Gluon mass scale through the Schwinger mechanism

J. Papavassiliou; M. N. Ferreira

REVIEW 3 major objections 3 minor 2 cited by

Gluon mass scale through the Schwinger mechanism

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Massless Schwinger poles in QCD vertices generate the gluon mass, yielding 367 MeV versus the lattice 354 MeV.

desk verdict A well-organized review of the authors' own Schwinger-mechanism program, but the 3.6% mass agreement is not a controlled prediction: the numerics mix a modified kernel with the lattice Lsg, breaking the identity behind Eq. (8.34). read the letter →

arxiv 2501.01080 v2 pith:PZB25S7P submitted 2025-01-02 hep-ph hep-lathep-thnucl-th

M. N. Ferreira , J. Papavassiliou This is my paper · ORCID

classification hep-phhep-lathep-thnucl-th PACS 12.38.-t14.70.Dj

keywords SchwingermechanismgluonmassscaleBethe-SalpeterequationmasslesspolesWardidentitydisplacementseagullSlavnov-TayloridentitieslatticeQCD

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

The paper makes the case that the infrared saturation of the gluon propagator observed in lattice QCD is not an input but a consequence of the Schwinger mechanism: the fundamental vertices of the theory develop colored, longitudinally coupled massless poles, whose residues transmit a pole to the vacuum polarization and thereby produce an effective gluon mass scale. It develops the full formalism — including the exact renormalization of the mass, the nonlinear structure of the pole equation, and the role of the Fredholm alternative theorem — and shows that the Ward identities obeyed by the pole-free parts of the vertices are displaced by precisely these residue functions. The quantitative payoff is a Bethe-Salpeter solution for the pole amplitude giving $m' = 367$ MeV, only $3.6\%$ away from the lattice benchmark $m_{\rm lat} = 354$ MeV. A second result is a 'smoking-gun' signal: the displacement function $C(r^2)$ extracted from lattice inputs is negative on the whole momentum range, so the null hypothesis $C(r^2)=0$ is excluded at the $5\sigma$ level. If the mechanism is right, the massless gluons of the Yang-Mills Lagrangian acquire their effective mass purely from nonperturbative dynamics, with a concrete numerical scale that can be checked against lattice data.

What carries the argument

The carrying object is the displacement/residue function $C(r^2)$ of the three-gluon vertex, defined by the residue of the longitudinally coupled massless pole $q_\alpha/q^2$ that appears when the momentum $q$ is the one entering the gluon self-energy. It does double duty: it acts as the bound-state amplitude for the colored scalar excitation and it shifts the soft-gluon Ward identity away from its pole-free form, thereby evading the seagull identity that would otherwise enforce a massless gluon. The supporting machinery consists of the seagull identity (the integral identity that kills naive mass terms), the displaced Ward identities, the Bethe-Salpeter equation for $B(r^2)$, the relation $m^2 = g^2 I^2$, and the Fredholm alternative theorem, which organizes the cancellation that would make $I$ vanish unless the nonlinear term $\omega$ is present.

