Conserved charge fluctuations at vanishing and non-vanishing chemical potential

TL;DR

This paper computes high-order cumulants of conserved charge fluctuations in lattice QCD to understand the transition from hadronic matter to quark-gluon plasma, highlighting deviations from the hadron resonance gas model at various chemical potentials.

Contribution

It provides detailed lattice QCD calculations of cumulants up to sixth order, extending understanding of conserved charge fluctuations at finite chemical potential.

Findings

Significant differences between QCD and HRG model at T > 155 MeV

Deviations increase with non-zero baryon chemical potential

Constraints on HRG applicability range at higher chemical potentials

Abstract

Up to 6th order cumulants of fluctuations of net baryon-number, net electric charge and net strangeness as well as correlations among these conserved charge fluctuations are now being calculated in lattice QCD. These cumulants provide a wealth of information on the properties of strong-interaction matter in the transition region from the low temperature hadronic phase to the quark-gluon plasma phase. They can be used to quantify deviations from hadron resonance gas (HRG) model calculations which frequently are used to determine thermal conditions realized in heavy ion collision experiments. Already some second order cumulants like the correlations between net baryon-number and net strangeness or net electric charge differ significantly at temperatures above 155 MeV in QCD and HRG model calculations. We show that these differences increase at non-zero baryon chemical potential constraining the applicability range of HRG model calculations to even smaller values of the temperature.

Figures

Equations

$${\chi }_n^X={\frac{1}{V{T}^3}\frac{{\partial }^n\ln Z(V,T,\vec{\mu })}{\partial X^n}|}_{\vec{\mu }=0},{\chi }_{nm}^{XY}={\frac{1}{V{T}^3}\frac{{\partial }^{(n+m)}\ln Z(V,T,\vec{\mu })}{\partial X^n\partial Y^m}|}_{\vec{\mu }=0},X,Y=B,Q,S,$$

$${\chi }_2^B(T,{\mu }_B)={\chi }_2^B+\frac{1}{2}{\chi }_4^B{\left(\frac{{\mu }_B}{T}\right)}^2+\mathcal{O}({\mu }_B^4),{\chi }_{11}^{BX}(T,{\mu }_B)={\chi }_{11}^{BX}+\frac{1}{2}{\chi }_{31}^{BX}{\left(\frac{{\mu }_B}{T}\right)}^2+\mathcal{O}({\mu }_B^4),X=Q,S.$$

$$\frac{{\chi }_4^B(T,{\mu }_B)}{{\chi }_2^B(T,{\mu }_B)}=\frac{{\chi }_4^B}{{\chi }_2^B}\left(1+\frac{1}{2}\left(\frac{{\chi }_6^B}{{\chi }_4^B}-\frac{{\chi }_4^B}{{\chi }_2^B}\right){\left(\frac{{\mu }_B}{T}\right)}^2+\mathcal{O}({\mu }_B^4)\right)\mathrm{for}{\mu }_Q={\mu }_S=0.$$

Full text

1 Introduction

The central goal of the beam energy scan (BES) program at RHIC is to seek evidence for the existence of a critical point in the phase diagram of strong-interaction matter. The hope is to detect this postulated second order phase transition point through the analysis of higher order cumulants of net charge fluctuations. Maxima of cumulants of net charge fluctuations, e.g. the $2^{nd}$ and $4^{th}$ order cumulants, trace the chiral crossover transition line at small values of the baryon chemical potential and diverge at a critical point.

In heavy ion experiments the observed net charge fluctuations are expected to reflect thermal conditions at the time of chemical-freeze out of various hadron species. If this freeze-out happens close to the pseudo-critical line for the chiral transition of QCD, where thermal fluctuations are large, the measured fluctuations have a chance to be indicative for the divergent fluctuations that will appear at a critical point in the QCD phase diagram.

A crucial anchor point for this scenario is to establish the relation between freeze-out and the QCD chiral transition at small or even vanishing net baryon chemical potential. In this case reliable theoretical calculations, based on Taylor series expansions in lattice QCD, exist and can be confronted with experimental findings at the LHC as well as the highest beam energies at RHIC. In experiments at the LHC one can analyze moments of charge fluctuations at almost vanishing baryon chemical potential ($\mu_{B}$) which allows a direct comparison with lattice QCD calculations performed at $\mu_{B}=0$.

