%Title: Leading Mass Correction to Cabibbo suppressed tau Decays: the Perturbation Theory Structure and Extraction of m_s
%Author: K. G. Chetyrkin, J.H. K\"uhn and A.A. Pivovarov
%Published: * Proceedings of the 10th International Seminar Quarks 98 * (Suzdal, Russia, May 17-24, 1998). Eds. F.L.Bezrukov, V.A.Matveev V.A.Rubakov, A.N.Tavkhelidze, S.V.Troitsky, Moscow 1999 vol. 1, p.58-70
%%Proceedings: version Sep 18; .to
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\newcommand{\dmu}{\mu^2\frac{d}{d\mu^2}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\kmu}{\dsp\frac{(m_i(\mu^2)\mp m_j(\mu^2))^2}{Q^2}}
\newcommand{\kq}{\dsp\frac{(m_i(Q^2)\mp m_j(Q^2))^2}{Q^2}}
\newcommand{\kmup}{\dsp\frac{(m_i(\mu^2)\pm m_j(\mu^2))^2}{Q^2}}
\newcommand{\kqp}{\dsp\frac{(m_i(Q^2)\pm m_j(Q^2))^2}{Q^2}}
\newcommand{\pva}{\Pi^{(0)}_{V/A}}
\newcommand{\pvaij}{\Pi^{(0)}_{V/A;ij}}
\newcommand{\pvatr}{\Pi^{(0+1)}_{V/A}}
\newcommand{\pvaijtr}{\Pi^{(0+1)}_{V/A;ij}}
\newcommand{\dvaijz}{\delta^{(D=2)}_{V/A;ij}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
\vspace{0.5cm}
\begin{center}
{\large \bf
Leading mass correction to Cabibbo suppressed $\tau$ decays:\\
the perturbation series structure and extraction of $m_s$}
\vspace{0.8cm}
\begin{large}
K.G.~Chetyrkin\footnote{On leave from Institute for Nuclear Research
of the Russian Academy of Sciences, Moscow, 117312, Russia.},
J.H.~K\"uhn \\
Institut f\"ur Theoretische Teilchenphysik\\
Universit\"at Karlsruhe\\
7500 Karlsruhe 1, Germany\\[5mm]
A.A.~Pivovarov\\
Institute for Nuclear Research
of the Russian Academy of Sciences\\
Moscow, 117312, Russia
\end{large}
\end{center}
\vspace{0.7cm}
\centerline{\bf Abstract}
In this talk the recent results on perturbative QCD corrections
to the hadronic decay rate of the
$\tau$ lepton are presented.
We discuss the structure of perturbation theory series
of the leading mass correction.
On the base of this analysis
the numerical value of the strange quark mass is extracted.
\vspace{1.7cm}
The determination of the
strong coupling constant $\alpha_s$ is a focal point in recent
investigations
on extraction of the standard model parameters and
high order perturbation theory calculations \cite{phys_report}.
The small reduction of the Cabibbo suppressed rate
of $\tau$ lepton (relative to the massless prediction) has been used recently
to extract a value for the strange quark mass
\cite{Chen}. The
analysis was based on the total rate within a theoretical calculation
of quadratic mass term with the coefficient
up to $\as^2$ order \cite{Chetyrkin93}.
Some
new development calls, however,
for a fresh look at the possibility for an $m_s$
determination from $\tau$ decays \cite{Chetyrkin93-update,Maltman}.
We report on the theoretical analysis extended and improved in
the several ways \cite{ChKuhnPiv}.
QCD corrections to the mass terms of order
$\alpha_s^3$ are calculated
for some of the moments of the spectral functions.
Analysis of different spin contributions is done
separately \cite{KM}.
Resummation of effects from the running
of both the coupling constant and the strange quark mass along the
contour of integration in the complex plane through the
renormalization group improvement is used for $\tau$ lepton
observables \cite{tau:resum}.
