%Title: Towards Order alpha_s^4 Accuracy in tau-decays
%Author: P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn
%Published: * Phys.Rev.D * ** 67 ** (2003) 074026-074039.
%hep-ph/0212299
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\begin{document}
\hfill THEP 02/17\\
\mbox{}
\hfill TTP02--41\\
\mbox{}
\hfill hep-ph/0212299 \\
\title{Towards Order $\alpha_s^4$ Accuracy in $\tau$-decays\\}
% Force line breaks with \\
\author{P.A.~Baikov}
\email{baikov@theory.sinp.msu.ru}
\affiliation{
Institute of Nuclear Physics,
Moscow State University
Moscow~119992, Russia\\
}%
\author{K.G.~Chetyrkin}
\altaffiliation[On leave from ]
{Institute for Nuclear Research of the Russian Academy of Sciences,
Moscow, 117312, Russia}
%Lines break automatically or can be forced with \\
\email{chet@particle.physik.uni-karlsruhe.de}
\affiliation{%
Fakult{\"a}t f{\"u}r Physik
Albert-Ludwigs-Universit{\"a}t Freiburg,
D-79104 Freiburg, Germany\\
}
\author{J.H.~K{\"u}hn}
\email{Johann.Kuehn@physik.uni-karlsruhe.de}
\affiliation{Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe,
D-76128 Karlsruhe, Germany\\
}%
\date{\today}% It is always \today, today,
% but any date may be explicitly specified
\begin{abstract}
Recently computed terms of orders $\mathcal{O}(\alpha_s^4 n_f^2)$ in
the perturbative series for the $\tau$ decay rate, and similar (new)
strange quark mass corrections, are used to discuss the validity of
various optimization schemes. The results are then employed to arrive at
improved predictions for the complete terms order
$\mathcal{O}(\alpha_s^4)$ and $\mathcal{O}(\alpha_s^5)$ in the
massless limit as well as for terms due to the strange quark mass.
Phenomenological implications are presented.
\end{abstract}
\pacs{12.38.Bx, 13.35.Dx, 13.66Bc }% PACS, the Physics and Astronomy
% Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
%display desired
\maketitle
\section{Introduction}\label{sec:intro}
The dependence of the $\tau$-decay rate on the strong coupling
$\alpha_s$ has been used for a determination of $ \alpha_s $ at lower
energies, with the most recent results of $0.334 \pm 0.007_\mathrm{exp} \pm
0.021_\mathrm{theo}$ and $0.3478\pm 0.009_\mathrm{exp} \pm 0.019_\mathrm{theo}$
by the ALEPH
\cite{exp1} and OPAL \cite{exp2} collaborations. After evolution up
to higher energies these results 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 $ \alpha_s $ 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
\cite{phys_report} is somewhat less or at most comparable
to the present experimental errors.
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 $ \alpha_s $ 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,Chetyrkin93-update,Chetyrkin:1998ej,pichprades,%
krajms,chenpich,Maltman:2002wb}.
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 to test
two popular optimization schemes --- based on the principles of
``minimal sensitivity'' (PMS) and of fastest apparent convergence
(FAC) \cite{ste,gru,krasn}
--- 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
(coinciding at $ \alpha_s ^4$) 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 the $\tau$ decay rate. Employing the four-loop QCD beta
function in combination with improved $ \alpha_s ^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
central values $ \alpha_s (M_{\tau})$ as obtained with FI and CI approaches
is observed after including order $ \alpha_s ^4$ and $ \alpha_s ^5$ corrections,
assuming for the moment that these estimates are indeed correct. On
the other hand, the theoretical uncertainty assigned to
$ \alpha_s (M_{\tau})$ {\em within} each method according to standard
techniques 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 will be presented for the terms of order $ n_f \alpha_s ^3\,
m_s^2$ 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 $ \alpha_s ^3 m_s^2$ term. Following an approach discussed in
\cite{Kataev:1994wd}, even a rough estimate $m_s^2\alpha_s^4$
terms can be obtained from these considerations.
However, the rapid increase of the coefficients indicates that
the inherent uncertainty of the present $m_s$ determinations will
not necessarily decrease with inclusion
of the higher orders. As we will see, the situation is
somewhat better for the spin one contribution if considered
separately.
\section{Generalities}
%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 radius
$|s|=M_{\tau}^2$
\begin{equation}
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}
%\label{nlab1}
{}.
\end{equation}
Here $\Pi^{[1]}$ and $\Pi^{[2]}$
are proper flavour combinations of the polarization operators
appearing in the decomposition of the correlators
of vector and axial vector currents of light quarks
\begin{equation}
\begin{array}{ll}
\Pi^{V/A}_{\mu\nu,ij}(q,m_i,m_j,m{},\mu,\alpha_s) & =
\displaystyle i \int dx e^{iqx}
\langle
T[\, j^{V/A}_{\mu,ij}(x) (j^{V/A}_{\nu,ij})^{\dagger} (0)\, ] \rangle
\\ & = \displaystyle
g_{\mu\nu} \Pi^{[1]}_{ij,V/A}(q^2)
+ q_{\mu}q_{\nu}
\Pi^{[2]}_{ij,V/A}(q^2)
{}
\end{array}
\label{correlator}
\end{equation}
with $m^2 = \sum_{f=u,d,s} m_f^2$
and $j^{V/A}_{\mu,ij} = \bar{q}_i\gamma_{\mu}(\gamma_5) q_j$.
The two (generically different) quarks with
masses $m_i$ and $m_j$ are denoted by $q_i$ and $q_j$ respectively.
For the case of the $\tau$-lepton the relevant combinations
of quark flavours are $ij = ud$ and $ij = us$.
The polarization functions $\Pi^{[l]}_{V/A}, \ \ l=1,2$
are conveniently represented in
the form ($Q^2 = -q^2$)
\begin{equation}
(Q^2)^{(l-2)}\Pi^{[l]}_{us,V/A}(q^2)
=
\frac{3}{16\pi^2}\Pi^{[l]}_{V/A,0}(\frac{\mu^2}{Q^2}, \alpha_s)
+
\frac{3}{16 \pi^2}
\sum_{D \ge 2} Q^{-D}\Pi^{[l]}_{V/A,D}(\frac{\mu^2}{Q^2},
m_s^2,\alpha_s)
{}.
\label{mass-exp}
\end{equation}
Here the first term on the rhs corresponds to the massless
limit while the first term in the sum
stands for quadratic mass corrections. We neglect the
masses of $u$ and $d$ quarks. Therefore in perturbative QCD
$\Pi^{[l]}_{V} = \Pi^{[l]}_{A}$
and we will often omit the subscript $V/A$ in the following.
Current conservation implies:
$
\Pi^{[1]}_{0} = \Pi^{[2]}_{0}
$.
The full tau-lepton hadron rate $R_{\tau}$ can be presented as a sum of
spin 1 and spin 0 parts, viz.
\begin{equation}
\label{spin:decomp}
\begin{array} {ll}
\displaystyle
R^{(1)}_{\tau} &=
\displaystyle
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]
{},
\\
\displaystyle
R^{(0)}_{\tau} &=
\displaystyle
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]
{}.
\end{array}
\end{equation}
where
\begin{equation}
\Pi^{(1)} = -\Pi^{[1]}/q^2, \ \
\Pi^{(0)} = \Pi^{[2]} + \Pi^{[1]}/q^2
{}.
\label{Pi12TOPi01}
\end{equation}
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
\begin{equation}
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}
\end{equation}
with $k \ge 0, \ \ l \ge 1$.
(Note that the moments introduced in \cite{DP}
are related to ours as
$
R^{kl}_\tau = R^{(1)k,l}_\tau + R^{(0)k,l}_\tau $.)
