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\begin{tabular}{l r}
HEP'99 \# 1\_224 & \hspace{6cm} DELPHI 99-122 CONF 309 \\
Submitted to Pa 1 & 15 June 1999 \\
\hspace{2.4cm} Pl 1 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Consistent Measurements of \boldmath $\alpha_s$ \\
from Precise Oriented \\
Event Shape Distributions } \\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\end{center}
\vfill\vfill
\begin{center}
Paper submitted to the HEP'99 Conference \\
Tampere, Finland, July 15-21
\end{center}
\vspace{\fill}
\end{titlepage}
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\def\nin{\noindent}
\newcommand{\ecm}{E_{cm}}
\newcommand{\ww}{{\rm WW}}
\newcommand{\zg}{{\rm Z}/\gamma}
\newcommand{\eps}{\varepsilon}
\newcommand{\fig}{Fig.~\ref}
\newcommand{\tab}{Table~\ref}
\newcommand{\das}{ $\Delta \alpha_s$ }
\newcommand{\asmz} {$\alpha_s(M_Z^2)$ }
\newcommand{\asmzx} {$\alpha_s(M_Z^2).$ }
\newcommand{\chindf} { $\rm \chi^2 / n_{df} $ }
\newcommand{\dasmz}{ $\Delta \alpha_s(M_Z^2)$ }
\newcommand{\as}{$\alpha_s$\hspace{0.1cm}}
\newcommand{\bfas}{\protect{ \boldmath $ \alpha_s $ \unboldmath } }
\newcommand{\bfasmz}{\protect{ \boldmath $ \alpha_s(M_Z^2) $ \unboldmath } }
\newcommand{\oas}{$\cal O$($\alpha_s$)\hspace{0.1cm}}
\newcommand{\oass}{$\cal O$($\alpha_s^2$)\hspace{0.1cm}}
\newcommand{\oassx}{$\cal O$($\alpha_s^2$)\hspace{0.05cm}.\hspace{0.1cm}}
\newcommand{\oasss}{$\cal O$($\alpha_s^3$)\hspace{0.1cm}}
\newcommand{\oasssx}{$\cal O$($\alpha_s^3$)\hspace{0.05cm}.\hspace{0.1cm}}
\newcommand{\rhoe}{\rho_{{\rm eff}}}
\newcommand{\gev}{\mbox{\,\,Ge\kern-0.2exV }}
\newcommand{\mev}{\mbox{\,\,Me\kern-0.2exV }}
\newcommand{\ee} {$\mbox{e}^+\mbox{e}^-$ }
\newcommand{\ddsigma} { \frac {1} {\sigma}
\frac {d^2\sigma} { dY d\cos\vartheta_T} }
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%========> now the second title page with abstract and authors <========%
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\begin{tabular*}{15.cm}{l@{\extracolsep{\fill}}r}
{\bf DELPHI Collaboration} & 99-122 CONF 309 \\
& 15 June 1999 \\
\hline
\end{tabular*}
\vspace*{2.0cm}
\begin{center} \Large {\bf Consistent Measurements of \boldmath $\alpha_s$ \\
from Precise Oriented \\
Event Shape Distributions } \\
\vspace*{2.0cm}
\normalsize {
{\bf S. Hahn $ ^{a}$, J. Drees $ ^{a}$ and G. Rodrigo $ ^{b}$} \\
\vspace{0.2 cm}
{\footnotesize $^{a}$ Fachbereich Physik, Bergische Universit{\"a}t-GH Wuppertal}\\
{\footnotesize Gau\ss{}stra\ss{}e 20, D-42097 Wuppertal, Germany}\\
\vspace{0.2 cm}
{\footnotesize $^{b}$ INFN - Sezione di Firenze }\\
{\footnotesize Largo E. Fermi 2, I-50125 Firenze, Italy }\\
}
\vspace*{1.5 cm}
\end{center}
\vspace{\fill}
\begin{abstract}
\noindent
An updated analysis using about 1.5 million events recorded at $ \sqrt{s} =
M_Z $ with the DELPHI detector in 1994 is presented. Eighteen infrared and
collinear safe event shape observables are measured as a function of the
polar angle of the thrust axis with respect to the $\mbox{e}^+\mbox{e}^-$
beam direction. The data are compared to theoretical calculations in
$ {\cal O} ( \alpha_s^2 )$ including the event orientation.
A combined fit of $\alpha_s$ and of the renormalization scale
$x_{\mu}$ in $\cal O$($\alpha_s^2$) yields an excellent description
of the high statistics data. \\
The weighted average from 18 observables including quark mass effects and
correlations is $ \rm \alpha_s(M_Z^2) = 0.1173 \pm 0.0026 $. The final
result, derived from the jet cone energy fraction, the observable
which commands the smallest theoretical and experimental uncertainty, is
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1180 \pm 0.0006 (exp.) \pm 0.0013 (hadr.)
\pm 0.0008 (scale) \pm 0.0007 (mass)$.
\end{center}
Further studies include an $\alpha_s$ determination using theoretical
predictions in the next to leading log approximation (NLLA),
matched NLLA and $\cal O$($\alpha_s^2$) predictions as well as theoretically
motivated optimized scale setting methods. The influence of higher order
contributions was also investigated by using the method of Pad\'{e} approximants.
Average $\alpha_s$ values derived from the different approaches are in
good agreement.
\end{abstract}
\vspace{\fill}
\begin{center}
Conference contribution for TAMPERE 1999
\end{center}
\end{titlepage}
\pagebreak
%=======================> now the paper itself <========================%
\section{Introduction}
The new attempt to perform a highly improved test of second order
perturbation theory and to provide an improved measurement of \asmz
as presented in this paper is based on recent progress in
next-to-leading order QCD calculations of oriented event shape
distributions \cite{GenAlg}. Furthermore, the most recent DELPHI data
used in this analysis are much improved in both their statistical
and systematic precision compared with those of previous DELPHI publications
\cite{DelPubs1,DelPubs2}. The distributions of 18 different
infrared and collinear safe hadronic event observables are determined
from 1.4 Million hadronic Z-decays at various values of the polar angle
$ \vartheta_T $ of the thrust axis with respect to the \ee beam direction.
The $ \vartheta_T $ dependence of all detector properties are fully
taken into account, thus achieving the best possible experimental
precision. \\
The precise experimental data are fully consistent with the expectation
from second order QCD. A two parameter fit to each of the distributions
measured at different polar angles $ \vartheta_T $ allowing an experimental
optimization of the \oass renormalization scale, results in a fully
consistent set of 18 \asmz values. For most of the distributions the
main error on the \asmz values is due to hadronization corrections
and not to renormalization scale errors. Any artificial increase
\cite{scaledisk} of the uncertainty of \asmz due to a large variation
of the renormalization scale is avoided so that the degree of
precision to which QCD can be tested remains transparent.
An average value of \asmz is derived taking account of the correlations
between the values obtained from the 18 distributions. \\
A number of additional studies have been performed to check the reliability
of the \asmz results obtained from experimentally optimized scales. In one of
these studies `optimized' renormalization scales as discussed in the
literature are used to determine \asmz in second order pertubation theory.
The different methods applied for choosing an optimized scale lead to
consistent results for the average value of \asmzx However, the scatter among
\asmz values from the individual distributions is smaller for the experimentally
optimized scales than that obtained using theoretically
motivated scale evaluation methods. The correlation between the renormalization
scales obtained with the different methods is also investigated. \\
Further determinations of \asmz are performed by using all orders resummed
calculations in next-to-leading logarithmic approximation (NLLA). Here
two different methods are applied. In the first case the pure NLLA
predictions are confronted with the data in a limited fit range. In
the second method \as is determined using matched NLLA and \oass
calculations. For both methods the renormalization scale is chosen to
be $ \mu = M_Z $. Both methods lead to average \as values consistent
with the average value obtained in \oass with experimentally optimized
renormalization scales. The very good agreement between the results of
the pure NLLA fits and those of the \oass is emphasized. A closer
inspection of the fits in matched NLLA and \oass to the highly
accurate data reveals a so far unreported problem with this method.
The trend of the data deviates systematically from the
expectation of the matched theory. \\
The selection of hadronic events and the correction procedures applied
to the data are described in Section \ref{selhadrsect}. Section
\ref{shapedefsect} introduces the investigated event shapes and compares
the expectations from various fragmentation models.
Section \ref{exoptsect} contains the comparison with angular dependent
second order QCD and a detailed discussion of the determination of \asmz.
Section \ref{theooptsect} summarises determinations of \asmz using
the renormalization scales discussed in the literature. Section \ref{padesect}
discusses results obtained by applying Pad\'{e} approximants for the
extrapolation of the pertubative predictions to higher orders.
Section \ref{NLLAsect} discusses results from applying NLLA.
The heavy quark mass correction of the \asmz values derived
from experimentally optimized renormalization scales is described in Section
\ref{qmassect}. The final results are summarized in the last section.
%=========================================================================%
\section{Detector and Data Analysis}
\label{selhadrsect}
%-------------------------------------------------------------------------%
In this analysis the reprocessed data measured with the DELPHI detector
in 1994 at a center of mass energy of $\rm \sqrt{s} = M_Z $ are used.
DELPHI is a hermetic detector with a solenoidal magnetic field of 1.2 T.
The detector and its performance have been described in detail in \cite{Detector}.
The following components are relevant to this analysis:
\begin{itemize}
\item
the Vertex Detector, VD, measuring charged particle coordinates in the
plane perpendicular to the beam with three layers of silicon micro-strip
detectors at radii between 6.3 and 11 cm and covering polar angles, $ \vartheta $,
with respect to the \ee beam between $ 37^{\circ} $ and $ 143^{\circ} $;
\item
the Inner Detector, ID, a cylindrical jet chamber with a polar angle
coverage from $ 17^{\circ} $ to $ 137^{\circ} $;
\item
the Time Projection Chamber, TPC, the principal tracking detector of DELPHI,
which has 6 sector plates on both sides, each with 16 pad rows and 192 sense wires,
in the forward and backward hemispheres, inner and outer radii of 30 cm and 122 cm
and covers polar angles from $ 20^{\circ} $ to $ 160^{\circ} $;
\item
the Outer Detector, OD, a five layer drift chamber at 192 cm radius covering
polar angles between $ 43^{\circ} $ and $ 137^{\circ} $;
\item
two sets of forward planar drift chambers, FCA and FCB, with 6 and 12 layers
respectively and overall polar angle coverages of $ 11^{\circ} $ to
$ 35^{\circ} $ and $ 145^{\circ} $ to $ 169^{\circ} $;
\item
the High Density Projection Chamber, HPC, a lead-glass electromagnetic
calorimeter with a very good spatial resolution located inside the DELPHI
coil between 208 cm and 260 cm radius and covering polar angles between
$ 43^{\circ} $ and $ 137^{\circ} $;
\item
the Forward Electromagnetic Calorimeter, FEMC, comprising two lead-glass
arrays, one in each endcap, each consisting of 4500 lead-glass blocks
with a projective geometry, and covering polar angles from $ 10^{\circ} $ to
$ 36.5^{\circ} $ and from $ 143.5^{\circ} $ to $ 170^{\circ} $;
\item
The hadron calorimeter, HAC, an iron-gas hadronic calorimeter outside the coil,
consisting of at least 19 layers of streamer tubes and 5 cm thick iron plates
also used as flux return, whose overall angular coverage is from
$ 11.2^{\circ} $ to $ 168.8^{\circ} $.
\end{itemize}
\subsection{Event Selection}
Only charged particles in hadronic events were used.
They were required to pass the following selection criteria:
\begin{itemize}
\item
momentum, $ p $, greater than 0.4 $ GeV/c $,
\item
$\Delta p / p $ less than 1,
\item
measured track length greater than 30 cm,
\item
track polar angle between $ 16^{\circ} $ and $ 164^{\circ} $,
\item
impact parameter with respect to the nominal interaction point
within 4 cm perpendicular and 10 cm along the beam.
\end{itemize}
\nin Hadronic events were selected by requiring:
\begin{itemize}
\item
at least 5 charged particles selected,
\item
the total energy of charged particles greater than $ 12\% \sqrt{s} $
where the pion mass has been assumed for all particles,
\item
the charged energy in each hemisphere of the detector, defined by
the (x,y) plane, $ \rm E_{hemis} $, greater than $ 3\% \sqrt{s} $,
\item
the polar angle of the thrust axis, $ \vartheta_T $, between
$\rm 90.0^{\circ}$ and $ 16.3^{\circ} $.
\end{itemize}
In total about 1.4 million events satisfy these cuts. The selection
efficiency is 92 \%. Since the thrust axis does not distinguish
between forward and backward directions, it is chosen such that
$\rm \cos \vartheta_T \ge 0 $. $\rm \vartheta_T $ is called
the event orientation. The data are binned
according to the event orientation into eight bins
\footnote{For the comparison of event shape observables with QCD predictions
in all orders resummed next-to-leading-log approximation, the
distributions have been integrated over $ \vartheta_T $. Differing
from the event selection criteria listed above, the hadronic
events were selected if the polar angle of the thrust axis satisfied
$\rm 90.0^{\circ} \le \vartheta_T < 40.0^{\circ} $ for these
angular integrated distributions.}
of $\rm \cos \vartheta_T $ between 0.0 and 0.96.
With the exception of the eighth bin, the thrust axis is well contained
within the detector acceptance.
%=========================================================================%
\subsection{Correction Procedure}
The contamination of beam gas events, $\gamma \gamma $ events and leptonic
events other than $ \tau^+ \tau^- $, is expected to be less than 0.1 \%
and has been neglected. The influence of $ \tau^+ \tau^- $ events which
have a pronounced 2-jet topology and contain high momentum particles has been
determined by the KORALZ model \cite{Koralz} treated by the full simulation
of the DELPHI detector DELSIM \cite{Dperf} and the standard data
reconstruction chain. The $ \tau^+ \tau^- $ contributions have been
subtracted from the measured data according to the relative abundance of
$\tau^+ \tau^- $ $ ( 0.46 \% \pm 0.03\%) $ and hadronic events. \\
The observed data distributions were corrected for kinematic cuts,
limited acceptance and resolution of the detector as well
as effects due to reinteractions of particles inside the detector material.
The simulated data were processed in the same way as the real data.
The correction for initial state photon radiation has been determined
using events generated by JETSET 7.3 PS \cite{Jetset} with and
without initial state radiation as predicted by DYMU3 \cite{Dymu}.
