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\noindent
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%{\it {\large version of \today}} \\[.3em] 
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%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
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Submitted to & & &
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\multicolumn{4}{l}{{\bf
                32nd International Conference 
                on High Energy Physics, ICHEP04},
                August~16,~2004,~Beijing} \\
                 & Abstract:        & {\bf 5-0167}    &\\
                 & Parallel Session & {\bf 5}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
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\begin{center}
\begin{Large}
  
  {\boldmath {\bf Measurement of $F_2^{c\bar{c}}$ and $F_2^{b\bar{b}}$
      at High $Q^2$ using the H1 Vertex Detector at HERA}}

\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{2cm}

\begin{abstract}
  \noindent
  Measurements are presented of inclusive charm and beauty cross sections in
  $e^+p$ collisions for values of $Q^2 > 110~{\rm GeV}^2$ 
  and $0.05 < y < 0.7$ , using a
  method based on the distance of closest approach, in the 
  transverse plane, of tracks to the 
  primary vertex as measured by the H1 vertex detector.  The data are
  divided into four intervals in $Q^2$ and Bjorken $x$ and values for the
  structure functions $F_2^{c\bar{c}}$ and $F_2^{b\bar{b}}$ are
  obtained.  The results are found to be compatible with the
  predictions of perturbative quantum chromodynamics.  
\end{abstract}

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\section{Introduction}
Heavy quark production is an important process to study quantum
chromodynamics (QCD). It is expected that perturbative QCD (pQCD) at 
next-to-leading order (NLO) should give a good description of heavy flavour
production in deep-inelastic scattering (DIS), especially at values of
the negative square of the four momentum of the exchanged boson $Q^2$
greater than the square of the heavy quark masses.  Measurements of the
open charm ($c$) cross section in DIS at HERA have mainly been of exclusive
$D$ or $D^*$ meson production\cite{H1ZEUSDstar,Chekanov:2003rb}.
The contribution of charm to the proton structure function,
$F_2^{c\bar{c}}$, is derived by correcting for the fragmentation 
fraction $f(c \rightarrow D)$ and the unmeasured phase space 
(mainly at low values of transverse
momentum of the meson). The results have been found to be in good agreement 
with QCD predictions.  The measurement of the beauty ($b$) cross section is
particularly challenging since $b$ events comprise only a small
fraction (typically $< 5\%$) of the total cross section. The $b$ cross section has been
measured in DIS ($Q^2 > 2~{\rm GeV^2}$) by ZEUS\cite{zeusBdis} and in
photoproduction ($Q^2 \simeq 0~{\rm GeV^2}$) by H1\cite{Adloff:1999nr} and
ZEUS\cite{zeusBgammap}, using the transverse momentum distribution of
muons relative to the $b$ jet in semi-muonic decays.
Measurements of the $b$ cross section have also been made in $p
\bar{p}$\cite{hadronb} and $\gamma \gamma$ collisions\cite{ggb}.

The analysis presented in this paper is of inclusive $c$ and
$b$ cross sections in $e^+p$ scattering at HERA in the 
range $Q^2 > 110~{\rm GeV}^2$.
Events containing heavy quarks can be distinguished from light quark
events by the long lifetimes of $c$ and $b$ hadrons,
which lead to displacements of tracks from the primary vertex. 
The contribution of light quark events with 
long lived decays may be reduced by
restricting the size of the maximum allowed displacements. 
The distance of a track to the primary vertex is reconstructed using 
precise spatial information from the H1 vertex 
detector. The
lifetimes of the heavy mesons and the quark fragmentation functions
are taken from measurements made by other
experiments\cite{Hagiwara:fs}. The results presented in this paper are
made in kinematic regions where 
there is little extrapolation needed to correct to the
full phase space.
The charm structure function $F_2^{c\bar{c}}$ and the corresponding
structure function for $b$ quarks $F_2^{b\bar{b}}$ are obtained after
small corrections for the longitudinal structure functions
$F_L^{c\bar{c}}$ and $F_L^{b\bar{b}}$ (taken from NLO predictions).
This is the first measurement of $F_2^{b\bar{b}}$ and an   
extension to high $Q^2$ of previous H1 $F_2^{c\bar{c}}$ measurements.

\section{Heavy Flavour Production}
\label{sec:theory}
In pQCD, in the region where $Q^2$ is much larger than the squared
mass $M^2$ of the heavy flavour quark, the production of heavy flavour
quarks is expected to be insensitive to threshold effects and the
quarks may be treated as massless partons.
At leading order (LO), in the `massless' scheme, the quark parton
model (QPM) process ($\gamma q \rightarrow q$) is the dominant
contribution. At NLO, the photon gluon fusion ($\gamma g \rightarrow
q\bar{q}$) and QCD Compton ($\gamma q \rightarrow qg$) processes also
contribute.
The approach is often referred to as the zero mass variable flavour
number scheme (ZM-VFNS)\cite{cteq4,Martin:1994kn}.


