QCD crossover line and critical endpoint at baryon chemical potentials beyond the reach of analytic continuation ($\mu_B \gtrsim 300$ MeV)

Quantum chromodynamics (QCD) at physical quark masses has a smooth crossover, not a phase transition, at zero baryon chemical potential, and first-principles lattice simulations anchored at imaginary $\mu_B$ determine the crossover line $T_c(\mu_B)$ reliably only up to $\mu_B \approx 300$ MeV (Borsányi et al., PRL 125, 052001 (2020): $\kappa_2 = 0.0153(18)$, $\kappa_4 = 0.00032(67)$, no sign of criticality). The open problem is to determine, from first-principles lattice QCD with controlled systematics, the phase boundary of 2+1-flavor QCD at larger real baryon chemical potential: the higher-curvature coefficients of the crossover line, whether the crossover strengthens into a first-order transition, and whether a chiral critical endpoint (CEP) exists in the $(T, \mu_B)$ plane — and if so where; if not, to what region it can be excluded. As of 2026 the CEP question remains open: the strongest result (Adam et al., PRD 113, 074525 (2026)) gives, on a single lattice spacing, an 84% probability that the CEP is either below $T \approx 103$ MeV or does not exist, with method caveats explicitly stated by the authors.

Background

Lattice QCD reformulates QCD as a statistical ensemble of gauge fields on a discrete space-time grid so that thermodynamics can be computed from first principles. At nonzero baryon chemical potential $\mu_B$ the Euclidean path-integral weight (the fermion determinant) becomes complex — the 'sign problem' — so direct importance sampling at real $\mu_B$ fails. The state of the art therefore reaches real $\mu_B$ only indirectly: (a) Taylor expansion of observables in $\mu_B$ around $\mu_B = 0$, with coefficients obtained at $\mu_B = 0$; and (b) simulation at purely imaginary $\mu_B$ (where the weight is real) followed by analytic continuation — in practice an extrapolation of fits in $\mu_B^2$ to positive $\mu_B^2$, whose reliability shrinks with distance from $\mu_B = 0$. At $\mu_B = 0$ and physical quark masses the transition is an analytic crossover with no singularity, so the 'crossover temperature' $T_c$ requires a definition; the chiral susceptibility (the peak, scanned over temperature, of the derivative of the chiral condensate $\langle \bar\psi \psi \rangle$, the order parameter of spontaneous chiral-symmetry breaking in the massless limit) is a standard choice. The crossover line is parameterized as $T_c(\mu_B)/T_c(0) = 1 - \kappa_2 (\mu_B/T_c)^2 - \kappa_4 (\mu_B/T_c)^4 - \cdots$, traditionally computed along the strangeness-neutral line $n_S = 0$ relevant for heavy-ion collisions. It is widely conjectured that at larger $\mu_B$ the crossover turns into a first-order transition line; the separating second-order point is the critical endpoint (CEP), expected to lie in the 3D Ising universality class. Borsányi et al. (PRL 125, 052001 (2020)) used rooted 4stout staggered fermions ('rooting' is the procedure that constructs physical-sea-taste fermions from the fourth root of the staggered determinant, a possible source of systematics at finite $\mu_B$), with $N_t = 10, 12, 16$ continuum extrapolation, and obtained the then-most-precise $\kappa_2 = 0.0153(18)$ and $\kappa_4 = 0.00032(67)$, $T_c(0) = 158.0 \pm 0.6$ MeV, controlled extrapolation to $\mu_B \approx 300$ MeV, no sign of the transition strengthening, and no sign of criticality. The Lee–Yang edge singularity — the closest zero of the partition function in the complex $\mu_B$ plane, which at a CEP would approach the real axis with 3D-Ising scaling — and rational (Padé-type) resummations of the free energy respecting charge conjugation and Roberge–Weiss periodicity (the exact periodicity of QCD in imaginary $\mu_B/T$ with period $2\pi/3$) have become the main tools for extrapolating beyond the Taylor radius. The canonical formulation, in which fixed baryon number (not $\mu_B$) is the simulation parameter, has recently been demonstrated as a direct, extrapolation-free route.

Problem Statement

From first-principles lattice QCD with physical quark masses and controlled systematics (continuum extrapolation, finite-volume control, and cross-checks between fermion discretizations), determine the phase boundary of 2+1-flavor QCD at real baryon chemical potential beyond the currently controlled analytic-continuation regime of $\mu_B \approx 300$ MeV. Concretely: (a) determine the crossover (pseudocritical) line $T_c(\mu_B)$ — along the phenomenologically relevant strangeness-neutral line $n_S = 0$ — to larger $\mu_B$, including decisive constraints on the higher-curvature coefficients $\kappa_4$ and $\kappa_6$ that currently dominate the uncertainty at intermediate $\mu_B$; and (b) determine whether the crossover strengthens into a first-order transition at larger $\mu_B$, i.e. whether a chiral critical endpoint exists in the $(T, \mu_B)$ plane and where it lies, or, if it does not exist, establish a principled exclusion region with quantified statistical and systematic confidence. These two deliverables jointly serve one objective — establishing the nature of strongly interacting matter at finite baryon density on the QCD phase diagram — and a complete answer must address both.

References

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