For an observable Y the bin-by-bin correction factor
$\rm C(Y,\cos \vartheta_T) $ is calculated as:
\begin{equation}
C(Y,\cos \vartheta_T) =
\frac { \left( \ddsigma \right)^{DELSIM}_{generated} }
{ \left( \ddsigma \right)^{DELSIM}_{reconstructed} }
\cdot
\frac { \left( \ddsigma \right)^{noISR} }
{ \left( \ddsigma \right)^{ISR} }
\end{equation}
\hspace{0.1 cm} \\
Particles with a lifetime larger than 1 ns were considered as stable in the
generated distributions. The bin widths were chosen on the basis of the
estimated experimental resolution as to minimize bin-to-bin migration
effects. \\
For the evaluation of systematic errors the cuts for the track and event
selections were varied over a wide range, including additional cuts on
the momentum imbalance, etc. From the stability of the measured
distributions a systematic uncertainty has been computed as the variance
with respect to the central value. As the systematic error is expected to grow
in proportion to the deviation of the overall correction factor from unity
and an additional relative systematic uncertainty of 10\% of this deviation
has been added quadratically to the above value.
%=========================================================================%
\section{Measured Event Shapes and Comparison with Fragmentation Models}
\label{shapedefsect}
%-------------------------------------------------------------------------%
\subsection{Definition of the Observables}
Thrust $\rm T$ is defined by \cite{Thrust} :
\begin{equation}
T = \max_{\vec{n}_{T}} \frac
{\sum_{i} \left| \vec{p}_{i} \cdot \vec{n}_{T} \right| }
{\sum_{i} \left| \vec{p}_{i} \right| }
\end{equation}
\nin where $\rm \vec{p}_{i}$ is the momentum vector of particle $\rm i$, and
$\rm \vec{n}_{T}$ is the thrust axis to be determined. \\
Major $\rm M$ and Minor $\rm m$ are defined similarly, replacing
$\rm \vec{n}_{T}$ in the expression above by the Major axis
$\rm \vec{n}_{Maj}$, which maximizes the momentum sum transverse to
$\vec{n}_{T}$ or the Minor axis
$\rm \vec{n}_{Min} \, = \, \vec{n}_{Maj} \times \vec{n}_{T}$
respectively. \\
The oblateness $\rm O$ is then defined by \cite{Obl}:
\begin{equation}
O=M-m
\end{equation}
\pagebreak
The C-parameter $\rm C$ is derived from the eigenvalues $\rm \lambda$
of the infrared-safe linear momentum tensor $\rm \Theta^{i,j}$ \cite{CPar}:
\begin{equation}
\Theta^{i,j} = \frac
{1}
{\sum_{k} \left|\vec{p}_{k}\right|}
\cdot \sum_{k} \frac
{p^{i}_{k}p^{j}_{k}}
{\left|\vec{p}_{k}\right|}
\end{equation}
\begin{equation}
C=3\cdot(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+
\lambda_{3}\lambda_{1})
\end{equation}
Events can be divided into two hemispheres, $\rm a$ and $\rm b$,
by a plane perpendicular to the thrust axis $\rm \vec{n}_{T}$. With
$\rm M_{a}$ and $\rm M_{b}$ denoting the invariant masses of the two
hemispheres, the normalized heavy jet mass $\rm \rho_{H}$, light jet mass
$\rm \rho_{L}$, the sum of the jet masses $\rm \rho_{S}$
and their difference $\rm \rho_{L}$ can be defined as
\begin{equation}
\rho_{H}= \frac
{\max(M_{a}^{2},M_{b}^{2})} {E_{vis}^{2}}
\end{equation}
\begin{equation}
\rho_{L}= \frac
{\min(M_{a}^{2},M_{b}^{2})} {E_{vis}^{2}}
\end{equation}
\begin{equation}
\rho_{S}= \rho_{H} + \rho_{L}
\end{equation}
\begin{equation}
\label{rhod}
\rho_{D}= \rho_{H} - \rho_{L}
\end{equation}
\nin where $\rm E_{vis} $ is the total visible energy measured in a hadronic
event. \\
Jet broadening measures have been proposed in \cite{JBroad}. In both
hemispheres $\rm a $ and $\rm b $ the transverse momenta of the particles
are summed thus:
\begin{equation}
B_{a,b} = \frac
{ \sum_{i \in a,b } \left| \vec{p}_{i}\times \vec{n}_{T}\right| }
{ 2 \sum_{i}\left| \vec{p}_{i}\right| }
\end{equation}
The wide jet broadening $\rm B_{max} $, the narrow jet broadening
$\rm B_{min} $, and the total jet broadening $\rm B_{sum} $
are then defined by
\begin{equation}
B_{max}=\max(B_{a},B_{b})
\end{equation}
\begin{equation}
B_{min}=\min(B_{a},B_{b})
\end{equation}
\begin{equation}
B_{sum}= B_{max} + B_{min}
\end{equation}
The first order prediction in perturbative QCD vanishes for both
$\rm \rho_{L} $ and $\rm B_{min} $. Therefore these observables
cannot be used for the determination of $\rm \alpha_{s} $. \\
Jet rates are commonly obtained using iterative clustering algorithms
\cite{clusalg} in which a distance criterion or a metric $\rm y_{ij}$,
such as the scaled invariant mass, is computed for all pairs of particles
$\rm i $ and $\rm j $. The pair with the smallest $\rm y_{ij} $ is
combined into a pseudoparticle (cluster) according to one of several
recombination schemes. The clustering procedure is repeated until all of
the $\rm y_{ij} $ are greater than a given threshold, the jet resolution
parameter $\rm y_{cut} $. The jet multiplicity of the event is defined
as the number of clusters remaining; the $\rm n$-jet rate
$\rm R_{n}(y_{cut}) $ is the fraction of events classified as
$\rm n$-jet, and the differential two-jet rate is defined as
\begin{equation}
D_{2}(y_{cut})=\frac
{R_{2}(y_{cut})-R_{2}(y_{cut}-\Delta y_{cut}) } {\Delta y_{cut} }
\end{equation}
Several algorithms have been proposed differing from each other in
their definition of $\rm y_{ij} $ and their recombination procedure.
We apply the $\rm E0 $, $\rm P$, $\rm P0 $, JADE \cite{Jade},
Durham \cite{Durham}, Geneva \cite{clusalg} and the Cambridge
algorithm \cite{Camjet}.
The definitions of the metrics $\rm y_{ij} $ and the recombination
schemes for the different algorithms are given below. \\
In the $\rm E0 $ algorithm $ \rm y_{ij} $ is defined as the square of
the scaled invariant mass of the pair of particles
$\rm i $ and $\rm j $:
\begin{equation}
\label{invmass}
y_{ij} = \frac
{(p_{i} + p_{j})^2 } {E_{vis}^2}
\end{equation}
\nin while the recombination is defined by:
\begin{equation}
E_{k} = E_{i} + E_{j}
\hspace{.2 cm} , \hspace{.2 cm}
p_{k} = \frac
{E_{k}}{\left| \vec{p}_{i} + \vec{p}_{j} \right|}
(\vec{p}_{i} + \vec{p}_{j})
\end{equation}
\nin where $\rm E_{i} $ and $\rm E_{j} $ are the energies and $\rm \vec{p}_{i} $
and $\rm \vec{p}_{j} $ are the momenta of the particles. \\
In the $\rm P $ algorithm $\rm y_{ij} $ is defined by Eq. (\ref{invmass}),
and the recombination is defined by
\begin{equation}
\label{mrecomb}
\vec{p}_{k} = \vec{p}_{i} + \vec{p}_{j}
\hspace{.2 cm} , \hspace{.2 cm}
E_{k} = \left| \vec{p}_{k} \right|
\end{equation}
The $\rm P0 $ algorithm is defined similarly to the $\rm P $ algorithm,
but the total energy $\rm E_{vis} $ is recalculated at each iteration
according to
\begin{equation}
E_{vis} = \sum_{i} E_{i}
\end{equation}
In the JADE algorithm, the definition of $\rm y_{ij} $ is
\begin{equation}
y_{ij}= \frac
{2 E_{i} E_{j}(1 - \cos \theta_{ij} ) }
{E_{vis}^2}
\end{equation}
\nin where $\rm \theta_{ij} $ is the angle between the pair of particles
$\rm i$ and $\rm j$. The recombination is defined by Eq. (\ref{mrecomb}). \\
In the Durham algorithm, the recombination is also defined by
Eq. (\ref{mrecomb}), and
\begin{equation}
\label{durhamyij}
y_{ij}= \frac
{2 \min (E_{i}^2 ,E_{j}^2)(1 - \cos \theta_{ij} ) }
{E_{vis}^2}
\end{equation}
In the Geneva algorithm,
\begin{equation}
y_{ij}= \frac
{8 E_{i} E_{j}(1 - \cos \theta_{ij} ) }
{9(E_{i}+E_{j})^2}
\end{equation}
\nin with the recombination defined by Eq. (\ref{mrecomb}). \\
The recently proposed Cambridge algorithm \cite{Camjet} introduces an ordering
of the particles $\rm i $ and $\rm j $ according to their opening angle,
using the ordering variable
\begin{equation}
\nu_{ij}= 2 ( 1 - \cos \theta_{ij} )
\end{equation}
\nin $ \rm y_{ij} $ is defined by Eq. (\ref{durhamyij}). The algorithm starts
clustering from a table of $\rm N_{obj} $ primary objects, which are
the particles four-momenta, and proceeds as follows:
\begin{enumerate}
\item
If only one object remains, store this as a jet and stop.
\item
Select the pair of objects $ \rm i $ and $\rm j $ that have the minimal
value of the ordering variable $\rm \nu_{ij} $ and calculate $\rm y_{ij} $
for that pair.
\item
If $\rm y_{ij} < y_{cut} $ then remove the objects $\rm i $ and $\rm j $
from the table and add the combined object with four-momentum
$\rm p_i + p_j $. If $\rm y_{ij} \ge y_{cut} $ then store the object
$\rm i $ or $\rm j $ with the smaller energy as a separated jet and remove
it from the table. The higher energy object remains in the table.
\item
go to 1.
\end{enumerate}
The energy-energy correlation EEC \cite{EEC} is defined in terms of the
angle $ \rm \chi_{ij} $ between two particles $\rm i $ and $\rm j $ in an
hadronic event:
\begin{equation}
EEC(\chi)= \frac {1}{N}
\frac {1}{\Delta \chi}
\sum\limits_{N} \sum\limits_{i,j}
\frac {E_{i} E_{j}} {E_{vis}^2}
\int\limits_{\chi-\frac{\Delta\chi}{2}}^{\chi+\frac{\Delta\chi}{2}}
\delta(\chi' - \chi_{ij}) d \chi'
\end{equation}
\nin where $\rm N $ is the total number of events, $\rm \Delta \chi $ is the
angular bin width and the angle $\rm \chi $ is taken from
$\rm \chi = 0^{\circ} $ to $\rm \chi = 180^{\circ} $. \\
The asymmetry of the energy-energy correlation AEEC is defined as
\begin{equation}
AEEC(\chi) = EEC(180^{\circ}-\chi) - EEC(\chi).
\end{equation}
The jet cone energy fraction JCEF \cite{JCEF} integrates the energy within
a conical shell of an opening angle $\rm \chi $ about the thrust axis. It
is defined as
\begin{equation}
JCEF(\chi)= \frac {1}{N}
\frac {1}{\Delta \chi}
\sum\limits_{N} \sum\limits_{i}
\frac {E_{i}} {E_{vis}}
\int\limits_{\chi-\frac{\Delta\chi}{2}}^{\chi+\frac{\Delta\chi}{2}}
\delta(\chi' - \chi_{i}) d \chi'
\end{equation}
\nin where
\begin{equation}
\chi_{i} = \arccos \left( \frac
{\vec{p}_{i} \vec{n}_{T}}
{ \left| \vec{p}_{i} \right| }
\right)
\end{equation}
\nin is the opening angle between a particle and the thrust axis vector
$\rm \vec{n}_{T} $, whose direction is defined here to point from
the heavy jet mass hemisphere to the light jet mass hemisphere.
%-------------------------------------------------------------------------%
\subsection{Fragmentation Models}
\label{fragmod}
QCD based hadronization models, which describe well the distributions
of the event shape observables in the hadronic final state of \ee
annihilation, are commonly used for modeling the transition from the
primary quarks to the hadronic final state. Perturbative QCD can describe
only a part of this transition, the radiation of hard gluons or the
evolution of a parton shower.
For a determination of the strong coupling constant $\rm \alpha_s $
one has to take account of the the so-called fragmentation or
hadronization process, which is characterized by a small momentum transfer
and hence a breakdown of perturbation theory.
Several Monte Carlo models are in use to estimate the size of the
hadronization effects and the corresponding uncertainty.
The most frequently used fragmentation models, namely JETSET 7.3 PS
\cite{Jetset}, ARIADNE 4.06 \cite{Ariadne} and HERWIG 5.8c \cite{Herwig}
have been extensively studied and tuned to DELPHI data and to identified
particle spectra from all LEP experiments in \cite{Tuning}. As discussed
in detail in \cite{Tuning} all models describe the data well.
Examples of the measured hadron distributions are presented in Figures
\ref{MCData} and \ref{MCDaThe}. The increased systematic accuracy of
the data is partially due to the fact that the $ \rm \vartheta_T $
dependence of the detector corrections is explicitly taken into account.
The $ \vartheta_T $ dependence of the detector corrections is shown,
for instance, in Figure \ref{MCDaThe} for two of the observables studied.
Detailed tables of the individual event shape distributions including
their statistical and systematic errors will be made available in HEPDATA
database\cite{HEPDB}. For figures of the distributions see
also \cite{DelNoteData}. \\
%-------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\mbox{\epsfig{file=da_mc_thr.eps, width=8.8cm}}
\hspace{1. cm}
\mbox{\epsfig{file=da_mc_jcef.eps, width=8.8cm}}
\caption[]{ {\it left part:}
Measured 1-T distribution integrated over $ \rm \cos \vartheta_T $.
The upper part shows the detector correction including effects due
to initial state radiation.
The part below shows the size of the hadronization correction.
The width of the band indicates the uncertainty of the
correction. In the central part the measured 1-T distribution
is compared to the expectation from four hadronization
generators, JETSET 7.3 PS D with DELPHI modification of heavy
particle decays, JETSET 7.4 PS, ARIADNE 4.6 and HERWIG 5.8 C.
Also shown is the 1-T range used in the QCD fit. The lower part
shows the ratio (Monte Carlo simulation-data)/data for the
four hadronization generators. The width of the band indicates
the size of the experimental errors.
