pith. sign in

arxiv: 2606.28212 · v1 · pith:MKD2PTGPnew · submitted 2026-06-26 · ✦ hep-ph · hep-lat· hep-th

Real poles with opposite-sign residues in the non-perturbative quark propagator

R. Alkofer , M.N. Ferreira , A.S. Miramontes , J.M. Morgado , J. Papavassiliou This is my paper

Pith reviewed 2026-06-29 03:20 UTC · model grok-4.3

classification ✦ hep-ph hep-lathep-th
keywords quark propagatorLandau gaugeanalytic structurereal polesquark-gluon vertexcolor confinementpositivity violationgap equation
0
0 comments X

The pith

The quark propagator develops a pair of real poles with opposite-sign residues when the full non-perturbative quark-gluon vertex is dynamically included in the gap equation.

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

The paper couples the quark gap equation in Landau gauge to the complete non-perturbative quark-gluon vertex expressed in its full basis. For time-like momenta below 1 GeV this produces two real poles carrying residues of opposite sign. The result stands in contrast to the complex conjugate poles obtained in common truncations. The structure is shown to be compatible with positivity violation while generating a constituent quark mass near 350 MeV. Three specific vertex form factors, when tuned together, are responsible for moving the poles onto the real axis.

Core claim

Dynamically coupling the gap equation to the full quark-gluon vertex basis yields, for sub-GeV time-like momenta, a pair of real poles with opposite-sign residues in the quark propagator; no complex conjugate poles appear. This analytic structure is controlled by the joint action of the tree-level vertex component, the anomalous chromomagnetic moment, and the spin-momentum curvature form factor, whose combined strengths place the low-lying poles on the real axis while preserving a robust 350 MeV constituent mass and remaining consistent with color-confinement signals tied to positivity violation.

What carries the argument

The triplet of vertex form factors (tree-level component, anomalous chromomagnetic moment, and spin-momentum curvature) whose independent strength tuning, within a fixed truncation, moves the poles from the complex plane onto the real axis.

If this is right

  • The real-pole structure evades the conceptual difficulties often linked to complex conjugate poles.
  • Positivity violation, an aspect of color confinement, remains intact.
  • A stable constituent quark mass of approximately 350 MeV is obtained.
  • The three form factors must act jointly; altering any one individually changes the pole locations in distinct ways.

Where Pith is reading between the lines

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

  • Similar real-pole patterns may emerge in gluon or ghost propagators once their respective full vertices are retained.
  • Bound-state equations that employ this propagator could produce different spectral properties than those using complex-pole approximations.
  • The result suggests that truncation artifacts, rather than fundamental dynamics, are responsible for complex poles in many existing studies.

Load-bearing premise

That the three identified vertex form factors can be varied independently in strength while the rest of the vertex and the gap-equation truncation stay fixed, and that this variation alone suffices to place the poles on the real axis.

What would settle it

A recalculation of the same gap equation with the full vertex basis but with the three form factors held at their tree-level values, or with an independent non-perturbative vertex model, that nevertheless produces complex conjugate poles below 1 GeV.

Figures

Figures reproduced from arXiv: 2606.28212 by A.S. Miramontes, J.M. Morgado, J. Papavassiliou, M.N. Ferreira, R. Alkofer.

