Order of the QCD transition at large baryochemical potential and metastable supercooling of the early universe

The open problem is whether QCD matter at large baryochemical potential $\mu_B$ undergoes a first-order phase transition with a long-lived metastable false vacuum in the parameter range that an early, baryon-dominated universe would cross — the central untested premise of the QCD 'little inflation' scenario of Boeckel, Schettler and Schaffner-Bielich (2010–2012). At $\mu_B=0$ the transition is an established lattice crossover, and controlled finite-density lattice methods reach only $\mu_B/T \lesssim 2$–$3$; functional calculations predict a critical endpoint near $(T,\mu_B)=(107,635)$ MeV while recent lattice Lee-Yang analyses conclude the endpoint is probably absent or at very low temperature; RHIC beam-energy-scan fluctuation searches are so far null; and a 2026 model study finds the supercooling required by the scenario difficult to achieve. The order of the QCD transition in the cosmologically relevant large-$\mu_B/T$ region, and the false-vacuum lifetime there, remain open; they decide the viability of QCD-epoch primordial-black-hole production, magnetic-field seeding, nano-Hz gravitational-wave signals, and dark-matter power-spectrum modification.

Background

Quantum chromodynamics (QCD), the theory of quarks and gluons, changes its equilibrium state from a confined hadronic gas at low temperature to a deconfined quark-gluon plasma at high temperature. In the $(T,\mu_B)$ plane — temperature versus baryochemical potential $\mu_B$, the thermodynamic variable conjugate to net baryon number (the energy cost of adding baryon density) — the character of this change matters cosmologically. A smooth crossover produces no dramatic event, while a first-order transition (a discontinuous jump between two coexisting phases, with latent heat and a barrier separating them) allows the hot phase to persist as a supercooled metastable 'false vacuum' that can temporarily dominate the energy density and drive de Sitter-like exponential expansion.

The standard cosmological trajectory crosses the QCD scale ($T \approx 155$ MeV, $t \approx 10^{-5}$ s after the big bang) at $\mu_B/T \sim 10^{-9}$, and lattice QCD has established that at $\mu_B = 0$ and physical quark masses the transition is a crossover, not first order (Aoki et al., Nature 2006). Historically, a first-order cosmological QCD transition had been assumed (most famously by Witten's 1984 'cosmic separation of phases'), motivating studies of primordial black holes (PBHs), magnetic-field seeds, and gravitational waves from bubble collisions, before the lattice crossover result removed these effects from standard cosmology.

Boeckel, Schettler and Schaffner-Bielich (2010–2012) revived these cosmological consequences by moving to large $\mu_B/T$. In their 'little inflation' scenario, an early baryogenesis mechanism — e.g. Affleck–Dine baryogenesis, in which a dynamical scalar condensate generates a very large baryon asymmetry — produces a baryon-to-photon ratio of order unity, so that $\mu_B/T = \mathcal{O}(1)$. The universe then crosses the QCD phase boundary in the large-baryon-density region, where effective models of QCD (the linear sigma model with quarks, NJL-type models) suggest the transition is first order. If the universe is trapped in the metastable quark-gluon false vacuum, a few ($\sim 7$) e-folds of exponential expansion dilute the baryon and dark-matter densities by $(a_f/a_i)^3 \sim 10^9$ down to the observed baryon-to-photon ratio while $\mu_B/T$ stays constant; the transition ends by bubble nucleation, producing potentially $\sim 1 M_\odot$ PBHs, seeds of galactic magnetic fields, a stochastic gravitational-wave signal peaked near $\nu \sim 10^{-8}$ Hz (in the band of pulsar timing arrays, SKA and LISA), and a suppression of the cold-dark-matter power spectrum up to mass scales $10^6$–$10^9 M_\odot$ depending on the background composition (the exponent $q = 3$–$4$ in $\dot{H} \propto (a_i/a)^q$). The entire scenario rests on one microphysical premise, which Boeckel, Schettler and Schaffner-Bielich explicitly flag as not established: that QCD at large baryochemical potential does exhibit a first-order transition whose false vacuum is sufficiently long-lived in the parameter range a cosmological trajectory would actually cross.

Testing this premise is hard. First-principles lattice QCD at real $\mu_B$ faces the fermion sign problem — the quark determinant becomes complex, invalidating the importance sampling underlying Monte Carlo simulation — so controlled results (Taylor expansion, analytic continuation from imaginary $\mu_B$) reach only $\mu_B/T \lesssim 2$–$3$ near the $\mu_B = 0$ axis and show a crossover there. Ultrarelativistic heavy-ion collisions freeze out at similar or smaller $\mu_B/T$. The cold, dense, low-temperature region is probed only indirectly through neutron-star physics. Functional methods (functional renormalization group and Dyson–Schwinger equations, continuum analogues of lattice QCD) predict a critical endpoint (CEP — the second-order point terminating a first-order line) at $(T,\mu_B) \approx (107, 635)$ MeV with a first-order line extending to larger $\mu_B$, while competing lattice-based Lee-Yang-zero analyses question whether a CEP exists at all; no experiment has confirmed either conclusion.

Problem Statement

Determine, from first-principles nonperturbative QCD or by direct experimental or astrophysical constraint, the order of the QCD transition in the region of the $(T,\mu_B)$ plane at large baryochemical potential that a baryon-dominated early universe would traverse, and the metastability of the hot phase there. Concretely: (i) does a first-order quark-hadron transition line exist at large $\mu_B$ — is there a critical endpoint, where does the first-order region lie, and does it extend to low enough temperature that constant-$\mu_B/T$ trajectories with $\mu_B/T = \mathcal{O}(1)$–$\mathcal{O}(10)$ cross it as a genuine phase transition rather than an analytic crossover; and (ii) if it exists, is the metastable false vacuum sufficiently long-lived — is the bubble nucleation rate per Hubble volume small enough — for the universe to supercool by a scale-factor factor of order $10^3$ (several e-folds) at the QCD epoch, as required by the little-inflation dilution scenario, together with the equilibrium and nucleation inputs (latent heat, surface tension, nucleation rate) on those trajectories. An answer that establishes the converse — that no first-order line, or no adequate metastability, exists in this region — equally resolves the problem by refuting the scenario's premise.

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