https://epub.uni-regensburg.de/32653/7/main.pdf

Full result for the QCD equation of state with 2+1 flavors

We review collective flow, its anisotropies, and its event-to-event fluctuations in relativistic heavy-ion collisions, as well as the extraction of the specific shear viscosity of quark–gluon plasma from collective flow data collected in heavy-ion collision experiments at RHIC and the LHC. We emphasize the similarities between the Big Bang of our universe and the Little Bangs created in heavy-ion collisions.

https://www.annualreviews.org/doi/pdf/10.1146/annurev-nucl-102212-170540

Collective Flow and Viscosity in Relativistic Heavy-Ion Collisions

The QCD transition temperature: Results with physical masses in the continuum limit

The present paper concludes our investigation on the QCD equation of state with 2+1 staggered flavors and one-link stout improvement. We extend our previous study [JHEP 0601:089 (2006)] by choosing even finer lattices. Lattices with $N_t=6,8$ and 10 are used, and the continuum limit is approached by checking the results at $N_t=12$. A Symanzik improved gauge and a stout-link improved staggered fermion action is utilized. We use physical quark masses, that is, for the lightest staggered pions and kaons we fix the $m_\pi/f_K$ and $m_K/f_K$ ratios to their experimental values. The pressure, the interaction measure, the energy and entropy density and the speed of sound are presented as functions of the temperature in the range $100 ...1000 \textmd{MeV}$. We give estimates for the pion mass dependence and for the contribution of the charm quark. We compare our data to the equation of state obtained by the "hotQCD" collaboration.

The QCD equation of state with dynamical quarks

The present paper concludes our investigations on the QCD cross-over transi- tion temperatures with 2+1 staggered avours and one-link stout improvement. We extend our previous two studies (Phys. Lett. B643 (2006) 46, JHEP 0906:088 (2009)) by choosing even ner lattices ( Nt=16) and we work again with physical quark masses. The new results on this broad cross-over are in complete agreement with our earlier ones. We compare our ndings with the published results of the hotQCD collaboration. All these results are con- fronted with the predictions of the Hadron Resonance Gas model and Chiral Perturbation Theory for temperatures below the transition region. Our results can be reproduced by using the physical spectrum in these analytic calculations. The ndings of the hotQCD collaboration can be recovered by using a distorted spectrum which takes into account lat- tice discretization artifacts and heavier than physical quark masses. This analysis provides a simple explanation for the observed discrepancy in the transition temperatures between our and the hotQCD collaborations.

/pdf/is-there-still-any-tc-mystery-in-lattice-qcd-results-with-65pweld3vz.pdf

Is there still any Tc mystery in lattice QCD? Results with physical masses in the continuum limit III

The standard model of particle physics predicts two phase transitions that are relevant for the evolution of the early Universe. One, the quantum chromodynamics transition, involves the strong force that binds quarks into protons and neutrons. Despite much theoretical effort, the nature of this transition remains ambiguous. Now Aoki et al. report computationally demanding calculations that suggest that there was no true phase transition. Instead, an analytic crossover took place, involving a rapid, continuous change with temperature as opposed to a jump. This means that it will be difficult to find experimental evidence of a transition from astronomical observations. The standard model of particle physics predicts two transitions that are relevant for the evolution of the early Universe. Computationally demanding calculations now reveal that a real phase transition did not occur, but rather an analytic crossover, involving a rapid change (as opposed to a jump) as the temperature varies. Quantum chromodynamics (QCD) is the theory of the strong interaction, explaining (for example) the binding of three almost massless quarks into a much heavier proton or neutron—and thus most of the mass of the visible Universe. The standard model of particle physics predicts a QCD-related transition that is relevant for the evolution of the early Universe. At low temperatures, the dominant degrees of freedom are colourless bound states of hadrons (such as protons and pions). However, QCD is asymptotically free, meaning that at high energies or temperatures the interaction gets weaker and weaker1,2, causing hadrons to break up. This behaviour underlies the predicted cosmological transition between the low-temperature hadronic phase and a high-temperature quark–gluon plasma phase (for simplicity, we use the word ‘phase’ to characterize regions with different dominant degrees of freedom). Despite enormous theoretical effort, the nature of this finite-temperature QCD transition (that is, first-order, second-order or analytic crossover) remains ambiguous. Here we determine the nature of the QCD transition using computationally demanding lattice calculations for physical quark masses. Susceptibilities are extrapolated to vanishing lattice spacing for three physical volumes, the smallest and largest of which differ by a factor of five. This ensures that a true transition should result in a dramatic increase of the susceptibilities. No such behaviour is observed: our finite-size scaling analysis shows that the finite-temperature QCD transition in the hot early Universe was not a real phase transition, but an analytic crossover (involving a rapid change, as opposed to a jump, as the temperature varied). As such, it will be difficult to find experimental evidence of this transition from astronomical observations.