What would settle it

Compute the soft-gluon three-gluon form factor $L_{\rm sg}(r^2)$ and all ingredients of $L_0(r^2)$ on the lattice at higher precision and lower momenta; if $C(r^2)=L_{\rm sg}(r^2)-L_0(r^2)$ turns out to be consistent with zero over the whole momentum range, the Schwinger mechanism as formulated here is excluded. A second decisive test is to obtain the four-gluon kernel nonperturbatively from its own equations of motion (or from lattice four-point functions) and solve the BSE of Eq. (8.53) without the fitted parameterization: the 367 MeV prediction would then stand or fall on its own.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the gluon mass scale emerges from massless scalar colored excitations, $\Phi^a$, formed as composite bound states of gluons, and that the residue of the resulting Schwinger pole, $C(r^2)$, is simultaneously the displacement of the soft-gluon Ward identity and the bound-state amplitude $B(r^2)$. The mass is carried by the transition amplitude $I$ through the exact relation $m^2 = g^2 I^2$, where $I$ is obtained from a renormalized integral equation. The renormalization is implemented exactly by an 'exceptional cancellation' whose mathematical origin is the Fredholm alternative theorem: if the Bethe-Salpeter kernel and the vertex SDE kernel were identical, the theorem would force $I=0$ and hence $m=0$; only the nonlinear term $\omega$ (quadratic in $B$) breaks the equality of kernels and lets the mass survive. Numerically, with the kernel modeled as one-gluon exchange modified by the effective propagator of Eq. (8.66), the paper obtains $m'=367$ MeV, in $3.6\%$ agreement with the lattice value, and a $C'(r^2)$ that is negative throughout and qualitatively similar to the lattice-extracted $C_{\rm WI}(r^2)$.

Load-bearing premise

The load-bearing premise is that the modified four-gluon kernel of Eq. (8.66), with three parameters chosen within stated intervals to bring the mass close to the lattice value, represents the omitted nonperturbative dynamics rather than encoding the answer; the qualitative mechanism separately assumes that the Bethe-Salpeter equation admits an exactly massless bound-state solution.

Editorial extensions

If this is right

If the mechanism is correct, the gluon propagator's finite value at zero momentum follows from a pole in the vacuum polarization rather than from a Lagrangian mass term, so no new scalar field is added to the QCD spectrum.
The displacement function $C(r^2)$ is predicted to be negative at all momenta, in line with the lattice-derived curve, and the null hypothesis $C(r^2)=0$ is excluded.
The gluon mass scale is fixed dynamically by the bound-state amplitude and survives renormalization exactly, so $m^2 = g^2 I^2$ is a finite, renormalization-group-invariant relation.
The Fredholm alternative theorem acts as a selection rule: without the nonlinear term $\omega$, the transition amplitude $I$ — and therefore the mass — would vanish even though the BSE admits a nontrivial solution for $B(r^2)$.

Reading between the lines

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

Editorial extension: the fitted kernel of Eq. (8.66) could be replaced by a four-gluon kernel computed from its own equations of motion, turning 367 MeV into a parameter-free prediction rather than a consistency check.
Editorial extension: if the massless bound-state solution persists at other gauge groups, the same mechanism would generate effective gauge-boson masses in SU(2) and similar non-Abelian theories, where lattice data already show infrared saturation.
Editorial extension: higher-precision or lower-momentum lattice data for $L_{\rm sg}(r^2)$ would either sharpen the $5\sigma$ exclusion of $C(r^2)=0$ or reveal where the WI-derived null hypothesis fails.
Editorial extension: the Fredholm cancellation suggests a truncation criterion for Schwinger-Dyson studies: any truncation that makes the BSE and vertex-SDE kernels identical forces $I=0$, so the distinction between the kernels $T$ and $K$ must be preserved.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a thorough review of the PT-BFM Schwinger-pole program, and it is honestly labeled as such: most key equations are traced to earlier papers by the same group. That is not a flaw for a review, but it means the bar should be set by clarity and by whether the advertised quantitative claims hold up. On the qualitative side, the paper does well. The derivations connecting massless poles in vertices to the gluon mass scale are laid out in unusual detail, the Ward-identity displacement is explained carefully, and the Fredholm-alternative discussion in Sec. 8.5 is genuinely instructive. The lattice-based extraction of C(r2) in Sec. 6.3 is the strongest piece: it is an independent, data-driven signal, and even accounting for the SDE input W(r2), it deserves to be taken seriously.

