%Title: The Strong Coupling Constant at Low and High Energies
%Author: J.H. K&uuml;hn
%Published:     Contribution to "Continuous Advances in QCD, 2008" Minnepolis, USA, May 15 - 18, 2008; to be published in the proceedings. 
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\title{
\vskip-3cm{\baselineskip14pt
\centerline{\normalsize TTP 08-28\hfill}
\centerline{\normalsize SFB/CPP-08-48\hfill}}
\vskip1.5cm
The Strong Coupling Constant at Low and High Energies}

\author{J.~H.~K\"uhn
\address{Institut f\"ur Theoretische Teilchenphysik,
  Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany}
}
       
\begin{document}

\begin{abstract}

Recent results for the
cross section of electron-positron annihilation into hadrons and for the
decay rates of the $Z$ boson and the  $\tau$ lepton into hadrons including 
corrections of order  $\alpha_s^4$ are reviewed.
The consistency between two values  of $\alpha_s$ measured 
at vastly different energies constitutes 
a striking test of asymptotic freedom and requires the proper matching conditions
at charm and bottom thresholds.
Combining the results from $Z$ and $\tau$ decays
leads to $\alpha_s(M_Z)=0.1198 \pm  0.0015$ as one of the most precise and presently only
NNNLO result for the strong coupling constant. We report a recent determination of $\alpha_s$
which is based on a lattice evaluation of the pseudoscalar correlator for the charm quarks combined
with continuum perturbation theory.

\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle


The strong coupling constant $\alpha_s$ is one of the three
fundamental gauge couplings constants of the Standard Model (SM) of
particle physics.  Its precise determination is one of the most
important aims of particle physics. Experiments at different energies
allow to test the predictions for its energy dependence based on the
renormalization group equations, the comparison of the results
obtained from different processes leads to critical tests of the
theory and potentially to the discovery of physics beyond the Standard
Model. Last not least, the convergence of the three gauge coupling
constants related by  SU(3)$\times$SU(2)$\times$U(1) to a common value, after
evolving them to high energies, allows to draw conclusions about the
possibility of embedding the SM in the framework of a Grand Unified
Theory.

A characteristical example is shown in Fig.1, which demonstrates in particular the anticipated
improvement in our information about coupling constant unification at the GUT scale, if the
experimental uncertanty in $\alpha_s$ will be reduced from 0.0027 (corresponding to the value 
quoted by the ''Electroweak Working Group`` \cite{Alcaraz:2007ri}) to 0.0009, 
a precision that might be within reach with $10^9$ $Z$-events
at a GIGA-Z \cite{Winter2001}. In this latter case it may well be possible to deduce non-trivial 
constraints on threshold corrections and the structure of symmetry breaking at the GUT-scale.

One of the most precise and theoretically safe determination 
of $\alpha_s$ is based on
measurements of the cross section for electron-positron annihilation
into hadrons. These have been performed in the low-energy region between 2~GeV
and 10~GeV and, in particular, at and around the $Z$ resonance at 
%PDG07: Mass m = 91.1876 ± 0.0021 GeV  
91.2~GeV. 
Conceptually closely related is the measurement of the semileptonic
decay rate of the $\tau$-lepton, leading to a determination of $\alpha_s$ at
a scale below 2 GeV.

\begin{figure}[t]
      \leavevmode
      \epsfxsize=.36\textwidth
      \epsffile{Alpha_GUT.eps}
  \caption{\label{fig::al}  Renormalization group equation of the three coupling constants 
  $\alpha_1$, $\alpha_2$ and $\alpha_3$  \cite{Allanach:2004ud}       }
\end{figure}

From the theoretical side, in the framework of perturbative QCD, these rates
and cross sections are evaluated as inclusive rates into massless quarks and gluons 
\cite{ChKK:Report:1996,Davier:2005xq}.
(Power suppressed  mass effects are  well under control for
$e^+e^-$-annihilation,  both at low energies and around the $Z$ resonance, 
and for $\tau$ decays 
\protect\cite{Chetyrkin:1996ii,ChetKuhn90,ChKH:mq4as3,Baikov:2004ku}, 
and the same applies to mixed QCD and electroweak corrections \cite{Czarnecki:1996ei,Harlander:1997zb}).

