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\pagestyle{empty}
\begin{titlepage}

\noindent
% {\bf H-UM \& JS: version of \today} \\[.3em] 
Submitted to the 30th International Conference on 
High-Energy Physics ICHEP2000, \\ 
Osaka, Japan, July 2000


\vspace*{3cm}

\begin{center}
  \Large
  {\bf 
    A Measurement of Single-Inclusive\\
    Jet Cross Sections in the Central\\
    and Forward Region at low x in DIS at HERA\\}
  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent
We present a measurement of single-inclusive jet cross sections as a
function of the transverse energy of the jet,
$E_T$, the ratio of the transverse energy squared over the photon virtuality, $E_T^2/Q^2$, and the 
Bjorken scaling variable, $x_{Bj}$, for jets with a transverse energy
$E_T >$~5~GeV in the Breit reference frame and a pseudorapidity in 
the laboratory frame $\eta_{lab}$ between -1.0 and 2.8.
The analysis is performed on data corresponding to an integrated luminosity
of about 21~$\mathrm{pb^{-1}}$ taken with the H1 detector in 1996/97 in the region 5~$< Q^2 <$~100~$\mathrm{GeV^2}$
and $\mathrm{10^{-4}} \lsim x_{Bj} \lsim \mathrm{5\cdot10^{-3}}$.
The distributions are measured in different ranges of $Q^2$ and $\eta_{lab}$.
The data are compared to fixed order QCD calculations using $E_T^2$ or $Q^2$ as renormalization scale $\mu_R^2$.
None of the two choices is able to describe the data over the full phase space.
Especially for $\mu_R^2 = E_T^2$
large discrepancies are observed in the forward direction for low $Q^2 \lsim$~20~$\mathrm{GeV^2}$, 
low $E_T \lsim$~20~GeV and low $x_{Bj} \lsim$~$\mathrm{10^{-3}}$.
The observed discrepancies leave room for 
additional cross section contributions from higher orders, resolved photon processes or modified parton dynamics.
\end{abstract}


\vfill
\begin{flushleft}
  {\bf Abstract: 999} \\
  {\bf Parallel session: 3} \\
  {\bf Plenary talk: 11 } 
\end{flushleft}

\end{titlepage}

\pagestyle{plain}

\section{Introduction}
\noindent
The HERA machine, colliding electrons or positrons on protons at a center-of-mass energy of about 300~GeV, offers 
an ideal testing ground for the theory of strong interactions in deep-inelastic 
scattering (DIS). The large available phase space allows for the production of highly energetic
hadrons which emerge from short-distance processes and to probe in a previously unexplored kinematic domain
the detailed characteristics of the hadronic final state in which clear jet structures can be observed.
These jets - sprays of almost collinear hadrons - form a link between the 
fundamental objects of the theory of strong interactions - quarks and gluons - and 
the hadrons observed in the final state.

In the Breit reference frame, where the parton and the 
photon collide head-on, the transverse energy $E_T$ of jets in the final state is exclusively produced by 
the underlying hard QCD process and reflects its hardness\footnote{The Breit frame is defined by $\vec{q}+2x_{Bj}\vec{P} = 0$, where $\vec{q}$ and $\vec{P}$ are the momenta
of the exchanged photon and the incoming proton.}. $E_T$ therefore is a natural scale in the 
process. However, also the photon virtuality $Q^2$
can be a hard scale, as for example in inclusive measurements of the
total DIS cross section. 
It is therefore interesting to study the 
interplay of these two QCD scales.
Aside from $E_T$ also the pseudorapidity $\eta$ and the Bjorken scaling variable $x_{Bj}$ characterize 
the short distance scattering process.

Jet cross sections can be calculated using the
$\mathcal{O}(\alpha^2_s)$ matrix elements together with parton density functions based on the DGLAP 
(Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) parton evolution formalism\cite{dglap}.
This formalism works remarkably well for totally inclusive measurements
like that of the proton structure function $F_2$ at HERA\cite{f2h1,f2zeus}. 
However, the assumptions made to derive the DGLAP equations should loose
their validity in the low-$x_{Bj}$ regime at HERA. In addition, in the forward region 
towards the proton remnant the production of high transverse energy jets should be suppressed
according to the DGLAP formalism.
Therefore a 'forward jet' regime was suggested by several authors\cite{muller,kwiec,bartels}
as a possible way to observe the onset of an alternative
parton evolution scheme, the BFKL (Balitsky-Fadin-Kuraev-Lipatov) formalism\cite{bfkl}. 
Recent results on forward jets 
from the HERA experiments\cite{h1forward,zeusforward} show discrepancies between 
data and LO QCD Monte Carlo models and $\mathcal{O}(\alpha_S^2)$ QCD calculations.
However, it was not studied whether the models and calculations are able to describe the data in a wider, 
more general phase space.

There are other possible explanations for the discrepancies observed in\cite{h1forward,zeusforward}.
First, the calculations might have
significant corrections from higher orders in the perturbation series, e.g. $\mathcal{O}(\alpha_S^3)$, which 
have not yet been calculated.
Second, the effects of a resolved hadronic substructure of the 
virtual photon can play an important role for the cross section if $Q^2 \lsim E^2_T$, i.e. if a 
parton emitted from the proton can act as a probe on the photon.

In this analysis we measure jet cross sections 
in a large phase space (5~$< Q^2 <$~100~$\mathrm{GeV^2}$, $\mathrm{10^{-4}}$~$\lsim x_{Bj} \lsim$~$\mathrm{5 \cdot 10^{-3}}$, 
-1.0~$< \eta_{lab} <$~2.8, with $\eta_{lab}$ the pseudorapidity in the laboratory reference frame) 
as a function of $E_T$, $E_T^2/Q^2$ and $x_{Bj}$.
The jet measurements are done single-inclusively, i.e. by counting single 
jets that pass certain selection criteria.
This seems to be the most basic way to 
test the predictions of perturbative QCD for jet production.
After measurements in the wide phase space we use the highly
increased amount of available data as compared to\cite{h1forward} to perform a detailed analysis in
specific phase space regions. In particular we study the forward jet
regime and compare the data to 
NLO QCD calculations.

