1 Introduction

The multiplicity of charged particles produced in high-energy pp collisions is one of the key observables to describe the global properties of the interactions and has been the subject of long standing experimental and theoretical investigations. The pp multiplicity distributions of primary-charged particles have been measured for five increasingly wider pseudorapidity ranges. A primary-charged particle is a charged particle with a mean proper lifetime \(\tau \) larger than 1 cm/c, which is either produced directly in the interaction, or from decays of particles with \(\tau \) smaller than 1 cm/c, excluding particles produced in interactions with material [1]. The results are determined using both the Silicon Pixel Detector (SPD) and the Forward Multiplicity Detector (FMD) in ALICE to widen the pseudorapidity coverage with respect to previous ALICE results [2,3,4], which made exclusive use of the SPD. The extension of the pseudorapidity coverage allows us to increase the high-multiplicity reach of the distributions by around 70–90\(\%\) with respect to the previous ALICE publication [4], exploring a wider phase space.

The multiplicity distribution of charged particles produced in high-energy pp collisions is sensitive to the number of interactions between quarks and gluons contained in the protons and to underlying mechanisms of particle production. At LHC energies, the particle production is dominated by soft QCD processes, which cannot be treated perturbatively and can only be modeled phenomenologically. On the other hand, as the colliding energy grows, the particle production receives increased contributions from hard scattering, which can be treated perturbatively.

We have compared directly our data to previous measurements from CMS [5]. ATLAS and LHCb use different \(p_{\text {T}}\) and \(\eta \) ranges [6, 7], making the direct comparison impossible. This manuscript presents also an overview of the parameters obtained when fitting multiplicity distributions with the sum of two Negative Binomial Distributions (NBDs). Additionally, the results have been compared to simulations regularly used at LHC [8,9,10,11] and calculations based on saturation density of gluons in the colliding hadrons [12, 13].

The manuscript is organized in the following way: Sect. 2 describes the detectors used to measure the charged-particle multiplicity distributions. Section 3 explains the analysis procedure in detail. The systematic uncertainties are described in Sect. 4 and the results along with comparisons to models are presented in Sect. 5, which contains also the analysis of the NBD fits. A brief summary and conclusions are finally given in Sect. 6.

2 Experimental setup

Full details of the sub-detectors are given elsewhere [14]. ALICE is designed to measure particles over a wide kinematic range \(-\,3.4<\eta <5.0\). Only the sub-detectors used in this analysis are described, namely the V0 scintillation counters, the SPD, and the FMD.

2.1 V0 detector

The V0 detector [15] is composed of two arrays of 32 scintillators positioned at 330 cm (V0-A) and −90 cm (V0-C) from the nominal interaction point (IP) along the beam axis. Each array has a ring structure segmented into 4 radial and 8 azimuthal sectors. The detector has full azimuthal coverage in the pseudorapidity ranges \(2.8< \eta < 5.1\) and \(-3.7< \eta < -1.7\). The signal amplitudes and times are recorded for each of the 64 scintillators. The V0 is appropriate for triggering, thanks to the good timing resolution of each scintillator (1 ns) along with its large acceptance for detecting charged particles. Different V0 trigger settings are used in the analysis.

2.2 Silicon pixel detector

The SPD are the two innermost cylindrical layers of the ALICE Inner Tracking System (ITS) [14] surrounding the beam line. The layers have full azimuthal coverage and radii of 3.9 and 7.6 cm with \(9.8\times 10^6\) silicon diodes, each of size \(50\times 425\) \({\upmu } \mathrm{m}^2\). The first layer of the SPD has the largest pseudorapidity coverage of the ITS (\(\vert \eta \vert <1.98\) for collisions at the nominal IP). Besides the readout of individual pixels with signals above a certain threshold, each SPD chip provides a fast signal every 100 ns, indicating a presence of fired pixels (FastOR), making it suitable for triggering. Charged particles can deposit energy in more than one pixel of the SPD. The offline reconstruction combines such neighboring signals into a a single cluster. The charged-particle multiplicity can then be estimated by counting the number of clusters detected in a SPD layer. This analysis uses only clusters from the inner layer of the SPD to provide the largest pseudorapidity coverage for particle detection. Alternatively, clusters from the two SPD layers together with the primary vertex can be combined to form tracklets [4], allowing to select primary particles with very high efficiency. The charged-particle multiplicity is then estimated by counting the number of tracklets.

