%Title: Photon Fragmentation at LEP
%Author: A. Gehrmann-De Ridder (TTP, Karlsruhe)
%Published: * Proceedings of the workshop on photon interactions and the photon structure * 10-12 September 1998, Lund, Sweden, eds. G. Jarlskog and T. Sjostrand, p.147.
%hep-ph/9810507
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\parskip 2ex
\def\e{\epsilon}
\def\d{{\rm d}}
\def\Li{{\rm Li}}
\def\zcut{z_{\rm cut}}
\def\ycut{y_{\rm cut}}
\def\ymin{y_{\rm min}}
\def\scut{y_{\rm cut}}
\def\smin{s_{\rm min}}
\def\as{\left(\frac{\alpha_s}{2\pi}\right)}
\def\aqed{\left(\frac{\alpha e_q^2}{2\pi}\right)}
\def\Dq{D_{q\to \gamma}}
\def\Pqpzero{P^{(0)}_{q\to \gamma}}
\def\Pqpone{P^{(1)}_{q\to \gamma}}
\def\Pqqzero{P^{(0)}_{q\to q}}
\begin{document}
\pagestyle{plain}
\begin{titlepage}
\vspace*{-1cm}
\begin{flushright}
TTP98-38 \\
October 1998 \\
\end{flushright}
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\begin{center}
{\Large\bf
Photon Fragmentation at LEP
\footnote{Talk presented at the Workshop on photon interactions and the
photon structure, Lund, Sweden, September 10-13,1998.}}
\vskip 1.3cm
{\large A.~Gehrmann--De Ridder}
\vskip .2cm
{\it Institut f\"{u}r Theoretische Teilchenphysik, Universit\"{a}t
Karlsruhe, \\ D-76128 Karlsruhe, Germany}
\vskip 2.3cm
\end{center}
\begin{abstract}
The production of final state photons in hadronic $Z$-boson decays
can be used to study the quark-to-photon fragmentation function
$D_{q\to \gamma}(z,\mu_{F})$. Currently, two different observables
are used at LEP to probe this function: the `photon' +~1 jet rate
and the inclusive photon energy distribution. We outline the results
of a calculation of the `photon' +~1 jet rate
at fixed ${\cal O}(\alpha \alpha_{s})$, which yield
a next-to-leading order determination of the quark-to-photon
fragmentation function $D_{q\to \gamma}(z,\mu_{F})$. The resulting
predictions for the isolated photon rate and the inclusive photon
spectrum at the same, fixed order, are found to be in good agreement
with experimental data. Furthermore, we outline the main features of
conventional approaches using parameterizations of the resummed
solutions of the evolution equation and
point out deficiencies of these currently available
parameterizations in the large $z$-region. We
finally demonstrate that the ALEPH data on the
`photon' +~1 jet rate are able to
discriminate between different parameterizations
of the quark-to-photon fragmentation function, which are equally
allowed by the OPAL photon energy distribution data.
\end{abstract}
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\section{Introduction}
The production of final state photons at large transverse momenta is
one of the key observables studied in hadronic collisions.
Data on high-$p_T$ photon production
yield valuable information on the gluon distribution in the proton,
while the presence of photons in the final state
represents an important background source in many searches
for new physics. A good understanding
of direct photon production within the context of
the Standard model is therefore essential.
Photons produced in hadronic collisions arise essentially
from two different sources: the {\em direct} production of a
photon off a primary parton or
through the {\em fragmentation} of a hadronic jet into a single photon
carrying a large fraction of the jet energy.
The former gives rise to perturbatively calculable short-distance
contributions whereas the latter is primarily a long
distance process which cannot be calculated within perturbative QCD.
It is described by the process-independent parton-to-photon
fragmentation function~\cite{phofrag}
which must be determined from experimental data.
Its evolution with the factorization scale $\mu_{F}$ can however
be determined by perturbative methods.
Directly emitted photons are usually well separated
from all hadron jets produced in a particular event, while photons
originating from fragmentation processes are primarily to be found
inside hadronic jets. Consequently, by imposing some isolation
criterion on the photon, one is in principle able to suppress (but
not to eliminate) the fragmentation contribution to final state
photon cross sections, and thus to define {\it isolated} photons.
However, recent analyses of the production of isolated photons
in electron-positron and proton-antiproton collisions
have shown that
the application of a geometrical isolation cone surrounding the
photon does not lead to a reasonable agreement between theoretical
prediction and experimental data.
