arxiv: 2604.02050 · v1 · submitted 2026-04-02 · ⚛️ nucl-th · hep-ph

Recognition: 1 theorem link

· Lean Theorem

Gauge invariant momentum broadening of hard probes in glasma

Margaret E. Carrington , Bryce T. Friesen , Stanislaw Mrowczynski

Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:53 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph

keywords glasmajet quenchingmomentum broadeningtransport coefficientheavy-ion collisionsgauge invarianceproper-time expansion

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The pith

A gauge-invariant calculation of momentum broadening in the glasma produces results nearly identical to previous approximations, confirming the glasma's important role in jet quenching.

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

This paper computes the transport coefficient that measures how much transverse momentum hard probes gain while traveling through the glasma formed in the first moments of heavy-ion collisions. It employs a proper-time expansion to handle the very early stage and adopts a fully gauge-invariant approach, unlike an earlier simplified calculation that broke gauge invariance. The new results turn out to be quantitatively very similar to the old ones. This similarity supports the claim that the glasma stage contributes meaningfully to the observed jet quenching in these collisions. Readers should care because accurate modeling of early-time dynamics affects predictions for particle spectra and flow in quark-gluon plasma studies.

Core claim

The authors calculate the momentum broadening coefficient q-hat for hard probes in the evolving glasma using a proper-time expansion in a gauge-invariant formulation. The computed values are quantitatively very close to those obtained in a prior simplified calculation that violated gauge invariance, thereby confirming the earlier conclusion that the glasma plays an important role in jet quenching.

What carries the argument

The gauge-invariant proper-time expansion method for computing the transport coefficient q-hat in the glasma.

If this is right

The glasma contributes substantially to transverse momentum broadening of hard probes.
The gauge-invariant results validate the use of the simplified approximation for quantitative estimates.
Models of jet quenching must account for the early glasma phase to be accurate.
Proper-time expansion provides a reliable tool for early-time glasma dynamics.

Where Pith is reading between the lines

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

Gauge invariance appears to play a minor role in the leading momentum broadening effects at the earliest times.
This approach could be extended to include later evolution stages or hydrodynamic matching.
Experimental measurements of jet quenching at high transverse momenta might be used to constrain glasma parameters.

Load-bearing premise

The proper-time expansion accurately captures the glasma dynamics at the earliest times, and the gauge-invariant formulation does not introduce new approximations that change the results significantly.

What would settle it

If a more complete calculation beyond the proper-time expansion or a non-perturbative simulation yields a momentum broadening coefficient differing by more than 20-30% from these values, the conclusion on glasma importance would be weakened.

Figures

Figures reproduced from arXiv: 2604.02050 by Bryce T. Friesen, Margaret E. Carrington, Stanislaw Mrowczynski.

**Figure 2.** Figure 2: FIG. 2. 1 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The momentum broadening coefficient ˆq [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. A close-up of the central region in Fig. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The momentum broadening coefficient ˆq [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

read the original abstract

We compute the transport coefficient $\hat q$ which quantifies the transverse momentum broadening of hard probes passing through the evolving glasma from the earliest stage of relativistic heavy-ion collisions. We use a proper-time expansion method which is designed to study the glasma at very early times. In our earlier calculations of $\hat q$ we used an approximation that greatly simplifies the complexity of the calculation but introduces a violation of gauge invariance. Based on these results we argued that the glasma plays an important role in jet quenching. In this paper we have used a gauge invariant formulation to calculate $\hat q$. The results for the momentum broadening coefficient are quantitatively very close to those of our previous simplified version of the calculation and confirm our earlier conclusion about the importance of the glasma contribution to jet quenching.

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.

Desk Editor's Note private letter to a colleague

Gauge-invariant recalculation of glasma q-hat gives numbers close to the prior approximation, which keeps the early-time jet quenching claim intact but does not independently test the shared proper-time expansion.

read the letter

The main takeaway is that replacing the gauge-violating approximation with a gauge-invariant formulation for transverse momentum broadening in the glasma produces results that are quantitatively very close to the earlier ones. This keeps the conclusion that the glasma contributes measurably to jet quenching at the earliest times on the same footing as before. The authors have extended their own proper-time expansion method to enforce gauge invariance while staying inside the same framework, and the abstract presents the numerical agreement as the central outcome. That consistency check addresses a clear technical gap in the prior work. The limitation is that both calculations rely on the same proper-time expansion at small tau, so the closeness could reflect common truncation effects rather than robustness. The abstract does not supply error estimates, convergence tests, or direct comparisons to full Yang-Mills evolution, which leaves the accuracy of the expansion at the earliest times unverified beyond the internal check. This paper is for specialists working on jet quenching phenomenology and the pre-equilibrium stage in heavy-ion collisions. It refines an existing line of calculation rather than introducing a new method or broad new physics. I would send it for peer review. The gauge-invariant version is a legitimate technical improvement worth documenting, and referees can evaluate the details of the formulation and the strength of the supporting evidence.

Referee Report

2 major / 1 minor

Summary. The manuscript computes the jet quenching transport coefficient q-hat for hard probes traversing the evolving glasma at the earliest proper times in relativistic heavy-ion collisions. It employs a proper-time expansion method in a gauge-invariant formulation, reports that the resulting momentum broadening is quantitatively close to the values obtained in an earlier gauge-violating approximation, and concludes that this confirms the glasma's important role in jet quenching.

