%================================================================
% 
% Conference contribution  ***   D-mesons in DIS   ***
% written by C.Grab,  ETHZ
% 
% v01 : first draft
% v02 : added first corrections of Gassner and myself (CG)
% v03 : added all corrections of Behnke and Gassner
% v04 : added all corrections of Waugh
% v05 : added corrections of Joe 5.7.02
% v06 : added corrections of EE + BW  11.7.02
% v07 : added corrections of eff.tables by Joe
% 
%================================================================
\documentclass[11pt]{article}
\usepackage{epsfig}
\usepackage{amsmath}
\usepackage{hhline}
\usepackage{amssymb}
\usepackage{times}
\usepackage{cite}
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%

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% Some useful tex commands - C.Grab additions
%
\newcommand{\GeV}{\mathrm{GeV}}
\newcommand{\TeV}{\rm{TeV}}
\newcommand{\pb}{\rm pb}
\newcommand{\cm}{\rm cm}
\newcommand{\hdick}{\noalign{\hrule height1.4pt}}
\newcommand{\Dstar}{\particle{D^{*}}}
\newcommand{\Dstarplus}{\particle{D^{*+}}}
\newcommand{\Dsubs}{\particle{D_S}}
\newcommand{\Dsubsplus}{\particle{D^{+}_S}}
\newcommand{\Dzero}{\particle{D^{0}}}
\newcommand{\Dplus}{\particle{D}^+}
\newcommand{\BR}{\rm{Br}}
\newcommand{\gev}{\mathrm{GeV}}
\newcommand{\mev}{\mathrm{MeV}}
%\newcommand{\qsq}{\rm{Q}^2}
%
\newcommand{\dst}{$D^{*+}$}
\newcommand{\dc}{$D^+$}
\newcommand{\dn}{$D^0$}
\newcommand{\ds}{$D^+_s$}
\newcommand{\dstdec}{$D^{*+} \rightarrow D^0 \pi^+ \rightarrow (K^- \pi^+) \pi^+$}
\newcommand{\dcdec}{$D^+ \rightarrow K^- \pi^+ \pi^+$}
\newcommand{\dndec}{$D^0 \rightarrow K^- \pi^+$}
\newcommand{\dsdec}{$D^+_s \rightarrow \phi \pi^+ \rightarrow (K^+ K^-) \pi^+$}
%
\newcommand{\particle}[1]{#1}
\newcommand{\cbar}{\particle{\bar{c}}}
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%===============================title page=============================


\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=figs/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                International Conference on High Energy Physics},
                July~25,~2002,~Amsterdam} \\
 {\bf ICHEP 2002:} 
                 & Abstract:        & {\bf 1015}    &\\
                 & Parallel Session & {\bf 5}   &\\
%                 & Plenary Session  & {\bf xx}   &\\[.7em]
\hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
%{\large Authors: J. Gassner, C. Grab  }  \\
%{\large Referees: O. Behnke, B. Waugh } \\
%{\large Preliminary version V0.6} \\
{\large Date: July 11, 2002}  \\
\end{tabular}
\end{small}
\end{center}

\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
Measurement of Inclusive {\boldmath $D$}-meson Production \\
in Deep Inelastic Scattering at HERA}

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

\begin{abstract}

\noindent

The inclusive production of charmed mesons in deep 
inelastic scattering is studied with the H1 detector at HERA\@.
%
Inclusive production cross sections are measured for the 
vector $D^{*+}$ and for the
pseudoscalar mesons $\Dzero$, $\Dsubs$ and, for the first 
time at HERA, also $\Dplus$.
% through their decay $D^+ \to K^- \pi^+ \pi^+$.
The finite lifetimes of 0.4 to 1 ps for the pseudoscalar mesons
%$\Dzero$ and $\Dplus$ 
lead to a separation of their production and decay vertices, which
is exploited to distinguish signal and background processes and to 
substantially improve the signal qualities. 
%The reconstruction of separation distances of some
%1/10 mm is made possible by exploiting the high-precision tracking
%capabilities of the central H1 silicon detector.
Differential distributions are measured for the $\Dplus$ and $\Dzero$
mesons and compared with predictions 
based on LO Monte Carlo simulations.

The measured production cross sections are used to test the isospin
invariance of the fragmentation process and to extract 
the strangeness suppression factor $\gamma_s$ and 
the fraction $P_V$ of $D$ mesons produced in a vector state.
The results are compared with values measured e.g.\ at
$e^+ e^-$ colliders and allow tests of the assumed 
universality of the charm fragmentation process.

\end{abstract}

\end{titlepage}

\pagestyle{plain}
%_________________________________________________________________________________
\large


\section{Introduction}


The production of heavy quarks in electron-proton interactions
proceeds, in QCD, almost exclusively via photon-gluon fusion, where 
a photon coupling to the incoming electron interacts 
with a gluon in the proton to form a quark-antiquark pair.
This holds both for deep inelastic scattering (DIS)
and for photoproduction, where the exchanged photon is almost real.
%
Differential charm production cross section 
measurements~\cite{h1-dgluon,h1-f2c,zeus-f2c}
in the range of experimental acceptance were found to be reasonably well
reproduced by a description based on perturbative QCD (pQCD)
done at Leading Order (LO) 
%(within MC programs)
and at Next to Leading Order (NLO)~\cite{dstargp}.
%
However, all these calculations are performed at the parton level, and the
conversion of the parton to hadrons -- the hadronisation process itself -- is
modelled by means of fragmentation functions:

\begin{eqnarray}
    d\sigma_h(p) = \int d\hat{\sigma}(p/z,\mu) \cdot D_h^q( z,\mu )\ dz .
\end{eqnarray}

Here $d\hat{\sigma}(p/z,\mu)$ describes the partonic cross section and
$\mu$ the factorization scale.
The fragmentation function $D_h^q( z,\mu )$ describes the
probability for a quark $q$ with momentum  $p$ to produce a hadron $h$ with momentum
fraction $z$, 
and contains both a perturbative part
(following an evolution in $\mu$ within the DGLAP formalism) 
and a non-perturbative part $D_{\it NP}(z)$,  
which is conventionally assumed to be universal.
Under the assumption of factorisation and universality, this $D_{\it NP}(z)$
is tuned at $e^+e^-$ colliders to hadron decay spectra and then 
applied to hadronisation processes at HERA\@.