2 Thermal conditions at vanishing net baryon number

The basic bulk thermodynamic observables, pressure ($P$), energy ($\epsilon$) and entropy ($s$) density of strong-interaction matter at vanishing baryon chemical potential, have been calculated in lattice QCD (for a recent review see: [1]). These calculations have recently been extended to non-vanishing baryon number densities using analytic continuation of calculations performed at imaginary values of $\mu_{B}$ [2] and Taylor expansions in $\mu_{B}$ [3]. In Fig. 1 (left) we show results for lines of constant $P$, $\epsilon$ and $s$, in the $T$-$\mu_{B}$ plane (phase diagram) obtained from a Taylor series up to ${\cal O}(\mu_{B}^{4})$ [3]. Lines are drawn for three values of these observables in the crossover region for the QCD chiral transition, which is well characterized by the current uncertainty on the chiral transition temperature, $T_{c}=154(9)$ MeV, at $\mu_{B}=0$ [4]. As can be seen in Fig. 1 (right) in this temperature interval the energy density changes by about a factor three, $\epsilon_{c}=(0.34\pm 0.16)$ GeV/fm3 [5].

Also shown in Fig. 1 (left) are experimental results for freeze-out parameters determined by the ALICE Collaboration at the LHC [6] and the STAR Collaboration from the BES at RHIC [7] by comparing measured particle yields with predictions from a statistical hadronization model, which utilizes the thermodynamics of a hadron resonance gas. Obviously, there is a significant difference in the determination of the freeze-out temperature at $\mu_{B}\simeq 0$. While the ALICE result for the freeze-out temperature ($T_{f}$) agrees well with the central value of the pseudo-critical temperature ($T_{c}$), the STAR results favor a larger value, $T_{f}\sim 165$ MeV, which is close to the hadronization temperature obtained by Becattini et al. [8].

A 10 MeV accuracy for the determination of the freeze-out temperature, which anyhow is not considered to be a temperature uniquely defined for all particle species, but rather a statistical average, may be considered to be appropriate for many purposes. However, in the search for evidence for a critical point such a difference has substantial consequences for expected properties of net charge fluctuations as the size of the critical region, in which charge fluctuations may become large, may well be only of that order [9]. A 10 MeV difference between $T_{c}$ and $T_{f}$ thus may decide whether or not freeze-out happens in the critical region.

Cumulants of net charge fluctuations and correlations among fluctuations of different conserved charges, i.e. baryon number ($B$), electric charge ($Q$) and strangeness ($S$) can be obtained as derivatives of the logarithm of the QCD partition function [3],

[TABLE]

with $\vec{\mu}\equiv(\mu_{B},\ \mu_{Q},\ \mu_{S})$ denoting the three chemical potentials connected with the conserved charges. As can be seen in Fig. 2 ratios of cumulants of net charge fluctuations change rapidly in the crossover region. The ratio of $4^{th}$ and $2^{nd}$ order net baryon-number fluctuations, $\chi_{4}^{B}/\chi_{2}^{B}$, changes from almost unity at $T=145$ MeV to about 0.5 at $T=165$ MeV (see Fig. 2 (left)). In the temperature range identified by ALICE as the freeze-out region, $T_{f}=156(2)$ MeV, the ratio is $\chi_{4}^{B}/\chi_{2}^{B}\simeq 0.75$. The situation is similar for ratios of conserved charge correlations. For the ratio of $4^{th}$ and $2^{nd}$ order cumulants characterizing correlations between net strangeness and net baryon-number fluctuations one finds $\chi_{31}^{BS}/\chi_{11}^{BS}\simeq 0.74$ at $T=155$ MeV, while this ratio drops to $\sim 0.36$ at $T=165$ MeV. This is in contrast to HRG model calculations with point-like non-interacting hadrons, where these ratios are unity irrespective of the particle content in the hadron spectrum.

