Novel scheme for parametrizing the chemical freeze-out surface in Heavy Ion Collision Experiments

TL;DR

This paper presents a new method for determining chemical freeze-out parameters in heavy-ion collisions using the Hadron Resonance Gas model, which accurately predicts hadron yield ratios without prior use in the parametrization.

Contribution

The novel scheme reliably estimates freeze-out parameters and predicts hadron yields, improving upon existing methods in heavy-ion collision analysis.

Findings

Accurately estimates chemical freeze-out parameters.

Predicts hadron yield ratios without prior data.

Demonstrates reliability in heavy-ion collision modeling.

Abstract

We introduce a new prescription for obtaining the chemical freeze-out parameters in the heavy-ion collision experiments using the Hadron Resonance Gas model. The scheme is found to reliably estimate the freeze-out parameters and predict the hadron yield ratios, which themselves were never used in the parametrization procedure.

Equations

$$\ln Z^{ideal}=\sum _i\ln Z_i^{ideal},$$

$$\ln Z_i^{ideal}=\pm \frac{V{g}_i}{(2\pi {)}^3}\int d^{3}p\ln [1\pm \exp (-(E_i-{\mu }_i)/T)],$$

$$n_i(T,{\mu }_B,{\mu }_Q,{\mu }_S)=\frac{g_i}{{(2\pi )}^3}\int \frac{d^{3}p}{\exp [(E_i-{\mu }_i)/T]\pm 1}.$$

$$\frac{d{N}_i}{dy}{|}_{Det}=\frac{dV}{dy}{n}_i^{Tot}{|}_{Det}$$

$$n_i^{Tot}$$

$$\frac{{\sum }_i{n}_i(T,{\mu }_B,{\mu }_Q,{\mu }_S)Q_i}{{\sum }_i{n}_i(T,{\mu }_B,{\mu }_Q,{\mu }_S)B_i}=r$$

$$\sum _i{n}_i(T,{\mu }_B,{\mu }_S,{\mu }_Q)S_i=0$$

$${\chi }^2=\sum _i\frac{(Rati{o}_i^{Model}-Rati{o}_i^{Expt}{)}^2}{{{\sigma }_i}^2}$$

$$\frac{\partial {\chi }^2}{\partial x}=0;\mathrm{for}x\subset \{T,{\mu }_B\}.$$

$$\frac{{\sum }_i^{Det}{B}_i\frac{d{N}_i}{dY}}{{\sum }_i^{Det}|B_i|\frac{d{N}_i}{dY}}$$

$$\frac{{\sum }_i^{Det}{B}_i\frac{d{N}_i}{dY}}{{\sum }_i^{Det}\frac{d{N}_i}{dY}}$$

Full text

Heavy Ion collision, Chemical freeze-out, Hadron Resonance Gas model

Introduction: $-$ Strongly interacting matter is expected to exhibit a rich phase structure under extreme conditions of temperature and density. Exotic phases with quasi-free quarks and gluons may have existed at high temperatures in the very early universe Kolb and Turner (1990). Even today, in the core of compact stars with high baryon densities, various exotic phases like color superconductivity, color superfluidity, may be present Rajagopal and Wilczek .

Direct signatures of these phases can only be accessed in experiments with relativistic nuclear collisions that are being pursued at CERN (France/Switzerland) and BNL (USA), and also to be carried out at GSI (Germany) and JINR (Russia). In the canonical picture of heavy-ion collision (HIC), the high density fireball formed, is expected to thermalize rapidly, expand out fast and then cool down quickly. As the system expands, the inter-particle distances increase, and subsequently all thermal and chemical interactions freeze-out. Finally the detected strongly interacting particles are the hadrons and their resonances which may be in chemical equilibrium Hagedorn (1965); Hagedorn and Rafelski (1980). Later, it has been argued that a transient partonic phase is more likely to drive the onset of equilibration in the system Anishetty et al. (1980); Rafelski and Müller (1982); Koch et al. (1986).

