Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV

We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to measure the space-time characteristics of particle production from the effects of quantum statistics and final-state interactions in two-particle correlations. Kaon femtoscopy is an important supplement to that of pions because it allows one to distinguish between different model scenarios working equally well for pions. In particular, we compare the measured 3D kaon radii with a purely hydrodynamical calculation and a model where the hydrodynamic phase is followed by a hadronic rescattering stage. The former predicts an approximate transverse mass ($m_{\mathrm{T}}$) scaling of source radii obtained from pion and kaon correlations. This $m_{\mathrm{T}}$ scaling appears to be broken in our data, which indicates the importance of the hadronic rescattering phase at LHC energies. A $k_{\mathrm{T}}$ scaling of pion and kaon source radii is observed instead. The time of maximal emission of the system is estimated using the three-dimensional femtoscopic analysis for kaons. The measured emission time is larger than that of pions. Our observation is well supported by the hydrokinetic model predictions.


Introduction
Extremely high energy densities achieved in heavy-ion collisions at the Large Hadron Collider (LHC) are expected to lead to the formation of the quark-gluon plasma (QGP), a state characterized by partonic degrees of freedom [1,2].The systematic study of many observables (transverse momentum spectra, elliptic flow, jets, femtoscopy correlations) measured at the Relativistic Heavy Ion Collider (RHIC) and the LHC confirmed the presence of strong collective motion and the hydrodynamic behavior of the system (see e.g.[3][4][5][6] and [7][8][9][10]).Whereas since quite a long time hydrodynamics describes momentum based observables, it could not describe spatial distributions at decoupling.Correlation femtoscopy (commonly referred to as femtoscopy or HBT, Hanbury Brown and Twiss interferometry), measures the space-time characteristics of particle production using particle correlations due to the effects of quantum statistics and strong and Coulomb final-state interactions [11][12][13][14][15].The problem to describe the spatiotemporal scales derived from femtoscopy in heavy-ion collisions at RHIC was solved only a few years ago, strongly constraining the hydrodynamical models [16][17][18].The following factors were understood to be important: existence of prethermal transverse flow, a crossover transition between quark-gluon and hadron matter, non-hydrodynamic behavior of the hadron gas at the latest stage (hadronic cascade phase), and correct matching between hydrodynamic and non-hydrodynamics phases (see e.g.[16]).
New challenges for hydrodynamics appeared when data were obtained at the LHC: the large statistics now allows one to investigate not only pion femtoscopy, which is the most common femtoscopic analysis, but also femtoscopy of heavier particles in differential analyses with high precision.
The main objective of ALICE [19] at the LHC is to study the QGP.ALICE has excellent capabilities to study femtoscopy observables due to good track-by-track particle identification (PID), particle acceptance down to low transverse momenta p T , and good resolution of secondary vertices.We already studied pion correlation radii in Pb-Pb collisions at 2.76 TeV [20,21].Pion femtoscopy showed genuine effects originating from collective flow in heavy-ion collisions, manifesting as a decrease of the source radii with increasing pair transverse mass m T = k 2 T + m 2 [15,22], where k T = |p T,1 + p T,2 |/2 is the average transverse momentum of the corresponding pair and m is the particles mass.
The next most numerous particle species after pions are kaons.The kaon analyses are expected to offer a cleaner signal compared to pions, as they are less affected by resonance decays.Studying charged and neutral kaon correlations together provides a convenient experimental consistency check, since they require different detection techniques.The theoretical models which describe pion femtoscopy well, should describe kaon results with equal precision.
