Continuum-limit, large-volume, and strangeness-neutral extensions of direct canonical-ensemble lattice QCD at finite baryon density with physical quark masses

A canonical-ensemble scheme recently provided the first direct, extrapolation-free lattice-QCD thermodynamics at real baryon chemical potential with physical quark masses, but only at a single finite lattice spacing ($N_t=8$), in a small volume ($16^3\times8$), at zero strangeness chemical potential, and up to $\mu_B\approx500$ MeV where statistics still tame the sign problem. The open problem is to extend this direct determination of finite-density QCD bulk thermodynamics to the regime required by heavy-ion and critical-point phenomenology: a continuum extrapolation with a numerically stable determinant construction, substantially larger physical volumes despite exponentially suppressed canonical signals, multiple conserved-charge constraints (strangeness neutrality and electric charge), and a quantified, controlled sign/overlap-problem reach in $\mu_B$ — or to establish rigorously where the direct canonical route ceases to be viable.

Background

Quantum chromodynamics (QCD) at nonzero baryon density cannot be simulated directly with standard importance sampling: after integrating out the quarks, the fermion determinant is complex at real baryon chemical potential $\mu_B$ (the complex-action or sign problem). Lattice results at finite density have therefore been indirect for decades: Taylor expansion of observables in $\mu_B$ around $\mu_B=0$ (continuum-extrapolated up to sixth order, and up to eighth and tenth order at finite lattice spacing), and analytic continuation from simulations at purely imaginary $\mu_B$. Both deliver truncated extrapolation formulae whose range of validity is hard to control, and whose high-order coefficients suffer a cancellation problem growing with volume and order. Straight reweighting from $\mu_B=0$ ensembles to real $\mu_B$ is direct but hits a sign problem and an overlap problem, and for staggered quarks (the formulation in which physical-mass thermodynamics is cheapest) the rooting procedure used to remove the taste degeneracy is ambiguous at real $\mu_B$, producing severe lattice artifacts including a spurious transition at $\mu_B\approx 3m_\pi/2$.

The canonical formulation offers a different handle: instead of fixing $\mu_B$ (grand canonical ensemble), one fixes the integer net-baryon number $N$ (canonical ensemble), where results are direct rather than extrapolated. Since a gauge configuration contributes to every $N$-sector after the quarks are integrated out, canonical partition functions $Z(N)$ are extracted by a Fourier transform of the grand-canonical determinant in an imaginary chemical potential (equivalently a Fourier-projector in the baryon-number phase angle), an idea developed by Hasenfratz and Toussaint, Alexandru et al., de Forcrand and Kratochvila, and Ejiri, in all cases before physical quark masses became reachable in this framework. A further exact limit procedure recovers the grand-canonical pressure from the canonical $Z(N)$ at non-integer density, with a known asymptotic form fixed by a Boltzmann hadron-resonance gas (the thermodynamics of an ideal gas of the known hadron species and their resonances).

Adam, Borsányi, Fodor, Guenther, Pirelli, Parotto, Pásztor, and Wong (arXiv:2604.14117) combined these ingredients with a high-statistics $16^3\times8$, $N_t=8$ data set of 2+1-flavor rooted-staggered QCD with the 4HEX action (a smeared staggered discretization) at physical quark masses generated at $\mu_B=0$, plus a partitioning of configuration space into the three sectors of the $Z_3$ center symmetry (all three sectors contribute equally to the canonical ensemble, but a $\mu_B=0$ simulation populates them very unequally), and obtained — for the first time with physical masses — direct results for the pressure, baryon density, and second-order baryon susceptibility up to $\mu_B\approx500$ MeV, with no extrapolation in $\mu_B$. The authors state the remaining limitations explicitly: the work sits at a single finite lattice spacing, the canonical signal is exponentially suppressed with volume ($Z(N{=}5)\sim10^{-7}$ on their volume, projected $Z(N{=}10)\sim10^{-14}$ on a doubled volume), only the baryon charge is treated ($\mu_S=0$), and the sign and overlap problems are merely overcome by brute-force statistics within the studied range. The same collaboration's Lattice-2025 proceedings (arXiv:2512.09415) already demonstrate a two-dimensional Fourier transform giving canonical partition functions at fixed baryon and strangeness numbers $(B,S)$ at finite lattice spacing, while Kashiwa and Kouno (arXiv:2605.19287) clarified that an apparent high-temperature inconsistency of the canonical approach (tied to the Roberge–Weiss transition, i.e. the $Z_3$-periodic structure of QCD at imaginary chemical potential) is smeared out at finite volume, validating the approach's use in lattice simulations.

