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\begin{document}
\thispagestyle{empty} 
\title{
\vskip-3cm
{\baselineskip14pt
\centerline{\normalsize DESY 03--014 \hfill ISSN 0418--9833}
\centerline{\normalsize MZ-TH/03--03 \hfill} 
\centerline{\normalsize  \hfill} 
\centerline{\normalsize February 2003 \hfill}} 
\vskip1.5cm
{\bf Inclusive $D^{*}$ Production in $\gamma \gamma$ Collisions:}\\
{\bf Including the Single-Resolved Contribution} \\
{\bf with Massive Quarks}
\author{G.~Kramer$^1$ and H.~Spiesberger$^2$
\vspace{2mm} \\
{\normalsize $^1$ II. Institut f\"ur Theoretische
  Physik,
%    \thanks{Supported by Bundesministerium f\"ur Forschung und
%    Technologie, Bonn, Germany, under Contract 05~HT9~GA3, and
%    by EU Fourth Framework Program {\it Training and Mobility of
%    Researchers} through Network {\it Quantum Chromodynamics and
%    Deep Structure of Elementary Particles}
%    under Contract ERBFMRX--CT98--0194 (DG12 MIHT).}
  Universit\"at Hamburg,}\\ 
\normalsize{Luruper Chaussee 149, D-22761 Hamburg, Germany} \vspace{2mm}
\\ 
\normalsize{$^2$ Institut f\"ur Physik,
  Johannes-Gutenberg-Universit\"at,}\\ 
\normalsize{Staudinger Weg 7, D-55099 Mainz, Germany} \vspace{2mm} \\
} }

\date{}
\maketitle
\begin{abstract}
\medskip
\noindent
We have calculated the next-to-leading order cross section for the
inclusive production of charm quarks as a function of the transverse
momentum $p_T$ and the rapidity in two approaches using massive or
massless charm quarks. For the single-resolved cross section we have
derived the massless limit from the massive theory. We find that this
limit differs from the genuine massless version with $\overline{\rm MS}$
factorization by finite corrections. By adjusting subtraction terms we
establish a massive theory with $\overline{\rm MS}$ subtraction which
approaches the massless theory very fast with increasing transverse
momentum. With these results and including the equivalent results for
the direct cross section obtained previously as well as double-resolved
contributions, we calculate the inclusive $D^{*\pm}$ cross section in
$\gamma \gamma$ collisions using realistic evolved non-perturbative
fragmentation functions and compare with recent data from the LEP
collaborations ALEPH, L3 and OPAL. We find good agreement. 
\\
%\begin{center}
%\today
%\end{center}
\end{abstract}

%**********************************************************************
%**********************************************************************
\clearpage

\section{Introduction}

Recently there has been quite some interest in the study of charm
production in two-photon collisions at high energy $e^+e^-$ colliders,
both experimentally and theoretically. The three LEP collaborations
ALEPH, L3 and OPAL have presented cross section data for inclusive $D^*$
production in two-photon collisions at $e^+e^-$ center-of-mass energies
close to $\sqrt{s} = 189$ GeV \cite{aleph2002,L32002,opal}. Besides the
total cross section for $\gamma + \gamma \rightarrow D^* + X$, also
differential cross sections with respect to the $D^*$ transverse
momentum, $d\sigma/dp_T$, and the pseudo-rapidity, $d\sigma/d\eta$, have
been measured.

On the theoretical side two distinct approaches for next-to-leading
order (NLO) calculations in perturbative QCD have been used for
comparison with the experimental data. In the so-called massless scheme
(ZM scheme) \cite{BKK}, which is the conventional parton model approach,
the zero-mass parton approximation is applied also to the charm quark,
although its mass $m$ is certainly much larger than $\Lambda_{QCD}$. In
this approach the charm quark is also an ingoing parton originating from
the photon, leading to additional single- and double-resolved
contributions (besides those from $u$, $d$, $s$ quarks and the gluon
$g$).  The charm quark fragments into the $D^*$ meson similarly as the
light quarks and the gluon with a fragmentation function (FF) known from
other processes.  The well-known factorization theorem then provides a
straightforward procedure for order-by-order perturbative calculations.
Although this approach can be used as soon as the factorization scales
of initial and final state are above the starting scale of the parton
distribution function (PDF) of the photon and of the FF of the $D^*$,
the predictions are reliable only in the region of large transverse
momenta $p_T \gg m$, where terms of the order of $m^2/p_T^2$ can safely
be neglected.

The other calculational scheme which has been applied to the process
$\gamma + \gamma \rightarrow D^* + X$ in \cite{FKL} is the so-called
massive scheme, also called fixed flavor-number scheme (FFN), in which
the number of active flavors in the initial state for the resolved
contributions is limited to $n_f=3$ and the charm quark appears only in
the final state of the direct, single-resolved and double-resolved
contributions. In this case, the $c$ quark is always treated as a heavy
particle and never as a parton. The actual mass parameter $m$ is
explicitly taken into account along with $p_T$ as if they were of the
same order, irrespective of their real relative magnitudes. In this
scheme the charm mass acts as a cutoff for the initial- and final-state
collinear singularities and sets the scale for the perturbative
calculations.  However, in NLO, terms $\propto \alpha_s \ln(p_T^2/m^2)$
arise from collinear emissions of a gluon by the charmed quark at large
transverse momenta or from almost collinear branchings of photons or
gluons into $c \overline{c}$ pairs. These terms are of order $O(1)$ for
large $p_T$ and with the choice $\mu_R \sim p_T$ for the renormalization
scale they spoil the convergence of the perturbation series. The FFN
approach with $n_f=3$ should thus be limited to a rather small range of
$p_T \sim m$.  Nevertheless, predictions in this approach have been
compared to experimental data up to $p_T = 12$ GeV
\cite{aleph2002,L32002}.

As has been explained at many places in the literature, mostly in the
context of charm production in deep inelastic $ep$ scattering (for a
very recent review see \cite{TKS}), the correct approach for $p_T \gg m$
is to absorb the potentially large logarithms into the charm PDF of the
photon and the FF of the $c$ into $D^*$. Then, large logarithms $\propto
\ln(M^2/m^2)$ defined with the factorization scale $M$ determine the
evolution to higher scales and can be resummed by virtue of the
Altarelli-Parisi equations. The unsubtracted terms $\propto
\ln(p_T^2/M^2)$ are of order $O(1)$ for the appropriate choice $M$ of
order $p_T$. After factorizing the $\ln m^2$ terms, the hard cross
section is infrared safe and $n_f=4$ is taken in the evolution equations
and the equation for $\alpha_s$. The remaining dependence on $m$, i.e.\ 
the terms proportional to $m^2/p_T^2$, can be kept in the hard cross
section to achieve better accuracy in the intermediate region $p_T \gsim
m$.  The factorization of mass divergent terms can be extended
consistently to higher orders in $\alpha_s$, as has been shown by
Collins in the context of heavy quark production in high-$Q^2$ $ep$
collisions \cite{Coll}.

Now it is well known that the subtraction of just the collinearly, i.e.\ 
mass, singular terms does not define a unique factorization
prescription. Also finite terms must be specified.  In the conventional
ZM calculations the mass $m$ is put to zero from the beginning and the
collinearly divergent terms are defined with the help of dimensional
regularization. This fixes the finite terms in a specific way and their
form is inherent to the chosen regularization procedure.  If one starts
with $m \neq 0$ and performs the limit $m \rightarrow 0$ afterwards, the
finite terms can be different. These terms have to be removed by
subtraction together with the $\ln m^2$ terms in such a way that in the
limit $p_T \rightarrow \infty$ the known massless $\overline{\rm MS}$
expressions are recovered. This requirement is actually unavoidable
since all existing PDF's and FF's, including those for heavy quarks, are
defined in this particular scheme (or sometimes in the DIS scheme which
can be derived from the $\overline{\rm MS}$ scheme). It is clear that a
subtraction scheme defined in this way is a correct extension of the
conventional zero-mass scheme to include charm mass effects in a
consistent way.

In a recent work we applied this finite charm mass scheme with
$\overline{\rm MS}$ subtraction to the calculation of the cross section
for $\gamma + \gamma \rightarrow D^* + X$ \cite{KS}. As a first step we
considered only the direct cross section with $m \neq 0$. In the
calculation of the full cross section needed for comparison with
experimental data, i.e.\ in the sum of the direct, single-resolved and
double-resolved parts, the latter two contributions were still treated
in the ZM 4-flavor scheme. It is the purpose of this work to extend the
finite charm mass calculation to the single-resolved cross section. This
cross section plays an important part due to the partonic subprocess
$\gamma + g \rightarrow c + \overline{c}$ with charm quarks in the final
state and due to the process $\gamma + q \rightarrow c + \overline{c} +
q$, where $q$ is one of the light (massless) quarks $q=u$, $d$, $s$.
These contributions and their NLO corrections should be computed with
massive charm quarks in the same way as the direct cross section due to
the partonic subprocess $\gamma + \gamma \rightarrow c + \overline{c}$
and its higher order corrections.  The double-resolved part with charm
quarks in the final state originating from $q + \overline{q} \rightarrow
c + \overline{c}$ and $g + g \rightarrow c + \overline{c}$ and the
corresponding NLO corrections will still be considered in the ZM
4-flavor approach since it contributes very little to the complete
double-resolved cross section \cite{CGKKKS,FKL,KS}.

Starting with $\gamma + g \rightarrow c + \overline{c}$, the NLO
corrections for the single-resolved cross section can be split into an
Abelian and a non-Abelian part. The Abelian part is, up to a constant
factor, identical to the NLO corrections to $\gamma + \gamma \rightarrow
c + \overline{c}$.  For this part, the terms in the massive theory
surviving in the limit $m \rightarrow 0$, which are not present in the
ZM approach, have been identified in our earlier work \cite{KS}.
Therefore, only the non-Abelian part of the NLO corrections to the
photon-gluon fusion cross section and the cross section for $\gamma + q
\rightarrow c + \overline{c} + q$ have to be investigated.  The
single-inclusive charm cross section with $m \neq 0$ has been calculated
recently by Merebashvili et al.\ \cite{MCG}. We can use these results to
derive the limit $m \rightarrow 0$ and establish the subtraction terms
by comparing to the $\overline{\rm MS}$ factorized cross section derived
in \cite{LG}, in the same way as we did in \cite{KS} for the Abelian
part. With this knowledge we can compute the finite mass corrections for
the full NLO single-resolved cross section with $\overline{\rm MS}$
factorization.

The outline of our work is as follows. In Section 2 we describe the
formulae which we use to calculate the non-Abelian part of the cross
section for $\gamma + g \rightarrow c (\overline{c}) + X$ and for
$\gamma + q \rightarrow c (\overline{c}) + X$ with non-zero charm mass
using the results of \cite{MCG}.  From these cross sections we derive
the limit $m \rightarrow 0$ and compare with the ZM theory of \cite{LG}.
The results are reported in Section 3, where also numerical tests for
checking the subtraction terms are presented. Here we show how the
various terms in the NLO cross section approach their corresponding
massless limits for large $p_T$. After adding the already known Abelian
part and the direct contribution with $\overline{\rm MS}$ subtraction,
as well as the double-resolved contribution, we compare our results to
recent experimental data from LEPII in Section 4.  A summary and
conclusions are given in Section 5.

%**********************************************************************

\section{Calculation of the LO and NLO Differential Cross Section
  \label{section2}} 

The single-resolved contribution to the process $\gamma + \gamma
\rightarrow D^{*} + X$ has many pieces. In this section we concentrate
on those contributions where the charm quark appears only in the final
state. We study the mass dependence in order to obtain the massless
limit which is then compared with the genuine massless theory. There is
only one leading-order parton process $\gamma + g \rightarrow c +
\bar{c}$ with the initial gluon originating from the resolved photon.
The NLO corrections to $\gamma + g \rightarrow c + \bar{c}$ are the
virtual corrections and the gluonic bremsstrahlung contributions $\gamma
+ g \rightarrow c + \bar{c} + g$. In addition, the subprocesses $\gamma
+ q \rightarrow c + \bar{c} + q$ and $\gamma + \bar{q} \rightarrow c +
\bar{c} + \bar{q}$, where $q$ denotes a light quark, must be taken into
account in NLO. The NLO correction to $\gamma + g \rightarrow c +
\bar{c}$ has an Abelian and a non-Abelian part. The Abelian part is up
to a trivial factor identical to the NLO corrections for the process
$\gamma + \gamma \rightarrow c + \bar{c}$, which has been considered in
our previous work \cite{KS}. So, we need to calculate only the
non-Abelian contribution. In the following subsection we present the LO
cross section in order to fix the notation. Then we proceed to the
calculation of the NLO corrections to the non-Abelian part and of the
cross section for $\gamma + q (\bar{q}) \rightarrow c + \bar{c} + q
(\bar{q})$.