An alternative approach to study final state photons produced in
a hadronic environment is obtained by applying the so-called democratic
clustering procedure \cite{andrew}. In this approach, the photon is
treated like any other hadron and is clustered simultaneously
with the other hadrons into jets. Consequently, one of the
jets in the final state contains a photon and is labelled `photon jet'
if the fraction of electromagnetic energy within the jet is
sufficiently large,
\begin{equation}
z=\frac{E_{EM}}{E_{EM} +E_{HAD}}>\zcut,
\label{eq:zdef}
\end{equation}
with $\zcut$ determined by the experimental conditions.
This photon is called
{\it isolated} if it carries more than a certain fraction, typically 95\%,
of the jet energy and said to be non-isolated otherwise.
Note that this separation is made by studying the experimental $z$
distribution and is usually such that hadronisation effects,
which tend to reduce $z$, are minimized.
This democratic procedure has been applied by the ALEPH
collaboration at CERN
in an analysis of two jet events in electron-positron annihilation
in which one of the jets contains a highly energetic photon
\cite{aleph}.
In this analysis,
ALEPH made a leading order determination of
the quark-to-photon fragmentation function
by comparing the
photon +~1 jet rate calculated up to ${\cal O}(\alpha)$
\cite{andrew} with the data. The theoretical basis on which
the measurement of the `photon' +~1 jet rate
relies, is an explicit counting of powers of the strong coupling $\alpha_{s}$
in both the direct and the fragmentation contributions, no resummation of $\ln \mu_{F}^2$ is performed.
We shall refer to this theoretical framework as the fixed order
approach. In Section \ref{sec:gammanlo}, we will describe the main features
of the leading and next-to-leading order calculation of the
photon +~1 jet rate following
this fixed order approach and see
how the obtained predictions compare with the available data.
More recently, the OPAL collaboration has measured the inclusive
photon distribution for final state photons
with energies as small as 10 GeV \cite{OPAL}.
This corresponds to the photon carrying a fraction of the
beam momentum, $x_{\gamma}$, to be as low as 0.2.
They have compared their results with the two model-dependent predictions
of GRV \cite{grv} and BFG \cite{bfg} and found a reasonable agreement
in both cases when
choosing the factorization scale $\mu_F=M_Z$.
In Section \ref{sec:gammanlo} we shall compare the predictions
obtained for the inclusive rate within our fixed order approach
with the OPAL data too.
The model predictions \cite{grv,bfg}
mentioned above are based on a resummation of
the logarithms of the factorization scale $\mu_{F}$
and naturally associate an inverse power of $\alpha_{s}$
with all fragmentation contributions. We shall
refer to this resummation procedure
as the conventional approach. In Section \ref{sec:gammabll} we shall present
the main features of this approach, outline the results
obtained for the photon +~1 jet rate in this approach while using either the
GRV or BFG parameterizations for the photon fragmentation function
and show how these compare with the ALEPH data.
Finally Section \ref{sec:Conclusions} contains our conclusions.
\section{The photon +~1 jet rate in the fixed order framework }
\label{sec:gammanlo}
\subsection{The photon +~1 jet rate at ${\cal O}(\alpha)$}
In the fixed order framework, the
cross section for the production of isolated photons
receives sizeable contributions from both direct photon and
fragmentation processes. More precisely,
the distribution of electromagnetic energy within
the photon jet of photon +~1 jet events, for a single quark of
charge $e_q$, at ${\cal O}(\alpha)$ in the $\overline{{\rm MS}}$-scheme,
can be written as \cite{andrew},
\begin{eqnarray}
\frac{1}{\sigma_0} \frac{{\rm d}\sigma^{(LO)}}{dz}
&=& 2 D_{q\to \gamma}(z,\mu_F)
+ \frac{\alpha e_q^2}{\pi}
P_{q \gamma}^{(0)}(z) \log \left(\frac{s_{\rm cut}}{\mu_F^2}\right) \,+\,
R_{\Delta}\delta(1-z) + \ldots, \nonumber \\
&=& 2 D_{q\to \gamma}(z,\mu_F) +C_{\gamma}^{(0)}(z,\mu_F).
\label{eq:sig0}
\end{eqnarray}
The $\ldots$ represent terms which are
well behaved as $z \to 1$.
$C^{(0)}_{\gamma}$ is the coefficient function corresponding to
the lowest order process $e^+e^- \to q \bar{q} \gamma$.
It is defined after the leading quark-photon singularity
has been subtracted and factorized in the bare quark-to-photon
fragmentation function in the $\overline{{\rm MS}}$ scheme.
