\documentclass[12pt]{article} \usepackage{lineno} %\usepackage{graphicx} \usepackage{epsfig} \usepackage{amsmath} \usepackage{hhline} \usepackage{amssymb} \usepackage{times} \usepackage{cite} \usepackage{array} \usepackage{multirow} \usepackage{color} \definecolor{rltred}{rgb}{0.75,0,0} \definecolor{rltgreen}{rgb}{0,0.5,0} \definecolor{rltblue}{rgb}{0,0,0.75} %\newif\ifpdf %\ifx\pdfoutput\undefined % \pdffalse % we are not running PDFLaTeX %\else % \pdfoutput=1 % we are running PDFLaTeX % \pdftrue %\fi % %\ifpdf %\linenumbers % Remove for final publication!!!!! %\usepackage{thumbpdf} %\usepackage[pdftex, % colorlinks=true, % urlcolor=rltblue, % \href{...}{...} external (URL) % filecolor=rltgreen, % \href{...} local file % linkcolor=rltred, % \ref{...} and \pageref{...} % pdftitle={Example File for pdflatex}, % pdfauthor={H1}, % pdfsubject={Template file}, % pdfkeywords={High-Energy Physics, Particle Physics}, % pagebackref, % pdfpagemode=None, % bookmarksopen=true]{hyperref} % %\fi \renewcommand{\topfraction}{1.0} \renewcommand{\bottomfraction}{1.0} \renewcommand{\textfraction}{0.0} \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 \begin{document} %\linenumbers %\renewcommand\linenumberfont{\normalfont\small\bfseries} \setlength\linenumbersep{1cm} \resetlinenumber % Some useful tex commands \newcommand{\pom}{{I\!\!P}} \def\lsim{\,\lower.25ex\hbox{$\scriptstyle\sim$}\kern-1.30ex% \raise 0.55ex\hbox{$\scriptstyle <$}\,} \newcommand{\qsq}{\ensuremath{Q^2} } \def\submJournal#1{{#1}} \def\Journal#1#2#3#4{{#1} {\bf #2} (#3) #4} \def\EJC{{\em Eur. Phys. J.} {\bf C}} \def\NIM{\em Nucl. Instrum. Methods} \def\CPC{\em Comp. Phys. Commun.} \def\EJ{{\em Eur. Phys. J.}} \def\NP{{\em Nucl. Phys.}} \def\PL{{\em Phys. Lett. }} \def\PR{{\em Phys. Rev. }} \def\PRL{\em Phys. Rev. Lett.} \def\ZP{{\em Z. Phys.}} \def\RMP{{\em Rev. Mod. Phys. }} \def\JHEP{{\em J. High Energy Phys.} {\bf JHEP}} \def\SJNP{{\em Sov. J. Nucl. Phys.}} \def\SPJ{{\em Sov. Phys. JETP}} \newcommand{\etal}{{\em et al.}} \newcommand{\arxiv}[1]{% %\ifpdf %\href{http://arXiv.org/ps/#1}{\mbox{[#1]}} %\else \mbox{[#1]}} %\fi} \newcommand{\arxivnobrackets}[1]{% %\ifpdf %\href{http://arXiv.org/ps/#1}{\mbox{#1}} %\else \mbox{#1}} %\fi} \newcommand\figuresname{Figures} \newcommand\tablesname{Tables} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})} \newcommand{\figref}[1]{\figurename~\ref{#1}} \newcommand{\figsref}[2]{\figuresname~\ref{#1} and~\ref{#2}} \newcommand{\figrange}[2]{\figuresname~\ref{#1}--\ref{#2}} \newcommand{\tabref}[1]{\tablename~\ref{#1}} \newcommand{\tabrange}[2]{\tablesname~\ref{#1}--\ref{#2}} \newcommand{\secref}[1]{Section~\ref{#1}} \newcommand{\secrange}[2]{Sections~\ref{#1}--\ref{#2}} \fboxsep5pt \fboxrule0.4pt \newcommand{\Pom}{{I\hspace{-0.75ex} P}} \newcommand{\etamax}{% \ensuremath{% {\eta_{\rm max}}}} \newcommand{\xpom}{% \ensuremath{% {x_\pom}}} \newcommand{\xpomeron}{% \ensuremath{% {x_\pom}}} \newcommand{\logxpomeron}{% \ensuremath{% {\log_{10}(x_\pom)}}} \newcommand{\ftd}{% \ensuremath{% {F_2^D}}} \newcommand{\pthat}{% \ensuremath{% {\hat{p}_T}}} \newcommand{\zpom}{% \ensuremath{% {z_\pom}}} \newcommand{\zpomeron}{% \ensuremath{% {z_\pom}}} \newcommand{\zpomeronjets}{% \ensuremath{% {z_\pom^{\rm jets}}}} \newcommand{\xgam}{% \ensuremath{% {x_\gamma}}} \newcommand{\xgamma}{% \ensuremath{% {x_\gamma}}} \newcommand{\xgammajets}{% \ensuremath{% {x_\gamma^{\rm jets}}}} \newcommand{\xgammajetspar}{% \ensuremath{% {x_{\gamma, {\rm PL}}^{\rm jets}}}} \newcommand{\ptjet}{% \ensuremath{% {E_T^{\rm jet1}}}} \newcommand{\etjetone}{% \ensuremath{% {E_T^{\rm *,jet1}}}} \newcommand{\ptjetone}{% \ensuremath{% {E_T^{\rm *,jet1}}}} \newcommand{\ppbar}{% \ensuremath{% {p\bar{p}}}} \newcommand{\etjettwo}{% \ensuremath{% {E_T^{\rm *,jet2}}}} \newcommand{\deltaeta}{% \ensuremath{% {\left| \Delta\eta_{\rm jet}\right|}}} \newcommand{\deltaetastar}{% \ensuremath{% {\left| \Delta\eta^*_{\rm jet}\right|}}} \newcommand{\etajetlab}{% \ensuremath{% {\eta_{\rm jet}^{\rm lab}}}} \newcommand{\meanetajetlab}{% \ensuremath{% {\langle \eta_{\rm jet}^{\rm lab}\rangle}}} \newcommand{\meanetajet}{% \ensuremath{% {\langle \eta_{\rm jet}\rangle}}} \newcommand{\mjj}{% \ensuremath{% {M_{12}}}} \begin{titlepage} \noindent DESY 07-018 \\ February 2007 \vspace{2cm} \begin{center} \begin{Large} {\bf Tests of QCD Factorisation in the Diffractive Production of Dijets in Deep-Inelastic Scattering and Photoproduction at HERA} \vspace{2cm} H1 Collaboration \end{Large} \end{center} \vspace{2cm} \begin{abstract} \noindent Measurements are presented of differential dijet cross sections in diffractive photoproduction ($Q^2<0.01$~GeV$^2$) and deep-inelastic scattering processes (DIS, $40.01$, small additional contributions from sub-leading meson (`reggeon') exchange have to be taken into account to describe the data. The H1 Collaboration has published QCD fits to two different data sets of inclusive diffractive DIS events. In a first analysis~\cite{h1f2d94}, data taken in the year 1994 were used to extract the LO `H1 fit 2' parton densities which have been used previously in comparisons with diffractive dijet production in DIS at HERA and at the Tevatron. A second analysis was based on the larger data samples of the years 1997--2000~\cite{h1f2d97}. The fit in~\cite{h1f2d97} led to the NLO `H1~2006~Fit~A' and NLO `H1~2006~Fit~B' DPDFs which both give a good description of inclusive diffraction, and which are the basis of the dijet predictions in this paper. The two sets of parton densities differ mainly in the gluon density at high fractional parton momentum, which is poorly constrained by the inclusive diffractive scattering data. The gluon density of Fit A is peaked at the starting scale at high fractional momentum and that of Fit B is flat. \section{Next-to-leading Order QCD Calculations} \label{sec_nlo} Existing programs which calculate NLO QCD partonic cross sections for dijet production in inclusive DIS and photoproduction can be adapted to calculate cross sections in diffraction. For DIS, the DISENT \cite{disent} program is used, as suggested in \cite{hautmann}. It was demonstrated in \cite{poetter,nlocomp,nlojet} that dijet calculations using this program agree very well with the results from other programs~\cite{disaster,jetvip,mepjet,nlojet}. The program by Frixione et~al.