arXiv subjects

Wolfgang Cassing

Publications and source records attributed to Wolfgang Cassing.

At least 19 recordsLinked to original sources

arXiv2025

Quantum correlation dynamics and in-medium 3$\leftrightarrow$3 collisions of fermions

In this study we aim for quantifying the role of in-medium 3$\leftrightarrow$3 collisions for systems of fermions which initially are out-off equilibrium. The formulation of the 3-body dynamics is based on the equations of motion method for identical fermions -- also denoted as quantum correlation dynamics -- and presented in detail. The on-shell 2-body collision integral is briefly reviewed and the on-shell 3-body collision integral is derived on the basis of the same two-body interaction in leading order. The resulting equations obey particle number as well as energy-momentum conservation. For a quantification of the relative impact of 3-body interactions we employ a model study for a homogeneous system in space in a finite box with periodic boundary conditions. We address spin-isospin symmetric nuclear matter systems with momentum distributions that are given by shifted Fermi spheres (without overlap) as encountered in the initial phase of nucleus-nucleus collisions after contact. The results for the relaxation times -- employing an effective 2-body interaction -- are compared to Boltzmann-Uehling-Uhlenbeck (BUU) transport calculations in the continuum limit for the same bombarding energies and are found to agree on the level of a few percent. We find that the additional 3-body interactions reduce the relaxation times up to a factor of 3 at 130 A$\cdot$MeV. Furthermore, it is shown in BUU transport calculations that an enhanced stopping by 3$\leftrightarrow$3 collisions shows up in the angular distribution of energetic nucleons ($>$ 60 MeV) e.g. in central $^{97}_{45}Rh$ collisions at 40 A$\cdot$MeV that lead to the formation of a compound nucleus. The angular distribution of the energetic nucleons changes from a slightly forward peaked angular distribution to a slightly sidewards peaked angular distribution which might be controlled experimentally.

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arXiv2022

Transport Model Comparison Studies of Intermediate-Energy Heavy-Ion Collisions

Transport models are the main method to obtain physics information from low to relativistic-energy heavy-ion collisions. The Transport Model Evaluation Project (TMEP) has been pursued to test the robustness of transport model predictions in reaching consistent conclusions from the same type of physical model. Calculations under controlled conditions of physical input and set-up were performed with various participating codes. These included both calculations of nuclear matter in a box with periodic boundary conditions, and more realistic calculations of heavy-ion collisions. In this intermediate review, we summarize and discuss the present status of the project. We also provide condensed descriptions of the 26 participating codes, which contributed to some part of the project. These include the major codes in use today. We review the main results of the studies completed so far. They show, that in box calculations the differences between the codes can be well understood and a convergence of the results can be reached. These studies also highlight the systematic differences between the two families of transport codes, known as BUU and QMD type codes. However, when the codes were compared in full heavy-ion collisions using different physical models, as recently for pion production, they still yielded substantially different results. This calls for further comparisons of heavy-ion collisions with controlled models and of box comparisons of important ingredients, like momentum-dependent fields, which are currently underway. We often indicate improved strategies in performing transport simulations and thus provide guidance to code developers. Results of transport simulations of heavy-ion collisions from a given code will have more significance if the code can be validated against benchmark calculations such as the ones summarized in this review.

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arXiv2019

Exploring the partonic phase at finite chemical potential within an extended off-shell transport approach

We extend the Parton-Hadron-String Dynamics (PHSD) transport approach in the partonic sector by explicitly calculating the total and differential partonic scattering cross sections as a function of temperature $T$ and baryon chemical potential $μ_B$ on the basis of the effective propagators and couplings from the Dynamical QuasiParticle Model (DQPM) that is matched to reproduce the equation of state of the partonic system above the deconfinement temperature $T_c$ from lattice QCD. The ratio of shear viscosity $η$ over entropy density $s$, i.e. $η/s$, is evaluated using the collisional widths and compared to lQCD calculations for $μ_B$ = 0 as well. We find only a very modest change of $η/s$ with the baryon chemical $μ_B$. This also holds for a variety of hadronic observables from central A+A and C+Au collisions in the energy range 5 GeV $\leq \sqrt{s_{NN}} \leq$ 200 GeV when implementing the differential cross sections into the PHSD approach. We only observe small differences in the strangeness and antibaryon sector with practically no sensitivity of rapidity and $p_T$ distributions to the $μ_B$ dependence of the partonic cross sections. Since we find only small traces of a $μ_B$-dependence in heavy-ion observables - although the effective partonic masses and widths as well as their partonic cross sections clearly depend on $μ_B$ - this implies that one needs a sizable partonic density and large space-time QGP volume to explore the dynamics in the partonic phase. These conditions are only fulfilled at high bombarding energies where $μ_B$ is, however, rather low. On the other hand, when decreasing the bombarding energy and thus increasing $μ_B$, the hadronic phase becomes dominant and accordingly, it will be difficult to extract signals from the partonic dynamics based on "bulk" observables.