The non-perturbative fragmentation function
is an exact solution at ${\cal O}(\alpha)$ of
the evolution equation in the factorization scale $\mu_{F}$,
\begin{equation}
D_{q\to \gamma}(z,\mu_{F}) =
\frac{\alpha e_q^2}{2 \pi} P_{q \gamma}^{(0)}(z)
\log\left(\frac{\mu_{F}^2}{\mu_{0}^2}\right) + D_{q\to \gamma}(z,\mu_{0}).
\end{equation}
In this equation, all unknown long-distance effects are related
to the behaviour
of $D_{q\to \gamma}(z,\mu_{0})$, the initial value of
this fragmentation function which has been fitted to the data
at some initial scale $\mu_{0}$ in~\cite{aleph}.
As $D_{q\to \gamma}(z,\mu_{F})$ is exact,
this solution does not take the commonly implemented
\cite{grv,bfg} resummations of $\log(\mu_{F}^2)$ into account and when
used to evaluate the photon +~1 jet rate at ${\cal O}(\alpha)$ yields a
factorization scale independent prediction for the cross section.
In the Durham jet algorithm, $s_{\rm cut}
\sim sz(1-z)^2/(1+z) \sim p_T^2$ \cite{durham} where $p_T$ is the
transverse momentum of the photon with respect to the cluster.
For $z<1$, we find that $\mu_F^2 \sim s_{\rm cut}$
and $\mu_F^2 \gg \mu_{0}^2$ . The `direct' contribution
in eq.~(\ref{eq:sig0}) is therefore suppressed relative
to the fragmentation contribution.
The conventional assignment of a power
of $1/\alpha_s$ to the fragmentation function can in this case be
motivated, this contribution is indeed more significant.
However, as $z\to 1$,
we see that the transverse size of the photon jet cluster
decreases such that $s_{\rm cut}\to 0$.
The hierarchy $s_{\rm cut} \sim \mu_F^2$ and $\mu_F^2 \gg \mu_0^2$ is
no longer preserved and both contributions in
eq.~(\ref{eq:sig0}) are important.
Large logarithms of $(1-z)$ become the most dominant contributions.
Being primarily interested in the high $z$ region, in \cite{andrew}
it was chosen not to impose the conventional prejudice
and resum the logarithms of $\mu_F$ {\em a priori} but to work
within a fixed order framework, to isolate the relevant large logarithms.
We have performed the calculation of the
${\cal O}(\alpha_s)$ corrections to the
`photon' +~1 jet rate using the same democratic procedure
to define the photon as in~\cite{andrew,aleph}.
The details of this fixed order calculation
have been presented in ~\cite{big}. In the following,
we shall limit ourselves to outline the main characteristics
of this calculation,
to summarize the results and
to show how these compare with the available experimental data from ALEPH.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The `photon' +~1 jet rate at ${\cal O}(\alpha \alpha_s)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
~ \epsfig{file=class.ps,width=14cm}
\caption{Parton level subprocesses contributing to the photon +~1 jet
rate at ${\cal O}(\alpha\alpha_s)$.}
\label{fig:class}
\end{center}
\end{figure}
The `photon' +~1 jet rate in $e^+e^-$ annihilation at ${\cal O} (\alpha
\alpha_s)$ receives contributions from five parton-level subprocesses
displayed in Fig.~\ref{fig:class}.
Although the `photon' +~1 jet cross section
is finite at ${\cal O}(\alpha \alpha_{s})$,
all these contributions
contain divergences (when the photon
and/or the gluon are collinear with one of the quarks,
when the gluon is soft or since the bare quark-to-photon fragmentation
function contains infinite counter terms).
All these divergences have to be isolated and cancelled analytically
before the `photon' +~1 jet cross section can be evaluated numerically.
The various configurations where the tree level process
$\gamma^* \to q \bar{q}g\gamma$ contributes
to the photon +~1 jet rate are illustrated in Fig.~\ref{fig:clus4}.
\begin{figure}[t]
\vspace{10cm}\begin{center}
~ \special{psfile=clus4.ps angle=00 hscale=60 vscale=60 hoffset=-200
voffset=-120}
\caption{Different final state `photon' + 1~jet topologies arising
from the tree level $\gamma^* \to q
\bar{q}\gamma g$ process. The `photon' jet is moving to the left while the recoiling hadronic jet moves to the right.
