Pith. sign in

REVIEW 3 major objections 5 minor 58 references

This paper constructs the complete nonredundant basis of lepton-number-violating HEFT operators through O(p^4) and gives the full tree-level matching of the type-II seesaw model onto that basis.

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

2026-08-01 00:32 UTC pith:D36BCO7C

load-bearing objection First complete tree-level HEFT matching of type-II seesaw plus a complete LNV basis; verification gaps but no demonstrated errors — worth refereeing. the 3 major comments →

arxiv 2607.26184 v1 pith:D36BCO7C submitted 2026-07-28 hep-ph

A HEFT Perspective on the Type-II Seesaw Model and the Complete Basis of Lepton-Number-Violating Operators

Zizhou Ge , Huayang Song , Xia Wan This is my paper
classification hep-ph
keywords type-II seesawHiggs effective field theorylepton-number violationoperator basisspuriontree-level matchingneutrinoless double beta decaycustodial symmetry
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 aims to establish a complete tree-level dictionary between the type-II seesaw model and the Higgs effective field theory (HEFT) at next-to-leading chiral order, O(p^4). It also constructs, for the first time, a complete and nonredundant basis of lepton-number-violating (LNV) HEFT operators through that order, including all flavor multiplicities. The matching keeps explicit dependence on the five physical scalar masses, the doublet–triplet mixing angle, and the triplet vacuum expectation value, without expanding in the triplet VEV or mixing angle. A sympathetic reader should care because the resulting Wilson coefficients and operator basis give one consistent framework for translating type-II seesaw parameters into predictions for Higgs couplings, electroweak precision, vector-boson scattering, neutrinoless double-beta decay, and charged-lepton flavor violation.

Core claim

The paper's central discovery is that the low-energy effects of the complex scalar triplet in the type-II seesaw model can be organized exactly, at tree level, into HEFT through O(p^4): integrating out the heavy K, η, H±, and H±± states generates definite Wilson coefficients for both the lepton-number-conserving and lepton-number-violating sectors, listed in full. In the LNV sector, all such operators are generated from a single dressed spurion, Ŝ = U* ε σ− U†, which carries B−L = +2 and fixes the custodial direction. The resulting basis is shown to be complete and nonredundant by Hilbert-series counting, and the overlap with a previously proposed broken-phase EFT is recovered after expandin

What carries the argument

The central object is the dressed LNV spurion Ŝ = U* ε S_R U† with S_R = S σ−, a complex 2×2 matrix that transforms as a symmetric SU(2)_L triplet and carries B−L = +2. It encodes both the lepton-number-violating insertion and the custodial orientation, and together with the standard HEFT building blocks V_μ = U D_μ U† and T = U σ3 U† it generates every LNV invariant. The second key ingredient is the primary-HEFT power-counting scheme, where the heavy scalar squared masses are assigned O(t^{-1}) while sin α, v_H, and ξ remain O(1); this lets the tree-level matching be performed without expanding in the mixing angle or triplet VEV. The matching is done by solving the classical equations of mo

Load-bearing premise

The load-bearing premise is that the heavy triplet scalars are all uniformly heavy, so that a single O(p^4) truncation captures the leading low-energy effects, and that tree-level matching is enough; if one heavy state is much lighter than the others, or if the triplet VEV is not small, higher-order terms can dominate.

What would settle it

Independently recompute the spurion-graded Hilbert series for LNV HEFT operators with three fermion generations and the same B−L and custodial grading, for example by symbolic component-level linear-independence checks; if the flavor multiplicities differ from those in Table VI, the basis is not complete or not nonredundant.

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

If this is right

  • The matched Wilson coefficients give concrete predictions for the rho parameter, Higgs couplings κ_W, κ_Z, and κ_f, and quartic gauge couplings that can be compared with LHC and future collider data.
  • The complete LNV basis provides a common operator language for neutrinoless double-beta decay, μ→3e, and same-sign dilepton searches; for example, the tree-level μ→3e rate is fixed by |y_Δ^{eμ} y_Δ^{ee*}|^2 / m_{H±±}^4.
  • Recovering the broken-phase EFT results in the small-triplet-VEV limit confirms the internal consistency of the matching and shows that the HEFT description contains the unexpanded nonlinear structure as a superset.
  • The spurion identity Ŝ† Ŝ = (1+T)/2 implies that, in this minimal HEFT, lepton-number violation and custodial-symmetry breaking are tied together, so correlated deviations in ρ, κ_Z/κ_W, and LNV rates should appear.
  • Because the basis is complete through O(p^4), other seesaw or LNV ultraviolet completions can be mapped onto the same operator set, provided their heavy states generate the same spurion orientation.