Of particular interest is the study of the m T -dependence of pion and kaon source radii.It was shown that the hydrodynamic picture of nuclear collisions for the particular case of small transverse flow leads to the same m T behavior of the longitudinal radii (R long ) for pions and kaons [23].This common m T -scaling for π and K is an indication that the thermal freeze-out occurs simultaneously for π and K and that these two particle species are subject to the same velocity boost from collective flow.Previous kaon femtoscopy studies carried out in Pb-Pb collisions at the SPS by the NA44 and NA49 Collaborations [24,25] reported the decrease of R long with m T as ∼ m −0.5 T as a consequence of the boost-invariant longitudinal flow.Subsequent studies carried out in Au-Au collisions at RHIC [26][27][28][29] have shown the same power in the m T -dependencies for π and K radii, consistent with a common freeze-out hypersurface.Like in the SPS data, no exact universal m T -scaling for the 3D radii was observed at RHIC, but still these experiments observed an approximate m T -scaling for pions and kaons.The recent study of the m T -dependence of kaon three-dimensional radii performed by the PHENIX collaboration [30] demonstrated breaking of this scaling especially for the "long" direction.PHENIX reported that the Hydro-Kinetic Model (HKM) describes well the overall trend of femtoscopic radii for pions and kaons [31,32].
We have published previously the study of one-dimensional correlation radii of different particle species: , pp, and pp correlations in Pb-Pb collisions at √ s NN = 2.76 TeV for several intervals of centrality and transverse mass [33].The decrease of the source radii with increasing transverse mass was observed for all types of particles, manifesting a fingerprint of collective flow in heavy-ion collisions.The one-dimensional femtoscopic radii demonstrated the approximate m T -scaling as it was expected by hydrodynamic model considerations [15].
Recent calculations made within a 3+1D hydrodynamical model coupled with a statistical hadronization code taking into account the resonance contribution, THERMINATOR-2, showed the approximate scaling of the three dimensional radii with transverse mass for pions, kaons and protons [34].An alternative calculation, that is the Hydro-Kinetic Model, including a hydrodynamic phase as well as a hadronic rescattering stage, predicts violation of such a scaling between pions and kaons at LHC energies [35].Both models observe approximate scaling if there is no rescattering phase.It is suggested in [35] that rescattering has significantly different influence on pions and kaons and is responsible for the violation of m T -scaling at the LHC energies.Moreover, the analysis of the emission times of pions and kaons obtained within HKM in [36] showed that kaons are emitted later than pions due to rescattering through the rather long-lived K * (892) resonance.This effect can explain the m T -scaling violation predicted in [35].
In [35] it was found that immediately decaying the K * (892) and φ (1020) resonances at the chemical freeze-out hypersurface has only a negligible influence on the kaon radii.In this scenario, resonances were allowed to be regenerated in the hadronic phase.Further analysis in [36] showed that it is indeed the regeneration of the K * (892) resonance through hadronic reactions which is responsible for the m Tscaling violation predicted in [35].This mechanism clearly manifests itself in the prolonged emission time of kaons caused by the rather long lifetime of the K * (892) resonance [35].
The approximate scaling of pion and kaon radii was predicted by investigating 3+1D hydrodynamical model + THERMINATOR-2 in [34] to hold for each of the three-dimensional radii separately.The scaling of one-dimensional pion and kaon radii was also studied in [34].It was shown that after averaging the three-dimensional radii and taking into account a mass-dependent Lorentz-boost factor, a deviation between one-dimensional pion and kaon radii appeared.These circumstances made it impossible to discriminate between THERMINATOR-2 [34] and the HKM calculation [35] in the earlier published one-dimensional analysis of pion and kaon radii by ALICE [33].The three-dimensional study presented here is not impeded by these effects and allows one to discriminate between the hypothesis of approximate scaling of three-dimensional radii predicted in [34] and strong scaling violation proposed in [35].Thus the study of the m T -dependence of three-dimensional pion and kaon radii can unambiguously distinguish between the different freeze-out scenarios and clarify the existence of a significant hadronic phase.