This paper is organized as follows: Section 2 gives an overview over the theoretical framework of
jet cross sections and parton evolution schemes and introduces the notion of forward jets. Section 3 
describes the QCD fixed order calculations and QCD Monte 
Carlo generators used in this analysis. Section 
4 gives an overview of the components of the H1 detector which are particularly important
for this analysis. Section 
5 describes the data selection, section 6 presents the results.

\section{Theoretical Background}
\noindent

For the analytic derivation of jet cross sections matrix elements of the hard
scattering and input parton density functions have to be considered.
The hard scattering process can be calculated using a 
perturbative expansion in the strong coupling constant $\alpha_S$. The Feynman 
graphs of fig.~\ref{feyn} show different contributions to the hard 
scattering. Fig.~\ref{feyn}a represents an example of a process of
leading order (LO) or $\mathcal{O}(\alpha_S)$. In fig.~\ref{feyn}b,
an additional gluon is radiated, corresponding to
$\mathcal{O}(\alpha^2_S)$ or next-to-leading order (NLO). Even more partons 
can be emitted from either the original parton emitted by the proton or 
from any other parton of the final state. Such higher orders (NNLO, NNNLO,...)
cannot yet be calculated exactly.

The input parton density functions are obtained from fits to
a variety of data sets according to the DGLAP evolution formalism\cite{dglap}.
The DGLAP formalism is based on the assumption of strong $k_T$ ordering of
the partons emitted along a parton ladder (see fig.~\ref{ladder}):
$k^2_{Tn} \gg k^2_{Tn-1} \gg ... \gg k^2_{T1}$. There is only a soft ordering
of the fractional longitudinal parton momenta, $x_n < x_{n-1} < ... < x_1$.
The formalism corresponds to a resummation of terms of the form
$\alpha_S \cdot \log Q^2$.
An alternative evolution scheme is based on the BFKL equation\cite{bfkl}
which should be valid at small $x_{Bj}$ and which assumes a strong ordering
in the fractional longitudinal parton momenta, 
$x_n \ll x_{n-1} \ll ... \ll x_1$. The BFKL formalism leads to a random 
distribution of the transverse momenta $k_{Ti}$; it corresponds to a resummation of 
terms of the form $\alpha_S \cdot \log \frac{1}{x}$.

Both evolution equations are complementary approximations to 
a more complete calculation. In the low-$x$ regime the $\alpha_S \cdot \log \frac{1}{x}$ 
terms get large and the observation of a deviation from the
DGLAP formalism should be possible, in particular in the region as close
as possible to the proton remnant. Here the emission of partons with 
high $E_T$ and therefore also the jet cross sections are
suppressed in the DGLAP picture.
This 'forward jet region' is particularly suited to
observe BFKL effects\cite{muller,kwiec,bartels,h1forward,zeusforward}. 
In more detail this region is defined by large
jet pseudorapidities $\eta_{lab}$, large scaled 
jet energies $x_{Jet} \equiv E_{Jet}/E_p$ (with $E_p$ the proton energy)
and a scaled transverse jet energy $E^2_T/Q^2$ of the order of 1.
The first and the third condition suppress DGLAP contributions due to the strong $k_T$
ordering which does not allow high $E_T$ partons close to the proton remnant; for low $x_{Bj}$ the 
second requirement provides a large phase space for the emission of partons along the ladder.

For sufficiently small $Q^2$ or $Q^2 \lsim E^2_T$ the parton emitted by the
proton is able to probe the hadronic substructure of the photon. In this case
the photon can be considered as a source of partons and additional ('resolved') 
processes enhancing the cross section can occur, see fig.~\ref{feyn}c and \ref{feyn}d
for examples of LO and NLO resolved processes. It was argued
in~\cite{kp99} that the NLO direct and resolved diagrams can be taken
as an approximation of the $\mathcal{O}(\alpha_S^3)$ direct calculation.

\section{QCD Calculations and Models}

\subsection{$\mathcal{O}(\alpha_S^2)$ QCD Calculations}
\noindent
There are currently several computer programs which implement calculations of jet cross sections up to
$\mathcal{O}(\alpha^2_s)$ like
MEPJET\cite{mepjet}, DISENT\cite{disent}, DISASTER++\cite{disaster}
and JetViP\cite{jetvip}.
Comparisons of these programs can be found in~\cite{nlocompare}. In
this analysis we compare our data to the DISENT and JetViP programs.

The DISENT program\cite{disent} is based on the subtraction method
for the cancelation of divergencies which occur in the calculation of the hard scattering 
cross section when soft or collinear partons are emitted.
The program calculates direct photon processes only, i.e. 
processes where the photon is regarded as pointlike to the order $\mathcal{O}(\alpha_S^2)$.
In this analysis we have set the renormalization scale $\mu_R^2$ either to $Q^2$ or $E^2_T$
and the factorization scale $\mu_F^2$ to $Q^2$. The influence of the
factorization scale is usually small for the jet cross sections
presented here.

The JetViP program\cite{jetvip} takes, in addition to direct 
processes, also resolved photon processes into account.
The parton density functions of the photon evolved in the DGLAP formalism are used.
JetViP is based on the phase-space slicing method for the treatment of infrared and 
collinear divergencies. This method introduces an unphysical arbitrary parameter $y_{cut}$ to isolate
these infrared and collinear divergencies associated with regions 
where two partons cannot be resolved. Two partons are unresolvable if their invariant mass
$s_{ij} \equiv 2\cdot p_i \cdot p_j$ is smaller than $y_{cut}$.
The renormalization 
and factorization scales are always set to the same value which was chosen to be $Q^2+E^2_T$.
This choice assures that the hard scattering scale is always larger than the photon virtuality.

