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Interferometry Radii at RHIC BES Energies within the integrated HydroKinetic Model

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Pith's one-line read Pion interferometry radii at RHIC BES energies favor a crossover QCD transition over a first-order phase transition.

desk verdict A genuine prediction of HBT radii from an already-calibrated model, but the central EoS-sensitivity claim is confounded because the crossover and first-order scenarios also differ in eta/s and eps_sw. read the letter →

arxiv 2506.19101 v2 pith:IO3O5OLD submitted 2025-06-23 hep-ph nucl-th

Vladislav Naboka , Yuri Sinyukov , Musfer Adzhymambetov , Hanna Zbroszczyk This is my paper · ORCID
classification hep-phnucl-th
keywords pionfemtoscopyHBTradiiintegratedHydroKineticModelRHICBeamEnergyScanequationofstateQCDphasetransitionrelativisticheavy-ioncollisionstwo-pioncorrelations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that pion interferometry radii measured in RHIC Beam Energy Scan collisions can be reproduced by the extended integrated HydroKinetic Model (iHKMe), and that these radii are sensitive enough to the equation of state to tell a smooth crossover from a first-order phase transition. The model couples UrQMD initial conditions, a gradual relaxation stage, viscous hydrodynamics, and a hadronic cascade, and the authors compute three-dimensional Gaussian radii $R_{out}$, $R_{side}$, $R_{long}$ for 5% central Au+Au collisions at $\sqrt{s_{NN}}=7.7$–$39$ GeV. They find the crossover equation of state gives better agreement with measured data overall, while the first-order equation of state predicts a noticeably larger $R_{long}$ because the mixed-phase stage lengthens the system's lifetime; the discrepancy grows with collision energy. At 7.7 GeV the first-order scenario still gives reasonable agreement, leaving the low-energy transition type undecided.

What carries the argument

The central object is the extended integrated HydroKinetic Model (iHKMe), which joins five stages: UrQMD-generated initial conditions, a pre-equilibrium relaxation stage in which the energy-momentum tensor and baryon current interpolate between UrQMD and hydrodynamic forms through a weight function $P(\tau)$, viscous hydrodynamics with a chosen equation of state, Cooper-Frye particlization on a constant-energy-density hypersurface, and a final UrQMD hadron cascade that runs to $t=400$ fm/c. The free parameters ($R$, $\tau_0$, $\tau_{rel}$, $\tau_{th}$, $\eta/s$, $\varepsilon_{sw}$) are taken from earlier spectra calibrations, and the interferometry radii are extracted from Gaussian fits to the two-pion correlation function in the out–side–long (Bertsch–Pratt) parametrization. Comparing the crossover and first-order equations of state is what carries the argument: the first-order scenario's mixed phase lengthens the emission duration, and $R_{long}$ is the observable that registers this lengthening.

What would settle it

A decisive check is to rerun the 11.5 and 14.5 GeV cases with the crossover equation of state but the first-order scenario's viscosity and particlization density (and vice versa); if $R_{long}$ follows those parameters rather than the equation of state, the paper's EoS-sensitivity conclusion would collapse. A new measurement of interferometry radii at 14.5 GeV would also test the model's prediction at an energy now interpolated between 11.5 and 19.6 GeV.

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Extended reading notes

Core claim

The central claim is that femtoscopic radii at intermediate collision energies can discriminate between equations of state. Within iHKMe, the crossover chiral equation of state reproduces the measured $R_{out}$, $R_{side}$, and $R_{long}$ across $\sqrt{s_{NN}}=7.7$–$39$ GeV, while the first-order phase transition equation of state, calibrated separately to match particle spectra, yields $R_{long}$ values noticeably above the data at 11.5 and 14.5 GeV and above. The authors trace this excess to a prolonged mixed-phase stage that extends emission duration. They conclude that the crossover scenario describes the interferometry radii better, especially at higher BES energies, and that both scenarios remain compatible with the 7.7 GeV data.

Load-bearing premise

The claim that the equation of state drives the better fit assumes that the two scenarios' other parameter differences—shear viscosity $\eta/s$ of 0.14 vs 0.08 and the energy density $\varepsilon_{sw}$ of 0.50 vs 0.35 at which the fluid becomes a hadron cascade—do not themselves explain the $R_{long}$ difference.

