Measurement of dijet production in diffractive deep-inelastic scattering with a leading proton at HERA

The cross section of diffractive deep-inelastic scattering ep→eXp is measured, where the system X contains at least two jets and the leading final state proton is detected in the H1 Forward Proton Spectrometer. The measurement is performed for fractional proton longitudinal momentum loss xℙ<0.1 and covers the range 0.1<|t|<0.7 GeV2 in squared four-momentum transfer at the proton vertex and 4<Q2<110 GeV2 in photon virtuality. The differential cross sections extrapolated to |t|<1 GeV2 are in agreement with next-to-leading order QCD predictions based on diffractive parton distribution functions extracted from measurements of inclusive and dijet cross sections in diffractive deep-inelastic scattering. The data are also compared with leading order Monte Carlo models.

Abstract The cross section of diffractive deep-inelastic scattering ep → eXp is measured, where the system X contains at least two jets and the leading final state proton is detected in the H1 Forward Proton Spectrometer. The measurement is performed for fractional proton longitudinal momentum loss x P < 0.1 and covers the range 0.1 < |t| < 0.7 GeV 2 in squared four-momentum transfer at the proton vertex and 4 < Q 2 < 110 GeV 2 in photon virtuality. The differential cross sections extrapolated to |t| < 1 GeV 2 are in agreement with next-to-leading order QCD predictions based on diffractive parton distribution functions extracted from measurements of inclusive and dijet cross sections in diffractive deep-inelastic scattering. The data are also compared with leading order Monte Carlo models.

Introduction
Diffractive processes such as ep → eXY , where the systems X and Y are separated in rapidity, have been studied extensively in deep-inelastic scattering (DIS) at the electron 1 -proton collider HERA [1][2][3][4][5][6][7][8]. Diffractive DIS events can be viewed as resulting from processes in which the photon probes a net colour singlet combination of exchanged partons. The photon virtuality Q 2 , the high transverse momentum of jets or a heavy quark mass can provide a hard scale for perturbative QCD calculations. For semi-inclusive DIS processes such as ep → eXp the hard scattering QCD collinear factorisation theorem [9] allows the definition of diffractive parton distribution functions (DPDFs). The dependence of diffractive DIS on a hard scale can thus be treated in a manner similar to the treatment of inclusive DIS, for example through the application of the DGLAP parton evolution equations [10][11][12][13][14]. DPDFs have been determined from QCD fits to diffractive DIS measurements at HERA [2,3,8]. The inclusive diffractive DIS cross section is directly proportional to the sum of the quark DPDFs and constrains the gluon DPDF via scaling violations. The production of diffractive hadronic final states containing heavy quarks or jets proceeds mainly via boson gluon fusion (BGF) and therefore directly constrains the diffractive gluon density [3,8].
In previous analyses at HERA, diffractive DIS events have been selected on the basis of the presence of a large rapidity gap (LRG) between system Y , which consists of the outgoing proton or its dissociative excitations, and the hadronic final state, system X [3,4]. The main advantage of the LRG method is its high acceptance for diffractive processes. A complementary way to study diffraction is by direct measurement of the outgoing proton, which remains in-tact in elastic interactions. This is achieved by the H1 experiment using the Forward Proton Spectrometer (FPS) [15,16], which is a set of tracking detectors along the proton beam line. Despite the low geometrical acceptance of the FPS, this method of selecting diffractive events has several advantages. The squared four-momentum transfer at the proton vertex, t, can be reconstructed with the FPS, while this is only possible in exclusive final states in the LRG case. The FPS method selects events in which the proton scatters elastically, whereas the LRG method does not distinguish between the case where the scattered proton remains intact or where it dissociates into a system of low mass M Y . The FPS method also allows measurements to be performed at higher values of fractional proton longitudinal momentum loss, x P , than possible using the LRG method.
This paper presents the first measurement of the cross section for the diffractive DIS process ep → ejj X p, with two jets and a leading proton in the final state. The diffractive dijet cross sections are compared with next-to-leading order (NLO) QCD predictions based on DPDFs from H1 [2,3] and with leading order (LO) Monte Carlo (MC) simulations based on different models.
The dijet cross sections are measured for two event topologies: for a topology where two jets are found in the central pseudorapidity range, labelled as 'two central jets', and for a topology where one jet is central and one jet is more forward 2 , labelled as 'one central + one forward jet'. The universality of DPDFs is studied using events with two central jets. The distributions of the proton vertex variables x P and t are compared to those of the inclusive diffractive DIS case. This comparison tests the proton vertex factorisation hypothesis which assumes that the DIS variable factorise from the four-momentum of the final state proton. The data are also compared directly with the LRG measurement of the dijet cross section in diffractive DIS [3] in order to test the compatibility of the two experimental techniques. Finally, events with one central and one forward jet are used to investigate diffractive DIS in a region of phase space where effects beyond DGLAP parton evolution may be enhanced. This topology is not accessible with the LRG method since the rapidity gap requirement limits the pseudorapidity of the reconstructed jets to the central region. Figure 1 illustrates the dominant process for diffractive dijet production in DIS. The incoming electron with fourmomentum k interacts with the proton with four-momentum P via the exchange of a virtual photon with four-momentum q. The DIS kinematic variables are defined as:

