Path-length dependence of parton and jet energy loss from universal scaling laws

Fran\c{c}ois Arleo; Guillaume Falmagne

REVIEW 2 major objections 4 minor 12 references

Path-length dependence of parton and jet energy loss from universal scaling laws

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Parton and jet energy loss in quark-gluon plasma share a common, nearly linear path-length scaling.

desk verdict The one new result here is the jet application of the v2/e rule, and its quoted beta_jets=1.03 is a fit-only, three-point estimate resting on the optical-Glauber ansatz; the hadron scaling is clean and the paper is honest about its own limits. read the letter →

arxiv 2411.13258 v1 pith:F3CYWSBG submitted 2024-11-20 hep-ph hep-ex

François Arleo , Guillaume Falmagne This is my paper · ORCID

classification hep-phhep-ex

keywords partonenergylossjetquenchingquark-gluonplasmanuclearmodificationfactorellipticflowpath-lengthdependenceBDMPSformalismRHICandLHCheavy-iondata

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

By using the observed universality of the nuclear modification factor $R_{\rm AA}(p_\perp)$ at large transverse momentum, the paper extracts an average parton energy-loss scale $\bar\epsilon$ in quark-gluon plasma and finds that all available light-hadron measurements from RHIC and the LHC follow $\bar\epsilon \propto L^\beta$ with $\beta=1.02^{+0.09}_{-0.06}$, the value expected when a longitudinally expanding medium controls the loss. It then derives, from the same scaling shape, a linear relation between the azimuthal anisotropy $v_2/e$ and the logarithmic slope $\mathrm{d}\ln R_{\rm AA}/\mathrm{d}\ln p_\perp$ that does not require knowing $\bar\epsilon$ itself. Applying that relation to data gives $\beta$ values for hadrons and for jets that agree with each other and with the multiplicity-scaling value, which the paper reads as evidence that parton and jet energy loss share the same parametric path-length dependence. If the claim holds, the relation furnishes a direct, nearly model-independent experimental handle on the medium-size dependence of jet quenching.

What carries the argument

The central objects are the universal scaling shape $R_{h,\rm AA}(p_\perp,\bar\epsilon,n)\simeq f(u\equiv p_\perp/(n\bar\epsilon))$, where $n$ is the spectral index of the hadron $p_\perp$ spectrum, and the BDMPS-inspired relation $\bar\epsilon = K\,(1/A_\perp)(\mathrm{d}N_{\rm ch}/\mathrm{d}y)\,L^\beta$ with $\beta=2-\alpha$ and $\alpha$ the expansion exponent of the jet-quenching transport coefficient. Carrying the argument is the azimuthal path-length profile $L(\phi)=L(1-e\cos 2\phi)$ from an optical Glauber model, which converts the universal shape $f$ into a prediction for the azimuthal modulation of $R_{\rm AA}$. Expanding that prediction to first order in eccentricity $e$ yields $v_2/e \simeq (\beta/2)\,\partial\ln f/\partial\ln u$, which becomes, for physical $p_\perp$, the data-only relation $v_2/e \simeq (\beta/2)\,\mathrm{d}\ln R_{\rm AA}/\mathrm{d}\ln p_\perp$; it is this identity that makes $\beta$ measurable independently of the absolute energy-loss scale.

What would settle it

Measure $v_2/e$ and $\mathrm{d}\ln R_{\rm AA}/\mathrm{d}\ln p_\perp$ in the same centrality classes with the high statistics of LHC Run 3 and 4; if the data depart from the straight line $y=(\beta/2)x$ by more than the quoted uncertainties, or if the hadron and jet $\beta$ values disagree, the proposed universal scaling is ruled out. A separate decisive check is to recompute $v_2/e$ using fluctuating event-by-event initial conditions instead of the optical-Glauber profile $L(\phi)=L(1-e\cos2\phi)$; if the predicted slope shifts substantially, the geometric ansatz, not the physics of energy loss, is the source of the extracted $\beta$.