In a pioneering work Cleymans and Redlich (1998) using the Hadron Resonance Gas (HRG) model, it was argued that the freeze-out surface may be universally characterized by the average energy per hadron to have a value of 1 GeV. Subsequently there has been a huge interest in studying the properties of strongly interacting matter using HRG model Rischke et al. (1991); Cleymans et al. (1993); Braun-Munzinger et al. (1995); Cleymans et al. (1997); Yen et al. (1997); Heinz (1999); Braun-Munzinger et al. (1999); Cleymans and Redlich (1999); Braun-Munzinger et al. (2001); Becattini et al. (2004); Braun-Munzinger et al. (2003); Karsch et al. (2003); Tawfik (2005); Becattini et al. (2006); Andronic et al. (2006, 2009); Manninen and Becattini (2008); Tiwari et al. (2012); Begun et al. (2013); Andronic et al. (2012); Fu (2013); Tawfik (2013); Garg et al. (2013); Bhattacharyya et al. (2014); Albright et al. (2014); Kadam and Mishra (2015, 2016); Kadam (2015); Albright et al. (2015); Bhattacharyya et al. (2015, 2016); Begun (2016); Bhattacharyya et al. (2017); Andronic et al. (2018); Dash et al. (2018). This model has successfully described hadron yields from AGS to LHC energies Braun-Munzinger et al. (1995); Cleymans et al. (1997); Yen et al. (1997); Braun-Munzinger et al. (1999); Cleymans and Redlich (1999); Braun-Munzinger et al. (2001, 2003); Becattini et al. (2004); Cleymans et al. (2005); Letessier and Rafelski (2008); Becattini et al. (2006); Cleymans et al. (2006); Andronic et al. (2006); Manninen and Becattini (2008); Andronic et al. (2009); Tiwari et al. (2012); Chatterjee et al. (2013); Alba et al. (2014); Chatterjee et al. (2015); Adak et al. (2017). In this context the discussions of multi-strange enhancement Chin and Kerman (1979) and saturation of strangeness Rafelski and Müller (1982) in a quark-gluon phase came up. Some authors also considered possible under-saturation of strangeness in the observed spectrum Koch et al. (1986), Becattini et al. (2004). Bulk properties of hadronic matter have also been studied in this model Karsch et al. (2003); Tawfik (2005); Andronic et al. (2012). Moreover, role of various undiscovered resonance states in determining the freeze-out surface has been investigated Chatterjee et al. (2017). There has been recent attempts to address the freeze-out conditions even from the first principle lattice QCD Gupta et al. (2011); Bazavov et al. (2012). The general perception from all these studies is that at freeze-out the hadrons are in thermodynamic equilibrium.

Here we propose a novel fitting procedure for extraction of thermal parameters from experimental hadron yields with the ideal HRG model and show that the ratios of the yields are very well reproduced for experiments over a wide range of collision energies.

HRG Model: $-$ The grand canonical partition function of the hadron resonance gas is given by,

[TABLE]

The sum runs over all hadrons and resonances. The thermodynamic potential for $i$’th species is given as,

[TABLE]

where the upper sign is for baryons and lower for mesons. Here $V$ is the volume, $T$ is the temperature, and for the $i^{th}$ species of hadron, $g_{i}$, $E_{i}$ and $m_{i}$ are respectively the degeneracy factor, energy and mass, while $\mu_{i}=B_{i}\mu_{B}+Q_{i}\mu_{Q}+S_{i}\mu_{S}$ is the chemical potential, with $B_{i}$, $Q_{i}$ and $S_{i}$ denoting the baryon number, electric charge and strangeness respectively. Here $\mu_{B}$, $\mu_{Q}$ and $\mu_{S}$ are the baryon, electric and strangeness chemical potentials respectively. For a thermalized system the number density $n_{i}$ can be calculated from partition function, which is given as,

[TABLE]

The thermal parameters $(T,\mu_{B},\mu_{Q},\mu_{S})$ may then be obtained by fitting experimental hadron yields to the model parametrization via the relation between the rapidity density for $i$’th detected hadron to the corresponding number density in the HRG model Manninen and Becattini (2008),

[TABLE]

where the subscript $Det$ denotes the detected hadrons. Here,

[TABLE]

where the summation is over the heavier resonances $j$ that decay to the $i^{th}$ hadron. Usually the systematics due to the volume factor is removed by considering hadron yield ratios. Thereafter one needs four equations to solve for the four freeze-out parameters $T$, $\mu_{B}$, $\mu_{Q}$ and $\mu_{S}$. The $\mu_{Q}$ and $\mu_{S}$ are fixed by imposing the constraints Alba et al. (2014),

[TABLE]

and

[TABLE]

where $r$ is net-charge to net-baryon number ratio of the colliding nuclei. For example, in Au + Au collisions $r=N_{p}/(N_{p}+N_{n})=0.4$, with $N_{p}$ and $N_{n}$ denoting the number of protons and neutrons in the colliding nuclei. Thereafter the $T$ and $\mu_{B}$ are conventionally fitted by optimizing the $\chi^{2}$ of the multiplicity ratios with respect to $T$ and $\mu_{B}$. Here the $\chi^{2}$ is defined as,

[TABLE]

and the two minimization equation correspond to,

[TABLE]

Thus the freeze-out parameters are obtained from Eq. (6$-$9). A satisfactory solution is obtained if $\chi^{2}$ over degrees of freedom (dof) is close to 1 Press et al. (1996).

