Flavour and Energy Dependence of Chemical Freeze-out Temperatures in Relativistic Heavy Ion Collisions from RHIC-BES to LHC Energies
FFlavor and Energy Dependence of Chemical Freeze-out Temperatures in RelativisticHeavy Ion Collisions from RHIC-BES to LHC Energies
Fernando Antonio Flor, ∗ Gabrielle Olinger, † and René Bellwied ‡ Department of Physics, University of Houston, Houston, Texas 77204, USA (Dated: October 1, 2020)We present calculations of the chemical freeze-out temperature ( T ch ) based on particle yieldsfrom STAR and ALICE measured at collision energies ranging from √ s NN = T L = 150 . ± . MeV and T S = 165 . ± . MeV, respectively.
I. INTRODUCTION
In depth determinations of a pseudo-critical tempera-ture based on continuum extrapolations of temperaturedependence of the chiral susceptibilities on the lattice, incomparison to calculations using Statistical Hadroniza-tion Models (SHM) using particle yields from experi-ments at the Relativistic Heavy Ion Collider (RHIC) andthe Large Hadron Collider (LHC), indicate that chemi-cal freeze-out and hadronization coincide near the phaseboundary in the Quantum Chromodynamics (QCD)phase diagram [1–5]. Whether this transition from quarkto hadron degrees of freedom occurs at a uniform temper-ature for all quark flavors remains a question of interest.SHMs have been successful in adequately reproducinghadronic particle abundances over nine orders of magni-tude in high energy collisions of heavy ions over a widerange in energy [6, 7]. In these calculations, assuminga thermally equilibrated system, experimental particleyields in relativistic heavy ion collisions serve as an an-chor for the determination of common freeze-out param-eters in the QCD phase diagram – namely, the baryo-chemical potential ( µ B ) and the chemical freeze-out tem-perature ( T ch ). The resulting parameters can also becompared with independently obtained results from ei-ther lattice QCD based susceptibility calculations of con-served quantum numbers or measurements of higher or-der fluctuations of net-particle distributions. II. SEQUENTIAL FLAVOR FREEZE-OUT
Continuum extrapolated susceptibility calculations ofsingle flavor quantum numbers on the lattice [8, 9] haveshown a difference in the determined freeze-out tempera-tures between flavors in the crossover region of the QCD ∗ fafl[email protected] † [email protected] ‡ [email protected] phase diagram. This effect is likely due to the differ-ence in the bare quark masses which is not negligible in athermally equilibrated deconfined system near the phaseboundary. In particular, a comparison of the flavor spe-cific susceptibility ratios χ /χ , suggested as an observ-able for directly determining freeze-out temperatures [10],show a deviation of the lattice and Hadron ResonanceGas (HRG) model calculations coinciding at the peaks ofthe lattice data, which occur at flavor specific tempera-tures differing by − MeV from light to strange quarks[9]. Net-particle fluctuation measurements by the STARcollaboration have also shown comparable temperaturedifferences between the light and strange mesons [11, 12].Thermal fits to experimental data via SHMs haveshown similar results, depicting a difference in the freeze-out temperatures between flavors in the crossover region.The STAR Collaboration recently published the depen-dence of their thermal fits to the yields on the particlespecies included in the fit rendering a freeze-out temper-ature about − MeV lower for a fit pions, kaonsand protons than a common freeze-out temperature ex-tracted from fits to yields of all measured particle species[13]. Moreover, it has been shown that assuming twodistinct freeze-out temperatures improves the overall fitto ALICE data. Thus, a point of interest arises whencomparing the extracted freeze-out parameters obtainedusing different sets of particles in the SHM calculation.A first study of this approach was performed in Ref. [14].