~\cite{frixione} is used for photoproduction. The two NLO programs are adapted to calculate diffractive cross sections according to the following procedure. The cross section at fixed $\xpom$ and $t=0$ is calculated by reducing the nominal proton beam energy by a factor $\xpom$. Since the $x_\pom$ and $t$ dependences of the DPDFs are assumed to factorise from the $\zpom$ and $\mu_F$ dependences, the proton PDFs can be replaced by the parton densities of the diffractive exchange $f_{i,\pom}(\zpom,\mu_F^2)$. The cross sections are multiplied by $f_{\pom}(x_\pom,t)$, integrated between $t=-1$~GeV$^2$ and the maximum kinematically allowed value of $t$. In the same way, a $\approx 3\%$ contribution from Reggeon exchange is calculated. Kinematic effects on the partonic configurations arising from finite values of $t$ are neglected. To compare the results with the measured cross sections in the region $\xpom<0.03$, the results are integrated over $\xpom$. The diffractive dijet cross sections of the modified programs have been compared at the LO tree level with predictions of the Monte Carlo generator RAPGAP~\cite{RAPGAP} (see also \secref{sec_rapgap}). Good agreement has been found for both DIS and photoproduction, indicating that the diffractive extension works correctly. The diffractive NLO predictions agree with independent calculations in both DIS and photoproduction~\cite{kkdis,kkgp}. For the NLO predictions in this paper, the recent H1 2006 DPDFs are used and the 2-loop strong coupling $\alpha_s(M_Z)$ is set to $0.118$; the same value is used in the evolution of the parton densities~\cite{h1f2d97}. The renormalisation scale is set to the transverse energy of the leading parton jet in the photon-proton centre-of-mass frame. In DISENT it is not possible to change the factorisation scale on an event-by-event basis. It is therefore set to the average $E_T$ of the leading jet observed in the DIS measurement ($6.2$~GeV). Variations of the QCD renormalisation scale by factors $0.5$ and $2$ in DISENT result in changes of the predicted dijet cross section by approximately $+24\%$ and $-17\%$, respectively, integrated over the DIS kinematic range specified in \tabref{tab_xsdef}. Varying the factorisation scale by factors $0.5$ and $2$ leads to changes of the predicted dijet cross section by approximately $+8\%$ and $-7\%$, respectively. In the Frixione program for photoproduction, the factorisation and renormalisation scales are fixed to be equal. Variations of the scales by factors $0.5$ and $2$ change the predicted cross section by approximately $+33\%$ and $-21\%$, respectively, integrated over the photoproduction kinematic range specified in \tabref{tab_xsdef}. In photoproduction, the GRV HO photon PDFs~\cite{GRVgamma} are used. Photon parton densities are not used in DISENT. The calculated NLO parton jet cross sections are corrected for the effects of hadronisation. The corrections, defined as \begin{equation} \left( 1+\delta_{\rm had} \right)_i = \left( \frac{\sigma^{\rm hadron}_{\rm dijet}}{\sigma^{\rm parton}_{\rm dijet}} \right)_{\!\!\!i}, \end{equation} are determined for both DIS and photoproduction in every measurement bin $i$ using the two Monte Carlo generators RAPGAP with Lund string fragmentation and HERWIG~\cite{herwig} with cluster fragmentation. The HERWIG program was extended to diffraction in the manner described above for the NLO programs and uses LO diffractive parton densities. For the parton level cross section $\sigma^{\rm parton}_{\rm dijet}$ the jet algorithm operates on the final state partons after the parton shower cascade. The hadronisation correction is calculated as the mean of the corrections obtained from RAPGAP and HERWIG. The difference between the two corrections serves as an error estimate. In DIS, the hadron level dijet cross section does not differ significantly from the cross section at the parton level. In photoproduction, the hadron level cross section is lower than the parton level cross section by $10\%$ on average. The correction is particularly large at high $\xgamma$ where contributions with $\xgamma \approx 1$ at the parton level are smeared towards lower values due to hadronisation. The estimated uncertainty on $(1+\delta_{\rm had})$ is $20\%$ for $\zpomeronjets>0.8$ in DIS and less than $10\%$ in all other measurement bins. It is listed in \tabrange{tab_xsdis1}{tab_xsgp2}. The uncertainty on the parton densities arising from experimental and theoretical uncertainties in the fit to inclusive diffractive data are much smaller than the QCD scale uncertainties of the dijet predictions and are neglected. The NLO corrections increase the LO cross section by factors $1.9$ and $1.7$ on average in DIS and photoproduction, respectively. This large correction is due to the low transverse energy of the jets. \section{Soft Colour Neutralisation} An approach conceptually different from that of diffractive parton densities is provided by soft colour neutralisation models. In these models, diffractive scattering is described by DIS or photoproduction hard scattering processes with subsequent colour rearrangements between the final state partons. This soft reconfiguration leaves the parton momenta unchanged and can produce colour singlet systems which are separated by a large rapidity gap. The Soft Colour Interaction model (SCI)~\cite{sci} contains one free parameter, the colour rearrangement probability, which was fitted to \ftd{} measurements. A refined version of the model (GAL)~\cite{gal} uses a generalised area law for the colour rearrangement probability. Both versions of the model give a reasonably good description~\cite{scitev} of HERA inclusive diffractive cross sections and of diffractive processes at the Tevatron~\cite{tevjets,scidata1,scidata2,scidata3,scidata4,scidata5}. Predictions for diffractive dijet production in the SCI and GAL models are obtained using the LO generator programs LEPTO~\cite{leptosci} and PYTHIA~\cite{PYTHIA} for the DIS and photoproduction kinematic regions, respectively. Higher order QCD effects are simulated using parton showers. The calculations are based on the CTEQ5L LO parton densities of the proton~\cite{cteq5l}. \section{Experimental Procedure} \subsection{H1 detector} A detailed description of the H1 detector can be found in~\cite{H1det}. Here, a brief account of the components most relevant to the present analysis is given. The H1 coordinate system convention defines the outgoing proton beam direction as the positive $z$ axis, also referred to as the `forward' direction. The polar angle $\theta$ is measured relative to this axis and the pseudorapidity is defined as $\eta \equiv -\ln \tan (\theta/2)$. The central $ep$ interaction region is surrounded by two large concentric jet drift chambers, two $z$ chambers, and two multi-wire proportional chambers (MWPCs), located inside a $1.15$~T solenoidal magnetic field. Charged particle momenta are measured by the drift chambers in the range \mbox{$-1.5<\eta<1.5$} with a resolution of \mbox{$\sigma(p_T)/p_T \simeq 0.005\, p_T/$GeV~$\oplus~0.015$.