\documentclass[12pt,a4paper,epsf,portrait,times,epsfig]{article}
\usepackage[dvips]{graphics}
\usepackage{epsfig}
\newlength{\dinwidth}
\newlength{\dinmargin}
\setlength{\dinwidth}{21.0cm}
\textheight23.5cm \textwidth16.0cm
\setlength{\dinmargin}{\dinwidth}
\setlength{\unitlength}{1mm}
\addtolength{\dinmargin}{-\textwidth}
\setlength{\dinmargin}{0.5\dinmargin}
\oddsidemargin -1.0in
\addtolength{\oddsidemargin}{\dinmargin}
\setlength{\evensidemargin}{\oddsidemargin}
\setlength{\marginparwidth}{0.9\dinmargin}
\marginparsep 8pt \marginparpush 5pt
\topmargin -42pt
\headheight 12pt
\headsep 30pt \footskip 24pt
\parskip 3mm plus 2mm minus 2mm
 
\newcommand{\qsq}{\ensuremath{Q^2}}
\newcommand{\gevsq}{\ensuremath{\mathrm{GeV}^2}}
\newcommand{\et}{\ensuremath{E_t^*} }
 
\begin{document}
\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
                & & & \\
\multicolumn{4}{l}{{\bf
                XXII International Symposium on 
                Lepton-Photon Interactions at High Energy, LP2005}} \\ 
\multicolumn{4}{l}{{
                June~30,~2005,~Uppsala}} \\
                 & Abstract:        & {\bf 390}    &\\
                 & Session: & {\bf QCD and hadron structure}   &\\ 
                 & & & \\
%\multicolumn{4}{l}{{\bf
%                International Europhysics Conference 
%                on High Energy Physics, EPS2005},
%                July~21,~2005,~Lisbon} \\
%                 & Abstract:        & {\bf xx-xxx}    &\\
%                 & Parallel Session & {\bf Hard QCD}   &\\ 
%                 & & & \\
                \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{1cm}

\begin{center}
\begin{Large}
{\bf Multi-jet production in high Q\textsuperscript{2} neutral current \linebreak deeply inelastic scattering at HERA and \linebreak
determination of $\alpha_{s}$}

\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{2cm}

%\title{Multi-jet production in high Q\textsuperscript{2} neutral current \linebreak deeply inelastic scattering at HERA and \linebreak
%determination of $\alpha_{s}$}
%\date{ }
%\author{H1 Collaboration}
%
%\maketitle

\begin{abstract}
\noindent
Deep-inelastic $e^{+}p$ scattering data, taken with the H1 detector at HERA, are used to investigate jet production over a range of four-momentum transfers $150 <$ Q\textsuperscript{2} $< 15000~$GeV\textsuperscript{2} and transverse jet energies $5 < E_{T} < 50$ GeV. The analysis is based on data corresponding to an integrated luminosity of $\cal L$$_{int}$ = 65.4 pb\textsuperscript{-1} taken in the years 1999-2000 at a centre-of-mass energy $\surd{s} = 319$ GeV.  Jets are defined by the inclusive $k_{\perp}$ algorithm in the Breit frame of reference.  Dijet and trijet cross sections are measured with respect to the exchanged boson virtuality and in addition the ratio of the trijet to the dijet cross section $R_{3/2}$ is investigated.  The results are compared to the predictions of perturbative QCD calculations in next-to-leading order in the strong coupling $\alpha_{s}$.  The value of $\alpha_{s}(M_{Z})$ determined from the study of $R_{3/2}$, is $\alpha_{s}(M_{Z})$ = 0.1175 $\pm$ 0.0017 (stat.) $\pm$ 0.0050 (syst.) $^{+0.0054}_{-0.0068}$ (th.).
\end{abstract}

\vspace{1.5cm}

\begin{center}
%To be submitted to Phys. Lett. B
\end{center}

\end{titlepage}

\section{Introduction}

\noindent
Deeply inelastic lepton-proton scattering (DIS) experiments have been vital to our understanding of the nature
 of the strong force.  This has resulted in quantum chromodynamics (QCD) becoming established as the theory to 
describe the strong force in terms of interactions between quarks and gluons that carry different \emph{colour} 
charges.  QCD contains just one free parameter, the strong coupling $\alpha_s$.  The value of this coupling is
 known to run with the interaction energy scale and so $\alpha_{s}$ is usually quoted as its value at a scale that is 
very precisely known - the mass of the Z$^{0}$ boson.  There are numerous experimental methods for the determination
 of $\alpha_{s}$, for example by the consideration of fragmentation functions or by studying violations of 
Bj\"orken scaling.  Furthermore, the production rates of DIS events containing more than one hard (high $E_{T}$)
 jet are directly sensitive to $\alpha_s$\cite{DetAlphS}.  By measuring the ratio of three and two jet event 
cross-sections ($R_{3/2}$), a value for $\alpha_{s}$ can be determined\cite{ZEUS3jet}.  For this method of extracting
 $\alpha_s$, precise values for both the inclusive dijet and trijet cross-sections are required.  This 
document presents a measurement of the inclusive dijet and trijet cross-sections, as well as an 
extraction of $\alpha_s$ from a study of R$_{3/2}$.
 
%\vspace{2 mm}
%\noindent
%The HERA $ep$ collider has a centre-of-mass energy $\surd{s} \sim 319$ GeV, which allows the study of 
%multi-jet production in DIS over a large region of phasespace.  The phasespace region specified constrains 
%the analysis to consider only events with high values of 
%virtuality ($Q^{2} > 150$GeV\textsuperscript{2}) for the exchanged boson.  The data are compared to next-to-leading order
%(NLO) perturbative QCD (pQCD) predictions that are $\cal O$($\alpha_{s}^{2}$) for dijets and $\cal O$($\alpha_{s}^{3}$) for trijets.
% A factor two increase in integrated luminosity, compared with several previous investigations\cite{H1Wobisch,H13jet,ZEUS}, 
%allows for some reduction of statistical uncertainties.

\section{Theoretical Framework}
%\subsection{Deeply Inelastic Scattering}
%\noindent
%The proton is believed to be composed of quarks and gluons (partons) that carry non-zero colour charge.  
%The process of neutral current deeply inelastic scattering (NC DIS) is the name given to the $ep$ collision 
%process where the incident electron\footnote{In this document, \emph{electron} refers to either an electron or a
% positron.} emits a high energy vector boson (either a $\gamma \ \rm or \ Z^{0}$) that interacts with a parton
% within the target proton, causing the proton to disintegrate (figure \ref{NCEvent1}).  The struck parton
% rapidly converts into a collimated spray of colourless hadrons known as a \emph{jet}.  Due to the confinement
% of the colour charge there can be no unique association of a jet with a single initial quark or gluon.  At 
%leading order in $\alpha_s$, dijet production takes place in NC DIS interactions via the QCD-Compton and 
%boson-gluon fusion processes as illustrated in figure \ref{NCEvent2}.  The contribution from the latter of these processes 
%results in the multi-jet cross-section being directly sensitive to the gluon density within the proton.  Three jet production can 
%be modeled in a similar way as dijet production, with the third jet being due to an additional non-collinear bremsstralung gluon 
%(figure \ref{NCEvent3}).

