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
$$\begin{aligned} Y_{\mathrm {corr}} = \frac{Y_{\mathrm {raw}}}{\Delta p_{\mathrm {T}} \,\Delta {y}}\times \frac{f_{\mathrm {SL}}}{A\times \varepsilon }\times (1-f_{\mathrm {cont}})\times f_{\mathrm {cross. sec.}} ~. \end{aligned}$$
(1)
\(Y_{\mathrm {corr}}\) 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.
Table 2 Summary of the kinematic ranges (\(p_\mathrm{T}\) (GeV/c) and \(\eta \) or y) covered by the individual analyses for the measurement of \(\pi ^{\pm }\), \(\text {K}^{\pm }\) and \((\bar{\text {p}})\text {p}\) in pp collisions at \(\sqrt{s} = 13 \, \text { TeV}\)
Table 3 Summary of the main sources and values of the relative systematic uncertainties (expressed in %) for the \(\pi ^{+} +\pi ^{-} \), \(\mathrm {K}^{+} +\mathrm {K}^{-} \), and \(\mathrm{p} +\mathrm{\overline{p}} \) \(p_{\mathrm {T}}\)-differential yields. A single value between two or three columns indicates that no \(p_{\mathrm {T}}\) dependence is observed. Values are reported for low, intermediate (wherever they are available) and high \(p_{\mathrm {T}}\). The abbreviation “negl.” indicates a negligible value