%Title: Three aspects of tau decays
%Author: P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn
%Published:     <EM>Proceedings of the Workshop on Heavy Quarks and Leptons, Vietri sul Mare, Italy, May 2002,</EM> FRASCATI PHYSICS SERIES, Vol. XXVIII, 453-468. 
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\begin{document}

%
\title{ 
Three aspects of tau decays
%\footnote{Presented by J.H.~K\"uhn}
}
\author{
P.~A.~Baikov
\\
{\em Institute of Nuclear Physics,}
\\
{ \em Moscow State University,
Moscow~119899, Russia}
\\
K.G.Chetyrkin 
\\
{\em 
Fakult{\"a}t f{\"u}r Physik  Albert-Ludwigs-Universit{\"a}t Freiburg,
}
\\
{\em
D-79104 Freiburg, Germany 
}
\\
J.H.~K\"uhn         \\
{\em Institut f\"ur Theoretische Teilchenphysik,}
\\{\em
     Universit\"at Karlsruhe,
     D-76128 Karlsruhe, Germany
} 
}
\maketitle
\baselineskip=11.6pt
\begin{abstract}

Three topics of relevance for $\tau$-decays will be presented: 
i) the possibility to
search for CP violation in the Cabbibo suppressed channels
$\tau \to \nu K \pi \ \ \mbox{and} \ \ \nu K \pi \pi$,  
ii) recent partial results on the
semileptonic decay rate in $\as^4$ accuracy,  
and iii) the possibility to determine the strange quark mass in inclusive
Cabbibo suppressed decays.
\end{abstract}
%
\baselineskip=14pt
%
\section{Introduction}
The analysis of $\tau$-lepton production and decay offers the unique
possibility for a multitude of precision measurements, in the context
of electroweak interactions as well as for QCD\cite{Review}. One of
the most precise determination of the vector and axial parts of the Z
coupling to leptons originates from the analysis of Z decays to $\tau$
leptons and fairly restrictive limits on anomalous charged current
couplings have been set by studies of the leptonic decays.  
First steps towards the precise determination of the strange quark
mass in Cabbibo-suppressed decays have been undertaken.  Last not
least, one of the most precise determinations of the strong coupling
constant is derived from the semileptonic decay rate.  From the
experimental side the final LEP results are expected soon, and the
same is true for the CLEO data. The yield of $\tau$-leptons at
B-factories is enormous: The integrated luminosity of 100 fb${}^{-1}$
which will soon be collected both at BaBar and BELLE, corresponds to
about hundred million pairs for each experiment. In principle this
huge sample could open the door to qualitatively new precision studies
of the lepton, and thus give important information on this member of
the third quark-lepton-family.  In this contribution we will discuss
three characteristic examples for the enormous potential of the
$\tau$-lepton:

\begin{enumerate}
\item

Properly chosen angular distributions in the decay modes $\tau \to \nu
K \pi $ \mbox{or} $ \nu K \pi \pi$ allow to test for CP violating
scalar couplings the of the type $(\bar{\nu}\tau) (\bar{s} u) $
mediated by charged Higgs decay as discussed in section 2.


\item 
The determination of $\as$ from the decays rate has been one of the
central themes in $\tau$ physics.  At present the error in $\as$ is
largely determinated by theoretical uncertainties, in particular by
the present ignorance on the behaviour of the perturbative series
beyond the $\as^3$ term.
Recently a   subset of the $\as^4$ terms has become
available. We discuss the magnitude of these terms and the
implications for the full $\as^4$  result.  

\item 

During the past  years  the determination
of the strange quark mass has been addressed in the various
theoretical and experimental analyses.  The Cabbibo suppressed decay
rate
$\Gamma(\tau \to \nu s \bar{u})$ depends  on $m_s$  
and the quadratic term in the expansion in $m_s^2/M_{\tau}^2$
has been evaluated in pQCD up to order $\as^2$, and for some moments even
up to $\as^3$.  The large coefficients of the series  in  $\as$ lead to a fairly
poor convergence and shed doubt on the validity of the entire
approach. In section five  we argue, on the basis of the newly
calculated terms of order $\as^3 n_f m_s^2$  
(together with those of order $\as^3 n_f^2 m_s^2$), that
the complete $\as^3$ term exhibits again a very large coefficient indicating
poor convergence of the complete series.
\end{enumerate}

