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\appendix
\section{Formalism for Inclusive DIS Cross Section}

\label{sec:formalism}

In this appendix, we give a concise and careful presentation
of the general formalism, including the relatively simple operational procedure
for calculating its various components. We hope this will help fill the gap
between the original, relatively sketchy, ACOT paper \cite{ACOT} and the
recent, more technical, all-order proof of the formalism by Collins
\cite{Collins97}.

The basis for all discussions is the factorization theorem in the presence of
non-zero quark masses \cite{Collins97}, Eq.\ \ref{eq:HadXsec0}. To
establish this theorem in PQCD (cf.\ Eq.~\ref{eq:ParXsec0} below),
and to give precise meaning to the various
factors, one must work with \emph{partonic cross-sections} and \emph{parton
distributions inside partons}, rather than the corresponding physical
quantities which appear in Eq.\ \ref{eq:HadXsec0}.  In this concise summary, we
proceed as follows: (i) spell out the operational procedure to establish the
factorization formula and calculate the
hard cross-sections in any scheme; (ii) describe the specifics of the 3-flavor
and 4-flavor schemes respectively; (iii) discuss the matching conditions
between the two; and (iv) consider the transition
from one to the other in the composite scheme, which constitutes the general
formalism of Collins \cite{Collins97}.  We finish with some remarks on the
meaning of ``LO'' and ``NLO'' calculations in different schemes.

\subsection{Procedure to define the factorization scheme and
calculate the hard cross-sections}%

\label{sec:procedure}

\noindent Given the QCD Lagrangian, with non-zero masses for the heavy quarks,
one arrives at the general factorization formula for partonic cross-sections
and parton distributions as follows:
%\footnote{%
%All non-essential indices are suppressed in the following discussions.}\ $%
%$\widehat{\omega }_{a}$ are obtained as follows:%
\footnote{%
This procedure may sound familiar because it is based on the basic principles
of PQCD. We include it here because the details, especially concerning the
heavy quark mass dependence, are quite distinct from conventional practices.
Thus, it is essential to explicitly spell out the steps involved.}%

\begin{Simlis}[]{0em}
\item
(i) Start with a set of relevant \emph{partonic structure functions}\ $\omega
_{a}$ similar to the left-hand side of Eq.~(\ref {eq:HadXsec0}) but with
on-shell parton targets and calculate them in perturbation theory in a given
renormalization scheme (i.e. with specific
ultra-violet counter-terms) to a given order in $\alpha_s$.
The result\ $\omega _{a}(\frac{Q}{\mu },x,\frac{%
m_{c}}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu \right) )$ will
depend on the renormalization scale
$\mu $ and will contain collinear singularities (represented
by $\frac{1}{\epsilon }$) as well as potentially large logarithm terms of the
form $(\alpha _{s}\ln (\frac{\mu }{m_{c}}))^{n}$.
(For simplicity we do not differentiate between the renormalization scale $%
\mu _{R}$ and the factorization scale $\mu _{f},$ both of which are taken in
practice to be of order $Q$.)%

\item
(ii) Independently, calculate the set of
process-independent \emph{perturbative partonic distribution functions } $%
\tilde{f}_{a}^{b}$ in the same renormalization scheme, using either the
(moment-space) operator-product expansion or, equivalently, the ($x$-space)
bi-local operator definition of the distribution functions. Both
ultra-violet and collinear singularities appear in this calculation. The
ultra-violet singularities are removed by additional counter-terms which,
along with the coupling constant renormalization counter-terms, define the
factorization scheme. The result takes the form $\tilde{f}_{a}^{b}(x,\frac{%
m_{c}}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu \right) )$.

\item
(iii) Confirm that all collinear singularities in the form of
$\frac{1}{\epsilon } $
terms, appearing in $\omega _{a}(\frac{Q}{\mu },x,\frac{m_{c}}{\mu
},\frac{1}{\epsilon },\alpha
_{s}\left( \mu \right) )$ appear in the universal form given in the
process-independent functions $\tilde{f}%
_{a}^{b}(x,\frac{m_{c}}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu
\right) )$, so that they can be factorized out in the manner of Eq.~(\ref
{eq:HadXsec0}),
\begin{equation}
\omega _{a}(\frac{Q}{\mu },x,\frac{m_{c}}{\mu },\frac{1}{\epsilon },\alpha
_{s}\left( \mu \right) )=\stackunder{b}{\sum }\tilde{f}_{a}^{b}(x,\frac{m_{c}%
}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu \right) )\otimes \widehat{%
\omega }_{b}(\frac{Q}{\mu },x, \frac{m_c}{\mu } ,\alpha _{s}\left( \mu
\right) )\;,  \label{eq:ParXsec0}
\end{equation}
with $\widehat{\omega }_{a}$ being fully infra-red safe in the sense that it
is free of all $\frac{1}{\epsilon }$ dependence.  In the 4-flavor scheme
defined below, the functions $\tilde{f}%
_{a}^{b}(x,\frac{m_{c}}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu
\right) )$ will also contain the same large logarithmic terms as
$\omega _{a}(\frac{Q}{\mu },x,\frac{m_{c}}{\mu
},\frac{1}{\epsilon },\alpha
_{s}\left( \mu \right) )$, so that these too factorize in Eq.~(\ref{eq:ParXsec0}) with the result that $\widehat{\omega }_{a}$ is free of
all $(\alpha _{s}\ln (\frac{\mu }{%
m_{c}}))^{n}$ terms, and it is well-behaved as $m_{c}\rightarrow 0.$
%(iii) Confirm that all collinear singularities, in the form of
%$\frac{1}{\epsilon } $ as well as $(\alpha _{s}\ln (\frac{\mu }{m_{c}}))^{n}$
%terms, appearing in $\omega _{a}(\frac{Q}{\mu },x,\frac{m_{c}}{\mu
%},\frac{1}{\epsilon },\alpha
%_{s}\left( \mu \right) )$ also appear in the universal functions $\tilde{f}%
%_{a}^{b}(x,\frac{m_{c}}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu
%\right) )$, so that they can be factorized out in the manner of Eq.~(\ref
%{eq:HadXsec0}),
%\begin{equation}
%\omega _{a}(\frac{Q}{\mu },x,\frac{m_{c}}{\mu },\frac{1}{\epsilon },\alpha
%_{s}\left( \mu \right) )=\stackunder{b}{\sum }\tilde{f}_{a}^{b}(x,\frac{m_{c}%
%}{\mu },\frac{1}{\epsilon },\alpha _{s}\left( \mu \right) )\otimes \widehat{%
%\omega }_{b}(\frac{Q}{\mu },x, \frac{m_c}}{\mu } ,\alpha _{s}\left( \mu
%\right) )\;,  \label{eq:ParXsec0}
%\end{equation}
%with $\widehat{\omega }_{a}$ being fully infra-red safe in the sense that it
%is free of $\frac{1}{\epsilon }$ as well as $(\alpha _{s}\ln (\frac{\mu }{%
%m_{c}}))^{n}$ terms, and it is well-behaved as $m_{c}\rightarrow 0,$ as
%indicated by the subscript $R$ in the $ \frac{m_c}}{\mu } $ dependence
%of $\widehat{\omega }_{a}$.

\item
(iv) Systematically invert Eq.~(\ref{eq:ParXsec0}%
) to solve for the set of finite hard cross-sections $\widehat{\omega }_{a},$
which are then used in Eq.~(\ref{eq:HadXsec0}) for calculating physical
structure functions.

\end{Simlis}

\noindent There are two points to note: (a) The inversion of Eq.~(\ref
{eq:ParXsec0}) order-by-order in the perturbation series is equivalent to
\emph{subtracting} the singularities contained in $\tilde{f}_{a}^{b}$ from $%
\omega _{a};$ (b) There is no need to set the quark mass(es) to zero anywhere
in the above procedure.

In the following, we apply the above procedure to define the two simple
renormalization schemes, involving \emph{3 or 4 active quark flavors,} which
underlies the general approach of Refs.~\cite{ColTun,Collins97}; and combine
them to define the latter in the subsection under the heading of \emph{the
generalized \msbar\ formalism}. These discussions are applicable to all orders
in perturbation theory. Throughout these discussions, the three
quarks $\left\{ u,d,s\right\} ,$ with masses comparable to or less than $%
\Lambda ,$ will be referred to as \emph{light quarks}, and denoted collectively
by $q$. The collection of light quarks plus the gluon $g$ will be referred to
as \emph{light partons}, and denoted by $l$. As mentioned earlier, although the
formalism applies to all heavy quarks $\left\{ c,b,t\right\} ,$ we shall use
the case of charm as a generic representative, for concreteness and clarity --
hence the 3- and 4-flavors. \ Because the real charm quark mass $m_{c}$ is not
large compared to the on-set of the region of applicability of PQCD, the
4-flavor scheme plays a more prominent role in practical applications
discussed in the main body of this paper. For a heavier quark, the two
corresponding schemes and their proper matching, as discussed in the rest of
this (theoretical) section, will be more relevant.

\subsection{Three-flavor Scheme}

\label{sec:3flv}

The 3-flavor scheme is precisely defined by choosing to work with only 3
active quark flavors, consisting of the light quarks, and using the
subtraction procedure of Ref.~\cite{CWZ}. The prescription for subtracting
ultra-violet divergences encountered in the calculation of the partonic
structure functions depends on the particle that produces the divergence.
Broadly speaking, divergences due to the light partons $l,$ are removed
using $\overline{\mbox{MS}}$ counter terms, whereas those due to the charm
quark $c$ are removed by BPH zero-momentum subtraction counter terms. The
precise definition can be found in Ref.~\cite{Collins97}. This ultra-violet
subtraction scheme has the nice feature that the charm quark explicitly
decouples as its mass becomes large. In particular, the operators which make
up the charm quark distribution function are suppressed by powers of order $%
\Lambda ^2/m_c^2$. Since these terms are power-suppressed in the ``heavy
quark'' mass, they are usually excluded from the 3-flavor scheme parton
picture. %, retaining only leading-twist dynamics.

In practice then the partonic calculations in this scheme are done by
considering diagrams where the massive charm quark can only appear in the
final state, and there are no charm quark distribution functions, cf.\
Fig.~\ref{fig:Threefl}. The light
parton distributions always evolve according to the 3-flavor DGLAP equation,
for all values of the renormalization scale $\mu $---both below and above
the heavy quark production threshold. The parton distribution functions
defined in this scheme will be restricted to the light parton $l=\left\{ g,q,%
\bar{q}\right\} $ sector, and they will be denoted by $^{3}f_{A}^{l}$. In
the perturbative calculation, $^{3}\tilde{f}_{l}^{l^{\prime }}$ contains $%
\epsilon ^{-1}$ pole terms which are due to collinear singularities. The
lowest order (LO, $\mathcal{O}(\alpha _{s}^{1})$) partonic process in which
the charm quark appears in this scheme is the $\gamma ^{\ast }g\rightarrow c%
\bar{c}$ ``heavy-flavor creation'' (HC) process (also known as boson-gluon
fusion), corresponding to the diagrams of Fig.(\ref{fig:Threefl}a). The
associated partonic structure function, denoted by $\omega _{g}^{c\bar{c}},$
is finite. The next-to-leading order (NLO) contribution includes the 1-loop
virtual corrections to $\gamma ^{\ast }g\rightarrow c\bar{c}$ (cf. Fig.(\ref
{fig:Threefl}b)), plus the real partonic HC processes $\gamma ^{\ast
}l\rightarrow c\bar{c}l$ (cf.\ Fig.(\ref{fig:Threefl}c)). The collinear
divergences which appear in the calculation of the $\mathcal{O}(\alpha
_{s}^{2}) $ partonic structure functions $^{3}\omega _{g}^{c\bar{c}}$ and $%
^{3}\omega _{l}^{c\bar{c}l}$ arise from splitting of massless light partons
in the collinear configuration, and take the form of $\epsilon ^{-1}$ pole
terms, precisely corresponding to those appearing in $^{3}\tilde{f}%
_{l}^{l^{\prime }}$ mentioned above. That is, the partonic structure
functions have the factorized structure shown in Eq.~(\ref{eq:HadXsec0}),
and the hard cross-section functions $\hat{\omega}_{l}$ will be free from $%
\epsilon ^{-1}$ collinear singularities. This 3-flavor scheme is the one
used by Ref.~\cite{SvNorg} to calculate charm production to NLO, i.e. $%
\mathcal{O}(\alpha _{s}^{2})$.\footnote{%
To be consistent, the virtual correction to the process $\gamma ^{\ast
}q\rightarrow qg$, which contains a charm quark loop, must also be included
at this order in the 3-flavor scheme calculation of the total inclusive
structure functions.~\cite{MSvN}.}

At high energies the hard cross-sections calculated in this scheme contain
powers of $\ln (Q^2/m_c^2),$ as mentioned in the introduction. The
perturbative expansion should be accurate at energy scales not too far above
threshold, or $Q^2\sim m_c^2,$ where $\ln (Q^2/m_c^2)$ is of order $1$.
However, at high $Q^2\gg m_c$ the perturbative expansion parameter is
effectively $\alpha _s\ln (Q^2/m_c^2),$ and the large logarithm factor
spoils the convergence of the perturbative series. In other words, the
``hard cross-sections'' $\hat{\omega}_a$ defined in this scheme are finite,
but \emph{not infra-red safe }in the limit $\frac{m_c}Q\rightarrow 0$.
%---they contain ``mass singularities'' in this sense.

