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Efficient on-shell matching

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proposes a numerical on-shell matching procedure that computes one-loop Wilson coefficients directly in a physical basis, including evanescent shifts, by solving the matching equations at rational on-shell kinematic points.

desk verdict A genuinely useful on-shell matching algorithm with one real soft spot: the evanescent-shift prescription is validated on a single example and deserves a referee's push before the finite-matching part is taken as general. read the letter →

arxiv 2411.12798 v1 pith:F3EKRNW2 submitted 2024-11-19 hep-ph hep-th

Mikael Chala , Javier López Miras , José Santiago , Fuensanta Vilches This is my paper · ORCID
classification hep-phhep-th
keywords on-shellmatchingeffectivefieldtheorySMEFTevanescentoperatorsWilsoncoefficientsGreen'sbasisreductionmethodofregionsrationalkinematics
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

Matching is the step in which a high-energy "full" theory is replaced by an effective field theory: the heavy particles are removed and their effects are encoded in the coefficients of local operators. The standard path computes off-shell Green's functions and then reduces a redundant Green's basis to a physical one, a tedious and error-prone job. This paper claims that the whole reduction can be skipped: compute physical on-shell amplitudes in the full and effective theories at randomly chosen rational kinematic points, and solve the matching condition $\mathcal{M}^{(0)}_{\mathrm{EFT}} = \mathcal{M}^{(1),\mathrm{hard}}_{\mathrm{full}} + \mathcal{M}^{(1),\mathrm{soft}}_{\mathrm{full}}\big|_{\mathrm{UV}} - \mathcal{M}^{(1),\mathrm{soft}}_{\mathrm{EFT}}\big|_{\mathrm{UV}}$ numerically. The non-local terms cancel automatically in the difference, and evanescent shifts come out of the ultraviolet-pole parts of soft-region amplitudes without ever constructing evanescent operators. If correct, this makes EFT matching, renormalization, and basis reduction much easier to automate and apply to arbitrary physical bases.

What carries the argument

The carrying object is the on-shell matching equation, Eq. (9), together with the rule for extracting evanescent shifts from ultraviolet poles of soft-region amplitudes, Eq. (7). Rational kinematic configurations, generated in $\mathbb{Q}$ via spinor-helicity variables, make the amplitudes exactly evaluable numbers, so the cancellation of non-local terms and the solution for the Wilson coefficients are both exact. The ultraviolet-pole extraction replaces scaleless integrals with $1/\epsilon_{\mathrm{UV}}$ and keeps the Dirac algebra in $d$ dimensions, treating the tensor integral's $1/d$ factor carefully so that the product of an $\epsilon$ term with a $1/\epsilon$ pole is retained.

What would settle it

Run the numerical on-shell algorithm on a one-loop matching problem with a known evanescent shift, for instance the heavy scalar that generates $R_{\ell e}$ rather than $O_{\ell e}$ in Section 3.4, and compare the extracted finite coefficient with the closed-form result; if the $\epsilon$-pole-times-$\epsilon$ finite part does not match, the ultraviolet-pole-only soft-region rule is wrong.

Watch

Extended reading notes

Core claim

The authors' central claim is that one-loop on-shell matching can be performed directly in a physical basis through a numerical solution of the on-shell matching equations. The matching condition, Eq. (9), combines the hard-region contribution of the full theory with the difference of the ultraviolet-pole parts of the soft-region contributions of the full and effective theories; this combination is local and includes the evanescent shifts. Rational on-shell kinematics, generated with spinor-helicity variables, make every amplitude evaluation an exact rational number, so the cancellation of light-bridge non-localities and the solution for Wilson coefficients are exact rather than approximate. The paper demonstrates the procedure on a heavy-Higgs model with leptons, a $Z_2$-symmetric scalar theory to dimension 8, and SMEFT dimension-8 examples, and cross-checks the outputs against existing results.

Load-bearing premise

The load-bearing premise is that the finite evanescent shift is correctly captured by the ultraviolet-pole parts of the soft-region amplitudes, with scaleless integrals replaced by $1/\epsilon_{\mathrm{UV}}$ and the Dirac algebra done in $d$ dimensions before $\epsilon \to 0$; the paper validates this on one example rather than proving it in general.

