Pith. sign in

REVIEW 3 major objections 6 minor 36 references

Full three-loop electron EDM is exactly three times the electroweak-Weinberg-only estimate.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 23:55 UTC pith:BWN2QGPF

load-bearing objection A substantial direct three-loop EDM calculation with a plausible factor-of-three result, but the m_W expansion in Eq. (3.2) leaves a real gap between the massless-W integrals and the Weinberg-operator contribution that a referee should force the authors to close. the 3 major comments →

arxiv 2602.11888 v3 pith:BWN2QGPF submitted 2026-02-12 hep-ph

Full Three-Loop Electroweak Multiplet Contributions to the Electron Electric Dipole Moment

Tatsuya Banno , Junji Hisano , Teppei Kitahara , Kiyoto Ogawa , Naohiro Osamura This is my paper
classification hep-ph
keywords electron electric dipole momentCP violationthree-loop radiative correctionselectroweak Weinberg operatorSU(2)L multipletsdark matterYukawa couplingseffective field theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper targets a specific gap in estimates of the electron electric dipole moment (EDM) induced by new SU(2)_L fermion multiplets with CP-violating Yukawa couplings. Earlier effective-field-theory estimates kept only the two-loop-generated electroweak-Weinberg operator and then matched it to the electron EDM at one loop; this paper shows that the direct three-loop contribution from the CP-odd electron dipole operator enters at the same order and must be included. The central result is a full three-loop evaluation of the twelve contributing diagrams, giving an EDM that is exactly three times the Weinberg-only value at leading order in the W-boson mass expansion, independent of the multiplet representation. Because current and next-generation EDM experiments can already probe TeV-scale masses in these models, the factor of three changes the mass reach and strengthens the case that future experiments can test dark-matter-motivated multiplets.

Core claim

For two fermion multiplets ψ_A and ψ_B in the same representation r of SU(2)_L and a complex scalar S that is a singlet, with CP-violating Yukawa couplings containing scalar and pseudoscalar components s and a, the paper claims the full three-loop electron EDM (in units of e) is d_e^Full/e = (α_2^2 m_e/(16π²)²)(r(r²−1)/12) Im(s a*) m_A m_B B(m_A, m_B, m_S, m_W). Expanding B in powers of m_W², B = B0 + m_W² B1 + …, the coefficient B0 is exactly three times the coefficient obtained by routing the same model through the electroweak-Weinberg operator and its one-loop matching, so d_e^Full ≈ 3 d_e^CW at leading order. The twelve-diagram calculation shows that only the effective WWγ vertex survive

What carries the argument

The load-bearing object is the three-loop integral combination B(m_A,m_B,m_S,m_W), assembled from twelve Feynman diagrams and expanded as B = B0 + m_W² B1 + O(m_W⁴/Λ⁴). B0 and B1 are written as finite combinations of three-loop vacuum master integrals; in the degenerate case m_A = m_B they reduce to seven master integrals with analytic expressions given in the paper. The representation dependence is packaged in the group identity Tr[Q[T+,T−]] = r(r²−1)/12, which makes the leading ratio to the Weinberg-only contribution independent of r. The argument works by comparing the analytic B0/B1 with the two-loop Weinberg coefficient derived from the same Yukawa couplings, isolating the factor of thr

Load-bearing premise

The factor of three rests on treating B0 + m_W² B1 as the complete leading behavior; if the neglected O(m_W⁴/Λ⁴) terms are not negligible at the TeV masses considered, the leading-order 3:1 ratio shifts.

