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\newcommand{\ccb}{c\bar{c}}
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\newcommand{\del}{$\delta\;$}
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\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
\begin{small}
\begin{tabular}{llrr}
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                32nd International Conference 
                on High Energy Physics, ICHEP04},
                August~16,~2004,~Beijing} \\
                 & Abstract:        & {\bf 5-0164}    &\\
                 & Parallel Session & {\bf 5}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
    Measurement of Beauty Photoproduction at HERA \\ Using Semi-Muonic Decays }

  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent

Differential measurements of beauty photoproduction cross sections 
in $ep$ collisions performed with the H1 detector at HERA are presented. 
%
The data were collected at an $ep$
centre-of-mass energy of 319 GeV in the years 1999-2000 
and correspond to an integrated luminosity of 48 pb$^{-1}$.
%
Events are selected by requiring at least two high-transverse momentum
jets, $p_t^{jet_{1(2)}}>7(6)$ GeV, jet pseudorapidities 
$|\eta^{jet}|<2.5$, and a muon in the final state.
%
Both the lifetime signature and the large mass of $b$ flavoured hadrons
are exploited to determine the fraction of events 
in the sample containing beauty.
%
Cross sections are measured in the region 
\mbox{$Q^2<1$ GeV$^2$} with inelasticity $0.2<y<0.8$ for muons with 
$-0.55<\eta^{\mu}<1.1$ and $p_t^{\mu}>2.5$ GeV.
%
The visible dijet-muon production cross section is measured to be 
$ \sigma_{vis}(ep \rightarrow e b\bar{b} X
\rightarrow e jj\mu X) = (42.5 \pm 3.4 (stat.) \pm 8.9 (sys.)) {\rm pb}.
$
Differential measurements are presented as
a function of the transverse momentum of the muon, 
the pseudorapidity of the muon and the quantity \xgobs.
%
The results are compared with Monte Carlo models based on leading 
order QCD and with next-to-leading order QCD calculations. 

\end{abstract}


\end{titlepage}

\pagestyle{plain}

\section{Introduction}
%
This paper presents measurements of 
open beauty production in $ep$ collisions in the
photoproduction regime, where a quasi-real photon ($Q^2\sim 0$) 
is emitted by the incoming lepton and interacts with the proton.
%
The large mass of the $b$ quark provides a hard scale, which
makes the study of $b$ quark production in photoproduction an excellent
testing ground for perturbative QCD (pQCD).

In pQCD, at leading order (LO), two processes can be distinguished
which contribute to the photoproduction of heavy quarks.
%
In {\em direct} photon processes
the quasi-real photon from the positron enters directly
in the hard process, e.g.\,$\gamma g \rightarrow b\bar{b}$, while
%
in {\em resolved} photon processes the photon fluctuates 
into a hadronic state before the hard interaction
and acts as a source of partons, one of 
which takes part in the hard interaction. 
%
Previous measurements, 
both in photoproduction~\cite{Adloff:1999nr,h1bprelim,Breitweg:2000nz,zeusbprelim}
and in deep inelastic scattering, 
have shown that the
beauty production cross section lies significantly
above the next-to-leading order QCD (NLO) expectations.
Similar observations have been made in hadron-hadron collisions~\cite{hadronb},
and also in two-photon interactions~\cite{ggb}.
%
%Zeus-sentence:
%This paper presents measurements by H1 with increased precision due to 
%using newer data with three times the luminosity than
%in previous H1 analyses~\cite{Adloff:1999nr}. 
%

This paper presents a new photoproduction measurement using
a data sample three times larger than in the previous
analysis~\cite{Adloff:1999nr}.
%
Events with two jets and a muon in the final state are used
to measure the beauty photoproduction cross section
$$
e^+p \rightarrow e^+ b\bar{b} X \rightarrow e^+ + jj + \mu^{\pm} + X.
$$
%
The cross section is measured 
differentially as a function of the muon transverse momentum,
\ptmu , muon pseudorapidity, \etamu , and \xgobs,
where \xgobs is defined as the 
fraction of the $(E-p_z$) of the hadronic system
that is carried by the two highest $p_T$ jets:
%.
$$
\xgobsm = \frac{ (E-p_z)_{jet_1} +(E-p_z)_{jet_2}}{(E-p_z)_h}.
$$  
In LO QCD, \xgobs is the fraction of the photon energy 
that enters the hard interaction.
%For {\em direct} photon processes it is expected that $\xgobsm \sim 1$
%while for {\em resolved} processes  \xgobs  can reach to much lower values. 