Where Pith is reading between the lines

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

  • Implicit in the spurion construction is that a full custodial multiplet of LNV spurions, Ŝ_a = U* ε σ_a U†, would decouple lepton-number violation from custodial breaking; introducing such charged sources would produce a qualitatively different, less-correlated pattern than the minimal type-II spurion.
  • Because the tree-level matching generates no independent field-strength operators at O(p^4), any anomalous triple-gauge couplings induced by the complex triplet must arise at loop level; a combined fit to quartic and triple gauge couplings could test that expectation.
  • The authors restrict to tree-level matching; one-loop matching would add dipole operators for radiative charged-lepton-flavor violation and extend the short-range neutrinoless-double-beta-decay sector, so the present basis will likely need augmentation at higher orders.
  • The paper also notes that a quantitative comparison of HEFT predictions with full type-II seesaw cross sections for specific observables has not been performed; until such a comparison exists, the numerical impact of the matched coefficients on actual collider observables remains unquantified.

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 / 5 minor

Summary. This paper performs the tree-level matching of the type-II seesaw model onto HEFT through O(p^4) in the chiral expansion, using the authors' nonlinear field representation and primary-HEFT power counting. The heavy scalar masses, the doublet–triplet mixing angle, and the triplet VEV are retained without additional expansions. The paper also constructs a complete, nonredundant basis of lepton-number-violating HEFT operators through O(p^4), based on a new complex dressed spurion that encodes B−L charge and custodial orientation, and validates the basis with Hilbert-series counting and component-level independence checks. Matching results are presented for both lepton-number-conserving and LNV sectors, compared with the real-triplet extension and with a broken-phase EFT, and used for qualitative phenomenological discussions of Higgs, VBS, top, 0νββ, µ→3e, and same-sign dilepton processes.

Significance. If the central claims are correct, this is the first systematic HEFT matching of the type-II seesaw model at tree level through O(p^4) and the first explicit, complete LNV HEFT operator basis at this order. The paper makes good use of external anchors: the Hilbert-series counting of Ref. [26], the broken-phase EFT of Ref. [24], and the LEFT basis of Ref. [31]. The constructed spurion identity Ŝ†Ŝ = (1+T)/2 and the explicit redundancy proof in Appendix A are concrete and checkable. The work should be useful as a reference for collider and low-energy LNV phenomenology within HEFT, provided the matching tables are reduced to the stated minimal basis in a fully transparent way.

