Different Non-extensive Models for heavy-ion collisions

TL;DR

This paper compares various non-extensive statistical models for describing transverse momentum spectra in heavy-ion collisions, exploring their relations and revisiting the Soft+Hard model with new assumptions.

Contribution

It introduces two different linear relations between temperature and Tsallis parameter based on different assumptions and revisits the Soft+Hard model with a T-independent average p_T^2.

Findings

Qualitative agreement with Wilk's results

Derivation of two linear T-q relations under different assumptions

Comparison with Kaniadakis' κ distribution

Abstract

The transverse momentum ($p_T$) spectra from heavy-ion collisions at intermediate momenta are described by non-extensive statistical models. Assuming a fixed relative variance of the temperature fluctuating event by event or alternatively a fixed mean multiplicity in a negative binomial distribution (NBD), two different linear relations emerge between the temperature, $T$, and the Tsallis parameter $q-1$. Our results qualitatively agree with that of G.~Wilk. Furthermore we revisit the "Soft+Hard" model, proposed recently by G.~G.~Barnaf\"oldi \textit{et.al.}, by a $T$-independent average $p_T^2$ assumption. Finally we compare results with those predicted by another deformed distribution, using Kaniadakis' $\kappa$ parametrization.

Tables

Table 1. Table 1: Fit parameters to p T subscript 𝑝 𝑇 p_{T} spectra in P ​ b ​ P ​ b 𝑃 𝑏 𝑃 𝑏 PbPb collisions within the ’soft + hard’ model of Tsallis − q 𝑞 -q .

Centrality,   Parameters:	$q_{soft}$	$T_{D,soft}$	$q_{hard}$	$T_{D,hard}$
$0\sim 5$	1.036	0.317	1.170	0.091
$5\sim 10$	1.037	0.318	1.167	0.097
$10\sim 20$	1.040	0.315	1.166	0.102
$20\sim 30$	1.044	0.309	1.165	0.107
$30\sim 40$	1.045	0.308	1.159	0.116
$40\sim 50$	1.051	0.296	1.157	0.121
$50\sim 60$	1.054	0.288	1.155	0.124
$60\sim 70$	1.059	0.279	1.150	0.129
$70\sim 80$	1.067	0.266	1.149	0.128

Table 2. Table 2: Fit parameters of p T subscript 𝑝 𝑇 p_{T} spectra in P ​ b ​ P ​ b 𝑃 𝑏 𝑃 𝑏 PbPb collisions with the ’soft + hard’ model using Kaniadakis’ κ 𝜅 \kappa .

Centrality,   Parameters	${\kappa }_{soft}$	$T_{D,soft}$	${\kappa }_{hard}$	$T_{D,hard}$
$0\sim 5$	0.088	0.358	0.190	0.126
$5\sim 10$	0.091	0.359	0.188	0.134
$10\sim 20$	0.097	0.359	0.189	0.132
$20\sim 30$	0.103	0.355	0.187	0.141
$30\sim 40$	0.108	0.354	0.184	0.150
$40\sim 50$	0.113	0.347	0.188	0.129
$50\sim 60$	0.127	0.330	0.183	0.154
$60\sim 70$	0.152	0.279	0.161	0.153
$70\sim 80$	0.154	0.264	0.165	0.145

Equations

$$e_q(x):=[1+(1-q)x{]}^{\frac{1}{1-q}}.$$

$$e_q(x)\cdot e_q(-x)=1.$$

$$n_{B,F}=\frac{1}{e_q(x)\mp 1}$$

$$e_{\kappa }(x)\cdot e_{\kappa }(-x)=1,$$

$$e_{\kappa }(x):=[\sqrt{1+(\kappa x{)}^2}+\kappa x{]}^{\frac{1}{\kappa }}.$$

$$q=1-\frac{1}{C}+\frac{\Delta T^2}{T^2},$$

$${\sigma }^2=\frac{\Delta T^2}{T^2}\approx \mathrm{const}.$$

$$T=E[{\sigma }^2-(q-1)]$$

$$P_{n}=\left(\begin{array}[]{c}n+k\\ n\\ \end{array}\right)f^{n}(1+f)^{-n-k-1}$$

$$G(t)=\sum P_n{e}^{nt}=-(k+1)\ln \left(1+f-f{e}^t\right).$$

$$⟨n⟩$$

$$\Delta n^2$$

$$T=\frac{E}{f}(q-1).$$

$$p^0\frac{dN}{d^{3}p}=(p^0\frac{dN}{d^{3}p}{)}^{hard}+(p^0\frac{dN}{d^{3}p}{)}^{soft}.$$

