Role of system size on freezeout conditions extracted from transverse momentum spectra of hadrons

The data on hadron transverse momentum spectra in different centrality classes of p+Pb collisions at $\sqrt{s}_{NN} = 5.02$ TeV has been analysed to extract the freezeout hypersurface within a simultaneous chemical and kinetic freezeout scenario. The freezeout hypersurface has been extracted for three different freezeout schemes that differ in the way strangeness is treated: i. unified freezeout for all hadrons in complete thermal equilibrium (1FO), ii. unified freezeout for all hadrons with an additional parameter $\gamma_S$ which accounts for possible out-of-equilibrium production of strangeness (1FO$+\gamma_S$), and iii. separate freezeout for hadrons with and without strangeness content (2FO). Unlike in heavy ion collisions where 2FO performs best in describing the mean hadron yields as well as the transverse momentum spectra, in p+Pb we find that 1FO$+\gamma_S$ with one less parameter than 2FO performs better. This confirms expectations from previous analysis on the system size dependence in the freezeout scheme with mean hadron yields: while heavy ion collisions that are dominated by constituent interactions prefer 2FO, smaller collision systems like proton + nucleus and proton + proton collisions with lesser constituent interaction prefer a unified freezeout scheme with varying degree of strangeness equilibration.


I. INTRODUCTION
The knowledge of the surface of last scattering of the hadrons produced in a heavy ion collision (HIC) event is of utmost significance as it contributes to the calibration of the hadronic physics baseline to be contrasted with data to extract information of the quark gluon plasma (QGP) phase [1,2] as well as those of the QCD critical point [3,4]. The hadron resonance gas model has been the main phenomenological model to extract the freezeout hypersurface by comparing to the data of hadron yields [5][6][7][8][9][10] as well as spectra [11][12][13][14]. The surface where the hadrons cease to interact inelastically is known as the chemical freezeout surface. The hadron yields freeze here. The surface where the hadrons cease to interact even elastically is known as the kinetic freezeout surface. The shape of the transverse momentum spectra of hadrons get fixed here. Depending on the model assumptions, the chemical and kinteic freezeout surfaces could be separate [15][16][17] or together [11,12,[18][19][20]. In this study, we have worked with the THERMINATOR event generator where a combined frezeout of both yields as well as spectra at the same surface is implemented [21,22].
Traditionally, a single unified freezeout of all hadrons have been studied (1FO) [7][8][9]. However, the data from LHC have thrown open the interpretation of freezeout * ajayd@niser.ac.in † ranbir.singh@niser.ac.in ‡ Sandeep.Chatterjee@fis.agh.edu.pl § cjena@iisertirupati.ac.in ¶ bedanga@niser.ac.in and several alternate schemes have been proposed [23][24][25][26][27][28][29][30][31][32]. In the standard picture, freezeout is interpreted as a competition between fireball expansion and interaction of the constituents. Thus it is natural to expect system size dependence in freezeout conditions, since constituent interactions decrease as one goes from nucleusnucleus (A+A) to proton-nucleus (p+A) and protonproton (p+p) collisions. On the contrary, it was found that 1FO provides equally good description of data on mean hadron yields of e + + e − , p+p and A+A [33]. This lack of sensitivity of the 1FO approach on the varying rate of interaction amongst the constituents and fireball expansion across system size raises doubt on the standard interpretation of freezeout as a competition between expansion and interaction.
In Ref. [34], the yield data was analysed within three different approaches: i. 1FO, ii. single unified freezeout of all hadrons with an additional parameter γ S accounting for non-equilibrium production of strangeness (1FO+γ S ), and iii. separate freezeout surface for hadrons with and without strangeness content (2FO). The data on hadron yield was analysed across systems: p+p, p+Pb and Pb+Pb enabling one to study the freezeout condition for mid-rapidity charged particle multiplicity as well as the system volume varying over three orders of magnitude. It was found that while 1FO and 1FO+γ S schemes are blind to system size, 2FO exhibits a strong system size dependence. While for central and mid-central collisions, 2FO provides the least chi-square per degree of freedom, for peripheral Pb+Pb to all centralities of p+Pb and min bias p+p, 1FO+γ S provides a better description. This emphasizes a plausible freezeout scenario: in case of large system sizes, the freezeout dynamics is dominated by hadron interactions and hence flavor dependence in hadron-hadron cross sections play a role resulting in 2FO being the preferred freezeout scheme. On the other hand, in small systems the freezeout is mostly driven by rapid expansion and little interaction resulting in a sudden and rapid freezeout and hence disfavoring 2FO.
In this paper, we extend the above line of argument by studying the data on hadron spectra. The 2FO prescription has been already demonstrated to describe better the data on hadron spectra than 1FO in Pb+Pb at √ s NN = 2.76 TeV [14]. Here we study the data on hadron spectra in p+Pb at √ s NN = 5.02 TeV [35][36][37] and finally connect to our previous findings with the spectra data in Pb+Pb [14]. The spectra of π + + π − , K + +K − , p+p, φ, Λ+Λ, Ξ+Ξ and Ω+Ω are used for this study which are measured in the mid rapidity (0<y cm <0.5) by the ALICE collaboration. We have performed the centrality dependence of this study by analyzing the data in seven centrality classes, 0-5%, 5-10%, 10-20%, 20-40%, 40-60%, 60-80% and 60-100%.
The paper is arranged in the following way. In Sec. II we discussed about the model used for this study. The results from the model and data are compared in Sec. III. Finally we summarize our findings in Sec. IV.

