Determining the Duration of the Hadronic Stage at RHIC-BES Energies via Resonance Suppression Using a Full Set of Rate Equations

Amine Chabane; Apiwit Kittiratpattana; Carsten Greiner; Marcus Bleicher; Tim Neidig; Tom Reichert

REVIEW 3 major objections 3 minor 44 references

Determining the Duration of the Hadronic Stage at RHIC-BES Energies via Resonance Suppression Using a Full Set of Rate Equations

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The hadronic stage in heavy-ion collisions lasts 2-4 times longer than STAR's estimate because resonance regeneration makes the K*/K ratio fall linearly rather than exponentially.

desk verdict Regeneration makes K*/K decay roughly linear and implies a 2-4x longer hadronic stage, but Eq. (12)'s universal slope is asserted, not demonstrated, and the advertised collision-loss term is absent from the solved network. read the letter →

arxiv 2501.03893 v1 pith:UG4NACT7 submitted 2025-01-07 nucl-th hep-ph

Tim Neidig , Apiwit Kittiratpattana , Tom Reichert , Amine Chabane , Carsten Greiner , Marcus Bleicher This is my paper · ORCID

classification nucl-thhep-ph

keywords hadronicstagedurationresonancesuppressionK*/Kratiorateequationspartialchemicalequilibriumheavy-ioncollisionsRHICbeamenergyscanregeneration

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 argues that the duration of the hadronic stage in central Au+Au collisions at RHIC-BES energies has been substantially underestimated. By running a full set of coupled rate equations that include both resonance decay and regeneration, the authors find that the $K^*/K$ ratio falls essentially linearly in time, not exponentially as previously assumed. The slope is nearly energy-independent, $d(K^*/K)/dt \approx -0.02~\mathrm{fm}^{-1}$, giving a direct formula for the hadron phase lifetime from the ratio difference between chemical and kinetic freeze-out. The resulting lifetimes are 2-4 times larger than STAR's estimates and fall in line with independent methods.

What carries the argument

The machinery is a network of 23 coupled rate equations for hadron species in partial chemical equilibrium, where each resonance's multiplicity evolves under detailed balance between its decay $h^* \to h + X$ and its regeneration $h + X \to h^*$. Equilibrium multiplicities are fixed at chemical freeze-out using a standard parametrization of $T_{\rm ch}$ and $\mu_{B,\rm ch}$, and the fireball expansion enters through a parametrized volume $V(t)$ with conserved entropy, baryon number, and strangeness. The load-bearing output is the nearly energy-independent slope of the $K^*/K$ ratio versus time, which converts a measured ratio difference into a hadron phase lifetime through Eq. (12).

What would settle it

Measure the $K^*/K$ ratio in a centrality or system where the hadron phase duration is independently known (e.g., from two-particle HBT radii) and check whether Eq. (12) with the universal slope reproduces that duration; a systematic mismatch beyond the stated uncertainties would show that the slope is not universal.

Watch

Extended reading notes

Core claim

The central finding is that the time evolution of the $K^*/K$ ratio after chemical freeze-out is controlled by a counterbalance between resonance decay and regeneration, and this balance makes the ratio decrease approximately linearly at a slope that is essentially independent of collision energy. Because regeneration compensates for decays, the ratio does not follow the simple exponential decay law used in earlier estimates. The difference between the $K^*/K$ ratio at chemical and kinetic freeze-out, divided by the universal slope of about $0.02~\mathrm{fm}^{-1}$, yields the lifetime of the hadronic stage, which the authors find to be 2-4 times larger than the STAR collaboration's previous extraction.

Load-bearing premise

The slope of about $-0.02~\mathrm{fm}^{-1}$ is extracted for central Au+Au collisions and then used as a universal constant for all centralities and energies, with no demonstration that it is centrality-independent and no estimate of its uncertainty.

Editorial extensions

If this is right

The hadronic phase at RHIC-BES energies is roughly 2-4 times longer than the STAR collaboration's estimate, resolving the tension between resonance suppression and other lifetime probes.
Analyses that treat resonance suppression as a pure exponential decay will systematically underestimate hadron phase durations, so the linear formula provides a corrected tool for future measurements.
The energy independence of the slope means a single constant can be used across the beam-energy scan, simplifying extraction of lifetimes at NA61, STAR, and FAIR.
The agreement with independent 4-8 fm/c estimates suggests that the rate-equation network captures the dominant physics of the late hadronic stage, at least for the $K^*$ channel.

