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arxiv: 2510.06390 · v2 · submitted 2025-10-07 · ⚛️ nucl-th

Proper time expansions and glasma dynamics

Margaret E Carrington , Bryce T. Friesen , Doug Pickering , Shane Sangster , Kaene Soopramania This is my paper

Pith reviewed 2026-05-18 09:08 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords glasmaproper time expansionYang-Mills equationsheavy ion collisionsearly time dynamicsclassical field theoryultrarelativistic collisions
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The pith

New methods extend reliable proper time expansions for glasma to 0.08 fm/c

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

The paper explores ways to push the limits of proper time expansions used to solve the classical Yang-Mills equations describing the glasma in the earliest moments of heavy ion collisions. The expansion is inherently limited because the parameter is time itself, and practical computations stop at eighth order due to resource constraints. This restricts accurate results to very early times around 0.05 fm/c. By testing several alternative methods, the authors show that reliable results can be obtained up to approximately 0.08 fm/c, extending the reach by about 1.5 times depending on the specific quantity.

Core claim

The paper claims that by using several different methods to increase the maximum time that can be reached with proper time expansions in solving the classical Yang-Mills equations for the glasma, the latest time for reliable results is extended approximately 1.5 times, from ∼0.05 fm/c to about 0.08 fm/c, depending slightly on the quantity calculated.

What carries the argument

The proper time expansion of solutions to the classical Yang-Mills equation, with new methods to extend its practical range beyond standard eighth-order limits.

Load-bearing premise

The new methods preserve the accuracy of the original proper-time series without introducing uncontrolled errors or instabilities, and that the criteria used to judge reliable results remain valid at the extended times.

What would settle it

A direct numerical integration of the full Yang-Mills equations at t ≈ 0.08 fm/c compared against the extended expansion results to verify agreement within expected errors.

Figures

Figures reproduced from arXiv: 2510.06390 by Bryce T. Friesen, Doug Pickering, Kaene Soopramania, Margaret E Carrington, Shane Sangster.

Figure 1
Figure 1. Figure 1: FIG. 1. The quantity [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The quantity [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The quantity [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The quantity [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The energy density at eight order (red markers) and the result from the second and fourth [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The quantity [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The coefficients [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
read the original abstract

The earliest phase of an ultrarelativistic heavy ion collision can be described as a highly populated system of gluons called glasma. The system's dynamics is governed by the classical Yang-Mills equation. Solutions can be found at early times using a proper time expansion. Since the expansion parameter is the time, this method is necessarily limited to the study of early time dynamics. In addition compute time and memory limitations restrict practical calculations to no more than eighth order in the expansion. The result is that the method produces reliable results only for very early times. In this paper we explore several different methods to increase the maximum time that can be reached. We find that, depending slightly on the quantity being calculated, the latest time for which reliable results are obtained can be extended approximately 1.5 times (from $\sim0.05$~fm/$c$ using previous methods to about $0.08$~fm/$c$).

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.

Referee Report

1 major / 2 minor

Summary. The manuscript explores algorithmic modifications to proper-time expansions of the classical Yang-Mills equations for the glasma in ultrarelativistic heavy-ion collisions. Building on an eighth-order truncation, the authors test several approaches and report that the window of reliable results can be extended by a factor of approximately 1.5, from ~0.05 fm/c to ~0.08 fm/c, with the precise gain depending modestly on the observable considered.

Significance. If the reported extension holds under the internal consistency checks already used for the baseline method, the work supplies a practical, incremental advance that enlarges the temporal overlap between the early-time analytic expansion and later numerical or hydrodynamic stages without requiring prohibitive higher-order terms. The modest, quantity-dependent character of the gain is consistent with the asymptotic nature of the series and is explicitly qualified.

major comments (1)
  1. [Numerical results and reliability criteria] The manuscript should supply a quantitative comparison of the internal consistency measures (e.g., residual norms or convergence indicators) between the baseline eighth-order truncation and each new method at times beyond 0.05 fm/c. Without such side-by-side data it remains unclear whether the 1.5× extension preserves the same accuracy standard or merely postpones the onset of uncontrolled growth.
minor comments (2)
  1. [Abstract] The abstract lists 'several different methods' but does not name them; a brief enumeration in the abstract or introduction would improve readability.
  2. [Figures] Figure captions should explicitly state the order of the truncation and the precise definition of 'reliable' used for each curve.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and for the constructive suggestion regarding internal consistency checks. We agree that a direct quantitative comparison will clarify the reliability of the extended time window and will incorporate the requested data in the revised manuscript.

read point-by-point responses

  1. Referee: [Numerical results and reliability criteria] The manuscript should supply a quantitative comparison of the internal consistency measures (e.g., residual norms or convergence indicators) between the baseline eighth-order truncation and each new method at times beyond 0.05 fm/c. Without such side-by-side data it remains unclear whether the 1.5× extension preserves the same accuracy standard or merely postpones the onset of uncontrolled growth.

    Authors: We agree that side-by-side quantitative comparisons are needed to substantiate the claimed extension. In the revised manuscript we will add figures or tables that directly compare residual norms, truncation-error estimates, and other convergence indicators for the baseline eighth-order expansion against each of the new methods at several times τ > 0.05 fm/c. These data will be presented for the same set of observables already shown in the original manuscript, allowing the reader to verify that the 1.5× gain preserves the original accuracy standard. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained numerical extension

full rationale

The paper describes practical algorithmic modifications to a truncated proper-time series solution of classical Yang-Mills equations for early-time glasma evolution. Reliability is assessed via internal consistency checks already used for the baseline eighth-order truncation, and the reported 1.5× extension of the usable time window is an empirical outcome of those tests rather than a quantity derived from the paper's own equations or prior self-citations. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-definitional relation, or an ansatz smuggled through citation; the work remains a direct computational exploration without tautological closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the glasma is accurately described by the classical Yang-Mills equation at early times and that 'reliable results' can be objectively identified and extended through alternative expansion or resummation techniques. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The system's dynamics is governed by the classical Yang-Mills equation.
    Stated directly in the abstract as the governing equation for glasma evolution.

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Reference graph

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