The matrix element calculation and the kinematics of resolved photon processes in JetViP  
are done in the limit of vanishing quark masses and for on-shell
partons, i.e. zero virtuality. This is consistent with the construction of the parton
density functions  which have been derived for on-shell partons.
However, for photon virtualities $Q^2 \gg$~1~$\mathrm{GeV^2}$ mass and
$k_T$ effects in the parton emission from the photon will become
important, but they are not included in the original JetViP
program\cite{jetvip}. It is therefore not clear whether the JetViP
calculations for the resolved processes will work well for larger
$Q^2$. One way to improve the situation for larger $Q^2$ studied
in~\cite{kp99} is to approximately include the mass effects by
combining virtual quarks with the massless matrix elements. This,
however, leads to a dependence of the cross sections on the unphysical
$y_{cut}$ parameter. We will study both versions (`on-shell scheme'
and `off-shell scheme') in the following.

Both the DISENT and the JetViP calculations where performed using the
CTEQ4M\cite{cteq} proton parton density functions. The strong
coupling constant is taken to be consistent with the one used in the
parton density function, where $\Lambda^{(5)}_{\overline{MS}}$~=~0.202~GeV
for five flavours.


\subsection{QCD Monte Carlo Models}

Simulations of the detailed properties of the hadronic final state are available 
in the form of Monte Carlo event generators.

The LEPTO 6.5\cite{lepto} and RAPGAP 2.06\cite{rapgap} Monte Carlos
use $\mathcal{O}(\alpha_S)$ matrix elements for the calculation of the hard 
subprocess and leave the simulation of higher orders to matched parton showers 
which take into account leading logarithms to all orders. The ARIADNE 4.08\cite{ariadne} Monte Carlo 
simulates the parton cascade
by a chain of independently radiating color dipoles according to the Colour Dipole Model (CDM)\cite{cdm}.
Unlike in LEPTO, the parton emissions in ARIADNE are not ordered in $k_T$\cite{cdmkt}. 
The $\mathcal{O}(\alpha_S)$ matrix elements for photon-gluon fusion events have to be added
to the CDM.

The hadronisation of the partonic system for all Monte Carlo generators was modelled 
using the Lund colour string model as implemented in JETSET\cite{jetset}. 

\section{The H1 Detector}
A detailed description of the H1 detector can be found elsewhere\cite{h1det}. The following section briefly 
describes the components of the detector relevant to this analysis: The SPACAL and LAr calorimeters 
and the tracking system.

The scattered lepton is measured with the backward optical scintillator calorimeter SPACAL\cite{spacal}.
The SPACAL covers the polar range $\mathrm{153}^\circ < \theta < \mathrm{177.8}^\circ$. 
It provides an electromagnetic energy resolution
of $\sigma(E)/E = 7.5\%/\sqrt{E/GeV} \oplus 2\%$ and has in addition a hadronic section resulting in
two interaction lengths in total.

The hadronic energy flow is mainly measured with the liquid argon (LAr) calorimeter\cite{lar} which
extends over the polar range $\mathrm{4}^\circ < \theta < \mathrm{154}^\circ$ with full azimuthal coverage.
It consists of an  electromagnetic section with lead absorbers and a hadronic section with steel absorbers.
Both sections are highly segmented in polar and azimuthal angle, having about 45 000 channels in total.
Test beam measurements of LAr calorimeter modules have shown an energy resolution of 
$\sigma(E)/E \approx 0.12/\sqrt{E/GeV} \oplus 0.01$ for electrons\cite{larem}
and of $\sigma(E)/E \approx 0.50/\sqrt{E/GeV} \oplus 0.02$ for charged pions\cite{larpio}. 
The hadronic energy measurement
is performed by applying  a weighting technique in order to account for the non-compensating
nature of the calorimeter.
The total depth of the LAr calorimeter varies between 4.5 and 8 interactions lengths in the angular range 
$\mathrm{4}^\circ < \theta < \mathrm{128}^\circ$. 

The calorimeters are surrounded by a superconducting solenoid providing a uniform magnetic field
of 1.15 T oriented parallel to the beam axis. Charged particles are measured in a central tracking device (CT)
which covers the polar range $\mathrm{20}^\circ < \theta < \mathrm{160}^\circ$\cite{track}. 
The CT consists of inner and outer 
cylindrical jet chambers, z-drift chambers and proportional chambers. The jet chambers are mounted around the beam pipe
and provide a maximum of 65 space points in the radial plane for tracks with sufficiently large transverse momentum.
A backward drift chamber (BDC) in front of the SPACAL improves the identification of 
the scattered lepton.

The luminosity is measured from elastic Bethe-Heitler events. The final state positron and photon are
detected in calorimeters situated at 33~m and 103~m from the interaction vertex in the positron beam direction.

\section{Data Selection}

The data presented in this paper have been taken in the years 1996/1997 and correspond to an integrated 
luminosity of about 21 $\mathrm{pb^{-1}}$. During this run period HERA collided 27.5~GeV positrons on
820~GeV protons.

