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Modified Coherence and the Transverse Extent of Jets

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Pith's one-line read The paper claims that the effective jet-medium coupling weakens continuously with parton virtuality because the medium's gluon transverse-momentum distribution cannot keep up with the shrinking radiating antenna.

desk verdict A useful, clearly written first pass at a qhat–TMDPDF matching, with a new coordinate-space shower-size estimate, but the central relation is a matching definition and the numerical weakening is illustrative rather than established. read the letter →

arxiv 2501.07823 v1 pith:4FMDGMX2 submitted 2025-01-14 hep-ph nucl-th

Amit Kumar , Abhijit Majumder , Chathuranga Sirimanna , Yasuki Tachibana This is my paper · ORCID
classification hep-phnucl-th
keywords jetquenchingtransportcoefficienttransversemomentumdependentPDFmodifiedcoherencehigher-twistformalismpartonshowersizequantumuncertaintydeepinelasticscattering
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 tries to establish that the apparent suppression of high-virtuality jet interactions with a medium is not an all-or-nothing coherence threshold but a continuous, calculable weakening. It derives the quantum uncertainty size of a splitting parton, shows that full jet showers are wider than classical antenna estimates, and links the jet transport coefficient to a gluon transverse-momentum-dependent parton distribution function (TMDPDF). Solving that link with a simple Gaussian TMDPDF model yields a transport coefficient that drops steeply as the splitting scale rises from about 3 to 9 GeV. The authors name this gradual scale-dependent suppression "modified coherence."

What carries the argument

The load-bearing identity is Eq. (49), a quotient: in the numerator the hard emission factor $h(\vec{k}_\perp, q^-, y, \vec{\ell}_\perp)$ from the higher-twist kernel is integrated against the medium's gluon TMDPDF $G(x_L, \vec{k}_\perp, \mu^2)$, while the denominator is $\vec{\nabla}^2_{k_\perp} h$ evaluated at $k_\perp = 0$. This equation converts the TMDPDF into a scale-dependent $\hat{q}$, and solving it produces the falling $f(\mu^2)$. A second piece of machinery is the wave-packet re-derivation of single-gluon emission, which extracts the uncertainty width $\delta z_\perp \sim 1/\ell_\perp$ at the split point; that width is inserted into a Monte-Carlo shower to obtain the transverse-size distributions. The final piece is the Gaussian ansatz for the gluon TMDPDF with a momentum-fraction-dependent width, which supplies the numerical input that makes the drop quantitative.

What would settle it

Compute Eq. (49) with a fully QCD-evolved gluon TMDPDF and the complete next-to-leading-twist hard kernel; if the resulting $f(\mu^2)$ no longer falls steeply between 3.1 and 9 GeV, or if it depends strongly on the chosen form of the hard function $h$, the modified-coherence interpretation fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is Eq. (49): the jet transport coefficient $\hat{q}(x_L, \ell_\perp^2)$ equals the full induced-gluon kernel, integrated against the gluon TMDPDF $G$, divided by the second derivative of the hard factor $h$ at zero exchanged momentum $k_\perp$. Equating the complete kernel with its Taylor-expanded form defines $\hat{q}$ as a scale- and momentum-fraction-dependent quantity rather than a constant medium parameter. Solving this relation with a Gaussian gluon TMDPDF and a leading-order gluon PDF gives a scaling function $f(\mu^2)$ that drops steeply as the splitting scale rises from 3.1 to 9 GeV. The accompanying wave-packet calculation gives the quantum uncertainty size $\delta z_\perp \sim 1/\ell_\perp$ for a splitting parton, and a Monte-Carlo shower carrying that size produces transverse shower radii around 2 fm at 5 fm/c, with no clean relation between the full shower width and the originating parton's virtuality.

Load-bearing premise

The load-bearing premise is that the quantity defined by Eq. (49) is a genuine, process-independent medium transport coefficient, not a ratio that merely absorbs whatever the chosen splitting kernel and Gaussian gluon model happen to put there.