Figure 1
Figure 1. Figure 1: Upper panel: Diagrammatic representation of the quark gap equation. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Real (top) and imaginary (bottom) parts of the quark-gluon vertex form factors [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the real parts of the Dirac component [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Upper panels: Individual contributions of the TLC, ACM, and SMC to the quark propagator functions [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Contour plot of σV (p 2 ) showing the evolution of the pole structure as the coefficient η is varied for the vertex in Eq. (16). For small η, the propagator features a pair of complex conjugate poles, which become real as η exceeds a critical value ηcr . The position of p 2 1 and p 2 2 remains stable, varying by less than 5% under changes in both the data sets and the number of points used to construct the… view at source ↗
Figure 6
Figure 6. Figure 6: Left panel: Imaginary part of the quark propagator pole, [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We investigate the analytic structure of the quark propagator in the Landau gauge by dynamically coupling the standard gap equation to the non-perturbative quark-gluon vertex. Employing the full vertex basis, we demonstrate that for sub-GeV time-like momenta, the proper inclusion of the underlying dynamics leads to a pair of real poles with opposite-sign residues. In particular, in stark contradistinction to the results obtained in widely used approximations, we see no sign of complex conjugate poles. This distinctive analytic structure evades conceptual shortcomings frequently associated with complex conjugate poles while remaining fully compatible with the aspects of color confinement related to positivity violation. Crucially, this novel behavior is governed by a dominant triplet of vertex form factors: the tree-level component, the anomalous chromomagnetic moment, and a component we label as "spin-momentum curvature". By gradually tuning the individual strengths of these components, we demonstrate that while they contribute in distinct ways to the quark propagator, their joint action is vital for stabilizing the system. Together, they place the low-lying poles onto the real axis while producing a robust constituent quark mass of $350$ MeV.

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

2 major / 0 minor

Summary. The paper investigates the analytic structure of the Landau-gauge quark propagator by dynamically coupling the gap equation to the full non-perturbative quark-gluon vertex. It claims that proper inclusion of the vertex dynamics produces, for sub-GeV time-like momenta, a pair of real poles with opposite-sign residues and a constituent mass of 350 MeV, with no complex-conjugate poles; this structure is attributed to the joint action of three dominant vertex form factors (tree-level, anomalous chromomagnetic moment, and spin-momentum curvature), whose strengths are gradually tuned while the rest of the vertex basis and truncation are held fixed.

Significance. If the real-pole structure survives a fully self-consistent determination of the vertex, the result would be significant for non-perturbative QCD: it supplies an analytic continuation of the propagator that evades difficulties associated with complex poles while remaining compatible with positivity violation as a confinement signature. The identification of a specific triplet of vertex components as the controlling mechanism provides a concrete, testable handle on the quark-gluon vertex that could inform future truncations.

major comments (2)
  1. [Abstract] Abstract (paragraph beginning 'By gradually tuning the individual strengths...'): The central claim—that the real poles with opposite-sign residues emerge from the underlying dynamics—rests on independent variation of the three vertex form factors while the remainder of the basis and the gap-equation truncation are fixed. In a consistent treatment these form factors are generated by the same vertex equation and must satisfy the same Slavnov-Taylor identities and renormalization conditions; the independent tuning therefore introduces an extra degree of freedom whose effect on pole locations is not guaranteed to persist once the form factors are determined self-consistently.
  2. [Results/numerical sections] Results and numerical implementation sections: No information is supplied on numerical convergence, error control, or the concrete procedure used to verify the absence of complex poles over the full momentum range. The reported 350 MeV mass and real-pole placement are obtained only after the tuning step, so the robustness of the claimed analytic structure cannot be assessed from the given material.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond point by point to the major concerns below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'By gradually tuning the individual strengths...'): The central claim—that the real poles with opposite-sign residues emerge from the underlying dynamics—rests on independent variation of the three vertex form factors while the remainder of the basis and the gap-equation truncation are fixed. In a consistent treatment these form factors are generated by the same vertex equation and must satisfy the same Slavnov-Taylor identities and renormalization conditions; the independent tuning therefore introduces an extra degree of freedom whose effect on pole locations is not guaranteed to persist once the form factors are determined self-consistently.

    Authors: We agree that independent tuning of the three form factors introduces an extra degree of freedom absent from a fully self-consistent vertex solution. Our procedure is explicitly exploratory: it isolates the minimal triplet whose joint action stabilizes real poles, thereby furnishing a concrete target for future self-consistent calculations. We will revise the abstract and add a dedicated paragraph in the conclusions to state this limitation clearly and to emphasize that the reported structure is a diagnostic result rather than a claim of dynamical self-consistency. revision: yes

  2. Referee: [Results/numerical sections] Results and numerical implementation sections: No information is supplied on numerical convergence, error control, or the concrete procedure used to verify the absence of complex poles over the full momentum range. The reported 350 MeV mass and real-pole placement are obtained only after the tuning step, so the robustness of the claimed analytic structure cannot be assessed from the given material.

