1 Introduction

The study of high-multiplicity events in proton–proton (pp) and proton–nucleus high-energy collisions reveals striking similarities with respect to the observations made for larger systems like a nucleus–nucleus collision, which are interpreted in terms of the creation of a strongly-interacting, fluid-like QCD medium: the quark–gluon plasma (QGP). The ridge structure arising from long-range azimuthal correlations observed in pp data [1,2,3] is also found in p–Pb collisions [4,5,6,7], where the presence of double-ridge structures is reported [4]. More recently, an ALICE measurement reported an enhancement in the relative production of (multi-) strange particles with respect to primary charged particles as a function of multiplicity in pp collisions [8]. This suggests that some observables related to the QGP formation might be driven just by the multiplicity regardless of collision systems at LHC energies.

In pp and p–Pb collisions, the selection of events with large hadronic final-state multiplicities biases the sample towards a large average number of Multiple Parton Interactions (MPIs) at the LHC [9, 10]. In the description provided by the colour reconnection (CR) mechanism [11, 12], CR in MPIs is expected to be particularly pronounced at high multiplicity. The effects of prominent CR at high multiplicity are supposed to account for basic observables like the correlations between the average momentum and the multiplicity of charged particles [13] as well as for the shape of their pseudorapidity distribution [14]. Indeed, the transverse momentum (\(p_\mathrm{T}\)) spectra of charged particles at high multiplicity [15, 16] can be attributed, in pp collisions, to a CR mechanism, while until now no multiplicity dependence study of charged particle pseudorapidity density has been published.

This document provides a large set of charged-particle multiplicity density measurements as a function of event multiplicity in pp collisions at different centre-of-mass energies. This work could shed light on the phenomenon of MPIs that is a key ingredient of models attempting to describe large-multiplicity events. In any collision system, the event-averaged pseudorapidity density of primary charged particles [17], \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \), is a key observable characterising the global properties of the collision. Especially in pp interactions, the \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) is described by the combination of the perturbative hard partonic processes and the underlying event [18, 19]. The underlying event includes various phenomena like initial- and final-state radiation, colour-connected beam remnants, and infrared MPIs. In particular, its normalisation is directly connected to the MPI cross section determined by the low-x behaviour of the gluon parton-distribution function and by the consequent colour screening effects at the \(p_\mathrm{T}\) cut-off, while its multiplicity distribution is more influenced by correlations within MPI in the fragmentation stage.

The methods adopted in this analysis are based on those used in the inclusive \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) (\(\mathrm {d}N_\mathrm {ch}^\mathrm {incl.}/\mathrm {d}\eta \)) measurements of ALICE [20,21,22,23,24]. This study introduces exclusive event classes for two complementary multiplicity estimators defined in the midrapidity and in the forward regions and exploiting high-multiplicity triggers to record a large sample of events for the highest multiplicity classes. The results are provided for an event selection defined in a fully experimental way. Measurements are performed for inelastic collisions with at least one charged particle produced in \(|\eta |<1\) (\(\mathrm {INEL}_{>0}\)), corresponding to about 75% of the total inelastic cross section [13, 23, 25, 26].

2 Experimental setup

The full description and performance of the ALICE detectors can be found elsewhere [27, 28]. The detectors used in this analysis are briefly presented below.

The V0 detector [29] is made of two arrays (V0A and V0C) of 32 scintillating counters each. The V0A is located at a distance of 329 cm away from the interaction point (IP) along the beam direction (z) and it covers the pseudorapidity range \(2.8< \eta <5.1\). The V0C is installed at \(z~=~-88\,{\hbox {cm}}\), covering the pseudorapidity range \(-3.7< \eta < -1.7\). Both counters cover the full azimuth. The V0 detector provides the minimum-bias and beam-gas removal trigger to ALICE. It measures the signal amplitude created by charged particles and their arrival times with a time resolution better than 1 ns.

