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\begin{document}
\newcommand{\x}{\mbox{$\mathrm{XXX}$}}
%\newcommand{\Rbb}{\mbox{R_{4\mathrm{b}}}}
\newcommand{\Rbb}{R_{4\mathrm{b}}}
\newcommand{\Rb}{\mbox{$R_{\mathrm{b}}$}}
\newcommand{\gbb}{\mbox{$g_{\mathrm{bb}}$}}
\newcommand{\gcc}{\mbox{$g_{\mathrm{cc}}$}}
\newcommand{\ebb}{\epsilon_{4\mathrm{b}}}
\newcommand{\ec}{\epsilon_{\mathrm{c}}}
\newcommand{\eb}{\epsilon_{2\mathrm{b}}}
\newcommand{\eq}{\epsilon_{\mathrm{q}}}
\newcommand{\Pbb}{\mathrm{b\bar{b}}}
\newcommand{\Pqq}{\mathrm{q\bar{q}}}
\newcommand{\Pqb}{\mathrm{qb}}
\newcommand{\PZz}{\mathrm{Z}}
\newcommand{\Pg}{\mathrm{g}}
\newcommand{\Pep}{\mathrm{e^+}}
\newcommand{\Pem}{\mathrm{e^-}}
\newcommand{\Pee}{\mathrm{e^+e^-}}
\newcommand{\ra}{\rightarrow}
\newcommand{\boh}{[...]}
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#776 & \hspace{6cm} DELPHI 98-77 CONF 145 \\
Submitted to Pa 1, 3 & 22 June, 1998 \\
\hspace{2.4cm} Pl 1, 4 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Measurement of the Rate of \boldmath $\mathrm{b} \overline{\mathrm{b}}
\mathrm{b} \overline{\mathrm{b}}$ events in Hadronic Z Decays and
the extraction of the gluon splitting into
$\mathrm{b} \overline{\mathrm{b}}$ }\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf N.~Amapane} $^1$,
{\bf C.~Mariotti} $^{2,3}$,
{\bf E.~Migliore} $^{1}$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
$\PZz \rightarrow \Pbb\Pbb$
was measured using about $2 \times 10^6$ hadronic
decays collected by the DELPHI experiment in 1994 and 1995.
\noindent
The analysis was performed forcing events to a 3-jet topology with
\mbox{$y_{min}>0.06$} and requiring every jet to pass the b-tagging
selection.
The rate was measured to be:
\begin{center}
$\Rbb= \frac{ \mathrm{BR}(\PZz\rightarrow \Pbb\Pbb) }
{ \mathrm{BR}(\PZz\rightarrow hadrons)}= (5.4 \pm 1.8 (stat.) \pm 1.5
(syst.))\times 10^{-4}$
\end{center}
\noindent where the minimum invariant mass of every $\Pbb$ system is
$ > 2\times m_b$ in the simulation.
Using the value of $\Rbb$ the rate of gluon splitting to $\Pbb$ can be
extracted and is found to be:
\begin{center}
$\gbb = (3.0\pm 1.0 (stat.) \pm 0.8 (syst.))\times 10^{-3}$. %@@@
\end{center}
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$ INFN Torino, Italy}
\par {\footnotesize $^2$ CERN, Geneva, Switzerland}
\par {\footnotesize $^3$ On leave of absence from INFN Torino, Italy}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
\section{Introduction}
The production of b quarks in $\Pee$ annihilation receives contributions
from two sources, namely the annihilation itself and the splitting of
perturbatively produced gluons,
{$\Pee\rightarrow\Pqq\Pg,\Pg\rightarrow\Pbb$}. The latter is usually
called {\sl secondary} b production.
While the ratio
$\Rb=\sigma(\Pee\rightarrow\Pbb)/\sigma(\Pee\rightarrow~hadrons)$ has been
measured with very high precision \cite{rb,otherrb},
the secondary production of b quarks is
comparatively poorly known \cite{gbb}.
