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%\documentclass[draft]{ws-procs9x6}  
\documentclass{ws-procs9x6}
\usepackage{axodraw}
%\usepackage{amsmath}
%\usepackage{amssymb}

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% Some definitions

\newcommand{\lamQCD}{\Lambda_{\rm QCD}}
\def\Mathematica{{\tt Mathematica}}
\def\Tracer{{\tt Tracer}}

% Schemes
\def\msbar{\ensuremath{{\rm{\overline{MS}}}}} 
\def\disg{\ensuremath{{\rm{DIS_\gamma}}}} 

% units
\newcommand{\mev    }{\ensuremath{\mathrm{MeV}}}
\newcommand{\mevsq  }{\ensuremath{\mathrm{MeV^2}}}
\newcommand{\gev    }{\ensuremath{\mathrm{GeV}}}
\newcommand{\gevsq  }{\ensuremath{\mathrm{GeV^2}}}
%

% Misc
\newcommand{\der}{\ensuremath{{\operatorname{d}}}}
\newcommand{\Ord}{\ensuremath{{O}}}
\newcommand{\imag}{\ensuremath{{\operatorname{Im}}}}
\newcommand{\mi}{\ensuremath{\mathrm{i}}}
\newcommand{\DISeg}{\ensuremath{\mathrm{DIS}_{e\gamma}}}

% States
\newcommand{\braket}[2]{\ensuremath{\left\langle{#1}|{#2}\right\rangle}}
\newcommand{\bra}[1]{\ensuremath{\left\langle{#1}| \right.}}
\newcommand{\ket}[1]{\ensuremath{\left. |{#1}\right\rangle}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Special definitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\cbar}{\ensuremath{\bar{c}}}
\newcommand{\qbar}{\ensuremath{\bar{q}}}

\newcommand{\mui}{\ensuremath{\mu_f}}
\newcommand{\muf}{\ensuremath{\mu_f^\prime}}
\newcommand{\mur}{\ensuremath{\mu_r}}

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

\begin{document}

\title{Open heavy-flavour photoproduction at NLO
%\footnote{\uppercase{T}his work is supported by etc, etc.}
}

\author{I.~SCHIENBEIN 
%\footnote{\uppercase{W}ork partially supported by ...}
}

\address{DESY, \\
Notkestrasse 85, \\ 
22603 Hamburg, Germany\\ 
E-mail: schien@mail.desy.de}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle
                      
\abstracts{
I review theoretical approaches to open heavy flavour
photoproduction and discuss theoretical aspects of a 
new calculation in a massive theory with $\msbar$ subtraction
based on the factorization theorem of John Collins with heavy
quarks.
}

\section{Introduction}
Heavy quarks are those with masses
$m_h \gg \Lambda_{\rm QCD}$ such that $\alpha_s(m_h^2) \propto
\ln^{-1}(\tfrac{m_h^2}{\Lambda^2_{\rm QCD}}) \ll 1$ and 
hence, according to this definition, the charm, bottom and top quarks 
($c, b, t$)
are heavy whereas the up, down and strange quarks ($u, d, s$) 
are light.
%
A lot of reasons can be found in the literature why heavy
quark production is interesting.
%
Most importantly, such processes are fundamental elementary
particle processes which take place with substantial rates
at high energy colliders.
%
Therefore a good phenomenological understanding of heavy quark
production is very important.
%
Moreover, there are more theoretical reasons:
The heavy quark mass $m_h$ acts as a physical long-distance cut-off
such that heavy quark production is an ideal testing ground
of perturbative QCD. Very often in addition to $m_h$ another
large scale is involved in the hard scattering process such that
one has to deal with multi-scale processes which complicate the
perturbative analysis.
% as can be seen below.

\begin{figure}
\begin{center}
\fbox{\begin{minipage}[]{0.45\linewidth}
\begin{picture}(140,100)(-10,-10)
\SetScale{1.0}
\SetWidth{1.0}
% Proton
\Line(0,10)(15,10)
\Line(20,7)(35,7)
\Line(20,3)(35,3)
\Line(20,12)(50,30)
%\IfColor{\COval(20,10)(10,5)(0){Black}{Yellow}}{\GOval(20,10)(10,5)(0){0.5}}
{\GOval(20,10)(10,5)(0){0.5}}