What would settle it

Evaluate the exact three-loop integral combination B(m_A,m_B,m_S,m_W) numerically at representative masses such as m_A = m_B = m_S = 1 TeV with m_W = 80 GeV, without expanding in m_W, and compare the ratio d_e^Full/d_e^CW with 3; a deviation larger than the estimated O(m_W⁴/Λ⁴) fraction would falsify the leading-order claim. A cheaper check is to evaluate B0 from the paper's analytic expression and verify the factor of three against the Weinberg coefficient in both degenerate and hierarchical limits.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The effective-field-theory estimate under-predicts the electron EDM by a factor of three; for a fixed experimental sensitivity, the reach in multiplet mass is wider by about a factor of 1.7 in the degenerate-mass regime.
  • A fermionic quintuplet (r = 5) dark-matter candidate with CP-violating Yukawa couplings and mass below about 350 GeV is already excluded by the current electron EDM bound; an order-of-magnitude sensitivity improvement would probe into the TeV region.
  • The factor of three holds at leading order for degenerate masses and in both hierarchical limits m_A = m_B ≪ m_S and m_S ≪ m_A = m_B, as well as being independent of the representation r.
  • Accurate predictions for future experiments must include both the electroweak-Weinberg route and the direct three-loop dipole contribution; neglecting the direct term is not a small correction but a missing factor of three.

Where Pith is reading between the lines

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

  • The 3:1 ratio is established only through the leading two terms of the m_W expansion; a numerical evaluation of the unexpanded three-loop integrals at TeV-scale masses would show whether the neglected O(m_W⁴/Λ⁴) terms preserve the clean factor.
  • Because the calculation is restricted to a scalar that is an SU(2)_L singlet and equal fermion representations, the 3× rule may not survive in more general representations where gauge-boson insertions on the scalar line add new diagrams; computing those would map out where the factor applies.
  • The r(r²−1) growth implies that even higher-dimensional multiplets beyond the plotted r = 5 would be more strongly constrained or discoverable at higher masses by the same experiments, a natural next target not developed in the paper.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript computes the electron EDM generated at three-loop order by CP-violating Yukawa interactions among SU(2)_L fermion multiplets ψ_A, ψ_B and a complex scalar S, restricted to the representation choice (A,B,S)=(r,r,1). It evaluates the twelve contributing three-loop diagrams using integration-by-parts reduction (Kira/Fermat) and analytic master integrals, and organizes the result as an expansion B(m_A,m_B,m_S,m_W) ≃ B0 + m_W^2 B1 in the W-boson mass. The central claim is that the full three-loop result is exactly three times larger than the previous EFT result mediated by the electroweak-Weinberg operator alone, at leading order in the mass expansion, independent of the representation r. The paper provides analytic expressions for the degenerate case m_A=m_B=m_S, asymptotic formulas for hierarchical masses, and numerical prospects for future EDM experiments.

Significance. If correct, this is a useful and nontrivial result: it identifies a factor-of-three correction to EDM predictions for a motivated class of TeV-scale SU(2)_L multiplet dark-matter models and highlights the r(r^2-1) representation enhancement. The calculation uses public IBP tools and analytic master integrals, and the on-shell WWγ consistency check after Eq. (3.1) is a genuine cross-check of the two-loop heavy subdiagram. The paper also provides an ancillary file for the non-degenerate case, which is a positive step for reproducibility. The main caveat is that the factor-3 claim rests on the m_W^2 expansion of Eq. (3.2), and the paper does not demonstrate that the soft-W region is included in B0+B1; this needs to be addressed before the result is fully established.

