## Abstract

The production of \(\pi ^{\pm }\), \(\mathrm{K}^{\pm }\), \(\mathrm{K}^{0}_{S}\), \(\mathrm{K}^{*}(892)^{0}\), \(\mathrm{p}\), \(\phi (1020)\), \(\Lambda \), \(\Xi ^{-}\), \(\Omega ^{-}\), and their antiparticles was measured in inelastic proton–proton (pp) collisions at a center-of-mass energy of \(\sqrt{s}\) = 13 TeV at midrapidity (\(|y|<0.5\)) as a function of transverse momentum (\(p_{\mathrm{T}}\)) using the ALICE detector at the CERN LHC. Furthermore, the single-particle \(p_{\mathrm{T}}\) distributions of \(\mathrm{K}^{0}_{S}\), \(\Lambda \), and \(\overline{\Lambda }\) in inelastic pp collisions at \(\sqrt{s} = 7\) TeV are reported here for the first time. The \(p_{\mathrm{T}}\) distributions are studied at midrapidity within the transverse momentum range \(0\le p_{\mathrm{T}}\le 20\) GeV/*c*, depending on the particle species. The \(p_{\mathrm{T}}\) spectra, integrated yields, and particle yield ratios are discussed as a function of collision energy and compared with measurements at lower \(\sqrt{s}\) and with results from various general-purpose QCD-inspired Monte Carlo models. A hardening of the spectra at high \(p_{\mathrm{T}}\) with increasing collision energy is observed, which is similar for all particle species under study. The transverse mass and \(x_{\mathrm{T}}\equiv 2p_{\mathrm{T}}/\sqrt{s}\) scaling properties of hadron production are also studied. As the collision energy increases from \(\sqrt{s}\) = 7–13 TeV, the yields of non- and single-strange hadrons normalized to the pion yields remain approximately constant as a function of \(\sqrt{s}\), while ratios for multi-strange hadrons indicate enhancements. The \(p_\mathrm{{T}}\)-differential cross sections of \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\) and \(\mathrm {p}\) (\(\overline{\mathrm{p}}\)) are compared with next-to-leading order perturbative QCD calculations, which are found to overestimate the cross sections for \(\pi ^{\pm }\) and \(\mathrm{p}\) (\(\overline{\mathrm{p}}\)) at high \(p_\mathrm{{T}}\).

## Introduction

Identified particle spectra and yields, which are among the most fundamental physical observables in high-energy hadronic collisions, have been intensively studied in hadron-collider and cosmic-ray physics for many decades [1]. Hadron production at collider energies originates from soft and hard scattering processes at the partonic level. Hard scatterings, where two partons interact with a large momentum transfer, are responsible for the production of particles with high transverse momenta. This process is theoretically described by perturbative Quantum Chromodynamics (pQCD) calculations based on the factorization theorem [2]. In this approach, the cross section is a convolution of the parton distribution function (PDF), the partonic QCD matrix elements, and the fragmentation function (FF). The PDFs describe the probability densities of finding a parton with a specific flavor carrying fraction *x* of the proton momentum, whereas the FFs encode the probability densities that the parton with a specific flavor fragments into a hadron carrying a fraction of the parton’s longitudinal momentum; both considered at a given energy scale. At the LHC, with increasing center-of-mass collision energy (\(\sqrt{s}\)), the lower *x* regime is probed and contributions from hard-scattering processes increase. In the kinematic region probed by these measurements, high-\(p_{\mathrm {T}}\) particles dominantly originate from the fragmentation of gluons [3, 4]. Parameterizations of both the PDFs and FFs are derived from global analyses [5, 6] based on fits to experimental data at various \(\sqrt{s}\) with next-to-leading order (NLO) accuracy. These include single-inclusive hadron production in semi-inclusive electron-positron annihilation data, semi-inclusive deep-inelastic scattering, and single inclusive hadron spectra at high \(p_{\mathrm {T}}\), notably including results at LHC energies. Results presented in this paper can be used as further input for these studies. In particular, identified particle spectra provide new constraints on the gluon-to-pion and, especially, gluon-to-kaon fragmentation functions [7,8,9,10]. While particle production at high \(p_{\mathrm {T}}\) is expected to be calculable with pQCD, the LHC results are in general not well reproduced by pQCD calculations, see Ref. [6] and references therein. Charged particle production at high \(p_{\mathrm {T}}\) is known to scale with \(x_{\mathrm {T}} \equiv 2p_{\mathrm {T}}/\sqrt{s}\), as observed in a wide energy range up to \(\sqrt{s} = 7 \, \text { TeV}\). This has been observed by the CDF Collaboration in p\(\bar{\text {p}}\) collisions at the Tevatron [11, 12], by the UA1 Collaboration at the CERN SPS [13], by the STAR Collaboration in pp collisions at RHIC [14], and by the CMS Collaboration [15] at the CERN LHC. Above \(x_{\mathrm {T}} \simeq 10^{-2}\), significant deviations from the leading-twist NLO pQCD predictions have been reported in Ref. [16] and are investigated in this paper.