% Photon
%\Line(0,70)(15,70)
\Photon(0,70)(15,70){2}{3}
\Line(20,73)(35,73)
\Line(20,77)(35,77)
\Line(20,68)(50,50)
%\IfColor{\COval(20,70)(10,5)(0){Black}{Yellow}}{\GOval(20,70)(10,5)(0){0.5}}
{\GOval(20,70)(10,5)(0){0.5}}

% Hard Scattering
\Line(60,37)(85,30)
\Line(60,33)(85,26)
\Line(60,29)(85,22)
\Line(60,44)(100,62)
%\IfColor{\COval(60,40)(15,15)(0){Black}{White}}{\GOval(60,40)(15,15)(0){0.5}}
{\GOval(60,40)(15,15)(0){1.0}}

% Fragmentation
\Line(105,59)(120,57)
\Line(105,55)(120,53)
\Line(105,65)(125,74)
%\IfColor{\COval(105,62)(10,5)(0){Black}{Yellow}}{\GOval(105,62)(10,5)(0){0.5}}
{\GOval(105,62)(10,5)(0){0.5}}

\Text(30,25)[l]{\normalsize $  j$}
\Text(30,54)[l]{\normalsize $  i$}
\Text(85,52)[l]{\normalsize $  k$}
\Text(85,26)[l]{\large $  X$}

\Text(60,40)[c]{\large $  \hat{\sigma}$}

\Text(20,-6)[c]{\normalsize $  f^p$}
\Text(20,86)[c]{\normalsize $  f^\gamma$}
\Text(105,82)[c]{\normalsize $  D_k^{D^\star}$}

\Text(132,74)[c]{\normalsize $  D^\star$}
\Text(-7,10)[c]{\normalsize $  p$}
\Text(-7,70)[c]{\normalsize $  \gamma$}
\end{picture}
\end{minipage}} 
\end{center}
\caption{Open charm photoproduction, $\gamma p \to D^\star X$, according
to Eq.\ \protect\eqref{eq:fact}.}
\label{fig:fact}
\end{figure}

In this talk I will restrict to open charm photoproduction,
$\gamma p \to D^\star X$, and describe theoretical aspects of
a new calculation in a massive variable flavour number scheme (VFNS)
based on the factorization theorem of John Collins including 
heavy quark masses \cite{Collins:1998rz}.
%
This work is along the lines of previous studies
of the process $\gamma \gamma \to D^\star X$ for the
direct contribution \cite{Kramer:2001gd} and
the single-resolved part \cite{Kramer:2003cw}.
Note also that the latter constitutes the direct contribution to
$\gamma p \to D^\star X$ \cite{Kramer:2003jw} considered here.
%
A calculation of the resolved contribution is in progress 
\cite{resolved} which will allow to perform a complete NLO
analysis of inclusive photoproduction of $D^\star$ mesons
in a massive VFNS which extends the successful massless
calculation 
\cite{BKK-D}
%\cite{Binnewies:1997gz,Binnewies:1998xq}
by terms of the order $\Ord(m_c^2/p_T^2)$ where $p_T$
is the transverse momentum of the $D^\star$ meson.
% 
A different approach to construct a massive VFNS has been
adopted in Refs.\ \refcite{Cacciari:2001td,Frixione:2002zv}
('FONLL').

