Properties of the QGP created in heavy-ion collisions
P. Moreau, O. Soloveva, I. Grishmanovskii, V. Voronyuk, L. Oliva, T. Song, V. Kireyeu, G. Coci, E. Bratkovskaya
RReceived 14 January 2020; Revised xxx; Accepted xxxDOI: xxx/xxxx
ARTICLE TYPE
Properties of the QGP created in heavy-ion collisions
P. Moreau | O. Soloveva | I. Grishmanovskii | V. Voronyuk | L. Oliva | T. Song | V.Kireyeu | G. Coci | E. Bratkovskaya* Department of Physics, Duke University,Durham, NC 27708, USA Institute for Theoretical Physics, Universityof Frankfurt, Frankfurt, Germany Joint Institute for Nuclear Research,Joliot-Curie 6, 141980 Dubna, MoscowRegion, Russia GSI, Helmholtzzentrum fรผrSchwerionenforschung GmbH, Darmstadt,Germany
Correspondence * GSI, Darmstadt, Germany Email:[email protected]
We review the properties of the strongly interacting quark-gluon plasma (QGP)at ๏ฌnite temperature ๐ and baryon chemical potential ๐ ๐ต as created in heavy-ioncollisions at ultrarelativistic energies. The description of the strongly interacting(non-perturbative) QGP in equilibrium is based on the e๏ฌective propagators andcouplings from the Dynamical QuasiParticle Model (DQPM) that is matched toreproduce the equation-of-state of the partonic system above the decon๏ฌnement tem-perature ๐ ๐ถ from lattice QCD. Based on a microscopic transport description ofheavy-ion collisions we discuss which observables are sensitive to the QGP creationand its properties. KEYWORDS: heavy-ions, quark-gluon plasma, transport models quark-gluon plasma
An understanding of the structure of our universe is an intrigu-ing topic of research in our Millennium, which combinesthe e๏ฌorts of physicists working in di๏ฌerent ๏ฌelds of astro-physics, cosmology and heavy-ion physics (Strassmeier et al.,2019). The common theoretical e๏ฌorts and modern achieve-ments in experimental physics allowed to make a substantialprogress in understanding the properties of the nuclear mat-ter and extended our knowledge of the QCD phase diagramwhich contains the information about the properties of our uni-verse from the early beginning โ directly after the Big Bangโ when the matter was in a quark-gluon plasma (QGP) phaseat very high temperature ๐ and practically zero baryon chem-ical potential ๐ ๐ต , to the later stages of the universe, wherein the expansion phase stars and galaxies have been formed.In the later phase, the matter is at low temperature, however,at very large baryon densities or baryon chemical potential ๐ ๐ต . The range of the phase diagram at large ๐ ๐ต and low tem-perature ๐ can also be explored in the astrophysical context(Klahn et al., 2006), such as in the dynamics of supernovaeor in the dynamics of neutron-star mergers and gravitationalwaves. Collisions of heavy-ions at ultra-relativistic energies โ e.g. at the Relativistic Heavy Ion Collider (RHIC) or the LargeHadron Collider (LHC) โ provide the possibility to reproduceon Earth the conditions closer to the Big Bang, when the matterwas in a QGP phase of unbound quarks and gluons at very hightemperature and almost vanishing ๐ ๐ต . Indeed, in experiments itis possible to achieve a QGP only in extremely small volumes,moreover, the fast expansion leads to a fast hadronization ofthe QGP such that only ๏ฌnal hadrons and leptons are measured.Thus, it is quite challenging to investigate the properties of theQGP experimentally.The region of phase diagram at ๏ฌnite ๐ and ๐ ๐ต is of spe-cial interest nowadays. According to lattice calculations ofquantum chromodynamics (lQCD) (Aoki et al., 2009; Aoki,Endrodi, Fodor, Katz, & Szabo, 2006; Bazavov et al., 2012;Bernard et al., 2005; J. Guenther et al., 2017), the phase tran-sition from hadronic to partonic degrees of freedom at smallbaryon chemical potential ๐ ๐ต โค MeV is a crossover.According to phenomenological models (e.g. recent PNJL cal-culations (Fuseau, Steinert, & Aichelin, 2020)), the crossoveris expected to turn into a ๏ฌrst order transition at some criticalpoint ( ๐ ๐๐ , ๐ ๐๐ ) in the phase diagram with increasing baryonchemical potential ๐ ๐ต . However, the theoretical predictionsconcerning the location of the critical point are rather uncer-tain. From the experimental side the beam energy scan (BES) a r X i v : . [ nu c l - t h ] J a n P. Moreau at al. program at RHIC aims to ๏ฌnd the critical point and the phaseboundary by scanning the collision energy (Kumar, 2011;Mohanty, 2011). Moreover, new facilities such as FAIR (Facil-ity for Antiproton and Ion Research) and NICA (Nuclotron-based Ion Collider fAcility) are presently under construction;they will explore the intermediate energy regime of ratherlarge baryon densities and moderate temperatures where onemight also study the competition between chiral symmetryrestoration and decon๏ฌnement as advocated in Refs. (Cass-ing, Palmese, Moreau, & Bratkovskaya, 2016; Palmese et al.,2016).From the theoretical side it is quite challenging to investigatethe "intermediate" region of the QCD phase diagram of non-vanishing quark (or baryon) densities from ๏ฌrst principles dueto the so called "sign problem" which prevents the lQCD calcu-lations to be extended to large ๐ ๐ต (cf. (J. N. Guenther, 2020)).Our present knowledge on QCD in Minkowski space for non-vanishing ๐ ๐ต are based