\section{CP-violation in $\tau$ -decays}
\subsection{Preliminaries}

CP violation has been experimentally observed only in the
$K$ meson system. The effect can be explained by a nontrivial complex
phase in the CKM flavour mixing matrix\cite{kmmat}. 
However, the fundamental origin of this CP violation
is still unknown.
In particular the CP properties of the third fermion
family are largely unexplored.
Production and decay of $\tau$ leptons might offer a particularly clean
laboratory to study these effects.
In this contribution, we investigate the effects of
CP violation\cite{cp1} to be observed in semileptonic $\tau$ decays
which could arise in a framework outside the mechanism prosposed
by Kobayashi and Maskawa.
We show that the structure function
formalism of Ref.\cite{km1} allows for a systematic analysis
of possible CP violation effects in the two and three
meson channels. Special emphasis is put on the $\Delta S=1$ transition
$\tau\rightarrow K\pi\nu_\tau$ where possible CP violating signals from
multi Higgs boson models   would be signaled by a nonvanishing
difference between the structure functions
$W_{SF}[\tau^-\rightarrow (K\pi)^-\nu_\tau]$ and
$W_{SF}[\tau^+\rightarrow (K\pi)^+\nu_\tau]$.
Such a measurement is possible for unpolarized single $\tau$'s without
reconstruction of the $\tau$ rest frame and without
polarized incident $e^+e^-$ beams.
It is shown that this CP violation requires both
nonvanishing hadronic phases and CP violating phases in the
Hamiltonian,
where the hadronic phases arise from the interference of complex 
Breit-Wigner propagators, whereas the CP violating phases could 
arise from an exotic charged Higgs boson.
An additional independent test of CP violation
in the two meson case is possible, but
would require the knowledge of the full kinematics and  $\tau$
polarization. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{CP Violating Signals in the $\tau\rightarrow K\pi\nu_\tau$ Decay Mode}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Transitions from the vacuum to two pseudoscalar mesons $h_1$ and
$h_2$ 
are induced through vector and scalar currents only.
%  , where the latter can 
%  be related to the former with the help of Eq.~(\ref{motion}).
Expanding these hadronic matrix elements along the set of independent momenta
$(q_1-q_2)_\beta$ and $Q_\beta=(q_1+q_2)_\beta$ we define
$(T_{\alpha\beta}=  g_{\alpha \beta} - (Q_\alpha Q_\beta)/Q^2)$
%
%
\begin{eqnarray}
%J_\beta=\langle h_1(q_1)h_2(q_2)|\bar{u}\gamma_\beta d|0 \rangle=
\langle h_1(q_1)h_2(q_2)|\bar{u}\gamma_\beta d|0 \rangle&=&
(q_1-q_2)^\alpha\,T_{\alpha\beta}\,\,F(Q^2) \,+\,
Q_\beta\, F_S(Q^2)\\
\langle h_1(q_1)h_2(q_2)|\bar{u} d|0 \rangle &=& F_H(Q^2).
\label{jdef}
\end{eqnarray}
%
%
The representation of the hadronic amplitude
$\langle h_1h_2|\bar{u}\gamma_\beta d|0\rangle $
corresponds to a decomposition into spin one and spin
zero contributions,
{\it e.g.} the vector form factor $F(Q^2)$
corresponds to the $J^P = 1^-$ component
of the  weak charged current, and the scalar form factor 
$F_S(Q^2)$ to the $J^P = 0^+$ component. 
A fictitious scalar Higgs exchange contribution is proportional to $\eta_S F_H$.
The general amplitude for the $\Delta S=1$ decay (where 
$h_1\equiv K, h_2\equiv \pi$)
%
%
\begin{equation}
\tau^-(l,s) \rightarrow \nu(l^\prime,s^\prime) + h_1(q_1,m_1)+h_2(q_2,m_2)\>,
\label{twoh}
\end{equation}
%
%
can thus be written as
\begin{equation}
{\cal{M}}=
\sin\theta_{c}\frac{\,G}{\sqrt{2}}
  \bar{u}(l^\prime,s^\prime)\,\gamma_{\alpha}
           (1-\gamma_5)\,u(l,s) 
  \left[(q_1-q_2)_\beta\,T^{\alpha\beta}\, F   \, + \,
  Q^\alpha\,\tilde{F}_S\right]
\label{me2h}
\end{equation}
%
%
\vspace*{-3mm}
\begin{equation}
\mbox{with}\hspace{1cm}\tilde{F}_S = F_S \,+\, \frac{\eta_S}{m_\tau}\,F_H
\label{fs}
\end{equation}


