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\title*{Open Heavy-Flavour Photoproduction at NLO}
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\toctitle{Open Heavy-Flavour Photoproduction at NLO}
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\titlerunning{Open Heavy-Flavour Photoproduction at NLO}
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\author{Gustav Kramer}
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\authorrunning{G.\ Kramer}
\institute{II.\ Institut f\"ur Theoretische Physik, Universit\"at Hamburg,
           Luruper Chaussee 149, D-22761 Hamburg, Germany}

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\begin{abstract}
We review the construction of non-pertubative fragmentation functions for
$D^{*\pm}$ mesons, both at leading and at next-to-leading order in two
different subtraction schemes in the so-called massless approach. The
fragmentation functions are determined by fitting to OPAL and ALEPH data
on inclusive $D^{*\pm}$ production in $e^+e^-$ annihilation. It is shown,
how recent data on inclusive $D^{*\pm}$ photoproduction in $ep$ collisions
at HERA are reproduced with two sets of fragmentation functions differing
in the subtraction schemes. The influence of different assumptions on the
charm distribution of the photon is pointed out.
\end{abstract}

\section{Introduction}

The production of heavy-flavoured hadrons (with charm or bottom) in the
colliding beam experiments with HERA provides new opportunities to study
the dynamics of perturbative QCD and to obtain information on the proton
and photon structure.
Heavy-flavour production at HERA is dominated by photoproduction events,
where the electron (or positron) is scattered by a small angle producing
photons of almost zero virtuality. In this case the electron scattering is
equivalent to the scattering of on-shell photons carrying a fraction of the
electron energy distributed according to the Weizs\"acker-Williams spectrum
\cite{1}. The dynamics of charm and bottom photoproduction can be probed
at HERA in a large kinematical region, the available center-of-mass energy
being one order of magnitude larger than in fixed target experiments.
As is well known, photoproduction at HERA is described with two mechanisms,
direct photoproduction, where the photon interacts directly with the quarks
coming out of the proton, and via resolved photon interactions, where the
photon fluctuates into parton constituents (quarks or gluons), which undergo
hard scattering with the partons from the proton.
Two types of approaches have been pursued  in the calculation of 
photoproduction cross sections for heavy-flavour production in next-to-leading
order (NLO). In the massive approach \cite{2} one assumes that light quarks and
gluons are the only active flavours within the ingoing proton and photon.
In this approach the heavy quark mass $m_Q \gg \Lambda_{QCD}$ acts as a cutoff
for initial and final state singularities and sets the scale for the
perturbative calculation. The cross section factorizes into a partonic 
hard-scattering cross section multiplied by light-quark and gluon densities.
In this so-called massive quark scheme, the number of active flavours in the
initial state is $n_f=3$, if charm and bottom are treated as massive, 
and, for example, $n_f=4$, if only the bottom quark is considered
massive. For the prediction of the total charm or bottom photoproduction
cross section this is the only possibility. The massive approach is also the
correct approach for the production of heavy flavoured hadrons with transverse
momenta $p_T^2 \simeq m_Q^2$. If $p_T^2 \gg m_Q^2$, mass terms in the hard
scattering cross sections are suppressed by powers of $m_Q/p_T$, but
potentially large terms $\propto \ln(p_T^2/m_Q^2)$ arise from the collinear
branching of gluons or photons into heavy quark-antiquark pairs or from the
collinear emission of a gluon by a heavy quark at large transverse momentum.
For $p_T \gg m_Q$ these terms spoil the convergence of the pertubation series
and must be summed by absorbing the collinear $\ln(p_T^2/m_Q^2)$ singular
terms into heavy-quark parton density functions (PDF's) of the proton or 
photon, respectively, or into fragmentation functions (FF's) of the heavy
quark into the heavy hadron. An alternative way to the absorption of the
large logarithm into parton densities and fragmentation functions is to treat
the heavy quark $Q$ as massless from the start and to absorb the collinear 
singularities into PDF's and FF's in the same way as it is done for the
lighter $u,~d$ and $s$ quarks using the standard $\overline{MS}$ subtraction
scheme. This allows for the consistent usage of the standard NLO parton 
densities with $n_f=4~(n_f=5)$ flavours for the proton or photon which have 
been constructed in this scheme. This way of calculating heavy-flavour 
production has been denoted the massless scheme \cite{3}. The mass of the 
heavy quark appears only in the starting scale of the c or b quark parton 
distribution functions of the proton and the photon and the fragmentation 
functions of the heavy quarks, c or b, into heavy flavoured hadrons. In the 
case of hadrons containing bottom quarks only the total photoproduction cross 
section has been measured so far \cite{40}, so that the massless approach has 
not become relevant yet.

For the production of charmed mesons, both H1 \cite{4} and ZEUS \cite{5,6}
have presented experimental results for $D^{*\pm}$ photoproduction. This
channel is favoured because of its clear experimental signal coming from the
$D^* \rightarrow D\pi$ decay with subsequent decay of the $D$ into $K\pi$ and
$K\pi\pi\pi$ channels. The data cover the $p_T$ range up to $p_T \simeq 10.5~
GeV$ \cite{4} and $p_T \simeq 12~GeV$ \cite{5,6}, so that at least for the
data in the high $p_T$ range the massless scheme should be applicable. In the
medium $p_T$ range it is conceivable that the massive (fixed-order) and 
the massless (resummed) approach are equally reliable, and only in the
low-$p_T$ range $p_T \simeq m_c$ the massive scheme is preferred.