major comments (3)
  1. [§3, Eq. (3.2)] The expansion B(m_A,m_B,m_S,m_W) ≃ B0 + m_W^2 B1 + O(m_W^4/Λ^4) is presented as a Taylor expansion of the three-loop integral. This is a hard-region expansion of the W propagators, valid when all loop momenta are much larger than m_W. The region where W momenta are of order m_W generates, via the one-loop electron-neutrino subdiagram, precisely the electroweak-Weinberg matching contribution d_CW of Eq. (2.9), which is of order 1/Λ^2 rather than O(m_W^4/Λ^4). The paper does not provide a method-of-regions analysis or a numerical evaluation of the unexpanded integrals at finite m_W, so the residual in Eq. (3.2) could be O(1/Λ^2) and the factor-3 ratio would not follow. The issue is not academic: at m_A=0.1 TeV, m_W^2/m_A^2≃0.64, so the truncation is also numerically questionable at the low end of the plotted range. I request an explicit region decomposition or a numerical cross-check (e.g.
  2. [§4.2, Eqs. (4.3)–(4.4)] The claim that d_Full/d_CW = 3 holds for all m_A=m_B≠m_S is asserted, but the only closed-form evidence displayed in the text is the two extreme limits m_A≪m_S and m_S≪m_A. The intermediate regime is relegated to the ancillary file solB.txt. Because the exact factor-3 relation is the central result, the main text should present the analytic ratio as a function of x=m_S/m_A (or at least a table of values) and describe how the ancillary file was checked. As written, the 'exactly three' statement is stronger than what the written derivation supports.
  3. [§3, text after Eq. (3.1)] The consistency check that the two-loop WWγ vertex reproduces the Wilson coefficient C_W when the two W bosons are on-shell is a useful check of the heavy subdiagram, but it does not validate the off-shell WWγ vertex at W momenta of order m_W, which is what enters the one-loop EDM matching. Thus this check is necessary but not sufficient to establish that the full three-loop integral in Eq. (3.1) contains both the direct dipole contribution and the Weinberg-operator-mediated contribution. The on-shell check should be supplemented by an off-shell/momentum-expansion analysis or by a direct comparison of the full and expanded integrals as in the previous comment.
minor comments (6)
  1. [Eq. (3.2)] The scale Λ in the remainder O(m_W^4/Λ^4) is not defined; it should be specified as, e.g., the largest of m_A,m_B,m_S.
  2. [§2] Minor grammatical issues: 'the later' should be 'the latter'; 'not occurred' should be 'does not occur'.
  3. [§4.2] The sentence 'we give Eq. (3.2) as solB.txt' is unclear; presumably the coefficients B0 and B1 for the non-degenerate case are provided, not Eq. (3.2) itself.
  4. [Appendix A] The integrals J with negative powers, e.g. J[2,2,3,1,-1,1] in Eq. (A.1), should be explicitly defined as integrals with inverse propagators (polynomials) so that readers do not misinterpret them as ordinary propagators.
  5. [§4.1] The statement 'the full contribution is exactly three times larger, independent of the representation' is slightly imprecise because it refers to the leading-order terms with the NLO term excluded; consider rephrasing as 'at leading order in m_W^2/m_A^2'.
  6. [Figures 4 and 6] The caption 'degenerated mass' should read 'degenerate mass'. Also, the figures would benefit from error bands or a clear statement of the expansion-truncation uncertainty in the plotted curves.

Circularity Check

0 steps flagged

No significant circularity: the three-loop integrals are computed independently and the factor-3 comparison is a genuine cross-check, not an input.

full rationale

The central derivation is self-contained. The full three-loop EDM in Eq. (3.1) is obtained by direct evaluation of B0 and B1 through IBP reduction with Kira/Fermat and analytic master integrals from Refs. [22,37], which are external rather than derived from the paper's own conclusions. No parameter is fitted to the electroweak-Weinberg contribution. The on-shell WW-gamma check after Eq. (3.1) is presented as a consistency test, not as a normalization condition. The factor-of-three result comes from comparing the independently computed coefficient in Eq. (4.1) with the prior EFT result in Eq. (4.2); the ratio is not imposed by construction. The paper does rely on the authors' earlier Ref. [15] for the electroweak-Weinberg contribution, but that is an independently published result and is also re-derived in Sec. 2 with Eqs. (2.5) and (2.9). The uniqueness statement about Im(sa*)mAmB is an elementary rephasing-invariance argument supported by Refs. [17,18]; it does not carry the factor-3 claim. The expansion in Eq. (3.2) is a physical approximation about the heavy-mass limit and is a correctness/regime assumption, not a circular reduction. No fitted-input-called-prediction, ansatz-smuggling, or renaming pattern is present. Minor self-citations occur but they are not load-bearing, so the paper is not circular in the sense relevant here.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The central claim rests on the validity of three-loop integration techniques (IBP, master integrals) and on the model assumption that only the specified diagrams contribute. The main free parameters are the Yukawa coupling and masses, which are scanned over benchmark values. The paper does not introduce any new ad hoc entities.

