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\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em]
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%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline
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\multicolumn{4}{l}{{\bf
                 14th International Workshop on Deep Inelastic Scattering,
                 DIS2006, April 20-24, 2006, Tsukuba, Japan}} \\
%                 & Abstract:        & {\bf xx-xxx}    &\\
%                 & Parallel Session & {\bf x}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
 & \multicolumn{3}{r}{H1prelim-06-011} \\[.2em]
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\vspace*{2cm}

\begin{center}
  \Large
  {\bfseries
Threejet production in deep inelastic $e$-$p$ scattering and
 low-$x$ parton dynamics at HERA
  }

  \vspace*{1cm}
    {\Large H1 Collaboration}
\end{center}

\begin{abstract}

\noindent
Di{f}ferential threejet cross sections in  deep inelastic scattering processes
at low $x$ and $Q^2$ have been measured with the H1 detector using an
 an integrated luminosity of $44{.}2\ {\rm pb}^{-1}$.
Threejet events are identified
using the inclusive $k_\perp$ cluster algorithm in the $\gamma^*p$ rest frame. The cross
sections are given at the level of parton jets and correspond to the kinematic range
\mbox{$ 10^{-4} < x < 10^{-2}$},  \mbox{$ 5 \ {\rm GeV}^2 \  < \ Q^2 \  < 80 \ {\rm GeV}^2 $},
\mbox{$E^*_{\perp,\ \rm jet} > 4 \ {\rm GeV}$} and \mbox{$ -1 < \eta_{\rm jet} < 2.5 $}.
Three phase-space regions have been selected
in order to study parton dynamics from the most global to the most restrictive regions of forward going
 (close to the proton-direction) jets.
The measurements of threejet cross sections  are
 compared  with fixed order QCD predictions of ${\mathcal{O}}(\alpha_{\rm s}^2)$ and
${\mathcal{O}}(\alpha_{\rm s}^3) $
 and with two leading order MC predictions where additional partons are
generated by initial state radiation
and by the color dipole model respectively. A good  description is given by the
${\mathcal{O}}(\alpha_{\rm s}^3) $ prediction for which remaining differences
  are concentrated at low $x$ and topologies with two forward jets, which are
most sensitive to unordered gluon radiation.
The LO MC with unordered  gluon radiation modeled by the color dipole model gives a good
 overall description
 of the data.  We interpret these results as strong hints for the presence
of large contributions
from non-$k_{\perp}$-ordered gluon radiation needed to describe threejet production at
 low $x$ and forward rapidities.
\end{abstract}

  \vspace*{1cm}

\end{titlepage}

\pagestyle{plain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
The study of parton dynamics at low $x$ has been an active area of research since
early HERA operation because HERA has entered a new kinematic regime where the approximations
 of the DGLAP evolution equations may no longer be valid. These approximations neglect terms
proportional to $\alpha_{\rm s}\cdot\ln(1/x)$ which are naturally expected to become large 
at small $x$. These effects are best studied in the hadronic final state
since these terms may, compared to DGLAP,  lead to a significant enhancement  of gluon radiation
 unordered in transverse momentum.
This effect should be 
 largest for high $P_\perp$ forward jets (near to the proton direction). Various measurements
\cite{forwardjet}--\cite{forward}  have shown, that the rate of  forward jets is indeed higher
 than predicted by
LO QCD predictions including initial state radiation. However, predictions including 
a resolved photon contribution  or unordered gluon radiation as implemented in the
 color dipole model were shown to give a decent description of forward jet data and of
 dijet cross sections at low $x$. Measurements of the ZEUS collaboration \cite{zeusforward} have shown
that the description of inclusive forward jet production is significantly improved 
by up to a factor 10 by the  fixed order $\alpha_{\rm s}^2 $ compared to the order  $\alpha_{\rm s}$
prediction.
This paper concentrates on threejet events which require at least one radiated gluon in
addition to the two hard scattered partons and the comparison with fixed order QCD predictions
which for threejet events  are either LO or NLO. In addition  two  LO Monte
 Carlo generators which were able to describe forward jet and dijet production at low $x$ 
 are also tested. These are the generator RAPGAP with initial state radiation including a
 resolved photon contribution and DJANGOH which uses the color dipole model  to produce
 additional gluon radiation.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{kinematics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Kinematics}
Figure \ref{feynman} shows two of the  hard DIS processes dominating at low $x$, which  lead 
 to three or more  jets in the final
 state. The diagrams correspond to order
$\alpha_{\rm s}^2$ and $\alpha_{\rm s}^3$ contributions to the cross section. The radiated
gluons are predominantly emitted
into the forward direction whereas the quarks from the hard scattering process are
mostly  central.
%%%
 