Physical $\tau$ lepton observables
are related to correlators of
vector and axial vector currents of light quarks
that are defined as follows
\begin{equation}
\Pi^{V/A}_{\mu\nu,ij}(q) =
\displaystyle i \int dx e^{iqx}
\langle
T j^{V/A}_{\mu,ij}(x) (j^{V/A}_{\nu,ij})^{\dagger} (0) \rangle
= g_{\mu\nu} \Pi^{[1]}_{ij,V/A}(q^2) + q_{\mu}q_{\nu}
\Pi^{[2]}_{ij,V/A}(q^2)
\label{correlator}
\end{equation}
with
$j^{V/A}_{\mu,ij} = \bar{q}_i\gamma_{\mu}(\gamma_5) q_j$ \cite{BraNarPic92}.
Here $q_i$ and $q_j$ are two quarks with
masses $m_i$ and $m_j$.
Another useful representation of the tensor $\Pi^{V/A}_{\mu\nu,ij}(q)$
in terms of scalar functions reads
\begin{equation}
\Pi^{V/A}_{\mu\nu,ij}(q,m_i,m_j,m{},\mu,\alpha_s) =
(q_{\mu}q_{\nu}-g_{\mu\nu}q^2) \Pi^{(1)}_{ij,V/A}(q^2)
+ q_{\mu}q_{\nu} \Pi^{(0)}_{ij,V/A}(q^2)
{}
\label{correlator2}
\end{equation}
where the
correlator is decomposed into the components
$ \Pi^{(0,1)}_{ij,V/A}(q^2)$
that contain contributions of the states with the angular momentum
$J=0$ and $J=1$ respectively.
A direct comparison of (\ref{correlator}) and (\ref{correlator2}) leads
to the relations
\beq
\Pi^{(1)} = -\Pi^{[1]}/q^2, \ \
\Pi^{(0)} = \Pi^{[2]} + \Pi^{[1]}/q^2
{}.
\label{Pi12TOPi01}
\eeq
In general $\Pi^{[1]}(0)$ may be different from zero
which implies a kinematical
singularity in both $\Pi^{(1)}$ and $\Pi^{(0)}$.
The hadronic decay rate of the $\tau$ lepton is obtained by integrating
the absorptive parts of the spectral functions with respect to the
invariant hadronic mass.
Corresponding to two different tensor decompositions \re{correlator}
and
\re{correlator2}, two different integral
representations for the decay rate can be obtained.
The first one displays the structure of hadronic contributions
classified according to their spin
\beq \EQN{4}
R_{\tau}= R^{(1)}_{\tau} + R^{(0)}_{\tau}
=
\int_0^{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) R^{(1)}(s)
+ R^{(0)}(s)\right]
\eeq
where
\beq \EQN{5}
R^{(J)}=|V_{ud}|^2(R^{(J)}_{ud,V}+R^{(J)}_{ud,A})
+ |V_{us}|^2(R^{(J)}_{us,V}+R^{(J)}_{us,A}), \quad J=0,1.
\eeq
The second one uses the absorptive parts of the structure functions
$\Pi^{[1]}$ and $\Pi^{[2]}$ and is
more suitable for analytic continuation into the complex plane
\beq \EQN{4b}
R_{\tau}=
\int_0^{M_{\tau}^2}\frac{ds}{M_{\tau}^2}
\left(1-\frac{s}{M_{\tau}^2}\right)^2
\left[\frac{2 s }{M_\tau^2} R^{(1)}(s)
+ R^{(2)}(s)\right]
{}.
\label{equivalent.repr}
\eeq
Due to the analyticity of $\Pi^{[1,2]}$
in the cut complex
$s$-plane (the absence of singularities away from the physical
cut, even of kinematical singularities at the origin)
$R_{\tau}$ can be expressed as the contour integral along a
circle C of the radius
$|s|=M_{\tau}^2$
\beq \EQN{6}
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
Along ``the large circle'' of radius $|s| = M^2_\tau$,
the functions $\Pi^{[1,2]}(s)$ are
assumed to be reliably evaluated within pQCD.
Therefore, representation (\ref{pi1andpi2}) is a
well-defined pQCD prediction
for $R_\tau$.
