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\title{ A measurement of multi-jet rates \newline in deep-inelastic
scattering at HERA}
\titlerunning{ }
\authorrunning{~}
\subtitle{\rm H1 Collaboration}
\author {\rm I.~Abt\inst{7} \and T.~Ahmed\inst{3} \and
V.~Andreev\inst{24}\and B.~Andrieu\inst{27}\and R.-D. Appuhn
\inst{11}\and M.~Arpagaus \inst{34}\and A.~Babaev \inst{25}\and H.~%
B\"arwolff \inst{33}\and J.~B\'an \inst{17}\and P.~Baranov \inst{24}\and
E.~Barrelet \inst{28}\and W.~Bartel\inst{11}, U.~Bassler \inst{28}\and
H.P.~ Beck \inst{35}\and H.-J.~Behrend \inst{11}\and A.~Belousov
\inst{24}\and Ch.~Berger \inst{1}\and H.~Bergstein \inst{1}\and
G.~Bernardi \inst{28}\and R.~Bernet \inst{34}\and G.~Bertrand-Coremans
\inst{4}\and M.~Besan\c con \inst{9}\and P.~Biddulph \inst{22}\and
E.~Binder \inst{11}\and A.~Bischoff \inst{33}\and J.C.~Bizot
\inst{26}\and V.~Blobel \inst{13}\and K.~Borras \inst{8}\and
P.C.~Bosetti \inst{2}\and V.~Boudry \inst{27}\and C.~Bourdarios
\inst{26}\and F.~Brasse \inst{11}\and U.~Braun \inst{2}\and W.~%
Braunschweig \inst{1}\and V.~Brisson \inst{26}\and D.~Bruncko
\inst{17}\and L.~B\"ungener \inst{13}\and J.~B\"urger \inst{11}\and
F.W.~B\"usser \inst{13}\and A.~Buniatian \inst{11,37}\and S.~Burke
\inst{19}\and G.~ Buschhorn \inst{25}\and A.J.~Campbell \inst{1}\and
T.~Carli \inst{25}\and F.~Charles \inst{28}\and D.~Clarke \inst{5}\and
A.B.~Clegg \inst{18}\and M.~Colombo \inst{8}\and J.A.~Coughlan
\inst{5}\and A.~Courau \inst{26}\and Ch.~Coutures \inst{9}\and
G.~Cozzika \inst{9}\and L.~Criegee \inst{11}\and J.~Cvach \inst{27}\and
S.~Dagoret \inst{28}\and J.B.~Dainton \inst{19}\and M.~Danilov
\inst{23}\and A.W.E.~Dann \inst{22}\and W.D.~Dau \inst{16}\and M.~David
\inst{9}\and E.~Deffur \inst{11}\and B.~Delcourt \inst{26}\and L.~%
Del~Buono \inst{28}\and M.~Devel \inst{26}\and A.~De~Roeck \inst{11}\and
P.~Dingus \inst{27}\and C.~Dollfus \inst{35}\and J.D.~Dowell
\inst{3}\and H.B.~Dreis \inst{2}\and A.~Drescher \inst{8}\and J.~Duboc
\inst{28}\and D.~ D\"ullmann \inst{13}\and O.~D\"unger \inst{13}\and
H.~Duhm \inst{12}\and R.~Ebbinghaus \inst{8}\and M.~Eberle \inst{12}\and
J.~Ebert \inst{32}\and T.R.~Ebert \inst{19}\and G.~Eckerlin
\inst{11}\and V.~Efremenko \inst{23} \and S.~Egli \inst{35}\and
S.~Eichenberger \inst{35}\and R.~Eichler \inst{34}\and F.~Eisele
\inst{14}\and E.~Eisenhandler \inst{20}\and N.N.~ Ellis \inst{3}\and
R.J.~Ellison \inst{22}\and E.~Elsen \inst{11}\and M.~ Erdmann
\inst{14}\and E.~Evrard \inst{4}\and L.~Favart \inst{4}\and A.~ Fedotov
\inst{23}\and D.~Feeken \inst{13}\and R.~Felst \inst{11}\and J.~%
Feltesse \inst{9}\and I.F.~Fensome \inst{3}\and J.~Ferencei
\inst{11}\and F.~Ferrarotto \inst{31}\and K.~Flamm \inst{11}\and
W.~Flauger \inst{11}\fnmsep\thanks{Deceased}\and M.~Fleischer
\inst{11}\and M.~Flieser \inst{25}\and G.~Fl\"ugge \inst{2}\and
A.~Fomenko \inst{24}\and B.~Fominykh \inst{23} \and M.~Forbush
\inst{7}\and J.~Form\'anek \inst{30}\and J.M.~Foster \inst{22}\and
G.~Franke \inst{11}\and E.~Fretwurst \inst{12}\and P.~ Fuhrmann
\inst{1}\and E.~Gabathuler \inst{19}\and K.~Gamerdinger \inst{25} \and
J.~Garvey \inst{3}\and J.~Gayler \inst{11}\and A.~Gellrich \inst{13}
\and M.~Gennis \inst{11}\and H.~Genzel \inst{1}\and R.~Gerhards
\inst{11} \and L.~Godfrey \inst{7}\and U.~Goerlach \inst{11}\and
L.~Goerlich \inst{6} \and N.~Gogitidze \inst{24}\and M.~Goldberg
\inst{28}\and A.M.~Goodall \inst{19}\and I.~Gorelov \inst{23}\and
P.~Goritchev \inst{23}\and C.~Grab \inst{34}\and H.~Gr\"assler
\inst{2}\and R.~Gr\"assler \inst{2}\and T.~ Greenshaw \inst{19}\and
H.~Greif \inst{25}\and G.~Grindhammer \inst{25} \and C.~Gruber
\inst{16}\and J.~Haack \inst{33}\and D.~Haidt \inst{11}\and L.~Hajduk
\inst{6}\and O.~Hamon \inst{28}\and D.~Handschuh \inst{11}\and
E.M.~Hanlon \inst{18}\and M.~Hapke \inst{11}\and J.~Harjes \inst{11}\and
R.~Haydar \inst{26}\and W.J.~Haynes \inst{5}\and J.~Heatherington
\inst{20} \and V.~Hedberg \inst{21}\and G.~Heinzelmann \inst{13}\and
R.C.W.~Henderson \inst{18}\and H.~Henschel \inst{33}\and R.~Herma
\inst{1}\and I.~Herynek \inst{29}\and W.~Hildesheim \inst{11}\and
P.~Hill \inst{11}\and C.D.~ Hilton \inst{22}\and J.~Hladk\'y
\inst{29}\and K.C.~Hoeger \inst{22}\and Ph.~Huet \inst{4}\and
H.~Hufnagel \inst{14}\and N.~Huot \inst{28}\and M.~ Ibbotson
\inst{22}\and H.~Itterbeck \inst{1}\and M.-A.~Jabiol \inst{9}\and
A.~Jacholkowska \inst{26}\and C.~Jacobsson \inst{21}\and M.~Jaffre
\inst{26}\and T.~Jansen \inst{11}\and L.~J\"onsson \inst{21}\and K.~%
Johannsen \inst{13}\and D.P.~Johnson \inst{4}\and L.~Johnson