In the massless scheme it is also easy to incorporate information on the 
fragmentation of $c$ quarks into $D^*$ mesons from other processes, as for
example $e^++e^- \rightarrow D^*+X$, into photoproduction of $D^*$'s at
HERA, since, according to the general factorization theorems, this information
is universal and can be used for the prediction of any inclusive $D^*$ 
production process in terms of universal FF's for $c \rightarrow D^*$. This
information on the $c \rightarrow D^*$ FF's comes from high-energy $e^+e^-$ 
data from LEP, where several data sets for $e^+e^- \rightarrow D^*X$, older
\cite{7,8} and newer \cite{9,10} ones, exist, which have been used to determine
the FF's of $D^*$'s employing two different subtraction schemes. 

In this report I shall review the influence of different descriptions of the
fragmentation process inside the resummed approach, valid for $p_T \gg m_c$,
on the predictions of inclusive $D^*$ production in $ep$ collisions at HERA. 
I shall give a comparison with ZEUS data only in cases relevant for the
comparison of these different approaches. A more complete confrontation of
the recent H1 \cite{4} and ZEUS \cite{5,6} data with the different theoretical
approaches, massive and massless, has been presented elsewhere at this workshop
\cite{11} and in the H1 \cite{4} and ZEUS \cite{5,6} papers.

In the next section I shall give a short account of the NLO formalism for
massless partons. In sect. 3 the calculation of the fragmentation functions
for $D^*$ mesons based on two different schemes is reviewed. Sect. 4 contains
a discussion of their consequences for the inclusive photoproduction of 
$D^*$'s at HERA. In sect. 5 I finish with some conclusions and an outlook
to possible work in the future.

\section{NLO Formalism for Massless Partons}

The calculation of the NLO inclusive cross section for the production of a
hadron $h$ with transverse momentum $p_T$ and rapidity $y$ in the resolved
mode starts from the formula
\begin{eqnarray}
   \frac{d^3\sigma}{dy\,d^2p_T}&=&\frac{1}{\pi}
\sum_{a,b,c}\int dx_\gamma dx_p{dx_h\over x_h^2}\,
F_a^\gamma(x_\gamma,M_\gamma^2)F_b^p(x_p,M_p^2)\, D_c^h(x_h,M_h^2)
\nonumber \\
&&\left[\frac{d\sigma_{ab\to c}^0}{dt}(s,t,u,\mu^2)
\,\delta\left(1+\frac{t+u}{s}\right)\right.\nonumber\\
&&\vphantom{\frac{\alpha_s(\mu^2)}{2\pi}}
  +\left.
\frac{\alpha_s(\mu^2)}{2\pi}K_{ab\to c}(s,t,u,\mu^2,M_\gamma^2,M_p^2,M_h^2)
\,\theta\left(1+\frac{t+u}{s}\right)\right],
\end{eqnarray}
where $s=(p_a+p_b)^2$, $t=(p_a-p_c)^2$ and $u=(p_b-p_c)^2$. The NLO corrections
\cite{12} $K_{ab \rightarrow c}$ comprise 16 channels and their $t \rightarrow
u$ crossed counterparts. $d\sigma^{0}_{ab \rightarrow c}/dt$ stands for the
LO hard-scattering cross sections for the transition $ab \rightarrow cX$,
where $X=d$ is a single parton in LO. $F_a^{\gamma}$ and $F_b^p$ are the 
PDF's of the photon and proton, respectively, which must be calculated at
scales $M_{\gamma}$ and $M_p$ with the help of the NLO evolution equations
derived for the $\overline{MS}$ subtraction scheme. Similarly, $D_c^h$
denots the FF for $c \rightarrow h$ (in our case $h=D^{*\pm}$) evolved up to 
the scale $M_h$, also in the $\overline{MS}$ subtraction scheme. We emphasize
that the summation over parton labes $a,b$ and $c$ includes also the charm
quark. This means, that the charm quark is also present in the initial state
inside the proton and particularly inside the photon.

The formula for $\gamma + p \rightarrow h + X$ via direct photoproduction
can be deduced from (1) by the replacements
\begin{eqnarray}
F_a^\gamma(x_\gamma,M_\gamma^2)&\to&\delta(1-x_\gamma),\nonumber \\
d\sigma_{ab\to c}^0&\to&d\sigma_{\gamma b\to c}^0,\nonumber \\\
K_{ab\to c}&\to&K_{\gamma b\to c}.
\end{eqnarray}
At LO, only the photon-gluon fusion process $\gamma g \rightarrow q\bar{q}$
and the Compton process $\gamma q \rightarrow gq$ contribute. NLO corrections
\cite{13} $K_{\gamma b \rightarrow c}$ include eight channels. It is well
known that NLO corrections lead to a reduction of the dependence on the
renormalization scale $\mu$ and the factorization scales $M_{\gamma}$,
$M_p$ and $M_h$ This way, they reduce the theoretical uncertainty and lead in
certain regions of phase space to sizable shifts in the cross section due to
additional parton emissions in the final state. It should also be mentioned,
that in NLO the definition of direct and resolved cross section looses its 
meaning. At higher orders only their sum is uniquely defined and can be 
compared to experimental data.