Square brackets denote theoretically
unresolved particles, round brackets experimental clusters.}
\label{fig:clus4}
\end{center}
\end{figure}
The associated real contributions can be separated into three
categories: either
theoretically resolved, single unresolved or double unresolved.
Within each singular region which is defined
using a theoretical criterion $\smin$,
the matrix elements are approximated and
the unresolved variables analytically integrated.
The evaluation of the singular contributions
associated with the process $\gamma ^* \to q \bar{q}g \gamma$ is of
particular interest as it contains various ingredients
which could directly be applied to the calculation of jet observables
at next-to-next-to-leading order.
Indeed, besides the contributions arising when
one final state gluon is collinear or soft (single unresolved
contributions, see fig.\ref{fig:clus4}(b)),
there are also contributions where {\em two} of the
final state partons are theoretically unresolved, see fig.\ref{fig:clus4}(c) .
The three different double unresolved contributions which
occur in this calculation are:
the {\it triple collinear} contributions,
arising when the photon and the gluon are simultaneously collinear
to one of the quarks, the {\it soft/collinear} contributions
arising when the photon is collinear to one of the quarks
while the gluon is soft and the {\it double single collinear} contributions,
resulting when the photon is collinear to one of the quarks while the gluon
is collinear to the other.
A detailed derivation of each of these singular real contributions
and of the singular contributions arising in the processes depicted
in Fig.~\ref{fig:class}(b)-(d) has been presented in~\cite{big}.
Combining all unresolved contributions present in the
processes shown in Fig.~\ref{fig:class}(a)-(d)
yields a result
that still contains single and double poles in $\e$.
These pole terms are however proportional
to the universal next-to-leading order splitting function
$P_{q\gamma}^{(1)}$ ~\cite{curci} and a convolution of two
lowest order splitting functions,
$(P_{qq}^{(0)}\otimes P_{q \gamma}^{(0)})$.
Hence, they can be factorized into the next-to-leading order
counterterm of the bare quark-to photon fragmentation function
\cite{fac} present in the contribution depicted in Fig.~\ref{fig:class}(e),
yielding a finite and factorization scale ($\mu_{F}$) dependent
result \cite{big}.
We have then chosen to evaluate the remaining
finite contributions numerically using the {\it hybrid subtraction} method,
a generalization of the {\it phase space slicing} procedure \cite{gg,kramer}.
The latter procedure turns out to be inappropriate
when more than one particle is potentially unresolved.
Indeed, in our calculation we found areas in the four parton phase space
which belong simultaneously to two different single collinear regions.
Those areas cannot be treated correctly within the phase space
slicing procedure.
Within the {\it hybrid subtraction} method developed in \cite{eec},
a parton resolution criterion $\smin$
is used to separate the phase space into different resolved and
unresolved regions.
Phase space slicing and hybrid subtraction methods vary only
in the numerical treatment of the unresolved regions.
While the matrix elements are set to zero in the former method, one
considers the difference between the full matrix
element and its approximation in all unresolved regions in the latter.
The non-singular contributions
are calculated using Monte Carlo methods like in the phase space
slicing scheme.
The numerical program finally evaluating the `photon' +~1 jet rate
at ${\cal O}(\alpha\alpha_s)$ contains four separate contributions.
\begin{figure}[t]
\begin{center}
~ \epsfig{file=ymin.ps,angle=-90,width=16cm}
\caption{${\cal O}(\alpha\alpha_s)$ individual contributions (left)
and sum of all ${\cal O}(\alpha\alpha_s)$ contributions (right) to the
photon +~1 jet rate for a single quark of charge $e_q$ such that
$\alpha e_q^2 = 2\pi$, $\alpha_s (N^2-1)/2N = 2\pi$ using
the Durham jet algorithm with $y_{{\rm cut}}=0.1$,
and integrated for $z>0.7$.}
\label{fig:ymin2}
\end{center}
\end{figure}
Each of them depends logarithmically (in fact
as $\log^3 (y_{{\rm min}})$) on
the theoretical resolution parameter $y_{{\rm min}}=s_{{\rm min}}/Q^2$.
The physical `photon' + 1~jet cross section,
which is the sum of all four contributions,
{\it must} of course be independent of the choice of $y_{{\rm min}}$,
the latter being just an artefact of the theoretical calculation.
In Fig.~\ref{fig:ymin2}, we see that the cross section approaches (within
numerical errors) a constant value provided that $y_{{\rm min}}$
is chosen small enough, indicating
that a complete cancellation of all powers of $\log (y_{{\rm min}})$
takes place. This
provides a strong check on the correctness of our results
and on the consistency of our approach.