The soft spot is the numerics, and I think the stress-test concern lands. Equation (8.34) is obtained by assuming the same kernel K appears in the homogeneous BSE for B and in the inhomogeneous equation for Lsg. In the numerical implementation, the kernel is modified to K' via Eq. (8.66), and the BSE is solved with K', but Lsg is not recomputed with K'; instead, the lattice Lsg is used. The substitution that eliminates Z3 then ceases to be an identity, and the dropped term is never estimated. On top of that, the three parameters c0, c1, c2 are varied to bring the output close to mlat = 354 MeV. So m' = 367 MeV is an instructive illustration of what the formalism can produce, not a controlled prediction with predictive weight. The paper is transparent about the tuning, but the abstract and Sec. 8.6 still present the 3.6% agreement as the main numerical result, which overstates its status.

For a review, this is acceptable provided the framing is corrected. The authors should either recompute Lsg with the modified kernel and bound the leftover term, or explicitly state that m' is a benchmark-tuned illustration. I would send it to peer review: the review is useful, the formal parts are coherent, and the lattice C(r2) signal deserves expert scrutiny. But I would not accept it with the current claim that m' = 367 MeV is a prediction. If the authors rework that section honestly, the paper becomes a solid reference for the field.

Referee Report

3 major / 3 minor

Summary. The paper presents a comprehensive review of the Schwinger mechanism as an explanation for the infrared gluon mass scale in QCD. It develops the formalism of massless poles in the fundamental vertices, derives the resulting displacement of Ward identities, extracts the displacement function C(r^2) from lattice inputs, and constructs a Bethe-Salpeter equation for the pole formation whose nonlinearity fixes the scale of the solution. The central quantitative claims are that the BSE yields a gluon mass m'=367 MeV, within 3.6% of the lattice benchmark m_lat=354 MeV, and that the lattice-based function C(r^2) is negative over the whole momentum range, excluding the null hypothesis C=0 at the 5-sigma level.

Significance. If the central claims are correct, the paper provides a coherent field-theoretic picture in which the saturation of the gluon propagator is not an input but a consequence of composite massless colored excitations. The formal machinery is impressive: the seagull-identity evasion, the WI displacement, the exact multiplicative-renormalization cancellation via the Fredholm alternative, and the nonlinear scale-fixing of the BSE are presented with considerable care and internal consistency. The paper is also transparent about the role of the kernel modification and about the fact that the final mass is compared with a lattice benchmark. The weakness is that the quantitative mass result is not a controlled prediction of the formal derivation, because the numerical kernel used in the BSE is not the same kernel that enters the cancellation underlying Eq. (8.34), and the kernel parameters are adjusted to approach the benchmark. The qualitative mechanism is defensible and interesting, but the advertised 3.6% agreement should not be presented as a parameter-free success.

major comments (3)

[Sec. 8.6, Eq. (8.66)] The numerical value m'=367 MeV is not controlled by the formalism as presented. The derivation of Eq. (8.34) in Sec. 8.4 relies on the toy-model cancellation in Eq. (8.38) with the identifications of Eq. (8.40), which require that the same kernel K(r,k) appear in the BSE, Eq. (8.25), and in the Lsg SDE, Eq. (7.49)/Eq. (8.26). In the numerical implementation, the kernel is replaced by K' through the substitution Delta(u^2) -> Delta'(u^2) in Eq. (8.66), while Lsg is taken from the lattice fit and is not recomputed with K'. Consequently, the identity Z3 = Lsg - alpha_s ∫ k^2 Delta'^2 K' Lsg used implicitly in the substitution is not satisfied, and a residual term of the form Lsg - Z3 - alpha_s ∫ k^2 Delta'^2 K' Lsg is dropped without an estimate. The reported 3.6% agreement with m_lat is therefore not a prediction of the formalism; it is an output of a modified kernel whose consistency with the derivation of Eq. (8.34) is not established.
[Sec. 8.6, Eqs. (8.64)-(8.66)] The parameters c0, c1, c2 are varied 'within certain intervals', and the set c0=0.503 GeV^-2, c1=0.00667 GeV^-2, c2=0.0486 GeV^-4 is selected because it brings m' close to m_lat=354 MeV. No prior distribution, sensitivity study, or goodness-of-fit measure is reported for this three-parameter adjustment. As a result, the 3.6% agreement is a fit to the benchmark rather than a falsifiable prediction. This does not invalidate the qualitative Schwinger-pole mechanism, but it removes the quantitative mass value as independent evidence for it.
[Sec. 6.3, Eq. (6.20)] The 'smoking-gun' claim that C(r^2) is negative over the whole momentum range and that C=0 is excluded at the 5-sigma level should be qualified. The extraction of C(r^2) uses L0(r^2), which depends on W(r^2) computed from an SDE in App. G and on eZ1 determined from a coupled SDE system, not directly on lattice data. The lattice inputs enter through Lsg, Delta, and F(0), but the systematic uncertainty of the SDE determinations of W and eZ1 is not propagated into the stated significance. The 5-sigma statement therefore reflects the statistical propagation of the lattice errors only, not the model dependence of the SDE ingredients.