At values of $Q^2 \sim M^2$,
the `massive' scheme\cite{massive}, in which the heavy flavour partons
are treated as massive quarks is more appropriate.  The heavy quarks
are produced perturbatively with their mass 
providing the hard scale.  The dominant LO process is photon
gluon fusion (PGF) and the NLO diagrams are of order $\alpha_s^2$.
The scheme is often referred to as the fixed flavour number scheme
(FFNS).  As $Q^2$ becomes large compared to $M^2$, the FFNS approach is
unreliable due to large logarithms in $\ln (Q^2/M^2)$, which lead to a
divergence in the perturbative series.

In order to provide reliable pQCD predictions for the description of
heavy flavour production, over the whole range in $Q^2$, composite
schemes which provide a smooth transition from the massive description
at $Q^2 \sim M^2$ to the massless behaviour at $Q^2 \gg M^2$ have been
developed\cite{VFNS1,VFNS2}.
The scheme is commonly referred to as the variable flavour number
scheme (VFNS).
The approach has been incorporated in various different forms to order
$\alpha_s$\cite{VFNS1}
and to order $\alpha_s^2$\cite{VFNS2}.


\section{H1 Detector}
Only a short description of the H1 detector is given here; a full
description may be found in\cite{Abt:1997xv}. A right handed
coordinate system is employed at H1 that has its $z$-axis pointing in
the proton beam, or forward, direction and $x$($y$) pointing in
the horizontal(vertical) direction.


Charged particles are measured in the central tracking detector (CTD).
This device consists of two cylindrical drift chambers interspersed with
$z$-chambers to improve the $z$-coordinate reconstruction and
multi--wire proportional chambers mainly used for triggering. The CTD
is situated in a uniform $1.15\,{\rm T}$ magnetic field, enabling
momentum measurement of charged particles over the pseudo-rapidity
range $-1.74< \eta<1.74$~\footnote{\noindent{ The pseudo-rapidity ($\eta=- \ln
  \tan \theta/2$) coverage of each detector component is given for the
  vertex in its nominal position.}}.  


The CTD tracks are linked to hits in the vertex detector (central
silicon tracker CST)\cite{cst}, to provide precise spatial track
reconstruction. The CST consists of two cylindrical layers of
double-sided silicon strip detectors surrounding the beam pipe,
covering a pseudo-rapidity range of $-1.32< \eta<1.32$. This detector
provides hit resolutions of $12 \mu$m in $r$--$\phi$ and 
$25 \mu$m in $z$. 
For CTD tracks with CST hits in both layers the transverse distance of closest
approach to the nominal vertex in $x$--$y$ can be measured with a resolution of
$ 33\;\mu\mbox{m} \oplus 90 \;\mu\mbox{m} /p_T
[\mbox{GeV}]$, where the first term represents the intrinsic
resolution (including alignment uncertainty) and the second term is
the contribution from multiple scattering in the beam pipe
and the CST; $p_T$ is
the transverse momentum of the track.
In this analysis, the measurement of the $z$ coordinate by the
CST is not used for determining the distance of closest approach
of a track. 
%

The track detectors are surrounded in the forward and central directions
($-1.5<\eta<3.4$) by a fine grained liquid argon calorimeter (LAr) and
in the backward region ($-4.0<\eta<-1.4$) by a
lead--scintillating fibre calorimeter\cite{Nicholls:1996di} with
electromagnetic and hadronic sections. These calorimeters provide
energy and angular reconstruction for final state particles
from the hadronic system. The LAr
is also used in this analysis to measure and identify the scattered
positron.

An electromagnetic calorimeter situated downstream in the positron
beam direction measures photons from the bremsstrahlung
process $ep\rightarrow ep\gamma$ for the purpose of luminosity
determination.



\section{Experimental Method}
The analysis is based on a high $Q^2$ sample of neutral current (NC)
events, taken in the years 1999-2000, when HERA was operated in
unpolarised $e^+p$ mode, with an $ep$ centre of mass
energy squared $s = 101200~{\rm GeV^2}$.  The events are selected in a
similar manner
to that described in\cite{H19900NCCC}. Additional requirements are made on the
performance of the CTD, yielding an integrated luminosity of
57.4 ${\rm pb}^{-1}$. 
The positron is identified and
measured in the LAr calorimeter, which restricts the measurement to
$Q^2>110~{\rm GeV}^2$.  The event kinematics $Q^2$ and the
inelasticity variable $y$ are reconstructed using 
the scattered positron.  The Bjorken scaling variable $x$
is obtained from $x = Q^2/sy$. After the inclusive selection 
the total number of events is around 121,000.

\subsection{Monte Carlo Simulation}
The data are corrected for the effects of detector resolution,
acceptance and efficiency by the use of Monte Carlo simulations.
The Monte Carlo program RAPGAP~\cite{Jung:1993gf} is used to generate
high $Q^2$ NC DIS events for the processes 
$ep \rightarrow eb\bar{b}X$, $ep \rightarrow ec\bar{c}X$ and
light quark production.
RAPGAP combines  $\cal{O}$($\alpha_s$) matrix
elements with higher order QCD effects modelled by the emission of
parton showers. The heavy flavour event samples are generated
according to the massive PGF matrix element with the mass of the $c$
and $b$ quarks set to $m_c=1.5 \ {\rm GeV}$ and $m_b=5.0 \ {\rm GeV}$,
respectively.  The partonic system is fragmented according to the LUND
string model implemented within the JETSET
program~\cite{Sjostrand:1993yb}.  The HERACLES
program~\cite{Kwiatkowski:1990es} is used to calculate single photon
radiative emissions off the lepton line and virtual electroweak
corrections.
% rg doesn't have z-exchange
In the event generation, the DIS cross section is calculated with a LO
parton density function (PDF) in the DIS scheme~\cite{Martin:1994kn}.
In order to improve the description of the data by the simulation, the
simulated cross sections are reweighted in $x$ and $Q^2$ using a NLO QCD 
fit (H1 PDF2000) to the H1 data~\cite{H19900NCCC}.