Editorial extensions

If this is right

  • The effective $\hat{q}$ used in jet-quenching simulations should inherit a falling $f(\mu^2)$ from the gluon TMDPDF rather than being treated as a constant fit parameter.
  • Above roughly $\mu \sim 9$ GeV for the kinematics studied, the medium-induced emission rate is strongly suppressed, so the high-virtuality stage of a jet is nearly vacuum-like in a continuous way rather than because of a hard resolution threshold.
  • The radiating system's size $\delta z_\perp \sim 1/\ell_\perp$ makes jets substantially wider objects than antenna estimates, so a larger portion of the medium is probed by the soft periphery of the jet.
  • Because shower width does not track initial virtuality in a simple way, observables that measure jet broadening cannot be inverted straightforwardly to infer the originating parton's virtuality.
  • Because Eq. (49) is presented as true for any choice of the hard function, the same method can be recycled with different emission kernels and different media to generate scale-dependent transport coefficients.

Reading between the lines

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

  • If Eq. (49) survives more realistic input, the same relation can be run in reverse: the measured scale dependence of $\hat{q}$ in a quark-gluon plasma becomes a probe of that plasma's effective gluon transverse-momentum distribution, which has no direct definition today.
  • The Monte-Carlo result that jets are several fm wide at 5 fm/c suggests that the soft periphery of a jet deposits energy into a much larger volume of the medium than antenna-based simulations assume, so jet wakes and medium response should be correspondingly broader.
  • A discriminating test, not performed in the paper, is to solve Eq. (49) with a fully QCD-evolved gluon TMDPDF: if the steep drop in $f(\mu^2)$ persists, modified coherence is a robust kinematic feature of the TMDPDF; if it mostly disappears, much of the advertised weakening is an artifact of the unevolved Gaussian model.
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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 / 4 minor

Summary. The paper has two main parts. In Secs. II–III the authors re-derive the single-gluon emission cross section in deep-inelastic scattering using incoming wave-packets, extract the transverse quantum-uncertainty size of the splitting quark, δz⊥ ~ 1/ℓ⊥, and implement this size in the MATTER Monte-Carlo generator to compute the transverse radius of 100 GeV quark showers at t = 5 fm/c. They find that the quantum-uncertainty size is larger than the classical antenna size and that the total shower size is not tightly correlated with the initial parton virtuality. In Secs. IV–V the authors reconsider the medium-modified emission kernel at next-to-leading twist, retain the k⊥ and xL dependence of the gluon TMDPDF, and, by comparing the full kernel with its collinear expansion, obtain Eq. (49) relating the jet transport coefficient q̂ to the gluon TMDPDF. Using a non-evolved BLNY/MSTW Gaussian TMDPDF at fixed y = 1/2 and q− = 200√2 GeV, they compute a scale-dependent factor f(μ²) that drops steeply from μ = 3.1 GeV to 9 GeV, a continuous weakening that the paper calls “modified coherence.”

Significance. If Eq. (49) were a robust q̂–TMDPDF relation and if the numerical drop in f(μ²) reflected a genuine medium property, the paper would be a valuable contribution: it would connect q̂ to the gluon TMDPDF in a way that can be improved with standard TMD evolution, and it would offer an alternative to the threshold-like coherence picture of jet quenching. The wave-packet derivation of the uncertainty size and the explicit Monte-Carlo implementation are also useful, and the authors are transparent that the final numerical model is simplified for reproducibility. The main strength is that Eq. (49) is stated in a form that can be tested with different hard kernels and different TMDPDF inputs; the main weakness is that the manuscript does not yet perform those tests, so the central claim currently rests on a matching definition and a single non-evolved input model.