In Eq.~(\ref{me2h}) $s$ 
denotes the polarization four-vector of the $\tau$ lepton.
The complex parameter $\eta_S$ in Eq.~(\ref{fs}) 
transforms like 
\begin{eqnarray}
\eta_S & \stackrel{\mbox{CP}}{\longrightarrow}&\eta^\ast_S
\end{eqnarray}
and thus allows for the parametrization of possible CP violation.
%
%
Up to the small isospin breaking terms, induced for example
by the small quark mass difference, CVC implies the
vanishing of $F_S$ for the two pion ($h_1\equiv\pi^-,h_2\equiv\pi^0$) case. 
For the
transition $\tau\rightarrow K\pi\nu$ the $J=1$ form factor $F$
is dominated by the $K^*(892)$ vector resonance contribution. 
The scalar form factor $F_S$ is expected\cite{kpi} to
receive a sizable resonance contribution 
($\sim 5\%$ to the decay rate)
from the $K_0^*(1430)$ with $J^P=0^+$.
The corresponding $\tau^+$ decay is obtained from 
Eq.~(\ref{me2h})
through the substitutions 
%
%
\begin{equation}
(1-\gamma_5)\, \rightarrow \, (1+\gamma_5), \hspace{1cm}
%\chi      \, \rightarrow \, \chi^*,     \hspace{1cm}
  \eta_S      \, \rightarrow \, \eta_S^*.     \hspace{1cm}
%\eta_S^d      \, \rightarrow \, \eta_S^{d *}.     \hspace{1cm}
\end{equation}
%
%
Reaction (\ref{twoh}) is most easily analyzed in the hadronic rest 
frame\cite{km1}
$\vec{q}_{1}+\vec{q}_{2}=~0$.
After integration over the unobserved
neutrino direction,
the differential decay rate in the
rest frame of $h_{1}+h_{2}$ is given   by\cite{kpi,km1}
%
%
\begin{eqnarray}
d\Gamma(\tau^-\rightarrow 2h\nu_\tau)&=&
 \left\{
    \bar{L}_{B} {W}_{B} 
%  + \bar{L}_{SA}{W}_{SA}(1+\tilde\eta_S^d)^2
  + \bar{L}_{SA}{W}_{SA}
  + \bar{L}_{SF}{W}_{SF} 
%  + \bar{L}_{SG}{W}_{SG})(1+\tilde\eta_S^d)  \right\}\nonumber\\[2mm]
  + \bar{L}_{SG}{W}_{SG}  \right\}\nonumber\\[3mm]
&&
\hspace{-2cm}
\frac{G^{2}}{2\mt} 
\sin^2\theta_{c}
\frac{1}{(4\pi)^{3}}
\frac{(\mt^{2}-Q^{2})^{2}}{\mt^{2}}\,
\,|\vec{q}_{1}|\,\,
     \frac{dQ^{2}}{\sqrt{Q^{2}}}  \, \frac{d\cos\theta}{2}\,
     \frac{d\alpha}{2\pi}\,
     \frac{d\cos\beta}{2}\label{gamma}\>.
\end{eqnarray}
%
%
For the definition and discussion of the angles and leptonic
coefficients $\bar{L}_X$ in Eq.~{(\ref{gamma})}
we refer the reader to Ref.\cite{cp1}.
The coefficients $\bar{L}_X$ contain all  $\alpha, \beta$ and $\theta$
angular  and $\tau$-polarization dependence.
The hadronic structure functions $W_X$,
$X\in\{B,SA,SF,SG\}$, depend only on
$Q^2$ and the form factors $F$ and $\tilde{F}_S$ of the hadronic current.
One has\cite{kpi,km1,cp1}:
%
%
\begin{equation}
\begin{array}{ll}
W_B[\tau^-]    = 4 (\vec{q}_1)^2\,|F|^2 &
\hspace{1cm}W_{SF}[\tau^-]
 = 4\sqrt{Q^2}|\vec{q}_1|\, \mbox{Re}\left[F\tilde{F}_S^*\right]       \\
W_{SA}[\tau^-] = Q^2\,  |\tilde{F}_S|^2 
&
\hspace{1cm}W_{SG}[\tau^-]
 =-4\sqrt{Q^2}|\vec{q}_1|\, \mbox{Im}\left[F\tilde{F}_S^*\right]
\end{array}
\label{wsg}
\end{equation}
%
%
The hadronic structure functions $W_X[\tau^+]$  are
obtained by the replacement $\eta_S\rightarrow\eta_S^*$
in $\tilde{F}_S$ in Eqs.~(\ref{wsg},\ref{fs}).
CP conservation implies that all four structure functions are identical
for $\tau^+$ and $\tau^-$. With the ansatz 
for the form factors formulated in Eq.~(\ref{me2h})
CP violation can be present in $W_{SF}$ and $W_{SG}$
only and requires complex $\eta_S$.


As demonstrated in Ref.\cite{cp1} $W_{SF}$ can be measured in 
$e^+e^-$ annihilation experiments in the study of single unpolarized
$\tau$ decays even if the $\tau$ rest frame cannot be reconstructed.
In this respect the result differ from earlier studies of the two 
meson modes where either polarized beams and reconstruction
of the full kinematics\cite{tsaicp} or correlated fully reconstructed
$\tau^-$ and $\tau^+$ decays were required\cite{nelson1}.
The determination of $W_{SG}$, however, requires the knowledge of the full
$\tau$ kinematics and $\tau$  polarization\cite{cp1}
which is possible with the help of vertex detectors.

The crucial observation made in Ref.\cite{cp1} is that
one can measure the following CP-violating differences
\begin{equation}
\Delta W_{SF} = \frac{1}{2}
\left( W_{SF}[\tau^-] -  W_{SF}[\tau^+]\right)
\hspace{5mm}
\Delta W_{SG} = \frac{1}{2}
\left( W_{SG}[\tau^-] -  W_{SG}[\tau^+]\right)
\end{equation}
under the above mentioned conditions.
The hadronic structure functions
$W_X[\tau^-]$ and $W_X[\tau^+]$ differ only in the phase os
$\eta_S$ and one obtains
%
%
\begin{equation}
\Delta W_{SF}
\,\,= \,\,\, 4\sqrt{Q^2}|\vec{q}_1|\, \,
%\frac{Q^2}{m_{\tau}(m_u-m_d)}\,\,\,\mbox{Im}\left(FF_S^*\right)\,\,
\frac{1}{m_{\tau}}\,\,\mbox{Im}\left(F F_H^*\right)\,\,
              \,     \mbox{Im}\left(\,\eta_S \right) \>.
\label{dwsf}
\end{equation}
%
% 
In essence the measurement on $\Delta W_{SF}$ 
analyses the difference in the
correlated energy distribution
of the mesons $h_1$ and $h_2$ from $\tau^+$ and $\tau^-$
decay in the laboratory. 
As already mentioned, $\Delta W_{SF}$ is observable for single
$\tau^+$ and $\tau^-$ decays without knowledge of the
$\tau$ rest frame. Any nonvanishing experimental result for 
$\Delta W_{SF}$ would be a clear signal of CP violation.
Note that a nonvanishing $\Delta W_{SF}$ requires nontrivial hadronic phases
(in addition to the CP violating phases $\eta_S$) in the form factors
$F$ and $F_H$. Such hadronic phases in $F$ ($F_H$) originate
in the $K\pi\nu_\tau$ decay mode from complex Breit Wigner
propagators for the $K^*$ ($K_0^*$) resonance. Sizable effects
of these hadronic phases are expected in this decay mode\cite{kpi}.