\inst{18}\and H.~Jung \inst{2}\and P.I.P.~Kalmus \inst{20}\and
S.~Kasarian \inst{11}\and R.~Kaschowitz \inst{2}\and P.~Kasselmann
\inst{12}\and U.~Kathage \inst{16} \and H.~H. Kaufmann \inst{33}\and
I.R.~Kenyon \inst{3}\and S.~Kermiche \inst{26}\and C.~Keuker
\inst{1}\and C.~Kiesling \inst{25}\and M.~Klein \inst{33}\and
C.~Kleinwort \inst{13}\and G.~Knies \inst{11}\and W.~Ko \inst{7}\and
T.~K\"ohler \inst{1}\and H.~Kolanoski \inst{8}\and F.~Kole \inst{7}\and
S.D.~Kolya \inst{22}\and V.~Korbel \inst{11}\and M.~Korn \inst{8}\and
P.~Kostka \inst{33}\and S.K.~Kotelnikov \inst{24}\and M.W.~ Krasny
\inst{6,28}\and H.~Krehbiel \inst{11}\and D.~Kr\"ucker \inst{2}\and
U.~Kr\"uger \inst{11}\and J.P.~Kubenka \inst{25}\and H.~K\"uster
\inst{2} \and M.~Kuhlen \inst{25}\and T.~Kur\v{c}a \inst{17}\and
J.~Kurzh\"ofer \inst{8}\and B.~Kuznik \inst{32}\and D.~Lacour
\inst{28}\and F.~Lamarche \inst{27}\and R.~Lander \inst{7}\and
M.P.J.~Landon \inst{20}\and W.~Lange \inst{33}\and R.~Langkau
\inst{12}\and P.~Lanius \inst{25}\and J.F.~ Laporte \inst{9}\and
A.~Lebedev \inst{24}\and A.~Leuschner \inst{11}\and C.~Leverenz
\inst{11}\and S.~Levonian \inst{11,24}\and D.~Lewin \inst{11} \and
Ch.~Ley \inst{2}\and A.~Lindner \inst{8}\and G.~Lindstr\"om \inst{12}
\and F.~Linsel \inst{11}\and J.~Lipinski \inst{13}\and P.~Loch \inst{11}
\and H.~Lohmander \inst{21}\and G.C.~Lopez \inst{20}\and D.~L\"uers
\inst{25}\fnmsep$^\star$ \and N.~Magnussen \inst{32}\and E.~ Malinovski
\inst{24}\and S.~Mani \inst{7}\and P.~Marage \inst{4}\and J.~ Marks
\inst{10}\and R.~Marshall \inst{22}\and J.~Martens \inst{32}\and R.~%
Martin \inst{19}\and H.-U.~Martyn \inst{1}\and J.~Martyniak \inst{6}\and
S.~Masson \inst{2}\and A.~Mavroidis \inst{20}\and S.J.~Maxfield
\inst{19} \and S.J.~McMahon \inst{19}\and A.~Mehta \inst{22}\and
K.~Meier \inst{15} \and D.~Mercer \inst{22}\and T.~Merz \inst{11}\and
C.A.~Meyer \inst{35} \and H.~Meyer \inst{32}\and J.~Meyer \inst{11}\and
S.~Mikocki \inst{6,26} \and V.~Milone \inst{31}\and E.~Monnier
\inst{28}\and F.~Moreau \inst{27} \and J.~Moreels \inst{4}\and
J.V.~Morris \inst{5}\and K.~M\"uller \inst{35} \and P.~Mur\'\i n
\inst{17}\and S.A.~Murray \inst{22}\and V.~Nagovizin \inst{23}\and
B.~Naroska \inst{13}\and Th.~Naumann \inst{33}\and P.R.~ Newman
\inst{3}\and D.~Newton \inst{18}\and D.~Neyret \inst{28}\and H.K.~%
Nguyen \inst{28}\and F.~Niebergall \inst{13}\and C.~Niebuhr
\inst{11}\and R.~Nisius \inst{1}\and G.~Nowak \inst{6}\and G.W.~Noyes
\inst{3}\and M.~ Nyberg \inst{21}\and H.~Oberlack \inst{25}\and
U.~Obrock \inst{8}\and J.E.~ Olsson \inst{11}\and S.~Orenstein
\inst{27}\and F.~Ould-Saada \inst{13} \and C.~Pascaud \inst{26}\and
G.D.~Patel \inst{19}\and E.~Peppel \inst{11} \and S.~Peters
\inst{25}\and H.T.~Phillips \inst{3}\and J.P.~Phillips \inst{22}\and
Ch.~Pichler \inst{12}\and W.~Pilgram \inst{2}\and D.~Pitzl \inst{34}\and
S.~Prell \inst{11}\and R.~Prosi \inst{11}\and G.~R\"adel \inst{11}\and
F.~Raupach \inst{1}\and K.~Rauschnabel \inst{8}\and P.~ Reimer
\inst{29}\and S.~Reinshagen \inst{11}\and P.~Ribarics \inst{25}\and
V.~Riech \inst{12}\and J.~Riedlberger \inst{34}\and S.~Riess
\inst{13}\and M.~Rietz \inst{2}\and S.M.~Robertson \inst{3}\and
P.~Robmann \inst{35}\and R.~Roosen \inst{4}\and A.~Rostovtsev
\inst{23}\and C.~Royon \inst{9}\and M.~Rudowicz \inst{25}\and M.~Ruffer
\inst{12}\and S.~Rusakov \inst{24}\and K.~Rybicki \inst{6}\and
N.~Sahlmann \inst{2}\and E.~Sanchez \inst{25}\and D.P.C.~Sankey
\inst{5}\and M.~Savitsky \inst{11}\and P.~Schacht \inst{25} \and
P.~Schleper \inst{14}\and W.~von~Schlippe \inst{20}\and C.~Schmidt
\inst{11}\and D.~Schmidt \inst{32}\and W.~Schmitz \inst{2}\and A.~%
Sch\"oning \inst{11}\and V.~Schr\"oder \inst{11}\and M.~Schulz \inst{11}
\and B.~Schwab \inst{14}\and A.~Schwind \inst{33}\and W.~Scobel
\inst{12} \and U.~Seehausen \inst{13}\and R.~Sell \inst{11}\and
A.~Semenov \inst{23} \and V.~Shekelyan \inst{23}\and I.~Sheviakov
\inst{24}\and H.~Shooshtari \inst{25}\and L.N.~Shtarkov \inst{24}\and
G.~Siegmon \inst{16}\and U.~ Siewert \inst{16}\and Y.~Sirois
\inst{27}\and I.O.~Skillicorn \inst{10} \and P.~Smirnov \inst{24}\and
J.R.~Smith \inst{7}\and L.~Smolik \inst{11} \and Y.~Soloviev
\inst{24}\and H.~Spitzer \inst{13}\and P.~Staroba \inst{29}\and
M.~Steenbock \inst{13}\and P.~Steffen \inst{11}\and R.~ Steinberg
\inst{2}\and B.~Stella \inst{31}\and K.~Stephens \inst{22}\and J.~Stier
\inst{11}\and U.~St\"osslein \inst{33}\and J.~Strachota \inst{11} \and
U.~Straumann \inst{35}\and W.~Struczinski \inst{2}\and J.P.~Sutton