} The \mbox{MWPCs} provide fast trigger information based on the signals of charged particles. In the central and forward region the track detectors are surrounded by a finely segmented Liquid Argon calorimeter (LAr). It consists of an electromagnetic section with lead absorbers and a hadronic section with steel absorbers and covers the range \mbox{$-1.5 <\eta <3.4$.} The energy resolution is $\sigma(E)/E \simeq 0.11/\sqrt{E/{\rm GeV}}$ for electromagnetic showers and $\sigma(E)/E \simeq 0.50/\sqrt{E/{\rm GeV}}$ for hadrons, as measured in test beams. The backward region $-4 < \eta < -1.4$ is covered by a lead/scintillating fibre calorimeter (SPACAL)~\cite{spacal} consisting of an electromagnetic and a hadronic section. The electromagnetic part is used to identify and measure the scattered positron in DIS events with an energy resolution of $\sigma(E)/E \simeq 0.07/\sqrt{E/{\rm GeV}}~\oplus~0.01$. In front of the SPACAL, the Backward Drift Chamber (BDC) provides track segments of charged particles with a resolution of $\sigma(r)=0.4$~mm and $r \sigma(\phi)=0.8$~mm. The forward region is instrumented with the Forward Muon Detector (FMD) and the Proton Remnant Tagger (PRT). Three double layers of drift chambers of the FMD are used to detect particles with pseudorapidities in the range $1.9 < \eta < 3.7$. The FMD can also detect particles from larger pseudorapidities which reach the detector after undergoing secondary scattering with the beam-pipe. The PRT consists of a set of scintillators surrounding the beam pipe at $z=26$~m and covers the region $6 < \eta < 7.5$. The $ep$ luminosity is measured with a precision of $1.5\%$ via the Bethe-Heitler Bremsstrahlung process $ep \rightarrow ep \gamma$, the photon being detected in a crystal calorimeter at $z=-103$ m. A further crystal calorimeter at $z=-33$~m is used as a small angle positron detector to measure the scattered positron in photoproduction events. \subsection{Event selection} The data correspond to an integrated luminosity of $18$~pb$^{-1}$ and were taken in the 1996 and 1997 running periods, in which HERA collided $820$~GeV protons with $27.5$~GeV positrons. The measurements are described in detail in~\cite{sesphd}. The photoproduction data are collected using a trigger which requires the scattered positron to be measured in the small angle positron detector, at least three tracks to be reconstructed in the central jet chambers and an event vertex to be identified. A veto cut requiring less than $0.5$~GeV of energy deposited in the photon detector of the luminosity system suppresses initial state radiation and coincidences with Bremsstrahlung events. The geometrical acceptance of the small scattering angle positron detector limits the photon virtuality to $Q^2<0.01$~GeV$^2$ and the photon-proton centre-of-mass energy to $165 0.8$~GeV) to be measured in the central jet chambers. Several cuts are applied on the SPACAL positron candidate to reduce background from photons and hadrons. The electromagnetic cluster energy is required to be larger than $8$~GeV and requirements are imposed on the width of the electromagnetic shower, the containment in the electromagnetic section of the SPACAL and an associated track segment in the BDC. DIS events with initial state QED radiation are suppressed by requiring the summed $E-p_z$ of all final state particles including the positron to be greater than $35$~GeV. The range in the photon virtuality is restricted to $4 3.2$. These selection criteria ensure that the gap between the systems $X$ and $Y$ spans at least the region $3.2 < \eta < 7.5$, and restrict $M_Y$ and $t$ to approximately $M_Y < 1.6$~GeV and \mbox{$-t < 1$~GeV$^2$.} A cut $\xpom < 0.03$ further reduces non-diffractive contributions. The hadronic system $X$ is measured in the LAr and SPACAL calorimeters and the central tracking system. Calorimeter cluster energies and track momenta are combined into hadronic objects using an algorithm which avoids double counting~\cite{combobj}. Jets are formed from the hadronic objects, using the inclusive $k_T$ cluster algorithm~\cite{kt} with a distance parameter of unity in the photon-proton rest frame, which is identical to the laboratory frame for photoproduction up to a Lorentz boost along the beam axis. The $p_T$ recombination scheme is used, which leads to massless jets. At least two jets are required, with transverse energies $\etjetone > 5$~GeV and $\etjettwo > 4$~GeV for the leading and sub-leading jet, respectively.\footnote{The `*' denotes variables in the photon-proton rest frame.} The jet axes of the two leading jets are required to lie within the region $-1 < \etajetlab < 2$, well within the acceptance of the LAr calorimeter. The final selection yields $1365$ events in photoproduction and $322$ events in DIS. \subsection{Kinematic reconstruction} \label{sec_rec} \subsubsection{Reconstruction of DIS events} In the DIS analysis, the energy $E_e$ and the polar angle $\theta_e$ of the scattered positron are measured in the backward calorimeter SPACAL and $y$ and $Q^2$ are reconstructed according to \begin{equation} y = 1- \frac{E_e}{E^0_e}\, \sin^2\frac{\theta_e}{2}\ , \qquad Q^2=4 E^0_e E_e\cos^2\frac{\theta_e}{2}\ , \end{equation} in which $E^0_e$ is the positron beam energy. The invariant mass $M_X$ of the hadronic system $X$ is reconstructed from the energies $E_i$ and the momenta $\vec{p_i}$ of all hadronic objects: \begin{equation} M_X^2 = (\sum_{i \in X} E_i)^2 - ( \sum_{i \in X} \vec{p}_i)^2\ . \label{eq_mx} \end{equation} The photon-proton centre-of-mass energy $W$ is reconstructed according to \eqref{eq_W} and the variable \xpom{} is given by \begin{equation} \xpom = \frac {Q^2 + M_X^2 }{Q^2 + W^2}\ . \end{equation} The estimators \xgammajets{} and \zpomeronjets{} of the fractional momenta of the partons entering the hard sub-process are reconstructed as \begin{equation} \xgammajets = \frac{\sum_{i=1}^2\, \left( E_{{\rm jet}\,i}^*-p^*_{z, {\rm jet}\,i} \right)}{\sum_{i \in X }\, \left( E_i^*-p^*_{z,i} \right)}\ , \qquad \zpomeronjets = \frac{ \qsq + M^2_{12}}{\qsq + M^2_X}\ . \end{equation} \subsubsection{Reconstruction of photoproduction events} In the photoproduction analysis, the energy $E_e$ of the scattered positron is measured in the small scattering angle positron detector and $y$ is reconstructed according to \begin{equation} y= 1-E_e/E^0_e. \end{equation} The estimators \xgammajets{} and \zpomeronjets{} are reconstructed as \begin{equation} \xgammajets = \frac{\sum_{i=1}^2\, \left( E_{{\rm jet}\,i}-p_{z, {\rm jet}\,i} \right)}{2\,y\,E^0_e}, \qquad \zpomeronjets = \frac{\sum_{i=1}^2\, \left( E_{{\rm jet}\,i}+p_{z, {\rm