\subsection{DIS Event Selection}
\noindent
The data were triggered by requiring an electromagnetic energy deposition of at least 6~GeV in the
backward calorimeter SPACAL, indicating a positron candidate, 
and some loose track requirements. 
The scattered positron is defined as the highest energetic electromagnetic
shower in the SPACAL  within a polar angle of $\mathrm{156^\circ} < \theta < \mathrm{176^\circ}$ 
and with a minimum energy of 10~GeV.
The analysis was performed in the phase space 5~$< Q^2 <$~100~$\mathrm{GeV^2}$ and 0.2~$< y <$~0.6. 
For $Q^2 \gsim$~60~$\mathrm{GeV^2}$ the cut $\theta > \mathrm{156^\circ}$ further limits the phase space.
These requirements lead to a range $\mathrm{10^{-4}}$~$\lsim x_{Bj} \lsim$~$\mathrm{5 \cdot 10^{-3}}$ 
in the Bjorken scaling variable $x_{Bj}$.
The kinematic quantities were reconstructed from the polar angle of the positron $\theta_e$ 
(measured with respect to the proton beam direction) and from 
the energy of the scattered positron $E'_e$ using $Q^2=4E_eE'_e\cos^2(\theta_e/2)$, 
$y=1-(E'_e/E_e)\sin^2(\theta_e/2)$ and $x_{Bj}=Q^2/(ys)$, were $E_e$ is the incident positron beam energy
and $s$ the square of the center-of-mass energy. In this phase space the 
energy of the scattered positron $E'_e$ is always larger than 10~GeV.

The hadronic final state is reconstructed from low momentum tracks and energy depositions in the
SPACAL and LAr calorimeters. Double counting of energy depositions between clusters and tracks is avoided by
neglecting the energy measured in the electromagnetic (hadronic) calorimeter in a cylinder with radius
25 (50)~cm around the axis given by the direction of a low momentum track. However, if the total energy 
in such a cylinder is larger than the track energy, only calorimeter information is considered\cite{f2h1}.
Using this method energy losses in the calorimeter due to dead material and noise suppression are compensated.

Further suppression of misidentified photoproduction events, initial state QED 
radiation background and badly measured events was achieved by a cut on the quantity 
$\Sigma_i (E_i - p_{z,i})$ where the sum runs over all final state objects. This
quantity had to be between 45 and 65~GeV. 
The events were required to have an event vertex within 35~cm of the nominal vertex position.

The selected inclusive DIS data sample consists of approximately 820.000 events.
The photoproduction background in this sample was estimated to be smaller than 3~$\%$ and was neglected in the 
following analysis.

\subsection{Jet Selection and Definition of Observables}
\noindent
Jets are defined with the inclusive $k_T$ cluster algorithm\cite{ktalgo} which is applied to 
all objects of the hadronic final state discussed above, to stable hadrons 
or to partons. The scattered lepton is excluded from the jet finding. 
For a complete description and an exact definition of how this algorithm has been used in this analysis 
see \cite{h1jetshapes}. The jet finding was performed in the Breit reference frame.

The jets were selected by demanding a minimum transverse energy $E_T$ of 5~GeV in the Breit frame and a 
pseudorapidity in the laboratory frame well within the LAr calorimeter acceptance, 
-1.0 $< \eta_{lab} <$~2.8. The lower limit corresponds to the backward
direction; the upper limit marks the very forward detector region towards the proton remnant
and corresponds to approximately 7 degree. There are 104.000 events with at least one such
jet; 34.000 events have two such jets, 4.600 have three and 530 events have even four such jets.

Each jet which fulfills the above requirements enters in the measurements of single-inclusive jet cross sections.
The following single-inclusive jet cross sections are measured: 
\vspace*{-0.35cm}
\begin{itemize}
\item the cross section $d\sigma_{Jet}/dE_T$ as a function
of the transverse energy $E_T$ in different ranges of the 
pseudorapidity $\eta_{lab}$ and of the photon virtuality
$Q^2$,
\item the cross section $d\sigma_{Jet}/d(E^2_T/Q^2)$ as a function of the ratio $E^2_T/Q^2$, also in 
ranges of $\eta_{lab}$ and $Q^2$, and finally
\item the cross section  $d\sigma_{Jet}/dx_{Bj}$ as a function of $x_{Bj}$ for 
two rapidity regions. An additional cut on the fractional jet energy 
$x_{Jet} \equiv E_{Jet}/E_p$ (with $E_p$ the proton beam energy and both $E_{Jet}$ and $E_p$ 
measured in the laboratory frame) is imposed on these distributions. However, in 
contrast to a former H1 analysis\cite{h1forward} no cut on the ratio $E_T^2/Q^2$ is performed. 
This brings the analysis closer to a recent analysis of forward $\pi^0$ production\cite{h1pio}.
\end{itemize}

\subsection{Correction Procedure and Systematic Errors}
\noindent
The QCD Monte Carlo models RAPGAP and ARIADNE are well able to describe
distributions of the kinematic variables $Q^2$, $x_{Bj}$ or $y$ in the inclusive 
DIS data sample. Also variables like the ratio of the measured positron energy and the 
positron energy reconstructed from the double angle method\cite{doubleangle} and quantities 
describing the hadronic final state
like the balance between hadronic and positron transverse energy in the event can be well 
reproduced, showing an overall good understanding of the calibration of the calorimeter.

All data cross sections shown in this paper have been corrected bin-by-bin for detector and QED radiation 
effects using the ARIADNE and RAPGAP event generators and the simulation of the H1 detector. 
HERACLES\cite{herakles} was used to correct for QED radiation.
The migration between the bins due to the limited detector resolution
was found to be small.
The quality  of the description of jet cross sections by the two models depends on the transverse energy $E_T$.
The ARIADNE model describes the data at low values of the transverse energy whereas it is too high at 
$E_T$ values larger than 20~GeV. The RAPGAP model undershoots the data significantly at 
low $E_T$, but it agrees quite well with the data at high $E_T \gsim$~20~GeV.
Although the jet cross section estimates of the two models differ considerably, the
factors used to correct for detector effects derived from the two 
models agree well and deviate from unity by usually less than  20~$\%$. 
The difference between the correction factors from the two models is taken as an estimate of the systematic
uncertainty of the correction procedure.
We also investigated jet profiles in order to check the 
simulation of the hadronic energy flow in the events (see also \cite{h1jetshapes}) and found them
to be well described by the Monte Carlo models.  

