Deep inelastic inclusive and diffractive scattering at $Q^2$ values from 25 to 320 GeV$^2$ with the ZEUS forward plug calorimeter

Deep inelastic scattering and its diffractive component, $ep \to e^{\prime}\gamma^* p \to e^{\prime}XN$, have been studied at HERA with the ZEUS detector using an integrated luminosity of 52.4 pb$^{-1}$. The $M_X$ method has been used to extract the diffractive contribution. A wide range in the centre-of-mass energy $W$ (37 -- 245 GeV), photon virtuality $Q^2$ (20 -- 450 GeV$^2$) and mass $M_X$ (0.28 -- 35 GeV) is covered. The diffractive cross section for $2<M_X<15$ GeV rises strongly with $W$, the rise becoming steeper as $Q^2$ increases. The data are also presented in terms of the diffractive structure function, $F^{\rm D(3)}_2$, of the proton. For fixed $Q^2$ and fixed $M_X$, $\xpom F^{\rm D(3)}_2$ shows a strong rise as $\xpom \to 0$, where $\xpom$ is the fraction of the proton momentum carried by the Pomeron. For Bjorken-$x<1 \cdot 10^{-3}$, $\xpom F^{\rm D(3)}_2$ shows positive $\log Q^2$ scaling violations, while for $x \ge 5 \cdot 10^{-3}$ negative scaling violations are observed. The diffractive structure function is compatible with being leading twist. The data show that Regge factorisation is broken.

, of the proton. For fixed Q 2 and fixed M X , x I P F shows a strong rise as x I P → 0, where x I P is the fraction of the proton momentum carried by the Pomeron. For Bjorken-x < 1 · 10 −3 , x I P F D(3) 2 shows positive log Q 2 scaling violations, while for x ≥ 5 · 10 −3 negative scaling violations are observed. The diffractive structure function is compatible with being leading twist. The data show that Regge factorisation is broken.

Introduction
The observation of events with a large rapidity gap in deep inelastic electron (positron) proton scattering (DIS) at HERA by the ZEUS experiment [1] has paved the way for a systematic study of diffraction at large centre-of-mass energies with a variable hard scale provided by the mass squared, −Q 2 , of the virtual photon. Diffraction is defined by the property that the cross section does not decrease as a power of the centre-of-mass energy. This can be interpreted as the exchange of a colourless system, the Pomeron, which leads to the presence of a large rapidity gap between the proton and the rest of the final state, which is not exponentially suppressed.
Before HERA came into operation, Ingelman and Schlein [2], based on data from UA8 [3,4], had suggested that the Pomeron may have a partonic structure. Since then, the H1 and ZEUS experiments at HERA have presented results on diffractive scattering in photoproduction and deep inelastic electron-proton scattering for many different final states. In parallel, a number of theoretical ideas and models have been developed in order to understand the data within the framework of Quantum Chromodynamics (QCD) [5].
Several methods have been employed by H1 and ZEUS for isolating diffractive contributions experimentally. In the case of exclusive vector-meson production, resonance signals in the decay mass spectrum combined with the absence of other substantial activity in the detector have been used [?, 6,8]. The contribution from inclusive diffraction has been extracted using the presence of a large rapidity gap (η max method [11]), the detection of the leading proton [?, 10] or the hadronic mass spectrum observed in the central detector (M X method [12,13]). The selections based on η max or on a leading proton may include additional contributions from Reggeon exchange. Such contributions are exponentially suppressed when using the M X method.
In this paper, inclusive processes ( Fig. 1), and diffractive processes (Fig. 2), where N is a proton or a low-mass nucleonic state and X is the hadronic system without N, are studied. The contribution from diffractive scattering is extracted with the M X method. Results on the proton structure function F 2 and on the diffractive cross section and structure function are presented for a wide range of centre-of-mass energies, photon virtualities −Q 2 and of mass M X of the diffractively produced hadronic system, using the data from the ZEUS experiment collected in 1999 and 2000. The results, which will be referred to as FPC II, are based on integrated luminosities of 11.0 pb −1 for Q 2 = 20 − 40 GeV 2 and 52.4 pb −1 for Q 2 = 40 − 450 GeV 2 .
In a previous study, which will be referred to as FPC I [13], results on inclusive and diffractive scattering were presented for the Q 2 values between 2.7 and 55 GeV 2 using an integrated luminosity of 4.2 pb −1 . The combined data from the FPC I and FPC II analyses provide a measurement of the Q 2 dependence of diffraction over a range of two orders of magnitude.

Experimental set-up and data set
The data used for this measurement were taken with the ZEUS detector in 1999-2000 when positrons of 27.5 GeV collided with protons of 920 GeV. The detector as well as the analysis methods are identical to those used for the FPC I study [13]. A detailed description of the ZEUS detector can be found elsewhere [14,15]. A brief outline of the components that are most relevant for this analysis is given below.
Charged particles were tracked in the central tracking detector (CTD) [16], which operated in a magnetic field of 1.43 T provided by a thin superconducting solenoid. The CTD consisted of 72 cylindrical drift chamber layers, organised in 9 superlayers covering the polar-angle region 15 • < θ < 164 • . The transverse momentum resolution for full-length tracks was σ(p T )/p T = 0.0058p T ⊕ 0.0065 ⊕ 0.0014/p T , with p T in GeV.
The high-resolution uranium-scintillator calorimeter (CAL [17]) consisted of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each part was subdivided transversely into towers and longitudinally into one electromagnetic section (EMC) and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections (HAC). The smallest division of the calorimeter was called a cell. The CAL energy resolutions, as measured under test beam conditions, were σ(E)/E = 0.18/ (E) for electrons and σ(E)/E = 0.35/ (E) for hadrons, with E in GeV.
The position of electrons scattered at small angles to the electron-beam direction was determined including the information from the SRTD [18,19] which was attached to the front face of the RCAL and consisted of two planes of scintillator strips. The rear hadronelectron separator (RHES [20]) was inserted in the RCAL.
In 1998, the forward-plug calorimeter (FPC) [21] was installed in the 20×20 cm 2 beam hole of the FCAL. The FPC was used to measure the energy of particles in the pseudorapidity 1 range η ≈ 4.0 − 5.0. The FPC was a lead-scintillator sandwich calorimeter read out by wavelength-shifter (WLS) fibres and photomultipliers (PMT). A hole of 3.15 cm radius was provided for the passage of the beams. In the FPC, 15 mm thick lead plates alternated with 2.6 mm thick scintillator layers. The scintillator layers consisted of tiles forming towers that were read out individually. The tower cross sections were 24 × 24 mm 2 in the electromagnetic and 48×48 mm 2 in the hadronic section. The measured energy resolution for positrons was σ E /E = (0.41±0.02)/ √ E⊕0.062±0.002, with E in GeV. When installed in the FCAL, the energy resolution for pions was σ E /E = (0.65 ± 0.02)/ √ E ⊕ 0.06 ± 0.01, with E in GeV, and the e/h ratio was close to unity.
The luminosity was measured from the rate of the bremsstrahlung process ep → eγp. The resulting small-angle energetic photons were measured by the luminosity monitor [22], a lead-scintillator calorimeter placed in the HERA tunnel at Z = −107 m.
A three-level trigger system was used to select events online [14,15,23]. The first-and second-level trigger selections were based on the identification of a scattered positron with impact point on the RCAL surface outside an area of 36×36 cm 2 centred on the beam axis ("set 1", integrated luminosity 11.0 pb −1 ), or outside a radius of 30 cm centred on the beam axis ("set 2", integrated luminosity 41.4 pb −1 ). In the offline analysis the reconstructed impact point had to lie outside an area of 40×40 cm 2 (set 1) or outside a radius of 32 cm (set 2).