The decay rate
$R_{\tau}$ may be expressed as the sum of different contributions
corresponding to Cabibbo suppressed or allowed decay modes, vector or
axial vector contributions and the mass dimension of the corrections
\begin{equation} \label {7}
R_{\tau} = R_{\tau,V} + R_{\tau,A} + R_{\tau,S}
\end{equation}
with
\begin{equation} \label {8}
\begin{array} {ll} \displaystyle
R_{V} =
& \displaystyle
\frac{3}{2}|V_{ud}|^2 \left( 1 + \delta_{0} + \sum_{D=2,4,\dots}
\delta_{V,ud,{D}} \right), \\
R_{A} =
& \displaystyle
\frac{3}{2}|V_{ud}|^2 \left( 1 + \delta_{0} + \sum_{D=2,4,\dots}
\delta_{A,ud,{D}} \right), \\
R_{S} =
& \displaystyle
3|V_{us}|^2 \left( 1 + \delta_{0} + \sum_{D=2,4,\dots}
\delta_{us,{D}} \right).
\end{array}
\end{equation}
Here $D$ indicates the mass dimension of the fractional corrections
$\delta_{V/A,ij,{D},}$ and $\delta_{ij,{D}}$ denotes the average of
the vector and the axial vector contributions:
$\delta_{ij,{D}}=(\delta_{V,ij,{D}}+\delta_{A,ij,{D}})/2$. If
a decomposition into different spin/parity contributions is made or a
particular pattern of moments is considered then we will use the
corresponding obvious generalization of (\ref{8}).
For instance,
\begin{equation}
R^{(1)kl}_{S,V} = a_{kl} \, |V_{us}|^2 \left( 1 + \delta^{kl}_0 + \sum_{D=2,4,\dots}
\delta^{(1),kl}_{V,us,D} \right)
{},
\end{equation}
and
\begin{equation}
R^{(0)kl}_{S,V} =
|V_{us}|^2 \left( \sum_{D=2,4,\dots}
\delta^{(0),kl}_{V,us,D} \right).
\end{equation}
Thus, in our notation we have the relation
\begin{equation}
\delta^{kl}_{V,us,{2}}
= a_{kl} \delta^{(1),kl}_{V,us,2} + \delta^{(0),kl}_{V,us,2}
\label{}
{}.
\end{equation}
The integral in
eq.~(\ref{pi1andpi2})
%eq.~(\ref{nlab1})
is, obviously, insensitive,
to the $Q^2$-independent terms in the polarization functions
$ \Pi^{[1]}_{0}$ and $\Pi^{[1]}_{2} {}$.
This means that without loss of generality we may
deal with the corresponding (Adler) $D$-functions, viz.
\[
D^{[1]}_0(Q^2) \equiv -\frac{3}{4}Q^2 \frac{ {\rm d}}{{\rm d} Q^2}
\Pi^{[1]}_{0} ,
\ \ \
D^{[1]}_2(Q^2) \equiv -\frac{1}{2}Q^2 \frac{ {\rm d}}{{\rm d} Q^2}
\Pi^{[1]}_{2}
{}.
\]
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.
The Adler functions $ D^{[1]}_0$ and $D^{[1]}_2$ have been
calculated with $\mathcal{O}(\alpha_s^3)$ accuracy, the polarization
function $\Pi^{[2]}_{2}$, however, to $\mathcal{O}(\alpha_s^2)$ only
(see \cite{Chetyrkin:1998ej} and references therein).
The (apparent) convergence of the perturbative series for $ D^{[1]}_0$
is acceptable, the one for $D^{[1]}_2$ and $ \Pi^{[2]}_{2} $
is at best marginal. This has led to significant theoretical
uncertainties in extracting $\alpha_s$ and to a fairly unstable
behaviour in extraction of $m_s$ from
$\tau$-decays.
% (see, e.g. \cite{}).
To improve the situation, we have computed the two leading terms in the
large $n_f$ expansion of the next order, i.e. terms of order
$\mathcal{O}(n_f^3\alpha_s^4)$, $\mathcal{O}(n_f^2\alpha_s^4)$ to $
D^{[1]}_0$, $ D^{[1]}_2$ and $\mathcal{O}(n_f^2\alpha_s^3)$,
$\mathcal{O}(n_f\alpha_s^3)$ to $\Pi^{[2]}_2$. Our results are
described in the following section.
\section{Fixed order results in $ \alpha_s ^3$ and $ \alpha_s ^4$}
Using the technique described in \cite{bai1,bai2,Baikov:2001aa}
and the parallel version of FORM \cite{Fliegner:2000uy,Vermaseren},
the leading and subleading (in $n_f$) terms of the next order in the
perturbative series for $ D^{[1]}_0$, $ D^{[1]}_2$ and $\Pi^{[2]}_2$
have been obtained in the standard $\msbar$ renormalization scheme \cite{ms,msbar}:
%They are displayed below in three equations.
%\input{tau1}
\bea
D^{[1]}_{0} &=& 1 +\, a_s + a_s^2 \left\{
\lsb -\frac{11}{12}+\frac{2}{3} \,\zeta_{3} \rsb \,n_f +\frac{365}{24}-11 \,\zeta_{3}\right\}
\nonumber
\\ &+& a_s^3 \left\{
\lsb \frac{151}{162}-\frac{19}{27} \,\zeta_{3} \rsb \, n_f^2 +\lsb -\frac{7847}{216}+\frac{262}{9} \,\zeta_{3}-\frac{25}{9} \,\zeta_{5} \rsb \,n_f \BreakI
% \phantom{ a_s^3 ||}
\hspace{1cm}
+\frac{87029}{288}-\frac{1103}{4} \,\zeta_{3}+\frac{275}{6} \,\zeta_{5}\right\}\nonumber
\\ &+&a_s^4\left\{
\lsb -\frac{6131}{5832}+\frac{203}{324} \,\zeta_{3}+\frac{5}{18} \,\zeta_{5} \rsb \, n_f^3 +\lsb \frac{1045381}{15552}+\frac{5}{6} \,\zeta_3^2\BreakII
% \phantom{a_s^4||}
\hspace{1cm}
-\frac{40655}{864} \,\zeta_{3}-\frac{260}{27} \,\zeta_{5} \rsb \, n_f^2 + d^{[1]4}_{0,1} \,n_f +d^{[1]4}_{0,0}\right\} \nonumber
\\ &=& 1 +\, a_s + a_s^2 \left\{
-0.1153 \,n_f +1.986\right\}
\nonumber
\\ &+& a_s^3 \left\{
0.08621 \, n_f^2-4.216 \,n_f +18.24\right\}\nonumber
\\ &+&a_s^4\left\{
-0.01009 \, n_f^3 + 1.875 \, n_f^2 + d^{[1]4}_{0,1} \,n_f +d^{[1]4}_{0,0}\right\} {},
\label{D1m0:exact} \end{eqnarray}
\bea
D^{[1]}_{2} = m_s^2 \left(\rule{0mm}{7mm} \right. 1 &+& \frac{5}{3}\, a_s + a_s^2 \left\{
\lsb -\frac{11}{8}+\frac{2}{3} \,\zeta_{3} \rsb \,n_f +\frac{5185}{144}-\frac{39}{2} \,\zeta_{3}\right\}
\nonumber
\\ &+& a_s^3 \left\{
\lsb \frac{8671}{11664}-\frac{13}{27} \,\zeta_{3} \rsb \, n_f^2 +\lsb -\frac{44273}{972}+\frac{3257}{81} \,\zeta_{3}\BreakII
% \phantom{ a_s^3 ||}
\hspace{1cm}
-\frac{5}{6} \,\zeta_{4}-\frac{1265}{81} \,\zeta_{5} \rsb \,n_f +\frac{2641517}{5184}-\frac{131275}{216} \,\zeta_{3}+\frac{12845}{36} \,\zeta_{5}\right\}\nonumber
\\ &+&a_s^4\left\{