major comments (3)
  1. [Sec. IV.D / Table XI] The L4 entry in Table XI is printed as the monomial (L^T C Ŝ L)(L^T C Ŝ L) with coefficient y_pr y_st. However, the independent basis element in Table VI is the flavor-antisymmetrized combination Y[pr;st] with n_f^2(n_f^2−1)/12 components. As written, the table entry is not reduced to the minimal basis: for example, pr=st=11 would appear nonvanishing although the basis element identically vanishes. The paper states that matching results are projected onto the basis (Sec. III.B), but Table XI does not display the projection. Please specify the projection convention explicitly or provide the reduced Wilson coefficients; without this, the LNV matching results are not uniquely defined and cannot be reproduced.
  2. [Sec. IV.C / footnote 5] The validation of the matching against the broken-phase EFT is only fully shown for dimension-three and dimension-four operators; the claimed agreement for dimension-five and dimension-six operators is asserted in a footnote without details. Since the bEFT comparison is one of the main external checks of the matching, the dimension-five and dimension-six comparisons should be documented in an appendix or auxiliary file. This is a reproducibility gap rather than a demonstrated error, but it is load-bearing for the claim of complete O(p^4) matching.
  3. [Sec. IV.C / Tables VIII–X] The spurion-refined Hilbert-series results in Tables VIII–X are presented without derivation or an explicit statement of the generating functions used. The completeness claim rests on agreement between these counts and the independent counting of Ref. [26]. To make the validation self-contained, the authors should state how these polynomials were computed (e.g., the Hilbert-series algorithm, spurion grading conventions, and any code used) or at least give the generating-function form for one nontrivial class such as L4 or L̄Q3.
minor comments (5)
  1. [Sec. V.A / Eq. (44)] The tree-level ρ parameter is standard and correct in the small-ξ limit. The paper should clarify that the exact formula (1+2ξ^2)/(1+4ξ^2) is used in the matched HEFT and that the O(ξ^4) term shown is only for orientation.
  2. [Sec. IV.D / Table XI] For operators with σ^{μν} or ϵ insertions, the flavor counts in Table VI sometimes require antisymmetry in p↔r. The table entries should indicate explicitly when the displayed coefficient is to be understood with the appropriate projector, to avoid confusion with the unprojected monomials.
  3. [Sec. II.B / Eq. (16)] The input set includes both v_H and ξ while v_EW^2 = v_H^2(1+2ξ^2). This is not an inconsistency if v_EW is treated as derived, but the authors should say so explicitly, since v_EW is later used as the fixed electroweak scale.
  4. [Appendix B] The notation '→0' in the last column of Table XIV is terse. A brief explanation that this refers to the vanishing of the generic-flavor matching after specializing to the 0νββ flavor assignment p=r=e would improve readability.
  5. [General] There are a number of formatting artifacts in the tables (e.g., 'dimTSpurion' and '42 + 2T 2' in Table VIII) that should be corrected in the published version.

Circularity Check

0 steps flagged

No significant circularity; central matching and LNV-basis claims are self-contained under the stated power-counting and independently benchmarked.

full rationale

I find no circular step. The matching in Sec. III is a direct solution of the heavy-field equations of motion (Sec. III.B), with Wilson coefficients expressed in terms of the input parameter set (16) under the explicit primary-HEFT scaling (17). No LNV or LNC coefficient is fitted to data, and the results are checked against the independent broken-phase EFT of Ref. [24] (Tables XII-XIII) and the real-triplet result of Ref. [45]. The LNV basis in Sec. IV is constructed from a spurion whose transformation properties are motivated by the Weinberg operator, but its completeness and nonredundancy are established independently by component-level linear-independence reduction and by agreement with the external Hilbert-series enumeration of Ref. [26]; the unresolved correspondence between the T-spurion and (S-hat,S-hat-dagger) refinements in B-L-violating sectors (Sec. IV.C) is openly stated and does not affect the validation via direct massive-particle counting. Ref. [30] (same authors) is cited for the primary-HEFT scheme, and Refs. [29,45] for the nonlinear representation and real-triplet comparison, but these are framework/benchmark choices, not load-bearing proofs of the matching or basis results; the calculation is self-contained once the stated power-counting assumption (17) is adopted. The remaining caveats - tree-level truncation (Sec. VI) and the assertion, rather than display, of the dimension-five/six bEFT agreement (footnote 5) - are verification gaps, not circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

The central claim depends on the chosen UV model, the primary-HEFT power-counting scheme, external completeness results, the minimal spurion assumption, and the tree-level truncation. No physical free parameter is fitted to data, and the invented entity is a nonphysical spurion. The main structural input from outside the paper is the Hilbert-series counting of Ref. [26] plus the pHEFT framework from the authors' own companion papers.