The bulk of particles produced at low transverse momenta (\(p_{\mathrm {T}} <2 \, \text { GeV/c} \)) originate from soft scattering processes involving small momentum transfers. In this regime, particle production cannot be calculated from first principles. Instead, calculations rely on QCD inspired phenomenological models, which are tuned to reproduce previous measurements. Hence, measurements at low \(p_{\mathrm {T}}\) provide further important constraints on such models. The universal transverse mass, \(m_{\mathrm {T}} \equiv \sqrt{m^{2}+p_{\mathrm {T}} ^{2}}\), scaling originally proposed by Hagedorn [17] was first seen to hold approximately at ISR energies [18]. It was then observed by the PHENIX [19, 20] and STAR [21] collaborations to hold only separately for mesons and baryons at RHIC energies, by applying the approximate \(m_{\mathrm {T}}\) scaling relation respectively for pions and protons. At \(\sqrt{s} = 900 \, \text { GeV} \) a disagreement was observed for charged kaons and \(\phi (1020)\) mesons, which indicated a breaking of the generalized scaling behavior [22]. Moreover, recent studies, based on identified particle spectra measured in pp collisions at \(\sqrt{s} = 7 \, \text { TeV}\) by ALICE, indicate that \(m_{\mathrm {T}}\) scaling also breaks in the low-\(p_{\mathrm {T}}\) region [23]. These observations motivate studies of the applicability of \(m_{\mathrm {T}}\) scaling of particle production through the precise measurement of identified hadrons at \(\sqrt{s} = 13 \, \text { TeV}\).

The results reported in this paper are the measurements of the production of \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\), \(\mathrm {K_{S}^{0}}\), \(\mathrm {K^{*}(892)^{0}}\), \(\mathrm {\overline{K}^{*}(892)^{0}}\), \(\mathrm{p}\), \(\mathrm{\overline{p}}\), \(\phi (1020)\), \(\Lambda \), \(\overline{\Lambda }\), \(\Xi ^{-}\), \(\overline{\Xi }^{+}\), \(\Omega ^{-}\) and \(\overline{\Omega }^{+}\) at the highest collision energies, and therefore extend the studies of the energy dependence of the production of light-flavor hadrons into new territory. The study of the production of the \(\mathrm {K^{*}(892)^{0}}\) and \(\phi (1020)\) resonances, containing respectively one and two strange valence quarks, contributes to a better understanding of strange particle production mechanisms. Because of their short lifetimes (\(\sim \,4 \,\text { fm/c} \) for \(\mathrm {K^{*}(892)^{0}}\) and \(\sim \,46 \,\text { fm/c} \) for \(\phi (1020)\)), their decay daughters may undergo re-scattering and/or regeneration processes that affect their yields and the shapes of their \(p_{\mathrm {T}}\) distributions. In addition, multi-strange baryons, \(\Omega ^{-}\) (\(\overline{\Omega }^{+}\)) and \(\Xi ^{-}\) (\(\overline{\Xi }^{+}\)), are of crucial importance due to their dominant strange (s) quark content. Furthermore, the production of \(\mathrm {K_{S}^{0}}\), \(\Lambda \), and \(\overline{\Lambda }\) in pp collisions at \(\sqrt{s} = 7 \, \text { TeV}\) is reported here for the first time, completing the set of reference measurements at that energy [24,25,26,27].