%\vspace{2mm}
%\noindent
%The lowest (leading) order dijet processes illustrated in figure \ref{NCEvent2} each 
%contain one strong coupling vertex, indicating that LO calculations for dijets contain terms $\cal O$($\alpha_{s}$).  
%The lowest order three jet processes (figure \ref{NCEvent3}) contain two such strong vertices, meaning that LO
% calculations for three jet events contain terms $\cal O$($\alpha_{s}^{2}$).  In this document, the order stated refers 
%to the order of the process being described and not the order of $\alpha_s$ utilised in the calculations.

%\subsection{DIS Kinematics}

%The kinematics for a NC DIS event may be completely specified by any two of a range of kinematic 
%variables\cite{GaugeTheories, QFT}.  The simplest DIS event is illustrated in figure \ref{NCEvent1}.  The 
%incident electron has a four-momentum $k^{\mu}$ and the target proton has a four-momentum $P^{\mu}$ given by:

%\begin{equation}
%k^{\mu} = (E_{e}, {\bf{p_{e}}})\hspace{3mm} ,\hspace{3mm}P^{\mu} = (E_{p}, {\bf{p_{p}}})
%\end{equation}

%\noindent
%where E and p represent energy and momentum respectively. The four-momenta of the scattered electron is 
%given by: 

%\begin{equation}
%{k'^\mu} = ({E'_e}, {\bf{{p'_e}}})
%\end{equation}

%\vspace{2mm}
%\noindent
%We define the four momentum transfer $Q^{2}$, the Bj\"orken scaling variable $x_{Bj}$ and the inelasticity $y$ 
%as:

%\begin{equation}
%Q^{2} = -q^{2} = -(k^\mu -{k'^\mu})^{2}
%\end{equation}

%\begin{equation}
%x_{Bj} = \frac{Q^{2}}{2P^{\mu}q_{\mu}}
%\end{equation}

%\begin{equation}
%y = \frac{P^{\mu}q_{\mu}}{P^{\mu}k_{\mu}}
%\end{equation}

%\vspace{2mm}
%\noindent
%In the limit $\bf{p_{p}}\to\infty$, the variable $x_{Bj}$ corresponds to the fraction of the proton momentum 
%carried by the struck parton.  

\subsection{The Breit Frame}
\noindent
This analysis has been conducted in the Breit inertial frame of reference\cite{BreitFrame}.   The Breit frame is defined as the
 frame in which the exchanged boson is completely spacelike such that the relation $2x_{Bj}P^{\mu} + q = 0$
holds, where the incident proton four-vector is given by $P^{\mu} = (E_{p}, {\bf{p_{p}}})$ and $q$ is the momentum
transfer.  The velocity of the Breit frame with respect to the laboratory frame is given by: 
\begin{equation}
{\bf{\beta}} = \frac{({\bf{{q}}} + 2x_{Bj}{\bf{{p_p}}})}{(q_{0} + 2x_{Bj}E_{p})}
\end{equation}

\vspace{2mm}
\noindent
The fourvectors describing the DIS events are boosted and rotated such that the exchanged virtual boson is 
orientated along the negative $z$ axis and such that the scattered lepton is directed in the positive $x$ 
direction.  It can be seen from the definition of the Breit frame that the transverse\footnote{Transverse 
refers to the component perpendicular to the $z$-axis.} energy of the hadronic final state does not have to 
balance the transverse energy of the scattered lepton, as is required in the laboratory frame.  Instead, it 
directly reflects the hardness of the underlying QCD process.  At lowest order the 
quark from the proton is backscattered and no transverse energy is produced.  Jets with high $E_{T}$ in this 
frame can only be accounted for by hard QCD processes whose contribution is at least $\cal O$($\alpha_{s}$) 
(figure \ref{breit})\cite{Breit}.

\section{Event Simulation}

\noindent
The order of magnitude and general features of the distributions for the jet cross-sections are well described
 by leading order (LO) calculations.  More accurate predictions are required to include next-to-leading order (NLO) processes.
  Full NLO event simulations are not yet available for DIS events.    An
 alternative method to obtain an approximation to higher order calculations is to use programs based on the 
parton cascade models.  Programs such as RAPGAP\cite{RAPGAP}, HERWIG\cite{HERWIG} and LEPTO\cite{LEPTO} 
contain additions to the LO matrix element based on soft and collinear emission approximations, described by 
parton shower models.  The program ARIADNE\cite{ARIADNE} utilises the colour dipole model for these 
approximations.  Programs to perform parton level cross-section calculations, at NLO,
in DIS for jet multiplicities upto and including three jets are available and are employed in this study.

\vspace{2mm}
\noindent
In this analysis the programs DJANGOH (version 1.2)\cite{DJANGO} (used as an interface for ARIADNE) and RAPGAP (version 28)
are utilised.  These programs simulate the hard (DIS) scattering process for leptons on protons, and include first order 
QED radiative corrections (via the HERACLES program\cite{HERACLES}) as well as initial and final state QCD radiation ($\cal O$($
\alpha_{s}$)).  
The parton density function (PDF) used in these 
programs is CTEQ5L. DJANGOH and RAPGAP account for the running of $\alpha_{s}$ to second order.  The hadronisation of
 the final state following the breakup of the proton is not described by ARIADNE or RAPGAP.  The hadronisation process is
 simulated using the Lund string fragmentation model\cite{Lund} implemented in the program JETSET 
(version 7.4).  Previous publications\cite{JetStruct} have shown that the event generators used in this 
analysis provide an acceptable description of the internal structure of jets.


%\section{The H1 Detector}

%The H1 detector is situated at the HERA\footnote{Hadron Electron Ring Anlage.} $ep$ collider at DESY, Hamburg, which 
%collides 27.6 GeV electrons with
% 920 GeV protons at a center of mass energy of $\sqrt{s}\sim319 GeV$ in order to probe the structure of the 
%proton at the most fundamental distance scales.  Only the components of H1 relevant for this analysis are 
%described here.  A detailed description of the H1 detector may be found elsewhere \cite{H1Detector}.  
%Figure \ref{Detector1} shows a schematic of the H1 detector.  Due to the asymmetry in the energy of the 
%electron and proton beams at HERA, the centre of mass for $ep$ collisions is strongly boosted along the proton
% direction.  To account for this the H1 detector is asymmetric in design\cite{H1Detector}, being considerably 
%more massive and highly segmented in the forward direction. This may be seen in figure \ref{Detector1}, which
% shows a cut along the beam axis.

%\vspace{2mm}
%\noindent
%The polar angle $\theta$ is measured with respect to the direction of the proton beam.  The H1 coordinate 
%system is a right handed Cartesian system, with the Z axis pointing in the proton beam direction, known as the
% ``forward direction'', and the X axis pointing left towards the centre of HERA.  The coordinate origin is at 
%the nominal interaction point, which is at the centre of the inner silicon tracker.

