Gauge-parameter cancellation and finite form factors in one-loop quark scattering in the refined Gribov-Zwanziger framework

Wan Wu

REVIEW

Gauge-parameter cancellation and finite form factors in one-loop quark scattering in the refined Gribov-Zwanziger framework

Wan Wu

High Energy Physics - Phenomenology · Submitted 28 Oct 2026 · 18 references

Pith's verdict: A real first calculation in the RGZ matter sector, carefully bookkept and honestly caveated, but with no numerical handle anyone else can check against.

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arXiv ID2610.38463 (version 1)

Pith IDIUOJ2WK5, Pith's permanent record of this paper and its review

CitationWan Wu. "Gauge-parameter cancellation and finite form factors in one-loop quark scattering in the refined Gribov-Zwanziger framework." arXiv:2610.38463 (2026). https://pith.science/paper/IUOJ2WK5

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Claim paper

Why Pith says this

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Editor's note

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New object, honest about its own limits, but nobody but the author can check the numbers yet. That's the short version.

What's actually new: this is the first complete one-loop distinct-flavor four-quark amplitude in the local BRST-invariant RGZ action — constraint determinant, both flavors in closed loops, mixed self-energy attachment, full auxiliary-field vertices, both box routings, external residues. The nearest prior object in the cited literature is the one-loop quark propagator of Ref. [7]. That is a real gap in the literature being filled, and I don't know of another paper that has done it.

What's done well: the bookkeeping is more careful than this genre usually gets. The point in Sec. V A that a dimensionally regulated scaleless zero does not mean the UV and IR residues vanish separately (Eqs. 34–35) is exactly the failure mode that bites these calculations, and the paper makes it explicit instead of burying it. The UV-pole closure in Eq. (42) is a genuine cross-check: only the octet S1 carries a pole, matching the color structure of the tree counterterm. Eq. (20) keeping D before the pole expansion is the right call, and they show the lazy order loses 3 g_s^2 C_F/(32 pi^2). The 144 coefficients ship with a Zenodo DOI, checksums and a versioned GPL release, which is more than most papers in this area bother with.

Soft spots, two of them, one bigger than the other.

First, everything is evaluated at M=0. The paper says at finite M the source slice is not BRST closed (sL_reg = M^4 xi^a c^a), so no symmetry protects the cancellation; it rests on a graph-weighted UV/IR-transfer check asserted in a single sentence in Sec. V A with no derivation displayed. The stress-test note is right that this is the least secured pillar. But note the authors say so themselves: App. S6 states the reconstruction check is export fidelity, not verification of the originating diagrams, and that no numerical evaluation is implied. That is an acknowledged gap, not a concealed one, and I would not weight it as if it were.

Second, and worse for actual use: there is no numerical evaluation of the finite functions. Between Eq. (31), Eq. (42) and the Born ratio R(Q^2), the only quantitative handles are hard logarithms and the UV pole. Anyone who wants these form factors has to build an evaluator on the same branch. The QCD massless limit is explicitly unavailable because poles coalesce, so there is no external benchmark. The coefficients can be internally consistent and still wrong in a way nothing shipped would catch.

Who it's for: people working on the RGZ matter sector, or on off-shell quark amplitudes with modified gluon propagators. Nobody else needs it.

Recommendation: send it out. It is a real calculation with real internal checks and unusually candid caveats. Ask the referee specifically to press on the M→0 regulator ordering and on whether even one numerical point in the finite functions can be produced.

What would settle it

Re-run the paper's symbolic steps in the opposite regulator order — expanding the small-mass regions in $\epsilon_M=M^2$ before taking the $D\to4$ limit — and check whether the sixteen coefficients of $\delta_\alpha M^{(1)}$ still vanish; the paper itself warns that the two orders need not agree term by term, so a nonzero residual would show Eq. (23) holds only for the chosen prescription. A second concrete check: at sample parameters such as $s=1$, $t=-1/2$, $N_c=3$ with positive RGZ masses, verify the three closures the paper asserts — the symbolic zero of Eq. (23), the match between the unrenormalized UV pole and the counterterm amplitude of Eq. (41) with $U_d$ from Eq. (42), and the cancellation of IR single and double poles in every form factor once the scaleless UV/IR ledger is restored.