minor comments (3)

[Sec. 6.3, item (v)] The renormalization point is stated as mu = 4.3 MeV; this should read mu = 4.3 GeV.
[Caption of Fig. 6.3] The text 'left panel of Eq. (6.3)' should read 'left panel of Fig. 6.3'.
[Sec. 2.2] There is a typo, 'gluon propapagator', in the introductory paragraph of Sec. 2.2; it should be 'gluon propagator'.

Circularity Check

1 steps flagged · score 7.0 of 10

The advertised 3.6% mass agreement is a fit, not a prediction: the kernel parameters c0,c1,c2 are tuned so that the BSE output approaches the lattice benchmark mlat, so the derived mass reproduces its input by construction.

fitted input called prediction [Sec. 8.6, Eqs. (8.64)-(8.66) and paragraph after Eq. (8.66)]
"We next vary the ci in Eq. (8.66) within certain intervals, and consider the resulting values for m. Our analysis reveals that the “optimal” set of values is given by c0 = 0.503 GeV−2, c1 = 0.00667 GeV−2 and c2 = 0.0486 GeV−4. ... The repetition of the steps (ii)-(iv) furnishes for the gluon mass scale the value m′ = 367 MeV, which differs by only 3.6% from mlat = 354 MeV."

The three parameters c0,c1,c2 in the product ansatz Δ′(u2)=Δ(u2)(1+c0u2)/(1+c1u2+c2u4) are not fixed by the formalism; they are varied 'within certain intervals' and then selected as 'optimal' precisely because the BSE output approaches the lattice benchmark mlat=354 MeV. The mass scale is then computed from the tuned kernel, so the advertised 3.6% agreement is a measure of the quality of the parameter search, not an independent prediction. The central numerical result of the paper therefore reduces by construction to reproducing its input benchmark.

full rationale

The formal development in Sections 2-7 is largely self-contained: the seagull identity, Ward-identity displacement, and the Bethe-Salpeter setup are derived from stated assumptions, and the 5σ displacement signal C(r2) in Sec. 6.3 is extracted from external lattice inputs, so those parts are not circular. The circularity is concentrated in the quantitative claim of Sec. 8.6. The one-gluon-exchange kernel Koge already yields a mass moge=1.27 GeV, far from the lattice value; the paper then replaces Δ(u2) by the ad hoc Δ′(u2) with three free parameters c0,c1,c2 and varies them 'within certain intervals' until the output m′=367 MeV lands within 3.6% of mlat=354 MeV. That agreement is therefore the success of a three-parameter fit, not a parameter-free prediction. In addition, the derivation of Eq. (8.34) (the 'exceptional cancellation') assumes the same kernel K in the BSE for B and the SDE for Lsg, whereas the numerics use the modified K′ in the BSE while retaining the lattice Lsg; this is a separate consistency gap, but it is an approximation error rather than a circularity. The score of 7 reflects that the central numerical prediction is fitted, while the qualitative mechanism and the lattice-based signal retain independent content.