Editorial extensions

If this is right

  • Green's basis reductions that previously required field redefinitions and equations of motion can be reproduced by solving on-shell matching equations, for any user-chosen physical basis.
  • Anomalous dimensions and beta functions can be computed directly in a physical basis, without redundant operators or the background-field method for gauge invariance.
  • Finite one-loop matching, including evanescent shifts, is obtained automatically because the algorithm never needs to construct the evanescent operators explicitly.
  • A single high-multiplicity on-shell amplitude can determine many Wilson coefficients at once, reducing the number of amplitudes needed to complete a matching.
  • The same procedure can translate between arbitrary physical bases and can renormalize any effective Lagrangian directly in terms of physical operators.

Reading between the lines

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

  • If the procedure scales beyond one loop, the hardest part of two-loop on-shell matching—the analytic cancellation of non-localities between full and effective theories—would also become a numerical routine, though the soft-region ultraviolet-pole extraction would need a two-loop analogue.
  • The evanescent-shift extraction is scheme-dependent through the choice of Dirac algebra and $\gamma_5$ treatment; a user working in a different scheme would have to convert the resulting Wilson coefficients rather than take them as scheme-independent.
  • The rational-kinematics core suggests a natural interface with modern amplitude methods such as spinor-helicity and numerical unitarity, which could make full-model Feynman-diagram generation unnecessary for matching.
  • Operator classes containing Levi-Civita tensors or other genuinely $d$-dimensional structures are the most likely to expose whether the ultraviolet-pole-only soft-region rule is general; testing those would be a direct extension of the paper's example.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proposes a numerical on-shell matching algorithm for effective field theories. The key idea is to evaluate renormalized physical amplitudes for rational on-shell kinematic configurations, thereby avoiding the analytic cancellation of non-local light-bridge contributions. For one-loop matching, Eq. (9) combines the hard-region contribution of the full theory with the UV-pole parts of the soft-region amplitudes in the full and effective theories; the difference of the latter is claimed to reproduce evanescent shifts. The method is illustrated by three applications: tree-level reduction of a scalar Green's basis to a physical basis up to dimension 8, reduction of bosonic SMEFT Green's-basis operators at dimension 8, one-loop anomalous dimensions in the SMEFT, and finite one-loop matching in a model with a heavy Higgs-like scalar, with cross-checks against Matchete, MatchMakerEFT, and the evanescent-operator literature.

Significance. If correct, the method is a genuinely useful complement to existing automated matching tools: it avoids Green's bases and field redefinitions, works in a user-chosen physical basis, and the rational-kinematics trick makes the cancellation of non-local terms nearly trivial. The scalar example is cross-checked both against explicit field redefinitions and against Matchete, and the finite matching example is checked against MatchMakerEFT and the published evanescent-shift result; these are concrete strengths. The main weakness is that the evanescent-shift prescription in Eq. (7) is the load-bearing element of the finite-matching claim, and it is validated only by a single heavy-Higgs example involving one Fierz identity. A general proof or an independent operator-class test is needed before the method can be regarded as a general algorithm for finite matching.