It has been argued by Nason {\it et al.}\ \cite{f-mellin,n-mellin},
however, that these assumptions may not hold, and that in fact
different processes may be sensitive
to different aspects of fragmentation. Thus it is necessary to test
the hypotheses of universality and factorisation in many different
processes.

This paper describes one way of doing this, namely through the comparison of the 
production rates of the various different charmed meson states. 
These $D$-meson states are identified by means of a lifetime tag, i.e.\ by
measuring the decay length distributions. 


The paper describes a measurement of the H1 Collaboration and is
organized in four sections: first
the established measurement of the $\Dstar$ production cross section is
repeated using only the central tracking chamber (CJC) information and
shown to be consistent with previous publications.
%
Next, confidence in the understanding of the new H1 silicon tracker
(CST), used subsequently for the lifetime tagging method, is
established by showing the high level of agreement with which the MC
simulations describe the data.  For this purpose ``tagged
$\Dzero$ mesons'' are used; these are those $\Dzero$s unambiguously
identified, using the $\Delta m$ method, as originating
from a $\Dstar$ decay.  Then the determination of the cross sections
of the different new
decay channels, based on lifetime tagging, is presented.  Finally,
because the  results of the different channels are extracted within
the same visible kinematic regime,  they allow the
determination of ratios sensitive to fragmentation.

%_________________________________________________________________________________


\section{Experimental Aspects and Data Analysis}

%_________________________________________________________________________________


\subsection{Detector, Kinematics and Simulation}

The data have been collected with the H1 detector at {\sc Hera} and
correspond in total to integrated luminosities of $(47.8 \pm
0.7)~\pb^{-1} $ of $e^+p$ interactions  at $\sqrt{s} = 318~\GeV$.  The
charmed mesons are detected through their decay products, in
particular \dcdec, \dndec, \dsdec and \dstdec.
\footnote{Hereafter, the charge conjugate states are always implicitly
included.}

%H1 detector
The H1 detector and its trigger capabilities have been 
described in detail elsewhere~\cite{H1det}.
Charged particles are 
measured by a set of tracking devices:
two cylindrical jet drift chambers (CJC) \cite{cjc,cjc-res}, 
mounted concentrically around the
beam-line inside a homogeneous magnetic field of 1.15~T,
yielding particle charge and momentum from the track curvature
in the polar angular range
of $20^{\circ}<\theta<160^{\circ}$,
where $\theta$ is measured using the CJC and
two polygonal $z$ drift chambers and is defined with respect to the 
incident proton beam direction. 
These central tracking devices are complemented by the 
central silicon tracker (CST) \cite{h1-cst} with two layers
at 57.5 mm and 97.5 mm radii. 
The CST is centred at the nominal interaction point and 
has an active length of 356 mm in the $z$-direction.
Its intrinsic spatial
resolution is  $12~\mu$m in $r\phi$  and
 $22~\mu$m in $z$, and serves for the vertex separation 
measurement of the long lived particles.
%
One double layer of cylindrical multi-wire proportional 
chambers (MWPC) \cite{mwpc} with pad readout
for triggering purposes is positioned between the CST and the CJC, and
another between the two jet chambers.
%
%The tracking detector is surrounded by a fine grained liquid
%argon calorimeter \cite{calo}, consisting of an electromagnetic
%section with lead absorbers and a hadronic section with
%steel absorbers.
%It covers polar angles between $4^{\circ}$ and $154^{\circ}$.
The backward region ($153^{\circ}<\theta<177.8^{\circ}$) of H1 
is equipped with 
a lead scintillator ``Spaghetti" calorimeter (SpaCal)~\cite{spacal},
which is optimized for the detection of the scattered electron in the 
DIS kinematic range under consideration here
and provides time-of-flight functionality 
for trigger purposes. 
It consists of an electromagnetic and a more coarsely 
segmented hadronic section.
A four-layer drift chamber (BDC)~\cite{bdc}
mounted in front of the SpaCal is used to reject neutral particle background.


The events used here have been triggered by an electromagnetic cluster
in the SpaCal of at least 6.0 GeV energy 
in coincidence with a charged track signal
from the MWPC and central drift chamber trigger.

%_________________________________________________________________________________


\subsection{Event Selection and Reconstruction}

The analysis covers the kinematic regime of $2 < Q^2 < 100 \ \gev^2$ and 
$0.05 < y < 0.7$, where
the scattered electron is used to determine the event kinematic quantities.
The identification of electrons is similar to the procedure
used in the inclusive structure function measurement~\cite{h1-f2}.
Scattered electrons are identified as clusters in the SpaCal with 
energy $E_{e^{\prime}}>8$ GeV, with a cluster radius $<$3.5~cm consistent with
electromagnetic energy deposition, and with a cluster centre matched 
by a charged track candidate in the BDC within 2.5 cm. 
The scattering angle is required to be 
$153^{\circ} < \theta_{e^{\prime}}<177^{\circ}$. 

The DIS kinematic variables are 
reconstructed using the ``electron ($e$)'' me\-thod~\cite{bbkine}, where
%
\begin{equation}
Q^2  = 4 E_e E_{e^{\prime}} \cos ^2 \frac{\theta_{e^{\prime}}}{2} ,
{\rm \, \, and \, \, } 
y_e = 1 - (E_{e^{\prime}}/E_e) \sin ^2 (\theta_{e^{\prime}} /2)\;\; 
\end{equation}
with $E_e, E_{e^{\prime}}$ denoting the energies of the incoming and scattered electron
respectively.
The results obtained with the $e$~method have been checked with another
($''\Sigma''$ \cite{bbkine}) method and are found to be in very good agreement. 
The absolute energy scale of the SpaCal is known to a precision 
of $\pm 4\%$.