Kinematics
where Q 2 is the photon virtuality, x is the longitudinal momentum fraction of the proton carried by the struck quark and y is the inelasticity of the process. These three variables are related via Q 2 = xys, where s denotes the ep centre-ofmass energy squared. The hadronic final state of diffractive events consists of two systems X and Y , separated by a gap in rapidity. In general, the system Y is the outgoing proton or one of its low mass excitations. In events where the outgoing proton remains intact, M Y = m p , the mass of the proton. The kinematics of diffractive DIS are described by: where x P denotes the longitudinal momentum fraction of the proton carried by the colour singlet exchange, t is the squared four-momentum transfer at the proton vertex and β is the fractional momentum of the diffractive exchange carried by the struck parton. The longitudinal momentum fraction of the diffractive exchange carried by the parton entering the hard scatter is where v is the four-momentum of the parton.

Theoretical framework and Monte Carlo models
Within Regge phenomenology, cross sections at high energies are described by the exchange of Regge trajectories. The diffractive cross section is dominated by a trajectory usually called the Pomeron (P). In analyses of HERA data [2,3,8], diffractive DIS cross sections are interpreted assuming 'proton vertex factorisation' which provides a description of diffractive DIS in terms of a resolved Pomeron [17,18]. The QCD factorisation theorem and DGLAP parton evolution equations are applied to the dependence of the cross section on Q 2 and β, while a Regge inspired approach is used to express the dependence on x P and t. The resolved Pomeron (RP) model [17] is implemented in the RAPGAP event generator [19]. RAPGAP implements both a leading Pomeron (P) trajectory and a sub-leading 'Reggeon' (R). In this analysis the DPDF H1 2006 Fit B [2] is used, which employs the Owens pion PDFs [20] for the partonic content of the Reggeon. The Reggeon contribution is significant for x P > 0.01. Higher order QCD radiation is modelled by parton showers. Processes with a resolved virtual photon are also included, with the photon structure function given by the SaS-G 2D LO parameterisation [21].
In the two-gluon Pomeron (TGP) model [22,23], the diffractive exchange is modelled at LO as the interaction of a colourless pair of gluons with a qq or qqg configuration emerging from the photon. The model is implemented in the RAPGAP generator. Higher order effects are simulated using parton showers. The unintegrated gluon PDF of set A0 [24] is used.
In the soft colour interaction (SCI) model [25,26], the diffractive exchange is modelled via non-diffractive DIS scattering with subsequent colour rearrangement between the partons in the final state, which can produce a colour singlet system separated by a large gap in pseudorapidity. A refined version of the SCI model which uses a generalised area law (GAL) for the probability of having a soft colour interaction [27] is used in this analysis (SCI+GAL). Predictions for diffractive dijet production within the SCI+GAL model are obtained using the leading order generator program LEPTO [28]. Higher order effects are simulated using parton showers [29,30]. The calculations are based on the CTEQ6L [31] proton PDFs. The probability for a soft colour interaction, R, has been tuned to 0.3 to describe the total diffractive dijet cross section as measured using the 'two central jets' topology.
In all three models hadronisation is simulated using the Lund string model [32] implemented within the PYTHIA program [33,34].
In this analysis the dijet cross section is also compared to NLO QCD calculations. Assuming proton vertex factorisation, NLO QCD predictions for the diffractive partonic dijet cross section are calculated in bins of x P using the NLO-JET++ [35] program and integrated over the full x P range of the measurement. The renormalisation and factorisation scales are set to μ r = μ f = Q 2 + P * T 2 , where P * T is the mean of the transverse momenta of the two leading jets in the hadronic centre-of-mass frame. In order to estimate the uncertainties of the NLO QCD calculations due to missing higher orders, the factorisation scale μ f and renormalisation scale μ r are varied simultaneously by factors of 0.5 and 2. The average uncertainty arising from the variation of the scale is about 33 %. The DPDFs used in the NLO QCD calculations are H1 2006 Fit B [2] and H1 2007 Jets [3]. The H1 2007 Jets fit is based on the diffractive inclusive and dijet data while H1 2006 Fit B is based on inclusive diffractive data only. The uncertainty of the NLO QCD calculations due to DPDFs is estimated by propagating the DPDF errors. The DPDF errors are available only for the DPDF set H1 2006 Fit B. The average uncertainty resulting from the DPDF errors is about 7 % which is much smaller than the scale uncertainty. In the NLOJET++ calculations the strong coupling is set via Λ (4) MS = 340 ± 37 MeV for four flavours, which corresponds to the value of α (5) s (M Z ) = 0.119 ± 0.002 for five flavours in the 2-loop approximation [36,37]. The average uncertainty resulting from the variation of α s (M Z ) is about 1.5 %. In order to demonstrate the size of the NLO corrections, the QCD calculations are also performed at leading order.
The NLO QCD partonic cross sections are corrected to the level of stable hadrons by evaluating effects due to initial and final state parton showering, fragmentation and hadronisation. The hadronisation corrections are defined in each bin as a ratio of the cross section obtained at the level of stable hadrons to the partonic cross sections. Two sets of hadronisation corrections have been obtained using the RAPGAP generator using two different parton shower models: parton showers based on leading logarithm DGLAP splitting functions in leading order α s [10][11][12][13] and parton showers based on the colour dipole model as implemented in ARIADNE [38]. The nominal set of corrections (1 + δ had ) is taken as the average of the two sets, while the difference between them is considered as the hadronisation uncertainty. The average hadronisation corrections are of about 0.9 with an estimated uncertainty of about 7 %. Uncertainties of the NLO QCD predictions arising due to scale variations and hadronisation corrections are added in quadrature.
In order to compare with the results of the FPS measurements, NLO QCD predictions as well as predictions of the RP model are scaled down by a factor of 1.20 [16] due to the fact that the DPDF sets H1 2006 Fit B and H1 2007 Jets use LRG data which contain a proton dissociation contribution. The t-dependence of the P and R fluxes implemented in the H1 DPDF sets and the RP model are tuned to reproduce the t-dependence measured in inclusive diffractive DIS with a leading proton in the final state [15].

