Locating the QCD critical point through contours of constant entropy density

Abstract

We propose a new method to investigate the existence and location of the conjectured high-temperature critical point of strongly interacting matter via contours of constant entropy density. By approximating these lines as a power series in the baryon chemical potential $μ_B$, one can extrapolate them from first-principle results at zero net-baryon density, and use them to locate the QCD critical point, including the associated first-order and spinodal lines. As a proof of principle, we employ currently available continuum-extrapolated first-principle results from the Wuppertal--Budapest collaboration to find a critical point at a temperature and a baryon chemical potential of $T_c = 114.3 \pm 6.9$ MeV and $μ_{B,c} = 602.1 \pm 62.1$ MeV, respectively. We advocate for a more precise determination of the required expansion coefficients via lattice QCD simulations as a means of pinpointing the location of the critical endpoint in the phase diagram of strongly interacting matter.

Figures (8)

• Figure 1: Left: entropy as a function of the temperature, at three different chemical potentials. In this scheme, $\mu_1<\mu_{B,c}<\mu_2$. Right: corresponding constant-entropy contours in the $(T,~\mu_B)$ plane. The blue star indicates the critical point, while the shaded area corresponds to the spinodal region. The red dots indicate the spinodal points for $\mu_B=\mu_2$.
• Figure 2: Left panel: Scaled entropy density as a function of the temperature for different slices of constant $\mu_B/T$. The curves are multiplied by different factors to avoid plotting overlap. Lattice QCD data points with error bars from Ref. Borsanyi:2021sxv are also shown by the dots and vertical lines. Right panel: Scaled entropy density as a function of the temperature for different slices of constant baryochemical potential, $\mu_B = 450$ MeV (red lines, crossover regime), $\mu_B = 602$ MeV (blue lines, critical regime), $\mu_B = 750$ MeV (orange lines, mixed phase regime). The blue circle shows the critical point. In both panels, the solid curves correspond to the mean parametrization of lattice QCD results, while the translucent curves reflect the error propagation of the lattice QCD input via Monte Carlo sampling.
• Figure 3: The solid points depict the location of the QCD critical point extracted using the contours of constant entropy density based on lattice QCD data using either parametrization (black) or smoothing splines (blue). The dashed ellipse corresponds to the 68% confidence interval and reflects the error propagation of the lattice input. The hazed gray points show the scatter of the critical points obtained by sampling the lattice QCD input. The solid and dashed black lines depict the coexistence curve and the spinodals, respectively. The green band shows the chiral crossover line from Borsanyi:2020fev. The orange line corresponds to the heavy-ion chemical freeze-out bound on the CP from Lysenko:2024hqp.
• Figure A.1: Scaled entropy density $s/T^3$ (left panel) and second order baryon number susceptibility $\chi_2^B$ (right panel) as functions of the temperature at $\mu_B = 0$. The orange symbols with error bars depict the lattice QCD data of the Wuppertal-Budapest collaboration Borsanyi:2013biaBorsanyi:2021sxv. The solid black line corresponds to the parameterization of the lattice data using the mean parameter values, while the hazed black lines correspond to different Monte Carlo samples of the parameters reflecting the uncertainties in the lattice data.
• Figure A.2: Temperature dependence of the second order expansion coefficient $\alpha_{2}$ (left panel), and its first (middle panel) and second (right panel) temperature derivatives obtained using the parametrization of the lattice data. The black lines correspond to mean parameter values while hazy gray lines correspond to the Monte Carlo sampling of parameter values.