II. MODEL
We have studied the data on hadron spectra in 3 schemes: 1FO, 1FO+γ S and 2FO using the THERMI-NATOR event generator [21,22]. While 1FO is implemented in the standard version of THERMINATOR, in Ref. [14] the standard version of THERMINATOR was extended to include the 2FO scheme. We now briefly describe the implementation of the freezeout hypersurface and the relevant parameters to be extracted in this approach.
The Cooper-Frye prescription provides the hadron spectra emanating from a freezeout hypersurface where T is the temperature, µ = {µ B , µ Q , µ S } refer to the three chemical potentials corresponding to the three conserved charges of QCD, u µ is the 4-velocity, dΣ µ is the differential element of the freezeout hypersurface over which the integration in Eq. 1 is supposed to be, p is the four momentum. There could be different choices for the parametrization of the freezeout hypersurface and u µ . We have worked within the Krakow model [11] whereby the freezeout is assumed to occur at a constant proper time while u µ is chosen to be where (t, x, y, z) is the space-time cooridinate.
THERMINATOR accounts for both primary production as well as secondary contribution from resonance decays when evaluating the distribution function f . The integration in Eq. 1 occurs over the freezeout hypersurface coordinates, namely the spacetime rapidity η s whose integration range is from minus infinity to plus infinity, the azimuthal angle φ which is integrated from 0 to 2π and ρ = x 2 + y 2 , the perpendicular distance between the Z-axis and the freezeout hypersurface. ρ is integrated from 0 to ρ max . Thus, we have 3 parameters within the 1FO scheme: T , τ f and ρ max to be extracted by comparison with data.
The choice of the thermodynamic ensemble is a relevant topic whenever one discusses system size dependence. In p+p collisions at the highest SPS and RHIC energies, the use of canonical ensemble or strangeness canonical ensemble has been suggested [38,39]. At the LHC energies, grand canonical ensemble was found to work best in describing the hadron yields [40]. Similar recent studies on the role of thermodynamic ensemble in small systems can be found in Refs. [41][42][43]. Here, we work with the grand canonical ensemble as well. Since we work with the LHC data that shows a very good particleantiparticle symmetry, we have set all the chemical potentials to zero. In 1FO+γ S , there is also the additional parameter γ S in f that accounts for out-of-equilibrium production of strangeness. In 2FO, we have different parameter sets for parametrising the non-strange (T ns , τ f ns and ρ max ns ) and strange (T s , τ f s and ρ max s ) freezeout hypersurfaces.

