Multiplicity and rapidity dependence of strange hadron production in pp, pPb, and PbPb collisions at the LHC

Measurements of strange hadron (K0s, Lambda + anti-Lambda, and Xi+ + Xi-) transverse momentum spectra in pp, pPb, and PbPb collisions are presented over a wide range of rapidity and event charged-particle multiplicity. The data were collected with the CMS detector at the CERN LHC in pp collisions at sqrt(s) = 7 TeV, pPb collisions at sqrt(s[NN]) = 5.02 TeV, and PbPb collisions at sqrt(s[NN]) = 2.76 TeV. The average transverse kinetic energy is found to increase with multiplicity, at a faster rate for heavier strange particle species in all systems. At similar multiplicities, the difference in average transverse kinetic energy between different particle species is observed to be larger for pp and pPb events than for PbPb events. In pPb collisions, the average transverse kinetic energy is found to be slightly larger in the Pb-going direction than in the p-going direction for events with large multiplicity. The spectra are compared to models motivated by hydrodynamics.


Introduction
Studies of strange-particle production in high energy collisions of protons and heavy ions provide important means to investigate the dynamics of the collision process.Earlier studies of relativistic heavy ion collisions at the BNL RHIC and CERN SPS colliders indicated an enhancement of strangeness production with respect to proton-proton (pp) collisions [1,2], which was historically interpreted to be due to the formation of a high-density quark-gluon medium [3].The abundance of strange particles at different center-of-mass energies is in line with calculations from thermal statistical models [4][5][6].In gold-gold (AuAu) collisions at RHIC, strong azimuthal correlations of final-state hadrons were observed, suggesting that the produced medium behaves like a near-perfect fluid undergoing a pressure-driven anisotropic expansion [2].Studies of strangeness and light flavor production and dynamics in heavy ion collisions have provided further insight into the medium's fluid-like nature and evidence for its partonic collectivity [2,7].
In recent years, the observation of a long-range "ridge" at small azimuthal separations in twoparticle correlations in pp [8] and proton-lead (pPb) [9][10][11] collisions with high event-by-event charged-particle multiplicity (referred to hereafter as "multiplicity") has provided an indication for collective effects in systems that are an order of magnitude smaller in size than heavy ion collisions.The nature of the observed long-range particle correlations in high multiplicity pp and pPb collisions is still under intense debate [12].While the collective flow of a fluid-like medium provides a natural interpretation [13][14][15][16], other models attribute this behavior to the initial correlation of gluons [17][18][19][20][21], or the anisotropic escape of particles [22].
Studies of identified particle production and correlations in high multiplicity pp and pPb collisions provide detailed information about the underlying particle production mechanism.Identified particle (including strange-hadron) transverse momentum (p T ) spectra and azimuthal anisotropies in lead-lead (PbPb) collisions at the CERN LHC have been studied [23,24] and described by hydrodynamic models [25,26].Similar measurements have been performed in pPb collisions as a function of multiplicity, where an indication of a common velocity boost to the produced particles, known as "radial flow" [27,28], and for a mass dependence of the anisotropic flow [29,30] have been observed.When comparing pPb and PbPb systems at similar multiplicities, a stronger radial velocity boost is seen in the smaller pPb collision system [27,30].This could be related to a much higher initial energy density in a high multiplicity but smaller system, resulting in a larger pressure gradient outward along the radial direction, as predicted in Ref. [31].To perform a quantitative comparison, a common average radial-flow velocity from different collision systems can be extracted from a simultaneous fit to the spectra of various particle species, based on the blast-wave model [32].Inspired by hydrodynamics, the blast-wave model assumes a common kinetic freeze-out temperature and radialflow velocity for all particles during the expansion of the system.The dependence of spectral shapes for identified hadrons on the multiplicity has been observed in high energy electron and proton-antiproton collisions [33,34], but this observation was not explored extensively in the hydrodynamic context.The blast-wave fit has been studied in pp, deuterium-gold, and AuAu collisions at RHIC [35].In pp collisions, it has been shown through studies with simulation that color reconnection processes could describe the observed multiplicity dependence of identified particle spectra [23,36].
It is of interest to study possible collective phenomena in very high multiplicity pp collisions, as demonstrated by the observation of long-range particle correlations in these events [8].Since pp events represent an even smaller system than pPb events, a stronger radial-flow boost might be present compared to pPb and PbPb events at a comparable multiplicity [31].Furthermore,

