%Title: Perturbative QCD and  tau-decays
%Author: P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn 
%Published:     <EM> Nucl. Phys. Proc. Suppl. </EM> <B> 144 </B> (2005) 81-87.
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\title{
\vspace*{-3cm}
\centerline{\normalsize\hfill  SSFB/CPP-04-75}
\centerline{\normalsize\hfill  TTP04-30      }
%\centerline{\normalsize\hfill hep-ph/}
\vspace{2cm} 
%\\
Perturbative QCD and  tau-decays\thanks{
Invited talk given by K.G.Ch.  at ``TAU-04, 14-17 September 2004, Nara, Japan''.
}}

\author{P.~A.~Baikov\address{Institute of Nuclear Physics,
        Moscow State University, \\
        Moscow~~119992, Russia}%
       ,
        K.G.Chetyrkin\address[KUNI]{Institut f\"ur Theoretische Teilchenphysik, \\
        Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany
        }%
\thanks{On leave from Institute for Nuclear Research
of the Russian Academy of Sciences, Moscow, 117312, Russia.}
        and
        J.H.~K\"uhn\addressmark[KUNI]}



\begin{document}

\begin{abstract}
The current status of the higher order QCD corrections to the width of
the tau is discussed. We report on the result of  the 
first complete  calculation of the quadratic
quark mass correction to the correlator of the two  currents
relevant for the strangeness-changing  semihadronic tau-decay rate 
{\em including its real} part at the four-loop level. 
We discuss  phenomenological implications   for determination of $m_s$.

\end{abstract}
% typeset front matter (including abstract)
\maketitle

\section{Introduction}

Hadronic $\tau$ decays present a handle to deal with the correlator of
two charged weak currents in an interesting energy region just above
the main low-laying resonances. 
The decay proceeds into vector and axial current induced final states,
which can be further separated experimentally into the spin zero and
spin one components \cite{Kuhn:1992nz}. In addition to the total rate also various
moments of the distribution in $s$, the invariant mass of the hadronic
state, can be considered \cite{LeDiberder:1992fr}.

With significant number of $\tau$
leptons detected in each LEP experiments and CLEO a lot of activity
Has been recently devoted to the theoretical analysis of the measured
$\tau$ structure functions within the QCD frameworks as it was
outlined in the pioneering works
\cite{Schilcher:1983ae,Braaten:1988ea,Narison:1988ni,Braaten:1991qm}.

The main outcome of the study of the total hadronic $\tau$ width
(normalized to the known leptonic width)
\begin{equation}
R_\tau=\frac{\Gamma(\tau\rightarrow{\rm hadrons}
  +\nu_\tau)}{\Gamma(\tau\rightarrow l+\bar\nu_l+\nu_\tau)}
= R_{\tau,NS}  + R_{\tau,S}
\label{Rtau1}
\end{equation}
has been a remarkable confirmation of QCD. It was found that
the value of the strong coupling constant $\alpha_s$ as obtained from
$R_{\tau}$ is in good agreement to those obtained from completely
different experiments such as the $Z$ boson decay into hadrons
\cite{Barate:1998uf,Ackerstaff:1998yj,Chetyrkin:1996ia,EWWG:2003}.


