Rapidity gap cross sections measured with the ATLAS detector in pp collisions at √ s = 7 TeV

Pseudorapidity gap distributions in proton-proton collisions at √ s = 7 TeV are studied using a minimum bias data sample with an integrated luminosity of 7.1 µb − 1 . Cross sections are measured differentially in terms of Δη F , the larger of the pseudorapidity regions extending to the limits of the ATLAS sensitivity, at η = ± 4 . 9, in which no ﬁnal state particles are produced above a transverse momentum threshold p cutT . The measurements span the region 0 < Δη F < 8 for 200 MeV < p cutT < 800 MeV. At small Δη F , the data test the reliability of hadronisation models in describing rapidity and transverse momentum ﬂuctuations in ﬁnal state particle production. The measurements at larger gap sizes are dominated by contributions from the single diffractive dissociation process ( pp → Xp ), enhanced by double dissociation ( pp → XY ) where the invariant mass of the lighter of the two dissociation systems satisﬁes M Y (cid:2) 7 GeV. The resulting cross section is d σ/ d Δη F ≈ 1 mb for Δη F (cid:3) 3. The large rapidity gap data are used to constrain the value of the Pomeron intercept appropriate to triple Regge models of soft diffraction. The cross section integrated over all gap sizes is compared with other LHC inelastic cross section measurements.

The ATLAS Collaboration January 13, 2012 Abstract Pseudorapidity gap distributions in protonproton collisions at √ s = 7 TeV are studied using a minimum bias data sample with an integrated luminosity of 7.1 µb −1 . Cross sections are measured differentially in terms of ∆η F , the larger of the pseudorapidity regions extending to the limits of the ATLAS sensitivity, at η = ±4.9, in which no final state particles are produced above a transverse momentum threshold p cut T . The measurements span the region 0 < ∆η F < 8 for 200 < p cut T < 800 MeV. At small ∆η F , the data test the reliability of hadronisation models in describing rapidity and transverse momentum fluctuations in final state particle production. The measurements at larger gap sizes are dominated by contributions from the single diffractive dissociation process (pp → Xp), enhanced by double dissociation (pp → XY ) where the invariant mass of the lighter of the two dissociation systems satisfies M Y < ∼ 7 GeV. The resulting cross section is dσ/d∆η F ≈ 1 mb for ∆η F > ∼ 3. The large rapidity gap data are used to constrain the value of the Pomeron intercept appropriate to triple Regge models of soft diffraction. The cross section integrated over all gap sizes is compared with other LHC inelastic cross section measurements.

Introduction
When two protons collide inelastically at a centre-ofmass energy of 7 TeV in the Large Hadron Collider (LHC), typically around six charged particles are produced with transverse momentum 1 p T > 100 MeV per 1 In the ATLAS coordinate system, the z-axis points in the direction of the anti-clockwise beam viewed from above. unit of pseudorapidity in the central region [1][2][3]. On average, the rapidity difference between neighbouring particles is therefore around 0.15 units of rapidity, with larger gaps occurring due to statistical fluctuations in the hadronisation process. Such random processes lead to an exponential suppression with gap size [4], but very large gaps are produced where a t-channel colour singlet exchange takes place. This may be due to an electroweak exchange, but occurs much more frequently through the exchange of strongly interacting states. At high energies such processes are termed 'diffractive' and are associated with 'Pomeron' exchange [5,6].
The total cross section in hadronic scattering experiments is commonly decomposed into four main components: elastic (pp → pp in the LHC context), singlediffractive dissociation (SD, pp → Xp or pp → pX, Figure 1a), double-diffractive dissociation (DD, pp → XY , Figure 1b) and non-diffractive (ND) contributions. The more complex central diffractive configuration (CD, pp → pXp, Figure 1c), in which final state particles are produced in the central region with intact protons on both sides, is suppressed relative to the SD process by a factor of around 10 at high energies [7]. Together, the diffractive channels contribute approximately 25-30% of the total inelastic cross section at LHC energies [8]. Following measurements at the LHC of the elastic [9], total [10] and total inelastic [8,10] cross sections, this article contains the first detailed exploration of diffractive dissociation processes.
Polar angles θ and transverse momenta p T are measured with respect to this axis. The pseudorapidity η = − ln tan(θ/2) is a good approximation to the rapidity of a particle whose mass is negligible compared with its energy and is used here, relative to the nominal z = 0 point at the centre of the apparatus, to describe regions of the detector.  Understanding diffractive processes is important in its own right, as they are the dominant contribution to high energy quasi-elastic scattering between hadrons and, via ideas derived from the optical theorem [11], are also related to the total cross section. They are often interpreted at the parton level in terms of the exchange of pairs of gluons [12,13] and are thus sensitive to possible parton saturation effects in the low Bjorken-x regime of proton structure [14][15][16]. Diffractive cross sections also have relevance to cosmic ray physics [17] and may be related to the string theory of gravity [18]. At the LHC, diffractive dissociation must be well understood for a good description of the additional inelastic proton-proton interactions (pile-up) which accompany most events. It also produces a significant uncertainty in approaches to luminosity monitoring which rely on measurements of the total, or total inelastic, cross section [19].
Diffractive dissociation cross sections have been measured previously over a wide range of centre-of-mass energies. Early measurements are reviewed in [20][21][22][23][24]. SD measurements have been made in pp scattering at the SPS [25,26] and the Tevatron [27,28], and also in photoproduction [29,30] and deep inelastic scattering [31][32][33] at HERA. Limited high energy DD [26,29,34] and CD [7,35,36] data are also available. In most cases, the momentum transfer is too small to permit an interpretation in terms of partonic degrees of freedom [37]. Instead, phenomenological models such as those based on Regge theory have been developed [22,38,39], which underlie the Monte Carlo generators typically used to predict the properties of soft inelastic collisions [40][41][42]. Mixed approaches have also been developed which employ perturbative QCD where possible [43,44]. Large theoretical uncertainties remain in the detailed dynamics expected at the LHC.
Direct measurements of the masses M X and M Y of the dissociated systems are difficult at ATLAS, since many of the final state particles are produced beyond the acceptance of the detector. However, the dissociation masses are closely correlated with the size of the rapidity region in which particle production is completely suppressed due to the net colour-singlet Pomeron exchange. This correlation is exploited in this paper, with cross sections reported as a function of the size of a pseudorapidity region which is devoid of final state particle production. These unpopulated pseudorapidity regions are referred to in the following as 'rapidity gaps', or simply 'gaps'.
To maximise the pseudorapidity coverage and sensitivity to charged and neutral particle production, rapidity gaps are identified using both the ATLAS calorimeters and tracking detectors. The specific observable studied is ∆η F , the larger of the two 'forward' pseudorapidity regions extending to at least η = ±4.9 in which no particles are produced with p T > p cut T , where p cut silicon strip and straw tube detectors and is enclosed within a uniform 2 T solenoidal magnetic field. The calorimeters lie outside the tracking system. A highly segmented electromagnetic (EM) liquid argon sampling calorimeter covers the range |η| < 3.2. The EM calorimeter also includes a pre-sampler covering |η| < 1.8. The hadronic end-cap (HEC, 1.5 < |η| < 3.2) and forward (FCal, 3.1 < |η| < 4.9) calorimeters also use liquid argon technology, with granularity decreasing with increasing |η|. Hadronic energy in the central region (|η| < 1.7) is reconstructed in a steel/scintillatortile calorimeter.
Minimum bias trigger scintillator (MBTS) detectors are mounted in front of the end-cap calorimeters on both sides of the interaction point and cover the pseudorapidity range 2.1 < |η| < 3.8. The MBTS is divided into inner and outer rings, both of which have eight-fold segmentation. The MBTS is used to trigger the events analysed here.
In 2010, the luminosity was measured using aČerenkov light detector which is located 17 m from the interaction point. The luminosity calibration is determined through van der Meer beam scans [19,46].