mainly on e๏ฌective approaches. Usinge๏ฌective models, one can study the properties of QCD inequilibrium, i.e., thermodynamic quantities, the Equation-of-State (EoS) of the QGP as well as transport coe๏ฌcients. Thedynamical quasiparticle model (DQPM) has been introducedfor that aim (Berrehrah, Bratkovskaya, Steinert, & Cassing,2016; Cassing, 2007a, 2007b; Linnyk, Bratkovskaya, & Cass-ing, 2016; Peshier & Cassing, 2005). It is based on partonicpropagators with sizeable imaginary parts of the self-energiesincorporated. Whereas the real part of the self-energies canbe attributed to a dynamically generated mass (squared), theimaginary parts contain the information about the interactionrates of the degrees-of-freedom. Moreover, the imaginary partsof the propagators de๏ฌne the spectral functions of the โparti-clesโ which might show narrow (or broad) quasiparticle peaks.An important advantage of a propagator based approach is thatone can formulate a consistent thermodynamics (Vanderhey-den & Baym, 1998) and a causal theory for non-equilibriumstates on the basis of Kadano๏ฌโBaym equations (Kadano๏ฌ &Baym, 1962).Since the QGP is formed for a short time in a ๏ฌnite vol-ume in heavy-ion collisions (HICs), it is very important tounderstand the time evolution of the expanding system. Thus,it is mandatory to have a proper non-equilibrium descrip-tion of the entire dynamics through di๏ฌerent phases - startingwith impinging nuclei in their groundstates, going throughthe QGP phase (showing some approach to equilibration)and end up with the ๏ฌnal asymptotic hadronic states. To thisaim, the PartonโHadronโString Dynamics (PHSD) transportapproach (Bratkovskaya, Cassing, Konchakovski, & Linnyk,2011; Cassing, 2009; Cassing & Bratkovskaya, 2008, 2009;Linnyk et al., 2016) has been formulated more than a decadeago (on the basis of the Hadron-String-Dynamics (HSD)approach (Cassing & Bratkovskaya, 1999)), and it was found to well describe observables from p+A and A+A collisions fromSIS (Schwerionensynchrotron) to LHC energies for the bulkdynamics, electromagnetic probes such as photons and dilep-tons as well as open cham hadrons (Bratkovskaya, Kostyuk,Cassing, & Stoecker, 2004; Linnyk et al., 2016; Linnyk, Kon-chakovski, Cassing, & Bratkovskaya, 2013; T. Song, Cassing,Moreau, & Bratkovskaya, 2018).In order to explore the partonic systems at ๏ฌnite ๐ ๐ต , thePHSD approach has been extended to incorporate partonicquasiparticles and their di๏ฌerential cross sections that dependnot only on temperature ๐ as obtained in Ref. (Ozvenchuk,Linnyk, Gorenstein, Bratkovskaya, & Cassing, 2013) andemployed in the previous PHSD studies, but also on chemi-cal potential ๐ ๐ต and center-of-mass energy of colliding partons โ ๐ and their angular distributions explicitly (Moreau, 2019;Moreau et al., 2019). Within this extended approach we havestudied the โbulkโ observables such as rapidity distributionsand transverse momentum spectra in heavy-ion collisions fromAGS (Alternating Gradient Synchrotron) to RHIC energies forsymmetric and asymmetric (light + heavy nuclear) systems.However, we have found only a small in๏ฌuence of ๐ ๐ต depen-dences of parton properties (masses and widths) and partonicinteraction cross sections in the bulk observables (Moreau etal., 2019).Recently we studied the collective ๏ฌow ( ๐ฃ , ๐ฃ ) coe๏ฌcientsfor di๏ฌerent identi๏ฌed hadrons and their sensitivity to the ๐ ๐ต dependences of partonic cross sections (Soloveva, Moreau,Oliva, Song, et al., 2020; Soloveva, Moreau, Oliva, Voronyuk,et al., 2020). We have found that the ๏ฌow coe๏ฌcients show asmall, but visible sensitivity to the ๐ ๐ต dependence of partonicinteractions.In this work we study the properties of the QGP created inheavy-ion collisions and show the time evolution of the ( ๐ , ๐ ๐ต ) distribution, as probed in HICs, and relate it to observablessuch as multiplicities and collective ๏ฌow ( ๐ฃ , ๐ฃ ) coe๏ฌcients. In this Section we recall the basic ideas of the PHSDtransport approach and the dynamical quasiparticle model(DQPM). The PartonโHadronโString Dynamics (PHSD)transport approach (Bratkovskaya et al., 2011; Cassing, 2009;Cassing & Bratkovskaya, 2008, 2009; Linnyk et al., 2016) isa microscopic o๏ฌ-shell transport approach for the descriptionof strongly interacting hadronic and partonic matter in andout-of equilibrium. It is based on the solution of Kadano๏ฌโBaym equations in ๏ฌrst-order gradient expansion in phasespace (Cassing, 2009) which allows to describe in a causal . Moreau at al. way the time evolution of nonperturbative interacting systems.Consequently they are applicable for the dynamics of the QGPwhich approximately behaves as a strongly interacting liquid at๏ฌnite ๐ due to the growing of the QCD coupling constant in thevicinity of the critical temperature ๐ ๐ถ , where pQCD methodsare not applicable.The partonic phase in the PHSD is modelled based onthe dynamical quasiparticle model (DQPM) (Cassing, 2007a,2007b; Peshier & Cassing, 2005). The DQPM has been intro-duced in Refs. (Cassing, 2007a, 2007b; Peshier & Cassing,2005) for the e๏ฌective description of the QGP in terms ofstrongly interacting quasiparticles - quarks and gluons, wherethe properties and interactions are adjusted to reproduce lQCDresults for