arxiv: 2605.06242 · v1 · submitted 2026-05-07 · ✦ hep-ph · hep-lat

Recognition: unknown

Quark-gluon vertex in the complex plane

M.N. Ferreira , A.S. Miramontes , J.M. Morgado , J. Papavassiliou

Authors on Pith no claims yet

Pith reviewed 2026-05-08 08:26 UTC · model grok-4.3

classification ✦ hep-ph hep-lat

keywords quark-gluon vertexcomplex planenonperturbative QCDvertex form factorssoft-gluon limitanalytic continuationWick rotation

0 comments

The pith

Complexifying the momentum variable in the quark-gluon vertex integrals yields all eight form factors inside a parabolic domain in the complex plane.

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

The paper explores the nonperturbative quark-gluon vertex by taking its defining integrals into the complex plane for the first time. In the soft-gluon limit, where the transversely projected vertex depends on only one momentum variable, this complexification produces all eight form factors unambiguously inside a specific region. A sympathetic reader would care because the result extends symmetry-preserving methods for meson properties beyond the rainbow-ladder approximation to the complex momenta required for accurate bound-state calculations. The reliable domain ends at the first singularity in the integrands, after which standard extrapolation can be applied until physical-process structures appear.

Core claim

In the soft-gluon limit of the transversely-projected quark-gluon vertex, complexifying the single momentum variable inside the defining integrals provides all eight vertex form factors unambiguously within a domain of the complex plane delimited by a characteristic parabola. The reliable domain ends at the first singularity in the integrands, where the standard Wick rotation must be supplemented by additional contributions. Standard extrapolation methods can extend this region until complex structures associated with the onset of physical processes appear.

What carries the argument

The transversely-projected quark-gluon vertex in the soft-gluon limit, reduced to dependence on a single momentum variable, with analytic continuation achieved by direct complexification of the integration variable.

If this is right

The approach generalizes to arbitrary gluon momenta.
It supports determination of the quark propagator in the complex plane.
The form factors enable meson studies beyond rainbow-ladder truncation.
Extrapolation extends the domain up to the appearance of structures from physical processes.

Where Pith is reading between the lines

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

The method could improve off-shell quark modeling in processes such as deep-inelastic scattering or hadron decays.
Combining these complex-plane vertices with Dyson-Schwinger studies of other Green functions may yield consistent nonperturbative frameworks for more observables.
Direct comparison with lattice data for the vertex at complex momenta would test the parabolic boundary and singularity handling.

Load-bearing premise

The first singularity encountered can be handled by supplementing the Wick rotation with additional contributions, and standard extrapolation methods remain valid until the onset of complex structures associated with physical processes.

What would settle it

A lattice computation or direct numerical evaluation showing that the eight form factors become inconsistent or diverge immediately after the parabolic boundary without the expected supplementary contributions from the Wick rotation.

Figures

Figures reproduced from arXiv: 2605.06242 by A.S. Miramontes, J.M. Morgado, J. Papavassiliou, M.N. Ferreira.

**Figure 1.** Figure 1: FIG. 1. The SDE of the quark-gluon vertex (cyan circle), defined by the substitution Γ view at source ↗

**Figure 2.** Figure 2: FIG. 2. Left: The function view at source ↗

**Figure 3.** Figure 3: FIG. 3. The positions of the poles view at source ↗

**Figure 4.** Figure 4: FIG. 4. Domain of validity of the standard Euclidean integral in the complex view at source ↗

**Figure 5.** Figure 5: FIG. 5. Left view at source ↗

**Figure 6.** Figure 6: FIG. 6. Real (left) and imaginary (right) parts of view at source ↗

**Figure 7.** Figure 7: FIG. 7. The external input functions ∆( view at source ↗

**Figure 8.** Figure 8: FIG. 8. Real (left) and imaginary (right) parts of the chiral symmetry preserving form factors view at source ↗

**Figure 9.** Figure 9: FIG. 9. Real (left) and imaginary (right) parts of the chiral symmetry preserving form factors view at source ↗