Referee report

Significance

If correct, this is the first complete one-loop on-shell four-quark amplitude in the RGZ framework with matter coupling, and the calculation is unusually explicit: the gauge comparison in Eq. (21) is symbolic across all sixteen comparison coefficients; the UV counterterms are quoted from published algebraic renormalization results rather than fitted; the IR endpoint analysis of Sec. V B is given in parametric form; and all 144 finite-function coefficients of Eq. (26) are shipped with an exact reconstruction check. The authors also state their limitations honestly, particularly in Sec. II B (finite-M slice not BRST closed) and App. S6 (export fidelity does not constitute independent verification of the originating diagrams). The main risk is not circularity or fitted parameters, but verifiability of the regulator-limit and scaleless-bookkeeping steps on which Eqs. (23) and (26) rest, and the absence of any numerical spot-check of the finite multiscale functions.

Major objections (3)

[Secs. II B and V A; Eqs. (23), (34)-(35)] The gauge-parameter cancellation is verified only at M=0, and the justification for removing the Stueckelberg regulator before comparing gauges rests on an internally applied 'graph-weighted check' that the net unapplied UV/IR transfer vanishes, together with the IR endpoint analysis of Sec. V B. The paper itself states (Sec. II B) that at finite M the source slice is not BRST closed, so no symmetry enforces delta_alpha M^(1)=0 away from M=0. As the authors note in Eqs. (34)-(35), the scaleless-zero failure mode is precisely that a mis-assignment at the M->0/epsilon->0 intersection leaves the formal identities intact while shifting finite terms. Because the graph-weighted check is described but not shipped, a reader cannot independently confirm the regulator-order claim, and any scaleless mis-assignment would survive the released export-fidelity check while changing the finite form factors in Eq. (26). I request that the M->0 transfer ledger (or the graph-weighted check in executable form) be included in the Supplemental Material, or, if it cannot be released, that the sections claiming Eq. (23) identify explicitly which part of the cancellation is a computed result and which part is the internal-check assumption.
[Secs. V B, App. C, App. S6; Eq. (26)] The central claims of finiteness (IR pole cancellation in every form factor, Sec. V B) and of UV pole closure (Eq. (42)) are supported only by symbolic assertions plus an electronic comparison that, by the authors' own statement in App. S6, 'establishes fidelity of the electronic presentation; it is distinct from independent verification of the originating diagrams.' No numerical evaluation of the 31 finite functions is supplied, so no independent reader can spot-check the cancellation of the last IR/UV remnants at a single kinematic point. Because these cancellations are load-bearing for Eq. (26), I ask for the electronic artifacts to include (i) the UV pole comparison for all eight color-spin rows, not merely the C_d S1 example displayed in Eq. (42), and (ii) a benchmark evaluation of Eq. (26) at one or two specified (s,t) points with stated RGZ masses and scale, so that finiteness is testable without redoing the full symbolic reduction. The authors acknowledge this: “No numerical evaluation of the multiscale finite functions, inclusive real-radiation cross section, or general proof of colored asymptotic states is implied.”
[Sec. III D and App. C] Equation (23) is presented as the exact symbolic vanishing of sixteen collected color-spin-scalar coefficients, but no coefficient or intermediate Ward-identity reduction is shown in the text, and the reconstruction check in App. C verifies only that the exported coefficients match the source representation. This is standard practice for large symbolic computations, and I do not object to the assertion itself; however, since the gauge-independence claim is the paper's headline result, a compact demonstration for one non-trivial comparison coefficient (e.g., the double-longitudinal double-counting point described near Eq. (22) and the NA V remainder containing its internal quark pole) would materially increase confidence that the sixteen-coefficient statement is not a bookkeeping artifact of the comparison representation.

Minor objections (5)

[Sec. V C, Eq. (42)] The UV-pole closure is displayed only for the C_d S1 coefficient; the text asserts the other seven coefficients vanish or match. Please state explicitly where the remaining rows are checked (e.g., an electronic file row-by-row comparison of the unrenormalized one-loop UV poles against Eq. (41)).
[Sec. V A] The paragraph describing the expansion in e_M = M^2 > 0 says the negative powers of e_M are bounded 'independently of the saved mass series.' A sentence clarifying what 'independently' means operationally (e.g., a separate dominant-tracking pass on the original scalar coefficients) would help a reader judge the strength of the check.
[Sec. III D] A short explicit example of a finite-M residual surviving (i.e., demonstrating that the naive M=0 substitution and the prescribed limit genuinely differ before the graph-weighted transfer is applied) would illustrate why the regulator-order prescription is physically necessary; currently the reader is asked to take the M=0 evaluation on trust.
[Abstract and Sec. I] The abstract states the result is 'finite after removal of the Stueckelberg regulator in the specified dimensional prescription' without mentioning the nonzero-mass IR domain restriction. Consider adding a scope phrase such as 'in the stated nonzero-mass domain' to the abstract, matching Sec. V B, to avoid over-reading.
[Various] Minor editorial points: the sentence in Sec. III D listing 'sixteen collected color-spin-scalar coefficients' should reference the comparison representation immediately (it does later in the same paragraph, but the opening sentence reads as if these were physical helicity amplitudes); and the caption of Fig. 1 would benefit from an explicit statement that the curves are tree-level only (this is stated in the text but not in the caption).