{\it right part:}
Same curves as shown in the left part but for JCEF
integrated over $ \rm \cos \vartheta_T $. }
\label{MCData}
\end{center}
\end{sidewaysfigure}
\begin{sidewaysfigure}
\begin{center}
\mbox{\epsfig{file=da_mc_the_thr.eps, width=8.8cm}}
\hspace{1. cm}
\mbox{\epsfig{file=da_mc_the_jcef.eps, width=8.8cm}}
\caption[]{ {\it left part:}
Measured 1-T distribution in two bins of
$ \rm \cos \vartheta_T $. The upper part shows the detector
corrections in the two $ \rm \cos \vartheta_T $ bins.
The part below shows the size of the relative hadronization
correction in the two $ \rm \cos \vartheta_T $ bins with
respect to the average correction. In the central part
the measured 1-T distributions are compared to JETSET 7.3 PS D.
The lower part shows the ratio
(Monte Carlo simulation-data)/data for the two
$ \rm \cos \vartheta_T $ bins.
{\it right part:}
Same curves as shown in the left part but for
JCEF in two bins of $ \rm \cos \vartheta_T $. }
\label{MCDaThe}
\end{center}
\end{sidewaysfigure}
%-------------------------------------------------------------------------%
Before theoretical expressions describing parton distributions can be
compared with experimental data, corrections have to be made for
hadronization effects, i.e. effects resulting from the transition of the
parton state into the observed hadronic state.
For the global event shape observables this transition is performed
by a matrix P, where $\rm P_{ij} $ is the probability that a parton state
contributing to the bin $\rm j $ of the partonic distribution will
contribute to the bin $\rm i $ in the hadronic distribution and is computed
from a Monte Carlo model. This probability matrix has been applied
to the distributions from \oass perturbative theory
$\rm D_{pert.} (Y,\cos\vartheta_T) $ to obtain the distributions for
the predictions of the observed final state
$\rm D_{hadr.} (Y,\cos\vartheta_T) $:
\begin{equation}
D_{hadr.} (Y,\cos\vartheta_T)_i =
\sum_{j} P_{ij}(Y,\cos\vartheta_T) D_{pert.} (Y,\cos\vartheta_T)_j
\end{equation}
In the case of the $\rm JCEF $, $\rm EEC $ and $\rm AEEC $, which are
defined in terms of single particles and pairs of particles, respectively,
bin-by-bin correction factors $\rm C_{Hadr.} $ similar to that described
above for the detector effects have been computed such as:
\begin{equation}
D_{hadr.} (Y,\cos\vartheta_T)_i =
C_{Hadr.}(Y,\cos\vartheta_T)_i D_{pert.} (Y,\cos\vartheta_T)_i
\end{equation}
Our reference model for evaluating hadronization effects is the JETSET 7.3
Parton Shower (PS) Generator, which has been modified with respect to the heavy
particle decays to obtain a better description of the heavy particle
branching fraction. The tuned parameters have been taken from \cite{Tuning},
where the updated tuning procedure is described in detail. \\
In order to estimate the systematic error of the hadronization correction
the analysis was repeated using alternative Monte Carlo generators with
different hadronization models. In addition the parameters for the
JETSET PS were varied. A description of the models and
their differences can be found e.g. in \cite{Tuning}.
The alternative models used are ARIADNE 4.06, HERWIG 5.8 c as well as
version 7.4 JETSET PS \cite{Jetset}.
All these models have been tuned to DELPHI data \cite{Tuning}.
For our standard Monte Carlo Program we applied also an alternative tuning
to the DELPHI data which includes Bose-Einstein correlations,
not included in the reference tuning.
Additionally the default setting of the JETSET PS parameter $\rm Q_{0} $
has been varied, which controls the parton virtuality at which the
parton shower is stopped. The $\rm Q_{0} $ value was varied from
$\rm Q_{0} = 0.5 $ $\rm GeV $ to $\rm Q_{0} = 4.0 $ $\rm GeV $.
The systematic error of $ \alpha_s $ originating from hadronization
corrections is then estimated as the variance of the fitted
$\rm \alpha_{s} $ values obtained by using all the hadronization corrections
mentioned above. \\
%=========================================================================%
\section{Comparison with Angular Dependent Second Order QCD using
Experimentally Optimized Scales}
\label{exoptsect}
The evaluation of the $\rm {\cal O} ( \alpha_s^2 )$ coefficients is
performed by using EVENT2 \cite{EVENT2}, a program for the integration of
the \oass matrix elements. The algorithm is described in \cite{GenAlg}. \\
Using this program, one can calculate the double differential cross
section for any infrared and collinear safe observable $\rm Y $ in
\ee annihilation as a function of the event orientation:
\begin{eqnarray}
\frac { 1 } { \sigma_{tot} }
\frac { d^{2} \sigma (Y, \cos \vartheta_T ) }
{ dY d \cos \vartheta_T } &
= &
\bar{ \alpha_s } ( \mu^2 ) \cdot
A (Y, \cos \vartheta_T ) \nonumber \\
&
+ &
\bar{ \alpha_s^2 } ( \mu^2 ) \cdot
\Bigg[ B (Y, \cos \vartheta_{T} ) +
\Big( 2 \pi \beta_{0} \ln ( x_{\mu} ) -2 \Big)
A(Y,\cos\vartheta_{T}) \Bigg]
\label{thseco}
\end{eqnarray}
\nin where $\rm \bar{ \alpha_s } = \alpha_s / 2\pi $ and
$\rm \beta_0 = (33 - 2n_f) / 12 \pi $.
$ \sigma_{tot} $ is the one loop corrected cross section for the process
\ee $\rm \to $ hadrons. The event orientation enters via $\rm \vartheta_T $,
which denotes the polar angle of the thrust axis with respect to the
$\mbox{e}^+\mbox{e}^-$ beam direction.
The renormalization scale factor $\rm x_{\mu} $ is defined by
$\rm \mu^2 = x_{\mu} Q^2 $ where $\rm Q = M_Z $ is the
center of mass energy. $\rm A $ and $\rm B $ denote the \oas and \oass
QCD coefficients, respectively. \\
Alternatively, the double differential cross section can be normalized
to the partial cross section in each $\rm \cos \vartheta_T $ interval:
\begin{equation}
R(Y,\cos \vartheta_T) =
\left( \frac {d\sigma} { d \cos \vartheta_T } \right)^{-1}
\frac { d^2 \sigma (Y,\cos \vartheta_T) } {dY d \cos \vartheta_T }
\end{equation}
\nin which is more appropriate for the study of residual QCD effects. \\
The running of the strong coupling $\rm \alpha_s $ at the renormalization
scale $\rm \mu $ is in second order perturbative QCD expressed as
\begin{equation}
\alpha_s( \mu ) = \frac
{1} { \beta_0 \ln \frac { \mu^2 } { \Lambda^2 } }
\left( 1 - \frac { \beta_1 } { \beta_0^2 }
\frac { \ln \ln \frac { \mu^2 } { \Lambda^2 } }
{ \ln \frac { \mu^2 } { \Lambda^2 } }
\right)
\end{equation}
\nin where $\rm \Lambda \equiv \Lambda_{ \overline{MS}}^{(5)} $ is the QCD scale
parameter computed in the Modified Minimal Subtraction
$\rm (\overline{MS}) $ scheme for $\rm n_f = 5 $ flavors and
$\rm \beta_1 = (153-19 n_f) / 24 \pi^2 $. \\
The renormalization scale $\rm \mu $ is a formally unphysical parameter
and should not enter at all into an exact infinite order calculation
\cite{ApplicPQCD}. Within the context of a truncated finite order
perturbative expansion for any particular process under consideration,
the definition of $\rm \mu $ depends on the renormalization scheme
employed, and its value is in principle completely arbitrary. This
renormalization scale problem has been discussed extensively in the
literature\cite{scaledisk,ApplicPQCD,theoambig}. \\
The traditional experimental approach to account for this problem has been
to measure all observables at the same, fixed scale value, the so-called
physical scale $\rm x_{\mu} = 1 $ or equivalently $\rm \mu^2 = Q^2 $. The
scale dependence has been taken into account by varying $\rm \mu $
over some wide ad hoc range, quoting the resulting change in the QCD
predictions as theoretical uncertainty. \\
However, the approach of choosing $\ x_{\mu} =1 $ has a severe disadvantage.
If we consider the ratio of the \oas and the \oass contributions, defined
as
\begin{equation}
\label{asratio}
r_{NLO}(Y) = \frac
{ \alpha_s ( x_{\mu} )
\bigg[ B(Y) + A(Y)
\big( 2 \pi \beta_0 \ln ( x_{\mu} ) - 2
\big)
\bigg]
}
{ A(Y) }
\end{equation}
\nin we find for many observables quite large values for the
second order contributions, in some cases this ratio can have
a magnitude approaching unity, indicating a poor convergence
behavior of the \oass predictions in the $\rm \overline{MS} $ scheme
which would quite naturally result in a wide spread of the measured
$\rm \alpha_s $ values. This has indeed been observed in previous analyses
using \oass QCD\cite{DelPubs1,NLLAPubs}. \\
Several proposals have been made which resolve the problem by choosing
optimized scales according to different theoretical prescriptions
\cite{PMSScale,ECHScale,BLMScale}. These methods have been discussed
with some controversy \cite{ApplicPQCD} and until now no consensus
has been achieved. The only approach for determining
an optimized scale value, which does not rely on specific theoretical
assumptions, is the experimental evaluation of an optimized \oass scale value
$\rm x_{\mu} $ for each measured observable separately. This strategy
has therefore been chosen for our central method. The theoretical
approaches for choosing an optimized renormalization scale value are
studied in detail in Section \ref{theooptsect}. As will be shown later,
the approach of applying experimentally optimized scales yields an impressive
consistency of the \asmz measurements from different observables. \\
The procedure applied here, is a combined fit of $\rm \alpha_s $
and the scale parameter $\rm x_{\mu} $. In the past this strategy suffered
from a poor sensitivity of the fit with respect to $\rm x_{\mu} $ for most
of the observables. Due to the high statistics and high precision data
now available, one may expect a better sensitivity at least for some of
the observables under consideration. \\
We determined $\alpha_s(M_Z^2) $ and the renormalization scale factor
$x_{\mu}$ simultaneously by comparing the corrected distributions
for each observable Y with the perturbative QCD calculations corrected
for hadronization effects as described in the previous section. The
theoretical predictions have been fitted to the measured distributions
$ R(Y,\cos\vartheta_T) $ by minimizing $ \chi^2 $, defined by using the
sum of the squares of the statistical and systematic experimental errors,
with respect to the variation of
$\Lambda_{\overline{MS}} $ and $ x_{\mu} $. \\
The fit range was chosen according to the following considerations:
\begin{itemize}
\item
Requiring a detector acceptance larger than 80 \%, the last bin in
$\rm \cos \vartheta_T $ was excluded in general, i.e. the fit range
was restricted to the interval $\rm 0.\le \cos \vartheta_T < 0.84 $
which corresponds to the polar angle interval
$\rm 32.9^{\circ} < \vartheta_T \le 90.0^{\circ}. $
\item
Acceptance corrections were required to be below about 25\%
and the hadronization corrections to be below $ \sim $ 40\%.
\item
The contribution of the second order term as defined in Eq. (\ref{asratio})
was required to be $ \left| r_{NLO}(Y) \right| < 1.0 $ over the
whole fit range.
\item
The requirement that the data can be well described by the theoretical
prediction, i.e. $\rm \chi^2 / n_{df} \simeq 1.0. $
\item
Stability of the $\rm \alpha_s $ - measurement with respect to the variation
of the fit range.
\end{itemize}
%-------------------------------------------------------------------------%
\subsection{Systematic and Statistical Uncertainties}
For each observable the uncertainties from the fit of
$\rm \alpha_s(M_Z^2) $ and of $\rm x_{\mu} $ have been determined by
changing the parameters corresponding to a unit increase of $ \chi^2 $.
In the case of asymmetric errors the higher value is taken. \\
The systematic experimental uncertainty was estimated by repeating the
analysis with different selections to calculate the acceptance corrections
as described in Section \ref{selhadrsect}. Additionally, an analysis was
performed including neutral clusters measured with the hadronic or
electromagnetic calorimeters. The overall uncertainty was taken as the
variance of the individual $\alpha_s(M_Z^2) $ measurements. \\
An additional source of experimental uncertainty arises from the determination
of the fit range, which has been estimated by varying the lower and
the upper edge of the fit range by $ \pm 1 $ bin, respectively, while the
other edge is kept fixed. Half of the maximum deviation in $ \alpha_s(M_Z^2) $
has been taken as the error due to the variation of the fit range and has
been added in quadrature. \\
The hadronization uncertainty was determined as described in
Section \ref{shapedefsect}. \\
The total uncertainty on $ \alpha_s(M_Z^2) $ is determined from the sum
of the squares of the errors listed above. \\
\pagebreak
An additional source of theoretical uncertainty arises from the
renormalization scale dependence of \asmz. But for experimentally
optimized scales the argument for any additional scale variation seems
ad hoc. To be conservative also in this case the theoretical uncertainty
has been evaluated by varying $ x_{\mu} $ between $ 0.5 \cdot x_{\mu}^{exp} $
and $ 2 \cdot x_{\mu}^{exp} $.