\begin{figure}[htb]
\begin{center}
\hspace*{0.1\linewidth}
$\mathcal{O}(\alpha_{\rm s}^2)$\hspace*{0.4\linewidth}$\mathcal{O}(\alpha_{\rm s}^3)$
\end{center}
\begin{center}
\includegraphics[width=0.40\linewidth]{dia0.pdf}
\hspace*{5mm}
\includegraphics[width=0.40\linewidth]{dia1.pdf}
\vspace{0.5 cm}
                     \caption[]
                          {Leading and next to leading order diagrams to 3-jet
                          production in DIS  at HERA with one and two radiated gluons, respectively.
                                   }
          \label{feynman}
        \end{center}
      \end{figure}
%%%%%%
%=======================
The relevant kinematic variables used are the Bj{\o}rken variable
$x = Q^2/Q^2_{\rm max}$ and $Q^2$, the
momentum transfer squared. Jets are selected using the inclusive $k_\perp$ algorithm in the
$\gamma^*p$ center of mass
system. They will be  characterised by their transverse momentum
$P_\perp^*$\footnote{Observables in the $\gamma^*p$ center of mass frame
carry a $^*$.} and their
pseudorapidity $\eta$ 
in the laboratory system. The 3-jet topology is fully described by the variables defined in
\cite{tevatron}. We use only a subset of these variables namely the normalised energy of the
jets $X_i = E_i'/\sum_jE_j'$ and the two angles $\theta'$ and $\psi'$ as defined in
figure \ref{Tevatronvar}. These observables are measured in the 3-jet center of mass frame.
%%%
 
 \begin{figure}[htb]
         \begin{center}
          \includegraphics[width=0.50\linewidth]{tevatron.pdf}
            \vspace{0.5 cm}
                     \caption[]
                          {Definition of the angles $\theta'$ and $\psi'$
in the threejet rest system. The figure was taken from
\cite{tevatron} }
          \label{Tevatronvar}
        \end{center}
      \end{figure}
%%%%%%

\section{Monte Carlo Simulations and Fixed Power QCD Predictions}
In the analysis two Monte Carlo programs were used to correct
the data for
detector inefficiencies and migrations, and to compare the measured
cross sections with model predictions.
Both use
leading order matrix elements for the hard QCD 2$\rightarrow$2
subprocess which are convoluted
with parton distributions of the proton and the photon, taken
at the scale $\mu^2=Q^2$ (DJANGOH) and \ $\mu^2=Q^2+\hat{p}_\perp^2$ (RAPGAP), respectively,
where $\hat{p}_\perp$ is the transverse momentum of the emerging hard partons.
The RAPGAP 2.08 Monte Carlo program \cite{RAPGAP} adds a
resolved photon component for which the SaS 2D
parton distribution functions are used, which were
found to give a good description of the effective photon structure
function as measured by H1.
Higher order effects are simulated using parton showers
in the leading $\log(\mu)$ approximation (MEPS), and the Lund string
model is used for hadronisation. The DJANGOH \cite{DJANGOH} Monte Carlo program on the other hand 
uses the color dipole model (CDM) \cite{CDM} which creates additional gluon radiation not ordered 
in $k_\perp$. Again the Lund string fragmentation is used for hadronisation. Radiative corrections are
 applied using the HERACLES program.\par
Fixed order QCD predictions at parton level are calculated using the NLOjet++ program \cite{NLO++} which
 is able to predict threejet parton cross sections in leading (LO) and next to leading (NLO) order. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{expproc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental Procedure}
\subsection{H1 Detector}
A detailed description of the H1 detector can be found in \cite{H1det}.
Here, a brief account of the components most relevant to the present 
analysis is given. The H1 coordinate system convention defines the
outgoing proton beam direction as the positive $z$ axis and the polar 
scattering angle $\theta$ such that the pseudorapidity 
$\eta = -\ln \tan (\theta/2)$ increases along $z$. 