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The paper

The paper claims

The paper claims the complete one-loop distinct-flavor massless quark scattering amplitude in the refined Gribov–Zwanziger theory is finite and gauge-parameter independent, delivered as eight color–spin form factors.

The reading

This paper attempts to establish that the complete one-loop amplitude for scattering two distinct massless quark flavors in the local BRST-invariant refined Gribov–Zwanziger (RGZ) theory is finite after renormalization and independent of the gauge-fixing parameter, provided the temporary Stueckelberg mass is removed in a precisely specified order. The calculation includes pieces earlier RGZ computations left out: the constraint-determinant Grassmann loops, both quark flavors in closed loops, the complete mixed self-energy attachment, and the full auxiliary-field vertices. The result is a closed form: the renormalized amplitude equals the tree term plus eight color–spin form factors built from 31 finite scalar functions, with all 144 rational coefficients supplied electronically. A curious reader should care because this isolates the theory's genuine mass dependence from gauge artifacts and supplies the virtual partonic four-point object that later steps toward cross sections would build on.

What carries the argument

The carrier of the argument is the gauge-difference construction of Sec. III D: rather than expand two separately reduced four-point amplitudes, the paper evaluates $\delta_\alpha M^{(1)}$ at $M=0$ on complete vertex and box integrands and combines it with the mixed self-energy and external-residue differences before any tensor reduction or pole decomposition. That ordering lets on-shell Ward identities act, exemplified by $\slash{q} = S_0^{-1}(r) - S_0^{-1}(r-q)$ on an external spinor, which is why the complete non-Abelian vertex set is required and why replacing it by only its pinched part would not reproduce Eq. (19). Two further pieces of machinery carry the load. The free-field matrix of Appendix A inverts the full quadratic form, resumming the Stueckelberg and localizing fields together with the RGZ masses $m^2$, $\mu^2$, $m_G^4$ into the propagators that enter every diagram. The UV/IR ledger of Sec. V A records the separate residues of scaleless integrals — $B_0(0;0,0)=\Delta_{\mathrm{UV}}-\Delta_{\mathrm{IR}}=0$ can hide a genuine pole — so that poles are attributed to the right origin before the $\epsilon_M=M^2\to0$ and $D\to4$ limits are taken in the stated order.

Extended reading notes

Core claim

The paper's central claim is that Eq. (26), $M_{\mathrm{ren}} = M^{(0)} + \sum_{A=1,8}\sum_{r=1}^4 C_A S_r F^{(1),\mathrm{fin}}_{Ar} + O(\hbar^2)$, is the complete, finite, gauge-parameter-independent one-loop amplitude for distinct-flavor massless quark scattering in the stated local refined Gribov–Zwanziger action. The argument proceeds in three steps. First, the difference $\delta_\alpha M^{(1)} = M^{(1)}(\alpha)-M^{(1)}(1)$ between a general linear covariant gauge and a reference gauge is formed at the integrand level, before tensor reduction and pole decomposition, and its sixteen color–spin–scalar coefficients vanish symbolically (Eq. (23)) once the Stueckelberg regulator is removed in the specified dimensional prescription; no numerical parameter assignment is used. Second, the ultraviolet poles are closed by local modified-minimal-subtraction counterterms, with the entire UV pole residing in the octet $S_1$ form factor, as the color structure of the tree counterterm requires; the compact check is the coefficient $U_d$ of Eq. (42). Third, after scaleless integrals are split into their separate UV and IR residues rather than discarded as zeros, the infrared single and double poles cancel in every form factor on the stated nonzero-mass domain. The output is eight form factors over a thirty-entry scalar integral inventory, with the analytic branch, kinematic domain, and regulator order explicitly retained.