\inst{3}\and R.E.~Taylor \inst{36,26}\and V.~Tchernyshov \inst{23}\and
C.~ Thiebaux \inst{27}\and G.~Thompson \inst{20}\and I.~Tichomirov
\inst{23} \and P.~Tru\"ol \inst{35}\and J.~Turnau \inst{6}\and J.~Tutas
\inst{14} \and L.~Urban \inst{25}\and A.~Usik \inst{24}\and S.~Valkar
\inst{30}\and A.~Valkarova \inst{30}\and C.~Vall\'ee \inst{28}\and
P.~Van~Esch \inst{4} \and A.~Vartapetian \inst{11}\fnmsep\thanks{Visitor
from Yerevan Phys.~Inst., Armenia}\and Y.~Vazdik\inst{24}\and M.~Vecko
\inst{29}\and P.~Verrecchia \inst{9}\and R.~Vick \inst{13}\and G.~Villet
\inst{9}\and E.~Vogel \inst{1}\and K.~Wacker \inst{8}\and I.W.~Walker
\inst{18}\and A.~Walther \inst{8}\and G.~Weber \inst{13}\and D.~Wegener
\inst{8}\and A.~Wegner \inst{11}\and H.~P.Wellisch \inst{25}\and
L.R.~West \inst{3} \and S.~Willard \inst{7}\and M.~Winde \inst{33}\and
G.-G.~Winter \inst{11}\and Th.~Wolff \inst{34}\and L.A.~Womersley
\inst{19}\and A.E.~ Wright \inst{22}\and N.~Wulff \inst{11}\and
T.P.~Yiou \inst{28}\and J.~ \v{Z}\'a\v{c}ek \inst{30}\and P.~Z\'avada
\inst{29}\and C.~Zeitnitz \inst{12}\and H.~Ziaeepour \inst{26}\and
M.~Zimmer \inst{11}\and W.~ Zimmermann \inst{11} \and F.~Zomer
\inst{26}\newpage}
\institute{\renewcommand{\thefootnote}{{\rm\alph{footnote}}}%
\setcounter{footnote}{0}%
I. Physikalisches Institut der RWTH, Aachen, Germany\thanks{Supported by
the Bundesministerium f\"ur Forschung und Technologie, FRG under
contract numbers 6AC17P, 6AC47P, 6DO57I, 6HH17P, 6HH27I, 6HD17I, 6HD27I,
6KI17P, 6MP17I, and 6WT87P} \and III. Physikalisches Institut der RWTH,
Aachen, Germany$^{\rm a}$ \and School of Physics and Space Research,
University of Birmingham, Birmingham, UK\thanks{Supported by the UK
Science and Engineering Research Council} \and Inter-University
Institute for High Energies ULB-VUB, Brussels, Belgium \thanks{Supported
by IISN-IIKW, NATO CRG-890478} \and Rutherford Appleton Laboratory,
Chilton, Didcot, UK$^{\rm b}$ \and Institute for Nuclear Physics,
Cracow, Poland\thanks{Supported by the Polish State Committee for
Scientific Research, grant No. 204209101} \and Physics Department and
IIRPA, University of California, Davis, California,
USA\footnote{Supported in part by USDOE grant DE F603 91ER40674} \and
Institut f\"ur Physik, Universit\"at Dortmund, Dortmund, Germany$^{\rm
a}$ \and DAPNIA, Centre d'Etudes de Saclay, Gif-sur-Yvette, France \and
Department of Physics and Astronomy, University of Glasgow, Glasgow,
UK$^{\rm b}$ \and DESY, Hamburg, Germany$^{\rm a}$ \and I. Institut
f\"ur Experimentalphysik, Universit\"at Hamburg, Hamburg, Germany$^{\rm
a}$\and II. Institut f\"ur Experimentalphysik, Universit\"at Hamburg,
Hamburg, Germany$^{\rm a}$ \and Physikalisches Institut, Universit\"at
Heidelberg, Heidelberg, Germany$^{\rm a}$ \and Institut f\"ur
Hochenergiephysik, Universit\"at Heidelberg, Heidelberg, Germany$^{\rm
a}$ \and Institut f\"ur Reine und Angewandte Kernphysik, Universit\"at
Kiel, Kiel, Germany$^{\rm a}$ \and Institute of Experimental Physics,
Slovak Academy of Sciences, Ko\v{s}ice, Slovak Republik \and School of
Physics and Materials, University of Lancaster, Lancaster, UK$^{\rm b}$
\and Department of Physics, University of Liverpool, Liverpool, UK$^{\rm
b}$ \and Queen Mary and Westfield College, London, UK$^{\rm b}$ \and
Physics Department, University of Lund, Lund, Sweden\footnote{Supported
by the Swedish Natural Science Research Council} \and Physics
Department, University of Manchester, Manchester, UK$^{\rm b}$ \and
Institute for Theoretical and Experimental Physics, Moscow, Russia \and
Lebedev Physical Institute, Moscow, Russia \and Max-Planck-Institut
f\"ur Physik, M\"unchen, Germany$^{\rm a}$ \and LAL, Universit\'{e} de
Paris-Sud, IN2P3-CNRS, Orsay, France \and LPNHE, Ecole Polytechnique,
IN2P3-CNRS, Palaiseau, France \and LPNHE, Universit\'{e}s Paris VI and
VII, IN2P3-CNRS, Paris, France \and Institute of Physics, Czech Academy
of Sciences, Praha, Czech Republik \and Nuclear Center, Charles
University, Praha, Czech Republik \and INFN Roma and Dipartimento di
Fisica, Universita "La Sapienzia", Roma, Italy \and Fachbereich Physik,
Bergische Universit\"at Gesamthochschule Wuppertal, Wuppertal,
Germany$^{\rm a}$ \and DESY, Institut f\"ur Hochenergiephysik, Zeuthen,
Germany$^{\rm a}$ \and Institut f\"ur Mittelenergiephysik, ETH,
Z\"urich, Switzerland\footnote{Supported by the Swiss National Science
Foundation} \and Physik-Institut der Universit\"at Z\"urich, Z\"urich,
Switzerland$^{\rm g}$ \and Stanford Linear Accelerator Center, Stanford
California, USA}
\date{Received 6 October 1993}
\maketitle