major comments (2)
  1. [Sec. 2.1, Eq. (7)] The evanescent-shift prescription is not derived. The replacement of scaleless tensor integrals by their UV pole, with d-dependent prefactors evaluated at d=4 before completing the Dirac algebra, is presented as a rule, but no general argument shows that this replacement commutes with tensor reduction, with different gamma5/reading-point prescriptions, or with the method of regions, nor that it equals the difference of the corresponding d-dimensional renormalized amplitudes. This matters because Eq. (9) relies on exactly this difference for finite matching. The only validation is the heavy-Higgs example in Sec. 3.4, which is a simple Fierz identity. I request either a general proof of the prescription or an independent check on a second operator class (for example, one involving both chiral and tensor structures, or a different heavy particle), since the advertised capability of 'finite matching, including evanescent contributions' depends on this point.
  2. [Sec. 2.1, Eq. (7)] The extraction of only the UV pole from scaleless integrals requires a separation of UV and IR poles, but the paper does not justify that evanescent O(epsilon) structures cannot multiply 1/epsilon_IR pieces of massless on-shell integrals and produce additional finite terms in Eq. (9). The statement that the effect 'is not affected by IR poles' is asserted rather than shown. Since the soft-region integrals in the on-shell amplitudes of Sec. 3.4 are scaleless, the 1/epsilon_IR partners are present, and the paper should demonstrate that they either cancel between the full and effective theories or are absent for evanescent insertions; otherwise the 'finite part from UV poles' is not uniquely defined.
minor comments (3)
  1. [Appendix C, In[21]] In the printed code, the line defining ampPhysBasis uses ampRedBasis on the right-hand side rather than ampPhysBasis; as written, the subsequent equations would be trivial identities. The ancillary file is presumably correct, but the listing should be fixed to avoid confusing readers who reproduce the code from the paper.
  2. [Sec. 3.4, Eqs. (65)-(74)] The text says that evanescent shifts are 'explicitly shown in red', but the arXiv plain-text rendering and monochrome print do not preserve this distinction. Please add a typographic marker (for example, a superscript or footnote) so that the evanescent contributions are identifiable in all formats.
  3. [Sec. 3.2, Eqs. (24)-(38)] The dimension-8 SMEFT reduction is presented as an extension beyond the state of the art, but the full operator definitions and the extended results are partly delegated to a GitHub notebook. Please give a versioned citation or DOI for the notebook and state explicitly which of the displayed formulas were cross-checked with independent methods; as it stands, the reproducibility of this section depends on external, unversioned material.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the matching condition and evanescent prescription are stated as explicit scheme choices, and the central results are cross-checked against external tools and benchmarks; overlapping-author citations are cross-checks, not load-bearing inputs.

full rationale

The derivation chain is self-contained. The matching condition Eq. (9) is the standard one-loop matching statement: tree-level EFT amplitudes are equated to the hard-region full-theory amplitude plus the UV-pole-induced finite difference of the soft-region amplitudes; neither side is fitted to the other. The evanescent prescription in Eq. (7) is an explicitly declared scheme choice (NDR for gamma5, a reading-point prescription, and extraction of UV poles before the d-dimensional Dirac algebra), not a parameter fitted to the target shifts. The finite shifts in Eqs. (65)-(74) are computed from that scheme and then compared with Ref. [42], an external benchmark. The scalar Green's-basis reduction in Sec. 3.1 is checked independently against explicit field redefinitions and Matchete, and the finite-matching example in Sec. 3.4 is checked against Matchete, MatchMakerEFT, and Ref. [42]. Some cited references involve overlapping authors ([37], [47], [49], [52], [56]), but they supply conventions, operator definitions, or confirmatory cross-checks; the central algorithmic claim does not reduce to those citations. In Sec. 3.3, the comparison with Ref. [52] confirms an independently computed on-shell result rather than providing its input. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The possible scheme-dependence of the Eq. (7) UV-pole/Dirac-algebra prescription for other operator classes is a correctness or robustness risk, not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters and no new entities. The method leans on standard EFT, dimReg, method of regions, and spinor-helicity inputs, plus a specific evanescent scheme choice; the most paper-specific assumption is the UV-pole extraction for the soft-region difference.

assumptions (6)
  • domain assumption Integrating out heavy fields yields a local EFT whose physical observables are captured by a finite set of physical operators.
    Standard EFT decoupling and power counting, used throughout Section 2 without proof.
  • standard math The method of regions correctly separates loop integrals into hard and soft contributions, and UV poles can be extracted from scaleless integrals by the 1/epsilon_UV substitution of Eq. (7).
    Section 2.1 and Section 3.4 rely on this for the evanescent shift calculation.
  • domain assumption Naive dimensional regularization with a reading-point prescription for gamma5 defines the evanescent scheme, matching the scheme of [42].
    Section 2.1 states the procedure 'automatically follow[s] the scheme advocated in [42]'.
  • standard math Rational on-shell kinematics can be generated for arbitrary masses and particle numbers using the spinor-helicity formalism of [54,55].
    Appendix A and the code in Appendix C depend on this to produce exact rational solutions.
  • domain assumption The on-shell matching equations are nonsingular for generic random rational kinematics, so the Wilson coefficients can be solved exactly.
    Step 3 of the algorithm in Section 2.2 assumes a sufficient number of independent kinematic configurations; the paper does not prove nonsingularity.
  • ad hoc to paper The soft-region difference M(1),soft_full|_UV - M(1),soft_EFT|_UV equals the evanescent shift even though the full and EFT amplitudes contain different d-dimensional operator structures (e.g., Rle vs. Ole).
    This is the load-bearing step for one-loop finite matching; it is motivated in Section 2.1 and tested only on the example in Section 3.4.