The present measurements serve as important baselines for studies of particle production as a function of the charged-particle multiplicity [28] or event shape (e.g. spherocity) [29] and also provide input to tune the modeling of several contributions in Monte Carlo (MC) event generators such as PYTHIA [30, 31] and EPOS-LHC [32]. In addition, measurements in minimum bias pp collisions reported in this paper serve as reference data to study nuclear effects in proton–lead (p–Pb) and lead–lead (Pb–Pb) collisions.

The paper is organized as follows. In Sect. 2 the ALICE experimental apparatus and the analyzed data samples are described, focusing on the detectors which are relevant for the presented measurements. In Sect. 3 the details of the event and track selection criteria and of the Particle IDentification (PID) techniques are discussed. The results are given in Sect. 4, in which the \(p_{\mathrm {T}}\) spectra and the extraction procedures for the \(p_{\mathrm {T}}\)-integrated yield and average \(p_{\mathrm {T}}\) are presented. Section 5 discusses the results, followed by a summary in Sect. 6. For the remainder of this paper, the masses will be omitted from the symbols of the strongly decaying particles, which will be denoted as \(\mathrm {K^{*0}}\), \(\mathrm {\overline{K}^{*0}}\), and \(\phi \).

## Experimental setup

A detailed description of the ALICE detector and its performance can be found in Refs. [33, 34]. The main subsystems of the ALICE detector used in this analysis are the V0 detector, the Inner Tracking System (ITS), the Time Projection Chamber (TPC), the Time of Flight (TOF) detector, and the High-Momentum Particle Identification Detector (HMPID).

The V0 detector [35] is used for triggering and beam background suppression. It is made up of two scintillator arrays placed along the beam axis on each side of the interaction point (IP) at \(z =340\) cm and \(z=-90\) cm, covering the pseudorapidity regions \(2.8<\eta <5.1\) (V0A) and \(-3.7<\eta <-1.7\) (V0C), respectively.

In the measurements of light-flavor hadrons, primary charged particles are considered. Primary particles are defined as particles with a mean proper lifetime \(\tau \) that is larger than 1 cm/*c*, which are either produced directly in the interaction or from decays of particles produced at the interaction vertex with \(\tau \) shorter than 1 cm/*c*. This excludes particles produced in interactions with the detector material [36]. Primary charged-hadron tracks are reconstructed by the ITS and TPC detectors, which have full azimuthal acceptance within \(|\eta |<0.8\) for full-length tracks. They are located inside a solenoidal magnet providing a magnetic field of \(B=0.5\) T.

The ITS [33, 37] is a silicon tracking detector made up of six concentric cylindrically-shaped layers, measuring high-resolution space points near the collision vertex. The two innermost layers consist of Silicon Pixel Detectors (SPD) used to reconstruct the primary vertex of the collision and short track segments called “tracklets”. The four outer layers are equipped with silicon drift (SDD) and strip (SSD) detectors and allow measurement of the specific energy loss (\(\text {d}E/\text {d}x\)) with a relative resolution of about 10%. The ITS is also used as a stand-alone tracking detector to reconstruct charged particles with momenta below \(200 \, \text { MeV/c}\) that are deflected or decay before reaching the TPC.

The TPC [38] is the main tracking detector of ALICE. It is a large volume cylindrical drift detector spanning the approximate radial and longitudinal ranges \(85< r < 250\) cm and \(-\,250< z < 250\) cm, respectively. The endcaps of the TPC are equipped with multiwire proportional chambers (MWPCs) segmented radially into pad rows. Together with the measurement of the drift time, the TPC provides three dimensional space point information, with up to 159 tracking points. Charged tracks originating from the primary vertex can be reconstructed down to \(p\sim \,100 \, \text { MeV/c} \) [34], albeit with a lower tracking efficiency for identified charged hadrons with \(p_{\mathrm {T}} <200 \, \text { MeV/c} \). Combining information from the ITS and TPC allows the momenta of charged particles to be measured for momenta from 0.05 to \(100 \, \text { GeV/c}\) with a resolution of 1–10%, depending on \(p_{\mathrm {T}}\). The TPC provides charged-hadron identification via measurement of the specific energy loss \(\text {d}E/\text {d}x\) in the fill gas, with a resolution of \(\sim \,5\%\) [38].

