Reentrant first-order deconfinement transition at high baryon density in heavy-quark lattice QCD and the connection between heavy- and light-quark first-order boundaries

In finite-temperature lattice QCD, the deconfinement transition is first order in the static heavy-quark limit at zero chemical potential and weakens to a crossover as the baryon chemical potential $\mu$ increases; effective-theory simulations (Ejiri, PoS LATTICE2024) found that at very large $\mu/T$ the plaquette (the elementary square Wilson loop of the lattice, i.e. a local gauge-energy density) distribution steepens again, suggesting a reentrant first-order region. Two coupled questions remain open: (i) whether this reappearance is a genuine feature of heavy-quark QCD once the full complex fermion determinant is included and the known systematics (Gaussian truncation of the phase, hopping-expansion convergence, fixed $N_t=6$ and finite volume) are controlled; and (ii) whether the first-order boundary of the heavy-quark region connects continuously to the first-order boundary of the light-quark (chiral) region in the quark-mass–chemical-potential plane, which would let the calculable heavy-quark critical $\mu$ constrain the critical endpoint at physical quark masses. Follow-up work in 2026 mapped the heavy-dense limit onto a three-dimensional three-state Potts model and confirmed a first-order–crossover–first-order pattern, but located the second critical point near quark saturation and conjectured it is a lattice-spacing-dependent effect unrelated to the physical critical point, leaving both questions unresolved. The problem is to establish the phase structure of heavy-quark QCD at high $\mu/T$ with controlled approximations and to determine the topology of the first-order boundary with respect to the light-quark region.

Background

Quantum chromodynamics (QCD) at finite temperature undergoes a transition from confined hadronic matter to a quark-gluon plasma, and the nature of this transition (first order versus smooth crossover) depends on the quark masses and the baryon chemical potential $\mu$ (the Lagrange multiplier fixing net baryon density). The 'Columbia plot' summarizes this in the plane of light and strange quark masses: at $\mu=0$, lattice simulations show a first-order transition in two opposite corners — near the chiral (massless) limit for three or more flavors, and in the heavy-quark corner — separated from a crossover region by critical lines whose endpoints are critical points in the three-dimensional Ising universality class (the standard class of thermal critical points with a single scalar order parameter). In the heavy-quark corner the first-order line and its critical point have been located precisely by reweighting (recasting quenched ensembles with the weights of the full determinant) and finite-size-scaling analyses (exploiting the spatial-volume dependence of observables near a critical point) (Saito et al., Phys. Rev. D 84 (2011) 054502; Ejiri et al., Phys. Rev. D 101 (2020) 054505; Ashikawa et al., Phys. Rev. D 110 (2024) 074508). Because direct simulation at finite $\mu$ is blocked by the 'sign problem' (the fermion determinant becomes complex, so its Boltzmann weight cannot be used as a probability), heavy-quark studies use an effective action built from the hopping parameter expansion (HPE) — an expansion in the hopping parameter $\kappa \simeq 1/(2am_q+8)$ (with $a$ the lattice spacing and $m_q$ the quark mass), whose leading term couples Polyakov loops (the trace of the ordered product of time-like links, an order parameter of the $Z_3$ center symmetry (the symmetry of pure-gauge QCD under center rotations of the temporal links, whose spontaneous breaking signals deconfinement), exact in the static-quark limit) to the gauge action. At finite $\mu$ the HPE coefficients alternate with the winding number of the Polyakov loop, and the effective theory remains numerically tractable because, as found by Ejiri, Kanaya and Kitazawa (PoS LATTICE2023 (2024) 174), the critical $\kappa$ decreases exponentially with $\mu$ so the first-order region narrows while the sign problem stays mild. A standard reference treatment is 'phase quenching' (dropping the complex phase of the determinant, equivalent for two degenerate flavors to an isospin chemical potential), with the phase effect then estimated by a Gaussian cumulant truncation — keeping the leading (second-order) connected cumulants of the phase distribution and neglecting cumulants of order four and higher. In this setting Ejiri (arXiv:2501.18828) found at $N_t=6$ (temporal lattice extent, setting the temperature $T=1/(a N_t)$) that the first-order transition at $\mu=0$ turns into a crossover as $\mu$ grows, but at much larger $\mu/T$ the histogram of the plaquette steepens again, and the Gaussian phase estimate strengthens this steepening, suggesting a reentrant first-order region. The phenomenological stakes come from the light-quark side: several studies expect the chiral first-order region to expand with density (though the shift of the chiral critical line measured by de Forcrand and Philipsen, JHEP 01 (2007) 077, on coarse lattices was tiny and of the opposite sign, so this expectation is itself unsettled), and if the expanding light-quark first-order region and the heavy-quark first-order boundary were continuously connected in the $(m_q,\mu)$ plane, the accurately calculable heavy-quark critical $\mu$ would constrain the critical $\mu$ at physical quark masses — a target otherwise blocked by the severe sign problem (compare Ejiri, Phys. Rev. D 77 (2008) 014508, for the general argument that finite density strengthens the transition). A 2026 development (Ejiri and Koiida, Phys. Rev. D 113 (2026) 074515, arXiv:2601.06446; proceedings version Ejiri, Koiida and Sato, PoS LATTICE2025 (2026) 107, arXiv:2601.15720) mapped the heavy-dense limit of QCD — static quarks, with the spatial links absorbed into an effective $\kappa_\mathrm{eff}$ — onto the three-dimensional three-state Potts model (a $Z_3$ spin model) with a complex external field controlled by the single parameter $C=(2\kappa)^{N_t}e^{\mu/T}$, and studied it with finite-size scaling and the tensor renormalization group (a deterministic coarse-graining method without a sign problem). It found the transition is first order at small $C$, becomes a crossover above a first critical point, and is first order again above a second critical point $C_c=1/[1.195(7)\times10^{-4}]$, both critical points being 3D Ising; the reappearance follows from an exact $C \leftrightarrow 1/C$ symmetry, since at $C\to\infty$ (quark saturation, the regime where quarks occupy essentially all lattice sites) the determinant becomes constant and the theory reduces to quenched QCD (QCD without dynamical-quark loops), which has a first-order deconfinement transition. However, the authors themselves caution that this second critical point is tied to quark saturation and lattice-scale physics and is 'not thought to be directly connected' to the critical point at physical quark masses. Thus the reappearance is so far established only within the static effective theory, not in full heavy-quark QCD, and the connectivity of the heavy- and light-quark first-order boundaries remains undetermined.