The New Approach: $-$ The value of $\chi^{2}$ and extracted set of parameters strongly depend on the set of multiplicity ratios chosen. Selecting a particular set of ratios may bias the minimization process Andronic et al. (2006). This is possibly due to the fact that the individual multiplicities or their ratios are not independent quantities. If they were then one could replace Eq.(8$-$9) by equating any two multiplicity ratios from the model and experimental data. These along with Eq. (6$-$7) would then give the freeze-out parameters. Usually if a solution is obtained in this method, the prediction of other multiplicity ratios are found to be significantly away from the experimental data. This is why a $\chi^{2}$ analysis is done including as many independent individual multiplicity ratios as available from experimental data. As a result the prediction of hadron yields in the $\chi^{2}$ fit seems to be built in by default. Nonetheless the $\chi^{2}$ fit does produce a set of freeze-out parameters commensurate with all the multiplicity ratios to some extent.

Here we ask if it is possible to consider any truly independent observable multiplicity ratios so that simply equating them between the model and the experimental data one can obtain the freeze-out parameters. And finally whether they predict all individual hadron multiplicity ratios satisfactorily. In this direction we again note that for strong interactions to be in chemical equilibrium, there are five independent thermodynamic variables $-$ $V$, $T$, $\mu_{B}$, $\mu_{Q}$ and $\mu_{S}$. Obviously the three net conserved charges are independent, but are not enough to determine the five thermodynamic parameters. The three corresponding total charges though not conserved are however independent. We therefore perform our analysis of freeze-out data based on this observation.

Considering the three net charges and three total charges we have one more independent quantity than that required to determine five thermodynamic variables. We can then assume a constrain like entropy conservation. In terms of the observed hadrons this imposes a constrain on the total number of particles Landau (1953). We can then uniquely determine all the five thermodynamic variables of the system. Following the general practice we considered ratios of the different charges to scale out the volume and any other systematics. We thus need to consider four independent ratios. Fortunately Eq. (6$-$7) are already of the desired form (Eq.(7) may be read as the ratio of net strangeness to total strangeness to be zero). Here, instead of the two optimization equations Eq. 9, with respect to $T$ and $\mu_{B}$ we introduce two new independent equations. We choose the net baryon number normalized to the total baryon number and the net baryon number normalized to the total hadron yield, to form the other two equations, as given below.

[TABLE]

The ratios on the left hand side of the above equations consists of the rapidity density of hadron yields measured in the HIC experiments and those on the right are the number densities calculated in the HRG model. The sum runs only over the identified hadrons for which the yield data are available. The equations are clearly unique and independent of each other if a sufficient number of identified hadrons are involved.

Data Analysis: $-$ We have used AGS Ahle et al. (2000a, b); Klay et al. (2003, 2002); Back et al. (2001a); Blume and Markert (2011); Back et al. (2001b); Barrette et al. (2000); Back et al. (2004), SPS Alt et al. (2008a, 2005); Afanasiev et al. (2002); Afanasev et al. (2000); Bearden et al. (2002); Anticic et al. (2004); Antinori et al. (2004, 2006); Alt et al. (2008b, c); Anticic et al. (2004), RHIC Kumar (2013); Das (2013); Adler et al. (2002a); Adams et al. (2004); Zhu (2012); Zhao (2014); Kumar (2014); Das (2014); Abelev et al. (2009a); Aggarwal et al. (2011); Abelev et al. (2009b); Adcox et al. (2002); Adams et al. (2004); Adler et al. (2002b); Adams et al. (2007, 2005); Kumar (2012); Adams et al. (2007) and LHC Abelev et al. (2012, 2013a, 2014, 2013b) data for our analysis. STAR BES data has been used following Chatterjee et al. (2015); Nasim et al. (2015); Adamczyk et al. (2017). In the present study we have only taken mid-rapidity data for the most central collisions.

