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\newcommand{\xgg}{x_{g}^{\gamma}}
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%\begin{document}
%
\title*{Forward Jet Production \protect\newline
as a Probe of Low $\mathbf{x}$ Dynamics}
%
%
\toctitle{}
% allows explicit linebreak for the table of content
%
%
\author{John Outhwaite}
%
%
\institute{Department of Physics, University of Durham,  Durham, DH1 3LE, UK.}

\maketitle              % typesets the title of the contribution

\begin{abstract}
%
The HERA facility at Hamburg has
unveiled a wealth of physics
spanning a broad kinematic domain. 
%
The question of how best to describe data obtained at low $x$ remains 
open, the main problem being the difficulty of cleanly disentangling
perturbative and non-perturbative effects.
%
A recent set of measurements, looking at final states with high transverse
momentum produced close to the proton remnant 
offers fresh insight into low $x$ dynamics.
%
The reasons why these
configurations are especially interesting are motivated,
and complementary 
approaches to describing the physics involved are outlined.
%
We illustrate this by comparing predictions based on 
small $x$ (QCD) dynamics with recent data
for deep inelastic events containing forward jets or 
forward $\pi^{0}$ mesons.
%
We quantify the effect of imposing the (higher order) consistency
condition on the BFKL equation and study uncertainties inherent in the QCD
predictions.
%
\end{abstract}
%
\section{Introduction}
%
%
The ability of perturbative QCD to describe a wide variety of HERA 
phenomena over a broad span of ($x,Q^{2}$) kinematic space has
been well demonstrated. 
%
There is general consensus that the mechanisms which 
underpin the description of physics
within the large $Q^{2}$ region, are based on the DGLAP
evolution equations which effectively resum the $\alpha_{s}\log Q^{2}$
contributions. The small $x$ limit of this approach resums the 
$\alpha_{s}\log Q^{2}\log(1/x)$ contributions and is known as the
Double Leading Log Approximation (DLLA).
%

%
In recent years, the focus of experimental and theoretical 
efforts have shifted to the region of low $x$ and with medium $Q^{2}$,
which goes beyond DLLA in that we must resum $\alpha_{s}\log(1/x)$
terms for all $Q^{2}$, and not just take the $\log Q^{2}$ limit.
%
As HERA began to probe at ever decreasing values of Bjorken
$x$ ($1/x\sim s$), it became apparent that at high energies, 
a perturbative expansion containing terms of the form 
$\alpha_{s}\log(1/x)$ 
at {\it all} $Q^{2}$, and not simply the $\log Q^{2}$ limit,
is necessary.
%
That is, for very low $x$,
the small size of the strong coupling $\alpha_{s}$
is not sufficient to tame large $\log(1/x)$
terms (unaccompanied by $\log Q^{2}$).
%
These would need to be resummed to all orders.
%
The failure of the standard DGLAP approach to DIS phenomena,
wherein large logarithms of $(\alpha_{s}\log Q^{2})^{2}$ 
and also $\alpha_{s}(\alpha_{s}\log Q^{2})^{2}$ are systematically
resummed to all orders,
would be a signal for the onset of a distinct low $x$ dynamics.
Is there experimental evidence for such effects?
This question is not so easy to answer as it first appears.
%

%
An early idea, first mooted by Mueller \cite{mueller}, 
in which one 
scatters inelastically from a parton, and selects 
kinematic configurations that can isolate clearly the low $x$ effects,
was enthusiastically taken up by the experimental 
community at HERA \cite{xh1,xzeus}.
%
Data of sufficient quality have now been obtained
and it is now an interesting phenomenological challenge to
see what the measurements imply about the underlying small $x$
dynamics. This is the subject of the current review.
%

%
Resumming the large $\log (1/x)$ terms to leading order is performed within the
Balitskij, Fadin, Kuraev, Lipatov (BFKL) formalism, first
outlined in the late 1970's \cite{xbfkl,lipatov}.
%
Since then, leading log BFKL based phenomenology was shown to
be insufficient, since the next-to-leading corrections were
calculated \cite{nlo1,ross,salam} and found to be large. 
%
A phenomenological modification to the leading
BFKL formalism \cite{sutton1,kmo} has provided 
an economical, physically based,
description of a variety of phenomena within the low $x$ QCD
framework.
%
An alternative formalism \cite{jung,kramer}, based on the concept of 
the parton content of the resolved photon, has shown similar success,
over a similarly broad range of phenomena. 
%

