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\begin{titlepage}
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\begin{tabular}{l r}
HEP'99 \# 1\_223 & \hspace{6cm} DELPHI 99-121 CONF 308 \\
Submitted to Pa 1, 3 & 15 June 1999 \\
\hspace{2.4cm} Pl 1, 3 & \\
\end{tabular}
\vspace*{0.5cm}
\begin{center}
\Huge {\bf Three- and four-jet heavy \boldmath $b$-quark production
rates in $e^+e^-$ annihilation at the Z peak }\\
\vspace*{0.3cm}
\centerline{\large Preliminary}
\vspace*{0.3cm}
\large {DELPHI Collaboration \\}
\vspace*{0.2cm}
\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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\begin{titlepage}
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\begin{center}
\Huge {\bf Three- and four-jet heavy \boldmath $b$-quark production
rates in $e^+e^-$ annihilation at the Z peak }\\
\end{center}
\vspace*{1.5cm}
\centerline{{\large\bf S. Cabrera and J. Fuster}}
\centerline{{\small IFIC, Val\`encia}}
\vspace*{0.2cm}
\centerline{{\large\bf S. Mart\'{\i} i Garc\'{\i}a}}
\centerline{{\small CERN, EP}}
\vspace*{1.5cm}
%\centerline{\bf Abstract}
\abstract{
Three- and four-jet topologies are studied for hadronic events
originated by $b$ (massive) and $uds$ (light) quarks using the DELPHI
detector at LEP. The data were collected during the years 1994 and 1995
at a center of mass energy corresponding to $\sqrt{s} \approx M_Z$.
The experimental results
are compared to theoretical predictions including NLO radiative
corrections with mass effects. As a result of this study and using
{\sc Cambridge} as the algorithm for reconstructing jets, the
measured ratio of the normalized three jet
rate of $b$ quark with respect to that of light ($\ell$) quark events is
\[
R_{3}^{b{\ell}}(y_c=0.005)= 0.965 \pm 0.004~({\rm stat.})
\pm 0.011~({\rm frag.}) \pm 0.001~({\rm tag.})
\]
This ratio is in agreement with the theoretical predictions based on
the running $b$-quark mass, $m_b(M_Z)$, at both LO and NLO, whereas only
a reasonable agreement is found at NLO when the prediction is based on
the pole $b$-quark mass, M$_b$. The LO prediction
in terms of M$_b$ is almost 5$\sigma$ away from the measured $R_3^{b\ell}$.
The value for the $b$-quark mass
\[
m_b(M_Z) = 2.61 \pm 0.18~({\rm stat}) ^{+0.45}_{-0.49}~({\rm frag.})
\pm 0.04~({\rm tag.}) \pm 0.07~({\rm theo.})~GeV/c^2
\]
is extracted from the NLO massive calculations of the $R_3^{b\ell}$ and its
measurement.
A test on the universality of the strong coupling constant is also performed
with high precision leading to
\[
\frac{\alpha_s^b}{\alpha_s^{\ell}} = 1.005
\pm 0.012 \ ({\rm stat.+ frag. + theo.})
\]
}
\end{titlepage}
\newpage
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%==================> Put the text =====> To be filled <======%
\section{\bf Introduction}
The effects of the b-quark mass in the production of three jet event topologies
at the Z peak in $e^+e^-$ annihilations have recently been observed
\cite{delmbmz,sldmb,opalmb}. There exist also theoretical studies on this
topic based on Next-to-Leading Order (NLO) calculations including mass
corrections \cite{nlo1,nlo2,nlo3}. These analyses \cite{delmbmz,sldmb,opalmb}
made possible to perform an improved experimental test of the universality
of strong interactions and the first measurement of the
b-quark mass far from the $b\bar{b}$ production threshold
\cite{delmbmz,sldmbmz}.
The previous DELPHI analysis on the subject \cite{delmbmz} was based on
the {\sc Durham} jet reconstruction algorithm \cite{durham} whereas
the present one uses also the more recent
{\sc Cambridge} jet reconstruction procedure \cite{camjet} with improved
understanding of the soft gluon emission. The normalized
three- and four-jet production rates of $b$-quarks with respect to that of
uds-quarks ($\ell$) using this latter algorithm has been recently studied
in reference \cite{nosaltres}. The {\sc Cambridge} jet finder has smaller NLO
corrections as well as a reasonable stable hadronization correction as a
function of the jet resolution parameter down to very low values.
This study presents the analysis of the DELPHI data collected during the
years 1994 and 1995 at a center of mass energy of $\sqrt{s} \approx M_Z$.