% define signa/backgroundl (?)
The samples of events generated for the $uds$, $c$, and $b$ 
processes are passed through a detailed simulation of the detector 
response based on the GEANT3
program~\cite{Brun:1978fy}, and through the same reconstruction
software as is used for the data.

% NC backgrounds gamma-p, lpair, QED-C, prompt photon

 
\subsection{Track, Vertex and Jet Reconstruction}
\label{sec:trackjetvertex}
The analysis is based on CTD tracks which are linked to 
$r$--$\phi$ hits in both
planes of the CST in order to improve the precision of the track
parameters.  In this paper, the CST-improved CTD tracks are referred
to as `CST tracks'.  Only those events which have at least 1
reconstructed CST track with polar angle $30<\theta_{\rm
  track}<150^\circ$ and a minimum transverse momentum of $0.5 \ {\rm
  GeV}$ are used.  The reconstructed $z$ position of the interaction
vertex must be within $\pm 20~{\rm cm}$ of the centre of the detector to
match the acceptance of the CST. At low values of $y$, the hadronic
final state (HFS) tends to go forward and outside the acceptance of
the CST.  Therefore, the analysis is restricted to $0.07<y<0.7$.  The
upper $y$ cut ensures a good trigger acceptance for the scattered
positron. In this
kinematic range, 
studies from Monte Carlo simulations show that
$93\%$ of $c$ events and $96\%$ of
$b$ events are expected to have at least one charged particle, 
with a $p_T>0.5 \ {\rm GeV}$ in the angular range $30<\theta <150^\circ$,
produced from the decay of a heavy hadron. 
The restriction to this kinematic range thus ensures that the
extrapolation to the full phase space, needed to calculate
$F_2^{c\bar{c}}$ and $F_2^{b\bar{b}}$, is small.

The  efficiency for a single charged particle with $p_T>0.5$~GeV
to produce a CST track  varies between $60\%$ and $75\%$
depending on the $z$ position of the track in the CST.  The polar angle and
transverse momentum distributions of CST tracks are compared to the
Monte Carlo simulation in figure~\ref{fig:thetatracks}. The simulation
gives a reasonable description of these distributions.

The primary event vertex in $r$--$\phi$ is 
reconstructed from all tracks (with or
without CST hits) and the position and spread of the beam interaction
region (referred to as the `beam-spot'). The beam-spot extension is
measured to be $\sim 145~{\rm \mu m}$ in $x$ and $\sim 25~{\rm \mu m}$
in $y$ for the data period considered here.  The position of the
beam-spot
is measured as the average over many events and the
resulting error on the position is small in comparison to the size of
the beam-spot, with a typical uncertainty of $\sim 5~{\rm \mu m}$.

In this analysis the transverse distance of closest approach
(DCA) of the track to the primary vertex point is used to separate
the different quark flavours (see section
\ref{quarkflavourseparation}).  The uncertainty of the measurement of
the DCA receives contributions from the position 
of the primary vertex
discussed above, the intrinsic resolution of the track and
distortions due to multiple scattering in the beam-pipe and
surrounding material.  In order to provide a successful description 
of the DCA in
the data by the Monte Carlo simulation program, the Monte Carlo
parameters for the beam-spot size, tracking resolution and inactive
material are adjusted to those observed in the data.

To identify long lived hadrons a `jet axis' is defined 
for each event 
in order to calculate a signed DCA for every track. 
Jets with a minimum $p_T$ of $5 \ {\rm GeV}$,
in the angular range $10^{\rm o} < \theta < 170^{\rm o}$, are reconstructed
using the invariant $k_T$ algorithm\cite{KTJET} run in the laboratory
frame over all reconstructed HFS particles.  HFS particles are
reconstructed using a combination of tracks and calorimeter energy
deposits\cite{Adloff:1997mi}. The jet axis is defined as the direction
of the jet with the highest transverse momentum or, if there is no jet
reconstructed in the event, as the direction of the struck quark in the
quark parton model~\cite{Ahmed:1992sk} as 
reconstructed from the scattered electron and HFS particles.  
In the $Q^2$ range of this
paper, the vector sum of all HFS particles in the laboratory frame always 
has a transverse momentum greater than $5~{\rm GeV}$ and 
$97\%$ of the events have the jet axis defined by a
reconstructed jet.