One more interesting feature of femtoscopy studies of heavy-ion collisions concerns the ratio of radius components in the transverse plane.The strong hydrodynamic flow produces significant positive spacetime correlations during the evolution of the freeze-out hypersurface.This influences the extracted radius parameters of the system in the plane perpendicular to the beam axis.The radius along the pair transverse momentum is reduced by the correlation with respect to the perpendicular one in the transverse plane.This effect appears to be stronger at LHC than at RHIC energies [37,38].It was studied by the ALICE collaboration for pions in Pb-Pb collisions at 2.76 TeV [21] at different centralities.This work extends this study to kaons and compares the obtained transverse radii with those found in the analysis for pions and to the model calculations discussed above.The paper is organized as follows.Section 2 explains the data selection and describes the identification of charged and neutral kaons.In Section 3 the details of the analysis of the correlation functions are discussed together with the investigation of the systematic uncertainties.Section 4 presents the measured source radii as well as the extracted emission times and compares them to model predictions.Finally, Section 5 summarizes the obtained results and discusses them within the hydrokinetic approach.√ s NN = 2.76 TeV ALICE Collaboration
Charged particle tracking is generally performed using the Time Projection Chamber (TPC) [40] and the Inner Tracking System (ITS) [19].The ITS also provides high spatial resolution in determining the primary collision vertex.
Particle identification (PID) for reconstructed tracks was carried out using both the TPC and the Time-of-Flight (TOF) detector [41].For TPC PID, a parametrization of the Bethe-Bloch formula was employed to calculate the specific energy loss (dE/dx) in the detector expected for a particle with a given mass and momentum.For PID with TOF, the particle mass hypothesis was used to calculate the expected timeof-flight as a function of track length and momentum.For each PID method, a value N σ was assigned to each track denoting the number of standard deviations between the measured track dE/dx or timeof-flight and the calculated one as described above.Different cut values of N σ were chosen based on detector performance for various particle types and track momenta (see Table 1 for specific values used in both analyses).More details on PID can be found in Secs.7.2-7.5 of [42].
The analysis details for charged and neutral kaons are discussed separately below.All major selection criteria are also listed in Table 1.

Charged kaon selection
Track reconstruction for the charged kaon analysis was performed using the tracks' signal in the TPC.The TPC is divided by the central electrode into two halves, each of them is composed of 18 sectors (covering the full azimuthal angle) with 159 padrows placed radially in each sector.A track signal in the TPC consists of space points (clusters), each of which is reconstructed in one of the padrows.A track was required to be composed out of at least 70 such clusters.The parameters of the track are determined by performing a Kalman fit to a set of clusters with an additional constraint that the track passes through the primary vertex.The quality of the fit is requested to have χ 2 /NDF better than 2. The transverse momentum of each track was determined from its curvature in the uniform magnetic field.The momentum from this fit in the TPC was used in the analysis.Tracks were selected based on their distance of closest approach (DCA) to the primary vertex, which was required to be less than 2.4 cm in the transverse and less than 3.0 cm in the longitudinal direction.K ± identification was performed using the TPC (for all momenta) and the TOF detector (for p > 0.5 GeV/c).The use of different values for N σ ,TPC and N σ ,TOF was the result of studies to obtain the best kaon purity, defined as the fraction of accepted kaon tracks that correspond to true kaon particles, while retaining a decent efficiency.The estimation of purity for p < 0.5 GeV/c was performed by parametrizing the TPC dE/dx distribution in momentum slices for the contributing species [42].The dominant contamination for charged kaons comes from e ± in the momentum range 0.4 < p < 0.5 GeV/c.The purity for p > 0.5 GeV/c, where the TOF information was employed, was studied with HIJING [43] simulations using GEANT [44] to model particle transport through the detector; the charged kaon purity was estimated to be greater than 99%.The momentum dependence of the single kaon purity is shown in Fig. 1(a).The pair purity is calculated as the product of two single-particle purities, where the momenta are taken from the experimentally determined distribution.The K ± pair purity as a function of k T at three different centralities is shown in Fig. 1(b).Kaon pair transverse momentum is an averaged p T of single kaons taken from the whole p T range, which is the reason why the pair purities are larger than single particle ones.