For the measurement presented here both the lifetime signature and 
the large mass of $b$ flavoured hadrons
are used to determine the fraction of beauty quark events in the sample.
%
Experimentally, the beauty quark mass and lifetime are reflected in a 
broad $p_T^{rel}$ distribution, the transverse muon momentum 
with respect to the jet direction and a large impact parameter 
\del of the muon track relative to the primary vertex.
%
The measurement of the impact parameter is facilitated
by the high precision tracking made possible with
the H1 central silicon tracker (CST).
%
The fraction of $b$ quark events in the final sample is determined 
by a fit to the two-dimensional distribution of the \ptrel and \del
observables in the data with adjustable fractions of beauty,
charm and light-quark components, the shapes of which are 
taken from Monte Carlo (MC) simulations.
%

This paper is organised as follows:
In sections \ref{sec:det} and \ref{sec:sel} the
detector components used for this analysis and the 
event selection are described.
The Monte Carlo simulations and data sets used to model the signal
and background components of the data are 
described in section \ref{sec:mc}.
In section \ref{sec:nlo} the calculations in perturbative QCD 
performed at next-to-leading order are explained.
% 
The key observables \ptrel and \del are introduced in 
section \ref{sec:fit} and the fit procedure used to determine 
the relative fractions of signal and 
background components is explained.
The measured cross sections are presented in section \ref{sec:results}
and compared with predictions from Monte Carlo simulations and pQCD.
%
\section{Detector Description}
\label{sec:det}

The H1 detector is described in detail in \cite{Abt:1997xv}.
Charged particles emerging from the interaction region
are measured by the central tracking detector (CTD) in the pseudorapidity
range $-1.74 < \eta < 1.74$\footnote{The pseudorapidity 
$\eta$ of an object with polar angle $\theta$ is given by
$\eta = -\ln \; \tan (\theta/2)$, where $\theta$ is
measured with respect to the $z$-axis given by the proton beam direction.}.
The CTD comprises two large
cylindrical central jet drift chambers (CJC) and two $z$-chambers 
arranged concentrically around the beam-line 
within a solenoidal magnetic field of 1.15 T. 
The CTD also provides triggering
information  based on track segments in the $r$-$\phi$ plane 
from the CJC and the $z$-position of the vertex from a double 
layer of multi-wire proportional chambers.

The CJC tracks are linked with hits in the Central
Silicon Tracking Detector
(CST) \cite{cst}, which consists of two cylindrical
layers of 
%double-sided 
silicon strip detectors, 
surrounding the beam pipe  
at radii of $R=57.5$ mm and $R=97 $ mm from the beam axis. 
%
%With an effective length of 358 mm the CST covers 
%a large part of the $ep$ interaction region
%and has a polar angle acceptance of $30^0 <\theta< 150^0$
%for the outer layer.
%
These double-sided silicon detectors with readout strip pitches 
of 50 $\mu$m and 88 $\mu$m provide
resolutions of 12 $\mu$m in $r$-$\phi$ and 25 $\mu$m in $z$.
Average hit efficiencies reach values of 97 (92)\% in 
$r$-$\phi$ ($z$).
%
For tracks with CST hits in both layers
the transverse distance of closest approach ($dca$) of the track to the interaction
point can be measured with a resolution of 
$\sigma_{dca} \approx 33\;\mu\mbox{m} 
\oplus 90 \;\mu\mbox{m} /p_T [\mbox{GeV}]$.
%
% See plot for this resolution vs pt
% in ~obehnke/hd/bphan/paw/dcapres.ps
% (from bbbar mc with 
% cutting on 50 cm cjc tracklength and requiring two cst hits
%
%
The first term represents the intrinsic resolution
and includes the uncertainty of the CST alignment.
%
The second term gives the contribution
from multiple scattering in the beam pipe.
%

The energies of final state particles are measured using
CTD+CST track information and measurements of the energy deposited
in the liquid 
argon (LAr) calorimeter, which surrounds the tracking chambers and 
covers the range $-1.5 < \eta < 3.4$. 
The backward region ($-4.0 < \eta < -1.4$) is covered by
a lead--scintillating fibre calorimeter (SPACAL~\cite{Nicholls:1996di}) with 
electromagnetic and hadronic sections.  
The calorimeters are surrounded by the iron return yoke of the solenoidal 
magnet.
The tracks of muons which penetrate the main detector are reconstructed using
limited streamer tubes placed within the iron in the range 
$-2.5 < \eta < 3.4$.
The luminosity is measured using the small angle Bremsstrahlung process
($ep\rightarrow ep\gamma$) in which the final state photon is
detected in a calorimeter close to the beam-pipe at
$103 \ {\rm m}$ from the interaction region.