%
Unequivocably describing the physics of low $x$ 
region, within the bounds of perturbative QCD, would 
certainly be a worthwhile achievement.
%
%
\section{BFKL $\log(1/x)$ resummation}
%
%
The BFKL prescription for the resummation of small $x$ 
contributions has been performed for both leading log (LL) 
\cite{xbfkl,lipatov}, terms
of the form $\alpha_{s}^{n}\log^{n}(1/x)$, and next-to-leading log 
(NLL) \cite{nlo1,ross,salam}, terms of the form $\alpha_{s}^{n+1}\log^{n}(1/x)$. 
%

%
All $\log(1/x)$ contributions have to be considered,
and so we must retain the full transverse momentum dependence
of the gluon distribution, and not simply the $\log Q^{2}$
dependence arising from the strong ordering in $k_{T}$.
%
The ordering in the transverse momentum
is relaxed, and 
%
an unintegrated gluon distribution, $\phi_{i}$, is introduced,
related to the usual gluon distribution, $g_{i}$, of a photon of virtuality $Q^{2}$
and polarisation $i=T,L$,
probed at a scale $\bar{Q}^{2}$ via:
\begin{displaymath}
xg_{i}(x,Q^{2},\bar{Q}^{2}) = \int^{\bar{Q}^{2}} {dk_{T}^{2} 
\over k_{T}^{2}} \phi_{i}(x,k_{T}^{2},Q^{2})
\end{displaymath}
%
$x$ is the longitudinal momentum fraction of the virtual photon
carried by the gluon, and $k_{T}$ denotes its transverse momentum.
%

%
We begin with a brief overview of the BFKL technique.
%
It is assumed that the dominant process 
in the high energy limit is gluon exchange in the $t$-channel.
%
Higher order QCD corrections are collected together to form an
`effective' diagram.
%
This comprises an effective gluon vertex, which 
subsumes real gluon emissions, whilst the
$t$-channel gluon reggeizes with the virtual corrections.
%
Then, using this `rung', one can recursively build up 
a ladder diagram with reggeized t-channel gluonic
propagators emitting regular gluons from effective vertices.
%
The BFKL equation represents the sum of all such ladder diagrams.
%
\pagebreak
%
\begin{displaymath}
\phi_{i}(x, k_{T}^{2}, Q^{2}) = \phi_{i}^{(0)}(x, k_{T}^{2}, Q^{2}) +
\end{displaymath}
%
\begin{equation}
 k_{T}^{2} \int\limits_{x}^{1} 
{dz \over z} \int\limits_{k_{0}^{2}}^{\infty}
{d k_{T}^{\prime 2} \over k_{T}^{\prime 2} }
\bar{\alpha_{s}} \left( {
\phi_{i}( {x/z}, k_{T}^{\prime 2}, Q^{2} ) - \phi_{i}
( {x/z}, k_{T}^{2}, Q^{2} )
\over \mid k_{T}^{2} - k_{T}^{\prime 2} \mid }
+ { \phi_{i}( {x/z}, k_{T}^{2}, Q^{2} ) \over
\left( 4 k_{T}^{\prime 4} + k_{T}^{4} \right)^{ 1 \over 2 } } \right)
\label{eqn:bfkl}
\end{equation}
%
where $i=T$ or $L$, the $\phi_{i}^{(0)}$ represent the driving terms
of the equation, $\bar{\alpha}_{s}=3\alpha_{s}/\pi$ and $x$ is the fraction
of the virtual photon's longitudinal momentum carried by the gluon.
%
The term proportional to $\phi_i(x/z,k_{T}^{\prime 2},Q^{2})$
corresponds to the real gluon emissions and the terms proportional
to $\phi_i(x/z,k_{T}^{2},Q^{2})$ describe the virtual corrections
generating reggeization of the gluons.
%
We see that real and 
virtual contributions to the integrand cancel in the potentially
singular limit 
of $k_{T}^{\prime 2} \longrightarrow k_{T}^{2}$.
%
Notice the equation is invariant under a scaling of the transverse
momentum, which allows an analytic solution in the case of fixed $\alpha_{s}$
to be calculated using Mellin transforms.
%