Load-bearing premise

The result stands or falls on the order of limits: the Stueckelberg regulator mass must be removed in the specific dimensional prescription of Sec. V A before the gauge comparison is evaluated at $M=0$, because at finite $M$ the regulator term is not BRST-closed and no symmetry enforces the cancellation.

Editorial extensions

If this is right

The eight form factors of Eq. (26) become the reference one-loop virtual amplitude for distinct-flavor quark scattering in this RGZ action; later RGZ calculations may take them as given within the stated branch and domain.
Gauge independence holds separately for the tree and loop terms, so the surviving dependence on $m^2$, $\mu^2$, and $m_G^4$ is a property of the theory itself, not an artifact of the gauge choice.
The two-flavor local counterterms close the ultraviolet poles with the pole confined to a single color–spin channel, so renormalizing the four-point amplitude adds no fitted parameters beyond the known RGZ renormalization relations.
The QCD massless limit is not a point inside the nonzero-mass IR domain; comparisons with QCD must be made on complete integral combinations sharing one IR regulator, not by setting $r_\pm$ and $\mu^2$ to zero in the coefficient table.
The amplitude alone does not define an inclusive next-to-leading-order cross section; real emission, an IR-safe observable, and a factorization prescription are distinct ingredients the paper explicitly leaves to separate work.

Reading between the lines

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

The same integrand-level gauge-difference construction could be reapplied to other processes in this action, such as quark–gluon or four-gluon scattering; the completeness counting of Sec. III B, not the algebra itself, looks like the transferable step.
Because the rational coefficients encode the pole separation $r_+-r_- = 2i\Omega$ and the residues $a_\pm$, the form factors could in principle be inverted: numerical evaluation of the finite $B_0$, $C_0$, $D_0$ functions at sample kinematics would constrain the condensate and horizon parameters $m^2,\mu^2,m_G^4$, a use the paper does not pursue.
A natural next benchmark is to evaluate the 24 finite scalar functions on the designated analytic sheet and verify numerically that the infrared poles cancel at nonexceptional kinematics; this would convert the symbolic claim into a reproducible datum and test the branch prescription.
The UV/IR ledger discipline generalizes beyond RGZ: in any theory with Stueckelberg-type regulators, interchanging mass and dimensional limits can misassign the separate UV and IR residues of scaleless integrals, so the paper's bookkeeping is a template for other regulator-limit calculations.

Original abstract

We calculate the one-loop amplitude for distinct-flavor massless quark scattering in a local BRST-invariant refined Gribov-Zwanziger theory with minimally coupled quarks. The constraint determinant and two dynamical quark flavors are included in the action and in the loop sum. Retaining the refined Gribov-Zwanziger masses in the free field matrix, we combine the mixed self-energy insertion, complete vertex and box corrections, and external-leg residues. The difference between a general linear covariant gauge and a reference gauge vanishes symbolically after removal of the Stueckelberg regulator in the specified dimensional prescription. We separate ultraviolet and infrared poles, retaining contributions hidden by scaleless integrals and by the order of mass and dimensional limits. Local modified-minimal-subtraction counterterms close the ultraviolet poles, and the remaining infrared poles cancel in the stated nonzero-mass domain. The finite answer is supplied as eight color-spin form factors over a thirty-entry scalar integral inventory, with all 144 finite-function coefficients provided electronically. The result defines a partonic amplitude on a specified analytic branch; it does not assume colored asymptotic states or establish an inclusive next-to-leading-order cross section.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 6,324 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.58 · compares with prior results 0.12 · mentions code or data availability 0.80.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We calculate the one-loop amplitude for distinct-flavor massless quark scattering in a local BRST-invariant refined Gribov-Zwanziger theory with minimally coupled quarks. The constraint determinant and two dynamical quark flavors are included in the action and in the loop sum. Retaining the refined Gribov-Zwanziger masses in the free field matrix, we combine the mixed self-energy insertion, complete vertex and box corrections, and external-leg residues. The difference between a general linear covariant gauge and a reference gauge vanishes symbolically after removal of the Stueckelberg regulator in the specified dimensional prescription. We separate ultraviolet and infrared poles, retaining contributions hidden by scaleless integrals and by the order of mass and dimensional limits. Local modified-minimal-subtraction counterterms close the ultraviolet poles, and the remaining infrared poles cancel in the stated nonzero-mass domain. The finite answer is supplied as eight color-spin form factors over a thirty-entry scalar integral inventory, with all 144 finite-function coefficients provided electronically. The result defines a partonic amplitude on a specified analytic branch; it does not assume colored asymptotic states or establish an inclusive next-to-leading-order cross section.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We calculate the one-loop amplitude for distinct-flavor massless quark scattering in a local BRST-invariant refined Gribov-Zwanziger theory with minimally coupled quarks.	contribution 0.99	0.96	0.07	asserted 1.00
s1	The constraint determinant and two dynamical quark flavors are included in the action and in the loop sum.	method 0.92	0.80	0.02	asserted 0.99
s2	Retaining the refined Gribov-Zwanziger masses in the free field matrix, we combine the mixed self-energy insertion, complete vertex and box corrections, and external-leg residues.	method 0.99	0.93	0.04	asserted 0.97
s3	The difference between a general linear covariant gauge and a reference gauge vanishes symbolically after removal of the Stueckelberg regulator in the specified dimensional prescription.	result 0.98	0.82	0.39	asserted 1.00
s4	We separate ultraviolet and infrared poles, retaining contributions hidden by scaleless integrals and by the order of mass and dimensional limits.	method 0.97	0.93	0.05	asserted 0.99
s5	Local modified-minimal-subtraction counterterms close the ultraviolet poles, and the remaining infrared poles cancel in the stated nonzero-mass domain.	result 0.48	0.92	0.20	asserted 1.00
s6	The finite answer is supplied as eight color-spin form factors over a thirty-entry scalar integral inventory, with all 144 finite-function coefficients provided electronically.	result 0.96	0.93	0.20	asserted 1.00
s7	The result defines a partonic amplitude on a specified analytic branch; it does not assume colored asymptotic states or establish an inclusive next-to-leading-order cross section.	limitation 0.96	0.83	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Born exchange ratio, Eq. (

