1 Introduction

The production rates of strange and multi-strange hadrons in high-energy hadronic interactions are important observables for the study of the properties of Quantum Chromodynamics (QCD) in the non-perturbative regime. In the simplest case, strange quarks (s) in proton-proton (pp) collisions can be produced from the excitation of the sea partons. Indeed, in the past decades a significant effort has been dedicated to the study of the actual strangeness content of the nuclear wave function [1]. In QCD-inspired Monte Carlo generators based on Parton Showers (PS) [2] the hard (perturbative) interactions are typically described at the Leading Order (LO). In this picture the s quark can be produced in the hard partonic scattering via flavour creation and flavour excitation processes as well as in the subsequent shower evolution via gluon splitting. At low transverse momentum, \(s\bar{s}\) pairs can be produced via non-perturbative processes, as described for instance in string fragmentation models, where the production of strangeness is suppressed with respect to light quark production due to the larger strange quark mass [3]. However, these models fail to quantitatively describe strangeness production in hadronic collisions [4,5,6].

An enhanced production of strange hadrons in heavy-ion collisions was suggested as a signature for the creation of a Quark-Gluon Plasma (QGP) [7, 8]. The main argument in these early studies was that the mass of the strange quark is of the order of the QCD deconfinement temperature, allowing for thermal production in the deconfined medium. The lifetime of the QGP was then estimated to be comparable to the strangeness relaxation time in the plasma, leading to full equilibration. Strangeness enhancement in heavy-ion collisions was indeed observed at the SPS [9] and higher energies [10, 11]. However, strangeness enhancement is no longer considered an unambiguous signature for deconfinement (see e.g. [12]). Strange hadron production in heavy-ion collisions is currently usually described in the framework of statistical-hadronisation (or thermal) models [13, 14]. In central heavy-ion collisions, the yields of strange hadrons turn out to be consistent with the expectation from a grand-canonical ensemble, i.e. the production of strange hadrons is compatible with thermal equilibrium, characterised by a common temperature. On the other hand, the strange hadron yields in elementary collisions are suppressed with respect to the predictions of the (grand-canonical) thermal models. The suppression of the relative abundance of strange hadrons with respect to lighter flavours was suggested to be, at least partially, a consequence of the finite volume, which makes the application of a grand-canonical ensemble not valid in hadron-hadron and hadron-nucleus interactions (canonical suppression) [15,16,17]. However, this approach cannot explain the observed particle abundances if the same volume is assumed for both strange and non-strange hadrons [18] and does not describe the system size dependence of the \(\phi \) meson, a hidden-strange hadron [19, 20].

The ALICE Collaboration recently reported an enhancement in the relative production of (multi-) strange hadrons as a function of multiplicity in pp collisions at \(\sqrt{s}\) = 7 TeV [21] and in p–Pb collisions at \(\sqrt{s_\mathrm{NN}}\) = 5.02 TeV [22, 23]. In the case of p-Pb collisions, the yields of strange hadrons relative to pions reach values close to those observed in Pb–Pb collisions at full equilibrium. These are surprising observations, because thermal strangeness production was considered to be a defining feature of heavy-ion collisions, and because none of the commonly-used pp Monte Carlo models reproduced the existing data [3, 21]. The mechanisms at the origin of this effect need to be understood, and then implemented in the state-of-the-art Monte Carlo generators [3].

In this paper, strangeness production in pp interactions is studied at the highest energy reached at the LHC, \(\sqrt{s}\) = 13 TeV. We present the measurement of the yields and transverse momentum (\(p_\mathrm{T}\)) distributions of single-strange (\(\mathrm{K}^{0}_{S}\), \(\Lambda \), \(\overline{\Lambda }\)) and multi-strange (\(\Xi ^{-}\), \(\overline{\Xi }^{+}\), \(\Omega ^{-}\), \(\overline{\Omega }^{+}\)) particles at mid-rapidity, \(\left| y\right| < 0.5\), with the ALICE detector [24]. The comparison of the present results with the former ones for pp and p–Pb interactions allows the investigation of the energy, multiplicity and system size dependence of strangeness production. Schematically, the multiplicity of a given pp event depends on (i) the number of Multiple Parton Interactions (MPI), (ii) the momentum transfer of those interactions, (iii) fluctuations in the fragmentation process. A systematic study of the biases induced by the choice of the multiplicity estimator along with the specific connections to the underlying MPI are also discussed in this paper.

