~{\rm GeV}]}^{\rm had}=(14.44 \pm 0.62)\times 10^{-10}$,
only a slight improvement is observed.

\section{Final Results}

Table~\ref{tab_alphares} shows the experimental and theoretical 
evaluations of \daqedhZ, \amuhad\ and \aehad\ for the respective 
energy regimes\footnote
{
   The evaluation of \aehad\ follows the same procedure as \amuhad.
}. 
Experimental errors between dif\/ferent lines are
assumed to be uncorrelated, whereas theoretical errors, but
those from $c\bar{c}$ and $b\bar{b}$ thresholds which are quark mass
dominated, are added linearily.

According to Table~\ref{tab_alphares}, the combination of the theoretical
and experimental evaluations of the integrals~(\ref{eq_int_alpha}) 
and (\ref{eq_int_amu}) yields the f\/inal results
\beqns
\label{eq_res3}
   \Delta\alpha_{\rm had}(M_{\rm Z}^2) 
      &=& (276.3 \pm 1.1_{\rm exp} \pm 1.1_{\rm theo})\times10^{-4}~, \nonumber\\
   \alpha^{-1}(M_{\rm Z}^2) 
      &=& 128.933 \pm 0.015_{\rm exp} \pm 0.015_{\rm theo}~, \nonumber\\[0.3cm]
   a_\mu^{\rm had}
      &=& (692.4 \pm 5.6_{\rm exp} \pm 2.6_{\rm theo})\times10^{-10}~, \nonumber\\
   a_\mu^{\rm SM}
      &=& (11\,659\,159.6 \pm 5.6_{\rm exp} \pm 3.7_{\rm theo})\times10^{-10}~,
\eeqns
and $a_e^{\rm had}=(187.5\pm1.7_{\rm exp}\pm0.7_{\rm theo})\times10^{-14}$ 
for the leading order hadronic contribution to $a_e$. 
The improvement for \aqedZ\ compared to the previous 
results~(\ref{eq_res2}) amounts to $40\%$ and $a_\mu^{\rm had}$ is 
about $17\%$ more precise than~(\ref{eq_res2}).

The total $a_\mu^{\rm SM}$ value includes an additional contribution
from non-leading order hadronic vacuum polarization summarized in 
Refs.~\cite{krause2,g_2pap} to be 
$a_\mu^{\rm had}[(\alpha/\pi)^3]=(-10.0\pm0.6)\times10^{-10}$.
Also the light-by-light scattering (LBLS) contribution has recently been 
reevaluated in Refs.~\cite{kinolight} and \cite{light2} of which the 
average $\langle a_\mu^{\rm had}[{\rm LBLS}]\rangle=(-8.5\pm2.5)\times10^{-10}$
is used here.

Figures~\ref{fig_results_alpha} and \ref{fig_results_amu} show a 
compilation of published results for the hadronic contributions 
\begin{figure}[t]
\psfig{figure=results_alpha.eps,height=1.86in}
\caption[.]{\label{fig_results_alpha}
            Comparison of \daqedhZ\ evaluations. The values are
            taken from Refs.~\rm
 \cite{lynn,eidelman,burkhardt,martin,swartz,g_2pap,alphapap,kuhnstein,erler,apap}.}
%\end{figure}
%\begin{figure}[t]
\vspace{0.cm}
\hspace{-1cm}\psfig{figure=results_amu.eps,height=1.86in}
\caption[.]{\label{fig_results_amu}
            Comparison of $a_\mu^{\rm had}$ evaluations. The values are
            taken from Refs.~\rm
            \cite{barkov,kinoshita,casas,eidelman,worstell,g_2pap,alphapap,apap}.}
\end{figure}
to \aqedZ\ and $a_\mu$. Some authors give the contribution 
for the f\/ive light quarks only and add the top quark part separately. 
This has been corrected for in F\/ig.~\ref{fig_results_alpha}.

The new result for \aqed\ is exploited in Ref.~\cite{apap}
to repeat the global electroweak f\/it in order to adjust the mass of the
Standard Model Higgs boson, $M_{\rm{Higgs}}$. The prediction
of the Standard Model is obtained from the ZFITTER electroweak 
library~\cite{leplib}. The f\/it yields 
$M_{\rm{Higgs}}=83^{+61}_{-38}~{\rm GeV}$
and an upper mass limit of 202~GeV at 95\pc\ CL~\cite{paus,apap}.

\section{Conclusions}

This note summarizes the recent ef\/fort that has been
undertaken in order to ameliorate the theoretical predictions for 
\aqedZ\ and \amuhad, crucially necessary to maintain the 
sensitivity of the diverse experimental improvements on the
Standard Model Higgs mass, on the one hand, and tests of the
electroweak theory on the other hand. The new value of 
$\alpha^{-1}(M_{\rm Z}^2)=128.933\pm0.021$ for the running f\/ine
structure constant proves as being suf\/f\/iciently accurate
to unaf\/fect the global Standard Model f\/it. On the contrary,
more ef\/fort is needed to further improve the precision of the
hadronic contribution to the anomalous magnetic moment of the muon
below the intended experimental accuracy at BNL-E821, which is 
about $4\times10^{-10}$~\cite{bnl}. Fortunately, new low energy 
data are expected in the near future from $\tau$ decays (CLEO, OPAL, 
DELPHI, BABAR) and from \eee\ annihilation (BES, CMD~II, DA$\Phi$NE).
Additional support might come from the theoretical side
using chiral perturbation theory to access the low energy inverse 
moment sum rules (\ref{eq_int_alpha}) and (\ref{eq_int_amu}) in 
applying, \eg, a similar procedure as the one suggested in Ref.~\cite{stern}.

\section*{Acknowledgements}
It is indeed a pleasure to thank my collaborator and friend Michel Davier
for the exciting work we are doing together at LAL. I gratefully acknowledge
numerous interesting discussions with J.~K\"uhn. I am indebted to all the 
women and men of the local organization committee for the most interesting 
and beautiful time I was able to spent in Vancouver.