\section{Event Selection}
\label{sec:sel}

The data were recorded
in 1999 and 2000 and correspond
to an integrated luminosity of $48\,$pb$^{-1}$.
%
The events were triggered by requiring the coincidence of signals 
from the muon system, the central drift chambers and the 
multi-wire proportional chambers. 
%
Photoproduction events are selected by requiring that there be no 
high energy
electromagnetic cluster in the backward calorimeter.
The accepted range of negative four-momentum
transfer squared is restricted to $Q^2<1$ GeV$^2$.  
An inelasticity  cut \mbox{0.2 $<y<$ 0.8}, 
where $y$ is calculated using the Jacquet-Blondel method~\cite{JB}, 
further reduces remaining background from deep inelastic scattering. 
%
Jets are reconstructed using the inclusive $k_t$ algorithm~\cite{kt} 
with radius $R=1$ in the $\eta$-$\phi$ plane.
% 
The $E_T$-recombination scheme is applied 
giving massless jets.
%which leads to massless jets.
% 
The selection requires at least two jets with 
transverse energy $p_t^{jet_{1(2)}}>7(6)$ GeV,
of which at least one contains a muon candidate.
% 
Muons are identified in the barrel part of the instrumented iron,
corresponding to polar angles $35^{\circ} < \theta (\mu)<130^{\circ}$,
and are required to have transverse momenta $p_t^{\mu}>2.5$ GeV.
At least two CST-$r$-$\phi$-hits must be associated with the 
muon candidate track, measured in the central drift chambers.  
The combined CJC-CST $r$-$\phi$-track fit probability 
must exceed 10\%.  
%
The final event sample consists of 1452 events.

%
\section{Monte Carlo Simulations and Control Data Samples}
\label{sec:mc}
%
Monte Carlo event samples for the processes 
$ep\ra e\bbb X$, $ep\ra e\ccb X$ and light quark production
are generated using the PYTHIA program~\cite{PYTHIA}
which is based on leading order QCD and parton showers.  
%
PYTHIA simulates direct and resolved photon processes and 
also includes excitation processes, in
which one heavy quark ($c$ or $b$) originates from the
resolved photon or the proton. 
%
PYTHIA is run in an inclusive mode and generates all the above 
processes using massless matrix elements.
%
The CTEQ5L~\cite{cteq5l} parton densities are used
% hep-ph/9903282
for the proton and those of
GRVG-LO~\cite{grvg} 
% M.~Gluck, E.~Reya and A.~Vogt, Phys. Rev. D46, 1973 (1992)
for the photon. 
%
The light quark sample is used to simulate
the background from fake muons, i.e. hadrons misidentified as muons, 
and decays of light mesons into muons.
The program CASCADE \cite{Jung:casc},
%\cite{Jung:2000hk,Jung:2001hx},
a Monte Carlo generator which
implements the CCFM parton evolution equation  
\cite{ccfm}
%\cite{Ciafaloni:1987ur,Catani:1989yc,Marchesini:1994wr}
is used for cross checks and for comparisons 
with the measured cross sections.

Dijet event samples 
which fulfill the same selection criteria as 
the signal sample, but without the muon trigger 
and muon-identification requirements, are used to 
study the tracking and jet reconstruction resolutions.
%
The resolutions of the Monte Carlo simulations
are tuned so that the PYTHIA light quark event sample
accurately describes these data samples.

%The selection efficiency (10\%) has been obtained from these Monte Carlo 
%simulations. 
%The trigger efficiency (79\%) was determined from data, 
%using an independent, calorimeter-based trigger. 
%-------------------------------------------------------
\section{Predictions Based on QCD NLO Calculations}
\label{sec:nlo}

The program for fixed order massive NLO calculations by 
Frixione et al. \cite{Frixione:1994dv} was modified to 
facilitate the comparison of the calculation with the 
visible cross sections in the experimentally accessible kinematic range.
The outgoing partons ($b$ quark, $\bar{b}$ quark and the gluon) are combined
into jets using the inclusive $k_t$ jet-algorithm (in the $E_t$-scheme).
The $b$ quark is then fragmented to a $B$ Hadron using the 
Peterson fragmentation function \cite{peterson}
with a fragmentation parameter
$\epsilon = 0.0033$ which subsequently decays into a muon.
%
The muon decay spectrum takes both direct and 
cascade decays via charm into account.

The calculation is performed for a $b$ quark mass of 4.75 GeV with
factorisation and renormalisation scales 
%in terms of the transverse $b$ mass 
defined as 
$\mu_R = \mu_F =  
\sqrt{m_b^2 + (p_t^b)^2}$. 
%
Systematic errors are 
estimated by varying the $b$ quark mass up and
down by 0.25 GeV and  
$\mu_R$ and $\mu_F$ up and down by factors
of two.
%
These variations are performed simultaneously
and lead to  cross section changes of 
\mbox{$\sim$ 25\%.}
%
%The systematic error is estimated by simultaneous
%variation of the $b$ quark mass, 
%$\mu_R$ and $\mu_F$ 
%up and down by 0.25 GeV and factors 
%For determination of the systematic error the 
%$b$ quark mass is varied up and down by 0.25 GeV
%and simultaneously 
%The systematic error from the scale uncertainties 
%is estimated by simultaneous variation 
%of the $b$ quark mass up and down by 0.25 GeV 
%by $\pm 0.25$ GeV 
%and 
%
%down by a factor of two. 
For the structure functions the DIS-scheme parametrisations
CTEQ5D \cite{cteq5l} for the proton and GRV-G HO \cite{grvg} for the photon
are taken.
The cross section variation when using
other proton structure functions such as MRSG or MRST1 \cite{mrs}
is less than 8\% in all regions of the measurement.
The uncertainty due to variations of the fragmentation parameter 
$\epsilon$ by $25\%$ is below 3\%.

