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Investigating QCD Dynamical Entropy in high-energy nuclear collisions

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Pith's one-line read This paper claims that the QCD dynamical entropy of proton-nucleus collisions is almost independent of the nucleus mass number A, and that geometric scaling makes it exactly equal to the proton result.

desk verdict The pA extension has one genuinely new numerical branch (Glauber-Gribov), but the headline A-independence claim rests on an unproven numerical collapse and a tautological scaling assumption. read the letter →

arxiv 2507.09349 v1 pith:HZQ3K5WK submitted 2025-07-12 hep-ph hep-exnucl-th

G. S. Ramos , L. S. Moriggi , M.V. T. Machado This is my paper
classification hep-phhep-exnucl-th PACS 12.38.-t24.85.+p25.75.Dw05.70.-a
keywords QCDdynamicalentropyproton-nucleuscollisionsnuclearunintegratedgluondistributiongeometricscalingGlauber-Gribovformalismrelativeinitialdensitysaturationscale
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

The paper extends the QCD dynamical entropy—a relative-entropy measure of the disorder generated by nonlinear QCD evolution from rapidity $Y_0$ to rapidity $Y$—to proton-nucleus and heavy-ion collisions. It constructs nuclear unintegrated gluon distributions by two routes: geometric scaling, where the nuclear saturation scale $Q_{s,A}$ inherits its $A$-dependence through a single prefactor, and the Glauber-Gribov formalism, where the proton dipole cross-section is replaced by a nuclear one. The central result is that after normalization the dynamical entropy $\Sigma_{Y_0\to Y}$ is essentially independent of the mass number $A$: in the geometric scaling case the $A$-dependence cancels completely and the nuclear entropy equals the proton entropy, while in the Glauber-Gribov case the $A$-independence holds numerically and the entropy is larger. If correct, this provides a weak-coupling estimate of the initial entropy density of heavy-ion collisions in which the nucleus enters only through the transverse area $R_A^2$.

What carries the argument

The load-bearing object is the nuclear unintegrated gluon distribution $\varphi_A(k,Y)$, normalized to a transverse momentum probability $P(k,Y)$ that enters the relative-entropy definition of the dynamical entropy. For the geometric scaling strategy, the mechanism is the scaling variable $\tau_A = k^2/Q_{s,A}^2$ together with the ratio identity $Q_{s,A}^2(Y)/Q_{s,A}^2(Y_0)=e^{\lambda \Delta Y}$, which holds because the same $A$-dependent prefactor appears at both rapidities and cancels. For the Glauber-Gribov strategy, the central object is the nuclear dipole cross-section built from the Woods-Saxon thickness function, whose nuclear UGD, when inserted in the same entropy formula, shows the $A$-cancellation numerically.

What would settle it

Compute the same dynamical entropy with a nuclear saturation scale that does not factor as a single $A$-dependent prefactor, for instance one with impact-parameter dependence or a running $x$-dependent power $\Delta$, and check whether a residual $A$-dependence reappears in $\Sigma_{Y_0\to Y}$; experimentally, extract the initial entropy density from charged-particle multiplicities for several nuclei and test whether $dS_D/dy$ follows $R_A^2$ with no additional $\ln A$ or $A^{1/3}$ corrections.

Watch

Extended reading notes

Core claim

The central claim is that, after the normalization of the nuclear unintegrated gluon distribution into a transverse momentum probability, the QCD dynamical entropy in proton-nucleus collisions is fixed by the rapidity interval $\Delta Y$ alone and not by the nuclear species. For the geometric scaling strategy this follows analytically: with $Q_{s,A}^2(Y) = (R_{pA}^2/R_A^2)^{\Delta} Q_s^2(Y)$, the same $A$-dependent prefactor appears at $Y$ and at $Y_0$, so the ratio $Q_{s,A}^2(Y)/Q_{s,A}^2(Y_0)$ reduces to $e^{\lambda \Delta Y}$ and every model probability yields the same $\Sigma_{Y_0\to Y}$ as in proton-proton collisions. The entropy density then scales as $dS_D/dy \propto R_A^2$, the nuclear area. In the Glauber-Gribov framework the numerical results also show no $A$-dependence in $\Sigma_{Y_0\to Y}$, although the entropy is larger than the proton value and does not reduce to it. The paper contrasts this with CGC entanglement and Wehrl entropies, which grow linearly with $A$, and argues that the dynamical entropy, being a relative entropy, is not extensive in the nuclear mass number.

