ATLAS Document

A search for a Higgs boson produced via vector-boson fusion and decaying into invisible particles is presented, using 20.3 fb−1 of proton–proton collision data at a centre-of-mass energy of 8 TeV recorded by the ATLAS detector at the LHC. For a Higgs boson with a mass of 125 GeV, assuming the Standard Model production cross section, an upper bound of 0.28 is set on the branching fraction of H → invisible at 95% confidence level, where the expected upper limit is 0.31. The results are interpreted in models of Higgs-portal dark matter where the branching fraction limit is converted into upper bounds on the dark-matter– nucleon scattering cross section as a function of the dark-matter particle mass, and compared to results from the direct dark-matter detection experiments. c © 2015 CERN for the benefit of the ATLAS Collaboration. Reproduction of this article or parts of it is allowed as specified in the CC-BY-3.0 license. ar X iv :1 50 8. 07 86 9v 1 [ he pex ] 3 1 A ug 2 01 5


Introduction
Proton-nucleus (p+A) collisions at the Large Hadron Collider (LHC) [1] provide an opportunity to probe the physics of the initial state of ultra-relativistic heavy-ion (A+A) collisions without the obscuring effects of thermalisation and collective evolution [2]. In particular, p+A measurements can provide insight into the effect of an extended nuclear target on the dynamics of soft and hard scattering processes and subsequent particle production. Historically, measurements of the average charged-particle multiplicity as a function of pseudorapidity, dN ch /dη, where pseudorapidity is defined as η = − ln tan(θ/2) with θ the particle angle with respect to the beam direction, have yielded important insight into soft particle production dynamics in proton-and deuteron-nucleus (p/d+A) collisions [3][4][5][6][7][8] and provided essential tests of models of inclusive soft hadron production.
Additional information is obtained if measurements of the charged-particle multiplicities are presented as a function of centrality, an experimental quantity that characterises the p/d+A collision geometry. Previous measurements in d+Au collisions at the Relativistic Heavy Ion Collider (RHIC) [9] have characterised the centrality using particle multiplicities at large pseudorapidity, either symmetric around midrapidity [10] or in the Au fragmentation direction [11]. These measurements have shown that the rapidityintegrated particle multiplicity in d+Au collisions scales with the number of inelastically interacting, or "participating", nucleons, N part . This scaling behaviour has been interpreted as the result of coherent multiple soft interactions of the projectile nucleon in the target nucleus, and is known as the woundednucleon (WN) model [12]. The charged-particle multiplicity distributions as a function of pseudorapidity measured in central d+Au collisions are asymmetric and peaked in the Au-going direction [7]. This observation has been explained using well-known phenomenology of soft hadron production [13].
There are alternative descriptions of the centrality dependence of the dN ch /dη distribution in d+Au collisions at RHIC [14,15] and p+Pb collisions at the LHC [15,16] based on parton saturation models. Measurements of the centrality dependence of dN ch /dη distributions in p+Pb collisions provide an essential test of soft hadron production mechanisms at the LHC. Such tests have become of greater importance given the observation of two-particle [17][18][19][20] and multi-particle [20,21] correlations in the final state of p+Pb collisions at the LHC. These correlations are currently interpreted as resulting from either initialstate saturation effects [15,22,23] or from the collective dynamics of the final state [24][25][26][27][28]. For either interpretation, information on the centrality dependence of dN ch /dη can provide important input for determining the mechanism responsible for these structures.
Recent measurements from the ALICE experiment [29] show behaviour in the centrality dependence of the charged-particle pseudorapidity distributions, which is qualitatively similar to that observed at RHIC. That analysis compared different methods for characterising centrality and suggested that the method used to define centrality may have a significant impact on the centrality dependence of the measured dN ch /dη distribution.
An important component of any centrality-dependent analysis is the geometric model used to relate experimental observables to the geometry of the nuclear collision. Glauber Monte Carlo models [30], which simulate the interactions of the incident nucleons using a semi-classical eikonal approximation, have been successfully applied to many different A+A measurements at RHIC and the LHC. A key parameter of such models is the inelastic nucleon-nucleon cross-section, which is taken to be 70 mb for this analysis [29]. However, the Glauber multiple-scattering approximation assumes that the nucleons remain on the mass shell between successive scatterings, and this assumption is badly broken in ultra-relativistic collisions. Corrections to the Glauber model [31], hereafter referred to as "Glauber-Gribov," are needed to account for the off-shell propagation of the nucleons between collisions.
A particular implementation of the Glauber-Gribov approach is provided by the colour-fluctuation model [32][33][34][35]. That model accounts for event-to-event fluctuations in the configuration of the incoming proton that are assumed to be frozen over the timescale of a collision and that can change the effective crosssection with which the proton scatters off nucleons in the nucleus. These event-by-event fluctuations in the cross-section can be represented by a probability distribution P(σ). The width of that distribution can be characterised by a parameter ω σ , which is the relative variance of the σ distribution, ω σ ≡ (σ/σ tot − 1) 2 . The usual total cross-section, σ tot , is the event-averaged cross-section, or, equivalently, the first moment of the P(σ) distribution, σ tot = model [30] and the GGCF model [34,35] with ω σ = 0.11 and 0.2 are used to estimate N part for each centrality interval, allowing a measurement of the N part dependence of the charged-particle multiplicity.
The paper is organised as follows. Section 2 describes the subdetectors of the ATLAS experiment relevant for this measurement. Section 3 describes the event selection. Section 4 describes the Monte Carlo simulations used to understand the performance and derive the corrections to the measured quantities. Section 5 describes the choice of centrality variable. Section 6 describes the measurement of the charged-particle multiplicity and Sect. 7 describes the estimation of the systematic uncertainties. Section 8 presents the results of the measurement, and the interpretation of the yields of charged particles per participant is discussed in Sect. 9. Section 10 concludes the paper. The estimation of the geometric parameters in each centrality interval for the Glauber and GGCF models is presented in detail in the Appendix.

