%Title: Precise Determinations of the Strong Coupling Constant
%Author: J.H. K&uuml;hn
%Published: <EM> Nucl. Phys. B (Proc.Suppl.) </EM> <B> 181 </B> (2008) (141-145). <EM> Contribution to "PHIPSI08, International Workshop on e+e- collisions from  Phi to Psi." Frascati, Italy, 7-10 April 2008.</EM> 
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\title{
\vskip-3cm{\baselineskip14pt
\centerline{\normalsize TTP 08-21\hfill}
\centerline{\normalsize SFB/CPP-08-37\hfill}}
\vskip1.5cm
Precise Determinations of the Strong Coupling Constant}

\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.
Combining the results from $Z$ and $\tau$ decays
we find $\alpha_s(M_Z)=0.1198 \pm  0.0015$ as one of the most precise and presently only
NNNLO result for the strong coupling constant.

\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

%    \begin{flushright}
%      \normalsize TTP/08-11\\
%      \normalsize SFB/CPP-08-18
%    \end{flushright}

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.

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 in \cite{PDG}) 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
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}, 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} 
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}, 
this direct and largely automatic procedure required enormous computing resources.

The so-called ``non-singlet'' diagrams, 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
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}) 
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
\label{r5_pred}\,.\nonumber
\eea
%$d_5=275, 152, 89$ and $r_5= -505, -134, 168$. 
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.
%
\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).

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 by BES~\cite{Bai:2001ct}, 
MD-1~\cite{Blinov:1993fw} and CLEO~\cite{Ammar:1997sk}.
Adding the correlated and uncorrelated errors of
the different experiments in
quadrature, % \footnote{We treat half of the systematic uncertainty
%  quoted in Ref.~\cite{Ammar:1997sk} as correlated to the new
%  measurements~\cite{unknown:2007qw} and the other half as
%  uncorrelated.},
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
measurements in the region below the bottom threshold. The shift in
$\alpha_s$ from the inclusion of the $\alpha_s^4$ term amounts to
$\delta \alpha_s(9 \mbox{GeV}) = 0.002$ 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 situation is different for  $Z$ decays. 
The analysis of the electroweak working  group \cite{Alcaraz:2007ri} 
is based on eq.~(\ref{R_numerical_nl})
with  $n_f=5$, including term  up to ${\cal O}(\alpha_s^3)$ and leads to
%The analysis of the $Z$ hadronic decays has led to a very precise prediction \cite{EWWG}
\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 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 
%$$R_{\tau} = \Gamma_{\rm semilept}/\Gamma_e $$
\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}.
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)
are summarized in  Table I.
%
\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
\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} 
to (\ref{als_tau}) we
arrive 
\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.

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}
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\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}}

\bibitem{PDG}
%  Particle Data Group
W.-M. Yao et al, Journal of Physics G33, 1 (2006)

\bibitem{Winter2001}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{M.~Winter}}
  (\bibinfo{year}{2001}), \eprint{ECFA-DESY Linear Collider Note
  LC-PHSM-2001-016}.

\bibitem{Allanach:2004ud}
  B.~C.~Allanach, G.~A.~Blair, S.~Kraml, H.~U.~Martyn, G.~Polesello, W.~Porod and P.~M.~Zerwas,
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