%**********************************************************************

\subsection{Leading-Order Cross Section \label{section21}}

We start with the process
\begin{equation}
 \gamma (p_1) + g (p_2) \rightarrow c(p_3) + \bar{c}(p_4) + [g(k)]
\end{equation}

where $p_i$, $i = 1$, 2, 3, 4 and $k$ denote the momenta of the incoming
photon, the incoming gluon, the outgoing $c$ and $\bar{c}$ quarks, and a
possible gluon in the final state (in square brackets), which is present
in the NLO corrections. We have the following invariants
\begin{equation}
 s = (p_1+p_2)^2, ~~
 t = T-m^2 = (p_1-p_3)^2 - m^2, ~~
 u = U-m^2 = (p_2-p_3)^2 - m^2
\end{equation}
and
\begin{equation}
 s_2 = S_2-m^2 = (p_1+p_2-p_3)^2-m^2 = s+t+u \, .
\end{equation}
We define the dimensionless variables $v$ and $w$ as usual by
\begin{equation}
 v = 1 +\frac{t}{s} ,~~
 w =-\frac{u}{s+t}
\end{equation}
so that
\begin{equation}
 t = -s(1-v), ~~
 u = -svw, ~~
 s_2 = sv(1-w) \, .
\end{equation}

The leading-order cross section is
\begin{eqnarray}
 \frac{d\sigma_{\rm LO}^{\gamma g}}{dvdw} = \frac{c(s)}{2} 
  \delta (1-w) 
  \left(\frac{t}{u}+\frac{u}{t} + 4\frac{sm^2}{tu} -
        4\left(\frac{sm^2}{tu}\right)^2
  \right)
\label{sigma_LO}
\end{eqnarray}
where
\begin{equation}
 c(s) = \frac{2\pi \alpha \alpha_s e_c^2}{s} \, .
\label{cs}
\end{equation} 
In (\ref{cs}), $e_c$ is the electric charge of the charm quark. If we
compare with the leading-order cross section $d\sigma_{\rm LO}/dv dw$
for $\gamma + \gamma \rightarrow c + \bar{c}$ in \cite{KS}, we have
\begin{eqnarray}
 \frac{d\sigma_{\rm LO}^{\gamma g}}{dvdw} = 
 \kappa C_F \frac{d\sigma_{\rm LO}}{dvdw} 
\label{sigmalogg}
\end{eqnarray}
where $\kappa = \alpha_s / (8\alpha e_c^2)$. From (\ref{sigma_LO}) the
finite charm mass corrections are clearly visible. Numerical results for
the direct contribution to $\gamma + \gamma \rightarrow c/\bar{c}
\rightarrow D^{*} + X$ can be found in our previous paper (Fig.\ 1 of
\cite{KS}).

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

\subsection{The Next-to-Leading-Order Cross Section \label{section22}}

\begin{figure}[b!] 
\unitlength 1mm
\begin{picture}(160,80)
\put(0,40){\epsfig{file=ggres1a.eps,width=35mm}}
\put(40,40){\epsfig{file=ggres1b.eps,width=40mm}}
\put(80,40){\epsfig{file=ggres1c.eps,width=40mm}}
\put(120,40){\epsfig{file=ggres1d.eps,width=40mm}}
\put(0,0){\epsfig{file=ggres1e.eps,width=40mm}}
\put(40,2.5){\epsfig{file=ggres1f.eps,width=35mm}}
\put(80,0.5){\epsfig{file=ggres1g.eps,width=35mm}}
\put(120,1){\epsfig{file=ggres1h.eps,width=35mm}}
\end{picture}
\caption{Feynman diagrams for the virtual NLO corrections to $\gamma + g
  \rightarrow c + \bar{c}$.}
\label{fig1}
\end{figure}

The NLO corrections for $\gamma + g \rightarrow c + \bar{c}$ with
non-zero charm quark mass $m$ have been calculated by several groups
\cite{EN,SvN,BS,MCG}. Only in the publication of Merebashvili et al.\ 
explicit formulae for the separate contributions due to one-loop
diagrams and due to gluonic bremsstrahlung are given in a form which
allows us to derive the massless limit $m \rightarrow 0$, in order to
establish the subtraction terms which are needed for the cross section
in the $\overline{\rm MS}$ factorization scheme. The results in Ref.\ 
\cite{MCG} are subdivided into four parts for the virtual corrections,
$d\sigma^{\gamma g}_{\rm a-e}$, $d\sigma^{\gamma g}_{\rm f}$,
$d\sigma^{\gamma g}_{\rm g}$, and $d\sigma^{\gamma g}_{\rm h}$, and the
bremsstrahlung cross section $d\sigma^{\gamma g}_{\rm Br}$.  These
separate contributions have the following structure: $d\sigma^{\gamma
  g}_{\rm a-e}$ stands for the contribution of the graphs (a)--(e) in
Fig.\ \ref{fig1}. It is
\begin{equation}
\frac{d\sigma^{\gamma g}_{\rm a-e}}{dvdw}
= \kappa C_F 
\frac{d\sigma_{\rm VSE}}{dvdw}
-\frac{N_C}{2} \frac{d\sigma_{\rm a-e}}{dv} \delta(1-w)
\label{dsae0}
\end{equation}
where $d\sigma_{\rm VSE}/dvdw$ determines the Abelian part and is given
in eq.\ (16) of \cite{KMC} and
\begin{eqnarray}
\frac{d\sigma_{\rm a-e}}{dv} 
& = & \frac{C(s)}{8} 
\left( 2 \tilde{A}_1 \left\{ 2 
     \left[ \zeta_2 - {\rm Li}_2\left(\frac{T}{m^2}\right)\right]
     \left(1 + 3 \frac{m^2}{t}\right)
- \ln\left(\frac{-t}{m^2}\right)\left(1 + \frac{m^2}{t}\right) + 2
\right\}
\right.
\nonumber
\\
& & 
%
+ \tilde{A}_2 \ln\left(\frac{-t}{m^2}\right)
%
+ \tilde{A}_3 \left[{\rm Li}_2\left(\frac{-T}{m^2}\right) - \zeta_2
\right] 
%
+ \tilde{A}_4
+ \left(t \leftrightarrow u \right)
\Biggr)
\label{dsae}
\end{eqnarray}
with
\begin{equation}
 C(s) = \frac{\alpha_s}{2\pi} c(s) \, .
\label{capcs}
\end{equation}
The coefficients $\tilde{A}_i$ ($i = 1$, 2, 3, 4) are functions of
$m^2$, $s$, $t$ and $u$. They are given in appendix A of \cite{MCG}. The
result of graph (f), i.e.\ the contribution of the box diagram to the
virtual corrections, can be written as
\begin{equation}
\frac{d\sigma^{\gamma g}_{\rm f}}{dvdw}
= \kappa \left(C_F - \frac{N_C}{2}\right) 
\frac{d\sigma_{\rm Box}}{dvdw}
\label{dsbox}
\end{equation}
where $d\sigma_{\rm Box}/dvdw$ is defined in eq.\ (22) of \cite{KMC}. Of
course, the term proportional to $C_F$ is the Abelian part and only the
term proportional to $N_C$ is relevant in this work.

The diagram (g) in Fig.\ \ref{fig1} is one of the contributions with the
three-gluon coupling, which are not present in the Abelian theory.  Its
contribution is written as
\begin{equation}
\frac{d\sigma^{\gamma g}_{\rm g}}{dvdw}
= - \frac{N_C}{2} 
\frac{d\sigma_{\rm g}}{dv} \delta(1-w)
\end{equation}
where
\begin{eqnarray}
\frac{d\sigma_{\rm g}}{dv} 
& = & \frac{C(s)}{8} 
\Biggl( 2 A_1  
     \left[ {\rm Li}_2\left(\frac{T}{m^2}\right) 
           + \ln^2\left(\frac{-t}{m^2}\right) - 2 \right]
%
+ A^{\prime}_1 \left[ 4 {\rm Li}_2\left(\frac{T}{m^2}\right) 
                    + 4 \ln^2\left(\frac{-t}{m^2}\right) \right]
%
\nonumber
\\
& &
+ A^{\prime}_2 \ln\left(\frac{-t}{m^2}\right)
%
+ A^{\prime}_3
+ \left(t \leftrightarrow u \right)
\Biggr) \, .
\label{dsg}
\end{eqnarray}
In eq.\ (\ref{dsg}), $A_1$ is given in appendix B of \cite{KMC} and the
coefficients $A^{\prime}_i$, $i = 1$, 2, 3 are written in appendix A of
\cite{MCG}. 

The contribution of the non-Abelian diagram (h) in Fig.\ \ref{fig1} is
\begin{equation}
\frac{d\sigma^{\gamma g}_{\rm h}}{dvdw}
= - \frac{N_C}{2} 
\frac{d\sigma_{\rm h}}{dv} \delta(1-w)
\end{equation}
where
\begin{eqnarray}
\frac{d\sigma_{\rm h}}{dv} 
& = & \frac{C(s)}{8} 
\Biggl( A_1  
     \left[ - \frac{35}{4}\zeta_2 
           - {\rm Li}_2\left(\frac{T}{m^2}\right) 
           + 4 \ln\left(\frac{-t}{m^2}\right) 
               \ln\left(\frac{-u}{m^2}\right) 
           - \ln^2\left(\frac{-t}{m^2}\right) \right]
%
\nonumber
\\
& &
+ B^{\prime}_1 {\rm Li}_2\left(\frac{T}{m^2}\right) 
%
+ \left(B^{\prime}_2 + A^{\prime}_1\right) \zeta_2
%
+ B^{\prime}_3 \ln^2\left(\frac{-t}{m^2}\right)
%
+ B^{\prime}_4 \ln\left(\frac{-t}{m^2}\right)
%
\nonumber
\\
& &
+ B^{\prime}_5 \ln\left(\frac{-t}{m^2}\right)
               \ln\left(\frac{-u}{m^2}\right)
%
+ B^{\prime}_6
+ \left(t \leftrightarrow u \right)
\Biggr) \, .
\label{dsh}
\end{eqnarray}

The coefficients $B^{\prime}_i$, $i = 1$, 2, $\ldots$, 6 are given in
appendix A of \cite{MCG}. In eqs.\ (\ref{dsae}), (\ref{dsg}) and
(\ref{dsh}), we left out the singular pieces in dimensional
regularization proportional to $1/\epsilon^2$ ($4-D = 2\epsilon$), and
$1/\epsilon$ ($d\sigma_{\rm Box}/dvdw$ in eq.\ (\ref{dsbox}) is
understood without the $1/\epsilon$-term). The infrared and collinearly
singular contributions cancel against the singular terms from the gluon
bremsstrahlung contributions, except one remaining term which is
canceled by a factorization counterterm corresponding to the final gluon
emitted collinearly with the initial gluon (see Fig.\ 2a) which is
written in the $\overline{\rm MS}$ factorization scheme. Further details
are found in \cite{MCG}, from which the eqs.\ (\ref{dsae0}) to
(\ref{dsh}) were taken.

\begin{figure}[th!] 
\unitlength 1mm
\begin{picture}(160,80)
\put(0,40){\epsfig{file=ggres2a1.eps,width=40mm}}
\put(40,40){\epsfig{file=ggres2a2.eps,width=40mm}}
\put(80,38.5){\epsfig{file=ggres2a3.eps,width=40mm}}
\put(120,40){\epsfig{file=ggres2a4.eps,width=40mm}}
\put(0,0){\epsfig{file=ggres2b1.eps,width=40mm}}
\put(40,0){\epsfig{file=ggres2b2.eps,width=40mm}}
\put(80,4){\epsfig{file=ggres2b3.eps,width=40mm}}
\put(120,4){\epsfig{file=ggres2b4.eps,width=40mm}}
\end{picture}
\caption{Feynman diagrams for the bremsstrahlung contributions 
  $\gamma + g \rightarrow c + \bar{c} + g$ (a) and for the process
  $\gamma + q \rightarrow c + \bar{c} + q$ (b). }
\label{fig2}
\end{figure}