Once the $\tau$ rest frame is known and a preferred direction
of polarization exists
one may also determine $\Delta W_{SG}$ which is
theoretically given by
%
% 
\begin{equation}
\Delta W_{SG}
\,\,= \,\,\, 4\sqrt{Q^2}|\vec{q}_1|\,\,\,
%\frac{Q^2}{m_{\tau}(m_u-m_d)}
\frac{1}{m_{\tau}}
\,\,\, \mbox{Re}\left(F F_H^*\right)
              \,\,     \mbox{Im}\left(\eta_S \right) \>.
\label{dwsg}
\end{equation}
%
% 
(This preferred direction of polarization could be substituted by the
analysis of correlations between $\tau^+$ and $\tau^+$ decay modes in
one and the same event.)
Any observed nonzero value of $\Delta W_{SF}$ or $\Delta W_{SG}$ would signal a 
true CP violation. Eqs.(\ref{dwsf}) and (\ref{dwsg}) show that the sensitivity
to CP violating effects in $\Delta W_{SF}$ and $\Delta W_{SG}$
can be fairly different depending on the hadronic phases.
Whereas $\Delta W_{SF}$ requires nontrivial hadronic phases
$\Delta W_{SG}$ is maximal for fixed $\eta_S$ in the absence of hadronic
phases.
\vspace*{-2mm}

\subsection{Three  Meson Decays}
\vspace*{-2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The structure function formalism\cite{km1}
allows also for a systematic analysis of possible CP
violation effects in the three meson case.
Some of these effects have already been briefly discussed in
Ref.\cite{argonne}. The $K\pi\pi$ and $KK\pi$ decay modes
with nonvanishing vector \underline{and} axial vector
current are of particular importance for the detection of possible
CP violation originating from exotic intermediate vector bosons.
This would be signalled by a nonvanishing difference between
the structure functions $W_X(\tau^-)$ and $W_X(\tau^+)$
with $X\in\{F,G,H,I\}$. A difference in the structure functions
with $X\in\{SB,SC,SD,SE,SF,SG\}$ can again be induced through
a CP violating scalar exchange.


\section{Towards Order   $\alpha_s^4$ Accuracy in  $\tau$-decays}

The dependence of the $\tau$-decay rate on the strong coupling
$\alpha_s$ has been used for a determination of $\as$ at lower
energies, with the most recent results of $0.334 \pm 0.007_{exp} \pm
0.021_{theo}$ and  $0.3478\pm 0.009_{exp} \pm 0.019_{theo}$ by the 
ALEPH\cite{exp1} and OPAL\cite{exp2} collaborations, which  after evolution up
to higher energies agree remarkably well with determinations based on
the hadronic $Z$ decay rate. In view of the relatively large value of
$\alpha_s(M_{\tau}) $ estimates for the yet unknown terms of higher
orders play an important role in current determinations of $\as$ at
low energies. This is in contrast to high energy measurements, where
the uncertainty from terms of order $ \alpha_s^4 $ of 0.001 to 0.002
 is somewhat less or at most comparable to 
to the present experimental errors\cite{phys_report}.

The situation is even more problematic for the determination of the
strange quark mass from the Cabbibo suppressed $\tau$
decays. Perturbative QCD corrections affecting the $m_s^2$ term are
extremely large and contributions from increasing powers of $\as$ are
barely decreasing, which casts doubts on our ability to extract a
reliable result for $m_s$ from this (in principle) clean and
straightforward measurement\cite{Maltman:1998qz,Chetyrkin93,%
Chetyrkin:1998ej,pichprades}.

% [we, mAltman, Aleph, Pich, ...].

Partial results of order $\alpha_s^4$ for the absorptive part of the
massless vector and scalar correlators have been obtained recently\cite{Baikov:2001aa}, 
namely terms proportional to $n_f^2$, where
$n_f$ denotes the number of massless fermion species. These allow test
two popular optimization schemes --- based on the principles of
``minimal sensitivity'' (PMS) and of fastest apparent convergence\cite{ste,gru}
(FAC) --- which have been used to predict yet
uncalulated higher order terms\cite{KS,Kataev:1994wd}.


It will be demonstrated that the predictions of both schemes
 for the coefficient of order
$n_f^2\alpha_s^4$ are in reasonable agreement with our calculations,
which are then used to predict the complete fixed order (FO) and the
"contour improved" (CI)\cite{tau:resum,DP} $\mathcal{O}(\alpha_s^4)$
contributions to $\tau$ decay rate. Employing the four-loop QCD beta
function in combination with improved $\as^4$ terms a rough estimate
even for $\mathcal{O}(\alpha_s^5)$ terms can be obtained.  The results
lead to fairly stable values for $\alpha_s$ consistent with current
analysises.