\begin{abstract}
Multi-jet production is observed in deep-inelastic electron proton
scattering with the H1~detector at HERA. Jet rates for momentum
transfers squared up to 500 GeV$^{2}$ are determined using the JADE jet
clustering algorithm. They are found to be in agreement with predictions
from QCD based models.
\end{abstract}

% The following code preserves some white space where the footnotes
% of the topmatter page can go.
\begin{figure}[b]\vspace{3.2cm}\end{figure}\setcounter{figure}{0}

\section{Introduction}
The study of deep-inelastic lepton scattering (DIS) off protons has been
extensively used for quantitative tests of QCD. To date most experiments
have studied the evolution of the inclusive structure functions with
momentum transfer $Q^2$, which is predicted in perturbative QCD. First
measurements of QCD effects in the hadronic final states have been
performed by fixed target experiments\cite{EMC0} and jet analyses have
been carried out ~\cite{E665}. These measurements suffer, however, from
the limited available energy and therefore limited phase space in which
to observe multi-jet production. This situation has changed with the
advent of the HERA electron-proton collider. With 26.7 GeV electrons
incident on 820 GeV protons, the accessible range in $Q^2$ is about two
orders of magnitude larger than in fixed target experiments. With our
1992 data sample and an integrated luminosity of 22.5 nb$^{-1}$ we give
results on jet rates for values of $Q^2$ up to 500~GeV$^{2}$, including
events with $Q^2$ up to 2600 GeV$^{2}$. The invariant mass $W$ of the
final state hadrons ranges from about 70 to 230 GeV, in contrast with
$e^+e^-$ experiments which typically measure at fixed $W$.

DIS from a proton constituent produces one or more partons with large
transverse momentum ($p_t$), which manifest themselves as high $p_t$
jets plus a proton remnant jet. In lowest order QCD, equivalent to the
quark parton model, a single quark (antiquark) is scattered, leading to
one current and one spectator jet --- a (1+1) jet configuration. To next
order in $\alpha_{\rm s}$, 2~jets are produced by gluon emission from
the scattering quark (QCD Compton) and by the photon-gluon fusion
process ($\gamma g \rightarrow q \bar{q}$)\footnote{For presently
accessible $Q^2$, $Z^0$ and $W^\pm$ exchange do not contribute
significantly}, leading to a (2+1) configuration. This makes for a more
complex situation than in $e^+e^-$ collisions where no strongly
interacting particles are present in the initial state. Higher order QCD
effects can be approximately described by parton showers as demonstrated
in $e^+e^-$ experiments. Exact QCD calculations to ${\cal O}(\alpha_{\rm
s}^2)$ have been performed for jet production in DIS~ \cite{order2} and
so, with good event statistics, new precision tests of QCD are possible
at HERA.

The H1 and ZEUS experiments have recently reported on first studies of
global properties of hadronic final states in deep-inelastic scattering
at HERA~\cite{H1HAD,ZHAD}. The ZEUS collaboration also presented results
on the characteristics of jet production based on a cone
algorithm~\cite{ZJET}. In this letter we study the multi-jet production
rates based on the JADE jet reconstruction algorithm~\cite{jade} and
compare the measurements with the predictions of QCD models. Preliminary
results of the present analysis have already been presented~
\cite{H1jet}.

\section{Detector description}
A detailed discussion of the H1 apparatus can be found
elsewhere~\cite{H1NIM}. Here we describe briefly the components of the
detector relevant to this analysis, which primarily makes use of the
calorimeters, and to a lesser extent the central and backward tracking
systems.

The scattered electrons and the hadronic energy flow are measured in a
liquid argon~(LAr) calorimeter and a lead-scintillator calorimeter
(BEMC). Leaking hadronic showers are measured in a surrounding
instrumented iron system. For the present analysis, the major part of
the hadronic energy flow ($> 90\%$) is measured in the LAr calorimeter.
The range $-3 < \eta < 3.2$ of pseudorapidity $\eta=-\ln(\tan
\frac{\theta}{2})$ is used for the energy flow measurement. Here and
elsewhere polar angles $\theta$ are defined with respect to the proton
beam direction ($z$ axis).

The liquid argon~(LAr) calorimeter~\cite{LARC} extends over a polar
angle range from $4^{0} < \theta < 153^{0}$ with full azimuthal
coverage. The calorimeter consists of an electromagnetic section with
lead absorbers (EMC), corresponding to a depth between 20 and 30
radiation lengths, and a hadronic section with steel absorbers (HAC).
The total depth of the LAr~calorimeter varies between 4.5 and 8 hadronic
interaction lengths. The calorimeter is highly segmented in both
sections with a total of around 45000 geometric cells. The EMC (HAC) has
a 3 to 4 (4 to 6) fold longitudinal segmentation. The lateral cell size
in the forward region corresponds to about 1 Moli\`{e}re radius in the
EMC and 1 interaction length in the HAC. The cells in the barrel region
are approximately twice this size. The electronic noise per channel is
typically between 10 and 30 MeV (1 $\sigma$ equivalent energy). The
method used to reconstruct the energy flow is described in
~\cite{H1NIM,H1PI}.

Test beam measurements of the LAr~calorimeter modules have demonstrated
that the energy resolution is $\sigma_{E}/E\approx
12\%/\sqrt{E\comm{(GeV)}}$ for electrons and $\approx
50\%/\sqrt{E\comm{(GeV)}}$ for charged pions~
\mbox{\cite{H1NIM,LARC,H1PI}}. Both the hadronic energy scale and
resolution from these beam tests have been verified from the balance of
transverse momentum between hadronic jets and the scattered electron in
DIS events, and they are known to a precision of $7\%$ and $20\%$
respectively. The uncertainty of the absolute scale for electrons is
$\approx 3\%$.

The lead-scintillator calorimeter BEMC, with a thickness of 22 radiation
lengths, covers the backward region of the detector ($151^\circ < \theta
< 177^\circ$). The main task of the BEMC, besides the contribution to
the energy flow measurement, is to trigger on and to measure electrons
scattered from DIS processes with $Q^2$ values ranging from 5 to 100
GeV$^{2}$. The resolution is presently
10\%/$\sqrt{E\comm{(GeV)}}\;\oplus\;3\%$. From a comparison of the
measured electron energy in the BEMC with a determination using only the
angles of both the hadronic system and the electron, the energy scale of
the BEMC is known to $2\%$.