%-------------------------------------------------------------------------%
\subsection{Results}
\label{exoptressubsect}
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular} { l c c c c c}
\hline
\hline
Observable & Fit Range & $ \cos\vartheta_T $ Range &
$ x_{\mu} $ & $\rm \chi^2 / n_{df} $ & $ \rm n_{df} $ \\
\hline
$\rm EEC $ & $28.8^{\circ} - 151.2^{\circ}$ & 0.0 - 0.84 &
$ 0.0112 \pm 0.0006 $ & 1.02 & 236 \\
$\rm AEEC $ & $25.2^{\circ} - 64.8^{\circ}$ & 0.0 - 0.84 &
$ 0.0066 \pm 0.0018 $ & 0.98 & 75 \\
$\rm JCEF $ & $104.4^{\circ} - 169.2^{\circ}$ & 0.0 - 0.84 &
$ 0.0820 \pm 0.0046 $ & 1.05 & 124 \\
$\rm 1-T $ & 0.05 - 0.30 & 0.0 - 0.84 &
$ 0.0033 \pm 0.0002 $ & 1.24 & 89 \\
$\rm O $ & 0.24 - 0.44 & 0.0 - 0.84 &
$ 2.30 \pm 0.40 $ & 0.90 & 33 \\
$\rm C $ & 0.24 - 0.72 & 0.0 - 0.84 &
$ 0.0068 \pm 0.0006 $ & 1.02 & 82 \\
$\rm B_{Max} $ & 0.10 - 0.24 & 0.0 - 0.84 &
$ 0.0204 \pm 0.0090 $ & 0.89 & 47 \\
$\rm B_{Sum} $ & 0.12 - 0.24 & 0.0 - 0.84 &
$ 0.0092 \pm 0.0022 $ & 1.19 & 40 \\
$\rm \rho_H $ & 0.03 - 0.14 & 0.0 - 0.84 &
$ 0.0036 \pm 0.0004 $ & 0.63 & 54 \\
$\rm \rho_S $ & 0.10 - 0.30 & 0.0 - 0.36 &
$ 0.0027 \pm 0.0019 $ & 0.82 & 16 \\
$\rm \rho_D $ & 0.05 - 0.30 & 0.0 - 0.84 &
$ 2.21 \pm 0.38 $ & 1.02 & 68 \\
$\rm D_2^{E0} $ & 0.07 - 0.25 & 0.0 - 0.84 &
$ 0.048 \pm 0.020 $ & 0.85 & 68 \\
$\rm D_2^{P0} $ & 0.05 - 0.18 & 0.0 - 0.84 &
$ 0.112 \pm 0.048 $ & 1.02 & 68 \\
$\rm D_2^{P} $ & 0.10 - 0.25 & 0.0 - 0.84 &
$ 0.0044 \pm 0.0004 $ & 1.00 & 47 \\
$\rm D_2^{Jade} $ & 0.06 - 0.25 & 0.0 - 0.84 &
$ 0.126 \pm 0.049 $ & 1.05 & 75 \\
$\rm D_2^{Durham} $ & 0.015 - 0.16 & 0.0 - 0.84 &
$ 0.0126 \pm 0.0015 $ & 0.92 & 96 \\
$\rm D_2^{Geneva} $ & 0.015 - 0.03 & 0.0 - 0.84 &
$ 7.10 \pm 0.28 $ & 0.84 & 19 \\
$\rm D_2^{Cambridge} $ & 0.011 - 0.18 & 0.0 - 0.84 &
$ 0.066 \pm 0.019 $ & 0.98 & 145 \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Observables used in the \oass QCD fits. For each of the observables
the fit range, the range in $ \cos \vartheta_T $,
the measured renormalization scale factor $ x_{\mu} $ together
with the uncertainty as determined from the fit\footnotemark[2],
the $\rm \chi^2 / n_{df} $ and the
number of degrees of freedom $ \rm n_{df} $ are shown. }
\label{results}
\end{table}
\footnotetext[2]{In the case of asymmetric errors the higher value is given.}
%--------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular} { l c c c c c }
\hline
\hline
Observable & \asmz & \das (Exp.) & \das (Hadr.)
& \das (Scale.) & \das (Tot.) \\
\hline
$\rm EEC $
& 0.1142 & $\pm$ 0.0007 & $\pm$ 0.0023
& $\pm$ 0.0014 & $\pm$ 0.0028 \\
$\rm AEEC $
& 0.1150 & $\pm$ 0.0037 & $\pm$ 0.0029
& $\pm$ 0.0100 & $\pm$ 0.0111 \\
$\rm JCEF $
& 0.1169 & $\pm$ 0.0006 & $\pm$ 0.0013
& $\pm$ 0.0008 & $\pm$ 0.0017 \\
$\rm 1-T $
& 0.1132 & $\pm$ 0.0009 & $\pm$ 0.0026
& $\pm$ 0.0023 & $\pm$ 0.0036 \\
$\rm O $
& 0.1171 & $\pm$ 0.0028 & $\pm$ 0.0030
& $\pm$ 0.0038 & $\pm$ 0.0056 \\
$\rm C $
& 0.1153 & $\pm$ 0.0021 & $\pm$ 0.0023
& $\pm$ 0.0017 & $\pm$ 0.0036 \\
$\rm B_{Max} $
& 0.1215 & $\pm$ 0.0022 & $\pm$ 0.0031
& $\pm$ 0.0013 & $\pm$ 0.0041 \\
$\rm B_{Sum} $
& 0.1138 & $\pm$ 0.0030 & $\pm$ 0.0032
& $\pm$ 0.0030 & $\pm$ 0.0053 \\
$\rm \rho_H $
& 0.1215 & $\pm$ 0.0014 & $\pm$ 0.0029
& $\pm$ 0.0050 & $\pm$ 0.0060 \\
$\rm \rho_S $
& 0.1161 & $\pm$ 0.0014 & $\pm$ 0.0018
& $\pm$ 0.0016 & $\pm$ 0.0033 \\
$\rm \rho_D $
& 0.1172 & $\pm$ 0.0013 & $\pm$ 0.0034
& $\pm$ 0.0007 & $\pm$ 0.0038 \\
$\rm D_2^{E0} $
& 0.1165 & $\pm$ 0.0027 & $\pm$ 0.0029
& $\pm$ 0.0017 & $\pm$ 0.0044 \\
$\rm D_2^{P0} $
& 0.1210 & $\pm$ 0.0018 & $\pm$ 0.0026
& $\pm$ 0.0009 & $\pm$ 0.0033 \\
$\rm D_2^{P} $
& 0.1187 & $\pm$ 0.0019 & $\pm$ 0.0021
& $\pm$ 0.0036 & $\pm$ 0.0046 \\
$\rm D_2^{Jade} $
& 0.1169 & $\pm$ 0.0011 & $\pm$ 0.0020
& $\pm$ 0.0028 & $\pm$ 0.0040 \\
$\rm D_2^{Durham} $
& 0.1169 & $\pm$ 0.0013 & $\pm$ 0.0016
& $\pm$ 0.0015 & $\pm$ 0.0026 \\
$\rm D_2^{Geneva} $
& 0.1178 & $\pm$ 0.0052 & $\pm$ 0.0075
& $\pm$ 0.0295 & $\pm$ 0.0309 \\
$\rm D_2^{Cambridge} $
& 0.1164 & $\pm$ 0.0008 & $\pm$ 0.0023
& $\pm$ 0.0004 & $\pm$ 0.0025 \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Individual sources of errors of the \asmz measurement. For each
observable, the value of \asmz, the experimental uncertainty
(statistical and systematic), the uncertainty resulting
from hadronization corrections, the theoretical uncertainty due to
scale variation around the central value $ x_{\mu}^{exp} $
in the range
$ 0.5 \cdot x_{\mu}^{exp} \le x_{\mu} \le 2 \cdot x_{\mu}^{exp} $
and the total uncertainty are shown. }
\label{expres}
\end{table}
\hspace{-1. cm}
%--------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular} { l c c c c }
\hline
\hline
Observable & \asmz & \das(Scale) & \das (Tot.) & $\rm \chi^2 / n_{df} $ \\
\hline
$\rm EEC $ & 0.1297 & $\pm$ 0.0037 & $\pm$ 0.0042 & 10.7 \\
$\rm AEEC $ & 0.1088 & $\pm$ 0.0015 & $\pm$ 0.0050 & 2.04 \\
$\rm JCEF $ & 0.1191 & $\pm$ 0.0012 & $\pm$ 0.0024 & 7.7 \\
$\rm 1-T $ & 0.1334 & $\pm$ 0.0042 & $\pm$ 0.0051 & 25.9 \\
$\rm O $ & 0.1211 & $\pm$ 0.0065 & $\pm$ 0.0077 & 2.38 \\
$\rm C $ & 0.1352 & $\pm$ 0.0043 & $\pm$ 0.0053 & 12.0 \\
$\rm B_{Max} $ & 0.1311 & $\pm$ 0.0073 & $\pm$ 0.0083 & 1.67 \\
$\rm B_{Sum} $ & 0.1403 & $\pm$ 0.0056 & $\pm$ 0.0071 & 8.1 \\
$\rm \rho_H $ & 0.1325 & $\pm$ 0.0036 & $\pm$ 0.0049 & 5.1 \\
$\rm \rho_S $ & 0.1441 & $\pm$ 0.0055 & $\pm$ 0.0062 & 2.16 \\
$\rm \rho_D $ & 0.1181 & $\pm$ 0.0012 & $\pm$ 0.0039 & 1.54 \\
$\rm D_2^{E0} $ & 0.1267 & $\pm$ 0.0033 & $\pm$ 0.0052 & 1.35 \\
$\rm D_2^{P0} $ & 0.1265 & $\pm$ 0.0026 & $\pm$ 0.0041 & 1.31 \\
$\rm D_2^{P} $ & 0.1154 & $\pm$ 0.0019 & $\pm$ 0.0036 & 5.35 \\
$\rm D_2^{Jade} $ & 0.1249 & $\pm$ 0.0030 & $\pm$ 0.0042 & 1.53 \\
$\rm D_2^{Durham} $ & 0.1222 & $\pm$ 0.0034 & $\pm$ 0.0046 & 3.47 \\
$\rm D_2^{Geneva} $ & 0.0735 & $\pm$ 0.0071 & $\pm$ 0.0116 & 120. \\
$\rm D_2^{Cambridge} $ & 0.1202 & $\pm$ 0.0021 & $\pm$ 0.0033 & 1.32 \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Results of the \asmz measurements using a fixed
renormalization scale $ x_\mu = 1 $. For each observable,
the value of \asmz, the uncertainty from the variation
of the scale between $ 0.5 \le x_\mu \le 2 $, the total
uncertainty and the $\rm \chi^2 / n_{df} $ of the fit are shown. }
\label{fixres}
\end{table}
%--------------------------------------------------------------------------%
The results of the fits to the 18 event shape
distributions are summarized in tables \ref{results} and \ref{expres}
and shown in Figure \ref{uaverexp}. It should be noted that for all
observables the normalized $ \chi^{2} $ is about one for typically
more than 50 degrees of freedom. The individual errors contributing
to the total error of the $\alpha_s $ are listed in table \ref{expres}.
For comparison reasons, the data have also been fitted in \oass applying
a fixed renormalization scale value $ x_{\mu} = 1 $. The fit range
has been chosen to be the identical to that for the experimentally optimized
scales. The results of these fits are summarized in table \ref{fixres}
and shown in Figure \ref{uaverfix}. As can be seen from the
$ \chi^2 / n_{df} $ values of the fits, the choice of $ x_{\mu} = 1 $
yields only a poor description of the data for most of the observables. \\
\pagebreak
More details concerning the QCD fits are presented in Figures \ref{Scale}
to \ref{DataD2Jad}. Figure \ref{Scale} shows the values
of \asmz and the corresponding values of $ \rm \Delta \chi^2 $ for the fits
as a function of the scale $ \rm \lg (x_{\mu}) $ for some of the investigated
observables. The shape of the $ \rm \Delta \chi^2 $ curves indicates
that for most distributions the renormalization scale has to be fixed to a
rather narrow range of values in order to be consistent with the data.
For most of the observables the renormalization scale dependence of \asmz
is significantly smaller in the region of the scale value for the minimum
in $ \chi^2 / n_{df} $ than for the region around $ x_{\mu} = 1 $.
It should however be noted, that even for observables
exhibiting a strong scale dependence of \asmz, e.g. $ D_2^{Geneva} $,
the \asmz value for the experimentally optimized scale value is perfectly
consistent with the average value. \\
Figures \ref{DataThr} to \ref{DataD2Jad} contain a direct comparison of the
data measured at various bins in $ \rm \cos \vartheta_T $ with the
results of the QCD fits. The measured dependence on both
$ \rm \cos \vartheta_T $ and the studied observable are precisely
reproduced by the fits. \\
Combining the 18 individual results using an unweighted average yields \\
\begin{center}
\asmz = 0.1170 $\pm$ 0.0025 \\
\end{center}
\nin whereas the corresponding average for the fixed scale measurements
yields \asmz $ = 0.1234 \pm 0.0154 $. For the experimentally optimized scales
the scatter of the individual measurements is significantly reduced.
Also for most observables the \asmz value is systematically lower than for
the fixed scale measurements. \\
\pagebreak
The consistency of the individual measurements is clearly shown by
a $\chi^2$ value for the unweighted average computed
on the basis of the total uncertainty without considering an additional
uncertainty due to the renormalization scale dependence.
This yields s $\chi^2 = 9.6 $, showing that the individual measurements are
fully compatible with each other without assuming an additional theoretical
uncertainty due to the scale dependence of \asmz. The corresponding $\chi^2 $
for the fixed scale measurements is $\chi^2 = 168 $, i.e. the individual
measurements are clearly inconsistent in this case. \\
The idea behind the common analysis of such a large number of observables is
to optimize the use of the information contained in the complex structure of
multi-hadron events. Errors due to the corrections for hadronization
effects may be expected to cancel to some extent in the averaging procedure.
To test this expectation the analysis of each of the individual 18 observables
is repeated by performing hadronization corrections with all hadronization
generators described in Section 3.2. This results in 7 times 18 individual
$\rm \alpha_s $ values. As a first test for each of the 18 observables the
unweighted average value of $ \rm \alpha_s $ from the seven models is
evaluated. The average value of the 18 $ \rm \alpha_s $ values is
\asmz = 0.1177 $ \pm $ 0.0029.
In a second step for each of the 7 hadronization
models an unweighted average of the corresponding 18 $\rm \alpha_s $ values is
calculated. Finally an unweighted average of the 7 average values
for each hadronization models is computed resulting in
\asmz = 0.1177 $\pm$ 0.0016. The result confirms that the scatter of the
average values due to different assumptions for hadronization corrections
is significantly smaller than the uncertainty of $ \pm 0.0025 $ of the
mean value from 18 individual observables. \\
The correlations between the \as values obtained from the
different observables must be taken into account in order
to calculate their weighted average. Since the correlations
are mostly unknown, the exact correlation pattern can hardly
be worked out reliably. Therefore, we use a recently proposed method
\cite{coraver}, which makes use of a robust
estimation of the covariance matrix for a conservative estimation
of the correlated errors. Here it is assumed that different measurements
$\rm i $ and $\rm j $ are correlated with a fixed fraction $ \rhoe $
of the maximum possible correlation $\rm C_{ij}^{max} $:
\begin{equation}
C_{ij} = \rhoe C_{ij}^{max} \quad i \neq j,
\quad \mbox{with} \quad C_{ij}^{max} = \sigma_i \sigma_j
\end{equation}
For $ \rhoe = 0 $ the measurements are treated as uncorrelated,
for $ \rhoe = 1 $ as 100\% correlated entities. When $ \chi^2 < n_{df} $
the measurements are assumed to be correlated and
the value $\rm \rho_{{\rm eff}} $ then is adjusted such that the
$\chi^2 $ is equal to the number of degrees of freedom $\rm n_{df} $:
\begin{equation}
\chi^2(\rhoe ) = \sum_{i,j}
(x_i - \overline{x}) (x_j - \overline{x}) (C^{-1})_{ij} = n - 1 = n_{df}
\end{equation}
When $ \chi^2 > n_{df} $ it is assumed that the errors of the
measurement are underestimated and will therefore be scaled until
$ \chi^2 = n_{df} $ is satisfied. \\
Applying this method to 17 observables ($ \rho_H $ is related to $ \rho_D $
and $ \rho_S $ by Eq. (\ref{rhod}) )
the weighted average (see also Fig. \ref {uaverexp}) yields \\
\begin{center}
\asmz = 0.1167 $\pm$ 0.0026. \\
\end{center}
%--------------------------------------------------------------------------%
\clearpage
%--------------------------------------------------------------------------%
\begin{figure}
\vspace{-1.5 cm}
\hspace{-1.5 cm}
\begin{center}
\mbox{\epsfig{file=alphas_exp.eps,width=14.0cm}}
\caption[]{Results of the QCD fits applying experimentally optimized scales
for 18 event shape distributions.