%_________________________________________________________________________________
\subsection{Signal Determination}

The reconstruction of the various $D$ mesons proceeds as follows:
charged particle tracks passing some standard quality cuts
and fulfilling transverse momentum cuts as listed in table~\ref{tab:selection}
are combined, and the resulting combinations loosely tested for 
compatibility with possible $D$ decay hypotheses. 
No particle identification is applied at any stage.
Assuming each track to be appropriately either a kaon or a pion,
the tracks of correct sign charges are combined and 
% fitted to a common secondary vertex.
%Candidate combinations resulting in a reasonable secondary vertex fit 
%(the vertexing will be explained below)
taken to determine the number of signal events in each channel as follows:

%% --------------------------------------------------------------

a)  \dcdec : the 3-particle invariant mass distribution $m(K \pi \pi)$ is directly fitted by a 
   Gaussian signal and a linear background shape (see fig.~\ref{fig:dplus-cst}).

% %% D+ from CST

\begin{figure}[p] 
  \begin{center}
    \epsfig{file=figs/dc2dc_signal.eps,width=8cm}
  \end{center}
  \caption{ \large
 Invariant mass distribution  $m(K \pi \pi)$ of $\Dplus$ candidates.}
  \label{fig:dplus-cst}
\end{figure} 

%% --------------------------------------------------------------

b) \dndec: the signal is directly extracted from a fit of the $K^-\pi^+$ candidate
   invariant mass distribution $m(K \pi)$ with
   a function composed of a Gaussian signal $G_{\it sig}$, an exponential 
   combinatorial background $B$ and
   a wrong-charge combination contribution $G_{\it wc}$,
   approximated by a broad Gaussian. 
   The mean position and the width of $G_{\it wc}$ as well as
   the normalisation ratio of $G_{\it sig}$ to  $G_{\it wc}$ 
   have been determined from the MC simulation 
   and subsequently kept fixed in the fit (see fig.~\ref{fig:dzero-cst}).

% D0 from CST
\begin{figure}[p] 
  \begin{center}
    \epsfig{file=figs/do2dc_signal_decomposed.eps,width=8cm}
  \end{center}
  \caption{ \large
Invariant mass distribution $m(K \pi)$ of $\Dzero$ candidates. 
   The decomposition of the data into the correct charge combination signal 
   (light shading),
   the wrong charge combination (dark shading) and the exponential background
   is separately indicated.}
  \label{fig:dzero-cst}
\end{figure} 

%% --------------------------------------------------------------

c) \dsdec: because the $\Dsubs$ decays through the intermediate
  vector-meson $\phi$~resonance, it lends  itself to further specific
  kinematic constraints.  In particular, the two particle combination
  $m(K^+ K^-)$ is required to lie within $2 \sigma$ of the nominal
  $\phi$ mass (i.e.\ within 11 MeV of 1020 MeV).  Furthermore, the
  angular distribution of the $K$-meson  flight direction transformed into
  the $\phi$ rest frame (the helicity angle $\theta^*$) follows  a
  $\cos ^2 \theta^*$ shape. To reduce combinatorial background,  $|
  \cos \theta^* | > 0.4$ is required.  Both the $\Dsubs$ and the
  $\phi$ distributions are shown in fig.~\ref{fig:dsubs-cst}.

% %% Ds from CST

\begin{figure}[p] 
  \begin{center}
    \epsfig{file=figs/ds2dc_signal.eps,width=16cm}
  \end{center}
  \caption{ \large
Invariant mass distributions  $m(KK \pi)$ of $\Dsubs$ candidates (left) and
  $m(KK)$ of the intermediate $\phi$-resonance candidates (right).}
  \label{fig:dsubs-cst}
\end{figure} 


%% --------------------------------------------------------------

d) \dstdec : the well-known $\Delta m$-tagging technique 
is applied \cite{deltam-feldmann}.
% those $K^-\pi^+$ pairs with an invariant mass 
% consistent with the $D^0$ mass within $\pm 80$ MeV
% are combined with a third track - hypothetically assumed to be a pion
% (``$\pi^+_{slow}$'') - of charge opposite in sign to that of the kaon candidate.
 For all candidate combinations lying within $\pm 3 \sigma$
 of the nominal value of $\Delta m$, the invariant mass 
 $m(K \pi)$ distribution is fitted with a Gaussian signal and a
 exponential background shape, 
 and $N_{\Dstar}$ thus determined (see fig.~\ref{fig:dstar-cst}).
 $\Dzero$ mesons from an identified \dstdec chain will be referred to as
 tagged $\Dzero$s.


% %% Dstar from CST
\begin{figure}[p] 
  \begin{center}
    \epsfig{file=figs/dstar_vfit2dc_signal.eps,width=16cm}
  \end{center}
  \caption{ \large $\Dstar$ candidate distributions:
  left: $K \pi$ invariant mass after a $3\sigma$ cut in $\Delta m$; right: the
   $\Delta m$ distribution after a $2\sigma$ cut in $m(K \pi)$ around the 
   nominal value.}
  \label{fig:dstar-cst}
\end{figure} 

%% --------------------------------------------------------------


In the case of $D$-meson differential distributions, the inclusive data
sample is divided into bins and the number of signal events extracted
in each bin separately. 
The position and width of the Gaussian signal shape are fixed to the values
found in the inclusive sample.
The normalizations of the signal and the background are left
as free parameters in the various fits.
Other methods of determining the number of candidates have also been applied 
and the variation entered into the systematic error.
Possible uncertainties due to assumptions about the background shapes have been estimated
by changing the background shape (from linear to exponential), and 
are also taken into account in the systematic errors listed in table~\ref{tab:syserr}.
Contributions to the signals due to other charm decays
(so-called reflections) are estimated 
from Monte Carlo simulations to be at most $3\%$, and are included in the
systematic errors.


