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arxiv: 2604.17345 · v1 · submitted 2026-04-19 · ✦ hep-ph · hep-ex

Deciphering the universal scaling of particle transverse momentum spectra in heavy-ion collisions

Xi-Yao Guo , Hua Zheng , Wenchao Zhang , Li-Lin Zhu , Xing-Quan Liu , Zhi-Guang Tan , Dai-Mei Zhou , Ben-Hao Sa This is my paper

Pith reviewed 2026-05-10 05:57 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords heavy-ion collisionstransverse momentum spectrauniversal scalingCooper-Frye formulaHwa-Yang scalinghydrodynamicsRHICparticle spectra
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The pith

Universal scaling appears in particle transverse momentum spectra when divided by multiplicity and mean pT, explained by the Cooper-Frye formula and matching Hwa-Yang scaling.

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

The paper investigates the transverse momentum spectra of pions, kaons, and protons produced in gold and uranium collisions at the Relativistic Heavy Ion Collider. Scaling each spectrum by the average total particle multiplicity and the mean transverse momentum causes the data from different energies and centralities to fall on a common curve. This universal behavior confirms similar findings from LHC experiments and breaks down at high transverse momenta and in peripheral collisions. The authors derive this scaling from the Cooper-Frye formula that describes how hadrons are emitted from a hydrodynamic fluid at freeze-out. They further show that the observed scaling is equivalent to the Hwa-Yang scaling introduced two decades ago.

Core claim

Transverse momentum spectra scaled by the product of total multiplicity and mean transverse momentum exhibit a universal functional form across a wide range of collision energies and centralities in heavy-ion data, arising naturally from the Cooper-Frye hadronization in hydrodynamics and being identical to the Hwa-Yang scaling.

What carries the argument

The Cooper-Frye formula, which computes the particle yield from the thermal distribution on the freeze-out hypersurface in hydrodynamic simulations, allowing the spectra to be expressed as a universal function modulated by global multiplicity and mean pT.

If this is right

  • The scaling should be valid for other hadron species under similar conditions.
  • Breakdown occurs when non-hydrodynamic processes like jet fragmentation become important at high pT.
  • The equivalence links recent experimental observations to established theoretical scaling from the early 2000s.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the Cooper-Frye mechanism is responsible, the scaling could serve as a diagnostic for the applicability of hydrodynamics in a given collision system.
  • Similar scaling might be testable in high-multiplicity proton-proton collisions if they exhibit collective behavior.
  • Using the scaled spectra could help constrain the freeze-out temperature and flow velocity parameters more effectively.

Load-bearing premise

The universal scaling is caused by the Cooper-Frye formula within a hydrodynamic description of the collision rather than by other dynamical mechanisms or biases in the data selection.

What would settle it

A clear failure of the scaled spectra to collapse onto a single curve in central collisions at low to moderate pT, or the appearance of the scaling in systems where hydrodynamic evolution is not expected.

Figures

Figures reproduced from arXiv: 2604.17345 by Ben-Hao Sa, Dai-Mei Zhou, Hua Zheng, Li-Lin Zhu, Wenchao Zhang, Xing-Quan Liu, Xi-Yao Guo, Zhi-Guang Tan.

Figure 1
Figure 1. Figure 1: FIG. 1. For each row, the top panel shows the scaled spectra [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. For each row, the top panel shows the scaled spectra [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. For each row, the top panel shows the scaled spectra [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Panel (a) The scaled spectra [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Panel (a) shows a direct comparison between [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

We systematically investigate the scaling properties of the transverse momentum spectra for pions, kaons, and protons in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV, as well as in U+U collisions at $\sqrt{s_{NN}}$ = 193 GeV, across different centrality classes, using experimental data from the collaborations at the Relativistic Heavy Ion Collider (RHIC). Universal scaling emerges when the particle transverse momentum spectra are scaled by global physical quantities, i.e., the average total particle multiplicity and mean transverse momentum, confirming recent scaling findings from the data at the Large Hadron Collider (LHC) by the ExTrEMe collaboration. The scaling behavior breaks down in the high $p_{T}$ region and in peripheral collisions. We provide a natural explanation for these observations by invoking the Cooper-Frye formula, which is used for hadronization in hydrodynamics. Furthermore, we demonstrate the equivalence between the scaling found by the ExTrEMe collaboration and the Hwa-Yang scaling which was proposed two decades ago.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript systematically analyzes transverse momentum spectra of pions, kaons, and protons in Au+Au collisions at RHIC energies 7.7–200 GeV and U+U at 193 GeV across centralities using published data. It reports that scaling the spectra by average total particle multiplicity and mean p_T produces a universal collapse, confirming LHC ExTrEMe results, with breakdowns at high p_T and in peripheral collisions. The authors invoke the Cooper-Frye formula for a hydrodynamic explanation and algebraically demonstrate equivalence to the two-decade-old Hwa-Yang scaling.

