Nonequilibrium kinetic freeze-out properties in relativistic heavy ion collisions from energies employed at the RHIC beam energy scan to those available at the LHC

BGBW is a phenomenological model for hadron spectra based on flowing local thermal sources with global variables of temperature $T$ and transverse flow profile $\beta$ Schnedermann:1993ws ; Schnedermann:1994gc . $T$ is the temperature of the local thermal sources which particles radiate from. While the longitudinal expansion is assumed to be boost invariant, the transverse radial flow velocity of the thermal source is parametrized as $\beta(r)=\beta_{S}(\frac{r}{R})^{n}$ at radius $0\leqslant r\leqslant R$ with surface velocity $\beta_{S}$ and exponent $n$. The average radial flow velocity then can be written as $\langle\beta\rangle=\beta_{S}\cdot 2/(2+n)$ Ristea:2013ara . For an emitting source with Boltzmann-Gibbs distribution, the produced particle spectrum is therefore written in the form of

where $A$ is a normalization factor. $m_{T}=\sqrt{p^{2}_{T}+m^{2}}$ is the transverse mass of a particle. $I_{0}$ and $K_{1}$ are the modified Bessel functions. $\rho=\tanh^{-1}\beta$. $T$ is the kinetic freeze-out temperature. In order to compare with TBW results, we take $n=1$ for the BGBW model in this paper. With common freeze-out temperature $T$ and average radial flow velocity $\langle\beta\rangle$, the shape of the spectrum for each particle species is determined by its mass in BGBW.

TBW Tang:2008ud ; Shao:2009mu ; Tang:2011xq ; Ristea:2013ara is modified from the standard BGBW model when a Tsallis statistics replaces the conventional Boltzmann-Gibbs statistics for the particle emission distribution. The invariant differential particle yield in TBW is then written in the form of

where

$y_{s}$ is the source rapidity. $y_{b}$ is the beam rapidity. $\phi$ is the particle emission angle in the rest frame of the thermal source. $q$ is the parameter characterizing the degree of non-equilibrium of the produced system, which is the new parameter introduced in TBW compared to the BGBW model. Although the applicability of such a model to high energy nuclear collisions is still under investigation, possible physics implications are available in the literature. The initial energy density in heavy ion collision has multiple hot spots caused by Color-Glass Condensate formation in a nucleon or individual nucleon-nucleon collisions. Those hot spots are dissipated into the system, producing more particles, generating collective flow, and resulting in a temperature fluctuation at the initial state Wilk:2008ue ; Wilk:2009nn . The initial state fluctuation is not guaranteed to be completely washed out by the medium, in QGP or hadron gas phases. The survived fluctuation will leave imprints in spectra at low and intermediate $p_{T}$ range. Such features in spectra will lead to $q$ values larger than 1 in the TBW model Ristea:2013ara . When $q=1$, Eq. (2) recovers its familiar Boltzmann-Gibbs form.

In this paper, we also use a Tsallis blast-wave model with four fit parameters with different $q$ for mesons and baryons separately, referred to as TBW4. TBW4 was first proposed in reference Tang:2008ud for a better description of meson and baryon spectra at $p$+$p$ collisions while the TBW fits with one single $q$ for all particles gave a very poor $\chi^{2}/nDoF$. TBW without further description in this paper refers to the default one with three fit parameters that is the one using the same $q$ for both mesons and baryons.

This section compares three blast-wave model fits of the transverse momentum spectra. Table 1 lists the particle spectra data used in this paper. Those particle spectra form two species groups for the fit procedure: one with all available hadrons, and another with charged pion, kaon, proton, and antiproton only. The first group aims to identify the common freeze-out properties for all particles. The latter is chosen to be consistent with previous publications Adamczyk:2017iwn for an apple-to-apple comparison. The reported experimental systematic and statistical uncertainties are combined as a quadratic sum for the fitting procedures. For all fit procedures, the average flow velocity $\langle\beta\rangle$ is limited to the range of $0\leqslant\langle\beta\rangle\leqslant 2/3$ Abelev:2008ab for a better convergence in fitting and to avoid nonphysical results (negative value or faster than the speed of light) Tang:2008ud . Furthermore, spectra fit range is limited to $p_{T}\leqslant$ 3 GeV/$c$ in order to have a comparable $p_{T}$ range for all energies and to focus on the bulk properties. For the sake of concision, this section only shows the fits to spectra for all particles in most central and most peripheral centrality classes at four collision energies as examples in Figures 1, 2, 3 and 4. Figure 1 (Figure 2) shows blast-wave fits to identified particle transverse momentum spectra in most central (most peripheral) collisions, with corresponding deviations of those fits to experimental data divided by data uncertainties shown in Figure 3 (Figure 4). The fit results of kinetic freeze-out parameters for both species groups at various centrality classes and collision energies are discussed in the next section. Those extracted fit parameters and $\chi^{2}/nDoF$ of TBW models are also summarized in Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13.

