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\begin{document}

\title{Open charm and beauty production}
\author{SANJAY PADHI}
\author{(On behalf of the H1 and ZEUS collaborations)}
\address{Deutsches Elektronen-Synchrotron, \\ 
Notkestrasse 85, 22607 Hamburg, Germany\\
E-mail: spadhi@mail.desy.de}  
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% You may repeat \author \address as often as necessary      %
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\maketitle
\abstracts{
New results in open heavy flavour production in $ep$
collisions are presented. Aspects sensitive to the various stages of
the production mechanism are compared to the experimental data. Based
on the measurements and the comparison with leading-order parton
shower Monte Carlo models and next-to-leading order QCD
predictions, conclusions related to data alone as well as model
initiated issues are drawn.}
%\vspace{-0.95cm}
%
\section{Introduction}
%
Open heavy quark production at HERA offers a novel way of testing both
perturbative and non-perturbative aspects of quantum chromodynamics
(QCD). For the quark to be ``heavy'', its mass\footnote{The top quark,
  however, is too heavy and decays weakly before hadronizing.} has to be larger than the QCD
scale $\Lambda_{\text QCD} \sim$ 200 - 300~MeV. Hence throughout this
article, ``heavy quark'' will refer to a charm or a bottom
quark. The following steps in 
heavy quark production can be
distinguished (See Fig.~\ref{fig:tags}~a): 
\begin{itemize}
\item Hard subprocess: basic $2 \rightarrow 2$ process, e.g $\gamma g
\rightarrow Q \bar{Q}$, boson gluon fusion (BGF) to form the heavy
quark $Q\bar{Q}$ pair. 
\item Initial/Final state parton radiation.
\item Fragmentation: transition of final state coloured partons to colourless
  hadrons.
\item Unstable hadrons decay according to their branching ratios.
\end{itemize}
The first two stages are based on perturbative QCD (pQCD)
calculations, while the last two rely on non-perturbative models. 
The cross section of a generic collision of two hadrons
producing heavy quark pairs at HERA can be written as: 
\begin{equation}
\sigma(S) = \sum_{i,j} \int dx_1 \int dx_2 f_i^{\gamma}(x_1, \mu_F)
f_j^{p}(x_2, \mu_F) \hat{\sigma}_{ij}(x_1 x_2 S, m_Q^2, \mu_R^{2})
\label{eq:partonic}
\end{equation}
where the cross section is a convolution of parton densities,
$f_i^{\gamma}(x_1, \mu_F)$ and $f_j^{p}(x_2, \mu_F)$ of the photon and
proton with the short distance cross section $\hat{\sigma}_{ij}$. The
fractional hadronic momenta carried by the interacting partons are represented
by $x_1$ and $x_2$ with $\mu_F$ and  $\mu_R$ as the factorisation and
renormalisation scales. The partonic cross section presented 
above further needs to be convoluted with a fragmentation function to
obtain the hadronic cross section. 
 
Two different collinear schemes and their combination are used for 
heavy quark production. In the massive-quark or fixed-order
next-to-leading order scheme (FO NLO), the heavy quark $Q$ is
on-mass-shell and 
it only appears in the final state, but not as an active parton inside
the incoming photon or proton. The calculation is expected to be valid for
$p_T \sim m_Q$. However in the massless-quark scheme or zero-mass
variable-flavour-number scheme (VFNS), the quark $Q$, (with $p_T
> m_Q$) is treated as massless, such that it appears as an active
parton of 
the incoming photon or proton, with a non perturbative parton density
function. The attempt made to combine the results from FO
NLO with suitably subtracted VFNS and with perturbative fragmentation
functions (PFFs), resulted in the fixed order next-to-leading logarithm
(FONLL) scheme\cite{mca}. The FONLL thus is expected to give more
reliable predictions for heavy quark production. 
%
\subsection{Experimental techniques}\label{subsec:experitech}
The reconstruction of heavy quark mesons in the two HERA experiments 
H1 and ZEUS is based on either mass or lifetime tags. Due to
the large $b$-quark mass H1 and ZEUS consider muons from
semi-leptonic $b$-decays, which usually have high values of transverse
momentum $p_T^{rel}$, with respect to the closest $b$ quark initiated
jet axis. The muons coming from charm and light quark decays have low
values of $p_T^{rel}$ as shown in Fig.~\ref{fig:tags}(b). Also, the decay
length of beauty is longer than that for 
charm and other light quarks. Hence the silicon vertex detector in
H1 was used to tag the lifetime (see Fig.~\ref{fig:tags}(c)) to
distinguish beauty from other quarks. 
\begin{figure}[ht]
\centerline{\epsfxsize=4.7cm\epsfbox{spadhi_hard_jet.eps}
\hspace{-0.7cm}
\epsfxsize=4.2cm\epsfbox{spadhi_ptrelh1.eps}
\hspace{-0.7cm}
\epsfxsize=4.2cm\epsfbox{spadhi_deltah1.eps} 
 \put(-295,0){\bf (a)}
 \put(-195,0){\bf (b)}
 \put(-90,0){\bf (c)}
}
\caption{(a) Representation of the various stages of production and
  fragmentation of heavy quarks. The distribution of (b) the muon $p_T^{rel}$
  relative to the jet axis and (c) the impact parameter $\delta$, of the
  muon track, are compared with the Monte Carlo simulations.\label{fig:tags}}
\end{figure}