2.3 Forward multiplicity detector

The purpose of the FMD is to extend, with high spatial resolution, the charged-particle detection acceptance beyond the reach of the SPD and central detectors in ALICE [16]. The FMD is a silicon strip detector and consists of three sub-detectors placed at 320 cm (FMD1), 79 cm (FMD2), and \(-69\) cm (FMD3) from the nominal IP along the beam pipe. FMD2 and FMD3 contain both inner and outer rings of silicon strips. FMD1 is located farther from the IP and has only one inner ring. Inner rings consist of 10 sensors, each with two azimuthal sectors and 512 strips with radii from 4.2 to 17.2 cm. Outer rings contain 20 sensors each again with two azimuthal sectors, but with 256 strips, with radii from 15.4 to 28.4 cm. Each ring (inner or outer), therefore, contains 10,240 strips giving in total 51,200 strips. The FMD has full azimuthal coverage in the pseudorapidity ranges \(-\,3.4<\eta <-\,1.7\) and \(1.7<\eta <5.0\).

The FMD records, for each strip, the energy deposited by charged particles traversing the detector. Various selection criteria, see [17, 18] for details, are applied to the energy measured in each strip to determine if the signal corresponds to a single particle traversing only this strip or also a neighboring strip. The number of particles traversing the FMD is determined taking into account only the signals which pass these selection criteria. The majority of the particles which reach the FMD, however, are secondary particles produced in interactions with the beam pipe, the material of the ITS, cables and support structures [17]. Therefore, a detailed Monte Carlo simulation is needed to determine the number of primary particles produced in the collision.

3 Analysis procedure

The multiplicity distribution of the primary particles is affected by many detector effects, such as dead detector regions and secondary particle production. These detector effects must be minimized and corrected for as they have increasing effects when determining accurately the probability of progressively higher-multiplicity events. The unfolding method is used to correct for the detector effects, as will be described in the following.

3.1 Event selection

Collisions at three different center of mass energies (0.9, 7, and 8 TeV) are analyzed. The data used for the analysis were collected at low beam currents and low pileup during three data taking periods: the 0.9 and 7 TeV samples were acquired in 2010, while the 8 TeV sample was collected in 2012. The last sample is the most affected by the pileup contamination. For this reason we selected few specific runs with low contamination from pileup events for this energy, and used data taken with interaction rate not exceeding 1 kHz. A pileup is defined as more than one collision occurring during the readout time of the detector (300 ns, for the SPD, and 2 \(\upmu \)s sampling time, for the FMD). Such events produce a bias towards larger multiplicity that enhance mostly the tail of the multiplicity distribution. Table 1 shows the number of selected events at each energy and the average number of interactions per bunch crossing, \(\langle \mu \rangle \), measured by the experiment [19]. This parameter is determined experimentally and for this measurements, in which \(\langle \mu \rangle \ll 1\), the average probability of having more than one interaction in a single bunch crossing, where at least one interaction occurs, is around \(1-2\)% (\(\langle \mu \rangle /2\)).

Table 1 Data samples used in this analysis. For each center-of-mass energy, the total number of selected minimum-bias (MB) events along with the average number of interactions per bunch crossing, \(\langle \mu \rangle \), are listed

Inelastic non-diffractive scatterings are the dominant processes in pp collisions, for which most of the hadrons are produced as a consequence of an exchange of color charge. On the contrary, diffractive events can be single-, double-, or central-diffractive. In Regge theory [20], diffraction occurs when the Pomeron interacts with the proton and produces a system of particles, called the diffractive system. The case in which only one of the protons dissociates is called single-diffractive.

The signals of the V0 and SPD are used to select events where at least one interaction occurred, which are triggered by requiring the detection of at least one particle in either the V0-A, V0-C, or SPD (\(\hbox {MB}_{\text {OR}}\)). Events are divided into three classes depending on further requirements. The first class includes all inelastic events (the INEL class), which is the same condition as used to select events where an interaction occurred (\(\hbox {MB}_{\text {OR}}\) trigger). The second class (the INEL > 0) requires the presence of at least one charged particle (tracklet) in the region \(\vert \eta \vert <1.0\) in addition to the INEL condition. This class has higher trigger efficiency and, therefore, reduced corrections relative to the INEL event class. The third class requires charged particles to be detected in both the V0-A and the V0-C (\(\hbox {MB}_{\text {AND}}\)). This class is used to remove the majority of the single-diffractive events and is, therefore, called the non-single-diffractive (NSD) event class.

To remove interactions of the beam with residual gas in the beam pipe, further selection criteria are applied to the event sample. Since these interactions can occur anywhere along the beam line, the most efficient way to reject them is to require that the interaction occurs close to the expected bunch-crossing position. The position of the collision along the beam pipe is determined from the vertex position reconstructed correlating SPD tracklets, with a precision of about 0.2 cm. Beam-gas interactions far from the IP are vetoed by the time difference in the V0-A and V0-C detectors. The vertex is required to be within 4 cm of the nominal IP position to reduce the contribution from beam-gas interactions and to remove acceptance gaps in the pseudorapidity coverage of the SPD and FMD, since the acceptance depends on the vertex position.