%
\begin{table}[htb]
 \large
 \renewcommand{\arraystretch}{1.0}
\begin{center}
 \begin{tabular}{|c|c|c|c|c|}
  \hline
  {\bf Selection criteria} & \dc & \dn & \ds & \dst \\
  \hline
  min $p_t(K)\ [ \mev /c ]\ $     & 500 & 800 & 400 & 250 \\
  min $p_t(\pi)\ \ [ \mev /c ]\ $ & 400 & 800 & 400 & 250 \\
  min $p_t(\pi_s)\ [ \mev /c ]\ $ &   - &   - &   - & 140 \\
  min $|\cos(\theta^*)|\ $      &   - &   - & 0.4 &   - \\
  \hline
%  $\max(p_t)\ >$             & \multicolumn{4}{c|}{$800\,MeV/c$}  \\
  $\theta_{\it track}\ $          & \multicolumn{4}{c|}{$[\ 20^\circ,\,160^\circ\,]$} \\
  max decay length error $ \sigma_l  $              & \multicolumn{4}{c|}{$300~\mu$m} \\
  min fit probability ${\cal P}_{\it VF}  $          & \multicolumn{4}{c|}{$ 0.05$} \\
  \hline
   min decay length sig. $ S_l = l/\sigma_l >$       &   5 &   3 &   2 &   1 \\
  min impact par. sig. $(2\times S_d) >$ & 2.5 & 2 &   1 &  -1 \\
  \hline
 \end{tabular}
\end{center}
\caption{  \large
  Selection criteria for the charged daughter tracks for each decay mode.
  The requirements on the vertexing parameters are also listed. }
\label{tab:selection}
\end{table}
        
%_________________________________________________________________________________

\subsection{Vertexing Issues}


The finite lifetime of 0.4 to 1 ps of the $D$ mesons leads to a
spatial separation between their production (assumed to be the
primary) and decay (secondary) vertices, which is measured in
terms of the radial decay length $l$ with error $\sigma_l$.
% measures the distance between these two vertices
Its significance $S_l = l/\sigma_l$ constitutes one of the most
powerful quantities to distinguish long-lived hadrons from background
events, which originate predominantly  from random tracks from the
primary vertex.  The calculation of $S_l$ depends on the understanding
of the combination of CST and  CJC information.

To obtain tracks with the desired precision, the well-established CJC
tracks \cite{cjc} are geometrically extrapolated into the CST\@.  The
closest CST hits available within a $5\sigma$ search window are
assigned to the tracks and these CJC-track / CST-hit combinations are
then refitted.  Each track can thus have up to 2 CST hits assigned.

Combinations of these CST-improved tracks (2 or 3 depending on the
decay mode) are then fitted in the $r\phi$ plane to a common secondary
vertex under the assumption that the $D$-meson mother particle decayed
into (2 or 3) daughter particles at this secondary vertex.
A combination of tracks is taken  if at least two of the tracks have an
impact parameter d with significance $S_d = d/\sigma_d$ larger than a
minimal value.  Of these tracks only one is allowed to have at most
one CST hit missing.
%if all but one of the daughter tracks have two (of the possible two) 
%CST-hits assigned.
This effectively restricts the $D$ mesons to lie in a polar angular range of 
typically $25^\circ < \theta < 155 ^\circ$.
%
%, well detached from the primary event vertex.
In the fit, the mother particle $D$ is constrained to have originated
from the primary vertex.
%i.e.\ the vertex-separation vector must be parallel to the mother 
%particle momentum vector.

$D$-meson decay candidates are selected by quality requirements on the
vertexing,  in particular on the probability of the vertex fit ${\cal
P}_{\it VF}$ and on the vertex separation significance $S_l = l/\sigma_l$.
%For further background rejection, combinations where only ONE track
%contains lifetime information (i.e.\ has a reasonable impact parameter)
%are discarded
%by requiring the impact parameter significances $S_d = d/\sigma_d$ of 
%at least two tracks to be larger than a minimal value.
The cut values actually applied on $\sigma_l$, ${\cal P}_{\it VF}$, $S_l$
and $S_d$  are listed in table~\ref{tab:selection}.  The degree of
improvement is demonstrated in fig.~\ref{fig:phenix} in a comparison
of the $\Dplus$ signal before and after a cut on $S_l$.

%% --------------------------------------------------------------

% %% Phenix plot for D+

\begin{figure}[h]
  \begin{center}
    \epsfig{file=figs/dc_phoenix.eps,width=16. cm}
\caption{ \large
 Comparison of the invariant mass distributions $m(K \pi \pi)$ for
 $D^+ \rightarrow K^- \pi^+ \pi^+$ decay candidates
(a) before and (b) after a cut on the decay length significance $S_l = l/\sigma_l > 8$.
%where $l$ is the radial separation distance between the meson production and its decay vertex, 
%and $\sigma_l$ the error of $l$.
The background contribution is suppressed by $\cal O$(300) and the signal to background ratio
is improved by a factor $\cal O$(50) when vertexing information measured with the 
H1 central silicon vertex detector CST is exploited.}
  \label{fig:phenix}
  \end{center}
\end{figure}


%% --------------------------------------------------------------


The simulation of the silicon tracker CST has been scrutinized 
in detailed comparisons with data distributions by using among others
tagged $\Dzero$ events. 