Reconstruction of kinematics and event selection
The methods for extracting the inclusive DIS and diffractive data samples are identical to those applied in the FPC I study [13] and will be described only briefly.
The reaction e(k) p(P ) → e(k ′ ) + anything, see Fig. 1, at fixed squared centre-of-mass energy, s = (k + P ) 2 , is described in terms of Q 2 ≡ −q 2 = −(k − k ′ ) 2 , Bjorken-x = Q 2 /(2P · q) and s ≈ 4E e E p , where E e and E p denote the positron and proton beam energies, respectively. For these data, √ s = 318 GeV. The fractional energy transferred to the proton in its rest system is y ≈ Q 2 /(sx). The centre-of-mass energy of the hadronic final state, W , is given by W 2 = [P + q] 2 = m 2 p + Q 2 (1/x − 1) ≈ Q 2 /x = ys, where m p is the mass of the proton. In diffraction, proceeding via see Fig. 2, the incoming proton undergoes a small perturbation and emerges either intact (N = p), or as a low-mass nucleonic state N, in both cases carrying a large fraction, x L , of the incoming proton momentum. The virtual photon dissociates into a hadronic system X.
Diffraction is parametrised in terms of the mass M X of the system X, and the mass M N of the system N. Since t, the four-momentum transfer squared between the incoming proton and the outgoing system N, t = (P −N) 2 , was not measured, the results presented are integrated over t. The measurements performed by ZEUS with the leading proton spectrometer [10] show that the diffractive contribution has a steeply falling t distribution with typical |t| values well below 0.5 GeV 2 .
Diffraction was also analysed in terms of the momentum fraction x I P of the proton carried by the Pomeron exchanged in the t-channel, x I P = [(P − N) · q]/(P · q) ≈ (M 2 X + Q 2 )/(W 2 + Q 2 ), and the fraction of the Pomeron momentum carried by the struck quark, β = Q 2 /[2(P − N) · q] ≈ Q 2 /(M 2 X + Q 2 ). The variables x I P and β are related to the Bjorken scaling variable, x, via x = βx I P .
The events studied are of the type ep → e ′ X + rest, (4) where X denotes the hadronic system observed in the detector and 'rest' the particle system escaping detection through the forward and/or rear beam holes.
The coordinates of the event vertex were determined with tracks reconstructed in the CTD. Scattered positrons were identified with an algorithm based on a neural network [24]. The direction and energy of the scattered positron were determined from the combined information given by CAL, SRTD, RHES and CTD. Fiducial cuts on the impact point of the reconstructed scattered positron on the CAL surface were imposed to ensure a reliable measurement of the positron energy.
The hadronic system was reconstructed from energy-flow objects (EFO) [25,26] which combine the information from CAL and FPC clusters and from CTD tracks, and which were not assigned to the scattered positron.
If a scattered-positron candidate was found, the following criteria were imposed to select the DIS events: • the scattered-positron energy, E ′ e , be at least 10 GeV; • the total measured energy of the hadronic system be at least 400 MeV; • y FB JB > 0.006, where y FB JB = h (E h − P Z,h )/(2E e ) summed over all hadronic EFOs in FCAL plus BCAL; or at least 400 MeV be deposited in the BCAL or in the RCAL outside of the ring of towers closest to the beamline; • −54 < Z vtx < 50 cm, where Z vtx is the Z-coordinate of the event vertex; • 43 < i=e,h (E i −P Z,i ) < 64 GeV, where the sum runs over both the scattered positron and all hadronic EFOs. This cut reduces the background from photoproduction and beam-gas scattering and removes events with large initial-state QED radiation; • candidates for QED-Compton (QEDC) events, consisting of a scattered-positron candidate and a photon candidate, with mass M eγ less than 0.25 GeV and total transverse momentum less than 1.5 GeV, were removed. A Monte Carlo (MC) study showed that the number of remaining QEDC events was negligible.
The value of Q 2 was reconstructed from the measured energy E ′ e and scattering angle θ e of the positron, Q 2 = 2E e E ′ e (1 + cos θ e ). In the FPC I analysis, which covered lower Q 2 values, the value of W was determined as the weighted average of the values given by the positron and hadron measurement. Here, the value of W was reconstructed with the double-angle algorithm (DA) [27] which relies only on the measurement of the angles of the scattered positron and of the hadronic system.
The mass of the system X was determined by summing over all hadronic EFOs, where P h = (p X,h , p Y,h , p Z,h , E h ) is the four-momentum vector of a hadronic EFO. All kinematic variables used to describe inclusive and diffractive scattering were derived from M X , W and Q 2 .
A total of 60 events were found without a vertex, which were due either to cosmic radiation (45) or to an overlay of cosmic radiation with DIS (15); these events were discarded.
About 630k events for data set 1 and 1.4M events for data set 2 passed the selection cuts. The kinematic range for inclusive and diffractive events was chosen taking into account detector resolution and statistics. About 930k events were retained which satisfied 37 < W < 245 GeV and 20 < Q 2 < 450 GeV 2 .
The resolutions of the reconstructed kinematic variables were estimated using MC simulation of diffractive events of the type γ * p → XN (see Section 4). For the M X , W and Q 2 bins considered in this analysis, the resolutions are approximately the same as for the FPC I analysis: σ(W )