Problem Statement

Establish direct, extrapolation-free (in $\mu_B$) lattice-QCD bulk thermodynamics at finite baryon density with physical quark masses in the regime that heavy-ion and critical-point phenomenology actually requires. The source scheme (Adam et al., arXiv:2604.14117) achieves this only at one finite lattice spacing, one small volume, zero strangeness chemical potential, and $\mu_B\lesssim500$ MeV. The problem is to extend it — or an equivalently direct canonical-route realization — along the four jointly required, scientifically linked directions the authors themselves identify as open:

(i) Continuum limit: perform the canonical computation at finer lattice spacings ($N_t\ge12$, with at least three lattice spacings in total so that a controlled continuum extrapolation can be formed) and carry out that extrapolation, which requires a numerically stable and affordable construction of the canonical (eigenvalue-based) fermion determinant, whose current cost scales as $\sim N_t^9$ and whose stability at $N_t=12$ is unproven (possibly requiring stochastic estimators).

(ii) Larger physical volumes: push beyond $16^3\times8$, reaching at least the doubled source volume ($20^3\times8$) and ideally larger, toward volumes where chiral observables and heavy-ion thermodynamics are meaningful, despite the exponentially suppressed canonical signal ($Z(N)\sim10^{-7}$ at $N=5$ on the current volume, projected $\sim10^{-14}$ at $N=10$ on a doubled volume), e.g. via density-of-states-type sampling (sampling driven by a global constraint such as net baryon number) or an equivalent enrichment of the sampled high-density configuration space, and determine whether sufficient statistics are reachable at all.

(iii) Multiple conserved charges: incorporate strangeness neutrality ($n_S=0$, with the induced $\mu_S(\mu_B)$) and the electric-charge constraint ($n_Q/n_B$ fixed at the heavy-ion value $n_Q/n_B\approx0.4$) into the canonical-to-grand-canonical reconstruction — building on the demonstrated $(B,S)$-canonical two-dimensional Fourier transform but completing it with the physical neutrality constraints and the third charge. An optional auxiliary methodological extension, not required for (iii), is to generalize the configuration-space partitioning so the method can also use ensembles simulated directly at imaginary chemical potential.

(iv) Sign/overlap reach: quantify and control the sign and overlap problems as functions of volume and $\mu_B$, establishing the density range over which the direct determination remains viable beyond the current $\mu_B\approx500$ MeV.

The single overarching objective is a continuum-extrapolated, strangeness-neutral, extrapolation-free QCD equation of state at real baryon chemical potential with quantified statistical and systematic errors; items (i)–(iv) are the interlocking components that jointly deliver it, and none of them is optional. The task is resolved in exactly one of two ways: (a) by achieving (i)–(iv) jointly in the target regime above, or (b) by a rigorous, quantitative demonstration that the direct canonical route cannot attain the overarching objective — a no-go that identifies the failing direction (volume, lattice spacing, or $\mu_B$ reach) and establishes, at a stated physical volume, lattice spacing, and statistics budget, that the corresponding canonical signal or its systematic control is unattainable there, so that the boundary of viability is fixed quantitatively rather than merely inferred from the fact that the required statistics grow rapidly. Positive progress on a proper subset of (i)–(iv), or an unquantified assertion of unattainability that does not meet the quantitative bar of (b), does not resolve the task. The direct (non-extrapolated in $\mu_B$) character of the computation must be preserved: answers that return to Taylor-type or analytic-continuation extrapolation in $\mu_B$ do not address this problem.

References

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