%% --------------------------------------------------------------


% Decay length significance for tagged D0  and
% %% SL cut efficiency

\begin{figure}[ht]
  \begin{center}
    \epsfig{file=figs/dstar_decay_length_sig1.eps,width=7. cm}
    \epsfig{file=figs/dst2dc_sigcut_eff.eps,width=7. cm}
\caption{ \large 
a) Distribution of the  decay length significance $S_l$ for data
(solid dots), and the fitted decomposition into signal (hatched) and 
background (shaded) contributions.
b) Efficiency as a function of the cut on the
  decay length significance $S_l = l/\sigma_l$; 
  the efficiency is defined as the number of fitted $\Dzero$ mesons 
  after applying the cut on $S_l$ divided by the total number of $\Dzero$ mesons
  without $S_l$ cuts.
  Data (solid dots) are compared with the expectations from the simulation
(open squares). }
  \label{fig:eff-sl}
  \end{center}
\end{figure}

%% -------------------

To indicate the level of understanding of the vertexing simulation,
fig.~\ref{fig:eff-sl}a shows the measured decay length significance
distribution  $S_l$ of tagged $D^0$ candidates (solid
dots). Separately shown is its fitted decomposition into a signal
(hatched) and a background (shaded)  contribution.
%and the fit combining the two (solid line).  
The functional form of the signal
distribution $F_S$ is taken from  the simulation, whereas the
background shape $F_{B}$  is extracted from the 250-MeV-wide sideband
region  in the $m(K\pi)$ spectrum of the data.
%
In the fit, only the normalisations of the signal and of the
background are left as free parameters.
The excellent $\chi^2/{\it ndf} = 34/32$ indicates that the MC simulation 
describes the signal shape very well. 
Furthermore, the number of $\Dzero$ candidates extracted by means of the
$S_l$ fit is fully consistent within errors with the one 
determined from the fit of the invariant mass distribution $m(K\pi)$. 

The efficiency $\epsilon (S_l) $ as a function of the cut
on $S_l$  as measured in data  (solid dots) is shown 
in fig.~\ref{fig:eff-sl}b to be very reasonably described by the
MC simulation (open squares).


%% --------------------------------------------------------------

\subsection{Acceptance and efficiency determination}

A Monte Carlo simulation is used to determine the detector acceptance
and the efficiency of the reconstruction and the selection cuts.
Electroproduction events were generated in Leading Order with the
AROMA 2.2~\cite{aroma} program, combined with parton showering
\cite{pythia}.  The hadronization step was performed according to the
Lund string model, tuned to the world average fragmentation factors
$f_{w.a.}(c \to D)$, which are dominated by the LEP results. The generated
events were then processed by the H1 detector simulation program, and
were subjected to the same reconstruction and analysis chain as the
real data.

For the different decay channels $\Dplus$, $\Dzero$, $\Dsubs$ and $\Dstar$, 
the efficiencies and acceptances are given in table~\ref{tab:efficiencies}.
%overall detector efficiencies
%are determined to be 58.3, 68.4, 57.2 and 59.8 $\%$ within the 
%acceptances of 36.4, 43.8, 49.2 and 61.9 $\%$.
%The vertexing efficiencies are found to be 19.3, 11.3, 21.0 and 39.4 $\%$,
%espectively.

%---------------
\begin{table}[ht]
 \large
 \renewcommand{\arraystretch}{1.2}
\begin{center}
 \begin{tabular}{|c|c|c|c|c|}
  \hline
  {\bf } & \dc & \dn & \ds & \dst \\
  \hline
overall detector efficiency  (\%) &  50.8 &   61.2 &  48.8 &  52.1    \\
vertexing efficiency (\%) &  19.9 &  12.4 &  22.2 &  39.4 \\
acceptance  	     (\%) &  36.4 &  43.8 &  49.2 &  61.9  \\
  \hline
 \end{tabular}
\end{center}
   \caption{ \large
Efficiency and acceptance values in $(\%)$ for the four meson
   states.  }
\label{tab:efficiencies}
\end{table}

%The changes found when varying the shape of 
%the charm fragmentation function by using a Peterson 
%function with $\epsilon_c$ ranging from 0.026 to 0.046
%are accounted for in the systematic error. 
The dependence of the simulated acceptances and efficiencies on parameter 
choices made for the simulation (charm mass, parton density distributions,
fragmentation parameters and QCD scales) was found to be
less than $\pm 2\% $ in all cases and is included in the 
theoretical systematic error.

Contributions from $b\bar{b}$ production, with subsequent decays of
$b$ flavoured hadrons into $D$ mesons, have been calculated  using the
AROMA generator.  No subtraction is made: the quoted $\Dstar$ cross
sections thus include any $b$~contributions.
A systematic error incorporates the change in efficiency brought about
by a variation of the $b\bar{b}$ contribution by $\pm 30\%$ around the
published $b\bar{b}$ cross section~\cite{h1-bxsec}, which shows an unexpectedly
high excess over QCD expectation.


%% --------------------------------------------------------------

\section{Results}
%_________________________________________________________________________________

\subsection{Production Cross Sections}

The production cross sections are determined for all four decay
channels for the identical visible  kinematic range of
$2~\gev^2 \le Q^2 \le 100~\gev^2$,
\ $0.05 \le y \le 0.7$,
\ $p_t(D) \ge 2.5~\GeV/c$,
\ $|\eta(D)| \le 1.5$.
%
Here $\sigma_{\it vis}$ is defined as the sum of both particle and
antiparticle meson states, and is given by 

\begin{eqnarray}
 \begin{array}{lll}
  \nonumber
 \sigma_{\it vis}(ep \rightarrow eDX) 
% & = &  \sigma_{vis}(ep \rightarrow eDX) + \sigma_{vis}(ep \rightarrow e\bar{D}X) \hspace{1cm} \\
  & = &    \{\sigma_{\it vis}(ep \rightarrow ec\bar{c}X)\cdot 2\cdot f(c \rightarrow D)\}_{\it vis} \hspace{1cm} \\
 & \quad + & \{\sigma_{\it vis}(ep \rightarrow eb\bar{b}X)\cdot 2\cdot f(b \rightarrow D)\}_{\it vis} \hspace{1cm}
 \end{array}
\end{eqnarray}
where $ f(c \rightarrow D)$ and $ f(b \rightarrow D)$ are the charm
and beauty fragmentation fractions into the respective $D$ meson
states.  The factor of two corresponds to the fact that both $c$ and
$\bar{c}$ may decay into the final state under consideration.  The
measured values of the visible cross sections for the four meson decay
channels are summarized in table~\ref{tab:xsection}.