Experimental setup
The ATLAS detector is described in detail in Ref. [36]. The data selection and analysis presented in this paper is performed using the ATLAS inner detector (ID), calorimeters, minimum-bias trigger scintillators (MBTS), and the trigger system The inner detector measures charged-particle tracks using a combination of silicon pixel detectors, silicon microstrip detectors (SCT), and a straw-tube transition-radiation tracker (TRT), all immersed in a 2 T axial magnetic field. The pixel detector is divided into "barrel" and "endcap" sections. For collisions occurring at the nominal interaction point, 1 the barrel section of the pixel detector allows measurements of charged-particle tracks over |η| < 2.2. The endcap sections extend the detector coverage, spanning the pseudorapidity interval 1.6 < |η| < 2.7. The SCT and TRT detectors cover |η| < 2.5 and |η| < 2, respectively, also through a combination of barrel and endcap sections.
The barrel section of the pixel detector consists of three layers of staves at radii of 50.5 mm, 88.5 mm, and 122.5 mm from the nominal beam axis, and extending ±400.5 mm from the centre of the detector in the z direction. The endcap consists of three disks placed symmetrically on each side of the interaction region at z locations of ±493 mm, ±578 mm and ±648 mm from the centre of the detector. All pixel sensors in the pixel detector, in both the barrel and endcap regions, are identical and have a nominal size of 50 µm × 400 µm.
The MBTS detect charged particles in the range 2.1 < |η| < 3.9 using two hodoscopes, each of which is subdivided into 16 counters positioned at z = ±3.6 m. The ATLAS calorimeters cover the full azimuth and the pseudorapidity range |η| < 4.9 with the forward part (FCal) consisting of two modules positioned on either side of the interaction region and covering 3.1 < |η| < 4.9. The FCal modules are composed of tungsten and copper absorbers with liquid argon as the active medium, which together provide 10 interaction lengths of material.
The LHC delivered its first proton-nucleus collisions in a short p+Pb "pilot" run at √ s NN = 5.02 TeV in September 2012. During that run the LHC was configured with a clockwise 4 TeV proton beam and an anti-clockwise 1.57 TeV per-nucleon Pb beam that together produced collisions with a nucleon-nucleon centre-of-mass energy of √ s NN = 5.02 TeV and a longitudinal rapidity boost of 0.465 units with respect to 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2). the ATLAS laboratory frame. Following a common convention used for p+A measurements, the rapidity is taken to be positive in the direction of the proton beam, i.e. opposite to the usual ATLAS convention for pp collisions. With this convention, the ATLAS laboratory frame rapidity y and the p+Pb centre-of-mass system rapidity y cm are related as y cm = y − 0.465.

Event selection
Minimum-bias p+Pb collisions were selected by a trigger that required a signal in at least two MBTS counters. The p+Pb events selected for analysis are required to have at least one hit in each side of the MBTS, a difference between the times measured in the two MBTS hodoscopes of less than 10 ns, and a reconstructed collision vertex in longitudinal direction, z vtx , within 175 mm of the nominal centre of the ATLAS detector. Collision vertices are defined using charged-particle tracks reconstructed by an algorithm optimised for pp minimum-bias measurements [38]. Reconstructed vertices are required to have at least two tracks with transverse momentum p T > 0.4 GeV. Events containing multiple p+Pb collisions are rare due to very low instantaneous luminosity during the pilot run and are further suppressed in the analysis by rejecting events with two collision vertices that are separated in z by more than 15 mm. Applying this selection reduces the fraction of events with multiple collisions from less than 0.07% to below 0.01%.
To remove potentially significant contributions from electromagnetic and diffractive processes, the topology of the events was first analysed in a manner similar to that performed in a measurement of rapidity gap cross-sections in 7 TeV proton-proton collisions [39]. The pseudorapidity coverage of the calorimeter, −4.9 < η < 4.9, is divided into ∆η = 0.2 intervals, and each interval containing one or more clusters with p T greater than 0.2 GeV is considered as occupied. To suppress the contributions from noise, clusters are considered only if they contained at least one cell with an energy at least four times the standard deviation of the cell noise distribution.
Then, the edge-gap on the Pb-going side of the detector is calculated as the distance in pseudorapidity between the detector edge η = −4.9 and the nearest occupied interval. Events with edge-gaps larger than two units of pseudorapidity typically result from electromagnetic or diffractive excitation of the proton and are removed from the analysis. The effect of this selection is identical to the requirement of a cluster with transverse energy E T > 0.2 GeV to be present in the region η < −2.9. No requirement is imposed on edge-gaps on the proton-going side. The gap requirement removes, with good efficiency, a sample of events which are not naturally described in a Glauber picture of p+Pb collisions. Of the events passing the vertex and MBTS cuts, this requirement removes a further fraction f gap = 1% of the events, yielding a total of 2.1 million events for this analysis. The result of this event selection is to isolate a fiducial class of p+Pb events, defined as inelastic p+Pb events that have a suppressed contribution from diffractive proton excitation events.

Monte Carlo simulation
The response of the ATLAS detector and the performance of the charged-particle reconstruction algorithms are evaluated using one million minimum-bias 5.02 TeV Monte Carlo (MC) p+Pb events, produced by version 1.38b of the Hijing event generator [40] with diffractive processes disabled. The fourmomentum of each generated particle is longitudinally boosted by a rapidity of 0.465 to match the beam conditions in the data. The detector response to these events is fully simulated using Geant4 [41,42]. The resulting events are digitised using conditions appropriate for the pilot p+Pb run and fully reconstructed using the same algorithms that are applied to the experimental data. This MC sample is primarily used to evaluate the efficiency of the ATLAS detector for the charged-particle measurements.
