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\begin{document}
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#152 & \hspace{6cm} DELPHI 98-129 CONF 190 \\
Submitted to Pa 3, 7 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4, 9 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf New determination of the $b$-quark mass using improved jet
clustering\\ algorithms}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
{\bf S. Cabrera$^a$, J. Fuster$^a$, S. Mart\'{\i} i Garc\'{\i}a$^b$ \\ }
}
\vspace{0.5cm}
\noindent
{\footnotesize {\sl
$^a$ IFIC, Centro Mixto Universitat de Val\`encia -- CSIC, Spain \\
$^b$ CERN, Switzerland \\ }}
\end{center}
\vspace*{1cm}
\begin{abstract}
\noindent
Mass effects in the three-jet production of $b$-quarks have been
investigated using the new Cambridge jet algorithm at the $M_Z$ scale
in $e^+e^-$ annihilation. This preliminary study only includes data
collected by DELPHI during the year 1994 which contains about 1.2 million
hadronic Z decays. The predictions of the recent Next-to-Leading order
QCD corrections for massive quarks are discussed and compared with
data. The data are reasonably well described by the Next-to-Leading
calculations in terms of both the $b$-quark running mass at the $M_Z$ scale
($m_b(M_Z)$) and the $b$-quark pole mass ($M_b$). The description at Leading
Order is however only acceptable for $m_b(M_Z)$.
\end{abstract}
%\vfill
\vspace*{1.3cm}
\begin{center}
Paper to be submitted to the HEP'98 Conference \\
Vancouver (Canada)
\end{center}
\end{titlepage}
\pagebreak
\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\bf Introduction}
The new theoretical calculations at Next-to-Leading order (NLO) including mass
effects for three-jet event topologies in $e^+e^-$ annihilation processes
have enabled new experimental activities to test QCD
\cite{val_nlo_dur,aachen_nlo,nason_nlo,val_nlo_cam}: determinations of the
$b$-quark running mass, $m_b(M_Z)$ as defined in the $\overline{MS}$ scheme
at the $M_Z$ scale by DELPHI \cite{delphi_mb} and precise studies of the
flavour independence of the strong coupling constant by DELPHI
\cite{delphi_mb}, SLD \cite{sld_alphas} and OPAL \cite{opal_alphas} collaborations.
The DELPHI analysis confirmed that the QCD radiative corrections including
mass effects were needed to correctly describe the data and verified the
flavour independence of the strong coupling constant for $b$ and light quarks
($\ell=u,d,s$) within $\sim$1\% accuracy. It was also observed an evidence for
the $running$ of the $b$-quark mass which was quantified as an effect of 2-3
standard deviations.
In the present work we study further and discuss the $b$-quark mass effects to
the three-jet production rate as reconstructed using either the Durham \cite{durham}
or the recently introduced Cambridge \cite{camjet1} algorithms. A comparison
with the theoretical expectations calculated at Next-to-Leading order for both
algorithms is also performed \cite{val_nlo_dur,val_nlo_cam}.
\section{\bf The experimental strategy}
The experimental method for this analysis is based on the same particle
and event selection as in \cite{delphi_mb}. The data analysis is the same
as the one developed in \cite{delphi_mb} though it uses the Cambridge jet
algorithm for the clustering of jets \cite{camjet1} and a more sophisticated
technique for the quark flavour tagging \cite{old_btag,new_btag}. The present
work only includes $\sim1.2$ million hadronic Z decays collected during the
year 1994 and a further study containing all data collected by DELPHI at the Z
peak will follow.