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\bibitem{krause1}    A.~Czarnecki, B.~Krause and W.J.~Marciano,
                     \PRL\ {\bf 76} (1995) 3267; \PR\ {\bf D52} (1995) 2619
\bibitem{peris}      S.~Peris, M.~Perrottet and E.~de Rafael, 
                     \PL\ {\bf B355} (1995) 523
\bibitem{weinberg}   R.~Jackiw and S.~Weinberg, \PR\ {\bf D5} (1972) 2473
\bibitem{rafael}     M.~Gourdin and E.~de~Rafael, \NP\ {\bf B10} (1969) 667
\bibitem{rafael2}    S.J.~Brodsky and E.~de Rafael, \PR\ {\bf 168} (1968) 1620
\bibitem{aleph_vsf}  ALEPH \Cl\ (R.~Barate \ea)
                     \ZP\ {\bf C76} (1997) 15
\bibitem{etapi}      S.~Tisserant and T.N.~Truong,
                     {\it Phys. Lett.} {\bf B115} (1982) 264; \\
                     A.~Pich, {\it Phys. Lett.} {\bf B196} (1987) 561; \\
                     H.~Neufeld and H.~Rupertsberger,
                     {\it Z. Phys.} {\bf C68} (1995) 91
\bibitem{pdg}         C.~Caso \ea\ (Particle Data Group),\EPJ\ {\bf C3} (1998) 1
\bibitem{ichep_bes}  Z.~Zhao, \ICHEP, Vancouver (1998)
\bibitem{aleph_asf}  ALEPH \Cl\ (R.~Barate \ea), ``{\it
                     Measurement of the Spectral Functions of Axial-Vector 
                     Hadronic $\tau$ Decays and Determination of 
                     $\alpha_s(M_\tau^2)$}'',
                     Report CERN PPE/98-012 (1998)
\bibitem{opal_asf}   OPAL \Cl\ (K.~Ackerstaf\/f \ea), ``{\it
                     Measurement of the Strong Coupling Constant $\alpha_s$ and the 
                     Vector and Axial-Vector Spectral Functions in
                     Hadronic Tau Decays}'', CERN-EP/98-102 (1998)
\bibitem{ichep_opal} A.~Stahl,  \ICHEP, Vancouver (1998)
\bibitem{wilson}     K.G.~Wilson, \PR\ {\bf 179} (1969) 1499
\bibitem{svz}        M.A.~Shifman, A.L.~Vainshtein and V.I.~Zakharov,
                     \NP\ {\bf B147} (1979) 385, 448, 519
\bibitem{alphapap}   M.~Davier and A.~H\"ocker, \PL\ {\bf B419} (1998) 419
\bibitem{adler}      S.~Adler, \PR\ {\bf D10} (1974) 3714
\bibitem{3loop}      L.R.~Surguladze and M.A.~Samuel, 
                     \PRL\ {\bf 66} (1991) 560; \\
                     S.G.~Gorishny, A.L.~Kataev and S.A.~Larin,
                     \PL\ {\bf B259} (1991) 144
\bibitem{kataev}     S.G.~Gorishny, A.L.~Kataev and S.A.~Larin,
                     {\it Il Nuovo Cim.} {\bf 92A} (1986) 119
\bibitem{bnp}        E.~Braaten, S.~Narison and A.~Pich, 
                     \NP\ {\bf B373} (1992) 581
\bibitem{apap}       M.~Davier and A.~H\"ocker, ``{\it
                     New Results on the Hadronic Contribution to 
                     \aqedZ\ and to $(g-2)_\mu$}'',
                     Report LAL 98-38 (1998)
\bibitem{kuhn1}      K.G.~Chetyrkin, J.H.~K\"uhn and M.~Steinhauser,
                     \NP\ {\bf B482} (1996) 213
\bibitem{reinders}   L.J.~Reinders, H.~Rubinstein and S.~Yazaki,
                     \PRep\ {\bf 127} (1985) 1
\bibitem{bertlmann}  R.A.~Bertlmann, \ZP\ {\bf C39} (1988) 231
\bibitem{E_78}       C.~Bacci \ea\ ($\gamma\gamma2$ \Cl),
                     \PL\ {\bf B86} (1979) 234
\bibitem{E_96}       J.L.~Siegrist \ea\ (MARK~I \Cl),
                     \PR\ {\bf D26} (1982) 969
\bibitem{CB}         Z.~Jakubowski \ea\ (Crystal Ball \Cl),
                     \ZP\ {\bf C40} (1988) 49; \\
                     C.Edwards \ea\ (Crystal Ball \Cl),
                     SLAC-PUB-5160 (1990)
\bibitem{MD1}        A.E.~Blinov \ea\ (MD-1 \Cl),
                     \ZP\ {\bf C49} (1991) 239; \\
                     A.E.~Blinov \ea\ (MD-1 \Cl),
                     \ZP\ {\bf C70} (1996) 31
\bibitem{kuhnstein}  J.H.~K\"uhn and M.~Steinhauser,
                     ``{\it 
 	               A theory driven analysis of the effective QED
                       coupling at $M_{\rm Z}$}'', MPI-PHT-98-12 (1998)
\bibitem{erler}      J.~Erler, ``{\it 
                     \MSbar\ Scheme Calculation of the QED
                     Coupling $\hat{\alpha}(M_Z)$}'', UPR-796-T (1998)
\bibitem{schilcher}  S.~Groote, J.G.~K\"orner, N.F.~Nasrallah and K.~Schilcher,
                     ``{\it 
                       QCD Sum Rule Determination of $\alpha(M_{\rm Z})$
                       with Minimal Data Input}'', 
                     Report MZ-TH-98-02 (1998)
\bibitem{krause2}    B.~Krause, \PL\ {\bf B390} (1997) 392
\bibitem{kinolight}  M.~Hayakawa, T.~Kinoshita, \PR\ {\bf D57} (1998) 465
\bibitem{light2}     J.~Bijnens, E.~Pallante and J.~Prades,
                     \NP\ {\bf B474} (1996) 379
\bibitem{lynn}       B.W.~Lynn, G.~Penso and C.~Verzegnassi,
                     \PR\ {\bf D35} (1987) 42
\bibitem{burkhardt}  H.~Burkhardt and B.~Pietrzyk, \PL\ {\bf B356} (1995) 398
\bibitem{martin}     A.D.~Martin and D.~Zeppenfeld, \PL\ {\bf B345} (1995) 558
\bibitem{swartz}     M.L.~Swartz, \PR\ {\bf D53} (1996) 5268
\bibitem{barkov}     L.M.~Barkov {\it et al.} (OLYA, CMD Collaboration),
                     \NP\ {\bf B256} (1985) 365
\bibitem{kinoshita}  T.~Kinoshita, B.~Ni\v{z}i\'c and Y.~Okamoto,
                     \PR\ {\bf D31} (1985) 2108
\bibitem{casas}      J.A.~Casas, C.~L\'opez and F.J.~Yndur\'ain, 
                     \PR\ {\bf D32} (1985) 736
\bibitem{worstell}   D.H.~Brown and W.A.~Worstell, \PR\ {\bf D54} (1996) 3237
\bibitem{leplib}     D.~Bardin \ea, ``{\it
                     Reports of the working group on precision calculations
                     for the Z resonance''}, CERN-PPE 95-03
\bibitem{paus}       C.~Paus, {\it private communication}, June 1998
\bibitem{stern}      M.~Davier, L.~Girlanda, A.~H\"ocker and J.~Stern,
                     ``{\it 
                        Finite Energy Chiral Sum Rules and Tau Spectral Functions}'',
                     IPNO-TH-98-05, LAL-98-05,  
                     {\it to be published in Phys. Rev. D}

