\documentclass[12pt]{article}
\usepackage{a4}
\usepackage{latexsym} % Gets \Box etc
\usepackage{amssymb}  % \gtrsim, \geqslant, etc etc: 
\usepackage{amsmath} 
\usepackage{graphicx}
\usepackage{epsfig,wrapfig}
\usepackage[ usenames]{color}
\usepackage[rflt]{floatflt} 
    \textheight     23cm
    \textwidth      16.5cm
    \marginparwidth 0.80in
    \oddsidemargin  -0.5cm
    \evensidemargin -0.5cm
    \topmargin -1.cm

%    \textheight     8.66in
%    \textwidth      5.70in
%    \marginparwidth 0.80in
%    \oddsidemargin  0.20in
%    \evensidemargin 0.20in
%        
\begin{document}

%\voffset-3cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% several bibliographies in one text file
% the series of new commands, tested on LaTeX2e and on Latex2.09
% via the compatibility mode.
% version redefining thebibliography and cite.
% J. Bijnens 18 Sep. 1998
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\talk}[3]
{\noindent{#1}\\ \mbox{}\ \ \ {\it #2} \dotfill {\pageref{#3}}\\[1.8mm]}
\newcommand{\stalk}[3]
{{#1} & {\it #2} & {\pageref{#3}}\\}
\newcommand{\snotalk}[3]
{{#1} & {\it #2} & {{#3}n.r.}\\}
\newcommand{\notalk}[3]
{\noindent{#1}\\ \mbox{}\ \ \ {\it #2} \hfill {{#3}n.r.}\\[1.8mm]}
\newcounter{zyxabstract}     %  step by one when going to new abstract
\newcounter{zyxrefers}        %  counts the references in every bibliography

\newcommand{\newabstract}
{\clearpage\stepcounter{zyxabstract}\setcounter{equation}{0}
\setcounter{footnote}{0}\setcounter{figure}{0}\setcounter{table}{0}}

\newcommand{\talkin}[2]{\newabstract\label{#2}\input{#1}}                

\newcommand{\rlabel}[1]{\label{zyx\arabic{zyxabstract}#1}}
\newcommand{\rref}[1]{\ref{zyx\arabic{zyxabstract}#1}}

\renewenvironment{thebibliography}[1] % as standard,the width of largest number
{\section*{References}\setcounter{zyxrefers}{0}
\begin{list}{ [\arabic{zyxrefers}]}{\usecounter{zyxrefers}}}
{\end{list}}
% for those who really need every possible mm:
\newenvironment{thebibliographynotitle}[1] % as 
%                              standard,the width of largest number
{\setcounter{zyxrefers}{0}
\begin{list}{ [\arabic{zyxrefers}]}
{\usecounter{zyxrefers}\setlength{\itemsep}{-2mm}}}
{\end{list}}

\renewcommand{\bibitem}[1]{\item\rlabel{y#1}}% extra y to avoid duplication
\renewcommand{\cite}[1]{[\rref{y#1}]}      %of labels in text and bibliography
\newcommand{\citetwo}[2]{[\rref{y#1},\rref{y#2}]}
\newcommand{\citethree}[3]{[\rref{y#1},\rref{y#2},\rref{y#3}]}
\newcommand{\citefour}[4]{[\rref{y#1},\rref{y#2},\rref{y#3},\rref{y#4}]}
\newcommand{\citefive}[5]
{[\rref{y#1},\rref{y#2},\rref{y#3},\rref{y#4},\rref{y#5}]}
\newcommand{\citesix}[6]
{[\rref{y#1},\rref{y#2},\rref{y#3},\rref{y#4},\rref{y#5},\rref{y#6}]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\documentclass[12pt]{article}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% PROCEEDINGS
%
% All review speakers and the speakers who gave individual talks shall provide a
% maximal 2 page proceedings contribution.  Review speakers should give a brief
% introduction to their subject. Individual speakers should submit  in a concise
% form the relevant results and statements of the author and/or the group he or
% she represents.   We also invite the session conveners to submit a 1 or 2 page
% summary of the most important points that came out of their session, including
% places where there are still disagreements.  Everyone is welcome to include
% sentences that comment about the results of other groups. These contributions
% will be collected in a web proceedings that shall be submitted to the eprint
% arXiv, with the organizers plus all people who make proceedings contributions
% listed alphabetically as authors.  
% 
% Please use the liked template to prepare your contribution and send your
% contribution back to the alphas mail account (alphas@mpp.mpg.de) by March 15, 2011.  
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%     RULES:
%     
%     Page limit is 2 (two) for each contribution.
%
%     Do not use sections or subsections.
%
%     If you really need to include pictures use the includegraphics command
%     such as (figure size is up to you)
%
%      \begin{center}
%      \includegraphics[height=5.cm]{epsfilename.eps} 
%      \end{center}
%
%     You can create tables as you like.
% 
%     Please do you use self-created makros.
%    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%            DO NOT CHANGE ANYTHING ABOVE THIS LINE !!!!!
%
%           YOUR CONTRIBUTION STARTS ONLY BELOW THIS LINE !!!!!
%         
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{center}
{\large\bf Running and decoupling of $\alpha_s$}\\[0.5cm]
Matthias Steinhauser\\[0.3cm]
Karlsruhe Institute of Technology
\end{center}