Monte Carlo simulations
The data were corrected for detector acceptance and resolution, and for radiative effects, with suitable combinations of several MC models, following the same procedure and using the same MC models as in the FPC I [13] analysis.
Events from inclusive DIS, including radiative effects, were simulated using the HERA-CLES 4.6.1 [?] program with the DJANGOH 1.1 [29] interface to ARIADNE 4 [30] and using the CTEQ4D next-to-leading-order parton distribution functions [31]. In HERA-CLES, O(α) electroweak corrections are included. The colour-dipole model of ARIADNE, including boson-gluon fusion, was used to simulate the O(α S ) plus leading-logarithmic corrections to the quark-parton model. The Lund string model as implemented in JETSET 7.4 [32] was used by ARIADNE for hadronisation.
Diffractive DIS in which the proton does not dissociate, ep → eXp (including the production of ω and φ mesons via ep → eV p, V = ω, φ but excluding ρ 0 production), were simulated with SATRAP, which is based on a saturation model [33] and is interfaced to the RAPGAP 2.08 framework [34]. SATRAP was reweighted as a function of Q 2 /(Q 2 + M 2 X ) and W . The production of ρ 0 mesons, ep → eρ 0 p, was simulated with ZEUSVM [35], which uses a parametrisation of the measured ρ 0 cross sections as well as of the production and decay angular distributions [8,36,37]. The QED radiative effects were simulated with HERACLES. The QCD parton showers were simulated with LEPTO 6.5 [38].
Diffractive DIS in which the proton dissociates, ep → eXN, was simulated with SATRAP interfaced to the model called SANG [39], which also includes the production of ρ 0 mesons. Following the previous experience (FPC I), the mass spectrum of the system N was generated according to dσ/dM 2 GeV. This parametrisation was found to fit the data in the FPC I analysis. The fragmentation of the system N was simulated using JETSET 7.4.
The parameters of SANG, in particular those determining the shape of the M N spectrum and the overall normalisation, were checked with a subset of the data. Events in this subset were required to have a minimum rapidity gap ∆η > η min between at least one EFO and its nearest neighbours, all with energies greater than 400 MeV. Good sensitivity for double dissociation was obtained with four event samples for the kinematic regions η min = 3.0 , W = 55 − 135 GeV, and η min = 4.0, W = 135 − 245 GeV, for Q 2 = 40 − 80 GeV 2 and 80 − 450 GeV 2 . The mass of the hadronic system reconstructed from the energy deposits in FPC+FCAL, M FFCAL , depends approximately linearly on the mass M gen N of the generated system N. Thus, the M FFCAL distribution is sensitive to those proton dissociative events in which considerable energy of the system N is deposited in FPC and FCAL. The study showed that this is the case, broadly speaking, when the mass of N taken at the generator level is M N > 2.3 GeV. At large M FFCAL , the distribution is dominated by double dissociation. Figure 3 presents the M FFCAL distributions in four representative (Q 2 , W ) regions for the data compared to the Monte Carlo expectations for Xp, ρ 0 p, XN and non-diffractive processes. The contribution expected from XN as predicted by SANG is dominant. Good agreement between the number of events measured and those predicted is obtained. Since the contribution from diffraction with M N > 2.3 GeV can affect the determination of the slope b for the non-diffractive contribution (see Section 5) it was subtracted statistically from the data as a function of M X , W and Q 2 .
Background from photoproduction, estimated with PYTHIA 5.7 [32], was negligible and was neglected.
The ZEUS detector response was simulated using a program based on GEANT 3.13 [40]. The generated events were passed through the detector and trigger simulations and processed by the same reconstruction and analysis programs as the data.
The measured hadronic energies for data and MC were increased by a factor of 1.065 in order to achieve an average transverse momentum balance between the scattered positron and the hadronic system. The mass M X reconstructed from the energy-corrected EFOs, in the M X region analysed, required an additional correction factor of 1.10 as determined from MC simulation 2 .
Good agreement between data and simulated event distributions was obtained for both the inclusive and the diffractive samples.