The obervable studied in the present analysis is analogous to that introduced
in the previous DELPHI analysis whose definition is:
\begin{equation}
R_3^{b{\ell}}(y_c) = \frac{\Gamma_{3j}^{Z\rightarrow b\bar{b}g}(y_c)/
\Gamma_{tot}^{Z\rightarrow b\bar{b}}}
{\Gamma_{3j}^{Z\rightarrow {\ell}\bar{{\ell}}g}(y_c)/
\Gamma_{tot}^{Z\rightarrow {\ell}\bar{{\ell}}}}
= 1 + r_b(\mu) \cdot \left( b_I(y_c,r_b(\mu)) +
\frac{\alpha_s(\mu)}{\pi} \cdot b_{II}(y_c,r_b(\mu)) \right)
\label{eq:r3bl}
\end{equation}
where $\Gamma_{3j}^{Z\rightarrow q\bar{q}g}$ and
$\Gamma_{tot}^{Z\rightarrow q\bar{q}}$ are the differential three-jet and total
cross-sections, respectively, for the $b$ ($q=b$) and light ($q= \ell$) quarks.
The functions $b_I$ which mainly contains the LO corrections and $b_{II}$ were
calculated at NLO in reference~\cite{val_nlo_cam} using the Cambridge jet
algorithm. The term $r_b(\mu)$ is just $m_b^2(\mu)/M_Z^2$.
%%\hspace{14pt}
\subsection{\bf The jet reconstruction}
The recently introduced Cambridge jet clustering algorithm is applied to group
the selected charged and neutral particles in jets. This Cambridge
algorithm represents an improvement of the original Durham scheme \cite{durham}
in the effort of having a better understanding of the processes involving soft
gluon radiation \cite{camjet1,camjet2}.
The Cambridge method involves an extension of the general
definition of the clustering jet algorithms, with the incorporation
of a new ordering variable different from the test variable
and with a more complex iterative process of recombination.
Starting with an initial set of objects (particles),
Cambridge uses an ordering variable $v_{ij}=2(1-\cos\theta_{ij})$
to clusterize the pair with the minimum relative angle.
This angular ordering is the proposed cure to reduce the
formation of spurious jets formed with low momentum particles.
The test variable of this pair, defined like in Durham:
\[
y_{ij}= \frac{\min(E_i^2,E_j^2)}{E_{vis}^2}v_{ij}
\]
is compared with the resolution scale $y_{c}$. If $y_{ij}~{<}~y_{c}$ the
pair is recombined in a new object with a four-momentum $p_{i}+p_{j}$. If
$y_{ij}~{>}~y_{c}$ the softer particle will be a resolved jet and the other
will remain in the table. This mechanims is called soft freezing and eliminates
the tendency of soft 'resolved' jets of attracting extra wide-angle
particles. The iterative procedure will finish when only one object remains
in the table \cite{camjet1}.
\vspace*{0.7cm}
\begin{figure}[hbt]
\epsfverbosetrue
\vskip -1.5 cm
\begin{center}\mbox{\epsfxsize=12.cm\epsfysize=10.cm
\epsffile{system98.eps}}
\end{center}
\vspace*{0.7cm}
\caption[]{Hadronization corrections to the $R_3^{b{\ell}}$ observable
calculated with HERWIG and JETSET generators, and Durham and Cambridge
jet algorithms}
\label{fig:frag}
\end{figure}
\vspace*{0.7cm}
The distinction between the ordering variable and the test variable,
and the incorporation of the soft freezing mechanics makes the
sequence of clustering dependent of the external $y_{c}$. This fact
leads to the developement of a more complex software to determine
the transition values of $y_c$ for which a given event clusterized in
$n$-jets passes to $m$-jets, $y^{m{\leftarrow}n}$ (see Ref. \cite{camjet2}
for a detailed discussion on the subject).