Even though runs with very low \(\langle \mu \rangle \) (average 0.04) were chosen, a residual background from pileup events remains. The majority of pileup events are identified and removed by searching for additional vertices in the same event. It is required that the uncertainty on the measurement of the longitudinal vertex position is less than 0.2 cm to have the most accurate determination of the vertex. Events with an additional vertex separated by more than 0.8 cm from the main one and containing at least three attached tracklets are tagged as pileup and removed from the analysis. Dedicated simulations show that the probability for the pileup event to pass this selection criteria is at most 10\(\%\) and the residual pileup does not exceed 10\(\%\) up to the highest multiplicities kept in this analysis. Therefore, the overall pileup contribution does not exceed 0.2\(\%\) for \(\langle \mu \rangle =0.04\) and, because it is covered by systematic uncertainties for all multiplicities, no correction is applied for this bias.

3.2 Unfolding

The FMD had nearly 100% azimuthal acceptance, but the SPD had a significant number of modules excluded from read-out that must be accounted for. On the other hand, interactions in detector material increase the detected number of charged particles, in particular in the FMD. A good understanding of the detector acceptance and of the number of secondary particles which hit the FMD and the SPD is crucial.

The main ingredients necessary to evaluate the primary multiplicity distributions are the raw (detected) multiplicity distributions and a matrix, which maps the measured multiplicity to the number of charged-primary particles distributions, called true. The raw multiplicity distributions are determined by counting the number of clusters in the SPD acceptance, the number of signals passing selection criteria in the FMD, or the average between the two if the acceptance of the SPD and FMD overlaps. The response of the detector is determined by the matrix \(R_{mt}\), which corresponds to the probability that an event with true multiplicity t and measured multiplicity m occurs. This matrix is obtained using PYTHIA ATLAS-CSC flat tune [21] simulations in which the generated particles are transported through the experimental setup using the GEANT3 [22] software package. The same reconstruction algorithm is used for simulations of real data. Experimental conditions and detector settings at the time of data-taking at a center-of-mass energy of \(\sqrt{s}=\) 0.9, 7, and 8 TeV are simulated when evaluating the response matrices. Figure 1 shows two different response matrices for different pseudorapidity ranges. The left panel of Fig. 1 shows the response matrix obtained for the \(\vert \eta \vert <2.0\). In this range, the unfolding increases the multiplicity on average because of the acceptance gaps in the SPD. When the extended pseudorapidity range, \(\vert \eta \vert <3.4\), is used, the number of detected counts exceeds on average the number of true counts as the secondary particles in the FMD dominate the bias. This is shown in the right panel of Fig. 1.

Fig. 1
figure 1

Response matrices obtained propagating Monte Carlo generated events, in this case with the PYTHIA ATLAS-CSC flat tune [21] for the non-single-diffractive event class selection. Left: Matrix including the overlap region between SPD and FMD. Right: Matrix for the region where the majority of the counts are from the FMD. The diagonal (generated=reconstructed) is plotted as a black dotted line

A method based on Bayes’ Theorem [23] is used to derive the final multiplicity distributions. Bayes’ Theorem states that the conditional probability \(\text {P}(A\vert B)\) (probability of A if B is true) can be written as

$$ \text {P}(A\vert B)=\frac{\text {P}(B\vert A)\text {P}(A)}{\text {P}(B)}, $$

in which \(\text {P}(A)\) and \(\text {P}(B)\) are the independent probabilities of A and B, and \(\text {P}(B\vert A)\) is the probability of B if A is true. A can be identified as a certain true multiplicity, while B is the measured multiplicity. The conditional probability \(\text {P}(B\vert A)\) is the response matrix of the detector, and can then be computed.

Equation 1 is restated as

$$ {\widetilde{R}}_{tm}=\frac{R_{mt}{\text {P}}_{t}}{\sum _{\text{t'}}R_{mt'}{\text {P}}_{t'}}, $$

where \(\text {P}_{t}\) is an a priori guess of the true distribution and \(\widetilde{R}_{tm}\) is the matrix of probabilities that allows one to compute the true multiplicity distribution from the measured one. The unfolded distribution, \(U_{t}\), is then obtained from

$$ U_{t}=\sum _{m}\widetilde{R}_{tm}M_{m}, $$

in which \(M_{m}\) is the measured distribution. The obtained \(U_{t}\) is used as a priori probability for the next iteration. The number of iterations is fixed to 10. This parameter has been chosen by examining the optimal performance obtained from simulation studies, performing closure tests using different number of iterations.