The derivation of these formulas is done in dimensional regularization, which
is needed to extract the ultraviolet, infrared and collinear divergencies.
Since the charm quark is considered massless as all the other quark flavours,
divergencies originating from the $cg$ collinear singularities are encountered,
which are absorbed into the bare PDF's and FF of the photon, proton and $D^*$
meson. At this step finite terms in the NLO matrix elements
$K_{ab \rightarrow c}$ remain, which are fixed by selecting the 
$\overline{MS}$ subtraction scheme. In the massive scheme these collinear
singularities are cut off by the finite quark mass $m_c$ and no subtraction
is necessary. In addition, there are no bare PDF's or FF's in this scheme,
which could absorb the subtraction terms.

\section{Fragmentation Functions for $D^*$}

Several different procedures to describe the fragmentation of $c$ quarks into
$D^*$'s have been developed. They differ whether and how information for the 
production of massive quarks in $e^+e^-$ annihilation
is included in the transition $c \rightarrow D^*$. They are described by
introducing the function $D_a^C(x,\mu^2,m_c^2)$ for the transition of a 
massless parton a (including the massless charm quark $c$)  
to a massive $c$ quark (denoted $C$). These functions
have been derived by Mele and Nason \cite{14} by matching the cross section
$d\sigma (e^+e^- \rightarrow C+X)/dx$, where all powers of $m_c^2/s$
have been neglected, in the massive scheme ($m_c \neq 0$) to the cross section
$d\sigma (e^+e^- \rightarrow a\bar{a}+X)$ in the massless scheme
($m_c=0$ from the beginning). The result is
\begin{eqnarray}
\frac{d\sigma_{e^+e^-\to C +X}}{dx}(x,s,m_c^2)
 = \sum_a\int_x^1\frac{dz}{z}\,D_a^C\left(\frac{x}{z},\mu^2,m_c^2\right)
   \frac{d\sigma_{e^+e^-\to a+X}}{dz}(z,s,\mu^2)
\end{eqnarray}
where
\begin{eqnarray}
D_a^C(x,\mu^2,m_c^2) 
 &=& \delta_{aC}\delta(1-x)
-\frac{\alpha_s(\mu^2)}{2\pi}d_{a\to C}(x,\mu^2,m_c^2)\\
d_{a\to C}(x,\mu^2,m_c^2)
 &=& -P_{q\to q}^{(0,T)}(x)\ln\frac{\mu^2}{m_c^2}+\frac{4}{3}
\left\{-2\delta(1-x)\vphantom{\left(\frac{1}{1-x}\right)_+}\right.\nonumber \\
&+&\left.2\!\left(\frac{1}{1-x}\right)_+\!\!\!
+\!4\!\left[\frac{\ln(1-x)}{1-x}\right]_+\!\!\!
-\!(1+x)\!\left[1+2\ln(1-x)\right]\!\right\}\quad \\
d_{g\to C}(x,\mu^2,m_c^2)
 &=& -P_{g\to q}^{(0,T)}(x)\ln\frac{\mu^2}{m_c^2}\\
d_{q,\bar q,\bar c\to C}(x,\mu^2,m_c^2) 
 &=& 0
\end{eqnarray}
This formula can be interpreted in many ways. Of course, when
$D_a^C(x,\mu^2,m_c^2)=\delta_{aC}\delta (1-x)$, the left hand side of (3) is 
just the cross section  for $e^+e^- \rightarrow c\bar{c}+X$ with massless $c$ 
quarks. If, on the other hand, the remaining terms in $D_a^C$ are moved to the
hard scattering factor $d\sigma (e^+e^- \rightarrow a+X)/dx$, we recover the
formula for the production of massive $c$ quarks at NLO up to the neglected
terms proportional to $m_c$. $\mu$ is the factorization scale for the massless
cross section, which is usually chosen as $\mu=\sqrt{s}$, where $s$ is the
squared center-of-mass energy. When $\mu=\sqrt{s} \simeq 2m_c$ the formula (3)
with (4) and (5) can be used directly as it stands. When $\mu=\sqrt{s} \gg m_c$
large logarithms appear which must be summed. To preserve the structure
contained in $d_{a \rightarrow C}(x,\mu^2,m_c^2)$ for finite $m_c$ the
function (4) is assumed to determine the FF for $a \rightarrow C$ at the 
starting scale $\mu_0 \simeq m_c$, and one calculates the FF at larger scales
$\mu$ via the usual Altarelli- Parisi equations \cite{15}. It follws from (5)
that the evolution is given by the well known massless evolution kernel
$P^{(0,T)}_{q \rightarrow q}(x)$. In this way these so-called perturbative FF's
are fully determined concerning the $\mu$ dependence and also the initial
conditions. Only the starting scale $\mu_0$ has to be chosen appropriately.
This approach with pertubative FF's was first applied to the calculation
of cross sections for the production of $b$ quarks in hadron-hadron collisions
\cite{16} and later to the production of $c$ quarks in $\gamma p$ \cite{17}
and $\gamma \gamma $ collisions \cite{18} based on the massless scheme
described in the previous section. Later it became clear that the pertubative
FF as just described is not sufficient to predict cross sections for the 
production of heavy flavoured hadrons, like $D^*$ mesons, and 
$D_a^C(x,\mu^2,m_c^2)$ must be convoluted with a non-pertubative FF 
$D_C^{D^*}(x,\mu_0^2)$ which accounts for the transition $C \rightarrow D^*$
\cite{17}. This FF can not be calculated in pertubation theory. It is asumed to
be universal, so that it can be fixed by comparing with one set of
experimental data and then used to predict other $D^*$ production cross 
sections. It has been obtained from inclusive $e^++e^- \rightarrow D^*+X$
cross sections measured at DORIS and LEP \cite{17}. It is clear that with this
procedure the total fragmentation $a \rightarrow D^*$, starting from the 
massless $c$ quark $a$, is a convolution of the pertubative FF with the
non-pertubative FF.