The obtained parton level cross sections are corrected to the hadron level
using the PYTHIA Monte Carlo generator. The corrections are 
smaller than $20\%$ in all bins of the measurement except 
for the region $0.5<$\xgobs$<0.75$ where the correction 
is $\sim 40\%$. If the Monte Carlo generator CASCADE is used 
the calculated cross sections at hadron level 
are generally $15\%-20\%$ smaller than those obtained from PYTHIA.


\section{Determination of Signal and Background Components}
\label{sec:fit}
%
For each muon candidate, the impact parameter \del is calculated
in the plane transverse to the beam axis.
%
Its magnitude is given by the distance 
of closest approach of the track
to the primary event vertex.
%
Its sign is positive if the intercept of the track with the jet axis 
is downstream of the primary vertex, and negative 
otherwise. 
%
Decays of long-lived particles are signalled by positive 
impact parameters, whereas the finite track resolution yields 
a symmetric distribution centered on zero.
%
The transverse profile of the beam interaction region at HERA
has a Gaussian width of about $145 \; \mu\mbox{m}$  in the horizontal
and of about $25\; \mu\mbox{m}$ in the vertical direction.
%
The average $x$ and $y$ coordinates are determined
by the information collected from many events recorded 
within the same time intervals.
%
For each event the knowledge of the
$ep$ collision point is significantly improved
by applying a primary vertex fit to selected tracks.
%with CST information.
%
The muon track candidate under consideration is excluded from this fit.
%
% Only those tracks enter the event vertex fit 
% that have an impact parameter relative 
% to the run-averaged vertex smaller than two standard deviations.
%
An average muon impact pa\-ra\-me\-ter resolution
of 80 $\mu$m is achieved with comparable contributions
from the muon track resolution and the primary event vertex 
position uncertainty.
%
The transverse momentum \ptrel of the muon track is calculated 
relative to the momentum of the associated jet
%jet with which it is associated 
after subtraction of the muon momentum.

The two observables $\delta$ and $p_T^{rel}$
are complementary in the discrimination
of the beauty component in the data
from the background sources.
%
The fraction of beauty events in the data is determined from
a combined fit to the two-dimensional distribution of
$\delta$ and $p_T^{rel}$. 
%
The fit uses the shapes 
of the distributions
of beauty, charm and light quark events from the PYTHIA Monte
Carlo simulation.  
%
The relative weights of all three components are adjusted 
such that the likelihood is maximized.
%
The overall normalisation of the summed contributions
is adjusted to match the data.
%
% Detailed result:
% FITRES( 1) = 10002
% FITRES( 2) = 20002
% FITRES( 3) = 30002
% FITRES( 4) = 1.06E+06
% FITRES( 5) = 1
% FITRES( 6) = 2
% FITRES( 7) = 0
% FITRES( 8) = 30.7299
% FITRES( 9) = 60.5673
% FITRES(10) = 8.70264
% FITRES(11) = 2.45345
% FITRES(12) = 4.5079
% FITRES(13) = 4.01832
% FITRES(14) = 382.086
% FITRES(15) = 331
%
The fit yields a sample composition of 
$f_b = (30.7 \pm 2.5)\,\%$ (beauty),
$f_c = (60.6 \pm 4.5)\,\%$ (charm) and
$f_{uds} = (8.7 \pm 4.0)\,\%$ (uds). Here the errors refer to the statistical
uncertainties.
%
The quality of the description of the data sample
using the fractions obtained with the two-dimensional 
fit is demonstrated using the one-dimensional \del and \ptrel projections.
%

Figure~\ref{fig:signal-delta} shows the measured 
impact parameter distribution in the data together 
with histograms indicating the contributions from 
$b$ production and from the $c$ and $uds$ backgrounds using the relative
fractions obtained in the two-dimensional fit.
%
% To get to the following numbers 
% do the following
% run mfinal and stop after delta fit and then do:
% fitd3 10000 20000 30000 60000 1 2 0 0 0 
% FITRES( 1) = 10000
% FITRES( 2) = 20000
% FITRES( 3) = 30000
% FITRES( 4) = 1.06E+06
% FITRES( 5) = 1
% FITRES( 6) = 2
% FITRES( 7) = 0
% FITRES( 8) = 27.9647
% FITRES( 9) = 51.8947
% FITRES(10) = 20.1435
% FITRES(11) = 4.17161
% FITRES(12) = 9.5968
% FITRES(13) = 8.14872
% FITRES(14) = 47.5061
% FITRES(15) = 39
%
The data are well described by the sum of the 
estimated contributions.
%
As a cross check a free fit to the \del distribution alone
yields a sample composition of
$f_b = (28.0 \pm 4.2)\%$,
$f_c = (51.9 \pm 9.6)\%$
and $f_{uds} = (20.1 \pm 8.1)\%$, 
in good agreement with the above fit results.
%