Determination of the diffractive contribution
The diffractive contribution was extracted from the data using the M X method, which has been described elsewhere [12] and which has also been used for the FPC I analysis [13].
In the QCD picture of non-peripheral DIS, γ * p → X + rest, the hadronic system X measured in the detector is related to the struck quark and 'rest' to the proton remnant, both of which are coloured states. The final-state particles are expected to be uniformly emitted in rapidity along the γ * p collision axis and to uniformly populate the rapidity gap between the struck quark and the proton remnant, as described elsewhere [41]. In this case, the mass M X is distributed as where b and c are constants 3 . DJANGOH predicts, for non-peripheral DIS, b ≈ 1.9.
The diffractive reaction, γ * p → XN, on the other hand, has different characteristics. Diffractive scattering shows up as a peak near x L = 1, the mass of the system X being limited by kinematics to M 2 X /W 2 < ∼ 1 − x L . Moreover, the distance in rapidity between the outgoing nucleon system N and the system X is ∆η ≈ ln(1/(1 − x L )), becoming large when x L is close to one. Combined with the limited values of M X and the peaking of the diffractive cross section near x L = 1, this leads to a large separation in rapidity between N and any other hadronic activity in the event. For the vast majority of diffractive events, the decay particles from the system N leave undetected through the forward beam hole. For a wide range of M X values, the particles of the system X are emitted entirely within the acceptance of the detector. Monte Carlo studies show that X can be reliably reconstructed over the full M X range of this analysis.
Regge phenomenology predicts the shape of the M X distribution for peripheral processes. Diffractive production by Pomeron exchange in the t-channel, which dominates x L values close to unity, leads to an approximately constant ln M 2 X distribution (b ≈ 0). Figure 4 shows distributions of ln M 2 X for the data (from which the contribution from double dissociation with M N > 2.3 GeV, as predicted by SANG, has been subtracted) for low-and high-W bins at low and high Q 2 together with the expectations from MC simulations for non-peripheral DIS (DJANGOH) and for diffractive processes (SATRAP + ZEUSVM and SANG for M N < 2.3 GeV). The observed distributions agree well with the expectation for a non-diffractive component giving rise to an exponentially growing ln M 2 X distribution, and for a diffractive component producing an almost constant distribution in a substantial part of the ln M 2 X range. At high W there is a clear signal for a contribution from diffraction. At low W the diffractive contribution is seen to be small.
The ln M 2 X spectra for all (W, Q 2 ) bins studied in this analysis are displayed in Fig. 5. They are compared with the MC predictions for the contributions from non-peripheral and diffractive production. The MC simulations are in good agreement with the data. It can be seen that the events at low and medium values of ln M 2 X originate predominantly from diffractive production.
The assumption of an exponential rise of the ln M 2 X distribution for non-diffractive processes permits the subtraction of this component and, therefore, the extraction of the diffractive contribution without assumptions about its exact M X dependence. The distribution is of the form: with M X in GeV, D is the diffractive contribution and the second term represents the non-diffractive contribution. The quantity (ln W 2 − η 0 ) specifies the maximum value of ln M 2 X up to which the exponential behaviour of the non-diffractive contribution holds. A value of η 0 = 2.2 was found from the data. Equation (6) was fitted to the data in the limited range ln W 2 − 4.4 < ln M 2 X < ln W 2 − η 0 in order to determine the parameters b and c. The parameter D was assumed to be constant over the fit range, which is suggested by Figs. 4 and 5 where at high W and high Q 2 , dN/ ln M 2 X is a slowly varying function when M 2 X > Q 2 [42,43]. However, the diffractive contribution was not taken from the fit but was obtained from the observed number of events after subtracting the non-diffractive contribution determined using the fitted values of b and c.
The fit range chosen is smaller than that used for the FPC I analysis (viz. for FPC I: . This change takes account of the observation that at high Q 2 and low values of M X diffraction is suppressed, as seen in Fig. 5. The non-diffractive contribution in the (W, Q 2 ) bins was determined by fitting for every (W, Q 2 ) interval the ln M 2 X distribution of the data from which the contribution of γ * p → XN with M N > 2.3 GeV as given by SANG, has been subtracted (see Appendix A and  Tables 2 and 3). A fit of the form of Eq. (6) treating b, c and D as fit variables, was used. Note that this is a difference compared to the FPC I analysis, where for each (W, Q 2 ) interval, the same value of b, obtained as an average over all W , Q 2 values, was used. Good fits with χ 2 per degree of freedom of about unity were obtained. The value of the slope b varied typically between 1.4 and 1.9. The statistical error of the diffractive contribution includes the uncertainties on b and c.
Only bins of M X , W, Q 2 , for which the non-diffractive background was less than 50%, were kept for further analysis.
The M X method used for extracting the diffractive contribution was tested by performing a "Monte Carlo experiment" in which a sample of simulated non-peripheral DIS events (DJANGOH) and diffractive events with (SATRAP+ZEUSVM+SANG) and without proton dissociation (SATRAP + ZEUSVM) was analysed as if it were the data. The resulting diffractive structure function (as defined in Section 9 below) is shown in Fig. 6 as a function of x I P for the β and Q 2 values used in the analysis. Only the statistical uncertainties are shown. The extracted structure function agrees with the diffractive structure function used for generating the events which validates the self consistency of the analysis procedure.
The extraction of the diffractive contribution was also studied for the case of a possible contribution from Reggeon exchange interfering with the contribution from diffraction. The amount of Reggeon-Pomeron interference allowed by the data [10] was found to be smaller than the combined statistical and systematic uncertainties in the present mea-surement, see Appendix B.

Evaluation of cross sections and systematic uncertainties
The total and diffractive cross sections for ep scattering in a given (W, Q 2 ) bin were determined from the integrated luminosity, the number of observed events corrected for background, acceptance and smearing, and corrected to the QED Born level.
The cross sections and structure functions are presented at chosen reference values M Xref , W ref and Q 2 ref . This was achieved as follows: first, the cross sections and structure functions were determined at the weighted average of each (M X , W , Q 2 ) bin. They were then transported to the reference position using the ZEUS NLO QCD fit [44] in the case of the proton structure function F 2 , and the result of the BEKW(mod) fit (see Section 9.4) for the diffractive cross sections and structure functions. The resulting changes to the cross section and structure function values from the average to those at the reference positions were at the 5 -15% level.