% Factorization Formula
\section{Factorization Formula}
\label{sec:fact}
Generically, the theoretical description of open charm photoproduction
is based on the following factorization formula [see Fig.\ \ref{fig:fact}]
\begin{eqnarray}
\der \sigma(\gamma p \to D^\star X) 
& =& 
\sum_{i,j,k} \int dx_1\ dx_2\ dz\
f_i^\gamma(x_1,\mui)\ f_j^p(x_2,\mui)\ \times
\nonumber\\
& &
\phantom{\sum_{i,j,k}\int}
\der \hat{\sigma}(i j \to k X)\  
D_k^{D^\star}(z,\muf)
+ \Ord(\alpha_s^{n+1},\tfrac{\Lambda^2}{p_T^2})\, ,
\label{eq:fact}
\end{eqnarray}
where $\Lambda$ is a typical hadronic scale of a few hundred $\mev$,
$p_T$ is the transverse momentum of the $D^\star$ meson
and
where $f_i^\gamma(x_1,\mui)$ and $f_j^p(x_2,\mui)$ 
are (universal) parton distribution functions (PDFs) of the photon and the
proton, respectively.
%
The partons of the proton are gluons and quarks,
$j= g, q, \ldots$ ($q=u, d, s$)
while the
partons of the photon are $i= g, q, \ldots$
('resolved contribution') and
$ i=\gamma$ ('direct contribution'\footnote{$ f_\gamma^\gamma(x,\mu_f) 
= \delta(1-x)$}).
%
The ellipses indicate that the question whether heavy quarks 
are included as partons
depends on the factorization scale $\mui$
and the heavy flavour scheme to be discussed in the
next section.
%
Furthermore, the fragmentation of the final state parton 
$k=g, q, \ldots$ into the
$D^\star$ meson is described by (universal) fragmentation functions (FFs)
$D_k^{D^\star}(z,\muf)$.
The ellipses again refer to the heavy quark treatment.
\footnote{It should be noted that
in any case charm quarks (heavy quarks) will be produced in
the final state of the partonic subprocess ($k=c$).
%
However, if the final state charm quark is not
treated as a parton it is nevertheless phenomenologically 
necessary to include a scale-independent
fragmentation function $D_c^{D^\star}(z)$.
Such a function is in principle
not related to the factorization formula in Eq.\ \eqref{eq:fact} and
it is not clear whether it has a universal meaning.
See also Sec.\ \ref{sec:schemes}.}
%
Finally, the hard scattering cross sections
$\der \hat{\sigma}(i j \to k X)(\mui,\muf,\alpha_s(\mu),[\tfrac{m_c}{p_T}])$ 
are pertubatively calculable and depend on the factorization
scales $\mui$ and $\muf$ and the renormalization scale $\mu$.
As has been proved to all orders of perturbation theory 
in Ref.\ \refcite{Collins:1998rz} heavy quark
mass effects can be consistently included in the hard part.
The square bracket around $\tfrac{m_c}{p_T}$ indicates 
that, depending on the heavy flavour scheme, these 
terms are included or neglected.
%
Of course, in order to obtain a physically meaningful 
(scheme independent) result the hard part and the PDFs and FFs
have to be employed in the same factorization scheme.

% Errors and Schemes
Before turning to a discussion of heavy flavour schemes it is
instructive to consider the theoretical uncertainties of 
predictions based on Eq.\ \eqref{eq:fact}.
The sources are:
\begin{itemize}
\item Uncertainties in the PDFs and FFs.
\item The error of the factorization formula which is
of the order $\Ord(\tfrac{\Lambda^2}{p_T^2})$.
\item The order of the perturbative calculation of the hard part.
An NLO calculation includes terms of the order $\Ord(\alpha_s^2)$ 
($n=2$ in Eq.\ \eqref{eq:fact}).
Usually the error due to higher order terms is estimated by
varying the factorization and renormalization scales.
\end{itemize}

\section{Heavy Flavour Schemes}
\label{sec:schemes}
In this section I will give a basic overview
over the main heavy flavour schemes used in the literature.
For a more detailed review see, e.g., 
Ref.\ \refcite{Tung:2001mv}.
%
References to NLO calculations of $\gamma p \to D^\star X$
based on Eq.\ \eqref{eq:fact}
and using a given heavy flavour scheme
can be found in the corresponding section headlines.

%
\subsection{Conventional massless Parton Model: ZM-VFNS 
\protect\cite{BKK-D}}
\label{sec:mp}
The conventional (massless) parton model is also known
as 'zero mass variable flavour number scheme' (ZM-VFNS).
In this approach the charm mass $m_c$ is set to zero 
in the calculation
of the hard scattering cross section. 
%
The number of partons is variable depending on the factorization
scale with
the following finite transformations (valid up to NLO)
at the so-called transition scale $Q_0 = m_c$:
\footnote{For a discussion of the general case $Q_0=\Ord(m_c)$ 
see, e.g., Ref.\ \protect\refcite{Kretzer:1999zd}. Therein, also
references to the literature of the matching conditions of
the PDFs and $\alpha_s$ can be found.}
\begin{eqnarray}
n_f=3 & \rightarrow & n_f=4
\nonumber\\
\alpha_s^{({ 3})} &\rightarrow& 
\alpha_s^{({ 4})}=\alpha_s^{({ 3})}
+\Ord(\alpha_s^3) 
\nonumber\\
%f_i^{({ 3})}(x,Q_0^2=m_c^2) &\rightarrow& 
%f_i^{({ 4})}(x,Q_0^2=m_c^2)=f_i^{({ 3})}(x,Q_0^2=m_c^2)
f_i^{({ 3})} &\rightarrow& 
f_i^{({ 4})} = f_i^{({ 3})}
+\Ord(\alpha_s^2)
\end{eqnarray}
and the perturbative boundary condition for the charm quark
\begin{equation}
f_c^{({ 4})}(x,Q_0^2=m_c^2)=0\ .
\end{equation}