jet}\,i} \right)}{2\,\xpom\,E_p}, \end{equation} in which $E_p$ is the incident proton beam energy. The variable \xpom{} is reconstructed according to \begin{equation} \xpom = \frac{\sum_{i \in X} \left(E_i+p_{z,i}\right)}{2\, E_p}. \end{equation} The reconstruction of $\zpomeronjets$ and $\xpom$ is different from the DIS case due to the large contribution of resolved photon processes. \subsection{Monte Carlo simulations} \label{sec_rapgap} Monte Carlo programs are used in the analysis to correct the measured distributions for detector effects. The H1 detector response is simulated using detailed detector simulation programs based on GEANT~\cite{geant3}. The Monte Carlo events are subjected to the same analysis chain as the data. The main Monte Carlo generator used to correct the data distributions is RAPGAP~\cite{RAPGAP}. Events are generated according to a convolution of LO diffractive parton densities with LO QCD matrix elements for the hard $2 \rightarrow 2$ subprocess. The `H1 fit 2' DPDFs of~\cite{h1f2d94} are used. RAPGAP includes resolved photon processes for which the partonic cross sections are also convoluted with the parton densities of the photon. In photoproduction, the leading order GRV '94 parton distribution functions~\cite{GRVgamma} are used, which were found to give a good description of the effective photon structure function as measured by H1~\cite{Kaufmann}. For DIS, processes with a resolved virtual photon are generated using the SAS-2D parameterisation~\cite{sas2d}, which leads to a reasonable description of inclusive dijet production~\cite{lowq2} in a similar $Q^2$ and $E_T$ range to that studied here. The PDFs are taken at the scale $\mu_F^2=\hat{p}_T^2 + 4 m_q^2$, where $\hat{p}_T$ is the transverse momentum of the emerging hard partons and $m_q$ is the mass of the quarks produced. Higher order effects are simulated using parton showers~\cite{PS} in the leading log($\mu$) approximation. The Lund string model~\cite{LUND} is used for hadronisation. Photon radiation from the positron lines is simulated using the program HERACLES~\cite{heracles}. The used RAPGAP version simulates only processes in which the proton stays intact. \subsection{Cross section measurement} The data are first corrected for losses at the trigger level. The trigger efficiency is approximately $90\%$ in DIS, the losses being mainly due to the tracking requirements. In photoproduction the efficiency also depends on the energy of the positron detected in the small scattering angle detector and varies between $\approx 90\%$ at low $y$ and $\approx 50\%$ at high $y$. Non-diffractive background migrating into the measurement region from $M_Y>5~$GeV and large \xpom{} is statistically subtracted using inclusive dijet production simulations (RAPGAP in DIS and PYTHIA in photoproduction). The subtracted background amounts to $3\%$ in photoproduction and $5\%$ in DIS. Due to the limited geometrical detector acceptance in the forward direction it is not possible to distinguish an intact final state proton from one which dissociates into a low-mass system $Y$. Thus the measured cross section is defined to include proton dissociation with $M_Y<1.6$~GeV. The correction factor for migrations about the measurement boundary $M_Y=1.6$~GeV is determined using the DIFFVM~\cite{DIFFVM} simulation of proton dissociation in the range $m_p 5$ GeV} \\ \hline \multicolumn{2}{|c|}{$\etjettwo > 4$ GeV} \\ \hline $-1 < \eta_{\rm jet(1,2)} < 2$ & $-3 < \eta_{\rm jet(1,2)}^* < 0$ \\ \hline \multicolumn{2}{|c|}{$\xpom <$ 0.03} \\ \hline \multicolumn{2}{|c|}{$M_Y < 1.6$ GeV} \\ \hline \multicolumn{2}{|c|}{$-t < 1$ GeV$^2$} \\ \hline \end{tabular} \end{center} \caption{The kinematic ranges of the measured hadron level $ep$ cross sections.} \label{tab_xsdef} \end{table} \subsection{Analysis of systematic uncertainties} The following systematic errors on the measured cross sections arise from experimental sources such as detector calibration uncertainties. The cross section errors are estimated by repeating the analysis with variations in the reconstruction of detector-simulated Monte Carlo events. \begin{itemize} \item A $4\%$ uncertainty in the absolute energy scale of the hadronic LAr calorimeter in the jet $E_T$ range considered here~\cite{frankphd} affects the reconstruction of the hadronic final state. The resulting uncertainty on the measured cross section is $4\%$ in DIS and $8\%$ in photoproduction and is strongly correlated between the data points. The influence of this uncertainty in DIS and in photoproduction is different due to the different reconstruction of $\xpom$. A $7\%$ uncertainty in the SPACAL hadronic energy scale affects the cross sections by $1\%$. The uncertainty in the fraction of the energy of the reconstructed hadronic objects which is carried by tracks is $3\%$ and gives rise to errors on the cross section of $4\%$ in photoproduction and $3\%$ in DIS, again strongly correlated between data points. \item The absolute SPACAL electromagnetic energy scale is known to $0.3\%$ for scattered posi\-trons with $E_e=27.5$~GeV and $2.0\%$ at $E_e=8$~GeV. The polar scattering angle of the positron is measured to $1$~mrad precision. The uncertainties of the positron energy and angle measurements in DIS result in cross section errors in the range of $4$ to $5\%$ for the energy uncertainty and $2\%$ for the scattering angle. In photoproduction, the uncertainty in the knowledge of the acceptance and efficiency of the small angle positron detector results in a cross section error of $5\%$ on average. \item The uncertainties on the trigger efficiencies and the luminosity measurement give rise to cross section uncertainties of $6\%$ and $1.5\%$, respectively. \item An uncertainty of $25\%$ in the fraction of events lost due to noise in the FMD translates into a $1.3\%$ normalisation error on the cross section. \end{itemize} Systematic errors arising from uncertainties in the acceptance and migration corrections are estimated by repeating the measurements with variations in the kinematic dependences and other details of the Monte Carlo models within experimentally allowed limits. \begin{itemize} \item The shapes of the following distributions in the RAPGAP simulation have been varied: a) the $\zpomeron$ distribution in photoproduction has been reweighted by factors $\zpomeron^{\pm 0.3}$ and \mbox{$(1-\zpomeron)^{\pm 0.3}$}; b) the $\ptjet$ distribution by $\hat{p}_T^{\pm 0.5}$ in both photoproduction and DIS; c) the $\xpomeron$ distribution by $\xpomeron^{\pm 0.2}$ in photoproduction and $\xpomeron^{\pm 0.3}$ in DIS; d) the $\xgamma$ distribution by $\xgamma^{\pm 0.3}$ and $(1-\xgamma)^{\pm 0.3}$ in both kinematic