^{1,2}

^{2,3}

^{1}

^{1,2,4}

^{3}.

Thermodynamic functions, the (higher-order) fluctuations and correlations of conserved charges at

The empirically known spectrum of hadrons suggests a rapid, possibly exponential, increase of the density of states at large masses

Hagedorn states are possibly created in multiparticle reactions, e.g., during heavy-ion collisions

In the simplest Hagedorn model all hadrons are treated as point particles. Due to the exponentially increasing hadron mass spectrum, the Hagedorn temperature

The temperature dependence of thermodynamic functions at zero chemical potential within the gas of extended quark-gluon bags with a crossover transition was considered in Ref.

The model assumes a multicomponent system of color neutral objects. These objects, henceforth referred to as particles, have finite sizes—the eigenvolumes. The particles can carry three Abelian charges—baryon number, electric charge, and strangeness. These three charges are characterized by the corresponding chemical potentials

First, we consider a system with a finite number of different components

In the thermodynamic limit,

In the isobaric ensemble, the input particle spectrum enters through Eq.

Finally, let us rewrite Eq.

We follow the picture presented in Refs.

the established, low mass hadrons and resonances listed in Particle Data Group (PDG)

an exponential Hagedorn spectrum of the heavy quark-gluon bags.

Therefore,
^{1}

We neglect here the effects of finite resonance widths. These can have an important effect in precision thermal model applications, such as thermal fits

Each of the PDG hadrons is assumed to have a finite eigenvolume parameter

The mass-volume spectrum of the quark-gluon bags,

Equation

The values of parameters

The accuracy of Eq.

The pressure function [Eq.

Let us compute the function

The quark-gluon bags in Eq.

The expression for

The integration over

Let us note that

Applying Laplace's method to Eq.

Recalling the definition of the “upper” incomplete gamma function,

The function ^{2}

Note that

Lattice QCD simulations at physical quark masses reveal that the transition from hadronic to partonic degrees of freedom at

The farthest-right singularity of the isobaric partition function

The first term in Eq.

One should note that the application of Laplace's method used to perform the mass integration in the derivation of Eq.

It should be pointed out that the mass-volume density

The particle number density is the average number of particles per unit volume. It is the sum (integral) of individual densities corresponding to different particle species:

The individual densities can be computed by introducing the fictitious fugacities

The total hadron density reads

For the particle spectrum

Another interesting quantity is the filling fraction (f.f.)—the ratio between the average total volume occupied by hadrons over the system volume. The definition of this quantity reads

The average eigenvolume of a particle in the thermal system can be computed as follows:

The average mass of a particle in the thermal system is given by

The explicit calculation, employing Laplace's method to integrate over

Calculations here are performed for the following set of parameters:

All model parameters are fixed; the only exception is the

The value of the bag constant

The constant

The parameter

The temperature dependence of the scaled pressure ^{3}

Similar lattice results were also obtained by the HotQCD collaboration

Temperature dependence of (a) scaled pressure

The auxiliary quantities introduced in Sec.

The temperature dependence of (a) the filling fraction (f.f.), (b) the average particle mass

The particle chemistry at different temperatures can be clarified by studying the temperature dependence of the mean hadron mass

The temperature dependence of the mean hadron volume

The nontrivial behavior of

Fluctuations and correlations of conserved charges are other observables, accessible with lattice QCD, suggested long ago to be sensitive to the parton-hadron transition

The matrix of the second order conserved charge susceptibilities has been studied in lattice QCD simulations at the physical point in Refs.

The temperature dependencies of the second order diagonal susceptibilities,

While the simple bag model picture above appears to describe many qualitative features seen in lattice data, the quantitative description of the main thermodynamical functions, such as pressure, energy density, interaction measure, and the speed of sound, is obviously not very good. This description cannot be notably improved solely by a variation of the parameters in Eq.

Equation

Here we adopt a similar strategy and consider constant, finite values of quark and gluon masses:

Equation

Equation

For

The temperature dependence of the scaled pressure

The decreased value of the bag constant necessitates an increase of the value of the parameter

As before, we set

The temperature dependence of the scaled pressure

The temperature dependence of (a) the scaled pressure

The model describes the lattice data on a semiquantitative level. Sizeable deviations are seen close to

We turn now to the behavior of the susceptibilities of conserved charges. The temperature dependence of the matrix of the second order conserved charge susceptibilities in the present model is depicted in Fig.