% Pro and Contra
The advantage of this approach is that it is simple
and large collinear logarithms $\ln \tfrac{\mu^2}{m_c^2}$
are resummed to all orders in perturbation theory 
in the evolved distributions $f_c(x,\mu^2)$ and 
$D_c^{D^\star}(x,\mu^2)$.
On the other hand, charm mass terms
$\tfrac{m_c^2}{p_T^2}$ are neglected in the hard part, such
that theoretical predictions are good only for
$ p_T^2 \gg m_c^2$.

%
\subsection{($n_f=3$)-Fixed Flavour Number Scheme (3-FFNS) 
\protect\cite{Frixione:1995dv,Frixione:1995qc}}
In this approach $m_c \ne 0$ and the number of 
flavours is fixed to $n_f = 3$.
The partonic subprocesses involve
gluons and light quarks as partons in the initial state
whereas no charm parton distribution
is taken into account ($f_c(x,\mui) =0$). 
%
On the other hand charm quarks are produced in the partonic
subprocesses and hence appear in the final state.
Here it is phenomenologically unavoidable 
to include a non-perturbative function, for example of Peterson type,
describing the transition from the charm
quark to the observable $D^\star$ meson.
It should be noted that this fragmentation function 
is {\em not}
related to the fragmentation functions in the 
factorization theorem in Eq.\ \eqref{eq:fact} which are
universal and factorization scale dependent.
%

%
% Pro and Contra:
In the fixed flavour scheme perturbative calculations are more
involved due to $m_c \ne 0$.
The advantage is that
$\tfrac{m_c^2}{p_T^2}$ terms are included
and a correct threshold behaviour is guaranteed which is 
especially important in deep inelastic scattering.
%
On the other hand, the fixed order logarithms $\ln \tfrac{p_T^2}{m_c^2}$
can become large for $p_T^2 \gg m_c^2$ such that a resummation
of these large logarithms is necessary.
Therefore, calculations within the FFNS are expected to be
good mainly in the kinematical region $p_T^2 \simeq m_c^2$.
%
\footnote{
Of course it is a matter of debate when 
conditions like $p_T^2 \gg m_c^2$ are actually satisfied
and the answer certainly also depends on the process
under consideration.}
%
A major drawback is that the charm fragmentation function
$D_c^{D^\star}(z)$ is not based on
the factorization theorem and hence not universal.
Therefore there is no deep theoretical reason to fix this 
function in one experiment and to use it making predictions
for other experiments.
%
Finally, it is noteworthy that it is not possible to
include a non-perturbative charm component in the proton in this scheme.

%
\subsection{Massive VFNS (GM-VFNS) 
\protect\cite{Cacciari:2001td,Frixione:2002zv,resolved}}
The massive VFNS (GM-VFNS)
is a variable flavour number scheme
as introduced in Sec.\ \ref{sec:mp}
which includes heavy quark mass effects in the hard scattering
cross sections/coefficients.
%
Besides the usual gluon and light quark parton distributions
a charm quark parton distribution is taken into account.
In order to avoid double counting collinear logarithms
$\ln \tfrac{\mu^2}{m_c^2}$ are subtracted from the
partonic subprocesses and are resummed to all orders with help
of the DGLAP evolution equations ($\rightarrow$ $ f_c \ne 0$).
%
The same is true for the final state, where 
collinear logarithms are subtracted from the hard part and
are resummed to all orders by introducing a charm quark fragmentation
function $D_c^{D^\star}(z,{\muf}^2)$.