The ratio 
$R(s)\equiv \sigma(e^+e^-\to {\rm hadrons}) / \sigma(e^+e^-\to\mu^+\mu^-)$ 
is expressed through the absorptive part of the 
correlator of the electromagnetic current $j_\mu$:
\begin{eqnarray}
R(s) &=& 12\pi\, \Im\Pi(-s - i\epsilon)
,
\\
3\, Q^2\, \Pi(Q^2) &=& 
i\int \dd^4 xe^{iq\cdot x}\langle 0|{\rm  T}j_\mu(x)j^\mu(0)|0\rangle
{},\nn\\
\end{eqnarray}  
with $Q^2 = -q^2$. 
It is also convenient to introduce  the Adler function as
\beq
{D}(Q^2) =  -12\, \pi^2
Q^2\, \frac{\mathrm{d}}{\mathrm{d} Q^2} \Pi(Q^2)
=
\int_0^\infty \frac{ Q^2\ R(s) d s }{(s+Q^2)^2}
\nonumber
{},
\eeq
%\beq
%R(s) = \ice{\frac{1}{2\pi i} \int_{-s -i\ep}^{-s +i\ep} \frac{d\, Q^2}{Q^2}=}
%D(s) - \pi^2 \beta_0^2 \{ \frac{ d_1}{3} a_s^3 + (d_2 + \frac{5}{6\,\beta_0 }d_1\,\beta_1)\,a_s^4\} + \dots
%{}\ .
%\eeq
Conversely, in the perturbative expansion $R(s)$ is given by
\bea
R(s) &=& \ice{\frac{1}{2\pi i} \int_{-s -i\ep}^{-s +i\ep} \frac{d\, Q^2}{Q^2}=}
D(s)- \pi^2 \beta_0^2 \{ \frac{ d_1}{3} a_s^3 \nn\\&& +(d_2 + \frac{5}{6\,\beta_0 }d_1\,\beta_1)\,a_s^4\}
 + \dots
{}\ .
\eea
Here we define  the perturbative  expansions 
\beq
{D}(Q^2) = \sum_{i=0}^{\infty} \  {d}_i a_s^i(Q^2),  \ \
{R}(s) =  \sum_{i=0}^{\infty} \  {r}_i a_s^i(s)
{},  
%\nonumber
\eeq
where $a_s \equiv \alpha_s/\pi$ and  we have set the normalization scale $\mu^2=Q^2$ or to $\mu^2= s$
for the Euclidian and Minkowskian functions respectively.
The results
for generic values of $\mu$ can be easily recovered with  standard
RG techniques.

Note that the first three terms of the perturbative series for $D$ and $R$ 
coincide. Starting from $r_3$, terms proportional $\pi^2$ arise which
can be predicted from those of lower order.
It has been speculated that these ``$\pi^2$-terms'', also  called ``kinematical  terms'', 
 might constitute a major
part of the full higher order corrections (see, e.g. \cite{Shirkov:2000iu}); 
however, the validity of this
hypothesis can only be established by the full calculation. Indeed, for the
scalar correlator this assumption has been shown to fail completely 
\cite{Baikov:2005rw}.

For the vector correlator the terms of order $a_s^2$ and $a_s^3$ have been
evaluated nearly thirty and about fifteen years ago 
\cite{Chetyrkin:1979bj,Gorishnii:1991vf,Chetyrkin:1996ez}, respectively. 
The $a_s^4$ corrections are conveniently classified according to their 
power of $n_f$, with $n_f$ denoting the number of light quarks.  
The $a_s^4 n_f^3$ term is part of the ``renormalon chain'', the evaluation of
the next term, of order $a_s^4 n_f^2$, was a test case for the techniques used
extensively in this paper and, furthermore, led to useful
insights into the structure of the perturbative series already 
\cite{ChBK:vv:as4nf2}.