regions and e) the $y$ distribution by $y^{\pm 0.5}$ and $(1-y)^{\pm 0.5}$ in both kinematic regions. In DIS, the largest deviation ($9\%$) is due to the $\xpomeron$ reweighting. In photoproduction the largest error ($6\%$) arises from the $\pthat$ reweighting. \item The $t$ distribution is varied by factors $e^{\pm 2t/{\rm GeV}^2}$ as constrained by inclusive measurements~\cite{tslope,kapishin} leading to cross section errors of $2$ to $3\%$. \item The estimated number of non-diffractive background events which migrate into the sample from the unmeasured region $\xpomeron > 0.03$ or $M_Y>5$~GeV is varied by $\pm 50\%$, leading to a mean cross section uncertainty of $2\%$ in photoproduction and $3\%$ in DIS. \item A $7\%$ error arises from uncertainties in the migrations about the $M_Y$ boundary of the measurement. It is estimated by varying the simulated efficiencies of the forward detectors FMD and PRT by $\pm 4\%$ and $\pm 25\%$, respectively, and by variations in the DIFFVM simulation of a) the ratio of elastic proton to proton dissociation cross sections between $1:2$ and $2:1$, b) the generated $M_Y$ distribution within $M_Y^{-2.0\pm 0.3}$, c) the $t$ dependence in the proton dissociation simulation by factors $e^{\pm t/{\rm GeV}^2}$. \item The loss of diffractive events due to the $\etamax$ cut and the cuts on the FMD and PRT is corrected using the RAPGAP simulation. By studying jet events with an elastically scattered proton (measured in a Roman pot detector) in the range $\xpom < 0.05$, it is established that the RAPGAP simulation describes the loss seen in the data within a $10\%$ and $14\%$ statistical uncertainty for photoproduction and DIS, respectively~\cite{schenk}. This uncertainty is used to estimate the uncertainty on the rapidity gap selection in the present analysis and translates into cross section errors of $1\%$ in both photoproduction and DIS. \end{itemize} The largest errors in photoproduction arise from the uncertainty in the LAr energy scale and the migrations about the $M_Y$ boundary. In DIS, the largest error arises from the $\xpomeron$ reweighting of RAPGAP. The uncertainties due to the LAr hadronic energy scale, the energy contribution of tracks, the luminosity, the FMD noise, the estimated number of background events and the positron energy in the SPACAL for DIS are correlated between cross section bins. Both for the bin-to-bin correlated and the uncorrelated errors all individual contributions are added in quadrature to obtain the full uncertainties. \section{Results} The measurement results are presented in \figrange{fig_diszpomeron}{fig_scigp} and are listed in \tabrange{tab_xsdis1}{tab_xsgp2} as bin-averaged differential hadron level cross sections for a set of kinematic variables which characterise the scattering process. The measurements are compared with next-to-leading order QCD predictions based on the factorisation approach in \secrange{sec_dis}{sec_xgamdep} and to leading order soft colour neutralisation models in \secref{sec_sci}. \subsection{Diffractive dijet production in DIS} \label{sec_dis} In \figsref{fig_diszpomeron}{fig_disjets}, the differential cross sections are shown as functions of \zpomeronjets, $\logxpomeron$, $W$, $Q^2$, \ptjetone, \meanetajetlab, and $\deltaetastar$. The data are compared with NLO QCD predictions obtained using the DISENT program with the `H1 2006 Fit A' and `H1 2006 Fit B' diffractive parton densities. The NLO prediction based on the `H1 2006 Fit A' parton densities (only shown in \figref{fig_diszpomeron}) overestimates the measured cross section, in particular at high $\zpomeronjets$. The NLO prediction based on the `H1 2006 Fit B' parton densities agrees well with the distributions of all variables within the given errors. Hence the dijet cross sections distinguish between the two parton density sets which describe inclusive diffractive DIS similarly well. The good description of the differential cross section as a function of $\logxpomeron$ indicates that the $\xpomeron$ dependence of $f_{\pom}(x_\pom,t)$ is compatible with the dijet production mechanism within the shown errors. The agreement between predicted and measured differential cross sections as functions of $\ptjetone$ and $\deltaetastar$ suggests that the NLO QCD matrix element describes the hard scatter correctly within the uncertainties shown. The good description of both inclusive diffractive scattering and diffractive dijet production obtained from the `H1 2006 Fit B' parton densities supports the validity of QCD hard scattering factorisation in diffractive DIS. In the following discussion of diffractive dijet photoproduction, only the `H1 2006 Fit B' densities are considered. \subsection{Diffractive photoproduction of dijets} \label{sec_gp} Differential cross sections measured for photoproduction are shown in~\figref{fig_gpzpomeron} as functions of \zpomeronjets{} and $\xgammajets$. The measurements are compared with NLO predictions obtained with the Frixione et al. program, interfaced to the `H1~2006~Fit~B' diffractive parton densities. The NLO prediction overestimates the measured dijet cross section by a factor of approximately $2$. Diffractive dijet photoproduction thus cannot be described using the parton densities which lead to a good description of diffractive scattering in DIS. QCD hard scattering factorisation is therefore broken in photoproduction. A more detailed comparison of the cross sections in DIS and photoproduction is given in the next section. \subsection{Ratio of dijet cross sections in diffractive photoproduction and DIS} \label{sec_ratio} A reliable method to test QCD factorisation is obtained by dividing the ratio of measured to predicted cross sections in photoproduction by the corresponding ratio in DIS. In this double ratio many experimental errors and also theoretical scale errors cancel to a large extent. The double ratio is shown in \figref{fig_ratio} as a function of the photon-proton centre-of-mass energy $W$. The two NLO calculations are based on the `H1 2006 Fit B' diffractive parton densities and are corrected for hadronisation. The double ratio is rather insensitive to the detailed shape of the diffractive gluon density and the conclusions remain unchanged if the `H1 2006 Fit A' parton densities are used. The double ratio is $\approx 0.5$ throughout the measured $W$ range, indicating a suppression factor which is independent of the centre-of-mass energy within the uncertainties. Integrated over the measured kinematic range the ratio of data to NLO expectation for photoproduction is a factor $0.5\pm 0.1$ smaller than the same ratio in DIS where the error includes scale uncertainties. This confirms