Two kinds of two-track effects have been investigated: splitting, where a signal produced by one particle is incorrectly reconstructed as two tracks, and merging, where two particles are reconstructed as only one track.These detector inefficiencies can be suppressed by employing specific pair selection criteria.We used the same procedure as in [21] which works here as well with slightly modified cut values.Charged kaon pairs were required to have a separation of |∆ϕ * | > 0.04 and |∆η| > 0.02.Here, ϕ * is the azimuthal position of the track in the TPC at R = 1.2 m, taking into account track curvature in the magnetic field, and η is the pseudorapidity.Also, all track pairs sharing more than 5% of TPC clusters were rejected.

Neutral kaon selection
The decay channel K 0 S → π + π − was used for the identification of neutral kaons.The secondary pion tracks were reconstructed using TPC and ITS information.The single-particle cuts for parents (K 0 S ) and daughters (π ± ) used in the decay-vertex reconstruction are shown in Table 1.The daughter-daughter DCA, that is the distance of closest approach of the two daughter pions from a candidate K 0 S decay, proved useful in rejecting background topologies.PID for the pion daughters was performed using both TPC (for all momenta) and TOF (for p > 0.8 GeV/c).The very good detector performance is reflected in the FWHM of the K 0 S peak of only 8 MeV/c 2 .The selection criteria used in this analysis were chosen as a compromise to maximize statistics while keeping a high signal purity.The neutral kaon purity (defined as Sig./[Sig.+Bkg.]for 0.480 < m π + π − < 0.515 GeV/c 2 ) was larger than 0.95.Two main two-particle cuts were used in the neutral kaon analysis.To resolve two-track inefficiencies associated with the daughter tracks, such as the splitting or merging of tracks discussed above, a separation cut was employed in the following way.For each kaon pair, the spatial separation between the same-sign pion daughters was calculated at several points throughout the TPC (every 20 cm radially from 85 cm to 245 cm) and averaged.If the average separation of either pair of tracks was below 5 cm, the kaon pair was not used.Another cut was used to prevent two reconstructed kaons from using the same daughter track.If two kaons shared a daughter track, one of them was excluded using a procedure which compared the two K 0 S candidates and kept the candidate whose reconstructed parameters best matched those expected for a true K 0 S particle in two of three categories (smaller K 0 S DCA to primary vertex, smaller daughter-daughter DCA, and K 0 S mass closer to the PDG value [45]).This procedure was shown, using HIJING+GEANT simulations, to have a success rate of about 95% in selecting a true K 0 S particle over a fake one.More details about the K 0 S K 0 S analysis can be found in Refs.[46,47].K 0 S candidate selection criteria developed in other works [33] were used here as well; they are included in Table 1.

Correlation functions
The femtoscopic correlation function C is constructed experimentally as the ratio C(q) = A(q)/B(q), where A(q) is the measured distribution of the difference q = p 2 − p 1 between the three-momenta of the two particles p 1 and p 2 taken from the same event, B(q) is a reference distribution of pairs of particles taken from different events (mixed).For a detailed description of the formalism, see e.g.[14].The pairs in the denominator distribution B(q) are constructed by taking a particle from one event and pairing it with a particle from another event with a similar centrality and primary vertex position along the beam direction.Each event is mixed with five (ten) others for the K 0 S (K ± ) analysis.The numerator and denominator are normalized in the full q = |q| 2 − q 2 0 range used (0-0.3GeV/c) such that C(q) → 1 means no correlation.Pair cuts have been applied in exactly the same way for the same-event (signal) and mixed-event (background) pairs.The momentum difference is calculated in the longitudinally co-moving system (LCMS), where the longitudinal pair momentum vanishes, and is decomposed into (q out , q side , q long ), with the "long" axis going along the beam, "out" along the pair transverse momentum, and "side" perpendicular to the latter in the transverse plane (Bertsch-Pratt convention).