The complete five-loop calculation has been performed in \cite{PRL}. 
It required the evaluation of 
about twenty thousand diagrams generated automatically via QGRAF \cite{Nogueira:1991ex}. 
Using ``infrared rearrangement''
\cite{Vladimirov:1980zm}, the
$R^*$ operation \cite{ChS:R*} and the prescriptions formulated in
\cite{gvvq} to algorithmically resolve the necessary combinatorics, it
was possible to express the absorptive part of the five-loop diagrams
in terms of four-loop massless propagator integrals. 
Using then a representation for Feynman integrals
proposed in \cite{Baikov:tadpoles:96,Baikov:explit_solutions:97},  these
integrals were reduced to a sum of 28 master integrals, with
coefficients which are rational functions of the space-time dimension
$D$. These coefficients were, in a first step, evaluated in the
large-$D$ limit, and, after evaluating sufficiently many terms of the
$1/D$ expansion, fully reconstructed \cite{Baikov:2005nv}.  This direct and
largely automatic procedure required enormous computing resources and
was performed on a 32+8-node SGI parallel computer with SMP architecture
and on the HP XC4000 supercomputer of the federal state Baden-W\"urttemberg 
using a parallel version \cite{Tentyukov:2004hz} of FORM 
\cite{Vermaseren:2000nd}.


The so-called ``non-singlet'' diagrams, 
where one and the same closed  quark line is connected to the external
currents, and which were evaluated in \cite{PRL},
are sufficient for a complete description of
$\tau$-decays. For $e^+e^-$ annihilation, they correspond to the dominant terms
proportional $\sum_i Q_i^2$. The singlet contributions proportional
$(\sum_i Q_i)^2$  arise for the first time in ${\cal O}(\alpha_s^3)$ and 
are known to be small. Similar comments apply to the singlet contributions in 
$Z$ decays.

The analytic result for the Adler function contains
transcendental numbers up to $\zeta_7$, with $\zeta_4$ and $\zeta_6$ being absent.
It leads straightforwardly  to  $R$ at order $\alpha_s^4$.
In  numerical form
\bea
R &=& 1  + a_s +  (1.9857 - 0.1152\, n_f)\, a_s^2 
\nonumber
\\
&+&
(-6.63694 - 1.20013 n_f - 0.00518 n_f^2 ) \, a_s^3
\label{R_numerical_nl}
\nonumber
\\
&+&(-156.61 + 18.77\, n_f - 0.7974\, n_f^2 
\\
\nonumber
&+& 0.0215\,  n_f^3 ) \, a_s^4
{}.
%
\eea


It is also instructive to explicitly display  the genuine five-loop contributions to  $d_4$
(underlined in (\ref{r_3_separated},\ref{r_4_separated}) below) and  the  ``kinematical'' terms
originating from the analytic continuation:
\bea
r_3 &=& \unl{18.2} - 24.9  + (\unl{-4.22} + 3.02)\, n_f 
\label{r_3_separated}
\\
&{+}& (\unl{-0.086} + 0.091)\, n_f^2
\nonumber
{},
\\
r_4 &=& \unl{135.8} - 292.4  + (\unl{-34.4} + 53.2)\, n_f
\label{r_4_separated} 
\\
&+&(\unl{1.88} - 2.67)\, n_f^2 + (\unl{-0.010} + 0.032)\, n_f^3
\nonumber
{}.
\eea