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arXiv2018

Discrepancy in low transverse momentum dileptons from relativistic heavy-ion collisions

The dilepton transverse momentum spectra and invariant mass spectra for low $p_T <0.15$~GeV/c in Au+Au collisions of different centralities at $\sqrt{s_{NN}}$ = 200 GeV are studied within the parton-hadron-string dynamics (PHSD) transport approach. The PHSD describes the whole evolution of the system on a microscopic basis, incorporates hadronic and partonic degrees-of-freedom, the dynamical hadronization of partons and hadronic rescattering. For dilepton production in p+p, p+A and A+A reactions the PHSD incorporates the leading hadronic and partonic channels (also for heavy flavors) and includes in-medium effects such as a broadening of the vector meson spectral functions in hadronic matter and a modification of initial heavy-flavor correlations by interactions with the partonic and hadronic medium. The transport calculations reproduce well the momentum integrated invariant mass spectra from the STAR Collaboration for minimum bias Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV, while the description of the STAR data - when gating on low $p_T < 0.15$ GeV/c - is getting worse when going from central to peripheral collisions. An analysis of the transverse momentum spectra shows that the data for peripheral (60-80\%) collisions are well reproduced for $p_T>0.2$ GeV/c while the strong peak at low $p_T < 0.15$ GeV/c, that shows up in the experimental data for the mass bins ($0.4 < M < 0.7$ GeV and $1.2 < M < 2.6$ GeV), is fully missed by the PHSD and cannot be explained by the standard in-medium effects. This provides a new puzzle for microscopic descriptions of low $p_T$ dilepton data from the STAR Collaboration.

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arXiv2018

Covariant interacting Hadron-Resonance Gas model

The Hadron-Resonance Gas (HRG) approach - used to model hadronic matter at small baryon potentials $μ_B$ and finite temperature $T$ - is extended to finite and large chemical potentials by introducing interactions between baryons in line with relativistic mean-field theory defining an interacting HRG (IHRG). Using lattice data for $μ_B=0$ as well as information on the nuclear equation of state at $T=0$ we constrain the attractive and repulsive interactions of the IHRG such that it reproduces the lattice equation of state at $μ_B=0$ and the nuclear equation of state at $T=0$ and finite $μ_B$. The formulated covariant approach is thermodynamically consistent and allows us to provide further information on the phase boundary between hadronic and partonic phases of strongly interacting matter by assuming constant thermodynamic potentials.

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arXiv2018

Open charm and dileptons from relativistic heavy-ion collisions

Dileptons are considered as one of the cleanest signals of the quark-gluon plasma (QGP), however, the QGP radiation is masked by many 'background' sources from either hadronic decays or semileptonic decays from correlated charm pairs. In this study we investigate the relative contribution of these channels in heavy-ion collisions from $\sqrt{s_{\rm NN}}=$ 8 GeV to 5 TeV with a focus on the competition between the thermal QGP radiation and the semileptonic decays from correlated $D-$meson pairs. As a 'tool' we employ the parton-hadron-string dynamics (PHSD) transport approach to study dilepton spectra in Pb+Pb (Au+Au) collisions in a wide energy range incorporating for the first time a fully microscopic treatment of the charm dynamics and their semileptonic decays. We find that the dileptons from correlated $D-$meson decays dominate the 'thermal' radiation from the QGP in central Pb+Pb collisions at the intermediate masses (1.2 GeV $< M <$ 3 GeV) for $\sqrt{s_{\rm NN}} > $ 40 GeV, while for $\sqrt{s_{\rm NN}}=$ 8 to 20 GeV the contribution from $D,{\bar D}$ decays to the intermediate mass dilepton spectra is subleading such that one should observe a rather clear signal from the QGP radiation. We, furthermore, study the $p_T$-spectra and the $R_{AA}(p_T)$ of single electrons at different energies as well as the excitation function of the inverse slope of the $m_T$- spectra for intermediate-mass dileptons from the QGP and from charm decays. We find moderate but characteristic changes in the inverse slope parameter for $\sqrt{s_{\rm NN}} > $ 20 GeV which can be observed experimentally in high statistics data. Additionally, we provide detailed predictions for dilepton spectra from Pb+Pb collisions at $\sqrt{s_{\rm NN}} = $ 5.02 TeV.