The fraction of events from $b$-decays in a specific data
sample\cite{h1a} extracted on a statistical basis by adjusting the relative
fractions of the simulated bottom, charm and light quarks decays to
the measured $p_T^{rel}$ and impact parameter $\delta$
distribution\footnote{The impact parameter $\delta$ is calculated in
  the plane transverse to the beam axis. Its magnitude is 
  the distance of the closest approach of the track to the
  primary event vertex.} is respectively found to be $f_b = 28.8 \pm
2.8\%$ and $f_b = 28.0 \pm 4.2\%$ in the H1 analysis. The two independent methods are found
to be consistent in predicting the beauty fraction. 

The charmed hadrons on the other hand are generally observed by forming the
invariant mass of the tracks identified with a specific decay
channel. In the presence of a
vertex detector backgrounds can be reduced by
measuring the displaced vertices from the primary interaction
point. The recent measurement\cite{h1b} by H1 for the charmed meson $D^+$, with
a tag on the decay significance length, shows an improvement in
signal to background ratio by a factor of $\sim \mathcal{O}(50)$.  

In the following sections, results of open heavy quark production from
HERA are discussed. Beauty, due to its large mass, is expected to have a
faster convergence in the perturbative expansion in powers of the
strong coupling constant $\alpha_s$. Thus QCD calculations are expected
to provide more reliable predictions for beauty than for charm, and
will be discussed 
in the next section followed by the experimental results for open
charm production.
%
\section{Measurements of open beauty cross
  sections}\label{sec:openbeauty}
One of the primary limitations of experimental measurements
is the restriction of kinematic region, which is usually defined by
the detector acceptance. Measurements 
performed in the restricted region are often extrapolated to the full
phase space using a particular theoretical framework (either Monte
Carlo models or next-to-leading order calculations), in order to
compare between different experiments. As long as the model
assumptions are not absorbed in the experimental results, but in the
corresponding QCD predictions, the published measurements can be
considered to be ``safe''. 

On the other hand in the case of extrapolation to
an unmeasured kinematic region, the extrapolation done should not only
be quoted, but should also be compared within the same
theoretical framework. For example the extrapolated cross section to
an unmeasured phase space using a given Monte Carlo simulation cannot
be reliably compared to the cross section predicted by a NLO
calculation. Thus
only those beauty measurements are considered here, where the
measurements are made in the visible kinematic region without
extrapolation.