free parameters (4)
  • Yukawa coupling Im(s a*) = 0.25 (reference value used in figures)
    CP-violating source strength. Set to 0.25 to illustrate numerical reach; EDM scales linearly with it. Not fitted to data but a benchmark choice.
  • m_A = m_B = m_S (degenerate mass case) = Varies 0.1-10 TeV in figures
    New-physics mass scale. Scanned over a relevant range, not fitted. However the central analytic result is derived in this restricted mass configuration.
  • m_S (non-degenerate case) = 1 TeV in figures
    Scalar mass fixed to 1 TeV for illustration. The analytic result in Eq. (4.3) covers the hierarchy limits.
  • SU(2)_L representation r = r = 2,3,4,5
    Model parameter controlling the EDM through r(r^2-1)/12. Not fitted; choices motivated by dark matter candidates.
axioms (5)
  • domain assumption Standard Model Effective Field Theory and the BMHV scheme give the correct matching for the electroweak-Weinberg operator.
    Used in Sec. 2 to relate the full-theory result to the EFT operator. This is a standard assumption in the field, but the paper itself notes the subtlety with evanescent operators.
  • standard math The three-loop master integrals from Refs. [22,37] are correct and applicable.
    Essential for evaluating B0 and B1. These are external results from the literature.
  • standard math The IBP reduction performed with Kira and Fermat is correct.
    Assumed to be bug-free and correctly applied. This is standard practice but is a potential source of error.
  • domain assumption The expansion in m_W^2/Lambda^2 is valid for the mass ranges considered.
    Used in Eq. (3.2). For the lightest masses considered (m_A ~ 0.1 TeV), m_W^2/m_A^2 ~ 0.64, so the O(m_W^4/m_A^4) terms could be non-negligible, though the paper includes the m_W^2 B1 term and shows some numerical results. It assumes truncation is justified.
  • domain assumption Only the twelve Feynman diagrams in Fig. 3 contribute.
    The paper argues that WW/ZZ two-point functions and ZZ gamma/gamma gamma gamma vertices do not contribute because of Levi-Civita structure or cancellation. This is plausible but stated briefly.
invented entities (1)
  • None no independent evidence
    purpose: N/A
    No new particles, forces, or entities are introduced. The model uses standard SU(2)_L multiplets, which are well-motivated by dark matter considerations.

pith-pipeline@v1.3.0-alltime-deepseek · 12671 in / 9005 out tokens · 64500 ms · 2026-08-02T23:55:16.412118+00:00 · methodology

0 comments
read the original abstract

Experimental sensitivity to the electric dipole moment (EDM) of the electron has improved remarkably in recent years. Consequently, future prospects could probe new physics whose contribution to the electron EDM first arises at three-loop order. Additional SU(2)$_L$ multiplets with CP-violating Yukawa interactions, which contribute to the electron EDM at three-loop level, is one such testable new physics scenario. In this scenario, the electron EDM is radiatively induced from two contributions: the CP-odd trilinear $W$-boson coupling, called the electroweak-Weinberg operator, and the CP-odd dipole operator of electron. The former and the latter operators are generated at two-loop and three-loop levels, respectively, after integrating out the SU(2)$_L$ multiplets. Within the same models, according to an analysis based on the Standard Model Effective Field Theory (SMEFT), we previously found that the contribution to the electron EDM from the electroweak-Weinberg operator can be probed in future experiments. However, the one-loop matching condition between the electron EDM and the electroweak-Weinberg operator does not receive a large logarithmic enhancement because the associated anomalous dimension is zero. The CP-odd dipole operator of the electron would contribute to the electron EDM at the same three-loop order as the contribution through the electroweak-Weinberg operator. In this paper, we directly calculate the electron EDM induced by the CP-violating Yukawa interactions of the SU(2)$_L$ multiplets at full three-loop level. A central result is that the full three-loop calculation is a factor of three larger than that of the electroweak-Weinberg operator alone.