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arXiv2018

Baryon-antibaryon dynamics in relativistic heavy-ion collisions

The dynamics of baryon-antibaryon annihilation and reproduction ($B{\bar B} \leftrightarrow 3 M$) is studied within the Parton-Hadron-String Dynamics (PHSD) transport approach for Pb+Pb and Au+Au collisions as a function of centrality from lower Super Proton Synchrotron (SPS) up to Large Hadron Collider (LHC) energies on the basis of the quark rearrangement model (QRM). At Relativistic Heavy-Ion Collider (RHIC) energies we find a small net reduction of baryon-antibaryon ($B {\bar B}$) pairs while for the LHC energy of $\sqrt{s_{NN}}$ = 2.76 GeV a small net enhancement is found relative to calculations without annihilation (and reproduction) channels. Accordingly, the sizeable difference between data and statistical calculations in Pb+Pb collisions at $\sqrt{s_{NN}}$= 2.76 TeV for proton and antiproton yields \cite{53}, where a deviation of 2.7 $σ$ was claimed by the ALICE Collaboration, should not be attributed to a net antiproton annihilation. This is in line with the observation that no substantial deviation between the data and statistical hadronization model (SHM) calculations is seen for antihyperons, since according to the PHSD analysis the antihyperons should be modified by the same amount as antiprotons. As the PHSD results for particle ratios are in line with the ALICE data (within error bars) this might point towards a deviation from statistical equilibrium in the hadronization (at least for protons/antiprotons). Furthermore, we find that the $B {\bar B} \leftrightarrow 3 M$ reactions are more effective at lower SPS energies where a net suppression for antiprotons and antihyperons up to a factor of 2 -- 2.5 can be extracted from the PHSD calculations for central Au+Au collisions.

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arXiv2017

Single electrons from heavy-flavor mesons in relativistic heavy-ion collisions

We study the single electron spectra from $D-$ and $B-$meson semileptonic decays in Au+Au collisions at $\sqrt{s_{\rm NN}}=$200, 62.4, and 19.2 GeV by employing the parton-hadron-string dynamics (PHSD) transport approach that has been shown to reasonably describe the charm dynamics at RHIC and LHC energies on a microscopic level. In this approach the initial heavy quarks are produced by using the PYTHIA which is tuned to reproduce the FONLL calculations. The produced heavy quarks interact with off-shell massive partons in QGP with scattering cross sections which are calculated in the dynamical quasi-particle model (DQPM). At energy densities close to the critical energy density the heavy quarks are hadronized into heavy mesons through either coalescence or fragmentation. After hadronization the heavy mesons interact with the light hadrons by employing the scattering cross sections from an effective Lagrangian. The final heavy mesons then produce single electrons through semileptonic decay. We find that the PHSD approach well describes the nuclear modification factor $R_{\rm AA}$ and elliptic flow $v_2$ of single electrons in d+Au and Au+Au collisions at $\sqrt{s_{\rm NN}}=$ 200 GeV and the elliptic flow in Au+Au reactions at $\sqrt{s_{\rm NN}}=$ 62.4 GeV from the PHENIX collaboration, however, the large $R_{\rm AA}$ at $\sqrt{s_{\rm NN}}=$ 62.4 GeV is not described at all. Furthermore, we make predictions for the $R_{\rm AA}$ of $D-$mesons and of single electrons at the lower energy of $\sqrt{s_{\rm NN}}=$ 19.2 GeV. Additionally, the medium modification of the azimuthal angle $ϕ$ between a heavy quark and a heavy antiquark is studied. We find that the transverse flow enhances the azimuthal angular distributions close to $ϕ=$ 0 because the heavy flavors strongly interact with nuclear medium in relativistic heavy-ion collisions and almost flow with the bulk matter.

