^{3}.

The behavior of strange matter in the frame of the SU(3) Polyakov-loop extended Nambu–Jona-Lasinio (PNJL) model including the

The study of matter formed during the collision of heavy ions at high energies is currently of significant interest in high-energy physics. Much interest still focuses on the search of the critical endpoint and phase transition in hot and dense matter. The search for quark-gluon plasma (QGP) where hadrons dissolve into interacting gluons and quarks is difficult due to the short lifetime of the QCD phase. It is needed to find sensible probes for the transition to the QGP phase (i.e., deconfinement transition and the chiral symmetry restoration). One of the suggested signals was the strangeness enhancement which was explained through the interactions between partons in QGP.

Intriguing results were obtained from PbPb and AuAu collisions: a structure in the ratio of the positive charged kaon to the positive charged pion, named the “horn” (see Fig.

The strangeness suppression in

An exact theoretical reproduction of the horn in the

The

The statistical model of early stages (SMES) considers a slow increase and the following jump in the ratio of strange-to-nonstrange particle production as a result of the deconfinement transition. According to the SMES, at low collision energies confined matter is produced, and the increase in the ratio is due the low

Success in the description of experimental data was achieved when the partial restoration of chiral symmetry

The qualitative reproduction of the peak in the energy dependence of the kaon to pion ratio was obtained in the statistical model, which includes hadron resonances and the

The SU(3) Nambu–Jona-Lasinio (NJL) model with the Polyakov loop (PNJL model) seems to be most promising as an instrument for describing the chiral phase transition, the deconfinement properties, and the existence of quarks and hadron states

We address this paper to the problem of the kaon to pion ratio in the context of the SU(3) PNJL model. In Sec.

We consider the Polyakov-loop extended SU(3) Nambu–Jona-Lasinio model with scalar-pseudoscalar interaction and the t'Hooft interaction, which breaks

The effective potential

To obtain the value of the Polyakov loop field

The meson masses in NJL-like models are defined by the Bethe–Salpeter equation at

The mass spectrum for zero chemical potential is shown in Fig.

(top panel) The mass spectra at

The normalized quark condensate of light and strange quarks and the Polyakov loop field

The calculations in SU(3) NJL-like models are complicated by the need to introduce the strange quark chemical potential. As a rule, when only thermodynamics is considered, the chemical potential of the strange quark is supposed equal to zero,

matter with equal chemical potentials

matter with

For both cases the phase diagrams are similar and are shown in Fig.

(top panel) The phase diagram in the

In Fig.

At nonzero chemical potential and low

The spectra of meson masses as a function of the normalized baryon density at

To discuss the horn problem, we have to consider the ratio of the number of kaons to the number of pions. Within the PNJL model the number densities of mesons

If all experimental data are taken from various experiments, it is shown in the statistical model that for each experiment the temperature and the baryon chemical potential of freeze-out

It is evident that the experimental data at higher energy correspond to high temperature and low density or chemical potential, and the data at low energy correspond to high density or chemical potential and low temperature.

(top panel)

The calculated (top panel) and rescaled experimental (bottom panel) ratios

It is clearly seen from the figure that, in the region of high temperature and low density (high values of

At low values of

The quark-matter properties in the frame of the PNJL model are formed by the choice of different assumptions for the environment: Case I and Case II. Generally speaking, both invented assumptions cannot reproduce the properties of the medium in a real collision of heavy ions. Nevertheless, the choice of these two cases can illustrate that the peak position is connected with the position of the critical endpoint. In Fig.

The

The contour plot of the

The strangeness enhancement in heavy-ion collisions has been suggested as the QGP signal long ago

The dependence of the strangeness to nonstrangeness ratio was discussed in Refs.

The main idea of this work was to show that the cause of the appearance of the horn at energies

In the PNJL model, the masses of positive and negative mesons are split at high densities. This splitting can explain the difference in the behavior of the

We are thankful to A. Khvorostukhin and E. Kolomeitsev for useful advice. We also thank W. Cassing and D. Blaschke for discussions. The work of A.F. was supported by the Russian Science Foundation under Grant No. 17-12-01427.