This is not true for the
spin-separated parts.
Indeed, the direct use of (\ref{Pi12TOPi01}) leads to
\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 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 certainly a nonperturbative
constant. Thus, within pQCD we cannot predict
the decay rates $R^{(1,0)}_{\tau}$ separately.
In the massless limit $\Pi^{(0)} = 0$ within perturbation
theory
and $R^{(0)}$ is saturated
by $\Pi^{[1]}(0)$ corresponding to the massless pion (kaon) pole
for the axial part of the correlator.
The unknown constant drops out from 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 $k \ge 0, \ \ l \ge 1$.
(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 $.)
We stress that inclusion of Cabibbo suppressed modes
into the analysis gives
not only an additional set of experimental data but open conceptually
new possibilities because the massive piece can be measured in
conjunction with the massless contribution thus providing a strict
normalization and reducing the systematic errors of the experimental
data.
The quark mass corrections for the vector and axial correlators are
identical for the case under consideration with
$m_i = m_s\neq 0$ and $m_j=m_u=m_d=0$.
The perturbative prediction for the quadratic mass corrections up to
order $\alpha_s^3$ and for arbitrary quark masses has been presented in
\cite{Wboson}.
The results for the mass correction to
both polarization functions read
\begin{eqnarray}
\lefteqn{\Pi^{[1]}_{V,2} = 2 m_s^2\left\{
\rule{0.mm}{6mm}
\right.
l_{\mu Q}
%zero == 0
{+} \frac{\alpha_s}{\pi}
\left[
\frac{25}{4}
-4 \,\zeta(3)
+\frac{5}{3} l_{\mu Q}
+ l_{\mu Q}^2
%zero == 0
\right] }
\nonumber\\
&{+}&\left(\frac{\alpha_s}{\pi}\right)^2
\left[
\frac{18841}{432}
-\frac{1}{360} \pi^4
-\frac{3607}{54} \,\zeta(3)
+\frac{1265}{27} \,\zeta(5)
\Break
\phantom{+\left(\frac{\alpha_s}{\pi}\right)^2}
+\frac{4591}{144} l_{\mu Q}
-\frac{35}{2} \,\zeta(3)l_{\mu Q}
+\frac{22}{3} l_{\mu Q}^2
+\frac{17}{12} l_{\mu Q}^3
%zero == 0
\right]
\nonumber\\
&{+}&\left(\frac{\alpha_s}{\pi}\right)^3
\left[
l_{\mu Q}
\left(
\rule{0.mm}{5mm}
\right.
\frac{1967833}{5184}
-\frac{1}{36} \pi^4
-\frac{11795}{24} \,\zeta(3)
+\frac{33475}{108} \,\zeta(5)
\right. \nonumber \\ &{}& \left.
\phantom{\left(\frac{\alpha}{\pi}\right)}
+\frac{4633}{36} l_{\mu Q}
-\frac{475}{8} \,\zeta(3)l_{\mu Q}
+\frac{79}{4} l_{\mu Q}^2
+\frac{221}{96} l_{\mu Q}^3
%zero == 0
\right)
+k^{[1]}_3
\left.
\rule{0.mm}{6mm}
\right]
\left.
\rule{0.mm}{6mm}
\right\}{},
\label{pi1}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\lefteqn{\Pi^{[2]}_{V,2} = - 4 m_s^2\left\{
\rule{0.mm}{6mm}
\right.
1
{+} \frac{\alpha_s}{\pi}
\left[
\frac{7}{3}
+2 l_{\mu Q}
%zero == 0
\right]}
\nonumber\\
&{+}&\left(\frac{\alpha_s}{\pi}\right)^2
\left[
\frac{13981}{432}
+\frac{323}{54} \,\zeta(3)
-\frac{520}{27} \,\zeta(5)
+\frac{35}{2} l_{\mu Q}
+\frac{17}{4} l_{\mu Q}^2
%zero == 0
\right]
\nonumber\\
&{+}&\left(\frac{\alpha_s}{\pi}\right)^3
\left[
l_{\mu Q}
\left(
\frac{14485}{54}
+\frac{3659}{108} \,\zeta(3)
-\frac{3380}{27} \,\zeta(5)
+\frac{1643}{24} l_{\mu Q}
\right.