%
Taking $\alpha_{s}$ fixed, it is possible to calculate the 
characteristic exponent of the singular power behaviour in $x$ of the 
unintegrated gluon distribution:
%
\begin{displaymath}
\phi \sim x^{-\lambda}
\end{displaymath}
%
with exponents at leading and next-to-leading 
log, in the $\overline{\rm{MS}}$ scheme,
given by
%
\begin{displaymath}
\lambda_{LL}=\bar{\alpha}_{s} 4 \log 2
\end{displaymath}
%
\begin{equation}
\lambda_{NLL} = \lambda_{LL}\left(1 - 3.4\bar{\alpha}_{s}-0.15{N_{f}\alpha_{s} \
\over \pi}\right)
\label{eqn:lam}
\end{equation}
%
[A brief aside: The BFKL singular behaviour of $g_{i}$ as $x\longrightarrow0$
cannot be the complete picture of low $x$ physics. 
On general conceptual grounds, the Froissart bound forbids the
growth of the cross-section faster than $\log^{2}(s)$. 
We can motivate this physically by recognising that as we go to 
higher energies, the density of gluons within the proton will increase
to the point that they can no longer be treated as free partons. The gluons
will recombine, and the cross-section will be suppressed.
At current HERA energies there seems to be no evidence yet 
for the presence of these {\it `shadowing'} 
or {\it `unitarity'} corrections \cite{shadow}.]
%

%
It is clear that the NLL corrections are substantial and thus that
LL based phenomenology is strictly unreliable.
%
It is possible to impose higher order, physically motivated restrictions
in the governing evolution equation {\it to all orders}.
%
One important example is that
the LL BFKL equation does not respect energy-momentum conservation,
the violation being produced by subleading terms. 
%
In fact Orr and Stirling \cite{wjs} are developing a BFKL Monte Carlo
which will allow the effect of energy-momentum conservation to 
be quantified.
%

%
There exists a further physical constraint that is more stringent than
energy-momentum conservation, and implemented in the BFKL equation
rather simply.
%
The {\it consistency constraint} imposes the physical condition
that the virtuality of exchanged gluons along a BFKL chain should
arise predominantly from the transverse component of momentum \cite{sutton1}.
%
\begin{displaymath}
k_{T}^{\prime 2} < {k_{T}^{2} \over z}.
\end{displaymath}
%

%
This ensures the validity of the small $x$ approximation, 
by restricting the maximum transverse momentum of an emitted gluon.
%
The BFKL equation was derived assuming this constraint, and in the
region of phase space beyond that delimited by the constraint
the BFKL equation is inappropriate.
%
Note that, in particular, the gluon propagators which appear
in the BFKL equation contain only the transverse momenta squared.
%
It has been shown \cite{sutton1} that including this condition 
implicitly subsumes the neglected energy-momentum conservation
over the greater part
of the small $x$ kinematic space.
%

%
It can be neatly encapsulated, and incorporated into the LL BFKL
equation, through the inclusion of a $\Theta$-function on the kernel
component governing real gluon emission:
%
\begin{displaymath}
\Theta(k_{T}^{2}/k_{T}^{\prime 2}-z)
\end{displaymath}
%
It transpires
that limiting the LL BFKL formalism in this way induces 
an effect at all orders which if truncated to NLL gives the
dominant part of the NLL expression of (\ref{eqn:lam}).
%
One might conjecture that these represent the dominant higher
order effects, and that the modified BFKL evolution resums these.
%
Is it possible to obtain a good description of the HERA data by 
implementing this type of formalism?
%
%
\section{Experimental signatures of BFKL-type dynamics}
%
%
A definitive experimental identification of the 
low $x$ physics `smoking gun' has proven
somewhat elusive. There exist several variations on the original
`gold plated' measurements proposed by Mueller\cite{mueller}, that in principle 
might exhibit the characteristic signs of a rapid 
rise in gluon distribution at low $x$,
but uniquely separating the $\log(1/x)$ from non-perturbative
effects is certainly non-trivial.
%
\subsection{Inclusive quantities}
%
The first dramatic evidence for the effects of small $x$ resummations
came from HERA
measurements of the proton structure function $F_{2}(x,Q^{2})$, 
which exhibit a strong rise with decreasing $x$. 
%
In \cite{stasto} a quantitative analysis of the $F_{2}$ data within 
a united modified BFKL and DGLAP framework was presented. 
%