Furthermore, the strangeness changing 
(Cabbibo-suppressed) part of the decomposition of the decay
rate and the moments 
\bea                                                                   
\label{Rtaukl}                                                                     
R_\tau^{kl} &\equiv& \!\int\limits_0^{M_\tau^2}\! ds\, \biggl( 1 -                   
\frac{s}{M_\tau^2} \biggr)^{\!k}\!\biggl(\frac{s}{M_\tau^2}\biggr)^{\!l}           
\frac{d R_\tau}{ds} 
\nonumber
\\
&=& R_{\tau,NS}^{kl} + R_{\tau,S}^{kl} \,,                       
\\
R_\tau^{00} &\equiv& R_\tau, \ \ R_{\tau,NS}^{00} \equiv R_{\tau,NS},
\ \ R_{\tau,S}^{00} \equiv R_{\tau,S}
\nonumber
{}
\eea
can be used to determine the strange quark mass, one of the
fundamental parameters of the Standard Model
\cite{Barate:1999hj,Abbiendi:2004xa,Chetyrkin:1993hi,Maltman:1998qz,%
Chetyrkin:1998ej,Pich:1998yn,Pich:1999hc,%
Korner:2000wd,Chen:2001qf,Kambor:2000dj,Maltman:2000wg}. 
The moments as
introduced above can be experimentally determined from the measured
distribution in the invariant mass of the final state hadrons. Hadronic physics is 
encoded in  the quantities  $r^{kl}_{ij}$ defined through 
\bea
&{}&R_{\tau,NS}^{kl}  + R_{\tau,S}^{kl}=
\nnb
\\
&{}&
 3\, S_{\rm EW}
  \left(|V_{ud}|^2 \, r_{ud}^{kl} +|V_{us}|^2 \, r_{us}^{kl}
\right)
{},
\label{Rtau2}
\eea
where $V_{ij}$ stands for the CKM matrix and 
$S_{EW}$ for the universal electroweak
correction \cite{Marciano:1988vm,Braaten:1990ef}.  
The function  $r_{ij}^{kl}$ is directly related to 
the correlator of the weak charged current\footnote{
The combinations $(q)$ and $(0)$ are equivalent to $L+T$ and $L$ in the
notation  of ref.~\cite{Pich:1998yn}.} 
$j_\mu(x) =\bar{\psi_i} \g_\mu(1-\g_5)\psi_j $
\bea
 i \int \mathrm{d} x\,  e^{iqx} 
\langle  
T\left[
j_\mu(x) j^{\dagger}_\nu(0)
\right]
\rangle
&=&
\label{corr:def}
\\
(-g_{\mu\nu} q^2 +  q_{\mu}q_{\nu}) \Pi^{(q)}_{ij}(q^2)
      &+&  g_{\mu\nu} q^2 \Pi^{(0)}_{ij}
\nnb 
{}.
\eea
through 
%\long
\bea
\label{rij}
\lefteqn{r_{ij}^{k,l} =  r_{ij}^{(q)k,l} +  r_{ij}^{(0)k,l} = 
6i\pi 
{\displaystyle
\oint_{|s|=M_{\tau}^2}
}
\frac{ds}{M_{\tau}^2}
}
\\
%&{}&
&\times&\left[
w^{(q)k,l}(x)
\Pi^{(q)}_{ij}(s)
+
w^{(0)k,l}(x)
\Pi^{(0)}_{ij}(s)\right]
{}.
\nonumber
\eea
where $x=s/M^2_{\tau}$, and the weight functions 
\[
w^{(q)k,l} =(1+2x)(1-x)^{k+2}\,x^l, 
\]
\[
w^{(0)k,l} = 2x^{l+1}(1-x)^{k+2} 
{}.
\]
The behaviour of the perturbative series for the part of the integral
arising from $\Pi^{(q)}$ is more stable than the one from $\Pi^{(0)}$.
However, the latter can either be modelled theoretically and
phenomenologically on the basis of scalar and pseudoscalar  resonance physics
\cite{Maltman:1998qz,Kambor:2000dj,Maltman:2001sv,Gamiz:2002nu,Gamiz:2004ar},
or, being solely determined  by spin zero contribution in the $\tau$ decays,
determined experimentally through  the analysis of angular distribution of the 
$K\pi$ and $K\pi\pi$, thus separating spin zero and spin one 
contributions\footnote{A  related
discussion along these lines, which deals with $J=1$ and $J=0$
spectral functions and the resulting moments separately and   the
corresponding pQCD predictions can be found in
\cite{Chetyrkin:1998ej}.}.

Since $m_u \ll m_s$ and $m_s \ll M_\tau$,
the function $r_{us}^{k,l}$ is well described by setting $m_u=0$ and
keeping only the leading and quadratic contributions to the correlator
eq.~(\ref{corr:def}),
viz.
\beq
\Pi^{(q)}_{ij}(q^2,m_s) = \frac{3}{16\pi^2} \left(
\Pi^{(q)}_0(q^2) + \frac{m_s^2}{Q^2} \Pi^{(q)}_{2,ij}(q^2) 
\right)
\label{Pi:q}
\eeq
\beq
\Pi^{(0)}_{ij}(q^2,m_s) =
 \frac{3}{16\pi^2} \frac{m_s^2}{Q^2} \Pi^{(0)}_{2,ij}(q^2) 
{},
\label{Pi:0}
{}.
\eeq
Here $Q^2 = -q^2$.  In the following, we limit ourselves by the
perturbative contributions\footnote{The small non-perturbative terms, as well
as the $m_s^4$ terms will be effectively included in the
phenomenological analysis at the end. For a recent discussion of power-suppressed
contributions, see \cite{Gorbunov:2004wy}.}.