that QCD hard scattering factorisation is broken for diffractive dijet production in photoproduction with respect to the same process in DIS. The suppression in photoproduction is much smaller than the suppression in diffractive dijet production at the Tevatron~\cite{tevjets}. \subsection{Study of QCD factorisation breaking in photoproduction} \label{sec_xgamdep} The simple assumption that the suppression factor in photoproduction does not depend on any kinematic variable is studied by scaling the NLO predictions by an overall suppression factor $0.5$. Using such a global factor for both resolved and direct photon processes leads to a good description of all measured distributions as shown in \figsref{fig_gpzpomeronscaled}{fig_gpjetsscaled}. Whilst a suppression of resolved photoproduction is generally expected, a suppression of the direct photon contribution is in contradiction to theoretical expectations~\cite{Collins}. At NLO, the contributions of direct and resolved photon processes to the dijet cross section cannot be calculated separately. The following discussion therefore focuses on the dependence of the suppression on the variable $\xgammajetspar$, reconstructed at the parton level (PL) from parton jets before hadronisation, which is related to the fraction of the photon energy entering the jet system. In events with $\xgammajetspar>0.9$ almost the entire photon energy enters the jet system, whereas for events with $\xgammajetspar<0.9$ a significant photon remnant system is present which may lead to secondary interactions and rapidity gap destruction. A fit of the NLO prediction to the cross section differential in $\xgammajets$ with two free normalisation parameters for contributions from $\xgammajetspar<0.9$ and $\xgammajetspar>0.9$ yields suppression factors of $0.47 \pm 0.16$ and $0.53 \pm 0.14$, respectively. This result indicates again that the suppression is independent of $\xgammajetspar$ and that both direct and resolved contributions have to be suppressed by the same factor. Finally an investigation is performed of how well the data can be described under the assumption that the NLO calculation with $\xgammajetspar>0.9$ is not suppressed. The best agreement in a $\chi^2$ fit is reached for a suppression factor $0.44$ for the NLO calculation with $\xgammajetspar<0.9$ and the resulting distributions are shown for $\xgammajets$, $W$, $\meanetajet$ and $\ptjet$ in \figref{fig_resscale}. This prediction is incompatible with the measured cross sections. The assumption that the direct part obeys QCD factorisation is therefore strongly disfavoured by the present analysis. \subsection{Leading order soft colour neutralisation models} \label{sec_sci} The predictions of the soft colour interaction models SCI and GAL using the CTEQ5L LO parton densities of the proton are compared with the measurements in \figref{fig_scidis} in the DIS kinematic region. The SCI model describes the dijet cross section reasonably well. If the GRV '94 HO proton parton densities~\cite{grv5007} are used the cross sections are underestimated by a factor of approximately $2$ in agreement with the conclusions drawn in~\cite{disjets}. The GAL model overestimates the dijet rate by about $65\%$ on average. It gives a good description of the shapes of the differential cross sections as functions of $W$ and $\meanetajetlab$ but not as functions of $\zpomeronjets$ and $\logxpomeron$. The predictions for photoproduction are shown in \figref{fig_scigp}. The normalisation of the cross section is underestimated by factors of approximately $2.2$ for the SCI model and $1.5$ in the case of the GAL model. Both models describe the shapes of the differential cross sections reasonably well for $\logxpomeron$, $W$ and $\xgammajets$ but fail for $\zpomeronjets$. In summary, neither of the two models which describe diffractive dijet production in $p\bar{p}$ collisions is able to describe it in both DIS and photoproduction. \section{Summary} Diffractive dijet production is measured in deep-inelastic scattering and photoproduction in the same kinematic range \mbox{$165 < W < 242$~GeV}, \mbox{$\xpom < 0.03$,} \mbox{$\etjetone > 5$~GeV} and \mbox{$\etjettwo > 4$ GeV}, with limits on the photon virtuality $4 0.9$, where $\xgammajetspar$ is the fraction of the photon momentum entering the hard scatter and is reconstructed at the parton level from parton jets before hadronisation. The two factors are compatible with each other, indicating that the suppression is independent of $\xgammajetspar$. Direct photon processes contribute primarily at highest values of $\xgammajetspar$ and the present analysis therefore indicates that they are suppressed by a similar factor as resolved photon processes. A suppression of direct photon processes cannot be explained by models which base the rapidity gap survival probability on the presence of photon spectator interactions. The dijet cross sections are also compared with predictions of two soft colour neutralisation models. The SCI model which describes diffractive structure functions at HERA and diffractive dijet production at the Tevatron reproduces DIS dijet cross sections reasonably well but fails for photoproduction both in normalisation and in the shape of the differential cross section in $\zpomeronjets$. The GAL model is incompatible with both data sets. \section*{Acknowledgements} We are grateful to the HERA machine group whose outstanding efforts have made this experiment possible. 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The predictions based on the `H1 2006 Fit B' DPDFs are shown both with hadronisation corrections (solid white line) and at the parton level (solid black line). The inner band around the Fit B predictions indicates the uncertainty resulting from the variation of the renormalisation scale by factors $0.5$ and $2$ and the full band includes the uncertainty due to the hadronisation corrections added linearly.} \label{fig_diszpomeron} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f5.eps} \caption{Differential cross sections for the diffractive production of two jets in DIS in the kinematic region specified in \tabref{tab_xsdef} as a function of the variables a) \ptjetone{}, b) \meanetajetlab{} and c) $\deltaetastar$. The DISENT prediction based on the `H1~2006~Fit B' DPDFs at NLO with (white line) and without (black line) hadronisation corrections is also shown. For details about the errors see the caption of \figref{fig_diszpomeron}.