The Time of Flight detector (TOF) [39,40,41] is a cylindrical array of multi-gap resistive plate chambers which sits outside the TPC. It covers the pseudorapidity range \(|\eta |<0.9\) with (almost) full azimuthal acceptance. The total time-of-flight resolution, including the resolution on the collision time, is about 90 ps in pp collisions.

The HMPID consists of seven proximity focusing Ring Imaging Cherenkov (RICH) counters. Primary charged particles penetrate the radiator volume, filled with liquid \(\text {C}_{6}\text {F}_{14}\), and generate Cherenkov photons that are converted into photoelectrons in thin CsI-coated photocathodes. Photo-electron clusters, together with pad clusters (also called “MIP” clusters) associated with the primary ionization of a particle, form Cherenkov rings. The amplified signal is read out by MWPCs, filled with \(\text {CH}_{4}\). The detector covers \(|\eta |<0.5\) and \(1.2^{\circ }<\varphi <58.5^{\circ }\), which corresponds to \(\sim \,5\%\) of the TPC geometrical acceptance.

## Event and track selection

### Event selection

The measurements at \(\sqrt{s} = 13 \, \text { TeV}\) are obtained from a minimum bias data sample of pp collisions collected in June 2015 during a period of low pileup in LHC Run 2. The minimum bias trigger required at least one hit in both of the V0 scintillator arrays in coincidence with the arrival of proton bunches from both directions along the beam axis. The mean number of inelastic proton–proton interactions per single bunch crossing ranges between 2 and 14%. A requirement of a coincidence of signals in both V0A and V0C detectors removes contamination from single-diffractive and electromagnetic events. Contamination arising from beam-induced background events, produced outside the interaction region, is removed offline by using timing information from the V0 detector, which has a time resolution better than 1 ns. Background events are further rejected by exploiting the correlation between the number of clusters and the multiplicity of tracklets in the SPD. From the triggered events, only events with a reconstructed primary vertex are considered for the analyses. Additionally, the position of the primary vertex along the beam axis is required to be within \(\pm 10\) cm with respect to the nominal interaction point (center of the ALICE barrel). This requirement ensures that the vast majority of reconstructed tracks are within the central barrel acceptance (\(|\eta |<0.8\)) and it reduces background events by removing unwanted collisions from satellite bunches. Contamination from pileup events, which have more than one pp collision per bunch crossing, were rejected offline by excluding events with multiple primary vertices reconstructed in the SPD [34]. The pileup-rejected events are less than 1% of the total sample of minimum bias events. The size of the analyzed sample after selections ranges between 40 and 60 million events (corresponding to an integrated luminosity 0.74–1.1 \(\mathrm{nb}^{-1}\)), depending on the requirements of the analyses of the different particle species.

The measurements of \(\mathrm {K_{S}^{0}}\), \(\Lambda \), and \(\overline{\Lambda }\) at \(\sqrt{s} = 7 \, \text { TeV}\) are obtained by analyzing a sample of about 150 million events (corresponding to an integrated luminosity \(2.41\,\mathrm {nb}^{-1}\)) collected in 2010 during the LHC Run 1 data taking period. The corresponding trigger and event selection criteria applied were very similar to those used for the measurements at \(\sqrt{s} = 13 \, \text { TeV}\); see Refs. [24, 42] for details on the triggering and event selection for these periods.

All corrections are calculated using Monte Carlo events from PYTHIA 6 and PYTHIA 8. The PYTHIA 6.425 (Perugia 2011 tune) and PYTHIA 8.210 (Monash 2013 tune) event generators were used for \(\sqrt{s} = 13 \, \text { TeV}\). PYTHIA 6.421 (Perugia 0 tune) was used for \(\mathrm {K_{S}^{0}}\) and \(\Lambda \) at \(\sqrt{s} = 7 \, \text { TeV}\) because that production was used for correcting the other \(\sqrt{s} = 7 \, \text { TeV}\) analyses. The particles produced using these event generators were propagated through a simulation of the ALICE detector using GEANT3 [43].