III. MODEL AND DATA PREPARATION
The entirety of our analysis was performed using theopen source Thermal FIST (The FIST) thermal modelpackage [15]. Without loss of generality, The FIST isa user-friendly package within the family of HRG Mod-els. Although there exists a wide range of options for theHRG model within The FIST framework, we restrictedour analysis to the default – namely, modeling an idealnon-interacting gas of hadrons and resonances within aGrand Canonical Ensemble (GCE). a r X i v : . [ nu c l - e x ] S e p All our calculations used the PDG2016 + hadronic spec-trum [16] as the HRG input list, including a total of 783states (i.e. *, **, *** and **** states from the 2016Particle Data Group Data Book [17]). Deviations of theHRG calculations from the lattice curves of flavor specificsusceptibilities at specific temperatures in the crossoverregion may be affected by the inclusion of certain states[18–20], thus a realistic determination of the underlyinghadronic spectrum is key to this study. The PDG2016 + hadronic spectrum has been shown to be an optimizedcompromise between too few (found) and too many (froma simple Quark Model) excited states when compared toa large number of lattice QCD predictions [16].Yield data for π + , π - , K + , K - , p , ¯ p , Λ , ¯Λ , Ξ − , ¯Ξ + , Ω − , ¯Ω + , K , and φ for ALICE PbPb collisions at √ s NN =2 . TeV [21–24] and preliminary results at . TeV [25]in the 0 - 10% centrality class, as well as STAR AuAucollisions at √ s NN = 11 . , . , . , . , . and GeV [13, 26–30] were used. We excluded AuAu collisionsat √ s NN = 7 . GeV due to the wider centrality binningof the data, particularly for multi-strange baryons.For the sake of brevity, we introduced a shorthand no-tation when naming our fits with (anti)particle species(e.g. Ω refers to both Ω − and ¯Ω + , etc.). This shorthandis used for the remainder of this letter.(Anti)proton yields for the STAR data in Refs. [13, 28,29] are all inclusive. In order to correct for weak-decayfeed-down contributions from ( ¯Λ ) Λ , we interpolated thecontributions to (anti)proton yields based on the methodsuggested in Ref. [31]. Table I summarizes our interpola-tion results. The contribution from ( ¯Λ ) Λ to (anti)protonyields are labeled as δ . These δ values were subtractedfrom unity and multiplied by their respective (anti)protonyields. The resulting (anti)proton yields were then usedfor the entirety of this analysis. This procedure should beconsidered an upper limit for the feed-down contributionsince the experiment imposes an, albeit loose, primaryvertex cut on the ( ¯Λ ) Λ decay daughter candidates. Thesepercentages are in general agreement with estimates inthe aforementioned STAR papers, though. TABLE I: Interpolation results from methods used in Ref. [31]for weak-decay feed-down contributions to (anti)proton yieldsat STAR AuAu Collisions from √ s NN = √ s NN (GeV) δ δ % % % % % % % % % %
200 34.00 % % All our thermal fits were performed with T ch (MeV) and V (fm ) as free parameters, setting γ S and γ q to unity.Our analysis focused on varying the particle species in-cluded in the fit. The particle species included in ourtemperature fits were πKp (light), πKp ΛΞΩ K φ (all)and K ΛΞΩ K φ (strange), respectively. The inclusion ofthe kaons in the light fit was done in order to avoid toofew degrees of freedom in the fit and has no effect on theextracted freeze-out temperature, since the kaon yield israther insensitive to the temperature, as was also shownpreviously in Ref. [32]. For the two ALICE energies, welet µ B = 0 . For all fits to STAR AuAu data, we let µ B be a free parameter in the fits to also gauge its sensitivityto the flavor-specific fits. IV. RESULTS AND DISCUSSION