%
The steep increase 
was initially lauded as a clear demonstration of the BFKL small $x$
resummation effects, but it soon became apparent that
it was not possible to ascribe this feature unambiguously to
BFKL-type physics.
%
It proved possible to adequately describe the unexpected features
in the data as a consequence of DLLA resummations, in which terms
of the form $\alpha^{n}_{s}\log^{n}(1/x)\log^{n}(Q^{2})$ are 
resummed to all orders.
%
There exist sufficient provisos in the theoretical descriptions of 
$F_{2}$ to compromise its discriminatory powers:
%
\begin{itemize}
%
\item
DGLAP evolution, with a sufficiently `tuned' 
non-perturbative input distribution
can mimic the inverse power behaviour predicted by the BFKL approach.
%
\item
The disorder in $k_{T}$ corresponds to a random walk in transverse momentum
along the gluon chain. The kinematic bounds imposed by 
the $F_{2}$  measurement allow for this transverse momentum to 
significantly diffuse into the infra-red region, blurring
clear interpretation.
%
\item
The parton distribution functions necessary to an analysis
are not sufficiently well determined at all the required kinematical
configurations for a quantitative analysis.
\end{itemize}
A more stringent process is required in order to draw firm conclusions.
%
\subsection{Exclusive quantities}
%
It is clear that inclusive measurements ($F_{2}$ etc...) 
are unable to disentangle cleanly low $x$ parton dynamics from
non-perturbative effects.
In attempting to do so, model dependences are necessarily introduced.
%
One should try and utilise more exclusive measurements. More specifically, 
final states restricted in kinematic ($x,Q^{2}$) space by the requirement
that they are produced close to the proton remnant, prove 
to be of particular interest.
%

%
Mueller proposed that DIS events containing an identified forward jet
would prove an excellent discriminator for low $x$ dynamics.
%
The detailed formalism was given in \cite{sutton2}.
%
These particular measurements were made by the H1 and ZEUS collaborations at 
HERA \cite{xh1,xzeus}. Recent analyses can be found in 
\cite{sutton2,sabine}.
%
Subsequently, it was realised that this process could be extended, 
and in some sense refined, by
studying energetic forward $\pi^{0}$'s produced close to the proton
direction \cite{sabine},
and also that the forward dijet rate \cite{claire} would yield 
another possible window on the low $x$ physics. 
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\textwidth]{outhwaite1.eps}
\end{center}
\caption{Generic BFKL formalism for the Mueller process. 
Virtual photon-gluon fusion realised through the quark box, and 
soft gluon radiation between that and the struck parton
produced close to the proton remnants.}
\end{figure}
%

%
These processes share the same basic properties that make them a useful
probe of low $x$ dynamics:
%
\begin{itemize}
\item
%
Experimentally, one requires that $Q^{2} \sim k_{T}^{2}$. 
This ensures that DGLAP type evolution is suppressed, and that 
features in the data arise solely from small $x$ dynamics.
%
\item
%
By restricting the study to configurations where the final state is 
produced close to the beam pipe, one insists that the jet carries
a large fraction of the longitudinal momentum of the parent proton. 
This ensures the evolution parameter $x/x_{j}$ is small, 
necessary for the validity of the BFKL approach, and also that $x_{j}$
is sufficiently large for the parton distribution functions to be 
determined at regions in the phase space where they are well determined
from global analyses.
%
\item
In principle, the large $k_{T}^{2}$ of the process suppresses the
diffusion into the infra-red region due to 
the random walk in transverse momentum
along the gluon chain.
%
\end{itemize}
%
In summary, Mueller's proposal was to perform deep inelastic 
scattering on parton constituent of the proton, and so to 
neatly sidestep non-perturbative hadronization problems.
%

