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\title*{Forward Jet and Hadron Production \protect\newline 
at Low $\mathbf{x}$ and $\mathbf{Q^2}$}
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\toctitle{Forward Jet and Hadron Production 
at Low $\mathbf{x}$ and $\mathbf{Q^2}$}
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\titlerunning{Forward Jet and Hadron Production}
% allows abbreviation of title, if the full title is too long
% to fit in the running head
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\author{Timothy Greenshaw}
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\authorrunning{T. Greenshaw}
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\institute{Oliver Lodge Laboratory, Liverpool University, 
           Liverpool L69 7ZE, ENGLAND}

\maketitle              % typesets the title of the contribution

\begin{abstract}
Recent measurements of forward jet and single forward hadron
production at HERA are described. 
An introduction is given to some of the 
consequences of various QCD approximation schemes used in the 
calculation of forward jet and particle production cross sections. 
\end{abstract}

\section{Introduction}
%
\subsection{HERA measurements and QCD}
%
The H1 and ZEUS experiments at the HERA electron-proton 
($ep$) collider have
made impressive progress since HERA delivered first luminosity
in 1992. Measurements of many aspects of $ep$ interactions have been 
made with high precision over a large 
kinematic range. Here, we 
concentrate on measurements of jet and particle 
production in the forward region\footnote{The HERA 
co-ordinate system has its origin at the nominal 
interaction point, its $z$ axis in the proton direction and its
$y$ axis vertical; the $x$ axis completes the right-handed orthogonal 
co-ordinate system. Polar angles are denoted by $\theta$ and azimuthal
angles by $\phi$. The term ``forward'' 
refers to the region for which $\theta < 90^\circ$.}
and the description of these measurements
provided by QCD. We shall work in the 
deep-inelastic scattering (DIS) regime, the 
regime in which both the negative squared four-momentum transferred 
to the proton from the electron $Q^2 = -q^2 \gg m_p^2$
and the final state hadronic mass squared 
$W^2 \gg m_p^2$,
where $m_p$ is the mass of the proton. 
Figure~\ref{fig:disfig01} illustrates the definition of
some of these variables.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.3\textwidth]{greenshaw.1.eps}
\end{center}
\caption[]{A generic Feynman diagram 
for deep-inelastic $ep$ scattering, illustrating the 
definition of some of the commonly used variables.}
\label{fig:disfig01}
\end{figure}
For the measurements discussed here $Q^2 \ll m_Z^2$,
where $m_Z$ is the $Z^0$ mass, so the interactions 
can be considered to be mediated by virtual photons. 
In addition $W^2 \gg Q^2$, so the 
Bjorken scaling variable $x = Q^2/(Q^2+W^2) \ll 1$. 
Recall that, in the frame in which the proton has infinite momentum, 
$x$ is the fraction of the proton's momentum carried
by the quark struck in the deep-inelastic interaction.

The HERA data are, on the whole, described extremely well
by QCD. 
For example, the dijet cross section measurements presented 
in~\cite{Joensson}
are beautifully reproduced by NLO QCD calculations. 

\subsection{First forward jet measurements}
% 
If we now move to consider explicitly the forward 
region, the above no longer holds; NLO QCD~\cite{Mirkes}
fails to describe the first 
HERA forward jet cross section measurements~\cite{h1fjet1},
as shown in 
figure~\ref{fig:h1fjet1}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.7\textwidth]{greenshaw.2.eps}
\end{center}
\caption[]{The forward jet cross section in the 
kinematic regon described  the text.
The error bars 
illustrate the quadratic sum of the statistical and 
the systematic errors. The curves show NLO QCD calculations
with an error band corresponding to a 
scale variation of two orders of magnitude, LO QCD calculations
and the results of a BFKL calculation.} 
\label{fig:h1fjet1}
\end{figure}
What is happening here? 