\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                      END OF proc.tex                                  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Psfig/TeX 
\def\PsfigVersion{1.10}
\def\setDriver{\DvipsDriver} % \DvipsDriver or \OzTeXDriver
%
% All software, documentation, and related files in this distribution of
% psfig/tex are Copyright 1993 Trevor J. Darrell
%
% Permission is granted for use and non-profit distribution of psfig/tex 
% providing that this notice is clearly maintained. The right to
% distribute any portion of psfig/tex for profit or as part of any commercial
% product is specifically reserved for the author(s) of that portion.
%
% To use with LaTeX, use \documentstyle[psfig,...]{...}
% To use with TeX, use \input psfig.sty
%
% Bugs and improvements to trevor@media.mit.edu.
%
% Thanks to Ned Batchelder, Greg Hager (GDH), J. Daniel Smith (JDS),
% Tom Rokicki (TR), Robert Russell (RR), George V. Reilly (GVR),
% Ken McGlothlen (KHC), Baron Grey (BG), Gerhard Tobermann (GT).
% and all others who have contributed code and comments to this project!
%
% ======================================================================
% Modification History:
%
%  9 Oct 1990   JDS	used more robust bbox reading code from Tom Rokicki
% 29 Mar 1991   JDS	implemented rotation= option
% 25 Jun 1991   RR	if bb specified on cmd line don't check
%			for .ps file.
%  3 Jul 1991	JDS	check if file already read in once
%  4 Sep 1991	JDS	fixed incorrect computation of rotated
%			bounding box
% 25 Sep 1991	GVR	expanded synopsis of \psfig
% 14 Oct 1991	JDS	\fbox code from LaTeX so \psdraft works with TeX
%			changed \typeout to \ps@typeout
% 17 Oct 1991	JDS	added \psscalefirst and \psrotatefirst
% 23 Jun 1993   KHC     ``doclip'' must appear before ``rotate''
% 27 Oct 1993   TJD	removed printing of filename to avoid 
%			underscore problems. changed \frame to \fbox.
%			Added OzTeX support from BG. Added new
%			figure search path code from GT.
%
% ======================================================================
%
% Command synopsis:
%
% \psdraft	draws an outline box, but doesn't include the figure
%		in the DVI file.  Useful for previewing.
%
% \psfull	includes the figure in the DVI file (default).
%
% \psscalefirst width= or height= specifies the size of the figure
% 		before rotation.
% \psrotatefirst (default) width= or height= specifies the size of the
% 		 figure after rotation.  Asymetric figures will
% 		 appear to shrink.
%
% \psfigurepath{dir:dir:...}  sets the path to search for the figure
%
% \psfig
% usage: \psfig{file=, figure=, height=, width=,
%			bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=, angle=, silent=}
%
%	"file" is the filename.  If no path name is specified and the
%		file is not found in the current directory,
%		it will be looked for in directory \psfigurepath.
%	"figure" is a synonym for "file".
%	By default, the width and height of the figure are taken from
%		the BoundingBox of the figure.
%	If "width" is specified, the figure is scaled so that it has
%		the specified width.  Its height changes proportionately.
%	If "height" is specified, the figure is scaled so that it has
%		the specified height.  Its width changes proportionately.
%	If both "width" and "height" are specified, the figure is scaled
%		anamorphically.
%	"bbllx", "bblly", "bburx", and "bbury" control the PostScript
%		BoundingBox.  If these four values are specified
%               *before* the "file" option, the PSFIG will not try to
%               open the PostScript file.
%	"rheight" and "rwidth" are the reserved height and width
%		of the figure, i.e., how big TeX actually thinks
%		the figure is.  They default to "width" and "height".
%	The "clip" option ensures that no portion of the figure will
%		appear outside its BoundingBox.  "clip=" is a switch and
%		takes no value, but the `=' must be present.
%	The "angle" option specifies the angle of rotation (degrees, ccw).
%	The "silent" option makes \psfig work silently.
%
% ======================================================================
% check to see if macros already loaded in (maybe some other file says
% "\input psfig") ...
\ifx\undefined\psfig\else\endinput\fi
%
% from a suggestion by eijkhout@csrd.uiuc.edu to allow
% loading as a style file. Changed to avoid problems
% with amstex per suggestion by jbence@math.ucla.edu

\let\LaTeXAtSign=\@
\let\@=\relax
\edef\psfigRestoreAt{\catcode`\@=\number\catcode`@\relax}
%\edef\psfigRestoreAt{\catcode`@=\number\catcode`@\relax}
\catcode`\@=11\relax
\newwrite\@unused
\def\ps@typeout#1{{\let\protect\string\immediate\write\@unused{#1}}}

\def\DvipsDriver{
	\ps@typeout{psfig/tex \PsfigVersion -dvips}
\def\PsfigSpecials{\DvipsSpecials} 	\def\ps@dir{/}
\def\ps@predir{} }
\def\OzTeXDriver{
	\ps@typeout{psfig/tex \PsfigVersion -oztex}
	\def\PsfigSpecials{\OzTeXSpecials}
	\def\ps@dir{:}
	\def\ps@predir{:}
	\catcode`\^^J=5
}

%% Here's how you define your figure path.  Should be set up with null
%% default and a user useable definition.