Detector and data samples
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, which provides an axial field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker (with 13 and 14 layers in the central and endcap regions, respectively), a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections.The tracker covers the pseudorapidity range |η| < 2.5.Reconstructed tracks with 1 < p T < 10 GeV typically have resolutions of 1.5-3% in p T and 25-90 (45-150) µm in the transverse (longitudinal) impact parameter [38].The ECAL and HCAL each cover |η| < 3.0 while forward hadron calorimeters (HF) cover 3 < |η| < 5. Muons with |η| < 2.4 are measured with gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid.A more detailed description of the CMS detector, together with a definition of the coordinate system and the relevant kinematic variables, can be found in Ref. [39].The Monte Carlo (MC) simulation of the particle propagation and detector response is based on the GEANT4 [40] program.
The data samples used in this analysis are as follows: pp collisions collected in 2010 at √ s = 7 TeV, pPb collisions collected in 2013 at √ s = 5.02 TeV, and PbPb collisions collected in 2011 at √ s NN = 2.76 TeV, with integrated luminosities of 6.2 pb −1 , 35 nb −1 , and 2.3 µb −1 , respectively.
For the pPb data, the beam energies are 4 TeV for the protons and 1.58 TeV per nucleon for the lead nuclei.The data were collected in two different run periods: one with the protons circulating in the clockwise direction in the LHC ring, and one with them circulating in the counterclockwise direction.By convention, the proton beam rapidity is taken to be positive when combining the data from the two run periods.Because of the asymmetric beam conditions, the nucleon-nucleon center-of-mass in the pPb collisions moves with speed β = 0.434 in the laboratory frame, corresponding to a rapidity of 0.465.As a consequence, the rapidity of a particle in the nucleon-nucleon center-of-mass frame (y cm ) is detected in the laboratory frame