The experimental results are compared to the predictions described
in \cite{nosaltres} for the observables:
\begin{equation}
R_{j}^{b{\ell}}(y_c) =
\frac{[\Gamma_{j}(y_c)/\Gamma_{tot}] ^{Z\rightarrow b\bar{b}}}
{[\Gamma_{j}(y_c)/\Gamma_{tot}]^{Z\rightarrow {\ell}\bar{{\ell}}}}~~~~~~j=3,4~\rm{jets}
\label{eq:r3bl}
\end{equation}
where $[\Gamma_{j}(y_c)/\Gamma_{tot}] ^{Z\rightarrow b\bar{b}}$ and
$[\Gamma_{j}(y_c)/\Gamma_{tot}]^{Z\rightarrow {\ell}\bar{{\ell}}}$ represent
the normalized three- ($j=3$) and four- ($j=4$) jet cross sections for $b$-
and $\ell$-quarks.
The theoretical predictions include up to $O (\alpha_s^2)$ terms which
means that NLO corrections are considered for the 3-jet rates, while only
LO terms enter on 4-jet rates.
\section{\bf The selection procedure}
The present study was based on a sample of $\sim 1.33 \times 10^6$ hadronic
decays of the $Z^0$ boson recorded with the DELPHI detector \cite{delphi} at
LEP during the years 1994 and 1995 and corresponded to centre of mass energies
of $\sqrt{s} \approx M_Z$.
\subsection{Event selection}
In the first stage of the selection procedure quality cuts
were applied to select charged and neutral particles
in order to ensure a reliable determination
of their kinematic variables: momenta and energies.
Hadronic $Z^0$ decays were selected demanding at least
5 well measured ($\Delta$($p$)/$p\leq$1) charged particles
% in order to reduce the ${\tau}^+{\tau}^-$ background
and 15 GeV of visible energy carried by charge particles.
Events having poorly measured particles were reduced with a charged balance.
Those events containing particles with an energy higher than 40 GeV
were discarded in order to avoid the participation of those particles
in the thrust or jet axis determination. The retained data sample contained
$\sim 1.33 \times 10^6$ hadronic $Z^0$ decays ($\sim 9 \times 10^5$ in
the year 1994 and $\sim 4.3 \times 10^5$ in
1995) with a small contamination from ${\tau}^+{\tau}^-$ pairs
($\sim$0.1\%) and a negligible background from beam-gas scattering
and $\gamma\gamma$ interactions.
%The efficiency of the hadronic selection was estimated from
%the Monte Carlo simulation giving a result of: $(55.14\pm0.03)\%$. The cut of
%${\theta}_{thrust}$ was the most restrictive cut in the hadronic selection.
\subsection{Jet reconstruction}
The jet clustering algorithm {\sc Durham}, the most extended in LEP physics,
and the recently introduced {\sc Cambridge} jet algorithm were applied to
group the selected charge and neutral particles in jets. The new ingredients
in the {\sc Cambridge} method: the angular ordering and the {\it soft freezing}
improve the understanding of soft gluon radiation.
For each pair of particles $ij$, the ordering variable $v_{ij}$ and the
resolution variable $y_{ij}$ were calculated from their respective
four-momentum vectors. The pair with the smallest $v_{ij}$ was combined
to form a new pseudo-particle with four-momentum $p_k=p_i+p_j$
if the resolution variable $y_{ij}$ did not exceed the jet resolution
parameter $y_c$, otherwise the {\it soft freezing} mechanism would be
activated: the particle $i$ with $E_i~<~E_j$ would
constitute a jet and the particle $j$ would return to the binary procedure.
The {\it soft freezing} eliminated the tendency of soft `resolved' jets
attracting extra wide-angle particles.
The procedure was iterated until no further pairs of particles
or pseudo-particles could be recombined. The number of remaining objects
added to the jets formed by {\it soft freezing} determined the class of
the event: two-jet, three-jet, etc.
The {\sc Durham} sequence of clustering was independent of the external $y_c$
parameter because both the ordering variable and the resolution variable
were the same (see table \ref{tab:alg}) and the {\it soft freezing}
mechanism was absent. The number of jets was monotonically decreasing for
increasing $y_c$. The transition values $y^{n{\leftarrow}n+1}$ or $y_c$
parameter values of change between a $n+1$ jet configuration to a $n$ jet
configuration were determined for $n=3,4$ in each hadronic event to classify
the event as a three- or four-jet event and to study the b-quark mass effects
in the jet production rates $R_3$ and $R_4$. For the {\sc Cambridge}
reconstruction algorithm the number of jets is not necessarily monotonically
decreasing for increasing $y_c$. This is due to its definition
and in some circumstances certain jet topologies were not present for a
specific event. In the case of three jets this affected ${\sim}1\%$ of the
events in the range $y_{c}~{\ge}~0.01$.