Experimental technique
The e ± p data used in this analysis were collected with the H1 detector in the years 2005 to 2007 and correspond to an integrated luminosity of 156.6 pb −1 . During this period the HERA collider was operated at electron and proton beam energies of E e = 27.6 GeV and E p = 920 GeV respectively, corresponding to an ep centre-of-mass energy of √ s = 319 GeV.

H1 detector
A detailed description of the H1 detector can be found elsewhere [39][40][41]. Here, the components most relevant for the presented measurement are described briefly. A righthanded coordinate system is employed with the origin at the nominal interaction point, where the z-axis pointing in the proton beam or forward direction and the x(y) axis points in the horizontal (vertical) direction. The polar angle θ is measured with respect to the proton beam axis and the pseudorapidity is defined as η = − ln tan(θ/2). The Central Tracking Detector (CTD), with a polar angle coverage of 20 • < θ < 160 • , is used to reconstruct the interaction vertex and to measure the momenta of charged particles from the curvature of their trajectories in the 1.16 T field provided by a superconducting solenoid. Scattered electrons with polar angles in the range 154 • < θ e < 176 • are measured in a lead/scintillatingfibre calorimeter, the SpaCal [41]. The energy resolution is σ (E)/E ≈ 7 %/ √ E[GeV] ⊕ 1 % as determined from the test beam measurement [42,43]. A Backward Proportional Chamber (BPC) in front of the SpaCal is used to measure the electron polar angle. The finely segmented Liquid Argon (LAr) sampling calorimeter surrounds the tracking system and covers the range in polar angle 4 • < θ < 154 • corresponding to a pseudorapidity range −1.5 < η < 3.4. The LAr calorimeter consists of an electromagnetic section with lead as the absorber and a hadronic section with steel as the absorber. The total depth varies with θ between 4.5 and 8 interaction lengths. The energy resolution, determined from test beam measurements [42,43], is σ (E)/E ≈ 11 %/ √ E[GeV] ⊕ 1 % for electrons and σ (E)/E ≈ 50 %/ √ E[GeV] ⊕ 2 % for hadrons. The hadronic final state is reconstructed using an energy flow algorithm which combines charged particles measured in the CTD with information from the SpaCal and LAr calorimeters [44].
The luminosity is determined by measuring the rate of the Bethe-Heitler process ep → epγ detected in a photon detector located at z = −103 m.
The energy and scattering angle of the leading proton are obtained from track measurements in the FPS [45]. Protons scattered at small angles are deflected by the proton beamline magnets into a system of detectors placed within the proton beam pipe inside two movable stations, known as Roman Pots. Both Roman Pot stations contain four planes, where each plane consists of five layers of scintillating fibres, which together measure two orthogonal coordinates in the (x, y) plane. The fibre coordinate planes are sandwiched between planes of scintillator tiles used for the trigger. The stations approach the beam horizontally and are positioned at z = 61 m and z = 80 m. The detectors are sensitive to scattered protons which lose less than 10 % of their energy in the ep interaction and are scattered through angles below 1 mrad.
The energy resolution of the FPS is approximately 5 GeV within the measured range. The absolute energy scale uncertainty is 1 GeV. The effective resolution in the reconstruction of the transverse momentum components of the scattered proton with respect to the incident proton is determined to be ∼50 MeV for P x and ∼150 MeV for P y , dominated by the intrinsic transverse momentum spread of the proton beam at the interaction point. The scale uncertainties in the transverse momentum measurements are 10 MeV for P x and 30 MeV for P y . Further details of the analysis of the FPS resolution and scale uncertainties can be found elsewhere [16]. For a leading proton which passes through both FPS stations, the track reconstruction efficiency is 48 % on average.