\def\figurepath{./:}
\def\psfigurepath#1{\edef\figurepath{#1:}}

%%% inserted for Searching Unixpaths
%%% (the path must end with :)
%%% (call: \DoPaths\figurepath )
%%%------------------------------------------------------
\def\DoPaths#1{\expandafter\EachPath#1\stoplist}
%
\def\leer{}
\def\EachPath#1:#2\stoplist{% #1 part of the list (delimiter :)
  \ExistsFile{#1}{\SearchedFile}
  \ifx#2\leer
  \else
    \expandafter\EachPath#2\stoplist
  \fi}
%
% exists the file (does not work for directories!)
%
\def\ps@dir{/}
\def\ExistsFile#1#2{%
   \openin1=\ps@predir#1\ps@dir#2
   \ifeof1
       \closein1
       %\ps@typeout{...not: \ps@predir#1\ps@dir#2}
   \else
       \closein1
       %\ps@typeout{...in:  \ps@predir#1\ps@dir#2}
        \ifx\ps@founddir\leer
          %\ps@typeout{set founddir #1}
           \edef\ps@founddir{#1}
        \fi
   \fi}
%------------------------------------------------------
%
% Get dir in path or error
%
\def\get@dir#1{%
  \def\ps@founddir{}
  \def\SearchedFile{#1}
  \DoPaths\figurepath
%  \fi
}
%------------------------------------------------------
%%% END of Searching Unixpaths


%
% @psdo control structure -- similar to Latex @for.
% I redefined these with different names so that psfig can
% be used with TeX as well as LaTeX, and so that it will not 
% be vunerable to future changes in LaTeX's internal
% control structure,
%
\def\@nnil{\@nil}
\def\@empty{}
\def\@psdonoop#1\@@#2#3{}
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}
% 
% \fbox is defined in latex.tex; so if \fbox is undefined, assume that
% we are not in LaTeX.
% Perhaps this could be done better???
\ifx\undefined\fbox
% \fbox code from modified slightly from LaTeX
\newdimen\fboxrule
\newdimen\fboxsep
\newdimen\ps@tempdima
\newbox\ps@tempboxa
\fboxsep = 3pt
\fboxrule = .4pt
\long\def\fbox#1{\leavevmode\setbox\ps@tempboxa\hbox{#1}\ps@tempdima\fboxrule
    \advance\ps@tempdima \fboxsep \advance\ps@tempdima \dp\ps@tempboxa
   \hbox{\lower \ps@tempdima\hbox
  {\vbox{\hrule height \fboxrule
          \hbox{\vrule width \fboxrule \hskip\fboxsep
          \vbox{\vskip\fboxsep \box\ps@tempboxa\vskip\fboxsep}\hskip 
                 \fboxsep\vrule width \fboxrule}
                 \hrule height \fboxrule}}}}
\fi
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% file reading stuff from epsf.tex
%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%   Portions modified/removed for use in PSFIG package by
%      J. Daniel Smith, 9 October 1990.
%
\newread\ps@stream
\newif\ifnot@eof       % continue looking for the bounding box?
\newif\if@noisy        % report what you're making?
\newif\if@atend        % %%BoundingBox: has (at end) specification
\newif\if@psfile       % does this look like a PostScript file?
%
% PostScript files should start with `%!'
%
{\catcode`\%=12\global\gdef\epsf@start{%!}}
\def\epsf@PS{PS}
%
\def\epsf@getbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
\openin\ps@stream=\ps@predir#1
\ifeof\ps@stream\ps@typeout{Error, File #1 not found}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that doesn't
%   start with %. We're looking for the bounding box comment.
%
   {\not@eoftrue \chardef\other=12
    \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
    \loop
       \if@psfile
	  \read\ps@stream to \epsf@fileline
       \else{
	  \obeyspaces
          \read\ps@stream to \epsf@tmp\global\let\epsf@fileline\epsf@tmp}
       \fi
       \ifeof\ps@stream\not@eoffalse\else
%
%   Check the first line for `%!'.  Issue a warning message if its not
%   there, since the file might not be a PostScript file.
%
       \if@psfile\else
       \expandafter\epsf@test\epsf@fileline:. \\%
       \fi
%
%   We check to see if the first character is a % sign;
%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:' and the `(atend)' specification was not found.
%   That is, the only way to stop is when the end of file is reached,
%   or a `%%BoundingBox: llx lly urx ury' line is found.
%
          \expandafter\epsf@aux\epsf@fileline:. \\%
       \fi
   \ifnot@eof\repeat
   }\closein\ps@stream\fi}%
%
% This tests if the file we are reading looks like a PostScript file.
%
\long\def\epsf@test#1#2#3:#4\\{\def\epsf@testit{#1#2}
			\ifx\epsf@testit\epsf@start\else
\ps@typeout{Warning! File does not start with `\epsf@start'.  It may not be a PostScript file.}
			\fi
			\@psfiletrue} % don't test after 1st line
%
%   We still need to define the tricky \epsf@aux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12\global\let\epsf@percent=%\global\def\epsf@bblit{%BoundingBox}}
%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.  We continue searching if `(at end)'
%   was found after the `%BoundingBox:'.
%
\long\def\epsf@aux#1#2:#3\\{\ifx#1\epsf@percent
   \def\epsf@testit{#2}\ifx\epsf@testit\epsf@bblit
	\@atendfalse
        \epsf@atend #3 . \\%
	\if@atend	
	   \if@verbose{
		\ps@typeout{psfig: found `(atend)'; continuing search}
	   }\fi
        \else
        \epsf@grab #3 . . . \\%
        \not@eoffalse
        \global\no@bbfalse
        \fi
   \fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsf@grab #1 #2 #3 #4 #5\\{%
   \global\def\epsf@llx{#1}\ifx\epsf@llx\empty
      \epsf@grab #2 #3 #4 #5 .\\\else
   \global\def\epsf@lly{#2}%
   \global\def\epsf@urx{#3}\global\def\epsf@ury{#4}\fi}%
%
% Determine if the stuff following the %%BoundingBox is `(atend)'
% J. Daniel Smith.  Copied from \epsf@grab above.
%
\def\epsf@atendlit{(atend)} 
\def\epsf@atend #1 #2 #3\\{%
   \def\epsf@tmp{#1}\ifx\epsf@tmp\empty
      \epsf@atend #2 #3 .\\\else
   \ifx\epsf@tmp\epsf@atendlit\@atendtrue\fi\fi}