\Break
\phantom{+\left(\frac{\alpha_s}{\pi}\right)^3l_{\mu Q}}
+\frac{221}{24} l_{\mu Q}^2
%zero == 0
\right)
\left.
\left.
+k^{[2]}_3
\rule{0.mm}{6mm}
\right]
\right\}{}.
\label{pi2}
\end{eqnarray}
Here $l_{\mu Q} = {\mathrm{ln}}(\mu^2/Q^2)$,
the mass $m_s$ and QCD coupling constant $\alpha_s$ are
taken at a t' Hooft~mass $\mu$. All
correlators are renormalized within
$\overline{\mbox{MS}}$ scheme.
Note that terms of order $\alpha_s^3$ are known only with
``logarithmic'' accuracy, that is the constant parts $k_3^{[1]}$
and $k_3^{[2]}$ in the large $Q$
behavior of the corresponding correlators are not
available\footnote{In fact the very calculation of the constant parts
is well beyond the present calculational techniques.}.
Thus
we do not know the $O(m_s^2\alpha_s^3)$
contributions to $R_\tau$; however
these constants (as well as the also unknown ``low-energy'' constant
$\Pi^{[1]}(0)$) do not appear in the moments $R^{(1,0)k,l}_{\tau}$
with $k \ge 0, \ \ l \ge 1$.
We present some concrete results.
The perturbative
result for the $m^2$ terms does not differentiate between vector
and axial vector channels and we consider
their sum.
The mass corrections to the moments of spin $0$ and spin
$1$ final state distributions and with or without resummation
of effects from running are now cast into a general form
($m_s = m_s(M_\tau)$, $\alpha_s = \alpha_s(M_\tau)$)
\beq
\delta_{us} =
b_0\frac{m_s^2}{M_\tau^2}
\{
1
+ b_1 \frac{\alpha_s}{\pi}
+ b_2 \left(\frac{\alpha_s}{\pi}\right)^2
+ b_3 \left(\frac{\alpha_s}{\pi}\right)^3
\}
{}
\label{form1}
\eeq
Coefficients $b_0,\, b_1, \, b_2,\, b_3$
take some numerical values depending on the parity and spin
of the channel as well as
the method (with or without resummation).
In view of the additional
nonperturbative contribution $\sim \Pi^{[1]}(0)$ which appears
in the lowest order moment ($ l=0 $) due to the spin separation
only the results for $ l \ge 1$ can be obtained.
The rapid change of coefficients of perturbation theory expansions
for different moments is
caused by the running of the coupling constant and the mass along the
contour of integration. The resummation of these effects can be
performed in all orders of $\alpha_s$.
The technique for the massless case is described in the literature
\cite{tau:resum,DP,resum}
so we concentrate on the massive case.
The appearance of mass introduces another freedom in the choice of
the basic quantities that accumulate the perturbative information.
As basic quantities we choose
$ \Pi^{[1]}_{us}(q^2) $ and
$ \Pi^{[2]}_{us}(q^2)$.
The renormalization of the pieces proportional to $m_s^2$
is different for
$ \Pi^{[1]} $ and
$ \Pi^{[2]}$.
$ \Pi^{[2]}_{us,2}(Q^2)$
is scale-invariant and the renormalization group
improvement can be performed directly
\begin{equation}
\label{piq}
Q^2 \Pi^{[2]}_{us,2}(Q^2) = k^{[2]}_0 m_s^2(Q^2)(1 +k^{[2]}_1 a(Q^2)+\ldots)
\end{equation}
with $a=\alpha_/\pi$ and $k^{[2]}_i$ being the known coefficients.
$\Pi^{[1]}$ is
not renormalized multiplicatively.
The problem is solved by
introducing the corresponding $D$-function $D^{[1]}(Q^2)$
by one differentiation with respect to $Q^2$.