Kinematic reconstruction
The inclusive DIS variables Q 2 , x and the inelasticity y are reconstructed by combining information from the scattered electron and the hadronic final state using the following method [1]: Here, y e and y d denote the values of y obtained from the scattered electron only (electron method) and from the angles of the electron and the hadronic final state (double angle method), respectively [46,47].
The observable x P is reconstructed as: where E p is the measured energy of the leading proton in the FPS. The quantity β is reconstructed as β = x/x P . The squared four-momentum transfer at the proton vertex is reconstructed using the transverse momentum P T of the leading proton measured with the FPS and x P as described above, such that: where |t min | is the minimum kinematically accessible value of |t|. The absolute resolution in t varies over the measured range from 0.06 GeV 2 at |t| = 0.1 GeV 2 to 0.17 GeV 2 at |t| = 0.7 GeV 2 .
An estimator for the momentum fraction z P is defined at the level of stable hadrons as: where M jj denotes the invariant mass of the dijet system. The cross sections are studied in terms of the DIS variables y, Q 2 , β, z P , the proton vertex variables x P and t, the jet variables P * T and η, and where P * T ,1 , η * 1 , φ * 1 and P * T ,2 , η * 2 , φ * 2 are transverse momenta, pseudorapidities and azimuthal angles of the axes of the leading and next-to-leading jets, respectively, reconstructed in the hadronic centre-of-mass frame. The indices 1, 2 stand for the two jets used in the specific analyses.

Event selection
The events used in the 'two central jets' and 'one central + one forward jet' analyses are triggered on the basis of a coincidence of a signal in the FPS trigger scintillator tiles and in the electromagnetic SpaCal. The trigger efficiency, calculated using events collected with independent triggers, is found to be 99 % on average and is independent of kinematic variables.

DIS selection
The selection of DIS events is based on the identification of the scattered electron as the most energetic electromagnetic cluster in the SpaCal calorimeter. The energy E e and polar angle θ e of the scattered electron are determined from the SpaCal cluster and the interaction vertex reconstructed in the CTD. The electron candidate is required to be in range 154 • < θ e < 176 • and E e > 10 GeV. In order to improve background rejection, an additional requirement on the transverse cluster radius, estimated using square root energy weighting [48], of less then 4 cm is imposed.
The reconstructed z coordinate of the event vertex is required to be within ±35 cm of the mean position. At least one track originating from the interaction vertex and reconstructed in the CTD is required to have a transverse momentum above 0.1 GeV.
The quantity (E − P z ), summed over the energies and longitudinal momenta of all reconstructed particles including the electron, is required to be between 35 GeV and 70 GeV. For neutral current DIS events this quantity is expected to be twice the electron beam energy when neglecting detector effects and QED radiation. This requirement is applied to remove radiative DIS events and photoproduction background.
In order to ensure a good detector acceptance the measurement is restricted to the ranges 4 < Q 2 < 110 GeV 2 and 0.05 < y < 0.7.

Leading proton selection
A high FPS acceptance is ensured by requiring the energy of the leading proton E p to be greater than 90 % of the proton beam energy E p and the horizontal and vertical projections of the transverse momentum to be in the ranges −0.63 < P x < −0.27 GeV and |P y | < 0.8 GeV, respectively. Additionally, t is restricted to the range 0.1 < |t| < 0.7 GeV 2 .
The quantity (E + P z ), summed over all reconstructed particles including the leading proton, is required to be below 1880 GeV. For neutral current DIS events this quantity is expected to be twice the proton beam energy. This requirement is applied to suppress cases where a DIS event reconstructed in the central detector coincides with background in the FPS, for example due to interactions between offmomentum protons from the beam halo with residual gas within the beampipe.
Previous diffractive dijet DIS measurements [3,4,6] and DPDF fits [2,3,8] have been performed for |t min | < |t| < 1 GeV 2 . To compare with these results, the cross sections are extrapolated to the range |t min | < |t| < 1 GeV 2 using the t dependence measured in inclusive diffractive DIS with a leading proton in the final state [15].