Also on the level of second order cumulants, which currently get measured by the ALICE Collaboration [10], differences between HRG model calculations and QCD results are already significant. In Fig. 2 (right) we show the ratio, $\chi_{11}^{BQ}/\chi_{2}^{B}$, which in resonance gas models for non-interacting, point-like hadrons has the interpretation of the relative contribution of charged baryons to the total baryon contribution of the pressure. As can be seen at $T=165$ MeV the ratio $\chi_{11}^{BQ}/\chi_{2}^{B}$ is about 50% smaller than predicted by HRG model calculations based on the experimentally known hadron spectrum (PDG-HRG). Model calculations that include additional strange baryon resonances predicted in QCD motivated quark model calculations (QM-HRG) provide a better approximation to QCD calculations but still over-predict the ratio $\chi_{11}^{BQ}/\chi_{2}^{B}$ for T\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}150 MeV. A similar behavior has been found for the correlations between net baryon-number and net strangeness fluctuations, $\chi_{11}^{BS}$, normalized to either $\chi_{2}^{B}$ or $\chi_{2}^{S}$. This may be taken as evidence for the importance of additional strange baryon resonance contributing significantly to the thermodynamics close to the chiral transition.

There is, of course, also the possibility that additional interactions among hadrons need to be taken into account in order to improve agreement between QCD and hadron gas calculations at low temperature. However, as is also evident from Fig. 2, it seems that neither taking into account repulsive interactions through the finite volume of baryons [11] nor an additional attractive contribution through a van der Waals interaction (vdW-HRG) can improve the validity range of HRG model based calculations [12].

3 Cumulants of conserved charge fluctuations at non-zero net

baryon number

Ratios of $4^{th}$ and $2^{nd}$ order cumulants such as $\chi_{4}^{B}/\chi_{2}^{B}$ (Fig. 2 (left)) or $\chi_{31}^{BX}/\chi_{11}^{BX}$, $X=Q,\ S$, also give the leading ${\cal O}(\mu_{B}^{2})$ corrections to the $2^{nd}$ order cumulants themselves. For $\mu_{Q}=\mu_{S}=0$ one has

[TABLE]

As these ratios are smaller than the corresponding HRG values, this also means that differences between lattice QCD results for these cumulants and HRG model calculations increase with $\mu_{B}$. Some results for $\chi_{11}^{BS}$ and $\chi_{11}^{BQ}$ at three values of the temperature are shown in Fig. 3. Obviously deviations from HRG model calculations become large for \mu_{B}/T\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}1.5, which in the BES at RHIC corresponds to \sqrt{s_{NN}}\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}15 GeV.

The situation is similar for ratios of cumulants of net baryon-number fluctuations. For instance,

[TABLE]

is unity in HRG model calculations with non-interacting, point-like hadrons for all values of $\mu_{B}$. In QCD, however, the ${\cal O}(\mu_{B}^{2})$ expansion coefficient in Eq. 3 is negative for T\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}150 MeV [13]. This indicates that the ratio $\chi_{4}^{B}(T,\mu_{B})/\chi_{2}^{B}(T,\mu_{B})$ becomes smaller with increasing $\mu_{B}$. Even if the ratio is close to HRG model results at $\mu_{B}=0$, it thus will further deviate from this with increasing $\mu_{B}$. This is consistent with the trend found by STAR for the corresponding kurtosis ratio of net proton-number fluctuations, $\kappa_{P}\sigma^{2}_{P}=\chi_{4}^{P}/\chi_{2}^{P}$ [14].

4 Conclusions

Many ratios of $4^{th}$ and $2^{nd}$ order cumulants of conserved charge fluctuations calculated in QCD agree with HRG model calculations within 25% or better at temperatures below $T=155$ MeV. For observables including strangeness fluctuations this often requires to take into account contributions from additional strange hadrons not listed in the Particle Data Tables. At $T=165$ MeV HRG model calculations often deviate from QCD results by more than 50% even for observables that only involve quadratic charge fluctuations.

Acknowledgements: This work was supported through Contracts No. DE-SC001270 with the U.S. Department of Energy and No. 05P15PBCAA with the German Bundesministerium für Bildung und Forschung.