    Authors: The referee correctly notes the absence of numerical details. In the revised manuscript we will insert a new subsection that specifies the momentum-grid discretization, the convergence threshold for the iterative gap-equation solver, the ultraviolet cutoff dependence, and the explicit algorithm used to locate poles (analytic continuation of the denominator function on a dense complex-plane grid together with a systematic search for conjugate-pair solutions). These additions will allow independent assessment of the robustness of the 350 MeV mass and the real-pole placement. revision: yes

Circularity Check

1 steps flagged

Tuning of three vertex form factors places poles on real axis by construction

specific steps
  1. fitted input called prediction [Abstract]
    "By gradually tuning the individual strengths of these components, we demonstrate that while they contribute in distinct ways to the quark propagator, their joint action is vital for stabilizing the system. Together, they place the low-lying poles onto the real axis while producing a robust constituent quark mass of 350 MeV."

    The paper states that the dynamics lead to real poles, yet the placement of poles on the real axis and the specific mass value are obtained by tuning the three form-factor strengths; the reported outcome is therefore the direct result of this parameter adjustment rather than a prediction independent of the tuning.

full rationale

The paper's central claim that proper inclusion of dynamics produces real poles with opposite-sign residues is achieved explicitly by gradually tuning the strengths of the tree-level, anomalous chromomagnetic moment, and spin-momentum curvature form factors while holding the rest of the vertex basis and gap-equation truncation fixed. This tuning directly determines the pole locations and the 350 MeV mass, making the reported analytic structure dependent on the chosen parameter values rather than an independent, untuned outcome of the equations. No self-citation chains or other enumerated patterns appear in the provided text, but the explicit dependence on adjustable strengths constitutes partial circularity of the fitted-input type.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard Landau-gauge Dyson-Schwinger framework plus the specific choice to retain only three dominant vertex form factors whose strengths are adjusted to produce the reported pole locations and mass value.

free parameters (1)
  • strengths of the three vertex form factors
    Tuned independently to move poles onto the real axis and produce the 350 MeV constituent mass.
axioms (2)
  • domain assumption Landau gauge is the appropriate choice for studying the non-perturbative quark propagator
    Standard in the field but restricts the result to one gauge.
  • domain assumption The gap equation truncation with the chosen vertex basis captures the dominant dynamics
    Invoked when the authors state that the triplet governs the analytic structure.

pith-pipeline@v0.9.1-grok · 5752 in / 1540 out tokens · 21788 ms · 2026-06-29T03:20:43.532646+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

104 extracted references · 65 canonical work pages · 3 internal anchors

  1. [1]

    Alkofer, L

    R. Alkofer, L. von Smekal, Phys. Rept. 353 (2001) 281. doi:10.1016/S0370-1573(01)00010-2

    work page doi:10.1016/s0370-1573(01)00010-2 2001

  2. [2]

    J. M. Pawlowski, D. F. Litim, S. Nedelko, L. von Smekal, Phys. Rev. Lett. 93 (2004) 152002. doi:10. 1103/PhysRevLett.93.152002

    2004

  3. [3]

    Binosi, J

    D. Binosi, J. Papavassiliou, Phys. Rept. 479 (2009) 1–

    2009

  4. [4]

    doi:10.1016/j.physrep.2009.05.001

    work page doi:10.1016/j.physrep.2009.05.001 2009

  5. [5]

    Maas, Phys

    A. Maas, Phys. Rept. 524 (2013) 203–300. doi:10. 1016/j.physrep.2012.11.002

    2013

  6. [6]

    I. C. Cloet, C. D. Roberts, Prog. Part. Nucl. Phys. 77 (2014) 1–69. doi:10.1016/j.ppnp.2014.02.001

    work page doi:10.1016/j.ppnp.2014.02.001 2014

  7. [7]

    M. Q. Huber, Phys. Rept. 879 (2020) 1–92. doi:10. 1016/j.physrep.2020.04.004

    2020

  8. [8]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, N. Wschebor, Phys. Rept. 910 (2021) 1–114. doi:10.1016/j.physrep.2021. 01.001

    work page doi:10.1016/j.physrep.2021 2021

  9. [9]