In figure~\ref{fig:signal-ptrel} the observed 
$p_t^{rel}$ distribution is shown.
The histogram represents the summed contributions from 
$b$ production and from the backgrounds, using
the fractions determined above.
%
The data distribution is reasonably well described by the sum of the
estimated contributions.
%
As a cross check a free fit is performed 
to the \ptrel distribution alone. 
%
The shapes of the \ptrel distributions
for charm and light quark events are very similar 
%and
%consequently the relative contribution of these two 
%components cannot be precisely determined from this fit to the \ptrel 
%distribution alone.
%
%Hence the charm and light quark components 
and hence these two contributions are combined
using the prediction from the PYTHIA Monte Carlo simulation.
%
The fit yields a sample composition of
%  
%  h/file 1 plotsfinal/hist-30622-110103-1230-b1690-7-2comp-lsq2.hbook  
%  fitd3 10001 20001 30001 0 1 2 0 0 0 0 
% FITRES( 1) = 10001
% FITRES( 2) = 20001
% FITRES( 3) = 30001
% FITRES( 4) = 0
% FITRES( 5) = 1
% FITRES( 6) = 2
% FITRES( 7) = 0
% FITRES( 8) = 28.8383
% FITRES( 9) = 71.1611
% FITRES(10) = 0
% FITRES(11) = 2.78049
% FITRES(12) = 3.22386
% FITRES(13) = 0
% FITRES(14) = 28.127
% FITRES(15) = 24 
%
$f_b = (28.8 \pm 2.8)\%$,
$f_c + f_{uds} = (71.2 \pm 3.2)\%$,
%
also in good agreement with the above fit results.
%
% chisquare is 42/24
%
%The fit gives a good description of the data.
%
% HIGH PURITY STUFF, TAKEN OUT
%
%At \ptrel $> 1.5$ GeV the beauty component is the dominant component.
%
%Figure~\ref{fig:high-purity} shows the observed 
%distributions of the impact parameter $\delta$ and
%the relative transverse muon momentum $p_t^{rel}$ 
%with cuts of $p_t^{rel}>1.4$ GeV and $\delta>0.03$ cm, respectively,
%thus enhancing the fraction of $b$ events.
%
%To further elucidate the consistency of the two observables
%$\delta$ and $p_T^{rel}$ one can enrich the beauty component
%in the events by restricting the range of one variable and then studying
%the distribution of the other quantity.
%
%Figure~\ref{fig:high-purity-delta} 
%shows the observed $\delta$ spectrum obtained
%after a cut $p_T^{rel} >1.2\; $GeV.
%The different contributions,
%shown in Figure~\ref{fig:high-purity-delta},
%are the absolute predictions from the $\delta$ fit
%for the limited $p_T^{rel}$ region.
%
%The observed impact parameter spectrum and the fit prediction,
%with a dominating beauty component, agree within the errors.
%
%Figure~\ref{fig:high-purity-ptrel} 
%shows the $p_T^{rel}$ spectrum after a
%cut on $\delta > 300\;\mu$m.
%
%The spectrum
%also agrees within the statistical
%errors with the fit prediction.
%
%The distributions of high $b$ purity 
%add to the confidence that the data are well understood.

%\section{Comparison of Data with Monte Carlo Simulations}
%\label{sec:control}

Figure~\ref{fig:control} shows the distributions of the 
muon transverse momentum \ptmu, the
pseudorapidity \etamu, the transverse momentum
$p_t^{jet_{1(2)}}$ for the highest  and second-highest-$p_t$
jet and of the observable \xgobs obtained from data
together with the expectation of the PYTHIA Monte Carlo simulation
using the fractions of beauty, charm and light quarks obtained 
from the two-dimensional fit.
%
The overall normalisation of the number of events in the 
Monte Carlo simulation is adjusted to the data. 
%
The data are adequately described by the simulations.
%
%\section{Systematic Uncertainties}
%\label{sec:errors}
%% The sources of systematic errors and the size of the resulting uncertainties are
%% given in table \ref{tab:syserr}. 
%% \begin{table}[htb]
%% \begin{center}
%% \begin{tabular}{lr}
%% \hline
%% Track Reconstruction and CST Hit Linking Efficiency &  5\% \\
%% Muon Identification Efficiency &  10\% \\
%% Track Resolutions &  10\% \\
%% Jet Reconstruction &  7\% \\
%% Trigger & 5\% \\
%% Luminosity & 2\% \\
%% MC statistics & 5\% \\ 
%% % Fit Procedure (2-Component Fit) &  5\% \\
%% Parton Evolution (PYTHIA / CASCADE) & 7\% \\
%% Fragmentation (Lund / Peterson) & 7\% \\
%% $K,\pi$-inflight decays ($\pm$ Factor 2) & 2\% \\
%% \hline
%% \hline
%% Total & $\sim$ 21\% \\
%% 
%% \end{tabular}
%% \end{center}
%% \caption{List of systematic uncertainties}
%% \label{tab:syserr}
%% \end{table}
%
%-------------------------------------------------------