Jet selection
Reconstructed hadronic final state objects are used as input to the longitudinally invariant k T jet algorithm [49] using the p T recombination scheme with a jet radius of 1.0 as implemented in the FastJet package [50]. The jet finding algorithm is applied in the photon-proton centre-of-mass system (γ * p frame). The jet variables in the γ * p frame are denoted by a asterisk.
In the 'two central jets' analysis, the requirements are P * T ,1 > 5 GeV and P * T ,2 > 4 GeV for the leading and nextto-leading jet, respectively. Asymmetric cuts are placed on the jet transverse momenta to restrict the phase space to a region where NLO calculations are reliable. The axes of the jets are required to lie within the pseudorapidity range −1 < η 1,2 < 2.5 in the laboratory frame. The selected event topology is similar to that in the LRG dijet data used in the DPDF fits [3,8]. This data selection is used for testing the proton vertex factorisation hypothesis and the DPDFs in processes with a leading proton in the final state.
The selection of the 'one central + one forward jet' topology is motivated by the study of diffractive DIS processes in a phase space where deviations from DGLAP parton evolution may be present. The requirement of a forward jet suppresses the parton p T ordering which is assumed by DGLAP Leading Proton evolution. At least one central jet with −1 < η c < 2.5 and one forward jet with 1 < η f < 2.8, where η f > η c , are required with P * T > 3.5 GeV. In addition, the invariant mass of the central-forward jet system is required to be larger than 12 GeV to avoid the phase space region in which NLO QCD calculations are unreliable.
The selection criteria for the two analyses are summarised in Table 1. The 'two central jets' data sample contains 581 events and the 'one central + one forward jet' data sample contains 309 events.

Background subtraction
The selected data samples contain background events arising from random coincidences of non-diffractive DIS events, with off-momentum beam-halo protons producing a signal in the FPS. The beam-halo background contribution is estimated statistically by combining the quantity (E + P z ) summed over all reconstructed particles in the central detector in DIS events (without the requirement of a track in the FPS) with the quantity (E + P z ) for beam-halo protons from randomly triggered events. The (E + P z ) spectra for leading proton and beam-halo DIS events for both dijet event topologies are shown in Fig. 2. The background distribution is normalised to the FPS DIS data distribution in the range (E + P z ) > 1880 GeV where the beam-halo background dominates. The ratio of signal to background depends on the signal cross section and is found to be considerably larger than in the inclusive diffractive DIS processes measured with the FPS detector [16]. After the selection cut (E + P z ) < 1880 GeV the remaining background amounts on average to about 5 %. The background is determined and subtracted bin-by-bin using this method.

Detector simulation
Monte Carlo simulations are used to correct the data for the effects of detector acceptance, inefficiencies, migrations between measurement intervals due to finite resolution and QED radiation. The response of the H1 detector is simulated in detail using the GEANT3 program [51] and the events are passed through the same analysis chain as is used for the data. The reaction ep → eXp is simulated with the RAPGAP program [19] using the RP model and the DPDF set H1 2006 Fit B as described in Sect. 3. QED radiative effects are simulated using the HERACLES [52] program within the RAPGAP event generator. In the 'two central jets' analysis the η * 2 distribution of the Monte Carlo simulation is reweighted in order to describe the experimental data. A similar procedure is applied to the η * f distribution in the 'one central + one forward jet' sample. More details of the analysis can be found elsewhere [53].
A comparison of the FPS data and the RAPGAP simulation is presented in Fig. 3 for the variables x P and |t| reconstructed with the FPS detector. The contributions of light quarks (uds) to P and R exchanges and of charm quarks to P exchange are also shown in the log 10 (x P ) distribution. Figure 4 presents the data and the Monte Carlo distribu-tions of the variables P * T ,1 , | η * | and z P for the 'two central jets' sample and of the variables P * T , η f and z P for the 'one central + one forward jet' topology. For this comparison z P is reconstructed from the scattered electron and the hadronic final state in the H1 detector. The MC simulation reproduces the data within the experimental systematic uncertainties. The average detector resolutions on the reconstructed jet variables η, P * T and z P are 7 %, 13 % and 32 %, respectively.

