Improved Regge Kinematics and All-Path-Length Corrections to Momentum Broadening in the Quark-Gluon Plasma

Dario van den Berg; Isobel Kolbe

REVIEW 4 major objections 6 minor 15 references

Improved Regge Kinematics and All-Path-Length Corrections to Momentum Broadening in the Quark-Gluon Plasma

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper extends the GLV formalism with all-path-length and improved sub-Regge corrections and shows these can substantially change the momentum broadening distribution and the jet transport coefficient in small collision systems.

desk verdict Timely idea, but the two correction factors are imported from the authors' own unpublished Ref [14] and the numerics lack inputs—send back for derivation and reproducibility, not desk reject. read the letter →

arxiv 2608.08894 v1 pith:OJCJBY5D submitted 2026-08-09 hep-ph nucl-th

Dario van den Berg , Isobel Kolbe This is my paper · ORCID

classification hep-phnucl-th

keywords quark-gluonplasmatransversemomentumbroadeningjetquenchingGLVopacityexpansionall-path-lengthcorrectionssub-Reggekinematicstransportcoefficientsmallcollisionsystems

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 standard Gyulassy-Levai-Vitev (GLV) description of how a high-energy parton gains transverse momentum while crossing a quark-gluon plasma. It relaxes two approximations of that formalism: the assumption that the first scattering happens far from the production point and the assumption that the exchanged momentum is tiny compared with the parton's longitudinal momentum. The paper derives analytic first-order expressions for the broadening distribution and the transport coefficient qhat with these relaxations. These corrections matter because in small collision systems the relevant length and coherence scales are comparable, so the standard approximations may fail. The paper finds that the all-path-length correction suppresses broadening while the improved sub-Regge kinematics enhance it, and that the two partially cancel when combined.

What carries the argument

The central object is the first-order GLV opacity expansion for transverse momentum broadening, truncated at one hard scattering with the medium. The argument is carried by two multiplicative correction factors that modify the standard GLV integrand: the all-path-length factor $(1 - \frac{1}{2} e^{-\mu_\perp \Delta z})^2$ from relaxing the large-separation-distance approximation, and the improved sub-Regge factor $4/(1+\gamma)^2$ with $\gamma = \sqrt{1 - 2q_\perp^2/P^{+2}}$ from retaining subleading contributions in the $1/P^+$ expansion. These factors show how finite path length and finite exchanged momentum alter the broadening distribution and $\hat{q}$, and they reduce to the conventional GLV results in the appropriate limits.

What would settle it

Evaluate the first-order GLV broadening amplitude numerically with the full kinematic replacement of Eq. (12), without expanding in $1/P^+$, and check whether the ratio to the standard GLV integrand is exactly $4/(1+\gamma)^2$ with $\gamma = \sqrt{1 - 2q_\perp^2/P^{+2}}$; repeat for the full $\Delta z$ dependence and check whether the coefficient of $e^{-\mu_\perp\Delta z}$ is exactly $1/2$. If either factor differs, the correction is not what the paper claims.

Watch

Extended reading notes

Core claim

The paper's central claim is that, to first order in the opacity expansion, the standard GLV momentum broadening distribution is changed by two independent factors. Relaxing the large-separation-distance approximation produces the all-path-length factor $(1 - \frac{1}{2} e^{-\mu_\perp \Delta z})^2$, which suppresses both the distribution and the transport coefficient $\hat{q}$ at short path lengths. Retaining subleading terms in the $1/P^+$ expansion replaces the strict Regge kinematics by the sub-Regge factor $4/(1+\gamma)^2$ with $\gamma = \sqrt{1 - 2q_\perp^2/P^{+2}}$, which enhances both observables when the exchanged momentum is a sizable fraction of the parton's longitudinal momentum. The combined scheme interpolates between these limits, and the paper argues that this may alleviate the negative energy loss found when only APL corrections are included.

Load-bearing premise

The load-bearing assumption is that the two analytic correction factors, the all-path-length factor $(1 - \frac{1}{2} e^{-\mu_\perp \Delta z})^2$ and the sub-Regge factor $\gamma = \sqrt{1 - 2q_\perp^2/P^{+2}}$, are the correct consequences of relaxing the stated approximations; the paper does not derive these factors from the modified amplitudes, so an error in either coefficient would change the magnitude and possibly the sign of the reported corrections.

Editorial extensions

If this is right

In small systems where $\Delta z$ is comparable to the inverse Debye mass, the all-path-length factor suppresses $\hat{q}$ relative to the standard GLV value, so extractions of medium properties from such systems must account for it.
When the transverse momentum exchange is a non-negligible fraction of the parton's longitudinal momentum, the sub-Regge factor enhances the broadening distribution and $\hat{q}$, an effect that grows with $p_\perp/P^+$.
The combined correction lies between the APL-only and sub-Regge-only predictions, so the two effects partially cancel and the net deviation from GLV is smaller than either correction alone.
The analytic formulas provide a direct extension path to the full GLV radiative energy-loss framework, where the sub-Regge enhancement may offset the negative energy loss previously seen from short-path-length corrections.