Since it will presumably take a long time until the next term of the
perturbative series will be evaluated,
%exactly,
it is of interest to investigate
the predictive power of various optimization schemes empirically. Using the
principles of ``Fastest Apparent Convergence'' (FAC)  \cite{Grunberg:1984fw} 
or of ``Minimal
Sensitivity'' (PMS) \cite{Stevenson:1981vj}, which happen to coincide in this order, the central  values of 
the  predictions
\cite{Kataev:1995vh,ChBK:tau:as4nf2}
$$
d_4^{\rm pred}(n_f=3,4,5) = 27 \pm 16, 8 \pm 18,-8 \pm 44
$$
differ significantly from the exact result 
\beq
d_4^{\rm exact}(n_f=3,4,5) = 49.08,\ \ 27.39,\ \ 9.21
{}.
\eeq
However, within the error estimates \cite{ChBK:tau:as4nf2} predicted and exact values are  
in agreement. 
The picture changes, once these estimates are used to predict the
coefficient $r_4$. Although sizable cancellations between ``dynamical'' and
``kinematical'' terms are observed for the individual $n_f$ coefficients in
(\ref{r_4_separated})
the predictions  for the final results
are significantly closer (in  relative sense) to the results of the exact calculation:
\bea
r_4^{\rm pred}(n_f=3,4,5) &=& 
-129 \pm 16,\   -112 \pm 30, \nn\\&&-97 \pm 44
{},
\nonumber
\\
\nonumber
r_4^{\rm exact}(n_f=3,4,5) &=& - 106.88, \ \ \  -92.898, \nn\\&& -79.98
{}.
\eea
(This is in striking contrast to the case  of the scalar correlator, where the
predictions for the dynamical terms work well, but, as a consequence of the
strong cancellations between dynamical and kinematical terms fail completely
in the Minkowskian region.) Using FAC and the exact result for $d_4$, the
coefficients $d_5$ and $r_5$ can be predicted (following \cite{ChBK:tau:as4nf2,Kataev:1995wd}) 
for $n_f=3,4,5$, namely 
%\cite{Kataev:1995wd,ChBK:tau:as4nf2}
\bea
d_5^{\rm pred}(n_f=3,4,5) &=& 
275,\ \  152,  \ \ 89
{},
\label{d5_pred}
\\
r_5^{\rm pred}(n_f=3,4,5) &=& -505, \ \ -134,\ \  168\nn\\
\label{r5_pred}
{}.
\eea
These terms may become of relevance for the
International Linear Collider (ILC) running in the GIGA-Z mode with an
anticipated precision of \mbox{$\delta\alpha_s = 0.0005$ --- $0.0007$ \cite{Winter2001}}, and 
already today for the analysis of $\tau$-decays.

%
\begin{figure}[t]
  \begin{center}
    \begin{tabular}{c}
      \leavevmode
      \epsfxsize=.5\textwidth
      \epsffile{r.eps}
    \end{tabular}
  \end{center}
  \caption{\label{fig::R}
    $R(s)$ as measured by
    CLEO ~\cite{Ammar:1997sk,unknown:2007qw}.
    The full solid line and the hatched band correspond to
    the theory prediction based on $\alpha_s(M_Z^2) = 0.110^{-0.014}_{-0.017}$.
    The dashed line represents theory prediction where results from the
    CLEO analysis ~\cite{unknown:2007qw}, $\alpha_s^{(5)}(M_Z)=0.126$, has
    been used.
          }
\end{figure}

Let us now move to the analysis of present data for $e^+e^-$ annihilation and $\tau$ decays.
Measurements of $R(s)$ at lower energies, with their correspondingly larger values of 
$\alpha_s$, are in principle more sensitive to $\alpha_s(M_Z)$ if the same relative precision
could be obtained. At present, however, the systematic experimental error of 2\% is a limiting
element for a competitive measurement. Nevertheless, it is interesting, that the result of a recent 
analysis $\alpha_s(9^2~\mathrm{GeV}^2)=0.160\pm 0.034$ \cite{Kuhn:2007tc} is a clear $3\sigma$ signal for the influence
of QCD corrections also in this energy range. Furthermore, as shown in \cite{Kuhn:2007tc}, proper 
inclusion of quark mass effects and matching at flavour threshold is required to obtain the
correct result (Fig. 2). A convenient FORTRAN program, 
called RHAD, which predicts $R_{\rm had}(s)$ and where all quark mass effects and
higher order QCD corrections
are implemented, is described in \cite{Harlander:2002ur}.
Ignoring quark mass effects and proper matching conditions \cite{Ammar:1997sk}, leads to a shift
of $\alpha_s$ by one standard deviation, as shown in Fig. 2 (dashed line). 