Load-bearing premise

For the geometric scaling result, everything rests on the assumption that the nuclear saturation scale differs from the proton one by a single $A$-dependent factor that is the same at the initial and final rapidities; if the real nuclear saturation scale carries impact-parameter dependence, running-coupling effects, or an $x$-dependent power $\Delta$, the cancellation and with it the $A$-independence would disappear.

Editorial extensions

If this is right

  • For the geometric scaling models, the dynamical entropy in proton-nucleus collisions is exactly the proton entropy, so the nucleus enters only through the $R_A^2$ factor in the entropy density.
  • The entropy density $dS_D/dy$ scales with the nuclear area, meaning heavier nuclei produce more total dynamical entropy per unit rapidity even though the per-cell entropy is $A$-independent.
  • In the Glauber-Gribov framework the dynamical entropy is also $A$-independent, but its magnitude is larger than in the geometric scaling treatment and does not reduce to the proton result.
  • The dynamical entropy does not scale extensively with $A$, in contrast to CGC entanglement and Wehrl entropies, which grow as $A\,e^{\lambda Y}$.
  • Because the thermalization time is controlled by the initial entropy density, an $A$-independent per-cell entropy with an $R_A^2$ prefactor gives a concrete weak-coupling baseline for initial conditions in heavy-ion phenomenology.

Reading between the lines

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

  • A natural next test would be to repeat the Glauber-Gribov computation with an impact-parameter-dependent nuclear saturation scale; if the $A$-independence persists there, it is a genuine property of the entropy functional, whereas if it breaks, the factorization assumption is the true source.
  • Because the geometric scaling strategy makes nuclear and proton entropies exactly equal at the same $\Delta Y$, any measured difference in entropy production between proton-proton and proton-nucleus collisions would directly probe violations of geometric scaling.
  • As a relative entropy, the dynamical entropy is connected to the Fisher information metric on the space of gluon distributions; the $A$-independence suggests that the family of final distributions over different nuclei is indexed by $\Delta Y$ alone and could serve as a natural coordinate chart for that space.
  • If the Glauber-Gribov result is closer to reality, its larger entropy indicates that the normalization procedure hides part of the nuclear dynamics, and comparing the two strategies against measured multiplicities in proton-nucleus collisions could identify which nuclear gluon distribution is phenomenologically favored.
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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

4 major / 6 minor

Summary. This paper extends the QCD dynamical entropy formalism of Peschanski to proton-nucleus collisions. Two strategies are used to construct nuclear unintegrated gluon distributions: (i) a geometric-scaling ansatz in which the nuclear saturation scale is Q_{s,A}^2(Y)=(R_{pA}^2/R_A^2)^Δ Q_s^2(Y), applied to the GBW, MPM, and L V models; and (ii) a Glauber-Gribov framework with a Woods-Saxon thickness profile and the GBW nuclear UGD. The central reported result is that, after the normalization procedure, the dynamical entropy Σ_{Y0→Y} is almost independent of the mass number A, while the entropy density dS_D/dy scales with R_A^2. The paper also compares the dynamical entropy with CGC entanglement and Wehrl entropies for nuclear targets.

Significance. If established, the claimed A-independence would be a striking and useful property: the nuclear transverse-momentum probability at fixed τ_A=k^2/Q_{s,A}^2, and hence the relative entropy of the rapidity evolution, would be universal in the rapidity difference ΔY, with all nuclear dependence residing in the R_A^2 prefactor of the entropy density. The paper is transparent that Strategy I relies on the factorization ansatz Eq. (10), and it provides a closed-form demonstration that the A-dependence cancels in that case. However, the independent support from the Glauber-Gribov branch is only a visual coincidence of curves, with no analytic explanation or numerical quantification. The comparison with other entropy measures is a useful phenomenological addition. On balance, the result is promising but not yet established to the standard claimed in the abstract.