Instead of introducing the concept of a pertubative component for the FF one 
can incorporate the difference between the massless and massive cross section
for $e^++e^- \rightarrow C+X$ in a different way, namely, by combining the 
terms in $D_a^C$ proportional to $d_{a \rightarrow C}$ with the hard
scattering cross section $d\sigma (e^+e^- \rightarrow a+X)/dz$ in the sense
of changing the final-state factorization scheme. This amounts to substituting
in the hard-scattering cross section
\begin{eqnarray}
\alpha_s(\mu^2)P_{a\to c}^{(0,T)}(x)\ln\frac{s}{\mu^2} \to
\alpha_s(\mu^2)P_{a\to c}^{(0,T)}(x)\ln\frac{s}{\mu^2}
 -\alpha_s(\mu_0^2)d_{a\to C}(x,\mu_0^2,m_c^2)
\end{eqnarray}
One should note that the coupling in front of $d_{a \rightarrow C}$ is
evaluated at the input scale $\mu_0$. This shift leaves the cross section for
massive charm production for scales $\mu \simeq \mu_0 \simeq m_c$ unchanged,
when $D_a^C(x,\mu^2,m_c^2)=\delta_{aC}\delta_{aC}(1-x)$ is retained. In 
\cite{19} it was shown that this way of incorporating the differnce of the 
massless and massive hard-scattering cross section is consistent with the cross
section for $e^++e^- \rightarrow C+X$ including all terms up to O($\alpha_s^2$)
in the leading logarithmic approximation. It is clear that when this change of
final-state factorization scheme is performed for calculating the
$e^+e^- \rightarrow D^*X$ cross section, in order to construct the FF's for
$a \rightarrow D^*$ from experimental data, it must be intoduced also in all 
other hard-scattering cross sections for predicting inclusive $D^*$ production.
This change of the final-state factorization has been used for studying the
influence of various approximations in the fragmenation of $D^*$ mesons in 
\cite{20} and somewhat later for predicting photoproduction cross sections 
with fragmentation information coming from LEP experiments in \cite{21}.

The method of combining pertubative and non-pertubative FF's in the analysis
of $D^*$ production at DORIS and  LEP for the calculation of NLO inclusive
$D^*$ photoproduction cross sections has been applied in \cite{17} and
reviewed by Cacciari at the 1997 Workshop \cite{22}. As already said, it
turned out that the fragmentation of charm or bottom quarks into $D^*$ mesons
cannot be calculated from first principles in pertubation theory. In fact, in
order to realistically desribe the formation of the $D^*$ meson, a 
non-pertubative component, which is not known theoretically, is always needed.
Hence, it is appropriate to give up the pertubative component of the FF input
altogether and to describe the $c \rightarrow D^*$ and $b \rightarrow D^*$
transitions entirely by non-pertubative FF's, as it also done for the
fragmentation of $u$, $d$ and $s$ quarks into light mesons. This is the basis 
of our recent work \cite{23}, which has been compared with the new ZEUS 
and H1 experimental results in \cite{5,6} and \cite{4} (see also \cite{11}). It
should be clear that the parameters of the non-pertubative FF's differ,
depending on whether a pertubative component of the FF is introduced 
(via the initial conditions respectively via a change of the subtraction 
scheme) or not.

The construction of the non-pertubative FF's for $c \rightarrow D^*$ by
fitting large-s data on $e^++e^- \rightarrow D^*+X$ is complicated since in
hadronic $Z$ decays charmed mesons are expected to be produced either directly
through the hadronization of charm quarks in the process $Z \rightarrow 
c\bar{c}$ or via weak decays of $B$ hadrons produced in $Z \rightarrow 
b\bar{b}$ with approximately equal rate. Charmed mesons from $Z \rightarrow
c\bar{c}$ allow us to measure the $c$-quark FF. The main task of the
experimental analysis was to disentangle these two main sources of $D^{*\pm}$
production at the $Z$ boson resonance. In our first analysis, in which the
$d_{a \rightarrow C}$ subtraction scheme was used, we made use of the LEP
data from ALEPH \cite{7} and OPAL \cite{8}.