Though being interesting in its own, it is
usually considered as a source of background for the study of
$\Pee\rightarrow\Pbb$ processes and is, in facts, one of the main
sources of uncertainty on the measurement of $\Rb$ \cite{rb,otherrb}.
In this analysis,
$\Rbb= \frac{ \mathrm{BR}(\PZz\rightarrow \Pbb\Pbb) } {\mathrm{BR}(\mathrm{Z}^0\rightarrow hadrons)}$
, i.e. the rate of events showing both primary and
secondary production of b quarks, is measured for the first time at LEP.
From this measurement we can estimate the rate of events with secondary
b-quark production $\gbb$.
Interference and mass effects between the four massive b-quarks,
absent at leading order for $\Pqq\Pbb$ events (q massless)~\cite{seymour},
are taken into account using theoretical computations.
%It is in agreement with theoretical computation of $\gbb=0.175\%$ \cite{Sey}.
%The value of $\gbb$ can be extracted from $\Rbb$ providing that
%both the interference between primary and secondary
%production and the effect of having different masses of the primary
%produced quarks are negligible. The effect of interference has been
%calculated to be small~\cite{Sey} \boh.
In 1994 and 1995 a large sample of $Z$ decays was collected by
the DELPHI experiment
with a new Vertex Detector~\cite{new_vd} capable of measuring
the coordinates of points on tracks in three dimensions, thus improving
considerably the b-tagging performance. In addition, major
improvements in
the reconstruction programs enhanced the tracking efficiency.
The improved b-tagging efficiency allows the measurement of $\Rbb$ with
the identification of at least three jets as produced by the
hadronization of a b quark.
%======================================================================
\section{The DELPHI Detector}
\label{sec:vd}
The DELPHI detector and its performance have been described in detail in
ref.~\cite{perf}. Here only the new Vertex Detector (VD)
~\cite{new_vd}, the most relevant detector used in this analysis,
will be described.
The VD is the innermost detector in DELPHI. It
is located between the LEP beam pipe and the Inner Detector.
In 1994 the DELPHI Vertex Detector~\cite{vdpaper} was upgraded
to provide a three-dimensional readout~\cite{new_vd}.
It consists of three concentric
layers of silicon microstrip detectors at radii of 6.3, 9 and 11 cm
from the beam line, called the closer, inner and outer layer respectively.
The microstrip detectors of the closer and outer layers
provide hits in the $R\Phi$ and the $Rz$-plane
\footnote{In the DELPHI Cartesian coordinate system $z$ is along the beam line,
$\Phi$ is the azimuthal angle in the $xy$ plane, $R$ is the radius
and $\theta$ is the polar angle with respect to the $z$ axis.},
while for the inner layer only the $R\Phi$ coordinate is measured.
For polar angles of $44^\circ \le \theta \le 136^\circ$ a track crosses
all the three silicon layers of the VD. The closer layer covers
the polar region between $25^\circ$ and $155^\circ$.
The measured intrinsic precision is about 8~$\mu$m for the $R\Phi$
measurement while for $z$ it depends on the polar angle of the
incident track, and goes from about 10~$\mu$m for tracks perpendicular
to the modules to 20~$\mu$m for tracks with a polar angle of
$25^\circ$. For charged particle tracks with hits in all three
$R\Phi$ VD layers, the impact parameter\footnote{The impact parameter is
defined as the distance of closest approach of a charged particle to
the reconstructed primary vertex.}
resolution is
$\sigma_{R\Phi}^2=[61 / (p \sin ^{3/2} \theta)]^2+ 20^2 \mu$m$^2$
while for tracks with hits in both the $Rz$ layers it is
$\sigma_{z}^2=[67 / (p \sin ^{5/2} \theta )]^2+ 33^2 \mu$m$^2$, where
$p$ is the momentum in GeV/c.
%======================================================================
\section{Analysis and results }
\subsection{The identification of b jets}
\label{sec:btag}
The b-tagging method used in this analysis is described in detail
in ref.\cite{gb}.
