1 Introduction

The unprecedented energies available at the Large Hadron Collider (LHC) provide unique opportunities to investigate the properties of strongly-interacting matter. Particle production at large transverse momenta (\(p_{\text {T}}\)) is well-described by perturbative Quantum Chromodynamics (pQCD). The soft regime (\(p_{\text {T}}\) \(\lesssim \) 2 GeV/c), in which several collective phenomena are observed in proton-proton (pp), proton-lead (p–Pb), and heavy-ion (A–A) collisions, is not calculable from first principles of QCD. Instead, in order to describe bulk particle production in A–A collisions, one usually relies on hydrodynamic and thermodynamic modelling, which assumes the system to be in kinetic and chemical equilibrium [1, 2]. On the other hand, the description of low-\(p_{\text {T}}\) particle spectra in smaller systems such as pp collisions is often based on phenomenological modelling of multi-partonic interactions (MPI) and color reconnection (CR) [3, 4] or overlapping strings [5].

Recent reports on the enhancement of (multi)strange hadrons [6], double-ridge structure [7, 8], non-zero \(v_{\text {2}}\) coefficients [9], mass ordering in hadron \(p_{\text {T}}\) spectra, and characteristic modifications of baryon-to-meson ratios [10] suggest that collective phenomena are present at the LHC energies also in p–Pb collisions. This is further extended to even smaller systems, such as pp collisions at \(\sqrt{s}\ =\ 7\ \text {TeV}\), where similar observations have been reported in high multiplicity events, indicating that the collective effects are not characteristic of heavy-ion collisions only. Furthermore, a continuous transition of light-flavor hadron to pion ratios as a function of charged-particle multiplicity density \(\text {d}^{}N_{\text {ch}}/\text {d}\eta \) from pp to p–Pb and then to Pb–Pb collisions was found [11,12,13]. The observed similarities suggest the existence of a common underlying mechanism determining the chemical composition of particles produced in these three collision systems.

Results from pp [11] and p–Pb [10] collisions indicate that particle production scales with \(\text {d}^{}N_{\text {ch}}/\text {d}\eta \) independent of the colliding system. Measurements reported in previous multiplicity-dependent studies have considered different colliding systems, each at a different center-of-mass energy. In this work, we extend the existing observations by performing a detailed study of pp collisions at \(\sqrt{s}\ =\ 13\ \text {TeV}\). A similar study has been reported by the CMS Collaboration, albeit in a limited \(p_{\text {T}}\) range [14]. Thanks to the availability of Run 2 data from the LHC, for the first time, in pp collisions, we can disentangle the effect of center-of-mass energy from the multiplicity dependence of \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\)and p (\(\overline{\mathrm{p}} \)) production in a wide \(p_{\text {T}}\) range.

In this paper, we report on the multiplicity dependence of the production of primary \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\)and \(\text {p}\) (\(\overline{\mathrm{p}} \)) at \(\sqrt{s}\ =\ 13\ \text {TeV}\). Particles are considered as primary if their mean proper decay length \(c\tau \) is larger than 1 cm and they are created in the collision (including products of strong and electromagnetic decays), but not from a weak decay of other light-flavor hadrons or muons. An exception to this are products of weak decays, where \(c\tau \) of the weakly decaying particle is less than 1 cm [15]. The reported particle spectra are measured in the rapidity region \(|y|<0.5\) with the ALICE detector [16], which offers excellent tracking and particle identification capabilities from \(p_{\text {T}} =0.1\) \(\text {GeV}/c\) to several tens of \(\text {GeV}/c\)  [17]. As particles and anti-particles are produced roughly in equal amounts at LHC energies [18], we adopt a notation where \(\pi \), \(\text {K}\), and \(\text {p}\) refer to \((\pi ^{+}+ \pi ^{-})\), \((\mathrm {K}^{+}+ \mathrm {K}^{-})\), and \((\text {p} \) + \(\overline{\mathrm{p}} \)) unless stated otherwise. This paper is organized as follows. In Sect. 2, the details on particle identification techniques, systematic uncertainties, spectra corrections and normalization are provided. The results are presented and discussed in Sect. 3, together with comparisons to Monte Carlo model predictions. Finally, the most important findings are summarized in Sect. 4.

