%Title: Multi-Loop Calculations, Quark Masses and the Strong Coupling Constant
%Author: J.H. K&uuml;hn
%Published:     <EM> Nucl. Phys. B (Proc.Suppl.), 183, 97-102, 2008 Contribution to "Loops and Legs in Quantum Field Theory, 9 th DESY Workshop on Elementary Particle Theory." Sondershausen, Germany, 20 - 25 April 2008 </EM> 
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\title{
\vskip-3cm{\baselineskip14pt
\centerline{\normalsize TTP 08-22\hfill}
\centerline{\normalsize SFB/CPP-08-36\hfill}}
\vskip1.5cm
Multi-Loop Calculations, Quark Masses and the Strong Coupling Constant}


\author{J.~H.~K\"uhn
\address{Institut f\"ur Theoretische Teilchenphysik,
  Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany}}
       
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\runtitle{Multi-Loop Calculations, Quark Masses and the Strong Coupling Constant}
\runauthor{J.~H.~K\"uhn}

\begin{document}

\begin{abstract}

The impact of multi-loop calculations on precise determinations of charm- and bottom-quark masses
and the strong coupling constant is discussed. Recent $N^3LO$ calculations, combined with new 
precision data, lead to a significant reduction of the errors. The results of these analyses, 
$m_c(3~\mbox{GeV}) = 0.986(13)~\mbox{GeV}$, $m_b(10~\mbox{GeV}) = 3.609(19)~\mbox{GeV}$ 
and $\alpha_s(M_Z)=0.1198(15)$
constitute the most precise and presently only $N^3LO$ determinations of these fundamental quantities.

\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

The strong coupling constant and the quark masses are the fundamental input
parameters of the theory of strong interaction.
Quark masses are an essential input for the evaluation of weak decay
rates of heavy mesons and for quarkonium spectroscopy. 
Decay rates and branching ratios of a light Higgs boson, suggested by
electroweak precision measurements, depend critically on the masses of
the charm and bottom quarks, $m_c$ and $m_b$. Last not least, 
confronting the predictions for these masses with experiment is an 
important task for all variants of Grand Unified Theories.

Similarly 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)xSU(2)xU(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.

Precise theoretical calculations, combined with excellent measurements, are
the decisive prerequisite for an optimal determination of these fundamental 
parameters. Advanced techniques of perturbative QCD have been developed 
during the last years, which have led to significantly improved determinations
of quark masses and the strong coupling constant. These techniques and a few
selected results will be presented in the following.\\[-8pt]

{\it Quark masses}\\[-10pt]

A detailed analysis of $m_c$ and $m_b$ based on the ITEP sum rules 
\cite{Novikov:1977dq} has been
performed several years ago \cite{Kuhn:2001dm} and lead to 
$m_c(m_c)=1.304(27)$~GeV and $m_b(m_b)=4.191(51)$~GeV. 
During the past years new and more precise data for 
$\sigma (e^+e^-\to\mbox{hadrons})$ 
have become available in the low energy region,
in particular for the parameters of the charmonium and bottomonium
resonances. Furthermore, the error in the strong coupling constant
$\alpha_s(M_Z)$ has been reduced. Last not least, the 
vacuum polarization induced by massive quarks has been
computed in four-loop
approximation \cite{Chetyrkin:2006xg,Boughezal:2006px}; more 
precisely: its first derivative at $q^2=0$ has been evaluated, which
corresponds to the lowest moment of the familiar $R$-ratio.
Based on these developments a new determination of the
quark masses has been performed in Ref.~\cite{Kuhn:2007vp}.