It combines, in a linear way, four different variables defined for each
jet.
The first variable $P^+_J$, originally proposed by ALEPH~\cite{aleph}
and further developed by DELPHI~\cite{rb}, represents the probability
that, in a given jet, all the tracks
with positive impact parameter
originate from the primary vertex.
The track impact parameters are computed separately in the $R\Phi$ plane and
along the $z$ direction~\cite{tuning}.
The sign of the impact parameter is defined with respect to the jet direction.
It is positive if the point of closest approach of the track to the jet axis
is downstream of the primary vertex along the jet direction,
and negative if it is upstream. In this way the same sign is assigned
to the $R\Phi$ and $z$ impact parameters.
Additional selection variables are defined only for the jets
where a secondary vertex is reconstructed.
They are: the effective mass, the rapidity
with respect to the jet direction and the energy
of the charged particles included in the secondary vertex.
Reconstructed secondary vertices are accepted
if $L/ \sigma_L \ge 4$ where $L$ is the
distance from the primary vertex and $\sigma_L$ is its uncertainty,
which happens in about 55\% of the hemispheres with b-quarks.
Whenever a secondary vertex is reconstructed, the
jet direction is recomputed as the direction from the primary
vertex to the secondary vertex and the sign of the impact parameter is
redefined accordingly.
For a given selection variable $x$
the ratio of the probability density function for background $f^B(x)$
and for signal events $f^S(x)$ is defined to be $y = f^B(x)/f^S(x)$ \cite{gb}.
In the case of several independent variables the definition of this ratio is:
\begin{eqnarray}
y={\frac{f^B(x_1,...x_n)}{f^S(x_1,...x_n)}}
= \prod {\frac {f^B(x_i)}{f^S(x_i)}} = \prod y_i \, .
\label{eq:xeffj}
\end{eqnarray}
The jet is tagged as containing a b quark if
the discriminating variable \mbox{$\eta=-log_{10}y>\eta_0$}; the choice of
$\eta_0$ defines the efficiency and purity of the sample.
%======================================================================
\subsection{Event Selection}
The criteria to identify hadronic Z decays are the following.
Charged tracks were accepted if:
\begin{itemize}
\item{their length was larger than 30~cm;}
\item{their impact parameter
was less than 4~cm in the $R\Phi$ plane and less than 10~cm in $z$};
\item{their momentum was larger than 400~MeV/$c$.}
\item{The relative error ${\sigma_p}/p$ was less than 1, where $p$ is the track momentum.}
\end{itemize}
Since the VD dominates the impact parameter resolution, only tracks
with VD information were used for the b-tagging. In particular, both
for the probability computation and the secondary vertex
reconstruction, tracks were accepted only if they had at least one
$R\Phi$ VD hit or at least one $Rz$ VD hit.
Neutral particles were accepted if their energy deposition was larger
than 700 MeV
in the barrel electromagnetic calorimeter HPC
or 400 MeV in the forward calorimeter FEMC. %~\cite{perf}
Neutral particles were used in the jet axis and direction reconstruction;
their selection was therefore optimised for this pourpose.
Hadronic Z decays were selected by requiring:
\begin{itemize}
\item{at least six charged particles;}
\item{the summed energy of the charged particles to be larger than 12\%
of the center-of-mass energy.}
\end{itemize}
Only events collected while the Vertex Detector and the TPC were
fully operational were accepted.
With these requirements, about 1,400,000 and 600,000 $\PZz$ events were
selected in the 1994 and 1995 data samples respectively.
Simulated events were generated using the JETSET 7.3 parton shower
(PS) Monte Carlo program~\cite{jetset} tuned for the DELPHI data.
The response of the DELPHI detector was simulated in full detail using
the DELSIM program~\cite{delsim}.
A good simulation of the impact parameter and the b-tagging variables for
$\PZz$ decays into light quarks (udsc) is very important in this analysis;
for this reason a fine tuning of the $R\Phi$ and $z$ impact
parameter resolutions has been developed and applied~\cite{tuning}.
