The equation of state in (2+1)-flavor QCD

Abstract: We present results for the equation of state in (2+1)-flavor QCD using the highly improved staggered quark action and lattices with temporal extent $N_{\tau}=6,~8,~10$, and $12$. We show that these data can be reliably extrapolated to the continuum limit and obtain a number of thermodynamic quantities and the speed of sound in the temperature range $(130-400)$ MeV. We compare our results with previous calculations, and provide an analytic parameterization of the pressure, from which other thermodynamic quantities can be calculated, for use in phenomenology. We show that the energy density in the crossover region, $145~ {\rm MeV} \leq T \leq 163$ MeV, defined by the chiral transition, is $\epsilon_c=(0.18-0.5)~{\rm GeV}/{\rm fm}^3$, $i.e.$, $(1.2-3.1)\ \epsilon_{\rm nuclear}$. At high temperatures, we compare our results with resummed and dimensionally reduced perturbation theory calculations. As a byproduct of our analyses, we obtain the values of the scale parameters $r_0$ from the static quark potential and $w_0$ from the gradient flow.

• The paper significantly refines the QCD equation of state using the HISQ action, reducing lattice discretization errors.
• It extrapolates results from lattices with Nτ=6 to 12 to yield precise measurements of pressure, energy density, and sound speed.
• The study provides an analytic formula bridging the hadron resonance and high-temperature regimes for improved heavy-ion simulations.

Analyzing the Equation of State in (2+1)-Flavor QCD

The study of thermodynamic properties of QCD at high temperatures is crucial for understanding the phase transition between hadronic matter and the quark-gluon plasma (QGP), a novel state of matter predicted by lattice QCD calculations. The discussed paper presents a refined investigation into the equation of state (EoS) of (2+1)-flavor QCD using the Highly Improved Staggered Quark (HISQ) action, which significantly reduces lattice artifacts.

Technical Overview and Methodology

The work utilizes lattices of temporal extent $N_\tau=6, 8, 10,$ and $12$, achieving robust results through careful extrapolation to the continuum limit. The study covers a temperature range of 130 to 400 MeV. Notably, the HISQ action enhances the continuum potential's precision by mitigating $O(a^2)$ discretization errors and reducing taste violations, which plagued earlier lattice studies.

The EoS values stem from the trace of the energy-momentum tensor $\Theta^{\mu\mu}$, which is measured through sophisticated lattice simulations. The key thermodynamic quantities, such as pressure, energy density, and the speed of sound, are derived through integration techniques and supplemented by an analytic parameterization for use in phenomenological models.

Results and Their Implications

The study provides several pivotal outcomes:

1. Trace Anomaly: The HISQ action results in significantly reduced peaks compared to previous estimates using p4 and asqtad actions. Below the crossover temperature, results align well with Hadron Resonance Gas (HRG) model predictions, demonstrating consistency in this phase.
2. Continuum Extrapolation: By comparing results across varying $N_\tau$, the authors achieved highly accurate continuum values, especially for the peak region. These extrapolated results in turn offer a refined EoS compared to earlier studies.
3. Equation of State: The document introduces an analytic formula for the EoS that smoothly interpolates between the HRG regime of low temperatures and the perturbed high-temperature phase, providing a practical resource for simulations in heavy ion phenomenology.
4. Comparison with Stout Action: The results from HISQ are compared favorably with those using the stout action. Although deviations occur at higher temperatures, the main properties show consistent trends, aiding in verifying the applicability of these actions in predicting QCD thermodynamics.
5. Speed of Sound and Specific Heat: The calculations of specific heat and speed of sound are insightful for hydrodynamic models of the QGP, confirming the minimum sound speed near the predicted crossover temperature, a characteristic useful for modeling the softest point of the EoS in hydrodynamic expansions.

Implications and Future Directions

This detailed assessment of the QCD EoS using HISQ further enhances our capability to accurately model the quark-gluon plasma and to predict the behavior of strongly interacting matter under extreme conditions. The results have relevance for interpreting data from heavy-ion collision experiments like those at RHIC and LHC, which probe the early universe's high-energy conditions.

The findings underscore the necessity of continual refinement in lattice QCD methods and call for additional research, especially to extend comparisons at even higher temperatures where perturbative QCD techniques might become applicable. Further collaborative benchmarks with different Lattice QCD actions are recommended to resolve the subtle tensions at high temperature scales, which may significantly enhance predictive precision.

The paper's results reflect substantial progress in the field, establishing a strong foundation for theoretical predictions and expanding opportunities for experimental validation, consequently advancing our understanding of fundamental interactions in particle physics.