Unfortunately, no essential decrease of the difference between the
values $\as(M_{\tau})$ as obtained with FI and CI approaches
correspondingly is observed after including higher order $\as^4$ and
$\as^5$ corrections, assuming for the moment that these estimates are
indeed correct.  On the other hand, the theoretical uncertainty
assigned to $\as(M_{\tau})$ {\em within} each method does decrease
significantly. 
 



The implication of this approach for the extraction of $m_s$ from
Cabbibo suppressed decays will be investigated along the same lines.
New results,  based on Refs.\cite{Baikov:2001aa,preparations} will be
presented for the terms of order $m_s^2 n_f \as^3$ in the total rate.
In this case the agreement between FAC/PMS predictions and our results
is quite encouraging and naturally suggests the use of the former as a
reliable prediction for the complete $m_s^2 \as^3$ term. Following an
approach discussed in \cite{Kataev:1994wd}, even a rough estimate
$m_s^2\alpha_s^4 n_f^2$ terms can be obtained from these
considerations.


However, the rapid increase of the coefficients indicates, that
inherent uncertainty of the present $m_s$ determinations will be
worsened by the higher orders. As we will see, the situation is
somewhat better for the spin one contribution if considered
separately.

%Our starting point will be
We start with the well-known representation
%\cite{SchTra84,Bra88,NarPic88,Bra89,BraNarPic92} 
of the tau-lepton hadronic rate
 as the contour integral along a
circle C of the radius
$|s|=M_{\tau}^2$
\beq 
R_{\tau}=6i\pi\int_{|s|=M_{\tau}^2}\frac{ds}{M_{\tau}^2}
\left(1-\frac{s}{M_{\tau}^2}\right)^2
\left[\Pi^{[2]}(s)
-\frac{2}{M_{\tau}^2}\Pi^{[1]}(s)\right]
\label{pi1andpi2}
{}.
\eeq

Here $\Pi^{[1]}$ and $\Pi^{[2]}$
are proper flavour combinations of the polarization   operators
appearing in the following decomposition of the  correlators 
of  vector and axial vector currents of light quarks
proportional to $g_{\mu\nu}$ and $q_{\mu} q_{\nu}$ respectively.


The full tau-lepton hadron rate  $R_{\tau}$ can be presented as a sum of 
spin 1 and spin 0 parts, viz.
\beq \label{6b}
\ba{ll}
\dsp
R^{(1)}_{\tau} &= 
\dsp
6i\pi\int_{|s|=M_{\tau}^2}\frac{ds}{M_{\tau}^2}
\left(1-\frac{s}{M_{\tau}^2}\right)^2
\left[ \left(1+2\frac{s}{M_{\tau}^2}\right) \Pi^{(1)}(s)
+\Pi^{[1]}(0)/s
\right]
{},
\\
\dsp
R^{(0)}_{\tau} &= 
\dsp
6i\pi\int_{|s|=M_{\tau}^2}\frac{ds}{M_{\tau}^2}
\left(1-\frac{s}{M_{\tau}^2}\right)^2
\left[\Pi^{(0)}(s)
-\Pi^{[1]}(0)/s
\right]
{}.
\ea
\eeq
where 
\beq
\Pi^{(1)}  = -\Pi^{[1]}/q^2, \ \  
\Pi^{(0)}  = \Pi^{[2]} +    \Pi^{[1]}/q^2
{}.
\label{Pi12TOPi01}
\eeq
and the contribution of the singularity at the origin 
(proportional to $\Pi^{[1]}(0)$) has to be included.
A nonvanishing  value of $\Pi^{[1]}(0)$   
is   a nonperturbative 
constant. 

On the other hand, the unknown constant drops out if one
considers moments 
\beq
R^{(1,0)k,l}_{\tau}(s_0) =  
\int_0^{s_0}
\frac{ds}{M_{\tau}^2}
\left(1-\frac{s}{M_{\tau}^2}\right)^{k} 
\left(\frac{s}{M_\tau^2}\right)^l
\frac{d R^{(1,0)}_\tau }{ds}
{},
\label{def:moments}
\eeq
with $k \ge 0, \ \ l \ge 1$.

The integral in eq. (\ref{pi1andpi2}) is, obviously, insensitive,
to  the $Q^2$-independent terms in the polarization functions
$ \Pi^{[1]}_{0}/Q^2$ and $\Pi^{[1]}_{2} {}$.
This means that without  loss of generality we may 
deal with the corresponding (Adler) $D$-functions, viz.
\[
D^{[1]}_0(Q^2) \equiv \frac{1}{4}Q^2 \frac{ {\rm d}}{{\rm d}\  Q^2}
\Pi^{[1]}_{0}/Q^2 , 
\   \  \
D^{[1]}_2(Q^2) \equiv -\frac{1}{2}Q^2 \frac{ {\rm d}}{{\rm d}\  Q^2}
\Pi^{[1]}_{2} 
{}.
\]
An important property of the functions  $ D^{[1]}_0, D^{[1]}_2$ and
$ \Pi^{[2]}_{2} $ is their scale independence,  which  implies that 
they are directly related to measurements.