    M. N. Ferreira, J. Papavassiliou, Particles 6 (2023) 312–

    2023

  10. [10]

    doi:10.3390/particles6010017

    work page doi:10.3390/particles6010017

  11. [11]

    M. N. Ferreira, J. Papavassiliou, Prog. Part. Nucl. Phys. 144 (2025) 104186. doi:10.1016/j.ppnp.2025. 104186

    work page doi:10.1016/j.ppnp.2025 2025

  12. [12]

    M. Q. Huber, 2025.arXiv:2510.18960

    Pith/arXiv arXiv 2025

  13. [13]

    Alkofer, W

    R. Alkofer, W. Detmold, C. S. Fischer, P. Maris, Phys. Rev. D 70 (2004) 014014. doi:10.1103/PhysRevD.70. 014014

    work page doi:10.1103/physrevd.70 2004

  14. [14]

    Strauss, C

    S. Strauss, C. S. Fischer, C. Kellermann, Phys. Rev. Lett. 109 (2012) 252001. doi:10.1103/PhysRevLett.109. 252001

    work page doi:10.1103/physrevlett.109 2012

  15. [15]

    Frederico, G

    T. Frederico, G. Salmè, M. Viviani, Phys. Rev. D 89 (2014) 016010. doi:10.1103/PhysRevD.89.016010

    work page doi:10.1103/physrevd.89.016010 2014

  16. [16]

    Dudal, O

    D. Dudal, O. Oliveira, P. J. Silva, Phys. Rev. D 89 (2014) 014010. doi:10.1103/PhysRevD.89.014010

    work page doi:10.1103/physrevd.89.014010 2014

  17. [17]

    de Paula, T

    W. de Paula, T. Frederico, G. Salmè, M. Viviani, Phys. Rev. D 94 (2016) 071901. doi:10.1103/PhysRevD.94. 071901

    work page doi:10.1103/physrevd.94 2016

  18. [18]

    El-Bennich, G

    B. El-Bennich, G. Krein, E. Rojas, F. E. Serna, Few Body Syst. 57 (2016) 955–963. doi:10.1007/ s00601-016-1133-x

    2016

  19. [19]

    Tripolt, P

    R.-A. Tripolt, P. Gubler, M. Ulybyshev, L. V on Smekal, Comput. Phys. Commun. 237 (2019) 129–142. doi:10. 1016/j.cpc.2018.11.012

    2019

  20. [20]

    Binosi, R.-A

    D. Binosi, R.-A. Tripolt, Phys. Lett. B 801 (2020) 135171. doi:10.1016/j.physletb.2019.135171

    work page doi:10.1016/j.physletb.2019.135171 2020

  21. [21]

    Dudal, O

    D. Dudal, O. Oliveira, M. Roelfs, P. Silva, Nucl. Phys. B 952 (2020) 114912. doi:10.1016/j.nuclphysb. 2019.114912

    work page doi:10.1016/j.nuclphysb 2020

  22. [23]

    S. W. Li, P. Lowdon, O. Oliveira, P. J. Silva, Phys. Lett. B 803 (2020) 135329. doi:10.1016/j.physletb.2020. 135329. 9

    work page doi:10.1016/j.physletb.2020 2020

  23. [24]

    S. W. Li, P. Lowdon, O. Oliveira, P. J. Silva, Phys. Lett. B 823 (2021) 136753. doi:10.1016/j.physletb.2021. 136753

    work page doi:10.1016/j.physletb.2021 2021

  24. [25]

    C. S. Fischer, M. Q. Huber, Phys. Rev. D 102 (2020) 094005. doi:10.1103/PhysRevD.102.094005

    work page doi:10.1103/physrevd.102.094005 2020

  25. [26]

    Horak, J

    J. Horak, J. Papavassiliou, J. M. Pawlowski, N. Wink, Phys. Rev. D 104 (2021) 074017. doi:10.1103/ PhysRevD.104.074017

    2021

  26. [27]

    Horak, J

    J. Horak, J. M. Pawlowski, J. Rodríguez-Quintero, J. Turnwald, J. M. Urban, N. Wink, S. Zafeiropou- los, Phys. Rev. D 105 (2022) 036014. doi:10.1103/ PhysRevD.105.036014