The key to understanding the problem lies 
in the definition of the kinematic region in which the 
forward jet measurement was made; it was required that 
the squared transverse momentum of the jet
be similar to $Q^2$, 
$0.5 < p_{T{\rm jet}}^2/Q^2 < 4$, and that 
the jets have a large longitudinal momentum in the 
proton direction. This 
latter condition arises from the requirements that the 
jets be found in the forward region,
$6^\circ < \theta_{\rm jet} < 20^\circ$, 
and that the ratio $x_{\rm jet} = E_{\rm jet}/E_p > 0.025$,
Here, $E_{\rm jet}$ and $E_p$ are the jet 
and proton energies, respectively.
Hence,
the parton emitted
from the proton which caused the production of the 
forward jet
must have carried a large proportion of the 
proton's momentum, and yet the quark 
struck by the virtual photon carried
only a very small proportion of
the proton's momentum, as the Bjorken $x$ range considered 
was $2 \times 10^{-4} < x < 2 \times 10^{-3}$. 
The diagram in 
figure~\ref{fig:disfig01} may well be misleading. 
The phase space for the parton emerging from the proton to
emit further partons
is particularly large in this case,
so the situation could be more like that illustrated in 
figure~\ref{fig:disfig02}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.3.eps}
\end{center}
\caption[]{A Feynman diagram illustrating low $x$ DIS
with the production of a forward jet carrying a large proportion 
of the proton's momentum, containing an energetic $\pi^0$.} 
\label{fig:disfig02}
\end{figure}
Calculating to order $\alpha_s^2$
allows us a maximum of $3$ partons, 
as shown in figure~\ref{fig:disfig02}, 
and such calculations are unlikely to be 
able to describe situations in which 
larger numbers of partons  
play a significant role.
Is it possible for us to include the effects 
of these? Here, we use the Uncertainty 
Principle to try and 
construct an approximate picture of how we 
might expect these partons to behave,
before going on compare the results
of QCD calculations, made using 
various approximation schemes, 
to the HERA data. 
The calculations are discussed in
more detail in other contributions to these 
proceedings~\cite{theorists}.

Consider a fast parton, 1, which emits a further 
parton, 2, as illustrated in 
figure~\ref{fig:disfig03}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.4\textwidth]{greenshaw.4.eps}
\end{center}
\caption[]{A parton 1 emitting a parton 2, with four-momenta
as illustrated.}
\label{fig:disfig03}
\end{figure}
The initially massless system
has now acquired a mass given by:
\begin{eqnarray*}
M^2  =  (p+k)^2 =  2(PK-PK \cos \alpha) \approx PK \alpha^2. 
\end{eqnarray*}
The magnitude of the resulting energy change, which 
determines the length of time the virtual system 
can exist before 2 is re-absorbed by 1, is
$\Delta E = M$ in the rest frame of the system. Including
the time dilation factor $\gamma \approx P/M $ gives 
the time for which this state can exist in the laboratory:
\begin{eqnarray*}
\Delta t & = & \frac{1}{\Delta E} =  \frac{1}{M} \frac{P}{M} \\
         & = & \frac{P}{PK \alpha^2}  =  \frac{1}{K \alpha^2}.
\end{eqnarray*}
Using the fact that $\alpha \approx k_T/K$ we have 
$\Delta t \approx K/k_T^2$. Such emissions 
and re-absorptions will take 
place continuously in a proton prior to a deep-inelastic
interaction. The interaction will ``free'' all the partons that 
exist at the time it takes place. 
We see that there
will be some which can have existed for a long time 
before the interaction, those with large 
$\Delta t \approx K/k_T^2 \approx k_L/k_T^2$,
and others that can only have existed for a very short time 
before the interaction, those with small $\Delta t \approx k/k_T^2$; 
it will appear that there is shower of 
time ordered partons $1, 2, 3\dots$ such that 
$K_1/k_{T1}^2 > K_2/k_{T2}^2 > K_3/k_{T3}^2 \dots$