Event selection and backgrounds
The data used in this analysis were collected during the first LHC run at √ s = 7 TeV in March 2010, when the LHC was filled with two bunches per beam, one pair colliding at the ATLAS interaction point. The peak instantaneous luminosity was 1.1×10 27 cm −2 s −1 . Events were collected from colliding proton bunch crossings in which the MBTS trigger recorded one or more inner or outer segments above threshold on at least one side of ATLAS. After reconstruction, events are required to have hits in at least two of the MBTS segments above a threshold of 0.15 pC. This threshold cut suppresses contributions from noise, which are well modelled by a Gaussian with 0.02 pC width. No further event selection requirements are applied. The data sample corresponds to an integrated luminosity of 7.1 ± 0.2 µb −1 and the number of recorded events is 422776. The mean number of interactions per bunch crossing is below 0.005, which is consistent with the approximately 400 events which have multiple reconstructed vertices. Pile-up contamination is thus negligible.
The data sample contains a contribution from beaminduced background, mainly due to scattering of beam protons from residual gas particles inside the detector region. This contamination is estimated using events collected in unpaired bunches and is subtracted statistically in each measurement interval. Averaged over the full measurement region, it amounts to 0.2% of the sample. More complex backgrounds in which beam-induced background is overlaid on a physics event are negligible.

Reconstruction of rapidity gaps
The analysis of final state activity in the central region (|η| < 2.5) is based on combined information from inner detector tracks and calorimeter modules. In the region 2.5 < |η| < 4.9, beyond the acceptance of the inner detector, calorimeter information alone is used. The track selection is as detailed in [1]. Energy deposits from final state particles in the calorimeters are identified using a topological clustering algorithm [47,48], with a further requirement to improve the control over noise contributions, as described below.
The identification of rapidity gap signatures relies crucially on the suppression of calorimeter noise contributions. The root-mean-squared cell energies due to noise vary from around 20 MeV in the most central region to around 200 MeV for the most forward region [49]. The shapes of the cell noise distributions in each calorimeter are well described by Gaussian distributions of standard deviation σ noise , with the exception of the tile calorimeter, which has extended tails. The default clustering algorithm [48] is seeded by cells for which the significance of the measured energy, E, is S = E/σ noise > 4. However, with this threshold there are on average six clusters reconstructed per empty event due to fluctuations in the noise distributions. To suppress noise contributions to acceptable levels for gap finding, clusters of calorimeter energy deposits are thus considered only if they contain at least one cell outside the tile calorimeter with an energy significance above an η-dependent threshold, S th . This threshold is determined separately in pseudorapidity slices of size 0.1 such that the probability of finding at least one noisy cell in each η-slice has a common value, 1.4×10 −4 . This choice optimises the resolution of the reconstructed gap sizes with respect to the gaps in the generated final state particle distributions according to MC studies. Since the number of cells in an η-slice varies from about 4000 in the central region to 10 in the outer part of the FCal, the cell thresholds vary between S th = 5.8 in the central region and S th = 4.8 at the highest |η| values in the FCal.
The level of understanding of the calorimeter noise is illustrated in Figure 2, which shows the distributions of the cell significance S for each of the liquid argon modules. MBTS-triggered data from colliding bunch crossings are compared with a Monte Carlo simulation  MBTS-triggered minimum bias data (points) are compared with events randomly triggered on empty bunch crossings (histograms) and with a Monte Carlo simulation (shaded areas). and with events which are required to exhibit no activity in the non-calorimeter components of the detector, triggered randomly on empty bunch crossings. The signal from pp collisions is clearly visible in the long positive tails, which are well described by the simulation. The data from the empty bunch crossings show the shape of the noise distribution with no influence from physics signals. The empty bunch crossing noise distributions are symmetric around zero and their negative sides closely match the negative parts of the MBTStriggered data distributions. The noise distribution is well described over seven orders of magnitude by the MC simulation, the small residual differences at positive significances being attributable to deficiencies in the modelling of pp collision processes.
The measured energies of calorimeter clusters which pass the noise requirements are discriminated using a given value of p cut T , neglecting particle masses. The calorimeter energy scale for electromagnetic showers is determined from electron test-beam studies and Z → e + e − data [50], confirmed at the relatively small energies relevant to the gap finding algorithm through a dedicated study of π 0 → γγ decays. The calorimeter response to hadronic showers is substantially lower than that to electromagnetic showers. In the central region, the scale of the hadronic energy measurements is determined relative to the electromagnetic scale through comparisons between the calorimeter and inner detector measurements of single isolated hadrons [51][52][53]. Beyond the acceptance region of the tracking detectors, the difference between the electromagnetic and the hadronic response is determined from test-beam results [54][55][56]. For the purposes of discriminating against thresholds in the gap finding algorithm, all cluster energy measurements are taken at this hadronic scale. An interval in η is deemed to contain final state particles if at least one cluster in that interval passes the noise suppression requirements and has a transverse momentum above p cut T , or if there is at least one good inner detector track with transverse momentum above p cut T .