major comments (4)
  1. [Sec. IV.C, Eq. (49)] The derivation of the q̂–TMDPDF relation is a matching construction rather than an independent derivation. The full kernel in Eq. (46) and its second-order Taylor expansion in Eq. (48) are set equal and solved for q̂; consequently Eq. (49) is true by construction for any h and any G. The statement in Sec. IV.C that the relation is “true for any choice of h” therefore does not by itself establish that the object in Eq. (49) is a universal transport coefficient. Since xL = ℓ⊥²/[2q−P+y(1−y)] enters G(xL, k⊥, ℓ⊥²), and h depends on y and q−, q̂ in Eq. (49) manifestly depends on the hard-splitting kinematics. The paper fixes y = 1/2 and q− = 200√2 GeV in Sec. V and provides no check that q̂ at fixed μ² is independent of y, q−, or of the infrared cutoff ΛQCD = 0.2 GeV used in the k⊥ integral. Without such a check, the computed f(μ²) cannot be interpreted as a scale-dependent property of the medium rather than a property of the chosen matching point.
  2. [Sec. V, Eqs. (51)–(53) and Fig. 10] The numerical evidence for the central “modified coherence” claim rests on an un-evolved Gaussian TMDPDF whose x-dependence and width dominate the result. The drop in f(μ²) for μ ≳ 6 GeV is driven by xL = ℓ⊥²/100 rising toward the steeply falling large-x region of the MSTW input F(xL), while the width w(xL, μ²) is fixed by the BLNY ansatz with no Collins–Soper evolution. The authors assert that evolution would produce “a quantitative, but not a qualitative, shift,” but no calculation or argument is supplied; for μ² between roughly 10 and 80 GeV², TMD evolution is not negligible. The conclusion that the weakening is a universal medium effect would require repeating the calculation with an evolved TMDPDF, with alternative PDF inputs, and with the kinematics varied so that the result can be examined at fixed xL rather than at fixed y = 1/2.
  3. [Sec. IV.C, Eq. (50)] The denominator of Eq. (49) is evaluated with the approximate result Eq. (50), which follows from the phase-derivative-neglecting collinear expansion of Ref. [24], while the numerator uses the full h from the second line of Eq. (41). The manuscript cites Ref. [30] to argue that the method of Ref. [24] is close to the complete next-to-leading-twist result at τ = 2τf, but it does not demonstrate that Eq. (49) is stable when the full kernel of Ref. [30] is used consistently in both numerator and denominator. Because Eq. (49) is a ratio of a k⊥-integral of hG to ∇²h|_{k⊥=0}, a partial use of an approximate h could bias the scale dependence of q̂.
  4. [Sec. III, Eq. (33) and Figs. 4–7] The Monte-Carlo claim that quantum uncertainty makes jets several femtometers wide depends on assigning the single-split uncertainty δr⊥ to every parton at the observation time, including partons that have not split and on-shell daughters after the split. Equation (31) is derived for the parent quark at the moment of a split, not as a time-dependent size of arbitrary partons. The measure in Eq. (33) also combines this uncertainty with classical antenna positions, so the statement that there is no clear relation between virtuality and transverse size is tied to this particular prescription. The authors should state more carefully what fraction of the width in Figs. 4–7 comes from the added δr⊥ term and what fraction comes from the antenna coordinates.
minor comments (4)
  1. [General] The text contains several typos (“spilts,” “preceeding,” “naïve,” “vanishies”) and Ref. [49] duplicates Ref. [29]; these should be corrected in a revision.
  2. [Sec. II, Eqs. (27)–(31)] The infrared regulator ε appears as log(1/ε) in Eqs. (27)–(30) and is said to have no significance, but the final relation Eq. (31) is obtained by cancelling these logarithms; a sentence explaining the cancellation would improve clarity.
  3. [Fig. 5 caption] The caption of Fig. 5 should state whether the plotted distributions are normalized to unit area; without this information it is difficult to compare the widths across virtuality bins.
  4. [Sec. II, text below Eq. (22)] The assumption that the parent quark transverse momentum distribution is much narrower than the radiated gluon’s is stated in the text, but it is a strong approximation; the derivation of the uncertainty size should note where this assumption breaks down at small ℓ⊥.

Circularity Check

1 steps flagged · score 6.0 of 10

Eq. (49) defines ĥ as the ratio that forces the Taylor-expanded kernel Eq. (48) to equal the full kernel Eq. (46); the plotted drop of f(μ^2) is a consequence of that definition plus the chosen TMD/h, not an independent first-principles prediction.