Cross section determination
In order to account for migration and smearing effects and to evaluates the dijet cross sections at the level of stable hadrons, matrix unfolding of the reconstructed data is performed [54]. The resolution and acceptance of the H1 detector is reflected in the unfolding matrix A which relates reconstructed variables y rec with variables on the level of stable hadrons x true via the formula A x true = y rec . The matrix A, obtained for each measured distribution using the RAPGAP simulation, is constructed within an enlarged phase space in order to take into account possible migrations from outside of the measured kinematic range. The following sources of migrations to the analysis phase space Fig. 2 The distribution of (E + P z ) for FPS DIS events (points with error bars) and for beam-halo DIS events (histogram) Fig. 3 The distributions of the variables x P (a) and |t| (b) reconstructed using the FPS (points with error bars) for events with two central jets. The beam-halo background is subtracted from the data. The RAPGAP Monte Carlo simulation, reweighted to describe the η * 2 distribution, is shown as a histogram. Contributions from sub-processes are illustrated in the x P distribution as areas filled with different colours Fig. 4 The distributions of the variables P * T ,1 , | η * | and z P for events with two central jets and of the variables P * T , η 2 and z P for events with one central and one forward jet (points with the error bars). The beam-halo background is subtracted from the data. The Rapgap Monte Carlo simulation is shown as histogram are considered: migrations from low Q 2 , from low y, from large x P , from low P T jets, from the single jet topology, fulfilling the P T requirements for the leading jet as given in Table 1, and in case of the 'one central + one forward jet' analysis from large η f . In order to treat the contamination of the measurement by these migrations correctly the analysis is performed in an extended phase space which includes side-bins in y rec and x true for each of the migration sources listed above.
The unfolded true distribution on the level of stable hadrons is obtained from the measured one by minimising a χ 2 function defined as where χ 2 A is a measure of a deviation of A x true from the data bins y rec . The matrix V is the covariance matrix of the data, based on the statistical uncertainties. In order to avoid statistical fluctuations, the regularisation term χ 2 L is implemented into the χ 2 function and defined as χ 2 L = ( x true ) 2 . The regularisation parameter τ is tuned in order to minimise the bin-to-bin correlations of the covariance matrix V. Further details of the unfolding method can be found in [55,56].
The Born level cross section is calculated in each bin i according to the formula: where x i is the number of background subtracted events as obtained with the unfolding procedure described above, L is the total integrated luminosity and (1 + δ rad ) are the QED radiative corrections which amount to about 5 % on average. The differential cross sections are obtained by dividing by the bin width.

Systematic uncertainties on the measured cross sections
The systematic uncertainties are implemented into the response matrix A and propagated through the unfolding procedure. They are considered from the following sources listed below.
• • The systematic uncertainty arising from the hadronic final state reconstruction is determined by varying the energy scale of the hadronic final state by ±2 % as obtained using a dedicated calibration [57]. The 2 % uncertainty of the calibration is confirmed by studies in the region of low jet transverse momenta and low photon virtuality. This source leads to an average uncertainty of the cross section measurements of 6.2 % for production of two central jets and 9.5 % for production of one central and one forward jet. • The model dependence of the acceptance and migration corrections is estimated by varying the shapes of the distributions in the kinematic variables P * T , η * 2 , η * f , x P , β and Q 2 in the RAPGAP simulation within the constraints imposed on those distributions by the presented data. The η * 2 and η * f reweightings are varied within the errors of the parameters of the reweighting function, which amount up to a factor 4. The P * T distribution is reweighted by P * T ±0. 15 , the x P distribution by (1/x P ) ±0.05 , the β distribution by β ±0.05 and (1 − β) ∓0.05 and the Q 2 distribution by log(Q 2 ) ±0.2 . For the 'two central jets' selection the largest uncertainty is introduced by the η * 2 reweighting (4 %), followed by β (2.7 %), while the reweights in x P , P * T and Q 2 result in an overall uncertainty of 2.3 %. The uncertainties for the 'one central + one forward jet' topology are 12.8 % for the η * f reweighting, followed by P * T (2.1 %), while the reweights in x P , β and Q 2 result in an overall uncertainty of 1.8 %.
• Reweighting the t distribution by e ±t results in a normalisation uncertainty of 4.2 % for the extrapolation in t from the measured range of 0.1 < |t| < 0.7 GeV 2 to the region |t min | < |t| < 1 GeV 2 covered by the LRG data [3]. The uncertainty arising from the t reweighting within the FPS acceptance range of 0.1 < |t| < 0.7 GeV 2 is on average 1.4 %.
The following uncertainties are considered to influence the normalisation of all measured cross sections in a correlated way: • Two sources of systematics related to the background subtraction are taken into account: the energy scale uncertainty and the limited statistics in the data sample without the (E + p z ) cut. Firstly, the beam-halo spectrum is shifted within the quoted uncertainties of the hadronic energy scale and proton energy scale. Secondly, the normalisation of the background spectrum is shifted by 1 ± 1/ N bkg , where N bkg is the number of events in the FPS data sample in the range (E +P z ) > 1880 GeV. The uncertainties from these two sources are combined in quadrature. The uncertainty of the proton beam-halo background is considered as a normalisation error and found to be 3.5 % for the production of two central jets and 1.5 % for the production of one central and one forward jet.
• A normalisation uncertainty of 1 % is attributed to the trigger efficiencies, evaluated using event samples obtained with independent triggers. • The uncertainty in the FPS track reconstruction efficiency results in a normalisation uncertainty of 2 %. • A normalisation uncertainty of 3.7 % arises from the luminosity measurement.
The systematic errors shown in the figures are obtained by adding in quadrature all the contributions except for the normalisation uncertainties, leading to an average uncertainty of 11 % for 'two central jets' and 17 % for 'one central + one forward jet'. The overall normalisation uncertainty of the cross section measurement obtained by adding in quadrature all normalisation uncertainties is 7 % for 'two central jets' and 6.2 % for 'one central + one forward jet'. The cross section measurement in t has a normalisation uncertainty 4.6 %.