Significance. If the data collapse is robust, the work provides a compact, parameter-free characterization of spectra across energies and systems, linking empirical observations to hydrodynamics and reviving Hwa-Yang scaling. The strength lies in the broad, systematic comparison of RHIC datasets; the hydrodynamic interpretation, however, remains underdeveloped relative to the empirical claim.

major comments (2)
  1. [Abstract and hydrodynamic discussion] Abstract and discussion of hydrodynamic explanation: the assertion that the Cooper-Frye formula naturally yields the observed scaling lacks an explicit reduction. The integral form dN/dp_T ∝ ∫ (p·u) exp(−p·u/T) dΣ does not automatically become a universal function of p_T / <p_T> after normalization by total multiplicity (volume factor) and <p_T> (T- and flow-dependent) without additional assumptions on constant T, boost invariance, or hypersurface details; these steps are not performed in the manuscript.
  2. [Equivalence to Hwa-Yang scaling] Section demonstrating equivalence to Hwa-Yang scaling: while the algebraic equivalence is stated, the manuscript should specify whether the equivalence follows from the same Cooper-Frye assumptions used for the hydrodynamic explanation or is a purely formal identity independent of the underlying dynamics.
minor comments (1)
  1. [Data analysis and figures] The manuscript would benefit from explicit statements of how uncertainties are propagated through the scaling procedure and from tabulated centrality bin definitions for each energy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and hydrodynamic discussion] Abstract and discussion of hydrodynamic explanation: the assertion that the Cooper-Frye formula naturally yields the observed scaling lacks an explicit reduction. The integral form dN/dp_T ∝ ∫ (p·u) exp(−p·u/T) dΣ does not automatically become a universal function of p_T / <p_T> after normalization by total multiplicity (volume factor) and <p_T> (T- and flow-dependent) without additional assumptions on constant T, boost invariance, or hypersurface details; these steps are not performed in the manuscript.

    Authors: We agree that an explicit reduction from the Cooper-Frye integral to the observed scaling form would improve clarity. In the revised manuscript we add an appendix that performs this derivation under the standard assumptions of boost-invariant hydrodynamics with constant freeze-out temperature and a simple cylindrical hypersurface. The steps show how normalization by total multiplicity (volume factor) and mean p_T (which absorbs the combined effects of T and collective flow) yields an approximate universal function of p_T/<p_T> in the hydrodynamic regime, while the observed breakdowns at high p_T and in peripheral collisions remain outside this regime. revision: yes

  2. Referee: [Equivalence to Hwa-Yang scaling] Section demonstrating equivalence to Hwa-Yang scaling: while the algebraic equivalence is stated, the manuscript should specify whether the equivalence follows from the same Cooper-Frye assumptions used for the hydrodynamic explanation or is a purely formal identity independent of the underlying dynamics.

    Authors: The algebraic equivalence between the ExTrEMe scaling and Hwa-Yang scaling is a formal mathematical identity obtained by re-expressing the scaling variables; it does not depend on the Cooper-Frye formula or any hydrodynamic assumptions. We will revise the relevant section to state this explicitly, noting that the physical motivation for the scaling can be linked to hydrodynamics via Cooper-Frye, while the equivalence itself is independent of the underlying dynamics. revision: yes

Circularity Check

0 steps flagged

No significant circularity: data collapse and algebraic equivalence are independent of fitted inputs

full rationale

The paper reports an empirical observation of data collapse when transverse momentum spectra are normalized by measured global quantities (average multiplicity and mean pT) extracted directly from the same RHIC datasets. This normalization is the standard procedure for testing scaling and does not involve fitting model parameters that are then relabeled as predictions. The claimed equivalence to Hwa-Yang scaling is described as an algebraic demonstration, not a reduction to self-citation or ansatz. Invocation of the Cooper-Frye formula is presented as a qualitative explanation rather than a load-bearing derivation that forces the observed scaling by construction. No self-citation chains, self-definitional loops, or fitted-input-as-prediction patterns appear in the central claims. The result remains falsifiable against independent datasets and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard hydrodynamic modeling assumptions and experimental data without introducing new free parameters or postulated entities beyond those already in the cited literature.

axioms (1)
  • domain assumption The Cooper-Frye formula provides the correct mapping from hydrodynamic fluid elements to observed particle spectra at freeze-out.
    Invoked in the abstract to explain the origin of the observed scaling.

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