Systematic uncertainties
A study of the main sources contributing to the systematic uncertainties of the measurements was performed. The systematic uncertainties were calculated by varying the cuts or modifying the analysis procedure and repeating the full analysis for every variation. The size of the variations of cuts and the changes of the energy scales were chosen commensurate with the resolutions or the uncertainties of the relevant variables: • the acceptance depends on the position measurement of the scattered positron. For set 1 the cut was increased from 40 × 40 cm 2 to 41 × 41 cm 2 (systematic uncertainty 1a) and decreased to 39 × 39 cm 2 (systematic uncertainty 1b). For set 2, the radius cut was increased from 32 cm to 33 cm (systematic uncertainty 1a) and decreased to 31 cm (systematic uncertainty 1b). This affected the low-Q 2 region. Changes below 1% were observed; • the measured energy of the scattered positron was increased (decreased) by 2% in the data, but not in the MC (systematic uncertainties 2a,b). In most cases the changes were smaller than 1%. For a few bins changes up to 3% were observed. For one bin at high Q 2 and high W , a change of 7% was found; • the lower cut for the energy of the scattered positron was lowered to 8 GeV (raised to 12 GeV) (systematic uncertainties 3a,b). In most cases the changes were smaller than 1%. For a few bins changes up to 3% were found. For one bin at high Q 2 and high W , a change of 7% was found; • to estimate the systematic uncertainties due to the uncertainty in the hadronic energy, the analysis was repeated after increasing (decreasing) the hadronic energy measured by the CAL by 2% [19] in the data but not in MC (systematic uncertainties 4a,b).
The changes were below 3%; • the energies measured by the FPC were increased (decreased) by 10% in the data but not in MC (systematic uncertainties 5a,b). The changes were below 1%; • to estimate the uncertainties when the hadronic system h is in one of the transition regions: beam/(FPC+FCAL) (polar angle of the hadronic system θ h < 8 • ); FCAL/BCAL (27 • < θ h < 40 • ) or BCAL/RCAL (128 • < θ h < 138 • ), the energy of h was increased in the data by 10% but not in MC (systematic uncertainty 6). This led to changes below 1%; • the minimum hadronic energy cut of 400 MeV as well as the cut y JB > 0.006 were increased by 50% (systematic uncertainty 7). In most cases the changes were below 1%. For a few bins at Q 2 ≤ 35 GeV 2 , changes up to 3% were found; • in order to check the simulation of the hadronic final state, the selection on i=e,h (E i − P Z,i ) was changed from 43 -64 GeV to 35 -64 GeV (systematic uncertainty 8), leading for Q 2 = 25, 35 GeV 2 to maximum changes at the level of 4%, and to changes up to 6% for Q 2 = 320 GeV 2 .
The above systematic tests apply to the total as well as to the diffractive cross sections.
The following systematic tests apply to the diffractive cross section only: • the reconstructed mass M X of the system X was increased (decreased) by 5% in the data but not in the MC (systematic uncertainties 9a,b). Changes below 1% were observed except for Q 2 = 25, 35 GeV 2 , where decreasing M X led to changes up to 5% at high y; • the contribution from double dissociation with M N > 2.3 GeV was determined with the reweighted SANG simulation and was subtracted from the data. The diffractive cross section was redetermined by increasing (decreasing) the predicted contribution from SANG by 30% (systematic uncertainties 10a,b). The resulting changes in the diffractive cross section were well below the statistical uncertainty; • the slope b describing the ln M 2 X dependence of the non-diffractive contribution (see Eq. (6)) was increased (decreased) by 0.2 units (systematic uncertainties 11a,b); this led to an increase (decrease) of the number of diffractive events for the highest M X value at a given W, Q 2 by 1 (1.5) times the size of the statistical uncertainty. For the lower M X values the changes were smaller.
The uncertainty in the luminosity measurement was 2% and was neglected. The major sources of systematic uncertainties for the diffractive cross section, dσ diff /dM X , were found to be the uncertainties 4a,b; 8; 9a,b, 10a,b; and 11a,b for the largest M X value at a given value of W . The total systematic uncertainty for each bin was determined by adding the individual contributions in quadrature.
7 Proton structure function F 2 and the total γ * p cross section The differential cross section for inclusive ep scattering mediated by virtual photon exchange is given in terms of the structure functions F i of the proton by where Y = 1 + (1 − y) 2 , F 2 is the main component of the cross section which in the DIS factorisation scheme corresponds to the sum of the momentum densities of the quarks and antiquarks weighted by the squares of their charges, F L is the longitudinal structure function and δ r is a term accounting for radiative corrections.
In the Q 2 range considered in this analysis, Q 2 ≤ 450 GeV 2 , the contributions from Z 0 exchange and Z 0γ interference are at most of the order of 0.4% and were ignored. The contribution of F L to the cross section relative to that from F 2 is given by (y 2 /Y )·(F L /F 2 ). For the determination of F 2 , the F L contribution was taken from the ZEUS NLO QCD fit [44]. The contribution of F L to the cross section in the highest y (= lowest x) bin of this analysis was 3.2%, decreasing to 1.3% for the next highest y bin. For the other bins, the F L contribution is below 1%. The resulting uncertainties on F 2 are below 1%.
The measured F 2 values are listed in Table 4, and are shown in Fig. 7 together with those from the FPC I analysis. Here, the F 2 values of FPC I measured at Q 2 = 27 GeV 2 were transported to Q 2 = 25 GeV 2 . Good agreement is observed between the measurements done at the same values of Q 2 , namely 25 and 55 GeV 2 . The data are compared to the predictions of the ZEUS NLO QCD fit [44] obtained from previous ZEUS F 2 measurements [19]. The fit describes the data well.
The proton structure function, F 2 , rises rapidly as x → 0 for all values of Q 2 , the slope increasing as Q 2 increases. The form was fitted for every Q 2 bin to the F 2 data, requiring x < 0.01 to exclude the region where valence quarks may dominate. Since, for fixed Q 2 , the x dependence of F 2 is related to the W dependence of the total γ * p cross section, the power λ can be related to the intercept of the Pomeron trajectory, λ = α IP (0) − 1 (see Section 8.1). For later comparison with the diffractive results, these α IP values will be referred to as α tot IP . The resulting values for c and α tot IP (0) are listed in Table 5. Figure 8 presents the results from this study together with those from the FPC I analysis. The parameter α tot IP (0) rises approximately linearly with ln Q 2 from α tot IP (0) = 1.155 ± 0.011(stat.) +0.007 −0.011 (syst.) at Q 2 = 2.7 GeV 2 , to 1.322 ± 0.017 (statistical and systematic uncertainties added in quadrature) at Q 2 = 70 GeV 2 , substantially above the 'soft Pomeron' value of 1.096 +0.012 −0.009 deduced from hadronhadron scattering data [45]. This is in agreement with previous observations [13,46,47]. Since the Pomeron intercept is changing with Q 2 , the assumption of single Pomeron exchange cannot be sustained.
The total cross section for virtual photon-proton scattering, σ tot γ * p ≡ σ T (x, Q 2 ) + σ L (x, Q 2 ), where T (L) stands for transverse (longitudinal) photons, was extracted from the measurement of F 2 using the relation which is valid for 4m 2 p x 2 ≪ Q 2 [48]. The total cross section values are listed in Table 6 for fixed Q 2 as a function of W .
The total cross section multiplied by Q 2 is shown in Fig. 9 together with the results from the FPC I analysis. For fixed value of Q 2 , Q 2 σ tot γ * p rises rapidly with W . For Q 2 ≤ 14 GeV 2 , the rise becomes steeper with increasing Q 2 , while for Q 2 ≥ 70 GeV 2 the rise becomes less steep as Q 2 increases. The W behaviour of σ tot γ * p reflects the x dependence of F 2 as x → 0, viz. σ tot γ * p ∝ W 2(α tot I P (0)−1) .

Diffractive cross section
The cross section for diffractive scattering via ep → eXN can be expressed in terms of the transverse (T) and longitudinal (L) cross sections, σ diff T and σ diff L , for γ * p → XN as Here, a term (1 − has been neglected [13,[48][49][50]. Since y = W 2 /s, this approximation reduces the diffractive cross section for M X < 2 GeV by at most 8% at W < 200 GeV, and by at most 22% in the highest W bin, 200 -245 GeV, under the assumption that only longitudinal photons contribute. Since the reduction is always smaller than the total uncertainty of the diffractive cross section given by the statistical and systematic uncertainties added in quadrature: no correction was applied.