the QGP in equilibrium at ๏ฌnite temperature ๐ andbaryon chemical potential ๐ ๐ต (Aoki et al., 2009, 2006; Baza-vov et al., 2012; Bernard et al., 2005; J. Guenther et al., 2017).In the DQPM, the quasiparticles are characterized by single-particle Greenโs functions (in propagator representation) withcomplex self-energies. The real part of the self-energies isrelated to the dynamically generated parton masses (squared),whereas the imaginary part provides information about thelifetime and/or reaction rates of the degrees-of-freedom. Thus,in the DQPM the properties of the partons (quarks and gluons)are characterized by broad spectral functions ๐ ๐ ( ๐ = ๐, ฬ๐, ๐ ),i.e. the partons are o๏ฌ-shell. This di๏ฌerentiates the PHSD fromconventional cascade or transport models dealing with on-shell particles, i.e., ๐ฟ -functions in the invariant mass squared.The quasiparticle spectral functions are assumed to have aLorentzian form (Linnyk et al., 2016), which are speci๏ฌed bythe parton masses and width parameters: ๐ ๐ ( ๐, ๐ฉ ) = ๐พ ๐ ๐ธ ๐ ( ๐ โ ๐ธ ๐ ) + ๐พ ๐ โ 1( ๐ + ๐ธ ๐ ) + ๐พ ๐ ) (1)separately for quarks/antiquarks and gluons ( ๐ = ๐, ฬ๐, ๐ ). Withthe convention ๐ธ ( ๐ฉ ) = ๐ฉ + ๐ ๐ โ ๐พ ๐ , the parameters ๐ ๐ and ๐พ ๐ are directly related to the real and imaginary parts of theretarded self-energy, e.g., ฮ ๐ = ๐ ๐ โ 2 ๐๐พ ๐ ๐ . The functionalforms, i.e. the ( ๐ , ๐ ๐ต )-dependences of the dynamical partonmasses and widths are chosen in the spirit of hard-thermal loop(HTL) calculations by introducing three parameters. They aredetermined by comparison to the calculated entropy density ๐ ,pressure ๐ and energy density ๐ from the DQPM to those fromlQCD at ๐ ๐ต = 0 from Ref. (Borsanyi et al., 2012, 2014).The DQPM also allows to de๏ฌne a scalar mean-๏ฌeld ๐ ๐ ( ๐ ๐ ) for quarks and antiquarks which can be expressed by thederivative of the potential energy density with respect to thescalar density ๐ ๐ ( ๐ , ๐ ๐ต ) , ๐ ๐ ( ๐ ๐ ) = ๐๐ ๐ ( ๐ ๐ ) ๐๐ ๐ , (2) which is evaluated numerically within the DQPM. Here, thepotential energy density is evaluated by: ๐ ๐ ( ๐ , ๐ ๐ต ) = ๐ ๐ โ ( ๐ , ๐ ๐ต ) + ๐ ๐ โ ( ๐ , ๐ ๐ต ) + ๐ ฬ๐ โ ( ๐ , ๐ ๐ต ) , (3)where the di๏ฌerent contributions ๐ ๐ โ correspond to the space-like part of the energy-momentum tensor component ๐ ๐ ofparton ๐ = ๐, ๐, ฬ๐ (cf. Section 3 in Ref. (Cassing, 2007a)). Thescalar mean-๏ฌeld ๐ ๐ ( ๐ ๐ ) for quarks and antiquarks is repulsiveas a function of the parton scalar density ๐ ๐ and shows thatthe scalar mean-๏ฌeld potential is in the order of a few GeVfor ๐ ๐ > fm โ3 . The mean-๏ฌeld potential (2) determines theforce on a partonic quasiparticle ๐ , i.e., โผ ๐ ๐ โ ๐ธ ๐ โ ๐ ๐ ( ๐ฅ ) = ๐ ๐ โ ๐ธ ๐ ๐๐ ๐ โ ๐๐ ๐ โ ๐ ๐ ( ๐ฅ ) , where the scalar density ๐ ๐ ( ๐ฅ ) isdetermined numerically on a space-time grid in PHSD.The quasiparticles - quarks and gluons - which are mov-ing in the self-generated scalar mean-๏ฌeld potential (2) caninteract via elastic and inelastic scattering. A two-body inter-action strength can be extracted from the DQPM as well fromthe quasiparticle widths in line with Ref. (Peshier & Cassing,2005). In the latest version of PHSD (v. 5.0) the follow-ing elastic and inelastic interactions are included ๐๐ โ ๐๐ , ฬ๐ ฬ๐ โ ฬ๐ ฬ๐ , ๐๐ โ ๐๐ , ๐๐ โ ๐ , ๐ ฬ๐ โ ๐ , ๐๐ โ ๐๐ , ๐ ฬ๐ โ ๐ ฬ๐ . The backward reactions are de๏ฌned using โdetailed-balanceโ with cross sections calculated from the leading orderFeynman diagrams employing the e๏ฌective propagators andcouplings ๐ ( ๐ โ ๐ ๐ , ๐ ๐ต ) from the DQPM (Moreau et al., 2019).In Ref. (Moreau et al., 2019), the di๏ฌerential and total o๏ฌ-shellcross sections have been evaluated as a function of the invari-ant energy of the colliding o๏ฌ-shell partons โ ๐ for each ๐ , ๐ ๐ต .We recall that in the early PHSD studies (using version 4.0 andbelow) the cross sections depend only on ๐ (cf. the detailedevaluation in Ref. (Ozvenchuk et al., 2013)).In the PHSD5.0 the quasiparticle properties (masses andwidths) as well as their interactions, de๏ฌned by the di๏ฌer-ential cross sections and mean-๏ฌeld potential, depend on theโLagrange parametersโ ๐ and ๐ ๐ต in each computational cell inspace-time. The evaluation of ๐ and ๐ ๐ต , as calculated in thePHSD from the local energy density ๐ and baryon density ๐ ๐ต ,is realized by employing the lattice equation of state and and bya diagonalization of the energy-momentum tensor from PHSDas described in Ref. (Moreau et al., 2019).The hadronization, i.e. the transition from partonic tohadronic degrees-of-freedom (and vice versa) is described bycovariant transition rates for the fusion of quarkโantiquarkpairs or three quarks (antiquarks), respectively. The full micro-scopic description allows to obey ๏ฌavor currentโconservation,color neutrality as well as energyโmomentum conserva-tion (Cassing & Bratkovskaya, 2009). Since the dynamicalquarks and antiquarks become very massive close to thephase transition, the formed resonant โprehadronicโ color-dipole states ( ๐ ฬ๐ or ๐๐๐ ) are of high invariant mass, too, and P. Moreau at al. sequentially decay to the groundstate meson and baryon octets,thus increasing the total entropy during hadronization.On the hadronic side, PHSD includes explicitly the baryonoctet and decouplet, the โ - and โ -meson nonets as wellas