\subsection{Four-flavor scheme with non-zero $m_c$}

\label{sec:4flv}

In order to better deal with these logarithms at high energies it is more
useful to use the \emph{4-flavor scheme}, in which the renormalization of $%
\omega _{a}$ and $\tilde{f}_{a}^{b}$\ is carried out using dimensional
regularization and the $\overline{\mbox{MS}}$ counter terms for all Feynman
diagrams, \emph{while keeping the full quark mass dependence} ($m_c$)
of the Lagrangian.

Charm distribution functions calculated in this scheme, $^{4}\tilde{f}%
_{a}^{c}$ ($a=l,c$), are not suppressed as in the 3-flavor scheme, but
contain powers of $\ln (m_{c}/\mu )$, along with possible $\epsilon ^{-1}$
poles. Because of the different subtraction procedures used in the two
schemes, even the light parton distributions $^{4}\tilde{f}_{l}^{l^{\prime
}} $ will differ from $^{3}\tilde{f}_{l}^{l^{\prime }}$ by a finite
renormalization in general. (We will return to this point later.) Because
renormalization constants in the $\overline{\mbox{MS}}$ subtraction
procedure are independent of mass, the evolution kernels for the $^{4}\tilde{%
f}_{a}^{b}$ parton distributions will be the same as the corresponding ones
in the familiar zero-mass 4-flavor case. %\footnote{%
%Evolution kernels (or splitting functions) correspond to anomalous dimension
%constants in moment space, which are derivatives of renormalization
%constants of the relevant operators.}
This is a significant convenience. The perturbative parton distribution
functions $^{4}\tilde{f}_{a}^{b}$ have been calculated to NLO in Ref.~\cite
{MSvN}.

Since charm also has a parton interpretation in this scheme, the set of
partonic processes are expanded to include those involving charm initial
states. The LO partonic process that involves the charm quark in the
4-flavor scheme is the $\gamma ^{\ast }c\rightarrow c$ ``heavy-quark
excitation'' (HE) process (Fig.(\ref{fig:Fourfl}a)). NLO charm quark
contributions in the 4-flavor scheme come from the 1-loop virtual
corrections to HE $\gamma ^{\ast }c\rightarrow c$ (Fig.(\ref{fig:Fourfl}b)),
and from the real HE $\gamma ^{\ast }c\rightarrow gc$ and HC $\gamma ^{\ast
}g\rightarrow c\bar{c}$ processes (Fig.(\ref{fig:Fourfl}c,d)). Partonic
structure functions $\omega _{a}$ calculated beyond LO in this subtraction
scheme contain both $\epsilon ^{-1}$ poles (due
to collinear singularities associated with light degrees of freedom) and
powers of mass-logarithms, $\ln (Q/m_{c})$, (due to collinear configurations
associated with the heavy degree of freedom), just as in the 3-flavor scheme.
The important difference
compared to the latter case is that these potentially large logarithm
terms also appear in the 4-flavor parton distributions $^{4}\tilde{f}%
_{a}^{b} $. Consequently, \emph{they are systematically factored out from} $\omega _{a}$
when we obtain the hard cross-sections $\hat{\omega}_{a}$ by inverting the
factorization formula Eq.~(\ref{eq:ParXsec0}). The charm distribution
function represents the \emph{resummed} contribution of all the large
(infra-red unsafe) logarithm terms in $\omega _{a}.$ As a result, $\hat{%
\omega}_{a}$ is free from both types of collinear ``singularities'' (in
quotes since the logarithms become singular only in the zero-mass limit). In
effect, all logarithm factors $\ln (Q/m_{c})$ in $\omega _{a}$ are replaced
by $\ln (Q/\mu )$ in $\hat{\omega}_{a}$, (with accompanying finite
subtractions), and the latter is \emph{infra-red safe} in the $\frac{m_{c}}{Q%
}\rightarrow 0$ limit.\footnote{%
The validity of these statements to order $\alpha _{s}^{2}$ can be inferred
from the explicit calculations of Refs.~\cite
{SvNorg,MSvN,CSvN}. The proof to all orders of perturbation
theory has been given in Ref.~\cite{Collins97}.} Thus, the 4-flavor scheme
has a well-defined high energy limit, and is expected to give a much more
reliable description of the physics of charm production at large $Q$ than
the 3-flavor scheme.

As formulated above, the hard cross-sections still contain finite charm-mass
dependence, i.e. $\hat{\omega}_{a}=\hat{\omega}_{a}(x,\frac{Q}{\mu },\frac{%
m_{c}}{Q},\mu )$. Being infra-red safe, as $m_{c}/Q\rightarrow 0,$ the
limit $\hat{\omega}_{a}(x,\frac{Q}{\mu },\frac{m_{c}}{Q},\mu ){%
\rightarrow }\hat{\omega}_{a}^{m_{c}=0}(x,\frac{Q}{\mu },\mu )$ is well
defined. In this limit, the 4-flavor scheme with non-zero charm mass reduces
to the conventional zero-mass (ZM) 4-flavor parton scheme%
\footnote{%
In conventional zero-mass (ZM) 4-flavor theory, collinear singularities due
to charm appear as $\epsilon ^{-1}$ poles along with those from other
flavors, and are regulated accordingly. When properly calculated, the
massless limit of our ($m_{c}\neq 0$) Wilson coefficients,
$\hat{\omega}_{a}^{m_{c}=0}(x,Q,\mu ),$
should agree with the standard zero-mass results.}%
, as mentioned in the introduction. As emphasized in Ref.~\cite{ACOT},
however, the factorization of potentially dangerous $\ln (m_{c})$ terms does
not require taking the $m_{c}\rightarrow 0$ limit in the infra-red safe
coefficient functions. The conventional practice of always setting $m=0$ in
the hard cross-section $\hat{\omega}_{a}(x,Q,\mu )$ is a convenience, not a
necessity; it results from the use of dimensional regularization of the
zero-mass theory as a simple and efficient way to classify and to remove the
collinear singularities. For a ``heavy quark'' with non-zero mass $m_{c},$
this convenient method of achieving infra-red safety is not a natural one
(as it is for light flavors), since $m_{c}$ itself already provides a
natural cutoff. In other words, the theory has no real collinear
``singularities'' associated with the charm quark, and the universal (i.e.\
process-independent) and potentially large mass-logarithms can be factorized
systematically as outlined above. In fact, by keeping the charm quark mass
dependence, this scheme can be extended down to lower values of $Q$ with
much more reliable results than in the zero-mass case.  This is possible
because of the well-defined relation between the 4-flavor calculation with
non-zero $m_c$ and the 3-flavor (FFN) calculation; e.g.\ at
order $\alpha _{s}$, Ref.~\cite{ACOT} showed that, for given {$x,Q$}
\begin{equation}
{}^{4}F_{2}^{c}(x,\frac{Q}{\mu },\frac{m_{c}}{\mu })\,\stackunder{\lim \,\mu
\rightarrow m_{c}}{\longrightarrow }\,{}^{3}F_{2}^{c}(x,\frac{Q}{\mu },\frac{%
m_{c}}{\mu })\rule{4em}{0ex}{\cal{O}}( \alpha _{s})  \label{eq:4flvF2MuLim}
\end{equation}
where the superscripts {3,4} refer to the 3- and 4-flavor scheme
calculations respectively. To distinguish this more general 4-flavor scheme from the
conventional zero-mass (ZM) 4-flavor scheme, we can refer it as the
general-mass (GM) 4-flavor scheme.

The theoretical result (Eq.~\ref{eq:4flvF2MuLim}) does \emph{not}, however,
constrain the threshold behavior of the predicted physical structure function
in the limit of $\lim \thinspace Q \,\,({\rm or}\, W) \rightarrow m_{c}$; to
make a physical prediction, one needs to first choose $\mu $ as a function of
the physical variables \{{$x,Q,m_{c}$\}}.
This is related to the well-known scale dependence of PQCD prediction in
general.  We shall return to this problem at the end of the next subsection.

There is one additional advantage of the 4-flavor scheme. Since the charm
quark distribution is explicitly included in the 4-flavor scheme, and
since $m_c$ is not much larger than a typical non-perturbative scale such as
the nucleon mass, one can allow for the existence of a possible
nonperturbative (``intrinsic'') charm component inside a hadron at a low
energy scale, say $Q_0$--- as the boundary condition for evolution to higher
scales, just like the other light flavors. This is a possibility not
permitted in the 3-flavor scheme by assumption.

\subsection{The generalized \msbar\ formalism with non-zero $m_{c}$}

\label{sec:CompSch}

Both the 3-flavor and the 4-flavor schemes described above are valid schemes
for defining the perturbative series of the inclusive cross section in
principle. They are equivalent if both are carried out to all orders in the
perturbation series. At a given finite order, they differ by a finite
renormalization\footnote{%
The magnitude of the ``finite'' renormalization depends on the renormalization
scale: e.g. $\ln (m_{c}/\mu )$ factors are finite, but can be numerically large
if $\mu \gg m_{c}.$} of the distribution functions, as well as the strong
coupling $\alpha _{s}$. From the physics point of view, when calculated to the
appropriate order (cf. below), the 3-flavor scheme provides a more natural and
accurate description of the charm production mechanism near the threshold
($Q^{2}\sim m_{c}^{2}$), whereas the 4-flavor scheme does the same in the high
energy regime ($Q^{2}\gg m_{c}^{2}$), as shown in Fig.~\ref{fig:cartoon}.

The precise definitions given in the above subsections provide the means to
implement the intuitive ideas discussed in the introduction. \emph{A unified
program} to calculate the inclusive structure functions, including charm, which
maintains uniform accuracy over the full energy range, must be a composite
scheme consisting of:
\begin{Simlis}[]{1em}
\item
(i) the 3-flavor scheme, applied from low energy scales, of the order of $m_c$,
and extended up;

\item
(ii) the 4-flavor scheme, applied from high energy scales on down; and

\item
(iii) a set of matching conditions which define the perturbative relation between the two
schemes applied at a specific matching scale $\mu_m$.

\end{Simlis}

\noindent
It is useful to explicitly discuss all the elements of this composite scheme which
link the component 3-flavor and 4-flavor calculations discussed in previous
subsections to physics predictions of the general formalism:

%+++++++++++++
\begin{Simlis}{1em}
%
\item  \textbf{Choice of scale:}
Within each scheme ($i=3$ or $4$), one needs to specify $\mu $ as a
function of the physical variables in order to make a {\em physical
prediction}, i.e.\
\[
F_{A,\lambda }^{phys}(Q^{2},x,m_{c})=F_{A,\lambda }^{(i)}(Q^{2},x,m_{c},\mu
(x,Q,m_{c}))
\]
Although there is considerable freedom in choosing $\mu (x,Q,m_{c})$, two
conditions should be met so that the prediction can be reliable: (i) $\mu $
must be of the order of $Q$ or $m_{c}$ so that PQCD applies, and (ii) $%
F_{A,\lambda }^{(i)}(Q^{2},x,m_{c},\mu )$ must be relatively stable with
respect to variations of $\mu $ for the ($x,Q$) of interest. This is the
well-known {\em scale-dependence} of any PQCD calculation. In
Fig.~\ref{fig:cartoon}, the presence of the uncertainty associated with the
choice of $\mu(x,Q,m_{c})$ in each scheme is represented by the respective
bands.