  1. self definitional [Sec. IV.C, Eq. (49); used in Sec. V, Eq. (54) and Fig. 10.]
    "Thus, to obtain the full scale (size) dependence of ĥ, we simply compare Eq. (46) with Eq. (48), to obtain, ĥ(x_L, ℓ_⊥^2) = [∫ d^2k_⊥ ρ x_L/(4 k_⊥^2) h(k⃗_⊥, q^-, y, ℓ⃗_⊥) G(x_L, k⃗_⊥, ℓ_⊥^2)] / [∇⃗^2_{k_⊥} h(k⃗_⊥, q^-, y, ℓ⃗_⊥)|_{k_⊥=0}]. The equation above, as written, is more general than the higher-twist approach used to derive it. It directly relates the TMDPDF of a given medium with the transverse momentum coefficient ĥ, and is true for any choice of h."

    By construction, ĥ is the ratio that makes the Taylor-expanded spectrum Eq. (48) exactly equal to the unexpanded kernel Eq. (46); it is not defined from any independent broadening measurement. Thus the f(μ^2) obtained by “solving” Eq. (49) is a bookkeeping identity for the chosen h and G. The statement that the relation is “true for any choice of h” is the signature of a definition, not a derived universal law. The drop with μ is driven by the input x_L = ℓ_⊥^2/[2q^-P^+y(1-y)], the non-evolved Gaussian width, and F(x_L) at fixed y=1/2, q^-=200√2 GeV. No h-independence or kinematic independence is shown, so the “modified coherence” weakening reduces to the defining relation and inputs.

full rationale

The load-bearing circular step is the construction of ĥ in Eq. (49). Equating the full induced-gluon kernel Eq. (46) with the Taylor-expanded form Eq. (48) defines ĥ as the matching ratio between the two; the subsequent evaluation of this ratio with a Gaussian BLNY/MSTW TMD and the fixed h of Ref. [24] yields the f(μ^2) curve. The paper presents this as solving for the scale dependence of the transport coefficient, but the quantity plotted is an effective coefficient defined to make the collinear expansion exact, so the rapid drop with μ is not an independent prediction. The paper’s caveats—no Collins-Soper evolution, only a non-evolved TMD, and a limited 3.1–9 GeV window—reinforce that the curve is model-limited. Other parts of the paper are not circular: the transverse uncertainty size δz_⊥ is computed in a wave-packet calculation, and the larger Monte-Carlo transverse size follows by including the positive uncertainty term in the definition of r_⊥; this is presentation rather than a hidden circular claim. The consistency citations to Refs. [38,44] (partially overlapping authorship) are supportive but not load-bearing for the derivation of Eq. (49). The main defect is definitional: if one calls the matching ratio in Eq. (49) ĥ, then its weakening is automatic, while the physically relevant universality of ĥ across h, y, and q^- is asserted but not established. This warrants a partial-circularity score of 6.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central numerical results rest on externally fitted TMDPDF and PDF parameters, a Gaussian ansatz, and a matching condition that defines qhat through the ratio of the full kernel to its collinear expansion. No new physical entities are introduced; the wave-packet width choices drop out or are controlled by assumptions, but the BLNY/MSTW parameters and the IR cutoff directly set the shape of f(mu^2).