%
In addition one can examine the subset of events that 
produce $\pi^{0}$'s in the forward direction \cite{xh1,new_h1}. 
These measurements possess additional qualities that complement the 
basic Mueller process:
%
\begin{itemize}
%
\item
%
The $\pi^{0}$ measurement samples more energetic forward jets
than could be measured explicitly in jets. By allowing the struck parton 
fragment collinearly into a pion with longitudinal momentum fraction
$x_{\pi} < x_{j}$ and transverse momentum $p_{\pi T} < k_{jT}$ we sample
a greater amount of the relevant phase space.
%
\item
%
Experiments take measurements at the {\it hadron} level, theoreticians
make calculations at the {\it parton} level,
so there is a certain amount of ambiguity involved in comparing
the predictions with the data... 
In studying final state particles, hadronisation uncertainties 
are swept into parameterisations of fragmentation functions, and we can
be certain of comparing like quantities.
%
\item
%
Experimentally, it is easier to unambiguously identify a $\pi^{0}$ 
rather than a jet. The weak model dependence from the algorithms
used to specify exactly what constitutes a `jet' is also eliminated
by looking at a specific final state.
%
\item
The conceptual `gains' outlined above must be tempered with a practical 
`loss'. By requiring a single energetic fragment of the jet, we
reduce the cross-section and so the observed event rate.
%
\end{itemize}
%
Finally, we can obtain an additional measure of the low $x$
region by looking at the subset of forward jet events that
contain {\it two} jets in the forward region.
%
%
\section{The BFKL approach to forward jet production}
%
Unlike the previous analyses \cite{sutton2,sabine}, 
ref.~\cite{kmo} uses the modified BFKL equation
to evolve the gluon distribution from the photon end down to
the struck parton.
%
The virtual photon is connected to the gluon chain through a 
quark box, which is calculable in perturbative QCD. 
%
This comprises the driving term for the unintegrated gluon distribution,
$\phi^{(0)}$ in the BFKL equation.
%
The modified BFKL equation is solved numerically
for the unintegrated gluon distribution
of the virtual photon, with the quark box as 
the inhomogeneous driving terms,
and allowing $\alpha_{s}$ to run.
%
Longitudinal momentum fractions are related at the jet vertex
through the {\it strong ordering} condition
\begin{displaymath}
k_{aT}^{2} \ll k_{jT}^{2},\,\, x_{j}^{P} \gg x
\end{displaymath}
and utilising the jet on-shell condition $k_{j}^{2}=0$
%
The singular behaviour of the unintegrated gluon distribution 
in the BFKL approach then feeds through into observables 
via $k_{T}$ factorisation
%
\begin{equation}
{ \partial^{2} F_{i} \over \partial \xjp \partial k_{jT}^{2} } =
{\bar{\alpha}_{s} k_{jT}^{4} }
\left( \sum_{a} f_{a}(\xjp,k_{jT}^{2}) \right) \Phi_{i}
\left({x \over \xjp}, k_{jT}^{2},Q^{2} \right)
\label{eqn:ktfac}
\end{equation}
%
where $i=1,2$ and $\sum_{a} f_{a}$ is the sum over parton 
distribution functions 
in the effective structure function approximation.
A recent leading order paramaterisation of the 
parton distributions is used \cite{mrst}.
%

%
These differential structure functions are related to differential 
cross-sections in the usual way
%
\begin{displaymath}
{ \partial \sigma_{j} \over
\partial x \partial Q^{2} \partial \xjp \partial k_{jT}^{2} } =
{ 4 \pi \alpha^{2} \over x Q^{4} } \left( (1-y)
{\partial^{2} F_{2} \over \partial{\xjp} \partial k_{jT}^{2} } + {1 \over 2}y^{2}
{\partial^{2} F_{T} \over \partial{\xjp} \partial k_{jT}^{2} } \right)
\end{displaymath}
%
The experimental cuts applied by the H1 
\cite{xh1} and ZEUS \cite{xzeus} collaborations 
in order to restrict the sampling to purely forward jet events are
imposed on the numerical evaluation of the cross-sections.
%