Using the technique described in\cite{Baikov:2001aa}
the leading and subleading (in $n_f$) terms of the next  order in the
perturbative series for $ D^{[1]}_0$, $ D^{[1]}_2$ are obtained:
%They are displayed below in three equations. 

\bea
D^{[1]}_0 &=& 1 + \apiq + \left(\apiq\right)^2( - 0.115295 n_f^2 +1.98571 )
\nonumber
\\
&+&\left(\apiq\right)^3( 0.0862069 n_f^3  - 4.21585 n_f +18.2427 )
\label{D1m0:exact}
\\
&+&\left(\apiq\right)^4( - 0.0100928 n_f^3 + 1.87525 n_f^2 
+ d^{[1]4}_{0,1} n_f +  d^{[1]4}_{0,0})
\nonumber
\eea 



\bea
D^{[1]}_2 &=& 1 + \apiq 1.66667 + 
\left(\apiq\right)^2(- 0.573629 n_f + 2.5668 )
\nonumber
\\
&+&\left(\apiq\right)^3(0.16463 n_f^2   - 14.3097 n_f + 148.978)
\\
\label{D1m2:exact}
&+&\left(\apiq\right)^4( - 0.0156289 n_f ^3 + 6.06677 n_f^2 
+ d^{[1]4}_{2,1} n_f +  d^{[1]4}_{2,0})
\eea 



\bea
\Pi^{[2]}_2 &=& -4 m_s^2 \left(1 + \apiq \frac{7}{3}  + 
\left(\apiq\right)^2(23.5094 - 1.30879 n_f )
\nonumber
\right.
\\
&+&\left(\apiq\right)^3
\left.
(0.464663 n_f^2 - 32.0843 n_f  + k^{[2]3}_{2,0})\right)
\label{Pi2m2as3}
\eea 
The result for  the  $\as^4$ terms in   $D^{1]}_0$  has been already  
given in\cite{Baikov:2001aa}, the coefficients of the $\as^4$  and  $\as^3$ 
terms in  in $D^{[1]}_2$ and $\Pi^{[2]}_2$ respectively have been 
obtained in\cite{preparations}.

\section{Massless case: phenomenological implications}

Both FAC (Fastest Apparent Convergence) and PMS (Principle of Minimal
Sensitivity) methods have been used to "predict" (at least roughly)
the magnitude of higher order corrections.  
% \cite{Kataev:1994wd}.
For the function $D^{[1]}_0$ the result is known since long from the
work by Kataev and Starshenko\cite{KS}.



\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$n_f$ & $d_{3} ^{\rm exact}$ & $d_3^{\rm FAC}$ & $d_3^{\rm PMS}$ &
$d_4^{\rm FAC/PMS}$  & $d_5^{\rm FAC}$\\
\hline\hline
3 & $ 6.371$ & $ 5.604$ & $ 6.394 $ & $ 27 \pm 16 $ & $ 145 \pm 100$\\
\hline
4 & $ 2.758$ & $ 4.671$ & $ 5.264 $ & $ 8  \pm 28 $ & $ 40  \pm  160$\\
\hline
5 & $ -0.68$ & $ 3.762$ & $ 4.160 $ & $ -8 \pm 44 $ & $ -3   \pm 230$\\
\hline
\end{tabular}
\end{center}
\caption{\label{t1}
Estimates for  the  coefficients $d_3 = d^{[1]3}_{0}$ and $d_4 = d^{[1]4}_{0}$ 
in the function    $D^{[1]}_0 $
based on  FAC and PMS optimizations. 
The estimate for  of $d^{[1]4}_{0}$ employs the exact value of $d^{[1]3}_{0}$.
The last column contains the FAC predictions for the coefficient  $d_5^{\rm FAC}$
which was obtained assuming the value for $d^{[1]4}_{0}$ as given in 
the fifth column; the corresponding uncertainties  have been  estimated as 
described in the  text.} 
\end{table}



From the predictions corresponding to  $n_f = 3,4$ and $5$ one easily
restores the FAC/PMS  prediction   for the $n_f$ dependence
of the $\as^4$ term in  $D^{[1]}_{0}$  
\beq                                     
d^{[1]4}_{0} = 127.58 - 44.211 n_f + 3.6439 n_f^2   - 0.0100928 n_f^3 
{}.
\label{K4:nf}
\eeq

It is also interesting to compare the FAC/PMS predictions  
for the $n_f$ dependence of coefficient $d^{[1]3}_{0}$ with the exact result 
given in  eq. \re{D1m0:exact}. The results  of both estimates are 
\beq                                       
d^{[1]3}_{0}(FAC) =   8.54 - 1.013 n_f + 0.0116 n_f^2
\label{d1m0as3:FAC}
{},
\eeq
\beq                                   
d^{[1]3}_{0}(PMS) =   9.93 - 1.23 n_f + 0.0125 n_f^2 
\label{d1m0as3:PMS}
{}.
\eeq

The comparison  of the complete $\as^3$ and partial $\as^4$ results with 
FAC and PMS  estimates leads to following observations:
\begin{itemize}
\item  Starting from $\as^3$, the  terms of
order $\as^3 n^2_f$ and $\as^4 n_f^3$ 
 are numerically quite small (at least for $n_f \le 6$)
and, thus, should have a negligibly small influence on the
coefficients of the $\as$ expansion.  On the other hand, the
subleading in $n_f$ term, say,  of order $\as^3 n_f$ is comparable in size
%with the remaning unknown terms of
with the term of order $\as^3 n_f^0$. 
Similarly, the $\as^4 n_f^3$ term is significantly smaller than  the 
$\as^4 n_f^2$ one,  whereas the $\as^4 n_f$ and $\as^4 n_f^0$
terms are expected to be of similar magnitude.   
%One observes a similar picture at $\as^3$ order.