The detector response and event selection efficiencies for peripheral and diffractive p+Pb events have properties similar to those for inelastic or diffractive pp collisions, respectively. To evaluate these responses and efficiencies, the pp samples are generated at √ s = 5.02 TeV with particle kinematics boosted to match the p+Pb beam conditions. Separate samples of minimum-bias, single-diffractive, and doublediffractive pp collisions with one million events each are produced using both Pythia6 [43] (version 6.425, AMBT2 parameter set (tune) [44], CTEQ6L1 PDF [45]) and Pythia8 [46] (version 8.150, 4C tune [47], MSTW2008LO PDF [48]), and simulated, digitised and reconstructed in the same manner as the p+Pb events. These six samples are primarily used for the Glauber model analysis described in the Appendix.

Centrality selection
For Pb+Pb collisions, the ATLAS experiment uses the total transverse energy, E T , measured in the two forward calorimeter sections to characterise the collision centrality [49]. However, the intrinsic asymmetry of the p+Pb collisions and the rapidity shift of the centre-of-mass causes an asymmetry in the soft particle production measured on the two sides of the calorimeter. Figure 1 shows the correlation between the summed transverse energies measured in the proton-going (3.1 < η < 4.9) and Pb-going (−4.9 < η < −3.1) directions, E p T and E Pb T , respectively. The transverse energies are evaluated at an energy scale calibrated for electromagnetic showers and have not been corrected for hadronic response. that E p T is less sensitive than E Pb T to the increased particle production expected to result from multiple interactions of the proton in the target nucleus in central collisions. Thus, E Pb T is chosen as the primary quantity used to characterise p+Pb collision centrality for the measurement presented in this paper.
The distribution of E Pb T for events passing the p+Pb analysis selection is shown in Fig. 2. The following centrality intervals are defined in terms of percentiles of the E Pb T distribution: 0-1%, 1-5%, 5-10%, 10-20%, 20-30%, 30-40%, 40-60%, and 60-90%. The E Pb T ranges corresponding to these centrality intervals are indicated by the alternating filled and unfilled regions in Fig. 2, with the 0-1% interval, containing the most central collisions, being rightmost. Since the composition of the events in the most peripheral 90-100% interval is not well constrained, these events are excluded from the analysis. The definitions of the centrality intervals accounted for a (2±2)% inefficiency as described in the Appendix for the fiducial class of p+Pb events defined above to pass the applied event selections. While the inefficiency is confined to the 90-100% interval, it influences the E Pb T ranges associated with each centrality interval. Potential hard scattering contributions to E Pb T have been evaluated in a separate analysis [50] by explicitly subtracting the contributions from reconstructed jets that fall partly or completely in the Pb-going FCal acceptance. That analysis showed negligible impact from hard scattering processes on the measured E Pb T distribution.  To test the sensitivity of the results of this analysis to the choice of pseudorapidity interval used for the E T measurement, two alternative E T quantities are defined. The first, E η<−4 T , is defined as the total transverse energy in FCal cells with η < −4.0. The second, E 3.6<|η cm |<4.4 T , is defined as the total transverse energy in the two intervals 4.0 < η < 4.9 and −4.0 < η < −3.1. These intervals are approximately symmetric when expressed in pseudorapidity in the centre of mass system η cm . The first of these alternatives is used to evaluate the potential auto-correlation between the measured charged-particle multiplicities and the centrality observable by increasing the rapidity gap between the two measurements. The second is used to evaluate the differences between an asymmetric (Pb-going) and symmetric (both sides) centrality observable. The effect of these alternative definitions is discussed in Sect. 8.
The Glauber analysis [30] was applied to estimate N part for each of the centrality intervals used in this analysis. A detailed description is given in the Appendix; only a brief summary of the method is given here. The PHOBOS MC program [51] was used to simulate the geometry of inelastic p+Pb collisions using both the standard Glauber and GGCF models. The resulting N part distributions are convolved with a model of the N part -dependent E Pb T distributions, the parameters of which are obtained by fitting the measured E Pb T distribution. The average N part associated with each centrality interval is obtained with systematic uncertainties. The results are shown in Fig. 3 for the Glauber model and for the GGCF model with ω σ = 0.11 and 0.2. 6 Measurement of charged-particle multiplicity 6

.1 Two-point tracklet and pixel track methods
The measurement of the charged-particle multiplicity is performed using only the pixel detector to maximise the efficiency for reconstructing charged particles with low transverse momenta. Two approaches are used in this analysis. The first is the two-point tracklet method commonly used in heavy-ion collision experiments [37,52,53]. Two variants of this method are implemented in this analysis to construct the dN ch /dη distribution and to estimate the systematic uncertainties, as described below. The second method uses "pixel tracks", obtained by applying the full track reconstruction algorithm [54] only to the pixel detector. The pixel tracking is less efficient than the tracklet method as is justified later in the text, but provides measurements of the particle p T . The dN ch /dη distribution measured using pixel tracks provides a cross-check on the primary measurement that is performed using the two-point tracklets.
In the two-point tracklet algorithm, the event vertex and clusters [55] on an inner pixel layer define a search region for clusters in the outer layers. The algorithm uses all clusters, except the clusters which have low energy deposits inconsistent with minimum-ionising particles originating from the primary vertex. The algorithm also rejects duplicate clusters resulting from the overlap of the pixel sensors or arising from a small set of pixels at the centre of the pixel modules that share readout channels [56]. Two clusters in a given layer of the pixel detector are considered as one if they have an angular separation (δφ) 2 + (δη) 2 < 0.02. Such clusters likely result from the passage of a single particle.