Definition of forward rapidity gap observable
The reconstructed forward gap size ∆η F is defined by the larger of the two empty pseudorapidity regions extending between the edges of the detector acceptance at η = 4.9 or η = −4.9 and the nearest track or calorimeter cluster passing the selection requirements at smaller |η|. No requirements are placed on particle production at |η| > 4.9 and no attempt is made to identify gaps in the central region of the detector. The rapidity gap size relative to η = ±4.9 lies in the range 0 < ∆η F < 8, such that for example ∆η F = 8 implies that there is no reconstructed particle with p T > p cut T in one of the regions −4.9 < η < 3.1 or −3.1 < η < 4.9. The upper limit on the gap size is constrained via the requirement of a high trigger efficiency by the acceptance of the MBTS detector.
The measurement is performed in ∆η F intervals of 0.2, except at the smallest values ∆η F < 2.0, where the differential cross section varies fastest with ∆η F and the gap end-point determination is most strongly dependent on the relatively coarse cell granularity of the FCal. The bin sizes in this region are increased to 0.4 pseudorapidity units, commensurate with the resolution.
The default value of the transverse momentum threshold is chosen to be p cut T = 200 MeV. This value lies within the acceptance of the track reconstruction for the inner detector and ensures that the efficiency of the calorimeter cluster selection is greater than 50% throughout the η region which lies beyond the tracking acceptance.
As described in Section 3.4, the data are fully corrected for experimental effects using the Monte Carlo simulations introduced in Section 3.2. The rapidity gap observable defining the measured differential cross sections are thus specified in terms of stable (proper lifetime > 10 ps) final state particles (hereafter referred to as the 'hadron level'), with transverse momentum larger than the threshold, p cut T , used in the gap reconstruction algorithm.

Kinematic Variables and Theory
As illustrated in Figure 1a and b, diffractive dissociation kinematics can be described in terms of the invariant masses M X and M Y of the dissociation systems X and Y , respectively (with M Y = M p in the SD case), and the squared four-momentum transfer t. In the following, the convention M Y < M X is adopted. The cross section is vastly dominated by small values of |t| < ∼ 1 GeV 2 , such that the intact proton in SD events is scattered through only a small angle, gaining transverse momentum p T ≃ √ |t|. Further commonly used kinematic variables are defined as where s is the square of the centre-of-mass energy. Diffractive dissociation cross sections can be modelled using Regge phenomenology [38,39,57], with Pomeron exchange being the dominant process at small ξ X values. For the SD case, the amplitude is factorised into a Pomeron flux associated with the proton which remains intact, and a total probability for the interaction of the Pomeron with the dissociating proton. The latter can be described in terms of a further Pomeron exchange using Muller's generalisation of the optical theorem [11], which is applicable for s ≫ M 2 X ≫ m 2 p . The SD cross section can then be expressed as a triple Pomeron (IPIPIP) amplitude, where G 3IP (0) is a product of couplings and α IP (t) = α IP (0) + α ′ IP t is the Pomeron trajectory. The term f (t) is usually taken to be exponential such that dσ/dt ∝ e B(s,M 2 X ) t at fixed s and M X , B being the slope parameter. With α IP (0) close to unity and |t| small, equation (2) leads to an approximately constant dσ/d ln ξ X at fixed s. The DD cross section follows a similar dependence at fixed ξ Y . The deviations from this behaviour are sensitive to the intercept α IP (0) of the Pomeron trajectory [58,59] and to absorptive corrections associated with unitarity constraints [43,44].
The rapidity gap size and its location are closely correlated with the variables ξ X and ξ Y . For the SD process, the size ∆η of the rapidity gap between the final state proton and the X system satisfies The ∆η F observable studied here differs from ∆η in that ∆η F takes no account of particle production at |η| > 4.9. For the SD process, where the intact proton has η ≃ ± 1 2 ln(s/m 2 p ) ≃ ±8.9, the gap variables are related by ∆η F ≃ ∆η − 4. Equations (2) and (3) thus lead to approximately constant predicted cross sections dσ/d∆η F for SD and low M Y DD events. With the high centre-of-mass energy of the LHC and the extensive acceptance of the ATLAS detector, events with ξ X between around 10 −6 and 10 −2 can be selected on the basis of their rapidity gap signatures, corresponding approximately to 7 < M X < 700 GeV.
Previous proton-proton scattering [25] and photoproduction [29,30] experiments have observed enhancements relative to triple-Pomeron behaviour at the smallest M X values in the triple Regge region. This effect has been interpreted in terms of a further triple Regge term (IPIPIR) in which the reaction still proceeds via Pomeron exchange, but where the total Pomeron-proton cross section is described by a sub-leading Reggeon (IR) with intercept α IR (0) ≃ 0.5 [58]. This leads by analogy with equation (2) to a contribution to the cross section which falls as dσ/dM 2 X ∝ 1/M 3 X . In the recent model of Ryskin, Martin and Khoze (RMK) [43], a modified triple-Pomeron approach to the large ξ X region is combined with a dedicated treatment of low mass diffrac- 6 The ATLAS Collaboration tive dissociation, motivated by the original s-channel picture of Good and Walker [60], in which proton and excited proton eigenstates scatter elastically from the target with different absorption coefficients. This leads to a considerable enhancement in the low ξ X cross section which is compatible with that observed in the pre-LHC data [25].