free parameters (2)
  • primary-HEFT input set (m_h, m_K, m_η, m_+, m_++, sin α, v_H, ξ)
    Eq. (16) chooses these as independent UV scalar-sector inputs after the stationarity conditions; they are not fitted in this paper. All Wilson coefficients are functions of them, so the paper's claims are parameter-map claims rather than data-driven predictions.
  • auxiliary expansion parameter t
    Eq. (17) uses t as a bookkeeping order parameter for the primary-HEFT power counting; observable results are t-independent after re-expression in physical masses. It is an ordering device, not a physical constant.
axioms (5)
  • domain assumption The type-II seesaw UV Lagrangian in Eqs. (6)-(9), with a complex SU(2)_L triplet of hypercharge 1, Yukawa y_Δ, and trilinear μ, is the correct starting point.
    The entire matching is built on this specific UV model; the paper does not derive it from a more fundamental theory.
  • domain assumption The nonlinear U representation and primary-HEFT power counting of Eq. (17) organize the low-energy expansion, with no additional expansion in sin α or ξ.
    This is the framework inherited from the same group's companion papers; the validity of the claimed O(p^4) completeness is conditional on this counting.
  • standard math The external LNC HEFT basis of Ref. [25] and the Hilbert-series counting of Ref. [26] are correct and complete.
    The LNV basis completeness/nonredundancy argument depends directly on the Hilbert-series dimensions of Ref. [26]; the paper does not re-derive those results.
  • domain assumption The minimal neutral LNV spurion orientation Ŝ ∝ U* ε S σ_- U† with X=+1 is sufficient to span all LNV operators through O(p^4) in the minimal HEFT.
    The spurion construction in Sec. IV.A assumes no additional charged LNV source or enlarged low-energy field content; the paper notes that a complete custodial multiplet would require charged sources absent from the minimal HEFT.
  • domain assumption Tree-level matching, with IBP reduction and local field redefinitions, is sufficient at O(p^4); one-loop effects are neglected.
    The paper explicitly restricts to tree level and lists loop-level operators (e.g., dipoles relevant to radiative charged-lepton-flavor violation) as beyond scope.
invented entities (1)
  • Dressed LNV spurion Ŝ = U* ε S σ_- U† (X=+1) no independent evidence
    purpose: Encodes the B−L charge and custodial orientation of LNV insertions; used to construct invariant LNV HEFT operators and a spurion-refined Hilbert series.
    This is a formal spurion, not a propagating particle or force. It has no direct experimental handle, but it is an invented bookkeeping construct that carries the operator-basis construction and the LNV/custodial analysis.

pith-pipeline@v1.3.0-alltime-deepseek · 36252 in / 16296 out tokens · 168643 ms · 2026-08-01T00:32:44.528804+00:00 · methodology

0 comments
read the original abstract

We perform the complete tree-level matching of the type-II seesaw model onto the Higgs effective field theory (HEFT) through $\mathcal{O}(p^4)$ in the chiral expansion. Employing a nonlinear field representation and the primary-HEFT power-counting scheme, we retain the dependence on the independent heavy-scalar masses, the neutral-scalar mixing angle, and the triplet vacuum expectation value without introducing additional expansions in these parameters. To systematically describe lepton-number violation, we construct a complete and nonredundant basis of LNV HEFT operators through $\mathcal{O}(p^4)$, including their full flavor multiplicities, using a complex dressed spurion that encodes the $B-L$ charge and custodial orientation of the LNV insertion. The basis is independently validated by Hilbert-series counting. We then compare our matching results with those for the corresponding real-triplet extension and with a previously proposed broken-phase effective field theory. These comparisons identify the effects of the additional CP-odd and doubly charged scalar states and show that the overlapping broken-phase results are recovered after the appropriate parameter expansion, while HEFT retains the unexpanded nonlinear electroweak structure. We further discuss representative implications for Higgs and electroweak precision observables, vector-boson scattering, multi-Higgs production, anomalous gauge couplings, top-quark processes, neutrinoless double-beta decay, charged-lepton flavor violation, and same-sign dilepton production.

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

58 extracted references · 46 linked inside Pith

  1. [1]

    The leading and next-to-leading contributions occur atO(t −1)andO(t 0), respectively

    Bosonic operators The HEFT Lagrangian is organized as an expansion in the auxiliary parametert. The leading and next-to-leading contributions occur atO(t −1)andO(t 0), respectively. The corresponding bosonic terms are L(t−1) HEFT ⊃ v2 H (4ξ2 + 1)m2 K(ξc+s) 2 −ξ 2m2 η 8(4ξ2 + 1) − h3s2m2 η(2ξc+s) 2ξ(4ξ 2 + 1)vH + h4s2m2 η 8ξ2(4ξ2 + 1)v2 H m2 η(3sc+ 4ξ−6ξs ...