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arXiv2017

Evidence for chiral symmetry restoration in heavy-ion collisions

We study the effect of the chiral symmetry restoration (CSR) on heavy-ion collisions observables in the energy range $\sqrt{s_{NN}}$ = 3-20 GeV within the Parton-Hadron-String Dynamics (PHSD) transport approach. The PHSD includes the deconfinement phase transition as well as essential aspects of CSR in the dense and hot hadronic medium, which are incorporated in the Schwinger mechanism for particle production. Our systematic studies show that chiral symmetry restoration plays a crucial role in the description of heavy-ion collisions at $\sqrt{s_{NN}}$ = 3-20 GeV, realizing an increase of the hadronic particle production in the strangeness sector with respect to the non-strange one. Our results provide a microscopic explanation for the "horn" structure in the excitation function of the $K^+/π^+$ ratio: the CSR in the hadronic phase produces the steep increase of this particle ratio up to $\sqrt{s_{NN}} \approx $ 7 GeV, while the drop at higher energies is associated to the appearance of a deconfined partonic medium. Furthermore, the appearance/disappearance of the "horn" structure is investigated as a function of the system size. We additionally present an analysis of strangeness production in the ($T,μ_B$)-plane (as extracted from the PHSD for central Au+Au collisions) and discuss the perspectives to identify a possible critical point in the phase diagram.

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arXiv2016

A dynamical quasiparticle approach for the Quark-Gluon-Plasma bulk and transport properties

The properties of quantum-chromo dynamics (QCD) nowadays are accessable by lattice QCD calculations at vanishing quark chemical potential $μ_q$=0 but often lack a transparent physical interpretation. In this review we report about results from an extended dynamical quasiparticle model (DQPM$^*$) in which the effective parton propagators have a complex selfenergy that depends on the temperature $T$ of the medium as well as on the chemical potential $μ_q$ and the parton three-momentum ${\boldsymbol p}$ with respect to the medium at rest. It is demonstrated that this approach allows for a good description of QCD thermodynamics with respect to the entropy density, pressure etc. above the critical temperature $T_c \approx$ 158 MeV. Furthermore, the quark susceptibility $χ_q$ and the quark number density $n_q$ are found to be reproduced simultaneously at zero and finite quark chemical potential. The shear and bulk viscosities $η, ζ$, and the electric conductivity $σ_e$ from the DQPM$^*$ also turn out in close agreement with lattice results for $μ_q$ =0. The DQPM$^*$, furthermore, allows to evaluate the momentum $p$, $T$ and $μ_q$ dependencies of the partonic degrees of freedom also for larger $μ_q$ which are mandatory for transport studies of heavy-ion collisions in the regime 5 GeV $< \sqrt{s_{NN}} <$ 10 GeV. We finally calculate the charm quark diffusion coefficient $D_s$ -- evaluated from the differential cross sections of partons in the medium for light and heavy quarks by employing the propagators and couplings from the DQPM -- and compare to the available lattice data. It is argued that the complete set of observables allows for a transparent interpretation of the properties of hot QCD.

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arXiv2016

What are the early degrees of freedom in ultra-relativistic nucleus-nucleus collisions?

The Parton-Hadron-String-Dynamics (PHSD) transport model is used to study the impact on the choice of initial degrees of freedom on the final hadronic and electromagnetic observables in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. We find that a non-perturbative system of massive gluons (scenario I) and a system dominated by quarks and antiquarks (scenario II) lead to different hadronic observables when imposing the same initial energy-momentum tensor $T_{μν}(x)$ just after the passage of the impinging nuclei. In case of the gluonic initial condition the formation of $s,{\bar s}$ pairs in the QGP proceeds rather slow such that the anti-strange quarks and accordingly the $K^+$ mesons do not achieve chemical equilibrium even in central Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. Accordingly, the $K^+$ rapidity distribution is suppressed in the gluonic scenario and in conflict with the data from the BRAHMS Collaboration. The proton and antiproton rapidity distributions also disfavor the scenario I. Furthermore, a clear suppression of direct photon and dilepton production is found for the pure gluonic initial conditions which is not so clearly seen in the present photon and dilepton spectra from Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV due to a large contribution from other channels. It is argued that dilepton spectra in the invariant mass range 1.2 GeV $< M <$ 3 GeV will provide a definitive answer once the background from correlated $D$-meson decays is subtracted experimentally.