Reading between the lines

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

Because the correction factors are multiplicative and depend on $\mu_\perp\Delta z$ and $q_\perp^2/P^{+2}$, the predicted deviation of $\hat{q}$ from GLV should vary in a specific way with jet energy and medium size; comparing proton-nucleus and light-nucleus collisions at different beam energies would probe that scaling.
The square root in $\gamma$ implies a kinematic bound $q_\perp \leq P^+/\sqrt{2}$; beyond it the eikonal parameterization becomes unphysical, so an extension dropping the eikonal approximation would be needed to describe the most extreme momentum transfers.
The same correction factors could be inserted into implementations of jet quenching beyond first order in opacity, offering a test of whether the short-path and sub-Regge effects remain as simple multiplicative factors when multiple scatterings are included.

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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: the paper addresses a legitimate problem in an important direction, but in its current form it is a results memo that leans entirely on the authors' own unpublished Ref [14] for the two factors that do all the work. If you want to evaluate the actual physics, you need that other paper in hand; the manuscript itself does not stand alone.

What is genuinely useful: the motivation is timely. With the LHC now running O-O and p-O/Ne-Ne, the assumptions that formation and coherence scales are small relative to the medium are worth questioning, and the GLV broadening setup is the right place to ask. The paper also does a good job of laying out the four approximation schemes in Table 1, so the reader can see exactly where each correction enters. The qualitative finding—APL suppresses, sub-Regge enhances, and the two partially cancel—is plausible and connects to an existing puzzle (negative energy loss in Ref [15]). That is a reasonable physics story.

The soft spots are real, and they are load-bearing, not cosmetic. The two central factors—(1 − 1/2 e^{−µ⊥ ∆z})^2 in Table 1 and γ = sqrt(1 − 2q⊥²/P^{+2}) in Eq. (13)—are introduced without derivation. The text says 'retaining the leading subleading contributions' and 'this yields', but no amplitude-level expression or expansion is shown. The paper itself points to Ref [14] for further details, and that reference has nearly the same title as this manuscript; it is a self-citation that carries the core of the derivation. That is not acceptable in a standalone result. If [14] is available, the authors should either reproduce the derivation in an appendix or give equation-level pointers; otherwise the central claim is unverifiable from this paper alone.

The numerics are also not reproducible: Fig. 2 has no α_s, µ_D, medium density profile, or even a statement of whether the medium is static or Bjorken-expanding. Those parameters are essential because the size of the corrections depends on them. This is minor in the sense that it can be fixed by stating inputs, but it should be fixed.

Net: I would not desk-reject this; the topic is relevant and the corrections, if correct, matter for small-system jet tomography. But I would send it back with a strong request for the derivation or a precise mapping to [14], plus full numerical inputs. With those, this could be a useful contribution to the jet quenching literature.

Referee Report

4 major / 6 minor

Summary. The manuscript extends the first-order Gyulassy-Levai-Vitev (GLV) opacity expansion for transverse momentum broadening to include all-path-length (APL) corrections, which relax the large separation distance approximation, and improved sub-Regge kinematics, which go beyond the strict Regge limit. The paper states analytic expressions for the first-order broadening distribution for four approximation schemes in Table 1, defines the corresponding transport coefficient qhat, and illustrates in Fig. 2 that these corrections can be sizable for short path lengths and large transverse momentum. The central formulas are presented as results, but their derivation is largely deferred to an unpublished self-citation, Ref. [14], and the route from the stated kinematic modifications to the specific correction factors is not shown in this manuscript.

Significance. If the formulas in Table 1 are correct, the paper addresses a timely question: how momentum broadening in the QGP is modified when the medium length is not asymptotically large and when the exchanged transverse momentum is not far below the projectile longitudinal momentum. The observation that APL suppression is partially compensated by sub-Regge enhancement is a potentially useful contribution, and the concise tabulation of the four approximation schemes is convenient. However, the significance is presently conditional: the two multiplicative factors that carry the entire effect, the APL factor (1 - 1/2 e^{-mu_perp Delta z})^2 and the sub-Regge factor 4/(1+gamma)^2 with gamma from Eq. (13), are stated without derivation, and gamma becomes imaginary in an important kinematic region. The paper cannot be fully evaluated until these points are resolved.

major comments (4)