\item \textbf{Matching conditions and choice of matching scale:}
For a given set of arguments, $F_{A,\lambda }^{(3)}(Q^{2},x,m_{c},\mu
)$ and $F_{A,\lambda }^{(4)}(Q^{2},x,m_{c},\mu )$ are not independent. Being
the same physical quantity calculated in two different schemes (cf.\
Sec.~\ref{sec:3flv} and \ref{sec:4flv}), they are related by a finite
renormalization:
\begin{equation}
\begin{array}{c}
^{4}\alpha _{s}(\mu )\,=\,^{3}\alpha _{s}(\mu )+\Delta \alpha  \\
^{4}f^{a}(x,\mu )\,=\,^{3}f^{a}(x,\mu )+\Delta f^{a}(x,\mu )
\end{array}
\;\mbox{\rm applied at }\mu =\mu _{m}  \label{eq:match1}
\end{equation}
where $\Delta \alpha $ and $\Delta f^{a}(x,\mu )$\ are fully calculable once
the two schemes are defined. They have been calculated to order $\alpha
_{s}^{2}$ \cite{MSvN}. Specifically, the simpler results at order
$\alpha_s$ are \cite{ColTun}:
\begin{equation}
^{4}\alpha _{s}(\mu )=\,^{3}\alpha _{s}(\mu )\left[ 1+\frac{^{3}\alpha
_{s}(\mu )}{6\pi }\ln \frac{\mu ^{2}}{m_{c}^{2}}+\mathcal{O}(\alpha _{s}^{2})%
\right]  \label{eq:Matcha}
\end{equation}
\begin{equation}
\begin{array}{ccccccc}
^{4}f^{q}(x,\mu ) & = & ^{3}f^{q}(x,\mu ) & + & 0 & + & \mathcal{O}(\alpha
_{s}^{2}) \\
^{4}f^{g}(x,\mu ) & = & ^{3}f^{g}(x,\mu ) & - & \frac{^{3}\alpha _{s}(\mu )}{%
6\pi }\ln \frac{\mu ^{2}}{m_{c}^{2}}\ ^{3}f^{g}(x,\mu ) & + & \mathcal{O}%
(\alpha _{s}^{2}) \\
^{4}f^{c}(x,\mu ) & = & 0 & + & \frac{^{3}\alpha _{s}(\mu )}{4\pi }\ln \frac{%
\mu ^{2}}{m_{c}^{2}}\,\int \frac{dz}{z}\,(z^{2}+\left( 1-z\right) ^{2})\
^{3}f^{g}(\frac{x}{z},\mu ) & + & \mathcal{O}(\alpha _{s}^{2})
\end{array}
\label{eq:Matchf}
\end{equation}


The scale at which these two schemes are matched will be called the {\em %
matching point}, and denoted by $\mu _{m}.$\ Note that either scheme can
still be used with $\mu$ above or below the matching point, it is just that
the equations (\ref{eq:match1}) are only enforced at $\mu_m$.
In principle, $\mu _{m}$ can be
chosen at {\em any} value -- different choices lead to the same overall
results, up to higher order corrections. As can be seen in Eqs.~\ref{eq:Matcha} and
\ref{eq:Matchf}, in the generalized \msbar\ scheme,
$\Delta \alpha (\mu )$ and $\Delta f^{a}(x,\mu )$ are both of the form
$\alpha _{s}(\mu )\ln (\mu /m_{c})\ C_{1}+ {\cal O}(\alpha _{s}^{2})$.
Thus, if one chooses $\mu _{m}=m_{c}$, both
functions $\alpha (\mu )$ and $f^{a}(x,\mu)$ in the 3-flavor scheme are
equal to their counterpart in the 4-flavor scheme to first order in
$\alpha _{s}$ at the matching point. \ Most recent works adopt this choice
\cite{ColTun,MRST,CSvN}; we do the same in this paper. \ Although this
choice is convenient, it is {\em not} required in the general formalism. The
ideas behind the matching conditions, Eq.~\ref{eq:match1}, are illustrated in
Fig.~\ref{fig:match} which show two possible matching points, the first being
the special one $\mu _{m}=m_{c}$. This plot also shows that $\mu_m$ should not
be chosen too far above $m_c$, lest the factor $\alpha_s \ln(\mu/m_c) $ in the
discontinuity ceases to be perturbative.
%
\figmatch
%
\item \textbf{Choice of transition scale:}
With $F_{A,\lambda }^{(i)}(Q^{2},x,m_{c},\mu (x,Q,m_{c}))$, $i=3,4$,
we have {\em two} sets of calculations, one for each scheme. For physics
applications, we need to specify which of these to use, say
\begin{equation}
F_{A,\lambda }^{phys}(Q^{2},x,m_{c})=\{
\begin{array}{lll}
F_{A,\lambda }^{(3)}(Q^{2},x,m_{c},\mu (x,Q,m_{c})) &  & Q<\mu _{t} \\
F_{A,\lambda }^{(4)}(Q^{2},x,m_{c},\mu (x,Q,m_{c})) &  & Q>\mu _{t}
\end{array}
\label{eq:transition}
\end{equation}
where we have introduced another scale $\mu _{t}$ -- the {\em transition point}
-- where one switches from one scheme to the other, according to which one is
more appropriate, as discussed in the introduction, cf.\ Fig.\ref{fig:cartoon}
and the previous subsections of this appendix.

Conceptually, the transition point is distinct from the matching point, as
should be clear from their defining equations, \ref{eq:match1} and \ref
{eq:transition}.%
\footnote{ This distinction was first made in Collins' paper on the general
formalism \cite{Collins97}. }
%
The guiding principle for choosing the optimal $\mu _{t}$ is that it should be
within the region where the $F_{A,\lambda }^{(3)}$ and $F_{A,\lambda }^{(4)}$
calculations are both valid, and that their differences within this region are
small. For instance, in the idealized situation depicted in
Fig.~\ref{fig:cartoon}, $\mu _{t}$ is best chosen to be in the middle of the
$Q$ range where both uncertainty bands are relatively narrow.

%+++++++++++

In practice, one can only estimate the range of uncertainties of the 3-flavor
and 4-flavor calculations, say by examining the scale-dependence of the
respective calculations and then make a judicious choice of $\mu_{t}$. In the
case of inclusive charm production discussed in Sec.~\ref{sec:inclusive},
Fig.~\ref{fig:F2cTh} shows that the transition scale can be chosen at a relatively
low value, close to $m_c$.  For this choice, the composite
scheme calculation reduces, in practice, to just the 4-flavor calculation.

\end{Simlis}

\subsection{What do ``LO'' and ``NLO'' mean?}

\label{sec:LoNlo}

As already mentioned in the body of this paper, in a multi-scale
problem such as heavy quark production, the designation of
``LO'' and ``NLO'' to a given calculation
can be rather misleading in conventional fixed-order calculations, due to the
presence of large logarithms which vitiates the naive counting of powers of
$\alpha_s$.
In the composite scheme, which has a wider range of applicability than FFN schemes,
the meaning of ``LO'' and ``NLO'' can be better defined, provided the relative
magnitudes of the large scales are properly kept in mind.  We elaborate a
little bit.

\begin{Simlis}{1em}
\item
In the 3-flavor scheme, ``LO'' consists of the $%
\mathcal{O}(\alpha _{s}^{1})$ HC $\gamma ^{\ast }g\rightarrow c\bar{c}$
process; whereas ``NLO'' involves $\mathcal{O}(\alpha _{s}^{2})$ processes such as
$\gamma ^{\ast }g\rightarrow gc\bar{c}$. This formal designation makes physical
sense only in the threshold region.
\item In the 4-flavor scheme, the ``LO'' process is represented
by the $\mathcal{O}(\alpha _{s}^{0})$ HE $\gamma ^{\ast }c\rightarrow c$
process; and the ``NLO'' ones
consist of the $\mathcal{O}(\alpha _{s}^{1})$ HC process as well as the $%
\mathcal{O}(\alpha _{s}^{1})$ HE $\gamma ^{\ast }c\rightarrow gc$ process. This
formal designation coincides with the familiar one in the conventional treatment
of DIS structure functions; it makes physical sense only when $Q^2 \gg m_c^2$.
\end{Simlis}%
The apparent mismatch of orders in $\alpha _{s}$ (e.g.\ order $\alpha_s$ being
``LO'' in the former, but ``NLO'' in the latter) can be understood within the
composite scheme which takes into account the order of magnitudes of the other
relevant quantities in the factorization formula at the appropriate energy ranges.

Specifically, the $\mathcal{O}(\alpha _{s}^{1})$ and $\mathcal{O}(\alpha
_{s}^{2})$ 3-flavor calculations lose their ``LO'' and ``NLO'' meaning as $%
Q^{2}$ becomes very large compared to $m_{c}^{2},$ since each power of $%
\alpha _{s}$ is accompanied (and neutralized) by a large logarithm $\ln
(Q^{2}/m_{c}^{2})$ factor in the hard cross-section. Consequently, all terms
become effectively $\mathcal{O}(\alpha _{s}^{0})$! In fact, the resummation
of these large logarithms to all orders of $\alpha _{s}$ gives rise
precisely to the charm distribution, resulting in the $\mathcal{O}(\alpha
_{s}^{0})$ charm excitation process $\gamma ^{\ast }c\rightarrow c$ of the
4-flavor scheme. \ Conversely, in the 4-flavor scheme, although the charm
quark distribution $f_{A}^{c}(x,Q)$ can be considered to be $\mathcal{O}%
(\alpha _{s}^{0})$ at high $Q^{2}$ (where all flavors are on an equal
footing), as we go down to a lower energy range $Q^{2}\sim m_{c}^{2}$, one
finds $f_{A}^{c}(x,Q)\sim \alpha _{s}^{1}\ln (Q^{2}/m_{c}^{2})\sim \mathcal{O%
}(\alpha _{s}^{1})$ compared to the dominant gluon and light quark
distribution functions (assuming there is no large non-perturbative charm
component). Therefore, to consistently match the 4-flavor calculation onto
the LO ($\mathcal{O}(\alpha _{s}^{1})$) 3-flavor calculation, one must
include both the ``LO'' $\mathcal{O}(\alpha _{s}^{0})$ $\gamma ^{\ast
}c\rightarrow c$ and the ``NLO'' $\mathcal{O}(\alpha _{s}^{1})$ $\gamma
^{\ast }g\rightarrow c\bar{c}$ contributions, along with the associated
subtraction term in the 4-flavor calculation (Cf. Ref.~\cite{ACOT}). This
implies that
our 4-flavor calculation, which is NLO at high energy scales,
becomes effectively ``LO'' near the threshold, because both the 
formally $\mathcal{O}%
(\alpha _{s}^{0})$ and $\mathcal{O}(\alpha _{s}^{1})$ contributions become
of the same order of magnitude, $\mathcal{O}(\alpha _{s}^{1})$ --  as in the
LO 3-flavor scheme. (The calculations in the main part of this paper show
that, with the natural choice of scale $\mu =\sqrt{Q^{2}+m_{c}^{2}},$ the
numerical predictions of this calculation are actually fairly close to those
of the NLO 3-flavor calculation, which is consistent with this observation
because the size of the NLO correction is within the range of uncertainty of
a LO result.)

This mixing of terms with different \emph{apparent} powers of $\alpha _{s}$ is
physically natural ( cf.\ Fig.~\ref{fig:cartoon} ) and logically consistent -- it
is a necessary feature of switching between different primary schemes, since any
finite renormalization always entails a \emph{resummation} (i.e.\
re-organization) of the perturbation series to all orders. In more concrete
terms, the need for mixing terms of different apparent powers of $\alpha _{s}$
arises when:%

\begin{Simlis}[]{0em}
\item%
(i) the LO diagrams for different subprocesses start at different orders of $
\alpha _{s};$%
\item%
(ii) the associated parton densities are of different numerical orders of
magnitude (such as between $g,q,c$);%
\item%
(iii) the order of magnitude of a parton distribution changes as it evolves
with $Q$ (such as for $c$ in the region above the threshold); and%
\item%
(iv) the hard cross-section
contains logarithms of ratios of energy scales which become large.%
\footnote{%
For these reasons, to require a naive uniform counting of powers of $\alpha
_{s}$ over a wide range of $Q,$ when a composite scheme must be used, would
miss the basic tenet of adapting the renormalization scheme to the
appropriate number of active flavors as the physical scale varies. \cite
{RobertsT}}%
\end{Simlis}%
The generalized \msbar\ formalism, by keeping the physical $%
m_{c}, $ provides the appropriate scheme to describe the underlying physical
processes in the different regions encountered in heavy quark production.