The hadronisation corrections were estimated using the LEPTO and ARIADNE Monte Carlos. The 
mean of the two predictions was taken to be the 
correction factor which was later applied to the parton-level QCD calculations; 
the difference served as an estimate for the 
uncertainty of our knowledge of the hadronisation corrections.

We tested another method to reconstruct the event kinematics which involves
in addition to the positron also the hadronic final state. The results where 
compatible; the differences were treated as a systematic uncertainty.
Further systematic uncertainties were determined by varying the 
energy depositions in the LAr (SPACAL) calorimeters
by $\pm$4~$\%$ ($\pm$7~$\%$). In addition, the track momenta entering in 
the definition of the hadronic final state were 
changed by $\pm$3~$\%$. The uncertainty on 
the energy measurement of the scattered positron is taken to be $\pm$1~$\%$. Additional 
uncertainties come from the resolution of the angle of the scattered 
positron ($\pm$2~mrd) and from 
the total luminosity normalization ($\pm$1.5~$\%$). All errors are added in 
quadrature and are included in all the 
following results. The total systematic error is typically of the order of 15~$\%$, the 
main contribution coming from the uncertainty of the hadronic energy scale.

\section{Results}
\noindent
All following cross sections are measured for the kinematic range 5~$< Q^2 <$~100~$\mathrm{GeV^2}$, 0.2~$< y <$~0.6; 
the transverse energy $E_T$ is always measured in the Breit reference frame. 
The inner error bars always indicate the statistical errors only, the total error bars include also the 
systematic errors added in quadrature. All NLO calculations are based on DISENT if not stated otherwise.


The measured jet cross section $d\sigma_{Jet}/dE_T$ as a function of the transverse jet energy $E_T$ 
in three regions
of the pseudorapidity for the total $Q^2$ range from 5 to 100~$\mathrm{GeV^2}$ 
is shown in fig.~\ref{etinetaet}. 
The cross section falls over four orders of magnitude. $E_T$ values up
to 60~GeV are reached.
In the top part, the cross section is compared to NLO QCD calculations using $E^2_T$ as 
renormalization scale (dotted line). Also shown is the 
uncertainty of the NLO prediction (shaded band) which is estimated by varying the renormalization scale
$\mu^2_R$ a factor of four up and down, henceforth referred to as renormalization scale uncertainty. 
The LO prediction is shown as a dashed line.
The hadronisation corrections and their uncertainties are indicated 
in the center of the figure, they are typically of the order of 10-15~$\%$.
The hadronisation corrections lower the QCD predictions. In the bottom part of the figure the 
ratio $(NLO-Data)/Data$ is displayed, where $NLO$ now is the QCD prediction including hadronisation
corrections. The inner, darker shaded band
reflects the uncertainty of the hadronisation correction, the total shaded area also includes
the uncertainty from the variation of the renormalization scale $\mu^2_R$.
It can be observed that at high $E_T \gsim$~20~GeV the data are well described by the
NLO QCD calculations in all regions of $\eta_{lab}$, whereas at low $E_T$ the QCD 
prediction is up to a factor of 2 below the data for the forward
and slightly below the data in the central rapidity region. The NLO predictions generally
describe the data much better than the LO calculations. 
In regions with discrepancies between data and the theoretical predictions the estimated 
renormalization scale uncertainties, the difference between the LO and the NLO predictions and the 
hadronisation corrections are found to be large.

Fig.~\ref{etinetaq2} shows the same data, but now $\mu^2_R = Q^2$ is chosen 
as renormalization scale. Now for $E_T \lsim$~20~GeV the data are described by 
the calculation for all values of $\eta_{lab}$.
However, there is a slight tendency of the calculations to overshoot 
the data at high values of the transverse energy,
$E_T \gsim$~20~GeV. The obtained renormalization scale uncertainties are much larger
for this choice of scale than for $\mu_R^2 = E_T^2$, especially for high 
values of $E_T$ where they are up to twice as large.

Figs.~\ref{etinetasumet} and \ref{etinetasumq2} present the ratio $(NLO-Data)/Data$ for
the cross section $d\sigma_{Jet}/dE_T$ as a function of $E_T$ in the two ranges 5~$< Q^2 <$~20~$\mathrm{GeV^2}$ and 
20~$< Q^2 <$~100~$\mathrm{GeV^2}$
for $\mu_R^2 = E_T^2$ and $\mu_R^2 = Q^2$, respectively.
Again, the dark shaded band reflects the hadronisation uncertainty; the total shaded area includes also the 
uncertainty from the variation of the renormalization scale.
For $\mu_R^2 = E_T^2$, the discrepancies between data 
and the NLO calculation for low $E_T$ in the forward direction get significant for the lower $Q^2$ region. 
For $\mu_R^2 = Q^2$, the excess of the NLO calculation over the data for
20~$\lsim E_T \lsim$~40~GeV is also more pronounced for 
the lower $Q^2$ range.

Fig.~\ref{etinq2etfw} shows the cross section $d\sigma_{Jet}/dE_T$ 
as a function of $E_T$ for the forward direction 
1.5~$< \eta_{lab} <$~2.8
in 5 different ranges of the photon virtuality $Q^2$. The data are
compared to NLO QCD predictions using 
$E^2_T$ as renormalization scale.
Only for low $E_T \lsim$~20~GeV and $Q^2 \lsim$~40~$\mathrm{GeV^2}$ 
there is a discrepancy between data and the predictions.
The picture changes again when we consider the renormalization scale $Q^2$ (see fig.~\ref{etinq2q2fw}).
In this case the description of the data by
the calculation is satisfactory everywhere when all errors and uncertainties are taken into
account, except for the range in transverse energy from 20 to 40~GeV where the calculations overshoot
the data for low $Q^2 <$~40~$\mathrm{GeV^2}$.
The estimated scale uncertainties are again up to almost twice as large as for a choice $\mu^2_R = E_T^2$.