The quantities $R_n$ were properly normalized in all cases and this
property was also considered in the theoretical calculations.
%In the case of the {\sc Cambridge} method its complexity caused the sequence of
%clustering to be dependent of the external $y_c$: the number of jets was not
%monotonically decreasing for increasing $y_c$ and it was not posible to
%reconstruct a three-jet topology in the $(5.11~\pm~0.01)\%$ of the hadronic
%sample in all $y_{c}$ range. Those discontinuities were concentrated on
%a very low $y_c$ region, and were reduced to ${\sim}1\%$ in the range $y_{c}~{\ge}~0.01$.
\begin{table}[hbt]
\centering
\vspace{7mm}
\begin{tabular}{cccc}
\hline
& & & \\
{\rm Algorithm} & {\rm Resolution} & {\rm Ordering} & Recombination \\
& & & \\
\hline
& & & \\
{\rm {\sc Durham}}~\cite{durham} &
$y_{ij}$ = $ {2 \cdot {\rm min}(E_i\sp{2},E_j\sp{2}) \cdot (1-\cos\theta_{ij})\over
E_{vis}\sp{2} } $
& $v_{ij}$ = $y_{ij}$ & $ p_k = p_i + p_j $ \\
& & & \\
{\rm {\sc Cambridge}}~\cite{camjet} &
$y_{ij}$ = $ {2 \cdot {\rm min}(E_i\sp{2},E_j\sp{2}) \cdot (1-\cos\theta_{ij})\over
E_{vis}\sp{2} } $
& $v_{ij}$ = $2 \cdot (1-\cos\theta_{ij})$
& $p_k = p_i + p_j$ \\
\hline
\end{tabular}
\vspace{0.5cm}
\caption{Definition of the jet resolution variable $y_{ij}$,
ordering variable and recombination procedure of the {\sc Durham}
and {\sc Cambridge} jet finders.
$E_{vis}$ is the total visible energy of the event,
$p_{i} \equiv (E_{i},\vec{p}_{i})$ denotes a 4-vector
and $\theta_{ij}$ is the angle between $\vec{p}_{i}$ and
$\vec{p}_{j}$.}
\label{tab:alg}
\end{table}
\subsection{Selection on the reconstruction event quality}
In order to ensure a good energy balance in the event all of them were
reconstructed as three-jet and additional cuts were applied using the
reconstructed jet information. For those cases were {\sc Cambridge} was
unable to find exactly three jets (5\% of the cases), {\sc Durham} was
used instead. All the three jets in the event were demanded to comply with a
minimum charge multiplicity per jet of $N_{ch} \geq 1$, at least 1 GeV of
visible energy carried by charged particles belonging to the jet,
the jet polar angle to be well contained within the detector volume and
the three jet axis to be in a planar configuration.
A total of $\sim 1.15 \times 10^6$
hadronic events passed these criteria ($\sim 7.7 \times 10^5$ in 1994 year
and $\sim 3.7 \times 10^5$ in 1995 year).
\subsection{The quark flavour tag}
The selection of ${\ell}=uds$ and b quark initiated events was performed
using two different methods: the signed impact parameter of all charge
particles in the event \cite{ip} and the combined tagging technique
developed by DELPHI \cite{comb}.
The first method was based on the construction of a function, $P^{+}_{E}$,
in order to estimate the probability of having all particles compatible with
being generated in the events' Interaction Point (IP). This method was already used in our
former publication \cite{delmbmz}.
The decays of long lived B hadrons led to particles generated in secondary vertices far away
from the IP, biasing $P^{+}_{E}$ towards low values, while uds events have an
uniform distribution of $P^+_E$. Consequently, b-quark events were
selected by requiring $P^{+}_{E}<5{\cdot}10^3$ and ${\ell}$-quark events with
$P^{+}_{E}>0.2$. The $P^+_E$ distribution and the flavour tagging regions are
presented in figure \ref{fig:ip}.
In the second method, an optimal combination of a set of
discriminating variables defined for each reconstructed jet was performed.
The secondary vertices reconstructed in each jet were required to have at
least two tracks not compatible with the primary vertex and to have
$L/{{\sigma}_L}~>~4$ where L is the distance from primary to secondary
vertex and ${\sigma}_L$ is its error.
The quantities combined per jet were: the jet lifetime probability,
${ P}^+_j$; 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 of the method see reference \cite{comb}. Finally the
discriminating variables per jet were combined in a single variable
per event $X_{\rm effev}$. Candidates for b-quark events were selected by
requiring $X_{\rm effev}>-0.2$ and ${\ell}$-quark events with
$X_{\rm effev}<-1.$ Figure \ref{fig:comb} presents the $X_{\rm effev}$
distribution as well as the flavour tagging regions.