### Track selection

Tracks from charged particles are reconstructed in the TPC and ITS detectors and then propagated to the outer detectors and matched with reconstructed points in the TOF and HMPID. Additionally, in the analysis of \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\) and \(\mathrm{p}(\mathrm{\overline{p}})\), a dedicated tracking algorithm based only on ITS information (ITS stand-alone, ITS-sa) was used to reconstruct low momentum tracks. In the measurements, global tracks, which are reconstructed using the combined ITS and TPC information, are distinguished from ITS-sa tracks.

For analyses using global tracks, track selection criteria are applied to limit the contamination due to secondary particles, to maximize tracking efficiency and improve the \(\text {d}E/\text {d}x\) and momentum resolution for primary charged particles, and to guarantee an optimal PID quality. The number of crossed pad rows in the TPC is required to be at least 70 (out of a maximum possible of 159); the ratio of the number of crossed pad rows to the number of findable clusters (that is the number of geometrically possible clusters which can be assigned to a track) is restricted to be greater than 0.8, see Ref. [34] for the details. The goodness-of-fit values \(\chi ^{2}\) per cluster (\(\chi ^{2}/N_\mathrm{clusters}\)) of the track fit in the TPC must be less than 4. Tracks must be associated with at least one cluster in the SPD and the \(\chi ^{2}\) values per cluster in the ITS are restricted in order to select high-quality tracks. The distance of closest approach (DCA) to the primary vertex in the plane perpendicular to the beam axis (DCA\(_{xy}\)) is required to be less than 7 times the resolution of this quantity; this selection is \(p_{\mathrm {T}}\) dependent, i.e. \(\text {DCA}_{xy} < 7 \times (0.0015+0.05\times (p_{\mathrm {T}}/( \, \text { GeV/c}))^{-1.01})\) cm. A loose selection criterion is also applied on the DCA in the beam direction (DCA\(_{z}\)), by rejecting tracks with DCA\(_{z}\) larger than 2 cm, to remove tracks from possible residual pileup events. The transverse momentum of each track must be greater than 150 \(\mathrm {MeV}/c\) and the pseudorapidity is restricted to the range \(|\eta |<0.8\) to avoid edge effects in the TPC acceptance. Additionally, tracks produced by the reconstructed weak decays of pions and kaons (the “kink” decay topology) are rejected.

For the topological reconstruction of weakly decaying particles, the selected global tracks are combined using specific algorithms, as described in Sect. 4.3. Track selection criteria are the same applied for global tracks with a few exceptions: for tracks used in the reconstruction of \(\mathrm {K_{S}^{0}}\), \(\Lambda \), \(\overline{\Lambda }\), \(\Xi ^{-}\), \(\overline{\Xi }^{+}\), \(\Omega ^{-}\), and \(\overline{\Omega }^{+}\), no ITS information is required and special selection criteria are applied on the DCA to the collision vertex, as shown in Table 4. The kink topology tracks that are used to reconstruct the weak decays of \(\mathrm {K}^{\pm }\) do not have ITS information. For the latter, removal of contributions from pileup collisions outside the trigger proton bunch (“out-of-bunch pileup”) is achieved by requiring that at least one charged decay track matches a hit in a “fast” detector (either the ITS or the TOF detector).

ITS stand-alone tracking uses similar selection criteria to those mentioned above. Tracks are required to have at least four ITS clusters, with at least one in the SPD, three in the SSD and SDD and \(\chi ^{2}/N_\mathrm{clusters}<2.5\). This further reduces contamination from secondary tracks and provides high resolution for the track impact parameter and optimal resolution for \(\text {d}E/\text {d}x\). Similar to global tracks, a \(p_{\mathrm {T}}\)-dependent parameterization of the DCA\(_{xy}\) selection is used, but with different parameters to account for the different resolution. For the \(p_{\mathrm {T}}\) ranges used in this analysis, the selected ITS-sa tracks have the same \(p_{\mathrm {T}}\) resolution as those measured in pp collisions at \(\sqrt{s} = 7 \, \text { TeV}\): 6% for pions, 8% for kaons, and 10% for protons [24].