%++++++++++++++++++++++
\begin{thebibliography}{99}

\bibitem{H1charm96}
H1 collaboration: C.\ Adloff et al., \emph{Zeit. Phys.}
\textbf{C72} (1996) 593.

\bibitem{ZeusCharm}
 ZEUS collaboration: J. Breitweg et al., \emph{Phys. Lett.}
\textbf{B407} (1997) 402.

\bibitem{EHLQ}
E.\ Eichten et al., {\em Rev. Mod. Phys.} {\bf 56}, 579 (1984), Erratum 58, 1065 (1986).

\bibitem{MRS}
%\cite{Martin:1996as}
A.~D.~Martin, R.~G.~Roberts and W.~J.~Stirling,
%``Parton distributions: A study of the new HERA data, alpha(s),  the gluon and p anti-p jet production,''
Phys.\ Lett.\  {\bf B387}, 419 (1996)
.
%%CITATION = ;%%

\bibitem{CTEQ}
%\cite{Lai:1997mg}
H.~L.~Lai {\it et al.},
%``Improved parton distributions from global analysis of recent deep  inelastic scattering and inclusive jet data,''
Phys.\ Rev.\  {\bf D55}, 1280 (1997)
.
%%CITATION = ;%%

\bibitem{NDE}
P. Nason, S. Dawson, R.K. Ellis, {\em Nucl. Phys.} {\bf B327}, 49 (1989),
Erratum: B335, 260 (1990).

\bibitem{SvNorg}
W.~Beenakker, H.~Kuijf, W.~L.~van Neerven and J.~Smith,
%``QCD Corrections To Heavy Quark Production In P Anti-P Collisions,''
Phys.\ Rev.\  {\bf D40}, 54 (1989);
%%CITATION = PHRVA,D40,54;%%
E.~Laenen, S.~Riemersma, J.~Smith and W.~L.~van Neerven,
%``Complete O (alpha-s) corrections to heavy flavor structure functions in electroproduction,''
Nucl.\ Phys.\  {\bf B392}, 162 (1993);
%%CITATION = NUPHA,B392,162;%%
E.~Laenen, S.~Riemersma, J.~Smith and W.~L.~van Neerven,
%``O(alpha-s) corrections to heavy flavor inclusive distributions in electroproduction,''
Nucl.\ Phys.\  {\bf B392}, 229 (1993).
%%CITATION = NUPHA,B392,229;%%
%\href{\wwwspires?j=NUPHA\%2cB392\%2c229}{SPIRES}

\bibitem{FMNR97}
S.~Frixione, M.~L.~Mangano, P.~Nason and G.~Ridolfi,
%``Heavy-quark production,'' .
%%CITATION = ;%%
in \emph{Heavy Flavors II}, Edited by A.J. Buras and M. Lindner. World Scientific, 1998.
.

\bibitem{ColTun}
J.~C.~Collins and W.~Tung,
%``Calculating Heavy Quark Distributions,''
Nucl.\ Phys.\  {\bf B278}, 934 (1986).
%%CITATION = NUPHA,B278,934;%%

\bibitem{OlnTun87}
F.~I.~Olness and W.~Tung,
%``When Is A Heavy Quark Not A Parton? Charged Higgs Production And Heavy Quark Mass Effects In The QCD Based Parton Model,''
Nucl.\ Phys.\  {\bf B308}, 813 (1988).
%%CITATION = NUPHA,B308,813;%%

\bibitem{AOT94a}
M.~A.~Aivazis, F.~I.~Olness and W.~Tung,
%``Leptoproduction of heavy quarks. 1. General formalism and kinematics of charged current and neutral current production processes,''
Phys.\ Rev.\  {\bf D50}, 3085 (1994)
.
%%CITATION = ;%%

\bibitem{ACOT}
M.~A.~Aivazis, J.~C.~Collins, F.~I.~Olness and W.~Tung,
%``Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies,
Phys.\ Rev.\  {\bf D50}, 3102 (1994)
.
%%CITATION = ;%%

\bibitem{Collins97}
J.~C.~Collins,
%``Hard-scattering factorization with heavy quarks: A general treatment,''
Phys.\ Rev.\  {\bf D58}, 094002 (1998)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{dis97tung}
W.~K.~Tung, in
\emph{Proceedings of the 5th International Workshop on Deep Inelastic
Scattering}, Chicago, IL, eds. J Repond and D Krakauer, American
Institute of Physics, 1997
e-Print : .

\bibitem{RobertsT}
R.~S.~Thorne and R.~G.~Roberts,
%``An ordered analysis of heavy flavour production in deep inelastic  scattering,''
Phys.\ Rev.\  {\bf D57}, 6871 (1998)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{BMSvN}
M.~Buza, Y.~Matiounine, J.~Smith and W.~L.~van Neerven,
%``Charm electroproduction viewed in the variable-flavour number scheme  versus fixed-order perturbation theory,''
Eur.\ Phys.\ J.\  {\bf C1}, 301 (1998)
;
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}
M.~Buza, Y.~Matiounine, J.~Smith and W.~L.~van Neerven,
%``Comparison between the various descriptions for charm electroproduction  and the HERA-data,''
Phys.\ Lett.\  {\bf B411}, 211 (1997)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{NasonEtAl}
%\cite{Cacciari:1998it}
M.~Cacciari, M.~Greco and P.~Nason,
%``The p(T) spectrum in heavy-flavour hadroproduction,''
JHEP {\bf 9805}, 007 (1998)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{CSvN}
A.~Chuvakin, J.~Smith and W.~L.~van Neerven,
%``Comparison between variable flavor number schemes for charm quark  electroproduction,''
Phys.\ Rev.\  {\bf D61}, 096004 (2000)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{CWZ}
%``Low-Energy Manifestations of Heavy Particles: Application to the
%Neutral Current'',
%COO-2220-127, (Received Apr 1978). 21pp.
J.~Collins, F.~Wilczek, and A.~Zee, Phys.\ Rev.\ D {\bf 18}, 242 (1978).

\bibitem{ZeroMfact}
Cf.\ for instance,
R.~K.~Ellis, H.~Georgi, M.~Machacek, H.~D.~Politzer and G.~G.~Ross,
%``Perturbation Theory And The Parton Model In QCD,''
Nucl.\ Phys.\  {\bf B152}, 285 (1979).
%%CITATION = NUPHA,B152,285;%%
%\href{\wwwspires?j=NUPHA\%2cB152\%2c285}{SPIRES}

\bibitem{KretzerS}
%\cite{Kretzer:1998ju}
S.~Kretzer and I.~Schienbein,
%``Heavy quark initiated contributions to deep inelastic structure  functions,''
Phys.\ Rev.\  {\bf D58}, 094035 (1998)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{KOS}
%\cite{Kramer:2000hn}
M.~Kr\"amer, F.~I.~Olness and D.~E.~Soper,
%``Treatment of heavy quarks in deeply inelastic scattering,''
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{ColMarRys}
%\cite{Collins:1997fy}
J.~C.~Collins, A.~D.~Martin and M.~G.~Ryskin,
%``Charm in deep inelastic scattering,''
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}
in Proceedings of \emph{Madrid Workshop on Low X Physics}, Edited by F. Barreiro,
L.\ Labarga, J.\ Del Peso. Madrid, Spain, 1997,World Scientific, 1998 .

\bibitem{CTEQ5}
%\cite{Lai:2000wy}
H.~L.~Lai {\it et al.}  [CTEQ Collaboration],
%``Global {QCD} analysis of parton structure of the nucleon: CTEQ5 parton  distributions,''
Eur.\ Phys.\ J.\  {\bf C12}, 375 (2000)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}
\bibitem{Xiaoning}
Xiaoning Wang,
{\it Heavy Quark Production in Perturbative QCD at HERA},
Michigan State University Ph.D.\ thesis, UMI-99-22388, (1998).

\bibitem{HoffMoor}
%\cite{Hoffmann:1983ah}
E.~Hoffmann and R.~Moore,
%``Subleading Contributions To The Intrinsic Charm Of The Nucleon,''
Z.\ Phys.\  {\bf C20}, 71 (1983).
%%CITATION = ZEPYA,C20,71;%%
%\href{\wwwspires?j=ZEPYA\%2cC20\%2c71}{SPIRES}

\bibitem{RiemS}
%\cite{Riemersma:1995hv}
S.~Riemersma, J.~Smith and W.~L.~van Neerven,
%``Rates for inclusive deep inelastic electroproduction of charm quarks at HERA,''
Phys.\ Lett.\  {\bf B347}, 143 (1995)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{ZEUScharm97}
%\cite{Breitweg:2000ad}
J.~Breitweg {\it et al.}  [ZEUS Collaboration],
%``Measurement of D*+- production and the charm contribution to F2 in  deep inelastic scattering at HERA,''
Eur.\ Phys.\ J.\  {\bf C12}, 35 (2000)
;
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}
See also,  B.~W.~Harris, .

\bibitem{MelNas}
%THE FRAGMENTATION FUNCTION FOR HEAVY QUARKS IN QCD.
%CERN-TH-5972-90, Dec 1990. 21pp.
B.~Mele and P.~Nason, Nucl.\ Phys.\ {\bf B361}, 626 (1991).


\bibitem{Peterson}
%\cite{Peterson:1983ak}
C.~Peterson, D.~Schlatter, I.~Schmitt and P.~Zerwas,
%``Scaling Violations In Inclusive E+ E- Annihilation Spectra,''
Phys.\ Rev.\  {\bf D27}, 105 (1983).
%%CITATION = PHRVA,D27,105;%%
%\href{\wwwspires?j=PHRVA\%2cD27\%2c105}{SPIRES}

\bibitem{Cacciari:1997ad}
M.~Cacciari,
%``On heavy quarks photoproduction and c --> D* fragmentation functions,''
%%CITATION = ;%%
in \emph{Proceedings of Ringberg Workshop on New Trends in HERA Physic}, Ringberg Castle,
   Tegernsee, Germany, 1997, Edited by B.A.
   Kniehl, G. Kramer, A. Wagner. World Scientific, 1998 .

\bibitem{OPAL95}
K.~Ackerstaff {\it et al.}  [OPAL Collaboration],
%``Determination of the production rate of D*0 mesons and of the ratio  V/(V+P) in Z0 --> c anti-c decays,''
Eur.\ Phys.\ J.\  {\bf C5}, 1 (1998)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{JaffRand}
%\cite{Jaffe:1994ie}
R.~L.~Jaffe and L.~Randall,
%``Heavy quark fragmentation into heavy mesons,''
Nucl.\ Phys.\  {\bf B412}, 79 (1994)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{HarrSmith}
%\cite{Harris:1998zq}
B.~W.~Harris and J.~Smith,
%``Heavy quark correlations in deep inelastic electroproduction,''
Nucl.\ Phys.\  {\bf B452}, 109 (1995)

%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}
;%``Charm quark and D*+- cross sections in deeply inelastic scattering at  HERA,''
Phys.\ Rev.\  {\bf D57}, 2806 (1998)

%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}
; and B.~W.~Harris, .