    2022

  27. [28]

    Horak, J

    J. Horak, J. M. Pawlowski, N. Wink, SciPost Phys. Core 8 (2025) 048. doi:10.21468/SciPostPhysCore.8.3. 048

    work page doi:10.21468/scipostphyscore.8.3 2025

  28. [29]

    Eichmann, E

    G. Eichmann, E. Ferreira, A. Stadler, Phys. Rev. D 105 (2022) 034009. doi:10.1103/PhysRevD.105.034009

    work page doi:10.1103/physrevd.105.034009 2022

  29. [30]

    Horak, J

    J. Horak, J. M. Pawlowski, N. Wink, SciPost Phys. 15 (2023) 149. doi:10.21468/SciPostPhys.15.4.149

    work page doi:10.21468/scipostphys.15.4.149 2023

  30. [31]

    M. Q. Huber, W. J. Kern, R. Alkofer, Phys. Rev. D 107 (2023) 074026. doi:10.1103/PhysRevD.107.074026

    work page doi:10.1103/physrevd.107.074026 2023

  31. [32]

    M. Q. Huber, W. J. Kern, R. Alkofer, Symmetry 15 (2023) 414. doi:10.3390/sym15020414

    work page doi:10.3390/sym15020414 2023

  32. [33]

    J. M. Pawlowski, J. Wessely, Eur. Phys. J. C 85 (2025)

    2025

  33. [34]

    doi:10.1140/epjc/s10052-025-14683-z

    work page doi:10.1140/epjc/s10052-025-14683-z

  34. [35]

    M. N. Ferreira, A. S. Miramontes, J. M. Morgado, J. Pa- pavassiliou (2026).arXiv:2605.06242

    Pith/arXiv arXiv 2026

  35. [36]

    Maris, P

    P. Maris, P. C. Tandy, Phys. Rev. C 61 (2000) 045202. doi:10.1103/PhysRevC.61.045202

    work page doi:10.1103/physrevc.61.045202 2000

  36. [37]

    The $\pi$, $K^+$, and $K^0$ electromagnetic form factors

    P. Maris, P. C. Tandy, Phys. Rev. C 62 (2000) 055204. doi:10.1103/PhysRevC.62.055204. arXiv:nucl-th/0005015

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.62.055204 2000

  37. [38]

    S. R. Cotanch, P. Maris, Phys. Rev. D 68 (2003) 036006. doi:10.1103/PhysRevD.68.036006. arXiv:nucl-th/0308008

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.68.036006 2003

  38. [39]

    Nicmorus, G

    D. Nicmorus, G. Eichmann, A. Krassnigg, R. Alkofer, Phys. Rev. D 80 (2009) 054028. doi:10.1103/ PhysRevD.80.054028

    2009

  39. [40]

    Hilger, M

    T. Hilger, M. Gomez-Rocha, A. Krassnigg, Phys. Rev. D 91 (2015) 114004. doi:10.1103/PhysRevD.91. 114004

    work page doi:10.1103/physrevd.91 2015

  40. [41]

    Hilger, C

    T. Hilger, C. Popovici, M. Gomez-Rocha, A. Krass- nigg, Phys. Rev. D 91 (2015) 034013. doi:10.1103/ PhysRevD.91.034013

    2015

  41. [42]

    Eichmann, H

    G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, C. S. Fischer, Prog. Part. Nucl. Phys. 91 (2016) 1–100. doi:10.1016/j.ppnp.2016.07.001

    work page doi:10.1016/j.ppnp.2016.07.001 2016

  42. [44]

    Sanchis-Alepuz, R

    H. Sanchis-Alepuz, R. Alkofer, C. S. Fischer, Eur. Phys. J. A 54 (2018) 41. doi:10.1140/epja/ i2018-12465-x

    work page doi:10.1140/epja/ 2018

  43. [45]

    E. Weil, G. Eichmann, C. S. Fischer, R. Williams, Phys. Rev. D 96 (2017) 014021. doi:10.1103/PhysRevD.96. 014021

    work page doi:10.1103/physrevd.96 2017

  44. [46]