\lsb -\frac{396781}{559872}+\frac{461}{1296} \,\zeta_{3}-\frac{1}{48} \,\zeta_{4}+\frac{5}{18} \,\zeta_{5} \rsb \, n_f^3 +\lsb \frac{61913567}{1119744}-\frac{59}{54} \,\zeta_3^2\BreakII
% \phantom{a_s^4||}
\hspace{1cm}
-\frac{352549}{7776} \,\zeta_{3}+\frac{67}{96} \,\zeta_{4}+\frac{22859}{3888} \,\zeta_{5} \rsb \, n_f^2 + d^{[1]4}_{2,1} \,n_f +d^{[1]4}_{2,0}\right\} \left.\rule{0mm}{7mm} \right)
\nonumber
\\ \phantom{\Pi^{[2]}_{2}} = m_s^2 \left(\rule{0mm}{7mm} \right.1 &+&1.667\, a_s + a_s^2 \left\{
-0.5736 \,n_f +12.57\right\}
\nonumber
\\ &+& a_s^3 \left\{
0.1646 \, n_f^2-14.31 \,n_f +149.\right\}\nonumber
\\ &+&a_s^4\left\{
-0.01563 \, n_f^3 + 6.067 \, n_f^2 + d^{[1]4}_{2,1} \,n_f +d^{[1]4}_{2,0}\right\} \left.\rule{0mm}{7mm} \right) {},
\label{D1m2:exact} \end{eqnarray}
\bea
\Pi^{[2]}_{2} = -4 m_s^2 \left(\rule{0mm}{7mm} \right. 1 &+&\frac{7}{3}\, a_s + a_s^2 \left\{
\lsb -\frac{25}{24}-\frac{2}{9} \,\zeta_{3} \rsb \,n_f +\frac{15331}{432}+\frac{359}{54} \,\zeta_{3}-\frac{520}{27} \,\zeta_{5}\right\}
\nonumber
\\ &+& a_s^3 \left\{
\lsb \frac{2131}{11664}+\frac{19}{81} \,\zeta_{3} \rsb \, n_f^2 +\lsb -\frac{68135}{1944}-\frac{52}{27} \,\zeta_3^2\BreakII
% \phantom{ a_s^3 ||}
\hspace{1cm}
-\frac{3997}{486} \,\zeta_{3}-\frac{5}{6} \,\zeta_{4}+\frac{3875}{243} \,\zeta_{5} \rsb \,n_f +k^{[2]3}_{2,0}\right\} \left.\rule{0mm}{7mm} \right)
\nonumber
\\
%\phantom{\Pi^{[2]}_{2}}
\hspace{1cm}
= -4 m_s^2 \left(\rule{0mm}{7mm} \right. 1 &+&2.333\, a_s + a_s^2 \left\{
-1.309 \,n_f +23.51\right\}
\nonumber
\\ &+& a_s^3 \left\{
0.4647 \, n_f^2-32.08 \,n_f +k^{[2]3}_{2,0}\right\} \left.\rule{0mm}{7mm} \right) {}.
\label{Pi2m2:exact}
\end{eqnarray}
Here we have used $a_s = \frac{\alpha_s(Q^2)}{\pi}, m_s = m_s(Q^2) $
and set the normalization scale $\mu^2 = Q^2$; results for generic
values of $\mu$ can be easily recovered with the standard
renormalization group techniques.
The result for the $ \alpha_s ^4$
terms in $D^{[1]}_0$ has been already presented in
\cite{Baikov:2001aa}, the coefficients of the $ \alpha_s ^4$ and $
\alpha_s ^3$ terms in in $D^{[1]}_2$ and $\Pi^{[2]}_2$ respectively
are new.
\section{Implications for the $ \alpha_s ^4$ predictions and phenomenological analysis}
\subsection{Massless case}
FAC (Fastest Apparent Convergence) and PMS (Principle of Minimal
Sensitivity) methods are both based eventually on the concept of
scheme-invariant properties and the idea of the choice of an
"optimal" scheme to provide better convergence of the resulting
perturbative series. For both methods the optimal scheme depends on
the physical observable we are dealing with. With FAC it should be a
scheme which minimizes (set to zero by construction) all the terms of
order $ \alpha_s ^2$ and higher, while the PMS scheme is fixed by the requirement
that the perturbative expansion for the observable is as insensitive as
possible to a change in the scheme. The assumption that a
renormalization scheme is in a sense optimal sets certain constraints on not yet
computed higher order corrections in any other scheme. These
constraints can be used to "predict" (at least roughly) the magnitude of
these corrections.
For the function $D^{[1]}_0$ the result is known since long from
Ref.~\cite{KS} (see Table~1, column 5).
{}From the three entries corresponding to $n_f = 3,4$ and $5$ one easily
restores the FAC/PMS prediction for the $n_f$ dependence
of the $ \alpha_s ^4$ term in $D^{[1]}_{0}$ (the term of order $n_f^3$ was fixed to its
computed value):
\begin{equation}
d^{[1]4}_{0} = 127.6 - 44.2 \, n_f + 3.64 \, n_f^2 - 0.0100928 \, n_f^3
d^{[1]4}_{0} = 127.58 - 44.211 \, n_f + 3.6439 \, n_f^2 - 0.0100928 \, n_f^3
{}.
\label{K4:nf}
\end{equation}
(Note that FAC and PMS predictions happen to coincide for the $ \alpha_s ^4$ term.)
\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$ & $ 6.39 $ & $ 27 \pm 16 $ & $ 145 \pm 100$\\
\hline
4 & $ 2.758$ & $ 4.671$ & $ 5.26 $ & $ 8 \pm 28 $ & $ 40 \pm 160$\\
\hline
5 & $ -0.68$ & $ 3.762$ & $ 4.16 $ & $ -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}
It is 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.~\ref{D1m0:exact}. The results of both estimates are
\begin{equation}
d^{[1]3}_{0}(FAC) = 8.54 - 1.013 \, n_f + 0.0116 \, n_f^2
\label{d1m0as3:FAC}
{},
\end{equation}
\begin{equation}
d^{[1]3}_{0}(PMS) = 9.93 - 1.23 \, n_f + 0.0125 \, n_f^2
\label{d1m0as3:PMS}
{}.
\end{equation}
The comparison of the complete $ \alpha_s ^3$ and partial $ \alpha_s ^4$ results with
FAC and PMS estimates leads to the following observations:
\begin{itemize}
\item Starting from $ \alpha_s ^3$, the leading in $ \alpha_s $ {\em and} $n_f$ terms of
order $ \alpha_s ^3 n^2_f$ and $ \alpha_s ^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 $ \alpha_s $ expansion. On the other hand, the term
subleading in $n_f$, say, of order $ \alpha_s ^3 n_f$ is comparable in size
with the term of order
$ \alpha_s ^3 n_f^0$.
Similarly, the $ \alpha_s ^4 n_f^3$ term is significantly smaller than the
$ \alpha_s ^4 n_f^2$ one, whereas the $ \alpha_s ^4 n_f$ and $ \alpha_s ^4 n_f^0$
terms are expected to be of similar magnitude.
%One observes a similar picture at $ \alpha_s ^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).
\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 $O( \alpha_s ^n)$ 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.