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Extended reading notes

Core claim

The central claim is that the path-length dependence of parton energy loss is fixed by data to be essentially linear, $\langle\epsilon\rangle \propto L^\beta$ with $\beta=1.02^{+0.09}_{-0.06}$, and that the same exponent controls the elliptic-flow anisotropy $v_2/e$ through the relation $v_2/e \simeq (\beta/2)\,\mathrm{d}\ln R_{\rm AA}/\mathrm{d}\ln p_\perp$. The paper demonstrates this by fitting energy-loss scales extracted from universal $R_{\rm AA}$ shapes against $(1/A_\perp)\,\mathrm{d}N_{\rm ch}/\mathrm{d}y\,L^\beta$, then by showing that both hadron data from CMS and jet data from ATLAS fall on the predicted straight line with $\beta_{\rm hadrons}=0.94\pm0.04$ and $\beta_{\rm jets}=1.03\pm0.06$, consistent with the density-scaling value. This agreement is presented as evidence that parton and jet energy loss in quark-gluon plasma have the same parametric dependence on path length.

Load-bearing premise

The whole extraction assumes that the way suppression varies with angle is fully described by a smooth, fixed path-length profile $L(\phi)=L(1-e\cos 2\phi)$ from optical Glauber geometry; if event-by-event shape fluctuations or non-eikonal effects break that relation, the measured $v_2$-to-$R_{\rm AA}$ slope is not a clean measure of $\beta$.

Editorial extensions

If this is right

The linear relation $v_2/e \simeq (\beta/2)\,\mathrm{d}\ln R_{\rm AA}/\mathrm{d}\ln p_\perp$ can be used at the LHC to extract the path-length exponent directly from data, with normalization uncertainties in $R_{\rm AA}$ cancelling.
Because hadrons and jets return consistent $\beta$, inclusive jet suppression and single-hadron suppression probe the same parametric medium-size dependence, a constraint that jet-quenching models with explicit $\Delta E \propto L^a$ must reproduce.
The extracted $\beta\simeq1$ supports the picture of a longitudinally expanding QGP in which the transport coefficient falls as $\hat q(\tau)\propto1/\tau$.
Deviations from the straight line, such as the 50--60% centrality hadron point, signal additional physics such as back-to-back jet contamination or eccentricity mis-estimation, making the relation a diagnostic as well as a measurement.

Reading between the lines

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

One could test the relation separately for heavy-flavor hadrons and prompt photons; a different $\beta$ would expose color-charge or quark-mass dependence of the path-length exponent that the present light-hadron and jet samples cannot resolve.
If event-by-event eccentricity fluctuations are included, the slope relation may acquire corrections whose size could explain the outlier centralities without invoking new physics; this is a concrete check once fluctuating initial conditions are available.
The same logic could be applied to $v_3$ or to other harmonic coefficients, whose ratios to $R_{\rm AA}$ would provide independent cross-checks of the geometry ansatz rather than additional parameters.
The agreement between hadron and jet $\beta$ suggests that coherent, multi-parton effects in jet energy loss do not alter the parametric length dependence, a statement that contradicts some explicit models quoted in the paper and that Run 3/4 precision can settle.

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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 is a conference proceedings whose only genuinely new piece is the application of the v2/e versus d ln R_AA/d ln pT relation to ATLAS jets, giving beta_jets = 1.03 ± 0.06. Read that number for what it is: a three-point fit, without propagated uncertainties, resting on the optical-Glauber ansatz of Eq. (5). The headline claim that hadrons and jets share the same path-length exponent is a model-dependent indication, not a measurement. To the paper's credit, it mostly says so itself.

The paper does real things well. The density-scaling fit of Eq. (4) is clean, with chi2/ndf = 0.51, and the derivation of Eq. (7) from the stated geometric ansatz is straightforward and correct. The v2/e relation is a genuinely useful observable: it removes the energy-loss scale and the R_AA normalization uncertainty, and it converts a slope into beta without model machinery beyond the ansatz. The authors also cite the competing results (L^0.59, L^1.4, L^2) and do not hide the 50–60% hadron overshoot or the exclusion of central bins. That is honest scholarship.