Evolving the result from 9 GeV to $M_Z$ leads to $\alpha_s(M_Z^2)=0.110^{+0.014}_{-0.017}$.
It is instructive to combine this value with
the one obtained in Ref.~\cite{Kuhn:2001dm}, 
$\alpha_s^{(4)}(5^2\mbox{GeV}^2)=0.235 ^{+0.047}_{-0.047}$, which
was based on earlier measurements by BES~\cite{Bai:2001ct}, 
MD-1~\cite{Blinov:1993fw} and CLEO~\cite{Ammar:1997sk} in the region
between 3 and 10 GeV.
Adding the correlated and uncorrelated errors of
the different experiments in
quadrature, the final result
\beq
\alpha_s(9^2\mbox{GeV}^2)^{NNLO}=0.182 ^{+0.022}_{-0.025}
\label{alpha_9gev}
\eeq
represents the combined information on the strong coupling from
measurements in the region below the bottom threshold. The shift in
$\alpha_s$ from the inclusion of the $\alpha_s^4$ term amounts to
$\delta \alpha_s(9^2 \mbox{GeV}^2) = 0.002$ and is thus irrelevant compared to
the large experimental error.

In contrast to the situation below 10 GeV the extraction of $\alpha_s$ from $Z$ decays is 
affected by the $\alpha_s^4$ terms.
The analysis of the electroweak working  group \cite{Alcaraz:2007ri} 
is based on eq.~(\ref{R_numerical_nl})
with  $n_f=5$, including terms  up to ${\cal O}(\alpha_s^3)$ and leads to
\beq
 \alpha_s(M_Z)^{NNLO} = 0.1185 \pm 0.0026^{\rm exp}{}.
 \label{alpha_MZ_old}
\eeq
Including  the $\alpha_s^4$ term leads to a shift   
$\delta \alpha_s(M_Z) = 0.0005$ 
\beq
 \alpha_s(M_Z)^{NNNLO} =  0.1190 \pm {0.0026}^{\rm exp}
{}.
\label{alpha_MZ_new}
\eeq

The theory  error may either be conservatively based on the shift produced  by the last term  (0.0005) or
on the scale variation with $\mu/M_Z = \frac{1}{3} \div 3$,
leading to $\pm 0.0002$  and can now be neglected  in both cases.

Higher orders are of particular relevance in the low-energy region, for example
in $\tau$ decays. The correction   from perturbative QCD to the ratio 
%$$R_{\tau} = \Gamma_{\rm semilept}/\Gamma_e $$
\beq
\begin{array}{c}
R_{\tau,V+A} = \frac{\Gamma(\tau\rightarrow{\rm hadrons}_{S=0}
  +\nu_\tau)}{\Gamma(\tau\rightarrow l+\bar\nu_l+\nu_\tau)}
\\
= 3 |V_{ud}|^2 S_{EW}(1+\delta_0 + \delta'_{EW}+ \delta_2+ \delta_{NP})
%= R_{\tau,NS}  + R_{\tau,S}
\end{array}
\label{Rtau1}
\eeq
is given by 
%(see, e.g. \cite{Davier:2005xq})
\beq     
1+\delta_0 =
2\int_0^{M_{\tau}^2}\frac{ds}{M_{\tau}^2}
\left(1-\frac{s}{M_{\tau}^2}\right)^2
\left( 1+ \frac{2 s }{M_\tau^2}
\right)
\, R(s)
{}.
\label{equivalent.repr}
\eeq