In the previous DELPHI analysis on the measurement of the $b$-quark mass effects
the impact of the fragmentation process on the observable $R_3^{b{\ell}}$ was
studied \cite{delphi_mb} using the Durham clustering algorithm in two ways. One
was by varying the most relevant parameters of the string fragmentation model
incorporated in JETSET \cite{jetset} within an interval of $\pm 2 \sigma$ from
its central value as tuned by DELPHI \cite{delphi_tuning}. The second was to analyze
the dependence on the fragmentation model itself by comparing the HERWIG \cite{herwig}
model with JETSET \cite{jetset}. The different fragmentation correction factors
obtained for each model were considered as a source of systematic errors. This
in fact was the largest contribution to the total error of the measurement. The
final correction adopted was the average of those two models
and the $fragmentation$ $model$ uncertainty was taken to be half of their
difference. The decision of the valid $y_c$ interval region, $y_c>0.015$ was
also connected to the fragmentation correction which was required to be flat
and the four-jet contribution small ($\leq 2$\%) as is shown in
Fig. \ref{fig:frag}. When adopting the same arguments to the new analysis
based on the Cambridge jet reconstruction algorithm the valid $y_c$
region can be extended towards lower values, $y_c>0.005$, for which the
four jet contribution is still small ($\leq$5\%) and the sensitivity to the
mass effects considerable enlarged \cite{val_nlo_cam} (see Fig. \ref{fig:frag}).
\subsection{\bf The quark flavour tag}
The previous DELPHI analysis \cite{delphi_mb} used the signed impact
parameter of all charged particles in the event \cite{old_btag},
this new analysis incorporates the last combined tagging technique
developed by DELPHI \cite{new_btag}. To tag the $b$-quark and
$\ell$-quark samples an optimal combination of a set of discriminating
variables defined for each reconstructed jet is performed. The quantities
combined per jet are: the jet lifetime probability, ${\cal P}^+_j$; the effective
mass distribution of particles included in the secondary
vertex, $M_s$; the rapidity distribution of tracks included in
the secondary vertex with respect to the jet direction, $R_s^{tr}$ and
the fraction of the charged energy distribution of a jet
included in the secondary vertex, $X_{s}^{ch}$. For a more detailed
description see Ref. \cite{new_btag}. As a result of this new tagging
technique the purities of the tagged $b$-quark and $\ell$-quark samples
have been increased preserving about the same
efficiencies (see Tab. \ref{tab:qtag}).
\vspace*{0.7cm}
\begin{table}[htb]
\centering
\vspace{7mm}
\begin{tabular}{|c|c|c|c|}
\hline
~$q$-type~ &
~${\ell} \rightarrow q$-type (\%)~ & ~$c \rightarrow q$-type (\%)~ &
~$b\rightarrow q$-type (\%)~\\
\hline
${\ell}$ &
86.30 $\pm$ 0.04 & 11.92 $\pm$ 0.04 & ~1.78$\pm$0.02 \\
$b$ &
~3.71 $\pm$ 0.04 & 7.07 $\pm$ 0.05 & 89.21 $\pm$0.04 \\
\hline
\end{tabular}
\vspace{0.7cm}
\caption{Flavour compositions of the samples tagged as
light ($\ell \equiv u,d,s$) and $b$-quark events extracted from
the DELPHI simulation as they were obtained in the present analysis.}
\label{tab:qtag}
\end{table}
\vspace*{0.7cm}
\section{\bf Results and discussion}
The DELPHI observation of the $b$-quark mass effects in the observable
$R_3^{b{\ell}}$ using the Durham reconstruction algorithm are reproduced
in Fig. \ref{fig:durham}. The corrected data values for
$R_3^{b{\ell}}(y_{cut})$ are shown in comparison with the
LO and NLO theoretical expectations
\cite{val_nlo_dur,aachen_nlo} in Fig. \ref{fig:durham}. The values used
for the $b$-mass in the theoretical predictions are extracted from the
$\Upsilon$ resonances and correspond to a $b$-quark pole
mass of $M_b=4.6$ GeV/c$^2$ (dashed curves) and accordingly to
a $b$-quark running mass of $m_b(M_Z)=2.8$ GeV/c$^2$ (solid curves)
\cite{low_mb}\footnote{These values are taken as a conservative numerical
example to compare with. No average value extracted from the present existing
literature has been used.}. It is observed\footnote{Notice however that the data
points are not independent though they provide a consistency check as a
function of $y_c$.} that the NLO corrections
are larger for the parametrization of $R_3^{b{\ell}}$ in terms of the
running mass than for the pole mass though the data prefers to be
described in terms of the running mass better than of the pole mass. None of
the LO predictions describes the data reasonable well. This result was
interpreted in Ref. \cite{delphi_mb} as an evidence of the $running$ of the
$b$ mass and quantified as 2-3 sigma effect.