The error bars indicated by the solid lines
are the quadratic sum of the experimental and the hadronization
uncertainty. The error bars indicated by the dotted lines include
also the additional uncertainty due to the variation of the
renormalization scale due to scale variation around the central
value $ x_{\mu}^{exp} $ in the range
$ 0.5 \cdot x_{\mu}^{exp} \le x_{\mu} \le 2 \cdot x_{\mu}^{exp} $.
Also shown is the correlated weighted average (see text).
$\rm \rho_H $ has been excluded from the
averaging procedure, since it is fully correlated with the other
two jet mass observables $\rm \rho_S $ and $ \rm \rho_D $. }
\label{uaverexp}
\end{center}
\end{figure}
%--------------------------------------------------------------------------%
\begin{figure}
\vspace{-1.5 cm}
\hspace{-1.5 cm}
\begin{center}
\mbox{\epsfig{file=alphas_fix.eps,width=14.0cm}}
\caption[]{Results of the QCD fits applying a fixed renormalization scale
$ x_\mu = 1 $ .
The error bars indicated by the solid lines
are the quadratic sum of the experimental and the hadronization
uncertainty. The error bars indicated by the dotted lines include
also the additional uncertainty due to the variation of the
renormalization scale around the central value $ x_{\mu}^{exp} $ from
$ 0.5 \cdot x_{\mu}^{exp} \le x_{\mu} \le 2 \cdot x_{\mu}^{exp} $.
Also shown is the correlated weighted average. It has been
calculated assuming the same effective correlation
$ \rhoe = 0.65 $ as for the fit results applying experimentally
optimized scales. The resulting \chindf is 68 / 16. \\
In order to yield $ \chi^2 / n_{df} = 1 $, the errors have to be scaled
by a factor $ f_{err} =3.5 $. }
\label{uaverfix}
\end{center}
\end{figure}
%--------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{-1. cm}
%\vspace{1. cm}
\mbox{\epsfig{file=scaledep_1.eps, width=12.0cm, height=16.0cm }}
\hspace{-1. cm}
\mbox{\epsfig{file=scaledep_2.eps, width=12.0cm, height=16.0cm }}
\hspace{-1. cm}
%\vspace{1. cm}
\caption[]{ ({\it left side}) \asmz and $\rm \Delta \chi^2 = \chi^2 - \chi^2_{min} $
from \oass fits to the double differential distributions
in $\rm \cos \vartheta_T $ and $\rm 1-T $, $\rm C $,
$\rm O $, $ \rm EEC $, $ \rm JCEF $, $\rho_D $ .
Additionally, the $ \chi^2 $ minima
are indicated in the \asmz curves.
({\it right side}) The same for the double differential
distributions in $\rm \cos \vartheta_T $ and the
differential 2-jet rate applying the Durham, Cambridge,
Geneva, Jade, P and E0 jet algorithm. }
\label{Scale}
\end{center}
\end{sidewaysfigure}
%--------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{1. cm}
\vspace{1. cm}
\mbox{\epsfig{file=data_thr.eps, width=11.0cm}}
\mbox{\epsfig{file=theta_thr.eps, width=11.0cm}}
\caption[]{(a) QCD fits to the measured thrust distribution for two bins
in $ \rm \cos \vartheta_T $. (b) Measured thrust distribution
at various fixed values of $\rm 1-T $ as a function of
$ \rm \cos \vartheta_T $. The solid lines represent the QCD fit. }
\label{DataThr}
\end{center}
\end{sidewaysfigure}
%--------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{1. cm}
\vspace{1. cm}
\mbox{\epsfig{file=data_jcef.eps, width=11.0cm}}
\mbox{\epsfig{file=theta_jcef.eps, width=11.0cm}}
\caption[]{Same as Figure 6 but for the jet cone energy fraction $\rm JCEF $ }
\label{DataJCEF}
\end{center}
\end{sidewaysfigure}
%--------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{1. cm}
\vspace{1. cm}
\mbox{\epsfig{file=data_eec.eps, width=11.0cm}}
\mbox{\epsfig{file=theta_eec.eps, width=11.0cm}}
\caption[]{Same as Figure 6 but for the energy energy correlation $\rm EEC $.}
\label{DataEEC}
\end{center}
\end{sidewaysfigure}
%--------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{1. cm}
\vspace{1. cm}
\mbox{\epsfig{file=data_d2jad.eps, width=11.0cm}}
\mbox{\epsfig{file=theta_d2jad.eps, width=11.0cm}}
\caption[]{Same as Figure 6 but for the differential two jet rate
with the Jade Algorithm $\rm D_2^{Jade} $.}
\label{DataD2Jad}
\end{center}
\end{sidewaysfigure}
%--------------------------------------------------------------------------%
\clearpage
%=========================================================================%
\section{Scale Setting Methods from Theory}
\label{theooptsect}
Several methods for the choice of an optimized value for the renormalization
scale value have been suggested by theory. See e.g. \cite{theoambig} for
an overview. In this section we compare the \asmz measurements, using
renormalization scales predicted by three different approaches with the
\asmz measurements using experimentally optimized scales: \\
({\it i}) The principle of minimal sensitivity (PMS) : Since all order
predictions should be independent of the renormalization scale, Stevenson
\cite{PMSScale} suggests choosing the scale to be least sensitive with
respect to its variation, i.e. from the solution of
\begin{equation}
\frac {\partial \sigma } {\partial x_{\mu} } = 0 \hspace{0.2 cm}.
\end{equation}
({\it ii}) The method of effective charges (ECH) : The basic idea of this
approach \cite{ECHScale} is to choose the renormalization scheme in such
a way that the relation between the physical quantity and the coupling is
the simplest possible one. In \oass, where the ECH approach is equivalent
to the method of fastest apparent convergence (FAC) \cite{ECHScale} the
scale is chosen in such a way that the second order term in Eq.
(\ref{thseco}) vanishes:
\begin{equation}
B (Y, \cos \vartheta_{T} )
+ 2 \pi \beta_{0} \ln ( x_{\mu} ) A(Y,\cos\vartheta_{T}) = 0 \hspace{0.2 cm}.
\end{equation}
({\it iii}) The method of Brodsky, Lepage and MacKenzie (BLM) \cite{BLMScale}:
This method follows basic ideas in QED, where the renormalized
electric charge is fully given by the vacuum polarization due to charged
fermion-antifermion pairs \cite{theoambig}. In QCD it is suggested to fix
the scale with the requirement that all the effects of quark pairs be
absorbed in the definition of the renormalized coupling itself. In \oass
this amounts to the requirement that $ x_{\mu} $ is chosen in such a way,
that the flavor dependence $ n_{f} $ of the second order term in Eq.
\ref{thseco} is removed:
\begin{equation}
\left. \frac {\partial } {\partial n_{f} } \hspace {0.1 cm}
\Bigg\{ B (Y, n_{f}) + 2 \pi \beta_{0} \ln ( x_{\mu} ) A(Y)\Bigg\}
\right|_{n_f=5} = 0 \hspace{0.2 cm}.
\end{equation}
%-------------------------------------------------------------------------%
\begin{table} [tbph]
\begin{center}
\begin{tabular} { l c c c c }
\hline
\hline
\hspace{0.1 cm}
Observable & $\alpha_s^{EXP}(M_Z^2)$ & $\alpha_s^{PMS}(M_Z^2)$ &
$\alpha_s^{ECH}(M_Z^2)$ & $\alpha_s^{BLM}(M_Z^2)$ \\
\hline
$\rm EEC $ & 0.1142 & 0.1133 & 0.1135 & 0.1142 \\
$\rm AEEC $ & 0.1150 & 0.1063 & 0.1064 & 0.1179 \\
$\rm JCEF $ & 0.1169 & 0.1168 & 0.1169 & \\
$\rm 1-T $ & 0.1132 & 0.1101 & 0.1111 & 0.1133 \\
$\rm O $ & 0.1171 & 0.1128 & 0.1124 & \\
$\rm C $ & 0.1153 & 0.1119 & 0.1124 & 0.1144 \\
$\rm B_{Max} $ & 0.1215 & 0.1222 & 0.1217 & 0.1268 \\
$\rm B_{Sum} $ & 0.1138 & 0.1023 & 0.1021 & 0.1118 \\
$\rm \rho_H $ & 0.1215 & 0.1197 & 0.1198 & 0.1258 \\
$\rm \rho_S $ & 0.1161 & 0.1154 & 0.1149 & 0.1169 \\
$\rm \rho_D $ & 0.1172 & 0.1190 & 0.1203 & \\
$\rm D_2^{E0} $ & 0.1165 & 0.1145 & 0.1142 & 0.1143 \\
$\rm D_2^{P0} $ & 0.1210 & 0.1204 & 0.1202 & 0.1232 \\
$\rm D_2^{P} $ & 0.1187 & 0.1110 & 0.1108 & 0.1118 \\
$\rm D_2^{Jade} $ & 0.1169 & 0.1137 & 0.1134 & 0.1137 \\
$\rm D_2^{Durham} $ & 0.1169 & 0.1162 & 0.1159 & 0.1241 \\
$\rm D_2^{Geneva} $ & 0.1178 & 0.1064 & 0.1171 & \\
$\rm D_2^{Cambridge} $ & 0.1164 & 0.1164 & 0.1163 & 0.1124 \\
\hline
w. average & $ 0.1167 \pm 0.0026 $ & $ 0.1154 \pm 0.0046 $ &
$ 0.1154 \pm 0.0046 $ & $ 0.1174 \pm 0.0070 $ \\
$ \chi^2 / n_{df} $ & 5.6 / 16 & 19 / 16 &
19 / 16 & 27 / 13 \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Comparison of the \asmz values obtained using the different
methods for evaluating the renormalization scale suggested
by theory. For each observable the \asmz values using
experimentally optimized scales and \asmz values for
the scales predicted by the
PMS, ECH and BLM methods are shown. The errors for the \asmz
measurements are assumed to be identical for all methods
(see table \ref{expres}).
The weighted averages are calculated using $ \rhoe = 0.65 $
and scaling the errors to yield $ \chi^2 / n_{df} = 1 $ in the
case of the PMS, ECH and the BLM methods (see text).
The fits using the scales predicted by BLM did not converge
for the observables JCEF, O, $ \rho_D $ and $ D_2^{Geneva} $. }
\label{theores}
\end{table}
%-------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{-1. cm}
\mbox{\epsfig{file=x_corel_ech.eps,width=8.0cm}}
\hspace{-0.5 cm}
\mbox{\epsfig{file=x_corel_pms.eps,width=8.0cm}}
\hspace{-0.5 cm}
\mbox{\epsfig{file=x_corel_blm.eps,width=8.0cm}}
\hspace{-1. cm}
\vspace{1. cm}
\caption[] {Correlation between experimentally optimized renormalization
scales and the scales predicted by ECH, PMS and the BLM method.
}
\label{x_corel}
\end{center}
\end{sidewaysfigure}
%-------------------------------------------------------------------------%
The results of the \asmz measurements for the individual observables
applying the different scale setting prescriptions are listed in table
\ref{theores}. The weighted averages for the different methods yield: \\
({\it i}) ECH method :
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1154 \pm 0.0046
\hspace{1. cm} ( \chi^2 / n_{df} = 19 / 16 ) $
\end{center}
({\it ii}) PMS method :
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1154 \pm 0.0046
\hspace{1. cm} ( \chi^2 / n_{df} = 29 / 16 ) $
\end{center}
({\it iii}) BLM method :
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1174 \pm 0.0070
\hspace{1. cm} ( \chi^2 / n_{df} = 27 / 13 ) $
\end{center}
\nin to be compared with $ \rm \alpha_s(M_Z^2) = 0.1167 \pm 0.0026 $
( $ \chi^2 / n_{df} = 5.6 / 16 $) using the experimentally
optimized scales. \\
\pagebreak
The weighted averages for the different theoretical methods are in
agreement with the average using experimentally optimized
scales. The scatter of the individual measurements is lowest for
the experimentally optimized scales and highest for the BLM method.
Whereas for the experimentally optimized scales the fit values for
the individual \asmz measurements are perfectly consistent, the
consistency for the ECH and the PMS methods is only moderate.
The results for the ECH and the PMS methods are very similar, the
correlation $ \rho $ between ECH and PMS scales is almost 1.
In the case of the BLM method the $ \chi^2 / n_{df} $
indicates that the individual \asmz measurements are inconsistent.
Moreover, the fits using the scales predicted by the BLM method did not
converge at all for the observables $ JCEF $, $ O $, $ \rho_D $ and
$ D_2^{Geneva} $. Figure \ref{x_corel} shows the correlation between
the experimentally optimized scales and the scales predicted by the ECH, PMS
and the BLM methods. For the ECH and the PMS method the correlations are
$ \rho = 0.77 $ in the case of BLM we find no positive correlation with
the experimentally optimized scales. Our results indicate that the ECH
and the PMS methods are useful in the case where an experimental optimization
can not be performed, whereas the BLM method does not seem to be suitable
for the determination of \asmz .
%=========================================================================%
\section{Pad\'{e} Approximation}
\label{padesect}
An approach for estimating higher-order contributions to a perturbative
QCD series is based on Pad\'{e} Approximations. The Pad\'{e} Approximant $ [N/M] $
to the series
\begin{equation}
S = S_0 + S_1 x + S_2 x^2 + \ldots + S_n x^n
\end{equation}
\nin is defined \cite{PAbas} by
\begin{equation}
[N/M] \equiv \frac {a_0 + a_1 x + a_2 x^2 + \ldots + a_N x^N }
{1 + b_1 x + b_2 x^2 + \ldots + b_M x^M }
\hspace{1. cm} ; \hspace{0.5 cm} N + M = n
\end{equation}
\nin and
\begin{equation}
\label{PaSet}
[N/M] = S + {\cal{O}} (x^{N+M+1}) \hspace{0.2 cm}.
