Mapping the chiral-spin-symmetric (stringy-fluid) band of the QCD phase diagram: the quantitative upper boundary at finite baryon chemical potential, its fate at large density, and the order of the inter-regime transitions

Above the chiral restoration crossover at $T_{ch}\approx 155$ MeV, lattice QCD reveals an intermediate regime — a 'stringy fluid' — characterized by an approximate chiral spin symmetry of spatial and temporal correlators and by hadron-like rather than partonic degrees of freedom, which is assumed, but not established, to give way smoothly to a quark-gluon plasma above an upper boundary $T_s\sim 500$ MeV at vanishing baryon chemical potential. This regime is known only qualitatively. The problem is to map its extent and establish its nature at finite baryon chemical potential $\mu_B$: (i) a quantitative lattice determination of the upper boundary $T_s(\mu_B)$ (via screening masses at imaginary $\mu_B$ with analytic continuation, or Taylor expansion in $\mu_B/T$) does not exist; (ii) it is unknown whether the band persists to large $\mu_B$ or terminates — for example by its upper and lower boundaries merging at a critical endpoint of a first-order chiral transition, itself unconfirmed; and (iii) whether the boundaries separating the hadron-resonance-gas, stringy-fluid, and quark-gluon-plasma regimes are genuine non-analytic phase transitions or only smooth crossovers has not been settled, and requires finite-size-scaling evidence. All three prongs support one objective: determining the extent and nature of the intermediate chirally symmetric but confining regime of the QCD phase diagram.

Background

Quantum chromodynamics (QCD) at finite temperature undergoes, at physical quark masses and zero baryon chemical potential $\mu_B$, a smooth crossover at $T_{ch}\approx 155$ MeV at which the approximate chiral (flavor) symmetry of the light-quark sector is restored and hadronic matter is conventionally said to become a quark-gluon plasma (QGP). A series of lattice studies starting in 2017 showed, however, that above $T_{ch}$ the thermal medium is not yet a weakly coupled partonic gas: spatial correlation functions of mesons with total spin $J=1$ become approximately degenerate in a temperature window extending to roughly $3T_{ch}$. This degeneracy signals an approximate chiral spin symmetry $SU(2)_{CS}$ — a symmetry that rotates quark spin and flavor together, which is a symmetry of the color-charge (chromoelectric) part of the QCD interaction but not of the full Dirac Lagrangian — together with a larger $SU(2N_F)$ symmetry (with $N_F$ the number of light quark flavors). Because the quark chemical potential term $\bar{\Psi}\mu\gamma_4\Psi$ in the action is manifestly invariant under these groups, the symmetry is expected to persist at finite density. The intermediate regime, in which chirally symmetric quarks are still bound into color-singlet compounds by a confining chromoelectric field, has been dubbed a 'stringy fluid'; supporting evidence includes the breakdown of thermal perturbation theory in the equation of state and in meson screening masses (the exponential decay slopes of spatial correlators, which are even functions of $\mu_B/T$ by $CP$ symmetry) at $T\approx 2\text{--}3\,T_{ch}$, and pion-like and bottomonium excitations surviving above $T_{ch}$. At $\mu_B=0$ the band is bounded below by the chiral crossover line and above by $T_s$, operationally defined for example by the bend of the vector screening masses at $T_s\sim 500$ MeV; Glozman, Philipsen and Pisarski (2022) proposed this three-regime picture and sketched the band's expected downward bending with $\mu_B$, since the chiral crossover curve is known from lattice Taylor-expansion and imaginary-$\mu_B$ calculations to decrease as $T_{ch}(\mu_B)/T_{ch}(0)=1-0.016(5)(\mu_B/T_{ch}(0))^2+\cdots$ and the same Taylor logic applied to a fixed screening-mass contour $m_V/T=C_0+C_2(\mu_B/T)^2+\cdots$ with $C_2>0$ bends the upper boundary downwards. Lattice calculations at real $\mu_B$ face the sign problem (the fermion determinant becomes complex), so finite-density results rely on simulations at imaginary $\mu_B$ followed by analytic continuation, or on Taylor expansions of limited convergence radius; whether a critical endpoint (CEP) of a first-order chiral transition exists at larger $\mu_B$ remains one of the open pillars of QCD thermodynamics. Distinguishing a genuine phase transition from a smooth crossover requires finite-size scaling — demonstrating whether a discontinuity or non-analyticity develops as the lattice volume grows toward the thermodynamic limit. Recent 2025–2026 reviews relabel the upper boundary $T_d$ (a deconfinement temperature), and new evidence from center vortices (topological objects carrying the confining flux), from large-$N_c$ scaling (the number of colors $N_c$), and from holographic (gauge–gravity duality) model analyses has made the question of the band's extent and the order of its boundaries an actively contested issue rather than a settled one.

Problem Statement

Determine the extent and nature of the approximately chiral-spin-symmetric ('stringy fluid') regime of QCD above the chiral crossover, as a function of baryon chemical potential $\mu_B$, by answering three linked questions that jointly fix the regime's place in the QCD phase diagram: (i) Boundaries: determine quantitatively how the upper boundary $T_s(\mu_B)$ — equivalently $T_d(\mu_B)$ in later notation, $T_s\sim 500$ MeV at $\mu_B=0$, e.g. at the bend of the vector screening masses — moves with $\mu_B$, stating the operational definition used, with controlled systematic uncertainties. (ii) Fate at large density: establish whether the chiral-spin-symmetric band persists to large $\mu_B$ or terminates — in particular whether its upper and lower boundaries merge at a critical endpoint of a first-order chiral phase transition, whose own existence and location are unconfirmed, or whether the band connects to other dense-matter phases (e.g. quarkyonic or inhomogeneous matter). (iii) Order of the boundaries: establish, by finite-size scaling (or an equivalently decisive non-analyticity test) of the thermodynamic limit, whether the boundaries separating the hadron-resonance-gas, stringy-fluid, and quark-gluon-plasma regimes are genuine phase transitions or smooth crossovers, at $\mu_B=0$ and, to the extent the sign problem permits, at nonzero $\mu_B$.

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