selected higher resonances as in the HadronโStringโDynamics (HSD) approach (Cassing & Bratkovskaya, 1999).Note that PHSD and HSD (without explicit partonic degrees-of-freedom) merge at low energy density, in particular belowthe local critical energy density ๐ ๐ โ as extractedfrom the lQCD results in Ref. (Borsanyi et al., 2012, 2014). The transport properties of the QGP close to equilibrium arecharacterized by various transport coe๏ฌcients. The transportcoe๏ฌcients play an important role for the hydrodynamic mod-els since they de๏ฌne the properties of the propagating ๏ฌuid.The shear viscosity ๐ and bulk viscosity ๐ describe the ๏ฌuidโsdissipative corrections at leading order. In the hydrodynamicequations, the viscosities appear as dimensionless ratios, ๐ โ ๐ and ๐ โ ๐ , where ๐ is the ๏ฌuid entropy density. Such speci๏ฌcviscosities are more meaningful than the unscaled ๐ and ๐ values because they describe the magnitude of stresses insidethe medium relative to its natural scale. Both coe๏ฌcients ๐ and ๐ are generally expected to depend on the temperature ๐ and baryon chemical potential ๐ ๐ต . While for the hydrody-namic models the transport coe๏ฌcient are an โinputโ and haveto be adopted from some models, a fully miscoscopic descrip-tion of the QGP dynamics by transport approaches allows tode๏ฌne them from the interactions of the underlying degrees offreedom.One way to evaluate the viscosity coe๏ฌcients of partonicmatter is the Kubo formalism (Aarts & Martinez Resco, 2002;Fernandez-Fraile & Gomez Nicola, 2006; Kubo, 1957; Lang,Kaiser, & Weise, 2015), which was used to calculate the vis-cosities for a previous version of the DQPM within the PHSDtransport approach in a box with periodic boundary conditionsin Ref. (Ozvenchuk et al., 2013) as well as in the more recentstudy with the DQPM model in Refs. (Moreau et al., 2019;Soloveva, Moreau, & Bratkovskaya, 2020). Another way tocalculate transport coe๏ฌcients is to use the relaxationโtimeapproximation (RTA) as incorporated in Refs. (Albright &Kapusta, 2016; Chakraborty & Kapusta, 2011; Gavin, 1985;Hosoya & Kajantie, 1985). This strategy has been used in ourrecent studies where we have investigated the transport prop-erties of the QGP in the ( ๐ , ๐ ๐ต ) plane based on the DQPM(Moreau et al., 2019; Soloveva, Moreau, & Bratkovskaya,2020; Soloveva et al., 2019). We ๏ฌnd that the ratios of transport coe๏ฌcients ๐ โ ๐ and ๐ โ ๐ evaluated within DQPM are in a good agreement withthe available lQCD results for pure SU(3) gauge theory(Astrakhantsev, Braguta, & Kotov, 2017, 2018). The ๐ โ ๐ ratioshows a minimum at ๐ ๐ถ and slightly rises with ๐ , while ๐ โ ๐ grows at ๐ ๐ถ in line with the lQCD data. This shows that theQGP in the PHSD behaves as a strongly interacting nonper-turbative system rather then a dilute gas of weakly interactingpartons. In this Section we discuss the properties of the QGP createdin heavy-ion collisions in terms of achievable energy densitiesand temperatures as well as baryon chemical potentials. Westart with an illustration in Fig. 1 of a time evolution of cen-tral Au+Au collisions (upper row, section view) at a collisionalenergy of โ ๐ ๐๐ = 19 . GeV within the PHSD (Moreau,2019). The snapshots are taken at times ๐ก = 0 . , , , and8 fm/c. The baryons, antibaryons, mesons, quarks and gluonsare shown as colored dots.We show the local temperature ๐ (middle row) and baryonchemical potential ๐ ๐ต (lower row), as extracted from the PHSDin the region with ๐ฆ โ 0 . The black lines (middle row) indicatethe critical temperature ๐ ๐ = 0 . GeV.As follows from the upper part of Fig. 1 , the QGP is cre-ated in the early phase of collisions and when the systemexpands, hadronization occurs. One can see that during theoverlap phase the ๐ and ๐ ๐ต are very large (we give a warn-ing that an extraction of the thermodynamic quantities for thestrongly non-equilibrium initial stage is not a consistent pro-cedure) and they decrease with time. However even at 8 fm/cthere are "hot spots" of QGP at front surfaces of high rapidity.We investigate now the time evolution of the ๐ and ๐ ๐ต dis-tribution in for cells having a temperature ๐ > ๐ ๐ ( ๐ ๐ต ) atmidrapidity ( | ๐ฆ cell | <
1) for 5% central Pb+Pb collisions at 158A GeV. Fig. 2 shows this distribution for times ๐ก < < ๐ก < ๐ก > ๐ก < < ๐ก < โผ ๐ก > ๐ ๐ ) essentially stays around ๐ ๐ต โ 0 . GeV.We mention that the values of ๐ ๐ต probed around the transi-tion temperature ๐ ๐ in the PHSD are in accordance with theexpectation from statistical models, which for central Pb+Pbcollisions at 158 A GeV quote a value of ๐ ๐ต = 0 . GeV(Cleymans, Oeschler, Redlich, & Wheaton, 2006). . Moreau at al. FIGURE 1