%\vspace{2mm}
%\noindent
%In the angular range $4^{\circ} < \theta < 154^{\circ}$, electromagnetic and hadronic energies deposited by 
%particles from the $ep$ collision are measured by the Liquid Argon (LAr) calorimeter, consisting of an 
%electromagnetic section with lead absorbers and a hadronic section with steel absorbers.  The backward region 
%($153^{\circ} < \theta < 177^{\circ}$) is covered by a lead-fiber calorimeter, known as SPACAL. 
%Charged particle tracks are measured in two concentric drift chamber modules (CJC), covering the polar angular 
%range $25^{\circ} < \theta < 165^{\circ}$.
%  A forward tracking detector covers $7^{\circ} < \theta < 25^{\circ}$ and consists of drift chambers with 
%alternating planes of parallel wires and others with wires in the radial direction.  The calorimeters and 
%tracking chambers are surrounded by a superconducting solenoid providing a uniform magnetic field of 1.15T 
%parallel to the beam axis in the tracking region.

%\vspace{2mm}
%\noindent
%Luminosity is measured using the Bethe-Heitler process $ep \to e\gamma p$.  The final state positron and 
%photon are detected in calorimeters situated close to the beam pipe\cite{Lumi}.

\section{Event selection and jet search}
\vspace{2mm}
\noindent
The differential cross-sections presented in this report are given for the kinematic region defined by 
$150 < Q^{2} < 15000\ GeV^{2}$ and $0.2 < y < 0.6$.
%%%% and $s = 4E_{p}E_{e}$.

\subsection{Data sample}
\noindent
The data sample was collected with the H1 detector during the 1999-2000 
$e^+p$ data taking period at HERA and 
corresponds to an integrated luminosity of 65.4 pb\textsuperscript{-1}.  
%%%%Data taken during this time using
%%%%electrons instead of positrons are not considered.  
The events chosen for this analysis are 
characterised by a high-energy isolated positron in the central region of the detector.  The event selection criteria closely 
follow that used in recent measurements of the inclusive DIS cross-section\cite{DISCuts}, the dijet cross 
section\cite{H1Wobisch} and the trijet cross section\cite{H13jet}.  %The following selection criteria were applied:
%\begin{itemize}
%\item A reconstructed vertex with $-32.2 < Z < 37.8\ cm$ was required to ensure events are consistent with inelastic $ep$ collisions;
%\item  In order to suppress photoproduction events, $E - P_{Z}$ was required to be greater than 45 GeV.  An 
%upper limit of 65 GeV was also applied to $E - P_{Z}$ to reduce any cosmic ray background;
%\item The scattered positron was required to have an energy of at least 11 GeV with a polar angle 
%$\theta_e < 153^{\circ}$. For $\theta_e < 35^{\circ}$ the positron candidate is only valid if it can be 
%associated with a reconstructed track pointing in the direction of the positron cluster;
%\item  Fiducial cuts were made to avoid cracks in the detector acceptance in both $\phi$ and $Z$ directions.
%Regions of the LAr, where the trigger is $<$ 95$\%$ efficient, are also removed.
%The remaining events were reweighted to account for the event loss, using the concept of phi symmetry;
%\item To ensure sound event reconstruction, the sub-detectors CJC1, CJC2, CIP, COP, TOF, LAr, SPACAL and Lumi system
%are required to have been enabled;
%\item The event interaction vertex must be found in the central region of the detector and the scattered electron
%must be found in the LAr detector;
%\item A requirement on the CJCT0 of $\pm 20$ ns is made for data events, to remove non $ep$ background;
%\item The polar angle of the hadronic final state must be $>$8$^{\circ}$ due to the geometric acceptance of the LAr;
%\item In order to reduce the number of charged current events that make it into the event sample, a cut of $P_{Tmiss} <$ 15 GeV
%is applied.
%Standard cuts to remove background from lepton pairs and cosmic rays were also applied. 
%\end{itemize}
%\vspace{2mm}
%\noindent
%The subtrigger combination (S67 $\|$ S75 $\|$ S77) is used for triggering events, with S71 used as a monitor trigger.
%The trigger efficiencies for the final jet event samples are above 99$\%$ % (figure \ref{trigEff}).
%
%\vspace{2mm}
%\noindent
The variables $Q^{2}$, $x_{Bj}$ and $y$ were determined using the electron-sigma method \cite{ElecSig}.

\subsection{Jet search}
\noindent
The inclusive $k_{\perp}$-cluster algorithm was used to reconstruct jets in both data and simulated events.  The 
algorithm is defined by a radius parameter $R_o$, which defines the minimal separation of jets in 
pseudorapidity\footnote{Pseudorapidity is defined as ${\eta} = -ln(tan\frac{\theta}{2})$ where $\theta$ is the
 polar angle.} and azimuth space.  $R_o$ was set to unity as in \cite{Kt}.  A complete description of the 
inclusive $k_{\perp}$-clustering algorithm can be found in \cite{H1Wobisch, Kt}.

\vspace{2mm}
\noindent
Only jets with $E_{T} > 5$ GeV are considered for the jet cross section measurements.  Jets must also fall within the 
pseudorapidity range $-1 < \eta^{jet}_{Lab} < 2.5$. In order to reduce infra-red sensitive regions of phase space, and 
thus ensure the stability of the NLO calculations, events considered for the trijet (dijet) cross section are included only
if the three (two) jets of highest $E_{T}$ have an invariant mass $M_{3jet} > 25$~GeV  ($M_{2jet} > 25$~GeV).

\vspace{2mm}
\noindent
The range of the inelasticity variable, $y$, was chosen to exclude regions with large $x_{Bj}$ (where jets tend to be produced in the 
forward direction, at the edge of detector acceptance) and to ensure large energies of the scattered positron
 respectively.   In this document the proton remnant is not included in the jet count and is neglected.

\vspace{2mm}
\noindent
After all cuts have been made a total of 5460 dijet events and 1757 trijet events remained.

\section{Correction Procedure} 
\noindent
The data are corrected for limiting features of the detector, such as resolution and acceptance.  Samples of 
simulated events were created using the generators ARIADNE, which is implemented via the DJANGOH program (version 
1.2), and RAPGAP (version 28).  Both DJANGOH and RAPGAP provide a good description of the distributions
of the experimental data for the DIS, dijet and trijet samples\footnote{The jet samples from RAPGAP require an aditional renormalisation factor of $\sim$ 1.2.}
 (figures \ref{DISCont1} - \ref{3jetCont}) and so the weighted average of the correction factors from both simulations 
were utilised in the correction procedure.

\vspace{2mm}
\noindent
%The number of events, per $Q^{2}$ bin, generated by the simulations under the phasespace conditions specified was recorded and is 
%referred to as the \emph{generator} level set. The event sample is then subjected to a simulation of the H1 
%detector.  The simulation of the H1 detector and reconstruction of the events were performed using the 
%standard H1 software packages H1sim and H1rec\cite{GEANT}.  The number of events per bin following the 
%detector simulation is referred to as the \emph{reconstructed} level set.  
\noindent
The correction functions are determined on a bin-by-bin basis as the ratio of the value of the generator and reconstructed level sets. 
It was found that the values of the correction functions are typically $\sim$ 1.10 for dijet events and $\sim$ 0.95 for trijet 
events %(figure \ref{detCorr}).  
The ratio of generator level events without QED radiation to those with, yields QED correction 
factors that are typically $\sim$ 1.05 for dijet and trijet events.