The  perturbative quantity $\delta_0 $ can be evaluated in Fixed Order perturbation theory
 or with ``Contour Improvement'' as proposed in \cite{Pivovarov:1991rh,LeDiberder:1992te}
\bea
\delta_0^{FO} &=&   a_s + 5.202 \ a_s^2 +  26.366 \ a_s^3  +127.079 \, a_s^4 
\label{Rtau_FO}
{},\nonumber\\
\\
\delta_0^{CI} &=& 
 1.364 \, a_s+ 2.54 \, a_s^2 + 9.71 \, a_s^3
 + 64.29 \, a_s^4
{}.\nonumber\\
\label{Rtau_CI}
\eea
From the experimental side
$\delta_0^{\rm exp}= 0.1998  \pm 0.0043_{\rm exp}
%{0.199773, 0.00426348}
\label{delta_0_exp}$
has been used \cite{Davier:2007ym}.
More recent evaluations of the experimental input correspond to slightly larger values
$\delta_0^{\rm exp}=0.2066\pm 0.0070$ \cite{Davier:2008sk} 
and $\delta_0^{\rm exp}=0.2042\pm 0.0050$ \cite{Beneke:2008ad}, but
are still consistent with this previous value.
Starting from $\delta_0^{\rm exp}$ and using eq.~(\ref{Rtau_FO}) and eq.~(\ref{Rtau_CI})
respectively, one easily obtains $\alpha_s(M_\tau)$. Its   
dependence on the choice of $d_4$ (previous estimate and the new exact result)
is summarized in  Table I.
%
\begin{table}[h]
\caption{\label{tab1}{\small
Results for  $\alpha_s(M_{\tau})$ for different  values of  $d_4$, extracted from eq.~(\ref{Rtau_FO}).
The first line displays the $\alpha_s^3$ results (with the $\alpha_s^4$ terms
set to zero in eqs.~(\ref{Rtau_FO},\ref{Rtau_CI})).
The second  line  uses the previously predicted  value for $d_4$,
the last  one uses  the exact result (\ref{R_numerical_nl}).     
The first error is the experimental one;
the second (theoretical) uncertainty in the value of $\alpha_s$ corresponds
to changing the  renormalization scale  $\mu$ as follows $\mu^2/M_{\tau}^2 = 0.4 - 2$.
}}
{\footnotesize
\begin{tabular*}{\hsize}{l@{\extracolsep{0ptplus1fil}}c@{\extracolsep{0ptplus1fil}}r}
  \hline\hline       
        &
          $\alpha_s^{FO}(M_{\tau})$
        &
          $\alpha_s^{CI}(M_{\tau})$

          \\
        \hline
          %-\!\!-
          &
           $0.337 \pm 0.004\ice{_{\rm exp}} \pm 0.03$ \hspace*{5pt}
          &
          $0.354  \pm 0.006\ice{_{\rm exp}} \pm 0.02$ 
          \\
%        \hline
         $d_4 = 25$ \hspace*{3pt}
          &
        $0.325 \pm 0.004\ice{_{\rm exp}} \pm 0.02$ \hspace*{5pt}
          &
         $0.347 \pm 0.006\ice{_{\rm exp}} \pm 0.009$ 
       \\
       $d_4 = 49.08$  \hspace*{3pt}    
        &
     $0.322 \pm 0.004\ice{_{\rm exp}} \pm  0.02 $ \hspace*{5pt}
         &
     $0.342  \pm 0.005\ice{_{\rm exp}} \pm  0.01$ 
       \\
  \hline\hline
\end{tabular*}
}
\end{table}

As stated above the theory error for $\alpha_s$ from Z  decays is small compared to the experimental
uncertainties.  The situation is more problematic for $\tau$ decays and to some extent the 
theory error remains to be  matter of choice.  As anticipated in \cite{ChBK:tau:as4nf2} it
decreases significantly, once $\alpha_s^4$
terms are included. However, the difference between the two methods stabilizes
(this was checked in \cite{ChBK:tau:as4nf2} by adding an estimate for the   $\alpha_s^5$ term) 
and must be considered as irreducible uncertainty, with the final result
\beq
\alpha_s(M_\tau) =0.332 \pm 0.005_{\rm exp}      \pm 0.015_{\rm th} 
{}.
\label{als_tau}
\eeq
For the central value the mean value of FO and CI has been used,
for the theory error half of the difference between
two methods (0.01) plus (module of) the estimated correction from $\alpha_s^5$ term 
(-0.005), the latter being based on $d_5 = 275$ (see eq.~(\ref{d5_pred})).  