Assumptions & free parameters 3 free parameters · 7 assumptions · 2 invented entities

The quantitative claim is a composite of postulates: massless poles, longitudinal vertex decomposition, existence of massless colored bound states, and a truncation of the SDE/BSE tower. The only numbers introduced ad hoc are the kernel parameters c0, c1, c2; alpha_s and eZ1 come from prior lattice and continuum analyses. The massless scalar Phi is an invented composite entity, with an indirect lattice signature. Overall the ledger is heavy, meaning the computation is not self-contained.

free parameters (3)

Kernel modification parameters c0, c1, c2 = c0=0.503 GeV^-2, c1=0.00667 GeV^-2, c2=0.0486 GeV^-4
Chosen in Eq. (8.66) to bring the BSE mass close to the lattice benchmark; this is parameter fitting to the target observable.
Strong coupling alpha_s at mu=4.3 GeV = 0.27
Input from the asymmetric MOM scheme and lattice renormalization; the BSE eigenvalue and mass scale depend on this choice.
Ghost-gluon renormalization constant eZ1 = 0.9333 +/- 0.0075
Determined by minimizing chi^2 between the ghost SDE solution and lattice data, so it is fitted to an external curve.

assumptions (7)

domain assumption The vacuum polarization develops a pole at q^2=0 with positive residue, giving the gauge boson a mass (Eq. 4.1).
This is the core postulate of the Schwinger mechanism; all subsequent mass formulas follow from it.
domain assumption The vertices split into pole-free and pole parts, with the pole parts strictly longitudinal (Eq. 5.12).
Needed so that poles decouple from physical amplitudes and act as displacement functions.
domain assumption Massless colored scalar bound states Phi^a exist as solutions of the BSE (Eq. 8.18).
The existence of a massless bound state at zero mass is assumed; the BSE eigenvalue problem is solved with a given kernel, not derived from first principles.
standard math The seagull identity (Eq. 3.12) holds for the dressed propagators Delta and D.
Used to show that without poles the gluon remains massless.
ad hoc to paper The four-gluon kernel is represented by one-gluon exchange modified by the fitted function Delta'(u^2) (Eq. 8.66), with ghost and four-gluon pole contributions omitted.
The truncation and kernel modification are introduced for this calculation, not derived, and the parameters are tuned to the target mass.
domain assumption The three-gluon vertex obeys the 'planar degeneracy' approximation Lsg(s^2) (Eq. 8.58).
Used to reduce the kernel to a function of a single variable; supported by lattice studies but an approximation.
standard math The Fredholm alternative theorem applies to the symmetric rescaled kernel eK(x,y) (Eq. 8.48).
Relied on to show that with omega=0 the transition amplitude I and thus m vanish; the paper argues nonlinearity does not change this.

invented entities (2)

Massless colored scalar composite Phi^a independent evidence
purpose: Provides the massless pole in the three-gluon and ghost-gluon vertices that drives the Schwinger mechanism (Sec. 7.2).
Not an elementary field; its displacement function C(r^2) is claimed to be measurable from lattice inputs, but the extraction relies on the continuum-computed W(r^2), so the handle is indirect.
Double (mixed) Schwinger poles in the three-gluon vertex
purpose: Maintains Slavnov-Taylor identity consistency when the gluon propagator is infrared finite; these poles are inert to mass generation (Sec. 6.4).
No separately observable signature; their coefficient V9 is fixed by the STI consistency condition and is not independently measured.