The extraction of $m_Q$ from low moments of the
cross section $\sigma(e^+e^-\to Q\bar{Q})$ exploits its sharp rise
close to the threshold for open charm and bottom production and the
importance of the contributions from the narrow quarkonium resonances. 
By evaluating the moments
\begin{eqnarray}
  {\cal M}_n^{\rm exp} &\equiv& \int \frac{{\rm d}s}{s^{n+1}} R_Q(s)
  \,,
  \label{eq::Mexp}
\end{eqnarray}
with low values of $n$, the long distance contributions are averaged
out and ${\cal M}_n$ involves short distance physics only, with a
characteristic scale of order $E_{\rm threshold}=2 m_Q$.
Through dispersion relations the moments are directly related to
derivatives of the vacuum polarization function at
$q^2=0$,
\begin{eqnarray}
  {\cal M}_n^{\rm theor} &=& \frac{12\pi^2}{n!}
  \left(\frac{{\rm d}}{{\rm d}q^2}\right)^n
  \Pi_Q(q^2)\Bigg|_{q^2=0}
  \,,
\label{eq::Mexp1}
\end{eqnarray}
which can be evaluated in perturbative QCD. 

In its domain of analyticity $\Pi_Q(q^2)$ can be cast into the form
\begin{eqnarray}
  \Pi_Q(q^2) &=& Q_Q^2 \frac{3}{16\pi^2} \sum_{n\ge0}
                       \bar{C}_n \left( \frac{q^2}{4m_Q^2} \right)^n
  \,,
  \label{eq::pimom}
\end{eqnarray}
where $m_Q=m_Q(\mu)$ is the $\overline{\rm MS}$ 
mass at the scale $\mu$.
The perturbative series for the coefficients $\bar{C}_n$ 
in order $\alpha_s^2$ was evaluated in 
Ref.~\cite{Chetyrkin:1995ii}, %Chetyrkin:1997mb}, 
the four-loop contributions
to $\bar{C}_0$ and $\bar{C}_1$ in
Refs.~\cite{Chetyrkin:2006xg,Boughezal:2006px}.
More recently even $\bar{C}_2$ has been evaluated \cite{Marq}.
Combining Eqs. (\ref{eq::Mexp}), (\ref{eq::Mexp1}) and  (\ref{eq::pimom}),
the charm quark mass can be obtained:
\begin{eqnarray}
  m_c(\mu) &=& \frac{1}{2} 
  \left(\frac{\bar{C}_n}{{\cal M}_n^{\rm exp}}\right)^{1/(2n)} 
  \,.
  \label{eq::mc1}
\end{eqnarray}

The moment with $n=1$ is least sensitive to non-perturbative
contributions from condensates, to the Coulombic higher order effects, the
variation of $\mu$ and the parametric $\alpha_s$ dependence \cite{Kuhn:2007vp}:
\begin{eqnarray}
  m_c(3~\mbox{GeV}) &=& 0.986(13)~\mbox{GeV}
  \,,
  \label{eq::mc3final}
\end{eqnarray}
Transforming this to the scale-invariant mass $m_c(m_c)$ one finds
$  m_c(m_c) = 1.286(13)~\mbox{GeV}$.
Using the three-loop 
relation~\cite{Chetyrkin:1999ys,Melnikov:2000qh}
between pole- and $\overline{\rm MS}$-mass this leads to
$M_c^{\rm(3-loop)} = 1.666~\mbox{GeV}$.

Instead of measuring the moments ${\cal M}_n^{\rm exp}$ in $e^+e^-$ 
annihilation they can also be determined in lattice simulations. This
approach has recently been pioneered in \cite{Allison:2008xk} using the Highly Improved 
Staggered Quarks (HISQ) discretization of the quark action \cite{Follana:2006rc,Follana:2007uv} to
simulate the moments of the pseudoscalar correlator. The final result,
$m_c(3~\mbox{GeV}) = 0.984(16)~\mbox{GeV}$ is in excellent agreement with the determinations based on 
$e^+e^-$ data.