 As a result one has a
convenient decomposition
\beq
r_{u
d} = r_0 + \delta_{ud}, \ \ \ r_{us} = r_0 + \delta_{us} \ \ \
{}.
\eeq
Let us in addition define the  difference
\begin{equation}                                                                
\label{delRtaukl}                                                               
\delta R_\tau^{kl} \equiv \frac{R_{\tau,NS}^{kl}}{|V_{ud}|^2} -                 
\frac{R_{\tau,S}^{kl}}{|V_{us}|^2} =                                            
3\, S_{{\rm EW}}
 \,  \delta r^{kl}
%\nonumber
{}.                                                    
\end{equation}  
which is a usefull  combination to probe the $SU(3)$ breaking
effects as 
$\delta r^{kl} \equiv \Big(\delta_{ud}^{kl} -\delta_{us}^{kl}\Big)$  
vanishes in the limit of  exact $SU(3)$ flavour
symmetry.  In the spirit of the decomposition (\ref{rij}) one similarly defines 
related quantities $\delta R_\tau^{(q)kl}$, $\delta R_\tau^{(0)kl}$, 
$\delta r^{(q)kl}$ and $\delta r_\tau^{(0)kl}$
such that
$
\delta R_\tau^{kl}  = \delta R_\tau^{(q)kl} + \delta R_\tau^{(0)kl}
\nonumber
$
,
$
\delta r^{kl}  = \delta r^{(q)kl} + \delta r^{(0)kl}
\nonumber
{}.
$

The currently known  pQCD predictions for $r_0$, $\delta_{us}$  and $\delta_{ud}$ can be
shortly summarized as follows\footnote{Displayed below are so called 
Fixed Order Perturbation Theory results.
A procedure of ``improving'' of pQCD predictions by a partial resummation of
higher order terms \cite{Pivovarov:1991rh,LeDiberder:1992te} will be discussed later.}
\bea
r_0^{\mathrm{FOPT}} &=&  
\label{Rtau:FO}
\left(
1 + \, a_s+ 5.202 \, a^2_s+ 26.37 \, a_s^3 
    \right)
{},
\\
\delta_{us}^{\mathrm{FOPT}} &=&  
\frac{m_s^2}{M_{\tau}^2}
\left(
1 + 5.33\, a_s+ 46.0\, a^2_s 
    \right)
{},
\label{delta_us:FO}
\\
\delta_{ud}^{\mathrm{FOPT}} &=&  
-0.35\  a^2_s \ 
\frac{m_s^2}{M_{\tau}^2}
\label{delta_ud:FO}
{},
\eea
where $a_s = \als(M_\tau)/\pi$.  Now, with $a_s \approx .1$ one
observes that the ``apparent convergency'' of the  series is
acceptable for $r_0$ but should be considered  at best as marginal for
$\delta_{us}$.  Clearly, the next term in (\ref{delta_us:FO})  is
important for the interpretation of the measurements. Partial
results for the $\as^4$ term in eq.~(\ref{Rtau:FO}) and the $\as^3$
term in eq.~(\ref{delta_us:FO}) have been published
\cite{ChBK:vv:as4nf2,ChBK:tau:as4nf2}. We will describe below the results of the
calculation of the missing term as well as its phenomenological
implications.

\section{Calculation}
At first sight, with the first ${\cal O}(\als^3)$ calculations of the
absorptive part of two-point correlators being completed in the
beginning of 90-ties \cite{Gorishnii:1991vf},
it should have been not very difficult to compute
the ${\cal O}(\als^3)$ contribution to  $\delta^{(q)}_2$
and $\delta^{(0)}_2$ long ago.
This is indeed the case for $\delta^{(0)}$ \cite{Chetyrkin:1993hi}.
However, it does not apply for $\delta^{(q)}$, as a consequence of the (purely kinematical)
appearance of the  factor  $1/Q^2$ in eq.~(\ref{Pi:q}). Indeed, in
performing the contour integration over $s$ in eq.~(\ref{rij}) one
may freely ignore any s-independent term in the function
$\Pi^{(0)}_2$, however,  {\em not} in $\Pi^{(q)}_2$: in the latter case the
$1/Q^2$ factor  produces a non-zero contribution from a constant
term in $\Pi^{(q)}_2$. One could rephrase this  as
follows: the function $\delta r^{kl}$ depends on the absorptive part of
$\Pi^{(g)}_2(q^2)$ only but on {\em both}  absorptive {\em and}
real parts  of the function $\Pi^{(q)}_2(q^2)$.

Now, the ${\cal O}(\als^3)$ contributions to eq.~(\ref{rij}) are
represented by four-loop propagators.  The calculation of the absorptive
part of (massless) four-loop propagators can be reduced to the evaluation
of a  properly constructed set of {\em three}-loop massless propagator via
so-called  $R^*$ operation \cite{ChS:R*,gvvq}.
The resulting three-loop diagrams can be conveniently computed with
the help of the algebraic program MINCER \cite{mincer:91} written in
FORM \cite{Vermaseren:91}, which implements the traditional
integration-by-parts algorithm to evaluate massless three loop
propagators.  Naturally, the absorptive part of five-loop propagators
(which correspond to the ${\cal O}(\als^4)$ contribution to $r_0$) can be
also reduced to the calculation of the real parts of the four-loop
propagator, and so on.