  2. [2]

    +O(h 2),(20) Four-fermion operators first arise atO(t)

    Fermionic operators in the LNC sector Operators involving two fermions first appear atO(t0): L(t0) HEFT ⊃ −vH +hc√ 2 ¯QLU  yu 0 0y d   QR + ¯LLU  0 0 0y e   LR +h.c. +O(h 2),(20) Four-fermion operators first arise atO(t). The two-fermion current operators are collected in TABLE III, while the four-quark and the leptonic and semileptonic four-fermi...

  3. [3]

    Konetschny and W

    W. Konetschny and W. Kummer,Nonconservation of Total Lepton Number with Scalar Bosons,Phys. Lett. B70(1977) 433–435

    1977

  4. [4]

    Magg and C

    M. Magg and C. Wetterich,Neutrino Mass Problem and Gauge Hierarchy,Phys. Lett. B94 (1980) 61–64

    1980

  5. [5]

    T. P. Cheng and L.-F. Li,Neutrino Masses, Mixings and Oscillations in SU(2) x U(1) Models of Electroweak Interactions,Phys. Rev. D22(1980) 2860

    1980

  6. [6]

    R. N. Mohapatra and G. Senjanovic,Neutrino Masses and Mixings in Gauge Models with Spontaneous Parity Violation,Phys. Rev. D23(1981) 165

    1981

  7. [7]

    Arhrib, R

    A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka, M. C. Peyranere, L. Rahili, and J. Ramadan,The Higgs Potential in the Type II Seesaw Model,Phys. Rev. D84(2011) 095005, [arXiv:1105.1925]

    Pith/arXiv arXiv 2011

  8. [8]

    Mitra, S

    M. Mitra, S. Niyogi, and M. Spannowsky,Type-II Seesaw Model and Multilepton Signatures at Hadron Colliders,Phys. Rev. D95(2017), no. 3 035042, [arXiv:1611.09594]

    Pith/arXiv arXiv 2017

  9. [9]

    Y. Du, A. Dunbrack, M. J. Ramsey-Musolf, and J.-H. Yu,Type-II Seesaw Scalar Triplet Model at a 100 TeVppCollider: Discovery and Higgs Portal Coupling Determination,JHEP 01(2019) 101, [arXiv:1810.09450]

    Pith/arXiv arXiv 2019

  10. [10]

    O. A. Ducu, A. E. Dumitriu, A. Jinaru, R. Kukla, E. Monnier, G. Moultaka, A. Tudorache, and H. Xu,Type-II Seesaw Higgs triplet productions and decays at the LHC,JHEP06(2025) 020, [arXiv:2410.14830]

    arXiv 2025

  11. [11]

    Weinberg,Phenomenological Lagrangians,Physica A96(1979), no

    S. Weinberg,Phenomenological Lagrangians,Physica A96(1979), no. 1-2 327–340

    1979

  12. [12]

    Georgi,Effective field theory,Ann

    H. Georgi,Effective field theory,Ann. Rev. Nucl. Part. Sci.43(1993) 209–252

    1993

  13. [13]

    Henning, X

    B. Henning, X. Lu, and H. Murayama,How to use the standard model effective field theory, JHEP01(2016), no. 023 [arXiv:1412.1837]

    Pith/arXiv arXiv 2016

  14. [14]

    Brivio and M

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

    Pith/arXiv arXiv 2019

  15. [15]

    Feruglio,The Chiral approach to the electroweak interactions,Int

    F. Feruglio,The Chiral approach to the electroweak interactions,Int. J. Mod. Phys. A8 (1993) 4937–4972, [hep-ph/9301281]

    Pith/arXiv arXiv 1993

  16. [16]

    Grinstein and M

    B. Grinstein and M. Trott,A Higgs-Higgs bound state due to new physics at a TeV,Phys. Rev. D76(2007) 073002, [arXiv:0704.1505]. 45

    Pith/arXiv arXiv 2007

  17. [17]

    Alonso, M

    R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin, and J. Yepes,The Effective Chiral Lagrangian for a Light Dynamical ”Higgs Particle”,Phys. Lett. B722(2013) 330–335, [arXiv:1212.3305]. [Erratum: Phys.Lett.B 726, 926 (2013)]

    Pith/arXiv arXiv 2013

  18. [18]