} \label{fig_disjets} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f6.eps} \caption{Differential cross sections for the diffractive production of two jets in photoproduction in the kinematic region specified in \tabref{tab_xsdef} as a function of a) \zpomeronjets{} and b) \xgammajets. The inner error bars represent the statistical errors, the outer error bars include the uncorrelated systematic errors added in quadrature. The shaded band around the data points indicates an additional systematic uncertainty which is correlated between the data points. The NLO QCD predictions based on the Frixione et al. program (FR) and using the `H1 2006 Fit B' diffractive parton densities are shown with hadronisation corrections (white line) and at the parton level (black line). The inner band around the NLO prediction indicates the uncertainty resulting from simultaneous variations of the renormalisation and factorisation scales by factors $0.5$ and $2$ and the full band includes the uncertainty due to the hadronisation corrections added linearly.} \label{fig_gpzpomeron} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.45\textwidth,keepaspectratio]{d07-018f7.eps} \caption{Cross section double ratio of data to NLO prediction for photoproduction and DIS as a function of the photon-proton centre-of-mass energy $W$. The error bars indicate uncorrelated experimental uncertainties. The error bands around the ratio points show systematic uncertainties which are correlated between the ratio points. The inner band shows experimental uncertainties. The full band shows the quadratic sum of the correlated experimental uncertainties and NLO QCD uncertainties, estimated from variations of the factorisation and renormalisation scales. The nominal QCD scale $E_T$ is varied by the same factors ($0.5$ and $2$) and simultaneously in the same direction for the DIS and photoproduction calculations. The two NLO predictions are based on the same `H1~2006~Fit~B' diffractive parton densities and are corrected for hadronisation effects.} \label{fig_ratio} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f8.eps} \caption{Differential cross sections for the diffractive production of two jets in the photoproduction kinematic region specified in \tabref{tab_xsdef} as a function of a) \zpomeronjets, b) \xgammajets{}, c) \logxpomeron{} and d) $W$. The NLO prediction of the Frixione et al. program interfaced to the `H1~2006~Fit~B' DPDFs with and without hadronisation corrections, scaled by an overall normalisation factor $0.5$ is also shown. For details about the errors see the caption of \figref{fig_gpzpomeron}. } \label{fig_gpzpomeronscaled} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f9.eps} \caption{Differential cross sections for the diffractive production of two jets in photoproduction in the kinematic region specified in \tabref{tab_xsdef} as a function of the jet variables a) \ptjet, b) \meanetajetlab, c) \deltaeta{} and d) \mjj. The NLO prediction of the Frixione et al. program interfaced to the `H1~2006~Fit~B' DPDFs with and without hadronisation corrections, scaled by an overall normalisation factor $0.5$ is also shown. For details about the errors see the caption of \figref{fig_gpzpomeron}.} \label{fig_gpjetsscaled} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f10.eps} \caption{Differential cross sections for the diffractive production of two jets in photoproduction in the kinematic region specified in \tabref{tab_xsdef} as a function of a) \xgammajets, b) $W$, c) $\meanetajet$ and d) $\ptjet$. The NLO prediction of the Frixione et al. program interfaced to the `H1~2006~Fit~B' DPDFs with hadronisation corrections is also shown. The part of the NLO calculation for which $\xgammajetspar<0.9$ at the parton level is scaled by $0.44$. For details about the errors see the caption of \figref{fig_gpzpomeron}.} \label{fig_resscale} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f11.eps} \caption{Differential cross sections for the diffractive production of two jets in DIS in the kinematic region specified in \tabref{tab_xsdef} as a function of the variables a) $W$, b) \meanetajetlab, c) \zpomeronjets{} and d) \logxpomeron. Leading order predictions of the soft colour neutralisation models SCI and GAL as implemented in LEPTO are also shown, based on the CTEQ5L leading order proton parton densities. For details about the errors see the caption of~\figref{fig_diszpomeron}.} \label{fig_scidis} \end{figure} \begin{figure}[hhh] \centering \includegraphics[width=0.8\textwidth,keepaspectratio]{d07-018f12.eps} \caption{Differential cross sections for the diffractive production of two jets in photoproduction in the kinematic region specified in \tabref{tab_xsdef} as a function of the variables a) \zpomeronjets, b) \logxpomeron, c) $W$ and d) \xgammajets. Leading order predictions of the soft colour neutralisation models SCI and GAL as implemented in PYTHIA are also shown, based on the CTEQ5L leading order proton parton densities.} \label{fig_scigp} \end{figure} \setlength{\tabcolsep}{0.2cm} \begin{table} \renewcommand{\arraystretch}{1.3} \begin{center} {\textbf{Diffractive DIS Dijet Cross Sections}} \vglue1em \begin{tabular}{|c|c|c|c|c|c|} \hline $\zpomeronjets$ & ${\rm d}\sigma/{\rm d}\zpomeronjets$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[0.1,0.4)$ & $59$ & $5$ & $5$ & $10$ & $1.00 \pm 0.03$ \\ \hline $[0.4,0.6)$ & $34$ & $4$ & $3$ & $8$ & $0.97 \pm 0.02$ \\ \hline $[0.6,0.8)$ & $16$ & $3$ & $2$ & $4$ & $0.95 \pm 0.02$ \\ \hline $[0.8,1)$ & $5.2$ & $1.2$ & $0.6$ & $2.0$ & $1.0 \pm 0.2$ \\ \hline \hline $\logxpomeron$ & ${\rm d}\sigma/{\rm d}\logxpomeron$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[-2.3,-1.9)$ & $20.3$ & $2.6$ & $1.5$ & $4.3$ & $1.01 \pm 0.02$ \\ \hline $[-1.9,-1.7)$ & $43$ & $5$ & $3$ & $7$ & $0.99 \pm 0.01$ \\ \hline $[-1.7,-1.5)$ & $62$ & $7$ & $8$ & $13$ & $0.98 \pm 0.04$ \\ \hline \hline $W$ (GeV) & ${\rm d}\sigma/{\rm d}W$ & $\Delta_{\rm stat}$ & $\Delta_{\rm corr}$ & $\Delta_{\rm tot}$ & $1+\delta_{\rm had}$ \\ & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & \\ \hline $[165,185)$ & $0.45$ & $0.06$ & $0.04$ & $0.10$ & $0.95 \pm 0.03$ \\ \hline $[185,205)$ & $0.40$ & $0.05$ & $0.03$ & $0.08$ & $1.00 \pm 0.03$ \\ \hline $[205,225)$ & $0.37$ & $0.05$ & $0.03$ & $0.07$ & $1.00 \pm 0.02$ \\ \hline $[225,242)$ & $0.33$ & $0.05$ & $0.04$ & $0.07$ & $1.03 \pm 0.02$ \\ \hline \hline $Q^2$ (GeV$^2$) & ${\rm d}\sigma/{\rm d}Q^2$ & $\Delta_{\rm stat}$ & $\Delta_{\rm corr}$ & $\Delta_{\rm tot}$ & $1+\delta_{\rm had}$ \\ & (pb GeV$^{-2}$) & (pb GeV$^{-2}$) & (pb GeV$^{-2}$) & (pb GeV$^{-2}$) & \\ \hline $[4,6)$ & $3.4$ & $0.5$ & $0.2$ & $0.7$ & $0.97 \pm 0.02$ \\ \hline $[6,8)$ & $2.4$ & $0.4$ & $0.2$ & $0.5$ & $0.99 \pm 0.02$ \\ \hline $[8,12)$ & $0.83$ & $0.17$ & $0.08$ & $0.21$ & $0.98 \pm 0.02$ \\ \hline $[12,20)$ & $0.63$ & $0.10$ & $0.06$ & $0.14$ & $1.01 \pm 0.06$ \\ \hline $[20,30)$ & $0.47$ & $0.08$ & $0.06$ & $0.12$ & $0.96 \pm 0.03$ \\ \hline $[30,40)$ & $0.18$ & $0.05$ & $0.02$ & $0.06$ & $1.03 \pm 0.05$ \\ \hline $[40,80)$ & $0.081$ & $0.021$ & $0.009$ & $0.026$ & $1.03 \pm 0.02$ \\ \hline \end{tabular} \end{center} \caption{The hadron level differential cross section of diffractive dijet production in $ep$ collisions in the DIS kinematic range specified in \tabref{tab_xsdef}. The quoted cross section is the average value over the bin specified in the first column. The quantity $\Delta_{\rm stat}$ is the statistical uncertainty, $\Delta_{\rm corr}$ the bin-correlated systematic uncertainty and $\Delta_{\rm tot}$ the total quadratic sum of statistical and systematic errors including $\Delta_{\rm corr}$. The quantity $1+\delta_{\rm had}$ is the factor by which the parton level NLO calculation is multiplied to correct for hadronisation effects.