Selection of events and tracks
The triggers, event reconstruction, and event selection are the same as those discussed for pp, pPb, and PbPb collisions in Refs.[8,41].They are briefly outlined in the following paragraphs for pp and pPb collisions, which are the main focus of this Letter.A subset of peripheral PbPb data collected in 2011 with a minimum-bias trigger is reprocessed using the same event selection and track reconstruction algorithm as for the present pPb and pp analyses, in order to more directly compare the three systems at the same multiplicity.Details of the 2011 PbPb analysis can be found in Refs.[41,42].
Minimum-bias pPb events are triggered by requiring at least one track with p T > 0.4 GeV to be found in the pixel tracker.Because of hardware limitations in the data acquisition rate, only a small fraction (∼10 −3 ) of triggered minimum-bias events are recorded.In order to collect a large sample of high-multiplicity pPb collisions, a dedicated high-multiplicity trigger is implemented using the CMS Level-1 (L1) and high-level trigger (HLT) systems [43].At L1, the total transverse energy summed over the ECAL and HCAL is required to exceed either 20 or 40 GeV, depending on the multiplicity requirement as specified below.Charged particles are reconstructed at the HLT level using the pixel detectors.It is required that these tracks originate within a cylindrical region (30 cm in length along the direction of the beam axis and 0.2 cm in radius in the direction perpendicular to that axis) centered on the nominal interaction point.For each event, the number of pixel tracks (N online trk ) with |η| < 2.4 and p T > 0.4 GeV is determined for each reconstructed vertex.Only tracks with a distance of closest approach 0.4 cm or less to one of the vertices are included.The HLT selection requires N online trk for the vertex with the largest number of tracks to exceed a specific value.Data are collected in pPb collisions with thresholds N online trk > 100 and 130 for events with an L1 transverse energy threshold of 20 GeV, and N online trk > 160 and 190 for events with an L1 threshold of 40 GeV.While all events with N online trk > 190 are accepted, only a fraction of the events from the other thresholds are retained.This fraction is dependent on the instantaneous luminosity.Data from both the minimumbias trigger and the high-multiplicity trigger are retained for offline analysis.Similar highmultiplicity triggers, with different thresholds, were developed for pp collisions, with details given in Ref. [8].
In the subsequent analysis of all collision systems, hadronic events are selected by requiring the presence of at least one energy deposit larger than 3 GeV in each of the two HF calorimeters.Events are also required to contain a primary vertex within 15 cm of the nominal interaction point along the beam axis and 0.15 cm in the transverse direction, where the primary vertex is the reconstructed vertex with the largest track multiplicity.At least two reconstructed tracks are required to be associated with this primary vertex, a condition that is important only for minimum-bias events.Beam-related background is suppressed by rejecting events in which less than 25% of all reconstructed tracks satisfy the high-purity selection defined in Ref. [38].In the pPb data sample, there is a 3% probability to have at least one additional interaction in the same bunch crossing (pileup).The procedure used to reject pileup events in pPb collisions is described in Ref. [41].It is based on the number of tracks associated with each reconstructed vertex and the distance between different vertices.A purity of 99.8% for single pPb collision events is achieved for the highest multiplicity pPb range studied in this Letter.For the pp data, the average number of collisions per bunch crossing is 1.2.However, pp interactions that are well separated from each other do not interfere.Thus, among events identified as containing pileup, the event is retained if the separation between the primary vertex and any other vertex exceeds 1 cm.In such events, only tracks from the highest multiplicity vertex are used.
The event-by-event charged-particle multiplicity N offline trk is defined using primary tracks, i.e, tracks that satisfy the high-purity criteria of Ref. [38] and, in addition, the following criteria designed to improve track quality and ensure the tracks emanate from the primary vertex.The impact parameter significance of the track with respect to the primary vertex in the direction along the beam axis, d z /σ(d z ), is required to be less than 3, as is the corresponding impact parameter in the transverse plane, d T /σ(d T ).The relative p T uncertainty, σ(p T )/p T , must be less than 10%.To ensure high tracking efficiency and to reduce the rate of misreconstructed tracks, the tracks are required to satisfy |η| < 2.4 and p T > 0.4 GeV.Based on simulated samples generated with the HIJING program, the efficiency for primary track reconstruction is found to be greater than 80% for charged particles with p T > 0.6 GeV and |η| < 2.4.For the multiplicity range studied in this Letter, no dependence of the tracking efficiency on multiplicity is found and the rate of misreconstructed tracks is 1-2%.
The pp, pPb, and PbPb data are divided into classes based on N offline trk .The quantity N corrected trk is the corresponding multiplicity corrected for detector and algorithm inefficiencies in the same kinematic region (|η| < 2.4 and p T > 0.4 GeV).The fraction of the total multiplicity found in each interval and the average number of tracks both before and after accounting for the corrections are listed in Table 1 for the pp data and in Ref. [41] for the pPb and PbPb data.The uncertainty in the average value N corrected trk is evaluated from the uncertainty in the tracking efficiency, which is 3.9% for a single track [48].For the pp data, six multiplicity intervals, indicated in Table 1, are defined, which are inclusive for the lower bounds and exclusive for the upper bounds, as indicated in Table 1.The average N offline trk value of minimum-bias events is similar to that for the multiplicity range N offline trk < 35.For the pPb and PbPb data, eight intervals are defined.These eight intervals are indicated, e.g., in the legend of Fig. 2. Note that, unlike pp and PbPb collisions, N offline trk for pPb collisions is not determined in the center-of-mass frame.However, the difference in the N offline trk definition between the laboratory and the centerof-mass frames is found to be minimal and so this difference is ignored.The detector condition has been checked to be stable for events with different multiplicities.4 The K 0 S , Λ, and Ξ − reconstruction and yields The reconstruction and selection procedures for K 0 S , Λ, and Ξ − candidates are presented in Refs.