Editorial extensions

If this is right

  • Above roughly 11.5 GeV, the crossover equation of state describes the measured interferometry radii better than the first-order phase transition, which overpredicts $R_{long}$.
  • At 7.7 GeV both equations of state remain consistent with data, so low-energy femtoscopy alone cannot yet rule out a first-order transition.
  • The model reproduces the observed pattern of $R_{long}$ rising with collision energy while $R_{side}$ and $R_{out}$ stay roughly flat, supporting the picture that higher energies mainly extend emission duration.
  • The successful description from 7.7 to 39 GeV indicates that iHKMe's smooth stage coupling handles long nuclear overlap times and partial thermalization adequately.
  • The trend seen in the comparison suggests that at even lower energies (around 2–4 GeV) the first-order equation of state could become the preferred description.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's EoS comparison is partly entangled with other parameter differences: the first-order sets use $\eta/s=0.08$ and $\varepsilon_{sw}=0.35$ while the crossover sets use $0.14$ and $0.50$, so a decisive test would vary the equation of state while holding these parameters fixed.
  • The paper's own trend implies a testable prediction for future low-energy collider programs: below 7.7 GeV the first-order scenario should make $R_{long}$ rise more steeply with energy, which precise low-energy femtoscopy could confirm.
  • Because no experimental 14.5 GeV data exist, the model's prediction at that energy is a direct, currently untested discriminator between the two equations of state.
  • Combining $R_{long}$ with $R_{out}$ and $R_{side}$ as a lifetime-versus-geometry diagnostic may sharpen EoS discrimination in future correlation analyses, including non-Gaussian source shapes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper applies the extended integrated HydroKinetic Model (iHKMe) to Au+Au collisions at RHIC BES energies, sqrt(s_NN) = 7.7-39 GeV, and computes the pion interferometry radii R_out, R_side, and R_long. The model combines UrQMD initial conditions, a relaxation stage, viscous hydrodynamics, particlization, and a hadronic cascade. The parameters are taken from a prior spectra-calibration study [22], so the HBT radii are genuine predictions rather than fitted quantities. Two equations of state are compared at 7.7, 11.5, and 14.5 GeV: a crossover EoS and a first-order phase-transition EoS. The central claim is that the crossover EoS describes the STAR interferometry data better, especially at higher BES energies, and that the first-order EoS produces too large R_long because of an extended mixed-phase stage; the paper also notes that the model overestimates the energy growth of R_long at high energies.

Significance. If the central claim were established, this would be a valuable result: femtoscopic radii from a realistic dynamical model, with parameters fixed by spectra, would provide independent evidence about the nature of the QCD transition at moderate baryon density. The work covers a broad energy range, uses two EoS scenarios, and reports a substantial number of events per parameter set. The main strength is that the HBT radii are computed from previously calibrated parameters rather than fitted to the femtoscopic data, giving the comparison with STAR a predictive character. However, the comparison is qualitative, and the two EoS scenarios differ in several other model parameters, so the attribution of the R_long difference to the EoS is not currently supported.

major comments (1)
  1. [Sec. III, Fig. 3; Sec. IV] The model "noticeably overestimates the increase of the long radii with energy" at 19.6, 27, and 39 GeV, yet Sec. IV summarizes the results as giving a "satisfactory description ... across the entire energy range." This acknowledged discrepancy is not analyzed in terms of its physical origin, nor is its impact on the overall conclusion discussed. The authors should report a quantitative measure for the high-energy R_long deviation and state explicitly whether this discrepancy weakens the claim that the crossover EoS scenario reproduces the femtoscopic data.
minor comments (1)
  1. [Sec. III, centrality definition] The 5% most central events are selected using 20,000 standalone UrQMD events, while the hydrodynamic simulations are a separate batch of 150 events. The possible bias from this two-step centrality selection is not quantified; a brief estimate of its effect on the extracted radii would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: HBT radii are genuine predictions computed from a full simulation chain, with model parameters taken from an earlier spectra-calibration study.

full rationale

The paper's central claim is that pion interferometry radii computed with iHKMe discriminate between a crossover and a first-order phase-transition equation of state. The radii are obtained from a multi-stage dynamical simulation (UrQMD initial conditions, relaxation, viscous hydrodynamics, particlization, and hadronic cascade) and compared with STAR data. No interferometry radius is fitted in this paper. The model parameters in Table I are taken from the authors' previous work [22], where they were calibrated using transverse-momentum spectra and compared with experimental data. That calibration is a separate observable and is externally falsifiable, so the self-citation provides independent support rather than circular input. A limitation exists: the CO and PT parameter sets differ not only in the EoS but also in eta/s, eps_sw, and sometimes tau_th, so the attribution of the R_long excess specifically to the first-order mixed phase is not uniquely isolated. However, this is a parameter-identifiability or confounding issue, not circularity: the paper does not define the EoS scenarios in terms of the HBT outcome, does not fit the radii, and does not rename a known result as a prediction. Therefore no specific circular step can be exhibited, and the circularity score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central predictions depend on five model parameters previously calibrated to particle spectra (R, tau0, tau_th, eta/s, eps_sw), all of which differ between the two EoS scenarios. No invented entities are introduced. The assumptions are standard modeling choices in heavy-ion hybrid models.