BGBW and TBW models are compared in the left and middle panels of Figures 1, 2, 3 and 4. From top to bottom panels of each figure, the spectra (Figures 1 and 2) or difference between model and experiment data divided by the error of data (Figures 3 and 4) in Pb + Pb or Au+Au collisions at $\sqrt{s_{\rm{NN}}}=$ 2.76 TeV, 200 GeV, 62.4 GeV and 7.7 GeV are presented. At LHC and top RHIC energies, the deviation of the BGBW model from experimental data is larger in peripheral than in central collisions. As beam energy decreases, the deviation of BGBW model fit from data decreases. We note that there are fewer experimental data at $p_{T}>2$ GeV/$c$ for energies below 39 GeV. Overall, TBW has better fits and has smaller $\chi^{2}/nDoF$ than BGBW. TBW agrees with most data points within three-$\sigma$ standard deviation from experimental data. TBW yields a smaller $q$ toward lower beam energy, which indicates that the system is closer to equilibrium state toward lower energy. For all LHC and RHIC energies, TBW fits in peripheral collisions have larger $q$ values than those in central collisions at the same collision energy. In short, the TBW model performs much better than BGBW and the non-equilibrium seems to be necessary for peripheral collisions at high energies. As the BGBW model assumes thermal equilibrium and the TBW model uses non-equilibrium statistics, the above observations suggest that the collision system deviates more from thermal equilibrium at higher energy, especially in peripheral collisions.

The comparison of TBW (with a single $q$) and TBW4 (with separate $q$ for meson and baryon) is shown in the middle and right panels of Figs. 1, 2, 3 and 4. TBW4 has even smaller $\chi^{2}/nDoF$ than TBW for all LHC and RHIC energies, while the improvement is larger in peripheral than central collisions. For TBW4, the non-equilibrium parameter $q$ of baryons is found to be smaller than that of mesons, as baryons have steeper spectra than mesons. Baryons used in the fitting include mostly strange ($\Lambda$) and multi-strange ($\Xi$ and $\Omega$) particle species and strangeness has smaller $q$ value and higher freeze-out temperature. More details on different freeze-out will be discussed later in Section III.2.

The extracted results for temperature $T$ and average radial flow velocity $\langle\beta\rangle$ from BGBW, TBW, and TBW4 are compared in Figures 5 and 6. The beam energy, centrality, and particle species dependences of $T$, $\langle\beta\rangle$, and $q$ from the TBW model are investigated in Figures 7 and 8.