Monte Carlo Simulations
Triple Pomeron-based parameterisations are implemented in the commonly used Monte Carlo (MC) event generators, pythia [40,41] and phojet [42,61]. These generators are used to correct the data for experimental effects and as a means of comparing the corrected data with theoretical models.
By default, the pythia model of diffractive dissociation processes uses the Schuler and Sjöstrand parameterisation [62] of the Pomeron flux, which assumes a Pomeron intercept of unity and an exponential t dependence e B(ξX ,ξY )t . Three alternative flux models are also implemented. The Bruni and Ingelman version [63] is similar to Schuler and Sjöstrand, except that its t dependence is given by the sum of two exponentials. In the Berger and Streng [64,65] and Donnachie and Landshoff [66] models, the Pomeron trajectory is linear, with variable parameters, the default being α IP (t) = 1.085+0.25t [67], consistent with results from fits to total [58,59] and elastic [68] hadronic cross section data. Whilst the model attributed to Berger and Streng has an exponential t dependence, the Donnachie and Landshoff version is based on a dipole model of the proton elastic form factor. For all flux parameterisations in pythia, additional factors are applied to modify the distributions in kinematic regions in which a triple-Pomeron approach is known to be inappropriate. Their main effects are to enhance the low mass components of the dissociation spectra, to suppress the production of very large masses and, in the DD case, to reduce the probability of the systems X and Y overlapping in rapidity space [41,62].
Above the very low mass resonance region, dissociation systems are treated in the pythia6 generator using the Lund string model [69], with final state hadrons distributed in a longitudinal phase space with limited transverse momentum. In pythia8, diffractive parton distribution functions from HERA [31] are used to include diffractive final states which are characteristic of hard partonic collisions, whilst preserving the ξ X , ξ Y , s and t dependences of the diffractive cross sections from the pythia6 model [70]. This approach yields a significantly harder final state particle transverse momentum spectrum in SD and DD processes in pythia8 compared with pythia6, in better agreement with the present data. The default pythia multiple parton interaction model is applied to ND events and, in the case of pythia8, also within the dissociated systems in SD and DD events.
The specific versions used to correct the data are pythia6.4.21 (with the AMBT1 tune performed by ATLAS [71]) and pythia8.145 (with the 4C tune [72]). Updated versions, pythia8.150 and pythia6.4.25 (using the 4C and AMBT2B tunes, respectively), are used for comparisons with the corrected data (see Table 1). The 4C tune of pythia8 takes account of the measurement of the diffractive fraction f D of the inelastic cross section in [8], whilst keeping the total cross section fixed, resulting in a somewhat smaller diffractive cross section than in pythia6.
The phojet model uses the two component dual parton model [73] to combine features of Regge phenomenology with AGK cutting rules [74] and leading order QCD. Diffractive dissociation is described in a two-channel eikonal model, combining a triple Regge approach to soft processes with lowest order QCD for processes with parton scattering transverse momenta above 3 GeV. The Pomeron intercept is taken to be α IP (0) = 1.08 and for hard diffraction, the diffractive parton densities are taken from [75,76]. Hadronisation follows the Lund string model, as for pythia. The CD process is included at the level of 1.7% of the total inelastic cross section. The specific version used is pho-jet1.12.1.35, with fragmentation and hadronisation as in pythia6.1.15.
After integration over t, ξ X and ξ Y , the cross sections for the diffractive processes vary considerably between the default MC models, as shown in Table 1. The DD variation is particularly large, due to the lack of experimental constraints. For use in the data correction procedure, the overall fractional non-diffractive (f ND ) and diffractive (f D = f SD + f DD + f CD = 1 − f ND ) contributions to the total inelastic cross section are modified to match the results obtained in the context of each model in a previous ATLAS analysis [8]. Despite the close agreement between the diffractive fractions f D ∼ 30% determined for the three default models (see the 'Tuned' fractions in Table 1), the f D parameter is rather sensitive to the choice of Pomeron flux model and to the value of α IP (0), for example reaching f D ∼ 25% for the Bruni and Ingelman flux in pythia8 [8].
The default phojet and pythia models do not take into account Tevatron data which are relevant to the decomposition of the diffractive cross section into SD, DD and CD components, so these fractions are also adjusted for the present analysis. Based on CDF SD [28] and DD [34] cross section data, extrapolated to the full diffractive kinematic ranges in each of the mod-√ s = 7 TeV 7 els, constraints of 0.29 < σ DD /σ SD < 0.68 and 0.44 < σ DD /σ SD < 0.94 are derived for the pythia and phojet models of diffraction, respectively. The tuned ratios used in the correction procedure are taken at the centres of these bounds. The CD contribution in phojet is compatible with the measured Tevatron value of 9.3% of the SD cross Section [7] and σ CD /σ SD is therefore kept fixed, with f CD increasing in proportion to f SD . Table 1 summarises the tuned decomposition of the inelastic cross section for each MC model.
Despite the substantial differences between the approaches to diffraction taken in phojet and pythia, the two models both employ the Lund string model [69] of hadronisation. In order to investigate the sensitivity of the data at small gap sizes to the hadronisation model for ND processes, comparisons of the measured cross sections are also made with the herwig++ generator [77] (version 2.5.1 with the UE7-2 tune [78,79]), which uses an alternative cluster-based model. The herwig++ minimum bias generator takes the total inelastic cross section to be 81 mb, based on a Donnachie-Landshoff model [80]. Perturbatively treated semi-hard processes are distinguished from soft processes according to whether they produce objects with transverse momentum above a fixed threshold which is taken to be 3.36 GeV. Partons produced from the parton shower are combined into colour singlet pairs called clusters, which can be interpreted as excited hadronic resonances. The clusters are then successively split into new clusters until they reach the required mass to form hadrons. The most recent herwig++ versions contain a mechanism to reconnect partons between cluster pairs via a colour reconnection (CR) algorithm, which improves the modelling of charged particle multiplicities in pp collisions [81]. Similarly to pythia, herwig++ contains an eikonalised underlying event model, which assumes that separate scatterings in the same event are independent. At fixed impact parameter, this leads to Poisson distributions for both the number of soft scatters and the number of semi-hard processes per event. There is thus a small probability for 'empty' events to occur with no scatterings of either type. Under these circumstances, particle production occurs only in association with the dissociation of the beam protons, in a manner which is reminiscent of diffractive dissociation processes.

Comparisons between Monte Carlo simulations and uncorrected data
For use in the correction procedure, MC events are processed through the ATLAS detector simulation program [82], which is based on geant4 [83]. They are then subjected to the same reconstruction and analysis chain as is used for the data.
The quality of the MC description of the most important distributions for the correction procedure is tested through a set of control plots which compare the uncorrected data and MC distributions. These include energy flows, track and calorimeter cluster multiplicities and transverse momentum distributions, as well as leading cell energy significances in different pseudorapidity regions. All such distributions are reasonably well described. Examples are shown in Figures 3a-d, where the total multiplicities of calorimeter clusters which pass the selection described in Section 2.3 are shown for events in four different regions of reconstructed forward rapidity gap size. Whilst none of the MC models gives a perfect description, particularly at small multiplicities, the three models tend to bracket the data, with pythia6 showing an excess at low multiplicities and pythia8 and phojet showing a deficiency in the same region.
A further example control distribution is shown in Figure 3e. The probability of detecting at least one calorimeter cluster passing the noise requirements with p T > p cut T = 200 MeV in the most central region (|η| < 0.1) is shown as a function of the p T of the leading track reconstructed in the same η region. In cases where this track has p T below around 400 MeV, it spirals in the solenoidal field outside the acceptance of the EM calorimeter. The plotted quantity then corresponds to the detection probability for neutral particles in the vicinity of a track. Good agreement is observed between MC and data. The shape of the uncorrected ∆η F distribution for p cut T = 200 MeV is compared between the data and the MC models in Figure 3f. The binning reflects that used in the final result (Section 2.4) except that contributions with ∆η F > 8, where the trigger efficiency becomes small, are also shown. None of the models considered are able to describe the data over the full ∆η F range, with the largest deviations observed for small non-zero gaps in phojet. All of the models give an acceptable description of the shape of the distribution for large gaps up to the limit of the measurement at ∆η F = 8 and beyond.
Considering all control plots together, pythia8 provides the best description of the shapes of the distributions. Hence this generator is chosen to correct the data. The deviations from pythia8 of pythia6 and phojet, which often lie in opposite directions and tend to enclose the data, are used to evaluate the systematic uncertainties on the unfolding procedure.