free parameters (5)
  • R (UrQMD smearing) = 0.5 fm
    Gaussian smearing of UrQMD initial conditions; calibrated to spectra in reference [22].
  • tau0 (relaxation start) = 0.6 to 2.6 fm/c depending on energy and EoS
    Starting time of pre-equilibrium relaxation; calibrated to spectra in reference [22].
  • tau_th (thermalization time) = 1.4 to 3.6 fm/c depending on energy and EoS
    End of relaxation stage; calibrated to spectra in reference [22].
  • eta/s = 0.08 (PT), 0.14 (CO)
    Shear viscosity to entropy ratio; calibrated to spectra in reference [22]; differs between EoS scenarios.
  • eps_sw (particlization energy density) = 0.35 GeV/fm3 (PT), 0.50 GeV/fm3 (CO)
    Freeze-out energy density; calibrated to spectra in reference [22]; differs between EoS scenarios.
assumptions (5)
  • domain assumption UrQMD transport model provides realistic initial conditions and evolved final-state cascade at BES energies
    Used for initial state, centrality selection, and afterburner; cited references [20,21], not independently validated here.
  • ad hoc to paper The relaxation stage ansatz P(tau) in Eq. (3) with tau_rel = tau_th - tau0 describes the approach to thermal equilibrium
    A phenomenological interpolation between UrQMD and hydrodynamics.
  • domain assumption Cooper-Frye particlization on a constant energy-density hypersurface with the Cornelius algorithm yields the correct hadron emission function
    Standard in hybrid models; assumed valid.
  • standard math The smoothness and mass-shell approximations leading to Eq. (8) are valid for pion pairs at these energies
    Standard femtoscopy approximations.
  • domain assumption Interferometry radii vary smoothly with collision energy, used to compare 14.5 GeV results with 11.5 and 19.6 GeV data
    Explicitly assumed in Section III.

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Cite this review

Pith. "Pith review of Interferometry Radii at RHIC BES Energies within the integrated HydroKinetic Model." pith.science (2026). https://pith.science/paper/IO3O5OLD

@misc{pith2026250619101,
  author       = {Pith},
  title        = {Pith review of: Interferometry Radii at RHIC BES Energies within the integrated HydroKinetic Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IO3O5OLD}},
  note         = {Machine review of arXiv:2506.19101}
}
abstract

The work is devoted to research of pion femtoscopic correlations in relativistic heavy-ion collisions across the RHIC Beam Energy Scan range using the extended integrated HydroKinetic Model (iHKMe). The model provides a comprehensive description of the system's dynamical evolution, starting from the initial collision state of colliding nuclei, passing through a possible thermalization process and hydrodynamic expansion to the hadron interacting cascade and formation of the observed particle spectra. The model smoothly couples all these stages of the matter evolution, ensuring a smooth transition between the stages. A primary focus of the current work, in contrast to the similar investigation within iHKM for high energies, concentrated in the energy region where extended nuclear overlap times and incomplete thermalization significantly influence the system's expansion comparing with very high energies. We extract the three-dimensional interferometry radii ($R_{out}$, $R_{side}$, $R_{long}$) in the region from 7.7 to 39 GeV per nucleon pair and evaluate their sensitivity to the features of the equation of state (EoS), specifically comparing crossover and first-order phase transition scenarios. The model demonstrates good agreement with experimental measurements in the crossover case. As for the first-order phase transition scenario, the $R_{long}$ component, at the optimal model parameters for particle spectra, is noticeably higher than the experimental data, and the difference is more pronounced with the growth of collision energy. Such a behavior is caused by the increased system's lifetime during the mixed-phase stage. The corresponding analysis within iHKMe below $\sqrt{s_{NN}} = 7.7$ GeV will be presented within a separate investigation.

Figures

Figures reproduced from arXiv: 2506.19101 by the authors.

Figure 1
Figure 1. Results of calculations for Rlong, Rside and Rout for π- pairs of 5% most central collisions of Au+Au at energies 7.7, 11.5 and 14.5 GeV. In these calculations, Chiral (crossover) EoS is used. The experimental data are taken from [43]. 0.25 0.3 0.35 0.4 0.45 0.5 0.55 mT , GeV 1 2 3 4 5 6 7 8 9 10 , fm long R iHKMe 7.7 GeV, PT iHKMe 11.5 GeV, PT iHKMe 14.5 GeV, PT STAR 7.7 GeV STAR 11.5 GeV STAR 19.6 GeV ), Phase tra… view at source ↗
Figure 2
Figure 2. Results of calculations for Rlong, Rside and Rout for π- pairs of 5% most central collisions of Au+Au at energies 7.7, 11.5 and 14.5 GeV. In these calculations, the first-order phase transition EoS is used. The experimental data is taken from [43] [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Results of calculations for Rlong, Rside and Rout for π- pairs of 5% most central collisions of Au+Au at energies 19.6, 27 and 39 GeV. In these calculations, Chiral (crossover) EoS is used. The experimental data are taken from [43]. We find that the crossover EoS yields a better description of the interferometry radii, especially at higher BES energies, while at lower energies, such as 7.7 GeV, the first-order phase… view at source ↗

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