Finite density QCD phase structure from strangeness fluctuations

TL;DR

This work tackles the finite-density QCD phase structure by leveraging simulations at imaginary baryon chemical potential and enforcing strangeness neutrality. It introduces two high-signal proxies for the crossover, the second strange-quark susceptibility $\chi_2^S$ and the ratio $\mu_S/\mu_B$, and demonstrates that they remain remarkably stable along the chiral transition, both at imaginary and real chemical potentials. Through extensive lattice simulations with the 4HEX improved staggered action and a two-dimensional Taylor expansion, the authors obtain continuum-extrapolated contours up to $\mu_B \approx 400$ MeV (and up to $\approx 550$ MeV on the coarser lattice), finding striking agreement with the chiral crossover and with HRG predictions. The results imply that the crossover line defined by these proxies can be mapped to higher densities than previously accessible and suggest that deviations from the chemical freeze-out curve may occur around $\mu_B \sim 4$–$5\times10^2$ MeV, while the critical endpoint is expected near $\mu_B \sim 6$–$6.5\times10^2$ MeV. These findings provide a practical pathway to chart the QCD phase diagram and validate HRG-based expectations at finite density.

Abstract

Charting the phase diagram of Quantum Chromodynamics (QCD) at large density is a challenging task due to the complex action problem in lattice simulations. Through simulations at imaginary baryon chemical potential $μ_B$ we observe that, if the strangeness neutrality condition is imposed, both the strangeness chemical potential $μ_S/μ_B$ and the strangeness susceptibility $χ_2^S$ take on constant values at the chiral transition for varying $μ_B$. We present new lattice data to extrapolate contours of constant $μ_S/μ_B$ or $χ_2^S$ to finite baryon chemical potential. We argue that they are good proxies for the QCD crossover because, as we show, they are only mildly influenced by criticality and by finite volume effects. We obtain continuum limits for these proxies up to $μ_B = 400$ MeV, through a next-to-next-to-leading order (N$^2$LO) Taylor expansion based on large-statistics data on $16^3 \times 8$, $20^3 \times 10$ and $24^3 \times 12$ lattices with our 4HEX improved staggered action. We show that these are in excellent agreement with existing results for the chiral transition and, strikingly, also with analogous contours obtained with the hadron resonance gas (HRG) model. On the $16^3 \times 8$ lattice, we carry out the expansion up to next-to-next-to-next-to-next-to-leading order (N$^4$LO), and extend the extrapolation beyond $μ_B=500$ MeV, again finding perfect agreement with the HRG model. This suggests that the crossover line constructed from this proxy starts deviating from the chemical freeze-out line near $μ_B\approx500$ MeV, as expected but not yet observed.

Figures (11)

• Figure 1: The chiral susceptibility in the case of strangeness neutrality, as a function of the strangeness susceptibility (top) and the normalized strangeness chemical potential (bottom), on a ${{48}^3\!\!\times\!\!{12}}$ lattice for different imaginary values of the baryon chemical potential.
• Figure 2: The contour obtained through the conditions $\chi^S_2={\rm const.}$ or $\mu_S/\mu_B={\rm const.}$ in continuum extrapolated results in the $T^\prime$ expansion from Ref. Borsanyi:2022qlh, compared to the chiral transition line from Ref. Borsanyi:2020fev.
• Figure 3: Different definitions and proxies of the QCD crossover temperature as functions of the lattice volume in temperature units (the aspect ratio) on $N_\tau=12$ lattices.
• Figure 4: Diagrams contributing to the second order fluctuations of net-proton number. Figure from Ref. Stephanov:1999zu.
• Figure 5: The extrapolated $\chi_2^S$ at different values of $\mu_B$ and different orders in the Taylor expansion, on our ${{16}^3\!\!\times\!\!{8}}$ lattice. The strangeness chemical potential $\mu_S$ was set for each $T,\mu_B$ pair to match the strangeness neutrality condition.