## Data analysis techniques

Table 1 lists the basic characteristics of the particles studied in this paper. This section describes the techniques used to measure the yields of the various hadron species. In Sect. 4.1, aspects common to all analyses are described, including the correction and normalization procedure and the common sources of systematic uncertainties. Next, the analysis of each hadron species is described in detail. The measurements of charged pions, charged kaons, and (anti)protons, which are performed using several different PID techniques, are described in Sect. 4.2. It is worth noting that charged kaons are also identified using the kink topology of their two-body decays. The measurements of weakly decaying strange hadrons (\(\mathrm {K_{S}^{0}}\), \(\Lambda \), \(\Xi ^{-}\), \(\Omega ^{-}\) and their antiparticles) are reported in Sect. 4.3, followed by the strongly decaying resonances (\(\mathrm {K^{*0}}\), \(\mathrm {\overline{K}^{*0}}\), and \(\phi \)) in Sect. 4.4.

### Common aspects of all analyses

In several of the analyses presented below, the measured PID signal is compared to the expected value based on various particle mass hypotheses. The difference between the measured and expected values is expressed in terms of \(\sigma \), the standard deviation of the corresponding measured signal distribution. The size of this difference, in multiples of \(\sigma \), is denoted \(n_{\sigma }\). In the following, the \(\sigma \) values accounting for the resolution of the PID signals measured in the TPC and TOF detectors are denoted as \(\sigma _{\mathrm {TPC}}\) and \(\sigma _{\mathrm {TOF}}\), respectively.

The corrected yield of each hadron species as a function of \(p_{\mathrm {T}}\) is

\(Y_{\mathrm {corr}}\) is obtained by following the procedure described in previous publications. Here, \(Y_{\mathrm {raw}}\) is the number of particles measured in each \(p_{\mathrm {T}}\) bin and \(A\times \varepsilon \) is the product of the acceptance and the efficiency (including PID efficiency, matching efficiency, detector acceptance, reconstruction, and selection efficiencies). Monte Carlo simulations are used to evaluate \(A\times \varepsilon \), which takes on similar values to those found in our previous analyses. The factor \(f_{\mathrm {SL}}\), also known as the “signal-loss” correction, accounts for reductions in the measured particle yields due to event triggering and primary vertex reconstruction. Such losses are more important at low \(p_{\mathrm {T}}\), since events that fail the trigger conditions or fail to have a reconstructible primary vertex tend to have softer particle \(p_{\mathrm {T}}\) spectra than the average inelastic collision. For \(\sqrt{s} = 13 \, \text { TeV}\), \(f_{\mathrm {SL}}\) deviates from unity by a few percent at low \(p_{\mathrm {T}}\) to less than one percent for \(p_{\mathrm {T}} \gtrsim 2 \, \text { GeV/c} \). The trigger configuration used for \(\sqrt{s} = 7 \, \text { TeV}\) resulted in negligible signal loss, thus \(f_{\mathrm {SL}}\) is set to unity for this energy. The factor \((1-f_{\mathrm {cont}})\) is used to correct for contamination from secondary and misidentified particles; \(f_{\mathrm {cont}}\) is non-zero only for the measurements of \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\), \(\mathrm{p}(\mathrm{\overline{p}})\), \(\Lambda \), and \(\overline{\Lambda }\), and it is more important at low \(p_{\mathrm {T}}\). The computation of \(f_{\mathrm {cont}}\) for those species is described further in the relevant sections below. The factor \(f_{\mathrm {cross. sec.}}\) corrects for inaccuracies in the hadronic production cross sections in GEANT3, which is used in the calculation of \(A\times \varepsilon \) to describe the interactions of hadrons with the detector material of ALICE. GEANT4 and FLUKA [45], which have more accurate descriptions of the hadronic cross sections, are used to calculate the correction factor, which can be different from unity by up to a few percent. The correction \(f_{\mathrm {cross. sec.}}\) is applied only for the analyses of \(\mathrm {K}^{-}\), \(\mathrm{\overline{p}}\), \(\overline{\Lambda }\), \(\overline{\Xi }^{+}\), and \(\overline{\Omega }^{+}\).