Recently it has been argued \cite{Davier:2008sk}, that predictions based on FO are unstable and should
therefore be abandoned. Using CI only and the new value for $d_4$, and assuming small higher
order terms in the expansion of the Adler function, leads to a relatively large value
$\alpha_s(m_\tau)=0.344\pm 0.005_{\rm exp}\pm 0.007_{\rm th}$ and an optimistic theory error.
This line of reasoning is in striking contrast with a new study performed in \cite{Beneke:2008ad}. 
In this paper the Adler function series is
described by a model which incorporates the exactly known coefficients up to $d_4$ and theory
constraints on the large-order behaviour and it is argued, that FO smoothly approaches the
correct asymptotic result. The result 
$\alpha_s(m_\tau)=0.320\pm 0.003_{\rm exp}{}^{+0.011}_{-0.006}$
is significantly lower than the previous one. As expected the two results, taken at face value, 
are in agreement with the result for FO and CI respectively, as listed in Table 1. On the other hand,
they are mutually exclusive. Individually, after evolving $\alpha_s$ to $M_Z$ the associated error 
corresponds to about or less than $10^{-4}$ and the respective result would completely dominate
all determinations of $\alpha_s$. For the time being it seems prudent to stay with  eq.~(\ref{als_tau})
which takes the difference between FO and CI into account.

Applying  four-loop running and matching \cite{vanRitbergen:1997va,Czakon:2004bu,Chetyrkin:2005ia,Schroder:2005hy} 
as implemented in the updated version of the MATHEMATICA-routine RunDec 
\cite{Chetyrkin:2000yt} to (\ref{als_tau}) we arrive at
\bea
\label{eq:asres_mz}
   \as(M_Z) &=& 0.1202 \pm 0.0006_{\rm exp}
                                       \pm 0.0018_{\rm th}\nonumber\\
                                       &&\pm 0.0003_{\rm evol}~, \nonumber \\
                     &=& 0.1202  \pm  0.0019 
{}.
\eea
Here the evolution error receives 
contributions from the uncertainties in the $c$-quark mass (0.00003,
$m_c(m_c)=1.286(13)$ GeV \cite{Kuhn:2007vp}) and the $b$-quark mass 
(0.00001, $m_b(m_b)= 4.164(25)$ GeV \cite{Kuhn:2007vp}), 
the matching scale (0.0001, $\mu$ varied between
$0.7\,m_q(m_q)$ and $3.0\,m_q(m_q)$), the four-loop truncation in the
matching expansion (0.0001) and the four-loop truncation in the RGE
equation (0.0003). (For the last two errors the size of the shift due
the highest known perturbative term was treated as systematic
uncertainty.)  The errors had been added in quadrature.

A recent, completely different determination of $\alpha_s$ is based on the combination of
lattice simulation and continuum pertubation theory \cite{Allison:2008xk}. 
Derivatives of the correlator of two %pseudoscalar 
charm-quark currents $m_c\overline{\psi}\gamma_5\psi$ were evaluated at  
vanishing momentum $q^2$. Relating lattice and perturbative results for the same quantity
then allowed for a determination of $m_c$ and $\alpha_s$. Specifically,
higher moments, corresponding to higher 
derivatives with respect to $q^2$, were used for an accurate determination of $m_c$.
From the second derivative, being dimensionless, a value $\alpha_s(M_Z)=0.113(4)$ was obtained.
Although this method cannot yet compete with the determinations discussed above, the combination
of lattice and continuum methods may well lead to precission determinations of $\alpha_s$.