% Pro and Contra:
This scheme is most ambitious aiming at combining the virtues
of both the massless VFNS and the FFNS
and therefore it is technically more involved.
%
A calculation requires to consider all subprocesses
occurring already in the FFNS with $m_c \ne 0$ and to
subtract the collinear parts in order to obtain
'IR-safe' hard parts.
Technically this is equivalent to mass factorization
within a massive regularization scheme.
%kinematics: factorization with massive partons $\rightarrow$ '{ ACOT}-$ \chi$'
%
In addition the subprocesses with charm quarks in the initial
state (occurring also in the massless VFNS) have to be included.
%
However, all desired features are fulfilled in this approach:
The large collinear logarithms $ \ln \tfrac{\mu^2}{m_c^2}$
are resummed in terms of evolved charm quark PDFs and FFs 
$f_c(x,\mu^2)$ and $D_c^{D^\star}(x,\mu^2)$, respectively,
and charm quark mass terms $\Ord(\tfrac{m_c^2}{p_T^2})$ are included
in the hard parts.
Therefore, calculations in this scheme are expected to be valid
for both $p_T^2 \simeq m_c^2$ and $p_T^2 \gg m_c^2$ such that
for $p_T^2 \lesssim m_c^2$ the transition to the 3-FFNS is achieved
and asymptotically, for $p_T^2 \gg m_c^2$, the ZM-VFNS is reproduced.
%
Again it should be noted that as in the massless VFNS a 
non-perturbative charm component in the proton can be included.

\section{Theoretical Aspects of Open Charm Photoproduction in a
massive VFNS}
In this section some theoretical issues relevant for open charm
photoproduction in a massive VFNS will be addressed.
%
One of the key ingredients in establishing such a scheme
is the removal of collinear would-be singularities in order
to avoid double counting.
As already mentioned, this is technically nothing else 
but mass factorization within
a massive regularization scheme and will be discussed in
Sec.\ \ref{sec:massfact}.
%
Then, in Sec.\ \ref{sec:implementation} the residual implementation
freedom concerning finite mass terms will be discussed 
which is the main reason for the different approaches to
massive VFNS existing in the literature.
%
Our procedure adopted for establishing a massive VFNS 
which fixes these ambiguities and which is applicable
to all kinds of processes will be described.

\subsection{Mass Factorization}
\label{sec:massfact}
Before turning to mass factorization in a massively regularized
theory it is helpful to recap some facts about 
$\msbar$ mass factorization in a dimensionally regularized theory.
%
As is well-known, mass divergences ($\equiv$ 'collinear divergences')
occur when an
internal propagator gets on its mass-shell 
due to a {\em collinear} splitting process as 
exemplified in Fig.\ \ref{fig:collinear}.
%
\begin{figure}
\begin{center}
{\begin{minipage}[]{4.3cm}
\begin{picture}(120,60)(-5,10)
\SetOffset(-8,0)
\Gluon(35,20)(35,40){2}{4}
\Vertex(35,40){2.0}
%\SetColor{Blue}
\CArc(35,40)(15,0,90)
\SetColor{Black}
\Text(40,47)[]{ $\theta$}
\Line(35,40)(35,60)
\Line(35,40)(55,40)
\DashLine(55,40)(65,40){1}
\DashLine(35,60)(35,70){1}
\DashLine(35,10)(35,20){1}
%\Text(0,40)[l]{ \scriptsize FOPT}
\Text(30,20)[]{ \scriptsize $g$}
\Text(30,60)[]{ \scriptsize $c$}
\Text(55,35)[]{ \scriptsize ${\bar c}$}
%
\SetOffset(40,0)
\Line(35,20)(35,40)
\Vertex(35,40){2.0}
%\SetColor{Blue}
\CArc(35,40)(15,0,90)
\SetColor{Black}
\Text(40,47)[]{ $\theta$}
\Line(35,40)(35,60)
\Gluon(35,40)(55,40){2}{4}
\DashLine(55,40)(65,40){1}
\DashLine(35,60)(35,70){1}
\DashLine(35,10)(35,20){1}
\Text(30,20)[]{ \scriptsize $c$}
\Text(30,60)[]{ \scriptsize $c$}
\Text(55,35)[]{ \scriptsize $g$}
\end{picture}
\end{minipage}}
%\fbox{\begin{minipage}[b]{2.7cm}
{\begin{minipage}[]{2.7cm}
\begin{flushright}
{\small 
collinear: { $\theta \simeq 0$} }
\end{flushright}
\end{minipage}}
\end{center}
\caption{Collinear splitting processes $g \to c \bar{c}$ and $c \to c g$.}
\label{fig:collinear}
\end{figure}
%
The associated regularized collinear 'divergences' are subtracted
from the (partonic) cross sections and are absorbed into PDFs and
FFs.
%
Along with the genuine (regularized) poles {\em finite terms} 
can be subtracted leading to
different factorization schemes; among them the
$\msbar$ scheme considered here.
%
It should be noted that
even for the same factorization scheme
the {\em subtraction terms} do depend on 
the {\em regularization procedure}.
%
Usually the $\msbar$ (factorization/renormalization) scheme 
is introduced or defined within 
{\em dimensional regularization} ($m_c = 0$)
such that collinear divergences occur as $\tfrac{1}{\epsilon}$ poles.