The measurement of the beauty cross section by ZEUS\cite{zeusa} in the
deep inelastic scattering 
(DIS) regime, using the $p_T^{rel}$ method, as a function of $Q^2$ is shown in
Fig.~\ref{fig:B_inphpdis}(a). The total visible cross section
$\sigma$ was determined in the kinematic range $Q^2
>2$~GeV$^2$, inelasticity between $0.05 < y < 0.7$, with at least one
jet in the Breit frame with $E_{T,jet}^{Breit} > 6$~GeV and
pseudorapidity $\eta_{jet}^{Lab} < 2.5$, and with a muon between
$30^\circ < \theta^{\mu} < 160^{\circ}$ and $p_T^\mu >
2$~GeV. The measured cross section is 
\begin{displaymath}
\sigma = 38.7 \pm 7.7 \hspace{0.2cm}
(\text{stat.})^{+6.1}_{-5.0}(\text{syst.}) \hspace{0.2cm}{\text{pb.}}  
\end{displaymath} 
about a factor of 1.4 above the NLO QCD predictions of
$28.1^{+5.3}_{-3.5}$~pb. It is, however, consistent with the measurement within the
large theoretical and experimental uncertainties. The errors on the
prediction were estimated by varying the factorisation and
renormalisation scales by a factor of 2 and the b-quark mass between
4.5 and 5.0~GeV. CASCADE, based on the CCFM evolution equations, is
in agreement with the measured cross section. The RAPGAP Monte carlo
simulation based on DGLAP on the other hand lies below the data.
\begin{figure}[ht]
\centerline{\hspace{0.0cm}\epsfxsize=6.0cm\epsfbox{spadhi_diszeusB.ps}
\hspace{-0.5cm}
\epsfxsize=6.4cm\epsfbox{spadhi_zeush1_bphppt.eps} 
 \put(-320,35){\bf (a)}
 \put(-140,35){\bf (b)}
}
\caption{Beauty differential cross section (a) in DIS as a function of
  $Q^2$ with at least one reconstructed jet in the Breit frame in
  association with a muon and (b) the ratio of measured cross section
  over theoretical expectation, in the photoproduction regime 
  as a function of $p_T^{\mu}$, for events with two jets and a muon
  with $p_T^{\mu} > 2.5$.\label{fig:B_inphpdis}}
\end{figure}

In order to obtain a unified picture on the beauty production H1\cite{h1a} and
ZEUS\cite{zeusb} recently measured the beauty cross section in the photoproduction
regime within almost identical phase
spaces. Fig.~\ref{fig:B_inphpdis}(b) shows the ratio of the measured cross
section over theoretical expectations as a function of $p_T^{\mu}$
obtained either using the $p_T^{rel}$ and $\delta$ method (H1) or the $p_T^{rel}$
method (ZEUS). The low
virtuality ($Q^2 < 1$~GeV$^2$), dijet photoproduction samples, are
considered with $E_T^{jet1(2)} > 7(6)$~GeV, $|\eta^{jet}| < 2.5$ and
$0.2 < y < 0.8$. The differential cross section as a function of the 
transverse momentum of the associated muons with $p_T^{\mu} > 2.5$~GeV
and the pseudorapidity $-0.55 < \eta^{\mu} < 1.1$~(H1) or $-1.6 <
\eta^{\mu} < 2.3$~(ZEUS) are found to be in agreement within 
the two experiments. The FO NLO QCD calculation is in reasonable
agreement with the data within the large experimental and theoretical
uncertainties, which are of similar size, except for the lowest
$p_T^\mu$ H1 measured bin.
%
\section{Measurements of open charm cross sections}\label{sec:opencharm}
%
Inclusive production of the charmed mesons $D^+$, $D^0$, $D_s$ and $D^+$
in DIS was measured\cite{h1b} with the H1 detector, using an integrated
luminosity of $48$~pb$^{-1}$. In these measurements the charm lifetime
properties are exploited using a two layer silicon vertex
detector. Differential cross sections are measured for the $D^+$ and
$D^0$ mesons and compared with the AROMA MC
predictions in the kinematic range $2 < Q^2 < 100$~GeV$^2$, $0.05 < y
< 0.7$, $p_t(D) > 2.5$~GeV and $|\eta(D)| < 1.5$. Results for the $D^+$
channel are shown in Fig.~\ref{fig:c_inclu}(A). The distributions are
well described both in shape and normalisation at LO with a parton
shower QCD simulation. The lower shaded band indicates the AROMA beauty
contribution, scaled by a factor of 4.3, according to the H1
published values\cite{h1c}.
\begin{figure}[ht]
\centerline{\hspace{0.0cm}\epsfxsize=6.2cm\epsfbox{spadhi_H1prelim-02-076.fig8.eps}
\hspace{-0.45cm}
\epsfxsize=6.0cm\epsfbox{spadhi_nll22.eps} 
 \put(-320,-2){\bf (A)}
 \put(-148,-2){\bf (B)}
}
\caption{Differential cross section for (A) $D^+$ as a function of
  $p_t(D^+)$, $Q^2$, $\eta(D^+)$ and electron inelasticity $y$
  in the DIS regime, compared with the AROMA predictions and (B) inclusive
  $D^*$ photoproduction events, as a function of a) $p_T(D^*)$, b) $\eta(D^*)$, c) $W(D^*)$
  and d) $z(D^*)$. FO NLO predictions with nominal parameters are given by
  the solid histogram. The VFNS predictions (labeled as NLL) are shown
  by the solid curves. The GRV photon parameterisation and direct
  photon predictions are given by dash-dotted and dotted lines.\label{fig:c_inclu}}
\end{figure}
 \begin{figure}[ht]
\centerline{\hspace{0.0cm}\epsfxsize=6.2cm\epsfysize=8.0cm\epsfbox{spadhi_fonll1.ps}
\hspace{0.2cm}
\epsfxsize=6.0cm\epsfysize=8.0cm\epsfbox{spadhi_zeush1sf.eps} 
 \put(-340,-2){\bf (A)}
 \put(-155,-2){\bf (B)}
}
\caption{(A) Differential cross section $d\sigma/d\eta$ for inclusive
  $D^*$ photoproduction in four $p_T$ regions, compared to FO NLO
  (histograms) and FONLL (curves) predictions. (B) Charm
  contribution to the proton structure function for different $Q^2$ at
  a function of $x$.\label{fig:c_insf}}
\end{figure}