Significance. If the quantitative agreement survives scrutiny, the work supplies a gauge-invariant corroboration of earlier claims that glasma fields contribute substantially to transverse momentum broadening at early times. This would strengthen the case for including glasma dynamics in phenomenological jet-quenching models. The methodological advance of a gauge-invariant proper-time expansion is a clear technical improvement over the prior approximation.

major comments (2)

[Abstract] Abstract: the central claim that the gauge-invariant results are 'quantitatively very close' to the previous approximation and thereby confirm the glasma's importance rests on the proper-time expansion remaining accurate at the smallest tau. No error estimate, convergence test, or comparison to full Yang-Mills evolution is supplied to exclude the possibility that the agreement reflects shared truncation error rather than robustness to gauge violation.
[Method] Method (proper-time expansion): the formulation is stated to be 'designed to study the glasma at very early times,' yet the manuscript provides no independent cross-check (e.g., comparison of leading-order versus next-order terms or benchmark against known limits) that would establish the truncation error at the earliest times where the glasma contribution is claimed to be dominant.

minor comments (1)

[Abstract] Notation for the transport coefficient should be introduced once and used consistently; the symbol q-hat appears without an explicit definition equation in the abstract.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful review and the positive evaluation of the significance of our work. We address the major comments below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the gauge-invariant results are 'quantitatively very close' to the previous approximation and thereby confirm the glasma's importance rests on the proper-time expansion remaining accurate at the smallest tau. No error estimate, convergence test, or comparison to full Yang-Mills evolution is supplied to exclude the possibility that the agreement reflects shared truncation error rather than robustness to gauge violation.

Authors: We acknowledge that an explicit error estimate or convergence test would provide stronger support for the accuracy of the proper-time expansion at the smallest proper times. The expansion is organized in powers of tau, with higher-order terms becoming negligible as tau approaches zero, which is the regime where the glasma contribution is most relevant. The fact that the gauge-invariant calculation yields results quantitatively close to the earlier gauge-violating approximation indicates that the momentum broadening is not sensitive to the gauge violation in this limit. In the revised version, we will add a paragraph discussing the expected size of the truncation error based on the structure of the expansion. revision: partial
Referee: [Method] Method (proper-time expansion): the formulation is stated to be 'designed to study the glasma at very early times,' yet the manuscript provides no independent cross-check (e.g., comparison of leading-order versus next-order terms or benchmark against known limits) that would establish the truncation error at the earliest times where the glasma contribution is claimed to be dominant.

Authors: The proper-time expansion method has been developed and tested in previous works on glasma dynamics, where it has been shown to reproduce known analytic limits at early times, such as the initial energy density and field configurations from the McLerran-Venugopalan model. While a direct comparison of leading versus next-to-leading order terms for the specific observable q-hat is not included in the current manuscript, the agreement between the gauge-invariant and approximate calculations serves as an indirect consistency check. We will include additional references to the validation of the method in the revised manuscript and clarify the regime of validity. revision: partial

standing simulated objections not resolved

A direct numerical comparison to full Yang-Mills evolution to quantify the truncation error at the smallest tau

Circularity Check

0 steps flagged

Gauge-invariant recalculation is independent; closeness to prior results is an outcome, not an input

full rationale

The paper executes a fresh gauge-invariant computation of the momentum broadening coefficient using the proper-time expansion and reports that the numerical values are close to those from the authors' earlier gauge-violating approximation. This closeness is presented as a result of the new calculation rather than being imposed by any equation or fit. The derivation chain does not reduce to a self-citation or to a previously fitted parameter by construction. Self-citation to the prior work exists but is not load-bearing; the central claim rests on the new, gauge-invariant evaluation itself. No self-definitional, fitted-input, or ansatz-smuggling steps are present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit list of free parameters or axioms; the calculation is described only at the level of method choice and gauge invariance.

pith-pipeline@v0.9.0 · 5434 in / 1028 out tokens · 24201 ms · 2026-05-13T20:53:32.537500+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We use a proper-time expansion method which is designed to study the glasma at very early times... gauge invariant formulation to calculate ˆq

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Light-front Hamiltonian jet evolution in the Glasma
hep-ph 2026-05 unverdicted novelty 7.0

A light-front Hamiltonian method evolves a quark through Glasma fields to obtain transverse momentum broadening and jet quenching consistent with classical scaling in saturation momentum.
Quantum simulating multi-particle processes in high energy nuclear physics: dijet production and color (de)coherence
hep-ph 2026-04 unverdicted novelty 6.0

A framework is developed that encodes leading-order QCD antenna and dipole processes as quantum circuits, with benchmarks against analytic limits in simplified media.
Kinetic and canonical momentum broadening in the Glasma
hep-ph 2026-04 unverdicted novelty 6.0

Derives gauge-invariant equations of motion for kinetic and canonical momentum of particles in a classical non-Abelian background, finding that transverse fields contribute to kinetic momentum broadening even in the e...

Reference graph

Works this paper leans on

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These hard partons propagate through the system that is produced in the collision and lose a substantial fraction of their initial energy

INTRODUCTION Hard probes are created through high momentum transfer processes at the earliest stage of a heavy-ion collision. These hard partons propagate through the system that is produced in the collision and lose a substantial fraction of their initial energy. This energy loss is responsible for the phenomenon of jet quenching which is treated as a si...

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FORMULATION OF THE PROBLEM We calculate the coefficient ˆqthat determines the transverse momentum broadening of a hard probe as ˆq= 2 v δαβ − vαvβ v2 X αβ(⃗ v) (2.1) where⃗ vis the velocity of the probe. The tensorX αβ is defined as X αβ(⃗ v) =g2 2Nc Z t 0 dt′ F i a(t, ⃗ x)Wab(t, ⃗ x|t′, ⃗ y)Fj b (t′, ⃗ y) ,(2.2) where⃗ x−⃗ y=⃗ v(t−t′), ⃗Fa(t, ⃗ x) =g ⃗Ea...

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