% Example
\begin{figure}
\begin{center}
{\begin{minipage}[]{3.6cm}
\begin{picture}(110,60)(-2,2)
%\SetOffset(-8,0)
\Gluon(5,5)(15,15){1.5}{3}
\ArrowLine(45,15)(15,15)
\Line(15,15)(15,25)
\Vertex(15,15){1.0}
%
\Line(12,26)(18,26)
\Line(12,28)(18,28)
%
\Line(15,29)(15,45)
\Gluon(15,35)(35,35){1.5}{3}
\Vertex(15,35){1.0}

\Vertex(15,45){1.0}
\ArrowLine(15,45)(45,45)
\Photon(5,60)(15,45){1.5}{3}
\Text(0,5)[]{ \scriptsize $p_2$}
\Text(0,60)[]{ \scriptsize $p_1$}
\Text(14,20)[r]{ \scriptsize $x_2$}

%\SetColor{Blue}
%\SetColor{Black}
%\Text(55,35)[]{ \scriptsize ${\bar c}$}
%
\SetOffset(55,0)
\Gluon(5,5)(15,15){1.5}{3}
\ArrowLine(45,15)(15,15)
\Line(15,15)(15,25)
\Vertex(15,15){1.0}
%
\Line(12,26)(18,26)
\Line(12,28)(18,28)
%
\ArrowLine(15,29)(15,45)

\Gluon(25,45)(45,35){-1.5}{3}
\Vertex(25,45){1.0}

\Vertex(15,45){1.0}
\Line(15,45)(45,45)
\Photon(5,60)(15,45){1.5}{3}
\Text(0,5)[]{ \scriptsize $p_2$}
\Text(0,60)[]{ \scriptsize $p_1$}
\Text(14,20)[r]{ \scriptsize $x_2$}

\end{picture}
\end{minipage}}
\end{center}
\caption{Diagrammatical representation of the subtraction term in
Eq.\ \protect\eqref{eq:fact_example}.}
\label{fig:fact_example}
\end{figure}
As an example consider Fig.\ \ref{fig:fact_example}.
%
The collinear part to be subtracted via mass factorization is 
given by
\begin{equation}
\der \sigma_{\rm SUB}({\mui}^2,\mu^2)
= \int_0^1 \der {x_2}\ G^1_{cg}({ x_2}) 
\der \hat{\sigma}^{(0)}_{\gamma c \to c g}(p_1,{ x_2} p_2)
%\\
\equiv 
%G^1_{cg}(x_2) \otimes 
G^1_{cg} \otimes 
\der \hat{\sigma}^{(0)}_{\gamma c \to c g}
\label{eq:fact_example}
\end{equation}
with the universal function [$ n=4-2 \epsilon$]
%
\begin{align}
G^1_{cg}(x,{\mui}^2,\mu^2) =& \tfrac{\alpha_s(\mu^2)}{2 \pi}
\Big[
{P^{(0)}_{cg}(x) }
{
(-\tfrac{1}{\epsilon} + \gamma_E - \ln 4\pi)}
(\tfrac{\mu^2}{{\mui}^2})^\epsilon 
%\\
%&
%\phantom{\tfrac{\alpha_s(\mu^2)}{2 \pi}\Big[}
+
\underbrace{g^1_{cg}(x)}_{\msbar = 0}\Big]
\label{eq:Gcg}
\end{align}
containing the 
$\tfrac{1}{\epsilon}$ pole and the finite part
$g^1_{cg}(x)$ being zero in the $\msbar$ scheme.
%
A few remarks are in order:
(i) The variable $x_2$ in Eq.\ \eqref{eq:fact_example}
can be calculated in terms of the external Mandelstam variables,
$x_2 = x_2(s,t,u)$, since the Born cross section is proportional
to the Delta-function 
$\delta(\hat{s}+\hat{t}+\hat{u})$ where $\hat{s},\hat{t},\hat{u}$
are obtained in the usual way from the 4-momenta $p_1,x_2 p_2$ 
entering the Born subprocess.
(ii) The Born cross section in Eq.\ \eqref{eq:fact_example}
is usually taken to be $n$ dimensional which generates finite
terms when multiplied with the $\tfrac{1}{\epsilon}$ pole
in the function $G^1_{cg}$.
(iii) Finally it is interesting to keep track of the logarithms
involved in the mass factorization procedure.
The collinear part of the unsubtracted partonic cross section 
is proportional to 
$-\tfrac{1}{\epsilon}(\mu^2/s)^\epsilon = -\tfrac{1}{\epsilon}+
\ln \tfrac{s}{\mu^2}$ from which 
$-\tfrac{1}{\epsilon}(\mu^2/{\mui}^2)^\epsilon =-\tfrac{1}{\epsilon}+
\ln \tfrac{{\mui}^2}{\mu^2}$ is subtracted [see Eq.\ \eqref{eq:Gcg}] 
generating a logarithm
$\ln \tfrac{s}{{\mui}^2}$ in the IR-safe hard part which depends on
the factorization scale.