2 Data set and experimental setup

The dataset used for this study was recorded by the ALICE Experiment during the 2016 LHC pp data taking period. Overall \(\sim \)143M events have been analysed, corresponding to an integrated luminosity of \(2.47\text {\ nb}^{-1}\) considering the visible cross-section measured with the V0 detector [19]. A detailed description of the ALICE detector and its performance is provided in [16, 17]. Measurements of identified particle spectra have been performed by using the central barrel detectors: the Inner Tracking System (ITS) (Sect. 3.1 of [16]), the Time Projection Chamber (TPC) [20] and the Time-of-Flight detector (TOF) [21]. The charged-particle multiplicity estimation is done by the V0 detector (Sect. 5.4 of [16]), which consists of two arrays of 32 scintillators each, positioned in the forward (V0A, \(2.8< \eta < 5.1\)) and backward (V0C, \(-3.7< \eta < -1.7\)) rapidity regions. In addition, the V0 is also used for triggering purposes as well as background rejection. The determination of the event collision time [22] is performed by the T0 detector as well as the TOF detector. The former consists of two arrays of Cherenkov counters, positioned on both sides of the interaction region, and covering a pseudorapidity range of \(-3.3< \eta < -2.9\) (T0-C) and \(4.5<\eta <5\) (T0-A). The central barrel detectors are placed inside a solenoidal magnet, which provides a field strength of 0.5 T.

The ITS is the innermost detector and consists of six concentric cylindrical layers of high-resolution silicon detectors based on different technologies, covering pseudorapidity region \(|\eta |<0.9\). The two innermost layers form the Silicon Pixel Detector (SPD), which features binary readout and is also used as a trigger detector. The Silicon Drift Detector (SDD) and the Silicon Strip Detector (SSD), which form the four outer layers of the ITS, provide the amplitude of the charge signal, which is used for particle identification through the measurement of specific energy loss at low transverse momenta (\(p_{\text {T}} \gtrsim 100\) MeV/c).

The TPC, which is the main tracking detector of the ALICE central barrel, is based on a cylindrical gaseous chamber with radial and longitudinal dimensions of \(85\mathrm{\,cm}< r < 247\mathrm{\,cm}\) and \(-250\mathrm{\,cm}<z<250\mathrm{\,cm}\), respectively. The TPC is read out by multi-wire proportional chambers (MWPC) with cathode pad readout, located at its endplates. With the measurement of drift time, the TPC provides three-dimensional space-point information for each charged track in pseudorapidity range \(|\eta |<0.8\) with up to 159 samples per track. In the TPC, the identification of charged particles is based on the measurement of the specific energy loss, which in pp collisions is performed with a resolution of \(5.2\%\) [17].

The TOF is a large-area array of multigap resistive plate chambers (MRPC), formed into a \(\sim{4}\) m radius cylinder around the interaction point and covering the pseudorapidity region \(|\eta |<0.9\) with full-azimuth coverage. The time-of-flight is measured as the difference between the particle arrival time and the event collision time, enabling particle identification at intermediate transverse momenta, \(0.5 \lesssim p_{\text {T}} \lesssim 4\) \(\text {GeV}/c\). The arrival time is measured by the MRPCs with an intrinsic time resolution of 50 ps, while the event collision time is determined by combining the T0 detector measurement with the estimate using the particle arrival times at the TOF [22].

2.1 Event selection, classification and normalization

The analysed data were recorded using a minimum-bias trigger requiring signals in both V0A and V0C scintillators in coincidence with the arrival of the proton bunches from both directions. The background events produced outside the interaction region are rejected using the correlation between the SPD clusters and the tracklets reconstructed in SPD. The out-of-bunch pileup was rejected offline using the timing information from the V0 counter. The primary vertex was reconstructed either using global tracks (reconstructed using ITS and TPC information) or SPD tracklets (reconstructed using only the SPD information) with \(|z_{\text {vtx}}| < 10\) cm along the beam axis. Events with in-bunch pileup were removed if a second vertex was reconstructed within \(8\mathrm{\,mm}\) of the primary vertex in the beam direction. The typical interaction rate of pp collisions in the 2016 data taking periods was around 120 kHz while beam-gas interactions occurred at a rate of 1.2 kHz.

In the analysis presented in this paper, we consider the event class INEL>0 with at least one charged particle produced in the pseudorapidity region \(|\eta |<1\), which corresponds to \(\sim{75\%}\) of the total inelastic scattering cross-section [23]. To avoid auto-correlation biases [