Illustration of the time evolution of central Au+Au collisions (upper row, section view) at a collisional energy of โ ๐ ๐๐ = 19 . GeV within the PHSD (Moreau, 2019). The local temperature ๐ (middle row), baryon chemical potential ๐ ๐ต (lower row), as extracted from the PHSD for ๐ฆ โ 0 . The black lines (middle row) indicate the critical temperature ๐ ๐ = 0 . GeV.By varying the collisional energy of the initial nuclei, onecan increase or decrease the volume of the QGP produced inthe heavy-ion collisions. In Fig. 3 we show the QGP energyfraction versus the total energy for Au+Au at di๏ฌerent colli-sional energies โ ๐ ๐๐ accounting only the midrapidity region | ๐ฆ | < . . One can see that for high energies the QGP frac-tion is large compared to lower collisional energies where thevolume of QGP is small. While at high energy heavy-ion colli-sions the QGP phase appears suddenly after the initial primaryNN collision, at low energies it appearance is smoother sincethe passing time of the nuclei is longer. Correspondingly, atlow energies the QGP lifetime is large, however, the QGP vol-ume is very small; thus it in๏ฌuence on the dynamics is muchreduced compared to high energy collisions where practically90% of matter at midrapidity is in the QGP phase (at least fora short time).In Fig. 4 we show the average baryon chemical potential(from PHSD simulations) taken around the chemical freeze-out temperature (given by the statistical analysis from Refs.(Adamczyk et al., 2017; Andronic et al., 2010; Cleymans et al., 2006)). We look only at cells in the midrapidity regionto compare with the results from the statistical models. Oneshould stress that we compare two di๏ฌerent quantities here:one is ๐ ๐ต obtained from a statistical analysis of the ๏ฌnal parti-cle spectra, and the other is the average ๐ ๐ต probed in a PHSDsimulation around the chemical freeze-out temperature (thelatter itself being obtained by a statistical analysis). As onecan see in Fig. 4 , these two quantities are in a fairly goodagreement especially at high energies where ๐ ๐ต is rather small.At low collisional energies the extracted value of ๐ ๐ต fromthe transport calculation (from cells at midrapidity) becomesslightly larger; this might partly be related to the fact that our ๐ , ๐ ๐ต extraction procedure, optimized for the QGP is not suf-๏ฌciently accurate for such large values of ๐ ๐ต โ ๐ as probed inthis regime. The work on improving the ๐ , ๐ ๐ต extraction forhadronic dominated matter is in progress. P. Moreau at al. 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FIGURE 2
Distributions in ๐ and ๐ ๐ต as extracted from theDQPM equation of state in a PHSD simulation of a centralPb+Pb collision at 158 A GeV for cells with a temperature ๐ > ๐ ๐ ( ๐ ๐ต ) at midrapidity ( | ๐ฆ cell | < ๐ โ ๐ ๐ต divided by the total number of cells in the corre-sponding time window (see legend). The solid black line is theDQPM phase boundary for orientation; the gray dashed linesindicate ratios of ๐ ๐ต โ ๐ ranging from 1 to 5 while the verticalline corresponds to ๐ ๐ต = 0. In our recent study (Moreau et al., 2019) we have investi-gated the sensitivity of โbulkโ observables, such as rapidity andtransverse momentum distributions of di๏ฌerent hadrons pro-duced in symmetric and asymmetric heavy-ion collisions fromAGS to top RHIC energies, on the details of the QGP inter-actions and the properties of the partonic degrees-of-freedom.For that, we have considered the following three cases: (1)โPHSD4.0โ: the quasiparticle properties (masses and widthsof quarks and gluons) and partonic interaction cross sections
FIGURE 3
The QGP energy fraction from PHSD as a func-tion of time ๐ก in central (impact parameter b = 2 fm) Au+Aucollisions for di๏ฌerent collisional energies โ ๐ ๐๐ , taking intoaccount only the midrapidity region | ๐ฆ | < . (Moreau, 2019). (cid:4) (cid:7) (cid:4) (cid:2) (cid:7) (cid:2) (cid:4) (cid:2) (cid:2)(cid:3) (cid:2) (cid:3) (cid:2) (cid:2)(cid:2) (cid:1) (cid:2)(cid:2) (cid:1) (cid:3)(cid:2) (cid:1) (cid:4)(cid:2) (cid:1) (cid:5)(cid:2) (cid:1) (cid:6)(cid:2) (cid:1) (cid:7)(cid:2) (cid:1) (cid:8)(cid:2) (cid:1) (cid:9)(cid:2) (cid:1) (cid:10) (cid:1) (cid:1) (cid:1) (cid:11) (cid:10) (cid:12) (cid:9) (cid:1) (cid:24) (cid:23) (cid:1) (cid:2) (cid:3)(cid:3) (cid:24) (cid:1) (cid:7) (cid:1) (cid:5) (cid:4) (cid:6)(cid:5) (cid:3) (cid:6) (cid:2) (cid:1) (cid:14) (cid:15) (cid:18) (cid:22) (cid:20) (cid:13) (cid:17) (cid:1) (cid:8) (cid:3) (cid:8) (cid:1) (cid:14) (cid:19) (cid:17) (cid:17) (cid:21) (cid:16) (cid:19) (cid:18) (cid:21) (cid:1) (cid:8) (cid:9) (cid:3) (cid:7)(cid:1) (cid:8) (cid:6) (cid:8)(cid:1) (cid:3) (cid:5) (cid:8)(cid:1) (cid:3) (cid:16) (cid:12) (cid:18) (cid:17) (cid:16) (cid:14) (cid:11) (cid:1) (cid:13) (cid:19) (cid:1) (cid:10) (cid:15) (cid:2)(cid:1) (cid:4) (cid:13) (cid:11) (cid:10) (cid:19) (cid:19) (cid:14) (cid:16) (cid:14) (cid:1) (cid:13) (cid:19) (cid:1) (cid:10) (cid:15) (cid:2) (cid:1) (cid:1) (cid:1) (cid:4) (cid:2) (cid:6) (cid:3) (cid:5) (cid:7) (cid:2) (cid:1)(cid:3)(cid:4) (cid:4) (cid:1) (cid:4) (cid:2) (cid:6) (cid:3) (cid:5) FIGURE 4