\bibitem{Brodsky:1981se}
%\cite{Brodsky:1981se}
S.~J.~Brodsky, C.~Peterson and N.~Sakai,
%``Intrinsic Heavy Quark States,''
Phys.\ Rev.\  {\bf D23}, 2745 (1981).
%%CITATION = PHRVA,D23,2745;%%
%\href{\wwwspires?j=PHRVA\%2cD23\%2c2745}{SPIRES}

\bibitem{HSV96}
%\cite{Harris:1996jx}
B.~W.~Harris, J.~Smith and R.~Vogt,
%``Reanalysis of the EMC charm production data with extrinsic and intrinsic charm at NLO,''
Nucl.\ Phys.\  {\bf B461}, 181 (1996)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{GV97}
%\cite{Gunion:1997mx}
J.~F.~Gunion and R.~Vogt,
%``Intrinsic charm at high-Q**2 and HERA data,''
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{MT97}
%\cite{Melnitchouk:1997ig}
W.~Melnitchouk and A.~W.~Thomas,
%``HERA anomaly and hard charm in the nucleon,''
Phys.\ Lett.\  {\bf B414}, 134 (1997)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{Steffens}
F.~M.~Steffens, W.~Melnitchouk and A.~W.~Thomas,
%``Charm in the nucleon,''
Eur.\ Phys.\ J.\  {\bf C11}, 673 (1999)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{MSvN}
M.~Buza, Y.~Matiounine, J.~Smith, R.~Migneron and W.~L.~van Neerven,
%``Heavy quark coefficient functions at asymptotic values $Q~2 \gg m~2$,''
Nucl.\ Phys.\  {\bf B472}, 611 (1996)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\bibitem{MRST}
%\cite{Martin:1998sq}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``Parton distributions: A new global analysis,''
Eur.\ Phys.\ J.\  {\bf C4}, 463 (1998)
.
%%CITATION = ;%%
%\href{\wwwspires?eprint={SPIRES}

\end{thebibliography}
\newcommand{\msbar}{$\overline{{\rm MS}}$}

\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}

\newcommand{\text}[1]{{\rm #1}}

%               Usage: \begin{Simlis}[opt-label]{left-margin}
%                        \item ...
%                      \end{Symlis}
\newenvironment{Simlis}[2][$\bullet$]
{\begin{list}{#1}
 {
  \settowidth{\labelwidth}{#1}
  \setlength{\labelsep}{0.5em}
  \setlength{\leftmargin}{#2}
  \setlength{\rightmargin}{0em}
  \setlength{\itemsep}{0ex}
  \setlength{\topsep}{0ex}
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}
{\end{list}}


\newcommand \End {\end{document}}
%TCIDATA{LaTeXparent=0,0,lhk4.TEX}

%TCIDATA{ChildDefaults=chapter:1,page:1}

\section{Introduction}

\label{sec:Intro}

Recent measurements of charm production in deep inelastic scattering (DIS)
at HERA \cite{H1charm96,ZeusCharm} have shown that up to 25\% of the total
cross-section at small-x contains charm in the final state. This is within
the expectations of perturbative QCD based on conventional parton
distributions. We are now in a position to utilize this process to study
details of the production mechanism of heavy quarks in general, and to
extract useful information on the charm and gluon structure of the proton in
particular.

Conventional perturbative QCD (PQCD) theory is formulated in terms of
zero-mass quark-partons. For processes depending on one hard scale $Q,$ the
well-known factorization theorem then provides a straightforward procedure
for order-by-order perturbative calculations, as well as an associated
intuitive parton picture interpretation of the perturbation series. Heavy
quark production presents a challenge in PQCD because the heavy quark mass, $%
m{H}$ $(H=c,b,t),$ provides an additional hard scale which complicates the
situation -- it requires a different organization of the perturbative series
depending on the relative magnitudes of $m{H}$ and $Q$.

The two standard methods for PQCD calculation of heavy quark processes
represent two diametrically opposite ways of reducing the two-scale problem to
a one-scale problem. (i) In the conventional \emph{parton model approach} used
in many global QCD analyses of parton distributions \cite {EHLQ,MRS,CTEQ} and
Monte Carlo programs, the zero-mass parton approximation is applied to a heavy
quark calculation
as soon as the typical energy scale\footnote{%
We use $Q$ as the generic name for a typical kinematic physical scale. It could
be $Q$, $W$, or $p{T}$, depending on the process.} of the physical process $Q$
is above the mass threshold $m{H}$. This leaves $Q$ as the only apparent hard
scale in the problem. (ii) In the \emph{heavy quark approach} which played a
dominant role in ``NLO calculations'' of the production of heavy quarks
\cite{NDE,SvNorg,FMNR97}, the quark $H$ is always treated as a ``heavy''
particle and never as a parton. The mass parameter $m{H}$ is explicitly kept
along with $Q$ as if they are of the same order, irrespective of their real
relative magnitudes.

The co-existence of these two opposite approaches represents an uneasy
dichotomy in the current literature. On physical grounds, the zero-mass parton
picture of heavy quarks should be applicable at energy scales very much larger
than the relevant quark mass, $m{H}$ $\ll $ $Q$, whereas the heavy quark
approach (often referred to as the fixed-flavor-number (FFN) scheme) should be
more appropriate at energy scales comparable to the quark mass $m{H}$ $\sim $
$Q$. The actual experimental regime often lies \emph{in between} these two
extreme regions, where the validity of either approach can be called into
question. There is, however, a natural way to incorporate both approaches in a
unified framework in PQCD which provides a smooth transition between the two.
This has been formulated in a series of papers over the years,
Refs.\ \cite{ColTun,OlnTun87,AOT94a,ACOT,Collins97}%
\footnote{%
See Ref.~\cite{dis97tung} for a brief review.}, %
which has now been adopted, in different guises, by most recent literature on
heavy quark production in PQCD. \cite{RobertsT,BMSvN,NasonEtAl,CSvN}

To see the basic ideas behind this unified picture, let us focus explicitly on
the production of charm ($H=c$) in deep inelastic scattering. All
considerations
apply to a generic heavy quark. Consider the PQCD calculation of the $%
F{2}(x,Q)$ structure function which receives substantial contribution from
charm production as mentioned earlier. The underlying physical ideas are
illustrated graphically in Fig.\ \ref{fig:cartoon} where the charm contribution
to this structure function, denoted $F{2}^{c}(x,Q)$, is
plotted as a function of $Q$ at some fixed value of $x$.%
\figcartoon%
Near threshold $Q\sim m{c}$, it is natural to consider the charm quark as a
heavy particle, and to adopt the 3 active-parton-flavor scheme of calculation
(the ``heavy quark approach''). As $Q$ becomes large compared to $m{c}$, the
fixed-order calculation in this approach becomes unreliable since the
perturbative expansion contains terms of the form $\alpha {s}^{n}\log
^{n}\left( m{c}^{2}/Q^{2}\right) $ at any order $n$, which ruin the
convergence of the series---these terms are \emph{not infra-red safe} as
$m{c}$ $\rightarrow 0$ or $Q\rightarrow \infty $.
Thus the uncertainty of the 3-flavor calculation grows as $Q/m{c}$ becomes
large. This is illustrated in Fig.\ \ref{fig:cartoon} as an error
band marked by horizontal hashes which is narrow near threshold but becomes
ever wider as $Q/m{c}$ increases.
On the other hand, starting from the high energy end
($Q\gg m{c}$), the most natural calculational scheme to adopt is the
conventional 4-flavor scheme with active charm partons. (In this approach, the
infra-red unsafe large logarithms mentioned earlier are ``resummed" and
absorbed into the finite charm parton distributions.) However, as we go down
the energy scale toward the charm production threshold region, the 4-flavor
calculation becomes unreliable because the approximation $m{c}=0$ deteriorates
as $Q\rightarrow m{c}$. The uncertainty band of such a calculation is outlined
in Fig.\ \ref{fig:cartoon} by the vertical hashes -- it is narrow at high
energies, but becomes increasingly wider as one approaches the threshold region.

The intuitive ideas embodied in Fig\ \ref{fig:cartoon} illustrate that: (i)
these two conventional approaches are individually unsatisfactory over the full
energy range, but are mutually complementary; and (ii) the most reliable PQCD
prediction for the physical $F{2}(x,Q)$, at a given order of calculation, can
be obtained by utilizing the most appropriate scheme at that energy scale $Q$,
resulting in a composite scheme, as represented by the cross-hashed region in
Fig.\ \ref{fig:cartoon}.
The use of a composite scheme consisting of different numbers of flavors in
different energy ranges, rather than a fixed number of flavors, is familiar in
the conventional zero-mass parton picture.  The new formalism espoused in
Refs.\
\cite{ColTun,OlnTun87,AOT94a,ACOT} provides a quantum field
theoretical basis \cite{CWZ,Collins97}
for this intuitive picture in the presence of non-zero quark mass.

This generalization brings
about several important distinguishing features and insights. First, after the
hard cross-section is rendered infra-red safe by factorizing out the $\ln
(m{c})$ terms, the remaining finite dependence on $m{c}$ can be kept in the
hard cross-sections to
maintain better accuracy in the intermediate energy region.%
\footnote{%
By contrast, in the standard literature the $m{c}\rightarrow 0$ limit is tied
to the proof of factorization \cite{ZeroMfact} in the first place. This
association is not needed, cf.\ \cite{Collins97}}%
One can then show that the
unified formalism reproduces the two conventional
approaches in well-defined ways in their respective regions of applicability
\cite{ColTun}-\cite{Collins97}.
Secondly, it has become clear recently that there is much inherent flexibility in the
choice of the transition energy scale (cf.\ Fig.~\ref{fig:cartoon}) as well as
the detailed matching condition between the 3- and 4-flavor calculations.
This
makes it possible to have a variety of different implementations of the new
formalism \cite{RobertsT,BMSvN,NasonEtAl,CSvN}, with different emphases. Properly
understood, this feature can be exploited to lend strength to the formalism;
but, during the development of the theory, the subtleties have given rise to
much misunderstanding and confusion in existing literature.

In this paper, we study charm production in DIS using the general mass
formalism, and compare the results with recent HERA data. The main goals of
this work are:

\begin{Simlis}[]{0em}
\item%
(i) Through this specific example, we give a concise and careful presentation
of the general formalism, including the relatively simple operational procedure
for calculating its various components at any order of $\alpha_s$.
We hope this will help fill the gap
between the original, relatively sketchy, ACOT paper \cite{ACOT} and the
recent, more technical, all-order proof of the formalism by Collins
\cite{Collins97}.
%
\item%
(ii) We carry out the numerical calculations to make concrete the intuitive
ideas illustrated in Fig.~\ref{fig:cartoon}; and to demonstrate the validity of
the physical principles underlying the composite scheme.
%
\item%
(iii) We show that the flexibility of the general formalism mentioned above can
result in efficient PQCD calculations of inclusive quantities at relatively low
order in $\alpha_s$ compared to FFN calculations -- because the relevant
physics has been effectively ``resummed'' by the appropriate scheme adopted for
the given energy scale.

\end{Simlis}

Item (i) is basic; it forms the foundation for the other two parts. However,
since this part is about the clarification of the existing theoretical
formalism rather than new work, and since not all readers are concerned with
theoretical precision, we have elected to place it in the Appendix. Hopefully, this will
increase the accessibility of the main body of the paper. In regards to the
underlying physics, we believe the discussion in the appendix should make a useful
contribution to clear the confusion and misunderstanding among the various
approaches that have been proposed in the recent literature following
\cite{ACOT}.

Based on the terminologies discussed in this Appendix, in Sec.~\ref{sec:TotInc}
we describe the complete order $\alpha_s$ calculation carried out in this paper
in relation to previous work on this subject. The new calculation extends the
validity of the original ACOT results to NLO in the high energy regime -- on
the same level as the conventional zero-mass total inclusive structure
functions. In addition, the new perspective, as illustrated in Fig.~\ref{fig:cartoon}
above and discussed quantitatively in the paper proper, allows a re-assessment
of the physical predictions of the order $\alpha_s$ calculation near the
threshold region, making it a viable alternative to the order $\alpha_s^2$ FFN
calculation. In Sec.~\ref{sec:inclusive} we present the numerical results on
inclusive charm production, and demonstrate that the validity and the
efficiency of the general formalism, as described in (ii) and (iii) above, are
indeed seen at this order. We show that very good agreement with recent HERA
data on inclusive $F{2}^{c}$ is obtained in practice. For this study, we have
developed a new implementation of the generalized \msbar\ formalism using Monte Carlo
methods. This implementation allows the computation of differential
distributions, with kinematic cuts, such as $d\sigma ^{D^{\ast
}}/dp{t},\;d\sigma ^{D^{\ast }}/dQ,$ and $d\sigma ^{D^{\ast }}/d\eta $ which
we present in Sec.~\ref{sec:tagged}. These results, are in qualitative
agreement with available data from HERA.  However, they are not as good as
those of the order $\alpha {s}^{2}$ calculation in the 3-flavor scheme. This
is to be expected for differential distributions with experimental cuts, since
the (resummed) low-order calculation contains more severe approximations to the
kinematics of the final state partons. For future quantitative studies, the
general formalism needs to be expanded to incorporate higher-order results
(adaptable from existing FFN calculations). This point is discussed in the
concluding section, along with other observations.
%TCIDATA{LaTeXparent=0,0,lhk4.J2.TEX}
                      