The result is
%\begin{equation}
% \label{gpart}
\[
D^{[1]}_{us,2}(Q^2)= -\frac{Q^2}{2}\frac{d}{d Q^2} \Pi^{[1]}_{us,2}(Q^2)
= m_s^2(Q^2)(1 + d^{[1]}_1 a(Q^2)+\ldots).
\]
%\end{equation}
The running of the mass as taken into account
through the renormalization group equation
\begin{equation}
\label{mrg}
Q^2\frac{d}{d Q^2}m_s(Q^2)= \gamma_m(a(Q^2))m_s(Q^2) , \quad
\gamma_m(a)=-\gamma_0 a-\gamma_1 a^2-\ldots
\end{equation}
with the solution
\begin{equation}
\label{rgmasssol}
m_s(Q^2)=m_s(\mu^2)\exp\int_{\alpha_s(\mu^2)}^{\alpha_s(Q^2)}
{\gamma(x)dx\over \beta(x)}
{}.
\end{equation}
Subsequently, the integration along the contour can be performed
directly.
The explicit formula in the leading order of
the $\beta$-function for $\Pi^{[2]}(Q^2)$
is easily found.
We have
\begin{equation}
\label{piqimp}
Q^2 \Pi^{[2]}_{us,2}(Q^2)
= k^{[2]}_0 m_s^2(\mu^2)\left(a(Q^2)\over
a(\mu^2)\right)^\frac{2\gamma_0}{\beta_0}
(1 + k^{[2]}_1 a(Q^2)+\ldots)
\end{equation}
where $a(Q^2)$ is the running coupling. For
another amplitude the result is
\[
\Pi^{[1]}_{us,2}(Q^2)\!
%\]
%\begin{equation}
% \label{pigimp}
=\!-{m_s^2(\mu^2)\over \beta_0}
\!\!\left(\!{a(Q^2)\over a(\mu^2)}\right)^\frac{2\gamma_0}{\beta_0}
\!\!\left(\!{1\over{\left(\frac{2\gamma_0}{\beta_0}-1\right)}}\frac{1}{a(Q^2)}
+{d^{[1]}_1\over \frac{2\gamma_0}{\beta_0}}
+{d^{[1]}_2\over \frac{2\gamma_0}{\beta_0}+1}a(Q^2)
+\ldots\!\!\right)\! .
\]
%\end{equation}
These formulae should be substituted in \re{pi1andpi2},\re{6b} and
integrated along the contour. The generalization to higher orders of
the $\beta$- and $\gamma$-functions is straightforward.
The coefficients
$d^{[1]}_i$ and $k^{[2]}_i$
can be inferred from the explicit expression for the
polarization function \re{pi1}. They accumulate the whole
perturbative information because the $\beta$-function and the quark
anomalous dimension necessary for the restoration of the full
expressions \re{pi1} and
\re{pi2} are known.
We perform the analysis along these lines for the same moments as in
the finite order of perturbation theory with the available
coefficients for $\Pi^{[2]}$ and for the $D$-function in the case of
$\Pi^{[1]}$ and with $\beta$- and $\gamma$-functions from first to
fourth order \cite{beta4,gamma4,gamma4full}.
The main problem of the strange quark determination and even the
total perturbative analysis of the Cabibbo suppressed $\tau$ lepton
decays is the interpretation of the perturbation theory series for
$m_s^2$ corrections. In passing we note that also perturbative
corrections to the power suppressed terms proportional to
$m_s^4$ and $m_s^6$
might provide us with another example
of bad behaviour of higher order terms. However, since
the ${\cal O }(\alpha_s^2)$ corrections to the spin one
are available we
discuss the $m_s^2$ terms only.
The problem is fairly obvious from a consideration of the two lowest
order moments. We use $\alpha_s(M_\tau) = 0.334$
\cite{hocker}
in computing
the perturbative contributions.
The order of magnitude of the unknown coefficient $k_3^{[2]}$
in Eq.~(\ref{pi2})
is estimated on the basis of a geometric series to amount to
$k_3^{[2]} = 0 \pm (19.6)^2/2.33
= 0 \pm (k_2^{[2]})^2/k_1^{[2]} = 0 \pm 164.4$.