Each value of $P^{+}_{E}$ and $X_{\rm effev}$ corresponded to a well
determined combination of purity and efficiency. The ${\ell}$-quark
selection efficiency obtained with both methods was ${\sim}60\%$.
The $b$-quark selection efficiency
attained was ${\sim}67\%$ for the impact parameter method and ${\sim}53\%$
with the combined method of tagging. The combined method reached a higher
purity ${\sim}85\%$ in the $b$ sample in relation to the impact parameter
method ${\sim}81\%$ with a lower level of contamination for $c$-quark
${\sim}10\%$ while maintaining at the same time enough efficiency
(see table \ref{tab:ciq}).
\begin{table}[hbt]
\centering
\vspace{7mm}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
b-tag & ~~\# ev. in data~~ & ~$q$-type &
~${\ell} \rightarrow q$-type (\%)~ & ~$c \rightarrow q$-type (\%)~ &
~$b\rightarrow q$-type (\%)~\\
\hline
Imp. par. & 606831 & ${\ell}$ & 85. & 13. & \phantom{0}2. \\
Imp. par. & 262252 & $b$ & \phantom{0}4. & 15. & 81. \\
\hline
Combined & 754729 & ${\ell}$ & 82. & \phantom{5}15.5 & \phantom{2.5}2.5 \\
Combined & 245124 & $b$ & \phantom{0}4. & 10. & 86. \\
\hline
\end{tabular}
\vspace{0.5cm}
\caption{Event statistics and flavour compositions of the samples tagged as
light-quark
($\ell \equiv u,d,s$) and $b$-quark events in real data and for each tagging
technique.}
\label{tab:ciq}
\end{table}
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=16.cm\epsfysize=16.cm
\epsffile{ichep99eps/ip_logp.eps}}
\end{center}
\caption[]{Event distribution of the probability P
to contain no secondary vertices. The data (points) and
the Monte Carlo (histogram) are compared. The specific
contribution of each quark flavour is displayed as derived from the
Monte Carlo. The cuts used to tag the $b$-quark and $\ell$-quark
($\ell \equiv uds$ samples are also indicated).}
\label{fig:ip}
\end{figure}
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=16.cm\epsfysize=16.cm
\epsffile{ichep99eps/combined_xeffev.eps}}
\end{center}
\caption[]{Event distribution of the combined variable $X_{\rm effev}$.
The data (points) and
the Monte Carlo (histogram) are compared. The specific
contribution of each quark flavour is displayed as derived from the
Monte Carlo. The cuts used to tag the $b$-quark and $\ell$-quark
($\ell \equiv u,d,s$) samples are also indicated}
\label{fig:comb}
\end{figure}
\section{Experimental three- and four-jet event rates}
The raw distributions of the $R^{b{\ell}}_3$ and $R^{b{\ell}}_4$ observables
(eq. \ref{eq:r3bl}) were corrected using a sample of $\sim 4.3 \times 10^6$
events generated with JETSET 7.3
Parton Shower (PS) Monte Carlo \cite{jetset}. These Monte Carlo events were
processed by the full DELPHI simulation program, and then passed through
the same reconstruction and analysis chain than the real data events.
The flavour assignment of the simulated events was defined to be
that of the pair of quarks coupled to the Z which initiated the parton shower.
The same convention was considered in the theoretical calculation \cite{nlo1,nosaltres} thus
allowing a consistent comparison.
The experimental method to correct the measured $R^{b{\ell}}_3$ and $R^{b{\ell}}_4$
quantities for detector acceptance effects, kinematic biases introduced in the
two tagging procedures, and the hadronization process was the same as
described in \cite{delmbmz}. Three- and four- jet topologies had different
hadronization correction factors as well as detector, acceptance and
tagging correction factors, but the same flavour compositions were used in
the correction procedure.
Both tagging methods described in the previous section have been considered in the analysis. Therefore
the average of the experimental results of $R^{b{\ell}}_3$ obtained with both methods has been taken
as the experimental result of the partonic $R^{b{\ell}}_3$. A new additional systematic uncertainty is assigned to
$R^{b{\ell}}_3$ as half of the difference between the results with both methods and referred to as $tag.$
The $R^{b{\ell}}_3$ corrected experimental result is shown for both algorithms
{\sc Durham} and {\sc Cambridge} in figures \ref{r3bl_d} and \ref{r3bl_c}
respectively.
Figure \ref{r3bl_d} shows that when using Durham as jet finder, neither the
LO calculations in terms of the pole mass ($M_b=4.6~GeV/{c}^{2}$)
nor those in terms of the running mass
($m_b(M_Z)=2.8~GeV/{c}^{2}$)\footnote{These
values for $M_b$ and $m_b(M_Z)$ are taken from analyses based on data
collected at the $\Upsilon$ production threshold as published in references
\cite{upsilon}} provide an acceptable description of the data points.