[Section 2 / Table 1] The central results of the paper, the APL factor (1 - 1/2 e^{-mu_perp Delta z})^2 in Table 1 and the sub-Regge factor 4/(1+gamma)^2 with gamma from Eq. (13), are stated without derivation. The text only says that the derivation "closely follows [14]" (Section 2) and that retaining subleading contributions "gives rise to" gamma (Section 3.2). Since Ref. [14] is an unpublished self-citation with a nearly identical title, the present manuscript is not self-contained and the central claim cannot be verified by the reader. Please provide the amplitude-level derivation, or at least the key intermediate steps, showing how these factors arise from the first-order GLV amplitude and the kinematics of Eqs. (11)-(12).
[Section 3.2, Eq. (13)] The gamma factor becomes imaginary for q_perp > P^+/sqrt(2), so the integrals in Table 1 are not defined over the full q_perp plane unless a kinematic cutoff or analytic continuation is specified. This is not a peripheral issue because the paper emphasizes corrections for q_perp ~ P^+; the domain of validity of Eq. (13) and of the final expressions must be stated explicitly.
[Section 4, Fig. 2b] With qhat defined in Eq. (7) as a single number <p_perp^2>/L, the ordinate of Fig. 2b as a function of p_perp is undefined. Please specify the accumulated or differential observable being plotted and the integration limits; otherwise the quantitative statement that qhat is enhanced or suppressed cannot be evaluated.
[Table 1] In all four rows of Table 1 the integrand contains 1/mu_perp^4 and no explicit q_perp dependence, whereas the interaction in Eq. (3) has a momentum-dependent potential 4 pi alpha_s / (q^2 + mu_D^2). Please clarify how the potential has been processed (for example, whether part of the q_perp integral has already been carried out) and whether mu_perp in the table is the same as mu_D; as written, the connection to Eq. (3) is not transparent.

minor comments (6)

[Eq. (4)] The quantity dT in the denominator is not defined; please define it when it is first used (for example, as the dimension of the representation).
[Section 3.1 / Table 1] The text uses mu_D in the exponential e^{-mu_D Delta z}, while Table 1 uses mu_perp; please use a single symbol or define the relation between them.
[Eq. (5)] The source J(p) is referred to but not explicitly defined; since Table 1 uses |J(p-q)|^2 - |J(p)|^2, please specify J or state that it represents a sharply peaked initial distribution.
[Fig. 1 caption] The caption says "at leading order in the opacity" but should read "in the opacity expansion" for consistency with the text.
[Section 4] The word "dissapear" should be "disappear" in the sentence about the corrections disappearing in the appropriate limits.
[Section 4] The final sentence connects the combined correction to the negative energy loss of Ref. [15] in a speculative way; consider labeling this explicitly as a future direction rather than a conclusion of this paper.

Circularity Check

2 steps flagged · score 6.0 of 10

Central derivation is deferred to an unpublished self-citation; the correction factors that drive all results are not derived in this paper.

self citation load bearing [Section 2, after Eq. (7)]
"The derivation presented here closely follows [14], to which we refer the reader for further details."

This sentence transfers the derivation of the paper's central output, the APL and sub-Regge corrected broadening distributions and qhat, entirely to Ref. [14], a companion paper by the same authors. The abstract claims 'we derive analytic expressions', but the text does not reproduce the steps leading to Eq. (13) or Table 1. Ref. [14] is not machine-checked, code-reproduced, or externally verified here, so the only support offered for the load-bearing formulas is the authors' own prior claim. The numerical results and phenomenological conclusions in Figs. 2(a,b) and Section 4 all depend on those formulas; without [14] there is no derivation in this manuscript.
self citation load bearing [Section 3.2, Eq. (13), and Section 4/Table 1]
"Retaining this additional kinematical contribution gives rise to the sub-Regge factor γ = sqrt(1 − 2q_⊥^2 / P^{+2}) ... The APL correction introduces the finite path-length factor (1 − 1/2 e^{−μ_⊥ Δz})^2, while the improved sub-Regge kinematics introduce the multiplicative factor 4/(1+γ)^2."

These two factors are the entire quantitative content of the claimed extension, but they are stated as outcomes rather than derived from the amplitude-level GLV expressions or from the stated momentum replacement Eq. (12). The paper gives no calculation connecting the retained e^{-μ_D Δz} terms to the squared exponential factor, nor the P^+ replacement to the specific square-root gamma. Since the only derivation pointer is the self-citation in Ref. [14], the final expressions in Table 1 are effectively inputs: the standard GLV distribution multiplied by two asserted factors. All reported corrections (suppression/enhancement, sign, p_perp dependence) are consequences of those asserted factors, not of a derivation shown in this paper.