%
\begin{table}[ht]
 \large
 \renewcommand{\arraystretch}{1.2}
\begin{center}
 \begin{tabular}{|c|c|c|c|c|}
  \hline
  {\bf Cross section} & \dc & \dn & \ds & \dst \\
  \hline 
$\sigma_{\it vis}(ep \rightarrow eDX)$ (nb)  &
 2.16      & 6.53     & 1.67 & 2.90   \\
stat\ error on  $ \sigma_{\it vis}$          &
$\pm 0.19$ & $\pm 0.49$ & $\pm 0.41$ & $\pm 0.20$  \\
syst\ error on $ \sigma_{\it vis}$           &
$^{+0.46}_{-0.35}$ & $^{+1.06}_{-1.30}$ & $^{+0.54}_{-0.54}$ & $^{+0.58}_{-0.44}$ \\
  \hline  
AROMA LO prediction $ \sigma_{\it vis}$ &
2.45 & 5.54 & 1.15 & 2.61   \\
 \  uncertainty          &
$\pm 0.30$ & $\pm 0.69$ &  $\pm 0.30$&  $\pm 0.31 $  \\
  \hline
 \end{tabular}
\end{center}
   \caption{ \large
  Inclusive charmed meson electroproduction cross sections for the four 
  meson states in the visible kinematic range, defined by 
  $2\, \le \,Q^2 \le 100\, \GeV^2,\
\ 0.05 \le y  \le  0.7, \ p_t(D) \ge 2.5\,\GeV/c$ and $|\eta(D)|  \le 1.5. $    }
\label{tab:xsection}
\end{table}
%% --------------------------------------------------------------

\subsection{Systematic Errors}

The various contributions to the systematic errors on $\sigma_{\it vis}$
are summarized in table~\ref{tab:syserr}. 
They are dominated by CJC tracking efficiency (($^{+5}_{-1})\% $ per track)
and by the vertexing uncertainty of $10 \% $. 
A variation of the CST spatial resolution of up to 20\% shows only a minor effect
on $\sigma_{\it vis}$ of order $< 4\%$.
The SpaCal energy scale uncertainty is taken into account by a variation of
$\pm 4\% $, and it contributes a change in $\sigma_{\it vis}$ of $(^{+5}_{-9})\%$.

Included in the table is 
the uncertainty due to variations  in the Monte Carlo
acceptance, which itself is dominated by the charm mass variation
$(1.4 < m_c < 1.6\  \GeV $) and the fragmentation uncertainties.
Separately indicated is the error due to initial
state radiation (ISR) corrections.

\begin{table}
 \large
 \renewcommand{\arraystretch}{1.4}
 \centering
 \begin{tabular}{|c|c|c|c|c|}
%  \multicolumn{5}{l}{relative systematic errors on the inclusive cross sections $[\%]$}\\
  \hline
  {\bf Source of Uncertainty} & \dc & \dn & \ds & \dst \\
\hline
Acceptance & \multicolumn{4}{c|}{$< 2.0$}\\
\hline
CJC efficiency  & $^{+15.0}_{\ -3.0}$  & $^{+10.0}_{\ -2.0}$  & $^{+15.0}_{\ -3.0}$  & $^{+15.0}_{\ -3.0}$  \\
\hline
CJC resolution ($\pm10\%$) & $^{+3.0}_{-1.5}$  &  $^{+0.6}_{-0.7}$  &$^{+1.8}_{-3.7}$  & $^{+2.4}_{-0.5}$  \\
\hline
CST efficiency  & $\pm 5.6$  &  $\pm 3.6$  & $\pm 5.4$  & $\pm 3.6$  \\
\hline
CST resolution ($\pm20\%$)  & $^{+2.0}_{-1.0}$ & $^{+0.4}_{-0.1}$  & $^{+0.1}_{-3.6}$  & $^{+1.5}_{-0.2}$ \\
\hline
vertexing & \multicolumn{4}{c|}{$\pm 10.0$}  \\
\hline
SpaCal calibration ($\pm4\%$) & $^{+3.6}_{-7.3}$  & $^{+3.4}_{-8.1}$  & $^{+4.5}_{-7.8}$  & $^{+4.5}_{-9.0}$  \\
\hline
signal extraction & $^{+1.7}_{-0.4}$  & $^{\ +4.9}_{-13.4}$  & $^{\ +1.4}_{-12.1}$  & $^{+3.7}_{-3.2}$  \\
\hline
branching ratio & $\pm6.7$ & $\pm2.3$ & $\pm24.7$ \ & $\pm2.3$ \\
\hline
ISR correction & \multicolumn{4}{c|}{$\pm2.6$} \\
\hline
trigger efficiency  & \multicolumn{4}{c|}{$\pm1.0$} \\
\hline
\hline
total systematic error & $^{+21.2}_{-16.0}$ & $^{+16.5}_{-19.9}$ & $^{+32.0}_{-31.8}$ & $^{+20.0}_{-15.2}$ \\

\hline

 \end{tabular}
\caption{ \large
  Relative systematic errors on the inclusive cross sections in $\%$
  for the different $D$-meson decay modes.}
\label{tab:syserr}
\end{table}

%% --------------------------------------------------------------


\subsection{Comparison of Cross Sections with Predictions}

The measured cross sections
have been compared in detail with LO calculations.
The corresponding predictions 
using the AROMA MC generator (using GRV-LO-98 proton structure
functions \cite{grv-lo98} and a charm mass of 1.5 GeV) are included in
table~\ref{tab:xsection}, and are in good agreement with the 
measured values for all decay channels.
A comparison of the $\Dstar$ production cross section with predictions
using NLO calculations based on the HVQDIS program \cite{bh111}, supplemented
by a Peterson \cite{peterson} fragmentation parametrisation (with $\epsilon_c = 0.036$
according to \cite{nason-oleari})
yields $\sigma_{\it vis} = 2.55\,$~nb for charm only, in very good
agreement with the LO calculations.