To illustrate this feature, let us consider the worst case: the
FAC/PMS prediction for the $ \alpha_s ^3 n_f$ term in the function
$d^{[1]}_0$. Here the ratio of the exact result relative to the predicted one
is quite large (about 4). Without knowledge of the $ \alpha_s ^3 n_f^0$
contribution one would expect that the uncertainty of the prediction
for the full $ \alpha_s ^3$ coefficient should be at least around
\[
(d^{[1]3}_{0,1}|_{exact} -d^{[1]3}_{0,1}|_{FAC}) \ n_f .\]
For $n_f =
3,4,5 $ this amounts to 9, 12 and 15, which should be compared to the
corresponding differences of the full order $O( \alpha_s ^3)$ (summed over
all contributing power of $n_f$) coefficients, viz. $1, 2, 4$. This
example demonstrates that the deviation of FAC/PMS predictions for
subleading in $n_f$ terms may well serve as conservative
estimate of the accuracy in the prediction of the
complete terms of orders $ \alpha_s ^3$ and $ \alpha_s ^4$.
\end{itemize}
These observations motivate the
assumption that the prediction for the coefficient
$d^{[1]4}_0
(\equiv d^{[1]4}_{0,0} + d^{[1]4}_{0,1}n_f +d^{[1]4}_{0,2}n_f^2
+ d^{[1]4}_{0,3}n_f^3)
$
should also be correct within
\[
\pm (d^{[1]4}_{0}|_{exact} -d^{[1]4}_{0}|_{FAC}) \ n_f^2
\]
Thus, in our phenomenological analysis of the $\tau$-lepton
decays the estimate
\begin{equation}
d^{[1]4}_0 |_{n_f=3}= 27 \pm 16
\label{d1m0as4nf3:num}
\end{equation}
will be used. On the basis of this improved estimate and the four loop $\beta$
function \cite{vanRitbergen:1997va} one may even speculate about the $\alpha_s^5$ term
(whose exact evaluation is completely out of reach in the foreseeable future).
Following the discussion of Kataev and Starshenko \cite{Kataev:1994wd}, one obtains
\begin{equation}
d_0^{[1]5}|_{n_f=3} = 145 \pm 100
\label{d1m0as5nf3:num}
{},
\end{equation}
not far from the previous estimates of Ref.~\cite{Kataev:1994wd}. The variation of
$d^{[1]4}_0$ by $\pm 16$ leads to the variation of $ d^{[1]5}_0 $ by $\pm 100$.
For other values of $n_f$, the corresponding predictions can be
obtained in the same way. They are listed in Table~1.
The FAC/PMS prediction (\ref{K4:nf}) for the
$n_f$ dependence of the coefficient
$d^{[1]4}_{0}$ does not take into account the available knowledge of
the corresponding $n_f^2$ part.
One can easily include this by fitting
the FAC/PMS predictions for only ${\it two}$ values of $n_f$ with
a ${\it linear}$ function of $n_f$. As a result one
obtains \footnote{We thank Matthias Steihnauser
and Robert Harlander for a useful discusson of this
point.}
(we have boxed the predicted coefficients in order to separate them
clearly from the input)
\bea
d^{[1]4}_{0} ( \mathrm{FAC/PMS} , n_f = 3,4 ) &=&
\fbox{$105.7 - 31.8\, n_f$} + 1.875 \, n_f^2 -0.01009 \, n_f^3
{},
\label{3,4->5}
\\
d^{[1]4}_{0} ( \mathrm{FAC/PMS} , n_f = 4,5 ) &=&
\fbox{$107.7 - 32.3\, n_f$} + 1.875 \, n_f^2 -0.01009 \, n_f^3
{},
\label{4,5->3}
\\
d^{[1]4}_{0} ( \mathrm{FAC/PMS} , n_f = 3,5 ) &=&
\fbox{$106.4 - 32.0 \,n_f$} + 1.875 \, n_f^2 -0.01009 \, n_f^3
{}.
\label{3,5->4}
\eea
One could now perform a self-consistency check of Table I by
predicting, say, $d^{[1]4}_{0}$ for $n_f=5$ from eq.~(\ref{3,4->5}).
The result -- (-7.5) -- is compared successfully to the
value listed in the table, viz. -8.
The corresponding predictions from eqs.~(\ref{4,5->3}) and (\ref{3,5->4})
are also in very good agreement to Table~I.
An instructive example of how knowledge and inclusion of the
subleading $n_f$ term can improve FAC/PMS predictions is provided by
the (exactly known) coefficient $d^{[1]3}_{0}$. Indeed, assuming the
knowledge of
$d^{[1]3}_{0,2}$ and $d^{[1]3}_{0,1}$ and using the values of
$d_3^{\mathrm{FAC}}$ and $d_3^{\mathrm{PMS}}$
(at $n_f = 3$) as given by Table 1, one easily arrives at
\begin{equation}
d^{[1]3}_{0}(\mathrm{FAC}) = \fbox{17.48} -4.216\, n_f + 0.08621\, n_f^2
\label{d1m0as3:FAC:nf->3}
{}
\end{equation}
and
\begin{equation}
d^{[1]3}_{0}(\mathrm{PMS}) = \fbox{18.27} -4.216\, n_f + 0.08621\, n_f^2
\label{d1m0as3:PMS:nf->3}
{},
\end{equation}
which should be compared to the exact value $d^{[1]3}_{0,0} = 18.24$.
Repeating the same exercise for $n_f=4,5$
we get 20.16 (FAC) and 20.75 (PMS) for $n_f=4$ as well as $22.69$ (FAC)
and 23.08 (PMS) for $n_f=5$ respectively.
Bearing in mind that in many cases FAC/PMS predictions made for $n_f= 3$
are in better agreement with the exact results (see, e.g. Table~I and tables
from Ref.~ \cite{Kataev:1994wd}), we suggest eq.~(\ref{3,4->5}) as the best FAC/PMS
prediction for the constant term and the term linear in $n_f$ in the coefficient
$d^{[1]4}_{0}$.
Eqs. (\ref{d1m0as4nf3:num},\ref{d1m0as5nf3:num}) can be used to predict
$R_{\tau}$ in the massless limit first in
fixed order perturbation theory (FOPT)
\begin{eqnarray}
&R_{\tau}^{\mathrm{FOPT}} = &
\label{Rtau:FO}
\\
& 3 \left(
1 + \, a_s+ 5.202 \, a^2_s+ 26.37 \, a_s^3 + a_s^4 \ (105 \pm 16) +
\, a_s^5 (138 \pm 230 \pm 100)
\right)
&
\nonumber
{}.
\end{eqnarray}
Here $a_s= \alpha_s(M_{\tau})/\pi$.
The first uncertainty in the $ \alpha_s^5$ term comes from that of
$d^{[1]4}_0$ while the second is our estimation of the error in
the very coefficient $d^{[1]5}_0$. (Of course, within this approach
they are strongly correlated).
Similarly, we can use "contour improved" (CI) formulae \cite{tau:resum,DP}
(assuming as reference value $ \alpha_s (M_{\tau}) = 0.334$ \cite{exp1})
to get
\begin{equation}
R_{\tau}^{CI} = 3 \left(
1 + 1.364 \, a_s+ 2.54 \, a_s^2 + 9.71 \, a_s^3
+ 1.31 \, a_s^4 d^{[1]4}_0 + 0.95 \, a_s^5 d^{[1]5}_0
\right)
{}
\label{Rtau:CIa}
\end{equation}
or, equivalently,
\begin{equation}
R_{\tau}^{CI} = 3 \left(
1+ 1.364\, a_s+ 2.54 \, a_s^2 + 9.71 \, a_s^3 + \, a_s^4 (35 \pm 20) + a_s^5 (138 \pm 95)
\right)
\label{Rtau:CI}
{}.
\end{equation}
Let us compare our new value for the coefficient $d^{[1]4}_0$
in (\ref{d1m0as4nf3:num}) with the ones used in extracting
$\alpha_s(M_{\tau})$ from $\tau$ data by the OPAL \cite{exp2} and ALEPH
\cite{exp1} collaborations, namely
\begin{equation} d^{[1]4}_0 |_{n_f=3} = 25 \pm 25
\ \mbox{(OPAL)} , \ \ \ 50 \pm 50 \ \mbox{(ALEPH)}
\label{d4:OPAL+ALEPH} {}.