    F. E. Serna, B. El-Bennich, G. Krein, Phys. Rev. D 96 (2017) 014013. doi:10.1103/PhysRevD.96.014013

    work page doi:10.1103/physrevd.96.014013 2017

  45. [47]

    P. C. Wallbott, G. Eichmann, C. S. Fischer, Phys. Rev. D100 (2019) 014033. doi:10.1103/PhysRevD. 100.014033

    work page doi:10.1103/physrevd 2019

  46. [48]

    Santowsky, G

    N. Santowsky, G. Eichmann, C. S. Fischer, P. C. Wall- bott, R. Williams, Phys. Rev. D 102 (2020) 056014. doi:10.1103/PhysRevD.102.056014

    work page doi:10.1103/physrevd.102.056014 2020

  47. [49]

    Miramontes, A

    Á. Miramontes, A. Bashir, K. Raya, P. Roig, Phys. Rev. D 105 (2022) 074013. doi:10.1103/PhysRevD.105. 074013

    work page doi:10.1103/physrevd.105 2022

  48. [50]

    F. Gao, A. S. Miramontes, J. Papavassiliou, J. M. Pawlowski, Phys. Lett. B 863 (2025) 139384. doi:10. 1016/j.physletb.2025.139384

    arXiv 2025

  49. [51]

    Xu, JHEP 2024 (2024) 118

    Y .-Z. Xu, JHEP 2024 (2024) 118. doi:10.1007/ JHEP07(2024)118

    2024

  50. [52]

    Hoffer, G

    J. Hoffer, G. Eichmann, C. S. Fischer, Phys. Rev. D 111 (2025) 054028. doi:10.1103/PhysRevD.111.054028

    work page doi:10.1103/physrevd.111.054028 2025

  51. [53]

    Eichmann, 2025.arXiv:2503.10397

    G. Eichmann, 2025.arXiv:2503.10397

    arXiv 2025

  52. [54]

    Hagel, C

    S. Hagel, C. S. Fischer, M. Q. Huber, J. Y . Yigzaw, 2025. arXiv:2510.27423

    arXiv 2025

  53. [55]

    C. Shi, L. Lu, I. C. Cloët, W. Jia, P. C. Tandy, Phys. Rev. D 113 (2026) L111501. doi:10.1103/yklv-ntq9

    work page doi:10.1103/yklv-ntq9 2026

  54. [57]

    Windisch, Phys

    A. Windisch, Phys. Rev. C 95 (2017) 045204. doi:10. 1103/PhysRevC.95.045204.arXiv:1612.06002

    Pith/arXiv arXiv 2017

  55. [58]

    Sanchis-Alepuz, R

    H. Sanchis-Alepuz, R. Williams, Comput. Phys. Com- mun. 232 (2018) 1–21. doi:10.1016/j.cpc.2018.05. 020

    work page doi:10.1016/j.cpc.2018.05 2018

  56. [59]

    Á. S. Miramontes, H. Sanchis-Alepuz, Eur. Phys. J. A 55 (2019) 170. doi:10.1140/epja/i2019-12847-6. 10

    work page doi:10.1140/epja/i2019-12847-6 2019

  57. [60]

    Alkofer, C

    R. Alkofer, C. S. Fischer, F. Zierler, Phys. Rev. D 113 (2026) 094002. doi:10.1103/q3kq-s8qn

    work page doi:10.1103/q3kq-s8qn 2026

  58. [61]

    Maris, P

    P. Maris, P. C. Tandy, Phys. Rev. C60 (1999) 055214. doi:10.1103/PhysRevC.60.055214

    work page doi:10.1103/physrevc.60.055214 1999

  59. [62]

    Chang, C

    L. Chang, C. D. Roberts, Phys. Rev. Lett. 103 (2009) 081601. doi:10.1103/PhysRevLett.103.081601

    work page doi:10.1103/physrevlett.103.081601 2009

  60. [63]

    Kondo, M

    K.-I. Kondo, M. Murakami, Phys. Rev. D 101 (2020) 074044

    2020

  61. [64]