CST tracks are associated to the jet axis if they lie within a cone of
size 1 in $\eta$--$\phi$ space centred about the jet axis.  Approximately
$65\%$ of events within the kinematic range have at least one CST
track which is matched to the jet axis.  Figure~\ref{fig:jettheta}
shows the polar angle and $p_T$ distributions of the jets which
contain one or more CST tracks.  Figure~\ref{fig:csttracks} shows the
number of reconstructed CST tracks assigned to the jet axis. The
simulation gives a reasonable description of these distributions. The
deviations observed at high track multiplicities are most likely due to an
incomplete description of light quark multiplicities in the Monte
Carlo. This has negligible effect on the measurements.


\subsection{Quark Flavour Separation}
\label{quarkflavourseparation}

%Add lep references (aleph - MIP papers 1 and 3, OPAL is NN, 
%delphi uses ALEPH method, L3 too, SLD is SV finding- to be done by someone
%with a cut and paste ability )

The different quark flavours that contribute to the DIS cross section
are distinguished on the basis of the different lifetimes of the
produced hadrons. Due to the relatively low cross sections and modest
CST track reconstruction efficiency the decay length of the heavy
hadrons is not reconstructed directly, but the DCA of tracks is used
instead.  The chosen heavy flavour tagging method
also allows events with only 1 CST track to be used, for which it is
not possible to reconstruct a secondary vertex. The DCA is defined as
positive if the angle between the jet axis and the line joining the
vertex to the point of DCA is less than $90^\circ$, and is defined as
negative otherwise. Tracks from the decays of long lived particles
will mainly have a positive true DCA, whilst those produced at the
primary vertex will have zero true DCA. Tracks reconstructed with
negative DCA values mainly result from detector resolution.


Figure~\ref{fig:dca}(a) shows the DCA distribution of CST tracks
associated to the jet axis.  The data are seen to be asymmetric with
positive values in excess of negative values indicating the presence
of long lived particles. The simulation gives a reasonable
description of the data. The component of the simulation that arises
from light quarks is almost symmetric at low DCA.  The
asymmetry at DCA $\gapprox 0.1~{\rm cm}$ is mainly due to long
lived strange particles such as $K_s$. The $c$ component exhibits
a moderate asymmetry and the $b$ components shows a marked
asymmetry. The differences are due to the different lifetimes of the
produced hadrons.  In order to reduce the effects of the strange
component, a cut of $|$DCA$|< $0.1~${\rm cm}$ is imposed on all
tracks used in the analysis.


In order to optimise the separation of the quark flavours use is made
of the significance~\cite{Buskulic:1993ka}, defined as the ratio of
the DCA to its error. This distribution is shown for all tracks in
figure~\ref{fig:dca}(b), where a good description of the data by
the simulation is observed apart from the tails, where the data are
observed to lie above the simulation. For this reason, the
significance distributions are only fitted in the range
$-10 <$ significance $< 10$. 
 A further optimisation is made by using different
significance distributions for events with different multiplicities.
The first significance distribution $S_1$ is defined for events where
only 1 reconstructed CST track is linked to the jet, and is simply the
significance of the track.  The second significance distribution $S_2$
is defined for events with two or more tracks associated with the jet
and is the significance of the track with the second highest absolute
significance.  
% new sentence
%The second highest significance track is chosen 
%in preference to the highest, in order
%to reduce the contribution from light quark events where there is less 
%chance that two tracks are produced with large significance.
%
%Furthermore, events in which the tracks with the first and second
%highest absolute significance have different signs are removed from
%the $S_2$ distribution.  This selection removes around $50\%$ of
%events from the $S_2$ distribution, which are dominated by light
%quarks.

The $S_1$ and $S_2$ distributions are shown in figure~\ref{fig:s1s2}.
The distribution of $S_2$ gives a better separation power of light to
heavy quarks, since  for heavy quarks
$\ge 2$ tracks are usually produced with high significance,
whereas for light quarks the chances of two tracks being produced at
large significance due to resolution effects are small. 
%By requiring the two most significant tracks to have the same 
%significance sign further suppresses the background contribution.  
Events with 1
CST track are retained to improve the statistical precision of the
measurements.

In order to substantially reduce the uncertainty due to the DCA 
resolution and
the light quark normalisation the negative bins in the $S_1$ and $S_2$
distributions are subtracted from the positive. The subtracted
distributions are shown in figure~\ref{fig:s1negsub}. It can be seen
that the resulting distributions are expected to be dominated by $c$
quark events\footnote{\noindent Events that contain $c$ hadrons via decay of
  $b$ hadrons are not included in the definition of $c$ quark
  events.}, with an increasing $b$ fraction with increasing
significance. The light quarks contribute around $15\%$ or less
for all values of significance.