\end{equation}
While OPAL's central value is basically the same as ours, their error
bar is somewhat larger. In the case of ALEPH both the
central value and the uncertainty assigned to it are significantly
larger than our numbers. In this connection, we would like to stress
that eq.~(\ref{d1m0as4nf3:num}) utilizes completely new non-trivial
information given in (\ref{D1m0:exact}): the subleading term
in $n_f$ term of order $ \alpha_s ^4$.
It is of interest to see in detail which accuracy in the determination
of $\alpha_s(M_{\tau})$ one could achieve assuming
eqs. (\ref{d1m0as4nf3:num},\ref{d1m0as5nf3:num}).
Let us introduce the
quality $\delta_P$ as follows:
\begin{equation}
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}
\end{equation}
with
\[
R_{\tau} = 3 (1 + \delta_P + \delta_{EW} + \delta_{NP})
{}.
\]
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 $\delta_{NP} = -.003 \pm 0.003$
(see, e.g. \cite{NarPic88}).
Here the flavour mixing matrix element
$|V_{ud}|^2 = 0.9475 \pm 0.0016$ \cite{PDG}.
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 electroweak correction \cite{ewcorr2}.
Using for definiteness the result of ALEPH~\cite{exp1}
\begin{equation}
R_{\tau S=0} = 3.492 \pm 0.016
{},
\label{Rtau:ALEPH}
\end{equation}
one arrives at
\begin{equation}
\delta_P^\mathrm{exp} = 0.207 \pm 0.007
\label{delta_P:exp}
{}.
\end{equation}
To get a value for $\alpha_s(M_{\tau})$ one should simply
fit $\delta_P^\mathrm{exp}$ against $R_{\tau}/3 - 1$ as given by eq.~(\ref{Rtau:FO}) or
by eq.~(\ref{Rtau:CI}) to get a result corresponding
FOPT or CIPT (Fixed Order or Contour Improved PT).
Unfortunately, there is no
unique way to assign a
theoretical uncertainty $\delta { \alpha_s }$ 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 $\delta { \alpha_s }$ is a half of the shift in $\alpha_s$ induced by the last
fully computed
term in the PT (that is by the one of order $ \alpha_s ^3$ at present).
\item $\delta { \alpha_s }$ 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 $ \alpha_s ^4$ and higher in
eq.~(\ref{Rtau:FO}) and in the $D$-function have been set to zero)
\begin{eqnarray}
\alpha_s ^{\mathrm{FOPT}}(M_{\tau}) &=& 0.345 \pm (0.025| 0.037 )
{},
\label{asFOas3}
\\
\alpha_s ^{\mathrm{CIPT}}(M_{\tau}) &=& 0.364 \pm (0.012| 0.021 )
{}.
\label{asCIas3}
\end{eqnarray}
Here the first(second) value in brackets correspond to the
use of the first(second) suggestion for the error estimation.
After evolution from $M_{\tau}$ to $M_Z$ this corresponds to
\begin{eqnarray}
\alpha_s ^{\mathrm{FOPT}}(M_{Z}) &=& 0.1209 \pm (0.0024 | 0.0037 )
{},
\label{asFOas3_MZ}
\\
\alpha_s ^{\mathrm{CIPT}}(M_{Z}) &=& 0.1229 \pm (0.0011 | 0.0020 )
{}.
\label{asCIas3_MZ}
\end{eqnarray}
\item
$\delta { \alpha_s }$ is equal to the change in $\alpha_s$ caused by the uncertainty in the
predicted (that is not yet completely known) higher order
terms in the perturbative series for $R_{\tau}$;
\item $\delta { \alpha_s }$ is a half of the difference in the $\alpha_s(M_{\tau})$ as
obtained within FOPT and CIPT. This difference
comes from different handling of higher order terms.
\end{enumerate}
In order to quantify the error estimates according to 3. and 4. we
have shown in Table~\ref{t2} the results for $ \alpha_s (M_{\tau})$ obtained with
various choices for $d^{[1]4}_0$, $d^{[1]5}_0$ and $\mu$. The entries
with the choices $\pm 100$ for the coefficients illustrate the large
change in $ \alpha_s $ which would result from a failure of PMS and
FAC once higher order terms are included.
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 $ \alpha_s $ series.
Our final predictions
for $ \alpha_s ^{\mathrm{FOPT}}(M_{\tau})$ and $ \alpha_s ^{\mathrm{CIPT}}(M_{\tau})$
are given in the first column of Table~\ref{alpha_fin},
together with experimental error\footnote{
According to eq.~(\ref{delta_P:exp}) the latter
includes (small) uncertainties in the values of $|V_{ud}|$ and the nonperturbative
correction $\delta_{NP}$ in addition to the experimental error per se as displayed
in~eq.~(\ref{Rtau:ALEPH}).}
and the combined theory uncertainty. The values of
theory uncertainties are listed separately in columns 3,4 and 5.
The corresponding values at the scale
of $M_Z$ are
\begin{eqnarray}
\alpha_s ^{\mathrm{FOPT}}(M_{Z}) &=& 0.1192 \pm 0.0007 \pm 0.002
{},
\label{asFOas3_MZ_fin}
\\
\alpha_s ^{\mathrm{CIPT}}(M_{Z}) &=& 0.1219 \pm 0.001 \pm 0.0006
{}.
\label{asCIas3_MZ_fin}
\end{eqnarray}
\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$d^{[1]4}_0$ & $ \alpha_s ^4$ & $d^{[1]5}_0$ & $ \alpha_s ^5$
\\
\hline
27 & $0.331 \pm 0.02$ & 145 & $0.33 \pm 0.02$
\\
\cline{2-2}\cline{4-4}
& $0.357 \pm 0.009$ & &$0.354 \pm 0.004$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
43 & $0.329 \pm 0.02$ & 245 & $0.324 \pm 0.01$
\\
\cline{2-2}\cline{4-4}
& $0.353 \pm 0.01$ & &$0.348 \pm 0.005$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
11 & $0.333 \pm 0.02$ & 45 & $0.335 \pm ?$
\\
\cline{2-2}\cline{4-4}
& $0.361 \pm 0.008$ & &$0.360 \pm 0.002$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
100 & $0.324 \pm 0.02$ & 100 & $0.314 \pm 0.01$
\\
\cline{2-2}\cline{4-4}
& $0.340 \pm 0.01$ & &$0.338 \pm 0.007$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
-100 & $0.348 \pm ?$ & -100 & $0.349 \pm ?$
\\
\cline{2-2}\cline{4-4}
& $0.398 \pm 0.03$ & &$0.401 \pm ? $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
\end{tabular}
\end{center}
\caption{\label{t2}
The predicted value of $ \alpha_s (M_{\tau})$ in dependence of
chosen values for the coefficients
$d^{[1]4}_0$, $d^{[1]5}_0$.
The second and the forth columns differ in the the number of terms in the
perturbative series included.
The upper value of $ \alpha_s $ is the one predicted within FOPT, the
lower corresponds to CIPT. The uncertainty in the value of $ \alpha_s $ corresponds
to changing the normalization point $\mu$ as follows $\mu^2/M_{\tau}^2 = 0.4 - 2.$
The entry with question mark means that an equation for $ \alpha_s (\mu)$ does
not have a solution for some value of $\mu$ within the interval.