% End of file reading stuff from epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% trigonometry stuff from "trig.tex"
\chardef\psletter = 11 % won't conflict with \begin{letter} now...
\chardef\other = 12

\newif \ifdebug %%% turn me on to see TeX hard at work ...
\newif\ifc@mpute %%% don't need to compute some values
\c@mputetrue % but assume that we do

\let\then = \relax
\def\r@dian{pt }
\let\r@dians = \r@dian
\let\dimensionless@nit = \r@dian
\let\dimensionless@nits = \dimensionless@nit
\def\internal@nit{sp }
\let\internal@nits = \internal@nit
\newif\ifstillc@nverging
\def \Mess@ge #1{\ifdebug \then \message {#1} \fi}

{ %%% Things that need abnormal catcodes %%%
	\catcode `\@ = \psletter
	\gdef \nodimen {\expandafter \n@dimen \the \dimen}
	\gdef \term #1 #2 #3%
	       {\edef \t@ {\the #1}%%% freeze parameter 1 (count, by value)
		\edef \t@@ {\expandafter \n@dimen \the #2\r@dian}%
				   %%% freeze parameter 2 (dimen, by value)
		\t@rm {\t@} {\t@@} {#3}%
	       }
	\gdef \t@rm #1 #2 #3%
	       {{%
		\count 0 = 0
		\dimen 0 = 1 \dimensionless@nit
		\dimen 2 = #2\relax
		\Mess@ge {Calculating term #1 of \nodimen 2}%
		\loop
		\ifnum	\count 0 < #1
		\then	\advance \count 0 by 1
			\Mess@ge {Iteration \the \count 0 \space}%
			\Multiply \dimen 0 by {\dimen 2}%
			\Mess@ge {After multiplication, term = \nodimen 0}%
			\Divide \dimen 0 by {\count 0}%
			\Mess@ge {After division, term = \nodimen 0}%
		\repeat
		\Mess@ge {Final value for term #1 of 
				\nodimen 2 \space is \nodimen 0}%
		\xdef \Term {#3 = \nodimen 0 \r@dians}%
		\aftergroup \Term
	       }}
	\catcode `\p = \other
	\catcode `\t = \other
	\gdef \n@dimen #1pt{#1} %%% throw away the ``pt''
}

\def \Divide #1by #2{\divide #1 by #2} %%% just a synonym

\def \Multiply #1by #2%%% allows division of a dimen by a dimen
       {{%%% should really freeze parameter 2 (dimen, passed by value)
	\count 0 = #1\relax
	\count 2 = #2\relax
	\count 4 = 65536
	\Mess@ge {Before scaling, count 0 = \the \count 0 \space and
			count 2 = \the \count 2}%
	\ifnum	\count 0 > 32767 %%% do our best to avoid overflow
	\then	\divide \count 0 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 0 < -32767
		\then	\divide \count 0 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\ifnum	\count 2 > 32767 %%% while retaining reasonable accuracy
	\then	\divide \count 2 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 2 < -32767
		\then	\divide \count 2 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\multiply \count 0 by \count 2
	\divide \count 0 by \count 4
	\xdef \product {#1 = \the \count 0 \internal@nits}%
	\aftergroup \product
       }}

\def\r@duce{\ifdim\dimen0 > 90\r@dian \then   % sin(x+90) = sin(180-x)
		\multiply\dimen0 by -1
		\advance\dimen0 by 180\r@dian
		\r@duce
	    \else \ifdim\dimen0 < -90\r@dian \then  % sin(-x) = sin(360+x)
		\advance\dimen0 by 360\r@dian
		\r@duce
		\fi
	    \fi}

\def\Sine#1%
       {{%
	\dimen 0 = #1 \r@dian
	\r@duce
	\ifdim\dimen0 = -90\r@dian \then
	   \dimen4 = -1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 90\r@dian \then
	   \dimen4 = 1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 0\r@dian \then
	   \dimen4 = 0\r@dian
	   \c@mputefalse
	\fi
%
	\ifc@mpute \then
        	% convert degrees to radians
		\divide\dimen0 by 180
		\dimen0=3.141592654\dimen0
%
		\dimen 2 = 3.1415926535897963\r@dian %%% a well-known constant
		\divide\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2
		\Mess@ge {Sin: calculating Sin of \nodimen 0}%
		\count 0 = 1 %%% see power-series expansion for sine
		\dimen 2 = 1 \r@dian %%% ditto
		\dimen 4 = 0 \r@dian %%% ditto
		\loop
			\ifnum	\dimen 2 = 0 %%% then we've done
			\then	\stillc@nvergingfalse 
			\else	\stillc@nvergingtrue
			\fi
			\ifstillc@nverging %%% then calculate next term
			\then	\term {\count 0} {\dimen 0} {\dimen 2}%
				\advance \count 0 by 2
				\count 2 = \count 0
				\divide \count 2 by 2
				\ifodd	\count 2 %%% signs alternate
				\then	\advance \dimen 4 by \dimen 2
				\else	\advance \dimen 4 by -\dimen 2
				\fi
		\repeat
	\fi		
			\xdef \sine {\nodimen 4}%
       }}