    Buchalla, O

    G. Buchalla, O. Catà, and C. Krause,Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO,Nucl. Phys. B880(2014) 552–573, [arXiv:1307.5017]. [Erratum: Nucl.Phys.B 913, 475–478 (2016)]

    Pith/arXiv arXiv 2014

  19. [19]

    Dawson and C

    S. Dawson and C. W. Murphy,Standard Model EFT and Extended Scalar Sectors,Phys. Rev. D96(2017), no. 1 015041, [arXiv:1704.07851]

    Pith/arXiv arXiv 2017

  20. [20]

    Corbett, A

    T. Corbett, A. Joglekar, H.-L. Li, and J.-H. Yu,Exploring Extended Scalar Sectors with Di-Higgs Signals: A Higgs EFT Perspective,JHEP05(2018) 061, [arXiv:1705.02551]

    Pith/arXiv arXiv 2018

  21. [21]

    Dawson, S

    S. Dawson, S. Homiller, and S. D. Lane,Putting standard model EFT fits to work,Phys. Rev. D102(2020), no. 5 055012, [arXiv:2007.01296]

    Pith/arXiv arXiv 2020

  22. [22]

    Du, X.-X

    Y. Du, X.-X. Li, and J.-H. Yu,Neutrino seesaw models at one-loop matching: discrimination by effective operators,JHEP09(2022) 207, [arXiv:2201.04646]

    Pith/arXiv arXiv 2022

  23. [23]

    X. Li, D. Zhang, and S. Zhou,One-loop matching of the type-II seesaw model onto the Standard Model effective field theory,JHEP04(2022) 038, [arXiv:2201.05082]

    Pith/arXiv arXiv 2022

  24. [24]

    Zhang,Complete one-loop structure of the type-(I+II) seesaw effective field theory,JHEP 03(2023) 217, [arXiv:2208.07869]

    D. Zhang,Complete one-loop structure of the type-(I+II) seesaw effective field theory,JHEP 03(2023) 217, [arXiv:2208.07869]

    Pith/arXiv arXiv 2023

  25. [25]

    Adhikary, T

    N. Adhikary, T. Biswas, J. Chakrabortty, C. Englert, and M. Spannowsky,Electroweak scalar effects beyond dimension-six in SMEFT,Phys. Rev. D113(2026), no. 3 036003, [arXiv:2501.12160]

    Pith/arXiv arXiv 2026

  26. [26]

    Liao, X.-D

    Y. Liao, X.-D. Ma, and Y. Uchida,Effective field theory for a type II seesaw model: Symmetric phase vs broken phase,Phys. Rev. D112(2025), no. 3 035005, [arXiv:2504.02580]

    Pith/arXiv arXiv 2025

  27. [27]

    Sun, M.-L

    H. Sun, M.-L. Xiao, and J.-H. Yu,Complete NLO operators in the Higgs effective field theory, JHEP05(2023) 043, [arXiv:2206.07722]

    Pith/arXiv arXiv 2023

  28. [28]

    L. Gráf, B. Henning, X. Lu, T. Melia, and H. Murayama,Hilbert series, the Higgs mechanism, and HEFT,JHEP02(2023) 064, [arXiv:2211.06275]

    Pith/arXiv arXiv 2023

  29. [29]

    Scholer, J

    O. Scholer, J. de Vries, and L. Gráf,νDoBe — A Python tool for neutrinoless double beta decay,JHEP08(2023) 043, [arXiv:2304.05415]. 46

    Pith/arXiv arXiv 2023

  30. [30]

    S. R. Coleman, J. Wess, and B. Zumino,Structure of phenomenological Lagrangians. 1., Phys. Rev.177(1969) 2239–2247

    1969

  31. [31]

    Song and X

    H. Song and X. Wan,A Non-linear Representation of General Scalar Extensions of the Standard Model for HEFT Matching,arXiv:2412.00355

    Pith/arXiv arXiv

  32. [32]

    Z. Ge, H. Song, and X. Wan,Establishing the primary HEFT as a precision benchmark for UV-HEFT matching,Phys. Rev. D114(2026), no. 1 015013, [arXiv:2602.14418]

    arXiv 2026

  33. [33]

    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

  34. [34]