Three samples consisting of $4.3\times10^6$ ~hadronic Z decays,
$2.1\times10^6$ Z$\rightarrow\Pbb$
events and $2\times10^4$ Z$\rightarrow\Pbb\Pbb$ events were
used\footnote{Since
the detector did not change between 1994 and 1995 and the
efficiencies of selecting signal and background events
obtained with the description of the detector for those years
are compatible within their error, events simulated for 1994 and 1995
are considered together here and in the following.}.
Two additional samples of $10^4~\Pbb\Pbb$ events each were produced
with the JETSET Matrix Element (ME) generator~\cite{ME} and with the
WPHACT Monte Carlo generator \cite{wphact} and used
to test the model dependence of the result.
%*****Herwig ecc of bbbb ********
Simulated events were reweighted according to the latest LEPHF
recommendations~\cite{EWHFWG}. In particular, the rate of gluon
splitting to c quarks $\gcc$ was set to 2.38\%.
%and the rate of gluon splitting to b quarks $\gbb$ to $0.13\times\gcc$.
\subsection{The Method}
Events can be grouped in four categories:
\begin{itemize}
\label{evttype}
\item Signal events, called \textbf{\textsl{4B}} events in the following;
\item Events with primary b production and gluon splitting to
$\mathrm{c\bar{c}}$, called \textbf{\textsl{ C}} events in the following;
\item Events with primary b production but no gluon splitting to heavy
quarks, called \textbf{\textsl{2B}} events in the following;
\item All other events (i.e. without primary b production), hereafter called
\textbf{\textsl{Q}} events.
\end{itemize}
The efficiencies of selecting these classes of events after all cuts will be
indicated with $\ebb, \ec, \eb, \eq$ respectively.
The rate of
$\Pbb\Pbb$ events can then be extracted from the relation
\begin{equation}
\label{theformula}
\Rbb=\frac{f_d-\eq-\Rb[\gcc(\ec-\eb)+\eb-\eq]}{\ebb-\eb}
\end{equation}
\noindent where $f_d$ is the fraction of events selected in data, $\Rb$ is set
to the world average value \cite{pdg}
of $0.2170\pm0.0009$ and
$\gcc=(2.38\pm0.48)\%$ is the value of the gluon splitting probability
to c quarks measured by OPAL~\cite{opal}.
The selection is based on the identification of 3 b-originated jets in
the event.
First, reconstructed tracks were clustered in each event into 3 jets using
the DURHAM~\cite{durham} algorithm.
Genuine two-jet-like events were rejected with a cut on the variable
$y_{min}$\footnote{The variable $y_{min}$ is defined as the DURHAM
distance between the two nearest jets in the event.}.
%This approach was preferred to the more conventional one which consists in
%discarding two-jet events after a clusterization with a fixed cutoff
%on the value of $y_{min}$,
%as it allows a better control on the
%jet clusterization itself.
%This is a critical point since the present analysis relies
%on the b-tagging of jets and therefore on the correct association of
%tracks.
%Moreover,
Fixing to three the number of jets has the result, for
events with higher jet multiplicity,
%> {\tt[cite:ca. 15\% of the total]}
of joining the nearest jets, which in most cases are those
produced by the gluon splitting process.
%>{\tt [show plots: angles of generated bs: g->bb, Z->bb, all couples of bs]}
This increases the efficiency of the b-tagging selection which grows
with the number of used tracks~\cite{gb}.%@@@cite?
After this selection, jets were sorted using the
b-tagging variable $\eta$ (cf. Eq.~\ref{eq:xeffj}), jet 1 being
the one with the higher probability to contain tracks from b
decay. This allows a separate choice of the three cut values.
%======================================================================
\subsection{Results}
\label{sec:results}
Figure~\ref{fig:ymin} shows the normalised distributions
of $y_{min}$ for data and simulated events.