\subsection{Fit Procedure}
\label{sec:fit}
The fractions of $c$, $b$ and light quarks of the data, 
are extracted in each $x$--$Q^2$ interval
using a simultaneous fit to 
the subtracted $S_1$ and $S_2$ distributions 
(as in figure~\ref{fig:s1negsub}) and
the total number of inclusive events before track selection,
using the $c$, $b$ and $uds$ Monte Carlo samples as templates. 
A standard least squares fit is used.  The Monte
Carlo $c$, $b$ and $uds$ distributions in each $x$--$Q^2$ interval 
are allowed to be modified by the scale factors $P_c$, $P_b$ and $P_l$, 
respectively. The $\chi^2$ to be minimised in each interval is thus:

\begin{equation*}
  \chi^2=\sum_{\rm i} \frac{\left(N_i^{\rm data}-P_c N^{\rm MC}_{ci}-P_b N^{\rm MC}_{bi}
-P_l N^{\rm MC}_{li}\right)^2}
{\sigma^2 \left(N_i^{\rm data}\right)+\left(P_c\sigma\left(N^{\rm MC}_{ci}\right)\right)^2+\left(P_b\sigma\left(N^{\rm MC}_{bi}\right)\right)^2+\left(P_l\sigma\left(N^{\rm MC}_{li}\right)\right)^2} 
\end{equation*}
\begin{equation}
+\frac {\left(N^{\rm data}_{\rm tot}-P_c N^{\rm MC}_{ {\rm  tot} c}
-P_bN^{\rm MC}_{{\rm  tot} b} - P_l N^{\rm MC}_{{\rm  tot} l}\right)^2 }
{\sigma^2 \left(N^{\rm data}_{\rm tot}\right)+\left(P_c\sigma\left(N^{\rm MC}_{{\rm  tot} c}\right)\right)^2+\left(P_b\sigma\left(N^{\rm MC}_{{\rm  tot} b}\right)\right)^2+\left(P_l\sigma\left(N^{\rm MC}_{{\rm  tot} l}\right)\right)^2},
\end{equation}
where the sum runs over all bins with significance $<10$ in both 
the $S_1$ and $S_2$ histograms; $N_i^{\rm data}$ is the number of data 
events in
each bin with corresponding error $\sigma (N_i^{\rm data})$; $N^{\rm
  MC}_{ci}$, $N^{\rm MC}_{bi}$ and $N^{\rm MC}_{li}$ is the number of
Monte Carlo $c$, $b$ and light quark events in each bin respectively,
with corresponding errors $\sigma(N^{\rm MC}_{ci})$, $\sigma(N^{\rm
  MC}_{bi})$ and $\sigma(N^{\rm MC}_{li})$; $N^{\rm data}_{\rm tot}$
is the total number of events in each $x$--$Q^2$ interval with
corresponding error $\sigma(N^{\rm data}_{\rm tot})$; $N^{\rm
  MC}_{{\rm tot} c}$, $N^{\rm MC}_{{\rm tot} b}$ and $N^{\rm MC}_{{\rm
    tot} l}$ is the total number of $c$, $b$ and light quark Monte
Carlo events before track selection 
in each $x$--$Q^2$ interval respectively, with
corresponding statistical errors $\sigma(N^{\rm MC}_{{\rm tot} c})$,
$\sigma(N^{\rm MC}_{{\rm tot} b})$ and $\sigma(N^{\rm MC}_{{\rm tot}
  l})$.  The first term of the $\chi^2$ gives information on the
$P_c$ and $P_b$ fractions from the difference in shape of the
significance distributions, whilst the second term constrains
the overall normalisation and, therefore, determines $P_l$.


The results of the fit to the complete data sample are shown in
figure~\ref{fig:s1negsub}.
% stability
The fits give acceptable $\chi^2$ values for all $x$--$Q^2$ intervals
and for the total.  Consistent results, all within two standard
deviations of the statistical error,  are found when fitting
different significance distributions, for example fitting the $S_1$ or
$S_2$ distributions alone; fitting the highest absolute significance
track distribution for all events; and fitting the distribution for
the track with the third highest absolute significance. Consistent
results are also obtained fitting the significance distributions
without subtraction of the negative bins from the positive, 
and also when varying the range of significance to be fitted
within the range $<8$ to $<13$. 

The results are converted to a measurement of the differential $c$
cross section using:
\begin{equation}
\frac{{\rm d}\sigma^{c\bar{c}}}{{\rm d} x{\rm d} Q^2} = 
\frac{{\rm d}\sigma}{{\rm d} x{\rm d} Q^2} \frac{P_c N^{\rm MC gen}_c}{P_c N^{\rm MC gen}_c+P_b N^{\rm MC gen}_b+P_l N^{\rm MC gen}_l},
\end{equation}
where ${\rm d} \sigma / {\rm d} x {\rm d} Q^2$ is the measured
inclusive differential cross section from H1~\cite{H19900NCCC} and
$N^{\rm MC gen}_c$, $N^{\rm MC gen}_b$ and $N^{\rm MC gen}_l$ are the
generated number of $c$, $b$ and light quark events from the Monte
Carlo in each bin, respectively. 
A small ( $\le 5\%$ ) 
bin centre correction  is applied using the NLO QCD expectation 
(see section~\ref{results}) to
convert the bin averaged measurement into a measurement at a single
$x$--$Q^2$ point. The cross section is defined so as to include a
correction for pure QED initial and final state radiative effects, but
not electroweak corrections (see \cite{H19900NCCC} for a more complete
discussion).