\begin{figure}[bth]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=10.cm\epsfysize=10.cm
\epsffile{r3bd_durham.eps}}
\end{center}
\caption[]{Corrected data values of $R_3^{b{\ell}}(y_c)$ Durham
(grey points) in comparison with the LO and NLO theoretical
results for this observable expressed in terms of the pole mass, $M_b$
(dashed curves) or the running mass $m_b(M_Z)$ (solid curves).}
\label{fig:durham}
\end{figure}
In case that the same analysis is performed using the Cambridge algorithm the
result obtained is shown in Fig. \ref{fig:cam}. The errors drawn in this
figure correspond to the quadratic sum of statistical error (indicated as an
horizontal small bar on top of the error) and the total hadronization
error, 1\%-1.2\% depending on $y_c$. The theoretical calculation
indicates a different situation \cite{val_nlo_cam} with respect to that
of Fig. \ref{fig:durham}: both the LO and the NLO curves in terms of
$m_b(M_Z)$ are very close with respect to one each other and, on the contrary,
the two curves corresponding to the $b$-quark pole mass differ
significantly. The NLO curve approximates to those in terms of
$m_b(M_Z)$. The data\footnote{Notice again that the data
points are not independent though they provide a consistency check as a
function of $y_c$.} lie in the intermmediate region where
the two NLO curves are though they are still compatible with the
LO-$m_b(M_Z)$.
\begin{figure}[th]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=10.cm\epsfysize=10.cm
\epsffile{r3bd_cambridge.eps}}
\end{center}
\caption[]{Corrected data values of $R_3^{b{\ell}}(y_c)$ Cambridge
(grey points) in comparison with the LO and NLO theoretical
results for this observable expressed in terms of the pole mass, $M_b$ (dashed lines)
or the running mass $m_b(M_Z)$ (solid lines).}
\label{fig:cam}
\end{figure}
In summary, as seen in Fig. \ref{fig:cam} the bulk of the NLO corrections
is mainly described by the running of the $b$-quark mass at the $M_Z$
scale, indicating that the situation for the Cambridge jet algorithm is
more similar to the inclusive $\Gamma (Z) \rightarrow b\overline{b}$
or $\Gamma (H) \rightarrow b\overline{b}$ decay widths\footnote{These
quantities show a better convergence when they are expressed in terms
of the running mass than in terms of the pole mass \cite{running}} than
it is for the Durham jet algorithm \cite{val_nlo_dur,val_nlo_cam}. The NLO
calculation of the observable defined in Eq. \ref{eq:r3bl} expressed
in terms of the $b$-quark running mass at $M_Z$ is the curve
which shows the best agreement with the data for both Durham and Cambridge
algorithms. In the case of the Cambridge algorithm the data is also well
described by the predictions at NLO in terms of the $b$-quark pole
mass, $M_b=4.6$ GeV/c$^2$. Also the LO theoretical calculation for
$m_b(M_Z)$ GeV/c$^2$ is reasonably well supported by the data.
%The
%combination of all DELPHI data will contribute to consistently check
%this observation for a wider range of $y_c$.
\subsection*{Acknowledgements}
\vskip 3 mm
We are greatly indebted to our technical
collaborators and to the funding agencies for their
support in building and operating the DELPHI detector, and to the members
of the CERN-SL Division for the excellent performance of the LEP collider.
We also thank M. Bilenky, G. Rodrigo and A. Santamaria for their
collaboration and very helpful discussions on the subject.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\newpage
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%
\end{document}