The $\beta$ function governs the renormalization scale dependence of the
strong coupling and is known to four-loop
accuracy since almost 14~years~\cite{Vermaseren:1997fq}\cite{Czakon:2004bu}. 
There are different ways to solve the corresponding differential
equation. The prefered method is the numerical solution with truncation of 
$\beta(\alpha_s)$ at the desired order. There are also several approximate
(analytical) expressions, e.g., the one based on the iterative (perturbative)
solution where the result for $\alpha_s(\mu)$ is given as an expansion in
$1/L = 1/\ln(\mu^2/\Lambda^2)$~\cite{PDG}. This formula should be used with
care, in particular for small renormalization scales $\mu$. If one considers,
e.g., $\mu=M_\tau$ one observes a shift of $+0.004$ after including the
four-loop corrections and negative shift of approximately the same order of
magnitude at five-loop level. These numbers have to be compared with the
current experimental precision which is cited as $\pm0.005$ in
Ref.~\cite{Baikov:2008jh} (see also the other contributions on $\alpha_s$ from
$\tau$ decays in these proceedings).

Next to the running itself also the decoupling of heavy quarks form the
running of the strong coupling constant is a crucial ingredient of the
precision determination of $\alpha_s$. Every time a heavy quark threshold is
crossed one has to apply the decoupling constants which relate $\alpha_s$ with
$n_f$ active quark flavours, usually denoted by $\alpha_s^{(n_f)}$,
to the coupling with only $n_f-1$ active quark flavours. 
The decoupling constants are obtained by matching $n_f$-flavour QCD
to the effective theory with the number of quarks equal to $n_f-1$. The theoretical
framework for the calculation of the decoupling constants has been set up in 
Ref.~\cite{Chetyrkin:1997un} where formulae are given relating 
$l$-loop corrections to $l$-loop vacuum integrals.

%\begin{table}[hb]
  \begin{center}
    \includegraphics[width=.9\textwidth,height=.4\textwidth]{steinhauser_fig1.eps} 
  \end{center}
%  \caption{\label{fig::dec}}
%\end{table}

As a consequence of the decoupling relations $\alpha_s(\mu)$ is not a
continuous function of $\mu$ but has finite steps at the energy scale where
the heavy quark is integrated out, $\mu_{\rm dec}$. This energy is not fixed
by theory, should, however, be in the vicinity of the heavy quark mass. On
general grounds the dependence on $\mu_{\rm dec}$ should become weaker if
higher order perturbative corrections are included in the analysis. This is
demonstrated in the figure above
where $\alpha_s^{(5)}(M_Z)$ is computed using $\alpha_s^{(3)}(M_\tau)$ as a
starting point. The decoupling of the charm quark is performed at the fixed
scale $\mu_c=3$~GeV and the decoupling scale of the the bottom quark
$\mu_b$ is varied in the broad range between 1~GeV and 100~GeV.
$N$-loop running goes along with $(N-1)$-loop decoupling. Results are shown
for $N=1$ (upper right dotted line), $N=2$ (steep
dashed line), $N=3$ (lower dashed line) and $N=4$ (dash-dotted line). One
observes a dramatic reduction of the $\mu_b$ dependence with increasing $N$
resulting in a quite flat four-loop result (Note that the scale on the
ordinate only varies by 0.0009.).