Average baryon chemical potential ๐ ๐ต extractedfrom PHSD simulations (Moreau, 2019) around the chemi-cal freeze-out temperature ๐ ๐โ in comparison with the val-ues obtained from statistical models (Adamczyk et al., 2017;Andronic et al., 2010; Cleymans et al., 2006).depend only on ๐ as calculated in Ref. (Ozvenchuk et al.,2013). (2) โPHSD5.0 - ๐ ๐ต โ: the masses and widths of quarksand gluons depend on ๐ and ๐ ๐ต explicitly; the di๏ฌerential andtotal partonic cross sections are obtained by calculations of theleading order Feynman diagrams from the DQPM and explic-itly depend on invariant energy โ ๐ , temperature ๐ and baryonchemical potential ๐ ๐ต (Moreau et al., 2019). (3) โPHSD5.0 - ๐ ๐ต = 0 โ: the same at (2), but for ๐ ๐ต = 0 .The comparison of the โbulkโ observables for A+A colli-sions within the three cases of PHSD in Ref. (Moreau et al.,2019) has illuminated that they show a very low sensitivity to . Moreau at al. the ๐ ๐ต dependences of parton properties (masses and widths)and their interaction cross sections such that the results fromPHSD5.0 with and without ๐ ๐ต were very close to each other.Only in the case of kaons, antiprotons ฬ๐ and antihyperons ฬ ฮ + ฬ ฮฃ , a small di๏ฌerence between PHSD4.0 and PHSD5.0could be seen at top SPS and top RHIC energies. This can beunderstood as follows: as has been illustrated in Section 4, athigh energies such as top RHIC where the QGP volume is verylarge in central collisions, the ๐ ๐ต is very small, while, whendecreasing the collision energy (and consequently increase ๐ ๐ต ) the fraction of the QGP is decreasing such that the ๏ฌnalobservables are dominated by the hadronic phase. Thus, theprobability for the hadrons created at the QGP hadronizationto rescatter, decay, or be absorbed in hadronic matter increasesstrongly; as a result the sensitivity to the properties of the QGPis washed out to a large extend.In Fig. 5 we present the excitation function of hadronmultiplicities at midrapidity ๐๐ โ ๐๐ฆ | ๐ฆ =0 as a function of thecollisional energy โ ๐ ๐๐ . The symbols correspond to theexperimental data, while the lines correspond to the PHSD5.0calculations including ( ๐ , ๐ ๐ต ) for mesons - ๐ ยฑ , ๐พ ยฑ and (anti-)baryons - ๐, ฬ๐, ฮ + ฮฃ , ฬ ฮ + ฬ ฮฃ , ฮ โ . One can see that themesons and (anti-)baryons multiplicities grow rapidly withincreasing energy. The production of particles at high energiesis practically identical with respect to particles-antiparticlesand mainly originates from the hadronization process of theQuark-Gluon Plasma. With decreasing energy one can seea separation between baryons and antibaryons which comesfrom baryon stopping, from possible ๐ต โ ฬ๐ต annihilations,and from the fact that antibaryons (especially multi-strangebaryons) are preferentially produced by the hadronization pro-cess from the QGP and, thus, more sensitive to the ( ๐ , ๐ ๐ต ) properties of the QGP partons. As follows from hydrodynamical calculations (Bernhard,Moreland, Bass, Liu, & Heinz, 2016; Marty, Bratkovskaya,Cassing, Aichelin, & Berrehrah, 2013; Romatschke &Romatschke, 2007; H. Song & Heinz, 2008) and the Bayesiananalysis (Bernhard et al., 2016), the results for the ๏ฌow har-monics ๐ฃ ๐ are sensitive to the transport coe๏ฌcients Recentlywe investigated the sensitivity of the collective ๏ฌow ( ๐ฃ , ๐ฃ )coe๏ฌcients to the ๐ ๐ต dependences of partonic cross sectionsof DQPM which provide the ๐ โ ๐ and ๐ โ ๐ consistent withlQCD results as discussed in Section 2. For that we calcu-lated ๐ฃ , ๐ฃ for di๏ฌerent identi๏ฌed hadrons - mesons and(anti-) baryons (Soloveva, Moreau, Oliva, Song, et al., 2020;Soloveva, Moreau, Oliva, Voronyuk, et al., 2020).We start with presenting the results of our study (Soloveva,Moreau, Oliva, Song, et al., 2020; Soloveva, Moreau, Oliva, -3 -2 -1 p p ๏ + ๏ ๏ + ๏ ๏ - ๏ฐ + ๏ฐ - K + K - BES RHICSPS
PHSD d N / d y | y = s [GeV] A+A 0-5% & |y| < 0.5
AGS
FIGURE 5
Excitation function ( ๐๐ โ ๐๐ฆ ) | ๐ฆ =0 of particles pro-duced in central heavy-ion collisions at midrapidity from thePHSD (Moreau, 2019) in comparison with experimental datafrom E895 (Klay et al., 2002, 2003), E866-E917 (Ahle et al.,2000a, 2000b; Back et al., 2001), E896 (Albergo et al., 2002)and E895 (Pinkenburg et al., 2002) collaborations at AGS ener-gies, from the NA49 (Afanasiev et al., 2002; Alt et al., 2005,2006, 2008a, 2008b) and the NA57 (Antinori et al., 2004) col-laborations for SPS energies, and from the STAR collaborationfor the Beam-Energy-Scan (Adamczyk et al., 2017; Ashraf,2016) and top RHIC regimes (Abelev et al., 2009; Adcox etal., 2002; Aggarwal et al., 2011).Voronyuk, et al., 2020) for the traces of ๐ ๐ต dependences ofthe QGP interaction cross sections in the directed ๏ฌow. Fig. 6depicts the directed ๏ฌow ๐ฃ of identi๏ฌed hadrons ( ๐พ ยฑ , ๐, ฬ๐, ฮ+ฮฃ , ฬ ฮ+ ฬ ฮฃ ) versus rapidity for Au+Au collisions at โ ๐ ๐๐ = 27GeV. One can see a good agreement between PHSD results andexperimental data from the STAR collaboration (Adamczyk etal., 2018). However, the di๏ฌerent versions of PHSD for the ๐ฃ coe๏ฌcients show a quite similar behavior; only antihyperonsindicate a slightly di๏ฌerent ๏ฌow. This supports again the ๏ฌnd-ing that strangeness, and in particular anti-strange hyperons,are the most sensitive probes for the QGP properties.We continue with presenting the results for the elliptic๏ฌow of charged hadrons from heavy-ion collisions within thePHSD5.0. In Fig. 7 we display the elliptic ๏ฌow ๐ฃ of identi๏ฌedhadrons ( ๐พ ยฑ , ๐, ฬ๐, ฮ + ฮฃ , ฬ ฮ + ฬ ฮฃ ) as a function of ๐ ๐ at โ ๐ ๐๐ = 27 GeV for PHSD4.0 (green lines), PHSD5.0 with partoniccross sections and parton masses calculated for ๐ ๐ต = 0 (bluedashed lines) and with cross sections and parton masses eval-uated at the actual chemical potential ๐ ๐ต in each individualspace-time cell (red lines) in comparison to the experimen-tal data of the STAR Collaboration (Adamczyk et al., 2016).Similar to the directed ๏ฌow shown in Fig. 6 , the elliptic ๏ฌow P. Moreau at