Finally, after factorization of the quark-photon singularities, the
${\cal O}(\alpha \alpha_s)$ cross section takes the following form,
\begin{equation}
\frac{1}{\sigma_{0}}\frac {{\rm d} \sigma^{(NLO)}}{{\rm d}z}
=
\frac{1}{\sigma_0}\frac{{\rm d}\sigma^{(LO)}}{{\rm d}z}
+ \as \aqed
C^{(1)}_{\gamma}(z,\mu_{F})+
C_{q}^{(0)}
\otimes D_{q\to \gamma}(z,\mu_{F}).
\label{eq:signlo}
\end{equation}
The lowest order cross section has been given in eq.(\ref{eq:sig0}) while
the hard scattering coefficient functions $C_{i}^{(n)}$ appearing
explicitly in this equation are defined as follows.
The (finite) next-to-leading order coefficient function
$C^{(1)}_{\gamma}$ is obtained numerically
after the next-to-leading quark-photon singularity has been subtracted.
More precisely, $C^{(1)}_{\gamma}$ is obtained after summing
all contributions which are independent
of $D_{q\to \gamma}(z,\mu_{F})$
arising from the Feynman diagrams depicted
in Fig.~\ref{fig:class} together.
A detailed description of the evaluation of $C^{(1)}_{\gamma}$ has been given
in \cite{big}.
The coefficient function $C_{q}^{(0)}$ is the finite part
associated with the sum of real and virtual gluon contributions
to the process $e^+e^- \to q \bar q$. It is straightforward to evaluate,
and can be found for example in \cite{kt}.
\subsection{Comparison with Experimental Data }
A comparison between the measured `photon' +~1 jet rate \cite{aleph}
and our calculation yielded a first determination of
the quark-to-photon fragmentation function accurate up to
${\cal O}(\alpha \alpha_s)$ \cite{letter}.
This function, which parameterizes the
perturbatively incalculable long-distance effects,
has to satisfy a perturbative evolution equation in the factorization
scale $\mu_F$.
At next-to-leading order (${\cal O}(\alpha \alpha_{s})$)
this equation reads,
\begin{equation}
\frac{\partial \Dq(z,\mu_{F})}
{\partial \log(\mu_{F}^2)}=
\aqed \Pqpzero(z) \,+\,\aqed \as \Pqpone(z)
+\as \Pqqzero\otimes \Dq(z,\mu_{F}).
\label{eq:evolnlo}
\end{equation}
$\Pqqzero$ and $\Pqpone$ are respectively the lowest order quark-to-quark
and the next-to-leading order quark-to-photon universal splitting
functions \cite{curci,rijken,AP}.
The next-to-leading order fragmentation
function can be expressed as an {\it exact} solution of
this evolution equation
up to ${\cal O}(\alpha \alpha_s)$ \cite{big},
\begin{eqnarray}
\Dq(z,\mu_{F})&=&
\frac{\alpha e_{q}^2}{2\pi}P^{(0)}_{q \gamma}(z)
\log\left(\frac{\mu^2_{F}}{\mu_{0}^2}\right)
+\frac{\alpha e_{q}^2}{2\pi} \frac{\alpha_{s}}{2 \pi}
\left(\frac{N^2 -1}{2N}\right)P_{q \gamma}^{(1)}(z)
\log \left(\frac{\mu^2_{F}}{\mu_{0}^2}\right)
\nonumber\\
& & +
\frac{\alpha_{s}}{2 \pi}
\left(\frac{N^2 -1}{2N}\right)
\log \left(\frac{\mu^2_{F}}{\mu_{0}^2}\right) P_{qq}^{(0)}(z)\otimes
\frac{\alpha e_{q}^2}{2 \pi}\frac{1}{2}P_{q \gamma}^{(0)}(z)
\log \left(\frac{\mu^2_{F}}{\mu_{0}^2}\right)
\nonumber\\
& &
+\frac{\alpha_{s}}{2 \pi}
\left(\frac{N^2 -1}{2N}\right)
\log \left(\frac{\mu^2_{F}}{\mu_{0}^2}\right) P_{qq}^{(0)}(z)\otimes
D(z,\mu_{0}) \,+D(z,\mu_{0}).