For comparison we show in the figure two more curves which correspond
to $N=5$. They incorporate the four-loop decoupling
relations~\cite{Schroder:2005hy}\cite{Chetyrkin:2005ia}. For the unknown five-loop
coefficient of the $\beta$ 
function we have chosen $\beta_4=0$ (solid line) and $\beta_4=150$ (dashed
line parallel to the solid one; the normalization corresponding to 
$\{\beta_0,\beta_1,\beta_2,\beta_3\} \approx \{1.92,2.42,2.83,18.85\}$
has been chosen).

From the figure above it is possible to estimate an uncertainty on
$\alpha_s^{(5)}(M_Z)$ as obtained from $\alpha_s^{(3)}(M_\tau)$ due to missing
higher order corrections. If we restrict ourselves to a range of $\mu_b$
between 2~GeV and 10~GeV and take the difference between the three- and
four-loop curve as an estimate for the uncertainty we obtain
$\delta\alpha_s^{(5)}(M_Z)\approx 0.0002$. The difference between the
four- and five-loop (dashed) curve would even lead to
$\delta\alpha_s^{(5)}(M_Z)\approx 0.0003$. The variation of
$\alpha_s^{(5)}(M_Z)$ due to the variation of $\mu_b$ leads to an additional
uncertainty of $\delta\alpha_s^{(5)}(M_Z)\approx 0.0002$. A similar
uncertainty is obtained from the variation of $\mu_c$ between 2~GeV and 5~GeV.
(This can easily be checked with the program {\tt RunDec}~\cite{Chetyrkin:2000yt}.)
Thus a total uncertainty of $\pm0.0004$
(obtained by adding the three uncertainties in quadrature) should be assigned
to $\alpha_s^{(5)}(M_Z)$. The uncertainties induced by the errors in the quark
masses are much smaller.


\begin{thebibliography}{99}

\bibitem{Vermaseren:1997fq}
  J.~A.~M.~Vermaseren, S.~A.~Larin and T.~van Ritbergen,
  Phys.\ Lett.\ B {\bf 405} (1997) 327
  [hep-ph/9703284].
  %%CITATION = HEP-PH 9703284;%%

\bibitem{Czakon:2004bu}
  M.~Czakon,
  Nucl.\ Phys.\ B {\bf 710} (2005) 485
  [hep-ph/0411261].
  %%CITATION = HEP-PH 0411261;%%

\bibitem{PDG}
  K. Nakamura et al. (Particle Data Group), J. Phys. {\bf G 37} (2010) 075021.

%\cite{Baikov:2008jh}
\bibitem{Baikov:2008jh}
  P.~A.~Baikov, K.~G.~Chetyrkin and J.~H.~Kuhn,
%  ``Order $\alpha^4_s$ QCD Corrections to $Z$ and $\tau$ Decays,''
  Phys.\ Rev.\ Lett.\  {\bf 101} (2008) 012002
  [arXiv:0801.1821 [hep-ph]].
  %%CITATION = PRLTA,101,012002;%%

%\cite{Chetyrkin:1997un}
\bibitem{Chetyrkin:1997un}
  K.~G.~Chetyrkin, B.~A.~Kniehl and M.~Steinhauser,
  %``Decoupling relations to O(alpha(s)**3) and their connection to
  %low-energy
  %theorems,''
  Nucl.\ Phys.\  B {\bf 510} (1998) 61
  [arXiv:hep-ph/9708255].
  %%CITATION = NUPHA,B510,61;%%

\bibitem{Schroder:2005hy}
  Y.~Schr\"oder and M.~Steinhauser,
  %``Four-loop decoupling relations for the strong coupling,''
  JHEP {\bf 0601} (2006) 051
  [arXiv:hep-ph/0512058].
  %%CITATION = JHEPA,0601,051;%%

%\cite{Chetyrkin:2005ia}
\bibitem{Chetyrkin:2005ia}
  K.~G.~Chetyrkin, J.~H.~K\"uhn and C.~Sturm,
  %``QCD decoupling at four loops,''
  Nucl.\ Phys.\  B {\bf 744} (2006) 121
  [arXiv:hep-ph/0512060].
  %%CITATION = NUPHA,B744,121;%%

%\cite{Chetyrkin:2000yt}
\bibitem{Chetyrkin:2000yt}
  K.~G.~Chetyrkin, J.~H.~Kuhn 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;%%

\end{thebibliography}




%%*********************************************************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%            DO NOT CHANGE ANYTHING BELOW THIS LINE !!!!!
%         
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}
