The QCD Equation of State to $\mathcal{O}(μ_B^6)$ from Lattice QCD (1701.04325v3)

Abstract: We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range $T\in [135~{\rm MeV}, 330~{\rm MeV}]$ using up to four different sets of lattice cut-offs corresponding to lattices of size $N_\sigma^3\times N_\tau$ with aspect ratio $N_\sigma/N_\tau=4$ and $N_\tau =6-16$. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios $m_s/m_l=20$ and $27$, which in the continuum limit correspond to a pion mass of about $160$ MeV and $140$ MeV espectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature ($\mu_B\le 2T$). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to $\sqrt{s_{NN}}\sim 12$ GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the $T$-$\mu_B$ plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for $\mu_B/T\le 2$ and $T/T_c(\mu_B=0) > 0.9$.

• The paper presents a sixth-order Taylor expansion of μB in lattice QCD to precisely estimate pressure, energy, and entropy densities.
• It applies the HISQ action over a 135–330 MeV temperature range, effectively reducing truncation errors up to μB≤2T.
• These results refine QCD phase diagram models and inform hydrodynamic simulations by disfavoring a critical point for μB/T≤2.

Overview of the QCD Equation of State to $O(\mu_B^6)$ from Lattice QCD

The paper "The QCD Equation of State to $O(\mu_B^6)$ from Lattice QCD" offers a meticulous analysis of the pressure, baryon density, energy density, and entropy density of QCD matter using lattice QCD simulations. This investigation pushes the boundaries of the Taylor expansion technique to sixth order in baryon chemical potential $\mu_B$, providing insights into the behavior of strong-interaction matter under various thermal conditions up to $\sqrt{s_{NN}\sim 12}$ GeV.

Taylor Expansion Approach

The authors employ a Taylor expansion approach to express bulk thermodynamic quantities with respect to the conserved charges' chemical potentials. Specifically, they explore expansions concerning baryon ($\mu_B$), strangeness ($\mu_S$), and electric charge ($\mu_Q$). The computations are based on the Highly Improved Staggered Quark (HISQ) action, working over a temperature range of $T\in [135~{\rm MeV}, 330~{\rm MeV}]$ using lattices with varied temporal extents.

The inputs from sixth-order Taylor expansion provide estimates crucial for gauging truncation errors prevalent in lower-order expansions, particularly the fourth order. The paper reveals that truncation errors are minimal for baryon chemical potentials up to twice the temperature ($\mu_B\le 2T$). This strict evaluation helps in modeling dense matter characteristic of heavy ion collisions at lower energies.

Thermodynamic Parametrization and Implications

The paper meticulously evaluates thermodynamic quantities, offering parameterizations adaptable to hydrodynamic simulations. Up to sixth-order coefficient results on bulk thermodynamics facilitate the elucidation of lines of constant pressure, energy, and entropy densities in the temperature-$\mu_B$ plane. These lines are indispensable for understanding conditions in heavy ion collisions, especially concerning the freeze-out parameters observed experimentally.

Impactfully, this research explores the critical point speculation in the QCD phase diagram, proposing a clear disfavor of a critical point's existence for $\mu_B/T\le 2$ and $T/T_c(\mu_B=0) > 0.9$. Such claims are pivotal for theoretical frameworks predicting QCD phase transitions, influencing future event simulations, and guiding experimental setups.

Future Directions

This paper's detailed measurements and results from sixth-order contributions provoke queries about higher-order expansions and critical phenomena connectivity. While current findings oppose the critical point existence in the discussed regime, further investigations with enhanced lattice resolutions could afford more definitive conclusions and refine our understanding of QCD thermodynamics.

Moreover, expanding the evaluation to include larger chemical potentials may reveal additional transitional behavior, providing an empirical foundation for extreme state modeling in astrophysics and beyond. Future work would benefit from focusing on improving numerical precision, exploring larger lattice volumes, and incorporating additional fluctuation observables to comprehensively map the QCD landscape.

By providing detailed parameterizations applicable to hydrodynamic codes and systematically estimating truncation errors, this paper sets a benchmark for theoretical studies in QCD thermodynamics under realistic heavy ion conditions.