[30,49].To increase the efficiency for tracks with low momenta and large impact parameters, both characteristic of the strange-particle decay products, the loose selection of tracks, as defined in Ref. [38], is used.The K 0 S and Λ candidates (generically referred to as "V 0 s") are reconstructed, by combining oppositely charged particles to define a secondary vertex.Each of the two tracks must have hits in at least four layers of the silicon tracker, and transverse and longitudinal impact parameter significances with respect to the primary vertex greater than 1.The distance of closest approach of the pair of tracks to each other is required to be less than 0.5 cm.The fitted three-dimensional vertex of the pair of tracks is required to have a χ 2 value divided by the number of degrees of freedom less than 7.Each of the two tracks is assumed to be a pion in the case of the K 0 S reconstruction.As the proton carries nearly all of the momentum in the Λ decay, the higher-momentum track is assumed to be a proton and the other track a pion in the case of the Λ reconstruction.To reconstruct Ξ − particles, a Λ candidate is combined with an additional charged particle carrying the correct sign, to define a common secondary vertex.This additional track is required to have hits in at least four layers of the silicon tracker, and both the transverse and longitudinal impact parameter significances with respect to the primary vertex are required to exceed 3. Due to the long lifetime of the K 0 S and Λ particles, the significance of the V 0 decay length, which is the three-dimensional distance between the primary and V 0 vertices divided by its uncertainty, is required to exceed 5. To remove K 0 S candidates misidentified as Λ particles and vice versa, the Λ (K 0 S ) candidate mass assuming both tracks to be pions (the lower-momentum track to be a pion and the higher-momentum track a proton) must differ by more than 20 (10) MeV from the nominal [50] K 0 S (Λ) mass value.To remove photon conversions to an electron-positron pair, the mass of a K 0 S or Λ candidate assuming both tracks to have the electron mass must exceed 15 MeV.The angle θ point between the V 0 momentum vector and the vector connecting the primary and V 0 vertices is required to satisfy cos θ point > 0.999.This reduces the contributions of particles from nuclear interactions, random combinations of tracks, and secondary Λ particles originating from the weak decays of Ξ and Ω particles.
To optimize the reconstruction of Ξ − particles, requirements on the three-dimensional impact parameter significance of its decay products with respect to the primary vertex are applied.This significance must be larger than 3 (4) for the proton (pion) tracks from the Λ decay, and larger than 5 for the direct pion candidate from the Ξ − decay.To further reduce the background from random combinations of tracks, the corresponding impact parameter significance of Ξ − candidates cannot exceed 2.5.The three-dimensional decay length significance, with respect to the primary vertex, of the Ξ − candidate and the associated Λ candidate must exceed 3 and 12, respectively.
The K 0 S , Λ, and Ξ − reconstruction efficiencies are about 15, 5, and 0.7% for p T ≈ 1 GeV, and 20, 10, and 2% for p T > 3 GeV, averaged over |η| < 2.4.These efficiencies account for the effects of acceptance, and for the branching fractions of the decay modes in which the strange particles are reconstructed.The invariant mass distributions of reconstructed K 0 S , Λ, and Ξ − candidates with 1 < p T < 3 GeV are shown in Fig. 1 for pPb events with 220 ≤ N offline trk < 260.Prominent mass peaks are visible, with little background.The solid lines show the result of a maximum likelihood fit.In this fit, the strange-particle peaks are modeled as the sum of two Gaussian functions with a common mean.The "average σ" values in Fig. 1 are the square root of the weighted average of the variances of the two Gaussian functions.The background is modeled with a quadratic function for the K 0 S results, with the analytic form Aq 1/2 + Bq 3/2 with q = m − (m π + m p ) for the Λ results, and with the form Cq D with q = m − (m Λ + m π ) for the Ξ − results, where A, B, C, and D are fitted parameters.These fit functions are found to provide a good description of the signal and background with relatively few free parameters.The fits are performed over the ranges of strange-particle invariant masses indicated in Fig. 1 to obtain the raw strange-particle yields N raw K 0 S /Λ/Ξ − .The raw strange-particle yields are corrected to account for the branching fraction of the reconstructed decay mode, and for the acceptance and reconstruction efficiency of the strange particle, using simulated event samples based on the PYTHIA 6 (pp) or EPOS (pPb and PbPb) event generator and GEANT4 modeling of the detector: where R corr is a correction factor from simulation given by the ratio of the raw reconstructed yield to the total generated yield for the respective strange particle, with N corr K 0 S /Λ/Ξ − the corrected yield.
The raw Λ particle yield includes contributions from the decays of Ξ − and Ω particles.This "nonprompt" contribution is largely determined by the relative Ξ − to Λ yield (because the contribution from Ω particles is negligible).The stringent requirements placed on cos θ point remove a large fraction of the nonprompt Λ component but, from simulation, up to 10% of the Λ candidates at high p T are nonprompt.If the relative Ξ − to Λ yield in simulation is modeled precisely, the contamination from nonprompt Λ particles will be removed by the correction procedure of Eq. ( 1).Otherwise, an additional correction to account for the residual contamination is necessary.As the Ξ − particle yields are explicitly measured in this analysis, this residual correction factor can be determined directly from the data as: where f raw,MC Λ,np denotes the fraction of nonprompt Λ particles in the raw reconstructed Λ sample as determined from simulation, while N corr Ξ − /N corr Λ and N MC Ξ − /N MC Λ are the Ξ − -to-Λ yield ratios from the data after applying the corrections of Eq. ( 1), and from generator-level simulation, respectively.The final prompt Λ particle yield is given by . Based on EPOS MC studies, which has a similar Ξ − /Λ ratio to the data, the residual nonprompt contributions to the Λ yields are found to be negligible in pPb and PbPb collisions, while in pp collisions the correction is 1-3% depending on the p T value of the Λ particle.Note that N corr Λ in Eq. ( 2) is derived using Eq. ( 1), which in principle contains the residual nonprompt Λ contributions.Nonetheless, by applying Eq. ( 2) in an iterative fashion, we expect N corr Λ to approach a result corresponding to prompt Λ particles only.A second iteration of correction is found to have an effect of less than 0.1% on the Λ particle yield.As a cross-check we treat the sample of simulated events generated with the HIJING program like data and verify that we obtain the correct yields at the generator level after applying the correction procedure described above.