After correction, the yields are normalized to the number of inelastic pp collisions using the ratio of the ALICE visible cross section to the total inelastic cross section. This ratio is \(0.852^{+0.062}_{-0.030}\) for \(\sqrt{s} = 7 \, \text { TeV}\) [46] and \(0.7448 \pm 0.0190\) for \(\sqrt{s} = 13 \, \text { TeV}\) [47, 48].

The procedures for the estimation of systematic uncertainties strictly follow those applied in our measurements from LHC Run 1. All described uncertainties are assumed to be strongly correlated among adjacent \(p_{\mathrm {T}}\) bins. For the evaluation of the total systematic uncertainty in every analysis, all contributions originating from different sources are considered to be uncorrelated and summed in quadrature. Components of uncertainties related to the ITS-TPC matching efficiency correction and to the event selection are considered correlated among different measurements. The systematic uncertainty due to the normalization to the number of inelastic collisions is \(\pm ~2.6\%\) for \(\sqrt{s} = 13 \, \text { TeV}\) and \(^{+7.3\%}_{-3.5\%}\) for \(\sqrt{s} = 7 \, \text { TeV}\) independent of \(p_{\mathrm {T}}\). This uncertainty is common to all measured \(p_{\mathrm {T}}\) spectra and \(\text {d}N/\text {d}y\) values (see Sect. 5.1) at a given energy. The systematic uncertainty associated to possible residual contamination from pileup events was estimated varying pileup rejection criteria and was found to be of 1%. The signal loss correction has a small dependence on the Monte Carlo event generator used to calculate it. These variations result in \(p_{\mathrm {T}}\)-dependent uncertainties that are largest at low \(p_{\mathrm {T}}\), where they have values of \(0.2\%\) for \(\Omega \), \(\sim \,1\%\) for \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\), \(\mathrm{p}(\mathrm{\overline{p}})\), and \(\Xi \), and \(\sim \,2\%\) for \(\mathrm {K_{S}^{0}}\), \(\overline{\Lambda }\), \(\mathrm {K^{*0}}\), and \(\phi \).

The systematic uncertainty accounting for the limited knowledge of the material budget is estimated by varying the amount of detector material in the MC simulations within its expected uncertainties [34]. For the analysis of \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\), \(\mathrm{p}(\mathrm{\overline{p}})\), \(\mathrm {K^{*0}}\), and \(\phi \), the values are taken from the studies reported in Refs. [49] and [50]. This uncertainty is estimated to be around \(3.3\%\) for \(\mathrm {K}^{\pm }\), \(1.1\%\) for \(\pi ^{\pm }\), \(1.8\%\) for \(\mathrm{p}(\mathrm{\overline{p}})\), \(3\%\) for \(\mathrm {K^{*0}}\), and \(2\%\) for \(\phi \); it is largest at low momenta and tends to be negligible towards higher momenta. For the measurement of \(\mathrm {K_{S}^{0}}\) and \(\Lambda \) at \(\sqrt{s} = 7 \, \text { TeV}\), the material budget uncertainty is estimated to be \(4\%\), independent of \(p_{\mathrm {T}}\). For the measurements of \(\mathrm {K_{S}^{0}}\), \(\Lambda \), \(\Xi \) and \(\Omega \) at \(\sqrt{s} = 13 \, \text { TeV}\), the material budget uncertainty is \(p_{\mathrm {T}}\) dependent for low \(p_{\mathrm {T}}\) (\(\lesssim 2 \, \text { GeV/c} \)) and constant at higher \(p_{\mathrm {T}}\). For low \(p_{\mathrm {T}}\), the uncertainty reaches maximum values of about 4.7% for \(\mathrm {K_{S}^{0}}\), 6.7% for \(\Lambda \), 6% for \(\Xi \), and 3.5% for \(\Omega \); at high \(p_{\mathrm {T}}\), the uncertainty is less than 1% for \(\mathrm {K_{S}^{0}}\), \(\Lambda \), and \(\Xi \), and about 1.5% for \(\Omega \).