full rationale

The paper's baseline GLV formalism is standard and properly cited to Refs. [12,13], so the baseline is not circular. The circularity concern is concentrated in the new corrections. The manuscript states that the derivation 'closely follows [14]', where Ref. [14] is an unpublished companion paper by the same two authors, and it then presents the corrected formulas (Table 1, Eq. 13) without deriving them. The two correction factors — (1 − 1/2 e^{−μ_⊥ Δz})^2 and 4/(1+γ)^2 with γ = sqrt(1 − 2 q_⊥^2/P^{+2}) — are load-bearing: the figures, the sign of the corrections, and the conclusions about qhat all follow from them. For this paper to be self-contained, it would need to show how these factors arise from the Feynman amplitudes in Fig. 1 after retaining e^{-μ_D Δz} and sub-Regge terms. Instead the only evidence offered for those factors is the self-citation. This is not a case of independent support, because Ref. [14] is not machine-checked, code-reproduced, or shown to rest on assumptions that exclude the target result. I do not assign a higher score (8 or 10) because the formulas could in principle be derived correctly in Ref. [14], and the paper does present a concrete starting point (Eqs. (11)-(12), the retained exponential terms) from which such a derivation could proceed. As written, however, the central 'derivation' reduces to the authors' prior work, making the central claim partially circular and unverifiable from this manuscript alone.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities. The central claim rests on the specific forms of the APL and sub-Regge correction factors, which are asserted without derivation and depend on unstated medium parameters for numerical evaluation.

free parameters (3)

alpha_s (strong coupling) = unspecified
Needed to evaluate the numerical results in Fig. 2, but the paper does not state the value used.
mu_D (Debye screening mass) = unspecified
Needed to evaluate the numerical results in Fig. 2, but the paper does not state the value used.
medium density profile rho(Delta z) = unspecified
The integral over Delta z in Table 1 requires the density profile, which is not specified in the paper.

assumptions (3)

domain assumption Static Debye-screened Gyulassy-Wang potential
Eqs. (2)-(3) in Section 2 model the interaction with scattering centres as a static, Debye-screened Yukawa potential.
domain assumption First-order opacity expansion is sufficient
Section 2 states the calculation is to first order in the GLV opacity expansion, i.e., a single hard scattering with the medium.
domain assumption Eikonal kinematics with modified p^+ assignment
Section 4 replaces the final state momentum p^+ with P^+ + q_z/sqrt(2) in Eq. (12), but the on-shell relation between q_z and q_perp is not derived.

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Cite this review

Pith. "Pith review of Improved Regge Kinematics and All-Path-Length Corrections to Momentum Broadening in the Quark-Gluon Plasma." pith.science (2026). https://pith.science/paper/OJCJBY5D

@misc{pith2026260808894,
  author       = {Pith},
  title        = {Pith review of: Improved Regge Kinematics and All-Path-Length Corrections to Momentum Broadening in the Quark-Gluon Plasma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJCJBY5D}},
  note         = {Machine review of arXiv:2608.08894}
}

abstract

We present a study of transverse momentum broadening for high-energy partons propagating through the quark-gluon plasma (QGP), extending the Gyulassy-Levai-Vitev (GLV) formalism to include both all-path-length (APL) corrections and improved sub-Regge kinematics. The standard GLV framework relies on the large separation distance and large formation time approximations, which are well justified in large nuclei but may become unreliable in small collision systems, where the relevant length and coherence scales are comparable. Working to first order in the opacity expansion, we derive analytic expressions for the transverse momentum broadening distribution and the corresponding momentum transport coefficient, $\hat{q}$, while systematically relaxing these approximations. The APL correction arises from relaxing the large separation distance approximation, whereas the sub-Regge and combined sub-Regge-APL corrections are obtained using an improved set of Regge kinematics beyond the strict Regge limit. We find that these corrections can substantially modify both the transverse momentum broadening distribution and the momentum transport coefficient, particularly for short path lengths and high exchanged momentum, providing a more complete theoretical description of momentum broadening in the QGP.

Figures

Figures reproduced from arXiv: 2608.08894 by the authors.

**Figure 1.** The relevant Feynman diagrams for momentum broadening at leading order in the opacity. allowing the jet transport coefficient to be calculated qˆ = ⟨p 2 ⊥⟩ L . (7) The derivation presented here closely follows [14], to which we refer the reader for further details. 3 Beyond the Standard GLV Approximations The standard GLV formalism relies on several simplifying approximations. In this work we consider two of these: … view at source ↗

**Figure 2.** (a) Momentum broadening distribution and (b) qˆ as functions of the transverse momentum p⊥ for different approximation schemes. 4 Results Using the setup described in the previous section, we derive analytic expressions for the transverse momentum broadening distribution to first order in the opacity expansion for four approximation schemes: the standard GLV result, GLV with all-path-length (APL) corrections, GLV wi… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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