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arxiv: 2606.19221 · v1 · pith:E6N7OMMWnew · submitted 2026-06-17 · ✦ hep-lat · hep-ph· hep-th

The Collins-Soper kernel from a vacuum soft function

Anthony Francis , C.-J. David Lin , Wayne Morris , Yong Zhao This is my paper

Pith reviewed 2026-06-26 18:15 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-th
keywords Collins-Soper kernelvacuum soft functionlattice QCDWilson linesrapidity evolutiontransverse momentum distributions
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The pith

Collins-Soper kernel extracted from vacuum soft function on Euclidean lattice using complex-directional Wilson lines

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

This paper shows that the Collins-Soper kernel can be obtained by computing a vacuum soft function on the lattice with space-like Wilson lines that have complex directions. Pure gauge calculations reach high statistical precision, and the soft function's dependence on rapidity differences follows the expected Collins-Soper evolution over a wide range. The extracted kernel reaches error levels comparable to those from hadronic observables, though it displays saturated behavior at large transverse separations. The approach works entirely in pure gauge theory without hadronic matrix elements.

Core claim

The Collins-Soper kernel is calculated from a vacuum soft function using space-like Wilson lines with complex-directional vectors on the Euclidean lattice. Pure gauge calculations achieve high statistical precision in computing the soft function, whose rapidity dependence is well described by Collins-Soper evolution across a wide range of rapidity differences. The extracted kernel contains errors comparable to those achieved in state-of-the-art lattice calculations based on hadronic observables, but exhibits saturated behavior at large transverse Wilson-line separations.

What carries the argument

Vacuum soft function computed from space-like Wilson lines with complex-directional vectors on the Euclidean lattice

If this is right

  • High statistical precision is reached in pure gauge theory calculations of the soft function.
  • The soft function's rapidity dependence matches Collins-Soper evolution across wide ranges of differences.
  • The kernel errors are comparable to those from lattice calculations that use hadronic observables.
  • Saturated behavior appears at large transverse Wilson-line separations.

Where Pith is reading between the lines

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

  • The method could be extended to full QCD simulations that include dynamical quarks.
  • It offers a route to TMD calculations that avoids constructing hadronic matrix elements.
  • The observed saturation at large separations could be examined in other formulations of the soft function to test its origin.

Load-bearing premise

The rapidity dependence of the computed vacuum soft function follows the Collins-Soper evolution equation sufficiently well to permit reliable extraction of the kernel.

What would settle it

A measurement in which the soft function's rapidity dependence deviates from the Collins-Soper evolution prediction by more than the achieved statistical errors would show that the kernel extraction is not reliable.

Figures

Figures reproduced from arXiv: 2606.19221 by Anthony Francis, C.-J. David Lin, Wayne Morris, Yong Zhao.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the Wilson loops in Eqs. (3), and (12). [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Time dependence of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The CS kernel determined on the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Our result for the CS kernel compared with other [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Feynman diagrams relevant at one-loop in perturba [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Top plot: Double ratio after interpolating data to [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Extracting the CS kernel using two ensembles. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

The Collins-Soper kernel is calculated from a vacuum soft function using space-like Wilson lines with complex-directional vectors on the Euclidean lattice. Our pure gauge calculations with this method achieve high statistical precision in computing the soft function, whose rapidity dependence is well described by Collins-Soper evolution across a wide range of rapidity differences. The extracted kernel contains errors comparable to those achieved in state-of-the-art lattice calculations based on hadronic observables, but exhibits saturated behavior at large transverse Wilson-line separations.

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 / 0 minor

Summary. The manuscript claims to compute the Collins-Soper kernel K(b) from the rapidity dependence of a vacuum soft function S(b,y) evaluated on the Euclidean lattice with space-like Wilson lines using complex directional vectors in pure gauge theory. It reports high statistical precision in the soft function, that its rapidity dependence is well described by Collins-Soper evolution over a wide range, and that the resulting kernel has errors comparable to state-of-the-art hadronic lattice calculations, while noting saturated behavior at large transverse Wilson-line separations.

Significance. If the central assumption holds, the approach supplies a new lattice route to the Collins-Soper kernel that dispenses with hadronic matrix elements and achieves high statistical precision in the underlying soft function; this could simplify future calculations while maintaining competitive errors.

major comments (1)
  1. The extraction of K(b) rests on the assumption that the computed vacuum soft function obeys the Collins-Soper evolution equation d log S / dy = K(b) (plus known perturbative pieces) with sufficient fidelity across the simulated rapidity range. The reported saturation at large transverse separations is not the expected large-b form of the kernel; without explicit tests (e.g., consistency of extracted K(b) across sub-ranges of rapidity or comparison to perturbative expectations at small b) it remains possible that lattice artifacts or the complex Wilson-line prescription introduce systematic deviations that bias the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: The extraction of K(b) rests on the assumption that the computed vacuum soft function obeys the Collins-Soper evolution equation d log S / dy = K(b) (plus known perturbative pieces) with sufficient fidelity across the simulated rapidity range. The reported saturation at large transverse separations is not the expected large-b form of the kernel; without explicit tests (e.g., consistency of extracted K(b) across sub-ranges of rapidity or comparison to perturbative expectations at small b) it remains possible that lattice artifacts or the complex Wilson-line prescription introduce systematic deviations that bias the result.

    Authors: The manuscript already reports that the rapidity dependence of the soft function is well described by Collins-Soper evolution over a wide range. To strengthen the validation against possible systematics from lattice artifacts or the Wilson-line prescription, we will add two explicit tests in the revised manuscript: (i) extraction of K(b) from independent sub-ranges of the simulated rapidity values, demonstrating consistency within statistical errors, and (ii) direct comparison of the extracted kernel at small b against perturbative expectations. These additions will quantify the fidelity of the evolution assumption. We will also expand the discussion of the observed saturation at large transverse separations to note explicitly that it deviates from the expected large-b asymptotic form of the kernel and to comment on possible origins without claiming agreement with that form. revision: yes

Circularity Check

0 steps flagged

No significant circularity: kernel extraction applies standard definition to independently computed lattice soft function

full rationale

The paper computes the vacuum soft function S(b,y) directly on the Euclidean lattice using space-like Wilson lines, then extracts K(b) from the observed rapidity dependence. The abstract states that this dependence 'is well described by Collins-Soper evolution', indicating an empirical check rather than a definitional assumption that forces the result. No quoted equations or self-citations reduce the extracted kernel to a fit or prior result by construction; the lattice computation of S is independent of the evolution equation used for extraction. This is the normal, non-circular case for a first-principles lattice calculation that invokes a known evolution relation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters or invented entities are mentioned. The central extraction step invokes one domain assumption.

axioms (1)
  • domain assumption The rapidity dependence of the soft function follows Collins-Soper evolution
    Used to extract the kernel from the computed soft function across a range of rapidity differences.

pith-pipeline@v0.9.1-grok · 5602 in / 1136 out tokens · 29230 ms · 2026-06-26T18:15:38.900363+00:00 · methodology

discussion (0)

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Reference graph

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