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Pith. "Pith review of Efficient on-shell matching." pith.science (2026). https://pith.science/paper/F3EKRNW2

@misc{pith2026241112798,
  author       = {Pith},
  title        = {Pith review of: Efficient on-shell matching},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3EKRNW2}},
  note         = {Machine review of arXiv:2411.12798}
}
read the original abstract

We propose an efficient method to perform on-shell matching calculations in effective field theories. The standard off-shell approach to matching requires the use of a Green's basis that includes redundant and evanescent operators. The reduction of such a basis to a physical one is often highly non-trivial, difficult to automate and error prone. Our proposal is based on a numerical solution of the corresponding on-shell matching equations, which automatically implements in a trivial way the delicate cancellation between the non-local terms in the full theory and those in the effective one. The use of rational on-shell kinematics ensures an exact analytic solution despite the numerical procedure. In this way we only need a physical basis to perform the matching. Our procedure can be used to reduce any Green's basis to an arbitrary physical one, or to translate between physical bases; to renormalize arbitrary effective Lagrangians, directly in terms of a physical basis; and to perform finite matching, including evanescent contributions, as we discuss with explicit examples.

Figures

Figures reproduced from arXiv: 2411.12798 by the authors.

Figure 1
Figure 1. Tree-level topologies contributing to 4-scalar amplitudes (left) and 6-scalar amplitudes (center and right) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Tree-level topologies contributing to 8-scalar amplitudes. dimension d, respectively. In this example we will use on-shell matching at tree level to reduce the Green’s basis onto the physical one. In order to do that we use the complete Green’s basis as full model (LFull = L) while the physical basis plays the role of the EFT (LEFT = L[βdi = 0]). Step 1 of our algorithm tells us to compute the 1PI contribution to th… view at source ↗
Figure 3
Figure 3. Representative diagrams contributing to the renormalization of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Representative diagams contributing to the on-shell renormalization of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Tree-level topologies contributing to the physical amplitude of ν¯LeRH0H¯ 0H− (all incoming) in the EFT involving one insertion of a one-loop sized WC (denoted by a square vertex). Recall that, following our numerical procedure, we now have to replace (no renormalizati…
Figure 6
Figure 6. Figure 6: One loop topologies contributing to the physical amplitude of ν¯LeRH0H¯ 0H− (all in￾coming) in the full theory involving, at least, one heavy propagator. matching, for which only local contributions have to be considered in the EFT and therefore, this amplitude would o…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

58 extracted references · 4 canonical work pages · cited by 5 Pith papers

  1. [42]

    Fuentes-Mart ´ ın, M

    J. Fuentes-Mart ´ ın, M. K¨ onig, J. Pag` es, A. E. Thomsen and F. Wilsch,Evanescent operators in one-loop matching computations , JHEP 02 (2023) 031, [ 2211.09144]

    arXiv 2023

  2. [1]

    N. P. Hartland, F. Maltoni, E. R. Nocera, J. Rojo, E. Slade, E. Vryonidou et al., A Monte Carlo global analysis of the Standard Model Effective Field Theory: the top quark sector , JHEP 04 (2019) 100, [ 1901.05965]

    arXiv 2019

  3. [2]

    Brivio, S

    I. Brivio, S. Bruggisser, F. Maltoni, R. Moutafis, T. Plehn, E. Vryonidou et al., O new physics, where art thou? A global search in the top sector , JHEP 02 (2020) 131, [ 1910.03606]

    arXiv 2020

  4. [3]

    Banerjee, R

    S. Banerjee, R. S. Gupta, J. Y. Reiness, S. Seth and M. Spannowsky, Towards the ultimate differential SMEFT analysis , JHEP 09 (2020) 170, [ 1912.07628]

    arXiv 2020

  5. [4]