The starting scales for the $D^*$ FF's of the gluon and the $u$, $d$, $s$ and
$c$ quarks and antiquarks was set to $\mu_0=2m_c$ with $m_c=1.5~GeV$, whereas
for bottom it was set to $\mu_0=2m_b$ with $m_b= 5~GeV$. The FF's of the
gluon and the first three quark flavours are assumed to be zero at the
starting scale. For the parametrization of charm- and bottom quark FF's at 
their respective starting scales two different forms were employed, the 
so-called standard (S) and mixed (M) sets \cite{21}. Here we review only the
results with the mixed set. The results with the standard set are very similar
and lead to almost identical predictions for the photoproduction of $D^*$'s.
In the mixed set the FF of the bottom quark is assumed to be of the form
\begin{equation}
D_b^{D^{*\pm}}(x,\mu_0^2)=Nx^\alpha(1-x)^\beta
\end{equation}
which contains three parameters and is best suited for FF's which peak at 
smaller $x$. The charm-quark FF at the initial scale is equal to the Peterson
distribution \cite{24}
\begin{eqnarray}
D_c^{D^{*\pm}}(x,\mu_0^2) =
N\frac{x(1-x)^2}{[(1-x)^2+\epsilon x]^2}
%\frac{N}{x[1-1/x-\epsilon/(1-x)]^2}
\end{eqnarray}
which is particularly suitable to describe FF's that peak at large $x$. It
depends only on two free parameters, $N$ and $\epsilon$. Both ALEPH \cite{7}
and OPAL \cite{8} extracted the total $D^{*\pm}$ momentum distribution as well
as the individual contributions due to $Z \rightarrow c\bar{c}$ and
$Z \rightarrow b\bar{b}$. In the performed LO (NLO) analysis the one-loop
(two-loop) formula for $\alpha_s$ with $\Lambda^{(5)}_{\overline{MS}}=108~MeV~
(227~MeV)$ was used. The obtained values of $N$, $\alpha$, $\beta$ and 
$\epsilon$ resulting from the combined LO (NLO) fits to the ALEPH and OPAL
data are given in \cite{21}. The $\epsilon$ parameter in the LO (NLO) fit is
$\epsilon = 0.0856~(0.0204)$. In the NLO fit $\epsilon$ is very much reduced
since part of the fragmentation is described by the pertubative component.  
In Fig. 1 the fits are compared to the OPAL \cite{8} data. For the fits to the 
ALEPH data see \cite{21}.
Both, the LO and NLO results are shown. Except for very small $x$, the
LO and NLO curves are very similar. At small $x$ large differences occur, 
indicating that in this region the pertubative treatment ceases to be valid.
Also, in this region, the massless approximation is not valid anymore. 
In any case, due to the finite $D^*$ mass, the $D^*$ can only be 
produced for $x>x_{min}=2m(D^*)/M_Z=0.046$.
This is the region, where the NLO results turn negative. Therefore the results
are meaningful only for $x>x_{cut}=0.1$. For the fitting actually only the $x$
bins in the interval $[0.1,0.9]$ were used. The quality of the fit measured 
in terms of average $\chi^2$ for all selected data points is quite good. 
Details are found in \cite{21}. Also the branching fractions of the two
transitions $c \rightarrow D^*$ and $b \rightarrow D^*$ which are calculated 
from the FF's over $x$ in the interval $x_{cut}<x<1$ came out in agreement 
with the measured values: $B_c=0.259$ and $B_b=0.252$ \cite{8} with errors of
$25\%$. The corresponding theoretical values are $0.231~(0.221)$ and 
$0.215~(0.219)$ for the LO (NLO) fit. These numbers show that we have the 
correct normalization. Another quantity of interest is the mean momentum 
fraction $<x>_Q(\mu)$ where $Q=c,b$. This depends on the scale, where it is 
evaluated. For $\mu=M_Z$ we have $<x>_c=0.490~(0.556)$ for the LO (NLO) FF, 
which has to be compared to $<x>_c=0.495^{+0.010}_{-0.011} \pm 0.007$ \cite{7} 
and $<x>_c=0.515^{+0.008}_{-0.005}\pm 0.010$ \cite{8}.

Looking at the curves in Fig. 1 we notice that the NLO fits lead to negative
cross sections in the vicinity of $x=1$. This is directly conneted with the
subtraction of the pertuabative $d_{a \rightarrow C}$ functions. To remove
this unphysical behaviour one must resum the pertubative contributions for
$x \rightarrow 1$ originating from the finite mass cross section. Also due to 
this extra complication the description in terms of pure non-pertubative FF's
is preferred. Negative cross sections at $x \leq 1$ occur also in all the 
other formulations with pertubative pieces in the FF's \cite{17,25}. 
\begin{figure}[tb]
\begin{center}
\epsfig{file=kramer.1,height=6cm}
\caption{Comparison of the OPAL data \cite{9} on $e^++e^-\to D^{*\pm}+X$ with
the LO~M (dashed) and NLO~M (solid) calculations. The three sets of curves
and data correspond to the $Z \rightarrow c\bar{c}$ and
$Z \rightarrow b\bar{b}$ samples as well as their combination.}
\end{center}
\end{figure}
With the same ALEPH \cite{7} and OPAL \cite{8} data we performed fits also in
the truly massless scheme discarding the pertubative contributions in form
of the $d_{a \to C}$ functions. Here the fit could be done in the interval
$[0.1,1]$ in contrast to the fit with the $d_{a \to C}$ functions, which was
limited to $[0.1,0.9]$. The main difference  in the NLO fit was a change of
the Peterson parameter to $\epsilon =0.116$. The fit was equally good, the
$\chi^2$ changed only from $0.96$ to $1.01$. Of course, also the fragmentation
parameters for $b \to D^*$ fragmentation changed somewhat, but much less than
the parameters for the $c$ fragmentation.