%>fare con eventi ripesati, anche plot segnale
The cut value $y_{cut} = 0.06$ was chosen in order to reject as much as
possible the events which contained no gluon splitting, while
retaining a good fraction the signal.
According to the simulation, the efficiency of this selection was
$(47.2\pm0.3)\%$ for signal events. The discrepancy between data and
simulation in the fraction of selected events was found to be
\begin{equation}
\label{eq:rdm}
\frac{f^{DATA}_{3-jet}}{f^{SIM}_{3-jet}}=0.949\pm0.002
\end{equation}
\noindent Figure~\ref{fig:xeffj} shows the distributions of $\eta$ for the
three ordered jets for data and simulation.
The values for the three $\eta$ cuts were chosen in order to
minimise the final relative error on $\Rbb$
and were 0.9, 0.2, and -0.1 for the three ordered jets respectively.
With this requirement, 140 events were selected in the data sample.
\label{sec:rdm}
The data-Monte Carlo discrepancy of Eq.~\ref{eq:rdm} was taken into
account attributing
it to \textbf{\textsl{2B}} and \textbf{\textsl{Q}} background events
(99.4\% of the initial Monte Carlo sample).
The efficiencies $\eb$ and $\eq$ were therefore rescaled from
$(0.0162\pm0.0007)\%$ to $(0.0155\pm0.0007)\%$ and from
$(0.000090\pm0.000052)\%$ to $(0.000086\pm0.000051)\%$ respectively.
For the efficiencies of tagging b-jets which correspond to our cuts,
the analysis of $\Rb$~\cite{rb} found a 3\% discrepancy
between data and simulation. This discrepancy is stable within $1\%$ as a
function of the jet momentum.
The signal efficiency $\ebb$ was therefore corrected by a
factor $1.09\pm0.03$ and the background efficiencies
$\ec$ and $\eb$ by a factor $1.06\pm0.02$. The uncertainties on these
coefficients were used to estimate a systematic error
(cf. Sec.~\ref{sec:sys}).
The efficiencies of the different classes of events are
summarised in Table~\ref{tab:eff}.
\begin{table}[htbp]
\begin{center}
\begin{tabular} {|| l | c ||} \hline
Event type & Efficiency (\%) \\ \hline
\textbf{\textsl{4B}} & $\ebb =3.20\pm0.11$ \\ \hline
\textbf{\textsl{ C}} & $\ec =0.326\pm0.027$ \\ \hline
\textbf{\textsl{2B}} & $\eb =0.0164\pm0.0007$ \\ \hline
\textbf{\textsl{Q}} & $\eq =0.00009\pm0.00005$ \\ \hline
\end{tabular}
\end{center}
\caption{\sl Total efficiencies for the different classes of events
(cf. Sec.~\ref{evttype}) in the simulation (after rescaling).}
\label{tab:eff}
\end{table}
The measured value is
\begin{equation}
\Rbb = (5.4\pm1.8(stat.))\times10^{-4}
\label{eq:r4b}
\end{equation}
%The result is very stable as a function of the cuts.
\noindent Fig.~\ref{fig:stab_ymin} shows the stability of the result as a
function of the cut on $y_{min}$ and
Fig.~\ref{fig:stab_xeffj} as a
function of the cuts on the b-tagging variables of the three ordered jets.
%======================================================================
\subsection{Systematic uncertainties}
\label{sec:sys}
The following contributions were considered:
{\bf Data-Monte Carlo discrepancy in the $y_{min}$ cut}.
The rescaling of $\eb$ and $\eq$ according to Eq.~\ref{eq:rdm} was made
under the assumption that the gluon
splitting process to b and c quarks is correctly described by the
JETSET PS Monte Carlo generator. A systematic uncertainty was
therefore estimated assigning the discrepancy to both signal and
background.
This gave a contribution of $\pm0.71\times10^{-4}$
to the systematic error.
{\bf B-tagging}.
As described in Sec.~\ref{sec:results}, the signal
efficiency $\ebb$ was rescaled by $1.09\pm0.03$
and the background efficiencies $\ec$ and $\eb$ by $1.06\pm0.02$.
