Freezeout systematics due to the hadron spectrum
Sandeep Chatterjee Debadeepti Mishra Bedangadas Mohanty Subhasis Samanta




 






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    ⟨⟩ł()̊ł()̊Sandeep.Chatterjee@fis.agh.edu.plAGH University of Science and Technology,
         
        Faculty of Physics and Applied Computer Science,
        
        al. Mickiewicza 30, 30-059 Krakow, Polanddebadeepti.m@niser.ac.inSchool of Physical Sciences,
         National Institute 
        of Science Education and Research, HBNI,
         Jatni-752050, Indiabedanga@niser.ac.inSchool of Physical Sciences,
         National Institute 
        of Science Education and Research, HBNI,
         Jatni-752050, Indiasubhasis@niser.ac.inSchool of Physical Sciences,
         National Institute 
        of Science Education and Research, HBNI,
         Jatni-752050, India
        
        We investigate systematics of the freezeout surface in heavy ion collisions due to the hadron 
        spectrum. The role of suspected resonance states that are yet to be confirmed experimentally 
        in identifying the freezeout surface has been investigated. We have studied two different 
        freezeout schemes - unified freezeout scheme where all hadrons are assumed to freezeout at the 
        same thermal state and a flavor dependent sequential freezeout scheme with different freezeout 
        thermal states for hadrons with or without valence strange quarks. The data of mean hadron yields 
        as well as scaled variance of net proton and net charge distributions have been analysed. We find 
        the freezeout temperature T to drop by ∼5% while the dimensionless freezeout parameters 
        μ_B/T and VT^3 (μ_B and V are the baryon chemical potential and the volume at freezeout 
        respectively) are insensitive to the systematics of the input hadron spectrum. The observed hint 
        of flavor hierarchy in T and VT^3 with only confirmed resonances survives the systematics 
        of the hadron spectrum. It is more prominent between ∼10 - 100 GeV where the maximum 
        hierarchy in T∼10% and VT^3∼40%. However, the uncertainties in the thermal parameters 
        due to the systematics of the hadron spectrum and their decay properties do not allow us to make 
        a quantitative estimate of the flavor hierarchy yet.
        
        Freezeout systematics due to the hadron spectrum
            Subhasis Samanta
            December 30, 2023
        ================================================
        
        
        
        § INTRODUCTION
        
        The determination of the last surface of inelastic scattering, the chemical freezeout surface (CFO), 
        is an integral part of the standard model of heavy ion collision <cit.>. An ideal gas of all the confirmed hadrons and resonances as listed by 
        the Particle Data Group (PDG) <cit.> forms the Hadron Resonance Gas (HRG) model that has met 
        with considerable success across a broad range of beam energies in describing the mean hadron yields <cit.> and more recently moments of conserved charges of QCD like baryon number (B), strangeness 
        (S) and charge (Q) <cit.> with a few 
        thermal parameters. Such an analysis gives us an access to the thermodynamic state of the fireball just prior to 
        freezeout. The ongoing hunt for the QCD critical point crucially depends on our knowledge of the background 
        dominated by the thermal hadronic physics close to freezeout. 
        
        The HRG partition function Zł T,μ_B,Q,S$̊ for a thermal state atł T,μ_B,μ_Q,μ_S$̊ 
        where T is the temperature and μ_B, μ_Q and μ_S are the chemical potentials corresponding 
        to the three conserved charges B, Q and S respectively, can be written as
        ln Z    =   ∑_iln Z_ił T,μ_B,Q,Seq.lnZ
        where Z_i is the single particle partition function corresponding to the ith hadron species written as
        ln Z_ił T,μ_B,Q,S=    VT^3ag_i/2π^2∫ dpp^2/T^3
              ×lnł1+ae^-ł√(ł p^2+m_i^2)+μ_i/̊Teq.lnZi
        where a=-1(+1) for mesons (baryons), g_i and m_i refer to the degeneracy factor and mass of the ith 
        hadron species and μ_i is its hadron chemical potential which within a complete chemical equilibrium 
        scenario is written as 
        μ_i    =    B_iμ_B+Q_iμ_Q+S_iμ_S
        where B_i, Q_i and S_i are the baryon number, charge and strangeness of the ith hadron species. 
        
        The sum in Eq. <ref> runs over all the established resonances from the PDG. However, quark 
        models <cit.> and studies on the lattice <cit.> predict many more 
        resonances than that have been confirmed so far in experiments. It has been pointed out in studies based on 
        comparison between QCD thermodynamics on the lattice and HRG that these resonances could have significant 
        contribution to several thermodynamic quantities <cit.> and influence the extraction 
        of the freezeout parameters within the HRG framework <cit.>. There have been interesting 
        studies on the status and influence of the systematics of the hadron spectrum on several quanitites <cit.>. In this work, we have studied the systematics in the determination of 
        the freezeout surface within the HRG framework due to the uncertainties over the hadron spectrum.
        
        
        
        § EXTRACTING CHEMICAL FREEZEOUT SURFACE
        
        The standard pratice has been to extract the CFO parameters by fitting the mean hadron yields. The primary 
        yields N^p_i are obtained from Eq. <ref> as follows
        
        N^p_i    =   ∂/∂μ_iln Z
        while the total yields of the stable hadrons that are fitted to data are obtained after adding the secondary 
        contribution from the resonance decays to their primary yields
        
        N^t_i    =    N^p_i + ∑_j BR_j→ iN^p_j
        where BR_j→ i refers to the branching ratio (BR) of the decay of the jth to ith hadron species.
        