The dependence of $T$ on $\langle\beta\rangle$ of BGBW, TBW, and TBW4 is shown in Figure 5 for charged pions, kaons, and protons, and in Figure 6 for all available hadrons including strange and multi-strange particles. Symbols with same color represent A+A collisions at same beam energy for different centrality classes. In general, fit parameters in Figure 6 for all particles have smaller fit uncertainties than those in Figure 5 for charged pions, kaons, and protons only, as more particles are used to study their common freeze-out properties. Other than that, these two species groups give similar results for $T$ dependence on $\langle\beta\rangle$. BGBW results shown on the left panel display an anti-correlation between $T$ and $\langle\beta\rangle$. At the same collision energy, $T$ decreases and $\langle\beta\rangle$ increases as the system moves from peripheral to central collisions. As collision energy increases, the anti-correlation curve moves toward high $\langle\beta\rangle$. Such anti-correlation behavior was also reported in Ref. Adamczyk:2017iwn . TBW results in the middle panel, however, are different from BGBW ones. $T$ from TBW has much weaker dependence on centrality than the BGBW one. For example, at the LHC energies, the increase of $T$ from most central to most peripheral collisions is around 40% in BGBW. In contrast, the variation of $T$ is only about 5% in TBW. Similar behavior of weak centrality dependence for temperature was also reported in Ref. Mazeliauskas:2019ifr , when further resonance decay after sudden equilibrium freeze-out is considered in BGBW. The resonance decay is one of the microscopic non-equilibrium sources in the macroscopic TBW approach. The common observation from these two studies supports the expectation that the strong centrality dependence of freeze-out temperature in the BGBW model is rooted in its incapability to describe the non-equilibrium system. The parameter $\langle\beta\rangle$ in most central collisions from TBW is between 0.4 and 0.5 for RHIC energies and around 0.6 at the LHC, similar with those in BGBW. In peripheral collisions $\langle\beta\rangle$ is lower in TBW than that in BGBW, while it even reaches zero value for the most peripheral collisions at RHIC energies. It seems that from the TBW model’s viewpoint hadron scatterings (or QGP droplets if any) are not sufficient to produce a large collective radial flow or to maintain a thermal equilibrium in peripheral collisions at RHIC. The BGBW model, while lacking a knob for non-equilibrium degree, has to boost its radial flow parameter in a struggle to fit the high yields of the spectra at intermediate $p_{T}$ in the peripheral collisions. In Figures 5 and 6, $T$ and $\langle\beta\rangle$ from TBW4 on the right panels are similar to those from TBW with one single $q$ in the middle panels and different from those from BGBW on the left panels. There is a weaker centrality dependence for $T$ and lower $\langle\beta\rangle$ values at peripheral collisions for both TBW models than BGBW. We observed that at the LHC energies, $T$ and $\langle\beta\rangle$ from TBW4 tend to have a positive correlation rather than anti-correlation as in BGBW fits or a lack of correlation as in default TBW. The fit parameter values are slightly different for the two TBW models as discussed below. In Figure 5, for charged pions, kaons, and protons, $T$ from TBW4 is lower than the one in default TBW at $\sqrt{s_{\rm{NN}}}$ above 62.4 GeV. For lower beam energies, within the uncertainties, $T$ values from these two models appear to be consistent. For the cases including all hadrons and shown in Figure 6, the results are in general with smaller uncertainties of all the fit parameters, and $T$ from the TBW4 are lower than the default TBW for all the energies. The main observation from the comparison of BGBW and two TBW models is that $T$ in TBW models has weaker centrality dependence and $\langle\beta\rangle$ at peripheral collisions is lower than those in BGBW.

Figure 7 shows the energy and centrality dependence of kinetic freeze-out parameters $T$, $\langle\beta\rangle$, and $(q-1)$ from the TBW model for charged pions, kaons, and protons in the left column and for all particles in the right column. The kinetic freeze-out temperature $T$ in panel (a) and (b) of Figure 7 shows weak collision energy dependence at $\sqrt{s_{\rm{NN}}}$ of 7.7 - 39 GeV, while it drops from $\sqrt{s_{\rm{NN}}}=$ 62.4 GeV to 5.02 TeV in panel (a) and 2.76 TeV in panel (b). At 7.7 - 39 GeV, at any given collision energy, $T$ only decreases marginally from peripheral to central collisions. At 62.4 GeV to 5.02 TeV, the centrality dependence of $T$ is even smaller. In contrast, as discussed in the previous section, in BGBW, $T$ decreases notably from peripheral to central collisions. In most of the peripheral collisions, BGBW deviates significantly from data with larger $\chi^{2}/nDoF$. Meanwhile, the non-equilibrium parameter $q$ in TBW for peripheral collisions also increases with increasing collision energy as shown in the bottom two panels of Figure 7. The strong centrality dependence of $T$ in BGBW may be synthetic to the model’s incapacity to incorporate the large non-equilibrium effect of the system in the peripheral collisions. The average transverse radial flow velocity $\langle\beta\rangle$ shown in panel (c) and (d) of Figure 7, for most central collisions, is between 0.4 and 0.5 at RHIC energies, and around 0.6 at the LHC energies. $\langle\beta\rangle$ decreases from central to peripheral collisions. In most peripheral collisions, $\langle\beta\rangle$ drops to zero at RHIC energies and is less than 0.3 at the LHC energies. For the most peripheral collisions at RHIC, the system in general fails to generate a rapid radial expansion. The non-equilibrium parameter ($q-1$) in panel (e) and (f) of Figure 7 is small in central Au+Au collisions, suggesting that the produced particles are approaching thermal equilibrium. In peripheral collisions, ($q-1$) increases from less than 0.04 at 7.7 GeV to more than 0.1 at 5.02 TeV, indicating an increasing deviation from Boltzmann statistics as collision energy increases. The centrality dependence of the ($q-1$) parameter suggests an evolution from an almost thermalized system in the central collisions towards a highly off-equilibrium system in the peripheral collisions. Such large ($q-1$) is also found in the study of $p$+$p$ collision Jiang:2013gxa . This may be because the energy density fluctuations at initial state due to Color-Glass Condensate formation or individual hard scattering (minijets) inside a nucleus-nucleus collision increase as collision energy increases. Such fluctuations are not completely washed out by subsequent QGP evolution or hadronic interactions and leave footprints in final state particle spectra at the $p_{T}$ range in our paper Tang:2008ud .