A systematic uncertainty on the b-tagging efficiency
was estimated varying these factors of their full error.
This gives a contribution of $\pm0.015\times10^{-4}$ to the final error.
This rescaling is accurate if the correlation between the b-tagging
efficiencies of the three jets is small.
As shown in Ref.~\cite{rb}, the major source of correlation among the
b-tagging efficiencies is the common primary vertex.
The measurement was therefore repeated fitting a different primary
vertex for each jet. This approach increases the purity of the
selected sample, but reduces considerably the b-tagging efficiency,
resulting in 78 selected events in the data sample.
The result is \mbox{$\Rbb =
(5.2\pm2.2(stat.)\pm1.4(syst.))\times10^{-4}$}, in good agreement
with Eq.~\ref{eq:r4b}, showing that our error does not allow to
be sensitive to
the effect of jet-jet correlations in the b-tagging efficiencies.
% record 5069 (rel.49.4%)
{\bf $\gamma$, $K^0$ and $\Lambda$ production rates}.
Jets with non reconstructed $\gamma$, $K^0$ and $\Lambda$ can be wrongly
identified as b-jets.
The rates of these particles were varied in the simulation by $\pm 50\%$,
$\pm 10\%$ and $\pm 15\%$ respectively, i.e. by the amount
of the difference of reconstruction efficiency in data and simulation.
This leads to a contribution of $\pm0.15\times10^{-4}$ to the systematic
error.
{\bf Value of $\gcc$ and $\Rb$}. Varying
$\Rb$ and $\gcc$ in Eq.~\ref{theformula} by their full error according
to the LEPHF recommendations %~\cite{EWHFWG}
results in contributions of
$\pm0.067\times10^{-4}$ and $\pm1.0.\times10^{-4}$ to the systematic
error respectively.
{\bf Model Dependence}.
To test the dependence of the result on the model used to
simulate signal events, the same selection was repeated on two
dedicated samples of $10^5$ signal
events produced with the JETSET Matrix Element Monte Carlo
generator~\cite{ME} and with the WPHACT generator~\cite{wphact}. The resulting
efficiencies were
$\ebb^{ME}=(3.65\pm0.19)\%$ and $\ebb^{WP}=(3.77\pm0.19)\%$
respectively. The spread of these efficiencies
%The larger between the differences \mbox{$|\ebb-\ebb^{ME}|$},
%%\mbox{$|\ebb-\ebb^{WP}|$}, and their error
was taken as systematic uncertainty
on $\ebb$. This gives a contribution of $\pm0.53\times10^{-4}$ to the
systematic error.
{\bf Monte Carlo statistics}.The limited Monte Carlo
statistics gives a contribution of $\pm0.70\times10^{-4}$ to the
systematic error.
The total systematic error amounts to $\pm 1.5 \times 10^{-4}$.
All systematic errors considered are summarised in table \ref{error}.
\begin{table}[htbp]
\begin{center}
\begin{tabular} {|| l | c | c ||} \hline
Source of systematics & Range & $\Delta\Rbb\times 10^{-4} $\\ \hline \hline
Data/MC ($y_{min}$) & & $\pm 0.71$ \\ \hline
b-tagging & & $\pm 0.015$ \\ \hline
$\gamma$, $K^0$, $\Lambda$ & $\pm 50\%$, $\pm 10\%$ , $\pm15\%$ & $\pm 0.15$ \\ \hline
$\gcc$ &$(2.38\pm0.48)\%$ & $\pm 1.0$ \\ \hline
$\Rb$ &$(21.70\pm0.09)\%$ & $\pm 0.067$ \\ \hline
MC model & & $\pm 0.53$ \\ \hline
MC statistics & & $\pm 0.70$ \\ \hline
\hline
\multicolumn{2}{||l|}{Total} & $\pm 1.51$ \\ \hline
\end{tabular}
\end{center}
\caption{\sl Systematic errors on the measurement of $\Rbb$}
\label{error}
\end{table}
%\subsection{Systematic checks}
%\boh
%
%{\sl We will redo the analysis fitting one primary vertex per jet
%to estimate the effect of jet-jet correlation in the b-efficiency}
%
%{\sl We wil use HERWIG mc}
%
%
%{\tt
%-> VXeffj (decidere se usare per l'analisi od il sistematico)
%->Suggerimento Drees = fare l'analisisi solo su eventi in cui il
%3. jet in ordine di b-tagging NON e' il meno energetico (e quindi si
%suppone venga da produzione primaria). Per farlo correttamente
%occorrerebbe richiedere la conservazione del quadrimpulso dei jet .