\item  In general FAC/PMS methods correctly reproduce sign and order of
magnitude of the higher order coefficients. The agreement is getting
better when the coefficients happen to be large, as is the case for
the quadratic mass corrections (see below).

%The predictions should in all
%probability be correct within $\pm 50\%$.
    
\item  Taken separately, the FAC/PMS estimates of the coefficients of
the $n_f$ expansion could deviate rather strongly from the true
result.  However, for a given $n_f$, the deviation in the predicted
value of the full $O(\as^n)$ term tends to be significantly smaller
than what could be expected from summing individual terms of the $n_f$
expansion.  In addition, for the particular point $n_f = 3$  very good
agreement is observed.
\end{itemize}

These  coefficients  can be used to predict 
$R_{\tau}$  in the massless limit  first  in 
fixed order  perturbation theory ($a = \alpha_s(M_{\tau})/\pi$)
\beq
R_{\tau}^{FOPT} = 3 \left(  
1 + a + 5.202 a + 26.37 a^3 +  a^4 \ (105 \pm 16) + a^5 (138 \pm 213 \pm 100) 
    \right) 
\label{Rtau:FO}
{}.
\eeq

Here the first uncertainty in the $\as^5$  term comes from that of 
$d^{[1]4}_0$  while the second is our estimation\cite{preparations}
of the error in 
the very  coefficient  $d^{[1]5}_0$. 
Similarly, we can use  "contour improved" formulae\cite{tau:resum,DP} 
to get 
\beq
R_{\tau}^{CI} = 3 \left(  
1+ 1.364a + 2.54 a^2 + 9.71 a^3 + a^4 (35 \pm 20) + a^5 (138  \pm 95)
                    \right) 
\label{Rtau:CI}
{}.
\eeq
\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|} 
\hline
$d^{[1]4}_0$  & $\as^4$ & $d^{[1]5}_0$ & $\as^5$ 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\ 
\hline
 27 & $0.327 \pm 0.012$ & 145 & $0.326 \pm 0.004$ 
\\ 
\cline{2-2}\cline{4-4} 
  &   $0.351 \pm 0.005$ &  &$0.349 \pm 0.0015$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\ 
\hline
 43 & $0.326 \pm 0.014$ & 245 & $0.321 \pm 0.006$ 
\\ 
\cline{2-2}\cline{4-4} 
  &   $0.347 \pm 0.0065$ &  &$0.343 \pm 0.002$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\ 
\hline
 11 & $0.329 \pm 0.011$ & 45 & $0.332 \pm 0.0095$ 
\\ 
\cline{2-2}\cline{4-4} 
  &   $0.355 \pm 0.0035$ &  &$0.355 \pm 0.001$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline 
\end{tabular}
\end{center}
\caption{\label{t2}
The predicted value of $\as(M_{\tau})$ in dependence of 
chosen  values for the coefficients 
$d^{[1]4}_0$, $d^{[1]5}_0$.
The third and the forth  columns differ in the the number of terms in the
perturbative series included. 
The upper value of $\as$ is the one predicted within FOPT, the
lower  corresponds CIPT. The uncertainty in the value of $\as$ corresponds 
to changing the  normalization point $\mu$ as follows $\mu^2/M_{\tau}^2 = 0.4 - 2.$
} 
\end{table}


It is of interest to see in detail which accuracy in the determination
of $\alpha_s(M_{\tau})$ one could achieve assuming
that the term of order $\as^4 $ and $\as^5 $ are known. 
%eqs. (\ref{d1m0as4nf3:num},\ref{d1m0as5nf3:num}).
It is natural to introduce the 
quality $\delta_P$ as follows:  
\beq
R_{\tau S=0} = \
\frac{\Gamma(\tau \to h_{S=0} \nu)}{\Gamma(\tau \to l \bar{\nu}\nu)}
=  |V_{ud}|^2 S_{EW} R_{\tau}
{},
\label{delta_exp}
\eeq
with (in obvious notataion)
$
R_{\tau} = 3 (1 + \delta_P + \delta_{EW} + \delta_{NP}) 
$
The first term here is the parton result,  the second  stands for pQCD effects.
The non-perturbative corrections  represented by  $\delta_{NP}$
happens to be  rather small and  will be negelected   in the  following. 
The factor $S_{EW} = 1.0194 $  is the electroweak correction   
which collects the large logarithmic terms\cite{ewcorr1}, 
while $\delta_{EW} = 0.001$ is an additive not logarithmically enhanced 
electroweak correction\cite{ewcorr2}.  
Using for definiteness the result of ALEPH 
one arrives at
$\delta_P^{exp} =  0.203 \pm 0.007 {}$. 