The analytic results for the NLO corrections arising from the gluonic
bremsstrahlung have also been presented in \cite{MCG}. The corresponding
diagrams are shown in Fig.\ \ref{fig2}a. The squared sum of the
amplitudes (plus those arising from $p_1 \leftrightarrow p_2$) after
summing over final spins and colors and averaging over initial colors
is written as (see \cite{MCG})
\begin{equation}
4m^2 \left| M^{\gamma g}_{2 \rightarrow 3} \right|^2 
= \kappa_B(\epsilon) \left( \frac{C_F}{2} G^{\gamma \gamma} 
                          - \frac{N_C}{2} G^{\gamma g} \right)
\end{equation}
where $\kappa_B(\epsilon) = (4\pi)^3 \alpha \alpha_s^2 e_c^2
\mu^{6\epsilon}$ and $G^{\gamma \gamma}$ is the expression in the square
bracket of eq.\ (24) of \cite{KMC} (plus $p_1 \leftrightarrow p_2$).
$G^{\gamma \gamma}$ has been treated already in our previous work
\cite{KS}. $G^{\gamma g}$ represents the new non-Abelian part with the
factor $-N_C$ instead of $C_F$. The non-Abelian bremsstrahlung
contributions have been integrated also in $D \neq 4$ dimensions. The
details are reported in \cite{KMC} and \cite{MCG}. Using $\bar{y} =
\sqrt{(t+u)^2 - 4m^2 s}$ and $x = (1-\beta)/(1+\beta)$ where $\beta =
\sqrt{1 - 4m^2/s}$, the final result for $d\sigma^{\gamma g}_{\rm
  Br}/dvdw$ is given in eq.\ (4.5) of \cite{MCG}. For later use we
subdivide this cross section into six parts
\begin{equation}
\frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw}
= \sum_{i=1}^6  
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(i)}
\end{equation}
where
\begin{eqnarray}
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(1)}
& = &
- \frac{C(s) N_C}{64} \frac{v s_2 s}{S_2}
\Biggl\{ 
e_1 
+ \frac{2S_2}{s_2(s+u)} e_2 \ln \frac{S_2}{m^2}
+ \frac{4S_2}{m^2(s+t)^2} e_3
\nonumber
\\
& &
+ \frac{2S_2}{s_2(s+t)} e_4 \ln \frac{S_2}{m^2}
+ e_7 I_8 + e_8 I_{10} + e_{10} I_{16}  
\nonumber
\\
& &
+ e_{11} I_1(t \leftrightarrow u) 
+ f_1 F_1 + f_2 F_2 
\Biggr\} \, ;
\label{br1}
\end{eqnarray}
%
\begin{equation}
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(2)}
= 
- \frac{C(s) N_C}{64} \frac{1}{(1-w)_+} \frac{s_2^2}{S_2}
\left\{ 
  \frac{2S_2}{s_2\bar{y}} \tilde{e}_5 \ln \frac{T+U-\bar{y}}{T+U+\bar{y}}
+ \tilde{e}_6 I_{11}(t \leftrightarrow u) 
+ \tilde{e}_9 I_{11} 
\right\} \, ;
\label{br2}
\end{equation}
%
\begin{eqnarray}
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(3)}
& = &
- \frac{C(s) N_C}{64} \frac{v s s_2}{S_2}
\Biggr\{ 
  \tilde{f}_{3} F_{3}^c + \tilde{f}_{4} F_{4}^c
+ \tilde{f}_{7} F_{7}^c + \tilde{f}_{8} F_{8}^c
+ \tilde{f}_{10} F_{10}^c + \tilde{f}_{11} F_{11}^c
\nonumber
\\
& &
- \left(2 \ln \frac{s_2}{m^2} + \ln \frac{m^2}{S_2} \right)
  \left(\tilde{f}_{3} F_{3}^s + \tilde{f}_{4} F_{4}^s
       + \tilde{f}_{8} F_{8}^s + \tilde{f}_{10} F_{10}^s
       + \tilde{f}_{11} F_{11}^s
  \right)
\Biggr\} \, ;
\label{br3}
\end{eqnarray}
%
\begin{eqnarray}
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(4)}
& = &
- \frac{C(s) N_C}{64} \frac{1}{(1-w)_+} \frac{s_2^2}{S_2}
\Biggl\{ 
  \tilde{f}_{6} F_{6}^c + \tilde{f}_{9} F_{9}^c
\nonumber
\\
& &
- \left(2 \ln \frac{s v}{m^2} + \ln \frac{m^2}{S_2} \right)
  \left(\tilde{f}_{5} F_{5}^s + \tilde{f}_{6} F_{6}^s
       + \tilde{f}_{9} F_{9}^s
  \right)
\Biggr\} \, ;
\label{br4}
\end{eqnarray}
%
\begin{equation}
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(5)}
= 
\frac{C(s) N_C}{32} \left(\frac{\ln(1-w)}{1-w}\right)_+ 
\frac{s_2^2}{S_2}
\left\{\tilde{f}_{5} F_{5}^s + \tilde{f}_{6} F_{6}^s
       + \tilde{f}_{9} F_{9}^s
\right\} \, ;
\label{br5}
\end{equation}
%
\begin{eqnarray}
\left( \frac{d\sigma^{\gamma g}_{\rm Br}}{dvdw} \right)^{(6)}
& = &
\frac{\alpha_s}{2\pi} N_C \frac{d\sigma^{\gamma g}_{\rm LO}}{dvdw}
\Biggl\{ 
2 \ln^2 \frac{s v}{m^2} - \ln^2 x 
+ \frac{1}{2} \ln^2 \left(\frac{tx}{u}\right)
+ 2 \ln \frac{u}{t} \ln \frac{s v}{m^2} 
\nonumber
\\
& &
+ {\rm Li}_2 \left( 1 - \frac{u}{xt} \right)
- {\rm Li}_2 \left( 1 - \frac{t}{xu} \right)
- 2 \zeta_2
\nonumber
\\
& &
-\frac{2m^2 - s}{s\beta} \left[ 
    \left(2 \ln \frac{sv}{m^2} - \ln x \right) \ln x 
    - {\rm Li}_2 \left( \frac{-4 \beta}{(1-\beta)^2} \right)
    \right]
\Biggr\} \, .
\label{br6}
\end{eqnarray}
The coefficients $e_i$, $f_i$, $\tilde{e}_i$ and $\tilde{f}_i$ used in
eqs.\ (\ref{br1}) - (\ref{br6}) are given in appendix B of \cite{MCG}.
The integrals $I_i$ and $F_i$ are found in appendix C of \cite{KMC} and
appendix C of \cite{MCG}, respectively\footnote{In the course of our
  work we found several misprints in Ref.\ \protect\cite{MCG} which are
  not mentioned in detail here, but can be found in
  \protect\cite{MCGErr}.}. 

The six pieces in $d\sigma^{\gamma g} / dvdw$ differ in the singular
behavior for $w \rightarrow 1$. The second and the fourth term are
proportional to $1 / (1-w)_+$, whereas the fifth term contains the
factor $\left( \ln(1-w) / (1-w) \right)_+$. The sixth term, being
proportional to $d\sigma^{\gamma g}_{\rm LO} / dvdw$, has the factor
$\delta(1-w)$. The remaining terms are finite for $w \rightarrow 1$ as
long as $m \neq 0$. In the limit $m \rightarrow 0$ they give rise to
additional terms proportional to $\delta(1-w)$, $1 / (1-w)_+$ and
$\left( \ln(1-w) / (1-w) \right)_+$.

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

\subsection{The Subprocess $\gamma + q \rightarrow c + \bar{c} + q$
  \label{section23}} 

The diagrams contributing to this process are shown in Fig.\ 
\ref{fig2}b. The squared sum of the amplitudes after summing and
averaging over spins and colors is written as in \cite{MCG}:
\begin{equation}
4m^2 \left| M^{\gamma q}_{2 \rightarrow 3} \right|^2 
= \frac{2}{N_C} (4\pi)^3 \alpha \alpha_s^2 
\left( e_c^2 Q_1 + e_q^2 Q_2 - e_c e_q Q_3 \right) 
\label{q4}
\end{equation}
where $e_q$ is the charge of the light quark $q$. The squared sum for
the process $\gamma + \bar{q} \rightarrow c + \bar{c} + \bar{q}$ is
given by the same expression with an opposite sign in the last term of
eq.\ (\ref{q4}). The result for the corresponding cross sections
$d\sigma^{\gamma q}_{Q_i} / dvdw$ ($i = 1$, 2, 3) have been worked out
in \cite{MCG} with the following result:

\begin{eqnarray}
\frac{d\sigma^{\gamma q}_{Q_1}}{dvdw}
& = &
L e_c^2 
\Biggl\{
e_1 
+ \frac{4S_2}{m^2(s+t)^2} e_3
+ \frac{2S_2}{s_2(s+t)} e_4 \ln \frac{S_2}{m^2}
\nonumber
\\
& &
+ e_8 I_{10} 
+ \tilde{f}_{4} F_{4}^c 
+ \tilde{f}_{6} F_{6}^c 
+ \tilde{f}_{8} F_{8}^c 
+ \tilde{f}_{10} F_{10}^c 
+ \tilde{f}_{11} F_{11}^c 
\Biggr\} \, ;
\label{q4q1}
\end{eqnarray}
%
\begin{equation}
\frac{d\sigma^{\gamma q}_{Q_2}}{dvdw}
= 
L e_q^2 
\left\{
e_1 
+ f_{12} F_{12}
+ f_{13} F_{13}
+ f_{14} F_{14}
+ \tilde{f}_{16} F_{16}^c 
+ \tilde{f}_{17} F_{17}^c 
+ \tilde{f}_{20} F_{20}^c 
\right\} \, ;
\label{q4q2}
\end{equation}
%
\begin{eqnarray}
\frac{d\sigma^{\gamma q}_{Q_3}}{dvdw}
& = &
L e_c e_q 
\Biggl\{
e_1 
+ \frac{2S_2}{s_2(s+t)} e_4 \ln \frac{S_2}{m^2}
+ e_8 I_{10} 
+ \tilde{f}_{4} F_{4}^c 
+ \tilde{f}_{6} F_{6}^c 
+ f_{12} F_{12}
+ f_{14} F_{14}
\nonumber
\\
& &
+ \tilde{f}_{16} F_{16}^c 
+ f_{18} F_{18}
+ \tilde{f}_{19} F_{19}^c 
+ \tilde{f}_{20} F_{20}^c 
+ \tilde{f}_{21} F_{21}^c 
\Biggr\} \, ,
\label{q4q3}
\end{eqnarray}
%
with
\begin{equation}
L = \frac{\alpha \alpha_s^2}{4 N_C} \frac{v s_2}{S_2} \, .
\label{q4norm}
\end{equation}
%which differs from $L$ in \cite{MCG} by the factor $2\pi$
%and in eq.\ (\ref{q4}) the sign of the last term has been reversed). 
The coefficients $e_i$, $f_i$, and $\tilde{f}_i$ as well as expressions
for the integrals $F_i^c$ and $F_i$ are all given in appendix B and C of
\cite{MCG}. The integral $I_{10}$ can be obtained from appendix C of
\cite{KMC}.