We extracted freeze-out parameters, T ch and V ,for the full ( πKp ΛΞΩ K φ ), light ( πKp ), and strange( K ΛΞΩ K φ ) particle thermal fits. Figure 1 shows T ch from the full fits as a function of µ B . The magenta linerepresents the spline fit function with three nodes (PbPbat √ s NN = 5 . TeV and AuAu at √ s NN = 27 . and . GeV) to the freeze-out parameters. The magenta bandrepresents the residual to the spline fit and is determinedby the mean square error of the spline function. (MeV) B m (MeV) ch T ALICE PbPb @ 5.02 TeVSTAR AuAu @ 200 GeVSTAR AuAu @ 62.4 GeVSTAR AuAu @ 39.0 GeVSTAR AuAu @ 27.0 GeVSTAR AuAu @ 19.6 GeVSTAR AuAu @ 11.5 GeV f S0 K W X L K p p Thermal FIST: PDG2016+
Grand Canonical Ensemble
FIG. 1:
Full
GCE fits to STAR and ALICE data measuredat collision energies ranging from √ s NN = + hadronicspectrum. Magenta bands shows a spline fit to the points. Figure 2 shows T ch from the light and strange fit as afunction of µ B ; both fits are compared to Lattice QCDcalculations in Ref. [1]. Each flavor dependent spline fitand error band was determined in the same manner as inFigure 1. The width of the lattice curve is based on thewidth ( σ ) of the chiral susceptibility [1].Detailed fit results for each energy including V and χ /dof , are shown in Table II. Generally the separationinto light and strange particles improves the quality ofthe fits by at least a factor two at all energies. (MeV) B m (MeV) ch T f S0 K W X L K K p p Thermal FIST: PDG2016+
Grand Canonical Ensemble
Lattice QCD
WB Collaboration (2015)
FIG. 2:
Strange (blue points) and light (red points) GCEfits to STAR and ALICE data measured at collision energiesranging from √ s NN = % ) via TheFIST using the PDG2016 + hadronic spectrum. In Figure 2, the light and strange fits consistently fallwithin the lattice QCD crossover width as long as thatwidth is defined by the pseudo-critical temperature ofall possible order parameters. Our flavor-dependent fitsagree with the calculated freeze-out temperatures fromnet-proton, net-charge and net-kaon fluctuations up to µ B (cid:39) MeV [12, 33]. T ch remains constant with in-creasing µ B until µ B (cid:39) MeV, where the strange andlight fit begin to approach. The two fits converge withinerrors at µ B (cid:39) MeV; therefore, we propose that aseparate treatment of strange and light particles mightnot be meaningful at µ B ≥ MeV. The convergence ofthe two flavor-dependent temperatures is expected in thevicinity of a critical point in the QCD phase diagram. Yields / dof = 5.639 c : T = 150.4 MeV, = 2.446 L / dof) c = 142.4 MeV, ( L : T = 0.163 S / dof) c = 164.4 MeV, ( S : T + p - p + K - K S0 K p p
L L f X X W W - - - Data s (Model-Data)/ - Yields / dof = 1.407 c : T = 161.8 MeV, = 0.866 L / dof) c = 151.0 MeV, ( L : T = 0.961 S / dof) c = 164.4 MeV, ( S : T + p - p + K - K S0 K p p
L L f X X W W - - - Data s (Model-Data)/ FIG. 3: Top and bottom panels show GCE fits to ALICE PbPb at √ s NN = 5 . TeV (0 - 10%) and STAR AuAu at √ s NN = 39 . GeV (0 - 10%), respectively, via The FIST using the PDG2016 + hadronic spectrum. Single temperature (1CFO) yield calculationsare shown in magenta. Two temperature (2CFO) yield calculations are shown in dashed blue lines. Experimental values [25, 26]are shown in green. TABLE II: The FIST Grand Canonical Ensemble Yield Fits via the PDG2016 + hadronic spectrum for collision energies rangingfrom √ s NN = 11 . to GeV. The top, middle and bottom sections show the full ( πKp