al. (cid:1)(cid:3) (cid:2)(cid:3) (cid:6)(cid:1)(cid:3) (cid:2)(cid:3) (cid:5)(cid:1)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:3)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:5)(cid:3) (cid:2)(cid:3) (cid:6) (cid:1)(cid:4) (cid:2)(cid:7) (cid:1)(cid:4) (cid:2)(cid:3) (cid:1)(cid:3) (cid:2)(cid:7) (cid:3) (cid:2)(cid:3) (cid:3) (cid:2)(cid:7) (cid:4) (cid:2)(cid:3) (cid:4) (cid:2)(cid:7)(cid:1)(cid:3) (cid:2)(cid:3) (cid:6)(cid:1)(cid:3) (cid:2)(cid:3) (cid:5)(cid:1)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:3)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:5)(cid:3) (cid:2)(cid:3) (cid:6) (cid:1)(cid:4) (cid:2)(cid:7) (cid:1)(cid:4) (cid:2)(cid:3) (cid:1)(cid:3) (cid:2)(cid:7) (cid:3) (cid:2)(cid:3) (cid:3) (cid:2)(cid:7) (cid:4) (cid:2)(cid:3) (cid:4) (cid:2)(cid:7) (cid:1)(cid:3) (cid:2)(cid:3) (cid:5)(cid:1)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:3)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:5) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:6) (cid:3)(cid:1) (cid:5) (cid:2) (cid:4) (cid:1) (cid:6) (cid:3)(cid:1) (cid:5) (cid:2) (cid:4) (cid:1)(cid:10) (cid:9) (cid:12) (cid:8) (cid:4)(cid:1)(cid:10) (cid:9) (cid:12) (cid:8) (cid:5) (cid:1)(cid:2)(cid:1)(cid:1) (cid:1) (cid:6) (cid:3)(cid:1)(cid:10) (cid:9) (cid:12) (cid:8) (cid:5) (cid:1)(cid:2)(cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:12) (cid:13) (cid:7) (cid:11) (cid:1) (cid:2) (cid:11) (cid:23) (cid:3) (cid:11) (cid:23) (cid:1) (cid:1) (cid:21) (cid:1) (cid:1) (cid:1) (cid:10) (cid:1) (cid:7) (cid:9) (cid:1) (cid:12) (cid:16) (cid:13) (cid:1) (cid:6) (cid:5) (cid:4) (cid:8) (cid:5) (cid:2) (cid:1) (cid:15) (cid:16) (cid:19) (cid:22) (cid:20) (cid:14) (cid:18) (cid:17) (cid:22) (cid:24)
FIGURE 6
Directed ๏ฌow of identi๏ฌed hadrons as a func-tion of rapidity for Au+Au collisions at โ ๐ ๐๐ = 27 GeV forPHSD4.0 (green lines), PHSD5.0 with partonic cross sectionsand parton masses calculated for ๐ ๐ต = 0 (blue dashed lines)and with cross sections and parton masses evaluated at theactual chemical potential ๐ ๐ต in each individual space-time cell(red lines) in comparison to the experimental data of the STARCollaboration (Adamczyk et al., 2018).from all three scenarios in the PHSD shows a rather similarbehavior; the di๏ฌerences are very small (within the statisticsachieved here). Only antiprotons and antihyperons show a ten-dency for a small decrease of the ๐ฃ at larger ๐ ๐ for PHSD5.0compared to PHSD4.0, which can be attributed to the explicit โ ๐ -dependence and di๏ฌerent angular distribution of partoniccross sections in the PHSD5.0. We note that the underestima-tion of ๐ฃ for protons and ฮ โs might be attributed to detailsof the hadronic vector potentials involved in this calculationswhich seem to underestimate the baryon repulsion. We alsoattribute the slight overestimation of antibaryon ๐ฃ to the lackof baryon potential.In this study we also explore the lower collision energy โ ๐ ๐๐ = 14.5 GeV and present in Fig. 8 the results for the ๐ฃ of identi๏ฌed hadrons ( ๐ ยฑ , ๐พ ยฑ , ๐, ฬ๐, ฮ + ฮฃ , ฬ ฮ + ฬ ฮฃ ) as a func-tion of ๐ ๐ . We ๏ฌnd the same tendency as in Fig. 7 and showexplicitly the PHSD4.0 (orange dashed lines) and the PHSD5.0calculations with cross sections and parton masses dependingon ๐ ๐ต (blue solid lines) in comparison to the experimental dataof the STAR Collaboration (Adam et al., 2020). (cid:4) (cid:2)(cid:4) (cid:4)(cid:4) (cid:2)(cid:4) (cid:6)(cid:4) (cid:2)(cid:5) (cid:4)(cid:4) (cid:2)(cid:4) (cid:4)(cid:4) (cid:2)(cid:4) (cid:6)(cid:4) (cid:2)(cid:5) (cid:4)(cid:4) (cid:2)(cid:5) (cid:6)(cid:4) (cid:2)(cid:4) (cid:4) (cid:2)(cid:6) (cid:5) (cid:2)(cid:4) (cid:5) (cid:2)(cid:6)(cid:4) (cid:2)(cid:4) (cid:4)(cid:4) (cid:2)(cid:4) (cid:6)(cid:4) (cid:2)(cid:5) (cid:4)(cid:4) (cid:2)(cid:5) (cid:6) (cid:4) (cid:2)(cid:4) (cid:4) (cid:2)(cid:6) (cid:5) (cid:2)(cid:4) (cid:5) (cid:2)(cid:6) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:12) (cid:23) (cid:3) (cid:12) (cid:23) (cid:1) (cid:11) (cid:1) (cid:1) (cid:21) (cid:1) (cid:10) (cid:1) (cid:7) (cid:9) (cid:1) (cid:13) (cid:17) (cid:14) (cid:1) (cid:6) (cid:5) (cid:4) (cid:8) (cid:5) (cid:2) (cid:1) (cid:16) (cid:17) (cid:19) (cid:22) (cid:20) (cid:15) (cid:18) (cid:1) (cid:2) (cid:2) (cid:1) (cid:1) (cid:1)(cid:4) (cid:5) (cid:2) (cid:3)(cid:1) (cid:1)(cid:10) (cid:9) (cid:11) (cid:8) (cid:1)(cid:5) (cid:3)(cid:4)(cid:1) (cid:1)(cid:10) (cid:9) (cid:11) (cid:8) (cid:1)(cid:6) (cid:3)(cid:4) (cid:1)(cid:2)(cid:1)(cid:1) (cid:1) (cid:7) (cid:4) (cid:1)(cid:1) (cid:1)(cid:10) (cid:9) (cid:11) (cid:8) (cid:1)(cid:6) (cid:3)(cid:4) (cid:1)(cid:2)(cid:1)(cid:1) (cid:1) (cid:2)(cid:1) (cid:1) (cid:1) (cid:13) (cid:1) (cid:1)(cid:9)(cid:7) (cid:12) (cid:8) (cid:3)(cid:11) (cid:10) (cid:1) (cid:13) (cid:1) (cid:1)(cid:9)(cid:7) (cid:12) (cid:8) (cid:3)(cid:11) (cid:10) FIGURE 7
