The QCD Critical Point

The question is: What evidence constrains the existence and location of a QCD critical point, and what result would materially settle the present ambiguity? It matters because a critical endpoint would organize the finite-temperature, finite-baryon-density phase diagram of quantum chromodynamics (QCD), fix a new universality regime, and leave correlated signatures in heavy-ion collisions. This assessment concerns physical $2+1$-flavor QCD in thermal equilibrium and the dynamical fireballs used to probe it; it does not treat a critical point inserted by construction into an effective equation of state as evidence that QCD has one.

Evidence cutoff. 11 August 2026.

Required background. QCD Phase Structure and Crossover Thermodynamics fixes the thermodynamic limits and the established crossover at small baryon chemical potential. Conserved-Charge Fluctuations and Criticality supplies the susceptibility and cumulant observables used below.

Helpful background. Reweighting, Taylor Expansion, and Imaginary Density explains why analytic continuation has a bounded reach; Dynamic Scaling and Critical Slowing Down connects equilibrium singularities to finite-time collision signals; and Dense QCD Regimes, Access, and Evidence Boundaries separates sign-problem-free information from model-dependent extrapolation.

The endpoint that QCD and collisions must identify together

A QCD critical point is the second-order endpoint of a first-order transition line in the $(T,\mu_B)$ plane. In the usual scenario its long-distance static behavior is in the three-dimensional Ising universality class. Neither that scenario nor the endpoint follows from the observed crossover at $\mu_B=0$.

Coordinate	Included here	Not answered here
Theory	QCD with physical light and strange quark masses	Two-color QCD, large-isospin QCD, or effective models as substitutes for physical QCD
Equilibrium	Singularities of the pressure and conserved-charge susceptibilities at real $\mu_B$	A model equation of state in which an endpoint was chosen as an input
Experiment	Au+Au fluctuation data from the Relativistic Heavy Ion Collider (RHIC), with acceptance and dynamical evolution explicit	Equating net-proton cumulants directly with grand-canonical net-baryon susceptibilities
Location	A point quoted with a method domain and uncertainty region	A single best-fit coordinate assembled from mutually dependent extrapolations

The relevant map is not one-to-one. A collision follows a finite, inhomogeneous trajectory; baryon number is only partially observed through protons; conservation laws, volume fluctuations, hadronic rescattering, detector efficiency, and critical slowing down all reshape cumulants. Conversely, a smooth cumulant curve over a finite beam-energy interval cannot exclude a small or inaccessible critical region.

Direct assessment

Regime	Status at the evidence cutoff	Basis and limit
$\mu_B=0$ and small real $\mu_B$	Partially resolved: a crossover and its leading curvature are established	Continuum-extrapolated lattice calculations with physical masses determine a crossover near $156$ MeV and extend it perturbatively away from zero density; they do not establish the nearest real-axis singularity. HotQCD 2019
Moderate $\mu_B/T$ reached by expansions	Constrained, not resolved	Resummed and alternative Taylor schemes produce controlled thermodynamics over tested intervals, but truncation and analytic-continuation errors grow before the most interesting high-density region. Borsányi et al. 2021
RHIC Beam Energy Scan	Constrained and still ambiguous	High-order proton cumulants show nontrivial energy dependence, but current precision and noncritical baselines do not select a unique critical interpretation. The 2026 BES-II fifth- and sixth-order result reports no sign-alternating two-component pattern within its uncertainties. STAR 2026
Physical QCD at larger real $\mu_B$	Open	The fermion sign problem prevents direct importance-sampling lattice calculations in the target region; functional and effective descriptions are informative but not theorem-level determinations of QCD.

Scope-qualified conclusion. No included result demonstrates that physical QCD has a critical point, and no included result excludes one throughout the experimentally and astrophysically relevant phase diagram. The existence question is open; the location question is therefore not yet well posed as a parameter-estimation problem.

Facts, assumptions, and interpretations

Kind	Statement	Evidential weight
Established fact	At physical masses and $\mu_B=0$, thermal QCD undergoes a crossover rather than a first-order transition.	Direct continuum lattice result.
Established fact	Ordinary finite-density importance sampling has an exponentially severe complex-weight problem.	A method obstruction, not evidence for either existence or absence.
Established fact	BES-II measures proton rather than total baryon cumulants in finite kinematic acceptance.	An essential qualification on the comparison with equilibrium susceptibilities.
Assumption	The endpoint, if present, has a sufficiently large Ising scaling region and the collision trajectory passes close enough to it.	Required to turn universal critical scaling into an observable signal.
Assumption	Nonequilibrium evolution, particlization, and hadronic transport can be modeled accurately enough to propagate critical fluctuations.	Present implementations remain model dependent.
Interpretation	A nonmonotonic cumulant ratio is a critical signature.	Plausible only after correlated noncritical baselines and analysis choices fail.