%\boh
%}
\subsection{The extraction of $\gbb$}
As in the present analysis primary and secondary heavy quark production
are not separated, the interference term between them should be carefully
evaluated in the interpretation of the result.
In particular it is not possible to express the gluon splitting
probability $\gbb$ simply as
the ratio between $\Rbb$ and $\Rb$.
A consistent definition of $\gbb$ is given by:
\begin{equation}
\gbb = \frac{\sigma(Z \ra \Pqq g,g\ra\Pbb)}{\sigma(Z \ra \Pbb \Pbb)}
\times \Rbb
= R_{th} \times \Rbb
\label{ballest}
\end{equation}
\noindent
where the first term is calculated summing on all flavours (q=u,d,s,c,b)
and the second one is measured in this analysis.
The term $R_{th}$ of Eq.\ref{ballest} depends weakly on the
theoretical parameters $\alpha_S$ and $m_{\mathrm{b}}$ since the
dependence is suppressed in the ratio of the two cross sections.
%In the first term of Eq.\ref{ballest} the dependence on the
%theoretical
%parameters $\alpha_s$ and $m_b$ is very weak because of the ratio
%of cross sections.
%OR:
%The two cross sections of Eq.\ref{ballest} show a dependence on the
%theoretical parameters $\alpha_s$ and $m_b$, which however cancels in
%their ratio.
%The cross section for a final state with 4 quarks varies with the mass of the
%quark involved because of the interference between primary and secondary
%quarks with non zero mass.
%The interference is negligible between the primary $\Pbb$ quark and
%the secondary $Pqq$ quark, where the mass of the quark q can be set to zero.
The theoretical estimation of the cross section depends on the minimum
parton virtuality $Q$ from which the integration is performed.
The cutoff values in the simulation are:
\begin{itemize}
\item 2.25 GeV for the light quarks. This value is set by the DELPHI
tuning of the JETSET $Q_0$ parameter.
\item twice the mass of the free b quark $m_b$
for the $\Pbb$ system. The value of $m_b$ is set to 4.7 GeV in the
JETSET PS Monte Carlo generator.
\end{itemize}
The value of the cutoff $Q_0$ could affect the measurement in two
different aspects. First, a different choice
of $Q_0$ could affect the estimation of non-\textbf{\textsl{4B}}
events in the measurement of $\Rbb$. Moreover, in the extraction of
$\gbb$ the cross section
$\sigma(Z \ra\Pqq g,g\ra\Pbb)$ is actually defined by $Q_0$.
%In this analysis the efficiency as a function of $Q$ was found to be flat
%down to the cutoff set in the simulation.
The signal efficiency as a function of $Q$ is shown in fig.\ref{fig:q0}.
It can be seen that the analysis is fully sensitive down to the cutoff value
set in the simulation.
%To better control this value an explicit cut was performed clustering each
%event into a 4 jets topology with DURHAM and selecting events with
%$y_{min} > \ldots$. As stated before this cut does not affect the counting of
%$\Pbb \Pbb$ events.
%{\tt The granularity of the detector could put a more stringent cutoff
% on $Q_0$.
%From the simulation we can estimate this for the light and the b quarks.