Therefore, and as already indicated by
the previous DELPHI analysis \cite{delmbmz}, QCD radiative corrections
including mass effects are necessary to properly describe
the data. As shown in figure \ref{r3bl_d}, the {\sc Durham} NLO corrections
are larger in terms of the running mass $m_b(M_Z)$ than in terms of the
pole mass $M_b$ (figure \ref{r3bl_d}),
however the data points are closer to the prediction based on the
running mass. This fact indicates that in this scheme the convergence of
the higher order terms is faster. This was observed for the first time
in the previous DELPHI publication \cite{delmbmz} using data from the
years 1992-1994 and quantified as a 2-3$\sigma$ effect. The same conclusion
can be draw from the present analysis, which uses different sets of data
(collected during the years 1994 and 1995) analyzed with two different
quark tagging methods.
The $R_3^{b\ell}$ NLO corrections for different jet algorithms have been
studied in a recent work \cite{newarca}. There the NLO corrections are
compared for {\sc Durham} and {\sc Jade-like} algorithms.
In reference \cite{nosaltres} the {\sc Cambridge} algorithm
NLO corrections have been studied and found to be smaller when the calculation
is expressed in terms of $m_b(M_Z)$ instead of $M_b$. The question then is
whether the data follows (or not) this behaviour as this would be the first
time in which theory and experiment would prefer (or not) a description of
jet production rates based on the running mass. This may be an scenario in
which the LO calculation as a function of $m_b(M_Z)$ would give a reasonable
description of the experimental results.
For this purpose in figure \ref{r3bl_c},
the $R^{b{\ell}}_3$ LO and NLO predictions using the {\sc Cambridge}
algorithm are compared to the data points. The NLO calculations
in terms of the running mass match the experimental points more accurately
than the NLO calculations in terms of the pole mass do.
At LO only the $m_b(M_Z)$ provides a rough description of the
experimental results, as the curve based on $M_b$
is 4-5$\sigma$ away from data.
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=15.cm\epsfysize=15.cm
\epsffile{ichep99eps/rn3_durham94y95_cor_ip_1_deft.eps}}
\end{center}
\caption[]{Corrected data values of $R_3^{b{\ell}}$ using {\sc Durham} algorithm
compared with the theoretical predictions from reference \cite{nosaltres} at LO
and NLO in terms of the pole mass $M_b=4.6~GeV/{c}^{2}$ (dashed lines) and
in terms of the running mass $m_b(M_Z)=2.8~GeV/{c}^{2}$ (solid lines).}
\label{r3bl_d}
\end{figure}
%
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=15.cm\epsfysize=15.cm
\epsffile{ichep99eps/rn3_camjet94y95_cor_ip_1_deft.eps}}
\end{center}
\caption[]
{Corrected data values of $R_3^{b{\ell}}$ using {\sc Cambridge} algorithm
compared with the theoretical predictions from reference \cite{nosaltres} at LO
and NLO in terms of the pole mass $M_b=4.6~GeV/{c}^{2}$ (dashed lines) and
in terms of the running mass $m_b(M_Z)=2.8~GeV/{c}^{2}$ (solid lines).}
\label{r3bl_c}
\end{figure}
The b-quark mass effects have been observed also in 4 jets topologies at the Z peak.
In the case of 4 jets topologies, theoretical calculations including b-quark mass effects
are available only at LO \cite{4jt}. Figure \ref{r4bl_d} presents the corrected $R_4^{b{\ell}}$
observable as a function of the {\sc Durham} $y_c$. The massive LO calculations
with $m_b=2.8~GeV/{c}^{2}$ and $4.6~GeV/{c}^{2}$ are included for
comparison. Since none of them provides a successful description of the
data, QCD radiative corrections with mass effects are required to describe
the results.
NLO calculations would be needed if one plans to use the mass effects
in the 4-jet rate in order to extract a reliable $m_b$ value.
$R_4^{b{\ell}}$ data points lay in the band limited by the LO massive
calculations with $m_b=2.8~GeV/{c}^{2}$ and $4.6~GeV/{c}^{2}$.
This situation is analogous to that found in the case of $R_3^{b{\ell}}$
(figure \ref{r3bl_d}).
However for the $R_4^{b{\ell}}$ the net effect of the mass is considerably larger (${\sim}8\%$ for $y_c=0.02$
compared to a ${\sim}3\%$ for $R_3^{b{\ell}}$ at the same $y_c$ value), so the mass effects become more apparent. A
similar study using the {\sc Cambridge} algorithm will follow.