https://arxiv.org/pdf/hep-lat/0611014.pdf

The order of the quantum chromodynamics transition predicted by the standard model of particle physics

We determine the equation of state of QCD for nonzero chemical potentials via a Taylor expansion of the pressure. The results are obtained for N_f=2+1 flavors of quarks with physical masses, on various lattice spacings. We present results for the pressure, interaction measure, energy density, entropy density, and the speed of sound for small chemical potentials. At low temperatures we compare our results with the Hadron Resonance Gas model. We also express our observables along trajectories of constant entropy over particle number. A simple parameterization is given (the Matlab/Octave script parameterization.m, submitted to the arXiv along with the paper), which can be used to reconstruct the observables as functions of T and mu, or as functions of T and S/N.

QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order mu^2

We present the equation of state (pressure, trace anomaly, energy density and entropy density) of the SU(3) gauge theory from lattice field theory in an unprecedented precision and temperature range. We control both finite size and cut-off effects. The studied temperature window (0.7...1000 T_c) stretches from the glueball dominated system into the perturbative regime, which allows us to discuss the range of validity of these approaches. We also determine the preferred renormalization scale of the Hard Thermal Loop scheme and we fit the unknown g^6 order perturbative coefficient at extreme high temperatures T>100 T_c. We furthermore quantify the nonperturbative contribution to the trace anomaly using a simple functional form. Our high precision data allows one to have a complete theoretical description of the equation of state from T=0 all the way to the phase transition, through the transition region into the perturbative regime up to the Stefan-Boltzmann limit. We will discuss this description, too.

Precision SU(3) lattice thermodynamics for a large temperature range

We determine the equation of state of 2+1-flavor QCD with physical quark masses, in the presence of a constant (electro)magnetic background field on the lattice. To determine the free energy at nonzero magnetic fields we develop a new method, which is based on an integral over the quark masses up to asymptotically large values where the effect of the magnetic field can be neglected. The method is compared to other approaches in the literature and found to be advantageous for the determination of the equation of state up to large magnetic fields. Thermodynamic observables including the longitudinal and transverse pressure, magnetization, energy density, entropy density and interaction measure are presented for a wide range of temperatures and magnetic fields, and provided in ancillary files. The behavior of these observables confirms our previous result that the transition temperature is reduced by the magnetic field. We calculate the magnetic susceptibility and permeability, verifying that the thermal QCD medium is paramagnetic around and above the transition temperature, while we also find evidence for weak diamagnetism at low temperatures.

/pdf/the-qcd-equation-of-state-in-background-magnetic-fields-2guxd49o8s.pdf

The QCD equation of state in background magnetic fields

We study the influence of strong external magnetic fields on gluonic and fermionic observables in the QCD vacuum at zero and nonzero temperatures, via lattice simulations with N_f=1+1+1 staggered quarks of physical masses. The gluonic action density is found to undergo magnetic catalysis at low temperatures and inverse magnetic catalysis near and above the transition temperature, similar to the quark condensate. Moreover, the gluonic action develops an anisotropy: the chromo-magnetic field parallel to the external field is enhanced, while the chromo-electric field in this direction is suppressed. We demonstrate that the same hierarchy is obtained using the Euler-Heisenberg effective action. Conversely, the topological charge density correlator does not reveal a significant anisotropy up to magnetic fields eB~1 GeV^2. Furthermore, we show that the pressure remains isotropic even for nonzero magnetic fields, if it is defined through a compression of the system at fixed external field. In contrast, if the flux of the field is kept fixed during the compression -- which is the situation realized in the lattice simulation -- the pressure develops an anisotropy. We estimate the quark and gluonic contributions to this anisotropy, and relate them to the magnetization of the QCD vacuum. After performing electric charge renormalization, we obtain an estimate for the magnetization, which indicates that QCD is paramagnetic.

G. Endrodi

Papers

The order of the quantum chromodynamics transition predicted by the standard model of particle physics

QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order mu^2

Precision SU(3) lattice thermodynamics for a large temperature range

The QCD equation of state in background magnetic fields

Magnetic field-induced gluonic (inverse) catalysis and pressure (an)isotropy in QCD