The structure function $F_2^{c\bar{c}}$ is then evaluated
from the expression

\begin{equation}
\frac{{\rm d}\sigma^{c\bar{c}} }{{\rm d} x{\rm d} Q^2} = \frac {2 \pi \alpha^2}{x Q^4 }  ((1+ (1-y)^2) F_2^{c\bar{c}}   - y^2  F_L^{c\bar{c}}),
\end{equation}
where  the longitudinal structure function $F_L^{c\bar{c}}$ is
estimated from the NLO QCD expectation\cite{H19900NCCC}.
The differential $b$ cross section
and $F_2^{b\bar{b}}$ are evaluated in the same manner.
The maximum contribution of the longitudinal structure function
is $2.3\%$ of $F_2^{c\bar{c}}$ and $4.6\%$ of $F_2^{b\bar{b}}$.
It is also convenient to express the cross section as a `reduced cross
section' defined as
\begin{equation}
\tilde{\sigma}^{c\bar{c}} = \frac{{\rm d}\sigma^{c\bar{c}} }{{\rm d} x{\rm d} Q^2}  \frac {x Q^4 } {2 \pi \alpha^2 (1+ (1-y)^2)}.
\end{equation}



\subsection{Systematic Errors}
\label{systematics}
The sources of systematic uncertainty considered are as follows:
\begin{itemize}
\item A track efficiency uncertainty of $\pm 3.6\%$.
\item An uncertainty in the DCA resolution of the tracks is estimated by
  varying the resolution in the Monte Carlo Simulation 
  by an amount that encompasses the differences 
  between the data and simulation.
  An additional Gaussian smearing of $200$~$\mu{\rm m}$ to
  $5\%$ of the tracks and $25$~$\mu{\rm m}$ to the rest
  is introduced.
\item A $4\%$ uncertainty on the hadronic energy scale.
\item An error on the jet axis is estimated by introducing an
  additional Gaussian smearing of 2$^\circ$.
\item The uncertainty on the asymmetry of the light quark DCA is
  estimated by repeating the fits with the light quark $S_1$ and $S_2$
  distributions (figure~\ref{fig:s1negsub}) set to zero.
\item The uncertainties on the various $D$ and $B$ meson lifetimes,
  decay branching fractions and mean charge multiplicities are
  estimated by varying the input values of the Monte Carlo
  simulation  by the errors on the
  world average measurements, or by adjusting the simulation to the world
  average value depending on which variation is larger. For the
  branching fractions of quarks to mesons and the lifetimes of the
  mesons the central values and errors on the world averages are taken
  from\cite{Hagiwara:fs}.  For the mean charged track multiplicities the
  values and uncertainties for $c$ and $b$ quarks are taken from
  MarkIII~\cite{Coffman:1991ud} and LEP/SLD~\cite{lepjetmulti}
  measurements, respectively.
\item An uncertainty on the fragmentation function of the heavy quarks
  used in the Monte Carlo is estimated by repeating the fits with the
  Monte Carlo templates generated using the Peterson fragmentation
  function\cite{peterson} with parameters
  $\epsilon_c = 0.058$ and $\epsilon_b = 0.0069$. 
\item An uncertainty on the bin centre correction leading to a
$5\%$ error on the measured cross section.
\item An uncertainty in the QCD model of heavy quark production
  used in the Monte Carlo is
  estimated by using RAPGAP in a mode where heavy quarks are generated
  with a $1:1$ ratio of QPM to PGF induced events.
\item Other sources of systematic error pertaining to the NC selection
  were also considered\cite{H19900NCCC}: a $1.5\%$ uncertainty on the
  luminosity measurement; an uncertainty on the scattered positron
  polar angle of $1$--$3$~${\rm mrad}$ and energy of $0.7$--$3.0\%$
  depending on the polar angle; a $0.5\%$ uncertainty on the scattered
  positron identification efficiency; a $0.5\%$ uncertainty on the
  positron track-cluster link efficiency; a $0.3\%$ uncertainty on the
  trigger efficiency
%; a $X\%$ uncertainty on the hadronic energy scale
  and a $1\%$ uncertainty on the cross section
  evaluation due to QED radiative corrections.

\end{itemize}

%The uncertainties that contribute most to the total systematic error
%on $\tilde{\sigma}^{c\bar{c}}$ are the track resolution, the 
%light quark DCA asymmetry and the bin centre correction,
% leading to errors on
%the cross section of $9\%$, $9\%$ and $5\%$, respectively. Those that
%contribute most to the total systematic error on
%$\tilde{\sigma}^{b\bar{b}}$ are the track resolution, the track
%efficiency, and the QCD model leading to errors on the cross section
%of $14\%$, $11\%$ and $8\%$, respectively.
The total systematic error is obtained by adding
all individual contributions in quadrature and is around $15\%$
for $\tilde{\sigma}^{c\bar{c}}$ and $25\%$ for $\tilde{\sigma}^{b\bar{b}}$.




\subsection{Results}
\label{results}
The measurements of $F_2^{c\bar{c}}$ and $F_2^{b\bar{b}}$ are shown in
figure~\ref{fig:f2c} as a function of $x$ for two
values of $Q^2$. 
The H1 data for $F_2^{c\bar{c}}$ are compared with the
results of the ZEUS collaboration~\cite{Chekanov:2003rb}, 
where the cross sections were
obtained from the measurement of $D^{*\pm}$ mesons. 
The results of the two measurements for $F_2^{c\bar{c}}$ are in good
agreement.