Reading between the lines

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

The same detailed-balance mechanism should apply to other resonance ratios such as $\rho/\pi$ or $\Delta/N$; if those ratios also evolve linearly, the universal-slope method could be extended to multi-resonance consistency checks.
One could test the universality claim directly by comparing central and peripheral collisions: if the slope changes with system size or centrality, Eq. (12) would need a centrality-dependent slope.
Because the slope is anchored to the assumed volume expansion law and the chemical freeze-out time from UrQMD, a direct measurement of the time dependence of $K^*/K$ (for instance through femtoscopy) could validate the linear evolution independently.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper makes a real correction to the standard resonance-suppression estimate. STAR's exponential-decay formula misses regeneration; with the rate-equation network the K*/K ratio falls approximately linearly and the extracted hadronic stage is 2-4 times longer, bringing it into line with HBT/transport estimates. The qualitative point survives.

The new result is not the rate-equation framework itself — that is from the authors' 2022 paper — but the application to RHIC-BES energies, the linear-decrease finding, and the improved lifetime relation Eq. (12). Fig. 1's reproduction of the STAR centrality and energy dependence is a genuine benchmark, and the authors are clear that regeneration counteracts decay.

Soft spots. First, the stress-test note is correct: Eq. (2) advertises a collision loss term L_coll, but the equations actually solved, Eq. (4) plus the 23-species network, contain only the decay/regeneration channel h* ⇌ X + h. K* + N or K* + π inelastic dissociation is not in the network. Missing loss would make the slope more negative than -0.02 fm^-1, so the claimed factor 2-4 is an upper bound, not a robust number. This is a real omission.

Second, the universal slope is asserted with no uncertainty and no centrality check. It is read off central collisions in Fig. 3 and applied to all centralities via Eq. (12), even though the volume expansion, chemical freeze-out time, and V_ch parametrization all enter. The circularity concern also has bite: the model is benchmarked against STAR K*/K and then used to extract the lifetimes from the same data. Not fatal, but it means the 2-4x number is not an independent measurement.

Third, the 'now in line with other estimates' comparison cites several papers with overlapping authors; not disqualifying, but the agreement is less independent than it looks. And no code or data are released, so sensitivity analysis is impossible.

Who is this for? Practitioners analyzing K*/K at RHIC-BES, NA61, or FAIR. They should read it, and they should not take 0.02 fm^-1 as a universal constant until the collisional loss and centrality dependence are quantified. I would send it to a serious referee: the linear-regeneration point is important enough to put on record, and the missing L_coll needs to be either implemented or explicitly argued away. I would not desk-reject it.

Referee Report

3 major / 3 minor

Summary. The manuscript studies the time evolution of the K*/K ratio during the hadronic phase in Au+Au collisions at RHIC-BES energies using a network of 23 coupled rate equations with partial chemical equilibrium and detailed balance. The authors report that the K*/K ratio decreases approximately linearly rather than exponentially, because regeneration counteracts decay, and they extract an essentially energy-independent slope d(K*/K)/dt ≈ -0.02 fm^-1 (Section IV, Fig. 3). They propose an improved relation, Eq. (12), between the K*/K suppression and the hadronic-phase lifetime, and apply it to STAR data to obtain lifetimes that are 2-4 times larger than the STAR estimates based on the exponential formula in Eq. (1).

Significance. If the central claims hold, the paper offers a more realistic method for extracting the hadronic-stage duration from resonance suppression and identifies a systematic bias in previous exponential-decay estimates. The explicit rate-equation framework with detailed balance, the reproduction of centrality and energy dependence of the STAR K*/K data in Fig. 1, and the clear linear behavior shown in Fig. 3 are genuine strengths. The proposed relation Eq. (12) is simple and directly usable by STAR, NA61, and future FAIR experiments. However, the quantitative validity of the result depends on the completeness of the loss terms and on the universality of the slope, both of which need additional support before the factor-2-4 statement can be considered robust.

major comments (3)