    Brivio, J

    I. Brivio, J. Gonzalez-Fraile, M. C. Gonzalez-Garcia, and L. Merlo,The complete HEFT Lagrangian after the LHC Run I,Eur. Phys. J. C76(2016), no. 7 416, [arXiv:1604.06801]

    Pith/arXiv arXiv 2016

  35. [35]

    de Blas, O

    J. de Blas, O. Eberhardt, and C. Krause,Current and Future Constraints on Higgs Couplings in the Nonlinear Effective Theory,JHEP07(2018) 048, [arXiv:1803.00939]

    Pith/arXiv arXiv 2018

  36. [36]

    O. J. P. Eboli, M. C. Gonzalez-Garcia, and M. Martines,Electroweak Higgs effective field theory after LHC run 2,Phys. Rev. D105(2022), no. 5 053003, [arXiv:2112.11468]

    Pith/arXiv arXiv 2022

  37. [37]

    M.-C. Chen, S. Dawson, and T. Krupovnickas,Higgs triplets and limits from precision measurements,Phys. Rev. D74(2006) 035001, [hep-ph/0604102]

    Pith/arXiv arXiv 2006

  38. [38]

    M. Aoki, S. Kanemura, M. Kikuchi, and K. Yagyu,Radiative corrections to the Higgs boson couplings in the triplet model,Phys. Rev. D87(2013), no. 1 015012, [arXiv:1211.6029]

    Pith/arXiv arXiv 2013

  39. [39]

    Ashanujjaman, P

    S. Ashanujjaman, P. S. B. Dev, J. Huang, and S. Zhou,Precision Higgs boson probe of type-II seesaw models,Phys. Rev. D113(2026), no. 5 L051704, [arXiv:2512.07532]

    arXiv 2026

  40. [40]

    M. Aiko, S. Kanemura, M. Kikuchi, K. Sakurai, S. Taniguchi, and K. Yagyu,Radiative corrections to decays of the 125 GeV Higgs boson in the complex Higgs triplet model,JHEP 05(2026) 136, [arXiv:2601.15983]

    Pith/arXiv arXiv 2026

  41. [41]

    Gómez-Ambrosio, F

    R. Gómez-Ambrosio, F. J. Llanes-Estrada, A. Salas-Bernárdez, and J. J. Sanz-Cillero, Distinguishing electroweak EFTs with WLWL→n×h,Phys. Rev. D106(2022), no. 5 053004, [arXiv:2204.01763]

    Pith/arXiv arXiv 2022

  42. [42]

    Gómez-Ambrosio, F

    R. Gómez-Ambrosio, F. J. Llanes-Estrada, A. Salas-Bernárdez, and J. J. Sanz-Cillero, SMEFT is falsifiable through multi-Higgs measurements (even in the absence of new light particles),Commun. Theor. Phys.75(2023), no. 9 095202, [arXiv:2207.09848]. 47

    Pith/arXiv arXiv 2023

  43. [43]

    M. J. Herrero and R. A. Morales,One-loop corrections for WW to HH in Higgs EFT with the electroweak chiral Lagrangian,Phys. Rev. D106(2022), no. 7 073008, [arXiv:2208.05900]

    Pith/arXiv arXiv 2022

  44. [44]

    Domenech, C

    Anisha, D. Domenech, C. Englert, M. J. Herrero, and R. A. Morales,Bosonic multi-Higgs correlations beyond leading order,Phys. Rev. D110(2024), no. 9 095016, [arXiv:2405.05385]

    Pith/arXiv arXiv 2024

  45. [45]

    Domenech, C

    Anisha, D. Domenech, C. Englert, M. J. Herrero, and R. A. Morales,HEFT’s appraisal of triple (versus double) Higgs weak boson fusion,Phys. Rev. D111(2025), no. 5 055004, [arXiv:2407.20706]

    Pith/arXiv arXiv 2025

  46. [46]

    Braun, P

    J. Braun, P. Bredt, G. Heinrich, and M. Höfer,Double Higgs production in vector boson fusion at NLO QCD in HEFT,JHEP07(2025) 209, [arXiv:2502.09132]

    Pith/arXiv arXiv 2025

  47. [47]

    Song and X

    H. Song and X. Wan,Matching the real Higgs triplet extension of Standard Model to HEFT, arXiv:2503.00707

    Pith/arXiv arXiv

  48. [48]