%
The analgous cross-sections for $\pi^{0}$ production are constructed by
convoluting the DIS + forward jet differential cross-sections with 
the $\pi^{0}$ fragmentation function parameterisations.
%
These are parameterisations of the non-perturbative
hadronization of the parton into a final state $\pi^{0}$. 
%
We use the LO $\pi^{\pm}$ fragmentaion functions of Binnewies et al. 
\cite{binnewies2},
and the $\pi^{0}$ functions are then taken to be 
${1\over 2}(\pi^{+}+\pi^{-})$ distributions.
%
The $\pi^{0}$ is assumed to carry a fraction $z=x_{\pi}/x_{j}^{P}$ of the jet's
longitudinal momentum in a direction collinear to the parent parton.
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{outhwaite2.eps}
\end{center}
\caption{The DIS + forward jet differtential cross-section versus 
Bjorken-$x$ as measured at the hadron level by the H1 \cite{xh1}
and ZEUS \cite{zeus} collaborations. The curves are predictions,
at the parton level, based on the BFKL formalism including 
sub-leading corrections, corresponding to the three choices of the
scales and infrared cut-off given in (\ref{eqn:par}).} 
%
\label{fig:jetx}
\end{figure}
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{outhwaite3.eps}
\end{center}
\caption{The $\pi^{0}$ differential cross-section versus Bjorken-$x$
obtained from H1 1996 data \cite{new_h1}. The curves are predictions,
based on the BFKL formalism including sub-leading corrections,
corresponding to the three choices of the 
scales and infrared cut-off given in (\ref{eqn:par}).}
\label{fig:pix}
\end{figure}
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{outhwaite4.eps}
\end{center}
\caption{The differential cross-section for (a) DIS + forward jet, (b)
DIS + forward $\pi^{0}$, events. The continuous and dashed curves 
correspond to the inclusion and omission of the consistency constraint.
The lower set of curves corresponds to the scales and infrared cut-off
of (i) in (\ref{eqn:par}), and the upper set of curves to
(iii). The jet data are at hadron level; converting
to the parton level is expected to lower the jet data points by
15-20\%.}
\label{fig:ho}
\end{figure}
%
%
A sample of the results are shown in 
Figs.~\ref{fig:jetx} and \ref{fig:pix}.
Plots are shown for different choices of the
scales 
(in the driving term $\phi^{(0)}$ of (\ref{eqn:bfkl}), and in $\alpha_{s}$
of (\ref{eqn:ktfac})) and of the infrared cut-off $k_{0}^{2}$ in
(\ref{eqn:bfkl}), taken respectively to be:
%
\begin{equation}
\begin{array}{ccccc}
%
\mathrm{(i)} & (k_{T}^{2} + m_{q}^{2})/4,\; & k_{T}^{2}/4,\;
& k_{0}^{2} = 0.5\mathrm{GeV}^{2}
& \mathrm{(upper \; dashed)} \\
%
\mathrm{(ii)} & (k_{T}^{2} + m_{q}^{2})/4,\; & k_{T}^{2}/4\;,
& k_{0}^{2} = 1\mathrm{GeV}^{2}
& \mathrm{(continuous)} \\
%
\mathrm{(iii)} & (k_{T}^{2} + m_{q}^{2}),\; & k_{T}^{2},\;
& k_{0}^{2} = 0.5\mathrm{GeV}^{2}
& \mathrm{(lower \; dashed)}.
%
\end{array}
\label{eqn:par}
\end{equation}
%
We note that the calculations describe 
the forward jet, forward $\pi^{0}$ and forward dijet measurements well,
for physically reasonably scales and cut-off. 
%
We see that they exhibit a weak residual 
dependence on a transverse momentum cut-off parameter
and a stronger dependence on changes in the choice of scale.
%

%
In Fig.~\ref{fig:ho} we contrast the predictions of the modified 
BFKL formalism with that of the leading-$\log(1/x)$ approximation.
The data prefer the inclusion
of the resummation to all orders of subleading terms through
the consistency constraint, especially since converting the data
down to the parton level is expected to lead to a decrease
in cross-section of some 15-20\%.
%
%
\section{The resolved and direct photon description of forward jet production}
%
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\textwidth]{outhwaite5.eps}
\end{center}
\caption{Deep inelastic scattering and a forward jet 
within the LO resolved + direct photon
formalism of \cite{jung}, with DGLAP evolution from 
both the proton and virtual photon
ends towards a $qg \longrightarrow qg$ subprocess.}
\end{figure}

%
The resolved and direct photon approach splits the process into 
two distinct mechanisms according to the size of the characteristic
scale $\mu^{2}$ of the subprocess involved.
%
When $\mu^{2} < Q^{2}$, the inverse size of the photon, a standard
perturbative approach, with point-like photon interactions is taken.
However, when $\mu^{2} >  Q^{2}$,
the partonic structure of the virtual photon is said to have
been resolved. This is then parameterised by a 
set of parton density functions
analgous to the familiar proton parton density functions. 
%