W dependence of the diffractive cross section
The diffractive cross section dσ diff /dM X for γ * p → XN, M N < 2.3 GeV, corrected for radiative effects and after transporting the measured values to the reference values (M X , W, Q 2 ) using the BEKW(mod) fit (see Section 9.4), is presented in Tables 7 -12 and Figs. 10 and 11 as a function of W . The results from the FPC I and FPC II analyses are shown. Where measurements at the same values of Q 2 are available, agreement is observed between the two data sets.
The diffractive cross section dσ diff /dM X varies with M X , W and Q 2 . For M X = 1.2 GeV, the diffractive cross section shows a moderate increase with increasing W and a steep reduction with Q 2 , approximately proportional to 1/Q 4 . For larger M X values, the diffractive cross section exhibits a substantial rise with increasing W and a less steep decrease with Q 2 roughly proportional to 1/Q 2 . The diffractive cross section is significant up to Q 2 = 320 GeV 2 , provided M X = 11 − 30 GeV.
The W dependence was quantified by fitting the form to the data for each (M X , Q 2 ) bin with M X < 15 GeV; here W 0 = 1 GeV and h, a diff are free parameters. The a diff values from the FPC I and II analyses are shown in Fig. 12 as a function of Q 2 for different M X intervals. For M X > 4 GeV they range from 0.3 to 0.8 with a trend for a diff to be larger by about 0.2 -0.4 units when Q 2 is above 20 GeV 2 .
Under the assumption that the diffractive cross section can be described by the exchange of a single Pomeron, the parameter a diff is related to the Pomeron trajectory averaged over t: α IP = 1 + a diff /4. In the present measurement, the diffractive cross section is integrated over t, providing t-averaged values of α IP . In the framework of Regge phenomenology, the cross section for diffractive scattering can be written as [51], where f (t) characterises the t dependences of the (γ * IP γ * ) and (pIP N) vertices. Assuming dσ/dt ∝ e A·t and α IP (t) = α IP (0) + α ′ IP · t leads to α IP (0) = α IP + α ′ IP /A. Taking A = 7.9 ± 0.5(stat.) +0. 9 −0.5 (syst.) GeV −2 , as measured by ZEUS with the leading proton spectrometer (LPS) [10] 4 , and α ′ IP = 0.25 GeV −2 [45], gives α IP (0) ≈ α IP + 0.03 = 1.03 + a diff /4. The α IP (0) values deduced from diffractive cross sections are denoted as α diff IP (0). The α diff IP (0) values for individual M X bins are given in Table 13. The combined results from FPC I and FPC II for 2 < M X < 15 GeV are given in Table 14 and are shown in Fig. 8 as a function of Q 2 for α ′ is compatible with the soft-Pomeron value, while a substantial rise with Q 2 above the soft-Pomeron value is observed for Since the Pomeron intercept is changing with Q 2 , the Pomeron observed in deep inelastic scattering does not correspond to a simple pole in the angular momentum plane.
8.2 M X and Q 2 dependences of the diffractive cross section at fixed W Figure 13 shows the diffractive cross section multiplied by a factor of Q 2 as a function of M X for W = 220 GeV. For Q 2 values up to about 55 GeV 2 masses M X below 5 GeV are prevalent. As Q 2 increases, the maximum shifts to larger values of M X .
The Q 2 dependence of diffraction was studied in terms of the diffractive cross section multiplied by the factor Q 2 · (Q 2 + M 2 X ) since scaling of the diffractive structure function implies that the quantity X (see below) should be independent of Q 2 , up to logarithmic terms. Figure 14 and Tables 15, 16, 17 as a function of Q 2 separately for M X = 1.2, 3, 6 GeV and M X = 11, 20, 30 GeV. In both cases the data lie within a band of about ±25% width for fixed Q 2 for the M X values given. For the lower M X region, is approximately constant up to Q 2 ≈ 30 − 40 GeV 2 , followed by a decrease proportional to log Q 2 . For larger M X values, the data show a weak dependence on log Q 2 . A similar behaviour is observed for lower values of W . Thus, the scaling behaviour of dσ diff

Diffractive contribution to the total cross section
The relationship between the total and diffractive cross sections can be derived under certain assumptions. For instance, the imaginary part of the amplitude for elastic scat-tering, A γ * p→γ * p (t, W, Q 2 ), at t = 0 can be assumed to be linked to the total cross section by a generalisation of the optical theorem to virtual photon scattering. Assuming that σ tot γ * p ∝ W 2λ and that the elastic and inclusive diffractive amplitudes at t = 0 are purely imaginary and have the same W and Q 2 dependences, then A γ * p→γ * p (t = 0, W, Q 2 ) is proportional to W 2λ . Neglecting the real part of the scattering amplitudes, the rise of the diffractive cross section with W should then be proportional to W 4λ , so that the ratio of the diffractive cross section to the total γ * p cross section, should behave as r diff tot ∝ W 2λ . The ratio r diff tot was determined for all M a < M X < M b intervals, with the σ tot γ * p values taken from this analysis. The ratio r diff tot is listed in Tables 18 -23 and is shown in Fig. 15 for the FPC II data, and in Fig. 16 for those from the FPC I analysis. The relative contribution of diffraction to the total cross section is approximately independent of W . It is substantial when M 2 X > Q 2 . For Q 2 = 25 -320 GeV 2 , diffraction with M X < 2 GeV accounts for about 0.1 to 0.4% of the total cross section, while the M X intervals 15 -25 GeV and 25 -35 GeV together account for 3 -4%.
The ratio r = σ diff (0.28 < M X < 35 GeV, M N < 2.3 GeV)/σ tot was evaluated as a function of Q 2 for the highest W bin (200 < W < 245 GeV) which provides the best coverage in M X . Both FPC I and FPC II data are listed in Table 24 and shown in Fig. 17. The ratio r is 15.8 +1.1 −1.0 % at Q 2 = 4 GeV 2 , decreasing to 5.0 +0.9 −0.9 % at Q 2 = 190 GeV 2 . The data are well described by the form r = a − b · ln(1 + Q 2 ). Considering both statistical and systematic uncertainties, the fit yielded a = 0.2069 ± 0.0075 and b = 0.0320 ± 0.0020, which is shown by the line in Fig. 17. The figure shows that the ratio r of the diffractive to the total cross section is decreasing logarithmically with Q 2 .