As shown in Fig. \ref{fig::al_s}, it is instructive to compare the results of $\alpha_s$ 
measurements at $m_\tau$, at intermediate energies around 9 GeV and at $M_Z$ directly with the
predictions based on four-loop running and matching, starting from a value 
$\alpha_s(M_Z)=0.1198\pm 0.0015$. Theory prediction and the three experimental results
are in remarkable agreement.

\begin{figure}[t]
  \begin{center}
    \begin{tabular}{c}
      \leavevmode
      \epsfxsize=.5\textwidth
      \epsffile{as.eps}
    \end{tabular}
  \end{center}
  \caption{\label{fig::al_s}
    Comparison of the experimental results eqs.~(\ref{alpha_9gev}), (\ref{alpha_MZ_new}) and (\ref{als_tau})  
    with the four-loop evaluation
    and a starting value $\alpha_s(M_Z)=0.1198\pm 0.0015$
          }
\end{figure}


{\em Summary}: The exact result for the $\alpha_s^4$ term in the Adler
function allows to extract the strong coupling constant from $Z$ and
$\tau$ decays with high precision. Including the exact $\alpha_s^4$
%and an estimate for the $\alpha_s^5$ term 
leads to small shifts of the
central value and to a significant reduction of the theory
uncertainty. Note that the shifts in $\alpha_s(M_Z)$
from Z and $\tau$ decays,
are opposite in sign and move  the values in the  {\em proper} direction,  
decreasing, thus, the current slight mismatch between
two independent  determinations of $\alpha_s$. The final results 
\bea
\alpha_s(M_Z)|_{Z}  &=& 0.1190 \pm 0.0026\,,\\
\label{alpha_MZ_new_fin}
\alpha_s(M_Z)|_{\tau }  &=& 0.1202 \pm 0.0019\,,
\label{alpha_Tau_new_fin}
 \eea
from these two observables, although based on measurements of
vastly different energy scales, are in remarkable agreement. This constitutes 
a striking test    of asymptotic freedom in QCD. The two values can be combined to
\[
\alpha_s(M_Z) =0.1198 \pm  0.0015
{}.
 \] 
This  is one of the most precise and presently only $N^3LO$
result. \\

{\it Acknowledgement}\\
I would like to thank P. Baikov and K. Chetyrkin for a fruitful and pleasant collaberation.
Work supported by SFB/TR-9 ``Computational Particle Physics'' and by the High Performance 
Computing Center Stuttgart (HLRS) under the grant ``ParFORM''.


\begin{thebibliography}{42}
%\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}


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  \bibinfo{author}{\bibfnamefont{J.~H.} \bibnamefont{K{\"u}hn}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Sturm}},
  Nucl. Phys. B{\bf 744} (2006) 121,
  %(\bibinfo{year}{2005}), 
  \eprint{hep-ph/0512060}.

\bibitem{Schroder:2005hy}
\bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Schroder}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Steinhauser}},
  \bibinfo{journal}{JHEP} \textbf{\bibinfo{volume}{01}}, \bibinfo{pages}{051}
  (\bibinfo{year}{2006}), 
  \eprint{hep-ph/0512058}.

%\cite{Chetyrkin:2000yt}
\bibitem{Chetyrkin:2000yt}
  K.~G.~Chetyrkin, J.~H.~K\"uhn and M.~Steinhauser,
  %``RunDec: A Mathematica package for running and decoupling of the strong
  %coupling and quark masses,''
  Comput.\ Phys.\ Commun.\  {\bf 133} (2000) 43
  [arXiv:hep-ph/0004189].
  %%CITATION = CPHCB,133,43;%%

%50
\bibitem{Kuhn:2007vp}
\bibinfo{author}{\bibfnamefont{J.~H.} \bibnamefont{K{\"u}hn}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Steinhauser}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Sturm}},
  \bibinfo{journal}{Nucl. Phys.} \textbf{\bibinfo{volume}{B778}},
  \bibinfo{pages}{192} (\bibinfo{year}{2007}{\natexlab{b}}),
  \eprint{hep-ph/0702103}.

%\cite{Allison:2008xk}
\bibitem{Allison:2008xk}
  I.~Allison {\it et al.},
  %``High-Precision Charm-Quark Mass from Current-Current Correlators in Lattice
  %and Continuum QCD,''
  arXiv:0805.2999 [hep-lat].
  %%CITATION = ARXIV:0805.2999;%%

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