In 1997 more precise data for $e^++e^- \rightarrow D^{*\pm}+X$ from ALEPH
\cite{9} and OPAL \cite{10} became available. These new samples are compatible
with the respective older samples, but not with each other. Both ALEPH \cite{9}
and OPAL \cite{10} presented momentum distributions for their full
$Z \rightarrow D^{*\pm}+X$ sample and for the $Z \rightarrow b\bar{b}$
subsample, but not separately for $Z \rightarrow c\bar{c}$ as in the previous
measurements. We used the new data to perform only fits without the 
$d_{a \rightarrow C}$ subtraction. The fit parameters for the two data sets in
LO and NLO and for the two flavours $c$ and $b$ are displayed
in Tab. I of \cite{23}. The obtained parameters for ALEPH and OPAL come out
different indicating that the two data sets do not 
overlap inside their errors. Both sets have much larger $\epsilon$
parameters in NLO compared to the earlier fits with the massive subtraction.
In Fig. 2 the OPAL \cite{10} data are compared with the sets LO O and NLO O 
(O stands for OPAL). The comparison of the corresponding LO and NLO fits to 
the ALEPH data can be found in Fig. 1b of \cite{23}. Except at very small $x$, 
the LO and NLO results are again very similar.
There are large differences between the LO and NLO
parameters obtained from the same data. The value $\epsilon = 0.0851$ of set
LO O is very similar to $\epsilon =0.0856$ of the set obtained from the older 
data \cite{7,8}. Compared to the NLO fit with the $d_{a \rightarrow C}$ 
subtraction shown in Fig. 1, the new NLO fit in Fig. 2 stays positive
also for $x \geq 0.9$, coincides there with the LO result, and nicely 
describes the data.

\begin{figure}[tb]
\begin{center}
\epsfig{file=kramer.2,height=6cm}
\caption{Comparison of the OPAL data \cite{11} on $e^+e^-\to D^{*\pm}+X$ with
the LO~O (dashed) and NLO~O (solid) calculations. The three sets of curves
correspond to the $Z \rightarrow c\bar{c},~ Z \rightarrow b\bar{b}$ and full
samples.}
\end{center}
\end{figure}
As with the fits to the older data, we studied also the branching fractions
and the mean momentum fractions for the two sets and compared them with the 
available experimental information. Both quantities turned out to be in 
reasonable agreement with the data, details are found in \cite{23}. We tested 
also, whether the FF's constructed from LEP data lead to a consistent
description of $e^+e^- \rightarrow D^*X$-data at lower energies. This presents
a direct test of the underlying scaling violations of fragmentation via the 
time-like Altarelli-Parisi equations. The comparison with data from ARGUS
\cite{26} at $\sqrt s = 10.49~GeV$, from HRS \cite{27} at $\sqrt s= 29~GeV$
and from TASSO \cite {28} at $\sqrt s = 34.2~GeV$ calculated with the LO and
NLO OPAL and ALEPH sets can be seen in in Fig. 2a,b of \cite{23} together with
the OPAL and ALEPH data, respectively. The agreement with all the data is
quite good. These comparisons assure us that the $D^{*\pm}$ FF's, although
constructed from data at $\sqrt{s} = M_Z$, lead to a valid description of 
$D^{*\pm}$ fragmentation at lower scales.

Based on the universality of fragmentation we have made predictions for
inclusive $D^{*\pm}$ photoproduction at HERA. The results for the FF's based on
the older LEP data and empoying the massive charm subtraction scheme are found 
in \cite{21} as well as the comparison with the ZEUS data \cite{29} published
in 1997. The more recent H1 \cite{4} and ZEUS \cite{5,6} have been compared 
only with the more recent predictions based on FF's obtained with the recent
ALEPH and OPAL results and without the massive subtraction. Such comparisons
have been shown also at the workshop in Coldewey's contribution \cite{11}. In
\cite{23} we compared with preliminary ZEUS data from their 1996 analysis 
\cite{30}. In the next section I shall present some of these results with
particular emphasis on how the two fragmentation schemes, with and without the
massive subtraction, influence the predictions. In addition I shall discuss
the contribution of the resolved cross section.

\section{$D^{*\pm}$ Production in Low-$Q^2$ ep Collisions}

It is clear that the cross section for inclusive $D^{*\pm}$ production in
low-$Q^2$ ep collisions at HERA depends on input from different sources.
First, it depends on the way the FF for the $D^{*\pm}$ is constructed. In
\cite{20} several versions of approximating the $D^*$ fragmentation have been
investigated. For example, in \cite{20} the $\epsilon $ parameter in the
Peterson fragmentation formula was varied by $\pm 0.02$. This changed the cross
section by $\pm 15\%$ with the branching fraction fixed to its experimental 
value. This is not a very large change. Since in our more recent work
\cite{21,23}
the $D^*$ FF was fixed by LEP data this dependence on the FF is largely
eliminated. The final prediction, though, may still depend on the scheme used
to construct the fragmentation in NLO. i.e. with or without the massive
subtraction terms. When we fitted to the LEP data, we also had to take into 
account $D^{*\pm}$ production by bottom-quark fragmentation. In ep collisions,
however, the bottom quark does not contribute to $D^{*\pm}$ production below 
the threshold at $m_T=\sqrt{p_T^2+m_c^2}=m_b$. In contrast to $e^+e^-$-
annihilation the bottom quark does not contribute significantly even above
threshold. In \cite{21} we found that the relative contribution due to the
$b$-quark FF's is below $5\%$ and decreases with increasing $p_T$.