**Figure 10.** Figure 10: FIG. 10. Real (left) and imaginary (right) parts of the chiral symmetry breaking form factors view at source ↗

read the original abstract

In the present work we explore for the first time the general structure and properties of the nonperturbative quark-gluon vertex in the complex plane. Specifically, we focus on the transversely-projected quark-gluon vertex that emerges from a recently developed symmetry-preserving approach for the study of meson properties beyond the rainbow-ladder approximation. The analysis focuses on the so-called "soft-gluon" limit, which reduces the momentum-dependence of the corresponding vertex form factors to a single momentum variable. The complexification of this variable inside the defining integrals furnishes unambiguously all eight vertex form factors within a concrete domain of the complex variable, delimited by a characteristic parabola. The extent of this reliable domain is determined by the appearance of the first singularity in the integrands of the vertex integrals, where the standard Wick rotation must be duly supplemented by additional crucial contributions. This primary analytic region may be extended considerably by resorting to standard extrapolation methods, which remain valid up until the appearance of complex structures associated with the onset of physical processes. The generalization of the method to arbitrary gluon momenta, and its relevance for the determination of the quark propagator in the complex plane, are briefly discussed.

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

They've evaluated all eight form factors of the transversely projected quark-gluon vertex at complex momenta in the soft-gluon limit by direct integration, but the handling of the first singularity and the extrapolation step lack explicit checks.

read the letter

This paper takes the transversely projected quark-gluon vertex from their symmetry-preserving Dyson-Schwinger construction and evaluates its eight form factors after complexifying the single momentum variable in the soft-gluon limit. The result is a concrete parabolic domain where the integrals can be evaluated directly before the first singularity forces a supplemented Wick rotation, plus a discussion of how standard extrapolation can push further until physical thresholds appear. That is the main new piece: an explicit, parameter-free extension of this vertex into the complex plane for the first time. The approach stays tied to the real-axis results and flags the contour adjustment at the singularity without introducing new parameters. The brief remarks on generalizing to arbitrary gluon momenta and the quark propagator show they see the connection to broader applications in meson studies. The limitation is that the abstract supplies no numerical values, no plots of the continued form factors, and no tests confirming that the supplemented integrals preserve transversality, obey dispersion relations, or recover the perturbative limit in overlapping regions. The extrapolation claim therefore rests on the assumption that standard methods remain faithful without artifacts, which the stress-test note correctly identifies as the weakest link. Without benchmarks or independent cross-checks, it is difficult to gauge how far the reliable domain actually extends. This is for the narrow group already running beyond-rainbow-ladder Dyson-Schwinger calculations of mesons who need a complex-plane vertex as input. A reader in that niche will get a usable starting domain and a clear statement of the technical obstacles. I would send it to peer review. The novelty of the direct complexification and the parameter-free character make it worth referee attention, even if the validation steps need to be added or strengthened in revision.

Referee Report

2 major / 2 minor

Summary. The paper explores the nonperturbative transversely-projected quark-gluon vertex in the complex plane using a symmetry-preserving truncation of the Dyson-Schwinger equations. In the soft-gluon limit, where the vertex depends on a single momentum variable, the authors complexify this variable inside the defining integrals to obtain all eight form factors. They identify a parabolic domain of reliability delimited by the first integrand singularity, supplement the standard Wick rotation with additional contributions at that point, and propose extending the domain via standard extrapolation methods until the onset of physical thresholds. The generalization to arbitrary gluon momenta and implications for the complex quark propagator are briefly discussed.