%\vspace{2mm}
%\noindent
%The measurements of the cross-sections are further influenced by the effects of QED radiation and so to correct the measured 
%values for these effects a further correction factor is necessary.  The QED correction factor C$_{QED}$ can be calculated by 
%using a version of the simulated events with the effects of QED radiation turned off.  The ratio of generator level events 
%without QED radiation to those with, yields QED correction factors that are typically $\sim$ 1.05 for dijet and trijet events.

\section{NLO Predictions}
\noindent
There are several programs available to perform dijet cross-section calculations in next-to-leading order.  DISENT\cite{DISENT},
and DISASTER have been used in many previous analyses and provide consistent results\cite{DISENTcompDISAST}.  However, only
NLOJET++ allows the three jet cross-section from $ep$ collisions to be calculated to NLO.  NLOJET++ has been shown to 
be consistent with both DISENT and DISASTER for the dijet cross-section at LO and NLO\cite{H13jet, NLOJET}.

\vspace{2mm}
\noindent
For the region of $Q^{2}$ considered in this analysis, it has been shown that there is little difference in the size
of the NLO corrections to the born level, for dijets, between the choice of $Q^{2}$ or 
$\overline{E}_{T}^{2}$\footnote{Mean $E_{T}$ of the leading jets} for the renormalisation/factorisation
scale used in the NLO calculations\cite{H1Wobisch}.  It has been observed that the choice of $Q^{2}$ produces marginally smaller
scale uncertainty for the value of $R_{3/2}$ and so this has been used for the renormalisation/factorisation scale in this
analysis.  Values for the k-factor, which represents the size of the NLO correction to the born level, for dijets are 
$\sim$ 1.2 - 1.05 and for trijets are $\sim$ 1.25 (figure \ref{kFactor}).

\vspace{2mm}
\noindent
The NLO predictions use the $\overline{MS}$ scheme for five massless quark flavours.  The parton distribution function (PDF) used in this analysis 
for comparison with data is CTEQ5M (the CTEQ4A\cite{cteq} PDF is also employed for the QCD analysis).  The value of $\alpha_{s}(M_{Z})$ used 
with this PDF was $\alpha_{s}(M_{Z})$ = 0.118, and this value
was subsequently evolved according to the two-loop solution of the renormalisation group equation.  NLOJET++ produces cross-sections
at the parton level and as such requires a correction for hadronisation effects before a comparison with data can be made.  This was
performed bin-by-bin, where hadronisation correction factors C$_{had}$ were defined as the ratio of the hadron-to-parton level 
cross-sections and were taken to be the average of values found using the programs DJANGOH and RAPGAP.  The hadronisation correction
 factors were typically $\sim$ 0.93 for dijets and $\sim$ 0.75 for trijets %(figure \ref{hadCorr} as well as tables 2 and 3).

\vspace{2mm}
\noindent
The influence of heavy quark masses and electroweak effects are not included in the calculations from NLOJET++.  It has been
shown that at LO these effects are small ($ < 1\%$) for the kinematic region considered in this analysis, with the exception
of the highest $Q^{2}$ bin (5000 $<$ $Q^{2}$ $<$ 15000 GeV$^{2}$), where the expected change in the dijet cross-section is
$\sim$ 25$\%$.  For this reason, this bin is not used 
in the QCD analysis presented in this document.  It may
be possible to include this region at a future time using an electroweak correction procedure, using a method similar to 
that employed for detector, QED and hadronisation corrections.

\section{Systematic and theoretical uncertainties}
\noindent
The effect of several sources of systematic uncertainties are considered.  %They are given below, along with the 
%typical change of the cross-sections [dijet;trijet]:

\begin{itemize}
\item The calibration of the positron energy %in the LAr 
is varied by $\pm$ (0.7-3)\%, depending on the position of the positron within the detector %(figure \ref{elecCali}) %[1$\%$ ; 1.5$\%$].
\item The positron polar angle is varied by $\pm$ 1-3 mrad, depending on the position of the positron within the detector %[1$\%$ ; 1.5$\%$].
\item The positron azimuthal angle is varied by $\pm$ 3 mrad %[$< 1\%$ ; $< 1\%$].
%\item A simultaneous shift of the hadronic energy scale of the LAr calorimeter ($\pm 2\%$), the track momenta of the hadronic 
%final state ($\pm 3\%$) and the hadronic energy scale of the SPACAL ($\pm 5\%$), producing an overall hadronic uncertainty of
%$\sim \pm 2\%$ %(figure \ref{hadCali}) %[2$\%$ ; 3$\%$].
\item  An overall hadronic uncertainty of 2\% is taken into account by shifts up and down of the hadronic energy scale of the H1 calorimeters as well as the track momenta of the hadronic final state. %(figure \ref{hadCali}) %[2$\%$ ; 3$\%$].
\item The difference between using the average correction factor and that of DJANGOH or RAPGAP alone as an estimate of the model
uncertainty %[3.5$\%$ ; 5$\%$].
\item The uncertainty due to the measurement of the luminosity is also considered %[1.02$\%$ ; 1.02$\%$].
\end{itemize}

\noindent
The systematic uncertainties are combined in quadrature with the statistical uncertainties of the data and the
 correction function to produce a total estimation of the error on the measurement of the cross-sections.

\vspace{2mm}
\noindent
The main contributions to the uncertainty associated with the theoretical cross-section predictions are:
\begin{itemize}
\item Uncertainties due to the hadronisation correction factor, estimated as the difference between using the average
correction factor and that of DJANGOH or RAPGAP alone %[1$\%$ ; 4$\%$].
\item Uncertainties due to terms beyond NLO and the choice of scale, estimated by varying the renormalisation/factorisation
scale by a factor of two %[3$\%$ ; 10$\%$].
\end{itemize}

\noindent
The total theoretical uncertainty was obtained by adding the individual contributions in quadrature.

\section{Experimental Results}
\subsection{Cross-sections}
\noindent
Values for the inclusive dijet and trijet cross-sections show good agreement with published measurements 
\cite{H1Wobisch, H13jet}, taking into account slight differences due to the increase in the HERA beam
energy.  Owing to the size of the dataset employed in this study, the range of $Q^{2}$ has been extended compared with 
existing three jet studies\cite{H13jet}.  The highest $Q^{2}$ point has large statistical uncertainties due to the low 
event rate in this kinematic region, but represents the first measurement of the three jet cross-section in this kinematic region 
for $ep$ collisions.  The cross-sections show good agreement with the predictions of NLOJET++ in the phasespace region 
where electroweak effects can be neglected (figures \ref{XSections},\ref{Data2Theo}).%, tables 1 and 2).