\section{Results}
\label{sec:results}

The cross section measurements reported here are obtained from the
likelihood fit to the two-dimensional distribution of \ptrel and $\delta$.
%
The number of beauty events in the data, as estimated from the fit,
is translated into a cross section by dividing by the 
detector acceptance, the efficiency and the integrated luminosity.
%
The detector acceptances and efficiencies are determined 
from the PYTHIA Monte Carlo simulation.  
%

The dijet-muon beauty production cross section, 
$\sigma_{vis}(ep \rightarrow e b\bar{b} X \rightarrow e jj\mu X)$, is measured
in the visible range 
$Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$. 
The cross section is defined for jets which include 
all final state particles. 
The muon must be associated with 
one of the two jets.
The visible cross section is measured as
$$
 \sigma_{vis}(ep \rightarrow e b\bar{b} 
 \rightarrow e jj\mu X) = (42.5 \pm 3.4 (stat.) \pm 8.9 (sys.)) {\rm pb}.
$$

The dominant sources of the systematic uncertainty are
the muon identification efficiency and
the modeling of the tracking and vertexing
resolutions. 
%
%comparing the data with 
%the PYTHIA and 
Dependence on the
physics model is studied by
using for the modeling of beauty and charm events   
alternatively the CASCADE Monte Carlo simulation 
and by using either Peterson \cite{peterson} 
or Lund \cite{jetset} fragmentation.
% both programs. 
%for the charm and beauty components.
%Each of the above sources contributes approximately 10\% to the
%systematic error, which, together with uncertainties from
%other sources, such as trigger efficiencies and 
%jet reconstruction, is estimated to amount to 22\% in total.
%

In comparison the prediction from the NLO QCD calculation including
fragmentation and hadronisation corrections as described in section 
\ref{sec:nlo} is $(24.1 ^{+7.2}_{-5.1}) {\rm pb}$.
Taking the errors of both
measurement and theory into account the calculation is below the
measurement by 1.5 $\sigma$.

The differential cross sections are measured
as functions of the pseudorapidity 
\etamu\ (figures~\ref{fig:xsec-eta} and \ref{fig:xsec-eta-nlo}), 
the transverse momentum of the muon \ptmu\ 
(figures~\ref{fig:xsec-ptmu} and \ref{fig:xsec-ptmu-nlo}) 
and of \xgobs\ (figures~\ref{fig:xsec-xg} and \ref{fig:xsec-xg-nlo}).
%
The cross section values are determined 
separately for each bin, using the beauty 
fraction from the fit to the 
two-dimensional distribution of \ptrel and \del
in that bin.
%
The cross section is obtained by dividing 
the number of beauty events from the fit 
by the detector acceptance and efficiency,
the integrated luminosity and the width of the bin.
%
%correcting 
%for detector efficiencies and acceptances.

The data are compared with expectations obtained from 
the PYTHIA and CASCADE generators and 
the prediction from the QCD NLO calculation. 
%
For PYTHIA the contribution from resolved events is displayed separately
in the figures.
%
While both generators, PYTHIA and CASCADE, give a good
description of the shapes of the \etamu and \xgobs distributions
observed in the data, 
their predictions lie significantly too low. 
%
The disagreement in normalization appears to decrease towards larger
values of \ptmu\  (figure~\ref{fig:xsec-ptmu}).
%
The comparison of the data with the QCD NLO prediction shows that the
\etamu\  and \xgobs\ distributions are well described in shape.
In contrast the \ptmu dependence of the prediction appears 
to be somewhat harder than the measured 
leading to good agreement at large \ptmu\ 
(figure~\ref{fig:xsec-ptmu-nlo}).

Using the same data sample as described above, 
including the jet requirements, an inclusive muon 
cross section $\sigma(ep \rightarrow b\bar{b}X \rightarrow \mu X)$
is determined. The same kinematic region as in previous 
analyses \cite{h1bprelim} is chosen, $Q^2<1$ GeV$^2$, $0.1<y<0.8$, 
$p_t^{\mu}>2$ and $35^{\circ}<\theta<130^{\circ}$, and
the AROMA program \cite{AROMA} is used for
the extrapolation into the unmeasured phase 
space\footnote{The extrapolation 
and hence the resulting uncertainties can be 
large, as has been discussed in \cite{Jung:2001rp,tankredi}}.
%
The result is $\sigma = (177 \pm 14 (stat.) \pm 37 (sys.))$ pb.
%
The systematic error does not include a contribution from the 
uncertainty of the extrapolation.
%
The result is good in agreement with previous measurements of the 
inclusive $b \rightarrow \mu X$ cross section.