Diffractive structure function of the proton
The diffractive structure function of the proton, F D(3) 2 (β, x I P , Q 2 ), is related to the diffractive cross section for W 2 ≫ Q 2 as follows: With this definition, F will include also contributions from longitudinal photons. If F is interpreted in terms of quark densities, it specifies the probability to find, in a proton undergoing a diffractive reaction, a quark carrying a fraction x = βx I P of the proton momentum. for the FPC II data set as a function of x I P for fixed Q 2 and fixed M X , or, equivalently fixed β: rises approximately proportional to ln 1/x I P as x I P → 0. This rise reflects the increase of the diffractive cross section dσ diff /dM X with W . Figures 19 and 20 show that the combined FPC I and FPC II data exhibit this rise for most Q 2 values from 2.7 to 320 GeV 2 . The data are also provided in Tables 25 -29.
for fixed β and x I P is provided in Tables 30 -38 and is presented in Fig. 21 for the FPC I and FPC II data. Fits of the form yielded the values of c and a given in Table 39 for selected values of x I P , β with six or more data points. Figure 21 and the fit results show that with increasing β the slope a changes from positive values, corresponding to positive logarithmic scaling violations, to constancy or negative logarithmic scaling violations. The data are dominated by positive scaling violations in the region characterised roughly by x I P β = x < 1 · 10 −3 , by negative scaling violations for x ≥ 5 · 10 −3 , and by constancy in between.
The data contradict the assumption of Regge factorisation [2], that the diffractive structure function x I P F D(3) 2 (β, x I P , Q 2 ) factorises into a term that depends only on x I P and a second term that depends only on β and Q 2 . This can be seen in Table 39 which gives the fit results for fixed β = 0.4 and β = 0.7, where the term a shows a strong dependence on x I P . The was also studied for selected values of x I P = 0.0001, 0.0003, 0.001, 0.003, 0.01 and of β. These choices of x I P and β values were made for the purpose of comparison with the results from H1 [?]. The values of the diffractive structure function at these values of x I P and β were obtained from those at the measured x I P , β values by using the BEKW(mod) fit to the combined FPC I and FPC II data with a total of 427 measured points (see below). Only points for which the ratio of the transported to the measured value of x I P F D(3) 2 was within 0.75 -1.33 were retained, corresponding to about half of the data sample. Since the x I P F D(3) 2 data from H1 had been determined for M N < 1.6 GeV while those from this measurement are presented for M N < 2.3 GeV, the H1 data may have to be increased by a factor of 1.1 to 1.2 for an absolute comparison; no correction has been applied. at the chosen x I P values were obtained from those at the measured x I P values using the BEKW(mod) fit (see below). The diffractive structure function exhibits a fall towards β = 1 and a broad maximum around β = 0.5. The broad maximum is approximately of the form β(1 − β) as expected when the virtual photon turns into a qq system. For x I P ≥ 0.005, x I P F D(3) 2 rises as β → 0 which is suggestive for the formation of qqg states via gluon radiation. For x I P = 0.0025 and 0.005 there is some excess at high β ≥ 0.95. Since here the qq contribution from transverse photons is expected to be small, the excess suggests diffractive contributions from longitudinal photons.

Comparison with the BEKW parametrisation
data can be gained with the help of the BEKW parametrisation [?] which considers the contributions from the transitions: transverse photon → qq, longitudinal photon → qq and transverse photon → qqg. In the BEKW parametrisation, the incoming virtual photon fluctuates into a qq or qqg dipole which interacts with the target proton via two-gluon exchange. The β spectrum and the scaling behaviour in Q 2 are derived from the wave functions of the incoming transverse (T ) or longitudinal (L) photon on the light cone in the non-perturbative limit. The x I P dependence of the cross section is not predicted by BEKW but is to be determined by experiment. Specifically where The contribution from longitudinal photons coupling to qq is limited to β values close to unity. The qq contribution from transverse photons is expected to have a broad maximum around β = 0.5, while the qqg contribution becomes important at small β, provided the power γ is large. The original BEKW parametrisation also includes a higher-twist term for qq produced by transverse photons. The present data are insensitive to this term, and it has, therefore, been neglected.
For F L qq , the term ( Q 2 ) provided by BEKW was replaced by the factor ( ) to avoid problems as Q 2 → 0. The powers n T,L,g (Q 2 ) were assumed by BEKW to be of the form n(Q 2 ) = n 0 + n 1 · ln[1 + ln( Q 2 ]. The rise of α IP (0) with ln Q 2 observed in the present data suggested using the form n(Q 2 ) = n 0 + n 1 ln(1 + Q 2 ). This modified BEKW form will be referred to as BEKW(mod). Taking x 0 = 0.01 and Q 2 0 = 0.4 GeV 2 , the BEKW(mod) form gives a good description of the data. According to the fit, the coefficients n 0 can be set to zero, and the coefficient n 1 can be assumed to be the same for T , L and g.
The fits of BEKW(mod) to the data from this analysis (FPC II), to the data from the FPC I analysis and to the combined FPC I and FPC II data led to the results shown in Table 40. (β, x I P , Q 2 ) data from the FPC I and FPC II analyses with the BEKW(mod) fit. The fit gives a good description of the total of 427 data points.
The measured Q 2 and β dependences of the diffractive structure function are also well reproduced by the BEKW(mod) fit, see Figs. 21, 25 -27. Based on the BEKW(mod) fit, the data show that the (qq) T contribution from transverse photons dominates the diffractive structure function for 0.2 < β < 0.9. In the region β > 0.95, the contribution from longitudinal photons, (qq) L , is dominant. This reflects, at least in part, the increase of the contribution from longitudinal compared to transverse photons in the production of ρ 0 mesons [8]. For β ≤ 0.15, the largest contribution is due to gluon emission as described by the term (qqg) T . These conclusions hold for all Q 2 values studied.