The silicon pixel detector (SPD) [30, 31] is the innermost detector of ALICE. It is located inside a large solenoid that produces a homogeneous magnetic field of 0.5 T. The SPD consists of two cylindrical layers coaxial to the beam line at radii 3.9 and 7.6 cm. It is made of 10 million pixels distributed on 240 sensors that cover the pseudorapidity range \(|\eta |<2\) for the first layer and \(|\eta |<1.4\) for the second layer for particles that originate from collisions at the nominal interaction point. An enlarged pseudorapidity coverage of \(|\eta |<2\) is reached using events whose primary vertex is not at zero, but within \(\pm 10\) cm from the nominal interaction point. The SPD provides a precise measurement of the position of the primary interaction vertex with a spatial resolution of on average \(30\,\upmu \mathrm {m}\) in the beam direction [23, 31]. The multiplicity measurement of this analysis relies on the reconstruction of tracklets, which are track segments connecting hits on the two SPD layers and pointing to the primary vertex. Due to the bending in the magnetic field and multiple scattering, the reconstruction efficiency of tracklets is limited to \(p_\mathrm{T}>50\,{\mathrm{Mev}}/c\).

Fig. 1
figure 1

The distribution of the V0M amplitude (total energy deposition in the region \(-3.7<\eta <-1.7\) and \(2.8<\eta <5.1\)) scaled by its average value \(\langle \mathrm V0M \rangle \) that is used to determine the forward multiplicity classes (a) and the distribution of the total number of SPD tracklets in an event (\(N_{\mathrm {SPD\,\,Tracklet}}\), \(-2<\eta <2\)) scaled by its average value \(\langle N_{\mathrm {SPD\,\,Tracklet}} \rangle \) that is used to determine the midrapidity multiplicity classes (b) in pp collisions at \(\sqrt{s}~=~13\,{\mathrm{TeV}}\). Note that the percentile values of the multiplicity classes are fractions of the visible cross section \(\Delta \sigma /\sigma _\mathrm{MB_{AND>0}}\) (see text for details)

3 Data sample and analysis

The minimum-bias pp data samples at \(\sqrt{s}\) = 5.02, 7 and 13 TeV used in this analysis correspond to the integrated luminosities \(\mathcal {L}_{\mathrm {int}}~=~12.4\pm 0.3\), \(3.78\pm 0.13\) and \(0.946\pm 0.020\,\hbox {nb}^{-1}\), respectively [28, 32, 33]. The data sample at \(\sqrt{s}\) = 13 TeV benefits from a high-multiplicity trigger that was implemented in ALICE at the beginning of the LHC Run 2.

The minimum-bias trigger (\({\mathrm{MB}}_{AND}\)) requires hits in both the V0A and V0C detectors in coincidence of a beam crossing. The contribution from diffractive interactions is minimised by requiring at least one SPD tracklet in \(|\eta |<1\); the resulting data sample is called \(\mathrm{MB}_{AND>0}\). The contamination from beam-induced background is removed by using the timing information of the V0 detectors and taking into account the correlation between tracklets and clusters in the SPD detector [28]. The events used for the analysis are required to have a primary vertex in the fiducial region \(|z|<10\) cm. The primary vertex is reconstructed by correlating hits in the two SPD layers. The contamination from in-bunch pile-up events is removed offline excluding events with multiple vertices reconstructed in the SPD [23]. The pile-up probability estimated considering the beam conditions ranges from \(10^{-3}\) to \(10^{-2}\). After the offline rejection, the remaining pile-up has a negligible impact on the final results. This was verified by analysing data samples separately with high and low initial pile-up contamination.

Multiplicity classes are defined by a probability (percentile) range that is interpreted as a fractional cross section \(\Delta \sigma /\sigma _\mathrm{MB_{AND>0}}\), with the visible cross section in pp collisions, \(\sigma _{\mathrm {MB_{AND>0}}}\), constituting 100%. Percentile values for higher multiplicity collisions are close to 0% and for lower ones close to 100%. Forward multiplicity classes are estimated by V0M, which is the sum of the energy deposition measured by the V0A and V0C scintillators. The distribution of the V0M amplitude scaled by its average value \(\langle {\mathrm{V0M}} \rangle \) (self-normalised V0M) is shown in Fig. 1a for \({\mathrm { MB}}_{{\mathrm {AND}}>0}\) pp collisions at \(\sqrt{s} = 13\,{\mathrm{TeV}}\). A dedicated high-multiplicity trigger is defined by the threshold \({\mathrm{V0M}} / \langle {\mathrm{V0M}} \rangle > \sim 4.9\), corresponding to \(\sigma /\sigma _\mathrm{MB_{AND>0}}~=~0.1\%\). The SPD tracklets are used to define multiplicity classes in the midrapidity region \(|\eta | < 2\). The distribution of the self-normalised number of SPD tracklets for \({\mathrm { MB}}_{{\mathrm {AND}}>0}\) pp collisions in \(|\eta |<2\) is shown in Fig. 1b. For all the midrapidity multiplicity classes, only the minimum-bias trigger is used because the high-multiplicity trigger relying on V0M amplitudes would give an additional bias. The data analysis is performed by classifying \({\mathrm { MB}}_{{\mathrm {AND}}>0}\) data samples using the mid and forward multiplicity estimators.