However, any direct generalization of the MINCER algorithm to the four
loop case is far from being trivial, if possible at all.  Instead we
have used a new method to reduce a four-loop propagator to the master
integrals, namely an expansion of the corresponding coefficient functions in
the limit of large space-time dimension $D$ \cite{Baikov:2003zq}. The method has
demonstrated its feasibility for a restricted set of topologies of
four-loop propagators in \cite{ChBK:vv:as4nf2,ChBK:tau:as4nf2,ChBK:vv:mq2as4nf2}.
The present calculation deals, for the
first time, with a complete problem including all possible topologies.

To compute the real part of $\Pi^{(q)}_2(q^2)$ we proceed as
follows.
First, using the criterion of irreducibility of Feynman
integrals \cite{Baikov:criterion:00}, the set of irreducible integrals involved in
the problem was constructed.  Second, the
coefficients multiplying  these integrals were calculated as series in
the $1/D\rightarrow0$ expansion.
Third, the exact answer, i.e.  a rational function of $D$,
 was reconstructed from this expansion.
And, finally,  extensive testing of the answer was performed.
%\input{japan_table1}
%\clearpage
\begin{table*}[htb!]
\caption{\label{tab1}
Contributions of successive orders of PT  to $\delta \, r^{(q)kl}$
\ice{evaluated within the standard fixed order PT}
for the value of  $\alpha_s(M_{tau}) = .334$, and normalized 
to the value of $\delta \, r^{(q)kl}$ in the Born approximation.
First five lines:  fixed order PT, second five lines: contour improved version,
with RG improvement  for the Adler  $D^{(q)}$ ; last five lines: contour improved
version with RG summation made for the function $\Pi^{(q)}$.
}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{lll}
\hline\\
$(kl) $ & Perturbative series &
          \\
\hline
(0,0)  &  1\ + \ 0.425\ + \ 0.283\ + \ 0.178\ + \ 0.0857 &= \ 1.97	 
\\    	  						       	 
(1,0)  &  1\ + \ 0.532\ + \ 0.458\ + \ 0.423\ + \ 0.424 &= \ 2.84 	 
\\    	  						       	 
(2,0)  &  1\ + \ 0.606\ + \ 0.589\ + \ 0.623\ + \ 0.721 &= \ 3.54 	 
 \\   	  						       	 
(3,0)  &  1\ + \ 0.663\ + \ 0.695\ + \ 0.793\ + \ 0.987 &= \ 4.14 	 
\\    	  						       	 
(4,0)  &  1\ + \ 0.708\ + \ 0.784\ + \ 0.943\ + \ 1.23 &= \ 4.67    
\\    
\hline
(0,0) &  0.753\ + \ 0.214\ + \ 0.065\  - \ 0.0611\ - \ 0.213 &= \  0.75
\\    							      
(1,0) & 0.912\ + \ 0.334\ + \ 0.192\ + \ 0.0675\   -  \ 0.0969 &= \  1.41
\\    							      
(2,0) & 1.05\ + \ 0.451\ + \ 0.33\ + \ 0.228\ + \ 0.0802 &= \ 2.14     
 \\   							      
(3,0) & 1.19\ + \ 0.571\ + \ 0.484\ + \ 0.425\ + \ 0.33 &= \ 3.	      
\\    							      
(4,0) & 1.32\ + \ 0.697\ + \ 0.657\ + \ 0.665\ + \ 0.664 &= \ 4.01     
\\
\hline								 
(0,0) & 0.857\ + \ 0.122\   - \ 0.00901\  - \ 0.158\ - \ 0.34 &= \ 0.473	 
\\     									 
(1,0) & 1.11\ + \ 0.232\ + \ 0.11\  - \ 0.0417\  - \ 0.268 &= \ 1.14	 
\\     									 
(2,0) & 1.35\ + \ 0.347\ + \ 0.251\ + \ 0.124\  - \ 0.115 &= \ 1.95	 
 \\    									 
(3,0) & 1.59\ + \ 0.471\ + \ 0.42\ + \ 0.35\ + \ 0.139 &= \ 2.97		 
\\     									 
(4,0) & 1.83\ + \ 0.61\ + \ 0.623\ + \ 0.648\ + \ 0.521 &= \ 4.23         
\\    
\hline
\end{tabular}\\[2pt]
\end{table*}


\section{Results}

Our results for the functions
$\Pi^{(q)}_{2,us}$ and $\Pi^{(q)}_{2,ud}$ read:



\bea
&{}&\Pi^{(q)}_{2,us} =
 -4 -\frac{28}{3}\, a_s 
\label{Pi2m2us:exact}
\\
&+& a_s^2 \left\{
 -\frac{13981}{108} - \frac{646}{27} \, \zeta_{3}  
+\frac{2080}{27}
 \,\zeta_{5}\right\}
\nonumber  
\\
&{+}& a_s^3 \left\{
 -\frac{2092745}{1296}-\frac{14713}{162} \,\zeta_{3} 
-122 \,\zeta_3^2
+10 \,\zeta_{4} 
%%%%%%%%%%
\right. 
\nonumber 
\\ &{}& 
\left.
%\left. 
%\hspace{-1.3cm}
%%%%%%%%%%%%%%%%
\  \ \ \ \ \ 
+\frac{41065}{27} \,\zeta_{5}  
-\frac{79835}{162}\, \zeta_7
\right\}
\nnb
\\
&{}& \hspace{-.6cm}= -4  \left(  1 +2.333\, a_s +
19.58\  a_s^2 
+   202.309\  a_s^3
%\left\{(k^{{(q)}3}_{2,us} = 202.38)\right\} 
 \right)  {},
\nonumber
%\label{Pi2m2us:exact}
\end{eqnarray}

\bea
&{}& \Pi^{(q)}_{2,ud} =
 a_s^2 \left\{
\frac{128}{9} - \frac{32}{3} \, \zeta_{3}  
\right\}
\\
&+& a_s^3 \left\{
\frac{6392}{27} - \frac{4496}{27} \,\zeta_{3} 
-
16 \,\zeta_3^2
+\frac{320}{27}\,\zeta_{5} 
\right\}
\nonumber
\\
%\hspace{-.6cm}
&=& 
-4  \left(  -0.35\, a_s^2 
-6.437\  a_s^3 
 \right) 
\nonumber
\label{Pi2m2ud:exact}
{}.
\end{eqnarray}
where we have set the normalization scale $\mu^2=-q^2=Q^2$ 
and $a_s = \as(Q^2)$. The results for generic values 
of $\mu$ can be easily recovered with the standard RG techniques.  

%\input{japan_tables}

It is instructive to compare the exact result for the ${\cal
O}(\als^3)$ contribution to (\ref{Pi2m2us:exact}) with the recently obtained
predictions (\cite{ChBK:tau:as4nf2}) 
based on the  optimization schemes as PMS and FAC:
\[
k^{(q)3}_{2,us} =  202.309\,(\mbox{exact}),  
\  201 \,(\mbox{PMS}),  \ 199 \,(\mbox{FAC})
\label{Kq3_2:predcitions}
{}.
\]
This astonishingly good agreement (a similar phenomenon   has been observed for 
the prediction of the $O(\alpha_s^3)$ term for the correlator of the diagonal currents
\cite{ChBK:LL04}  can  be considered as  a strong argument to repeat
the procedure  and predict, starting from the now completely known
$k_2^{(q)3}$,  the corresponding result for one loop
more, that is for $k_2^{(q)4}$.  To be
definite, we use  the PMS predictions (again for $n_f=3$; the FAC
result is  very similar)
\beq
k^{(q)4}_{2,us} =  2276 \pm 200 
\eeq 
 and 
\beq
k^{(q)4}_{2,us} - k^{(q)4}_{2,ud} = 2378 \pm 200 
{}.
\eeq
It is, of course, difficult to estimate 
uncertainty in the above predictions; however, the simple
comparison with  eq.~(\ref{Kq3_2:predcitions}) clearly demonstrates that an error of about
10\% should be considered quite conservative.