The term $R_{th}$ in Eq.\ref{ballest} was computed using the WPHACT
program~\cite{wphact}
with the following cuts on the invariant masses of quark pairs:
2.25 GeV, 5.82 GeV, 9.4 GeV for $\Pqq$, $\Pqb$ and $\Pbb$ respectively.
For $m_b = 4.7$ GeV and $\alpha_S$ running we obtain
\mbox{$R_{th} = 5.457 \pm 0.008$}
where the error takes into account the numerical accuracy of the
calculation only.
Inserting this value in Eq.\ref{ballest} we get
\begin{eqnarray}
\gbb = (3.0\pm1.0(stat.)\pm0.8(syst.)\times10^{-3}
\end{eqnarray}
%New systematic: 1.532E-4
The effect of the value of $m_b$ on the extraction of $\gbb$ was
investigated recomputing $R_{th}$ for $m_b = 3$ GeV and with
the same set of cuts on the invariant masses of quark pairs.
The value obtained is
$R_{th} = 5.660\pm0.010$. From this we get
\mbox{$\gbb = (3.1\pm1.0(stat.)\pm0.9(syst.))\times10^{-3}$}.
%{\tt This value is compatible with the measurements published}
%======================================================================
\section{Conclusions}
The rate of events with four b quarks in the final state was measured.
Events were selected clustering tracks into three jets, each of those was
required to pass the b-tagging selection. Two-jet like events were
discarded using a cut on the $y_{min}$ variable.
As result, we obtained
$$ \Rbb = (5.4 \pm 1.8 (stat.) \pm 1.5 (syst.))\times10^{-4}$$
where the first error is statistical and the second includes all systematic
effects.
From $\Rbb$ we can estimate the rate of secondary b quarks to be:
$$ \gbb = (3.0 \pm 1.0 (stat.) \pm 0.8 (syst.))\times10^{-3}$$ %@@@
This value is in agreement with the previous measured values.
\section{Acknowledgements}
It is a pleasure to thank A. Ballestrero for
the many fruitful discussions on the theoretical issues on
this measurement and for providing us a dedicated version of WPHACT.
We would also like to thank J. Fuster, K. Hamacher, K. Moenig, P. Nason and
C. Oleari for the very useful comments and suggestions.
One of us (N.A.) would like to thank the University of Wuppertal for
the help and hospitality during his stay in Wuppertal.
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\newpage
\begin{figure}
\begin{center}
\mbox{\epsfig{file=ymin.eps,width=13.cm}}
\end{center}
\caption{ {\sl Distribution of the $y_{min}$ variable for real data (dots) and
simulation (histogram).}}
\label{fig:ymin}
\end{figure}
\begin{figure}
\begin{center}
\mbox{\epsfig{file=xeffj.eps,width=15.cm}}
\end{center}
\caption{ {\sl Distribution of the b-tagging variables for the three
ordered jets for data (dots) and simulation (histogram).}}
\label{fig:xeffj}
\end{figure}
\begin{figure}
\begin{center}
\mbox{\epsfig{file=stab_ymin.eps,width=13.cm}}
\end{center}
\caption{ {\sl Stability of the result as a function of the $y_{min}$
cut. The bars represent the uncorrelated errors referred to the
central cut at 0.06.}}
\label{fig:stab_ymin}
\end{figure}
\begin{figure}
\begin{center}
\mbox{\epsfig{file=stab_xeffj.eps,width=15.cm}}
\end{center}
\caption{ {\sl Stability of the result as a function of the b-tagging
variables for the three ordered jets. The bars represent the
uncorrelated errors referred to the central cuts at 0.9, 0.2, -0.1
respectively.}}
\label{fig:stab_xeffj}
\end{figure}
\begin{figure}
\begin{center}
\mbox{\epsfig{file=minv.eps,width=13.cm}}
\end{center}
\caption{ {\sl The signal efficiency $\epsilon_{4b}$ as a function of the
minimum invariant mass between b quark pairs. The histogram represents
the generated spectrum in JETSET PS simulation.}}
\label{fig:q0}
\end{figure}
\end{document}