\subsection{R$_{3/2}$ and determination of $\alpha_{s}$}
\vspace{2mm}
\noindent
The distribution of R$_{3/2}$ with respect to $Q^{2}$ is well described by the NLO predictions in the region where electroweak 
effects can be neglected.  Data lie within one standard deviation of the central NLO prediction, and well within the theoretical 
uncertainty associated with the renormalisation/factorisation scale, across almost two orders of magnitude in $Q^{2}$ 
(figure \ref{R32}). %, table 3).  
Many systematic uncertainties cancel in the ratio as a large proportion of the trijet sample is 
contained within the dijet sample.  %The total experimental uncertainty on R$_{3/2}$ is typically $\sim$ 6$\%$.

\vspace{2mm}
\noindent
The distribution of R$_{3/2}$ with respect to $Q^{2}$, excluding the final $Q^{2}$ bin, was used to extract $\alpha_{s}(M_{Z})$.
The PDF CTEQ4A provides PDF sets obtained with different $\alpha_{s}(M_{Z})$ values\footnote{$\alpha_{s}(M_{Z})$ values
available in the CTEQ4A PDF are 0.110, 0.113, 0.116, 0.119 and 0.122}.  It has been shown to be consistent with
the CTEQ5 PDFs for the $Q^{2}$ range considered in this analysis.  %(figures \ref{ASBin12} and \ref{ASBin34}).
The method for extraction is similar to that employed in a recent 
publication by the ZEUS collaboration\cite{ZEUS3jet}.  The procedure is as follows:
\begin{itemize}
\item   The NLO calcultions are repeated for each available value of
$\alpha_{s}(M_{Z})$. %The theoretical sensitivity to $\alpha_{s}(M_{Z})$ is shown in figure \ref{cteq4BW}).
\item  The function $C_{1}\alpha_{s}(M_{Z}) + C_{2}\alpha_{s}^{2}(M_{Z})$ is fitted to the theoretical predictions from NLOJET++
 using the CTEQ4A PDF, for each bin in $Q^{2}$.
\item A value of $\alpha_{s}(M_{Z})$ is determined for each bin in $Q^{2}$ and an overall value calculated using a $\chi^{2}$
minimisation over the entire $Q^{2}$ range.
\end{itemize}

\noindent
Taking into account only the statistical uncertainties on the measured values of R$_{3/2}$, $\alpha_{s}(M_{Z})$ is determined
to be $\alpha_{s}(M_{Z})$ = 0.1175 $\pm$ 0.0017 (stat.)

\vspace{2mm}
\noindent
The systematic and theoretical uncertainties of the extracted value of the strong coupling are found by repeating the
above procedure using the cross-sections shifted by the total systematic and theoretical uncertainties as described above. % (table 4).
  The value of $\alpha_{s}(M_{Z})$ extracted from R$_{3/2}$ is thus:\newline \newline
\centerline{$\alpha_{s}(M_{Z})$ = 0.1175 $\pm$ 0.0017 (stat.) $\pm$ 0.0050 (syst.) $^{+0.0054}_{-0.0068}$ (th.).}
\newline \newline
\noindent
This result is in good agreement with the current world average (figure \ref{alphaS}) and with the evolution of $\alpha_{s}$
predicted by pQCD (figure \ref{alphaSRun}).  The value is competitive with results from other experimental procedures from different
processes %(figure \ref{alphaSComp}) 
and is in agreement with the result recently obtained from R$_{3/2}$ by the ZEUS 
collaboration\cite{ZEUS3jet}.

\section{Summary}
\noindent
Differential cross-sections, with respect to $Q^{2}$, have been measured for dijets and trijets in neutral current DIS for 
$150 <$ Q\textsuperscript{2} $< 15000~$GeV\textsuperscript{2} at HERA using the H1 detector.  The ratio R$_{3/2}$ of the 
trijet to the dijet cross-section has been measured as a function of $Q^{2}$.  NLO pQCD predictions provide a good description
of the cross-sections and R$_{3/2}$ in the phasespace region where electroweak effects can be neglected.  $\alpha_{s}(M_{Z})$ is
extracted using the distribution of R$_{3/2}$ for $150 <$ Q\textsuperscript{2} $< 5000 $GeV\textsuperscript{2}, with the
value of the strong coupling, at the mass of the Z boson, found to be $\alpha_{s}(M_{Z})$ = 0.1175 $\pm$ 0.0017 (stat.) $\pm$ 0.0050 
(syst.) $^{+0.0054}_{-0.0068}$ (th.) in good agreement with the current world average and previous HERA results.
%experimental determinations.

\section{Acknowledgements}
\noindent
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.


\section{References}

\begin{thebibliography}{99}
\bibitem{DetAlphS}{H1 Collaboration, S. Aid et al., Nucl. Phys. B 449 (1995) 3; \newline ZEUS Collaboration, M. Derrick et al., Phys. Lett. B 363 (1995) 201; \newline H1 Collaboration, C. Adloff et al., Eur. Phys. J. C5 (1998) 625; \newline H1 Collaboration, C. Adloff et al., Eur Phys. J. C6 (1999) 575.}
\bibitem{ZEUS3jet}{ZEUS Collaboration, S. Chekanov et al., hep-ex/0502007 (2005)}
\bibitem{BreitFrame}{R.P. Feynman, ``Photon-Hadron Interactions''[Benjamin, NY (1972)]\newline K.H.Streng, T.F.Walsh, P.M. Zerwas, Z. Phys. C 2, 237 (1979).}
\bibitem{Breit}{B.R. Webber, J. Phys. G19 (1993) 1567.}
\bibitem{RAPGAP}{H. Jung, Comp. Phys. Comm. 86 (1995) 147.}
\bibitem{HERWIG}{G. Marchesini et al., Comp. Phys. Comm. 67 (1992) 465.}
\bibitem{LEPTO}{G. Ingelman, A. Edin and J. Rathsman, Comp Phys. Comm, 101 (1997) 108.}
\bibitem{ARIADNE}{L. L\"onnblad, Comp. Phys. Comm. 71 (1992) 15.}
\bibitem{DJANGO}{K. Charchula, G. Schuler and H. Spiesberger, Comp. Phys. Comm. 81 (1994) 381.}
\bibitem{HERACLES}{A. Kwiatkowski, H. Spiesberger and H-J. M\"ohring, Comp. Phys. Comm. 69, 155 (1992)}
\bibitem{Lund}{B. Andersson et al., Phys. Rep 97 (1983) 31.}
\bibitem{JetStruct}{H1 Collaboration, C. Adloff et al., Nucl. Phys. B 545 (1999) 3.}
\bibitem{DISCuts}{H1 Collaboration, C. Adloff et al., Eur. Phys. J. C 13 (2000) 609.}
\bibitem{H1Wobisch}{H1 Collaboration, C. Adloff et al., hep-ex/0010054v1 (2000)}
\bibitem{H13jet}{H1 Collaboration, C. Adloff et al., Phys. Lett. B515 (2001) 17-29.}
\bibitem{ElecSig}{U. Bassler and G. Bernardi, Nucl. Instr. Meth. A 426 (1999) 583.}
\bibitem{Kt}{S.D. Ellis and D.E. Soper, Phys. Rev. D 48 (1993) 3160; \newline S. Catani, Yu.L. Dokshitzer, M.H. Seymour and B.R. Webber, Nucl. Phys. B 406 (1993) 187.}
\bibitem{DISENT}{S. Cantani and M.H. Seymour, Nucl. Phys. B 485 (1997) 291, Erratum-ibid. B 510 (1997) 503.}
\bibitem{DISENTcompDISAST}{C. Duprel, Th. Hadig, N. Kauer and M. Wobisch, Proceedings of the HERA Monte Carlo Workshop, eds. G. Grindhammer, G. Ingelman, H. Jung, T. Doyle, DESY-PROC-02-1999 (1999) 142.}
\bibitem{NLOJET}{Z. Nagy and Z. Trocsanyi, Phys. Rev. Lett. 87 (2001) 082001.}
\bibitem{cteq}{H.L. Lai et al., Phys. Rev. D. 55 (1997) 1280.}
\bibitem{PDG}{Particle Data Group, S. Eitelman et al., Phys. Lett. B 592 (2004) 1.}
\end{thebibliography}