Unfortunately, there is no 
unique way to assign a
theoretical uncertainty $\delta {\as}$ to the obtained value of  
$\alpha_s(M_{\tau})$. In the literature one finds several suggestions.
Let us consider them in turn. 

\begin{enumerate}

\item  $\delta {\as}$ is  a half of the shift in $\alpha_s$ induced by the last computed
        term in the PT (that is by the one of order $\as^3$ at present).

\item    $\delta {\as}$ is equal to the  change  in $\alpha_s$ caused by varying the normalization point
     $\mu$ around $M_{\tau}$, typically within the range of $1.1 -- 2.5$ GeV.

The corresponding results read (terms of order $\as^4$ and higher in 
 eq.     \re{Rtau:FO}  and in    the $D$-function have been set to zero)
\bea
\as^{FOPT} &=&   0.340 \pm   (0.027|0.033   ) 
{},
\label{asFOas3}
\\
\as^{CIPT} &=&    0.358  \pm  (0.011| 0.016  ) 
{}.
\label{asCIas3}
\eea
%{$0.34 \pm 0.028$, $0.358 \pm 0.013$}
Here the first(second) value in brackets  correspond to the
use of the first(second) suggestion for the error  estimation.   

\item  
 $\delta {\as}$ is  equal to the  change in $\alpha_s$ caused by the uncertainity in the 
    predicted (that is not yet completely known) higher order 
    terms in the perturbative series for $R_{\tau}$;

\item  $\delta {\as}$ is  a  half of the difference  in the    $\alpha_s(M_{\tau})$  as 
    obtained within FOPT and CIPT. This difference 
    comes from different suggestions about  the behaviour of PT in higher orders.
\end{enumerate}
In order to quantify the error estimates according to 2. -- 4.  we
have shown in Table \ref{t2} the results for $\as$ obtained with
various choices for $d^{[1]4}_0$, $d^{[1]5}_0$ and $\mu$. 
For the plausible values of $d_0^{[1]4}$ and $d_0^{[1]5}$ we
observe an significant  decrease of the $\mu$ dependence  after
inclusion  of  additional terms in the $\as$ series. However, as discussed
before, the large difference between FOPT and CIPT results remains unaffected 
if $\as^4$ terms and even $\as^5$ based on FAC  estimate are included. 

\section{Quadratic mass corrections}

A comparison of estimates and the exact results reveals for the mass
corrections a picture qualitatively similar to the massless case but
with some modifications.  Let us just present the final result of
thsese considerations. In a fixed order calculation
one obtaines for the mass correction to the total rate 
\bea
\delta^{00}_{us,2}  &=& -8 \frac{m_s^2}{M_{\tau}^2}(1+0.567 +0.520 + 0.521 + 0.592)
\nonumber
\\
&=& 
-8 \frac{m_s^2}{M_{\tau}^2} (3.2 \pm 0.6)
\eea
where we have displayed separately  the  terms of increasing  order in $\as$.
However, one must admit  that   terms  of order
$\as$, $\as^2$, $\as^3$  and $\as^4$   
are all of comparable  size, casting doubt on the stability of
the series!





\sloppy
\raggedright
\def\app#1#2#3{{{\it} Act. Phys. Pol. }{{\bf} B #1} (#2) #3}
\def\apa#1#2#3{{{\it} Act. Phys. Austr.}{{\bf} #1} (#2) #3}
\def\lhc{Proc. LHC Workshop, CERN 90-10}
\def\npb#1#2#3{{{\it} Nucl. Phys. }{{\bf} B #1} (#2) #3}
\def\plb#1#2#3{{{\it} Phys. Lett. }{{\bf} B #1} (#2) #3}
\def\partial#1#2#3{{{\it} Phys. Rev. }{{\bf} D #1} (#2) #3}
\def\pR#1#2#3{{{\it} Phys. Rev. }{ {\bf} #1} (#2) #3}
\def\prl#1#2#3{{{\it} Phys. Rev. Lett. }{{\bf} #1} (#2) #3}
\def\prc#1#2#3{{{\it} Phys. Reports }{{\bf} #1} (#2) #3}
\def\cpc#1#2#3{{{\it} Comp. Phys. Commun. }{{\bf} #1} (#2) #3}
\def\nim#1#2#3{{{\it} Nucl. Inst. Meth. }{{\bf} #1} (#2) #3}
\def\pr#1#2#3{{{\it} Phys. Reports }{{\bf} #1} (#2) #3}
\def\sovnp#1#2#3{{{\it} Sov. J. Nucl. Phys. }{{\bf} #1} (#2) #3}
\def\jl#1#2#3{{{\it} JETP Lett. }{{\bf} #1} (#2) #3}
\def\jet#1#2#3{{{\it} JETP Lett. }{{\bf} #1} (#2) #3}
\def\zpc#1#2#3{{{\it} Z. Phys. }{\bf C #1}, #3  (#2) }
\def\ptp#1#2#3{{{\it} Prog.~Theor.~Phys.~}{{\bf} #1} (#2) #3}
\def\nca#1#2#3{{{\it} Nouvo~Cim.~}{{\bf} #1A} (#2) #3}
\def\mpl#1#2#3{{{\it} Mod. Phys. Lett.~}{{\bf} A #1} (#2) #3}



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