Corrections for Experimental Effects
After the statistical subtraction of the beam-induced background in each interval of ∆η F (Section 2.2), the data are corrected for the influence of the limited acceptance and small particle detection inefficiencies of the MBTS using the MC simulation. For the ND, SD and DD processes, the trigger efficiency is close to 100% for ∆η F < 7, dropping to around 80% at ∆η F = 8. Since the topology of CD events sometimes involves hadronic activity in the central region of the detector, with gaps on either side, a larger fraction fail the trigger requirement, with efficiencies of close to 100% for ∆η F < 3 and between 85% and 95% for 3 < ∆η F < 8.
The data are corrected for migrations between the reconstructed and hadron level ∆η F values, due to missed or spurious activity and cases where a final state particle is observed in a different η interval from that in which it is produced. The migration corrections are obtained using a Bayesian unfolding method [84] with a single iteration. The priors for the unfolding procedure with each MC model are taken after tuning the diffractive cross sections as described in Section 3.2. The migration matrix between the reconstructed and hadron level forward gap distributions according to the pythia8 MC is shown for p cut T = 200 MeV in Figure 4. An approximately diagonal matrix is obtained. MeV, according to pythia8. The distribution is normalised to unity in columns and is shown to beyond the limit of the measurement at ∆η F = 8.

Systematic Uncertainties
The sources of systematic uncertainty on the measurement are outlined below.
MC Model and Unfolding Method Dependence: The trigger efficiency and migration correction procedure is carried out using each of the pythia6, pythia8 and phojet models. The deviation of the data unfolded with phojet from those obtained with pythia8 is used to obtain a systematic uncertainty due to the assumed ξ X , ξ Y and t dependences in the unfolding procedure. The model dependence due to the details of the final state particle production is obtained from the difference between the results obtained with pythia6 and pythia8. Both of these model dependences are evaluated separately in each measurement interval and are applied symmetrically as upward and downward uncertainties. They produce the largest uncertainty on the measurement over most of the measured range. For p cut T = 200 MeV, the contributions from the pythia6 and phojet variations are of similar size. Their combined effect is typically at the 6% level for large ∆η F , growing to 20% for gaps of around 1.5 pseudorapidity units. At larger p cut T values, the pythia6 source becomes dominant. The dependence on the unfolding technique has also been studied by switching between the default Bayesian method [84], a method using a Singular Value Decomposition of the unfolding matrix [85] and a simple bin-to-bin method. The resulting variations in the measured cross section are always within the systematic uncertainty defined by varying the MC model.

Modelling of Diffractive Contributions:
In addition to the differences between the Monte Carlo generators, additional systematic uncertainties are applied on the modelling of the fractional diffractive cross sections. The SD and DD cross sections in the pythia8 model are each varied to the limits of the constraints from Tevatron data described in Section 3.2. The fraction f DD is enhanced to 11.3% of the total inelastic cross section, with f SD reduced to 18.5% to compensate. At the opposite extreme, f SD is enhanced to 23.2% of the cross section, with f DD reduced to 6.6%. These changes result in an uncertainty at the 1% level for ∆η F > 3. A systematic uncertainty on the CD cross section is obtained by varying the CD and SD cross sections in phojet between the tuned values and σ CD /σ SD = 0.093, corresponding to the CDF measurement in [7]. This variation also results in a 1% uncertainty in the large gap region.
Calorimeter Energy Scale: The uncertainty on the calorimeter energy scale is constrained to be below the 5% level down to energies of a few hundred MeV in the central region, |η| < 2.3, through comparisons between isolated calorimeter cluster energy measurements and momentum determinations of matched tracks in the inner detector [51][52][53]. This method is not available for larger |η| values beyond the tracking acceptance. However, as |η| grows, the default p cut T = 200 MeV threshold corresponds to increasingly large energies, reaching beyond 10 GeV at the outer limits of the FCal. The uncertainty on the response to electromagnetic showers in this energy range is determined as a function of |η| from the maximum observed deviations between the data and the MC simulation in the peaks of π 0 → γγ signals, under a variety of assumptions on background shapes and cluster energy resolutions. The relative response to charged pions compared with the electromagnetic scale has been studied in the relevant energy range for the FCal [55,86] and HEC [54,86] test-beam data, with systematic uncertainties of 8% and 4%, respectively, determined from the difference between data and MC. Adding the uncertainties in the electromagnetic scale and in the relative response to hadrons in quadrature, energy scale uncertainties of 5% for |η| < 1.37, 21% for 1.37 < |η| < 1.52 (transition region between barrel and end-cap), 5% for 1.52 < |η| < 2.3, 13% for 2.3 < |η| < 3.2 and 12% for 3.2 < |η| < 4.9 are ascribed. In addition to the absolute calorimeter response, these values account for systematic effects arising from dead material uncertainties and from the final state decomposition into different particle species. In the unfolding procedure, the corresponding systematic variation is applied to energy depositions from simulated final state particles in MC, with noise contributions left unchanged. The clustering algorithm is then re-run over the modified calorimeter cells. The scale uncertainty variation is thus considered both in the application of the p cut T threshold to the clusters and in the discrimination of cells within selected clusters against the significance cut used to veto noise. The resulting fractional uncertainties on the differential cross sections at the default p cut T = 200 MeV are largest (reaching ∼ 12%) in the region ∆η F < ∼ 3, where the gap identification relies most strongly on the calorimeter information. For larger gaps, the well measured tracks play an increasingly important role in defining the gap size and the cross section is dominated by low ξ X diffractive events for which particle production in the gap region is completely suppressed. The sensitivity to the calorimeter scale is correspondingly reduced to a few percent.

MBTS Efficiency:
The description of the MBTS efficiency in the MC models leads to a potential systematic effect on the trigger efficiency and on the off-line MBTS requirement. Following [8], the associated uncertainty is evaluated by increasing the thresholds of all MBTS counters in the simulation to match the maximum variation in the measured response in data according to studies with particles extrapolated from the tracker or FCal. This systematic error amounts to typically 0.5 − 1% for ∆η F > 2 and is negligible at the smallest ∆η F .

