The QCD phase diagram within effective models Thorsten Steinert Dissertation Institut für Theoretische Physik Fachbereich 07 Mathematik und Informatik, Physik, Geographie Justus-Liebig-Universität Giessen Contents 1 Introduction 1 2 Overview of many-body physics 5 2.1 Thermodynamic relations . .5 2.2 Systems in equilibrium . .9 2.3 Fundamental properties of QCD . 13 2.4 Lattice QCD . 17 2.5 Systems out-of equilibrium . 24 2.5.1 Non-relativistic transport . 24 2.5.2 Relativistic transport . 25 3 The Dynamical QuasiParticle Model 29 3.1 DQPM . 29 3.2 DQPM* . 39 3.3 The DQPM at finite chemical potential . 47 3.3.1 Scaling hypothesis . 48 3.3.2 Flow equation . 49 3.4 Transport coefficients . 59 4 The effective Nambu Jona-Lasinio model 73 4.1 The Nambu Jona-Lasinio model . 73 4.2 The Polyakov NJL model . 82 4.3 Quark effects on the Polyakov potential . 87 4.4 Accessing the equation of state via the quark condensate . 95 5 Thermodynamics of hadronic systems 99 5.1 Hadron-Resonance Gas . 99 5.2 Nuclear equation of state . 104 5.3 Interacting Hadron-Resonance Gas . 114 5.4 Chiral condensate . 131 5.5 Probing the chiral condensate in relativistic heavy-ion collisions . 136 6 The QCD phase boundary 145 6.1 A universal hadronization condition . 145 6.2 The phase boundary between the DQPM∗ and the IHRG . 148 6.3 Partonic quasiparticle models at low temperatures . 152 6.4 Probing the phase diagram in relativistic heavy-ion collisions . 158 7 Summary and Outlook 167 A Appendix 173 A.1 Grand-canonical potential in propagator representation . 173 A.2 DQPM thermodynamics . 178 A.3 Curvature parameter . 183 A.4 Thermodynamic consistent scaling hypothesis . 184 A.5 Thermodynamic potential of the NJL model . 186 A.6 Polyakov loop in the PNJL . 191 A.7 Hadronic degrees of freedom . 194 A.8 Density-dependent relativistic mean-field theory . 195 A.9 Thermodynamic consistency of relativistic mean-field theory . 199 A.10 Pion-nucleon σ-term . 203 Abstract We study the QCD phase diagram using effective theories with the respective degrees of freedom for the different phases of QCD. In the deconfined phase we employ the dynam- ical quasiparticle model (DQPM), that is able to describe the dynamics of hot QCD at vanishing chemical potential. We extend to model to momentum-dependent selfenergies in order to match the correct perturbative limit of the propagators at high momenta. Within this generalized quasiparticle approach, denoted as DQPM∗, we can simultane- ously reproduce the lattice QCD (lQCD) equation of state (EoS) and baryon number susceptibility. Using thermodynamic consistency we extend the model to finite baryon chemical potential exceeding the application range of lQCD by far. We give predictions for the EoS and the most important transport coefficients. In the confined phase the medium is composed of hadrons. At large temperatures they interact predominantly by resonant scatterings, which can be well described in terms of a hadron-resonance gas (HRG). At large chemical potential and low temperature the nature of the interaction changes from resonant scatterings to meson exchange as described by relativistic mean- field theories. We combine both approaches to get an interacting HRG (IHRG), that is compatible to the lQCD EoS (µ ≈ 0; T > 0) and the nuclear EoS (T ≈ 0; µ > 0). For a complete description of the phase diagram we have to switch between the partonic and the hadronic model. In accordance with heavy-ion simulations we define the transition at lines of constant thermodynamics. The resulting EoS is valid up to µB ≈ 450 MeV. We perform heavy-ion simulations with the PHSD transport approach and determine the region in the QCD phase diagram that is probed by different collision energies. The ∗ EoS constructed from the DQPMp and the IHRG can be used to describe collisions at low beam energies down to s ≈ 7:7 GeV. Using simulations at even lower beam ener- gies we determine the conditions necessary for the discovery of the critical point in the QCD phase diagram. Abstract Wir untersuchen das QCD-Phasendiagramm unter Verwendung verschiedener Effektiver Theorien. Wir beschreiben die deconfinierte Phase mit einem partonischen Quasi- teilchenmodell, dem DQPM, das erfolgreich die Dynamik heißer QCD-Materie repro- duzieren kann. Wir erweitern das Modell auf impulsabhängige Selbstenergien um den korrekten störungstheoretischen Grenzwert der Propagatoren zu gewährleisten. Mit diesem generalisierten Quasiteilchenmodell, dem DQPM∗, können wir gleichzeitig die von Gitter-QCD Rechnungen prognostizierte Zustandsgleichung sowie die Suszeptibi- lität beschreiben. Wir nutzen thermodynamische Konsistenz und erweitern das Modell auf endliche chemische Potentiale, die die Anwendbarkeit von Gitter-QCD Rechnun- gen bei weitem übersteigen, und bestimmen die Zustandsgleichung