Problem Statement

Determine the finite-temperature phase structure of heavy-quark QCD at high baryon density, with controlled approximations, and the topology of its first-order boundary. Concretely: (i) establish whether the deconfinement transition that is first order at $\mu=0$ in the heavy-quark region and weakens to a crossover at moderate $\mu$ reappears as a first-order transition at very large $\mu/T$ in full heavy-quark lattice QCD — that is, including the full complex fermion determinant (or a validated beyond-Gaussian treatment of its phase, with the quantitative effect of connected cumulants of order four and higher assessed), including higher-order hopping-expansion terms nonperturbatively, and with the fixed-$N_t=6$ and finite-volume systematics controlled by studies at other temporal extents and larger volumes; and (ii) map the resulting first-order boundary in the quark-mass–chemical-potential plane and determine, positively or negatively with evidence, whether it connects continuously to the first-order boundary of the light-quark (chiral) region — the condition under which the accurately calculable heavy-quark critical $\mu$ would constrain the critical chemical potential at physical quark masses. A complete answer states the location and order of the transition line(s) across the heavy-quark region up to large $\mu/T$ with quantified systematics, and gives an evidence-backed determination of the connectivity (or disconnection, e.g. via a saturation/lattice-scale mechanism as conjectured in the effective-model literature) of the heavy- and light-quark first-order boundaries.

References

1. S. Ejiri, 'First-order phase transitions in the heavy quark region of lattice QCD at high temperatures and high densities', PoS LATTICE2024 (2025) 162, arXiv:2501.18828, doi:10.22323/1.466.0162, https://arxiv.org/abs/2501.18828
2. S. Ejiri, M. Koiida, 'Phase structure of heavy dense lattice QCD and the three-state Potts model', Phys. Rev. D 113 (2026) 074515, arXiv:2601.06446, doi:10.1103/xbd7-v9yw, https://arxiv.org/abs/2601.06446
3. S. Ejiri, M. Koiida, T. Sato, 'Phase structure of lattice QCD in the heavy quark high-density region and the three-state Potts model', PoS LATTICE2025 (2026) 107, arXiv:2601.15720, doi:10.22323/1.518.0107, https://arxiv.org/abs/2601.15720
4. S. Ejiri, K. Kanaya, M. Kitazawa, 'Chemical potential dependence of the endpoint of first-order phase transition in heavy-quark region of finite-temperature lattice QCD', PoS LATTICE2023 (2024) 174, arXiv:2311.11508, doi:10.22323/1.453.0174, https://arxiv.org/abs/2311.11508
5. R. Ashikawa, M. Kitazawa, S. Ejiri, K. Kanaya, 'High-precision analysis of the critical point in heavy-quark QCD at $N_t=6$', Phys. Rev. D 110 (2024) 074508, arXiv:2407.09156, doi:10.1103/PhysRevD.110.074508, https://arxiv.org/abs/2407.09156
6. S. Ejiri, S. Itagaki, R. Iwami, K. Kanaya, M. Kitazawa, A. Kiyohara, M. Shirogane, T. Umeda [WHOT-QCD], 'End point of the first-order phase transition of QCD in the heavy quark region by reweighting from quenched QCD', Phys. Rev. D 101 (2020) 054505, arXiv:1912.10500, doi:10.1103/PhysRevD.101.054505, https://arxiv.org/abs/1912.10500
7. H. Saito, S. Ejiri, S. Aoki, T. Hatsuda, K. Kanaya, Y. Maezawa, H. Ohno, T. Umeda, 'Phase structure of finite temperature QCD in the heavy quark region', Phys. Rev. D 84 (2011) 054502, arXiv:1106.0974, doi:10.1103/PhysRevD.84.054502, https://arxiv.org/abs/1106.0974
8. P. de Forcrand, O. Philipsen, 'The chiral critical line of $N_f=2+1$ QCD at zero and non-zero baryon density', JHEP 01 (2007) 077, arXiv:hep-lat/0607017, doi:10.1088/1126-6708/2007/01/077, https://arxiv.org/abs/hep-lat/0607017
9. S. Ejiri, 'On the existence of the critical point in finite density lattice QCD', Phys. Rev. D 77 (2008) 014508, arXiv:0706.3549, doi:10.1103/PhysRevD.77.014508, https://arxiv.org/abs/0706.3549