The correlation functions have been corrected for momentum resolution effects, by using the HIJING event generator and assigning a quantum-statistical weight to each particle pair.Further, these modified events were propagated through the full simulation of the ALICE detectors [19].The ratios of the correlation functions obtained before and after this full event simulation have been taken as the correction factors.The correlation function from the data has been divided by this q-dependent factor.The correction increases the obtained radii by 3-5%.√ s NN = 2.76 TeV ALICE Collaboration Fig. 2: A sample projected K ± K ± correlation function with fit.The error bars are statistical only.Systematic uncertainties on the points are equal to or less than the statistical error bars shown.

Charged kaon
The three-dimensional correlation functions were fitted by the Bowler-Sinyukov formula [48,49]: where R out , R side , and R long are the Gaussian femtoscopic radii in the LCMS frame, N is the normalization factor, and q is the momentum difference in the pair rest frame (PRF) 1 .The λ parameter, which characterizes the correlation strength, can be affected by long-lived resonances, coherent sources [50][51][52], and non-Gaussian features of the particle-emission distribution.We account for Coulomb effects through K(q), calculated according to Ref. [49,51] as Here, the theoretical correlation function C(QS) takes into account quantum statistics only and C(QS + Coulomb) considers quantum statistics and the Coulomb final-state interaction (FSI) contribution to the wave function [14].
The experimental correlation functions have been corrected for purity according to: where ζ is the pair purity taken from Fig. 1.
Figure 2 shows a sample projected K ± K ± correlation function with a fit performed according to Eq. 1.
When the 3D correlation function is projected on one axis, the momentum differences in the two other directions are required to be within (-0.04,0.04)GeV/c.The f 0 and a 0 masses and coupling parameters, all in GeV.

Neutral kaon
K 0 S K 0 S correlation functions were fitted using a parametrization which includes Bose-Einstein statistics as well as strong final-state interactions [28,53].Strong final-state interactions have an important effect on K 0 S K 0 S correlations.Particularly, the K 0 K 0 channel is affected by the near-threshold resonances f 0 (980) and a 0 (980).Using the equal emission time approximation in the pair rest frame (PRF) [53], the elastic K 0 K 0 transition is written as a stationary solution Ψ − k * ( r * ) of the scattering problem in the PRF, where k * and r * represent the momentum of a particle and the emission separation of the pair in the PRF (the − k * subscript refers to a reversal of time from the emission process), which at large distances has the asymptotic form of a superposition of a plane wave and an outgoing spherical wave, where g(k * ) is the s-wave scattering amplitude for a given system.For K 0 K 0 , g(k * ) is dominated by the f 0 and a 0 resonances and written in terms of the resonance masses and decay couplings [28]: Here, s = 4(m 2 K + k * 2 ); γ r (γ r ) refers to the couplings of the resonances to the f 0 → K 0 K 0 (f 0 → ππ) and a 0 → K 0 K 0 (a 0 → πη) channels; m r is the resonance mass; and k r refers to the momentum in the PRF of the second decay channel (f 0 → ππ or a 0 → πη) with the corresponding partial width Γ r = γ r k r /m r .The amplitudes g I of isospin I = 0 and I = 1 refer to the f 0 and a 0 , respectively.The parameters associated with the resonances and their decays are taken from several experiments [54][55][56][57], and the values are listed in Table 2.