The hadronic final state $X$ is measured with a tracking and a
calorimeter system.
 The central $ep$ interaction region is surrounded by two large concentric drift
chambers, located inside a 1.15 T solenoidal magnetic field. Charged particle momenta are measured
 in the range
$-1.5< \eta <1.5$ with a resolution of $\sigma/p_T=0.01\, p_T/$GeV. A finely segmented electromagnetic
 and hadronic liquid argon calorimeter (LAr) covers the
range $-1.5 < \eta < 3.4$. The energy resolution is
$\sigma/E=0.11/\sqrt{E/{\rm GeV}}$
 for electromagnetic showers and $\sigma/E=0.50/\sqrt{E/{\rm GeV}}$ for hadrons, as
measured in test beams. A lead/scintillating fibre calorimeter (SPACAL) covers the backward 
region $-4 < \eta < -1.4$. It is used to detect the scattered positron.

\subsection{Event Selection}
The data used in this analysis were taken in the 1999 and 2000 running
periods, in which HERA collided $920\ {\rm GeV}$ protons with $27{.}5\ {\rm GeV}$
positrons. The data are collected using a trigger which requires the 
scattered electron to be measured in the electron detector 
and at least one high transverse momentum track ($p_\perp>800\ {\rm MeV}$)
in the central jet chamber. The scattered positron is required to be measured in the backward
electromagnetic calorimeter with an energy $E'_e>9\ {\rm GeV}$. The kinematic range is chosen
to be $5\ {\rm GeV}^2<Q^2<80\ {\rm GeV}^2$ and $0.1 < y < 0.7$. 

Jets are formed from the tracks and clusters of the hadronic final state , 
 using the
inclusive $k_\perp$ cluster algorithm
(with a distance parameter of 1.0) in the $\gamma^*p$ rest frame.
%, which
At least 3 jets are required, with transverse energies $E_\perp^*>4\ {\rm GeV}$
for all jets and ${E_\perp^*}_1+{E_\perp^*}_2>9\ {\rm GeV}$ for the sum of the  leading and
subleading jet, respectively. 
The jet axes are required to lie within the region $-1<\eta_{\rm jet}< 2{.}5$, well within
the acceptance of the LAr calorimeter except for one jet which is required to be central e.g.\
in the range $-1<\eta_{\rm jet}<1{.}3$
in order to guarantee a good trigger efficiency. After all cuts we are left with 38400 events 
with at least three jets of which 6000 have more than 3 jets. This is a very large statistical
improvement compared to earlier studies.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{reconstruction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Kinematic Reconstruction}
The energy $E'_e$ of the scattered electron is measured directly in
the backward electromagnetic calorimeter and the inelasticity $y$ is derived from it
\begin{equation*}
y = 1-\frac{E'_e}{E_e}\sin^2\frac{\theta_e'}{2} 
\end{equation*}
where $E_e$ is the electron beam energy. The virtuality of the exchanged virtual
Boson and the Bj{\o}rken scaling variable is reconstructed from
\begin{equation*}
Q^2=4E_eE_e'\cos^2\frac{\theta_e'}{2}\;,\qquad x=\frac{Q^2}{ys}\;,
\end{equation*}
where $\theta_e'$ is the angle of the scattered positron and $s$ is the $ep$ center of
mass energy squared.