}
\end{table}
Thus we observe that the total uncertainty based on a combination of
not yet calculated higher order terms, $\mu$-dependence and scheme
dependence is reduced, once $ \alpha_s ^4$ terms are available. However, the
difference between FOPT and CIPT results of roughly 0.02 is a remaining,
at the moment irreducible uncertainty\footnote{
This is in agreement to the analysis of
Ref.~\cite{Groote:1997cn}, where it has been concluded that
``the resummed values of $\alpha_s$ from $\tau$
decay lie outside the convergence radii and can therefore not be obtained
from a power series expansion. Regular perturbation series do not converge
to their resummed counterparts. The experimental value of $R_{\tau}$ appears
to be too large for a fixed order perturbation analysis to apply''.
See also \cite{krajals}.}.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Method & $ \alpha_s (M_{\tau})$ & $\Delta \, \delta^\mathrm{exp}_P$ &
$\Delta \, \mu $ & $\Delta \, d_0^{[1]4}$ & $\Delta \, d_0^{[1]5}$
\\
\hline
FOPT & $0.330 \pm 0.006 \pm 0.02$ & 0.006 & 0.019 & 0.0045 & 0.0011
\\
\hline
CIPT & $0.354\pm 0.009 \pm 0.006 $ & 0.009 & 0.0036 & 0.0042 & 0.0019
\\
\hline
\end{tabular}
\end{center}
\caption{\label{alpha_fin}
The value of $ \alpha_s (M_{\tau})$ obtained with
$ \delta^\mathrm{exp}_P$, $d_0^{[1]4}$ and $d_0^{[1]5}$ fixed to their
central values according to
eqs.~(\ref{delta_P:exp},\ref{d1m0as4nf3:num},\ref{d1m0as5nf3:num})
together with corresponding errors.
}
\end{table}
It is thus of interest to study this difference as a function of
$m_{\tau}$. In practice, this could be applied to sum rules for
spectral functions as determined in $e^+ e^-$ annihilation.
Therefore, let us consider a hypothetical case of $\tau$ lepton with
the mass equal 3 GeV. Assuming $ \alpha_s (1.77 \,\mathrm{GeV}) = 0.334$ and running
this value to 3 GeV via standard 4-loop evolution equation one gets
$ \alpha_s (3 \,\mathrm{GeV}) = 0.2558$ and predicts
\[ \delta_P^\mathrm{exp} = 0.1353 \]
which
corresponds to eq.~(\ref{Rtau:FO}) with the $ \alpha_s ^4$ and $ \alpha_s ^5$ terms fixed
to their central values (see eqs.~(\ref{d1m0as4nf3:num}--\ref{d1m0as5nf3:num})). \
Let us now investigate the results for $ \alpha_s $
and the theory error that would result from $\delta_P^\mathrm{exp} = 0.1353$
as a starting point.
The corresponding analogs
of eqs.
%(\ref{asFOas3}, \ref{asCIas3},re{asFOas3_MZ},(\ref{asCIas3_MZ}))
(\ref{asFOas3}--\ref{asCIas3_MZ})
and Table~(\ref{t2}) are displayed
below as eqs. ((\ref{asFOas3_3GeV},\ref{asCIas3_3GeV})) and Table
(\ref{t2_Mtau_3}) correspondingly. The difference between FOPT and CIPT
decreases significantly, and this remains true even after extrapolating
to $ \alpha_s (M_Z)$.
\begin{eqnarray}
\alpha_s ^{\mathrm{FOPT}}(3 \mbox{GeV}) &=& 0.263 \pm (0.013|0.014 )
{},
\label{asFOas3_3GeV}
\\
\alpha_s ^{\mathrm{CIPT}}(3 \mbox{GeV} &=& 0.265 \pm (0.005| 0.008 )
{},
\label{asCIas3_3GeV}
\\
\alpha_s ^{\mathrm{FOPT}}(M_{Z}) &=& 0.1198 \pm (0.002|0.003 )
{},
\label{asFOas3_3GeV_MZ}
\\
\alpha_s ^{\mathrm{CIPT}}(M_{Z}) &=& 0.1203 \pm (0.009| 0.0016 )
{}.
\label{asCIas3_3GeV_MZ}
\end{eqnarray}
\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$d^{[1]4}_0$ & $ \alpha_s ^4$ & $d^{[1]5}_0$ & $ \alpha_s ^5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
27 & $0.256 \pm 0.007$ & 145 & $0.256 \pm 0.003$
\\
\cline{2-2}\cline{4-4}
& $0.262 \pm 0.004$ & &$0.261 \pm 0.002$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
43 & $0.256 \pm 0.007$ & 245 & $0.254 \pm 0.003$
\\
\cline{2-2}\cline{4-4}
& $0.26 \pm 0.004$ & &$0.259 \pm 0.002$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
11 & $0.257 \pm 0.006$ & 45 & $0.258 \pm 0.005$
\\
\cline{2-2}\cline{4-4}
& $0.264 \pm 0.003$ & &$0.263 \pm 0.001$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
100 & $0.253 \pm 0.008$ & 100 & $0.249 \pm 0.004$
\\
\cline{2-2}\cline{4-4}
& $0.255 \pm 0.008$ & &$0.254 \pm 0.002$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
-100 & $0.264 \pm 0.01$ & -100 & $0.277 \pm \, ?$
\\
\cline{2-2}\cline{4-4}
& $0.279 \pm 0.02$ & &$0.28 \pm \, ?$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\hline
\end{tabular}
\end{center}
\caption{\label{t2_Mtau_3}
The predicted value of $ \alpha_s (M_{\tau})$ in dependence of
chosen values for the coefficients
$d^{[1]4}_0$, $d^{[1]5}_0$ for a hypothetical case
of $M_{\tau} = 3$ GeV.
The third and the forth columns differ in the the number of terms in the
perturbative series included.
The upper value of $ \alpha_s $ is the one predicted within FOPT, the
lower corresponds CIPT. The uncertainty in the value of $ \alpha_s $ corresponds
to changing the normalization point $\mu$ as follows
$\mu^2/M_{\tau}^2 = 0.4 - 2.$
The entry with question mark means that an equation for $ \alpha_s (M_{\mu})$ does
not have a solution for some value of $\mu$ within the interval.
}
\end{table}
The same coefficients
$d^{[1]4}_0 $ and $ d^{[1]5}_0$
can be used to predict \cite{KS,Kataev:1994wd}
(the non-singlet part of) corrections of orders $\alpha_s^4$ and
$\alpha_s^5$ to the $R$-ratio in Z decays:
\[
R(n_f=5) = 1+ a_s+ 1.409 \, a_s^2 -12.77 \, a_s^3 + (-97 \pm 44) \, a_s^4 +(76 \pm 230) \, a_s^5
{}.
\]
It is also of interest to display $R(s)$ for $n_f =3$ and 4,
which is accessible at $e^+ e^-$ colliders at lower energies.
\[
R(n_f = 4) = 1+ a_s+ 1.525 \, a_s^2 - 11.52 \, a_s^3 + (-112 \pm 30) \, a_s^4
+(-245 \pm 160) a_s^5
{}.
\]
\[
R(n_f = 3) = 1+ \, a_s+ 1.640 \, a_s^2 -10.28 \, a_s^3 + (-129 \pm 16) \, a_s^4 + (-635 \pm 100) a_s^5
{}.
\]
Our results are close to those of \cite{KS,Kataev:1994wd},
they employ, however, the additional information from
\cite{Baikov:2001aa,vanRitbergen:1997va}.
These formulae demonstrate rather
good convergency for $n_f =5 $ and a reasonably good one for $n_f =3$
and $4$ if our predictions for the coefficients $d^{[1]4}_0$,
$d^{[1]5}_0$ deviate from the true values within the assumed
error margins.
\subsection{Quadratic mass corrections}
Let us first discuss the function $ D^{[1]}_2 $.