% 
\begin{figure}[tb]
\begin{center}
\epsfig{file=kramer.3,height=5cm}
\caption{Comparison of the ZEUS data on $e+p\to D^{*\pm}+X$ with the NLO
calculations according to cases I (dashed) and II (solid).}
\end{center}  
\end{figure}
%
Another issue is the problem of the scheme dependence in the construction of
the FF's. Above we presented the results for $D^{*\pm}$ FF's obtained from
fitting the same LEP data, but with the different schemes, one with and one 
without the massive subtraction $d_{a \rightarrow C}$. As we have seen, these
two schemes lead to completely different non-pertubative components in NLO,
expressed by two completely different $\epsilon $ parameters, $\epsilon=0.0204$
and $\epsilon = 0.116$, respectively. In Fig. 3 I show the predictions
for the $D^{*\pm}$ photoproduction cross section with the two schemes. The
dashed curve (case I) is with $d_{a \rightarrow C}$ subtraction and the full
curve (case II) without. In Fig. 3 the $p_T$ distributions $d\sigma /dp_T$  is
compared to the older ZEUS \cite{29} data. We observe, that the two predictions
differ approximately by $25\%$. We think that this difference is 
characteristically the error caused by using two different 
fragmentation approaches. We note that
the cross section with the massive subtraction lies higher than without the
subtraction, which is based on the pure $\overline{MS}$ factorization. This is
in contrast to the predictions of Cacciari et al. \cite{17}, where the
$d_{a \rightarrow C}$ functions are taken into account via the initial 
conditions of the FF's. Although these two approaches are equivalent up to 
O($\alpha_s^2$) in the logarithmic terms, they differ in nonlogarithmic terms 
in the same order. We assume that this difference in the treatment of
the $d_{a \rightarrow C}$ terms is the explanation for the smaller $D^{*}$
photoproduction cross sections in \cite{17}.

The largest uncertainty in the calculation of the $D^{*\pm}$ photoproduction
cross section is related to the charm content of the resolved photon. The most
sensitive distributions to study this dependence are the differential rapidity
distributions for fixed $p_T$. To illustrate the sensitivity to the charm PDF
of the photon, the NLO analysis (here with the NLO S fragmentation function and
with the FF in the massive subtraction scheme from \cite{21}) with this PDF
switched off is shown in Fig. 4 and compared to the full calculation. We 
observe that $73\%$~to~$100\%$ ($50\%$~to~$89\%$) of the resolved (full) cross
section is induced by the charm content of the photon, which is most important
in the backward direction. In Fig. 4 the cross section is plotted as a 
function of $y_{lab}$, integrated over $3<p_T<12~GeV$. We emphasize, the lower
$p_T$ limit may be somewhat too small to justify the massless scheme. A larger
$p_T$ limit should be suited better. However, here the influence of the 
resolved component diminishes somewhat. The three experimental points are 
from the first ZEUS measurements \cite{29}.

% 
\begin{figure}[tb]
\begin{center}
\epsfig{file=kramer.4,height=5cm}
\caption{Sensitivity to the charm content of the photon.}
\end{center}  
\end{figure}
%

It is well known, that the pure massive scheme \cite{2} contains a contribution
from the charm component in the photon. This is hidden in the NLO corrections 
to the direct cross section. As long as all $m_c^2/p_T^2$ terms are negligible
this contribution, in a first approximation, should be equivalent in the
massless scheme to the LO resolved component. Therefore it is of interest to
see the influence of the NLO corrections to the resolved component since this
contribution is certainly absent in the massive scheme. This is shown
in Fig. 5, where the NLO O calculation of $d\sigma/dp_T$ is compared to the
case, where the NLO terms in the partonic cross sections are switched off.
For $p_T \geq 5~GeV$ we observe an increase of up to $50\%$ due to the NLO
corrections. This is almost the difference between the results of the massive 
scheme and the massless scheme, as can be seen in Fig. 2 of \cite{5}. In Fig. 5
the most recent ZEUS data \cite{5} are compared to the full NLO massless 
calculation of \cite{23}. The experimental
data lie somewhat above the predicted results, in particular for the larger 
$p_T$ values, where the agreement actually should be better. The difference 
amounts to approximately $30\%$ (not considering the experimental errors) for 
$p_T > 5~GeV$. In Fig. 5 we plotted also the LO O prediction. It is very 
similar to the NLO O result showing good pertubative stability. We emphasize 
that the LO curves are obtained with LO PDF's, FF's and the one-loop formula 
for $\alpha_s$.
%
\begin{figure}[tb]
\begin{center}
\epsfig{file=kramer.5,width=6cm}
\caption{Comparison of the ZEUS data on $e+p\to D^{*\pm}+X$ with the LO~O
(dashed) and NLO~O (solid) calculations. The dot-dashed curve emerges from the
solid one by switching off the NLO terms in the partonic cross sections of the
resolved processes.}
\end{center}  
\end{figure}
%