The hadronic system $X$, containing the jets,
is measured in the LAr and SPACAL calorimeters
and the central tracking system. Calorimeter cluster energies and
track momenta are combined using algorithms which avoid double
counting. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{xsmeasurement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cross Section Measurement}
%\vspace{0.5cm}
\setlength{\tabcolsep}{0.5cm}
\begin{table}                                                                  
\begin{center}
{\textbf{Cross Section Definition}}\\[1em] 
\begin{tabular}{|c|}
\hline
$0.1 < y < 0.7$ \\
$ 4 \ {\rm GeV}^2 < Q^2 < 80 \ {\rm GeV}^2$ \\
\hline
$N_{\rm jet} \ge 3$\\
$E_{\perp,\ {\rm jet}}^* > 4\ {\rm GeV}$ \\
$ E_{\perp,\ {\rm jet 1}}^* + E_{\perp,\ {\rm jet 2}}^*>9\ {\rm GeV}$ \\
$-1 < \eta_{\rm jet}^{\rm lab} < 2.5$\\
\hline
one jet in the range \\
 $-1 < \eta_{\rm jet}^{\rm lab} < 1.3$ \\
\hline
\end{tabular}
\end{center}
\caption[Definition of Cross Sections]{The kinematic domain in which the
cross sections are measured. The jets are reconstructed using the
inclusive $k_\perp$ algorithm with the distance parameter set to 1.0
for detector jets as well as for the jets on parton or hadron level}
\label{tab:xsdef}
\end{table}
%\vspace{0.5cm}

The measured cross sections are defined at the level
of parton jets for comparisons with fixed order QCD predictions and at the level of stable hadrons
 for the comparison with LO Monte Carlo generators. The data are corrected in two steps. The first step
 corrects from detector jets to jets at 
stable hadron level accounting for detector inefficiencies
and migrations of kinematic quantities in the reconstruction (detector correction). The second step corrects for the effects of hadronisation
from hadron jets to parton jets ('inverse' hadronisation correction). Since the two steps are logically independent,
the uncertainties of both
correction factors are added in quadrature to determine the uncertainty of the cross sections at the parton jet level.
  The correction factors for both steps are determined  using the 
DJANGOH Monte Carlo program which offers a better description of the parton distributions than RAPGAP
 which is therefore only used for the estimation of systematic effects. For generated events,
the H1 detector response is simulated in detail and the Monte Carlo events
are subjected to the same analysis chain as the data. A reliable determination of the correction factors
 requires a good description of both the kinematic distributions and the energy flow in the events. Since 
both generators show significant deviations from the data distributions, the events are weighted in a few 
variables to adjust their kinematic distributions to the data. These variables are the $P_{\perp}$ of the leading jet,
$\eta_1 - \eta_2$, $\eta_1 + \eta_2 $ and $Q^2$.
The weighted simulations give a reasonable description of the shapes of all data  distributions.
According to the simulations, the detector level observables are well
correlated with the hadron and parton level quantities. Remaining differences are considered in the estimate
 of the  systematic uncertainties of the correction factors.