Lee–Yang-edge analyses are a promising bridge because a critical point is a singularity approaching the real chemical-potential axis. Current extractions use finite lattice orders, Padé choices, imaginary-density data, and scaling ansätze; the approach of a fitted singularity is therefore qualifying evidence, not a direct observation. Clarke et al. 2024

Competing positions and discriminating evidence

Position in its strongest form	Evidence for	Evidence against or qualifying it
A critical point lies in the RHIC-accessible region.	BES-I and BES-II cumulant structures motivate targeted searches; Ising-mapped equations of state can reproduce characteristic nonmonotonic patterns.	Those equations of state assume a critical point. Conservation, stopping, volume, acceptance, and hadronic dynamics can also generate structure; current high-order BES-II data are not a decisive universal pattern.
A point exists, but at larger $\mu_B$ or lower $T$ than present collisions cleanly access.	Small-density lattice results neither locate nor exclude a more distant singularity; analytic-continuation radii are finite.	It is difficult to falsify without extending first-principles reach or a new experimental facility, so this is a viable possibility rather than positive evidence.
Physical QCD has no critical endpoint.	Smooth thermodynamics throughout every controlled small-density domain and the absence of a unique experimental signature are compatible with no endpoint.	A finite smooth domain cannot prove analyticity over the whole real-density axis.

Model, functional-renormalization-group, and holographic calculations are useful for response patterns and dynamical tests. Their agreement on a coordinate is not independent QCD evidence when calibrations, universality maps, or lattice inputs are shared.

Obstructions and uncertainty

• Analytic reach: finite Taylor order and continuation from imaginary $\mu_B$ cannot distinguish a remote real singularity from complex singularities without stability under order, volume, lattice spacing, and ansatz changes.
• Finite-size and finite-time suppression: a collision cannot develop an infinite correlation length; critical slowing down limits the growth and can preserve memory of the trajectory.
• Observable translation: efficiency-corrected proton cumulants require a covariance-aware map through baryon conservation, isospin randomization, resonance decays, and acceptance.
• Dependence: several phenomenological studies reuse the same STAR data, lattice coefficients, hydrodynamic code families, and Ising equation-of-state parametrizations. They are not independent confirmations.

What would materially resolve the ambiguity?

A persuasive existence result would be either:

1. a controlled first-principles calculation at real $\mu_B$ showing a thermodynamic singularity after continuum and infinite-volume limits, with the expected scaling and independent reproduction; or
2. a pre-registered, covariance-preserving experimental pattern across several conserved-charge cumulants, acceptances, centralities, and beam energies that one dynamical critical model predicts jointly and validated noncritical baselines cannot reproduce.

A persuasive exclusion would require a sign-problem solution or convergent analytic control proving analyticity across a declared real-$\mu_B$ region, together with collision limits excluding every remaining accessible critical region. A narrower result—such as a robust zero-free domain—is already valuable if its domain is explicit.

Connected methods and tests

• Thermal and Nonequilibrium Field Theory organizes the equilibrium-to-collision inference chain.
• Lattice and Hamiltonian Field Theory explains continuum and volume control; Euclidean Lattice Inference and Continuum Extrapolation compares its error budget.
• Functional Equations and Functional Renormalization Group assesses truncation-dependent access beyond small density.
• Real-Time QFT, Kinetic Theory, and Hydrodynamics treats propagation of fluctuations through an expanding medium.
• Finite-Temperature QCD Equation of State isolates the small-density thermodynamic evidence.
• Finite-order continuation should be stress-tested on mock functions with known complex singularities while varying expansion order, resummation, and conformal or Padé ansatz. Stochastic critical-dynamics studies should vary expansion rate, relaxation time, conservation laws, and acceptance cuts to quantify finite-time distortion. These checks calibrate inference procedures; they do not prove that QCD contains a critical endpoint.

Evidence boundary

This assessment uses the finite source set below, selected by targeted searches of arXiv, INSPIRE, APS, journal records, and official STAR records for continuum lattice thermodynamics, finite-density singularity searches, and high-order BES data. Public results through 11 August 2026 were eligible. Conference-only forecasts, model coordinates with an endpoint inserted as input, and inaccessible claims were not used to set status; the selection is not exhaustive.

References

• Borsányi, S., et al. (2021). “Lattice QCD Equation of State at Finite Chemical Potential from an Alternative Expansion Scheme.” Physical Review Letters 126, 232001. DOI.
• Clarke, D. A., Dimopoulos, P., Di Renzo, F., Goswami, J., Schmidt, C., Singh, S., and Zambello, K. (2024). “Searching for the QCD Critical Point Using Lee–Yang Edge Singularities.” arXiv:2401.08820.
• HotQCD Collaboration (Bazavov, A., et al.) (2019). “Chiral Crossover in QCD at Zero and Non-Zero Chemical Potentials.” Physics Letters B 795, 15–21. DOI.
• STAR Collaboration (2026). “Measurement of Fifth- and Sixth-Order Fluctuations of (Net-)Proton Number in Au+Au Collisions from Phase II of the Beam Energy Scan Program at RHIC.” Physical Review C 113, L051901. DOI; arXiv:2512.19352.