The paper is organised as follows. In Sect. 2 we discuss the data set and detectors used for the measurement; in Sect. 3 we describe the analysis techniques; in Sect. 4 we cover the evaluation of the systematic uncertainties; in Sect. 5 we present and discuss the results; in Sect. 6 we report our conclusions.

2 Experimental setup and data selection

A detailed description of the ALICE detector and its performance can be found in [24, 25]. In this section, we briefly outline the main detectors used for the measurements presented in this paper. The ALICE apparatus comprises a central barrel used for vertex reconstruction, track reconstruction and charged-hadron identification, complemented by specialised forward detectors. The central barrel covers the pseudorapidity region \(\left| \eta \right| < 0.9\). The main central-barrel tracking devices used for this analysis are the Inner Tracking System (ITS) and the Time-Projection Chamber (TPC), which are located inside a solenoidal magnet providing a 0.5 T magnetic field. The ITS is composed of six cylindrical layers of high-resolution silicon tracking detectors. The innermost layers consist of two arrays of hybrid Silicon Pixel Detectors (SPD), located at an average radial distance r of 3.9 and 7.6 cm from the beam axis and covering \(\left| \eta \right| < 2.0\) and \(\left| \eta \right| < 1.4\), respectively. The SPD is also used to reconstruct tracklets, short two-point track segments covering the pseudorapidity region \(\left| \eta \right| < 1.4\). The outer layers of the ITS are composed of silicon strips and drift detectors, with the outermost layer having a radius \(r = 43~\mathrm {cm}\). The TPC is a large cylindrical drift detector of radial and longitudinal sizes of about \(85<r<250~\hbox {cm}\) and \(-250<z<250~\hbox {cm}\), respectively. It is segmented in radial “pad rows”, providing up to 159 tracking points. It also provides charged-hadron identification information via specific ionisation energy loss in the gas filling the detector volume. The measurement of strange hadrons is based on “global tracks”, reconstructed using information from the TPC as well as from the ITS, if the latter is available. Further outwards in radial direction from the beam-pipe and located at a radius of approximately 4 m, the Time of Flight (TOF) detector measures the time-of-flight of the particles. It is a large-area array of multigap resistive plate chambers with an intrinsic time resolution of 50 ps. The V0 detectors are two forward scintillator hodoscopes employed for triggering, background suppression and event-class determination. They are placed on either side of the interaction region at \(z=-0.9~\hbox {m}\) and \(z=3.3~\hbox {m}\), covering the pseudorapidity regions \(-3.7<\eta <-1.7\) and \(2.8<\eta <5.1\), respectively.

The data considered in the analysis presented in this paper were collected in 2015, at the beginning of Run 2 operations of the LHC at \(\sqrt{s}\) = 13 TeV. The sample consists of 50 M events collected with a minimum bias trigger requiring a hit in both V0 scintillators in coincidence with the arrival of proton bunches from both directions. The interaction probability per single bunch crossing ranges between 2 and 14%.

The contamination from beam-induced background is removed offline by using the timing information in the V0 detectors and taking into account the correlation between tracklets and clusters in the SPD detector, as discussed in detail in [25]. The contamination from in-bunch pile-up events is removed offline excluding events with multiple vertices reconstructed in the SPD. Part of the data used in this paper were collected in periods in which the LHC collided “trains” of bunches each separated by 50 ns from its neighbours. In these beam conditions most of the ALICE detectors have a readout window wider than a single bunch spacing and are therefore sensitive to events produced in bunch crossings different from those triggering the collision. In particular, the SPD has a readout window of 300 ns. The drift speed in the TPC is about \(2.5~\hbox {cm}/{\upmu }\hbox {s}\), which implies that events produced less than about \(0.5~{\upmu }\hbox {s}\) apart cannot be resolved. Pile-up events produced in different bunch crossings are removed exploiting multiplicity correlations in detectors having different readout windows.