The FAC/PMS predictions can be easily obtained following
\cite{CheKniSir97}; they are listed in Table~\ref{t3}.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$n_f$ & $d_{3} ^{\rm exact}$ & $d_3^{\rm FAC}$ & $d_3^{\rm PMS}$ &
$d_4^{\rm FAC/PMS}$ \\
\hline\hline
3 & $ 107.5$ & $ 89.86$ & $ 91.17 $ & $ 1200 \pm 400$ \\
\hline
4 & $ 94.37$ & $ 79.13$ & $ 80.11 $ & $ 950 \pm 300$ \\
\hline
5 & $ 81.54$ & $ 68.66$ & $ 69.33 $ & $ 750 \pm 200$ \\
\hline
\end{tabular}
\end{center}
\caption{\label{t3}
Estimations of the coefficients $d_3 = d^{[1]3}_{2}$ and $d_4 = d^{[1]4}_{2}$
in the function $D^{[1]}_2$ based on the FAC and PMS optimizations.
The estimation of $d^{[1]4}_{2}$ employs the exact value of $d^{[1]3}_{2}$.}
\end{table}
We again restore the $n_f$ dependence of the coefficient
$d^{[1]4}_{2}$ as predicted by FAC/PMS:
\begin{equation}
d^{[1]4}_{2}(\mbox{FAC/PMS}) = 1931.44 - 281.956 \, n_f + 9.0294 \, n_f^2 - 0.0156289 \, n_f^3
\label{d1m2as4:FAC,PMS}
{}
\end{equation}
as well as that of $d^{[1]3}_{2}$:
\begin{equation}
d^{[1]3}_{2}(\,\mbox{FAC}) = \ \ 123.654 - 11.6638 \, n_f + 0.133293 \, n_f^2
\label{Pi2m2as3:FAC}
{},
\end{equation}
\begin{equation}
d^{[1]3}_{2}(\,\mbox{PMS}) = \ \ 125.975 - 12.0028 \, n_f + 0.134769 \, n_f^2
\label{Pi2m2as3:PMS}
{}
\end{equation}
to be compared with
\begin{equation}
d^{[1]3}_{2}(\,\mbox{exact}) = 148.978 - 14.3097 \, n_f +0.16463 \, n_f^2
\nonumber
{}.
\end{equation}
The comparison of estimates and exact results reveals a picture
qualitatively similar to the massless case but with some modifications.
%Let us consider the (completely known) $ \alpha_s ^3$ order first.
A few important observations are in order.
1. All three terms of the $n_f$ expansion of $d_2^{[1]3}$ are
successfully predicted within about 20 \% accuracy.
2. Unlike the massless case the agreement between FAC/PMS predictions
for the coefficient $d_2^{[1]3}$ for $n_f = 3,4,5$ and the
corresponding exact numbers is within the range of 15\%--20\%. On the
other hand, the estimation of accuracy of the $ \alpha_s ^3$ fixed $n_f$
predictions obtained {\em exclusively} from the knowledge of
the subleading contribution of $O( \alpha_s ^3 n_f )$ is of the right
order of magnitude but somewhat less. All this is probably a
consequence of a significantly larger $n_f$ independent contribution.
3. At ${\mathcal{O}}( \alpha_s ^4)$ the exact result for the full
coefficient
\[
d_2^{[1]4} =d^{[1]4}_{2,0} + d^{[1]4}_{2,1} n_f
+ d^{[1]4}_{2,2} n_f^2 + d^{[1]4}_{2,3} n_f^3
\]
is unknown, apart from its leading and subleading terms in $n_f$. The
prediction for the subleading coefficient
$d^{[1]4}_{2,2} = 9.0 $ is by 50\% larger than the exact
value 6.07.
The predicted values
for $d^{[1]4}_{2,0}$ and $d^{[1]4}_{2,1}$ are {\em very} large. In
view of this largeness, the estimate $ (d^{[1]4}_{2,2}|_{exact}
-d^{[1]4}_{2,2}|_{FAC}) \ n_f^2 $ (= 30, 50, 80 for $n_f = 3,4,5$
respectively) looks somewhat too optimistic. Therefore we
assign a conservative 30\% uncertainty to the fixed $n_f$
predictions listed in the fifth column of the Table~\ref{t3}.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$n_f$ & $k_3^{\rm FAC}$ & $k_3^{\rm PMS}$ &
$k_4^{\rm PMS}$ \\
3 & $ 199.1$ & $ 200 \pm 60 $ & $ 2200 \pm 1500 $\\
\hline
4 & $ 171.2$ & $ 170 \pm 50 $ & $ 1800 \pm 1100 $\\
\hline
5 & $ 144.7$ & $ 145 \pm 40 $ & $ 1400 \pm 900 $\\
\hline
\end{tabular}
\end{center}
\caption{\label{t5}
Estimates of the coefficients $k_3 = k^{[2]3}_{2}$ and $k_4 = k^{[2]4}_{2}$
in the function $\Pi^{[2]}_2$
based on the FAC and PMS optimizations.
The estimate of $k^{[1]4}_{2}$ employs the predicted value for $k_3$;
the corresponding uncertainties include only the ones induced by $k_3$.}
\end{table}
At last, we repeat the analysis for the function $ \Pi^{[2]}_2$.
The results of FAC/PMS optimization methods are given in Table~\ref{t5}.
Using the values of $k_3^{\rm PMS}$ from the table for
$n_f=3,4,5$ we reconstruct the corresponding full $n_f$ dependence:
\[ k^{[1]3}_{2} = 294.472 - 33.2429 \, n_f + 0.696598 \, n_f^2
{}.
\]
The comparison with the known terms of order $n_f$ and $n_f^2$
($- 32.0843 \, n_f , \ \ 0.464663 \, nf^2 $) demonstrates a remarkably good
agreement for the subleading $n_f$ contribution.
The 50\% error in the predicted value of the $n_f^2$ contribution
looks natural as the corresponding coefficient is
small. Following the same line of reasoning as above we have assigned a
30\% uncertainty to the $O( \alpha_s ^3)$ fixed $n_f$ result.
To get a general idea about the size of the $ \alpha_s ^4$ contribution to
eq.~(\ref{Pi2m2:exact}) we used FAC/PMS and the predicted $ \alpha_s ^3$ coefficient.
The results are listed in the forth column of Table~\ref{t5}.
Let us now consider the effect of $ \alpha_s ^3$ and $ \alpha_s ^4$ corrections on
the determination of the strange quark mass\footnote{For a recent
review of various attempts to extract the strange quark mass from
$\tau$ data see Ref.~\cite{Maltman:2002wb}.}.
The mass correction to the $R_{\tau}$ depend on both functions
$D^{[1]}_2$ and $ \Pi^{[2]}_2$. Let us use the central ALEPH value
of $ \alpha_s (M_{\tau}) = 0.334 $
when estimating the size of the perturbative corrections. For fixed order
one finds the mass correction to the total rate:
\begin{eqnarray}
{\delta}^{00}_{us,2} &=&
-8\frac{m_s^2}{M_\tau^2}(
1. + 5.33 \, a_s+ 46.0 \, a_s^2 + 284 \, a_s^3
\nonumber\\
&+& 0.75 \, a_s^3 k^{[2]3}_2
+ a_s^4(723. + 0.25 \ d^{[1]4}_2 + 9.84 \ k^{[2]3}_2 + 0.75\ k^{[2]4}_2 )
)
\nonumber
\\
&{=}&
-8\frac{m_s^2}{M_\tau^2}(1. + 0.567 + 0.520 + 0.521\pm 0.05 + 0.593 )
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}
(3.2 \pm 0.6)
\label{num:disc:1.3}
{},
\end{eqnarray}
where in the last line we have assumed the (maximal!) value of
the ${\cal{O}}(\alpha_s^4)$
term as an estimate of the theoretical uncertainty
(this convention will be used also below).