Inclusive photoproduction of $D^*$ mesons has been measured\cite{zeusc} with
the ZEUS detector in the kinematic region $Q^2 < 1$~GeV$^2$,
$\gamma p$ centre-of-mass energies $130 < W < 280$~GeV, $1.9 <
p_T(D^*) < 20$~GeV and $|\eta(D^*)| < 1.6$ using an integrated
luminosity of $79$~pb$^{-1}$. The measured differential cross sections
were compared with FO NLO, VFNS and FONLL QCD predictions. In
Fig.~\ref{fig:c_inclu}(B) the differential cross section as a function
of a) $p_T(D*)$, b) $\eta(D^*)$, c) $W(D^*)$ and d) $z(D^*)$ are
compared with the FO~NLO and VFNS QCD calculations, where $z(D^*)$ is
the fraction of the photon energy carried by the charmed meson in the
proton rest frame. The precision of the data is enormously better than the
theoretical uncertainties. The uncertainties for VFNS are
larger than for the FO NLO calculations, which were obtained by
varying the charm mass and the renormalisation scales
simultaneously. The central FO NLO predictions are below the data,
mainly in the proton direction ($\eta > 0$) and the low $z$
region. The VFNS prediction 
is closer to the data, in particular it is better than FO NLO for
$d\sigma/dz$ and for positive pseudorapidity. The direct photon
processes alone in VFNS cannot describe the data distribution and hence a
significant resolved contribution is required, which as expected shows
sensitivity to the variation in the photon structure function
parameterisations.