3 Analysis details

The results are presented for primary strange hadrons.Footnote 1 The measurements reported here have been obtained for events having at least one charged particle produced with \(p_\mathrm{T} >0\) in the pseudorapidity interval \(\left| \eta \right| < 1\) (\(\mathrm {INEL}>0\)), corresponding to about 75% of the total inelastic cross-section. In order to study the multiplicity dependence of strange and multi-strange hadrons, for each multiplicity estimator the sample is divided into event classes based on the total charge deposited in the V0 detectors (V0M amplitude) or on the number of tracklets in two pseudorapidity regions: \(\left| \eta \right| < 0.8\) and \(0.8< \left| \eta \right| < 1.5\). The event classes are summarised in Table 1. Since the measurement of strange hadrons is performed in the region \(\left| y\right| < 0.5\), the usage of these three estimators allows one to evaluate potential biases on particle production, arising from measuring the multiplicity in a pseudorapidity region partially overlapping with the one of the reconstructed strange hadrons (\(N_{\mathrm{tracklets}}^{\left| \eta \right| < 0.8}\)), or disjoint from it (V0M and \(N_\mathrm{tracklets}^{0.8< \left| \eta \right| < 1.5}\)). In particular, the effect of fluctuations can be expected to be stronger if the multiplicity estimator and the observable of interest are measured in the same pseudorapidity region. The usage of two different non-overlapping estimators allows the study of the effect of a rapidity gap between the multiplicity estimator and the measurement of interest.

The events used for the data analysis are required to have a reconstructed vertex in the fiducial region \(\left| z\right| <10~\hbox {cm}\). As mentioned in the previous section, events containing more than one distinct vertex are tagged as pile-up and discarded. For each event class and each multiplicity estimator, the average pseudorapidity density of primary charged-particles \(\langle \mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \rangle \) is measured at mid-rapidity (\(\left| \eta \right| < 0.5\)) using the technique described in [27]. The \(\langle \mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \rangle \) values, corrected for acceptance and efficiency as well as for contamination from secondary particles and combinatorial background, are listed in Table 1. When multiplicity event classes are selected outside the \(\left| \eta \right| < 0.8\) region, the corresponding charged particle multiplicity at mid-rapidity is characterized by a large variance. In the case of the V0M estimator, the variance ranges between 30 and 70% of the mean \(\mathrm {d}N_{\mathrm{ch}}/\mathrm {d}\eta \) for the highest and lowest multiplicity classes, respectively.

Table 1 Event classes selected according to different multiplicity estimators (see text for details). For each estimator, the second column summarises the relative multiplicity w.r.t. the INEL > 0 event class and the third column represents the corresponding fraction of the INEL > 0 cross-section. The average charged particle multiplicity density is reported in the last column for all event classes. For all the multiplicity estimators, the charged particle multiplicity density is quoted in \(\left| \eta \right| < 0.5\)

The strange hadrons \(\mathrm{K}^{0}_{S}\), \(\Lambda \), \(\overline{\Lambda }\), \(\Xi ^{-}\), \(\overline{\Xi }^{+}\), \(\Omega ^{-}\) and \(\overline{\Omega }^{+}\) are reconstructed at mid-rapidity (\(\left| y\right| < 0.5\)) with an invariant mass analysis, exploiting their specific weak decay topology. The following decay channels are studied [28]:

$$\begin{aligned} \begin{array}{ll} \mathrm{K}^{0}_{S} \rightarrow {\pi }^{+} + {\pi }^{-} &{} \hbox {B.R.} = (69.20 \pm 0.05) \% \\ \Lambda (\overline{\Lambda }) \rightarrow \mathrm p (\overline{\mathrm{p}}) + {\pi }^{-} ({\pi }^{+}) &{} \hbox {B.R.} = (63.9 \pm 0.5) \% \\ \Xi ^{-} (\overline{\Xi }^{+}) \rightarrow \Lambda (\overline{\Lambda }) + {\pi }^{-} ({\pi }^{+}) &{} \hbox {B.R.} = (99.887 \pm 0.035) \% \\ \Omega ^{-} (\overline{\Omega }^{+}) \rightarrow \Lambda (\overline{\Lambda }) + \mathrm{K}^{-} (\mathrm{K}^{+}) &{} \hbox {B.R.} = (67.8 \pm 0.7) \% \end{array} \end{aligned}$$

In the following, we refer to the sum of particles and anti-particles, \(\Lambda +\overline{\Lambda } \), \(\Xi ^{-} +\overline{\Xi }^{+} \) and \(\Omega ^{-} +\overline{\Omega }^{+} \), simply as \(\Lambda \), \(\Xi \) and \(\Omega \).

