QCD Thermodynamics from the Lattice

TL;DR

This paper surveys lattice QCD thermodynamics, focusing on how ab initio simulations illuminate the phase structure, equation of state, and in-medium hadron properties of QCD at finite temperature and density. It surveys the theoretical framework (lattice gauge theory, various fermion formulations, and cutoff effects) and the main observables used to locate the transition temperature, such as the Polyakov loop and chiral condensate, along with their susceptibilities. It synthesizes current understanding of the zero-density phase diagram (crossover at physical masses, with a richer structure in massless limits) and the challenges of nonzero density due to the sign problem, cataloguing multiple approaches (reweighting, imaginary chemical potential, Taylor expansion, canonical methods, and stochastic quantization). The review also covers the equation of state, its methods (derivative, integral, and step-scaling), and nonzero-density extensions, as well as in-medium properties, such as screening masses and the persistence of charmonium near Tc, and progress toward determining transport coefficients. Overall, lattice QCD thermodynamics provides a quantitative, first-principles window into the quark–gluon plasma, guiding phenomenology for heavy-ion experiments and informing the connection to perturbative QCD at very high temperatures, while highlighting key open questions and methodological challenges ahead.

Abstract

We review the current methods and results of lattice simulations of quantum chromodynamics at nonzero temperatures and densities. The review is intended to introduce the subject to interested nonspecialists and beginners. It includes a brief overview of lattice gauge theory, a discussion of the determination of the crossover temperature, the QCD phase diagram at zero and nonzero densities, the equation of state, some in-medium properties of hadrons including charmonium, and some plasma transport coefficients.

Figures (31)

• Figure 1: The bare-parameter phase diagram for two flavors of clover-improved Wilson fermions and an improved gauge action for zero and nonzero temperatures, illustrating the mapping necessary for thermodynamics studies with Wilson fermions. In this plot the hopping parameter $\kappa$ is denoted by $K$. The line of chiral critical hopping parameters $\kappa_c(T=0)$ was determined from the vanishing of the pion mass. The line $\kappa_t$ indicates the high temperature crossover at $N_\tau=4$. It was determined from the Polyakov line (see section \ref{['sec:Polyakov']}). The region "chiral phase transition" shows where the thermal crossover happens for small pion masses. The parity broken phases come from lattice artifacts of Wilson fermions. The data are from the CP-PACS collaboration, AliKhan:2000iz, as shown in Ejiri:2007qk.
• Figure 2: Plot showing that the lattice artifact taste splitting of pion masses vanishes as $\alpha_s^2 a^2$ in the continuum limit. The splitting is measured as the difference of the squared masses of the multiplet member and the Goldstone pion member. It is given in units of $r_1 \approx 0.318$ fm. The plot symbols distinguish the members of the multiplet. (The subscripts in the legend denote the Dirac-gamma-matrix-style classification of the pion tastes, ranging from singlet ($s$) and $\gamma_0$ to $\gamma_0\gamma_5$.) The line is drawn with unit log-log slope to test proportionality to $\alpha_s^2 a^2$.
• Figure 3: Sketch, courtesy of Taku Izubuchi, of the domain-wall fermion setup. Left and right handed modes are exponentially bound to the left and right domain walls. The residual mass $m_{res}$ is determined from an axial Ward identity applied in the center slice.
• Figure 4: Plot of the residual mass $m_{res}$ as a function of $L_s$ showing its desired suppression with increasing $L_s$ and increasing inverse gauge coupling $\beta$. Also shown are fits to an exponential fall-off plus a $1/L_s$ contribution, from a recent $2+1$ flavor dynamical domain-wall fermion simulation Antonio:2008zz.
• Figure 5: Residual mass $m_{res}$ for the recent nonzero-temperature simulations on $N_\tau=8$ lattices with $L_s=32$Cheng:2008ge. At lower $\beta$, corresponding to lower temperatures, $m_{res}$ increases rapidly, and is larger than the input light quark mass already in the transition region.