Figures reproduced from arXiv:2610.38463 by the authors.

References

Works cited (18)

[7]

D. Binosi and J. Papavassiliou, Phys. Rept.479, 1 (2009), arXiv:0909.2536

View on Pith Pith review arXiv 2009
[1]

V. N. Gribov, Nucl. Phys. B139, 1 (1978)

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[2]

D. Zwanziger, Nucl. Phys. B323, 513 (1989)

View on Pith 1989
[3]

D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, and H. Verschelde, Phys. Rev. D78, 065047 (2008), arXiv:0806.4348

View on Pith Pith review arXiv 2008
[4]

and an algebraic renormalization analysis [5]. These developments make it possible to ask a more restrictive question than the behavior of an individual propagator: how do the localizing fields, the constraint sector, and the ordinary quark interactions combine in a complete on-shell four-point amplitude? A replacement of the gluon propagator inside famil...

View on Pith
[5]

M. A. L. Capri, D. Dudal, D. Fiorentini, M. S. Guimaraes, I. F. Justo, A. D. Pereira, B. W. Mintz, L. F. Palhares, R. F. Sobreiro, and S. P. Sorella, Phys. Rev. D 94, 025035 (2016), arXiv:1605.02610

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[6]

M. A. L. Capri, D. Fiorentini, A. D. Pereira, and S. P. Sorella, Phys. Rev. D96, 054022 (2017), arXiv:1708.01543

View on Pith Pith review arXiv 2017
[8]

G. P. de Brito, P. D. Fabritiis, and A. D. Pereira, Quark propagator at one-loop in the Refined Gribov-Zwanziger framework (2025), arXiv:2505.10602

View on Pith Pith review arXiv 2025
[9]

G. Passarino and M. J. G. Veltman, Nucl. Phys. B160, 151 (1979)

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[10]

A. Denner and S. Dittmaier, Nucl. Phys. B844, 199 (2011), arXiv:1005.2076

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[11]

H. H. Patel, Comput. Phys. Commun.197, 276 (2015), arXiv:1503.01469

View on Pith Pith review arXiv 2015
[12]

V. Shtabovenko, Comput. Phys. Commun.218, 48 (2017), arXiv:1611.06793

View on Pith Pith review arXiv 2017
[13]

W. Wu, RGZWorkflow, Version 1.0.0, open-source soft- ware (2026), released 28 October 2026; GPL-3.0-or-later

View on Pith 2026
[14]

A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, Comput. Phys. Commun.185, 2250 (2014), arXiv:1310.1921