Tracking Efficiency:
The dominant uncertainty in the charged particle track reconstruction efficiency arises due to possible inadequacies in the modelling of the material through which the charged particles pass [1]. This uncertainty is quantified by studying the influence on the data correction procedure of using an MC sample produced with a 10% enhancement in the support material in the inner detector. The resulting uncertainty is smaller than 3.5% throughout the measured distribution.
Luminosity: Following the van der Meer scan results in [46], the normalisation uncertainty due to the uncertainty on the integrated luminosity is 3.4%.
Each of the systematic uncertainties is determined with correlations between bins taken into account in the unfolding by repeating the full analysis using data or MC distributions after application of the relevant systematic shift. The final systematic error on the differential cross section is taken to be the sum in quadrature of all sources. Compared with the systematic uncertainties, the statistical errors are negligible at the smallest gap sizes, where the differential cross section is largest. For gap sizes ∆η F > ∼ 3, the statistical errors are at the 1% level and are typically smaller than the systematic errors by factors between five and ten.

Differential cross section for forward rapidity gaps
In this section, measurements are presented of the inelastic cross section differential in forward rapidity gap size, ∆η F , as defined in Section 2.4. The data cover the range 0 < ∆η F < 8. In the large gap region which is populated by diffractive processes, the cross section corresponds to a t-integrated sum of SD events in which either of the colliding protons dissociates and DD events with ξ Y < ∼ 10 −6 (M Y < ∼ 7 GeV). The data span the range ξ X > ∼ 10 −5 . Diffractive events with smaller ξ X values are subject to large MBTS trigger inefficiencies and thus lie beyond the kinematic range of the measurement.
As discussed in Section 2.4, the lowest transverse momentum requirement for the gap definition which is directly accessible experimentally is p cut T = 200 MeV. Figure 5a shows the differential gap cross section for this choice of p cut T , which is also given numerically in Table 2. The uncertainty on the measurement is typically less than 8% for ∆η F > 3, growing to around 20% at ∆η F = 1.5 before improving to around 10% for events with little or no forward gap. The data are compared with the predictions of the default settings of the pythia6 (labelled 'pythia6 atlas ambt2b') pythia8 ('pythia8 4c') and phojet models. In Figures 5b-d, the results are compared with each of the MC models separately, with the default decomposition of the cross section into ND, SD, DD and CD contributions according to the models (Table 1) also indicated.

Small gap sizes and constraints on hadronisation models
At ∆η F < ∼ 2, all models agree that the ND process is dominant and the expected [4] exponential decrease of the cross section with increasing gap size, characteristic of hadronisation fluctuations, is the dominant feature of the data. According to the models, this region also contains DD events which have ξ Y > ∼ 10 −6 , such that the Y system extends into the ATLAS detector acceptance, as well as both SD and DD events with very large ξ X , such that no large rapidity gap is present within the region |η| < 4.9. The default MC models tend to lie above the data in this region, a result which is consistent with the overestimates of the total inelastic cross section observed for the same models in [8]. The pythia8 model is closest in shape to the data, which is partly due to the modification of f D in the most recent versions made in light of the previous ATLAS data [8]. Both pythia models are closer to the small ∆η F data than phojet, which exhibits an excess of almost a factor of two for ∆η F ∼ 1.
As can be inferred from comparisons between the predicted shapes of the ND contributions in the different MC models (Figures 5b-d), there are considerable uncertainties in the probability of obtaining large hadronisation fluctuations among low transverse momentum final state particles [87]. Studying the dependence of the measured differential cross section on p cut T provides a detailed probe of fluctuations in the hadronisation process in soft scattering and of hadronisation models in general. The measurement is thus repeated with different choices of p cut T , applied both in the rapidity gap reconstruction and in the definition of the measured hadron level cross section. To avoid cases where the largest gap switches from one side of the detector to the other when low p T particles are excluded by the increased p cut T choice, the side of the detector on which the gap is located is fixed to that determined at p cut T = 200 MeV for all measured cross sections.
A comparison between the results with p cut T = 200, 400, 600 and 800 MeV is shown in Figure 6a. Figures 6bd show the results for p cut T = 400, 600 and 800 MeV, respectively, compared with the pythia8, pythia6 and phojet MC models. The ND contributions according to each of the models are also shown. As p cut T increases, the exponential fall becomes less steep, so larger ∆η F values become more heavily populated and the nondiffractive and diffractive contributions in the models become similar. Also, the uncertainties due to the MC model dependence of the unfolding procedure grow.
The influence of changing from p cut T = 200 MeV to p cut T = 400 MeV is small at large ∆η F , where the cross section is dominated by small ξ X diffractive events and particle production is kinematically forbidden over a wide range of pseudorapidity. For ∆η F < ∼ 4, where ND contributions become important, a significant fraction of events are assessed as having larger gaps for p cut T = 400 MeV than for p cut T = 200 MeV. As the value of p cut T increases to 600 and 800 MeV, soft ND events migrate to larger ∆η F values, giving significant contributions throughout most of the distribution and confirming [1] that the production of final state particles with more than a few hundred MeV is rare in minimum  bias events, even at LHC energies. All MC models are able to reproduce the general trends of the data, though none provides a full description.
It is interesting to investigate the extent to which the alternative cluster-based approach to hadronisation in the non-diffractive herwig++ model is able to describe the data at small gap sizes, where the contribution from ND processes is dominant. A comparison of the data at each of the p cut T values with herwig++ is shown in Figure 7. Despite not containing an explicit diffractive component, the default herwig++ minimum bias model produces a sizeable fraction of events with large gaps, overshooting the measured cross section by up to factor of four in the interval 2 < ∆η F < 7 and producing an enhancement centred around ∆η F = 6. When colour reconnection is switched off, this large gap contribution is reduced considerably, but remains at a similar level to that measured in the range 3 < ∆η F < 5. The enhancement near ∆η F ≈ 6 is still present. The events with zero scatters in the herwig++ underlying event model provide a partial explanation for the large gap contribution. Removing this contribution reduces the predicted large gap cross section, but the non-exponential tail and large ∆η F enhancement persist. For all scenarios considered, the alternative cluster based hadronisation model in herwig++ shows structure which is incompatible with the data.