The expressions (\ref{q4q1}) and (\ref{q4q2}) do not contain terms with
$1/\epsilon$ poles since they are equal with opposite sign to the
corresponding counterterms. In connection with these terms there appears
an additional contribution originating from the factor $\left(S_2 m^2 /
s_2^2 \right)^{\epsilon}$. This yields the following contribution to the
$Q_1$ part of the cross section:
\begin{equation}
\left(\frac{d\sigma^{\gamma q}_{Q_1}}{dvdw}\right)_{\rm eps}
= \frac{C(s)}{4} \ln \frac{s_2^2}{S_2 m^2} \frac{2sv}{1-vw}
P_{gq}(x_2) B(x_2 s, t, x_2 u)
\label{q4q1eps}
\end{equation}
where $B(s,t,u)$ corresponds to the LO matrix element for $\gamma + g
\rightarrow c + \bar{c}$ (see eq.\ (\ref{sigma_LO}))
\begin{equation}
B(s, t, u) = \frac{1}{s} \left[ \frac{t}{u} + \frac{u}{t}
+ 4 \frac{s m^2}{tu} \left( 1 - \frac{s m^2}{tu} \right) \right] \, .
\end{equation}
$x_2 = (1-v)/(1-vw)$ and $P_{gq}(x)$ is the well-known splitting
function,
\begin{equation}
P_{gq}(x) = C_F \left( \frac{ 1 + (1-x)^2}{x} - \epsilon x \right)
\, .
\label{pgq}
\end{equation}
%(in eqs.\ (5.8) and (5.9) of \cite{MCG} $F(0)$ must be replaced by
%$2F(0)$). 
A similar term connected with the $1/\epsilon$ pole is present in the
$Q_2$ part of the subprocess $\gamma + q \rightarrow c + \bar{c} + q$.
This yields the following additional contribution:
\begin{equation}
\left(\frac{d\sigma^{\gamma q}_{Q_2}}{dvdw}\right)_{\rm eps}
= \frac{4 C(s)}{9} \frac{e_q^2}{e_c^2} \ln \frac{s_2^2}{S_2 m^2} 
P_{q \gamma }(w) B_{q \bar{q}}(ws, wt, u)
\label{q4q2eps}
\end{equation}
where $B_{q\bar{q}}(s,t,u)$ is the LO matrix element for $q + \bar{q}
\rightarrow c + \bar{c}$, 
\begin{equation}
B_{q\bar{q}}(s, t, u) = \frac{1}{s} \left[ \frac{t^2}{s^2} +
\frac{u^2}{s^2} + 2 \frac{m^2}{s} \right] \, .
\label{bqqbar}
\end{equation}
The splitting function $P_{q \gamma }(x)$ is given by
\begin{equation}
P_{q\gamma}(x) = \frac{N_C}{2} \left[ x^2 + (1-x)^2 - 2 \epsilon x (1-x)
\right] \, .
\label{pqg}
\end{equation}
%(In \cite{MCG} the factor $N_C$ in (\ref{pqg}) is missing). 

According to \cite{MCG} the expressions for the $Q_1$ and $Q_2$ parts of
the cross section are not complete if one needs them as derived in
dimensional regularization. The results in \cite{MCG} were obtained in
dimensional reduction where the $\epsilon$-dependent terms of the
splitting functions in eqs.\ (\ref{q4q1eps}) and (\ref{q4q2eps}) vanish.
These terms induce extra finite terms in the limit $\epsilon \rightarrow
0$. Therefore the transition from dimensional reduction to dimensional
regularization requires the addition of the following conversion terms
to the $Q_1$ and $Q_2$ parts, respectively:
\begin{equation}
\left(\frac{d\sigma^{\gamma q}_{Q_1}}{dvdw}\right)_{\rm conv}
= - \frac{C(s)}{4} \frac{2sv}{1-vw}
P_{gq}^{\epsilon}(x_2) B(x_2 s, t, x_2 u)
\label{q4q1conv}
\end{equation}
and
\begin{equation}
\left(\frac{d\sigma^{\gamma q}_{Q_2}}{dvdw}\right)_{\rm conv}
= - \frac{16}{9} \frac{C(s)}{4} \frac{e_q^2}{e_c^2} 
P_{q \gamma }^{\epsilon}(w) B_{q \bar{q}}(ws, wt, u) 
\label{q4q2conv}
\end{equation}
where $P_{gq}^{\epsilon}$ and $P_{q\gamma}^{\epsilon}$ are the
$\epsilon$-dependent parts of the splitting functions (without the
factor $\epsilon$) given in (\ref{pgq}) and (\ref{pqg}), respectively.

The contribution $Q_3$, which arises from the interference of the
diagrams in (b1) and (b2) with (b3) and (b4) in Fig.\ \ref{fig2}b, has
no $1/\epsilon$ pole.  Therefore there is no counterterm proportional to
$e_q e_c$ and no additional contribution proportional to $\ln (s_2^2 /
S_2 m^2)$ like eqs.\ (\ref{q4q1eps}) and (\ref{q4q2eps}).

This completes the collection of formulae for the NLO corrections to
$\gamma + g \rightarrow c + \bar{c}$ and the cross section for the
subprocess $\gamma + q \rightarrow c + \bar{c} + q$, all based on the
work of Merebashvili et al. \cite{MCG} where many details must still be
looked up. In order to obtain the subtraction terms for the calculation
of the cross section in the $\overline{\rm MS}$ factorization scheme, we
need to know the limit $m \rightarrow 0$ of the expressions given above.
Of course, for the LO contribution the result is easily obtained. For
the NLO corrections, however, and for the subprocess $\gamma + q
\rightarrow c + \bar{c} + q$, the derivation of the limit $m \rightarrow
0$ is quite lengthy. In the next section we shall report the result of
this computation and some numerical tests which we performed to check
that the massless limits, from which the subtraction terms are obtained,
are correct.

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

\section{Zero-Mass Limit of the Massive Cross Sections \label{section3}}

In this section we present cross sections in the limit $m \rightarrow 0$
for the NLO corrections to $\gamma + g \rightarrow c + \bar{c}$ and for
the cross section $\gamma + q \rightarrow c + \bar{c} + q$ given in
section \ref{section2}. The result for this limit will in general be
different from the cross section obtained in the approach where the mass
of the charm quark is neglected from the beginning. In the genuine
massless calculation, worked out in Ref.\ \cite{ABDFS} and confirmed
later in Ref.\ \cite{LG}, the collinear singularities connected with the
charm quark appear as $1/\epsilon$ poles in dimensional regularization.
In the massive theory, they appear as terms proportional to $\ln
(m^2/s)$, instead. So, in this theory the collinear singularities are
regularized with a finite, although very small, charm mass. Due to this
different procedure for regularizing the collinear divergent
contributions, also different finite terms appear. The origin of
different finite terms in these two regularization schemes lies in the
fact that the two limits, $m \rightarrow 0$ and $\epsilon \rightarrow
0$, are not interchangeable.  The different finite terms must be
subtracted, if one wants to perform the factorization of these singular
terms in the so-called $\overline{\rm MS}$ scheme which is based on
dimensional regularization with massless quarks from the start. Such
finite terms have been found already in our previous work for the case
of NLO corrections for $\gamma + \gamma \rightarrow c + \bar{c}$. These
results for the subtraction terms are, up to a common constant $\kappa
C_F$ (see eq.\ (\ref{sigmalogg})), identical to the Abelian
contributions (proportional to $C_F$) of the NLO corrections to $\gamma
+ g \rightarrow c + \bar{c}$. Therefore in this paper we need to
establish the finite subtraction terms only for the non-Abelian part
proportional to $N_C$ and for the cross section of $\gamma + q
\rightarrow c + \bar{c} + q$.  We write the result again in a form which
has been introduced in the calculation for massless quarks by Gordon
\cite{LG}.  This will allow us to identify the subtraction terms we are
looking for.