\subsection{The DGLAP, BFKL and CCFM equations}
%
The above argument suggests that we can think of the 
initial state partons as forming a cascade in 
which either the $k_T$ increases strongly as we move towards
the photon-quark vertex, with the longitudinal 
momentum relatively constant, or the longitudinal 
momentum decreases and the $k_T$ remains approximately 
constant. A further alternative 
is to consider there to be an ordering of the 
emission angles\footnote{For completeness sake, note that the 
emissions in the final state occur in order of decreasing
angle.},
as illustrated in 
figure~\ref{fig:disfig04}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.6\textwidth]{greenshaw.5.eps}
\end{center}
\caption[]{Angular ordering in initial and final 
state parton cascades.}
\label{fig:disfig04}
\end{figure}
These three possibilities 
correspond roughly 
to three approximation schemes within QCD which 
allow the inclusion of the effects of certain terms 
in the perturbative expansion of DIS processes to all 
orders. The case in which the cascade is 
considered as being composed of partons strongly ordered
in $k_T$, $k_{T1}^2 \ll k_{T2} \ll k_{T3}^2 \dots $,
corresponds to using the Dokshitzer, Gribov, Lipatov, 
Altarelli and Parisi (DGLAP) equations~\cite{DGLAP2}, 
which sum to all orders, $n$,  
the terms $\sim \log^n Q^2$. How this is done
is discussed in~\cite{qandl}.
The case in which the cascade is considered to be 
composed of partons with decreasing longitudinal 
momenta, $x_1 \gg x_2 \gg x_3 \dots $,
corresponds to using the Balitski, 
Fadin, Kadeev and Lipatov
equations~\cite{BFKL0}, which are discussed 
in~\cite{Jeff}. 
These sum to all orders the terms $\sim \log^n(1/x)$. 
Thinking of the partons as being 
ordered in emission angle, $\alpha_1 < \alpha_2 < \alpha_3 \dots$,
corresponds to using the 
Ciafaloni, Catani, Fiorani and Marchesini
(CCFM) equations~\cite{CCFM}.

The success of the DGLAP equations is well known. 
For example, they 
give an excellent description of HERA $F_2(x,Q^2)$ measurements
despite the range of more than four 
orders of magnitude in both $Q^2$ and $x$ spanned by the HERA data,
as is discussed in~\cite{F2talk}.
Can they describe the forward jet measurements? Looking at the 
kinematic region chosen, we must conclude that it is unlikely.
The DGLAP equations
describe data in which the parton cascade can be thought of 
as being ordered in $k_T$, the forward jet measurement, in which
it is required that 
$k_{T{\rm jet}}^2 \sim Q^2$ precludes the development of such 
cascades as the $k_T$ at both ends of the cascade is similar. 
Figures~\ref{fig:zfjetres}, \ref{fig:h1fjet2}a) 
and~\ref{fig:h1fjet2}c) confirm our suspicions.

Let us turn instead to the BFKL equation. This should 
give accurate results where the cascade can be thought of
as consisting of partons with strongly decreasing longitudinal
momenta. This is exactly the situation here. The requirement that
$x_{\rm jet}$ be large and $x$ small ensures that the longitudinal
momentum at the start of the cascade is large and that at the end small.
Indeed, this 
is precisely the reason why these 
conditions were chosen for the measurement~\cite{muellerfjet}, the hope
being that a kinematic region could be found in which there was 
clear evidence for the applicability of the BFKL equation.
The evidence from the early forward jet measurements,
as shown in figure~\ref{fig:h1fjet1},
is that the BFKL solution~\cite{Bartels} does a good job.