We have seen above that the charm component of the resolved photon makes an
important contribution to the $D^{*\pm}$ photoproduction cross section. All
the results shown so far are obtained with the parametrization of GRV in LO or
NLO \cite{31}. It is interesting to see the results for other photon PDF's.
Such comparisons are shown in Fig. 6, where the differential rapidity 
distributions are plotted after integration over four different $p_T$ ranges
$p_{T,min} < p_T < 12~GeV$ with $p_{T,min}=2,3,4$ and $6~GeV$. Besides GRV
the photon PDF's of Gordon and Storrow (denoted GS92) \cite{32} and two others
from Aurenche et al. for massless charm quarks \cite{33} denoted AFG and for 
massive charm quarks \cite{34} denoted ACFGP($m_c$) are employed. We observe 
a significant variation of the cross sections, especially also in the 
backward direction. The difference is particularly marked for the curves with $p_{T,min}=2~GeV$. For $p_{T,min}=6~GeV$ the differences are much less. In this
figure the various predictions are also compared to the recent ZEUS data 
\cite{5}. Of course, for $p_{T,min}=2~GeV$ we would not expect that the 
massless scheme yields reliable predictions, although the GS92 curve comes 
close to the data. All other PDF's give too large cross sections in this case.
Also for the other $p_{T,min}=3,4$ and $6~GeV$ the GS92 prediction agrees 
reasonably well with the data points, but also GRV and AFG agree for the two 
largest $p_{T,min}$'s. The GS92 prediction is also better in the forward 
direction where the deviation between the data and the prediction is largest.

%
\begin{figure}[tb]
\begin{tabular}{cc}
\parbox{7cm}{
\epsfig{file=kramer.6,width=5cm}
}
&
\parbox{7cm}{
\epsfig{file=kramer.7,width=5cm}
}
\\
\parbox{7cm}{
\epsfig{file=kramer.8,width=5cm}
}
&
\parbox{7cm}{
\epsfig{file=kramer.9,width=5cm}
}
\end{tabular}
\caption{Comparison of the ZEUS data on $e+p\to D^{*\pm}+X$ with the NLO~O
calculation for different photon PDF's.}
\end{figure}
%

Very recently the ZEUS collaboration also presented data in a smaller total
center-of-mass energy region $W$, $80<W<120~GeV$ \cite{6} compared to the
high $W$ region $130<W<280~GeV$ already published \cite{5}. The $p_T$ and
$y_{lab}$ intervals of these data are $2<p_T<8~GeV$ and $-1<y_{lab}<1.5$. The
predicted $d\sigma/dp_T$ distribution based on our fragmentation approach
without the massive subtraction and $\epsilon =0.116$ is in good agreement 
with the data. The cross section as a function of $y_{lab}$ integrated over
the region $3.25<p_T<8~GeV$ agrees best with the newer Gordon-Storrow photon 
PDF \cite{35} (see \cite{6}), although some access
in the forward region is still evident if compared to the prediction.  

\section{Conclusions}

The study of inclusive $D^{*\pm}$ production in $e^+e^-$ and $ep$ collisions
allows for a meaningful test of the QCD-improved parton model. Inside the
massless approach, where the charm-quark mass is neglected, we discussed
two different subtraction schemes for the fragmentation functions in NLO.
Both can be used to fit $D^{*\pm}$ production at LEP and lead to comparable
photoproduction cross sections at HERA. It is emphazised that the fit results
for the input parameter $\epsilon$ of the Peterson et al. parametrization
is highly scheme dependent at NLO and must not be naively compared
disregarding the theoretical framework which it refers to.

A large part of the $D^{*\pm}$ photoproduction cross section originates from 
the resolved process, eventually allowing to determine the charm distribution
function of the photon more accurately. The reasonable agreement between the
NLO masslesss theory and recent $D^{*\pm}$ photoproduction data from ZEUS is
quite encouraging. This is also true for the
$\gamma \gamma \rightarrow D^{*\pm}X$ process, for which new data were shown 
at this workshop \cite{36}. It is pointed out, that the theoretical cross 
section is particularly enhanced by the NLO corrections to the resolved 
component, a contibution, which is absent in any pure massive calculation. 
It is understood, that the massless-quark approximation is only appropriate 
for $D^{*\pm}$ production at large enough $p_T$, say $p_T>4~GeV$. So it 
should not be expected to yield meaningful predictions for the photoproduction
at small-$p_T$ $D^{*\pm}$ mesons at HERA. The next step in theory would be to 
retain the finite charm-quark mass and to subtract in the NLO corections 
to the direct process the colllinear logarithmic mass
divergences in such a way that the $\overline{MS}$ subtracted massless theory
is aproached in the limit $m_c \rightarrow 0$. This formulation should be
applicable at small and large $p_T$. 

%*********************************************************************
\subsection*{Acknowledgements}
 This work was supported by Bundesministerium f\"ur Forschung und
 Technologie, Bonn, Germany, under Contract 05~7~HH~92P~(0), and
 by the EU Fourth Framework Program {\it Training and Mobility of
 Researchers} through Network {\it Quantum Chromodynamics and
 Deep Structure of Elementary Particles}
 under Contract FMRX--CT98--0194 (DG12 MIHT). I would like to thank
J.\ Binnewies, B.\ A.\ Kniehl and M.\ Spira for their collaboration and B.\ A.\
Kniehl for help with the manuscript.

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\addcontentsline{toc}{section}{Index}
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%\end{document}