Similar conclusions can be drawn from the 
cross section $d\sigma_{Jet}/(dE_T^2/Q^2)$ as a function of $E^2_T/Q^2$ in different ranges of the pseudorapidity 
for the complete range of $Q^2$ (fig.~\ref{et2q2inetaet}).
In the top part, the data are compared to a NLO
calculation of the cross section using $\mu^2_R = E_T^2$. Good
agreement between data and calculations for the backward part of the detector is observed, but
for the central and forward direction for values of 
the ratio $E_T^2/Q^2$ between 2 and 50 discrepancies are revealed. 
In the central and forward region also the corrections
between LO and NLO predictions get very large up to a factor 5.
However, for a choice $\mu_R^2 = Q^2$, fig.~\ref{et2q2inetaq2}, 
the data are quite well described 
by the NLO calculation, except for the largest values of $E_T^2/Q^2 >$~50 
where the calculation overshoots the data slightly. This may reflect
the discrepancies observed in the cross section $d\sigma_{Jet}/dE_T$ for high values of $E_T$
(compare fig.~\ref{etinetaq2}).

Fig.~\ref{et2q2sumet} and \ref{et2q2sumq2} show, for the cross section $d\sigma_{Jet}/(dE_T^2/Q^2)$ 
as a function of $E^2_T/Q^2$,
the ratio $(NLO-Data)/Data$ in the two ranges 5~$< Q^2 <$~20~$\mathrm{GeV^2}$ and 
20~$< Q^2 <$~100~$\mathrm{GeV^2}$ for 
renormalization scales $E_T^2$ and $Q^2$, 
respectively.
Especially for $\mu^2_R = E^2_T$ the discrepancies between data and 
NLO QCD calculations are more pronounced in the lower $Q^2$ range. As for the $E_T$ distributions the estimated scale
uncertainties are much larger for $\mu_R^2 = Q^2$.

Fig.~\ref{xinetaet} illustrates the cross section $d\sigma_{Jet}/dx_{Bj}$ as a function of $x_{Bj}$ in two 
different regions of $\eta_{lab}$ for the total range 5~$< Q^2 <$~100~$\mathrm{GeV^2}$. 
For these distributions an additional cut $x_{Jet} \equiv E_{Jet}/E_p >$~0.035 was applied to the data.
The NLO calculations are
based on the renormalization scale $E_T^2$. 
 In the central rapidity region (left column) the data are described by the calculation within the errors. But
in the forward region (right column) for low values of $x_{Bj} \lsim 10^{-3}$ the data are 
up to twice as high as the theoretical predictions, with differences increasing as $x_{Bj}$ decreases. 
This effect was also observed in an H1 analysis
of forward $\pi^0$ production\cite{h1pio}. 
As for the distributions of $d\sigma/dE_T$, the situation changes when we
assume $Q^2$ to be the right scale, see fig.~\ref{xinetaq2}. Here the data are described by the calculations
for all values of the Bjorken scaling variable within the errors and uncertainties. The differences are largest for 
low values of $x_{Bj}$ in the central rapidity region. However, the
obtained renormalization scale uncertainties are up to three times as large as for $\mu_R^2 = E_T^2$.

Fig.~\ref{xsum} summarizes the results for the ratio $(NLO-Data)/Data$ 
for the $x_{Bj}$ distributions for $\mu_R^2 = E^2_T$ and for $\mu_R^2 = Q^2$
and two regions of the pseudorapidity.
The different sizes of the renormalization scale uncertainties are clearly visible.

We also compared our data to the cross section predictions of the JetViP program for direct and resolved photon 
processes.
Fig.~\ref{jvplot} shows the cross section $d\sigma_{Jet}/d(E^2_T/Q^2)$ as a function of $E^2_T/Q^2$ in 
three different regions of the pseudorapidity. The data points are
compared to three different JetViP predictions, one according to the on-shell scheme and two according
to the off-shell scheme for two different values of the parameter
$y_{cut}$, see section 3. It can be seen that the three
JetViP predictions differ considerably, as may be expected from the discussion in section~3. 
We also note that none of the different choices is able to describe the data over the whole phase space.
Therefore, a physical interpretation of the observed discrepancies between direct NLO calculations 
and the data as a sign for the need of resolved photon processes in DIS has to wait for a better
understanding of the ambiguities in NLO calculations of resolved
processes for $Q^2 \gsim$~5~$\mathrm{GeV^2}$.
A recently published comparison of ZEUS data on $E^2_T/Q^2$ distributions of forward jets and JetViP calculations
of direct and resolved photon processes\cite{zeuset2q2} obtained good agreement in a particular
phase space region and a specific choice in the calculations.

Summarizing the jet cross section measurements we can state
that using $Q^2$ as renormalization scale the data are not well described at high
values of $E_T$ or $E_T^2/Q^2$. Since
in these regions the calculations tend to be too high, 
it may be concluded that in these parts of the phase space $Q^2$ is too low to be 
the dominant hard scale. In addition, there are very large scale uncertainties
observed for this choice of scale.
If, however, $E_T$ is chosen as scale, there are various 
possible interpretations of the observed discrepancies at low $Q^2$ and low $E_T$:
 
First, the calculations might be lacking higher orders
in the perturbative series. When implemented, these could increase 
the cross section predictions in the according to the DGLAP formalism 
kinematically suppressed forward direction.
Cross section calculations of the order
$\mathcal{O}(\alpha_S^3)$ are, however, not yet available.
Second, the parton evolution in the low $Q^2$, low $x_{Bj}$ region might
require to take BFKL-type dynamics into account.
However, up to now there is no BFKL QCD calculation available for our data.
Third, in this analysis it could not be clarified whether resolved photon contributions to the cross section
would be sufficient to remove the disagreement between data and direct NLO calculations. This is because
the only available NLO QCD calculation of such processes,
implemented in JetViP, gives no consistent description of the data over the full phase space 
(in the 'on-shell' scheme) or depends 
on the unphysical resolution parameter
$y_{cut}$ which limits the predictive power of this model (in the 'off-shell' scheme).