% General case
The above example can be systematically generalized using universal
functions of the form
\begin{equation}
G^1_{ij}(x,{\mui}^2,\mu^2) = \tfrac{\alpha_s(\mu^2)}{2 \pi}
\Big[
{ P^{(0)}_{ij}(x) }
{
(-\tfrac{1}{\epsilon} + \gamma_E - \ln 4\pi)}
(\tfrac{\mu^2}{{\mui}^2})^\epsilon + 
\underbrace{g^1_{ij}(x)}_{\msbar = 0}\Big]\ .
\end{equation}

% Massive
In the case of heavy quark production it is
natural to use the heavy quark mass $m$ as 
collinear regulator such that the collinear
poles appear as logarithms $\ln \tfrac{s}{m^2}$.
The {\em massive} $\msbar$ subtraction terms
$\der \sigma_{\rm SUB}(\mui,m)$ 
are identified by the condition
\begin{equation}
\lim_{m \to 0}\ (\der \sigma(m) - \der \sigma_{\rm SUB}(m)) 
\overset{!}{=}
\der \hat{\sigma}({\scriptsize {\msbar}})
\label{eq:condition}
\end{equation}
As before the subtraction terms can be factorized into
universal functions
$G^1_{ij}(x,{\mui}^2,\mu^2)$ and $ D^1_{ij}(x,{\mui}^2,\mu^2)$
for the initial and final state, respectively, and Born cross
sections:
$\der \sigma^1_{\rm SUB}({\mui}^2,\mu^2)= G^1(x)\otimes
\der \hat{\sigma}^{(0)}\, , \, \der \hat{\sigma}^{(0)}\otimes D^1(z)$.
The hard (IR safe) cross sections entering the factorization formula
in Eq.\ \eqref{eq:fact} are then obtained by removing
% subtracting
the mass singular parts from the partonic cross section
\begin{equation}
\der \hat{\sigma}^{(1)} = \der \sigma^1 - \der \sigma^1_{\rm SUB}\ .
\end{equation}

%
Setting $ \mui = \mu$, the massively regularized functions
$G^1_{ij}$ (initial state) are given in the $\msbar$ scheme by
\begin{align}
G^1_{cg}(x,\mu^2) & = 
\tfrac{\alpha_s(\mu)}{2 \pi} P^{(0)}_{cg}(x) 
\ln \tfrac{\mu^2}{m^2}\ ,
\nonumber\\
G^1_{cc}(x,\mu^2) & = 
\tfrac{\alpha_s(\mu)}{2 \pi} P^{(0)}_{cc}(x) 
\ln \tfrac{\mu^2}{m^2}\ ,
\label{eq:initial}
\\
G^1_{gg}(x,\mu^2) & = 
\tfrac{\alpha_s(\mu)}{2 \pi}
\ln \tfrac{\mu^2}{m^2} \delta(1-x)\ ,
\nonumber
\end{align}
and the functions $D^1_{ij}$ (final state) read
\begin{align}
D^1_{cg}(z,\mu^2) & = 
 \tfrac{\alpha_s(\mu)}{2 \pi} P^{(0)}_{cg}(z) 
\ln \tfrac{\mu^2}{m^2}\ ,
\nonumber\\
D^1_{cc}(z,\mu^2) & = 
C_F \tfrac{\alpha_s(\mu)}{2 \pi} 
\big[\tfrac{1+z^2}{1-z} 
(\ln \tfrac{\mu^2}{m^2}- 2 \ln(1-z)-1)\big]_+ \ .
\label{eq:final}
\end{align}
The function $D^1_{cc}(z,\mu^2)$ has been derived for the 
first time from the process \mbox{$\gamma^\star \to c \bar{c} g$} with
a timelike photon \cite{Mele:1991cw} and has also been found
by taking the massless limit of the process 
$\gamma^\star c \to c g$ with a spacelike photon
\cite{Kretzer:1998nt} 
and also in Ref.\ \refcite{Kramer:2001gd} for the process 
$\gamma \gamma \to c \bar{c} g$
demonstrating the universality of 
this function.
All other functions not listed here are zero to this order
in the coupling constant $\alpha_s$. 
Furthermore, analogous results for processes involving
photon splittings can be found by some obvious replacements
($g \to \gamma$, $\alpha_s \to \alpha$ and appropriate 
modifications of colour factors) in Eqs.\ \eqref{eq:initial}
and \eqref{eq:final}.