The pseudorapidity and azimuthal angle of the cluster in the innermost layer (η, φ) and their differences between the outer and inner layers (∆η, ∆φ) are taken as the parameters of the reconstructed tracklet. The ∆η of a tracklet is largely determined by the multiple scattering of the incident particles in the material of the beam pipe and detector. This effect plays a less significant role in the ∆φ of a tracklet, which is driven primarily by the bending of charged particles in the magnetic field, and hence one expects ∆φ to be larger. The tracklet selection cuts are: Keeping tracklets with |∆φ| < 0.1 corresponds to accepting particles with p T 0.1 GeV. The selection in Eq.
(2) accounts for the momentum dependence of charged-particle multiple scattering.
The Monte Carlo simulation for the dN ch /dη analysis is based on the Hijing event generator, which is described in Sect. 4. The Hijing event generator is known to not accurately reproduce the measured particle p T distributions. This is addressed by reweighting the Hijing p T distribution using the ratio of reconstructed spectra measured with the pixel track method in the data and in the MC simulation. The reweighting function is extrapolated below p T = 0.1 GeV and applied to all generated particles and their decay products. This is done in intervals of centrality and pseudorapidity. Generator-level primary particles are defined as particles with a mean lifetime τ > 0.3 · 10 −10 s either directly produced in p+Pb interactions or from subsequent decays of particles with a shorter lifetime. All other particles are defined as secondaries. Tracklets are classified as primary or secondary depending on whether the associated generator-level charged particle is primary or secondary. Association between the tracklets and the generator-level particles is based on the Geant4 information about hits produced by these particles. Tracklets that are formed from the random association of hits produced by unrelated particles, or hits in the detector which are not matched to any generated particle are referred to as "fake" tracklets.
The contribution of fake tracklets is relatively difficult to model in the simulation, because of the a priori unknown contributions of multiple sources, such as noisy clusters or very low energy particles. To address this problem, the tracklet algorithm is used in two different implementations referred to as "Method 1" and "Method 2". In Method 1, at most one tracklet is reconstructed for each cluster on the first pixel layer. If multiple clusters on the second pixel layer fall within the search region, the resulting tracklets are merged into a single tracklet. This approach reduces, but does not eliminate, the contribution of fake tracklets that are then accounted for using an MC-based correction. Method 2 reconstructs tracklets for all combinations of clusters in only two pixel layers, the innermost and the next-to-innermost detector layers. To account for the fake tracklets arising from random combinations of clusters, the same analysis is performed after inverting the x and y positions of all clusters on the second layer with respect to the . The tracklet yield from this "flipped" analysis, N fl tr , is then subtracted from the original tracklet yield, N ev tr to obtain an estimated yield of true tracklets N tr , Distributions of ∆η and ∆φ of reconstructed tracklets using Method 1 for data and simulated events are shown in Fig. 4 for the barrel (upper plots) and endcap (lower plots) parts of the pixel detector. The simulation results show the three contributions from primary, secondary and fake tracklets. The selection criteria specified by Eq. (1) are shown in Fig. 4 as vertical lines and applied in ∆φ for ∆η plots and vice versa. Outside those lines, the contributions from secondary and fake tracklets are more difficult to take into account, especially in the endcap region. These contributions partially arise from low-p T particles on  spiral trajectories and their description in the MC simulation is therefore very sensitive to the amount of detector material. The ratio between simulation and the data is also shown for each plot. These ratios are closer to unity in the barrel region than in the endcap region, where they deviate by up to 5% except at very low |∆φ|. At low |∆φ| corresponding to high p T , the MC deviates from the data even after reweighing procedure based on pixel tracks. This is due to low resolution of pixel track at high p T , however, the contribution of high-p T particles to dN ch /dη is small.
The top left panel of Fig. 5 shows the pseudorapidity distribution of tracklets reconstructed with Method 2 and satisfying the criteria of Eqs. (1) and (2) in the 0-10% centrality interval for data (markers) and for the simulation (lines). The results of flipped reconstruction are also shown in the plot. Data and MC In the 0-10% centrality interval, the Method 2 fake tracklet contribution amounts to 8% of the yield at midpseudorapidity and up to 16% at large pseudorapidity. In the same centrality interval, the fake tracklet contribution using Method 1 (pixel tracks) varies from 2% to 10% (0.2% to 1.5%). The multiplicity measurements using Method 1 and the pixel track method rely on the MC simulation to correct for the contribution from fake tracks, and the results from all three methods rely on the MC simulation to correct for the contribution of secondary particles.

Extraction of the charged-particle distribution
The data analysis and corresponding corrections are performed in eight intervals of detector occupancy (O) parameterised using the number of reconstructed clusters in the first pixel layer and chosen to cor- respond to the eight p+Pb centrality intervals, and in seven intervals of z vtx , each 50 mm wide. For each analysis method, a set of multiplicative correction factors is obtained from MC simulations according to Here, N pr and N rec represent the number of primary charged particles at the generator level and the number of tracks or tracklets at the reconstruction level, respectively. These correction factors account for several effects: inactive areas in the detector and reconstruction efficiency, contributions of residual fake and secondary particles, and losses due to track or tracklet selection cuts including particles with p T below 0.1 GeV. They are evaluated as a function of O, z vtx , and η both for the fiducial region, p T > 0.1 GeV, and for full acceptance, p T > 0 GeV. The results are presented in η-intervals of 0.1 unit width. Due to the excellent η-resolution of the tracklets, as seen from Fig. 4, migration of tracklets between neighbouring bins is negligible.