\section{Phenomenology}

Apart of evaluations using fixed order perturbation theory two formally equivalent
versions of the contour improved procedure can be found in the
literature. The first \cite{Chetyrkin:1998ej} 
is based directly on the integration of the
polarization function $\Pi^{(q)}_2$, the second \cite{Pich:1998yn} 
is based on the
integration of the Adler function 
\[ D^{(q)}_2 \equiv s\frac{d}{d  s}\Pi^{(q)}_2 \]
and is obtained from the first one by partial
integration.  
Partial integration and renormalization group
improvement do not commute as long as finite orders are
considered. Since the second procedure moves part  of the lower
order input to higher orders (contrary to the spirit of CIPT) and
since, furthermore, the first procedure leads to a somewhat more stable
perturbation series \footnote{At least this is true for small values of $\as$; for
realistic higher value $\as = .334$ both series start to oscillate.}, 
we consider the first of the two choices
as preferable. 
%%%%%%%% insertion 3 starts %%%%%%%%%%%%%
The three options for the perturbative series are displayed in 
Table~\ref{tab1}. As a consequence of the large value of $\alpha_s$ and the rapidly
growing coefficients of the perturbative series it seems at first
\newpage
\clearpage
%\input{japan_tables}
% Ininitially I have started from tables_short.tex as of 20.11.04
\begin{table*}[h!tb]    
\caption{
Result for $m_s$, derived from different levels of approximation, based on contour
improvement from \cite{Pich:1998yn}, and a list of different conributions to the associated  error. 
\ice{
An update of the Table 1 of\  \  \cite{Gamiz:2004ar}. 
It shows the influence of
the newly computed $\alpha_s^3$ term on the value of
$m_s(M_{\tau})$. The contour improvement has been done according to
\cite{Pich:1998yn}.  For comparison we have shown the results of \cite{Gamiz:2004ar}
based on the ``geometrical'' estimation of the coefficient $d_2^{3}$.
Central results for $m_s(M_\tau)$ extracted from the different moments
are shown, as well as the uncertainties for $m_s$. The latter are
composed from two parts. The first comes from variation of the last
accounted term in the PT expansion: it has been either doubled or
discarded. By ``others'' we mean  all uncertainties (added in quadrature) of the input
parameters different from the ${\cal O}(\as^3)$ (or ${\cal O}(\as^4)$) terms in the 
perturbative contribution. They include experimental errors in the moments as reported in
\cite{Abbiendi:2004xa}, $|V_{us}| = .2208 \pm 0.0034$  from  \cite{Gamiz:2004ar}, 
$\alpha_s(M_{\tau}) = 0.334 \pm 0.022$ as well as uncertainties in the
parameters related to construction of  the subtracted longitudinal
part.  We have computed all these other uncertainties using numbers
from \cite{Gamiz:2004ar}. The last column  shows a weighted average over the
different moments (as the individual error for a given moment we have
chosen the larger one).  
}}
\label{tab2}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
%\begin{center}                                                                         
\begin{tabular}{crrrr}                                                                 
\hline                                                                                 
Parameter  & Value & \quad(2,0)\quad & \quad(3,0)\quad & \quad(4,0)  
\\  
\\
\hline  
$m_s({\cal O}(\as^3),\mbox{est.})$ &    &   124.  &   107.  &   94.5  
 \\ 
\hline                             
${\cal O}(\as^3)$ & $\asthree$ &        ${}^{-5.4}_{6.2}$  &   ${}^{-6.6}_{8.1}$  &   ${}^{-7.2}_{9.3}$       
 \\ 
%\hline                             
$\xi$ & ${}^{1.5}_{1}$ &         ${}^{3.3}_{0}$  &   ${}^{6.2}_{0}$  &   ${}^{8.3}_{0}$          
\\  
%\hline                             
$\alpha_s(M_{\tau})$ & $.334 \pm 0.022$ &    ${}^{0.99}_{0.14}$  &   ${}^{-0.94}_{1.8}$  &   ${}^{-2.2}_{3.}$
\\                 
%\hline                                       
others \cite{Abbiendi:2004xa,Gamiz:2004ar}& & ${}^{+22.6}_{-26.}$ & ${}^{+17.6}_{-19.7}$ & ${}^{+14.9}_{-16.5}$
\\                        
%\hline                                                                                 
Total    & & ${}^{+23.7}_{-26.5}$ & ${}^{+20.4}_{-20.8}$ & ${}^{+19.6}_{-18.1}$
\\  
\hline
$m_s({\cal O}(\as^3),\mbox{exact})$ &    &   123.  &   106.  &   92.8 
\\
\hline                             
${\cal O}(\as^3)$ & $\asthree$ &  ${}^{-6.3}_{7.4}$  &   ${}^{-7.5}_{9.6}$  &   ${}^{-8.}_{11.}$  
 \\ 
%\hline                             