major comments (4)
  1. [II, Eqs. (10) and (18)] The A-independence of the dynamical entropy in Strategy I is a consequence of the factorization assumed in Eq. (10), not an emergent property. Since the factor (R_{pA}^2/R_A^2)^Δ multiplies both Q_{s,A}^2(Y) and Q_{s,A}^2(Y0), it cancels identically in Eq. (18), so any distribution that depends only on τ_A yields e^{λΔY}. The abstract's phrase 'almost independent' should be stated as 'exactly independent under the ansatz (10)', and the paper should discuss the validity of this factorization, including impact-parameter dependence and a possible x-dependent Δ.
  2. [III, Figs. 3 and 4] The Glauber-Gribov A-independence is presented as a visual coincidence of curves. Equation (16), with B=(1/2)T_A(b)σ_0, does not obviously reduce to a function of τ_A after normalization, because the b-integral over the Woods-Saxon profile samples different B ranges for different A. To make the central claim load-bearing, the authors should provide an analytic argument for the cancellation or, at minimum, a quantitative collapse test: e.g., report max_A |Σ_A(ΔY)-⟨Σ(ΔY)⟩|/⟨Σ(ΔY)⟩ and a plot of the residuals for Si, Ca, Au, and Pb. Without this, the second strategy does not independently support the headline result.
  3. [III, Figs. 3 and 4] Strategy II is implemented only for the GBW nuclear UGD of Eq. (16); no Glauber-Gribov results are shown for the MPM or L V models. The summary statement that 'both the nuclear transverse probability density and the total entropy ... were larger' overgeneralizes, since these comparisons are not shown for all models. If the A-independence is expected to be model-independent, this should be demonstrated; otherwise the claims should be restricted to the GBW case.
  4. [III, after Eq. (20)] The statement that the Glauber-Gribov dynamical entropy 'does not reduce to the proton case, and it is substantially larger' is unexplained. Since the normalization procedure in Eq. (2) removes the overall A-dependent prefactor, the origin of the larger entropy should be traced to the shape of P_A(k,Y) produced by the b-integral. Providing this explanation would turn a numerical observation into a physical result.
minor comments (6)
  1. [II, Eq. (9)] Equation (9) labels both branches as 'τ<1'; the second branch should read 'τ≥1'.
  2. [II, Eqs. (14) and (15)] The symbol σ_{dA} is used both for the b-dependent differential cross-section and for the integrated one; please introduce distinct notation, e.g., σ_{dA}(x,r)=∫ d²b σ_{dA}(x,r,b).
  3. [III, Fig. 1] The sentence 'For heavy nuclei, the peak is lower for the smaller ones' is confusing; since k_max ∼ R_A^{-2Δ}, the peak occurs at smaller k for heavier nuclei, and the sentence should be rephrased accordingly.
  4. [Abstract and Sec. III] The term 'almost independent' is imprecise: within Strategy I the relative entropy is exactly independent of A under Eq. (10), while the entropy density dS_D/dy depends on A through R_A^2. Please specify in the abstract which quantity is claimed to be A-independent.
  5. [III, all figures] No numerical uncertainties are given for the fitted parameters λ, x0, σ0, a, b, Δ, ε, and γ_s; at least a propagation of these uncertainties into Σ(ΔY) would support the 'almost' in the abstract.
  6. [References] Several references are incomplete (e.g., [3], [37], [50]); please provide published journal and DOI information where available.

Circularity Check

1 steps flagged · score 5.0 of 10

Strategy I's A-independence is the cancellation, in Eq. (18), of the A-factor that Eq. (10) itself inserts; Strategy II supplies only an unquantified 'it appears' numerical support.

  1. self definitional [Sec. II, Strategy I (Eqs. 10, 17-18); Sec. III; Abstract]
    ""It is shown that the normalization procedure and the geometric scaling property make the dynamical entropy almost independent of the nucleus mass number A." (Abstract) ... "Specifically, for the nuclear saturation scale Qs,A(Y): Q^2_{s,A}(Y) = (R^2_{pA}/R^2_A)^\Delta Q^2_s(Y), (10)" ... "Q^2_{s,A}(Y)/Q^2_{s,A}(Y0) = ... = e^{\lambda\Delta Y} \equiv s, (18)" ... "Moreover, we have shown that the relation in Eq. (18) makes the nuclear result identical to the proton one.""