The kinematic region for which the cross sections are measured
is given in table \ref{tab:xsdef}.
An analysis of systematic uncertainties has been performed in which
the sensitivity of the measurement to variations of the detector
calibration and the Monte Carlo
Models used for correction are evaluated. 
The dominant systematic error on the cross sections arises from the 
uncertainty in the LAr calorimeter energy scale and from the uncertainty of the corrections needed to go from
detector to hadron jets and from hadron to parton jets. The latter has been estimated by using RAPGAP in addition to
DJANGOH to
determine the cross sections and using 50 \% of the difference of the correction factors evaluated by the
 two generators as estimate of their systematic error. All cross sections have an absolute normalisation error of 18\%
which is normally not shown.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
In Figures \ref{fig:njet} to \ref{fig:cospsi}, the  differential cross sections are presented 
 for  events with  3 or more jets for the number of jets ($N_{\rm jet}$), $x$, the pseudorapidities of the three jets
and the 3-jet variables describing relative jet energy and angles. The  kinematic range for which the cross
 sections are determined
is  specified in table \ref{tab:xsdef}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_n_jet.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_x.pdf}
\end{center}
\caption{
Differential cross sections in the number of jets $N_{\rm jet}$ and the Bj{\o}rken
scaling variable $x$. The inner error bars represent the statistical error of the data, the total error bars
correspond to the statistical and uncorrelated sytematic errors added in quadrature. The (orange) hatched
 error bands show the estimate of the correlated systematic uncertainties. The data has an additional overall
 normalisation error of 19 \%.  The shaded (red) band shows the NLO prediction where the size of the band
indicates the scale uncertainty of the NLO calculation, the 
dashed dotted line represents the LO prediction. The data for $N_{\rm jet}$ are also compared to
the two LO Monte Carlo programs RAPGAP  (dotted line) and DJANGOH (CDM)
(solid  line). }
\label{fig:njet}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_eta1.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_eta2.pdf}
\end{center}
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_eta3.pdf}
\hspace*{5mm}
\hspace*{0.4\linewidth}
\end{center}
\caption{Differential cross sections in the pseudorapidity $\eta_i$
for each of the three jets
(${P^*_\perp}_1>{P^*_\perp}_2>{P^*_\perp}_3$) compared to the LO (dashed-dotted line) and to the NLO
 (shaded (red) error band) prediction.
Other details are as in the caption to fig.\ \ref{fig:njet}}
\label{fig:eta}
\end{figure}
\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_x1.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_x2.pdf}
\end{center}
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_costhet.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot_cospsi.pdf}
\end{center}
\caption{Differential cross sections of the normalised energies $X_i'$ of the
two leading jets ($E_1'>E_2'>E_3'$ in the 3-jet
center of mass frame) and the two angles $\theta'$ and $\psi'$ as defined in figure
\ref{Tevatronvar} compared to the LO and the NLO prediction.
Other details are as in the caption to fig.\ \ref{fig:njet}}
\label{fig:cospsi}
\end{figure}
The figures also show the predictions of the NLO++ fixed order QCD prediction in LO
(dashed-dotted histogram) and NLO
(solid  line and  shaded  error bands).
The error bands of the NLO predictions are scale errors obtained by varying the renormalisation
and factorisation scale $\mu$  in the QCD prediction by a factor 2 and 0.5,  respectively. The
value of $\alpha_{\rm s}(m_{Z^0})$ has been fixed to 0.118 and the parton parametrisation
CTEQ6 has been used.  Figure \ref{fig:njet} shows the jet multiplicity distribution which
extends up to $N_{\rm jet} = 7$. For this
distributions also the predictions of the two LO Monte Carlo programs are shown. It can be noted
that the color dipole model
(DJANGOH + CDM) gives an excellent description of this distribution while 
RAPGAP fails. The  NLO prediction agrees for $N_{\rm jet}=3$, misses a fraction of 4-jet events
 and of course does not produce any events with more than 4 jets. In total it  misses 18\% of events 
with four or more jets.
\par
The kinematic distributions shown in figures \ref{fig:njet}--\ref{fig:cospsi} are not described
by the LO (${\mathcal{O}}(\alpha_{\rm s}^2)$) 
QCD predictions neither in shape nor in magnitude. Main discrepancies are seen at low $x$ and
for forward 
jets (large $\eta$) where by far too few events are predicted. The NLO prediction improves
the situation dramatically in all regions where deviations are observed. At lowest $x$ the discrepancy
 diminishes from a factor of about $3{.}3$ to $1{.}7$. The threejet variables of figure \ref{fig:cospsi}
are well described by the NLO prediction apart from an 18 \% difference in normalisation.
The conclusion is therefore that events with more than
3 jets are missing mainly at low $x$ and large $\eta$. 

