Constraining the Phase-Transition EoS using the Energy Dependence of Directed Flow

We propose a hybrid equation of state (VDF+MIT EoS) to describe the hadron-quark phase transition in dense nuclear matter. By coupling this EoS with the AMPT-HC transport model and comparing to recent experimental data on proton and $Λ$ directed flow $v_1$, we constrain the transition to likely occur near $5ρ_0$–$6ρ_0$, ruling out transitions below $3ρ_0$. Furthermore, we introduce the energy derivative of the mid-rapidity $v_1$ slope, $d(dv_1/dy)/d(\sqrt{s_{NN}})$, as a weakly model-dependent observable. Its zero crossing provides a direct signature of the phase transition critical point, offering a new tool for mapping the QCD phase diagram in future experiments.

💡 Research Summary

The authors develop a hybrid equation of state (EoS) that combines a relativistic vector‑density‑functional (VDF) description of hadronic matter with the MIT bag model for deconfined quark matter. Three VDF parameter sets (VDF1, VDF2, VDF3) are constructed, each reproducing normal nuclear saturation properties but differing in high‑density stiffness (incompressibility K₀ ranging from ~260 to ~360 MeV). To avoid causality violations at very high density, the MIT bag model (massless u and d quarks, αₛ = 0.1) is grafted onto the VDF pressure curve at a density where the spinodal region of the QCD transition begins. By adjusting the bag constant B, a smooth transition from the hadronic branch to the quark branch is achieved, yielding three hybrid EoSs: VDF1+MIT, VDF2+MIT, and VDF3+MIT.

These hybrid EoSs are embedded in the AMPT‑HC transport framework, which simulates heavy‑ion collisions using only hadronic cascades (no partonic stage). The model initializes Au+Au collisions at beam energies √sₙₙ ≈ 1–10 GeV, evolves nucleons, resonances, strange hadrons, and their mean‑field potentials, and records the maximum baryon density reached during the compression phase. The simulations show that the peak density ρ_max reaches 5–7 ρ₀, precisely the region where VDF2+MIT and VDF3+MIT predict the hadron‑to‑quark transition. VDF1+MIT, whose transition occurs at lower density, fails to reproduce the observed dynamics.

The hybrid EoSs are confronted with experimental directed‑flow data (v₁) for protons and Λ hyperons measured in the Beam Energy Scan (BES) program. The slope of v₁ at mid‑rapidity, dv₁/dy, is highly sensitive to the stiffness of the EoS. VDF3+MIT reproduces the characteristic dip in dv₁/dy around √sₙₙ ≈ 4–7 GeV, indicating a softening of the EoS consistent with a phase transition near 5–6 ρ₀. VDF2+MIT yields a moderate agreement, while VDF1+MIT is incompatible with the data. Consequently, the authors rule out a transition occurring below 3 ρ₀.

Astrophysical constraints are also examined. Solving the Tolman‑Oppenheimer‑Volkoff equations with each hybrid EoS produces mass‑radius (M‑R) curves. VDF3+MIT satisfies the NICER/XMM‑Newton measurements of PSR J0030+0451 and PSR J0740+6620, as well as the GW190425 binary‑neutron‑star merger constraints. The predicted maximum mass is ≈ 2.1 M⊙, comfortably below the theoretical upper bound (~2.3 M⊙). Small residual tension between the calculated maximum mass and the theoretical limit is discussed as a possible hint of exotic components (e.g., dark matter admixture) or strong‑gravity effects, but the authors emphasize that this does not undermine the main conclusion regarding the phase transition.

A novel observable is proposed: the energy derivative of the directed‑flow slope, d(dv₁/dy)/d(√sₙₙ). The simulations reveal a zero‑crossing of this quantity precisely when the system traverses the softening region of the EoS. Because the derivative emphasizes the curvature of dv₁/dy versus √sₙₙ, it is less sensitive to model‑specific details (e.g., initial‑state fluctuations) and more directly reflects the underlying thermodynamic response. The authors argue that measuring this derivative across a fine beam‑energy scan (as planned for BES‑II, HIAF, FAIR) would provide a robust, quasi‑model‑independent signature of the QCD critical point and the first‑order phase boundary.

In summary, the paper delivers three major contributions: (1) a physically consistent hybrid VDF+MIT EoS that respects causality and matches both nuclear and astrophysical constraints; (2) a systematic transport‑model study linking high‑density compression in low‑energy heavy‑ion collisions to directed‑flow observables, thereby pinpointing the hadron‑quark transition around 5–6 ρ₀; and (3) the introduction of the derivative observable d(dv₁/dy)/d(√sₙₙ) as a weakly model‑dependent, experimentally accessible probe of the QCD phase transition. This work thus advances the methodology for mapping the QCD phase diagram and offers concrete guidance for upcoming experimental programs.