[Paper Review] The QCD EoS from simulations on BlueGene L Supercomputers at LLNL and NYBlue

Rajan Gupta|arXiv (Cornell University)|Oct 9, 2008

High-Energy Particle Collisions Research6 references5 citations

TL;DR

This paper presents high-precision lattice QCD simulations of the QCD equation of state (EoS) using improved asqtad and p4fat3 staggered fermions on BlueGene/L supercomputers at LLNL and NYBlue. The study achieves <20% uncertainty in the trace anomaly on $N_{\tau}=8$ lattices and finds a consistent crossover transition temperature of $185-195$ MeV, establishing that lattice EoS estimates for energy density and pressure are reliable for phenomenological analysis of RHIC and LHC data.

ABSTRACT

We present results for the QCD Equation of State (EoS) obtained using simulations of lattice QCD at zero chemical potential. Our high statistics results compare improved asqtad and p4fat3 staggered quarks on lattices with a temporal extent N_tau = 6 and 8 and light quark masses approximately one fifth and one tenth the strange quark mass. We find that the two actions give consistent results and estimate that the trace anomaly (epsilon - 3p)/T^4 obtained on N_tau = 8 lattices represents the continuum value to better than 20% uncertainty over the temperature range 140-700 MeV. The precision in the estimates of energy density and pressure are better, therefore, we conclude that lattice estimates of the energy density and pressure should be used in the phenomenological analysis of RHIC and LHC data. We also find a consistent picture of the crossover temperature from all observables studied, with the best estimated range to be 185-195 MeV. These calculations are being carried out on the IBM BlueGene/L supercomputer at Lawrence Livermore National Laboratory and at the New York Center for Computational Science (NYBlue).

Motivation & Objective

• To determine the QCD equation of state (EoS) with high precision over the temperature range 140–700 MeV, relevant for RHIC and LHC experiments.
• To assess systematic uncertainties in lattice QCD calculations, particularly discretization errors and continuum extrapolation, using two improved staggered fermion actions.
• To identify a robust and consistent estimate of the crossover transition temperature in the $2+1$ flavor QCD system.
• To evaluate the reliability of lattice QCD results for energy density and pressure in phenomenological hydrodynamic models of the quark-gluon plasma.
• To enable future continuum extrapolation by simulating at $N_{\tau}=12$ and extending to lighter quark masses including $m_{\ell}/m_s = 0.05$.

Proposed method

• Simulating $2+1$ flavor QCD using two $O(a^2)$-improved staggered fermion actions: asqtad and p4fat3, on $N_\tau=6$ and $8$ lattices with spatial extent $32^3$.
• Performing simulations along lines of constant physics by fixing the strange quark mass to $M_{\bar{s}s} \approx 686$ MeV via the $r_0$ scale, and varying light quark masses at $m_\ell/m_s = 0.2$, $0.1$, and $0.05$.
• Using high-statistics ensembles with up to 60,000 trajectories per $\beta$ value to reduce statistical errors and ensure convergence.
• Calculating the trace anomaly $(\varepsilon - 3p)/T^4$ as the primary EoS observable, with ultraviolet divergences subtracted using $T=0$ data from $32^4$ lattices.
• Employing multiple deconfinement and chiral transition probes: renormalized Polyakov loop, strange quark number susceptibility $\chi_s/T^2$, chiral condensate $\langle\bar{\psi}\psi\rangle$, and its susceptibility $\chi_{\rm disc}$.
• Performing continuum extrapolation using $N_\tau=6$, $8$, and $12$ lattices, and chiral extrapolation using data at $m_\ell/m_s = 0.2$, $0.1$, and $0.05$.

Experimental results

Research questions

• RQ1 To what extent do the asqtad and p4fat3 staggered fermion actions yield consistent results for the QCD EoS at $N_\tau=6$ and $8$?
• RQ2 What is the uncertainty in the trace anomaly $(\varepsilon - 3p)/T^4$ on $N_\tau=8$ lattices, and does it represent the continuum limit within $20\%$?
• RQ3 What is the most reliable estimate of the crossover transition temperature $T_c$ in $2+1$ flavor QCD, based on multiple observables?
• RQ4 How do systematic errors from discretization, chiral extrapolation, and finite $N_\tau$ affect the precision of the EoS and transition temperature?
• RQ5 Can the lattice QCD EoS, particularly energy density and pressure, be reliably used in hydrodynamic phenomenological models of the quark-gluon plasma at RHIC and LHC?

Key findings

• The trace anomaly $(\varepsilon - 3p)/T^4$ on $N_\tau=8$ lattices shows $<20\%$ uncertainty across the temperature range 140–700 MeV, indicating that the $N_\tau=8$ results are close to the continuum limit.
• Energy density and pressure estimates on $N_\tau=8$ lattices have even better precision than the trace anomaly, supporting their use in phenomenological analyses of RHIC and LHC data.
• The crossover transition temperature is consistently estimated at $185-195$ MeV across all observables—trace anomaly, Polyakov loop, quark number susceptibility, and chiral susceptibility—despite differing functional forms.
• The chiral susceptibility $\chi_{\rm disc}$ shows a broadening peak with decreasing quark mass, and extrapolation to the chiral limit suggests $T_{\text{transition}} > 180$ MeV for $N_\tau=8$, consistent with the $185-195$ MeV band.
• The inflection point in the renormalized Polyakov loop and the peak in the strange quark number susceptibility both locate at $185-195$ MeV, confirming consistency across independent probes.
• The $N_\tau=6$ data show a shift of $\approx 5$ MeV toward lower $T$ when compared to $N_\tau=8$, indicating that $N_\tau=8$ is necessary to avoid significant $O(a^2)$ discretization errors.