The systematic uncertainty due to the limited description of the hadronic interaction cross sections in the transport code is evaluated using GEANT4 and FLUKA. This leads to uncertainties of up to \(2.8\%\) for \(\pi ^{\pm }\), \(2.5\%\) for \(\mathrm {K}^{\pm }\), \(0.8\%\) for \(\mathrm{p}\), and \(5\%\) for \(\mathrm{\overline{p}}\) [49]. It is at most \(3\%\) for \(\mathrm {K^{*0}}\), \(2\%\) for \(\phi \) and 1–2% for the strange baryons. It is negligible for \(\mathrm {K_{S}^{0}}\) at both reported collision energies. In the following sections, details are given on the contributions (specific to each analysis) related to track or topological selections and signal extraction methods, as well as those related to feed-down.

### Identification of primary charged pions, charged kaons, and (anti)protons

To measure the production of primary charged pions, kaons, and (anti)protons over a wide range of \(p_{\mathrm {T}}\), five analyses using distinct PID techniques were carried out. The individual analyses follow the techniques adopted in previous measurements based on data collected at lower center-of-mass energies and for different collision systems during LHC Run 1 [24, 25, 51,52,53]. The \(p_{\mathrm {T}}\) spectra have been measured from \(p_{\mathrm {T}} =0.1 \, \text { GeV/c} \) for pions, \(p_{\mathrm {T}} =0.2 \, \text { GeV/c} \) for kaons, and \(p_{\mathrm {T}} =0.3 \, \text { GeV/c} \) for protons, up to \(20 \, \text { GeV/c}\) for all three species. The individual analyses with their respective \(p_{\mathrm {T}}\) reaches are summarized in Table 2. All the analysis techniques are extensively described in Refs. [7, 24, 49, 51]. Each procedure is discussed separately in Sects. 4.2.1–4.2.5, with special emphasis on those aspects that are relevant for the current measurements. The results for the different analyses are then combined as described in Sect. 4.2.6.

The calculation of \(f_{\mathrm {cont}}\) in Eq. 1 at low \(p_{\mathrm {T}}\) is performed by subtracting the secondary \(\pi ^{\pm }\), \(\mathrm {K}^{\pm }\), and \(\mathrm{p}(\mathrm{\overline{p}})\) from the primary particle sample. This method is data-driven and it is based on the measured distance of closest approach to the primary vertex in the plane transverse to the beam direction (\(\text {DCA}_{xy}\)), following the same procedure adopted in Ref. [24]. The \(\text {DCA}_{xy}\) distribution of the selected tracks was fitted in every \(p_{\mathrm {T}}\) bin with Monte Carlo templates composed of three ingredients: primary particles, secondaries from material and secondaries from weak decays, each accounting for the expected shapes of the distribution. Because of the different track and PID selection criteria, the contributions are different for each analysis. The resulting corrections are significant at low \(p_{\mathrm {T}}\) and decrease towards higher \(p_{\mathrm {T}}\) due to decay kinematics. Up to \(p_{\mathrm {T}} =2 \, \text { GeV/c} \), the contamination is 2–10% for pions, up to \(20\%\) for kaons (in the narrow momentum range where the \(\text {d}E/\text {d}x\) response for kaons and secondary electrons overlap), and 15–20% for protons.

The main sources of systematic uncertainties for each analysis are summarized in Table 3, including contributions common to all analyses. The systematic uncertainty due to the subtraction of secondary particles is estimated by changing the fit range of the \(\text {DCA}_{xy}\) distribution, resulting in uncertainties of up to \(4\%\) for protons and \(1\%\) for pions, with negligible uncertainties for kaons. The uncertainty due to the matching of TPC tracks with ITS hits is estimated to be in the range \(\sim \) 1–5% for \(p_{\mathrm {T}} \lesssim 3 \, \text { GeV/c} \) depending on \(p_{\mathrm {T}}\), while it takes values around \(6\%\) at higher \(p_{\mathrm {T}}\). This uncertainty together with that resulting from the variation of the track quality selection criteria lead to the systematic uncertainty of the global tracking efficiency that varies from 2.2 to 7.3% from low to high \(p_{\mathrm {T}}\), independent of particle species.