    The central claim for Strategy I, that geometric scaling makes the dynamical entropy almost independent of A, is produced by construction. The nuclear UGD is defined as a function of τ_A = k^2/Q^2_{s,A}(Y), and Q^2_{s,A}(Y) is assumed to factorize as (R^2_{pA}/R^2_A)^Δ Q^2_s(Y) (Eq. 10), a purely A-dependent factor times the proton scale. In the relative-entropy integral (17), the Jacobian πQ^2_{s,A}(Y) cancels the 1/Q^2_{s,A}(Y) in the normalized P_A, and the ratio Q^2_{s,A}(Y)/Q^2_{s,A}(Y0) cancels that same A-factor identically (Eq. 18), leaving only e^{λΔY}. The paper states the mechanism: "the relation in Eq.

full rationale

Strategy I's A-independence is by construction, and the paper's own algebra displays the reduction: Eq. (10) puts the A-dependent factor (R^2_{pA}/R^2_A)^Δ into Q^2_{s,A}(Y) with the same power at Y and Y0, Eq. (18) cancels it identically, and the text says "the relation in Eq. (18) makes the nuclear result identical to the proton one." Since any multiplicative c_A Q_s(Y) would cancel the same way, the abstract's claim that geometric scaling "make[s] the dynamical entropy almost independent of A" is a corollary of the ansatz, which is itself just the geometric-scaling absorption property cited from [37]. This is a partial self-definitional reduction of the central claim, though an openly derived one. The paper is not wholly circular. Strategy II (Glauber-Gribov, Eq. 16) uses the proton Q_s with A entering only through B = (1/2)T_A(b)σ0, so no A-factor is present to cancel; its A-independence (Figs. 3-4) is a genuine numerical consequence, not a construction. Its weakness is absence of proof, not circularity: the paper itself concedes "it appears that geometric scaling and the normalization procedure also eliminate the A dependence," with no analytic argument, error estimate, or collapse criterion. Notably, this branch does not reduce to the proton case and is "substantially larger," confirming that Strategy I's proton-identical result is dictated by the multiplicative construction. Fitted parameters (λ, x0, σ0, a, b, ε, γs) are inherited from published UGD models fitted to external DIS and LHC data and are not renamed as predictions, since no data comparison is claimed. Self-citations ([25], [32,33], [12]) are present but not load-bearing in a circular way: the dynamical-entropy formalism originates externally (Peschanski, Ref. [19]), and the MPM model is externally validated against charged-particle spectra. No uniqueness theorem is imported from the authors' prior work. Weighing the one by-construction pillar against the independent but only numerically evidenced Glauber-Gribov pillar, and the paper's full disclosure of the cancellation mechanism, the appropriate score is 5: the central claim's Strategy I support reduces to its input assumption, while Strategy II carries independent but unterived numerical content.

Assumptions & free parameters 14 free parameters · 7 assumptions · 0 invented entities

The central claim rests on imported fitted parameters (lambda, x0, sigma0, a, b, Delta, epsilon, gamma_s) and on scaling/postulate-level assumptions about saturation and nuclear geometry. No new particles, forces, or dynamical entities are introduced. The geometric-scaling A-independence is essentially a direct consequence of Eq. (10) and the cancellation in Eq. (18), so the ledger is dominated by the factorization assumption for Q_{s,A}.