For fixed order we thus find series
\bea
{\delta}^{00}_{us,2} &=&
-8\frac{m_s^2}{M_\tau^2}(
1. + 5.33 a + 46.0 a^2 + 284 a^3 + 0.75 a^3 k_3^{[2]} )
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}(1. + 0.567 + 0.520 + 0.341 \pm 0.148 )
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}
( 2.4 \pm 0.5)
\label{num:disc:1.3}
{},
\eea
where we have assumed the (maximal!) value of the ${\cal{O}}(\alpha_s^3)$
term as an estimate of the theoretical uncertainty.
For the ``contour improved'' series (which is marked with a tilde
in our notation) one obtains
\bea
\tilde{\delta}^{00}_{us,2} &=&
-8\frac{m_s^2}{M_\tau^2}(
1.44 + 3.65 a + 30.9 a^2 + 72.2 a^3 + 1.18 a^3 k_3^{[2]}
)
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}(1.44 + 0.389 + 0.349 + 0.0867 \pm 0.234)
\nonumber \\
&{=}&
-8\frac{m_s^2}{M_\tau^2}
( 2.26 \pm 0.32)
\label{num:disc:2.3}
{}.
\eea
Comparing the ``improved'' with the finite order analysis one
observes that higher orders give numerically smaller contributions
although the apparent convergence is still rather marginal.
The ratio between two correction terms
$\tilde{\delta}^{00}_{us,2}/{\delta}^{00}_{us,2} \approx 0.95$
shows a relatively stable behaviour.
Eq.~(\ref{num:disc:2.3}), with the uncertainty increased by perhaps
a factor 2 could be considered as a reasonable estimate of the strange
mass corrections.
In fixed order approximation the moments with $l \ge 1$ are independent
of the constant $k_3^{[2]}$; a residual dependence remains,
however,
in the ``contour improved'' treatment.
For the fixed order we find
\bea
{\delta}^{01}_{us,2} &=&
-\frac{5}{3} \frac{m_s^2}{M_\tau^2}
(1 - 4.17 a - 113. a^2 - 1820. a^3) \nonumber \\
&{=}&
-\frac{5}{3} \frac{m_s^2}{M_\tau^2}
(1 - 0.443 - 1.27 - 2.19 ) \nonumber \\
&{=}&-\frac{5}{3} \frac{m_s^2}{M_\tau^2}(-2.9 \pm 2.2 )
\label{num:disc:3.3}
{},
\eea
while the ``contour improved'' method gives
\bea
\tilde{\delta}^{01}_{us,2} &=&
-\frac{5}{3} \frac{m_s^2}{M_\tau^2}
( -2.26 - 14.7 a - 204. a^2 - 171. a^3 - 13.3 a^3 k_3^{[2]})
\nonumber \\
&{=}&
-\frac{5}{3} \frac{m_s^2}{M_\tau^2}(-2.26 - 1.56 - 2.3 - 0.206 \mp 2.62)
\nonumber \\
&{=}&
-\frac{5}{3} \frac{m_s^2}{M_\tau^2}(-6.3 \pm 2.8 )
\label{num:disc:4.3}
{}.
\eea
The $(0,1)$ moments
thus
exhibit a rapid growth of the coefficients and, at the same time,
with $\tilde{\delta}^{01}_{us,2}/{\delta}^{01}_{us,2} =2.2$
a strong dependence on the improvement procedure.
This comparison shows that
there is no consistent prediction in $\msbar$ scheme for
this observable --
the first moment of the differential rate.
Now we turn to the contributions of spin one and spin zero
separately.
As stated before, the lowest moments $(l=0)$
of the spin-dependent functions depend on the nonperturbative quantity
$\Pi^{[1]}(0)$.