$$T:=T_D\sqrt{\frac{1-v}{1+v}},$$

$$\omega :=\gamma (m_T-v{p}_T)-m,$$

$$p^0\frac{dN}{d^{3}p}=\sum _i{C}_i\cdot e_{q_i}(-{\omega }_i/T_i)$$

$$\frac{\omega }{T}\to \frac{\gamma (1-v)p_T}{T}=\frac{p_T}{T_D}.$$

$$p^{0}\frac{dN}{d^{3}p}\Bigg{|}_{i}\>=\>C_{i}\cdot e_{q_{i}}(-p_{T}/T_{Di}).$$

$$⟨p_T^2⟩\approx \frac{\int p_T^{3}d{p}_T(1+\frac{q-1}{T_D}{p}_T{)}^{-\frac{1}{q-1}}}{\int p_{T}d{p}_T(1+\frac{q-1}{T_D}{p}_T{)}^{-\frac{1}{q-1}}}=\frac{6T_D^2}{(4-3q)(5-4q)}.$$

$$p^{0}\frac{dN}{d^{3}p}\Bigg{|}_{i}\>=\>C_{i}\cdot e_{\kappa_{i}}(-p_{T}/T_{Di})$$

$$⟨p_T^2⟩=\frac{6T_D^2}{1-16{\kappa }^2}.$$

Full text

††journal: Journal of LaTeX Templates

1 Introduction

Non-extensive statistics, as the simplest and most natural generalization of the canonical Boltzmann–Gibbs statistics, can be applied widely. Heavy-ion collisions also offer an excellent field for such applications. The transverse momentum spectra exhibit power-law-tailed behavior, that can be described with non-extensive distribution functions better than with Boltzmann distributions, $\exp(-\beta E)\to\exp_{q}(-\beta E)$. Here the notion of $q-$exponential, $e_{q}(x)$, stands for

[TABLE]

This form was first suggested by V. Pareto in 1896 [1] for describing the distribution of wealth, and it has been recently promoted by C. Tsallis [2] in connection with non-extensive entropy. The use of a cut power-law in high energy physics was first introduced by R. Hagedorn [3], describing the data of the invariant cross section of hadrons as a function of $p_{T}$ over a wide range.

In order to gain an improved insight into this generalized non-extensive statistics, as well as to value its application to $p_{T}$ spectra on the top of pQCD motivated power-laws, ever used for high-energy jets, the meaning of its key parameter, $q$, must inevitably be studied. Considering the thermodynamical properties of hot and dense nuclear matter formed in heavy-ion collisions, on the other hand it is worth to study the fluctuations of the relevant physical observables, with a large number of particles produced. For example the multiplicity fluctuations in heavy-ion collisions have been investigated by several authors, including G. Wilk et al.[4]. Connecting to such studies, we have obtained Tsallis’ distribution for particular particle number fluctuation patterns due to a finite heat bath, for details see [5].

On the other hand, further non-extensive distribution functions were being proposed and studied to describe the data on $p_{T}$ spectra in heavy-ion collisions. K. Urmossy et al. proposed a model in which the hadrons produced in heavy-ion collisions stem from a quark-gluon plasma (QGP), referred to as ’soft’ yields, on top of those stemming from jets, called ’hard’ yields [6]. Both of these contributions are distributed according to respective cut power-laws, $\exp_{q}(-\beta E)$. Based on this view in this paper we investigate the scatter of the parameter $q$ with respect to the fitted temperatures $T$. This investigation should differentiate between hard and soft origin of power laws and should look different for heavy-ion as for $pp$ data.

Considering quantum-statistics even further constraints arise from particle-hole CPT symmetry. They lead to a special requirement on the cut power-law function, alternatively called deformed exponential [7]:

[TABLE]

For the Pareto-Hagedorn-Tsallis distribution, $\exp_{q}(x)$, it is easy to see that it does not satisfy this relation, therefore it is inadequate to apply naively extended quantum statistical distributions in the form

[TABLE]

with $x=(\omega-\mu)/T$, where $\omega$, $\mu$ and $T$ are the energy, chemical potential and temperature of the system, respectively, following the notation used in [8]. In the present paper we also shall apply the $\kappa$-exponential distribution, proposed by G. Kaniadakis for the description of relativistic plasmas [9]. We compare our corresponding analysis with the ones using Tsallis’ distribution, discussed above. The $\kappa$-exponential function has the advantage that it readily satisfies the particle-hole symmetry demand:

[TABLE]

based on the definition

[TABLE]

2 $p_{T}$ spectra with finite heat capacity

Particle transverse momentum ($p_{T}$) spectra provide a tool for measuring thermal properties of the QGP formed in ultra-relativistic heavy-ion collisions. Nowadays more and more experimental data show that the $p_{T}$ spectrum exhibits power-like rather than the earlier expected exponential behavior, and multiparticle distributions are also broader than expected [10]. The number of particles in such processes is not that large ($N\sim 10^{3}-10^{5}~{}\ll 10^{23}$, much less than Avogadro’s number). All of these warrant a suitable modification of the thermal model by accounting for possible intrinsic fluctuations in observables measured in high-energy collisions.