The LO cross section for the process $\gamma + g \rightarrow c +
\bar{c}$ with $m=0$ has the simple form (see eq.\ (\ref{sigma_LO}))
\begin{eqnarray}
\lim_{m\rightarrow 0}\frac{d\sigma^{\gamma g}_{\rm LO}}{dvdw} 
= \frac{c(s)}{2} \delta(1-w) \tau_0(v)
~~~~~ {\rm with} ~~~~~ 
\tau_0(v) = \frac{v}{1-v} + \frac{1-v}{v} \, .
\label{sigma_LO_massless}
\end{eqnarray}
The NLO cross section and the cross section for $\gamma + q \rightarrow
c + \bar{c} + q$ in the limit $m \rightarrow 0$ is decomposed in the
following form:
\begin{eqnarray}
\lim_{m\rightarrow 0}\frac{d\sigma}{dvdw}
& = &
  \left(c_1 + \tilde{c}_1 \ln\frac{m^2}{s}\right) \delta(1-w)
\nonumber 
\\
& &
+ \left(c_2 + \tilde{c}_2 \ln\frac{m^2}{s}
  \right) \left(\frac{1}{1-w}\right)_{+}
+ c_3\left(\frac{\ln (1-w)}{1-w}\right)_{+} 
\nonumber
\\
& &
+ c_5 \ln v + c_6\ln (1-vw) + c_7 \ln (1-v+vw) + c_8\ln (1-v) 
\nonumber 
\\
& &
+ c_9 \ln w + c_{10}\ln (1-w) + c_{11} 
+ \tilde{c}_{11} \ln \frac{m^2}{s} 
\nonumber 
\\
& &
+ c_{12}\frac{\ln (1-v+vw)}{1-w} + c_{13}\frac{\ln w}{1-w}
+ c_{14}\frac{\ln (\frac{1-v}{1-vw})}{1-w} \, .
\label{sigma_massless}
\end{eqnarray}

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

\subsection{Massless Limit for NLO Corrections to $\gamma + g
\rightarrow c + \bar{c}$ \label{section31}} 

Here we report the results for the various coefficients $c_i$ which are
written in the form
\begin{equation}
c_i = \hat{c}_i + \Delta c_i
\end{equation}
where $\hat{c}_i$ are the results of \cite{LG} in the $\overline{\rm
MS}$ factorization with massless quarks and the $\Delta c_i$ are the
subtraction terms we need to know.

To shorten the expressions, we make use of the abbreviations
\begin{equation}
X = 1 - vw \, , ~~~~~Y = 1 - v + vw \, , ~~~~~ v_i = i -v \, .
\end{equation}

We shall give the result only for the non-Abelian contributions
proportional to $N_C$. The Abelian terms proportional to $C_F$ are
equal, up to a common factor, to the coefficients derived in our earlier
work \cite{KS} and can be taken from this reference. Terms proportional
to $\beta_0 = 11 - \frac{2}{3}n_f$ due to coupling constant
renormalization are not shown explicitly, although they are, of course,
included in our numerical evaluations.  For the $c_i$ we found the
following results:

\begin{eqnarray}
c_1 & = & - C(s) \frac{N_C}{2}
\Biggl\{
     4 \ln v \ln v_1 \tau_0(v) + \ln v + \ln v_1
\nonumber 
\\
& &
     + \left(\frac{17}{2} - \frac{7}{2v_1} - \frac{4}{v}\right) \ln^2 v 
     + \frac{1}{2} \left(\frac{1}{v} + 1 \right) \ln^2 v_1 
\Biggr\}
     + \Delta c_1 \, ,
\label{ci_first}
\end{eqnarray}
where
\begin{eqnarray}
\Delta c_1 = - C(s) \frac{N_C}{4} \frac{\ln v}{v v_1} \, ; 
\end{eqnarray}
%
\begin{eqnarray}
\tilde{c}_1 & = & C(s) N_C \left\{ \ln v_1 - \ln v \right\}
           \tau_0(v) + \Delta \tilde{c}_1 \, ,
\end{eqnarray}
where
\begin{eqnarray}
\Delta \tilde{c}_1 = C(s) \frac{N_C}{4} \frac{1}{v v_1} \, ; 
\end{eqnarray}
%
\begin{eqnarray}
c_2 = - C(s) N_C \left\{ \ln v_1 - 3 \ln v \right\} \tau_0(v) 
+ \Delta c_2 \, ,
\end{eqnarray}
where
\begin{eqnarray}
\Delta c_2 = - C(s) \frac{N_C}{4} \frac{1}{v v_1} \, ;
\end{eqnarray}
%
\begin{eqnarray}
\tilde{c}_2 = - C(s) N_C \tau_0(v) \, ;
\end{eqnarray}
%
\begin{eqnarray}
\tilde{c}_3 = 2 C(s) N_C \tau_0(v) \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_5 & = & C(s) \frac{N_C}{2} 
\Biggl\{ 
-\frac{2v v_1}{X^3} + \frac{2 v v_2}{X^2} - \frac{6}{v_1 X}
+ \frac{4 v^2 + 6 v_1}{X}
+ \frac{1}{w} \left[ \frac{4}{v_1^2} + \frac{6}{v} - \frac{2}{v_1}
                     - 3 + 4v \right]
\nonumber
\\
& &
+ w \left[ \frac{8}{v_1^2} - \frac{12}{v_1} + 6 \right]
- \frac{8}{v_1^2} + \frac{8}{v_1} - 2
\Biggr\}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_6 = C(s) \frac{N_C}{2} 
\left\{
- \frac{1}{w} \left[ \frac{4}{v_1^2} + \frac{1}{v} - \frac{1}{v_1} + 2
\right] 
- w \left[ \frac{8}{v_1^2} - \frac{16}{v_1} + 8 \right]
+ \frac{8}{v_1^2} - \frac{9}{v_1} + 1
\right\}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_7 = C(s) \frac{N_C}{2} \left\{ \frac{1}{vw} + \frac{v}{v_1} \right\}
\, ; 
\end{eqnarray}
%
\begin{eqnarray}
c_8 = - c_6 - C(s) \frac{N_C}{2} 
\left\{
\frac{3}{wv_1} - \frac{3}{v_1} - \frac{3}{w} + \frac{2}{vw} + 3
\right\} 
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_9 & = & - C(s) \frac{N_C}{2} 
\Biggl\{
\frac{2}{v_1 X} + \frac{2v - 2 - 2v^2}{X}
- \frac{1}{w} \left[ \frac{4}{v_1^2} + \frac{2}{v} - \frac{4}{v_1} + 2v
+ 1 \right]
\nonumber
\\
& &
- w \left[ \frac{8}{v_1^2} - \frac{16}{v_1} + 8 \right]
+ \frac{8}{v_1^2} - \frac{11}{v_1} + 3
\Biggr\}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{10} = c_5 + C(s) \frac{N_C}{2} 
\left\{
\frac{2}{v_1 X} + \frac{2v - 2v^2 - 2}{X} + \frac{1}{w} \left[
\frac{1}{v_1} - \frac{2}{v} - 2v + 1 \right] + \frac{v}{v_1}
\right\}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{11} & = & C(s) \frac{N_C}{2} 
\Biggl\{
- \frac{2v}{Y} + \frac{6vv_1}{X^3} + \frac{v(2v-7)}{X^2} +
\frac{1}{v_1X} + \frac{v-v_1}{X} 
\nonumber
\\
& &
- w \left(\frac{4}{v_1^2} -
\frac{2}{v_1} + 4\right) + \frac{4}{v_1^2} - \frac{4}{v_1} + 4
\Biggr\}
+ \Delta c_{11}
\, ,
\end{eqnarray}
where
\begin{eqnarray}
\Delta c_{11} = C(s) \frac{N_C}{2} 
\left\{
\frac{2vv_1}{X^3} + \frac{vv_1}{X^2} + \frac{1}{2v_1X} +
\frac{3v-2v^2-1}{2X} - \frac{1}{2w} \left(\frac{1}{v} + 2v -3 \right)
\right\} \, .
\end{eqnarray}
We remark that $c_{11} - \Delta c_{11}$ agrees (up to the factor $C(s)$,
which is common to all $c_i$) with $\hat{c}_{11}$ in \cite{LG} for
$\lambda = 1$, i.e.\ for $D$-dimensional spin averaging.
\begin{eqnarray}
\tilde{c}_{11} = - c_5 + C(s) \frac{N_C}{2} 
\left\{
- \frac{2}{v_1X} + \frac{2+2v^2-2v}{X} + \frac{1}{w} \left[
\frac{2}{v_1} + \frac{4}{v} - 4 + 2v \right] - \frac{4}{v_1} + 4
\right\}\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{12} = C(s) \frac{N_C}{2} 
\left\{
\frac{1}{v_1} + \frac{1}{v} + 2
\right\}\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{13} = C(s) \frac{N_C}{2} 
\left\{
\frac{1}{v_1} + \frac{2}{v} - 5
\right\}\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{14} = C(s) \frac{N_C}{2} 
\left\{
\frac{1}{v} + \frac{2}{v_1} - 5
\right\}\, .
\label{ci_last}
\end{eqnarray}

We found subtraction terms in $c_1$, $\tilde{c}_1$, $c_2$ and $c_{11}$.
$\Delta \tilde{c}_1$ is remarkable since it means that there is a term
proportional to $\ln (m^2/s)$ which is not connected with a collinear
divergent term as obtained with massless quarks in $\overline{\rm MS}$
factorization. No such term was found in our previous work. The origin
of $\Delta \tilde{c}_1$ remains unclear to us. In any case, its presence
is not relevant for our purpose since it will be subtracted to obtain
the massive theory with $\overline{\rm MS}$ factorization. In order to
check the subtraction terms $\Delta c_1$, $\Delta \tilde{c}_1$, $\Delta
c_2$ and $\Delta c_{11}$, we made numerical tests which will be
described in subsection \ref{section33}.

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

\subsection{Massless Limit for the Cross Section of $\gamma + q
\rightarrow c + \bar{c} + q$ \label{section32}} 

We shall present the cross section limits for the contributions $Q_1$,
$Q_2$ and $Q_3$ separately. For all three terms there is no LO cross
section of the order $O(\alpha \alpha_s)$. Therefore we have $c_1 =
\tilde{c}_1 = c_2 = \tilde{c}_2 = c_3 = c_{12} = c_{13} = c_{14} = 0$
for $Q_1$, $Q_2$ and $Q_3$.

The other coefficients for $d\sigma^{\gamma g}_{Q_1}/dvdw$, which is
decomposed as in (\ref{sigma_massless}), are:
%
\begin{eqnarray}
c_{5} = C(s) C_F 
\left\{
\frac{v}{2X^2} - \frac{v v_3}{2Xv_1} - \frac{(1+3v^2)(1-w)}{v_1^2} +
\frac{1+v^2}{v_1^2w} 
\right\}\, ;
\label{c4q_5}
\end{eqnarray}
%
\begin{eqnarray}
c_{6} = C(s) C_F 
\left\{
\frac{2v(1+v)}{v_1^2} - \frac{1+v^2}{v_1^2w} - \frac{4v^2w}{v_1^2}
\right\}\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{7} = 0 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{8} = - c_6 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{9} = - c_6 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{10} = c_5 \, ;
\end{eqnarray}
%
\begin{eqnarray}
\tilde{c}_{11} = - c_5 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{11} = C(s) C_F 
\left\{
- \frac{v}{2X^2} + \frac{v}{2v_1X} + \frac{1}{2w} - \frac{2w}{v_1^2} +
\frac{2v^2}{v_1^2} + \frac{2}{v_1} + \frac{w}{v_1} - 2w
\right\}\, .
\label{c4q_11}
\end{eqnarray}

We found no additional terms compared to \cite{LG}. Therefore in the
$Q_1$ part all coefficients obtained with mass regularization agree in
the limit $m \rightarrow 0$ with the massless calculation in
$\overline{\rm MS}$ factorization. We remark that the result eq.\ 
(\ref{c4q_11}) for $c_{11}$ corresponds to $D$-dependent spin averaging
of the photon and gluon ($\lambda = 1$ in appendix B8 of \cite{LG}). In
order to obtain the results (\ref{c4q_5}--\ref{c4q_11}), i.e.\ without
additional terms $\Delta c_i$, it was essential to incorporate the
conversion term (\ref{q4q1conv}).

The coefficients for $d\sigma^{\gamma g}_{Q_2}/dvdw$ are the following:
%
\begin{eqnarray}
c_{5} = C_q(s) C_F 
\left\{
\frac{(3-2v)v}{2Y} - \frac{vv_1}{2Y^2} - \left(3v^2 + v_1^2\right)(1-w)
+ \frac{1-2vv_1}{w}
\right\}\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{6} = 0 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{7} = - C_q(s) C_F 
\frac{v^2w}{Y^2}
\left(
1 - 2v + 2vw
\right)\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{8} = 0 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{9} = C_q(s) C_F 
\left\{
\frac{1-2vv_1}{w} + 2v - 4v^2(1-w)
\right\}\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{10} = c_5 \, ;
\end{eqnarray}
%
\begin{eqnarray}
\tilde{c}_{11} = - c_5 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{11} & = & C_q(s) C_F 
\Biggl\{
\frac{(3v-4)v}{2Y} + \frac{vv_1}{Y^2} 
+ \frac{1}{w} \left(1 + v - v^2\right)
\nonumber
\\
& &
+ w \left(3v - 1 - 4 v^2\right) + 4 v^2 - 2 v
\Biggr\}
+ \Delta c_{11} \, , 
\end{eqnarray}
where
\begin{eqnarray}
\Delta c_{11} = - C_q(s) C_F 
\frac{1}{w}
\left(
w^2 + (1-w)^2
\right)\, .
\end{eqnarray}
%
Here we have introduced the factor
\begin{equation}
C_q(s) = \frac{2\pi \alpha \alpha_s e_q^2}{s} \frac{\alpha_s}{2\pi} =
\frac{e_q^2}{e_c^2} C(s) \, .
\end{equation}

In the contribution $Q_2$ we found one extra term in $c_{11}$, which
must be subtracted when going over to the $\overline{\rm MS}$
factorization scheme. The result for $\tilde{c}_{11}$ agrees with the
result in \cite{LG}, if one compares to $\tilde{c}_{11} +
\tilde{\tilde{c}}_{11}$, where $\tilde{c}_{11}$
($\tilde{\tilde{c}}_{11}$) gives the terms originating from the
collinear singularities of the initial state (final state). Since in the
massive theory both have the scale $m$, we have no possibility, based on
the explicit results of \cite{MCG}, to distinguish initial and final
state. We have calculated $\Delta c_{11}$ using the result in \cite{LG}
for $\lambda = 1$, i.e.\ with $D$-dimensional spin averaging for the
photon.

The massless limit for the $Q_3$ term is obtained in the same way. The
corresponding coefficients have the prefactor
\begin{equation}
C_{cq}(s) = \frac{2\pi \alpha \alpha_s e_q e_c}{s} \frac{\alpha_s}{2\pi}
= \frac{e_q}{e_c} C(s)
\end{equation}
and the following form:
%
\begin{eqnarray}
c_{5} = 0 \, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{6} = C_{cq}(s) C_F 
\frac{v^2(1-2w)}{v_1}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{7} = C_{cq}(s) C_F 
\frac{v^2(1-2w)}{v_1}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{8} = C_{cq}(s) C_F 
2v(1-2w)
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{9} = C_{cq}(s) C_F 
\left\{
\frac{3v^2}{v_1} + 2 - \frac{6v^2w}{v_1} + \frac{8v^2w^2}{v_1}
\right\}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{10} = - C_{cq}(s) C_F 
\left\{
2 + \frac{5v^2}{v_1} - \frac{10v^2w}{v_1} + \frac{8v^2w^2}{v_1}
\right\}
\, ;
\end{eqnarray}
%
\begin{eqnarray}
c_{11} = C_{cq}(s) C_F 
\left\{
\frac{v}{Y} - \frac{v}{X} - 4w + 2
\right\}
\, .
\end{eqnarray}
As expected, there are no contributions to $\tilde{c}_{11}$ since $Q_3$
does not have any collinear singularities. Since there are no $\Delta
c_i$ terms, all coefficients agree with the results in \cite{LG}. In
total, we have found only one $\Delta c_i$, namely $\Delta c_{11}$ in
