[Paper Review] QCD equation of state matched to lattice data and exhibiting a critical point singularity

Paolo Parotto, Marcus Bluhm+7 more|arXiv (Cornell University)|May 14, 2018

High-Energy Particle Collisions Research19 citations

TL;DR

This paper constructs a family of QCD equations of state that match lattice QCD data up to O(μ⁴_B) and incorporate a critical point in the 3D Ising model universality class, positioned within the Beam Energy Scan II (BES-II) range at RHIC. By mapping the Ising model's universal scaling behavior onto QCD via non-universal variables, the model captures the correct critical singularity and enables hydrodynamic simulations with fluctuation observables for direct comparison with BES-II data to constrain the critical point's location.

ABSTRACT

We construct a family of equations of state for QCD in the temperature range 30 MeV $\leq T\leq$ 800 MeV and in the chemical potential range $0\leq \mu_B \leq$ 450 MeV. These equations of state match available lattice QCD results up to $\mathcal{O}(\mu_B^4)$ and in each of them we place a critical point in the 3D Ising model universality class. The position of this critical point can be chosen in the range of chemical potentials covered by the second Beam Energy Scan at RHIC. We discuss possible choices for the free parameters, which arise from mapping the Ising model onto QCD. Our results for the pressure, entropy density, baryon density, energy density and speed of sound can be used as inputs in the hydrodynamical simulations of the fireball created in heavy ion collisions. We also show our result for the second cumulant of the baryon number in thermal equilibrium, displaying its divergence at the critical point. In the future, comparisons between RHIC data and the output of the hydrodynamic simulations, including calculations of fluctuation observables, built upon the model equations of state that we have constructed may be used to locate the critical point in the QCD phase diagram, if there is one to be found.

Motivation & Objective

• To develop a family of QCD equations of state that incorporate a critical point singularity in the 3D Ising universality class.
• To ensure compatibility with lattice QCD results up to O(μ⁴_B) at finite baryon chemical potential.
• To enable hydrodynamic simulations of heavy-ion collisions that include critical point effects for comparison with BES-II experimental data.
• To provide a parametric framework where the critical point's location can be constrained by matching model predictions to experimental fluctuation observables.

Proposed method

• Choose a critical point location in the (μ_B, T) plane within the BES-II accessible region.
• Use a parametrization of the 3D Ising model's universal scaling behavior near the critical point.
• Map the Ising model's thermodynamics onto QCD via a non-universal, parametric transformation of variables.
• Compute the critical contribution to the pressure and its derivatives up to O(μ⁴_B) using Ising model scaling functions.
• Reconstruct the full equation of state by combining the lattice QCD result at μ_B = 0 with the critical contribution, ensuring continuity and correct singularity behavior.
• Validate the model by showing divergence of the second baryon number cumulant at the critical point, consistent with universal critical behavior.

Experimental results

Research questions

• RQ1 Can a QCD equation of state be constructed that matches lattice QCD data up to O(μ⁴_B) while incorporating a critical point in the 3D Ising universality class?
• RQ2 Where in the QCD phase diagram can a critical point be placed such that it lies within the accessible range of the BES-II program at RHIC?
• RQ3 How can the universal scaling behavior of the 3D Ising model be consistently mapped onto QCD thermodynamics to reproduce the correct critical singularity?
• RQ4 What are the implications of the critical point for fluctuation observables, such as the second cumulant of baryon number, in thermal equilibrium?
• RQ5 Can this model framework be used to constrain the location of a QCD critical point by comparing hydrodynamic simulations with BES-II data?

Key findings

• The constructed equations of state match lattice QCD results for the pressure and its derivatives up to O(μ⁴_B) at vanishing baryon chemical potential.
• The model successfully incorporates a critical point in the 3D Ising universality class, with its location tunable within the BES-II accessible region of the QCD phase diagram.
• The second cumulant of the baryon number diverges at the critical point, as required by universal scaling laws, confirming the correct critical singularity.
• The model provides a consistent framework to compute thermodynamic quantities such as pressure, entropy density, energy density, baryon density, and speed of sound across the (μ_B, T) plane.
• The model enables future hydrodynamic simulations with critical point effects, allowing direct comparison with BES-II data to constrain the critical point's parameters.
• The parametric map between Ising variables and QCD coordinates introduces free parameters, including the critical point's (μ_B, T) location, which can be constrained by experimental data.