Probing of EoS with clusters and hypernuclei

The study of the nuclear equation-of-state (EoS) is a one of the primary goals of experimental and theoretical heavy-ion physics. The comparison of recent high statistics data from the STAR Collaboration with transport models provides a unique possibility to address this topic in a yet unexplored energy domain. Employing the microscopic N-body Parton-Hadron-Quantum-Molecular Dynamics (PHQMD) transport approach, which allows to describe the propagation and interactions of hadronic and partonic degrees of freedom including cluster and hyper-nucleus formation and dynamics, we investigate the influence of different EoS on bulk observables, the multiplicity, $p_T$ and rapidity distributions of protons, $Λ$s and clusters up to A=4 as well as their influence on the collective flow. We explore three different EoS: two static EoS, dubbed ‘soft’ and ‘hard’, which differ in the compressibility modulus, as well as a soft momentum dependent EoS. We find that a soft momentum dependent EoS reproduces most baryon and cluster observables, including the flow observables, quantitatively, however, hard EOS show a similar trend.

💡 Research Summary

The paper investigates the nuclear equation of state (EoS) by confronting high‑statistics STAR data from Au+Au collisions at √sₙₙ = 3 GeV with results from the Parton‑Hadron‑Quantum‑Molecular Dynamics (PHQMD) transport model. PHQMD is an N‑body approach that merges Quantum Molecular Dynamics (QMD) for baryon propagation with the Parton‑Hadron‑String‑Dynamics (PHSD) treatment of partonic and hadronic interactions, and it explicitly includes the formation and dynamics of light clusters (A ≤ 4) and hypernuclei (³ΛH, ⁴ΛH).

Three different EoS parametrizations are employed: (i) a static “soft” Skyrme‑type potential with compressibility K ≈ 200 MeV, (ii) a static “hard” version with K ≈ 380 MeV, and (iii) a “soft‑MD” potential that adds a momentum‑dependent term derived from proton‑nucleus elastic scattering data (optical potential U_opt(p)). Because experimental information on U_opt(p) is limited to p ≲ 1 GeV/c, the authors construct three extrapolations (parameterizations I, II, III) for higher momenta and test their impact on observables.

Key findings:

1. Bulk yields and spectra – The soft‑MD EoS reproduces the measured rapidity distributions, transverse‑momentum spectra, and total yields of protons, Λ hyperons, and light clusters (deuteron, triton, ³He, ⁴He) with the smallest χ². The hard EoS yields similar shapes but over‑predicts yields by roughly 10–15 % in several channels, especially at higher p_T.

2. Hypernuclei – Within the limited statistics, the soft‑MD scenario also matches the observed ³ΛH and ⁴ΛH yields, whereas the hard EoS tends to produce slightly too many hypernuclei.

3. Collective flow – Directed flow v₁ and elliptic flow v₂ are highly sensitive to the compressibility. The soft‑MD EoS gives v₁ slopes and v₂ magnitudes for both protons and Λ’s that are in quantitative agreement with STAR. The hard EoS generates a steeper v₁ and a reduced v₂, deviating from the data.

4. Momentum‑dependence sensitivity – Comparing the three high‑momentum extrapolations shows that differences affect early‑time high‑p_T particle distributions but have negligible influence on final bulk observables, because the system equilibrates by ~7 fm/c and the average field becomes dominated by low‑momentum nucleons where the potentials are constrained by data.

5. Model implementation – The authors incorporate Lorentz contraction of the incoming nuclei via modified Gaussian Wigner functions, but find that at √sₙₙ = 3 GeV relativistic corrections are minor.

Overall, the study demonstrates that a soft EoS with a realistic momentum‑dependent mean field best describes the full set of STAR observables at this beam energy, providing a tighter constraint on the nuclear EoS in the density region up to ~3 ρ₀ and temperatures around 120 MeV. The work also highlights the importance of momentum‑dependent interactions for correctly reproducing flow observables, a feature often omitted in earlier transport studies. Future measurements with higher statistics, especially of multi‑hypernuclei, will allow further refinement of the momentum dependence and the high‑density behavior of the nuclear EoS.