The data are also compared with the prediction from 
the H1 PDF 2000 fit\cite{H19900NCCC} in which the $c$ and
$b$ quarks are treated in the ZM-VFNS scheme.
The QCD prediction is compatible with the data.
There is no evidence for an excess of
the $b$ cross section compared with QCD predictions as has been
reported in other analyses~\cite{Adloff:1999nr,zeusBdis,hadronb,ggb}.
The errors on the data 
do not yet allow the different QCD schemes (see section ~\ref{sec:theory}) 
to be distinguished.  


The measurements are also shown figure~\ref{fig:fraccb} 
in the form of the fractional contribution to the total $ep$ cross
section:
\begin{equation}
f^{c\bar{c}} =  \frac{
{\rm d} \sigma^{c\bar{c}} / {\rm d} x {\rm d} Q^2
}
{
{\rm d} \sigma / {\rm d} x {\rm d} Q^2
}.
\end{equation}
 
The $b$ fraction $f^{b\bar{b}}$ is defined in the same manner.  NLO
QCD is found to give a good description of the data, as shown by
comparison with the ZM-VFNS prediction from the H1
fit~\cite{H19900NCCC}.


The $c$ and $b$ fractions and cross sections are also measured
integrated over the range \linebreak
$Q^2>150$~${\rm GeV}^2$ and $0.1<y<0.7$. 
This is a more restricted range
than for the differential measurements, in order 
to minimise the extrapolations, and
the following values are found:

\begin{equation*}
 \sigma^{c\bar{c}} =  431 \pm 59 \pm 69 \ {\rm pb}
\end{equation*}
\begin{equation*}
 \sigma^{b\bar{b}} = 45 \pm 11 \pm 11 \ {\rm pb}
\end{equation*}

NLO QCD is found to agree well with the data. For
example, the VFNS prediction from MRST~\cite{Martin:2003sk} gives:
\begin{equation*}
 \sigma^{c\bar{c}} =  426  \ {\rm pb}
\end{equation*}
\begin{equation*}
 \sigma^{b\bar{b}} =  47 \ {\rm pb}.
\end{equation*}



\section{Conclusion}
The production of $c$ and $b$ quarks at HERA has been studied using
precise tracking information from the H1 vertex detector.  The
inclusive $c$ and $b$ cross sections are measured using a technique
based on the lifetime of the heavy quark hadrons. The
measurements are made using all events containing tracks with vertex
detector information eliminating the need for large model dependent
extrapolations to the full cross section.  The predictions from
perturbative QCD are found
to agree well with both the integrated and differential cross
sections.

\section*{Acknowledgements}

We are grateful to the HERA machine group whose outstanding efforts
have made this experiment possible.  We thank the engineers and
technicians for their work in constructing and maintaining the H1
detector, our funding agencies for financial support, the DESY
technical staff for continual assistance and the DESY directorate for
support and for the hospitality which they extend to the non DESY
members of the collaboration.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%%CITATION = PHLTA,B313,535;%%

%\cite{Coffman:1991ud}
\bibitem{Coffman:1991ud}
D.~Coffman {\it et al.}  [MARK-III Collaboration],
%``Measurement of the inclusive decay properties of charmed mesons,''
Phys.\ Lett.\ B {\bf 263} (1991) 135.
%%CITATION = PHLTA,B263,135;%%

%\cite{lepjmulti}
\bibitem{lepjetmulti}
D.~Abbaneo {\it et al.}  [LEP/SLD Heavy Flavour Working Group],
 LEPHF 2001-01, available from 
\begin{verb}
http://lepewwg.web.cern.ch/LEPEWWG/heavy/
\end{verb}.



\bibitem{peterson}
% Peterson
C.~Peterson, D.~Schlatter, I.~Schmitt, and P.M.~Zerwas,
Phys.\,Rev.\ D 27 (1983) 105.


\bibitem{ref:adaptive}
R.~Fr\"{u}hwirth,et al., 
%New developments in vertex reconstruction for CMS, Nucl. Instr. Methods A502(2003)
Nucl.\ Instrum.\ Meth.\ A {\bf 502}, 446 (2003).

\bibitem{ref:annealing}
R.~Fr\"{u}hwirth, A.~Strandlie, 
%Adaptive multi-track fitting, 
Comput. Phys. Commun. 120 (1999) 197.
%K. 




%% CTEQ6 - add H1 as well?
%\bibitem{Pumplin:2002vw}
%J.~Pumplin, D.~R.~Stump, J.~Huston, H.~L.~Lai, P.~Nadolsky and W.~K.~Tung,
%%``New generation of parton distributions with uncertainties from global  QCD
%%analysis,''
%JHEP {\bf 0207} (2002) 012
%[hep-ph/0201195].
%%%CITATION = HEP-PH 0201195;%%



%\cite{Martin:2003sk}
\bibitem{Martin:2003sk}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``Uncertainties of predictions from parton distributions. II: Theoretical
%errors,''
[hep-ph/0308087].
%%CITATION = HEP-PH 0308087;%%


\end{thebibliography}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig3.eps}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig2.eps}