3.3 Event selection efficiency

The probability that an event is triggered depends on the multiplicity of charged particles. At high multiplicities, it is more probable that one of the trigger detectors is fired. At low multiplicities large trigger inefficiencies for finding events exist and must be corrected for. The event selection efficiency, \(\epsilon _{\text {TRIG}}\), is defined via simulations as

$$ \epsilon _{\text {TRIG}}=\frac{N_{\text {ev,reco}}(\text {TRIG}\, \,\vert \text {v}_{\text {z,reco}}\vert<4\,\mathrm{\text {cm}})}{N_{\text {ev,gen}}(\text {TRIG}\, \,\vert \text {v}_{\text {{z,gen}}}\vert <4\,\mathrm{\text {cm}})}, $$

where the numerator is the number of reconstructed events with the selected hardware trigger condition (MB\(_{\text {AND}}\) or MB\(_{\text {OR}}\)) and with the reconstructed vertex less than 4 cm from the nominal IP, in longitudinal direction. There is a dependence in the z vertex distribution and selecting \( \text {v}_{\text {z,reco}}\) introduces a bias in the efficiency. The effect is visible only for narrow vertex selections, and it is not relevant for \(\vert \text {v}_{\text {z}}\vert <4\,\mathrm{\text {cm}}\). The denominator is a similar quantity, but for the generated sample (inelastic or non-single-diffractive events). The unfolded distribution is corrected for the vertex and trigger inefficiency by dividing each multiplicity bin by its \(\epsilon _{\text {TRIG}}\) value.

The efficiencies used are shown in Fig. 2 for 0.9 and 7 TeV for the range \(\vert \eta \vert <3.0\). Both the INEL and NSD efficiencies are displayed. The points are obtained by averaging the efficiencies found with the PYTHIA Perugia 0 [8] and the PHOJET  [9] diffraction tuned event generators. Diffraction was accounted for using the Kaidalov–Poghosyan model [24] to tune the diffractive processes. The event generators are adjusted to reproduce the measured diffraction cross-sections and the shapes of the diffractive masses. The cross-section ratios are \(\sigma _{\text {SD}}/\sigma _{\text {INEL}}\backsim 0.20\) for upper diffractive mass limit of \(M_{\text {X}}<200\) GeV\(/c^{2}\), and \(\sigma _{\text {DD}}/\sigma _{\text {INEL}}\backsim 0.11\) for a pseudorapidity gap of \(\Delta \eta >3\), as measured at the LHC [25]. The uncertainties are estimated by evaluating the difference between the two event generators and are only relevant at low multiplicity. The efficiency of NSD trigger requiring signal in V0-A and V0-C detectors, on both sides of the IP, is lower at low multiplicities than that of INEL trigger, which requires response of at least one V0. For \(N_{\text {ch}}\gtrsim 20\) at the widest pseudorapidity ranges probed, both efficiencies reach 100\(\%\) and the corresponding systematic uncertainty becomes negligible.

Fig. 2
figure 2

Event selection efficiencies for 0.9 and 7 TeV for both INEL and NSD event samples as a function of the number of primary-charged particles for the \(\vert \eta \vert <3.0\) range

4 Systematic uncertainties

The steps involved in the analysis depend on the knowledge of the detector response to charged particles. The uncertainties in Table 2 are purely model dependent and related to how diffraction, and soft QCD in general, are processed in the two generators used to determine the efficiency uncertainty. The difference between PYTHIA Perugia 0 and PHOJET diffraction tuned generators, used to determine this uncertainty, is larger for small values of \(N_{\text {ch}}\). Therefore, the uncertainty mostly influences the first bins of the multiplicity distributions. Table 2 reports the values for charged-multiplicity of 0, 1, and 2. In general, the Lorentz boost of the diffracted system increases with increasing center-of-mass energies, and single and double diffraction contributions are smaller when going to higher energies. At wider pseudorapidity ranges there are higher chances of including diffractive events in the distribution. We observe that the uncertainty for NSD events at \(-\,3.4<\eta <+\,5.0\) is higher for lower energy in one multiplicity bin, where the description of diffraction differs the most among PYTHIA and PHOJET.

Table 2 Systematic uncertainties (in percent) for the efficiency correction, for the INEL, NSD and INEL\(>0\) event classes. Numbers are given for multiplicity 0, 1, and 2

Systematic effects from different sources related to run conditions could produce biases in the number of detected particles. To investigate such effects, the fluctuations in the results are examined for all three energies by splitting the data set into two separate samples with similar beam conditions, which are then unfolded with two different response matrices. The response matrices are calculated from simulations relative to the conditions of the runs that are used to