Given the precision of the ZEUS data, the differential cross sections 
for different $p_T(D^*)$ regions as a function of $\eta(D^*)$ are
compared to the FO NLO and FONLL predictions in
Fig.~\ref{fig:c_insf}(A). This comparison allows to identify the
region of phase  
space where the discrepancy between the data and the calculations can
be localized. The data is close to the upper limit of the uncertainties of the
predictions and is significantly above both the FO NLO and FONLL
predictions at medium $p_T$ and positive $\eta$. As expected
due to the inherited properties from FO NLO, the FONLL is close to the
former at low $p_T$, but is surprisingly below FO NLO at large
$p_T$. 
\subsection{Measurements sensitive to structure functions and hard
  scattering}
The photon acts as a pointlike probe of the proton in DIS, where the
BGF (Fig.~\ref{fig:charmdiagrams}(e)) process dominates at HERA. This provides an unique opportunity to
study the charm contribution $F_2^{c\bar{c}}$, to the proton structure
$F_2$, which in turn is sensitive to the gluon density of
the proton. The production of $D^*$ mesons has been studied in DIS at
HERA\cite{zeusd} in the kinematic region $1.5 < Q^2 < 1000$~GeV$^2$, $0.02 < y <
0.5$, $1.5 < p_T(D^*) < 15 $~GeV and $|\eta(D^*)| < 1.5$. Predictions
from NLO QCD are found to be in reasonable agreement with the measured
differential cross
section as a function of $Q^2$ and $x$. Hence this calculation was then used to extract
$F_2^{c\bar{c}}$ using:
\begin{equation}
F_{2,\text{meas}}^{c\bar{c}} =
\frac{\sigma(x,Q^2)_{\text{meas}}}{\sigma(x,Q^2)_{\text{theo}}} \hspace{0.2cm}F_{2,\text{theo}}^{c\bar{c}}.
\end{equation} 
The extrapolation factors to the full phase space vary between 4.7 and
1.5 at low and high $Q^2$ respectively.
The extracted $F_2^{c\bar{c}}$ are shown compared with the NLO QCD
prediction in Fig.~\ref{fig:c_insf}(B). A remarkable confirmation of
the gluon density obtained from scaling violations of the inclusive
$F_2$ can be observed within $10\%$ accuracy. The
data shows a steep rise at low $x$, indicating a large contribution of
the gluon density in the proton. Since at low $Q^2$, the uncertainties of
the data are comparable to those from the PDF fit, the measured
differential cross section $\sigma(x,Q^2)$ should be used in future
fits to constrain the gluon density of the proton.
\begin{figure}[ht]
\centerline{\epsfxsize=2.5cm\epsfbox{spadhi_cg1.ps}
\hspace{0.0cm}
\epsfxsize=2.5cm\epsfbox{spadhi_cq.ps}
\hspace{0.0cm}
\epsfxsize=2.5cm\epsfbox{spadhi_cg2.ps}
 \hspace{0.0cm}
\epsfxsize=2.5cm\epsfbox{spadhi_gg-cc.ps}
\put(-260,3){\small (a)}
\put(-185,3){\small (b)}
\put(-105,3){\small (c)}
\put(-30,3){\small (d)}
\put(-255,-10){\line(0,0){6}}
\put(-255,-10){\line(1,0){155}}
\put(-180,-10){\line(0,1){6}}
\put(-100,-10){\line(0,1){6}}
\put(-210,-20){\small {\it Charm from photon}}
}
\centerline{\hspace{-8.0cm}\epsfxsize=2.5cm\epsfbox{spadhi_bgf.ps}
\put(-30,-5){\small (e)}
\put(30,50){\small (a), (b), (c), (d) : {\it Resolved photon processes} }
\put(85,30){\small (e) : {\it Direct photon process}}}
\caption{Various LO sub-processes with charm, dominant in
      the HERA region of phase space.\label{fig:charmdiagrams}}
\end{figure}

In an analogous way, dijet photoproduction at HERA allows to study
the structure of the photon. In leading order (LO) QCD the underlying
sub-processes (Fig.~\ref{fig:charmdiagrams}) can be divided into
either direct photon or resolved photon processes. In direct photon
processes the photon 
participates in the hard scatter predominantly via the boson-gluon
fusion process. Hence the fraction of the photon energy participating in
the formation of two highest transverse energy jets, experimentally defined as
$x_\gamma^{\rm obs}$, is close to 1. On the other hand in resolved photon
processes the photon acts like a source of incoming partons (quarks
and gluons) and only a fraction of its momentum participates in the
hard scatter, giving rise to $x_\gamma^{\rm obs} < 1$. 

ZEUS recently measured such a photon energy fraction associated with
at least one charm quark for dijet photoproduction events\cite{zeuse}. 
Fig.~\ref{fig:xgamma}(a) shows the differential cross section as a
function of $x_\gamma^{\rm obs}$ in
comparison with PYTHIA, HERWIG and CASCADE. Not only a rise at high
$x_\gamma^{\rm obs}$ is observed, there is a substantial tail at low
$x_\gamma^{\rm obs}$, which requires a significant ($\approx 40\%$)
LO-resolved contribution to describe the data. The low $x_\gamma^{\rm
  obs}$ data are consistent with the presence of resolved photon processes,
and can also be simulated in CASCADE via initial-state radiation
based on CCFM evolution. However, CASCADE overestimates the high
$x_\gamma^{\rm obs}$ region.