    Aoude, T

    R. Aoude, T. Hurth, S. Renner and W. Shepherd, The impact of flavour data on global fits of the MFV SMEFT , JHEP 12 (2020) 113, [ 2003.05432]

    arXiv 2020

  6. [5]

    Dawson, S

    S. Dawson, S. Homiller and S. D. Lane, Putting standard model EFT fits to work , Phys. Rev. D 102 (2020) 055012, [ 2007.01296]

    arXiv 2020

  7. [6]

    Das Bakshi, J

    Anisha, S. Das Bakshi, J. Chakrabortty and S. K. Patra, Connecting electroweak-scale observables to BSM physics through EFT and Bayesian statistics , Phys. Rev. D 103 (2021) 076007, [2010.04088]

    arXiv 2021

  8. [7]

    Falkowski, M

    A. Falkowski, M. Gonz´ alez-Alonso and O. Naviliat-Cuncic,Comprehensive analysis of beta decays within and beyond the Standard Model , JHEP 04 (2021) 126, [ 2010.13797]

    arXiv 2021

Show all 58 references
  1. [8]

    Ellis, M

    J. Ellis, M. Madigan, K. Mimasu, V. Sanz and T. You, Top, Higgs, Diboson and Electroweak Fit to the Standard Model Effective Field Theory , JHEP 04 (2021) 279, [ 2012.02779]

    2021 arXiv

  2. [9]

    SMEFiT collaboration, J. J. Ethier, G. Magni, F. Maltoni, L. Mantani, E. R. Nocera, J. Rojo et al., Combined SMEFT interpretation of Higgs, diboson, and top quark data from the LHC , JHEP 11 (2021) 089, [ 2105.00006]

    2021 arXiv

  3. [10]

    de Blas, Y

    J. de Blas, Y. Du, C. Grojean, J. Gu, V. Miralles, M. E. Peskin et al., Global SMEFT Fits at Future Colliders, in Snowmass 2021 , 6, 2022. 2206.08326

    2021 arXiv

  4. [11]

    Grzadkowski, M

    B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, [ 1008.4884]

    2010 arXiv

  5. [12]

    Brivio and M

    I. Brivio and M. Trott, The Standard Model as an Effective Field Theory , Phys. Rept. 793 (2019) 1–98, [ 1706.08945]

    2019 arXiv

  6. [13]

    Isidori, F

    G. Isidori, F. Wilsch and D. Wyler, The standard model effective field theory at work , Rev. Mod. Phys. 96 (2024) 015006, [ 2303.16922]. 26

    2024 arXiv

  7. [14]

    del Aguila, S

    F. del Aguila, S. Bar-Shalom, A. Soni and J. Wudka, Heavy Majorana Neutrinos in the Effective Lagrangian Description: Application to Hadron Colliders , Phys. Lett. B 670 (2009) 399–402, [0806.0876]

    2009 arXiv

  8. [15]

    Aparici, K

    A. Aparici, K. Kim, A. Santamaria and J. Wudka, Right-handed neutrino magnetic moments , Phys. Rev. D 80 (2009) 013010, [ 0904.3244]

    2009 arXiv

  9. [16]

    Bhattacharya and J

    S. Bhattacharya and J. Wudka, Dimension-seven operators in the standard model with right handed neutrinos, Phys. Rev. D 94 (2016) 055022, [ 1505.05264]

    2016 arXiv

  10. [17]

    Liao and X.-D

    Y. Liao and X.-D. Ma, Operators up to Dimension Seven in Standard Model Effective Field Theory Extended with Sterile Neutrinos , Phys. Rev. D 96 (2017) 015012, [ 1612.04527]

    2017 arXiv

  11. [18]

    Aebischer, T

    J. Aebischer, T. Kapoor and J. Kumar, wilson: A package for renormalization group running in the SMEFT with Sterile Neutrinos , 2411.07220

    arXiv

  12. [19]

    E. E. Jenkins, A. V. Manohar and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching , JHEP 03 (2018) 016, [ 1709.04486]

    2018 arXiv

  13. [20]

    Bischer and W

    I. Bischer and W. Rodejohann, General neutrino interactions from an effective field theory perspective, Nucl. Phys. B 947 (2019) 114746, [ 1905.08699]

    2019 arXiv

  14. [21]