The fully corrected, per-event charged-particle pseudorapidity distributions are calculated according to where ∆N tr indicates either the number of reconstructed pixel tracks or two-point tracklets, and N evt (z vtx ) is the number of analysed events in the intervals of the primary vertex along the z direction. The sum in Eq. (5) runs over primary vertex intervals, the number of which varies from seven for |η| < 2.2 for twopoint tracklets and |η| < 2 for pixel tracks to two at the edges of the measured pseudorapidity range of |η| < 2.7 for two-point tracklets and |η| < 2.5 for pixel tracks respectively. The primary vertex intervals used in the analysis are chosen such that C(O, z vtx , η) does not vary by more than 20% over the included bins. By construction, the correction factors for each centrality were determined using only the corresponding occupancy bin.  Fig. 6 show the reconstructed distribution from the data and the filled markers are the corresponding distribution for the three methods after applying corrections derived from the simulation. The lower panel shows the ratio of the results obtained from Method 2 and the pixel track method to those obtained using Method 1. The three methods agree within 2% in the barrel region of the detector and within 3% in the endcap region. This agreement demonstrates that the rejection of fake track or tracklets and the correction procedure are well understood. For this paper, Method 1 is chosen as the default result for dN ch /dη, Method 2 is used when evaluating systematic uncertainties, and the pixel track method is used primarily as a consistency test, as discussed in detail below.

Systematic uncertainties
The systematic uncertainties on the dN ch /dη measurement arise from three main sources: inaccuracies in the simulated detector geometry, sensitivity to selection criteria used in the analysis including the residual contributions of fake tracklets and secondary particles, and differences between the generated particles used in the simulation and the data. The different components of the systematic uncertainty are described below. To determine the systematic uncertainties, the analysis is repeated in full for different variations of parameters or methods and the results are compared to the standard Method 1 results. A summary of the results are presented in Table 1.
The uncertainty due to the simulated detector geometry arises primarily from the details of the pixel detector acceptance and efficiency. The locations of the inactive pixel modules are matched between the data and simulation. Areas smaller than a single module that are found to have intermittent inefficiencies are estimated to contribute less than 1.7% uncertainty to the final result. This uncertainty has no centrality dependence, and is approximately independent of pseudorapidity.
The amount of inactive detector material in the tracking system is known with a precision of 5% in the central region and up to 15% in the forward region. In order to study the effect on the tracking efficiency, samples generated with increased material are used. The net effect on the final result is found to be 0.5-3% independent of centrality.
Uncertainties due to tracklet selection cuts are evaluated by independently varying the cuts on |∆η| and |∆φ| up and down by 40%. The effect of these variations is less than 1%, except at large values of |η| where it is 1.5%, and has only a weak centrality dependence.
The systematic uncertainty due to applying the p T reweighting procedure to the generated particles is taken from the difference in dN ch /dη between applying and not applying the reweighting procedure. The uncertainty is less than 0.5% for |η| < 1.5 and grows to 3.0% towards the edges of the η acceptance. The uncertainty has a centrality dependence because the p T distributions in central and peripheral collisions are different.
Tracklets are reconstructed using Method 1 for particles with p T > 0.1 GeV. The unmeasured region of the spectrum contributes approximately 6% to the final dN ch /dη distribution. The systematic uncertainty on the number of particles with p T ≤ 0.1 GeV is partially included in the variation of the tracklet ∆φ selection criteria. An additional uncertainty is evaluated by varying the shape of the spectra below 0.1 GeV. This uncertainty is estimated to be as much as 2.5% at large values of |η| and has a weak centrality dependence.
To test the sensitivity to the particle composition in Hijing, the fraction of pions, kaons and protons in Hijing are varied within a range based on measured differences in particle composition between pp and Pb+Pb collisions [57,58]. The resulting changes in dN ch /dη are found to be less than 1% for all centrality intervals.
Systematic uncertainties due to the fake tracklets are estimated by comparing the results of the two tracklet methods. The differences in the most central collisions are found to vary with pseudorapidity from 1.5% in the barrel region to about 2.5% at the ends of the measured pseudorapidity range. The uncertainty associated with the event selection efficiency for the fiducial class of p+Pb events is evaluated by defining new E Pb T centrality ranges after accounting for an increase (decrease) in the efficiency by 2% and repeating the full analysis. This resulting change of the dN ch /dη distribution is less than 0.5% in central collisions; it increases to 6% in peripheral collisions.
Systematic uncertainties depend on the pseudorapidity and the centrality intervals and are correlated between different points. The impact in different regions of pseudorapidity and centrality are shown in different columns of Table 1. Uncertainties coming from different sources and listed in the same column are treated as independent. The resulting total systematic uncertainty shown in the lower line of the table is the sum in quadrature of the individual contributions. the dN ch /dη distribution measured in the fiducial acceptance of the ATLAS detector, detecting particles with p T > 0.1 GeV. The results for the dN ch /dη distribution with p T > 0 GeV are shown in the right panel of Fig. 7. The charged-particle pseudorapidity distribution increases by typically 5%, consistent with extrapolation of spectra measured in pp collisions to zero p T [38]. At the edges of the measured pseudorapidity interval, it increases dN ch /dη by 11%.

Results
In the most peripheral collisions with a centrality of 60-90%, the dN ch /dη distribution has a doublypeaked shape similar to that seen in pp collisions [38,59]. In collisions that are more central, the shape of dN ch /dη becomes progressively more asymmetric, with more particles produced in the Pb-going direction than in the proton-going direction. To investigate further the centrality evolution, the dN ch /dη distributions in each centrality interval are divided by the dN ch /dη distribution for the 60-90% interval. The results are shown in Fig. 8, where the double-peak structure disappears in the ratios. The ratios are observed to grow nearly linearly with decreasing pseudorapidity, with a slope whose magnitude increases from peripheral to central collisions. In the 0-1% centrality interval, the ratio changes by almost a factor of two over the measured η-range. The greatest increase in multiplicity between adjacent centrality intervals occurs between the 1-5% and 0-1% intervals. Averaged over the η-interval of the measurement, the dN ch /dη distribution increases by more than 25% between the 1-5% and 0-1% intervals.
