QCD Matter and Multistage Collision Inference

Thermal-QCD phenomenology is an inference chain: Euclidean QCD constrains equilibrium thermodynamics; dynamical models convert fluctuating nuclear initial states into hydrodynamic fields and particles; probes sample different parts of that history; and statistical calibration asks which parameters survive correlated experimental and model uncertainties. This chapter keeps every link typed and dated so that a successful fit is not mistaken for a unique microscopic explanation.

Helpful background. The QCD perturbative domain supplies the underlying theory. Thermal partition functions supplies equilibrium observables, and inverse-problem error budgets supplies the statistical language.

The research-sensitive evidence on this chapter overview was checked through 10 August 2026. Every quoted constraint remains conditional on the cited action, dataset, model, and analysis version.

Enter this chapter

The first four pages concern equilibrium and near-equilibrium QCD:

• The QCD equation of state explains continuum thermodynamics and presents the equation-of-state provenance table.
• Crossover thermodynamics separates chiral, deconfinement, and screening diagnostics and presents the phase-and-charge provenance table.
• Conserved-charge fluctuations connects equilibrium susceptibilities to finite-acceptance collision cumulants without identifying the two.
• Dense-QCD access distinguishes asymptotically controlled cold quark matter from sign-problem-limited and model-dependent intermediate density.

The remaining pages build the collision chain:

$$\text{nuclei} \longrightarrow T^{\mu\nu},J^\mu \longrightarrow \text{pre-equilibrium} \longrightarrow \text{viscous fluid} \longrightarrow \text{hadrons} \longrightarrow \text{detector-level observables}.$$

Initial conditions and hydrodynamization define the first two interfaces. Viscous hydrodynamics, particlization, and afterburners carries the bulk evolution to stable particles. Jets, electromagnetic radiation, open heavy flavor, and quarkonium are typed probes with distinct baselines and kernels. Global Bayesian inference presents the common provenance table and explains identifiability.

Three forms of evidence

First-principles equilibrium evidence. At zero baryon chemical potential, lattice QCD can approach the physical-mass, infinite-volume, continuum theory. Its outputs still depend on an action, scale setting, interpolation, and continuum analysis. At real baryon chemical potential, the fermion sign problem sharply limits direct Monte Carlo access.

Effective dynamical evidence. Relativistic hydrodynamics is an expansion in gradients and nonhydrodynamic amplitudes. Kinetic theory and effective field theories control some matching limits. Hadronic cascades approximate dilute late dynamics. Each framework has a validity domain; none is a first-principles solution of the full real-time collision at all stages.

Phenomenological inference. A posterior such as

$$p(\theta\mid y,M) \propto p(y\mid\theta,M)\,p(\theta\mid M)$$

is conditional on model class $M$, prior $p(\theta\mid M)$, likelihood and covariance, emulator, selected data, and discrepancy model. It quantifies uncertainty inside that analysis. It is not a model-independent probability distribution for nature.

These distinctions are why current lattice calculations establish a smooth crossover near zero baryon density but do not establish a QCD critical point at larger density, and why global collision analyses constrain transport combinations without uniquely reconstructing every microscopic stage.

Matching rules

Every interface conserves the quantities the downstream theory evolves. A switching surface should carry a renormalized stress tensor and all relevant currents, together with fluctuations and correlations:

$$n_\mu T_{\rm upstream}^{\mu\nu} =n_\mu T_{\rm downstream}^{\mu\nu},\qquad n_\mu J_{\rm upstream}^{\mu} =n_\mu J_{\rm downstream}^{\mu}.$$

Equality of fluxes does not imply equality of microscopic distributions. The residual shear stress, bulk pressure, diffusion currents, and nonhydrodynamic modes determine whether the downstream truncation is adequate.

At particlization, Cooper–Frye sampling should reproduce the same fluxes in expectation. At detector comparison, theory and experiment must share centrality selection, kinematic acceptance, particle definitions, feed-down convention, and correlated uncertainties. For rare probes, the $pp$ or nuclear baseline and its covariance are part of the model, not external decoration.

Current evidence boundaries

Continuum lattice calculations determine the zero-density equation of state across the crossover and increasingly far above it; a 2025 three-flavor calculation extended nonperturbative thermodynamics from 3 to 165 GeV but is not a replacement for physical $2+1$-flavor crossover data Bresciani et al. 2025. Beam-energy-scan fluctuation measurements and finite-density lattice constraints are informative, yet as of the cutoff they do not constitute discovery of a critical point. Multistage Bayesian analyses have constrained shear/bulk-viscosity parameterizations, but alternative initial conditions, particlization corrections, and model discrepancy remain material Everett et al. 2021.

Review the chapter

1. Conditional posterior. Two collision models give different posteriors for $\eta/s(T)$ from the same measurements. Must one analysis be statistically wrong?

Solution

No. Each posterior is conditional on its model, priors, covariance, and discrepancy treatment. The difference may reveal model dependence or non-identifiability. Closure tests, posterior prediction, and explicit model comparison are needed before locating the cause.

2. Conserved matching. Why is matching only the local energy density insufficient at a pre-equilibrium-to-hydrodynamic switch?

Solution

Energy density is only one projection of $T^{\mu\nu}$. Momentum density, longitudinal/transverse stresses, shear tensor, bulk pressure, and conserved currents affect subsequent evolution. Dropping them can violate flux conservation and manufacture artificial entropy or flow.

References

• Bresciani, Matteo, Mattia Dalla Brida, Leonardo Giusti, and Michele Pepe. “QCD Equation of State with $N_f=3$ Flavors up to the Electroweak Scale.” Physical Review Letters 134, no. 20 (2025): 201904. DOI.
• Everett, D., et al. (JETSCAPE Collaboration). “Multisystem Bayesian Constraints on the Transport Coefficients of QCD Matter.” Physical Review C 103, no. 5 (2021): 054904. DOI.
• Heinz, Ulrich, and Raimond Snellings. “Collective Flow and Viscosity in Relativistic Heavy-Ion Collisions.” Annual Review of Nuclear and Particle Science 63 (2013): 123–151. DOI.
• Paquet, Jean-François. “Applications of Emulation and Bayesian Methods in Heavy-Ion Physics.” Journal of Physics G: Nuclear and Particle Physics 51, no. 10 (2024): 103001. DOI.