The details of the analysis have been discussed in earlier ALICE publications [5, 6, 18, 22]. The tracks retained in the analysis are required to cross at least 70 TPC readout pads out of a maximum of 159. Tracks are also required not to have large gaps in the number of expected tracking points in the radial direction. This is achieved by checking that the number of clusters expected based on the reconstructed trajectory and the measurements in neighbouring TPC pad rows do not differ by more than 20%.

Table 2 Track, topological and candidate selection criteria applied to \(\mathrm{K}^{0}_{S}\), \(\Lambda \) and \(\overline{\Lambda }\) candidates. DCA stands for “distance of closest approach”, PV represents the “primary event vertex” and \(\theta \) is the angle between the momentum vector of the reconstructed \(V^0\) and the displacement vector between the decay and primary vertices. The selection on DCA between \(V^0\) daughter tracks takes into account the corresponding experimental resolution
Table 3 Track, topological and candidate selection criteria applied to charged \(\Xi ^{-}\), \(\overline{\Xi }^{+}\), \(\Omega ^{-}\) and \(\overline{\Omega }^{+}\) candidates. DCA stands for “distance of closest approach”, PV represents the “primary event vertex” and \(\theta \) is the angle between the momentum vector of the reconstructed \(V^0\) or cascade and the displacement vector between the decay and primary vertices. The selection on DCA between \(V^0\) daughter tracks takes into account the corresponding experimental resolution

Each decay product arising from \(V^0\) (\(\mathrm{K}^{0}_{S}\), \(\Lambda \), \(\overline{\Lambda }\)) and cascade (\(\Xi ^{-}\), \(\overline{\Xi }^{+}\), \(\Omega ^{-}\), \(\overline{\Omega }^{+}\)) candidates is verified to lie within the fiducial tracking region \(\left| \eta \right| < 0.8\). The specific energy loss (\(\mathrm{d}E/\mathrm{d}x\)) measured in the TPC, used for the particle identification (PID) of the decay products, is also requested to be compatible within \(5\sigma \) with the one expected for the corresponding particle species’ hypothesis. The \(\mathrm{d}E/\mathrm{d}x\) is evaluated as a truncated mean using the lowest 60% of the values out of a possible total of 159. This leads to a resolution of about 6%. A set of “geometrical” selections is applied in order to identify specific decay topologies (topological selection), improving the signal/background ratio. The topological variables used for \(V^{0}\hbox {s}\) and cascades are described in detail in [6]. In addition, in order to reject the residual out-of-bunch pile-up background on the measured yields, it is requested that at least one of the tracks from the decay products of the (multi-)strange hadron under study is matched in either the ITS or the TOF detector. The selections used in this paper are summarised in Table 2 for the \(V^0\)s and in Table 3 for the cascades.

Strange hadron candidates are required to be in the rapidity window \(\left| y\right| < 0.5\). \(\mathrm{K}^{0}_{S}\) (\(\Lambda \)) candidates compatible with the alternative \(V^0\) hypothesis are rejected if they lie within \(\pm 5~\mathrm {MeV}/c^2\) (\(\pm 10~\mathrm {MeV}/c^2\)) of the nominal \(\Lambda \) (\(\mathrm{K}^{0}_{S}\)) mass. A similar selection is applied to the \(\Omega \), where candidates compatible within \(\pm 8~\mathrm {MeV}/c^2\) of the nominal \(\Xi \) mass are rejected. The width of the rejected region was determined according to the invariant mass resolution of the corresponding competing signal. Furthermore, candidates whose proper lifetimes are unusually large for their expected species are also rejected to avoid combinatorial background from interactions with the detector material. The signal extraction is performed as a function of \(p_\mathrm{T}\). A preliminary fit is performed on the invariant mass distribution using a Gaussian plus a linear function describing the background. This allows for the extraction of the mean (\(\mu \)) and width (\(\sigma _{\mathrm{G}}\)) of the peak. A “peak” region is defined within \(\pm 6(4)\sigma _{\mathrm{G}}\) for \(V^{0}\hbox {s}\) (cascades) with respect to \(\mu \) for any measured \(p_\mathrm{T}\) bin. Adjacent background bands, covering a combined mass interval as wide as the peak region, are defined on both sides of that central region. The signal is then extracted with a bin counting procedure, subtracting counts in the background region from those of the signal region. Alternatively, the signal is extracted by fitting the background with a linear function extrapolated under the signal region. This procedure is used to compute the systematic uncertainty due to the signal extraction. Examples of the invariant mass peaks for all particles are shown in Fig. 1.