\subsection{Monte Carlo Programs}
%
Monte Carlo (MC) programs provide an
alternative means of comparing QCD calculations with 
measurements. The programs of interest here are the following:
\begin{itemize}
\item LEPTO~\cite{Lepto1} matches LO QCD matrix elements 
with DGLAP inspired parton showers;
the showers display $k_T$ ordering.
\item RAPGAP~\cite{Rapgap}, originally developed to simulate 
diffractive DIS events, is very similar to LEPTO; 
the facility to simulate the effects of allowing the 
virtual photon to develop hadronic structure is of interest here, 
as is discussed in the following. 
\item HERWIG~\cite{Herwig} produces DGLAP-like showers using
angular ordering. The showers are corrected 
to incorporate the correct LO QCD results
for the photon-parton interaction.
\item ARIADNE~\cite{Ariadne} 
is an implementation of the Colour Dipole 
Model~\cite{CDM} (CDM),
which treats QCD radiation as arising from the colour dipole 
generated in the deep-inelastic
scattering process; the resulting
parton cascades do not display $k_T$ ordering and in this 
sense the results of the CDM are similar to those of 
the BFKL equation.
\item The Linked Dipole Chain~\cite{LDC}
(LDC) model relies on a re-formulation 
of the CCFM equations and should produce DGLAP-like 
and BFKL-like showers in the appropriate high $Q^2$ and low $x$ 
limits, or, more accurately in the situations in which there 
is a large $k_T$ or $x$ ``lever arm'', respectively. 
It has been shown that, at the level of the leading log (LL) 
terms $\sim \log^n(1/x)$,
the predictions of the CCFM 
equations for
the hadronic final state in DIS at small $x$ 
are identical to those of the BFKL equation~\cite{Salamfs}.

\end{itemize}
One advantage of using MCs to compare theoretical predictions
with data is that the MCs incorporate hadronisation models.
This allows direct comparison of the predictions with data;
parton level jet cross sections are likely to differ by 
$15$ to $20\%$ from the results expected following
hadronisation.
All the above models, barring HERWIG, use the JETSET~\cite{Jetset}
implementation of colour string fragmentation 
to perform hadronisation,
following their various parton cascade schemes. The HERWIG model
forms low mass colourless ``pre-clusters'' following its cascade 
procedure. These are then allowed to decay into hadrons.

\section{New forward jet measurements and BFKL calculations}
%
The H1 and ZEUS Collaborations have both produced 
new forward jet measurements since those discussed above.
Both groups used a ``cone'' algorithm
to identify the jets in the laboratory frame of reference.
These algorithms 
maximise the transverse energy within a 
cone of radius
$\sqrt{\Delta \eta^2 + \Delta \phi^2} = 1$,
where $\Delta \eta$ is the difference in 
pseudorapidity, $\eta = -\ln(\tan(\theta/2))$, between 
the cone's axis and the calorimeter energy deposits under 
consideration, and $\Delta \phi $ is the 
difference in azimuthal angle.
The algorithm used by ZEUS is described in~\cite{Snowmass}.
The ZEUS measurement requires in addition
that $x_{\rm jet}^\prime = p_{z{\rm jet}}/E_p > 0.036$, 
$0.5 < E_{T{\rm jet}}^2/Q^2 < 2$, $\eta_{\rm jet} < 2.6$
({\it i.e.}\ $\theta_{\rm jet} > 8.5^\circ$) and that 
$E_{T{\rm jet}} > 5\,$GeV. 
Further, to explicitly 
remove events in which the forward jet is caused by
the quark scattered in the deep-inelastic interaction, 
ZEUS require that the longitudinal momentum 
of the jet in the Breit frame of reference\footnote{The 
Breit frame is the frame of reference in which 
the four-momenta of the virtual photon and the proton are
given by $(E,p_T,p_L) = (0,0,-Q)$ and $(Q/2x, 0, Q/2x)$, 
respectively.}
$p_{z{\rm jet}}^{\rm Breit} > 0$.
The Bjorken $x$ region studied is
$4.5 \times 10^{-4} < x < 4.5 \times 10^{-2}$. 
Despite the experimental difficulties involved in making this
measurement, particularly the problems of measuring energy 
deposits close to the outgoing proton remnant 
and the forward beamhole in the calorimeter,
the distribution of transverse energy about the 
forward jet axis following
the above selection does reveal jet structure, as is 
illustrated in 
figure~\ref{fig:zfjetprof}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.6.eps}
\end{center}
\caption[]{The transverse energy flow around the forward jet 
axis averaged over all forward jets for various $\eta_{\rm jet}$
regions; grey indicates the $E_T$ 
attributed to the jet, black the $E_T$ around the 
calorimeter beam hole and white other $E_T$ deposits.}
\label{fig:zfjetprof}
\end{figure}