    O. J. P. Eboli, M. C. Gonzalez-Garcia, and M. Martines,Bounds on quartic gauge couplings in HEFT from electroweak gauge boson pair production at the LHC,Phys. Rev. D109 (2024), no. 3 033007, [arXiv:2311.09300]

    Pith/arXiv arXiv 2024

  49. [49]

    Castillo, R

    A. Castillo, R. L. Delgado, A. Dobado, and F. J. Llanes-Estrada,Top–antitop production fromW + L W − L andZ LZL scattering under a strongly interacting symmetry-breaking sector, Eur. Phys. J. C77(2017), no. 7 436, [arXiv:1607.01158]

    Pith/arXiv arXiv 2017

  50. [50]

    Chen and D

    M. Chen and D. Liu,Top Yukawa coupling measurement at the muon collider,Phys. Rev. D 109(2024), no. 7 075020, [arXiv:2212.11067]

    Pith/arXiv arXiv 2024

  51. [51]

    Liu, K.-F

    Z. Liu, K.-F. Lyu, I. Mahbub, and L.-T. Wang,Top Yukawa coupling determination at high energy muon collider,Phys. Rev. D109(2024), no. 3 035021, [arXiv:2308.06323]

    Pith/arXiv arXiv 2024

  52. [52]

    R. D. Matheus, O. J. P. Eboli, R. Rahaman, and A. S. Navarro,Exploring Higgs EFT in tthhat high luminosity LHC,JHEP07(2026) 156, [arXiv:2602.03842]

    Pith/arXiv arXiv 2026

  53. [53]

    A. G. Akeroyd, M. Aoki, and H. Sugiyama,Lepton Flavour Violating Decays tau —>anti-l ll and mu —>e gamma in the Higgs Triplet Model,Phys. Rev. D79(2009) 113010, [arXiv:0904.3640]

    Pith/arXiv arXiv 2009

  54. [54]

    D. N. Dinh, A. Ibarra, E. Molinaro, and S. T. Petcov,Theµ−eConversion in Nuclei, µ→eγ, µ→3eDecays and TeV Scale See-Saw Scenarios of Neutrino Mass Generation, JHEP08(2012) 125, [arXiv:1205.4671]. [Erratum: JHEP 09, 023 (2013)]. 48

    Pith/arXiv arXiv 2012

  55. [55]

    N. D. Barrie and S. T. Petcov,Lepton Flavour Violation tests of Type II Seesaw Leptogenesis,JHEP01(2023) 001, [arXiv:2210.02110]

    Pith/arXiv arXiv 2023

  56. [56]

    P. S. B. Dev, M. J. Ramsey-Musolf, and Y. Zhang,Doubly-Charged Scalars in the Type-II Seesaw Mechanism: Fundamental Symmetry Tests and High-Energy Searches,Phys. Rev. D 98(2018), no. 5 055013, [arXiv:1806.08499]. [55]SINDRUMCollaboration, U. Bellgardt et al.,Search for the Decayµ + →e +e+e−,Nucl. Phys. B299(1988) 1–6. [56]Mu3eCollaboration, K. Arndt et al...

    Pith/arXiv arXiv 2018

  57. [57]

    B. Fuks, J. Neundorf, K. Peters, R. Ruiz, and M. Saimpert,Probing the Weinberg operator at colliders,Phys. Rev. D103(2021), no. 11 115014, [arXiv:2012.09882]. [58]CMSCollaboration, A. Tumasyan et al.,Search for long-lived heavy neutral leptons with displaced vertices in proton-proton collisions at√s=13 TeV,JHEP07(2022) 081, [arXiv:2201.05578]

    Pith/arXiv arXiv 2021

  58. [59]

    B. Fuks, M. Nemevšek, and R. Ruiz,Doubly Charged Higgs Boson Production at Hadron Colliders,Phys. Rev. D101(2020), no. 7 075022, [arXiv:1912.08975]. [60]ATLASCollaboration, G. Aad et al.,Search for doubly charged Higgs boson production in multi-lepton final states using 139 fb−1 of proton–proton collisions at√s= 13 TeV with the ATLAS detector,Eur. Phys. J...

    Pith/arXiv arXiv 2020