A similar study in both the $\gamma p$ and DIS regimes\cite{zeusf}
with two high 
$E_T^{jet}$ jets was 
done in order to probe the structure of the virtual photon. The ratio
of the low and high $x_\gamma^{\rm obs}$ regions is shown in
Fig.~\ref{fig:xgamma}(b) and compared to CASCADE and AROMA
expectations. The AROMA model using the DGLAP evolution scheme lies below
the data, whereas CASCADE with CCFM evolution is much closer to the data. The
fact that the ratio does not change significantly with $Q^2$ is in
marked contrast to the case in which the presence of charm is not
required\cite{zeusg}, suggesting that the suppression of the low $x_\gamma^{\rm
  obs}$ cross section due to the charm mass and the
suppression due to photon virtuality as the scales are not independent. 
\begin{figure}[ht]
\centerline{\hspace{0.0cm}\epsfxsize=5.0cm\epsfbox{spadhi_xgz1.ps}
\hspace{-0.2cm}
\epsfxsize=7.0cm\epsfbox{spadhi_result.casaro.eps} 
 \put(-314,32){\bf (a)}
 \put(-170,32){\bf (b)}
}
\caption{(a) Differential cross section $d\sigma/dx_{\gamma}^{\rm
    obs}$ for photoproduction events compared with MC
  simulations. Each MC distribution is normalised to the data, as
  indicated in the brackets. (b) Ratio of low to
  high $x_{\gamma}^{obs}$ for events with a $D^*$ compared to the
  predictions of AROMA and CASCADE, which do not include photon
  structure.\label{fig:xgamma}}
\end{figure}

In order to probe the dynamics of the hard scattering in the
sub-process (see Fig.~\ref{fig:charmdiagrams}) and in particular to
study the charm content 
of the photon, the differential cross section
$d\sigma/d|\cos\theta^*|$ as shown in Fig.~\ref{fig:cos}(a), as a function of
$|\cos\theta^*|$ was measured\cite{zeuse}, where $\theta^*$ is the angle
between the jet-jet axis and the beam direction in the dijet rest
frame. The distribution was measured for resolved-enriched
($x_\gamma^{\rm obs} < 0.75$) and direct-enriched ($x_\gamma^{\rm obs}
> 0.75$) samples. The cross section for the sample enriched with
resolved photon events exhibits a more rapid rise towards high
values of $|\cos\theta^*|$ than does the cross section for the
direct-enriched sample. The measured differential cross section in
comparison to the LO partonic matrix elements reflects the different
spins of the quark and gluon propagators in the dominant
diagrams. Consequently the subprocess $gg \rightarrow c\bar{c}$ 
cannot be the dominant resolved photon process for the charm dijet
events. 

Independent of any model assumptions, the Rutherford
scattering cross section for a $q$-exchange, ($\approx (1 -
|\cos\theta^*|)^{-1}$) should show a mild rise, in contrast to the
$g$-exchange diagram $\approx (1 - |\cos\theta^*|)^{-2}$. To quantify this rise,
the measured cross section for the direct and resolved enriched
samples were fitted to a function $1/(1 - |\cos\theta^*|)^\kappa$. The
resultant values of $\kappa$ are:
\begin{equation}
\begin{array}{rcl}
\kappa(x_\gamma^{\rm obs} < 0.75) = 1.735 \pm 0.176;  \\[1pt]
\kappa(x_\gamma^{\rm obs} > 0.75) = 0.743 \pm 0.107
\end{array}\label{eq:spa}
\end{equation}
Given the fact that the shapes are expected to be distorted due to the
additional parton radiation and hadronisation corrections, the results
are consistent with the Rutherford scattering expectations. 
\begin{figure}[ht]
\centerline{\hspace{0.0cm}\epsfxsize=6.0cm\epsfbox{spadhi_partonic.ps}
\hspace{-0.26cm}
\epsfxsize=6.0cm\epsfbox{spadhi_DESY-03-015_5b.ps} 
 \put(-314,-7){\bf (a)}
 \put(-145,-7){\bf (b)}
}
\caption{(a) Differential cross section $d\sigma/d|\cos\theta^*|$ for
  samples enriched in resolved and direct photon events, compared to
  leading order $2 \rightarrow 2$ matrix element with at least a charm
  in the final state. All matrix element distributions are normalised
  to the first data bin. (b) Differential cross section as a function
  of $\cos\theta^*$ for samples enriched in resolved and direct photon
  events compared with FO NLO predictions. The NLO uncertainty after
  hadronisation correction is given by the shaded band.\label{fig:cos}}
\end{figure}

The two jets are then distinguished into a charm-initiated jet ($D^*$
jet), and the other jet in $\eta - \phi$ space (the variable $\Delta R_i \equiv
\sqrt{(\phi_{jet_i} - \phi_{D^*})^2 +(\eta_{jet_i} - \eta_{D^*})^2
}$) is used with the $D^*$ jet having the smallest $\Delta R_{i(1=1,2)} <
1.0$. Thus the sign of the unfolded $\cos\theta^*$ distribution is
given by the direction of the $D^*$ meson. It is positive for the proton
direction and negative for the photon
direction. 