\end{equation}
\hspace{0.1 cm} \\
The set of equations (\ref{PaSet}) can be solved, and by consideration
of the terms of $ {\cal{O}} (x^{N+M+1}) $ one can obtain an estimate
of the next order term $ S_{N+M+1} $ of the original series.
This is called the PA method.
Furthermore, for an asymptotic series $ [N/M]$ can be taken as an
estimate of the sum (PS) of the series to all orders.
The PA method has been used successfully to estimate coefficients in statistical
physics \cite{PAbas}, and various quantum field theories including QCD
\cite{PAellis}. Justifications for some of these successes have been found
in mathematical theorems on the convergence and renormalization scale
invariance of PAs \cite{PAellis}. In many cases the PAs yield
predictions for the higher order coefficients in perturbative series with
high accuracy, whereas this accuracy is not expected for the lower order
predictions as \oasss. For the application of the \asmz determination from
event shapes, the Pad\'{e} Approximation can serve as a reasonable estimate of the
errors due to higher order corrections \cite{PAedisk}. \\
For each bin of our observables an estimate for the \oasss coefficient
$ C(y) $ can be derived from $ [0/1] $:
\begin{equation}
C^{Pad\acute{e}}(y) = \frac { B^2(y) } { A(y) } .
\end{equation}
\hspace{0.1 cm} \\
It should be noted that the PA predictions, $ C^{Pad\acute{e}}(y) $ are positive
by construction which will result in large errors for kinematical regions
where the \oasss contribution is negative. The fit range has therefore been
determined in the following way: Starting from the same fit range as was used in
\oass, the fit has been accepted if the $ \chi^2 / n_{df} $ was
$ \chi^2 / n_{df} \le 5 $. Otherwise the fit range was reduced bin by bin
until the fit yielded $ \chi^2 / n_{df} \le 5 $. \\
In addition to the \oasss fits in the Pad\'{e} Approximation, the PS method has been
used as an estimate of the sum of the perturbative series and \asmz has been
extracted by fitting the [0/1] approximation directly to the data. Here,
the fit range has been chosen to be the same than for the fits in the \oasss Pad\'{e}
approximation. The $ \chi^{2} $ dependence of the \asmz fits applying the Pad\'{e}
Approximation as a function of the renormalization scale value $ x_{\mu} $ is
quite small, especially for the PS method. For most of the observables
\asmz and $ x_{\mu} $ could not be determined in a simultaneous fit. Therefore,
the fits have been done choosing a fixed renormalization scale value $ x_{\mu} = 1 $.
The uncertainty due to the scale dependence of \asmz has been estimated
by varying $ x_{\mu} $ between $ 0.5 \le x_{\mu} \le 2 $. \\
%-------------------------------------------------------------------------%
\begin{table} [bthp]
\begin{center}
\begin{tabular} { l c c c c }
\hline
\hline
Observable & Fit Range & \asmz & \das (Scale.) & \das(Tot.) \\
\hline
$\rm EEC $ & $28.8^{\circ} - 151.2^{\circ}$ & 0.1189 &
$ \pm 0.0016 $ & $ \pm 0.0026 $ \\
$\rm AEEC $ & $25.2^{\circ} - 64.8^{\circ}$ & 0.1074 &
$ \pm 0.0030 $ & $ \pm 0.0056 $ \\
$\rm JCEF $ & $104.4^{\circ} - 169.2^{\circ}$ & 0.1169 &
$ \pm 0.0006 $ & $ \pm 0.0016 $ \\
$\rm 1-T $ & 0.07 - 0.30 & 0.1207 &
$ \pm 0.0023 $ & $ \pm 0.0036 $ \\
$\rm O $ & 0.24 - 0.32 & 0.1098 &
$ \pm 0.0014 $ & $ \pm 0.0044 $ \\
$\rm C $ & 0.32 - 0.72 & 0.1208 &
$ \pm 0.0023 $ & $ \pm 0.0039 $ \\
$\rm B_{Max} $ & 0.10 - 0.24 & 0.1183 &
$ \pm 0.0016 $ & $ \pm 0.0042 $ \\
$\rm B_{Sum} $ & 0.14 - 0.18 & 0.1127 &
$ \pm 0.0016 $ & $ \pm 0.0068 $ \\
$\rm \rho_H $ & 0.03 - 0.14 & 0.1230 &
$ \pm 0.0015 $ & $ \pm 0.0036 $ \\
$\rm \rho_S $ & 0.10 - 0.30 & 0.1252 &
$ \pm 0.0024 $ & $ \pm 0.0038 $ \\
$\rm \rho_D $ & 0.07 - 0.30 & 0.1045 &
$ \pm 0.0015 $ & $ \pm 0.0040 $ \\
$\rm D_2^{E0} $ & 0.05 - 0.18 & 0.1159 &
$ \pm 0.0014 $ & $ \pm 0.0042 $ \\
$\rm D_2^{P0} $ & 0.05 - 0.18 & 0.1199 &
$ \pm 0.0011 $ & $ \pm 0.0034 $ \\
$\rm D_2^{P} $ & 0.10 - 0.20 & 0.1128 &
$ \pm 0.0008 $ & $ \pm 0.0030 $ \\
$\rm D_2^{Jade} $ & 0.06 - 0.25 & 0.1142 &
$ \pm 0.0014 $ & $ \pm 0.0032 $ \\
$\rm D_2^{Durham} $ & 0.015 - 0.16 & 0.1170 &
$ \pm 0.0009 $ & $ \pm 0.0023 $ \\
$\rm D_2^{Cambridge} $ & 0.011 - 0.18 & 0.1164 &
$ \pm 0.0007 $ & $ \pm 0.0026 $ \\
\hline
average & & \spc $ 0.1169 \pm 0.0053 $ &
\multicolumn{2}{l} { \hspace{0.5 cm} \chindf = 27 / 15 } \\
%w. average & & $ 0.1165 \pm 0.0032 $ &
% & \\
%$ \chi^2 / n_{df} $ & & 13 / 16 &
% & \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Results on \asmz for QCD-Fits including the \oasss Term
in the Pad\'{e} Approximation (PA). For each of the observables
the fit range, \asmz , the uncertainty due to scale variation
between $ 0.5 \le x_{\mu} \le 2 $ and the total uncertainty are
shown. The experimental errors and the uncertainties due to the
hadronization corrections are assumed to be the same as
for the \oass measurements.
The weighted average is calculated using $ \rhoe = 0.65 $
and scaling the errors to yield $ \chi^2 / n_{df} = 1 $ (see text).
The fit for $ D_2^{Geneva} $ did not converge. }
\label{paderes}
\end{table}
%-------------------------------------------------------------------------%
\begin{table} [bthp]
\begin{center}
\begin{tabular} { l c c c }
\hline
\hline
Observable & \asmz & \das (Scale.) & \das(Tot.) \\
\hline
$\rm EEC $ & 0.1147 & $ \pm 0.0003 $ & $ \pm 0.0021 $ \\
$\rm AEEC $ & 0.1070 & $ \pm 0.0002 $ & $ \pm 0.0048 $ \\
$\rm JCEF $ & 0.1169 & $ \pm 0.0003 $ & $ \pm 0.0015 $ \\
$\rm 1-T $ & 0.1165 & $ \pm 0.0003 $ & $ \pm 0.0028 $ \\
$\rm O $ & 0.1135 & $ \pm 0.0003 $ & $ \pm 0.0042 $ \\
$\rm C $ & 0.1150 & $ \pm 0.0003 $ & $ \pm 0.0032 $ \\
$\rm B_{Max} $ & 0.1196 & $ \pm 0.0003 $ & $ \pm 0.0039 $ \\
$\rm \rho_H $ & 0.1219 & $ \pm 0.0004 $ & $ \pm 0.0033 $ \\
$\rm \rho_S $ & 0.1161 & $ \pm 0.0003 $ & $ \pm 0.0029 $ \\
$\rm \rho_D $ & 0.1098 & $ \pm 0.0003 $ & $ \pm 0.0037 $ \\
$\rm D_2^{E0} $ & 0.1136 & $ \pm 0.0003 $ & $ \pm 0.0040 $ \\
$\rm D_2^{P0} $ & 0.1198 & $ \pm 0.0003 $ & $ \pm 0.0032 $ \\
$\rm D_2^{P} $ & 0.1124 & $ \pm 0.0003 $ & $ \pm 0.0029 $ \\
$\rm D_2^{Jade} $ & 0.1123 & $ \pm 0.0003 $ & $ \pm 0.0029 $ \\
$\rm D_2^{Durham} $ & 0.1164 & $ \pm 0.0003 $ & $ \pm 0.0021 $ \\
$\rm D_2^{Cambridge} $ & 0.1162 & $ \pm 0.0003 $ & $ \pm 0.0025 $ \\
\hline
average & \spc $ 0.1154 \pm 0.0048 $ &
\multicolumn{2}{l} { \hspace{0.5 cm} \chindf = 25 / 14 } \\
%w. average & & $ 0.1165 \pm 0.0032 $ &
% & \\
%$ \chi^2 / n_{df} $ & & 13 / 16 &
% & \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Results on \asmz for QCD-Fits applying the
Pad\'{e} Sum Approximation (PS). For each of the observables
\asmz , the uncertainty due to scale variation
between $ 0.5 \le x_{\mu} \le 2 $ and the total uncertainty are
shown. The experimental errors and the uncertainties due to the
hadronization corrections are assumed to be the same as
for the \oass measurements.
The weighted average is calculated using $ \rhoe = 0.65 $
and scaling the errors to yield $ \chi^2 / n_{df} = 1 $ (see text).
The fits for $ B_{Sum} $ and $ D_2^{Geneva} $ did not converge. }
\label{pasures}
\end{table}
%--------------------------------------------------------------------------%
\begin{figure}
\begin{center}
\hspace{-1. cm}
\mbox{\epsfig{file=scale_jcef_pade.eps, width=14.cm }}
\hspace{-1. cm}
\caption[]{ \asmz and $\rm \Delta \chi^2 = \chi^2 - \chi^2_{min} $
for the distribution of the Jet Cone Energy Fraction
as a function of $ x_{\mu} $
from QCD fits applying \oass prediction, \oasss in
Pad\'{e} Approximation and the Pad\'{e} Sum Approximation.
Additionally, the $ \chi^2 $ minimum for the \oass fit
and the renormalization scale value $ x_{\mu} = 1 $
have been indicated in the \asmz curves. }
\label{Scale-pade}
\end{center}
\end{figure}
%--------------------------------------------------------------------------%
The fit results for the individual observables are listed in tables
\ref{paderes} and \ref{pasures}. The fit applying PS to the $ B_{Sum} $
distribution did not converge for any fit range chosen. For the
$ D_2^{Geneva} $ distribution, the fits did not converge for either method.
Comparing the fit results of the \oasss fits in Pad\'{e} Approximation
with the fit results in \oass applying $ x_{\mu} = 1 $, as given in table
\ref{fixres} , the scale dependence of
\asmz is reduced for most of the observables, as one would expect from measurements
using exact calculations in \oasssx For the PS method, the reduction of the
scale dependence is even larger. Here, \asmz is less scale dependent than
in the \oass fits for all observables considered. Figure \ref{Scale-pade}
shows the scale dependence of \asmz applying the different QCD predictions
to the distribution of the Jet Cone Energy Fraction as an example.
There is almost no $ \chi^{2} $ dependence of the \asmz fits as a function of the
renormalization scale for the PS prediction.
For the fits applying \oasss in the Pad\'{e} Approximation, the $ \chi^{2} $ dependence
is less than for the \oass prediction. However, the JCEF is one of the few
observables, where a simultaneous fit of \asmz and $ x_{\mu} $ is possible. \\
Assuming the same correlation as in the \oass fits, the weighted averages
of \asmz over the observables used have been calculated as:
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1169 \pm 0.0053 $
\end{center}
\nin for the PA fits and
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1154 \pm 0.0048 $
\end{center}
\nin for the PS fits. The averages are in excellent agreement with the \oass
value of $ \rm \alpha_s(M_Z^2) = 0.1167 \pm 0.0026 $ using optimized scales,
whereas the scatter is somewhat larger than in the \oass case.
Also, the $ \chi^{2} / n_{df} $ for the average values of 27/16 and 25/15
respectively indicate less consistency than for the \oass fits.
%=========================================================================%
\section{QCD in the Next-to-Leading Log Approximation}
\label{NLLAsect}
All orders resummed QCD calculations in the Next-to-Leading Log Approximation
(NLLA) matched with \oass calculations have been used widely to measure \asmz
from event shape observables \cite{DelPubs2,NLLAPubs}. \\
For a generic event shape observable $\rm y $ for which the theoretical
prediction can be exponentiated, the cumulative cross
section at the scale $\rm Q^2 \equiv s $ is defined by:
\begin{equation}
R(y,\alpha_s)= \frac {1} {\sigma}
\int\limits_{0}^{y}
\frac {d \sigma} {dy'} dy'
\end{equation}
\nin and can be expanded in the form
\begin{equation}
R(y,\alpha_s)= C(\alpha_s)\exp {\Sigma(\alpha_s,L)} + F(\alpha_s,y) \hspace{0.2 cm},
\end{equation}
\nin where $ L \equiv \ln(1/y) $ and
\begin{equation}
C(\alpha_s) = 1 + \sum\limits_{i=1}^{\infty} C_{i} \bar{\alpha}_s^i
\end{equation}
\begin{equation}
\label{sigma}
\Sigma (\alpha_s,L) = \sum\limits_{i=1}^{\infty} \bar{\alpha}_s^i
\sum\limits_{j=1}^{i+1} G_{ij} L^m
\end{equation}
\begin{equation}
F(\alpha_s,y) = \sum\limits_{i=1}^{\infty} F_{i}(y) \bar{\alpha}_s^i \hspace{0.2 cm}.