20
Inclusive and diffractive scattering has been measured with data taken in 1999-2000 with the ZEUS detector augmented by the forward-plug calorimeter (FPC), for Q 2 between 25 and 320 GeV 2 using an integrated luminosity of 52.4 pb −1 . Where appropriate, the results from a previous study (FPC I) using 4.2 pb −1 and covering the region Q 2 = 2.7 -55 GeV 2 , were included.
The proton structure function, F 2 (x, Q 2 ), shows a rapid rise as x → 0 at all Q 2 values. The rise for the region x < 0.01 has been parametrised in terms of the Pomeron trajectory α tot IP (0), showing a rapid increase of α tot IP (0) ∝ ln Q 2 for Q 2 values between 2.7 and 70 GeV 2 . The total cross section for virtual-photon proton scattering multiplied by Q 2 , Q 2 σ tot γ * p , shows a rapid rise with increasing W , reflecting the rise of F 2 as x → 0; at lower Q 2 values (2.7 -55 GeV 2 ), this rise becomes steeper as Q 2 increases. At higher Q 2 values, the trend is reversed.
3 GeV, was studied as a function of the hadronic centre-of-mass energy W , of the mass M X of the diffractively produced system X and for different Q 2 values. For M X = 1.2 GeV, the cross section decreases rapidly with increasing Q 2 . For larger M X values a strong rise with W is observed up to M X values of 11 GeV. The intercept of the Pomeron trajectory deduced from the data rises with increasing Q 2 but its size is not as large as observed for For fixed Q 2 , the ratio of the diffractive cross section for 0.28 < M X < 35 GeV to the total cross section is independent of W . For W = 200 − 245 GeV this ratio decreases ∝ ln(1 + Q 2 ) from 15.8 ± 0.7(stat.) +0. 9 −0.7 (syst.)% at Q 2 = 4 GeV 2 to 5.0 ± 0.4(stat.) +0. 8 −0.8 (syst.)% at Q 2 = 190 GeV 2 . Diffraction has also been studied in terms of the diffractive structure function of the proton, F shows a strong rise as x I P → 0 for all Q 2 between 2.7 and 320 GeV 2 . The x I P dependence of varies only modestly with Q 2 . The data show positive scaling violations proportional to ln Q 2 in the region x I P β = x < 2 · 10 −3 , and constancy with Q 2 or negative scaling violations proportional to ln Q 2 for x ≥ 2 · 10 −3 . Therefore, in the Q 2 region studied, the diffractive structure function is consistent with being of leading twist.
The data contradict Regge factorisation: the diffractive structure function F does not factorise into a term which depends only on x I P and a second term which depends only on β and Q 2 .
as a function of x I P , β and Q 2 has been obtained by fitting the data with the BEKW(mod) parametrisation. This fit implies that the region 0.25 < β < 0.9 is dominated by the γ * → (qq) T contribution, the region β > 0.95 is 21 dominated by the γ * → (qq) L term, while the rise of x I P F D(3) 2 as β → 0 results from gluon emission described by the γ * → (qqg) T term.

Appendix
A Subtraction of the contribution from proton dissociation with M N > 2.3 GeV The contribution from proton dissociation with M N > 2.3 GeV to the diffractive data sample was determined with SANG and subtracted from the data sample. Tables 2 and 3 give for every Q 2 , W , M X bin, for which diffractive cross sections are quoted in Tables 7 -12, the fraction of events from M N > 2.3 GeV: For 84% of the bins, the fraction of events for proton dissociation with M N > 2.3 GeV that are subtracted, is less than or equal to 20%.

B Extracting the diffractive contribution in the presence of Reggeon exchange
For this analysis the effect of Reggeon exchange interfering with the diffractive component was studied. A positive interference between Pomeron (IP ) and Reggeon exchange (IR), which reproduces the rise observed in the LPS data [10] for x IP F D(3) 2 as x I P > 0.03, can be achieved by the exchange of the f -meson trajectory. The LPS data were fit to the form is taken from the fit to the FPC I and FPC II data, see Section 9.5.1, and the second term represents the Reggeon contribution. The fit to the LPS data yielded d 1 = 0.768 ± 0.020 and d 2 = 0.0177 ± 0.0019, with χ 2 = 135 for 78 degrees of freedom.
In order to determine the possible contribution from Reggeon exchange and Reggeon-Pomeron interference (IR 2 + 2 · IP · IR) to the diffractive data, Monte Carlo (MC) events were generated according to These MC events were subjected to the same analysis procedure as the data. The Reggeon plus Reggeon-Pomeron interference contribution (IR 2 + 2 · IP · IR) to the diffractive cross section dσ diff /dM X was found to be smaller than the combined statistical and systematic uncertainty for all but 3 of the 166 data points. No correction was applied to the data.         Table 5: The results of the fits of F 2 data for x < 0.01 in bins of Q 2 to F 2 (x, Q 2 ) = c · x −λ , where α tot IP (0) = 1 + λ. The errors give the statistical and systematic uncertainties added in quadrature.  Table 6: Total γ * p cross section σ tot γ * p . 32 (GeV) (GeV 2 ) (GeV) (nb/GeV) (GeV) (GeV 2 ) (GeV) (nb/GeV) GeV, for W = 220 GeVas a function of Q 2 for M X = 1.2 and 3.0 GeV. The first uncertainties are statistical and the second are the systematic uncertainties.
GeV, for W = 220 GeVas a function of Q 2 for M X = 6 and 11 GeV. The first uncertainties are statistical and the second are the systematic uncertainties.
GeV, for W = 220 GeVas a function of Q 2 for M X = 20 and 30 GeV. The first uncertainties are statistical and the second are the systematic uncertainties.
± stat. ± syst.  (β, x I P , Q 2 ), for diffractive scattering, γ * p → XN, M N < 2.3 GeV, for Q 2 = 320 GeV 2 , in bins of β and x I P .  (β, x I P , Q 2 ), for diffractive scattering, γ * p → XN, M N < 2.3 GeV, for fixed x I P = 0.00015, 0.0003, 0.0006 and fixed β. The errors are the statistical and systematic uncertainties added in quadrature.  (β, x I P , Q 2 ), for diffractive scattering, γ * p → XN, M N < 2.3 GeV, for fixed x I P = 0.0012 and fixed β. The errors are the statistical and systematic uncertainties added in quadrature.   (β, x I P , Q 2 ), for diffractive scattering, γ * p → XN, M N < 2.3 GeV, for fixed x I P = 0.005 and fixed β. The errors are the statistical and systematic uncertainties added in quadrature. 58