View on Pith Pith review arXiv 2014
[15]

T. Hahn, Comput. Phys. Commun.140, 418 (2001), arXiv:hep-ph/0012260

View on Pith Pith review arXiv 2001
[16]

V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun.256, 107478 (2020), arXiv:2001.04407

View on Pith Pith review arXiv 2020
[17]

V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun.306, 109357 (2025), arXiv:2312.14089

View on Pith Pith review arXiv 2025
[18]

R. N. Lee and A. A. Pomeransky, JHEP (11), 165, arXiv:1308.6676. 10 Supplemental Material Gauge-parameter cancellation and finite form factors in one-loop quark scattering in the refined Gribov–Zwanziger framework Wan Wu — Independent Researcher wuw20phy@gmail.com Appendix S1: Exact electronic result and conventions This supplement specifies the finite on...

View on Pith Pith review arXiv 2016

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [7]

D. Binosi and J. Papavassiliou, Phys. Rept.479, 1 (2009), arXiv:0909.2536

View on Pith Pith review arXiv 2009

[2] [1]

V. N. Gribov, Nucl. Phys. B139, 1 (1978)

View on Pith 1978

[3] [2]

D. Zwanziger, Nucl. Phys. B323, 513 (1989)

View on Pith 1989

[4] [3]

D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, and H. Verschelde, Phys. Rev. D78, 065047 (2008), arXiv:0806.4348

View on Pith Pith review arXiv 2008

[5] [4]

and an algebraic renormalization analysis [5]. These developments make it possible to ask a more restrictive question than the behavior of an individual propagator: how do the localizing fields, the constraint sector, and the ordinary quark interactions combine in a complete on-shell four-point amplitude? A replacement of the gluon propagator inside famil...

View on Pith

[6] [5]

M. A. L. Capri, D. Dudal, D. Fiorentini, M. S. Guimaraes, I. F. Justo, A. D. Pereira, B. W. Mintz, L. F. Palhares, R. F. Sobreiro, and S. P. Sorella, Phys. Rev. D 94, 025035 (2016), arXiv:1605.02610

View on Pith Pith review arXiv 2016

[7] [6]

M. A. L. Capri, D. Fiorentini, A. D. Pereira, and S. P. Sorella, Phys. Rev. D96, 054022 (2017), arXiv:1708.01543

View on Pith Pith review arXiv 2017

[8] [8]

G. P. de Brito, P. D. Fabritiis, and A. D. Pereira, Quark propagator at one-loop in the Refined Gribov-Zwanziger framework (2025), arXiv:2505.10602

View on Pith Pith review arXiv 2025

[9] [9]

G. Passarino and M. J. G. Veltman, Nucl. Phys. B160, 151 (1979)

View on Pith 1979

[10] [10]

A. Denner and S. Dittmaier, Nucl. Phys. B844, 199 (2011), arXiv:1005.2076

View on Pith Pith review arXiv 2011

[11] [11]

H. H. Patel, Comput. Phys. Commun.197, 276 (2015), arXiv:1503.01469

View on Pith Pith review arXiv 2015

[12] [12]

V. Shtabovenko, Comput. Phys. Commun.218, 48 (2017), arXiv:1611.06793

View on Pith Pith review arXiv 2017

[13] [13]

W. Wu, RGZWorkflow, Version 1.0.0, open-source soft- ware (2026), released 28 October 2026; GPL-3.0-or-later

View on Pith 2026

[14] [14]

A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, Comput. Phys. Commun.185, 2250 (2014), arXiv:1310.1921

View on Pith Pith review arXiv 2014

[15] [15]

T. Hahn, Comput. Phys. Commun.140, 418 (2001), arXiv:hep-ph/0012260

View on Pith Pith review arXiv 2001

[16] [16]

V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun.256, 107478 (2020), arXiv:2001.04407

View on Pith Pith review arXiv 2020

[17] [17]

V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun.306, 109357 (2025), arXiv:2312.14089

View on Pith Pith review arXiv 2025

[18] [18]

R. N. Lee and A. A. Pomeransky, JHEP (11), 165, arXiv:1308.6676. 10 Supplemental Material Gauge-parameter cancellation and finite form factors in one-loop quark scattering in the refined Gribov–Zwanziger framework Wan Wu — Independent Researcher wuw20phy@gmail.com Appendix S1: Exact electronic result and conventions This supplement specifies the finite on...

View on Pith Pith review arXiv 2016