Table 1 Correspondence of the multiplicity classes between \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0})\) and \(\mathrm {P}(\mathrm {INEL}_{>0})\). The trigger efficiency is estimated using PYTHIA 8 Monash 2013  [34,35,36] and GEANT 3 [37]

The multiplicity percentile intervals of the visible cross section \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0}) = \Delta \sigma /\sigma _\mathrm{MB_{AND>0}}\) can be converted to fractional intervals with respect to the \(\mathrm {INEL}_{>0}\) cross section \(\mathrm {P}(\mathrm {INEL}_{>0}) = \Delta \sigma /\sigma _{\mathrm {INEL}_{>0}}\) in pp collisions by following the conversion rule

$$\begin{aligned} \mathrm {P}_i(\mathrm {INEL}_{>0}) = \frac{\mathrm {P}_i({\mathrm { MB}}_{{\mathrm {AND}}>0}) / \epsilon _i }{ \sum _j \left( \mathrm {P}_j({\mathrm { MB}}_{{\mathrm {AND}}>0}) / \epsilon _j\right) }\quad , \end{aligned}$$

where i indicates a specific multiplicity class, j runs over all multiplicity classes for a given collision energy and multiplicity estimator, and \(\epsilon _i\) (\(\epsilon _j\)) is the \({\mathrm { MB}}_{{\mathrm {AND}}>0}\) trigger efficiency for the \(\mathrm {INEL}_{>0}\) event sample \(N_{{\mathrm { MB}}_{{\mathrm {AND}}>0}} / N_{\mathrm {INEL}_{>0}}\) for the \(i\mathrm{th}\) (\(j\mathrm{th}\)) multiplicity class. The correspondence between \(\mathrm {P}(\mathrm {INEL}_{>0})\) and \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0})\) is reported in Table 1. In this document, multiplicity classes for the results of ALICE are represented with \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0})\), which is a quantity defined using detector-level variables. In order to perform precise comparisons of particle-level simulations with the ALICE data, the \(\mathrm {P}(\mathrm {INEL}_{>0})\) intervals corresponding to a given \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0})\) interval for each centre-of-mass energy and multiplicity class reported in Table 1 need to be used in the particle-level simulations.

Table 2 The correction factors of \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) from the multiplicity classes \(\mathrm {P}(\mathrm {INEL}_{>0})\) in Table 1 to those of \(\mathrm {P}(\mathrm {INEL}_{>0})\) in the leftmost column of this table. The correction factors are estimated for the generated values of \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) using PYTHIA 8 Monash 2013 [34,35,36]

Alternatively, the values of \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) with ALICE data for the multiplicity classes of \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0})\) in Table 1 can be corrected such that they correspond to the multiplicity classes of \(\mathrm {P}(\mathrm {INEL}_{>0})\) given in the leftmost column of Table 2. For example, the correction factor of 0.9995 for the \(\mathrm {P}(\mathrm {INEL}_{>0}) =\) 0–0.01% interval of the forward multiplicity estimator at \(\sqrt{s}= 5.02\) TeV is the ratio of the generated values of \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) between \(\mathrm {P}(\mathrm {INEL}_{>0}) =\) 0–0.01% and 0–0.0091% with PYTHIA 8 Monash 2013 [34,35,36]. The data measurement of \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) for \(\mathrm {P}({\mathrm { MB}}_{{\mathrm {AND}}>0}) =\) 0–0.01% would therefore need to be multiplied by this factor in order to compare directly with a generated interval of \(\mathrm {P}(\mathrm {INEL}_{>0}) =\) 0–0.01%.

The value of \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \) is obtained by correcting the number of SPD tracklets for detector acceptance as well as reconstruction and selection efficiency following the procedure developed earlier [23, 24, 38,