The non-extensive form of statistical mechanics, proposed by Tsallis [2], has found applications in many fields including high energy physics [11, 12]. In several situations the $p_{T}$ spectra can be best described by a Tsallis distribution, characterized by a non-extensive parameter $q$ and a scale parameter $T$ [13]. G. Wilk et al. have demonstrated that $q$ turns to be a function of fluctuations of temperature $T$, together with fluctuations of other variables, from an analysis of different experimental data [14]. Moreover, considering the statistical power-law tailed distributions as canonical distributions in a thermal system connected to a heat reservoir with finite heat capacity[5], we have the combined expression

[TABLE]

as next to leading order approximation in the total size of the system. Here $C$ denotes the heat capacity of the reservoir, and $1/C=dT/dE$. It is connected with the temperature and the underlying equation of state, $S(E)$, of the system via the relations $\langle S^{\prime}(E)\rangle=1/T$ and $\langle S^{\prime\prime}(E)\rangle=-1/CT^{2}$. Here the averaging is done over the ensemble of different collision events with fluctuating parameters. For temperature fluctuations with the Gaussian variance, $\Delta T/T=1/\sqrt{C}$, following from traditional consideration, one arrives at $q=1$: nothing but the classical Boltzmann–Gibbs case.

Based on this more generalized function of the Tsallis parameter $q$, we consider two different situations:

Constant Relative Variance due to $\beta=S^{\prime}(E)$-Fluctuations. In multi-particle production processes in heavy-ion collisions, it has been shown that the local temperature, $1/\beta$, fluctuates from point to point around some equilibrium value, $T=\langle 1/\beta\rangle$. However, for different collisions the relative variance due to its fluctuations stays approximately constant, namely

[TABLE]

Thus we conclude from Eq.(6) that

[TABLE]

where we approximate the finite heat capacity by the ideal gas ratio $C=E/T$. 2. 2.

Constant Average Occupancy. For elementary $pp$ collisions on the other hand, the measurements of charged particle multiplicity distributions are well described by the negative binomial distribution (NBD) for the full rapidity region.[15] Within the NBD scenario we have the probability

[TABLE]

therefore we obtain the generating function

[TABLE]

For the expectation value and variance in the number we get

[TABLE]

Assuming now that the average occupancy of phase space by the newly produced hadrons, $f$, is constant, we arrive at another relation between $T$ and $q$:

[TABLE]

Again we used relatrions valid for ideal gases, $C=E/T=\langle n\rangle$ and $\Delta T^{2}/T^{2}=\Delta n^{2}/\langle n\rangle^{2}$.

In Fig.1 the comparison between our model predictions and G Wilk’s collection is shown. From it we conclude that (i) For the fitting parameters with a Tsallis distribution in $AA$ collisions, assuming constant relative variance of temperature for different collisions works well with experimental data. (ii) For $pp$ collisions the relative fluctuations are no longer the same but the average occupancy parameter in the NBD looks constant for the full rapidity region. This makes another line of correlation between $T$ and $q$ as seen in Fig.1. There is a remarkable agreement between our simple model predictions and G. Wilk’s extensive data fits.

3 $p_{T}$ spectra within A ’Soft+Hard’ Model

Recently a so-called ’soft+hard’ model was proposed in Ref.[6]. According to this model the $p_{T}$ spectra are divided into hadron yields stemming from a supposedly 3d-isotropic QGP (soft) and from quasi 1d-jets (hard), respectively:

[TABLE]

For both yields the classical Tsallis distribution, $e_{q}(-\omega/T)$, has been applied with independently fitted $T$ and $q$ parameters [16].

We apply in the present work a radial blast wave flow based Doppler factor to the heavy-ion data, interpreting the spectral temperature as

[TABLE]

with $T_{D}$ Doppler-shifted Temperature. This is a simplification compared to the classical Tsallis distribution, that interprets the single particle energy as a relativistic expression with a given mass, $m$:

[TABLE]

with $m_{T}=\sqrt{m^{2}+p_{T}^{2}}$ transverse mass and $\gamma=1/\sqrt{1-v^{2}}$. For $p_{T}\gg m$ these expressions agree. Then we inspect the $p_{T}$ spectra of charged hadrons as a sum of hard and soft yields

[TABLE]

for $i=soft,~{}hard$ and the $C_{i}$-s being normalization constants.