\subsection{Forward Jet Selections}
Here we look at a restricted sample of events with forward jets where the observed
differences for the global selection were largest 
and where the largest sensitivity to unordered gluon radiation is expected. A forward
jet is defined in
agreement with earlier publications \cite{forward} by
$$ \theta_{\rm jet} < 20^\circ \ \hbox {\rm and}
\  x_{\rm jet} = \frac{E^*_{\rm jet}}{E_{\rm p,\ beam}} > 0.035$$ 
The forward jet sample is further divided into two subsamples. Sample 1 requires
two central jets in the range
$-1<\eta_{\rm jet}<1$ and one forward jet, sample 2 requires one central jet and two
forward jets (one forward jet and one additional jet with $\eta >1$). It is expected that
forward jets are predominantly produced by gluon radiation while central jets should predominantly originate
 from the hard scattering process.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth]{plot2cen_x.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth]{plot2fwd_x.pdf}
\end{center}
\caption{Differential cross sections in the Bj{\o}rken scaling variable $x$ for the samples
with two central (left) and with two forward jets (right).
Other details are as in the caption to fig.\ \ref{fig:njet}}
\label{fig:fwdx}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth]{plot2cen_eta1.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth]{plot2fwd_eta1.pdf}
\end{center}
\begin{center}
\includegraphics[width=0.4\linewidth]{plot2cen_pt1.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth]{plot2fwd_pt1.pdf}
\end{center}
\caption{Differential cross sections of the pseudorapidity (top) and of the transverse
momentum (bottom) of the leading jet for the samples with two central
(left) resp.\ two forward jets (right).
Other details are as in the caption to fig.\ \ref{fig:njet}}
\label{fig:fwdeta}
\end{figure}
This is confirmed by a study of the parton composition of threejet events using the
DJANGOH (CDM) Monte Carlo program. Therefore sample 1 will have many events with 
only one gluon radiation while sample 2 with two forward jets will have
a larger fraction of events with 
two radiated gluons thus being  more sensitive to unordered gluon radiation.
Results are shown 
in figures \ref{fig:fwdx} and \ref{fig:fwdeta} for the variables $x$, $\eta_1$ and
${P^*_\perp}_1$. The fixed order NLO prediction  gives a rather
good description for the sample with two central jets, where the step from LO to NLO
improves the agreement a low $x$ and large rapidity dramatically by more than
a factor 2, missing only about 30\% of events.
The sample with two forward jets on the other hand gives an even more dramatic
change reducing the discrepancy at small $x$ from a factor of 10 to $3{.}5$
when going from the LO to the NLO prediction, but a large discrepancy remains.
In summary the main discrepancies beween data and the fixed order NLO prediction are found at
small $x$ and large rapidities for the sample with two forward jets.
This is exactly the kinematic region where unordered
gluon radiation is expected to make a large contribution.
It should be noted that the DGLAP evolution equation in order $\alpha_{\rm s}^3$
include additional terms $\propto\alpha_{\rm s}\cdot\ln(1/x)$ not present in leading order.

\section{Comparison to the LO Monte Carlo Programs}
In a last step we compare to the two LO Monte Carlo Programs RAPGAP (with resolved photon contribution) 
 and DJANGOH (with color dipole model)
which have both been successful in describing the forward jet data and the dijet cross
sections to the measured cross
sections at hadron jet level, especially to the angular distributions of the the threejet system.
Here the comparison is made to the non-weighted  predictions of the MC generators.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo_x.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo_deta12.pdf}
\end{center}
\caption{Differential cross sections at hadron jet level in the Bj{\o}rken scaling variable $x$ and
the difference of the pseudorapidities of the two leading jets.
The inner error bars represent the statistical error of the data,
the total error bars correspond to the statistical and uncorrelated
systematic errors added in quadrature.
The correlated systematic errors are 
shown by the hatched error band. The data are
 compared to the two LO Monte Carlo programs DJANGO (CDM) (dashed line) and
RAPGAP ( dashed dotted line). 
Both Monte Carlo cross sections are scaled to the data cross sections by factors of
$1{.}08$ (DJANGOH) resp. 1{.}74 (RAPGAP)}
\label{fig:lo}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo_x1.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo_x2.pdf}
\end{center}
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo_costhet.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo_cospsi.pdf}
\end{center}
\caption{Differential cross sections of the normalised energies $X_i'= \frac {E_i'}{E_1' +E_2' + E_3'}$ of the
two leading jets ($E_1'>E_2'>E_3')$ in the 3-jet
center of mass frame and the two angles $\theta'$ and $\psi'$ as defined in figure
\ref{Tevatronvar} at hadron jet level  compared to the two LO Monte Carlo programs.
Other details are as in the caption to fig.\ \ref{fig:lo}}
\label{fig:loteva}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo2fwd_costhet.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot2fwd_costhet.pdf}
\end{center}
\begin{center}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plotlo2fwd_cospsi.pdf}
\hspace*{5mm}
\includegraphics[width=0.4\linewidth,keepaspectratio]{plot2fwd_cospsi.pdf}
\end{center}
\caption{Differential cross sections in the two angles as introduced in figure
\ref{Tevatronvar} for the sample with two forward jets. Top: $\cos\theta'$, bottom:
$\cos\psi'$. On the left side, the data are compared to both LO Monte Carlo
programs (for details see caption to figure \ref{fig:lo}), on the right side
the parton jet level cross sections are compared to the LO and NLO prediction 
 (details see figure \ref{fig:njet}),
normalised to the data cross
section by applying a factor or $1{.}34$ to both the LO and the NLO prediction}
\label{fig:lofwd}
\end{figure}
Figures \ref{fig:lo} and \ref{fig:loteva} show the comparison for the  threejet cross sections in 
the Bj{\o}rken scaling
variable $x$, the difference of the pseudorapidity of the two leading $P^*_\perp$ jets
($\eta_1-\eta_2$), the variables $X_1, X_2$  and the two threejet angles $\cos\theta'$
and $\cos\psi$.
Figure \ref{fig:lofwd} shows the two angles for the 2 forward jet sample compared to the LO MC predictions
at hadron jet level and  also the parton jet level cross sections 
compared to the fixed order 
NLO++ QCD predictions.
Since the absolute normalisation of the RAPGAP and DJANGOH predictions are too
low by 74\% and 8\% respectively, the Monte Carlo predictions are scaled up to the data
in order to compare only the shape of the distributions. The same is done for the NLO comparison. 
RAPGAP fails to describe the data. DJANGOH+CDM on the other hand compares
remarkably well. It starts to fail however in describing the threejet angles where especially for the 2 forward jet sample the NLO prediction is significantly better. 