Where the reader's skepticism lands and where it does not: the circularity worry mostly does not land. The beta in Eq. (4) is fitted to epsilon values, but the v2/e test is then checked against independent CMS and ATLAS v2 data not used in that fit; the grey band in Fig. 3 is a prediction. The real weakness is the Eq. (5) ansatz itself: phi-dependence as a deterministic power law with L(phi) from optical Glauber. Event-by-event eccentricity fluctuations, non-eikonal corrections, or back-to-back jet contamination would break the clean identification of the fitted slope with beta. The central-bin exclusion matters precisely because that is where fluctuations dominate. The jet panel of Fig. 4 has only three centrality points, and the x-axis itself carries uncertainty from the agnostic R_AA slope fits, none of which is propagated into beta_jets; the ±0.06 is understated.

This is a legitimate conference contribution for people following the path-length dependence debate in jet quenching. It deserves a referee, not a desk reject. A full journal version should propagate the slope and v2 uncertainties and quantify the eccentricity-model sensitivity. Send it out.

Referee Report

2 major / 4 minor

Summary. This proceedings contribution proposes a data-driven determination of the path-length exponent beta for parton energy loss in quark-gluon plasma. It builds on the universal quenching function R_AA(p_T) ~ f(p_T/(n epsilon_bar)) to extract energy-loss scales from light-hadron R_AA data across RHIC and LHC systems, then fits these scales to the parametric form epsilon_bar = K (1/A_perp) (dN_ch/dy) L^beta, obtaining beta = 1.02 +0.09/-0.06 with chi2/ndf = 0.51. It next introduces an azimuthal ansatz R_AA(u,n,phi) = f(u (L/L(phi))^beta) with L(phi) = L(1 - e cos(2phi)), derives v2/e ~ (beta/2) d ln R_AA/d ln p_T, and tests this relation against CMS hadron and ATLAS jet data. The paper concludes that hadron and jet data give consistent beta values, pointing to the same parametric path-length dependence of parton and jet energy loss.

Significance. If the central claim is correct, the result that mean parton energy loss scales approximately linearly with L is a clean and physically important confirmation of the BDMPS expectation for a longitudinally expanding QGP. The density-scaling fit in Fig. 2 is compact and exhibits excellent chi2/ndf = 0.51, and the relation in Eq. (7) is elegant because it cancels normalization uncertainties and does not require the absolute energy-loss scale. I do not see a circularity problem: the v2 data used in Fig. 4 are independent of the epsilon_bar values extracted in Fig. 2. The main limitation is model dependence through the Eq. (5) ansatz, and the jet beta determination is far less precise than the quoted error suggests; this affects the strength of the headline jet/hadron consistency claim, but the underlying density-scaling result is solid.

major comments (2)

[Fig. 4 (right), text after Eq. (7)]
[Eqs. (5)–(7), Fig. 4, abstract]

minor comments (4)

[Fig. 4 caption]
[Footnote 2]
[References]
[Abstract]

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the fitted beta values are cross-checked against independent CMS and ATLAS v2 data; self-citations are disclosed and not used as unverifiable forcing of the result.

full rationale

The central β=1.02 result is obtained by extracting energy-loss scales ε̄ from fits to external hadron R_AA data (Eq. 1) and then fitting Eq. (4) with β as a free parameter. This is an honestly reported fit, not a quantity predicted from the same data. The v2/e relation, Eq. (7), is derived explicitly from the stated ansatz Eq. (5) and is then tested against CMS v2 data (Fig. 3) and ATLAS jet R_AA and v2 data (Fig. 4); those v2 measurements were not used in the Fig. 2 beta fit, so the agreement is a genuine cross-check. Self-citations to Refs. [1,2] supply the universal scaling function f and the BDMPS-based relation Eq. (2), but the function is empirically constrained by data, and the derivation steps are reproduced in the text. No equation reduces to its inputs by construction, and no fitted parameter is relabeled as a prediction. The caveats noted in the paper, such as excluding central bins, the hadron overshoot near 50-60%, and the jet fit being based on few points with fit uncertainties only, affect precision and model dependence but do not constitute circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central result is obtained by fitting beta and K to energy-loss scales that are themselves extracted using the authors' universal scaling function. It relies on BDMPS power-law scaling, Bjorken multiplicity-to-density conversion, and deterministic optical Glauber geometry. No new entities are postulated. The key burden is whether these modeling assumptions, especially the eccentricity ansatz, hold.