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Cite this review

Pith. "Pith review of Gluon mass scale through the Schwinger mechanism." pith.science (2026). https://pith.science/paper/PZB25S7P

@misc{pith2026250101080,
  author       = {Pith},
  title        = {Pith review of: Gluon mass scale through the Schwinger mechanism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZB25S7P}},
  note         = {Machine review of arXiv:2501.01080}
}

read the original abstract

It has long been argued that the action of the Schwinger mechanism in the gauge sector of Quantum Chromodynamics leads to the generation of a gluon mass scale. Within this scenario, the analytic structure of the fundamental vertices is modified by the creation of scalar colored excitations with vanishing mass. In the limit of zero momentum transfer, these terms act as massless poles, providing the required conditions for the infrared stabilization of the gluon propagator, and producing a characteristic displacement to the associated Ward identities. In this article we offer an extensive overview of the salient notions and techniques underlying this dynamical picture. We place particular emphasis on recent developments related to the exact renormalization of the mass, the nonlinear nature of the pole equation, and the key role played by the Fredholm alternative theorem.

Figures

Figures reproduced from arXiv: 2501.01080 by the authors.

**Figure 1.1.** Gluon propagator, ∆(q 2 ), obtained from large volume lattice simulations, all displaying a saturation at the origin. Upper left: Quenched SU(3) Landau gauge results from various lattice setups of [13, 21, 23]. Upper right: Quenched SU(3) data for various values of the gauge fixing parameter, ξ, from [18]. Lower left: Landau gauge SU(3) data for different numbers of dynamical quark flavors, Nf , namely: Nf = 0 (blue… view at source ↗

**Figure 2.1.** Diagrammatic conventions for the fully dressed three-gluon vertex (left) and its tree-level counterpart (right). [PITH_FULL_IMAGE:figures/full_fig_p011_2_1.png] view at source ↗

**Figure 2.2.** Diagrammatic representation of the ghost-gluon scattering kernel, [PITH_FULL_IMAGE:figures/full_fig_p012_2_2.png] view at source ↗

Figures from the paper (24 more)

**Figure 2.3.** Figure 2.3: Diagrammatic representation of the ghost SDE given in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p015_2_3.png]

**Figure 2.4.** Figure 2.4: Diagrammatic representations of the gluon self-energy (top panel), [PITH_FULL_IMAGE:figures/full_fig_p015_2_4.png]

**Figure 2.5.** Figure 2.5: Diagrammatic representation of the Qa µ(q)Bb ν (−q) self-energy δ abΠeµν(q); the small grey circles at the end of the gluon lines indicate a background gluon. The corresponding Feynman rules are given in Appendix B of [111]. The BFM is a powerful quantization framewo…

**Figure 3.1.** Figure 3.1: One-loop dressed diagrams of the photon self-energy in scalar QED. Exceptionally in this figure, dashed lines represent electrically [PITH_FULL_IMAGE:figures/full_fig_p023_3_1.png]

**Figure 3.2.** Figure 3.2: The full photon self-energy in QED4. The full photon self-energy, Πµν(q), is given by the single diagram shown in [PITH_FULL_IMAGE:figures/full_fig_p024_3_2.png]

**Figure 5.1.** Figure 5.1: The diagrammatic representation of the three-gluon and ghost-gluon vertices introduced in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p029_5_1.png]

**Figure 5.2.** Figure 5.2: Cancellation of longitudinally coupled poles when contracted with conserved currents. [PITH_FULL_IMAGE:figures/full_fig_p030_5_2.png]

**Figure 6.1.** Figure 6.1: SDE for the ghost-gluon kernel, Hνµ(p, q, r), in compact form. If we now combine Eqs. (5.1) and (5.18), it is clear that P µ µ′ (r)P ν ν′ (p) [q α IΓαµν(q, r, p)] = P µ µ′ (r)P ν ν′ (p)[q αΓαµν(q, r, p) + Cµν(q, r, p)] , (6.4) 42 [PITH_FULL_IMAGE:figures/full_fig_p0…

**Figure 6.2.** Figure 6.2: Upper panel: The gluon propagator (left) and the first derivative of its inverse (right). Lower panel: The ghost dressing function (left) and the soft gluon form factor Lsg(r 2 ) of the three-gluon vertex (right). All items are taken from [23], and have been cured fr…