Fig. 7
The differential cross section for the production of two central jets shown as a function of Q 2 , y, log 10 (x P ) and z P . The inner error bars represent the statistical errors. The outer error bars indicate the statistical and systematic errors added in quadrature. The RP, SCI+GAL and TPR models are shown as solid, dotted and dashed-dotted lines, respectively. R denotes the ratio of the measured cross sections and MC model predictions to the nominal values of the measured cross sections. The total normalisation error of 7.0 % is not shown on the leading-logarithm approximation and parton showers. he ratios of the measured cross sections to the MC predictions show that the RP model gives a good description of the shape, but underestimates the dijet cross section by a factor of 1.5. For this comparison the reweighting with respect to the η * 2 distribution specified in Sect. 5.2 is not applied to the RP model. Since the P and R fluxes which determine the x P dependence in the RP model has been tuned to the inclusive diffractive DIS LRG data [2] the good agreement in shape of the RP model with the dijet data supports the hypothesis of the proton vertex factorisation. Both the SCI+GAL and TGP models fail to describe the data. The SCI+GAL model predicts harder spectra in Q 2 and z P and a softer spectrum in log 10 (x P ) than are seen in the data. It should be noted that the probability of soft colour in-teractions and hence the normalisation of diffractive processes in the SCI+GAL model is adjusted to the measured dijet cross section. The TGP model is in agreement with the data only at low x P but underestimates the data significantly at larger x P sub-leading contributions are expected to be large. Figure 8 shows the differential cross sections in P *

T ,1
and | η * | for the data and the MC models. The shapes of these distributions are again well described by the RP model. Although the SCI+GAL model is not able to describe the differential cross sections as a function of the diffractive kinematic variables x P and z P and of the DIS kinematic variable Q 2 this model reproduces reasonably well the measurements as a function of the jet variables P *