the $Q_2$ contribution. This completes the calculation of the
subtraction terms needed for the $\overline{\rm MS}$ factorization
scheme with massive charm quarks.

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

\subsection{Numerical Test of the Subtraction Terms \label{section33}}

The calculation of the subtraction terms, in particular those in
subsection \ref{section31}, was rather involved. Special care had to be
exercised in order to recover all the terms proportional to
$\delta(1-w)$, $1/(1-w)_+$, $(\ln(1-w)/(1-w))_+$ and the remaining terms
in the decomposition (\ref{sigma_massless}). In order to check that the
$\Delta c_i$ presented in subsection \ref{section31} are correct, and
also to see how the various contributions to $d\sigma^{\gamma g}/dvdw$
written down in subsection \ref{section22} behave as a function of the
transverse momentum $p_T$ of the charm quark in comparison with the
cross section for massless quarks, we have calculated the NLO
corrections in two ways. First, we have taken the six terms from the
bremsstrahlung contribution as given in eqs.\ (\ref{br1}) -- (\ref{br6})
and the contributions of the virtual corrections $d\sigma^{\gamma
  g}_{\rm a-e}$, $d\sigma^{\gamma g}_{\rm f}$, $d\sigma^{\gamma g}_{\rm
  g}$ and $d\sigma^{\gamma g}_{\rm h}$ in eqs.\ (\ref{dsae0}--\ref{dsh})
and calculated these parts as a function of $p_T$ for charm quarks with
mass $m = 1.5$ GeV, and also from formulae derived directly from the
expressions in subsection \ref{section22} in the limit $m \rightarrow
0$. We normalize these parts to the LO cross section for massless charm
quarks calculated from the formula (\ref{sigma_LO}) in subsection
\ref{section21} and denote them by $T_i(N_C)$:
\begin{equation}
T_i(N_C) = \frac{d^2\sigma^{\gamma g}_i/dydp_T}{d^2\sigma^{\gamma
    g}_{\rm LO}/dydp_T (m = 0)} \, .
\label{tratio}
\end{equation}

The cross sections in the numerator and denominator of $T_i$ are
actually the cross section for $e^+ + e^- \rightarrow e^+ + e^- + c +
X$, i.e.\ the photon-photon cross section folded with both the
Weizs\"acker-Williams spectrum (with $\theta_{\rm max} = 0.033$) and the
distribution of gluons inside the photon (taken from \cite{GRV}). For
the present purpose it is sufficient to evaluate the cross sections at
rapidity $y = 0$.  All other input data are chosen as in our previous
work \cite{KS}: $\sqrt{s} = 193$ GeV, $\Lambda_{(n_f = 4)} = 328$ MeV
(i.e., $\alpha_s(m_Z) = 0.1181$). The charm quarks are not fragmented
and $BR(c \rightarrow D^*) = 1$. The renormalization scale is $\mu_R =
\sqrt{p_T^2 + m^2}$ and the factorization scales are equal to $M_I = M_F
= m$, as inherent to the formalism in subsection \ref{section22}.
Details for how $d\sigma / dy dp_T$ is calculated from $d\sigma / dv dw$
have been given for $\gamma \gamma$ scattering in eq.\ (45) in
\cite{KS}. In this equation, the distribution function of the second
photon is replaced by the convolution of the Weizs\"acker-Williams
spectrum with the distribution of a gluon in the photon.

%----------------------------------
\begin{figure}[p!] 
\unitlength 1mm
\begin{picture}(160,210)
\put(2,140){\begin{minipage}[b][68mm][b]{68mm}
\include{npt3a}
\end{minipage}}
\put(60,208){(a)}
\put(82,140){\begin{minipage}[b][68mm][b]{68mm}
\include{npt4a}
\end{minipage}}
\put(142,208){(b)}
\put(0,66){\begin{minipage}[b][68mm][b]{68mm}
\include{npt5a}
\end{minipage}}
\put(60,133){(c)}
\put(82,66){\begin{minipage}[b][68mm][b]{68mm}
\include{npt6a}
\end{minipage}}
\put(142,133){(d)}
\put(0,-8){\begin{minipage}[b][68mm][b]{68mm}
\include{npt7a}
\end{minipage}}
\put(18,58){(e)}
\put(84,-8){\begin{minipage}[b][68mm][b]{68mm}
\include{npt8a}
\end{minipage}}
\put(101,58){(f)}
\end{picture}
\caption{Comparison of the massive and massless calculations for the
  bremsstrahlung contributions $T_1-T_6$ (a$-$f), see text.}
\label{fig3}
\end{figure}
%----------------------------------

%----------------------------------
\begin{figure}[t!] 
\unitlength 1mm
\begin{picture}(160,142)
\put(0,69){\begin{minipage}[b][68mm][b]{68mm}
\include{npt9a}
\end{minipage}}
\put(18,135){(a)}
\put(79,69){\begin{minipage}[b][68mm][b]{68mm}
\include{npt12a}
\end{minipage}}
\put(100,135){(b)}
\put(-2,-6){\begin{minipage}[b][68mm][b]{68mm}
\include{npt10a}
\end{minipage}}
\put(60,60){(c)}
\put(79,-7){\begin{minipage}[b][68mm][b]{68mm}
\include{npt11a}
\end{minipage}}
\put(141,60){(d)}
\end{picture}
\caption{Comparison of the massive and massless calculations for the
  virtual contributions $T_{\rm a-e}-T_{\rm h}$ (a$-$d), see text.}
\label{fig4}
\end{figure}
%----------------------------------

%----------------------------------
\begin{figure}[th!] 
\unitlength 1mm
\begin{picture}(158,80)
\put(-2,-5){\begin{minipage}[b][70mm][b]{70mm}
\include{npt2a}
\end{minipage}}
\put(38,75){(a)}
\put(81,-5){\begin{minipage}[b][70mm][b]{70mm}
\include{npt2b}
\end{minipage}}
\put(121,75){(b)}
\end{picture}
\caption{Comparison of the massive calculation (dashed lines) with its
  massless limit (full lines) for the terms proportional to $N_C$ for
  the process $\gamma + g \rightarrow c + \bar{c} + X$. In (a), the sum
  of all ratios defined in eq.\ (\ref{tratio}) is shown, in (b) the
  difference of the sums for the massless limit and the calculation with
  $m\neq 0$.  The dotted lines represent the results of the massless
  limit with subtracted $\Delta c_i$, i.e.\ the calculation with
  $\overline{\rm MS}$ factorization \protect\cite{LG}.  }
\label{fig5}
\end{figure}
%----------------------------------

%----------------------------------
\begin{figure}[ht!] 
\unitlength 1mm
\begin{picture}(158,78)
\put(-2,-5){\begin{minipage}[b][70mm][b]{70mm}
\include{npt1a}
\end{minipage}}
\put(38,75){(a)}
\put(81,-5){\begin{minipage}[b][70mm][b]{70mm}
\include{npt1b}
\end{minipage}}
\put(121,75){(b)}
\end{picture}
\caption{Comparison of massless (full lines) and massive (dashed lines)
  calculations for the sum of all contributions proportional to $C_F$
  for the process $\gamma + g \rightarrow c + \bar{c} + X$ as in Fig.\ 
  \protect\ref{fig5}.}
\label{fig7}
\end{figure}
%----------------------------------

The results for the ratios $T_i(N_C)$ ($i = 1, \ldots 6$), $T_{\rm
  a-e}(N_C)$, $T_{\rm f}(N_C)$, $T_{\rm g}(N_C)$ and $T_{\rm h}(N_C)$,
are shown in Figs.\ \ref{fig3} and \ref{fig4}. The dashed lines give the
contribution from the massive theory ($m \neq 0$) and the full lines are
the massless limit of the massive calculation. We see that the dashed
curves approach the full curves for increasing $p_T$ quite nicely. This
means that the extraction of additional terms proportional to
$\delta(1-w)$ in the $T_i(N_C)$ which appear in the massless limit, have
been extracted correctly. We observe that above $p_T = 5$ GeV the two
curves, dashed and full, are already very close to each other.  Larger
deviations are seen only for $p_T < 5$ GeV which originate from the
terms $\propto m^2/p_T^2$ we are interested in. In Fig.\ \ref{fig5}a we
have plotted the sum of all the $T_i(N_C)$. We see that the results of
the massive theory (dashed curve) approaches the massless limit (full
curve) quite well. For $p_T > 10$ GeV the difference between the two
curves is negligibly small.  The dotted curve gives the result where the
$\Delta c_i$ terms given in subsection \ref{section31} are subtracted
from the massless limit. This cross section is the same as the one from
the massless theory with $\overline{\rm MS}$ factorization as given in
\cite{LG}. We see that the difference to the full curve is
non-negligible and increases with increasing $p_T$. This difference will
be added later to the massive theory in order to adjust the massive
theory to the $\overline{\rm MS}$ factorization. The difference between
the massless limit and the massive cross section is more evident in
Fig.\ \ref{fig5}b, where we have plotted the difference of the sum of
all $T_i$ terms for the massless limit (the massless theory without
$\Delta c_i$) and the massive calculation.

%----------------------------------
\begin{figure}[th!] 
\unitlength 1mm
\begin{picture}(158,156)
\put(-3,69){\begin{minipage}[b][80mm][b]{80mm}
\include{qpq2}
\end{minipage}}
\put(39,149){(a)}
\put(80,69){\begin{minipage}[b][80mm][b]{80mm}
\include{qpq3}
\end{minipage}}
\put(122,149){(b)}
\put(40,-8){\begin{minipage}[b][80mm][b]{80mm}
\include{qpq4}
\end{minipage}}
\put(81,72){(c)}
\end{picture}
\caption{Comparison of the massless (full lines) and massive 
  calculations (dashed lines) for the contributions to the process
  $\gamma + u \rightarrow u + c + \bar{c}$ normalized as in eq.\ 
  (\ref{tratio}).  In (b) the dotted line shows the $Q_2$-part with
  subtracted $\Delta c_{11}$.}
\label{fig8}
\end{figure}
%----------------------------------

The sum $\sum T_i(N_C)$ in the massless limit has been calculated
independently by going back to the decomposition (\ref{sigma_massless})
with the $c_i$ given in eqs.\ (\ref{ci_first}) -- (\ref{ci_last}), i.e.\ 
with the subtraction terms $\Delta c_i$ included as compared to Gordon's
result. For this calculation we obtained the same full curves in Figs.\ 
\ref{fig5}a and b. This demonstrates that the extraction of the terms
$\Delta c_1$, $\Delta \tilde{c}_1$, $\Delta c_2$ and $\Delta c_{11}$ as
given in subsection \ref{section31} is correct.

For comparison we have performed the same calculation for the Abelian
part of the cross section. Here we have calculated $\sum T_i(C_F)$
directly from the massless limit in terms of the coefficients $c_i$,
written in \cite{KS} for the process $\gamma + \gamma \rightarrow c +
\bar{c}$. The results, together with the cross section for the massless
theory of \cite{LG} ($\Delta c_i$ subtracted), are shown in Fig.\ 
\ref{fig7}a, b.  We see that the massive result approaches the massless
limit for large $p_T$.  The difference between the two curves (see Fig.\ 
\ref{fig7}b) above $p_T = 5$ GeV is insignificant. We observe that the
effect of the $\Delta c_i$ terms found in \cite{KS} on the $C_F$
contribution is very important.

The calculation of the subtraction term for $\gamma + q \rightarrow c +
\bar{c} + q$ was much easier. So we show numerical results only for the
massless limit obtained from the results in subsection \ref{section32}.
The comparison with the massive cross section for the cases $Q_1$, $Q_2$
and $Q_3$ is shown in Figs.\ \ref{fig8}a--c, again normalized to the LO
cross section for $\gamma + g \rightarrow c + \bar{c}$ with $m=0$ as in
eq.\ (\ref{tratio}). We see that the massless limit approaches the
massive cross section quite nicely. We see also that the effect of the
only $\Delta c_i$ term, namely $\Delta c_{11}$ in case of $Q_2$ (dotted
curve in Fig.\ \ref{fig8}b, note the scale), has an insignificant
effect.


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

\section{Comparison with LEPII Data \label{section4}}

In this section we compare our results with the experimental data from
the ALEPH \cite{aleph2002}, the L3 \cite{L32002} and the OPAL
\cite{opal} collaborations. We do this for every experiment separately,
since the three collaborations analyzed their measurements with
different average center-of-mass energies $\sqrt{s}$, different
anti-tagging conditions (limit on $Q^2$), and within slightly different
rapidity ranges. For the rapidity distributions $d\sigma/dy$, the
average over the $p_T$ range is also not identical for the three data
sets.  The values used for the average $\sqrt{s}$, for $Q_{max}^2$, for
$|y_{max}|$ and the $p_T$ range used for the comparison are given in
Tab.\ 1.

\begin{table}[h]
\begin{center}
\begin{tabular}{rccc}
\hline\hline
\rule[0mm]{0mm}{5mm}
& OPAL & L3 & ALEPH \\[1mm]
\hline
\rule[0mm]{0mm}{5mm}
$\sqrt{s}$ [GeV] & 193 & 197 & 197 \\[1mm]
tagging & $\theta < 0.033$ & $Q<m_{\rho}$ & $Q^2<4.5 {\rm GeV}^2$ \\[1mm]
$|y|_{\rm max}$ & 1.5 & 1.4 & 1.5 \\[1mm]
$p_T$-range [GeV] & $2-12$ & $1-12$ & $2-12$ \\[1mm]
\hline
\end{tabular}
\caption{Experimental conditions for the measurement of inclusive
  $D^{\ast}$ production.}
\end{center}
\end{table}

When comparing predictions for the $p_T$-distribution with the
experimental results, we show both the differential cross section
$d\sigma/dp_T$ as a function of $p_T$, as well as values averaged over
$p_T$-bins as used by the experimental collaborations. These bins are