Systematic uncertainties
Table 2 summarizes the different sources of systematic uncertainty in the yields of each strange particle species.The values in parentheses correspond to the systematic uncertainties in the forward rapidity regions (−2.4 < y cm < −1.5 and 0.8 < y cm < 1.5) for pPb data, if they differ from those at mid-rapidity.The dominant sources of systematic uncertainty are associated with the strange-particle reconstruction, especially the efficiency determination.The systematic uncertainty in determining the efficiency of a single track is 3.9% [48].The tracking efficiency is strongly correlated with the lifetime of a particle because when and where a particle decays determine how efficiently the detector captures its decay products.We observe agreement of the K 0 S lifetime distribution (cτ) between data and simulation, and similarly for the Λ and Ξ − , which provides a cross-check of the systematic uncertainty.This translates into a systematic uncertainty in the reconstruction efficiency of 7.8% for the K 0 S and Λ particles, and 11.7% for the Ξ − particles.Different background fit functions and methods to extract the yields for the K 0 S , Λ, and Ξ − are compared.The background fit function is varied to a fourth-order polynomial for the K 0 S and Λ studies, and to a linear function for the Ξ − study.The yields are obtained by integrating over a region that is ±5 times the average resolution and centered at the mean, rather than over the entire fitted mass range.Possible contamination by residual misidentified V 0 candidates (i.e., a K 0 S particle misidentified as a Λ particle, or vice versa) is investigated by varying the invariant mass range used to reject misidentified V 0 candidates.On the basis of these studies we assign systematic uncertainties of 2-4% to the yields.Systematic effects related to the selection of the strange-particle candidates are evaluated by varying the selection criteria, resulting in an uncertainty of 1-7%.The impact of finite momentum resolution on the spectra is estimated using the EPOS event generator.Specifically, the generator-level p T spectra of the strange particles are smeared by the momentum resolution, which is determined through comparison of the generator-level and matched reconstructed-level particle information.The difference between the smeared and original spectra is less than 2%.The systematic uncertainty associated with nonprompt Λ corrections to the Λ spectra is evaluated through propagation of the systematic uncertainty in the N corr Ξ − /N corr Λ ratio in Eq. ( 2) to the f residual Λ,np factor, and is found to be less than 2%.Systematic uncertainties introduced by possible residual pileup effects for pp data are estimated to be 1-3%.This uncertainty is evaluated through both tightening (only one reconstructed vertex allowed per event) and loosening (no event rejection on the basis of the number of vertices) the pileup rejection criteria [41].The uncertainty associated with pileup is negligible for the pPb and PbPb data since there are very few events in those samples with more than one reconstructed vertex.In pPb collisions, the direction of the p and Pb beams were reversed during course of the data collection, as mentioned in Section 2. Comparison of the particle p T spectra with and without the beam reversal yields an uncertainty of 2-5% for all particle types.The effect of the choice of the rapidity bins is assessed by dividing each bin into two, thereby doubling the number of bins, resulting in a systematic uncertainty of 1-3% for the p T spectra.For the Ξ − , the reconstruction efficiency correction is smoothed by averaging adjacent bins in order to compensate for the limited statistical precision of the MC sample.Variations in the smoothing procedure lead to a systematic uncertainty of 5% for the p T spectra of the Ξ − .All sources of systematic uncertainty are uncorrelated and summed in quadrature to define the total systematic uncertainties in the p T spectra of each strange particle.The total systematic uncertainties between the pp, pPb, and PbPb systems are similar and largely correlated.When calculating ratios of particle yields, most of the systematic uncertainties partially or entirely cancel.For example, the systematic uncertainties due to tracking efficiency and pileup for the Λ/2K 0 S ratio are negligible.