$\xi$ & ${}^{1.5}_{1}$ &  ${}^{3.6}_{0}$  &   ${}^{6.7}_{0}$  &   ${}^{8.8}_{0}$ 
\\ 
%\hline                             
$\alpha_s(M_{\tau})$  &$.334 \pm 0.022$ &  ${}^{1.1}_{0.1}$  &   ${}^{-0.93}_{1.9}$  &   ${}^{-2.2}_{3.}$ 
\\                
%\hline                                                                                 
others \cite{Abbiendi:2004xa,Gamiz:2004ar} & &${}^{+22.4}_{-25.7}$ & ${}^{+17.4}_{-19.5}$ & ${}^{+14.8}_{-16.3}$ 
\\                        
%\hline                                                                                 
Total    &  & ${}^{+23.8}_{-26.5}$ & ${}^{+21.}_{-20.9}$ & ${}^{+20.6}_{-18.3}$ 	
\\
\hline  
$m_s({\cal O}(\as^4),\mbox{PMS})$ &    &       121.  &   99.9  &   84.7  
 \\ 
\hline                             
${\cal O}(\as^4)$ & $\asfour$ &               ${}^{-2.2}_{2.3}$  &   ${}^{-5.1}_{6.}$  &   ${}^{-6.3}_{8.}$  	  
 \\ 
%\hline                             
$\xi$ & ${}^{1.5}_{1}$ &                  ${}^{-2.}_{0}$  &   ${}^{3.7}_{0}$  &   ${}^{6.8}_{0}$      
\\  
%\hline                             
$\alpha_s(M_{\tau})$ &  $.334 \pm 0.022$ &    ${}^{4.}_{-1.4}$  &   ${}^{0.26}_{1.4}$  &   ${}^{-1.8}_{3.}$  
\\                 
%\hline                                       
others \cite{Abbiendi:2004xa,Gamiz:2004ar,Czarnecki:2004cw}& & ${}^{+22.}_{-25.3}$ & ${}^{+16.7}_{-18.7}$ & ${}^{+14.}_{-15.5}$ 
\\                        
%\hline                                                                                 
Total    &  &${}^{+22.5}_{-25.5}$ & ${}^{+18.1}_{-19.4}$ & ${}^{+17.8}_{-16.8}$
\\  
\hline
\end{tabular}                                                                          
%\end{center}                                                                           
\end{table*}                                                                            
\samepage
\nopagebreak
\nobreak
\begin{table*}[h!tb]                                                                       
\caption{
The same as Table \ref{tab2} but with the contour 
improvement  done according to \cite{ChBK:tau:as4nf2}.
}\label{tab3}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2}                                                      
\begin{tabular}{crrrr}                                                                 
\hline                                                                                 
Parameter   & Value & \quad(2,0)\quad & \quad(3,0)\quad & \quad(4,0)  
\\
\hline  
$m_s({\cal O}(\as^3),\mbox{exact})$ &    &   123.  &   103.  &   88.        
 \\ 
\hline                             
${\cal O}(\as^3)$ & $\asthree$ &         ${}^{-3.5}_{3.9}$  &   ${}^{-5.8}_{7.}$  &   ${}^{-6.8}_{8.9}$      
 \\ 
%\hline                             
$\xi$ & ${}^{1.5}_{1}$ &               ${}^{-1.4}_{20.}$  &   ${}^{4.5}_{9.1}$  &   ${}^{7.5}_{2.7}$     
\\  
%\hline                             
$\alpha_s(M_{\tau})$ & $.334 \pm 0.022$ &     ${}^{3.8}_{-1.4}$  &   ${}^{0.44}_{1.2}$  &   ${}^{-1.5}_{2.8}$  
\\                 
%\hline                                       
others \cite{Abbiendi:2004xa,Gamiz:2004ar}& &${}^{+22.3}_{-25.7}$ & ${}^{+17.}_{-19.1}$ & ${}^{+14.3}_{-15.8}$
\\                        
%\hline                                                                                 
Total    &  & ${}^{+23.}_{-26.}$ & ${}^{+19.}_{-20.}$ & ${}^{+18.6}_{-17.3}$
\\  
\hline
$m_s({\cal O}(\as^4),\mbox{PMS})$ &    & 126.  &   100.  &   82.4      
 \\ 
%\hline                             
${\cal O}(\as^4)$ & $\asfour$ &             ${}^{3.9}_{-3.6}$  &   ${}^{-2.3}_{2.4}$  &   ${}^{-4.6}_{5.6}$      
 \\ 
%\hline                             
$\xi$ & ${}^{1.5}_{0}$ &        ${}^{-8.9}_{0}$  &   ${}^{-2.}_{0}$  &   ${}^{4.3}_{0}$          
\\  
%\hline                             
$\alpha_s(M_{\tau})$ & $.334 \pm 0.022$ &  ${}^{13.}_{-5.4}$  &   ${}^{4.2}_{-0.65}$  &   ${}^{0.24}_{2.1}$   
\\                 
%\hline                                       
others \cite{Abbiendi:2004xa,Gamiz:2004ar,Czarnecki:2004cw}& &${}^{+22.9}_{-26.4}$ & ${}^{+16.7}_{-18.8}$ & ${}^{+13.8}_{-15.3}$
\\                        
%\hline                                                                                 
Total    &  & ${}^{+26.4}_{-28.6}$ & ${}^{+17.4}_{-19.}$ & ${}^{+15.7}_{-16.}$
\\  
\hline
\end{tabular}                                                                          
\end{table*}                                                                            