[Section II, Eqs. (2) and (4)] The loss rate due to collisions, L_coll, announced in Eq. (2) is never implemented in the equations actually solved. The evolution equation Eq. (4) contains only the decay/regeneration channel h* ⇌ X + h, and no collisional dissociation term is defined or discussed in Section II. In the dense hadronic medium at RHIC-BES energies, channels such as K* + N → K + N or K* + π → K_1 → ... would add an extra loss mechanism that makes d(K*/K)/dt more negative than -0.02 fm^-1. Since Eq. (12) divides the measured suppression by this slope, omitting collisional losses could make the inferred Δt_hadronic systematically too large, and the claimed factor of 2-4 relative to STAR would rest on an unverified completeness assumption. The authors should either implement L_coll in the network or provide a quantitative estimate of its magnitude and demonstrate that it is negligible for the slope.
[Section IV, Eq. (12), Figs. 3 and 4] The slope s = d(K*/K)/dt ≈ -0.02 fm^-1 is extracted from central Au+Au collisions in Fig. 3, but Eq. (12) is then applied to all centralities and energies in Fig. 4 without demonstrating that the slope is centrality independent. The slope also depends on the volume parametrization Vch in Eq. (9), the chemical freeze-out time tch taken from UrQMD, and the transverse expansion time t_perp in Eq. (11); none of these dependencies is quantified. The paper provides no uncertainty on s and no sensitivity analysis, although any change in s linearly rescales all extracted lifetimes. The statement in Section II that t_perp has only minor influence is not supported by a calculation. Please show the slope as a function of centrality, provide an uncertainty band for s, and propagate that band into Fig. 4.
[Section III, Fig. 1] The benchmarking against STAR data in Fig. 1 does not independently validate the slope used in Eq. (12). The kinetic freeze-out point in Fig. 3 is read off when the model's K*/K matches the same STAR data that enter the numerator of Eq. (12), and the slope is taken from the same model run. This consistency is necessary but not sufficient to certify the slope as a universal constant. An independent cross-check, such as comparing the model's K*/K evolution against a transport-code calculation with controlled resonance processes, would substantially strengthen the claim that the slope is a reliable input for lifetime extraction.

minor comments (3)

[Fig. 1 caption] The caption states that filled symbols with bands denote the rate-equation calculations, but the meaning of the band is not defined in the text; please specify whether it reflects numerical uncertainty, centrality-bin width, or a model variation.
[Fig. 4] The vertical axis extends to negative values and some STAR points appear at negative hadronic lifetimes, but the text does not explain how a negative duration is to be interpreted; please address this for clarity.
[Section II] The paper refers to a network of 23 rate equations and to 'the specific rate equations described in [24]', but the included species and the relevant decay/regeneration channels are not itemized; a table listing the 23 species and their principal channels would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the slope in Eq. (12) is a model output, not a fit to the target lifetimes.

full rationale

This paper's central relation (Eq. 12) converts the measured K*/K suppression into a hadronic-phase duration using a slope d(K*/K)/dt ≈ -0.02 fm^-1 that is read off the rate-equation time evolution in Fig. 3. This is a model output, not a fitted parameter: the rate equations are stated in Section II with decay widths from PDG [37], initial conditions from the Cleymans parametrization [38], the volume from Pan-Pratt [40], and tch from UrQMD [41]; no parameter is adjusted to reproduce the STAR K*/K data. The STAR data are used only to identify the kinetic freeze-out time (circles in Fig. 3) and as the numerator in Eq. 12. The linear time evolution and the value of the slope are consequences of the regeneration term in Eq. (4), independent of the final lifetime values. The self-citations to [24] (rate-equation framework), [41] (UrQMD tch), and [10,42,43] (comparison estimates) provide modeling inputs or external benchmarks rather than a uniqueness theorem or a forced conclusion. The omission of Lcoll from Eq. (2) is a completeness/sensitivity concern, not a circular reduction. Therefore no load-bearing argument reduces to its own inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central result depends on a chain of model choices inherited from previous papers (PCE, rate equations, volume parametrization) plus a newly introduced universal slope. The main free parameters are the chemical freeze-out volume parametrization, t_ch, t_perp, and the effective slope. No invented entities.

free parameters (4)