Large gap sizes and sensitivity to diffractive dynamics
At large ∆η F , the differential cross section exhibits a plateau, which is attributed mainly to diffractive processes (SD events, together with DD events at ξ Y < ∼ 10 −6 ) and is shown in detail in Figure 8. According to the phojet MC model, the CD contribution is also distributed fairly uniformly across this region. Over a wide range of gap sizes with ∆η F > ∼ 3, the differential cross section is roughly constant at around 1 mb per unit of rapidity gap size. Given the close correlation between ∆η F and − ln ξ (Section 3.1), this behaviour is expected as a consequence of the dominance of soft diffractive processes. All MC models roughly reproduce the diffractive plateau, though none gives a detailed description of the shape as a function of ∆η F . When absolutely normalised, the pythia predictions overshoot the data throughout most of the diffractive region, despite the tuning of f D to previous ATLAS data [8] in these models. The excess here is partially a reflection of the 10% overestimate of the pythia prediction in the total inelastic cross section and may also be associated with the large DD cross section in the measured region, which exceeds that expected based on Tevatron data [34] and gives rise to almost equal SD and DD contributions at large ∆η F . For phojet, the underestimate of the diffractive fraction f D is largely compensated by the excess in the total inelastic cross section, such that the large gap cross section is in fair agreement with the measurement up to ∆η F ≈ 6. The DD contribution to the cross section in phojet is heavily suppressed compared with that in the pythia models.
Integrated over the diffractive-dominated region 5 < ∆η F < 8, corresponding approximately to −5.1 < ∼ log 10 (ξ X ) < ∼ −3.8 according to the MC models, the measured cross section is 3.05 ± 0.23 mb, approximately 4% of the total inelastic cross section. This can be compared with 3.58 mb, 3.89 mb and 2.71 mb for the default versions of pythia8, pythia6 and phojet, respectively.  As can be seen in Figure 8, the differential cross section rises slowly with increasing ∆η F for ∆η F > ∼ 5. Non-diffractive contributions in this region are small and fall with increasing ∆η F according to all models, so this rise is attributable to the dynamics of the SD and DD processes. Specifically the rising cross section is as expected from the IPIPIP term in triple Regge models with a Pomeron intercept in excess of unity (see equation (2)). In Figure 8a, a comparison is made with the pythia8 model, after replacing the default Schuler and Sjöstrand Pomeron flux with the Donnachie and Landshoff (DL) version using the default Pomeron trajectory, α IP (t) = 1.085+0.25t ('pythia8 dl'). It is clear that the data at large ∆η F are not perfectly described with this choice.
Whilst the data are insensitive to the choice of α IP ′, there is considerable sensitivity to the value of α IP (0). The data in the cleanest diffractive region ∆η F > 6 are used to obtain a best estimate of the appropriate choice of the Pomeron intercept to describe the data. SD and DD pythia8 samples are generated with the DL Pomeron flux for a range of α IP (0) values. In each case, the default α IP ′ value of 0.25 GeV −2 is taken and the tuned ratios of the SD and DD contributions appropriate to pythia8 from Table 1 are used. The χ 2 value for the best fit to the data in the region 6 < ∆η F < 8 is obtained for each of the samples with different α IP (0) values, with the cross section integrated over the fitted region allowed to float as a free parameter. The optimum α IP (0) is determined from the minimum of the resulting χ 2 parabola.
The full procedure is repeated for data points shifted according to each of the systematic effects described in Section 4, such that correlations between the uncertainties on the data points are taken into account in evaluating the uncertainties. The systematic uncertainty is dominated by the MC model dependence of the data correction procedure, in particular the effect of unfolding using pythia6 in place of pythia8, which leads to a significantly flatter dependence of the data on ∆η F at large gap sizes.
The data are thus compatible with a value of α IP (0) which matches that appropriate to the description of total hadronic cross sections [58,59]. When the Berger-Streng Pomeron flux, which differs from the DL version in the modelling of the t dependence, is used in the fit procedure, the result is modified to α IP (0) = 1.056. The effects of varying α IP ′ between 0.1 GeV −2 and 0.4 GeV −2 and of varying the f SD and f DD fractions assumed in the fit in the ranges given in Section 4 are also smaller than the statistical uncertainty. Compatible results are obtained by fitting the higher p cut T data.
A comparison between the data and a modified version of pythia8, with α IP (0) as obtained from the fit, is shown in Figure 9. Here, the diffractive contribution to the inelastic cross section f D = 25.6% is matched 2 to the fitted value of α IP (0) using the results in [8]. Together with the cross section integrated over the region 6 < ∆η F < 8 as obtained from the fit and the tuned ratio f DD /f SD from Table 1, this fixes the normalisation of the full distribution. The description of the data at large ∆η F is excellent and the exponential fall at small ∆η F is also adequately described. There is a discrepancy in the region 2 < ∆η F < 4, which may be a consequence of the uncertainty in modelling large hadronisation fluctuations in ND events (compare the ND tails to large ∆η F in Figure 8b, c and d). It may also be attributable to sub-leading trajectory exchanges [29,31] or to the lack of a CD component in the pythia model. 2 Since only data at large ∆η F are included in the fit, the result for α IP (0) is insensitive to systematic variations in f D .