\section{Conclusions}
\noindent
We have measured single-inclusive jet cross sections in the low $Q^2$ regime 
(5~$< Q^2 <$~100~$\mathrm{GeV^2}$) in DIS as a function of $E_T$, $E^2_T/Q^2$ and
$x_{Bj}$ in different regions of the pseudorapidity and of the photon virtuality. A comparison 
of the data to NLO QCD calculations showed in general good agreement for the backward and central
detector regions.
However, when going to the forward direction, i.e. to large pseudorapidities, 
discrepancies between 
data and NLO calculations are observed. 
The choice $\mu^2_R = E_T^2$ for the renormalization scale 
leads to discrepancies of up to a factor 2 at low values of the 
transverse energy $E_T \lsim$~20~GeV, for values of the ratio $E_T^2/Q^2$ between 2 and 50 and 
for low values of the Bjorken scaling variable $x_{Bj} \lsim 10^{-3}$.
The discrepancies are usually largest in regions where also
the difference between leading and next-to-leading order QCD predictions, the hadronisation corrections 
and the renormalization scale uncertainties are large. They are prominent for low values of $Q^2 <$~20~$\mathrm{GeV^2}$.
Choosing however $\mu^2_R = Q^2$ the data are in overall good agreement 
with the theoretical predictions. Only for high values of $E_T$ between 20 and 40~GeV and
for $E^2_T/Q^2 >$~50 the QCD calculations overshoot the data. However, for $\mu_R^2 = Q^2$
the obtained renormalization scale dependencies are larger than for $\mu_R^2 = E_T^2$ by up to a factor of 3.

A more satisfactory interpretation of the present results might be possible when higher order calculations 
based on the DGLAP equations, BFKL-type calculations and also a consistent treatment of resolved photon 
processes in the kinematic range of our data will be available.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}

We are grateful to the HERA machine group whose outstanding
efforts have made and continue to make this experiment possible. 
We thank
the engineers and technicians for their work in constructing and now
maintaining the H1 detector, our funding agencies for 
financial support, the
DESY technical staff for continual assistance, 
and the DESY directorate for the
hospitality which they extend to the non DESY 
members of the collaboration.
We also would like to thank M.H. Seymour and B. P\"otter for invaluable
discussions and help with their QCD programs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\clearpage

%\include{firstplots}
%\include{lastplots}

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig1.eps,
 bbllx=0,bblly=0,bburx=882,bbury=792,clip,width=0.9\textwidth}}
 \end{picture}
\caption{\label{feyn}Examples of contributions to the inclusive jet 
cross section at different orders of the strong coupling
constant for direct and resolved photon processes; a: direct process at $\mathcal{O}(\alpha_s)$ (LO); 
b: direct process at $\mathcal{O}(\alpha^2_s)$ (NLO); c: resolved process at $\mathcal{O}(\alpha^2_s)$ (LO); 
d: resolved process at $\mathcal{O}(\alpha^3_s)$ (NLO); } 
\end{figure}
\end{center}

\clearpage

\begin{figure}[c] \centering
\epsfig{file=H1prelim-00-032.fig2.eps,%
width=0.5\textwidth}
\caption{\label{ladder}Example of a ladder diagram. Emitted partons are characterised by their transverse momentum
$k_{Ti}$. The longitudinal components of the momenta are labelled $x_i$.}
\end{figure}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
\begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig3.eps,
 bbllx=0,bblly=30,bburx=620,bbury=824,clip,width=0.95\textwidth}}
\end{picture}
\caption{\label{etinetaet}Single-inclusive jet cross section $d\sigma_{Jet}/dE_T$ as a function of $E_T$ 
in three regions of the pseudorapidity.
In the top part the data are compared to LO (dashed line) and NLO QCD predictions (dotted line) obtained with DISENT 
using $\mu^2_R = E_T^2$; the shaded band 
indicates the uncertainty on the NLO QCD prediction obtained by varying the squared renormalization scale by a factor
4 up and down. The center part displays the hadronisation corrections. In the bottom part the expression 
$(NLO-Data)/Data$ is shown, where $NLO$ means NLO QCD predictions with hadronisation corrections applied.}
\end{figure}
\end{center}
\clearpage

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig4.eps,
 bbllx=0,bblly=30,bburx=620,bbury=824,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{etinetaq2}As fig.~\ref{etinetaet}, but for a renormalization scale $\mu_R^2 = Q^2$.}
\end{figure}
\end{center}
\clearpage