\section{Summary}
In contrast to inclusive and dijet jet studies, threejet events in DIS require at
least the radiation of one hard gluon in addition to the two partons from the dominating 
hard gluon-fusion  scattering process. They are therefore ideally suited to study gluon
radiation at low $x$ and to search for unordered radiation since kinematic
regimes can be selected which
lead to a good separation of partons from the hard scattering process and the radiated hard gluons. 
As known since a long time from various dijet and inclusive
forward jet measurements, the data show a large excess of forward jets at low $x$
compared to LO DGLAP predictions. Such an excess is also
seen in this analysis of threejet events. However, this
excess can neither be explained by additional $k_\perp$ ordered initial state radiation 
nor by the addition of a resolved photon contribution which still worked for inclusive
forward jets and
dijets. The addition of non $k_\perp$-ordered gluon radiation as implemented
in the color dipole model (CDM) on the
other hand gives a remarkably good description of the threejet events and even
of higher multiplicities.
The most remarkable result of the present analysis however is the success of the fixed
order QCD prediction in order $\alpha_{\rm s}^3$. The addition of diagrams which allow
two gluons to be radiated improves the agreement between data and the prediction
dramatically and closes most of the gap between
the measured cross sections and the LO (${\mathcal{O}}(\alpha_s^2)$) prediction,
especially for the kinematic selection with two central and one forward jet. The NLO
prediction is especially good in 
describing the relative angles of the threejet topology where it is significantly
better than the
description by the color dipole model. Remaining discrepancies are concentrated at $x$
values below $10^{-3}$
and events where two jets are going forward. This is a topology which is most sensitive to gluon
radiation because
the two forward jets are predominantly due to two radiated gluons.
We conclude therefore that unordered gluon emission plays
a significant role at low $x$. It is important to note that
the NLO prediction includes for the first time also additional terms
$\propto\alpha_{\rm s}\cdot\ln(1/x)$, which lead to unordered gluon
emission over the full phase space.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 thank J. Bartels for useful discussions. 

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%\input{bibliography}
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\begin{thebibliography}{99}
\def\Journal#1#2#3#4{{#1}{\bf #2} (#3) #4}
\def\NCA{Nuovo Cim.}
\def\NIM{Nucl. Instrum. Methods}
\def\NIMA{{ Nucl. Instrum. Methods} {\bf A}}
\def\NPB{{Nucl. Phys.}   {\bf B}}
\def\PLB{{Phys. Lett.}   {\bf B}}
\def\PRL{Phys. Rev. Lett. }
\def\PRD{{Phys. Rev.}    {\bf D}}
\def\ZPC{{Z. Phys.}      {\bf C}}
\def\EJC{{Eur. Phys. J.} {\bf C}}
\def\CPC{Comp. Phys. Commun. }
\def\PR{{Phys. Rev. }}
\def\PL{{Phys. Lett. }}
\def\RMP{{Rev. Mod. Phys. }}
\newcommand{\etal}{{\em et al.}}
%\bibitem{Pom93} P.~Bruni, G.~Ingelman, {\bibtitlefont Proc. of
%the Europhysics Conference, Marseilles, France, July 1993} 595; siehe
%auch {\tt http://www3.tsl.uu.se/thep/pompyt/}