ΛΞΩ K S φ ), light ( πKp ), and strange( K ΛΞΩ K S φ ) particle fits, respectively. For all fits, µ B , T ch , and V were used as free parameters. πKp ΛΞΩ K S φ √ s NN (GeV) µ B (MeV) T ch (MeV) V ( fm ) χ /dof ± ± ± ± ± ± ± . ± ± ± ± ± ± ± ± ± . ± ± ± . ± ± ± πKp √ s NN (GeV) µ B (MeV) T ch (MeV) V ( fm ) χ /dof ± ± ± ± ± ± ± . ± ± ± ± ± ± ± ± ± . ± ± ± . ± ± ± K ΛΞΩ K S φ √ s NN (GeV) µ B (MeV) T ch (MeV) V ( fm ) χ /dof ± ± ± ± ± ± ± . ± ± ± ± ± ± ± ± ± . ± ± ± . ± ± ± The greatest difference between the two temperaturesis seen at the highest energy, namely the ALICE top en-ergy. To address the impact of a flavor dependent freeze-out on the particle abundances, we calculated yields forthe full particle set ( πKp
ΛΞΩ K φ ) for ALICE PbPb 5.02TeV with a one chemical freeze-out (1CFO) approach andcompared them with yields calculated for light ( πKp )and strange ( K ΛΞΩ K φ ) particles separately with a twochemical freeze-out (2CFO) approach. We fixed the tem-perature(s) and volume(s) to the T ch and V values shownin Table II for √ s NN = 5020 GeV. In the 1CFO approach,we calculated yields using a temperature of 150.4 MeV.In the 2CFO approach, our light and strange particleyield calculations were done with temperatures fixed to142.4 MeV and 164.4 MeV, respectively. We note thatour 1CFO temperature differs from the value quoted byALICE, which is based on the Heidelberg-GSI model, byabout 4 MeV, most likely due to the difference in the hadronic input spectrum [25].Figure 3 shows this comparison of 1CFO and 2CFOapproaches for full datasets at two exemplary energies,namely the preliminary 5.02 TeV central PbPb data fromALICE and the 39.0 GeV central AuAu data from STAR.The deviations of each yield calculation from the exper-imental value are shown at the bottom of each plot. Weobserve that the 2CFO approach provides an excellentand much improved description of the experimental data;rendering yields within one standard deviation of the ex-perimental measurements for most particle species. The2CFO treatment all but eliminates the tension betweenlight and strange baryons, the so-called proton anomaly,seen in the 1CFO approach. It should also be notedthat alternative methods to treat interactions in the SHMvia the S-matrix approach [34] impact in particular theproton yields and improve the performance of the 1CFOmethod in the Heidelberg-GSI fits [7].
V. CONCLUSION
We presented calculations of the chemical freeze-outtemperature ( T ch ) based on particle yields from STARand ALICE measured at collision energies ranging from √ s NN = T L = 150 . ± . MeV and a strange flavor freeze-out temperature T S =165 . ± . MeV at vanishing µ B , employing the GCEapproach within the framework of the The FIST HRGmodel package. We showed evidence for flavor-dependentchemical freeze-out temperatures in the crossover regionof the QCD phase diagram, which start to convergeabove µ B (cid:39) MeV. We employed the flavor-dependenttwo temperature approach via The FIST to successfully model and reproduce experimental yields at top ALICEenergies. Thus, at the highest energies at RHIC and theLHC a separation of the hadronization temperature oflight and strange particles seems likely. Furthermore, ourresults suggest the existence of a critical point in the QCDphase diagram above µ B (cid:39) MeV and a temperaturebelow T ch (cid:39) MeV.