Multiplicity dependence at mid-rapidity
The p T spectra of K 0 S , Λ, and Ξ − particles with |y cm | < 1 in pp collisions at √ s = 7 TeV (top), pPb collisions at √ s = 5.02 TeV (middle), and PbPb collisions at √ s NN = 2.76 TeV (bottom) are presented in Fig. 2, for different multiplicity intervals.Due to details in the implementation of the dedicated high-multiplicity trigger thresholds used to select the pp events, the multiplicity intervals for pp events differ slightly from those for pPb and PbPb events.The p T differential yield is defined as dN 2 /(2π p T )dp T dy.For the purpose of better visibility, the data are scaled by factors of 2 −n , as indicated in the figure legend.A clear evolution of the spectral shape with multiplicity can be seen for each particle species in each collision system.For higher multiplicity events, the spectra tend to become flatter (i.e., "harder"), indicating a larger KE T value.Within each collision system, heavier particles (e.g., Ξ − ) exhibit a harder spectrum than lighter particles (K 0 S ), especially for high-multiplicity events.To examine the differences in the multiplicity dependence of the spectra in greater detail, the ratios Λ/2K 0 S and Ξ − /Λ of the yields are shown in Fig. 3 as a function of p T for different multiplicity ranges in the pp, pPb, and PbPb systems.The results for the Λ/2K 0 S ratio are shown in Fig. 3 (top).For p T 2 GeV, the Λ/2K 0 S ratio is seen to be smaller in high-multiplicity events   than in low-multiplicity events for a given p T value.In pp and pPb collisions, this trend is similar to what has been observed between peripheral and central PbPb collisions [23]; this trend is not as evident for the PbPb data in Fig. 3 (top right) because in the present study only PbPb events of 50-100% centrality are considered.At higher p T , this multiplicity ordering of the Λ/2K 0 S ratio is reversed.In hydrodynamic models such as those presented in Refs.[51,52], this behavior can be interpreted as the effect of radial flow.A stronger radial flow is developed in higher-multiplicity events, which boosts heavier particles (e.g., Λ) to higher p T , resulting in a suppression of the Λ/2K 0 S ratio at low p T .Comparing the various collision systems at low p T , the difference in the Λ/2K 0 S ratio between low-and high-multiplicity events is seen to be largest for the pp data.In the hydrodynamic model of Ref. [31], smaller collision systems like pp produce a larger radial-flow effect than larger systems like pPb or PbPb, for similar multiplicities, which could explain this observation.For p T > 2 GeV, the baryon enhancement could be explained by recombination models, in which free quarks recombine to form hadrons [53].In previous studies (e.g., Ref. [54]), it has been shown that the average p T value of various particle species has only a slight center-of-mass energy dependence (10% at high multiplicity).This dependence is not sufficient to explain the differences observed in Fig. 3 between the various systems.
For each multiplicity interval, the Λ/2K 0 S ratio reaches a maximum that has a similar value for all three collision processes, and then decreases at higher p T .The location of the maximum increases with multiplicity from around p T = 2 to 3 GeV.
The results for the Ξ − /Λ ratio are shown in Fig. 3 (bottom).In this case, the difference between the low-and high-multiplicity events is much smaller than for the Λ/2K 0 S ratio, for all three collisions systems.For all systems, the Ξ − /Λ ratio increases with p T and reaches a plateau at around p T = 3 GeV.Due to the large systematic uncertainty, it is not possible to draw a conclusion with respect to the radial-flow interpretation.The extracted kinetic freeze-out temperature, T kin , versus the average radial-flow velocity, β T , from a simultaneous blast-wave fit to the K 0 S and Λ p T spectra at |y cm | < 1 for different multiplicity intervals in pp, pPb, and PbPb collisions.The six pp and eight pPb and PbPb multiplicity intervals are indicated in the legend of Fig. 2. For the results in this plot, the multiplicity increases from left to right.The correlation ellipses represent the statistical uncertainties.Systematic uncertainties, which are evaluated to be on the order of a few percent, are not shown.
Motivated by the hydrodynamic model, we perform a simultaneous fit of a blast-wave function [32] to the K 0 S and Λ spectra in Fig. 2. The fits are restricted to low p T because that is the region in which the blast-wave model is valid.The blast-wave model is strictly appropriate only for directly produced particles, while about 1/3 of the K 0 S mesons may be from higher mass resonances [55].The Ξ − particle is not used in the fit as there are not many Ξ − at low p T .The fits are performed for each collision system separately.The fit ranges are 0.1 < p T < 1.5 GeV for the K 0 S and 0.6 < p T < 3.0 GeV for the Λ.The fitted function is: where ρ = tanh −1 β T = tanh −1 (β s (r/R) n ) is the velocity profile, R is the radius of the medium (set to unity in the fit), r is the radial distance from the center of the medium in the transverse plane, n is the exponent of the velocity profile, β T is the transverse expansion velocity (also known as the radial-flow velocity), β s is the transverse expansion velocity on the surface of the medium, T kin is the kinetic freeze-out temperature, and I 0 and K 1 are modified Bessel functions.The fitted parameters that govern the shape are n, β s , and T kin .
In the blast-wave model, common values of T kin and average radial-flow velocity β T are assumed for all particle species, as is expected if the system is locally thermalized and undergoes a radial-flow expansion.It is useful to directly compare the extracted values of T kin and β T from the different systems to study the system-size dependence at similar multiplicities.
( Data/Fit 0.8 0.9 Figure 5: Examples of simultaneous blast-wave fits of the p T spectra of K 0 S and Λ particles in low-and high-multiplicity pPb events.The inclusion of the charge-conjugate states is implied for Λ particles.The ratios of the fits to the data as a function of p T are shown in the bottom panels.The uncertainties are statistical only and are too small to be visible for most of the points.
The extracted values of T kin and β T are shown in Fig. 4 for the six pp and for the eight pPb and PbPb multiplicity intervals.In this figure, the multiplicity increases from left to right.The ellipses correspond to one standard deviation statistical uncertainties, which for pp collisions are smaller at low and high multiplicity due to the use of events collected with minimum bias and high-multiplicity triggers.Systematic uncertainties, which are evaluated by propagating the systematic uncertainties from the spectra to the blast-wave fits and altering the fit ranges, are on the order of a few percent and are not shown.Examples of the fits are shown in Fig. 5 for a low-and high-multiplicity range in pPb collisions.In general, the fit quality is good for high-multiplicity events except for the lowest p T range, while for low-multiplicity events there are discrepancies on the order of 5%.However, the discrepancies between the fit and data lie within the systematic uncertainty.
The precise meaning of the T kin and β T parameters is model dependent, and they should not be interpreted literally as the kinetic freeze-out temperature and radial-flow velocity of the system.The main purpose of Fig. 4 is to provide a qualitative comparison of the spectral shapes in the three systems.In the context of the blast-wave model, when comparing at similar multiplicities, the T kin parameter has the same value within 15% among the three systems, while the β T parameter is larger when the system is smaller, i.e., β T pp > β T pPb > β T PbPb .This is qualitatively consistent with the prediction of Ref. [31].The results of blast-wave fits are known to depend on the particle species.Due to the limited set of particles in this analysis, future studies will be needed to further substantiate the conclusions.The evolution of the p T spectra with multiplicity can be compared more directly between the three systems through examination of the KE T value.The KE T values at |y cm | < 1 for K 0 S , Λ, and Ξ − particles as a function of multiplicity are shown in Fig. 6.Extrapolation of the p T spectra down to p T = 0 GeV is a crucial step in extracting the KE T values, while the impact of the extrapolation up to p T ≈ ∞ is negligible, both on the value of KE T and its uncertainty.For the Ξ − particle, only results in pPb collisions are shown due to the limitation of the low-p T reach in pp and PbPb collisions, as can be seen from Fig. 2. Blast-wave fits to the individual spectra, which only consider the spectrum shape but do not impose any physics constraint, are used to obtain the extrapolation.The fraction of the extrapolated yield with respect to the total yield is about 1.2-2.5% for the K 0 S , 5.8-15.1% for the Λ, and 5.4-20.4% for the Ξ − particles, depending on the multiplicity.Alternative methods to perform the extrapolation are used to evaluate a systematic uncertainty, including use of the predictions from the simultaneous blastwave fit to the K 0 S and Λ p T spectra, and a linear extrapolation from the yields in a low range of p T .The systematic uncertainties from Table 2 are also included in the evaluation of the KE T uncertainties.
For the lowest multiplicity range, the KE T values for each particle species are seen to be similar.For all particle species, KE T increases with increasing multiplicity.However, the slope of the increase differs for different particles, with the heavier particles exhibiting a faster growth in KE T for all systems.For a given multiplicity range, the KE T value is roughly proportional to the particle's mass.In PbPb collisions, this can be understood to be due to the onset of radial flow [2,7].The observed difference between particle species at high multiplicity is seen to be larger for pp and pPb events than for PbPb events.Note, however, the difference in the center-of-mass energies between the three systems.