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=15.cm\epsfysize=15.cm
\epsffile{ichep99eps/rn4_durham94_cor_ip.eps}}
\end{center}
\caption[]{Corrected data values of $R_4^{b{\ell}}$ with their statistical errors
using {\sc Durham} algorithm compared with the theoretical predictions
from reference \cite{4jt} at LO
with $M_b=2.8~GeV/{c}^{2}$ (solid line) and $M_b=4.6~GeV/{c}^{2}$ (dashed line).}
\label{r4bl_d}
\end{figure}
\section{\bf Hadronization corrections and systematic uncertainties}
The impact of the fragmentation process on the observable $R_3^{b{\ell}}$
was studied and quantified by adding in quadrature two different source of
errors: $\sigma_{tun}$, uncertainty with origin in the lack of an exact
knowledge of the main fragmentation parameters in JETSET \cite{jetset}
and $\sigma_{mod}$, uncertainty due to the dependence of the hadronization
correction factors with the two fragmentation
models considered: cluster fragmentation in HERWIG \cite{herwig} and
Lund string fragmentation in JETSET.
Complete details of the evaluation of these two uncertainties are given in reference \cite{delmbmz}.
The total error due to the lack of knowledge on the
hadronization process can be expressed as:
\be
\sigma_{had}(y_c) = \sqrt{\sigma_{tun}^2(y_c)+\sigma_{mod}^2(y_c)}
\ee
Figure \ref{fig:had} presents the size of the hadronization correction uncertainty for {\sc Durham} and {\sc Cambridge} algorithms.
This figure shows a larger flat $y_c$-region in the case of {\sc Cambridge}
with respect to {\sc Durham} which can be extended down to $y_c=0.004$ while keeping the four jet contribution $\leq$10\%.
However, in this flat region the total hadronization error of the {\sc Cambridge} algorithm is higher than
in the plateau reached with {\sc Durham} algorithm.
Nevertheless the relative sensitivity to the mass correction is larger for
{\sc Cambridge} at
$y_c$ values close to 0.005 than for {\sc Durham} at $y_c$ values close to
0.02. For comparison purposes the difference between the theoretical prediction
of $R_3^{bd}$ at LO in terms of the pole mass, $M_b=4.6$ GeV/$c^2$ with
respect to that obtained using the running mass $m_b(M_Z)=2.8$ GeV/$c^2$
is also shown. A higher sensitivity to this difference is again found for
the new {\sc Cambridge} jet algorithm in the valid, flat, $y_c$-region
($y_c>0.004$) thus enabling a more significant test of the mass effects in
the jet rates.
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=15.cm\epsfysize=15.cm
\epsffile{ichep99eps/hadcam.eps}}
\end{center}
\caption[]
{Evolution of the total hadronization error of $R_3^{b{\ell}}$ with the resolution
parameter $y_c$ in comparison with the LO prediction in terms of the pole mass, $M_b=4.6~GeV/{c}^{2}$
and the difference between the LO prediction using $M_b=4.6~GeV/{c}^{2}$ and $m_b(M_Z)=2.8~GeV/{c}^{2}$.}
\label{fig:had}
\end{figure}
\section{\bf Results and discussion}
The values of $R_3^{b\ell}$ and $m_b(M_Z)$ obtained with {\sc Durham}
algorithm at $y_c=0.02$ and the breakdown of all affecting uncertainties
to both quantities are summarized in table \ref{tabdur}. The result of
the $R_3^{b\ell}$ observable and $m_b(M_Z)$ are
fully compatible with the previous results published in \cite{delmbmz}
(see table \ref{res_r3bl} for $R_3^{b\ell}$ and table \ref{res_mb}
for $m_b(M_Z)$).
The argument followed for the election of the $y_c$ value to perform
the $m_b(M_Z)$ measurement is such that allows a more accurate
determination of $R_3^{b\ell}$, thus the optimization of all the statistical
and systematic uncertainties.
The values of $R_3^{b\ell}$ and $m_b(M_Z)$ obtained with {\sc Cambridge}
algorithm at $y_c=0.005$ and the listing of all affecting uncertainties
to both quantities are summarized in table \ref{tabcam}.
The total error of $m_b(M_Z)$ is ${\sim}0.52~GeV/c^2$ at $y_c=0.005$.
The experimental result obtained with {\sc Cambridge} algorithm at $y_c=0.005$
for the $R_3^{b\ell}$ observable is: $0.965\pm0.012\ {\rm (stat.+syst.)}$. This value compared with
the rough LO prediction with a pole mass $M_b=4.6~{\rm GeV/{c}^{2}}$, $R_{3}^{b{\ell}}(y_c=0.005)=0.910$
differs by almost 5$\sigma$. As the experimental value is in agreement with the LO and the NLO calculation
in terms of the running b-quark mass, the present result exhibits the effects of the running mass of
the b-quark in the 3-jet rate. This represents a net improvement with
respect to the former \cite{delmbmz} and current results
with the {\sc Durham} algorithm where only a 2-3$\sigma$ deviation was seen.