The correlation function is then calculated by integrating Ψ − k * ( r * ) in the Koonin-Pratt equation [58,59] where S K ( r * ) is the Gaussian source distribution in terms of R out , R side , and R long , K is the average pair momentum, and Ψ S − k * ( r * ) is the symmetrized version of Ψ − k * ( r * ) for bosons.Although Eq. 7 can be integrated analytically for K 0 S K 0 S correlations with FSI for the one-dimensional case [28], for the three-dimensional case this integration cannot be performed analytically.In order to form the 3D correlation function, we combine a Monte Carlo emission simulation with a calculation of the twoparticle wavefunction, thus performing a numerical integration of Eq. 7. The Monte Carlo emission simulation consists of generating the pair positions sampled from a three-dimensional Gaussian in the PRF, with three input radii as the width parameters, and generating the particle momenta sampled from a distribution taken from data.Using the MC-sampled positions and momenta, we calculate Ψ S − k * ( r * ).We then build a correlation function using the wavefunction weights to form the signal distribution, and an unweighted distribution acts as a background.This theoretical correlation function is then used to fit the data.Finally, we make a Lorentz boost, γ, of R out from the PRF to the LCMS frame (R side and R long are not affected by the boost).More details on the 3D fitting procedure can be found in Ref. [46].
Figure 3 shows a sample projected K 0 S K 0 S correlation function with fit.Also shown is the contribution to the fit from the quantum statistics part only.As seen, the FSI part produces a significant depletion of the correlation function in the q range 0-0.1 GeV/c in each case.

Systematic uncertainties
The effects of various sources of systematic uncertainty on the extracted fit parameters were studied as functions of centrality and k T .For each source, we take the maximal deviation and apply it symmetrically as the uncertainty.Table 3 shows minimum and maximum uncertainty values for various sources of systematic uncertainty for charged and neutral kaons.The systematic errors are summed up quadratically.The values of the total uncertainty are not necessarily equal to the sum of the individual uncertainties, as the latter can come from different centrality or k T bins.Both analyses studied the effects of changing the selection criteria used for the events, particles, and pairs (variation of cut values up to ±25%) and varying the range of q values over which the fit is performed (variation of q limits up to ±25%).Uncertainties associated with momentum resolution corrections are included into the K ± analysis; for the K 0 S analysis, these uncertainties are found to be small compared to other contributions.Both analyses were performed separately for the two different polarities of the ALICE solenoid magnetic field, the difference was found to be negligible.
For the K 0 S fitting procedure, the mean γ value is calculated for each centrality and k T selection and used to scale R out .However, each bin has a spread of γ values associated with it.The standard deviation of the mean γ value for each k T bin was used as an additional source of systematic error for R out .For K 0 S , an uncertainty on the strong FSI comes from the fact that several sets of f 0 (980) and a 0 (980) parameters are available [54][55][56][57]; each set is used to fit the data, the results are averaged, and the maximal difference was taken as a systematic error.
The K ± analysis has uncertainties associated with the choice of the radius for the Coulomb function.For each correlation function it is set to the value from the one-dimensional analysis [33].Its variation by ±1 fm is a source of systematic uncertainty.Another source of systematic uncertainty is misidentification of particles and the associated purity correction.A 10% variation of the parameters in the purity correction was performed.We also incorporated sets with a reduced electron contamination by I) tightening the PID criteria, in particular extending the momentum range where the TOF signal was used and requiring the energy-loss measurement to be consistent with the kaon hypothesis within one sigma, and Table 3: Minimum and maximum uncertainty values for various sources of systematic uncertainty for charged and neutral kaons (in percent).Note that each value is the maximum uncertainty from a specific source, but can pertain to a different centrality or k T bin.Thus, the maximum total uncertainties are smaller than (or equal to) the quadratic sum of the maximum individual uncertainties.

Results and discussion
Figure 4 shows the m T -dependence of the extracted femtoscopic radii R out , R side , and R long in three centrality selections for pions [21] and charged and neutral kaons.The obtained radii are smaller for more peripheral collisions than for central ones.The radii decrease with increasing m T and each particle species roughly follows an m −1/2 T dependence.The radii in "out" and "long" directions exhibit larger values for kaons than for pions at the same transverse mass demonstrating that the m T -scaling is broken.This difference increases with centrality and is maximal for the most central collisions.Also presented in Fig. 4 are the predictions of the (3+1)D hydrodynamical model coupled with the statistical hadronization code THERMINATOR-2 [34].The model describes well the m T -dependence of pion radii, but underestimates kaon radii.Consistent with the data, the (3+1)D Hydro+THERMINATOR-2 model shows mild breaking in the "long" direction for central collisions, but it underestimates the breaking in the "out" direction.The significance of this breaking of the scaling is discussed further in this section.