In addition to the results presented in Figs. 7 and 8, the dN ch /dη measurement is repeated using the alternative definitions of the event centrality variables defined in Sect. 5. Figure 9 demonstrates the sensitivity  centrality definition, the dN ch /dη distributions change in an approximately η-independent fashion by −3% and +3% for the 0-1% and 60-90% intervals, respectively. The dN ch /dη distributions in the other centrality intervals change in a manner that effectively interpolates between these extremes. As a result, the increase in dN ch /dη between the most peripheral and most central collisions would be reduced by 6% relative to the nominal measurement. Using the symmetric, E 3.6<|η cm |<4.4 T centrality definition, the dN ch /dη distribution in each interval changes in an η-dependent way such that the ratio is consistent with a linear function of η. In particular, the dN ch /dη values in the different centrality intervals move closer together at η = −2.7 and separate at η = +2.7. The change is at most 6% at the ends of the η range in the most central and most peripheral centrality intervals, and smaller elsewhere. Thus, for the symmetric centrality selection the ratios in Fig. 8 for the 0-1% bin would increase by 9% at η = 2.7, and decrease by 6% at η = −2.7. Generally, the alternative centrality definitions considered in this analysis yield no qualitative and only modest quantitative changes in the centrality dependence of the dN ch /dη distributions. These variations should not be considered a systematic uncertainty on the dN ch /dη measurement but do indicate that the particular centrality method used in the analysis must be accounted for when interpreting the results of the measurement. Figure 10 shows a comparison, where possible, of the measurements presented in this paper to results from the ALICE experiment [29] using the "V0A" centrality selection, which is based on the detector covering the pseudorapidity region −5.1 < η < −2.8, similar to the E Pb T -based selection used in this measurement. The ATLAS results for 0-1% and 1-5% centrality intervals are combined to match the ALICE experiment result for 0-5% interval. Similarly, the 20-30% and 30-40% intervals are combined to match the ALICE experiment result for 20-40% interval. The results from the two experiments are consistent with each other.  Figure 10: Charged-particle pseudorapidity distribution dN ch /dη measured in different centrality intervals compared to similar results from the ALICE experiment [29] using the "V0A" centrality selection. The ATLAS centrality intervals have been combined, where possible, to match the ALICE centrality selections.

Particle multiplicities per participant pair
A common way of representing the centrality dependence of particle yields in A+A and p+A collisions is by showing the yield per participant or per participant pair, N part /2, which is determined for each centrality interval and each geometrical model as shown in Fig. 3. Figure 11 shows dN ch /dη per participant pair for the most central and most peripheral intervals of centrality measured in the analysis as a function of η for three different models of the collisions geometry: the standard Glauber model and the GGCF model with ω σ = 0.11 and 0.2 in the top, middle and lower panels, respectively. The results for the most Figure 11: Charged-particle pseudorapidity distribution dN ch /dη per pair of participants as a function of η for 0-1% and 60-90% centrality intervals for the three models used to calculate N part . The standard Glauber calculation is shown in the top panel, the GGCF model with ω σ = 0.11 in the middle and ω σ = 0.2 is in the lowest panel. The bands shown with thin lines represent the systematic uncertainty of the dN ch /dη measurement, the shaded bands indicate the total systematic uncertainty including the uncertainty on N part . Statistical uncertainties, shown with vertical bars are typically smaller than the marker size. peripheral (60-90%) centrality interval, shown with circles, are similar between all three panels. This is due to relatively small difference between the calculations of N part for Glauber and GGCF models in this centrality interval. The shape of the distribution indicates more abundant particle production in the proton-going direction in comparison to the Pb-going. This can be explained by the higher energy of the proton, compared to the nucleon energy in lead in the laboratory system. In the most central collisions (0-1%), shown with diamond markers in all three panels, this trend is reversed. Conversely, the magnitude of dN ch /dη per participant pair strongly depends on the geometric model used to calculate N part . The point at which the central and peripheral scaled distributions cross each other also depends on the choice of geometric model. Figure 12 shows the dN ch /dη distribution per participant pair as a function of N part for the three different models of the collisions geometry. Since the charged-particle yields have significant pseudorapidity dependence, dN ch /dη/( N part /2) is presented in five η intervals including the full pseudorapidity interval, −2.7 < η < 2.7. In the region 0 < η < 1, the dN ch /dη distribution is consistent with an empirical fit to inelastic pp data that suggest dN ch /dη increases with centre-of-mass energy, √ s, as ∝ s 0.10 [16]. . The open boxes represent the systematic uncertainty of the dN ch /dη measurement only, and the width of the box is chosen for better visibility (they are not shown for −1.0 < η < 0 and 0 < η < 1). The shaded boxes represent the total uncertainty (they are shown only on −2.7 < η < 2.7 interval for visibility) which is dominated by the uncertainty of the N part given in Table 4 and Fig. 3. This uncertainty is asymmetric due to the asymmetric uncertainties on N part . The statistical uncertainties are smaller than the marker size for all points.
The dN ch /dη/( N part /2) values from the standard Glauber model are approximately constant up to N part ≈ 10 and then increase for larger N part . This trend is absent in the GGCF model with ω σ = 0.11, which shows a relatively constant behaviour for the integrated yield divided by the number of participant pairs. The dN ch /dη/( N part /2) values from the GGCF model with ω σ = 0.2 show a slight decrease with N part in all η intervals.