\end{equation}
\nin The $ C_i $ are constant and the $ F_i(y) $ vanish in the infrared
limit $ y \rightarrow 0 $. The factor $ \Sigma $ to be exponentiated
can be written
\begin{equation}
\Sigma(\alpha_s,L) = L f_{LL}(\alpha_s L ) + f_{NLL}(\alpha_s L)
+ \mbox{subleading terms}
\end{equation}
\nin where $ f_{LL} $ and $ f_{NLL} $ represent the leading and the next-to-leading
logarithms. They have been calculated for a number of observables, including
$ 1-T $ \cite{NLLAT}, $ C $ \cite{NLLAC}, $ B_{max} $\cite{NLLAB},
$ B_{sum} $\cite{NLLAB}, $ \rho_H $ \cite{NLLAR} and $ D_2^{Durham} $
\cite{NLLAD}, where the NLLA predictions for $ B_{max} $ and $ B_{sum} $
entering into this analysis are the recently improved calculations by
Yu. L. Dokshitzer et al. \cite{NLLABNEW}. \\
Pure NLLA calculations can be used to measure \asmz in a limited kinematic
region close to the infrared limit, where $\rm L $ becomes large. In order
to achieve a prediction where the kinematical range can be extended
towards the 3 jet region, several procedures have been suggested
\cite{NLLAComb} to match the NLLA calculations with the calculations
in \oassx \\
The \oass QCD formula can be written in the integrated form:
\begin{equation}
R_{ \mbox{$\cal{O}$} (\alpha_s^2) }(y,\alpha_s)
= 1 + \mbox{$\cal{A}$} (y) \bar{\alpha}_s
+ \mbox{$\cal{B}$} (y) \bar{\alpha}_s^2 \hspace{0.2 cm},
\end{equation}
\nin where $\cal A $ $ (y) $ and $ \cal B $ $ (y) $ are the cumulative forms
of $ A(y,\cos\vartheta_{T})$ and $ B(y,\cos\vartheta_{T}) $ in
Eq. (\ref{thseco}), integrated over $ \vartheta_{T} $. Together with
the first and second order part of Eq. (\ref{sigma})
\begin{equation}
g_1(L) = G_{12} L^2 + G_{11} L
\end{equation}
\begin{equation}
g_2(L) = G_{23} L^3 + G_{22} L^2 + G_{21} L \hspace{0.2 cm},
\end{equation}
\nin the $ \ln R $ matching scheme can be defined as:
\begin{equation}
\ln R(y,\alpha_s) = \Sigma(\alpha_s,L)
+ H_1(y) \bar{\alpha}_s + H_2(y) \bar{\alpha}_s^2 \hspace{0.2 cm},
\end{equation}
\nin where
\begin{equation}
H_1(y) = \mbox{$\cal{A}$} (y) - g_1(L)
\end{equation}
\begin{equation}
H_2(y) = \mbox{$\cal{B}$} (y)
- \frac {1} {2} \mbox{$\cal{A}$}^2 (y) - g_2(L) .
\end{equation}
When combining \oass predictions with NLLA calculations one has to take
into account that the resummed terms do not vanish at the upper kinematic
limit $ y_{max} $ of the event shape distributions. In order to correct
for this the resummed logarithms are redefined
\cite{NLLAJets} by:
\begin{equation}
L = \ln ( 1/y - 1/y_{max} + 1 ) .
\end{equation}
%--------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular}{ l c c }
\hline
\hline
Observable & Fit Range (NLLA) & Fit Range (matched) \\
\hline
$ \rm 1-T $ & 0.04 - 0.09 & 0.04 - 0.30 \\
$ \rm C $ & 0.08 - 0.16 & 0.08 - 0.72 \\
$ \rm B_{max} $ & 0.02 - 0.04 & 0.02 - 0.24 \\
$ \rm B_{sum} $ & 0.06 - 0.08 & 0.06 - 0.24 \\
$ \rm \rho_H $ & 0.03 - 0.06 & 0.03 - 0.30 \\
$ \rm D_2^{Durham} $ & & 0.015 - 0.16 \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{Fit range for the observables in pure NLLA and
matched NLLA fits. The observable $ D_2^{Durham} $
has not been used for pure NLLA fits, since no fit range can
be found, where the resummed logarithms dominate the
predictions (see text). }
\label{nllafitran}
\end{table}
%--------------------------------------------------------------------------%
Several other matching schemes can be defined which differ in the treatment
of the subleading terms, thus introducing a principal ambiguity in the
matching procedure. The $ \ln R $ matching scheme has become
the preferred one, because it includes the $ C_2 $ and the $ G_{21} $
coefficients implicitly and uses only those NLLA terms which are known
analytically. It yields the best description of the
data in terms of $ \chi^2 / n_{df} $ in most cases \cite{DelPubs2}.
Therefore, we quote the results for the matched predictions
using the $ \ln R $ matching scheme and use the $ R $ and $ R-G21 $
matching schemes as defined e.g. in \cite{DelPubs2} for the estimation
of the uncertainty due to the matching ambiguity.
%--------------------------------------------------------------------------%
\subsection{Measurement of \bfasmz using pure NLLA predictions}
To measure \asmz from pure NLLA calculations the fit range has to be
restricted to the extreme 2-jet region, where L becomes large
and the resummed logarithms dominate. We define $ \omega $ as the ratio
of the resummed logarithms to the non-exponentiating second order
contributions as follows :
\begin{equation}
\omega = \frac { \Sigma(\alpha_s,L) }
{ H_1 (y) \bar{\alpha}_s + H_2 (y) \bar{\alpha}_s^2 }
\end{equation}
In addition to the fit range criteria listed in Section \ref{exoptsect}
we require the minimum of the ratio $ \omega $ over the fit range not
to fall below 5 for the fits in pure NLLA. This leads to
the fit ranges listed in table \ref{nllafitran}. For the observable
$ D_2^{Durham} $ the ratio $ \omega $ remains small even for small
values of $ y_{cut} $. No fit range can be found where the resummed
logarithms dominate the prediction. Therefore $ D_2^{Durham} $ has not
been used for the fits in pure NLLA. \\
Contrary to \oass predictions an optimization procedure
for the renormalization scale is unknown for the predictions in NLLA.
Therefore the scale was fixed to $ x_{\mu} = 1 $. The uncertainty due
to the scale dependence of \asmz
was estimated by varying the scale between $ 0.5 \le x_{\mu} \le 2 $. \\
The fit results for the individual observables are listed in table
\ref{nllares}. The weighted average of \asmz for the 5 observables is \\
\begin{center}
\asmz = $ 0.116 \pm 0.006 $ \\
\end{center}
\nin which is in excellent agreement with the average value for the \oass fits
of \asmz = $ 0.1167 \pm 0.0026 $.
%--------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular} { l c c c c c c }
\hline
\hline
Observable \spb & \spb \asmz & \das \spv (exp.) & \das \spv (had.) &
\das \spv (scal.) & \das \spv (tot.) & \chindf \\
\hline
$ \rm 1-T $ & \spb 0.120 &
$ \pm 0.001 $ &
$ \pm 0.004 $ &
$ \pm 0.004 $ &
$ \pm 0.006 $ &
0.59 \\
$ \rm C $ & \spb 0.116 &
$ \pm 0.002 $ &
$ \pm 0.003 $ &
$ \pm 0.004 $ &
$ \pm 0.006 $ &
0.53 \\
$ \rm B_{max} $ & \spb 0.111 &
$ \pm 0.004 $ &
$ \pm 0.003 $ &
$ \pm 0.002 $ &
$ \pm 0.006 $ &
2.37 \\
$ \rm B_{sum} $ & \spb 0.116 &
$ \pm 0.003 $ &
$ \pm 0.004 $ &
$ \pm 0.002 $ &
$ \pm 0.006 $ &
1.24 \\
$ \rm \rho_H $ & \spb 0.117 &
$ \pm 0.004 $ &
$ \pm 0.006 $ &
$ \pm 0.004 $ &
$ \pm 0.009 $ &
0.43 \\
\hline
average & \spc $ 0.116 \pm 0.006 $ &
\multicolumn{5}{l}{
\hspace{1.0 cm} \chindf = 1.2 / 4
\hspace{1.0 cm} $ \rhoe $ = 0.71 } \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Results for the \asmz fits in pure NLLA for the individual
observables together with the individual sources of
uncertainties and the $ \chi^2 / n_{df} $ for the NLLA fits.
The total error on \asmz listed, is the quadratic sum of the
experimental error (statistical and systematic uncertainty),
the uncertainty due to the hadronization correction and the
uncertainty due to the scale dependence of \asmz.
Also listed is the weighted average of \asmz for the
5 observables together with the $ \chi^2 / n_{df} $ for the
averaging procedure and the correlation parameter
$ \rhoe $. }
\label{nllares}
\end{table}
%--------------------------------------------------------------------------%
\subsection{Measurement of \bfasmz using NLLA predictions matched with \oass }
For the QCD fits using NLLA theory matched with \oass predictions the
fit range has been chosen as the combined fit range for the pure NLLA and
the \oass fits. The results for the individual observables in the
$\ln R $ matching scheme are listed in table \ref{lnrres}. The additional
uncertainty due to the matching ambiguity has been estimated as the
maximum difference of \asmz in the $ \ln R $ matching scheme and the
two alternative matching schemes, $ R $ and $ R-G21 $. \\
The average value of \asmz in the $ \ln R $ matching scheme is
\begin{center}
\asmz = $ 0.119 \pm 0.005 $ \\
\end{center}
\nin which is in good agreement with the average value for the \oass fits
of \asmz = $ 0.1167 \pm 0.0026 $. \\
%--------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular}{ l c c c c c c c }
\hline
\hline
Observable \spc & \spd \asmz &
\spd \das \spv (exp.) &
\spb \das \spv (had.) &
\spb \das \spv (scal.) &
\spb \das \spv (mat.) &
\spb \das \spv (tot.) &
\spb \chindf \\
\hline
$ \rm 1-T $ & \spd 0.124 &
\spd $ \pm 0.002 $ &
\spb $ \pm 0.003 $ &
\spb $ \pm 0.004 $ &
\spb $ \pm 0.003 $ &
\spb $ \pm 0.007 $ &
\spb 9.5 \\
$ \rm C $ & \spd 0.120 &
\spd $ \pm 0.002 $ &
\spb $ \pm 0.002 $ &
\spb $ \pm 0.004 $ &
\spb $ \pm 0.004 $ &
\spb $ \pm 0.007 $ &
\spb 15.2 \\
$ \rm B_{max} $ & \spd 0.113 &
\spd $ \pm 0.002 $ &
\spb $ \pm 0.002 $ &
\spb $ \pm 0.003 $ &
\spb $ \pm 0.003 $ &
\spb $ \pm 0.005 $ &
\spb 8.4 \\
$ \rm B_{sum} $ & \spd 0.122 &
\spd $ \pm 0.002 $ &
\spb $ \pm 0.003 $ &
\spb $ \pm 0.004 $ &
\spb $ \pm 0.005 $ &
\spb $ \pm 0.008 $ &
\spb 11.9 \\
$ \rm \rho_H $ & \spd 0.119 &
\spd $ \pm 0.002 $ &
\spb $ \pm 0.002 $ &
\spb $ \pm 0.003 $ &
\spb $ \pm 0.006 $ &
\spb $ \pm 0.008 $ &
\spb 1.33 \\
$ \rm D_2^{Durham} $ & \spd 0.121 &
\spd $ \pm 0.001 $ &
\spb $ \pm 0.002 $ &
\spb $ \pm 0.002 $ &
\spb $ \pm 0.005 $ &
\spb $ \pm 0.006 $ &
\spb 1.70 \\
\hline
average & \spd $ 0.119 \pm 0.005 $ &
\multicolumn{5}{l}{
\hspace{1.0 cm} \chindf = 2.3 / 5
\hspace{1.0 cm} $ \rhoe $ = 0.57 } \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Results of the QCD fits in the $\ln R $ matching scheme for
the individual observables together with the individual sources
of uncertainties and the $ \chi^2 / n_{df} $ for the \asmz fits.
The total error is the quadratic sum of the
experimental error (statistical and systematic uncertainty),
the uncertainty due to the hadronization correction, the
uncertainty due to the scale dependence of \asmz and the
uncertainty due to the matching ambiguity.
Also listed is the weighted average of \asmz for the
6 observables together with the $ \chi^2 / n_{df} $ for the
averaging procedure and the correlation parameter
$ \rhoe $. }
\label{lnrres}
\end{table}
%-------------------------------------------------------------------------%
Looking at the individual fit results, one finds from the $\chi^2 / n_{df} $
that the shape distributions are only poorly described by the combined
theory. The \asmz values are higher than for the fits in pure NLLA for
all observables considered. In the case of $ 1-T $, $ C $ and $ B_{sum} $
the measured \asmz values are even above the values for both the pure NLLA
fits and the \oass fits using experimentally optimized scales, where one
naively might expect the matched predictions to be a kind of `average' of the
individual theories. \\
In order to investigate this result further it is instructive
to compare the theoretical predictions of the shape distributions for
the different methods with the data distributions. Figure \ref{lnrvgl}
shows experimental distributions for $ 1-T $ and $ C $ in comparison with the
fitted curves for three different types of QCD fits, namely \oass using
experimentally optimized scales, \oass using a fixed renormalization scale
$\rm x_{\mu} $ = 1 and the fits in the $\ln R $ matching scheme.
For the fits in \oass using experimentally optimized scales, the data are
described well over the whole fit range. For the fits in \oass using a fixed
renormalization scale and the fits in the $\ln R $ matching scheme,
we find only a poor description and the slope of both
curves show a similar systematic distortion with respect to the data.
In the case of \oass applying $ x_{\mu} = 1 $ the distortion
arises from the wrong choice of the renormalization scale. Since the
scale value for the matched predictions is also chosen to be
$ x_{\mu} = 1 $, the similarity of the curves indicates that the
subleading and non-logarithmic terms originating from the \oass part
of the matched theory and introduced using the scale value
$ x_{\mu} = 1 $ dominate the $ \ln R $ predictions.
It should be noted that the matched theory requires a renormalization
scale value of $\cal O $ (1). Unlike the \oass case, 2 parameter fits in
\asmz and $ x_{\mu} $ do not converge for most of the observables; for such
low scale values as in \oass the data can not be described at all in the
matched theory.
It seems that the combination of all orders resummed predictions and
terms only known in \oass results in a systematic shift in \asmz due
to the impossibility of choosing an appropriate renormalization scale
value. Although the average values for the \oass fits, the fits in pure
NLLA and the fits in the $\ln R $ matching scheme are in good agreement,
the matched results should be considered less reliable than those of the
\oass and pure NLLA analyses.
%-------------------------------------------------------------------------%
\begin{sidewaysfigure}
\begin{center}
\hspace{-1. cm}
\mbox{\epsfig{file=thr_lnr_vgl.eps,width=10.0cm,height=14.cm}}
\mbox{\epsfig{file=cpar_lnr_vgl.eps,width=10.0cm,height=14.cm}}
\hspace{-1. cm}
\caption[]{ {\it left part:}
Comparison of DELPHI data with three different QCD Fits:
{\it i }) \oass using an experimentally optimized
renormalization scale, {\it ii}) \oass using a fixed
renormalization scale $\rm x_{\mu} = 1 $ and
{\it iii}) $ \ln R $ matched NLLA ($\rm x_{\mu}=1 $)
for the Thrust Distribution.
The lower part shows the relative difference (Fit-Data)/Fit.
Whereas the \oass curve describes the data over the whole
fit range, the slope of the curves for the fixed scale
and $ \ln R $ matching show a similar
systematic distortion with respect to the data.