%TCIDATA{ChildDefaults=chapter:2,page:1}


%\input lhk4b.tex

\section{Total Inclusive Structure Functions in the general formalism}

\label{sec:TotInc}

We consider the inclusive DIS structure functions, such as $F_{2}$, focusing
on the contribution of a massive quark. For definiteness, we assume that the
only relevant quark with non-zero mass is the charm quark. The generic
leptoproduction process is depicted in Fig.~\ref{fig:lhProc}: 
\begin{equation}
\ell _{1}+A\longrightarrow \ell _{2}+X,  \label{eq:LepProc}
\end{equation}
where $A$ is a hadron, $\ell _{1,2}$ are leptons, and $X$ represents the
summed-over final state hadronic particles. Note that $X$ may or may not
contain a visible heavy-flavor hadron. \figlhProc After the calculable
leptonic part of the cross-section has been factored out, we work with the
hadronic process induced by the virtual vector boson $\gamma ^{\ast }$ of
momentum $q$ and polarization $\lambda $: 
\begin{equation}
\gamma ^{\ast }(q,\lambda )+A(P)\longrightarrow X(P_{X}).
\label{eq:BosonProc}
\end{equation}
Although our considerations apply to DIS processes induced by $W$ and $Z$ as
well, we shall explicitly refer to the neutral current interaction with the
exchange of a virtual photon $\gamma ^{\ast }$ in order to be concrete. The
cross-section is expressed in terms of the hadronic tensor 
\begin{equation}
W_{\lambda \sigma }(q,P)=\frac{1}{4\pi }\mathrel{\mathop{\overline{\sum
}}\limits_{X(P_{X}),spin}}\langle P|e_{\lambda }^{\ast }\cdot J^{\dagger
}|P_{X}\rangle (2\pi )^{4}{\delta ^{(4)}\left( P+q-P_{X}\right) }\langle
P_{X}|e_{\sigma }\cdot J|P\rangle .  \label{eq:HadTensor}
\end{equation}
where $\overline{{\sum }}$ denotes a sum over all final hadronic states. In
most cases, it suffices to consider the diagonal elements of the tensor $%
F^{\lambda }\equiv W_{\lambda \lambda }(q,P).$

The factorization theorem in the presence of non-zero quark masses --
assumed in \cite{ACOT} and established to all orders in PQCD \cite{Collins97}
-- states that the inclusive cross-section can be written as a convolution: 
\begin{equation}
F_{A}^{\lambda }(Q^{2},x,..)=\stackunder{a}{\sum }f_{A}^{a}(x,\mu )\otimes 
\widehat{\omega }_{a,\lambda }(x,Q/\mu ,Q/m_{c},\alpha _{s}\left( \mu
\right) )\;+\;\mathcal{O}\Bigl(\Lambda /Q\Bigr)^{p}  \label{eq:HadXsec0}
\end{equation}
where $f_{A}^{a}$ is the distribution of parton $a$ inside the hadron $A$, $%
\widehat{\omega }_{a,\lambda }$ is the perturbatively calculable hard
cross-section for $\gamma ^{\ast }+a\rightarrow X$, $p$ is some positive
number, $\mu$ denotes collectively the renormalization and factorization
scales, and a convolution in the $x$ variable is implied. The helicity
structure functions $F^{\lambda }$ are simply related to the familiar $%
F_{1,2,3}$ \cite{AOT94a}.

The exact way that the physical structure function factorizes into the
long-distance ($f_{A}^{a}$) and the short-distance ($\widehat{\omega }%
_{a,\lambda }$) pieces on the right-hand side of Eq.~(\ref{eq:HadXsec0})
depends on the scheme used to define the parton distributions. The physical
structure function $F_A^\lambda$ should be independent of any calculational
scheme; therefore, the definition of the hard cross-sections is determined
by the subtraction procedure used to define the parton distributions. As
discussed in the Introduction, the general formalism consists of the
3-flavor scheme at low energy scales, the 4-flavor scheme at high energy
scales, and a suitably chosen transition region where matching conditions
between the two schemes are applied. The precise description of these
elements of the formalism is given in the Appendix. Operational definitions
of quantities needed in subsequent discussions are also discussed in some
detail there.

\subsection{Previous calculations}

\label{sec:PrevCal}

To put the current calculation in context, we first summarize the existing
calculations of leptoproduction of charm using the precise definitions given
in the Appendix.

\begin{Simlis}{1em}
\item  \textbf{NLO 3-flavor }(3$\alpha _{s}^{2}$) \textbf{calculation}
\textbf{\cite{SvNorg}}: Most dedicated calculations of heavy flavor production
in recent years have been carried out in this scheme. The LO process is
$\mathcal{O}(\alpha _{s}^{1})$ heavy-flavor creation (HC), $\gamma ^{\ast
}g\rightarrow c \bar{c}$. The NLO processes consist of the $\mathcal{O}(\alpha
_{s}^{2})$ virtual corrections to $\gamma ^{\ast }g\rightarrow c\bar{c}$ as
well as the real HC $\gamma ^{\ast }l\rightarrow c\bar{c}l$ process, where $l$
denotes any light parton. Cf.\ Fig.~\ref{fig:Threefl}. This calculation becomes
questionable when $Q\gg m_{c}$ -- {\it i.e.,} it ceases to be ``NLO''
in accuracy, as indicated in Fig.\ \ref{fig:cartoon},
because the perturbative expansion is
actually in $\alpha_s ln(Q^2/ m_{c}^2)$ for large $Q$.
\figThreefl

\item  \textbf{Zero-mass 4-flavor} (ZM) (4$\alpha _{s}^{1};$ $m_{c}=0$) \textbf{%
calculation:} This is the formalism used in most conventional QCD parton model
calculations and popular Monte Carlo programs. It represents an approximation
to the general mass (GM) 4-flavor scheme by setting $m_{c}=0$
in the hard cross-sections $\hat{\omega}_{a}(x,\frac{Q}{\mu },\frac{m_{c}}{Q}%
,\mu )\stackrel{m_{c}/Q\rightarrow 0}{\longrightarrow }\hat{\omega}%
_{a}^{m_{c}=0}(x,\frac{Q}{\mu },\mu ).$ The LO contribution consists of the
$\mathcal{O}(\alpha _{s}^{0})$ $\gamma ^{\ast }c\rightarrow c$ heavy-flavor
excitation (HE) process. The NLO contribution consists of the
$\mathcal{O}(\alpha _{s}^{1})$ virtual corrections to $\gamma^{\ast
}c\rightarrow c$ plus the real HC $\gamma ^{\ast }g\rightarrow c\bar{c}$ and,
$\gamma ^{\ast }c\rightarrow gc$ HE processes. Cf.\ Fig.~\ref{fig:Fourfl}. This
calculation is unreliable in the threshold region, as mentioned in the
introduction.

\figFourfl

\item  \textbf{LO generalized \msbar\ calculation} (ACOT) \cite{ACOT}:
This represents the simplest implementation of the generalized formalism
\cite{ACOT}. It emphasizes the overlapping physics underlying the 3-flavor and
4-flavor calculations in the region not far above threshold. The 4-flavor
calculation consists of HE $\gamma^{\ast}c\rightarrow c$
(Fig.~\ref{fig:Fourfl}a) plus $m_c \neq 0$ HC $\gamma ^{\ast}g\rightarrow
c\bar{c}$ (Fig.~\ref{fig:Fourfl}d), with the requisite subtraction term which
removes the $m_{c}$ mass-logarithm. Mathematically, the result on
$F_2(x,Q,\mu)$ can be shown to match that of the $\mathcal{O}(\alpha _{s}^{1})$
3-flavor scheme calculation (HC $\gamma ^{\ast }g\rightarrow c \bar{c}$) as the
(unphysical) factorization scale $\mu$ approaches $m_{c}$ from above
\cite{ACOT}. Physically, the predicted behavior of $F_2^c$ for $Q \sim m_c$
will depend on the choice of $\mu$ as a function of the physical variables,
cf.\ next section. At a high energy scale, $\mu \sim Q\gg m_{c}$, this
calculation only approximates the 4-flavor NLO results: it contains the most
important $\mathcal{O}(\alpha _{s}^{1})$ term, HC $\gamma ^{\ast }g\rightarrow
c\bar{c}$ (because of the large gluon distribution and the need for matching),
but does not include the smaller $\mathcal{O}(\alpha _{s}^{1})$ terms
represented by Fig.~\ref{fig:Fourfl}b,c. This calculation has been further
studied by Kretzer and Schienbein \cite{KretzerS} and Kr\"amer {\it et al}
\cite{KOS}.

\item \textbf{Variations on the ``variable flavor number'' theme:}
In recent years, the general approach proposed in \cite{ColTun,ACOT} has been
adopted by other groups, starting from different historical perspectives,
\cite{RobertsT,BMSvN,NasonEtAl,CSvN}. In contrast to the fixed flavor-number
(FFN) approach, these are usually referred to as being in the variable
flavor-number (VFN) scheme. This terminology has caused some confusion, since
although the common theme is that of \cite{ColTun,ACOT}, as shown in Fig.\
\ref{fig:cartoon}, the various implementations differ considerably. Whether a
particular implementation is self-consistent within the general framework of
PQCD, or whether two implementations are compatible within the accuracy of the
perturbative expansion, are often unclear or controversial (e.g.\
\cite{ColMarRys}) because of the complexity of the multi-scale problem and
because of possible misunderstandings. It is beyond the scope of this paper to
critically review these approaches. By pursuing the goals described in the
introduction, we hope to provide a clearer picture of the general formalism of
\cite{Collins97}, hence a better basis to help address some of the
controversial issues in the future.
\end{Simlis}

\subsection{The full order $\protect\alpha_s$ generalized \msbar\ calculation%
}

The calculation of charm contribution to the inclusive structure functions
reported in this paper completes the order $\alpha_{s}$ calculation in the
general formalism, initiated in \cite{ACOT}, including all the hard
processes of Fig.~\ref{fig:Fourfl}. The additional terms, although
relatively small numerically at current energies, are required to make the
4-flavor part of this calculation truly NLO at high energies, so that it
becomes equivalent to the conventional zero-mass NLO theory used in most
modern analyses of precision DIS data. In the remainder of this section we
discuss the theoretical issues and uncertainties in the order $\alpha_s$
4-flavor calculation at all energy scales, in order to address the issues
highlighted at the end of the introduction, Sec.~\ref{sec:Intro}.

At order $\alpha _{s}^{1}$ the 3-flavor component of the composite scheme
consists of only the (unsubtracted) $\mathcal{O}(\alpha _{s}^{1})$ HC $%
\gamma ^{\ast }g\rightarrow c\bar{c}$ process. The result is standard.
Therefore, our main calculation concerns the \emph{4-flavor scheme component}%
. At order $\alpha _{s}^{1}$ the right-hand side of Eq.~(\ref{eq:HadXsec0})
consists of three terms 
\begin{equation}
\begin{array}{lll}
F^{(4)}_{A,\lambda }(Q^{2},x,m_c,\mu) & = & \;\;\;f_{A}^{c}\otimes \,^{0}%
\widehat{\omega }_{c,\lambda }^{c} \\ 
&  & +\;f_{A}^{g}\otimes \,^{1}\widehat{\omega }_{g,\lambda }^{c\bar{c}} \\ 
&  & +\;f_{A}^{c}\otimes \,^{1}\widehat{\omega }_{c,\lambda }^{cX} \\ 
&  & +\;\text{light-parton\ contributions}\ ,
\end{array}
\label{eq:HadXsec4fl0}
\end{equation}
where the superscript (4) indicates that this is a 4-flavor calculation, and
the \emph{hard-scattering cross-sections} $^{i}\widehat{\omega}_{a,\lambda
}^{X}$ for the various subprocesses are calculated from the corresponding 
\emph{partonic cross-sections} $^i\omega_{a,\lambda}^X$ (without the hat)
according to the procedures described in the Appendix. A description of each of the terms follows:

\begin{Simlis}{1em}
\item  The leading order $\gamma ^{*}c\rightarrow c$ (Fig.~\ref{fig:Fourfl}a)
partonic cross-section ${}^0\omega _{c,\lambda }^c$ is
infra-red safe, thus
\begin{equation}
^0\widehat{\omega }_{c,\lambda }^c=\,^0\omega _{c,\lambda }^c\ .
\label{eq:Hardxsc0}
\end{equation}

\item  The $\gamma ^{*}g\rightarrow c\bar{c}$ (Fig.~\ref{fig:Fourfl}d)
partonic cross-section contains a single power of $\ln
\mu ^2/m_c^2$ which can be factorized into the charm distribution function by
the subtraction \cite{ACOT}
\begin{equation}
^1\widehat{\omega }_{g,\lambda }^{c\bar{c}}=\,^1\omega _{g,\lambda }^{c\bar{c%
}}-\,^1\tilde{f}_g^c\otimes \,^0\omega _{c,\lambda }^c\ ,
\label{eq:Hardxsc1}
\end{equation}
where the cancelling logarithm with mass-singularity resides in the
$\mathcal{O}(\alpha _s)$ perturbative parton distribution function
\begin{equation}
^1\tilde{f}_g^c=(\alpha _s/2\pi )P_{g\rightarrow q}(x)\ln (\mu ^2/m_c^2)
\label{eq:fgc1}
\end{equation}