Figure~\ref{fig:zfjetres} shows the forward jet 
cross section~\cite{zfjet} obtained by the ZEUS 
Collaboration\footnote{In this and 
subsequent figures in which
experimental data are shown, the inner error bars correspond to 
the statistical errors, the outer error bars to the quadratic 
sum of the statistical and systematic errors.}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.6\textwidth]{greenshaw.7.eps}
\end{center}
\caption[]{The forward jet cross section as a function of $x$ in 
the kinematic region described in the text with the additional 
requiremnts that the scattered electron energy $> 10\,$GeV and 
that $y > 0.1$. The curves show the predictions of various 
MC programs.}
\label{fig:zfjetres}
\end{figure}
Also shown in
figure~\ref{fig:zfjetres}
are the predictions of several MC programs.
Neither the DGLAP based MCs, LEPTO and HERWIG,  
nor the LDC MC are able to describe the data. 
The ``pseudo-BFKL'' MC prediction,
that of ARIADNE, is the best of those shown.

The new H1 forward jet measurements~\cite{h1fjet2}
are shown in 
figure~\ref{fig:h1fjet2}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.8.eps}
\end{center}
\caption[]{The forward jet cross section as a function of $x$
for $p_{T{\rm jet}} > 3.5\,$GeV 
and $p_{T{\rm jet}} > 5\,$GeV; the curves in a) and c) are 
MC predictions, in b) and d) the full lines are analytic LO
BFKL calculations without inclusion of the effects of 
the jet algorihm, the dashed lines are
NLO QCD calculations made using DISENT~\cite{Disent} and 
include the effects of the jet algorithm at the parton level.}
\label{fig:h1fjet2}
\end{figure}
The cone algorithm used here is described in~\cite{h1cone}.
The requirements made on the forward jets were
that $x_{\rm jet} > 0.035$, 
$0.5 < p_{T{\rm jet}}^2/Q^2 < 2$ and 
$7^\circ < \theta_{\rm jet} < 20^\circ$.
Results are shown in two regions of $p_{T{\rm jet}}$, 
namely $p_{T{\rm jet}} > 3.5\,$GeV and $p_{T{\rm jet}} > 5\,$GeV.
As in the above case, 
all MC predictions fall below the data, including those
of the LDC model, with ARIADNE providing the best description of
the data. NLO QCD calculations are again seen to fall significantly 
below the data, with a LO BFKL calculation 
tending to be too high. The discrepancy between 
the latter calculation and the LDC MC is intriguing given 
the formal equivalence of the CCFM and BFKL equations
at the LL level. Are there significant sub-leading terms?
Have we still to reach ``low'' $x$? 

As mentioned above, and described in more detail in~\cite{Outhwaite},
the BFKL results used to this point rely on a summation of the 
LL terms.
Recently, a calculation of 
the effects of the next-to-leading log (NLL)
terms, $\sim \log^{n-1}(1/x)$, has become available~\cite{NLLBFKL}
which has showed that they can be 
as large as the effects of the LL terms.
The resulting debate on whether and how, despite this,
BFKL-based phenomenology may be possible
has yet to be 
resolved~\cite{Ross,Salam,Forshaw1}\footnote{For discussion of the
problems associated with the running of $\alpha_S$ in the 
BFKL formalism see, for example,~\cite{Armesto}}.
Here, we consider modified 
LL BFKL
calculations, incorporating a ``consistency constraint''
which effectively includes
part of the NLL terms~\cite{fpicalc}. 
The forward jet cross section 
determined using this 
approach is  
shown in 
figure~\ref{fig:h1zfjet}, for various choices of scale, 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.9.eps}
\end{center}
\caption[]{The forward jet cross section 
measurements of ZEUS and H1 
as a function of $x$ compared with 
BFKL calculations including a ``consistency constraint'',
for various choices of scale.}
\label{fig:h1zfjet}
\end{figure}
with the forward jet measurements of the
ZEUS and H1 Collaborations. The modified LL BFKL 
calculation is significantly better than the unadulterated
LL BFKL result shown in 
figure~\ref{fig:h1fjet2}.