In general, the group of charged pion, kaon, and proton and the group of all particles as shown in Figure 7 produce similar kinetic freeze-out parameters in TBW fits. Small difference can be identified with slightly higher $T$ and lower $q$ for the group with all particles than that with only the $\pi/K/p$. Such difference may come from the influence of particle species as the group of all particles contains more strange particles. The direct comparison of non-strange and strangeness in Figure 8 confirms that the strange hadrons have higher temperature ($T$) and a smaller non-equilibrium degree ($q$) than those of non-strange hadrons, while their radial flow values ($\langle\beta\rangle$) are similar. A higher temperature indicates an earlier decoupling of strange hadrons from the system. The smaller $q$ in the strangeness group than the non-strangeness group and a similar $\langle\beta\rangle$ between those two groups suggest that the system is closer to an equilibrium state when the strangeness hadrons decouple from the system and further hadronic interactions do not increase the system’s radial flow velocity. A possible conclusion is that the hadronic phase does not increase radial flow of light hadrons significantly at RHIC and LHC energies, and instead drives the system toward non-equilibrium: the system in central collisions has approached thermal equilibrium at the partonic phase, and the later hadronic scattering drives the system off equilibrium and does not increase the radial flow of copiously produced light hadrons Shao:2009mu . Another interesting observation is that in Figure 8 (b) for non-strange particles the kinetic freeze-out temperature of the central collisions decreases from RHIC to LHC energies in the TBW model, while in Figure 8 (a) strangeness does not show this behavior. A possible explanation is that the system at the LHC has higher flow velocity and larger volume than that at RHIC and maybe needs more time for all particles to kinetic freeze-out (“cool”) in the expansion during the hadronic phase.

It has been argued within the framework of non-equilibrium statistics that the dependence of temperature and flow velocity on the non-equilibrium factor ($q-1$) is related to the shear and bulk $\xi$ viscosity in linear or quadratic proportion Wilk:1999dr ; Wilk:2008ue . This hypothesis is examined by quadratic fits of $\langle\beta\rangle=\langle\beta\rangle_{0}-a(q-1)^{2}$ and $T=T_{0}+b(q-1)-d\xi(q-1)^{2}$ (where $\xi$ is the bulk viscosity) to the inclusive hadron group as shown in Figures 9 and 10. Data at 7.7 GeV are close to equilibrium and do not provide a significant variation of the parameters, and are not included in this examination. From 11.5 to 2.76 TeV collision energy, there displays a clear evolution of $\langle\beta\rangle$ vs $(q-1)$ and $T$ vs $(q-1)$ relationships on collision energy. A summary of parameters $\langle\beta\rangle_{0}$, $a$, $T_{0}$, $b$, and $d\xi$ dependence on collision energy is depicted in Figures 11 and 12. As energy increases, $\langle\beta\rangle_{0}$ increases and the coefficient of the squared term $a$ decreases. A similar feature is observed for the $T$ vs $(q-1)$. With only three available centrality classes, the fitting procedure at 62.4 GeV is found to be not constrained. The relationship of $T$ vs $(q-1)$ was previously inspected in Ref. Tang:2008ud for 200 GeV where only a squared term (with a constant) is used. Our paper shows that both linear and quadratic terms are needed to describe the $T$ vs $(q-1)$ relationship for lower collision energies. The linear term parameter $b$ and quadratic term related to viscosity parameter $d\xi$ show a trend of decrease with collision energy. It is interesting to note that it has been argued that the bulk viscosity increases dramatically toward the phase transition Kharzeev:2007wb ; Karsch:2007jc , coinciding with the feature we observed of $d\xi$ shown in Figure 12.

In a short summary, BGBW and TBW have been used to explore the beam energy dependence of kinetic freeze-out properties of the system created in relativistic heavy ion collisions. The BGBW model is designed to describe the system in local thermal equilibrium. However, as collision energy increases, the produced system in peripheral collisions deviates far from equilibrium state, and can not be described well with a BGBW fit. An additional parameter $q$ is introduced in the TBW model to characterize the degree of non-equilibrium. The divergence between BGBW and TBW escalates with an increasing $q$ value as collision energy increases, especially in peripheral collisions. For 7.7 - 39 GeV collision energies, the increase of temperature in the TBW model from central to peripheral collisions is much less than that in BGBW. For 62.4 GeV to 5.02 TeV, the temperature in the TBW model stays almost constant from central to peripheral collisions. Meanwhile, the radial flow value in central collisions is around 0.4-0.5$c$ at RHIC energies, and becomes larger at the LHC energies.