sowie die wichtigs- ten Transportkoeffizienten. In der confinierten Phase besteht die Materie nicht aus Partonen sondern aus Hadronen. Bei großen Temperaturen wechselwirken die Hadro- nen hauptsächlich durch resonante Streuung miteinander. Dies kann durch ein sim- ples Hadron-Resonanz Gas (HRG) beschrieben werden. Bei kleinen Temperaturen und großen chemischen Potentialen dominiert der Austausch von Mesonen die Wechsel- wirkung. Dieser Mechanismus wird in relativistischen Modellen für unendlich aus- gedehnte Kernmaterie beschrieben. Wir kombinieren die beiden Modelle und definieren ein wechselwirkendes HRG (IHRG), das mit der Zustandsgleichung von Gitter-QCD- Rechnungen (µ ≈ 0; T > 0) sowie der Zustandsgleichung von unendlich ausgedehnter Kernmaterie (T ≈ 0; µ > 0) übereinstimmt. Für eine vollständige Beschreibung des QCD-Phasendiagramms müssen wir an der Phasengrenze von dem partonischen auf das hadronische Modell wechseln. Wir nutzen Erkenntnisse aus Simulationen von Schwer- ionenkollisionen und definieren die Phasengrenze bei konstanten thermodynamischen Bedingungen. Die resultierende Zustandsgleichung ist bis zu einem Baryonchemischen Potential von µB ≈ 450 MeV gültig. Wir simulieren Schwerionenkollisionen mit dem PHSD Transportmodell und untersuchen die Regionen des QCD Phasendiagramms die in tatsächlichen Kollisionen zugänglich sind. Die durch das DQPM∗ und das IHRG pdefinierte Zustandsgleichung kann für Kollisionen mit Schwerpunktsenergien von über s = 7:7 GeV verwendet werden. Wir nutzen Simulationen bei noch geringeren En- ergien und untersuchen die Bedingungen die nötig sind um den kritischen Punkt des QCD Phasendiagramms nachzuweisen. 1 Introduction The different phases of matter and their phase diagrams are among the most interesting and challenging fields of modern physics. Phase transition are important for many differ- ent phenomena from ultra-cold atoms and solid-states to nuclear matter and cosmology. Especially the early universe features several phase transitions that are connected to the most fundamental aspects of physics like the separation of the four fundamental forces of nature or the decoupling of photons. The conditions in the early universe can be recreated in ultra-relativistic heavy-ion collisions. The matter in the collisions gets com- pressed and heated up until it reaches temperatures similar to the first few microseconds after the big bang. Heavy-ion collisions probe the properties of quantum chromodynam- ics (QCD), the theory of the strong interaction, and are the only possible way to create hot and dense QCD matter and to investigate the phase diagram of QCD. Heavy-ion collisions -performed in the early 2000s at the Relativistic Heavy Ion Collider (RHIC)- reached collision energies not possible in previous heavy-ion experiments. The created matter showed properties never seen at lower beam energies and challenged the current understanding of heavy-ion physics [1, 2, 3, 4]. It was assumed that the collisions created a long predicted state of matter where quarks and gluons have been liberated from confinement [5, 6]. This new phase should appear once the density becomes large enough that individual hadrons overlap each other and the quarks -usually confined in hadrons- could then move freely in the hot and dense medium. This state of matter is called a Quark-Gluon Plasma (QGP). The QGP can not be studied directly and exists only for a short period of time as an intermediate state in the heavy-ion collision. Once the fireball expands and the density decreases, individual hadrons will form again. Nevertheless, the existence of the QGP has consequences for the dynamics of the medium and will influence the final particle spectra. Possible signals are anomalies in the flow [7, 8, 9, 10], J=Ψ suppression [11], jet quenching [12, 13, 14] and variations in the strangeness production [15, 16, 17]. All these are indirect signals that get affected by the interactions in the hadronic medium. Further important signals are electromagnetic probes like photon and dilepton radia- tion, because they do practically not interact with the surrounding medium and leave the collision undisturbed [18, 19, 20, 21]. Originally it was believed that the QGP resembles a weakly interacting gas of massless partons, however, the matter created at RHIC, and later also at the Large Hadron Col- lider (LHC) at CERN, showed properties of a fluid. Indeed, relativistic hydrodynamics has been successful in describing the experimental data [22]. Moreover, viscous hydrody- 1 2 CHAPTER 1. INTRODUCTION namics found an almost vanishing ratio of shear viscosity over entropy density η=s close to the theoretical limit (η=s)KSS = 1=4π [23, 24, 25], which indicates that the QGP is an almost perfect fluid and that the partonic medium is much