Fig.~\ref{fig:cos}(b) shows the differential cross section
as a function of $\cos\theta^*$ for the resolved- and direct-enriched
samples. The $\cos\theta^*$ distribution for direct-enriched sample is
almost symmetric, as expected from the BGF process. The slight
asymmetry can be explained by the PYTHIA estimation\cite{zeuse} of the resolved
process contribution to the direct enriched sample. The strong rise in
the cross section towards the photon direction for the resolved-enriched
sample is clear evidence for charm originating from the photon
with a gluon propagator. Although the shape for low $x_\gamma^{\rm
  obs}$ events is reproduced by both
CASCADE and the FO NLO predictions, the large uncertainties in the FO
NLO prediction localized in the photon direction indicate  
non-convergent terms in the perturbative 
expansion due to collinear emissions of partons causing a large scale
dependence. These terms then need to be absorbed either in the
fragmentation or in the charm distribution functions of the photon and
proton. Of course, to perform this absorption a charm contribution
in the structure function is needed.
\subsection{Charm fragmentation}
The non-perturbative aspect of QCD was studied in terms of charm
fragmentation. The data comparison to the
theoretical predictions in all hadronic colliders is done based on the
assumption of universality in charm fragmentation. This assumption
then allows to use the parameterisations of the 
fragmentation characteristics obtained in $e^+e^-$ annihilations in 
the calculations for charm production in $ep$ scattering. The
measurements of these fragmentation characteristics at HERA will not
only permit the verification of the universality of charm
fragmentation, but also can significantly contribute to the
understanding of the non-perturbative process.

The H1 and ZEUS measurements in the DIS\cite{h1b} and photoproduction\cite{zfr}
regime respectively, were used to obtain preliminary results on
the transfer of the quark's energy to a given meson. The fragmentation
fractions of the charm quark to the various mesons are shown in
Table~\ref{tab1}. In addition H1 and ZEUS measured cross sections,
which were used to test the isospin invariance of the fragmentation
process by calculating the ratio of neutral ($c\bar{u}$) to charged
$(c\bar{d})$ $D$ meson production. The measured ratio $R_{u/d}$ is
consistent 
with 1, confirming the isospin invariance. The $s$ quark production is
measured to be suppressed by a factor $\approx 3.5$ as shown by
$\gamma_s$. Both H1 and ZEUS rule out the naive spin counting
prediction of 0.75 for the fraction of $D$ mesons produced in a vector
state $P_V$. The results in the table are compared with the $e^+e^-$ world
average values\cite{leo}.
\begin{table}[ph]
\tbl{The fractions of $c$ quarks hadronising to a specific charmed
  meson. The results for $R_{u/d}$, $\gamma_s$ and $P_V$ are also
  provided, compared to the world average.\vspace*{1pt}}
{\footnotesize
\begin{tabular}{|c|r|r|r|}
\hline
{} &{} &{} &{}\\[-1.5ex]
{} & H1 (DIS)& ZEUS ($\gamma p$)& Combined
$e^+e^-$\\[1ex]
\hline
{} &{} &{} &{}\\[-1.5ex]
$f(c \rightarrow D^+)$ &$0.20 \pm
0.02^{+0.04+0.03}_{-0.03-0.02}$ & $0.249 \pm
0.014^{+0.004}_{-0.008}$& $0.232 \pm 0.010$\\[1ex]
$f(c \rightarrow D^0)$ &$0.66 \pm 0.05^{+0.12+0.09}_{-0.14-0.05}$
&$0.557 \pm 0.019^{+0.005}_{-0.013}$ &$0.549 \pm 0.023$ \\[1ex]
$f(c \rightarrow D^+_s)$ &$0.16 \pm 0.04^{+0.04+0.05}_{-0.04-0.05}$&
$0.107 \pm 0.009^{+0.005}_{-0.005}$ &$0.101 \pm 0.009$ \\[1ex]
$f(c \rightarrow \Lambda_c^+)$ &  &$0.076 \pm 0.020^{+0.017}_{-0.001}$
& $0.076 \pm 0.007$ \\[1ex]
$f(c \rightarrow D^{*+})$ &$0.26 \pm 0.02^{+0.06+0.03}_{-0.04-0.02}$ &
$0.223 \pm 0.009^{+0.003}_{-0.005}$ & $0.235 \pm 0.007$\\[1ex]
$R_{u/d}$ &$1.26 \pm 0.20^{+0.11+0.04}_{-0.11-0.04}$ &
$1.014 \pm 0.068^{+0.024}_{-0.031}$ & $1.00 \pm 0.09$\\[1ex]
$\gamma_s$ &$0.36 \pm 0.10^{+0.01+0.08}_{-0.01-0.08}$ &
$0.266 \pm 0.023^{+0.014}_{-0.012}$ & $0.26 \pm 0.03$\\[1ex]
$P_V$ &$0.69 \pm 0.045^{+0.004+0.009}_{-0.004-0.009}$ &
$0.554 \pm 0.019^{+0.008}_{-0.004}$ & $0.60 \pm 0.03$\\[1ex]
\hline
\end{tabular}\label{tab1} }
\vspace*{-13pt}
\end{table}