The calorimeters are surrounded by a superconducting solenoid providing
a uniform magnetic field of $1.15$ T along $z$ in the tracking region.
The return yoke surrounding this coil is fully instrumented with
streamer tubes to measure leakage of hadronic showers and to recognize
muons.

The main components of the tracking systems, which are located inside
the calorimeters, are central drift and proportional chambers, a forward
track detector, and a backward multiwire proportional chamber, covering
the polar angle ranges $ 25^\circ < \theta < 155^\circ$, $ 7^\circ <
\theta < 25^\circ$ and $155^\circ < \theta < 175^\circ$ respectively.

A scintillator hodoscope situated behind the BEMC is used to veto
proton-induced background events based on their early time of arrival
compared with nominal electron-proton collisions.

\section{Data selection}\label{sec_sele}

The analysis is split into a ``low~$Q^2$~sample" $(Q^2 < 80$ GeV$^2$)
where the electron is measured in the BEMC, and a ``high~$Q^2$~sample"
$(Q^2 >$ 100 GeV$^2)$ where the electron is measured in the
LAr~calorimeter. The systematic uncertainties and the event selection
criteria are different for these samples. The hardware trigger for both
samples is based only on the scattered electron so as to avoid any
trigger bias in the hadronic final state.

Events in the ``low~$Q^2$~sample" satisfy the following requirements:
\begin{itemize}
\item an electron candidate in the BEMC, defined as a cluster associated
with a space point in the backward proportional chamber, with angle
$160^\circ < \theta_e< 172.5^\circ $ with respect to the proton beam
axis,
\item momentum transfer squared $Q^2 = 4 E_e E_e' \cos^2(\theta_e/2)$
between 12 and 80 GeV$^2$, where $E_e$ and $E_e'$ are the energies of
the incoming and scattered electron, and
\item $E_e' > 14$ GeV, corresponding to $y$ $\lsim$ 0.5, where the
scaling variable $y = 1-E_e'/E_e\sin^2(\theta_e/2)$, to eliminate
possible background from photoproduction.
\end{itemize}

A more detailed discussion of triggering and event selection for this
sub-sample can be found in~\cite{F2pap}.

The ``high~$Q^2$~sample" consists of events with the scattered electron
detected in the LAr calorimeter. The electromagnetic
cluster~\cite{H1NIM,H1PI} in the LAr~calorimeter with the largest
transverse energy is considered as a candidate electron if it fulfills
the following criteria:
\begin{itemize}
\item $Q^2 > 100$ GeV$^2$ and $y < 0.7$, \item it should be fully
contained in the LAr~calorimeter,
\item it must not be associated (to within a cone of half opening angle
of $5^{\circ}$) with a muon track candidate reconstructed in the
instrumented iron (to reject cosmic showers),
\item the energy deposited in the electromagnetic section in a cylinder
of radius between 15 and 30 cm around the electron direction must be
less than 1.2 GeV, and in the hadronic section within 30 cm around the
electron direction must be less than 0.5 GeV ( $\lsim 1\%$ of clusters
are lost by this isolation cut).
\end{itemize}
The following additional requirements are common to both samples:
\begin{itemize}
\item at least
one charged particle track is demanded in the central tracking system to
define an event vertex within $\pm 50$ cm in $z$ from the nominal
interaction point, and
\item $W^2 > 5000$ GeV$^2$, to ensure large enough hadronic energy flow
into the detector. The invariant mass squared of the hadronic system,
given by $W^2 = (p + q)^2$, with $p$ and $q$ the four-momenta of the
incoming proton and virtual photon respectively, is evaluated using the
angles of the electron and of the hadronic system (double angle
method~\cite{dameth}).
\end{itemize}

\noindent The final low and high $Q^2$ samples contain 769 and 47 events
respectively.

\section{Jet reconstruction}\label{sec_jetrec}
In the present analysis jets are reconstructed using the JADE
algorithm~\cite{jade} which was developed originally for $e^+e^-$
interactions. The two main reasons for this choice are a) in Monte Carlo
(MC) studies we find good correspondence between the jet rates
calculated at the parton level and at the level of fully simulated and
reconstructed events (see Sect.~\ref{sec_res}), and b) higher order QCD
predictions, to be compared with future larger data samples, are
available on the basis of the JADE clustering scheme~\cite{order2}.

We reconstruct ``particle" four-vectors using the energy in individual
calorimeter cells and the vector joining the reconstructed event vertex
with the centre of the cells. Use of clusters of calorimeter cells~
\cite{H1NIM} as ``particle" four-vectors instead of individual cells,
yielded equivalent results. The invariant mass of all ``particle" pairs
(or combined objects)~$(i,j)$ is then calculated in the JADE algorithm
using $m_{ij}^{2} = 2 E_{i} E_{j} (1-\cos \theta_{ij})$, thus neglecting
the masses of ``particles" $i$ and $j$, and the pair with the smallest
mass is taken to form a new combined object $k$ by adding the
four-vectors, $p_k=p_i+p_j$. The procedure is repeated until all
remaining pairs have masses $m_{ij}$ exceeding a cut-off which is
defined by $m_{ij}^2 > y_{\rm cut} M^{2}$ where $y_{\rm cut}$ is a
resolution parameter and $M$ is a mass scale taken to be the invariant
mass of all objects entering the cluster algorithm~ ($\approx W$,
including the beam pseudoparticle, see below). The energy deposition
attributed to the scattered electron is excluded from this procedure. To
account for the loss in the beam pipe of most hadrons from the proton
remnant we introduce a pseudoparticle with a longitudinal momentum given
by the missing longitudinal momentum of the event and no transverse
momentum~\cite{HERAWS}. The final results given in Sect.~\ref{sec_res}
are obtained with $y_{\rm cut} = 0.02$, corresponding to a cut in
$m_{ij}$ of 10 to 30 GeV depending on W.

We investigated implementations of the JADE algorithm in alternative
Lorentz frames and for different recombination schemes. We also tested
other jet algorithms~\cite{Durham,arclus}. No significantly better
agreement of the jet rates at the parton level with those obtained at
the hadron level was achieved, both before and after full simulation of
the detector.

\section{QCD models and event simulation}\label{sec_models}
In QCD to ${\cal O}(\alpha_{\rm s})$ the 2+1 jet configuration is
expected from the \mbox{$\gamma$-gluon} fusion process and the QCD
Compton diagrams. Some of the models with which we will compare the data
make explicit use of these matrix elements; others approximate the QCD
effects with parton showers.