{\it right part:}
The same for the C-Parameter. Here the distortion
is even stronger. }
\label{lnrvgl}
\end{center}
\end{sidewaysfigure}
%-------------------------------------------------------------------------%
%--------------------------------------------------------------------------%
\clearpage
%=========================================================================%
\section{Heavy Quark Mass Effects}
\label{qmassect}
Studies of the influence of quark mass effects on jet
cross sections \cite{impr-bmass} have shown that these effects can be important
in the study of event shape observables and hence in the measurement
of the strong coupling. For a natural mixture of quark flavors, the
influence of the quark masses on the measurement of \asmz from event shapes
is expected to be $ \sim 1 \% $ \cite{vancouver}.
In order to derive a high precision value of \asmz, the \oass measurement
with experimentally optimized scales has been refined by considering
the influence of $ b $ quark mass effects in leading order.
%---------------------------------------------------------------------------------%
\begin{table} [b]
\begin{center}
\begin{tabular} { l c c c }
\hline
\hline
Observable & \asmz & \das (Mass) & \das (Tot.) \\
\hline
$\rm EEC $ & 0.1145 & $\pm$ 0.0001 & $\pm$ 0.0028 \\
$\rm AEEC $ & 0.1149 & $\pm$ 0.0001 & $\pm$ 0.0111 \\
$\rm JCEF $ & 0.1180 & $\pm$ 0.0007 & $\pm$ 0.0018 \\
$\rm 1-T $ & 0.1136 & $\pm$ 0.0001 & $\pm$ 0.0036 \\
$\rm O $ & 0.1184 & $\pm$ 0.0006 & $\pm$ 0.0057 \\
$\rm C $ & 0.1156 & $\pm$ 0.0002 & $\pm$ 0.0036 \\
$\rm B_{Max} $ & 0.1223 & $\pm$ 0.0002 & $\pm$ 0.0041 \\
$\rm B_{Sum} $ & 0.1144 & $\pm$ 0.0003 & $\pm$ 0.0053 \\
$\rm \rho_H $ & 0.1216 & $\pm$ 0.0001 & $\pm$ 0.0060 \\
$\rm \rho_S $ & 0.1160 & $\pm$ 0.0001 & $\pm$ 0.0029 \\
$\rm \rho_D $ & 0.1196 & $\pm$ 0.0008 & $\pm$ 0.0039 \\
$\rm D_2^{E0} $ & 0.1165 & $\pm$ 0.0002 & $\pm$ 0.0044 \\
$\rm D_2^{P0} $ & 0.1207 & $\pm$ 0.0002 & $\pm$ 0.0033 \\
$\rm D_2^{P} $ & 0.1186 & $\pm$ 0.0001 & $\pm$ 0.0046 \\
$\rm D_2^{Jade} $ & 0.1182 & $\pm$ 0.0005 & $\pm$ 0.0041 \\
$\rm D_2^{Durham} $ & 0.1172 & $\pm$ 0.0003 & $\pm$ 0.0026 \\
$\rm D_2^{Geneva} $ & 0.1216 & $\pm$ 0.0013 & $\pm$ 0.0310 \\
$\rm D_2^{Cambridge} $ & 0.1176 & $\pm$ 0.0006 & $\pm$ 0.0026 \\
\hline
average & \spb $ 0.1173 \pm 0.0026 $ &
\multicolumn{2}{l} { \hspace{0.3 cm} \chindf = 6.10 / 17
\hspace{0.5 cm} $\rhoe $ = 0.62 } \\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{ Results of the refined \oass measurement of \asmz including
$ b $ quark mass effects in leading order. The central value
of \asmz quoted is the average of \as derived from applying
the $ b $ pole mass $ M_b $ and the $ b $ running mass
$ m_b(M_Z) $ definition. $ \Delta \alpha_s(Mass) =
\left| \alpha_s(M_b) - \alpha_s(m_b(M_Z)) \right| / 2 $ has been
taken as an estimate of the uncertainty due to quark mass effects
of higher orders.
The total error displayed is the quadratic sum of the
experimental uncertainty, the hadronization uncertainty,
the scale uncertainty and the uncertainty due to quark mass
effects. }
\label{alphas-b}
\end{table}
%-----------------------------------------------------------------------------------%
\begin{equation}
\frac { 1 } { \sigma_{tot} }
\frac { d^{2} \sigma (Y, \cos \vartheta_T ) }
{ dY d \cos \vartheta_T } =
\bar{ \alpha_s } ( \mu^2 ) \cdot
A (Y, \cos \vartheta_T ) \big[ 1 + \Delta m \big] +
\bar{ \alpha_s^2 } ( \mu^2 ) \cdot
B (Y, \cos \vartheta_{T}, x_{\mu} )
\label{thmass}
\end{equation}
\pagebreak
\nin where
\begin{equation}
\Delta m =
\frac{\displaystyle \sum_{q,m_b\ne 0} d^2 \sigma_q (Y,\cos \theta_T)/
\sum_{q,m_b\ne 0} \sigma_q}
{\displaystyle \sum_{q,m_b=0} d^2 \sigma_q (Y,\cos \theta_T)/
\sum_{q,m_b=0} \sigma_q} - 1
\label{qmassratio}
\end{equation}
The coefficients $ \Delta m $ for the 18 observables studied have been
computed numerically \cite{german} for two different definitions of the
b-quark mass: The $ b $ pole mass of
$ M_b \thickapprox 4.6 \hspace{0.1 cm} {\rm GeV/c^2} $
and the $ b $ running mass at the $ M_Z $,
$ m_b(M_Z) \thickapprox 2.8 \hspace{0.1 cm} {\rm GeV/c^2} $ \cite{impr-bmass}.
All definitions of the quark mass are equivalent to leading order;
differences are entirely due to higher orders in $ \alpha_s $.
\asmz has been determined applying both mass definitions.
The average of \asmz derived from the pole mass definition and the running mass
definition has been taken as the central value, and half the difference
has been taken as an estimate of the uncertainty due to higher order mass effects.
The results for the individual observables are listed in table \ref{alphas-b}.
%=========================================================================%
\section{Summary}
>From 1.4 Million hadronic $\rm Z_0 $ decays recorded with the DELPHI detector
and reprocessed with improved analysis software, the distributions of 18
infrared and collinear safe observables have been precisely measured
at various values of the polar angle $\rm \vartheta_T $ of the thrust axis
with respect to the beam direction. The $\rm \vartheta_T $ dependence of all
detector properties has been taken fully into account to achieve the best
possible experimental precision. In order to compare with QCD
calculations in \oass, hadronization corrections to the data are evaluated
from recently tuned fragmentation models. \\
The precise data are used to measure \asmz applying a number of different
methods described in the literature. The most detailed studies have been
performed in second order pertubative QCD. Fits taking explicit account
of the \as dependent event orientation as predicted by QCD with
experimental acceptance corrections less than $ \sim 25 \% $ and
hadronization corrections less than $ \sim 40 \% $ yield the surprising result
that the data can be accurately described in \oass by using a common value
of \asmz with a small uncertainty. \\
Taking full account of the correlation among
the observables an average value of $\alpha_s(M_Z^2) = 0.1167 \pm 0.0026 $
can be derived. The consistency of the individual \asmz determinations
from the 18 shape distributions is achieved by using the different values
of the renormalization scales as obtained from the individual fits,
i.e. applying the so called experimental optimization method.
The significance of the fits is improved due to the large number
of data points per distribution. Thus, definite results concerning
the choice of the renormalization scale value become possible.
It should be pointed out that for most of the investigated observables
the scale dependence of \as is very small in the vicinity of the
experimentally optimized scale. The quoted error
of \asmz includes the uncertainty due to a variation of the
experimentally optimized scale in the range between
$ 0.5 \cdot x_{\mu}^{exp} $ and $ 2. \cdot x_{\mu}^{exp} $. \\
To check the reliability of the \as results obtained from the
experimentally optimized scales three further approaches for
choosing an optimized value of the renormalization scale have been
investigated: The principle of minimal sensitivity (PMS), the method
of effective charges (ECH), and the method of Brodsky, Lepage and
MacKenzie (BLM). The weighted averages of \as from the three methods
are in excellent agreement with the weighted average of \as obtained
from the experimental optimization, but their scatter is larger.
The scatter is largest for BLM. A close correlation between the
renormalization scale values evaluated with ECH and PMS with the
experimentally optimized scale values is observed. No such correlation
exists for the BLM scales. \\
Further approaches to estimate the influence of higher order contributions
to the perturbative QCD series are based on Pad\'{e} approximants. The ${\rm PA} [0/1] $
has been used as an estimate of the sum of the perturbative series as well as
for the extrapolation of the unknown \oasss coefficients for the 18 distributions.
In both studies the renormalization scale has been set
to $ x_{\mu} = 1 $. Again the average values of \asmz are consistent with the
average value from the experimental scale optimization in \oass. \\
While all above mentioned determinations of \asmz use fixed order
perturbation theory the last part of the paper describes measurements of
\asmz using all orders resummed calculations in the next-to-leading
logarithmic approximation (NLLA). In a first step, pure NLLA predictions
have been confronted with the data in a limited fit range where the ratio
of the resummed next-to-leading logarithms to the non-exponentiating
\oass contributions is large. The very good agreement between the
average value of \asmz obtained from the pure NLLA fits with a
renormalization scale value $ x_{\mu} =1 $ and the \oass fits using
experimentally optimized scales is remarkable. In a further step
NLLA matched to \oass calculations have been applied. The corresponding
average value of \asmz is again consistent with the \oass result though
the \as values from all investigated observables are systematically
higher. More importantly, the application of matched NLLA to the high
precision data reveals that the trend of the data deviates in a systematic
fashion from the predictions of the matched theory. This problem has not
previously been observed. \\
The \as values derived from the different approaches considered
are in very good agreement. The \oass analysis applying a simultaneous
fit of $\alpha_s$ and $ x_{\mu} $ to the experimental data yields superior
results in all respects. \\
In a final step the influence of heavy quark mass
effects on the measurement of \asmz has been studied. The weighted average
of \asmz from the \oass measurements using experimentally optimized
renormalization scale values and corrected for the $b-\rm{mass}$ to
leading order yields \\
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1173 \pm 0.0026 $.
\end{center}
Among the observables studied, the Jet Cone Energy Fraction (JCEF) \cite{JCEF}
reveals some superior properties. First, the size of the hadronization
correction is extremely small, the average correction within the applied fit range
being only about $ 3.5 \% $. Furthermore, the second order
contribution to the cross section is quite small. Within the applied fit range,
the average ratio of the second to first order contributions is
$ \left< r_{NLO} \right> \sim 25 \% $ if $ x_{\mu} $ is fixed at 1 and
only $ \left< r_{NLO} \right> \sim 6 \% $ if the experimentally
optimized scale value is applied. This indicates a good convergence behavior of the
corresponding perturbative series. Also the scale
dependence of \asmz derived from JCEF is very small. Both experimental
and theoretical uncertainties are smallest for the \asmz measurement from the
JCEF distribution. \\
%------------------------------------------------------------------------------------%
\begin{table} [tbp]
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
prediction & \asmz & $ x_{\mu} $ & \chindf \\
\hline
\oass exp. opt. scale & $ 0.1169 \pm 0.0017 $ & 0.0820 & 1.05 \\
ECH/FAC & $ 0.1169 \pm 0.0017 $ & 0.2189 & 2.75 \\
PMS & $ 0.1168 \pm 0.0017 $ & 0.1576 & 1.91 \\
Pad\'{e} \oasss (fixed scale) & $ 0.1169 \pm 0.0016 $ & 1.0 & 3.12 \\
Pad\'{e} Sum (fixed scale) & $ 0.1169 \pm 0.0015 $ & 1.0 & 3.05 \\
Pad\'{e} \oasss (exp. opt. scale) & $ 0.1164 \pm 0.0015 $ & 0.1814 & 2.45 \\
\oass (fixed scale) & $ 0.1191 \pm 0.0024 $ & 1.0 & 7.7 \\
\hline
\oass exp. opt. scale & $ 0.1180 \pm 0.0018 $ & & \\
+ LO quark mass effects & & & \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Summary of \asmz measurements from the distribution of the
Jet Cone Energy Fraction (JCEF). }
\label{jcefsummar}
\end{table}
%------------------------------------------------------------------------------------%
Table \ref{jcefsummar} shows a summary of \as measurements from
the JCEF distribution for the different methods. The \oass fit
of \asmz applying a fixed renormalization scale value $ x_{\mu} = 1 $ clearly
fails to describe the data with \chindf = 965/125.
However, even for this method, the deviation of \asmz
from the value obtained using the experimentally optimized scale
value is only about 2 \%. All other approaches yield nearly identical
\as values within a few per mille. The deviation of the \as values
derived from the different methods, which can serve as an estimate of the theoretical
uncertainty due to missing higher order terms \cite{asstatus}, is clearly smaller
than the uncertainty of $ \pm 0.0008 $ derived from the variation of the
renormalization scale value. Due to the outstanding qualities of this observable,
the JCEF is considered as best suited for a precise determination of \asmzx
After correcting the measured value for heavy quark mass effects, the final result is \\
\begin{center}
$ \rm \alpha_s(M_Z^2) = 0.1180 \pm 0.0006 (exp.) \pm 0.0013 (hadr.)
\pm 0.0008 (scale) \pm 0.0007 (mass)$.
\end{center}
Comparing this result with other recent precision measurements of \asmz, there is
very good agreement with the determination
of \asmz = $ 0.1174 \pm 0.0024 $ from
Lattice Gauge Theory \cite{LGTres} and the recent result from an NNLO analysis of
$ ep $ deep inelastic scattering data
of \asmz $ = 0.1163 \pm 0.0023 $ \cite{epdisres}.
The result is also in good agreement with the result from the LEP electroweak
working group of \asmz $ = 0.119 \pm 0.004 $ \cite{ewres} from the standard
model fit to the full set of electroweak precision data.
Compared with recent results from spectral functions in hadronic tau decays, \as
is significantly smaller than the central value of \asmz $ = 0.1219 \pm 0.0020 $
quoted in \cite{taures}, but in very good agreement with the value of
\asmz $ = 0.1169 \pm 0.0017 $ derived in \cite{taures}
by using an alternative analysis method considering renormalon chains.
%=========================================================================%
\section*{Acknowledgements}
We thank M. Seymour for providing us with the EVENT2 generator and for
useful discussions. We further thank P. Aurenche, S. Catani, J. Ellis
and P. Zerwas for critical comments and stimulating discussions.\\
%=========================================================================%
% Das Quellenverzeichnis
\newpage
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%=======================================================================%
\end{document}