III. RESULT
We have varied the T in the range from 145 to 162 MeV in the steps of 1 − 2 MeV whereas ρ max and τ f are varied in the range 1.5 to 4.1 fm and 1.5 to 3.1 fm, respectively in steps of 0.1 fm. The goodness of the parameter set in describing the data is ascertained from the χ 2 /ndf , where and ndf = Number of data points -Number of free parameters.
The sum goes over all available p+Pb data points up to p T = 2.5 GeV/c [35][36][37]. For 1FO, we have varied all the three parameters T , ρ max and τ f to arrive at the best  [21,22] and data [35][36][37] is shown for three centralities: 0 − 5%, 10 − 20% and 60 − 80% in three different FO schemes at 0 < ycm < 0.5. The gross features of the spectra comparison seems to be independent of the freezeout scheme. The bottom panels show the ratio of data to model calculation. TeV. The average error on T and γs is 2 MeV and 0.02, respectively, whereas the error on ρmax and τ f is around 15% for all centrality classes.
Centrality parameter set. In 1FO+γ S , we have also varied γ S in the range 0.7 to 1.0 in steps of 0.2 while for 2FO, we have varied T , ρ max and τ f for both, non-strange as well as strange freezeout hypersurfaces. The p T spectra as obtained in the model for the different freezeout schemes have been compared with data in Fig. 1. In the bottom panel, we have shown the ratio of data to model. Unlike in Pb+Pb, where there are noticeable diagreement between 1FO and data referred to as proton anomaly which goes away on extending 1FO to 2FO, in p+Pb we don't find any such noteworthy tensions in 1FO. The quality of description of the spectra seems similar overall.
The χ 2 /ndf obtained in the different freezeout schemes across various centralities have been compared in Fig. 2 and the respective values of χ 2 and ndf are given in Table I. For all centralities, 1FO+γ S provides the least χ 2 /ndf . The improvement over 1FO and 2FO grows as one goes from central to peripheral collisions. This is driven by the strange sector which is more sensitive to the three different freezeout schemes studied here that differ in the treatment of the freezeout of strange hadrons. The yields in the non-strange sector receives a partial contribution from the decays of strange resonances. This leads to a small sensitivity in the fit quality of the non strange sector to the different freezeout schemes studied here. The improvement in the non-strange sector with 1FO+γ S is mild and uniform across centralities. We have enlisted the best parameter values that describe the transverse momentum spectra across different centralities within the three freezeout schemes in Table II. Fig. 3 we have plotted the extracted freezeout parameters corresponding to the least χ 2 /ndf with event multiplicity across different centralities in p+Pb and Pb+Pb that vary over three orders of magnitude. While the T remains mostly flat between 145 − 160 MeV, ρ max and τ f show a growth of 5-7 times. The growth rate is smooth across system size. We also note that the difference between the non-strange and strange freezeout parameters systematically increase as we go to events with higher multiplicity, signifying the role of interaction. However, currently the uncertainties over the extracted parameters in the non-strange and strange sectors are large and does not allow us to futher quantify the magnitude of the hierarchy in freezeout of the strange and non-strange flavors. γ S in p+Pb steadily grows from 0.74 to about 0.94 across peripheral to central collisions. The approach to strangeness equilibration with more central p+Pb events could be related to the larger entropy deposition in the initial state in central p+Pb collisions as opposed to peripheral events [44]. We use similar errors on T, ρ max and τ f as for Pb+Pb results [14] since the errors are mostly system size independent.

IV. SUMMARY AND OUTLOOK
The hadron yields and p T spectra are the standard observables to throw light on freezeout dynamics.
Contrary to expectations, the 1FO scheme is known to be blind to system size dependence in freezeout [33]. However, simultaneous analysis of the hadron yields in Pb+Pb, p+Pb and p+p revealed an interesting system size dependence of the preferred freezeout scheme-2FO is preferred over 1FO and 1FO+γ S in Pb+Pb while in small systems like p+Pb and p+p, 1FO+γ S is preferred [34]. In order to put this hypothesis on a more strong footing, here we extend the previous analysis to hadron spectra. While 2FO is known to describe the hadron spectra better in Pb+Pb, here we analyse the data for different centralities in p+Pb. We find that allowing for a different hypersurface for the freezeout of the strange hadrons do not improve the quality of the fits. This is in accordance to our previous study with the hadron yields [34]. Thus, our current analysis with the data on hadron spectra reaffirms the hypothesis on the system size dependence of freezeout scheme: flavor dependent freezeout scheme is preferred in large systems while unified freezeout is preferred in small systems. Thus, the role of interaction in larger system is mostly to delay the freezeout of the non-strange hadrons. The extracted freezeout parameters with best goodness of fit for different centralities in p+Pb and Pb+Pb collisions. There is a gradual preference for sequential freezeout of the strange and non-strange flavors as we go to higher multiplicity events. The γS value is fixed at one for Pb+Pb collisions at √ s NN = 2,76 TeV.