The model MEPS is based on the ${\cal O}(\alpha_{\rm s})$ matrix
elements and an approximate treatment of higher order effects with
leading-log parton showers. The model MEPS is implemented in the Monte
Carlo generator LEPTO 6.1~\cite{Lepto}. We also use this generator
without the parton showers (ME), and without parton showers and QCD
matrix elements (QPM), to compare the data also with predictions from a
simple quark parton model. Further we study in the frame of LEPTO a
model (PSWQ) in which QCD effects are taken only to be leading-log
parton showers with a scale $W\cdot\sqrt{Q^2}$. This scale fixes the
maximum allowed virtuality at the beginning of the parton cascade.
Extreme choices of scales such as $W^2$ and $Q^2$, are already excluded
by previous data analyses~ \mbox{\cite{H1HAD,ZHAD}}. In MEPS the scale
for the parton showers is considerably restricted by the matrix element.
The hadronization of partons follows the Lund string model as
implemented in JETSET~\cite{jetset}. The PSWQ model includes QED
radiation from the electron, which is achieved by combining the
appropriate version of LEPTO with the radiative correction program
HERACLES~\cite{Heracles}. The radiative corrections for the (2+1) jet
rates at $y_{\rm cut}=0.02$ are less then 5\%.

In contrast with the Bremsstrahlung-like parton shower model used in
LEPTO, the colour dipole model does not distinguish between initial and
final state radiation and includes interference effects between them. It
describes gluon emission by a chain of radiating colour dipoles,
starting with a dipole formed between the scattered point-like quark and
the extended proton remnant. We use the implementation of this model in
ARIADNE~\cite{ariadne} in which the scale is given by the $p_t^2$ of the
radiated gluon with an upper limit proportional to $W^{4/3}$. Events are
generated with LEPTO 6.1 for the electroweak interaction and for the
\mbox{$\gamma$-gluon} fusion, followed by ARIADNE 4.03 which includes
QCD Compton and further gluon emission in the framework of the colour
dipole model. Such event simulations are henceforth referred to as CDM.

The data are also compared with the HERWIG event generator~\cite{Herwig}
where the QCD processes, without explicit ${\cal O}(\alpha_{\rm s})$
matrix elements, are based on leading-log parton showers which differ
from those in LEPTO. Here the scale is essentially $Q^2$. It sets the
upper limit for the shower evolution variables, which in HERWIG, in
contrast to LEPTO, are related to energies and angles rather than to
parton virtualities. As a result a more complete description of
coherence effects is possible with HERWIG. For the final hadronization
step, HERWIG uses a cluster fragmentation model.

No attempt has been made to tune the parameters of the models to the
data in the new kinematical region at HERA. For LEPTO and ARIADNE the
default parameters are used with the exceptions of $\Lambda_{\rm QCD}=
200$ MeV and of the $y_{\rm cut}$ at the parton level (PARL(8)=0.0025 in
case of MEPS and ME to stay well below the experimental jet resolution
parameter $y_{\rm cut}=0.02$). The decrease of $\Lambda_{\rm QCD}$ by 50
MeV lowers the (2+1) jet rate by less than 10\%. For HERWIG we use a
version specified by parameters suggested in~ \cite{MCgen}. None of the
models, besides PSWQ (see above), contains radiative corrections.

The jet fractions predicted by the models depend also on the proton
structure function used in the event generation. In particular the gluon
distribution is important at low $Q^2$ due to the contribution of
\mbox{$\gamma$-gluon} fusion to the jet production. In general we have
assumed the MRSD0 parametrization~\cite{MRSD}. The parton distributions
in such a parametrization enter the prediction for the fraction of (2+1)
jet events, $R_{2+1}$ in two ways. For the production of two non-remnant
jets with $m_{ij}^2 > y_{\rm cut} W^{2}$, the parton momentum fraction
$x$ of the proton which enters the hard scattering process must be
larger than $y_{\rm cut}$ ($=0.02$). For such parton $x$, present parton
density parametrizations differ only marginally. On the other hand, the
total number of events (in the denominator of $R_{2+1}$) averages over a
Bjorken $x$ range between $10^{-4}$ and $10^{-2}$ (for the
low~$Q^2$~sample) where the predictions of parton density
parametrizations, based on extrapolations of fixed target data, are
quite different. The predicted $R_{2+1}$ at $y_{\rm cut}=0.02$ using the
$MRSD^-$ parametrization~ \cite{MRSD} which is closer to the recent
measurements of the proton structure function $F_2$ at HERA
\cite{F2pap,ZF2} is less than that using the MRSD0 parametrization by
0.025 in the lowest $Q^2$ bin of our event sample. At higher $Q^2$ this
difference is smaller.

\begin{figure*}
\vspace{18.0cm}
{\bf Fig.1a-f.}Transverse energy flow for the low~$Q^2$~sample as a
function of $\Delta\phi$ and $\Delta\eta$, each with respect to the \rm
jet axis of non-remnant jets, for (1+1) jet events ({\bf a,b}
respectively) and for (2+1) \rm jet events ({\bf c,d} and {\bf e,f} for
the jets with smaller and larger $\eta$ respectively). The sense of
$\phi$ is away from the scattered electron for the jet at smallest
$\eta$. The points~ ($\bullet$) are the data, and the dashed and dotted
lines show the $E_t$ flow attributed by the JADE algorithm to the \rm
jet at smaller or larger $\eta$ respectively. The open and shaded
histograms are Monte Carlo simulations using MEPS and HERWIG
respectively
\end{figure*}
\setcounter{figure}{1}
\begin{figure*}
\def\thefigure{\arabic{figure}a-d}
\sidecaption
\fboxrule=0pt
\mpicplace{11.4cm}{8.9cm}
\caption {Fractions of N+1 jets ($R_{N+1}$) versus the cut variable of
the jet algorithm for $12 < Q^2 < 80$ GeV$^2$ ({\bf a,c}) and $Q^2 >
100$~GeV$^2$ ({\bf b,d}), compared in {\bf a, b} with simulation with
MEPS at the detector level, at the hadron level and at the parton level,
and in {\bf c, d} to predictions from the \rm QCD based models MEPS,
CDM, PSWQ and HERWIG}
\end{figure*}

\section{Results}\label{sec_res}

In Fig. 1 we show, for the low $Q^2$ sample and $y_{\rm cut} = 0.02$,
the flow of transverse energy, $E_t$, (transverse with respect to the
beam) as a function of $\Delta\eta$ and $\Delta\phi$. Here $\Delta\eta$
and $\Delta\phi$ are the distance of the calorimeter cells from the jet
axis in pseudorapidity $\eta$ and azimuth $\phi$~(radians) for
non-remnant jets found with the JADE algorithm. The angle $\phi$ is
taken to increase away from the scattered electron for the jet at
smallest $\eta$. The energy flow is shown for the (1+1) jet and the
(2+1) jet events. For the (2+1) jet events, the profiles are shown
separately for the jet at smaller (Fig. 1c,d) and larger (Fig. 1e,f)
pseudorapidity $\eta$ respectively. Also shown are the $E_t$ flows
actually attributed by the JADE algorithm to the jets. Distinct jet
structures are observed for both event classes. The jets are
predominantly at nearly opposite azimuth $\phi$ in the (2+1) jet event
class (Fig. 1c,e), as expected at low $Q^2$.