The presence or absence of N part scaling does not in itself suggest a preference for one or another of the geometric models. However, this study emphasises that considering fluctuations of the nucleon-nucleon cross-section in the GGCF model may lead to significant changes in the N part scaling behaviour of the p+Pb dN ch /dη data and, thus, their interpretations.

Conclusions
This paper presents a measurement of the centrality dependence of the charged-particle pseudorapidity distribution, dN ch /dη, measured in approximately 1 µb −1 of p+Pb collisions at a nucleon-nucleon centreof-mass energy of √ s NN =5.02 TeV collected by the ATLAS detector at the LHC. The fully corrected measurements are presented for the fiducial acceptance of the ATLAS detector (p T > 0.1 GeV) and in the full acceptance (p T > 0 GeV). The dN ch /dη distributions are presented as a function of pseudorapidity over the range −2.7 < η < 2.7 and as a function of collision centrality for the 0-90% p+Pb collisions. The centrality is characterised using the energy deposited in the forward calorimeter covering −4.9 < η < −3.1 in the Pb-going direction.
The shape of dN ch /dη evolves gradually with centrality from an approximately symmetric shape in the most peripheral collisions to a highly asymmetric distribution in the most central collisions. The ratios of dN ch /dη measured in different centrality intervals to the dN ch /dη distribution in the most peripheral interval are approximately linear in η with a slope that is strongly dependent on centrality. It is noteworthy that the greatest increase in charged-particle multiplicity between successive centrality bins occurs between the 1-5% and 0-1% centrality bins.
The results are also interpreted using models of the underlying collision geometry. The average number of participants in each centrality interval, N part , is estimated using a standard Glauber model Monte Carlo simulation with a fixed nucleon-nucleon cross-section, as well as with two Glauber-Gribov colour fluctuation models which allow the nucleon-nucleon cross-section to fluctuate event-by-event. The N part dependence of dN ch /dη/( N part /2) is found to be sensitive to the modelling of the p+Pb collision geometry, especially in the most central collisions: while the standard Glauber modelling leads to a strong increase in the multiplicity per participant pair for collisions in the centrality range (0-30)% the GGCF model produces a much milder centrality dependence.
These results point to the importance of understanding not just the initial state of the nuclear wave function, but also the fluctuating nature of nucleon-nucleon collisions themselves.

Appendix: Glauber model analysis
The PHOBOS Glauber MC program [51] is used to perform the standard Glauber model calculations used in this analysis. The Pb nucleon density is taken to be a Woods-Saxon distribution with radius and skin depth parameters, R = 6.62 fm and a = 0.546 fm [60], respectively. The nucleon-nucleon inelastic cross-section is taken to be 70 mb. The resulting probability distribution, P(N part ), of the number of participating nucleons N part -nucleons that undergo at least one hadronic scattering during the p+Pb collision -is shown in Fig. 13. The GGCF model is implemented in a modified version of the PHOBOS MC program. Following Ref. [34], the probability distribution to find the nucleons in a configuration having a nucleon-nucleon scattering cross-section, σ, is taken to be Here, ρ is a normalisation constant, Ω controls the width of the P(σ) distribution, and σ 0 determines the configuration-averaged total cross-section σ tot ≡ σ . The inelastic cross-section, σ NN , is taken to be a constant fraction, λ, of the total cross-section [35] so the probability distribution of σ NN is given by The values used in this analysis for Ω, σ 0 , σ tot and λ corresponding to ω σ = 0.11, 0.2 are shown in Table 2. The σ tot values differ because the analyses leading to the two estimates for ω σ were done at different times. The first analysis yielding ω σ = 0.11 [34] assumed σ tot = 86 mb, consistent with the Donnachie and Landshoff [61] parameterisation of σ tot (s). The second analysis yielding ω σ = 0.2 used the results of recent measurements of the pp total cross-section at the LHC [62] to set σ tot = 94.8 mb.
Modifying the parameters for the ω σ = 0.11 case to be consistent with this improved knowledge of σ tot produces a negligible change in the resulting P(σ) distribution. The values for λ are chosen to produce the above-quoted nucleon-nucleon inelastic cross-section of 70 mb. The GGCF P H (σ NN ) distributions are shown in the inset of Fig. 13, while the resulting P(N part ) distributions are shown in the main panel of the figure.
To connect an experimental measurement of collision centrality such as E Pb T to the results of the Glauber or GGCF Monte Carlo simulation, a model for the N part dependence of the E Pb T distribution is required. The usual basis for models previously applied to A+A and p/d+A collisions is the WN model [12], which predicts that the average E Pb T increases proportionally to N part with the proportionality constant equal to one half the corresponding average FCal E T in pp collisions. Under the WN model, the E Pb T distribution for fixed N part would be obtained from a N part -fold convolution of the corresponding distribution in pp collisions. This convolution is straightforward if the E T distribution in pp collisions is described by a gamma distribution [63] since gamma distributions have the property that an N-fold convolution of a gamma distribution with parameters k and θ yields another gamma distribution with the same θ and a modified k parameter, k = Nk.
Attempts to fit the measured E Pb T distribution using pure WN-convolved gamma distributions and the Glauber N part distribution yield unphysical results for the nucleon-nucleon parameters, k 0 and θ 0 , when those parameters are free parameters of the fit. In particular, k 0 is less than unity, which implies a E T distribution that increases with decreasing E T faster than e − E T /θ 0 , and θ 0 is unrealistically large. The resulting nucleon-nucleon E T distribution is also inconsistent with that measured in pp collisions [64]. The poor behaviour of the WN model is primarily due to the difference in shape between the Glauber N part distribution and the measured E Pb T distribution. To improve the description of the measured E Pb T distribution, a generalisation of the WN model is implemented that parameterises the N part dependence of the k and θ parameters of the gamma distribution as For k 1 = k 0 /2 and θ 1 = 0, this model reduces to the WN model. The log N part − 1 term allows for a possible variation in the effective acceptance of the FCal due to an N part -dependent backward shift in the p+Pb centre-of-mass system [65].