For the spin one part and for
$(k,l) = (0,1)$ we find within finite order perturbation theory
\bea
{\delta}^{(1)01}_{us,2} &=&
-5 \frac{m_s^2}{M_\tau^2}(1. + 4.83 a + 35.7 a^2 + 276. a^3)
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}(1. + 0.514 + 0.404 + 0.331 )
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}(2.25 \pm 0.33)
{}
\label{num:disc:5.3}
\eea
and with resummation
\bea
\tilde{\delta}^{(1)01}_{us,2} &=&
-5\frac{m_s^2}{M_\tau^2}(1.37 + 2.55 a + 16.1 a^2 + 135 a^3 )
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}( 1.37 + 0.271 + 0.182 + 0.163 )
\nonumber \\
&{=}&
-5 \frac{m_s^2}{M_\tau^2}( 2.0 \pm 0.2)
{}.
\label{num:disc:.6.3}
\eea
Note that the 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, is only moderately dependent on the improvement
prescription with
$\tilde{\delta}^{(1)01}_{us,2}/{\delta}^{(1)01}_{us,2} = 0.89$.
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 0 part is proportional to
$m_s^2$
and is ideal for a
measurement of $m_s$. However, the behaviour of the perturbative
series
\bea
\tilde{\delta}^{(0)01}_{us,2} &=&
\frac{3}{2}\frac{m_s^2}{M_\tau^2}
(3.19 + 11.2 a + 126. a^2 + 289. a^3 + 6.63 a^3 k_3^{[2]} )
\nonumber \\
&{=}&
\frac{3}{2}\frac{m_s^2}{M_\tau^2}
( 3.19 + 1.19 + 1.42 + 0.347 \pm 1.31 )
\nonumber \\
&{=}&
\frac{3}{2} \frac{m_s^2}{M_\tau^2}
( 6.14 \pm 1.6 )
\label{num:disc:.8.3}
\eea
shows a rapid growth of the coefficients. The series is not expected
to provide an accurate prediction for the mass effects.
Besides the series for observables themselves
one has also to look at convergence of
the $\beta$- and $\gamma$-functions that determine the running along the
contour.
In the present case
\[
\beta(a)=-2.25 a^2(1+1.78 a+ 4.47 a^2+ 21.0 a^3)
\]
or at $a=0.1$
\[
\beta(0.1)=-0.0225
(1+0.18 + 0.045+ 0.021)
\]
which is quite good.
For the $\gamma$-function, however,
\[
\gamma(a)=- a(1+3.79 a+ 12.4 a^2+ 44.3 a^3)
\]
and the convergence is marginally acceptable.
The resumed series behave in general better than those of finite
order.
However, for the mass corrections they still do not
satisfy the heuristic criteria
of convergence.
In practice the resummation maintains the convergence
pattern of the corresponding $D$-function. For
the mass corrections the $D$-functions themselves exhibit
rapidly growing coefficients of the perturbative series,
whence the resummation does not lead to a significant
improvement. Numerically the
convergence for $D$-functions in $\overline{\rm MS}$ scheme is
marginal. For the present case the $D$-function for
$\Pi^{[1]}(Q^2)$ is given by
\[
D^{[1]}(Q^2)= m_s^2(Q^2)(1+\frac{5}{3}a+a^2
\left(\frac{4591}{144} - \frac{35}{2}\zeta(3)\right)
\]
\[
+a^3
\left(\frac{1967833}{5184} - \frac{\pi^4}{36}
- \frac{11795}{24}\zeta(3) + \frac{33475}{108}\zeta(5)\right)
\]
\[
= m_s^2(Q^2)(1+1.67 a + 10.84 a^2+107.53 a^3)
\]
whereas a much better pattern
of convergence is observed for massless part
\beq
D_0(Q^2)=1+a+a^2
\left(\frac{299}{24} - 9\zeta(3)\right)
+a^3\left(\frac{58057}{288} -
\frac{779}{4}\zeta(3) + \frac{75}{2}\zeta(5)\right)
\label{m00}
\eeq
\[
=1+a+1.64 a^2 + 6.37 a^3 \,.
\]
This is the reason why the
precise determination of the strong coupling constant from the $\tau$
lepton lifetime is possible.