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

36 extracted references · 26 linked inside Pith

  1. [2]

    An improved bound on the electron’s electric dipole moment,

    T. S. Roussyet al., “An improved bound on the electron’s electric dipole moment,” Science 381(2023) adg4084[arXiv:2212.11841]

    Pith/arXiv arXiv 2023

  2. [3]

    Electric dipole moments and the search for new physics,

    R. Alarconet al., “Electric dipole moments and the search for new physics,” inSnowmass

  3. [4]

    SiPM module for the ACME III electron EDM search,

    A. Hiramotoet al., “SiPM module for the ACME III electron EDM search,” Nucl. Instrum. Meth. A1045(2023) 167513[arXiv:2210.05727]

    Pith/arXiv arXiv 2023

  4. [5]

    CKM benchmarks for electron electric dipole moment experiments,

    M. Pospelov and A. Ritz, “CKM benchmarks for electron electric dipole moment experiments,” Phys. Rev. D89(2014) 056006[arXiv:1311.5537]

    Pith/arXiv arXiv 2014

  5. [6]

    Standard Model Prediction for Paramagnetic Electric Dipole Moments,

    Y. Ema, T. Gao, and M. Pospelov, “Standard Model Prediction for Paramagnetic Electric Dipole Moments,” Phys. Rev. Lett.129(2022) 231801[arXiv:2202.10524]

    Pith/arXiv arXiv 2022

  6. [7]

    Minimal dark matter,

    M. Cirelli, N. Fornengo, and A. Strumia, “Minimal dark matter,” Nucl. Phys. B753(2006) 178–194[hep-ph/0512090]

    Pith/arXiv arXiv 2006

  7. [8]

    Non-perturbative effect on thermal relic abundance of dark matter,

    J. Hisano, S. Matsumoto, M. Nagai, O. Saito, and M. Senami, “Non-perturbative effect on thermal relic abundance of dark matter,” Phys. Lett. B646(2007) 34–38 [hep-ph/0610249]

    Pith/arXiv arXiv 2007

  8. [9]

    Cosmology and Astrophysics of Minimal Dark Matter,

    M. Cirelli, A. Strumia, and M. Tamburini, “Cosmology and Astrophysics of Minimal Dark Matter,” Nucl. Phys. B787(2007) 152–175[arXiv:0706.4071]

    Pith/arXiv arXiv 2007

  9. [10]

    Minimal Dark Matter: Model and results,

    M. Cirelli and A. Strumia, “Minimal Dark Matter: Model and results,” New J. Phys.11 (2009) 105005[arXiv:0903.3381]. [11]A TLASCollaboration, “Search for long-lived charginos based on a disappearing-track signature using 136 fb−1 of pp collisions at√s= 13 TeV with the ATLAS detector,” Eur. Phys. J. C82(2022) 606[arXiv:2201.02472]

    Pith/arXiv arXiv 2009

  10. [12]

    Electric Dipole Moment of the Electron and of the Neutron,

    S. M. Barr and A. Zee, “Electric Dipole Moment of the Electron and of the Neutron,” Phys. Rev. Lett.65(1990) 21–24. [Erratum: Phys.Rev.Lett. 65, 2920 (1990)]

    1990

  11. [13]