For light particles like the pion in the observed $p_{T}$ range the assumption $p_{T}\gg m$ can safely be made and therefore $\omega\rightarrow\gamma(1-v)p_{T}$ leads to

[TABLE]

Investigating pion yields from various centrality classes we fit therefore a sum of soft and hard $q-$deformed distributions using the variable $x=p_{T}/T_{D}$:

[TABLE]

Moreover, for $p_{T}$ spectra stemming from the same identified hadrons but just for different centralities, the average $p_{T}^{2}$ can also be considered as a constant; it is connected with nothing but the collision energy. In this model approximation we obtain

[TABLE]

In order to test these assumptions we fitted $p_{T}$ spectra measured in $PbPb$ collisions at $\sqrt{s}=2.76$ TeV by the ALICE experiment using the functional form in Eq.(16). We used identified hadron spectra and then the fitted parameters were compared with our results from above, cf. Fig.2.

In Fig.2, we present the ’soft + hard’ model fits for the $p_{T}$ spectra of charged pions in $PbPb$ collisions at $\sqrt{s}=2.76$ TeV, as seen in[16]. The fitted parameter values are listed in Tab.1; here fluctuations and normalization factors are not explicitly shown, for simplicity . As for the fitting parameters, we used our theoretical and phenomenological model with the constant average $p_{T}^{2}$ to describe the connection between $T_{Di}$ and $q_{i}$. Moreover, it is demonstrated that parameters fit the theoretical results quite well. In this fit for the soft part, we obtain $\langle p_{T}^{2}\rangle\approx 0.75$ GeV2, while $\langle p_{T}^{2}\rangle\approx 0.40$ GeV2 for the hard part. This also supports that $p_{T}$ spectra can be studied in two parts, like in the ’soft + hard’ model [6].

In the followings we demonstrate that using Kaniadakis’ kappa-distribution, fit for the non-extensive extension of quantum statistics, works equally well as the Tsallis one in the ’soft + hard’ model, cf. Fig.3. The spectra using the $\kappa$-deformed distribution Eq.(5) are given as

[TABLE]

Then the corresponding $\langle p_{T}^{2}\rangle$ is

[TABLE]

In Fig.3 it is shown that this model also works quite well with $\kappa$-deformed distributions. Within the constant $\langle p_{T}^{2}\rangle$ the fitting parameters $\kappa_{i}$ and $T_{Di}$ are related with each other too, their values are listed in Tab.2. Worth to be mentioned that the total sum of soft and hard contributions to $\langle p_{T}^{2}\rangle$ should be and is indeed model-independent: $0.75+0.40\approx 0.88+0.24$.

4 Conclusions and outlook

In order to interpret the transverse momentum spectrum in heavy-ion collisions in terms of thermal concepts, we have considered several models based on non-extensive statistical mechanics. Both the original and a generalized Tsallis-Pareto distributions were fitted. Firstly, as QGP is a well known candidate for a thermal system, we have reminded that Tsallis’ $q$ parameter is connected to the finite heat capacity of the reservoir as well as to the intrinsic fluctuations of temperature. Then we have demonstrated that for $pp$ and $AA$ collisions the non-extensive parameter $q$ and temperature $T$ must fulfill different relations. Different formulas have been derived and both results are well fitted with other phenomenological results, like G. Wilk’s data collection shown in a talk at the Erice School on Complexity, in 2015 [14].

As a further approach we have considered a constant $\langle p_{T}^{2}\rangle$ for various centrality collisions, and used this model to investigate the collection of fit parameters $q$ and $T$. Based on experimental data taken at different centralities in $PbPb$ collisions at $\sqrt{s}=2.76$ TeV, we fitted the ’soft + hard’ model to $p_{T}$ spectra. The Tsallis’ $q$ is found to be well connected with the temperature, $T$, gathering along lines following from simple arguments.

We have also investigated this correspondence using another deformed distribution, Kaniadakis’ $\kappa$-distribution. We have demonstrated that the ’soft+hard’ model works equally well with this deformed exponential function. This is important, because quantum statistical distributions, like the Fermi and Bose distributions, can be constructed by using this particle–hole symmetric deformation of the exponential easily. In the future we plan to explore the applicability of the $\kappa$-distribution to detailed spectra stemming from ultra-relativistic heavy-ion collisions.

Acknowledgement This work has been supported in part by MOST of China under 2014DFG02050, the Hungarian National Research Fund OTKA (K104260), a bilateral governmental Chinese-Hungarian agreement NIH TET$\_$12$\_$CN-1-2012-0016 and by the NSFC of China with Project Nos. 11322546, 11435004.