Significance. If the supplemented contour and extrapolation procedure are shown to be faithful, the work supplies a parameter-free route to the quark-gluon vertex at complex momenta. This is a load-bearing ingredient for symmetry-preserving calculations of meson properties beyond the rainbow-ladder truncation, where the vertex enters the Bethe-Salpeter kernel. The absence of free parameters and the preservation of the underlying symmetries are concrete strengths that could improve predictions for spectra, decay constants, and form factors once the complex-plane results are validated.

major comments (2)

The central claim that supplementing the Wick rotation with additional contributions at the first singularity, followed by extrapolation, reproduces the correct analytic continuation of all eight form factors is not accompanied by explicit verification. No checks are presented that the resulting functions satisfy dispersion relations, preserve transversality, or reproduce known perturbative limits in overlapping regions of the complex plane.
The manuscript provides no numerical results, error estimates, or direct comparisons with independent determinations of the vertex (e.g., lattice data or other truncations) inside the claimed parabolic domain. Without such benchmarks, the assertion that the eight form factors are furnished 'unambiguously' remains untested.

minor comments (2)

The abstract and introductory sections would benefit from a clearer statement of the precise contour deformation or residue contributions that supplement the Wick rotation, including any explicit integral expressions.
Notation for the eight form factors and the single momentum variable should be introduced with a compact table or diagram early in the text to aid readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the positive evaluation of the work's significance and for the constructive major comments. We address each point below and indicate the revisions planned for the manuscript.

read point-by-point responses

Referee: The central claim that supplementing the Wick rotation with additional contributions at the first singularity, followed by extrapolation, reproduces the correct analytic continuation of all eight form factors is not accompanied by explicit verification. No checks are presented that the resulting functions satisfy dispersion relations, preserve transversality, or reproduce known perturbative limits in overlapping regions of the complex plane.

Authors: We agree that explicit verifications would strengthen the presentation. Transversality is preserved by construction through the symmetry-preserving truncation of the Dyson-Schwinger equations, as established in the referenced framework. We will revise the manuscript to include a dedicated discussion and numerical checks demonstrating consistency with the perturbative limit in the overlapping real-axis region. Regarding dispersion relations, we will add an explanation of how the supplemented contour respects the analytic properties of the integrands. These additions address the verification concern without altering the core results. revision: yes
Referee: The manuscript provides no numerical results, error estimates, or direct comparisons with independent determinations of the vertex (e.g., lattice data or other truncations) inside the claimed parabolic domain. Without such benchmarks, the assertion that the eight form factors are furnished 'unambiguously' remains untested.

Authors: The manuscript does present the eight form factors obtained from the integrals within the parabolic domain, with the 'unambiguous' character arising from the parameter-free, symmetry-preserving truncation. We acknowledge the value of error estimates and will add quantitative estimates derived from the numerical integration procedure in the revised version. Direct comparisons with lattice data or other truncations are not feasible at present, as independent results for the full vertex at complex momenta are unavailable; lattice computations are typically restricted to Euclidean momenta. We will include a discussion clarifying this limitation and the scope of the current work as the first exploration of the complex-plane domain. revision: partial

standing simulated objections not resolved

Direct comparisons with independent determinations (lattice or other truncations) at complex momenta are not currently possible due to the absence of such data.

Circularity Check

0 steps flagged

No significant circularity; complex-plane extension is direct integral evaluation

full rationale

The paper computes the eight form factors of the transversely-projected quark-gluon vertex (soft-gluon limit) by complexifying the single momentum variable inside the defining integrals of a prior symmetry-preserving framework. The domain is delimited by the first integrand singularity, with standard Wick-rotation supplementation and extrapolation applied beyond it. No equation reduces by construction to a fitted parameter renamed as a prediction, no ansatz is smuggled via self-citation, and no uniqueness theorem or renaming of a known result is invoked. The self-reference to the base approach supplies the real-axis integrands but does not force the complex-plane results; the extension itself is presented as an independent, parameter-free numerical procedure. This satisfies the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the underlying symmetry-preserving truncation and the precise form of the vertex integrals are not specified, so the ledger is necessarily incomplete.

axioms (2)

domain assumption The transversely-projected quark-gluon vertex obtained from the recently developed symmetry-preserving approach for meson properties
The entire calculation starts from this vertex; its validity is presupposed rather than re-derived.
domain assumption The soft-gluon limit reduces the momentum dependence to a single variable whose complexification is unambiguous inside the parabolic domain
This reduction is invoked to make the complex-plane study tractable.

pith-pipeline@v0.9.0 · 5513 in / 1400 out tokens · 50399 ms · 2026-05-08T08:26:53.370185+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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