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arXiv2016

Hadron production within PHSD

We study the production of hadrons in nucleus-nucleus collisions within the Parton-Hadron-String Dynamics (PHSD) transport approach that is extended to incorporate essentials aspects of chiral symmetry restoration (CSR) in the hadronic sector (via the Schwinger mechanism) on top of the deconfinement phase transition as implemented in PHSD. The essential impact of CSR is found in the Schwinger mechanism (for string decay) which fixes the ratio of strange to light quark production in the hadronic medium. Our studies provide a microscopic explanation for the maximum in the $K^+/π^+$ ratio at about 30 A GeV which only shows up if in addition to CSR a deconfinement transition to partonic degrees-of-freedom is incorporated in the reaction dynamics.

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arXiv2016

Charm production in Pb+Pb collisions at the Large Hadron Collider energy

We study charm production in Pb+Pb collisions at $\sqrt{s_{\rm NN}}=$2.76 TeV in the Parton-Hadron-String-Dynamics transport approach and the charm dynamics in the partonic and hadronic medium. The charm quarks are produced through initial binary nucleon-nucleon collisions by using the PYTHIA event generator taking into account the (anti-)shadowing incorporated in the EPS09 package. The produced charm quarks interact with off-shell massive partons in the quark-gluon plasma and are hadronized into $D$ mesons through coalescence or fragmentation close to the critical energy density, and then interact with hadrons in the final hadronic stage with scattering cross sections calculated in an effective Lagrangian approach with heavy-quark spin symmetry. The PHSD results show a reasonable $R_{\rm AA}$ and elliptic flow of $D$ mesons in comparison to the experimental data for Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV from the ALICE Collaboration. We also study the effect of temperature-dependent off-shell charm quarks in relativistic heavy-ion collisions. We find that the scattering cross sections are only moderately affected by off-shell charm degrees of freedom. However, the position of the peak of $R_{\rm AA}$ for $D$ mesons depends on the strength of the scalar partonic forces which also have an impact on the $D$ meson elliptic flow. The comparison with experimental data on the $R_{\rm AA}$ suggests that the repulsive force is weaker for off-shell charm quarks as compared to that for light quarks. Furthermore, the effects from radiative charm energy loss appear to be low compared to the collisional energy loss up to transverse momenta of $\sim$ 15 GeV/c.

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arXiv2015

The quark susceptibility in a generalized dynamical quasiparticle model

The quark susceptibility $χ_q$ at zero and finite quark chemical potential provides a critical benchmark to determine the quark-gluon-plasma (QGP) degrees of freedom in relation to the results from lattice QCD (lQCD) in addition to the equation of state and transport coefficients. Here we extend the familiar dynamical-quasiparticle model (DQPM) to partonic propagators that explicitly depend on the three-momentum with respect to the partonic medium at rest in order to match perturbative QCD (pQCD) at high momenta. Within the extended dynamical-quasi-particle model (DQPM$^*$) we reproduce simultaneously the lQCD results for the quark number density and susceptibility and the QGP pressure at zero and finite (but small) chemical potential $μ_q$. The shear viscosity $η$ and the electric conductivity $σ_e$ from the extended quasiparticle model (DQPM$^*$) also turn out in close agreement with lattice results for $μ_q$ =0. The DQPM$^*$, furthermore, allows to evaluate the momentum $p$, temperature $T$ and chemical potential $μ_q$ dependencies of the partonic degrees of freedom also for larger $μ_q$ which are mandatory for transport studies of heavy-ion collisions in the regime 5 GeV $< \sqrt{s_{NN}} <$ 10 GeV.

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arXiv2015

(Anti-)strangeness production in heavy-ion collisions

The production and dynamics of strange and antistrange hadrons in heavy-ion reactions from $\sqrt{s_{NN}} \approx$ 3 GeV to 200 GeV is analyzed within the Parton-Hadron-String-Dynamics (PHSD) transport model. The PHSD results for strange baryon and antibaryon production are roughly consistent with the experimental data starting from upper SPS energies. Nevertheless, hadronic final state flavor-exchange reactions are important for the actual abundances, in particular at large rapidities where hadronic dynamics, parton fragmentation and string decay dominate. A striking disagreement between the PHSD results and the available data persists, however, for bombarding energies below $\sqrt{s_{NN}} \approx$ 8 GeV where the strangeness production is significantly underestimated as in earlier HSD studies. This finding implies that the strangeness enhancement seen experimentally at FAIR/NICA energies cannot be attributed to a deconfinement phase transition or crossover but probably involves the approximate restoration of chiral symmetry in the hadronic phase.