\item  The virtual correction to $\gamma ^{*}c\rightarrow c$
(Fig.~\ref{fig:Fourfl}b) plus the real $\gamma ^{*}c\rightarrow gc$
(Fig.~\ref{fig:Fourfl}c) partonic process also contain $\ln \left( \mu
^2/m_c^2\right) $ terms which are factorized into the charm distribution
function by the subtraction
\begin{equation}
^1\widehat{\omega }_{c,\lambda }^{cX}=\,^1\omega _{c,\lambda }^{cX}-\,^1%
\tilde{f}_c^c\otimes \,^0\omega _{c,\lambda }^c\ ,  \label{eq:Hardxsc2}
\end{equation}
where the logarithm appears in the $\mathcal{O}(\alpha _s^1)$ perturbative
parton distribution function%
\footnote{%
We have calculated these terms keeping a finite charm quark mass. As
discussed in Ref.~\cite{Collins97,KOS}, it would also have been consistent to
calculate diagrams with an initial state charm quark using a zero charm
quark mass. The errors near threshold in both methods of calculation are
comparable and of order $\alpha_s^2$.}%
\begin{equation}
^1\tilde{f}_c^c={\frac{\alpha _s}{2\pi }}{\frac 43}\Biggl[\biggl({\frac{1+x^2%
}{1-x}}\biggr)\biggl(\ln {\frac{\mu ^2}{m_c^2}}-1-2\ln (1-x)\biggr)\Biggr]%
_{+}\ .  \label{eq:fcc1}
\end{equation}
\end{Simlis}
Substituting Eqs.~(\ref{eq:Hardxsc0})-(\ref{eq:Hardxsc2}) into Eq.~(\ref
{eq:HadXsec4fl0}), the right-hand side can be re-organized as 
\begin{equation}
\begin{array}{lll}
F^{(4)}_{A,\lambda }(Q^{2},x,m_c,\mu) & = & \;\;f_A^g\otimes \,^1\omega
_{g,\lambda }^{c\bar{c}} \\ 
\; &  & +\,(f_A^c-\;f_A^g\otimes \,^1\tilde{f}_g^c-f_A^c\otimes \,^1\tilde{f}%
_c^c)\,\otimes \,^0\omega _{c,\lambda }^c \\ 
&  & +\;f_A^c\otimes \,^1\omega _{c,\lambda }^{cX} \\ 
&  & +\;\text{light-quark\ terms}\ ,
\end{array}
\label{eq:HadXsec4fl1}
\end{equation}
where the $\ln \left( \mu/ {m_c}\right) $ terms in the $\omega _{a,\lambda }$
factors are kept intact, and the needed subtraction terms are explicitly
grouped with the leading 2$\rightarrow $1 term with the same kinematics.
This is the form we use for the actual numerical calculations, which we
implement using a Monte Carlo approach.

It is useful to compare this calculation with the NLO 3-flavor calculation 
\cite{SvNorg} which can be written, in the same notation, as follows 
\begin{equation}
\begin{array}{lll}
F_{A,\lambda }^{(3)}(Q^{2},x,m_{c},\mu ) & = & \;\;f_{A}^{g}\otimes
\,^{1}\omega _{g,\lambda }^{c\bar{c}} \\ 
&  & +\;f_{A}^{g}\otimes \,^{2}\omega _{g,\lambda }^{gc\bar{c}} \\ 
&  & +\;f_{A}^{q}\otimes \,^{2}\omega _{q,\lambda }^{qc\bar{c}} \\ 
&  & +\;\text{light-quark\ terms}
\end{array}
\ .  \label{eq:HadXsec3fl}
\end{equation}
The term common to the two schemes is $\mathcal{O}\left( \alpha
_{s}^{1}\right) $ $\gamma ^{\ast }g\rightarrow c\bar{c}$, which appears as
the first line in both Eq.~\ref{eq:HadXsec4fl1} and Eq.~\ref{eq:HadXsec3fl}.
For this comparison, one can consider the rest of the terms in these
equations as complementary ``corrections'' to the first term. In particular,
the last two terms in the 3-flavor formula Eq.~\ref{eq:HadXsec3fl} are
genuine $\mathcal{O}\left( \alpha _{s}^{2}\right) $ corrections to the
common term in the threshold region; hence are commonly referred to as NLO.
But at high $Q^{2}\gg m_{c}^{2}$, these terms contain large logarithm
factors $\ln {Q^{2}}/{m_{c}^{2}}$ which vitiates the perturbation expansion.
They are \emph{not} NLO in this region.

In the 4-flavor calculation as organized in the form Eq.~\ref{eq:HadXsec4fl1}%
, the last two lines are also effectively $\mathcal{O}\left(
\alpha_{s}^{2}\right)$. This is because the distribution $f_{A}^{c}$ is
effectively $\mathcal{O}\left( \alpha _{s}^{1}\right)$ near threshold, and
there is a built-in cancellation between the leading order $\gamma ^{\ast
}c\rightarrow c$ partonic cross-section and the first subtraction term in
Eq.~\ref{eq:HadXsec4fl1}, as discussed in detail in Ref.\cite{ACOT}.
Although these $\mathcal{O}\left( \alpha_{s}^{2}\right)$ ``correction''
terms to the $\mathcal{O}\left( \alpha_{s}^{1}\right)$ common term (first
line) do not contain the full NLO $\mathcal{O}\left( \alpha _{s}^{2}\right)$
corrections at threshold, they do contain all the $\mathcal{O}\left(
\alpha_{s}^{2}\right)$ contributions which are enhanced by $\ln \left( {Q^{2}%
}/{m_{c}^{2}}\right)$ and quickly dominate as $Q^{2}$ increases. In fact,
these large logarithmic terms have been resummed to all orders in
perturbation theory via the DGLAP-evolved charm parton distribution
contributions in the second and third lines. Therefore, in the large $Q^{2}$
region the 4-flavor result represents a true NLO calculation.

In principle, if carried out to all orders in $\alpha _{s}$, the 3-flavor
and 4-flavor calculations in the form of $F_{A,\lambda
}^{(i)}(Q^{2},x,m_{c},\mu )$, $i=3,4$, would give exactly the same
prediction for all values of the arguments. The dependence on the scheme $%
(i) $ and the scale $(\mu )$ arises from the truncation of the perturbation
series. Therefore, in order to produce a physical prediction $F_{A,\lambda
}^{phys}(Q^{2},x,m_{c})$ and to relate the predictions in the two schemes, a
number of additional steps must be taken. In the presence of a non-zero
heavy quark mass, some of these steps are non-obvious; hence they can be the
source of confusion. An explicit discussion of these elements of the
calculation will make clear the flexibility as well as the uncertainties
inherent in the formalism. This we do in the Appendix, as part of the more
precise description of the formalism. Here, we mention only two features
which are particularly relevant for the subsequent discussions of physical
predictions.

First, within each scheme ($i=3$ or $4$), one needs to specify $\mu $ as a
function of the physical variables in order to make a \emph{physical
prediction}, i.e.\ 
\[
F_{A,\lambda }^{phys}(Q^{2},x,m_{c})=F_{A,\lambda }^{(i)}(Q^{2},x,m_{c},\mu
(x,Q,m_{c}))
\]
Although there is considerable freedom in choosing $\mu (x,Q,m_{c})$, two
conditions should be met so that the prediction can be reliable: (i) $\mu $
must be of the order of $Q$ or $m_{c}$ so that PQCD applies, and (ii) $%
F_{A,\lambda }^{(i)}(Q^{2},x,m_{c},\mu )$ must be relatively stable with
respect to variations of $\mu $ for the ($x,Q$)-range of interest. This is
the well-known \emph{scale-dependence} of any PQCD calculation. For the
problem at hand, a common choice for $\mu (x,Q,m_{c})$ is $\sqrt{%
Q^{2}+m_{c}^{2}}$: it represents the typical virtuality of the internal
parton lines in the important subprocesses. The presence of the uncertainty
associated with the choice of $\mu (x,Q,m_{c})$ in each scheme is
illustrated in Fig.~\ref{fig:cartoon} by the respective bands.\footnote{%
In the original ACOT paper, a different scale function $\mu (x,Q,m_{c})$ was
chosen: it was designed to enforce the condition $%
F^{(4)}(Q^{2},x,m_{c},\mu (x,Q,m_{c}))\rightarrow F_{LO}^{(3)}$ $%
(Q^{2},x,m_{c},m_{c})$ as $Q\rightarrow m_{c}.$ In retrospect, this choice
is artificial and unnecessary. The general formalism automatically ensures
that, $F^{(4)}(Q^{2},x,m_{c},\mu )\rightarrow F_{LO}^{(3)}$ $%
(Q^{2},x,m_{c},m_{c})$ as $\mu \rightarrow m_{c}$ for given $(Q^{2},x)$
to the order which we are working; but
it does not place any restriction on the behavior of $F^{phys}(Q^{2},x,m_{c})
$ as $Q\rightarrow m_{c}$ in any given scheme. As shown in Sec.\ \ref
{sec:inclusive}, the more natural choice of scale $\mu (x,Q,m_{c})=\sqrt{%
Q^{2}+m_{c}^{2}}$ leads to much improved physical predictions. \label%
{fn:AcotScale}}

Secondly, as shown in Fig.~\ref{fig:cartoon}, one needs to identify an
appropriate scale at which the 3-flavor and the 4-flavor scheme predictions
are both reasonable and mutually comparable, so that the transition from one
to the other in the composite scheme can be made smoothly. In the next
section we show that, for the inclusive charm production cross-section,
these conditions can be met over a rather large range of $Q$, extending down
to near the threshold region. One possibility then is to choose the
transition point (cf.\ Fig.~\ref{fig:cartoon}) at a low value, close to $m_c$%
, so that in effect the 4-flavor calculation by itself covers the full range
of physical interest.

In existing literature, it is already known that the 3-flavor order $%
\alpha_s^2$ (NLO) calculation can be extended to most of the currently
accessible energy scales without manifest ill-effects of the large
logarithms; and it agrees with data rather well. Its efficacy at very large $%
Q$ is not fully tested (cf.\ \cite{BMSvN} and results of next section). Our
calculation will demonstrate the robustness of the complementary, much
simpler (order $\alpha_s$) 4-flavor calculation. It is worthwhile pointing
out that, in the 4-flavor scheme, the order $\alpha_s$ calculation is, in
fact, also \emph{NLO} -- since the LO $\gamma^* c \rightarrow c$ term is of
order $\alpha_s^0$, as is the case in the standard QCD theory of inclusive
structure functions. This important point is discussed in detail in the
Appendix, Sec.\ \ref{sec:LoNlo}.
%TCIDATA{LaTeXparent=0,0,lhk4.TEX}

%TCIDATA{ChildDefaults=chapter:4,page:1}


%\input lhk4d.tex

\section{Semi-inclusive Cross Sections with Tagged Charm Hadrons}

\label{sec:tagged}

\subsection{General considerations}

Now we consider semi-inclusive cross sections, with a charm hadron tagged in
the final state. Naively, to compute this cross section, one simply convolves
the cross sections for parton final states, Eqs.~\ref {eq:HadXsec4fl1} and
\ref{eq:HadXsec3fl}, with a suitable fragmentation function of partons into the
final charm hadron. However, the factorization of the final state particles
through fragmentation functions is only rigorously defined in the limit
$Q^{2}\gg m_{c}^{2}$. Thus, the treatment of tagged charm particles in the
final state can only be systematically applied at high energies, using the
4-flavor scheme. However, it is a common practice to introduce fragmentation
functions into charm hadrons even in the 3-flavor scheme, and for energies not
far above threshold. This approach should be considered a convenient
phenomenological model of hadronization, perhaps adequate for current
experimental accuracy, rather than rigorous theory. We follow this practice in
our calculation, employing fragmentation functions over the full range of
$Q^{2}$, while maintaining the correct factorization-scheme implementation at
high $Q^{2}$.