%Figure~\ref{fig:summary} gives a summary of the measurements 
%of beauty production at HERA. 
%
In the earlier measurements
the measured cross sections were given after extrapolation
using LO QCD Monte Carlo simulations. 
%
The results showed large deviations from the NLO QCD expectation.
%
Good agreement of this analysis with the previous 
measurements is found when using the same extrapolation procedure,
as discussed in the previous paragraph.
%
More recently, the HERA beauty data
have been compared with NLO QCD predictions calculated in the 
experimentally accessible kinematic region 
as described 
%for this analysis 
in section~\ref{sec:nlo}.
%
%In figure~\ref{fig:summary} the result of this analysis is
%presented as the full point.
%
Using the latter procedure the predictions from NLO QCD 
are in significantly better agreement with the data, 
but still fall somewhat low.

\section{Conclusions}
\label{sec:conclusions}

New measurements of beauty production cross sections performed
with the H1 detector at HERA are presented.
%
The analysis uses semi-muonic decays of 
$b$ flavoured hadrons and exploits their lifetime and mass properties 
in a simultaneous fit to the 
impact parameter and relative transverse momentum distribution 
of the decay muons.
%
The total visible dijet-muon cross section, measured in the region
$Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet_{1,2}}|<2.5$, 
is a factor of 1.8 (1.5 $\sigma$) above the prediction from NLO QCD.

The cross sections are also measured differentially 
in \etamu, \ptmu, and \xgobs.
The excess above expectations from NLO QCD and also  
from the leading order parton shower Monte Carlo 
generators PYTHIA and CASCADE appears to decrease somewhat towards
larger values of \ptmu .