% factorization variables $ x = ?$, $ z = ?$

\subsection{Implementation Freedom and Adopted Procedure}
\label{sec:implementation}
Even fixing the factorization scheme to $\msbar$ leaves some
freedom in the implementation of a massive VFNS as has been
discussed for the case of deep inelastic scattering (DIS) in
Ref.\ \refcite{Tung:2001mv}.
This can be most easily seen from the condition in 
Eq.\ \eqref{eq:condition} which fixes the subtraction term 
$\der \sigma_{\rm SUB}(\tfrac{\mu}{m},\tfrac{m}{p_T})$
only up to terms $\tfrac{m}{p_T}$ vanishing in the limit
$m \to 0$. The precise treatment of such terms $\tfrac{m}{p_T}$
is not prescribed by factorization.

% Our Approach
In our approach to open charm photoproduction the following procedure
has been adopted \cite{Kramer:2001gd,Kramer:2003cw}:
\begin{itemize}
\item Derive the $m \to 0$ limit of the massive 3-FFNS calculation
only keeping $m$ as regulator in logarithms $\ln \tfrac{m^2}{s}$.
Here special care is required in order to recover certain distributions
($\delta(1-w)$, $(\tfrac{1}{1-w})_+$, $(\tfrac{\ln(1-w)}{1-w})_+$)
occurring in the massless $\msbar$ calculation. 
\item Compare the massless limit with the corresponding 
contribution in the (massless) $\msbar$ calculation in order to
identify the subtraction terms by the unique prescription
\begin{equation}
\der \sigma_{\rm SUB} = ( \lim_{m \to 0} \der \sigma(m) )
- \der \hat{\sigma}(\msbar)\ .
\end{equation}
By this procedure no mass terms are removed from the
partonic cross section apart from
the collinear logarithms $\ln \tfrac{m^2}{s}$.
\item Include contributions with charm quarks in the 
{\em initial state} in the massless approach \cite{Kramer:2000hn}.
This rule is of great practical importance and also has been employed
in the recent work \cite{Kretzer:2003it} of the CTEQ collaboration.
\end{itemize}

\section{Summary}
In this talk I have discussed theoretical issues relevant
for heavy flavour photoproduction, $\gamma p \to D^\star X$, 
in a massive VFNS.
%
A first study including mass effects for the direct 
contribution has been performed in Ref.\ \refcite{Kramer:2003jw}.
These results indicate an improved description of HERA
data at small transverse momenta of the $D^\star$ meson.
% reduced scale dependence ?
%
A calculation of the resolved 
contribution including mass effects 
is in progress allowing for a complete NLO analysis
of open heavy flavour production in a massive
VFNS rigorously based on the factorization theorem
with massive partons by John Collins \cite{Collins:1998rz}
%
which is the theoretical basis to incorporate information
on the FFs into $D^\star$ mesons from other reactions, as for 
example $e^+ e^- \to D^\star X$.

\section*{Acknowledgments}
I am grateful to the organizers for the kind invitation
to this very well organized workshop at such an inspiring 
location and to B.\ A.\ Kniehl, G.\ Kramer and H.\ Spiesberger 
for their collaboration.

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\expandafter\ifx\csname bibfnamefont\endcsname\relax
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\end{thebibliography}

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\end{document}