free parameters (4)

beta (path-length exponent) = 1.02 +0.09/-0.06 from Eq. (4); 0.94 +/- 0.04 (hadron v2/e); 1.03 +/- 0.06 (jet v2/e)
Central exponent in epsilon proportional to L^beta; fitted to data rather than derived. The v2/e values are fit-uncertainty only.
K (overall normalization) = 0.33 fm with asymmetric fit uncertainty (Fig. 2 caption)
Normalization of the epsilon-density scaling relation in Eq. (4); free in the fit.
Per-system energy-loss scales epsilon_bar = Numerous values across systems and centralities, shown in Fig. 1
Extracted by fitting R_AA with the universal function f; these are the y-values whose scaling yields beta. They are determined from data but are not externally fixed.
Parameters of universal quenching function f(u) = Not given in this paper; from Refs. [1,2]
The extraction of epsilon_bar for each system uses fits of R_AA with f; the shape of f is model output, not re-derived here.

assumptions (5)

domain assumption BDMPS scaling: epsilon_bar proportional to <z> C_k qhat0 tau0^alpha L^(2-alpha), Eq. (2)
Central input translating QGP transport to an energy-loss scale; assumes coherent BDMPS radiative energy loss and power-law time dependence qhat(tau) = qhat0 (tau0/tau)^alpha.
domain assumption Bjorken relation n0 proportional to dN_ch/dy divided by (A_perp tau0), Eq. (3)
Converts final charged multiplicity to initial parton density; uses a fixed 3/2 factor and neglects chemical composition and entropy production details.
domain assumption Optical Glauber geometry with hard-sphere nuclear densities determines L, A_perp, and e, with L(phi) = L(1 - e cos(2 phi))
All geometric inputs to Eqs. (4)-(7) come from a deterministic Glauber model; fluctuations in eccentricity are explicitly not accounted for, forcing exclusion of central bins.
domain assumption Universal scaling form R_h_AA(pT, epsilon, n) approximately f(pT/(n epsilon)), Eq. (1), holds across systems and for jets
The entire extraction assumes one common function f, a model output from Refs. [1,2], not independently derived here.
standard math First-order Taylor expansion in eccentricity e for v2/e, Eq. (6)
Derivation of v2/e approximately (beta/2) d ln f/d ln u assumes e is small; stated and standard, but marginal for peripheral collisions.

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Cite this review

Pith. "Pith review of Path-length dependence of parton and jet energy loss from universal scaling laws." pith.science (2026). https://pith.science/paper/F3CYWSBG

@misc{pith2026241113258,
  author       = {Pith},
  title        = {Pith review of: Path-length dependence of parton and jet energy loss from universal scaling laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3CYWSBG}},
  note         = {Machine review of arXiv:2411.13258}
}

abstract

The universal dependence of hadron suppression, $R_{\rm{AA}}(p_\perp)$, observed at large-$p_\perp$ in heavy ion collisions at RHIC and LHC allows for a systematic determination of the average parton energy loss $\langle \epsilon \rangle$ in quark-gluon plasma (QGP). A simple relation between $\langle \epsilon \rangle$ and the soft particle multiplicity allows for probing the dependence of parton energy loss on the medium path-length. We find that all the available measurements are consistent with $\langle \epsilon \rangle \propto L^\beta$ with $\beta=1.02\pm^{0.09}_{0.06}$, consistent with the pQCD expectation of parton energy loss in a longitudinally expanding QGP. We then show, based on the model predictions, that the data on the azimuthal anisotropy coefficient divided by the collision eccentricity, $v_2/\rm{e}$, follows the same scaling property as $R_{\rm{AA}}$. Finally, a linear relationship between $v_2/\rm{e}$ and the logarithmic derivative of $R_{\rm{AA}}$ at large $p_\perp$ offers a purely data-driven access to the $L$ dependence of parton energy loss. Quite remarkably, both hadron and jet measurements obey this latter relationship, moreover with consistent values of $\beta$. This points to the same parametric path-length dependence of parton and jet energy loss in QGP.