In our HRG spectrum we have used all hadrons up to 2 GeV which are confirmed with known degrees of freedom. The masses and branching ratios used are as given in Wheaton and Cleymans (2009); Tanabashi et al. (2018). But data are available for only a few hadrons at various collision energies. The identified hadrons used to obtain the freeze-out parameters are, $\pi^{\pm}$ (139.57 MeV), $k^{\pm}$ (493.68 MeV), $p$,$\bar{p}$ (938.27 MeV), $\Lambda$,$\bar{\Lambda}$ (1115.68 MeV), $\Xi$,$\bar{\Xi}$ (1321.71 MeV). All those hadrons reported at one collision energy may not be available in another. For example, we could not find the $\bar{\Lambda}$ yield at LHC. At this energy we assumed $\bar{\Lambda}$ yield to be same as that reported for ${\Lambda}$. Similarly, we did not use $\Omega$ data for any parametrization, as the individual yields of $\Omega^{+}$ and $\Omega^{-}$ are not available for most of the $\sqrt{s}$. Also, $\phi$ (1019.46 MeV) has been excluded from the fitting as it is already included in the model through its strong decay channel to kaon. For the lower AGS energies we could not find any anti-baryon data. There we have used $\Lambda$ to proton ratio and total baryon to total hadron yield as a substitution of Eq. 10 and Eq. 11.

The equations Eq. (6$-$7$-$10$-$11), are highly non-linear and are solved numerically using the Broyden’s method with a convergence criteria of $10^{-6}$ or better. We had to tune the initial conditions accordingly for different $\sqrt{s}$ to achieve desired convergence accuracy. The variances of the fitted parameters were obtained by extracting the freeze-out parameters at the extremum values of the hadron yields given by the experimental variances.

Freeze-out Parameters: $-$ The freeze-out parameters are depicted in Fig. 1. The general behavior as well as the quantitative estimates are commensurate with those in the existing literature. The variation of the freeze-out temperature with the center of mass energy $\sqrt{s}$ is shown in Fig. 1a. As expected, the temperature increases with increasing $\sqrt{s}$, and approaches a saturation Hagedorn (1965), except at the LHC energy where the temperature is lower. This is probably due to the lower yield of protons at LHC Abelev et al. (2013b).

In Fig. 1b the various chemical potentials are shown as functions of $\sqrt{s}$. The baryon chemical potential $\mu_{B}$ decreases with increasing $\sqrt{s}$, which is usually understood as follows. For low collision energies a significant amount of baryons may be deposited in the collision region (baryon stopping). On the other hand at high collision energies the colliding baryons may almost pass through each other and get deposited outside the collision region. Similarly the electric charge chemical potential $\mu_{Q}$ should have also followed the same trend as the colliding nuclei only consisted of positively charged protons. However $\mu_{Q}$ remains negative throughout, approaching zero for increasing $\sqrt{s}$. Here, the neutrons in the colliding nuclei (lead or gold), being more abundant than the protons, induce an isospin dominance in favor of $\pi^{-}$ than $\pi^{+}$. The pions being the lightest charged particles, dictates the sign of $\mu_{Q}$. On the other hand, strangeness production is expected to be dominant at higher baryon densities due to the possible redistribution of Fermi momentum among larger degrees of freedom lowering the Fermi energy Witten (1984). Though it is not clear whether this picture should hold in the HIC scenario, the fitted strangeness chemical potential $\mu_{S}$, does indeed show such a behavior.

In Fig.1a and Fig.1b we have also shown comparison of our results with those in literature. Specifically we have used results from Andronic et al. (2006); Becattini et al. (2006) for SPS and RHIC energies. For BES energy range we have compared our results with (Adamczyk et al., 2017). The general agreement in almost all the studies is apparant, though there are certain variations in the models. For example Ref. (Adamczyk et al., 2017) considered a strangeness suppression factor and have assumed $\mu_{Q}$ to be zero. Similarly Ref. Becattini et al. (2004, 2006); Cleymans et al. (2006); Manninen and Becattini (2008); Andronic et al. (2006) have considered a strangeness suppression factor. The system volume was extracted as a parameter in Ref. Andronic et al. (2006, 2009, 2012). In an other work Letessier and Rafelski (2008) a light quark fugacity factor has also been introduced, which may have significant effect on the value of extracted parameter set and $\chi^{2}/dof$.

Hadron Yield Ratios: $-$ With the freeze-out parameters obtained, we now discuss the various predicted hadron yield ratios. Though the hadronic yields were used in the analysis, none of their individual ratios were part of the equations solved, and are therefore quite independent predictions from the model. The only exception is the use of the single ratio $\Lambda/p$ for the lower AGS energies. From the experimental data the variations in the yield ratios are obtained from those of the individual yields using standard error propagation method Knoll (2000). We have considered both systematic and statistical errors and the total error for a particular yield was obtained in quadrature. Here we discuss some important representative hadron ratios. The predictions of other hadron yields also came out satisfactorily and will be presented elsewhere.