In addition to the aforementioned three-dimensional radii, here for the 0-5% most central events, Fig. 5 also shows the m T dependence of the ratio R out /R side for charged and neutral kaons in comparison with HKM predictions [35] with and without the hadronic rescattering phase.The HKM calculations without rescattering exhibit an approximate m T -scaling but do not describe the data, while the data are well reproduced by the full hydro-kinetic model calculations thereby showing the importance of the rescattering phase at LHC energies.The R out and R side radii are both influenced by flow and rescatterings, so their ratio is rather robust against these effects.The fact that R out /R side ratio of pions and kaons coincide in the HKM simulations (Fig. 5) is related to some underestimation of R side radii for pions while pion R out radii are slightly overestimated in the model.Fig. 4: The 3D LCMS radii vs. m T for charged (light green crosses) and neutral (dark green squares) kaons and pions [21] (blue circles) in comparison with the theoretical predictions of the (3+1)D Hydro + THERMINATOR-2 model [34] for pions (blue solid lines) and kaons (red solid lines).Fig. 5: The 3D LCMS radii vs. m T for 0-5% most central collisions in comparison with the theoretical predictions of HKM [35] for pions (blue lines) and kaons (red lines).Fig. 6: The 3D LCMS radii vs. k T for charged (light green crosses) and neutral (dark green squares) kaons and pions [21] (blue circles).
It was predicted in [35] that the radii scale better with k T at LHC energies as a result of the interplay of different factors in the model, including the particular initial conditions.Figure 6 illustrates the k Tdependence of the femtoscopic radii R out , R side , and R long .Unlike the m T -dependence, the radii seem to scale better with k T in accordance with this prediction.
The ratio R out /R side appears to be sensitive to the space-time correlations present at the freeze-out hypersurface [21,37,38].As it was observed in [21], the ratio for pions is consistent with unity, slowly decreasing for more peripheral collisions and higher k T .In Fig. 7, the ratio R out /R side is shown for pions and kaons at different centralities.The systematic uncertainties partially cancel in the ratio.Systematic uncertainties are correlated in m T for each type of particle pair; no correlation between the systematic uncertainties of the charged and neutral species exists.The measured R out /R side ratios are slightly larger for kaons than for pions.This is an indication of different space-time correlations for pions and kaons, and a more prolonged emission duration for kaons.
In our previous pion femtoscopy analysis [20] the information about the emission time (decoupling time) at kinetic freeze-out τ ∼ 10 fm/c was extracted by fitting the m T -dependence of R 2 long using the blast-wave expression [60]: where T kin is the temperature at kinetic freeze-out, and K n are the integer-order modified Bessel functions.We tried to use Eq. ( 8) to fit the R 2 long m T -dependence (Fig. 8) for pions and kaons taking the thermal freeze-out temperature T kin = 0.120 GeV as in [20] (dotted lines) and T kin = 0.144 GeV (dashed lines).The emission times extracted from the fit are presented in Table 4.However, though this formula works well for pions, it fails to describe kaon longitudinal radii.Large transverse flow may be partially responsible for this failure [36].The following analytical formula for the time of maximal emission,  : T = 0.144 GeV, Eq.( 9) π K: T = 0.144 GeV, Eq.( 9) : T = 0.144 GeV, BW Eq.( 8) π K: T = 0.144 GeV, BW Eq.( 8) : T = 0.120 GeV, BW Eq.( 8) π K: T = 0.120 GeV, BW Eq.( 8) Fig. 8: R2 long vs. m T for kaons and pions.The solid lines show the fit using Eq. ( 9) for pions and kaons to extract the emission times (τ); the dashed and dotted lines show the fit using Eq. ( 8) with T kin = 0.144 GeV and T kin = 0.120 GeV, respectively.For pions at small m T , the dashed and dotted line coincide.