%\begin{figure}[htb]
%\vspace{9pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
%\caption{Good sharp prints should be used and not (distorted) photocopies.}
%\label{fig:largenenough}
%\end{figure}

\begin{figure}[t]
  \begin{center}
    %\begin{tabular}{c}
      \leavevmode
      %\epsfxsize=0.95\textwidth
      \includegraphics[width=18pc]{mccmp.eps}   \\
      \includegraphics[width=18pc]{mbcmp.eps} 
      %\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
      %\epsffile{mbcmp.eps}
   % \end{tabular}
  \end{center}
  \vspace*{-2em}
  \caption{
    \label{fig::mb_compare}
    Comparison of recent determinations of $m_c(m_c)$ and $m_b(m_b)$. 
  }
\end{figure}

The approach based on $e^+e^-$ data is also applicable to the determination 
of $m_b$. The three results based on $n=1,2$ and 3 are of comparable
precision. The relative size of the contributions from the threshold
and the continuum region decreases for the moments $n=2$ and 3. On the
other hand, the theory uncertainty estimated from the variation of
$\mu$ and the unknown four-loop contribution
is still acceptable. Therefore 
the result from $n=2$ is taken as the final answer \cite{Kuhn:2007vp},
\begin{equation}
  m_b(10~{\rm GeV} ) = 3.609(25)~\mbox{GeV}\,,
  \label{eq::mbten}
\end{equation}
corresponding to  $m_b(m_b) = 4.164(25)~\mbox{GeV}$ and
a pole mass of $M_b^{\rm(3-loop)} = 4.800~\mbox{GeV}$.
The recent evaluation \cite{Marq} of $\bar{C}_2$ leads to a decrease
of the central value by $2~\mbox{MeV}$ and a reduction of the error from
$25~\mbox{MeV}$ to $19~\mbox{MeV}$.
A comparison of a few selected $m_b$- and $m_c$-determinations
is shown in Fig.~\ref{fig::mb_compare}.

For various applications, either related to $Z$-boson decays or in
connections to Grand Unified Theories (GUTs), the values of $m_b(\mu)$
at $M_Z$ and  $m_t(m_t) = 161.8 \pm 2.0~\mbox{GeV}$ (as derived from
$M_t=171.4\pm2.1~\mbox{GeV}$) are of interest: 
\bea
  m_b(M_Z) &=& 2.834 \pm 0.019 \pm 0.017~\mbox{GeV}\,\nonumber\\
  m_b(161.8) &=& 2.703 \pm 0.018 \pm 0.019~\mbox{GeV} \,.
\eea

{\it The strong coupling constant}\\[-10pt]

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.

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 
%conveniently 
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
%\eeq
\end{eqnarray}  
with $Q^2 = -q^2$. 
It is also convenient to introduce  the Adler function as
\bea
%\begin{array}{c}
%\nonumber
{D}(Q^2) =  -12\, \pi^2
Q^2\, \frac{\mathrm{d}}{\mathrm{d} Q^2} \Pi(Q^2)
=
\int \frac{ Q^2\ R(s) d s }{(s+Q^2)^2}\,,\nonumber
\eea
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.

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,Dine:1979,Gorishnii:1991vf,Surguladze:1990tg}, 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 next term of order $a_s^4 n_f^2$ was
evaluated in \cite{ChBK:vv:as4nf2}.

The complete five-loop calculation has been performed in \cite{PRL} 
using QGRAF \cite{Nogueira:1991ex} for the automatic generation
of Feynman diagrams and the field theoretical methods described in more detail in 
\cite{PRL,Vladimirov:1980zm,ChS:R*,gvvq,Baikov:tadpoles:96},
using a parallel version \cite{Tentyukov:2004hz} of FORM \cite{Vermaseren:2000nd}.

The so-called ``non-singlet'' diagrams, evaluated in \cite{PRL}, 
are sufficient for a complete description of
$\tau$-decays, for $e^+e^-$ annihilation and $Z$ decays they correspond to the dominant terms.

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
Using the Principle of Fastest Apparent Convergences \cite{Grunberg:1984fw} and the exact result 
for $d_4$, the coefficients
 $d_5$ and $r_5$ can be predicted (following \cite{Kataev:1995wd,ChBK:tau:as4nf2})  
%\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
\label{r5_pred}\,.\nonumber
\eea
%$d_5=275, 152, 89$ and $r_5= -505, -134, 168$.
% with an estimated error of $ \pm xx$. 
These estimates may become of relevance for the
International Linear Collider (ILC) running in the GIGA-Z mode with an
anticipated precision of \mbox{$\delta\alpha_s = 0.0005$ --- $0.0007$ \cite{Winter2001}}, and 
already today for the analysis of $\tau$-decays.