To compare the measurements with previous results, $\sigma(ep \to e
\Dstar X)$ was also determined for the same kinematic range as applied
in the H1 publication \cite{h1-dgluon}, where the only difference
was a lower $p_T(\Dstar)$ cut of $1.5$~GeV.  The present analysis
yields in that range a value of $\sigma_{\it vis} = (5.28
\pm0.42)$~nb, which agrees with the published H1 result of  $(5.75 \pm
0.35 \pm 0.79)$~nb and also with the LO QCD predictions of  5.53~nb
(with GRV-LO 98 proton pdf).  Furhermore, the present result is
consistent within one $\sigma$ error  with the H1 measurement
\cite{h1f2-dif},
%(a published $\sigma^{vis} = 4.17 \pm 0.9$~nb to
%agrees with a value of $3.82 \pm 0.23$~nb within the present analysis and 
%a LO prediction of 3.86~nb).
but it is slightly below the values quoted in \cite{h1-f2c}.


%The effects of limited detector resolution on the 
%variables are small with respect to the chosen bin sizes 
%and are corrected for using the Monte Carlo simulation. 

The differential production cross sections were also determined  for 
all decay channels, and results for the \dn\ and \dc\ channels are shown 
in fig.~\ref{fig:d0-xsec} and fig.~\ref{fig:dc-xsec} as a function 
of the various kinematic quantities.
The effects of limited detector resolution on the 
variables are small with respect to the chosen bin sizes 
and are corrected for using the Monte Carlo simulation. 
Overlaid in the figures are the predictions of the LO AROMA simulation 
(dark shaded bands), 
including a beauty contribution, which was scaled by a factor of 4.3,  according
to the H1 published values \cite{h1-bxsec}. The beauty contribution
itself is indicated separately by light shaded bands.
Both the $p_t(D)$ distribution and the $\eta(D)$ distribution 
are quite well described by the LO QCD simulation,
even in the forward direction in $\eta$.
This may be attributed to the improved signal-to-noise ratio 
achieved with the higher tracking precision even in the forward direction.

% %% D0 DIFFERENTIAL 

\begin{figure}[p] 
  \begin{center}
%    \epsfig{file=figs/d0_diff_xsections_all.eps,width=16cm}
    \epsfig{file=figs/d0_differential.eps,width=16cm}
  \end{center}
  \caption{ \large 
Differential production cross section for \dn \ as a function of
   the $\Dzero$ variables (left) and the event variables (right).
   The dark shaded bands indicate the AROMA MC predictions including a scaled 
   beauty contribution, which is shown separately as light shaded bands.}
  \label{fig:d0-xsec}
\end{figure} 

% %% Dplus DIFFERENTIAL 

\begin{figure}[p] 
  \begin{center}
%    \epsfig{file=figs/dc_diff_xsections_all.eps,width=16cm}
    \epsfig{file=figs/dc_differential.eps,width=16cm}
  \end{center}
  \caption{ \large 
Differential production cross section for \dc \ as a function of
   the $\Dplus$ variables (at left) and the event variables (at right).
   The dark shaded bands indicate the AROMA MC predictions including a scaled 
   beauty contribution, shown separately as light shaded bands.}
  \label{fig:dc-xsec}
\end{figure} 

 
%_________________________________________________________________________________
\subsection{Fragmentation Fractions and Isospin Ratios}

From the measured cross sections the fragmentation fractions $f_c$ can be deduced by 
subtracting the small beauty contribution and then 
removing the fragmentation dependence $f_{w.a.}$ from the simulation 
using the following relation:
\begin{eqnarray}
\nonumber
\displaystyle f^{}(c\rightarrow D) = 
% \frac{\displaystyle \sigma_{vis} - 2 \cdot \sigma^{MC}_{b\bar{b}} \cdot
% \displaystyle f^{}(b\rightarrow D)}
% {\displaystyle 2 \cdot \sigma^{MC}_{c\bar{c}}}
%
 \frac{\displaystyle \sigma_{\it vis}(ep \rightarrow eDX) - 
\sigma^{\it MC}_{\it vis} (ep \rightarrow b\bar{b} \rightarrow eDX)}
%
{  \frac{\displaystyle \sigma^{\it MC}_{\it vis} (ep \rightarrow c\bar{c} \rightarrow eDX)} 
        {\displaystyle f_{w.a.}(c \rightarrow D)}
}
\end{eqnarray}
%

The resulting values are listed in table~\ref{tab:ffrag}.

%
\begin{table}[htb]
 \large
 \renewcommand{\arraystretch}{1.3}
\begin{center}
\hspace*{-1.cm}
 \begin{tabular}{|c|c|c|c|c|}
  \hline
  {\bf Fragmentation factors} & \dc & \dn & \ds & \dst \\
  \hline 
  $ f(c \to D)$         & 0.202 &  0.658  & 0.156 & 0.263 \\
 stat\ error            &  $\pm 0.020$ &  $\pm 0.054$ &  $\pm 0.043$ &  $\pm 0.019 $ \\
 syst\ error            & $^{+0.045}_{-0.033}$ & $^{+0.117}_{-0.142}$ & $^{+0.036}_{-0.035}$ & $^{+0.056}_{-0.042}$\\
 theo\ error            & $^{+0.029}_{-0.021}$ & $^{+0.086}_{-0.048}$ & $^{+0.050}_{-0.046}$ & $^{+0.031}_{-0.022}$ \\
\hline
 $f_{w.a.}$ = world average          & 0.232 $\pm 0.018$ & 0.549 $\pm 0.026$ & 0.101  $\pm 0.027$  & 0.235  $\pm 0.010$  \\
  \hline
 \end{tabular}
\end{center}
   \caption{  \large
Fragmentation factors deduced from the measured cross sections. 
  The small $b$~contributions are subtracted. The  deduced values compare well with 
  present world average numbers. }
\label{tab:ffrag}
\end{table}