% Now the Cosine can be calculated easily by calling \Sine
\def\Cosine#1{\ifx\sine\UnDefined\edef\Savesine{\relax}\else
		             \edef\Savesine{\sine}\fi
	{\dimen0=#1\r@dian\advance\dimen0 by 90\r@dian
	 \Sine{\nodimen 0}
	 \xdef\cosine{\sine}
	 \xdef\sine{\Savesine}}}	      
% end of trig stuff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\psdraft{
	\def\@psdraft{0}
	%\ps@typeout{draft level now is \@psdraft \space . }
}
\def\psfull{
	\def\@psdraft{100}
	%\ps@typeout{draft level now is \@psdraft \space . }
}

\psfull

\newif\if@scalefirst
\def\psscalefirst{\@scalefirsttrue}
\def\psrotatefirst{\@scalefirstfalse}
\psrotatefirst

\newif\if@draftbox
\def\psnodraftbox{
	\@draftboxfalse
}
\def\psdraftbox{
	\@draftboxtrue
}
\@draftboxtrue

\newif\if@prologfile
\newif\if@postlogfile
\def\pssilent{
	\@noisyfalse
}
\def\psnoisy{
	\@noisytrue
}
\psnoisy
%%% These are for the option list.
%%% A specification of the form a = b maps to calling \@p@@sa{b}
\newif\if@bbllx
\newif\if@bblly
\newif\if@bburx
\newif\if@bbury
\newif\if@height
\newif\if@width
\newif\if@rheight
\newif\if@rwidth
\newif\if@angle
\newif\if@clip
\newif\if@verbose
\def\@p@@sclip#1{\@cliptrue}
%
%
\newif\if@decmpr
%
\def\@p@@sfigure#1{\def\@p@sfile{null}\def\@p@sbbfile{null}\@decmprfalse
   % look directly for file (e.g. absolute path)
   \openin1=\ps@predir#1
   \ifeof1
	\closein1
	% failed, search directories for file
	\get@dir{#1}
	\ifx\ps@founddir\leer
		% failed, search directly for file.bb
		\openin1=\ps@predir#1.bb
		\ifeof1
			\closein1
			% failed, search directories for file.bb
			\get@dir{#1.bb}
			\ifx\ps@founddir\leer
				% failed, lose.
				\ps@typeout{Can't find #1 in \figurepath}
			\else
				% found file.bb in search dir
				\@decmprtrue
				\def\@p@sfile{\ps@founddir\ps@dir#1}
				\def\@p@sbbfile{\ps@founddir\ps@dir#1.bb}
			\fi
		\else
			\closein1
			%found file.bb directly
			\@decmprtrue
			\def\@p@sfile{#1}
			\def\@p@sbbfile{#1.bb}
		\fi
	\else
		% found file in search dir
		\def\@p@sfile{\ps@founddir\ps@dir#1}
		\def\@p@sbbfile{\ps@founddir\ps@dir#1}
	\fi
   \else
	% found file directly
	\closein1
	\def\@p@sfile{#1}
	\def\@p@sbbfile{#1}
   \fi
}
%
%
%
\def\@p@@sfile#1{\@p@@sfigure{#1}}
%
\def\@p@@sbbllx#1{
		%\ps@typeout{bbllx is #1}
		\@bbllxtrue
		\dimen100=#1
		\edef\@p@sbbllx{\number\dimen100}
}
\def\@p@@sbblly#1{
		%\ps@typeout{bblly is #1}
		\@bbllytrue
		\dimen100=#1
		\edef\@p@sbblly{\number\dimen100}
}
\def\@p@@sbburx#1{
		%\ps@typeout{bburx is #1}
		\@bburxtrue
		\dimen100=#1
		\edef\@p@sbburx{\number\dimen100}
}
\def\@p@@sbbury#1{
		%\ps@typeout{bbury is #1}
		\@bburytrue
		\dimen100=#1
		\edef\@p@sbbury{\number\dimen100}
}
\def\@p@@sheight#1{
		\@heighttrue
		\dimen100=#1
   		\edef\@p@sheight{\number\dimen100}
		%\ps@typeout{Height is \@p@sheight}
}
\def\@p@@swidth#1{
		%\ps@typeout{Width is #1}
		\@widthtrue
		\dimen100=#1
		\edef\@p@swidth{\number\dimen100}
}
\def\@p@@srheight#1{
		%\ps@typeout{Reserved height is #1}
		\@rheighttrue
		\dimen100=#1
		\edef\@p@srheight{\number\dimen100}
}
\def\@p@@srwidth#1{
		%\ps@typeout{Reserved width is #1}
		\@rwidthtrue
		\dimen100=#1
		\edef\@p@srwidth{\number\dimen100}
}
\def\@p@@sangle#1{
		%\ps@typeout{Rotation is #1}
		\@angletrue
%		\dimen100=#1
		\edef\@p@sangle{#1} %\number\dimen100}
}
\def\@p@@ssilent#1{ 
		\@verbosefalse
}
\def\@p@@sprolog#1{\@prologfiletrue\def\@prologfileval{#1}}
\def\@p@@spostlog#1{\@postlogfiletrue\def\@postlogfileval{#1}}
\def\@cs@name#1{\csname #1\endcsname}
\def\@setparms#1=#2,{\@cs@name{@p@@s#1}{#2}}
%
% initialize the defaults (size the size of the figure)
%
\def\ps@init@parms{
		\@bbllxfalse \@bbllyfalse
		\@bburxfalse \@bburyfalse
		\@heightfalse \@widthfalse
		\@rheightfalse \@rwidthfalse
		\def\@p@sbbllx{}\def\@p@sbblly{}
		\def\@p@sbburx{}\def\@p@sbbury{}
		\def\@p@sheight{}\def\@p@swidth{}
		\def\@p@srheight{}\def\@p@srwidth{}
		\def\@p@sangle{0}
		\def\@p@sfile{} \def\@p@sbbfile{}
		\def\@p@scost{10}
		\def\@sc{}
		\@prologfilefalse
		\@postlogfilefalse
		\@clipfalse
		\if@noisy
			\@verbosetrue
		\else
			\@verbosefalse
		\fi
}
%
% Go through the options setting things up.
%
\def\parse@ps@parms#1{
	 	\@psdo\@psfiga:=#1\do
		   {\expandafter\@setparms\@psfiga,}}
%
% Compute bb height and width
%
\newif\ifno@bb
\def\bb@missing{
	\if@verbose{