\newpage
%\begin{tabular}{|c|c|c|c||c|c|}
%\hline
%$Q^{2}$& (d$\sigma$ / d$Q^{2}$)$_{dijet}$ & $\delta$ Statistical ($\%$) & $\delta$ Systematic ($\%$) & C$_{had}$\\
%$(GeV^{2})$&($pbGeV^{-2}$)& & & \\
%\hline
%\hline
%150 - 220&0.349&2.11&$^{+2.02}_{-2.39}$&0.93\\
%220 - 350&0.190&2.03&$^{+5.52}_{-5.72}$&0.93\\
%350 - 500&0.0673&2.28&$^{+2.76}_{-2.97}$&0.93\\
%500 - 5000&0.00525&2.48&$^{+4.08}_{-3.92}$&0.94\\
%5000 - 15000&0.000106&9.05&$^{+10.4}_{-10.9}$&0.93\\
%\hline
%\end{tabular}
%\newline
%
%\noindent
%{\bf{Table 1:}}
%\emph{ The dijet differential cross-section $d\sigma / dQ^{2}$ for hadronic jets in the Breit frame.  The
%statistical and systematic uncertainties are shown seperately as percentages.  The hadronisation correction factor C$_{had}$
%is shown in the last column.}
%
%\vspace{9mm}
%\begin{tabular}{|c|c|c|c||c|}
%\hline
%$Q^{2}$& (d$\sigma$ / d$Q^{2}$)$_{trijet}$& $\delta$ Statistical ($\%$) & $\delta$ Systematic ($\%$) & C$_{had}$\\
%$(GeV^{2})$&($pbGeV^{-2}$)& & & \\
%\hline
%\hline
%150 - 220&0.108&3.48&$^{+5.36}_{-5.06}$&0.76\\
%220 - 350&0.0539&3.58&$^{+5.47}_{-6.40}$&0.76\\
%350 - 500&0.0205&3.94&$^{+6.17}_{-5.52}$&0.75\\
%500 - 5000&0.00126&4.48&$^{+7.79}_{-8.53}$&0.75\\
%5000 - 15000&0.0000389&14.1&$^{+12.3}_{-10.4}$&0.74\\
%\hline
%\end{tabular}
%\newline
%
%\noindent
%{\bf{Table 2:}}
%\emph{The trijet differential cross-section $d\sigma / dQ^{2}$.  Other details as Table 1.}
%
%\vspace{9mm}
%\begin{tabular}{|c|c|c|c|}
%\hline
%$Q^{2} (GeV^{2})$& R$_{3/2}$ & $\delta$ Statistical ($\%$) & $\delta$ Systematic ($\%$)\\
%\hline
%\hline
%150 - 220&0.310&4.07&$^{+4.98}_{-4.76}$\\
%220 - 350&0.284&4.11&$^{+7.26}_{-7.64}$\\
%350 - 500&0.305&4.55&$^{+5.48}_{-5.42}$\\
%500 - 5000&0.241&5.12&$^{+8.39}_{-8.56}$\\
%5000 - 15000&0.368&16.8&$^{+14.0}_{-13.9}$\\
%\hline
%\end{tabular}
%\newline
%
%\noindent
%{\bf{Table 3:}}
%\emph{The ratio of trijet to dijet differential cross-sections.  Other details as Table 1.}
%
%\vspace{9mm}
%\begin{tabular}{|c|c|c|c|c|}
%\hline
%$Q^{2} (GeV^{2})$& $\alpha_{s}(M_{Z})$ & $\delta$ Statistical  & $\delta$ Systematic & $\delta$ Theoretical \\
%\hline
%\hline
%150 - 220&0.1166&$\pm$0.0028&$^{+0.0035}_{-0.0034}$&$^{+0.0035}_{-0.0053}$\\
%220 - 350&0.1131&$\pm$0.0035&$^{+0.0061}_{-0.0066}$&$^{+0.0052}_{-0.0074}$\\ 
%350 - 500&0.1239&$\pm$0.0036&$^{+0.0043}_{-0.0044}$&$^{+0.0059}_{-0.0070}$\\ 
%500 - 5000&0.1167&$\pm$0.0045&$^{+0.0074}_{-0.0077}$&$^{+0.0089}_{-0.0089}$\\
%\hline
%\hline
%150 - 5000&0.1175&$\pm$0.0017&$\pm$0.0050&$^{+0.0054}_{-0.0068}$\\
%\hline
%\end{tabular}
%\newline
%
%\noindent
%{\bf{Table 4:}}
%\emph{The $\alpha_{s}(M_{Z})$ values determined in this analysis.  Statistical, systematic and theoretical
%uncertainties are shown seperately.}

%\begin{figure}[!h]
%\centerline{\resizebox{9cm}{!}{\includegraphics{Plots/ncdisquark.eps}}}
%\caption{Feynman diagram showing tree-level neutral current DIS single jet event.}
%\label{NCEvent1}
%\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{9cm}{!}{\includegraphics{Plots/qcdcomp.eps}}\resizebox{9cm}{!}{\includegraphics{Plots/bgfusion.eps}}}
%\caption{Feynman diagrams showing LO QCD-Compton (left) and boson-gluon fusion (right) processes for dijets.}
%\label{NCEvent2}
%\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{13cm}{!}{\includegraphics{Plots/3jet.eps}}}
%\caption{Feynman diagram showing an example of a LO three jet process.}
%\label{NCEvent3}
%\end{figure}

\begin{figure}[!h]
\centerline{\resizebox{7cm}{!}{\includegraphics{H1prelim-05-033.fig1a.eps}}\hspace{1cm}\resizebox{6cm}{!}{\includegraphics{H1prelim-05-033.fig1b.eps}}}
\caption{Diagrams showing lowest order (left) and high $E_{T}$ (right) jet events in the Breit frame\cite{H1Wobisch}.}
\label{breit}
\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{15cm}{!}{\includegraphics[angle=270]{H1prelim-05-033.fig2.eps}}}
%  \caption{Cut along the beam axis of the H1 detector}
%  \label{Detector1}
%\end{figure}