The temperature dependence of the second order conserved charge susceptibilities: (a)

The net charge susceptibility

We also consider the baryon-strangeness correlator ratio,

The temperature dependence of the baryon-strangeness correlator ratio

Higher-order susceptibilities are expected to be particularly sensitive to crossing the crossover transition

The temperature dependence of the conserved charges susceptibilities: (a)

The so-called kurtosis of the net baryon number fluctuations—the

The temperature dependence of the net strangeness kurtosis—the

The behavior of the sixth and eighth order net baryon susceptibilities

The model can also be applied to study observables at imaginary chemical potentials. This is achieved through the analytic continuation. These observables can then be compared with lattice QCD data at imaginary chemical potentials. Such a comparison with an independent set of lattice observables provides an important cross-check of the model validity.

Here we consider the behavior of the model at the imaginary baryochemical potential ^{4}

The Roberge-Weiss transition is expected at

The QCD pressure at imaginary baryochemical potential can be written in terms of the Fourier series

The net baryon density at imaginary

The leading four Fourier coefficients

Comparison with the lattice data is presented in Fig.

Temperature dependence of the leading four Fourier coefficients of the net baryon density at imaginary

The present model agrees qualitatively with the available lattice data. It does appear to underestimate

As presented, the description of a hadronic gas together with fluctuating Hagedorn bag-like states within the pressure ensemble including their (repulsive) eigenvolume interactions shows many agreements with the current state of the art lattice QCD equation of state. On the other hand, the chiral transition, taking place with increasing temperature in the crossover region, has not been discussed, as the present model is not suited for a straightforward calculation of the chiral order parameter—the chiral condensate

The temperature dependence of the chiral transition in the presented picture can be characterized by the available volume fraction,

Temperature dependence of the available volume fraction,

In the further discussion of Sec.

If a true second order chiral phase transition occurs at vanishing baryon chemical potential at the critical temperature, the parameters

We have studied the behavior of thermodynamic functions, various conserved charges susceptibilities at zero chemical potentials, and the Fourier coefficients at imaginary

The Hagedorn quark-gluon bag-like model, introduced in Refs.

In the present work we have considered only the case where the crossover transition is realized, at all

The

We thank J. Steinheimer for a suggestion to compare the available volume fraction with the chiral condensate from lattice QCD. The work of M.I.G. was supported by the Alexander von Humboldt Foundation and by the Program of Fundamental Research of the Department of Physics and Astronomy of National Academy of Sciences of Ukraine. This work was also supported by the Bundesministerium für Bildung und Forschung (BMBF), the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse, and by the Collaborative Research Center CRC-TR 211 “Strong-interaction matter under extreme conditions” funded by DFG. H.S. acknowledges the support through the Judah M. Eisenberg Laureatus Chair at Goethe University, and the Walter Greiner Gesellschaft, Frankfurt.

In this Appendix we evaluate the mass distribution of PDG hadrons and of quark-gluon bags for the “massive quarks” parameter set used in the present work [Eq.

To calculate the mass spectrum of PDG hadrons we smear their masses with relativistic Breit-Wigner distributions. The Breit-Wigner widths are taken to be constant and correspond to the resonance widths listed in Particle Data Tables. We also assume a width of 10 MeV for all stable hadrons; this is done for presentation purposes. The mass spectrum of quark-gluon bags is calculated numerically from Eq.

Figure

The mass spectrum of the PDG hadrons (blue line) and of the QGP bags, using the Hagedorn mass spectrum form [Eq.

The picture for the massless quarks parameter sets [Eq.

In the present work the pressure has been determined as the solution of the transcendental equation

the nonrelativistic approximation [Eq.

the Laplace's method to perform the integration over the mass [Eqs.

Both approximations are expected to be rather accurate for the heavy quark-gluon bags, as discussed in the main text. Nevertheless, it can be useful to quantify the error introduced by these approximations. In order to do that, we consider the exact transcendental equation for the model pressure without approximations:

Figure

At large temperatures the system is dominated by heavy bags. As already elaborated on, the heavier the bags are, the more accurate are the approximations.

The temperature dependence of the ratio

At low temperatures the system is dominated by the PDG hadrons, which are evaluated without approximations. Therefore, possible inaccuracies in evaluating the contributions from the quark-gluon bags at these temperatures are irrelevant as these contributions are negligible anyhow.

If only one of the two approximations discussed is preserved, i.e., if only the nonrelativistic approximation is used but not Laplace's method, or vice versa, then the relative error in the calculated pressure is within 1%.

We have performed similar checks for the massless quarks parameter sets [Eq.