where cosh y , T max is the temperature at the hypersurface of maximal emission, β = 1/T max , and α is a free parameter determining the intensity of flow 2 .The advantage of the formula given in Eq. ( 9) is that it is derived for a scenario with transverse flow of any intensity, which  4: Emission times for pions and kaons extracted using the Blast-wave formula Eq. ( 8) and the analytical formula Eq. ( 9).
is especially important for LHC energies.
The analytical formula Eq. ( 9) was used to fit the m T -dependence of R 2 long (Fig. 8).The fit was performed using the following parameters determined in [36] by fitting light flavor particle spectra [61]: T max = 0.144 GeV, and α π = 5.0 and α K = 2.2.
The extracted times of maximal emission are presented in Table 4.
In order to estimate the systematic errors of the extracted times of maximal emission we also have performed fitting with T max , α π and α K varied within the range of their uncertainty [36]: ±0.03 GeV, ±3.5 and ±0.7, respectively.The maximum deviations from the central values appeared to be (+1.8,-0.5) fm/c for pions and (+0.5, -0.1) fm/c for kaons.These systematic errors are fully correlated.Regardless of the specific parameter choice, we consistently observe the time of maximal emission for kaons to be larger than the one for pions.The extracted times of maximal emission are rather close to those obtained within the HKM model [36]: τ π = 9.44 ± 0.02 fm/c, τ K = 12.40 ± 0.04 fm/c3 .There is evidence that the time of maximal emission for pions is smaller than the one for kaons.This observation can explain the observed breaking of m T -scaling between pions and kaons.It is interesting to note that in [36] this difference in the emission times is explained by the different influence of resonances on pions and kaons during the rescattering phase due to kaon rescattering through the K * (892) resonance (with lifetime of 4-5 fm/c).It was shown in [36] that a significant regeneration of the K * (892) takes place in full HKM simulations with rescatterings (UrQMD cascade), whereas this process is not present in a scenario where only resonance decays are taken into account.
Similar findings were reported in [62], where the production yield of K * (892) in heavy-ion collisions at the LHC was studied.Also there, the inclusion of a hadronic phase in the theoretical modeling of the production process proved to be essential in order to reproduce the experimentally found suppression pattern of K * (892) production when compared to pp collisions [63].

Summary
We presented the first results of three-dimensional femtoscopic analyses for charged and neutral kaons in Pb-Pb collisions at √ s NN = 2.76 TeV.
A decrease of source radii with increasing transverse mass and decreasing event multiplicity was observed.The m T scaling expected by pure hydro-dynamical models appears to be broken in our data.A scaling of pion and kaon radii with k T was observed instead.The measured ratio of transverse radii R out /R side is larger for kaons than for pions, indicating different space-time correlations.A new approach [36] for extracting the emission times for pions and especially for kaons was applied.It was shown that the measured time of maximal emission for kaons is larger than that of pions.
The comparison of measured three-dimensional radii with a model, wherein the hydrodynamic phase is followed by the hadronic rescattering phase [35], and pure hydrodynamical calculations [34,35] has shown that pion femtoscopic radii are well reproduced by both approaches while the behavior of the three-dimensional kaon radii can be described only if the hadronic rescattering phase is present in the model.

Fig. 1 :
Fig. 1: Single K ± purity (a) and pair purity for small relative momenta (b) for different centralities.In (b) the k T values for different centrality intervals are slightly offset for clarity.

Fig. 3 :
Fig. 3: A sample projected K 0 S K 0 S correlation function with fit.Also shown is the contribution to the fit from the quantum statistics part only.The error bars are statistical only.Systematic uncertainties on the points are equal to or less than the statistical error bars shown.

Table 1 :
Single particle selection criteria.