The H1 forward dijet rates, measured in the kinematic range 
given above with the requirement that both jets satisfy
$p_{T{\rm jet}} > 3.5\,$GeV, is 
$\sigma_{\rm 2 jet} = 6.0 \pm 0.8\,({\rm stat.}) 
\pm 3.2\,({\rm syst.})\,$pb, which compares favourably with the
prediction from the modified BFKL calculations of
$\sigma_{\rm 2 jet} = 2.7 \dots 5.2\,$pb, where the range 
results from varying the choice of scale.


\section{Forward particle production}
%
A problem that arises when trying to compare forward jet 
measurements with BFKL based calculations is the uncertainty
of $15$ to $20\%$ introduced by the effect of hadronisation. 
This can be circumvented by comparing predictions
for single forward $\pi^0$ production with measurement.
Here, measured $\pi$ fragmentation functions are folded with
the above modified BFKL calculations~\cite{fpicalc}, 
as illustrated in figure~\ref{fig:disfig02}.
The success of this approach is 
apparent in 
figure~\ref{fig:fpi}, 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.4\textwidth]{greenshaw.10.eps}
\end{center}
\caption[]{The H1 forward $\pi^0$ cross section  
as a function of $x$ compared with 
BFKL calculations including a ``consistency constraint'', 
and with two DGLAP MCs, one which allows for the 
effects of virtual photon structure (RAPGAP) and one
which does not (LEPTO).}
\label{fig:fpi}
\end{figure}
in which the calculations are compared with the
H1 forward $\pi^0$ measurements~\cite{h1fpi0}. 

\section{Forward jets and virtual photon structure}
%
Intriguingly, in figure~\ref{fig:fpi} it is not only 
the BFKL calculation that provides a reasonable description 
of the data, the results of the RAPGAP MC are also respectable. 
RAPGAP is a DGLAP based MC, but allows for the inclusion of the
effects of virtual photon structure. Such effects may become 
significant in the regime in which the $k_T^2$ at the photon 
end of the parton cascade exceeds the photon's virtuality
$Q^2$. Recall that the transverse dimensions of the 
photon are $\sim 1/ \sqrt{Q^2}$ and that the transverse wavelength of 
the partons $\lambda_T \sim 1/k_T$. If 
$\lambda_T < 1/ \sqrt{Q^2}$
the parton is able to resolve the hadronic structure of the virtual 
photon which arises through the $\gamma \rightarrow q \bar{q}$ 
splitting. Indeed, such events have been used to determine
the effective parton density in the virtual photon~\cite{vphot}. 
Is this relevant to forward 
particle and jet production? The answer is a resounding perhaps,
as can be seen by the following argument. The requirement 
that $0.5 < p_{T{\rm jet}}^2/Q^2 <  2$ allows events 
in which $k_T > Q^2$ into the sample studied, both explicitly
at the high end of the $p_{T{\rm jet}}^2/Q^2$ range
and implicitly
in that the correlation between $p_{T{\rm jet}}$ and $k_T$ is far from
$100\%$. This implies we are in a regime
in which virtual photon structure may be significant. 
In addition, the 
fact that the parton entering the hard 
scattering process from the 
photon side carries 
only a fraction $x_\gamma = p_{\rm part} /p_\gamma$
of the momentum $p_\gamma$ of the photon ensures that 
the resulting jets will tend to be more forward than in the 
situation in which $x_\gamma = 1$, as is illustrated in 
figure~\ref{fig:disfig05}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.11.eps}
\end{center}
\caption[]{A DIS event in the HERA laboratory frame 
in which the structure of the virtual photon is resolved
leading to the production of forward jets and particles, 
{\it i.e.}\ jets and particles adjacent to the proton remnant.}
\label{fig:disfig05}
\end{figure}
In terms of parton cascades, if we choose to 
explain the development of both the 
virtual photon structure ``within''
the photon and the structure within the proton
using the DGLAP prescription, we will have cascades of
partons with increasing $k_T$ developing from both the 
photon and proton ends of the interaction, as illustrated 
in figure~\ref{fig:disfig06}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.4\textwidth]{greenshaw.12.eps}
\end{center}
\caption[]{An illustration of a  DIS event
in which the 
structure of both the virtual photon 
and the proton 
develops via a parton cascade described by the DGLAP
prescription, with $k_T$ ordering as indicated by the 
increasing width of the relevant triangles.}
\label{fig:disfig06}
\end{figure}
Note that this picture 
bears some resemblance to the parton cascade 
produced by the ARIADNE MC, in which there is no $k_T$ ordering. 
This is a possible explanation for the success of ARIADNE in 
describing the forward jet and forward particle measurements. 