VI. ACKNOWLEDGMENTS
The authors acknowledge edifying discussions withVolodymyr Vovchenko, Claudia Ratti, Paolo Parotto,Jamie Stafford, Livio Bianchi and Boris Hippolyte.This work was supported by the DOE grant DEFG02-07ER4152. [1] R. Bellwied, S. Borsányi, Z. Fodor, J. Günther, S. Katz,C. Ratti, and K. Szabó, Physics Letters B , 559(2015).[2] S. Borsányi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg,C. Ratti, and K. K. Szabó, Journal of High EnergyPhysics , 73 (2010).[3] S. Borsányi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg,and K. K. Szabó, Physics Letters B , 99 (2014).[4] A. Bazavov et al. (HotQCD Collaboration), Phys. Rev.D , 094503 (2014).[5] J. Stachel, A. Andronic, P. Braun-Munzinger, andK. Redlich, Journal of Physics: Conference Series ,012019 (2014).[6] J. Cleymans and K. Redlich, Phys. Rev. Lett. , 5284(1998).[7] A. Andronic, P. Braun-Munzinger, K. Redlich, andJ. Stachel, Nature , 321 (2018).[8] C. Ratti, R. Bellwied, M. Cristoforetti, and M. Barbaro,Phys. Rev. D , 014004 (2012).[9] R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, andC. Ratti, Phys. Rev. Lett. , 202302 (2013).[10] F. Karsch, Central European Journal of Physics , 1234(2012).[11] L. Adamczyk et al. (STAR Collaboration), Phys. Rev.Lett. , 032302 (2014).[12] R. Bellwied, J. Noronha-Hostler, P. Parotto, I. Por-tillo Vazquez, C. Ratti, and J. M. Stafford, Phys. Rev.C , 034912 (2019).[13] L. Adamczyk et al. (STAR Collaboration), Phys. Rev. C , 044904 (2017).[14] S. Chatterjee, A. K. Dash, and B. Mohanty, Journalof Physics G: Nuclear and Particle Physics , 105106(2017).[15] V. Vovchenko and H. Stoecker, Computer Physics Com-munications , 295 (2019).[16] P. Alba, R. Bellwied, S. Borsányi, Z. Fodor, J. Gün-ther, S. D. Katz, V. Mantovani Sarti, J. Noronha-Hostler,P. Parotto, A. Pasztor, I. P. Vazquez, and C. Ratti, Phys. Rev. D , 034517 (2017).[17] C. Patrignani et al. (Particle Data Group), Chin. Phys. C40 , 100001 (2016).[18] J. Noronha-Hostler and C. Greiner, Nuclear Physics A , 1108 (2014).[19] P. Alba, V. M. Sarti, J. Noronha-Hostler, P. Parotto,I. Portillo-Vazquez, C. Ratti, and J. M. Stafford, Phys.Rev. C , 054905 (2020).[20] A. Bazavov, H.-T. Ding, P. Hegde, O. Kaczmarek,F. Karsch, E. Laermann, Y. Maezawa, S. Mukherjee,H. Ohno, P. Petreczky, C. Schmidt, S. Sharma, W. Soeld-ner, and M. Wagner, Phys. Rev. Lett. , 072001(2014).[21] B. Abelev et al. (ALICE Collaboration), Phys. Rev. C , 044910 (2013).[22] B. Abelev et al. (ALICE Collaboration), Phys. Rev. Lett. , 222301 (2013).[23] B. Abelev et al. (ALICE Collaboration), Phys. Rev. C , 024609 (2015).[24] B. Abelev et al. (ALICE Collaboration), Physics LettersB , 216 (2014).[25] F. Bellini (ALICE Collaboration), Nuclear Physics A , 427 (2019).[26] J. Adam et al. (STAR Collaboration), Phys. Rev. C ,034909 (2020).[27] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C , 034909 (2009).[28] M. M. Aggarwal et al. (STAR Collaboration), Phys. Rev.C , 024901 (2011).[29] G. Agakishiev et al. (STAR Collaboration), Phys. Rev.Lett. , 072301 (2012).[30] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. , 062301 (2007).[31] A. Andronic, P. Braun-Munzinger, and J. Stachel, Nu-clear Physics A , 167 (2006).[32] D. Magestro, Journal of Physics G: Nuclear and ParticlePhysics , 1745 (2002).[33] P. Alba, W. Alberico, R. Bellwied, M. Bluhm, V. M. Sarti, M. Nahrgang, and C. Ratti, Physics Letters B , 305 (2014).[34] A. Andronic, P. Braun-Munzinger, B. Friman, P. M. Lo, K. Redlich, and J. Stachel, Phys. Lett.