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig5.eps,
 bbllx=0,bblly=20,bburx=595,bbury=830,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{etinetasumet}$(NLO-Data)/Data$ for the single-inclusive jet cross section 
$d\sigma_{Jet}/dE_T$ as a function of $E_T$ 
in three regions of the pseudorapidity for DISENT predictions using $\mu_R^2 = E^2_T$. 
Top: 5~$< Q^2 <$~20~$\mathrm{GeV^2}$,
bottom: 20~$< Q^2 <$~100~$\mathrm{GeV^2}$.
$NLO$ includes hadronisation corrections.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig6.eps,
 bbllx=0,bblly=20,bburx=595,bbury=830,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{etinetasumq2}As fig.~\ref{etinetasumet}, but for a renormaliation scale $\mu_R^2 = Q^2$.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(0,8){\epsfig{file=H1prelim-00-032.fig7.eps,
 bbllx=0,bblly=0,bburx=830,bbury=595,clip,height=15cm,width=1.\textwidth}}
 \end{picture}
\caption{\label{etinq2etfw}Single-inclusive jet cross section $d\sigma_{Jet}/dE_T$ as a function of $E_T$ in five ranges 
of the photon virtuality for the forward detector region.
In the top part the data are compared to LO (dashed line) and NLO QCD predictions (dotted line) obtained with DISENT using 
$\mu^2_R = E_T^2$; the shaded band 
indicates the uncertainty on the NLO QCD prediction obtained by varying the squared renormalization scale by a factor
4 up and down. The center part displays the hadronisation corrections. In the bottom part the expression 
$(NLO-Data)/Data$ is shown, where $NLO$ means NLO QCD predictions with hadronisation corrections applied.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(0,8){\epsfig{file=H1prelim-00-032.fig8.eps,
 bbllx=0,bblly=0,bburx=830,bbury=595,clip,height=15cm,width=1.\textwidth}}
 \end{picture}
\caption{\label{etinq2q2fw}As fig.~\ref{etinq2etfw}, but for a renormaliation scale $\mu_R^2 = Q^2$.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,0){\epsfig{file=H1prelim-00-032.fig9.eps,
 bbllx=0,bblly=10,bburx=590,bbury=820,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{et2q2inetaet}Single-inclusive jet cross section $d\sigma_{Jet}/d(E^2_T/Q^2)$ as a function of $E^2_T/Q^2$
in three regions 
of the pseudorapidity.
In the top part the data are compared to LO (dashed line) and NLO QCD predictions (dotted line) obtained with DISENT using 
$\mu^2_R = E_T^2$; the shaded band 
indicates the uncertainty on the NLO QCD prediction obtained by varying the squared renormalization scale by a factor
4 up and down. The center part displays the hadronisation corrections. In the bottom part the expression 
$(NLO-Data)/Data$ is shown, where $NLO$ means NLO QCD predictions with hadronisation corrections applied.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,0){\epsfig{file=H1prelim-00-032.fig10.eps,
 bbllx=0,bblly=10,bburx=590,bbury=820,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{et2q2inetaq2}As fig.~\ref{et2q2inetaet}, but for a renormalization scale $\mu_R^2 = Q^2$.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig11.eps,
 bbllx=0,bblly=10,bburx=595,bbury=830,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{et2q2sumet}$(NLO-Data)/Data$ for the single-inclusive jet cross section 
$d\sigma_{Jet}/d(E^2_T/Q^2)$ as a function of $E^2_T/Q^2$ 
in three regions of the pseudorapidity for DISENT predictions using $\mu_R^2 = E^2_T$. 
Top: 5~$< Q^2 <$~20~$\mathrm{GeV^2}$,
bottom: 20~$< Q^2 <$~100~$\mathrm{GeV^2}$.
$NLO$ includes hadronisation corrections.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig12.eps,
 bbllx=0,bblly=10,bburx=595,bbury=830,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{et2q2sumq2}As fig.~\ref{et2q2sumet}, but for a renormalization scale $\mu_R^2 = Q^2$.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig13.eps,
 bbllx=0,bblly=80,bburx=555,bbury=830,clip,width=0.85\textwidth}}
 \end{picture}
\caption{\label{xinetaet}Single-inclusive jet cross section $d\sigma_{Jet}/dx_{Bj}$ as a function of $x_{Bj}$ 
in two regions of the pseudorapidity for 5~$< Q^2 <$~100~$\mathrm{GeV^2}$.
In the top part the data are compared to LO (dashed line) and NLO QCD predictions (dotted line) obtained with DISENT 
using $\mu^2_R = E_T^2$; the shaded band 
indicates the uncertainty on the NLO QCD prediction obtained by varying the squared renormalization scale by a factor
4 up and down. The center part displays the hadronisation corrections. In the bottom part the expression 
$(NLO-Data)/Data$ is shown, where $NLO$ means NLO QCD predictions with hadronisation corrections applied.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig14.eps,
 bbllx=0,bblly=80,bburx=555,bbury=830,clip,width=0.85\textwidth}}
 \end{picture}
\caption{\label{xinetaq2}As fig.~\ref{xinetaet}, but for a renormalization scale $\mu_R^2 = Q^2$.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,28){\epsfig{file=H1prelim-00-032.fig15.eps,
 bbllx=0,bblly=80,bburx=510,bbury=700,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{xsum}$(NLO-Data)/Data$ for the single-inclusive jet cross section 
$d\sigma_{Jet}/dx_{Bj}$ as a function of $x_{Bj}$ in two regions of the pseudorapidity for the DISENT predictions
and for  for 5~$< Q^2 <$~100~$\mathrm{GeV^2}$. 
Top: $\mu_R^2 = E^2_T$,
bottom: $\mu_R^2 = Q^2$.
$NLO$ includes hadronisation corrections.}
\end{figure}
\end{center}

\clearpage

\begin{center}
\begin{figure}[b] \unitlength 1mm
 \begin{picture}(80,100)
   \put(5,8){\epsfig{file=H1prelim-00-032.fig16.eps,%
 bbllx=0,bblly=30,bburx=450,bbury=430,clip,width=0.95\textwidth}}
 \end{picture}
\caption{\label{jvplot}Single-inclusive jet cross section $d\sigma_{Jet}/d(E^2_T/Q^2)$ as a function of $E^2_T/Q^2$ 
in three regions of the pseudorapidity. The data points are compared to three different JetViP calculations. 
The two scenarios labelled $y_{cut}=0.005$ and $y_{cut}=0.01$ correspond to the off-shell scheme. For the on-shell scheme
there is no dependence on the $y_{cut}$ parameter.}
\end{figure}
\end{center}



\end{document}