    Chala and A

    M. Chala and A. Titov, One-loop matching in the SMEFT extended with a sterile neutrino , JHEP 05 (2020) 139, [ 2001.07732]

    2020 arXiv

  15. [22]

    Li, X.-D

    T. Li, X.-D. Ma and M. A. Schmidt, General neutrino interactions with sterile neutrinos in light of coherent neutrino-nucleus scattering and meson invisible decays , JHEP 07 (2020) 152, [2005.01543]

    2020 arXiv

  16. [23]

    Li, X.-D

    T. Li, X.-D. Ma and M. A. Schmidt, Constraints on the charged currents in general neutrino interactions with sterile neutrinos , JHEP 10 (2020) 115, [ 2007.15408]

    2020 arXiv

  17. [24]

    Gripaios and D

    B. Gripaios and D. Sutherland, An operator basis for the Standard Model with an added scalar singlet, JHEP 08 (2016) 103, [ 1604.07365]

    2016 arXiv

  18. [25]

    Chala, G

    M. Chala, G. Guedes, M. Ramos and J. Santiago, Running in the ALPs , Eur. Phys. J. C 81 (2021) 181, [ 2012.09017]

    2021 arXiv

  19. [26]

    Bauer, M

    M. Bauer, M. Neubert, S. Renner, M. Schnubel and A. Thamm, The Low-Energy Effective Theory of Axions and ALPs , JHEP 04 (2021) 063, [ 2012.12272]

    2021 arXiv

  20. [27]

    Grojean, J

    C. Grojean, J. Kley and C.-Y. Yao, Hilbert series for ALP EFTs , JHEP 11 (2023) 196, [2307.08563]

    2023 arXiv

  21. [28]

    C. W. Murphy, Dimension-8 operators in the Standard Model Effective Field Theory , JHEP 10 (2020) 174, [ 2005.00059]

    2020 arXiv

  22. [29]

    C. W. Murphy, Low-Energy Effective Field Theory below the Electroweak Scale: Dimension-8 Operators, JHEP 04 (2021) 101, [ 2012.13291]

    2021 arXiv

  23. [30]

    H.-L. Li, Z. Ren, M.-L. Xiao, J.-H. Yu and Y.-H. Zheng, Low energy effective field theory operator basis at d ≤ 9, JHEP 06 (2021) 138, [ 2012.09188]

    2021 arXiv

  24. [31]

    H.-L. Li, Z. Ren, M.-L. Xiao, J.-H. Yu and Y.-H. Zheng, Operator bases in effective field theories with sterile neutrinos: d ≤ 9, JHEP 11 (2021) 003, [ 2105.09329]. 27

    2021 arXiv

  25. [32]

    H.-L. Li, Z. Ren, M.-L. Xiao, J.-H. Yu and Y.-H. Zheng, Operators for generic effective field theory at any dimension: on-shell amplitude basis construction , JHEP 04 (2022) 140, [2201.04639]

    2022 arXiv

  26. [33]

    R. V. Harlander and M. C. Schaaf, AutoEFT: Automated operator construction for effective field theories, Comput. Phys. Commun. 300 (2024) 109198, [ 2309.15783]

    2024 arXiv

  27. [34]

    Arzt, Reduced effective Lagrangians, Phys

    C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189–195, [ hep-ph/9304230]

    1995 arXiv

  28. [35]

    J. C. Criado and M. P´ erez-Victoria, Field redefinitions in effective theories at higher orders , JHEP 03 (2019) 038, [ 1811.09413]

    2019 arXiv

  29. [36]

    Fuentes-Mart ´ ın, M

    J. Fuentes-Mart ´ ın, M. K¨ onig, J. Pag` es, A. E. Thomsen and F. Wilsch,A proof of concept for matchete: an automated tool for matching effective theories , Eur. Phys. J. C 83 (2023) 662, [2212.04510]

    2023 arXiv

  30. [37]

    Carmona, A

    A. Carmona, A. Lazopoulos, P. Olgoso and J. Santiago, Matchmakereft: automated tree-level and one-loop matching , SciPost Phys. 12 (2022) 198, [ 2112.10787]

    2022 arXiv

  31. [38]

    L. F. Abbott, The Background Field Method Beyond One Loop , Nucl. Phys. B 185 (1981) 189–203

    1981

  32. [39]