        In this work we characterise the freezeout surface by three parameters of which only T has dimension. The 
        other two are suitably scaled dimensionless paramteters μ_B/T and VT^3. While μ_B/T controls the 
        baryon fugacity factor, VT^3 can be interpreted as the effective phase space volume occupied by the HRG at 
        freezeout. The masses of the hadrons are the relevant scales in this problem. They decide the freezeout T. 
        Thus, it is natural that systematic variation of the hadron spectrum will result in corresponding variation 
        of the freezeout T. On the other hand, the influence of the systematics of the hadron spectrum on the 
        dimensionless parameters μ_B/T and VT^3 is expected to be lesser. This motivates us to work with ł T, 
        μ_B/T, VT^3$̊ instead of the standard choice ofł T, μ_B, V$̊. The other parameter that is often used 
        in literature, the strangeness understauration factor γ_S has been taken to be unity here. μ_S and 
        μ_Q are solved consistently from the strangeness neutrality condition and demanding the ratio of net B to 
        Q be equal to that of the colliding nuclei (this ratio ∼2.5 for Au and Pb nuclei which we consider here)
        Net S    =    0
        Net B/Net Q    =    2.5
        As is evident from Eq. <ref>, knowledge of the BRs is essential to compute the contribution of 
        the secondary yield which is the feeddown from the heavier unstable resonances to the observed hadrons. As a result, 
        extraction of the freezeout surface based on the hadron yield data suffer from systematic uncertainties 
        of the decay properties of these addtional resonances.
        
        The freezeout surface can also be estimated by comparing higher moments of the conserved charges in 
        experiment and theory <cit.>. One of the important advantage in using the fluctuations of conserved charges over 
        hadron yields in estimating the freezeout surface is that it is enough to know only the quantum numbers of these 
        unconfirmed states. Decays under strong interactions should conserve the charges B, Q and S. Thus the conserved 
        charge susceptibilities are not influenced by the systematic uncertainties of the BRs of the unconfirmed resonances. 
        
        On the theoretical side, it is straightforward to compute the conserved charge susceptibilities χ^ijk_BQS 
        of order ł i+j+k$̊ from the partition function
        χ^ijk_BQS   =   ∂^i+j+kł P/T^4/∂^iłμ_B/T∂̊^jłμ_Q/T∂̊^kłμ_S/T
        where the pressurePis obtained from
        
        P    =   T/Vln Z
        The above susceptibilities computed in a model can then be converted easily to moments for a comparison 
        with the measured data. For example, the meanMand varianceσ^2of the conserved charge distribution 
        has one-to-one correspondence with the first two orders of susceptibility of the respective chargecM_c    =    N_c = VT^3χ^1_c
        σ^2_c    =   ł N_c- N_c^̊2 = VT^3χ^2_c
        withN_cbeing the observed net charge of typecin an event while N_cis the ensemble average.  
        We have evaluated the susceptibilities within HRG and estimated the influence of the missing resonances to 
        the extraction of freezeout parameters thereof. It has been found that scaled varianceσ^2/Mof netQand netBare well described within the HRG framework while higher moments like skewness and kurtosis show 
        up discrepancises, particularly at lower energies <cit.>. These higher 
        moments are also sensitive to non-ideal corrections like incorporating repulsive and attractive interactions 
        within the HRG framework <cit.>. Hence, 
        in this study we stick toσ^2/Mof netQandBto ascertain the influence of the systematics of 
        the hadron spectrum on the freezeout surface extracted from the data on conserved charge fluctuation. 
        
        On the experimental front, several uncertainties can creep into the measurement of conserved charge fluctuations. 
        The acceptance cuts in transverse momentum and rapidity is one of them <cit.>. Also, the neutral particles are 
        not detected which means net-proton fluctuations only act as an approximate proxy for netB<cit.>. 
        Currently, there is a tension between the PHENIX <cit.> and STAR <cit.> 
        measurements for net charge fluctuation. In this work, we have extracted the freezeout parameters for STAR data alone. 
        We expect the dependence of the freezeout parameters on the uncertainties of the hadron spectrum to be similar for 
        STAR and PHENIX data.
        
        While a single unified freezeout picture (1CFO) provides a good qualitative description across a broad range of beam 
        energies, recent studies have shown that a natural step beyond 1CFO would be to consider flavor hierarchy in 
        freezeout (2CFO), based on various arguments like flavor hierarchy in QCD thermodynamic quantities on the 
        lattice <cit.>, hadron-hadron cross sections <cit.> and melting 
        of in-medium hadron masses <cit.>. We have analysed the yield data in both the freezeout 
        schemes: 1CFO and 2CFO. However, for the conserved charge fluctuation study, currently only data for moment of net 
        proton (proxy for net baryon) <cit.> and net charge <cit.> are available. The netBand netQfluctuations are dominated by the non-strange sector as the lightest hadrons contributing to these 
        quantities are non-strange. Thus, the analysis of the data on the conserved charge fluctuations is not sensitive to 
        the thermal state of the strange sector. Hence, we have analysed the data on higher moments of the conserved charges 
        only within 1CFO.
        
        
        
        § HADRON SPECTRUM