%%%%%%%%%%%%%%%%%%%% Introduction
%\bibitem{Collins} 
%J.~Collins, \Journal{\PRD}{57}{1998}{3051} and erratum-ibid. {\bf D61}
%(2000) 019902.
\bibitem{forwardjet}
J.~Breitweg {\it et al.} [ZEUS Collaboration], {\it Measurement of dijet production in
neutral current deep inelastic scattering at high $Q^2$ and determination of
$\alpha_{\rm s}$\/}, Phys.\ Lett.\ B {\bf 507} (2001) 70 [arXiv:hep-ex/0102042],\\
A.~Aktas {\it et al.} [H1 Collaboration], {\it Inclusive dijet production at low Bj{\o}rken-x
in deep inelastic scattering\/}, Eur.\ Phys.\ J.\ C {\bf 33} (2004) 477
[arXiv:hep-ex/0310019],\\
A.~Aktas {\it et al.} [H1 Collaboration], {\it Measurement of dijet production at low
$Q^2$ at HERA\/}, Eur.\ Phys.\ J.\ C {\bf 37} (2004) 141 [arXiv:hep-ex/0401010].

\bibitem{forward}
A.~Aktas {\it et al.} [H1 Collaboration], {\it Forward jet production in deep inelastic
scattering at HERA\/}, arXiv:hep-ex/0508055.

\bibitem{zeusforward}
S.~Chekanov {\it et al.} [ZEUS Collaboration], {\it Forward jet production in deep inelastic
e p scattering and low-x  parton dynamics at HERA\/}, arXiv:hep-ex/0502029.

\bibitem{tevatron}
S.~Geer, T.~Asakawa, {\it The Analysis of Multijet Events Produced at High Energy
Hadron Colliders\/}, Phys.\ Rev.\ D {\bf 53} (1996) 4793 [arXiv:hep-ph/9510351].

\bibitem{RAPGAP}
H.~Jung, {\it Hard diffractive scattering in high-energy e p collisions and the Monte
Carlo generation RAPGAP\/}, Comput.\ Phys.\ Commun.\  {\bf 86} (1995) 147.

\bibitem{DJANGOH}
K.~Charchula, G.~A.~Schuler, H.~Spiesberger, {\it Combined QED and QCD radiative effects
in deep inelastic lepton - proton scattering: The Monte Carlo generator DJANGO6\/},
Comput.\ Phys.\ Commun.\  {\bf 81} (1994) 381;
http://wwwthep.physik.uni-mainz.de/~hspiesb/djangoh/djangoh.html.

\bibitem{CDM}
Y.~I.~Azimov, Y.~L.~Dokshitzer, V.~A.~Khoze, S.~I.~Troian, {\it The String Effect and QCD
Coherence\/}, Phys.\ Lett.\ B {\bf 165} (1985) 147;\\
G.~Gustafson, {\it Dual Description of a Confined Color Field\/},
Phys.\ Lett.\ B {\bf 175} (1986) 453;\\
G.~Gustafson, U.~Pettersson, {\it Dipole Formulation of QCD Cascades\/},
Nucl.\ Phys.\ B {\bf 306} (1988) 746;\\
B.~Andersson, G.~Gustafson, L.~Lonnblad, U.~Pettersson, {\it Coherence Effects in Deep
Inelastic Scattering\/}, Z.\ Phys.\ C {\bf 43} (1989) 625.

\bibitem{NLO++}
Z.~Nagy, Z.~Trocsanyi, {\it Multi-jet cross sections in deep inelastic scattering at
next-to-leading order\/}, Phys.\ Rev.\ Lett.\ {\bf 87} (2001), 082001
[hep-ph/0104315].

\bibitem{H1det}
I.~Abt {\em et al\/}. [H1 Collaboration], Nucl.\ Instrum.\ Meth.\ A {\bf 386}
(1997) 310;\\
I.~Abt {\em et al\/}. [H1 Collaboration], Nucl.\ Instrum.\ Meth.\ A {\bf 386}
(1997) 348.
\end{thebibliography}

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