T ,1
and | η * |. Fig. 8 The differential cross section for production of two central jets shown as a function of P * T ,1 and | η * |. For more details see Fig. 7 Fig. 9 The differential cross section for production of two central jets shown as a function of t (a), the corresponding t -slope (circle) shown as a function of x P (b). The result is compared to the H1 inclusive diffractive DIS data (triangles) [16]. The error bars indicate the statistical and systematic errors added in quadrature None of the LO Monte Carlo models are able to describe all features of the measured differential cross sections. The best shape description in all cases is provided by the RP model. However, this model is a factor of 1.5 below the data in normalisation. The TGP and SCI+GAL models fail to describe the shape of the differential cross sections.
The differential cross section in |t| shown in Fig. 9a is fit using an exponential form exp(Bt) motivated by Regge phenomenology. An iterative procedure is used to determine the slope parameter B, where bin centre corrections are applied to the differential cross section in t using the value of B extracted from the previous fit iteration. The final fit results in B = 5.89 ± 0.50 (exp.) GeV −2 , where the experimental uncertainty is defined as the quadratic sum of the statistical and systematic uncertainties and the full covariance matrix is taken into account in the fit. As shown in Fig. 9b, this t-slope parameter is consistent within the errors with the tslope measured in inclusive diffractive DIS with a leading proton in the final state [16] at the same value of x P . The consistency of the measured t dependence with that for the inclusive diffractive DIS cross sections supports the validity of the proton vertex factorisation hypothesis.
The cross section for the production of two central jets can be compared with the diffractive dijet measurement obtained using the LRG technique [3]. The LRG measurement includes proton dissociation to states Y with masses M Y < 1.6 GeV. To correct for the contributions of pro-ton dissociation processes, the LRG dijet data are scaled down by a factor of 1.20, taken from the diffractive inclusive DIS measurement [16]. To compare to the results of the LRG method, dijet events are selected in the same kinematic range. The DIS and jet variables Q 2 , y, P * T ,1 and η 1,2 are restricted to the ranges 4 < Q 2 < 80 GeV 2 , 0.1 < y < 0.7, P * T ,1 > 5.5 GeV, and −1 < η 1,2 < 2, respectively. The results are presented in Fig. 10. The comparison shows consistency of the results within the experimental errors. Compared to the LRG measurement, the phase space of the present analysis extends to x P values that are a factor of three larger. 7.2 Differential cross section for the production of one central + one forward jet Figure 11 shows the differential cross sections for the production of 'one central + one forward jet' as a function of | η * |, η f and the mean transverse momentum of the forward and central jets P * T together with the expectations from the NLO QCD. Within the errors, the measured data are described by NLO QCD predictions. The NLO QCD predictions are shown with the hadronisation uncertainties and the scale uncertainties, which dominate over the DPDF uncertainties.
In order to test the predictions in a wider kinematic range, the η f distribution of the forward jet shown in Fig. 11 is ex-Fig. 10 The differential cross section for the production of two central jets in the phase space of the LRG measurement [3] as described the text in Sect. 7.1. The cross section is shown as a function of log 10 (x P ). The inner error bars represent the statistical errors. The outer error bars indicate the statistical and systematic errors added in quadrature. The published LRG dijet data are scaled down by a factor of 1.20 to correct for the proton dissociation contribution are shown as open circles with the error bars indicating the statistical and systematic errors added in quadrature tended down to a minimum value of −0.6 where the prediction overshoots the data. LO QCD calculations, performed using the DPDF set H1 2007 Jets underestimate the measured cross section by a factor of about 2.5.
The differential cross sections measured as a function of z P , log 10 (β) and | φ * | are presented in Fig. 12. The data are well described by the NLO QCD predictions. In the BFKL approach [58][59][60], additional gluons can be emitted in the gap between the two jets, leading to a de-correlation in azimuthal angle | φ * |. The observed agreement between the measured cross sections and NLO DGLAP predictions in this distribution shows no evidence for such an effect in the kinematic region accessible in this analysis. Figure 13 presents the differential cross sections for the production of 'one central + one forward jet' as a function of the variables P * T , | η * | and η f . in the case of 'two central jets', The RP model is a factor of 2.2 below the data which is a larger discrepancy in normalisation than that observed in the 'two central jets' sample. A similar trend is seen for the LO QCD contributions in the two samples. The normalisation of the SCI+GAL model, tuned to 'two central jets', agrees with the cross section for 'one central + The total normalisation error of 6.2 % is not shown Fig. 12 The differential cross section for production of one central and one forward jet shown as a function of z P , log 10 (β) and | φ * |. For more details see Fig. 11 Fig. 13 The differential cross section for production of one central and one forward jet shown as a function of the mean transverse momentum of two jets P * T , | η * | and η f . The inner error bars represent the statistical errors. The outer error bars indicate the statistical and systematic errors added in quadrature. The RP and the SCI+GAL models are shown as solid and dotted lines, respectively. R denotes the ratio of the measured cross sections and MC model predictions to the nominal values of the measured cross sections. The total normalisation error of 6.2 % is not shown Fig. 14 The differential cross section for production of one central and one forward jet shown as a function of z P , log 10 (β) and | φ * |. The RP, SCI+GAL and TPG models are shown as full, dotted and dashed-dotted lines. For more details see Fig. 13 one forward jet'. The shapes of the distributions are reasonably well described by both the RP and SCI+GAL models.
The differential cross sections in z P , log 10 (β) and | φ * | are shown in Fig. 14. The shapes of all distributions are well described only by the RP model. As for the case of the 'two central jets' the SCI+GAL model is not able to describe the distributions of the diffractive kinematic variables but it well reproducing the shape of the | φ * | distribution. The TGP model completely fails again to describe the z P spectrum.

Summary
Integrated and differential cross sections are measured for dijet production in the diffractive DIS process ep → ejj X p. In the process studied, the scattered proton carries at least 90 % of the incoming proton momentum and is measured in the H1 Forward Proton Spectrometer. The presented results are compatible with the previous measurements based on the LRG method and explore a new domain at large x P .
Dijet cross sections are measured for an event topology with two jets produced in the central pseudorapidity region, where DGLAP parton evolution mechanism is expected to dominates, and for a topology with one jet in the central region and one jet in the forward region, where effects of non-DGLAP parton evolution may be observed. NLO QCD predictions based on the DGLAP approach and using DPDFs extracted from inclusive diffraction measurements describe the dijet cross sections within the errors for both event topologies, supporting the universality of DPDFs. The measured t-slope of the dijet cross section is consistent within uncertainties with the value measured in inclusive diffractive DIS with a leading proton in the final state. This confirms the validity of the proton vertex factorisation hypothesis for dijet production in diffractive DIS.
The measured cross sections are compared with predictions from Monte Carlo models based on leading order matrix elements and parton showers. The Resolved Pomeron model describes the shape of the cross sections well, but is too low in normalisation. This suggests that contributions from higher order processes are expected to be sizable in this approach. The SCI+GAL model is able to reproduce the normalisation of the cross section for both dijet topologies presented after tuning the model to the 'two central jets' data. The dependence of the diffractive dijet cross section on x P and z P is able to distinguish between the models. The SCI+GAL and Two Gluon Pomeron models fail to describe the shape of the distributions of the diffractive variables, while the Resolved Pomeron model describes the shape of these distributions well. maintaining the H1 detector, our funding agencies for financial support, the DESY technical staff for continual assistance and the DESY directorate for support and for the hospitality which they extend to the non-DESY members of the collaboration.
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