Rapidity dependence in pPb events
The rapidity dependence of the p T spectra of the K 0 S and Λ particles is studied in the pPb data.No results for Ξ − particles are presented due to statistical limitations.As a pPb collision is asymmetric in rapidity, it is interesting to compare the spectra along the Pb-going (y cm < 0) and p-going (y cm > 0) directions [37].The p T spectra of K 0 S and Λ particles in different y cm ranges are shown in Fig. 7 for small (top), intermediate (middle), and large (bottom) average multiplicities.The Λ/2K 0 S ratios from the −1.5 < y cm < −0.8 (Pb-going) and 0.8 < y cm < 1.5 (p-going) rapidity regions are compared in Fig. 8 for multiplicity ranges 0 ≤ N offline trk < 35 and 220 ≤ N offline trk < 260.For both the low-multiplicity and the high-multiplicity events, the Λ/2K 0 S ratio from the Pb-going direction lies above the results from the p-going direction, with the largest difference observed at high p T in the high-multiplicity sample.As a further study, we calculate KE T , following the procedure outlined in Section 6.1, and examine its dependence on y cm for K 0 S and Λ particles in the pPb collisions.The results are shown in Fig. 9.Although the systematic uncertainties at forward rapidities are large, the KE T values are seen to become slightly asymmetric as multiplicity increases.At low multiplicities (0 ≤ N offline trk < 35), the ratios of KE T between the Pb-going side (−1.5 < y cm < −0.8) and the p-going side (0.8 < y cm < 1.5) are 1.01 ± 0.01 (syst.)for K 0 S particles and 1.04 ± 0.05 (syst.)for Λ particles, both of which are consistent with unity within the systematic uncertainties (the statistical uncertainties are negligible).However, in the highest multiplicity range, 220 ≤ N offline trk < 260, the ratios become 1.06 ± 0.01 (syst) for K 0 S particles and 1.12 ± 0.06 (syst) for Λ particles, suggesting that an asymmetry in KE T is developed between the Pb-going and pgoing sides.This trend is qualitatively consistent with the hydrodynamic prediction for pPb collisions [37].