    \caption{The polar angle distribution  (a) and transverse momentum distribution (b) of all CST tracks.
      Included in the figure is the expectation from the RAPGAP Monte
      Carlo, showing the contribution from the various quark flavours after
      applying the scale factors obtained  from the fit to the
  subtracted significance distributions of the data.}
    \label{fig:thetatracks} 
  \end{center}
%  \vspace{-12.4cm} \hspace{1.1cm}  (a) 
%  \vspace{6.4cm}  \hspace{7.2cm} (b)
%  \vspace{6.4cm}

\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig4.eps}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig5.eps}
  \caption{The polar angle distribution (a) and transverse momentum 
 distribution (b) of the highest $p_T$ jet
  which contains at least 1
  reconstructed CST track within a cone of radius 1.0.  If there are
  no reconstructed jets the complete hadronic final state is
  used to define the jet axis. Included in the figure is the expectation from the 
  RAPGAP Monte Carlo
  after applying the scale factors obtained from the fit to the
  subtracted significance distributions of the data.}  \label{fig:jettheta}
  \end{center}
%  \vspace{-6.4cm} \hspace{1.1cm}  (a) 
%  \hspace{7.2cm} (b)
%  \vspace{6.4cm}

\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center} \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig1.eps}
  \caption{The number of reconstructed central silicon tracker (CST)
  tracks per event matched to the jet axis. Each CST track is required to 
  have at least two CST hits
  and $p_T>0.5$ GeV. Included in the figure is the expectation from the RAPGAP
  Monte Carlo, showing the contribution from the various quark
  flavours after applying the scale factors obtained from the fit of the
  subtracted significance distributions of the data (see section~\ref{sec:fit}).}
\label{fig:csttracks} 
  \end{center}

\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig6.eps}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig7.eps}
  \caption{The distance of closest approach of a track to the vertex
  (DCA) in the $x$--$y$ plane (a) and the signed significance (DCA/
  $\sigma$ (DCA) (b), where $\sigma$ (DCA) is the error on the DCA,
for all CST tracks.  
      The cut $|$DCA$|$ $<$ 0.1~${\rm cm}$ has been applied in figure (b).
Included in the figure is the expectation from the RAPGAP Monte Carlo, showing
      the contribution from the various quark flavours after
      applying the scale factors obtained  from the fit to the
  subtracted significance distributions of the data.}
    \label{fig:dca} 
  \end{center}
%  \vspace{-6.4cm} \hspace{1.1cm}  (a) 
%  \hspace{7.2cm} (b)
%  \vspace{6.4cm}

\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig8.eps}
    \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig9.eps}
    \caption{The signed significance $S_1=  DCA /\sigma (DCA) $
      distribution per event (a)  for events that contain 
       1 reconstructed CST track matched to the jet axis and the 
      signed significance $S_2=  DCA /\sigma (DCA) $  distribution per event 
      (b) of the track with the second highest absolute significance 
      for events with $\ge 2$ reconstructed CST tracks matched to the jet.
      Included in the figure is
      the expectation from the RAPGAP Monte Carlo, showing the
      contribution from the various quark flavours after applying the
      scale factors obtained from the fit to the subtracted significance
      distributions of the data.}
    \label{fig:s1s2} 
  \end{center}
%  \vspace{-6.4cm} \hspace{1.1cm}  (a) 
%  \hspace{7.2cm} (b)
%  \vspace{6.4cm}

\end{figure}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig10a.eps}
  \includegraphics[width=1.0\textwidth]{H1prelim-04-072.fig10b.eps}
  \caption{The distributions (a) $S_1$ and (b) $S_2$, after subtracting
  the bins with equal magnitude but negative sign from the positive.
  Included in the figure is the result from the fit to the data of the Monte Carlo 
 distributions of events arising from 
  $c$ quarks or $b$ quarks. The light quark contribution is also shown.}
  \label{fig:s1negsub} \end{center}
%  \vspace{-6.4cm} \hspace{1.1cm}  (a) 
%  \hspace{7.2cm} (b)
%  \vspace{6.4cm}

\end{figure}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=\textwidth]{H1prelim-04-072.fig14.eps}
    \includegraphics[width=\textwidth]{H1prelim-04-072.fig15.eps}
    \caption{The measured $F_2^{c\bar{c}}$ (a) and $F_2^{b\bar{b}}$ (b)
      shown as a function of $x$ for two different $Q^2$ values.
      The
      measurements of $F_2^{c\bar{c}}$ from ZEUS using $D^*$s to tag
      charm\cite{Chekanov:2003rb} and the prediction of 
      a NLO QCD fit are also shown.}
    \label{fig:f2c} 
  \end{center}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=\textwidth]{H1prelim-04-072.fig13.eps}
    \caption{The ratio $f^{c\bar{c}}$ and $f^{b\bar{b}}$ shown as a function
      of $x$ for two different $Q^2$ values. The
      measurements of $f^{c\bar{c}}$ from ZEUS using $D^*$s to tag
      charm\cite{Chekanov:2003rb} and the predictions of the
      H1 NLO QCD fit are also shown.}
    \label{fig:fraccb} 
  \end{center}
\end{figure}




\end{document}