To limit the number of free parameters when fitting the E Pb T distribution, k 0 and θ 0 are obtained by fitting the detector-level E A T distributions in Pythia6 and Pythia8 pp simulations. These simulations have been shown to give a reasonable description of the corresponding pp collision data at √ s = 7 TeV [64] although they both slightly under-predict the average forward transverse energy. The contribution of electronic noise to the simulated distribution was determined by examining the E Pb T distribution in empty beam bunch crossings in data. In the fit, the gamma distributions were convolved with the effects of this noise before comparison with the data. The Pythia8 fit results, k 0 = 1.40 and θ 0 = 3.41, are used for the default analysis. The Pythia6 fit results, k 0 = 1.23 and θ 0 = 2.68, are used to evaluate systematic uncertainties.
The measured E Pb T distribution is fitted with a distribution produced by summing the N part -dependent gamma distributions, after weighting them by P(N part ) and including an additional convolution to account for electronic noise. The model distribution is also re-weighted to properly describe the E Pb T -dependent event selection efficiency in the data, which is estimated using the Pythia MC samples under the assumption that the p+Pb inefficiency for a given E Pb T is the same as that in pp collisions. Results are shown in Fig. 14

E Pb
T distribution for E Pb T < 100 GeV for all three geometric models, although at higher E Pb T the Glauber fit describes the data better. The deviations of the GGCF fits from the data become significant near E Pb T = 120 GeV; the fraction of the total E Pb T distribution above this value is approximately 0.1%.
The parameters k 1 and θ 1 are obtained from fixing k 0 and θ 0 and fitting the E T distribution to the data. They are presented in Table 3 along with the ratios k 1 /k 0 for each of the geometric models. Pure WN behaviour would correspond to k 1 /k 0 = 0.5 and θ 1 = 0. The results indicate substantial deviations from WN behaviour for the Glauber and GGCF ω σ = 0.2 fits, while the GGCF ω σ = 0.11 fit yields both a k 1 /k 0 that is close to 0.5 and a small θ 1 . The success of the above-described fitting procedure in describing the measured E Pb T distributions using parameterisations of the N part dependence of the E Pb T response with only two free parameters is due to the similarity of the shapes of the P(N part ) distribution and the measured E Pb T distributions. However, the pronounced knee in the Glauber P(N part ) distribution requires more non-linearity in the N part dependence of the E Pb T response. In contrast, the lack of such a feature in the GGCF P(N part ) distributions allows a simpler description of the measured E Pb T distribution. The results of the fit procedure described above provide a data-driven estimate of the total p+Pb eventselection efficiency. For the default Pythia8-based results, the integral of the simulated E Pb T distribution is 2% higher than that in the data. The deficit in the data is concentrated at low E Pb T values, consistent with losses due to event selection. However, a detailed analysis of the residual differences between the best fit and measured distributions indicates an excess of very low E Pb T events in the data, which varies from ∼ 0% for the Glauber fit to 1.8% for the GGCF fit with ω σ = 0.2. These may arise from residual diffractive or photo-nuclear collisions and are considered background. For the purpose of defining E Pb T centrality intervals, this background effectively increases the event-selection efficiency by adding events that are all in the 90-100% centrality interval. For Pythia6, the E Pb T fits using the default model yield a total efficiency of 97% and up to 1% background. The alternative models for k N part and θ N part yield a similar total efficiency of 98% and a background rate as high as 2%, a rate that is compatible with independent estimates of the rate for collisions involving diffractive excitation of the proton to pass the applied event selections. Based on these results, the total effective efficiency including background is then taken to be 98% and the uncertainty is conservatively estimated to be 2%.
The N part values are obtained for each of the centrality intervals using the results of the fits to the E Pb T distributions. The N part results along with the total systematic uncertainties, which are described below, are shown in Fig. 3 and listed in Table 4.
To obtain systematic errors on N part in each centrality interval, the maximum positive and negative fractional variation in N part away from the default results is determined for different classes of variations, detailed below.
To evaluate the impact of the total event selection uncertainty, new centrality intervals are chosen assuming a total efficiency of 100% and 96% and the complete analysis is repeated. To account for possible inaccuracies in the Pythia8-simulated dE T /dη in the region of the FCal acceptance, the analysis is repeated separately under ±10% re-scalings of Pythia8 E Pb T values commensurate with the scale of the data-Pythia8 differences observed in Ref. [64]. Other variations for which the complete analysis is repeated are: (i) using the Pythia6 event generator to fix k 0 and θ 0 , (ii) alternative models for k N part and θ N part , (iii) ±5 mb changes in σ NN , and (iv) variations in the parameters of the nuclear density distribution. For the model uncertainty, two alternative parameterisations for k N part and θ N part are used.
One of these kept θ constant, θ N part = θ 0 while allowing for a quadratic dependence of k on N part . The other included both a quadratic term in k N part and the logarithmic term in θ N part but fixed k 1 = k 0 /2 to reduce the number of free parameters.
The resulting maximal variations are then summed in quadrature over the different classes separately for the positive and negative variations to produce the uncertainties listed in Table 4. The uncertainties   for most centrality intervals are dominated by comparable contributions from the choice of model parameterisation, differences between the Pythia6 and Pythia8 E T fits, and uncertainties in σ NN . For the 40-60% and 60-90% interval the uncertainty in event selection efficiency has a contribution to the N part systematic uncertainty that is similar in magnitude to these other three sources.
The ATLAS Collaboration