Using NLO calculations \cite{nosaltres} and the $R_3^{b\ell}$ measurement,
the $m_b(M_Z)$ value extracted with
{\sc Cambridge} algorithm at $y_c=0.005$ is:
\be
\label{eqdel}
m_b(M_Z) =
2.61 \pm 0.18~({\rm stat}) ^{+0.45}_{-0.49}~({\rm frag.})
\pm 0.04~({\rm tag.}) \pm 0.07~({\rm theo.})~{\rm GeV/c^2}
\ee
A high stability of the $m_b(M_Z)$ value versus $y_c$ is obtained with the {\sc Cambridge} algorithm,
given that none of the $m_b(M_Z)$ values extracted from $R_3^{b\ell}$ in the range
$0.005~{\le}~y_c~{\le}~0.025$ differs from the reference value by a quantity greater than 0.15 ${\rm GeV/c^2}$.
A direct measurement of the pole mass of the $b$-quark from the NLO
prediction gives
$M_b=4.1 \pm 0.5 (\mbox{stat.+frag.+theo.})~{\rm GeV/{c}^{2}}$ in
reasonable agreement with that obtained at the $\Upsilon$
($M_b=4.6~{\rm GeV/{c}^{2}}$) \cite{upsilon}. The
conversion of the value of running $b$-quark mass into the pole $b$-mass is
$M_b=4.3 \pm 0.5~(\mbox{stat.+frag.+theo.}) {\rm GeV/{c}^{2}}$ which is
compatible with the direct measurement. These results reveal a coherent
picture within errors of the mass effects at NLO at
the $M_Z$ scale. The use of the running $b$-quark mass shows its advantage.
Note that the theoretical error associated with the $m_b(M_Z)$ measurement is
smaller when using {\sc Cambridge} (table \ref{tabcam}) instead of
{\sc Durham} (table \ref{tabdur}). This is due to the weak dependence of the
$R_3^{b\ell}$ with the scale ($\mu$) of the process \cite{nosaltres}.
The present result of $m_b(M_Z)$ (eq. \ref{eqdel}) is nice agreement with
the former DELPHI measurement \cite{delmbmz}. It also agrees with the
result from the reference \cite{sldmb} extracted from SLD data of
$m_b(M_Z)$ obtained with 6 jet algorithms ({\sc E},{\sc E0},{\sc P},
{\sc P0},{\sc Durham} and {\sc Geneva}) and taking into account statistical
correlations as well as correlations in the systematic error and
hadronization uncertainties:
\be
\label{eqsld}
~\cite{sldmb}:~~ m_b(M_Z) = 2.52 \pm 0.27\ ({\rm stat.})
^{+0.33} _{-0.47} ({\rm frag.})
^{+0.54} _{-1.46} ({\rm theo.})
{\rm ~GeV/c^2}
\ee
A net change in the value of the running b-quark mass between the scales ${\mu}_1=M_Z$
and ${\mu}_2=M_{{\Upsilon}/2}$ is observed with almost 3 standard deviations
$m_b(m_\Upsilon/2)-m_b(M_Z)=1.55\pm0.54~{\rm GeV}/c^{2}$ (see figure \ref{fig:run}).
This result is in good agreement with that predicted from the QCD evolution.
The test of flavour independence of $\alpha_s$ is performed with the
experimental result of $R_3^{b\ell}$ using the {\sc Cambridge} algorithm
at $y_c=0.005$ and the NLO calculations \cite{nosaltres} and following the
same method as in \cite{delmbmz}. Taking as an input now the hypothesis that
the QCD prediction for $R_3^{b\ell}$
is governed by $m_b(M_Z)=2.8~{\rm GeV/c^2}$ the result is:
\begin{equation}
\label{eq:alfa_fl}
\frac{\alpha_s^b}{\alpha_s^{\ell}} = 1.005
\pm 0.012 \ ({\rm stat. + frag.+ theo.}),
\end{equation}
which verifies the flavour independence of the strong coupling constant for b and light quarks.
The b-quark mass effects have been also seen in the 4-jet rate and compared
with the LO massive calculations.
A better understanding of the hadronization effects would lead to very competitive values of the $b$ mass
with respect to those measured from the $\Upsilon$ resonance production with the additional benefit of being
extracted far from the $b\bar{b}$ production threshold.