\begin{figure}[!h]
  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{H1prelim-05-033.fig2.eps}}}
  \caption{Control distributions for the DIS event sample.  Distributions are normalised to equal number of events and show
comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are energy of scattered positron (GeV) 
(top left), log$_{10}$(Q$^{2}$) (top right), E-P$_{Z}$ (GeV) (bottom left) and polar angle of the scattered positron (degrees) 
(bottom right).}
  \label{DISCont1}
\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/DISCont2.eps}}}
%  \caption{Control distributions for the DIS event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are inelasticity (top left),
%azimuthal angle of the scattered positron (degrees) (top right), Z position of primary vertex (cm) (bottom left) and 
%Bj\"orken-x (bottom right).}
%  \label{DISCont2}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/DISCont3.eps}}}
%  \caption{Control distributions for the DIS event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are Ratio of hadron-to-electron P$_{T}$ \
%(top left), the difference from the ideal for E-P$_{Z}$ (GeV) (top right), azimuthal angle of total hadronic final state (degrees) 
%(bottom left) and event weights (bottom right).  Data points have some weight due to phi symmetry reweight. }
%  \label{DISCont3}
%\end{figure}

\clearpage

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/2jetCont1.eps}}}
%   \caption{Control distributions for the dijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are energy of scattered positron (GeV) 
%(top left), log$_{10}$(Q$^{2}$) (top right), E-P$_{Z}$ (GeV) (bottom left) and polar angle of the scattered positron (degrees) 
%(bottom right).}
%  \label{2jetCont1}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/2jetCont2.eps}}}
%   \caption{Control distributions for the dijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are inelasticity (top left),
%azimuthal angle of the scattered positron (degrees) (top right), Z position of primary vertex (cm) (bottom left) and 
%Bj\"orken-x (bottom right).}
%  \label{2jetCont2}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/2jetCont3.eps}}}
% \caption{Control distributions for the dijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are Ratio of hadron-to-electron P$_{T}$ \
%(top left), the difference from the ideal for E-P$_{Z}$ (GeV) (top right), azimuthal angle of total hadronic final state (degrees) 
%(bottom left) and event weights (bottom right).  Data points have some weight due to phi symmetry reweight.}  
%  \label{2jetCont3}
%\end{figure}


\begin{figure}[!h]
  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{H1prelim-05-033.fig3.eps}}}
 \caption{Control distributions for the diijet event sample.  Distributions are normalised to equal number of events and show
comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are energy of scattered positron (GeV)
(top left), log$_{10}$(Q$^{2}$) (top right), E$_{T}$ of highest E$_{T}$ jet (GeV) (bottom left) and 
E$_{T}$ of second highest E$_{T}$ jet (GeV) (bottom right).}
  \label{2jetCont}
\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/2jetCont5.eps}}}
%   \caption{Control distributions for the dijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are log$_{10}$ of the invariant mass of 
%two jets with highest  E$_{T}$ (GeV) (top left), total E$_{T}$ of all jets in event (GeV) (top right), total $\eta$ of all 
%jets in event, in the Breit frame (bottom left) and log$_{10}$ of the ratio of the mean  E$_{T}^{2}$ of the two highest 
%E$_{T}$ jets to $Q^{2}$.  (bottom right).}
%  \label{2jetCont5}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/2jetCont6.eps}}}
%   \caption{Control distributions for the dijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are the ratio of the difference
%in E$_{T}$ of the two jets with highest E$_{T}$ to the E$_{T}$ of the jet with highest E$_{T}$ (top left),
%$\xi$ of the event (top right), $\eta'$ = $\frac{1}{2} |\eta_{1} - \eta_{2}|$ (bottom left) 
%$\eta^{boost}$ = $\frac{1}{2} |\eta_{1} + \eta_{2}|$ and (bottom right).}
%  \label{2jetCont6}
%\end{figure}

\clearpage

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/3jetCont1.eps}}}
%   \caption{Control distributions for the trijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are energy of scattered positron (GeV)
%(top left), log$_{10}$(Q$^{2}$) (top right), E-P$_{Z}$ (GeV) (bottom left) and polar angle of the scattered positron (degrees) 
%(bottom right).}
%  \label{3jetCont1}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/3jetCont2.eps}}}
%   \caption{Control distributions for the trijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are inelasticity (top left),
%azimuthal angle of the scattered positron (degrees) (top right), Z position of primary vertex (cm) (bottom left) and 
%Bj\"orken-x (bottom right).}
%  \label{3jetCont2}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/3jetCont3.eps}}}
% \caption{Control distributions for the trijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are Ratio of hadron-to-electron P$_{T}$ \
%(top left), the difference from the ideal for E-P$_{Z}$ (GeV) (top right), azimuthal angle of total hadronic final state (degrees) 
%(bottom left) and event weights (bottom right). Data points have some weight due to phi symmetry reweight.} 
%  \label{3jetCont3}
%\end{figure}

\begin{figure}[!h]
  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{H1prelim-05-033.fig4.eps}}}
 \caption{Control distributions for the trijet event sample.  Distributions are normalised to equal number of events and show
comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are energy of scattered positron (GeV)
(top left), log$_{10}$(Q$^{2}$) (top right), E$_{T}$ of highest E$_{T}$ jet (GeV) (bottom left) and 
E$_{T}$ of second highest E$_{T}$ jet (GeV) (bottom right).}
  \label{3jetCont}
\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/3jetCont5.eps}}}
% \caption{Control distributions for the trijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line).  Plots are  log$_{10}$ of the invariant mass 
%of the three jets with highest  E$_{T}$ (GeV) (top left), total E$_{T}$ of all jets in event (GeV) (top right), total $\eta$ 
%of all jets in event, in the Breit frame (bottom left) and log$_{10}$ of the ratio of the mean  E$_{T}^{2}$ for the three 
%highest E$_{T}$ jets to $Q^{2}$.  (bottom right).}
%  \label{3jetCont5}
%\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{20cm}{!}{\includegraphics[angle=270]{Plots/3jetCont6.eps}}}
%\caption{Control distributions for the trijet event sample.  Distributions are normalised to equal number of events and show
%comparison of data (points) with RAPGAP (solid line) and DJANGOH (dashed line). Plots are the ratio of the difference
%in E$_{T}$ of the two jets with highest E$_{T}$ to the E$_{T}$ of the jet with highest E$_{T}$ (top left),
%$\xi$ of the event (top right),  $\eta'$ = $\frac{1}{2} |\eta_{1} - \eta_{2}|$ (bottom left) 
%$\eta^{boost}$ = $\frac{1}{2} |\eta_{1} + \eta_{2}|$ and (bottom right).}
%  \label{3jetCont6}
%\end{figure}