$2-3$, $3-5$ and $5-12$ GeV for all three experiments and the additional
bin $1-2$ GeV for L3. Since the $y$-distributions are very flat, there
is no need to average over bins and we will show only $d\sigma/dy$.

The theoretical predictions of the massive theory consist of three
parts, the direct cross section with $m \neq 0$ already presented in our
previous work \cite{KS}, the single-resolved contribution considered in
this paper, and the double-resolved cross section. The single-resolved
cross section has two components. First there are the contributions from
$\gamma g \rightarrow c\bar{c}$ together with the corresponding NLO
corrections and the cross section for the processes $\gamma q
\rightarrow qc\bar{c}$, both with $m \neq 0$ and after subtracting the
terms $\Delta c_i$ as given in sect.\ 3 and in our earlier work for the
Abelian part \cite{KS}. Second, we add the contributions coming from the
charm PDF in the photon. This part is evaluated with $m=0$ in the hard
scattering cross section including NLO corrections, using the formulae
given in Ref.\ \cite{LG}.

The particular prescription for the treatment of the incoming heavy
charm quark in the single-resolved contribution as a massless parton is
in fact mandatory. Since all available charm PDF's are determined with
$m=0$ in the hard scattering cross section, this is the only consistent
choice.  The finite charm quark mass appears only in the starting scale
$\mu_0 = m$ with the effect that the charm PDF vanishes below the scale
$\mu_0$ \cite{GRV}.  This prescription is usually also applied in the
treatment of heavy quarks in deeply inelastic scattering, as advocated
in \cite{Coll,KOS,TKS}.

We will also compare with predictions of the massless theory.  In this
case, all contributions are calculated with $m=0$, again using the
results of Ref.\ \cite{LG}. In both the massless and the massive
calculations, we add the double-resolved contributions evaluated with
$m=0$. The corresponding formulae are based on the work of
\cite{aversa}.

%----------------------------------
\begin{figure}[p!] 
\unitlength 1mm
\begin{picture}(158,175)
\put(-3,87){\begin{minipage}[b][84mm][b]{84mm}
\include{dft9a}
\end{minipage}}
\put(42,173){(a)}
\put(77,87){\begin{minipage}[b][84mm][b]{84mm}
\include{dft9b}
\end{minipage}}
\put(122,173){(b)}
\put(38,0){\begin{minipage}[b][84mm][b]{84mm}
\include{dft9c}
\end{minipage}}
\put(80,87){(c)}
\end{picture}
\caption{The single-resolved contribution: (a) shows the charm-initiated
  contribution (dotted line), the contribution due to gluons and light
  quarks (dashed line) and the sum (full line); (b) compares the sum of
  all contributions for the massless (full line) and the massive (dashed
  line) calculation; (c) shows the ratio of the massive over massless
  calculation with (full) and without (dashed line) charm in the
  photon.}
\label{fig9}
\end{figure}
%----------------------------------

In the following the fragmentation $c \rightarrow D^{*}$ is calculated
with the purely non-perturbative FF of Binnewies et al.\ \cite{BKK}
(OPAL set at NLO) determined from OPAL $e^+e^- \rightarrow D^{*}X$ data
as explained in detail in \cite{BKK}. We choose the renormalization
scale and $\alpha_s(m_Z)$ as stated above, include $n_f=4$ flavors and
choose the factorization scales $M_I = M_F = 2 \sqrt{p_T^2+m^2}$. The
transition from $M_I = M_F = m$ to this choice of scales is performed,
as described in our previous work (see eq.\ (42) in \cite{KS}), using the
coefficients $\tilde{c}_1$, $\tilde{c}_2$ and $\tilde{c}_{11}$, i.e.\ 
based on the massless calculation. The coefficients of the massless
calculation must be used here, since the evolution of PDF's and FF's is
also based on massless evolution kernels. This choice of scales allows
us to calculate $d\sigma/dp_T$ down to small $p_T$, since otherwise we
would come below the starting scale of the non-perturbative FF of
\cite{BKK}, which is approximately equal to $2m$.  We have calculated
$d^2\sigma/dydp_T$, integrated over the $y$ or $p_T$ ranges as given in
Tab.\ 1, for the three data sets. $y$ is the pseudo-rapidity as used in
the analysis of the experimental data \cite{aleph2002,L32002,opal}.  We
identify in our calculations the pseudo-rapidity of the $D^{*}$ with the
rapidity of the charm quark.  Finally, we note that we neglect the small
contribution from the fragmentation of gluons, $g \rightarrow D^*$.

In order to show the amount originating from the charm content in the
photon, we have plotted $d\sigma/dp_T$ as a function of $p_T$ for the
massive single-resolved contribution, i.e.\ without the charm
contribution in the photon, and the sum of both contributions in Fig.\ 
\ref{fig9}a. We see that for small $p_T$ the component due to light
quarks and gluons in the initial state is dominant.  For large $p_T$ the
contribution of the charm PDF increases, as to be expected, and amounts
to $68\,\%$ of the sum at $p_T=12$ GeV.  It is clear that due to this
component the influence of the charm quark mass, i.e.\ the correction
from the $m^2/p_T^2$ terms, diminishes.  This is shown in Fig.\ 
\ref{fig9}b, where the total single-resolved cross section is plotted
for the massive and the massless calculation. On this logarithmic plot,
the influence of the finite charm mass is visible only for small $p_T <
2$ GeV. At $p_T=2$ GeV, the $m \neq 0$ cross section is reduced by $\sim
16\, \%$ as compared to the massless approximation, as is seen more
clearly in Fig.\ \ref{fig9}c where the ratio is shown. The dashed curve
in this plot shows the ratio before the charm in the photon contribution
is added.  The strong increase of this ratio for $p_T < 2$ GeV is caused
by the large NLO corrections in the massless cross section. For
definiteness, the cross sections in Fig.\ \ref{fig9} have been
calculated with the ALEPH kinematical constraints (see Tab.\ 1).

%----------------------------------
\begin{figure}[b!] 
\unitlength 1mm
\begin{picture}(160,85)
\put(32,-5){\begin{minipage}[b][90mm][b]{90mm}
\include{dft10}
\end{minipage}}
\put(83,25){$-$}
\put(90,22){$+$}
\end{picture}
\caption{The double-resolved contribution. The contribution due to light
  quarks and gluons in the initial state is shown by the dash-dotted
  line. This part is negative at small $p_T$ ($p_T < 7$ GeV) where the
  absolute value is shown.}
\label{fig10}
\end{figure}
%----------------------------------

The influence of the $m^2/p_T^2$ terms is further diminished in the
small $p_T$ region, if we add the double-resolved cross section. This
part is almost entirely due to the charm component in the photon, which
we have calculated with $m=0$ in the hard scattering parton-parton cross
sections in the LO and NLO terms in the same way as we have done for the
corresponding single-resolved cross section. To demonstrate this, we
show in Fig.\ \ref{fig10} the double-resolved cross section
$d\sigma/dp_T$ due to light quarks and gluons in the initial state,
i.e.\ without the charm component, together with the sum of all
contributions. We see that the cross section without charm in the
initial state is indeed very small in its absolute value and the
double-resolved cross section is dominated by the charm component. The
light-quark plus gluon component becomes negative for $p_T < 7$ GeV due
to NLO corrections.  This negative contribution leads to an
insignificant decrease of the total double-resolved cross section for
$p_T > 2$ GeV. At LO, the cross section without charm in the photon is
essentially given by the $gg \rightarrow c\bar{c}$ component, which is
small, since the gluon component in the photon PDF is not large. Another
contribution at LO is from $qq \rightarrow c\bar{c}$, which also is
expected to be small. These non-charm contributions are computed in the
approximation with $m=0$. The fact that their contribution is negligible
is also true for $m \neq 0$ \cite{CGKKKS,FKL}.

To obtain an overview of the partition of $d\sigma/dp_T$ into direct and
resolved contributions and the influence of the massless approximation,
we show in Fig.\ \ref{fig12}a these cross sections with cuts as in the
ALEPH analysis.  All three cross sections are calculated up to NLO. At
$p_T=2$ GeV, the three contributions, direct, single-resolved and
double-resolved, amount to approximately $39\,\%$, $16\,\%$ and $45\,\%$
of the total sum, whereas at $p_T=12$ GeV the relative contributions are
$68\,\%$, $16\,\%$ and $16\,\%$, respectively. These numbers refer to
the cross section with $m \neq 0$ except for the contributions with
charm in the photon PDF. With increasing $p_T$, the resolved cross
sections decrease much stronger than the direct component.  In Fig.\ 
\ref{fig12}a we also show the direct and single-resolved cross sections
for $m \neq 0$ and for $m=0$ for comparison. We see that the finite
charm mass effects, i.e.\ the $m^2/p_T^2$ terms, are essential only for
small $p_T \leq 2$ GeV and are largest for the direct component. They
lead to a reduction of the direct cross section. With this reduction the
double-resolved cross section becomes dominant and therefore the
influence of the finite charm mass effects is very much reduced in the
total sum. For a clearer presentation of the three components in the
cross section with $m \neq 0$, we show them in Fig.\ \ref{fig12}b again,
where the dominance of the double-resolved part for small $p_T$ is
clearly visible.

%----------------------------------
\begin{figure}[t!] 
\unitlength 1mm
\begin{picture}(160,85)
\put(-3,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dft12a}
\end{minipage}}
\put(42.5,81){(a)}
\put(77,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dft12b}
\end{minipage}}
\put(122,81){(b)}
\end{picture}
\caption{The partition of the full calculation (full lines) into direct
  (dotted), single-resolved (dashed) and double-resolved (dash-dotted)
  contributions. In (a), upper curves correspond to the massless and
  lower curves to the massive calculation. The results of the massive
  calculation are shown separately in (b) again for clarity.}
\label{fig12}
\end{figure}
%----------------------------------

%----------------------------------
\begin{figure}[th!] 
\unitlength 1mm
\begin{picture}(158,165)
\put(-2,79){\begin{minipage}[b][80mm][b]{80mm}
\include{dft11a}
\end{minipage}}
\put(41,164){(a)}
\put(79,79){\begin{minipage}[b][80mm][b]{80mm}
\include{dft11c}
\end{minipage}}
\put(123,164){(b)}
\put(-2,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dft11b}
\end{minipage}}
\put(41,81){(c)}
\put(78,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dft11d}
\end{minipage}}
\put(121,81){(d)}
\end{picture}
\caption{The full calculation compared with experimental data from
  ALEPH (a), L3 (b) and OPAL (c). Full lines represent the massless,
  dashed lines the massive calculation. Histograms show $d\sigma/dp_T$
  averaged over the corresponding $p_T$-bins. The histograms in (d) are
  for the massive calculation (full line: ALEPH, dashed line: OPAL,
  dotted line: L3).}
\label{fig11}
\end{figure}
%----------------------------------

After these preparatory studies we are in the position to compare our
results with the data of the three LEP collaborations. This is shown in
Figs.\ \ref{fig11}a, b, c where we compare our results of the
calculation with massive ($m \neq 0$) and massless ($m=0$) charm quarks
with the experimental data of the ALEPH \cite{aleph2002}, L3
\cite{L32002} and OPAL \cite{opal}\footnote{We take the OPAL data from
  the second reference in \protect\cite{opal}.} collaborations.  The
data are always averages over the respective bins in $p_T$: $1-2$ (L3),
$2-3$, $3-5$, $5-12$ GeV (ALEPH, L3 and OPAL), but we show theoretical
predictions for both the differential cross section $d\sigma/dp_T$ as
well as for corresponding bin averages.  The agreement between data and
theory is quite satisfactory on average. The L3 point in the first bin
is lower than the theoretical prediction, the results in the bin $2-3$
GeV are in good agreement for all experiments, while our results for the
last two bins tend to be lower than the measured values. Only the L3
point in the highest $p_T$-bin is again in perfect agreement with the
predictions.  In Figs.\ \ref{fig11}a, b, c we have plotted also the pure