In Fig. 2 the $\pi^{-}$ to $\pi^{+}$ ratio is shown as a function of $\sqrt{s}$. As discussed earlier, this ratio is greater than 1 for low $\sqrt{s}$ and approaches 1 at higher collision energies. Rather than the +ve charge of the protons the higher neutron abundance seems to push for the isospin asymmetry in favor of $\pi^{-}$. The data are well reproduced by our analysis.

Similarly the $p/\pi^{+}$ and $\bar{p}/\pi^{-}$ variation with $\sqrt{s}$ is also well reproduced as shown in Fig 3. At lower $\sqrt{s}$, $p>\bar{p}$ and $\pi^{-}>\pi^{+}$, while at higher $\sqrt{s}$ the corresponding particles and antiparticles become equal. This explains the variations shown.

The $k/\pi$ ratio is considered to be an important observable for strangeness enhancement in high energy collisions. A ’$horn$’ in the $k^{+}/\pi^{+}$ ratio was originally suggested as a signature of QGP Gazdzicki and Gorenstein (1999); Gazdzicki (2004); Bratkovskaya et al. (2004); Koch et al. (2005). Several authors have tried to explain the behavior of $k^{+}/\pi^{+}$ and $k^{-}/\pi^{-}$ using different approaches (see Nayak et al. (2010) and references therein). The comparison between the experimental data for these ratios and the corresponding predictions from our model analysis is shown in Fig. 4. We find that irrespective of the underlying physical mechanism that gives rise to the $horn$, which is beyond the scope of the HRG model, the experimental ratios and model predictions agree quite well.

In Fig. 5 the ratios $\Lambda/p$ and $\Xi^{-}/p$ are shown. The agreement for $\Lambda/p$ is reasonable except for a slight down-shift of the predicted results as compared to the experimental data. Consideration of possible uncertainties in contribution from weak decays may remove this discrepancy Abelev et al. (2009a). Such uncertainties are not included in our analysis. The model predictions for $\Xi^{-}/p$ is found to agree well with the experimental data, as shown in Fig. 5b.

Finally we present some ratios for the $\phi$, and $\Omega$ particles whose yield data were never used in the analysis. The $\phi/\pi^{+}$ ratio is shown in Fig. 6. Since $\phi$ has no net charge of any kind, it is dependent only on the temperature. The predicted $\phi/\pi^{+}$ plot closely resembles the temperature plot. Again prediction from our model agrees reasonably with the experimental data.

The predictions for the $\Omega/p$ ratios are shown in Fig. 7. For this ratio the experimental data are available at only a few $\sqrt{s}$. The model predictions seem to agree quite well.

As mentioned earlier we have not done a $\chi^{2}$ fit. But it would be interesting to find out the values of $\chi^{2}/dof$ from our estimated freeze-out parameters. One can make various choices of independent particle ratios. We have used two sets $-$ one with various hadron yields in the numerator and $\pi^{+}$ yield in the denominator (Set 1), and another with different indepndent ratios $\pi^{-}/\pi^{+}$, $k^{+}/\pi^{-}$, $k^{-}/k^{+}$, $p/k^{-}$, $\bar{p}/p$, $\Lambda/\bar{p}$, $\bar{\Lambda}/\Lambda$, $\Xi^{-}/\bar{\Lambda}$, $\Xi^{+}/\Xi^{-}$ (Set 2). The variation of $\chi^{2}/dof$ with $\sqrt{s}$ is shown in Fig. 8, along with some available results in the literature Andronic et al. (2006); Becattini et al. (2006); Adamczyk et al. (2017) We find the reduced $\chi^{2}$ in various approaches to be generally in agreement with each other.

Summary and Outlook: $-$ Here we introduced a novel formalism for parametrizing the chemical freeze-out surface of hadrons identified from the heavy-ion collision experiments. We have demonstrated that the conserved charges of strong interactions may be utilized to obtain the freeze-out parameters. We explored the chemical freeze-out surface, in the central rapidity bins for the most central collisions in a wide range of center of mass energies. The various charge ratios agree in the model and data to the accuracy of $10^{-6}$ or better. This strongly confirms that the various charges themselves are in chemical equilibrium near freeze-out.

Our formalism allows us to successfully reproduce the various hadronic yield ratios. Even the predictions for multi-strange hadrons like $\Omega$, whose yields were never used in the parametrization, are predicted satisfactorily. This simple yet physically consistent approach could be a viable alternative over the conventional $\chi^{2}$ scheme for analyzing data from the upcoming heavy-ion collision experiments.