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 \cite{Kuhn:2007tc} $\alpha_s(9^2~\mathrm{GeV}^2)=0.160\pm 0.034$
 is a clear $3\sigma$ signal for the influence
of QCD corrections also in this energy range. Furthermore, as shown in this paper, proper 
inclusion of quark mass effects and matching at flavour threshold is required to obtain the
correct result.

Evolving this value 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 result 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 in the region between
3 and $10\mbox{GeV}$. The final result
$\alpha_s^{(4)}(9^2\mbox{GeV}^2)=0.182 ^{+0.022}_{-0.025}$
represents the combined information on the strong coupling from these $R$
measurements in the region below the bottom threshold. The shift in
$\alpha_s$ from the inclusion of the $\alpha_s^4$ term is small 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 situation is different for  $Z$ decays. 
The ${\cal O}(\alpha_s^3)$ analysis of the electroweak working  group \cite{Alcaraz:2007ri} 
finds
\beq
 \alpha_s(M_Z)^{NNLO} = 0.1185 \pm 0.0026^{\rm exp}
 \label{alpha_MZ_old}
\eeq
Including  the $\alpha_s^4$ term \cite{PRL} 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 can now safely be neglected.

Higher orders are of particular relevance in the low-energy region, for example
in $\tau$ decays. The correction   from perturbative QCD to the ratio 
$\Gamma(\tau\rightarrow {\rm hadrons}_{S=0}+\nu_\tau)/
\Gamma(\tau\rightarrow l+\bar{\nu}_l+\nu_\tau) $
is given by the factor 
%(see, e.g. \cite{Davier:2005xq})
\bea
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){}.\nn
\label{equivalent.repr}
\eea
%Using $\delta_2$ and $\delta_{NP}$

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}.
Starting from $\delta_0^{\rm exp}$%(\ref{delta_0_exp}) 
one easily obtains the  new values of $\alpha_s(M_\tau)$. Its   
dependence on the choice of $d_4$ (previously estimate and the new exact result)
is summarized in  Table 1.

\begin{table}[h]
\caption{\label{tab1}{\small
Results for  $\alpha_s(M_{\tau})$ for different  values of  $d_4$.
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 $\mu^2/M_{\tau}^2 = 0.4 - 2$.
}}
%\begin{ruledtabular}
{\footnotesize
\begin{tabular*}{\hsize}{l@{\extracolsep{0ptplus1fil}}c@{\extracolsep{0ptplus1fil}}r}
  \hline\hline       
        &
          $\alpha_s^{FO}(M_{\tau})$
        &
          $\alpha_s^{CI}(M_{\tau})$

          \\
%\colrule
        \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{ruledtabular}
\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 \cite{PRL}
\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})).  
Applying  four-loop running and matching \cite{vanRitbergen:1997va,Czakon:2004bu,Chetyrkin:2005ia,Schroder:2005hy} 
one finds 
\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 quark masses (
$m_c(m_c)=1.286(13)$ GeV and $m_b(m_b)= 4.164(25)$ GeV \cite{Kuhn:2007vp}), 
the matching scale, the four-loop truncation in the
matching expansion and in the RGE equation. 
 
 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}\nonumber\\
\alpha_s(M_Z)|_{\tau }  &=& 0.1202 \pm 0.0019
\label{alpha_Tau_new_fin}\nonumber
 \eea
from these two observables, 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, K. Chetyrkin, M. Steinhauser and C. Sturm 
for a fruitful and pleasant collaberation.
Work supported by SFB/TR-9 ``Computational Particle Physics'' .



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{thebibliography}{9}

%\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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\end{thebibliography}

\end{document}