Vch(x) coefficients = 2351.7, -10009.5, 13878.4, exponent -1.36
Volume at chemical freeze-out as a function of x = ln(sqrt(s_NN)/GeV), Eq. (9), fitted to the data compilation of [39]. This parametrization sets the initial volume and feeds into all rate equations.
t_ch (chemical freeze-out time) = Not quoted, taken from UrQMD [41]
Sets the start time for the hadronic stage; the paper states t_ch is fixed based on previous UrQMD calculations but does not give the value or the sensitivity.
t_perp (transverse expansion time) = 6.5 fm/c
Parameter in the volume expansion formula Eq. (11), kept constant; the paper claims minor influence but shows no sensitivity study.
Linear slope s = d(K*/K)/dt = 0.02 fm^-1
Read off the model time evolution in Fig. 3 and used as a universal constant in Eq. (12) to convert measured K*/K differences into hadronic lifetimes; no uncertainty or centrality dependence provided.

assumptions (6)

domain assumption Partial chemical equilibrium: total yields of stable hadrons are fixed by non-equilibrium chemical potentials, while resonances evolve by decays and regeneration.
Invoked in Section II, framework follows [28-31]. Underlies the rate equations and the initial conditions.
standard math Detailed balance: the cross section for h + X -> h* and the decay width of h* -> h + X are related by Eq. (6), with equilibrium multiplicities.
Eq. (6) in Section II; necessary to close the rate equations.
domain assumption Decay widths and cross sections are taken from Particle Data Tables and are thermally averaged.
Section II, used in the rate equations.
domain assumption Volume expansion follows V(t) = V_ch (t/t_ch) (t_perp^2 + t^2)/(t_perp^2 + t_ch^2) from Pan and Pratt [40].
Eq. (11), Section II; the time evolution of densities depends on this.
domain assumption Chemical freeze-out conditions are given by the Cleymans parametrizations, Eqs. (7) and (8), and the centrality scaling of Vch with charged particle multiplicity.
Section II, from [38,39]; sets initial T, mu_B, mu_S, V.
ad hoc to paper The slope d(K*/K)/dt is essentially independent of collision energy and centrality, so a single value can be used in Eq. (12).
Section IV; stated without a dedicated centrality test, and used to extract lifetimes for all N_part in Fig. 4.

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Pith. "Pith review of Determining the Duration of the Hadronic Stage at RHIC-BES Energies via Resonance Suppression Using a Full Set of Rate Equations." pith.science (2026). https://pith.science/paper/UG4NACT7

@misc{pith2026250103893,
  author       = {Pith},
  title        = {Pith review of: Determining the Duration of the Hadronic Stage at RHIC-BES Energies via Resonance Suppression Using a Full Set of Rate Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UG4NACT7}},
  note         = {Machine review of arXiv:2501.03893}
}

abstract

We present realistic estimates for the duration of the hadronic stage in central Au+Au reactions in the RHIC-BES energy regime. To this aim, we employ a full set of coupled rate equations to describe the time evolution of the system from chemical to kinetic freeze-out. Combined with the recently measured data by the STAR collaboration on $K^*/K$ ratios, we show that the previous estimates substantially underestimated the duration of this stage due to the omission of the regeneration of hadron resonances. We provide an improved relation between the $K^*/K$ ratio at chemical and kinetic freeze-out and the life time of the hadronic phase. The calculated improved life times are now in line with estimates from other methods and are relevant for the NA61 and STAR collaborations and for upcoming experiments at the FAIR facility.

Figures

Figures reproduced from arXiv: 2501.03893 by the authors.

**Figure 1.** [Color online] K∗ /K ratio in Au+Au reactions as a function of centrality in the energy range from √ sNN = 7.7 GeV to √ sNN = 39 GeV (shown from upper left to bottom right). Full colored symbols with band denote the rate equation network calculations, while the open black symbols show the experimental data from the STAR collaboration [26]. with t⊥ = 6.5 fm/c. We fix tch, based on previous UrQMD calculations [41]. T… view at source ↗

**Figure 3.** [Color online] Time evolution of the K∗ /K ratio starting from the chemical freeze-out time obtained from the full set of rate equations in central Au+Au reaction from √ sNN = 7.7 GeV to √ sNN = 39 GeV (denoted by differently colored lines). The time at which the ratio matches the measured K∗ /K value is denoted by a circle. teracts the decay of the K∗ . Such a behavior is of course expected by the principle of de… view at source ↗

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Reviewed August 10, 2026 · model on record in the stance chip above.

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