**Figure 6.3.** Figure 6.3: Left: Lattice data of [188] for Lsg(r 2 ) (points), the corresponding fit (blue continuous), and the null hypothesis prediction, L0(r 2 ), of Eq. (6.20) (green dot-dashed). Right: Result for C(r 2 ) (black continuous line) obtained from Eq. (6.20); the error error ba…

**Figure 6.4.** Figure 6.4: Left: Form factor A1(q 2 ) of the ghost-gluon scattering kernel in the soft-antighost limit, determined in [213], and converted to the asymmetric MOM scheme through rescaling by a factor of 0.933 [144]. Right: The function V9(q 2 ), associated with the mixed double p…

**Figure 7.1.** Figure 7.1: Left: The effective vertex Babc µν (q, r, p), with Lorentz, color, and momentum conventions indicated. Center: The propagator of the massless composite scalar, Dab Φ (q). Right: Gluon-scalar transition amplitude, I ab α (q). The emergence of these three items, (a)-(c…

**Figure 7.2.** Figure 7.2: Decomposition of the four-gluon scattering kernel, [PITH_FULL_IMAGE:figures/full_fig_p053_7_2.png]

**Figure 7.3.** Figure 7.3: First line: SDE for the three-gluon vertex. Second line: The pole induced to the three-gluon vertex due to the component M of T in Eq. (7.5) (see also [PITH_FULL_IMAGE:figures/full_fig_p053_7_3.png]

**Figure 7.4.** Figure 7.4: Relation between the gluon mass and the transition amplitude, [PITH_FULL_IMAGE:figures/full_fig_p054_7_4.png]

**Figure 7.5.** Figure 7.5: The general structure of the three-gluon vertex after the activation of the Schwinger mechanism. Note, in particular, that the [PITH_FULL_IMAGE:figures/full_fig_p055_7_5.png]

**Figure 7.6.** Figure 7.6: Diagrammatic representation of Eq. (7.20), corresponding to the first nonvanishing term in the Taylor expansion of Babc µν (q, r, p) around q = 0. Finally, combining Eqs. (5.20), (7.7) and (7.21), we obtain a relation between the displacement function, C(r 2 ), and t…

**Figure 7.7.** Figure 7.7: Diagrammatic definition of the scalar form factor, [PITH_FULL_IMAGE:figures/full_fig_p057_7_7.png]

**Figure 7.8.** Figure 7.8: SDE for the regular part of the three-gluon vertex, [PITH_FULL_IMAGE:figures/full_fig_p058_7_8.png]

**Figure 7.9.** Figure 7.9: Skeleton expansion of the scattering kernel, [PITH_FULL_IMAGE:figures/full_fig_p058_7_9.png]

**Figure 8.1.** Figure 8.1: BSE for the effective vertex [PITH_FULL_IMAGE:figures/full_fig_p064_8_1.png]

**Figure 8.2.** Figure 8.2: Diagrammatic illustration of the renormalization of [PITH_FULL_IMAGE:figures/full_fig_p068_8_2.png]

**Figure 8.3.** Figure 8.3: Upper left: The effective gluon propagator, corresponding to the two choices used in our analysis, namely ∆(u 2 ) (orange dashed) and ∆′ (u 2 ) (blue continuous); the inset shows the associated dressing functions. Upper right: The diagonal slice of the angle-integrat…

**Figure 8.4.** Figure 8.4: Left: The W(r 2 ) obtained from the SDE analysis of App. G, using diagrams (h1) and (h2) of Fig. G.1 (blue continuous), and the W ′ (r 2 ) (purple-dashed), obtained from Eq. (8.67), enforcing CWI(r 2 ) = C′ (r 2 ). Right: The contribution W3(r 2 ) obtained from Eq. (…

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Reviewed August 10, 2026 · model on record in the stance chip above.

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