%\begin{figure}[!h]
 %\centerline{\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/trigEffZimp.eps}} \resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/trigEffJetEt1.eps}}}
%  \caption{Trigger efficiencies for the dijet event sample.  Efficiencies are for the subtrigger combination 
%(S67 $\|$ S75 $\|$ S77) vs positron Z impact (cm) (left) and leading jet $E_{T}$ (GeV) (right)}
 % \label{trigEff}
%\end{figure}

%\begin{figure}[!h]
 % \centerline{\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/DetCorr2jet.eps}} \resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/DetCorr3jet.eps}}}
 % \caption{Detector level correction factors from DJANGOH (circles) and RAPGAP (triangles) for dijet (left) and trijet (right) events.}
 % \label{detCorr}
%\end{figure}


%\begin{figure}[!h]
%  \centerline{\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/hadCorr2jet.eps}} \resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/hadCorr3jet.eps}}}
%  \caption{Hadronisation correction factors vs $Q^{2}$ for dijet (left) and trijet (right) event samples.  Values from DJANGOH 
%(circles) and RAPGAP (triangles) are in close agreement for dijets, but differ by up to 8$\%$ for trijets due to the different parton
%cascade models employed.}
%  \label{hadCorr}
%\end{figure}

\begin{figure}[!h]
\centerline{\resizebox{15cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig5.eps}}}
\caption{k-Factors for dijets (solid line) and trijets (dashed line).  $Q^{2}$ is used as the choice of renormalisation/factorisation
scale.}
\label{kFactor}
\end{figure}

%\begin{figure}[!h]
%  \centerline{\resizebox{15cm}{!}{\includegraphics[angle=0]{Plots/elecCali.eps}}}
%  \caption{Uncertainty due to the calibration of the positron energy in the LAr.  The shaded band (left) shows the known H1 uncertainty.}
%  \label{elecCali}
%\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{15cm}{!}{\includegraphics[angle=0]{Plots/hadCali.eps}}}
%\caption{ The total hadronic final state uncertainty.}
%\label{hadCali}
%\end{figure}
%\clearpage


\begin{figure}[!h]
  \centerline{\resizebox{18cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig6.eps}}}
  \caption{NC dijet and trijet differential cross-sections, with respect to $Q^{2}$, shown with NLO pQCD predictions 
including hadronisation corrections.  The shaded bands show the effect of varying the renormalisation/factorisation scale 
by a factor of two.  The highest $Q^{2}$ bin is overestimated by the theoretical predictions due to the absence of electroweak effects in the NLO calculations.}
  \label{XSections}
\end{figure}
\vspace{5mm}

\begin{figure}[!h]
  \centerline{\resizebox{15cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig7a.eps}}}
  \centerline{\resizebox{15cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig7b.eps}}}
  \caption{Ratio of dijet (top) and trijet (bottom) differential cross-sections, to NLO predictions (including hadronisation corrections).  The light shaded bands show the effect of varying the renormalisation/factorisation scale by a factor of two.  The dark shaded bands show
the uncertainty associated with the hadronisation correction.}
  \label{Data2Theo}
\end{figure}

\begin{figure}[!h]
\centerline{\resizebox{18cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig8.eps}}}
\caption{Distribution of measured  values of $R_{3/2}$ against $Q^{2}$ compared with a NLO pQCD prediction, with
hadronisation corrections.  The light shaded
band shows the effect of varying the renormalisation/factorisation scale by a factor of two. The dark shaded band shows
the uncertainty associated with the hadronisation correction.}
\label{R32}
\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{18cm}{!}{\includegraphics[angle=0]{Plots/cteq4BW.eps}}}
%\caption{Distribution of measured  values of $R_{3/2}$ against $Q^{2} $ compared with NLO pQCD predictions, with
%hadronisation corrections, for five values of $\alpha_{s}(M_{Z})$.  The CTEQ4A PDF set features parton information
%calculated using different $\alpha_{s}(M_{Z})$ values.}
%\label{cteq4BW}
%\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/extract1.eps}}\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/extract2.eps}}}
%\caption{$R_{3/2}$ against $\alpha_{s}(M_{Z})$ for Q\textsuperscript{2} 150 - 220 GeV\textsuperscript{2} (left) and for Q\textsuperscript{2} 
%220 - 350 GeV\textsuperscript{2} (right). The function $C_{1}\alpha_{s}(M_{Z}) + C_{2}\alpha_{s}^{2}(M_{Z})$ is fitted to the theoretical predictions from NLOJET++ %using the CTEQ4A PDF (triangles).  The value obtained using the CTEQ5 PDF is also shown (square point).}
%\label{ASBin12}
%\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/extract3.eps}}\resizebox{9cm}{!}{\includegraphics[angle=0]{Plots/extract4.eps}}}
%\caption{$R_{3/2}$ against $\alpha_{s}(M_{Z})$ for Q\textsuperscript{2} 350 - 700 GeV\textsuperscript{2} (left) and for 
%Q\textsuperscript{2} 700 - 5000 GeV\textsuperscript{2} (right).}
%\label{ASBin34}
%\end{figure}

\begin{figure}[!h]
\centerline{\resizebox{18cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig9.eps}}}
\caption{$\alpha_s (M_{Z})$ values from each Q\textsuperscript{2} bin.  The averaged value, found using
a $\chi^{2}$ minimisation fit, is shown at the far right (empty diamond) and is disassociated with the $x$-axis.  Uncertainties 
shown are statistical up to the horizontal bar, and then the quadratic sum of statistical and systematic uncertainties.  
Theoretical uncertainties, associated with the extraction of $\alpha_{s}$, are offset right.  The world average 
is $\alpha_{s}(M_{Z})$ = 0.1187 $\pm 0.0020$.}
\label{alphaS}
\end{figure}

\begin{figure}[!h]
\centerline{\resizebox{18cm}{!}{\includegraphics[angle=0]{H1prelim-05-033.fig10.eps}}}
\caption{$\alpha_s (M_{Z})$ values from each Q\textsuperscript{2} bin evolved to their values at their respective
values of Q\textsuperscript{2} (triangles) using the two-loop solution of the renormalisation group equation.  The averaged value of
$\alpha_{s}(M_{Z})$, found using a $\chi^{2}$ minimisation fit, is shown at the far right (empty diamond).   
The evolution of the current world average value of $\alpha_{s}(M_{Z})$ is shown as a shaded
band.  Uncertainties shown are statistical up to the horizontal bar, and then the quadratic sum of statistical and systematic 
uncertainties.  .  The world average is $\alpha_{s}(M_{Z})$ = 0.1187 $\pm 0.0020$.}
\label{alphaSRun}
\end{figure}

%\begin{figure}[!h]
%\centerline{\resizebox{20cm}{!}{\includegraphics[angle=0]{Plots/alphaSComp.eps}}}
%\caption{Summary of $\alpha_s (M_{Z})$ values from various experimental methods.  Uncertainties shown are experimental and
%theoretical combined.  The world average is $\alpha_{s}(M_{Z})$ = 0.1187
%$\pm 0.0020$.}
%\label{alphaSComp}
%\end{figure}

\end{document}