		\ps@typeout{psfig: searching \@p@sbbfile \space  for bounding box}
	}\fi
	\no@bbtrue
	\epsf@getbb{\@p@sbbfile}
        \ifno@bb \else \bb@cull\epsf@llx\epsf@lly\epsf@urx\epsf@ury\fi
}	
\def\bb@cull#1#2#3#4{
	\dimen100=#1 bp\edef\@p@sbbllx{\number\dimen100}
	\dimen100=#2 bp\edef\@p@sbblly{\number\dimen100}
	\dimen100=#3 bp\edef\@p@sbburx{\number\dimen100}
	\dimen100=#4 bp\edef\@p@sbbury{\number\dimen100}
	\no@bbfalse
}
% rotate point (#1,#2) about (0,0).
% The sine and cosine of the angle are already stored in \sine and
% \cosine.  The result is placed in (\p@intvaluex, \p@intvaluey).
\newdimen\p@intvaluex
\newdimen\p@intvaluey
\def\rotate@#1#2{{\dimen0=#1 sp\dimen1=#2 sp
%            	calculate x' = x \cos\theta - y \sin\theta
		  \global\p@intvaluex=\cosine\dimen0
		  \dimen3=\sine\dimen1
		  \global\advance\p@intvaluex by -\dimen3
% 		calculate y' = x \sin\theta + y \cos\theta
		  \global\p@intvaluey=\sine\dimen0
		  \dimen3=\cosine\dimen1
		  \global\advance\p@intvaluey by \dimen3
		  }}
\def\compute@bb{
		\no@bbfalse
		\if@bbllx \else \no@bbtrue \fi
		\if@bblly \else \no@bbtrue \fi
		\if@bburx \else \no@bbtrue \fi
		\if@bbury \else \no@bbtrue \fi
		\ifno@bb \bb@missing \fi
		\ifno@bb \ps@typeout{FATAL ERROR: no bb supplied or found}
			\no-bb-error
		\fi
		%
%\ps@typeout{BB: \@p@sbbllx, \@p@sbblly, \@p@sbburx, \@p@sbbury} 
%
% store height/width of original (unrotated) bounding box
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\edef\ps@bbw{\number\count203}
		\edef\ps@bbh{\number\count204}
		%\ps@typeout{ psbbh = \ps@bbh, psbbw = \ps@bbw }
		\if@angle 
			\Sine{\@p@sangle}\Cosine{\@p@sangle}
	        	{\dimen100=\maxdimen\xdef\r@p@sbbllx{\number\dimen100}
					    \xdef\r@p@sbblly{\number\dimen100}
			                    \xdef\r@p@sbburx{-\number\dimen100}
					    \xdef\r@p@sbbury{-\number\dimen100}}
%
% Need to rotate all four points and take the X-Y extremes of the new
% points as the new bounding box.
                        \def\minmaxtest{
			   \ifnum\number\p@intvaluex<\r@p@sbbllx
			      \xdef\r@p@sbbllx{\number\p@intvaluex}\fi
			   \ifnum\number\p@intvaluex>\r@p@sbburx
			      \xdef\r@p@sbburx{\number\p@intvaluex}\fi
			   \ifnum\number\p@intvaluey<\r@p@sbblly
			      \xdef\r@p@sbblly{\number\p@intvaluey}\fi
			   \ifnum\number\p@intvaluey>\r@p@sbbury
			      \xdef\r@p@sbbury{\number\p@intvaluey}\fi
			   }
%			lower left
			\rotate@{\@p@sbbllx}{\@p@sbblly}
			\minmaxtest
%			upper left
			\rotate@{\@p@sbbllx}{\@p@sbbury}
			\minmaxtest
%			lower right
			\rotate@{\@p@sbburx}{\@p@sbblly}
			\minmaxtest
%			upper right
			\rotate@{\@p@sbburx}{\@p@sbbury}
			\minmaxtest
			\edef\@p@sbbllx{\r@p@sbbllx}\edef\@p@sbblly{\r@p@sbblly}
			\edef\@p@sbburx{\r@p@sbburx}\edef\@p@sbbury{\r@p@sbbury}
%\ps@typeout{rotated BB: \r@p@sbbllx, \r@p@sbblly, \r@p@sbburx, \r@p@sbbury}
		\fi
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\edef\@bbw{\number\count203}
		\edef\@bbh{\number\count204}
		%\ps@typeout{ bbh = \@bbh, bbw = \@bbw }
}
%
% \in@hundreds performs #1 * (#2 / #3) correct to the hundreds,
%	then leaves the result in @result
%
\def\in@hundreds#1#2#3{\count240=#2 \count241=#3
		     \count100=\count240	% 100 is first digit #2/#3
		     \divide\count100 by \count241
		     \count101=\count100
		     \multiply\count101 by \count241
		     \advance\count240 by -\count101
		     \multiply\count240 by 10
		     \count101=\count240	%101 is second digit of #2/#3
		     \divide\count101 by \count241
		     \count102=\count101
		     \multiply\count102 by \count241
		     \advance\count240 by -\count102
		     \multiply\count240 by 10
		     \count102=\count240	% 102 is the third digit
		     \divide\count102 by \count241
		     \count200=#1\count205=0
		     \count201=\count200
			\multiply\count201 by \count100
		 	\advance\count205 by \count201
		     \count201=\count200
			\divide\count201 by 10
			\multiply\count201 by \count101
			\advance\count205 by \count201
			%
		     \count201=\count200
			\divide\count201 by 100
			\multiply\count201 by \count102
			\advance\count205 by \count201
			%
		     \edef\@result{\number\count205}
}
\def\compute@wfromh{
		% computing : width = height * (bbw / bbh)
		\in@hundreds{\@p@sheight}{\@bbw}{\@bbh}
		%\ps@typeout{ \@p@sheight * \@bbw / \@bbh, = \@result }
		\edef\@p@swidth{\@result}
		%\ps@typeout{w from h: width is \@p@swidth}
}