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%
% JET Algorithms
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\newpage
\begin{table}[htb]
\caption{Values of $R_3^{b\ell}$ and $m_b(M_Z)$ obtained with {\sc Durham} algorithm and break-down
of their associated errors (statistical and systematic) for $y_c=0.02$}
\begin{tabular}{lrc}
\hline
{\sc Durham} (94+95) & $R_3^{bl}(y_c=0.02)$ & $m_b(M_Z)$ GeV/c$^2$ \\
\hline
& & \\
Value & 0.968 & 2.81 \\
& & \\
Statistical error & $\pm0.004$ & $\pm0.20$ \\
Flavour tag error & $\pm0.001$ & $\pm0.04$ \\
Simulation error & $\pm0.003$ & $\pm0.15$ \\
Fragmentation Model error & $\pm0.006$ & $\pm0.30$ \\
Fragmentation Tuning error & $\pm0.003$ & $\pm0.15$ \\
Mass Ambiguity error & --~~~ & $\pm0.25$ \\
$\mu$-scale error ($0.5 \leq \mu/M_Z \leq 2$) & --~~~ & $\pm0.10$ \\
\hline
\end{tabular}
\label{tabdur}
\end{table}
\begin{table}[htb]
\caption{Values of $R_3^{b\ell}$ and $m_b(M_Z)$ obtained with {\sc Cambridge} algorithm and break-down
of their associated errors (statistical and systematic) for $y_c=0.005$}
\begin{tabular}{lrc}
\hline
{\sc Cambridge} (94+95) & $R_3^{bl}(y_c=0.005)$ & $m_b(M_Z)$ GeV/c$^2$ \\
\hline
& & \\
Value & 0.965 & 2.61 \\
& & \\
Statistical error & $\pm0.003$ & $\pm0.13$ \\
Flavour tag error & $\pm0.001$ & $\pm0.04$ \\
Simulation error & $\pm0.003$ & $\pm0.13$ \\
Fragmentation Model error & $\pm0.011$ & $^{+0.45}_{-0.49}$ \\
Fragmentation Tuning error & $\pm0.003$ & $\pm0.12$ \\
Mass Ambiguity error & --~~~ & $\pm0.07$ \\
$\mu$-scale error ($0.1 \leq \mu/M_Z \leq 1$) & --~~~ & $\pm0.02$ \\
\hline
\end{tabular}
\label{tabcam}
\end{table}
\begin{table}[hbt]
\caption{Results of $R_3^{b\ell}$ obtained with {\sc Durham} and {\sc Cambridge} algorithms.}
\begin{tabular}{|l|l|l|c|c|c|c|}
\hline
$y_c$ & Algorithm & $R_3^{b\ell}$ & $\sigma_{stat}$ & $\sigma_{frag}$ & $\sigma_{b-tag}$ & Reference
\\
\hline
0.02 & {\sc Durham} & 0.971 & 0.005 & 0.007 & - & \cite{delmbmz} \\
\hline
0.02 & {\sc Durham} & 0.968 & 0.005 & 0.007 & 0.001 & present study \\
\hline
0.005 & {\sc Cambridge} & 0.965 & 0.004 & 0.011 & 0.001 & present study \\
\hline
\end{tabular}
\label{res_r3bl}
\end{table}
\begin{table}[hbt]
\caption{Results of $m_b(M_Z)$ obtained with {\sc Durham} and {\sc Cambridge} algorithms.}
\begin{tabular}{|l|l|l|c|c|c|c|c|}
\hline
$y_c$ & Algorithm & $m_b(M_Z)$ & $\sigma_{stat}$ & $\sigma_{frag}$ & $\sigma_{b-tag}$ & $\sigma_{theo}$ & Reference
\\
\hline
0.02 & {\sc Durham} & 2.67 & 0.25 & 0.34 & - & 0.27 & \cite{delmbmz} \\
\hline
0.02 & {\sc Durham} & 2.81 & 0.25 & 0.34 & 0.04 & 0.27 & present study \\
\hline
0.005 & {\sc Cambridge} & 2.61 & 0.18 & $~^{+0.45}_{-0.49}$ & 0.04 & 0.07 & present study \\
\hline
\end{tabular}
\label{res_mb}
\end{table}
\begin{figure}[hbt]
\epsfverbosetrue
\begin{center}\mbox{\epsfxsize=15.cm\epsfysize=15.cm
\epsffile{ichep99eps/mbmz.eps}}
\end{center}
\caption[]
{The running of $m_b({\mu})$ from the scale $M_{\Upsilon}/2$ up to the $M_Z$ scale
using the QCD renormalization group equations. The $m_b(M_Z)$ value obtained by
DELPHI and the value from reference \cite{sldmb} using SLD results
are displayed together with the statistical and total errors.}
\label{fig:run}
\end{figure}
\end{document}