Resummation leads to a modest improvement of the apparent convergence
of the series, such that the lowest moments can be predicted with
acceptable accuracy. Assuming sufficiently precise data, a combined
analysis of the spin zero and one moments might thus lead to a
reliable determination of the strange quark mass.
As we mentioned the inclusion of Cabibbo suppressed decay modes
into the analysis leads to much stronger effects
than only providing an additional set of observables.
The whole system of $\tau$ lepton physics becomes bound
stricter. Experimentally, new data on
the massive piece can be obtained in
conjunction with the massless one that
greatly reduces the systematic errors.
Theoretically, pQCD series for several independent
observables of the same system become available that
allows one to reduce the renormalization ambiguity
considerably.
Indeed, the prospective observables for experimental study are,
for instance,
the total rate $R_{tot}$ and the first moments of spin separated
rates
$R^{1}_{J=1}$ and $R^{1}_{J=0}$.
They have a representation in terms of theoretical pQCD series
\[
R_{tot}=C_0(\alpha_s)-\frac{m_s^2}{M_\tau^2}C_m^{tot}(\alpha_s),
\]
\[
R^{1}_{J=1}=C_0^{1}(\alpha_s)-\frac{m_s^2}{M_\tau^2}C_m^{J=1}(\alpha_s),
\]
\beq
R^{1}_{J=0}=\frac{m_s^2}{M_\tau^2}C_m^{J=0}(\alpha_s).
\label{inv}
\eeq
There are three independent perturbative coefficients
from the system of five $C_{\#}^{\#}(\alpha_s)$.
They are generated by (and reflect the perturbative structure of)
the massless part
and two pieces of the massive part of the
current correlator \re{correlator}.
Explicit expressions are given by
Eq.~\re{m00} for the massless part and by
Eqs.~\re{pi1}
and \re{pi2} for the massive contributions.
The remarkable features of theoretical analysis of the $\tau$ system
with mass corrections are now explicit.
Because the relative relations between $C_\#^{\#}(\alpha_s)$
are renormalization scheme independent one cannot make the
corresponding perturbative series well convergent (in a heuristic
sense)
by just playing with redefinition of the coupling constant and the
mass parameter $m_s$. The $\msbar$ scheme definition apparently
must not be the optimal (or in any sense preferable)
when mass corrections are included.
We remind the reader that the $\msbar$ scheme is essensially
the definition of the coupling constant for the massless approximation
in $e^+e^-$ annihilation.
With all these features in mind, one way to interprete the wild
behaviour of perturbation theory series in the system of $\tau$ lepton
observables would be to consider it as a revelation of the
real divergence of PT (not an artefact of unproperly chosen scheme)
and an indication to some nonperturbative dynamics.
This interpretation, though qualitatively attractive, requires further
quantitative study.
To conclude,
we note that the large value for the correction factor
$\tilde{\delta}^{00}_{us,2}/(-8)$ leads us to a reduction of the
$m_s$ value as determined from the data \cite{Chen} on the basis of
the earlier calculation \cite{Chetyrkin93}. The reduction by about
15\% leads to $m_s(M_\tau ) = (150 \pm 30_{exp} \pm 20_{th}$) MeV. In
view of the large corrections the theoretical uncertainity can just be
considered as a guess based on Eq.~(\ref{num:disc:2.3}) and the
subsequent discussion. This corresponds to
$m_s(\mbox{1 GeV} ) = 200 \pm 40_{exp} \pm 30_{th}$ MeV
in good agreement with other determinations
(see, e.g.~\cite{others,ms:Jamin95,ms:as^2-analysis}).
{\large \bf Acknowledgements}\\
This work is partially supported
by BMBF under Contract No. 057KA92P, DGF under Contract KU 502/8-1,
INTAS under Contract INTAS-93-744-ext and by Volkswagen Foundation
under Contract No.~I/73611.
A.A.Pivovarov is supported in part by the
Russian Fund for Basic Research under contracts Nos.~96-01-01860 and
97-02-17065.
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\end{document}