\def\compute@hfromw{
		% computing : height = width * (bbh / bbw)
	        \in@hundreds{\@p@swidth}{\@bbh}{\@bbw}
		%\ps@typeout{ \@p@swidth * \@bbh / \@bbw = \@result }
		\edef\@p@sheight{\@result}
		%\ps@typeout{h from w : height is \@p@sheight}
}
\def\compute@handw{
		\if@height 
			\if@width
			\else
				\compute@wfromh
			\fi
		\else 
			\if@width
				\compute@hfromw
			\else
				\edef\@p@sheight{\@bbh}
				\edef\@p@swidth{\@bbw}
			\fi
		\fi
}
\def\compute@resv{
		\if@rheight \else \edef\@p@srheight{\@p@sheight} \fi
		\if@rwidth \else \edef\@p@srwidth{\@p@swidth} \fi
		%\ps@typeout{rheight = \@p@srheight, rwidth = \@p@srwidth}
}
%		
% Compute any missing values
\def\compute@sizes{
	\compute@bb
	\if@scalefirst\if@angle
% at this point the bounding box has been adjsuted correctly for
% rotation.  PSFIG does all of its scaling using \@bbh and \@bbw.  If
% a width= or height= was specified along with \psscalefirst, then the
% width=/height= value needs to be adjusted to match the new (rotated)
% bounding box size (specifed in \@bbw and \@bbh).
%    \ps@bbw       width=
%    -------  =  ---------- 
%    \@bbw       new width=
% so `new width=' = (width= * \@bbw) / \ps@bbw; where \ps@bbw is the
% width of the original (unrotated) bounding box.
	\if@width
	   \in@hundreds{\@p@swidth}{\@bbw}{\ps@bbw}
	   \edef\@p@swidth{\@result}
	\fi
	\if@height
	   \in@hundreds{\@p@sheight}{\@bbh}{\ps@bbh}
	   \edef\@p@sheight{\@result}
	\fi
	\fi\fi
	\compute@handw
	\compute@resv}
%
%
%
\def\OzTeXSpecials{
	\special{empty.ps /@isp {true} def}
	\special{empty.ps \@p@swidth \space \@p@sheight \space
			\@p@sbbllx \space \@p@sbblly \space
			\@p@sbburx \space \@p@sbbury \space
			startTexFig \space }
	\if@clip{
		\if@verbose{
			\ps@typeout{(clip)}
		}\fi
		\special{empty.ps doclip \space }
	}\fi
	\if@angle{
		\if@verbose{
			\ps@typeout{(rotate)}
		}\fi
		\special {empty.ps \@p@sangle \space rotate \space} 
	}\fi
	\if@prologfile
	    \special{\@prologfileval \space } \fi
	\if@decmpr{
		\if@verbose{
			\ps@typeout{psfig: Compression not available
			in OzTeX version \space }
		}\fi
	}\else{
		\if@verbose{
			\ps@typeout{psfig: including \@p@sfile \space }
		}\fi
		\special{epsf=\@p@sfile \space }
	}\fi
	\if@postlogfile
	    \special{\@postlogfileval \space } \fi
	\special{empty.ps /@isp {false} def}
}
\def\DvipsSpecials{
	%
	\special{ps::[begin] 	\@p@swidth \space \@p@sheight \space
			\@p@sbbllx \space \@p@sbblly \space
			\@p@sbburx \space \@p@sbbury \space
			startTexFig \space }
	\if@clip{
		\if@verbose{
			\ps@typeout{(clip)}
		}\fi
		\special{ps:: doclip \space }
	}\fi
	\if@angle
		\if@verbose{
			\ps@typeout{(clip)}
		}\fi
		\special {ps:: \@p@sangle \space rotate \space} 
	\fi
	\if@prologfile
	    \special{ps: plotfile \@prologfileval \space } \fi
	\if@decmpr{
		\if@verbose{
			\ps@typeout{psfig: including \@p@sfile.Z \space }
		}\fi
		\special{ps: plotfile "`zcat \@p@sfile.Z" \space }
	}\else{
		\if@verbose{
			\ps@typeout{psfig: including \@p@sfile \space }
		}\fi
		\special{ps: plotfile \@p@sfile \space }
	}\fi
	\if@postlogfile
	    \special{ps: plotfile \@postlogfileval \space } \fi
	\special{ps::[end] endTexFig \space }
}
%
% \psfig
% usage : \psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=}
%
% "clip=" is a switch and takes no value, but the `=' must be present.
\def\psfig#1{\vbox {
	% do a zero width hard space so that a single
	% \psfig in a centering enviornment will behave nicely
	%{\setbox0=\hbox{\ }\ \hskip-\wd0}
	%
	\ps@init@parms
	\parse@ps@parms{#1}
	\compute@sizes
	%
	\ifnum\@p@scost<\@psdraft{
		\PsfigSpecials 
		% Create the vbox to reserve the space for the figure.
		\vbox to \@p@srheight sp{
		% 1/92 TJD Changed from "true sp" to "sp" for magnification.
			\hbox to \@p@srwidth sp{
				\hss
			}
		\vss
		}
	}\else{
		% draft figure, just reserve the space and print the
		% path name.
		\if@draftbox{		
			% Verbose draft: print file name in box
			% 10/93 TJD changed to fbox from frame
			\hbox{\fbox{\vbox to \@p@srheight sp{
			\vss
			\hbox to \@p@srwidth sp{ \hss 
			        % 10/93 TJD deleted to avoid ``_'' problems
				% \@p@sfile
			 \hss }
			\vss
			}}}
		}\else{
			% Non-verbose draft
			\vbox to \@p@srheight sp{
			\vss
			\hbox to \@p@srwidth sp{\hss}
			\vss
			}
		}\fi	



	}\fi
}}
\psfigRestoreAt
\setDriver
\let\@=\LaTeXAtSign