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arXiv2015

Observables in ultrarelativistic heavy-ion collisions from two different transport approaches for the same initial conditions

For nucleus-nucleus collisions at Relativistic-Heavy-Ion Collider (RHIC) energies we calculate observables in two conceptually transport theories, i.e. the n-body molecular dynamical model RSP and in the Parton-Hadron-String-Dynamics (PHSD), starting out from the same distribution of the initial energy density at the quark gluon plasma (QGP) formation time. The RSP is based on the Nambu Jona-Lasinio (NJL) Lagrangian whereas in PHSD the partons are described by the dynamical quasi-particle model (DQPM). Despite of the very different description of the parton properties, their interactions and of the hadronization in both approaches the final transverse momentum distributions of pions and kaons are rather similar. Thus it will be difficult to conclude from the final experimenal data on the details of the initial plasma properties.

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arXiv2015

Tomography of the Quark-Gluon-Plasma by Charm Quarks

We study charm production in ultra-relativistic heavy-ion collisions by using the Parton-Hadron-String Dynamics (PHSD) transport approach. The initial charm quarks are produced by the Pythia event generator tuned to fit the transverse momentum spectrum and rapidity distribution of charm quarks from Fixed-Order Next-to-Leading Logarithm (FONLL) calculations. The produced charm quarks scatter in the quark-gluon plasma (QGP) with the off-shell partons whose masses and widths are given by the Dynamical Quasi-Particle Model (DQPM) which reproduces the lattice QCD equation-of-state in thermal equilibrium. The relevant cross section are calculated in a consistent way by employing the effective propagators and couplings from the DQPM. Close to the critical energy density of the phase transition, the charm quarks are hadronized into $D$ mesons through coalescence and/or fragmentation depending on transverse momentum. The hadronized $D$ mesons then interact with the various hadrons in the hadronic phase with cross sections calculated in an effective lagrangian approach with heavy-quark spin symmetry. Finally, the nuclear modification factor $\rm R_{AA}$ and the elliptic flow $v_2$ of $D^0$ mesons from PHSD are compared with the experimental data from the STAR Collaboration for Au+Au collisions at $\sqrt{s_{\rm NN}}$ =200 GeV. We find that in the PHSD the energy loss of $D$ mesons at high $p_T$ can be dominantly attributed to partonic scattering while the actual shape of $\rm R_{AA}$ versus $p_T$ reflects the heavy quark hadronization scenario, i.e. coalescence versus fragmentation. Also the hadronic rescattering is important for the $\rm R_{AA}$ at low $p_T$ and enhances the $D$-meson elliptic flow $v_2$.

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arXiv2015

Strangeness production in heavy-ion collisions

A study of the "horn" in the particle ratio $K^+/π^+$ for central heavy-ion collisions as a function of the collision energy $\sqrt{s}$ is presented. We analyse two different interpretations: the onset of deconfinement and the transition from a baryon- to a meson-dominated hadron gas. We use a realistic equation of state (EOS), which includes both hadron and quark degrees-of-freedom. The Taub-adiabate procedure is followed to determine the system at the early stage. Our results do not support an explanation of the horn as due to the onset of deconfinement. Using only hadronic EOS we reproduced the energy dependence of the $K^+/π^+$ and $Λ/π^-$ ratios employing an experimental parametrisation of the freeze-out curve. We observe a transition between a baryon- and a meson-dominated regime; however, the reproduction of the $K^+/π^+$ and $Λ/π^-$ ratios as a function of $\sqrt{s}$ is not completely satisfying. We finally propose a new idea for the interpretation of the data, the roll-over scheme, in which the scalar meson field $σ$ has not reached the thermal equilibrium at freeze-out. The rool-over scheme for the equilibration of the $σ$-field is based on the inflation mechanism. The non-equilibrium evolution of the scalar field influences the particle production, e.g. $K^+/π^+$, however, the fixing of the free parameters in this model is still an open issue.

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