%_________________________________________________________________________________

Furthermore, the fraction of vector mesons $P_V$ produced in the fragmentation process
can be calculated in different ways 
%  !!!!!!!!!!!!!!!!!!!!!1
% for the different quark-antiquark systems (cd)  or (cu)
% PV uses only cd, whereas pv' uses both (cd) AND (cu) AND isospin invariance.
%
based on the extracted fragmentation fractions and using the known branching ratios.
The relations are given in the following equations, 
where {\it VM} denotes the total number of vector mesons
and {\it PS} the number of pseudoscalar $D$ mesons produced in the 
fragmentation process.
The value $P'_V$ is extracted under the assumption of
isospin invariance $f(c \rightarrow D^{*+}) = f(c \rightarrow D^{*0})$,
and includes directly the $D^0$ measurements.
%_
\begin{eqnarray}
 \renewcommand{\arraystretch}{1.8}
% \large
 \begin{array}{ll}
P_V \ \ & = \frac{\displaystyle \it VM}{\displaystyle \it PS+VM} \\
        & = \frac{\displaystyle f(c \rightarrow D^{*+})}
{\displaystyle f(c\rightarrow D^+) +f(c\rightarrow D^{*+})\ {\it BR}(D^{*+}\rightarrow D^0\pi^+)}\\
%P_V \ \ & = \frac{\displaystyle VM}{\displaystyle PS+VM} \\
%        & = \frac{\displaystyle f(c \rightarrow D^{*+})}
%{\displaystyle f(c\rightarrow D^+) +f(c\rightarrow D^{*+})\ BR(D^{*+}\rightarrow D^0\pi^+)}\\
P'_V    & = \frac{\displaystyle 2\ f(c \rightarrow D^{*+})}
{\displaystyle f(c\rightarrow D^+) +f(c\rightarrow D^o)}
 \end{array}
\end{eqnarray}

Inserting the numbers from table~\ref{tab:ffrag}, the following results are obtained:
$P_V = (0.693 \pm 0.045 \pm 0.004  \pm 0.009) $ and
$P'_V = (0.613 \pm 0.061 \pm 0.033  \pm 0.008)$,
 where the errors are of
 statistical, systematic and theoretical nature respectively. 
The theoretical errors include uncertainties due to branching fractions, 
MC parameter variations and the beauty subtraction.
The results compare favourably with the
present world average value \cite{gladilin} of $(0.601 \pm 0.032)$,
which is dominated by measurements performed at $e^+e^-$ experiments.

%________________________________________________________________________________
In addition, the fragmentation ratios may be determined according to:
\begin{eqnarray}
 \renewcommand{\arraystretch}{1.8}
% \large
 \begin{array}{ll}
R_{u/d} & =  \frac
{\displaystyle f(c\rightarrow D^0) -f(c\rightarrow D^{*+})\ {\it BR}(D^{*+}\rightarrow D^0\pi^+)}
{\displaystyle f(c\rightarrow D^+) +f(c\rightarrow D^{*+})\ {\it BR}(D^{*+}\rightarrow D^0\pi^+)}\\
\gamma_s & = \frac{\displaystyle 2\ f(c \rightarrow D_s^+)}
{\displaystyle f(c\rightarrow D^+) +f(c\rightarrow D^o)}\\
 \end{array}
\end{eqnarray}

The extracted numbers are 
$R_{u/d} =  (1.26\ \pm 0.20\ $({\it stat})$\ \pm
0.11\ $({\it syst})$ \pm 0.04\ $({\it theory})$)$, and 
$\gamma_s =  (0.36 \pm 0.10\ $({\it stat})$\ \pm
 0.01\ $({\it syst})$\ \pm 0.08 $({\it theory})$)$. 
They too are in agreement with the present
world average numbers \cite{gladilin} of 
 $ (1.00\ \pm 0.09)$ and $(0.26\ \pm 0.07)$ respectively.

%_________________________________________________________________________________

\section{Conclusions}

Production cross sections are measured for the vector $\Dstar$ and for the
pseudoscalar charmed mesons $\Dzero$, $\Dsubs$ and, for the first 
time at HERA, also $\Dplus$ mesons through their decay $D^+ \to K^- \pi^+ \pi^+$.
The measurements rely on the proper reconstruction of the vertex separation distance and
its error for the $D$-meson decays.

The inclusive  $\Dstar$ production cross sections are, within errors,
in agreement with  previous measurements and with Monte Carlo
predictions based on the leading order AROMA generator program
including parton shower modelling.

Differential cross sections are measured for $\Dzero$ and $\Dplus$
mesons as a function of meson transverse momentum and rapidity, and as a
function of the event kinematic variables $y$ and $Q^2$.  They are
found to be reasonably well described by the LO Monte Carlo
predictions.

Based on the measured cross sections, the fragmentation-sensitive
parameters  $P_V$, $R_{u/d}$ and $\gamma_{s}$ are extracted. They
compare favourably with the present world average values, and as such
support the hypothesis of universality of charm fragmentation.

%%_________________________________________________________________________________


\section*{Acknowledgments}

We are grateful to the HERA machine group.
We thank the engineers and technicians for their work in constructing and now
maintaining the H1 detector, our funding agencies for 
financial support, and the
DESY technical staff for continual assistance.

%% _________________________________________________________________________________


%
%   References for D-meson paper
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{thebibliography}

% ---------------------------------------------------------------------------------------------
\end{document}