Summary
Measurements of strange hadron (K 0 S , Λ+Λ, and Ξ − +Ξ + ) transverse momentum spectra in pp, pPb, and PbPb collisions are presented over a wide range of event charged-particle multiplicity and particle rapidity.The study is based on samples of pp collisions at √ s = 7 TeV, pPb collisions at √ s = 5.02 TeV, and PbPb collisions at √ s NN = 2.76 TeV, collected with the CMS detector at the LHC.In the context of hydrodynamic models, the measured particle spectra are fitted with a blast wave function, which describes an expanding fluid-like system.When comparing at a similar multiplicity, the extracted radial-flow velocity parameters are found to be larger in pp and pPb collisions than that in PbPb collisions.The average transverse kinetic energy KE T of strange hadrons is observed to increase with multiplicity, with a stronger increase for heavier particles.At similar multiplicities, the difference in KE T between the strange-particle species is larger in the smaller pp and pPb systems than in the PbPb system.For pPb collisions, KE T in the Pb-going direction for K 0 S (Λ+Λ) is 6% (12%) larger than in the p-going direction for events with the highest particle multiplicities.

Figure 1 :
Figure 1: Invariant mass distribution of K 0 S (left), Λ (middle), and Ξ − (right) candidates in the p T range 1-3 GeV for 220 ≤ N offline trk < 260 in pPb collisions.The inclusion of the chargeconjugate states is implied for Λ and Ξ − particles.The solid lines show the results of fits described in the text.The dashed lines indicate the fitted background component.

Figure 2 :
Figure 2: The p T spectra of K 0 S , Λ, and Ξ − particles in the center-of-mass rapidity range |y cm | < 1 in pp collisions at √ s = 7 TeV (top), pPb collisions at √ s = 5.02 TeV (middle), and PbPb collisions at √ s NN = 2.76 TeV (bottom) for different multiplicity intervals.The inclusion of the charge-conjugate states is implied for Λ and Ξ − particles.The data in the different multiplicity intervals are scaled by factors of 2 −n for better visibility.The statistical uncertainties are smaller than the markers and the systematic uncertainties are not shown.

Figure 3 :
Figure 3: Ratios of p T spectra for Λ/2K 0 S (top) and Ξ − /Λ (bottom) in the center-of-mass rapidity range |y cm | < 1.0 for pp collisions at √ s = 7 TeV (left), pPb collisions at √ s = 5.02 TeV (middle), and PbPb collisions at √ s NN = 2.76 TeV (right).Two (for pp) or three (for pPb and PbPb) representative multiplicity intervals are presented.The inclusion of the charge-conjugate states is implied for Λ and Ξ − particles.The error bars represent the statistical uncertainties, while the boxes indicate the systematic uncertainties.
Figure4: The extracted kinetic freeze-out temperature, T kin , versus the average radial-flow velocity, β T , from a simultaneous blast-wave fit to the K 0 S and Λ p T spectra at |y cm | < 1 for different multiplicity intervals in pp, pPb, and PbPb collisions.The six pp and eight pPb and PbPb multiplicity intervals are indicated in the legend of Fig.2.For the results in this plot, the multiplicity increases from left to right.The correlation ellipses represent the statistical uncertainties.Systematic uncertainties, which are evaluated to be on the order of a few percent, are not shown.

Figure 6 :
Figure 6: The average transverse kinetic energy, KE T , at |y cm | < 1 for K 0 S , Λ, and Ξ − particles as a function of multiplicity in pp, pPb, and PbPb collisions.The inclusion of the chargeconjugate states is implied for Λ and Ξ − particles.For the Ξ − , only results from pPb collisions are shown.The error bars represent the statistical uncertainties, while the boxes indicate the systematic uncertainties.

Figure 7 :
Figure 7: The p T spectra of K 0 S and Λ particles in different y cm ranges for pPb collisions at √ s = 5.02 TeV.The inclusion of the charge-conjugate states is implied for Λ and particles.Results are shown for three multiplicity ranges: 0 ≤ N offline trk

Figure 8 :
Figure 8: Ratios of p T spectra, Λ/2K 0 S , from the −1.5 < y cm < −0.8 (Pb-going) and 0.8 < y cm < 1.5 (p-going) rapidity regions in pPb collisions at √ s = 5.02 TeV.The inclusion of the charge-conjugate states is implied for Λ particles.Results are presented for two multiplicity ranges 0 ≤ N offline trk

8 ΛFigure 9 :
Figure 9: The average transverse kinetic energy, KE T , as a function of y cm for the K 0 S and Λ particles in three ranges of multiplicity in pPb collisions at √ s = 5.02 TeV.The inclusion of the charge-conjugate states is implied for Λ particles.The error bars represent the statistical uncertainties, while the boxes indicate the systematic uncertainties.
dustrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the Mobility Plus program of the Ministry of Science and Higher Education (Poland); the OPUS program of the National Science Center (Poland); the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the National Priorities Research Program by Qatar National Research Fund; the Programa Clarín-COFUND del Principado de Asturias; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University (Thailand); the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); and the Welch Foundation, contract C-1845.

Table 1 :
Fraction of the full event sample in each multiplicity interval and the average multiplicity per interval for pp data.The multiplicities N offline trk and N corrected trk are determined for |η| < 2.4 and p T > 0.4 GeV before and after efficiency corrections, respectively.The third and fourth columns list the average values of N offline

Table 2 :
Summary of systematic uncertainties for the p T spectra of K 0 S , Λ, and Ξ − particles in the center-of-mass rapidity range |y cm | < 1.0 (for pPb events, at forward rapidities, if different) for the three collision systems.