The simulations of the QCD models MEPS and CDM (the latter not shown in
Fig. 1) reproduce the gross features in shape and height of the observed
$\Delta\eta$ and $\Delta\phi$ jet profiles. Changing to the default
parameters of MEPS has little influence on the energy flow for the (2+1)
jet events. These distributions are also quite well described by the
parton shower model PSWQ which, however, fails to describe the forward
energy flow in the (1+1) jet events (not shown in Fig. 1b). For the
(2+1) jet events the HERWIG model, with present parameters, shows a lack
of $E_t$ flow in the region of the forward jet and between the jets
(Fig. 1 d to f), and fails to describe the separation in $\phi$ (Fig. 1
c,e) \footnote{Neither of these discrepancies are affected using the
available option of a ``soft underlying event"}. The level of agreement
between the data and the models in Fig. 1 is insensitive to variations
of $y_{\rm cut}$ in the range 0.01 to 0.03.

The dependence of the jet rates on the resolution parameter $y_{\rm
cut}$ (see Sect.~\ref{sec_jetrec}) is shown for the low and high~$Q^2$
data sample in Fig. 2a and b respectively. The data are not \rm
corrected for detector bias. $R_{N+1}$ is the fraction of $(N+1)$ jet
events in the sample. At $y_{\rm cut}=0.02$ the (1+1) and (2+1) classes
dominate, with $R_{2+1}\approx 10$~to~$20\%$. It should be noted that
the data points in Fig. 2 are strongly correlated as the same event
sample is used for each value of $y_{\rm cut}$. In this and the
following figures only the statistical errors are given. Systematic
effects are discussed in Sect.~\ref{sec_err}.

To check if these rates can be taken as a measure of jet multiplicities
at the parton level, the data in Fig. 2a and b are compared to Monte
Carlo predictions at the parton level (dotted curve), to the same model
after hadronization (dashed curve), and to the prediction including a
complete simulation of the H1~detector (full curve). In all cases the
MEPS model is used. The figure demonstrates that in the
high~$Q^2$~sample hadronization and detector effects are small ($\approx
10\%$). For the low~$Q^2$~sample the differences between these curves
are larger ($\approx 15\%$) for $y_{\rm cut} \gsim 0.02$. For lower
values of $y_{\rm cut}$ more event structure is resolved as ``jets"
because the jet algorithm then fails to include individual hadrons in
jet clusters.

It should be noted that the correspondence between parton and hadron
jets is not ``one to one" because there are migrations of events
classified as (2+1) jet at the parton level to the (1+1) class at the
hadron level and vice versa. This migration is because of the finite
resolution of the $y_{\rm cut}$ value at which a (2+1) jet event is
turned into a (1+1) event at the parton and hadron levels (see Table 1).

\begin{table}[h]
\caption{Probabilities for the events in the two $Q^2$ samples to be
classified as (2+1) jet or other (predominantly 1+1) at $y_{\rm{cut}} =
0.02$. The MEPS simulation has been used to correlate the multiplicities
a) at the hadron and detector level, and b) at the parton and hadron
level. }
\begin{tabular}{|c|c|c|c||c|c|}
\cline{3-6}
 \multicolumn{2}{c|}{ } &  \multicolumn{2}{c||}{low $Q^2$ sample } &
 \multicolumn{2}{c|}{high $Q^2$ sample} \\
 \hline
\multicolumn{2}{|l|}{a)} &\multicolumn{4}{c|}{ detector level} \\
\cline{2-6}
& \rm jet classes & not (2+1) &(2+1) & not (2+1) & (2+1) \\
\hline
hadron & not (2+1)   & 0.86   & 0.05        & 0.75   & 0.07     \\
\cline{2-6}
level & \rule{6.7mm}{0pt}(2+1)   & 0.04   & 0.05   & 0.06   & 0.12    \\
\hline
   \hline
 \multicolumn{2}{|l|}{b)} &\multicolumn{4}{c|}{ hadron level} \\
\cline{2-6}
& \rm jet classes & not (2+1) &(2+1) & not (2+1) & (2+1) \\
\hline
parton & not (2+1)   & 0.88   & 0.03   &   0.74   & 0.07      \\
\cline{2-6}
level & \rule{6.7mm}{0pt}(2+1) & 0.03   & 0.06   &  0.08   & 0.11   \\
\hline
\end{tabular}
\end{table}

All QCD-based models discussed in Sect.~\ref{sec_models} reproduce the
main dependence on $y_{\rm cut}$ of the jet rates for both data samples,
as shown in Figs. 2c,d where the statistical errors due to the MC
simulation are less than half those of the experimental data. The PSWQ
model prediction of the (2+1) jet rate in the region around $y_{\rm
cut}=0.02$ shows the most significant deviation from the data, namely by
$\approx +50\%$ at $Q^2 < 80$ GeV$^2$. The fraction of (0+1) jet events
(not shown in Fig. 2), for which all the hadronic energy is attributed
to the proton remnant jet, increases with $y_{\rm cut}$ somewhat more in
the data than in the simulated models.

The $Q^2$~dependence of $R_{2+1}$ (not corrected for detector bias) is
shown in Fig.3 for fixed $y_{\rm cut}=0.02$ together with the
expectation of the different models discussed above. The experimental
points are well described by the MEPS model. $R_{2+1}$ assuming the PSWQ
model systematically exceeds the measurement. $R_{2+1}$ using HERWIG and
especially CDM, with present parameters, shows little $Q^2$ dependence.
$R_{2+1}$ using the quark parton model (QPM, Sect.~\ref{sec_models} not
shown in Fig. 3) is $< 5\%$ of that measured.