The fragmentation functions $d_{a}^{H}(x,\mu )$ obey the standard
(mass-independent) QCD evolution equations, and are determined from suitable
initial functions at some given scale $\mu= Q_{0}$, of the order of $m_{c}$.
Following Mele and Nason \cite{MelNas}, for a given final-state charm hadron
${H},$ we write
\begin{equation}
d_{a}^{H}(x,Q_0)=d_{a}^{c}(x,Q_0)\otimes D_{c}^{H}(x,Q_0) \label{eq:HadFrag}
\end{equation}
where the partonic charm fragmentation functions $\left\{
d_{a}^{c};a=l,c\right\} $ are perturbatively calculable, and $%
D_{c}^{H}(x,Q_0)$ is considered non-perturbative and is to be obtained by
comparison with experiment.

For the perturbatively calculable fragmentation functions, Ref.~\cite{MelNas}
gives, to order $\alpha _{s}$:\
\begin{eqnarray}
d_{c}^{c}(x,Q_0) &=&\delta (1-x)+{\frac{{\alpha _{s}(Q_0)C_{F}}}{{%
2\pi }}}\left[ {\frac{{1+x^{2}}}{{1-x}}}\left( \ln {\frac{{Q_0^{2}}}{{%
m_{c}^{2}}}}-2\ln (1-x)-1\right) \right] _{+}  \nonumber \\
d_{g}^{c}(x,Q_0) &=&{\frac{{\alpha _{s}(Q_0)T_{F}}}{{2\pi }}}%
(x^{2}+(1-x)^{2})\ln {\frac{{Q_0^{2}}}{{m_{c}^{2}}}} \label{eq:pertFrag}
\\ d_{q,\bar{q},\bar{c}}^{c}(x,Q_0) &=&0  \nonumber
\end{eqnarray}
where $T_{F}=1/2$ and $C_{F}=4/3$. \ In keeping with the choice of the matching
scale in our overall calculation, we choose $Q_0=m_{c}$ for convenience in this
paper.

For the non-perturbative charm quark into charmed mesons fragmentation
function $D_{c}^{H}(z),$ we used the conventional Peterson form \cite
{Peterson},
\begin{equation}
D_{c}^{H}(z)=\frac{A}{z[1-1/z-\epsilon /(1-z)]^{2}},  \label{eq:peterson}
\end{equation}
For the charm meson $D^{(\ast )},$ which will be our focus because of available
experimental data, we take $\epsilon =0.02,$ cf.\ \cite{Cacciari:1997ad}, and a
value for $A$ such that the branching fraction $B(c\rightarrow D^{\ast })=0.22$
\cite{OPAL95}. We note that, although the perturbative fragmentation functions,
Eq.\ \ref {eq:pertFrag}, contain singular (generalized) functions, the overall
parton-to-charm-meson fragmentation functions $d_{a}^{H}(x,\mu),$ Eq.\
\ref{eq:HadFrag}, are well behaved after convolution with the above
non-perturbative fragmentation function $D_{c}^{D^{(\ast )}}(z)$.

In principle, after evolving to high enough $Q^{2}$ so that $\alpha _{s}\log
(Q^{2}/m_{c}^{2})$ is of order one, all of the fragmentation functions $%
d_{c}^{H},\ d_{g}^{H},\ d_{q,\bar{q},\bar{c}}^{H}$ eventually become of the
same size. \ In practice, however, at HERA energies we find $d_{c}^{H}\gg
d_{g}^{H}\gg d_{q,\bar{q},\bar{c}}^{H}$. For currently required accuracy, it
suffices to keep only the charm-to-hadron contributions, proportional to $%
d_{c}^{H}$. In this approximation, our calculation of the cross section with
a tagged charm hadron can be written in the 4-flavor scheme as
\begin{equation}
\begin{array}{lll}
F_{A,\lambda }^{H}(Q^{2},x,..) & = & \;\;f_{A}^{g}\otimes \,^{1}\omega
_{g,\lambda }^{c\bar{c}}\otimes \,d_{c}^{H} \\
\; &  & +\,(f_{A}^{c}-\;f_{A}^{g}\otimes \,^{1}\tilde{f}_{g}^{c}-f_{A}^{c}%
\otimes \,^{1}\tilde{f}_{c}^{c})\,\otimes \,^{0}\omega _{c,\lambda
}^{c}\otimes \,d_{c}^{H} \\
&  & +\;f_{A}^{c}\otimes \,^{1}\omega _{c,\lambda }^{cX}\otimes \,d_{c}^{H}\
.
\end{array}
\label{eq:HadXsec4fl1Hc}
\end{equation}

To ensure that this calculation is adequate, we have also calculated the
contribution from one of the more important remaining subprocesses: gluon
fragmentation in an order $\alpha _{s}$ light parton hard scattering, i.e.\ $%
\gamma ^{\ast }q\rightarrow gq\ ;\ g\rightarrow H$. It is given by: $%
f_{A}^{q}\otimes \,^{1}\widehat{\omega }_{q,\lambda }^{qg}\otimes \,d_{g}^{H}
$, where $d_{g}^{H}$ is the gluon fragmentation function computed from
Eqs.~\ref{eq:HadFrag} and \ref{eq:pertFrag}. We have verified that its
contribution remains small throughout the current energy range. It becomes more
noticeable only in the large $Q$ limit. However, in this limit, the gluon
fragmentation function term is not infra-red safe by itself. To insure
consistency at high energies, one needs to include a full set of infra-red safe
higher order subprocesses along with it. The full calculation is more
appropriately considered as part of the next order project.

At the same level of accuracy, it is also reasonable to ignore the evolution of
$d_{a}^{H}(x,\mu(x,Q))$, since the effect of QCD evolution is not significant
over the currently accessible HERA $Q$ range. One can use the un-evolved
$d_{c}^{H}(x,\mu=Q_0)$ in place of the fully evolved $d_{c}^{H}(x,\mu(x,Q,m_c))
$ with much gain in efficiency of calculation and little sacrifice in accuracy.
The error incurred is of the same order as that incurred by
neglecting the subleading fragmentation functions, $d_{g}^{H}$ and $d_{q,%
\bar{q},\bar{c}}^{H}$ ; and the comments on accounting for $\ln (m_{c})$
factors made there also apply here. To be sure about this, we have performed
the calculation both with and without evolving $d_{c}^{H}.$  The difference is
indeed small. Therefore, in all our subsequent plots we shall include only the
direct charm-to-hadron contributions of Eq.~(\ref{eq:HadXsec4fl1Hc}) with the
un-evolved fragmentation function $d_{c}^{H}(x,Q_{0})$.

%\figSmallFrag


\subsection{Differential distributions}

We employ the Monte Carlo method to carry out the numerical phase-space
integration; the new program package is implemented in the C++ programming
language. Therefore, we can generate differential distributions involving
final-state charm mesons, incorporating kinematic cuts appropriate for
specific experimental measurements, in addition to fully inclusive
cross-sections.

In working with on-mass-shell heavy flavor quarks and hadrons in the parton
language, there is an ambiguity in defining the momentum fraction variables $%
x\;(z)$ for the parton distribution (fragmentation) function. \ This problem
arises in all schemes; and it goes away at high energies ($Q\gg m_{c}$), where
the parton picture becomes accurate.\ Following the modern practice in proofs
of factorization, we define the momentum fraction variables as ratios of the
relevant light-cone momentum components, e.g.$\;p_{D}^{+}=zp_{c}^{+}$ for
fragmentation of a charm quark into a $D$ meson. Other authors, e.g.\
\cite{FMNR97,HarrSmith}, use the prescription $\vec{p}_{D}=z\vec{p}_{c}$ and
adjust the energy variable to enforce the mass-shell condition. At moderate
energies, any noticeable differences in results due to the choice of this
prescription signals that the calculation using fragmentation functions is
outside the region of applicability of the parton formalism. We have verified
that the results presented below are insensitive to the choice
between the two prescriptions.\footnote{%
There are some differential distributions, especially those associated with
the unphysical partons (such as momentum fraction carried by the charm
quark, sometimes seen in the literature) which are more sensitive to the
choice of definition of the momentum fraction variable.}

The QCD formula, Eq.\ \ref{eq:HadXsec4fl1Hc}, contains three scale choices in
principle: the renormalization scale, the factorization scale and the
fragmentation scale. For simplicity, we choose the same energy scale
$\mu(x,Q,m_c)$ for all three. As in the case of the inclusive $F_2^c(x,Q)$, for
results shown below, we choose the simple function $\mu=c (Q^2+m_c^2)^{1/2}$,
which characterizes the typical virtuality of the process. The constant $c$ is
of order 1; and is varied over same range when we try to estimate the
scale-dependence of the physical predictions. The magnitude of the charm
cross-section is sensitive to the value of $m_c$. For results presented here,
we use $m_c^{\overline{\rm MS}}=1.3 GeV$, the value used in the CTEQ5HQ parton
distribution analysis (which is in the middle of the range given by the PDG
review).

Fig.\ \ref{fig:DiffDis} shows plots of four differential distributions for $%
D^{\ast }$ production at HERA, calculated using the NLO ($\alpha _{s}$)
generalized \msbar\ 4-flavor formalism described above. \ The kinematic
variables and their ranges correspond to those of the 1996-97 ZEUS
data: $1<Q^{2}<600\;$GeV$^{2}\;;\;0.02<y<0.7\;;\;1.5<p_{T}^{D^{\ast }}<15\;$%
GeV ; $\left| \eta _{D^{\ast }}\right| <1.5.$ Each distribution contains two
curves obtained with two values of the constant $c=0.5,1$ in the definition of the
scale parameter described above. These predictions, using the CTEQ5HQ parton
distributions, are compared to the ZEUS data \cite {ZEUScharm97}. We observe a
%
rather large scale dependence in these results. This is not surprising, since
the compensation among the various subprocesses which underlie the
scale-independence of the physics predictions (up to some order of perturbation
theory), strictly speaking, only apply to the inclusive cross-section. The
experimental kinematical cuts implemented in these exclusive calculations to
some extent undermine the mutual cancellation between diagrams which are
necessary for relatively scale-independent predictions. For example, the order
$\alpha_s^0$
%
$\gamma^* c \rightarrow c$ HE term (which resums the logarithms arising from
the near-collinear configurations of an infinite tower of higher-order
diagrams) implements the full contribution in collinear kinematics, a clear
over-simplification.

Keeping this fact in mind, and with current relatively large experimental
errors, the results of Fig.\ \ref{fig:DiffDis} can be considered rather
encouraging: the $Q^2$ and $p_T$ distributions show very good general
agreement; while the $W$ and $\eta_D$ distributions are ``in the right ball
park'', the shapes are too scale-dependent to allow for meaningful
``predictions''. (A specific choice of scale, in between the two shown, will
actually yield theory curves in reasonable agreement with data, within errors.)
In order to make genuine predictions on differential distributions in the
4-flavor scheme, it is necessary to extend the calculation to order
$\alpha_s^2$, which would be NNLO in the 4-flavor scheme.  This can be done by
transforming already available NLO results for 3-flavor calculations into the
4-flavor scheme. At the same order in $\alpha_s$, the 4-flavor scheme
calculation is, of course, more involved than the (NLO) 3-flavor one because of
the need for including the necessary subtraction terms in a NNLO calculation --
such as those appearing in Eqs.\ \ref{eq:Hardxsc1}, \ref{eq:Hardxsc2}, and
\ref{eq:HadXsec4fl1}.

The calculation of these differential distributions in the $\alpha _{s}^{2}$
3-flavor FFN scheme was carried out by \cite{HarrSmith}. Generally good
agreement between these calculations and the recent ZEUS data, using specific
parton distributions, scale choices, etc.\ has been reported in Ref.\ \cite
{ZEUScharm97}. Although the dependence of the predictions to the scale choice
was not discussed in this comparison, it is relatively mild, according to
\cite{HarrSmith}. This is to be expected, because the sensitivity to cuts is
reduced with a better approximation of the final state particle configurations
provided by the order $\alpha _{s}^{2}$ calculation. This fact implies greater
predictive power for the differential distributions than the order $\alpha_s$
4-flavor calculation.

The $\eta _{D^{\ast }}$ distribution in both the 3-flavor and the 4-flavor
calculations appear to differ in shape compared to the existing data points.
This could be due to the inadequacy of applying the fragmentation function
approach at less than asymptotic region, as discussed in the beginning of this
section. In particular, if the $D^{\ast }$ is not collinear to the parton, as
assumed in this approach, the rapidity distribution will be affected. More
extensive study of this effect is obviously needed.

\figDiffDis