\clearpage
\noindent
glance difficult to consider any of the theoretical predictions as
truly preferable. Nevertheless, the results for moments $(2,0),
(3,0)$ and $(4,0)$ are at least in plausible agreement among the three
methods and exhibit acceptably decreasing subsequent terms. Since
these moments are also relatively most precise, as far as experiment
is concerned, they will be used in the subsequent analysis.
%\input{japan_tables}
The phenomenological analysis will be based on the most recent
evaluation  \cite{Czarnecki:2004cw} of $|V_{us}|$.
For the phenomenological description of the contribution due to 
$r^{(0)kl}$  we adopt the analysis presented in \cite{Gamiz:2004ar}
where the contour improvent for  $D^{(q)}_2$ was employed.

%for the second version of contour improvement. 



The results for $m_s$, derived from different moments and different
ways of implementing the contour improvement procedure are shown in
Table~\ref{tab3}. (Details about the error estimates and the
corresponding analysis will be given elsewhere.) The main difference,
compared to the previous analysis, is a downward shift of $m_s$ by
about 5 MeV from the inclusion of the $\als^4$ terms and an upward
shift by as much as 20 MeV from the new input for
$|V_{us}|$. The ambiguity in the determined value of $m_s$, including the
newly computed $\als^3$ term, and the estimate for the $\als^4$ term
is shown in Table~\ref{tab3}. The renormalzation scale
$\mu = \xi M_{\tau}$ is allowed to vary between $\xi = 1 - 1.5 $.
Values of $\mu$ lower than $M_{\tau}$ lead to a blow up of $\als$ and
destabilize the result.  By ``others'' we mean all uncertainties
(added in quadrature) of the input parameters different from the
${\cal O}(\as^3)$ (or ${\cal O}(\as^4)$) terms in the perturbative
contribution. They include experimental errors in the moments as
reported in \cite{Abbiendi:2004xa}, 
errors in   $|V_{us}| = 0.2259(23)$ from   \cite{Czarnecki:2004cw}             
as well as uncertainties in the                         
parameters related to construction of  the subtracted longitudinal                             
part.  We have computed the   uncertainties using numbers                            
from \cite{Gamiz:2004ar}.


From Table \ref{tab1} we find that the inclusion of the $\als^4$ term
leads for the third and the fourth moments 
to a better agreement between predictions based on two different
methods of implementing of the ``contour improvement'' approach.
Since the third  moment also exhibits
a smaller theoretical error  we choose it to derive our final
result for $m_s$ which will be given below.

In total we find 
\beq
%m_s(M_{\tau}) = 100 \pm 20 \  \mbox{MeV}
m_s(M_{\tau}) = 100 
+ \left({}^{+4.8}_{-3.1} \right)_{theo} \ \
+  \left( {}^{+17.}_{-19.}  \right)_{rest}
\label{our:fin}
{}.
\eeq 
If one compares (\ref{our:fin}) to the ${\cal O}(\als^3)$  result of \cite{Gamiz:2004ar} 
(corrected for a different $|V_{us}|$),namely  $m_s(M_{\tau}) = 106$ (with larger theoretical and
identical remaning errors) one sees 
an  essential but still not too large sensitivity to the
$\alpha_s^4$ contribution.  In fact, for the third moment the shift in
$m_s$ due inclusion of the $\alpha_s^4$ term (-2.5 MeV) is about a third of 
of the corresponding change (-7 MeV) due to the  $\alpha_s^3$
contribution.  Thus,  the purely theoretical uncertainty from not
yet computed higher orders could be estimated as roughly  3 MeV.
Unfortunately, one can hardly hope that the error from not yet
computed higher orders in $\alpha_s$ could be reduced further by means
of a direct calculation in any foreseeable future.

%\samepage
%\nopagebreak


\section{Acknowledgments}
The authors are grateful to A.~Pivovarov and K.~Maltman
for   very useful discussions.  This work was supported by the
the Deutsche Forschungsgemeinschaft in the Sonderforschungsbereich/Transregio 
SFB/TR-9 ``Computational Particle Physics'',  by INTAS (grant
03-51-4007), by RFBR (grant 03-02-17177) and by Volkswagen Foundation
(grant I/77788).

%\newpage

%\input{japan_talk2.bbl}

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\end{document}

%\bibliographystyle{unsrt}
\bibliographystyle{h-elsevier2}
\bibliography{chet,baikov,our_NPBtau,tau_PRD,jamin_ms_04_11,jamin_ms_04,%
optimization,kataev,vermaseren,pivovarov_tau.bib}



\end{document}