The final uncertainty in the non-perturbative part of the QCD
calculations is the fragmentation function, which affects both the
shape and the absolute normalisation of the 
predicted cross section. It was recently measured by the ZEUS
collaboration\cite{zff}. Charm associated jet events were used in the
photoproduction regime in order to measure the jet energy fraction
carried by a $D^*$ meson defined as $z = (E + p_{\parallel})_{D^*}/(E
+ p_{\parallel})_{jet})$, where the $p_{\parallel}$ is the
longitudinal momentum of the $D^*$ relative to the jet axis. The
relative cross section shown in Fig.~\ref{fig:frag}(a) is compared to the PYTHIA MC
simulations with different values of the free parameter $\epsilon$
within the Peterson fragmentation function\cite{pff}. Strong
sensitivity to the parameter $\epsilon$, varied between 0.01 to
0.1 was found. The fit using
$\chi^2$ minimisation yields $\epsilon = 0.064 \pm
0.006^{+0.011}_{-0.008}$ in comparison to 0.053 from LO fits\cite{ol} to the
LEP data. Although there are independent ways of measuring the
fragmentation function, in Fig.~\ref{fig:frag}(b) the OPAL and ARGUS
results with 
different $z$ definition are compared. They are similar in shape to the
HERA data. The
low-$z$ peak in the OPAL data arises from a significant
gluon-splitting due to the much higher centre-of-mass energy.
  \begin{figure}[ht]
\centerline{\hspace{0.0cm}\epsfxsize=6.5cm\epsfbox{spadhi_cfrag_2.eps}
\hspace{-0.9cm}
\epsfxsize=6.5cm\epsfbox{spadhi_cfrag_6.eps} 
 \put(-195,150){\bf (a)}
 \put(-150,150){\bf (b)}
}
\caption{Relative cross section $1/\sigma d\sigma/dz$ for the data
  compared (a) with PYTHIA predictions for different values of the
  parameter $\epsilon$ in the Peterson fragmentation function and (b)
  to $e^+e^-$ measurements, where the data sets are normalised to
  1/(bin width) for $z > 0.3$.\label{fig:frag}}
\end{figure}

The ZEUS fragmentation measurements are in good agreement with $e^+e^-$
annihilations, indicating the universality of charm fragmentation.
\section{Conclusions}
Open heavy flavour measurements provide an important tool for
understanding both perturbative and non-perturbative aspects of
QCD. The QCD calculations do a fair job in describing the
data, but fail within certain kinematic regions, and have large
theoretical uncertainties. Beauty measurements made
in almost identical kinematic phase spaces by the H1 and ZEUS
experiments agree with each other. Very precise open charm
measurements are made in both DIS and photoproduction.

Although the first HERA open charm signal consisted of $48 \pm 11$
events\cite{zeush}, with the enormous amount of available statistics,
measurements sensitive to each aspect of the convolution
($\text{PDF}_p \otimes \text{PDF}_\gamma \otimes \text{hard scatter}
\otimes \text{fragmentation}$) have been made. The recent trend at HERA
not only includes the open beauty measurements in the visible
range, but also structure function sensitive measurements like
$F_2^{c\bar{c}}$, evidence of charm originating from the photon,
  multiscale issues with charm in deeply inelastic virtual photon
  scattering and of course new measurements of the charm
  fragmentation confirming the evidence of its universality.
%\vspace*{8pt}

\enlargethispage{10mm}
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\end{document}