\label{eq:Dnlo}
\end{eqnarray}
The initial function $\Dq^{(NLO)}(z,\mu_{0})$ has been fitted to the ALEPH ~1 jet
data \cite{letter} for
$\frac{1}{\sigma_{0}}\frac{d\sigma}{dz}$,
for the jet resolution parameter $y_{{\rm cut}}=0.06$
and $\alpha_s(M_z^2) = 0.124$
to yield \footnote{Note that the logarithmic term proportional to
$P^{(0)}_{q \gamma}(z)$
is introduced to ensure that the predicted $z$ distribution is
well behaved as $z \to 1$ \cite{andrew}.},
\begin{equation}
D^{(NLO)}(z,\mu_{0})=\frac{\alpha e_{q}^2}{2 \pi}
\left(-P^{(0)}_{q \gamma}(z)\log(1-z)^2 \;+\,20.8\,(1-z)-11.07\right),
\label{eq:fitnlo}
\end{equation}
where $\mu_{0}=0.64$~GeV.
The next-to-leading order ($\overline{{\rm MS}}$)
quark-to-photon fragmentation function (for a quark of unit charge)
at a factorization scale $\mu_F=M_Z$ were shown
in \cite{letter} and
compared with the lowest order fragmentation function obtained
in~\cite{aleph}. A large difference between the leading and next-to-leading
order quark-to-photon fragmentation functions was observed only for
$z$ close to 1, indicating the presence of large $\log (1-z)$ terms.
Moreover, a comparison between
the ALEPH data and the results of the ${\cal O}(\alpha \alpha_s)$
calculation using the fitted next-to-leading order
fragmentation function for different values of $\ycut$
can be found in \cite{big,letter}.
The next-to-leading order corrections were found to be moderate
for all values of $\ycut$ demonstrating the
perturbative stability of our fixed order approach.
\begin{figure}[t]
\begin{center}
~ \epsfig{file=ycut.ps,width=9cm,angle=-90}
\caption{The integrated photon +~1 jet rate above $z=0.95$ as function of
$y_{{\rm cut}}$, compared with the ${\cal O}(\alpha)$
and ${\cal O}(\alpha \alpha_s)$
order calculations including the appropriate
quark-to-photon fragmentation functions.}
\label{fig:ycut}
\end{center}
\end{figure}
To test the generality of our results, we have considered
two further applications:
the `isolated' photon rate and the inclusive photon distribution
which we shall now briefly present.
Using the results of the calculation of the photon +~1 jet rate at
${\cal O}(\alpha \alpha_s)$ and
the fitted quark-to-photon fragmentation function, we have
determined the {\it isolated}
rate defined as the photon +~1 jet rate for $z>0.95$
in the democratic approach.
The result of this calculation compared with
data from ALEPH~\cite{aleph} and the leading order calculation~\cite{andrew}
is shown in Fig.~\ref{fig:ycut}. It can clearly be seen that inclusion of
the next-to-leading order corrections improves the agreement between
data and theory over the whole range of $y_{{\rm cut}}$.
It is also apparent that the next-to-leading order corrections
to the isolated photon +~1 jet rate obtained in this
democratic clustering approach are of reasonable size
indicating a good perturbative stability of this {\it isolated}
photon definition.
\begin{figure}[t]
\begin{center}
~ \epsfig{file=inclusive.ps,width=9cm}
\caption{The inclusive photon energy distribution
normalized to the hadronic cross section as measured by the OPAL Collaboration
compared with the ${\cal O}(\alpha)$ and ${\cal O}(\alpha \alpha_s)$
order calculations including the appropriate
quark-to-photon fragmentation functions
determined using the ALEPH photon +~1 jet data.}
\label{fig:inclusive}
\end{center}
\end{figure}
The OPAL collaboration has recently measured the inclusive photon distribution
for final state photons with energies between 10 and 42~GeV \cite{OPAL}.
Fig.~\ref{fig:inclusive} shows our (scale independent) predictions
for the inclusive photon energy distribution at both leading and
next-to-leading order.
We see good agreement with the data, even though the phase space
relevant for the OPAL data, which corresponds to $x_{\gamma}$
values as small as 0.2,
far exceeds that used to determine the fragmentation functions from the ALEPH
photon +~1 jet data.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The photon +~1 jet rate in the conventional approach}
\label{sec:gammabll}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the conventional approach, the parton-to-photon fragmentation function
$D_{i\to \gamma}$ satisfies an all order inhomogeneous
evolution equation \cite{AP}.
Usually, these equations can be diagonalized in
terms of the singlet and non-singlet
quark fragmentation functions as well as the gluon fragmentation function.
However when analyzing the global features of the solutions
of these evolution equations, as was discussed in \cite{papernew},
several simplifications can be consistently made.