free parameters (14)
  • lambda_GBW = 0.248
    Saturation-scale exponent fitted in Ref. [39]; sets the rapidity growth of Qs and therefore the absolute value of the GBW dynamical entropy in Eq. (20).
  • x0_GBW = 4.2e-5
    Reference Bjorken x fitted in Ref. [39]; fixes Y0 and the initial saturation scale.
  • Q0 = 1 GeV
    Reference saturation momentum in Q_s^2(Y) = Q0^2 exp(lambda (Y - Y0)); sets the momentum normalization.
  • sigma0_GBW = 27.32 mb
    Dipole-proton cross-section normalization fitted in Ref. [39]; enters the Glauber-Gribov nuclear UGD through B = (1/2) T_A sigma0 in Eq. (16).
  • lambda_MPM = 0.33
    Saturation-scale exponent used to fit the MPM parameters a and b in Refs. [32,33].
  • a_MPM = 0.055
    Tsallis-inspired UGD parameter fitted to pp charged-hadron data in Refs. [32,33]; enters the MPM probability distribution, Eq. (12).
  • b_MPM = 0.204
    Fitted UGD parameter in Refs. [32,33]; enters the MPM probability distribution, Eq. (12).
  • Delta = 1.27
    Nuclear geometric-scaling exponent from Ref. [37], Eq. (10); the entire A-dependence of Q_{s,A} rests on this value.
  • Rp = 3.56 GeV
    Proton radius in momentum units from Ref. [37], Eq. (10); part of the assumed nuclear saturation-scale factorization.
  • epsilon_LV = 0.2
    Saddle-point constant in the Levin-Tuchin/LV UGD, Eq. (8); controls the damping of the saturated branch.
  • gamma_s = 0.63
    Effective anomalous dimension near the saturation line in the LV UGD, Eq. (9).
  • d_LV = 0.5954
    Continuity constant joining the saturated and dilute branches of the LV UGD, Eq. (9).
  • mu = 3*pi/2
    Average number of gluonic degrees of freedom per transverse cell, identified by matching Eq. (3) to expression (25) of Ref. [31].
  • alpha_s = 1/5
    Strong coupling constant used in C_m and in the UGD normalizations; its value is chosen rather than derived here.
assumptions (7)
  • standard math The Kullback-Leibler divergence is a valid measure of distinguishability between the normalized gluon momentum distributions P(k,Y) and P(k,Y0).
    Used implicitly in the definition of Sigma, Eq. (1), and in the discussion of relative entropy in Section I.
  • domain assumption Geometric scaling holds: the proton and nuclear UGDs depend on transverse momentum only through tau = k^2 / Q_s^2.
    Invoked after Eq. (2) and used to rewrite the entropy as an integral over tau in Eq. (17). This is an approximate property of saturation models, not an exact QCD statement.
  • domain assumption The nuclear saturation scale factorizes as Q_{s,A}^2(Y) = (R_{pA}^2/R_A^2)^Delta Q_s^2(Y) with Delta = 1.27.
    This is Eq. (10), taken from Ref. [37]. It is the load-bearing premise for the A-independence of Strategy I.
  • domain assumption The Glauber-Gribov replacement sigma_dA(Y,r,b) = 2 (1 - exp(-(1/2) T_A(b) sigma_dip(Y,r))) is valid for nuclear dipole cross sections.
    Used in Eq. (14) to construct the nuclear UGD in Strategy II, with the Woods-Saxon profile for T_A.
  • domain assumption The saturation scale grows exponentially with rapidity, Q_s^2 proportional to exp(lambda Y) with lambda approximately 0.3.
    Stated in Section II as a general feature of leading-order solutions of nonlinear QCD evolution; this is what makes the ratio in Eq. (18) a simple exponential.
  • domain assumption The dynamical-entropy density formula dS_D/dy = C_m mu R_h^2 / R_0^2 Sigma from Ref. [19], with mu = 3 pi / 2, is accepted as the correct macroscopic relation.
    Equation (3) and the identification of mu are taken from Ref. [19] and Ref. [31]; no independent derivation is given here.
  • domain assumption The LV/Levin-Tuchin UGD forms in Eqs. (8) and (9) approximate the full leading-order Balitsky-Kovchegov solution.
    The paper states that the model provides a closer approximation to the numerical BK solution, which is a modeling assumption inherited from Refs. [34,35].

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Cite this review

Pith. "Pith review of Investigating QCD Dynamical Entropy in high-energy nuclear collisions." pith.science (2026). https://pith.science/paper/HZQ3K5WK

@misc{pith2026250709349,
  author       = {Pith},
  title        = {Pith review of: Investigating QCD Dynamical Entropy in high-energy nuclear collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZQ3K5WK}},
  note         = {Machine review of arXiv:2507.09349}
}
abstract

In this work, the concept of QCD dynamical entropy is extended to heavy ion systems. This notion of entropy can be understood as a relative entropy and can also be used to estimate the initial entropy density in ultra-relativistic heavy ion collisions. The key quantity used to calculate this entropy is the nuclear unintegrated gluon distribution (nUGD), which provides a transverse momentum probability density. In the numerical analysis, both the geometric scaling phenomenon and the Glauber-Gribov approach have been used to evaluate realistic models for the nUGD. It is shown that the normalization procedure and the geometric scaling property make the dynamical entropy almost independent of the nucleus mass number $A$. Results are presented for the dynamical entropy density, $dS_D/dy$, in terms of the rapidity.

Figures

Figures reproduced from arXiv: 2507.09349 by the authors.

Figure 1
Figure 1. FIG. 1: The nuclear transverse momentum probability, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Total QCD dynamical entropy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Transverse momentum probability distributions [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Total dynamical entropy in proton-nucleus [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Nuclear QCD dynamical entropy in proton-nucleus collisions corresponding to the QCD evolution in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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