%
Two groups \cite{jung,kramer} have implemented the concept of a resolved photon
contribution to describe forward jets in DIS.
Both have produced Monte Carlo event generators (RAPGAP and JetViP) 
that describe the surplus of events over the naive 
(direct, point-like photon) DIS expectation.
%

%
RAPGAP \cite{jung} uses purely leading order matrix elements, and allows
parton showering between both the hard subprocess and the resolved photon, and
the subprocess and proton. It is thought that these simulate
higher order processes, as the leading order matrix elements
alone prove insufficient to describe the data.
%

%
The total cross-section is then calculated as the sum 
of resolved and direct photon contributions. 
%
Whilst we have transverse momentum ordering within each
of the DGLAP chains, this formalism allows for an overall disordering in
$k_{T}^{2}$ between the virtual photon and proton.
%
This $k_{T}^{2}$ disorder appears to be a `necessary' condition in
producing the rise in the cross-sections at low $x$. 
It features in many of the models that best describe the DIS +
forward jet data (c.f. BFKL phenomenology, this approach and the
color dipole model (CDM) implemented in ARIADNE)
%

%
JetViP \cite{kramer} treats the 
resolved and direct contributions consistently
using NLO matrix elements, taking care to count only once those
parts from the NLO direct contribution which are implicitly 
included from the resolved photon. 
%
It was found that the significant part of the NLO corrections arose from
the NLO contribution from the resolved photon.
%
This piece corresponds to an extra rung (and so includes one `disorder')
on the gluon ladder from the photon remnant and is 
essentially calculable within the framework of 
perturbative QCD.
%

%
Both formalisms including a resolved photon contribution into
the DIS + forward jet process find good agreement between HERA 
forward jet data and calculation.
%

%
\section{Conclusions}
%
One cannot obtain a sufficient increase with decreasing $x$ of the
forward jet cross-section from fixed order QCD calculations. 
NLO predictions \cite{bartels,mirkes} show a large deficit compared with
the data.
%
It is clear that the modified BFKL, and the resolved photon 
(both NLO and LO + parton showers.) theoretical approaches
to processes designed to clarify low $x$ physics
provide a good description of HERA measurements. 
%
Resumming subleading effects to {\it all} orders in 
BFKL is clearly necessary given the large NLL corrections. 
%
Unfortunately, it is not known how to perform such a resummation
exactly, indeed it is not clear if such a process is a feasible
goal.
%

%
The consistency constraint
provides a route by which we can circumvent the need for
a full resummation, and instead implement the rather simpler
all-orders resummation of the {\it dominant} subleading corrections.
It is then possible to make 
phenomenological predictions for forward jets and $\pi^{0}$'s.
%

%
Alternatives to this approach exist, based on supplementing the
`traditional' direct photon contribution with a situation
where the photon itself exhibits partonic structure and is 
said to be resolved.
%
The different formalisms, although at first glance appear quite distinct, 
share some of their conceptual ideas, and it is 
an informative exercise to contrast them.
%
One might think of the modified BFKL approach, with
gluon evolution extending {\it downwards} from the
virtual photon-gluon fusion, as representing in some sense the
unintegrated gluon content of the virtual photon.
%

%
For the resolved photon approach, 
the parton showering included at LO level is thought to
approximate the explicit higher order effects in the full NLO calculation. 
%
In both cases significant enhancement of the cross-section arises from
additional gluon emission from virtual photon corrections.
%

%
The LO resolved photon formalism can be 
thought of as mimicking, in a limited way, the complete $k_{T}$
disordering of the BFKL approach, to the extent that the two
approaches are indistinguishable at HERA.
%

%
Finally it is worth emphasising here the chief conceptual distinctions.
%
The resolved photon analysis unilaterally 
divides the underlying mechanism  according
to the relative sizes of the scales ($Q^{2}$ and  $k_{T}^{2}$) involved,
whereas the modified BFKL approach treats in a unified way 
all of the possible kinematical configurations.
%
In the BFKL approach, the $Q^{2}$ dependence of 
the parton `content' of the virtual photon
is dynamically
generated through a perturbative calculation of the quark box, 
rather than paramaterised through the virtual photon
parton distribution functions.
%
%
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%\end{document}
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