Elliptic ๏ฌow of identi๏ฌed hadrons ( ๐พ ยฑ , ๐, ฬ๐, ฮ +ฮฃ , ฬ ฮ + ฬ ฮฃ ) as a function of ๐ ๐ for Au+Au collisions at โ ๐ ๐๐ = 27 GeV for PHSD4.0 (green lines), PHSD5.0 with partoniccross sections and parton masses calculated for ๐ ๐ต = 0 (bluedashed lines) and with cross sections and parton masses eval-uated at the actual chemical potential ๐ ๐ต in each individualspace-time cell (red lines) in comparison to the experimentaldata of the STAR Collaboration (Adamczyk et al., 2016). In this study we have reviewed the thermodynamic propertiesof the QGP as created in heavy-ion collisions and reportedon the in๏ฌuence of the baryon chemical potential ๐ ๐ต on theexperimental observables such as the directed and elliptic ๏ฌow.For the description of the QGP we have employed theextended Dynamical QuasiParticle Model (DQPM) that ismatched to reproduce the lQCD equation-of-state versus tem-perature ๐ at zero and ๏ฌnite baryon chemical potential ๐ ๐ต .We recall that the ratios ๐ โ ๐ and ๐ โ ๐ from the DQPM agreevery well with the lQCD results from Ref. (Astrakhantsev etal., 2018) and show a similar behavior as the ratio obtainedfrom a Bayesian ๏ฌt (Bernhard et al., 2016). As found pre-viously in Refs. (Moreau et al., 2019; Soloveva, Moreau, &Bratkovskaya, 2020) the transport coe๏ฌcients show only amild dependence on ๐ ๐ต .Our study of the heavy-ion collisions, i.e. non-equilibriumQGP, has been performed within the extended PartonโHadronโString Dynamics (PHSD) transport approach(Moreau et al., 2019) in which the properties of quarks andgluons, i.e. their masses and widths, depend on ๐ and ๐ ๐ต explicitly. Moreover, the partonic interaction cross sections . Moreau at al. v + PHSD5.0PHSD4.0STAR
PHSD5.0PHSD4.0STAR v K + PHSD5.0PHSD4.0STAR K PHSD5.0PHSD4.0STAR v p PHSD5.0PHSD4.0STAR p PHSD5.0PHSD4.0STAR p T [GeV]0.000.050.100.15 v + PHSD5.0PHSD4.0STAR p T [GeV] 0.000.050.100.15 + PHSD5.0PHSD4.0STAR
Au+Au @ s NN = 14.5 GeV, 10-40% central FIGURE 8
Elliptic ๏ฌow of identi๏ฌed hadrons( ๐ ยฑ , ๐พ ยฑ , ๐, ฬ๐, ฮ + ฮฃ , ฬ ฮ + ฬ ฮฃ ) as a function of ๐ ๐ for Au+Aucollisions at โ ๐ ๐๐ = 14.5 GeV for the PHSD4.0 (orangedashed lines) and the PHSD5.0 with cross sections and partonmasses depending on ๐ ๐ต (blue solid lines) in comparison tothe experimental data of the STAR Collaboration (Adam etal., 2020).are obtained by evaluation of the leading order Feynman dia-grams from the DQPM e๏ฌective propagators and couplingsand explicitly depend on the invariant energy โ ๐ as well as ๐ and ๐ ๐ต .We have demonstrated the time evolution of heavy-ion col-lisions and ๐ โ ๐ ๐ต trajectories as obtained from the PHSD5.0calculations for SPS energies. We have explored the energydependence of the averaged ๐ ๐ต around the chemical freeze-outtemperature and found a reasonable agreement with statisticalmodel results (Adamczyk et al., 2017; Andronic et al., 2010;Cleymans et al., 2006).We have explored the sensitivity of heavy-ion observablessuch as the excitation function of the hadron multiplicities andcollective ๏ฌow coe๏ฌcients ๐ฃ , ๐ฃ on the ( ๐ , ๐ ๐ต ) dependence ofthe QGP properties. After remanding that the sensitivity (w.r.t.the ๐ ๐ต -dependence) of hadronic rapidity and ๐ ๐ distributionsof hadrons for symmetric and asymmetric heavy-ion collisionsfrom AGS to RHIC energies was found to be very low (Moreau et al., 2019; Soloveva, Moreau, & Bratkovskaya, 2020), wehave discussed the sensitivity of the collective ๏ฌow of hadrons.As an example, we have shown the results for โ ๐ ๐๐ = 14.5and 27 GeV . As has been shown in Refs. (Soloveva, Moreau,Oliva, Song, et al., 2020; Soloveva, Moreau, Oliva, Voronyuk,et al., 2020) we ๏ฌnd only very small di๏ฌerences between theresults from PHSD4.0 and from PHSD5.0 on the hadronic ๏ฌowobservables at high as well as at intermediate energies. Thisis related to the fact that at high energies, where the matteris dominated by the QGP, one probes only a small baryonchemical potential in central collisions at midrapidity, whileat lower energies (and larger ๐ ๐ต ) the fraction of the QGPdrops rapidly (cf. Fig. 3 ) such that in total the ๏ฌnal observ-ables are dominated by the hadronic interactions and thus theinformation about the partonic properties and scatterings iswashed out. This ๏ฌnding is also consistent with the fact thatthe transport coe๏ฌcients - as evaluated in the DQPM - showonly a weak dependence on ๐ ๐ต . We have shown that the mild ๐ ๐ต -dependence of QGP interactions is more pronounced inobservables for strange hadrons (kaons and especially anti-strange hyperons) which provides an experimental hint for thesearch of ๐ ๐ต traces of the QGP for experiments at the futureFAIR/NICA facilities and the BESII program at RHIC. ACKNOWLEDGEMENTS
The authors acknowledge inspiring discussions with JรถrgAichelin, Wolfgang Cassing, Ilya Selyuzhenkov and ArkadiyTaranenko. The computational resources have been pro-vided by the LOEWE-Centerfor Scienti๏ฌc Computing and the"Green Cube" at GSI, Darmstadt. P.M. acknowledges supportby the U.S. D.O.E. under Grant No. DE-FG02-05ER41367.O.S., I.G. acknowledge support from HGS-HIRe for FAIR.L.O. has been ๏ฌnancially supported by the Alexander vonHumboldt Foundation. Furthermore, we acknowledge sup-port by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation): grant BR 4000/7-1, grant CRC-TR 211โStrong-interaction matter under extreme conditionsโ - Projectnumber 315477589 - TRR 211; by the Russian Science Foun-dation grant 19-42-04101; by the European Unionโs Horizon2020 research and innovation program under grant agreementNo 824093 (STRONG-2020) and by the COST Action THOR,CA15213.
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