%However, one must admit that terms of order
%$a_s$, $a_s^2$, $a_s^3$ and $a_s^4$
For the ``contour improved'' series one obtains
\begin{eqnarray}
\tilde{\delta}^{00}_{us,2} &=&
-8\frac{m_s^2}{M_\tau^2}(
1.44 + 3.65 \, a_s+ 30.9 \, a_s^2 + 72.2 \, a_s^3 + 1.18 \, a_s^3 k_2^{[2],3}
\nonumber
\\
&{}& \phantom{MMMMMMMxxl} + a_s^4 ( 0.678 \ d^{[1]4}_2 + 1.06 \ k^{[2]4}_2 )
)
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}(1.44 + 0.389 + 0.349 + 0.371 \pm 0.09 + 0.403)
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}
( 2.95 \pm 0.4)
\label{num:disc:2.3}
{}.
\end{eqnarray}
Now we consider the contributions of spin 1 and spin 0 separately.
The lowest moments ($L = 0$) of the spin-dependent functions depend on
a nonperturbative quantity and, thus, can not be treated
perturbatively in principle \cite{Chetyrkin:1998ej}.
For the spin one part and for $(k,l) = (0,1)$ we find
\begin{eqnarray}
{\delta}^{(1)01}_{us,2} &=&
-5 \frac{m_s^2}{M_\tau^2}(
1. + 4.83 a + 35.7 \, a_s^2 + 276. \, a_s^3 +a_s^4 (1350 + d^{[1]4}_2))
)
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}
(1. + 0.514 + 0.404 + 0.331 + 0.326)
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}
(2.58 \pm 0.33)
{}
\label{num:disc:5.3}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{\delta}^{(1)01}_{us,2} &=&
-5\frac{m_s^2}{M_\tau^2}(
1.37 + 2.55 a + 16.1 \, a_s^2 + 135 \, a_s^3 + 0.895 \, a_s^4 d_2^{[1]4})
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}( 1.37 + 0.271 + 0.182 + 0.163 + 0.137 )
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}
( 2.12 \pm 0.14)
{}.
\label{num:disc:.6.3}
\end{eqnarray}
Note that spin 1 contribution is determined by the component
$\Pi^{[1]}$ alone and is known up to third order.
Clearly, this series is decreasing in a reasonable way
(comparable to the behaviour of $\tilde{\delta}^{00}_{us,2}$)
and, at the same
time, only moderately dependent on the improvement
prescription with
$\tilde{\delta}^{(1)01}_{us,2}/{\delta}^{(1)01}_{us,2} = 0.82$.
On the basis of Eq.~(\ref{num:disc:.6.3})
this moment might well serve for a reliable $m_s$ determination, with
a sufficiently careful interpretation of the theoretical
uncertainty.
The corresponding spin zero part is, per se, proportional to
$m_s^2$ (not counting non-perturbative, so-called ``condensate'' contributions)
and thus could be considered as ideal for a
measurement of $m_s$. However, the behaviour of the perturbative
series
\begin{eqnarray}
{\delta}^{(0)01}_{us,2} &=&
\frac{3}{2} \frac{m_s^2}{M_\tau^2}(
1. + 9.33 a + 110 \, a_s^2 + 1323 \, \, a_s^3
\nonumber \\
&{}&\phantom{MMMMMM} + a_s^4 (12200 + d_2^{[1]4} + 17.5 k_2^{[2]3})
\nonumber \\
&{=}&
\frac{m_s^2}{M_\tau^2}
(1. + 0.992 + 1.24 + 1.59 + 2.16)
\nonumber \\
&{=}&
\frac{3}{2} \frac{m_s^2}{M_\tau^2}
(7.0 \pm 2 )
\label{num:disc:.7.3}
{}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{\delta}^{(0)01}_{us,2} &=&
\frac{3}{2}\frac{m_s^2}{M_\tau^2}(
3.19 + 11.2 a + 126. \, a_s^2 + 289. \, a_s^3 + 6.63 \, a_s^3 k_2^{[2]3}
\nonumber
\\
&{}&\phantom{MMMMMM} + a_s^4 (2.71 d_2^{[1]4} + 7.76 k_2^{[2]4})
)
\nonumber \\
&{=}&
\frac{3}{2}\frac{m_s^2}{M_\tau^2}(
3.19 + 1.19 + 1.42 + 1.94 + 2.6 \pm 1.31
)
\nonumber \\
&{=}&
\frac{3}{2} \frac{m_s^2}{M_\tau^2}( 10.3 \pm 2.6 )
\label{num:disc:.8.3}
{}
\end{eqnarray}
shows a rapid growth of the coefficients. The series is not expected
to provide an accurate prediction for the mass effects.
\section{Summary}
Implications of the newly calculated $\alpha_s^4 n_f^2$ terms together
with the $\alpha_s^4 n_f^3$ terms for an improved extraction of $ \alpha_s $
from the $\tau$ decay are presented. Arguments are presented in
support of predictions for the remaining terms of order $\alpha_s^4
n_f$ and $\alpha_s^4 n_f^0$ which are based on FAC or PMS
optimization. The complete calculation will lead to a reduction of the
theory uncertainty within the frameworks of FOPT or CIPT down to a
negligible amount. However, an irreducible difference between the
results from these two schemes of $\delta \alpha_s (M_{\tau}) \approx 0.02$
corresponding to $\delta \alpha_s (M_{Z}) \approx 0.002$ persists even
after inclusion of $\mathcal{O}(\alpha_s^4)$ (and even
$\mathcal{O}(\alpha_s^5)$ terms). Similar investigations, based on data
up to higher energies (e. g. for fictitious heavy lepton of 3 GeV
of for sum rules based on $e^+ e^-$ data) would
lead to significantly smaller errors.
New contributions of orders $\mathcal{O}(m_s^2\alpha_s^4 n_f^2)$ and
$\mathcal{O}(\alpha_s^4 n_f^2)$ to (axial)vector correlators relevant
for the QCD description of the semileptonic $\tau$ decay into hadrons
are obtained. The moments $R^{(0,0)}$ and $R^{(0,1)}$ are evaluated
separately for spin zero and spin one final states
\cite{Chetyrkin:1998ej}. The results are tested against predictions
of FAC/PMS optimization methods. Good agreement is found. This has
motivated us to take the full of FAC/PMS predictions as the basis for
a new extraction of $\alpha_s$ and $m_s$ from $\tau$-decays with
$\mathcal{O}(\alpha_s^4)$ accuracy. Using $\delta^\mathrm{exp}_P =
0.207$ we find 0.330 and 0.1192 for $ \alpha_s
^{\mathrm{FOPT}}(M_{\tau})$ and $ \alpha_s ^{\mathrm{FOPT}}(M_{Z})$
respectively. In the framework of contour improved evaluation these
values increase to 0.354 and 0.1219 respectively
In contrast to the massless
result, the PT series contributing to the $m_s^2$ dependent part seem
to be barely convergent and the additional higher order terms
seemingly do not lead to any significant improvement of the
theoretical accuracy in the determination of the strange quark mass
from the $\tau $ decays.
A slightly more favorable pattern of convergence is observed for the
moments of the spin one contribution separately.
\section{Acknowledgments}
The authors are grateful to A.A.~Pivovarov for a lot of useful
discussions. One of us (K.G.~Ch.) acknowledges useful
correspondence to K.~Maltman and A.L.~Kataev. This work was supported by the
DFG-Forschergruppe {\it ``Quantenfeldtheorie, Computeralgebra und
Monte-Carlo-Simulation''} (contract FOR 264/2-1), by INTAS (grant
00-00313), by RFBR (grant 01-02-16171), by Volkswagen Foundation and
by the European Union under contract HPRN-CT-2000-00149.
\sloppy
\raggedright
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