Fig. 1
figure 1

Invariant mass distributions for \(\mathrm{K}^{0}_{S}\), \(\Lambda \), \(\Xi ^{-}\), \(\Omega ^{-}\) in different V0M multiplicity and \(p_\mathrm{T}\) intervals. The candidates are reconstructed in \(\left| y\right| < 0.5\). The grey areas delimited by the short-dashed lines are used for signal extraction in the bin counting procedure. The red dashed lines represent the fit to the invariant mass distributions, shown for drawing purpose only

Only the \(\Lambda \) turns out to be affected by a significant contamination from secondary particles, coming from the decay of charged and neutral \(\Xi \). In order to estimate this contribution we use the measured \(\Xi ^{-}\) and \(\overline{\Xi }^{+}\) spectra, folded with a \(p_\mathrm{T}\)-binned 2D matrix describing the decay kinematics and secondary \(\Lambda \) reconstruction efficiencies. The \(\Xi \rightarrow \Lambda \pi \) decay matrix is extracted from Monte Carlo (see below for the details on the generator settings). The fraction of secondary \(\Lambda \) particles in the measured spectrum varies between 10 and 20%, depending on \(p_\mathrm{T}\) and multiplicity. Further details on the uncertainties characterising the feed-down contributions are provided in the next section.

The raw \(p_\mathrm{T}\) distributions are corrected for acceptance and efficiency using Monte Carlo simulated data. Events are generated using the PYTHIA 6.425, (Tune Perugia 2011) [29, 30] event generator, and transported through a GEANT 3 [31] (v2-01-1) model of the detector. With respect to previous GEANT 3 versions, the adopted one contains a more realistic description of (anti)proton interactions. The quality of this description was cross-checked comparing to the results obtained with the state-of-the-art transport codes FLUKA [32, 33] and GEANT 4.9.5 [34]. It was found that a correction factor \(<5\%\) is needed for the efficiency of \(\overline{\Lambda }\), \(\overline{\Xi }^{+}\), \(\overline{\Omega }^{+}\) for \(p_\mathrm{T} < 1~\mathrm{GeV}/c\), while the effect is negligible at higher \(p_\mathrm{T}\). Events generated using PYTHIA 8.210 (tune Monash 2013) [35, 36] and EPOS-LHC (CRMC package 1.5.4) [37] and transported in the same way are used for systematic studies, namely to compute the systematic uncertainties arising from the normalisation and from the closure of the correction procedure (details provided in the next sections).

The acceptance-times-efficiency changes with \(p_\mathrm{T}\), saturating at a value of about 40%, 30%, 30% and 20% at \(p_\mathrm{T}\) \(\simeq \) 2, 3, 3 and 4 \(\mathrm{GeV}/c\) for \(\mathrm{K}^{0}_{S}\), \(\Lambda \), \(\Xi \) and \(\Omega \), respectively. These values include the losses due to the branching ratio. They are found to be independent of the multiplicity class within 2%, limited by the available Monte Carlo simulated data. The dependence of the efficiency on the generated \(p_\mathrm{T}\) distributions was checked for all particle species. It is found to be relevant only in the case of the \(\Omega \), where large \(p_\mathrm{T}\) bins are used. This effect is removed by reweighting the Monte Carlo \(p_\mathrm{T}\) distribution with the measured one using an iterative procedure.

In order to compute \(\langle p_\mathrm{T} \rangle \) and the \(p_\mathrm{T}\)-integrated production yields, the spectra are fitted with a Tsallis–Lévy [38] distribution to extrapolate in the unmeasured \(p_\mathrm{T}\) region. The systematic uncertainties on this extrapolation procedure are evaluated using other fit functions, as discussed in Sec. 4.

4 Systematic uncertainties

Several sources of systematic effects on the evaluation of the \(p_\mathrm{T}\) distributions were investigated. The main contributions for three representative \(p_\mathrm{T}\) values are summarised in Table 4 for the INEL \(> 0\) data sample.

The stability of the signal extraction method was checked by varying the widths used to define the “signal” and “background” regions, expressed in terms of number of \(\sigma _{G}\)