free parameters (5)
  • BLNY TMDPDF width parameters = g1 = 0.21 GeV^2, g2 = 0.68 GeV^2, g3 = 1.6, Q0 = 1.6 GeV
    Eq. (52), taken from BLNY fits to Drell-Yan data; controls the Gaussian width w(x, mu^2) that drives the drop of f(mu^2) in Fig. 10.
  • MSTW leading-order PDF parameters = A = 0.0012216, delta = -1.83657, eta = 2.3882, epsilon = -38.997, g = 1445.5
    Eq. (53), from the MSTW global fit; the xL dependence of F(xL) controls the sharp drop at high mu.
  • k_perp infrared cutoff = Lambda_QCD = 0.2 GeV
    Sec. V, chosen by hand as the IR regulator for the d^2 k_perp integral in Eq. (49); affects the normalization of qhat.
  • Normalization scale for f(mu^2) = mu0 = 3.1 GeV
    Sec. V, f is defined to be 1 at mu = 3.1 GeV because lower scales are excluded as needing more careful numerical treatment.
  • Monte-Carlo initialization parameters = E = 100 GeV, Q_max ~ 50 GeV, Q0 ~ 1 GeV, t = 5 fm/c, pT cuts 0, 2, 10 GeV
    Sec. III, simulation parameters for the MATTER runs that set the transverse-size distributions in Figs. 4-7.
assumptions (6)
  • domain assumption Higher-twist factorization of the A-DIS hadronic tensor into a hard part and nuclear matrix elements
    Invoked in Sec. IV.A, Eq. (38), following Refs. [23,24,30]; the whole qhat derivation lives inside this factorization.
  • domain assumption Translation invariance of the nuclear correlator and decomposition into nucleon states via the approximation in Eq. (45)
    Used in Sec. IV.B, Eqs. (44)-(45); it lets the xi^- integral complete and turns the soft factor into a nucleon TMDPDF.
  • domain assumption The collinear expansion of h around k_perp = 0 is valid for the high-virtuality kernel
    Sec. IV.C, Eq. (47), uses only the Laplacian of h at k_perp = 0; the authors note criticism in Refs. [61,62] and rely on Ref. [30] that the method is close at tau = 2 tau_f.
  • ad hoc to paper The gluon TMDPDF is the non-evolved BLNY Gaussian ansatz with MSTW LO PDF input
    Sec. V, Eqs. (51)-(53); the parameters are externally fitted and no Collins-Soper evolution is applied. The authors state that evolution would shift results quantitatively but not qualitatively, without demonstrating this.
  • domain assumption The incoming proton is replaced by a Gaussian wave-packet with widths Delta_+ and Delta_perp, with Delta approximately lambda Q and lambda Q much larger than Lambda_QCD
    Sec. II.A, Eq. (11); this defines the uncertainty size used throughout the paper.
  • ad hoc to paper The parent quark's average transverse momentum distribution is much narrower than the radiated gluon's, so pbar_perp is ignored relative to l_perp and lq_perp
    Sec. II.B, text before Eq. (22), justified by choosing Delta_perp narrow; this controls the delta z_perp derivation.

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Pith. "Pith review of Modified Coherence and the Transverse Extent of Jets." pith.science (2026). https://pith.science/paper/4FMDGMX2

@misc{pith2026250107823,
  author       = {Pith},
  title        = {Pith review of: Modified Coherence and the Transverse Extent of Jets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4FMDGMX2}},
  note         = {Machine review of arXiv:2501.07823}
}
abstract

We present a study of the transverse size of parton showers and their diminishing interaction with the medium in the high virtuality stage of jet evolution. We consider the process of a hard quark produced in deep inelastic scattering off a large nucleus. Single gluon radiation from this quark, in the absence of scattering, is re-derived using wave-packets. This allows for a derivation of the quantum uncertainty size of the hard quark, at the point of splitting. This uncertainty size is then incorporated within a Monte-Carlo shower routine yielding transverse shower sizes noticeably larger than the classical antenna size of the shower. No clear relation is found between the full uncertainty size of the shower and the virtuality of the originating parton. The single gluon emission from the hard quark is then re-analysed for the case of single rescattering off the remainder of the nucleus. A relation is derived between the jet transport coefficient $\hat{q}$ and the gluon Transverse Momentum Dependent Parton Distribution Function (gTMDPDF). Solving this relation, for a simple case, clearly demonstrates the weakening of $\hat{q}$ with the virtuality of the hard splitting parton.

Figures

Figures reproduced from arXiv: 2501.07823 by the authors.

Figure 1
Figure 1. FIG. 1. Single gluon emission from the outgoing struck quark [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The antenna picture of a developing parton shower, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Comparison of the transverse size of a developing [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figures from the paper (4 more)
Figure 6
Figure 6. Figure 6: FIG. 6. The transverse size of the jet shower with a minimum [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Single gluon emission from the outgoing struck quark [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (Color Online) A plot of the hard portion (2nd line) [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. A plot of the scaling function for ˆq [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

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Forward citations

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