} \label{tab_xsdis1} \end{table} \setlength{\tabcolsep}{0.2cm} \begin{table} \renewcommand{\arraystretch}{1.3} \begin{center} {\textbf{Diffractive DIS Dijet Cross Sections}} \vglue1em \begin{tabular}{|c|c|c|c|c|c|} \hline $\ptjetone$ (GeV) & ${\rm d}\sigma/{\rm d}\ptjetone$ & $\Delta_{\rm stat}$ & $\Delta_{\rm corr}$ & $\Delta_{\rm tot}$ & $1+\delta_{\rm had}$ \\ & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & \\ \hline $[5,7)$ & $9.3$ & $0.8$ & $0.6$ & $1.5$ & $1.03 \pm 0.01$ \\ \hline $[7,9)$ & $4.1$ & $0.5$ & $0.4$ & $0.9$ & $0.96 \pm 0.01$ \\ \hline $[9,11)$ & $1.0$ & $0.2$ & $0.2$ & $0.4$ & $0.91 \pm 0.09$ \\ \hline \hline $\meanetajetlab$ & ${\rm d}\sigma/{\rm d}\meanetajetlab$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[-0.7,-0.4)$ & $21$ & $3$ & $2$ & $5$ & $0.97 \pm 0.06$ \\ \hline $[-0.4,-0.1)$ & $29$ & $4$ & $3$ & $6$ & $1.06 \pm 0.03$ \\ \hline $[-0.1,0.2)$ & $21$ & $3$ & $2$ & $5$ & $1.03 \pm 0.02$ \\ \hline \hline $\deltaetastar$ & ${\rm d}\sigma/{\rm d}\deltaetastar$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[0,0.5)$ & $30$ & $3$ & $2$ & $5$ & $0.96 \pm 0.04$ \\ \hline $[0.5,1)$ & $15$ & $2$ & $2$ & $3$ & $1.06 \pm 0.05$ \\ \hline $[1,2)$ & $5.9$ & $0.8$ & $0.5$ & $1.3$ & $1.01 \pm 0.06$ \\ \hline \end{tabular} \end{center} \caption{The hadron level differential cross section of diffractive dijet production in $ep$ collisions in the DIS kinematic range specified in \tabref{tab_xsdef} (continued). For details see the caption of \tabref{tab_xsdis1}.} \label{tab_xsdis2} \end{table} \setlength{\tabcolsep}{0.2cm} \begin{table} \renewcommand{\arraystretch}{1.3} \begin{center} {\textbf{Diffractive Photoproduction Dijet Cross Sections}} \vglue1em \begin{tabular}{|c|c|c|c|c|c|} \hline $\zpomeronjets$ & ${\rm d}\sigma/{\rm d}\zpomeronjets$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[0.1,0.4)$ & $290$ & $20$ & $30$ & $50$ & $0.85 \pm 0.01$ \\ \hline $[0.4,0.6)$ & $340$ & $20$ & $40$ & $70$ & $0.86 \pm 0.02$ \\ \hline $[0.6,0.8)$ & $310$ & $20$ & $30$ & $50$ & $0.95 \pm 0.06$ \\ \hline $[0.8,1)$ & $150$ & $10$ & $10$ & $30$ & $1.00 \pm 0.04$ \\ \hline \hline $\xgammajets$ & ${\rm d}\sigma/{\rm d}\xgammajets$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[0.1,0.4)$ & $150$ & $10$ & $20$ & $30$ & $0.69 \pm 0.02$ \\ \hline $[0.4,0.6)$ & $250$ & $20$ & $20$ & $50$ & $0.80 \pm 0.02$ \\ \hline $[0.6,0.8)$ & $370$ & $20$ & $20$ & $40$ & $1.36 \pm 0.05$ \\ \hline $[0.8,1)$ & $350$ & $20$ & $60$ & $70$ & $0.81 \pm 0.02$ \\ \hline \hline $\logxpomeron$ & ${\rm d}\sigma/{\rm d}\logxpomeron$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[-2.3,-2.1)$ & $110$ & $10$ & $10$ & $30$ & $1.13 \pm 0.05$ \\ \hline $[-2.1,-1.9)$ & $200$ & $10$ & $20$ & $30$ & $0.94 \pm 0.02$ \\ \hline $[-1.9,-1.7)$ & $350$ & $20$ & $30$ & $50$ & $0.88 \pm 0.02$ \\ \hline $[-1.7,-1.5)$ & $550$ & $30$ & $60$ & $100$ & $0.84 \pm 0.01$ \\ \hline \hline $W$ (GeV) & ${\rm d}\sigma/{\rm d}W$ & $\Delta_{\rm stat}$ & $\Delta_{\rm corr}$ & $\Delta_{\rm tot}$ & $1+\delta_{\rm had}$ \\ & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & \\ \hline $[165,185)$ & $3.8$ & $0.2$ & $0.4$ & $0.6$ & $0.87 \pm 0.02$ \\ \hline $[185,205)$ & $3.5$ & $0.2$ & $0.3$ & $0.5$ & $0.92 \pm 0.02$ \\ \hline $[205,225)$ & $3.0$ & $0.2$ & $0.3$ & $0.4$ & $0.93 \pm 0.02$ \\ \hline $[225,242)$ & $2.4$ & $0.2$ & $0.2$ & $0.4$ & $0.85 \pm 0.02$ \\ \hline \end{tabular} \end{center} \caption{The hadron level differential cross section of diffractive dijet production in $ep$ collisions in the photoproduction kinematic range specified in \tabref{tab_xsdef}. For details see the caption of \tabref{tab_xsdis1}.} \label{tab_xsgp1} \end{table} \setlength{\tabcolsep}{0.2cm} \begin{table} \renewcommand{\arraystretch}{1.3} \begin{center} {\textbf{Diffractive Photoproduction Dijet Cross Sections}} \vglue1em \begin{tabular}{|c|c|c|c|c|c|} \hline $\ptjet$ (GeV) & ${\rm d}\sigma/{\rm d}\ptjet$ & $\Delta_{\rm stat}$ & $\Delta_{\rm corr}$ & $\Delta_{\rm tot}$ & $1+\delta_{\rm had}$ \\ & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & \\ \hline $[5,7)$ & $85$ & $3$ & $7$ & $12$ & $0.85 \pm 0.01$ \\ \hline $[7,9)$ & $28$ & $2$ & $3$ & $4$ & $0.98 \pm 0.03$ \\ \hline $[9,11)$ & $7.3$ & $0.8$ & $1.0$ & $1.9$ & $1.02 \pm 0.05$ \\ \hline \hline $\meanetajetlab$ & ${\rm d}\sigma/{\rm d}\meanetajetlab$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[-0.7,-0.4)$ & $140$ & $10$ & $20$ & $30$ & $0.88 \pm 0.02$ \\ \hline $[-0.4,-0.1)$ & $230$ & $10$ & $20$ & $30$ & $1.01 \pm 0.02$ \\ \hline $[-0.1,0.2)$ & $190$ & $10$ & $20$ & $30$ & $0.99 \pm 0.02$ \\ \hline $[0.2,0.8)$ & $89$ & $6$ & $7$ & $13$ & $0.79 \pm 0.02$ \\ \hline \hline $\deltaeta$ & ${\rm d}\sigma/{\rm d}\deltaeta$ (pb) & $\Delta_{\rm stat}$ (pb) & $\Delta_{\rm corr}$ (pb) & $\Delta_{\rm tot}$ (pb) & $1+\delta_{\rm had}$ \\ \hline $[0,0.5)$ & $179$ & $9$ & $19$ & $29$ & $0.88 \pm 0.01$ \\ \hline $[0.5,1)$ & $157$ & $9$ & $14$ & $21$ & $0.90 \pm 0.02$ \\ \hline $[1,1.5)$ & $88$ & $6$ & $7$ & $13$ & $0.90 \pm 0.02$ \\ \hline $[1.5,2)$ & $55$ & $5$ & $6$ & $9$ & $0.90 \pm 0.03$ \\ \hline \hline $M_{12}$ (GeV) & ${\rm d}\sigma/{\rm d}M_{12}$ & $\Delta_{\rm stat}$ & $\Delta_{\rm corr}$ & $\Delta_{\rm tot}$ & $1+\delta_{\rm had}$ \\ & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & (pb GeV$^{-1}$) & \\ \hline $[9,17)$ & $25.6$ & $0.9$ & $2.3$ & $3.9$ & $0.88 \pm 0.01$ \\ \hline $[17,27)$ & $3.8$ & $0.3$ & $0.4$ & $0.7$ & $0.97 \pm 0.03$ \\ \hline \end{tabular} \end{center} \caption{The hadron level differential cross section of diffractive dijet production in $ep$ collisions in the photoproduction kinematic range specified in \tabref{tab_xsdef} (continued). For details see the caption of \tabref{tab_xsdis1}.} \label{tab_xsgp2} \end{table} \end{document}