Figures

Figures reproduced from arXiv: 2411.13258 by the authors.

**Figure 1.** Energy loss scales 𝜖¯ extracted from light (𝜋 0 , ℎ ± ) and heavy (𝐽/𝜓, 𝐷 in PbPb at 5.02 TeV) hadron data in various collision systems and centralities. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 3.** 𝑣2/e of light hadrons vs. 𝑝⊥ /𝑛𝜖¯. The band corresponds to 𝛽 = 1.02+0.09 −0.06. Once the dependence of 𝜖¯ with 𝐿 is empirically determined, it becomes possible to investigate the azimuthal dependence of hadron suppression, from which the 𝑣2 coefficient can be computed. Using the universal shape of 𝑅AA, its 𝜙 dependence can be modeled as 𝑅AA(𝑢, 𝑛, 𝜙) = 𝑓 𝑢 × (𝐿/𝐿(𝜙)) 𝛽 , 𝑛 , (5) 3 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

**Figure 4.** Relation between 𝑣2/e and d ln 𝑅AA/d ln 𝑝⊥ for light hadrons in CMS [3–6] (left) and inclusive jets in ATLAS [7, 8] (right). The bands 𝑦 = 𝛽 𝑥/2 show the expectation from (7), with the value 𝛽 = 1.02+0.09 −0.06 determined from the density scaling fit of hadrons of Eq. (4) (grey band, left), and the 𝛽 values extracted from linear fits of the displayed hadron or jet data (blue bands). dependence of 𝑅AA of both process… view at source ↗

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Works this paper leans on

12 extracted references · 1 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Arleo,Quenching of Hadron Spectra in Heavy Ion Collisions at the LHC,Phys

F. Arleo,Quenching of Hadron Spectra in Heavy Ion Collisions at the LHC,Phys. Rev. Lett. 119 (2017) 062302 [1703.10852]

arXiv 2017

[2] [2]

Arleo and G

F. Arleo and G. Falmagne,Probing the path-length dependence of parton energy loss via scaling properties in heavy ion collisions,Phys. Rev. D109(2024) L051503 [2212.01324]

arXiv 2024

[3] [3]

CMScollaboration,Azimuthal anisotropy of charged particles at high transverse momenta in PbPb collisions at√𝑠𝑁𝑁 = 2.76 TeV, Phys. Rev. Lett.109 (2012) 022301 [1204.1850]

arXiv 2012

[4] [4]

CMS collaboration,Azimuthal anisotropy of charged particles with transverse momentum up to 100 GeV/ c in PbPb collisions at√𝑠𝑁𝑁 =5.02 TeV, Phys. Lett. B776 (2018) 195 [1702.00630]

arXiv 2018

[5] [5]

CMScollaboration, Study of high-𝑝⊥ charged particle suppression in PbPb compared to pp collisions at√𝑠NN=2.76 TeV,Eur. Phys. J.C72(2012) 1945 [1202.2554]

arXiv 2012

[6] [6]

CMScollaboration, Charged-particle nuclear modification factors in PbPb and pPb collisions at√𝑠N N= 5.02 TeV, JHEP 04(2017) 039 [1611.01664]

arXiv 2017

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ATLAScollaboration,Measurement of the nuclear modification factor for inclusive jets in Pb+Pb collisions at√𝑠NN= 5.02TeV with the ATLAS detector,Phys. Lett. B790(2019) 108 [1805.05635]

arXiv 2019

[8] [8]

ATLAScollaboration,Measurements of azimuthal anisotropies of jet production in Pb+Pb collisions at√𝑠𝑁𝑁 =5.02 TeV with the ATLAS detector, Phys. Rev. C105 (2022) 064903 [2111.06606]

arXiv 2022