The integrated inelastic cross section
By summing over the ∆η F distribution from zero to a maximum gap size ∆η F Cut , the integrated inelastic cross section can be obtained, excluding the contribution from events with very large gaps ∆η F > ∆η F Cut . As discussed in Section 3.1, there is a strong correlation between the size of the gap and the kinematics of diffraction (see e.g. equation (3) for the SD process). The cross section integrated over a given range of gap size can thus be converted into an integral over the inelastic pp cross section down to some minimum value ξ Cut of ξ X . The variation in the integrated inelastic cross section with ∆η F Cut can then be used to compare inelastic cross section results with different lower limits, ξ Cut .
The integral of the forward gap cross section Cut values varying between 3 and 8 by cumulatively adding the cross section contributions from successive bins of the measured gap distribution. The correspondence between maximum gap size and minimum ξ X used here is determined from the pythia8 model to be log 10 ξ Cut = −0.45 ∆η F Cut − 1.52. The uncertainty on this correlation is small; for example the phojet model results in the same slope of −0.45 with an intercept of −1.56. This correlation is applied to convert to an integral 1 ξCut dσ dξ X dξ X . A small correction is applied to account for the fact that the gap cross section neglects particles with 3 p T < p cut T = 200 MeV and includes a contribution from ND processes. This correction factor is calculated using pythia8 with the DL flux, and the optimised α IP (0) and f D values, as determined in Section 5.3. The integration range is chosen such that the correction is always smaller than ±1.3%. The systematic uncertainty on the correction factor, evaluated by comparison with results obtained using phojet or pythia8 with the default Schuler and Sjöstrand flux, together with the systematic variations of the tuned fractions f SD and f DD as in Section 4, is also small. The integrated inelastic cross section is shown as a function of ξ Cut in Figure 10, where it is also compared with a previous ATLAS result [8] and with the TOTEM extraction of the full inelastic cross section [10], derived from a measurement of the elastic cross section via the optical theorem. The errors on all of the experimental data points are dominated by the luminosity uncertainties. The previous ATLAS result was also based on MBTS-triggered data, but is quoted at the ξ X value corresponding to 50% trigger efficiency, which is slightly beyond the range accessed here. Extrapolating according to the measured dependence on ξ Cut , the new data are in good agreement with the previous result, the small apparent difference being well within the uncertainty due to run-to-run luminosity measurement variations.
It is instructive to compare the TOTEM result with the ATLAS measurements, since the latter omit the poorly understood lowest ξ X region. By comparing the lowest ξ Cut data point from the present analysis with the TOTEM measurement and neglecting any correlations between the ATLAS and TOTEM uncertainties, the inelastic cross section integrated over ξ X < 8 × 10 −6 is inferred to be 14.5 +2.0 −1.5 mb. Significantly smaller contributions are predicted by the default versions of pythia (∼ 6 mb) and phojet (∼ 3 mb). Figure 10 also shows two versions of the RMK model (see Section 3.1), corresponding to versions (i) (upper curve) and (ii) (lower curve) in [43]. These versions differ in the radii attributed to the elastically scattered eigenstates comprising the low ξ X contribution which is added to the more standard triple Pomeron 3 The finite p cut T value in the measured gap cross sections tends to increase gap sizes slightly relative to p cut T = 0. However, MC studies indicate that this effect has the biggest influence on the exponentially falling distribution at small gap sizes, whereas the difference for the ∆η F values which are relevant to the integrated cross section are relatively small. According to the MC models, the cross section integrated over 5 < ∆η F < 8 decreases by 2% when changing from p cut T = 200 MeV to p cut T = 0.  Fig. 10: Inelastic cross section excluding diffractive processes with ξ X < ξ Cut , obtained by integration of the differential cross section from gap sizes of zero to a variable maximum. The results from the present analysis ('ATLAS L = 7.1 µb −1 ') are compared with a previous ATLAS result [8] and a TOTEM measurement integrated over all kinematically accessible ξ X values [10]. The predictions of the default versions of the pythia6, pythia8 and phojet models are also shown, along with two versions of the RMK model [43] (see text). The vertical error bars on the ATLAS measurements denote the systematic uncertainties excluding that on the luminosity measurement, whilst the shaded area represents the full systematic uncertainty. For the TOTEM point, the error bar represents the statistical uncertainty whilst the shaded area represents the full uncertainty. The uncertainties on the data points obtained in the present analysis are strongly correlated between neighbouring points. calculation in the model, (ii) being the favoured version and (i) being indicative of the flexibility in the model whilst preserving an acceptable description of pre-LHC data. The additional low ξ X processes enhance the inelastic cross section by 5.5 mb and 6.7 mb in versions (i) and (ii), respectively. Although the RMK model lies below the data in general, the low ξ X enhancement is compatible with that observed [88]. The shape of the distribution at low ξ X is not predicted in the model, but is compatible with the data if, as shown here [89], it is assumed to have the steep ξ X dependence associated with the IPIPIR, rather than the IPIPIP triple Regge term. Similar conclusions have been reached previously from proton-proton [25] and photoproduction [29,30] data.

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The ATLAS Collaboration

Summary
A novel algorithm has been devised for identifying rapidity gaps in the final state of minimum bias ATLAS data, leading to measurements in which particle production is considered down to transverse momentum thresholds p cut T between 200 MeV and 800 MeV. The differential cross section dσ/d∆η F is measured for forward rapidity gaps of size 0 < ∆η F < 8, corresponding to the larger of the two gaps extending to η = ±4.9, with no requirements on activity at |η| > 4.9. An exponentially falling non-diffractive contribution is observed at small gap sizes, which is also a feature of the pythia, phojet and herwig++ Monte Carlo models. However, none of the models describes the ∆η F or p cut T dependence of this region in detail. At large gap sizes, the differential cross section exhibits a plateau, which corresponds to a mixture of the single-diffractive dissociation process and double dissociation with ξ Y < ∼ 10 −6 . This plateau amounts to a cross section close to 1 mb per unit of gap size and its magnitude is roughly described by the pythia and phojet Monte Carlo models. None of the default models reproduce the rise of the differential cross section as a function of gap size at the largest ∆η F values. This rise is interpreted within the triple Pomeron-based approach of the pythia8 model with a Donnachie-Landshoff Pomeron flux in terms of a Pomeron intercept of α IP (0) = 1.058 ± 0.003(stat.) +0.034 −0.039 (syst.). Since the bulk of the inelastic pp cross section is contained within the measured range, integrated cross sections are also obtained and compared with previous measurements. The contribution to the total inelastic cross section from the region ξ X < 10 −5 is determined to be around 20%, which is considerably larger than is predicted by most models.  Table 2: The measured differential cross section data points for p cut T = 200 MeV, with each value corresponding to an average over the given ∆η F range. Also quoted are the percentage statistical (δ stat ) uncertainty and the upward (+δ tot ) and downward (−δ tot ) total uncertainties, obtained by adding all uncertainties in quadrature. The remaining columns contain the percentage shifts due to each of the contributing systematic sources, which are correlated between data points. Those due to the modelling of final state particle production (δ py6 ), the modelling of the ξ X , ξ Y and t dependences (δ pho ) and variation of the CD (δ cd ) cross section in the unfolding procedure are applied symmetrically as upward and downward uncertainties, as are those due to the dead material budget in the tracking region (δ mat ) and the MBTS response (δ mbts ). The uncertainties due to variations in the relative energy scale in data and MC are evaluated separately for upward (δ e+ ) and downward (δ e− ) shifts, as are the modelling uncertainties due to enhancing (δ sd ) or reducing (δ dd ) the σ SD /σ DD cross section ratio. Minus signs appear where the shift in a variable is anti-correlated rather than correlated with the shift in the differential cross section. The 3.4% normalisation uncertainty due to the luminosity measurement is also included in the ±δ tot values. These data points can be obtained from the HEPDATA database [90], along with their counterparts for p cut T = 400, 600 and 800 MeV. A Rivet [91] routine is also available.  22 The ATLAS Collaboration The ATLAS Collaboration