For example, the gluon-to-photon fragmentation function is by orders
of magnitude smaller than the quark-to-photon fragmentation functions,
as was shown in \cite{papernew}. Its contribution to the photon
cross section can safely be ignored.
Consequently, the flavour singlet and
non-singlet quark-to-photon fragmentation functions become equal to a unique
fragmentation function $\Dq$ which satisfies
an evolution equation having a similar form than
the next-to-leading order evolution
valid in the fixed order approach, see eq.~(\ref{eq:evolnlo}).
Unlike in
eq.~(\ref{eq:evolnlo}) though, the strong coupling $\alpha_{s}$
is now a function of the factorization scale, it runs.
The full solution $\Dq$ of the inhomogeneous evolution equation is given by
the sum of two contributions; a pointlike (or perturbative) part $\Dq^{pl}$
which is a solution of the inhomogeneous equation follows
eq.(\ref{eq:evolnlo})
and a hadronic (or non-perturbative) part $\Dq^{had}$
which is the solution of the corresponding homogeneous equation.
In the conventional approach, approximate solutions of these evolution
equations are commonly obtained as follows \cite{grv,bfg}.
First an analytic solution in moment space is
obtained in the leading logarithm (LL) or beyond leading logarithm (BLL)
approximations. These are then inverted numerically to give
the fragmentation function in $x$-space.
At LL only terms of the form $(\alpha_s^n \ln^{n+1} \mu_F^2)$
are kept while at BLL both leading
$(\alpha_s^n \ln^{n+1} \mu_F^2)$ and subleading $(\alpha_s^n \ln^n \mu_F^2)$
logarithms of the mass factorization scale $\mu_F$ are resummed
to all orders in the strong coupling $\alpha_{s}$.
It is worth noting that both LL and BLL solutions have an asymptotic
behaviour given by,
\begin{equation}
\Dq^{asympt}(z,\mu_F)=\aqed \frac{2 \pi}{\alpha_{s}(\mu_{F}^2)}\,a(z),
\label{eq:asympt}
\end{equation}
where $a(z)$ contains the splitting function $\Pqpzero$.
This asymptotic form lends support to
the common assumption that the
quark-to-photon fragmentation function $\Dq$ is
${\cal O}\left( \alpha/\alpha_{s} \right)$.
This assumption is in contrast with
that adopted in the fixed order approach (cf. Section~\ref{sec:gammanlo})
where the quark-to-photon fragmentation function is ${\cal O}(\alpha)$.
It leads to significant differences in the respective expressions
of the one-photon production cross sections.
Indeed, the LL and BLL expression of the cross section in the
$\overline{{\rm MS}}$ scheme arising when one uses the corresponding
resummed LL or BLL fragmentation functions in this approach
reads
\begin{eqnarray}
\frac{1}{\sigma_{0}}\frac {{\rm d} \sigma^{LL}}{{\rm d}z} &=&
\;D_{q\to \gamma}(z,\mu_{F}),
\nonumber\\
\frac{1}{\sigma_{0}}\frac {{\rm d} \sigma^{BLL}}{{\rm d}z}
&=&
D_{q\to \gamma}(z,\mu_{F})\,
+\as C_{q}^{(0)}
\otimes D_{q\to \gamma}(z,\mu_{F})
+ \aqed C^{(0)}_{\gamma}(z,\mu_{F}).
\label{eq:sigbll}
\end{eqnarray}
As can be seen from these equations, no direct term contributes to the
cross section at the LL level, while only the ${\cal O}(\alpha)$ direct
term $C^{(0)}_{\gamma}$ contributes at the BLL level to it.
At the BLL level, contributions arising
from \ref{fig:class}(a)-(b) do not enter in the cross section.
As explained at length in~\cite{letter}, this conventional
procedure of associating an inverse power of $\alpha_{s}$
with the fragmentation function
is clearly appropriate when the logarithms of the factorization scale
$\mu_{F}$ are the {\em only} potentially large logarithms but
is problematic when different classes of large
logarithms can occur as it is the case in the `photon' +~1 jet cross section.
All unknown long-distance effects are
related to the behaviour of the input fragmentation function
$D_{q\to \gamma}^{np}(z,\mu_{0})$ implicitly present in eq.(\ref{eq:sigbll}).
In the approaches of GRV or BFG,
the non perturbative input function
$D_{q \to \gamma}^{np}(z,\mu_{0})$ is treated with only
minor differences. Those have been detailed in \cite{papernew}.
We shall here concentrate on the major common points in these approaches.