    Higgsino Dark Matter in High-Scale Supersymmetry,

    N. Nagata and S. Shirai, “Higgsino Dark Matter in High-Scale Supersymmetry,” JHEP01 (2015) 029[arXiv:1410.4549]

    Pith/arXiv arXiv 2015

  12. [14]

    Effective Interaction of Electroweak-Interacting Dark Matter with Higgs Boson and Its Phenomenology,

    J. Hisano, D. Kobayashi, N. Mori, and E. Senaha, “Effective Interaction of Electroweak-Interacting Dark Matter with Higgs Boson and Its Phenomenology,” Phys. Lett. B742(2015) 80–85[arXiv:1410.3569]

    Pith/arXiv arXiv 2015

  13. [15]

    Impact of the electroweak Weinberg operator on the electric dipole moment of electron,

    T. Banno, J. Hisano, T. Kitahara, K. Ogawa, and N. Osamura, “Impact of the electroweak Weinberg operator on the electric dipole moment of electron,” JHEP02(2025) 082 [arXiv:2408.02375]

    Pith/arXiv arXiv 2025

  14. [16]

    Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions,

    E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions,” JHEP01(2018) 084[arXiv:1711.05270]

    Pith/arXiv arXiv 2018

  15. [17]

    Closer look at the matching condition for radiative QCDθparameter,

    T. Banno, J. Hisano, T. Kitahara, and N. Osamura, “Closer look at the matching condition for radiative QCDθparameter,” JHEP02(2024) 195[arXiv:2311.07817]

    Pith/arXiv arXiv 2024

  16. [18]

    Two-loop corrections to QCDθangle from evanescent operator in the BMHV scheme,

    T. Banno, J. Hisano, T. Kitahara, K. Ogawa, and N. Osamura, “Two-loop corrections to QCDθangle from evanescent operator in the BMHV scheme,” JHEP09(2025) 135 [arXiv:2502.14500]

    arXiv 2025

  17. [19]

    Model independent evaluation of the Wilson coefficient of the Weinberg operator in QCD,

    T. Abe, J. Hisano, and R. Nagai, “Model independent evaluation of the Wilson coefficient of the Weinberg operator in QCD,” JHEP03(2018) 175[arXiv:1712.09503]. [Erratum: JHEP 09, 020 (2018)]. – 14 –

    Pith/arXiv arXiv 2018

  18. [20]

    The Standard model effective potential at two loops,

    C. Ford, I. Jack, and D. R. T. Jones, “The Standard model effective potential at two loops,” Nucl. Phys. B387(1992) 373–390[hep-ph/0111190]. [Erratum: Nucl.Phys.B 504, 551–552 (1997)]

    Pith/arXiv arXiv 1992

  19. [21]

    Complete two loop dominant corrections to the mass of the lightest CP even Higgs boson in the minimal supersymmetric standard model,

    J. R. Espinosa and R.-J. Zhang, “Complete two loop dominant corrections to the mass of the lightest CP even Higgs boson in the minimal supersymmetric standard model,” Nucl. Phys. B586(2000) 3–38[hep-ph/0003246]

    Pith/arXiv arXiv 2000

  20. [22]

    Two Loop Effective Potential for a General Renormalizable Theory and Softly Broken Supersymmetry,

    S. P. Martin, “Two Loop Effective Potential for a General Renormalizable Theory and Softly Broken Supersymmetry,” Phys. Rev. D65(2002) 116003[hep-ph/0111209]

    Pith/arXiv arXiv 2002

  21. [23]

    Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,

    E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,” JHEP03(2018) 016[arXiv:1709.04486]. [Erratum: JHEP 12, 043 (2023)]

    Pith/arXiv arXiv 2018

  22. [24]

    Regularization and Renormalization of Gauge Fields,

    G. ’t Hooft and M. J. G. Veltman, “Regularization and Renormalization of Gauge Fields,” Nucl. Phys. B44(1972) 189–213

    1972

  23. [25]