    Wieland, R

    G. Wieland, R. Alkofer, 2026.arXiv:2604.20235

    Pith/arXiv arXiv 2026

  62. [65]

    Chang, Y .-X

    L. Chang, Y .-X. Liu, C. D. Roberts, Phys. Rev. Lett. 106 (2011) 072001. doi:10.1103/PhysRevLett.106. 072001

    work page doi:10.1103/physrevlett.106 2011

  63. [66]

    Binosi, L

    D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, C. D. Roberts, Phys. Rev. D95 (2017) 031501. doi:10.1103/ PhysRevD.95.031501

    2017

  64. [68]

    Itzykson, J

    C. Itzykson, J. B. Zuber, Quantum Field Theory, Inter- national Series in Pure and Applied Physics, New York, USA: Mcgraw-Hill (1980) 705 p., 1980

    1980

  65. [69]

    A. C. Aguilar, M. N. Ferreira, B. M. Oliveira, J. Papavas- siliou, G. T. Linhares, Eur. Phys. J. C 84 (2024) 1231. doi:10.1140/epjc/s10052-024-13605-9

    work page doi:10.1140/epjc/s10052-024-13605-9 2024

  66. [70]

    Alkofer, C

    R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada, K. Schwenzer, Annals Phys. 324 (2009) 106–172. doi:10.1016/j.aop.2008.07.001

    work page doi:10.1016/j.aop.2008.07.001 2009

  67. [71]

    Williams, C

    R. Williams, C. S. Fischer, W. Heupel, Phys. Rev. D93 (2016) 034026. doi:10.1103/PhysRevD.93.034026

    work page doi:10.1103/physrevd.93.034026 2016

  68. [72]

    Cornwall, R

    J. Cornwall, R. Norton, Phys. Rev. D 8 (1973) 3338–

    1973

  69. [73]

    doi:10.1103/PhysRevD.8.3338

    work page doi:10.1103/physrevd.8.3338

  70. [74]

    J. M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D 10 (1974) 2428–2445. doi:10.1103/PhysRevD.10. 2428

    work page doi:10.1103/physrevd.10 1974

  71. [75]

    Berges, Phys

    J. Berges, Phys. Rev. D 70 (2004) 105010. doi:10.1103/ PhysRevD.70.105010

    2004

  72. [76]

    M. E. Carrington, Y . Guo, Phys. Rev. D 83 (2011) 016006. doi:10.1103/PhysRevD.83.016006

    work page doi:10.1103/physrevd.83.016006 2011

  73. [77]

    A. S. Miramontes, J. M. Morgado, J. Papavassiliou, J. M. Pawlowski, Eur. Phys. J. C 85 (2025) 1055. doi:10. 1140/epjc/s10052-025-14774-x

    2025

  74. [78]

    M. N. Ferreira, A. S. Miramontes, J. M. Morgado, J. Pa- pavassiliou, J. M. Pawlowski, Eur. Phys. J. C 86 (2026)

    2026

  75. [79]

    doi:10.1140/epjc/s10052-026-15487-5

    work page doi:10.1140/epjc/s10052-026-15487-5

  76. [80]

    M. N. Ferreira, A. S. Miramontes, M. J. M., J. Papavas- siliou, 2026.arXiv:2604.07221

    Pith/arXiv arXiv 2026

  77. [81]

    Eichmann, R

    G. Eichmann, R. Williams, R. Alkofer, M. Vuji- novic, Phys. Rev. D89 (2014) 105014. doi:10.1103/ PhysRevD.89.105014

    2014

  78. [82]

    A. Blum, M. Q. Huber, M. Mitter, L. von Smekal, Phys. Rev. D89 (2014) 061703. doi:10.1103/PhysRevD.89. 061703

    work page doi:10.1103/physrevd.89 2014

  79. [83]

    M. Q. Huber, Phys. Rev. D 93 (2016) 085033. doi:10. 1103/PhysRevD.93.085033

    2016

  80. [84]

    M. Q. Huber, Phys. Rev. D 101 (2020) 114009. doi:10. 1103/PhysRevD.101.114009

    2020

Showing first 80 references.