The MC predictions including the effects of
virtual photon structure not only describe 
forward particle production but also provide a 
reasonable description of the forward jet results, as is shown in
figure~\ref{fig:zfjetvp}. 
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.13.eps}
\end{center}
\caption[]{The ZEUS forward jet measurements as a function of
$x$ shown with predictions from MCs with, RAPGAP RES+DIR, and 
without, LEPTO and RAPGAP DIR, the effects of virtual photon 
structure.}
\label{fig:zfjetvp}
\end{figure}
In this figure comparison is made firstly
with LEPTO and RAPGAP allowing 
only direct photons to contribute, that is those in which the 
structure of the photon is not resolved. These models are very 
similar, as is borne out by the results shown, and are not able 
to describe the data. Allowing the inclusion of resolved photon 
processes, those in which the partonic structure of the 
virtual photon is resolved, significantly improves the quality 
of the description.

This approach, allowing the inclusion of virtual photon structure,
can also be taken in NLO QCD calculations, as is discussed 
in~\cite{Kramer,Poetter}. The results of such calculations are shown in
figure~\ref{fig:h1fjetvp}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.5\textwidth]{greenshaw.14.eps}
\includegraphics[width=.5\textwidth]{greenshaw.15.eps}
\end{center}
\caption[]{The H1 forward jet measurements as a function of
$x$ shown with the results of NLO QCD calculations 
(a) without and (b) with
the effects of virtual photon structure for various 
choices of scale.} 
\label{fig:h1fjetvp}
\end{figure}
together with the H1 forward jet measurements. Again, the agreement
is found to be good. 

\section{Summary}
%
HERA has provided new impetus to both theoretical 
and experimental investigations of low $x$ QCD and recent 
progress has been impressive.
We are now able to describe reasonably the 
measured rates of forward jets~\cite{h1fjet2,zfjet} using 
either NLO QCD calculations incorporating virtual 
photon structure~\cite{Kramer}, 
modified BFKL calculations~\cite{fpicalc} or MCs based on LO QCD 
allowing either directly, as in the case of RAPGAP~\cite{Rapgap}, 
or indirectly,
as in the case of ARIADNE~\cite{Ariadne},
for virtual photon structure. The relative success of 
ARIADNE may alternatively be considered to be 
associated with the BFKL-like nature of the parton showers
it produces.
It is likely that NNLO QCD calculations will provide 
as good a description of data as the above
without having to invoke virtual
photon structure~\cite{Poetter}.
Forward $\pi^0$ production is also reasonably described 
within the modified BFKL framework.
New low $x$ MC programs are being developed, 
such as the LDC MC~\cite{LDC},
which is based on a reformulation of the CCFM~\cite{CCFM} equations
and as such should reproduce BFKL predictions for
the hadronic final state in DIS at small $x$~\cite{Salamfs}. 
Many problems remain to be solved, however. 
The recent calculation of the NLL corrections to the 
BFKL equation~\cite{NLLBFKL}
raises difficulties associated with both the running of $\alpha_S$ 
and the unexpectedly large size of the corrections. 
The apparent disagreement between LDC MC predictions and LL BFKL
calculations is a puzzle. Experimentally, measurements in the
forward region remain a challenge. Improving the precision 
of current measurements and devising new ways of testing low $x$ 
QCD will tax the ingenuity of physicists for many years to come.


\section*{Acknowledgments}
%
The author would like to thank the organisers of Ringberg 99 for 
the invitation to spend a week in delightful surroundings in such 
stimulating company. Thanks also to Dr.\ S.J.\ Maxfield 
for his careful reading of this manuscript.


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\end{thebibliography}


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