    Dimensional Renormalization and the Action Principle,

    P. Breitenlohner and D. Maison, “Dimensional Renormalization and the Action Principle,” Commun. Math. Phys.52(1977) 11–38

    1977

  24. [26]

    A bound on CP violating couplings of the W boson

    F. Hoogeveen, “A bound on CP violating couplings of the W boson.”. MPI-PAE/PTh-25/87, 1987

    1987

  25. [27]

    One loop P and T odd W+- electromagnetic moments,

    D. Atwood, C. P. Burgess, C. Hamazaou, B. Irwin, and J. A. Robinson, “One loop P and T odd W+- electromagnetic moments,” Phys. Rev. D42(1990) 3770–3777

    1990

  26. [28]

    Anomalous moments of quarks and leptons from nonstandard W W gamma couplings,

    F. Boudjema, K. Hagiwara, C. Hamzaoui, and K. Numata, “Anomalous moments of quarks and leptons from nonstandard W W gamma couplings,” Phys. Rev. D43(1991) 2223–2232

    1991

  27. [29]

    Signets of CP violation,

    A. De Rujula, M. B. Gavela, O. Pene, and F. J. Vegas, “Signets of CP violation,” Nucl. Phys. B357(1991) 311–356

    1991

  28. [30]

    Effective Lagrangian approach to fermion electric dipole moments induced by a CP-violating WW gamma vertex,

    H. Novales-Sanchez and J. J. Toscano, “Effective Lagrangian approach to fermion electric dipole moments induced by a CP-violating WW gamma vertex,” Phys. Rev. D77(2008) 015011[arXiv:0712.2008]

    Pith/arXiv arXiv 2008

  29. [31]

    Low-energy effective field theory below the electroweak scale: matching at one loop,

    W. Dekens and P. Stoffer, “Low-energy effective field theory below the electroweak scale: matching at one loop,” JHEP10(2019) 197[arXiv:1908.05295]. [Erratum: JHEP 11, 148 (2022)]

    Pith/arXiv arXiv 2019

  30. [32]

    Composite dark matter with forbidden annihilation,

    T. Abe, R. Sato, and T. Yamanaka, “Composite dark matter with forbidden annihilation,” JHEP09(2024) 064[arXiv:2404.03963]

    Pith/arXiv arXiv 2024

  31. [33]

    Kira—A Feynman integral reduction program,

    P. Maierhöfer, J. Usovitsch, and P. Uwer, “Kira—A Feynman integral reduction program,” Comput. Phys. Commun.230(2018) 99–112[arXiv:1705.05610]

    Pith/arXiv arXiv 2018

  32. [34]

    Integral reduction with Kira 2.0 and finite field methods,

    J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, “Integral reduction with Kira 2.0 and finite field methods,” Comput. Phys. Commun.266(2021) 108024[arXiv:2008.06494]

    Pith/arXiv arXiv 2021

  33. [35]

    Kira 3: integral reduction with efficient seeding and optimized equation selection

    F. Lange, J. Usovitsch, and Z. Wu, “Kira 3: integral reduction with efficient seeding and optimized equation selection.”arXiv:2505.20197

    Pith/arXiv arXiv

  34. [36]

    Fermat, A Computer Algebra System for Polynomial and Matrix Computation,

    R. H. Lewis, “Fermat, A Computer Algebra System for Polynomial and Matrix Computation,”. URL:http://home.bway.net/lewis/

  35. [37]

    Evaluation of the general 3-loop vacuum Feynman integral,

    S. P. Martin and D. G. Robertson, “Evaluation of the general 3-loop vacuum Feynman integral,” Phys. Rev. D95(2017) 016008[arXiv:1610.07720]. – 15 – [38]Particle Data GroupCollaboration, “Review of particle physics,” Phys. Rev. D110 (2024) 030001. – 16 –

    Pith/arXiv arXiv 2017

  36. [2021]

    2022.arXiv:2203.08103

    Pith/arXiv arXiv 2022