\begin{figure}
\vspace{4.5cm}
\caption{Uncorrected (2+1) jet fraction $R_{2+1}$ at $y_{\rm{cut}}=0.02$
versus $Q^2$ compared with different QCD based models: MEPS, CDM, PSWQ
and HERWIG. The vertical error bars correspond to the statistical
errors. For systematic errors see Sect.~7}
\end{figure}
\begin{figure}
\vspace{4.5cm}
\caption{(2+1) jet fraction $R_{2+1}$ at $y_{\rm{cut}}=0.02$ versus
$Q^2$, corrected for detector bias for $W^2 > 5000~{\rm GeV}^2$ and
$y<0.5$, in comparison with the MEPS model with a running $\alpha_{\rm
s}$ and a constant $\alpha_{\rm s}=0.25$, and also with the matrix
elements but no parton showers~(ME~model). The data are plotted at the
\rm corrected $<Q^2>$ per bin. The error bars correspond to the
statistical errors of data and MC correction. For systematic errors see
Sect.~7}
\end{figure}

The models with ${\cal O}(\alpha_{\rm s})$ matrix elements, especially
MEPS, describe the observed energy flow and the jet fractions $R_{N+1}$
in a satisfactory way without any further adjustment of parameters. We
therefore correct the data in the above $Q^2$ bins for detector effects
using the MEPS model and the simple ansatz
\begin{eqnarray*}
R_{(2+1)}^{\rm corrected}(y_{\rm cut}=0.02,Q^2)=
\end{eqnarray*}
\begin{eqnarray*}
\frac{R_{(2+1)}^{\rm MC\ hadrons}}{R_{(2+1)}^{\rm MC\ recon.}}
R_{(2+1)}^{\rm data}(y_{\rm cut}=0.02,Q^2)
\end{eqnarray*}
where $R_{(2+1)}^{\rm MC\ hadrons}$ was obtained applying the JADE
algorithm to the hadrons generated by MC, with $W^2 > 5000$ GeV$^2$, and
for the low and the the high $Q^2$ sample $y < 0.5$. The fraction
$R_{(2+1)}^{MC\ recon.}$ was obtained from MC events, subjected to a
full simulation of the H1~detector and to the same analysis as the data.
The corrections for detector effects do not exceed $20\%$. The \rm
corrected (2+1) jet rate is given in Fig. 4 as a function of $Q^2$.

In the MEPS model, with parameters as specified in
Sect.~\ref{sec_models}, the ${\cal O}(\alpha_{\rm s})$ matrix elements
dominate the contribution to $R_{2+1}$ but the addition of parton
showers improves the description of the data. This can be seen in Fig. 4
by comparing the predictions of MEPS with the model ME which contains
the matrix elements but not the parton showers. The observed \rm jet
rate is mainly due to the \mbox{$\gamma$-gluon} fusion process. In the
low (high) $Q^2$ sample $\approx 75\%$ $(\approx 50\%)$ of (2+1) \rm jet
events simulated with MEPS are attributed to the \mbox{$\gamma$-gluon}
fusion graph.

Because of the $Q^2$ dependence of the coupling constant $\alpha_{\rm
s}$, the matrix element for \rm jet production decreases with increasing
$Q^2$. This decrease is not manifest directly in $R_{2+1}$ because of
the increase in phase space for \rm jet production at higher $Q^2$,
given the kinematic cuts of the present event sample and the \rm jet
definition. It is, however, interesting to test the sensitivity of the
$Q^2$ variation of the (2+1) \rm jet rate due to the running of
$\alpha_{\rm s}$ by comparing the data to a model with constant
$\alpha_{\rm s}=0.25$ (Fig. 4). The curve with a running coupling
constant ($\Lambda = 200$ MeV) gives a slightly better description of
the data. However the large statistical and systematic errors at this
stage allow no quantitative discrimination.

\section{Systematic errors}\label{sec_err}

The main sources of systematic uncertainty in the measured \rm jet
fractions are due to detector effects and the model dependent
description of the hadronic final state. To quantify these contributions
we use the variation of $R_{2+1}$ at $y_{\rm{cut}}= 0.02$.

The uncertainty of the energy calibration and resolution of the
LAr~calorimeter for this sample of low energy hadronic \rm jets ($7\%$
and $20\%$ respectively) leads to fractional errors of $7\%$ and $10\%$
for $R_{2+1}$ respectively. The limited hadronic response of the BEMC
calorimeter has little influence on $R_{2+1}$.Changing its calibration
by $50\%$ results in a fractional change of less than $5\%$ for
$R_{2+1}$.

Particles hitting material close to the beam in the forward region
outside the detector acceptance can generate additional energy flow in
the calorimeter. Increasing in simulation this additional energy flow by
$50 \%$ leads to a fractional change for $R_{2+1}$ of $\lsim 5\%$.
$R_{2+1}$ obtained with both the JADE algorithm and the assumption of a
pseudoparticle for the remnant \rm jet is also quite insensitive to the
calorimeter acceptance, despite the fact that in most cases one of the
two non-remnant \rm jets appears at small polar angle ($<\theta_{\rm
jet}>\approx 8^\circ$). For example a variation of the minimum polar
angle near the beam pipe between 3.7 and $6.7^\circ$ changes $R_{2+1}$
for data and simulation (MEPS) by less than $\pm 10\%$, and this is not
included in the overall systematic error.

The corrections for detector bias of the \rm jet fractions as a function
of $Q^2$ are determined using the MEPS model and found to be less than
$20\%$. A rough estimate of the model dependence is given by evaluating
the corrections with different models, whence an estimate of the
systematic uncertainty in using the MEPS model to correct for detector
bias is $14\%$.

Adding the errors above in quadrature, we estimate a total systematic
error of $\pm 14\%$ for the measured \rm jet fraction in Fig. 3 and an
uncertainty of $\pm 20\%$ for the \rm corrected \rm jet fraction in Fig.
4. These systematic errors will decrease with future larger data
samples, where it will be possible to improve the calorimeter
calibration and simulation and to tune the models.

\section{Conclusion}

Multi-jet production has been observed in the new kinematic domain of
lepton scattering at HERA. The \rm jets were reconstructed using the
JADE algorithm, requiring jet-jet masses above 10 GeV or more (depending
on W). Between 10 and $20\%$ of the events contain 2+1 jets (2 at high
$p_t$ and the proton remnant). Jet energy profiles and the dependence of
multi-jet production rates on the jet resolution parameter and on $Q^2$
have been presented. It has been shown that hadronization and detector
effects are small and controllable using the JADE jet reconstruction
algorithm. A good overall description of multi-jet production is
achieved using first order \rm QCD matrix elements with higher order
effects modelled by parton showers.

\begin{acknowledgement}
We are grateful to the HERA machine group whose outstanding efforts made
possible this experiment. We acknowlege the support of the DESY
technical staff. We are grateful for the immense contributions of the
engineers and technicians who constructed and who now maintain the
H1~detector. We thank the funding agencies for financial support. We
also wish to thank the DESY directorate for the hospitality extended to
the non-DESY members of the collaboration.
\end{acknowledgement}

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\end{document}
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