$m=0$ cross sections.  The results are not very different from the
massive cross sections, even at low $p_T$.  This is due to the large
contribution of the double-resolved cross section at small $p_T$, which
is there of the same order as the direct contribution. In general, the
massive theory is below the pure massless theory. But the reduction is
not very large and converges to zero with increasing $p_T$ by
construction. In Fig.\ \ref{fig11}d we have collected all three data
sets in one plot and compare them with the respective theoretical
predictions. We see that the theoretical cross sections with ALEPH and
OPAL constraints almost coincide.  Only the L3 prediction is lower due
to the different rapidity cut and the different anti-tagging condition.
We remark again that we have in general good agreement for $p_T \geq 2$
GeV. Below this $p_T$ value, the theoretical prediction lies higher by
an appreciable amount as compared to the L3 data point.

Since the double-resolved cross section is dominated by the contribution
due to the charm content in the photon, which is evaluated with $m =0$
for consistency as explained earlier, this part is divergent in the
limit $p_T \rightarrow 0$. This explains the strong increase towards
small $p_T$.  This strong increase in the small $p_T$ region could be
eliminated by switching to the fixed flavor theory with $n_f = 3$ below
some $p_T$ value.  This would eliminate the resolved contribution
completely and the whole cross section would be given by the
contributions with charm quarks only in the final state and not in the
initial state.  In the three-flavor theory, non-perturbative effects
from the charm distribution in the photon, and possibly from the
fragmentation $c \rightarrow D^*$, would not be present.  A similar
procedure has also been proposed in many papers on charm production in
DIS (see for example \cite{TKS}).  The problem, however, lies in the
fact that the precise position of the matching point is unknown and,
secondly, that there is not a unique presription for how to achieve a
continuous matching.  The matching point will certainly lie somewhere at
small $p_T$, say at $p_T \leq 2$ GeV. On the other hand, the smallest
possible value of $p_T$, where the matching must occur, is, of course,
the point where the photon PDF or the charm FF vanishes. With the
factorization scale considered so far, this is $p_T=0$. It is obvious
therefore that these naive considerations do not provide a criterium
which can be used to fix the matching scale. We might come back to this
question in a future study.

The partition of the cross section into the three separate parts depends
on the choice of the factorization scales. For example, for $M_I = M_F =
\sqrt{p_T^2+m^2}$ the direct contribution is larger by approximately
$10\,\%$ for all $p_T$ values, and the single-resolved contribution
becomes steeper (with an increase of $38\,\%$ at $p_T=2$ GeV and
$10\,\%$ at 12 GeV). In contrast, the double-resolved contribution is
reduced (by $60\,\%$ at $p_T = 2$ GeV and $34\,\%$ at $p_T = 12$ GeV).
Of course, the change of these separate parts is unphysical; only the
change of the sum is relevant. The cross section $d\sigma/dp_T$ for the
separate pieces and their sum with the new scale choice is shown in
Fig.\ \ref{fig13}a (for the ALEPH experimental setup) which can directly
be compared with Fig.\ \ref{fig12}b, where the results were given with
the scale $2\sqrt{p_T^2+m^2}$. This specific variation of the
factorization scales amounts to a $22\,\%$ decrease at $p_T=2$ GeV and a
$1\,\%$ increase at $p_T=12$ GeV of the physical cross section. As a
consequence, the agreement of theory and data in the first bin improves
with the new scale, but worsens slightly in the second bin (see Fig.\ 
\ref{fig13}b). One should keep in mind that continuous variations of the
renormalization and factorization scales would lead to an error band of
theoretical predictions which, however, we do not show.

%----------------------------------
\begin{figure}[t!] 
\unitlength 1mm
\begin{picture}(160,85)
\put(-2,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dx1t12a}
\end{minipage}}
\put(43,82){(a)}
\put(78,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dx1t11a}
\end{minipage}}
\put(121,82){(b)}
\end{picture}
\caption{Effect of varied factorization scales: $M_I = M_F =
  \sqrt{p_T^2+m^2}$ is chosen in this figure. (a) should be compared to
  Fig.\ \protect\ref{fig12}b, (b) to Fig.\ \protect\ref{fig11}a.}
\label{fig13}
\end{figure}
%----------------------------------

%----------------------------------
\begin{figure}[h!] 
\unitlength 1mm
\begin{picture}(158,175)
\put(0,85){\begin{minipage}[b][80mm][b]{80mm}
\include{dft14a}
\end{minipage}}
\put(40,172){(a)}
\put(80,85){\begin{minipage}[b][84mm][b]{80mm}
\include{dft14c}
\end{minipage}}
\put(121,172){(b)}
\put(40,-4){\begin{minipage}[b][80mm][b]{80mm}
\include{dft14b}
\end{minipage}}
\put(80,83){(c)}
\end{picture}
\caption{Theoretical predictions and experimental results for the
  $y$-distribution ((a): ALEPH, (b): L3, (c): OPAL). We show separately
  the direct (dotted), single-resolved (dashed) and double-resolved
  (dash-dotted) contributions and their sums (full lines). Upper curves
  are for the massless, lower curves for the massive calculation.}
\label{fig14}
\end{figure}
%----------------------------------

The LEP collaborations have measured also the cross section $d\sigma/dy$
as a function of $y$, where $d^2\sigma/dydp_T$ is integrated over the
$p_T$ regions $2-12$ GeV (ALEPH and OPAL) and $1-12$ GeV (L3). We have
calculated these rapidity distributions and compare them with the
respective data points from ALEPH, L3 and OPAL in Figs.\ \ref{fig14}a,
b, c. The ALEPH and OPAL points agree with the theoretical prediction
inside the experimental errors (the OPAL points are slightly above the
$m \neq 0$ prediction).  The L3 points, however, lie below the
theoretical curves. Since the cross section after integration over $p_T$
is dominated by the contribution from the lowest $p_T$ bin, this is
consistent with the comparison of the $p_T$ distribution shown earlier
at the smallest $p_T$ bin.  In the three figures we also show how the
total $y$-distribution is separated into the direct, single-resolved and
double-resolved parts for both the $m\neq 0$ and the $m=0$ theory. In
Fig.\ \ref{fig14}b one can see clearly that the cross section with $m=0$
(the upper full line) is not a reliable approximation if integrated down
to $p_{Tmin}=1$ GeV. The cross sections shown in Figs.\ \ref{fig14}a, b,
c are again calculated with the scale $2\sqrt{p_T^2+m^2}$.

%**********************************************************************

\section{Summary and Conclusions}

In this work we have compared two approaches for the calculation of
inclusive charm production. One is based on a calculation with massless
quarks and $\overline{\rm MS}$ factorization, the second on a
calculation with massive charm quarks. By considering the massless limit
of the massive calculation, we were able to derive subtraction terms
which allowed us to combine the massive calculation in a consistent way
with parton distribution and fragmentation functions defined in the
$\overline{\rm MS}$ factorization scheme.

The cross section for the direct component of the $\gamma \gamma$
reaction in NLO was studied already in a previous work. Here we extend
the study to the single-resolved contributions in NLO. The NLO
corrections to $\gamma g \rightarrow c\bar{c}$ consist of an Abelian
part and a non-Abelian part. The first part is identical, up to a
normalization factor, to the NLO corrections for the direct
contribution. For the second, the non-Abelian part, we found also that
the massless limit of the massive cross section differs from the
massless theory with $\overline{\rm MS}$ factorization by finite terms
which are non-singular for $m \rightarrow 0$. These finite terms must be
subtracted from the massive hard cross section, since the latter has to
be folded with a fragmentation function for the transition $c
\rightarrow D^*$ and with parton distribution functions of the photon
which are available only in the $\overline{\rm MS}$ factorization scheme
based on calculations with massless quarks.

It turns out that this massive theory with $\overline{\rm MS}$
factorization leads to cross sections which converge rapidly to their
massless limits with increasing $p_T$. Only at rather small $p_T$, terms
proportional to $m^2/p_T^2$ are important. These terms are more
important for the direct cross section than for the single-resolved one,
since the latter receives contributions also from the charm distribution
in the photon which must be evaluated with zero charm mass.

The double-resolved contribution has also two parts. One originates from
light quarks and gluons in the initial state, the other is due to an
initial charm quark in one of the two scattering photons. This latter
contribution overwhelms the double-resolved cross section by far and is
computed with massless quarks. Since the part with charm quarks in the
final state, on the other hand, is negligible except possibly for very
small $p_T$, the total double-resolved cross section can safely be
evaluated with massless charm quarks. 

For reliable predictions one needs a good description of the
fragmentation process. In our numerical evaluation of the inclusive
$D^*$ production we have taken the fragmentation functions from fits to
$D^*$ production in $e^+e^-$ annihilation at LEP1. To compare with
recent measurements of the $p_T$ and $y$ distributions in $\gamma
\gamma$ collisions at LEP2, we added the direct, single-resolved and
double-resolved contributions. The agreement of our predictions with the
data is quite good (see Fig.\ \ref{fig11}) down to $p_T \simeq 2$ GeV.
Finite charm mass effects are essential only for values of $p_T$ below 3
GeV. To improve the theory at very small $p_T$ it seems necessary to
switch from the four-flavor theory to the three-flavor theory at some
matching point.
\\

%**********************************************************************

\section*{Acknowledgement}

We thank Z.\ Merebashvili for clarifying communications about Refs.\ 
\cite{MCG,MCGErr} and B.A.\ Kniehl and M.\ Spira for providing us with
programs for the calculation of the double-resolved contributions.
\\

%**********************************************************************

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  abstract \# 582 ICHEP02 Amsterdam
  
\bibitem{L32002} P.\ Achard et al., L3 Collaboration, CERN-EP/2002-012,
  , Phys.\ Lett.\ B535 (2002) 59
% Inclusive D*+- production in two-photon collisions at LEP
% Abstract # 527, ICHEP02 Amsterdam
  
\bibitem{opal} G.\ Abbiendi et al., OPAL Collaboration, Eur.\ Phys.\ J
  C16 (2000) 579; OPAL Collaboration, OPAL PN 453

\bibitem{BKK} J.\ Binnewies, B.A.\ Kniehl and G.\ Kramer, Phys.\ Rev.\ D
  58 (1998) 014014; \\
  J.\ Binnewies, B.A.\ Kniehl and G.\ Kramer, Phys.\ Rev.\ D 53 (1996)
  6110
  
\bibitem{FKL} S.\ Frixione, M.\ Kr\"amer and E.\ Laenen, Nucl.\ Phys.\ 
  B571 (2000) 169 and earlier references given there

\bibitem{TKS} W.\ K.\ Tung, S.\ Kretzer and C.\ Schmidt, J.\ Phys G28
  (2002) 983 and earlier papers given there

\bibitem{Coll} J.\ Collins, Phys.\ Rev.\ D58 (1998) 094002
  
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\bibitem{CGKKKS} M.\ Cacciari, M.\ Greco, B.\ A.\ Kniehl, M.\ Kr\"amer,
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\bibitem{MCG} Z.\ Merebashvili, A.\ P.\ Contogouris and G.\ Grispos,
  Phys.\ Rev.\ D62 (2000) 114509
  
\bibitem{LG} L.\ E.\ Gordon, Phys.\ Rev.\ D50 (1994) 6753
  
\bibitem{EN} R.\ K.\ Ellis and P.\ Nason, Nucl.\ Phys.\ B312 (1989) 551
  
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\bibitem{MCGErr} Z.\ Merebashvili, A.\ P.\ Contogouris and G.\ Grispos,
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\bibitem{GRV}  M.\ Gl\"uck, E.\ Reya, A.\ Vogt, Phys.\ Rev.\ D45 (1992)
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\end{thebibliography}

\end{document}

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LT3
392 1312 M
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1.000 UL
LT1
418 310 M
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