At LL both GRV and BFG agree
that $D_{q \to \gamma}^{np}(z,\mu_{0})$
is negligible and can be described by a vector meson dominance model (VMD)
as explained in \cite{grv} and \cite{bfg} respectively.
However at BLL and in the $\overline{{\rm MS}}$ scheme,
the input fragmentation function cannot be negligible
due to the presence of the direct term $C_{\gamma}^{(0)}$ and
cannot be described by a VMD input alone.
Indeed, $C_{\gamma}^{(0)}(z)$ diverges as $z \to 1$ and would drive
the cross section to unacceptable negative values if a VMD input
alone is considered for the input fragmentation function.
Note that the requirement that the
cross section is positive led the authors in \cite{andrew, letter} to consider
in the fixed order approach a term proportional to $\Pqpzero \ln(1-z)^2$
in the expression of $D_{q \to \gamma}^{np}(z,\mu_{0})$.
In summary, in any resummed or fixed order approach, as soon
as the direct term $C_{\gamma}^{(0)}$ enters the cross section,
as it does in the $\overline{{\rm MS}}$ factorization scheme,
the input fragmentation function
$D_{q \to \gamma}^{np}(z,\mu_{0})$ must compensate the large $z$
behaviour of $C_{\gamma}^{(0)}$. Consequently,
this input fragmentation function
$D_{q \to \gamma}^{np}(z,\mu_{0})$ as well as the total solution
$D_{q \to \gamma}(z,\mu_{F})$ in this $\overline{{\rm MS}}$ scheme
should clearly exhibit a divergent behaviour as $z \to 1$ .
In Fig. \ref{fig:dfrag} we compare the analytic expression of the fragmentation
function obtained in the fixed order approach, eq.(\ref{eq:Dnlo}) with the BLL
parameterizations of GRV and BFG for the numerically resummed solutions.
We clearly see, that the fixed order solution
does diverge as $z \to 1$ while the numerical solutions do not.
This significant disagreement is mainly due to
deficiencies in the numerical parameterizations.
In fact, it can be traced back to the presence
of logarithms of $(1-z)$ that are explicit in the expanded result.
These logarithms should also be present in the numerical resummed results.
However, the parameterizations are necessarily
obtained by inverting only a finite number of moments
and it is a well known problem to describe a logarithmic behaviour with a
polynomial expansion.
This clearly indicates that the presently available
parameterizations for the resummed
fragmentation functions are not accurate at large $z$ and particularly
for $z > 0.95$.
\begin{figure}[t]
\begin{center}
~ \epsfig{file=dfrag.ps,width=9cm,angle=-90}
\caption{The quark-to-photon fragmentation function
$z D_{u \to \gamma}(z,\mu_F)$
evaluated at $\mu_F = M_Z$ in the $(\overline{{\rm MS}})$-scheme.
The NLO fit from the ALEPH `photon' + 1~jet data is shown
as solid line. The GRV (BFG) parameterization is shown
dashed (dotted).}
\label{fig:dfrag}
\end{center}
\end{figure}
Except in the very high $z$ region however,
we see that, the various fragmentation functions generally agree
well with each other in shape and magnitude.
Consequently we can expect, that predictions
for the inclusive photon cross sections
(which run over a wide range of $z$) will be largely in agreement, while
significant differences may be apparent in the `photon' + 1~jet
estimates which focus on the large $z$ region.
Indeed, we mentioned that the OPAL data were in agreement with predictions
using either (GRV or BFG) parameterizations in the conventional
approach, and we showed in Section \ref{sec:gammanlo} that these data were
also in agreement with the predictions obtained in our fixed order approach.
Let us now concentrate on the `photon' +~1 jet cross section, an
observable which is sensitive on the large $z$ region ($0.70.95$.
Ignoring this high $z$ region, the BLL predictions for the photon +~1
jet rate obtained using the BFG parameterization are found to be
in agreement with the ALEPH data, while the predictions using the
fragmentation function of GRV lie too high.
To summarize,
we have seen that the inclusive and `photon' +~1 jet data from LEP can
be described using either the ${\cal O}(\alpha \alpha_{s})$
fragmentation function
whose non-perturbative input is fitted to the
ALEPH data or using the BLL parameterization of BFG.
In the latter case,
the agreement needs however to be restricted to $z$-values below 0.95.
\section*{Acknowledgements}
I wish to thank G.~Jarlskog, L.~J\"onsson and T.~Sj\"ostrand for
organizing an interesting and pleasant workshop.
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