    Georgi, On-shell effective field theory , Nucl

    H. Georgi, On-shell effective field theory , Nucl. Phys. B 361 (1991) 339–350

    1991

  33. [40]

    Li and S

    X. Li and S. Zhou, One-loop Matching and Running via On-shell Amplitudes , 2309.10851

    arXiv

  34. [41]

    De Angelis and G

    S. De Angelis and G. Durieux, EFT matching from analyticity and unitarity , SciPost Phys. 16 (2024) 071, [ 2308.00035]

    2024 arXiv

  35. [43]

    Fuentes-Martin, J

    J. Fuentes-Martin, J. Portoles and P. Ruiz-Femenia, Integrating out heavy particles with functional methods: a simplified framework , JHEP 09 (2016) 156, [ 1607.02142]

    2016 arXiv

  36. [44]

    Beneke and V

    M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold , Nucl. Phys. B 522 (1998) 321–344, [ hep-ph/9711391]

    1998 arXiv

  37. [45]

    Gherardi, D

    V. Gherardi, D. Marzocca and E. Venturini, Matching scalar leptoquarks to the SMEFT at one loop, JHEP 07 (2020) 225, [ 2003.12525]

    2020 arXiv

  38. [46]

    Ren and J.-H

    Z. Ren and J.-H. Yu, A complete set of the dimension-8 Green ’s basis operators in the Standard Model effective field theory , JHEP 02 (2024) 134, [ 2211.01420]

    2024 arXiv

  39. [47]

    Chala, A

    M. Chala, A. D ´ ıaz-Carmona and G. Guedes,A Green ’s basis for the bosonic SMEFT to dimension 8 , JHEP 05 (2022) 138, [ 2112.12724]

    2022 arXiv

  40. [48]

    Banerjee, J

    U. Banerjee, J. Chakrabortty, C. Englert, S. U. Rahaman and M. Spannowsky, Integrating out heavy scalars with modified equations of motion: Matching computation of dimension-eight SMEFT coefficients, Phys. Rev. D 107 (2023) 055007, [ 2210.14761]

    2023 arXiv

  41. [49]

    S. D. Bakshi, M. Chala, A. D ´ ıaz-Carmona, Z. Ren and F. Vilches, Renormalization of the SMEFT to dimension eight: Fermionic interactions I , 2409.15408

    arXiv

  42. [50]

    L. F. Abbott, Introduction to the Background Field Method , Acta Phys. Polon. B 13 (1982) 33. 28

    1982

  43. [51]

    Alloul, N

    A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0 - A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250–2300, [1310.1921]

    2014 arXiv

  44. [52]

    Das Bakshi, M

    S. Das Bakshi, M. Chala, A. D ´ ıaz-Carmona and G. Guedes,Towards the renormalisation of the Standard Model effective field theory to dimension eight: bosonic interactions II , Eur. Phys. J. Plus 137 (2022) 973, [ 2205.03301]

    2022 arXiv

  45. [53]

    Rgb: On shell reduction of green’s bases in efts (to appear)

    J. L´ opez-Miras and F. Vilches, “Rgb: On shell reduction of green’s bases in efts (to appear).”

  46. [54]

    De Angelis, Amplitude bases in generic EFTs , JHEP 08 (2022) 299, [ 2202.02681]

    S. De Angelis, Amplitude bases in generic EFTs , JHEP 08 (2022) 299, [ 2202.02681]

    2022 arXiv

  47. [55]

    M. A. Huber, Spinorhelicity4d: a mathematica toolbox for the four-dimensional spinor-helicity formalism, 2023

    2023

  48. [56]

    Chala, G

    M. Chala, G. Guedes, M. Ramos and J. Santiago, Towards the renormalisation of the Standard Model effective field theory to dimension eight: Bosonic interactions I , SciPost Phys. 11 (2021) 065, [2106.05291]

    2021 arXiv

  49. [57]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: New features and improvements , Comput. Phys. Commun. 256 (2020) 107478, [ 2001.04407]

    2020 arXiv

  50. [58]

    Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3 , Comput

    T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3 , Comput. Phys. Commun. 140 (2001) 418–431, [ hep-ph/0012260]. 29

    2001 arXiv

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