%
%   References for Contact Interaction paper
%
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Nucl.\ Phys.\ B {\bf 336} (1990) 18;\\
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%\cite{Frixione:1994dv}
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% Peterson
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%% 
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%% 
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%% %\cite{Ciafaloni:1987ur}
%% %%%\bibitem{Ciafaloni:1987ur}
%% %%%M.~Ciafaloni,
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%% %%%Nucl.\ Phys.\ B {\bf 296} (1988) 49.
%% %%CITATION = NUPHA,B296,49;%%
%% %\cite{Catani:1989yc}
%% %%%\bibitem{Catani:1989yc}
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%% %%\cite{Marchesini:1994wr}
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\clearpage
\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig1.eps,width=15cm}}
\end{picture}
\caption{Distribution of impact parameter \del of the muon track.
The data (points) are compared with the Monte Carlo
simulation (solid line). The decomposition of the Monte Carlo
distribution into the $b$ (dashed line), the $c$ (dotted line)
and the light quark (dash-dotted line) components
is determined from a fit to the two-dimensional  
distribution of \ptrel and \del (see text).}
\label{fig:signal-delta}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig2.eps,width=15cm}}
\end{picture}
\caption{Distribution of transverse muon momentum \ptrel relative
to the jet axis.
The data (points) are compared with the Monte Carlo
simulation (solid line). The decomposition of the Monte Carlo
distribution into the $b$ (dashed line), the $c$ (dotted line)
and the light quark (dash-dotted line) components
is determined from a fit to the two-dimensional  
distribution of \ptrel and \del (see text).}
\label{fig:signal-ptrel}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,20.)
\put(-1.3,0.){\epsfig{file=H1prelim-03-072.fig3.eps,width=18cm}}
\end{picture}
\caption{
Distribution of the muon transverse momentum \ptmu,
pseudorapidity \etamu, the transverse momentum
$p_t^{jet_{1(2)}}$ for the highest and second-highest-$p_t$
jets, and the observable \xgobs .
%
The data are compared to the PYTHIA Monte Carlo simulation.
%
The estimated contributions of beauty, charm and light
quark events, taken from the fit result, 
are shown as separate curves.}
%
%Separately the estimated 
%The sum of the Monte Carlo contributions is shown
%and separately the 
%estimated beauty, charm and light
%quark components.}
% from the PYTHIA Monte Carlo.
%
\label{fig:control}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig4.eps,width=16cm}}
\end{picture}
\caption{Differential dijet muon beauty production cross section 
$d\sigma/d\eta_{\mu}(ep \rightarrow eb\bar{b}X \rightarrow ejj\mu X)$ as a function
of \etamu\ in 
the range $Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$.
The inner error bars show the statistical error, the outer error bars represent
the statistical and systematic uncertainty added in quadrature.
Also shown are the predictions from the Monte Carlo generator programs PYTHIA
(solid line), the resolved contribution to the PYTHIA cross section
(dashed-dotted line) and 
CASCADE (dashed line).}
\label{fig:xsec-eta}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig5.eps,width=16cm}}
\end{picture}
\caption{Differential dijet muon beauty production cross section 
$d\sigma/d\eta_{\mu}(ep \rightarrow eb\bar{b}X \rightarrow ejj\mu X)$ as a function
of \etamu\ in 
the range $Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$.
The inner error bars show the statistical error, the outer error bars comprise
the statistical and systematic uncertainty added in quadrature. 
Also shown is the prediction from a pQCD NLO calculation \cite{Frixione:1994dv}
at parton level (dashed line) and hadron level (solid line).
The band shows the uncertainty obtained from a
simultaneous variation of the $b$ quark mass, $\mu_r$ and $\mu_f$
(see text).}
\label{fig:xsec-eta-nlo}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig6.eps,width=16cm}}
\end{picture}
\caption{Differential dijet muon beauty production cross section 
$d\sigma/dp_t^{\mu}(ep \rightarrow eb\bar{b}X \rightarrow ejj\mu X)$ as a function
of \ptmu\ in 
the range $Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$.
The inner error bars show the statistical error, the outer error bars comprise
the statistical and systematic uncertainty added in quadrature.
Also shown are the predictions from the Monte Carlo 
generator programs PYTHIA (solid line), the resolved contribution to the
PYTHIA cross section
(dashed-dotted line) and 
CASCADE (dashed line).}
\label{fig:xsec-ptmu}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig7.eps,width=16cm}}
\end{picture}
\caption{Differential dijet muon beauty production cross section 
$d\sigma/dp_t^{\mu}(ep \rightarrow eb\bar{b}X \rightarrow ejj\mu X)$ as a function
of \ptmu\ in 
the range $Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$.
The inner error bars show the statistical error, the outer error bars comprise
the statistical and systematic uncertainty added in quadrature.
Also shown is the prediction from a pQCD NLO calculation \cite{Frixione:1994dv}
at parton level (dashed line) and hadron level (solid line).
The band shows the uncertainty obtained from a
simultaneous variation of the $b$ quark mass, $\mu_r$ and $\mu_f$
(see text).}
\label{fig:xsec-ptmu-nlo}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig8.eps,width=16cm}}
\end{picture}
\caption{Differential dijet muon beauty production cross section 
$d\sigma/d\xgobsm(ep \rightarrow eb\bar{b}X \rightarrow ejj\mu X)$ as a function
of \xgobs\ in 
the range $Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$.
The inner error bars show the statistical error, the outer error bars comprise
the statistical and systematic uncertainty added in quadrature.
Also shown are the predictions from the Monte Carlo generator programs PYTHIA
(solid line), the resolved contribution to the PYTHIA cross section
(dashed-dotted line) and 
CASCADE (dashed line).}
\label{fig:xsec-xg}
\end{figure} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\setlength{\unitlength}{1cm} 
\begin{picture}(14.0,15.)
\put(0.,0.){\epsfig{file=H1prelim-03-072.fig9.eps,width=16cm}}
\end{picture}
\caption{Differential dijet muon beauty production cross section 
$d\sigma/d\xgobsm(ep \rightarrow eb\bar{b}X \rightarrow ejj\mu X)$ as a function
of \xgobs\ in 
the range $Q^2<1$ GeV$^2$, $0.2<y<0.8$, \ptmu $>2.5$ GeV, $-0.55<\eta^{\mu}<1.1$,
$p_t^{jet_{1(2)}} > 7(6)$ GeV and $|\eta^{jet}|<2.5$.
The inner error bars show the statistical error, the outer error bars comprise
the statistical and systematic uncertainty added in quadrature.
Also shown is the prediction from a pQCD NLO calculation \cite{Frixione:1994dv}
at parton level (dashed line) and hadron level (solid line).
The band shows the uncertainty obtained from a
simultaneous variation of the $b$ quark mass, $\mu_r$ and $\mu_f$
(see text).}
\label{fig:xsec-xg-nlo}
\end{figure} 

%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \begin{figure}
%% \setlength{\unitlength}{1cm} 
%% \begin{picture}(14.0,16.)
%% \put(-2.,-1.){\epsfig{file=summary2.eps,width=19cm}}
%% \end{picture}
%% \caption{Summary of beauty production cross section measurements
%% at HERA in comparison with predictions from
%% NLO QCD. The full point shows the result of this analysis in 
%% comparison with other results
%% \cite{Adloff:1999nr,h1bprelim,Breitweg:2000nz,zeusbprelim}.
%% Note that the different measurements 
%% are defined for different regions of phase space.}
%% \label{fig:summary}
%% \end{figure} 
%% 

\end{document}